id	sid	tid	token	lemma	pos
ejpam-5240	1	1	european	european	PROPN
ejpam-5240	1	2	journal	journal	PROPN
ejpam-5240	1	3	of	of	ADP
ejpam-5240	1	4	pure	pure	ADJ
ejpam-5240	1	5	and	and	CCONJ
ejpam-5240	1	6	applied	apply	VERB
ejpam-5240	1	7	mathematics	mathematic	NOUN
ejpam-5240	1	8	vol	vol	NOUN
ejpam-5240	1	9	.	.	PROPN
ejpam-5240	2	1	17	17	NUM
ejpam-5240	2	2	,	,	PUNCT
ejpam-5240	2	3	no	no	INTJ
ejpam-5240	2	4	.	.	NOUN
ejpam-5240	2	5	3	3	NUM
ejpam-5240	2	6	,	,	PUNCT
ejpam-5240	2	7	2024	2024	NUM
ejpam-5240	2	8	,	,	PUNCT
ejpam-5240	2	9	1779	1779	NUM
ejpam-5240	2	10	-	-	SYM
ejpam-5240	2	11	1803	1803	NUM
ejpam-5240	2	12	issn	issn	VERB
ejpam-5240	2	13	1307	1307	NUM
ejpam-5240	2	14	-	-	SYM
ejpam-5240	2	15	5543	5543	NUM
ejpam-5240	2	16	–	–	PUNCT
ejpam-5240	2	17	ejpam.com	ejpam.com	X
ejpam-5240	2	18	published	publish	VERB
ejpam-5240	2	19	by	by	ADP
ejpam-5240	2	20	new	new	PROPN
ejpam-5240	2	21	york	york	PROPN
ejpam-5240	2	22	business	business	PROPN
ejpam-5240	2	23	global	global	PROPN
ejpam-5240	2	24	on	on	ADP
ejpam-5240	2	25	the	the	DET
ejpam-5240	2	26	k	k	ADV
ejpam-5240	2	27	-	-	PUNCT
ejpam-5240	2	28	restricted	restrict	VERB
ejpam-5240	2	29	intersection	intersection	NOUN
ejpam-5240	2	30	graph	graph	NOUN
ejpam-5240	2	31	mariane	mariane	PROPN
ejpam-5240	2	32	eliz	eliz	PROPN
ejpam-5240	2	33	d.	d.	PROPN
ejpam-5240	2	34	pelagio1∗	pelagio1∗	PROPN
ejpam-5240	2	35	,	,	PUNCT
ejpam-5240	2	36	kathlen	kathlen	VERB
ejpam-5240	2	37	c.	c.	PROPN
ejpam-5240	2	38	mendoza1	mendoza1	PROPN
ejpam-5240	2	39	,	,	PUNCT
ejpam-5240	2	40	neil	neil	PROPN
ejpam-5240	2	41	m.	m.	PROPN
ejpam-5240	2	42	mame1	mame1	PROPN
ejpam-5240	2	43	1	1	NUM
ejpam-5240	2	44	college	college	NOUN
ejpam-5240	2	45	of	of	ADP
ejpam-5240	2	46	arts	art	NOUN
ejpam-5240	2	47	and	and	CCONJ
ejpam-5240	2	48	sciences	sciences	PROPN
ejpam-5240	2	49	,	,	PUNCT
ejpam-5240	2	50	batangas	batangas	PROPN
ejpam-5240	2	51	state	state	PROPN
ejpam-5240	2	52	university	university	PROPN
ejpam-5240	2	53	the	the	DET
ejpam-5240	2	54	national	national	PROPN
ejpam-5240	2	55	engineering	engineering	PROPN
ejpam-5240	2	56	university	university	PROPN
ejpam-5240	2	57	,	,	PUNCT
ejpam-5240	2	58	batangas	batangas	PROPN
ejpam-5240	2	59	city	city	PROPN
ejpam-5240	2	60	,	,	PUNCT
ejpam-5240	2	61	batangas	batangas	PROPN
ejpam-5240	2	62	,	,	PUNCT
ejpam-5240	2	63	philippines	philippine	NOUN
ejpam-5240	2	64	abstract	abstract	ADJ
ejpam-5240	2	65	.	.	PUNCT
ejpam-5240	3	1	the	the	DET
ejpam-5240	3	2	problem	problem	NOUN
ejpam-5240	3	3	of	of	ADP
ejpam-5240	3	4	intersection	intersection	NOUN
ejpam-5240	3	5	graphs	graph	NOUN
ejpam-5240	3	6	was	be	AUX
ejpam-5240	3	7	introduced	introduce	VERB
ejpam-5240	3	8	by	by	ADP
ejpam-5240	3	9	szpilrajn	szpilrajn	NOUN
ejpam-5240	3	10	-	-	PUNCT
ejpam-5240	3	11	marczewski	marczewski	NOUN
ejpam-5240	3	12	in	in	ADP
ejpam-5240	3	13	1945	1945	NUM
ejpam-5240	3	14	.	.	PUNCT
ejpam-5240	4	1	this	this	DET
ejpam-5240	4	2	study	study	NOUN
ejpam-5240	4	3	introduces	introduce	VERB
ejpam-5240	4	4	a	a	DET
ejpam-5240	4	5	new	new	ADJ
ejpam-5240	4	6	variant	variant	NOUN
ejpam-5240	4	7	of	of	ADP
ejpam-5240	4	8	the	the	DET
ejpam-5240	4	9	intersection	intersection	NOUN
ejpam-5240	4	10	graph	graph	NOUN
ejpam-5240	4	11	,	,	PUNCT
ejpam-5240	4	12	called	call	VERB
ejpam-5240	4	13	the	the	DET
ejpam-5240	4	14	k	k	ADV
ejpam-5240	4	15	-	-	PUNCT
ejpam-5240	4	16	restricted	restrict	VERB
ejpam-5240	4	17	intersection	intersection	NOUN
ejpam-5240	4	18	graph	graph	NOUN
ejpam-5240	4	19	.	.	PUNCT
ejpam-5240	5	1	let	let	VERB
ejpam-5240	5	2	sn	sn	PROPN
ejpam-5240	5	3	be	be	AUX
ejpam-5240	5	4	a	a	DET
ejpam-5240	5	5	nonempty	nonempty	ADJ
ejpam-5240	5	6	n	n	CCONJ
ejpam-5240	5	7	-	-	PUNCT
ejpam-5240	5	8	element	element	NOUN
ejpam-5240	5	9	set	set	NOUN
ejpam-5240	5	10	,	,	PUNCT
ejpam-5240	5	11	for	for	ADP
ejpam-5240	5	12	some	some	DET
ejpam-5240	5	13	positive	positive	ADJ
ejpam-5240	5	14	integer	integer	NOUN
ejpam-5240	5	15	n	n	CCONJ
ejpam-5240	5	16	,	,	PUNCT
ejpam-5240	5	17	and	and	CCONJ
ejpam-5240	5	18	let	let	VERB
ejpam-5240	5	19	s(n	s(n	PROPN
ejpam-5240	5	20	,	,	PUNCT
ejpam-5240	5	21	k	k	NOUN
ejpam-5240	5	22	)	)	PUNCT
ejpam-5240	5	23	be	be	VERB
ejpam-5240	5	24	the	the	DET
ejpam-5240	5	25	set	set	NOUN
ejpam-5240	5	26	of	of	ADP
ejpam-5240	5	27	all	all	DET
ejpam-5240	5	28	the	the	DET
ejpam-5240	5	29	k	k	ADJ
ejpam-5240	5	30	-	-	ADJ
ejpam-5240	5	31	element	element	ADJ
ejpam-5240	5	32	subsets	subset	NOUN
ejpam-5240	5	33	of	of	ADP
ejpam-5240	5	34	sn	sn	PROPN
ejpam-5240	5	35	where	where	SCONJ
ejpam-5240	5	36	0	0	NUM
ejpam-5240	5	37	≤	≤	NUM
ejpam-5240	5	38	k	k	X
ejpam-5240	5	39	≤	≤	PROPN
ejpam-5240	5	40	n.	n.	NOUN
ejpam-5240	5	41	a	a	DET
ejpam-5240	5	42	k	k	ADV
ejpam-5240	5	43	-	-	ADJ
ejpam-5240	5	44	restricted	restricted	ADJ
ejpam-5240	5	45	intersection	intersection	NOUN
ejpam-5240	5	46	graph	graph	NOUN
ejpam-5240	5	47	,	,	PUNCT
ejpam-5240	5	48	denoted	denote	VERB
ejpam-5240	5	49	by	by	ADP
ejpam-5240	5	50	gs(n	gs(n	NOUN
ejpam-5240	5	51	,	,	PUNCT
ejpam-5240	5	52	k	k	NOUN
ejpam-5240	5	53	)	)	PUNCT
ejpam-5240	5	54	,	,	PUNCT
ejpam-5240	5	55	is	be	AUX
ejpam-5240	5	56	a	a	DET
ejpam-5240	5	57	graph	graph	NOUN
ejpam-5240	5	58	with	with	ADP
ejpam-5240	5	59	vertex	vertex	NOUN
ejpam-5240	5	60	set	set	NOUN
ejpam-5240	5	61	s(n	s(n	PROPN
ejpam-5240	5	62	,	,	PUNCT
ejpam-5240	5	63	k	k	NOUN
ejpam-5240	5	64	)	)	PUNCT
ejpam-5240	5	65	such	such	ADJ
ejpam-5240	5	66	that	that	SCONJ
ejpam-5240	5	67	two	two	NUM
ejpam-5240	5	68	vertices	vertex	NOUN
ejpam-5240	5	69	a	a	DET
ejpam-5240	5	70	,	,	PUNCT
ejpam-5240	5	71	b	b	PROPN
ejpam-5240	5	72	∈	∈	PROPN
ejpam-5240	5	73	s(n	s(n	PROPN
ejpam-5240	5	74	,	,	PUNCT
ejpam-5240	5	75	k	k	NOUN
ejpam-5240	5	76	)	)	PUNCT
ejpam-5240	5	77	are	be	AUX
ejpam-5240	5	78	adjacent	adjacent	ADJ
ejpam-5240	5	79	whenever	whenever	SCONJ
ejpam-5240	5	80	a	a	DET
ejpam-5240	5	81	∩	∩	ADJ
ejpam-5240	5	82	b	b	NOUN
ejpam-5240	5	83	̸=	̸=	PROPN
ejpam-5240	5	84	∅	∅	NOUN
ejpam-5240	5	85	and	and	CCONJ
ejpam-5240	5	86	a	a	DET
ejpam-5240	5	87	̸=	̸=	PROPN
ejpam-5240	5	88	b.	b.	NOUN
ejpam-5240	5	89	here	here	ADV
ejpam-5240	5	90	,	,	PUNCT
ejpam-5240	5	91	we	we	PRON
ejpam-5240	5	92	determined	determine	VERB
ejpam-5240	5	93	the	the	DET
ejpam-5240	5	94	order	order	NOUN
ejpam-5240	5	95	and	and	CCONJ
ejpam-5240	5	96	size	size	NOUN
ejpam-5240	5	97	of	of	ADP
ejpam-5240	5	98	gs(n	gs(n	NOUN
ejpam-5240	5	99	,	,	PUNCT
ejpam-5240	5	100	k	k	NOUN
ejpam-5240	5	101	)	)	PUNCT
ejpam-5240	5	102	.	.	PUNCT
ejpam-5240	6	1	moreover	moreover	ADV
ejpam-5240	6	2	,	,	PUNCT
ejpam-5240	6	3	some	some	DET
ejpam-5240	6	4	parameters	parameter	NOUN
ejpam-5240	6	5	such	such	ADJ
ejpam-5240	6	6	as	as	ADP
ejpam-5240	6	7	independence	independence	NOUN
ejpam-5240	6	8	number	number	NOUN
ejpam-5240	6	9	,	,	PUNCT
ejpam-5240	6	10	domination	domination	NOUN
ejpam-5240	6	11	number	number	NOUN
ejpam-5240	6	12	,	,	PUNCT
ejpam-5240	6	13	and	and	CCONJ
ejpam-5240	6	14	isolate	isolate	VERB
ejpam-5240	6	15	domination	domination	NOUN
ejpam-5240	6	16	number	number	NOUN
ejpam-5240	6	17	of	of	ADP
ejpam-5240	6	18	the	the	DET
ejpam-5240	6	19	k	k	ADV
ejpam-5240	6	20	-	-	PUNCT
ejpam-5240	6	21	restricted	restrict	VERB
ejpam-5240	6	22	intersection	intersection	NOUN
ejpam-5240	6	23	graph	graph	NOUN
ejpam-5240	6	24	were	be	AUX
ejpam-5240	6	25	established	establish	VERB
ejpam-5240	6	26	.	.	PUNCT
ejpam-5240	7	1	finally	finally	ADV
ejpam-5240	7	2	,	,	PUNCT
ejpam-5240	7	3	necessary	necessary	ADJ
ejpam-5240	7	4	and	and	CCONJ
ejpam-5240	7	5	sufficient	sufficient	ADJ
ejpam-5240	7	6	conditions	condition	NOUN
ejpam-5240	7	7	for	for	ADP
ejpam-5240	7	8	a	a	DET
ejpam-5240	7	9	gs(n	gs(n	NOUN
ejpam-5240	7	10	,	,	PUNCT
ejpam-5240	7	11	k	k	NOUN
ejpam-5240	7	12	)	)	PUNCT
ejpam-5240	7	13	to	to	PART
ejpam-5240	7	14	be	be	AUX
ejpam-5240	7	15	isomorphic	isomorphic	ADJ
ejpam-5240	7	16	to	to	ADP
ejpam-5240	7	17	the	the	DET
ejpam-5240	7	18	cycle	cycle	NOUN
ejpam-5240	7	19	graph	graph	NOUN
ejpam-5240	7	20	and	and	CCONJ
ejpam-5240	7	21	complete	complete	ADJ
ejpam-5240	7	22	graph	graph	NOUN
ejpam-5240	7	23	were	be	AUX
ejpam-5240	7	24	determined	determine	VERB
ejpam-5240	7	25	.	.	PUNCT
ejpam-5240	8	1	2020	2020	NUM
ejpam-5240	8	2	mathematics	mathematic	NOUN
ejpam-5240	8	3	subject	subject	NOUN
ejpam-5240	8	4	classifications	classification	NOUN
ejpam-5240	8	5	:	:	PUNCT
ejpam-5240	8	6	05c62	05c62	NUM
ejpam-5240	8	7	,	,	PUNCT
ejpam-5240	8	8	05c69	05c69	X
ejpam-5240	8	9	key	key	ADJ
ejpam-5240	8	10	words	word	NOUN
ejpam-5240	8	11	and	and	CCONJ
ejpam-5240	8	12	phrases	phrase	NOUN
ejpam-5240	8	13	:	:	PUNCT
ejpam-5240	8	14	intersection	intersection	NOUN
ejpam-5240	8	15	graph	graph	NOUN
ejpam-5240	8	16	,	,	PUNCT
ejpam-5240	8	17	k	k	ADJ
ejpam-5240	8	18	-	-	PUNCT
ejpam-5240	8	19	restricted	restricted	ADJ
ejpam-5240	8	20	intersection	intersection	NOUN
ejpam-5240	8	21	graph	graph	NOUN
ejpam-5240	8	22	,	,	PUNCT
ejpam-5240	8	23	k	k	ADJ
ejpam-5240	8	24	-	-	ADJ
ejpam-5240	8	25	element	element	ADJ
ejpam-5240	8	26	subsets	subset	NOUN
ejpam-5240	8	27	1	1	NUM
ejpam-5240	8	28	.	.	PUNCT
ejpam-5240	9	1	introduction	introduction	NOUN
ejpam-5240	9	2	graph	graph	NOUN
ejpam-5240	9	3	theory	theory	NOUN
ejpam-5240	9	4	has	have	AUX
ejpam-5240	9	5	been	be	AUX
ejpam-5240	9	6	linked	link	VERB
ejpam-5240	9	7	to	to	ADP
ejpam-5240	9	8	some	some	DET
ejpam-5240	9	9	areas	area	NOUN
ejpam-5240	9	10	of	of	ADP
ejpam-5240	9	11	mathematics	mathematic	NOUN
ejpam-5240	9	12	such	such	ADJ
ejpam-5240	9	13	as	as	ADP
ejpam-5240	9	14	set	set	NOUN
ejpam-5240	9	15	theory	theory	NOUN
ejpam-5240	9	16	.	.	PUNCT
ejpam-5240	10	1	specifically	specifically	ADV
ejpam-5240	10	2	,	,	PUNCT
ejpam-5240	10	3	the	the	DET
ejpam-5240	10	4	utilization	utilization	NOUN
ejpam-5240	10	5	of	of	ADP
ejpam-5240	10	6	family	family	NOUN
ejpam-5240	10	7	sets	set	NOUN
ejpam-5240	10	8	as	as	ADP
ejpam-5240	10	9	vertices	vertex	NOUN
ejpam-5240	10	10	of	of	ADP
ejpam-5240	10	11	graphs	graph	NOUN
ejpam-5240	10	12	was	be	AUX
ejpam-5240	10	13	one	one	NUM
ejpam-5240	10	14	of	of	ADP
ejpam-5240	10	15	the	the	DET
ejpam-5240	10	16	examples	example	NOUN
ejpam-5240	10	17	of	of	ADP
ejpam-5240	10	18	associating	associate	VERB
ejpam-5240	10	19	set	set	ADJ
ejpam-5240	10	20	theory	theory	NOUN
ejpam-5240	10	21	with	with	ADP
ejpam-5240	10	22	graph	graph	NOUN
ejpam-5240	10	23	theory	theory	NOUN
ejpam-5240	10	24	as	as	SCONJ
ejpam-5240	10	25	stated	state	VERB
ejpam-5240	10	26	by	by	ADP
ejpam-5240	10	27	golumbic	golumbic	ADJ
ejpam-5240	10	28	,	,	PUNCT
ejpam-5240	10	29	m.	m.	NOUN
ejpam-5240	10	30	(	(	PUNCT
ejpam-5240	10	31	1980	1980	NUM
ejpam-5240	10	32	)	)	PUNCT
ejpam-5240	10	33	in	in	ADP
ejpam-5240	10	34	his	his	PRON
ejpam-5240	10	35	study	study	NOUN
ejpam-5240	10	36	“	"	PUNCT
ejpam-5240	10	37	algorithmic	algorithmic	ADJ
ejpam-5240	10	38	graph	graph	NOUN
ejpam-5240	10	39	theory	theory	NOUN
ejpam-5240	10	40	and	and	CCONJ
ejpam-5240	10	41	perfect	perfect	ADJ
ejpam-5240	10	42	graphs	graph	NOUN
ejpam-5240	10	43	”	"	PUNCT
ejpam-5240	10	44	[	[	X
ejpam-5240	10	45	5	5	NUM
ejpam-5240	10	46	]	]	PUNCT
ejpam-5240	10	47	.	.	PUNCT
ejpam-5240	11	1	for	for	ADP
ejpam-5240	11	2	instance	instance	NOUN
ejpam-5240	11	3	,	,	PUNCT
ejpam-5240	11	4	some	some	PRON
ejpam-5240	11	5	of	of	ADP
ejpam-5240	11	6	the	the	DET
ejpam-5240	11	7	special	special	ADJ
ejpam-5240	11	8	types	type	NOUN
ejpam-5240	11	9	of	of	ADP
ejpam-5240	11	10	graphs	graph	NOUN
ejpam-5240	11	11	like	like	ADP
ejpam-5240	11	12	the	the	DET
ejpam-5240	11	13	hamming	hamming	NOUN
ejpam-5240	11	14	graph	graph	NOUN
ejpam-5240	11	15	by	by	ADP
ejpam-5240	11	16	richard	richard	NOUN
ejpam-5240	11	17	hamming	hamming	PROPN
ejpam-5240	11	18	and	and	CCONJ
ejpam-5240	11	19	the	the	DET
ejpam-5240	11	20	johnson	johnson	PROPN
ejpam-5240	11	21	graph	graph	NOUN
ejpam-5240	11	22	by	by	ADP
ejpam-5240	11	23	selmer	selmer	PROPN
ejpam-5240	11	24	m.	m.	PROPN
ejpam-5240	11	25	john	john	PROPN
ejpam-5240	11	26	were	be	AUX
ejpam-5240	11	27	both	both	PRON
ejpam-5240	11	28	derived	derive	VERB
ejpam-5240	11	29	from	from	ADP
ejpam-5240	11	30	the	the	DET
ejpam-5240	11	31	system	system	NOUN
ejpam-5240	11	32	of	of	ADP
ejpam-5240	11	33	sets	set	NOUN
ejpam-5240	11	34	and	and	CCONJ
ejpam-5240	11	35	were	be	AUX
ejpam-5240	11	36	being	be	AUX
ejpam-5240	11	37	used	use	VERB
ejpam-5240	11	38	in	in	ADP
ejpam-5240	11	39	coding	code	VERB
ejpam-5240	11	40	theory	theory	NOUN
ejpam-5240	11	41	–	–	PUNCT
ejpam-5240	11	42	which	which	PRON
ejpam-5240	11	43	was	be	AUX
ejpam-5240	11	44	also	also	ADV
ejpam-5240	11	45	a	a	DET
ejpam-5240	11	46	field	field	NOUN
ejpam-5240	11	47	of	of	ADP
ejpam-5240	11	48	mathematics	mathematic	NOUN
ejpam-5240	12	1	[	[	X
ejpam-5240	12	2	1	1	NUM
ejpam-5240	12	3	]	]	PUNCT
ejpam-5240	12	4	.	.	PUNCT
ejpam-5240	13	1	connecting	connect	VERB
ejpam-5240	13	2	the	the	DET
ejpam-5240	13	3	concept	concept	NOUN
ejpam-5240	13	4	of	of	ADP
ejpam-5240	13	5	a	a	DET
ejpam-5240	13	6	set	set	NOUN
ejpam-5240	13	7	to	to	PART
ejpam-5240	13	8	graph	graph	NOUN
ejpam-5240	13	9	theory	theory	NOUN
ejpam-5240	13	10	paved	pave	VERB
ejpam-5240	13	11	the	the	DET
ejpam-5240	13	12	way	way	NOUN
ejpam-5240	13	13	for	for	ADP
ejpam-5240	13	14	the	the	DET
ejpam-5240	13	15	introduction	introduction	NOUN
ejpam-5240	13	16	of	of	ADP
ejpam-5240	13	17	intersection	intersection	NOUN
ejpam-5240	13	18	graphs	graph	NOUN
ejpam-5240	13	19	.	.	PUNCT
ejpam-5240	14	1	an	an	DET
ejpam-5240	14	2	intersection	intersection	NOUN
ejpam-5240	14	3	graph	graph	NOUN
ejpam-5240	14	4	contained	contain	VERB
ejpam-5240	14	5	a	a	DET
ejpam-5240	14	6	family	family	NOUN
ejpam-5240	14	7	of	of	ADP
ejpam-5240	14	8	sets	set	NOUN
ejpam-5240	14	9	as	as	ADP
ejpam-5240	14	10	its	its	PRON
ejpam-5240	14	11	vertices	vertex	NOUN
ejpam-5240	14	12	and	and	CCONJ
ejpam-5240	14	13	each	each	DET
ejpam-5240	14	14	vertices	vertex	NOUN
ejpam-5240	14	15	were	be	AUX
ejpam-5240	14	16	connected	connect	VERB
ejpam-5240	14	17	by	by	ADP
ejpam-5240	14	18	an	an	DET
ejpam-5240	14	19	edge	edge	NOUN
ejpam-5240	14	20	whenever	whenever	SCONJ
ejpam-5240	14	21	the	the	DET
ejpam-5240	14	22	sets	set	NOUN
ejpam-5240	14	23	had	have	VERB
ejpam-5240	14	24	a	a	DET
ejpam-5240	14	25	nonempty	nonempty	ADJ
ejpam-5240	14	26	intersection	intersection	NOUN
ejpam-5240	14	27	.	.	PUNCT
ejpam-5240	15	1	this	this	DET
ejpam-5240	15	2	graph	graph	NOUN
ejpam-5240	15	3	was	be	AUX
ejpam-5240	15	4	introduced	introduce	VERB
ejpam-5240	15	5	by	by	ADP
ejpam-5240	15	6	szpilrajn	szpilrajn	NOUN
ejpam-5240	15	7	-	-	PUNCT
ejpam-5240	15	8	marczewski	marczewski	PROPN
ejpam-5240	15	9	(	(	PUNCT
ejpam-5240	15	10	1945	1945	NUM
ejpam-5240	15	11	)	)	PUNCT
ejpam-5240	15	12	in	in	ADP
ejpam-5240	15	13	their	their	PRON
ejpam-5240	15	14	paper	paper	NOUN
ejpam-5240	15	15	entitled	entitle	VERB
ejpam-5240	15	16	“	"	PUNCT
ejpam-5240	15	17	on	on	ADP
ejpam-5240	15	18	two	two	NUM
ejpam-5240	15	19	properties	property	NOUN
ejpam-5240	15	20	of	of	ADP
ejpam-5240	15	21	set	set	ADJ
ejpam-5240	15	22	classes”[14	classes”[14	PROPN
ejpam-5240	15	23	]	]	PUNCT
ejpam-5240	15	24	,	,	PUNCT
ejpam-5240	15	25	wherein	wherein	SCONJ
ejpam-5240	15	26	they	they	PRON
ejpam-5240	15	27	also	also	ADV
ejpam-5240	15	28	asserted	assert	VERB
ejpam-5240	15	29	that	that	SCONJ
ejpam-5240	15	30	all	all	DET
ejpam-5240	15	31	graphs	graph	NOUN
ejpam-5240	15	32	may	may	AUX
ejpam-5240	15	33	be	be	AUX
ejpam-5240	15	34	∗corresponding	∗corresponde	VERB
ejpam-5240	15	35	author	author	NOUN
ejpam-5240	15	36	.	.	PUNCT
ejpam-5240	16	1	doi	doi	NOUN
ejpam-5240	16	2	:	:	PUNCT
ejpam-5240	16	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5240	https://doi.org/10.29020/nybg.ejpam.v17i3.5240	NOUN
ejpam-5240	16	4	email	email	NOUN
ejpam-5240	16	5	addresses	address	VERB
ejpam-5240	16	6	:	:	PUNCT
ejpam-5240	16	7	marianeeliz.pelagio@g.batstate-u.edu.ph	marianeeliz.pelagio@g.batstate-u.edu.ph	NUM
ejpam-5240	16	8	(	(	PUNCT
ejpam-5240	16	9	m.e	m.e	PROPN
ejpam-5240	16	10	.	.	PROPN
ejpam-5240	16	11	pelagio	pelagio	PROPN
ejpam-5240	16	12	)	)	PUNCT
ejpam-5240	16	13	,	,	PUNCT
ejpam-5240	16	14	kathlen.mendoza@g.batstate-u.edu.ph	kathlen.mendoza@g.batstate-u.edu.ph	PROPN
ejpam-5240	16	15	(	(	PUNCT
ejpam-5240	16	16	k.	k.	PROPN
ejpam-5240	16	17	mendoza	mendoza	PROPN
ejpam-5240	16	18	)	)	PUNCT
ejpam-5240	16	19	,	,	PUNCT
ejpam-5240	16	20	neil.mame@g.batstate-u.edu.ph	neil.mame@g.batstate-u.edu.ph	PROPN
ejpam-5240	16	21	(	(	PUNCT
ejpam-5240	16	22	n.	n.	NOUN
ejpam-5240	16	23	mame	mame	PROPN
ejpam-5240	16	24	)	)	PUNCT
ejpam-5240	16	25	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5240	16	26	1779	1779	NUM
ejpam-5240	17	1	©	©	ADP
ejpam-5240	17	2	2024	2024	NUM
ejpam-5240	17	3	ejpam	ejpam	NOUN
ejpam-5240	17	4	all	all	DET
ejpam-5240	17	5	rights	right	NOUN
ejpam-5240	17	6	reserved	reserve	VERB
ejpam-5240	17	7	.	.	PUNCT
ejpam-5240	18	1	m.e	m.e	PROPN
ejpam-5240	18	2	.	.	PROPN
ejpam-5240	18	3	pelagio	pelagio	PROPN
ejpam-5240	18	4	,	,	PUNCT
ejpam-5240	18	5	n.	n.	NOUN
ejpam-5240	18	6	mame	mame	PROPN
ejpam-5240	18	7	,	,	PUNCT
ejpam-5240	18	8	k.	k.	PROPN
ejpam-5240	18	9	mendoza	mendoza	PROPN
ejpam-5240	18	10	/	/	SYM
ejpam-5240	18	11	eur	eur	PROPN
ejpam-5240	18	12	.	.	PUNCT
ejpam-5240	19	1	j.	j.	PROPN
ejpam-5240	19	2	pure	pure	PROPN
ejpam-5240	19	3	appl	appl	PROPN
ejpam-5240	19	4	.	.	PROPN
ejpam-5240	19	5	math	math	PROPN
ejpam-5240	19	6	,	,	PUNCT
ejpam-5240	19	7	17	17	NUM
ejpam-5240	19	8	(	(	PUNCT
ejpam-5240	19	9	3	3	NUM
ejpam-5240	19	10	)	)	PUNCT
ejpam-5240	19	11	(	(	PUNCT
ejpam-5240	19	12	2024	2024	NUM
ejpam-5240	19	13	)	)	PUNCT
ejpam-5240	19	14	,	,	PUNCT
ejpam-5240	19	15	1779	1779	NUM
ejpam-5240	19	16	-	-	SYM
ejpam-5240	19	17	1803	1803	NUM
ejpam-5240	19	18	1780	1780	NUM
ejpam-5240	19	19	represented	represent	VERB
ejpam-5240	19	20	as	as	ADP
ejpam-5240	19	21	an	an	DET
ejpam-5240	19	22	intersection	intersection	NOUN
ejpam-5240	19	23	graph	graph	NOUN
ejpam-5240	19	24	.	.	PUNCT
ejpam-5240	20	1	this	this	PRON
ejpam-5240	20	2	was	be	AUX
ejpam-5240	20	3	proven	prove	VERB
ejpam-5240	20	4	and	and	CCONJ
ejpam-5240	20	5	supported	support	VERB
ejpam-5240	20	6	by	by	ADP
ejpam-5240	20	7	erdős	erdős	PROPN
ejpam-5240	20	8	,	,	PUNCT
ejpam-5240	20	9	goodman	goodman	PROPN
ejpam-5240	20	10	,	,	PUNCT
ejpam-5240	20	11	and	and	CCONJ
ejpam-5240	20	12	pósa	pósa	VERB
ejpam-5240	20	13	in	in	ADP
ejpam-5240	20	14	1966	1966	NUM
ejpam-5240	20	15	in	in	ADP
ejpam-5240	20	16	their	their	PRON
ejpam-5240	20	17	study	study	NOUN
ejpam-5240	20	18	“	"	PUNCT
ejpam-5240	20	19	the	the	DET
ejpam-5240	20	20	representation	representation	NOUN
ejpam-5240	20	21	of	of	ADP
ejpam-5240	20	22	a	a	DET
ejpam-5240	20	23	graph	graph	NOUN
ejpam-5240	20	24	by	by	ADP
ejpam-5240	20	25	set	set	NOUN
ejpam-5240	20	26	intersections”[10	intersections”[10	NOUN
ejpam-5240	20	27	]	]	PUNCT
ejpam-5240	20	28	.	.	PUNCT
ejpam-5240	21	1	in	in	ADP
ejpam-5240	21	2	this	this	DET
ejpam-5240	21	3	study	study	NOUN
ejpam-5240	21	4	,	,	PUNCT
ejpam-5240	21	5	they	they	PRON
ejpam-5240	21	6	provided	provide	VERB
ejpam-5240	21	7	a	a	DET
ejpam-5240	21	8	more	more	ADV
ejpam-5240	21	9	efficient	efficient	ADJ
ejpam-5240	21	10	construction	construction	NOUN
ejpam-5240	21	11	of	of	ADP
ejpam-5240	21	12	an	an	DET
ejpam-5240	21	13	intersection	intersection	NOUN
ejpam-5240	21	14	graph	graph	NOUN
ejpam-5240	21	15	and	and	CCONJ
ejpam-5240	21	16	defined	define	VERB
ejpam-5240	21	17	the	the	DET
ejpam-5240	21	18	total	total	ADJ
ejpam-5240	21	19	number	number	NOUN
ejpam-5240	21	20	of	of	ADP
ejpam-5240	21	21	set	set	ADJ
ejpam-5240	21	22	elements	element	NOUN
ejpam-5240	21	23	which	which	PRON
ejpam-5240	21	24	required	require	VERB
ejpam-5240	21	25	a	a	DET
ejpam-5240	21	26	smaller	small	ADJ
ejpam-5240	21	27	number	number	NOUN
ejpam-5240	21	28	of	of	ADP
ejpam-5240	21	29	vertices	vertex	NOUN
ejpam-5240	21	30	.	.	PUNCT
ejpam-5240	22	1	furthermore	furthermore	ADV
ejpam-5240	22	2	,	,	PUNCT
ejpam-5240	22	3	variations	variation	NOUN
ejpam-5240	22	4	on	on	ADP
ejpam-5240	22	5	intersection	intersection	NOUN
ejpam-5240	22	6	graphs	graph	NOUN
ejpam-5240	22	7	was	be	AUX
ejpam-5240	22	8	also	also	ADV
ejpam-5240	22	9	introduced	introduce	VERB
ejpam-5240	22	10	as	as	SCONJ
ejpam-5240	22	11	most	most	ADJ
ejpam-5240	22	12	of	of	ADP
ejpam-5240	22	13	these	these	PRON
ejpam-5240	22	14	were	be	AUX
ejpam-5240	22	15	derived	derive	VERB
ejpam-5240	22	16	from	from	ADP
ejpam-5240	22	17	sets	set	NOUN
ejpam-5240	22	18	on	on	ADP
ejpam-5240	22	19	some	some	DET
ejpam-5240	22	20	kind	kind	NOUN
ejpam-5240	22	21	of	of	ADP
ejpam-5240	22	22	geometric	geometric	ADJ
ejpam-5240	22	23	configuration	configuration	NOUN
ejpam-5240	22	24	which	which	PRON
ejpam-5240	22	25	was	be	AUX
ejpam-5240	22	26	specified	specify	VERB
ejpam-5240	22	27	in	in	ADP
ejpam-5240	22	28	[	[	X
ejpam-5240	22	29	11	11	NUM
ejpam-5240	22	30	]	]	PUNCT
ejpam-5240	22	31	.	.	PUNCT
ejpam-5240	23	1	some	some	PRON
ejpam-5240	23	2	of	of	ADP
ejpam-5240	23	3	these	these	PRON
ejpam-5240	23	4	were	be	AUX
ejpam-5240	23	5	the	the	DET
ejpam-5240	23	6	circle	circle	NOUN
ejpam-5240	23	7	graph	graph	NOUN
ejpam-5240	23	8	(	(	PUNCT
ejpam-5240	23	9	or	or	CCONJ
ejpam-5240	23	10	the	the	DET
ejpam-5240	23	11	intersection	intersection	NOUN
ejpam-5240	23	12	graph	graph	NOUN
ejpam-5240	23	13	from	from	ADP
ejpam-5240	23	14	the	the	DET
ejpam-5240	23	15	chords	chord	NOUN
ejpam-5240	23	16	of	of	ADP
ejpam-5240	23	17	a	a	DET
ejpam-5240	23	18	circle	circle	NOUN
ejpam-5240	23	19	)	)	PUNCT
ejpam-5240	23	20	,	,	PUNCT
ejpam-5240	23	21	string	string	NOUN
ejpam-5240	23	22	graph	graph	NOUN
ejpam-5240	23	23	(	(	PUNCT
ejpam-5240	23	24	the	the	DET
ejpam-5240	23	25	intersection	intersection	NOUN
ejpam-5240	23	26	graph	graph	NOUN
ejpam-5240	23	27	of	of	ADP
ejpam-5240	23	28	curves	curve	NOUN
ejpam-5240	23	29	on	on	ADP
ejpam-5240	23	30	a	a	DET
ejpam-5240	23	31	plane	plane	NOUN
ejpam-5240	23	32	)	)	PUNCT
ejpam-5240	23	33	,	,	PUNCT
ejpam-5240	23	34	and	and	CCONJ
ejpam-5240	23	35	circular	circular	ADJ
ejpam-5240	23	36	arc	arc	NOUN
ejpam-5240	23	37	graph	graph	NOUN
ejpam-5240	23	38	(	(	PUNCT
ejpam-5240	23	39	intersection	intersection	NOUN
ejpam-5240	23	40	graph	graph	NOUN
ejpam-5240	23	41	derived	derive	VERB
ejpam-5240	23	42	from	from	ADP
ejpam-5240	23	43	the	the	DET
ejpam-5240	23	44	arcs	arcs	NOUN
ejpam-5240	23	45	of	of	ADP
ejpam-5240	23	46	the	the	DET
ejpam-5240	23	47	circle)[15	circle)[15	NOUN
ejpam-5240	23	48	]	]	PUNCT
ejpam-5240	23	49	,	,	PUNCT
ejpam-5240	23	50	to	to	PART
ejpam-5240	23	51	name	name	VERB
ejpam-5240	23	52	a	a	DET
ejpam-5240	23	53	few	few	ADJ
ejpam-5240	23	54	.	.	PUNCT
ejpam-5240	24	1	the	the	DET
ejpam-5240	24	2	assertion	assertion	NOUN
ejpam-5240	24	3	that	that	SCONJ
ejpam-5240	24	4	all	all	DET
ejpam-5240	24	5	graphs	graph	NOUN
ejpam-5240	24	6	can	can	AUX
ejpam-5240	24	7	be	be	AUX
ejpam-5240	24	8	represented	represent	VERB
ejpam-5240	24	9	as	as	ADP
ejpam-5240	24	10	an	an	DET
ejpam-5240	24	11	intersection	intersection	NOUN
ejpam-5240	24	12	graph	graph	NOUN
ejpam-5240	24	13	is	be	AUX
ejpam-5240	24	14	an	an	DET
ejpam-5240	24	15	interesting	interesting	ADJ
ejpam-5240	24	16	concept	concept	NOUN
ejpam-5240	24	17	.	.	PUNCT
ejpam-5240	25	1	since	since	SCONJ
ejpam-5240	25	2	the	the	DET
ejpam-5240	25	3	majority	majority	NOUN
ejpam-5240	25	4	of	of	ADP
ejpam-5240	25	5	the	the	DET
ejpam-5240	25	6	intersection	intersection	NOUN
ejpam-5240	25	7	graphs	graph	NOUN
ejpam-5240	25	8	that	that	PRON
ejpam-5240	25	9	are	be	AUX
ejpam-5240	25	10	discovered	discover	VERB
ejpam-5240	25	11	employ	employ	VERB
ejpam-5240	25	12	the	the	DET
ejpam-5240	25	13	usage	usage	NOUN
ejpam-5240	25	14	of	of	ADP
ejpam-5240	25	15	sets	set	NOUN
ejpam-5240	25	16	in	in	ADP
ejpam-5240	25	17	the	the	DET
ejpam-5240	25	18	field	field	NOUN
ejpam-5240	25	19	of	of	ADP
ejpam-5240	25	20	geometry	geometry	NOUN
ejpam-5240	25	21	,	,	PUNCT
ejpam-5240	25	22	this	this	PRON
ejpam-5240	25	23	gives	give	VERB
ejpam-5240	25	24	the	the	DET
ejpam-5240	25	25	motivation	motivation	NOUN
ejpam-5240	25	26	to	to	PART
ejpam-5240	25	27	introduce	introduce	VERB
ejpam-5240	25	28	and	and	CCONJ
ejpam-5240	25	29	explore	explore	VERB
ejpam-5240	25	30	a	a	DET
ejpam-5240	25	31	related	related	ADJ
ejpam-5240	25	32	study	study	NOUN
ejpam-5240	25	33	adopting	adopt	VERB
ejpam-5240	25	34	the	the	DET
ejpam-5240	25	35	area	area	NOUN
ejpam-5240	25	36	of	of	ADP
ejpam-5240	25	37	set	set	NOUN
ejpam-5240	25	38	theory	theory	NOUN
ejpam-5240	25	39	.	.	PUNCT
ejpam-5240	26	1	specifically	specifically	ADV
ejpam-5240	26	2	,	,	PUNCT
ejpam-5240	26	3	finding	find	VERB
ejpam-5240	26	4	an	an	DET
ejpam-5240	26	5	intersection	intersection	NOUN
ejpam-5240	26	6	graph	graph	NOUN
ejpam-5240	26	7	which	which	PRON
ejpam-5240	26	8	is	be	AUX
ejpam-5240	26	9	restricted	restrict	VERB
ejpam-5240	26	10	to	to	ADP
ejpam-5240	26	11	all	all	DET
ejpam-5240	26	12	the	the	DET
ejpam-5240	26	13	k	k	ADJ
ejpam-5240	26	14	-	-	ADJ
ejpam-5240	26	15	element	element	ADJ
ejpam-5240	26	16	subsets	subset	NOUN
ejpam-5240	26	17	of	of	ADP
ejpam-5240	26	18	an	an	DET
ejpam-5240	26	19	n	n	CCONJ
ejpam-5240	26	20	-	-	PUNCT
ejpam-5240	26	21	element	element	NOUN
ejpam-5240	26	22	set	set	NOUN
ejpam-5240	26	23	,	,	PUNCT
ejpam-5240	26	24	where	where	SCONJ
ejpam-5240	26	25	n	n	PRON
ejpam-5240	26	26	is	be	AUX
ejpam-5240	26	27	a	a	DET
ejpam-5240	26	28	positive	positive	ADJ
ejpam-5240	26	29	integer	integer	NOUN
ejpam-5240	26	30	and	and	CCONJ
ejpam-5240	26	31	k	k	PROPN
ejpam-5240	26	32	is	be	AUX
ejpam-5240	26	33	a	a	DET
ejpam-5240	26	34	nonnegative	nonnegative	ADJ
ejpam-5240	26	35	integer	integer	NOUN
ejpam-5240	26	36	such	such	ADJ
ejpam-5240	26	37	that	that	SCONJ
ejpam-5240	26	38	k	k	PROPN
ejpam-5240	26	39	≤	≤	PROPN
ejpam-5240	26	40	n.	n.	NOUN
ejpam-5240	26	41	hence	hence	ADV
ejpam-5240	26	42	,	,	PUNCT
ejpam-5240	26	43	by	by	ADP
ejpam-5240	26	44	using	use	VERB
ejpam-5240	26	45	the	the	DET
ejpam-5240	26	46	concepts	concept	NOUN
ejpam-5240	26	47	of	of	ADP
ejpam-5240	26	48	k	k	ADJ
ejpam-5240	26	49	-	-	ADJ
ejpam-5240	26	50	element	element	ADJ
ejpam-5240	26	51	subsets	subset	NOUN
ejpam-5240	26	52	and	and	CCONJ
ejpam-5240	26	53	intersection	intersection	NOUN
ejpam-5240	26	54	graphs	graph	NOUN
ejpam-5240	26	55	,	,	PUNCT
ejpam-5240	26	56	this	this	DET
ejpam-5240	26	57	paper	paper	NOUN
ejpam-5240	26	58	introduces	introduce	VERB
ejpam-5240	26	59	a	a	DET
ejpam-5240	26	60	k	k	ADV
ejpam-5240	26	61	-	-	ADJ
ejpam-5240	26	62	restricted	restricted	ADJ
ejpam-5240	26	63	intersection	intersection	NOUN
ejpam-5240	26	64	graph	graph	NOUN
ejpam-5240	26	65	,	,	PUNCT
ejpam-5240	26	66	which	which	PRON
ejpam-5240	26	67	is	be	AUX
ejpam-5240	26	68	an	an	DET
ejpam-5240	26	69	additional	additional	ADJ
ejpam-5240	26	70	variation	variation	NOUN
ejpam-5240	26	71	of	of	ADP
ejpam-5240	26	72	an	an	DET
ejpam-5240	26	73	intersection	intersection	NOUN
ejpam-5240	26	74	graph	graph	NOUN
ejpam-5240	26	75	.	.	PUNCT
ejpam-5240	27	1	to	to	PART
ejpam-5240	27	2	determine	determine	VERB
ejpam-5240	27	3	this	this	DET
ejpam-5240	27	4	graph	graph	NOUN
ejpam-5240	27	5	,	,	PUNCT
ejpam-5240	27	6	its	its	PRON
ejpam-5240	27	7	vertex	vertex	NOUN
ejpam-5240	27	8	set	set	NOUN
ejpam-5240	27	9	contains	contain	VERB
ejpam-5240	27	10	all	all	DET
ejpam-5240	27	11	the	the	DET
ejpam-5240	27	12	k	k	ADJ
ejpam-5240	27	13	-	-	ADJ
ejpam-5240	27	14	element	element	ADJ
ejpam-5240	27	15	subsets	subset	NOUN
ejpam-5240	27	16	of	of	ADP
ejpam-5240	27	17	an	an	DET
ejpam-5240	27	18	n	n	CCONJ
ejpam-5240	27	19	-	-	PUNCT
ejpam-5240	27	20	element	element	NOUN
ejpam-5240	27	21	set	set	NOUN
ejpam-5240	27	22	and	and	CCONJ
ejpam-5240	27	23	two	two	NUM
ejpam-5240	27	24	vertices	vertex	NOUN
ejpam-5240	27	25	are	be	AUX
ejpam-5240	27	26	adjacent	adjacent	ADJ
ejpam-5240	27	27	if	if	SCONJ
ejpam-5240	27	28	they	they	PRON
ejpam-5240	27	29	have	have	VERB
ejpam-5240	27	30	a	a	DET
ejpam-5240	27	31	nonempty	nonempty	ADJ
ejpam-5240	27	32	intersection	intersection	NOUN
ejpam-5240	27	33	.	.	PUNCT
ejpam-5240	28	1	with	with	ADP
ejpam-5240	28	2	these	these	DET
ejpam-5240	28	3	notions	notion	NOUN
ejpam-5240	28	4	,	,	PUNCT
ejpam-5240	28	5	it	it	PRON
ejpam-5240	28	6	can	can	AUX
ejpam-5240	28	7	be	be	AUX
ejpam-5240	28	8	gleaned	glean	VERB
ejpam-5240	28	9	that	that	SCONJ
ejpam-5240	28	10	the	the	DET
ejpam-5240	28	11	difference	difference	NOUN
ejpam-5240	28	12	between	between	ADP
ejpam-5240	28	13	an	an	DET
ejpam-5240	28	14	intersection	intersection	NOUN
ejpam-5240	28	15	graph	graph	NOUN
ejpam-5240	28	16	and	and	CCONJ
ejpam-5240	28	17	a	a	DET
ejpam-5240	28	18	k	k	ADV
ejpam-5240	28	19	-	-	PUNCT
ejpam-5240	28	20	restricted	restricted	ADJ
ejpam-5240	28	21	intersection	intersection	NOUN
ejpam-5240	28	22	graph	graph	NOUN
ejpam-5240	28	23	is	be	AUX
ejpam-5240	28	24	their	their	PRON
ejpam-5240	28	25	vertex	vertex	NOUN
ejpam-5240	28	26	set	set	VERB
ejpam-5240	28	27	wherein	wherein	SCONJ
ejpam-5240	28	28	the	the	DET
ejpam-5240	28	29	first	first	ADJ
ejpam-5240	28	30	graph	graph	NOUN
ejpam-5240	28	31	contains	contain	VERB
ejpam-5240	28	32	nonempty	nonempty	ADJ
ejpam-5240	28	33	family	family	NOUN
ejpam-5240	28	34	of	of	ADP
ejpam-5240	28	35	sets	set	NOUN
ejpam-5240	28	36	while	while	SCONJ
ejpam-5240	28	37	the	the	DET
ejpam-5240	28	38	second	second	ADJ
ejpam-5240	28	39	one	one	NOUN
ejpam-5240	28	40	involves	involve	VERB
ejpam-5240	28	41	the	the	DET
ejpam-5240	28	42	collection	collection	NOUN
ejpam-5240	28	43	of	of	ADP
ejpam-5240	28	44	all	all	DET
ejpam-5240	28	45	k	k	ADJ
ejpam-5240	28	46	-	-	ADJ
ejpam-5240	28	47	element	element	ADJ
ejpam-5240	28	48	subsets	subset	NOUN
ejpam-5240	28	49	of	of	ADP
ejpam-5240	28	50	a	a	DET
ejpam-5240	28	51	set	set	NOUN
ejpam-5240	28	52	.	.	PUNCT
ejpam-5240	29	1	this	this	DET
ejpam-5240	29	2	study	study	NOUN
ejpam-5240	29	3	introduces	introduce	VERB
ejpam-5240	29	4	a	a	DET
ejpam-5240	29	5	new	new	ADJ
ejpam-5240	29	6	variant	variant	NOUN
ejpam-5240	29	7	of	of	ADP
ejpam-5240	29	8	the	the	DET
ejpam-5240	29	9	intersection	intersection	NOUN
ejpam-5240	29	10	graph	graph	NOUN
ejpam-5240	29	11	which	which	PRON
ejpam-5240	29	12	is	be	AUX
ejpam-5240	29	13	the	the	DET
ejpam-5240	29	14	k	k	ADV
ejpam-5240	29	15	-	-	PUNCT
ejpam-5240	29	16	restricted	restricted	ADJ
ejpam-5240	29	17	intersection	intersection	NOUN
ejpam-5240	29	18	graph	graph	NOUN
ejpam-5240	29	19	.	.	PUNCT
ejpam-5240	30	1	the	the	DET
ejpam-5240	30	2	formal	formal	ADJ
ejpam-5240	30	3	definition	definition	NOUN
ejpam-5240	30	4	of	of	ADP
ejpam-5240	30	5	this	this	DET
ejpam-5240	30	6	graph	graph	NOUN
ejpam-5240	30	7	is	be	AUX
ejpam-5240	30	8	presented	present	VERB
ejpam-5240	30	9	in	in	ADP
ejpam-5240	30	10	chapter	chapter	NOUN
ejpam-5240	30	11	3	3	NUM
ejpam-5240	30	12	.	.	PUNCT
ejpam-5240	31	1	also	also	ADV
ejpam-5240	31	2	,	,	PUNCT
ejpam-5240	31	3	this	this	DET
ejpam-5240	31	4	study	study	NOUN
ejpam-5240	31	5	provides	provide	VERB
ejpam-5240	31	6	the	the	DET
ejpam-5240	31	7	conditions	condition	NOUN
ejpam-5240	31	8	when	when	SCONJ
ejpam-5240	31	9	a	a	DET
ejpam-5240	31	10	k	k	ADV
ejpam-5240	31	11	-	-	PUNCT
ejpam-5240	31	12	restricted	restricted	ADJ
ejpam-5240	31	13	intersection	intersection	NOUN
ejpam-5240	31	14	graph	graph	NOUN
ejpam-5240	31	15	is	be	AUX
ejpam-5240	31	16	isomorphic	isomorphic	ADJ
ejpam-5240	31	17	to	to	ADP
ejpam-5240	31	18	some	some	DET
ejpam-5240	31	19	special	special	ADJ
ejpam-5240	31	20	classes	class	NOUN
ejpam-5240	31	21	of	of	ADP
ejpam-5240	31	22	graph	graph	NOUN
ejpam-5240	31	23	.	.	PUNCT
ejpam-5240	32	1	lastly	lastly	ADV
ejpam-5240	32	2	,	,	PUNCT
ejpam-5240	32	3	some	some	PRON
ejpam-5240	32	4	of	of	ADP
ejpam-5240	32	5	the	the	DET
ejpam-5240	32	6	graph	graph	NOUN
ejpam-5240	32	7	parameters	parameter	NOUN
ejpam-5240	32	8	are	be	AUX
ejpam-5240	32	9	determined	determine	VERB
ejpam-5240	32	10	such	such	ADJ
ejpam-5240	32	11	as	as	ADP
ejpam-5240	32	12	the	the	DET
ejpam-5240	32	13	order	order	NOUN
ejpam-5240	32	14	,	,	PUNCT
ejpam-5240	32	15	size	size	NOUN
ejpam-5240	32	16	,	,	PUNCT
ejpam-5240	32	17	independence	independence	NOUN
ejpam-5240	32	18	number	number	NOUN
ejpam-5240	32	19	,	,	PUNCT
ejpam-5240	32	20	domination	domination	NOUN
ejpam-5240	32	21	number	number	NOUN
ejpam-5240	32	22	,	,	PUNCT
ejpam-5240	32	23	and	and	CCONJ
ejpam-5240	32	24	isolate	isolate	VERB
ejpam-5240	32	25	domination	domination	NOUN
ejpam-5240	32	26	number	number	NOUN
ejpam-5240	32	27	.	.	PUNCT
ejpam-5240	33	1	2	2	X
ejpam-5240	33	2	.	.	X
ejpam-5240	33	3	preliminaries	preliminary	NOUN
ejpam-5240	33	4	some	some	DET
ejpam-5240	33	5	necessary	necessary	ADJ
ejpam-5240	33	6	definitions	definition	NOUN
ejpam-5240	33	7	of	of	ADP
ejpam-5240	33	8	sets	set	NOUN
ejpam-5240	33	9	,	,	PUNCT
ejpam-5240	33	10	combinations	combination	NOUN
ejpam-5240	33	11	and	and	CCONJ
ejpam-5240	33	12	subsets	subset	NOUN
ejpam-5240	33	13	,	,	PUNCT
ejpam-5240	33	14	and	and	CCONJ
ejpam-5240	33	15	graph	graph	NOUN
ejpam-5240	33	16	theory	theory	NOUN
ejpam-5240	33	17	are	be	AUX
ejpam-5240	33	18	presented	present	VERB
ejpam-5240	33	19	in	in	ADP
ejpam-5240	33	20	this	this	DET
ejpam-5240	33	21	section	section	NOUN
ejpam-5240	33	22	.	.	PUNCT
ejpam-5240	34	1	also	also	ADV
ejpam-5240	34	2	,	,	PUNCT
ejpam-5240	34	3	the	the	DET
ejpam-5240	34	4	discussion	discussion	NOUN
ejpam-5240	34	5	includes	include	VERB
ejpam-5240	34	6	known	know	VERB
ejpam-5240	34	7	theorems	theorem	NOUN
ejpam-5240	34	8	from	from	ADP
ejpam-5240	34	9	combinatorics	combinatoric	NOUN
ejpam-5240	34	10	and	and	CCONJ
ejpam-5240	34	11	graph	graph	NOUN
ejpam-5240	34	12	theory	theory	NOUN
ejpam-5240	34	13	and	and	CCONJ
ejpam-5240	34	14	is	be	AUX
ejpam-5240	34	15	presented	present	VERB
ejpam-5240	34	16	without	without	ADP
ejpam-5240	34	17	proof	proof	NOUN
ejpam-5240	34	18	.	.	PUNCT
ejpam-5240	35	1	2.1	2.1	NUM
ejpam-5240	35	2	.	.	PUNCT
ejpam-5240	35	3	combination	combination	NOUN
ejpam-5240	35	4	and	and	CCONJ
ejpam-5240	35	5	subsets	subset	NOUN
ejpam-5240	35	6	this	this	DET
ejpam-5240	35	7	section	section	NOUN
ejpam-5240	35	8	contains	contain	VERB
ejpam-5240	35	9	the	the	DET
ejpam-5240	35	10	concept	concept	NOUN
ejpam-5240	35	11	of	of	ADP
ejpam-5240	35	12	applying	apply	VERB
ejpam-5240	35	13	the	the	DET
ejpam-5240	35	14	method	method	NOUN
ejpam-5240	35	15	of	of	ADP
ejpam-5240	35	16	combination	combination	NOUN
ejpam-5240	35	17	in	in	ADP
ejpam-5240	35	18	enumerating	enumerate	VERB
ejpam-5240	35	19	subsets	subset	NOUN
ejpam-5240	35	20	.	.	PUNCT
ejpam-5240	36	1	also	also	ADV
ejpam-5240	36	2	,	,	PUNCT
ejpam-5240	36	3	the	the	DET
ejpam-5240	36	4	notions	notion	NOUN
ejpam-5240	36	5	of	of	ADP
ejpam-5240	36	6	sets	set	NOUN
ejpam-5240	36	7	and	and	CCONJ
ejpam-5240	36	8	subsets	subset	NOUN
ejpam-5240	36	9	discussed	discuss	VERB
ejpam-5240	36	10	were	be	AUX
ejpam-5240	36	11	employed	employ	VERB
ejpam-5240	36	12	.	.	PUNCT
ejpam-5240	37	1	the	the	DET
ejpam-5240	37	2	references	reference	NOUN
ejpam-5240	37	3	used	use	VERB
ejpam-5240	37	4	are	be	AUX
ejpam-5240	37	5	found	find	VERB
ejpam-5240	37	6	in	in	ADP
ejpam-5240	37	7	[	[	X
ejpam-5240	37	8	3	3	NUM
ejpam-5240	37	9	]	]	PUNCT
ejpam-5240	37	10	,	,	PUNCT
ejpam-5240	37	11	[	[	X
ejpam-5240	37	12	4	4	NUM
ejpam-5240	37	13	]	]	PUNCT
ejpam-5240	37	14	,	,	PUNCT
ejpam-5240	37	15	and	and	CCONJ
ejpam-5240	37	16	[	[	X
ejpam-5240	37	17	8	8	NUM
ejpam-5240	37	18	]	]	PUNCT
ejpam-5240	37	19	.	.	PUNCT
ejpam-5240	38	1	a	a	DET
ejpam-5240	38	2	set	set	NOUN
ejpam-5240	38	3	s	s	PART
ejpam-5240	38	4	is	be	AUX
ejpam-5240	38	5	a	a	DET
ejpam-5240	38	6	collection	collection	NOUN
ejpam-5240	38	7	of	of	ADP
ejpam-5240	38	8	distinct	distinct	ADJ
ejpam-5240	38	9	well	well	ADV
ejpam-5240	38	10	-	-	PUNCT
ejpam-5240	38	11	defined	define	VERB
ejpam-5240	38	12	objects	object	NOUN
ejpam-5240	38	13	where	where	SCONJ
ejpam-5240	38	14	an	an	DET
ejpam-5240	38	15	‘	'	PUNCT
ejpam-5240	38	16	object	object	NOUN
ejpam-5240	38	17	’	'	PUNCT
ejpam-5240	38	18	is	be	AUX
ejpam-5240	38	19	a	a	DET
ejpam-5240	38	20	generic	generic	ADJ
ejpam-5240	38	21	term	term	NOUN
ejpam-5240	38	22	that	that	PRON
ejpam-5240	38	23	refers	refer	VERB
ejpam-5240	38	24	to	to	ADP
ejpam-5240	38	25	elements	element	NOUN
ejpam-5240	38	26	(	(	PUNCT
ejpam-5240	38	27	or	or	CCONJ
ejpam-5240	38	28	members	member	NOUN
ejpam-5240	38	29	)	)	PUNCT
ejpam-5240	38	30	of	of	ADP
ejpam-5240	38	31	the	the	DET
ejpam-5240	38	32	set	set	NOUN
ejpam-5240	38	33	.	.	PUNCT
ejpam-5240	39	1	the	the	DET
ejpam-5240	39	2	cardinality	cardinality	NOUN
ejpam-5240	39	3	of	of	ADP
ejpam-5240	39	4	s	s	PROPN
ejpam-5240	39	5	,	,	PUNCT
ejpam-5240	39	6	denoted	denote	VERB
ejpam-5240	39	7	by	by	ADP
ejpam-5240	39	8	|s|	|s|	PROPN
ejpam-5240	39	9	,	,	PUNCT
ejpam-5240	39	10	refers	refer	VERB
ejpam-5240	39	11	to	to	ADP
ejpam-5240	39	12	the	the	DET
ejpam-5240	39	13	number	number	NOUN
ejpam-5240	39	14	of	of	ADP
ejpam-5240	39	15	elements	element	NOUN
ejpam-5240	39	16	of	of	ADP
ejpam-5240	39	17	s.	s.	PROPN
ejpam-5240	39	18	if	if	SCONJ
ejpam-5240	39	19	x	x	PRON
ejpam-5240	39	20	is	be	AUX
ejpam-5240	39	21	an	an	DET
ejpam-5240	39	22	element	element	NOUN
ejpam-5240	39	23	of	of	ADP
ejpam-5240	39	24	s	s	PROPN
ejpam-5240	39	25	,	,	PUNCT
ejpam-5240	39	26	then	then	ADV
ejpam-5240	39	27	we	we	PRON
ejpam-5240	39	28	write	write	VERB
ejpam-5240	39	29	x	x	PUNCT
ejpam-5240	39	30	∈	∈	PROPN
ejpam-5240	39	31	s	s	PROPN
ejpam-5240	39	32	,	,	PUNCT
ejpam-5240	39	33	m.e	m.e	PROPN
ejpam-5240	39	34	.	.	PROPN
ejpam-5240	39	35	pelagio	pelagio	PROPN
ejpam-5240	39	36	,	,	PUNCT
ejpam-5240	39	37	n.	n.	NOUN
ejpam-5240	39	38	mame	mame	PROPN
ejpam-5240	39	39	,	,	PUNCT
ejpam-5240	39	40	k.	k.	PROPN
ejpam-5240	39	41	mendoza	mendoza	PROPN
ejpam-5240	39	42	/	/	SYM
ejpam-5240	39	43	eur	eur	PROPN
ejpam-5240	39	44	.	.	PUNCT
ejpam-5240	40	1	j.	j.	PROPN
ejpam-5240	40	2	pure	pure	PROPN
ejpam-5240	40	3	appl	appl	PROPN
ejpam-5240	40	4	.	.	PROPN
ejpam-5240	40	5	math	math	PROPN
ejpam-5240	40	6	,	,	PUNCT
ejpam-5240	40	7	17	17	NUM
ejpam-5240	40	8	(	(	PUNCT
ejpam-5240	40	9	3	3	NUM
ejpam-5240	40	10	)	)	PUNCT
ejpam-5240	40	11	(	(	PUNCT
ejpam-5240	40	12	2024	2024	NUM
ejpam-5240	40	13	)	)	PUNCT
ejpam-5240	40	14	,	,	PUNCT
ejpam-5240	40	15	1779	1779	NUM
ejpam-5240	40	16	-	-	SYM
ejpam-5240	40	17	1803	1803	NUM
ejpam-5240	40	18	1781	1781	NUM
ejpam-5240	40	19	otherwise	otherwise	ADV
ejpam-5240	40	20	,	,	PUNCT
ejpam-5240	40	21	x	x	PROPN
ejpam-5240	40	22	/∈	/∈	PUNCT
ejpam-5240	40	23	s.	s.	PROPN
ejpam-5240	41	1	also	also	ADV
ejpam-5240	41	2	,	,	PUNCT
ejpam-5240	41	3	if	if	SCONJ
ejpam-5240	41	4	s	s	PRON
ejpam-5240	41	5	has	have	VERB
ejpam-5240	41	6	no	no	DET
ejpam-5240	41	7	elements	element	NOUN
ejpam-5240	41	8	,	,	PUNCT
ejpam-5240	41	9	then	then	ADV
ejpam-5240	41	10	s	s	VERB
ejpam-5240	41	11	is	be	AUX
ejpam-5240	41	12	called	call	VERB
ejpam-5240	41	13	an	an	DET
ejpam-5240	41	14	empty	empty	ADJ
ejpam-5240	41	15	set	set	NOUN
ejpam-5240	41	16	and	and	CCONJ
ejpam-5240	41	17	is	be	AUX
ejpam-5240	41	18	written	write	VERB
ejpam-5240	41	19	as	as	ADP
ejpam-5240	41	20	s	s	NOUN
ejpam-5240	41	21	=	=	PUNCT
ejpam-5240	41	22	∅.	∅.	VERB
ejpam-5240	41	23	now	now	ADV
ejpam-5240	41	24	,	,	PUNCT
ejpam-5240	41	25	if	if	SCONJ
ejpam-5240	41	26	s	s	VERB
ejpam-5240	41	27	is	be	AUX
ejpam-5240	41	28	an	an	DET
ejpam-5240	41	29	n	n	CCONJ
ejpam-5240	41	30	-	-	PUNCT
ejpam-5240	41	31	element	element	NOUN
ejpam-5240	41	32	set	set	NOUN
ejpam-5240	41	33	,	,	PUNCT
ejpam-5240	41	34	then	then	ADV
ejpam-5240	41	35	we	we	PRON
ejpam-5240	41	36	can	can	AUX
ejpam-5240	41	37	rewrite	rewrite	VERB
ejpam-5240	41	38	this	this	PRON
ejpam-5240	41	39	as	as	ADP
ejpam-5240	41	40	sn	sn	PROPN
ejpam-5240	41	41	.	.	PUNCT
ejpam-5240	42	1	in	in	ADP
ejpam-5240	42	2	the	the	DET
ejpam-5240	42	3	succeeding	succeed	VERB
ejpam-5240	42	4	discussions	discussion	NOUN
ejpam-5240	42	5	,	,	PUNCT
ejpam-5240	42	6	sets	set	NOUN
ejpam-5240	42	7	with	with	ADP
ejpam-5240	42	8	indicated	indicate	VERB
ejpam-5240	42	9	cardinality	cardinality	NOUN
ejpam-5240	42	10	shall	shall	AUX
ejpam-5240	42	11	be	be	AUX
ejpam-5240	42	12	denoted	denote	VERB
ejpam-5240	42	13	as	as	ADP
ejpam-5240	42	14	sn	sn	PROPN
ejpam-5240	42	15	.	.	PUNCT
ejpam-5240	43	1	definition	definition	NOUN
ejpam-5240	43	2	1	1	NUM
ejpam-5240	43	3	.	.	PUNCT
ejpam-5240	44	1	let	let	VERB
ejpam-5240	44	2	s	s	PRON
ejpam-5240	44	3	and	and	CCONJ
ejpam-5240	44	4	t	t	PROPN
ejpam-5240	44	5	be	be	AUX
ejpam-5240	44	6	sets	set	NOUN
ejpam-5240	44	7	.	.	PUNCT
ejpam-5240	45	1	then	then	ADV
ejpam-5240	45	2	t	t	PROPN
ejpam-5240	45	3	is	be	AUX
ejpam-5240	45	4	a	a	DET
ejpam-5240	45	5	subset	subset	NOUN
ejpam-5240	45	6	of	of	ADP
ejpam-5240	45	7	s	s	PROPN
ejpam-5240	45	8	,	,	PUNCT
ejpam-5240	45	9	written	write	VERB
ejpam-5240	45	10	t	t	PROPN
ejpam-5240	45	11	⊆	⊆	NUM
ejpam-5240	45	12	s	s	NOUN
ejpam-5240	45	13	,	,	PUNCT
ejpam-5240	45	14	if	if	SCONJ
ejpam-5240	45	15	for	for	ADP
ejpam-5240	45	16	all	all	DET
ejpam-5240	45	17	x	x	SYM
ejpam-5240	45	18	∈	∈	PROPN
ejpam-5240	45	19	t	t	NOUN
ejpam-5240	45	20	,	,	PUNCT
ejpam-5240	45	21	then	then	ADV
ejpam-5240	45	22	x	x	SYM
ejpam-5240	45	23	∈	∈	PROPN
ejpam-5240	45	24	s.	s.	PROPN
ejpam-5240	45	25	a	a	DET
ejpam-5240	45	26	power	power	NOUN
ejpam-5240	45	27	set	set	NOUN
ejpam-5240	45	28	of	of	ADP
ejpam-5240	45	29	s	s	PROPN
ejpam-5240	45	30	,	,	PUNCT
ejpam-5240	45	31	denoted	denote	VERB
ejpam-5240	45	32	by	by	ADP
ejpam-5240	45	33	p(s	p(s	NOUN
ejpam-5240	45	34	)	)	PUNCT
ejpam-5240	45	35	,	,	PUNCT
ejpam-5240	45	36	is	be	AUX
ejpam-5240	45	37	the	the	DET
ejpam-5240	45	38	set	set	NOUN
ejpam-5240	45	39	containing	contain	VERB
ejpam-5240	45	40	all	all	DET
ejpam-5240	45	41	the	the	DET
ejpam-5240	45	42	subsets	subset	NOUN
ejpam-5240	45	43	of	of	ADP
ejpam-5240	45	44	s.	s.	PROPN
ejpam-5240	45	45	note	note	VERB
ejpam-5240	45	46	that	that	SCONJ
ejpam-5240	45	47	a	a	DET
ejpam-5240	45	48	set	set	NOUN
ejpam-5240	45	49	is	be	AUX
ejpam-5240	45	50	also	also	ADV
ejpam-5240	45	51	a	a	DET
ejpam-5240	45	52	subset	subset	NOUN
ejpam-5240	45	53	of	of	ADP
ejpam-5240	45	54	itself	itself	PRON
ejpam-5240	45	55	.	.	PUNCT
ejpam-5240	46	1	now	now	ADV
ejpam-5240	46	2	,	,	PUNCT
ejpam-5240	46	3	let	let	VERB
ejpam-5240	46	4	s	s	PRON
ejpam-5240	46	5	and	and	CCONJ
ejpam-5240	46	6	t	t	PROPN
ejpam-5240	46	7	be	be	AUX
ejpam-5240	46	8	nonempty	nonempty	ADJ
ejpam-5240	46	9	sets	set	NOUN
ejpam-5240	46	10	.	.	PUNCT
ejpam-5240	47	1	the	the	DET
ejpam-5240	47	2	difference	difference	NOUN
ejpam-5240	47	3	between	between	ADP
ejpam-5240	47	4	s	s	PRON
ejpam-5240	47	5	and	and	CCONJ
ejpam-5240	47	6	t	t	PROPN
ejpam-5240	47	7	,	,	PUNCT
ejpam-5240	47	8	written	write	VERB
ejpam-5240	47	9	s	s	PRON
ejpam-5240	47	10	\t	\t	NOUN
ejpam-5240	47	11	and	and	CCONJ
ejpam-5240	47	12	read	read	VERB
ejpam-5240	47	13	as	as	ADP
ejpam-5240	47	14	“	"	PUNCT
ejpam-5240	47	15	s	s	PROPN
ejpam-5240	47	16	minus	minus	ADP
ejpam-5240	47	17	t	t	PROPN
ejpam-5240	47	18	”	"	PUNCT
ejpam-5240	47	19	,	,	PUNCT
ejpam-5240	47	20	is	be	AUX
ejpam-5240	47	21	the	the	DET
ejpam-5240	47	22	set	set	NOUN
ejpam-5240	47	23	containing	contain	VERB
ejpam-5240	47	24	all	all	DET
ejpam-5240	47	25	elements	element	NOUN
ejpam-5240	47	26	of	of	ADP
ejpam-5240	47	27	s	s	PRON
ejpam-5240	47	28	that	that	PRON
ejpam-5240	47	29	are	be	AUX
ejpam-5240	47	30	not	not	PART
ejpam-5240	47	31	in	in	ADP
ejpam-5240	47	32	t	t	PROPN
ejpam-5240	47	33	.	.	PUNCT
ejpam-5240	48	1	there	there	PRON
ejpam-5240	48	2	is	be	VERB
ejpam-5240	48	3	no	no	DET
ejpam-5240	48	4	defined	define	VERB
ejpam-5240	48	5	cardinality	cardinality	NOUN
ejpam-5240	48	6	for	for	ADP
ejpam-5240	48	7	a	a	DET
ejpam-5240	48	8	generalized	generalized	ADJ
ejpam-5240	48	9	difference	difference	NOUN
ejpam-5240	48	10	between	between	ADP
ejpam-5240	48	11	two	two	NUM
ejpam-5240	48	12	sets	set	NOUN
ejpam-5240	48	13	.	.	PUNCT
ejpam-5240	49	1	however	however	ADV
ejpam-5240	49	2	,	,	PUNCT
ejpam-5240	49	3	if	if	SCONJ
ejpam-5240	49	4	we	we	PRON
ejpam-5240	49	5	get	get	VERB
ejpam-5240	49	6	the	the	DET
ejpam-5240	49	7	difference	difference	NOUN
ejpam-5240	49	8	between	between	ADP
ejpam-5240	49	9	a	a	DET
ejpam-5240	49	10	set	set	NOUN
ejpam-5240	49	11	and	and	CCONJ
ejpam-5240	49	12	its	its	PRON
ejpam-5240	49	13	subset	subset	NOUN
ejpam-5240	49	14	,	,	PUNCT
ejpam-5240	49	15	then	then	ADV
ejpam-5240	49	16	we	we	PRON
ejpam-5240	49	17	can	can	AUX
ejpam-5240	49	18	easily	easily	ADV
ejpam-5240	49	19	tell	tell	VERB
ejpam-5240	49	20	the	the	DET
ejpam-5240	49	21	cardinality	cardinality	NOUN
ejpam-5240	49	22	of	of	ADP
ejpam-5240	49	23	their	their	PRON
ejpam-5240	49	24	difference	difference	NOUN
ejpam-5240	49	25	.	.	PUNCT
ejpam-5240	50	1	remark	remark	PROPN
ejpam-5240	50	2	1	1	NUM
ejpam-5240	50	3	.	.	PUNCT
ejpam-5240	51	1	let	let	VERB
ejpam-5240	51	2	s	s	PRON
ejpam-5240	51	3	and	and	CCONJ
ejpam-5240	51	4	t	t	PROPN
ejpam-5240	51	5	be	be	AUX
ejpam-5240	51	6	nonempty	nonempty	ADJ
ejpam-5240	51	7	sets	set	NOUN
ejpam-5240	51	8	.	.	PUNCT
ejpam-5240	52	1	if	if	SCONJ
ejpam-5240	52	2	t	t	PROPN
ejpam-5240	52	3	⊆	⊆	NUM
ejpam-5240	52	4	s	s	NOUN
ejpam-5240	52	5	,	,	PUNCT
ejpam-5240	52	6	then	then	ADV
ejpam-5240	52	7	|s	|s	PROPN
ejpam-5240	52	8	\	\	PROPN
ejpam-5240	52	9	t	t	PROPN
ejpam-5240	52	10	|	|	NOUN
ejpam-5240	52	11	=	=	PUNCT
ejpam-5240	52	12	|s|	|s|	PROPN
ejpam-5240	52	13	−	−	PROPN
ejpam-5240	52	14	|t	|t	NOUN
ejpam-5240	52	15	|	|	ADV
ejpam-5240	52	16	.	.	PUNCT
ejpam-5240	53	1	consider	consider	VERB
ejpam-5240	53	2	the	the	DET
ejpam-5240	53	3	nonempty	nonempty	ADJ
ejpam-5240	53	4	sets	set	NOUN
ejpam-5240	53	5	s	s	PART
ejpam-5240	53	6	and	and	CCONJ
ejpam-5240	53	7	t	t	PROPN
ejpam-5240	53	8	.	.	PUNCT
ejpam-5240	54	1	the	the	DET
ejpam-5240	54	2	intersection	intersection	NOUN
ejpam-5240	54	3	of	of	ADP
ejpam-5240	54	4	s	s	PRON
ejpam-5240	54	5	and	and	CCONJ
ejpam-5240	54	6	t	t	PROPN
ejpam-5240	54	7	,	,	PUNCT
ejpam-5240	54	8	denoted	denote	VERB
ejpam-5240	54	9	by	by	ADP
ejpam-5240	54	10	s∩t	s∩t	PROPN
ejpam-5240	54	11	,	,	PUNCT
ejpam-5240	54	12	is	be	AUX
ejpam-5240	54	13	defined	define	VERB
ejpam-5240	54	14	as	as	ADP
ejpam-5240	54	15	the	the	DET
ejpam-5240	54	16	set	set	NOUN
ejpam-5240	54	17	containing	contain	VERB
ejpam-5240	54	18	all	all	DET
ejpam-5240	54	19	the	the	DET
ejpam-5240	54	20	elements	element	NOUN
ejpam-5240	54	21	that	that	PRON
ejpam-5240	54	22	belong	belong	VERB
ejpam-5240	54	23	to	to	ADP
ejpam-5240	54	24	both	both	DET
ejpam-5240	54	25	s	s	PROPN
ejpam-5240	54	26	and	and	CCONJ
ejpam-5240	54	27	t	t	PROPN
ejpam-5240	54	28	.	.	PUNCT
ejpam-5240	55	1	if	if	SCONJ
ejpam-5240	55	2	two	two	NUM
ejpam-5240	55	3	sets	set	NOUN
ejpam-5240	55	4	do	do	AUX
ejpam-5240	55	5	not	not	PART
ejpam-5240	55	6	have	have	VERB
ejpam-5240	55	7	any	any	DET
ejpam-5240	55	8	element(s	element(s	PROPN
ejpam-5240	55	9	)	)	PUNCT
ejpam-5240	55	10	in	in	ADP
ejpam-5240	55	11	common	common	ADJ
ejpam-5240	55	12	,	,	PUNCT
ejpam-5240	55	13	then	then	ADV
ejpam-5240	55	14	we	we	PRON
ejpam-5240	55	15	call	call	VERB
ejpam-5240	55	16	this	this	PRON
ejpam-5240	55	17	as	as	ADP
ejpam-5240	55	18	an	an	DET
ejpam-5240	55	19	empty	empty	ADJ
ejpam-5240	55	20	intersection	intersection	NOUN
ejpam-5240	55	21	,	,	PUNCT
ejpam-5240	55	22	denoted	denote	VERB
ejpam-5240	55	23	by	by	ADP
ejpam-5240	55	24	s	s	NOUN
ejpam-5240	55	25	∩	∩	PROPN
ejpam-5240	55	26	t	t	NOUN
ejpam-5240	55	27	=	=	PUNCT
ejpam-5240	55	28	∅.	∅.	VERB
ejpam-5240	55	29	otherwise	otherwise	ADV
ejpam-5240	55	30	,	,	PUNCT
ejpam-5240	55	31	we	we	PRON
ejpam-5240	55	32	write	write	VERB
ejpam-5240	55	33	as	as	ADP
ejpam-5240	55	34	s	s	PROPN
ejpam-5240	55	35	∩	∩	NOUN
ejpam-5240	55	36	t	t	PROPN
ejpam-5240	55	37	̸=	̸=	PROPN
ejpam-5240	55	38	∅	∅	NOUN
ejpam-5240	55	39	and	and	CCONJ
ejpam-5240	55	40	refer	refer	VERB
ejpam-5240	55	41	this	this	PRON
ejpam-5240	55	42	as	as	ADP
ejpam-5240	55	43	a	a	DET
ejpam-5240	55	44	nonempty	nonempty	ADJ
ejpam-5240	55	45	intersection	intersection	NOUN
ejpam-5240	55	46	.	.	PUNCT
ejpam-5240	56	1	we	we	PRON
ejpam-5240	56	2	now	now	ADV
ejpam-5240	56	3	introduce	introduce	VERB
ejpam-5240	56	4	the	the	DET
ejpam-5240	56	5	notion	notion	NOUN
ejpam-5240	56	6	of	of	ADP
ejpam-5240	56	7	binomial	binomial	ADJ
ejpam-5240	56	8	coefficient	coefficient	NOUN
ejpam-5240	56	9	.	.	PUNCT
ejpam-5240	57	1	for	for	ADP
ejpam-5240	57	2	the	the	DET
ejpam-5240	57	3	integers	integer	NOUN
ejpam-5240	57	4	n	n	CCONJ
ejpam-5240	57	5	,	,	PUNCT
ejpam-5240	57	6	k	k	PROPN
ejpam-5240	57	7	≥	≥	PROPN
ejpam-5240	57	8	0	0	NUM
ejpam-5240	57	9	,	,	PUNCT
ejpam-5240	57	10	and	and	CCONJ
ejpam-5240	57	11	0	0	NUM
ejpam-5240	57	12	≤	≤	NUM
ejpam-5240	57	13	k	k	X
ejpam-5240	57	14	≤	≤	PROPN
ejpam-5240	57	15	n	n	CCONJ
ejpam-5240	57	16	,	,	PUNCT
ejpam-5240	57	17	the	the	DET
ejpam-5240	57	18	binomial	binomial	ADJ
ejpam-5240	57	19	coefficient	coefficient	NOUN
ejpam-5240	57	20	is	be	AUX
ejpam-5240	57	21	given	give	VERB
ejpam-5240	57	22	by	by	ADP
ejpam-5240	57	23	(	(	PUNCT
ejpam-5240	57	24	n	n	X
ejpam-5240	57	25	k	k	NOUN
ejpam-5240	57	26	)	)	PUNCT
ejpam-5240	58	1	=	=	SYM
ejpam-5240	58	2	n	n	X
ejpam-5240	58	3	!	!	PUNCT
ejpam-5240	58	4	k!(n−	k!(n−	PROPN
ejpam-5240	59	1	k	k	X
ejpam-5240	59	2	)	)	PUNCT
ejpam-5240	59	3	!	!	PUNCT
ejpam-5240	59	4	.	.	PUNCT
ejpam-5240	60	1	note	note	VERB
ejpam-5240	60	2	that	that	SCONJ
ejpam-5240	60	3	the	the	DET
ejpam-5240	60	4	number	number	NOUN
ejpam-5240	60	5	(	(	PUNCT
ejpam-5240	60	6	n	n	NOUN
ejpam-5240	60	7	k	k	PROPN
ejpam-5240	60	8	)	)	PUNCT
ejpam-5240	60	9	,	,	PUNCT
ejpam-5240	60	10	read	read	VERB
ejpam-5240	60	11	as	as	ADP
ejpam-5240	60	12	“	"	PUNCT
ejpam-5240	60	13	n	n	CCONJ
ejpam-5240	60	14	choose	choose	VERB
ejpam-5240	60	15	k”and	k”and	NOUN
ejpam-5240	60	16	also	also	ADV
ejpam-5240	60	17	called	call	VERB
ejpam-5240	60	18	as	as	ADP
ejpam-5240	60	19	a	a	DET
ejpam-5240	60	20	combination	combination	NOUN
ejpam-5240	60	21	or	or	CCONJ
ejpam-5240	60	22	combinatorial	combinatorial	ADJ
ejpam-5240	60	23	formula	formula	NOUN
ejpam-5240	60	24	,	,	PUNCT
ejpam-5240	60	25	is	be	AUX
ejpam-5240	60	26	the	the	DET
ejpam-5240	60	27	number	number	NOUN
ejpam-5240	60	28	of	of	ADP
ejpam-5240	60	29	ways	way	NOUN
ejpam-5240	60	30	choosing	choose	VERB
ejpam-5240	60	31	k	k	PROPN
ejpam-5240	60	32	unordered	unordered	ADJ
ejpam-5240	60	33	outcomes	outcome	NOUN
ejpam-5240	60	34	from	from	ADP
ejpam-5240	60	35	n	n	DET
ejpam-5240	60	36	possibilities	possibility	NOUN
ejpam-5240	60	37	.	.	PUNCT
ejpam-5240	61	1	we	we	PRON
ejpam-5240	61	2	will	will	AUX
ejpam-5240	61	3	utilize	utilize	VERB
ejpam-5240	61	4	this	this	DET
ejpam-5240	61	5	formula	formula	NOUN
ejpam-5240	61	6	to	to	PART
ejpam-5240	61	7	count	count	VERB
ejpam-5240	61	8	how	how	SCONJ
ejpam-5240	61	9	many	many	ADJ
ejpam-5240	61	10	k	k	ADJ
ejpam-5240	61	11	-	-	ADJ
ejpam-5240	61	12	element	element	ADJ
ejpam-5240	61	13	subsets	subset	NOUN
ejpam-5240	61	14	can	can	AUX
ejpam-5240	61	15	be	be	AUX
ejpam-5240	61	16	formed	form	VERB
ejpam-5240	61	17	from	from	ADP
ejpam-5240	61	18	sn	sn	PROPN
ejpam-5240	61	19	.	.	PUNCT
ejpam-5240	61	20	remark	remark	PROPN
ejpam-5240	61	21	2	2	NUM
ejpam-5240	61	22	.	.	PUNCT
ejpam-5240	62	1	the	the	DET
ejpam-5240	62	2	binomial	binomial	ADJ
ejpam-5240	62	3	coefficient	coefficient	NOUN
ejpam-5240	62	4	(	(	PUNCT
ejpam-5240	62	5	0	0	NUM
ejpam-5240	62	6	0	0	NUM
ejpam-5240	62	7	)	)	PUNCT
ejpam-5240	62	8	is	be	AUX
ejpam-5240	62	9	equal	equal	ADJ
ejpam-5240	62	10	to	to	ADP
ejpam-5240	62	11	1	1	NUM
ejpam-5240	62	12	.	.	PUNCT
ejpam-5240	62	13	remark	remark	NOUN
ejpam-5240	62	14	2	2	NUM
ejpam-5240	62	15	discusses	discuss	VERB
ejpam-5240	62	16	the	the	DET
ejpam-5240	62	17	trivial	trivial	ADJ
ejpam-5240	62	18	case	case	NOUN
ejpam-5240	62	19	for	for	ADP
ejpam-5240	62	20	the	the	DET
ejpam-5240	62	21	binomial	binomial	ADJ
ejpam-5240	62	22	coefficient	coefficient	NOUN
ejpam-5240	62	23	.	.	PUNCT
ejpam-5240	63	1	now	now	ADV
ejpam-5240	63	2	,	,	PUNCT
ejpam-5240	63	3	consider	consider	VERB
ejpam-5240	63	4	the	the	DET
ejpam-5240	63	5	0	0	NUM
ejpam-5240	63	6	-	-	PUNCT
ejpam-5240	63	7	element	element	NOUN
ejpam-5240	63	8	set	set	VERB
ejpam-5240	63	9	s0	s0	NOUN
ejpam-5240	63	10	=	=	PUNCT
ejpam-5240	63	11	∅.	∅.	AUX
ejpam-5240	63	12	observe	observe	VERB
ejpam-5240	63	13	that	that	SCONJ
ejpam-5240	63	14	the	the	DET
ejpam-5240	63	15	only	only	ADJ
ejpam-5240	63	16	subset	subset	NOUN
ejpam-5240	63	17	that	that	SCONJ
ejpam-5240	63	18	every	every	DET
ejpam-5240	63	19	s0	s0	NOUN
ejpam-5240	63	20	has	have	VERB
ejpam-5240	63	21	empty	empty	ADJ
ejpam-5240	63	22	subset	subset	NOUN
ejpam-5240	63	23	.	.	PUNCT
ejpam-5240	64	1	by	by	ADP
ejpam-5240	64	2	this	this	PRON
ejpam-5240	64	3	,	,	PUNCT
ejpam-5240	64	4	we	we	PRON
ejpam-5240	64	5	will	will	AUX
ejpam-5240	64	6	also	also	ADV
ejpam-5240	64	7	consider	consider	VERB
ejpam-5240	64	8	the	the	DET
ejpam-5240	64	9	0	0	NUM
ejpam-5240	64	10	-	-	PUNCT
ejpam-5240	64	11	element	element	NOUN
ejpam-5240	64	12	set	set	VERB
ejpam-5240	64	13	s0	s0	NOUN
ejpam-5240	64	14	as	as	ADP
ejpam-5240	64	15	a	a	DET
ejpam-5240	64	16	trivial	trivial	ADJ
ejpam-5240	64	17	case	case	NOUN
ejpam-5240	64	18	.	.	PUNCT
ejpam-5240	65	1	remark	remark	NOUN
ejpam-5240	65	2	3	3	NUM
ejpam-5240	65	3	.	.	PUNCT
ejpam-5240	66	1	the	the	DET
ejpam-5240	66	2	only	only	ADJ
ejpam-5240	66	3	subset	subset	NOUN
ejpam-5240	66	4	of	of	ADP
ejpam-5240	66	5	s0	s0	PROPN
ejpam-5240	66	6	is	be	AUX
ejpam-5240	66	7	∅.	∅.	AUX
ejpam-5240	66	8	given	give	VERB
ejpam-5240	66	9	remark	remark	NOUN
ejpam-5240	66	10	3	3	NUM
ejpam-5240	66	11	,	,	PUNCT
ejpam-5240	66	12	sn	sn	NOUN
ejpam-5240	66	13	with	with	ADP
ejpam-5240	66	14	n	n	NOUN
ejpam-5240	66	15	=	=	SYM
ejpam-5240	66	16	0	0	NUM
ejpam-5240	66	17	will	will	AUX
ejpam-5240	66	18	be	be	AUX
ejpam-5240	66	19	regarded	regard	VERB
ejpam-5240	66	20	as	as	ADP
ejpam-5240	66	21	a	a	DET
ejpam-5240	66	22	trivial	trivial	ADJ
ejpam-5240	66	23	case	case	NOUN
ejpam-5240	66	24	for	for	ADP
ejpam-5240	66	25	the	the	DET
ejpam-5240	66	26	set	set	PROPN
ejpam-5240	66	27	sn	sn	NOUN
ejpam-5240	66	28	.	.	PUNCT
ejpam-5240	67	1	with	with	ADP
ejpam-5240	67	2	this	this	PRON
ejpam-5240	67	3	,	,	PUNCT
ejpam-5240	67	4	the	the	DET
ejpam-5240	67	5	proceeding	proceeding	NOUN
ejpam-5240	67	6	discussions	discussion	NOUN
ejpam-5240	67	7	shall	shall	AUX
ejpam-5240	67	8	now	now	ADV
ejpam-5240	67	9	be	be	AUX
ejpam-5240	67	10	focused	focus	VERB
ejpam-5240	67	11	on	on	ADP
ejpam-5240	67	12	the	the	DET
ejpam-5240	67	13	n	n	CCONJ
ejpam-5240	67	14	-	-	PUNCT
ejpam-5240	67	15	element	element	NOUN
ejpam-5240	67	16	set	set	NOUN
ejpam-5240	67	17	sn	sn	PROPN
ejpam-5240	67	18	where	where	SCONJ
ejpam-5240	67	19	n	n	PRON
ejpam-5240	67	20	is	be	AUX
ejpam-5240	67	21	a	a	DET
ejpam-5240	67	22	positive	positive	ADJ
ejpam-5240	67	23	integer	integer	NOUN
ejpam-5240	67	24	.	.	PUNCT
ejpam-5240	68	1	theorem	theorem	NOUN
ejpam-5240	68	2	1	1	NUM
ejpam-5240	68	3	.	.	X
ejpam-5240	69	1	for	for	ADP
ejpam-5240	69	2	any	any	DET
ejpam-5240	69	3	positive	positive	ADJ
ejpam-5240	69	4	integer	integer	NOUN
ejpam-5240	69	5	n	n	CCONJ
ejpam-5240	69	6	,	,	PUNCT
ejpam-5240	69	7	(	(	PUNCT
ejpam-5240	69	8	n	n	X
ejpam-5240	69	9	k	k	NOUN
ejpam-5240	69	10	)	)	PUNCT
ejpam-5240	69	11	=	=	SYM
ejpam-5240	69	12	1	1	NUM
ejpam-5240	69	13	if	if	SCONJ
ejpam-5240	69	14	and	and	CCONJ
ejpam-5240	69	15	only	only	ADV
ejpam-5240	69	16	if	if	SCONJ
ejpam-5240	69	17	k	k	PROPN
ejpam-5240	69	18	=	=	SYM
ejpam-5240	69	19	0	0	PROPN
ejpam-5240	69	20	or	or	CCONJ
ejpam-5240	69	21	k	k	PROPN
ejpam-5240	69	22	=	=	SYM
ejpam-5240	69	23	n.	n.	PROPN
ejpam-5240	69	24	theorem	theorem	NOUN
ejpam-5240	69	25	1	1	NUM
ejpam-5240	69	26	is	be	AUX
ejpam-5240	69	27	the	the	DET
ejpam-5240	69	28	boundary	boundary	ADJ
ejpam-5240	69	29	values	value	NOUN
ejpam-5240	69	30	for	for	ADP
ejpam-5240	69	31	the	the	DET
ejpam-5240	69	32	recursive	recursive	ADJ
ejpam-5240	69	33	formula	formula	NOUN
ejpam-5240	69	34	of	of	ADP
ejpam-5240	69	35	the	the	DET
ejpam-5240	69	36	binomial	binomial	ADJ
ejpam-5240	69	37	coefficient	coefficient	NOUN
ejpam-5240	69	38	.	.	PUNCT
ejpam-5240	70	1	the	the	DET
ejpam-5240	70	2	readers	reader	NOUN
ejpam-5240	70	3	may	may	AUX
ejpam-5240	70	4	refer	refer	VERB
ejpam-5240	70	5	to	to	ADP
ejpam-5240	70	6	[	[	X
ejpam-5240	70	7	2	2	X
ejpam-5240	70	8	]	]	PUNCT
ejpam-5240	70	9	for	for	ADP
ejpam-5240	70	10	the	the	DET
ejpam-5240	70	11	proof	proof	NOUN
ejpam-5240	70	12	of	of	ADP
ejpam-5240	70	13	theorem	theorem	NOUN
ejpam-5240	70	14	1	1	NUM
ejpam-5240	70	15	.	.	PUNCT
ejpam-5240	70	16	m.e	m.e	PROPN
ejpam-5240	70	17	.	.	PROPN
ejpam-5240	70	18	pelagio	pelagio	PROPN
ejpam-5240	70	19	,	,	PUNCT
ejpam-5240	70	20	n.	n.	NOUN
ejpam-5240	70	21	mame	mame	PROPN
ejpam-5240	70	22	,	,	PUNCT
ejpam-5240	70	23	k.	k.	PROPN
ejpam-5240	70	24	mendoza	mendoza	PROPN
ejpam-5240	70	25	/	/	SYM
ejpam-5240	70	26	eur	eur	PROPN
ejpam-5240	70	27	.	.	PUNCT
ejpam-5240	71	1	j.	j.	PROPN
ejpam-5240	71	2	pure	pure	PROPN
ejpam-5240	71	3	appl	appl	PROPN
ejpam-5240	71	4	.	.	PROPN
ejpam-5240	71	5	math	math	PROPN
ejpam-5240	71	6	,	,	PUNCT
ejpam-5240	71	7	17	17	NUM
ejpam-5240	71	8	(	(	PUNCT
ejpam-5240	71	9	3	3	NUM
ejpam-5240	71	10	)	)	PUNCT
ejpam-5240	71	11	(	(	PUNCT
ejpam-5240	71	12	2024	2024	NUM
ejpam-5240	71	13	)	)	PUNCT
ejpam-5240	71	14	,	,	PUNCT
ejpam-5240	71	15	1779	1779	NUM
ejpam-5240	71	16	-	-	SYM
ejpam-5240	71	17	1803	1803	NUM
ejpam-5240	71	18	1782	1782	NUM
ejpam-5240	71	19	definition	definition	NOUN
ejpam-5240	71	20	2	2	NUM
ejpam-5240	71	21	.	.	PUNCT
ejpam-5240	72	1	let	let	VERB
ejpam-5240	72	2	sn	sn	PROPN
ejpam-5240	72	3	be	be	AUX
ejpam-5240	72	4	an	an	DET
ejpam-5240	72	5	n	n	NOUN
ejpam-5240	72	6	-	-	PUNCT
ejpam-5240	72	7	element	element	NOUN
ejpam-5240	72	8	set	set	NOUN
ejpam-5240	72	9	where	where	SCONJ
ejpam-5240	72	10	n	n	PRON
ejpam-5240	72	11	is	be	AUX
ejpam-5240	72	12	a	a	DET
ejpam-5240	72	13	positive	positive	ADJ
ejpam-5240	72	14	integer	integer	NOUN
ejpam-5240	72	15	and	and	CCONJ
ejpam-5240	72	16	let	let	VERB
ejpam-5240	72	17	k	k	PRON
ejpam-5240	72	18	be	be	AUX
ejpam-5240	72	19	a	a	DET
ejpam-5240	72	20	nonnegative	nonnegative	ADJ
ejpam-5240	72	21	integer	integer	NOUN
ejpam-5240	72	22	such	such	ADJ
ejpam-5240	72	23	that	that	SCONJ
ejpam-5240	72	24	k	k	PROPN
ejpam-5240	72	25	≤	≤	PROPN
ejpam-5240	72	26	n.	n.	NOUN
ejpam-5240	72	27	then	then	ADV
ejpam-5240	72	28	the	the	DET
ejpam-5240	72	29	set	set	NOUN
ejpam-5240	72	30	containing	contain	VERB
ejpam-5240	72	31	all	all	DET
ejpam-5240	72	32	k	k	ADJ
ejpam-5240	72	33	-	-	ADJ
ejpam-5240	72	34	element	element	ADJ
ejpam-5240	72	35	subsets	subset	NOUN
ejpam-5240	72	36	of	of	ADP
ejpam-5240	72	37	sn	sn	PROPN
ejpam-5240	72	38	,	,	PUNCT
ejpam-5240	72	39	written	write	VERB
ejpam-5240	72	40	as	as	ADP
ejpam-5240	72	41	s(n	s(n	PROPN
ejpam-5240	72	42	,	,	PUNCT
ejpam-5240	72	43	k	k	NOUN
ejpam-5240	72	44	)	)	PUNCT
ejpam-5240	72	45	,	,	PUNCT
ejpam-5240	72	46	is	be	AUX
ejpam-5240	72	47	the	the	DET
ejpam-5240	72	48	set	set	NOUN
ejpam-5240	72	49	containing	contain	VERB
ejpam-5240	72	50	all	all	DET
ejpam-5240	72	51	the	the	DET
ejpam-5240	72	52	subsets	subset	NOUN
ejpam-5240	72	53	of	of	ADP
ejpam-5240	72	54	sn	sn	PROPN
ejpam-5240	72	55	having	have	VERB
ejpam-5240	72	56	k	k	NOUN
ejpam-5240	72	57	-	-	NOUN
ejpam-5240	72	58	elements	element	NOUN
ejpam-5240	72	59	.	.	PUNCT
ejpam-5240	73	1	since	since	SCONJ
ejpam-5240	73	2	this	this	DET
ejpam-5240	73	3	study	study	NOUN
ejpam-5240	73	4	does	do	AUX
ejpam-5240	73	5	not	not	PART
ejpam-5240	73	6	consider	consider	VERB
ejpam-5240	73	7	multiset	multiset	VERB
ejpam-5240	73	8	,	,	PUNCT
ejpam-5240	73	9	it	it	PRON
ejpam-5240	73	10	follows	follow	VERB
ejpam-5240	73	11	that	that	SCONJ
ejpam-5240	73	12	s(n	s(n	PROPN
ejpam-5240	73	13	,	,	PUNCT
ejpam-5240	73	14	k	k	NOUN
ejpam-5240	73	15	)	)	PUNCT
ejpam-5240	73	16	contains	contain	VERB
ejpam-5240	73	17	the	the	DET
ejpam-5240	73	18	distinct	distinct	ADJ
ejpam-5240	73	19	k	k	NOUN
ejpam-5240	73	20	-	-	PUNCT
ejpam-5240	73	21	subsets	subset	NOUN
ejpam-5240	73	22	of	of	ADP
ejpam-5240	73	23	sn	sn	PROPN
ejpam-5240	73	24	.	.	PUNCT
ejpam-5240	74	1	moreover	moreover	ADV
ejpam-5240	74	2	,	,	PUNCT
ejpam-5240	74	3	by	by	ADP
ejpam-5240	74	4	utilizing	utilize	VERB
ejpam-5240	74	5	the	the	DET
ejpam-5240	74	6	binomial	binomial	ADJ
ejpam-5240	74	7	coefficient	coefficient	NOUN
ejpam-5240	74	8	,	,	PUNCT
ejpam-5240	74	9	we	we	PRON
ejpam-5240	74	10	define	define	VERB
ejpam-5240	74	11	the	the	DET
ejpam-5240	74	12	cardinality	cardinality	NOUN
ejpam-5240	74	13	of	of	ADP
ejpam-5240	74	14	the	the	DET
ejpam-5240	74	15	set	set	NOUN
ejpam-5240	74	16	s(n	s(n	PROPN
ejpam-5240	74	17	,	,	PUNCT
ejpam-5240	74	18	k	k	NOUN
ejpam-5240	74	19	)	)	PUNCT
ejpam-5240	74	20	be	be	AUX
ejpam-5240	74	21	equal	equal	ADJ
ejpam-5240	74	22	to	to	ADP
ejpam-5240	74	23	the	the	DET
ejpam-5240	74	24	number	number	NOUN
ejpam-5240	74	25	(	(	PUNCT
ejpam-5240	74	26	n	n	NOUN
ejpam-5240	74	27	k	k	PROPN
ejpam-5240	74	28	)	)	PUNCT
ejpam-5240	74	29	where	where	SCONJ
ejpam-5240	74	30	n	n	PRON
ejpam-5240	74	31	is	be	AUX
ejpam-5240	74	32	a	a	DET
ejpam-5240	74	33	positive	positive	ADJ
ejpam-5240	74	34	integer	integer	NOUN
ejpam-5240	74	35	and	and	CCONJ
ejpam-5240	74	36	k	k	PROPN
ejpam-5240	74	37	is	be	AUX
ejpam-5240	74	38	a	a	DET
ejpam-5240	74	39	nonnegative	nonnegative	ADJ
ejpam-5240	74	40	integer	integer	NOUN
ejpam-5240	74	41	.	.	PUNCT
ejpam-5240	75	1	given	give	VERB
ejpam-5240	75	2	that	that	SCONJ
ejpam-5240	75	3	|s(n	|s(n	PROPN
ejpam-5240	75	4	,	,	PUNCT
ejpam-5240	75	5	k)|	k)|	NOUN
ejpam-5240	75	6	=	=	PUNCT
ejpam-5240	75	7	(	(	PUNCT
ejpam-5240	75	8	n	n	X
ejpam-5240	75	9	k	k	PROPN
ejpam-5240	75	10	)	)	PUNCT
ejpam-5240	75	11	,	,	PUNCT
ejpam-5240	75	12	it	it	PRON
ejpam-5240	75	13	can	can	AUX
ejpam-5240	75	14	be	be	AUX
ejpam-5240	75	15	observed	observe	VERB
ejpam-5240	75	16	that	that	SCONJ
ejpam-5240	75	17	we	we	PRON
ejpam-5240	75	18	can	can	AUX
ejpam-5240	75	19	not	not	PART
ejpam-5240	75	20	form	form	VERB
ejpam-5240	75	21	a	a	DET
ejpam-5240	75	22	k	k	ADJ
ejpam-5240	75	23	-	-	NOUN
ejpam-5240	75	24	element	element	NOUN
ejpam-5240	75	25	subset	subset	NOUN
ejpam-5240	75	26	from	from	ADP
ejpam-5240	75	27	sn	sn	PROPN
ejpam-5240	75	28	when	when	SCONJ
ejpam-5240	75	29	n	n	X
ejpam-5240	75	30	<	<	X
ejpam-5240	75	31	k.	k.	PROPN
ejpam-5240	75	32	thus	thus	ADV
ejpam-5240	75	33	s(n	s(n	PROPN
ejpam-5240	75	34	,	,	PUNCT
ejpam-5240	75	35	k	k	NOUN
ejpam-5240	75	36	)	)	PUNCT
ejpam-5240	75	37	=	=	SYM
ejpam-5240	75	38	∅	∅	NOUN
ejpam-5240	75	39	which	which	PRON
ejpam-5240	75	40	implies	imply	VERB
ejpam-5240	75	41	that	that	SCONJ
ejpam-5240	75	42	|s(n	|s(n	PROPN
ejpam-5240	75	43	,	,	PUNCT
ejpam-5240	75	44	k)|	k)|	NOUN
ejpam-5240	75	45	=	=	PUNCT
ejpam-5240	75	46	(	(	PUNCT
ejpam-5240	75	47	n	n	X
ejpam-5240	75	48	k	k	NOUN
ejpam-5240	75	49	)	)	PUNCT
ejpam-5240	75	50	=	=	SYM
ejpam-5240	75	51	0	0	X
ejpam-5240	75	52	.	.	PUNCT
ejpam-5240	76	1	equivalently	equivalently	ADV
ejpam-5240	76	2	,	,	PUNCT
ejpam-5240	76	3	we	we	PRON
ejpam-5240	76	4	have	have	VERB
ejpam-5240	76	5	remark	remark	NOUN
ejpam-5240	76	6	4	4	NUM
ejpam-5240	76	7	.	.	NOUN
ejpam-5240	76	8	remark	remark	NOUN
ejpam-5240	76	9	4	4	NUM
ejpam-5240	76	10	.	.	PUNCT
ejpam-5240	77	1	if	if	SCONJ
ejpam-5240	77	2	n	n	PRON
ejpam-5240	77	3	<	<	X
ejpam-5240	77	4	k	k	X
ejpam-5240	77	5	,	,	PUNCT
ejpam-5240	77	6	then	then	ADV
ejpam-5240	77	7	(	(	PUNCT
ejpam-5240	77	8	n	n	X
ejpam-5240	77	9	k	k	NOUN
ejpam-5240	77	10	)	)	PUNCT
ejpam-5240	78	1	=	=	PUNCT
ejpam-5240	78	2	0	0	X
ejpam-5240	78	3	.	.	X
ejpam-5240	79	1	2.2	2.2	NUM
ejpam-5240	79	2	.	.	PUNCT
ejpam-5240	80	1	graph	graph	NOUN
ejpam-5240	80	2	theory	theory	NOUN
ejpam-5240	80	3	this	this	DET
ejpam-5240	80	4	section	section	NOUN
ejpam-5240	80	5	discusses	discuss	VERB
ejpam-5240	80	6	some	some	DET
ejpam-5240	80	7	basic	basic	ADJ
ejpam-5240	80	8	concepts	concept	NOUN
ejpam-5240	80	9	in	in	ADP
ejpam-5240	80	10	graph	graph	NOUN
ejpam-5240	80	11	theory	theory	NOUN
ejpam-5240	80	12	.	.	PUNCT
ejpam-5240	81	1	we	we	PRON
ejpam-5240	81	2	adapt	adapt	VERB
ejpam-5240	81	3	the	the	DET
ejpam-5240	81	4	definitions	definition	NOUN
ejpam-5240	81	5	in	in	ADP
ejpam-5240	81	6	[	[	X
ejpam-5240	81	7	9	9	NUM
ejpam-5240	81	8	]	]	PUNCT
ejpam-5240	81	9	for	for	ADP
ejpam-5240	81	10	the	the	DET
ejpam-5240	81	11	concepts	concept	NOUN
ejpam-5240	81	12	used	use	VERB
ejpam-5240	81	13	here	here	ADV
ejpam-5240	81	14	.	.	PUNCT
ejpam-5240	82	1	also	also	ADV
ejpam-5240	82	2	,	,	PUNCT
ejpam-5240	82	3	the	the	DET
ejpam-5240	82	4	references	reference	NOUN
