id	sid	tid	token	lemma	pos
ejpam-4870	1	1	european	european	PROPN
ejpam-4870	1	2	journal	journal	PROPN
ejpam-4870	1	3	of	of	ADP
ejpam-4870	1	4	pure	pure	ADJ
ejpam-4870	1	5	and	and	CCONJ
ejpam-4870	1	6	applied	apply	VERB
ejpam-4870	1	7	mathematics	mathematic	NOUN
ejpam-4870	1	8	vol	vol	NOUN
ejpam-4870	1	9	.	.	PUNCT
ejpam-4870	2	1	16	16	NUM
ejpam-4870	2	2	,	,	PUNCT
ejpam-4870	2	3	no	no	INTJ
ejpam-4870	2	4	.	.	NOUN
ejpam-4870	2	5	4	4	NUM
ejpam-4870	2	6	,	,	PUNCT
ejpam-4870	2	7	2023	2023	NUM
ejpam-4870	2	8	,	,	PUNCT
ejpam-4870	2	9	2476	2476	NUM
ejpam-4870	2	10	-	-	SYM
ejpam-4870	2	11	2498	2498	NUM
ejpam-4870	2	12	issn	issn	PROPN
ejpam-4870	2	13	1307	1307	NUM
ejpam-4870	2	14	-	-	SYM
ejpam-4870	2	15	5543	5543	NUM
ejpam-4870	2	16	–	–	PUNCT
ejpam-4870	2	17	ejpam.com	ejpam.com	X
ejpam-4870	2	18	published	publish	VERB
ejpam-4870	2	19	by	by	ADP
ejpam-4870	2	20	new	new	PROPN
ejpam-4870	2	21	york	york	PROPN
ejpam-4870	2	22	business	business	PROPN
ejpam-4870	2	23	global	global	PROPN
ejpam-4870	2	24	on	on	ADP
ejpam-4870	2	25	the	the	DET
ejpam-4870	2	26	j	j	NOUN
ejpam-4870	2	27	-	-	PUNCT
ejpam-4870	2	28	edge	edge	NOUN
ejpam-4870	2	29	intersection	intersection	NOUN
ejpam-4870	2	30	graph	graph	NOUN
ejpam-4870	2	31	of	of	ADP
ejpam-4870	2	32	cycle	cycle	NOUN
ejpam-4870	2	33	graph	graph	NOUN
ejpam-4870	2	34	jhon	jhon	PROPN
ejpam-4870	2	35	cris	cris	PROPN
ejpam-4870	2	36	c.	c.	PROPN
ejpam-4870	2	37	bonifacio1	bonifacio1	PROPN
ejpam-4870	2	38	,	,	PUNCT
ejpam-4870	2	39	clarence	clarence	NOUN
ejpam-4870	2	40	joy	joy	NOUN
ejpam-4870	2	41	m.	m.	NOUN
ejpam-4870	2	42	andaya1	andaya1	PROPN
ejpam-4870	2	43	,	,	PUNCT
ejpam-4870	3	1	daryl	daryl	PROPN
ejpam-4870	3	2	m.	m.	PROPN
ejpam-4870	3	3	magpantay1,∗	magpantay1,∗	PROPN
ejpam-4870	3	4	1	1	NUM
ejpam-4870	3	5	college	college	NOUN
ejpam-4870	3	6	of	of	ADP
ejpam-4870	3	7	arts	art	NOUN
ejpam-4870	3	8	and	and	CCONJ
ejpam-4870	3	9	sciences	sciences	PROPN
ejpam-4870	3	10	,	,	PUNCT
ejpam-4870	3	11	batangas	batangas	PROPN
ejpam-4870	3	12	state	state	PROPN
ejpam-4870	3	13	university	university	PROPN
ejpam-4870	3	14	the	the	DET
ejpam-4870	3	15	national	national	PROPN
ejpam-4870	3	16	engineering	engineering	PROPN
ejpam-4870	3	17	university	university	PROPN
ejpam-4870	3	18	,	,	PUNCT
ejpam-4870	3	19	batangas	batangas	PROPN
ejpam-4870	3	20	city	city	PROPN
ejpam-4870	3	21	,	,	PUNCT
ejpam-4870	3	22	batangas	batangas	PROPN
ejpam-4870	3	23	,	,	PUNCT
ejpam-4870	3	24	philippines	philippine	NOUN
ejpam-4870	3	25	abstract	abstract	ADJ
ejpam-4870	3	26	.	.	PUNCT
ejpam-4870	4	1	this	this	DET
ejpam-4870	4	2	paper	paper	NOUN
ejpam-4870	4	3	defines	define	VERB
ejpam-4870	4	4	a	a	DET
ejpam-4870	4	5	new	new	ADJ
ejpam-4870	4	6	class	class	NOUN
ejpam-4870	4	7	of	of	ADP
ejpam-4870	4	8	graphs	graph	NOUN
ejpam-4870	4	9	using	use	VERB
ejpam-4870	4	10	the	the	DET
ejpam-4870	4	11	spanning	span	VERB
ejpam-4870	4	12	subgraphs	subgraph	NOUN
ejpam-4870	4	13	of	of	ADP
ejpam-4870	4	14	a	a	DET
ejpam-4870	4	15	cycle	cycle	NOUN
ejpam-4870	4	16	graph	graph	NOUN
ejpam-4870	4	17	as	as	ADP
ejpam-4870	4	18	vertices	vertex	NOUN
ejpam-4870	4	19	.	.	PUNCT
ejpam-4870	5	1	this	this	DET
ejpam-4870	5	2	class	class	NOUN
ejpam-4870	5	3	of	of	ADP
ejpam-4870	5	4	graphs	graph	NOUN
ejpam-4870	5	5	is	be	AUX
ejpam-4870	5	6	called	call	VERB
ejpam-4870	5	7	j	j	NOUN
ejpam-4870	5	8	-	-	PUNCT
ejpam-4870	5	9	edge	edge	NOUN
ejpam-4870	5	10	intersection	intersection	NOUN
ejpam-4870	5	11	graph	graph	NOUN
ejpam-4870	5	12	of	of	ADP
ejpam-4870	5	13	cycle	cycle	NOUN
ejpam-4870	5	14	graph	graph	NOUN
ejpam-4870	5	15	,	,	PUNCT
ejpam-4870	5	16	denoted	denote	VERB
ejpam-4870	5	17	by	by	ADP
ejpam-4870	5	18	ec(n	ec(n	PROPN
ejpam-4870	5	19	,	,	PUNCT
ejpam-4870	5	20	j	j	PROPN
ejpam-4870	5	21	)	)	PUNCT
ejpam-4870	5	22	.	.	PUNCT
ejpam-4870	6	1	the	the	DET
ejpam-4870	6	2	vertex	vertex	NOUN
ejpam-4870	6	3	set	set	NOUN
ejpam-4870	6	4	of	of	ADP
ejpam-4870	6	5	ec(n	ec(n	PROPN
ejpam-4870	6	6	,	,	PUNCT
ejpam-4870	6	7	j	j	PROPN
ejpam-4870	6	8	)	)	PUNCT
ejpam-4870	6	9	is	be	AUX
ejpam-4870	6	10	the	the	DET
ejpam-4870	6	11	set	set	NOUN
ejpam-4870	6	12	of	of	ADP
ejpam-4870	6	13	spanning	span	VERB
ejpam-4870	6	14	subgraphs	subgraph	NOUN
ejpam-4870	6	15	of	of	ADP
ejpam-4870	6	16	cycle	cycle	NOUN
ejpam-4870	6	17	graph	graph	NOUN
ejpam-4870	6	18	with	with	ADP
ejpam-4870	6	19	j	j	PROPN
ejpam-4870	6	20	edges	edge	NOUN
ejpam-4870	6	21	where	where	SCONJ
ejpam-4870	6	22	n	n	NUM
ejpam-4870	6	23	≥	≥	X
ejpam-4870	6	24	3	3	NUM
ejpam-4870	6	25	and	and	CCONJ
ejpam-4870	6	26	j	j	PROPN
ejpam-4870	6	27	is	be	AUX
ejpam-4870	6	28	a	a	DET
ejpam-4870	6	29	nonnegative	nonnegative	ADJ
ejpam-4870	6	30	integer	integer	NOUN
ejpam-4870	6	31	such	such	ADJ
ejpam-4870	6	32	that	that	SCONJ
ejpam-4870	6	33	1	1	NUM
ejpam-4870	6	34	≤	≤	NUM
ejpam-4870	6	35	j	j	PROPN
ejpam-4870	6	36	≤	≤	PROPN
ejpam-4870	6	37	n.	n.	PROPN
ejpam-4870	6	38	two	two	NUM
ejpam-4870	6	39	distinct	distinct	ADJ
ejpam-4870	6	40	vertices	vertex	NOUN
ejpam-4870	6	41	are	be	AUX
ejpam-4870	6	42	adjacent	adjacent	ADJ
ejpam-4870	6	43	if	if	SCONJ
ejpam-4870	6	44	they	they	PRON
ejpam-4870	6	45	have	have	VERB
ejpam-4870	6	46	exactly	exactly	ADV
ejpam-4870	6	47	one	one	NUM
ejpam-4870	6	48	edge	edge	NOUN
ejpam-4870	6	49	in	in	ADP
ejpam-4870	6	50	common	common	ADJ
ejpam-4870	6	51	.	.	PUNCT
ejpam-4870	7	1	ec(n	ec(n	PROPN
ejpam-4870	7	2	,	,	PUNCT
ejpam-4870	7	3	j	j	NOUN
ejpam-4870	7	4	)	)	PUNCT
ejpam-4870	7	5	is	be	AUX
ejpam-4870	7	6	considered	consider	VERB
ejpam-4870	7	7	as	as	ADP
ejpam-4870	7	8	a	a	DET
ejpam-4870	7	9	simple	simple	ADJ
ejpam-4870	7	10	graph	graph	NOUN
ejpam-4870	7	11	.	.	PUNCT
ejpam-4870	8	1	furthermore	furthermore	ADV
ejpam-4870	8	2	,	,	PUNCT
ejpam-4870	8	3	ec(n	ec(n	PROPN
ejpam-4870	8	4	,	,	PUNCT
ejpam-4870	8	5	j	j	NOUN
ejpam-4870	8	6	)	)	PUNCT
ejpam-4870	8	7	is	be	AUX
ejpam-4870	8	8	characterized	characterize	VERB
ejpam-4870	8	9	by	by	ADP
ejpam-4870	8	10	the	the	DET
ejpam-4870	8	11	value	value	NOUN
ejpam-4870	8	12	of	of	ADP
ejpam-4870	8	13	j	j	PROPN
ejpam-4870	8	14	that	that	PRON
ejpam-4870	8	15	is	be	AUX
ejpam-4870	8	16	when	when	SCONJ
ejpam-4870	8	17	j	j	PROPN
ejpam-4870	8	18	=	=	SYM
ejpam-4870	8	19	1	1	NUM
ejpam-4870	8	20	or	or	CCONJ
ejpam-4870	8	21	⌈n	⌈n	VERB
ejpam-4870	8	22	2	2	NUM
ejpam-4870	8	23	⌉	⌉	ADP
ejpam-4870	8	24	<	<	X
ejpam-4870	8	25	j	j	PROPN
ejpam-4870	8	26	≤	≤	PROPN
ejpam-4870	8	27	n	n	PRON
ejpam-4870	8	28	and	and	CCONJ
ejpam-4870	8	29	2	2	NUM
ejpam-4870	8	30	≤	≤	NUM
ejpam-4870	8	31	j	j	PROPN
ejpam-4870	8	32	≤	≤	PROPN
ejpam-4870	8	33	⌈n	⌈n	NOUN
ejpam-4870	8	34	2	2	NUM
ejpam-4870	8	35	⌉.	⌉.	ADV
ejpam-4870	8	36	when	when	SCONJ
ejpam-4870	8	37	j	j	PROPN
ejpam-4870	8	38	=	=	SYM
ejpam-4870	8	39	1	1	NUM
ejpam-4870	8	40	or	or	CCONJ
ejpam-4870	8	41	⌈n	⌈n	VERB
ejpam-4870	8	42	2	2	NUM
ejpam-4870	8	43	⌉	⌉	ADP
ejpam-4870	8	44	<	<	X
ejpam-4870	8	45	j	j	PROPN
ejpam-4870	8	46	≤	≤	NUM
ejpam-4870	8	47	n	n	CCONJ
ejpam-4870	8	48	,	,	PUNCT
ejpam-4870	8	49	the	the	DET
ejpam-4870	8	50	new	new	ADJ
ejpam-4870	8	51	graph	graph	NOUN
ejpam-4870	8	52	only	only	ADV
ejpam-4870	8	53	produced	produce	VERB
ejpam-4870	8	54	an	an	DET
ejpam-4870	8	55	empty	empty	ADJ
ejpam-4870	8	56	graph	graph	NOUN
ejpam-4870	8	57	.	.	PUNCT
ejpam-4870	9	1	hence	hence	ADV
ejpam-4870	9	2	,	,	PUNCT
ejpam-4870	9	3	the	the	DET
ejpam-4870	9	4	proponents	proponent	NOUN
ejpam-4870	9	5	only	only	ADV
ejpam-4870	9	6	considered	consider	VERB
ejpam-4870	9	7	the	the	DET
ejpam-4870	9	8	value	value	NOUN
ejpam-4870	9	9	when	when	SCONJ
ejpam-4870	9	10	2	2	NUM
ejpam-4870	9	11	≤	≤	NUM
ejpam-4870	9	12	j	j	PROPN
ejpam-4870	9	13	≤	≤	PROPN
ejpam-4870	9	14	⌈n	⌈n	NOUN
ejpam-4870	9	15	2	2	NUM
ejpam-4870	9	16	⌉	⌉	X
ejpam-4870	9	17	in	in	ADP
ejpam-4870	9	18	determining	determine	VERB
ejpam-4870	9	19	the	the	DET
ejpam-4870	9	20	order	order	NOUN
ejpam-4870	9	21	and	and	CCONJ
ejpam-4870	9	22	size	size	NOUN
ejpam-4870	9	23	of	of	ADP
ejpam-4870	9	24	ec(n	ec(n	PROPN
ejpam-4870	9	25	,	,	PUNCT
ejpam-4870	9	26	j	j	PROPN
ejpam-4870	9	27	)	)	PUNCT
ejpam-4870	9	28	.	.	PUNCT
ejpam-4870	10	1	moreover	moreover	ADV
ejpam-4870	10	2	,	,	PUNCT
ejpam-4870	10	3	this	this	DET
ejpam-4870	10	4	paper	paper	NOUN
ejpam-4870	10	5	discusses	discuss	VERB
ejpam-4870	10	6	necessary	necessary	ADJ
ejpam-4870	10	7	and	and	CCONJ
ejpam-4870	10	8	sufficient	sufficient	ADJ
ejpam-4870	10	9	conditions	condition	NOUN
ejpam-4870	10	10	where	where	SCONJ
ejpam-4870	10	11	the	the	DET
ejpam-4870	10	12	j	j	PROPN
ejpam-4870	10	13	-	-	PUNCT
ejpam-4870	10	14	edge	edge	NOUN
ejpam-4870	10	15	intersection	intersection	NOUN
ejpam-4870	10	16	graph	graph	NOUN
ejpam-4870	10	17	of	of	ADP
ejpam-4870	10	18	cn	cn	PROPN
ejpam-4870	10	19	is	be	AUX
ejpam-4870	10	20	isomorphic	isomorphic	ADJ
ejpam-4870	10	21	to	to	ADP
ejpam-4870	10	22	the	the	DET
ejpam-4870	10	23	cycle	cycle	NOUN
ejpam-4870	10	24	graph	graph	NOUN
ejpam-4870	10	25	.	.	PUNCT
ejpam-4870	11	1	furthermore	furthermore	ADV
ejpam-4870	11	2	,	,	PUNCT
ejpam-4870	11	3	the	the	DET
ejpam-4870	11	4	researchers	researcher	NOUN
ejpam-4870	11	5	determined	determine	VERB
ejpam-4870	11	6	a	a	DET
ejpam-4870	11	7	lower	lower	ADV
ejpam-4870	11	8	bound	bind	VERB
ejpam-4870	11	9	for	for	ADP
ejpam-4870	11	10	the	the	DET
ejpam-4870	11	11	independence	independence	NOUN
ejpam-4870	11	12	number	number	NOUN
ejpam-4870	11	13	,	,	PUNCT
ejpam-4870	11	14	and	and	CCONJ
ejpam-4870	11	15	an	an	DET
ejpam-4870	11	16	upper	upper	ADJ
ejpam-4870	11	17	bound	bind	VERB
ejpam-4870	11	18	for	for	ADP
ejpam-4870	11	19	the	the	DET
ejpam-4870	11	20	domination	domination	NOUN
ejpam-4870	11	21	number	number	NOUN
ejpam-4870	11	22	of	of	ADP
ejpam-4870	11	23	ec(n	ec(n	PROPN
ejpam-4870	11	24	,	,	PUNCT
ejpam-4870	11	25	j	j	PROPN
ejpam-4870	11	26	)	)	PUNCT
ejpam-4870	11	27	when	when	SCONJ
ejpam-4870	11	28	j	j	PROPN
ejpam-4870	11	29	=	=	PROPN
ejpam-4870	11	30	2	2	NUM
ejpam-4870	11	31	.	.	NOUN
ejpam-4870	11	32	2020	2020	NUM
ejpam-4870	11	33	mathematics	mathematic	NOUN
ejpam-4870	11	34	subject	subject	NOUN
ejpam-4870	11	35	classifications	classification	NOUN
ejpam-4870	11	36	:	:	PUNCT
ejpam-4870	11	37	05	05	NUM
ejpam-4870	11	38	key	key	ADJ
ejpam-4870	11	39	words	word	NOUN
ejpam-4870	11	40	and	and	CCONJ
ejpam-4870	11	41	phrases	phrase	NOUN
ejpam-4870	11	42	:	:	PUNCT
ejpam-4870	11	43	edge	edge	NOUN
ejpam-4870	11	44	intersection	intersection	NOUN
ejpam-4870	11	45	graph	graph	NOUN
ejpam-4870	11	46	,	,	PUNCT
ejpam-4870	11	47	cycle	cycle	NOUN
ejpam-4870	11	48	graph	graph	NOUN
ejpam-4870	11	49	,	,	PUNCT
ejpam-4870	11	50	special	special	ADJ
ejpam-4870	11	51	classes	class	NOUN
ejpam-4870	11	52	of	of	ADP
ejpam-4870	11	53	graphs	graph	NOUN
ejpam-4870	11	54	,	,	PUNCT
ejpam-4870	11	55	parameters	parameter	NOUN
ejpam-4870	11	56	of	of	ADP
ejpam-4870	11	57	graphs	graph	NOUN
ejpam-4870	11	58	,	,	PUNCT
ejpam-4870	11	59	spanning	span	VERB
ejpam-4870	11	60	subgraph	subgraph	NOUN
ejpam-4870	11	61	1	1	NUM
ejpam-4870	11	62	.	.	PUNCT
ejpam-4870	11	63	introduction	introduction	NOUN
ejpam-4870	11	64	graph	graph	NOUN
ejpam-4870	11	65	is	be	AUX
ejpam-4870	11	66	a	a	DET
ejpam-4870	11	67	very	very	ADV
ejpam-4870	11	68	effective	effective	ADJ
ejpam-4870	11	69	tool	tool	NOUN
ejpam-4870	11	70	to	to	PART
ejpam-4870	11	71	model	model	VERB
ejpam-4870	11	72	issues	issue	NOUN
ejpam-4870	11	73	that	that	PRON
ejpam-4870	11	74	have	have	VERB
ejpam-4870	11	75	their	their	PRON
ejpam-4870	11	76	origins	origin	NOUN
ejpam-4870	11	77	in	in	ADP
ejpam-4870	11	78	almost	almost	ADV
ejpam-4870	11	79	every	every	PRON
ejpam-4870	11	80	aspect	aspect	NOUN
ejpam-4870	11	81	of	of	ADP
ejpam-4870	11	82	human	human	ADJ
ejpam-4870	11	83	life	life	NOUN
ejpam-4870	11	84	.	.	PUNCT
ejpam-4870	12	1	studying	study	VERB
ejpam-4870	12	2	graphs	graph	NOUN
ejpam-4870	12	3	through	through	ADP
ejpam-4870	12	4	a	a	DET
ejpam-4870	12	5	framework	framework	NOUN
ejpam-4870	12	6	provides	provide	VERB
ejpam-4870	12	7	answers	answer	NOUN
ejpam-4870	12	8	to	to	ADP
ejpam-4870	12	9	many	many	ADJ
ejpam-4870	12	10	arrangement	arrangement	NOUN
ejpam-4870	12	11	,	,	PUNCT
ejpam-4870	12	12	networking	networking	NOUN
ejpam-4870	12	13	,	,	PUNCT
ejpam-4870	12	14	optimization	optimization	NOUN