ejpam-5240	82	5	[	[	X
ejpam-5240	82	6	6	6	NUM
ejpam-5240	82	7	]	]	PUNCT
ejpam-5240	82	8	,	,	PUNCT
ejpam-5240	82	9	[	[	X
ejpam-5240	82	10	7	7	NUM
ejpam-5240	82	11	]	]	PUNCT
ejpam-5240	82	12	,	,	PUNCT
ejpam-5240	82	13	[	[	X
ejpam-5240	82	14	12	12	NUM
ejpam-5240	82	15	]	]	PUNCT
ejpam-5240	82	16	,	,	PUNCT
ejpam-5240	82	17	and	and	CCONJ
ejpam-5240	82	18	[	[	X
ejpam-5240	82	19	13	13	NUM
ejpam-5240	82	20	]	]	PUNCT
ejpam-5240	82	21	are	be	AUX
ejpam-5240	82	22	utilized	utilize	VERB
ejpam-5240	82	23	.	.	PUNCT
ejpam-5240	83	1	a	a	DET
ejpam-5240	83	2	graph	graph	NOUN
ejpam-5240	83	3	,	,	PUNCT
ejpam-5240	83	4	denoted	denote	VERB
ejpam-5240	83	5	by	by	ADP
ejpam-5240	83	6	g	g	NOUN
ejpam-5240	83	7	,	,	PUNCT
ejpam-5240	83	8	is	be	AUX
ejpam-5240	83	9	an	an	DET
ejpam-5240	83	10	ordered	order	VERB
ejpam-5240	83	11	pair	pair	NOUN
ejpam-5240	83	12	g	g	NOUN
ejpam-5240	83	13	=	=	PUNCT
ejpam-5240	83	14	(	(	PUNCT
ejpam-5240	83	15	v	v	NOUN
ejpam-5240	83	16	(	(	PUNCT
ejpam-5240	83	17	g	g	NOUN
ejpam-5240	83	18	)	)	PUNCT
ejpam-5240	83	19	,	,	PUNCT
ejpam-5240	83	20	e(g	e(g	PROPN
ejpam-5240	83	21	)	)	PUNCT
ejpam-5240	83	22	)	)	PUNCT
ejpam-5240	83	23	where	where	SCONJ
ejpam-5240	83	24	the	the	DET
ejpam-5240	83	25	vertex	vertex	NOUN
ejpam-5240	83	26	set	set	VERB
ejpam-5240	83	27	v	v	NOUN
ejpam-5240	83	28	(	(	PUNCT
ejpam-5240	83	29	g	g	NOUN
ejpam-5240	83	30	)	)	PUNCT
ejpam-5240	83	31	is	be	AUX
ejpam-5240	83	32	a	a	DET
ejpam-5240	83	33	nonempty	nonempty	ADJ
ejpam-5240	83	34	set	set	NOUN
ejpam-5240	83	35	of	of	ADP
ejpam-5240	83	36	elements	element	NOUN
ejpam-5240	83	37	called	call	VERB
ejpam-5240	83	38	vertices	vertex	NOUN
ejpam-5240	83	39	and	and	CCONJ
ejpam-5240	83	40	the	the	DET
ejpam-5240	83	41	edge	edge	NOUN
ejpam-5240	83	42	set	set	VERB
ejpam-5240	83	43	e(g	e(g	PROPN
ejpam-5240	83	44	)	)	PUNCT
ejpam-5240	83	45	is	be	AUX
ejpam-5240	83	46	a	a	DET
ejpam-5240	83	47	set	set	NOUN
ejpam-5240	83	48	of	of	ADP
ejpam-5240	83	49	unordered	unordered	ADJ
ejpam-5240	83	50	pairs	pair	NOUN
ejpam-5240	83	51	of	of	ADP
ejpam-5240	83	52	vertices	vertex	NOUN
ejpam-5240	83	53	called	call	VERB
ejpam-5240	83	54	edges	edge	NOUN
ejpam-5240	83	55	.	.	PUNCT
ejpam-5240	84	1	we	we	PRON
ejpam-5240	84	2	write	write	VERB
ejpam-5240	84	3	the	the	DET
ejpam-5240	84	4	edges	edge	NOUN
ejpam-5240	84	5	of	of	ADP
ejpam-5240	84	6	a	a	DET
ejpam-5240	84	7	graph	graph	NOUN
ejpam-5240	84	8	g	g	NOUN
ejpam-5240	84	9	as	as	ADP
ejpam-5240	84	10	[	[	X
ejpam-5240	84	11	x	x	X
ejpam-5240	84	12	,	,	PUNCT
ejpam-5240	84	13	y	y	PROPN
ejpam-5240	84	14	]	]	X
ejpam-5240	84	15	for	for	ADP
ejpam-5240	84	16	x	x	X
ejpam-5240	84	17	,	,	PUNCT
ejpam-5240	84	18	y	y	PROPN
ejpam-5240	84	19	∈	∈	PROPN
ejpam-5240	84	20	v	v	NOUN
ejpam-5240	84	21	(	(	PUNCT
ejpam-5240	84	22	g	g	NOUN
ejpam-5240	84	23	)	)	PUNCT
ejpam-5240	84	24	.	.	PUNCT
ejpam-5240	85	1	the	the	DET
ejpam-5240	85	2	edges	edge	NOUN
ejpam-5240	85	3	of	of	ADP
ejpam-5240	85	4	g	g	NOUN
ejpam-5240	85	5	are	be	AUX
ejpam-5240	85	6	said	say	VERB
ejpam-5240	85	7	to	to	PART
ejpam-5240	85	8	be	be	AUX
ejpam-5240	85	9	unordered	unordered	ADJ
ejpam-5240	85	10	pairs	pair	NOUN
ejpam-5240	85	11	so	so	SCONJ
ejpam-5240	85	12	we	we	PRON
ejpam-5240	85	13	also	also	ADV
ejpam-5240	85	14	say	say	VERB
ejpam-5240	85	15	that	that	SCONJ
ejpam-5240	86	1	[	[	X
ejpam-5240	86	2	x	x	X
ejpam-5240	86	3	,	,	PUNCT
ejpam-5240	86	4	y	y	PROPN
ejpam-5240	86	5	]	]	X
ejpam-5240	86	6	=	=	PUNCT
ejpam-5240	87	1	[	[	X
ejpam-5240	87	2	y	y	PROPN
ejpam-5240	87	3	,	,	PUNCT
ejpam-5240	87	4	x	x	NOUN
ejpam-5240	87	5	]	]	X
ejpam-5240	87	6	.	.	PUNCT
ejpam-5240	88	1	moreover	moreover	ADV
ejpam-5240	88	2	,	,	PUNCT
ejpam-5240	88	3	the	the	DET
ejpam-5240	88	4	number	number	NOUN
ejpam-5240	88	5	|v	|v	NOUN
ejpam-5240	88	6	(	(	PUNCT
ejpam-5240	88	7	g)|	g)|	PROPN
ejpam-5240	88	8	is	be	AUX
ejpam-5240	88	9	called	call	VERB
ejpam-5240	88	10	the	the	DET
ejpam-5240	88	11	order	order	NOUN
ejpam-5240	88	12	of	of	ADP
ejpam-5240	88	13	g	g	NOUN
ejpam-5240	88	14	while	while	SCONJ
ejpam-5240	88	15	the	the	DET
ejpam-5240	88	16	number	number	NOUN
ejpam-5240	88	17	|e(g)|	|e(g)|	NOUN
ejpam-5240	88	18	is	be	AUX
ejpam-5240	88	19	referred	refer	VERB
ejpam-5240	88	20	to	to	ADP
ejpam-5240	88	21	as	as	ADP
ejpam-5240	88	22	the	the	DET
ejpam-5240	88	23	size	size	NOUN
ejpam-5240	88	24	of	of	ADP
ejpam-5240	88	25	g.	g.	PROPN
ejpam-5240	88	26	a	a	DET
ejpam-5240	88	27	graph	graph	NOUN
ejpam-5240	88	28	of	of	ADP
ejpam-5240	88	29	order	order	NOUN
ejpam-5240	89	1	n	n	PRON
ejpam-5240	89	2	≥	≥	NOUN
ejpam-5240	89	3	1	1	NUM
ejpam-5240	89	4	having	have	VERB
ejpam-5240	89	5	no	no	DET
ejpam-5240	89	6	edges	edge	NOUN
ejpam-5240	89	7	is	be	AUX
ejpam-5240	89	8	called	call	VERB
ejpam-5240	89	9	an	an	DET
ejpam-5240	89	10	empty	empty	ADJ
ejpam-5240	89	11	graph	graph	NOUN
ejpam-5240	89	12	,	,	PUNCT
ejpam-5240	89	13	denoted	denote	VERB
ejpam-5240	89	14	as	as	ADP
ejpam-5240	89	15	kn	kn	PROPN
ejpam-5240	89	16	.	.	PUNCT
ejpam-5240	90	1	furthermore	furthermore	ADV
ejpam-5240	90	2	,	,	PUNCT
ejpam-5240	90	3	a	a	DET
ejpam-5240	90	4	graph	graph	NOUN
ejpam-5240	90	5	with	with	ADP
ejpam-5240	90	6	only	only	ADV
ejpam-5240	90	7	one	one	NUM
ejpam-5240	90	8	vertex	vertex	NOUN
ejpam-5240	90	9	is	be	AUX
ejpam-5240	90	10	referred	refer	VERB
ejpam-5240	90	11	to	to	ADP
ejpam-5240	90	12	as	as	ADP
ejpam-5240	90	13	a	a	DET
ejpam-5240	90	14	trivial	trivial	ADJ
ejpam-5240	90	15	graph	graph	NOUN
ejpam-5240	90	16	.	.	PUNCT
ejpam-5240	91	1	it	it	PRON
ejpam-5240	91	2	was	be	AUX
ejpam-5240	91	3	taken	take	VERB
ejpam-5240	91	4	into	into	ADP
ejpam-5240	91	5	consideration	consideration	NOUN
ejpam-5240	91	6	that	that	SCONJ
ejpam-5240	91	7	a	a	DET
ejpam-5240	91	8	graph	graph	NOUN
ejpam-5240	91	9	with	with	ADP
ejpam-5240	91	10	an	an	DET
ejpam-5240	91	11	empty	empty	ADJ
ejpam-5240	91	12	edge	edge	NOUN
ejpam-5240	91	13	set	set	NOUN
ejpam-5240	91	14	is	be	AUX
ejpam-5240	91	15	still	still	ADV
ejpam-5240	91	16	considered	consider	VERB
ejpam-5240	91	17	as	as	ADP
ejpam-5240	91	18	a	a	DET
ejpam-5240	91	19	graph	graph	NOUN
ejpam-5240	91	20	.	.	PUNCT
ejpam-5240	92	1	however	however	ADV
ejpam-5240	92	2	,	,	PUNCT
ejpam-5240	92	3	note	note	VERB
ejpam-5240	92	4	that	that	SCONJ
ejpam-5240	92	5	if	if	SCONJ
ejpam-5240	92	6	v	v	X
ejpam-5240	92	7	(	(	PUNCT
ejpam-5240	92	8	g	g	NOUN
ejpam-5240	92	9	)	)	PUNCT
ejpam-5240	92	10	=	=	NOUN
ejpam-5240	92	11	∅	∅	NOUN
ejpam-5240	92	12	,	,	PUNCT
ejpam-5240	92	13	then	then	ADV
ejpam-5240	92	14	g	g	PROPN
ejpam-5240	92	15	here	here	ADV
ejpam-5240	92	16	is	be	AUX
ejpam-5240	92	17	undefined	undefined	ADJ
ejpam-5240	92	18	since	since	SCONJ
ejpam-5240	92	19	we	we	PRON
ejpam-5240	92	20	can	can	AUX
ejpam-5240	92	21	not	not	PART
ejpam-5240	92	22	form	form	VERB
ejpam-5240	92	23	a	a	DET
ejpam-5240	92	24	graph	graph	NOUN
ejpam-5240	92	25	with	with	ADP
ejpam-5240	92	26	no	no	DET
ejpam-5240	92	27	vertices	vertex	NOUN
ejpam-5240	92	28	.	.	PUNCT
ejpam-5240	93	1	the	the	DET
ejpam-5240	93	2	degree	degree	NOUN
ejpam-5240	93	3	of	of	ADP
ejpam-5240	93	4	vertex	vertex	NOUN
ejpam-5240	93	5	x	x	SYM
ejpam-5240	93	6	∈	∈	NOUN
ejpam-5240	93	7	v	v	ADP
ejpam-5240	93	8	(	(	PUNCT
ejpam-5240	93	9	g	g	NOUN
ejpam-5240	93	10	)	)	PUNCT
ejpam-5240	93	11	,	,	PUNCT
ejpam-5240	93	12	denoted	denote	VERB
ejpam-5240	93	13	by	by	ADP
ejpam-5240	93	14	deg(x	deg(x	NOUN
ejpam-5240	93	15	)	)	PUNCT
ejpam-5240	93	16	,	,	PUNCT
ejpam-5240	93	17	is	be	AUX
ejpam-5240	93	18	the	the	DET
ejpam-5240	93	19	number	number	NOUN
ejpam-5240	93	20	of	of	ADP
ejpam-5240	93	21	edges	edge	NOUN
ejpam-5240	93	22	adjacent	adjacent	ADJ
ejpam-5240	93	23	to	to	ADP
ejpam-5240	93	24	the	the	DET
ejpam-5240	93	25	vertex	vertex	NOUN
ejpam-5240	93	26	x.	x.	NOUN
ejpam-5240	94	1	if	if	SCONJ
ejpam-5240	94	2	the	the	DET
ejpam-5240	94	3	vertex	vertex	NOUN
ejpam-5240	94	4	x	x	PUNCT
ejpam-5240	94	5	has	have	VERB
ejpam-5240	94	6	deg(x	deg(x	ADV
ejpam-5240	94	7	)	)	PUNCT
ejpam-5240	95	1	=	=	SYM
ejpam-5240	95	2	0	0	NUM
ejpam-5240	95	3	,	,	PUNCT
ejpam-5240	95	4	then	then	ADV
ejpam-5240	95	5	x	x	PUNCT
ejpam-5240	95	6	is	be	AUX
ejpam-5240	95	7	called	call	VERB
ejpam-5240	95	8	an	an	DET
ejpam-5240	95	9	isolated	isolated	ADJ
ejpam-5240	95	10	vertex	vertex	NOUN
ejpam-5240	95	11	.	.	PUNCT
ejpam-5240	96	1	meaning	mean	VERB
ejpam-5240	96	2	to	to	PART
ejpam-5240	96	3	say	say	VERB
ejpam-5240	96	4	,	,	PUNCT
ejpam-5240	96	5	an	an	DET
ejpam-5240	96	6	isolated	isolated	ADJ
ejpam-5240	96	7	vertex	vertex	NOUN
ejpam-5240	96	8	is	be	AUX
ejpam-5240	96	9	a	a	DET
ejpam-5240	96	10	vertex	vertex	NOUN
ejpam-5240	96	11	that	that	PRON
ejpam-5240	96	12	is	be	AUX
ejpam-5240	96	13	not	not	PART
ejpam-5240	96	14	adjacent	adjacent	ADJ
ejpam-5240	96	15	to	to	ADP
ejpam-5240	96	16	any	any	DET
ejpam-5240	96	17	other	other	ADJ
ejpam-5240	96	18	vertex	vertex	NOUN
ejpam-5240	96	19	of	of	ADP
ejpam-5240	96	20	g.	g.	PROPN
ejpam-5240	96	21	there	there	PRON
ejpam-5240	96	22	are	be	VERB
ejpam-5240	96	23	times	time	NOUN
ejpam-5240	96	24	when	when	SCONJ
ejpam-5240	96	25	the	the	DET
ejpam-5240	96	26	degrees	degree	NOUN
ejpam-5240	96	27	of	of	ADP
ejpam-5240	96	28	every	every	DET
ejpam-5240	96	29	vertex	vertex	NOUN
ejpam-5240	96	30	help	help	NOUN
ejpam-5240	96	31	determine	determine	VERB
ejpam-5240	96	32	the	the	DET
ejpam-5240	96	33	size	size	NOUN
ejpam-5240	96	34	of	of	ADP
ejpam-5240	96	35	a	a	DET
ejpam-5240	96	36	graph	graph	NOUN
ejpam-5240	96	37	specifically	specifically	ADV
ejpam-5240	96	38	if	if	SCONJ
ejpam-5240	96	39	the	the	DET
ejpam-5240	96	40	degrees	degree	NOUN
ejpam-5240	96	41	are	be	AUX
ejpam-5240	96	42	the	the	DET
ejpam-5240	96	43	same	same	ADJ
ejpam-5240	96	44	in	in	ADP
ejpam-5240	96	45	number	number	NOUN
ejpam-5240	96	46	.	.	PUNCT
ejpam-5240	97	1	in	in	ADP
ejpam-5240	97	2	this	this	DET
ejpam-5240	97	3	regard	regard	NOUN
ejpam-5240	97	4	,	,	PUNCT
ejpam-5240	97	5	we	we	PRON
ejpam-5240	97	6	say	say	VERB
ejpam-5240	97	7	that	that	SCONJ
ejpam-5240	97	8	a	a	DET
ejpam-5240	97	9	graph	graph	NOUN
ejpam-5240	97	10	with	with	ADP
ejpam-5240	97	11	the	the	DET
ejpam-5240	97	12	same	same	ADJ
ejpam-5240	97	13	number	number	NOUN
ejpam-5240	97	14	of	of	ADP
ejpam-5240	97	15	degrees	degree	NOUN
ejpam-5240	97	16	for	for	ADP
ejpam-5240	97	17	every	every	DET
ejpam-5240	97	18	vertex	vertex	NOUN
ejpam-5240	97	19	is	be	AUX
ejpam-5240	97	20	a	a	DET
ejpam-5240	97	21	regular	regular	ADJ
ejpam-5240	97	22	graph	graph	NOUN
ejpam-5240	97	23	.	.	PUNCT
ejpam-5240	98	1	a	a	DET
ejpam-5240	98	2	graph	graph	NOUN
ejpam-5240	98	3	g	g	NOUN
ejpam-5240	98	4	is	be	AUX
ejpam-5240	98	5	regular	regular	ADJ
ejpam-5240	98	6	if	if	SCONJ
ejpam-5240	98	7	every	every	DET
ejpam-5240	98	8	vertex	vertex	NOUN
ejpam-5240	98	9	has	have	VERB
ejpam-5240	98	10	the	the	DET
ejpam-5240	98	11	same	same	ADJ
ejpam-5240	98	12	degree	degree	NOUN
ejpam-5240	98	13	.	.	PUNCT
ejpam-5240	99	1	moreover	moreover	ADV
ejpam-5240	99	2	,	,	PUNCT
ejpam-5240	99	3	g	g	PROPN
ejpam-5240	99	4	is	be	AUX
ejpam-5240	99	5	said	say	VERB
ejpam-5240	99	6	to	to	PART
ejpam-5240	99	7	be	be	AUX
ejpam-5240	99	8	regular	regular	ADJ
ejpam-5240	99	9	of	of	ADP
ejpam-5240	99	10	degree	degree	NOUN
ejpam-5240	99	11	r	r	NOUN
ejpam-5240	99	12	(	(	PUNCT
ejpam-5240	99	13	or	or	CCONJ
ejpam-5240	99	14	r	r	NOUN
ejpam-5240	99	15	-	-	ADJ
ejpam-5240	99	16	regular	regular	ADJ
ejpam-5240	99	17	)	)	PUNCT
ejpam-5240	99	18	if	if	SCONJ
ejpam-5240	99	19	deg(x	deg(x	ADV
ejpam-5240	99	20	)	)	PUNCT
ejpam-5240	100	1	=	=	SYM
ejpam-5240	100	2	r	r	NOUN
ejpam-5240	100	3	for	for	ADP
ejpam-5240	100	4	all	all	DET
ejpam-5240	100	5	vertices	vertex	NOUN
ejpam-5240	100	6	x	x	VERB
ejpam-5240	100	7	in	in	ADP
ejpam-5240	100	8	g.	g.	PROPN
ejpam-5240	100	9	the	the	DET
ejpam-5240	100	10	size	size	NOUN
ejpam-5240	100	11	of	of	ADP
ejpam-5240	100	12	an	an	DET
ejpam-5240	100	13	r	r	NOUN
ejpam-5240	100	14	-	-	PUNCT
ejpam-5240	100	15	regular	regular	ADJ
ejpam-5240	100	16	graph	graph	NOUN
ejpam-5240	100	17	can	can	AUX
ejpam-5240	100	18	be	be	AUX
ejpam-5240	100	19	determined	determine	VERB
ejpam-5240	100	20	as	as	ADP
ejpam-5240	100	21	follows:∑	follows:∑	PROPN
ejpam-5240	100	22	x∈v	x∈v	PROPN
ejpam-5240	100	23	(	(	PUNCT
ejpam-5240	100	24	g	g	NOUN
ejpam-5240	100	25	)	)	PUNCT
ejpam-5240	100	26	deg(x	deg(x	PROPN
ejpam-5240	100	27	)	)	PUNCT
ejpam-5240	100	28	=	=	SYM
ejpam-5240	101	1	2	2	NUM
ejpam-5240	101	2	m	m	NOUN
ejpam-5240	101	3	nr	nr	NOUN
ejpam-5240	101	4	=	=	ADJ
ejpam-5240	101	5	2	2	NUM
ejpam-5240	101	6	m	m	NOUN
ejpam-5240	101	7	m	m	VERB
ejpam-5240	101	8	=	=	ADJ
ejpam-5240	101	9	nr	nr	PROPN
ejpam-5240	101	10	2	2	NUM
ejpam-5240	101	11	(	(	PUNCT
ejpam-5240	101	12	1	1	NUM
ejpam-5240	101	13	)	)	PUNCT
ejpam-5240	101	14	m.e	m.e	PROPN
ejpam-5240	101	15	.	.	PROPN
ejpam-5240	101	16	pelagio	pelagio	PROPN
ejpam-5240	101	17	,	,	PUNCT
ejpam-5240	101	18	n.	n.	NOUN
ejpam-5240	101	19	mame	mame	PROPN
ejpam-5240	101	20	,	,	PUNCT
ejpam-5240	101	21	k.	k.	PROPN
ejpam-5240	101	22	mendoza	mendoza	PROPN
ejpam-5240	101	23	/	/	SYM
ejpam-5240	101	24	eur	eur	PROPN
ejpam-5240	101	25	.	.	PUNCT
ejpam-5240	102	1	j.	j.	PROPN
ejpam-5240	102	2	pure	pure	PROPN
ejpam-5240	102	3	appl	appl	PROPN
ejpam-5240	102	4	.	.	PROPN
ejpam-5240	102	5	math	math	PROPN
ejpam-5240	102	6	,	,	PUNCT
ejpam-5240	102	7	17	17	NUM
ejpam-5240	102	8	(	(	PUNCT
ejpam-5240	102	9	3	3	NUM
ejpam-5240	102	10	)	)	PUNCT
ejpam-5240	102	11	(	(	PUNCT
ejpam-5240	102	12	2024	2024	NUM
ejpam-5240	102	13	)	)	PUNCT
ejpam-5240	102	14	,	,	PUNCT
ejpam-5240	102	15	1779	1779	NUM
ejpam-5240	102	16	-	-	SYM
ejpam-5240	102	17	1803	1803	NUM
ejpam-5240	102	18	1783	1783	NUM
ejpam-5240	102	19	provided	provide	VERB
ejpam-5240	102	20	that	that	SCONJ
ejpam-5240	102	21	nr	nr	PRON
ejpam-5240	102	22	is	be	AUX
ejpam-5240	102	23	even	even	ADV
ejpam-5240	102	24	.	.	PUNCT
ejpam-5240	103	1	let	let	VERB
ejpam-5240	103	2	w	w	NOUN
ejpam-5240	103	3	:	:	PUNCT
ejpam-5240	103	4	x1	x1	ADJ
ejpam-5240	103	5	,	,	PUNCT
ejpam-5240	103	6	x2	x2	PROPN
ejpam-5240	103	7	,	,	PUNCT
ejpam-5240	103	8	...	...	PUNCT
ejpam-5240	103	9	,	,	PUNCT
ejpam-5240	103	10	xk	xk	PROPN
ejpam-5240	103	11	,	,	PUNCT
ejpam-5240	103	12	xk+1	xk+1	NUM
ejpam-5240	103	13	be	be	AUX
ejpam-5240	103	14	a	a	DET
ejpam-5240	103	15	walk	walk	NOUN
ejpam-5240	103	16	of	of	ADP
ejpam-5240	103	17	length	length	NOUN
ejpam-5240	104	1	k	k	PROPN
ejpam-5240	104	2	>	>	X
ejpam-5240	104	3	0	0	X
ejpam-5240	104	4	.	.	PUNCT
ejpam-5240	105	1	this	this	DET
ejpam-5240	105	2	walk	walk	NOUN
ejpam-5240	105	3	is	be	AUX
ejpam-5240	105	4	closed	close	VERB
ejpam-5240	105	5	if	if	SCONJ
ejpam-5240	105	6	x1	x1	PROPN
ejpam-5240	105	7	=	=	SYM
ejpam-5240	105	8	xk+1	xk+1	X
ejpam-5240	105	9	.	.	PUNCT
ejpam-5240	106	1	moreover	moreover	ADV
ejpam-5240	106	2	,	,	PUNCT
ejpam-5240	106	3	a	a	DET
ejpam-5240	106	4	closed	closed	ADJ
ejpam-5240	106	5	walk	walk	NOUN
ejpam-5240	106	6	is	be	AUX
ejpam-5240	106	7	called	call	VERB
ejpam-5240	106	8	a	a	DET
ejpam-5240	106	9	cycle	cycle	NOUN
ejpam-5240	106	10	if	if	SCONJ
ejpam-5240	106	11	the	the	DET
ejpam-5240	106	12	vertices	vertex	NOUN
ejpam-5240	106	13	x1	x1	PROPN
ejpam-5240	106	14	,	,	PUNCT
ejpam-5240	106	15	x2	x2	PROPN
ejpam-5240	106	16	,	,	PUNCT
ejpam-5240	106	17	...	...	PUNCT
ejpam-5240	106	18	,	,	PUNCT
ejpam-5240	106	19	xk	xk	PROPN
ejpam-5240	106	20	are	be	AUX
ejpam-5240	106	21	distinct	distinct	ADJ
ejpam-5240	106	22	.	.	PUNCT
ejpam-5240	107	1	a	a	DET
ejpam-5240	107	2	graph	graph	NOUN
ejpam-5240	107	3	g	g	NOUN
ejpam-5240	107	4	of	of	ADP
ejpam-5240	107	5	order	order	NOUN
ejpam-5240	107	6	n	n	PRON
ejpam-5240	107	7	≥	≥	NOUN
ejpam-5240	107	8	3	3	NUM
ejpam-5240	107	9	is	be	AUX
ejpam-5240	107	10	called	call	VERB
ejpam-5240	107	11	a	a	DET
ejpam-5240	107	12	cycle	cycle	NOUN
ejpam-5240	107	13	graph	graph	NOUN
ejpam-5240	107	14	of	of	ADP
ejpam-5240	107	15	order	order	NOUN
ejpam-5240	107	16	n	n	CCONJ
ejpam-5240	107	17	,	,	PUNCT
ejpam-5240	107	18	denoted	denote	VERB
ejpam-5240	107	19	by	by	ADP
ejpam-5240	107	20	cn	cn	PROPN
ejpam-5240	107	21	,	,	PUNCT
ejpam-5240	107	22	if	if	SCONJ
ejpam-5240	107	23	the	the	DET
ejpam-5240	107	24	vertices	vertex	NOUN
ejpam-5240	107	25	of	of	ADP
ejpam-5240	107	26	g	g	NOUN
ejpam-5240	107	27	are	be	AUX
ejpam-5240	107	28	labeled	label	VERB
ejpam-5240	107	29	x1	x1	PROPN
ejpam-5240	107	30	,	,	PUNCT
ejpam-5240	107	31	x2	x2	PROPN
ejpam-5240	107	32	,	,	PUNCT
ejpam-5240	107	33	...	...	PUNCT
ejpam-5240	107	34	,	,	PUNCT
ejpam-5240	107	35	xn	xn	PROPN
ejpam-5240	108	1	so	so	SCONJ
ejpam-5240	108	2	that	that	SCONJ
ejpam-5240	108	3	the	the	DET
ejpam-5240	108	4	edges	edge	NOUN
ejpam-5240	108	5	[	[	X
ejpam-5240	108	6	x1	x1	PROPN
ejpam-5240	108	7	,	,	PUNCT
ejpam-5240	108	8	x2	x2	PROPN
ejpam-5240	108	9	]	]	PUNCT
ejpam-5240	108	10	,	,	PUNCT
ejpam-5240	108	11	[	[	X
ejpam-5240	108	12	x2	x2	X
ejpam-5240	108	13	,	,	PUNCT
ejpam-5240	108	14	x3	x3	ADJ
ejpam-5240	108	15	]	]	PUNCT
ejpam-5240	108	16	,	,	PUNCT
ejpam-5240	108	17	...	...	PUNCT
ejpam-5240	108	18	,	,	PUNCT
ejpam-5240	109	1	[	[	X
ejpam-5240	109	2	xn−1	xn−1	PROPN
ejpam-5240	109	3	,	,	PUNCT
ejpam-5240	109	4	xn	xn	PROPN
ejpam-5240	109	5	]	]	PUNCT
ejpam-5240	109	6	,	,	PUNCT
ejpam-5240	109	7	[	[	X
ejpam-5240	109	8	xn	xn	X
ejpam-5240	109	9	,	,	PUNCT
ejpam-5240	109	10	x1	x1	PROPN
ejpam-5240	109	11	]	]	PUNCT
ejpam-5240	109	12	form	form	VERB
ejpam-5240	109	13	a	a	DET
ejpam-5240	109	14	cycle	cycle	NOUN
ejpam-5240	109	15	.	.	PUNCT
ejpam-5240	110	1	it	it	PRON
ejpam-5240	110	2	is	be	AUX
ejpam-5240	110	3	noted	note	VERB
ejpam-5240	110	4	that	that	SCONJ
ejpam-5240	110	5	the	the	DET
ejpam-5240	110	6	order	order	NOUN
ejpam-5240	110	7	and	and	CCONJ
ejpam-5240	110	8	size	size	NOUN
ejpam-5240	110	9	of	of	ADP
ejpam-5240	110	10	a	a	DET
ejpam-5240	110	11	cn	cn	PROPN
ejpam-5240	110	12	is	be	AUX
ejpam-5240	110	13	n.	n.	ADJ
ejpam-5240	110	14	a	a	DET
ejpam-5240	110	15	graph	graph	NOUN
ejpam-5240	110	16	of	of	ADP
ejpam-5240	110	17	order	order	NOUN
ejpam-5240	110	18	n	n	X
ejpam-5240	110	19	is	be	AUX
ejpam-5240	110	20	said	say	VERB
ejpam-5240	110	21	to	to	PART
ejpam-5240	110	22	be	be	AUX
ejpam-5240	110	23	a	a	DET
ejpam-5240	110	24	complete	complete	ADJ
ejpam-5240	110	25	graph	graph	NOUN
ejpam-5240	110	26	of	of	ADP
ejpam-5240	110	27	order	order	NOUN
ejpam-5240	110	28	n	n	CCONJ
ejpam-5240	110	29	,	,	PUNCT
ejpam-5240	110	30	denoted	denote	VERB
ejpam-5240	110	31	by	by	ADP
ejpam-5240	110	32	kn	kn	PROPN
ejpam-5240	110	33	,	,	PUNCT
ejpam-5240	110	34	if	if	SCONJ
ejpam-5240	110	35	every	every	DET
ejpam-5240	110	36	vertex	vertex	NOUN
ejpam-5240	110	37	is	be	AUX
ejpam-5240	110	38	adjacent	adjacent	ADJ
ejpam-5240	110	39	to	to	ADP
ejpam-5240	110	40	every	every	DET
ejpam-5240	110	41	other	other	ADJ
ejpam-5240	110	42	vertex	vertex	NOUN
ejpam-5240	110	43	.	.	PUNCT
ejpam-5240	111	1	since	since	SCONJ
ejpam-5240	111	2	every	every	DET
ejpam-5240	111	3	vertex	vertex	NOUN
ejpam-5240	111	4	of	of	ADP
ejpam-5240	111	5	kn	kn	PROPN
ejpam-5240	111	6	is	be	AUX
ejpam-5240	111	7	adjacent	adjacent	ADJ
ejpam-5240	111	8	to	to	ADP
ejpam-5240	111	9	every	every	DET
ejpam-5240	111	10	vertex	vertex	NOUN
ejpam-5240	111	11	of	of	ADP
ejpam-5240	111	12	this	this	DET
ejpam-5240	111	13	graph	graph	NOUN
ejpam-5240	111	14	it	it	PRON
ejpam-5240	111	15	follows	follow	VERB
ejpam-5240	111	16	that	that	SCONJ
ejpam-5240	111	17	the	the	DET
ejpam-5240	111	18	degrees	degree	NOUN
ejpam-5240	111	19	of	of	ADP
ejpam-5240	111	20	every	every	DET
ejpam-5240	111	21	vertex	vertex	NOUN
ejpam-5240	111	22	are	be	AUX
ejpam-5240	111	23	equal	equal	ADJ
ejpam-5240	111	24	to	to	ADP
ejpam-5240	111	25	n	n	PROPN
ejpam-5240	111	26	−	−	PROPN
ejpam-5240	111	27	1	1	NUM
ejpam-5240	111	28	,	,	PUNCT
ejpam-5240	111	29	since	since	SCONJ
ejpam-5240	111	30	we	we	PRON
ejpam-5240	111	31	consider	consider	VERB
ejpam-5240	111	32	a	a	DET
ejpam-5240	111	33	simple	simple	ADJ
ejpam-5240	111	34	graph	graph	NOUN
ejpam-5240	111	35	kn	kn	NOUN
ejpam-5240	111	36	having	have	VERB
ejpam-5240	111	37	no	no	DET
ejpam-5240	111	38	loops	loop	NOUN
ejpam-5240	111	39	and	and	CCONJ
ejpam-5240	111	40	multiple	multiple	ADJ
ejpam-5240	111	41	edges	edge	NOUN
ejpam-5240	111	42	.	.	PUNCT
ejpam-5240	112	1	this	this	PRON
ejpam-5240	112	2	implies	imply	VERB
ejpam-5240	112	3	that	that	SCONJ
ejpam-5240	112	4	kn	kn	PROPN
ejpam-5240	112	5	is	be	AUX
ejpam-5240	112	6	an	an	DET
ejpam-5240	112	7	(	(	PUNCT
ejpam-5240	112	8	n−1)-regular	n−1)-regular	ADJ
ejpam-5240	112	9	graph	graph	NOUN
ejpam-5240	112	10	.	.	PUNCT
ejpam-5240	113	1	now	now	ADV
ejpam-5240	113	2	,	,	PUNCT
ejpam-5240	113	3	to	to	PART
ejpam-5240	113	4	determine	determine	VERB
ejpam-5240	113	5	the	the	DET
ejpam-5240	113	6	size	size	NOUN
ejpam-5240	113	7	of	of	ADP
ejpam-5240	113	8	kn	kn	PROPN
ejpam-5240	113	9	,	,	PUNCT
ejpam-5240	113	10	by	by	ADP
ejpam-5240	113	11	utilizing	utilize	VERB
ejpam-5240	113	12	equation(1	equation(1	NOUN
ejpam-5240	113	13	)	)	PUNCT
ejpam-5240	113	14	,	,	PUNCT
ejpam-5240	113	15	we	we	PRON
ejpam-5240	113	16	have	have	VERB
ejpam-5240	113	17	m	m	NOUN
ejpam-5240	113	18	=	=	NOUN
ejpam-5240	113	19	nr	nr	PROPN
ejpam-5240	113	20	2	2	NUM
ejpam-5240	113	21	m	m	NOUN
ejpam-5240	113	22	=	=	NOUN
ejpam-5240	113	23	n(n−	n(n−	ADJ
ejpam-5240	113	24	1	1	NUM
ejpam-5240	113	25	)	)	PUNCT
ejpam-5240	113	26	2	2	NUM
ejpam-5240	113	27	.	.	PUNCT
ejpam-5240	114	1	observe	observe	VERB
ejpam-5240	114	2	that	that	SCONJ
ejpam-5240	114	3	m	m	NOUN
ejpam-5240	114	4	is	be	AUX
ejpam-5240	114	5	always	always	ADV
ejpam-5240	114	6	defined	define	VERB
ejpam-5240	114	7	since	since	SCONJ
ejpam-5240	114	8	the	the	DET
ejpam-5240	114	9	product	product	NOUN
ejpam-5240	114	10	of	of	ADP
ejpam-5240	114	11	the	the	DET
ejpam-5240	114	12	consecutive	consecutive	ADJ
ejpam-5240	114	13	integers	integer	NOUN
ejpam-5240	114	14	n	n	PRON
ejpam-5240	114	15	and	and	CCONJ
ejpam-5240	114	16	n−1	n−1	PROPN
ejpam-5240	114	17	is	be	AUX
ejpam-5240	114	18	even	even	ADV
ejpam-5240	114	19	.	.	PUNCT
ejpam-5240	115	1	any	any	DET
ejpam-5240	115	2	cycle	cycle	NOUN
ejpam-5240	115	3	graph	graph	NOUN
ejpam-5240	115	4	,	,	PUNCT
ejpam-5240	115	5	in	in	ADP
ejpam-5240	115	6	particular	particular	ADJ
ejpam-5240	115	7	,	,	PUNCT
ejpam-5240	115	8	c3	c3	X
ejpam-5240	115	9	whose	whose	DET
ejpam-5240	115	10	order	order	NOUN
ejpam-5240	115	11	and	and	CCONJ
ejpam-5240	115	12	size	size	NOUN
ejpam-5240	115	13	are	be	AUX
ejpam-5240	115	14	equal	equal	ADJ
ejpam-5240	115	15	to	to	ADP
ejpam-5240	115	16	3	3	NUM
ejpam-5240	115	17	,	,	PUNCT
ejpam-5240	115	18	is	be	AUX
ejpam-5240	115	19	a	a	DET
ejpam-5240	115	20	2	2	NUM
ejpam-5240	115	21	-	-	PUNCT
ejpam-5240	115	22	regular	regular	ADJ
ejpam-5240	115	23	graph	graph	NOUN
ejpam-5240	115	24	.	.	PUNCT
ejpam-5240	116	1	furthermore	furthermore	ADV
ejpam-5240	116	2	,	,	PUNCT
ejpam-5240	116	3	a	a	DET
ejpam-5240	116	4	complete	complete	ADJ
ejpam-5240	116	5	graph	graph	NOUN
ejpam-5240	116	6	of	of	ADP
ejpam-5240	116	7	order	order	NOUN
ejpam-5240	116	8	3	3	NUM
ejpam-5240	116	9	,	,	PUNCT
ejpam-5240	116	10	which	which	PRON
ejpam-5240	116	11	is	be	AUX
ejpam-5240	116	12	a	a	DET
ejpam-5240	116	13	(	(	PUNCT
ejpam-5240	116	14	3−1	3−1	NUM
ejpam-5240	116	15	)	)	PUNCT
ejpam-5240	116	16	=	=	PUNCT
ejpam-5240	116	17	2	2	NUM
ejpam-5240	116	18	-	-	PUNCT
ejpam-5240	116	19	regular	regular	ADJ
ejpam-5240	116	20	graph	graph	NOUN
ejpam-5240	116	21	,	,	PUNCT
ejpam-5240	116	22	has	have	VERB
ejpam-5240	116	23	size	size	NOUN
ejpam-5240	116	24	equal	equal	ADJ
ejpam-5240	116	25	to	to	ADP
ejpam-5240	116	26	3	3	NUM
ejpam-5240	116	27	.	.	PUNCT
ejpam-5240	117	1	now	now	ADV
ejpam-5240	117	2	,	,	PUNCT
ejpam-5240	117	3	it	it	PRON
ejpam-5240	117	4	can	can	AUX
ejpam-5240	117	5	be	be	AUX
ejpam-5240	117	6	observed	observe	VERB
ejpam-5240	117	7	that	that	SCONJ
ejpam-5240	117	8	c3	c3	PROPN
ejpam-5240	117	9	and	and	CCONJ
ejpam-5240	117	10	k3	k3	VERB
ejpam-5240	117	11	have	have	VERB
ejpam-5240	117	12	the	the	DET
ejpam-5240	117	13	same	same	ADJ
ejpam-5240	117	14	number	number	NOUN
ejpam-5240	117	15	of	of	ADP
ejpam-5240	117	16	order	order	NOUN
ejpam-5240	117	17	,	,	PUNCT
ejpam-5240	117	18	size	size	NOUN
ejpam-5240	117	19	,	,	PUNCT
ejpam-5240	117	20	and	and	CCONJ
ejpam-5240	117	21	degree	degree	NOUN
ejpam-5240	117	22	of	of	ADP
ejpam-5240	117	23	every	every	DET
ejpam-5240	117	24	vertex	vertex	NOUN
ejpam-5240	117	25	.	.	PUNCT
ejpam-5240	118	1	through	through	ADP
ejpam-5240	118	2	the	the	DET
ejpam-5240	118	3	notion	notion	NOUN
ejpam-5240	118	4	of	of	ADP
ejpam-5240	118	5	graph	graph	NOUN
ejpam-5240	118	6	isomorphism	isomorphism	NOUN
ejpam-5240	118	7	,	,	PUNCT
ejpam-5240	118	8	denoted	denote	VERB
ejpam-5240	118	9	by	by	ADP
ejpam-5240	118	10	≃	≃	NOUN
ejpam-5240	118	11	,	,	PUNCT
ejpam-5240	118	12	it	it	PRON
ejpam-5240	118	13	can	can	AUX
ejpam-5240	118	14	be	be	AUX
ejpam-5240	118	15	seen	see	VERB
ejpam-5240	118	16	that	that	SCONJ
ejpam-5240	118	17	there	there	PRON
ejpam-5240	118	18	exists	exist	VERB
ejpam-5240	118	19	an	an	DET
ejpam-5240	118	20	isomorphism	isomorphism	NOUN
ejpam-5240	118	21	regarding	regard	VERB
ejpam-5240	118	22	the	the	DET
ejpam-5240	118	23	two	two	NUM
ejpam-5240	118	24	graphs	graph	NOUN
ejpam-5240	118	25	.	.	PUNCT
ejpam-5240	119	1	remark	remark	NOUN
ejpam-5240	119	2	5	5	NUM
ejpam-5240	119	3	.	.	PUNCT
ejpam-5240	120	1	let	let	VERB
ejpam-5240	120	2	c3	c3	PROPN
ejpam-5240	120	3	and	and	CCONJ
ejpam-5240	120	4	k3	k3	VERB
ejpam-5240	120	5	be	be	AUX
ejpam-5240	120	6	a	a	DET
ejpam-5240	120	7	cycle	cycle	NOUN
ejpam-5240	120	8	and	and	CCONJ
ejpam-5240	120	9	complete	complete	ADJ
ejpam-5240	120	10	graphs	graph	NOUN
ejpam-5240	120	11	of	of	ADP
ejpam-5240	120	12	order	order	NOUN
ejpam-5240	120	13	3	3	NUM
ejpam-5240	120	14	,	,	PUNCT
ejpam-5240	120	15	respectively	respectively	ADV
ejpam-5240	120	16	.	.	PUNCT
ejpam-5240	121	1	then	then	ADV
ejpam-5240	121	2	c3	c3	PROPN
ejpam-5240	121	3	≃	≃	PROPN
ejpam-5240	121	4	k3	k3	PROPN
ejpam-5240	121	5	.	.	PUNCT
ejpam-5240	122	1	definition	definition	NOUN
ejpam-5240	122	2	3	3	NUM
ejpam-5240	122	3	.	.	PUNCT
ejpam-5240	123	1	let	let	VERB
ejpam-5240	123	2	t	t	NOUN
ejpam-5240	123	3	=	=	SYM
ejpam-5240	123	4	{	{	PUNCT
ejpam-5240	123	5	t1	t1	NOUN
ejpam-5240	123	6	,	,	PUNCT
ejpam-5240	123	7	t2	t2	NOUN
ejpam-5240	123	8	,	,	PUNCT
ejpam-5240	123	9	...	...	PUNCT
ejpam-5240	123	10	,	,	PUNCT
ejpam-5240	123	11	tm	tm	PROPN
ejpam-5240	123	12	}	}	PUNCT
ejpam-5240	123	13	be	be	AUX
ejpam-5240	123	14	a	a	DET
ejpam-5240	123	15	nonempty	nonempty	ADJ
ejpam-5240	123	16	collection	collection	NOUN
ejpam-5240	123	17	of	of	ADP
ejpam-5240	123	18	sets	set	NOUN
ejpam-5240	123	19	.	.	PUNCT
ejpam-5240	124	1	a	a	DET
ejpam-5240	124	2	graph	graph	NOUN
ejpam-5240	124	3	g	g	NOUN
ejpam-5240	124	4	is	be	AUX
ejpam-5240	124	5	called	call	VERB
ejpam-5240	124	6	an	an	DET
ejpam-5240	124	7	intersection	intersection	NOUN
ejpam-5240	124	8	graph	graph	NOUN
ejpam-5240	124	9	whose	whose	DET
ejpam-5240	124	10	v	v	NOUN
ejpam-5240	124	11	(	(	PUNCT
ejpam-5240	124	12	g	g	NOUN
ejpam-5240	124	13	)	)	PUNCT
ejpam-5240	124	14	=	=	SYM
ejpam-5240	125	1	t	t	PROPN
ejpam-5240	125	2	and	and	CCONJ
ejpam-5240	125	3	[	[	X
ejpam-5240	125	4	ti	ti	X
ejpam-5240	125	5	,	,	PUNCT
ejpam-5240	125	6	tj	tj	X
ejpam-5240	125	7	]	]	PUNCT
ejpam-5240	125	8	∈	∈	PROPN
ejpam-5240	125	9	e(g	e(g	PROPN
ejpam-5240	125	10	)	)	PUNCT
ejpam-5240	125	11	for	for	ADP
ejpam-5240	125	12	1	1	NUM
ejpam-5240	125	13	≤	≤	NOUN
ejpam-5240	125	14	i	i	PRON
ejpam-5240	125	15	,	,	PUNCT
ejpam-5240	125	16	j	j	PROPN
ejpam-5240	125	17	≤	≤	PROPN
ejpam-5240	125	18	m	m	PROPN
ejpam-5240	125	19	,	,	PUNCT
ejpam-5240	125	20	where	where	SCONJ
ejpam-5240	125	21	i	i	PRON
ejpam-5240	125	22	̸=	̸=	PROPN
ejpam-5240	125	23	j	j	PROPN
ejpam-5240	125	24	is	be	AUX
ejpam-5240	125	25	an	an	DET
ejpam-5240	125	26	edge	edge	NOUN
ejpam-5240	125	27	in	in	ADP
ejpam-5240	125	28	g	g	PROPN
ejpam-5240	125	29	if	if	SCONJ
ejpam-5240	125	30	they	they	PRON
ejpam-5240	125	31	have	have	VERB
ejpam-5240	125	32	a	a	DET
ejpam-5240	125	33	nonempty	nonempty	ADJ
ejpam-5240	125	34	intersection	intersection	NOUN
ejpam-5240	125	35	.	.	PUNCT
ejpam-5240	126	1	for	for	ADP
ejpam-5240	126	2	every	every	DET
ejpam-5240	126	3	intersection	intersection	NOUN
ejpam-5240	126	4	graph	graph	NOUN
ejpam-5240	126	5	g	g	NOUN
ejpam-5240	126	6	,	,	PUNCT
ejpam-5240	126	7	we	we	PRON
ejpam-5240	126	8	have	have	VERB
ejpam-5240	126	9	|v	|v	NOUN
ejpam-5240	126	10	(	(	PUNCT
ejpam-5240	126	11	g)|	g)|	NOUN
ejpam-5240	126	12	=	=	PUNCT
ejpam-5240	126	13	|t	|t	NOUN
ejpam-5240	127	1	|	|	ADV
ejpam-5240	127	2	,	,	PUNCT
ejpam-5240	127	3	where	where	SCONJ
ejpam-5240	127	4	t	t	PROPN
ejpam-5240	127	5	is	be	AUX
ejpam-5240	127	6	a	a	DET
ejpam-5240	127	7	nonempty	nonempty	ADJ
ejpam-5240	127	8	family	family	NOUN
ejpam-5240	127	9	of	of	ADP
ejpam-5240	127	10	sets	set	NOUN
ejpam-5240	127	11	.	.	PUNCT
ejpam-5240	128	1	to	to	ADP
ejpam-5240	128	2	date	date	NOUN
ejpam-5240	128	3	,	,	PUNCT
ejpam-5240	128	4	there	there	PRON
ejpam-5240	128	5	is	be	VERB
ejpam-5240	128	6	no	no	DET
ejpam-5240	128	7	defined	define	VERB
ejpam-5240	128	8	size	size	NOUN
ejpam-5240	128	9	for	for	ADP
ejpam-5240	128	10	a	a	DET
ejpam-5240	128	11	generalized	generalized	ADJ
ejpam-5240	128	12	intersection	intersection	NOUN
ejpam-5240	128	13	graph	graph	NOUN
ejpam-5240	128	14	.	.	PUNCT
ejpam-5240	128	15	example	example	NOUN
ejpam-5240	129	1	1	1	NUM
ejpam-5240	129	2	.	.	X
ejpam-5240	129	3	consider	consider	VERB
ejpam-5240	129	4	the	the	DET
ejpam-5240	129	5	intersection	intersection	NOUN
ejpam-5240	129	6	graph	graph	NOUN
ejpam-5240	129	7	g	g	PROPN
ejpam-5240	129	8	over	over	ADP
ejpam-5240	129	9	the	the	DET
ejpam-5240	129	10	set	set	NOUN
ejpam-5240	129	11	t	t	NOUN
ejpam-5240	129	12	=	=	SYM
ejpam-5240	129	13	{	{	PUNCT
ejpam-5240	129	14	{	{	PUNCT
ejpam-5240	129	15	1	1	NUM
ejpam-5240	129	16	,	,	PUNCT
ejpam-5240	129	17	2	2	NUM
ejpam-5240	129	18	,	,	PUNCT
ejpam-5240	129	19	3	3	NUM
ejpam-5240	129	20	}	}	PUNCT
ejpam-5240	129	21	,	,	PUNCT
ejpam-5240	129	22	{	{	PUNCT
ejpam-5240	129	23	2	2	NUM
ejpam-5240	129	24	,	,	PUNCT
ejpam-5240	129	25	4	4	NUM
ejpam-5240	129	26	}	}	PUNCT
ejpam-5240	129	27	,	,	PUNCT
ejpam-5240	129	28	{	{	PUNCT
ejpam-5240	129	29	1	1	NUM
ejpam-5240	129	30	,	,	PUNCT
ejpam-5240	129	31	3	3	NUM
ejpam-5240	129	32	}	}	PUNCT
ejpam-5240	129	33	}	}	PUNCT
ejpam-5240	129	34	.	.	PUNCT
ejpam-5240	130	1	it	it	PRON
ejpam-5240	130	2	can	can	AUX
ejpam-5240	130	3	be	be	AUX
ejpam-5240	130	4	observed	observe	VERB
ejpam-5240	130	5	that	that	SCONJ
ejpam-5240	130	6	the	the	DET
ejpam-5240	130	7	order	order	NOUN
ejpam-5240	130	8	of	of	ADP
ejpam-5240	130	9	g	g	PROPN
ejpam-5240	130	10	is	be	AUX
ejpam-5240	130	11	3	3	NUM
ejpam-5240	130	12	.	.	PUNCT
ejpam-5240	131	1	also	also	ADV
ejpam-5240	131	2	,	,	PUNCT
ejpam-5240	131	3	since	since	SCONJ
ejpam-5240	131	4	{	{	PUNCT
ejpam-5240	131	5	1	1	NUM
ejpam-5240	131	6	,	,	PUNCT
ejpam-5240	131	7	2	2	NUM
ejpam-5240	131	8	,	,	PUNCT
ejpam-5240	131	9	3	3	NUM
ejpam-5240	131	10	}	}	PUNCT
ejpam-5240	131	11	∩	∩	NOUN
ejpam-5240	131	12	{	{	PUNCT
ejpam-5240	131	13	2	2	NUM
ejpam-5240	131	14	,	,	PUNCT
ejpam-5240	131	15	4	4	NUM
ejpam-5240	131	16	}	}	PUNCT
ejpam-5240	131	17	=	=	SYM
ejpam-5240	131	18	{	{	PUNCT
ejpam-5240	131	19	2	2	NUM
ejpam-5240	131	20	}	}	PUNCT
ejpam-5240	131	21	and	and	CCONJ
ejpam-5240	131	22	{	{	PUNCT
ejpam-5240	131	23	1	1	NUM
ejpam-5240	131	24	,	,	PUNCT
ejpam-5240	131	25	2	2	NUM
ejpam-5240	131	26	,	,	PUNCT
ejpam-5240	131	27	3}∩{1	3}∩{1	NUM
ejpam-5240	131	28	,	,	PUNCT
ejpam-5240	131	29	3	3	NUM
ejpam-5240	131	30	}	}	PUNCT
ejpam-5240	131	31	=	=	SYM
ejpam-5240	131	32	{	{	PUNCT
ejpam-5240	131	33	1	1	NUM
ejpam-5240	131	34	,	,	PUNCT
ejpam-5240	131	35	3	3	X
ejpam-5240	131	36	}	}	PUNCT
ejpam-5240	131	37	both	both	PRON
ejpam-5240	131	38	contain	contain	VERB
ejpam-5240	131	39	nonempty	nonempty	ADJ
ejpam-5240	131	40	intersection	intersection	NOUN
ejpam-5240	131	41	it	it	PRON
ejpam-5240	131	42	follows	follow	VERB
ejpam-5240	131	43	that	that	SCONJ
ejpam-5240	131	44	[	[	X
ejpam-5240	131	45	{	{	PUNCT
ejpam-5240	131	46	1	1	NUM
ejpam-5240	131	47	,	,	PUNCT
ejpam-5240	131	48	2	2	NUM
ejpam-5240	131	49	,	,	PUNCT
ejpam-5240	131	50	3	3	NUM
ejpam-5240	131	51	}	}	PUNCT
ejpam-5240	131	52	,	,	PUNCT
ejpam-5240	131	53	{	{	PUNCT
ejpam-5240	131	54	2	2	NUM
ejpam-5240	131	55	,	,	PUNCT
ejpam-5240	131	56	4	4	NUM
ejpam-5240	131	57	}	}	PUNCT
ejpam-5240	131	58	]	]	PUNCT
ejpam-5240	131	59	and	and	CCONJ
ejpam-5240	131	60	[	[	X
ejpam-5240	131	61	{	{	PUNCT
ejpam-5240	131	62	1	1	NUM
ejpam-5240	131	63	,	,	PUNCT
ejpam-5240	131	64	2	2	NUM
ejpam-5240	131	65	,	,	PUNCT
ejpam-5240	131	66	3	3	NUM
ejpam-5240	131	67	}	}	PUNCT
ejpam-5240	131	68	,	,	PUNCT
ejpam-5240	131	69	{	{	PUNCT
ejpam-5240	131	70	1	1	NUM
ejpam-5240	131	71	,	,	PUNCT
ejpam-5240	131	72	3	3	NUM
ejpam-5240	131	73	}	}	PUNCT
ejpam-5240	131	74	]	]	PUNCT
ejpam-5240	131	75	are	be	AUX
ejpam-5240	131	76	edges	edge	NOUN
ejpam-5240	131	77	of	of	ADP
ejpam-5240	131	78	g.	g.	PROPN
ejpam-5240	131	79	moreover	moreover	ADV
ejpam-5240	131	80	,	,	PUNCT
ejpam-5240	131	81	{	{	PUNCT
ejpam-5240	131	82	2	2	NUM
ejpam-5240	131	83	,	,	PUNCT
ejpam-5240	131	84	4	4	NUM
ejpam-5240	131	85	}	}	PUNCT
ejpam-5240	131	86	and	and	CCONJ
ejpam-5240	131	87	{	{	PUNCT
ejpam-5240	131	88	1	1	NUM
ejpam-5240	131	89	,	,	PUNCT
ejpam-5240	131	90	3	3	NUM
ejpam-5240	131	91	}	}	PUNCT
ejpam-5240	131	92	has	have	VERB
ejpam-5240	131	93	no	no	DET
ejpam-5240	131	94	intersection	intersection	NOUN
ejpam-5240	131	95	,	,	PUNCT
ejpam-5240	131	96	so	so	SCONJ
ejpam-5240	131	97	[	[	X
ejpam-5240	131	98	{	{	PUNCT
ejpam-5240	131	99	2	2	NUM
ejpam-5240	131	100	,	,	PUNCT
ejpam-5240	131	101	4	4	NUM
ejpam-5240	131	102	}	}	PUNCT
ejpam-5240	131	103	,	,	PUNCT
ejpam-5240	131	104	{	{	PUNCT
ejpam-5240	131	105	1	1	NUM
ejpam-5240	131	106	,	,	PUNCT
ejpam-5240	131	107	3	3	NUM
ejpam-5240	131	108	}	}	PUNCT
ejpam-5240	131	109	]	]	PUNCT
ejpam-5240	131	110	is	be	AUX
ejpam-5240	131	111	not	not	PART
ejpam-5240	131	112	in	in	ADP
ejpam-5240	131	113	e(g	e(g	NOUN
ejpam-5240	131	114	)	)	PUNCT
ejpam-5240	131	115	.	.	PUNCT
ejpam-5240	132	1	by	by	ADP
ejpam-5240	132	2	these	these	PRON
ejpam-5240	132	3	,	,	PUNCT
ejpam-5240	132	4	then	then	ADV
ejpam-5240	132	5	e(g	e(g	PROPN
ejpam-5240	132	6	)	)	PUNCT
ejpam-5240	133	1	=	=	PRON
ejpam-5240	133	2	{	{	PUNCT
ejpam-5240	134	1	[	[	X
ejpam-5240	134	2	{	{	PUNCT
ejpam-5240	134	3	1	1	NUM
ejpam-5240	134	4	,	,	PUNCT
ejpam-5240	134	5	2	2	NUM
ejpam-5240	134	6	,	,	PUNCT
ejpam-5240	134	7	3	3	NUM
ejpam-5240	134	8	}	}	PUNCT
ejpam-5240	134	9	,	,	PUNCT
ejpam-5240	134	10	{	{	PUNCT
ejpam-5240	134	11	2	2	NUM
ejpam-5240	134	12	,	,	PUNCT
ejpam-5240	134	13	4	4	NUM
ejpam-5240	134	14	}	}	PUNCT
ejpam-5240	134	15	]	]	PUNCT
ejpam-5240	134	16	,	,	PUNCT
ejpam-5240	134	17	[	[	X
ejpam-5240	134	18	{	{	PUNCT
ejpam-5240	134	19	1	1	NUM
ejpam-5240	134	20	,	,	PUNCT
ejpam-5240	134	21	2	2	NUM
ejpam-5240	134	22	,	,	PUNCT
ejpam-5240	134	23	3	3	NUM
ejpam-5240	134	24	}	}	PUNCT
ejpam-5240	134	25	,	,	PUNCT
ejpam-5240	134	26	{	{	PUNCT
ejpam-5240	134	27	1	1	NUM
ejpam-5240	134	28	,	,	PUNCT
ejpam-5240	134	29	3	3	NUM
ejpam-5240	134	30	}	}	PUNCT
ejpam-5240	134	31	]	]	PUNCT
ejpam-5240	134	32	}	}	PUNCT
ejpam-5240	134	33	.	.	PUNCT
ejpam-5240	135	1	shown	show	VERB
ejpam-5240	135	2	in	in	ADP
ejpam-5240	135	3	figure	figure	NOUN
ejpam-5240	135	4	1	1	NUM
ejpam-5240	135	5	is	be	AUX
ejpam-5240	135	6	a	a	DET
ejpam-5240	135	7	pictorial	pictorial	ADJ
ejpam-5240	135	8	illustration	illustration	NOUN
ejpam-5240	135	9	of	of	ADP
ejpam-5240	135	10	the	the	DET
ejpam-5240	135	11	intersection	intersection	NOUN
ejpam-5240	135	12	graph	graph	NOUN
ejpam-5240	135	13	over	over	ADP
ejpam-5240	135	14	the	the	DET
ejpam-5240	135	15	set	set	NOUN
ejpam-5240	135	16	t	t	NOUN
ejpam-5240	135	17	=	=	SYM
ejpam-5240	135	18	{	{	PUNCT
ejpam-5240	135	19	{	{	PUNCT
ejpam-5240	135	20	1	1	NUM
ejpam-5240	135	21	,	,	PUNCT
ejpam-5240	135	22	2	2	NUM
ejpam-5240	135	23	,	,	PUNCT
ejpam-5240	135	24	3	3	NUM
ejpam-5240	135	25	}	}	PUNCT
ejpam-5240	135	26	,	,	PUNCT
ejpam-5240	135	27	{	{	PUNCT
ejpam-5240	135	28	2	2	NUM
ejpam-5240	135	29	,	,	PUNCT
ejpam-5240	135	30	4	4	NUM
ejpam-5240	135	31	}	}	PUNCT
ejpam-5240	135	32	,	,	PUNCT
ejpam-5240	135	33	{	{	PUNCT
ejpam-5240	135	34	1	1	NUM
ejpam-5240	135	35	,	,	PUNCT
ejpam-5240	135	36	3	3	NUM
ejpam-5240	135	37	}	}	PUNCT
ejpam-5240	135	38	}	}	PUNCT
ejpam-5240	135	39	.	.	PUNCT
ejpam-5240	136	1	m.e	m.e	PROPN
ejpam-5240	136	2	.	.	PROPN
ejpam-5240	136	3	pelagio	pelagio	PROPN
ejpam-5240	136	4	,	,	PUNCT
ejpam-5240	136	5	n.	n.	NOUN
ejpam-5240	136	6	mame	mame	PROPN
ejpam-5240	136	7	,	,	PUNCT
ejpam-5240	136	8	k.	k.	PROPN
ejpam-5240	136	9	mendoza	mendoza	PROPN
ejpam-5240	136	10	/	/	SYM
ejpam-5240	136	11	eur	eur	PROPN
ejpam-5240	136	12	.	.	PUNCT
ejpam-5240	137	1	j.	j.	PROPN
ejpam-5240	137	2	pure	pure	PROPN
ejpam-5240	137	3	appl	appl	PROPN
ejpam-5240	137	4	.	.	PROPN
ejpam-5240	137	5	math	math	PROPN
ejpam-5240	137	6	,	,	PUNCT
ejpam-5240	137	7	17	17	NUM
ejpam-5240	137	8	(	(	PUNCT
ejpam-5240	137	9	3	3	NUM
ejpam-5240	137	10	)	)	PUNCT
ejpam-5240	137	11	(	(	PUNCT
ejpam-5240	137	12	2024	2024	NUM
ejpam-5240	137	13	)	)	PUNCT
ejpam-5240	137	14	,	,	PUNCT
ejpam-5240	137	15	1779	1779	NUM
ejpam-5240	137	16	-	-	SYM
ejpam-5240	137	17	1803	1803	NUM
ejpam-5240	137	18	1784	1784	NUM
ejpam-5240	137	19	{	{	PUNCT
ejpam-5240	137	20	2	2	NUM
ejpam-5240	137	21	,	,	PUNCT
ejpam-5240	137	22	4	4	NUM
ejpam-5240	137	23	}	}	PUNCT
ejpam-5240	137	24	{	{	PUNCT
ejpam-5240	137	25	1	1	NUM
ejpam-5240	137	26	,	,	PUNCT
ejpam-5240	137	27	3}{1	3}{1	NUM
ejpam-5240	137	28	,	,	PUNCT
ejpam-5240	137	29	2	2	NUM
ejpam-5240	137	30	,	,	PUNCT
ejpam-5240	137	31	3	3	NUM
ejpam-5240	137	32	}	}	PUNCT
ejpam-5240	137	33	figure	figure	NOUN
ejpam-5240	137	34	1	1	NUM
ejpam-5240	137	35	:	:	PUNCT
ejpam-5240	137	36	an	an	DET
ejpam-5240	137	37	illustration	illustration	NOUN
ejpam-5240	137	38	of	of	ADP
ejpam-5240	137	39	the	the	DET
ejpam-5240	137	40	intersection	intersection	NOUN
ejpam-5240	137	41	graph	graph	NOUN
ejpam-5240	137	42	of	of	ADP
ejpam-5240	137	43	the	the	DET
ejpam-5240	137	44	set	set	PROPN
ejpam-5240	137	45	t	t	PROPN
ejpam-5240	137	46	.	.	PUNCT
ejpam-5240	138	1	consider	consider	VERB
ejpam-5240	138	2	a	a	DET
ejpam-5240	138	3	graph	graph	NOUN
ejpam-5240	138	4	g.	g.	VERB
ejpam-5240	138	5	the	the	DET
ejpam-5240	138	6	complement	complement	NOUN
ejpam-5240	138	7	of	of	ADP
ejpam-5240	138	8	g	g	NOUN
ejpam-5240	138	9	,	,	PUNCT
ejpam-5240	138	10	denoted	denote	VERB
ejpam-5240	138	11	by	by	ADP
ejpam-5240	138	12	g	g	NOUN
ejpam-5240	138	13	,	,	PUNCT
ejpam-5240	138	14	is	be	AUX
ejpam-5240	138	15	the	the	DET
ejpam-5240	138	16	graph	graph	NOUN
ejpam-5240	138	17	whose	whose	DET
ejpam-5240	138	18	vertex	vertex	NOUN
ejpam-5240	138	19	set	set	NOUN
ejpam-5240	138	20	is	be	AUX
ejpam-5240	138	21	v	v	NOUN
ejpam-5240	138	22	(	(	PUNCT
ejpam-5240	138	23	g	g	NOUN
ejpam-5240	138	24	)	)	PUNCT
ejpam-5240	138	25	and	and	CCONJ
ejpam-5240	138	26	such	such	ADJ
ejpam-5240	138	27	that	that	PRON
ejpam-5240	138	28	for	for	ADP
ejpam-5240	138	29	every	every	DET
ejpam-5240	138	30	pair	pair	NOUN
ejpam-5240	138	31	x	x	NOUN
ejpam-5240	138	32	,	,	PUNCT
ejpam-5240	138	33	y	y	PROPN
ejpam-5240	138	34	∈	∈	PROPN
ejpam-5240	138	35	v	v	NOUN
ejpam-5240	138	36	(	(	PUNCT
ejpam-5240	138	37	g	g	NOUN
ejpam-5240	138	38	)	)	PUNCT
ejpam-5240	138	39	,	,	PUNCT
ejpam-5240	139	1	[	[	X
ejpam-5240	139	2	x	x	X
ejpam-5240	139	3	,	,	PUNCT
ejpam-5240	139	4	y	y	PROPN
ejpam-5240	139	5	]	]	X
ejpam-5240	139	6	is	be	AUX
ejpam-5240	139	7	an	an	DET
ejpam-5240	139	8	edge	edge	NOUN
ejpam-5240	139	9	of	of	ADP
ejpam-5240	139	10	g	g	NOUN
ejpam-5240	139	11	if	if	SCONJ
ejpam-5240	139	12	and	and	CCONJ
ejpam-5240	139	13	only	only	ADV
ejpam-5240	139	14	if	if	SCONJ
ejpam-5240	139	15	[	[	X
ejpam-5240	139	16	x	x	X
ejpam-5240	139	17	,	,	PUNCT
ejpam-5240	139	18	y	y	PROPN
ejpam-5240	139	19	]	]	PUNCT
ejpam-5240	139	20	is	be	AUX
ejpam-5240	139	21	not	not	PART
ejpam-5240	139	22	an	an	DET
ejpam-5240	139	23	edge	edge	NOUN
ejpam-5240	139	24	of	of	ADP
ejpam-5240	139	25	g.	g.	PROPN
ejpam-5240	139	26	recall	recall	VERB
ejpam-5240	139	27	that	that	SCONJ
ejpam-5240	139	28	the	the	DET
ejpam-5240	139	29	degree	degree	NOUN
ejpam-5240	139	30	of	of	ADP
ejpam-5240	139	31	every	every	DET
ejpam-5240	139	32	vertex	vertex	NOUN
ejpam-5240	139	33	in	in	ADP
ejpam-5240	139	34	a	a	DET
ejpam-5240	139	35	regular	regular	ADJ
ejpam-5240	139	36	graph	graph	NOUN
ejpam-5240	139	37	is	be	AUX
ejpam-5240	139	38	equal	equal	ADJ
ejpam-5240	139	39	in	in	ADP
ejpam-5240	139	40	number	number	NOUN
ejpam-5240	139	41	.	.	PUNCT
ejpam-5240	140	1	this	this	PRON
ejpam-5240	140	2	can	can	AUX
ejpam-5240	140	3	infer	infer	VERB
ejpam-5240	140	4	that	that	SCONJ
ejpam-5240	140	5	the	the	DET
ejpam-5240	140	6	vertices	vertex	NOUN
ejpam-5240	140	7	in	in	ADP
ejpam-5240	140	8	a	a	DET
ejpam-5240	140	9	complement	complement	NOUN
ejpam-5240	140	10	graph	graph	NOUN
ejpam-5240	140	11	of	of	ADP
ejpam-5240	140	12	a	a	DET
ejpam-5240	140	13	regular	regular	ADJ
ejpam-5240	140	14	graph	graph	NOUN
ejpam-5240	140	15	yield	yield	NOUN
ejpam-5240	140	16	also	also	ADV
ejpam-5240	140	17	a	a	DET
ejpam-5240	140	18	degree	degree	NOUN
ejpam-5240	140	19	that	that	PRON
ejpam-5240	140	20	is	be	AUX
ejpam-5240	140	21	equal	equal	ADJ
ejpam-5240	140	22	in	in	ADP
ejpam-5240	140	23	number	number	NOUN
ejpam-5240	140	24	.	.	PUNCT
ejpam-5240	141	1	in	in	ADP
ejpam-5240	141	2	sum	sum	NOUN
ejpam-5240	141	3	,	,	PUNCT
ejpam-5240	141	4	the	the	DET
ejpam-5240	141	5	complement	complement	NOUN
ejpam-5240	141	6	graph	graph	NOUN
ejpam-5240	141	7	of	of	ADP
ejpam-5240	141	8	a	a	DET
ejpam-5240	141	9	regular	regular	ADJ
ejpam-5240	141	10	graph	graph	NOUN
ejpam-5240	141	11	is	be	AUX
ejpam-5240	141	12	considered	consider	VERB
ejpam-5240	141	13	a	a	DET
ejpam-5240	141	14	regular	regular	ADJ
ejpam-5240	141	15	graph	graph	NOUN
ejpam-5240	141	16	.	.	PUNCT
ejpam-5240	142	1	definition	definition	NOUN
ejpam-5240	142	2	4	4	NUM
ejpam-5240	142	3	.	.	PUNCT
ejpam-5240	143	1	let	let	VERB
ejpam-5240	143	2	g	g	PRON
ejpam-5240	143	3	be	be	AUX
ejpam-5240	143	4	a	a	DET
ejpam-5240	143	5	graph	graph	NOUN
ejpam-5240	143	6	.	.	PUNCT
ejpam-5240	144	1	the	the	DET
ejpam-5240	144	2	nonempty	nonempty	ADV
ejpam-5240	144	3	set	set	VERB
ejpam-5240	144	4	t	t	PROPN
ejpam-5240	144	5	⊆	⊆	NUM
ejpam-5240	144	6	v	v	NOUN
ejpam-5240	144	7	(	(	PUNCT
ejpam-5240	144	8	g	g	NOUN
ejpam-5240	144	9	)	)	PUNCT
ejpam-5240	144	10	is	be	AUX
ejpam-5240	144	11	called	call	VERB
ejpam-5240	144	12	an	an	DET
ejpam-5240	144	13	independent	independent	ADJ
ejpam-5240	144	14	set	set	NOUN
ejpam-5240	144	15	in	in	ADP
ejpam-5240	144	16	a	a	DET
ejpam-5240	144	17	graph	graph	NOUN
ejpam-5240	144	18	g	g	NOUN
ejpam-5240	144	19	if	if	SCONJ
ejpam-5240	144	20	for	for	ADP
ejpam-5240	144	21	every	every	DET
ejpam-5240	144	22	x	x	NOUN
ejpam-5240	144	23	,	,	PUNCT
ejpam-5240	144	24	y	y	PROPN
ejpam-5240	144	25	∈	∈	PROPN
ejpam-5240	144	26	t	t	PROPN
ejpam-5240	144	27	,	,	PUNCT
ejpam-5240	144	28	then	then	ADV
ejpam-5240	144	29	[	[	X
ejpam-5240	144	30	x	x	X
ejpam-5240	144	31	,	,	PUNCT
ejpam-5240	144	32	y	y	PROPN
ejpam-5240	144	33	]	]	PUNCT
ejpam-5240	144	34	/∈	/∈	PUNCT
ejpam-5240	144	35	e(g	e(g	PROPN
ejpam-5240	144	36	)	)	PUNCT
ejpam-5240	144	37	.	.	PUNCT
ejpam-5240	145	1	the	the	DET
ejpam-5240	145	2	independence	independence	NOUN
ejpam-5240	145	3	number	number	NOUN
ejpam-5240	145	4	of	of	ADP
ejpam-5240	145	5	a	a	DET
ejpam-5240	145	6	graph	graph	NOUN
ejpam-5240	145	7	,	,	PUNCT
ejpam-5240	145	8	denoted	denote	VERB
ejpam-5240	145	9	by	by	ADP
ejpam-5240	145	10	α(g	α(g	NOUN
ejpam-5240	145	11	)	)	PUNCT
ejpam-5240	145	12	,	,	PUNCT
ejpam-5240	145	13	is	be	AUX
ejpam-5240	145	14	the	the	DET
ejpam-5240	145	15	cardinality	cardinality	NOUN
ejpam-5240	145	16	of	of	ADP
ejpam-5240	145	17	the	the	DET
ejpam-5240	145	18	largest	large	ADJ
ejpam-5240	145	19	independent	independent	ADJ
ejpam-5240	145	20	set	set	NOUN
ejpam-5240	145	21	of	of	ADP
ejpam-5240	145	22	g.	g.	PROPN
ejpam-5240	145	23	in	in	ADP
ejpam-5240	145	24	a	a	DET
ejpam-5240	145	25	graph	graph	NOUN
ejpam-5240	145	26	g	g	NOUN
ejpam-5240	145	27	,	,	PUNCT
ejpam-5240	145	28	if	if	SCONJ
ejpam-5240	145	29	there	there	PRON
ejpam-5240	145	30	exists	exist	VERB
ejpam-5240	145	31	an	an	DET
ejpam-5240	145	32	independent	independent	ADJ
ejpam-5240	145	33	set	set	NOUN
ejpam-5240	145	34	t	t	PROPN
ejpam-5240	145	35	⊆	⊆	NUM
ejpam-5240	145	36	v	v	NOUN
ejpam-5240	145	37	(	(	PUNCT
ejpam-5240	145	38	g	g	NOUN
ejpam-5240	145	39	)	)	PUNCT
ejpam-5240	145	40	,	,	PUNCT
ejpam-5240	145	41	it	it	PRON
ejpam-5240	145	42	follows	follow	VERB
ejpam-5240	145	43	that	that	SCONJ
ejpam-5240	145	44	α(g	α(g	PROPN
ejpam-5240	145	45	)	)	PUNCT
ejpam-5240	145	46	≥	≥	PRON
ejpam-5240	145	47	|t	|t	VERB
ejpam-5240	145	48	|	|	ADV
ejpam-5240	145	49	.	.	PUNCT
ejpam-5240	146	1	the	the	DET
ejpam-5240	146	2	next	next	ADJ
ejpam-5240	146	3	theorem	theorem	NOUN
ejpam-5240	146	4	determines	determine	VERB
ejpam-5240	146	5	the	the	DET
ejpam-5240	146	6	independence	independence	NOUN
ejpam-5240	146	7	number	number	NOUN
ejpam-5240	146	8	of	of	ADP
ejpam-5240	146	9	a	a	DET
ejpam-5240	146	10	complete	complete	ADJ
ejpam-5240	146	11	graph	graph	NOUN
ejpam-5240	146	12	of	of	ADP
ejpam-5240	146	13	order	order	NOUN
ejpam-5240	146	14	n.	n.	NOUN
ejpam-5240	146	15	definition	definition	NOUN
ejpam-5240	146	16	5	5	NUM
ejpam-5240	146	17	.	.	PUNCT
ejpam-5240	147	1	let	let	VERB
ejpam-5240	147	2	g	g	PROPN
ejpam-5240	147	3	=	=	SYM
ejpam-5240	147	4	(	(	PUNCT
ejpam-5240	147	5	v	v	NOUN
ejpam-5240	147	6	(	(	PUNCT
ejpam-5240	147	7	g	g	NOUN
ejpam-5240	147	8	)	)	PUNCT
ejpam-5240	147	9	,	,	PUNCT
ejpam-5240	147	10	e(g	e(g	PROPN
ejpam-5240	147	11	)	)	PUNCT
ejpam-5240	147	12	)	)	PUNCT
ejpam-5240	148	1	be	be	AUX
ejpam-5240	148	2	a	a	DET
ejpam-5240	148	3	graph	graph	NOUN
ejpam-5240	148	4	.	.	PUNCT
ejpam-5240	149	1	a	a	DET
ejpam-5240	149	2	nonempty	nonempty	NOUN
ejpam-5240	149	3	subset	subset	VERB
ejpam-5240	149	4	t	t	NOUN
ejpam-5240	149	5	of	of	ADP
ejpam-5240	149	6	v	v	PROPN
ejpam-5240	149	7	(	(	PUNCT
ejpam-5240	149	8	g	g	NOUN
ejpam-5240	149	9	)	)	PUNCT
ejpam-5240	149	10	is	be	AUX
ejpam-5240	149	11	called	call	VERB
ejpam-5240	149	12	the	the	DET
ejpam-5240	149	13	dominating	dominating	NOUN
ejpam-5240	149	14	set	set	NOUN
ejpam-5240	149	15	of	of	ADP
ejpam-5240	149	16	g	g	PROPN
ejpam-5240	149	17	if	if	SCONJ
ejpam-5240	149	18	every	every	DET
ejpam-5240	149	19	element	element	NOUN
ejpam-5240	149	20	of	of	ADP
ejpam-5240	149	21	v	v	NOUN
ejpam-5240	149	22	(	(	PUNCT
ejpam-5240	149	23	g	g	NOUN
ejpam-5240	149	24	)	)	PUNCT
ejpam-5240	149	25	\	\	PROPN
ejpam-5240	149	26	t	t	PROPN
ejpam-5240	149	27	is	be	AUX
ejpam-5240	149	28	adjacent	adjacent	ADJ
ejpam-5240	149	29	to	to	ADP
ejpam-5240	149	30	some	some	DET
ejpam-5240	149	31	element	element	NOUN
ejpam-5240	149	32	of	of	ADP
ejpam-5240	149	33	t	t	PROPN
ejpam-5240	149	34	.	.	PUNCT
ejpam-5240	150	1	moreover	moreover	ADV
ejpam-5240	150	2	,	,	PUNCT
ejpam-5240	150	3	the	the	DET
ejpam-5240	150	4	domination	domination	NOUN
ejpam-5240	150	5	number	number	NOUN
ejpam-5240	150	6	,	,	PUNCT
ejpam-5240	150	7	written	write	VERB
ejpam-5240	150	8	as	as	ADP
ejpam-5240	150	9	γ(g	γ(g	PROPN
ejpam-5240	150	10	)	)	PUNCT
ejpam-5240	150	11	,	,	PUNCT
ejpam-5240	150	12	of	of	ADP
ejpam-5240	150	13	a	a	DET
ejpam-5240	150	14	graph	graph	NOUN
ejpam-5240	150	15	g	g	NOUN
ejpam-5240	150	16	is	be	AUX
ejpam-5240	150	17	the	the	DET
ejpam-5240	150	18	minimum	minimum	ADJ
ejpam-5240	150	19	cardinality	cardinality	NOUN
ejpam-5240	150	20	among	among	ADP
ejpam-5240	150	21	all	all	DET
ejpam-5240	150	22	the	the	DET
ejpam-5240	150	23	dominating	dominating	NOUN
ejpam-5240	150	24	sets	set	NOUN
ejpam-5240	150	25	of	of	ADP
ejpam-5240	150	26	g.	g.	PROPN
ejpam-5240	150	27	the	the	DET
ejpam-5240	150	28	vertex	vertex	NOUN
ejpam-5240	150	29	set	set	VERB
ejpam-5240	150	30	v	v	NOUN
ejpam-5240	150	31	(	(	PUNCT
ejpam-5240	150	32	g	g	NOUN
ejpam-5240	150	33	)	)	PUNCT
ejpam-5240	150	34	of	of	ADP
ejpam-5240	150	35	a	a	DET
ejpam-5240	150	36	graph	graph	NOUN
ejpam-5240	150	37	g	g	NOUN
ejpam-5240	150	38	is	be	AUX
ejpam-5240	150	39	a	a	DET
ejpam-5240	150	40	dominating	dominating	NOUN
ejpam-5240	150	41	set	set	NOUN
ejpam-5240	150	42	since	since	SCONJ
ejpam-5240	150	43	v	v	NOUN
ejpam-5240	150	44	(	(	PUNCT
ejpam-5240	150	45	g	g	NOUN
ejpam-5240	150	46	)	)	PUNCT
ejpam-5240	150	47	\	\	PROPN
ejpam-5240	150	48	v	v	X
ejpam-5240	150	49	(	(	PUNCT
ejpam-5240	150	50	g	g	NOUN
ejpam-5240	150	51	)	)	PUNCT
ejpam-5240	150	52	=	=	NOUN
ejpam-5240	150	53	∅	∅	NOUN
ejpam-5240	150	54	which	which	PRON
ejpam-5240	150	55	implies	imply	VERB
ejpam-5240	150	56	that	that	SCONJ
ejpam-5240	150	57	there	there	PRON
ejpam-5240	150	58	are	be	VERB
ejpam-5240	150	59	no	no	DET
ejpam-5240	150	60	other	other	ADJ
ejpam-5240	150	61	vertices	vertex	NOUN
ejpam-5240	150	62	that	that	PRON
ejpam-5240	150	63	are	be	AUX
ejpam-5240	150	64	needed	need	VERB
ejpam-5240	150	65	to	to	PART
ejpam-5240	150	66	be	be	AUX
ejpam-5240	150	67	considered	consider	VERB
ejpam-5240	150	68	.	.	PUNCT
ejpam-5240	151	1	a	a	DET
ejpam-5240	151	2	graph	graph	NOUN
ejpam-5240	151	3	h	h	NOUN
ejpam-5240	151	4	is	be	AUX
ejpam-5240	151	5	called	call	VERB
ejpam-5240	151	6	a	a	DET
ejpam-5240	151	7	subgraph	subgraph	NOUN
ejpam-5240	151	8	of	of	ADP
ejpam-5240	151	9	a	a	DET
ejpam-5240	151	10	graph	graph	NOUN
ejpam-5240	151	11	g	g	NOUN
ejpam-5240	151	12	,	,	PUNCT
ejpam-5240	151	13	written	write	VERB
ejpam-5240	151	14	h	h	NOUN
ejpam-5240	151	15	⊆	⊆	NUM
ejpam-5240	151	16	g	g	NOUN
ejpam-5240	151	17	,	,	PUNCT
ejpam-5240	151	18	if	if	SCONJ
ejpam-5240	151	19	v	v	X
ejpam-5240	151	20	(	(	PUNCT
ejpam-5240	151	21	h	h	NOUN
ejpam-5240	151	22	)	)	PUNCT
ejpam-5240	151	23	⊆	⊆	NUM
ejpam-5240	151	24	v	v	NOUN
ejpam-5240	151	25	(	(	PUNCT
ejpam-5240	151	26	g	g	NOUN
ejpam-5240	151	27	)	)	PUNCT
ejpam-5240	151	28	and	and	CCONJ
ejpam-5240	151	29	e(h	e(h	NOUN
ejpam-5240	151	30	)	)	PUNCT
ejpam-5240	151	31	⊆	⊆	NUM
ejpam-5240	151	32	e(g	e(g	PROPN
ejpam-5240	151	33	)	)	PUNCT
ejpam-5240	151	34	.	.	PUNCT
ejpam-5240	152	1	moreover	moreover	ADV
ejpam-5240	152	2	,	,	PUNCT
ejpam-5240	152	3	a	a	DET
ejpam-5240	152	4	subgraph	subgraph	NOUN
ejpam-5240	152	5	h	h	NOUN
ejpam-5240	152	6	of	of	ADP
ejpam-5240	152	7	a	a	DET
ejpam-5240	152	8	graph	graph	NOUN
ejpam-5240	152	9	g	g	NOUN
ejpam-5240	152	10	is	be	AUX
ejpam-5240	152	11	called	call	VERB
ejpam-5240	152	12	an	an	DET
ejpam-5240	152	13	induced	induced	ADJ
ejpam-5240	152	14	-	-	PUNCT
ejpam-5240	152	15	subgraph	subgraph	NOUN
ejpam-5240	152	16	(	(	PUNCT
ejpam-5240	152	17	also	also	ADV
ejpam-5240	152	18	called	call	VERB
ejpam-5240	152	19	a	a	DET
ejpam-5240	152	20	“	"	PUNCT
ejpam-5240	152	21	vertex	vertex	NOUN
ejpam-5240	152	22	-	-	PUNCT
ejpam-5240	152	23	induced	induce	VERB
ejpam-5240	152	24	subgraph	subgraph	NOUN
ejpam-5240	152	25	”	"	PUNCT
ejpam-5240	152	26	)	)	PUNCT
ejpam-5240	152	27	,	,	PUNCT
ejpam-5240	152	28	written	write	VERB
ejpam-5240	152	29	as	as	ADP
ejpam-5240	152	30	<	<	X
ejpam-5240	152	31	h	h	X
ejpam-5240	152	32	>	>	X
ejpam-5240	152	33	,	,	PUNCT
ejpam-5240	152	34	if	if	SCONJ
ejpam-5240	152	35	whenever	whenever	SCONJ
ejpam-5240	152	36	x	x	X
ejpam-5240	152	37	,	,	PUNCT
ejpam-5240	152	38	y	y	PROPN
ejpam-5240	152	39	∈	∈	PROPN
ejpam-5240	152	40	h	h	NOUN
ejpam-5240	152	41	and	and	CCONJ
ejpam-5240	152	42	[	[	X
ejpam-5240	152	43	x	x	X
ejpam-5240	152	44	,	,	PUNCT
ejpam-5240	152	45	y	y	PROPN
ejpam-5240	152	46	]	]	X
ejpam-5240	152	47	∈	∈	PROPN
ejpam-5240	152	48	e(g	e(g	PROPN
ejpam-5240	152	49	)	)	PUNCT
ejpam-5240	152	50	,	,	PUNCT
ejpam-5240	152	51	then	then	ADV
ejpam-5240	152	52	[	[	X
ejpam-5240	152	53	x	x	X
ejpam-5240	152	54	,	,	PUNCT
ejpam-5240	152	55	y	y	PROPN
ejpam-5240	152	56	]	]	X
ejpam-5240	152	57	is	be	AUX
ejpam-5240	152	58	an	an	DET
ejpam-5240	152	59	edge	edge	NOUN
ejpam-5240	152	60	of	of	ADP
ejpam-5240	152	61	<	<	X
ejpam-5240	152	62	h	h	X
ejpam-5240	152	63	>	>	PUNCT
ejpam-5240	152	64	.	.	PUNCT
ejpam-5240	153	1	definition	definition	NOUN
ejpam-5240	153	2	6	6	NUM
ejpam-5240	153	3	.	.	PUNCT
ejpam-5240	154	1	a	a	DET
ejpam-5240	154	2	dominating	dominating	NOUN
ejpam-5240	154	3	set	set	NOUN
ejpam-5240	154	4	t	t	PROPN
ejpam-5240	154	5	is	be	AUX
ejpam-5240	154	6	called	call	VERB
ejpam-5240	154	7	an	an	DET
ejpam-5240	154	8	isolate	isolate	NOUN
ejpam-5240	154	9	dominating	dominating	NOUN
ejpam-5240	154	10	set	set	NOUN
ejpam-5240	154	11	if	if	SCONJ
ejpam-5240	154	12	the	the	DET
ejpam-5240	154	13	subgraph	subgraph	NOUN
ejpam-5240	154	14	induced	induce	VERB
ejpam-5240	154	15	by	by	ADP
ejpam-5240	154	16	t	t	PROPN
ejpam-5240	154	17	on	on	ADP
ejpam-5240	154	18	graph	graph	NOUN
ejpam-5240	154	19	g	g	PROPN
ejpam-5240	154	20	has	have	VERB
ejpam-5240	154	21	at	at	ADV
ejpam-5240	154	22	least	least	ADJ
ejpam-5240	154	23	one	one	NUM
ejpam-5240	154	24	isolated	isolated	ADJ
ejpam-5240	154	25	vertex	vertex	NOUN
ejpam-5240	154	26	.	.	PUNCT
ejpam-5240	155	1	an	an	DET
ejpam-5240	155	2	isolate	isolate	ADJ
ejpam-5240	155	3	domination	domination	NOUN
ejpam-5240	155	4	number	number	NOUN
ejpam-5240	155	5	,	,	PUNCT
ejpam-5240	155	6	denoted	denote	VERB
ejpam-5240	155	7	by	by	ADP
ejpam-5240	155	8	γ0	γ0	NOUN
ejpam-5240	155	9	,	,	PUNCT
ejpam-5240	155	10	has	have	VERB
ejpam-5240	155	11	the	the	DET
ejpam-5240	155	12	minimal	minimal	ADJ
ejpam-5240	155	13	cardinality	cardinality	NOUN
ejpam-5240	155	14	among	among	ADP
ejpam-5240	155	15	all	all	DET
ejpam-5240	155	16	the	the	DET
ejpam-5240	155	17	isolate	isolate	ADJ
ejpam-5240	155	18	dominating	dominating	NOUN
ejpam-5240	155	19	sets	set	NOUN
ejpam-5240	155	20	of	of	ADP
ejpam-5240	155	21	g.	g.	PROPN
ejpam-5240	155	22	note	note	VERB
ejpam-5240	155	23	that	that	SCONJ
ejpam-5240	155	24	the	the	DET
ejpam-5240	155	25	domination	domination	NOUN
ejpam-5240	155	26	number	number	NOUN
ejpam-5240	155	27	must	must	AUX
ejpam-5240	155	28	be	be	AUX
ejpam-5240	155	29	the	the	DET
ejpam-5240	155	30	smallest	small	ADJ
ejpam-5240	155	31	among	among	ADP
ejpam-5240	155	32	the	the	DET
ejpam-5240	155	33	cardinality	cardinality	NOUN
ejpam-5240	155	34	of	of	ADP
ejpam-5240	155	35	the	the	DET
ejpam-5240	155	36	domination	domination	NOUN
ejpam-5240	155	37	sets	set	NOUN
ejpam-5240	155	38	of	of	ADP
ejpam-5240	155	39	v	v	NOUN
ejpam-5240	155	40	(	(	PUNCT
ejpam-5240	155	41	g	g	NOUN
ejpam-5240	155	42	)	)	PUNCT
ejpam-5240	155	43	.	.	PUNCT
ejpam-5240	156	1	by	by	ADP
ejpam-5240	156	2	this	this	PRON
ejpam-5240	156	3	,	,	PUNCT
ejpam-5240	156	4	the	the	DET
ejpam-5240	156	5	isolate	isolate	ADJ
ejpam-5240	156	6	domination	domination	NOUN
ejpam-5240	156	7	number	number	NOUN
ejpam-5240	156	8	is	be	AUX
ejpam-5240	156	9	always	always	ADV
ejpam-5240	156	10	equal	equal	ADJ
ejpam-5240	156	11	to	to	ADP
ejpam-5240	156	12	or	or	CCONJ
ejpam-5240	156	13	greater	great	ADJ
ejpam-5240	156	14	than	than	ADP
ejpam-5240	156	15	the	the	DET
ejpam-5240	156	16	domination	domination	NOUN
ejpam-5240	156	17	number	number	NOUN
ejpam-5240	156	18	.	.	PUNCT
ejpam-5240	157	1	also	also	ADV
ejpam-5240	157	2	,	,	PUNCT
ejpam-5240	157	3	note	note	VERB
ejpam-5240	157	4	that	that	SCONJ
ejpam-5240	157	5	if	if	SCONJ
ejpam-5240	157	6	there	there	PRON
ejpam-5240	157	7	exists	exist	VERB
ejpam-5240	157	8	an	an	DET
ejpam-5240	157	9	isolate	isolate	NOUN
ejpam-5240	157	10	dominating	dominating	NOUN
ejpam-5240	157	11	set	set	VERB
ejpam-5240	157	12	in	in	ADP
ejpam-5240	157	13	g	g	PROPN
ejpam-5240	157	14	,	,	PUNCT
ejpam-5240	157	15	say	say	VERB
ejpam-5240	157	16	t	t	PROPN
ejpam-5240	157	17	⊆	⊆	NUM
ejpam-5240	157	18	v	v	NOUN
ejpam-5240	157	19	(	(	PUNCT
ejpam-5240	157	20	g	g	NOUN
ejpam-5240	157	21	)	)	PUNCT
ejpam-5240	157	22	,	,	PUNCT
ejpam-5240	157	23	then	then	ADV
ejpam-5240	157	24	γ(g	γ(g	PROPN
ejpam-5240	157	25	)	)	PUNCT
ejpam-5240	157	26	≤	≤	NOUN
ejpam-5240	158	1	γ0(g	γ0(g	SYM
ejpam-5240	158	2	)	)	PUNCT
ejpam-5240	158	3	≤	≤	NOUN
ejpam-5240	158	4	|t	|t	VERB
ejpam-5240	158	5	|	|	INTJ
ejpam-5240	158	6	.	.	PUNCT
ejpam-5240	159	1	3	3	X
ejpam-5240	159	2	.	.	X
ejpam-5240	160	1	the	the	DET
ejpam-5240	160	2	k	k	ADV
ejpam-5240	160	3	-	-	PUNCT
ejpam-5240	160	4	restricted	restrict	VERB
ejpam-5240	160	5	intersection	intersection	NOUN
ejpam-5240	160	6	graph	graph	NOUN
ejpam-5240	160	7	recall	recall	NOUN
ejpam-5240	160	8	that	that	SCONJ
ejpam-5240	160	9	s(n	s(n	PROPN
ejpam-5240	160	10	,	,	PUNCT
ejpam-5240	160	11	k	k	NOUN
ejpam-5240	160	12	)	)	PUNCT
ejpam-5240	160	13	is	be	AUX
ejpam-5240	160	14	the	the	DET
ejpam-5240	160	15	collection	collection	NOUN
ejpam-5240	160	16	of	of	ADP
ejpam-5240	160	17	all	all	DET
ejpam-5240	160	18	k	k	ADJ
ejpam-5240	160	19	-	-	ADJ
ejpam-5240	160	20	element	element	ADJ
ejpam-5240	160	21	subsets	subset	NOUN
ejpam-5240	160	22	of	of	ADP
ejpam-5240	160	23	sn	sn	PROPN
ejpam-5240	160	24	.	.	PUNCT
ejpam-5240	161	1	from	from	ADP
ejpam-5240	161	2	remark	remark	NOUN
ejpam-5240	161	3	3	3	NUM
ejpam-5240	161	4	,	,	PUNCT
ejpam-5240	161	5	it	it	PRON
ejpam-5240	161	6	was	be	AUX
ejpam-5240	161	7	regarded	regard	VERB
ejpam-5240	161	8	that	that	SCONJ
ejpam-5240	161	9	s0	s0	PROPN
ejpam-5240	161	10	is	be	AUX
ejpam-5240	161	11	a	a	DET
ejpam-5240	161	12	trivial	trivial	ADJ
ejpam-5240	161	13	case	case	NOUN
ejpam-5240	161	14	such	such	ADJ
ejpam-5240	161	15	that	that	SCONJ
ejpam-5240	161	16	s(0,0	s(0,0	NOUN
ejpam-5240	161	17	)	)	PUNCT
ejpam-5240	161	18	=	=	SYM
ejpam-5240	161	19	{	{	PUNCT
ejpam-5240	161	20	∅	∅	NOUN
ejpam-5240	161	21	}	}	PUNCT
ejpam-5240	161	22	.	.	PUNCT
ejpam-5240	162	1	then	then	ADV
ejpam-5240	162	2	,	,	PUNCT
ejpam-5240	162	3	s(0,0	s(0,0	NOUN
ejpam-5240	162	4	)	)	PUNCT
ejpam-5240	162	5	is	be	AUX
ejpam-5240	162	6	considered	consider	VERB
ejpam-5240	162	7	a	a	DET
ejpam-5240	162	8	trivial	trivial	ADJ
ejpam-5240	162	9	case	case	NOUN
ejpam-5240	162	10	for	for	ADP
ejpam-5240	162	11	a	a	DET
ejpam-5240	162	12	k	k	ADV
ejpam-5240	162	13	-	-	ADJ
ejpam-5240	162	14	restricted	restricted	ADJ
ejpam-5240	162	15	intersection	intersection	NOUN
ejpam-5240	162	16	graph	graph	NOUN
ejpam-5240	162	17	.	.	PUNCT
ejpam-5240	163	1	as	as	ADP
ejpam-5240	163	2	a	a	DET
ejpam-5240	163	3	result	result	NOUN
ejpam-5240	163	4	,	,	PUNCT
ejpam-5240	163	5	the	the	DET
ejpam-5240	163	6	0	0	NUM
ejpam-5240	163	7	-	-	PUNCT
ejpam-5240	163	8	restricted	restrict	VERB
ejpam-5240	163	9	intersection	intersection	NOUN
ejpam-5240	163	10	graph	graph	NOUN
ejpam-5240	163	11	of	of	ADP
ejpam-5240	163	12	s0	s0	PROPN
ejpam-5240	163	13	is	be	AUX
ejpam-5240	163	14	a	a	DET
ejpam-5240	163	15	trivial	trivial	ADJ
ejpam-5240	163	16	graph	graph	NOUN
ejpam-5240	163	17	containing	contain	VERB
ejpam-5240	163	18	the	the	DET
ejpam-5240	163	19	vertex	vertex	NOUN
ejpam-5240	163	20	∅.	∅.	PROPN
ejpam-5240	163	21	m.e	m.e	PROPN
ejpam-5240	163	22	.	.	PROPN
ejpam-5240	163	23	pelagio	pelagio	PROPN
ejpam-5240	163	24	,	,	PUNCT
ejpam-5240	163	25	n.	n.	NOUN
ejpam-5240	163	26	mame	mame	PROPN
ejpam-5240	163	27	,	,	PUNCT
ejpam-5240	163	28	k.	k.	PROPN
ejpam-5240	163	29	mendoza	mendoza	PROPN
ejpam-5240	163	30	/	/	SYM
ejpam-5240	163	31	eur	eur	PROPN
ejpam-5240	163	32	.	.	PUNCT
ejpam-5240	164	1	j.	j.	PROPN
ejpam-5240	164	2	pure	pure	PROPN
ejpam-5240	164	3	appl	appl	PROPN
ejpam-5240	164	4	.	.	PROPN
ejpam-5240	164	5	math	math	PROPN
ejpam-5240	164	6	,	,	PUNCT
ejpam-5240	164	7	17	17	NUM
ejpam-5240	164	8	(	(	PUNCT
ejpam-5240	164	9	3	3	NUM
ejpam-5240	164	10	)	)	PUNCT
ejpam-5240	164	11	(	(	PUNCT
ejpam-5240	164	12	2024	2024	NUM
ejpam-5240	164	13	)	)	PUNCT
ejpam-5240	164	14	,	,	PUNCT
ejpam-5240	164	15	1779	1779	NUM
ejpam-5240	164	16	-	-	SYM
ejpam-5240	164	17	1803	1803	NUM
ejpam-5240	164	18	1785	1785	NUM
ejpam-5240	164	19	throughout	throughout	ADP
ejpam-5240	164	20	the	the	DET
ejpam-5240	164	21	succeeding	succeed	VERB
ejpam-5240	164	22	discussion	discussion	NOUN
ejpam-5240	164	23	,	,	PUNCT
ejpam-5240	164	24	we	we	PRON
ejpam-5240	164	25	shall	shall	AUX
ejpam-5240	164	26	consider	consider	VERB
ejpam-5240	164	27	n	n	PRON
ejpam-5240	164	28	as	as	ADP
ejpam-5240	164	29	a	a	DET
ejpam-5240	164	30	positive	positive	ADJ
ejpam-5240	164	31	integer	integer	NOUN
ejpam-5240	164	32	for	for	ADP
ejpam-5240	164	33	sn	sn	PROPN
ejpam-5240	164	34	and	and	CCONJ
ejpam-5240	164	35	a	a	DET
ejpam-5240	164	36	nonnegative	nonnegative	ADJ
ejpam-5240	164	37	integer	integer	NOUN
ejpam-5240	164	38	k	k	PROPN
ejpam-5240	164	39	such	such	ADJ
ejpam-5240	164	40	that	that	SCONJ
ejpam-5240	164	41	k	k	PROPN
ejpam-5240	164	42	≤	≤	PROPN
ejpam-5240	164	43	n.	n.	NOUN
ejpam-5240	164	44	definition	definition	NOUN
ejpam-5240	164	45	7	7	NUM
ejpam-5240	164	46	.	.	PUNCT
ejpam-5240	165	1	let	let	VERB
ejpam-5240	165	2	sn	sn	PROPN
ejpam-5240	165	3	be	be	AUX
ejpam-5240	165	4	an	an	DET
ejpam-5240	165	5	n	n	NOUN
ejpam-5240	165	6	-	-	PUNCT
ejpam-5240	165	7	element	element	NOUN
ejpam-5240	165	8	set	set	NOUN
ejpam-5240	165	9	where	where	SCONJ
ejpam-5240	165	10	n	n	PRON
ejpam-5240	165	11	is	be	AUX
ejpam-5240	165	12	a	a	DET
ejpam-5240	165	13	positive	positive	ADJ
ejpam-5240	165	14	integer	integer	NOUN
ejpam-5240	165	15	,	,	PUNCT
ejpam-5240	165	16	let	let	VERB
ejpam-5240	165	17	k	k	PROPN
ejpam-5240	165	18	≤	≤	PROPN
ejpam-5240	165	19	n	n	CCONJ
ejpam-5240	165	20	be	be	AUX
ejpam-5240	165	21	a	a	DET
ejpam-5240	165	22	nonnegative	nonnegative	ADJ
ejpam-5240	165	23	integer	integer	NOUN
ejpam-5240	165	24	,	,	PUNCT
ejpam-5240	165	25	and	and	CCONJ
ejpam-5240	165	26	let	let	VERB
ejpam-5240	165	27	s(n	s(n	PROPN
ejpam-5240	165	28	,	,	PUNCT
ejpam-5240	165	29	k	k	NOUN
ejpam-5240	165	30	)	)	PUNCT
ejpam-5240	165	31	be	be	VERB
ejpam-5240	165	32	the	the	DET
ejpam-5240	165	33	collection	collection	NOUN
ejpam-5240	165	34	of	of	ADP
ejpam-5240	165	35	all	all	DET
ejpam-5240	165	36	k	k	ADJ
ejpam-5240	165	37	-	-	ADJ
ejpam-5240	165	38	element	element	ADJ
ejpam-5240	165	39	subsets	subset	NOUN
ejpam-5240	165	40	of	of	ADP
ejpam-5240	165	41	sn	sn	PROPN
ejpam-5240	165	42	.	.	PUNCT
ejpam-5240	166	1	a	a	DET
ejpam-5240	166	2	k	k	ADV
ejpam-5240	166	3	-	-	PUNCT
ejpam-5240	166	4	restricted	restricted	ADJ
ejpam-5240	166	5	intersection	intersection	NOUN
ejpam-5240	166	6	graph	graph	NOUN
ejpam-5240	166	7	,	,	PUNCT
ejpam-5240	166	8	denoted	denote	VERB
ejpam-5240	166	9	by	by	ADP
ejpam-5240	166	10	gs(n	gs(n	NOUN
ejpam-5240	166	11	,	,	PUNCT
ejpam-5240	166	12	k	k	NOUN
ejpam-5240	166	13	)	)	PUNCT
ejpam-5240	166	14	,	,	PUNCT
ejpam-5240	166	15	is	be	AUX
ejpam-5240	166	16	the	the	DET
ejpam-5240	166	17	graph	graph	NOUN
ejpam-5240	166	18	whose	whose	DET
ejpam-5240	166	19	vertex	vertex	NOUN
ejpam-5240	166	20	set	set	NOUN
ejpam-5240	166	21	is	be	AUX
ejpam-5240	166	22	s(n	s(n	PROPN
ejpam-5240	166	23	,	,	PUNCT
ejpam-5240	166	24	k	k	NOUN
ejpam-5240	166	25	)	)	PUNCT
ejpam-5240	166	26	and	and	CCONJ
ejpam-5240	166	27	two	two	NUM
ejpam-5240	166	28	vertices	vertex	NOUN
ejpam-5240	166	29	a	a	PRON
ejpam-5240	166	30	and	and	CCONJ
ejpam-5240	166	31	b	b	NOUN
ejpam-5240	166	32	are	be	AUX
ejpam-5240	166	33	adjacent	adjacent	ADJ
ejpam-5240	166	34	whenever	whenever	SCONJ
ejpam-5240	166	35	a	a	DET
ejpam-5240	166	36	∩b	∩b	NOUN
ejpam-5240	166	37	̸=	̸=	NOUN
ejpam-5240	166	38	∅	∅	NOUN
ejpam-5240	166	39	and	and	CCONJ
ejpam-5240	166	40	a	a	DET
ejpam-5240	166	41	̸=	̸=	PROPN
ejpam-5240	166	42	b.	b.	NOUN
ejpam-5240	167	1	the	the	DET
ejpam-5240	167	2	elements	element	NOUN
ejpam-5240	167	3	of	of	ADP
ejpam-5240	167	4	v	v	NOUN
ejpam-5240	167	5	(	(	PUNCT
ejpam-5240	167	6	gs(n	gs(n	NOUN
ejpam-5240	167	7	,	,	PUNCT
ejpam-5240	167	8	k	k	NOUN
ejpam-5240	167	9	)	)	PUNCT
ejpam-5240	167	10	)	)	PUNCT
ejpam-5240	167	11	are	be	AUX
ejpam-5240	167	12	the	the	DET
ejpam-5240	167	13	k	k	NOUN
ejpam-5240	167	14	-	-	PUNCT
ejpam-5240	167	15	subsets	subset	NOUN
ejpam-5240	167	16	of	of	ADP
ejpam-5240	167	17	sn	sn	NOUN
ejpam-5240	167	18	so	so	ADV
ejpam-5240	167	19	for	for	ADP
ejpam-5240	167	20	every	every	DET
ejpam-5240	167	21	a	a	DET
ejpam-5240	167	22	∈	∈	PROPN
ejpam-5240	167	23	v	v	NOUN
ejpam-5240	167	24	(	(	PUNCT
ejpam-5240	167	25	gs(n	gs(n	NOUN
ejpam-5240	167	26	,	,	PUNCT
ejpam-5240	167	27	k	k	NOUN
ejpam-5240	167	28	)	)	PUNCT
ejpam-5240	167	29	)	)	PUNCT
ejpam-5240	167	30	,	,	PUNCT
ejpam-5240	167	31	|a|	|a|	PROPN
ejpam-5240	167	32	=	=	PROPN
ejpam-5240	167	33	k.	k.	PROPN
ejpam-5240	168	1	moreover	moreover	ADV
ejpam-5240	168	2	,	,	PUNCT
ejpam-5240	168	3	an	an	DET
ejpam-5240	168	4	unordered	unordered	ADJ
ejpam-5240	168	5	pair	pair	NOUN
ejpam-5240	168	6	of	of	ADP
ejpam-5240	168	7	vertex	vertex	NOUN
ejpam-5240	168	8	[	[	X
ejpam-5240	168	9	a	a	X
ejpam-5240	168	10	,	,	PUNCT
ejpam-5240	168	11	b	b	NOUN
ejpam-5240	168	12	]	]	X
ejpam-5240	168	13	∈	∈	PROPN
ejpam-5240	168	14	e(gs(n	e(gs(n	PROPN
ejpam-5240	168	15	,	,	PUNCT
ejpam-5240	168	16	k	k	NOUN
ejpam-5240	168	17	)	)	PUNCT
ejpam-5240	168	18	)	)	PUNCT
ejpam-5240	169	1	if	if	SCONJ
ejpam-5240	169	2	a	a	PRON
ejpam-5240	169	3	and	and	CCONJ
ejpam-5240	169	4	b	b	NOUN
ejpam-5240	169	5	have	have	VERB
ejpam-5240	169	6	a	a	DET
ejpam-5240	169	7	nonempty	nonempty	ADJ
ejpam-5240	169	8	intersection	intersection	NOUN
ejpam-5240	169	9	and	and	CCONJ
ejpam-5240	169	10	a	a	DET
ejpam-5240	169	11	andb	andb	NOUN
ejpam-5240	169	12	are	be	AUX
ejpam-5240	169	13	distinct	distinct	ADJ
ejpam-5240	169	14	.	.	PUNCT
ejpam-5240	170	1	although	although	SCONJ
ejpam-5240	170	2	for	for	ADP
ejpam-5240	170	3	every	every	DET
ejpam-5240	170	4	nonempty	nonempty	NOUN
ejpam-5240	170	5	subset	subset	VERB
ejpam-5240	170	6	a	a	DET
ejpam-5240	170	7	∈	∈	PROPN
ejpam-5240	170	8	v	v	NOUN
ejpam-5240	170	9	(	(	PUNCT
ejpam-5240	170	10	gs(n	gs(n	NOUN
ejpam-5240	170	11	,	,	PUNCT
ejpam-5240	170	12	k	k	NOUN
ejpam-5240	170	13	)	)	PUNCT
ejpam-5240	170	14	)	)	PUNCT
ejpam-5240	170	15	,	,	PUNCT
ejpam-5240	170	16	a	a	DET
ejpam-5240	170	17	∩	∩	ADJ
ejpam-5240	170	18	a	a	DET
ejpam-5240	170	19	=	=	SYM
ejpam-5240	170	20	a	a	NOUN
ejpam-5240	170	21	,	,	PUNCT
ejpam-5240	170	22	it	it	PRON
ejpam-5240	170	23	can	can	AUX
ejpam-5240	170	24	be	be	AUX
ejpam-5240	170	25	observed	observe	VERB
ejpam-5240	170	26	that	that	SCONJ
ejpam-5240	170	27	every	every	DET
ejpam-5240	170	28	edge	edge	NOUN
ejpam-5240	170	29	in	in	ADP
ejpam-5240	170	30	gs(n	gs(n	NOUN
ejpam-5240	170	31	,	,	PUNCT
ejpam-5240	170	32	k	k	NOUN
ejpam-5240	170	33	)	)	PUNCT
ejpam-5240	170	34	should	should	AUX
ejpam-5240	170	35	contain	contain	VERB
ejpam-5240	170	36	distinct	distinct	ADJ
ejpam-5240	170	37	vertices	vertex	NOUN
ejpam-5240	170	38	so	so	SCONJ
ejpam-5240	170	39	gs(n	gs(n	ADJ
ejpam-5240	170	40	,	,	PUNCT
ejpam-5240	170	41	k	k	NOUN
ejpam-5240	170	42	)	)	PUNCT
ejpam-5240	170	43	does	do	AUX
ejpam-5240	170	44	not	not	PART
ejpam-5240	170	45	contain	contain	VERB
ejpam-5240	170	46	any	any	DET
ejpam-5240	170	47	loops	loop	NOUN
ejpam-5240	170	48	.	.	PUNCT
ejpam-5240	171	1	now	now	ADV
ejpam-5240	171	2	,	,	PUNCT
ejpam-5240	171	3	given	give	VERB
ejpam-5240	171	4	in	in	ADP
ejpam-5240	171	5	example	example	NOUN
ejpam-5240	171	6	2	2	NUM
ejpam-5240	171	7	is	be	AUX
ejpam-5240	171	8	an	an	DET
ejpam-5240	171	9	example	example	NOUN
ejpam-5240	171	10	for	for	ADP
ejpam-5240	171	11	gs(n	gs(n	NOUN
ejpam-5240	171	12	,	,	PUNCT
ejpam-5240	171	13	k	k	NOUN
ejpam-5240	171	14	)	)	PUNCT
ejpam-5240	171	15	of	of	ADP
ejpam-5240	171	16	a	a	DET
ejpam-5240	171	17	4	4	NUM
ejpam-5240	171	18	-	-	PUNCT
ejpam-5240	171	19	element	element	NOUN
ejpam-5240	171	20	set	set	VERB
ejpam-5240	171	21	s4	s4	PROPN
ejpam-5240	171	22	where	where	SCONJ
ejpam-5240	171	23	k	k	PROPN
ejpam-5240	171	24	=	=	SYM
ejpam-5240	171	25	2	2	X
ejpam-5240	171	26	.	.	NOUN
ejpam-5240	171	27	example	example	NOUN
ejpam-5240	171	28	2	2	NUM
ejpam-5240	171	29	.	.	PUNCT
ejpam-5240	171	30	let	let	VERB
ejpam-5240	171	31	s4	s4	PROPN
ejpam-5240	171	32	=	=	SYM
ejpam-5240	171	33	{	{	PUNCT
ejpam-5240	171	34	x1	x1	PROPN
ejpam-5240	171	35	,	,	PUNCT
ejpam-5240	171	36	x2	x2	PROPN
ejpam-5240	171	37	,	,	PUNCT
ejpam-5240	171	38	x3	x3	ADJ
ejpam-5240	171	39	,	,	PUNCT
ejpam-5240	171	40	x4	x4	PROPN
ejpam-5240	171	41	}	}	PUNCT
ejpam-5240	171	42	and	and	CCONJ
ejpam-5240	171	43	let	let	VERB
ejpam-5240	171	44	k	k	NOUN
ejpam-5240	171	45	=	=	SYM
ejpam-5240	171	46	2	2	X
ejpam-5240	171	47	.	.	PUNCT
ejpam-5240	171	48	then	then	ADV
ejpam-5240	171	49	gs(4,2	gs(4,2	NUM
ejpam-5240	171	50	)	)	PUNCT
ejpam-5240	171	51	has	have	VERB
ejpam-5240	171	52	the	the	DET
ejpam-5240	171	53	vertex	vertex	NOUN
ejpam-5240	171	54	set	set	VERB
ejpam-5240	171	55	v	v	NOUN
ejpam-5240	171	56	(	(	PUNCT
ejpam-5240	171	57	gs(4,2	gs(4,2	NOUN
ejpam-5240	171	58	)	)	PUNCT
ejpam-5240	171	59	)	)	PUNCT
ejpam-5240	172	1	=	=	PRON
ejpam-5240	172	2	{	{	PUNCT
ejpam-5240	172	3	{	{	PUNCT
ejpam-5240	172	4	x1	x1	PROPN
ejpam-5240	172	5	,	,	PUNCT
ejpam-5240	172	6	x2	x2	PROPN
ejpam-5240	172	7	}	}	PUNCT
ejpam-5240	172	8	,	,	PUNCT
ejpam-5240	172	9	{	{	PUNCT
ejpam-5240	172	10	x1	x1	ADJ
ejpam-5240	172	11	,	,	PUNCT
ejpam-5240	172	12	x3	x3	ADJ
ejpam-5240	172	13	}	}	PUNCT
ejpam-5240	172	14	,	,	PUNCT
ejpam-5240	172	15	{	{	PUNCT
ejpam-5240	172	16	x1	x1	PROPN
ejpam-5240	172	17	,	,	PUNCT
ejpam-5240	172	18	x4	x4	PROPN
ejpam-5240	172	19	}	}	PUNCT
ejpam-5240	172	20	,	,	PUNCT
ejpam-5240	172	21	{	{	PUNCT
ejpam-5240	172	22	x2	x2	ADJ
ejpam-5240	172	23	,	,	PUNCT
ejpam-5240	172	24	x3	x3	ADJ
ejpam-5240	172	25	}	}	PUNCT
ejpam-5240	172	26	,	,	PUNCT
ejpam-5240	172	27	{	{	PUNCT
ejpam-5240	172	28	x2	x2	PROPN
ejpam-5240	172	29	,	,	PUNCT
ejpam-5240	172	30	x4	x4	PROPN
ejpam-5240	172	31	}	}	PUNCT
ejpam-5240	172	32	,	,	PUNCT
ejpam-5240	172	33	{	{	PUNCT
ejpam-5240	172	34	x3	x3	ADJ
ejpam-5240	172	35	,	,	PUNCT
ejpam-5240	172	36	x4	x4	PROPN
ejpam-5240	172	37	}	}	PUNCT
ejpam-5240	172	38	}	}	PUNCT
ejpam-5240	172	39	.	.	PUNCT
ejpam-5240	173	1	now	now	ADV
ejpam-5240	173	2	,	,	PUNCT
ejpam-5240	173	3	observe	observe	VERB
ejpam-5240	173	4	that	that	SCONJ
ejpam-5240	173	5	{	{	PUNCT
ejpam-5240	173	6	x2	x2	PROPN
ejpam-5240	173	7	,	,	PUNCT
ejpam-5240	173	8	x3}∩{x2	x3}∩{x2	NUM
ejpam-5240	173	9	,	,	PUNCT
ejpam-5240	173	10	x4	x4	PROPN
ejpam-5240	173	11	}	}	PUNCT
ejpam-5240	173	12	=	=	SYM
ejpam-5240	173	13	{	{	PUNCT
ejpam-5240	173	14	x2	x2	PROPN
ejpam-5240	173	15	}	}	PUNCT
ejpam-5240	173	16	,	,	PUNCT
ejpam-5240	173	17	so	so	CCONJ
ejpam-5240	173	18	[	[	X
ejpam-5240	173	19	{	{	PUNCT
ejpam-5240	173	20	x2	x2	ADJ
ejpam-5240	173	21	,	,	PUNCT
ejpam-5240	173	22	x3	x3	ADJ
ejpam-5240	173	23	}	}	PUNCT
ejpam-5240	173	24	,	,	PUNCT
ejpam-5240	173	25	{	{	PUNCT
ejpam-5240	173	26	x2	x2	PROPN
ejpam-5240	173	27	,	,	PUNCT
ejpam-5240	173	28	x4	x4	PROPN
ejpam-5240	173	29	}	}	PUNCT
ejpam-5240	173	30	]	]	PUNCT
ejpam-5240	173	31	∈	∈	NOUN
ejpam-5240	173	32	e(gs(4,2	e(gs(4,2	NOUN
ejpam-5240	173	33	)	)	PUNCT
ejpam-5240	173	34	)	)	PUNCT
ejpam-5240	173	35	.	.	PUNCT
ejpam-5240	174	1	also	also	ADV
ejpam-5240	174	2	,	,	PUNCT
ejpam-5240	174	3	{	{	PUNCT
ejpam-5240	174	4	x1	x1	ADJ
ejpam-5240	174	5	,	,	PUNCT
ejpam-5240	174	6	x2	x2	ADJ
ejpam-5240	174	7	}	}	PUNCT
ejpam-5240	174	8	∩	∩	NOUN
ejpam-5240	174	9	{	{	PUNCT
ejpam-5240	174	10	x1	x1	PROPN
ejpam-5240	174	11	,	,	PUNCT
ejpam-5240	174	12	x3	x3	ADJ
ejpam-5240	174	13	}	}	PUNCT
ejpam-5240	174	14	=	=	SYM
ejpam-5240	174	15	{	{	PUNCT
ejpam-5240	174	16	x1	x1	PROPN
ejpam-5240	174	17	}	}	PUNCT
ejpam-5240	174	18	which	which	PRON
ejpam-5240	174	19	implies	imply	VERB
ejpam-5240	174	20	that	that	SCONJ
ejpam-5240	174	21	[	[	X
ejpam-5240	174	22	{	{	PUNCT
ejpam-5240	174	23	x1	x1	PROPN
ejpam-5240	174	24	,	,	PUNCT
ejpam-5240	174	25	x2	x2	PROPN
ejpam-5240	174	26	}	}	PUNCT
ejpam-5240	174	27	,	,	PUNCT
ejpam-5240	174	28	{	{	PUNCT
ejpam-5240	174	29	x1	x1	ADJ
ejpam-5240	174	30	,	,	PUNCT
ejpam-5240	174	31	x3	x3	ADJ
ejpam-5240	174	32	}	}	PUNCT
ejpam-5240	174	33	]	]	PUNCT
ejpam-5240	174	34	is	be	AUX
ejpam-5240	174	35	also	also	ADV
ejpam-5240	174	36	an	an	DET
ejpam-5240	174	37	edge	edge	NOUN
ejpam-5240	174	38	of	of	ADP
ejpam-5240	174	39	gs(4,2	gs(4,2	NOUN
ejpam-5240	174	40	)	)	PUNCT
ejpam-5240	174	41	.	.	PUNCT
ejpam-5240	175	1	however	however	ADV
ejpam-5240	175	2	,	,	PUNCT
ejpam-5240	175	3	the	the	DET
ejpam-5240	175	4	vertices	vertex	NOUN
ejpam-5240	175	5	{	{	PUNCT
ejpam-5240	175	6	x1	x1	PROPN
ejpam-5240	175	7	,	,	PUNCT
ejpam-5240	175	8	x2	x2	PROPN
ejpam-5240	175	9	}	}	PUNCT
ejpam-5240	175	10	and	and	CCONJ
ejpam-5240	175	11	{	{	PUNCT
ejpam-5240	175	12	x3	x3	ADJ
ejpam-5240	175	13	,	,	PUNCT
ejpam-5240	175	14	x4	x4	PROPN
ejpam-5240	175	15	}	}	PUNCT
ejpam-5240	175	16	bear	bear	VERB
ejpam-5240	175	17	an	an	DET
ejpam-5240	175	18	empty	empty	ADJ
ejpam-5240	175	19	intersection	intersection	NOUN
ejpam-5240	175	20	so	so	SCONJ
ejpam-5240	175	21	they	they	PRON
ejpam-5240	175	22	are	be	AUX
ejpam-5240	175	23	not	not	PART
ejpam-5240	175	24	adjacent	adjacent	ADJ
ejpam-5240	175	25	to	to	ADP
ejpam-5240	175	26	each	each	DET
ejpam-5240	175	27	other	other	ADJ
ejpam-5240	175	28	.	.	PUNCT
ejpam-5240	176	1	to	to	PART
ejpam-5240	176	2	enumerate	enumerate	VERB
ejpam-5240	176	3	the	the	DET
ejpam-5240	176	4	edges	edge	NOUN
ejpam-5240	176	5	of	of	ADP
ejpam-5240	176	6	gs(4,2	gs(4,2	NOUN
ejpam-5240	176	7	)	)	PUNCT
ejpam-5240	176	8	,	,	PUNCT
ejpam-5240	176	9	we	we	PRON
ejpam-5240	176	10	have	have	VERB
ejpam-5240	176	11	:	:	PUNCT
ejpam-5240	176	12	e(gs(4,2	e(gs(4,2	NOUN
ejpam-5240	176	13	)	)	PUNCT
ejpam-5240	176	14	)	)	PUNCT
ejpam-5240	177	1	=	=	PRON
ejpam-5240	177	2	{	{	PUNCT
ejpam-5240	178	1	[	[	X
ejpam-5240	178	2	{	{	PUNCT
ejpam-5240	178	3	x1	x1	PROPN
ejpam-5240	178	4	,	,	PUNCT
ejpam-5240	178	5	x2	x2	PROPN
ejpam-5240	178	6	}	}	PUNCT
ejpam-5240	178	7	,	,	PUNCT
ejpam-5240	178	8	{	{	PUNCT
ejpam-5240	178	9	x1	x1	ADJ
ejpam-5240	178	10	,	,	PUNCT
ejpam-5240	178	11	x3	x3	ADJ
ejpam-5240	178	12	}	}	PUNCT
ejpam-5240	178	13	]	]	PUNCT
ejpam-5240	178	14	,	,	PUNCT
ejpam-5240	178	15	[	[	X
ejpam-5240	178	16	{	{	PUNCT
ejpam-5240	178	17	x1	x1	PROPN
ejpam-5240	178	18	,	,	PUNCT
ejpam-5240	178	19	x2	x2	PROPN
ejpam-5240	178	20	}	}	PUNCT
ejpam-5240	178	21	,	,	PUNCT
ejpam-5240	178	22	{	{	PUNCT
ejpam-5240	179	1	x1	x1	PROPN
ejpam-5240	179	2	,	,	PUNCT
ejpam-5240	179	3	x4	x4	PROPN
ejpam-5240	179	4	}	}	PUNCT
ejpam-5240	179	5	]	]	PUNCT
ejpam-5240	179	6	,	,	PUNCT
ejpam-5240	179	7	[	[	X
ejpam-5240	179	8	{	{	PUNCT
ejpam-5240	179	9	x1	x1	PROPN
ejpam-5240	179	10	,	,	PUNCT
ejpam-5240	179	11	x2	x2	PROPN
ejpam-5240	179	12	}	}	PUNCT
ejpam-5240	179	13	,	,	PUNCT
ejpam-5240	179	14	{	{	PUNCT
ejpam-5240	179	15	x2	x2	ADJ
ejpam-5240	179	16	,	,	PUNCT
ejpam-5240	179	17	x3	x3	ADJ
ejpam-5240	179	18	}	}	PUNCT
ejpam-5240	179	19	]	]	PUNCT
ejpam-5240	179	20	,	,	PUNCT
ejpam-5240	179	21	[	[	X
ejpam-5240	179	22	{	{	PUNCT
ejpam-5240	179	23	x1	x1	PROPN
ejpam-5240	179	24	,	,	PUNCT
ejpam-5240	179	25	x2	x2	PROPN
ejpam-5240	179	26	}	}	PUNCT
ejpam-5240	179	27	,	,	PUNCT
ejpam-5240	179	28	{	{	PUNCT
ejpam-5240	179	29	x2	x2	PROPN
ejpam-5240	179	30	,	,	PUNCT
ejpam-5240	179	31	x4	x4	PROPN
ejpam-5240	179	32	}	}	PUNCT
ejpam-5240	179	33	]	]	PUNCT
ejpam-5240	179	34	,	,	PUNCT
ejpam-5240	179	35	[	[	X
ejpam-5240	179	36	{	{	PUNCT
ejpam-5240	179	37	x1	x1	ADJ
ejpam-5240	179	38	,	,	PUNCT
ejpam-5240	179	39	x3	x3	ADJ
ejpam-5240	179	40	}	}	PUNCT
ejpam-5240	179	41	,	,	PUNCT
ejpam-5240	179	42	{	{	PUNCT
ejpam-5240	179	43	x1	x1	PROPN
ejpam-5240	179	44	,	,	PUNCT
ejpam-5240	179	45	x4	x4	PROPN
ejpam-5240	179	46	}	}	PUNCT
ejpam-5240	179	47	]	]	PUNCT
ejpam-5240	179	48	,	,	PUNCT
ejpam-5240	179	49	[	[	X
ejpam-5240	179	50	{	{	PUNCT
ejpam-5240	179	51	x1	x1	ADJ
ejpam-5240	179	52	,	,	PUNCT
ejpam-5240	179	53	x3	x3	ADJ
ejpam-5240	179	54	}	}	PUNCT
ejpam-5240	179	55	,	,	PUNCT
ejpam-5240	179	56	{	{	PUNCT
ejpam-5240	179	57	x2	x2	ADJ
ejpam-5240	179	58	,	,	PUNCT
ejpam-5240	179	59	x3	x3	ADJ
ejpam-5240	179	60	}	}	PUNCT
ejpam-5240	179	61	]	]	PUNCT
ejpam-5240	179	62	,	,	PUNCT
ejpam-5240	179	63	[	[	X
ejpam-5240	179	64	{	{	PUNCT
ejpam-5240	179	65	x1	x1	ADJ
ejpam-5240	179	66	,	,	PUNCT
ejpam-5240	179	67	x3	x3	ADJ
ejpam-5240	179	68	}	}	PUNCT
ejpam-5240	179	69	,	,	PUNCT
ejpam-5240	179	70	{	{	PUNCT
ejpam-5240	179	71	x3	x3	ADJ
ejpam-5240	179	72	,	,	PUNCT
ejpam-5240	179	73	x4	x4	PROPN
ejpam-5240	179	74	}	}	PUNCT
ejpam-5240	179	75	]	]	PUNCT
ejpam-5240	179	76	,	,	PUNCT
ejpam-5240	179	77	[	[	X
ejpam-5240	179	78	{	{	PUNCT
ejpam-5240	179	79	x1	x1	PROPN
ejpam-5240	179	80	,	,	PUNCT
ejpam-5240	179	81	x4	x4	PROPN
ejpam-5240	179	82	}	}	PUNCT
ejpam-5240	179	83	,	,	PUNCT
ejpam-5240	179	84	{	{	PUNCT
ejpam-5240	179	85	x2	x2	PROPN
ejpam-5240	179	86	,	,	PUNCT
ejpam-5240	179	87	x4	x4	PROPN
ejpam-5240	179	88	}	}	PUNCT
ejpam-5240	179	89	]	]	PUNCT
ejpam-5240	179	90	,	,	PUNCT
ejpam-5240	179	91	[	[	X
ejpam-5240	179	92	{	{	PUNCT
ejpam-5240	179	93	x1	x1	PROPN
ejpam-5240	179	94	,	,	PUNCT
ejpam-5240	179	95	x4	x4	PROPN
ejpam-5240	179	96	}	}	PUNCT
ejpam-5240	179	97	,	,	PUNCT
ejpam-5240	179	98	{	{	PUNCT
ejpam-5240	179	99	x3	x3	ADJ
ejpam-5240	179	100	,	,	PUNCT
ejpam-5240	179	101	x4	x4	PROPN
ejpam-5240	179	102	}	}	PUNCT
ejpam-5240	179	103	]	]	PUNCT
ejpam-5240	179	104	,	,	PUNCT
ejpam-5240	179	105	[	[	X
ejpam-5240	179	106	{	{	PUNCT
ejpam-5240	179	107	x2	x2	ADJ
ejpam-5240	179	108	,	,	PUNCT
ejpam-5240	179	109	x3	x3	ADJ
ejpam-5240	179	110	}	}	PUNCT
ejpam-5240	179	111	,	,	PUNCT
ejpam-5240	179	112	{	{	PUNCT
ejpam-5240	179	113	x2	x2	PROPN
ejpam-5240	179	114	,	,	PUNCT
ejpam-5240	179	115	x4	x4	PROPN
ejpam-5240	179	116	}	}	PUNCT
ejpam-5240	179	117	]	]	PUNCT
ejpam-5240	179	118	,	,	PUNCT
ejpam-5240	179	119	[	[	X
ejpam-5240	179	120	{	{	PUNCT
ejpam-5240	179	121	x2	x2	ADJ
ejpam-5240	179	122	,	,	PUNCT
ejpam-5240	179	123	x3	x3	ADJ
ejpam-5240	179	124	}	}	PUNCT
ejpam-5240	179	125	,	,	PUNCT
ejpam-5240	179	126	{	{	PUNCT
ejpam-5240	179	127	x3	x3	ADJ
ejpam-5240	179	128	,	,	PUNCT
ejpam-5240	179	129	x4	x4	PROPN
ejpam-5240	179	130	}	}	PUNCT
ejpam-5240	179	131	]	]	PUNCT
ejpam-5240	179	132	,	,	PUNCT
ejpam-5240	179	133	[	[	X
ejpam-5240	179	134	{	{	PUNCT
ejpam-5240	179	135	x2	x2	PROPN
ejpam-5240	179	136	,	,	PUNCT
ejpam-5240	179	137	x4	x4	PROPN
ejpam-5240	179	138	}	}	PUNCT
ejpam-5240	179	139	,	,	PUNCT
ejpam-5240	179	140	{	{	PUNCT
ejpam-5240	179	141	x3	x3	ADJ
ejpam-5240	179	142	,	,	PUNCT
ejpam-5240	179	143	x4	x4	PROPN
ejpam-5240	179	144	}	}	PUNCT
ejpam-5240	179	145	]	]	PUNCT
ejpam-5240	179	146	}	}	PUNCT
ejpam-5240	179	147	.	.	PUNCT
ejpam-5240	180	1	it	it	PRON
ejpam-5240	180	2	can	can	AUX
ejpam-5240	180	3	be	be	AUX
ejpam-5240	180	4	observed	observe	VERB
ejpam-5240	180	5	that	that	SCONJ
ejpam-5240	180	6	the	the	DET
ejpam-5240	180	7	order	order	NOUN
ejpam-5240	180	8	of	of	ADP
ejpam-5240	180	9	gs(4,2	gs(4,2	NOUN
ejpam-5240	180	10	)	)	PUNCT
ejpam-5240	180	11	is	be	AUX
ejpam-5240	180	12	6	6	NUM
ejpam-5240	180	13	and	and	CCONJ
ejpam-5240	180	14	its	its	PRON
ejpam-5240	180	15	size	size	NOUN
ejpam-5240	180	16	is	be	AUX
ejpam-5240	180	17	12	12	NUM
ejpam-5240	180	18	.	.	PUNCT
ejpam-5240	181	1	the	the	DET
ejpam-5240	181	2	graph	graph	NOUN
ejpam-5240	181	3	illustrated	illustrate	VERB
ejpam-5240	181	4	in	in	ADP
ejpam-5240	181	5	figure	figure	NOUN
ejpam-5240	181	6	2	2	NUM
ejpam-5240	181	7	is	be	AUX
ejpam-5240	181	8	a	a	DET
ejpam-5240	181	9	pictorial	pictorial	ADJ
ejpam-5240	181	10	representation	representation	NOUN
ejpam-5240	181	11	of	of	ADP
ejpam-5240	181	12	gs(4,2	gs(4,2	NOUN
ejpam-5240	181	13	)	)	PUNCT
ejpam-5240	181	14	.	.	PUNCT
ejpam-5240	182	1	since	since	SCONJ
ejpam-5240	182	2	[	[	X
ejpam-5240	182	3	a	a	X
ejpam-5240	182	4	,	,	PUNCT
ejpam-5240	182	5	a	a	PRON
ejpam-5240	182	6	]	]	X
ejpam-5240	182	7	/∈	/∈	PUNCT
ejpam-5240	182	8	e(gs(n	e(gs(n	NOUN
ejpam-5240	182	9	,	,	PUNCT
ejpam-5240	182	10	k	k	NOUN
ejpam-5240	182	11	)	)	PUNCT
ejpam-5240	182	12	)	)	PUNCT
ejpam-5240	182	13	,	,	PUNCT
ejpam-5240	182	14	it	it	PRON
ejpam-5240	182	15	follows	follow	VERB
ejpam-5240	182	16	that	that	SCONJ
ejpam-5240	182	17	a	a	DET
ejpam-5240	182	18	k	k	ADV
ejpam-5240	182	19	-	-	PUNCT
ejpam-5240	182	20	restricted	restricted	ADJ
ejpam-5240	182	21	intersection	intersection	NOUN
ejpam-5240	182	22	graph	graph	NOUN
ejpam-5240	182	23	does	do	AUX
ejpam-5240	182	24	not	not	PART
ejpam-5240	182	25	contain	contain	VERB
ejpam-5240	182	26	any	any	DET
ejpam-5240	182	27	loop	loop	NOUN
ejpam-5240	182	28	.	.	PUNCT
ejpam-5240	183	1	also	also	ADV
ejpam-5240	183	2	,	,	PUNCT
ejpam-5240	183	3	note	note	VERB
ejpam-5240	183	4	that	that	SCONJ
ejpam-5240	183	5	v	v	NOUN
ejpam-5240	183	6	(	(	PUNCT
ejpam-5240	183	7	gs(n	gs(n	NOUN
ejpam-5240	183	8	,	,	PUNCT
ejpam-5240	183	9	k	k	NOUN
ejpam-5240	183	10	)	)	PUNCT
ejpam-5240	183	11	)	)	PUNCT
ejpam-5240	183	12	is	be	AUX
ejpam-5240	183	13	the	the	DET
ejpam-5240	183	14	set	set	NOUN
ejpam-5240	183	15	containing	contain	VERB
ejpam-5240	183	16	the	the	DET
ejpam-5240	183	17	distinct	distinct	ADJ
ejpam-5240	183	18	k	k	NOUN
ejpam-5240	183	19	-	-	PUNCT
ejpam-5240	183	20	subsets	subset	NOUN
ejpam-5240	183	21	of	of	ADP
ejpam-5240	183	22	sn	sn	PROPN
ejpam-5240	183	23	,	,	PUNCT
ejpam-5240	183	24	this	this	PRON
ejpam-5240	183	25	means	mean	VERB
ejpam-5240	183	26	that	that	SCONJ
ejpam-5240	183	27	e(gs(n	e(gs(n	NOUN
ejpam-5240	183	28	,	,	PUNCT
ejpam-5240	183	29	k	k	NOUN
ejpam-5240	183	30	)	)	PUNCT
ejpam-5240	183	31	)	)	PUNCT
ejpam-5240	183	32	is	be	AUX
ejpam-5240	183	33	not	not	PART
ejpam-5240	183	34	a	a	DET
ejpam-5240	183	35	multiset	multiset	NOUN
ejpam-5240	183	36	.	.	PUNCT
ejpam-5240	184	1	hence	hence	ADV
ejpam-5240	184	2	,	,	PUNCT
ejpam-5240	184	3	gs(n	gs(n	NOUN
ejpam-5240	184	4	,	,	PUNCT
ejpam-5240	184	5	k	k	NOUN
ejpam-5240	184	6	)	)	PUNCT
ejpam-5240	184	7	has	have	VERB
ejpam-5240	184	8	no	no	DET
ejpam-5240	184	9	multiple	multiple	ADJ
ejpam-5240	184	10	edges	edge	NOUN
ejpam-5240	184	11	.	.	PUNCT
ejpam-5240	185	1	remark	remark	NOUN
ejpam-5240	185	2	6	6	NUM
ejpam-5240	185	3	.	.	PUNCT
ejpam-5240	186	1	a	a	DET
ejpam-5240	186	2	k	k	ADV
ejpam-5240	186	3	-	-	PUNCT
ejpam-5240	186	4	restricted	restricted	ADJ
ejpam-5240	186	5	intersection	intersection	NOUN
ejpam-5240	186	6	graph	graph	NOUN
ejpam-5240	186	7	gs(n	gs(n	NOUN
ejpam-5240	186	8	,	,	PUNCT
ejpam-5240	186	9	k	k	NOUN
ejpam-5240	186	10	)	)	PUNCT
ejpam-5240	186	11	is	be	AUX
ejpam-5240	186	12	a	a	DET
ejpam-5240	186	13	simple	simple	ADJ
ejpam-5240	186	14	graph	graph	NOUN
ejpam-5240	186	15	.	.	PUNCT
ejpam-5240	187	1	theorem	theorem	ADJ
ejpam-5240	187	2	2	2	NUM
ejpam-5240	187	3	determines	determine	VERB
ejpam-5240	187	4	the	the	DET
ejpam-5240	187	5	order	order	NOUN
ejpam-5240	187	6	of	of	ADP
ejpam-5240	187	7	a	a	DET
ejpam-5240	187	8	gs(n	gs(n	NOUN
ejpam-5240	187	9	,	,	PUNCT
ejpam-5240	187	10	k	k	NOUN
ejpam-5240	187	11	)	)	PUNCT
ejpam-5240	187	12	.	.	PUNCT
ejpam-5240	188	1	recall	recall	VERB
ejpam-5240	188	2	that	that	SCONJ
ejpam-5240	188	3	the	the	DET
ejpam-5240	188	4	order	order	NOUN
ejpam-5240	188	5	of	of	ADP
ejpam-5240	188	6	a	a	DET
ejpam-5240	188	7	graph	graph	NOUN
ejpam-5240	188	8	refers	refer	VERB
ejpam-5240	188	9	to	to	ADP
ejpam-5240	188	10	the	the	DET
ejpam-5240	188	11	cardinality	cardinality	NOUN
ejpam-5240	188	12	of	of	ADP
ejpam-5240	188	13	its	its	PRON
ejpam-5240	188	14	vertex	vertex	NOUN
ejpam-5240	188	15	set	set	NOUN
ejpam-5240	188	16	.	.	PUNCT
ejpam-5240	189	1	theorem	theorem	NOUN
ejpam-5240	189	2	2	2	NUM
ejpam-5240	189	3	.	.	PUNCT
ejpam-5240	189	4	let	let	VERB
ejpam-5240	190	1	gs(n	gs(n	NOUN
ejpam-5240	190	2	,	,	PUNCT
ejpam-5240	190	3	k	k	NOUN
ejpam-5240	190	4	)	)	PUNCT
ejpam-5240	190	5	be	be	VERB
ejpam-5240	190	6	a	a	DET
ejpam-5240	190	7	k	k	ADV
ejpam-5240	190	8	-	-	ADJ
ejpam-5240	190	9	restricted	restricted	ADJ
ejpam-5240	190	10	intersection	intersection	NOUN
ejpam-5240	190	11	graph	graph	NOUN
ejpam-5240	190	12	.	.	PUNCT
ejpam-5240	191	1	then	then	ADV
ejpam-5240	191	2	the	the	DET
ejpam-5240	191	3	order	order	NOUN
ejpam-5240	191	4	of	of	ADP
ejpam-5240	191	5	gs(n	gs(n	NOUN
ejpam-5240	191	6	,	,	PUNCT
ejpam-5240	191	7	k	k	NOUN
ejpam-5240	191	8	)	)	PUNCT
ejpam-5240	191	9	is	be	AUX
ejpam-5240	191	10	(	(	PUNCT
ejpam-5240	191	11	n	n	X
ejpam-5240	191	12	k	k	PROPN
ejpam-5240	191	13	)	)	PUNCT
ejpam-5240	191	14	.	.	PUNCT
ejpam-5240	192	1	proof	proof	NOUN
ejpam-5240	192	2	.	.	PUNCT
ejpam-5240	193	1	assume	assume	VERB
ejpam-5240	193	2	gs(n	gs(n	NOUN
ejpam-5240	193	3	,	,	PUNCT
ejpam-5240	193	4	k	k	NOUN
ejpam-5240	193	5	)	)	PUNCT
ejpam-5240	193	6	is	be	AUX
ejpam-5240	193	7	a	a	DET
ejpam-5240	193	8	k	k	ADV
ejpam-5240	193	9	-	-	ADJ
ejpam-5240	193	10	restricted	restricted	ADJ
ejpam-5240	193	11	intersection	intersection	NOUN
ejpam-5240	193	12	graph	graph	NOUN
ejpam-5240	193	13	.	.	PUNCT
ejpam-5240	194	1	by	by	ADP
ejpam-5240	194	2	definition	definition	NOUN
ejpam-5240	194	3	7	7	NUM
ejpam-5240	194	4	,	,	PUNCT
ejpam-5240	194	5	v	v	NOUN
ejpam-5240	194	6	(	(	PUNCT
ejpam-5240	194	7	gs(n	gs(n	NOUN
ejpam-5240	194	8	,	,	PUNCT
ejpam-5240	194	9	k	k	NOUN
ejpam-5240	194	10	)	)	PUNCT
ejpam-5240	194	11	)	)	PUNCT
ejpam-5240	194	12	is	be	AUX
ejpam-5240	194	13	equal	equal	ADJ
ejpam-5240	194	14	to	to	ADP
ejpam-5240	194	15	s(n	s(n	PROPN
ejpam-5240	194	16	,	,	PUNCT
ejpam-5240	194	17	k	k	NOUN
ejpam-5240	194	18	)	)	PUNCT
ejpam-5240	194	19	.	.	PUNCT
ejpam-5240	195	1	since	since	SCONJ
ejpam-5240	195	2	|s(n	|s(n	PROPN
ejpam-5240	195	3	,	,	PUNCT
ejpam-5240	195	4	k)|	k)|	NOUN
ejpam-5240	195	5	=	=	PUNCT
ejpam-5240	195	6	(	(	PUNCT
ejpam-5240	195	7	n	n	X
ejpam-5240	195	8	k	k	PROPN
ejpam-5240	195	9	)	)	PUNCT
ejpam-5240	195	10	,	,	PUNCT
ejpam-5240	195	11	it	it	PRON
ejpam-5240	195	12	follows	follow	VERB
ejpam-5240	195	13	that	that	SCONJ
ejpam-5240	195	14	|v	|v	PROPN
ejpam-5240	195	15	(	(	PUNCT
ejpam-5240	195	16	gs(n	gs(n	NOUN
ejpam-5240	195	17	,	,	PUNCT
ejpam-5240	195	18	k	k	NOUN
ejpam-5240	195	19	)	)	PUNCT
ejpam-5240	195	20	)	)	PUNCT
ejpam-5240	196	1	|	|	ADV
ejpam-5240	196	2	=	=	SYM
ejpam-5240	196	3	(	(	PUNCT
ejpam-5240	196	4	n	n	X
ejpam-5240	196	5	k	k	PROPN
ejpam-5240	196	6	)	)	PUNCT
ejpam-5240	196	7	.	.	PUNCT
ejpam-5240	197	1	the	the	DET
ejpam-5240	197	2	number	number	NOUN
ejpam-5240	197	3	(	(	PUNCT
ejpam-5240	197	4	n	n	NOUN
ejpam-5240	197	5	k	k	NOUN
ejpam-5240	197	6	)	)	PUNCT
ejpam-5240	197	7	provided	provide	VERB
ejpam-5240	197	8	that	that	SCONJ
ejpam-5240	197	9	n	n	PROPN
ejpam-5240	197	10	a	a	DET
ejpam-5240	197	11	positive	positive	ADJ
ejpam-5240	197	12	integer	integer	NOUN
ejpam-5240	197	13	and	and	CCONJ
ejpam-5240	197	14	k	k	PROPN
ejpam-5240	197	15	is	be	AUX
ejpam-5240	197	16	nonnegative	nonnegative	ADJ
ejpam-5240	197	17	,	,	PUNCT
ejpam-5240	197	18	has	have	VERB
ejpam-5240	197	19	a	a	DET
ejpam-5240	197	20	positive	positive	ADJ
ejpam-5240	197	21	integer	integer	NOUN
ejpam-5240	197	22	value	value	NOUN
ejpam-5240	197	23	.	.	PUNCT
ejpam-5240	198	1	meaning	mean	VERB
ejpam-5240	198	2	to	to	PART
ejpam-5240	198	3	say	say	VERB
ejpam-5240	198	4	,	,	PUNCT
ejpam-5240	198	5	v	v	NOUN
ejpam-5240	198	6	(	(	PUNCT
ejpam-5240	198	7	gs(n	gs(n	NOUN
ejpam-5240	198	8	,	,	PUNCT
ejpam-5240	198	9	k	k	NOUN
ejpam-5240	198	10	)	)	PUNCT
ejpam-5240	198	11	)	)	PUNCT
ejpam-5240	199	1	is	be	AUX
ejpam-5240	199	2	nonempty	nonempty	ADJ
ejpam-5240	199	3	for	for	ADP
ejpam-5240	199	4	all	all	DET
ejpam-5240	199	5	n	n	NOUN
ejpam-5240	199	6	and	and	CCONJ
ejpam-5240	199	7	k.	k.	PROPN
ejpam-5240	199	8	therefore	therefore	ADV
ejpam-5240	199	9	,	,	PUNCT
ejpam-5240	199	10	gs(n	gs(n	NOUN
ejpam-5240	199	11	,	,	PUNCT
ejpam-5240	199	12	k	k	NOUN
ejpam-5240	199	13	)	)	PUNCT
ejpam-5240	199	14	is	be	AUX
ejpam-5240	199	15	defined	define	VERB
ejpam-5240	199	16	for	for	ADP
ejpam-5240	199	17	all	all	DET
ejpam-5240	199	18	the	the	DET
ejpam-5240	199	19	given	give	VERB
ejpam-5240	199	20	values	value	NOUN
ejpam-5240	199	21	of	of	ADP
ejpam-5240	199	22	n	n	PRON
ejpam-5240	199	23	and	and	CCONJ
ejpam-5240	199	24	k.	k.	PROPN
ejpam-5240	199	25	m.e	m.e	PROPN
ejpam-5240	199	26	.	.	PROPN
ejpam-5240	199	27	pelagio	pelagio	PROPN
ejpam-5240	199	28	,	,	PUNCT
ejpam-5240	199	29	n.	n.	NOUN
ejpam-5240	199	30	mame	mame	PROPN
ejpam-5240	199	31	,	,	PUNCT
ejpam-5240	199	32	k.	k.	PROPN
ejpam-5240	199	33	mendoza	mendoza	PROPN
ejpam-5240	199	34	/	/	SYM
ejpam-5240	199	35	eur	eur	PROPN
ejpam-5240	199	36	.	.	PUNCT
ejpam-5240	200	1	j.	j.	PROPN
ejpam-5240	200	2	pure	pure	PROPN
ejpam-5240	200	3	appl	appl	PROPN
ejpam-5240	200	4	.	.	PROPN
ejpam-5240	200	5	math	math	PROPN
ejpam-5240	200	6	,	,	PUNCT
ejpam-5240	200	7	17	17	NUM
ejpam-5240	200	8	(	(	PUNCT
ejpam-5240	200	9	3	3	NUM
ejpam-5240	200	10	)	)	PUNCT
ejpam-5240	200	11	(	(	PUNCT
ejpam-5240	200	12	2024	2024	NUM
ejpam-5240	200	13	)	)	PUNCT
ejpam-5240	200	14	,	,	PUNCT
ejpam-5240	200	15	1779	1779	NUM
ejpam-5240	200	16	-	-	SYM
ejpam-5240	200	17	1803	1803	NUM
ejpam-5240	200	18	1786	1786	NUM
ejpam-5240	200	19	{	{	PUNCT
ejpam-5240	200	20	x1	x1	PROPN
ejpam-5240	200	21	,	,	PUNCT
ejpam-5240	200	22	x2	x2	PROPN
ejpam-5240	200	23	}	}	PUNCT
ejpam-5240	200	24	{	{	PUNCT
ejpam-5240	200	25	x3	x3	ADJ
ejpam-5240	200	26	,	,	PUNCT
ejpam-5240	200	27	x4	x4	PROPN
ejpam-5240	200	28	}	}	PUNCT
ejpam-5240	200	29	{	{	PUNCT
ejpam-5240	200	30	x1	x1	PROPN
ejpam-5240	200	31	,	,	PUNCT
ejpam-5240	200	32	x3	x3	ADJ
ejpam-5240	200	33	}	}	PUNCT
ejpam-5240	200	34	{	{	PUNCT
ejpam-5240	200	35	x2	x2	PROPN
ejpam-5240	200	36	,	,	PUNCT
ejpam-5240	200	37	x4	x4	PROPN
ejpam-5240	200	38	}	}	PUNCT
ejpam-5240	200	39	{	{	PUNCT
ejpam-5240	200	40	x1	x1	PROPN
ejpam-5240	200	41	,	,	PUNCT
ejpam-5240	200	42	x4	x4	PROPN
ejpam-5240	200	43	}	}	PUNCT
ejpam-5240	200	44	{	{	PUNCT
ejpam-5240	200	45	x2	x2	PROPN
ejpam-5240	200	46	,	,	PUNCT
ejpam-5240	200	47	x3	x3	ADJ
ejpam-5240	200	48	}	}	PUNCT
ejpam-5240	200	49	figure	figure	NOUN
ejpam-5240	200	50	2	2	NUM
ejpam-5240	200	51	:	:	PUNCT
ejpam-5240	200	52	a	a	DET
ejpam-5240	200	53	2	2	NUM
ejpam-5240	200	54	-	-	PUNCT
ejpam-5240	200	55	restricted	restrict	VERB
ejpam-5240	200	56	intersection	intersection	NOUN
ejpam-5240	200	57	graph	graph	NOUN
ejpam-5240	200	58	gs(4,2	gs(4,2	NOUN
ejpam-5240	200	59	)	)	PUNCT
ejpam-5240	200	60	.	.	PUNCT
ejpam-5240	201	1	illustration	illustration	NOUN
ejpam-5240	201	2	1	1	NUM
ejpam-5240	201	3	.	.	PUNCT
ejpam-5240	202	1	let	let	AUX
ejpam-5240	202	2	gs(4,2	gs(4,2	NOUN
ejpam-5240	202	3	)	)	PUNCT
ejpam-5240	202	4	be	be	AUX
ejpam-5240	202	5	a	a	DET
ejpam-5240	202	6	2	2	NUM
ejpam-5240	202	7	-	-	PUNCT
ejpam-5240	202	8	restricted	restrict	VERB
ejpam-5240	202	9	intersection	intersection	NOUN
ejpam-5240	202	10	graph	graph	NOUN
ejpam-5240	202	11	over	over	ADP
ejpam-5240	202	12	the	the	DET
ejpam-5240	202	13	4	4	NUM
ejpam-5240	202	14	-	-	PUNCT
ejpam-5240	202	15	element	element	NOUN
ejpam-5240	202	16	set	set	VERB
ejpam-5240	202	17	s4	s4	PROPN
ejpam-5240	202	18	=	=	SYM
ejpam-5240	202	19	{	{	PUNCT
ejpam-5240	202	20	x1	x1	PROPN
ejpam-5240	202	21	,	,	PUNCT
ejpam-5240	202	22	x2	x2	PROPN
ejpam-5240	202	23	,	,	PUNCT
ejpam-5240	202	24	x3	x3	ADJ
ejpam-5240	202	25	,	,	PUNCT
ejpam-5240	202	26	x4	x4	PROPN
ejpam-5240	202	27	}	}	PUNCT
ejpam-5240	202	28	whose	whose	DET
ejpam-5240	202	29	graph	graph	NOUN
ejpam-5240	202	30	is	be	AUX
ejpam-5240	202	31	pictorially	pictorially	ADV
ejpam-5240	202	32	represented	represent	VERB
ejpam-5240	202	33	in	in	ADP
ejpam-5240	202	34	figure	figure	NOUN
ejpam-5240	202	35	2	2	NUM
ejpam-5240	202	36	.	.	PUNCT
ejpam-5240	203	1	we	we	PRON
ejpam-5240	203	2	see	see	VERB
ejpam-5240	203	3	that	that	SCONJ
ejpam-5240	203	4	|v	|v	PROPN
ejpam-5240	203	5	(	(	PUNCT
ejpam-5240	203	6	gs(4,2	gs(4,2	NOUN
ejpam-5240	203	7	)	)	PUNCT
ejpam-5240	203	8	)	)	PUNCT
ejpam-5240	204	1	|	|	ADV
ejpam-5240	204	2	=	=	SYM
ejpam-5240	204	3	6	6	NUM
ejpam-5240	204	4	.	.	PUNCT
ejpam-5240	204	5	to	to	PART
ejpam-5240	204	6	verify	verify	VERB
ejpam-5240	204	7	this	this	PRON
ejpam-5240	204	8	by	by	ADP
ejpam-5240	204	9	utilizing	utilize	VERB
ejpam-5240	204	10	theorem	theorem	NOUN
ejpam-5240	204	11	2	2	NUM
ejpam-5240	204	12	,	,	PUNCT
ejpam-5240	204	13	since	since	SCONJ
ejpam-5240	204	14	n	n	NOUN
ejpam-5240	204	15	=	=	SYM
ejpam-5240	204	16	4	4	NUM
ejpam-5240	204	17	and	and	CCONJ
ejpam-5240	204	18	k	k	NOUN
ejpam-5240	204	19	=	=	SYM
ejpam-5240	204	20	2	2	NUM
ejpam-5240	204	21	,	,	PUNCT
ejpam-5240	204	22	it	it	PRON
ejpam-5240	204	23	follows	follow	VERB
ejpam-5240	204	24	that	that	SCONJ
ejpam-5240	204	25	|v	|v	PROPN
ejpam-5240	204	26	(	(	PUNCT
ejpam-5240	204	27	gs(4,2	gs(4,2	NOUN
ejpam-5240	204	28	)	)	PUNCT
ejpam-5240	204	29	)	)	PUNCT
ejpam-5240	205	1	|	|	ADV
ejpam-5240	205	2	=	=	SYM
ejpam-5240	205	3	(	(	PUNCT
ejpam-5240	205	4	4	4	NUM
ejpam-5240	205	5	2	2	NUM
ejpam-5240	205	6	)	)	PUNCT
ejpam-5240	205	7	=	=	SYM
ejpam-5240	206	1	6	6	X
ejpam-5240	206	2	.	.	X
ejpam-5240	207	1	there	there	PRON
ejpam-5240	207	2	are	be	VERB
ejpam-5240	207	3	instances	instance	NOUN
ejpam-5240	207	4	that	that	SCONJ
ejpam-5240	207	5	a	a	DET
ejpam-5240	207	6	gs(n	gs(n	NOUN
ejpam-5240	207	7	,	,	PUNCT
ejpam-5240	207	8	k	k	NOUN
ejpam-5240	207	9	)	)	PUNCT
ejpam-5240	207	10	is	be	AUX
ejpam-5240	207	11	a	a	DET
ejpam-5240	207	12	trivial	trivial	ADJ
ejpam-5240	207	13	graph	graph	NOUN
ejpam-5240	207	14	.	.	PUNCT
ejpam-5240	208	1	meaning	mean	VERB
ejpam-5240	208	2	to	to	PART
ejpam-5240	208	3	say	say	VERB
ejpam-5240	208	4	,	,	PUNCT
ejpam-5240	208	5	gs(n	gs(n	NOUN
ejpam-5240	208	6	,	,	PUNCT
ejpam-5240	208	7	k	k	NOUN
ejpam-5240	208	8	)	)	PUNCT
ejpam-5240	208	9	has	have	VERB
ejpam-5240	208	10	only	only	ADV
ejpam-5240	208	11	one	one	NUM
ejpam-5240	208	12	vertex	vertex	NOUN
ejpam-5240	208	13	.	.	PUNCT
ejpam-5240	209	1	note	note	VERB
ejpam-5240	209	2	that	that	SCONJ
ejpam-5240	209	3	from	from	ADP
ejpam-5240	209	4	theorem	theorem	NOUN
ejpam-5240	209	5	1	1	NUM
ejpam-5240	209	6	,	,	PUNCT
ejpam-5240	209	7	(	(	PUNCT
ejpam-5240	209	8	n	n	X
ejpam-5240	209	9	k	k	NOUN
ejpam-5240	209	10	)	)	PUNCT
ejpam-5240	209	11	=	=	SYM
ejpam-5240	209	12	|v	|v	PROPN
ejpam-5240	209	13	(	(	PUNCT
ejpam-5240	209	14	gs(n	gs(n	NOUN
ejpam-5240	209	15	,	,	PUNCT
ejpam-5240	209	16	k	k	NOUN
ejpam-5240	209	17	)	)	PUNCT
ejpam-5240	209	18	)	)	PUNCT
ejpam-5240	210	1	|	|	ADV
ejpam-5240	210	2	=	=	SYM
ejpam-5240	210	3	1	1	NUM
ejpam-5240	210	4	if	if	SCONJ
ejpam-5240	210	5	and	and	CCONJ
ejpam-5240	210	6	only	only	ADV
ejpam-5240	210	7	if	if	SCONJ
ejpam-5240	210	8	k	k	PROPN
ejpam-5240	210	9	=	=	SYM
ejpam-5240	210	10	0	0	PROPN
ejpam-5240	210	11	or	or	CCONJ
ejpam-5240	210	12	k	k	PROPN
ejpam-5240	210	13	=	=	SYM
ejpam-5240	210	14	n.	n.	PROPN
ejpam-5240	210	15	theorem	theorem	VERB
ejpam-5240	210	16	3	3	NUM
ejpam-5240	210	17	discusses	discusse	NOUN
ejpam-5240	210	18	gs(n	gs(n	NOUN
ejpam-5240	210	19	,	,	PUNCT
ejpam-5240	210	20	k	k	NOUN
ejpam-5240	210	21	)	)	PUNCT
ejpam-5240	210	22	when	when	SCONJ
ejpam-5240	210	23	k	k	PROPN
ejpam-5240	210	24	is	be	AUX
ejpam-5240	210	25	0	0	NUM
ejpam-5240	210	26	or	or	CCONJ
ejpam-5240	210	27	n.	n.	NOUN
ejpam-5240	210	28	theorem	theorem	NOUN
ejpam-5240	210	29	3	3	X
ejpam-5240	210	30	.	.	PUNCT
ejpam-5240	211	1	let	let	VERB
ejpam-5240	211	2	sn	sn	PROPN
ejpam-5240	211	3	be	be	AUX
ejpam-5240	211	4	an	an	DET
ejpam-5240	211	5	n	n	CCONJ
ejpam-5240	211	6	-	-	PUNCT
ejpam-5240	211	7	element	element	NOUN
ejpam-5240	211	8	set	set	NOUN
ejpam-5240	211	9	.	.	PUNCT
ejpam-5240	212	1	then	then	ADV
ejpam-5240	212	2	gs(n	gs(n	NOUN
ejpam-5240	212	3	,	,	PUNCT
ejpam-5240	212	4	k	k	NOUN
ejpam-5240	212	5	)	)	PUNCT
ejpam-5240	212	6	is	be	AUX
ejpam-5240	212	7	a	a	DET
ejpam-5240	212	8	trivial	trivial	ADJ
ejpam-5240	212	9	graph	graph	NOUN
ejpam-5240	212	10	if	if	SCONJ
ejpam-5240	213	1	and	and	CCONJ
ejpam-5240	213	2	only	only	ADV
ejpam-5240	213	3	if	if	SCONJ
ejpam-5240	213	4	k	k	PROPN
ejpam-5240	213	5	=	=	SYM
ejpam-5240	213	6	0	0	PROPN
ejpam-5240	213	7	or	or	CCONJ
ejpam-5240	213	8	k	k	NOUN
ejpam-5240	213	9	=	=	PUNCT
ejpam-5240	213	10	n.	n.	NOUN
ejpam-5240	213	11	proof	proof	NOUN
ejpam-5240	213	12	.	.	PUNCT
ejpam-5240	214	1	let	let	VERB
ejpam-5240	214	2	gs(n	gs(n	NOUN
ejpam-5240	214	3	,	,	PUNCT
ejpam-5240	214	4	k	k	NOUN
ejpam-5240	214	5	)	)	PUNCT
ejpam-5240	214	6	be	be	AUX
ejpam-5240	214	7	a	a	DET
ejpam-5240	214	8	trivial	trivial	ADJ
ejpam-5240	214	9	graph	graph	NOUN
ejpam-5240	214	10	.	.	PUNCT
ejpam-5240	215	1	hence	hence	ADV
ejpam-5240	215	2	,	,	PUNCT
ejpam-5240	215	3	we	we	PRON
ejpam-5240	215	4	have	have	VERB
ejpam-5240	215	5	|v	|v	X
ejpam-5240	215	6	(	(	PUNCT
ejpam-5240	215	7	gs(n	gs(n	NOUN
ejpam-5240	215	8	,	,	PUNCT
ejpam-5240	215	9	k	k	NOUN
ejpam-5240	215	10	)	)	PUNCT
ejpam-5240	215	11	)	)	PUNCT
ejpam-5240	216	1	|	|	ADV
ejpam-5240	216	2	=	=	SYM
ejpam-5240	216	3	1	1	X
ejpam-5240	216	4	.	.	PUNCT
ejpam-5240	217	1	since	since	SCONJ
ejpam-5240	217	2	the	the	DET
ejpam-5240	217	3	order	order	NOUN
ejpam-5240	217	4	of	of	ADP
ejpam-5240	217	5	gs(n	gs(n	NOUN
ejpam-5240	217	6	,	,	PUNCT
ejpam-5240	217	7	k	k	NOUN
ejpam-5240	217	8	)	)	PUNCT
ejpam-5240	217	9	is	be	AUX
ejpam-5240	217	10	equal	equal	ADJ
ejpam-5240	217	11	to	to	ADP
ejpam-5240	217	12	(	(	PUNCT
ejpam-5240	217	13	n	n	X
ejpam-5240	217	14	k	k	PROPN
ejpam-5240	217	15	)	)	PUNCT
ejpam-5240	217	16	,	,	PUNCT
ejpam-5240	217	17	hence	hence	ADV
ejpam-5240	217	18	we	we	PRON
ejpam-5240	217	19	have	have	VERB
ejpam-5240	217	20	(	(	PUNCT
ejpam-5240	217	21	n	n	X
ejpam-5240	217	22	k	k	NOUN
ejpam-5240	217	23	)	)	PUNCT
ejpam-5240	218	1	=	=	PUNCT
ejpam-5240	218	2	1	1	X
ejpam-5240	218	3	.	.	PUNCT
ejpam-5240	218	4	by	by	ADP
ejpam-5240	218	5	theorem	theorem	NOUN
ejpam-5240	218	6	1	1	NUM
ejpam-5240	218	7	,	,	PUNCT
ejpam-5240	218	8	k	k	X
ejpam-5240	218	9	is	be	AUX
ejpam-5240	218	10	either	either	CCONJ
ejpam-5240	218	11	0	0	NUM
ejpam-5240	218	12	or	or	CCONJ
ejpam-5240	218	13	n.	n.	NOUN
ejpam-5240	218	14	conversely	conversely	ADV
ejpam-5240	218	15	,	,	PUNCT
ejpam-5240	218	16	assume	assume	VERB
ejpam-5240	218	17	that	that	SCONJ
ejpam-5240	218	18	k	k	PROPN
ejpam-5240	218	19	=	=	PUNCT
ejpam-5240	218	20	0	0	PROPN
ejpam-5240	218	21	or	or	CCONJ
ejpam-5240	218	22	k	k	NOUN
ejpam-5240	218	23	=	=	SYM
ejpam-5240	218	24	n.	n.	NOUN
ejpam-5240	218	25	if	if	SCONJ
ejpam-5240	218	26	k	k	PROPN
ejpam-5240	218	27	=	=	SYM
ejpam-5240	218	28	0	0	PROPN
ejpam-5240	218	29	,	,	PUNCT
ejpam-5240	218	30	then	then	ADV
ejpam-5240	218	31	s(n,0	s(n,0	PROPN
ejpam-5240	218	32	)	)	PUNCT
ejpam-5240	219	1	=	=	PRON
ejpam-5240	219	2	{	{	PUNCT
ejpam-5240	219	3	∅	∅	NOUN
ejpam-5240	219	4	}	}	PUNCT
ejpam-5240	219	5	which	which	PRON
ejpam-5240	219	6	implies	imply	VERB
ejpam-5240	219	7	that	that	PRON
ejpam-5240	219	8	v	v	NOUN
ejpam-5240	219	9	(	(	PUNCT
ejpam-5240	219	10	gs(n,0	gs(n,0	PROPN
ejpam-5240	219	11	)	)	PUNCT
ejpam-5240	219	12	)	)	PUNCT
ejpam-5240	219	13	=	=	PUNCT
ejpam-5240	219	14	{	{	PUNCT
ejpam-5240	219	15	∅	∅	NOUN
ejpam-5240	219	16	}	}	PUNCT
ejpam-5240	219	17	.	.	PUNCT
ejpam-5240	220	1	an	an	DET
ejpam-5240	220	2	empty	empty	ADJ
ejpam-5240	220	3	subset	subset	NOUN
ejpam-5240	220	4	has	have	VERB
ejpam-5240	220	5	no	no	DET
ejpam-5240	220	6	element	element	NOUN
ejpam-5240	220	7	to	to	PART
ejpam-5240	220	8	consider	consider	VERB
ejpam-5240	220	9	so	so	SCONJ
ejpam-5240	220	10	it	it	PRON
ejpam-5240	220	11	does	do	AUX
ejpam-5240	220	12	not	not	PART
ejpam-5240	220	13	intersect	intersect	VERB
ejpam-5240	220	14	to	to	ADP
ejpam-5240	220	15	itself	itself	PRON
ejpam-5240	220	16	.	.	PUNCT
ejpam-5240	221	1	it	it	PRON
ejpam-5240	221	2	follows	follow	VERB
ejpam-5240	221	3	that	that	DET
ejpam-5240	221	4	e(gs(n,0	e(gs(n,0	NOUN
ejpam-5240	221	5	)	)	PUNCT
ejpam-5240	221	6	)	)	PUNCT
ejpam-5240	221	7	is	be	AUX
ejpam-5240	221	8	empty	empty	ADJ
ejpam-5240	221	9	.	.	PUNCT
ejpam-5240	222	1	thus	thus	ADV
ejpam-5240	222	2	,	,	PUNCT
ejpam-5240	222	3	gs(n,0	gs(n,0	PROPN
ejpam-5240	222	4	)	)	PUNCT
ejpam-5240	222	5	is	be	AUX
ejpam-5240	222	6	a	a	DET
ejpam-5240	222	7	trivial	trivial	ADJ
ejpam-5240	222	8	graph	graph	NOUN
ejpam-5240	222	9	.	.	PUNCT
ejpam-5240	223	1	moreover	moreover	ADV
ejpam-5240	223	2	,	,	PUNCT
ejpam-5240	223	3	if	if	SCONJ
ejpam-5240	223	4	k	k	PROPN
ejpam-5240	223	5	=	=	PUNCT
ejpam-5240	223	6	n	n	CCONJ
ejpam-5240	223	7	,	,	PUNCT
ejpam-5240	223	8	then	then	ADV
ejpam-5240	223	9	s(n	s(n	PROPN
ejpam-5240	223	10	,	,	PUNCT
ejpam-5240	223	11	n	n	CCONJ
ejpam-5240	223	12	)	)	PUNCT
ejpam-5240	223	13	=	=	SYM
ejpam-5240	223	14	{	{	PUNCT
ejpam-5240	223	15	sn	sn	NOUN
ejpam-5240	223	16	}	}	PUNCT
ejpam-5240	223	17	meaning	meaning	NOUN
ejpam-5240	223	18	,	,	PUNCT
ejpam-5240	223	19	v	v	NOUN
ejpam-5240	223	20	(	(	PUNCT
ejpam-5240	223	21	gs(n	gs(n	NOUN
ejpam-5240	223	22	,	,	PUNCT
ejpam-5240	223	23	n	n	CCONJ
ejpam-5240	223	24	)	)	PUNCT
ejpam-5240	223	25	)	)	PUNCT
ejpam-5240	224	1	=	=	PRON
ejpam-5240	224	2	{	{	PUNCT
ejpam-5240	224	3	sn	sn	X
ejpam-5240	224	4	}	}	PUNCT
ejpam-5240	224	5	.	.	PUNCT
ejpam-5240	225	1	this	this	PRON
ejpam-5240	225	2	implies	imply	VERB
ejpam-5240	225	3	that	that	SCONJ
ejpam-5240	225	4	gs(n	gs(n	NOUN
ejpam-5240	225	5	,	,	PUNCT
ejpam-5240	225	6	n	n	CCONJ
ejpam-5240	225	7	)	)	PUNCT
ejpam-5240	225	8	has	have	VERB
ejpam-5240	225	9	only	only	ADV
ejpam-5240	225	10	one	one	NUM
ejpam-5240	225	11	vertex	vertex	NOUN
ejpam-5240	225	12	.	.	PUNCT
ejpam-5240	226	1	by	by	ADP
ejpam-5240	226	2	remark	remark	NOUN
ejpam-5240	226	3	6	6	NUM
ejpam-5240	226	4	,	,	PUNCT
ejpam-5240	226	5	[	[	X
ejpam-5240	226	6	sn	sn	X
ejpam-5240	226	7	,	,	PUNCT
ejpam-5240	226	8	sn	sn	PROPN
ejpam-5240	226	9	]	]	PUNCT
ejpam-5240	226	10	/∈	/∈	PUNCT
ejpam-5240	226	11	e(gs(n	e(gs(n	NOUN
ejpam-5240	226	12	,	,	PUNCT
ejpam-5240	226	13	n	n	CCONJ
ejpam-5240	226	14	)	)	PUNCT
ejpam-5240	226	15	)	)	PUNCT
ejpam-5240	227	1	so	so	ADV
ejpam-5240	227	2	e(gs(n	e(gs(n	PROPN
ejpam-5240	227	3	,	,	PUNCT
ejpam-5240	227	4	n	n	CCONJ
ejpam-5240	227	5	)	)	PUNCT
ejpam-5240	227	6	)	)	PUNCT
ejpam-5240	228	1	=	=	PUNCT
ejpam-5240	228	2	∅.	∅.	VERB
ejpam-5240	228	3	therefore	therefore	ADV
ejpam-5240	228	4	,	,	PUNCT
ejpam-5240	228	5	gs(n	gs(n	NOUN
ejpam-5240	228	6	,	,	PUNCT
ejpam-5240	228	7	n	n	CCONJ
ejpam-5240	228	8	)	)	PUNCT
ejpam-5240	228	9	is	be	AUX
ejpam-5240	228	10	also	also	ADV
ejpam-5240	228	11	a	a	DET
ejpam-5240	228	12	trivial	trivial	ADJ
ejpam-5240	228	13	graph	graph	NOUN
ejpam-5240	228	14	.	.	PUNCT
ejpam-5240	229	1	illustration	illustration	NOUN
ejpam-5240	229	2	2	2	NUM
ejpam-5240	229	3	.	.	PUNCT
ejpam-5240	230	1	let	let	VERB
ejpam-5240	230	2	s4	s4	PROPN
ejpam-5240	230	3	=	=	SYM
ejpam-5240	230	4	{	{	PUNCT
ejpam-5240	230	5	x1	x1	PROPN
ejpam-5240	230	6	,	,	PUNCT
ejpam-5240	230	7	x2	x2	PROPN
ejpam-5240	230	8	,	,	PUNCT
ejpam-5240	230	9	x3	x3	ADJ
ejpam-5240	230	10	,	,	PUNCT
ejpam-5240	230	11	x4	x4	PROPN
ejpam-5240	230	12	}	}	PUNCT
ejpam-5240	230	13	.	.	PUNCT
ejpam-5240	231	1	if	if	SCONJ
ejpam-5240	231	2	k	k	PROPN
ejpam-5240	231	3	=	=	SYM
ejpam-5240	231	4	0	0	PROPN
ejpam-5240	231	5	,	,	PUNCT
ejpam-5240	231	6	then	then	ADV
ejpam-5240	231	7	v	v	X
ejpam-5240	231	8	(	(	PUNCT
ejpam-5240	231	9	gs(4,0	gs(4,0	PROPN
ejpam-5240	231	10	)	)	PUNCT
ejpam-5240	231	11	)	)	PUNCT
ejpam-5240	232	1	=	=	PUNCT
ejpam-5240	232	2	{	{	PUNCT
ejpam-5240	232	3	∅	∅	NOUN
ejpam-5240	232	4	}	}	PUNCT
ejpam-5240	232	5	.	.	PUNCT
ejpam-5240	233	1	furthermore	furthermore	ADV
ejpam-5240	233	2	,	,	PUNCT
ejpam-5240	233	3	if	if	SCONJ
ejpam-5240	233	4	k	k	PROPN
ejpam-5240	233	5	=	=	SYM
ejpam-5240	233	6	4	4	NUM
ejpam-5240	233	7	,	,	PUNCT
ejpam-5240	233	8	then	then	ADV
ejpam-5240	233	9	v	v	X
ejpam-5240	233	10	(	(	PUNCT
ejpam-5240	233	11	gs(4,4	gs(4,4	PROPN
ejpam-5240	233	12	)	)	PUNCT
ejpam-5240	233	13	)	)	PUNCT
ejpam-5240	234	1	=	=	PRON
ejpam-5240	234	2	{	{	PUNCT
ejpam-5240	234	3	{	{	PUNCT
ejpam-5240	234	4	x1	x1	PROPN
ejpam-5240	234	5	,	,	PUNCT
ejpam-5240	234	6	x2	x2	PROPN
ejpam-5240	234	7	,	,	PUNCT
ejpam-5240	234	8	x3	x3	ADJ
ejpam-5240	234	9	,	,	PUNCT
ejpam-5240	234	10	x4	x4	PROPN
ejpam-5240	234	11	}	}	PUNCT
ejpam-5240	234	12	}	}	PUNCT
ejpam-5240	234	13	.	.	PUNCT
ejpam-5240	235	1	pictorial	pictorial	ADJ
ejpam-5240	235	2	representations	representation	NOUN
ejpam-5240	235	3	of	of	ADP
ejpam-5240	235	4	gs(4,0	gs(4,0	PROPN
ejpam-5240	235	5	)	)	PUNCT
ejpam-5240	235	6	and	and	CCONJ
ejpam-5240	235	7	gs(4,4	gs(4,4	VERB
ejpam-5240	235	8	)	)	PUNCT
ejpam-5240	235	9	are	be	AUX
ejpam-5240	235	10	presented	present	VERB
ejpam-5240	235	11	in	in	ADP
ejpam-5240	235	12	figure	figure	NOUN
ejpam-5240	235	13	3	3	NUM
ejpam-5240	235	14	.	.	PUNCT
ejpam-5240	236	1	for	for	ADP
ejpam-5240	236	2	every	every	DET
ejpam-5240	236	3	gs(n	gs(n	NOUN
ejpam-5240	236	4	,	,	PUNCT
ejpam-5240	236	5	k	k	NOUN
ejpam-5240	236	6	)	)	PUNCT
ejpam-5240	236	7	,	,	PUNCT
ejpam-5240	236	8	if	if	SCONJ
ejpam-5240	236	9	k	k	PROPN
ejpam-5240	236	10	=	=	SYM
ejpam-5240	236	11	1	1	NUM
ejpam-5240	236	12	,	,	PUNCT
ejpam-5240	236	13	then	then	ADV
ejpam-5240	236	14	v	v	X
ejpam-5240	236	15	(	(	PUNCT
ejpam-5240	236	16	gs(n,1	gs(n,1	NOUN
ejpam-5240	236	17	)	)	PUNCT
ejpam-5240	236	18	)	)	PUNCT
ejpam-5240	236	19	contains	contain	VERB
ejpam-5240	236	20	the	the	DET
ejpam-5240	236	21	distinct	distinct	ADJ
ejpam-5240	236	22	1	1	NUM
ejpam-5240	236	23	-	-	PUNCT
ejpam-5240	236	24	element	element	NOUN
ejpam-5240	236	25	subsets	subset	NOUN
ejpam-5240	236	26	of	of	ADP
ejpam-5240	236	27	sn	sn	PROPN
ejpam-5240	236	28	.	.	PROPN
ejpam-5240	236	29	discussed	discuss	VERB
ejpam-5240	236	30	in	in	ADP
ejpam-5240	236	31	theorem	theorem	ADJ
ejpam-5240	236	32	4	4	NUM
ejpam-5240	236	33	is	be	AUX
ejpam-5240	236	34	a	a	DET
ejpam-5240	236	35	gs(n	gs(n	NOUN
ejpam-5240	236	36	,	,	PUNCT
ejpam-5240	236	37	k	k	NOUN
ejpam-5240	236	38	)	)	PUNCT
ejpam-5240	236	39	when	when	SCONJ
ejpam-5240	236	40	k	k	PROPN
ejpam-5240	236	41	=	=	SYM
ejpam-5240	236	42	1	1	X
ejpam-5240	236	43	.	.	PUNCT
ejpam-5240	236	44	theorem	theorem	NOUN
ejpam-5240	236	45	4	4	NUM
ejpam-5240	236	46	.	.	PUNCT
ejpam-5240	237	1	let	let	VERB
ejpam-5240	237	2	sn	sn	PROPN
ejpam-5240	237	3	=	=	PUNCT
ejpam-5240	237	4	{	{	PUNCT
ejpam-5240	237	5	x1	x1	PROPN
ejpam-5240	237	6	,	,	PUNCT
ejpam-5240	237	7	x2	x2	PROPN
ejpam-5240	237	8	,	,	PUNCT
ejpam-5240	237	9	...	...	PUNCT
ejpam-5240	237	10	,	,	PUNCT
ejpam-5240	237	11	xn	xn	PRON
ejpam-5240	237	12	}	}	PUNCT
ejpam-5240	237	13	be	be	VERB
ejpam-5240	237	14	an	an	DET
ejpam-5240	237	15	n	n	CCONJ
ejpam-5240	237	16	-	-	PUNCT
ejpam-5240	237	17	element	element	NOUN
ejpam-5240	237	18	set	set	NOUN
ejpam-5240	237	19	.	.	PUNCT
ejpam-5240	238	1	if	if	SCONJ
ejpam-5240	238	2	k	k	PROPN
ejpam-5240	238	3	=	=	SYM
ejpam-5240	238	4	1	1	NUM
ejpam-5240	238	5	,	,	PUNCT
ejpam-5240	238	6	then	then	ADV
ejpam-5240	238	7	gs(n,1	gs(n,1	NOUN
ejpam-5240	238	8	)	)	PUNCT
ejpam-5240	238	9	is	be	AUX
ejpam-5240	238	10	an	an	DET
ejpam-5240	238	11	empty	empty	ADJ
ejpam-5240	238	12	graph	graph	NOUN
ejpam-5240	238	13	of	of	ADP
ejpam-5240	238	14	order	order	NOUN
ejpam-5240	238	15	n.	n.	PROPN
ejpam-5240	238	16	m.e	m.e	PROPN
ejpam-5240	238	17	.	.	PROPN
ejpam-5240	238	18	pelagio	pelagio	PROPN
ejpam-5240	238	19	,	,	PUNCT
ejpam-5240	238	20	n.	n.	NOUN
ejpam-5240	238	21	mame	mame	PROPN
ejpam-5240	238	22	,	,	PUNCT
ejpam-5240	238	23	k.	k.	PROPN
ejpam-5240	238	24	mendoza	mendoza	PROPN
ejpam-5240	238	25	/	/	SYM
ejpam-5240	238	26	eur	eur	PROPN
ejpam-5240	238	27	.	.	PUNCT
ejpam-5240	239	1	j.	j.	PROPN
ejpam-5240	239	2	pure	pure	PROPN
ejpam-5240	239	3	appl	appl	PROPN
ejpam-5240	239	4	.	.	PROPN
ejpam-5240	239	5	math	math	PROPN
ejpam-5240	239	6	,	,	PUNCT
ejpam-5240	239	7	17	17	NUM
ejpam-5240	239	8	(	(	PUNCT
ejpam-5240	239	9	3	3	NUM
ejpam-5240	239	10	)	)	PUNCT
ejpam-5240	239	11	(	(	PUNCT
ejpam-5240	239	12	2024	2024	NUM
ejpam-5240	239	13	)	)	PUNCT
ejpam-5240	239	14	,	,	PUNCT
ejpam-5240	239	15	1779	1779	NUM
ejpam-5240	239	16	-	-	SYM
ejpam-5240	239	17	1803	1803	NUM
ejpam-5240	239	18	1787	1787	NUM
ejpam-5240	239	19	∅	∅	NOUN
ejpam-5240	239	20	{	{	PUNCT
ejpam-5240	239	21	x1	x1	PROPN
ejpam-5240	239	22	,	,	PUNCT
ejpam-5240	239	23	x2	x2	PROPN
ejpam-5240	239	24	,	,	PUNCT
ejpam-5240	239	25	x3	x3	ADJ
ejpam-5240	239	26	,	,	PUNCT
ejpam-5240	239	27	x4	x4	ADJ
ejpam-5240	239	28	}	}	PUNCT
ejpam-5240	239	29	figure	figure	VERB
ejpam-5240	239	30	3	3	NUM
ejpam-5240	239	31	:	:	PUNCT
ejpam-5240	239	32	illustrating	illustrate	VERB
ejpam-5240	239	33	gs(4,0	gs(4,0	PROPN
ejpam-5240	239	34	)	)	PUNCT
ejpam-5240	239	35	(	(	PUNCT
ejpam-5240	239	36	left	leave	VERB
ejpam-5240	239	37	)	)	PUNCT
ejpam-5240	239	38	and	and	CCONJ
ejpam-5240	239	39	gs(4,4	gs(4,4	VERB
ejpam-5240	239	40	)	)	PUNCT
ejpam-5240	239	41	(	(	PUNCT
ejpam-5240	239	42	right	right	NOUN
ejpam-5240	239	43	)	)	PUNCT
ejpam-5240	239	44	.	.	PUNCT
ejpam-5240	240	1	proof	proof	NOUN
ejpam-5240	240	2	.	.	PUNCT
ejpam-5240	241	1	if	if	SCONJ
ejpam-5240	241	2	k	k	PROPN
ejpam-5240	241	3	=	=	SYM
ejpam-5240	241	4	1	1	NUM
ejpam-5240	241	5	,	,	PUNCT
ejpam-5240	241	6	then	then	ADV
ejpam-5240	241	7	v	v	X
ejpam-5240	241	8	(	(	PUNCT
ejpam-5240	241	9	gs(n,1	gs(n,1	NOUN
ejpam-5240	241	10	)	)	PUNCT
ejpam-5240	241	11	)	)	PUNCT
ejpam-5240	242	1	=	=	PRON
ejpam-5240	242	2	{	{	PUNCT
ejpam-5240	242	3	{	{	PUNCT
ejpam-5240	242	4	x1	x1	PROPN
ejpam-5240	242	5	}	}	PUNCT
ejpam-5240	242	6	,	,	PUNCT
ejpam-5240	242	7	{	{	PUNCT
ejpam-5240	242	8	x2	x2	ADJ
ejpam-5240	242	9	}	}	PUNCT
ejpam-5240	242	10	,	,	PUNCT
ejpam-5240	242	11	...	...	PUNCT
ejpam-5240	242	12	,	,	PUNCT
ejpam-5240	242	13	{	{	PUNCT
ejpam-5240	242	14	xn	xn	X
ejpam-5240	242	15	}	}	PUNCT
ejpam-5240	242	16	}	}	PUNCT
ejpam-5240	242	17	.	.	PUNCT
ejpam-5240	243	1	it	it	PRON
ejpam-5240	243	2	can	can	AUX
ejpam-5240	243	3	be	be	AUX
ejpam-5240	243	4	observed	observe	VERB
ejpam-5240	243	5	that	that	SCONJ
ejpam-5240	243	6	for	for	ADP
ejpam-5240	243	7	all	all	DET
ejpam-5240	243	8	1	1	NUM
ejpam-5240	243	9	≤	≤	NUM
ejpam-5240	243	10	i	i	PRON
ejpam-5240	243	11	,	,	PUNCT
ejpam-5240	243	12	j	j	PROPN
ejpam-5240	243	13	≤	≤	PROPN
ejpam-5240	243	14	n	n	CCONJ
ejpam-5240	243	15	and	and	CCONJ
ejpam-5240	243	16	i	i	PRON
ejpam-5240	243	17	̸=	̸=	PROPN
ejpam-5240	243	18	j	j	PROPN
ejpam-5240	243	19	,	,	PUNCT
ejpam-5240	243	20	{	{	PUNCT
ejpam-5240	243	21	xi	xi	NOUN
ejpam-5240	243	22	}	}	PUNCT
ejpam-5240	243	23	∩	∩	NOUN
ejpam-5240	243	24	{	{	PUNCT
ejpam-5240	243	25	xj	xj	PROPN
ejpam-5240	243	26	}	}	PUNCT
ejpam-5240	243	27	=	=	PUNCT
ejpam-5240	243	28	∅.	∅.	ADP
ejpam-5240	243	29	this	this	PRON
ejpam-5240	243	30	implies	imply	VERB
ejpam-5240	243	31	that	that	SCONJ
ejpam-5240	243	32	every	every	DET
ejpam-5240	243	33	vertex	vertex	NOUN
ejpam-5240	243	34	in	in	ADP
ejpam-5240	243	35	gs(n,1	gs(n,1	NOUN
ejpam-5240	243	36	)	)	PUNCT
ejpam-5240	243	37	is	be	AUX
ejpam-5240	243	38	not	not	PART
ejpam-5240	243	39	adjacent	adjacent	ADJ
ejpam-5240	243	40	to	to	ADP
ejpam-5240	243	41	each	each	DET
ejpam-5240	243	42	other	other	ADJ
ejpam-5240	243	43	.	.	PUNCT
ejpam-5240	244	1	hence	hence	ADV
ejpam-5240	244	2	,	,	PUNCT
ejpam-5240	244	3	gs(n,1	gs(n,1	PROPN
ejpam-5240	244	4	)	)	PUNCT
ejpam-5240	244	5	is	be	AUX
ejpam-5240	244	6	an	an	DET
ejpam-5240	244	7	empty	empty	ADJ
ejpam-5240	244	8	graph	graph	NOUN
ejpam-5240	244	9	.	.	PUNCT
ejpam-5240	245	1	note	note	VERB
ejpam-5240	245	2	that	that	SCONJ
ejpam-5240	245	3	|v	|v	PROPN
ejpam-5240	245	4	(	(	PUNCT
ejpam-5240	245	5	gs(n,1	gs(n,1	NOUN
ejpam-5240	245	6	)	)	PUNCT
ejpam-5240	245	7	)	)	PUNCT
ejpam-5240	246	1	|	|	ADV
ejpam-5240	246	2	=	=	SYM
ejpam-5240	246	3	(	(	PUNCT
ejpam-5240	246	4	n	n	NOUN
ejpam-5240	246	5	1	1	NUM
ejpam-5240	246	6	)	)	PUNCT
ejpam-5240	246	7	=	=	SYM
ejpam-5240	246	8	n	n	CCONJ
ejpam-5240	246	9	,	,	PUNCT
ejpam-5240	246	10	which	which	PRON
ejpam-5240	246	11	implies	imply	VERB
ejpam-5240	246	12	that	that	SCONJ
ejpam-5240	246	13	gs(n,1	gs(n,1	PROPN
ejpam-5240	246	14	)	)	PUNCT
ejpam-5240	246	15	is	be	AUX
ejpam-5240	246	16	an	an	DET
ejpam-5240	246	17	empty	empty	ADJ
ejpam-5240	246	18	graph	graph	NOUN
ejpam-5240	246	19	of	of	ADP
ejpam-5240	246	20	order	order	NOUN
ejpam-5240	246	21	n.	n.	NOUN
ejpam-5240	246	22	note	note	VERB
ejpam-5240	246	23	that	that	SCONJ
ejpam-5240	246	24	all	all	DET
ejpam-5240	246	25	trivial	trivial	ADJ
ejpam-5240	246	26	graphs	graph	NOUN
ejpam-5240	246	27	are	be	AUX
ejpam-5240	246	28	empty	empty	ADJ
ejpam-5240	246	29	graphs	graph	NOUN
ejpam-5240	246	30	while	while	SCONJ
ejpam-5240	246	31	not	not	PART
ejpam-5240	246	32	every	every	DET
ejpam-5240	246	33	empty	empty	ADJ
ejpam-5240	246	34	graphs	graph	NOUN
ejpam-5240	246	35	are	be	AUX
ejpam-5240	246	36	trivial	trivial	ADJ
ejpam-5240	246	37	graphs	graph	NOUN
ejpam-5240	246	38	.	.	PUNCT
ejpam-5240	247	1	hence	hence	ADV
ejpam-5240	247	2	,	,	PUNCT
ejpam-5240	247	3	gs(n	gs(n	NOUN
ejpam-5240	247	4	,	,	PUNCT
ejpam-5240	247	5	n	n	CCONJ
ejpam-5240	247	6	)	)	PUNCT
ejpam-5240	247	7	and	and	CCONJ
ejpam-5240	247	8	gs(n,0	gs(n,0	PROPN
ejpam-5240	247	9	)	)	PUNCT
ejpam-5240	247	10	are	be	AUX
ejpam-5240	247	11	also	also	ADV
ejpam-5240	247	12	empty	empty	ADJ
ejpam-5240	247	13	graphs	graph	NOUN
ejpam-5240	247	14	.	.	PUNCT
ejpam-5240	248	1	illustration	illustration	NOUN
ejpam-5240	248	2	3	3	NUM
ejpam-5240	248	3	.	.	PUNCT
ejpam-5240	249	1	let	let	VERB
ejpam-5240	249	2	s4	s4	PROPN
ejpam-5240	249	3	=	=	SYM
ejpam-5240	249	4	{	{	PUNCT
ejpam-5240	249	5	x1	x1	PROPN
ejpam-5240	249	6	,	,	PUNCT
ejpam-5240	249	7	x2	x2	PROPN
ejpam-5240	249	8	,	,	PUNCT
ejpam-5240	249	9	x3	x3	ADJ
ejpam-5240	249	10	,	,	PUNCT
ejpam-5240	249	11	x4	x4	ADJ
ejpam-5240	249	12	}	}	PUNCT
ejpam-5240	249	13	and	and	CCONJ
ejpam-5240	249	14	consider	consider	VERB
ejpam-5240	249	15	k	k	NOUN
ejpam-5240	249	16	=	=	SYM
ejpam-5240	249	17	1	1	X
ejpam-5240	249	18	.	.	PUNCT
ejpam-5240	250	1	then	then	ADV
ejpam-5240	250	2	,	,	PUNCT
ejpam-5240	250	3	v	v	INTJ
ejpam-5240	250	4	(	(	PUNCT
ejpam-5240	250	5	gs(4,1	gs(4,1	NOUN
ejpam-5240	250	6	)	)	PUNCT
ejpam-5240	250	7	)	)	PUNCT
ejpam-5240	251	1	=	=	PRON
ejpam-5240	251	2	{	{	PUNCT
ejpam-5240	251	3	{	{	PUNCT
ejpam-5240	251	4	x1	x1	PROPN
ejpam-5240	251	5	}	}	PUNCT
ejpam-5240	251	6	,	,	PUNCT
ejpam-5240	251	7	{	{	PUNCT
ejpam-5240	251	8	x2	x2	ADJ
ejpam-5240	251	9	}	}	PUNCT
ejpam-5240	251	10	,	,	PUNCT
ejpam-5240	251	11	{	{	PUNCT
ejpam-5240	251	12	x3	x3	ADJ
ejpam-5240	251	13	}	}	PUNCT
ejpam-5240	251	14	,	,	PUNCT
ejpam-5240	251	15	{	{	PUNCT
ejpam-5240	251	16	x4	x4	ADV
ejpam-5240	251	17	}	}	PUNCT
ejpam-5240	251	18	}	}	PUNCT
ejpam-5240	251	19	.	.	PUNCT
ejpam-5240	252	1	it	it	PRON
ejpam-5240	252	2	can	can	AUX
ejpam-5240	252	3	be	be	AUX
ejpam-5240	252	4	observed	observe	VERB
ejpam-5240	252	5	that	that	SCONJ
ejpam-5240	252	6	the	the	DET
ejpam-5240	252	7	1	1	NUM
ejpam-5240	252	8	-	-	PUNCT
ejpam-5240	252	9	element	element	NOUN
ejpam-5240	252	10	subsets	subset	NOUN
ejpam-5240	252	11	have	have	VERB
ejpam-5240	252	12	no	no	DET
ejpam-5240	252	13	element	element	NOUN
ejpam-5240	252	14	in	in	ADP
ejpam-5240	252	15	common	common	ADJ
ejpam-5240	252	16	.	.	PUNCT
ejpam-5240	253	1	thus	thus	ADV
ejpam-5240	253	2	,	,	PUNCT
ejpam-5240	253	3	e(gs(4,1	e(gs(4,1	NOUN
ejpam-5240	253	4	)	)	PUNCT
ejpam-5240	253	5	)	)	PUNCT
ejpam-5240	254	1	=	=	NOUN
ejpam-5240	254	2	∅	∅	NOUN
ejpam-5240	254	3	which	which	PRON
ejpam-5240	254	4	means	mean	VERB
ejpam-5240	254	5	that	that	SCONJ
ejpam-5240	254	6	gs(4,1	gs(4,1	NOUN
ejpam-5240	254	7	)	)	PUNCT
ejpam-5240	254	8	is	be	AUX
ejpam-5240	254	9	an	an	DET
ejpam-5240	254	10	empty	empty	ADJ
ejpam-5240	254	11	graph	graph	NOUN
ejpam-5240	254	12	of	of	ADP
ejpam-5240	254	13	order	order	NOUN
ejpam-5240	254	14	4	4	NUM
ejpam-5240	254	15	.	.	PUNCT
ejpam-5240	254	16	shown	show	VERB
ejpam-5240	254	17	in	in	ADP
ejpam-5240	254	18	figure	figure	NOUN
ejpam-5240	254	19	4	4	NUM
ejpam-5240	254	20	is	be	AUX
ejpam-5240	254	21	a	a	DET
ejpam-5240	254	22	pictorial	pictorial	ADJ
ejpam-5240	254	23	representation	representation	NOUN
ejpam-5240	254	24	of	of	ADP
ejpam-5240	254	25	gs(4,1	gs(4,1	NOUN
ejpam-5240	254	26	)	)	PUNCT
ejpam-5240	254	27	of	of	ADP
ejpam-5240	254	28	s4	s4	PROPN
ejpam-5240	254	29	.	.	PUNCT
ejpam-5240	255	1	{	{	PUNCT
ejpam-5240	255	2	x1	x1	PROPN
ejpam-5240	255	3	}	}	PUNCT
ejpam-5240	255	4	{	{	PUNCT
ejpam-5240	255	5	x3}{x2	x3}{x2	PROPN
ejpam-5240	255	6	}	}	PUNCT
ejpam-5240	255	7	{	{	PUNCT
ejpam-5240	255	8	x4	x4	PROPN
ejpam-5240	255	9	}	}	PUNCT
ejpam-5240	255	10	figure	figure	VERB
ejpam-5240	255	11	4	4	NUM
ejpam-5240	255	12	:	:	PUNCT
ejpam-5240	255	13	pictorial	pictorial	ADJ
ejpam-5240	255	14	illustration	illustration	NOUN
ejpam-5240	255	15	of	of	ADP
ejpam-5240	255	16	gs(4,1	gs(4,1	NOUN
ejpam-5240	255	17	)	)	PUNCT
ejpam-5240	255	18	.	.	PUNCT
ejpam-5240	256	1	the	the	DET
ejpam-5240	256	2	degree	degree	NOUN
ejpam-5240	256	3	of	of	ADP
ejpam-5240	256	4	every	every	DET
ejpam-5240	256	5	vertex	vertex	NOUN
ejpam-5240	256	6	in	in	ADP
ejpam-5240	256	7	gs(n	gs(n	NOUN
ejpam-5240	256	8	,	,	PUNCT
ejpam-5240	256	9	k	k	NOUN
ejpam-5240	256	10	)	)	PUNCT
ejpam-5240	256	11	depends	depend	VERB
ejpam-5240	256	12	on	on	ADP
ejpam-5240	256	13	the	the	DET
ejpam-5240	256	14	value	value	NOUN
ejpam-5240	256	15	of	of	ADP
ejpam-5240	256	16	the	the	DET
ejpam-5240	256	17	nonnegative	nonnegative	ADJ
ejpam-5240	256	18	integer	integer	NOUN
ejpam-5240	256	19	k.	k.	PROPN
ejpam-5240	256	20	lemma	lemma	PROPN
ejpam-5240	256	21	1	1	NUM
ejpam-5240	256	22	determines	determine	VERB
ejpam-5240	256	23	the	the	DET
ejpam-5240	256	24	degree	degree	NOUN
ejpam-5240	256	25	of	of	ADP
ejpam-5240	256	26	every	every	DET
ejpam-5240	256	27	vertex	vertex	NOUN
ejpam-5240	256	28	in	in	ADP
ejpam-5240	256	29	a	a	DET
ejpam-5240	256	30	gs(n	gs(n	NOUN
ejpam-5240	256	31	,	,	PUNCT
ejpam-5240	256	32	k	k	NOUN
ejpam-5240	256	33	)	)	PUNCT
ejpam-5240	256	34	when	when	SCONJ
ejpam-5240	256	35	1	1	NUM
ejpam-5240	256	36	≤	≤	NUM
ejpam-5240	256	37	k	k	X
ejpam-5240	256	38	≤	≤	NUM
ejpam-5240	256	39	⌊	⌊	VERB
ejpam-5240	256	40	n	n	DET
ejpam-5240	256	41	2	2	NUM
ejpam-5240	256	42	⌋	⌋	NOUN
ejpam-5240	256	43	.	.	PUNCT
ejpam-5240	257	1	lemma	lemma	PROPN
ejpam-5240	257	2	1	1	X
ejpam-5240	257	3	.	.	PUNCT
ejpam-5240	258	1	let	let	VERB
ejpam-5240	258	2	sn	sn	PROPN
ejpam-5240	258	3	be	be	AUX
ejpam-5240	258	4	an	an	DET
ejpam-5240	258	5	n	n	NOUN
ejpam-5240	258	6	-	-	PUNCT
ejpam-5240	258	7	element	element	NOUN
ejpam-5240	258	8	set	set	NOUN
ejpam-5240	258	9	and	and	CCONJ
ejpam-5240	258	10	let	let	VERB
ejpam-5240	258	11	gs(n	gs(n	NOUN
ejpam-5240	258	12	,	,	PUNCT
ejpam-5240	258	13	k	k	NOUN
ejpam-5240	258	14	)	)	PUNCT
ejpam-5240	258	15	be	be	VERB
ejpam-5240	258	16	a	a	DET
ejpam-5240	258	17	k	k	ADV
ejpam-5240	258	18	-	-	ADJ
ejpam-5240	258	19	restricted	restricted	ADJ
ejpam-5240	258	20	intersection	intersection	NOUN
ejpam-5240	258	21	graph	graph	NOUN
ejpam-5240	258	22	.	.	PUNCT
ejpam-5240	259	1	then	then	ADV
ejpam-5240	259	2	for	for	ADP
ejpam-5240	259	3	all	all	DET
ejpam-5240	259	4	a	a	DET
ejpam-5240	259	5	∈	∈	PROPN
ejpam-5240	259	6	v	v	NOUN
ejpam-5240	259	7	(	(	PUNCT
ejpam-5240	259	8	gs(n	gs(n	NOUN
ejpam-5240	259	9	,	,	PUNCT
ejpam-5240	259	10	k	k	NOUN
ejpam-5240	259	11	)	)	PUNCT
ejpam-5240	259	12	)	)	PUNCT
ejpam-5240	259	13	,	,	PUNCT
ejpam-5240	259	14	deg(a	deg(a	PROPN
ejpam-5240	259	15	)	)	PUNCT
ejpam-5240	259	16	=	=	PUNCT
ejpam-5240	259	17	(	(	PUNCT
ejpam-5240	259	18	n	n	X
ejpam-5240	259	19	k	k	NOUN
ejpam-5240	259	20	)	)	PUNCT
ejpam-5240	259	21	−	−	PROPN
ejpam-5240	260	1	[	[	X
ejpam-5240	260	2	(	(	PUNCT
ejpam-5240	260	3	n−k	n−k	NOUN
ejpam-5240	260	4	k	k	PROPN
ejpam-5240	260	5	)	)	PUNCT
ejpam-5240	261	1	+	+	CCONJ
ejpam-5240	261	2	1	1	X
ejpam-5240	261	3	]	]	PUNCT
ejpam-5240	261	4	if	if	SCONJ
ejpam-5240	261	5	and	and	CCONJ
ejpam-5240	261	6	only	only	ADV
ejpam-5240	261	7	if	if	SCONJ
ejpam-5240	261	8	1	1	NUM
ejpam-5240	261	9	≤	≤	NUM
ejpam-5240	261	10	k	k	X
ejpam-5240	261	11	≤	≤	NUM
ejpam-5240	261	12	⌊	⌊	VERB
ejpam-5240	261	13	n	n	PRON
ejpam-5240	261	14	2	2	NUM
ejpam-5240	261	15	⌋	⌋	NOUN
ejpam-5240	261	16	.	.	PUNCT
ejpam-5240	262	1	proof	proof	NOUN
ejpam-5240	262	2	.	.	PUNCT
ejpam-5240	263	1	assume	assume	VERB
ejpam-5240	263	2	deg(a	deg(a	PROPN
ejpam-5240	263	3	)	)	PUNCT
ejpam-5240	263	4	=	=	PUNCT
ejpam-5240	264	1	(	(	PUNCT
ejpam-5240	264	2	n	n	X
ejpam-5240	264	3	k	k	NOUN
ejpam-5240	264	4	)	)	PUNCT
ejpam-5240	264	5	−	−	PROPN
ejpam-5240	265	1	[	[	X
ejpam-5240	265	2	(	(	PUNCT
ejpam-5240	265	3	n−k	n−k	NOUN
ejpam-5240	265	4	k	k	PROPN
ejpam-5240	265	5	)	)	PUNCT
ejpam-5240	266	1	+	+	CCONJ
ejpam-5240	266	2	1	1	X
ejpam-5240	266	3	]	]	PUNCT
ejpam-5240	266	4	for	for	ADP
ejpam-5240	266	5	all	all	DET
ejpam-5240	266	6	a	a	DET
ejpam-5240	266	7	in	in	ADP
ejpam-5240	266	8	v	v	NOUN
ejpam-5240	266	9	(	(	PUNCT
ejpam-5240	266	10	gs(n	gs(n	NOUN
ejpam-5240	266	11	,	,	PUNCT
ejpam-5240	266	12	k	k	NOUN
ejpam-5240	266	13	)	)	PUNCT
ejpam-5240	266	14	)	)	PUNCT
ejpam-5240	266	15	and	and	CCONJ
ejpam-5240	266	16	suppose	suppose	VERB
ejpam-5240	266	17	that	that	SCONJ
ejpam-5240	266	18	k	k	PROPN
ejpam-5240	266	19	=	=	PUNCT
ejpam-5240	266	20	0	0	NUM
ejpam-5240	266	21	or	or	CCONJ
ejpam-5240	266	22	⌊	⌊	X
ejpam-5240	266	23	n	n	ADV
ejpam-5240	266	24	2	2	NUM
ejpam-5240	266	25	⌋	⌋	NOUN
ejpam-5240	266	26	<	<	X
ejpam-5240	266	27	k	k	PROPN
ejpam-5240	266	28	≤	≤	PROPN
ejpam-5240	266	29	n.	n.	NOUN
ejpam-5240	266	30	if	if	SCONJ
ejpam-5240	266	31	k	k	PROPN
ejpam-5240	266	32	=	=	SYM
ejpam-5240	266	33	0	0	PROPN
ejpam-5240	266	34	,	,	PUNCT
ejpam-5240	266	35	(	(	PUNCT
ejpam-5240	266	36	n	n	X
ejpam-5240	266	37	0	0	NUM
ejpam-5240	266	38	)	)	PUNCT
ejpam-5240	266	39	−	−	PROPN
ejpam-5240	267	1	[	[	X
ejpam-5240	267	2	(	(	PUNCT
ejpam-5240	267	3	n−0	n−0	ADV
ejpam-5240	267	4	0	0	NUM
ejpam-5240	267	5	)	)	PUNCT
ejpam-5240	268	1	+	+	CCONJ
ejpam-5240	268	2	1	1	X
ejpam-5240	268	3	]	]	PUNCT
ejpam-5240	268	4	=	=	SYM
ejpam-5240	268	5	1	1	NUM
ejpam-5240	268	6	−	−	NOUN
ejpam-5240	268	7	(	(	PUNCT
ejpam-5240	268	8	1	1	NUM
ejpam-5240	268	9	+	+	NUM
ejpam-5240	268	10	1	1	NUM
ejpam-5240	268	11	)	)	PUNCT
ejpam-5240	268	12	=	=	SYM
ejpam-5240	268	13	−1	−1	NOUN
ejpam-5240	268	14	.	.	PUNCT
ejpam-5240	269	1	this	this	PRON
ejpam-5240	269	2	is	be	AUX
ejpam-5240	269	3	a	a	DET
ejpam-5240	269	4	contradiction	contradiction	NOUN
ejpam-5240	269	5	since	since	SCONJ
ejpam-5240	269	6	the	the	DET
ejpam-5240	269	7	degree	degree	NOUN
ejpam-5240	269	8	of	of	ADP
ejpam-5240	269	9	a	a	DET
ejpam-5240	269	10	vertex	vertex	NOUN
ejpam-5240	269	11	can	can	AUX
ejpam-5240	269	12	not	not	PART
ejpam-5240	269	13	be	be	AUX
ejpam-5240	269	14	a	a	DET
ejpam-5240	269	15	negative	negative	ADJ
ejpam-5240	269	16	integer	integer	NOUN
ejpam-5240	269	17	.	.	PUNCT
ejpam-5240	270	1	if	if	SCONJ
ejpam-5240	270	2	⌊	⌊	PROPN
ejpam-5240	270	3	n	n	ADV
ejpam-5240	270	4	2	2	NUM
ejpam-5240	270	5	⌋	⌋	NOUN
ejpam-5240	270	6	<	<	X
ejpam-5240	270	7	k	k	X
ejpam-5240	270	8	≤	≤	PROPN
ejpam-5240	270	9	n	n	CCONJ
ejpam-5240	270	10	,	,	PUNCT
ejpam-5240	270	11	then	then	ADV
ejpam-5240	270	12	n	n	CCONJ
ejpam-5240	270	13	−	−	PROPN
ejpam-5240	270	14	k	k	NOUN
ejpam-5240	270	15	≤	≤	NUM
ejpam-5240	270	16	⌊	⌊	VERB
ejpam-5240	270	17	n	n	PRON
ejpam-5240	270	18	2	2	NUM
ejpam-5240	270	19	⌋	⌋	NOUN
ejpam-5240	270	20	.	.	PUNCT
ejpam-5240	271	1	since	since	SCONJ
ejpam-5240	271	2	n	n	NUM
ejpam-5240	271	3	−	−	PROPN
ejpam-5240	271	4	k	k	PROPN
ejpam-5240	271	5	≤	≤	NUM
ejpam-5240	271	6	⌊	⌊	VERB
ejpam-5240	271	7	n	n	ADV
ejpam-5240	271	8	2	2	NUM
ejpam-5240	271	9	⌋	⌋	NOUN
ejpam-5240	271	10	<	<	X
ejpam-5240	271	11	k	k	X
ejpam-5240	271	12	which	which	PRON
ejpam-5240	271	13	implies	imply	VERB
ejpam-5240	271	14	that	that	SCONJ
ejpam-5240	271	15	n	n	PROPN
ejpam-5240	271	16	−	−	PROPN
ejpam-5240	271	17	k	k	X
ejpam-5240	271	18	<	<	X
ejpam-5240	271	19	k	k	X
ejpam-5240	271	20	,	,	PUNCT
ejpam-5240	271	21	by	by	ADP
ejpam-5240	271	22	remark	remark	NOUN
ejpam-5240	271	23	4	4	NUM
ejpam-5240	271	24	,	,	PUNCT
ejpam-5240	271	25	it	it	PRON
ejpam-5240	271	26	follows	follow	VERB
ejpam-5240	271	27	that	that	SCONJ
ejpam-5240	271	28	(	(	PUNCT
ejpam-5240	271	29	n−k	n−k	NOUN
ejpam-5240	271	30	k	k	X
ejpam-5240	271	31	)	)	PUNCT
ejpam-5240	272	1	=	=	PUNCT
ejpam-5240	272	2	0	0	X
ejpam-5240	272	3	.	.	PUNCT
ejpam-5240	273	1	hence	hence	ADV
ejpam-5240	273	2	,	,	PUNCT
ejpam-5240	273	3	deg(a	deg(a	PROPN
ejpam-5240	273	4	)	)	PUNCT
ejpam-5240	273	5	=	=	PUNCT
ejpam-5240	274	1	(	(	PUNCT
ejpam-5240	274	2	n	n	X
ejpam-5240	274	3	k	k	NOUN
ejpam-5240	274	4	)	)	PUNCT
ejpam-5240	275	1	−	−	PROPN
ejpam-5240	275	2	(	(	PUNCT
ejpam-5240	275	3	0	0	NUM
ejpam-5240	275	4	+	+	CCONJ
ejpam-5240	275	5	1	1	X
ejpam-5240	275	6	)	)	PUNCT
ejpam-5240	275	7	=	=	NOUN
ejpam-5240	275	8	(	(	PUNCT
ejpam-5240	275	9	n	n	X
ejpam-5240	275	10	k	k	NOUN
ejpam-5240	275	11	)	)	PUNCT
ejpam-5240	276	1	−	−	PROPN
ejpam-5240	277	1	1	1	X
ejpam-5240	277	2	.	.	PUNCT
ejpam-5240	278	1	this	this	PRON
ejpam-5240	278	2	is	be	AUX
ejpam-5240	278	3	a	a	DET
ejpam-5240	278	4	contradiction	contradiction	NOUN
ejpam-5240	278	5	to	to	ADP
ejpam-5240	278	6	the	the	DET
ejpam-5240	278	7	assumption	assumption	NOUN
ejpam-5240	278	8	that	that	SCONJ
ejpam-5240	278	9	deg(a	deg(a	PROPN
ejpam-5240	278	10	)	)	PUNCT
ejpam-5240	278	11	=	=	PUNCT
ejpam-5240	279	1	(	(	PUNCT
ejpam-5240	279	2	n	n	X
ejpam-5240	279	3	k	k	NOUN
ejpam-5240	279	4	)	)	PUNCT
ejpam-5240	279	5	−	−	PROPN
ejpam-5240	280	1	[	[	X
ejpam-5240	280	2	(	(	PUNCT
ejpam-5240	280	3	n−k	n−k	NOUN
ejpam-5240	280	4	k	k	PROPN
ejpam-5240	280	5	)	)	PUNCT
ejpam-5240	281	1	+	+	CCONJ
ejpam-5240	281	2	1	1	NUM
ejpam-5240	281	3	]	]	PUNCT
ejpam-5240	281	4	.	.	PUNCT
ejpam-5240	282	1	therefore	therefore	ADV
ejpam-5240	282	2	,	,	PUNCT
ejpam-5240	282	3	1	1	NUM
ejpam-5240	282	4	≤	≤	NUM
ejpam-5240	282	5	k	k	X
ejpam-5240	282	6	≤	≤	NUM
ejpam-5240	282	7	⌊	⌊	VERB
ejpam-5240	282	8	n	n	DET
ejpam-5240	282	9	2	2	NUM
ejpam-5240	282	10	⌋	⌋	NOUN
ejpam-5240	282	11	.	.	PUNCT
ejpam-5240	283	1	conversely	conversely	ADV
ejpam-5240	283	2	,	,	PUNCT
ejpam-5240	283	3	assume	assume	VERB
ejpam-5240	283	4	that	that	SCONJ
ejpam-5240	283	5	1	1	NUM
ejpam-5240	283	6	≤	≤	NUM
ejpam-5240	283	7	k	k	X
ejpam-5240	283	8	≤	≤	NUM
ejpam-5240	283	9	⌊	⌊	VERB
ejpam-5240	283	10	n	n	PRON
ejpam-5240	283	11	2	2	NUM
ejpam-5240	283	12	⌋	⌋	NOUN
ejpam-5240	283	13	and	and	CCONJ
ejpam-5240	283	14	let	let	VERB
ejpam-5240	283	15	a	a	PRON
ejpam-5240	283	16	be	be	AUX
ejpam-5240	283	17	an	an	DET
ejpam-5240	283	18	arbitrary	arbitrary	ADJ
ejpam-5240	283	19	vertex	vertex	NOUN
ejpam-5240	283	20	of	of	ADP
ejpam-5240	283	21	v	v	NOUN
ejpam-5240	283	22	(	(	PUNCT
ejpam-5240	283	23	gs(n	gs(n	NOUN
ejpam-5240	283	24	,	,	PUNCT
ejpam-5240	283	25	k	k	NOUN
ejpam-5240	283	26	)	)	PUNCT
ejpam-5240	283	27	)	)	PUNCT
ejpam-5240	283	28	.	.	PUNCT
ejpam-5240	284	1	since	since	SCONJ
ejpam-5240	284	2	a	a	DET
ejpam-5240	284	3	⊆	⊆	NUM
ejpam-5240	284	4	sn	sn	NOUN
ejpam-5240	284	5	,	,	PUNCT
ejpam-5240	284	6	then	then	ADV
ejpam-5240	284	7	by	by	ADP
ejpam-5240	284	8	remark	remark	NOUN
ejpam-5240	284	9	1	1	NUM
ejpam-5240	284	10	,	,	PUNCT
ejpam-5240	284	11	it	it	PRON
ejpam-5240	284	12	follows	follow	VERB
ejpam-5240	284	13	that	that	SCONJ
ejpam-5240	284	14	|sn	|sn	NUM
ejpam-5240	284	15	\	\	NOUN
ejpam-5240	284	16	a|	a|	PROPN
ejpam-5240	284	17	=	=	PROPN
ejpam-5240	284	18	n	n	PRON
ejpam-5240	284	19	−	−	PROPN
ejpam-5240	284	20	k.	k.	PROPN
ejpam-5240	284	21	now	now	ADV
ejpam-5240	284	22	,	,	PUNCT
ejpam-5240	284	23	let	let	VERB
ejpam-5240	284	24	s(n−k	s(n−k	VERB
ejpam-5240	284	25	,	,	PUNCT
ejpam-5240	284	26	k	k	NOUN
ejpam-5240	284	27	)	)	PUNCT
ejpam-5240	284	28	be	be	VERB
ejpam-5240	284	29	a	a	DET
ejpam-5240	284	30	set	set	NOUN
ejpam-5240	284	31	containing	contain	VERB
ejpam-5240	284	32	all	all	DET
ejpam-5240	284	33	the	the	DET
ejpam-5240	284	34	k	k	ADJ
ejpam-5240	284	35	-	-	ADJ
ejpam-5240	284	36	element	element	ADJ
ejpam-5240	284	37	subsets	subset	NOUN
ejpam-5240	284	38	of	of	ADP
ejpam-5240	284	39	sn	sn	PROPN
ejpam-5240	284	40	\	\	PROPN
ejpam-5240	284	41	a.	a.	NOUN
ejpam-5240	284	42	observe	observe	VERB
ejpam-5240	284	43	that	that	SCONJ
ejpam-5240	284	44	s(n−k	s(n−k	NOUN
ejpam-5240	284	45	,	,	PUNCT
ejpam-5240	284	46	k	k	NOUN
ejpam-5240	284	47	)	)	PUNCT
ejpam-5240	284	48	⊆	⊆	NUM
ejpam-5240	284	49	v	v	NOUN
ejpam-5240	284	50	(	(	PUNCT
ejpam-5240	284	51	gs(n	gs(n	NOUN
ejpam-5240	284	52	,	,	PUNCT
ejpam-5240	284	53	k	k	NOUN
ejpam-5240	284	54	)	)	PUNCT
ejpam-5240	284	55	)	)	PUNCT
ejpam-5240	284	56	.	.	PUNCT
ejpam-5240	285	1	if	if	SCONJ
ejpam-5240	285	2	1	1	NUM
ejpam-5240	285	3	≤	≤	NUM
ejpam-5240	285	4	k	k	X
ejpam-5240	285	5	≤	≤	NUM
ejpam-5240	285	6	⌊	⌊	VERB
ejpam-5240	285	7	n	n	DET
ejpam-5240	285	8	2	2	NUM
ejpam-5240	285	9	⌋	⌋	NOUN
ejpam-5240	285	10	,	,	PUNCT
ejpam-5240	285	11	then	then	ADV
ejpam-5240	285	12	⌊	⌊	VERB
ejpam-5240	285	13	n	n	ADV
ejpam-5240	285	14	2	2	NUM
ejpam-5240	285	15	⌋	⌋	NOUN
ejpam-5240	285	16	≤	≤	NUM
ejpam-5240	285	17	n−	n−	PROPN
ejpam-5240	286	1	k.	k.	PROPN
ejpam-5240	287	1	this	this	PRON
ejpam-5240	287	2	implies	imply	VERB
ejpam-5240	287	3	that	that	SCONJ
ejpam-5240	287	4	k	k	PROPN
ejpam-5240	287	5	≤	≤	PROPN
ejpam-5240	287	6	n−	n−	PROPN
ejpam-5240	287	7	k	k	PROPN
ejpam-5240	287	8	so	so	ADV
ejpam-5240	287	9	|s(n−k	|s(n−k	NOUN
ejpam-5240	287	10	,	,	PUNCT
ejpam-5240	287	11	k)|	k)|	NOUN
ejpam-5240	287	12	=	=	PUNCT
ejpam-5240	287	13	(	(	PUNCT
ejpam-5240	287	14	n−k	n−k	NOUN
ejpam-5240	287	15	k	k	PROPN
ejpam-5240	287	16	)	)	PUNCT
ejpam-5240	287	17	.	.	PUNCT
ejpam-5240	288	1	note	note	VERB
ejpam-5240	288	2	that	that	SCONJ
ejpam-5240	288	3	a	a	DET
ejpam-5240	288	4	∩	∩	NOUN
ejpam-5240	288	5	(	(	PUNCT
ejpam-5240	288	6	sn	sn	ADV
ejpam-5240	288	7	\a	\a	NUM