ejpam-4870	12	15	,	,	PUNCT
ejpam-4870	12	16	matching	matching	NOUN
ejpam-4870	12	17	,	,	PUNCT
ejpam-4870	12	18	and	and	CCONJ
ejpam-4870	12	19	operational	operational	ADJ
ejpam-4870	12	20	problems	problem	NOUN
ejpam-4870	12	21	.	.	PUNCT
ejpam-4870	13	1	the	the	DET
ejpam-4870	13	2	development	development	NOUN
ejpam-4870	13	3	,	,	PUNCT
ejpam-4870	13	4	computation	computation	NOUN
ejpam-4870	13	5	,	,	PUNCT
ejpam-4870	13	6	and	and	CCONJ
ejpam-4870	13	7	maintenance	maintenance	NOUN
ejpam-4870	13	8	of	of	ADP
ejpam-4870	13	9	multi	multi	ADJ
ejpam-4870	13	10	-	-	ADJ
ejpam-4870	13	11	part	part	ADJ
ejpam-4870	13	12	electric	electric	ADJ
ejpam-4870	13	13	circuits	circuit	NOUN
ejpam-4870	13	14	are	be	AUX
ejpam-4870	13	15	central	central	ADJ
ejpam-4870	13	16	to	to	ADP
ejpam-4870	13	17	the	the	DET
ejpam-4870	13	18	field	field	NOUN
ejpam-4870	13	19	of	of	ADP
ejpam-4870	13	20	electrical	electrical	ADJ
ejpam-4870	13	21	engineering	engineering	NOUN
ejpam-4870	13	22	,	,	PUNCT
ejpam-4870	13	23	and	and	CCONJ
ejpam-4870	13	24	these	these	DET
ejpam-4870	13	25	circuits	circuit	NOUN
ejpam-4870	13	26	are	be	AUX
ejpam-4870	13	27	frequently	frequently	ADV
ejpam-4870	13	28	represented	represent	VERB
ejpam-4870	13	29	graphically	graphically	ADV
ejpam-4870	13	30	using	use	VERB
ejpam-4870	13	31	graph	graph	NOUN
ejpam-4870	13	32	theory	theory	NOUN
ejpam-4870	13	33	techniques	technique	NOUN
ejpam-4870	13	34	.	.	PUNCT
ejpam-4870	14	1	since	since	SCONJ
ejpam-4870	14	2	graphs	graph	NOUN
ejpam-4870	14	3	are	be	AUX
ejpam-4870	14	4	very	very	ADV
ejpam-4870	14	5	helpful	helpful	ADJ
ejpam-4870	14	6	in	in	ADP
ejpam-4870	14	7	understanding	understand	VERB
ejpam-4870	14	8	things	thing	NOUN
ejpam-4870	14	9	,	,	PUNCT
ejpam-4870	14	10	many	many	ADJ
ejpam-4870	14	11	researchers	researcher	NOUN
ejpam-4870	14	12	have	have	AUX
ejpam-4870	14	13	created	create	VERB
ejpam-4870	14	14	their	their	PRON
ejpam-4870	14	15	own	own	ADJ
ejpam-4870	14	16	graphs	graph	NOUN
ejpam-4870	14	17	to	to	PART
ejpam-4870	14	18	become	become	VERB
ejpam-4870	14	19	a	a	DET
ejpam-4870	14	20	new	new	ADJ
ejpam-4870	14	21	field	field	NOUN
ejpam-4870	14	22	of	of	ADP
ejpam-4870	14	23	study	study	NOUN
ejpam-4870	14	24	.	.	PUNCT
ejpam-4870	15	1	all	all	DET
ejpam-4870	15	2	newly	newly	ADV
ejpam-4870	15	3	created	create	VERB
ejpam-4870	15	4	graphs	graph	NOUN
ejpam-4870	15	5	can	can	AUX
ejpam-4870	15	6	be	be	AUX
ejpam-4870	15	7	used	use	VERB
ejpam-4870	15	8	to	to	PART
ejpam-4870	15	9	better	well	ADV
ejpam-4870	15	10	understand	understand	VERB
ejpam-4870	15	11	a	a	DET
ejpam-4870	15	12	concept	concept	NOUN
ejpam-4870	15	13	.	.	PUNCT
ejpam-4870	16	1	a	a	DET
ejpam-4870	16	2	graph	graph	NOUN
ejpam-4870	16	3	is	be	AUX
ejpam-4870	16	4	an	an	DET
ejpam-4870	16	5	interesting	interesting	ADJ
ejpam-4870	16	6	concept	concept	NOUN
ejpam-4870	16	7	,	,	PUNCT
ejpam-4870	16	8	for	for	ADP
ejpam-4870	16	9	this	this	PRON
ejpam-4870	16	10	motivates	motivate	VERB
ejpam-4870	16	11	the	the	DET
ejpam-4870	16	12	proponents	proponent	NOUN
ejpam-4870	16	13	to	to	PART
ejpam-4870	16	14	create	create	VERB
ejpam-4870	16	15	their	their	PRON
ejpam-4870	16	16	own	own	ADJ
ejpam-4870	16	17	graphs	graph	NOUN
ejpam-4870	16	18	and	and	CCONJ
ejpam-4870	16	19	explore	explore	VERB
ejpam-4870	16	20	a	a	DET
ejpam-4870	16	21	related	related	ADJ
ejpam-4870	16	22	study	study	NOUN
ejpam-4870	16	23	.	.	PUNCT
ejpam-4870	17	1	in	in	ADP
ejpam-4870	17	2	the	the	DET
ejpam-4870	17	3	study	study	NOUN
ejpam-4870	17	4	entitled	entitle	VERB
ejpam-4870	17	5	“	"	PUNCT
ejpam-4870	17	6	on	on	ADP
ejpam-4870	17	7	the	the	DET
ejpam-4870	17	8	edge	edge	NOUN
ejpam-4870	17	9	-	-	PUNCT
ejpam-4870	17	10	intersection	intersection	NOUN
ejpam-4870	17	11	graphs	graph	NOUN
ejpam-4870	17	12	of	of	ADP
ejpam-4870	17	13	k	k	NOUN
ejpam-4870	17	14	-	-	PUNCT
ejpam-4870	17	15	bend	bend	ADJ
ejpam-4870	17	16	∗corresponding	∗corresponde	VERB
ejpam-4870	17	17	author	author	NOUN
ejpam-4870	17	18	.	.	PUNCT
ejpam-4870	18	1	doi	doi	NOUN
ejpam-4870	18	2	:	:	PUNCT
ejpam-4870	18	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4870	https://doi.org/10.29020/nybg.ejpam.v16i4.4870	NOUN
ejpam-4870	18	4	email	email	NOUN
ejpam-4870	18	5	addresses	address	NOUN
ejpam-4870	18	6	:	:	PUNCT
ejpam-4870	18	7	daryl.magpantay@g.batstate-u.edu.ph	daryl.magpantay@g.batstate-u.edu.ph	PROPN
ejpam-4870	18	8	(	(	PUNCT
ejpam-4870	18	9	d.	d.	PROPN
ejpam-4870	18	10	magpantay	magpantay	PROPN
ejpam-4870	18	11	)	)	PUNCT
ejpam-4870	18	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4870	18	13	2476	2476	NUM
ejpam-4870	18	14	©	©	ADP
ejpam-4870	18	15	2023	2023	NUM
ejpam-4870	18	16	ejpam	ejpam	NOUN
ejpam-4870	18	17	all	all	DET
ejpam-4870	18	18	rights	right	NOUN
ejpam-4870	18	19	reserved	reserve	VERB
ejpam-4870	18	20	.	.	PUNCT
ejpam-4870	19	1	j.c	j.c	PROPN
ejpam-4870	19	2	.	.	PROPN
ejpam-4870	19	3	bonifacio	bonifacio	PROPN
ejpam-4870	19	4	,	,	PUNCT
ejpam-4870	19	5	c.j	c.j	PROPN
ejpam-4870	19	6	.	.	PROPN
ejpam-4870	19	7	andaya	andaya	PROPN
ejpam-4870	19	8	,	,	PUNCT
ejpam-4870	19	9	d.	d.	PROPN
ejpam-4870	19	10	magpantay	magpantay	PROPN
ejpam-4870	19	11	/	/	SYM
ejpam-4870	19	12	eur	eur	PROPN
ejpam-4870	19	13	.	.	PUNCT
ejpam-4870	20	1	j.	j.	PROPN
ejpam-4870	20	2	pure	pure	PROPN
ejpam-4870	20	3	appl	appl	PROPN
ejpam-4870	20	4	.	.	PROPN
ejpam-4870	20	5	math	math	PROPN
ejpam-4870	20	6	,	,	PUNCT
ejpam-4870	20	7	16	16	NUM
ejpam-4870	20	8	(	(	PUNCT
ejpam-4870	20	9	4	4	NUM
ejpam-4870	20	10	)	)	PUNCT
ejpam-4870	20	11	(	(	PUNCT
ejpam-4870	20	12	2023	2023	NUM
ejpam-4870	20	13	)	)	PUNCT
ejpam-4870	20	14	,	,	PUNCT
ejpam-4870	20	15	2476	2476	NUM
ejpam-4870	20	16	-	-	SYM
ejpam-4870	20	17	2498	2498	NUM
ejpam-4870	20	18	2477	2477	NUM
ejpam-4870	20	19	paths	path	NOUN
ejpam-4870	20	20	in	in	ADP
ejpam-4870	20	21	grids”[2	grids”[2	PROPN
ejpam-4870	20	22	]	]	PUNCT
ejpam-4870	20	23	,	,	PUNCT
ejpam-4870	20	24	the	the	DET
ejpam-4870	20	25	motivation	motivation	NOUN
ejpam-4870	20	26	for	for	ADP
ejpam-4870	20	27	creating	create	VERB
ejpam-4870	20	28	this	this	DET
ejpam-4870	20	29	graph	graph	NOUN
ejpam-4870	20	30	is	be	AUX
ejpam-4870	20	31	an	an	DET
ejpam-4870	20	32	application	application	NOUN
ejpam-4870	20	33	in	in	ADP
ejpam-4870	20	34	conflict	conflict	NOUN
ejpam-4870	20	35	resolutions	resolution	NOUN
ejpam-4870	20	36	of	of	ADP
ejpam-4870	20	37	paths	path	NOUN
ejpam-4870	20	38	in	in	ADP
ejpam-4870	20	39	grid	grid	NOUN
ejpam-4870	20	40	networks	network	NOUN
ejpam-4870	20	41	.	.	PUNCT
ejpam-4870	21	1	moreover	moreover	ADV
ejpam-4870	21	2	,	,	PUNCT
ejpam-4870	21	3	one	one	NUM
ejpam-4870	21	4	interesting	interesting	ADJ
ejpam-4870	21	5	topic	topic	NOUN
ejpam-4870	21	6	in	in	ADP
ejpam-4870	21	7	graph	graph	NOUN
ejpam-4870	21	8	theory	theory	NOUN
ejpam-4870	21	9	is	be	AUX
ejpam-4870	21	10	the	the	DET
ejpam-4870	21	11	edge	edge	NOUN
ejpam-4870	21	12	intersection	intersection	NOUN
ejpam-4870	21	13	graph	graph	NOUN
ejpam-4870	21	14	.	.	PUNCT
ejpam-4870	22	1	in	in	ADP
ejpam-4870	22	2	an	an	DET
ejpam-4870	22	3	edge	edge	NOUN
ejpam-4870	22	4	intersection	intersection	NOUN
ejpam-4870	22	5	graph	graph	NOUN
ejpam-4870	22	6	,	,	PUNCT
ejpam-4870	22	7	the	the	DET
ejpam-4870	22	8	vertices	vertex	NOUN
ejpam-4870	22	9	of	of	ADP
ejpam-4870	22	10	a	a	DET
ejpam-4870	22	11	graph	graph	NOUN
ejpam-4870	22	12	are	be	AUX
ejpam-4870	22	13	usually	usually	ADV
ejpam-4870	22	14	represented	represent	VERB
ejpam-4870	22	15	by	by	ADP
ejpam-4870	22	16	the	the	DET
ejpam-4870	22	17	members	member	NOUN
ejpam-4870	22	18	of	of	ADP
ejpam-4870	22	19	some	some	DET
ejpam-4870	22	20	family	family	NOUN
ejpam-4870	22	21	of	of	ADP
ejpam-4870	22	22	sets	set	NOUN
ejpam-4870	22	23	;	;	PUNCT
ejpam-4870	22	24	and	and	CCONJ
ejpam-4870	22	25	two	two	NUM
ejpam-4870	22	26	vertices	vertex	NOUN
ejpam-4870	22	27	are	be	AUX
ejpam-4870	22	28	adjacent	adjacent	ADJ
ejpam-4870	22	29	if	if	SCONJ
ejpam-4870	22	30	the	the	DET
ejpam-4870	22	31	intersection	intersection	NOUN
ejpam-4870	22	32	of	of	ADP
ejpam-4870	22	33	their	their	PRON
ejpam-4870	22	34	corresponding	corresponding	ADJ
ejpam-4870	22	35	sets	set	NOUN
ejpam-4870	22	36	satisfies	satisfy	VERB
ejpam-4870	22	37	some	some	DET
ejpam-4870	22	38	specified	specified	ADJ
ejpam-4870	22	39	condition	condition	NOUN
ejpam-4870	22	40	.	.	PUNCT
ejpam-4870	23	1	the	the	DET
ejpam-4870	23	2	set	set	NOUN
ejpam-4870	23	3	of	of	ADP
ejpam-4870	23	4	rules	rule	NOUN
ejpam-4870	23	5	used	use	VERB
ejpam-4870	23	6	to	to	PART
ejpam-4870	23	7	define	define	VERB
ejpam-4870	23	8	the	the	DET
ejpam-4870	23	9	vertex	vertex	NOUN
ejpam-4870	23	10	and	and	CCONJ
ejpam-4870	23	11	edge	edge	NOUN
ejpam-4870	23	12	sets	set	NOUN
ejpam-4870	23	13	is	be	AUX
ejpam-4870	23	14	known	know	VERB
ejpam-4870	23	15	as	as	ADP
ejpam-4870	23	16	a	a	DET
ejpam-4870	23	17	model	model	NOUN
ejpam-4870	23	18	.	.	PUNCT
ejpam-4870	24	1	in	in	ADP
ejpam-4870	24	2	an	an	DET
ejpam-4870	24	3	intersection	intersection	NOUN
ejpam-4870	24	4	graph	graph	NOUN
ejpam-4870	24	5	model	model	NOUN
ejpam-4870	24	6	,	,	PUNCT
ejpam-4870	24	7	the	the	DET
ejpam-4870	24	8	choice	choice	NOUN
ejpam-4870	24	9	of	of	ADP
ejpam-4870	24	10	sets	set	NOUN
ejpam-4870	24	11	to	to	PART
ejpam-4870	24	12	represent	represent	VERB
ejpam-4870	24	13	the	the	DET
ejpam-4870	24	14	vertices	vertex	NOUN
ejpam-4870	24	15	of	of	ADP
ejpam-4870	24	16	a	a	DET
ejpam-4870	24	17	graph	graph	NOUN
ejpam-4870	24	18	pre	pre	NOUN
ejpam-4870	24	19	-	-	NOUN
ejpam-4870	24	20	determines	determine	VERB
ejpam-4870	24	21	the	the	DET
ejpam-4870	24	22	edges	edge	NOUN
ejpam-4870	24	23	and	and	CCONJ
ejpam-4870	24	24	the	the	DET
ejpam-4870	24	25	specific	specific	ADJ
ejpam-4870	24	26	sets	set	NOUN
ejpam-4870	24	27	corresponding	correspond	VERB
ejpam-4870	24	28	to	to	ADP
ejpam-4870	24	29	each	each	DET
ejpam-4870	24	30	vertex	vertex	NOUN
ejpam-4870	24	31	are	be	AUX
ejpam-4870	24	32	a	a	DET
ejpam-4870	24	33	representation	representation	NOUN
ejpam-4870	24	34	of	of	ADP
ejpam-4870	24	35	the	the	DET
ejpam-4870	24	36	graph	graph	NOUN
ejpam-4870	24	37	.	.	PUNCT
ejpam-4870	25	1	a	a	DET
ejpam-4870	25	2	graph	graph	NOUN
ejpam-4870	25	3	is	be	AUX
ejpam-4870	25	4	representable	representable	ADJ
ejpam-4870	25	5	with	with	ADP
ejpam-4870	25	6	respect	respect	NOUN
ejpam-4870	25	7	to	to	ADP
ejpam-4870	25	8	a	a	DET
ejpam-4870	25	9	given	give	VERB
ejpam-4870	25	10	model	model	NOUN
ejpam-4870	25	11	if	if	SCONJ
ejpam-4870	25	12	there	there	PRON
ejpam-4870	25	13	is	be	VERB
ejpam-4870	25	14	some	some	DET
ejpam-4870	25	15	representation	representation	NOUN
ejpam-4870	25	16	.	.	PUNCT
ejpam-4870	26	1	there	there	PRON
ejpam-4870	26	2	are	be	VERB
ejpam-4870	26	3	several	several	ADJ
ejpam-4870	26	4	studies	study	NOUN
ejpam-4870	26	5	in	in	ADP
ejpam-4870	26	6	relation	relation	NOUN
ejpam-4870	26	7	to	to	PART
ejpam-4870	26	8	edge	edge	VERB
ejpam-4870	26	9	intersection	intersection	NOUN
ejpam-4870	26	10	.	.	PUNCT
ejpam-4870	27	1	in	in	ADP
ejpam-4870	27	2	the	the	DET
ejpam-4870	27	3	paper	paper	NOUN
ejpam-4870	27	4	[	[	X
ejpam-4870	27	5	4	4	NUM
ejpam-4870	27	6	]	]	PUNCT
ejpam-4870	27	7	,	,	PUNCT
ejpam-4870	27	8	they	they	PRON
ejpam-4870	27	9	investigated	investigate	VERB
ejpam-4870	27	10	the	the	DET
ejpam-4870	27	11	class	class	NOUN
ejpam-4870	27	12	of	of	ADP
ejpam-4870	27	13	edge	edge	NOUN
ejpam-4870	27	14	intersection	intersection	NOUN
ejpam-4870	27	15	graphs	graph	NOUN
ejpam-4870	27	16	of	of	ADP
ejpam-4870	27	17	a	a	DET
ejpam-4870	27	18	collection	collection	NOUN
ejpam-4870	27	19	of	of	ADP
ejpam-4870	27	20	paths	path	NOUN
ejpam-4870	27	21	in	in	ADP
ejpam-4870	27	22	a	a	DET
ejpam-4870	27	23	tree	tree	NOUN
ejpam-4870	27	24	(	(	PUNCT
ejpam-4870	27	25	ept	ept	NOUN
ejpam-4870	27	26	graphs	graph	NOUN
ejpam-4870	27	27	)	)	PUNCT
ejpam-4870	27	28	where	where	SCONJ
ejpam-4870	27	29	two	two	NUM
ejpam-4870	27	30	paths	path	NOUN
ejpam-4870	27	31	edge	edge	VERB
ejpam-4870	27	32	intersect	intersect	ADJ
ejpam-4870	27	33	if	if	SCONJ
ejpam-4870	27	34	they	they	PRON
ejpam-4870	27	35	share	share	VERB
ejpam-4870	27	36	an	an	DET
ejpam-4870	27	37	edge	edge	NOUN
ejpam-4870	27	38	.	.	PUNCT
ejpam-4870	28	1	the	the	DET
ejpam-4870	28	2	cliques	clique	NOUN
ejpam-4870	28	3	of	of	ADP
ejpam-4870	28	4	an	an	DET