ejpam-5240	288	8	)	)	PUNCT
ejpam-5240	289	1	=	=	PUNCT
ejpam-5240	289	2	∅.	∅.	VERB
ejpam-5240	289	3	hence	hence	ADV
ejpam-5240	289	4	,	,	PUNCT
ejpam-5240	289	5	for	for	ADP
ejpam-5240	289	6	every	every	DET
ejpam-5240	289	7	b	b	PROPN
ejpam-5240	289	8	∈	∈	PROPN
ejpam-5240	289	9	s(n−k	s(n−k	NOUN
ejpam-5240	289	10	,	,	PUNCT
ejpam-5240	289	11	k	k	NOUN
ejpam-5240	289	12	)	)	PUNCT
ejpam-5240	289	13	,	,	PUNCT
ejpam-5240	289	14	a	a	DET
ejpam-5240	289	15	∩	∩	ADJ
ejpam-5240	289	16	b	b	NOUN
ejpam-5240	289	17	=	=	PUNCT
ejpam-5240	289	18	∅.	∅.	NOUN
ejpam-5240	289	19	this	this	PRON
ejpam-5240	289	20	means	mean	VERB
ejpam-5240	289	21	that	that	SCONJ
ejpam-5240	289	22	a	a	PRON
ejpam-5240	289	23	is	be	AUX
ejpam-5240	289	24	not	not	PART
ejpam-5240	289	25	adjacent	adjacent	ADJ
ejpam-5240	289	26	to	to	ADP
ejpam-5240	289	27	(	(	PUNCT
ejpam-5240	289	28	n−k	n−k	NOUN
ejpam-5240	289	29	k	k	NOUN
ejpam-5240	289	30	)	)	PUNCT
ejpam-5240	289	31	elements	element	NOUN
ejpam-5240	289	32	of	of	ADP
ejpam-5240	289	33	v	v	NOUN
ejpam-5240	289	34	(	(	PUNCT
ejpam-5240	289	35	gs(n	gs(n	NOUN
ejpam-5240	289	36	,	,	PUNCT
ejpam-5240	289	37	k	k	NOUN
ejpam-5240	289	38	)	)	PUNCT
ejpam-5240	289	39	)	)	PUNCT
ejpam-5240	289	40	.	.	PUNCT
ejpam-5240	290	1	by	by	ADP
ejpam-5240	290	2	theorem	theorem	NOUN
ejpam-5240	290	3	2	2	NUM
ejpam-5240	290	4	,	,	PUNCT
ejpam-5240	290	5	since	since	SCONJ
ejpam-5240	290	6	|v	|v	PROPN
ejpam-5240	290	7	(	(	PUNCT
ejpam-5240	290	8	gs(n	gs(n	NOUN
ejpam-5240	290	9	,	,	PUNCT
ejpam-5240	290	10	k	k	NOUN
ejpam-5240	290	11	)	)	PUNCT
ejpam-5240	290	12	)	)	PUNCT
ejpam-5240	290	13	|	|	ADV
ejpam-5240	290	14	=	=	SYM
ejpam-5240	290	15	(	(	PUNCT
ejpam-5240	290	16	n	n	X
ejpam-5240	290	17	k	k	PROPN
ejpam-5240	290	18	)	)	PUNCT
ejpam-5240	290	19	,	,	PUNCT
ejpam-5240	290	20	it	it	PRON
ejpam-5240	290	21	follows	follow	VERB
ejpam-5240	290	22	that	that	SCONJ
ejpam-5240	290	23	deg(a	deg(a	PROPN
ejpam-5240	290	24	)	)	PUNCT
ejpam-5240	290	25	=	=	PUNCT
ejpam-5240	291	1	(	(	PUNCT
ejpam-5240	291	2	n	n	X
ejpam-5240	291	3	k	k	NOUN
ejpam-5240	291	4	)	)	PUNCT
ejpam-5240	292	1	−	−	PROPN
ejpam-5240	292	2	(	(	PUNCT
ejpam-5240	292	3	n−k	n−k	NOUN
ejpam-5240	292	4	k	k	PROPN
ejpam-5240	292	5	)	)	PUNCT
ejpam-5240	292	6	.	.	PUNCT
ejpam-5240	293	1	additionally	additionally	ADV
ejpam-5240	293	2	,	,	PUNCT
ejpam-5240	293	3	by	by	ADP
ejpam-5240	293	4	remark	remark	NOUN
ejpam-5240	293	5	6	6	NUM
ejpam-5240	293	6	,	,	PUNCT
ejpam-5240	293	7	[	[	X
ejpam-5240	293	8	a	a	X
ejpam-5240	293	9	,	,	PUNCT
ejpam-5240	293	10	a	a	PRON
ejpam-5240	293	11	]	]	X
ejpam-5240	293	12	/∈	/∈	PUNCT
ejpam-5240	294	1	e(gs(n	e(gs(n	NOUN
ejpam-5240	294	2	,	,	PUNCT
ejpam-5240	294	3	k	k	NOUN
ejpam-5240	294	4	)	)	PUNCT
ejpam-5240	294	5	)	)	PUNCT
ejpam-5240	294	6	.	.	PUNCT
ejpam-5240	295	1	this	this	DET
ejpam-5240	295	2	m.e	m.e	PROPN
ejpam-5240	295	3	.	.	PROPN
ejpam-5240	295	4	pelagio	pelagio	PROPN
ejpam-5240	295	5	,	,	PUNCT
ejpam-5240	295	6	n.	n.	NOUN
ejpam-5240	295	7	mame	mame	PROPN
ejpam-5240	295	8	,	,	PUNCT
ejpam-5240	295	9	k.	k.	PROPN
ejpam-5240	295	10	mendoza	mendoza	PROPN
ejpam-5240	295	11	/	/	SYM
ejpam-5240	295	12	eur	eur	PROPN
ejpam-5240	295	13	.	.	PUNCT
ejpam-5240	296	1	j.	j.	PROPN
ejpam-5240	296	2	pure	pure	PROPN
ejpam-5240	296	3	appl	appl	PROPN
ejpam-5240	296	4	.	.	PROPN
ejpam-5240	296	5	math	math	PROPN
ejpam-5240	296	6	,	,	PUNCT
ejpam-5240	296	7	17	17	NUM
ejpam-5240	296	8	(	(	PUNCT
ejpam-5240	296	9	3	3	NUM
ejpam-5240	296	10	)	)	PUNCT
ejpam-5240	296	11	(	(	PUNCT
ejpam-5240	296	12	2024	2024	NUM
ejpam-5240	296	13	)	)	PUNCT
ejpam-5240	296	14	,	,	PUNCT
ejpam-5240	296	15	1779	1779	NUM
ejpam-5240	296	16	-	-	SYM
ejpam-5240	296	17	1803	1803	NUM
ejpam-5240	296	18	1788	1788	NUM
ejpam-5240	296	19	implies	imply	VERB
ejpam-5240	296	20	that	that	SCONJ
ejpam-5240	296	21	deg(a	deg(a	PROPN
ejpam-5240	296	22	)	)	PUNCT
ejpam-5240	296	23	=	=	PUNCT
ejpam-5240	297	1	(	(	PUNCT
ejpam-5240	297	2	n	n	X
ejpam-5240	297	3	k	k	NOUN
ejpam-5240	297	4	)	)	PUNCT
ejpam-5240	297	5	−	−	PROPN
ejpam-5240	298	1	[	[	X
ejpam-5240	298	2	(	(	PUNCT
ejpam-5240	298	3	n−k	n−k	NOUN
ejpam-5240	298	4	k	k	PROPN
ejpam-5240	298	5	)	)	PUNCT
ejpam-5240	299	1	+	+	CCONJ
ejpam-5240	299	2	1	1	NUM
ejpam-5240	299	3	]	]	PUNCT
ejpam-5240	299	4	.	.	PUNCT
ejpam-5240	300	1	since	since	SCONJ
ejpam-5240	300	2	a	a	PRON
ejpam-5240	300	3	was	be	AUX
ejpam-5240	300	4	arbitrarily	arbitrarily	ADV
ejpam-5240	300	5	chosen	choose	VERB
ejpam-5240	300	6	,	,	PUNCT
ejpam-5240	300	7	it	it	PRON
ejpam-5240	300	8	follows	follow	VERB
ejpam-5240	300	9	that	that	SCONJ
ejpam-5240	300	10	every	every	DET
ejpam-5240	300	11	vertex	vertex	NOUN
ejpam-5240	300	12	of	of	ADP
ejpam-5240	300	13	gs(n	gs(n	NOUN
ejpam-5240	300	14	,	,	PUNCT
ejpam-5240	300	15	k	k	NOUN
ejpam-5240	300	16	)	)	PUNCT
ejpam-5240	300	17	,	,	PUNCT
ejpam-5240	300	18	with	with	ADP
ejpam-5240	300	19	1	1	NUM
ejpam-5240	300	20	≤	≤	NUM
ejpam-5240	300	21	k	k	X
ejpam-5240	300	22	≤	≤	NUM
ejpam-5240	300	23	⌊	⌊	VERB
ejpam-5240	300	24	n	n	DET
ejpam-5240	300	25	2	2	NUM
ejpam-5240	300	26	⌋	⌋	NOUN
ejpam-5240	300	27	,	,	PUNCT
ejpam-5240	300	28	has	have	VERB
ejpam-5240	300	29	degree	degree	NOUN
ejpam-5240	300	30	equal	equal	ADJ
ejpam-5240	300	31	to	to	ADP
ejpam-5240	300	32	(	(	PUNCT
ejpam-5240	300	33	n	n	X
ejpam-5240	300	34	k	k	NOUN
ejpam-5240	300	35	)	)	PUNCT
ejpam-5240	300	36	−	−	PROPN
ejpam-5240	301	1	[	[	X
ejpam-5240	301	2	(	(	PUNCT
ejpam-5240	301	3	n−k	n−k	NOUN
ejpam-5240	301	4	k	k	PROPN
ejpam-5240	301	5	)	)	PUNCT
ejpam-5240	302	1	+	+	CCONJ
ejpam-5240	302	2	1	1	NUM
ejpam-5240	302	3	]	]	PUNCT
ejpam-5240	302	4	.	.	PUNCT
ejpam-5240	303	1	illustration	illustration	NOUN
ejpam-5240	303	2	4	4	NUM
ejpam-5240	303	3	.	.	PUNCT
ejpam-5240	304	1	let	let	VERB
ejpam-5240	304	2	s5	s5	PROPN
ejpam-5240	304	3	=	=	PUNCT
ejpam-5240	304	4	{	{	PUNCT
ejpam-5240	304	5	x1	x1	PROPN
ejpam-5240	304	6	,	,	PUNCT
ejpam-5240	304	7	x2	x2	PROPN
ejpam-5240	304	8	,	,	PUNCT
ejpam-5240	304	9	x3	x3	PROPN
ejpam-5240	304	10	,	,	PUNCT
ejpam-5240	304	11	x4	x4	PROPN
ejpam-5240	304	12	,	,	PUNCT
ejpam-5240	304	13	x5	x5	NOUN
ejpam-5240	304	14	}	}	PUNCT
ejpam-5240	304	15	and	and	CCONJ
ejpam-5240	304	16	let	let	VERB
ejpam-5240	304	17	k	k	NOUN
ejpam-5240	304	18	=	=	SYM
ejpam-5240	304	19	2	2	X
ejpam-5240	304	20	.	.	PUNCT
ejpam-5240	305	1	the	the	DET
ejpam-5240	305	2	vertex	vertex	NOUN
ejpam-5240	305	3	set	set	NOUN
ejpam-5240	305	4	of	of	ADP
ejpam-5240	305	5	gs(5,2	gs(5,2	NOUN
ejpam-5240	305	6	)	)	PUNCT
ejpam-5240	305	7	is	be	AUX
ejpam-5240	305	8	given	give	VERB
ejpam-5240	305	9	by	by	ADP
ejpam-5240	305	10	v	v	NOUN
ejpam-5240	305	11	(	(	PUNCT
ejpam-5240	305	12	gs(5,2	gs(5,2	NOUN
ejpam-5240	305	13	)	)	PUNCT
ejpam-5240	305	14	)	)	PUNCT
ejpam-5240	306	1	=	=	PRON
ejpam-5240	306	2	{	{	PUNCT
ejpam-5240	306	3	{	{	PUNCT
ejpam-5240	306	4	x1	x1	PROPN
ejpam-5240	306	5	,	,	PUNCT
ejpam-5240	306	6	x2	x2	PROPN
ejpam-5240	306	7	}	}	PUNCT
ejpam-5240	306	8	,	,	PUNCT
ejpam-5240	306	9	{	{	PUNCT
ejpam-5240	306	10	x1	x1	ADJ
ejpam-5240	306	11	,	,	PUNCT
ejpam-5240	306	12	x3	x3	ADJ
ejpam-5240	306	13	}	}	PUNCT
ejpam-5240	306	14	,	,	PUNCT
ejpam-5240	306	15	{	{	PUNCT
ejpam-5240	306	16	x1	x1	PROPN
ejpam-5240	306	17	,	,	PUNCT
ejpam-5240	306	18	x4	x4	PROPN
ejpam-5240	306	19	}	}	PUNCT
ejpam-5240	306	20	,	,	PUNCT
ejpam-5240	306	21	{	{	PUNCT
ejpam-5240	306	22	x1	x1	PROPN
ejpam-5240	306	23	,	,	PUNCT
ejpam-5240	306	24	x5	x5	PROPN
ejpam-5240	306	25	}	}	PUNCT
ejpam-5240	306	26	,	,	PUNCT
ejpam-5240	306	27	{	{	PUNCT
ejpam-5240	306	28	x2	x2	ADJ
ejpam-5240	306	29	,	,	PUNCT
ejpam-5240	306	30	x3	x3	ADJ
ejpam-5240	306	31	}	}	PUNCT
ejpam-5240	306	32	,	,	PUNCT
ejpam-5240	306	33	{	{	PUNCT
ejpam-5240	306	34	x2	x2	PROPN
ejpam-5240	306	35	,	,	PUNCT
ejpam-5240	306	36	x4	x4	PROPN
ejpam-5240	306	37	}	}	PUNCT
ejpam-5240	306	38	,	,	PUNCT
ejpam-5240	306	39	{	{	PUNCT
ejpam-5240	306	40	x2	x2	PROPN
ejpam-5240	306	41	,	,	PUNCT
ejpam-5240	306	42	x5	x5	PROPN
ejpam-5240	306	43	}	}	PUNCT
ejpam-5240	306	44	,	,	PUNCT
ejpam-5240	306	45	{	{	PUNCT
ejpam-5240	306	46	x3	x3	ADJ
ejpam-5240	306	47	,	,	PUNCT
ejpam-5240	306	48	x4	x4	PROPN
ejpam-5240	306	49	}	}	PUNCT
ejpam-5240	306	50	,	,	PUNCT
ejpam-5240	306	51	{	{	PUNCT
ejpam-5240	306	52	x3	x3	ADJ
ejpam-5240	306	53	,	,	PUNCT
ejpam-5240	306	54	x5	x5	PROPN
ejpam-5240	306	55	}	}	PUNCT
ejpam-5240	306	56	,	,	PUNCT
ejpam-5240	306	57	{	{	PUNCT
ejpam-5240	306	58	x4	x4	PROPN
ejpam-5240	306	59	,	,	PUNCT
ejpam-5240	306	60	x5	x5	PROPN
ejpam-5240	306	61	}	}	PUNCT
ejpam-5240	306	62	}	}	PUNCT
ejpam-5240	306	63	a	a	DET
ejpam-5240	306	64	pictorial	pictorial	ADJ
ejpam-5240	306	65	representation	representation	NOUN
ejpam-5240	306	66	of	of	ADP
ejpam-5240	306	67	the	the	DET
ejpam-5240	306	68	graph	graph	NOUN
ejpam-5240	306	69	gs(5,2	gs(5,2	NOUN
ejpam-5240	306	70	)	)	PUNCT
ejpam-5240	306	71	is	be	AUX
ejpam-5240	306	72	shown	show	VERB
ejpam-5240	306	73	in	in	ADP
ejpam-5240	306	74	figure	figure	NOUN
ejpam-5240	306	75	5	5	NUM
ejpam-5240	306	76	.	.	PUNCT
ejpam-5240	306	77	{	{	PUNCT
ejpam-5240	307	1	x1	x1	PROPN
ejpam-5240	307	2	,	,	PUNCT
ejpam-5240	307	3	x2	x2	PROPN
ejpam-5240	307	4	}	}	PUNCT
ejpam-5240	307	5	{	{	PUNCT
ejpam-5240	307	6	x3	x3	ADJ
ejpam-5240	307	7	,	,	PUNCT
ejpam-5240	307	8	x5	x5	PROPN
ejpam-5240	307	9	}	}	PUNCT
ejpam-5240	307	10	{	{	PUNCT
ejpam-5240	307	11	x1	x1	PROPN
ejpam-5240	307	12	,	,	PUNCT
ejpam-5240	307	13	x4	x4	PROPN
ejpam-5240	307	14	}	}	PUNCT
ejpam-5240	307	15	{	{	PUNCT
ejpam-5240	307	16	x3	x3	ADJ
ejpam-5240	307	17	,	,	PUNCT
ejpam-5240	307	18	x4	x4	PROPN
ejpam-5240	307	19	}	}	PUNCT
ejpam-5240	307	20	{	{	PUNCT
ejpam-5240	307	21	x1	x1	PROPN
ejpam-5240	307	22	,	,	PUNCT
ejpam-5240	307	23	x5	x5	PROPN
ejpam-5240	307	24	}	}	PUNCT
ejpam-5240	307	25	{	{	PUNCT
ejpam-5240	307	26	x2	x2	PROPN
ejpam-5240	307	27	,	,	PUNCT
ejpam-5240	307	28	x4	x4	PROPN
ejpam-5240	307	29	}	}	PUNCT
ejpam-5240	307	30	{	{	PUNCT
ejpam-5240	307	31	x1	x1	PROPN
ejpam-5240	307	32	,	,	PUNCT
ejpam-5240	307	33	x3	x3	ADJ
ejpam-5240	307	34	}	}	PUNCT
ejpam-5240	307	35	{	{	PUNCT
ejpam-5240	307	36	x2	x2	PROPN
ejpam-5240	307	37	,	,	PUNCT
ejpam-5240	307	38	x3	x3	ADJ
ejpam-5240	307	39	}	}	PUNCT
ejpam-5240	307	40	{	{	PUNCT
ejpam-5240	307	41	x4	x4	PROPN
ejpam-5240	307	42	,	,	PUNCT
ejpam-5240	307	43	x5	x5	PROPN
ejpam-5240	307	44	}	}	PUNCT
ejpam-5240	307	45	{	{	PUNCT
ejpam-5240	307	46	x2	x2	PROPN
ejpam-5240	307	47	,	,	PUNCT
ejpam-5240	307	48	x5	x5	ADJ
ejpam-5240	307	49	}	}	PUNCT
ejpam-5240	307	50	figure	figure	NOUN
ejpam-5240	307	51	5	5	NUM
ejpam-5240	307	52	:	:	PUNCT
ejpam-5240	307	53	a	a	DET
ejpam-5240	307	54	pictorial	pictorial	ADJ
ejpam-5240	307	55	representation	representation	NOUN
ejpam-5240	307	56	of	of	ADP
ejpam-5240	307	57	gs(5,2	gs(5,2	NOUN
ejpam-5240	307	58	)	)	PUNCT
ejpam-5240	307	59	.	.	PUNCT
ejpam-5240	308	1	it	it	PRON
ejpam-5240	308	2	can	can	AUX
ejpam-5240	308	3	be	be	AUX
ejpam-5240	308	4	observed	observe	VERB
ejpam-5240	308	5	that	that	SCONJ
ejpam-5240	308	6	the	the	DET
ejpam-5240	308	7	degree	degree	NOUN
ejpam-5240	308	8	of	of	ADP
ejpam-5240	308	9	every	every	DET
ejpam-5240	308	10	vertex	vertex	NOUN
ejpam-5240	308	11	in	in	ADP
ejpam-5240	308	12	gs(5,2	gs(5,2	NOUN
ejpam-5240	308	13	)	)	PUNCT
ejpam-5240	308	14	is	be	AUX
ejpam-5240	308	15	6	6	NUM
ejpam-5240	308	16	.	.	PUNCT
ejpam-5240	309	1	note	note	VERB
ejpam-5240	309	2	that	that	SCONJ
ejpam-5240	309	3	k	k	PROPN
ejpam-5240	309	4	=	=	SYM
ejpam-5240	309	5	2	2	NUM
ejpam-5240	309	6	which	which	PRON
ejpam-5240	309	7	implies	imply	VERB
ejpam-5240	309	8	that	that	SCONJ
ejpam-5240	309	9	k	k	PROPN
ejpam-5240	309	10	≤	≤	NUM
ejpam-5240	309	11	⌊	⌊	VERB
ejpam-5240	309	12	5	5	NUM
ejpam-5240	309	13	2	2	NUM
ejpam-5240	309	14	⌋	⌋	NOUN
ejpam-5240	309	15	=	=	SYM
ejpam-5240	309	16	2	2	X
ejpam-5240	309	17	.	.	PUNCT
ejpam-5240	309	18	by	by	ADP
ejpam-5240	309	19	using	use	VERB
ejpam-5240	309	20	lemma	lemma	PROPN
ejpam-5240	309	21	1	1	NUM
ejpam-5240	309	22	with	with	ADP
ejpam-5240	309	23	n	n	NOUN
ejpam-5240	309	24	=	=	SYM
ejpam-5240	309	25	5	5	NUM
ejpam-5240	309	26	and	and	CCONJ
ejpam-5240	309	27	k	k	NOUN
ejpam-5240	309	28	=	=	SYM
ejpam-5240	309	29	2	2	NUM
ejpam-5240	309	30	,	,	PUNCT
ejpam-5240	309	31	the	the	DET
ejpam-5240	309	32	degree	degree	NOUN
ejpam-5240	309	33	of	of	ADP
ejpam-5240	309	34	each	each	DET
ejpam-5240	309	35	a	a	DET
ejpam-5240	309	36	∈	∈	PROPN
ejpam-5240	309	37	v	v	NOUN
ejpam-5240	309	38	(	(	PUNCT
ejpam-5240	309	39	gs(5,2	gs(5,2	NOUN
ejpam-5240	309	40	)	)	PUNCT
ejpam-5240	309	41	)	)	PUNCT
ejpam-5240	309	42	is	be	AUX
ejpam-5240	309	43	given	give	VERB
ejpam-5240	309	44	by	by	ADP
ejpam-5240	309	45	deg(a	deg(a	PROPN
ejpam-5240	309	46	)	)	PUNCT
ejpam-5240	309	47	=	=	PUNCT
ejpam-5240	309	48	(	(	PUNCT
ejpam-5240	309	49	n	n	X
ejpam-5240	309	50	k	k	NOUN
ejpam-5240	309	51	)	)	PUNCT
ejpam-5240	309	52	−	−	PROPN
ejpam-5240	310	1	[	[	X
ejpam-5240	310	2	(	(	PUNCT
ejpam-5240	310	3	n−	n−	NOUN
ejpam-5240	310	4	k	k	NOUN
ejpam-5240	310	5	k	k	PROPN
ejpam-5240	310	6	)	)	PUNCT
ejpam-5240	311	1	+	+	CCONJ
ejpam-5240	311	2	1	1	X
ejpam-5240	311	3	]	]	PUNCT
ejpam-5240	311	4	=	=	PUNCT
ejpam-5240	311	5	(	(	PUNCT
ejpam-5240	311	6	5	5	NUM
ejpam-5240	311	7	2	2	NUM
ejpam-5240	311	8	)	)	PUNCT
ejpam-5240	311	9	−	−	PROPN
ejpam-5240	312	1	[	[	X
ejpam-5240	312	2	(	(	PUNCT
ejpam-5240	312	3	5−	5−	NUM
ejpam-5240	312	4	2	2	NUM
ejpam-5240	312	5	2	2	NUM
ejpam-5240	312	6	)	)	PUNCT
ejpam-5240	313	1	+	+	CCONJ
ejpam-5240	313	2	1	1	X
ejpam-5240	313	3	]	]	PUNCT
ejpam-5240	313	4	=	=	PUNCT
ejpam-5240	313	5	(	(	PUNCT
ejpam-5240	313	6	5	5	NUM
ejpam-5240	313	7	2	2	NUM
ejpam-5240	313	8	)	)	PUNCT
ejpam-5240	313	9	−	−	PROPN
ejpam-5240	314	1	[	[	X
ejpam-5240	314	2	(	(	PUNCT
ejpam-5240	314	3	3	3	NUM
ejpam-5240	314	4	2	2	NUM
ejpam-5240	314	5	)	)	PUNCT
ejpam-5240	314	6	+	+	CCONJ
ejpam-5240	314	7	1	1	X
ejpam-5240	314	8	]	]	PUNCT
ejpam-5240	314	9	=	=	SYM
ejpam-5240	314	10	10−	10−	X
ejpam-5240	314	11	(	(	PUNCT
ejpam-5240	314	12	3	3	NUM
ejpam-5240	314	13	+	+	NOUN
ejpam-5240	314	14	1	1	NUM
ejpam-5240	314	15	)	)	PUNCT
ejpam-5240	314	16	=	=	SYM
ejpam-5240	314	17	6	6	X
ejpam-5240	314	18	.	.	X
ejpam-5240	314	19	m.e	m.e	PROPN
ejpam-5240	314	20	.	.	PROPN
ejpam-5240	314	21	pelagio	pelagio	PROPN
ejpam-5240	314	22	,	,	PUNCT
ejpam-5240	314	23	n.	n.	NOUN
ejpam-5240	314	24	mame	mame	PROPN
ejpam-5240	314	25	,	,	PUNCT
ejpam-5240	314	26	k.	k.	PROPN
ejpam-5240	314	27	mendoza	mendoza	PROPN
ejpam-5240	314	28	/	/	SYM
ejpam-5240	314	29	eur	eur	PROPN
ejpam-5240	314	30	.	.	PUNCT
ejpam-5240	315	1	j.	j.	PROPN
ejpam-5240	315	2	pure	pure	PROPN
ejpam-5240	315	3	appl	appl	PROPN
ejpam-5240	315	4	.	.	PROPN
ejpam-5240	315	5	math	math	PROPN
ejpam-5240	315	6	,	,	PUNCT
ejpam-5240	315	7	17	17	NUM
ejpam-5240	315	8	(	(	PUNCT
ejpam-5240	315	9	3	3	NUM
ejpam-5240	315	10	)	)	PUNCT
ejpam-5240	315	11	(	(	PUNCT
ejpam-5240	315	12	2024	2024	NUM
ejpam-5240	315	13	)	)	PUNCT
ejpam-5240	315	14	,	,	PUNCT
ejpam-5240	315	15	1779	1779	NUM
ejpam-5240	315	16	-	-	SYM
ejpam-5240	315	17	1803	1803	NUM
ejpam-5240	315	18	1789	1789	NUM
ejpam-5240	315	19	furthermore	furthermore	ADV
ejpam-5240	315	20	,	,	PUNCT
ejpam-5240	315	21	lemma	lemma	PROPN
ejpam-5240	315	22	2	2	NUM
ejpam-5240	315	23	shows	show	VERB
ejpam-5240	315	24	the	the	DET
ejpam-5240	315	25	degree	degree	NOUN
ejpam-5240	315	26	of	of	ADP
ejpam-5240	315	27	every	every	DET
ejpam-5240	315	28	vertex	vertex	NOUN
ejpam-5240	315	29	in	in	ADP
ejpam-5240	315	30	gs(n	gs(n	NOUN
ejpam-5240	315	31	,	,	PUNCT
ejpam-5240	315	32	k	k	NOUN
ejpam-5240	315	33	)	)	PUNCT
ejpam-5240	315	34	if	if	SCONJ
ejpam-5240	315	35	and	and	CCONJ
ejpam-5240	315	36	only	only	ADV
ejpam-5240	315	37	if	if	SCONJ
ejpam-5240	315	38	k	k	PROPN
ejpam-5240	315	39	=	=	PUNCT
ejpam-5240	315	40	0	0	NUM
ejpam-5240	315	41	or	or	CCONJ
ejpam-5240	315	42	⌊	⌊	X
ejpam-5240	315	43	n	n	ADV
ejpam-5240	315	44	2	2	NUM
ejpam-5240	315	45	⌋	⌋	NOUN
ejpam-5240	315	46	<	<	X
ejpam-5240	315	47	k	k	PROPN
ejpam-5240	315	48	≤	≤	PROPN
ejpam-5240	315	49	n.	n.	PROPN
ejpam-5240	315	50	lemma	lemma	PROPN
ejpam-5240	316	1	2	2	X
ejpam-5240	316	2	.	.	PUNCT
ejpam-5240	316	3	let	let	VERB
ejpam-5240	316	4	sn	sn	PROPN
ejpam-5240	316	5	be	be	AUX
ejpam-5240	316	6	an	an	DET
ejpam-5240	316	7	n	n	NOUN
ejpam-5240	316	8	-	-	PUNCT
ejpam-5240	316	9	element	element	NOUN
ejpam-5240	316	10	set	set	NOUN
ejpam-5240	316	11	and	and	CCONJ
ejpam-5240	316	12	let	let	VERB
ejpam-5240	316	13	gs(n	gs(n	NOUN
ejpam-5240	316	14	,	,	PUNCT
ejpam-5240	316	15	k	k	NOUN
ejpam-5240	316	16	)	)	PUNCT
ejpam-5240	316	17	be	be	VERB
ejpam-5240	316	18	a	a	DET
ejpam-5240	316	19	k	k	ADV
ejpam-5240	316	20	-	-	ADJ
ejpam-5240	316	21	restricted	restricted	ADJ
ejpam-5240	316	22	intersection	intersection	NOUN
ejpam-5240	316	23	graph	graph	NOUN
ejpam-5240	316	24	.	.	PUNCT
ejpam-5240	317	1	then	then	ADV
ejpam-5240	317	2	deg(a	deg(a	PROPN
ejpam-5240	317	3	)	)	PUNCT
ejpam-5240	317	4	=	=	PUNCT
ejpam-5240	317	5	(	(	PUNCT
ejpam-5240	317	6	n	n	X
ejpam-5240	317	7	k	k	NOUN
ejpam-5240	317	8	)	)	PUNCT
ejpam-5240	318	1	−	−	ADP
ejpam-5240	318	2	1	1	NUM
ejpam-5240	318	3	for	for	ADP
ejpam-5240	318	4	all	all	DET
ejpam-5240	318	5	a	a	DET
ejpam-5240	318	6	∈	∈	PROPN
ejpam-5240	318	7	v	v	NOUN
ejpam-5240	318	8	(	(	PUNCT
ejpam-5240	318	9	gs(n	gs(n	NOUN
ejpam-5240	318	10	,	,	PUNCT
ejpam-5240	318	11	k	k	NOUN
ejpam-5240	318	12	)	)	PUNCT
ejpam-5240	318	13	)	)	PUNCT
ejpam-5240	319	1	if	if	SCONJ
ejpam-5240	319	2	and	and	CCONJ
ejpam-5240	319	3	only	only	ADV
ejpam-5240	319	4	if	if	SCONJ
ejpam-5240	319	5	k	k	PROPN
ejpam-5240	319	6	=	=	PUNCT
ejpam-5240	319	7	0	0	NUM
ejpam-5240	319	8	or	or	CCONJ
ejpam-5240	319	9	⌊	⌊	X
ejpam-5240	319	10	n	n	ADV
ejpam-5240	319	11	2	2	NUM
ejpam-5240	319	12	⌋	⌋	NOUN
ejpam-5240	319	13	<	<	X
ejpam-5240	319	14	k	k	PROPN
ejpam-5240	319	15	≤	≤	PROPN
ejpam-5240	319	16	n.	n.	NOUN
ejpam-5240	319	17	proof	proof	NOUN
ejpam-5240	319	18	.	.	PUNCT
ejpam-5240	320	1	assume	assume	VERB
ejpam-5240	320	2	that	that	SCONJ
ejpam-5240	320	3	deg(a	deg(a	PROPN
ejpam-5240	320	4	)	)	PUNCT
ejpam-5240	320	5	=	=	PUNCT
ejpam-5240	321	1	(	(	PUNCT
ejpam-5240	321	2	n	n	X
ejpam-5240	321	3	k	k	NOUN
ejpam-5240	321	4	)	)	PUNCT
ejpam-5240	321	5	−1	−1	NOUN
ejpam-5240	321	6	for	for	ADP
ejpam-5240	321	7	all	all	DET
ejpam-5240	321	8	a	a	DET
ejpam-5240	321	9	in	in	ADP
ejpam-5240	321	10	v	v	NOUN
ejpam-5240	321	11	(	(	PUNCT
ejpam-5240	321	12	gs(n	gs(n	NOUN
ejpam-5240	321	13	,	,	PUNCT
ejpam-5240	321	14	k	k	NOUN
ejpam-5240	321	15	)	)	PUNCT
ejpam-5240	321	16	)	)	PUNCT
ejpam-5240	322	1	and	and	CCONJ
ejpam-5240	322	2	suppose	suppose	VERB
ejpam-5240	322	3	that	that	SCONJ
ejpam-5240	322	4	1	1	NUM
ejpam-5240	322	5	<	<	X
ejpam-5240	322	6	k	k	PROPN
ejpam-5240	322	7	≤⌊	≤⌊	PROPN
ejpam-5240	322	8	n	n	NUM
ejpam-5240	322	9	2	2	NUM
ejpam-5240	322	10	⌋	⌋	NOUN
ejpam-5240	322	11	.	.	PUNCT
ejpam-5240	323	1	by	by	ADP
ejpam-5240	323	2	lemma	lemma	PROPN
ejpam-5240	323	3	1	1	NUM
ejpam-5240	323	4	,	,	PUNCT
ejpam-5240	323	5	since	since	SCONJ
ejpam-5240	323	6	deg(a	deg(a	PROPN
ejpam-5240	323	7	)	)	PUNCT
ejpam-5240	323	8	=	=	PUNCT
ejpam-5240	323	9	(	(	PUNCT
ejpam-5240	323	10	n	n	X
ejpam-5240	323	11	k	k	NOUN
ejpam-5240	323	12	)	)	PUNCT
ejpam-5240	323	13	−	−	PROPN
ejpam-5240	324	1	[	[	X
ejpam-5240	324	2	(	(	PUNCT
ejpam-5240	324	3	n−k	n−k	NOUN
ejpam-5240	324	4	k	k	PROPN
ejpam-5240	324	5	)	)	PUNCT
ejpam-5240	325	1	+	+	CCONJ
ejpam-5240	325	2	1	1	X
ejpam-5240	325	3	]	]	PUNCT
ejpam-5240	325	4	,	,	PUNCT
ejpam-5240	325	5	it	it	PRON
ejpam-5240	325	6	follows	follow	VERB
ejpam-5240	325	7	that	that	SCONJ
ejpam-5240	325	8	(	(	PUNCT
ejpam-5240	325	9	n	n	X
ejpam-5240	325	10	k	k	NOUN
ejpam-5240	325	11	)	)	PUNCT
ejpam-5240	325	12	−	−	PROPN
ejpam-5240	326	1	[	[	X
ejpam-5240	326	2	(	(	PUNCT
ejpam-5240	326	3	n−k	n−k	NOUN
ejpam-5240	326	4	k	k	PROPN
ejpam-5240	326	5	)	)	PUNCT
ejpam-5240	327	1	+	+	CCONJ
ejpam-5240	327	2	1	1	X
ejpam-5240	327	3	]	]	PUNCT
ejpam-5240	327	4	=	=	SYM
ejpam-5240	327	5	(	(	PUNCT
ejpam-5240	327	6	n	n	X
ejpam-5240	327	7	k	k	NOUN
ejpam-5240	327	8	)	)	PUNCT
ejpam-5240	327	9	−	−	PROPN
ejpam-5240	328	1	1	1	X
ejpam-5240	328	2	.	.	PUNCT
ejpam-5240	329	1	this	this	PRON
ejpam-5240	329	2	implies	imply	VERB
ejpam-5240	329	3	that	that	SCONJ
ejpam-5240	329	4	−	−	PROPN
ejpam-5240	329	5	(	(	PUNCT
ejpam-5240	329	6	n−k	n−k	NOUN
ejpam-5240	329	7	k	k	NOUN
ejpam-5240	329	8	)	)	PUNCT
ejpam-5240	329	9	=	=	SYM
ejpam-5240	329	10	0	0	NUM
ejpam-5240	329	11	which	which	PRON
ejpam-5240	329	12	can	can	AUX
ejpam-5240	329	13	be	be	AUX
ejpam-5240	329	14	equated	equate	VERB
ejpam-5240	329	15	to	to	ADP
ejpam-5240	329	16	(	(	PUNCT
ejpam-5240	329	17	n−k	n−k	NOUN
ejpam-5240	329	18	k	k	X
ejpam-5240	329	19	)	)	PUNCT
ejpam-5240	330	1	=	=	PUNCT
ejpam-5240	330	2	0	0	X
ejpam-5240	330	3	.	.	PUNCT
ejpam-5240	331	1	the	the	DET
ejpam-5240	331	2	only	only	ADJ
ejpam-5240	331	3	time	time	NOUN
ejpam-5240	331	4	that	that	SCONJ
ejpam-5240	331	5	(	(	PUNCT
ejpam-5240	331	6	n−k	n−k	NOUN
ejpam-5240	331	7	k	k	X
ejpam-5240	331	8	)	)	PUNCT
ejpam-5240	332	1	=	=	SYM
ejpam-5240	332	2	0	0	NUM
ejpam-5240	332	3	is	be	AUX
ejpam-5240	332	4	when	when	SCONJ
ejpam-5240	332	5	n	n	X
ejpam-5240	332	6	−	−	PROPN
ejpam-5240	332	7	k	k	X
ejpam-5240	332	8	<	<	X
ejpam-5240	332	9	k.	k.	PROPN
ejpam-5240	332	10	note	note	VERB
ejpam-5240	332	11	that	that	SCONJ
ejpam-5240	332	12	if	if	SCONJ
ejpam-5240	332	13	1	1	NUM
ejpam-5240	332	14	<	<	X
ejpam-5240	332	15	k	k	X
ejpam-5240	332	16	≤	≤	NUM
ejpam-5240	332	17	⌊	⌊	VERB
ejpam-5240	332	18	n	n	DET
ejpam-5240	332	19	2	2	NUM
ejpam-5240	332	20	⌋	⌋	NOUN
ejpam-5240	332	21	,	,	PUNCT
ejpam-5240	332	22	then	then	ADV
ejpam-5240	332	23	⌊	⌊	VERB
ejpam-5240	332	24	n	n	ADV
ejpam-5240	332	25	2	2	NUM
ejpam-5240	332	26	⌋	⌋	NOUN
ejpam-5240	332	27	≤	≤	NOUN
ejpam-5240	332	28	n	n	PRON
ejpam-5240	332	29	−	−	PROPN
ejpam-5240	332	30	k.	k.	PROPN
ejpam-5240	333	1	so	so	ADV
ejpam-5240	333	2	k	k	PROPN
ejpam-5240	333	3	≤	≤	PROPN
ejpam-5240	333	4	n	n	PRON
ejpam-5240	333	5	−	−	PROPN
ejpam-5240	334	1	k.	k.	PROPN
ejpam-5240	335	1	this	this	PRON
ejpam-5240	335	2	is	be	AUX
ejpam-5240	335	3	a	a	DET
ejpam-5240	335	4	contradiction	contradiction	NOUN
ejpam-5240	335	5	to	to	ADP
ejpam-5240	335	6	the	the	DET
ejpam-5240	335	7	fact	fact	NOUN
ejpam-5240	335	8	that	that	SCONJ
ejpam-5240	335	9	n	n	NOUN
ejpam-5240	335	10	−	−	PROPN
ejpam-5240	335	11	k	k	X
ejpam-5240	335	12	<	<	X
ejpam-5240	335	13	k.	k.	PROPN
ejpam-5240	336	1	therefore	therefore	ADV
ejpam-5240	336	2	,	,	PUNCT
ejpam-5240	336	3	k	k	PROPN
ejpam-5240	336	4	=	=	PUNCT
ejpam-5240	336	5	0	0	NUM
ejpam-5240	336	6	or	or	CCONJ
ejpam-5240	336	7	⌊	⌊	X
ejpam-5240	336	8	n	n	ADV
ejpam-5240	336	9	2	2	NUM
ejpam-5240	336	10	⌋	⌋	NOUN
ejpam-5240	336	11	<	<	X
ejpam-5240	336	12	k	k	PROPN
ejpam-5240	336	13	≤	≤	PROPN
ejpam-5240	336	14	n.	n.	NOUN
ejpam-5240	336	15	conversely	conversely	ADV
ejpam-5240	336	16	,	,	PUNCT
ejpam-5240	336	17	assume	assume	VERB
ejpam-5240	336	18	that	that	SCONJ
ejpam-5240	336	19	k	k	PROPN
ejpam-5240	336	20	=	=	PUNCT
ejpam-5240	336	21	0	0	NUM
ejpam-5240	336	22	or	or	CCONJ
ejpam-5240	336	23	⌊	⌊	X
ejpam-5240	336	24	n	n	ADV
ejpam-5240	336	25	2	2	NUM
ejpam-5240	336	26	⌋	⌋	NOUN
ejpam-5240	336	27	<	<	X
ejpam-5240	336	28	k	k	PROPN
ejpam-5240	336	29	≤	≤	PROPN
ejpam-5240	336	30	n.	n.	NOUN
ejpam-5240	336	31	if	if	SCONJ
ejpam-5240	336	32	k	k	PROPN
ejpam-5240	336	33	is	be	AUX
ejpam-5240	336	34	0	0	NUM
ejpam-5240	336	35	or	or	CCONJ
ejpam-5240	336	36	n	n	CCONJ
ejpam-5240	336	37	,	,	PUNCT
ejpam-5240	336	38	by	by	ADP
ejpam-5240	336	39	theorem	theorem	NOUN
ejpam-5240	336	40	3	3	NUM
ejpam-5240	336	41	,	,	PUNCT
ejpam-5240	336	42	gs(n	gs(n	NOUN
ejpam-5240	336	43	,	,	PUNCT
ejpam-5240	336	44	k	k	NOUN
ejpam-5240	336	45	)	)	PUNCT
ejpam-5240	336	46	is	be	AUX
ejpam-5240	336	47	a	a	DET
ejpam-5240	336	48	trivial	trivial	ADJ
ejpam-5240	336	49	graph	graph	NOUN
ejpam-5240	336	50	.	.	PUNCT
ejpam-5240	337	1	since	since	SCONJ
ejpam-5240	337	2	|v	|v	PROPN
ejpam-5240	337	3	(	(	PUNCT
ejpam-5240	337	4	gs(n	gs(n	NOUN
ejpam-5240	337	5	,	,	PUNCT
ejpam-5240	337	6	k	k	NOUN
ejpam-5240	337	7	)	)	PUNCT
ejpam-5240	337	8	)	)	PUNCT
ejpam-5240	338	1	|	|	ADV
ejpam-5240	338	2	=	=	SYM
ejpam-5240	338	3	(	(	PUNCT
ejpam-5240	338	4	n	n	X
ejpam-5240	338	5	k	k	NOUN
ejpam-5240	338	6	)	)	PUNCT
ejpam-5240	338	7	=	=	SYM
ejpam-5240	338	8	1	1	NUM
ejpam-5240	338	9	and	and	CCONJ
ejpam-5240	338	10	the	the	DET
ejpam-5240	338	11	degree	degree	NOUN
ejpam-5240	338	12	of	of	ADP
ejpam-5240	338	13	the	the	DET
ejpam-5240	338	14	vertex	vertex	NOUN
ejpam-5240	338	15	of	of	ADP
ejpam-5240	338	16	a	a	DET
ejpam-5240	338	17	trivial	trivial	ADJ
ejpam-5240	338	18	graph	graph	NOUN
ejpam-5240	338	19	is	be	AUX
ejpam-5240	338	20	0	0	NUM
ejpam-5240	338	21	,	,	PUNCT
ejpam-5240	338	22	it	it	PRON
ejpam-5240	338	23	follows	follow	VERB
ejpam-5240	338	24	that	that	SCONJ
ejpam-5240	338	25	deg(a	deg(a	PROPN
ejpam-5240	338	26	)	)	PUNCT
ejpam-5240	338	27	=	=	SYM
ejpam-5240	338	28	0	0	PUNCT
ejpam-5240	339	1	=	=	SYM
ejpam-5240	339	2	1	1	NUM
ejpam-5240	339	3	−	−	NUM
ejpam-5240	339	4	1	1	NUM
ejpam-5240	339	5	=	=	SYM
ejpam-5240	339	6	(	(	PUNCT
ejpam-5240	339	7	n	n	X
ejpam-5240	339	8	k	k	NOUN
ejpam-5240	339	9	)	)	PUNCT
ejpam-5240	340	1	−	−	PROPN
ejpam-5240	340	2	1	1	NUM
ejpam-5240	340	3	where	where	SCONJ
ejpam-5240	340	4	a	a	DET
ejpam-5240	340	5	∈	∈	PROPN
ejpam-5240	340	6	v	v	NOUN
ejpam-5240	340	7	(	(	PUNCT
ejpam-5240	340	8	gs(n	gs(n	NOUN
ejpam-5240	340	9	,	,	PUNCT
ejpam-5240	340	10	k	k	NOUN
ejpam-5240	340	11	)	)	PUNCT
ejpam-5240	340	12	)	)	PUNCT
ejpam-5240	340	13	.	.	PUNCT
ejpam-5240	341	1	now	now	ADV
ejpam-5240	341	2	,	,	PUNCT
ejpam-5240	341	3	let	let	VERB
ejpam-5240	341	4	⌊	⌊	VERB
ejpam-5240	341	5	n	n	ADV
ejpam-5240	341	6	2	2	NUM
ejpam-5240	341	7	⌋	⌋	NOUN
ejpam-5240	341	8	<	<	X
ejpam-5240	341	9	k	k	X
ejpam-5240	341	10	<	<	X
ejpam-5240	341	11	n	n	PROPN
ejpam-5240	341	12	and	and	CCONJ
ejpam-5240	341	13	let	let	VERB
ejpam-5240	341	14	a	a	PRON
ejpam-5240	341	15	be	be	AUX
ejpam-5240	341	16	an	an	DET
ejpam-5240	341	17	arbitrary	arbitrary	ADJ
ejpam-5240	341	18	vertex	vertex	NOUN
ejpam-5240	341	19	of	of	ADP
ejpam-5240	341	20	gs(n	gs(n	NOUN
ejpam-5240	341	21	,	,	PUNCT
ejpam-5240	341	22	k	k	NOUN
ejpam-5240	341	23	)	)	PUNCT
ejpam-5240	341	24	.	.	PUNCT
ejpam-5240	342	1	since	since	SCONJ
ejpam-5240	342	2	a	a	DET
ejpam-5240	342	3	⊆	⊆	NUM
ejpam-5240	342	4	sn	sn	NOUN
ejpam-5240	342	5	,	,	PUNCT
ejpam-5240	342	6	then	then	ADV
ejpam-5240	342	7	by	by	ADP
ejpam-5240	342	8	remark	remark	NOUN
ejpam-5240	342	9	1	1	NUM
ejpam-5240	342	10	,	,	PUNCT
ejpam-5240	342	11	it	it	PRON
ejpam-5240	342	12	follows	follow	VERB
ejpam-5240	342	13	that	that	SCONJ
ejpam-5240	342	14	|sn	|sn	NUM
ejpam-5240	342	15	\a|	\a|	NOUN
ejpam-5240	342	16	=	=	NOUN
ejpam-5240	342	17	n−k	n−k	NOUN
ejpam-5240	342	18	.	.	PUNCT
ejpam-5240	343	1	now	now	ADV
ejpam-5240	343	2	,	,	PUNCT
ejpam-5240	343	3	consider	consider	VERB
ejpam-5240	343	4	the	the	DET
ejpam-5240	343	5	set	set	NOUN
ejpam-5240	343	6	of	of	ADP
ejpam-5240	343	7	all	all	DET
ejpam-5240	343	8	k	k	ADJ
ejpam-5240	343	9	-	-	ADJ
ejpam-5240	343	10	element	element	ADJ
ejpam-5240	343	11	subsets	subset	NOUN
ejpam-5240	343	12	of	of	ADP
ejpam-5240	343	13	sn	sn	PROPN
ejpam-5240	343	14	\	\	PROPN
ejpam-5240	343	15	a	a	PRON
ejpam-5240	343	16	,	,	PUNCT
ejpam-5240	343	17	denoted	denote	VERB
ejpam-5240	343	18	by	by	ADP
ejpam-5240	343	19	s(n−k	s(n−k	PROPN
ejpam-5240	343	20	,	,	PUNCT
ejpam-5240	343	21	k	k	NOUN
ejpam-5240	343	22	)	)	PUNCT
ejpam-5240	343	23	.	.	PUNCT
ejpam-5240	344	1	observe	observe	VERB
ejpam-5240	344	2	that	that	SCONJ
ejpam-5240	344	3	s(n−k	s(n−k	NOUN
ejpam-5240	344	4	,	,	PUNCT
ejpam-5240	344	5	k	k	NOUN
ejpam-5240	344	6	)	)	PUNCT
ejpam-5240	344	7	⊆	⊆	NUM
ejpam-5240	344	8	v	v	NOUN
ejpam-5240	344	9	(	(	PUNCT
ejpam-5240	344	10	gs(n	gs(n	NOUN
ejpam-5240	344	11	,	,	PUNCT
ejpam-5240	344	12	k	k	NOUN
ejpam-5240	344	13	)	)	PUNCT
ejpam-5240	344	14	)	)	PUNCT
ejpam-5240	344	15	.	.	PUNCT
ejpam-5240	345	1	if	if	SCONJ
ejpam-5240	345	2	⌊	⌊	PROPN
ejpam-5240	345	3	n	n	ADV
ejpam-5240	345	4	2	2	NUM
ejpam-5240	345	5	⌋	⌋	NOUN
ejpam-5240	345	6	<	<	X
ejpam-5240	345	7	k	k	X
ejpam-5240	345	8	<	<	X
ejpam-5240	345	9	n	n	CCONJ
ejpam-5240	345	10	,	,	PUNCT
ejpam-5240	345	11	then	then	ADV
ejpam-5240	345	12	n	n	CCONJ
ejpam-5240	345	13	−	−	PROPN
ejpam-5240	345	14	k	k	NOUN
ejpam-5240	345	15	≤	≤	NUM
ejpam-5240	345	16	⌊	⌊	VERB
ejpam-5240	345	17	n	n	ADV
ejpam-5240	345	18	2	2	NUM
ejpam-5240	345	19	⌋	⌋	NOUN
ejpam-5240	345	20	which	which	PRON
ejpam-5240	345	21	means	mean	VERB
ejpam-5240	345	22	that	that	SCONJ
ejpam-5240	345	23	n	n	PROPN
ejpam-5240	345	24	−	−	PROPN
ejpam-5240	345	25	k	k	X
ejpam-5240	345	26	<	<	X
ejpam-5240	345	27	k.	k.	PROPN
ejpam-5240	345	28	thus	thus	ADV
ejpam-5240	345	29	by	by	ADP
ejpam-5240	345	30	remark	remark	NOUN
ejpam-5240	345	31	4	4	NUM
ejpam-5240	345	32	,	,	PUNCT
ejpam-5240	345	33	|s(n−k	|s(n−k	NOUN
ejpam-5240	345	34	,	,	PUNCT
ejpam-5240	345	35	k)|	k)|	NOUN
ejpam-5240	345	36	=	=	PUNCT
ejpam-5240	345	37	(	(	PUNCT
ejpam-5240	345	38	n−k	n−k	NOUN
ejpam-5240	345	39	k	k	X
ejpam-5240	345	40	)	)	PUNCT
ejpam-5240	346	1	=	=	SYM
ejpam-5240	346	2	0	0	X
ejpam-5240	346	3	.	.	X
ejpam-5240	346	4	note	note	VERB
ejpam-5240	346	5	that	that	SCONJ
ejpam-5240	346	6	a	a	DET
ejpam-5240	346	7	∩	∩	NOUN
ejpam-5240	346	8	(	(	PUNCT
ejpam-5240	346	9	sn	sn	ADV
ejpam-5240	346	10	\a	\a	NUM
ejpam-5240	346	11	)	)	PUNCT
ejpam-5240	347	1	=	=	NOUN
ejpam-5240	347	2	∅	∅	NOUN
ejpam-5240	347	3	,	,	PUNCT
ejpam-5240	347	4	so	so	ADV
ejpam-5240	347	5	for	for	ADP
ejpam-5240	347	6	every	every	DET
ejpam-5240	347	7	b	b	PROPN
ejpam-5240	347	8	∈	∈	PROPN
ejpam-5240	347	9	s(n−k	s(n−k	NOUN
ejpam-5240	347	10	,	,	PUNCT
ejpam-5240	347	11	k	k	NOUN
ejpam-5240	347	12	)	)	PUNCT
ejpam-5240	347	13	,	,	PUNCT
ejpam-5240	347	14	a	a	DET
ejpam-5240	347	15	∩	∩	ADJ
ejpam-5240	347	16	b	b	NOUN
ejpam-5240	347	17	=	=	SYM
ejpam-5240	347	18	∅.	∅.	NOUN
ejpam-5240	347	19	but	but	CCONJ
ejpam-5240	347	20	s(n−k	s(n−k	PROPN
ejpam-5240	347	21	,	,	PUNCT
ejpam-5240	347	22	k	k	NOUN
ejpam-5240	347	23	)	)	PUNCT
ejpam-5240	347	24	=	=	NOUN
ejpam-5240	347	25	∅	∅	NOUN
ejpam-5240	347	26	,	,	PUNCT
ejpam-5240	347	27	so	so	ADV
ejpam-5240	347	28	given	give	VERB
ejpam-5240	347	29	that	that	SCONJ
ejpam-5240	347	30	|v	|v	NOUN
ejpam-5240	347	31	(	(	PUNCT
ejpam-5240	347	32	gs(n	gs(n	NOUN
ejpam-5240	347	33	,	,	PUNCT
ejpam-5240	347	34	k	k	NOUN
ejpam-5240	347	35	)	)	PUNCT
ejpam-5240	347	36	)	)	PUNCT
ejpam-5240	348	1	|	|	ADV
ejpam-5240	348	2	=	=	SYM
ejpam-5240	348	3	(	(	PUNCT
ejpam-5240	348	4	n	n	X
ejpam-5240	348	5	k	k	PROPN
ejpam-5240	348	6	)	)	PUNCT
ejpam-5240	348	7	,	,	PUNCT
ejpam-5240	348	8	deg(a	deg(a	PROPN
ejpam-5240	348	9	)	)	PUNCT
ejpam-5240	348	10	=	=	PUNCT
ejpam-5240	348	11	(	(	PUNCT
ejpam-5240	348	12	n	n	X
ejpam-5240	348	13	k	k	NOUN
ejpam-5240	348	14	)	)	PUNCT
ejpam-5240	349	1	−	−	PROPN
ejpam-5240	349	2	0	0	X
ejpam-5240	349	3	.	.	PUNCT
ejpam-5240	350	1	also	also	ADV
ejpam-5240	350	2	,	,	PUNCT
ejpam-5240	350	3	since	since	SCONJ
ejpam-5240	350	4	[	[	X
ejpam-5240	350	5	a	a	X
ejpam-5240	350	6	,	,	PUNCT
ejpam-5240	350	7	a	a	PRON
ejpam-5240	350	8	]	]	X
ejpam-5240	350	9	/∈	/∈	PUNCT
ejpam-5240	350	10	e(gs(n	e(gs(n	NOUN
ejpam-5240	350	11	,	,	PUNCT
ejpam-5240	350	12	k	k	NOUN
ejpam-5240	350	13	)	)	PUNCT
ejpam-5240	350	14	)	)	PUNCT
ejpam-5240	350	15	it	it	PRON
ejpam-5240	350	16	follows	follow	VERB
ejpam-5240	350	17	that	that	SCONJ
ejpam-5240	350	18	deg(a	deg(a	PROPN
ejpam-5240	350	19	)	)	PUNCT
ejpam-5240	350	20	=	=	PUNCT
ejpam-5240	351	1	(	(	PUNCT
ejpam-5240	351	2	n	n	X
ejpam-5240	351	3	k	k	NOUN
ejpam-5240	351	4	)	)	PUNCT
ejpam-5240	352	1	−	−	PROPN
ejpam-5240	353	1	1	1	X
ejpam-5240	353	2	.	.	PUNCT
ejpam-5240	353	3	since	since	SCONJ
ejpam-5240	353	4	a	a	PRON
ejpam-5240	353	5	was	be	AUX
ejpam-5240	353	6	chosen	choose	VERB
ejpam-5240	353	7	arbitrarily	arbitrarily	ADV
ejpam-5240	353	8	,	,	PUNCT
ejpam-5240	353	9	it	it	PRON
ejpam-5240	353	10	follows	follow	VERB
ejpam-5240	353	11	that	that	SCONJ
ejpam-5240	353	12	every	every	DET
ejpam-5240	353	13	vertex	vertex	NOUN
ejpam-5240	353	14	of	of	ADP
ejpam-5240	353	15	gs(n	gs(n	NOUN
ejpam-5240	353	16	,	,	PUNCT
ejpam-5240	353	17	k	k	NOUN
ejpam-5240	353	18	)	)	PUNCT
ejpam-5240	353	19	,	,	PUNCT
ejpam-5240	353	20	with	with	ADP
ejpam-5240	353	21	⌊	⌊	PROPN
ejpam-5240	353	22	n	n	ADV
ejpam-5240	353	23	2	2	NUM
ejpam-5240	353	24	⌋	⌋	NOUN
ejpam-5240	353	25	<	<	X
ejpam-5240	353	26	k	k	X
ejpam-5240	353	27	<	<	X
ejpam-5240	353	28	n	n	CCONJ
ejpam-5240	353	29	,	,	PUNCT
ejpam-5240	353	30	has	have	VERB
ejpam-5240	353	31	a	a	DET
ejpam-5240	353	32	degree	degree	NOUN
ejpam-5240	353	33	equal	equal	ADJ
ejpam-5240	353	34	to	to	ADP
ejpam-5240	353	35	(	(	PUNCT
ejpam-5240	353	36	n	n	X
ejpam-5240	353	37	k	k	NOUN
ejpam-5240	353	38	)	)	PUNCT
ejpam-5240	354	1	−	−	PROPN
ejpam-5240	355	1	1	1	X
ejpam-5240	355	2	.	.	X
ejpam-5240	355	3	illustration	illustration	NOUN
ejpam-5240	355	4	5	5	NUM
ejpam-5240	355	5	provides	provide	VERB
ejpam-5240	355	6	an	an	DET
ejpam-5240	355	7	example	example	NOUN
ejpam-5240	355	8	for	for	ADP
ejpam-5240	355	9	the	the	DET
ejpam-5240	355	10	degree	degree	NOUN
ejpam-5240	355	11	of	of	ADP
ejpam-5240	355	12	every	every	DET
ejpam-5240	355	13	vertex	vertex	NOUN
ejpam-5240	355	14	of	of	ADP
ejpam-5240	355	15	a	a	DET
ejpam-5240	355	16	gs(n	gs(n	NOUN
ejpam-5240	355	17	,	,	PUNCT
ejpam-5240	355	18	k	k	NOUN
ejpam-5240	355	19	)	)	PUNCT
ejpam-5240	355	20	when	when	SCONJ
ejpam-5240	355	21	k	k	PROPN
ejpam-5240	355	22	=	=	PUNCT
ejpam-5240	355	23	0	0	NUM
ejpam-5240	355	24	or	or	CCONJ
ejpam-5240	355	25	⌊	⌊	X
ejpam-5240	355	26	n	n	ADV
ejpam-5240	355	27	2	2	NUM
ejpam-5240	355	28	⌋	⌋	NOUN
ejpam-5240	355	29	<	<	X
ejpam-5240	355	30	k	k	X
ejpam-5240	355	31	≤	≤	PROPN
ejpam-5240	355	32	n	n	CCONJ
ejpam-5240	355	33	with	with	ADP
ejpam-5240	355	34	n	n	NOUN
ejpam-5240	355	35	=	=	SYM
ejpam-5240	355	36	4	4	NUM
ejpam-5240	355	37	and	and	CCONJ
ejpam-5240	355	38	k	k	PROPN
ejpam-5240	355	39	is	be	AUX
ejpam-5240	355	40	0	0	NUM
ejpam-5240	355	41	,	,	PUNCT
ejpam-5240	355	42	4	4	NUM
ejpam-5240	355	43	and	and	CCONJ
ejpam-5240	355	44	3	3	NUM
ejpam-5240	355	45	.	.	X
ejpam-5240	355	46	illustration	illustration	NOUN
ejpam-5240	355	47	5	5	NUM
ejpam-5240	355	48	.	.	PUNCT
ejpam-5240	355	49	consider	consider	VERB
ejpam-5240	355	50	s4	s4	PROPN
ejpam-5240	355	51	=	=	SYM
ejpam-5240	355	52	{	{	PUNCT
ejpam-5240	355	53	x1	x1	PROPN
ejpam-5240	355	54	,	,	PUNCT
ejpam-5240	355	55	x2	x2	PROPN
ejpam-5240	355	56	,	,	PUNCT
ejpam-5240	355	57	x3	x3	ADJ
ejpam-5240	355	58	,	,	PUNCT
ejpam-5240	355	59	x4	x4	PROPN
ejpam-5240	355	60	}	}	PUNCT
ejpam-5240	355	61	.	.	PUNCT
ejpam-5240	356	1	to	to	PART
ejpam-5240	356	2	show	show	VERB
ejpam-5240	356	3	an	an	DET
ejpam-5240	356	4	illustration	illustration	NOUN
ejpam-5240	356	5	for	for	ADP
ejpam-5240	356	6	the	the	DET
ejpam-5240	356	7	case	case	NOUN
ejpam-5240	356	8	k	k	NOUN
ejpam-5240	356	9	=	=	PUNCT
ejpam-5240	356	10	0	0	NUM
ejpam-5240	356	11	or	or	CCONJ
ejpam-5240	356	12	⌊	⌊	X
ejpam-5240	356	13	n	n	ADV
ejpam-5240	356	14	2	2	NUM
ejpam-5240	356	15	⌋	⌋	NOUN
ejpam-5240	356	16	<	<	X
ejpam-5240	356	17	k	k	X
ejpam-5240	356	18	≤	≤	PROPN
ejpam-5240	357	1	n	n	CCONJ
ejpam-5240	357	2	,	,	PUNCT
ejpam-5240	357	3	we	we	PRON
ejpam-5240	357	4	have	have	VERB
ejpam-5240	357	5	the	the	DET
ejpam-5240	357	6	following	following	NOUN
ejpam-5240	357	7	:	:	PUNCT
ejpam-5240	357	8	let	let	VERB
ejpam-5240	357	9	k	k	PROPN
ejpam-5240	357	10	=	=	PUNCT
ejpam-5240	357	11	0	0	PUNCT
ejpam-5240	357	12	and	and	CCONJ
ejpam-5240	357	13	consider	consider	VERB
ejpam-5240	357	14	the	the	DET
ejpam-5240	357	15	pictorial	pictorial	ADJ
ejpam-5240	357	16	illustration	illustration	NOUN
ejpam-5240	357	17	of	of	ADP
ejpam-5240	357	18	gs(4,0	gs(4,0	PROPN
ejpam-5240	357	19	)	)	PUNCT
ejpam-5240	357	20	in	in	ADP
ejpam-5240	357	21	figure	figure	NOUN
ejpam-5240	357	22	3	3	NUM
ejpam-5240	357	23	which	which	PRON
ejpam-5240	357	24	is	be	AUX
ejpam-5240	357	25	a	a	DET
ejpam-5240	357	26	trivial	trivial	ADJ
ejpam-5240	357	27	graph	graph	NOUN
ejpam-5240	357	28	containing	contain	VERB
ejpam-5240	357	29	the	the	DET
ejpam-5240	357	30	vertex	vertex	NOUN
ejpam-5240	357	31	∅.	∅.	NOUN
ejpam-5240	357	32	it	it	PRON
ejpam-5240	357	33	can	can	AUX
ejpam-5240	357	34	be	be	AUX
ejpam-5240	357	35	observed	observe	VERB
ejpam-5240	357	36	that	that	SCONJ
ejpam-5240	357	37	deg(∅	deg(∅	VERB
ejpam-5240	357	38	)	)	PUNCT
ejpam-5240	357	39	=	=	SYM
ejpam-5240	358	1	0	0	X
ejpam-5240	358	2	.	.	PUNCT
ejpam-5240	358	3	to	to	PART
ejpam-5240	358	4	verify	verify	VERB
ejpam-5240	358	5	using	use	VERB
ejpam-5240	358	6	lemma	lemma	PROPN
ejpam-5240	358	7	2	2	NUM
ejpam-5240	358	8	with	with	ADP
ejpam-5240	358	9	n	n	NOUN
ejpam-5240	358	10	=	=	SYM
ejpam-5240	358	11	4	4	NUM
ejpam-5240	358	12	and	and	CCONJ
ejpam-5240	358	13	k	k	NOUN
ejpam-5240	358	14	=	=	SYM
ejpam-5240	358	15	0	0	PROPN
ejpam-5240	358	16	,	,	PUNCT
ejpam-5240	358	17	we	we	PRON
ejpam-5240	358	18	have	have	VERB
ejpam-5240	358	19	deg(∅	deg(∅	NOUN
ejpam-5240	358	20	)	)	PUNCT
ejpam-5240	359	1	=	=	PUNCT
ejpam-5240	359	2	(	(	PUNCT
ejpam-5240	359	3	4	4	NUM
ejpam-5240	359	4	0	0	NUM
ejpam-5240	359	5	)	)	PUNCT
ejpam-5240	359	6	−	−	PROPN
ejpam-5240	360	1	1	1	NUM
ejpam-5240	360	2	=	=	SYM
ejpam-5240	360	3	1−	1−	NUM
ejpam-5240	360	4	1	1	NUM
ejpam-5240	360	5	=	=	SYM
ejpam-5240	360	6	0	0	NUM
ejpam-5240	360	7	.	.	PUNCT
ejpam-5240	361	1	moreover	moreover	ADV
ejpam-5240	361	2	,	,	PUNCT
ejpam-5240	361	3	let	let	VERB
ejpam-5240	361	4	k	k	NOUN
ejpam-5240	361	5	=	=	NOUN
ejpam-5240	361	6	4	4	X
ejpam-5240	361	7	.	.	PUNCT
ejpam-5240	361	8	by	by	ADP
ejpam-5240	361	9	theorem	theorem	NOUN
ejpam-5240	361	10	3	3	NUM
ejpam-5240	361	11	,	,	PUNCT
ejpam-5240	361	12	gs(4,4	gs(4,4	X
ejpam-5240	361	13	)	)	PUNCT
ejpam-5240	361	14	is	be	AUX
ejpam-5240	361	15	a	a	DET
ejpam-5240	361	16	trivial	trivial	ADJ
ejpam-5240	361	17	graph	graph	NOUN
ejpam-5240	361	18	.	.	PUNCT
ejpam-5240	362	1	a	a	DET
ejpam-5240	362	2	pictorial	pictorial	ADJ
ejpam-5240	362	3	representation	representation	NOUN
ejpam-5240	362	4	of	of	ADP
ejpam-5240	362	5	gs(4,4	gs(4,4	PROPN
ejpam-5240	362	6	)	)	PUNCT
ejpam-5240	362	7	with	with	ADP
ejpam-5240	362	8	a	a	DET
ejpam-5240	362	9	vertex	vertex	NOUN
ejpam-5240	362	10	{	{	PUNCT
ejpam-5240	362	11	x1	x1	PROPN
ejpam-5240	362	12	,	,	PUNCT
ejpam-5240	362	13	x2	x2	PROPN
ejpam-5240	362	14	,	,	PUNCT
ejpam-5240	362	15	x3	x3	PROPN
ejpam-5240	362	16	,	,	PUNCT
ejpam-5240	362	17	x4	x4	PROPN
ejpam-5240	362	18	}	}	PUNCT
ejpam-5240	362	19	is	be	AUX
ejpam-5240	362	20	shown	show	VERB
ejpam-5240	362	21	in	in	ADP
ejpam-5240	362	22	figure	figure	NOUN
ejpam-5240	362	23	3	3	NUM
ejpam-5240	362	24	.	.	PUNCT
ejpam-5240	363	1	it	it	PRON
ejpam-5240	363	2	can	can	AUX
ejpam-5240	363	3	be	be	AUX
ejpam-5240	363	4	observed	observe	VERB
ejpam-5240	363	5	that	that	SCONJ
ejpam-5240	363	6	deg({x1	deg({x1	ADV
ejpam-5240	363	7	,	,	PUNCT
ejpam-5240	363	8	x2	x2	PROPN
ejpam-5240	363	9	,	,	PUNCT
ejpam-5240	363	10	x3	x3	ADJ
ejpam-5240	363	11	,	,	PUNCT
ejpam-5240	363	12	x4	x4	PROPN
ejpam-5240	363	13	}	}	PUNCT
ejpam-5240	363	14	)	)	PUNCT
ejpam-5240	363	15	=	=	SYM
ejpam-5240	364	1	0	0	X
ejpam-5240	364	2	.	.	PUNCT
ejpam-5240	364	3	to	to	PART
ejpam-5240	364	4	verify	verify	VERB
ejpam-5240	364	5	this	this	PRON
ejpam-5240	364	6	using	use	VERB
ejpam-5240	364	7	lemma	lemma	PROPN
ejpam-5240	364	8	2	2	NUM
ejpam-5240	364	9	,	,	PUNCT
ejpam-5240	364	10	since	since	SCONJ
ejpam-5240	364	11	n	n	NOUN
ejpam-5240	364	12	=	=	SYM
ejpam-5240	364	13	4	4	NUM
ejpam-5240	364	14	and	and	CCONJ
ejpam-5240	364	15	k	k	NOUN
ejpam-5240	364	16	=	=	SYM
ejpam-5240	364	17	4	4	NUM
ejpam-5240	364	18	,	,	PUNCT
ejpam-5240	364	19	it	it	PRON
ejpam-5240	364	20	follows	follow	VERB
ejpam-5240	364	21	that	that	SCONJ
ejpam-5240	364	22	deg({x1	deg({x1	ADV
ejpam-5240	364	23	,	,	PUNCT
ejpam-5240	364	24	x2	x2	PROPN
ejpam-5240	364	25	,	,	PUNCT
ejpam-5240	364	26	x3	x3	ADJ
ejpam-5240	364	27	,	,	PUNCT
ejpam-5240	364	28	x4	x4	PROPN
ejpam-5240	364	29	}	}	PUNCT
ejpam-5240	364	30	)	)	PUNCT
ejpam-5240	364	31	=	=	PUNCT
ejpam-5240	364	32	(	(	PUNCT
ejpam-5240	364	33	4	4	NUM
ejpam-5240	364	34	4	4	NUM
ejpam-5240	364	35	)	)	PUNCT
ejpam-5240	364	36	−	−	PROPN
ejpam-5240	365	1	1	1	NUM
ejpam-5240	365	2	=	=	SYM
ejpam-5240	365	3	1−	1−	NUM
ejpam-5240	365	4	1	1	NUM
ejpam-5240	365	5	=	=	SYM
ejpam-5240	365	6	0	0	NUM
ejpam-5240	365	7	.	.	PUNCT
ejpam-5240	366	1	now	now	ADV
ejpam-5240	366	2	,	,	PUNCT
ejpam-5240	366	3	let	let	VERB
ejpam-5240	366	4	k	k	PROPN
ejpam-5240	366	5	=	=	SYM
ejpam-5240	366	6	3	3	X
ejpam-5240	366	7	.	.	PUNCT
ejpam-5240	366	8	then	then	ADV
ejpam-5240	366	9	gs(4,3	gs(4,3	PROPN
ejpam-5240	366	10	)	)	PUNCT
ejpam-5240	366	11	has	have	VERB
ejpam-5240	366	12	the	the	DET
ejpam-5240	366	13	vertex	vertex	NOUN
ejpam-5240	366	14	set	set	VERB
ejpam-5240	366	15	v	v	NOUN
ejpam-5240	366	16	(	(	PUNCT
ejpam-5240	366	17	gs(4,3	gs(4,3	PROPN
ejpam-5240	366	18	)	)	PUNCT
ejpam-5240	366	19	)	)	PUNCT
ejpam-5240	367	1	=	=	PRON
ejpam-5240	367	2	{	{	PUNCT
ejpam-5240	367	3	{	{	PUNCT
ejpam-5240	367	4	x1	x1	PROPN
ejpam-5240	367	5	,	,	PUNCT
ejpam-5240	367	6	x2	x2	PROPN
ejpam-5240	367	7	,	,	PUNCT
ejpam-5240	367	8	x3	x3	ADJ
ejpam-5240	367	9	}	}	PUNCT
ejpam-5240	367	10	,	,	PUNCT
ejpam-5240	367	11	{	{	PUNCT
ejpam-5240	367	12	x1	x1	PROPN
ejpam-5240	367	13	,	,	PUNCT
ejpam-5240	367	14	x2	x2	PROPN
ejpam-5240	367	15	,	,	PUNCT
ejpam-5240	367	16	x4	x4	PROPN
ejpam-5240	367	17	}	}	PUNCT
ejpam-5240	367	18	,	,	PUNCT
ejpam-5240	367	19	{	{	PUNCT
ejpam-5240	367	20	x1	x1	ADJ
ejpam-5240	367	21	,	,	PUNCT
ejpam-5240	367	22	x3	x3	ADJ
ejpam-5240	367	23	,	,	PUNCT
ejpam-5240	367	24	x4	x4	PROPN
ejpam-5240	367	25	}	}	PUNCT
ejpam-5240	367	26	,	,	PUNCT
ejpam-5240	367	27	{	{	PUNCT
ejpam-5240	367	28	x2	x2	ADJ
ejpam-5240	367	29	,	,	PUNCT
ejpam-5240	367	30	x3	x3	ADJ
ejpam-5240	367	31	,	,	PUNCT
ejpam-5240	367	32	x4	x4	PROPN
ejpam-5240	367	33	}	}	PUNCT
ejpam-5240	367	34	}	}	PUNCT
ejpam-5240	367	35	.	.	PUNCT
ejpam-5240	368	1	a	a	DET
ejpam-5240	368	2	pictorial	pictorial	ADJ
ejpam-5240	368	3	representation	representation	NOUN
ejpam-5240	368	4	of	of	ADP
ejpam-5240	368	5	gs(4,3	gs(4,3	PROPN
ejpam-5240	368	6	)	)	PUNCT
ejpam-5240	368	7	is	be	AUX
ejpam-5240	368	8	illustrated	illustrate	VERB
ejpam-5240	368	9	in	in	ADP
ejpam-5240	368	10	figure	figure	NOUN
ejpam-5240	368	11	6	6	NUM
ejpam-5240	368	12	.	.	PUNCT
ejpam-5240	369	1	it	it	PRON
ejpam-5240	369	2	can	can	AUX
ejpam-5240	369	3	be	be	AUX
ejpam-5240	369	4	observed	observe	VERB
ejpam-5240	369	5	that	that	SCONJ
ejpam-5240	369	6	deg(a	deg(a	PROPN
ejpam-5240	369	7	)	)	PUNCT
ejpam-5240	369	8	=	=	SYM
ejpam-5240	369	9	3	3	NUM
ejpam-5240	369	10	for	for	ADP
ejpam-5240	369	11	all	all	DET
ejpam-5240	369	12	a	a	DET
ejpam-5240	369	13	∈	∈	PROPN
ejpam-5240	369	14	v	v	NOUN
ejpam-5240	369	15	(	(	PUNCT
ejpam-5240	369	16	gs(4,3	gs(4,3	PROPN
ejpam-5240	369	17	)	)	PUNCT
ejpam-5240	369	18	)	)	PUNCT
ejpam-5240	369	19	.	.	PUNCT
ejpam-5240	370	1	note	note	VERB
ejpam-5240	370	2	that	that	SCONJ
ejpam-5240	370	3	k	k	PROPN
ejpam-5240	370	4	=	=	SYM
ejpam-5240	370	5	3	3	NUM
ejpam-5240	370	6	and⌊	and⌊	NOUN
ejpam-5240	370	7	4	4	NUM
ejpam-5240	370	8	2	2	NUM
ejpam-5240	370	9	⌋	⌋	NOUN
ejpam-5240	370	10	=	=	SYM
ejpam-5240	370	11	2	2	NUM
ejpam-5240	370	12	<	<	X
ejpam-5240	370	13	k.	k.	NOUN
ejpam-5240	370	14	to	to	PART
ejpam-5240	370	15	verify	verify	VERB
ejpam-5240	370	16	using	use	VERB
ejpam-5240	370	17	lemma	lemma	PROPN
ejpam-5240	370	18	2	2	NUM
ejpam-5240	370	19	with	with	ADP
ejpam-5240	370	20	n	n	NOUN
ejpam-5240	370	21	=	=	SYM
ejpam-5240	370	22	4	4	NUM
ejpam-5240	370	23	and	and	CCONJ
ejpam-5240	370	24	k	k	NOUN
ejpam-5240	370	25	=	=	SYM
ejpam-5240	370	26	3	3	NUM
ejpam-5240	370	27	,	,	PUNCT
ejpam-5240	370	28	the	the	DET
ejpam-5240	370	29	degree	degree	NOUN
ejpam-5240	370	30	of	of	ADP
ejpam-5240	370	31	each	each	DET
ejpam-5240	370	32	m.e	m.e	PROPN
ejpam-5240	370	33	.	.	PROPN
ejpam-5240	370	34	pelagio	pelagio	PROPN
ejpam-5240	370	35	,	,	PUNCT
ejpam-5240	370	36	n.	n.	NOUN
ejpam-5240	370	37	mame	mame	PROPN
ejpam-5240	370	38	,	,	PUNCT
ejpam-5240	370	39	k.	k.	PROPN
ejpam-5240	370	40	mendoza	mendoza	PROPN
ejpam-5240	370	41	/	/	SYM
ejpam-5240	370	42	eur	eur	PROPN
ejpam-5240	370	43	.	.	PUNCT
ejpam-5240	371	1	j.	j.	PROPN
ejpam-5240	371	2	pure	pure	PROPN
ejpam-5240	371	3	appl	appl	PROPN
ejpam-5240	371	4	.	.	PROPN
ejpam-5240	371	5	math	math	PROPN
ejpam-5240	371	6	,	,	PUNCT
ejpam-5240	371	7	17	17	NUM
ejpam-5240	371	8	(	(	PUNCT
ejpam-5240	371	9	3	3	NUM
ejpam-5240	371	10	)	)	PUNCT
ejpam-5240	371	11	(	(	PUNCT
ejpam-5240	371	12	2024	2024	NUM
ejpam-5240	371	13	)	)	PUNCT
ejpam-5240	371	14	,	,	PUNCT
ejpam-5240	371	15	1779	1779	NUM
ejpam-5240	371	16	-	-	SYM
ejpam-5240	371	17	1803	1803	NUM
ejpam-5240	371	18	1790	1790	NUM
ejpam-5240	371	19	{	{	PUNCT
ejpam-5240	371	20	x1	x1	PROPN
ejpam-5240	371	21	,	,	PUNCT
ejpam-5240	371	22	x2	x2	PROPN
ejpam-5240	371	23	,	,	PUNCT
ejpam-5240	371	24	x3	x3	ADJ
ejpam-5240	371	25	}	}	PUNCT
ejpam-5240	371	26	{	{	PUNCT
ejpam-5240	371	27	x1	x1	PROPN
ejpam-5240	371	28	,	,	PUNCT
ejpam-5240	371	29	x2	x2	PROPN
ejpam-5240	371	30	,	,	PUNCT
ejpam-5240	371	31	x4	x4	PROPN
ejpam-5240	371	32	}	}	PUNCT
ejpam-5240	371	33	{	{	PUNCT
ejpam-5240	371	34	x1	x1	PROPN
ejpam-5240	371	35	,	,	PUNCT
ejpam-5240	371	36	x3	x3	ADJ
ejpam-5240	371	37	,	,	PUNCT
ejpam-5240	371	38	x4	x4	PROPN
ejpam-5240	371	39	}	}	PUNCT
ejpam-5240	371	40	{	{	PUNCT
ejpam-5240	371	41	x2	x2	PROPN
ejpam-5240	371	42	,	,	PUNCT
ejpam-5240	371	43	x3	x3	ADJ
ejpam-5240	371	44	,	,	PUNCT
ejpam-5240	371	45	x4	x4	ADJ
ejpam-5240	371	46	}	}	PUNCT
ejpam-5240	371	47	figure	figure	VERB
ejpam-5240	371	48	6	6	NUM
ejpam-5240	371	49	:	:	PUNCT
ejpam-5240	371	50	a	a	DET
ejpam-5240	371	51	pictorial	pictorial	ADJ
ejpam-5240	371	52	representation	representation	NOUN
ejpam-5240	371	53	of	of	ADP
ejpam-5240	371	54	gs(4,3	gs(4,3	PROPN
ejpam-5240	371	55	)	)	PUNCT
ejpam-5240	371	56	.	.	PUNCT
ejpam-5240	372	1	vertex	vertex	PROPN
ejpam-5240	372	2	a	a	PRON
ejpam-5240	372	3	in	in	ADP
ejpam-5240	372	4	gs(4,3	gs(4,3	NOUN
ejpam-5240	372	5	)	)	PUNCT
ejpam-5240	372	6	is	be	AUX
ejpam-5240	372	7	equal	equal	ADJ
ejpam-5240	372	8	to	to	ADP
ejpam-5240	372	9	deg(a	deg(a	PROPN
ejpam-5240	372	10	)	)	PUNCT
ejpam-5240	372	11	=	=	PUNCT
ejpam-5240	372	12	(	(	PUNCT
ejpam-5240	372	13	n	n	X
ejpam-5240	372	14	k	k	NOUN
ejpam-5240	372	15	)	)	PUNCT
ejpam-5240	373	1	−	−	PROPN
ejpam-5240	374	1	1	1	NUM
ejpam-5240	374	2	=	=	SYM
ejpam-5240	374	3	(	(	PUNCT
ejpam-5240	374	4	4	4	NUM
ejpam-5240	374	5	3	3	NUM
ejpam-5240	374	6	)	)	PUNCT
ejpam-5240	374	7	−	−	PROPN
ejpam-5240	374	8	1	1	NUM
ejpam-5240	374	9	=	=	SYM
ejpam-5240	374	10	4−	4−	NUM
ejpam-5240	374	11	1	1	NUM
ejpam-5240	374	12	=	=	SYM
ejpam-5240	374	13	3	3	X
ejpam-5240	374	14	.	.	PUNCT
ejpam-5240	375	1	lemma	lemma	PROPN
ejpam-5240	375	2	1	1	NUM
ejpam-5240	375	3	and	and	CCONJ
ejpam-5240	375	4	lemma	lemma	PROPN
ejpam-5240	375	5	2	2	NUM
ejpam-5240	375	6	discusses	discuss	VERB
ejpam-5240	375	7	the	the	DET
ejpam-5240	375	8	degree	degree	NOUN
ejpam-5240	375	9	of	of	ADP
ejpam-5240	375	10	every	every	DET
ejpam-5240	375	11	vertex	vertex	NOUN
ejpam-5240	375	12	in	in	ADP
ejpam-5240	375	13	a	a	DET
ejpam-5240	375	14	gs(n	gs(n	NOUN
ejpam-5240	375	15	,	,	PUNCT
ejpam-5240	375	16	k	k	NOUN
ejpam-5240	375	17	)	)	PUNCT
ejpam-5240	375	18	.	.	PUNCT
ejpam-5240	376	1	observe	observe	VERB
ejpam-5240	376	2	that	that	SCONJ
ejpam-5240	376	3	the	the	DET
ejpam-5240	376	4	vertices	vertex	NOUN
ejpam-5240	376	5	in	in	ADP
ejpam-5240	376	6	gs(n	gs(n	NOUN
ejpam-5240	376	7	,	,	PUNCT
ejpam-5240	376	8	k	k	NOUN
ejpam-5240	376	9	)	)	PUNCT
ejpam-5240	376	10	yield	yield	VERB
ejpam-5240	376	11	the	the	DET
ejpam-5240	376	12	same	same	ADJ
ejpam-5240	376	13	degree	degree	NOUN
ejpam-5240	376	14	.	.	PUNCT
ejpam-5240	377	1	by	by	ADP
ejpam-5240	377	2	this	this	PRON
ejpam-5240	377	3	,	,	PUNCT
ejpam-5240	377	4	then	then	ADV
ejpam-5240	377	5	gs(n	gs(n	NOUN
ejpam-5240	377	6	,	,	PUNCT
ejpam-5240	377	7	k	k	NOUN
ejpam-5240	377	8	)	)	PUNCT
ejpam-5240	377	9	is	be	AUX
ejpam-5240	377	10	a	a	DET
ejpam-5240	377	11	regular	regular	ADJ
ejpam-5240	377	12	graph	graph	NOUN
ejpam-5240	377	13	.	.	PUNCT
ejpam-5240	378	1	equivalently	equivalently	ADV
ejpam-5240	378	2	,	,	PUNCT
ejpam-5240	378	3	we	we	PRON
ejpam-5240	378	4	have	have	AUX
ejpam-5240	378	5	theorem	theorem	VERB
ejpam-5240	378	6	5	5	NUM
ejpam-5240	378	7	.	.	PUNCT
ejpam-5240	378	8	theorem	theorem	NOUN
ejpam-5240	378	9	5	5	NUM
ejpam-5240	378	10	.	.	PUNCT
ejpam-5240	379	1	let	let	VERB
ejpam-5240	379	2	gs(n	gs(n	NOUN
ejpam-5240	379	3	,	,	PUNCT
ejpam-5240	379	4	k	k	NOUN
ejpam-5240	379	5	)	)	PUNCT
ejpam-5240	379	6	be	be	VERB
ejpam-5240	379	7	a	a	DET
ejpam-5240	379	8	k	k	ADV
ejpam-5240	379	9	-	-	ADJ
ejpam-5240	379	10	restricted	restricted	ADJ
ejpam-5240	379	11	intersection	intersection	NOUN
ejpam-5240	379	12	graph	graph	NOUN
ejpam-5240	379	13	.	.	PUNCT
ejpam-5240	380	1	then	then	ADV
ejpam-5240	380	2	gs(n	gs(n	NOUN
ejpam-5240	380	3	,	,	PUNCT
ejpam-5240	380	4	k	k	NOUN
ejpam-5240	380	5	)	)	PUNCT
ejpam-5240	380	6	is	be	AUX
ejpam-5240	380	7	an	an	DET
ejpam-5240	380	8	r	r	NOUN
ejpam-5240	380	9	-	-	PUNCT
ejpam-5240	380	10	regular	regular	ADJ
ejpam-5240	380	11	graph	graph	NOUN
ejpam-5240	380	12	where	where	SCONJ
ejpam-5240	380	13	r	r	NOUN
ejpam-5240	380	14	=	=	SYM
ejpam-5240	380	15	{	{	PUNCT
ejpam-5240	380	16	(	(	PUNCT
ejpam-5240	380	17	n	n	X
ejpam-5240	380	18	k	k	NOUN
ejpam-5240	380	19	)	)	PUNCT
ejpam-5240	380	20	−	−	PROPN
ejpam-5240	381	1	[	[	X
ejpam-5240	381	2	(	(	PUNCT
ejpam-5240	381	3	n−k	n−k	NOUN
ejpam-5240	381	4	k	k	PROPN
ejpam-5240	381	5	)	)	PUNCT
ejpam-5240	382	1	+	+	CCONJ
ejpam-5240	382	2	1	1	X
ejpam-5240	382	3	]	]	PUNCT
ejpam-5240	382	4	if	if	SCONJ
ejpam-5240	382	5	and	and	CCONJ
ejpam-5240	382	6	only	only	ADV
ejpam-5240	382	7	if	if	SCONJ
ejpam-5240	382	8	1	1	NUM
ejpam-5240	382	9	≤	≤	NUM
ejpam-5240	382	10	k	k	X
ejpam-5240	382	11	≤	≤	NUM
ejpam-5240	382	12	⌊	⌊	VERB
ejpam-5240	382	13	n	n	DET
ejpam-5240	382	14	2	2	NUM
ejpam-5240	382	15	⌋	⌋	NOUN
ejpam-5240	382	16	;	;	PUNCT
ejpam-5240	382	17	(	(	PUNCT
ejpam-5240	382	18	n	n	X
ejpam-5240	382	19	k	k	NOUN
ejpam-5240	382	20	)	)	PUNCT
ejpam-5240	382	21	−	−	PROPN
ejpam-5240	382	22	1	1	NUM
ejpam-5240	383	1	if	if	SCONJ
ejpam-5240	383	2	and	and	CCONJ
ejpam-5240	383	3	only	only	ADV
ejpam-5240	383	4	if	if	SCONJ
ejpam-5240	383	5	k	k	PROPN
ejpam-5240	383	6	=	=	PUNCT
ejpam-5240	383	7	0	0	NUM
ejpam-5240	383	8	or	or	CCONJ
ejpam-5240	383	9	⌊	⌊	X
ejpam-5240	383	10	n	n	ADV
ejpam-5240	383	11	2	2	NUM
ejpam-5240	383	12	⌋	⌋	NOUN
ejpam-5240	383	13	<	<	X
ejpam-5240	383	14	k	k	PROPN
ejpam-5240	383	15	≤	≤	PROPN
ejpam-5240	383	16	n.	n.	NOUN
ejpam-5240	383	17	proof	proof	NOUN
ejpam-5240	383	18	.	.	PUNCT
ejpam-5240	384	1	this	this	PRON
ejpam-5240	384	2	is	be	AUX
ejpam-5240	384	3	the	the	DET
ejpam-5240	384	4	direct	direct	ADJ
ejpam-5240	384	5	consequence	consequence	NOUN
ejpam-5240	384	6	of	of	ADP
ejpam-5240	384	7	lemma	lemma	PROPN
ejpam-5240	384	8	1	1	NUM
ejpam-5240	384	9	and	and	CCONJ
ejpam-5240	384	10	lemma	lemma	PROPN
ejpam-5240	384	11	2	2	X
ejpam-5240	384	12	.	.	PUNCT
ejpam-5240	385	1	let	let	VERB
ejpam-5240	385	2	k	k	NOUN
ejpam-5240	385	3	=	=	NOUN
ejpam-5240	385	4	1	1	X
ejpam-5240	385	5	.	.	PUNCT
ejpam-5240	386	1	since	since	SCONJ
ejpam-5240	386	2	1	1	NUM
ejpam-5240	386	3	≤	≤	NUM
ejpam-5240	386	4	k	k	X
ejpam-5240	386	5	≤	≤	NUM
ejpam-5240	386	6	⌊	⌊	VERB
ejpam-5240	386	7	n	n	DET
ejpam-5240	386	8	2	2	NUM
ejpam-5240	386	9	⌋	⌋	NOUN
ejpam-5240	386	10	,	,	PUNCT
ejpam-5240	386	11	by	by	ADP
ejpam-5240	386	12	using	use	VERB
ejpam-5240	386	13	lemma	lemma	PROPN
ejpam-5240	386	14	1	1	NUM
ejpam-5240	386	15	,	,	PUNCT
ejpam-5240	386	16	then	then	ADV
ejpam-5240	386	17	deg(a	deg(a	PROPN
ejpam-5240	386	18	)	)	PUNCT
ejpam-5240	386	19	=	=	PUNCT
ejpam-5240	387	1	(	(	PUNCT
ejpam-5240	387	2	n	n	NOUN
ejpam-5240	387	3	1	1	NUM
ejpam-5240	387	4	)	)	PUNCT
ejpam-5240	387	5	−	−	PROPN
ejpam-5240	388	1	[	[	X
ejpam-5240	388	2	(	(	PUNCT
ejpam-5240	388	3	n−1	n−1	PROPN
ejpam-5240	388	4	1	1	NUM
ejpam-5240	388	5	)	)	PUNCT
ejpam-5240	388	6	+	+	CCONJ
ejpam-5240	388	7	1	1	X
ejpam-5240	388	8	]	]	PUNCT
ejpam-5240	388	9	=	=	SYM
ejpam-5240	388	10	n	n	PRON
ejpam-5240	388	11	−	−	PROPN
ejpam-5240	388	12	(	(	PUNCT
ejpam-5240	388	13	n	n	CCONJ
ejpam-5240	388	14	−	−	PROPN
ejpam-5240	388	15	1	1	NUM
ejpam-5240	388	16	+	+	NUM
ejpam-5240	388	17	1	1	NUM
ejpam-5240	388	18	)	)	PUNCT
ejpam-5240	388	19	=	=	SYM
ejpam-5240	388	20	0	0	NUM
ejpam-5240	388	21	for	for	ADP
ejpam-5240	388	22	all	all	DET
ejpam-5240	388	23	a	a	DET
ejpam-5240	388	24	∈	∈	PROPN
ejpam-5240	388	25	v	v	NOUN
ejpam-5240	388	26	(	(	PUNCT
ejpam-5240	388	27	gs(n,1	gs(n,1	NOUN
ejpam-5240	388	28	)	)	PUNCT
ejpam-5240	388	29	)	)	PUNCT
ejpam-5240	388	30	.	.	PUNCT
ejpam-5240	389	1	by	by	ADP
ejpam-5240	389	2	theorem	theorem	NOUN
ejpam-5240	389	3	5	5	NUM
ejpam-5240	389	4	,	,	PUNCT
ejpam-5240	389	5	it	it	PRON
ejpam-5240	389	6	follows	follow	VERB
ejpam-5240	389	7	that	that	SCONJ
ejpam-5240	389	8	gs(n,1	gs(n,1	PROPN
ejpam-5240	389	9	)	)	PUNCT
ejpam-5240	389	10	is	be	AUX
ejpam-5240	389	11	a	a	DET
ejpam-5240	389	12	0	0	NUM
ejpam-5240	389	13	-	-	PUNCT
ejpam-5240	389	14	regular	regular	ADJ
ejpam-5240	389	15	graph	graph	NOUN
ejpam-5240	389	16	.	.	PUNCT
ejpam-5240	390	1	moreover	moreover	ADV
ejpam-5240	390	2	,	,	PUNCT
ejpam-5240	390	3	observe	observe	VERB
ejpam-5240	390	4	that	that	SCONJ
ejpam-5240	390	5	gs(n	gs(n	NOUN
ejpam-5240	390	6	,	,	PUNCT
ejpam-5240	390	7	k	k	NOUN
ejpam-5240	390	8	)	)	PUNCT
ejpam-5240	390	9	is	be	AUX
ejpam-5240	390	10	also	also	ADV
ejpam-5240	390	11	a	a	DET
ejpam-5240	390	12	0	0	NUM
ejpam-5240	390	13	-	-	PUNCT
ejpam-5240	390	14	regular	regular	ADJ
ejpam-5240	390	15	graph	graph	NOUN
ejpam-5240	390	16	when	when	SCONJ
ejpam-5240	390	17	k	k	PROPN
ejpam-5240	390	18	=	=	PUNCT
ejpam-5240	390	19	0	0	PROPN
ejpam-5240	390	20	or	or	CCONJ
ejpam-5240	390	21	k	k	PROPN
ejpam-5240	390	22	=	=	SYM
ejpam-5240	390	23	n.	n.	PROPN
ejpam-5240	390	24	equivalently	equivalently	ADV
ejpam-5240	390	25	,	,	PUNCT
ejpam-5240	390	26	we	we	PRON
ejpam-5240	390	27	have	have	VERB
ejpam-5240	390	28	the	the	DET
ejpam-5240	390	29	following	follow	VERB
ejpam-5240	390	30	remark	remark	NOUN
ejpam-5240	390	31	.	.	PUNCT
ejpam-5240	391	1	remark	remark	PROPN
ejpam-5240	391	2	7	7	NUM
ejpam-5240	391	3	.	.	PUNCT
ejpam-5240	392	1	a	a	DET
ejpam-5240	392	2	k	k	ADV
ejpam-5240	392	3	-	-	PUNCT
ejpam-5240	392	4	restricted	restricted	ADJ
ejpam-5240	392	5	intersection	intersection	NOUN
ejpam-5240	392	6	graph	graph	NOUN
ejpam-5240	392	7	gs(n	gs(n	NOUN
ejpam-5240	392	8	,	,	PUNCT
ejpam-5240	392	9	k	k	NOUN
ejpam-5240	392	10	)	)	PUNCT
ejpam-5240	392	11	is	be	AUX
ejpam-5240	392	12	a	a	DET
ejpam-5240	392	13	0	0	NUM
ejpam-5240	392	14	-	-	PUNCT
ejpam-5240	392	15	regular	regular	ADJ
ejpam-5240	392	16	graph	graph	NOUN
ejpam-5240	392	17	if	if	SCONJ
ejpam-5240	392	18	an	an	DET
ejpam-5240	392	19	only	only	ADV
ejpam-5240	392	20	if	if	SCONJ
ejpam-5240	392	21	k	k	PROPN
ejpam-5240	392	22	is	be	AUX
ejpam-5240	392	23	0	0	NUM
ejpam-5240	392	24	,	,	PUNCT
ejpam-5240	392	25	1	1	NUM
ejpam-5240	392	26	,	,	PUNCT
ejpam-5240	392	27	or	or	CCONJ
ejpam-5240	392	28	n.	n.	VERB
ejpam-5240	392	29	the	the	DET
ejpam-5240	392	30	size	size	NOUN
ejpam-5240	392	31	of	of	ADP
ejpam-5240	392	32	a	a	DET
ejpam-5240	392	33	graph	graph	NOUN
ejpam-5240	392	34	is	be	AUX
ejpam-5240	392	35	a	a	DET
ejpam-5240	392	36	parameter	parameter	NOUN
ejpam-5240	392	37	that	that	PRON
ejpam-5240	392	38	is	be	AUX
ejpam-5240	392	39	also	also	ADV
ejpam-5240	392	40	helpful	helpful	ADJ
ejpam-5240	392	41	in	in	ADP
ejpam-5240	392	42	defining	define	VERB
ejpam-5240	392	43	a	a	DET
ejpam-5240	392	44	graph	graph	NOUN
ejpam-5240	392	45	.	.	PUNCT
ejpam-5240	393	1	since	since	SCONJ
ejpam-5240	393	2	gs(n	gs(n	NOUN
ejpam-5240	393	3	,	,	PUNCT
ejpam-5240	393	4	k	k	NOUN
ejpam-5240	393	5	)	)	PUNCT
ejpam-5240	393	6	is	be	AUX
ejpam-5240	393	7	a	a	DET
ejpam-5240	393	8	regular	regular	ADJ
ejpam-5240	393	9	graph	graph	NOUN
ejpam-5240	393	10	,	,	PUNCT
ejpam-5240	393	11	equation	equation	NOUN
ejpam-5240	393	12	1	1	NUM
ejpam-5240	393	13	,	,	PUNCT
ejpam-5240	393	14	it	it	PRON
ejpam-5240	393	15	follows	follow	VERB
ejpam-5240	393	16	that	that	SCONJ
ejpam-5240	393	17	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	393	18	,	,	PUNCT
ejpam-5240	393	19	k	k	NOUN
ejpam-5240	393	20	)	)	PUNCT
ejpam-5240	393	21	)	)	PUNCT
ejpam-5240	393	22	|	|	ADV
ejpam-5240	393	23	can	can	AUX
ejpam-5240	393	24	be	be	AUX
ejpam-5240	393	25	computed	compute	VERB
ejpam-5240	393	26	by	by	ADP
ejpam-5240	393	27	(	(	PUNCT
ejpam-5240	393	28	nk)(r	nk)(r	PROPN
ejpam-5240	393	29	)	)	PUNCT
ejpam-5240	393	30	2	2	NUM
ejpam-5240	393	31	where	where	SCONJ
ejpam-5240	393	32	r	r	NOUN
ejpam-5240	393	33	is	be	AUX
ejpam-5240	393	34	equal	equal	ADJ
ejpam-5240	393	35	to	to	ADP
ejpam-5240	393	36	either	either	CCONJ
ejpam-5240	393	37	(	(	PUNCT
ejpam-5240	393	38	n	n	X
ejpam-5240	393	39	k	k	NOUN
ejpam-5240	393	40	)	)	PUNCT
ejpam-5240	393	41	−	−	PROPN
ejpam-5240	394	1	[	[	X
ejpam-5240	394	2	(	(	PUNCT
ejpam-5240	394	3	n−k	n−k	NOUN
ejpam-5240	394	4	k	k	PROPN
ejpam-5240	394	5	)	)	PUNCT
ejpam-5240	395	1	+	+	CCONJ
ejpam-5240	395	2	1	1	X
ejpam-5240	395	3	]	]	PUNCT
ejpam-5240	395	4	or	or	CCONJ
ejpam-5240	395	5	(	(	PUNCT
ejpam-5240	395	6	n	n	X
ejpam-5240	395	7	k	k	NOUN
ejpam-5240	395	8	)	)	PUNCT
ejpam-5240	395	9	−	−	PROPN
ejpam-5240	395	10	1	1	X
ejpam-5240	395	11	.	.	PUNCT
ejpam-5240	395	12	theorem	theorem	NOUN
ejpam-5240	395	13	6	6	NUM
ejpam-5240	395	14	.	.	PUNCT
ejpam-5240	396	1	let	let	VERB
ejpam-5240	396	2	gs(n	gs(n	NOUN
ejpam-5240	396	3	,	,	PUNCT
ejpam-5240	396	4	k	k	NOUN
ejpam-5240	396	5	)	)	PUNCT
ejpam-5240	396	6	be	be	VERB
ejpam-5240	396	7	a	a	DET
ejpam-5240	396	8	k	k	ADV
ejpam-5240	396	9	-	-	ADJ
ejpam-5240	396	10	restricted	restricted	ADJ
ejpam-5240	396	11	intersection	intersection	NOUN
ejpam-5240	396	12	graph	graph	NOUN
ejpam-5240	396	13	.	.	PUNCT
ejpam-5240	397	1	then	then	ADV
ejpam-5240	397	2	the	the	DET
ejpam-5240	397	3	size	size	NOUN
ejpam-5240	397	4	of	of	ADP
ejpam-5240	397	5	gs(n	gs(n	NOUN
ejpam-5240	397	6	,	,	PUNCT
ejpam-5240	397	7	k	k	NOUN
ejpam-5240	397	8	)	)	PUNCT
ejpam-5240	397	9	is	be	AUX
ejpam-5240	397	10	given	give	VERB
ejpam-5240	397	11	by	by	ADP
ejpam-5240	397	12	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	397	13	,	,	PUNCT
ejpam-5240	397	14	k	k	NOUN
ejpam-5240	397	15	)	)	PUNCT
ejpam-5240	397	16	)	)	PUNCT
ejpam-5240	398	1	|	|	ADV
ejpam-5240	398	2	=	=	PUNCT
ejpam-5240	398	3			PUNCT
ejpam-5240	398	4	(	(	PUNCT
ejpam-5240	398	5	nk){(nk)−[(n−k	nk){(nk)−[(n−k	NOUN
ejpam-5240	398	6	k	k	X
ejpam-5240	398	7	)	)	PUNCT
ejpam-5240	398	8	+1	+1	PROPN
ejpam-5240	398	9	]	]	X
ejpam-5240	398	10	}	}	PUNCT
ejpam-5240	398	11	2	2	NUM
ejpam-5240	398	12	if	if	SCONJ
ejpam-5240	398	13	1	1	NUM
ejpam-5240	398	14	≤	≤	NUM
ejpam-5240	398	15	k	k	X
ejpam-5240	398	16	≤	≤	NUM
ejpam-5240	398	17	⌊	⌊	VERB
ejpam-5240	398	18	n	n	DET
ejpam-5240	398	19	2	2	NUM
ejpam-5240	398	20	⌋	⌋	NOUN
ejpam-5240	398	21	;	;	PUNCT
ejpam-5240	398	22	(	(	PUNCT
ejpam-5240	398	23	nk	nk	PROPN
ejpam-5240	398	24	)	)	PUNCT
ejpam-5240	398	25	[	[	X
ejpam-5240	398	26	(	(	PUNCT
ejpam-5240	398	27	n	n	PRON
ejpam-5240	398	28	k)−1	k)−1	NOUN
ejpam-5240	398	29	]	]	PUNCT
ejpam-5240	398	30	2	2	NUM
ejpam-5240	398	31	if	if	SCONJ
ejpam-5240	398	32	k	k	NOUN
ejpam-5240	398	33	=	=	PUNCT
ejpam-5240	398	34	0	0	NUM
ejpam-5240	398	35	or	or	CCONJ
ejpam-5240	398	36	⌊	⌊	X
ejpam-5240	398	37	n	n	ADV
ejpam-5240	398	38	2	2	NUM
ejpam-5240	398	39	⌋	⌋	NOUN
ejpam-5240	398	40	<	<	X
ejpam-5240	398	41	k	k	PROPN
ejpam-5240	398	42	≤	≤	PROPN
ejpam-5240	398	43	n.	n.	PROPN
ejpam-5240	398	44	m.e	m.e	PROPN
ejpam-5240	398	45	.	.	PROPN
ejpam-5240	398	46	pelagio	pelagio	PROPN
ejpam-5240	398	47	,	,	PUNCT
ejpam-5240	398	48	n.	n.	NOUN
ejpam-5240	398	49	mame	mame	PROPN
ejpam-5240	398	50	,	,	PUNCT
ejpam-5240	398	51	k.	k.	PROPN
ejpam-5240	398	52	mendoza	mendoza	PROPN
ejpam-5240	398	53	/	/	SYM
ejpam-5240	398	54	eur	eur	PROPN
ejpam-5240	398	55	.	.	PUNCT
ejpam-5240	399	1	j.	j.	PROPN
ejpam-5240	399	2	pure	pure	PROPN
ejpam-5240	399	3	appl	appl	PROPN
ejpam-5240	399	4	.	.	PROPN
ejpam-5240	399	5	math	math	PROPN
ejpam-5240	399	6	,	,	PUNCT
ejpam-5240	399	7	17	17	NUM
ejpam-5240	399	8	(	(	PUNCT
ejpam-5240	399	9	3	3	NUM
ejpam-5240	399	10	)	)	PUNCT
ejpam-5240	399	11	(	(	PUNCT
ejpam-5240	399	12	2024	2024	NUM
ejpam-5240	399	13	)	)	PUNCT
ejpam-5240	399	14	,	,	PUNCT
ejpam-5240	399	15	1779	1779	NUM
ejpam-5240	399	16	-	-	SYM
ejpam-5240	399	17	1803	1803	NUM
ejpam-5240	399	18	1791	1791	NUM
ejpam-5240	399	19	proof	proof	NOUN
ejpam-5240	399	20	.	.	PUNCT
ejpam-5240	400	1	note	note	VERB
ejpam-5240	400	2	that	that	SCONJ
ejpam-5240	400	3	gs(n	gs(n	NOUN
ejpam-5240	400	4	,	,	PUNCT
ejpam-5240	400	5	k	k	NOUN
ejpam-5240	400	6	)	)	PUNCT
ejpam-5240	400	7	is	be	AUX
ejpam-5240	400	8	a	a	DET
ejpam-5240	400	9	graph	graph	NOUN
ejpam-5240	400	10	of	of	ADP
ejpam-5240	400	11	order	order	NOUN
ejpam-5240	400	12	(	(	PUNCT
ejpam-5240	400	13	n	n	X
ejpam-5240	400	14	k	k	PROPN
ejpam-5240	400	15	)	)	PUNCT
ejpam-5240	400	16	.	.	PUNCT
ejpam-5240	401	1	by	by	ADP
ejpam-5240	401	2	theorem	theorem	NOUN
ejpam-5240	401	3	5	5	NUM
ejpam-5240	401	4	,	,	PUNCT
ejpam-5240	401	5	gs(n	gs(n	NOUN
ejpam-5240	401	6	,	,	PUNCT
ejpam-5240	401	7	k	k	NOUN
ejpam-5240	401	8	)	)	PUNCT
ejpam-5240	401	9	is	be	AUX
ejpam-5240	401	10	a	a	DET
ejpam-5240	401	11	regular	regular	ADJ
ejpam-5240	401	12	graph	graph	NOUN
ejpam-5240	401	13	.	.	PUNCT
ejpam-5240	402	1	now	now	ADV
ejpam-5240	402	2	,	,	PUNCT
ejpam-5240	402	3	using	use	VERB
ejpam-5240	402	4	equation	equation	NOUN
ejpam-5240	402	5	1	1	NUM
ejpam-5240	402	6	,	,	PUNCT
ejpam-5240	402	7	then	then	ADV
ejpam-5240	402	8	the	the	DET
ejpam-5240	402	9	size	size	NOUN
ejpam-5240	402	10	of	of	ADP
ejpam-5240	402	11	gs(n	gs(n	NOUN
ejpam-5240	402	12	,	,	PUNCT
ejpam-5240	402	13	k	k	NOUN
ejpam-5240	402	14	)	)	PUNCT
ejpam-5240	402	15	is	be	AUX
ejpam-5240	402	16	(	(	PUNCT
ejpam-5240	402	17	nk	nk	PROPN
ejpam-5240	402	18	)	)	PUNCT
ejpam-5240	402	19	(	(	PUNCT
ejpam-5240	402	20	r	r	NOUN
ejpam-5240	402	21	)	)	PUNCT
ejpam-5240	402	22	2	2	NUM
ejpam-5240	402	23	where	where	SCONJ
ejpam-5240	402	24	r	r	NOUN
ejpam-5240	402	25	is	be	AUX
ejpam-5240	402	26	the	the	DET
ejpam-5240	402	27	degree	degree	NOUN
ejpam-5240	402	28	of	of	ADP
ejpam-5240	402	29	every	every	DET
ejpam-5240	402	30	vertex	vertex	NOUN
ejpam-5240	402	31	in	in	ADP
ejpam-5240	402	32	gs(n	gs(n	NOUN
ejpam-5240	402	33	,	,	PUNCT
ejpam-5240	402	34	k	k	NOUN
ejpam-5240	402	35	)	)	PUNCT
ejpam-5240	402	36	.	.	PUNCT
ejpam-5240	403	1	if	if	SCONJ
ejpam-5240	403	2	1	1	NUM
ejpam-5240	403	3	≤	≤	NUM
ejpam-5240	403	4	k	k	X
ejpam-5240	403	5	≤	≤	NUM
ejpam-5240	403	6	⌊	⌊	VERB
ejpam-5240	403	7	n	n	PRON
ejpam-5240	403	8	2	2	NUM
ejpam-5240	403	9	⌋	⌋	NOUN
ejpam-5240	403	10	,	,	PUNCT
ejpam-5240	403	11	gs(n	gs(n	NOUN
ejpam-5240	403	12	,	,	PUNCT
ejpam-5240	403	13	k	k	NOUN
ejpam-5240	403	14	)	)	PUNCT
ejpam-5240	403	15	is	be	AUX
ejpam-5240	403	16	a	a	DET
ejpam-5240	403	17	{	{	PUNCT
ejpam-5240	403	18	(	(	PUNCT
ejpam-5240	403	19	n	n	X
ejpam-5240	403	20	k	k	NOUN
ejpam-5240	403	21	)	)	PUNCT
ejpam-5240	403	22	−	−	PROPN
ejpam-5240	404	1	[	[	X
ejpam-5240	404	2	(	(	PUNCT
ejpam-5240	404	3	n−k	n−k	NOUN
ejpam-5240	404	4	k	k	PROPN
ejpam-5240	404	5	)	)	PUNCT
ejpam-5240	405	1	+	+	CCONJ
ejpam-5240	405	2	1	1	NUM
ejpam-5240	405	3	]	]	PUNCT
ejpam-5240	405	4	}	}	ADV
ejpam-5240	405	5	-regular	-regular	ADJ
ejpam-5240	405	6	graph	graph	NOUN
ejpam-5240	405	7	.	.	PUNCT
ejpam-5240	406	1	thus	thus	ADV
ejpam-5240	406	2	,	,	PUNCT
ejpam-5240	406	3	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	406	4	,	,	PUNCT
ejpam-5240	406	5	k	k	NOUN
ejpam-5240	406	6	)	)	PUNCT
ejpam-5240	406	7	)	)	PUNCT
ejpam-5240	406	8	|	|	ADV
ejpam-5240	406	9	=	=	SYM
ejpam-5240	406	10	(	(	PUNCT
ejpam-5240	406	11	nk	nk	PROPN
ejpam-5240	406	12	)	)	PUNCT
ejpam-5240	406	13	{	{	PUNCT
ejpam-5240	406	14	(	(	PUNCT
ejpam-5240	406	15	nk)−	nk)−	X
ejpam-5240	406	16	[	[	PUNCT
ejpam-5240	406	17	(	(	PUNCT
ejpam-5240	406	18	n−k	n−k	NOUN
ejpam-5240	406	19	k	k	NOUN
ejpam-5240	406	20	)	)	PUNCT
ejpam-5240	406	21	+1	+1	PROPN
ejpam-5240	406	22	]	]	X
ejpam-5240	406	23	}	}	PUNCT
ejpam-5240	406	24	2	2	NUM
ejpam-5240	406	25	.	.	PUNCT
ejpam-5240	407	1	furthermore	furthermore	ADV
ejpam-5240	407	2	,	,	PUNCT
ejpam-5240	407	3	if	if	SCONJ
ejpam-5240	407	4	k	k	PROPN