ejpam-4870	28	5	ept	ept	NOUN
ejpam-4870	28	6	graph	graph	NOUN
ejpam-4870	28	7	are	be	AUX
ejpam-4870	28	8	characterized	characterize	VERB
ejpam-4870	28	9	and	and	CCONJ
ejpam-4870	28	10	shown	show	VERB
ejpam-4870	28	11	to	to	PART
ejpam-4870	28	12	have	have	VERB
ejpam-4870	28	13	strong	strong	ADJ
ejpam-4870	28	14	.	.	PUNCT
ejpam-4870	29	1	another	another	PRON
ejpam-4870	29	2	is	be	AUX
ejpam-4870	29	3	the	the	DET
ejpam-4870	29	4	study	study	NOUN
ejpam-4870	29	5	[	[	X
ejpam-4870	29	6	1	1	X
ejpam-4870	29	7	]	]	PUNCT
ejpam-4870	29	8	that	that	PRON
ejpam-4870	29	9	presents	present	VERB
ejpam-4870	29	10	some	some	DET
ejpam-4870	29	11	other	other	ADJ
ejpam-4870	29	12	results	result	NOUN
ejpam-4870	29	13	about	about	ADP
ejpam-4870	29	14	edge	edge	NOUN
ejpam-4870	29	15	intersection	intersection	NOUN
ejpam-4870	29	16	graphs	graph	NOUN
ejpam-4870	29	17	of	of	ADP
ejpam-4870	29	18	paths	path	NOUN
ejpam-4870	29	19	on	on	ADP
ejpam-4870	29	20	a	a	DET
ejpam-4870	29	21	grid	grid	NOUN
ejpam-4870	29	22	and	and	CCONJ
ejpam-4870	29	23	shows	show	VERB
ejpam-4870	29	24	several	several	ADJ
ejpam-4870	29	25	results	result	NOUN
ejpam-4870	29	26	of	of	ADP
ejpam-4870	29	27	the	the	DET
ejpam-4870	29	28	other	other	ADJ
ejpam-4870	29	29	clsses	clsse	NOUN
ejpam-4870	29	30	of	of	ADP
ejpam-4870	29	31	graphs	graph	NOUN
ejpam-4870	29	32	.	.	PUNCT
ejpam-4870	30	1	this	this	DET
ejpam-4870	30	2	paper	paper	NOUN
ejpam-4870	30	3	sought	seek	VERB
ejpam-4870	30	4	to	to	PART
ejpam-4870	30	5	introduce	introduce	VERB
ejpam-4870	30	6	the	the	DET
ejpam-4870	30	7	j	j	PROPN
ejpam-4870	30	8	-	-	PUNCT
ejpam-4870	30	9	edge	edge	NOUN
ejpam-4870	30	10	intersection	intersection	NOUN
ejpam-4870	30	11	graph	graph	NOUN
ejpam-4870	30	12	of	of	ADP
ejpam-4870	30	13	cn	cn	PROPN
ejpam-4870	30	14	.	.	PUNCT
ejpam-4870	31	1	this	this	DET
ejpam-4870	31	2	study	study	NOUN
ejpam-4870	31	3	aims	aim	VERB
ejpam-4870	31	4	to	to	PART
ejpam-4870	31	5	achieve	achieve	VERB
ejpam-4870	31	6	the	the	DET
ejpam-4870	31	7	following	following	ADJ
ejpam-4870	31	8	objectives	objective	NOUN
ejpam-4870	31	9	:	:	PUNCT
ejpam-4870	31	10	(	(	PUNCT
ejpam-4870	31	11	i	i	NOUN
ejpam-4870	31	12	)	)	PUNCT
ejpam-4870	31	13	to	to	PART
ejpam-4870	31	14	define	define	VERB
ejpam-4870	31	15	a	a	DET
ejpam-4870	31	16	j	j	NOUN
ejpam-4870	31	17	-	-	PUNCT
ejpam-4870	31	18	edge	edge	NOUN
ejpam-4870	31	19	intersection	intersection	NOUN
ejpam-4870	31	20	graph	graph	NOUN
ejpam-4870	31	21	of	of	ADP
ejpam-4870	31	22	cn	cn	PROPN
ejpam-4870	31	23	;	;	PUNCT
ejpam-4870	31	24	(	(	PUNCT
ejpam-4870	31	25	ii	ii	NOUN
ejpam-4870	31	26	)	)	PUNCT
ejpam-4870	31	27	to	to	PART
ejpam-4870	31	28	find	find	VERB
ejpam-4870	31	29	necessary	necessary	ADJ
ejpam-4870	31	30	and	and	CCONJ
ejpam-4870	31	31	sufficient	sufficient	ADJ
ejpam-4870	31	32	conditions	condition	NOUN
ejpam-4870	31	33	when	when	SCONJ
ejpam-4870	31	34	j	j	NOUN
ejpam-4870	31	35	-	-	PUNCT
ejpam-4870	31	36	edge	edge	NOUN
ejpam-4870	31	37	intersection	intersection	NOUN
ejpam-4870	31	38	graph	graph	NOUN
ejpam-4870	31	39	of	of	ADP
ejpam-4870	31	40	cn	cn	PROPN
ejpam-4870	31	41	is	be	AUX
ejpam-4870	31	42	isomorphic	isomorphic	ADJ
ejpam-4870	31	43	to	to	ADP
ejpam-4870	31	44	a	a	DET
ejpam-4870	31	45	special	special	ADJ
ejpam-4870	31	46	class	class	NOUN
ejpam-4870	31	47	of	of	ADP
ejpam-4870	31	48	graph	graph	NOUN
ejpam-4870	31	49	;	;	PUNCT
ejpam-4870	31	50	(	(	PUNCT
ejpam-4870	31	51	iii	iii	X
ejpam-4870	31	52	)	)	PUNCT
ejpam-4870	31	53	to	to	PART
ejpam-4870	31	54	identify	identify	VERB
ejpam-4870	31	55	some	some	PRON
ejpam-4870	31	56	of	of	ADP
ejpam-4870	31	57	the	the	DET
ejpam-4870	31	58	parameters	parameter	NOUN
ejpam-4870	31	59	of	of	ADP
ejpam-4870	31	60	a	a	DET
ejpam-4870	31	61	j	j	NOUN
ejpam-4870	31	62	-	-	PUNCT
ejpam-4870	31	63	edge	edge	NOUN
ejpam-4870	31	64	intersection	intersection	NOUN
ejpam-4870	31	65	graph	graph	NOUN
ejpam-4870	31	66	of	of	ADP
ejpam-4870	31	67	cn	cn	PROPN
ejpam-4870	31	68	such	such	ADJ
ejpam-4870	31	69	as	as	ADP
ejpam-4870	31	70	:	:	PUNCT
ejpam-4870	31	71	(	(	PUNCT
ejpam-4870	31	72	a	a	X
ejpam-4870	31	73	)	)	PUNCT
ejpam-4870	31	74	order	order	NOUN
ejpam-4870	31	75	;	;	PUNCT
ejpam-4870	31	76	and	and	CCONJ
ejpam-4870	31	77	(	(	PUNCT
ejpam-4870	31	78	b	b	NOUN
ejpam-4870	31	79	)	)	PUNCT
ejpam-4870	31	80	size	size	NOUN
ejpam-4870	31	81	.	.	PUNCT
ejpam-4870	32	1	(	(	PUNCT
ejpam-4870	32	2	iv	iv	X
ejpam-4870	32	3	)	)	PUNCT
ejpam-4870	32	4	to	to	PART
ejpam-4870	32	5	determine	determine	VERB
ejpam-4870	32	6	bounds	bound	NOUN
ejpam-4870	32	7	for	for	ADP
ejpam-4870	32	8	independence	independence	NOUN
ejpam-4870	32	9	number	number	NOUN
ejpam-4870	32	10	,	,	PUNCT
ejpam-4870	32	11	and	and	CCONJ
ejpam-4870	32	12	domination	domination	NOUN
ejpam-4870	32	13	number	number	NOUN
ejpam-4870	32	14	of	of	ADP
ejpam-4870	32	15	2	2	NUM
ejpam-4870	32	16	-	-	PUNCT
ejpam-4870	32	17	edge	edge	NOUN
ejpam-4870	32	18	intersection	intersection	NOUN
ejpam-4870	32	19	graph	graph	NOUN
ejpam-4870	32	20	of	of	ADP
ejpam-4870	32	21	cycle	cycle	NOUN
ejpam-4870	32	22	graph	graph	NOUN
ejpam-4870	32	23	.	.	PUNCT
ejpam-4870	33	1	2	2	X
ejpam-4870	33	2	.	.	X
ejpam-4870	33	3	preliminaries	preliminary	NOUN
ejpam-4870	33	4	graph	graph	NOUN
ejpam-4870	33	5	theory	theory	NOUN
ejpam-4870	33	6	and	and	CCONJ
ejpam-4870	33	7	the	the	DET
ejpam-4870	33	8	principle	principle	NOUN
ejpam-4870	33	9	of	of	ADP
ejpam-4870	33	10	counting	counting	NOUN
ejpam-4870	33	11	are	be	AUX
ejpam-4870	33	12	both	both	PRON
ejpam-4870	33	13	covered	cover	VERB
ejpam-4870	33	14	in	in	ADP
ejpam-4870	33	15	this	this	DET
ejpam-4870	33	16	chapter	chapter	NOUN
ejpam-4870	33	17	along	along	ADP
ejpam-4870	33	18	with	with	ADP
ejpam-4870	33	19	several	several	ADJ
ejpam-4870	33	20	other	other	ADJ
ejpam-4870	33	21	essential	essential	ADJ
ejpam-4870	33	22	ideas	idea	NOUN
ejpam-4870	33	23	.	.	PUNCT
ejpam-4870	34	1	for	for	ADP
ejpam-4870	34	2	the	the	DET
ejpam-4870	34	3	purpose	purpose	NOUN
ejpam-4870	34	4	of	of	ADP
ejpam-4870	34	5	further	further	ADJ
ejpam-4870	34	6	understanding	understanding	NOUN
ejpam-4870	34	7	concepts	concept	NOUN
ejpam-4870	34	8	,	,	PUNCT
ejpam-4870	34	9	examples	example	NOUN
ejpam-4870	34	10	,	,	PUNCT
ejpam-4870	34	11	and	and	CCONJ
ejpam-4870	34	12	illustrations	illustration	NOUN
ejpam-4870	34	13	are	be	AUX
ejpam-4870	34	14	given	give	VERB
ejpam-4870	34	15	.	.	PUNCT
ejpam-4870	35	1	also	also	ADV
ejpam-4870	35	2	,	,	PUNCT
ejpam-4870	35	3	some	some	DET
ejpam-4870	35	4	theorems	theorem	NOUN
ejpam-4870	35	5	are	be	AUX
ejpam-4870	35	6	presented	present	VERB
ejpam-4870	35	7	without	without	ADP
ejpam-4870	35	8	proof	proof	NOUN
ejpam-4870	35	9	.	.	PUNCT
ejpam-4870	36	1	the	the	DET
ejpam-4870	36	2	concepts	concept	NOUN
ejpam-4870	36	3	in	in	ADP
ejpam-4870	36	4	this	this	DET
ejpam-4870	36	5	section	section	NOUN
ejpam-4870	36	6	can	can	AUX
ejpam-4870	36	7	be	be	AUX
ejpam-4870	36	8	found	find	VERB
ejpam-4870	36	9	in	in	ADP
ejpam-4870	36	10	[	[	X
ejpam-4870	36	11	3	3	NUM
ejpam-4870	36	12	]	]	PUNCT
ejpam-4870	36	13	,	,	PUNCT
ejpam-4870	36	14	[	[	X
ejpam-4870	36	15	5	5	NUM
ejpam-4870	36	16	]	]	PUNCT
ejpam-4870	36	17	,	,	PUNCT
ejpam-4870	36	18	[	[	X
ejpam-4870	36	19	7	7	NUM
ejpam-4870	36	20	]	]	PUNCT
ejpam-4870	36	21	,	,	PUNCT
ejpam-4870	36	22	[	[	X
ejpam-4870	36	23	9	9	NUM
ejpam-4870	36	24	]	]	PUNCT
ejpam-4870	36	25	,	,	PUNCT
ejpam-4870	36	26	[	[	X
ejpam-4870	36	27	8	8	NUM
ejpam-4870	36	28	]	]	PUNCT
ejpam-4870	36	29	.	.	PUNCT
ejpam-4870	37	1	definition	definition	NOUN
ejpam-4870	37	2	1	1	NUM
ejpam-4870	37	3	.	.	PUNCT
ejpam-4870	38	1	a	a	DET
ejpam-4870	38	2	graph	graph	NOUN
ejpam-4870	38	3	,	,	PUNCT
ejpam-4870	38	4	denoted	denote	VERB
ejpam-4870	38	5	by	by	ADP
ejpam-4870	38	6	g	g	PROPN
ejpam-4870	38	7	is	be	AUX
ejpam-4870	38	8	an	an	DET
ejpam-4870	38	9	ordered	order	VERB
ejpam-4870	38	10	pair	pair	NOUN
ejpam-4870	38	11	g	g	NOUN
ejpam-4870	38	12	=	=	PUNCT
ejpam-4870	38	13	(	(	PUNCT
ejpam-4870	38	14	v	v	NOUN
ejpam-4870	38	15	(	(	PUNCT
ejpam-4870	38	16	g	g	NOUN
ejpam-4870	38	17	)	)	PUNCT
ejpam-4870	38	18	,	,	PUNCT
ejpam-4870	38	19	e(g	e(g	PROPN
ejpam-4870	38	20	)	)	PUNCT
ejpam-4870	38	21	)	)	PUNCT
ejpam-4870	38	22	where	where	SCONJ
ejpam-4870	38	23	the	the	DET
ejpam-4870	38	24	vertex	vertex	NOUN
ejpam-4870	38	25	set	set	VERB
ejpam-4870	38	26	v	v	NOUN
ejpam-4870	38	27	(	(	PUNCT
ejpam-4870	38	28	g	g	NOUN
ejpam-4870	38	29	)	)	PUNCT
ejpam-4870	38	30	is	be	AUX
ejpam-4870	38	31	a	a	DET
ejpam-4870	38	32	nonempty	nonempty	ADJ
ejpam-4870	38	33	set	set	NOUN
ejpam-4870	38	34	of	of	ADP
ejpam-4870	38	35	elements	element	NOUN
ejpam-4870	38	36	called	call	VERB
ejpam-4870	38	37	vertices	vertex	NOUN
ejpam-4870	38	38	,	,	PUNCT
ejpam-4870	38	39	and	and	CCONJ
ejpam-4870	38	40	the	the	DET
ejpam-4870	38	41	edge	edge	NOUN
ejpam-4870	38	42	set	set	VERB
ejpam-4870	38	43	e(g	e(g	PROPN
ejpam-4870	38	44	)	)	PUNCT
ejpam-4870	38	45	is	be	AUX
ejpam-4870	38	46	a	a	DET
ejpam-4870	38	47	set	set	NOUN
ejpam-4870	38	48	of	of	ADP
ejpam-4870	38	49	unordered	unordered	ADJ
ejpam-4870	38	50	pairs	pair	NOUN
ejpam-4870	38	51	of	of	ADP
ejpam-4870	38	52	distinct	distinct	ADJ
ejpam-4870	38	53	vertices	vertex	NOUN
ejpam-4870	38	54	called	call	VERB
ejpam-4870	38	55	edges	edge	NOUN
ejpam-4870	38	56	.	.	PUNCT
ejpam-4870	39	1	j.c	j.c	PROPN
ejpam-4870	39	2	.	.	PROPN
ejpam-4870	39	3	bonifacio	bonifacio	PROPN
ejpam-4870	39	4	,	,	PUNCT
ejpam-4870	39	5	c.j	c.j	PROPN
ejpam-4870	39	6	.	.	PROPN
ejpam-4870	39	7	andaya	andaya	PROPN
ejpam-4870	39	8	,	,	PUNCT
ejpam-4870	39	9	d.	d.	PROPN
ejpam-4870	39	10	magpantay	magpantay	PROPN
ejpam-4870	39	11	/	/	SYM
ejpam-4870	39	12	eur	eur	PROPN
ejpam-4870	39	13	.	.	PUNCT
ejpam-4870	40	1	j.	j.	PROPN
ejpam-4870	40	2	pure	pure	PROPN
ejpam-4870	40	3	appl	appl	PROPN
ejpam-4870	40	4	.	.	PROPN
ejpam-4870	40	5	math	math	PROPN
ejpam-4870	40	6	,	,	PUNCT
ejpam-4870	40	7	16	16	NUM
ejpam-4870	40	8	(	(	PUNCT
ejpam-4870	40	9	4	4	NUM
ejpam-4870	40	10	)	)	PUNCT
ejpam-4870	40	11	(	(	PUNCT
ejpam-4870	40	12	2023	2023	NUM
ejpam-4870	40	13	)	)	PUNCT
ejpam-4870	40	14	,	,	PUNCT
ejpam-4870	40	15	2476	2476	NUM
ejpam-4870	40	16	-	-	SYM
ejpam-4870	40	17	2498	2498	NUM
ejpam-4870	40	18	2478	2478	NUM
ejpam-4870	40	19	the	the	DET
ejpam-4870	40	20	edges	edge	NOUN
ejpam-4870	40	21	of	of	ADP
ejpam-4870	40	22	a	a	DET
ejpam-4870	40	23	graph	graph	NOUN
ejpam-4870	40	24	is	be	AUX
ejpam-4870	40	25	written	write	VERB
ejpam-4870	40	26	as	as	ADP
ejpam-4870	40	27	[	[	X
ejpam-4870	40	28	x1	x1	PROPN
ejpam-4870	40	29	,	,	PUNCT
ejpam-4870	40	30	x2	x2	PROPN
ejpam-4870	40	31	]	]	PUNCT
ejpam-4870	40	32	where	where	SCONJ
ejpam-4870	40	33	x1	x1	X
ejpam-4870	40	34	,	,	PUNCT
ejpam-4870	40	35	x2	x2	PROPN
ejpam-4870	40	36	∈	∈	PROPN
ejpam-4870	40	37	v	v	ADP
ejpam-4870	40	38	(	(	PUNCT
ejpam-4870	40	39	g	g	NOUN
ejpam-4870	40	40	)	)	PUNCT
ejpam-4870	40	41	.	.	PUNCT
ejpam-4870	41	1	two	two	NUM
ejpam-4870	41	2	vertices	vertex	NOUN
ejpam-4870	41	3	x1	x1	PROPN
ejpam-4870	41	4	and	and	CCONJ
ejpam-4870	41	5	x2	x2	PROPN
ejpam-4870	41	6	in	in	ADP
ejpam-4870	41	7	g	g	PROPN
ejpam-4870	41	8	are	be	AUX
ejpam-4870	41	9	connected	connect	VERB
ejpam-4870	41	10	by	by	ADP
ejpam-4870	41	11	a	a	DET
ejpam-4870	41	12	line	line	NOUN
ejpam-4870	41	13	segment	segment	NOUN
ejpam-4870	41	14	whenever	whenever	SCONJ
ejpam-4870	41	15	x1	x1	PROPN
ejpam-4870	41	16	is	be	AUX
ejpam-4870	41	17	adjacent	adjacent	ADJ
ejpam-4870	41	18	to	to	ADP
ejpam-4870	41	19	x2	x2	PROPN
ejpam-4870	41	20	or	or	CCONJ
ejpam-4870	41	21	x2	x2	PROPN
ejpam-4870	41	22	is	be	AUX
ejpam-4870	41	23	adjacent	adjacent	ADJ
ejpam-4870	41	24	to	to	ADP
ejpam-4870	41	25	x1	x1	PROPN
ejpam-4870	41	26	.	.	PUNCT
ejpam-4870	42	1	note	note	VERB
ejpam-4870	42	2	that	that	SCONJ
ejpam-4870	42	3	an	an	DET
ejpam-4870	42	4	edge	edge	NOUN
ejpam-4870	42	5	contains	contain	VERB
ejpam-4870	42	6	unordered	unordered	ADJ
ejpam-4870	42	7	pair	pair	NOUN
ejpam-4870	42	8	of	of	ADP
ejpam-4870	42	9	vertices	vertex	NOUN
ejpam-4870	42	10	,	,	PUNCT
ejpam-4870	42	11	so	so	SCONJ
ejpam-4870	42	12	[	[	X
ejpam-4870	42	13	x1	x1	X
ejpam-4870	42	14	,	,	PUNCT
ejpam-4870	42	15	x2	x2	PROPN
ejpam-4870	42	16	]	]	X
ejpam-4870	42	17	=	=	PUNCT
ejpam-4870	43	1	[	[	X
ejpam-4870	43	2	x2	x2	X
ejpam-4870	43	3	,	,	PUNCT
ejpam-4870	43	4	x1	x1	PROPN
ejpam-4870	43	5	]	]	X
ejpam-4870	43	6	.	.	PUNCT
ejpam-4870	44	1	if	if	SCONJ
ejpam-4870	44	2	[	[	X
ejpam-4870	44	3	x1	x1	X
ejpam-4870	44	4	,	,	PUNCT
ejpam-4870	44	5	x2	x2	PROPN
ejpam-4870	44	6	]	]	PUNCT
ejpam-4870	44	7	is	be	AUX
ejpam-4870	44	8	an	an	DET
ejpam-4870	44	9	element	element	NOUN
ejpam-4870	44	10	of	of	ADP
ejpam-4870	44	11	e(g	e(g	PROPN
ejpam-4870	44	12	)	)	PUNCT
ejpam-4870	44	13	,	,	PUNCT
ejpam-4870	44	14	then	then	ADV
ejpam-4870	44	15	the	the	DET
ejpam-4870	44	16	vertices	vertex	NOUN
ejpam-4870	44	17	x1	x1	PROPN
ejpam-4870	44	18	and	and	CCONJ
ejpam-4870	44	19	x2	x2	PROPN
ejpam-4870	44	20	are	be	AUX
ejpam-4870	44	21	said	say	VERB
ejpam-4870	44	22	to	to	PART
ejpam-4870	44	23	be	be	AUX
ejpam-4870	44	24	adjacent	adjacent	ADJ
ejpam-4870	44	25	in	in	ADP
ejpam-4870	44	26	g.	g.	PROPN
ejpam-4870	44	27	now	now	ADV
ejpam-4870	44	28	,	,	PUNCT
ejpam-4870	44	29	if	if	SCONJ
ejpam-4870	44	30	[	[	X
ejpam-4870	44	31	x1	x1	X
ejpam-4870	44	32	,	,	PUNCT
ejpam-4870	44	33	x2	x2	PROPN
ejpam-4870	44	34	]	]	PUNCT
ejpam-4870	44	35	/∈	/∈	PUNCT
ejpam-4870	45	1	e(g	e(g	PROPN
ejpam-4870	45	2	)	)	PUNCT
ejpam-4870	45	3	,	,	PUNCT
ejpam-4870	45	4	then	then	ADV
ejpam-4870	45	5	x1	x1	PROPN
ejpam-4870	45	6	and	and	CCONJ
ejpam-4870	45	7	x2	x2	PROPN
ejpam-4870	45	8	are	be	AUX
ejpam-4870	45	9	said	say	VERB
ejpam-4870	45	10	to	to	PART
ejpam-4870	45	11	be	be	AUX
ejpam-4870	45	12	non	non	ADJ
ejpam-4870	45	13	-	-	ADJ
ejpam-4870	45	14	adjacent	adjacent	ADJ
ejpam-4870	45	15	in	in	ADP
ejpam-4870	45	16	g.	g.	PROPN
ejpam-4870	45	17	moreover	moreover	ADV
ejpam-4870	45	18	,	,	PUNCT
ejpam-4870	45	19	edges	edge	NOUN
ejpam-4870	45	20	are	be	AUX
ejpam-4870	45	21	incident	incident	NOUN
ejpam-4870	45	22	if	if	SCONJ
ejpam-4870	45	23	there	there	PRON
ejpam-4870	45	24	is	be	VERB
ejpam-4870	45	25	a	a	DET
ejpam-4870	45	26	vertex	vertex	NOUN
ejpam-4870	45	27	between	between	ADP
ejpam-4870	45	28	these	these	DET
ejpam-4870	45	29	edges	edge	NOUN
ejpam-4870	45	30	.	.	PUNCT
ejpam-4870	46	1	the	the	DET
ejpam-4870	46	2	cardinality	cardinality	NOUN
ejpam-4870	46	3	of	of	ADP
ejpam-4870	46	4	v	v	NOUN
ejpam-4870	46	5	(	(	PUNCT
ejpam-4870	46	6	g	g	NOUN
ejpam-4870	46	7	)	)	PUNCT
ejpam-4870	46	8	and	and	CCONJ
ejpam-4870	46	9	e(g	e(g	PROPN
ejpam-4870	46	10	)	)	PUNCT
ejpam-4870	46	11	are	be	AUX
ejpam-4870	46	12	referred	refer	VERB
ejpam-4870	46	13	to	to	ADP
ejpam-4870	46	14	as	as	ADP
ejpam-4870	46	15	the	the	DET
ejpam-4870	46	16	order	order	NOUN
ejpam-4870	46	17	and	and	CCONJ
ejpam-4870	46	18	size	size	NOUN
ejpam-4870	46	19	of	of	ADP
ejpam-4870	46	20	g	g	NOUN
ejpam-4870	46	21	,	,	PUNCT
ejpam-4870	46	22	respectively	respectively	ADV
ejpam-4870	46	23	.	.	PUNCT
ejpam-4870	46	24	example	example	NOUN
ejpam-4870	47	1	1	1	NUM
ejpam-4870	47	2	.	.	PUNCT
ejpam-4870	47	3	let	let	VERB
ejpam-4870	47	4	g	g	PRON
ejpam-4870	47	5	be	be	AUX
ejpam-4870	47	6	a	a	DET
ejpam-4870	47	7	graph	graph	NOUN
ejpam-4870	47	8	such	such	ADJ
ejpam-4870	47	9	that	that	DET
ejpam-4870	47	10	v	v	NOUN
ejpam-4870	47	11	(	(	PUNCT
ejpam-4870	47	12	g	g	NOUN
ejpam-4870	47	13	)	)	PUNCT
ejpam-4870	47	14	=	=	SYM
ejpam-4870	47	15	{	{	PUNCT
ejpam-4870	47	16	x1	x1	PROPN
ejpam-4870	47	17	,	,	PUNCT
ejpam-4870	47	18	x2	x2	PROPN
ejpam-4870	47	19	,	,	PUNCT
ejpam-4870	47	20	x3	x3	ADJ
ejpam-4870	47	21	,	,	PUNCT
ejpam-4870	47	22	x4	x4	ADJ
ejpam-4870	47	23	}	}	PUNCT
ejpam-4870	47	24	and	and	CCONJ
ejpam-4870	47	25	e(g	e(g	PROPN
ejpam-4870	47	26	)	)	PUNCT
ejpam-4870	48	1	=	=	PRON
ejpam-4870	48	2	{	{	PUNCT
ejpam-4870	49	1	[	[	X
ejpam-4870	49	2	x1	x1	PROPN
ejpam-4870	49	3	,	,	PUNCT
ejpam-4870	49	4	x2	x2	PROPN
ejpam-4870	49	5	]	]	PUNCT
ejpam-4870	49	6	,	,	PUNCT
ejpam-4870	49	7	[	[	X
ejpam-4870	49	8	x1	x1	X
ejpam-4870	49	9	,	,	PUNCT
ejpam-4870	49	10	x4	x4	PROPN
ejpam-4870	49	11	]	]	PUNCT
ejpam-4870	49	12	,	,	PUNCT
ejpam-4870	49	13	[	[	X
ejpam-4870	49	14	x4	x4	PROPN
ejpam-4870	49	15	,	,	PUNCT
ejpam-4870	49	16	x3	x3	ADJ
ejpam-4870	49	17	]	]	PUNCT
ejpam-4870	49	18	,	,	PUNCT
ejpam-4870	50	1	[	[	X
ejpam-4870	50	2	x3	x3	ADJ
ejpam-4870	50	3	,	,	PUNCT
ejpam-4870	50	4	x2	x2	PROPN
ejpam-4870	50	5	]	]	X
ejpam-4870	50	6	}	}	PUNCT
ejpam-4870	50	7	.	.	PUNCT
ejpam-4870	51	1	then	then	ADV
ejpam-4870	51	2	|v	|v	PROPN
ejpam-4870	51	3	(	(	PUNCT
ejpam-4870	51	4	g)|	g)|	NOUN
ejpam-4870	51	5	=	=	PUNCT
ejpam-4870	51	6	4	4	NUM
ejpam-4870	51	7	and	and	CCONJ
ejpam-4870	51	8	|e(g)|	|e(g)|	ADJ
ejpam-4870	51	9	=	=	ADJ
ejpam-4870	51	10	4	4	NUM
ejpam-4870	51	11	.	.	PUNCT
ejpam-4870	52	1	the	the	DET
ejpam-4870	52	2	pictorial	pictorial	ADJ
ejpam-4870	52	3	representation	representation	NOUN
ejpam-4870	52	4	of	of	ADP
ejpam-4870	52	5	g	g	PROPN
ejpam-4870	52	6	is	be	AUX
ejpam-4870	52	7	shown	show	VERB
ejpam-4870	52	8	in	in	ADP
ejpam-4870	52	9	figure	figure	NOUN
ejpam-4870	52	10	1	1	NUM
ejpam-4870	52	11	.	.	PUNCT
ejpam-4870	53	1	x1	x1	PROPN
ejpam-4870	53	2	x4	x4	PROPN
ejpam-4870	54	1	x2	x2	PROPN
ejpam-4870	54	2	x3	x3	ADJ
ejpam-4870	54	3	figure	figure	VERB
ejpam-4870	54	4	1	1	NUM
ejpam-4870	54	5	:	:	PUNCT
ejpam-4870	54	6	example	example	NOUN
ejpam-4870	54	7	of	of	ADP
ejpam-4870	54	8	a	a	DET
ejpam-4870	54	9	graph	graph	NOUN
ejpam-4870	54	10	g	g	NOUN
ejpam-4870	54	11	it	it	PRON
ejpam-4870	54	12	can	can	AUX
ejpam-4870	54	13	be	be	AUX
ejpam-4870	54	14	noted	note	VERB
ejpam-4870	54	15	that	that	SCONJ
ejpam-4870	54	16	a	a	DET
ejpam-4870	54	17	pictorial	pictorial	ADJ
ejpam-4870	54	18	representation	representation	NOUN
ejpam-4870	54	19	of	of	ADP
ejpam-4870	54	20	a	a	DET
ejpam-4870	54	21	particular	particular	ADJ
ejpam-4870	54	22	graph	graph	NOUN
ejpam-4870	54	23	is	be	AUX
ejpam-4870	54	24	not	not	PART
ejpam-4870	54	25	unique	unique	ADJ
ejpam-4870	54	26	.	.	PUNCT
ejpam-4870	55	1	hence	hence	ADV
ejpam-4870	55	2	,	,	PUNCT
ejpam-4870	55	3	graph	graph	VERB
ejpam-4870	55	4	g	g	PROPN
ejpam-4870	55	5	in	in	ADP
ejpam-4870	55	6	figure	figure	NOUN
ejpam-4870	55	7	1	1	NUM
ejpam-4870	55	8	can	can	AUX
ejpam-4870	55	9	be	be	AUX
ejpam-4870	55	10	illustrated	illustrate	VERB
ejpam-4870	55	11	differently	differently	ADV
ejpam-4870	55	12	as	as	SCONJ
ejpam-4870	55	13	shown	show	VERB
ejpam-4870	55	14	in	in	ADP
ejpam-4870	55	15	figure	figure	NOUN
ejpam-4870	55	16	2	2	NUM
ejpam-4870	55	17	.	.	PUNCT
ejpam-4870	56	1	x1	x1	PROPN
ejpam-4870	56	2	x4	x4	PROPN
ejpam-4870	57	1	x2	x2	PROPN
ejpam-4870	57	2	x3	x3	VERB
ejpam-4870	58	1	x1	x1	NOUN
ejpam-4870	59	1	x3	x3	ADJ
ejpam-4870	59	2	x2	x2	PROPN
ejpam-4870	59	3	x4	x4	PROPN
ejpam-4870	59	4	figure	figure	NOUN
ejpam-4870	59	5	2	2	NUM
ejpam-4870	59	6	:	:	PUNCT
ejpam-4870	59	7	other	other	ADJ
ejpam-4870	59	8	pictorial	pictorial	ADJ
ejpam-4870	59	9	representations	representation	NOUN
ejpam-4870	59	10	of	of	ADP
ejpam-4870	59	11	g	g	NOUN
ejpam-4870	59	12	,	,	PUNCT
ejpam-4870	59	13	g1	g1	PROPN
ejpam-4870	59	14	and	and	CCONJ
ejpam-4870	59	15	g2	g2	PROPN
ejpam-4870	59	16	respectively	respectively	ADV
ejpam-4870	59	17	definition	definition	NOUN
ejpam-4870	59	18	2	2	NUM
ejpam-4870	59	19	.	.	PUNCT
ejpam-4870	59	20	an	an	DET
ejpam-4870	59	21	edge	edge	NOUN
ejpam-4870	59	22	of	of	ADP
ejpam-4870	59	23	the	the	DET
ejpam-4870	59	24	form	form	NOUN
ejpam-4870	59	25	[	[	X
ejpam-4870	59	26	x	x	X
ejpam-4870	59	27	,	,	PUNCT
ejpam-4870	59	28	x	x	X
ejpam-4870	59	29	]	]	X
ejpam-4870	59	30	is	be	AUX
ejpam-4870	59	31	called	call	VERB
ejpam-4870	59	32	a	a	DET
ejpam-4870	59	33	loop	loop	NOUN
ejpam-4870	59	34	.	.	PUNCT
ejpam-4870	60	1	moreover	moreover	ADV
ejpam-4870	60	2	,	,	PUNCT
ejpam-4870	60	3	multiple	multiple	ADJ
ejpam-4870	60	4	edges	edge	NOUN
ejpam-4870	60	5	are	be	AUX
ejpam-4870	60	6	edges	edge	NOUN
ejpam-4870	60	7	that	that	PRON
ejpam-4870	60	8	have	have	VERB
ejpam-4870	60	9	the	the	DET
ejpam-4870	60	10	same	same	ADJ
ejpam-4870	60	11	pair	pair	NOUN
ejpam-4870	60	12	of	of	ADP
ejpam-4870	60	13	vertices	vertex	NOUN
ejpam-4870	60	14	.	.	PUNCT
ejpam-4870	61	1	a	a	DET
ejpam-4870	61	2	graph	graph	NOUN
ejpam-4870	61	3	having	have	VERB
ejpam-4870	61	4	no	no	DET
ejpam-4870	61	5	loops	loop	NOUN
ejpam-4870	61	6	nor	nor	CCONJ
ejpam-4870	61	7	multiple	multiple	ADJ
ejpam-4870	61	8	edges	edge	NOUN
ejpam-4870	61	9	is	be	AUX
ejpam-4870	61	10	called	call	VERB
ejpam-4870	61	11	a	a	DET
ejpam-4870	61	12	simple	simple	ADJ
ejpam-4870	61	13	graph	graph	NOUN
ejpam-4870	61	14	.	.	PUNCT
ejpam-4870	62	1	the	the	DET
ejpam-4870	62	2	focus	focus	NOUN
ejpam-4870	62	3	of	of	ADP
ejpam-4870	62	4	this	this	DET
ejpam-4870	62	5	paper	paper	NOUN
ejpam-4870	62	6	is	be	AUX
ejpam-4870	62	7	on	on	ADP
ejpam-4870	62	8	finite	finite	ADJ
ejpam-4870	62	9	graphs	graph	NOUN
ejpam-4870	62	10	,	,	PUNCT
ejpam-4870	62	11	which	which	PRON
ejpam-4870	62	12	are	be	AUX
ejpam-4870	62	13	finite	finite	ADJ
ejpam-4870	62	14	in	in	ADP
ejpam-4870	62	15	both	both	DET
ejpam-4870	62	16	their	their	PRON
ejpam-4870	62	17	vertex	vertex	NOUN
ejpam-4870	62	18	and	and	CCONJ
ejpam-4870	62	19	edge	edge	NOUN
ejpam-4870	62	20	sets	set	NOUN
ejpam-4870	62	21	.	.	PUNCT
ejpam-4870	63	1	also	also	ADV
ejpam-4870	63	2	,	,	PUNCT
ejpam-4870	63	3	this	this	DET
ejpam-4870	63	4	paper	paper	NOUN
ejpam-4870	63	5	will	will	AUX
ejpam-4870	63	6	be	be	AUX
ejpam-4870	63	7	limited	limit	VERB
ejpam-4870	63	8	to	to	ADP
ejpam-4870	63	9	simple	simple	ADJ
ejpam-4870	63	10	finite	finite	ADJ
ejpam-4870	63	11	graphs	graph	NOUN
ejpam-4870	63	12	and	and	CCONJ
ejpam-4870	63	13	we	we	PRON
ejpam-4870	63	14	will	will	AUX
ejpam-4870	63	15	simply	simply	ADV
ejpam-4870	63	16	call	call	VERB
ejpam-4870	63	17	them	they	PRON
ejpam-4870	63	18	graphs	graph	NOUN
ejpam-4870	63	19	.	.	PUNCT
ejpam-4870	64	1	example	example	NOUN
ejpam-4870	64	2	2	2	NUM
ejpam-4870	64	3	.	.	X
ejpam-4870	64	4	graph	graph	VERB
ejpam-4870	64	5	g	g	PROPN
ejpam-4870	64	6	in	in	ADP
ejpam-4870	64	7	figure	figure	NOUN
ejpam-4870	64	8	3	3	NUM
ejpam-4870	64	9	is	be	AUX
ejpam-4870	64	10	an	an	DET
ejpam-4870	64	11	example	example	NOUN
ejpam-4870	64	12	of	of	ADP
ejpam-4870	64	13	a	a	DET
ejpam-4870	64	14	finite	finite	ADJ
ejpam-4870	64	15	graph	graph	NOUN
ejpam-4870	64	16	because	because	SCONJ
ejpam-4870	64	17	its	its	PRON
ejpam-4870	64	18	vertex	vertex	NOUN
ejpam-4870	64	19	and	and	CCONJ
ejpam-4870	64	20	edge	edge	NOUN
ejpam-4870	64	21	set	set	NOUN
ejpam-4870	64	22	is	be	AUX
ejpam-4870	64	23	finite	finite	ADJ
ejpam-4870	64	24	and	and	CCONJ
ejpam-4870	64	25	does	do	AUX
ejpam-4870	64	26	not	not	PART
ejpam-4870	64	27	contain	contain	VERB
ejpam-4870	64	28	any	any	DET
ejpam-4870	64	29	loops	loop	NOUN
ejpam-4870	64	30	or	or	CCONJ
ejpam-4870	64	31	multiple	multiple	ADJ
ejpam-4870	64	32	edges	edge	NOUN
ejpam-4870	64	33	so	so	SCONJ
ejpam-4870	64	34	it	it	PRON
ejpam-4870	64	35	is	be	AUX
ejpam-4870	64	36	a	a	DET
ejpam-4870	64	37	simple	simple	ADJ
ejpam-4870	64	38	graph	graph	NOUN
ejpam-4870	64	39	.	.	PUNCT
ejpam-4870	65	1	j.c	j.c	PROPN
ejpam-4870	65	2	.	.	PROPN
ejpam-4870	65	3	bonifacio	bonifacio	PROPN
ejpam-4870	65	4	,	,	PUNCT
ejpam-4870	65	5	c.j	c.j	PROPN
ejpam-4870	65	6	.	.	PROPN
ejpam-4870	65	7	andaya	andaya	PROPN
ejpam-4870	65	8	,	,	PUNCT
ejpam-4870	65	9	d.	d.	PROPN
ejpam-4870	65	10	magpantay	magpantay	PROPN
ejpam-4870	65	11	/	/	SYM
ejpam-4870	65	12	eur	eur	PROPN
ejpam-4870	65	13	.	.	PUNCT
ejpam-4870	66	1	j.	j.	PROPN
ejpam-4870	66	2	pure	pure	PROPN
ejpam-4870	66	3	appl	appl	PROPN
ejpam-4870	66	4	.	.	PROPN
ejpam-4870	66	5	math	math	PROPN
ejpam-4870	66	6	,	,	PUNCT
ejpam-4870	66	7	16	16	NUM
ejpam-4870	66	8	(	(	PUNCT
ejpam-4870	66	9	4	4	NUM
ejpam-4870	66	10	)	)	PUNCT
ejpam-4870	66	11	(	(	PUNCT
ejpam-4870	66	12	2023	2023	NUM
ejpam-4870	66	13	)	)	PUNCT
ejpam-4870	66	14	,	,	PUNCT
ejpam-4870	66	15	2476	2476	NUM
ejpam-4870	66	16	-	-	SYM
ejpam-4870	66	17	2498	2498	NUM
ejpam-4870	66	18	2479	2479	NUM
ejpam-4870	66	19	however	however	ADV
ejpam-4870	66	20	,	,	PUNCT
ejpam-4870	66	21	graph	graph	NOUN
ejpam-4870	66	22	h	h	NOUN
ejpam-4870	66	23	in	in	ADP
ejpam-4870	66	24	figure	figure	NOUN
ejpam-4870	66	25	3	3	NUM
ejpam-4870	66	26	contains	contain	VERB
ejpam-4870	66	27	the	the	DET
ejpam-4870	66	28	edge	edge	NOUN
ejpam-4870	66	29	[	[	X
ejpam-4870	66	30	x5	x5	NOUN
ejpam-4870	66	31	,	,	PUNCT
ejpam-4870	66	32	x5	x5	NOUN
ejpam-4870	66	33	]	]	PUNCT