ejpam-5240	407	5	=	=	PUNCT
ejpam-5240	407	6	0	0	NUM
ejpam-5240	407	7	or	or	CCONJ
ejpam-5240	407	8	⌊	⌊	X
ejpam-5240	407	9	n	n	ADV
ejpam-5240	407	10	2	2	NUM
ejpam-5240	407	11	⌋	⌋	NOUN
ejpam-5240	407	12	<	<	X
ejpam-5240	407	13	k	k	X
ejpam-5240	407	14	≤	≤	PROPN
ejpam-5240	407	15	n	n	CCONJ
ejpam-5240	407	16	,	,	PUNCT
ejpam-5240	407	17	gs(n	gs(n	NOUN
ejpam-5240	407	18	,	,	PUNCT
ejpam-5240	407	19	k	k	NOUN
ejpam-5240	407	20	)	)	PUNCT
ejpam-5240	407	21	is	be	AUX
ejpam-5240	407	22	a	a	DET
ejpam-5240	407	23	[	[	X
ejpam-5240	407	24	(	(	PUNCT
ejpam-5240	407	25	n	n	X
ejpam-5240	407	26	k	k	NOUN
ejpam-5240	407	27	)	)	PUNCT
ejpam-5240	408	1	−	−	PROPN
ejpam-5240	408	2	1	1	NUM
ejpam-5240	408	3	]	]	SYM
ejpam-5240	408	4	-regular	-regular	ADJ
ejpam-5240	408	5	graph	graph	NOUN
ejpam-5240	408	6	.	.	PUNCT
ejpam-5240	409	1	this	this	PRON
ejpam-5240	409	2	implies	imply	VERB
ejpam-5240	409	3	that	that	SCONJ
ejpam-5240	409	4	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	409	5	,	,	PUNCT
ejpam-5240	409	6	k	k	NOUN
ejpam-5240	409	7	)	)	PUNCT
ejpam-5240	409	8	)	)	PUNCT
ejpam-5240	410	1	|	|	ADV
ejpam-5240	410	2	=	=	SYM
ejpam-5240	410	3	(	(	PUNCT
ejpam-5240	410	4	nk	nk	PROPN
ejpam-5240	410	5	)	)	PUNCT
ejpam-5240	411	1	[	[	X
ejpam-5240	411	2	(	(	PUNCT
ejpam-5240	411	3	n	n	PRON
ejpam-5240	411	4	k)−1	k)−1	NOUN
ejpam-5240	411	5	]	]	PUNCT
ejpam-5240	411	6	2	2	NUM
ejpam-5240	411	7	.	.	PUNCT
ejpam-5240	411	8	illustration	illustration	NOUN
ejpam-5240	411	9	6	6	NUM
ejpam-5240	411	10	.	.	PUNCT
ejpam-5240	411	11	consider	consider	VERB
ejpam-5240	411	12	gs(5,2	gs(5,2	NOUN
ejpam-5240	411	13	)	)	PUNCT
ejpam-5240	411	14	shown	show	VERB
ejpam-5240	411	15	in	in	ADP
ejpam-5240	411	16	figure	figure	NOUN
ejpam-5240	411	17	5	5	NUM
ejpam-5240	411	18	.	.	PUNCT
ejpam-5240	411	19	observe	observe	VERB
ejpam-5240	411	20	that	that	SCONJ
ejpam-5240	411	21	k	k	PROPN
ejpam-5240	411	22	=	=	SYM
ejpam-5240	411	23	2	2	NUM
ejpam-5240	411	24	which	which	PRON
ejpam-5240	411	25	implies	imply	VERB
ejpam-5240	411	26	that	that	SCONJ
ejpam-5240	411	27	⌊	⌊	PROPN
ejpam-5240	411	28	5	5	NUM
ejpam-5240	411	29	2	2	NUM
ejpam-5240	411	30	⌋	⌋	NOUN
ejpam-5240	411	31	=	=	SYM
ejpam-5240	411	32	2	2	NUM
ejpam-5240	411	33	=	=	SYM
ejpam-5240	411	34	k.	k.	NOUN
ejpam-5240	411	35	since	since	SCONJ
ejpam-5240	411	36	gs(5,2	gs(5,2	NOUN
ejpam-5240	411	37	)	)	PUNCT
ejpam-5240	411	38	is	be	AUX
ejpam-5240	411	39	a	a	DET
ejpam-5240	411	40	graph	graph	NOUN
ejpam-5240	411	41	of	of	ADP
ejpam-5240	411	42	order	order	NOUN
ejpam-5240	411	43	10	10	NUM
ejpam-5240	411	44	and	and	CCONJ
ejpam-5240	411	45	a	a	DET
ejpam-5240	411	46	6	6	NUM
ejpam-5240	411	47	-	-	PUNCT
ejpam-5240	411	48	regular	regular	ADJ
ejpam-5240	411	49	graph	graph	NOUN
ejpam-5240	411	50	,	,	PUNCT
ejpam-5240	411	51	by	by	ADP
ejpam-5240	411	52	utilizing	utilize	VERB
ejpam-5240	411	53	theorem	theorem	NOUN
ejpam-5240	411	54	6	6	NUM
ejpam-5240	411	55	,	,	PUNCT
ejpam-5240	411	56	it	it	PRON
ejpam-5240	411	57	follows	follow	VERB
ejpam-5240	411	58	that	that	SCONJ
ejpam-5240	411	59	|e(gs(5,2	|e(gs(5,2	NOUN
ejpam-5240	411	60	)	)	PUNCT
ejpam-5240	411	61	)	)	PUNCT
ejpam-5240	412	1	|	|	ADV
ejpam-5240	412	2	=	=	SYM
ejpam-5240	412	3	10(6	10(6	ADJ
ejpam-5240	412	4	)	)	PUNCT
ejpam-5240	412	5	2	2	NUM
ejpam-5240	412	6	=	=	SYM
ejpam-5240	412	7	30	30	NUM
ejpam-5240	412	8	.	.	PUNCT
ejpam-5240	413	1	also	also	ADV
ejpam-5240	413	2	,	,	PUNCT
ejpam-5240	413	3	given	give	VERB
ejpam-5240	413	4	gs(4,3	gs(4,3	NOUN
ejpam-5240	413	5	)	)	PUNCT
ejpam-5240	413	6	in	in	ADP
ejpam-5240	413	7	figure	figure	NOUN
ejpam-5240	413	8	6	6	NUM
ejpam-5240	413	9	that	that	PRON
ejpam-5240	413	10	is	be	AUX
ejpam-5240	413	11	a	a	DET
ejpam-5240	413	12	3	3	NUM
ejpam-5240	413	13	-	-	PUNCT
ejpam-5240	413	14	regular	regular	ADJ
ejpam-5240	413	15	graph	graph	NOUN
ejpam-5240	413	16	of	of	ADP
ejpam-5240	413	17	order	order	NOUN
ejpam-5240	413	18	(	(	PUNCT
ejpam-5240	413	19	4	4	NUM
ejpam-5240	413	20	3	3	NUM
ejpam-5240	413	21	)	)	PUNCT
ejpam-5240	413	22	=	=	SYM
ejpam-5240	413	23	4	4	X
ejpam-5240	413	24	,	,	PUNCT
ejpam-5240	413	25	note	note	VERB
ejpam-5240	413	26	that	that	SCONJ
ejpam-5240	413	27	k	k	PROPN
ejpam-5240	413	28	=	=	PUNCT
ejpam-5240	413	29	3	3	NUM
ejpam-5240	413	30	which	which	PRON
ejpam-5240	413	31	is	be	AUX
ejpam-5240	413	32	⌊	⌊	NUM
ejpam-5240	413	33	4	4	NUM
ejpam-5240	413	34	2	2	NUM
ejpam-5240	413	35	⌋	⌋	NOUN
ejpam-5240	413	36	=	=	SYM
ejpam-5240	413	37	2	2	NUM
ejpam-5240	413	38	<	<	X
ejpam-5240	413	39	k.	k.	PROPN
ejpam-5240	414	1	so	so	ADV
ejpam-5240	414	2	,	,	PUNCT
ejpam-5240	414	3	we	we	PRON
ejpam-5240	414	4	have	have	VERB
ejpam-5240	414	5	|e(gs(4,3	|e(gs(4,3	PROPN
ejpam-5240	414	6	)	)	PUNCT
ejpam-5240	414	7	)	)	PUNCT
ejpam-5240	415	1	|	|	ADV
ejpam-5240	415	2	=	=	SYM
ejpam-5240	415	3	4(3	4(3	NUM
ejpam-5240	415	4	)	)	PUNCT
ejpam-5240	415	5	2	2	NUM
ejpam-5240	415	6	=	=	SYM
ejpam-5240	415	7	6	6	NUM
ejpam-5240	415	8	.	.	X
ejpam-5240	416	1	for	for	ADP
ejpam-5240	416	2	any	any	DET
ejpam-5240	416	3	positive	positive	ADJ
ejpam-5240	416	4	integer	integer	NOUN
ejpam-5240	416	5	n	n	CCONJ
ejpam-5240	416	6	,	,	PUNCT
ejpam-5240	416	7	we	we	PRON
ejpam-5240	416	8	have	have	AUX
ejpam-5240	416	9	identified	identify	VERB
ejpam-5240	416	10	that	that	SCONJ
ejpam-5240	416	11	gs(n	gs(n	NOUN
ejpam-5240	416	12	,	,	PUNCT
ejpam-5240	416	13	k	k	NOUN
ejpam-5240	416	14	)	)	PUNCT
ejpam-5240	416	15	is	be	AUX
ejpam-5240	416	16	a	a	DET
ejpam-5240	416	17	0	0	NUM
ejpam-5240	416	18	-	-	PUNCT
ejpam-5240	416	19	regular	regular	ADJ
ejpam-5240	416	20	graph	graph	NOUN
ejpam-5240	417	1	if	if	SCONJ
ejpam-5240	417	2	and	and	CCONJ
ejpam-5240	417	3	only	only	ADV
ejpam-5240	417	4	if	if	SCONJ
ejpam-5240	417	5	k	k	PROPN
ejpam-5240	417	6	=	=	SYM
ejpam-5240	417	7	0	0	PROPN
ejpam-5240	417	8	,	,	PUNCT
ejpam-5240	417	9	k	k	NOUN
ejpam-5240	417	10	=	=	SYM
ejpam-5240	417	11	1	1	NUM
ejpam-5240	417	12	,	,	PUNCT
ejpam-5240	417	13	or	or	CCONJ
ejpam-5240	417	14	k	k	NOUN
ejpam-5240	417	15	=	=	SYM
ejpam-5240	417	16	n	n	PROPN
ejpam-5240	417	17	by	by	ADP
ejpam-5240	417	18	remark	remark	NOUN
ejpam-5240	417	19	7	7	NUM
ejpam-5240	417	20	.	.	PUNCT
ejpam-5240	418	1	with	with	ADP
ejpam-5240	418	2	this	this	PRON
ejpam-5240	418	3	,	,	PUNCT
ejpam-5240	418	4	then	then	ADV
ejpam-5240	418	5	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	418	6	,	,	PUNCT
ejpam-5240	418	7	k	k	NOUN
ejpam-5240	418	8	)	)	PUNCT
ejpam-5240	418	9	)	)	PUNCT
ejpam-5240	418	10	|	|	ADV
ejpam-5240	418	11	=	=	SYM
ejpam-5240	418	12	(	(	PUNCT
ejpam-5240	418	13	nk)(0	nk)(0	NOUN
ejpam-5240	418	14	)	)	PUNCT
ejpam-5240	418	15	2	2	NUM
ejpam-5240	418	16	=	=	SYM
ejpam-5240	418	17	0	0	NUM
ejpam-5240	418	18	.	.	PUNCT
ejpam-5240	419	1	equivalently	equivalently	ADV
ejpam-5240	419	2	,	,	PUNCT
ejpam-5240	419	3	we	we	PRON
ejpam-5240	419	4	have	have	VERB
ejpam-5240	419	5	remark	remark	NOUN
ejpam-5240	419	6	8	8	NUM
ejpam-5240	419	7	.	.	PUNCT
ejpam-5240	419	8	remark	remark	PROPN
ejpam-5240	419	9	8	8	NUM
ejpam-5240	419	10	.	.	PUNCT
ejpam-5240	420	1	if	if	SCONJ
ejpam-5240	420	2	k	k	PROPN
ejpam-5240	420	3	=	=	SYM
ejpam-5240	420	4	0	0	PROPN
ejpam-5240	420	5	,	,	PUNCT
ejpam-5240	420	6	k	k	NOUN
ejpam-5240	420	7	=	=	SYM
ejpam-5240	420	8	1	1	NUM
ejpam-5240	420	9	,	,	PUNCT
ejpam-5240	420	10	or	or	CCONJ
ejpam-5240	420	11	k	k	NOUN
ejpam-5240	420	12	=	=	PUNCT
ejpam-5240	420	13	n	n	CCONJ
ejpam-5240	420	14	,	,	PUNCT
ejpam-5240	420	15	then	then	ADV
ejpam-5240	420	16	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	420	17	,	,	PUNCT
ejpam-5240	420	18	k	k	NOUN
ejpam-5240	420	19	)	)	PUNCT
ejpam-5240	420	20	)	)	PUNCT
ejpam-5240	421	1	|	|	ADV
ejpam-5240	421	2	=	=	SYM
ejpam-5240	421	3	0	0	NUM
ejpam-5240	421	4	.	.	NOUN
ejpam-5240	421	5	4	4	NUM
ejpam-5240	421	6	.	.	X
ejpam-5240	422	1	k	k	ADJ
ejpam-5240	422	2	-	-	PUNCT
ejpam-5240	422	3	restricted	restrict	VERB
ejpam-5240	422	4	intersection	intersection	NOUN
ejpam-5240	422	5	graph	graph	NOUN
ejpam-5240	422	6	as	as	ADP
ejpam-5240	422	7	a	a	DET
ejpam-5240	422	8	special	special	ADJ
ejpam-5240	422	9	class	class	NOUN
ejpam-5240	422	10	of	of	ADP
ejpam-5240	422	11	graph	graph	NOUN
ejpam-5240	422	12	the	the	DET
ejpam-5240	422	13	k	k	ADV
ejpam-5240	422	14	-	-	PUNCT
ejpam-5240	422	15	restricted	restrict	VERB
ejpam-5240	422	16	intersection	intersection	NOUN
ejpam-5240	422	17	graph	graph	NOUN
ejpam-5240	422	18	gs(n	gs(n	NOUN
ejpam-5240	422	19	,	,	PUNCT
ejpam-5240	422	20	k	k	NOUN
ejpam-5240	422	21	)	)	PUNCT
ejpam-5240	422	22	yields	yield	VERB
ejpam-5240	422	23	a	a	DET
ejpam-5240	422	24	complete	complete	ADJ
ejpam-5240	422	25	graph	graph	NOUN
ejpam-5240	422	26	and	and	CCONJ
ejpam-5240	422	27	cycle	cycle	NOUN
ejpam-5240	422	28	graph	graph	NOUN
ejpam-5240	422	29	depending	depend	VERB
ejpam-5240	422	30	on	on	ADP
ejpam-5240	422	31	the	the	DET
ejpam-5240	422	32	values	value	NOUN
ejpam-5240	422	33	of	of	ADP
ejpam-5240	422	34	the	the	DET
ejpam-5240	422	35	nonnegative	nonnegative	ADJ
ejpam-5240	422	36	integer	integer	NOUN
ejpam-5240	422	37	k.	k.	PROPN
ejpam-5240	423	1	furthermore	furthermore	ADV
ejpam-5240	423	2	,	,	PUNCT
ejpam-5240	423	3	the	the	DET
ejpam-5240	423	4	degree	degree	NOUN
ejpam-5240	423	5	of	of	ADP
ejpam-5240	423	6	every	every	DET
ejpam-5240	423	7	vertex	vertex	NOUN
ejpam-5240	423	8	for	for	ADP
ejpam-5240	423	9	the	the	DET
ejpam-5240	423	10	complement	complement	NOUN
ejpam-5240	423	11	graph	graph	NOUN
ejpam-5240	423	12	of	of	ADP
ejpam-5240	423	13	gs(n	gs(n	NOUN
ejpam-5240	423	14	,	,	PUNCT
ejpam-5240	423	15	k	k	NOUN
ejpam-5240	423	16	)	)	PUNCT
ejpam-5240	423	17	is	be	AUX
ejpam-5240	423	18	given	give	VERB
ejpam-5240	423	19	in	in	ADP
ejpam-5240	423	20	this	this	DET
ejpam-5240	423	21	section	section	NOUN
ejpam-5240	423	22	.	.	PUNCT
ejpam-5240	424	1	theorem	theorem	VERB
ejpam-5240	424	2	7	7	NUM
ejpam-5240	424	3	.	.	PUNCT
ejpam-5240	425	1	a	a	DET
ejpam-5240	425	2	k	k	ADV
ejpam-5240	425	3	-	-	PUNCT
ejpam-5240	425	4	restricted	restricted	ADJ
ejpam-5240	425	5	intersection	intersection	NOUN
ejpam-5240	425	6	graph	graph	NOUN
ejpam-5240	425	7	gs(n	gs(n	NOUN
ejpam-5240	425	8	,	,	PUNCT
ejpam-5240	425	9	k	k	NOUN
ejpam-5240	425	10	)	)	PUNCT
ejpam-5240	425	11	is	be	AUX
ejpam-5240	425	12	a	a	DET
ejpam-5240	425	13	complete	complete	ADJ
ejpam-5240	425	14	graph	graph	NOUN
ejpam-5240	425	15	of	of	ADP
ejpam-5240	425	16	order	order	NOUN
ejpam-5240	425	17	(	(	PUNCT
ejpam-5240	425	18	n	n	X
ejpam-5240	425	19	k	k	NOUN
ejpam-5240	425	20	)	)	PUNCT
ejpam-5240	426	1	if	if	SCONJ
ejpam-5240	426	2	and	and	CCONJ
ejpam-5240	426	3	only	only	ADV
ejpam-5240	426	4	if	if	SCONJ
ejpam-5240	426	5	k	k	PROPN
ejpam-5240	426	6	=	=	PUNCT
ejpam-5240	426	7	0	0	NUM
ejpam-5240	426	8	or	or	CCONJ
ejpam-5240	426	9	⌊	⌊	X
ejpam-5240	426	10	n	n	ADV
ejpam-5240	426	11	2	2	NUM
ejpam-5240	426	12	⌋	⌋	NOUN
ejpam-5240	426	13	<	<	X
ejpam-5240	426	14	k	k	PROPN
ejpam-5240	426	15	≤	≤	PROPN
ejpam-5240	426	16	n.	n.	NOUN
ejpam-5240	426	17	proof	proof	NOUN
ejpam-5240	426	18	.	.	PUNCT
ejpam-5240	427	1	assume	assume	VERB
ejpam-5240	427	2	gs(n	gs(n	NOUN
ejpam-5240	427	3	,	,	PUNCT
ejpam-5240	427	4	k	k	NOUN
ejpam-5240	427	5	)	)	PUNCT
ejpam-5240	427	6	is	be	AUX
ejpam-5240	427	7	a	a	DET
ejpam-5240	427	8	complete	complete	ADJ
ejpam-5240	427	9	graph	graph	NOUN
ejpam-5240	427	10	of	of	ADP
ejpam-5240	427	11	order	order	NOUN
ejpam-5240	427	12	(	(	PUNCT
ejpam-5240	427	13	n	n	X
ejpam-5240	427	14	k	k	PROPN
ejpam-5240	427	15	)	)	PUNCT
ejpam-5240	427	16	.	.	PUNCT
ejpam-5240	428	1	by	by	ADP
ejpam-5240	428	2	the	the	DET
ejpam-5240	428	3	definition	definition	NOUN
ejpam-5240	428	4	of	of	ADP
ejpam-5240	428	5	a	a	DET
ejpam-5240	428	6	complete	complete	ADJ
ejpam-5240	428	7	graph	graph	NOUN
ejpam-5240	428	8	,	,	PUNCT
ejpam-5240	428	9	gs(n	gs(n	NOUN
ejpam-5240	428	10	,	,	PUNCT
ejpam-5240	428	11	k	k	NOUN
ejpam-5240	428	12	)	)	PUNCT
ejpam-5240	428	13	is	be	AUX
ejpam-5240	428	14	an	an	DET
ejpam-5240	428	15	[	[	X
ejpam-5240	428	16	(	(	PUNCT
ejpam-5240	428	17	n	n	X
ejpam-5240	428	18	k	k	NOUN
ejpam-5240	428	19	)	)	PUNCT
ejpam-5240	429	1	−	−	PROPN
ejpam-5240	429	2	1	1	NUM
ejpam-5240	429	3	]	]	SYM
ejpam-5240	429	4	-regular	-regular	ADJ
ejpam-5240	429	5	graph	graph	NOUN
ejpam-5240	429	6	.	.	PUNCT
ejpam-5240	429	7	by	by	ADP
ejpam-5240	429	8	theorem	theorem	NOUN
ejpam-5240	429	9	5	5	NUM
ejpam-5240	429	10	,	,	PUNCT
ejpam-5240	429	11	then	then	ADV
ejpam-5240	429	12	k	k	PROPN
ejpam-5240	429	13	=	=	PUNCT
ejpam-5240	429	14	0	0	NUM
ejpam-5240	429	15	or	or	CCONJ
ejpam-5240	429	16	⌊	⌊	X
ejpam-5240	429	17	n	n	ADV
ejpam-5240	429	18	2	2	NUM
ejpam-5240	429	19	⌋	⌋	NOUN
ejpam-5240	429	20	<	<	X
ejpam-5240	429	21	k	k	PROPN
ejpam-5240	429	22	≤	≤	PROPN
ejpam-5240	429	23	n.	n.	NOUN
ejpam-5240	429	24	conversely	conversely	ADV
ejpam-5240	429	25	,	,	PUNCT
ejpam-5240	429	26	assume	assume	VERB
ejpam-5240	429	27	k	k	X
ejpam-5240	429	28	=	=	PUNCT
ejpam-5240	429	29	0	0	NUM
ejpam-5240	429	30	or	or	CCONJ
ejpam-5240	429	31	⌊	⌊	X
ejpam-5240	429	32	n	n	ADV
ejpam-5240	429	33	2	2	NUM
ejpam-5240	429	34	⌋	⌋	NOUN
ejpam-5240	429	35	<	<	X
ejpam-5240	429	36	k	k	PROPN
ejpam-5240	429	37	≤	≤	X
ejpam-5240	429	38	n.	n.	NOUN
ejpam-5240	429	39	by	by	ADP
ejpam-5240	429	40	theorem	theorem	NOUN
ejpam-5240	429	41	5	5	NUM
ejpam-5240	429	42	,	,	PUNCT
ejpam-5240	429	43	gs(n	gs(n	NOUN
ejpam-5240	429	44	,	,	PUNCT
ejpam-5240	429	45	k	k	NOUN
ejpam-5240	429	46	)	)	PUNCT
ejpam-5240	429	47	is	be	AUX
ejpam-5240	429	48	an	an	DET
ejpam-5240	429	49	[	[	X
ejpam-5240	429	50	(	(	PUNCT
ejpam-5240	429	51	n	n	X
ejpam-5240	429	52	k	k	NOUN
ejpam-5240	429	53	)	)	PUNCT
ejpam-5240	430	1	−	−	PROPN
ejpam-5240	430	2	1	1	NUM
ejpam-5240	430	3	]	]	SYM
ejpam-5240	430	4	-regular	-regular	ADJ
ejpam-5240	430	5	graph	graph	NOUN
ejpam-5240	430	6	.	.	PUNCT
ejpam-5240	431	1	since	since	SCONJ
ejpam-5240	431	2	|v	|v	PROPN
ejpam-5240	431	3	(	(	PUNCT
ejpam-5240	431	4	gs(n	gs(n	NOUN
ejpam-5240	431	5	,	,	PUNCT
ejpam-5240	431	6	k	k	NOUN
ejpam-5240	431	7	)	)	PUNCT
ejpam-5240	431	8	)	)	PUNCT
ejpam-5240	431	9	|	|	ADV
ejpam-5240	431	10	=	=	SYM
ejpam-5240	431	11	(	(	PUNCT
ejpam-5240	431	12	n	n	X
ejpam-5240	431	13	k	k	PROPN
ejpam-5240	431	14	)	)	PUNCT
ejpam-5240	431	15	,	,	PUNCT
ejpam-5240	431	16	it	it	PRON
ejpam-5240	431	17	follows	follow	VERB
ejpam-5240	431	18	that	that	SCONJ
ejpam-5240	431	19	every	every	DET
ejpam-5240	431	20	vertex	vertex	NOUN
ejpam-5240	431	21	of	of	ADP
ejpam-5240	431	22	gs(n	gs(n	NOUN
ejpam-5240	431	23	,	,	PUNCT
ejpam-5240	431	24	k	k	NOUN
ejpam-5240	431	25	)	)	PUNCT
ejpam-5240	431	26	is	be	AUX
ejpam-5240	431	27	adjacent	adjacent	ADJ
ejpam-5240	431	28	to	to	ADP
ejpam-5240	431	29	every	every	DET
ejpam-5240	431	30	other	other	ADJ
ejpam-5240	431	31	vertex	vertex	NOUN
ejpam-5240	431	32	of	of	ADP
ejpam-5240	431	33	gs(n	gs(n	NOUN
ejpam-5240	431	34	,	,	PUNCT
ejpam-5240	431	35	k	k	NOUN
ejpam-5240	431	36	)	)	PUNCT
ejpam-5240	431	37	.	.	PUNCT
ejpam-5240	432	1	hence	hence	ADV
ejpam-5240	432	2	,	,	PUNCT
ejpam-5240	432	3	gs(n	gs(n	NOUN
ejpam-5240	432	4	,	,	PUNCT
ejpam-5240	432	5	k	k	NOUN
ejpam-5240	432	6	)	)	PUNCT
ejpam-5240	432	7	is	be	AUX
ejpam-5240	432	8	a	a	DET
ejpam-5240	432	9	complete	complete	ADJ
ejpam-5240	432	10	graph	graph	NOUN
ejpam-5240	432	11	.	.	PUNCT
ejpam-5240	433	1	since	since	SCONJ
ejpam-5240	433	2	all	all	PRON
ejpam-5240	433	3	of	of	ADP
ejpam-5240	433	4	the	the	DET
ejpam-5240	433	5	vertex	vertex	NOUN
ejpam-5240	433	6	of	of	ADP
ejpam-5240	433	7	gs(n	gs(n	NOUN
ejpam-5240	433	8	,	,	PUNCT
ejpam-5240	433	9	k	k	NOUN
ejpam-5240	433	10	)	)	PUNCT
ejpam-5240	433	11	where	where	SCONJ
ejpam-5240	433	12	⌊	⌊	PROPN
ejpam-5240	433	13	n	n	ADV
ejpam-5240	433	14	2	2	NUM
ejpam-5240	433	15	⌋	⌋	NOUN
ejpam-5240	433	16	<	<	X
ejpam-5240	433	17	k	k	X
ejpam-5240	433	18	<	<	X
ejpam-5240	433	19	n	n	X
ejpam-5240	433	20	is	be	AUX
ejpam-5240	433	21	adjacent	adjacent	ADJ
ejpam-5240	433	22	to	to	ADP
ejpam-5240	433	23	each	each	DET
ejpam-5240	433	24	other	other	ADJ
ejpam-5240	433	25	,	,	PUNCT
ejpam-5240	433	26	it	it	PRON
ejpam-5240	433	27	follows	follow	VERB
ejpam-5240	433	28	that	that	SCONJ
ejpam-5240	433	29	every	every	DET
ejpam-5240	433	30	a	a	DET
ejpam-5240	433	31	∈	∈	PROPN
ejpam-5240	433	32	v	v	NOUN
ejpam-5240	433	33	(	(	PUNCT
ejpam-5240	433	34	gs(n	gs(n	NOUN
ejpam-5240	433	35	,	,	PUNCT
ejpam-5240	433	36	k	k	NOUN
ejpam-5240	433	37	)	)	PUNCT
ejpam-5240	433	38	)	)	PUNCT
ejpam-5240	433	39	bears	bear	VERB
ejpam-5240	433	40	an	an	DET
ejpam-5240	433	41	intersection	intersection	NOUN
ejpam-5240	433	42	to	to	ADP
ejpam-5240	433	43	any	any	DET
ejpam-5240	433	44	other	other	ADJ
ejpam-5240	433	45	vertex	vertex	NOUN
ejpam-5240	433	46	in	in	ADP
ejpam-5240	433	47	gs(n	gs(n	NOUN
ejpam-5240	433	48	,	,	PUNCT
ejpam-5240	433	49	k	k	NOUN
ejpam-5240	433	50	)	)	PUNCT
ejpam-5240	433	51	.	.	PUNCT
ejpam-5240	434	1	note	note	VERB
ejpam-5240	434	2	that	that	SCONJ
ejpam-5240	434	3	the	the	DET
ejpam-5240	434	4	trivial	trivial	ADJ
ejpam-5240	434	5	graph	graph	NOUN
ejpam-5240	434	6	is	be	AUX
ejpam-5240	434	7	a	a	DET
ejpam-5240	434	8	complete	complete	ADJ
ejpam-5240	434	9	graph	graph	NOUN
ejpam-5240	434	10	of	of	ADP
ejpam-5240	434	11	order	order	NOUN
ejpam-5240	434	12	1	1	NUM
ejpam-5240	434	13	.	.	PUNCT
ejpam-5240	434	14	meaning	mean	VERB
ejpam-5240	434	15	to	to	PART
ejpam-5240	434	16	say	say	VERB
ejpam-5240	434	17	,	,	PUNCT
ejpam-5240	434	18	gs(n,0	gs(n,0	PROPN
ejpam-5240	434	19	)	)	PUNCT
ejpam-5240	434	20	and	and	CCONJ
ejpam-5240	434	21	gs(n	gs(n	NOUN
ejpam-5240	434	22	,	,	PUNCT
ejpam-5240	434	23	n	n	CCONJ
ejpam-5240	434	24	)	)	PUNCT
ejpam-5240	434	25	are	be	AUX
ejpam-5240	434	26	complete	complete	ADJ
ejpam-5240	434	27	graphs	graph	NOUN
ejpam-5240	434	28	of	of	ADP
ejpam-5240	434	29	order	order	NOUN
ejpam-5240	434	30	1	1	NUM
ejpam-5240	434	31	.	.	PUNCT
ejpam-5240	434	32	illustration	illustration	NOUN
ejpam-5240	434	33	7	7	NUM
ejpam-5240	434	34	.	.	PUNCT
ejpam-5240	435	1	let	let	VERB
ejpam-5240	435	2	s5	s5	NOUN
ejpam-5240	435	3	be	be	AUX
ejpam-5240	435	4	equal	equal	ADJ
ejpam-5240	435	5	to	to	ADP
ejpam-5240	435	6	{	{	PUNCT
ejpam-5240	435	7	x1	x1	PROPN
ejpam-5240	435	8	,	,	PUNCT
ejpam-5240	435	9	x2	x2	PROPN
ejpam-5240	435	10	,	,	PUNCT
ejpam-5240	435	11	x3	x3	PROPN
ejpam-5240	435	12	,	,	PUNCT
ejpam-5240	435	13	x4	x4	PROPN
ejpam-5240	435	14	,	,	PUNCT
ejpam-5240	435	15	x5	x5	NOUN
ejpam-5240	435	16	}	}	PUNCT
ejpam-5240	435	17	and	and	CCONJ
ejpam-5240	435	18	let	let	VERB
ejpam-5240	435	19	k	k	PROPN
ejpam-5240	435	20	=	=	SYM
ejpam-5240	435	21	3	3	X
ejpam-5240	435	22	.	.	PUNCT
ejpam-5240	436	1	the	the	DET
ejpam-5240	436	2	vertex	vertex	NOUN
ejpam-5240	436	3	set	set	NOUN
ejpam-5240	436	4	of	of	ADP
ejpam-5240	436	5	the	the	DET
ejpam-5240	436	6	graph	graph	NOUN
ejpam-5240	436	7	gs(5,3	gs(5,3	NOUN
ejpam-5240	436	8	)	)	PUNCT
ejpam-5240	436	9	is	be	AUX
ejpam-5240	436	10	given	give	VERB
ejpam-5240	436	11	by	by	ADP
ejpam-5240	436	12	v	v	PROPN
ejpam-5240	436	13	(	(	PUNCT
ejpam-5240	436	14	gs(5,3	gs(5,3	NOUN
ejpam-5240	436	15	)	)	PUNCT
ejpam-5240	436	16	)	)	PUNCT
ejpam-5240	437	1	=	=	PRON
ejpam-5240	437	2	{	{	PUNCT
ejpam-5240	437	3	{	{	PUNCT
ejpam-5240	437	4	x1	x1	PROPN
ejpam-5240	437	5	,	,	PUNCT
ejpam-5240	437	6	x2	x2	PROPN
ejpam-5240	437	7	,	,	PUNCT
ejpam-5240	437	8	x3	x3	ADJ
ejpam-5240	437	9	}	}	PUNCT
ejpam-5240	437	10	,	,	PUNCT
ejpam-5240	437	11	{	{	PUNCT
ejpam-5240	437	12	x1	x1	PROPN
ejpam-5240	437	13	,	,	PUNCT
ejpam-5240	437	14	x2	x2	PROPN
ejpam-5240	437	15	,	,	PUNCT
ejpam-5240	437	16	x4	x4	PROPN
ejpam-5240	437	17	}	}	PUNCT
ejpam-5240	437	18	,	,	PUNCT
ejpam-5240	437	19	{	{	PUNCT
ejpam-5240	437	20	x1	x1	PROPN
ejpam-5240	437	21	,	,	PUNCT
ejpam-5240	437	22	x2	x2	PROPN
ejpam-5240	437	23	,	,	PUNCT
ejpam-5240	437	24	x5	x5	PROPN
ejpam-5240	437	25	}	}	PUNCT
ejpam-5240	437	26	,	,	PUNCT
ejpam-5240	437	27	{	{	PUNCT
ejpam-5240	437	28	x1	x1	ADJ
ejpam-5240	437	29	,	,	PUNCT
ejpam-5240	437	30	x3	x3	ADJ
ejpam-5240	437	31	,	,	PUNCT
ejpam-5240	437	32	x4	x4	PROPN
ejpam-5240	437	33	}	}	PUNCT
ejpam-5240	437	34	,	,	PUNCT
ejpam-5240	437	35	{	{	PUNCT
ejpam-5240	437	36	x1	x1	ADJ
ejpam-5240	437	37	,	,	PUNCT
ejpam-5240	437	38	x3	x3	ADJ
ejpam-5240	437	39	,	,	PUNCT
ejpam-5240	437	40	x5	x5	PROPN
ejpam-5240	437	41	}	}	PUNCT
ejpam-5240	437	42	,	,	PUNCT
ejpam-5240	437	43	{	{	PUNCT
ejpam-5240	437	44	x1	x1	PROPN
ejpam-5240	437	45	,	,	PUNCT
ejpam-5240	437	46	x4	x4	PROPN
ejpam-5240	437	47	,	,	PUNCT
ejpam-5240	437	48	x5	x5	PROPN
ejpam-5240	437	49	}	}	PUNCT
ejpam-5240	437	50	,	,	PUNCT
ejpam-5240	437	51	{	{	PUNCT
ejpam-5240	437	52	x2	x2	ADJ
ejpam-5240	437	53	,	,	PUNCT
ejpam-5240	437	54	x3	x3	ADJ
ejpam-5240	437	55	,	,	PUNCT
ejpam-5240	437	56	x4	x4	PROPN
ejpam-5240	437	57	}	}	PUNCT
ejpam-5240	437	58	,	,	PUNCT
ejpam-5240	437	59	{	{	PUNCT
ejpam-5240	437	60	x2	x2	ADJ
ejpam-5240	437	61	,	,	PUNCT
ejpam-5240	437	62	x3	x3	ADJ
ejpam-5240	437	63	,	,	PUNCT
ejpam-5240	437	64	x5	x5	PROPN
ejpam-5240	437	65	}	}	PUNCT
ejpam-5240	437	66	,	,	PUNCT
ejpam-5240	437	67	{	{	PUNCT
ejpam-5240	437	68	x2	x2	PROPN
ejpam-5240	437	69	,	,	PUNCT
ejpam-5240	437	70	x4	x4	PROPN
ejpam-5240	437	71	,	,	PUNCT
ejpam-5240	437	72	x5	x5	PROPN
ejpam-5240	437	73	}	}	PUNCT
ejpam-5240	437	74	,	,	PUNCT
ejpam-5240	437	75	{	{	PUNCT
ejpam-5240	437	76	x3	x3	ADJ
ejpam-5240	437	77	,	,	PUNCT
ejpam-5240	437	78	x4	x4	PROPN
ejpam-5240	437	79	,	,	PUNCT
ejpam-5240	437	80	x5	x5	PROPN
ejpam-5240	437	81	}	}	PUNCT
ejpam-5240	437	82	}	}	PUNCT
ejpam-5240	437	83	.	.	PUNCT
ejpam-5240	438	1	m.e	m.e	PROPN
ejpam-5240	438	2	.	.	PROPN
ejpam-5240	438	3	pelagio	pelagio	PROPN
ejpam-5240	438	4	,	,	PUNCT
ejpam-5240	438	5	n.	n.	NOUN
ejpam-5240	438	6	mame	mame	PROPN
ejpam-5240	438	7	,	,	PUNCT
ejpam-5240	438	8	k.	k.	PROPN
ejpam-5240	438	9	mendoza	mendoza	PROPN
ejpam-5240	438	10	/	/	SYM
ejpam-5240	438	11	eur	eur	PROPN
ejpam-5240	438	12	.	.	PUNCT
ejpam-5240	439	1	j.	j.	PROPN
ejpam-5240	439	2	pure	pure	PROPN
ejpam-5240	439	3	appl	appl	PROPN
ejpam-5240	439	4	.	.	PROPN
ejpam-5240	439	5	math	math	PROPN
ejpam-5240	439	6	,	,	PUNCT
ejpam-5240	439	7	17	17	NUM
ejpam-5240	439	8	(	(	PUNCT
ejpam-5240	439	9	3	3	NUM
ejpam-5240	439	10	)	)	PUNCT
ejpam-5240	439	11	(	(	PUNCT
ejpam-5240	439	12	2024	2024	NUM
ejpam-5240	439	13	)	)	PUNCT
ejpam-5240	439	14	,	,	PUNCT
ejpam-5240	439	15	1779	1779	NUM
ejpam-5240	439	16	-	-	SYM
ejpam-5240	439	17	1803	1803	NUM
ejpam-5240	439	18	1792	1792	NUM
ejpam-5240	439	19	moreso	moreso	NOUN
ejpam-5240	439	20	,	,	PUNCT
ejpam-5240	439	21	gs(5,3	gs(5,3	PROPN
ejpam-5240	439	22	)	)	PUNCT
ejpam-5240	439	23	is	be	AUX
ejpam-5240	439	24	pictorially	pictorially	ADV
ejpam-5240	439	25	illustrated	illustrate	VERB
ejpam-5240	439	26	in	in	ADP
ejpam-5240	439	27	figure	figure	NOUN
ejpam-5240	439	28	7	7	NUM
ejpam-5240	439	29	.	.	PUNCT
ejpam-5240	439	30	{	{	PUNCT
ejpam-5240	440	1	x1	x1	PROPN
ejpam-5240	440	2	,	,	PUNCT
ejpam-5240	440	3	x2	x2	PROPN
ejpam-5240	440	4	,	,	PUNCT
ejpam-5240	440	5	x3	x3	ADJ
ejpam-5240	440	6	}	}	PUNCT
ejpam-5240	440	7	{	{	PUNCT
ejpam-5240	440	8	x2	x2	PROPN
ejpam-5240	440	9	,	,	PUNCT
ejpam-5240	440	10	x4	x4	PROPN
ejpam-5240	440	11	,	,	PUNCT
ejpam-5240	440	12	x5	x5	PROPN
ejpam-5240	440	13	}	}	PUNCT
ejpam-5240	440	14	{	{	PUNCT
ejpam-5240	440	15	x1	x1	PROPN
ejpam-5240	440	16	,	,	PUNCT
ejpam-5240	440	17	x2	x2	PROPN
ejpam-5240	440	18	,	,	PUNCT
ejpam-5240	440	19	x5	x5	PROPN
ejpam-5240	440	20	}	}	PUNCT
ejpam-5240	440	21	{	{	PUNCT
ejpam-5240	440	22	x2	x2	PROPN
ejpam-5240	440	23	,	,	PUNCT
ejpam-5240	440	24	x3	x3	ADJ
ejpam-5240	440	25	,	,	PUNCT
ejpam-5240	440	26	x5	x5	NOUN
ejpam-5240	440	27	}	}	PUNCT
ejpam-5240	440	28	{	{	PUNCT
ejpam-5240	440	29	x1	x1	PROPN
ejpam-5240	440	30	,	,	PUNCT
ejpam-5240	440	31	x3	x3	ADJ
ejpam-5240	440	32	,	,	PUNCT
ejpam-5240	440	33	x4	x4	PROPN
ejpam-5240	440	34	}	}	PUNCT
ejpam-5240	440	35	{	{	PUNCT
ejpam-5240	440	36	x1	x1	PROPN
ejpam-5240	440	37	,	,	PUNCT
ejpam-5240	440	38	x4	x4	PROPN
ejpam-5240	440	39	,	,	PUNCT
ejpam-5240	440	40	x5	x5	PROPN
ejpam-5240	440	41	}	}	PUNCT
ejpam-5240	440	42	{	{	PUNCT
ejpam-5240	440	43	x1	x1	PROPN
ejpam-5240	440	44	,	,	PUNCT
ejpam-5240	440	45	x2	x2	PROPN
ejpam-5240	440	46	,	,	PUNCT
ejpam-5240	440	47	x4	x4	PROPN
ejpam-5240	440	48	}	}	PUNCT
ejpam-5240	440	49	{	{	PUNCT
ejpam-5240	440	50	x1	x1	PROPN
ejpam-5240	440	51	,	,	PUNCT
ejpam-5240	440	52	x3	x3	ADJ
ejpam-5240	440	53	,	,	PUNCT
ejpam-5240	440	54	x5	x5	NOUN
ejpam-5240	440	55	}	}	PUNCT
ejpam-5240	440	56	{	{	PUNCT
ejpam-5240	440	57	x3	x3	PROPN
ejpam-5240	440	58	,	,	PUNCT
ejpam-5240	440	59	x4	x4	PROPN
ejpam-5240	440	60	,	,	PUNCT
ejpam-5240	440	61	x5	x5	PROPN
ejpam-5240	440	62	}	}	PUNCT
ejpam-5240	440	63	{	{	PUNCT
ejpam-5240	440	64	x2	x2	PROPN
ejpam-5240	440	65	,	,	PUNCT
ejpam-5240	440	66	x3	x3	ADJ
ejpam-5240	440	67	,	,	PUNCT
ejpam-5240	440	68	x4	x4	ADJ
ejpam-5240	440	69	}	}	PUNCT
ejpam-5240	440	70	figure	figure	VERB
ejpam-5240	440	71	7	7	NUM
ejpam-5240	440	72	:	:	PUNCT
ejpam-5240	440	73	a	a	DET
ejpam-5240	440	74	pictorial	pictorial	ADJ
ejpam-5240	440	75	illustration	illustration	NOUN
ejpam-5240	440	76	of	of	ADP
ejpam-5240	440	77	gs(5,3	gs(5,3	PROPN
ejpam-5240	440	78	)	)	PUNCT
ejpam-5240	440	79	.	.	PUNCT
ejpam-5240	441	1	observe	observe	VERB
ejpam-5240	441	2	that	that	SCONJ
ejpam-5240	441	3	every	every	DET
ejpam-5240	441	4	vertex	vertex	NOUN
ejpam-5240	441	5	in	in	ADP
ejpam-5240	441	6	gs(5,3	gs(5,3	NOUN
ejpam-5240	441	7	)	)	PUNCT
ejpam-5240	441	8	is	be	AUX
ejpam-5240	441	9	adjacent	adjacent	ADJ
ejpam-5240	441	10	to	to	ADP
ejpam-5240	441	11	each	each	DET
ejpam-5240	441	12	other	other	ADJ
ejpam-5240	441	13	.	.	PUNCT
ejpam-5240	442	1	since	since	SCONJ
ejpam-5240	442	2	k	k	PROPN
ejpam-5240	442	3	=	=	SYM
ejpam-5240	442	4	3	3	NUM
ejpam-5240	442	5	and	and	CCONJ
ejpam-5240	442	6	⌊	⌊	VERB
ejpam-5240	442	7	5	5	NUM
ejpam-5240	442	8	2	2	NUM
ejpam-5240	442	9	⌋	⌋	NOUN
ejpam-5240	442	10	=	=	SYM
ejpam-5240	442	11	2	2	NUM
ejpam-5240	442	12	<	<	X
ejpam-5240	442	13	3	3	NUM
ejpam-5240	442	14	,	,	PUNCT
ejpam-5240	442	15	by	by	ADP
ejpam-5240	442	16	theorem	theorem	NOUN
ejpam-5240	442	17	7	7	NUM
ejpam-5240	442	18	,	,	PUNCT
ejpam-5240	442	19	it	it	PRON
ejpam-5240	442	20	follows	follow	VERB
ejpam-5240	442	21	that	that	SCONJ
ejpam-5240	442	22	gs(5,3	gs(5,3	PROPN
ejpam-5240	442	23	)	)	PUNCT
ejpam-5240	442	24	is	be	AUX
ejpam-5240	442	25	a	a	DET
ejpam-5240	442	26	complete	complete	ADJ
ejpam-5240	442	27	graph	graph	NOUN
ejpam-5240	442	28	of	of	ADP
ejpam-5240	442	29	order	order	NOUN
ejpam-5240	442	30	(	(	PUNCT
ejpam-5240	442	31	5	5	NUM
ejpam-5240	442	32	3	3	NUM
ejpam-5240	442	33	)	)	PUNCT
ejpam-5240	443	1	=	=	SYM
ejpam-5240	443	2	10	10	NUM
ejpam-5240	443	3	.	.	PUNCT
ejpam-5240	444	1	the	the	DET
ejpam-5240	444	2	next	next	ADJ
ejpam-5240	444	3	theorem	theorem	NOUN
ejpam-5240	444	4	imparts	impart	VERB
ejpam-5240	444	5	the	the	DET
ejpam-5240	444	6	necessary	necessary	ADJ
ejpam-5240	444	7	and	and	CCONJ
ejpam-5240	444	8	sufficient	sufficient	ADJ
ejpam-5240	444	9	conditions	condition	NOUN
ejpam-5240	444	10	for	for	ADP
ejpam-5240	444	11	gs(n	gs(n	NOUN
ejpam-5240	444	12	,	,	PUNCT
ejpam-5240	444	13	k	k	NOUN
ejpam-5240	444	14	)	)	PUNCT
ejpam-5240	444	15	to	to	PART
ejpam-5240	444	16	be	be	AUX
ejpam-5240	444	17	a	a	DET
ejpam-5240	444	18	cycle	cycle	NOUN
ejpam-5240	444	19	graph	graph	NOUN
ejpam-5240	444	20	.	.	PUNCT
ejpam-5240	445	1	theorem	theorem	ADJ
ejpam-5240	445	2	8	8	NUM
ejpam-5240	445	3	.	.	PUNCT
ejpam-5240	446	1	a	a	DET
ejpam-5240	446	2	k	k	ADV
ejpam-5240	446	3	-	-	PUNCT
ejpam-5240	446	4	restricted	restricted	ADJ
ejpam-5240	446	5	intersection	intersection	NOUN
ejpam-5240	446	6	graph	graph	NOUN
ejpam-5240	446	7	gs(n	gs(n	NOUN
ejpam-5240	446	8	,	,	PUNCT
ejpam-5240	446	9	k	k	NOUN
ejpam-5240	446	10	)	)	PUNCT
ejpam-5240	446	11	is	be	AUX
ejpam-5240	446	12	a	a	DET
ejpam-5240	446	13	cycle	cycle	NOUN
ejpam-5240	446	14	graph	graph	NOUN
ejpam-5240	446	15	of	of	ADP
ejpam-5240	446	16	order	order	NOUN
ejpam-5240	446	17	3	3	NUM
ejpam-5240	446	18	if	if	SCONJ
ejpam-5240	446	19	and	and	CCONJ
ejpam-5240	446	20	only	only	ADV
ejpam-5240	446	21	if	if	SCONJ
ejpam-5240	446	22	n	n	PROPN
ejpam-5240	446	23	=	=	SYM
ejpam-5240	446	24	3	3	NUM
ejpam-5240	446	25	and	and	CCONJ
ejpam-5240	446	26	k	k	NOUN
ejpam-5240	446	27	=	=	NOUN
ejpam-5240	446	28	2	2	X
ejpam-5240	446	29	.	.	PUNCT
ejpam-5240	446	30	proof	proof	NOUN
ejpam-5240	446	31	.	.	PUNCT
ejpam-5240	447	1	assume	assume	VERB
ejpam-5240	447	2	that	that	SCONJ
ejpam-5240	447	3	gs(n	gs(n	NOUN
ejpam-5240	447	4	,	,	PUNCT
ejpam-5240	447	5	k	k	NOUN
ejpam-5240	447	6	)	)	PUNCT
ejpam-5240	447	7	is	be	AUX
ejpam-5240	447	8	a	a	DET
ejpam-5240	447	9	cycle	cycle	NOUN
ejpam-5240	447	10	graph	graph	NOUN
ejpam-5240	447	11	of	of	ADP
ejpam-5240	447	12	order	order	NOUN
ejpam-5240	447	13	3	3	X
ejpam-5240	447	14	.	.	PUNCT
ejpam-5240	448	1	it	it	PRON
ejpam-5240	448	2	is	be	AUX
ejpam-5240	448	3	noted	note	VERB
ejpam-5240	448	4	that	that	SCONJ
ejpam-5240	448	5	every	every	DET
ejpam-5240	448	6	cycle	cycle	NOUN
ejpam-5240	448	7	graph	graph	NOUN
ejpam-5240	448	8	is	be	AUX
ejpam-5240	448	9	a	a	DET
ejpam-5240	448	10	2	2	NUM
ejpam-5240	448	11	-	-	PUNCT
ejpam-5240	448	12	regular	regular	ADJ
ejpam-5240	448	13	graph	graph	NOUN
ejpam-5240	448	14	.	.	PUNCT
ejpam-5240	448	15	suppose	suppose	VERB
ejpam-5240	448	16	that	that	SCONJ
ejpam-5240	448	17	n	n	PROPN
ejpam-5240	448	18	̸=	̸=	PROPN
ejpam-5240	448	19	3	3	NUM
ejpam-5240	448	20	or	or	CCONJ
ejpam-5240	448	21	k	k	PROPN
ejpam-5240	448	22	̸=	̸=	PROPN
ejpam-5240	448	23	2	2	NUM
ejpam-5240	448	24	.	.	PUNCT
ejpam-5240	449	1	if	if	SCONJ
ejpam-5240	449	2	n	n	X
ejpam-5240	449	3	<	<	X
ejpam-5240	449	4	3	3	NUM
ejpam-5240	449	5	,	,	PUNCT
ejpam-5240	449	6	then	then	ADV
ejpam-5240	449	7	(	(	PUNCT
ejpam-5240	449	8	n	n	X
ejpam-5240	449	9	k	k	NOUN
ejpam-5240	449	10	)	)	PUNCT
ejpam-5240	449	11	≤	≤	NOUN
ejpam-5240	449	12	2	2	NUM
ejpam-5240	449	13	,	,	PUNCT
ejpam-5240	449	14	which	which	PRON
ejpam-5240	449	15	is	be	AUX
ejpam-5240	449	16	a	a	DET
ejpam-5240	449	17	contradiction	contradiction	NOUN
ejpam-5240	449	18	to	to	ADP
ejpam-5240	449	19	the	the	DET
ejpam-5240	449	20	assumption	assumption	NOUN
ejpam-5240	450	1	that	that	SCONJ
ejpam-5240	450	2	|v	|v	PROPN
ejpam-5240	450	3	(	(	PUNCT
ejpam-5240	450	4	gs(n	gs(n	NOUN
ejpam-5240	450	5	,	,	PUNCT
ejpam-5240	450	6	k	k	NOUN
ejpam-5240	450	7	)	)	PUNCT
ejpam-5240	450	8	)	)	PUNCT
ejpam-5240	450	9	|	|	ADV
ejpam-5240	450	10	=	=	SYM
ejpam-5240	450	11	3	3	X
ejpam-5240	450	12	.	.	PUNCT
ejpam-5240	451	1	moreover	moreover	ADV
ejpam-5240	451	2	,	,	PUNCT
ejpam-5240	451	3	if	if	SCONJ
ejpam-5240	451	4	n	n	PROPN
ejpam-5240	451	5	>	>	X
ejpam-5240	451	6	3	3	NUM
ejpam-5240	451	7	,	,	PUNCT
ejpam-5240	451	8	then	then	ADV
ejpam-5240	451	9	(	(	PUNCT
ejpam-5240	451	10	n	n	X
ejpam-5240	451	11	k	k	NOUN
ejpam-5240	451	12	)	)	PUNCT
ejpam-5240	452	1	=	=	SYM
ejpam-5240	452	2	1	1	NUM
ejpam-5240	452	3	or	or	CCONJ
ejpam-5240	452	4	(	(	PUNCT
ejpam-5240	452	5	n	n	CCONJ
ejpam-5240	452	6	k	k	NOUN
ejpam-5240	452	7	)	)	PUNCT
ejpam-5240	452	8	≥	≥	NOUN
ejpam-5240	452	9	4	4	NUM
ejpam-5240	452	10	.	.	PUNCT
ejpam-5240	453	1	this	this	PRON
ejpam-5240	453	2	is	be	AUX
ejpam-5240	453	3	also	also	ADV
ejpam-5240	453	4	a	a	DET
ejpam-5240	453	5	contradiction	contradiction	NOUN
ejpam-5240	453	6	since	since	SCONJ
ejpam-5240	453	7	the	the	DET
ejpam-5240	453	8	order	order	NOUN
ejpam-5240	453	9	of	of	ADP
ejpam-5240	453	10	gs(n	gs(n	NOUN
ejpam-5240	453	11	,	,	PUNCT
ejpam-5240	453	12	k	k	NOUN
ejpam-5240	453	13	)	)	PUNCT
ejpam-5240	453	14	is	be	AUX
ejpam-5240	453	15	3	3	NUM
ejpam-5240	453	16	.	.	PUNCT
ejpam-5240	454	1	on	on	ADP
ejpam-5240	454	2	the	the	DET
ejpam-5240	454	3	other	other	ADJ
ejpam-5240	454	4	hand	hand	NOUN
ejpam-5240	454	5	,	,	PUNCT
ejpam-5240	454	6	if	if	SCONJ
ejpam-5240	454	7	k	k	PROPN
ejpam-5240	454	8	<	<	X
ejpam-5240	454	9	2	2	NUM
ejpam-5240	454	10	,	,	PUNCT
ejpam-5240	454	11	then	then	ADV
ejpam-5240	454	12	k	k	PROPN
ejpam-5240	454	13	is	be	AUX
ejpam-5240	454	14	0	0	NUM
ejpam-5240	454	15	or	or	CCONJ
ejpam-5240	454	16	1	1	NUM
ejpam-5240	454	17	.	.	PUNCT
ejpam-5240	454	18	by	by	ADP
ejpam-5240	454	19	theorem	theorem	NOUN
ejpam-5240	454	20	3	3	NUM
ejpam-5240	454	21	or	or	CCONJ
ejpam-5240	454	22	by	by	ADP
ejpam-5240	454	23	theorem	theorem	ADJ
ejpam-5240	454	24	4	4	NUM
ejpam-5240	454	25	,	,	PUNCT
ejpam-5240	454	26	gs(n	gs(n	NOUN
ejpam-5240	454	27	,	,	PUNCT
ejpam-5240	454	28	k	k	NOUN
ejpam-5240	454	29	)	)	PUNCT
ejpam-5240	454	30	is	be	AUX
ejpam-5240	454	31	a	a	DET
ejpam-5240	454	32	trivial	trivial	ADJ
ejpam-5240	454	33	graph	graph	NOUN
ejpam-5240	454	34	or	or	CCONJ
ejpam-5240	454	35	an	an	DET
ejpam-5240	454	36	empty	empty	ADJ
ejpam-5240	454	37	graph	graph	NOUN
ejpam-5240	454	38	of	of	ADP
ejpam-5240	454	39	order	order	NOUN
ejpam-5240	454	40	n	n	CCONJ
ejpam-5240	454	41	,	,	PUNCT
ejpam-5240	454	42	respectively	respectively	ADV
ejpam-5240	454	43	.	.	PUNCT
ejpam-5240	455	1	hence	hence	ADV
ejpam-5240	455	2	,	,	PUNCT
ejpam-5240	455	3	this	this	PRON
ejpam-5240	455	4	is	be	AUX
ejpam-5240	455	5	a	a	DET
ejpam-5240	455	6	contradiction	contradiction	NOUN
ejpam-5240	455	7	to	to	ADP
ejpam-5240	455	8	the	the	DET
ejpam-5240	455	9	assumption	assumption	NOUN
ejpam-5240	455	10	that	that	SCONJ
ejpam-5240	455	11	gs(n	gs(n	NOUN
ejpam-5240	455	12	,	,	PUNCT
ejpam-5240	455	13	k	k	NOUN
ejpam-5240	455	14	)	)	PUNCT
ejpam-5240	455	15	is	be	AUX
ejpam-5240	455	16	a	a	DET
ejpam-5240	455	17	cycle	cycle	NOUN
ejpam-5240	455	18	graph	graph	NOUN
ejpam-5240	455	19	of	of	ADP
ejpam-5240	455	20	order	order	NOUN
ejpam-5240	455	21	3	3	X
ejpam-5240	455	22	.	.	PUNCT
ejpam-5240	456	1	furthermore	furthermore	ADV
ejpam-5240	456	2	,	,	PUNCT
ejpam-5240	456	3	if	if	SCONJ
ejpam-5240	456	4	k	k	PROPN
ejpam-5240	456	5	>	>	X
ejpam-5240	456	6	2	2	NUM
ejpam-5240	456	7	,	,	PUNCT
ejpam-5240	456	8	then	then	ADV
ejpam-5240	456	9	(	(	PUNCT
ejpam-5240	456	10	n	n	X
ejpam-5240	456	11	k	k	NOUN
ejpam-5240	456	12	)	)	PUNCT
ejpam-5240	456	13	is	be	AUX
ejpam-5240	456	14	either	either	ADV
ejpam-5240	456	15	equal	equal	ADJ
ejpam-5240	456	16	to	to	ADP
ejpam-5240	456	17	1	1	NUM
ejpam-5240	456	18	or	or	CCONJ
ejpam-5240	456	19	greater	great	ADJ
ejpam-5240	456	20	than	than	ADP
ejpam-5240	456	21	or	or	CCONJ
ejpam-5240	456	22	equal	equal	ADJ
ejpam-5240	456	23	to	to	ADP
ejpam-5240	456	24	4	4	NUM
ejpam-5240	456	25	.	.	PUNCT
ejpam-5240	457	1	this	this	PRON
ejpam-5240	457	2	is	be	AUX
ejpam-5240	457	3	a	a	DET
ejpam-5240	457	4	contradiction	contradiction	NOUN
ejpam-5240	457	5	also	also	ADV
ejpam-5240	457	6	to	to	ADP
ejpam-5240	457	7	the	the	DET
ejpam-5240	457	8	assumption	assumption	NOUN
ejpam-5240	457	9	that	that	SCONJ
ejpam-5240	457	10	gs(n	gs(n	NOUN
ejpam-5240	457	11	,	,	PUNCT
ejpam-5240	457	12	k	k	NOUN
ejpam-5240	457	13	)	)	PUNCT
ejpam-5240	457	14	is	be	AUX
ejpam-5240	457	15	a	a	DET
ejpam-5240	457	16	cycle	cycle	NOUN
ejpam-5240	457	17	graph	graph	NOUN
ejpam-5240	457	18	of	of	ADP
ejpam-5240	457	19	order	order	NOUN
ejpam-5240	457	20	3	3	X
ejpam-5240	457	21	.	.	PUNCT
ejpam-5240	458	1	therefore	therefore	ADV
ejpam-5240	458	2	,	,	PUNCT
ejpam-5240	458	3	n	n	PROPN
ejpam-5240	458	4	=	=	SYM
ejpam-5240	458	5	3	3	NUM
ejpam-5240	458	6	and	and	CCONJ
ejpam-5240	458	7	k	k	NOUN
ejpam-5240	458	8	=	=	NOUN
ejpam-5240	458	9	2	2	X
ejpam-5240	458	10	.	.	PUNCT
ejpam-5240	458	11	conversely	conversely	ADV
ejpam-5240	458	12	,	,	PUNCT
ejpam-5240	458	13	assume	assume	VERB
ejpam-5240	458	14	that	that	SCONJ
ejpam-5240	458	15	n	n	NOUN
ejpam-5240	458	16	=	=	SYM
ejpam-5240	458	17	3	3	NUM
ejpam-5240	458	18	and	and	CCONJ
ejpam-5240	458	19	k	k	NOUN
ejpam-5240	459	1	=	=	SYM
ejpam-5240	459	2	2	2	X
ejpam-5240	459	3	.	.	X
ejpam-5240	459	4	observe	observe	VERB
ejpam-5240	459	5	that	that	SCONJ
ejpam-5240	459	6	⌊	⌊	PROPN
ejpam-5240	459	7	3	3	NUM
ejpam-5240	459	8	2	2	NUM
ejpam-5240	459	9	⌋	⌋	NOUN
ejpam-5240	459	10	=	=	PUNCT
ejpam-5240	459	11	1	1	NUM
ejpam-5240	459	12	<	<	X
ejpam-5240	459	13	2	2	NUM
ejpam-5240	459	14	,	,	PUNCT
ejpam-5240	459	15	so	so	ADV
ejpam-5240	459	16	by	by	ADP
ejpam-5240	459	17	theorem	theorem	ADJ
ejpam-5240	459	18	7	7	NUM
ejpam-5240	459	19	,	,	PUNCT
ejpam-5240	459	20	gs(3,2	gs(3,2	NOUN
ejpam-5240	459	21	)	)	PUNCT
ejpam-5240	459	22	is	be	AUX
ejpam-5240	459	23	a	a	DET
ejpam-5240	459	24	complete	complete	ADJ
ejpam-5240	459	25	graph	graph	NOUN
ejpam-5240	459	26	of	of	ADP
ejpam-5240	459	27	order	order	NOUN
ejpam-5240	459	28	(	(	PUNCT
ejpam-5240	459	29	3	3	NUM
ejpam-5240	459	30	2	2	NUM
ejpam-5240	459	31	)	)	PUNCT
ejpam-5240	459	32	=	=	SYM
ejpam-5240	460	1	3	3	X
ejpam-5240	460	2	.	.	PUNCT
ejpam-5240	460	3	by	by	ADP
ejpam-5240	460	4	remark	remark	NOUN
ejpam-5240	460	5	5	5	NUM
ejpam-5240	460	6	,	,	PUNCT
ejpam-5240	460	7	since	since	SCONJ
ejpam-5240	460	8	c3	c3	PROPN
ejpam-5240	460	9	≃	≃	NOUN
ejpam-5240	460	10	k3	k3	PROPN
ejpam-5240	460	11	,	,	PUNCT
ejpam-5240	460	12	it	it	PRON
ejpam-5240	460	13	follows	follow	VERB
ejpam-5240	460	14	that	that	SCONJ
ejpam-5240	460	15	gs(3,2	gs(3,2	NOUN
ejpam-5240	460	16	)	)	PUNCT
ejpam-5240	460	17	is	be	AUX
ejpam-5240	460	18	a	a	DET
ejpam-5240	460	19	cycle	cycle	NOUN
ejpam-5240	460	20	graph	graph	NOUN
ejpam-5240	460	21	of	of	ADP
ejpam-5240	460	22	order	order	NOUN
ejpam-5240	460	23	3	3	X
ejpam-5240	460	24	.	.	PUNCT
ejpam-5240	461	1	the	the	DET
ejpam-5240	461	2	only	only	ADJ
ejpam-5240	461	3	time	time	NOUN
ejpam-5240	461	4	that	that	SCONJ
ejpam-5240	461	5	a	a	DET
ejpam-5240	461	6	cycle	cycle	NOUN
ejpam-5240	461	7	graph	graph	NOUN
ejpam-5240	461	8	is	be	AUX
ejpam-5240	461	9	isomorphic	isomorphic	ADJ
ejpam-5240	461	10	to	to	ADP
ejpam-5240	461	11	a	a	DET
ejpam-5240	461	12	complete	complete	ADJ
ejpam-5240	461	13	graph	graph	NOUN
ejpam-5240	461	14	is	be	AUX
ejpam-5240	461	15	when	when	SCONJ
ejpam-5240	461	16	they	they	PRON
ejpam-5240	461	17	have	have	VERB
ejpam-5240	461	18	an	an	DET
ejpam-5240	461	19	order	order	NOUN
ejpam-5240	461	20	equal	equal	ADJ
ejpam-5240	461	21	to	to	ADP
ejpam-5240	461	22	3	3	NUM
ejpam-5240	461	23	.	.	PUNCT
ejpam-5240	462	1	it	it	PRON
ejpam-5240	462	2	can	can	AUX
ejpam-5240	462	3	be	be	AUX
ejpam-5240	462	4	verified	verify	VERB
ejpam-5240	462	5	that	that	SCONJ
ejpam-5240	462	6	when	when	SCONJ
ejpam-5240	462	7	n	n	X
ejpam-5240	462	8	=	=	SYM
ejpam-5240	462	9	3	3	NUM
ejpam-5240	462	10	and	and	CCONJ
ejpam-5240	462	11	k	k	NOUN
ejpam-5240	462	12	=	=	SYM
ejpam-5240	462	13	2	2	NUM
ejpam-5240	462	14	,	,	PUNCT
ejpam-5240	462	15	gs(3,2	gs(3,2	NOUN
ejpam-5240	462	16	)	)	PUNCT
ejpam-5240	462	17	is	be	AUX
ejpam-5240	462	18	a	a	DET
ejpam-5240	462	19	complete	complete	ADJ
ejpam-5240	462	20	m.e	m.e	PROPN
ejpam-5240	462	21	.	.	PROPN
ejpam-5240	462	22	pelagio	pelagio	PROPN
ejpam-5240	462	23	,	,	PUNCT
ejpam-5240	462	24	n.	n.	NOUN
ejpam-5240	462	25	mame	mame	PROPN
ejpam-5240	462	26	,	,	PUNCT
ejpam-5240	462	27	k.	k.	PROPN
ejpam-5240	462	28	mendoza	mendoza	PROPN
ejpam-5240	462	29	/	/	SYM
ejpam-5240	462	30	eur	eur	PROPN
ejpam-5240	462	31	.	.	PUNCT
ejpam-5240	463	1	j.	j.	PROPN
ejpam-5240	463	2	pure	pure	PROPN
ejpam-5240	463	3	appl	appl	PROPN
ejpam-5240	463	4	.	.	PROPN
ejpam-5240	463	5	math	math	PROPN
ejpam-5240	463	6	,	,	PUNCT
ejpam-5240	463	7	17	17	NUM
ejpam-5240	463	8	(	(	PUNCT
ejpam-5240	463	9	3	3	NUM
ejpam-5240	463	10	)	)	PUNCT
ejpam-5240	463	11	(	(	PUNCT
ejpam-5240	463	12	2024	2024	NUM
ejpam-5240	463	13	)	)	PUNCT
ejpam-5240	463	14	,	,	PUNCT
ejpam-5240	463	15	1779	1779	NUM
ejpam-5240	463	16	-	-	SYM
ejpam-5240	463	17	1803	1803	NUM
ejpam-5240	463	18	1793	1793	NUM
ejpam-5240	463	19	graph	graph	NOUN
ejpam-5240	463	20	of	of	ADP
ejpam-5240	463	21	order	order	NOUN
ejpam-5240	463	22	3	3	NUM
ejpam-5240	463	23	which	which	PRON
ejpam-5240	463	24	implies	imply	VERB
ejpam-5240	463	25	that	that	SCONJ
ejpam-5240	463	26	gs(3,2	gs(3,2	NOUN
ejpam-5240	463	27	)	)	PUNCT
ejpam-5240	463	28	is	be	AUX
ejpam-5240	463	29	also	also	ADV
ejpam-5240	463	30	a	a	DET
ejpam-5240	463	31	cycle	cycle	NOUN
ejpam-5240	463	32	graph	graph	NOUN
ejpam-5240	463	33	of	of	ADP
ejpam-5240	463	34	order	order	NOUN
ejpam-5240	463	35	3	3	X
ejpam-5240	463	36	.	.	PUNCT
ejpam-5240	464	1	hence	hence	ADV
ejpam-5240	464	2	,	,	PUNCT
ejpam-5240	464	3	for	for	ADP
ejpam-5240	464	4	any	any	DET
ejpam-5240	464	5	3	3	NUM
ejpam-5240	464	6	-	-	PUNCT
ejpam-5240	464	7	element	element	NOUN
ejpam-5240	464	8	set	set	NOUN
ejpam-5240	464	9	s3	s3	PROPN
ejpam-5240	464	10	,	,	PUNCT
ejpam-5240	464	11	gs(3,2	gs(3,2	NOUN
ejpam-5240	464	12	)	)	PUNCT
ejpam-5240	464	13	is	be	AUX
ejpam-5240	464	14	a	a	DET
ejpam-5240	464	15	cycle	cycle	NOUN
ejpam-5240	464	16	graph	graph	NOUN
ejpam-5240	464	17	as	as	ADV
ejpam-5240	464	18	well	well	ADV
ejpam-5240	464	19	as	as	ADP
ejpam-5240	464	20	a	a	DET
ejpam-5240	464	21	complete	complete	ADJ
ejpam-5240	464	22	graph	graph	NOUN
ejpam-5240	464	23	.	.	PUNCT
ejpam-5240	465	1	illustration	illustration	NOUN
ejpam-5240	465	2	8	8	NUM
ejpam-5240	465	3	.	.	PUNCT
ejpam-5240	466	1	consider	consider	VERB
ejpam-5240	466	2	the	the	DET
ejpam-5240	466	3	set	set	NOUN
ejpam-5240	466	4	s3	s3	NOUN
ejpam-5240	466	5	=	=	SYM
ejpam-5240	466	6	{	{	PUNCT
ejpam-5240	466	7	x1	x1	PROPN
ejpam-5240	466	8	,	,	PUNCT
ejpam-5240	466	9	x2	x2	PROPN
ejpam-5240	466	10	,	,	PUNCT
ejpam-5240	466	11	x3	x3	ADJ
ejpam-5240	466	12	}	}	PUNCT
ejpam-5240	466	13	and	and	CCONJ
ejpam-5240	466	14	let	let	VERB
ejpam-5240	466	15	k	k	NOUN
ejpam-5240	466	16	=	=	SYM
ejpam-5240	466	17	2	2	X
ejpam-5240	466	18	.	.	PUNCT
ejpam-5240	467	1	then	then	ADV
ejpam-5240	467	2	the	the	DET
ejpam-5240	467	3	vertex	vertex	NOUN
ejpam-5240	467	4	set	set	NOUN
ejpam-5240	467	5	of	of	ADP
ejpam-5240	467	6	gs(3,2	gs(3,2	NOUN
ejpam-5240	467	7	)	)	PUNCT
ejpam-5240	467	8	is	be	AUX
ejpam-5240	467	9	given	give	VERB
ejpam-5240	467	10	by	by	ADP
ejpam-5240	467	11	{	{	PUNCT
ejpam-5240	467	12	{	{	PUNCT
ejpam-5240	467	13	x1	x1	PROPN
ejpam-5240	467	14	,	,	PUNCT
ejpam-5240	467	15	x2	x2	PROPN
ejpam-5240	467	16	}	}	PUNCT
ejpam-5240	467	17	,	,	PUNCT
ejpam-5240	467	18	{	{	PUNCT
ejpam-5240	467	19	x1	x1	ADJ
ejpam-5240	467	20	,	,	PUNCT
ejpam-5240	467	21	x3	x3	ADJ
ejpam-5240	467	22	}	}	PUNCT
ejpam-5240	467	23	,	,	PUNCT
ejpam-5240	467	24	{	{	PUNCT
ejpam-5240	467	25	x2	x2	ADJ
ejpam-5240	467	26	,	,	PUNCT
ejpam-5240	467	27	x3	x3	ADJ
ejpam-5240	467	28	}	}	PUNCT
ejpam-5240	467	29	}	}	PUNCT
ejpam-5240	467	30	.	.	PUNCT
ejpam-5240	468	1	the	the	DET
ejpam-5240	468	2	order	order	NOUN
ejpam-5240	468	3	of	of	ADP
ejpam-5240	468	4	gs(3,2	gs(3,2	NOUN
ejpam-5240	468	5	)	)	PUNCT
ejpam-5240	468	6	is	be	AUX
ejpam-5240	468	7	3	3	NUM
ejpam-5240	468	8	.	.	PUNCT
ejpam-5240	469	1	since	since	SCONJ
ejpam-5240	469	2	k	k	PROPN
ejpam-5240	469	3	=	=	SYM
ejpam-5240	469	4	3	3	NUM
ejpam-5240	469	5	,	,	PUNCT
ejpam-5240	469	6	which	which	PRON
ejpam-5240	469	7	means	mean	VERB
ejpam-5240	469	8	that	that	SCONJ
ejpam-5240	469	9	⌊	⌊	VERB
ejpam-5240	469	10	3	3	NUM
ejpam-5240	469	11	2	2	NUM
ejpam-5240	469	12	⌋	⌋	NOUN
ejpam-5240	469	13	=	=	PUNCT
ejpam-5240	469	14	1	1	NUM
ejpam-5240	469	15	<	<	X
ejpam-5240	469	16	k	k	X
ejpam-5240	469	17	,	,	PUNCT
ejpam-5240	469	18	by	by	ADP
ejpam-5240	469	19	theorem	theorem	NOUN
ejpam-5240	469	20	7	7	NUM
ejpam-5240	469	21	,	,	PUNCT
ejpam-5240	469	22	it	it	PRON
ejpam-5240	469	23	follows	follow	VERB
ejpam-5240	469	24	that	that	SCONJ
ejpam-5240	469	25	gs(3,2	gs(3,2	NOUN
ejpam-5240	469	26	)	)	PUNCT
ejpam-5240	469	27	is	be	AUX
ejpam-5240	469	28	a	a	DET
ejpam-5240	469	29	complete	complete	ADJ
ejpam-5240	469	30	graph	graph	NOUN
ejpam-5240	469	31	of	of	ADP
ejpam-5240	469	32	order	order	NOUN
ejpam-5240	469	33	3	3	X
ejpam-5240	469	34	.	.	PUNCT
ejpam-5240	470	1	by	by	ADP
ejpam-5240	470	2	remark	remark	NOUN
ejpam-5240	470	3	5	5	NUM
ejpam-5240	470	4	,	,	PUNCT
ejpam-5240	470	5	c3	c3	PROPN
ejpam-5240	470	6	≃	≃	NOUN
ejpam-5240	470	7	k3	k3	VERB
ejpam-5240	470	8	.	.	PUNCT
ejpam-5240	471	1	hence	hence	ADV
ejpam-5240	471	2	,	,	PUNCT
ejpam-5240	471	3	gs(3,2	gs(3,2	NOUN
ejpam-5240	471	4	)	)	PUNCT
ejpam-5240	471	5	is	be	AUX
ejpam-5240	471	6	also	also	ADV
ejpam-5240	471	7	a	a	DET
ejpam-5240	471	8	cycle	cycle	NOUN
ejpam-5240	471	9	graph	graph	NOUN
ejpam-5240	471	10	of	of	ADP
ejpam-5240	471	11	order	order	NOUN
ejpam-5240	471	12	3	3	NUM
ejpam-5240	471	13	.	.	PUNCT
ejpam-5240	471	14	shown	show	VERB
ejpam-5240	471	15	in	in	ADP
ejpam-5240	471	16	figure	figure	NOUN
ejpam-5240	471	17	8	8	NUM
ejpam-5240	471	18	is	be	AUX
ejpam-5240	471	19	a	a	DET
ejpam-5240	471	20	pictorial	pictorial	ADJ
ejpam-5240	471	21	illustration	illustration	NOUN
ejpam-5240	471	22	of	of	ADP
ejpam-5240	471	23	gs(3,2	gs(3,2	NOUN
ejpam-5240	471	24	)	)	PUNCT
ejpam-5240	471	25	.	.	PUNCT
ejpam-5240	472	1	{	{	PUNCT
ejpam-5240	472	2	x1	x1	ADJ
ejpam-5240	472	3	,	,	PUNCT
ejpam-5240	472	4	x2	x2	PROPN
ejpam-5240	472	5	}	}	PUNCT
ejpam-5240	472	6	{	{	PUNCT
ejpam-5240	472	7	x1	x1	PROPN
ejpam-5240	472	8	,	,	PUNCT
ejpam-5240	472	9	x3	x3	ADJ
ejpam-5240	472	10	}	}	PUNCT
ejpam-5240	472	11	{	{	PUNCT
ejpam-5240	472	12	x2	x2	PROPN
ejpam-5240	472	13	,	,	PUNCT
ejpam-5240	472	14	x3	x3	ADJ
ejpam-5240	472	15	}	}	PUNCT
ejpam-5240	472	16	figure	figure	NOUN
ejpam-5240	472	17	8	8	NUM
ejpam-5240	472	18	:	:	PUNCT
ejpam-5240	472	19	pictorial	pictorial	ADJ
ejpam-5240	472	20	representation	representation	NOUN
ejpam-5240	472	21	of	of	ADP
ejpam-5240	472	22	gs(3,2	gs(3,2	NOUN
ejpam-5240	472	23	)	)	PUNCT
ejpam-5240	472	24	.	.	PUNCT
ejpam-5240	473	1	note	note	VERB
ejpam-5240	473	2	that	that	SCONJ
ejpam-5240	473	3	gs(n	gs(n	NOUN
ejpam-5240	473	4	,	,	PUNCT
ejpam-5240	473	5	k	k	NOUN
ejpam-5240	473	6	)	)	PUNCT
ejpam-5240	473	7	is	be	AUX
ejpam-5240	473	8	a	a	DET
ejpam-5240	473	9	simple	simple	ADJ
ejpam-5240	473	10	graph	graph	NOUN
ejpam-5240	473	11	.	.	PUNCT
ejpam-5240	474	1	now	now	ADV
ejpam-5240	474	2	,	,	PUNCT
ejpam-5240	474	3	we	we	PRON
ejpam-5240	474	4	consider	consider	VERB
ejpam-5240	474	5	the	the	DET
ejpam-5240	474	6	complement	complement	NOUN
ejpam-5240	474	7	of	of	ADP
ejpam-5240	474	8	gs(n	gs(n	NOUN
ejpam-5240	474	9	,	,	PUNCT
ejpam-5240	474	10	k	k	NOUN
ejpam-5240	474	11	)	)	PUNCT
ejpam-5240	474	12	,	,	PUNCT
ejpam-5240	474	13	denoted	denote	VERB
ejpam-5240	474	14	by	by	ADP
ejpam-5240	474	15	gs(n	gs(n	NOUN
ejpam-5240	474	16	,	,	PUNCT
ejpam-5240	474	17	k	k	NOUN
ejpam-5240	474	18	)	)	PUNCT
ejpam-5240	474	19	,	,	PUNCT
ejpam-5240	474	20	as	as	ADP
ejpam-5240	474	21	a	a	DET
ejpam-5240	474	22	simple	simple	ADJ
ejpam-5240	474	23	graph	graph	NOUN
ejpam-5240	474	24	.	.	PUNCT
ejpam-5240	475	1	the	the	DET
ejpam-5240	475	2	next	next	ADJ
ejpam-5240	475	3	theorem	theorem	ADJ
ejpam-5240	475	4	determines	determine	NOUN
ejpam-5240	475	5	gs(n	gs(n	NOUN
ejpam-5240	475	6	,	,	PUNCT
ejpam-5240	475	7	k	k	NOUN
ejpam-5240	475	8	)	)	PUNCT
ejpam-5240	475	9	as	as	ADV
ejpam-5240	475	10	well	well	ADV
ejpam-5240	475	11	as	as	ADP
ejpam-5240	475	12	the	the	DET
ejpam-5240	475	13	degree	degree	NOUN
ejpam-5240	475	14	of	of	ADP
ejpam-5240	475	15	its	its	PRON
ejpam-5240	475	16	vertices	vertex	NOUN
ejpam-5240	475	17	.	.	PUNCT
ejpam-5240	476	1	theorem	theorem	NOUN
ejpam-5240	476	2	9	9	NUM
ejpam-5240	476	3	.	.	PUNCT
ejpam-5240	477	1	let	let	VERB
ejpam-5240	477	2	gs(n	gs(n	NOUN
ejpam-5240	477	3	,	,	PUNCT
ejpam-5240	477	4	k	k	NOUN
ejpam-5240	477	5	)	)	PUNCT
ejpam-5240	477	6	be	be	VERB
ejpam-5240	477	7	a	a	DET
ejpam-5240	477	8	k	k	ADV
ejpam-5240	477	9	-	-	ADJ
ejpam-5240	477	10	restricted	restricted	ADJ
ejpam-5240	477	11	intersection	intersection	NOUN
ejpam-5240	477	12	graph	graph	NOUN
ejpam-5240	477	13	.	.	PUNCT
ejpam-5240	478	1	then	then	ADV
ejpam-5240	478	2	the	the	DET
ejpam-5240	478	3	complement	complement	NOUN
ejpam-5240	478	4	of	of	ADP
ejpam-5240	478	5	gs(n	gs(n	NOUN
ejpam-5240	478	6	,	,	PUNCT
ejpam-5240	478	7	k	k	NOUN
ejpam-5240	478	8	)	)	PUNCT
ejpam-5240	478	9	is	be	AUX
ejpam-5240	478	10	an	an	DET
ejpam-5240	478	11	r	r	NOUN
ejpam-5240	478	12	-	-	PUNCT
ejpam-5240	478	13	regular	regular	ADJ
ejpam-5240	478	14	graph	graph	NOUN
ejpam-5240	478	15	gs(n	gs(n	NOUN
ejpam-5240	478	16	,	,	PUNCT
ejpam-5240	478	17	k	k	NOUN
ejpam-5240	478	18	)	)	PUNCT
ejpam-5240	478	19	such	such	ADJ
ejpam-5240	478	20	that	that	SCONJ
ejpam-5240	478	21	r	r	NOUN
ejpam-5240	478	22	=	=	PUNCT
ejpam-5240	478	23	{	{	PUNCT
ejpam-5240	478	24	0	0	NUM
ejpam-5240	478	25	if	if	SCONJ
ejpam-5240	478	26	k	k	PROPN
ejpam-5240	478	27	=	=	PUNCT
ejpam-5240	478	28	0	0	NUM
ejpam-5240	479	1	or	or	CCONJ
ejpam-5240	479	2	⌊	⌊	X
ejpam-5240	479	3	n	n	ADV
ejpam-5240	479	4	2	2	NUM
ejpam-5240	479	5	⌋	⌋	NOUN
ejpam-5240	479	6	<	<	X
ejpam-5240	479	7	k	k	X
ejpam-5240	479	8	≤	≤	PROPN
ejpam-5240	479	9	n	n	CCONJ
ejpam-5240	480	1	;	;	PUNCT
ejpam-5240	480	2	(	(	PUNCT
ejpam-5240	480	3	n−k	n−k	NOUN
ejpam-5240	480	4	k	k	X
ejpam-5240	480	5	)	)	PUNCT
ejpam-5240	480	6	if	if	SCONJ
ejpam-5240	480	7	1	1	NUM
ejpam-5240	480	8	≤	≤	NUM
ejpam-5240	480	9	k	k	X
ejpam-5240	480	10	≤	≤	NUM
ejpam-5240	480	11	⌊	⌊	VERB
ejpam-5240	480	12	n	n	ADV
ejpam-5240	480	13	2	2	NUM
ejpam-5240	480	14	⌋	⌋	NOUN
ejpam-5240	480	15	proof	proof	NOUN
ejpam-5240	480	16	.	.	PUNCT
ejpam-5240	481	1	let	let	VERB
ejpam-5240	481	2	gs(n	gs(n	NOUN
ejpam-5240	481	3	,	,	PUNCT
ejpam-5240	481	4	k	k	NOUN
ejpam-5240	481	5	)	)	PUNCT
ejpam-5240	481	6	be	be	VERB
ejpam-5240	481	7	a	a	DET
ejpam-5240	481	8	k	k	ADV
ejpam-5240	481	9	-	-	ADJ
ejpam-5240	481	10	restricted	restricted	ADJ
ejpam-5240	481	11	intersection	intersection	NOUN
ejpam-5240	481	12	graph	graph	NOUN
ejpam-5240	481	13	.	.	PUNCT
ejpam-5240	482	1	by	by	ADP
ejpam-5240	482	2	theorem	theorem	NOUN
ejpam-5240	482	3	5	5	NUM
ejpam-5240	482	4	,	,	PUNCT
ejpam-5240	482	5	gs(n	gs(n	NOUN
ejpam-5240	482	6	,	,	PUNCT
ejpam-5240	482	7	k	k	NOUN
ejpam-5240	482	8	)	)	PUNCT
ejpam-5240	482	9	is	be	AUX
ejpam-5240	482	10	a	a	DET
ejpam-5240	482	11	regular	regular	ADJ
ejpam-5240	482	12	graph	graph	NOUN
ejpam-5240	482	13	.	.	PUNCT
ejpam-5240	483	1	note	note	VERB
ejpam-5240	483	2	that	that	SCONJ
ejpam-5240	483	3	its	its	PRON
ejpam-5240	483	4	complement	complement	NOUN
ejpam-5240	483	5	graph	graph	NOUN
ejpam-5240	483	6	,	,	PUNCT
ejpam-5240	483	7	gs(n	gs(n	NOUN
ejpam-5240	483	8	,	,	PUNCT
ejpam-5240	483	9	k	k	NOUN
ejpam-5240	483	10	)	)	PUNCT
ejpam-5240	483	11	is	be	AUX
ejpam-5240	483	12	also	also	ADV
ejpam-5240	483	13	a	a	DET
ejpam-5240	483	14	regular	regular	ADJ
ejpam-5240	483	15	graph	graph	NOUN
ejpam-5240	483	16	.	.	PUNCT
ejpam-5240	484	1	now	now	ADV
ejpam-5240	484	2	,	,	PUNCT
ejpam-5240	484	3	to	to	PART
ejpam-5240	484	4	find	find	VERB
ejpam-5240	484	5	the	the	DET
ejpam-5240	484	6	degree	degree	NOUN
ejpam-5240	484	7	of	of	ADP
ejpam-5240	484	8	every	every	DET
ejpam-5240	484	9	vertex	vertex	NOUN
ejpam-5240	484	10	in	in	ADP
ejpam-5240	484	11	gs(n	gs(n	NOUN
ejpam-5240	484	12	,	,	PUNCT
ejpam-5240	484	13	k	k	NOUN
ejpam-5240	484	14	)	)	PUNCT
ejpam-5240	484	15	,	,	PUNCT
ejpam-5240	484	16	consider	consider	VERB
ejpam-5240	484	17	the	the	DET
ejpam-5240	484	18	following	follow	VERB
ejpam-5240	484	19	cases	case	NOUN
ejpam-5240	484	20	:	:	PUNCT
ejpam-5240	484	21	case	case	NOUN
ejpam-5240	484	22	1	1	NUM
ejpam-5240	484	23	:	:	PUNCT
ejpam-5240	484	24	if	if	SCONJ
ejpam-5240	484	25	k	k	PROPN
ejpam-5240	484	26	=	=	PUNCT
ejpam-5240	484	27	0	0	NUM
ejpam-5240	484	28	or	or	CCONJ
ejpam-5240	484	29	⌊	⌊	X
ejpam-5240	484	30	n	n	ADV
ejpam-5240	484	31	2	2	NUM
ejpam-5240	484	32	⌋	⌋	NOUN
ejpam-5240	484	33	<	<	X
ejpam-5240	484	34	k	k	X
ejpam-5240	484	35	≤	≤	PROPN
ejpam-5240	484	36	n	n	CCONJ
ejpam-5240	484	37	,	,	PUNCT
ejpam-5240	484	38	by	by	ADP
ejpam-5240	484	39	theorem	theorem	NOUN
ejpam-5240	484	40	7	7	NUM
ejpam-5240	484	41	,	,	PUNCT
ejpam-5240	484	42	it	it	PRON
ejpam-5240	484	43	follows	follow	VERB
ejpam-5240	484	44	that	that	SCONJ
ejpam-5240	484	45	gs(n	gs(n	NOUN
ejpam-5240	484	46	,	,	PUNCT
ejpam-5240	484	47	k	k	NOUN
ejpam-5240	484	48	)	)	PUNCT
ejpam-5240	484	49	is	be	AUX
ejpam-5240	484	50	a	a	DET
ejpam-5240	484	51	complete	complete	ADJ
ejpam-5240	484	52	graph	graph	NOUN
ejpam-5240	484	53	.	.	PUNCT
ejpam-5240	485	1	the	the	DET
ejpam-5240	485	2	complement	complement	NOUN
ejpam-5240	485	3	of	of	ADP
ejpam-5240	485	4	a	a	DET
ejpam-5240	485	5	complete	complete	ADJ
ejpam-5240	485	6	graph	graph	NOUN
ejpam-5240	485	7	is	be	AUX
ejpam-5240	485	8	an	an	DET
ejpam-5240	485	9	empty	empty	ADJ
ejpam-5240	485	10	graph	graph	NOUN
ejpam-5240	485	11	which	which	PRON
ejpam-5240	485	12	implies	imply	VERB
ejpam-5240	485	13	that	that	SCONJ
ejpam-5240	485	14	gs(n	gs(n	NOUN
ejpam-5240	485	15	,	,	PUNCT
ejpam-5240	485	16	k	k	NOUN
ejpam-5240	485	17	)	)	PUNCT
ejpam-5240	485	18	is	be	AUX
ejpam-5240	485	19	an	an	DET
ejpam-5240	485	20	empty	empty	ADJ
ejpam-5240	485	21	graph	graph	NOUN
ejpam-5240	485	22	.	.	PUNCT
ejpam-5240	486	1	thus	thus	ADV
ejpam-5240	486	2	,	,	PUNCT
ejpam-5240	486	3	deg(a	deg(a	PROPN
ejpam-5240	486	4	)	)	PUNCT
ejpam-5240	486	5	=	=	NOUN
ejpam-5240	486	6	0	0	NUM
ejpam-5240	487	1	for	for	ADP
ejpam-5240	487	2	all	all	DET
ejpam-5240	487	3	a	a	DET
ejpam-5240	487	4	∈	∈	PROPN
ejpam-5240	487	5	v	v	NOUN
ejpam-5240	487	6	(	(	PUNCT
ejpam-5240	487	7	gs(n	gs(n	NOUN
ejpam-5240	487	8	,	,	PUNCT
ejpam-5240	487	9	k	k	NOUN
ejpam-5240	487	10	)	)	PUNCT
ejpam-5240	487	11	)	)	PUNCT
ejpam-5240	487	12	.	.	PUNCT
ejpam-5240	488	1	case	case	NOUN
ejpam-5240	488	2	2	2	NUM
ejpam-5240	488	3	:	:	PUNCT
ejpam-5240	488	4	let	let	VERB
ejpam-5240	488	5	a	a	PRON
ejpam-5240	488	6	be	be	AUX
ejpam-5240	488	7	an	an	DET
ejpam-5240	488	8	arbitrary	arbitrary	ADJ
ejpam-5240	488	9	element	element	NOUN
ejpam-5240	488	10	of	of	ADP
ejpam-5240	488	11	v	v	NOUN
ejpam-5240	488	12	(	(	PUNCT
ejpam-5240	488	13	gs(n	gs(n	NOUN
ejpam-5240	488	14	,	,	PUNCT
ejpam-5240	488	15	k	k	NOUN
ejpam-5240	488	16	)	)	PUNCT
ejpam-5240	488	17	)	)	PUNCT
ejpam-5240	488	18	.	.	PUNCT
ejpam-5240	489	1	if	if	SCONJ
ejpam-5240	489	2	1	1	NUM
ejpam-5240	489	3	≤	≤	NUM
ejpam-5240	489	4	k	k	X
ejpam-5240	489	5	≤	≤	NUM
ejpam-5240	489	6	⌊	⌊	VERB
ejpam-5240	489	7	n	n	DET
ejpam-5240	489	8	2	2	NUM
ejpam-5240	489	9	⌋	⌋	NOUN
ejpam-5240	489	10	,	,	PUNCT
ejpam-5240	489	11	by	by	ADP
ejpam-5240	489	12	lemma	lemma	PROPN
ejpam-5240	489	13	1	1	NUM
ejpam-5240	489	14	,	,	PUNCT
ejpam-5240	489	15	deg(a	deg(a	PROPN
ejpam-5240	489	16	)	)	PUNCT
ejpam-5240	489	17	=	=	PUNCT
ejpam-5240	490	1	(	(	PUNCT
ejpam-5240	490	2	n	n	X
ejpam-5240	490	3	k	k	NOUN
ejpam-5240	490	4	)	)	PUNCT
ejpam-5240	490	5	−	−	PROPN
ejpam-5240	491	1	[	[	X
ejpam-5240	491	2	(	(	PUNCT
ejpam-5240	491	3	n−k	n−k	NOUN
ejpam-5240	491	4	k	k	PROPN
ejpam-5240	491	5	)	)	PUNCT
ejpam-5240	492	1	+	+	CCONJ
ejpam-5240	492	2	1	1	X
ejpam-5240	492	3	]	]	PUNCT
ejpam-5240	492	4	for	for	ADP
ejpam-5240	492	5	all	all	DET
ejpam-5240	492	6	a	a	DET
ejpam-5240	492	7	∈	∈	PROPN
ejpam-5240	492	8	v	v	NOUN
ejpam-5240	492	9	(	(	PUNCT
ejpam-5240	492	10	gs(n	gs(n	NOUN
ejpam-5240	492	11	,	,	PUNCT
ejpam-5240	492	12	k	k	NOUN
ejpam-5240	492	13	)	)	PUNCT
ejpam-5240	492	14	)	)	PUNCT
ejpam-5240	492	15	.	.	PUNCT
ejpam-5240	493	1	it	it	PRON
ejpam-5240	493	2	can	can	AUX
ejpam-5240	493	3	be	be	AUX
ejpam-5240	493	4	observed	observe	VERB
ejpam-5240	493	5	that	that	SCONJ
ejpam-5240	493	6	a	a	PRON
ejpam-5240	493	7	is	be	AUX
ejpam-5240	493	8	not	not	PART
ejpam-5240	493	9	adjacent	adjacent	ADJ
ejpam-5240	493	10	to	to	ADP
ejpam-5240	493	11	(	(	PUNCT
ejpam-5240	493	12	n−k	n−k	NOUN
ejpam-5240	493	13	k	k	PROPN
ejpam-5240	493	14	)	)	PUNCT
ejpam-5240	494	1	+	+	CCONJ
ejpam-5240	494	2	1	1	NUM
ejpam-5240	494	3	vertices	vertex	NOUN
ejpam-5240	494	4	in	in	ADP
ejpam-5240	494	5	gs(n	gs(n	NOUN
ejpam-5240	494	6	,	,	PUNCT
ejpam-5240	494	7	k	k	NOUN
ejpam-5240	494	8	)	)	PUNCT
ejpam-5240	494	9	.	.	PUNCT
ejpam-5240	495	1	since	since	SCONJ
ejpam-5240	495	2	gs(n	gs(n	NOUN
ejpam-5240	495	3	,	,	PUNCT
ejpam-5240	495	4	k	k	NOUN
ejpam-5240	495	5	)	)	PUNCT
ejpam-5240	495	6	is	be	AUX
ejpam-5240	495	7	a	a	DET
ejpam-5240	495	8	simple	simple	ADJ
ejpam-5240	495	9	graph	graph	NOUN
ejpam-5240	495	10	,	,	PUNCT
ejpam-5240	495	11	it	it	PRON
ejpam-5240	495	12	follows	follow	VERB
ejpam-5240	495	13	that	that	SCONJ
ejpam-5240	495	14	deg(a	deg(a	PROPN
ejpam-5240	495	15	)	)	PUNCT
ejpam-5240	495	16	=	=	PRON
ejpam-5240	496	1	(	(	PUNCT
ejpam-5240	496	2	n−k	n−k	NOUN
ejpam-5240	496	3	k	k	PROPN
ejpam-5240	496	4	)	)	PUNCT
ejpam-5240	496	5	for	for	ADP
ejpam-5240	496	6	all	all	DET
ejpam-5240	496	7	a	a	DET
ejpam-5240	496	8	∈	∈	PROPN
ejpam-5240	496	9	v	v	NOUN
ejpam-5240	496	10	(	(	PUNCT
ejpam-5240	496	11	gs(n	gs(n	NOUN
ejpam-5240	496	12	,	,	PUNCT
ejpam-5240	496	13	k	k	NOUN
ejpam-5240	496	14	)	)	PUNCT
ejpam-5240	496	15	)	)	PUNCT
ejpam-5240	496	16	.	.	PUNCT
ejpam-5240	497	1	to	to	PART
ejpam-5240	497	2	illustrate	illustrate	VERB
ejpam-5240	497	3	theorem	theorem	NOUN
ejpam-5240	497	4	9	9	NUM
ejpam-5240	497	5	,	,	PUNCT
ejpam-5240	497	6	we	we	PRON
ejpam-5240	497	7	utilize	utilize	VERB
ejpam-5240	497	8	the	the	DET
ejpam-5240	497	9	graphs	graph	NOUN
ejpam-5240	497	10	gs(4,3	gs(4,3	NOUN
ejpam-5240	497	11	)	)	PUNCT
ejpam-5240	497	12	and	and	CCONJ
ejpam-5240	497	13	gs(4,2	gs(4,2	NOUN
ejpam-5240	497	14	)	)	PUNCT
ejpam-5240	497	15	shown	show	VERB
ejpam-5240	497	16	in	in	ADP
ejpam-5240	497	17	the	the	DET
ejpam-5240	497	18	previous	previous	ADJ
ejpam-5240	497	19	discussions	discussion	NOUN
ejpam-5240	497	20	.	.	PUNCT
ejpam-5240	498	1	m.e	m.e	PROPN
ejpam-5240	498	2	.	.	PROPN
ejpam-5240	498	3	pelagio	pelagio	PROPN
ejpam-5240	498	4	,	,	PUNCT
ejpam-5240	498	5	n.	n.	NOUN
ejpam-5240	498	6	mame	mame	PROPN
ejpam-5240	498	7	,	,	PUNCT
ejpam-5240	498	8	k.	k.	PROPN
ejpam-5240	498	9	mendoza	mendoza	PROPN
ejpam-5240	498	10	/	/	SYM
ejpam-5240	498	11	eur	eur	PROPN
ejpam-5240	498	12	.	.	PUNCT
ejpam-5240	499	1	j.	j.	PROPN
ejpam-5240	499	2	pure	pure	PROPN
ejpam-5240	499	3	appl	appl	PROPN
ejpam-5240	499	4	.	.	PROPN
ejpam-5240	499	5	math	math	PROPN
ejpam-5240	499	6	,	,	PUNCT
ejpam-5240	499	7	17	17	NUM
ejpam-5240	499	8	(	(	PUNCT
ejpam-5240	499	9	3	3	NUM
ejpam-5240	499	10	)	)	PUNCT
ejpam-5240	499	11	(	(	PUNCT
ejpam-5240	499	12	2024	2024	NUM
ejpam-5240	499	13	)	)	PUNCT
ejpam-5240	499	14	,	,	PUNCT
ejpam-5240	499	15	1779	1779	NUM
ejpam-5240	499	16	-	-	SYM
ejpam-5240	499	17	1803	1803	NUM
ejpam-5240	499	18	1794	1794	NUM
ejpam-5240	499	19	illustration	illustration	NOUN
ejpam-5240	499	20	9	9	NUM
ejpam-5240	499	21	.	.	PUNCT
ejpam-5240	500	1	let	let	VERB
ejpam-5240	500	2	s4	s4	PROPN
ejpam-5240	500	3	=	=	SYM
ejpam-5240	500	4	{	{	PUNCT
ejpam-5240	500	5	x1	x1	PROPN
ejpam-5240	500	6	,	,	PUNCT
ejpam-5240	500	7	x2	x2	PROPN
ejpam-5240	500	8	,	,	PUNCT
ejpam-5240	500	9	x3	x3	ADJ
ejpam-5240	500	10	,	,	PUNCT
ejpam-5240	500	11	x4	x4	PROPN
ejpam-5240	500	12	}	}	PUNCT
ejpam-5240	500	13	and	and	CCONJ
ejpam-5240	500	14	let	let	VERB
ejpam-5240	500	15	k	k	PROPN
ejpam-5240	500	16	=	=	SYM
ejpam-5240	500	17	3	3	X
ejpam-5240	500	18	.	.	PUNCT
ejpam-5240	500	19	a	a	DET
ejpam-5240	500	20	pictorial	pictorial	ADJ
ejpam-5240	500	21	representation	representation	NOUN
ejpam-5240	500	22	of	of	ADP
ejpam-5240	500	23	gs(4,3	gs(4,3	PROPN
ejpam-5240	500	24	)	)	PUNCT
ejpam-5240	500	25	is	be	AUX
ejpam-5240	500	26	presented	present	VERB
ejpam-5240	500	27	in	in	ADP
ejpam-5240	500	28	figure	figure	NOUN
ejpam-5240	500	29	9	9	NUM
ejpam-5240	500	30	.	.	PUNCT
ejpam-5240	501	1	since	since	SCONJ
ejpam-5240	501	2	k	k	PROPN
ejpam-5240	501	3	=	=	SYM
ejpam-5240	501	4	3	3	NUM
ejpam-5240	501	5	and	and	CCONJ
ejpam-5240	501	6	⌊	⌊	VERB
ejpam-5240	501	7	4	4	NUM
ejpam-5240	501	8	2	2	NUM
ejpam-5240	501	9	⌋	⌋	NOUN
ejpam-5240	501	10	=	=	SYM
ejpam-5240	501	11	2	2	NUM
ejpam-5240	501	12	<	<	SYM
ejpam-5240	501	13	3	3	NUM
ejpam-5240	501	14	it	it	PRON
ejpam-5240	501	15	follows	follow	VERB
ejpam-5240	501	16	that	that	SCONJ
ejpam-5240	501	17	gs(4,3	gs(4,3	PROPN
ejpam-5240	501	18	)	)	PUNCT
ejpam-5240	501	19	is	be	AUX
ejpam-5240	501	20	a	a	DET
ejpam-5240	501	21	complete	complete	ADJ
ejpam-5240	501	22	graph	graph	NOUN
ejpam-5240	501	23	of	of	ADP
ejpam-5240	501	24	order	order	NOUN
ejpam-5240	501	25	(	(	PUNCT
ejpam-5240	501	26	4	4	NUM
ejpam-5240	501	27	3	3	NUM
ejpam-5240	501	28	)	)	PUNCT
ejpam-5240	502	1	=	=	SYM
ejpam-5240	502	2	4	4	X
ejpam-5240	502	3	.	.	PUNCT
ejpam-5240	503	1	the	the	DET
ejpam-5240	503	2	complement	complement	NOUN
ejpam-5240	503	3	of	of	ADP
ejpam-5240	503	4	a	a	DET
ejpam-5240	503	5	complete	complete	ADJ
ejpam-5240	503	6	graph	graph	NOUN
ejpam-5240	503	7	is	be	AUX
ejpam-5240	503	8	an	an	DET
ejpam-5240	503	9	empty	empty	ADJ
ejpam-5240	503	10	graph	graph	NOUN
ejpam-5240	503	11	,	,	PUNCT
ejpam-5240	503	12	thus	thus	ADV
ejpam-5240	503	13	,	,	PUNCT
ejpam-5240	503	14	gs(4,3	gs(4,3	PROPN
ejpam-5240	503	15	)	)	PUNCT
ejpam-5240	503	16	is	be	AUX
ejpam-5240	503	17	an	an	DET
ejpam-5240	503	18	empty	empty	ADJ
ejpam-5240	503	19	graph	graph	NOUN
ejpam-5240	503	20	of	of	ADP
ejpam-5240	503	21	order	order	NOUN
ejpam-5240	503	22	(	(	PUNCT
ejpam-5240	503	23	4	4	NUM
ejpam-5240	503	24	3	3	NUM
ejpam-5240	503	25	)	)	PUNCT
ejpam-5240	503	26	=	=	SYM
ejpam-5240	504	1	4	4	X
ejpam-5240	504	2	.	.	NUM
ejpam-5240	504	3	presented	present	VERB
ejpam-5240	504	4	also	also	ADV
ejpam-5240	504	5	in	in	ADP
ejpam-5240	504	6	figure	figure	NOUN
ejpam-5240	504	7	9	9	NUM
ejpam-5240	504	8	is	be	AUX
ejpam-5240	504	9	a	a	DET
ejpam-5240	504	10	pictorial	pictorial	ADJ
ejpam-5240	504	11	representation	representation	NOUN
ejpam-5240	504	12	of	of	ADP
ejpam-5240	504	13	gs(4,3	gs(4,3	PROPN
ejpam-5240	504	14	)	)	PUNCT
ejpam-5240	504	15	.	.	PUNCT
ejpam-5240	505	1	{	{	PUNCT
ejpam-5240	505	2	x1	x1	PROPN
ejpam-5240	505	3	,	,	PUNCT
ejpam-5240	505	4	x2	x2	PROPN
ejpam-5240	505	5	,	,	PUNCT
ejpam-5240	505	6	x3	x3	ADJ
ejpam-5240	505	7	}	}	PUNCT
ejpam-5240	505	8	{	{	PUNCT
ejpam-5240	505	9	x1	x1	PROPN
ejpam-5240	505	10	,	,	PUNCT
ejpam-5240	505	11	x2	x2	PROPN
ejpam-5240	505	12	,	,	PUNCT
ejpam-5240	505	13	x4	x4	PROPN
ejpam-5240	505	14	}	}	PUNCT
ejpam-5240	505	15	{	{	PUNCT
ejpam-5240	505	16	x1	x1	PROPN
ejpam-5240	505	17	,	,	PUNCT
ejpam-5240	505	18	x3	x3	ADJ
ejpam-5240	505	19	,	,	PUNCT
ejpam-5240	505	20	x4	x4	PROPN
ejpam-5240	505	21	}	}	PUNCT
ejpam-5240	505	22	{	{	PUNCT
ejpam-5240	505	23	x2	x2	PROPN
ejpam-5240	505	24	,	,	PUNCT
ejpam-5240	505	25	x3	x3	ADJ
ejpam-5240	505	26	,	,	PUNCT
ejpam-5240	505	27	x4	x4	PROPN
ejpam-5240	505	28	}	}	PUNCT
ejpam-5240	505	29	{	{	PUNCT
ejpam-5240	505	30	x1	x1	PROPN
ejpam-5240	505	31	,	,	PUNCT
ejpam-5240	505	32	x2	x2	PROPN
ejpam-5240	505	33	,	,	PUNCT
ejpam-5240	505	34	x3	x3	ADJ
ejpam-5240	505	35	}	}	PUNCT
ejpam-5240	505	36	{	{	PUNCT
ejpam-5240	505	37	x1	x1	PROPN
ejpam-5240	505	38	,	,	PUNCT
ejpam-5240	505	39	x2	x2	PROPN
ejpam-5240	505	40	,	,	PUNCT
ejpam-5240	505	41	x4	x4	PROPN
ejpam-5240	505	42	}	}	PUNCT
ejpam-5240	505	43	{	{	PUNCT
ejpam-5240	505	44	x1	x1	PROPN
ejpam-5240	505	45	,	,	PUNCT
ejpam-5240	505	46	x3	x3	ADJ
ejpam-5240	505	47	,	,	PUNCT
ejpam-5240	505	48	x4	x4	PROPN
ejpam-5240	505	49	}	}	PUNCT
ejpam-5240	505	50	{	{	PUNCT
ejpam-5240	505	51	x2	x2	PROPN
ejpam-5240	505	52	,	,	PUNCT
ejpam-5240	505	53	x3	x3	ADJ
ejpam-5240	505	54	,	,	PUNCT
ejpam-5240	505	55	x4	x4	ADJ
ejpam-5240	505	56	}	}	PUNCT
ejpam-5240	505	57	figure	figure	VERB
ejpam-5240	505	58	9	9	NUM
ejpam-5240	505	59	:	:	PUNCT
ejpam-5240	505	60	pictorial	pictorial	ADJ
ejpam-5240	505	61	representations	representation	NOUN
ejpam-5240	505	62	of	of	ADP
ejpam-5240	505	63	gs(4,3	gs(4,3	NOUN
ejpam-5240	505	64	)	)	PUNCT
ejpam-5240	505	65	and	and	CCONJ
ejpam-5240	505	66	its	its	PRON
ejpam-5240	505	67	complement	complement	NOUN
ejpam-5240	505	68	graph	graph	NOUN
ejpam-5240	505	69	gs(4,3	gs(4,3	NOUN
ejpam-5240	505	70	)	)	PUNCT
ejpam-5240	505	71	,	,	PUNCT
ejpam-5240	505	72	respectively	respectively	ADV
ejpam-5240	505	73	.	.	PUNCT
ejpam-5240	506	1	it	it	PRON
ejpam-5240	506	2	can	can	AUX
ejpam-5240	506	3	be	be	AUX
ejpam-5240	506	4	observed	observe	VERB
ejpam-5240	506	5	that	that	SCONJ
ejpam-5240	506	6	deg(a	deg(a	PROPN
ejpam-5240	506	7	)	)	PUNCT
ejpam-5240	506	8	=	=	NOUN
ejpam-5240	506	9	0	0	NUM
ejpam-5240	506	10	for	for	ADP
ejpam-5240	506	11	all	all	DET
ejpam-5240	506	12	a	a	DET
ejpam-5240	506	13	∈	∈	PROPN
ejpam-5240	506	14	v	v	NOUN
ejpam-5240	506	15	(	(	PUNCT
ejpam-5240	506	16	gs(4,3	gs(4,3	PROPN
ejpam-5240	506	17	)	)	PUNCT
ejpam-5240	506	18	)	)	PUNCT
ejpam-5240	506	19	.	.	PUNCT
ejpam-5240	507	1	therefore	therefore	ADV
ejpam-5240	507	2	,	,	PUNCT
ejpam-5240	507	3	gs(4,3	gs(4,3	PROPN
ejpam-5240	507	4	)	)	PUNCT
ejpam-5240	507	5	is	be	AUX
ejpam-5240	507	6	a	a	DET
ejpam-5240	507	7	0	0	NUM
ejpam-5240	507	8	-	-	PUNCT
ejpam-5240	507	9	regular	regular	ADJ
ejpam-5240	507	10	graph	graph	NOUN
ejpam-5240	507	11	.	.	PUNCT
ejpam-5240	508	1	for	for	ADP
ejpam-5240	508	2	1	1	NUM
ejpam-5240	508	3	≤	≤	NOUN