ejpam-4870	66	34	and	and	CCONJ
ejpam-4870	66	35	there	there	PRON
ejpam-4870	66	36	are	be	VERB
ejpam-4870	66	37	multiple	multiple	ADJ
ejpam-4870	66	38	edges	edge	NOUN
ejpam-4870	66	39	[	[	X
ejpam-4870	66	40	x1	x1	PROPN
ejpam-4870	66	41	,	,	PUNCT
ejpam-4870	66	42	x2	x2	PROPN
ejpam-4870	66	43	]	]	PUNCT
ejpam-4870	66	44	so	so	CCONJ
ejpam-4870	66	45	it	it	PRON
ejpam-4870	66	46	is	be	AUX
ejpam-4870	66	47	not	not	PART
ejpam-4870	66	48	a	a	DET
ejpam-4870	66	49	simple	simple	ADJ
ejpam-4870	66	50	graph	graph	NOUN
ejpam-4870	66	51	.	.	PUNCT
ejpam-4870	67	1	x1	x1	PROPN
ejpam-4870	67	2	x4	x4	PROPN
ejpam-4870	68	1	x2	x2	PROPN
ejpam-4870	68	2	x3	x3	PROPN
ejpam-4870	68	3	x1	x1	PROPN
ejpam-4870	69	1	x4	x4	PROPN
ejpam-4870	69	2	x2	x2	PROPN
ejpam-4870	69	3	x3	x3	PROPN
ejpam-4870	69	4	x5	x5	PROPN
ejpam-4870	69	5	figure	figure	NOUN
ejpam-4870	69	6	3	3	NUM
ejpam-4870	69	7	:	:	PUNCT
ejpam-4870	69	8	simple	simple	ADJ
ejpam-4870	69	9	graph	graph	NOUN
ejpam-4870	69	10	g	g	NOUN
ejpam-4870	69	11	and	and	CCONJ
ejpam-4870	69	12	graph	graph	NOUN
ejpam-4870	69	13	h	h	PROPN
ejpam-4870	69	14	with	with	ADP
ejpam-4870	69	15	loop	loop	NOUN
ejpam-4870	69	16	and	and	CCONJ
ejpam-4870	69	17	multiple	multiple	ADJ
ejpam-4870	69	18	edges	edge	NOUN
ejpam-4870	69	19	definition	definition	NOUN
ejpam-4870	69	20	3	3	NUM
ejpam-4870	69	21	.	.	PUNCT
ejpam-4870	70	1	a	a	DET
ejpam-4870	70	2	graphg	graphg	NOUN
ejpam-4870	70	3	is	be	AUX
ejpam-4870	70	4	labeled	label	VERB
ejpam-4870	70	5	when	when	SCONJ
ejpam-4870	70	6	each	each	DET
ejpam-4870	70	7	vertex	vertex	NOUN
ejpam-4870	70	8	is	be	AUX
ejpam-4870	70	9	distinguished	distinguish	VERB
ejpam-4870	70	10	from	from	ADP
ejpam-4870	70	11	one	one	NUM
ejpam-4870	70	12	another	another	DET
ejpam-4870	70	13	by	by	ADP
ejpam-4870	70	14	symbols	symbol	NOUN
ejpam-4870	70	15	such	such	ADJ
ejpam-4870	70	16	as	as	ADP
ejpam-4870	70	17	x1	x1	PROPN
ejpam-4870	70	18	,	,	PUNCT
ejpam-4870	70	19	x2	x2	PROPN
ejpam-4870	70	20	,	,	PUNCT
ejpam-4870	70	21	...	...	PUNCT
ejpam-4870	70	22	,	,	PUNCT
ejpam-4870	70	23	xn	xn	PROPN
ejpam-4870	70	24	where	where	SCONJ
ejpam-4870	70	25	n	n	X
ejpam-4870	70	26	is	be	AUX
ejpam-4870	70	27	the	the	DET
ejpam-4870	70	28	order	order	NOUN
ejpam-4870	70	29	of	of	ADP
ejpam-4870	70	30	g.	g.	PROPN
ejpam-4870	70	31	otherwise	otherwise	ADV
ejpam-4870	70	32	,	,	PUNCT
ejpam-4870	70	33	it	it	PRON
ejpam-4870	70	34	is	be	AUX
ejpam-4870	70	35	called	call	VERB
ejpam-4870	70	36	unlabeled	unlabeled	ADJ
ejpam-4870	70	37	.	.	PUNCT
ejpam-4870	71	1	definition	definition	NOUN
ejpam-4870	71	2	4	4	NUM
ejpam-4870	71	3	.	.	PUNCT
ejpam-4870	72	1	a	a	DET
ejpam-4870	72	2	graph	graph	NOUN
ejpam-4870	72	3	of	of	ADP
ejpam-4870	72	4	order	order	NOUN
ejpam-4870	72	5	n	n	PRON
ejpam-4870	72	6	≥	≥	NOUN
ejpam-4870	72	7	1	1	NUM
ejpam-4870	72	8	having	have	VERB
ejpam-4870	72	9	no	no	DET
ejpam-4870	72	10	edges	edge	NOUN
ejpam-4870	72	11	is	be	AUX
ejpam-4870	72	12	called	call	VERB
ejpam-4870	72	13	an	an	DET
ejpam-4870	72	14	empty	empty	ADJ
ejpam-4870	72	15	graph	graph	NOUN
ejpam-4870	72	16	.	.	PUNCT
ejpam-4870	73	1	furthermore	furthermore	ADV
ejpam-4870	73	2	,	,	PUNCT
ejpam-4870	73	3	a	a	DET
ejpam-4870	73	4	graph	graph	NOUN
ejpam-4870	73	5	with	with	ADP
ejpam-4870	73	6	only	only	ADV
ejpam-4870	73	7	one	one	NUM
ejpam-4870	73	8	vertex	vertex	NOUN
ejpam-4870	73	9	is	be	AUX
ejpam-4870	73	10	referred	refer	VERB
ejpam-4870	73	11	to	to	ADP
ejpam-4870	73	12	as	as	ADP
ejpam-4870	73	13	a	a	DET
ejpam-4870	73	14	trivial	trivial	ADJ
ejpam-4870	73	15	graph	graph	NOUN
ejpam-4870	73	16	.	.	PUNCT
ejpam-4870	74	1	it	it	PRON
ejpam-4870	74	2	can	can	AUX
ejpam-4870	74	3	be	be	AUX
ejpam-4870	74	4	noted	note	VERB
ejpam-4870	74	5	that	that	SCONJ
ejpam-4870	74	6	it	it	PRON
ejpam-4870	74	7	can	can	AUX
ejpam-4870	74	8	not	not	PART
ejpam-4870	74	9	form	form	VERB
ejpam-4870	74	10	a	a	DET
ejpam-4870	74	11	graph	graph	NOUN
ejpam-4870	74	12	if	if	SCONJ
ejpam-4870	74	13	its	its	PRON
ejpam-4870	74	14	vertex	vertex	NOUN
ejpam-4870	74	15	set	set	NOUN
ejpam-4870	74	16	has	have	VERB
ejpam-4870	74	17	no	no	DET
ejpam-4870	74	18	elements	element	NOUN
ejpam-4870	74	19	.	.	PUNCT
ejpam-4870	75	1	also	also	ADV
ejpam-4870	75	2	,	,	PUNCT
ejpam-4870	75	3	it	it	PRON
ejpam-4870	75	4	can	can	AUX
ejpam-4870	75	5	be	be	AUX
ejpam-4870	75	6	observed	observe	VERB
ejpam-4870	75	7	that	that	SCONJ
ejpam-4870	75	8	all	all	DET
ejpam-4870	75	9	trivial	trivial	ADJ
ejpam-4870	75	10	graphs	graph	NOUN
ejpam-4870	75	11	are	be	AUX
ejpam-4870	75	12	empty	empty	ADJ
ejpam-4870	75	13	graphs	graph	NOUN
ejpam-4870	75	14	but	but	CCONJ
ejpam-4870	75	15	it	it	PRON
ejpam-4870	75	16	is	be	AUX
ejpam-4870	75	17	not	not	PART
ejpam-4870	75	18	always	always	ADV
ejpam-4870	75	19	true	true	ADJ
ejpam-4870	75	20	for	for	ADP
ejpam-4870	75	21	an	an	DET
ejpam-4870	75	22	empty	empty	ADJ
ejpam-4870	75	23	graph	graph	NOUN
ejpam-4870	75	24	to	to	PART
ejpam-4870	75	25	be	be	AUX
ejpam-4870	75	26	a	a	DET
ejpam-4870	75	27	trivial	trivial	ADJ
ejpam-4870	75	28	graph	graph	NOUN
ejpam-4870	75	29	.	.	PUNCT
ejpam-4870	75	30	example	example	NOUN
ejpam-4870	76	1	3	3	X
ejpam-4870	76	2	.	.	PUNCT
ejpam-4870	76	3	let	let	VERB
ejpam-4870	76	4	g1	g1	PROPN
ejpam-4870	76	5	be	be	AUX
ejpam-4870	76	6	a	a	DET
ejpam-4870	76	7	graph	graph	NOUN
ejpam-4870	76	8	where	where	SCONJ
ejpam-4870	76	9	v	v	X
ejpam-4870	76	10	(	(	PUNCT
ejpam-4870	76	11	g1	g1	PROPN
ejpam-4870	76	12	)	)	PUNCT
ejpam-4870	77	1	=	=	SYM
ejpam-4870	77	2	x1	x1	PROPN
ejpam-4870	77	3	and	and	CCONJ
ejpam-4870	77	4	e(g1	e(g1	ADJ
ejpam-4870	77	5	)	)	PUNCT
ejpam-4870	77	6	=	=	PUNCT
ejpam-4870	77	7	∅.	∅.	NOUN
ejpam-4870	77	8	since	since	SCONJ
ejpam-4870	77	9	v	v	PROPN
ejpam-4870	77	10	(	(	PUNCT
ejpam-4870	77	11	g1	g1	PROPN
ejpam-4870	77	12	)	)	PUNCT
ejpam-4870	77	13	has	have	VERB
ejpam-4870	77	14	only	only	ADV
ejpam-4870	77	15	an	an	DET
ejpam-4870	77	16	element	element	NOUN
ejpam-4870	77	17	,	,	PUNCT
ejpam-4870	77	18	it	it	PRON
ejpam-4870	77	19	follows	follow	VERB
ejpam-4870	77	20	that	that	SCONJ
ejpam-4870	77	21	g1	g1	PROPN
ejpam-4870	77	22	is	be	AUX
ejpam-4870	77	23	a	a	DET
ejpam-4870	77	24	trivial	trivial	ADJ
ejpam-4870	77	25	graph	graph	NOUN
ejpam-4870	77	26	.	.	PUNCT
ejpam-4870	78	1	let	let	VERB
ejpam-4870	78	2	g2	g2	PROPN
ejpam-4870	78	3	be	be	AUX
ejpam-4870	78	4	a	a	DET
ejpam-4870	78	5	graph	graph	NOUN
ejpam-4870	78	6	such	such	ADJ
ejpam-4870	78	7	that	that	DET
ejpam-4870	78	8	v	v	NOUN
ejpam-4870	78	9	(	(	PUNCT
ejpam-4870	78	10	g2	g2	PROPN
ejpam-4870	78	11	)	)	PUNCT
ejpam-4870	78	12	=	=	PRON
ejpam-4870	79	1	{	{	PUNCT
ejpam-4870	79	2	x1	x1	PROPN
ejpam-4870	79	3	,	,	PUNCT
ejpam-4870	79	4	x2	x2	PROPN
ejpam-4870	79	5	,	,	PUNCT
ejpam-4870	79	6	x3	x3	ADJ
ejpam-4870	79	7	,	,	PUNCT
ejpam-4870	79	8	x4	x4	ADJ
ejpam-4870	79	9	}	}	PUNCT
ejpam-4870	79	10	and	and	CCONJ
ejpam-4870	79	11	e(g2	e(g2	ADV
ejpam-4870	79	12	)	)	PUNCT
ejpam-4870	80	1	=	=	PUNCT
ejpam-4870	80	2	∅.	∅.	VERB
ejpam-4870	80	3	hence	hence	ADV
ejpam-4870	80	4	,	,	PUNCT
ejpam-4870	80	5	g2	g2	PROPN
ejpam-4870	80	6	is	be	AUX
ejpam-4870	80	7	said	say	VERB
ejpam-4870	80	8	to	to	PART
ejpam-4870	80	9	be	be	AUX
ejpam-4870	80	10	an	an	DET
ejpam-4870	80	11	empty	empty	ADJ
ejpam-4870	80	12	graph	graph	NOUN
ejpam-4870	80	13	.	.	PUNCT
ejpam-4870	81	1	shown	show	VERB
ejpam-4870	81	2	in	in	ADP
ejpam-4870	81	3	figure	figure	NOUN
ejpam-4870	81	4	4	4	NUM
ejpam-4870	81	5	are	be	AUX
ejpam-4870	81	6	pictorial	pictorial	ADJ
ejpam-4870	81	7	illustrations	illustration	NOUN
ejpam-4870	81	8	of	of	ADP
ejpam-4870	81	9	g1	g1	PROPN
ejpam-4870	81	10	and	and	CCONJ
ejpam-4870	81	11	g2	g2	PROPN
ejpam-4870	81	12	.	.	PUNCT
ejpam-4870	82	1	x1	x1	NUM
ejpam-4870	82	2	x1	x1	PROPN
ejpam-4870	83	1	x4	x4	PROPN
ejpam-4870	83	2	x2	x2	PROPN
ejpam-4870	83	3	x3	x3	ADJ
ejpam-4870	83	4	figure	figure	VERB
ejpam-4870	83	5	4	4	NUM
ejpam-4870	83	6	:	:	PUNCT
ejpam-4870	83	7	a	a	DET
ejpam-4870	83	8	trivial	trivial	ADJ
ejpam-4870	83	9	graph	graph	NOUN
ejpam-4870	83	10	g1	g1	NOUN
ejpam-4870	83	11	and	and	CCONJ
ejpam-4870	83	12	an	an	DET
ejpam-4870	83	13	empty	empty	ADJ
ejpam-4870	83	14	graph	graph	NOUN
ejpam-4870	83	15	g2	g2	PROPN
ejpam-4870	83	16	definition	definition	NOUN
ejpam-4870	83	17	5	5	NUM
ejpam-4870	83	18	.	.	PUNCT
ejpam-4870	84	1	the	the	DET
ejpam-4870	84	2	degree	degree	NOUN
ejpam-4870	84	3	of	of	ADP
ejpam-4870	84	4	a	a	DET
ejpam-4870	84	5	vertex	vertex	NOUN
ejpam-4870	84	6	x	x	NOUN
ejpam-4870	84	7	,	,	PUNCT
ejpam-4870	84	8	denoted	denote	VERB
ejpam-4870	84	9	by	by	ADP
ejpam-4870	84	10	deg(x	deg(x	NOUN
ejpam-4870	84	11	)	)	PUNCT
ejpam-4870	84	12	,	,	PUNCT
ejpam-4870	84	13	is	be	AUX
ejpam-4870	84	14	the	the	DET
ejpam-4870	84	15	number	number	NOUN
ejpam-4870	84	16	of	of	ADP
ejpam-4870	84	17	edges	edge	NOUN
ejpam-4870	84	18	incident	incident	NOUN
ejpam-4870	84	19	with	with	ADP
ejpam-4870	84	20	vertex	vertex	NOUN
ejpam-4870	84	21	x.	x.	PROPN
ejpam-4870	84	22	j.c	j.c	PROPN
ejpam-4870	84	23	.	.	PROPN
ejpam-4870	84	24	bonifacio	bonifacio	PROPN
ejpam-4870	84	25	,	,	PUNCT
ejpam-4870	84	26	c.j	c.j	PROPN
ejpam-4870	84	27	.	.	PROPN
ejpam-4870	84	28	andaya	andaya	PROPN
ejpam-4870	84	29	,	,	PUNCT
ejpam-4870	84	30	d.	d.	PROPN
ejpam-4870	84	31	magpantay	magpantay	PROPN
ejpam-4870	84	32	/	/	SYM
ejpam-4870	84	33	eur	eur	PROPN
ejpam-4870	84	34	.	.	PUNCT
ejpam-4870	85	1	j.	j.	PROPN
ejpam-4870	85	2	pure	pure	PROPN
ejpam-4870	85	3	appl	appl	PROPN
ejpam-4870	85	4	.	.	PROPN
ejpam-4870	85	5	math	math	PROPN
ejpam-4870	85	6	,	,	PUNCT
ejpam-4870	85	7	16	16	NUM
ejpam-4870	85	8	(	(	PUNCT
ejpam-4870	85	9	4	4	NUM
ejpam-4870	85	10	)	)	PUNCT
ejpam-4870	85	11	(	(	PUNCT
ejpam-4870	85	12	2023	2023	NUM
ejpam-4870	85	13	)	)	PUNCT
ejpam-4870	85	14	,	,	PUNCT
ejpam-4870	85	15	2476	2476	NUM
ejpam-4870	85	16	-	-	SYM
ejpam-4870	85	17	2498	2498	NUM
ejpam-4870	85	18	2480	2480	NUM
ejpam-4870	85	19	if	if	SCONJ
ejpam-4870	85	20	a	a	DET
ejpam-4870	85	21	vertex	vertex	NOUN
ejpam-4870	85	22	x	x	PRON
ejpam-4870	85	23	has	have	VERB
ejpam-4870	85	24	no	no	DET
ejpam-4870	85	25	degree	degree	NOUN
ejpam-4870	85	26	,	,	PUNCT
ejpam-4870	85	27	it	it	PRON
ejpam-4870	85	28	means	mean	VERB
ejpam-4870	85	29	that	that	SCONJ
ejpam-4870	85	30	it	it	PRON
ejpam-4870	85	31	is	be	AUX
ejpam-4870	85	32	not	not	PART
ejpam-4870	85	33	adjacent	adjacent	ADJ
ejpam-4870	85	34	to	to	ADP
ejpam-4870	85	35	any	any	DET
ejpam-4870	85	36	other	other	ADJ
ejpam-4870	85	37	vertices	vertex	NOUN
ejpam-4870	85	38	in	in	ADP
ejpam-4870	85	39	a	a	DET
ejpam-4870	85	40	graph	graph	NOUN
ejpam-4870	85	41	,	,	PUNCT
ejpam-4870	85	42	then	then	ADV
ejpam-4870	85	43	it	it	PRON
ejpam-4870	85	44	is	be	AUX
ejpam-4870	85	45	called	call	VERB
ejpam-4870	85	46	an	an	DET
ejpam-4870	85	47	isolated	isolated	ADJ
ejpam-4870	85	48	vertex	vertex	NOUN
ejpam-4870	85	49	.	.	PUNCT
ejpam-4870	86	1	thus	thus	ADV
ejpam-4870	86	2	,	,	PUNCT
ejpam-4870	86	3	deg(x	deg(x	X
ejpam-4870	86	4	)	)	PUNCT
ejpam-4870	86	5	=	=	SYM
ejpam-4870	86	6	0	0	NUM
ejpam-4870	86	7	for	for	ADP
ejpam-4870	86	8	every	every	DET
ejpam-4870	86	9	isolated	isolate	VERB
ejpam-4870	86	10	vertex	vertex	NOUN
ejpam-4870	86	11	x.	x.	NOUN
ejpam-4870	86	12	example	example	NOUN
ejpam-4870	86	13	4	4	X
ejpam-4870	86	14	.	.	PUNCT
ejpam-4870	87	1	let	let	VERB
ejpam-4870	87	2	g	g	PRON
ejpam-4870	87	3	be	be	AUX
ejpam-4870	87	4	a	a	DET
ejpam-4870	87	5	graph	graph	NOUN
ejpam-4870	87	6	where	where	SCONJ
ejpam-4870	87	7	v	v	NOUN
ejpam-4870	87	8	(	(	PUNCT
ejpam-4870	87	9	g	g	NOUN
ejpam-4870	87	10	)	)	PUNCT
ejpam-4870	87	11	=	=	SYM
ejpam-4870	88	1	{	{	PUNCT
ejpam-4870	88	2	x1	x1	PROPN
ejpam-4870	88	3	,	,	PUNCT
ejpam-4870	88	4	x2	x2	PROPN
ejpam-4870	88	5	,	,	PUNCT
ejpam-4870	88	6	x3	x3	PROPN
ejpam-4870	88	7	,	,	PUNCT
ejpam-4870	88	8	x4	x4	PROPN
ejpam-4870	88	9	,	,	PUNCT
ejpam-4870	88	10	x5	x5	NOUN
ejpam-4870	88	11	}	}	PUNCT
ejpam-4870	88	12	and	and	CCONJ
ejpam-4870	88	13	e(g	e(g	NOUN
ejpam-4870	88	14	)	)	PUNCT
ejpam-4870	89	1	=	=	PRON
ejpam-4870	89	2	{	{	PUNCT
ejpam-4870	90	1	[	[	X
ejpam-4870	90	2	x1	x1	PROPN
ejpam-4870	90	3	,	,	PUNCT
ejpam-4870	90	4	x2	x2	PROPN
ejpam-4870	90	5	]	]	PUNCT
ejpam-4870	90	6	,	,	PUNCT
ejpam-4870	90	7	[	[	X
ejpam-4870	90	8	x2	x2	X
ejpam-4870	90	9	,	,	PUNCT
ejpam-4870	90	10	x3	x3	ADJ
ejpam-4870	90	11	]	]	PUNCT
ejpam-4870	90	12	,	,	PUNCT
ejpam-4870	91	1	[	[	X
ejpam-4870	91	2	x2	x2	X
ejpam-4870	91	3	,	,	PUNCT
ejpam-4870	91	4	x4	x4	PROPN
ejpam-4870	91	5	]	]	PUNCT
ejpam-4870	91	6	,	,	PUNCT
ejpam-4870	91	7	[	[	X
ejpam-4870	91	8	x3	x3	ADJ
ejpam-4870	91	9	,	,	PUNCT
ejpam-4870	91	10	x5	x5	NOUN
ejpam-4870	91	11	]	]	X
ejpam-4870	91	12	,	,	PUNCT
ejpam-4870	91	13	[	[	X
ejpam-4870	91	14	x4	x4	PROPN
ejpam-4870	91	15	,	,	PUNCT
ejpam-4870	91	16	x5	x5	NOUN
ejpam-4870	91	17	]	]	X
ejpam-4870	91	18	}	}	PUNCT
ejpam-4870	91	19	.	.	PUNCT
ejpam-4870	92	1	given	give	VERB
ejpam-4870	92	2	in	in	ADP
ejpam-4870	92	3	figure	figure	NOUN
ejpam-4870	92	4	5	5	NUM
ejpam-4870	92	5	is	be	AUX
ejpam-4870	92	6	a	a	DET
ejpam-4870	92	7	pictorial	pictorial	ADJ
ejpam-4870	92	8	representation	representation	NOUN
ejpam-4870	92	9	of	of	ADP
ejpam-4870	92	10	a	a	DET
ejpam-4870	92	11	graph	graph	NOUN
ejpam-4870	92	12	g.	g.	NOUN
ejpam-4870	92	13	since	since	SCONJ
ejpam-4870	92	14	there	there	PRON
ejpam-4870	92	15	are	be	VERB
ejpam-4870	92	16	3	3	NUM
ejpam-4870	92	17	edges	edge	NOUN
ejpam-4870	92	18	incident	incident	NOUN
ejpam-4870	92	19	with	with	ADP
ejpam-4870	92	20	vertex	vertex	NOUN
ejpam-4870	92	21	x2	x2	NOUN
ejpam-4870	92	22	,	,	PUNCT
ejpam-4870	92	23	it	it	PRON
ejpam-4870	92	24	follows	follow	VERB
ejpam-4870	92	25	that	that	SCONJ
ejpam-4870	92	26	deg(x2	deg(x2	NOUN
ejpam-4870	92	27	)	)	PUNCT
ejpam-4870	92	28	=	=	SYM
ejpam-4870	93	1	3	3	X
ejpam-4870	93	2	.	.	PUNCT
ejpam-4870	93	3	shown	show	VERB
ejpam-4870	93	4	in	in	ADP
ejpam-4870	93	5	table	table	NOUN
ejpam-4870	93	6	1	1	NUM
ejpam-4870	93	7	is	be	AUX
ejpam-4870	93	8	the	the	DET
ejpam-4870	93	9	list	list	NOUN
ejpam-4870	93	10	for	for	ADP
ejpam-4870	93	11	the	the	DET
ejpam-4870	93	12	degree	degree	NOUN
ejpam-4870	93	13	of	of	ADP
ejpam-4870	93	14	every	every	DET
ejpam-4870	93	15	vertex	vertex	NOUN
ejpam-4870	93	16	in	in	ADP
ejpam-4870	93	17	g.	g.	PROPN
ejpam-4870	93	18	x1	x1	PROPN
ejpam-4870	94	1	x4	x4	PROPN
ejpam-4870	94	2	x2	x2	PROPN
ejpam-4870	94	3	x3	x3	PROPN
ejpam-4870	94	4	x5	x5	PROPN
ejpam-4870	94	5	figure	figure	NOUN
ejpam-4870	94	6	5	5	NUM
ejpam-4870	94	7	:	:	PUNCT
ejpam-4870	94	8	graph	graph	VERB
ejpam-4870	94	9	g	g	NOUN
ejpam-4870	94	10	of	of	ADP
ejpam-4870	94	11	order	order	NOUN
ejpam-4870	94	12	5	5	NUM
ejpam-4870	94	13	and	and	CCONJ
ejpam-4870	94	14	size	size	NOUN
ejpam-4870	94	15	5	5	NUM
ejpam-4870	94	16	table	table	NOUN
ejpam-4870	94	17	1	1	NUM
ejpam-4870	94	18	:	:	PUNCT
ejpam-4870	94	19	degrees	degree	NOUN
ejpam-4870	94	20	of	of	ADP
ejpam-4870	94	21	every	every	DET
ejpam-4870	94	22	vertex	vertex	NOUN
ejpam-4870	94	23	in	in	ADP
ejpam-4870	94	24	g	g	PROPN
ejpam-4870	94	25	in	in	ADP
ejpam-4870	94	26	figure	figure	NOUN
ejpam-4870	94	27	5	5	NUM
ejpam-4870	94	28	vertex	vertex	NOUN
ejpam-4870	94	29	deg(x	deg(x	PROPN
ejpam-4870	94	30	)	)	PUNCT
ejpam-4870	95	1	x1	x1	PROPN
ejpam-4870	95	2	1	1	NUM
ejpam-4870	95	3	x2	x2	SYM
ejpam-4870	95	4	3	3	NUM
ejpam-4870	95	5	x3	x3	PROPN
ejpam-4870	95	6	2	2	NUM
ejpam-4870	95	7	x4	x4	PROPN
ejpam-4870	95	8	2	2	NUM
ejpam-4870	95	9	x5	x5	NOUN
ejpam-4870	95	10	2∑	2∑	NUM
ejpam-4870	95	11	deg(x	deg(x	NOUN
ejpam-4870	95	12	)	)	PUNCT
ejpam-4870	95	13	10	10	NUM
ejpam-4870	95	14	j.c	j.c	PROPN
ejpam-4870	95	15	.	.	PROPN
ejpam-4870	95	16	bonifacio	bonifacio	PROPN
ejpam-4870	95	17	,	,	PUNCT
ejpam-4870	95	18	c.j	c.j	PROPN
ejpam-4870	95	19	.	.	PROPN
ejpam-4870	95	20	andaya	andaya	PROPN
ejpam-4870	95	21	,	,	PUNCT
ejpam-4870	95	22	d.	d.	PROPN
ejpam-4870	95	23	magpantay	magpantay	PROPN
ejpam-4870	95	24	/	/	SYM
ejpam-4870	95	25	eur	eur	PROPN
ejpam-4870	95	26	.	.	PUNCT
ejpam-4870	96	1	j.	j.	PROPN
ejpam-4870	96	2	pure	pure	PROPN
ejpam-4870	96	3	appl	appl	PROPN
ejpam-4870	96	4	.	.	PROPN
ejpam-4870	96	5	math	math	PROPN
ejpam-4870	96	6	,	,	PUNCT
ejpam-4870	96	7	16	16	NUM
ejpam-4870	96	8	(	(	PUNCT
ejpam-4870	96	9	4	4	NUM
ejpam-4870	96	10	)	)	PUNCT
ejpam-4870	96	11	(	(	PUNCT
ejpam-4870	96	12	2023	2023	NUM
ejpam-4870	96	13	)	)	PUNCT
ejpam-4870	96	14	,	,	PUNCT
ejpam-4870	96	15	2476	2476	NUM
ejpam-4870	96	16	-	-	SYM
ejpam-4870	96	17	2498	2498	NUM
ejpam-4870	96	18	2481	2481	NUM
ejpam-4870	96	19	by	by	ADP
ejpam-4870	96	20	the	the	DET
ejpam-4870	96	21	concept	concept	NOUN
ejpam-4870	96	22	of	of	ADP
ejpam-4870	96	23	degrees	degree	NOUN
ejpam-4870	96	24	of	of	ADP
ejpam-4870	96	25	a	a	DET
ejpam-4870	96	26	vertex	vertex	NOUN
ejpam-4870	96	27	,	,	PUNCT
ejpam-4870	96	28	the	the	DET
ejpam-4870	96	29	next	next	ADJ
ejpam-4870	96	30	theorem	theorem	NOUN
ejpam-4870	96	31	is	be	AUX
ejpam-4870	96	32	one	one	NUM
ejpam-4870	96	33	of	of	ADP
ejpam-4870	96	34	the	the	DET
ejpam-4870	96	35	fundamental	fundamental	ADJ
ejpam-4870	96	36	theorems	theorem	NOUN
ejpam-4870	96	37	in	in	ADP
ejpam-4870	96	38	graph	graph	NOUN
ejpam-4870	96	39	theory	theory	NOUN
ejpam-4870	96	40	.	.	PUNCT
ejpam-4870	97	1	this	this	DET
ejpam-4870	97	2	theorem	theorem	NOUN
ejpam-4870	97	3	represents	represent	VERB
ejpam-4870	97	4	the	the	DET
ejpam-4870	97	5	equality	equality	NOUN
ejpam-4870	97	6	between	between	ADP
ejpam-4870	97	7	the	the	DET
ejpam-4870	97	8	size	size	NOUN
ejpam-4870	97	9	of	of	ADP
ejpam-4870	97	10	a	a	DET
ejpam-4870	97	11	graph	graph	NOUN
ejpam-4870	97	12	g	g	NOUN
ejpam-4870	97	13	and	and	CCONJ
ejpam-4870	97	14	the	the	DET
ejpam-4870	97	15	totality	totality	NOUN
ejpam-4870	97	16	of	of	ADP
ejpam-4870	97	17	the	the	DET
ejpam-4870	97	18	degrees	degree	NOUN
ejpam-4870	97	19	of	of	ADP
ejpam-4870	97	20	all	all	DET
ejpam-4870	97	21	vertices	vertex	NOUN
ejpam-4870	97	22	of	of	ADP
ejpam-4870	97	23	g.	g.	PROPN
ejpam-4870	97	24	theorem	theorem	VERB
ejpam-4870	97	25	1	1	X
ejpam-4870	97	26	.	.	PUNCT
ejpam-4870	98	1	if	if	SCONJ
ejpam-4870	98	2	g	g	PROPN
ejpam-4870	98	3	is	be	AUX
ejpam-4870	98	4	a	a	DET
ejpam-4870	98	5	graph	graph	NOUN
ejpam-4870	98	6	with	with	ADP
ejpam-4870	98	7	size	size	NOUN
ejpam-4870	98	8	m	m	PROPN
ejpam-4870	98	9	,	,	PUNCT
ejpam-4870	98	10	then∑	then∑	VERB
ejpam-4870	98	11	x∈v	x∈v	PROPN
ejpam-4870	98	12	(	(	PUNCT
ejpam-4870	98	13	g	g	NOUN
ejpam-4870	98	14	)	)	PUNCT
ejpam-4870	98	15	deg(x	deg(x	PROPN
ejpam-4870	98	16	)	)	PUNCT
ejpam-4870	98	17	=	=	SYM
ejpam-4870	99	1	2	2	NUM
ejpam-4870	99	2	m.	m.	NOUN
ejpam-4870	99	3	the	the	DET
ejpam-4870	99	4	idea	idea	NOUN
ejpam-4870	99	5	of	of	ADP
ejpam-4870	99	6	the	the	DET
ejpam-4870	99	7	degrees	degree	NOUN
ejpam-4870	99	8	of	of	ADP
ejpam-4870	99	9	every	every	DET
ejpam-4870	99	10	vertex	vertex	NOUN
ejpam-4870	99	11	is	be	AUX
ejpam-4870	99	12	important	important	ADJ
ejpam-4870	99	13	in	in	ADP
ejpam-4870	99	14	finding	find	VERB
ejpam-4870	99	15	the	the	DET
ejpam-4870	99	16	order	order	NOUN
ejpam-4870	99	17	and	and	CCONJ
ejpam-4870	99	18	size	size	NOUN
ejpam-4870	99	19	of	of	ADP
ejpam-4870	99	20	a	a	DET
ejpam-4870	99	21	graph	graph	NOUN
ejpam-4870	99	22	especially	especially	ADV
ejpam-4870	99	23	if	if	SCONJ
ejpam-4870	99	24	the	the	DET
ejpam-4870	99	25	degrees	degree	NOUN
ejpam-4870	99	26	have	have	VERB
ejpam-4870	99	27	the	the	DET
ejpam-4870	99	28	same	same	ADJ
ejpam-4870	99	29	number	number	NOUN
ejpam-4870	99	30	.	.	PUNCT
ejpam-4870	100	1	that	that	PRON
ejpam-4870	100	2	is	be	AUX
ejpam-4870	100	3	,	,	PUNCT
ejpam-4870	100	4	a	a	DET
ejpam-4870	100	5	graph	graph	NOUN
ejpam-4870	100	6	with	with	ADP
ejpam-4870	100	7	the	the	DET
ejpam-4870	100	8	same	same	ADJ
ejpam-4870	100	9	number	number	NOUN
ejpam-4870	100	10	of	of	ADP
ejpam-4870	100	11	degrees	degree	NOUN
ejpam-4870	100	12	of	of	ADP
ejpam-4870	100	13	its	its	PRON
ejpam-4870	100	14	vertices	vertex	NOUN
ejpam-4870	100	15	forms	form	VERB
ejpam-4870	100	16	a	a	DET
ejpam-4870	100	17	regular	regular	ADJ
ejpam-4870	100	18	graph	graph	NOUN
ejpam-4870	100	19	.	.	PUNCT
ejpam-4870	101	1	definition	definition	NOUN
ejpam-4870	101	2	6	6	NUM
ejpam-4870	101	3	.	.	PUNCT
ejpam-4870	102	1	a	a	DET
ejpam-4870	102	2	graph	graph	NOUN
ejpam-4870	102	3	g	g	NOUN
ejpam-4870	102	4	is	be	AUX
ejpam-4870	102	5	regular	regular	ADJ
ejpam-4870	102	6	if	if	SCONJ
ejpam-4870	102	7	every	every	DET
ejpam-4870	102	8	vertex	vertex	NOUN
ejpam-4870	102	9	has	have	VERB
ejpam-4870	102	10	the	the	DET
ejpam-4870	102	11	same	same	ADJ
ejpam-4870	102	12	degree	degree	NOUN
ejpam-4870	102	13	.	.	PUNCT
ejpam-4870	103	1	moreover	moreover	ADV
ejpam-4870	103	2	,	,	PUNCT
ejpam-4870	103	3	g	g	PROPN
ejpam-4870	103	4	is	be	AUX
ejpam-4870	103	5	said	say	VERB
ejpam-4870	103	6	to	to	PART
ejpam-4870	103	7	be	be	AUX
ejpam-4870	103	8	regular	regular	ADJ
ejpam-4870	103	9	of	of	ADP
ejpam-4870	103	10	degree	degree	NOUN
ejpam-4870	103	11	r	r	NOUN
ejpam-4870	103	12	(	(	PUNCT
ejpam-4870	103	13	or	or	CCONJ
ejpam-4870	103	14	r	r	NOUN
ejpam-4870	103	15	-	-	ADJ
ejpam-4870	103	16	regular	regular	ADJ
ejpam-4870	103	17	)	)	PUNCT
ejpam-4870	103	18	if	if	SCONJ
ejpam-4870	103	19	deg(x	deg(x	ADV
ejpam-4870	103	20	)	)	PUNCT
ejpam-4870	104	1	=	=	SYM
ejpam-4870	104	2	r	r	NOUN
ejpam-4870	104	3	for	for	ADP
ejpam-4870	104	4	all	all	DET
ejpam-4870	104	5	vertices	vertex	NOUN
ejpam-4870	104	6	x	x	VERB
ejpam-4870	104	7	in	in	ADP
ejpam-4870	104	8	g.	g.	NOUN
ejpam-4870	104	9	by	by	ADP
ejpam-4870	104	10	using	use	VERB
ejpam-4870	104	11	theorem	theorem	NOUN
ejpam-4870	104	12	1	1	NUM
ejpam-4870	104	13	,	,	PUNCT
ejpam-4870	104	14	a	a	DET
ejpam-4870	104	15	formula	formula	NOUN
ejpam-4870	104	16	for	for	ADP
ejpam-4870	104	17	the	the	DET
ejpam-4870	104	18	order	order	NOUN
ejpam-4870	104	19	and	and	CCONJ
ejpam-4870	104	20	size	size	NOUN
ejpam-4870	104	21	of	of	ADP
ejpam-4870	104	22	an	an	DET
ejpam-4870	104	23	r	r	NOUN
ejpam-4870	104	24	-	-	PUNCT
ejpam-4870	104	25	regular	regular	ADJ
ejpam-4870	104	26	graph	graph	NOUN
ejpam-4870	104	27	can	can	AUX
ejpam-4870	104	28	be	be	AUX
ejpam-4870	104	29	derived	derive	VERB
ejpam-4870	104	30	.	.	PUNCT
ejpam-4870	105	1	given	give	VERB
ejpam-4870	105	2	n	n	ADV
ejpam-4870	105	3	as	as	ADP
ejpam-4870	105	4	its	its	PRON
ejpam-4870	105	5	order	order	NOUN
ejpam-4870	105	6	and	and	CCONJ
ejpam-4870	105	7	m	m	NOUN
ejpam-4870	105	8	as	as	ADP
ejpam-4870	105	9	its	its	PRON
ejpam-4870	105	10	size	size	NOUN
ejpam-4870	105	11	,	,	PUNCT
ejpam-4870	105	12	then∑	then∑	NOUN
ejpam-4870	105	13	x∈v	x∈v	PROPN
ejpam-4870	105	14	(	(	PUNCT
ejpam-4870	105	15	g	g	NOUN
ejpam-4870	105	16	)	)	PUNCT
ejpam-4870	105	17	deg(x	deg(x	PROPN
ejpam-4870	105	18	)	)	PUNCT
ejpam-4870	105	19	=	=	SYM
ejpam-4870	106	1	2	2	NUM
ejpam-4870	106	2	m	m	VERB
ejpam-4870	106	3	nr	nr	NOUN
ejpam-4870	106	4	=	=	SYM
ejpam-4870	106	5	2	2	NUM
ejpam-4870	106	6	m.	m.	NOUN
ejpam-4870	106	7	(	(	PUNCT
ejpam-4870	106	8	1	1	NUM
ejpam-4870	106	9	)	)	PUNCT
ejpam-4870	106	10	hence	hence	ADV
ejpam-4870	106	11	,	,	PUNCT
ejpam-4870	106	12	n	n	NOUN
ejpam-4870	106	13	=	=	SYM
ejpam-4870	106	14	2	2	NUM
ejpam-4870	106	15	m	m	NOUN
ejpam-4870	106	16	r	r	NOUN
ejpam-4870	106	17	and	and	CCONJ
ejpam-4870	106	18	m	m	PROPN
ejpam-4870	106	19	=	=	NOUN
ejpam-4870	106	20	nr	nr	PROPN
ejpam-4870	106	21	2	2	NUM
ejpam-4870	106	22	.	.	PUNCT
ejpam-4870	107	1	the	the	DET
ejpam-4870	107	2	next	next	ADJ
ejpam-4870	107	3	concept	concept	NOUN
ejpam-4870	107	4	is	be	AUX
ejpam-4870	107	5	the	the	DET
ejpam-4870	107	6	notion	notion	NOUN
ejpam-4870	107	7	of	of	ADP
ejpam-4870	107	8	a	a	DET
ejpam-4870	107	9	walk	walk	NOUN
ejpam-4870	107	10	which	which	PRON
ejpam-4870	107	11	is	be	AUX
ejpam-4870	107	12	a	a	DET
ejpam-4870	107	13	way	way	NOUN
ejpam-4870	107	14	of	of	ADP
ejpam-4870	107	15	traversing	traverse	VERB
ejpam-4870	107	16	a	a	DET
ejpam-4870	107	17	graph	graph	NOUN
ejpam-4870	107	18	by	by	ADP
ejpam-4870	107	19	moving	move	VERB
ejpam-4870	107	20	from	from	ADP
ejpam-4870	107	21	one	one	NUM
ejpam-4870	107	22	vertex	vertex	NOUN
ejpam-4870	107	23	to	to	ADP
ejpam-4870	107	24	another	another	PRON
ejpam-4870	107	25	through	through	ADP
ejpam-4870	107	26	the	the	DET
ejpam-4870	107	27	edges	edge	NOUN
ejpam-4870	107	28	of	of	ADP
ejpam-4870	107	29	the	the	DET
ejpam-4870	107	30	graph	graph	NOUN
ejpam-4870	107	31	.	.	PUNCT
ejpam-4870	108	1	definition	definition	NOUN
ejpam-4870	108	2	7	7	NUM
ejpam-4870	108	3	.	.	PUNCT
ejpam-4870	109	1	let	let	VERB
ejpam-4870	109	2	w	w	NOUN
ejpam-4870	109	3	:	:	PUNCT
ejpam-4870	109	4	x1	x1	ADJ
ejpam-4870	109	5	,	,	PUNCT