ejpam-5240	508	4	k	k	X
ejpam-5240	508	5	≤	≤	NUM
ejpam-5240	508	6	⌊	⌊	VERB
ejpam-5240	508	7	n	n	DET
ejpam-5240	508	8	2	2	NUM
ejpam-5240	508	9	⌋	⌋	NOUN
ejpam-5240	508	10	,	,	PUNCT
ejpam-5240	508	11	consider	consider	VERB
ejpam-5240	508	12	also	also	ADV
ejpam-5240	508	13	the	the	DET
ejpam-5240	508	14	set	set	NOUN
ejpam-5240	508	15	s4	s4	NOUN
ejpam-5240	508	16	=	=	SYM
ejpam-5240	508	17	{	{	PUNCT
ejpam-5240	508	18	x1	x1	PROPN
ejpam-5240	508	19	,	,	PUNCT
ejpam-5240	508	20	x2	x2	PROPN
ejpam-5240	508	21	,	,	PUNCT
ejpam-5240	508	22	x3	x3	ADJ
ejpam-5240	508	23	,	,	PUNCT
ejpam-5240	508	24	x4	x4	PROPN
ejpam-5240	508	25	}	}	PUNCT
ejpam-5240	508	26	and	and	CCONJ
ejpam-5240	508	27	let	let	VERB
ejpam-5240	508	28	k	k	NOUN
ejpam-5240	508	29	=	=	SYM
ejpam-5240	508	30	2	2	X
ejpam-5240	508	31	.	.	PUNCT
ejpam-5240	508	32	the	the	DET
ejpam-5240	508	33	graph	graph	NOUN
ejpam-5240	508	34	of	of	ADP
ejpam-5240	508	35	gs(4,2	gs(4,2	NOUN
ejpam-5240	508	36	)	)	PUNCT
ejpam-5240	508	37	is	be	AUX
ejpam-5240	508	38	previously	previously	ADV
ejpam-5240	508	39	shown	show	VERB
ejpam-5240	508	40	in	in	ADP
ejpam-5240	508	41	figure	figure	NOUN
ejpam-5240	508	42	2	2	NUM
ejpam-5240	508	43	.	.	PUNCT
ejpam-5240	509	1	now	now	ADV
ejpam-5240	509	2	,	,	PUNCT
ejpam-5240	509	3	shown	show	VERB
ejpam-5240	509	4	in	in	ADP
ejpam-5240	509	5	figure	figure	NOUN
ejpam-5240	509	6	10	10	NUM
ejpam-5240	509	7	,	,	PUNCT
ejpam-5240	509	8	is	be	AUX
ejpam-5240	509	9	a	a	DET
ejpam-5240	509	10	pictorial	pictorial	ADJ
ejpam-5240	509	11	representation	representation	NOUN
ejpam-5240	509	12	of	of	ADP
ejpam-5240	509	13	the	the	DET
ejpam-5240	509	14	complement	complement	NOUN
ejpam-5240	509	15	graph	graph	NOUN
ejpam-5240	509	16	of	of	ADP
ejpam-5240	509	17	gs(4,2	gs(4,2	NOUN
ejpam-5240	509	18	)	)	PUNCT
ejpam-5240	509	19	.	.	PUNCT
ejpam-5240	510	1	{	{	PUNCT
ejpam-5240	510	2	x1	x1	ADJ
ejpam-5240	510	3	,	,	PUNCT
ejpam-5240	510	4	x2	x2	PROPN
ejpam-5240	510	5	}	}	PUNCT
ejpam-5240	510	6	{	{	PUNCT
ejpam-5240	510	7	x3	x3	ADJ
ejpam-5240	510	8	,	,	PUNCT
ejpam-5240	510	9	x4	x4	PROPN
ejpam-5240	510	10	}	}	PUNCT
ejpam-5240	510	11	{	{	PUNCT
ejpam-5240	510	12	x1	x1	PROPN
ejpam-5240	510	13	,	,	PUNCT
ejpam-5240	510	14	x3	x3	ADJ
ejpam-5240	510	15	}	}	PUNCT
ejpam-5240	510	16	{	{	PUNCT
ejpam-5240	510	17	x2	x2	PROPN
ejpam-5240	510	18	,	,	PUNCT
ejpam-5240	510	19	x4	x4	PROPN
ejpam-5240	510	20	}	}	PUNCT
ejpam-5240	510	21	{	{	PUNCT
ejpam-5240	510	22	x1	x1	PROPN
ejpam-5240	510	23	,	,	PUNCT
ejpam-5240	510	24	x4	x4	PROPN
ejpam-5240	510	25	}	}	PUNCT
ejpam-5240	510	26	{	{	PUNCT
ejpam-5240	510	27	x2	x2	PROPN
ejpam-5240	510	28	,	,	PUNCT
ejpam-5240	510	29	x3	x3	ADJ
ejpam-5240	510	30	}	}	PUNCT
ejpam-5240	510	31	figure	figure	NOUN
ejpam-5240	510	32	10	10	NUM
ejpam-5240	510	33	:	:	PUNCT
ejpam-5240	510	34	pictorial	pictorial	ADJ
ejpam-5240	510	35	illustration	illustration	NOUN
ejpam-5240	510	36	of	of	ADP
ejpam-5240	510	37	the	the	DET
ejpam-5240	510	38	complement	complement	NOUN
ejpam-5240	510	39	graph	graph	NOUN
ejpam-5240	510	40	gs(4,2	gs(4,2	NOUN
ejpam-5240	510	41	)	)	PUNCT
ejpam-5240	510	42	.	.	PUNCT
ejpam-5240	511	1	now	now	ADV
ejpam-5240	511	2	,	,	PUNCT
ejpam-5240	511	3	by	by	ADP
ejpam-5240	511	4	connecting	connect	VERB
ejpam-5240	511	5	the	the	DET
ejpam-5240	511	6	vertices	vertex	NOUN
ejpam-5240	511	7	in	in	ADP
ejpam-5240	511	8	gs(4,2	gs(4,2	NOUN
ejpam-5240	511	9	)	)	PUNCT
ejpam-5240	511	10	with	with	ADP
ejpam-5240	511	11	empty	empty	ADJ
ejpam-5240	511	12	intersection	intersection	NOUN
ejpam-5240	511	13	,	,	PUNCT
ejpam-5240	511	14	we	we	PRON
ejpam-5240	511	15	have	have	VERB
ejpam-5240	511	16	the	the	DET
ejpam-5240	511	17	edge	edge	NOUN
ejpam-5240	511	18	set	set	NOUN
ejpam-5240	511	19	of	of	ADP
ejpam-5240	511	20	gs(4,2	gs(4,2	NOUN
ejpam-5240	511	21	)	)	PUNCT
ejpam-5240	511	22	equal	equal	ADJ
ejpam-5240	511	23	to	to	ADP
ejpam-5240	511	24	the	the	DET
ejpam-5240	511	25	set	set	NOUN
ejpam-5240	511	26	{	{	PUNCT
ejpam-5240	511	27	[	[	X
ejpam-5240	511	28	{	{	PUNCT
ejpam-5240	511	29	x1	x1	PROPN
ejpam-5240	511	30	,	,	PUNCT
ejpam-5240	511	31	x2	x2	PROPN
ejpam-5240	511	32	}	}	PUNCT
ejpam-5240	511	33	,	,	PUNCT
ejpam-5240	511	34	{	{	PUNCT
ejpam-5240	511	35	x3	x3	ADJ
ejpam-5240	511	36	,	,	PUNCT
ejpam-5240	511	37	x4	x4	PROPN
ejpam-5240	511	38	}	}	PUNCT
ejpam-5240	511	39	]	]	PUNCT
ejpam-5240	511	40	,	,	PUNCT
ejpam-5240	511	41	[	[	X
ejpam-5240	511	42	{	{	PUNCT
ejpam-5240	511	43	x1	x1	ADJ
ejpam-5240	511	44	,	,	PUNCT
ejpam-5240	511	45	x3	x3	ADJ
ejpam-5240	511	46	}	}	PUNCT
ejpam-5240	511	47	,	,	PUNCT
ejpam-5240	511	48	{	{	PUNCT
ejpam-5240	511	49	x2	x2	PROPN
ejpam-5240	511	50	,	,	PUNCT
ejpam-5240	511	51	x4	x4	PROPN
ejpam-5240	511	52	}	}	PUNCT
ejpam-5240	511	53	]	]	PUNCT
ejpam-5240	511	54	,	,	PUNCT
ejpam-5240	511	55	[	[	X
ejpam-5240	511	56	{	{	PUNCT
ejpam-5240	511	57	x2	x2	ADJ
ejpam-5240	511	58	,	,	PUNCT
ejpam-5240	511	59	x3	x3	ADJ
ejpam-5240	511	60	}	}	PUNCT
ejpam-5240	511	61	,	,	PUNCT
ejpam-5240	511	62	{	{	PUNCT
ejpam-5240	511	63	x1	x1	PROPN
ejpam-5240	511	64	,	,	PUNCT
ejpam-5240	511	65	x4	x4	PROPN
ejpam-5240	511	66	}	}	PUNCT
ejpam-5240	511	67	]	]	PUNCT
ejpam-5240	511	68	}	}	PUNCT
ejpam-5240	511	69	.	.	PUNCT
ejpam-5240	512	1	it	it	PRON
ejpam-5240	512	2	can	can	AUX
ejpam-5240	512	3	be	be	AUX
ejpam-5240	512	4	observed	observe	VERB
ejpam-5240	512	5	that	that	SCONJ
ejpam-5240	512	6	every	every	DET
ejpam-5240	512	7	element	element	NOUN
ejpam-5240	512	8	in	in	ADP
ejpam-5240	512	9	v	v	NOUN
ejpam-5240	512	10	(	(	PUNCT
ejpam-5240	512	11	gs(4,2	gs(4,2	NOUN
ejpam-5240	512	12	)	)	PUNCT
ejpam-5240	512	13	)	)	PUNCT
ejpam-5240	512	14	has	have	VERB
ejpam-5240	512	15	a	a	DET
ejpam-5240	512	16	degree	degree	NOUN
ejpam-5240	512	17	equal	equal	ADJ
ejpam-5240	512	18	to	to	ADP
ejpam-5240	512	19	1	1	NUM
ejpam-5240	512	20	.	.	PUNCT
ejpam-5240	513	1	note	note	VERB
ejpam-5240	513	2	that	that	SCONJ
ejpam-5240	513	3	k	k	PROPN
ejpam-5240	513	4	=	=	PUNCT
ejpam-5240	513	5	⌊	⌊	VERB
ejpam-5240	513	6	4	4	NUM
ejpam-5240	513	7	2	2	NUM
ejpam-5240	513	8	⌋	⌋	NOUN
ejpam-5240	513	9	=	=	SYM
ejpam-5240	513	10	2	2	X
ejpam-5240	513	11	.	.	X
ejpam-5240	513	12	using	use	VERB
ejpam-5240	513	13	theorem	theorem	NOUN
ejpam-5240	513	14	9	9	NUM
ejpam-5240	513	15	with	with	ADP
ejpam-5240	513	16	n	n	NOUN
ejpam-5240	513	17	=	=	SYM
ejpam-5240	513	18	4	4	NUM
ejpam-5240	513	19	and	and	CCONJ
ejpam-5240	513	20	k	k	NOUN
ejpam-5240	513	21	=	=	SYM
ejpam-5240	513	22	2	2	NUM
ejpam-5240	513	23	,	,	PUNCT
ejpam-5240	513	24	the	the	DET
ejpam-5240	513	25	degree	degree	NOUN
ejpam-5240	513	26	of	of	ADP
ejpam-5240	513	27	every	every	DET
ejpam-5240	513	28	vertex	vertex	NOUN
ejpam-5240	513	29	a	a	PRON
ejpam-5240	513	30	in	in	ADP
ejpam-5240	513	31	gs(4,2	gs(4,2	NOUN
ejpam-5240	513	32	)	)	PUNCT
ejpam-5240	513	33	we	we	PRON
ejpam-5240	513	34	have	have	VERB
ejpam-5240	513	35	:	:	PUNCT
ejpam-5240	513	36	m.e	m.e	PROPN
ejpam-5240	513	37	.	.	PROPN
ejpam-5240	513	38	pelagio	pelagio	PROPN
ejpam-5240	513	39	,	,	PUNCT
ejpam-5240	513	40	n.	n.	NOUN
ejpam-5240	513	41	mame	mame	PROPN
ejpam-5240	513	42	,	,	PUNCT
ejpam-5240	513	43	k.	k.	PROPN
ejpam-5240	513	44	mendoza	mendoza	PROPN
ejpam-5240	513	45	/	/	SYM
ejpam-5240	513	46	eur	eur	PROPN
ejpam-5240	513	47	.	.	PUNCT
ejpam-5240	514	1	j.	j.	PROPN
ejpam-5240	514	2	pure	pure	PROPN
ejpam-5240	514	3	appl	appl	PROPN
ejpam-5240	514	4	.	.	PROPN
ejpam-5240	514	5	math	math	PROPN
ejpam-5240	514	6	,	,	PUNCT
ejpam-5240	514	7	17	17	NUM
ejpam-5240	514	8	(	(	PUNCT
ejpam-5240	514	9	3	3	NUM
ejpam-5240	514	10	)	)	PUNCT
ejpam-5240	514	11	(	(	PUNCT
ejpam-5240	514	12	2024	2024	NUM
ejpam-5240	514	13	)	)	PUNCT
ejpam-5240	514	14	,	,	PUNCT
ejpam-5240	514	15	1779	1779	NUM
ejpam-5240	514	16	-	-	SYM
ejpam-5240	514	17	1803	1803	NUM
ejpam-5240	514	18	1795	1795	NUM
ejpam-5240	514	19	deg(a	deg(a	PROPN
ejpam-5240	514	20	)	)	PUNCT
ejpam-5240	514	21	=	=	PRON
ejpam-5240	514	22	(	(	PUNCT
ejpam-5240	514	23	n−	n−	NOUN
ejpam-5240	514	24	k	k	NOUN
ejpam-5240	514	25	k	k	PROPN
ejpam-5240	514	26	)	)	PUNCT
ejpam-5240	515	1	=	=	PUNCT
ejpam-5240	515	2	(	(	PUNCT
ejpam-5240	515	3	4−	4−	NOUN
ejpam-5240	515	4	2	2	NUM
ejpam-5240	515	5	2	2	NUM
ejpam-5240	515	6	)	)	PUNCT
ejpam-5240	515	7	=	=	SYM
ejpam-5240	515	8	(	(	PUNCT
ejpam-5240	515	9	2	2	NUM
ejpam-5240	515	10	2	2	NUM
ejpam-5240	515	11	)	)	PUNCT
ejpam-5240	515	12	=	=	SYM
ejpam-5240	515	13	1	1	X
ejpam-5240	515	14	.	.	PUNCT
ejpam-5240	515	15	hence	hence	ADV
ejpam-5240	515	16	,	,	PUNCT
ejpam-5240	515	17	gs(4,2	gs(4,2	PROPN
ejpam-5240	515	18	)	)	PUNCT
ejpam-5240	515	19	is	be	AUX
ejpam-5240	515	20	a	a	DET
ejpam-5240	515	21	1	1	NUM
ejpam-5240	515	22	-	-	PUNCT
ejpam-5240	515	23	regular	regular	ADJ
ejpam-5240	515	24	graph	graph	NOUN
ejpam-5240	515	25	.	.	PUNCT
ejpam-5240	516	1	it	it	PRON
ejpam-5240	516	2	can	can	AUX
ejpam-5240	516	3	be	be	AUX
ejpam-5240	516	4	perceived	perceive	VERB
ejpam-5240	516	5	from	from	ADP
ejpam-5240	516	6	case	case	NOUN
ejpam-5240	516	7	1	1	NUM
ejpam-5240	516	8	of	of	ADP
ejpam-5240	516	9	theorem	theorem	ADJ
ejpam-5240	516	10	9	9	NUM
ejpam-5240	516	11	gs(n	gs(n	NOUN
ejpam-5240	516	12	,	,	PUNCT
ejpam-5240	516	13	k	k	NOUN
ejpam-5240	516	14	)	)	PUNCT
ejpam-5240	516	15	is	be	AUX
ejpam-5240	516	16	an	an	DET
ejpam-5240	516	17	empty	empty	ADJ
ejpam-5240	516	18	graph	graph	NOUN
ejpam-5240	516	19	of	of	ADP
ejpam-5240	516	20	order	order	NOUN
ejpam-5240	516	21	(	(	PUNCT
ejpam-5240	516	22	n	n	X
ejpam-5240	516	23	k	k	PROPN
ejpam-5240	516	24	)	)	PUNCT
ejpam-5240	516	25	.	.	PUNCT
ejpam-5240	517	1	consequently	consequently	ADV
ejpam-5240	517	2	,	,	PUNCT
ejpam-5240	517	3	we	we	PRON
ejpam-5240	517	4	have	have	VERB
ejpam-5240	517	5	corollary	corollary	NOUN
ejpam-5240	517	6	1	1	NUM
ejpam-5240	517	7	which	which	PRON
ejpam-5240	517	8	is	be	AUX
ejpam-5240	517	9	a	a	DET
ejpam-5240	517	10	corollary	corollary	NOUN
ejpam-5240	517	11	to	to	AUX
ejpam-5240	517	12	theorem	theorem	ADJ
ejpam-5240	517	13	9	9	NUM
ejpam-5240	517	14	.	.	PUNCT
ejpam-5240	517	15	corollary	corollary	ADJ
ejpam-5240	517	16	1	1	NUM
ejpam-5240	517	17	.	.	PUNCT
ejpam-5240	518	1	let	let	VERB
ejpam-5240	518	2	gs(n	gs(n	NOUN
ejpam-5240	518	3	,	,	PUNCT
ejpam-5240	518	4	k	k	NOUN
ejpam-5240	518	5	)	)	PUNCT
ejpam-5240	518	6	be	be	VERB
ejpam-5240	518	7	a	a	DET
ejpam-5240	518	8	k	k	ADV
ejpam-5240	518	9	-	-	ADJ
ejpam-5240	518	10	restricted	restricted	ADJ
ejpam-5240	518	11	intersection	intersection	NOUN
ejpam-5240	518	12	graph	graph	NOUN
ejpam-5240	518	13	.	.	PUNCT
ejpam-5240	519	1	if	if	SCONJ
ejpam-5240	519	2	k	k	PROPN
ejpam-5240	519	3	=	=	PUNCT
ejpam-5240	519	4	0	0	NUM
ejpam-5240	519	5	or	or	CCONJ
ejpam-5240	519	6	⌊	⌊	X
ejpam-5240	519	7	n	n	ADV
ejpam-5240	519	8	2	2	NUM
ejpam-5240	519	9	⌋	⌋	NOUN
ejpam-5240	519	10	<	<	X
ejpam-5240	519	11	k	k	X
ejpam-5240	519	12	≤	≤	PROPN
ejpam-5240	519	13	n	n	CCONJ
ejpam-5240	519	14	,	,	PUNCT
ejpam-5240	519	15	then	then	ADV
ejpam-5240	519	16	gs(n	gs(n	NOUN
ejpam-5240	519	17	,	,	PUNCT
ejpam-5240	519	18	k	k	NOUN
ejpam-5240	519	19	)	)	PUNCT
ejpam-5240	519	20	is	be	AUX
ejpam-5240	519	21	an	an	DET
ejpam-5240	519	22	empty	empty	ADJ
ejpam-5240	519	23	graph	graph	NOUN
ejpam-5240	519	24	of	of	ADP
ejpam-5240	519	25	order	order	NOUN
ejpam-5240	519	26	(	(	PUNCT
ejpam-5240	519	27	n	n	X
ejpam-5240	519	28	k	k	PROPN
ejpam-5240	519	29	)	)	PUNCT
ejpam-5240	519	30	.	.	PUNCT
ejpam-5240	520	1	proof	proof	NOUN
ejpam-5240	520	2	.	.	PUNCT
ejpam-5240	521	1	assume	assume	VERB
ejpam-5240	521	2	that	that	SCONJ
ejpam-5240	521	3	k	k	PROPN
ejpam-5240	521	4	=	=	PUNCT
ejpam-5240	521	5	0	0	NUM
ejpam-5240	521	6	or	or	CCONJ
ejpam-5240	521	7	⌊	⌊	X
ejpam-5240	521	8	n	n	ADV
ejpam-5240	521	9	2	2	NUM
ejpam-5240	521	10	⌋	⌋	NOUN
ejpam-5240	521	11	<	<	X
ejpam-5240	521	12	k	k	PROPN
ejpam-5240	521	13	≤	≤	X
ejpam-5240	521	14	n.	n.	NOUN
ejpam-5240	521	15	by	by	ADP
ejpam-5240	521	16	theorem	theorem	NOUN
ejpam-5240	521	17	9	9	NUM
ejpam-5240	521	18	,	,	PUNCT
ejpam-5240	521	19	gs(n	gs(n	NOUN
ejpam-5240	521	20	,	,	PUNCT
ejpam-5240	521	21	k	k	NOUN
ejpam-5240	521	22	)	)	PUNCT
ejpam-5240	521	23	is	be	AUX
ejpam-5240	521	24	a	a	DET
ejpam-5240	521	25	complete	complete	ADJ
ejpam-5240	521	26	graph	graph	NOUN
ejpam-5240	521	27	of	of	ADP
ejpam-5240	521	28	order	order	NOUN
ejpam-5240	521	29	(	(	PUNCT
ejpam-5240	521	30	n	n	X
ejpam-5240	521	31	k	k	PROPN
ejpam-5240	521	32	)	)	PUNCT
ejpam-5240	521	33	.	.	PUNCT
ejpam-5240	522	1	since	since	SCONJ
ejpam-5240	522	2	the	the	DET
ejpam-5240	522	3	complement	complement	NOUN
ejpam-5240	522	4	of	of	ADP
ejpam-5240	522	5	a	a	DET
ejpam-5240	522	6	complete	complete	ADJ
ejpam-5240	522	7	graph	graph	NOUN
ejpam-5240	522	8	is	be	AUX
ejpam-5240	522	9	an	an	DET
ejpam-5240	522	10	empty	empty	ADJ
ejpam-5240	522	11	graph	graph	NOUN
ejpam-5240	522	12	,	,	PUNCT
ejpam-5240	522	13	it	it	PRON
ejpam-5240	522	14	follows	follow	VERB
ejpam-5240	522	15	that	that	SCONJ
ejpam-5240	522	16	gs(n	gs(n	NOUN
ejpam-5240	522	17	,	,	PUNCT
ejpam-5240	522	18	k	k	NOUN
ejpam-5240	522	19	)	)	PUNCT
ejpam-5240	522	20	is	be	AUX
ejpam-5240	522	21	an	an	DET
ejpam-5240	522	22	empty	empty	ADJ
ejpam-5240	522	23	graph	graph	NOUN
ejpam-5240	522	24	of	of	ADP
ejpam-5240	522	25	order	order	NOUN
ejpam-5240	522	26	(	(	PUNCT
ejpam-5240	522	27	n	n	X
ejpam-5240	522	28	k	k	PROPN
ejpam-5240	522	29	)	)	PUNCT
ejpam-5240	522	30	.	.	PUNCT
ejpam-5240	523	1	moreover	moreover	ADV
ejpam-5240	523	2	,	,	PUNCT
ejpam-5240	523	3	corollary	corollary	ADJ
ejpam-5240	523	4	2	2	NUM
ejpam-5240	523	5	determines	determine	VERB
ejpam-5240	523	6	the	the	DET
ejpam-5240	523	7	degree	degree	NOUN
ejpam-5240	523	8	of	of	ADP
ejpam-5240	523	9	every	every	DET
ejpam-5240	523	10	vertex	vertex	NOUN
ejpam-5240	523	11	in	in	ADP
ejpam-5240	523	12	the	the	DET
ejpam-5240	523	13	complement	complement	NOUN
ejpam-5240	523	14	graph	graph	NOUN
ejpam-5240	523	15	of	of	ADP
ejpam-5240	523	16	gs(n	gs(n	NOUN
ejpam-5240	523	17	,	,	PUNCT
ejpam-5240	523	18	k	k	NOUN
ejpam-5240	523	19	)	)	PUNCT
ejpam-5240	523	20	when	when	SCONJ
ejpam-5240	523	21	k	k	PROPN
ejpam-5240	523	22	=	=	NOUN
ejpam-5240	523	23	1	1	X
ejpam-5240	523	24	.	.	PUNCT
ejpam-5240	524	1	this	this	PRON
ejpam-5240	524	2	is	be	AUX
ejpam-5240	524	3	a	a	DET
ejpam-5240	524	4	corollary	corollary	NOUN
ejpam-5240	524	5	to	to	AUX
ejpam-5240	524	6	theorem	theorem	ADJ
ejpam-5240	524	7	9	9	NUM
ejpam-5240	524	8	.	.	PUNCT
ejpam-5240	524	9	corollary	corollary	ADJ
ejpam-5240	524	10	2	2	NUM
ejpam-5240	524	11	.	.	PUNCT
ejpam-5240	525	1	let	let	VERB
ejpam-5240	525	2	gs(n	gs(n	NOUN
ejpam-5240	525	3	,	,	PUNCT
ejpam-5240	525	4	k	k	NOUN
ejpam-5240	525	5	)	)	PUNCT
ejpam-5240	525	6	be	be	VERB
ejpam-5240	525	7	a	a	DET
ejpam-5240	525	8	k	k	ADV
ejpam-5240	525	9	-	-	ADJ
ejpam-5240	525	10	restricted	restricted	ADJ
ejpam-5240	525	11	intersection	intersection	NOUN
ejpam-5240	525	12	graph	graph	NOUN
ejpam-5240	525	13	.	.	PUNCT
ejpam-5240	526	1	if	if	SCONJ
ejpam-5240	526	2	k	k	PROPN
ejpam-5240	526	3	=	=	SYM
ejpam-5240	526	4	1	1	NUM
ejpam-5240	526	5	,	,	PUNCT
ejpam-5240	526	6	then	then	ADV
ejpam-5240	526	7	gs(n,1	gs(n,1	NOUN
ejpam-5240	526	8	)	)	PUNCT
ejpam-5240	526	9	is	be	AUX
ejpam-5240	526	10	a	a	DET
ejpam-5240	526	11	complete	complete	ADJ
ejpam-5240	526	12	graph	graph	NOUN
ejpam-5240	526	13	of	of	ADP
ejpam-5240	526	14	order	order	NOUN
ejpam-5240	526	15	n.	n.	NOUN
ejpam-5240	526	16	proof	proof	NOUN
ejpam-5240	526	17	.	.	PUNCT
ejpam-5240	527	1	assume	assume	VERB
ejpam-5240	527	2	that	that	SCONJ
ejpam-5240	527	3	k	k	PROPN
ejpam-5240	527	4	=	=	PUNCT
ejpam-5240	527	5	1	1	X
ejpam-5240	527	6	.	.	PUNCT
ejpam-5240	527	7	by	by	ADP
ejpam-5240	527	8	theorem	theorem	NOUN
ejpam-5240	527	9	9	9	NUM
ejpam-5240	527	10	,	,	PUNCT
ejpam-5240	527	11	gs(n,1	gs(n,1	NOUN
ejpam-5240	527	12	)	)	PUNCT
ejpam-5240	527	13	is	be	AUX
ejpam-5240	527	14	a	a	DET
ejpam-5240	527	15	[	[	X
ejpam-5240	527	16	(	(	PUNCT
ejpam-5240	527	17	n−k	n−k	NOUN
ejpam-5240	527	18	k	k	NOUN
ejpam-5240	527	19	)	)	PUNCT
ejpam-5240	527	20	]	]	PUNCT
ejpam-5240	527	21	-regular	-regular	ADJ
ejpam-5240	527	22	graph	graph	NOUN
ejpam-5240	527	23	.	.	PUNCT
ejpam-5240	528	1	now	now	ADV
ejpam-5240	528	2	,	,	PUNCT
ejpam-5240	528	3	setting	set	VERB
ejpam-5240	528	4	k	k	X
ejpam-5240	528	5	=	=	SYM
ejpam-5240	528	6	1	1	NUM
ejpam-5240	528	7	,	,	PUNCT
ejpam-5240	528	8	we	we	PRON
ejpam-5240	528	9	have	have	VERB
ejpam-5240	528	10	deg(a	deg(a	PROPN
ejpam-5240	528	11	)	)	PUNCT
ejpam-5240	528	12	=	=	PUNCT
ejpam-5240	529	1	(	(	PUNCT
ejpam-5240	529	2	n−1	n−1	PROPN
ejpam-5240	529	3	1	1	NUM
ejpam-5240	529	4	)	)	PUNCT
ejpam-5240	529	5	=	=	SYM
ejpam-5240	529	6	n	n	CCONJ
ejpam-5240	529	7	−	−	NUM
ejpam-5240	529	8	1	1	NUM
ejpam-5240	529	9	for	for	ADP
ejpam-5240	529	10	all	all	DET
ejpam-5240	529	11	a	a	DET
ejpam-5240	529	12	∈	∈	PROPN
ejpam-5240	529	13	v	v	NOUN
ejpam-5240	529	14	(	(	PUNCT
ejpam-5240	529	15	gs(n,1	gs(n,1	NOUN
ejpam-5240	529	16	)	)	PUNCT
ejpam-5240	529	17	)	)	PUNCT
ejpam-5240	529	18	.	.	PUNCT
ejpam-5240	530	1	since	since	SCONJ
ejpam-5240	530	2	|gs(n,1	|gs(n,1	PROPN
ejpam-5240	530	3	)	)	PUNCT
ejpam-5240	530	4	|	|	ADV
ejpam-5240	530	5	is	be	AUX
ejpam-5240	530	6	equal	equal	ADJ
ejpam-5240	530	7	to	to	ADP
ejpam-5240	530	8	|gs(n,1	|gs(n,1	NOUN
ejpam-5240	530	9	)	)	PUNCT
ejpam-5240	530	10	|	|	ADV
ejpam-5240	530	11	=	=	SYM
ejpam-5240	530	12	n	n	CCONJ
ejpam-5240	530	13	,	,	PUNCT
ejpam-5240	530	14	it	it	PRON
ejpam-5240	530	15	follows	follow	VERB
ejpam-5240	530	16	that	that	SCONJ
ejpam-5240	530	17	every	every	DET
ejpam-5240	530	18	vertex	vertex	NOUN
ejpam-5240	530	19	of	of	ADP
ejpam-5240	530	20	gs(n,1	gs(n,1	PROPN
ejpam-5240	530	21	)	)	PUNCT
ejpam-5240	530	22	is	be	AUX
ejpam-5240	530	23	adjacent	adjacent	ADJ
ejpam-5240	530	24	to	to	ADP
ejpam-5240	530	25	each	each	DET
ejpam-5240	530	26	other	other	ADJ
ejpam-5240	530	27	.	.	PUNCT
ejpam-5240	531	1	therefore	therefore	ADV
ejpam-5240	531	2	,	,	PUNCT
ejpam-5240	531	3	gs(n,1	gs(n,1	PROPN
ejpam-5240	531	4	)	)	PUNCT
ejpam-5240	531	5	is	be	AUX
ejpam-5240	531	6	a	a	DET
ejpam-5240	531	7	complete	complete	ADJ
ejpam-5240	531	8	graph	graph	NOUN
ejpam-5240	531	9	of	of	ADP
ejpam-5240	531	10	order	order	NOUN
ejpam-5240	531	11	n.	n.	NOUN
ejpam-5240	531	12	illustration	illustration	NOUN
ejpam-5240	531	13	10	10	NUM
ejpam-5240	531	14	.	.	PUNCT
ejpam-5240	532	1	let	let	VERB
ejpam-5240	532	2	s4	s4	PROPN
ejpam-5240	532	3	=	=	SYM
ejpam-5240	532	4	{	{	PUNCT
ejpam-5240	532	5	x1	x1	PROPN
ejpam-5240	532	6	,	,	PUNCT
ejpam-5240	532	7	x2	x2	PROPN
ejpam-5240	532	8	,	,	PUNCT
ejpam-5240	532	9	x3	x3	ADJ
ejpam-5240	532	10	,	,	PUNCT
ejpam-5240	532	11	x4	x4	PROPN
ejpam-5240	532	12	}	}	PUNCT
ejpam-5240	532	13	and	and	CCONJ
ejpam-5240	532	14	let	let	VERB
ejpam-5240	532	15	k	k	NOUN
ejpam-5240	532	16	=	=	NOUN
ejpam-5240	532	17	1	1	X
ejpam-5240	532	18	.	.	PUNCT
ejpam-5240	533	1	if	if	SCONJ
ejpam-5240	533	2	k	k	PROPN
ejpam-5240	533	3	=	=	SYM
ejpam-5240	533	4	1	1	NUM
ejpam-5240	533	5	,	,	PUNCT
ejpam-5240	533	6	then	then	ADV
ejpam-5240	533	7	by	by	ADP
ejpam-5240	533	8	theorem	theorem	ADJ
ejpam-5240	533	9	4	4	NUM
ejpam-5240	533	10	,	,	PUNCT
ejpam-5240	533	11	gs(4,1	gs(4,1	NOUN
ejpam-5240	533	12	)	)	PUNCT
ejpam-5240	533	13	is	be	AUX
ejpam-5240	533	14	an	an	DET
ejpam-5240	533	15	empty	empty	ADJ
ejpam-5240	533	16	graph	graph	NOUN
ejpam-5240	533	17	of	of	ADP
ejpam-5240	533	18	order	order	NOUN
ejpam-5240	533	19	4	4	X
ejpam-5240	533	20	.	.	PUNCT
ejpam-5240	534	1	we	we	PRON
ejpam-5240	534	2	know	know	VERB
ejpam-5240	534	3	that	that	SCONJ
ejpam-5240	534	4	the	the	DET
ejpam-5240	534	5	complement	complement	NOUN
ejpam-5240	534	6	graph	graph	NOUN
ejpam-5240	534	7	of	of	ADP
ejpam-5240	534	8	an	an	DET
ejpam-5240	534	9	empty	empty	ADJ
ejpam-5240	534	10	graph	graph	NOUN
ejpam-5240	534	11	is	be	AUX
ejpam-5240	534	12	a	a	DET
ejpam-5240	534	13	complete	complete	ADJ
ejpam-5240	534	14	graph	graph	NOUN
ejpam-5240	534	15	.	.	PUNCT
ejpam-5240	535	1	so	so	ADV
ejpam-5240	535	2	,	,	PUNCT
ejpam-5240	535	3	gs(4,1	gs(4,1	NOUN
ejpam-5240	535	4	)	)	PUNCT
ejpam-5240	535	5	is	be	AUX
ejpam-5240	535	6	a	a	DET
ejpam-5240	535	7	complete	complete	ADJ
ejpam-5240	535	8	graph	graph	NOUN
ejpam-5240	535	9	of	of	ADP
ejpam-5240	535	10	order	order	NOUN
ejpam-5240	535	11	4	4	NUM
ejpam-5240	535	12	.	.	PUNCT
ejpam-5240	535	13	a	a	DET
ejpam-5240	535	14	pictorial	pictorial	ADJ
ejpam-5240	535	15	representation	representation	NOUN
ejpam-5240	535	16	of	of	ADP
ejpam-5240	535	17	gs(4,1	gs(4,1	NOUN
ejpam-5240	535	18	)	)	PUNCT
ejpam-5240	535	19	and	and	CCONJ
ejpam-5240	535	20	gs(4,1	gs(4,1	NOUN
ejpam-5240	535	21	)	)	PUNCT
ejpam-5240	535	22	is	be	AUX
ejpam-5240	535	23	shown	show	VERB
ejpam-5240	535	24	in	in	ADP
ejpam-5240	535	25	figure	figure	NOUN
ejpam-5240	535	26	11	11	NUM
ejpam-5240	535	27	.	.	PUNCT
ejpam-5240	536	1	observe	observe	VERB
ejpam-5240	536	2	that	that	SCONJ
ejpam-5240	536	3	every	every	DET
ejpam-5240	536	4	a	a	DET
ejpam-5240	536	5	∈	∈	PROPN
ejpam-5240	536	6	v	v	NOUN
ejpam-5240	536	7	(	(	PUNCT
ejpam-5240	536	8	gs(4,1	gs(4,1	NOUN
ejpam-5240	536	9	)	)	PUNCT
ejpam-5240	536	10	)	)	PUNCT
ejpam-5240	536	11	,	,	PUNCT
ejpam-5240	536	12	deg(a	deg(a	PROPN
ejpam-5240	536	13	)	)	PUNCT
ejpam-5240	536	14	=	=	SYM
ejpam-5240	536	15	4	4	NUM
ejpam-5240	536	16	−	−	NOUN
ejpam-5240	536	17	1	1	NUM
ejpam-5240	536	18	=	=	SYM
ejpam-5240	536	19	3	3	NUM
ejpam-5240	536	20	.	.	PUNCT
ejpam-5240	537	1	thus	thus	ADV
ejpam-5240	537	2	,	,	PUNCT
ejpam-5240	537	3	every	every	DET
ejpam-5240	537	4	vertex	vertex	NOUN
ejpam-5240	537	5	of	of	ADP
ejpam-5240	537	6	gs(4,1	gs(4,1	NOUN
ejpam-5240	537	7	)	)	PUNCT
ejpam-5240	537	8	is	be	AUX
ejpam-5240	537	9	adjacent	adjacent	ADJ
ejpam-5240	537	10	to	to	ADP
ejpam-5240	537	11	each	each	DET
ejpam-5240	537	12	other	other	ADJ
ejpam-5240	537	13	.	.	PUNCT
ejpam-5240	538	1	therefore	therefore	ADV
ejpam-5240	538	2	,	,	PUNCT
ejpam-5240	538	3	gs(4,1	gs(4,1	NOUN
ejpam-5240	538	4	)	)	PUNCT
ejpam-5240	538	5	is	be	AUX
ejpam-5240	538	6	a	a	DET
ejpam-5240	538	7	complete	complete	ADJ
ejpam-5240	538	8	graph	graph	NOUN
ejpam-5240	538	9	of	of	ADP
ejpam-5240	538	10	order	order	NOUN
ejpam-5240	538	11	4	4	NUM
ejpam-5240	538	12	.	.	NOUN
ejpam-5240	538	13	5	5	NUM
ejpam-5240	538	14	.	.	PUNCT
ejpam-5240	538	15	additional	additional	ADJ
ejpam-5240	538	16	parameters	parameter	NOUN
ejpam-5240	538	17	of	of	ADP
ejpam-5240	538	18	gs(n	gs(n	NOUN
ejpam-5240	538	19	,	,	PUNCT
ejpam-5240	538	20	k	k	NOUN
ejpam-5240	538	21	)	)	PUNCT
ejpam-5240	538	22	parameters	parameter	NOUN
ejpam-5240	538	23	are	be	AUX
ejpam-5240	538	24	numerical	numerical	ADJ
ejpam-5240	538	25	values	value	NOUN
ejpam-5240	538	26	that	that	PRON
ejpam-5240	538	27	help	help	VERB
ejpam-5240	538	28	define	define	VERB
ejpam-5240	538	29	a	a	DET
ejpam-5240	538	30	graph	graph	NOUN
ejpam-5240	538	31	.	.	PUNCT
ejpam-5240	539	1	in	in	ADP
ejpam-5240	539	2	this	this	DET
ejpam-5240	539	3	section	section	NOUN
ejpam-5240	539	4	,	,	PUNCT
ejpam-5240	539	5	other	other	ADJ
ejpam-5240	539	6	graph	graph	NOUN
ejpam-5240	539	7	parameters	parameter	NOUN
ejpam-5240	539	8	such	such	ADJ
ejpam-5240	539	9	as	as	ADP
ejpam-5240	539	10	the	the	DET
ejpam-5240	539	11	independence	independence	NOUN
ejpam-5240	539	12	number	number	NOUN
ejpam-5240	539	13	,	,	PUNCT
ejpam-5240	539	14	domination	domination	NOUN
ejpam-5240	539	15	number	number	NOUN
ejpam-5240	539	16	,	,	PUNCT
ejpam-5240	539	17	and	and	CCONJ
ejpam-5240	539	18	isolate	isolate	VERB
ejpam-5240	539	19	domination	domination	NOUN
ejpam-5240	539	20	number	number	NOUN
ejpam-5240	539	21	are	be	AUX
ejpam-5240	539	22	presented	present	VERB
ejpam-5240	539	23	with	with	ADP
ejpam-5240	539	24	proofs	proof	NOUN
ejpam-5240	539	25	to	to	PART
ejpam-5240	539	26	determine	determine	VERB
ejpam-5240	539	27	a	a	DET
ejpam-5240	539	28	gs(n	gs(n	NOUN
ejpam-5240	539	29	,	,	PUNCT
ejpam-5240	539	30	k	k	NOUN
ejpam-5240	539	31	)	)	PUNCT
ejpam-5240	539	32	.	.	PUNCT
ejpam-5240	540	1	m.e	m.e	PROPN
ejpam-5240	540	2	.	.	PROPN
ejpam-5240	540	3	pelagio	pelagio	PROPN
ejpam-5240	540	4	,	,	PUNCT
ejpam-5240	540	5	n.	n.	NOUN
ejpam-5240	540	6	mame	mame	PROPN
ejpam-5240	540	7	,	,	PUNCT
ejpam-5240	540	8	k.	k.	PROPN
ejpam-5240	540	9	mendoza	mendoza	PROPN
ejpam-5240	540	10	/	/	SYM
ejpam-5240	540	11	eur	eur	PROPN
ejpam-5240	540	12	.	.	PUNCT
ejpam-5240	541	1	j.	j.	PROPN
ejpam-5240	541	2	pure	pure	PROPN
ejpam-5240	541	3	appl	appl	PROPN
ejpam-5240	541	4	.	.	PROPN
ejpam-5240	541	5	math	math	PROPN
ejpam-5240	541	6	,	,	PUNCT
ejpam-5240	541	7	17	17	NUM
ejpam-5240	541	8	(	(	PUNCT
ejpam-5240	541	9	3	3	NUM
ejpam-5240	541	10	)	)	PUNCT
ejpam-5240	541	11	(	(	PUNCT
ejpam-5240	541	12	2024	2024	NUM
ejpam-5240	541	13	)	)	PUNCT
ejpam-5240	541	14	,	,	PUNCT
ejpam-5240	541	15	1779	1779	NUM
ejpam-5240	541	16	-	-	SYM
ejpam-5240	541	17	1803	1803	NUM
ejpam-5240	541	18	1796	1796	NUM
ejpam-5240	541	19	{	{	PUNCT
ejpam-5240	541	20	x1	x1	PROPN
ejpam-5240	541	21	}	}	PUNCT
ejpam-5240	541	22	{	{	PUNCT
ejpam-5240	541	23	x2	x2	PROPN
ejpam-5240	541	24	}	}	PUNCT
ejpam-5240	541	25	{	{	PUNCT
ejpam-5240	541	26	x3	x3	PROPN
ejpam-5240	541	27	}	}	PUNCT
ejpam-5240	541	28	{	{	PUNCT
ejpam-5240	541	29	x4	x4	PROPN
ejpam-5240	541	30	}	}	PUNCT
ejpam-5240	541	31	{	{	PUNCT
ejpam-5240	541	32	x1	x1	PROPN
ejpam-5240	541	33	}	}	PUNCT
ejpam-5240	541	34	{	{	PUNCT
ejpam-5240	541	35	x2	x2	PROPN
ejpam-5240	541	36	}	}	PUNCT
ejpam-5240	541	37	{	{	PUNCT
ejpam-5240	541	38	x3	x3	PROPN
ejpam-5240	541	39	}	}	PUNCT
ejpam-5240	541	40	{	{	PUNCT
ejpam-5240	541	41	x4	x4	PROPN
ejpam-5240	541	42	}	}	PUNCT
ejpam-5240	541	43	figure	figure	NOUN
ejpam-5240	541	44	11	11	NUM
ejpam-5240	541	45	:	:	PUNCT
ejpam-5240	541	46	pictorial	pictorial	ADJ
ejpam-5240	541	47	illustrations	illustration	NOUN
ejpam-5240	541	48	of	of	ADP
ejpam-5240	541	49	gs(4,1	gs(4,1	NOUN
ejpam-5240	541	50	)	)	PUNCT
ejpam-5240	541	51	(	(	PUNCT
ejpam-5240	541	52	left	leave	VERB
ejpam-5240	541	53	)	)	PUNCT
ejpam-5240	541	54	and	and	CCONJ
ejpam-5240	541	55	its	its	PRON
ejpam-5240	541	56	complement	complement	NOUN
ejpam-5240	541	57	graph	graph	NOUN
ejpam-5240	541	58	gs(4,1	gs(4,1	NOUN
ejpam-5240	541	59	)	)	PUNCT
ejpam-5240	541	60	(	(	PUNCT
ejpam-5240	541	61	right	right	NOUN
ejpam-5240	541	62	)	)	PUNCT
ejpam-5240	541	63	.	.	PUNCT
ejpam-5240	542	1	5.1	5.1	NUM
ejpam-5240	542	2	.	.	PUNCT
ejpam-5240	543	1	independence	independence	NOUN
ejpam-5240	543	2	number	number	NOUN
ejpam-5240	543	3	of	of	ADP
ejpam-5240	543	4	a	a	DET
ejpam-5240	543	5	gs(n	gs(n	NOUN
ejpam-5240	543	6	,	,	PUNCT
ejpam-5240	543	7	k	k	NOUN
ejpam-5240	543	8	)	)	PUNCT
ejpam-5240	543	9	this	this	DET
ejpam-5240	543	10	subsection	subsection	NOUN
ejpam-5240	543	11	examines	examine	VERB
ejpam-5240	543	12	the	the	DET
ejpam-5240	543	13	independence	independence	NOUN
ejpam-5240	543	14	number	number	NOUN
ejpam-5240	543	15	of	of	ADP
ejpam-5240	543	16	gs(n	gs(n	NOUN
ejpam-5240	543	17	,	,	PUNCT
ejpam-5240	543	18	k	k	NOUN
ejpam-5240	543	19	)	)	PUNCT
ejpam-5240	543	20	given	give	VERB
ejpam-5240	543	21	the	the	DET
ejpam-5240	543	22	two	two	NUM
ejpam-5240	543	23	cases	case	NOUN
ejpam-5240	543	24	:	:	PUNCT
ejpam-5240	543	25	when	when	SCONJ
ejpam-5240	543	26	1	1	NUM
ejpam-5240	543	27	≤	≤	NUM
ejpam-5240	543	28	k	k	X
ejpam-5240	543	29	≤	≤	NUM
ejpam-5240	543	30	⌊	⌊	VERB
ejpam-5240	543	31	n	n	PRON
ejpam-5240	543	32	2	2	NUM
ejpam-5240	543	33	⌋	⌋	NOUN
ejpam-5240	543	34	and	and	CCONJ
ejpam-5240	543	35	k	k	NOUN
ejpam-5240	543	36	=	=	SYM
ejpam-5240	543	37	0	0	NUM
ejpam-5240	543	38	or	or	CCONJ
ejpam-5240	543	39	⌊	⌊	X
ejpam-5240	543	40	n	n	ADV
ejpam-5240	543	41	2	2	NUM
ejpam-5240	543	42	⌋	⌋	NOUN
ejpam-5240	543	43	<	<	X
ejpam-5240	543	44	k	k	PROPN
ejpam-5240	543	45	≤	≤	PROPN
ejpam-5240	543	46	n.	n.	NOUN
ejpam-5240	543	47	we	we	PRON
ejpam-5240	543	48	shall	shall	AUX
ejpam-5240	543	49	denote	denote	VERB
ejpam-5240	543	50	the	the	DET
ejpam-5240	543	51	independence	independence	NOUN
ejpam-5240	543	52	number	number	NOUN
ejpam-5240	543	53	of	of	ADP
ejpam-5240	543	54	gs(n	gs(n	NOUN
ejpam-5240	543	55	,	,	PUNCT
ejpam-5240	543	56	k	k	NOUN
ejpam-5240	543	57	)	)	PUNCT
ejpam-5240	543	58	as	as	ADP
ejpam-5240	543	59	α(gs(n	α(gs(n	NOUN
ejpam-5240	543	60	,	,	PUNCT
ejpam-5240	543	61	k	k	NOUN
ejpam-5240	543	62	)	)	PUNCT
ejpam-5240	543	63	)	)	PUNCT
ejpam-5240	543	64	.	.	PUNCT
ejpam-5240	544	1	the	the	DET
ejpam-5240	544	2	lemma	lemma	PROPN
ejpam-5240	544	3	below	below	ADV
ejpam-5240	544	4	determines	determine	VERB
ejpam-5240	544	5	the	the	DET
ejpam-5240	544	6	existence	existence	NOUN
ejpam-5240	544	7	of	of	ADP
ejpam-5240	544	8	an	an	DET
ejpam-5240	544	9	independent	independent	ADJ
ejpam-5240	544	10	set	set	NOUN
ejpam-5240	544	11	in	in	ADP
ejpam-5240	544	12	gs(n	gs(n	NOUN
ejpam-5240	544	13	,	,	PUNCT
ejpam-5240	544	14	k	k	NOUN
ejpam-5240	544	15	)	)	PUNCT
ejpam-5240	544	16	for	for	ADP
ejpam-5240	544	17	1	1	NUM
ejpam-5240	544	18	≤	≤	NOUN
ejpam-5240	544	19	k	k	X
ejpam-5240	544	20	≤	≤	NUM
ejpam-5240	544	21	⌊	⌊	VERB
ejpam-5240	544	22	n	n	DET
ejpam-5240	544	23	2	2	NUM
ejpam-5240	544	24	⌋	⌋	NOUN
ejpam-5240	544	25	.	.	PUNCT
ejpam-5240	545	1	lemma	lemma	PROPN
ejpam-5240	545	2	3	3	X
ejpam-5240	545	3	.	.	PUNCT
ejpam-5240	546	1	let	let	VERB
ejpam-5240	546	2	sn	sn	PROPN
ejpam-5240	546	3	=	=	PUNCT
ejpam-5240	546	4	{	{	PUNCT
ejpam-5240	546	5	x1	x1	PROPN
ejpam-5240	546	6	,	,	PUNCT
ejpam-5240	546	7	x2	x2	PROPN
ejpam-5240	546	8	,	,	PUNCT
ejpam-5240	546	9	...	...	PUNCT
ejpam-5240	546	10	,	,	PUNCT
ejpam-5240	546	11	xn	xn	PRON
ejpam-5240	546	12	}	}	PUNCT
ejpam-5240	546	13	be	be	VERB
ejpam-5240	546	14	an	an	DET
ejpam-5240	546	15	n	n	NOUN
ejpam-5240	546	16	-	-	PUNCT
ejpam-5240	546	17	element	element	NOUN
ejpam-5240	546	18	set	set	NOUN
ejpam-5240	546	19	and	and	CCONJ
ejpam-5240	546	20	let	let	VERB
ejpam-5240	546	21	gs(n	gs(n	NOUN
ejpam-5240	546	22	,	,	PUNCT
ejpam-5240	546	23	k	k	NOUN
ejpam-5240	546	24	)	)	PUNCT
ejpam-5240	546	25	be	be	VERB
ejpam-5240	546	26	a	a	DET
ejpam-5240	546	27	k	k	ADV
ejpam-5240	546	28	-	-	ADJ
ejpam-5240	546	29	restricted	restricted	ADJ
ejpam-5240	546	30	intersection	intersection	NOUN
ejpam-5240	546	31	graph	graph	NOUN
ejpam-5240	546	32	.	.	PUNCT
ejpam-5240	547	1	if	if	SCONJ
ejpam-5240	547	2	1	1	NUM
ejpam-5240	547	3	≤	≤	NUM
ejpam-5240	547	4	k	k	X
ejpam-5240	547	5	≤	≤	NUM
ejpam-5240	547	6	⌊	⌊	VERB
ejpam-5240	547	7	n	n	DET
ejpam-5240	547	8	2	2	NUM
ejpam-5240	547	9	⌋	⌋	NOUN
ejpam-5240	547	10	,	,	PUNCT
ejpam-5240	547	11	then	then	ADV
ejpam-5240	547	12	t	t	PROPN
ejpam-5240	547	13	=	=	SYM
ejpam-5240	547	14	{	{	PUNCT
ejpam-5240	547	15	{	{	PUNCT
ejpam-5240	547	16	x1	x1	PROPN
ejpam-5240	547	17	,	,	PUNCT
ejpam-5240	547	18	x2	x2	PROPN
ejpam-5240	547	19	,	,	PUNCT
ejpam-5240	547	20	...	...	PUNCT
ejpam-5240	547	21	,	,	PUNCT
ejpam-5240	547	22	xk	xk	PROPN
ejpam-5240	547	23	}	}	PUNCT
ejpam-5240	547	24	,	,	PUNCT
ejpam-5240	547	25	{	{	PUNCT
ejpam-5240	547	26	xk+1	xk+1	PROPN
ejpam-5240	547	27	,	,	PUNCT
ejpam-5240	547	28	...	...	PUNCT
ejpam-5240	547	29	,	,	PUNCT
ejpam-5240	547	30	x2k	x2k	NOUN
ejpam-5240	547	31	}	}	PUNCT
ejpam-5240	547	32	,	,	PUNCT
ejpam-5240	547	33	...	...	PUNCT
ejpam-5240	547	34	,	,	PUNCT
ejpam-5240	547	35	{	{	PUNCT
ejpam-5240	547	36	x(⌊n	x(⌊n	PROPN
ejpam-5240	547	37	k	k	PROPN
ejpam-5240	547	38	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	547	39	,	,	PUNCT
ejpam-5240	547	40	...	...	PUNCT
ejpam-5240	547	41	,	,	PUNCT
ejpam-5240	547	42	x(⌊n	x(⌊n	PROPN
ejpam-5240	547	43	k	k	PROPN
ejpam-5240	547	44	⌋)k	⌋)k	PROPN
ejpam-5240	547	45	}	}	PUNCT
ejpam-5240	547	46	}	}	PUNCT
ejpam-5240	547	47	is	be	AUX
ejpam-5240	547	48	an	an	DET
ejpam-5240	547	49	independent	independent	ADJ
ejpam-5240	547	50	set	set	NOUN
ejpam-5240	547	51	in	in	ADP
ejpam-5240	547	52	gs(n	gs(n	NOUN
ejpam-5240	547	53	,	,	PUNCT
ejpam-5240	547	54	k	k	NOUN
ejpam-5240	547	55	)	)	PUNCT
ejpam-5240	547	56	where	where	SCONJ
ejpam-5240	547	57	|t	|t	VERB
ejpam-5240	547	58	|	|	ADV
ejpam-5240	548	1	=	=	SYM
ejpam-5240	548	2	⌊	⌊	PROPN
ejpam-5240	548	3	n	n	PRON
ejpam-5240	548	4	k	k	NOUN
ejpam-5240	548	5	⌋	⌋	NOUN
ejpam-5240	548	6	.	.	PUNCT
ejpam-5240	549	1	proof	proof	NOUN
ejpam-5240	549	2	.	.	PUNCT
ejpam-5240	550	1	let	let	VERB
ejpam-5240	550	2	t	t	NOUN
ejpam-5240	550	3	=	=	PRON
ejpam-5240	550	4	{	{	PUNCT
ejpam-5240	550	5	{	{	PUNCT
ejpam-5240	550	6	x1	x1	PROPN
ejpam-5240	550	7	,	,	PUNCT
ejpam-5240	550	8	x2	x2	PROPN
ejpam-5240	550	9	,	,	PUNCT
ejpam-5240	550	10	...	...	PUNCT
ejpam-5240	550	11	,	,	PUNCT
ejpam-5240	550	12	xk	xk	PROPN
ejpam-5240	550	13	}	}	PUNCT
ejpam-5240	550	14	,	,	PUNCT
ejpam-5240	550	15	{	{	PUNCT
ejpam-5240	550	16	xk+1	xk+1	PROPN
ejpam-5240	550	17	,	,	PUNCT
ejpam-5240	550	18	...	...	PUNCT
ejpam-5240	550	19	,	,	PUNCT
ejpam-5240	550	20	x2k	x2k	NOUN
ejpam-5240	550	21	}	}	PUNCT
ejpam-5240	550	22	,	,	PUNCT
ejpam-5240	550	23	...	...	PUNCT
ejpam-5240	550	24	,	,	PUNCT
ejpam-5240	550	25	{	{	PUNCT
ejpam-5240	550	26	x(⌊n	x(⌊n	PROPN
ejpam-5240	550	27	k	k	PROPN
ejpam-5240	550	28	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	550	29	,	,	PUNCT
ejpam-5240	550	30	...	...	PUNCT
ejpam-5240	550	31	,	,	PUNCT
ejpam-5240	550	32	x(⌊n	x(⌊n	PROPN
ejpam-5240	550	33	k	k	PROPN
ejpam-5240	550	34	⌋)k	⌋)k	PROPN
ejpam-5240	550	35	}	}	PUNCT
ejpam-5240	550	36	}	}	PUNCT
ejpam-5240	550	37	.	.	PUNCT
ejpam-5240	551	1	it	it	PRON
ejpam-5240	551	2	can	can	AUX
ejpam-5240	551	3	be	be	AUX
ejpam-5240	551	4	observed	observe	VERB
ejpam-5240	551	5	that	that	SCONJ
ejpam-5240	551	6	t	t	PROPN
ejpam-5240	551	7	contains	contain	VERB
ejpam-5240	551	8	some	some	PRON
ejpam-5240	551	9	of	of	ADP
ejpam-5240	551	10	the	the	DET
ejpam-5240	551	11	partitions	partition	NOUN
ejpam-5240	551	12	of	of	ADP
ejpam-5240	551	13	sn	sn	PROPN
ejpam-5240	551	14	where	where	SCONJ
ejpam-5240	551	15	for	for	ADP
ejpam-5240	551	16	every	every	DET
ejpam-5240	551	17	a	a	DET
ejpam-5240	551	18	∈	∈	PROPN
ejpam-5240	551	19	t	t	NOUN
ejpam-5240	551	20	,	,	PUNCT
ejpam-5240	552	1	|a|	|a|	PROPN
ejpam-5240	552	2	=	=	PROPN
ejpam-5240	552	3	k.	k.	PROPN
ejpam-5240	552	4	this	this	PRON
ejpam-5240	552	5	implies	imply	VERB
ejpam-5240	552	6	that	that	SCONJ
ejpam-5240	552	7	t	t	PROPN
ejpam-5240	552	8	⊆	⊆	NUM
ejpam-5240	552	9	s(n	s(n	PROPN
ejpam-5240	552	10	,	,	PUNCT
ejpam-5240	552	11	k	k	NOUN
ejpam-5240	552	12	)	)	PUNCT
ejpam-5240	552	13	and	and	CCONJ
ejpam-5240	552	14	thus	thus	ADV
ejpam-5240	552	15	,	,	PUNCT
ejpam-5240	552	16	t	t	PROPN
ejpam-5240	552	17	⊆	⊆	NUM
ejpam-5240	552	18	v	v	NOUN
ejpam-5240	552	19	(	(	PUNCT
ejpam-5240	552	20	gs(n	gs(n	NOUN
ejpam-5240	552	21	,	,	PUNCT
ejpam-5240	552	22	k	k	NOUN
ejpam-5240	552	23	)	)	PUNCT
ejpam-5240	552	24	)	)	PUNCT
ejpam-5240	552	25	.	.	PUNCT
ejpam-5240	553	1	since	since	SCONJ
ejpam-5240	553	2	for	for	ADP
ejpam-5240	553	3	all	all	DET
ejpam-5240	553	4	a	a	DET
ejpam-5240	553	5	,	,	PUNCT
ejpam-5240	553	6	b	b	PROPN
ejpam-5240	553	7	∈	∈	PROPN
ejpam-5240	553	8	t	t	PROPN
ejpam-5240	553	9	and	and	CCONJ
ejpam-5240	553	10	a	a	DET
ejpam-5240	553	11	̸=	̸=	PROPN
ejpam-5240	553	12	b	b	PROPN
ejpam-5240	553	13	,	,	PUNCT
ejpam-5240	553	14	a	a	DET
ejpam-5240	553	15	∩	∩	ADJ
ejpam-5240	553	16	b	b	NOUN
ejpam-5240	553	17	=	=	SYM
ejpam-5240	553	18	∅	∅	NOUN
ejpam-5240	553	19	,	,	PUNCT
ejpam-5240	553	20	it	it	PRON
ejpam-5240	553	21	follows	follow	VERB
ejpam-5240	553	22	that	that	SCONJ
ejpam-5240	553	23	[	[	X
ejpam-5240	553	24	a	a	DET
ejpam-5240	553	25	,	,	PUNCT
ejpam-5240	553	26	b	b	NOUN
ejpam-5240	553	27	]	]	X
ejpam-5240	553	28	/∈	/∈	PUNCT
ejpam-5240	554	1	e(gs(n	e(gs(n	NOUN
ejpam-5240	554	2	,	,	PUNCT
ejpam-5240	554	3	k	k	NOUN
ejpam-5240	554	4	)	)	PUNCT
ejpam-5240	554	5	)	)	PUNCT
ejpam-5240	554	6	.	.	PUNCT
ejpam-5240	555	1	hence	hence	ADV
ejpam-5240	555	2	,	,	PUNCT
ejpam-5240	555	3	t	t	PROPN
ejpam-5240	555	4	is	be	AUX
ejpam-5240	555	5	an	an	DET
ejpam-5240	555	6	independent	independent	ADJ
ejpam-5240	555	7	set	set	NOUN
ejpam-5240	555	8	in	in	ADP
ejpam-5240	555	9	gs(n	gs(n	NOUN
ejpam-5240	555	10	,	,	PUNCT
ejpam-5240	555	11	k	k	NOUN
ejpam-5240	555	12	)	)	PUNCT
ejpam-5240	555	13	.	.	PUNCT
ejpam-5240	556	1	now	now	ADV
ejpam-5240	556	2	,	,	PUNCT
ejpam-5240	556	3	if	if	SCONJ
ejpam-5240	556	4	n	n	PRON
ejpam-5240	556	5	is	be	AUX
ejpam-5240	556	6	divisible	divisible	ADJ
ejpam-5240	556	7	by	by	ADP
ejpam-5240	556	8	k	k	PROPN
ejpam-5240	556	9	,	,	PUNCT
ejpam-5240	556	10	then	then	ADV
ejpam-5240	556	11	|t	|t	VERB
ejpam-5240	557	1	|	|	ADV
ejpam-5240	557	2	=	=	SYM
ejpam-5240	557	3	n	n	PROPN
ejpam-5240	557	4	k	k	NOUN
ejpam-5240	557	5	.	.	PUNCT
ejpam-5240	558	1	on	on	ADP
ejpam-5240	558	2	the	the	DET
ejpam-5240	558	3	other	other	ADJ
ejpam-5240	558	4	hand	hand	NOUN
ejpam-5240	558	5	,	,	PUNCT
ejpam-5240	558	6	if	if	SCONJ
ejpam-5240	558	7	n	n	PRON
ejpam-5240	558	8	is	be	AUX
ejpam-5240	558	9	not	not	PART
ejpam-5240	558	10	divisible	divisible	ADJ
ejpam-5240	558	11	by	by	ADP
ejpam-5240	558	12	k	k	PROPN
ejpam-5240	558	13	,	,	PUNCT
ejpam-5240	558	14	then	then	ADV
ejpam-5240	558	15	by	by	ADP
ejpam-5240	558	16	division	division	NOUN
ejpam-5240	558	17	algorithm	algorithm	NOUN
ejpam-5240	558	18	,	,	PUNCT
ejpam-5240	558	19	there	there	PRON
ejpam-5240	558	20	exist	exist	VERB
ejpam-5240	558	21	unique	unique	ADJ
ejpam-5240	558	22	integers	integer	NOUN
ejpam-5240	558	23	a	a	PRON
ejpam-5240	558	24	and	and	CCONJ
ejpam-5240	558	25	b	b	NOUN
ejpam-5240	558	26	with	with	ADP
ejpam-5240	558	27	0	0	NUM
ejpam-5240	558	28	<	<	X
ejpam-5240	558	29	b	b	X
ejpam-5240	558	30	<	<	X
ejpam-5240	558	31	k	k	X
ejpam-5240	558	32	such	such	ADJ
ejpam-5240	558	33	that	that	SCONJ
ejpam-5240	558	34	n	n	NOUN
ejpam-5240	558	35	=	=	PUNCT
ejpam-5240	558	36	ka+	ka+	PROPN
ejpam-5240	558	37	b.	b.	PROPN
ejpam-5240	558	38	meaning	mean	VERB
ejpam-5240	558	39	to	to	PART
ejpam-5240	558	40	say	say	VERB
ejpam-5240	558	41	,	,	PUNCT
ejpam-5240	558	42	there	there	PRON
ejpam-5240	558	43	are	be	VERB
ejpam-5240	558	44	b	b	NOUN
ejpam-5240	558	45	elements	element	NOUN
ejpam-5240	558	46	of	of	ADP
ejpam-5240	558	47	sn	sn	PROPN
ejpam-5240	558	48	that	that	PRON
ejpam-5240	558	49	are	be	AUX
ejpam-5240	558	50	not	not	PART
ejpam-5240	558	51	in	in	ADP
ejpam-5240	558	52	t	t	PROPN
ejpam-5240	558	53	.	.	PUNCT
ejpam-5240	559	1	we	we	PRON
ejpam-5240	559	2	can	can	AUX
ejpam-5240	559	3	not	not	PART
ejpam-5240	559	4	form	form	VERB
ejpam-5240	559	5	a	a	DET
ejpam-5240	559	6	k	k	ADJ
ejpam-5240	559	7	-	-	NOUN
ejpam-5240	559	8	element	element	NOUN
ejpam-5240	559	9	subset	subset	NOUN
ejpam-5240	559	10	from	from	ADP
ejpam-5240	559	11	the	the	DET
ejpam-5240	559	12	b	b	NOUN
ejpam-5240	559	13	remaining	remain	VERB
ejpam-5240	559	14	elements	element	NOUN
ejpam-5240	559	15	,	,	PUNCT
ejpam-5240	559	16	so	so	ADV
ejpam-5240	559	17	|t	|t	PROPN
ejpam-5240	560	1	|	|	ADV
ejpam-5240	560	2	=	=	SYM
ejpam-5240	560	3	⌊	⌊	PROPN
ejpam-5240	560	4	n	n	PRON
ejpam-5240	560	5	k	k	NOUN
ejpam-5240	560	6	⌋	⌋	PROPN
ejpam-5240	560	7	.	.	PUNCT
ejpam-5240	561	1	either	either	DET
ejpam-5240	561	2	way	way	NOUN
ejpam-5240	561	3	|t	|t	PROPN
ejpam-5240	561	4	|	|	ADV
ejpam-5240	561	5	=	=	SYM
ejpam-5240	561	6	⌊	⌊	PROPN
ejpam-5240	561	7	n	n	PRON
ejpam-5240	561	8	k	k	PROPN
ejpam-5240	561	9	⌋	⌋	PROPN
ejpam-5240	561	10	.	.	PUNCT
ejpam-5240	562	1	lemma	lemma	PROPN
ejpam-5240	562	2	3	3	NUM
ejpam-5240	562	3	identifies	identify	VERB
ejpam-5240	562	4	one	one	NUM
ejpam-5240	562	5	independent	independent	ADJ
ejpam-5240	562	6	set	set	NOUN
ejpam-5240	562	7	in	in	ADP
ejpam-5240	562	8	gs(n	gs(n	NOUN
ejpam-5240	562	9	,	,	PUNCT
ejpam-5240	562	10	k	k	NOUN
ejpam-5240	562	11	)	)	PUNCT
ejpam-5240	562	12	.	.	PUNCT
ejpam-5240	563	1	we	we	PRON
ejpam-5240	563	2	will	will	AUX
ejpam-5240	563	3	utilize	utilize	VERB
ejpam-5240	563	4	this	this	PRON
ejpam-5240	563	5	to	to	PART
ejpam-5240	563	6	determine	determine	VERB
ejpam-5240	563	7	α(gs(n	α(gs(n	NOUN
ejpam-5240	563	8	,	,	PUNCT
ejpam-5240	563	9	k	k	NOUN
ejpam-5240	563	10	)	)	PUNCT
ejpam-5240	563	11	)	)	PUNCT
ejpam-5240	563	12	for	for	ADP
ejpam-5240	563	13	1	1	NUM
ejpam-5240	563	14	≤	≤	NOUN
ejpam-5240	563	15	k	k	X
ejpam-5240	563	16	≤	≤	NUM
ejpam-5240	563	17	⌊	⌊	VERB
ejpam-5240	563	18	n	n	DET
ejpam-5240	563	19	2	2	NUM
ejpam-5240	563	20	⌋	⌋	NOUN
ejpam-5240	563	21	.	.	PUNCT
ejpam-5240	564	1	illustration	illustration	NOUN
ejpam-5240	564	2	11	11	NUM
ejpam-5240	564	3	.	.	PUNCT
ejpam-5240	565	1	let	let	VERB
ejpam-5240	565	2	s6	s6	PROPN
ejpam-5240	565	3	=	=	SYM
ejpam-5240	565	4	{	{	PUNCT
ejpam-5240	565	5	x1	x1	PROPN
ejpam-5240	565	6	,	,	PUNCT
ejpam-5240	565	7	x2	x2	PROPN
ejpam-5240	565	8	,	,	PUNCT
ejpam-5240	565	9	x3	x3	PROPN
ejpam-5240	565	10	,	,	PUNCT
ejpam-5240	565	11	x4	x4	PROPN
ejpam-5240	565	12	,	,	PUNCT
ejpam-5240	565	13	x5	x5	PROPN
ejpam-5240	565	14	,	,	PUNCT
ejpam-5240	565	15	x6	x6	PROPN
ejpam-5240	565	16	}	}	PUNCT
ejpam-5240	565	17	and	and	CCONJ
ejpam-5240	565	18	let	let	VERB
ejpam-5240	565	19	k	k	NOUN
ejpam-5240	565	20	=	=	SYM
ejpam-5240	565	21	2	2	X
ejpam-5240	565	22	.	.	PUNCT
ejpam-5240	566	1	then	then	ADV
ejpam-5240	566	2	the	the	DET
ejpam-5240	566	3	vertex	vertex	NOUN
ejpam-5240	566	4	set	set	NOUN
ejpam-5240	566	5	of	of	ADP
ejpam-5240	566	6	a	a	DET
ejpam-5240	566	7	gs(6,2	gs(6,2	PROPN
ejpam-5240	566	8	)	)	PUNCT
ejpam-5240	566	9	is	be	AUX
ejpam-5240	566	10	given	give	VERB
ejpam-5240	566	11	by	by	ADP
ejpam-5240	566	12	v	v	PROPN
ejpam-5240	566	13	(	(	PUNCT
ejpam-5240	566	14	gs(6,2	gs(6,2	PROPN
ejpam-5240	566	15	)	)	PUNCT
ejpam-5240	566	16	)	)	PUNCT
ejpam-5240	567	1	=	=	PRON
ejpam-5240	567	2	{	{	PUNCT
ejpam-5240	567	3	{	{	PUNCT
ejpam-5240	567	4	x1	x1	PROPN
ejpam-5240	567	5	,	,	PUNCT
ejpam-5240	567	6	x2	x2	PROPN
ejpam-5240	567	7	}	}	PUNCT
ejpam-5240	567	8	,	,	PUNCT
ejpam-5240	567	9	{	{	PUNCT
ejpam-5240	567	10	x1	x1	ADJ
ejpam-5240	567	11	,	,	PUNCT
ejpam-5240	567	12	x3	x3	ADJ
ejpam-5240	567	13	}	}	PUNCT
ejpam-5240	567	14	,	,	PUNCT
ejpam-5240	567	15	{	{	PUNCT
ejpam-5240	567	16	x1	x1	PROPN
ejpam-5240	567	17	,	,	PUNCT
ejpam-5240	567	18	x4	x4	PROPN
ejpam-5240	567	19	}	}	PUNCT
ejpam-5240	567	20	,	,	PUNCT
ejpam-5240	567	21	{	{	PUNCT
ejpam-5240	567	22	x1	x1	PROPN
ejpam-5240	567	23	,	,	PUNCT
ejpam-5240	567	24	x5	x5	PROPN
ejpam-5240	567	25	}	}	PUNCT
ejpam-5240	567	26	,	,	PUNCT
ejpam-5240	567	27	{	{	PUNCT
ejpam-5240	567	28	x1	x1	PROPN
ejpam-5240	567	29	,	,	PUNCT
ejpam-5240	567	30	x6	x6	PROPN
ejpam-5240	567	31	}	}	PUNCT
ejpam-5240	567	32	,	,	PUNCT
ejpam-5240	567	33	{	{	PUNCT
ejpam-5240	567	34	x2	x2	ADJ
ejpam-5240	567	35	,	,	PUNCT
ejpam-5240	567	36	x3	x3	ADJ
ejpam-5240	567	37	}	}	PUNCT
ejpam-5240	567	38	,	,	PUNCT
ejpam-5240	567	39	{	{	PUNCT
ejpam-5240	567	40	x2	x2	PROPN
ejpam-5240	567	41	,	,	PUNCT
ejpam-5240	567	42	x4	x4	PROPN
ejpam-5240	567	43	}	}	PUNCT
ejpam-5240	567	44	,	,	PUNCT
ejpam-5240	567	45	{	{	PUNCT
ejpam-5240	567	46	x2	x2	PROPN
ejpam-5240	567	47	,	,	PUNCT
ejpam-5240	567	48	x5	x5	PROPN
ejpam-5240	567	49	}	}	PUNCT
ejpam-5240	567	50	,	,	PUNCT
ejpam-5240	567	51	{	{	PUNCT
ejpam-5240	567	52	x2	x2	PROPN
ejpam-5240	567	53	,	,	PUNCT
ejpam-5240	567	54	x6	x6	PROPN
ejpam-5240	567	55	}	}	PUNCT
ejpam-5240	567	56	,	,	PUNCT
ejpam-5240	567	57	{	{	PUNCT
ejpam-5240	567	58	x3	x3	ADJ
ejpam-5240	567	59	,	,	PUNCT
ejpam-5240	567	60	x4	x4	PROPN
ejpam-5240	567	61	}	}	PUNCT
ejpam-5240	567	62	,	,	PUNCT
ejpam-5240	567	63	{	{	PUNCT
ejpam-5240	567	64	x3	x3	ADJ
ejpam-5240	567	65	,	,	PUNCT
ejpam-5240	567	66	x5	x5	PROPN
ejpam-5240	567	67	}	}	PUNCT
ejpam-5240	567	68	,	,	PUNCT
ejpam-5240	567	69	{	{	PUNCT
ejpam-5240	567	70	x3	x3	ADJ
ejpam-5240	567	71	,	,	PUNCT
ejpam-5240	567	72	x6	x6	PROPN
ejpam-5240	567	73	}	}	PUNCT
ejpam-5240	567	74	,	,	PUNCT
ejpam-5240	567	75	{	{	PUNCT
ejpam-5240	567	76	x4	x4	PROPN
ejpam-5240	567	77	,	,	PUNCT
ejpam-5240	567	78	x5	x5	PROPN
ejpam-5240	567	79	}	}	PUNCT
ejpam-5240	567	80	,	,	PUNCT
ejpam-5240	567	81	{	{	PUNCT
ejpam-5240	567	82	x4	x4	PROPN
ejpam-5240	567	83	,	,	PUNCT
ejpam-5240	567	84	x6	x6	PROPN
ejpam-5240	567	85	}	}	PUNCT
ejpam-5240	567	86	,	,	PUNCT
ejpam-5240	567	87	{	{	PUNCT
ejpam-5240	567	88	x5	x5	PROPN
ejpam-5240	567	89	,	,	PUNCT
ejpam-5240	567	90	x6	x6	PROPN
ejpam-5240	567	91	}	}	PUNCT
ejpam-5240	567	92	}	}	PUNCT
ejpam-5240	567	93	a	a	DET
ejpam-5240	567	94	pictorial	pictorial	ADJ
ejpam-5240	567	95	illustration	illustration	NOUN
ejpam-5240	567	96	of	of	ADP
ejpam-5240	567	97	gs(6,2	gs(6,2	PROPN
ejpam-5240	567	98	)	)	PUNCT
ejpam-5240	567	99	is	be	AUX
ejpam-5240	567	100	shown	show	VERB
ejpam-5240	567	101	in	in	ADP
ejpam-5240	567	102	figure	figure	NOUN
ejpam-5240	567	103	12	12	NUM
ejpam-5240	567	104	.	.	PUNCT
ejpam-5240	568	1	m.e	m.e	PROPN
ejpam-5240	568	2	.	.	PROPN
ejpam-5240	568	3	pelagio	pelagio	PROPN
ejpam-5240	568	4	,	,	PUNCT
ejpam-5240	568	5	n.	n.	NOUN
ejpam-5240	568	6	mame	mame	PROPN
ejpam-5240	568	7	,	,	PUNCT
ejpam-5240	568	8	k.	k.	PROPN
ejpam-5240	568	9	mendoza	mendoza	PROPN
ejpam-5240	568	10	/	/	SYM
ejpam-5240	568	11	eur	eur	PROPN
ejpam-5240	568	12	.	.	PUNCT
ejpam-5240	569	1	j.	j.	PROPN
ejpam-5240	569	2	pure	pure	PROPN
ejpam-5240	569	3	appl	appl	PROPN
ejpam-5240	569	4	.	.	PROPN
ejpam-5240	569	5	math	math	PROPN
ejpam-5240	569	6	,	,	PUNCT
ejpam-5240	569	7	17	17	NUM
ejpam-5240	569	8	(	(	PUNCT
ejpam-5240	569	9	3	3	NUM
ejpam-5240	569	10	)	)	PUNCT
ejpam-5240	569	11	(	(	PUNCT
ejpam-5240	569	12	2024	2024	NUM
ejpam-5240	569	13	)	)	PUNCT
ejpam-5240	569	14	,	,	PUNCT
ejpam-5240	569	15	1779	1779	NUM
ejpam-5240	569	16	-	-	SYM
ejpam-5240	569	17	1803	1803	NUM
ejpam-5240	569	18	1797	1797	NUM
ejpam-5240	569	19	{	{	PUNCT
ejpam-5240	569	20	x1	x1	PROPN
ejpam-5240	569	21	,	,	PUNCT
ejpam-5240	569	22	x2	x2	PROPN
ejpam-5240	569	23	}	}	PUNCT
ejpam-5240	569	24	{	{	PUNCT
ejpam-5240	569	25	x1	x1	PROPN
ejpam-5240	569	26	,	,	PUNCT
ejpam-5240	569	27	x3	x3	ADJ
ejpam-5240	569	28	}	}	PUNCT
ejpam-5240	569	29	{	{	PUNCT
ejpam-5240	569	30	x1	x1	PROPN
ejpam-5240	569	31	,	,	PUNCT
ejpam-5240	569	32	x4	x4	PROPN
ejpam-5240	569	33	}	}	PUNCT
ejpam-5240	569	34	{	{	PUNCT
ejpam-5240	569	35	x1	x1	PROPN
ejpam-5240	569	36	,	,	PUNCT
ejpam-5240	569	37	x5	x5	PROPN
ejpam-5240	569	38	}	}	PUNCT
ejpam-5240	569	39	{	{	PUNCT
ejpam-5240	569	40	x1	x1	PROPN
ejpam-5240	569	41	,	,	PUNCT
ejpam-5240	569	42	x6	x6	PROPN
ejpam-5240	569	43	}	}	PUNCT
ejpam-5240	569	44	{	{	PUNCT
ejpam-5240	569	45	x2	x2	PROPN
ejpam-5240	569	46	,	,	PUNCT
ejpam-5240	569	47	x3	x3	ADJ
ejpam-5240	569	48	}	}	PUNCT
ejpam-5240	569	49	{	{	PUNCT
ejpam-5240	569	50	x2	x2	PROPN
ejpam-5240	569	51	,	,	PUNCT
ejpam-5240	569	52	x4	x4	PROPN
ejpam-5240	569	53	}	}	PUNCT
ejpam-5240	569	54	{	{	PUNCT
ejpam-5240	569	55	x2	x2	PROPN
ejpam-5240	569	56	,	,	PUNCT
ejpam-5240	569	57	x5}{x2	x5}{x2	NOUN
ejpam-5240	569	58	,	,	PUNCT
ejpam-5240	569	59	x6	x6	PROPN
ejpam-5240	569	60	}	}	PUNCT
ejpam-5240	569	61	{	{	PUNCT
ejpam-5240	569	62	x3	x3	ADJ
ejpam-5240	569	63	,	,	PUNCT
ejpam-5240	569	64	x4	x4	PROPN
ejpam-5240	569	65	}	}	PUNCT
ejpam-5240	569	66	{	{	PUNCT
ejpam-5240	569	67	x3	x3	ADJ
ejpam-5240	569	68	,	,	PUNCT
ejpam-5240	569	69	x5	x5	PROPN
ejpam-5240	569	70	}	}	PUNCT
ejpam-5240	569	71	{	{	PUNCT
ejpam-5240	569	72	x3	x3	ADJ
ejpam-5240	569	73	,	,	PUNCT
ejpam-5240	569	74	x6	x6	PROPN
ejpam-5240	569	75	}	}	PUNCT
ejpam-5240	569	76	{	{	PUNCT
ejpam-5240	569	77	x4	x4	PROPN
ejpam-5240	569	78	,	,	PUNCT
ejpam-5240	569	79	x5	x5	PROPN
ejpam-5240	569	80	}	}	PUNCT
ejpam-5240	569	81	{	{	PUNCT
ejpam-5240	569	82	x4	x4	PROPN
ejpam-5240	569	83	,	,	PUNCT
ejpam-5240	569	84	x6	x6	PROPN
ejpam-5240	569	85	}	}	PUNCT
ejpam-5240	569	86	{	{	PUNCT
ejpam-5240	569	87	x5	x5	PROPN
ejpam-5240	569	88	,	,	PUNCT
ejpam-5240	569	89	x6	x6	PROPN
ejpam-5240	569	90	}	}	PUNCT
ejpam-5240	569	91	gs(6,2	gs(6,2	PROPN
ejpam-5240	569	92	)	)	PUNCT
ejpam-5240	569	93	figure	figure	NOUN
ejpam-5240	569	94	12	12	NUM
ejpam-5240	569	95	:	:	PUNCT
ejpam-5240	569	96	a	a	DET
ejpam-5240	569	97	pictorial	pictorial	ADJ
ejpam-5240	569	98	illustration	illustration	NOUN
ejpam-5240	569	99	of	of	ADP
ejpam-5240	569	100	gs(6,2	gs(6,2	PROPN
ejpam-5240	569	101	)	)	PUNCT
ejpam-5240	569	102	.	.	PUNCT
ejpam-5240	570	1	now	now	ADV
ejpam-5240	570	2	,	,	PUNCT
ejpam-5240	570	3	we	we	PRON
ejpam-5240	570	4	let	let	VERB
ejpam-5240	570	5	ti	ti	PROPN
ejpam-5240	570	6	⊆	⊆	NUM
ejpam-5240	570	7	v	v	NOUN
ejpam-5240	570	8	(	(	PUNCT
ejpam-5240	570	9	gs(6,2	gs(6,2	PROPN
ejpam-5240	570	10	)	)	PUNCT
ejpam-5240	570	11	)	)	PUNCT
ejpam-5240	570	12	for	for	ADP
ejpam-5240	570	13	1	1	NUM
ejpam-5240	570	14	≤	≤	NUM
ejpam-5240	570	15	i	i	PRON
ejpam-5240	570	16	≤	≤	NOUN
ejpam-5240	570	17	3	3	NUM
ejpam-5240	570	18	.	.	PUNCT
ejpam-5240	570	19	notice	notice	VERB
ejpam-5240	570	20	that	that	SCONJ
ejpam-5240	570	21	t1	t1	NOUN
ejpam-5240	570	22	=	=	PUNCT
ejpam-5240	570	23	{	{	PUNCT
ejpam-5240	570	24	{	{	PUNCT
ejpam-5240	570	25	xi	xi	PROPN
ejpam-5240	570	26	,	,	PUNCT
ejpam-5240	570	27	xj	xj	PROPN
ejpam-5240	570	28	}	}	PUNCT
ejpam-5240	570	29	}	}	PUNCT
ejpam-5240	570	30	where	where	SCONJ
ejpam-5240	570	31	1	1	NUM
ejpam-5240	570	32	≤	≤	PUNCT
ejpam-5240	570	33	i	i	PRON
ejpam-5240	570	34	,	,	PUNCT
ejpam-5240	570	35	j	j	PROPN
ejpam-5240	570	36	≤	≤	ADV
ejpam-5240	570	37	6	6	NUM
ejpam-5240	570	38	and	and	CCONJ
ejpam-5240	570	39	i	i	PRON
ejpam-5240	570	40	̸=	̸=	PROPN
ejpam-5240	570	41	j	j	PROPN
ejpam-5240	570	42	is	be	AUX
ejpam-5240	570	43	an	an	DET
ejpam-5240	570	44	independent	independent	ADJ
ejpam-5240	570	45	set	set	NOUN
ejpam-5240	570	46	since	since	SCONJ
ejpam-5240	570	47	gs(6,2	gs(6,2	PROPN
ejpam-5240	570	48	)	)	PUNCT
ejpam-5240	570	49	is	be	AUX
ejpam-5240	570	50	a	a	DET
ejpam-5240	570	51	simple	simple	ADJ
ejpam-5240	570	52	graph	graph	NOUN
ejpam-5240	570	53	.	.	PUNCT
ejpam-5240	571	1	moreover	moreover	ADV
ejpam-5240	571	2	,	,	PUNCT
ejpam-5240	571	3	the	the	DET
ejpam-5240	571	4	set	set	NOUN
ejpam-5240	571	5	t2	t2	NOUN
ejpam-5240	571	6	=	=	SYM
ejpam-5240	571	7	{	{	PUNCT
ejpam-5240	571	8	{	{	PUNCT
ejpam-5240	571	9	x1	x1	PROPN
ejpam-5240	571	10	,	,	PUNCT
ejpam-5240	571	11	x2	x2	PROPN
ejpam-5240	571	12	}	}	PUNCT
ejpam-5240	571	13	,	,	PUNCT
ejpam-5240	571	14	{	{	PUNCT
ejpam-5240	571	15	x3	x3	ADJ
ejpam-5240	571	16	,	,	PUNCT
ejpam-5240	571	17	x4	x4	PROPN
ejpam-5240	571	18	}	}	PUNCT
ejpam-5240	571	19	}	}	PUNCT
ejpam-5240	571	20	is	be	AUX
ejpam-5240	571	21	an	an	DET
ejpam-5240	571	22	independent	independent	ADJ
ejpam-5240	571	23	set	set	NOUN
ejpam-5240	571	24	also	also	ADV
ejpam-5240	571	25	since	since	SCONJ
ejpam-5240	571	26	{	{	PUNCT
ejpam-5240	571	27	x1	x1	ADJ
ejpam-5240	571	28	,	,	PUNCT
ejpam-5240	571	29	x2}∩{x3	x2}∩{x3	X
ejpam-5240	571	30	,	,	PUNCT
ejpam-5240	571	31	x4	x4	ADJ
ejpam-5240	571	32	}	}	PUNCT
ejpam-5240	571	33	=	=	PUNCT
ejpam-5240	571	34	∅	∅	NOUN
ejpam-5240	571	35	it	it	PRON
ejpam-5240	571	36	follows	follow	VERB
ejpam-5240	571	37	that	that	SCONJ
ejpam-5240	571	38	[	[	X
ejpam-5240	571	39	{	{	PUNCT
ejpam-5240	571	40	x1	x1	PROPN
ejpam-5240	571	41	,	,	PUNCT
ejpam-5240	571	42	x2	x2	PROPN
ejpam-5240	571	43	}	}	PUNCT
ejpam-5240	571	44	,	,	PUNCT
ejpam-5240	571	45	{	{	PUNCT
ejpam-5240	571	46	x3	x3	ADJ
ejpam-5240	571	47	,	,	PUNCT
ejpam-5240	571	48	x4	x4	PROPN
ejpam-5240	571	49	}	}	PUNCT
ejpam-5240	571	50	]	]	PUNCT
ejpam-5240	571	51	/∈	/∈	PUNCT
ejpam-5240	571	52	e(gs(6,2	e(gs(6,2	NOUN
ejpam-5240	571	53	)	)	PUNCT
ejpam-5240	571	54	)	)	PUNCT
ejpam-5240	571	55	.	.	PUNCT
ejpam-5240	572	1	this	this	PRON
ejpam-5240	572	2	implies	imply	VERB
ejpam-5240	572	3	that	that	SCONJ
ejpam-5240	572	4	there	there	PRON
ejpam-5240	572	5	exists	exist	VERB
ejpam-5240	572	6	an	an	DET
ejpam-5240	572	7	independent	independent	ADJ
ejpam-5240	572	8	set	set	NOUN
ejpam-5240	572	9	with	with	ADP
ejpam-5240	572	10	the	the	DET
ejpam-5240	572	11	cardinality	cardinality	NOUN
ejpam-5240	572	12	equal	equal	ADJ
ejpam-5240	572	13	to	to	ADP
ejpam-5240	572	14	2	2	NUM
ejpam-5240	572	15	.	.	PUNCT
ejpam-5240	573	1	on	on	ADP
ejpam-5240	573	2	the	the	DET
ejpam-5240	573	3	other	other	ADJ
ejpam-5240	573	4	hand	hand	NOUN
ejpam-5240	573	5	,	,	PUNCT
ejpam-5240	573	6	t3	t3	PROPN
ejpam-5240	573	7	=	=	PUNCT
ejpam-5240	573	8	{	{	PUNCT
ejpam-5240	573	9	{	{	PUNCT
ejpam-5240	573	10	x1	x1	PROPN
ejpam-5240	573	11	,	,	PUNCT
ejpam-5240	573	12	x2	x2	PROPN
ejpam-5240	573	13	}	}	PUNCT
ejpam-5240	573	14	,	,	PUNCT
ejpam-5240	573	15	{	{	PUNCT
ejpam-5240	573	16	x3	x3	ADJ
ejpam-5240	573	17	,	,	PUNCT
ejpam-5240	573	18	x4	x4	PROPN
ejpam-5240	573	19	}	}	PUNCT
ejpam-5240	573	20	,	,	PUNCT
ejpam-5240	573	21	{	{	PUNCT
ejpam-5240	573	22	x5	x5	PROPN
ejpam-5240	573	23	,	,	PUNCT
ejpam-5240	573	24	x6	x6	PROPN
ejpam-5240	573	25	}	}	PUNCT
ejpam-5240	573	26	}	}	PUNCT
ejpam-5240	573	27	is	be	AUX
ejpam-5240	573	28	also	also	ADV
ejpam-5240	573	29	an	an	DET
ejpam-5240	573	30	independent	independent	ADJ
ejpam-5240	573	31	set	set	NOUN
ejpam-5240	573	32	since	since	SCONJ
ejpam-5240	573	33	{	{	PUNCT
ejpam-5240	573	34	x1	x1	PROPN
ejpam-5240	573	35	,	,	PUNCT
ejpam-5240	573	36	x2}∩{x3	x2}∩{x3	X
ejpam-5240	573	37	,	,	PUNCT
ejpam-5240	573	38	x4	x4	ADJ
ejpam-5240	573	39	}	}	PUNCT
ejpam-5240	573	40	=	=	SYM
ejpam-5240	573	41	∅	∅	NOUN
ejpam-5240	573	42	,	,	PUNCT
ejpam-5240	573	43	{	{	PUNCT
ejpam-5240	573	44	x1	x1	PROPN
ejpam-5240	573	45	,	,	PUNCT
ejpam-5240	573	46	x2}∩{x5	x2}∩{x5	PROPN
ejpam-5240	573	47	,	,	PUNCT
ejpam-5240	573	48	x6	x6	PROPN
ejpam-5240	573	49	}	}	PUNCT
ejpam-5240	573	50	=	=	SYM
ejpam-5240	573	51	∅	∅	NOUN
ejpam-5240	573	52	,	,	PUNCT
ejpam-5240	573	53	and	and	CCONJ
ejpam-5240	573	54	{	{	PUNCT
ejpam-5240	573	55	x3	x3	ADJ
ejpam-5240	573	56	,	,	PUNCT
ejpam-5240	573	57	x4}∩	x4}∩	PROPN
ejpam-5240	573	58	{	{	PUNCT
ejpam-5240	573	59	x5	x5	PROPN
ejpam-5240	573	60	,	,	PUNCT
ejpam-5240	573	61	x6	x6	PROPN
ejpam-5240	573	62	}	}	PUNCT
ejpam-5240	573	63	=	=	PUNCT
ejpam-5240	573	64	∅	∅	NOUN
ejpam-5240	573	65	it	it	PRON
ejpam-5240	573	66	follows	follow	VERB
ejpam-5240	573	67	that	that	SCONJ
ejpam-5240	573	68	[	[	X
ejpam-5240	573	69	{	{	PUNCT
ejpam-5240	573	70	x1	x1	PROPN
ejpam-5240	573	71	,	,	PUNCT
ejpam-5240	573	72	x2	x2	PROPN
ejpam-5240	573	73	}	}	PUNCT
ejpam-5240	573	74	,	,	PUNCT
ejpam-5240	573	75	{	{	PUNCT
ejpam-5240	573	76	x3	x3	ADJ
ejpam-5240	573	77	,	,	PUNCT
ejpam-5240	573	78	x4	x4	PROPN
ejpam-5240	573	79	}	}	PUNCT
ejpam-5240	573	80	]	]	PUNCT
ejpam-5240	573	81	,	,	PUNCT
ejpam-5240	573	82	[	[	X
ejpam-5240	573	83	{	{	PUNCT
ejpam-5240	573	84	x1	x1	PROPN
ejpam-5240	573	85	,	,	PUNCT
ejpam-5240	573	86	x2	x2	PROPN
ejpam-5240	573	87	}	}	PUNCT
ejpam-5240	573	88	,	,	PUNCT
ejpam-5240	573	89	{	{	PUNCT
ejpam-5240	573	90	x5	x5	PROPN
ejpam-5240	573	91	,	,	PUNCT
ejpam-5240	573	92	x6	x6	PROPN
ejpam-5240	573	93	}	}	PUNCT
ejpam-5240	573	94	]	]	PUNCT
ejpam-5240	573	95	,	,	PUNCT
ejpam-5240	573	96	[	[	X
ejpam-5240	573	97	{	{	PUNCT
ejpam-5240	573	98	x3	x3	ADJ
ejpam-5240	573	99	,	,	PUNCT
ejpam-5240	573	100	x4	x4	PROPN
ejpam-5240	573	101	}	}	PUNCT
ejpam-5240	573	102	,	,	PUNCT
ejpam-5240	573	103	{	{	PUNCT
ejpam-5240	573	104	x5	x5	PROPN
ejpam-5240	573	105	,	,	PUNCT
ejpam-5240	573	106	x6	x6	PROPN
ejpam-5240	573	107	}	}	PUNCT
ejpam-5240	573	108	]	]	PUNCT
ejpam-5240	573	109	are	be	AUX
ejpam-5240	573	110	not	not	PART
ejpam-5240	573	111	edges	edge	NOUN
ejpam-5240	573	112	in	in	ADP
ejpam-5240	573	113	gs(6,2	gs(6,2	PROPN
ejpam-5240	573	114	)	)	PUNCT
ejpam-5240	573	115	.	.	PUNCT
ejpam-5240	574	1	observe	observe	VERB
ejpam-5240	574	2	that	that	SCONJ
ejpam-5240	574	3	⌊	⌊	PROPN
ejpam-5240	574	4	6	6	NUM
ejpam-5240	574	5	2	2	NUM
ejpam-5240	574	6	⌋	⌋	NOUN
ejpam-5240	574	7	=	=	SYM
ejpam-5240	574	8	3	3	NUM
ejpam-5240	574	9	and	and	CCONJ
ejpam-5240	574	10	|t3|	|t3|	NOUN
ejpam-5240	574	11	=	=	NOUN
ejpam-5240	574	12	3	3	X
ejpam-5240	574	13	.	.	PUNCT
ejpam-5240	575	1	hence	hence	ADV
ejpam-5240	575	2	,	,	PUNCT
ejpam-5240	575	3	there	there	PRON
ejpam-5240	575	4	exists	exist	VERB
ejpam-5240	575	5	an	an	DET
ejpam-5240	575	6	independent	independent	ADJ
ejpam-5240	575	7	set	set	NOUN
ejpam-5240	575	8	in	in	ADP
ejpam-5240	575	9	gs(6,2	gs(6,2	PROPN
ejpam-5240	575	10	)	)	PUNCT
ejpam-5240	575	11	with	with	ADP
ejpam-5240	575	12	the	the	DET
ejpam-5240	575	13	cardinality	cardinality	NOUN
ejpam-5240	575	14	equal	equal	ADJ
ejpam-5240	575	15	to	to	ADP
ejpam-5240	575	16	3	3	NUM
ejpam-5240	575	17	.	.	PUNCT
ejpam-5240	576	1	note	note	VERB
ejpam-5240	576	2	that	that	SCONJ
ejpam-5240	576	3	there	there	PRON
ejpam-5240	576	4	exists	exist	VERB
ejpam-5240	576	5	an	an	DET
ejpam-5240	576	6	independent	independent	ADJ
ejpam-5240	576	7	set	set	NOUN
ejpam-5240	576	8	in	in	ADP
ejpam-5240	576	9	gs(n	gs(n	NOUN
ejpam-5240	576	10	,	,	PUNCT
ejpam-5240	576	11	k	k	NOUN
ejpam-5240	576	12	)	)	PUNCT
ejpam-5240	576	13	.	.	PUNCT
ejpam-5240	577	1	with	with	ADP
ejpam-5240	577	2	this	this	PRON
ejpam-5240	577	3	,	,	PUNCT
ejpam-5240	577	4	we	we	PRON
ejpam-5240	577	5	can	can	AUX
ejpam-5240	577	6	now	now	ADV
ejpam-5240	577	7	compute	compute	VERB
ejpam-5240	577	8	for	for	ADP
ejpam-5240	577	9	α(gs(n	α(gs(n	NOUN
ejpam-5240	577	10	,	,	PUNCT
ejpam-5240	577	11	k	k	NOUN
ejpam-5240	577	12	)	)	PUNCT
ejpam-5240	577	13	)	)	PUNCT
ejpam-5240	577	14	.	.	PUNCT
ejpam-5240	578	1	theorem	theorem	VERB
ejpam-5240	578	2	10	10	NUM
ejpam-5240	578	3	determines	determine	VERB
ejpam-5240	578	4	the	the	DET
ejpam-5240	578	5	independence	independence	NOUN
ejpam-5240	578	6	number	number	NOUN
ejpam-5240	578	7	for	for	ADP
ejpam-5240	578	8	a	a	DET
ejpam-5240	578	9	gs(n	gs(n	NOUN
ejpam-5240	578	10	,	,	PUNCT
ejpam-5240	578	11	k	k	NOUN
ejpam-5240	578	12	)	)	PUNCT
ejpam-5240	578	13	if	if	SCONJ
ejpam-5240	578	14	1	1	NUM
ejpam-5240	578	15	≤	≤	NUM
ejpam-5240	578	16	k	k	X
ejpam-5240	578	17	≤	≤	NUM
ejpam-5240	578	18	⌊	⌊	VERB
ejpam-5240	578	19	n	n	DET
ejpam-5240	578	20	2	2	NUM
ejpam-5240	578	21	⌋	⌋	NOUN
ejpam-5240	578	22	.	.	PUNCT
ejpam-5240	579	1	theorem	theorem	ADJ
ejpam-5240	579	2	10	10	NUM
ejpam-5240	579	3	.	.	PUNCT
ejpam-5240	580	1	let	let	VERB
ejpam-5240	580	2	sn	sn	PROPN
ejpam-5240	580	3	=	=	PUNCT
ejpam-5240	580	4	{	{	PUNCT
ejpam-5240	580	5	x1	x1	PROPN
ejpam-5240	580	6	,	,	PUNCT
ejpam-5240	580	7	x2	x2	PROPN
ejpam-5240	580	8	,	,	PUNCT
ejpam-5240	580	9	...	...	PUNCT
ejpam-5240	580	10	,	,	PUNCT
ejpam-5240	580	11	xn	xn	PRON
ejpam-5240	580	12	}	}	PUNCT
ejpam-5240	580	13	be	be	VERB
ejpam-5240	580	14	an	an	DET
ejpam-5240	580	15	n	n	NOUN
ejpam-5240	580	16	-	-	PUNCT
ejpam-5240	580	17	element	element	NOUN
ejpam-5240	580	18	set	set	NOUN
ejpam-5240	580	19	and	and	CCONJ
ejpam-5240	580	20	let	let	VERB
ejpam-5240	580	21	gs(n	gs(n	NOUN
ejpam-5240	580	22	,	,	PUNCT
ejpam-5240	580	23	k	k	NOUN
ejpam-5240	580	24	)	)	PUNCT
ejpam-5240	580	25	be	be	VERB
ejpam-5240	580	26	a	a	DET
ejpam-5240	580	27	k	k	ADV
ejpam-5240	580	28	-	-	ADJ
ejpam-5240	580	29	restricted	restricted	ADJ
ejpam-5240	580	30	intersection	intersection	NOUN
ejpam-5240	580	31	graph	graph	NOUN
ejpam-5240	580	32	.	.	PUNCT
ejpam-5240	581	1	if	if	SCONJ
ejpam-5240	581	2	1	1	NUM
ejpam-5240	581	3	≤	≤	NUM
ejpam-5240	581	4	k	k	X
ejpam-5240	581	5	≤	≤	NUM
ejpam-5240	581	6	⌊	⌊	VERB
ejpam-5240	581	7	n	n	DET
ejpam-5240	581	8	2	2	NUM
ejpam-5240	581	9	⌋	⌋	NOUN
ejpam-5240	581	10	,	,	PUNCT
ejpam-5240	581	11	then	then	ADV
ejpam-5240	581	12	α(gs(n	α(gs(n	PROPN
ejpam-5240	581	13	,	,	PUNCT
ejpam-5240	581	14	k	k	NOUN
ejpam-5240	581	15	)	)	PUNCT
ejpam-5240	581	16	)	)	PUNCT
ejpam-5240	582	1	=	=	PUNCT
ejpam-5240	582	2	⌊	⌊	VERB
ejpam-5240	582	3	n	n	PRON
ejpam-5240	582	4	k	k	NOUN
ejpam-5240	582	5	⌋	⌋	NOUN
ejpam-5240	582	6	.	.	PUNCT
ejpam-5240	583	1	proof	proof	NOUN
ejpam-5240	583	2	.	.	PUNCT
ejpam-5240	584	1	let	let	VERB
ejpam-5240	584	2	t	t	NOUN
ejpam-5240	584	3	=	=	PRON
ejpam-5240	584	4	{	{	PUNCT
ejpam-5240	584	5	{	{	PUNCT
ejpam-5240	584	6	x1	x1	PROPN
ejpam-5240	584	7	,	,	PUNCT
ejpam-5240	584	8	x2	x2	PROPN
ejpam-5240	584	9	,	,	PUNCT
ejpam-5240	584	10	...	...	PUNCT
ejpam-5240	584	11	,	,	PUNCT
ejpam-5240	584	12	xk	xk	PROPN
ejpam-5240	584	13	}	}	PUNCT
ejpam-5240	584	14	,	,	PUNCT
ejpam-5240	584	15	{	{	PUNCT
ejpam-5240	584	16	xk+1	xk+1	PROPN
ejpam-5240	584	17	,	,	PUNCT
ejpam-5240	584	18	...	...	PUNCT
ejpam-5240	584	19	,	,	PUNCT
ejpam-5240	584	20	x2k	x2k	NOUN
ejpam-5240	584	21	}	}	PUNCT
ejpam-5240	584	22	,	,	PUNCT
ejpam-5240	584	23	...	...	PUNCT
ejpam-5240	584	24	,	,	PUNCT
ejpam-5240	584	25	{	{	PUNCT
ejpam-5240	584	26	x(⌊n	x(⌊n	PROPN
ejpam-5240	584	27	k	k	PROPN
ejpam-5240	584	28	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	584	29	,	,	PUNCT
ejpam-5240	584	30	...	...	PUNCT
ejpam-5240	584	31	,	,	PUNCT
ejpam-5240	584	32	x(⌊n	x(⌊n	PROPN
ejpam-5240	585	1	k	k	PROPN
ejpam-5240	585	2	⌋)k	⌋)k	PROPN
ejpam-5240	585	3	}	}	PUNCT
ejpam-5240	585	4	}	}	PUNCT
ejpam-5240	585	5	.	.	PUNCT
ejpam-5240	586	1	by	by	ADP
ejpam-5240	586	2	lemma	lemma	PROPN
ejpam-5240	586	3	3	3	NUM
ejpam-5240	586	4	,	,	PUNCT
ejpam-5240	586	5	t	t	PROPN
ejpam-5240	586	6	is	be	AUX
ejpam-5240	586	7	an	an	DET
ejpam-5240	586	8	independent	independent	ADJ
ejpam-5240	586	9	set	set	NOUN
ejpam-5240	586	10	in	in	ADP
ejpam-5240	586	11	gs(n	gs(n	NOUN
ejpam-5240	586	12	,	,	PUNCT
ejpam-5240	586	13	k	k	NOUN
ejpam-5240	586	14	)	)	PUNCT
ejpam-5240	586	15	with	with	ADP
ejpam-5240	586	16	|t	|t	PROPN
ejpam-5240	587	1	|	|	ADV
ejpam-5240	587	2	=	=	SYM
ejpam-5240	587	3	⌊	⌊	PROPN
ejpam-5240	587	4	n	n	PRON
ejpam-5240	587	5	k	k	NOUN
ejpam-5240	587	6	⌋	⌋	NOUN
ejpam-5240	587	7	.	.	PUNCT
ejpam-5240	588	1	since	since	SCONJ
ejpam-5240	588	2	there	there	PRON
ejpam-5240	588	3	exists	exist	VERB
ejpam-5240	588	4	an	an	DET
ejpam-5240	588	5	m.e	m.e	PROPN
ejpam-5240	588	6	.	.	PROPN
ejpam-5240	588	7	pelagio	pelagio	PROPN
ejpam-5240	588	8	,	,	PUNCT
ejpam-5240	588	9	n.	n.	NOUN
ejpam-5240	588	10	mame	mame	PROPN
ejpam-5240	588	11	,	,	PUNCT
ejpam-5240	588	12	k.	k.	PROPN
ejpam-5240	588	13	mendoza	mendoza	PROPN
ejpam-5240	588	14	/	/	SYM
ejpam-5240	588	15	eur	eur	PROPN
ejpam-5240	588	16	.	.	PUNCT
ejpam-5240	589	1	j.	j.	PROPN
ejpam-5240	589	2	pure	pure	PROPN
ejpam-5240	589	3	appl	appl	PROPN
ejpam-5240	589	4	.	.	PROPN
ejpam-5240	589	5	math	math	PROPN
ejpam-5240	589	6	,	,	PUNCT
ejpam-5240	589	7	17	17	NUM
ejpam-5240	589	8	(	(	PUNCT
ejpam-5240	589	9	3	3	NUM
ejpam-5240	589	10	)	)	PUNCT
ejpam-5240	589	11	(	(	PUNCT
ejpam-5240	589	12	2024	2024	NUM
ejpam-5240	589	13	)	)	PUNCT
ejpam-5240	589	14	,	,	PUNCT
ejpam-5240	589	15	1779	1779	NUM
ejpam-5240	589	16	-	-	SYM
ejpam-5240	589	17	1803	1803	NUM
ejpam-5240	589	18	1798	1798	NUM
ejpam-5240	589	19	independent	independent	ADJ
ejpam-5240	589	20	set	set	NOUN
ejpam-5240	589	21	in	in	ADP
ejpam-5240	589	22	gs(n	gs(n	NOUN
ejpam-5240	589	23	,	,	PUNCT
ejpam-5240	589	24	k	k	NOUN
ejpam-5240	589	25	)	)	PUNCT
ejpam-5240	589	26	with	with	ADP
ejpam-5240	589	27	a	a	DET
ejpam-5240	589	28	cardinality	cardinality	NOUN
ejpam-5240	589	29	equal	equal	ADJ
ejpam-5240	589	30	to	to	ADP
ejpam-5240	589	31	⌊	⌊	PROPN
ejpam-5240	589	32	n	n	CCONJ
ejpam-5240	589	33	k	k	NOUN
ejpam-5240	589	34	⌋	⌋	NOUN
ejpam-5240	590	1	,	,	PUNCT
ejpam-5240	590	2	it	it	PRON
ejpam-5240	590	3	follows	follow	VERB
ejpam-5240	590	4	that	that	SCONJ
ejpam-5240	590	5	α(gs(n	α(gs(n	PROPN
ejpam-5240	590	6	,	,	PUNCT
ejpam-5240	590	7	k	k	NOUN
ejpam-5240	590	8	)	)	PUNCT
ejpam-5240	590	9	)	)	PUNCT
ejpam-5240	590	10	≥	≥	PRON
ejpam-5240	590	11	⌊	⌊	PROPN
ejpam-5240	590	12	n	n	PRON
ejpam-5240	590	13	k	k	NOUN
ejpam-5240	590	14	⌋	⌋	NOUN
ejpam-5240	590	15	.	.	PUNCT
ejpam-5240	591	1	we	we	PRON
ejpam-5240	591	2	claim	claim	VERB
ejpam-5240	591	3	that	that	SCONJ
ejpam-5240	591	4	α(gs(n	α(gs(n	PROPN
ejpam-5240	591	5	,	,	PUNCT
ejpam-5240	591	6	k	k	NOUN
ejpam-5240	591	7	)	)	PUNCT
ejpam-5240	591	8	)	)	PUNCT
ejpam-5240	592	1	=	=	PUNCT
ejpam-5240	592	2	⌊	⌊	VERB
ejpam-5240	592	3	n	n	PRON
ejpam-5240	592	4	k	k	PROPN
ejpam-5240	592	5	⌋	⌋	NOUN
ejpam-5240	592	6	.	.	PUNCT
ejpam-5240	593	1	now	now	ADV
ejpam-5240	593	2	,	,	PUNCT
ejpam-5240	593	3	suppose	suppose	VERB
ejpam-5240	593	4	α(gs(n	α(gs(n	ADJ
ejpam-5240	593	5	,	,	PUNCT
ejpam-5240	593	6	k	k	NOUN
ejpam-5240	593	7	)	)	PUNCT
ejpam-5240	593	8	)	)	PUNCT
ejpam-5240	593	9	>	>	PUNCT
ejpam-5240	594	1	⌊	⌊	PROPN
ejpam-5240	594	2	n	n	X
ejpam-5240	594	3	k	k	PROPN
ejpam-5240	594	4	⌋	⌋	NOUN
ejpam-5240	594	5	.	.	PUNCT
ejpam-5240	595	1	this	this	PRON
ejpam-5240	595	2	implies	imply	VERB
ejpam-5240	595	3	that	that	SCONJ
ejpam-5240	595	4	there	there	PRON
ejpam-5240	595	5	exists	exist	VERB
ejpam-5240	595	6	w	w	PROPN
ejpam-5240	595	7	⊆	⊆	NUM
ejpam-5240	595	8	v	v	NOUN
ejpam-5240	595	9	(	(	PUNCT
ejpam-5240	595	10	gs(n	gs(n	NOUN
ejpam-5240	595	11	,	,	PUNCT
ejpam-5240	595	12	k	k	NOUN
ejpam-5240	595	13	)	)	PUNCT
ejpam-5240	595	14	)	)	PUNCT
ejpam-5240	595	15	with	with	ADP
ejpam-5240	595	16	|w	|w	ADJ
ejpam-5240	596	1	|	|	ADV
ejpam-5240	596	2	>	>	X
ejpam-5240	596	3	⌊	⌊	PROPN
ejpam-5240	596	4	n	n	CCONJ
ejpam-5240	596	5	k	k	NOUN
ejpam-5240	596	6	⌋	⌋	NOUN