ejpam-4870	109	6	x2	x2	PROPN
ejpam-4870	109	7	,	,	PUNCT
ejpam-4870	109	8	·	·	PUNCT
ejpam-4870	109	9	·	·	PUNCT
ejpam-4870	109	10	·	·	PUNCT
ejpam-4870	109	11	,	,	PUNCT
ejpam-4870	109	12	xk	xk	PROPN
ejpam-4870	109	13	,	,	PUNCT
ejpam-4870	109	14	xk+1	xk+1	NUM
ejpam-4870	109	15	be	be	AUX
ejpam-4870	109	16	a	a	DET
ejpam-4870	109	17	walk	walk	NOUN
ejpam-4870	109	18	of	of	ADP
ejpam-4870	109	19	length	length	NOUN
ejpam-4870	110	1	k	k	PROPN
ejpam-4870	110	2	>	>	X
ejpam-4870	110	3	0	0	X
ejpam-4870	110	4	.	.	PUNCT
ejpam-4870	111	1	a	a	DET
ejpam-4870	111	2	walk	walk	NOUN
ejpam-4870	111	3	is	be	AUX
ejpam-4870	111	4	closed	close	VERB
ejpam-4870	111	5	if	if	SCONJ
ejpam-4870	111	6	x1	x1	PROPN
ejpam-4870	111	7	=	=	PUNCT
ejpam-4870	111	8	xk+1	xk+1	X
ejpam-4870	111	9	.	.	PUNCT
ejpam-4870	112	1	a	a	DET
ejpam-4870	112	2	closed	closed	ADJ
ejpam-4870	112	3	walk	walk	NOUN
ejpam-4870	112	4	is	be	AUX
ejpam-4870	112	5	called	call	VERB
ejpam-4870	112	6	a	a	DET
ejpam-4870	112	7	cycle	cycle	NOUN
ejpam-4870	112	8	if	if	SCONJ
ejpam-4870	112	9	the	the	DET
ejpam-4870	112	10	vertices	vertex	NOUN
ejpam-4870	112	11	x1	x1	PROPN
ejpam-4870	112	12	,	,	PUNCT
ejpam-4870	112	13	x2	x2	PROPN
ejpam-4870	112	14	,	,	PUNCT
ejpam-4870	112	15	...	...	PUNCT
ejpam-4870	112	16	,	,	PUNCT
ejpam-4870	112	17	xk	xk	PROPN
ejpam-4870	112	18	are	be	AUX
ejpam-4870	112	19	distinct	distinct	ADJ
ejpam-4870	112	20	.	.	PUNCT
ejpam-4870	113	1	cycles	cycle	NOUN
ejpam-4870	113	2	are	be	AUX
ejpam-4870	113	3	special	special	ADJ
ejpam-4870	113	4	kinds	kind	NOUN
ejpam-4870	113	5	of	of	ADP
ejpam-4870	113	6	walks	walk	NOUN
ejpam-4870	113	7	in	in	ADP
ejpam-4870	113	8	graphs	graph	NOUN
ejpam-4870	113	9	such	such	ADJ
ejpam-4870	113	10	that	that	SCONJ
ejpam-4870	113	11	these	these	PRON
ejpam-4870	113	12	are	be	AUX
ejpam-4870	113	13	used	use	VERB
ejpam-4870	113	14	to	to	PART
ejpam-4870	113	15	name	name	VERB
ejpam-4870	113	16	a	a	DET
ejpam-4870	113	17	special	special	ADJ
ejpam-4870	113	18	class	class	NOUN
ejpam-4870	113	19	of	of	ADP
ejpam-4870	113	20	graph	graph	NOUN
ejpam-4870	113	21	.	.	PUNCT
ejpam-4870	114	1	there	there	PRON
ejpam-4870	114	2	are	be	VERB
ejpam-4870	114	3	numerous	numerous	ADJ
ejpam-4870	114	4	notable	notable	ADJ
ejpam-4870	114	5	graphs	graph	NOUN
ejpam-4870	114	6	that	that	PRON
ejpam-4870	114	7	have	have	AUX
ejpam-4870	114	8	been	be	AUX
ejpam-4870	114	9	discovered	discover	VERB
ejpam-4870	114	10	in	in	ADP
ejpam-4870	114	11	graph	graph	NOUN
ejpam-4870	114	12	theory	theory	NOUN
ejpam-4870	114	13	.	.	PUNCT
ejpam-4870	115	1	these	these	DET
ejpam-4870	115	2	graphs	graph	NOUN
ejpam-4870	115	3	have	have	VERB
ejpam-4870	115	4	special	special	ADJ
ejpam-4870	115	5	notations	notation	NOUN
ejpam-4870	115	6	that	that	PRON
ejpam-4870	115	7	are	be	AUX
ejpam-4870	115	8	used	use	VERB
ejpam-4870	115	9	exclusively	exclusively	ADV
ejpam-4870	115	10	to	to	PART
ejpam-4870	115	11	denote	denote	VERB
ejpam-4870	115	12	them	they	PRON
ejpam-4870	115	13	.	.	PUNCT
ejpam-4870	116	1	in	in	ADP
ejpam-4870	116	2	this	this	DET
ejpam-4870	116	3	section	section	NOUN
ejpam-4870	116	4	,	,	PUNCT
ejpam-4870	116	5	some	some	PRON
ejpam-4870	116	6	of	of	ADP
ejpam-4870	116	7	the	the	DET
ejpam-4870	116	8	common	common	ADJ
ejpam-4870	116	9	classes	class	NOUN
ejpam-4870	116	10	of	of	ADP
ejpam-4870	116	11	graphs	graph	NOUN
ejpam-4870	116	12	such	such	ADJ
ejpam-4870	116	13	as	as	ADP
ejpam-4870	116	14	the	the	DET
ejpam-4870	116	15	cycle	cycle	NOUN
ejpam-4870	116	16	graph	graph	NOUN
ejpam-4870	116	17	,	,	PUNCT
ejpam-4870	116	18	and	and	CCONJ
ejpam-4870	116	19	complete	complete	ADJ
ejpam-4870	116	20	graph	graph	NOUN
ejpam-4870	116	21	are	be	AUX
ejpam-4870	116	22	discussed	discuss	VERB
ejpam-4870	116	23	.	.	PUNCT
ejpam-4870	117	1	definition	definition	NOUN
ejpam-4870	117	2	8	8	NUM
ejpam-4870	117	3	.	.	PUNCT
ejpam-4870	118	1	a	a	DET
ejpam-4870	118	2	graph	graph	NOUN
ejpam-4870	118	3	g	g	NOUN
ejpam-4870	118	4	of	of	ADP
ejpam-4870	118	5	order	order	NOUN
ejpam-4870	118	6	n	n	PRON
ejpam-4870	118	7	≥	≥	NOUN
ejpam-4870	118	8	3	3	NUM
ejpam-4870	118	9	is	be	AUX
ejpam-4870	118	10	called	call	VERB
ejpam-4870	118	11	a	a	DET
ejpam-4870	118	12	cycle	cycle	NOUN
ejpam-4870	118	13	graph	graph	NOUN
ejpam-4870	118	14	of	of	ADP
ejpam-4870	118	15	order	order	NOUN
ejpam-4870	118	16	n	n	CCONJ
ejpam-4870	118	17	,	,	PUNCT
ejpam-4870	118	18	denoted	denote	VERB
ejpam-4870	118	19	by	by	ADP
ejpam-4870	118	20	cn	cn	PROPN
ejpam-4870	118	21	,	,	PUNCT
ejpam-4870	118	22	if	if	SCONJ
ejpam-4870	118	23	the	the	DET
ejpam-4870	118	24	vertices	vertex	NOUN
ejpam-4870	118	25	of	of	ADP
ejpam-4870	118	26	g	g	PROPN
ejpam-4870	118	27	is	be	AUX
ejpam-4870	118	28	labeled	label	VERB
ejpam-4870	118	29	x1	x1	PROPN
ejpam-4870	118	30	,	,	PUNCT
ejpam-4870	118	31	x2	x2	PROPN
ejpam-4870	118	32	,	,	PUNCT
ejpam-4870	118	33	...	...	PUNCT
ejpam-4870	118	34	,	,	PUNCT
ejpam-4870	118	35	xn	xn	PROPN
ejpam-4870	118	36	so	so	SCONJ
ejpam-4870	118	37	that	that	SCONJ
ejpam-4870	118	38	the	the	DET
ejpam-4870	118	39	edges	edge	NOUN
ejpam-4870	118	40	are	be	AUX
ejpam-4870	118	41	[	[	X
ejpam-4870	118	42	x1	x1	PROPN
ejpam-4870	118	43	,	,	PUNCT
ejpam-4870	118	44	x2	x2	PROPN
ejpam-4870	118	45	]	]	PUNCT
ejpam-4870	118	46	,	,	PUNCT
ejpam-4870	119	1	[	[	X
ejpam-4870	119	2	x2	x2	X
ejpam-4870	119	3	,	,	PUNCT
ejpam-4870	119	4	x3	x3	ADJ
ejpam-4870	119	5	]	]	PUNCT
ejpam-4870	119	6	,	,	PUNCT
ejpam-4870	119	7	...	...	PUNCT
ejpam-4870	119	8	,	,	PUNCT
ejpam-4870	119	9	[	[	X
ejpam-4870	119	10	xn−1	xn−1	PROPN
ejpam-4870	119	11	,	,	PUNCT
ejpam-4870	119	12	xn	xn	PROPN
ejpam-4870	119	13	]	]	PUNCT
ejpam-4870	119	14	,	,	PUNCT
ejpam-4870	119	15	[	[	X
ejpam-4870	119	16	xn	xn	X
ejpam-4870	119	17	,	,	PUNCT
ejpam-4870	119	18	x1	x1	PROPN
ejpam-4870	119	19	]	]	X
ejpam-4870	119	20	.	.	PUNCT
ejpam-4870	120	1	a	a	DET
ejpam-4870	120	2	cycle	cycle	NOUN
ejpam-4870	120	3	graph	graph	NOUN
ejpam-4870	120	4	is	be	AUX
ejpam-4870	120	5	said	say	VERB
ejpam-4870	120	6	to	to	PART
ejpam-4870	120	7	be	be	AUX
ejpam-4870	120	8	a	a	DET
ejpam-4870	120	9	2	2	NUM
ejpam-4870	120	10	-	-	PUNCT
ejpam-4870	120	11	regular	regular	ADJ
ejpam-4870	120	12	graph	graph	NOUN
ejpam-4870	120	13	.	.	PUNCT
ejpam-4870	121	1	so	so	ADV
ejpam-4870	121	2	,	,	PUNCT
ejpam-4870	121	3	by	by	ADP
ejpam-4870	121	4	using	use	VERB
ejpam-4870	121	5	equation	equation	NOUN
ejpam-4870	121	6	1	1	NUM
ejpam-4870	121	7	to	to	PART
ejpam-4870	121	8	determine	determine	VERB
ejpam-4870	121	9	the	the	DET
ejpam-4870	121	10	size	size	NOUN
ejpam-4870	121	11	m	m	PROPN
ejpam-4870	121	12	of	of	ADP
ejpam-4870	121	13	a	a	DET
ejpam-4870	121	14	cycle	cycle	NOUN
ejpam-4870	121	15	graph	graph	NOUN
ejpam-4870	121	16	of	of	ADP
ejpam-4870	121	17	order	order	NOUN
ejpam-4870	121	18	n	n	PRON
ejpam-4870	121	19	we	we	PRON
ejpam-4870	121	20	have	have	VERB
ejpam-4870	121	21	nr	nr	NOUN
ejpam-4870	121	22	=	=	SYM
ejpam-4870	121	23	2	2	NUM
ejpam-4870	121	24	m	m	NOUN
ejpam-4870	121	25	j.c	j.c	PROPN
ejpam-4870	121	26	.	.	PROPN
ejpam-4870	121	27	bonifacio	bonifacio	PROPN
ejpam-4870	121	28	,	,	PUNCT
ejpam-4870	121	29	c.j	c.j	PROPN
ejpam-4870	121	30	.	.	PROPN
ejpam-4870	121	31	andaya	andaya	PROPN
ejpam-4870	121	32	,	,	PUNCT
ejpam-4870	121	33	d.	d.	PROPN
ejpam-4870	121	34	magpantay	magpantay	PROPN
ejpam-4870	121	35	/	/	SYM
ejpam-4870	121	36	eur	eur	PROPN
ejpam-4870	121	37	.	.	PUNCT
ejpam-4870	122	1	j.	j.	PROPN
ejpam-4870	122	2	pure	pure	PROPN
ejpam-4870	122	3	appl	appl	PROPN
ejpam-4870	122	4	.	.	PROPN
ejpam-4870	122	5	math	math	PROPN
ejpam-4870	122	6	,	,	PUNCT
ejpam-4870	122	7	16	16	NUM
ejpam-4870	122	8	(	(	PUNCT
ejpam-4870	122	9	4	4	NUM
ejpam-4870	122	10	)	)	PUNCT
ejpam-4870	122	11	(	(	PUNCT
ejpam-4870	122	12	2023	2023	NUM
ejpam-4870	122	13	)	)	PUNCT
ejpam-4870	122	14	,	,	PUNCT
ejpam-4870	122	15	2476	2476	NUM
ejpam-4870	122	16	-	-	SYM
ejpam-4870	122	17	2498	2498	NUM
ejpam-4870	122	18	2482	2482	NUM
ejpam-4870	122	19	n(2	n(2	PROPN
ejpam-4870	122	20	)	)	PUNCT
ejpam-4870	122	21	=	=	PUNCT
ejpam-4870	122	22	2	2	NUM
ejpam-4870	122	23	m	m	NOUN
ejpam-4870	122	24	m	m	NOUN
ejpam-4870	122	25	=	=	ADJ
ejpam-4870	122	26	n.	n.	NOUN
ejpam-4870	123	1	hence	hence	ADV
ejpam-4870	123	2	,	,	PUNCT
ejpam-4870	123	3	if	if	SCONJ
ejpam-4870	123	4	n	n	PRON
ejpam-4870	123	5	is	be	AUX
ejpam-4870	123	6	the	the	DET
ejpam-4870	123	7	order	order	NOUN
ejpam-4870	123	8	of	of	ADP
ejpam-4870	123	9	cn	cn	PROPN
ejpam-4870	123	10	,	,	PUNCT
ejpam-4870	123	11	then	then	ADV
ejpam-4870	123	12	the	the	DET
ejpam-4870	123	13	size	size	NOUN
ejpam-4870	123	14	is	be	AUX
ejpam-4870	123	15	also	also	ADV
ejpam-4870	123	16	n.	n.	ADJ
ejpam-4870	123	17	example	example	NOUN
ejpam-4870	124	1	5	5	NUM
ejpam-4870	124	2	.	.	PUNCT
ejpam-4870	124	3	a	a	DET
ejpam-4870	124	4	cycle	cycle	NOUN
ejpam-4870	124	5	graph	graph	NOUN
ejpam-4870	124	6	c3	c3	PROPN
ejpam-4870	124	7	with	with	ADP
ejpam-4870	124	8	its	its	PRON
ejpam-4870	124	9	pictorial	pictorial	ADJ
ejpam-4870	124	10	representation	representation	NOUN
ejpam-4870	124	11	shown	show	VERB
ejpam-4870	124	12	in	in	ADP
ejpam-4870	124	13	figure	figure	NOUN
ejpam-4870	124	14	6	6	NUM
ejpam-4870	124	15	is	be	AUX
ejpam-4870	124	16	a	a	DET
ejpam-4870	124	17	cycle	cycle	NOUN
ejpam-4870	124	18	graph	graph	NOUN
ejpam-4870	124	19	of	of	ADP
ejpam-4870	124	20	order	order	NOUN
ejpam-4870	124	21	3	3	NUM
ejpam-4870	124	22	since	since	SCONJ
ejpam-4870	124	23	it	it	PRON
ejpam-4870	124	24	is	be	AUX
ejpam-4870	124	25	a	a	DET
ejpam-4870	124	26	closed	closed	ADJ
ejpam-4870	124	27	walk	walk	NOUN
ejpam-4870	124	28	that	that	PRON
ejpam-4870	124	29	contains	contain	VERB
ejpam-4870	124	30	distinct	distinct	ADJ
ejpam-4870	124	31	vertices	vertex	NOUN
ejpam-4870	124	32	.	.	PUNCT
ejpam-4870	125	1	x3	x3	VERB
ejpam-4870	125	2	x1	x1	NOUN
ejpam-4870	126	1	x2	x2	PROPN
ejpam-4870	126	2	figure	figure	VERB
ejpam-4870	126	3	6	6	NUM
ejpam-4870	126	4	:	:	PUNCT
ejpam-4870	126	5	cycle	cycle	NOUN
ejpam-4870	126	6	graph	graph	NOUN
ejpam-4870	126	7	c3	c3	PROPN
ejpam-4870	126	8	of	of	ADP
ejpam-4870	126	9	order	order	NOUN
ejpam-4870	126	10	3	3	NUM
ejpam-4870	126	11	considering	consider	VERB
ejpam-4870	126	12	cn	cn	PROPN
ejpam-4870	126	13	with	with	ADP
ejpam-4870	126	14	v	v	PROPN
ejpam-4870	126	15	(	(	PUNCT
ejpam-4870	126	16	cn	cn	PROPN
ejpam-4870	126	17	)	)	PUNCT
ejpam-4870	126	18	=	=	PRON
ejpam-4870	126	19	{	{	PUNCT
ejpam-4870	126	20	x1	x1	PROPN
ejpam-4870	126	21	,	,	PUNCT
ejpam-4870	126	22	x2	x2	PROPN
ejpam-4870	126	23	,	,	PUNCT
ejpam-4870	126	24	·	·	PUNCT
ejpam-4870	126	25	·	·	PUNCT
ejpam-4870	126	26	·	·	PUNCT
ejpam-4870	126	27	,	,	PUNCT
ejpam-4870	126	28	xn	xn	PROPN
ejpam-4870	126	29	}	}	PUNCT
ejpam-4870	126	30	.	.	PUNCT
ejpam-4870	127	1	throughout	throughout	ADP
ejpam-4870	127	2	the	the	DET
ejpam-4870	127	3	paper	paper	NOUN
ejpam-4870	127	4	,	,	PUNCT
ejpam-4870	127	5	vertices	vertice	VERB
ejpam-4870	127	6	x1	x1	PROPN
ejpam-4870	127	7	,	,	PUNCT
ejpam-4870	127	8	x2	x2	PROPN
ejpam-4870	127	9	,	,	PUNCT
ejpam-4870	127	10	·	·	PUNCT
ejpam-4870	127	11	·	·	PUNCT
ejpam-4870	127	12	·	·	PUNCT
ejpam-4870	127	13	,	,	PUNCT
ejpam-4870	127	14	xn	xn	PROPN
ejpam-4870	127	15	will	will	AUX
ejpam-4870	127	16	be	be	AUX
ejpam-4870	127	17	replaced	replace	VERB
ejpam-4870	127	18	by	by	ADP
ejpam-4870	127	19	vertices	vertex	NOUN
ejpam-4870	127	20	1	1	NUM
ejpam-4870	127	21	,	,	PUNCT
ejpam-4870	127	22	2	2	NUM
ejpam-4870	127	23	,	,	PUNCT
ejpam-4870	127	24	·	·	PUNCT
ejpam-4870	127	25	·	·	PUNCT
ejpam-4870	127	26	·	·	PUNCT
ejpam-4870	127	27	,	,	PUNCT
ejpam-4870	127	28	n	n	CCONJ
ejpam-4870	127	29	,	,	PUNCT
ejpam-4870	127	30	respectively	respectively	ADV
ejpam-4870	127	31	.	.	PUNCT
ejpam-4870	128	1	in	in	ADP
ejpam-4870	128	2	addition	addition	NOUN
ejpam-4870	128	3	,	,	PUNCT
ejpam-4870	128	4	an	an	DET
ejpam-4870	128	5	edge	edge	NOUN
ejpam-4870	128	6	[	[	X
ejpam-4870	128	7	x1	x1	PROPN
ejpam-4870	128	8	,	,	PUNCT
ejpam-4870	128	9	x2	x2	PROPN
ejpam-4870	128	10	]	]	PUNCT
ejpam-4870	128	11	will	will	AUX
ejpam-4870	128	12	be	be	AUX
ejpam-4870	128	13	denoted	denote	VERB
ejpam-4870	128	14	by	by	ADP
ejpam-4870	128	15	12	12	NUM
ejpam-4870	128	16	.	.	PUNCT
ejpam-4870	129	1	refer	refer	VERB
ejpam-4870	129	2	to	to	PART
ejpam-4870	129	3	figure	figure	VERB
ejpam-4870	129	4	7	7	NUM