ejpam-5240	596	7	that	that	PRON
ejpam-5240	596	8	is	be	AUX
ejpam-5240	596	9	an	an	DET
ejpam-5240	596	10	independent	independent	ADJ
ejpam-5240	596	11	set	set	NOUN
ejpam-5240	596	12	in	in	ADP
ejpam-5240	596	13	gs(n	gs(n	NOUN
ejpam-5240	596	14	,	,	PUNCT
ejpam-5240	596	15	k	k	NOUN
ejpam-5240	596	16	)	)	PUNCT
ejpam-5240	596	17	.	.	PUNCT
ejpam-5240	597	1	without	without	ADP
ejpam-5240	597	2	loss	loss	NOUN
ejpam-5240	597	3	of	of	ADP
ejpam-5240	597	4	generality	generality	NOUN
ejpam-5240	597	5	,	,	PUNCT
ejpam-5240	597	6	we	we	PRON
ejpam-5240	597	7	say	say	VERB
ejpam-5240	597	8	that	that	SCONJ
ejpam-5240	597	9	n	n	PRON
ejpam-5240	597	10	is	be	AUX
ejpam-5240	597	11	not	not	PART
ejpam-5240	597	12	divisible	divisible	ADJ
ejpam-5240	597	13	by	by	ADP
ejpam-5240	597	14	k.	k.	PROPN
ejpam-5240	597	15	hence	hence	ADV
ejpam-5240	597	16	,	,	PUNCT
ejpam-5240	597	17	by	by	ADP
ejpam-5240	597	18	the	the	DET
ejpam-5240	597	19	division	division	NOUN
ejpam-5240	597	20	algorithm	algorithm	NOUN
ejpam-5240	597	21	,	,	PUNCT
ejpam-5240	597	22	there	there	PRON
ejpam-5240	597	23	exist	exist	VERB
ejpam-5240	597	24	integers	integer	NOUN
ejpam-5240	597	25	a	a	PRON
ejpam-5240	597	26	and	and	CCONJ
ejpam-5240	597	27	b	b	NOUN
ejpam-5240	597	28	where	where	SCONJ
ejpam-5240	597	29	0	0	NUM
ejpam-5240	597	30	<	<	X
ejpam-5240	597	31	b	b	X
ejpam-5240	597	32	<	<	X
ejpam-5240	597	33	k	k	X
ejpam-5240	597	34	such	such	ADJ
ejpam-5240	597	35	that	that	SCONJ
ejpam-5240	597	36	n	n	PROPN
ejpam-5240	597	37	=	=	SYM
ejpam-5240	597	38	ka	ka	PROPN
ejpam-5240	598	1	+	+	PROPN
ejpam-5240	598	2	b.	b.	PROPN
ejpam-5240	598	3	however	however	ADV
ejpam-5240	598	4	|w	|w	PROPN
ejpam-5240	598	5	|	|	ADV
ejpam-5240	598	6	>	>	X
ejpam-5240	598	7	|t	|t	PROPN
ejpam-5240	598	8	|	|	ADV
ejpam-5240	598	9	.	.	PUNCT
ejpam-5240	599	1	thus	thus	ADV
ejpam-5240	599	2	,	,	PUNCT
ejpam-5240	599	3	the	the	DET
ejpam-5240	599	4	remaining	remain	VERB
ejpam-5240	599	5	b	b	NOUN
ejpam-5240	599	6	elements	element	NOUN
ejpam-5240	599	7	are	be	AUX
ejpam-5240	599	8	now	now	ADV
ejpam-5240	599	9	in	in	ADP
ejpam-5240	599	10	w	w	NOUN
ejpam-5240	599	11	in	in	ADP
ejpam-5240	599	12	the	the	DET
ejpam-5240	599	13	form	form	NOUN
ejpam-5240	599	14	of	of	ADP
ejpam-5240	599	15	k	k	NOUN
ejpam-5240	599	16	-	-	PUNCT
ejpam-5240	599	17	subsets	subset	NOUN
ejpam-5240	599	18	.	.	PUNCT
ejpam-5240	600	1	since	since	SCONJ
ejpam-5240	600	2	b	b	PROPN
ejpam-5240	600	3	<	<	X
ejpam-5240	600	4	k	k	X
ejpam-5240	600	5	,	,	PUNCT
ejpam-5240	600	6	it	it	PRON
ejpam-5240	600	7	follows	follow	VERB
ejpam-5240	600	8	that	that	SCONJ
ejpam-5240	600	9	these	these	DET
ejpam-5240	600	10	b	b	NOUN
ejpam-5240	600	11	elements	element	NOUN
ejpam-5240	600	12	are	be	AUX
ejpam-5240	600	13	paired	pair	VERB
ejpam-5240	600	14	with	with	ADP
ejpam-5240	600	15	some	some	DET
ejpam-5240	600	16	elements	element	NOUN
ejpam-5240	600	17	in	in	ADP
ejpam-5240	600	18	sn	sn	PROPN
ejpam-5240	600	19	to	to	PART
ejpam-5240	600	20	form	form	VERB
ejpam-5240	600	21	a	a	DET
ejpam-5240	600	22	k	k	ADJ
ejpam-5240	600	23	-	-	ADJ
ejpam-5240	600	24	element	element	NOUN
ejpam-5240	600	25	subset	subset	NOUN
ejpam-5240	600	26	.	.	PUNCT
ejpam-5240	601	1	by	by	ADP
ejpam-5240	601	2	this	this	PRON
ejpam-5240	601	3	,	,	PUNCT
ejpam-5240	601	4	it	it	PRON
ejpam-5240	601	5	can	can	AUX
ejpam-5240	601	6	be	be	AUX
ejpam-5240	601	7	observed	observe	VERB
ejpam-5240	601	8	that	that	SCONJ
ejpam-5240	601	9	there	there	PRON
ejpam-5240	601	10	will	will	AUX
ejpam-5240	601	11	exist	exist	VERB
ejpam-5240	601	12	a	a	DET
ejpam-5240	601	13	,	,	PUNCT
ejpam-5240	601	14	b	b	PROPN
ejpam-5240	601	15	∈	∈	PROPN
ejpam-5240	601	16	w	w	ADP
ejpam-5240	601	17	where	where	SCONJ
ejpam-5240	601	18	a	a	DET
ejpam-5240	601	19	̸=	̸=	PROPN
ejpam-5240	601	20	b	b	NUM
ejpam-5240	601	21	such	such	DET
ejpam-5240	601	22	that	that	SCONJ
ejpam-5240	601	23	a	a	DET
ejpam-5240	601	24	∩	∩	ADJ
ejpam-5240	601	25	b	b	NOUN
ejpam-5240	601	26	̸=	̸=	PROPN
ejpam-5240	601	27	∅.	∅.	ADP
ejpam-5240	601	28	this	this	PRON
ejpam-5240	601	29	is	be	AUX
ejpam-5240	601	30	a	a	DET
ejpam-5240	601	31	contradiction	contradiction	NOUN
ejpam-5240	601	32	to	to	ADP
ejpam-5240	601	33	the	the	DET
ejpam-5240	601	34	fact	fact	NOUN
ejpam-5240	601	35	that	that	SCONJ
ejpam-5240	601	36	w	w	NOUN
ejpam-5240	601	37	is	be	AUX
ejpam-5240	601	38	an	an	DET
ejpam-5240	601	39	independent	independent	ADJ
ejpam-5240	601	40	set	set	NOUN
ejpam-5240	601	41	.	.	PUNCT
ejpam-5240	602	1	therefore	therefore	ADV
ejpam-5240	602	2	,	,	PUNCT
ejpam-5240	602	3	α(gs(n	α(gs(n	PROPN
ejpam-5240	602	4	,	,	PUNCT
ejpam-5240	602	5	k	k	NOUN
ejpam-5240	602	6	)	)	PUNCT
ejpam-5240	602	7	)	)	PUNCT
ejpam-5240	602	8	=	=	PUNCT
ejpam-5240	602	9	⌊	⌊	VERB
ejpam-5240	602	10	n	n	PRON
ejpam-5240	602	11	k	k	PROPN
ejpam-5240	602	12	⌋	⌋	PROPN
ejpam-5240	602	13	.	.	PUNCT
ejpam-5240	603	1	illustration	illustration	NOUN
ejpam-5240	603	2	12	12	NUM
ejpam-5240	603	3	.	.	PUNCT
ejpam-5240	604	1	consider	consider	VERB
ejpam-5240	604	2	s6	s6	PROPN
ejpam-5240	604	3	=	=	SYM
ejpam-5240	604	4	{	{	PUNCT
ejpam-5240	604	5	x1	x1	PROPN
ejpam-5240	604	6	,	,	PUNCT
ejpam-5240	604	7	x2	x2	PROPN
ejpam-5240	604	8	,	,	PUNCT
ejpam-5240	604	9	x3	x3	PROPN
ejpam-5240	604	10	,	,	PUNCT
ejpam-5240	604	11	x4	x4	PROPN
ejpam-5240	604	12	,	,	PUNCT
ejpam-5240	604	13	x5	x5	PROPN
ejpam-5240	604	14	,	,	PUNCT
ejpam-5240	604	15	x6	x6	PROPN
ejpam-5240	604	16	}	}	PUNCT
ejpam-5240	604	17	and	and	CCONJ
ejpam-5240	604	18	let	let	VERB
ejpam-5240	604	19	k	k	NOUN
ejpam-5240	604	20	=	=	SYM
ejpam-5240	604	21	2	2	X
ejpam-5240	604	22	.	.	PUNCT
ejpam-5240	604	23	a	a	DET
ejpam-5240	604	24	pictorial	pictorial	ADJ
ejpam-5240	604	25	representation	representation	NOUN
ejpam-5240	604	26	of	of	ADP
ejpam-5240	604	27	gs(6,2	gs(6,2	PROPN
ejpam-5240	604	28	)	)	PUNCT
ejpam-5240	604	29	is	be	AUX
ejpam-5240	604	30	shown	show	VERB
ejpam-5240	604	31	in	in	ADP
ejpam-5240	604	32	figure	figure	NOUN
ejpam-5240	604	33	12	12	NUM
ejpam-5240	604	34	.	.	PUNCT
ejpam-5240	605	1	recall	recall	VERB
ejpam-5240	605	2	that	that	PRON
ejpam-5240	605	3	from	from	ADP
ejpam-5240	605	4	illustration	illustration	NOUN
ejpam-5240	605	5	11	11	NUM
ejpam-5240	605	6	,	,	PUNCT
ejpam-5240	605	7	t	t	NOUN
ejpam-5240	605	8	=	=	PRON
ejpam-5240	605	9	{	{	PUNCT
ejpam-5240	605	10	{	{	PUNCT
ejpam-5240	605	11	x1	x1	PROPN
ejpam-5240	605	12	,	,	PUNCT
ejpam-5240	605	13	x2	x2	PROPN
ejpam-5240	605	14	}	}	PUNCT
ejpam-5240	605	15	,	,	PUNCT
ejpam-5240	605	16	{	{	PUNCT
ejpam-5240	605	17	x3	x3	ADJ
ejpam-5240	605	18	,	,	PUNCT
ejpam-5240	605	19	x4	x4	PROPN
ejpam-5240	605	20	}	}	PUNCT
ejpam-5240	605	21	,	,	PUNCT
ejpam-5240	605	22	{	{	PUNCT
ejpam-5240	605	23	x5	x5	PROPN
ejpam-5240	605	24	,	,	PUNCT
ejpam-5240	605	25	x6	x6	PROPN
ejpam-5240	605	26	}	}	PUNCT
ejpam-5240	605	27	}	}	PUNCT
ejpam-5240	605	28	is	be	AUX
ejpam-5240	605	29	an	an	DET
ejpam-5240	605	30	independent	independent	ADJ
ejpam-5240	605	31	set	set	NOUN
ejpam-5240	605	32	in	in	ADP
ejpam-5240	605	33	gs(6,2	gs(6,2	PROPN
ejpam-5240	605	34	)	)	PUNCT
ejpam-5240	605	35	with	with	ADP
ejpam-5240	605	36	|t	|t	PROPN
ejpam-5240	605	37	|	|	ADV
ejpam-5240	605	38	=	=	SYM
ejpam-5240	605	39	3	3	X
ejpam-5240	605	40	=	=	SYM
ejpam-5240	605	41	⌊	⌊	PROPN
ejpam-5240	605	42	6	6	NUM
ejpam-5240	605	43	2	2	NUM
ejpam-5240	605	44	⌋	⌋	NOUN
ejpam-5240	605	45	.	.	PUNCT
ejpam-5240	606	1	note	note	VERB
ejpam-5240	606	2	that	that	SCONJ
ejpam-5240	606	3	{	{	PUNCT
ejpam-5240	606	4	x1	x1	ADJ
ejpam-5240	606	5	,	,	PUNCT
ejpam-5240	606	6	x2	x2	PROPN
ejpam-5240	606	7	}	}	PUNCT
ejpam-5240	606	8	,	,	PUNCT
ejpam-5240	606	9	{	{	PUNCT
ejpam-5240	606	10	x3	x3	ADJ
ejpam-5240	606	11	,	,	PUNCT
ejpam-5240	606	12	x4	x4	PROPN
ejpam-5240	606	13	}	}	PUNCT
ejpam-5240	606	14	,	,	PUNCT
ejpam-5240	606	15	{	{	PUNCT
ejpam-5240	606	16	x5	x5	PROPN
ejpam-5240	606	17	,	,	PUNCT
ejpam-5240	606	18	x6	x6	PROPN
ejpam-5240	606	19	}	}	PUNCT
ejpam-5240	606	20	are	be	AUX
ejpam-5240	606	21	2	2	NUM
ejpam-5240	606	22	-	-	PUNCT
ejpam-5240	606	23	element	element	NOUN
ejpam-5240	606	24	set	set	VERB
ejpam-5240	606	25	partitions	partition	NOUN
ejpam-5240	606	26	of	of	ADP
ejpam-5240	606	27	s6	s6	PROPN
ejpam-5240	606	28	.	.	PUNCT
ejpam-5240	607	1	thus	thus	ADV
ejpam-5240	607	2	adding	add	VERB
ejpam-5240	607	3	another	another	DET
ejpam-5240	607	4	element	element	NOUN
ejpam-5240	607	5	of	of	ADP
ejpam-5240	607	6	v	v	NOUN
ejpam-5240	607	7	(	(	PUNCT
ejpam-5240	607	8	gs(6,2	gs(6,2	PROPN
ejpam-5240	607	9	)	)	PUNCT
ejpam-5240	607	10	)	)	PUNCT
ejpam-5240	607	11	will	will	AUX
ejpam-5240	607	12	produce	produce	VERB
ejpam-5240	607	13	an	an	DET
ejpam-5240	607	14	intersection	intersection	NOUN
ejpam-5240	607	15	to	to	ADP
ejpam-5240	607	16	any	any	PRON
ejpam-5240	607	17	of	of	ADP
ejpam-5240	607	18	{	{	PUNCT
ejpam-5240	607	19	x1	x1	PROPN
ejpam-5240	607	20	,	,	PUNCT
ejpam-5240	607	21	x2	x2	PROPN
ejpam-5240	607	22	}	}	PUNCT
ejpam-5240	607	23	,	,	PUNCT
ejpam-5240	607	24	{	{	PUNCT
ejpam-5240	607	25	x3	x3	ADJ
ejpam-5240	607	26	,	,	PUNCT
ejpam-5240	607	27	x4	x4	PROPN
ejpam-5240	607	28	}	}	PUNCT
ejpam-5240	607	29	,	,	PUNCT
ejpam-5240	607	30	or	or	CCONJ
ejpam-5240	607	31	{	{	PUNCT
ejpam-5240	607	32	x5	x5	PROPN
ejpam-5240	607	33	,	,	PUNCT
ejpam-5240	607	34	x6	x6	PROPN
ejpam-5240	607	35	}	}	PUNCT
ejpam-5240	607	36	.	.	PUNCT
ejpam-5240	608	1	this	this	PRON
ejpam-5240	608	2	entails	entail	VERB
ejpam-5240	608	3	that	that	SCONJ
ejpam-5240	608	4	there	there	PRON
ejpam-5240	608	5	are	be	VERB
ejpam-5240	608	6	no	no	DET
ejpam-5240	608	7	independent	independent	ADJ
ejpam-5240	608	8	sets	set	NOUN
ejpam-5240	608	9	in	in	ADP
ejpam-5240	608	10	gs(6,2	gs(6,2	PROPN
ejpam-5240	608	11	)	)	PUNCT
ejpam-5240	608	12	of	of	ADP
ejpam-5240	608	13	cardinality	cardinality	NOUN
ejpam-5240	608	14	greater	great	ADJ
ejpam-5240	608	15	than	than	ADP
ejpam-5240	608	16	3	3	NUM
ejpam-5240	608	17	.	.	PUNCT
ejpam-5240	608	18	therefore	therefore	ADV
ejpam-5240	608	19	,	,	PUNCT
ejpam-5240	608	20	α(gs(6,2	α(gs(6,2	ADJ
ejpam-5240	608	21	)	)	PUNCT
ejpam-5240	608	22	)	)	PUNCT
ejpam-5240	609	1	=	=	SYM
ejpam-5240	609	2	3	3	X
ejpam-5240	609	3	.	.	PUNCT
ejpam-5240	609	4	now	now	ADV
ejpam-5240	609	5	,	,	PUNCT
ejpam-5240	609	6	using	use	VERB
ejpam-5240	609	7	theorem	theorem	NOUN
ejpam-5240	609	8	10	10	NUM
ejpam-5240	609	9	,	,	PUNCT
ejpam-5240	609	10	setting	set	VERB
ejpam-5240	609	11	n	n	X
ejpam-5240	609	12	=	=	SYM
ejpam-5240	609	13	6	6	NUM
ejpam-5240	609	14	and	and	CCONJ
ejpam-5240	609	15	k	k	NOUN
ejpam-5240	609	16	=	=	SYM
ejpam-5240	609	17	2	2	NUM
ejpam-5240	609	18	,	,	PUNCT
ejpam-5240	609	19	we	we	PRON
ejpam-5240	609	20	have	have	AUX
ejpam-5240	609	21	α(gs(6,2	α(gs(6,2	VERB
ejpam-5240	609	22	)	)	PUNCT
ejpam-5240	609	23	)	)	PUNCT
ejpam-5240	610	1	=	=	SYM
ejpam-5240	610	2	⌊n	⌊n	X
ejpam-5240	611	1	k	k	X
ejpam-5240	611	2	⌋	⌋	NOUN
ejpam-5240	612	1	=	=	PUNCT
ejpam-5240	612	2	⌊	⌊	VERB
ejpam-5240	612	3	6	6	NUM
ejpam-5240	612	4	2	2	NUM
ejpam-5240	612	5	⌋	⌋	NOUN
ejpam-5240	612	6	=	=	PUNCT
ejpam-5240	612	7	⌊3⌋	⌊3⌋	NOUN
ejpam-5240	612	8	=	=	SYM
ejpam-5240	612	9	3	3	X
ejpam-5240	612	10	.	.	PUNCT
ejpam-5240	612	11	now	now	ADV
ejpam-5240	612	12	,	,	PUNCT
ejpam-5240	612	13	by	by	ADP
ejpam-5240	612	14	theorem	theorem	NOUN
ejpam-5240	612	15	4	4	NUM
ejpam-5240	612	16	,	,	PUNCT
ejpam-5240	612	17	if	if	SCONJ
ejpam-5240	612	18	k	k	PROPN
ejpam-5240	612	19	=	=	SYM
ejpam-5240	612	20	1	1	NUM
ejpam-5240	612	21	,	,	PUNCT
ejpam-5240	612	22	then	then	ADV
ejpam-5240	612	23	gs(n,1	gs(n,1	NOUN
ejpam-5240	612	24	)	)	PUNCT
ejpam-5240	612	25	is	be	AUX
ejpam-5240	612	26	an	an	DET
ejpam-5240	612	27	empty	empty	ADJ
ejpam-5240	612	28	graph	graph	NOUN
ejpam-5240	612	29	of	of	ADP
ejpam-5240	612	30	order	order	NOUN
ejpam-5240	612	31	n.	n.	NOUN
ejpam-5240	612	32	note	note	VERB
ejpam-5240	612	33	that	that	SCONJ
ejpam-5240	612	34	there	there	PRON
ejpam-5240	612	35	is	be	VERB
ejpam-5240	612	36	no	no	DET
ejpam-5240	612	37	edge	edge	NOUN
ejpam-5240	612	38	connecting	connect	VERB
ejpam-5240	612	39	every	every	DET
ejpam-5240	612	40	vertex	vertex	NOUN
ejpam-5240	612	41	of	of	ADP
ejpam-5240	612	42	gs(n,1	gs(n,1	PROPN
ejpam-5240	612	43	)	)	PUNCT
ejpam-5240	612	44	and	and	CCONJ
ejpam-5240	612	45	v	v	X
ejpam-5240	612	46	(	(	PUNCT
ejpam-5240	612	47	gs(n,1	gs(n,1	NOUN
ejpam-5240	612	48	)	)	PUNCT
ejpam-5240	612	49	)	)	PUNCT
ejpam-5240	613	1	⊆	⊆	NUM
ejpam-5240	613	2	v	v	X
ejpam-5240	613	3	(	(	PUNCT
ejpam-5240	613	4	gs(n,1	gs(n,1	NOUN
ejpam-5240	613	5	)	)	PUNCT
ejpam-5240	613	6	)	)	PUNCT
ejpam-5240	613	7	,	,	PUNCT
ejpam-5240	613	8	so	so	CCONJ
ejpam-5240	613	9	the	the	DET
ejpam-5240	613	10	set	set	NOUN
ejpam-5240	613	11	v	v	NOUN
ejpam-5240	613	12	(	(	PUNCT
ejpam-5240	613	13	gs(n,1	gs(n,1	NOUN
ejpam-5240	613	14	)	)	PUNCT
ejpam-5240	613	15	)	)	PUNCT
ejpam-5240	613	16	is	be	AUX
ejpam-5240	613	17	an	an	DET
ejpam-5240	613	18	independent	independent	ADJ
ejpam-5240	613	19	set	set	NOUN
ejpam-5240	613	20	in	in	ADP
ejpam-5240	613	21	gs(n,1	gs(n,1	NOUN
ejpam-5240	613	22	)	)	PUNCT
ejpam-5240	613	23	.	.	PUNCT
ejpam-5240	614	1	since	since	SCONJ
ejpam-5240	614	2	|v	|v	PROPN
ejpam-5240	614	3	(	(	PUNCT
ejpam-5240	614	4	gs(n,1	gs(n,1	NOUN
ejpam-5240	614	5	)	)	PUNCT
ejpam-5240	614	6	)	)	PUNCT
ejpam-5240	614	7	|	|	ADV
ejpam-5240	614	8	=	=	SYM
ejpam-5240	614	9	n	n	CCONJ
ejpam-5240	614	10	,	,	PUNCT
ejpam-5240	614	11	it	it	PRON
ejpam-5240	614	12	follows	follow	VERB
ejpam-5240	614	13	that	that	DET
ejpam-5240	614	14	α(gs(n,1	α(gs(n,1	NOUN
ejpam-5240	614	15	)	)	PUNCT
ejpam-5240	614	16	)	)	PUNCT
ejpam-5240	615	1	=	=	PUNCT
ejpam-5240	615	2	n.	n.	NOUN
ejpam-5240	615	3	it	it	PRON
ejpam-5240	615	4	can	can	AUX
ejpam-5240	615	5	be	be	AUX
ejpam-5240	615	6	verified	verify	VERB
ejpam-5240	615	7	from	from	ADP
ejpam-5240	615	8	theorem	theorem	ADJ
ejpam-5240	615	9	10	10	NUM
ejpam-5240	615	10	that	that	SCONJ
ejpam-5240	615	11	when	when	SCONJ
ejpam-5240	615	12	k	k	PROPN
ejpam-5240	615	13	=	=	SYM
ejpam-5240	615	14	1	1	NUM
ejpam-5240	615	15	,	,	PUNCT
ejpam-5240	615	16	α(gs(n,1	α(gs(n,1	NOUN
ejpam-5240	615	17	)	)	PUNCT
ejpam-5240	615	18	)	)	PUNCT
ejpam-5240	616	1	=	=	PUNCT
ejpam-5240	616	2	⌊	⌊	VERB
ejpam-5240	616	3	n	n	ADV
ejpam-5240	616	4	1	1	NUM
ejpam-5240	616	5	⌋	⌋	NOUN
ejpam-5240	616	6	=	=	PUNCT
ejpam-5240	616	7	n.	n.	NOUN
ejpam-5240	616	8	correspondingly	correspondingly	ADV
ejpam-5240	616	9	,	,	PUNCT
ejpam-5240	616	10	we	we	PRON
ejpam-5240	616	11	have	have	VERB
ejpam-5240	616	12	remark	remark	NOUN
ejpam-5240	616	13	9	9	NUM
ejpam-5240	616	14	.	.	PUNCT
ejpam-5240	616	15	remark	remark	NOUN
ejpam-5240	616	16	9	9	NUM
ejpam-5240	616	17	.	.	PUNCT
ejpam-5240	617	1	let	let	VERB
ejpam-5240	617	2	gs(n	gs(n	NOUN
ejpam-5240	617	3	,	,	PUNCT
ejpam-5240	617	4	k	k	NOUN
ejpam-5240	617	5	)	)	PUNCT
ejpam-5240	617	6	be	be	VERB
ejpam-5240	617	7	a	a	DET
ejpam-5240	617	8	k	k	ADV
ejpam-5240	617	9	-	-	ADJ
ejpam-5240	617	10	restricted	restricted	ADJ
ejpam-5240	617	11	intersection	intersection	NOUN
ejpam-5240	617	12	graph	graph	NOUN
ejpam-5240	617	13	.	.	PUNCT
ejpam-5240	618	1	if	if	SCONJ
ejpam-5240	618	2	k	k	PROPN
ejpam-5240	618	3	=	=	SYM
ejpam-5240	618	4	1	1	NUM
ejpam-5240	618	5	,	,	PUNCT
ejpam-5240	618	6	then	then	ADV
ejpam-5240	618	7	α(gs(n,1	α(gs(n,1	PROPN
ejpam-5240	618	8	)	)	PUNCT
ejpam-5240	618	9	)	)	PUNCT
ejpam-5240	618	10	is	be	AUX
ejpam-5240	618	11	equal	equal	ADJ
ejpam-5240	618	12	to	to	AUX
ejpam-5240	618	13	n.	n.	VERB
ejpam-5240	618	14	we	we	PRON
ejpam-5240	618	15	have	have	AUX
ejpam-5240	618	16	now	now	ADV
ejpam-5240	618	17	identified	identify	VERB
ejpam-5240	618	18	the	the	DET
ejpam-5240	618	19	α(gs(n	α(gs(n	PROPN
ejpam-5240	618	20	,	,	PUNCT
ejpam-5240	618	21	k	k	NOUN
ejpam-5240	618	22	)	)	PUNCT
ejpam-5240	618	23	)	)	PUNCT
ejpam-5240	619	1	whenever	whenever	SCONJ
ejpam-5240	619	2	1	1	NUM
ejpam-5240	619	3	≤	≤	NUM
ejpam-5240	619	4	k	k	X
ejpam-5240	619	5	≤	≤	NUM
ejpam-5240	619	6	⌊	⌊	PROPN
ejpam-5240	619	7	n	n	PRON
ejpam-5240	619	8	k	k	PROPN
ejpam-5240	619	9	⌋	⌋	NOUN
ejpam-5240	619	10	.	.	PUNCT
ejpam-5240	620	1	now	now	ADV
ejpam-5240	620	2	,	,	PUNCT
ejpam-5240	620	3	theorem	theorem	VERB
ejpam-5240	620	4	11	11	NUM
ejpam-5240	620	5	determines	determine	VERB
ejpam-5240	620	6	the	the	DET
ejpam-5240	620	7	independence	independence	NOUN
ejpam-5240	620	8	number	number	NOUN
ejpam-5240	620	9	of	of	ADP
ejpam-5240	620	10	gs(n	gs(n	NOUN
ejpam-5240	620	11	,	,	PUNCT
ejpam-5240	620	12	k	k	NOUN
ejpam-5240	620	13	)	)	PUNCT
ejpam-5240	620	14	when	when	SCONJ
ejpam-5240	620	15	k	k	PROPN
ejpam-5240	620	16	=	=	PUNCT
ejpam-5240	620	17	0	0	NUM
ejpam-5240	620	18	or	or	CCONJ
ejpam-5240	620	19	⌊	⌊	AUX
ejpam-5240	620	20	n	n	ADV
ejpam-5240	620	21	2	2	NUM
ejpam-5240	620	22	⌋	⌋	NOUN
ejpam-5240	620	23	<	<	X
ejpam-5240	620	24	k	k	PROPN
ejpam-5240	620	25	≤	≤	PROPN
ejpam-5240	620	26	n.	n.	NOUN
ejpam-5240	620	27	theorem	theorem	VERB
ejpam-5240	620	28	11	11	NUM
ejpam-5240	620	29	.	.	PUNCT
ejpam-5240	621	1	let	let	VERB
ejpam-5240	621	2	gs(n	gs(n	NOUN
ejpam-5240	621	3	,	,	PUNCT
ejpam-5240	621	4	k	k	NOUN
ejpam-5240	621	5	)	)	PUNCT
ejpam-5240	621	6	be	be	VERB
ejpam-5240	621	7	a	a	DET
ejpam-5240	621	8	k	k	ADV
ejpam-5240	621	9	-	-	ADJ
ejpam-5240	621	10	restricted	restricted	ADJ
ejpam-5240	621	11	intersection	intersection	NOUN
ejpam-5240	621	12	graph	graph	NOUN
ejpam-5240	621	13	.	.	PUNCT
ejpam-5240	622	1	if	if	SCONJ
ejpam-5240	622	2	k	k	PROPN
ejpam-5240	622	3	=	=	PUNCT
ejpam-5240	622	4	0	0	NUM
ejpam-5240	622	5	or	or	CCONJ
ejpam-5240	622	6	⌊	⌊	X
ejpam-5240	622	7	n	n	ADV
ejpam-5240	622	8	2	2	NUM
ejpam-5240	622	9	⌋	⌋	NOUN
ejpam-5240	622	10	<	<	X
ejpam-5240	622	11	k	k	X
ejpam-5240	622	12	≤	≤	PROPN
ejpam-5240	622	13	n	n	CCONJ
ejpam-5240	622	14	,	,	PUNCT
ejpam-5240	622	15	then	then	ADV
ejpam-5240	622	16	α(gs(n	α(gs(n	PROPN
ejpam-5240	622	17	,	,	PUNCT
ejpam-5240	622	18	k	k	NOUN
ejpam-5240	622	19	)	)	PUNCT
ejpam-5240	622	20	)	)	PUNCT
ejpam-5240	623	1	=	=	PUNCT
ejpam-5240	623	2	1	1	X
ejpam-5240	623	3	.	.	PUNCT
ejpam-5240	623	4	proof	proof	NOUN
ejpam-5240	623	5	.	.	PUNCT
ejpam-5240	624	1	by	by	ADP
ejpam-5240	624	2	theorem	theorem	NOUN
ejpam-5240	624	3	7	7	NUM
ejpam-5240	624	4	,	,	PUNCT
ejpam-5240	624	5	gs(n	gs(n	NOUN
ejpam-5240	624	6	,	,	PUNCT
ejpam-5240	624	7	k	k	NOUN
ejpam-5240	624	8	)	)	PUNCT
ejpam-5240	624	9	is	be	AUX
ejpam-5240	624	10	a	a	DET
ejpam-5240	624	11	complete	complete	ADJ
ejpam-5240	624	12	graph	graph	NOUN
ejpam-5240	624	13	of	of	ADP
ejpam-5240	624	14	order	order	NOUN
ejpam-5240	624	15	(	(	PUNCT
ejpam-5240	624	16	n	n	X
ejpam-5240	624	17	k	k	PROPN
ejpam-5240	624	18	)	)	PUNCT
ejpam-5240	624	19	.	.	PUNCT
ejpam-5240	625	1	since	since	SCONJ
ejpam-5240	625	2	the	the	DET
ejpam-5240	625	3	independence	independence	NOUN
ejpam-5240	625	4	number	number	NOUN
ejpam-5240	625	5	of	of	ADP
ejpam-5240	625	6	a	a	DET
ejpam-5240	625	7	complete	complete	ADJ
ejpam-5240	625	8	graph	graph	NOUN
ejpam-5240	625	9	is	be	AUX
ejpam-5240	625	10	1	1	NUM
ejpam-5240	625	11	,	,	PUNCT
ejpam-5240	625	12	it	it	PRON
ejpam-5240	625	13	follows	follow	VERB
ejpam-5240	625	14	that	that	SCONJ
ejpam-5240	625	15	α(gs(n	α(gs(n	PROPN
ejpam-5240	625	16	,	,	PUNCT
ejpam-5240	625	17	k	k	NOUN
ejpam-5240	625	18	)	)	PUNCT
ejpam-5240	625	19	)	)	PUNCT
ejpam-5240	626	1	=	=	PUNCT
ejpam-5240	626	2	1	1	X
ejpam-5240	626	3	.	.	X
ejpam-5240	626	4	m.e	m.e	PROPN
ejpam-5240	626	5	.	.	PROPN
ejpam-5240	626	6	pelagio	pelagio	PROPN
ejpam-5240	626	7	,	,	PUNCT
ejpam-5240	626	8	n.	n.	NOUN
ejpam-5240	626	9	mame	mame	PROPN
ejpam-5240	626	10	,	,	PUNCT
ejpam-5240	626	11	k.	k.	PROPN
ejpam-5240	626	12	mendoza	mendoza	PROPN
ejpam-5240	626	13	/	/	SYM
ejpam-5240	626	14	eur	eur	PROPN
ejpam-5240	626	15	.	.	PUNCT
ejpam-5240	627	1	j.	j.	PROPN
ejpam-5240	627	2	pure	pure	PROPN
ejpam-5240	627	3	appl	appl	PROPN
ejpam-5240	627	4	.	.	PROPN
ejpam-5240	627	5	math	math	PROPN
ejpam-5240	627	6	,	,	PUNCT
ejpam-5240	627	7	17	17	NUM
ejpam-5240	627	8	(	(	PUNCT
ejpam-5240	627	9	3	3	NUM
ejpam-5240	627	10	)	)	PUNCT
ejpam-5240	627	11	(	(	PUNCT
ejpam-5240	627	12	2024	2024	NUM
ejpam-5240	627	13	)	)	PUNCT
ejpam-5240	627	14	,	,	PUNCT
ejpam-5240	627	15	1779	1779	NUM
ejpam-5240	627	16	-	-	SYM
ejpam-5240	627	17	1803	1803	NUM
ejpam-5240	627	18	1799	1799	NUM
ejpam-5240	627	19	5.2	5.2	NUM
ejpam-5240	627	20	.	.	PUNCT
ejpam-5240	628	1	domination	domination	NOUN
ejpam-5240	628	2	number	number	NOUN
ejpam-5240	628	3	of	of	ADP
ejpam-5240	628	4	gs(n	gs(n	NOUN
ejpam-5240	628	5	,	,	PUNCT
ejpam-5240	628	6	k	k	NOUN
ejpam-5240	628	7	)	)	PUNCT
ejpam-5240	628	8	this	this	DET
ejpam-5240	628	9	subsection	subsection	NOUN
ejpam-5240	628	10	further	far	ADV
ejpam-5240	628	11	exposes	expose	VERB
ejpam-5240	628	12	the	the	DET
ejpam-5240	628	13	domination	domination	NOUN
ejpam-5240	628	14	number	number	NOUN
ejpam-5240	628	15	of	of	ADP
ejpam-5240	628	16	a	a	DET
ejpam-5240	628	17	gs(n	gs(n	NOUN
ejpam-5240	628	18	,	,	PUNCT
ejpam-5240	628	19	k	k	NOUN
ejpam-5240	628	20	)	)	PUNCT
ejpam-5240	628	21	when	when	SCONJ
ejpam-5240	628	22	1	1	NUM
ejpam-5240	628	23	≤	≤	NUM
ejpam-5240	628	24	k	k	X
ejpam-5240	628	25	≤	≤	NUM
ejpam-5240	628	26	⌊	⌊	VERB
ejpam-5240	628	27	n	n	PRON
ejpam-5240	628	28	2	2	NUM
ejpam-5240	628	29	⌋	⌋	NOUN
ejpam-5240	628	30	and	and	CCONJ
ejpam-5240	628	31	k	k	NOUN
ejpam-5240	628	32	=	=	SYM
ejpam-5240	628	33	0	0	NUM
ejpam-5240	628	34	or	or	CCONJ
ejpam-5240	628	35	⌊	⌊	X
ejpam-5240	628	36	n	n	ADV
ejpam-5240	628	37	2	2	NUM
ejpam-5240	628	38	⌋	⌋	NOUN
ejpam-5240	628	39	<	<	X
ejpam-5240	628	40	k	k	PROPN
ejpam-5240	628	41	≤	≤	PROPN
ejpam-5240	628	42	n.	n.	NOUN
ejpam-5240	628	43	we	we	PRON
ejpam-5240	628	44	will	will	AUX
ejpam-5240	628	45	denote	denote	VERB
ejpam-5240	628	46	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	628	47	,	,	PUNCT
ejpam-5240	628	48	k	k	NOUN
ejpam-5240	628	49	)	)	PUNCT
ejpam-5240	628	50	)	)	PUNCT
ejpam-5240	628	51	as	as	ADP
ejpam-5240	628	52	the	the	DET
ejpam-5240	628	53	domination	domination	NOUN
ejpam-5240	628	54	number	number	NOUN
ejpam-5240	628	55	of	of	ADP
ejpam-5240	628	56	gs(n	gs(n	NOUN
ejpam-5240	628	57	,	,	PUNCT
ejpam-5240	628	58	k	k	NOUN
ejpam-5240	628	59	)	)	PUNCT
ejpam-5240	628	60	.	.	PUNCT
ejpam-5240	629	1	the	the	DET
ejpam-5240	629	2	next	next	ADJ
ejpam-5240	629	3	lemma	lemma	PROPN
ejpam-5240	629	4	establishes	establish	VERB
ejpam-5240	629	5	the	the	DET
ejpam-5240	629	6	existence	existence	NOUN
ejpam-5240	629	7	of	of	ADP
ejpam-5240	629	8	a	a	DET
ejpam-5240	629	9	dominating	dominating	NOUN
ejpam-5240	629	10	set	set	VERB
ejpam-5240	629	11	in	in	ADP
ejpam-5240	629	12	gs(n	gs(n	NOUN
ejpam-5240	629	13	,	,	PUNCT
ejpam-5240	629	14	k	k	NOUN
ejpam-5240	629	15	)	)	PUNCT
ejpam-5240	629	16	whenever	whenever	SCONJ
ejpam-5240	629	17	1	1	NUM
ejpam-5240	629	18	≤	≤	NUM
ejpam-5240	629	19	k	k	X
ejpam-5240	629	20	≤	≤	NUM
ejpam-5240	629	21	⌊	⌊	VERB
ejpam-5240	629	22	n	n	DET
ejpam-5240	629	23	2	2	NUM
ejpam-5240	629	24	⌋	⌋	NOUN
ejpam-5240	629	25	.	.	PUNCT
ejpam-5240	630	1	lemma	lemma	PROPN
ejpam-5240	630	2	4	4	X
ejpam-5240	630	3	.	.	PUNCT
ejpam-5240	631	1	let	let	VERB
ejpam-5240	631	2	gs(n	gs(n	NOUN
ejpam-5240	631	3	,	,	PUNCT
ejpam-5240	631	4	k	k	NOUN
ejpam-5240	631	5	)	)	PUNCT
ejpam-5240	631	6	be	be	VERB
ejpam-5240	631	7	a	a	DET
ejpam-5240	631	8	k	k	ADV
ejpam-5240	631	9	-	-	ADJ
ejpam-5240	631	10	restricted	restricted	ADJ
ejpam-5240	631	11	intersection	intersection	NOUN
ejpam-5240	631	12	graph	graph	NOUN
ejpam-5240	631	13	.	.	PUNCT
ejpam-5240	632	1	if	if	SCONJ
ejpam-5240	632	2	1	1	NUM
ejpam-5240	632	3	≤	≤	NUM
ejpam-5240	632	4	k	k	X
ejpam-5240	632	5	≤	≤	NUM
ejpam-5240	632	6	⌊	⌊	VERB
ejpam-5240	632	7	n	n	DET
ejpam-5240	632	8	2	2	NUM
ejpam-5240	632	9	⌋	⌋	NOUN
ejpam-5240	632	10	,	,	PUNCT
ejpam-5240	632	11	then	then	ADV
ejpam-5240	632	12	t	t	PROPN
ejpam-5240	632	13	=	=	SYM
ejpam-5240	632	14	{	{	PUNCT
ejpam-5240	632	15	{	{	PUNCT
ejpam-5240	632	16	x1	x1	PROPN
ejpam-5240	632	17	,	,	PUNCT
ejpam-5240	632	18	x2	x2	PROPN
ejpam-5240	632	19	,	,	PUNCT
ejpam-5240	632	20	...	...	PUNCT
ejpam-5240	632	21	,	,	PUNCT
ejpam-5240	632	22	xk	xk	PROPN
ejpam-5240	632	23	}	}	PUNCT
ejpam-5240	632	24	,	,	PUNCT
ejpam-5240	632	25	{	{	PUNCT
ejpam-5240	632	26	xk+1	xk+1	PROPN
ejpam-5240	632	27	,	,	PUNCT
ejpam-5240	632	28	...	...	PUNCT
ejpam-5240	632	29	,	,	PUNCT
ejpam-5240	632	30	x2k	x2k	NOUN
ejpam-5240	632	31	}	}	PUNCT
ejpam-5240	632	32	,	,	PUNCT
ejpam-5240	632	33	...	...	PUNCT
ejpam-5240	632	34	,	,	PUNCT
ejpam-5240	632	35	{	{	PUNCT
ejpam-5240	632	36	x(⌊n	x(⌊n	PROPN
ejpam-5240	632	37	k	k	PROPN
ejpam-5240	632	38	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	632	39	,	,	PUNCT
ejpam-5240	632	40	...	...	PUNCT
ejpam-5240	632	41	,	,	PUNCT
ejpam-5240	632	42	x(⌊n	x(⌊n	PROPN
ejpam-5240	632	43	k	k	PROPN
ejpam-5240	632	44	⌋)k	⌋)k	PROPN
ejpam-5240	632	45	}	}	PUNCT
ejpam-5240	632	46	}	}	PUNCT
ejpam-5240	632	47	,	,	PUNCT
ejpam-5240	632	48	is	be	AUX
ejpam-5240	632	49	a	a	DET
ejpam-5240	632	50	dominating	dominating	NOUN
ejpam-5240	632	51	set	set	VERB
ejpam-5240	632	52	in	in	ADP
ejpam-5240	632	53	gs(n	gs(n	NOUN
ejpam-5240	632	54	,	,	PUNCT
ejpam-5240	632	55	k	k	NOUN
ejpam-5240	632	56	)	)	PUNCT
ejpam-5240	632	57	where	where	SCONJ
ejpam-5240	632	58	|t	|t	VERB
ejpam-5240	632	59	|	|	ADV
ejpam-5240	632	60	=	=	SYM
ejpam-5240	632	61	⌊	⌊	PROPN
ejpam-5240	632	62	n	n	PRON
ejpam-5240	632	63	k	k	NOUN
ejpam-5240	632	64	⌋	⌋	NOUN
ejpam-5240	632	65	.	.	PUNCT
ejpam-5240	633	1	proof	proof	NOUN
ejpam-5240	633	2	.	.	PUNCT
ejpam-5240	634	1	let	let	VERB
ejpam-5240	634	2	t	t	NOUN
ejpam-5240	634	3	=	=	PRON
ejpam-5240	634	4	{	{	PUNCT
ejpam-5240	634	5	{	{	PUNCT
ejpam-5240	634	6	x1	x1	PROPN
ejpam-5240	634	7	,	,	PUNCT
ejpam-5240	634	8	x2	x2	PROPN
ejpam-5240	634	9	,	,	PUNCT
ejpam-5240	634	10	...	...	PUNCT
ejpam-5240	634	11	,	,	PUNCT
ejpam-5240	634	12	xk	xk	PROPN
ejpam-5240	634	13	}	}	PUNCT
ejpam-5240	634	14	,	,	PUNCT
ejpam-5240	634	15	{	{	PUNCT
ejpam-5240	634	16	xk+1	xk+1	PROPN
ejpam-5240	634	17	,	,	PUNCT
ejpam-5240	634	18	...	...	PUNCT
ejpam-5240	634	19	,	,	PUNCT
ejpam-5240	634	20	x2k	x2k	NOUN
ejpam-5240	634	21	}	}	PUNCT
ejpam-5240	634	22	,	,	PUNCT
ejpam-5240	634	23	...	...	PUNCT
ejpam-5240	634	24	,	,	PUNCT
ejpam-5240	634	25	{	{	PUNCT
ejpam-5240	634	26	x(⌊n	x(⌊n	PROPN
ejpam-5240	634	27	k	k	PROPN
ejpam-5240	634	28	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	634	29	,	,	PUNCT
ejpam-5240	634	30	...	...	PUNCT
ejpam-5240	634	31	,	,	PUNCT
ejpam-5240	634	32	x(⌊n	x(⌊n	PROPN
ejpam-5240	634	33	k	k	PROPN
ejpam-5240	634	34	⌋)k	⌋)k	PROPN
ejpam-5240	634	35	}	}	PUNCT
ejpam-5240	634	36	}	}	PUNCT
ejpam-5240	634	37	.	.	PUNCT
ejpam-5240	635	1	it	it	PRON
ejpam-5240	635	2	can	can	AUX
ejpam-5240	635	3	be	be	AUX
ejpam-5240	635	4	observed	observe	VERB
ejpam-5240	635	5	that	that	SCONJ
ejpam-5240	635	6	t	t	PROPN
ejpam-5240	635	7	contains	contain	VERB
ejpam-5240	635	8	some	some	PRON
ejpam-5240	635	9	of	of	ADP
ejpam-5240	635	10	the	the	DET
ejpam-5240	635	11	k	k	ADJ
ejpam-5240	635	12	-	-	ADJ
ejpam-5240	635	13	element	element	ADJ
ejpam-5240	635	14	partitions	partition	NOUN
ejpam-5240	635	15	of	of	ADP
ejpam-5240	635	16	sn	sn	PROPN
ejpam-5240	635	17	.	.	PUNCT
ejpam-5240	636	1	thus	thus	ADV
ejpam-5240	636	2	,	,	PUNCT
ejpam-5240	636	3	t	t	PROPN
ejpam-5240	636	4	⊆	⊆	NUM
ejpam-5240	636	5	s(n	s(n	PROPN
ejpam-5240	636	6	,	,	PUNCT
ejpam-5240	636	7	k	k	NOUN
ejpam-5240	636	8	)	)	PUNCT
ejpam-5240	636	9	which	which	PRON
ejpam-5240	636	10	implies	imply	VERB
ejpam-5240	636	11	that	that	SCONJ
ejpam-5240	636	12	t	t	PROPN
ejpam-5240	636	13	⊆	⊆	NUM
ejpam-5240	636	14	v	v	NOUN
ejpam-5240	636	15	(	(	PUNCT
ejpam-5240	636	16	gs(n	gs(n	NOUN
ejpam-5240	636	17	,	,	PUNCT
ejpam-5240	636	18	k	k	NOUN
ejpam-5240	636	19	)	)	PUNCT
ejpam-5240	636	20	)	)	PUNCT
ejpam-5240	636	21	.	.	PUNCT
ejpam-5240	637	1	if	if	SCONJ
ejpam-5240	637	2	n	n	PRON
ejpam-5240	637	3	is	be	AUX
ejpam-5240	637	4	divisible	divisible	ADJ
ejpam-5240	637	5	by	by	ADP
ejpam-5240	637	6	k	k	PROPN
ejpam-5240	637	7	,	,	PUNCT
ejpam-5240	637	8	then	then	ADV
ejpam-5240	637	9	t	t	PROPN
ejpam-5240	637	10	contains	contain	VERB
ejpam-5240	637	11	all	all	DET
ejpam-5240	637	12	the	the	DET
ejpam-5240	637	13	elements	element	NOUN
ejpam-5240	637	14	of	of	ADP
ejpam-5240	637	15	sn	sn	PROPN
ejpam-5240	637	16	that	that	PRON
ejpam-5240	637	17	are	be	AUX
ejpam-5240	637	18	partitioned	partition	VERB
ejpam-5240	637	19	into	into	ADP
ejpam-5240	637	20	k	k	NOUN
ejpam-5240	637	21	-	-	PUNCT
ejpam-5240	637	22	subsets	subset	NOUN
ejpam-5240	637	23	.	.	PUNCT
ejpam-5240	638	1	by	by	ADP
ejpam-5240	638	2	this	this	PRON
ejpam-5240	638	3	,	,	PUNCT
ejpam-5240	638	4	then	then	ADV
ejpam-5240	638	5	for	for	ADP
ejpam-5240	638	6	every	every	DET
ejpam-5240	638	7	a	a	DET
ejpam-5240	638	8	∈	∈	PROPN
ejpam-5240	638	9	v	v	NOUN
ejpam-5240	638	10	(	(	PUNCT
ejpam-5240	638	11	gs(n	gs(n	NOUN
ejpam-5240	638	12	,	,	PUNCT
ejpam-5240	638	13	k	k	NOUN
ejpam-5240	638	14	)	)	PUNCT
ejpam-5240	638	15	)	)	PUNCT
ejpam-5240	638	16	\t	\t	PUNCT
ejpam-5240	638	17	,	,	PUNCT
ejpam-5240	638	18	there	there	PRON
ejpam-5240	638	19	exists	exist	VERB
ejpam-5240	638	20	b	b	PROPN
ejpam-5240	638	21	∈	∈	PROPN
ejpam-5240	638	22	t	t	NOUN
ejpam-5240	638	23	such	such	ADJ
ejpam-5240	638	24	that	that	SCONJ
ejpam-5240	638	25	a	a	DET
ejpam-5240	638	26	∩	∩	ADJ
ejpam-5240	638	27	b	b	NOUN
ejpam-5240	638	28	̸=	̸=	PROPN
ejpam-5240	638	29	∅.	∅.	X
ejpam-5240	638	30	by	by	ADP
ejpam-5240	638	31	definition	definition	NOUN
ejpam-5240	638	32	7	7	NUM
ejpam-5240	638	33	,	,	PUNCT
ejpam-5240	638	34	[	[	X
ejpam-5240	638	35	a	a	X
ejpam-5240	638	36	,	,	PUNCT
ejpam-5240	638	37	b	b	NOUN
ejpam-5240	638	38	]	]	X
ejpam-5240	638	39	∈	∈	PROPN
ejpam-5240	638	40	e(gs(n	e(gs(n	PROPN
ejpam-5240	638	41	,	,	PUNCT
ejpam-5240	638	42	k	k	NOUN
ejpam-5240	638	43	)	)	PUNCT
ejpam-5240	638	44	)	)	PUNCT
ejpam-5240	638	45	.	.	PUNCT
ejpam-5240	639	1	hence	hence	ADV
ejpam-5240	639	2	,	,	PUNCT
ejpam-5240	639	3	t	t	PROPN
ejpam-5240	639	4	is	be	AUX
ejpam-5240	639	5	a	a	DET
ejpam-5240	639	6	dominating	dominating	NOUN
ejpam-5240	639	7	set	set	VERB
ejpam-5240	639	8	in	in	ADP
ejpam-5240	639	9	gs(n	gs(n	NOUN
ejpam-5240	639	10	,	,	PUNCT
ejpam-5240	639	11	k	k	NOUN
ejpam-5240	639	12	)	)	PUNCT
ejpam-5240	639	13	.	.	PUNCT
ejpam-5240	640	1	since	since	SCONJ
ejpam-5240	640	2	n	n	PRON
ejpam-5240	640	3	is	be	AUX
ejpam-5240	640	4	divisible	divisible	ADJ
ejpam-5240	640	5	by	by	ADP
ejpam-5240	640	6	k	k	PROPN
ejpam-5240	640	7	,	,	PUNCT
ejpam-5240	640	8	it	it	PRON
ejpam-5240	640	9	follows	follow	VERB
ejpam-5240	640	10	that	that	SCONJ
ejpam-5240	640	11	|t	|t	VERB
ejpam-5240	641	1	|	|	ADV
ejpam-5240	641	2	=	=	SYM
ejpam-5240	641	3	n	n	PROPN
ejpam-5240	641	4	k	k	NOUN
ejpam-5240	641	5	.	.	PUNCT
ejpam-5240	642	1	on	on	ADP
ejpam-5240	642	2	the	the	DET
ejpam-5240	642	3	other	other	ADJ
ejpam-5240	642	4	hand	hand	NOUN
ejpam-5240	642	5	,	,	PUNCT
ejpam-5240	642	6	if	if	SCONJ
ejpam-5240	642	7	n	n	PRON
ejpam-5240	642	8	is	be	AUX
ejpam-5240	642	9	not	not	PART
ejpam-5240	642	10	divisible	divisible	ADJ
ejpam-5240	642	11	by	by	ADP
ejpam-5240	642	12	k	k	PROPN
ejpam-5240	642	13	,	,	PUNCT
ejpam-5240	642	14	then	then	ADV
ejpam-5240	642	15	by	by	ADP
ejpam-5240	642	16	the	the	DET
ejpam-5240	642	17	division	division	NOUN
ejpam-5240	642	18	algorithm	algorithm	NOUN
ejpam-5240	642	19	,	,	PUNCT
ejpam-5240	642	20	there	there	PRON
ejpam-5240	642	21	exist	exist	VERB
ejpam-5240	642	22	integers	integer	NOUN
ejpam-5240	642	23	a	a	PRON
ejpam-5240	642	24	and	and	CCONJ
ejpam-5240	642	25	b	b	NOUN
ejpam-5240	642	26	with	with	ADP
ejpam-5240	642	27	0	0	NUM
ejpam-5240	642	28	<	<	X
ejpam-5240	642	29	b	b	X
ejpam-5240	642	30	<	<	X
ejpam-5240	642	31	k	k	X
ejpam-5240	642	32	such	such	ADJ
ejpam-5240	642	33	that	that	SCONJ
ejpam-5240	642	34	n	n	PROPN
ejpam-5240	642	35	=	=	SYM
ejpam-5240	642	36	ka	ka	PROPN
ejpam-5240	643	1	+	+	PROPN
ejpam-5240	643	2	b.	b.	PROPN
ejpam-5240	643	3	meaning	meaning	NOUN
ejpam-5240	643	4	to	to	PART
ejpam-5240	643	5	say	say	VERB
ejpam-5240	643	6	,	,	PUNCT
ejpam-5240	643	7	there	there	PRON
ejpam-5240	643	8	are	be	VERB
ejpam-5240	643	9	b	b	NUM
ejpam-5240	643	10	remaining	remain	VERB
ejpam-5240	643	11	elements	element	NOUN
ejpam-5240	643	12	that	that	PRON
ejpam-5240	643	13	are	be	AUX
ejpam-5240	643	14	not	not	PART
ejpam-5240	643	15	in	in	ADP
ejpam-5240	643	16	t	t	PROPN
ejpam-5240	643	17	since	since	SCONJ
ejpam-5240	643	18	we	we	PRON
ejpam-5240	643	19	can	can	AUX
ejpam-5240	643	20	not	not	PART
ejpam-5240	643	21	form	form	VERB
ejpam-5240	643	22	a	a	DET
ejpam-5240	643	23	k	k	NOUN
ejpam-5240	643	24	-	-	NOUN
ejpam-5240	643	25	subset	subset	NOUN
ejpam-5240	643	26	from	from	ADP
ejpam-5240	643	27	these	these	DET
ejpam-5240	643	28	elements	element	NOUN
ejpam-5240	643	29	.	.	PUNCT
ejpam-5240	644	1	observe	observe	VERB
ejpam-5240	644	2	that	that	SCONJ
ejpam-5240	644	3	the	the	DET
ejpam-5240	644	4	remaining	remain	VERB
ejpam-5240	644	5	b	b	NOUN
ejpam-5240	644	6	elements	element	NOUN
ejpam-5240	644	7	can	can	AUX
ejpam-5240	644	8	be	be	AUX
ejpam-5240	644	9	paired	pair	VERB
ejpam-5240	644	10	to	to	ADP
ejpam-5240	644	11	other	other	ADJ
ejpam-5240	644	12	elements	element	NOUN
ejpam-5240	644	13	of	of	ADP
ejpam-5240	644	14	sn	sn	PROPN
ejpam-5240	644	15	to	to	PART
ejpam-5240	644	16	form	form	VERB
ejpam-5240	644	17	a	a	DET
ejpam-5240	644	18	k	k	NOUN
ejpam-5240	644	19	-	-	NOUN
ejpam-5240	644	20	subset	subset	NOUN
ejpam-5240	644	21	such	such	ADJ
ejpam-5240	644	22	that	that	SCONJ
ejpam-5240	644	23	these	these	DET
ejpam-5240	644	24	subsets	subset	NOUN
ejpam-5240	644	25	are	be	AUX
ejpam-5240	644	26	elements	element	NOUN
ejpam-5240	644	27	of	of	ADP
ejpam-5240	644	28	v	v	NOUN
ejpam-5240	644	29	(	(	PUNCT
ejpam-5240	644	30	gs(n	gs(n	NOUN
ejpam-5240	644	31	,	,	PUNCT
ejpam-5240	644	32	k	k	NOUN
ejpam-5240	644	33	)	)	PUNCT
ejpam-5240	644	34	)	)	PUNCT
ejpam-5240	645	1	\	\	PROPN
ejpam-5240	645	2	t	t	PROPN
ejpam-5240	645	3	.	.	PUNCT
ejpam-5240	646	1	thus	thus	ADV
ejpam-5240	646	2	,	,	PUNCT
ejpam-5240	646	3	for	for	ADP
ejpam-5240	646	4	all	all	DET
ejpam-5240	646	5	a	a	DET
ejpam-5240	646	6	∈	∈	PROPN
ejpam-5240	646	7	v	v	NOUN
ejpam-5240	646	8	(	(	PUNCT
ejpam-5240	646	9	gs(n	gs(n	NOUN
ejpam-5240	646	10	,	,	PUNCT
ejpam-5240	646	11	k	k	NOUN
ejpam-5240	646	12	)	)	PUNCT
ejpam-5240	646	13	)	)	PUNCT
ejpam-5240	646	14	\	\	PROPN
ejpam-5240	646	15	t	t	PROPN
ejpam-5240	646	16	,	,	PUNCT
ejpam-5240	646	17	there	there	PRON
ejpam-5240	646	18	exists	exist	VERB
ejpam-5240	646	19	b	b	PROPN
ejpam-5240	646	20	∈	∈	PROPN
ejpam-5240	646	21	t	t	NOUN
ejpam-5240	646	22	where	where	SCONJ
ejpam-5240	646	23	a	a	DET
ejpam-5240	646	24	∩	∩	ADJ
ejpam-5240	646	25	b	b	X
ejpam-5240	646	26	̸=	̸=	PROPN
ejpam-5240	646	27	∅	∅	NOUN
ejpam-5240	646	28	that	that	PRON
ejpam-5240	646	29	implies	imply	VERB
ejpam-5240	646	30	[	[	X
ejpam-5240	646	31	a	a	X
ejpam-5240	646	32	,	,	PUNCT
ejpam-5240	646	33	b	b	NOUN
ejpam-5240	646	34	]	]	X
ejpam-5240	646	35	∈	∈	PROPN
ejpam-5240	646	36	e(gs(n	e(gs(n	PROPN
ejpam-5240	646	37	,	,	PUNCT
ejpam-5240	646	38	k	k	NOUN
ejpam-5240	646	39	)	)	PUNCT
ejpam-5240	646	40	)	)	PUNCT
ejpam-5240	646	41	.	.	PUNCT
ejpam-5240	647	1	by	by	ADP
ejpam-5240	647	2	this	this	PRON
ejpam-5240	647	3	,	,	PUNCT
ejpam-5240	647	4	t	t	PROPN
ejpam-5240	647	5	is	be	AUX
ejpam-5240	647	6	a	a	DET
ejpam-5240	647	7	dominating	dominating	NOUN
ejpam-5240	647	8	set	set	VERB
ejpam-5240	647	9	in	in	ADP
ejpam-5240	647	10	gs(n	gs(n	NOUN
ejpam-5240	647	11	,	,	PUNCT
ejpam-5240	647	12	k	k	NOUN
ejpam-5240	647	13	)	)	PUNCT
ejpam-5240	647	14	with	with	ADP
ejpam-5240	647	15	|t	|t	PROPN
ejpam-5240	648	1	|	|	ADV
ejpam-5240	648	2	=	=	SYM
ejpam-5240	648	3	⌊	⌊	PROPN
ejpam-5240	648	4	n	n	PRON
ejpam-5240	648	5	k	k	NOUN
ejpam-5240	648	6	⌋	⌋	PROPN
ejpam-5240	648	7	.	.	PUNCT
ejpam-5240	649	1	either	either	DET
ejpam-5240	649	2	way	way	NOUN
ejpam-5240	649	3	,	,	PUNCT
ejpam-5240	649	4	|t	|t	VERB
ejpam-5240	649	5	|	|	ADV
ejpam-5240	649	6	=	=	SYM
ejpam-5240	649	7	⌊	⌊	PROPN
ejpam-5240	649	8	n	n	PRON
ejpam-5240	649	9	k	k	NOUN
ejpam-5240	649	10	⌋	⌋	NOUN
ejpam-5240	649	11	.	.	PUNCT
ejpam-5240	650	1	the	the	DET
ejpam-5240	650	2	set	set	PROPN
ejpam-5240	650	3	t	t	PROPN
ejpam-5240	650	4	discussed	discuss	VERB
ejpam-5240	650	5	in	in	ADP
ejpam-5240	650	6	lemma	lemma	PROPN
ejpam-5240	650	7	4	4	NUM
ejpam-5240	650	8	is	be	AUX
ejpam-5240	650	9	one	one	NUM
ejpam-5240	650	10	of	of	ADP
ejpam-5240	650	11	the	the	DET
ejpam-5240	650	12	dominating	dominating	NOUN
ejpam-5240	650	13	sets	set	NOUN
ejpam-5240	650	14	in	in	ADP
ejpam-5240	650	15	gs(n	gs(n	NOUN
ejpam-5240	650	16	,	,	PUNCT
ejpam-5240	650	17	k	k	NOUN
ejpam-5240	650	18	)	)	PUNCT
ejpam-5240	650	19	.	.	PUNCT
ejpam-5240	651	1	this	this	DET
ejpam-5240	651	2	set	set	NOUN
ejpam-5240	651	3	shall	shall	AUX
ejpam-5240	651	4	be	be	AUX
ejpam-5240	651	5	employed	employ	VERB
ejpam-5240	651	6	to	to	PART
ejpam-5240	651	7	find	find	VERB
ejpam-5240	651	8	to	to	PART
ejpam-5240	651	9	find	find	VERB
ejpam-5240	651	10	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	651	11	,	,	PUNCT
ejpam-5240	651	12	k	k	NOUN
ejpam-5240	651	13	)	)	PUNCT
ejpam-5240	651	14	)	)	PUNCT
ejpam-5240	652	1	whenever	whenever	SCONJ
ejpam-5240	652	2	1	1	NUM
ejpam-5240	652	3	≤	≤	NUM
ejpam-5240	652	4	k	k	X
ejpam-5240	652	5	≤	≤	NUM
ejpam-5240	652	6	⌊	⌊	PROPN
ejpam-5240	652	7	n	n	PRON
ejpam-5240	652	8	k	k	PROPN
ejpam-5240	652	9	⌋	⌋	PROPN
ejpam-5240	652	10	.	.	PUNCT
ejpam-5240	653	1	illustration	illustration	NOUN
ejpam-5240	653	2	13	13	NUM
ejpam-5240	653	3	.	.	PUNCT
ejpam-5240	654	1	let	let	VERB
ejpam-5240	654	2	s6	s6	PROPN
ejpam-5240	654	3	=	=	SYM
ejpam-5240	654	4	{	{	PUNCT
ejpam-5240	654	5	x1	x1	PROPN
ejpam-5240	654	6	,	,	PUNCT
ejpam-5240	654	7	x2	x2	PROPN
ejpam-5240	654	8	,	,	PUNCT
ejpam-5240	654	9	x3	x3	PROPN
ejpam-5240	654	10	,	,	PUNCT
ejpam-5240	654	11	x4	x4	PROPN
ejpam-5240	654	12	,	,	PUNCT
ejpam-5240	654	13	x5	x5	PROPN
ejpam-5240	654	14	,	,	PUNCT
ejpam-5240	654	15	x6	x6	PROPN
ejpam-5240	654	16	}	}	PUNCT
ejpam-5240	654	17	and	and	CCONJ
ejpam-5240	654	18	k	k	X
ejpam-5240	654	19	=	=	SYM
ejpam-5240	654	20	2	2	X
ejpam-5240	654	21	.	.	PUNCT
ejpam-5240	654	22	to	to	PART
ejpam-5240	654	23	pictorially	pictorially	ADV
ejpam-5240	654	24	illustrate	illustrate	VERB
ejpam-5240	654	25	gs(6,2	gs(6,2	PROPN
ejpam-5240	654	26	)	)	PUNCT
ejpam-5240	654	27	,	,	PUNCT
ejpam-5240	654	28	refer	refer	VERB
ejpam-5240	654	29	to	to	PART
ejpam-5240	654	30	figure	figure	VERB
ejpam-5240	654	31	12	12	NUM
ejpam-5240	654	32	.	.	PUNCT
ejpam-5240	655	1	also	also	ADV
ejpam-5240	655	2	,	,	PUNCT
ejpam-5240	655	3	let	let	VERB
ejpam-5240	655	4	t	t	NOUN
ejpam-5240	655	5	=	=	PRON
ejpam-5240	655	6	{	{	PUNCT
ejpam-5240	655	7	{	{	PUNCT
ejpam-5240	655	8	x1	x1	PROPN
ejpam-5240	655	9	,	,	PUNCT
ejpam-5240	655	10	x2	x2	PROPN
ejpam-5240	655	11	}	}	PUNCT
ejpam-5240	655	12	,	,	PUNCT
ejpam-5240	655	13	{	{	PUNCT
ejpam-5240	655	14	x3	x3	ADJ
ejpam-5240	655	15	,	,	PUNCT
ejpam-5240	655	16	x4	x4	PROPN
ejpam-5240	655	17	}	}	PUNCT
ejpam-5240	655	18	,	,	PUNCT
ejpam-5240	655	19	{	{	PUNCT
ejpam-5240	655	20	x5	x5	PROPN
ejpam-5240	655	21	,	,	PUNCT
ejpam-5240	655	22	x6	x6	PROPN
ejpam-5240	655	23	}	}	PUNCT
ejpam-5240	655	24	}	}	PUNCT
ejpam-5240	655	25	.	.	PUNCT
ejpam-5240	656	1	it	it	PRON
ejpam-5240	656	2	can	can	AUX
ejpam-5240	656	3	be	be	AUX
ejpam-5240	656	4	observed	observe	VERB
ejpam-5240	656	5	that	that	SCONJ
ejpam-5240	656	6	t	t	PROPN
ejpam-5240	656	7	⊆	⊆	NUM
ejpam-5240	656	8	v	v	NOUN
ejpam-5240	656	9	(	(	PUNCT
ejpam-5240	656	10	gs(6,2	gs(6,2	PROPN
ejpam-5240	656	11	)	)	PUNCT
ejpam-5240	656	12	)	)	PUNCT
ejpam-5240	656	13	.	.	PUNCT
ejpam-5240	657	1	now	now	ADV
ejpam-5240	657	2	,	,	PUNCT
ejpam-5240	657	3	the	the	DET
ejpam-5240	657	4	set	set	NOUN
ejpam-5240	657	5	v	v	NOUN
ejpam-5240	657	6	(	(	PUNCT
ejpam-5240	657	7	gs(6,2	gs(6,2	PROPN
ejpam-5240	657	8	)	)	PUNCT
ejpam-5240	657	9	)	)	PUNCT
ejpam-5240	657	10	\	\	PROPN
ejpam-5240	657	11	t	t	PROPN
ejpam-5240	657	12	is	be	AUX
ejpam-5240	657	13	given	give	VERB
ejpam-5240	657	14	by	by	ADP
ejpam-5240	657	15	v	v	PROPN
ejpam-5240	657	16	(	(	PUNCT
ejpam-5240	657	17	gs(6,2	gs(6,2	PROPN
ejpam-5240	657	18	)	)	PUNCT
ejpam-5240	657	19	)	)	PUNCT
ejpam-5240	658	1	\	\	PROPN
ejpam-5240	658	2	t	t	PROPN
ejpam-5240	658	3	=	=	PRON
ejpam-5240	658	4	{	{	PUNCT
ejpam-5240	658	5	{	{	PUNCT
ejpam-5240	658	6	x1	x1	PROPN
ejpam-5240	658	7	,	,	PUNCT
ejpam-5240	658	8	x3	x3	ADJ
ejpam-5240	658	9	}	}	PUNCT
ejpam-5240	658	10	,	,	PUNCT
ejpam-5240	658	11	{	{	PUNCT
ejpam-5240	658	12	x1	x1	PROPN
ejpam-5240	658	13	,	,	PUNCT
ejpam-5240	658	14	x4	x4	PROPN
ejpam-5240	658	15	}	}	PUNCT
ejpam-5240	658	16	,	,	PUNCT
ejpam-5240	658	17	{	{	PUNCT
ejpam-5240	658	18	x1	x1	PROPN
ejpam-5240	658	19	,	,	PUNCT
ejpam-5240	658	20	x5	x5	PROPN
ejpam-5240	658	21	}	}	PUNCT
ejpam-5240	658	22	,	,	PUNCT
ejpam-5240	658	23	{	{	PUNCT
ejpam-5240	658	24	x1	x1	PROPN
ejpam-5240	658	25	,	,	PUNCT
ejpam-5240	658	26	x6	x6	PROPN
ejpam-5240	658	27	}	}	PUNCT
ejpam-5240	658	28	,	,	PUNCT
ejpam-5240	658	29	{	{	PUNCT
ejpam-5240	658	30	x2	x2	ADJ
ejpam-5240	658	31	,	,	PUNCT
ejpam-5240	658	32	x3	x3	ADJ
ejpam-5240	658	33	}	}	PUNCT
ejpam-5240	658	34	,	,	PUNCT
ejpam-5240	658	35	{	{	PUNCT
ejpam-5240	658	36	x2	x2	PROPN
ejpam-5240	658	37	,	,	PUNCT
ejpam-5240	658	38	x4	x4	PROPN
ejpam-5240	658	39	}	}	PUNCT
ejpam-5240	658	40	,	,	PUNCT
ejpam-5240	658	41	{	{	PUNCT
ejpam-5240	658	42	x2	x2	PROPN
ejpam-5240	658	43	,	,	PUNCT
ejpam-5240	658	44	x5	x5	PROPN
ejpam-5240	658	45	}	}	PUNCT
ejpam-5240	658	46	,	,	PUNCT
ejpam-5240	658	47	{	{	PUNCT
ejpam-5240	658	48	x2	x2	PROPN
ejpam-5240	658	49	,	,	PUNCT
ejpam-5240	658	50	x6	x6	PROPN
ejpam-5240	658	51	}	}	PUNCT
ejpam-5240	658	52	,	,	PUNCT
ejpam-5240	658	53	{	{	PUNCT
ejpam-5240	658	54	x3	x3	ADJ
ejpam-5240	658	55	,	,	PUNCT
ejpam-5240	658	56	x5	x5	PROPN
ejpam-5240	658	57	}	}	PUNCT
ejpam-5240	658	58	,	,	PUNCT
ejpam-5240	658	59	{	{	PUNCT
ejpam-5240	658	60	x3	x3	ADJ
ejpam-5240	658	61	,	,	PUNCT
ejpam-5240	658	62	x6	x6	PROPN
ejpam-5240	658	63	}	}	PUNCT
ejpam-5240	658	64	,	,	PUNCT
ejpam-5240	658	65	{	{	PUNCT
ejpam-5240	658	66	x4	x4	PROPN
ejpam-5240	658	67	,	,	PUNCT
ejpam-5240	658	68	x5	x5	PROPN
ejpam-5240	658	69	}	}	PUNCT
ejpam-5240	658	70	,	,	PUNCT
ejpam-5240	658	71	{	{	PUNCT
ejpam-5240	658	72	x4	x4	PROPN
ejpam-5240	658	73	,	,	PUNCT
ejpam-5240	658	74	x6	x6	PROPN
ejpam-5240	658	75	}	}	PUNCT
ejpam-5240	658	76	}	}	PUNCT
ejpam-5240	658	77	the	the	DET
ejpam-5240	658	78	vertices	vertex	NOUN
ejpam-5240	658	79	in	in	ADP
ejpam-5240	658	80	v	v	NOUN
ejpam-5240	658	81	(	(	PUNCT
ejpam-5240	658	82	gs(6,2	gs(6,2	PROPN
ejpam-5240	658	83	)	)	PUNCT
ejpam-5240	658	84	)	)	PUNCT
ejpam-5240	658	85	\t	\t	PUNCT
ejpam-5240	658	86	that	that	PRON
ejpam-5240	658	87	are	be	AUX
ejpam-5240	658	88	adjacent	adjacent	ADJ
ejpam-5240	658	89	to	to	ADP
ejpam-5240	658	90	{	{	PUNCT
ejpam-5240	658	91	x1	x1	PROPN
ejpam-5240	658	92	,	,	PUNCT
ejpam-5240	658	93	x2	x2	PRON
ejpam-5240	658	94	}	}	PUNCT
ejpam-5240	658	95	are	be	AUX
ejpam-5240	658	96	{	{	PUNCT
ejpam-5240	658	97	x1	x1	ADJ
ejpam-5240	658	98	,	,	PUNCT
ejpam-5240	658	99	x3	x3	ADJ
ejpam-5240	658	100	}	}	PUNCT
ejpam-5240	658	101	,	,	PUNCT
ejpam-5240	658	102	{	{	PUNCT
ejpam-5240	658	103	x1	x1	PROPN
ejpam-5240	658	104	,	,	PUNCT
ejpam-5240	658	105	x4	x4	PROPN
ejpam-5240	658	106	}	}	PUNCT
ejpam-5240	658	107	,	,	PUNCT
ejpam-5240	658	108	{	{	PUNCT
ejpam-5240	658	109	x1	x1	PROPN
ejpam-5240	658	110	,	,	PUNCT
ejpam-5240	658	111	x5	x5	PROPN
ejpam-5240	658	112	}	}	PUNCT
ejpam-5240	658	113	,	,	PUNCT
ejpam-5240	658	114	{	{	PUNCT
ejpam-5240	658	115	x1	x1	PROPN
ejpam-5240	658	116	,	,	PUNCT
ejpam-5240	658	117	x6	x6	PROPN
ejpam-5240	658	118	}	}	PUNCT
ejpam-5240	658	119	,	,	PUNCT
ejpam-5240	658	120	{	{	PUNCT
ejpam-5240	658	121	x2	x2	ADJ
ejpam-5240	658	122	,	,	PUNCT
ejpam-5240	658	123	x3	x3	ADJ
ejpam-5240	658	124	}	}	PUNCT
ejpam-5240	658	125	,	,	PUNCT
ejpam-5240	658	126	{	{	PUNCT
ejpam-5240	658	127	x2	x2	PROPN
ejpam-5240	658	128	,	,	PUNCT
ejpam-5240	658	129	x4	x4	PROPN
ejpam-5240	658	130	}	}	PUNCT
ejpam-5240	658	131	,	,	PUNCT
ejpam-5240	658	132	{	{	PUNCT
ejpam-5240	658	133	x2	x2	PROPN
ejpam-5240	658	134	,	,	PUNCT
ejpam-5240	658	135	x5	x5	PROPN
ejpam-5240	658	136	}	}	PUNCT
ejpam-5240	658	137	,	,	PUNCT
ejpam-5240	658	138	{	{	PUNCT
ejpam-5240	658	139	x2	x2	PROPN
ejpam-5240	658	140	,	,	PUNCT
ejpam-5240	658	141	x6	x6	PROPN
ejpam-5240	658	142	}	}	PUNCT
ejpam-5240	658	143	.	.	PUNCT
ejpam-5240	659	1	moreover	moreover	ADV
ejpam-5240	659	2	,	,	PUNCT
ejpam-5240	659	3	{	{	PUNCT
ejpam-5240	659	4	x3	x3	ADJ
ejpam-5240	659	5	,	,	PUNCT
ejpam-5240	659	6	x5	x5	PROPN
ejpam-5240	659	7	}	}	PUNCT
ejpam-5240	659	8	,	,	PUNCT
ejpam-5240	659	9	{	{	PUNCT
ejpam-5240	659	10	x3	x3	ADJ
ejpam-5240	659	11	,	,	PUNCT
ejpam-5240	659	12	x6	x6	PROPN
ejpam-5240	659	13	}	}	PUNCT
ejpam-5240	659	14	,	,	PUNCT
ejpam-5240	659	15	{	{	PUNCT
ejpam-5240	659	16	x4	x4	PROPN
ejpam-5240	659	17	,	,	PUNCT
ejpam-5240	659	18	x5	x5	PROPN
ejpam-5240	659	19	}	}	PUNCT
ejpam-5240	659	20	,	,	PUNCT
ejpam-5240	659	21	{	{	PUNCT
ejpam-5240	659	22	x4	x4	PROPN
ejpam-5240	659	23	,	,	PUNCT
ejpam-5240	659	24	x6	x6	PROPN
ejpam-5240	659	25	}	}	PUNCT
ejpam-5240	659	26	are	be	AUX
ejpam-5240	659	27	the	the	DET
ejpam-5240	659	28	vertices	vertex	NOUN
ejpam-5240	659	29	adjacent	adjacent	ADJ
ejpam-5240	659	30	to	to	AUX
ejpam-5240	659	31	{	{	PUNCT
ejpam-5240	659	32	x3	x3	VERB
ejpam-5240	659	33	,	,	PUNCT
ejpam-5240	659	34	x4	x4	PROPN
ejpam-5240	659	35	}	}	PUNCT
ejpam-5240	659	36	.	.	PUNCT
ejpam-5240	660	1	all	all	PRON
ejpam-5240	660	2	of	of	ADP
ejpam-5240	660	3	the	the	DET
ejpam-5240	660	4	elements	element	NOUN
ejpam-5240	660	5	in	in	ADP
ejpam-5240	660	6	v	v	NOUN
ejpam-5240	660	7	(	(	PUNCT
ejpam-5240	660	8	gs(6,2	gs(6,2	PROPN
ejpam-5240	660	9	)	)	PUNCT
ejpam-5240	660	10	)	)	PUNCT
ejpam-5240	660	11	\	\	PROPN
ejpam-5240	661	1	t	t	PROPN
ejpam-5240	661	2	are	be	AUX
ejpam-5240	661	3	adjacent	adjacent	ADJ
ejpam-5240	661	4	to	to	ADP
ejpam-5240	661	5	either	either	CCONJ
ejpam-5240	661	6	{	{	PUNCT
ejpam-5240	661	7	x1	x1	PROPN
ejpam-5240	661	8	,	,	PUNCT
ejpam-5240	661	9	x2	x2	PROPN
ejpam-5240	661	10	}	}	PUNCT
ejpam-5240	661	11	or	or	CCONJ
ejpam-5240	661	12	{	{	PUNCT
ejpam-5240	661	13	x3	x3	ADJ
ejpam-5240	661	14	,	,	PUNCT
ejpam-5240	661	15	x4	x4	PROPN
ejpam-5240	661	16	}	}	PUNCT
ejpam-5240	661	17	.	.	PUNCT
ejpam-5240	662	1	thus	thus	ADV
ejpam-5240	662	2	,	,	PUNCT
ejpam-5240	662	3	t	t	PROPN
ejpam-5240	662	4	is	be	AUX
ejpam-5240	662	5	a	a	DET
ejpam-5240	662	6	dominating	dominating	NOUN
ejpam-5240	662	7	set	set	NOUN
ejpam-5240	662	8	.	.	PUNCT
ejpam-5240	663	1	note	note	VERB
ejpam-5240	663	2	that	that	SCONJ
ejpam-5240	663	3	|t	|t	VERB
ejpam-5240	664	1	|	|	ADV
ejpam-5240	664	2	=	=	SYM
ejpam-5240	664	3	3	3	X
ejpam-5240	664	4	=	=	SYM
ejpam-5240	664	5	⌊	⌊	PROPN
ejpam-5240	664	6	6	6	NUM
ejpam-5240	664	7	2	2	NUM
ejpam-5240	664	8	⌋	⌋	NOUN
ejpam-5240	664	9	.	.	PUNCT
ejpam-5240	665	1	hence	hence	ADV
ejpam-5240	665	2	,	,	PUNCT
ejpam-5240	665	3	there	there	PRON
ejpam-5240	665	4	exists	exist	VERB
ejpam-5240	665	5	a	a	DET
ejpam-5240	665	6	dominating	dominating	NOUN
ejpam-5240	665	7	set	set	VERB
ejpam-5240	665	8	in	in	ADP
ejpam-5240	665	9	gs(6,2	gs(6,2	PROPN
ejpam-5240	665	10	)	)	PUNCT
ejpam-5240	665	11	with	with	ADP
ejpam-5240	665	12	a	a	DET
ejpam-5240	665	13	cardinality	cardinality	NOUN
ejpam-5240	665	14	of	of	ADP
ejpam-5240	665	15	⌊	⌊	PROPN
ejpam-5240	665	16	n	n	PROPN
ejpam-5240	665	17	k	k	NOUN
ejpam-5240	665	18	⌋	⌋	NOUN
ejpam-5240	666	1	=	=	PUNCT
ejpam-5240	666	2	⌊	⌊	VERB
ejpam-5240	666	3	6	6	NUM
ejpam-5240	666	4	2	2	NUM
ejpam-5240	666	5	⌋	⌋	NOUN
ejpam-5240	666	6	.	.	PUNCT
ejpam-5240	667	1	by	by	ADP
ejpam-5240	667	2	lemma	lemma	PROPN
ejpam-5240	667	3	4	4	NUM
ejpam-5240	667	4	,	,	PUNCT
ejpam-5240	667	5	it	it	PRON
ejpam-5240	667	6	can	can	AUX
ejpam-5240	667	7	be	be	AUX
ejpam-5240	667	8	observed	observe	VERB
ejpam-5240	667	9	that	that	SCONJ
ejpam-5240	667	10	there	there	PRON
ejpam-5240	667	11	exists	exist	VERB
ejpam-5240	667	12	a	a	DET
ejpam-5240	667	13	dominating	dominating	NOUN
ejpam-5240	667	14	set	set	VERB
ejpam-5240	667	15	in	in	ADP
ejpam-5240	667	16	gs(n	gs(n	NOUN
ejpam-5240	667	17	,	,	PUNCT
ejpam-5240	667	18	k	k	NOUN
ejpam-5240	667	19	)	)	PUNCT
ejpam-5240	667	20	.	.	PUNCT
ejpam-5240	668	1	hence	hence	ADV
ejpam-5240	668	2	,	,	PUNCT
ejpam-5240	668	3	we	we	PRON
ejpam-5240	668	4	can	can	AUX
ejpam-5240	668	5	now	now	ADV
ejpam-5240	668	6	compute	compute	VERB
ejpam-5240	668	7	for	for	ADP
ejpam-5240	668	8	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	668	9	,	,	PUNCT
ejpam-5240	668	10	k	k	NOUN
ejpam-5240	668	11	)	)	PUNCT
ejpam-5240	668	12	)	)	PUNCT
ejpam-5240	669	1	where	where	SCONJ
ejpam-5240	669	2	1	1	NUM
ejpam-5240	669	3	≤	≤	NUM
ejpam-5240	669	4	k	k	X
ejpam-5240	669	5	≤	≤	NUM
ejpam-5240	669	6	⌊	⌊	VERB
ejpam-5240	669	7	n	n	DET
ejpam-5240	669	8	2	2	NUM
ejpam-5240	669	9	⌋	⌋	NOUN
ejpam-5240	669	10	.	.	PUNCT
ejpam-5240	670	1	m.e	m.e	PROPN
ejpam-5240	670	2	.	.	PROPN
ejpam-5240	670	3	pelagio	pelagio	PROPN
ejpam-5240	670	4	,	,	PUNCT
ejpam-5240	670	5	n.	n.	NOUN
ejpam-5240	670	6	mame	mame	PROPN
ejpam-5240	670	7	,	,	PUNCT
ejpam-5240	670	8	k.	k.	PROPN
ejpam-5240	670	9	mendoza	mendoza	PROPN
ejpam-5240	670	10	/	/	SYM
ejpam-5240	670	11	eur	eur	PROPN
ejpam-5240	670	12	.	.	PUNCT
ejpam-5240	671	1	j.	j.	PROPN
ejpam-5240	671	2	pure	pure	PROPN
ejpam-5240	671	3	appl	appl	PROPN
ejpam-5240	671	4	.	.	PROPN
ejpam-5240	671	5	math	math	PROPN
ejpam-5240	671	6	,	,	PUNCT
ejpam-5240	671	7	17	17	NUM
ejpam-5240	671	8	(	(	PUNCT
ejpam-5240	671	9	3	3	NUM
ejpam-5240	671	10	)	)	PUNCT
ejpam-5240	671	11	(	(	PUNCT
ejpam-5240	671	12	2024	2024	NUM
ejpam-5240	671	13	)	)	PUNCT
ejpam-5240	671	14	,	,	PUNCT
ejpam-5240	671	15	1779	1779	NUM
ejpam-5240	671	16	-	-	SYM
ejpam-5240	671	17	1803	1803	NUM
ejpam-5240	671	18	1800	1800	NUM
ejpam-5240	671	19	theorem	theorem	NOUN
ejpam-5240	671	20	12	12	NUM
ejpam-5240	671	21	.	.	PUNCT
ejpam-5240	672	1	let	let	VERB
ejpam-5240	672	2	gs(n	gs(n	NOUN
ejpam-5240	672	3	,	,	PUNCT
ejpam-5240	672	4	k	k	NOUN
ejpam-5240	672	5	)	)	PUNCT
ejpam-5240	672	6	be	be	VERB
ejpam-5240	672	7	a	a	DET
ejpam-5240	672	8	k	k	ADV
ejpam-5240	672	9	-	-	ADJ
ejpam-5240	672	10	restricted	restricted	ADJ
ejpam-5240	672	11	intersection	intersection	NOUN
ejpam-5240	672	12	graph	graph	NOUN
ejpam-5240	672	13	.	.	PUNCT
ejpam-5240	673	1	if	if	SCONJ
ejpam-5240	673	2	1	1	NUM
ejpam-5240	673	3	≤	≤	NUM
ejpam-5240	673	4	k	k	X
ejpam-5240	673	5	≤	≤	NUM
ejpam-5240	673	6	⌊	⌊	VERB
ejpam-5240	673	7	n	n	DET
ejpam-5240	673	8	2	2	NUM
ejpam-5240	673	9	⌋	⌋	NOUN
ejpam-5240	673	10	,	,	PUNCT
ejpam-5240	673	11	then	then	ADV
ejpam-5240	673	12	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	673	13	,	,	PUNCT
ejpam-5240	673	14	k	k	NOUN
ejpam-5240	673	15	)	)	PUNCT
ejpam-5240	673	16	)	)	PUNCT
ejpam-5240	674	1	=	=	PUNCT
ejpam-5240	674	2	⌊	⌊	VERB
ejpam-5240	674	3	n	n	PRON
ejpam-5240	674	4	k	k	NOUN
ejpam-5240	674	5	⌋	⌋	NOUN
ejpam-5240	674	6	.	.	PUNCT
ejpam-5240	675	1	proof	proof	NOUN
ejpam-5240	675	2	.	.	PUNCT
ejpam-5240	676	1	let	let	VERB
ejpam-5240	676	2	t	t	NOUN
ejpam-5240	676	3	=	=	PRON
ejpam-5240	676	4	{	{	PUNCT
ejpam-5240	676	5	{	{	PUNCT
ejpam-5240	676	6	x1	x1	PROPN
ejpam-5240	676	7	,	,	PUNCT
ejpam-5240	676	8	x2	x2	PROPN
ejpam-5240	676	9	,	,	PUNCT
ejpam-5240	676	10	...	...	PUNCT
ejpam-5240	676	11	,	,	PUNCT
ejpam-5240	676	12	xk	xk	PROPN
ejpam-5240	676	13	}	}	PUNCT
ejpam-5240	676	14	,	,	PUNCT
ejpam-5240	676	15	{	{	PUNCT
ejpam-5240	676	16	xk+1	xk+1	PROPN
ejpam-5240	676	17	,	,	PUNCT
ejpam-5240	676	18	...	...	PUNCT
ejpam-5240	676	19	,	,	PUNCT
ejpam-5240	676	20	x2k	x2k	NOUN
ejpam-5240	676	21	}	}	PUNCT
ejpam-5240	676	22	,	,	PUNCT
ejpam-5240	676	23	...	...	PUNCT
ejpam-5240	676	24	,	,	PUNCT
ejpam-5240	676	25	{	{	PUNCT
ejpam-5240	676	26	x(⌊n	x(⌊n	PROPN
ejpam-5240	676	27	k	k	PROPN
ejpam-5240	676	28	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	676	29	,	,	PUNCT
ejpam-5240	676	30	...	...	PUNCT
ejpam-5240	676	31	,	,	PUNCT
ejpam-5240	676	32	x(⌊n	x(⌊n	PROPN
ejpam-5240	677	1	k	k	PROPN
ejpam-5240	677	2	⌋)k	⌋)k	PROPN
ejpam-5240	677	3	}	}	PUNCT
ejpam-5240	677	4	}	}	PUNCT
ejpam-5240	677	5	.	.	PUNCT
ejpam-5240	678	1	by	by	ADP
ejpam-5240	678	2	lemma	lemma	PROPN
ejpam-5240	678	3	4	4	NUM
ejpam-5240	678	4	,	,	PUNCT
ejpam-5240	678	5	t	t	PROPN
ejpam-5240	678	6	is	be	AUX
ejpam-5240	678	7	a	a	DET
ejpam-5240	678	8	dominating	dominating	NOUN
ejpam-5240	678	9	set	set	VERB
ejpam-5240	678	10	ings(n	ings(n	PROPN
ejpam-5240	678	11	,	,	PUNCT
ejpam-5240	678	12	k	k	NOUN
ejpam-5240	678	13	)	)	PUNCT
ejpam-5240	678	14	where	where	SCONJ
ejpam-5240	678	15	|t	|t	VERB
ejpam-5240	679	1	|	|	ADV
ejpam-5240	679	2	=	=	SYM
ejpam-5240	679	3	⌊	⌊	PROPN
ejpam-5240	679	4	n	n	PRON
ejpam-5240	679	5	k	k	NOUN
ejpam-5240	679	6	⌋	⌋	NOUN
ejpam-5240	679	7	.	.	PUNCT
ejpam-5240	680	1	since	since	SCONJ
ejpam-5240	680	2	there	there	PRON
ejpam-5240	680	3	exists	exist	VERB
ejpam-5240	680	4	a	a	DET
ejpam-5240	680	5	dominating	dominating	NOUN
ejpam-5240	680	6	set	set	VERB
ejpam-5240	680	7	in	in	ADP
ejpam-5240	680	8	gs(n	gs(n	NOUN
ejpam-5240	680	9	,	,	PUNCT
ejpam-5240	680	10	k	k	NOUN
ejpam-5240	680	11	)	)	PUNCT
ejpam-5240	680	12	with	with	ADP
ejpam-5240	680	13	a	a	DET
ejpam-5240	680	14	cardinality	cardinality	NOUN
ejpam-5240	680	15	equal	equal	ADJ
ejpam-5240	680	16	to	to	ADP
ejpam-5240	680	17	⌊	⌊	PROPN
ejpam-5240	680	18	n	n	CCONJ
ejpam-5240	680	19	k	k	NOUN
ejpam-5240	680	20	⌋	⌋	NOUN
ejpam-5240	680	21	,	,	PUNCT
ejpam-5240	680	22	it	it	PRON
ejpam-5240	680	23	follows	follow	VERB
ejpam-5240	680	24	that	that	SCONJ
ejpam-5240	680	25	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	680	26	,	,	PUNCT
ejpam-5240	680	27	k	k	NOUN
ejpam-5240	680	28	)	)	PUNCT
ejpam-5240	680	29	)	)	PUNCT
ejpam-5240	680	30	≤	≤	PUNCT
ejpam-5240	681	1	⌊	⌊	VERB
ejpam-5240	681	2	n	n	PRON
ejpam-5240	681	3	k	k	NOUN
ejpam-5240	681	4	⌋	⌋	NOUN
ejpam-5240	681	5	.	.	PUNCT
ejpam-5240	682	1	we	we	PRON
ejpam-5240	682	2	claim	claim	VERB
ejpam-5240	682	3	that	that	SCONJ
ejpam-5240	682	4	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	682	5	,	,	PUNCT
ejpam-5240	682	6	k	k	NOUN
ejpam-5240	682	7	)	)	PUNCT
ejpam-5240	682	8	)	)	PUNCT
ejpam-5240	683	1	=	=	PUNCT
ejpam-5240	683	2	⌊	⌊	VERB
ejpam-5240	683	3	n	n	PRON
ejpam-5240	683	4	k	k	PROPN
ejpam-5240	683	5	⌋	⌋	NOUN
ejpam-5240	683	6	.	.	PUNCT
ejpam-5240	684	1	now	now	ADV
ejpam-5240	684	2	,	,	PUNCT
ejpam-5240	684	3	suppose	suppose	VERB
ejpam-5240	684	4	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	684	5	,	,	PUNCT
ejpam-5240	684	6	k	k	NOUN
ejpam-5240	684	7	)	)	PUNCT
ejpam-5240	684	8	)	)	PUNCT
ejpam-5240	685	1	<	<	X
ejpam-5240	685	2	⌊	⌊	PROPN
ejpam-5240	685	3	n	n	X
ejpam-5240	685	4	k	k	PROPN
ejpam-5240	685	5	⌋	⌋	NOUN
ejpam-5240	685	6	.	.	PUNCT
ejpam-5240	686	1	thus	thus	ADV
ejpam-5240	686	2	,	,	PUNCT
ejpam-5240	686	3	there	there	PRON
ejpam-5240	686	4	exists	exist	VERB
ejpam-5240	686	5	w	w	PROPN
ejpam-5240	686	6	⊆	⊆	NUM
ejpam-5240	686	7	v	v	NOUN
ejpam-5240	686	8	(	(	PUNCT
ejpam-5240	686	9	gs(n	gs(n	NOUN
ejpam-5240	686	10	,	,	PUNCT
ejpam-5240	686	11	k	k	NOUN
ejpam-5240	686	12	)	)	PUNCT
ejpam-5240	686	13	)	)	PUNCT
ejpam-5240	686	14	such	such	ADJ
ejpam-5240	686	15	that	that	SCONJ
ejpam-5240	686	16	w	w	NOUN
ejpam-5240	686	17	is	be	AUX
ejpam-5240	686	18	a	a	DET
ejpam-5240	686	19	dominating	dominating	NOUN
ejpam-5240	686	20	set	set	VERB
ejpam-5240	686	21	in	in	ADP
ejpam-5240	686	22	gs(n	gs(n	NOUN
ejpam-5240	686	23	,	,	PUNCT
ejpam-5240	686	24	k	k	NOUN
ejpam-5240	686	25	)	)	PUNCT
ejpam-5240	686	26	and	and	CCONJ
ejpam-5240	686	27	|w	|w	NOUN
ejpam-5240	687	1	|	|	ADV
ejpam-5240	687	2	<	<	X
ejpam-5240	687	3	⌊	⌊	PROPN
ejpam-5240	687	4	n	n	X
ejpam-5240	687	5	k	k	NOUN
ejpam-5240	687	6	⌋	⌋	PROPN
ejpam-5240	687	7	.	.	PUNCT
ejpam-5240	688	1	without	without	ADP
ejpam-5240	688	2	loss	loss	NOUN
ejpam-5240	688	3	of	of	ADP
ejpam-5240	688	4	generality	generality	NOUN
ejpam-5240	688	5	,	,	PUNCT
ejpam-5240	688	6	assume	assume	VERB
ejpam-5240	688	7	that	that	SCONJ
ejpam-5240	688	8	n	n	PRON
ejpam-5240	688	9	is	be	AUX
ejpam-5240	688	10	not	not	PART
ejpam-5240	688	11	divisible	divisible	ADJ
ejpam-5240	688	12	by	by	ADP
ejpam-5240	688	13	k.	k.	PROPN
ejpam-5240	689	1	so	so	ADV
ejpam-5240	689	2	by	by	ADP
ejpam-5240	689	3	the	the	DET
ejpam-5240	689	4	division	division	NOUN
ejpam-5240	689	5	algorithm	algorithm	NOUN
ejpam-5240	689	6	,	,	PUNCT
ejpam-5240	689	7	there	there	PRON
ejpam-5240	689	8	exist	exist	VERB
ejpam-5240	689	9	unique	unique	ADJ
ejpam-5240	689	10	integers	integer	NOUN
ejpam-5240	689	11	a	a	DET
ejpam-5240	689	12	and	and	CCONJ
ejpam-5240	689	13	b	b	NOUN
ejpam-5240	689	14	where	where	SCONJ
ejpam-5240	689	15	n	n	NOUN
ejpam-5240	689	16	=	=	PROPN
ejpam-5240	689	17	ka	ka	PROPN
ejpam-5240	690	1	+	+	PROPN
ejpam-5240	690	2	b.	b.	PROPN
ejpam-5240	691	1	but	but	CCONJ
ejpam-5240	691	2	note	note	VERB
ejpam-5240	691	3	that	that	SCONJ
ejpam-5240	691	4	|w	|w	NOUN
ejpam-5240	692	1	|	|	ADV
ejpam-5240	692	2	<	<	X
ejpam-5240	692	3	|t	|t	VERB
ejpam-5240	692	4	|	|	INTJ
ejpam-5240	692	5	.	.	PUNCT
ejpam-5240	693	1	hence	hence	ADV
ejpam-5240	693	2	,	,	PUNCT
ejpam-5240	693	3	k	k	PROPN
ejpam-5240	693	4	≤	≤	PROPN
ejpam-5240	693	5	b	b	X
ejpam-5240	693	6	to	to	PART
ejpam-5240	693	7	make	make	VERB
ejpam-5240	693	8	|w	|w	ADJ
ejpam-5240	693	9	|	|	ADV
ejpam-5240	693	10	<	<	X
ejpam-5240	693	11	|t	|t	VERB
ejpam-5240	693	12	|	|	ADV
ejpam-5240	693	13	.	.	PUNCT
ejpam-5240	694	1	this	this	PRON
ejpam-5240	694	2	is	be	AUX
ejpam-5240	694	3	a	a	DET
ejpam-5240	694	4	contradiction	contradiction	NOUN
ejpam-5240	694	5	to	to	ADP
ejpam-5240	694	6	the	the	DET
ejpam-5240	694	7	fact	fact	NOUN
ejpam-5240	694	8	that	that	SCONJ
ejpam-5240	694	9	0	0	NUM
ejpam-5240	694	10	<	<	X
ejpam-5240	694	11	b	b	X
ejpam-5240	694	12	<	<	X
ejpam-5240	694	13	k.	k.	PROPN
ejpam-5240	694	14	therefore	therefore	ADV
ejpam-5240	694	15	,	,	PUNCT
ejpam-5240	694	16	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	694	17	,	,	PUNCT
ejpam-5240	694	18	k	k	NOUN
ejpam-5240	694	19	)	)	PUNCT
ejpam-5240	694	20	)	)	PUNCT
ejpam-5240	695	1	=	=	PUNCT
ejpam-5240	696	1	⌊	⌊	VERB
ejpam-5240	696	2	n	n	PRON
ejpam-5240	696	3	k	k	PROPN
ejpam-5240	696	4	⌋	⌋	PROPN
ejpam-5240	696	5	.	.	PUNCT
ejpam-5240	697	1	illustration	illustration	NOUN
ejpam-5240	697	2	14	14	NUM
ejpam-5240	697	3	.	.	PUNCT
ejpam-5240	698	1	let	let	VERB
ejpam-5240	698	2	s6	s6	PROPN
ejpam-5240	698	3	=	=	SYM
ejpam-5240	698	4	{	{	PUNCT
ejpam-5240	698	5	x1	x1	PROPN
ejpam-5240	698	6	,	,	PUNCT
ejpam-5240	698	7	x2	x2	PROPN
ejpam-5240	698	8	,	,	PUNCT
ejpam-5240	698	9	x3	x3	PROPN
ejpam-5240	698	10	,	,	PUNCT
ejpam-5240	698	11	x4	x4	PROPN
ejpam-5240	698	12	,	,	PUNCT
ejpam-5240	698	13	x5	x5	PROPN
ejpam-5240	698	14	,	,	PUNCT
ejpam-5240	698	15	x6	x6	PROPN
ejpam-5240	698	16	}	}	PUNCT
ejpam-5240	698	17	and	and	CCONJ
ejpam-5240	698	18	k	k	X
ejpam-5240	698	19	=	=	NOUN
ejpam-5240	698	20	2	2	X
ejpam-5240	698	21	.	.	PUNCT
ejpam-5240	699	1	in	in	ADP
ejpam-5240	699	2	illustration	illustration	NOUN
ejpam-5240	699	3	13	13	NUM
ejpam-5240	699	4	,	,	PUNCT
ejpam-5240	699	5	we	we	PRON
ejpam-5240	699	6	have	have	AUX
ejpam-5240	699	7	identified	identify	VERB
ejpam-5240	699	8	that	that	DET
ejpam-5240	699	9	t	t	NOUN
ejpam-5240	699	10	=	=	PRON
ejpam-5240	699	11	{	{	PUNCT
ejpam-5240	699	12	{	{	PUNCT
ejpam-5240	699	13	x1	x1	PROPN
ejpam-5240	699	14	,	,	PUNCT
ejpam-5240	699	15	x2	x2	PROPN
ejpam-5240	699	16	}	}	PUNCT
ejpam-5240	699	17	,	,	PUNCT
ejpam-5240	699	18	{	{	PUNCT
ejpam-5240	699	19	x3	x3	ADJ
ejpam-5240	699	20	,	,	PUNCT
ejpam-5240	699	21	x4	x4	PROPN
ejpam-5240	699	22	}	}	PUNCT
ejpam-5240	699	23	,	,	PUNCT
ejpam-5240	699	24	{	{	PUNCT
ejpam-5240	699	25	x5	x5	PROPN
ejpam-5240	699	26	,	,	PUNCT
ejpam-5240	699	27	x6	x6	PROPN
ejpam-5240	699	28	}	}	PUNCT
ejpam-5240	699	29	}	}	PUNCT
ejpam-5240	699	30	is	be	AUX
ejpam-5240	699	31	a	a	DET
ejpam-5240	699	32	dominating	dominating	NOUN
ejpam-5240	699	33	set	set	NOUN
ejpam-5240	699	34	.	.	PUNCT
ejpam-5240	700	1	we	we	PRON
ejpam-5240	700	2	can	can	AUX
ejpam-5240	700	3	not	not	PART
ejpam-5240	700	4	remove	remove	VERB
ejpam-5240	700	5	{	{	PUNCT
ejpam-5240	700	6	x5	x5	PROPN
ejpam-5240	700	7	,	,	PUNCT
ejpam-5240	700	8	x6	x6	PROPN
ejpam-5240	700	9	}	}	PUNCT
ejpam-5240	700	10	from	from	ADP
ejpam-5240	700	11	t	t	PROPN
ejpam-5240	700	12	since	since	SCONJ
ejpam-5240	700	13	{	{	PUNCT
ejpam-5240	700	14	x1	x1	PROPN
ejpam-5240	700	15	,	,	PUNCT
ejpam-5240	700	16	x2}∩{x5	x2}∩{x5	PROPN
ejpam-5240	700	17	,	,	PUNCT
ejpam-5240	700	18	x6	x6	PROPN
ejpam-5240	700	19	}	}	PUNCT
ejpam-5240	700	20	=	=	SYM
ejpam-5240	700	21	∅	∅	NOUN
ejpam-5240	700	22	and	and	CCONJ
ejpam-5240	700	23	{	{	PUNCT
ejpam-5240	700	24	x3	x3	ADJ
ejpam-5240	700	25	,	,	PUNCT
ejpam-5240	700	26	x4}∩{x5	x4}∩{x5	NUM
ejpam-5240	700	27	,	,	PUNCT
ejpam-5240	700	28	x6	x6	PROPN
ejpam-5240	700	29	}	}	PUNCT
ejpam-5240	700	30	=	=	SYM
ejpam-5240	700	31	∅	∅	NOUN
ejpam-5240	700	32	which	which	PRON
ejpam-5240	700	33	implies	imply	VERB
ejpam-5240	700	34	that	that	SCONJ
ejpam-5240	700	35	they	they	PRON
ejpam-5240	700	36	are	be	AUX
ejpam-5240	700	37	not	not	PART
ejpam-5240	700	38	adjacent	adjacent	ADJ
ejpam-5240	700	39	to	to	ADP
ejpam-5240	700	40	each	each	DET
ejpam-5240	700	41	other	other	ADJ
ejpam-5240	700	42	if	if	SCONJ
ejpam-5240	700	43	{	{	PUNCT
ejpam-5240	700	44	x5	x5	PROPN
ejpam-5240	700	45	,	,	PUNCT
ejpam-5240	700	46	x6	x6	PROPN
ejpam-5240	700	47	}	}	PUNCT
ejpam-5240	700	48	becomes	become	VERB
ejpam-5240	700	49	an	an	DET
ejpam-5240	700	50	element	element	NOUN
ejpam-5240	700	51	of	of	ADP
ejpam-5240	700	52	v	v	NOUN
ejpam-5240	700	53	(	(	PUNCT
ejpam-5240	700	54	gs(6,2	gs(6,2	PROPN
ejpam-5240	700	55	)	)	PUNCT
ejpam-5240	700	56	)	)	PUNCT
ejpam-5240	700	57	\t	\t	PROPN
ejpam-5240	700	58	.	.	PUNCT
ejpam-5240	701	1	thus	thus	ADV
ejpam-5240	701	2	,	,	PUNCT
ejpam-5240	701	3	there	there	PRON
ejpam-5240	701	4	are	be	VERB
ejpam-5240	701	5	no	no	DET
ejpam-5240	701	6	dominating	dominating	NOUN
ejpam-5240	701	7	sets	set	NOUN
ejpam-5240	701	8	in	in	ADP
ejpam-5240	701	9	gs(6,2	gs(6,2	PROPN
ejpam-5240	701	10	)	)	PUNCT
ejpam-5240	701	11	with	with	ADP
ejpam-5240	701	12	cardinality	cardinality	NOUN
ejpam-5240	701	13	less	less	ADJ
ejpam-5240	701	14	than	than	ADP
ejpam-5240	701	15	3	3	NUM
ejpam-5240	701	16	.	.	PUNCT
ejpam-5240	701	17	hence	hence	ADV
ejpam-5240	701	18	,	,	PUNCT
ejpam-5240	701	19	γ(gs(6,2	γ(gs(6,2	NOUN
ejpam-5240	701	20	)	)	PUNCT
ejpam-5240	701	21	)	)	PUNCT
ejpam-5240	702	1	=	=	SYM
ejpam-5240	702	2	3	3	X
ejpam-5240	702	3	.	.	PUNCT
ejpam-5240	702	4	now	now	ADV
ejpam-5240	702	5	,	,	PUNCT
ejpam-5240	702	6	by	by	ADP
ejpam-5240	702	7	using	use	VERB
ejpam-5240	702	8	theorem	theorem	NOUN
ejpam-5240	702	9	12	12	NUM
ejpam-5240	702	10	,	,	PUNCT
ejpam-5240	702	11	setting	set	VERB
ejpam-5240	702	12	n	n	X
ejpam-5240	702	13	=	=	SYM
ejpam-5240	702	14	6	6	NUM
ejpam-5240	702	15	and	and	CCONJ
ejpam-5240	702	16	k	k	NOUN
ejpam-5240	702	17	=	=	SYM
ejpam-5240	702	18	2	2	NUM
ejpam-5240	702	19	,	,	PUNCT
ejpam-5240	702	20	we	we	PRON
ejpam-5240	702	21	have	have	VERB
ejpam-5240	702	22	:	:	PUNCT
ejpam-5240	702	23	γ(gs(6,2	γ(gs(6,2	ADJ
ejpam-5240	702	24	)	)	PUNCT
ejpam-5240	702	25	)	)	PUNCT
ejpam-5240	703	1	=	=	SYM
ejpam-5240	703	2	⌊n	⌊n	X
ejpam-5240	704	1	k	k	X
ejpam-5240	704	2	⌋	⌋	NOUN
ejpam-5240	705	1	=	=	PUNCT
ejpam-5240	705	2	⌊	⌊	VERB
ejpam-5240	705	3	6	6	NUM
ejpam-5240	705	4	2	2	NUM
ejpam-5240	705	5	⌋	⌋	NOUN
ejpam-5240	705	6	=	=	PUNCT
ejpam-5240	705	7	⌊3⌋	⌊3⌋	NOUN
ejpam-5240	705	8	=	=	SYM
ejpam-5240	705	9	3	3	X
ejpam-5240	705	10	.	.	PUNCT
ejpam-5240	705	11	by	by	ADP
ejpam-5240	705	12	theorem	theorem	ADJ
ejpam-5240	705	13	4	4	NUM
ejpam-5240	705	14	,	,	PUNCT
ejpam-5240	705	15	gs(n,1	gs(n,1	NOUN
ejpam-5240	705	16	)	)	PUNCT
ejpam-5240	705	17	is	be	AUX
ejpam-5240	705	18	an	an	DET
ejpam-5240	705	19	empty	empty	ADJ
ejpam-5240	705	20	graph	graph	NOUN
ejpam-5240	705	21	of	of	ADP
ejpam-5240	705	22	order	order	NOUN
ejpam-5240	705	23	n.	n.	NOUN
ejpam-5240	705	24	since	since	SCONJ
ejpam-5240	705	25	every	every	DET
ejpam-5240	705	26	vertex	vertex	NOUN
ejpam-5240	705	27	of	of	ADP
ejpam-5240	705	28	an	an	DET
ejpam-5240	705	29	empty	empty	ADJ
ejpam-5240	705	30	graph	graph	NOUN
ejpam-5240	705	31	is	be	AUX
ejpam-5240	705	32	isolated	isolate	VERB
ejpam-5240	705	33	,	,	PUNCT
ejpam-5240	705	34	or	or	CCONJ
ejpam-5240	705	35	has	have	VERB
ejpam-5240	705	36	no	no	DET
ejpam-5240	705	37	adjacency	adjacency	NOUN
ejpam-5240	705	38	to	to	ADP
ejpam-5240	705	39	every	every	DET
ejpam-5240	705	40	other	other	ADJ
ejpam-5240	705	41	vertex	vertex	NOUN
ejpam-5240	705	42	,	,	PUNCT
ejpam-5240	705	43	it	it	PRON
ejpam-5240	705	44	follows	follow	VERB
ejpam-5240	705	45	that	that	SCONJ
ejpam-5240	705	46	v	v	X
ejpam-5240	705	47	(	(	PUNCT
ejpam-5240	705	48	gs(n,1	gs(n,1	NOUN
ejpam-5240	705	49	)	)	PUNCT
ejpam-5240	705	50	)	)	PUNCT
ejpam-5240	705	51	is	be	AUX
ejpam-5240	705	52	a	a	DET
ejpam-5240	705	53	dominating	dominating	NOUN
ejpam-5240	705	54	set	set	VERB
ejpam-5240	705	55	in	in	ADP
ejpam-5240	705	56	gs(n,1	gs(n,1	NOUN
ejpam-5240	705	57	)	)	PUNCT
ejpam-5240	705	58	.	.	PUNCT
ejpam-5240	706	1	it	it	PRON
ejpam-5240	706	2	can	can	AUX
ejpam-5240	706	3	be	be	AUX
ejpam-5240	706	4	verified	verify	VERB
ejpam-5240	706	5	that	that	SCONJ
ejpam-5240	706	6	there	there	PRON
ejpam-5240	706	7	are	be	VERB
ejpam-5240	706	8	no	no	DET
ejpam-5240	706	9	existing	exist	VERB
ejpam-5240	706	10	dominating	dominating	NOUN
ejpam-5240	706	11	set	set	VERB
ejpam-5240	706	12	in	in	ADP
ejpam-5240	706	13	gs(n,1	gs(n,1	NOUN
ejpam-5240	706	14	)	)	PUNCT
ejpam-5240	706	15	with	with	ADP
ejpam-5240	706	16	cardinality	cardinality	NOUN
ejpam-5240	706	17	less	less	ADJ
ejpam-5240	706	18	than	than	ADP
ejpam-5240	706	19	n	n	CCONJ
ejpam-5240	706	20	,	,	PUNCT
ejpam-5240	706	21	therefore	therefore	ADV
ejpam-5240	706	22	,	,	PUNCT
ejpam-5240	706	23	γ(gs(n,1	γ(gs(n,1	PROPN
ejpam-5240	706	24	)	)	PUNCT
ejpam-5240	706	25	)	)	PUNCT
ejpam-5240	707	1	=	=	PUNCT
ejpam-5240	707	2	n.	n.	NOUN
ejpam-5240	707	3	to	to	PART
ejpam-5240	707	4	verify	verify	VERB
ejpam-5240	707	5	using	use	VERB
ejpam-5240	707	6	theorem	theorem	NOUN
ejpam-5240	707	7	12	12	NUM
ejpam-5240	707	8	with	with	ADP
ejpam-5240	707	9	k	k	PROPN
ejpam-5240	707	10	=	=	SYM
ejpam-5240	707	11	1	1	NUM
ejpam-5240	707	12	,	,	PUNCT
ejpam-5240	707	13	then	then	ADV
ejpam-5240	707	14	we	we	PRON
ejpam-5240	707	15	have	have	VERB
ejpam-5240	707	16	γ(gs(n,1	γ(gs(n,1	PROPN
ejpam-5240	707	17	)	)	PUNCT
ejpam-5240	707	18	)	)	PUNCT
ejpam-5240	708	1	=	=	PUNCT
ejpam-5240	708	2	⌊	⌊	VERB
ejpam-5240	708	3	n	n	ADV
ejpam-5240	708	4	1	1	NUM
ejpam-5240	708	5	⌋	⌋	NOUN
ejpam-5240	708	6	=	=	PUNCT
ejpam-5240	708	7	n.	n.	PROPN
ejpam-5240	708	8	equivalently	equivalently	ADV
ejpam-5240	708	9	,	,	PUNCT
ejpam-5240	708	10	we	we	PRON
ejpam-5240	708	11	have	have	VERB
ejpam-5240	708	12	remark	remark	NOUN
ejpam-5240	708	13	10	10	NUM
ejpam-5240	708	14	.	.	PUNCT
ejpam-5240	709	1	remark	remark	PROPN
ejpam-5240	709	2	10	10	NUM
ejpam-5240	709	3	.	.	PUNCT
ejpam-5240	710	1	let	let	VERB
ejpam-5240	710	2	gs(n	gs(n	NOUN
ejpam-5240	710	3	,	,	PUNCT
ejpam-5240	710	4	k	k	NOUN
ejpam-5240	710	5	)	)	PUNCT
ejpam-5240	710	6	be	be	VERB
ejpam-5240	710	7	a	a	DET
ejpam-5240	710	8	k	k	ADV
ejpam-5240	710	9	-	-	ADJ
ejpam-5240	710	10	restricted	restricted	ADJ
ejpam-5240	710	11	intersection	intersection	NOUN
ejpam-5240	710	12	graph	graph	NOUN
ejpam-5240	710	13	.	.	PUNCT
ejpam-5240	711	1	if	if	SCONJ
ejpam-5240	711	2	k	k	PROPN
ejpam-5240	711	3	=	=	SYM
ejpam-5240	711	4	1	1	NUM
ejpam-5240	711	5	,	,	PUNCT
ejpam-5240	711	6	then	then	ADV
ejpam-5240	711	7	γ(gs(n,1	γ(gs(n,1	PROPN
ejpam-5240	711	8	)	)	PUNCT
ejpam-5240	711	9	)	)	PUNCT
ejpam-5240	711	10	is	be	AUX
ejpam-5240	711	11	equal	equal	ADJ
ejpam-5240	711	12	to	to	PART
ejpam-5240	711	13	n.	n.	VERB
ejpam-5240	711	14	moreover	moreover	ADV
ejpam-5240	711	15	,	,	PUNCT
ejpam-5240	711	16	theorem	theorem	VERB
ejpam-5240	711	17	13	13	NUM
ejpam-5240	711	18	determines	determine	VERB
ejpam-5240	711	19	the	the	DET
ejpam-5240	711	20	domination	domination	NOUN
ejpam-5240	711	21	number	number	NOUN
ejpam-5240	711	22	of	of	ADP
ejpam-5240	711	23	gs(n	gs(n	NOUN
ejpam-5240	711	24	,	,	PUNCT
ejpam-5240	711	25	k	k	NOUN
ejpam-5240	711	26	)	)	PUNCT
ejpam-5240	712	1	whenever	whenever	SCONJ
ejpam-5240	712	2	k	k	PROPN
ejpam-5240	712	3	=	=	PUNCT
ejpam-5240	712	4	0	0	NUM
ejpam-5240	712	5	or	or	CCONJ
ejpam-5240	712	6	⌊	⌊	X
ejpam-5240	712	7	n	n	ADV
ejpam-5240	712	8	2	2	NUM
ejpam-5240	712	9	⌋	⌋	NOUN
ejpam-5240	712	10	<	<	X
ejpam-5240	712	11	k	k	PROPN
ejpam-5240	712	12	≤	≤	PROPN
ejpam-5240	712	13	n.	n.	PROPN
ejpam-5240	712	14	theorem	theorem	VERB
ejpam-5240	712	15	13	13	NUM
ejpam-5240	712	16	.	.	PUNCT
ejpam-5240	713	1	let	let	VERB
ejpam-5240	713	2	gs(n	gs(n	NOUN
ejpam-5240	713	3	,	,	PUNCT
ejpam-5240	713	4	k	k	NOUN
ejpam-5240	713	5	)	)	PUNCT
ejpam-5240	713	6	be	be	VERB
ejpam-5240	713	7	a	a	DET
ejpam-5240	713	8	k	k	ADV
ejpam-5240	713	9	-	-	ADJ
ejpam-5240	713	10	restricted	restricted	ADJ
ejpam-5240	713	11	intersection	intersection	NOUN
ejpam-5240	713	12	graph	graph	NOUN
ejpam-5240	713	13	.	.	PUNCT
ejpam-5240	714	1	if	if	SCONJ
ejpam-5240	714	2	k	k	PROPN
ejpam-5240	714	3	=	=	PUNCT
ejpam-5240	714	4	0	0	NUM
ejpam-5240	714	5	or	or	CCONJ
ejpam-5240	714	6	⌊	⌊	X
ejpam-5240	714	7	n	n	ADV
ejpam-5240	714	8	2	2	NUM
ejpam-5240	714	9	⌋	⌋	NOUN
ejpam-5240	714	10	<	<	X
ejpam-5240	714	11	k	k	X
ejpam-5240	714	12	≤	≤	PUNCT
ejpam-5240	715	1	n	n	CCONJ
ejpam-5240	715	2	then	then	ADV
ejpam-5240	715	3	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	715	4	,	,	PUNCT
ejpam-5240	715	5	k	k	NOUN
ejpam-5240	715	6	)	)	PUNCT
ejpam-5240	715	7	)	)	PUNCT
ejpam-5240	716	1	=	=	PUNCT
ejpam-5240	716	2	1	1	X
ejpam-5240	716	3	.	.	PUNCT
ejpam-5240	716	4	proof	proof	NOUN
ejpam-5240	716	5	.	.	PUNCT
ejpam-5240	717	1	note	note	VERB
ejpam-5240	717	2	that	that	SCONJ
ejpam-5240	717	3	by	by	ADP
ejpam-5240	717	4	theorem	theorem	NOUN
ejpam-5240	717	5	7	7	NUM
ejpam-5240	717	6	,	,	PUNCT
ejpam-5240	717	7	gs(n	gs(n	NOUN
ejpam-5240	717	8	,	,	PUNCT
ejpam-5240	717	9	k	k	NOUN
ejpam-5240	717	10	)	)	PUNCT
ejpam-5240	717	11	is	be	AUX
ejpam-5240	717	12	a	a	DET
ejpam-5240	717	13	complete	complete	ADJ
ejpam-5240	717	14	graph	graph	NOUN
ejpam-5240	717	15	.	.	PUNCT
ejpam-5240	718	1	since	since	SCONJ
ejpam-5240	718	2	gs(n	gs(n	NOUN
ejpam-5240	718	3	,	,	PUNCT
ejpam-5240	718	4	k	k	NOUN
ejpam-5240	718	5	)	)	PUNCT
ejpam-5240	718	6	is	be	AUX
ejpam-5240	718	7	a	a	DET
ejpam-5240	718	8	complete	complete	ADJ
ejpam-5240	718	9	graph	graph	NOUN
ejpam-5240	718	10	,	,	PUNCT
ejpam-5240	718	11	since	since	SCONJ
ejpam-5240	718	12	the	the	DET
ejpam-5240	718	13	domination	domination	NOUN
ejpam-5240	718	14	number	number	NOUN
ejpam-5240	718	15	of	of	ADP
ejpam-5240	718	16	a	a	DET
ejpam-5240	718	17	complete	complete	ADJ
ejpam-5240	718	18	graph	graph	NOUN
ejpam-5240	718	19	is	be	AUX
ejpam-5240	718	20	1	1	NUM
ejpam-5240	718	21	,	,	PUNCT
ejpam-5240	718	22	it	it	PRON
ejpam-5240	718	23	follows	follow	VERB
ejpam-5240	718	24	that	that	SCONJ
ejpam-5240	718	25	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	718	26	,	,	PUNCT
ejpam-5240	718	27	k	k	NOUN
ejpam-5240	718	28	)	)	PUNCT
ejpam-5240	718	29	)	)	PUNCT
ejpam-5240	718	30	=	=	PUNCT
ejpam-5240	719	1	1	1	X
ejpam-5240	719	2	.	.	X
ejpam-5240	719	3	m.e	m.e	PROPN
ejpam-5240	719	4	.	.	PROPN
ejpam-5240	719	5	pelagio	pelagio	PROPN
ejpam-5240	719	6	,	,	PUNCT
ejpam-5240	719	7	n.	n.	NOUN
ejpam-5240	719	8	mame	mame	PROPN
ejpam-5240	719	9	,	,	PUNCT
ejpam-5240	719	10	k.	k.	PROPN
ejpam-5240	719	11	mendoza	mendoza	PROPN
ejpam-5240	719	12	/	/	SYM
ejpam-5240	719	13	eur	eur	PROPN
ejpam-5240	719	14	.	.	PUNCT
ejpam-5240	720	1	j.	j.	PROPN
ejpam-5240	720	2	pure	pure	PROPN
ejpam-5240	720	3	appl	appl	PROPN
ejpam-5240	720	4	.	.	PROPN
ejpam-5240	720	5	math	math	PROPN
ejpam-5240	720	6	,	,	PUNCT
ejpam-5240	720	7	17	17	NUM
ejpam-5240	720	8	(	(	PUNCT
ejpam-5240	720	9	3	3	NUM
ejpam-5240	720	10	)	)	PUNCT
ejpam-5240	720	11	(	(	PUNCT
ejpam-5240	720	12	2024	2024	NUM
ejpam-5240	720	13	)	)	PUNCT
ejpam-5240	720	14	,	,	PUNCT
ejpam-5240	720	15	1779	1779	NUM
ejpam-5240	720	16	-	-	SYM
ejpam-5240	720	17	1803	1803	NUM
ejpam-5240	720	18	1801	1801	NUM
ejpam-5240	720	19	5.3	5.3	NUM
ejpam-5240	720	20	.	.	PUNCT
ejpam-5240	720	21	isolate	isolate	VERB
ejpam-5240	720	22	domination	domination	NOUN
ejpam-5240	720	23	number	number	NOUN
ejpam-5240	720	24	of	of	ADP
ejpam-5240	720	25	gs(n	gs(n	NOUN
ejpam-5240	720	26	,	,	PUNCT
ejpam-5240	720	27	k	k	NOUN
ejpam-5240	720	28	)	)	PUNCT
ejpam-5240	720	29	this	this	DET
ejpam-5240	720	30	subsection	subsection	NOUN
ejpam-5240	720	31	analyzes	analyze	VERB
ejpam-5240	720	32	the	the	DET
ejpam-5240	720	33	isolate	isolate	ADJ
ejpam-5240	720	34	domination	domination	NOUN
ejpam-5240	720	35	of	of	ADP
ejpam-5240	720	36	gs(n	gs(n	NOUN
ejpam-5240	720	37	,	,	PUNCT
ejpam-5240	720	38	k	k	NOUN
ejpam-5240	720	39	)	)	PUNCT
ejpam-5240	720	40	given	give	VERB
ejpam-5240	720	41	the	the	DET
ejpam-5240	720	42	two	two	NUM
ejpam-5240	720	43	cases	case	NOUN
ejpam-5240	720	44	:	:	PUNCT
ejpam-5240	720	45	if	if	SCONJ
ejpam-5240	720	46	1	1	NUM
ejpam-5240	720	47	≤	≤	NUM
ejpam-5240	720	48	k	k	X
ejpam-5240	720	49	≤	≤	NUM
ejpam-5240	720	50	⌊	⌊	VERB
ejpam-5240	720	51	n	n	PRON
ejpam-5240	720	52	2	2	NUM
ejpam-5240	720	53	⌋	⌋	NOUN
ejpam-5240	720	54	and	and	CCONJ
ejpam-5240	720	55	if	if	SCONJ
ejpam-5240	720	56	k	k	PROPN
ejpam-5240	720	57	=	=	SYM
ejpam-5240	720	58	0	0	NUM
ejpam-5240	720	59	or	or	CCONJ
ejpam-5240	720	60	⌊	⌊	X
ejpam-5240	720	61	n	n	ADV
ejpam-5240	720	62	2	2	NUM
ejpam-5240	720	63	⌋	⌋	NOUN
ejpam-5240	720	64	<	<	X
ejpam-5240	720	65	k	k	PROPN
ejpam-5240	720	66	≤	≤	PROPN
ejpam-5240	720	67	n.	n.	NOUN
ejpam-5240	720	68	we	we	PRON
ejpam-5240	720	69	will	will	AUX
ejpam-5240	720	70	utilize	utilize	VERB
ejpam-5240	720	71	the	the	DET
ejpam-5240	720	72	notion	notion	NOUN
ejpam-5240	720	73	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	720	74	,	,	PUNCT
ejpam-5240	720	75	k	k	NOUN
ejpam-5240	720	76	)	)	PUNCT
ejpam-5240	720	77	)	)	PUNCT
ejpam-5240	720	78	to	to	PART
ejpam-5240	720	79	denote	denote	VERB
ejpam-5240	720	80	the	the	DET
ejpam-5240	720	81	isolate	isolate	ADJ
ejpam-5240	720	82	domination	domination	NOUN
ejpam-5240	720	83	number	number	NOUN
ejpam-5240	720	84	of	of	ADP
ejpam-5240	720	85	gs(n	gs(n	NOUN
ejpam-5240	720	86	,	,	PUNCT
ejpam-5240	720	87	k	k	NOUN
ejpam-5240	720	88	)	)	PUNCT
ejpam-5240	720	89	.	.	PUNCT
ejpam-5240	721	1	the	the	DET
ejpam-5240	721	2	next	next	ADJ
ejpam-5240	721	3	theorem	theorem	NOUN
ejpam-5240	721	4	determines	determine	VERB
ejpam-5240	721	5	the	the	DET
ejpam-5240	721	6	γ0	γ0	NOUN
ejpam-5240	721	7	of	of	ADP
ejpam-5240	721	8	a	a	DET
ejpam-5240	721	9	gs(n	gs(n	NOUN
ejpam-5240	721	10	,	,	PUNCT
ejpam-5240	721	11	k	k	NOUN
ejpam-5240	721	12	)	)	PUNCT
ejpam-5240	721	13	for	for	ADP
ejpam-5240	721	14	1	1	NUM
ejpam-5240	721	15	≤	≤	NOUN
ejpam-5240	721	16	k	k	X
ejpam-5240	721	17	≤	≤	NUM
ejpam-5240	721	18	⌊	⌊	VERB
ejpam-5240	721	19	n	n	DET
ejpam-5240	721	20	2	2	NUM
ejpam-5240	721	21	⌋	⌋	NOUN
ejpam-5240	721	22	.	.	PUNCT
ejpam-5240	722	1	the	the	DET
ejpam-5240	722	2	set	set	PROPN
ejpam-5240	722	3	t	t	PROPN
ejpam-5240	722	4	discussed	discuss	VERB
ejpam-5240	722	5	in	in	ADP
ejpam-5240	722	6	lemma	lemma	PROPN
ejpam-5240	722	7	3	3	NUM
ejpam-5240	722	8	and	and	CCONJ
ejpam-5240	722	9	lemma	lemma	PROPN
ejpam-5240	722	10	4	4	NUM
ejpam-5240	722	11	will	will	AUX
ejpam-5240	722	12	be	be	AUX
ejpam-5240	722	13	used	use	VERB
ejpam-5240	722	14	to	to	PART
ejpam-5240	722	15	define	define	VERB
ejpam-5240	722	16	the	the	DET
ejpam-5240	722	17	existence	existence	NOUN
ejpam-5240	722	18	of	of	ADP
ejpam-5240	722	19	an	an	DET
ejpam-5240	722	20	isolate	isolate	NOUN
ejpam-5240	722	21	dominating	dominating	NOUN
ejpam-5240	722	22	set	set	VERB
ejpam-5240	722	23	in	in	ADP
ejpam-5240	722	24	gs(n	gs(n	NOUN
ejpam-5240	722	25	,	,	PUNCT
ejpam-5240	722	26	k	k	NOUN
ejpam-5240	722	27	)	)	PUNCT
ejpam-5240	722	28	.	.	PUNCT
ejpam-5240	723	1	theorem	theorem	VERB
ejpam-5240	723	2	14	14	NUM
ejpam-5240	723	3	.	.	PUNCT
ejpam-5240	724	1	let	let	VERB
ejpam-5240	724	2	sn	sn	PROPN
ejpam-5240	724	3	=	=	PUNCT
ejpam-5240	724	4	{	{	PUNCT
ejpam-5240	724	5	x1	x1	PROPN
ejpam-5240	724	6	,	,	PUNCT
ejpam-5240	724	7	x2	x2	PROPN
ejpam-5240	724	8	,	,	PUNCT
ejpam-5240	724	9	...	...	PUNCT
ejpam-5240	724	10	,	,	PUNCT
ejpam-5240	724	11	xn	xn	PRON
ejpam-5240	724	12	}	}	PUNCT
ejpam-5240	724	13	be	be	VERB
ejpam-5240	724	14	an	an	DET
ejpam-5240	724	15	n	n	NOUN
ejpam-5240	724	16	-	-	PUNCT
ejpam-5240	724	17	element	element	NOUN
ejpam-5240	724	18	set	set	NOUN
ejpam-5240	724	19	and	and	CCONJ
ejpam-5240	724	20	let	let	VERB
ejpam-5240	724	21	gs(n	gs(n	NOUN
ejpam-5240	724	22	,	,	PUNCT
ejpam-5240	724	23	k	k	NOUN
ejpam-5240	724	24	)	)	PUNCT
ejpam-5240	724	25	be	be	VERB
ejpam-5240	724	26	a	a	DET
ejpam-5240	724	27	k	k	ADV
ejpam-5240	724	28	-	-	ADJ
ejpam-5240	724	29	restricted	restricted	ADJ
ejpam-5240	724	30	intersection	intersection	NOUN
ejpam-5240	724	31	graph	graph	NOUN
ejpam-5240	724	32	.	.	PUNCT
ejpam-5240	725	1	if	if	SCONJ
ejpam-5240	725	2	1	1	NUM
ejpam-5240	725	3	≤	≤	NUM
ejpam-5240	725	4	k	k	X
ejpam-5240	725	5	≤	≤	NUM
ejpam-5240	725	6	⌊	⌊	VERB
ejpam-5240	725	7	n	n	DET
ejpam-5240	725	8	2	2	NUM
ejpam-5240	725	9	⌋	⌋	NOUN
ejpam-5240	725	10	,	,	PUNCT
ejpam-5240	725	11	then	then	ADV
ejpam-5240	725	12	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	725	13	,	,	PUNCT
ejpam-5240	725	14	k	k	NOUN
ejpam-5240	725	15	)	)	PUNCT
ejpam-5240	725	16	)	)	PUNCT
ejpam-5240	726	1	=	=	PUNCT
ejpam-5240	726	2	⌊	⌊	VERB
ejpam-5240	726	3	n	n	PRON
ejpam-5240	726	4	k	k	NOUN
ejpam-5240	726	5	⌋	⌋	NOUN
ejpam-5240	726	6	.	.	PUNCT
ejpam-5240	727	1	proof	proof	NOUN
ejpam-5240	727	2	.	.	PUNCT
ejpam-5240	728	1	let	let	VERB
ejpam-5240	728	2	t	t	NOUN
ejpam-5240	728	3	=	=	PRON
ejpam-5240	728	4	{	{	PUNCT
ejpam-5240	728	5	{	{	PUNCT
ejpam-5240	728	6	x1	x1	PROPN
ejpam-5240	728	7	,	,	PUNCT
ejpam-5240	728	8	x2	x2	PROPN
ejpam-5240	728	9	,	,	PUNCT
ejpam-5240	728	10	...	...	PUNCT
ejpam-5240	728	11	,	,	PUNCT
ejpam-5240	728	12	xk	xk	PROPN
ejpam-5240	728	13	}	}	PUNCT
ejpam-5240	728	14	,	,	PUNCT
ejpam-5240	728	15	{	{	PUNCT
ejpam-5240	728	16	xk+1	xk+1	PROPN
ejpam-5240	728	17	,	,	PUNCT
ejpam-5240	728	18	...	...	PUNCT
ejpam-5240	728	19	,	,	PUNCT
ejpam-5240	728	20	x2k	x2k	NOUN
ejpam-5240	728	21	}	}	PUNCT
ejpam-5240	728	22	,	,	PUNCT
ejpam-5240	728	23	...	...	PUNCT
ejpam-5240	728	24	,	,	PUNCT
ejpam-5240	728	25	{	{	PUNCT
ejpam-5240	728	26	x(⌊n	x(⌊n	PROPN
ejpam-5240	728	27	k	k	PROPN
ejpam-5240	728	28	⌋−1)k+1	⌋−1)k+1	VERB
ejpam-5240	728	29	,	,	PUNCT
ejpam-5240	728	30	...	...	PUNCT
ejpam-5240	728	31	,	,	PUNCT
ejpam-5240	728	32	x(⌊n	x(⌊n	PROPN
ejpam-5240	729	1	k	k	PROPN
ejpam-5240	729	2	⌋)k	⌋)k	PROPN
ejpam-5240	729	3	}	}	PUNCT
ejpam-5240	729	4	}	}	PUNCT
ejpam-5240	729	5	.	.	PUNCT
ejpam-5240	730	1	note	note	VERB
ejpam-5240	730	2	that	that	SCONJ
ejpam-5240	730	3	by	by	ADP
ejpam-5240	730	4	lemma	lemma	PROPN
ejpam-5240	730	5	3	3	NUM
ejpam-5240	730	6	,	,	PUNCT
ejpam-5240	730	7	t	t	PROPN
ejpam-5240	730	8	is	be	AUX
ejpam-5240	730	9	an	an	DET
ejpam-5240	730	10	independent	independent	ADJ
ejpam-5240	730	11	set	set	NOUN
ejpam-5240	730	12	in	in	ADP
ejpam-5240	730	13	gs(n	gs(n	NOUN
ejpam-5240	730	14	,	,	PUNCT
ejpam-5240	730	15	k	k	NOUN
ejpam-5240	730	16	)	)	PUNCT
ejpam-5240	730	17	where	where	SCONJ
ejpam-5240	730	18	|t	|t	VERB
ejpam-5240	731	1	|	|	ADV
ejpam-5240	731	2	=	=	SYM
ejpam-5240	731	3	⌊	⌊	PROPN
ejpam-5240	731	4	n	n	PRON
ejpam-5240	731	5	k	k	NOUN
ejpam-5240	731	6	⌋	⌋	NOUN
ejpam-5240	731	7	.	.	PUNCT
ejpam-5240	732	1	it	it	PRON
ejpam-5240	732	2	can	can	AUX
ejpam-5240	732	3	be	be	AUX
ejpam-5240	732	4	verified	verify	VERB
ejpam-5240	732	5	that	that	SCONJ
ejpam-5240	732	6	<	<	X
ejpam-5240	732	7	t	t	X
ejpam-5240	732	8	>	>	X
ejpam-5240	732	9	is	be	AUX
ejpam-5240	732	10	an	an	DET
ejpam-5240	732	11	empty	empty	ADJ
ejpam-5240	732	12	graph	graph	NOUN
ejpam-5240	732	13	of	of	ADP
ejpam-5240	732	14	order	order	NOUN
ejpam-5240	732	15	⌊	⌊	VERB
ejpam-5240	732	16	n	n	PRON
ejpam-5240	732	17	k	k	NOUN
ejpam-5240	732	18	⌋	⌋	NOUN
ejpam-5240	732	19	.	.	PUNCT
ejpam-5240	733	1	thus	thus	ADV
ejpam-5240	733	2	,	,	PUNCT
ejpam-5240	733	3	for	for	ADP
ejpam-5240	733	4	all	all	DET
ejpam-5240	733	5	a	a	DET
ejpam-5240	733	6	∈	∈	PROPN
ejpam-5240	733	7	v	v	NOUN
ejpam-5240	733	8	(	(	PUNCT
ejpam-5240	733	9	<	<	X
ejpam-5240	733	10	t	t	PROPN
ejpam-5240	733	11	>	>	PUNCT
ejpam-5240	733	12	)	)	PUNCT
ejpam-5240	733	13	,	,	PUNCT
ejpam-5240	733	14	we	we	PRON
ejpam-5240	733	15	have	have	VERB
ejpam-5240	733	16	deg(a	deg(a	PROPN
ejpam-5240	733	17	)	)	PUNCT
ejpam-5240	733	18	=	=	SYM
ejpam-5240	733	19	0	0	NUM
ejpam-5240	733	20	which	which	PRON
ejpam-5240	733	21	implies	imply	VERB
ejpam-5240	733	22	that	that	SCONJ
ejpam-5240	733	23	every	every	DET
ejpam-5240	733	24	vertex	vertex	NOUN
ejpam-5240	733	25	in	in	ADP
ejpam-5240	733	26	<	<	X
ejpam-5240	733	27	t	t	X
ejpam-5240	733	28	>	>	X
ejpam-5240	733	29	is	be	AUX
ejpam-5240	733	30	an	an	DET
ejpam-5240	733	31	isolated	isolated	ADJ
ejpam-5240	733	32	vertex	vertex	NOUN
ejpam-5240	733	33	.	.	PUNCT
ejpam-5240	734	1	moreover	moreover	ADV
ejpam-5240	734	2	,	,	PUNCT
ejpam-5240	734	3	by	by	ADP
ejpam-5240	734	4	lemma	lemma	PROPN
ejpam-5240	734	5	4	4	NUM
ejpam-5240	734	6	,	,	PUNCT
ejpam-5240	734	7	the	the	DET
ejpam-5240	734	8	set	set	NOUN
ejpam-5240	734	9	t	t	PROPN
ejpam-5240	734	10	is	be	AUX
ejpam-5240	734	11	also	also	ADV
ejpam-5240	734	12	a	a	DET
ejpam-5240	734	13	dominating	dominating	NOUN
ejpam-5240	734	14	set	set	VERB
ejpam-5240	734	15	in	in	ADP
ejpam-5240	734	16	gs(n	gs(n	NOUN
ejpam-5240	734	17	,	,	PUNCT
ejpam-5240	734	18	k	k	NOUN
ejpam-5240	734	19	)	)	PUNCT
ejpam-5240	734	20	.	.	PUNCT
ejpam-5240	735	1	since	since	SCONJ
ejpam-5240	735	2	t	t	PROPN
ejpam-5240	735	3	is	be	AUX
ejpam-5240	735	4	an	an	DET
ejpam-5240	735	5	independent	independent	ADJ
ejpam-5240	735	6	set	set	NOUN
ejpam-5240	735	7	and	and	CCONJ
ejpam-5240	735	8	dominating	dominating	NOUN
ejpam-5240	735	9	set	set	VERB
ejpam-5240	735	10	in	in	ADP
ejpam-5240	735	11	gs(n	gs(n	NOUN
ejpam-5240	735	12	,	,	PUNCT
ejpam-5240	735	13	k	k	NOUN
ejpam-5240	735	14	)	)	PUNCT
ejpam-5240	735	15	,	,	PUNCT
ejpam-5240	735	16	it	it	PRON
ejpam-5240	735	17	follows	follow	VERB
ejpam-5240	735	18	that	that	SCONJ
ejpam-5240	735	19	t	t	PROPN
ejpam-5240	735	20	is	be	AUX
ejpam-5240	735	21	an	an	DET
ejpam-5240	735	22	isolate	isolate	ADJ
ejpam-5240	735	23	dominating	dominating	NOUN
ejpam-5240	735	24	set	set	VERB
ejpam-5240	735	25	in	in	ADP
ejpam-5240	735	26	gs(n	gs(n	NOUN
ejpam-5240	735	27	,	,	PUNCT
ejpam-5240	735	28	k	k	NOUN
ejpam-5240	735	29	)	)	PUNCT
ejpam-5240	735	30	with	with	ADP
ejpam-5240	735	31	|t	|t	PROPN
ejpam-5240	736	1	|	|	ADV
ejpam-5240	736	2	=	=	SYM
ejpam-5240	736	3	⌊	⌊	PROPN
ejpam-5240	736	4	n	n	PRON
ejpam-5240	736	5	k	k	PROPN
ejpam-5240	736	6	⌋	⌋	NOUN
ejpam-5240	736	7	.	.	PUNCT
ejpam-5240	737	1	hence	hence	ADV
ejpam-5240	737	2	,	,	PUNCT
ejpam-5240	737	3	there	there	PRON
ejpam-5240	737	4	exist	exist	VERB
ejpam-5240	737	5	an	an	DET
ejpam-5240	737	6	isolate	isolate	NOUN
ejpam-5240	737	7	dominating	dominating	NOUN
ejpam-5240	737	8	set	set	VERB
ejpam-5240	737	9	in	in	ADP
ejpam-5240	737	10	gs(n	gs(n	NOUN
ejpam-5240	737	11	,	,	PUNCT
ejpam-5240	737	12	k	k	NOUN
ejpam-5240	737	13	)	)	PUNCT
ejpam-5240	737	14	with	with	ADP
ejpam-5240	737	15	cardinality	cardinality	NOUN
ejpam-5240	737	16	equal	equal	ADJ
ejpam-5240	737	17	to	to	AUX
ejpam-5240	737	18	⌊	⌊	PROPN
ejpam-5240	737	19	n	n	PROPN
ejpam-5240	737	20	k	k	NOUN
ejpam-5240	737	21	⌋	⌋	NOUN
ejpam-5240	737	22	.	.	PUNCT
ejpam-5240	738	1	this	this	PRON
ejpam-5240	738	2	implies	imply	VERB
ejpam-5240	738	3	that	that	SCONJ
ejpam-5240	738	4	γ0(gs(n	γ0(gs(n	NOUN
ejpam-5240	738	5	,	,	PUNCT
ejpam-5240	738	6	k	k	NOUN
ejpam-5240	738	7	)	)	PUNCT
ejpam-5240	738	8	)	)	PUNCT
ejpam-5240	738	9	≤	≤	PUNCT
ejpam-5240	739	1	⌊	⌊	VERB
ejpam-5240	739	2	n	n	PRON
ejpam-5240	739	3	k	k	NOUN
ejpam-5240	739	4	⌋	⌋	NOUN
ejpam-5240	739	5	.	.	PUNCT
ejpam-5240	740	1	by	by	ADP
ejpam-5240	740	2	theorem	theorem	NOUN
ejpam-5240	740	3	12	12	NUM
ejpam-5240	740	4	,	,	PUNCT
ejpam-5240	740	5	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	740	6	,	,	PUNCT
ejpam-5240	740	7	k	k	NOUN
ejpam-5240	740	8	)	)	PUNCT
ejpam-5240	740	9	)	)	PUNCT
ejpam-5240	741	1	=	=	PUNCT
ejpam-5240	741	2	⌊	⌊	VERB
ejpam-5240	741	3	n	n	PRON
ejpam-5240	741	4	k	k	PROPN
ejpam-5240	741	5	⌋	⌋	PROPN
ejpam-5240	741	6	.	.	PUNCT
ejpam-5240	742	1	note	note	VERB
ejpam-5240	742	2	that	that	SCONJ
ejpam-5240	742	3	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	742	4	,	,	PUNCT
ejpam-5240	742	5	k	k	NOUN
ejpam-5240	742	6	)	)	PUNCT
ejpam-5240	742	7	)	)	PUNCT
ejpam-5240	743	1	≤	≤	NUM
ejpam-5240	744	1	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	744	2	,	,	PUNCT
ejpam-5240	744	3	k	k	NOUN
ejpam-5240	744	4	)	)	PUNCT
ejpam-5240	744	5	)	)	PUNCT
ejpam-5240	744	6	.	.	PUNCT
ejpam-5240	745	1	therefore	therefore	ADV
ejpam-5240	745	2	,	,	PUNCT
ejpam-5240	745	3	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	745	4	,	,	PUNCT
ejpam-5240	745	5	k	k	NOUN
ejpam-5240	745	6	)	)	PUNCT
ejpam-5240	745	7	)	)	PUNCT
ejpam-5240	746	1	=	=	PUNCT
ejpam-5240	746	2	⌊	⌊	VERB
ejpam-5240	746	3	n	n	PRON
ejpam-5240	746	4	k	k	PROPN
ejpam-5240	746	5	⌋	⌋	PROPN
ejpam-5240	746	6	.	.	PUNCT
ejpam-5240	747	1	theorem	theorem	VERB
ejpam-5240	747	2	15	15	NUM
ejpam-5240	747	3	.	.	PUNCT
ejpam-5240	748	1	let	let	VERB
ejpam-5240	748	2	gs(n	gs(n	NOUN
ejpam-5240	748	3	,	,	PUNCT
ejpam-5240	748	4	k	k	NOUN
ejpam-5240	748	5	)	)	PUNCT
ejpam-5240	748	6	be	be	VERB
ejpam-5240	748	7	a	a	DET
ejpam-5240	748	8	k	k	ADV
ejpam-5240	748	9	-	-	ADJ
ejpam-5240	748	10	restricted	restricted	ADJ
ejpam-5240	748	11	intersection	intersection	NOUN
ejpam-5240	748	12	graph	graph	NOUN
ejpam-5240	748	13	.	.	PUNCT
ejpam-5240	749	1	then	then	ADV
ejpam-5240	749	2	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	749	3	,	,	PUNCT
ejpam-5240	749	4	k	k	NOUN
ejpam-5240	749	5	)	)	PUNCT
ejpam-5240	749	6	)	)	PUNCT
ejpam-5240	750	1	=	=	SYM
ejpam-5240	750	2	1	1	NUM
ejpam-5240	750	3	if	if	SCONJ
ejpam-5240	750	4	k	k	PROPN
ejpam-5240	750	5	=	=	PUNCT
ejpam-5240	750	6	0	0	NUM
ejpam-5240	750	7	or	or	CCONJ
ejpam-5240	750	8	⌊	⌊	X
ejpam-5240	750	9	n	n	ADV
ejpam-5240	750	10	2	2	NUM
ejpam-5240	750	11	⌋	⌋	NOUN
ejpam-5240	750	12	<	<	X
ejpam-5240	750	13	k	k	PROPN
ejpam-5240	750	14	≤	≤	PROPN
ejpam-5240	750	15	n.	n.	NOUN
ejpam-5240	750	16	proof	proof	NOUN
ejpam-5240	750	17	.	.	PUNCT
ejpam-5240	751	1	by	by	ADP
ejpam-5240	751	2	theorem	theorem	NOUN
ejpam-5240	751	3	7	7	NUM
ejpam-5240	751	4	,	,	PUNCT
ejpam-5240	751	5	gs(n	gs(n	NOUN
ejpam-5240	751	6	,	,	PUNCT
ejpam-5240	751	7	k	k	NOUN
ejpam-5240	751	8	)	)	PUNCT
ejpam-5240	751	9	is	be	AUX
ejpam-5240	751	10	a	a	DET
ejpam-5240	751	11	complete	complete	ADJ
ejpam-5240	751	12	graph	graph	NOUN
ejpam-5240	751	13	of	of	ADP
ejpam-5240	751	14	order	order	NOUN
ejpam-5240	751	15	(	(	PUNCT
ejpam-5240	751	16	n	n	X
ejpam-5240	751	17	k	k	PROPN
ejpam-5240	751	18	)	)	PUNCT
ejpam-5240	751	19	.	.	PUNCT
ejpam-5240	752	1	since	since	SCONJ
ejpam-5240	752	2	gs(n	gs(n	NOUN
ejpam-5240	752	3	,	,	PUNCT
ejpam-5240	752	4	k	k	NOUN
ejpam-5240	752	5	)	)	PUNCT
ejpam-5240	752	6	is	be	AUX
ejpam-5240	752	7	a	a	DET
ejpam-5240	752	8	complete	complete	ADJ
ejpam-5240	752	9	graph	graph	NOUN
ejpam-5240	752	10	,	,	PUNCT
ejpam-5240	752	11	since	since	SCONJ
ejpam-5240	752	12	the	the	DET
ejpam-5240	752	13	isolate	isolate	ADJ
ejpam-5240	752	14	domination	domination	NOUN
ejpam-5240	752	15	number	number	NOUN
ejpam-5240	752	16	of	of	ADP
ejpam-5240	752	17	a	a	DET
ejpam-5240	752	18	complete	complete	ADJ
ejpam-5240	752	19	graph	graph	NOUN
ejpam-5240	752	20	is	be	AUX
ejpam-5240	752	21	1	1	NUM
ejpam-5240	752	22	,	,	PUNCT
ejpam-5240	752	23	it	it	PRON
ejpam-5240	752	24	follows	follow	VERB
ejpam-5240	752	25	that	that	SCONJ
ejpam-5240	752	26	γ0(gs(n	γ0(gs(n	NOUN
ejpam-5240	752	27	,	,	PUNCT
ejpam-5240	752	28	k	k	NOUN
ejpam-5240	752	29	)	)	PUNCT
ejpam-5240	752	30	)	)	PUNCT
ejpam-5240	753	1	=	=	PUNCT
ejpam-5240	754	1	1	1	NUM
ejpam-5240	754	2	.	.	NOUN
ejpam-5240	754	3	6	6	NUM
ejpam-5240	754	4	.	.	X
ejpam-5240	755	1	summary	summary	NOUN
ejpam-5240	755	2	,	,	PUNCT
ejpam-5240	755	3	conclusion	conclusion	NOUN
ejpam-5240	755	4	,	,	PUNCT
ejpam-5240	755	5	and	and	CCONJ
ejpam-5240	755	6	recommendations	recommendation	NOUN
ejpam-5240	755	7	this	this	DET
ejpam-5240	755	8	study	study	NOUN
ejpam-5240	755	9	introduces	introduce	NOUN
ejpam-5240	755	10	and	and	CCONJ
ejpam-5240	755	11	discusses	discuss	VERB
ejpam-5240	755	12	a	a	DET
ejpam-5240	755	13	k	k	ADV
ejpam-5240	755	14	-	-	ADJ
ejpam-5240	755	15	restricted	restricted	ADJ
ejpam-5240	755	16	intersection	intersection	NOUN
ejpam-5240	755	17	graph	graph	NOUN
ejpam-5240	755	18	including	include	VERB
ejpam-5240	755	19	some	some	PRON
ejpam-5240	755	20	of	of	ADP
ejpam-5240	755	21	the	the	DET
ejpam-5240	755	22	graph	graph	NOUN
ejpam-5240	755	23	’s	’s	PART
ejpam-5240	755	24	parameters.a	parameters.a	ADV
ejpam-5240	755	25	k	k	ADV
ejpam-5240	755	26	-	-	PUNCT
ejpam-5240	755	27	restricted	restrict	VERB
ejpam-5240	755	28	intersection	intersection	NOUN
ejpam-5240	755	29	graph	graph	NOUN
ejpam-5240	755	30	is	be	AUX
ejpam-5240	755	31	a	a	DET
ejpam-5240	755	32	simple	simple	ADJ
ejpam-5240	755	33	graph	graph	NOUN
ejpam-5240	755	34	whose	whose	DET
ejpam-5240	755	35	vertex	vertex	NOUN
ejpam-5240	755	36	set	set	NOUN
ejpam-5240	755	37	is	be	AUX
ejpam-5240	755	38	equal	equal	ADJ
ejpam-5240	755	39	to	to	ADP
ejpam-5240	755	40	s(n	s(n	PROPN
ejpam-5240	755	41	,	,	PUNCT
ejpam-5240	755	42	k	k	NOUN
ejpam-5240	755	43	)	)	PUNCT
ejpam-5240	755	44	and	and	CCONJ
ejpam-5240	755	45	two	two	NUM
ejpam-5240	755	46	vertices	vertex	NOUN
ejpam-5240	755	47	a	a	DET
ejpam-5240	755	48	,	,	PUNCT
ejpam-5240	755	49	b	b	NOUN
ejpam-5240	755	50	in	in	ADP
ejpam-5240	755	51	gs(n	gs(n	NOUN
ejpam-5240	755	52	,	,	PUNCT
ejpam-5240	755	53	k	k	NOUN
ejpam-5240	755	54	)	)	PUNCT
ejpam-5240	755	55	are	be	AUX
ejpam-5240	755	56	adjacent	adjacent	ADJ
ejpam-5240	755	57	whenever	whenever	SCONJ
ejpam-5240	755	58	a	a	DET
ejpam-5240	755	59	∩	∩	ADJ
ejpam-5240	755	60	b	b	NOUN
ejpam-5240	755	61	̸=	̸=	PROPN
ejpam-5240	755	62	∅	∅	NOUN
ejpam-5240	755	63	and	and	CCONJ
ejpam-5240	755	64	a	a	DET
ejpam-5240	755	65	̸=	̸=	PROPN
ejpam-5240	755	66	b.	b.	NOUN
ejpam-5240	755	67	the	the	DET
ejpam-5240	755	68	case	case	NOUN
ejpam-5240	755	69	s0	s0	NOUN
ejpam-5240	755	70	is	be	AUX
ejpam-5240	755	71	a	a	DET
ejpam-5240	755	72	trivial	trivial	ADJ
ejpam-5240	755	73	case	case	NOUN
ejpam-5240	755	74	for	for	ADP
ejpam-5240	755	75	gs(n	gs(n	NOUN
ejpam-5240	755	76	,	,	PUNCT
ejpam-5240	755	77	k	k	NOUN
ejpam-5240	755	78	)	)	PUNCT
ejpam-5240	755	79	.	.	PUNCT
ejpam-5240	756	1	the	the	DET
ejpam-5240	756	2	order	order	NOUN
ejpam-5240	756	3	of	of	ADP
ejpam-5240	756	4	gs(n	gs(n	NOUN
ejpam-5240	756	5	,	,	PUNCT
ejpam-5240	756	6	k	k	NOUN
ejpam-5240	756	7	)	)	PUNCT
ejpam-5240	756	8	is	be	AUX
ejpam-5240	756	9	given	give	VERB
ejpam-5240	756	10	by	by	ADP
ejpam-5240	756	11	(	(	PUNCT
ejpam-5240	756	12	n	n	X
ejpam-5240	756	13	k	k	PROPN
ejpam-5240	756	14	)	)	PUNCT
ejpam-5240	756	15	.	.	PUNCT
ejpam-5240	757	1	it	it	PRON
ejpam-5240	757	2	was	be	AUX
ejpam-5240	757	3	determined	determine	VERB
ejpam-5240	757	4	that	that	SCONJ
ejpam-5240	757	5	gs(n	gs(n	NOUN
ejpam-5240	757	6	,	,	PUNCT
ejpam-5240	757	7	k	k	NOUN
ejpam-5240	757	8	)	)	PUNCT
ejpam-5240	757	9	is	be	AUX
ejpam-5240	757	10	a	a	DET
ejpam-5240	757	11	trivial	trivial	ADJ
ejpam-5240	757	12	graph	graph	NOUN
ejpam-5240	757	13	if	if	SCONJ
ejpam-5240	758	1	and	and	CCONJ
ejpam-5240	758	2	only	only	ADV
ejpam-5240	758	3	if	if	SCONJ
ejpam-5240	758	4	k	k	PROPN
ejpam-5240	758	5	=	=	SYM
ejpam-5240	758	6	0	0	PROPN
ejpam-5240	758	7	or	or	CCONJ
ejpam-5240	758	8	k	k	PROPN
ejpam-5240	758	9	=	=	SYM
ejpam-5240	758	10	n.	n.	PROPN
ejpam-5240	758	11	also	also	ADV
ejpam-5240	758	12	,	,	PUNCT
ejpam-5240	758	13	if	if	SCONJ
ejpam-5240	758	14	k	k	PROPN
ejpam-5240	758	15	=	=	SYM
ejpam-5240	758	16	1	1	NUM
ejpam-5240	758	17	,	,	PUNCT
ejpam-5240	758	18	then	then	ADV
ejpam-5240	758	19	gs(n	gs(n	NOUN
ejpam-5240	758	20	,	,	PUNCT
ejpam-5240	758	21	k	k	NOUN
ejpam-5240	758	22	)	)	PUNCT
ejpam-5240	758	23	is	be	AUX
ejpam-5240	758	24	an	an	DET
ejpam-5240	758	25	empty	empty	ADJ
ejpam-5240	758	26	graph	graph	NOUN
ejpam-5240	758	27	of	of	ADP
ejpam-5240	758	28	order	order	NOUN
ejpam-5240	758	29	n.	n.	NOUN
ejpam-5240	758	30	in	in	ADP
ejpam-5240	758	31	addition	addition	NOUN
ejpam-5240	758	32	,	,	PUNCT
ejpam-5240	758	33	it	it	PRON
ejpam-5240	758	34	was	be	AUX
ejpam-5240	758	35	established	establish	VERB
ejpam-5240	758	36	that	that	SCONJ
ejpam-5240	758	37	gs(n	gs(n	NOUN
ejpam-5240	758	38	,	,	PUNCT
ejpam-5240	758	39	k	k	NOUN
ejpam-5240	758	40	)	)	PUNCT
ejpam-5240	758	41	is	be	AUX
ejpam-5240	758	42	a	a	DET
ejpam-5240	758	43	regular	regular	ADJ
ejpam-5240	758	44	graph	graph	NOUN
ejpam-5240	758	45	such	such	ADJ
ejpam-5240	758	46	that	that	PRON
ejpam-5240	758	47	for	for	ADP
ejpam-5240	758	48	any	any	DET
ejpam-5240	758	49	a	a	DET
ejpam-5240	758	50	∈	∈	PROPN
ejpam-5240	758	51	v	v	NOUN
ejpam-5240	758	52	(	(	PUNCT
ejpam-5240	758	53	gs(n	gs(n	NOUN
ejpam-5240	758	54	,	,	PUNCT
ejpam-5240	758	55	k	k	NOUN
ejpam-5240	758	56	)	)	PUNCT
ejpam-5240	758	57	)	)	PUNCT
ejpam-5240	758	58	,	,	PUNCT
ejpam-5240	758	59	deg(a	deg(a	PROPN
ejpam-5240	758	60	)	)	PUNCT
ejpam-5240	758	61	=	=	PUNCT
ejpam-5240	758	62	(	(	PUNCT
ejpam-5240	758	63	n	n	X
ejpam-5240	758	64	k	k	NOUN
ejpam-5240	758	65	)	)	PUNCT
ejpam-5240	758	66	−	−	PROPN
ejpam-5240	759	1	[	[	X
ejpam-5240	759	2	(	(	PUNCT
ejpam-5240	759	3	n−k	n−k	NOUN
ejpam-5240	759	4	k	k	PROPN
ejpam-5240	759	5	)	)	PUNCT
ejpam-5240	760	1	+	+	CCONJ
ejpam-5240	760	2	1	1	X
ejpam-5240	760	3	]	]	PUNCT
ejpam-5240	760	4	if	if	SCONJ
ejpam-5240	760	5	and	and	CCONJ
ejpam-5240	760	6	only	only	ADV
ejpam-5240	760	7	if	if	SCONJ
ejpam-5240	760	8	1	1	NUM
ejpam-5240	760	9	≤	≤	NUM
ejpam-5240	760	10	k	k	X
ejpam-5240	760	11	≤	≤	NUM
ejpam-5240	760	12	⌊	⌊	PROPN
ejpam-5240	760	13	n	n	PRON
ejpam-5240	760	14	k	k	PROPN
ejpam-5240	760	15	⌋	⌋	NOUN
ejpam-5240	760	16	.	.	PUNCT
ejpam-5240	761	1	furthermore	furthermore	ADV
ejpam-5240	761	2	,	,	PUNCT
ejpam-5240	761	3	deg(a	deg(a	PROPN
ejpam-5240	761	4	)	)	PUNCT
ejpam-5240	761	5	=	=	PUNCT
ejpam-5240	761	6	(	(	PUNCT
ejpam-5240	761	7	n	n	X
ejpam-5240	761	8	k	k	NOUN
ejpam-5240	761	9	)	)	PUNCT
ejpam-5240	762	1	−	−	PROPN
ejpam-5240	762	2	1	1	NUM
ejpam-5240	762	3	if	if	SCONJ
ejpam-5240	762	4	and	and	CCONJ
ejpam-5240	762	5	only	only	ADV
ejpam-5240	762	6	if	if	SCONJ
ejpam-5240	762	7	k	k	PROPN
ejpam-5240	762	8	=	=	PUNCT
ejpam-5240	762	9	0	0	NUM
ejpam-5240	762	10	or	or	CCONJ
ejpam-5240	762	11	⌊	⌊	PROPN
ejpam-5240	762	12	n	n	PRON
ejpam-5240	762	13	k	k	NOUN
ejpam-5240	762	14	⌋	⌋	NOUN
ejpam-5240	762	15	<	<	X
ejpam-5240	762	16	k	k	PROPN
ejpam-5240	762	17	≤	≤	PROPN
ejpam-5240	762	18	n.	n.	NOUN
ejpam-5240	762	19	with	with	ADP
ejpam-5240	762	20	these	these	PRON
ejpam-5240	762	21	,	,	PUNCT
ejpam-5240	762	22	the	the	DET
ejpam-5240	762	23	size	size	NOUN
ejpam-5240	762	24	of	of	ADP
ejpam-5240	762	25	gs(n	gs(n	NOUN
ejpam-5240	762	26	,	,	PUNCT
ejpam-5240	762	27	k	k	NOUN
ejpam-5240	762	28	)	)	PUNCT
ejpam-5240	762	29	is	be	AUX
ejpam-5240	762	30	proven	prove	VERB
ejpam-5240	762	31	to	to	PART
ejpam-5240	762	32	be	be	AUX
ejpam-5240	762	33	equal	equal	ADJ
ejpam-5240	762	34	to	to	ADP
ejpam-5240	762	35	|e(gs(n	|e(gs(n	PROPN
ejpam-5240	762	36	,	,	PUNCT
ejpam-5240	762	37	k	k	NOUN
ejpam-5240	762	38	)	)	PUNCT
ejpam-5240	762	39	)	)	PUNCT
ejpam-5240	763	1	|	|	ADV
ejpam-5240	763	2	=	=	SYM
ejpam-5240	763	3			PUNCT
ejpam-5240	763	4	(	(	PUNCT
ejpam-5240	763	5	nk	nk	PROPN
ejpam-5240	763	6	)	)	PUNCT
ejpam-5240	763	7	{	{	PUNCT
ejpam-5240	763	8	(	(	PUNCT
ejpam-5240	763	9	nk)−	nk)−	X
ejpam-5240	763	10	[	[	PUNCT
ejpam-5240	763	11	(	(	PUNCT
ejpam-5240	763	12	n−k	n−k	NOUN
ejpam-5240	763	13	k	k	NOUN
ejpam-5240	763	14	)	)	PUNCT
ejpam-5240	763	15	+1	+1	NOUN
ejpam-5240	763	16	]	]	X
ejpam-5240	763	17	}	}	PUNCT
ejpam-5240	763	18	2	2	NUM
ejpam-5240	763	19	if	if	SCONJ
ejpam-5240	763	20	1	1	NUM
ejpam-5240	763	21	≤	≤	NUM
ejpam-5240	763	22	k	k	X
ejpam-5240	763	23	≤	≤	NUM
ejpam-5240	763	24	⌊	⌊	VERB
ejpam-5240	763	25	n	n	DET
ejpam-5240	763	26	2	2	NUM
ejpam-5240	763	27	⌋	⌋	NOUN
ejpam-5240	763	28	;	;	PUNCT
ejpam-5240	763	29	(	(	PUNCT
ejpam-5240	763	30	nk	nk	PROPN
ejpam-5240	763	31	)	)	PUNCT
ejpam-5240	763	32	[	[	PUNCT
ejpam-5240	763	33	(	(	PUNCT
ejpam-5240	763	34	nk)−1	nk)−1	SYM
ejpam-5240	763	35	]	]	PUNCT
ejpam-5240	763	36	2	2	NUM
ejpam-5240	763	37	if	if	SCONJ
ejpam-5240	763	38	k	k	NOUN
ejpam-5240	763	39	=	=	PUNCT
ejpam-5240	763	40	0	0	NUM
ejpam-5240	763	41	or	or	CCONJ
ejpam-5240	763	42	⌊	⌊	X
ejpam-5240	763	43	n	n	ADV
ejpam-5240	763	44	2	2	NUM
ejpam-5240	763	45	⌋	⌋	NOUN
ejpam-5240	763	46	<	<	X
ejpam-5240	763	47	k	k	PROPN
ejpam-5240	763	48	≤	≤	PROPN
ejpam-5240	763	49	n.	n.	NOUN
ejpam-5240	763	50	references	reference	NOUN
ejpam-5240	763	51	1802	1802	NUM
ejpam-5240	763	52	additionally	additionally	ADV
ejpam-5240	763	53	,	,	PUNCT
ejpam-5240	763	54	the	the	DET
ejpam-5240	763	55	study	study	NOUN
ejpam-5240	763	56	disclosed	disclose	VERB
ejpam-5240	763	57	that	that	SCONJ
ejpam-5240	763	58	gs(n	gs(n	NOUN
ejpam-5240	763	59	,	,	PUNCT
ejpam-5240	763	60	k	k	NOUN
ejpam-5240	763	61	)	)	PUNCT
ejpam-5240	763	62	was	be	AUX
ejpam-5240	763	63	a	a	DET
ejpam-5240	763	64	complete	complete	ADJ
ejpam-5240	763	65	graph	graph	NOUN
ejpam-5240	763	66	if	if	SCONJ
ejpam-5240	763	67	and	and	CCONJ
ejpam-5240	763	68	only	only	ADV
ejpam-5240	763	69	if	if	SCONJ
ejpam-5240	763	70	k	k	PROPN
ejpam-5240	763	71	=	=	PUNCT
ejpam-5240	763	72	0	0	NUM
ejpam-5240	763	73	or	or	CCONJ
ejpam-5240	763	74	⌊	⌊	X
ejpam-5240	763	75	n	n	ADV
ejpam-5240	763	76	2	2	NUM
ejpam-5240	763	77	⌋	⌋	NOUN
ejpam-5240	763	78	<	<	X
ejpam-5240	763	79	k	k	PROPN
ejpam-5240	763	80	≤	≤	PROPN
ejpam-5240	763	81	n.	n.	NOUN
ejpam-5240	763	82	more	more	ADV
ejpam-5240	763	83	so	so	ADV
ejpam-5240	763	84	,	,	PUNCT
ejpam-5240	763	85	gs(n	gs(n	NOUN
ejpam-5240	763	86	,	,	PUNCT
ejpam-5240	763	87	k	k	NOUN
ejpam-5240	763	88	)	)	PUNCT
ejpam-5240	763	89	is	be	AUX
ejpam-5240	763	90	a	a	DET
ejpam-5240	763	91	cycle	cycle	NOUN
ejpam-5240	763	92	graph	graph	NOUN
ejpam-5240	763	93	of	of	ADP
ejpam-5240	763	94	order	order	NOUN
ejpam-5240	763	95	3	3	NUM
ejpam-5240	763	96	if	if	SCONJ
ejpam-5240	763	97	and	and	CCONJ
ejpam-5240	763	98	only	only	ADV
ejpam-5240	763	99	if	if	SCONJ
ejpam-5240	763	100	n	n	PROPN
ejpam-5240	763	101	=	=	SYM
ejpam-5240	763	102	3	3	NUM
ejpam-5240	763	103	and	and	CCONJ
ejpam-5240	763	104	k	k	NOUN
ejpam-5240	763	105	=	=	NOUN
ejpam-5240	763	106	2	2	X
ejpam-5240	763	107	.	.	PUNCT
ejpam-5240	763	108	the	the	DET
ejpam-5240	763	109	complement	complement	NOUN
ejpam-5240	763	110	graph	graph	NOUN
ejpam-5240	763	111	of	of	ADP
ejpam-5240	763	112	gs(n	gs(n	NOUN
ejpam-5240	763	113	,	,	PUNCT
ejpam-5240	763	114	k	k	NOUN
ejpam-5240	763	115	)	)	PUNCT
ejpam-5240	763	116	was	be	AUX
ejpam-5240	763	117	also	also	ADV
ejpam-5240	763	118	determined	determine	VERB
ejpam-5240	763	119	.	.	PUNCT
ejpam-5240	764	1	considering	consider	VERB
ejpam-5240	764	2	that	that	PRON
ejpam-5240	764	3	gs(n	gs(n	NOUN
ejpam-5240	764	4	,	,	PUNCT
ejpam-5240	764	5	k	k	NOUN
ejpam-5240	764	6	)	)	PUNCT
ejpam-5240	764	7	is	be	AUX
ejpam-5240	764	8	a	a	DET
ejpam-5240	764	9	simple	simple	ADJ
ejpam-5240	764	10	graph	graph	NOUN
ejpam-5240	764	11	,	,	PUNCT
ejpam-5240	764	12	then	then	ADV
ejpam-5240	764	13	its	its	PRON
ejpam-5240	764	14	complement	complement	NOUN
ejpam-5240	764	15	graph	graph	NOUN
ejpam-5240	764	16	is	be	AUX
ejpam-5240	764	17	also	also	ADV
ejpam-5240	764	18	a	a	DET
ejpam-5240	764	19	simple	simple	ADJ
ejpam-5240	764	20	graph	graph	NOUN
ejpam-5240	764	21	.	.	PUNCT
ejpam-5240	765	1	it	it	PRON
ejpam-5240	765	2	was	be	AUX
ejpam-5240	765	3	further	far	ADV
ejpam-5240	765	4	established	establish	VERB
ejpam-5240	765	5	that	that	SCONJ
ejpam-5240	765	6	gs(n	gs(n	NOUN
ejpam-5240	765	7	,	,	PUNCT
ejpam-5240	765	8	k	k	NOUN
ejpam-5240	765	9	)	)	PUNCT
ejpam-5240	765	10	is	be	AUX
ejpam-5240	765	11	a	a	DET
ejpam-5240	765	12	regular	regular	ADJ
ejpam-5240	765	13	graph	graph	NOUN
ejpam-5240	765	14	having	have	VERB
ejpam-5240	765	15	the	the	DET
ejpam-5240	765	16	following	follow	VERB
ejpam-5240	765	17	degree	degree	NOUN
ejpam-5240	765	18	for	for	ADP
ejpam-5240	765	19	all	all	DET
ejpam-5240	765	20	a	a	DET
ejpam-5240	765	21	∈	∈	PROPN
ejpam-5240	765	22	v	v	NOUN
ejpam-5240	765	23	(	(	PUNCT
ejpam-5240	765	24	gs(n	gs(n	NOUN
ejpam-5240	765	25	,	,	PUNCT
ejpam-5240	765	26	k	k	NOUN
ejpam-5240	765	27	)	)	PUNCT
ejpam-5240	765	28	)	)	PUNCT
ejpam-5240	765	29	,	,	PUNCT
ejpam-5240	765	30	depending	depend	VERB
ejpam-5240	765	31	on	on	ADP
ejpam-5240	765	32	each	each	DET
ejpam-5240	765	33	cases	case	NOUN
ejpam-5240	765	34	:	:	PUNCT
ejpam-5240	765	35	if	if	SCONJ
ejpam-5240	765	36	1	1	NUM
ejpam-5240	765	37	≤	≤	NUM
ejpam-5240	765	38	k	k	X
ejpam-5240	765	39	≤	≤	NUM
ejpam-5240	765	40	⌊	⌊	VERB
ejpam-5240	765	41	n	n	PRON
ejpam-5240	765	42	2	2	NUM
ejpam-5240	765	43	⌋	⌋	NOUN
ejpam-5240	765	44	,	,	PUNCT
ejpam-5240	765	45	deg(a	deg(a	PROPN
ejpam-5240	765	46	)	)	PUNCT
ejpam-5240	765	47	=	=	PRON
ejpam-5240	765	48	(	(	PUNCT
ejpam-5240	765	49	n−k	n−k	NOUN
ejpam-5240	765	50	k	k	PROPN
ejpam-5240	765	51	)	)	PUNCT
ejpam-5240	765	52	;	;	PUNCT
ejpam-5240	765	53	on	on	ADP
ejpam-5240	765	54	the	the	DET
ejpam-5240	765	55	other	other	ADJ
ejpam-5240	765	56	hand	hand	NOUN
ejpam-5240	765	57	if	if	SCONJ
ejpam-5240	765	58	k	k	PROPN
ejpam-5240	765	59	=	=	PUNCT
ejpam-5240	765	60	0	0	NUM
ejpam-5240	765	61	or	or	CCONJ
ejpam-5240	765	62	⌊	⌊	X
ejpam-5240	765	63	n	n	ADV
ejpam-5240	765	64	2	2	NUM
ejpam-5240	765	65	⌋	⌋	NOUN
ejpam-5240	765	66	<	<	X
ejpam-5240	765	67	k	k	X
ejpam-5240	765	68	≤	≤	PROPN
ejpam-5240	765	69	n	n	CCONJ
ejpam-5240	765	70	,	,	PUNCT
ejpam-5240	765	71	deg(a	deg(a	PROPN
ejpam-5240	765	72	)	)	PUNCT
ejpam-5240	765	73	=	=	SYM
ejpam-5240	766	1	0	0	X
ejpam-5240	766	2	.	.	PUNCT
ejpam-5240	767	1	finally	finally	ADV
ejpam-5240	767	2	,	,	PUNCT
ejpam-5240	767	3	other	other	ADJ
ejpam-5240	767	4	parameters	parameter	NOUN
ejpam-5240	767	5	of	of	ADP
ejpam-5240	767	6	gs(n	gs(n	NOUN
ejpam-5240	767	7	,	,	PUNCT
ejpam-5240	767	8	k	k	NOUN
ejpam-5240	767	9	)	)	PUNCT
ejpam-5240	767	10	such	such	ADJ
ejpam-5240	767	11	as	as	ADP
ejpam-5240	767	12	the	the	DET
ejpam-5240	767	13	independence	independence	NOUN
ejpam-5240	767	14	number	number	NOUN
ejpam-5240	767	15	,	,	PUNCT
ejpam-5240	767	16	domination	domination	NOUN
ejpam-5240	767	17	number	number	NOUN
ejpam-5240	767	18	,	,	PUNCT
ejpam-5240	767	19	and	and	CCONJ
ejpam-5240	767	20	the	the	DET
ejpam-5240	767	21	isolate	isolate	ADJ
ejpam-5240	767	22	domination	domination	NOUN
ejpam-5240	767	23	number	number	NOUN
ejpam-5240	767	24	were	be	AUX
ejpam-5240	767	25	specified	specify	VERB
ejpam-5240	767	26	.	.	PUNCT
ejpam-5240	768	1	the	the	DET
ejpam-5240	768	2	independence	independence	NOUN
ejpam-5240	768	3	number	number	NOUN
ejpam-5240	768	4	of	of	ADP
ejpam-5240	768	5	gs(n	gs(n	NOUN
ejpam-5240	768	6	,	,	PUNCT
ejpam-5240	768	7	k	k	NOUN
ejpam-5240	768	8	)	)	PUNCT
ejpam-5240	768	9	when	when	SCONJ
ejpam-5240	768	10	1	1	NUM
ejpam-5240	768	11	≤	≤	NUM
ejpam-5240	768	12	k	k	X
ejpam-5240	768	13	≤	≤	NUM
ejpam-5240	768	14	⌊	⌊	VERB
ejpam-5240	768	15	n	n	DET
ejpam-5240	768	16	2	2	NUM
ejpam-5240	768	17	⌋	⌋	NOUN
ejpam-5240	768	18	is	be	AUX
ejpam-5240	768	19	equal	equal	ADJ
ejpam-5240	768	20	to	to	PART
ejpam-5240	768	21	⌊	⌊	PROPN
ejpam-5240	768	22	n	n	CCONJ
ejpam-5240	768	23	k	k	NOUN
ejpam-5240	768	24	⌋	⌋	NOUN
ejpam-5240	768	25	while	while	SCONJ
ejpam-5240	768	26	if	if	SCONJ
ejpam-5240	768	27	k	k	PROPN
ejpam-5240	768	28	=	=	SYM
ejpam-5240	768	29	0	0	NUM
ejpam-5240	768	30	or	or	CCONJ
ejpam-5240	768	31	⌊	⌊	X
ejpam-5240	768	32	n	n	ADV
ejpam-5240	768	33	2	2	NUM
ejpam-5240	768	34	⌋	⌋	NOUN
ejpam-5240	768	35	<	<	X
ejpam-5240	768	36	k	k	X
ejpam-5240	768	37	≤	≤	PROPN
ejpam-5240	768	38	n	n	CCONJ
ejpam-5240	768	39	,	,	PUNCT
ejpam-5240	768	40	then	then	ADV
ejpam-5240	768	41	the	the	DET
ejpam-5240	768	42	independence	independence	NOUN
ejpam-5240	768	43	number	number	NOUN
ejpam-5240	768	44	is	be	AUX
ejpam-5240	768	45	1	1	NUM
ejpam-5240	768	46	.	.	PUNCT
ejpam-5240	768	47	further	far	ADV
ejpam-5240	768	48	,	,	PUNCT
ejpam-5240	768	49	it	it	PRON
ejpam-5240	768	50	was	be	AUX
ejpam-5240	768	51	found	find	VERB
ejpam-5240	768	52	out	out	ADP
ejpam-5240	768	53	that	that	SCONJ
ejpam-5240	768	54	for	for	ADP
ejpam-5240	768	55	any	any	DET
ejpam-5240	768	56	value	value	NOUN
ejpam-5240	768	57	of	of	ADP
ejpam-5240	768	58	k	k	NOUN
ejpam-5240	768	59	,	,	PUNCT
ejpam-5240	768	60	we	we	PRON
ejpam-5240	768	61	have	have	VERB
ejpam-5240	768	62	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	768	63	,	,	PUNCT
ejpam-5240	768	64	k	k	NOUN
ejpam-5240	768	65	)	)	PUNCT
ejpam-5240	768	66	)	)	PUNCT
ejpam-5240	769	1	=	=	PUNCT
ejpam-5240	769	2	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	769	3	,	,	PUNCT
ejpam-5240	769	4	k	k	NOUN
ejpam-5240	769	5	)	)	PUNCT
ejpam-5240	769	6	)	)	PUNCT
ejpam-5240	769	7	.	.	PUNCT
ejpam-5240	770	1	if	if	SCONJ
ejpam-5240	770	2	1	1	NUM
ejpam-5240	770	3	≤	≤	NUM
ejpam-5240	770	4	k	k	X
ejpam-5240	770	5	≤	≤	NUM
ejpam-5240	770	6	⌊	⌊	VERB
ejpam-5240	770	7	n	n	DET
ejpam-5240	770	8	2	2	NUM
ejpam-5240	770	9	⌋	⌋	NOUN
ejpam-5240	770	10	,	,	PUNCT
ejpam-5240	770	11	then	then	ADV
ejpam-5240	770	12	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	770	13	,	,	PUNCT
ejpam-5240	770	14	k	k	NOUN
ejpam-5240	770	15	)	)	PUNCT
ejpam-5240	770	16	)	)	PUNCT
ejpam-5240	771	1	=	=	PUNCT
ejpam-5240	771	2	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	771	3	,	,	PUNCT
ejpam-5240	771	4	k	k	NOUN
ejpam-5240	771	5	)	)	PUNCT
ejpam-5240	771	6	)	)	PUNCT
ejpam-5240	772	1	=	=	PUNCT
ejpam-5240	772	2	⌊	⌊	VERB
ejpam-5240	772	3	n	n	PRON
ejpam-5240	772	4	k	k	PROPN
ejpam-5240	772	5	⌋	⌋	NOUN
ejpam-5240	772	6	.	.	PUNCT
ejpam-5240	773	1	meanwhile	meanwhile	ADV
ejpam-5240	773	2	,	,	PUNCT
ejpam-5240	773	3	γ(gs(n	γ(gs(n	PROPN
ejpam-5240	773	4	,	,	PUNCT
ejpam-5240	773	5	k	k	NOUN
ejpam-5240	773	6	)	)	PUNCT
ejpam-5240	773	7	)	)	PUNCT
ejpam-5240	774	1	=	=	PUNCT
ejpam-5240	774	2	γ0(gs(n	γ0(gs(n	PROPN
ejpam-5240	774	3	,	,	PUNCT
ejpam-5240	774	4	k	k	NOUN
ejpam-5240	774	5	)	)	PUNCT
ejpam-5240	774	6	)	)	PUNCT
ejpam-5240	775	1	=	=	SYM
ejpam-5240	775	2	1	1	NUM
ejpam-5240	775	3	if	if	SCONJ
ejpam-5240	775	4	k	k	PROPN
ejpam-5240	775	5	=	=	PUNCT
ejpam-5240	775	6	0	0	NUM
ejpam-5240	775	7	or	or	CCONJ
ejpam-5240	775	8	⌊	⌊	X
ejpam-5240	775	9	n	n	ADV
ejpam-5240	775	10	2	2	NUM
ejpam-5240	775	11	⌋	⌋	NOUN
ejpam-5240	775	12	<	<	X
ejpam-5240	775	13	k	k	PROPN
ejpam-5240	775	14	≤	≤	PROPN
ejpam-5240	775	15	n.	n.	NOUN
ejpam-5240	775	16	in	in	ADP
ejpam-5240	775	17	conclusion	conclusion	NOUN
ejpam-5240	775	18	,	,	PUNCT
ejpam-5240	775	19	we	we	PRON
ejpam-5240	775	20	determine	determine	VERB
ejpam-5240	775	21	the	the	DET
ejpam-5240	775	22	order	order	NOUN
ejpam-5240	775	23	,	,	PUNCT
ejpam-5240	775	24	size	size	NOUN
ejpam-5240	775	25	,	,	PUNCT
ejpam-5240	775	26	independence	independence	NOUN
ejpam-5240	775	27	number	number	NOUN
ejpam-5240	775	28	,	,	PUNCT
ejpam-5240	775	29	domination	domination	NOUN
ejpam-5240	775	30	number	number	NOUN
ejpam-5240	775	31	,	,	PUNCT
ejpam-5240	775	32	and	and	CCONJ
ejpam-5240	775	33	isolated	isolated	ADJ
ejpam-5240	775	34	domination	domination	NOUN
ejpam-5240	775	35	number	number	NOUN
ejpam-5240	775	36	of	of	ADP
ejpam-5240	775	37	a	a	DET
ejpam-5240	775	38	k	k	ADV
ejpam-5240	775	39	-	-	ADJ
ejpam-5240	775	40	restricted	restricted	ADJ
ejpam-5240	775	41	intersection	intersection	NOUN
ejpam-5240	775	42	graph	graph	NOUN
ejpam-5240	775	43	in	in	ADP
ejpam-5240	775	44	this	this	DET
ejpam-5240	775	45	study	study	NOUN
ejpam-5240	775	46	.	.	PUNCT
ejpam-5240	776	1	also	also	ADV
ejpam-5240	776	2	,	,	PUNCT
ejpam-5240	776	3	we	we	PRON
ejpam-5240	776	4	provided	provide	VERB
ejpam-5240	776	5	the	the	DET
ejpam-5240	776	6	necessary	necessary	ADJ
ejpam-5240	776	7	and	and	CCONJ
ejpam-5240	776	8	sufficient	sufficient	ADJ
ejpam-5240	776	9	conditions	condition	NOUN
ejpam-5240	776	10	for	for	ADP
ejpam-5240	776	11	a	a	DET
ejpam-5240	776	12	gs(n	gs(n	NOUN
ejpam-5240	776	13	,	,	PUNCT
ejpam-5240	776	14	k	k	NOUN
ejpam-5240	776	15	)	)	PUNCT
ejpam-5240	776	16	to	to	PART
ejpam-5240	776	17	be	be	AUX
ejpam-5240	776	18	isomorphic	isomorphic	ADJ
ejpam-5240	776	19	to	to	ADP
ejpam-5240	776	20	a	a	DET
ejpam-5240	776	21	cycle	cycle	NOUN
ejpam-5240	776	22	graph	graph	NOUN
ejpam-5240	776	23	and	and	CCONJ
ejpam-5240	776	24	a	a	DET
ejpam-5240	776	25	complete	complete	ADJ
ejpam-5240	776	26	graph	graph	NOUN
ejpam-5240	776	27	.	.	PUNCT
ejpam-5240	777	1	from	from	ADP
ejpam-5240	777	2	the	the	DET
ejpam-5240	777	3	results	result	NOUN
ejpam-5240	777	4	drawn	draw	VERB
ejpam-5240	777	5	,	,	PUNCT
ejpam-5240	777	6	the	the	DET
ejpam-5240	777	7	researchers	researcher	NOUN
ejpam-5240	777	8	believed	believe	VERB
ejpam-5240	777	9	that	that	SCONJ
ejpam-5240	777	10	a	a	DET
ejpam-5240	777	11	parallel	parallel	ADJ
ejpam-5240	777	12	study	study	NOUN
ejpam-5240	777	13	may	may	AUX
ejpam-5240	777	14	be	be	AUX
ejpam-5240	777	15	done	do	VERB
ejpam-5240	777	16	to	to	PART
ejpam-5240	777	17	further	far	ADV
ejpam-5240	777	18	characterize	characterize	VERB
ejpam-5240	777	19	gs(n	gs(n	NOUN
ejpam-5240	777	20	,	,	PUNCT
ejpam-5240	777	21	k	k	NOUN
ejpam-5240	777	22	)	)	PUNCT
ejpam-5240	777	23	.	.	PUNCT
ejpam-5240	778	1	in	in	ADP
ejpam-5240	778	2	particular	particular	ADJ
ejpam-5240	778	3	,	,	PUNCT
ejpam-5240	778	4	we	we	PRON
ejpam-5240	778	5	recommend	recommend	VERB
ejpam-5240	778	6	that	that	SCONJ
ejpam-5240	778	7	future	future	ADJ
ejpam-5240	778	8	studies	study	NOUN
ejpam-5240	778	9	find	find	VERB
ejpam-5240	778	10	other	other	ADJ
ejpam-5240	778	11	parameters	parameter	NOUN
ejpam-5240	778	12	of	of	ADP
ejpam-5240	778	13	a	a	DET
ejpam-5240	778	14	gs(n	gs(n	NOUN
ejpam-5240	778	15	,	,	PUNCT
ejpam-5240	778	16	k	k	NOUN
ejpam-5240	778	17	)	)	PUNCT
ejpam-5240	778	18	such	such	ADJ
ejpam-5240	778	19	as	as	ADP
ejpam-5240	778	20	its	its	PRON
ejpam-5240	778	21	girth	girth	NOUN
ejpam-5240	778	22	,	,	PUNCT
ejpam-5240	778	23	clique	clique	ADJ
ejpam-5240	778	24	number	number	NOUN
ejpam-5240	778	25	,	,	PUNCT
ejpam-5240	778	26	chromatic	chromatic	ADJ
ejpam-5240	778	27	number	number	NOUN
ejpam-5240	778	28	,	,	PUNCT
ejpam-5240	778	29	and	and	CCONJ
ejpam-5240	778	30	locating	locate	VERB
ejpam-5240	778	31	domination	domination	NOUN
ejpam-5240	778	32	number	number	NOUN
ejpam-5240	778	33	to	to	PART
ejpam-5240	778	34	name	name	VERB
ejpam-5240	778	35	a	a	DET
ejpam-5240	778	36	few	few	ADJ
ejpam-5240	778	37	,	,	PUNCT
ejpam-5240	778	38	for	for	ADP
ejpam-5240	778	39	this	this	PRON
ejpam-5240	778	40	will	will	AUX
ejpam-5240	778	41	also	also	ADV
ejpam-5240	778	42	be	be	AUX
ejpam-5240	778	43	helpful	helpful	ADJ
ejpam-5240	778	44	in	in	ADP
ejpam-5240	778	45	determining	determine	VERB
ejpam-5240	778	46	the	the	DET
ejpam-5240	778	47	graph	graph	NOUN
ejpam-5240	778	48	.	.	PUNCT
ejpam-5240	779	1	moreover	moreover	ADV
ejpam-5240	779	2	,	,	PUNCT
ejpam-5240	779	3	it	it	PRON
ejpam-5240	779	4	is	be	AUX
ejpam-5240	779	5	recommended	recommend	VERB
ejpam-5240	779	6	to	to	PART
ejpam-5240	779	7	further	far	ADV
ejpam-5240	779	8	investigate	investigate	VERB
ejpam-5240	779	9	gs(n	gs(n	NOUN
ejpam-5240	779	10	,	,	PUNCT
ejpam-5240	779	11	k	k	NOUN
ejpam-5240	779	12	)	)	PUNCT
ejpam-5240	779	13	proposing	propose	VERB
ejpam-5240	779	14	the	the	DET
ejpam-5240	779	15	utilization	utilization	NOUN
ejpam-5240	779	16	of	of	ADP
ejpam-5240	779	17	binary	binary	ADJ
ejpam-5240	779	18	graph	graph	NOUN
ejpam-5240	779	19	operations	operation	NOUN
ejpam-5240	779	20	wherein	wherein	SCONJ
ejpam-5240	779	21	some	some	PRON
ejpam-5240	779	22	of	of	ADP
ejpam-5240	779	23	these	these	PRON
ejpam-5240	779	24	are	be	AUX
ejpam-5240	779	25	the	the	DET
ejpam-5240	779	26	sum	sum	NOUN
ejpam-5240	779	27	of	of	ADP
ejpam-5240	779	28	joint	joint	ADJ
ejpam-5240	779	29	,	,	PUNCT
ejpam-5240	779	30	cartesian	cartesian	ADJ
ejpam-5240	779	31	product	product	NOUN
ejpam-5240	779	32	,	,	PUNCT
ejpam-5240	779	33	composition	composition	NOUN
ejpam-5240	779	34	,	,	PUNCT
ejpam-5240	779	35	edge	edge	NOUN
ejpam-5240	779	36	gluing	gluing	NOUN
ejpam-5240	779	37	,	,	PUNCT
ejpam-5240	779	38	and	and	CCONJ
ejpam-5240	779	39	vertex	vertex	NOUN
ejpam-5240	779	40	gluing	gluing	NOUN
ejpam-5240	779	41	may	may	AUX
ejpam-5240	779	42	be	be	AUX
ejpam-5240	779	43	imperatively	imperatively	ADV
ejpam-5240	779	44	conducted	conduct	VERB
ejpam-5240	779	45	.	.	PUNCT
ejpam-5240	780	1	lastly	lastly	ADV
ejpam-5240	780	2	,	,	PUNCT
ejpam-5240	780	3	future	future	ADJ
ejpam-5240	780	4	researchers	researcher	NOUN
ejpam-5240	780	5	may	may	AUX
ejpam-5240	780	6	consider	consider	VERB
ejpam-5240	780	7	adapting	adapt	VERB
ejpam-5240	780	8	gs(n	gs(n	NOUN
ejpam-5240	780	9	,	,	PUNCT
ejpam-5240	780	10	k	k	NOUN
ejpam-5240	780	11	)	)	PUNCT
ejpam-5240	780	12	in	in	ADP
ejpam-5240	780	13	solving	solve	VERB
ejpam-5240	780	14	real	real	ADJ
ejpam-5240	780	15	-	-	PUNCT
ejpam-5240	780	16	world	world	NOUN
ejpam-5240	780	17	problems	problem	NOUN
ejpam-5240	780	18	for	for	ADP
ejpam-5240	780	19	some	some	PRON
ejpam-5240	780	20	of	of	ADP
ejpam-5240	780	21	the	the	DET
ejpam-5240	780	22	results	result	NOUN
ejpam-5240	780	23	established	establish	VERB
ejpam-5240	780	24	are	be	AUX
ejpam-5240	780	25	based	base	VERB
ejpam-5240	780	26	on	on	ADP
ejpam-5240	780	27	the	the	DET
ejpam-5240	780	28	binomial	binomial	ADJ
ejpam-5240	780	29	coefficient	coefficient	NOUN
ejpam-5240	780	30	or	or	CCONJ
ejpam-5240	780	31	the	the	DET
ejpam-5240	780	32	combination	combination	NOUN
ejpam-5240	780	33	formula	formula	NOUN
ejpam-5240	780	34	that	that	PRON
ejpam-5240	780	35	has	have	VERB
ejpam-5240	780	36	several	several	ADJ
ejpam-5240	780	37	real	real	ADJ
ejpam-5240	780	38	-	-	PUNCT
ejpam-5240	780	39	world	world	NOUN
ejpam-5240	780	40	applications	application	NOUN
ejpam-5240	780	41	.	.	PUNCT
ejpam-5240	781	1	acknowledgements	acknowledgement	NOUN
ejpam-5240	781	2	the	the	DET
ejpam-5240	781	3	authors	author	NOUN
ejpam-5240	781	4	would	would	AUX
ejpam-5240	781	5	like	like	VERB
ejpam-5240	781	6	to	to	PART
ejpam-5240	781	7	acknowledge	acknowledge	VERB
ejpam-5240	781	8	the	the	DET
ejpam-5240	781	9	following	follow	VERB
ejpam-5240	781	10	institutions	institution	NOUN
ejpam-5240	781	11	for	for	ADP
ejpam-5240	781	12	their	their	PRON
ejpam-5240	781	13	support	support	NOUN
ejpam-5240	781	14	on	on	ADP
ejpam-5240	781	15	the	the	DET
ejpam-5240	781	16	fulfilment	fulfilment	NOUN
ejpam-5240	781	17	of	of	ADP
ejpam-5240	781	18	this	this	DET
ejpam-5240	781	19	research	research	NOUN
ejpam-5240	781	20	project	project	NOUN
ejpam-5240	781	21	:	:	PUNCT
ejpam-5240	781	22	batangas	batangas	PROPN
ejpam-5240	781	23	state	state	PROPN
ejpam-5240	781	24	university	university	PROPN
ejpam-5240	781	25	the	the	DET
ejpam-5240	781	26	national	national	PROPN
ejpam-5240	781	27	engineering	engineering	PROPN
ejpam-5240	781	28	university	university	PROPN
ejpam-5240	781	29	(	(	PUNCT
ejpam-5240	781	30	batstateu	batstateu	PROPN
ejpam-5240	781	31	-	-	PUNCT
ejpam-5240	781	32	tneu	tneu	PROPN
ejpam-5240	781	33	)	)	PUNCT
ejpam-5240	781	34	and	and	CCONJ
ejpam-5240	781	35	department	department	NOUN
ejpam-5240	781	36	of	of	ADP
ejpam-5240	781	37	science	science	NOUN
ejpam-5240	781	38	and	and	CCONJ
ejpam-5240	781	39	technology	technology	NOUN
ejpam-5240	781	40	science	science	PROPN
ejpam-5240	781	41	education	education	PROPN
ejpam-5240	781	42	institute	institute	PROPN
ejpam-5240	781	43	,	,	PUNCT
ejpam-5240	781	44	science	science	NOUN
ejpam-5240	781	45	and	and	CCONJ
ejpam-5240	781	46	technology	technology	NOUN
ejpam-5240	781	47	regional	regional	ADJ
ejpam-5240	781	48	alliance	alliance	NOUN
ejpam-5240	781	49	of	of	ADP
ejpam-5240	781	50	universities	university	NOUN
ejpam-5240	781	51	for	for	ADP
ejpam-5240	781	52	national	national	ADJ
ejpam-5240	781	53	development	development	NOUN
ejpam-5240	781	54	(	(	PUNCT
ejpam-5240	781	55	dost	dost	NOUN
ejpam-5240	781	56	-	-	PUNCT
ejpam-5240	781	57	sei	sei	ADJ
ejpam-5240	781	58	,	,	PUNCT
ejpam-5240	781	59	strand	strand	NOUN
ejpam-5240	781	60	)	)	PUNCT
ejpam-5240	781	61	.	.	PUNCT
ejpam-5240	782	1	also	also	ADV
ejpam-5240	782	2	,	,	PUNCT
ejpam-5240	782	3	the	the	DET
ejpam-5240	782	4	authors	author	NOUN
ejpam-5240	782	5	would	would	AUX
ejpam-5240	782	6	like	like	VERB
ejpam-5240	782	7	to	to	PART
ejpam-5240	782	8	extend	extend	VERB
ejpam-5240	782	9	their	their	PRON
ejpam-5240	782	10	heartfelt	heartfelt	ADJ
ejpam-5240	782	11	gratitude	gratitude	NOUN
ejpam-5240	782	12	to	to	ADP
ejpam-5240	782	13	the	the	DET
ejpam-5240	782	14	anonymous	anonymous	ADJ
ejpam-5240	782	15	reviewers	reviewer	NOUN
ejpam-5240	782	16	of	of	ADP
ejpam-5240	782	17	this	this	DET
ejpam-5240	782	18	study	study	NOUN
ejpam-5240	782	19	and	and	CCONJ
ejpam-5240	782	20	the	the	DET
ejpam-5240	782	21	editors	editor	NOUN
ejpam-5240	782	22	of	of	ADP
ejpam-5240	782	23	this	this	DET
ejpam-5240	782	24	journal	journal	NOUN
ejpam-5240	782	25	,	,	PUNCT
ejpam-5240	782	26	for	for	ADP
ejpam-5240	782	27	their	their	PRON
ejpam-5240	782	28	efforts	effort	NOUN
ejpam-5240	782	29	in	in	ADP
ejpam-5240	782	30	reviewing	review	VERB
ejpam-5240	782	31	this	this	DET
ejpam-5240	782	32	publication	publication	NOUN
ejpam-5240	782	33	.	.	PUNCT
ejpam-5240	783	1	references	reference	NOUN
ejpam-5240	783	2	[	[	X
ejpam-5240	783	3	1	1	NUM
ejpam-5240	783	4	]	]	PUNCT
ejpam-5240	783	5	a.	a.	NOUN
ejpam-5240	783	6	neumaier	neumai	ADJ
ejpam-5240	783	7	a.	a.	PROPN
ejpam-5240	783	8	brouwer	brouwer	PROPN
ejpam-5240	783	9	,	,	PUNCT
ejpam-5240	783	10	a.	a.	PROPN
ejpam-5240	783	11	cohen	cohen	PROPN
ejpam-5240	783	12	.	.	PUNCT
ejpam-5240	784	1	distance	distance	NOUN
ejpam-5240	784	2	-	-	PUNCT
ejpam-5240	784	3	regular	regular	ADJ
ejpam-5240	784	4	graphs	graph	NOUN
ejpam-5240	784	5	.	.	PUNCT
ejpam-5240	785	1	springer	springer	PROPN
ejpam-5240	785	2	berlin	berlin	PROPN
ejpam-5240	785	3	,	,	PUNCT
ejpam-5240	785	4	heidelberg	heidelberg	PROPN
ejpam-5240	785	5	,	,	PUNCT
ejpam-5240	785	6	1989	1989	NUM
ejpam-5240	785	7	.	.	PUNCT
ejpam-5240	786	1	references	reference	NOUN
ejpam-5240	786	2	1803	1803	NUM
ejpam-5240	786	3	[	[	X
ejpam-5240	786	4	2	2	NUM
ejpam-5240	786	5	]	]	PUNCT
ejpam-5240	786	6	richard	richard	PROPN
ejpam-5240	786	7	a.	a.	PROPN
ejpam-5240	786	8	brualdi	brualdi	PROPN
ejpam-5240	786	9	.	.	PUNCT
ejpam-5240	787	1	introductory	introductory	ADJ
ejpam-5240	787	2	combinatorics	combinatoric	NOUN
ejpam-5240	787	3	.	.	PUNCT
ejpam-5240	788	1	prentice	prentice	NOUN
ejpam-5240	788	2	-	-	PUNCT
ejpam-5240	788	3	hall	hall	NOUN
ejpam-5240	788	4	,	,	PUNCT
ejpam-5240	788	5	2010	2010	NUM
ejpam-5240	788	6	.	.	PUNCT
ejpam-5240	789	1	[	[	X
ejpam-5240	789	2	3	3	X
ejpam-5240	789	3	]	]	X
ejpam-5240	789	4	j.	j.	PROPN
ejpam-5240	789	5	harris	harris	PROPN
ejpam-5240	789	6	et	et	PROPN
ejpam-5240	789	7	al	al	PROPN
ejpam-5240	789	8	.	.	PROPN
ejpam-5240	789	9	combinatorics	combinatoric	NOUN
ejpam-5240	789	10	and	and	CCONJ
ejpam-5240	789	11	graph	graph	NOUN
ejpam-5240	789	12	theory	theory	NOUN
ejpam-5240	789	13	.	.	PUNCT
ejpam-5240	790	1	springer	springer	NOUN
ejpam-5240	790	2	nature	nature	NOUN
ejpam-5240	790	3	,	,	PUNCT
ejpam-5240	790	4	2008	2008	NUM
ejpam-5240	790	5	.	.	PUNCT
ejpam-5240	791	1	[	[	X
ejpam-5240	791	2	4	4	X
ejpam-5240	791	3	]	]	X
ejpam-5240	791	4	d.	d.	PROPN
ejpam-5240	791	5	fowler	fowler	PROPN
ejpam-5240	791	6	.	.	PUNCT
ejpam-5240	792	1	the	the	DET
ejpam-5240	792	2	binomial	binomial	ADJ
ejpam-5240	792	3	coefficient	coefficient	NOUN
ejpam-5240	792	4	function	function	NOUN
ejpam-5240	792	5	.	.	PUNCT
ejpam-5240	793	1	the	the	DET
ejpam-5240	793	2	american	american	PROPN
ejpam-5240	793	3	mathematical	mathematical	PROPN
ejpam-5240	793	4	monthly	monthly	ADV
ejpam-5240	793	5	,	,	PUNCT
ejpam-5240	793	6	1996	1996	NUM
ejpam-5240	793	7	.	.	PUNCT
ejpam-5240	794	1	[	[	X
ejpam-5240	794	2	5	5	NUM
ejpam-5240	794	3	]	]	PUNCT
ejpam-5240	794	4	m.	m.	NOUN
ejpam-5240	794	5	golumbic	golumbic	NOUN
ejpam-5240	794	6	.	.	PUNCT
ejpam-5240	795	1	algorithmic	algorithmic	ADJ
ejpam-5240	795	2	graph	graph	NOUN
ejpam-5240	795	3	theory	theory	NOUN
ejpam-5240	795	4	and	and	CCONJ
ejpam-5240	795	5	perfect	perfect	ADJ
ejpam-5240	795	6	graphs	graph	NOUN
ejpam-5240	795	7	.	.	PUNCT
ejpam-5240	796	1	elsevier	elsevier	NOUN
ejpam-5240	796	2	,	,	PUNCT
ejpam-5240	796	3	2004	2004	NUM
ejpam-5240	796	4	.	.	PUNCT
ejpam-5240	797	1	[	[	X
ejpam-5240	797	2	6	6	NUM
ejpam-5240	797	3	]	]	PUNCT
ejpam-5240	797	4	d.	d.	PROPN
ejpam-5240	797	5	guichard	guichard	PROPN
ejpam-5240	797	6	.	.	PUNCT
ejpam-5240	798	1	graph	graph	NOUN
ejpam-5240	798	2	coloring	coloring	NOUN
ejpam-5240	798	3	and	and	CCONJ
ejpam-5240	798	4	independence	independence	NOUN
ejpam-5240	798	5	number	number	NOUN
ejpam-5240	798	6	of	of	ADP
ejpam-5240	798	7	a	a	DET
ejpam-5240	798	8	complete	complete	ADJ
ejpam-5240	798	9	graph	graph	NOUN
ejpam-5240	798	10	.	.	PUNCT
ejpam-5240	799	1	whitman	whitman	PROPN
ejpam-5240	799	2	college	college	PROPN
ejpam-5240	799	3	2017	2017	NUM
ejpam-5240	799	4	,	,	PUNCT
ejpam-5240	799	5	2017	2017	NUM
ejpam-5240	799	6	.	.	PUNCT
ejpam-5240	800	1	[	[	X
ejpam-5240	800	2	7	7	X
ejpam-5240	800	3	]	]	X
ejpam-5240	800	4	i.	i.	PROPN
ejpam-5240	800	5	hamid	hamid	PROPN
ejpam-5240	800	6	and	and	CCONJ
ejpam-5240	800	7	s.	s.	PROPN
ejpam-5240	800	8	balamurugan	balamurugan	VERB
ejpam-5240	800	9	.	.	PUNCT
ejpam-5240	801	1	isolate	isolate	VERB
ejpam-5240	801	2	domination	domination	NOUN
ejpam-5240	801	3	in	in	ADP
ejpam-5240	801	4	graphs	graph	NOUN
ejpam-5240	801	5	.	.	PUNCT
ejpam-5240	802	1	arab	arab	PROPN
ejpam-5240	802	2	journal	journal	PROPN
ejpam-5240	802	3	of	of	ADP
ejpam-5240	802	4	mathematical	mathematical	ADJ
ejpam-5240	802	5	sciences	science	NOUN
ejpam-5240	802	6	,	,	PUNCT
ejpam-5240	802	7	2016	2016	NUM
ejpam-5240	802	8	.	.	PUNCT
ejpam-5240	803	1	[	[	X
ejpam-5240	803	2	8	8	NUM
ejpam-5240	803	3	]	]	X
ejpam-5240	803	4	n.	n.	PROPN
ejpam-5240	803	5	helwig	helwig	PROPN
ejpam-5240	803	6	.	.	PUNCT
ejpam-5240	804	1	introduction	introduction	NOUN
ejpam-5240	804	2	to	to	PART
ejpam-5240	804	3	set	set	VERB
ejpam-5240	804	4	theory	theory	NOUN
ejpam-5240	804	5	.	.	PUNCT
ejpam-5240	805	1	university	university	NOUN
ejpam-5240	805	2	of	of	ADP
ejpam-5240	805	3	minnesota	minnesota	PROPN
ejpam-5240	805	4	,	,	PUNCT
ejpam-5240	805	5	2020	2020	NUM
ejpam-5240	805	6	.	.	PUNCT
ejpam-5240	806	1	[	[	X
ejpam-5240	806	2	9	9	NUM
ejpam-5240	806	3	]	]	X
ejpam-5240	806	4	d.	d.	PROPN
ejpam-5240	806	5	magpantay	magpantay	PROPN
ejpam-5240	806	6	j.c	j.c	PROPN
ejpam-5240	806	7	.	.	PROPN
ejpam-5240	806	8	bonifacio	bonifacio	PROPN
ejpam-5240	806	9	,	,	PUNCT
ejpam-5240	806	10	c.j	c.j	PROPN
ejpam-5240	806	11	.	.	PROPN
ejpam-5240	806	12	andaya	andaya	PROPN
ejpam-5240	806	13	.	.	PUNCT
ejpam-5240	807	1	on	on	ADP
ejpam-5240	807	2	the	the	DET
ejpam-5240	807	3	j	j	PROPN
ejpam-5240	807	4	-	-	PUNCT
ejpam-5240	807	5	edge	edge	NOUN
ejpam-5240	807	6	intersection	intersection	NOUN
ejpam-5240	807	7	graph	graph	NOUN
ejpam-5240	807	8	of	of	ADP
ejpam-5240	807	9	cycle	cycle	NOUN
ejpam-5240	807	10	graph	graph	NOUN
ejpam-5240	807	11	.	.	PUNCT
ejpam-5240	808	1	european	european	PROPN
ejpam-5240	808	2	journal	journal	PROPN
ejpam-5240	808	3	of	of	ADP
ejpam-5240	808	4	pure	pure	ADJ
ejpam-5240	808	5	and	and	CCONJ
ejpam-5240	808	6	applied	applied	ADJ
ejpam-5240	808	7	mathematics	mathematic	NOUN
ejpam-5240	808	8	,	,	PUNCT
ejpam-5240	808	9	2023	2023	NUM
ejpam-5240	808	10	.	.	PUNCT
ejpam-5240	809	1	[	[	X
ejpam-5240	809	2	10	10	NUM
ejpam-5240	809	3	]	]	X
ejpam-5240	809	4	l.	l.	NOUN
ejpam-5240	809	5	pósa	pósa	PUNCT
ejpam-5240	809	6	p.	p.	PROPN
ejpam-5240	809	7	erdős	erdős	PROPN
ejpam-5240	809	8	,	,	PUNCT
ejpam-5240	809	9	a.	a.	PROPN
ejpam-5240	809	10	goodman	goodman	PROPN
ejpam-5240	809	11	.	.	PUNCT
ejpam-5240	810	1	the	the	DET
ejpam-5240	810	2	representation	representation	NOUN
ejpam-5240	810	3	of	of	ADP
ejpam-5240	810	4	a	a	DET
ejpam-5240	810	5	graph	graph	NOUN
ejpam-5240	810	6	by	by	ADP
ejpam-5240	810	7	set	set	VERB
ejpam-5240	810	8	intersections	intersection	NOUN
ejpam-5240	810	9	.	.	PUNCT
ejpam-5240	811	1	canadian	canadian	ADJ
ejpam-5240	811	2	journal	journal	PROPN
ejpam-5240	811	3	of	of	ADP
ejpam-5240	811	4	mathematics	mathematics	PROPN
ejpam-5240	811	5	,	,	PUNCT
ejpam-5240	811	6	18(1):106–112	18(1):106–112	PROPN
ejpam-5240	811	7	,	,	PUNCT
ejpam-5240	811	8	1966	1966	NUM
ejpam-5240	811	9	.	.	PUNCT
ejpam-5240	812	1	[	[	X
ejpam-5240	812	2	11	11	NUM
ejpam-5240	812	3	]	]	PUNCT
ejpam-5240	812	4	m.	m.	NOUN
ejpam-5240	812	5	schaefer	schaefer	NOUN
ejpam-5240	812	6	.	.	PUNCT
ejpam-5240	813	1	complexity	complexity	NOUN
ejpam-5240	813	2	of	of	ADP
ejpam-5240	813	3	some	some	DET
ejpam-5240	813	4	geometric	geometric	ADJ
ejpam-5240	813	5	and	and	CCONJ
ejpam-5240	813	6	topological	topological	ADJ
ejpam-5240	813	7	problems	problem	NOUN
ejpam-5240	813	8	.	.	PUNCT
ejpam-5240	814	1	springer	springer	NOUN
ejpam-5240	814	2	link	link	PROPN
ejpam-5240	814	3	,	,	PUNCT
ejpam-5240	814	4	2010	2010	NUM
ejpam-5240	814	5	.	.	PUNCT
ejpam-5240	815	1	[	[	X
ejpam-5240	815	2	12	12	NUM
ejpam-5240	815	3	]	]	PUNCT
ejpam-5240	815	4	s.	s.	PROPN
ejpam-5240	815	5	selkow	selkow	PROPN
ejpam-5240	815	6	.	.	PUNCT
ejpam-5240	816	1	the	the	DET
ejpam-5240	816	2	independence	independence	NOUN
ejpam-5240	816	3	number	number	NOUN
ejpam-5240	816	4	of	of	ADP
ejpam-5240	816	5	graphs	graph	NOUN
ejpam-5240	816	6	in	in	ADP
ejpam-5240	816	7	terms	term	NOUN
ejpam-5240	816	8	of	of	ADP
ejpam-5240	816	9	degrees	degree	NOUN
ejpam-5240	816	10	.	.	PUNCT
ejpam-5240	817	1	discrete	discrete	ADJ
ejpam-5240	817	2	mathematics	mathematic	NOUN
ejpam-5240	817	3	,	,	PUNCT
ejpam-5240	817	4	122(1):343–348	122(1):343–348	NUM
ejpam-5240	817	5	,	,	PUNCT
ejpam-5240	817	6	1993	1993	NUM
ejpam-5240	817	7	.	.	PUNCT
ejpam-5240	818	1	[	[	X
ejpam-5240	818	2	13	13	NUM
ejpam-5240	818	3	]	]	PUNCT
ejpam-5240	818	4	a.	a.	NOUN
ejpam-5240	818	5	sugumaran	sugumaran	NOUN
ejpam-5240	818	6	and	and	CCONJ
ejpam-5240	818	7	e.	e.	PROPN
ejpam-5240	818	8	jayachandran	jayachandran	PROPN
ejpam-5240	818	9	.	.	PUNCT
ejpam-5240	819	1	domination	domination	NOUN
ejpam-5240	819	2	number	number	NOUN
ejpam-5240	819	3	of	of	ADP
ejpam-5240	819	4	some	some	DET
ejpam-5240	819	5	graphs	graph	NOUN
ejpam-5240	819	6	.	.	PUNCT
ejpam-5240	820	1	international	international	ADJ
ejpam-5240	820	2	journal	journal	NOUN
ejpam-5240	820	3	of	of	ADP
ejpam-5240	820	4	scientific	scientific	ADJ
ejpam-5240	820	5	development	development	NOUN
ejpam-5240	820	6	and	and	CCONJ
ejpam-5240	820	7	research	research	NOUN
ejpam-5240	820	8	(	(	PUNCT
ejpam-5240	820	9	ijsdr	ijsdr	PROPN
ejpam-5240	820	10	)	)	PUNCT
ejpam-5240	820	11	,	,	PUNCT
ejpam-5240	820	12	2018	2018	NUM
ejpam-5240	820	13	.	.	PUNCT
ejpam-5240	821	1	[	[	X
ejpam-5240	821	2	14	14	NUM
ejpam-5240	821	3	]	]	X
ejpam-5240	821	4	e.	e.	PROPN
ejpam-5240	821	5	szpilrajn	szpilrajn	PROPN
ejpam-5240	821	6	-	-	PUNCT
ejpam-5240	821	7	marczewski	marczewski	NOUN
ejpam-5240	821	8	.	.	PUNCT
ejpam-5240	822	1	on	on	ADP
ejpam-5240	822	2	two	two	NUM
ejpam-5240	822	3	properties	property	NOUN
ejpam-5240	822	4	of	of	ADP
ejpam-5240	822	5	set	set	ADJ
ejpam-5240	822	6	classes	class	NOUN
ejpam-5240	822	7	.	.	PUNCT
ejpam-5240	823	1	american	american	PROPN
ejpam-5240	823	2	mathematical	mathematical	PROPN
ejpam-5240	823	3	society	society	NOUN
ejpam-5240	823	4	,	,	PUNCT
ejpam-5240	823	5	33:303–307	33:303–307	NUM
ejpam-5240	823	6	,	,	PUNCT
ejpam-5240	823	7	1945	1945	NUM
ejpam-5240	823	8	.	.	PUNCT
ejpam-5240	824	1	[	[	X
ejpam-5240	824	2	15	15	NUM
ejpam-5240	824	3	]	]	X
ejpam-5240	824	4	f.r	f.r	PROPN
ejpam-5240	824	5	.	.	PROPN
ejpam-5240	824	6	mcmorris	mcmorris	PROPN
ejpam-5240	824	7	t.	t.	PROPN
ejpam-5240	824	8	mckee	mckee	PROPN
ejpam-5240	824	9	.	.	PUNCT
ejpam-5240	825	1	topics	topic	NOUN
ejpam-5240	825	2	in	in	ADP
ejpam-5240	825	3	intersection	intersection	NOUN
ejpam-5240	825	4	graph	graph	NOUN
ejpam-5240	825	5	theory	theory	NOUN
ejpam-5240	825	6	.	.	PUNCT
ejpam-5240	826	1	society	society	NOUN
ejpam-5240	826	2	for	for	ADP
ejpam-5240	826	3	industrial	industrial	ADJ
ejpam-5240	826	4	and	and	CCONJ
ejpam-5240	826	5	applied	applied	ADJ
ejpam-5240	826	6	mathematics	mathematic	NOUN
ejpam-5240	826	7	,	,	PUNCT
ejpam-5240	826	8	philadelphia	philadelphia	PROPN
ejpam-5240	826	9	,	,	PUNCT
ejpam-5240	826	10	united	united	PROPN
ejpam-5240	826	11	states	states	PROPN
ejpam-5240	826	12	,	,	PUNCT
ejpam-5240	826	13	1999	1999	NUM
ejpam-5240	826	14	.	.	PUNCT