ejpam-4870	129	5	.	.	NOUN
ejpam-4870	129	6	3	3	NUM
ejpam-4870	129	7	1	1	NUM
ejpam-4870	129	8	2	2	NUM
ejpam-4870	129	9	figure	figure	NOUN
ejpam-4870	129	10	7	7	NUM
ejpam-4870	129	11	:	:	PUNCT
ejpam-4870	129	12	cycle	cycle	NOUN
ejpam-4870	129	13	graph	graph	NOUN
ejpam-4870	129	14	c3	c3	PROPN
ejpam-4870	129	15	of	of	ADP
ejpam-4870	129	16	order	order	NOUN
ejpam-4870	129	17	3	3	NUM
ejpam-4870	129	18	definition	definition	NOUN
ejpam-4870	129	19	9	9	NUM
ejpam-4870	129	20	.	.	PUNCT
ejpam-4870	130	1	a	a	DET
ejpam-4870	130	2	graph	graph	NOUN
ejpam-4870	130	3	of	of	ADP
ejpam-4870	130	4	order	order	NOUN
ejpam-4870	130	5	n	n	X
ejpam-4870	130	6	is	be	AUX
ejpam-4870	130	7	said	say	VERB
ejpam-4870	130	8	to	to	PART
ejpam-4870	130	9	be	be	AUX
ejpam-4870	130	10	a	a	DET
ejpam-4870	130	11	complete	complete	ADJ
ejpam-4870	130	12	graph	graph	NOUN
ejpam-4870	130	13	of	of	ADP
ejpam-4870	130	14	order	order	NOUN
ejpam-4870	130	15	n	n	CCONJ
ejpam-4870	130	16	,	,	PUNCT
ejpam-4870	130	17	denoted	denote	VERB
ejpam-4870	130	18	by	by	ADP
ejpam-4870	130	19	kn	kn	PROPN
ejpam-4870	130	20	,	,	PUNCT
ejpam-4870	130	21	if	if	SCONJ
ejpam-4870	130	22	every	every	DET
ejpam-4870	130	23	vertex	vertex	NOUN
ejpam-4870	130	24	is	be	AUX
ejpam-4870	130	25	adjacent	adjacent	ADJ
ejpam-4870	130	26	to	to	ADP
ejpam-4870	130	27	every	every	DET
ejpam-4870	130	28	other	other	ADJ
ejpam-4870	130	29	vertex	vertex	NOUN
ejpam-4870	130	30	.	.	PUNCT
ejpam-4870	131	1	a	a	DET
ejpam-4870	131	2	complete	complete	ADJ
ejpam-4870	131	3	graph	graph	NOUN
ejpam-4870	131	4	with	with	ADP
ejpam-4870	131	5	one	one	NUM
ejpam-4870	131	6	vertex	vertex	NOUN
ejpam-4870	131	7	is	be	AUX
ejpam-4870	131	8	called	call	VERB
ejpam-4870	131	9	a	a	DET
ejpam-4870	131	10	singleton	singleton	NOUN
ejpam-4870	131	11	graph	graph	NOUN
ejpam-4870	131	12	and	and	CCONJ
ejpam-4870	131	13	is	be	AUX
ejpam-4870	131	14	denoted	denote	VERB
ejpam-4870	131	15	by	by	ADP
ejpam-4870	131	16	k1	k1	NOUN
ejpam-4870	131	17	.	.	PUNCT
ejpam-4870	132	1	since	since	SCONJ
ejpam-4870	132	2	all	all	DET
ejpam-4870	132	3	the	the	DET
ejpam-4870	132	4	vertices	vertex	NOUN
ejpam-4870	132	5	in	in	ADP
ejpam-4870	132	6	kn	kn	PROPN
ejpam-4870	132	7	are	be	AUX
ejpam-4870	132	8	adjacent	adjacent	ADJ
ejpam-4870	132	9	to	to	ADP
ejpam-4870	132	10	every	every	DET
ejpam-4870	132	11	other	other	ADJ
ejpam-4870	132	12	vertex	vertex	NOUN
ejpam-4870	132	13	,	,	PUNCT
ejpam-4870	132	14	it	it	PRON
ejpam-4870	132	15	can	can	AUX
ejpam-4870	132	16	be	be	AUX
ejpam-4870	132	17	observed	observe	VERB
ejpam-4870	132	18	that	that	SCONJ
ejpam-4870	132	19	deg(x	deg(x	ADV
ejpam-4870	132	20	)	)	PUNCT
ejpam-4870	132	21	=	=	SYM
ejpam-4870	132	22	n	n	CCONJ
ejpam-4870	132	23	−	−	NUM
ejpam-4870	132	24	1	1	NUM
ejpam-4870	132	25	for	for	ADP
ejpam-4870	132	26	all	all	PRON
ejpam-4870	132	27	x	x	SYM
ejpam-4870	132	28	∈	∈	NOUN
ejpam-4870	132	29	v	v	NOUN
ejpam-4870	132	30	(	(	PUNCT
ejpam-4870	132	31	g	g	NOUN
ejpam-4870	132	32	)	)	PUNCT
ejpam-4870	132	33	.	.	PUNCT
ejpam-4870	133	1	hence	hence	ADV
ejpam-4870	133	2	kn	kn	PROPN
ejpam-4870	133	3	is	be	AUX
ejpam-4870	133	4	an	an	DET
ejpam-4870	133	5	(	(	PUNCT
ejpam-4870	133	6	n	n	CCONJ
ejpam-4870	133	7	−	−	PROPN
ejpam-4870	133	8	1)-regular	1)-regular	NUM
ejpam-4870	133	9	graph	graph	NOUN
ejpam-4870	133	10	.	.	PUNCT
ejpam-4870	134	1	using	use	VERB
ejpam-4870	134	2	equation	equation	NOUN
ejpam-4870	134	3	1	1	NUM
ejpam-4870	134	4	,	,	PUNCT
ejpam-4870	134	5	the	the	DET
ejpam-4870	134	6	size	size	NOUN
ejpam-4870	134	7	of	of	ADP
ejpam-4870	134	8	a	a	DET
ejpam-4870	134	9	kn	kn	PROPN
ejpam-4870	134	10	is	be	AUX
ejpam-4870	134	11	given	give	VERB
ejpam-4870	134	12	by	by	ADP
ejpam-4870	134	13	nr	nr	NOUN
ejpam-4870	134	14	=	=	ADJ
ejpam-4870	134	15	2	2	NUM
ejpam-4870	134	16	m	m	NOUN
ejpam-4870	134	17	n(n−	n(n−	NOUN
ejpam-4870	134	18	1	1	NUM
ejpam-4870	134	19	)	)	PUNCT
ejpam-4870	134	20	=	=	PUNCT
ejpam-4870	135	1	2	2	NUM
ejpam-4870	135	2	m	m	NOUN
ejpam-4870	135	3	m	m	VERB
ejpam-4870	135	4	=	=	ADJ
ejpam-4870	135	5	n(n−	n(n−	ADJ
ejpam-4870	135	6	1	1	NUM
ejpam-4870	135	7	)	)	PUNCT
ejpam-4870	135	8	2	2	NUM
ejpam-4870	135	9	.	.	PUNCT
ejpam-4870	136	1	(	(	PUNCT
ejpam-4870	136	2	2	2	X
ejpam-4870	136	3	)	)	PUNCT
ejpam-4870	136	4	j.c	j.c	PROPN
ejpam-4870	136	5	.	.	PROPN
ejpam-4870	136	6	bonifacio	bonifacio	PROPN
ejpam-4870	136	7	,	,	PUNCT
ejpam-4870	136	8	c.j	c.j	PROPN
ejpam-4870	136	9	.	.	PROPN
ejpam-4870	136	10	andaya	andaya	PROPN
ejpam-4870	136	11	,	,	PUNCT
ejpam-4870	136	12	d.	d.	PROPN
ejpam-4870	136	13	magpantay	magpantay	PROPN
ejpam-4870	136	14	/	/	SYM
ejpam-4870	136	15	eur	eur	PROPN
ejpam-4870	136	16	.	.	PUNCT
ejpam-4870	137	1	j.	j.	PROPN
ejpam-4870	137	2	pure	pure	PROPN
ejpam-4870	137	3	appl	appl	PROPN
ejpam-4870	137	4	.	.	PROPN
ejpam-4870	137	5	math	math	PROPN
ejpam-4870	137	6	,	,	PUNCT
ejpam-4870	137	7	16	16	NUM
ejpam-4870	137	8	(	(	PUNCT
ejpam-4870	137	9	4	4	NUM
ejpam-4870	137	10	)	)	PUNCT
ejpam-4870	137	11	(	(	PUNCT
ejpam-4870	137	12	2023	2023	NUM
ejpam-4870	137	13	)	)	PUNCT
ejpam-4870	137	14	,	,	PUNCT
ejpam-4870	137	15	2476	2476	NUM
ejpam-4870	137	16	-	-	SYM
ejpam-4870	137	17	2498	2498	NUM
ejpam-4870	137	18	2483	2483	NUM
ejpam-4870	137	19	at	at	ADP
ejpam-4870	137	20	this	this	DET
ejpam-4870	137	21	point	point	NOUN
ejpam-4870	137	22	,	,	PUNCT
ejpam-4870	137	23	the	the	DET
ejpam-4870	137	24	notion	notion	NOUN
ejpam-4870	137	25	of	of	ADP
ejpam-4870	137	26	the	the	DET
ejpam-4870	137	27	subgraph	subgraph	NOUN
ejpam-4870	137	28	of	of	ADP
ejpam-4870	137	29	a	a	DET
ejpam-4870	137	30	graph	graph	NOUN
ejpam-4870	137	31	and	and	CCONJ
ejpam-4870	137	32	isomorphism	isomorphism	NOUN
ejpam-4870	137	33	among	among	ADP
ejpam-4870	137	34	graphs	graph	NOUN
ejpam-4870	137	35	is	be	AUX
ejpam-4870	137	36	discussed	discuss	VERB
ejpam-4870	137	37	.	.	PUNCT
ejpam-4870	138	1	definition	definition	NOUN
ejpam-4870	138	2	10	10	NUM
ejpam-4870	138	3	.	.	PUNCT
ejpam-4870	139	1	a	a	DET
ejpam-4870	139	2	graph	graph	NOUN
ejpam-4870	139	3	h	h	NOUN
ejpam-4870	139	4	=	=	PUNCT
ejpam-4870	139	5	(	(	PUNCT
ejpam-4870	139	6	v	v	NOUN
ejpam-4870	139	7	(	(	PUNCT
ejpam-4870	139	8	h	h	NOUN
ejpam-4870	139	9	)	)	PUNCT
ejpam-4870	139	10	,	,	PUNCT
ejpam-4870	139	11	e(h	e(h	PROPN
ejpam-4870	139	12	)	)	PUNCT
ejpam-4870	139	13	)	)	PUNCT
ejpam-4870	139	14	is	be	AUX
ejpam-4870	139	15	called	call	VERB
ejpam-4870	139	16	a	a	DET
ejpam-4870	139	17	subgraph	subgraph	NOUN
ejpam-4870	139	18	of	of	ADP
ejpam-4870	139	19	a	a	DET
ejpam-4870	139	20	graph	graph	NOUN
ejpam-4870	139	21	g	g	NOUN
ejpam-4870	139	22	=	=	PUNCT
ejpam-4870	139	23	(	(	PUNCT
ejpam-4870	139	24	v	v	NOUN
ejpam-4870	139	25	(	(	PUNCT
ejpam-4870	139	26	g	g	NOUN
ejpam-4870	139	27	)	)	PUNCT
ejpam-4870	139	28	,	,	PUNCT
ejpam-4870	139	29	e(g	e(g	PROPN
ejpam-4870	139	30	)	)	PUNCT
ejpam-4870	139	31	)	)	PUNCT
ejpam-4870	139	32	,	,	PUNCT
ejpam-4870	139	33	written	write	VERB
ejpam-4870	139	34	h	h	NOUN
ejpam-4870	139	35	⊆	⊆	NUM
ejpam-4870	139	36	g	g	NOUN
ejpam-4870	139	37	,	,	PUNCT
ejpam-4870	139	38	if	if	SCONJ
ejpam-4870	139	39	v	v	X
ejpam-4870	139	40	(	(	PUNCT
ejpam-4870	139	41	h	h	NOUN
ejpam-4870	139	42	)	)	PUNCT
ejpam-4870	139	43	⊆	⊆	NUM
ejpam-4870	139	44	v	v	NOUN
ejpam-4870	139	45	(	(	PUNCT
ejpam-4870	139	46	g	g	NOUN
ejpam-4870	139	47	)	)	PUNCT
ejpam-4870	139	48	and	and	CCONJ
ejpam-4870	139	49	e(h	e(h	NOUN
ejpam-4870	139	50	)	)	PUNCT
ejpam-4870	139	51	⊆	⊆	NUM
ejpam-4870	139	52	e(g	e(g	PROPN
ejpam-4870	139	53	)	)	PUNCT
ejpam-4870	139	54	.	.	PUNCT
ejpam-4870	140	1	it	it	PRON
ejpam-4870	140	2	can	can	AUX
ejpam-4870	140	3	be	be	AUX
ejpam-4870	140	4	noted	note	VERB
ejpam-4870	140	5	that	that	SCONJ
ejpam-4870	140	6	a	a	DET
ejpam-4870	140	7	graph	graph	NOUN
ejpam-4870	140	8	is	be	AUX
ejpam-4870	140	9	a	a	DET
ejpam-4870	140	10	subgraph	subgraph	NOUN
ejpam-4870	140	11	of	of	ADP
ejpam-4870	140	12	itself	itself	PRON
ejpam-4870	140	13	because	because	SCONJ
ejpam-4870	140	14	when	when	SCONJ
ejpam-4870	140	15	we	we	PRON
ejpam-4870	140	16	take	take	VERB
ejpam-4870	140	17	the	the	DET
ejpam-4870	140	18	entire	entire	ADJ
ejpam-4870	140	19	set	set	NOUN
ejpam-4870	140	20	of	of	ADP
ejpam-4870	140	21	vertices	vertex	NOUN
ejpam-4870	140	22	and	and	CCONJ
ejpam-4870	140	23	edges	edge	NOUN
ejpam-4870	140	24	from	from	ADP
ejpam-4870	140	25	a	a	DET
ejpam-4870	140	26	graph	graph	NOUN
ejpam-4870	140	27	,	,	PUNCT
ejpam-4870	140	28	we	we	PRON
ejpam-4870	140	29	get	get	VERB
ejpam-4870	140	30	the	the	DET
ejpam-4870	140	31	original	original	ADJ
ejpam-4870	140	32	graph	graph	NOUN
ejpam-4870	140	33	itself	itself	PRON
ejpam-4870	140	34	.	.	PUNCT
ejpam-4870	141	1	example	example	NOUN
ejpam-4870	142	1	6	6	NUM
ejpam-4870	142	2	.	.	PUNCT
ejpam-4870	142	3	consider	consider	VERB
ejpam-4870	142	4	the	the	DET
ejpam-4870	142	5	pictorial	pictorial	ADJ
ejpam-4870	142	6	representation	representation	NOUN
ejpam-4870	142	7	of	of	ADP
ejpam-4870	142	8	graphs	graph	NOUN
ejpam-4870	142	9	g	g	NOUN
ejpam-4870	142	10	,	,	PUNCT
ejpam-4870	142	11	h1	h1	NOUN
ejpam-4870	142	12	,	,	PUNCT
ejpam-4870	142	13	and	and	CCONJ
ejpam-4870	142	14	h2	h2	NOUN
ejpam-4870	142	15	shown	show	VERB
ejpam-4870	142	16	in	in	ADP
ejpam-4870	142	17	figure	figure	NOUN
ejpam-4870	142	18	8	8	NUM
ejpam-4870	142	19	.	.	PUNCT
ejpam-4870	143	1	the	the	DET
ejpam-4870	143	2	graph	graph	NOUN
ejpam-4870	143	3	h1	h1	PROPN
ejpam-4870	143	4	is	be	AUX
ejpam-4870	143	5	a	a	DET
ejpam-4870	143	6	subgraph	subgraph	NOUN
ejpam-4870	143	7	of	of	ADP
ejpam-4870	143	8	g	g	PROPN
ejpam-4870	143	9	since	since	SCONJ
ejpam-4870	143	10	v	v	PROPN
ejpam-4870	143	11	(	(	PUNCT
ejpam-4870	143	12	h1	h1	PROPN
ejpam-4870	143	13	)	)	PUNCT
ejpam-4870	143	14	⊆	⊆	NUM
ejpam-4870	143	15	v	v	NOUN
ejpam-4870	143	16	(	(	PUNCT
ejpam-4870	143	17	g	g	NOUN
ejpam-4870	143	18	)	)	PUNCT
ejpam-4870	143	19	and	and	CCONJ
ejpam-4870	143	20	e(h1	e(h1	NOUN
ejpam-4870	143	21	)	)	PUNCT
ejpam-4870	143	22	⊆	⊆	NUM
ejpam-4870	143	23	e(g	e(g	PROPN
ejpam-4870	143	24	)	)	PUNCT
ejpam-4870	143	25	.	.	PUNCT
ejpam-4870	144	1	however	however	ADV
ejpam-4870	144	2	,	,	PUNCT
ejpam-4870	144	3	graph	graph	NOUN
ejpam-4870	144	4	h2	h2	NOUN
ejpam-4870	144	5	is	be	AUX
ejpam-4870	144	6	not	not	PART
ejpam-4870	144	7	a	a	DET
ejpam-4870	144	8	subgraph	subgraph	NOUN
ejpam-4870	144	9	of	of	ADP
ejpam-4870	144	10	g	g	PROPN
ejpam-4870	144	11	since	since	SCONJ
ejpam-4870	144	12	v	v	PROPN
ejpam-4870	144	13	(	(	PUNCT
ejpam-4870	144	14	h2	h2	NOUN
ejpam-4870	144	15	)	)	PUNCT
ejpam-4870	144	16	⊆	⊆	NUM
ejpam-4870	144	17	v	v	NOUN
ejpam-4870	144	18	(	(	PUNCT
ejpam-4870	144	19	g	g	NOUN
ejpam-4870	144	20	)	)	PUNCT
ejpam-4870	144	21	but	but	CCONJ
ejpam-4870	144	22	e(h2	e(h2	NOUN
ejpam-4870	144	23	)	)	PUNCT
ejpam-4870	144	24	̸⊆	̸⊆	PROPN
ejpam-4870	144	25	e(g	e(g	PROPN
ejpam-4870	144	26	)	)	PUNCT
ejpam-4870	144	27	.	.	PUNCT
ejpam-4870	145	1	x1	x1	NUM
ejpam-4870	146	1	x2	x2	NOUN
ejpam-4870	146	2	x3	x3	PROPN
ejpam-4870	147	1	x4	x4	PROPN
ejpam-4870	147	2	x5	x5	PROPN
ejpam-4870	148	1	x1	x1	PROPN
ejpam-4870	148	2	x3	x3	PROPN
ejpam-4870	148	3	x4	x4	PROPN
ejpam-4870	148	4	x5	x5	PROPN
ejpam-4870	148	5	x1	x1	PROPN
ejpam-4870	148	6	x2	x2	PROPN
ejpam-4870	148	7	x3	x3	PROPN
ejpam-4870	148	8	x4	x4	PROPN
ejpam-4870	148	9	figure	figure	NOUN
ejpam-4870	148	10	8	8	NUM
ejpam-4870	148	11	:	:	PUNCT
ejpam-4870	148	12	subgraph	subgraph	NOUN
ejpam-4870	148	13	h1	h1	NOUN
ejpam-4870	148	14	of	of	ADP
ejpam-4870	148	15	graph	graph	NOUN
ejpam-4870	148	16	g	g	NOUN
ejpam-4870	148	17	,	,	PUNCT
ejpam-4870	148	18	and	and	CCONJ
ejpam-4870	148	19	not	not	PART
ejpam-4870	148	20	a	a	DET
ejpam-4870	148	21	subgraph	subgraph	NOUN
ejpam-4870	148	22	h2	h2	NOUN
ejpam-4870	148	23	of	of	ADP
ejpam-4870	148	24	graph	graph	NOUN
ejpam-4870	148	25	g	g	PROPN
ejpam-4870	148	26	all	all	DET
ejpam-4870	148	27	vertices	vertex	NOUN
ejpam-4870	148	28	and	and	CCONJ
ejpam-4870	148	29	edges	edge	NOUN
ejpam-4870	148	30	of	of	ADP
ejpam-4870	148	31	the	the	DET
ejpam-4870	148	32	graph	graph	NOUN
ejpam-4870	148	33	h1	h1	NOUN
ejpam-4870	148	34	are	be	AUX
ejpam-4870	148	35	all	all	PRON
ejpam-4870	148	36	subsets	subset	NOUN
ejpam-4870	148	37	of	of	ADP
ejpam-4870	148	38	the	the	DET
ejpam-4870	148	39	vertices	vertex	NOUN
ejpam-4870	148	40	and	and	CCONJ
ejpam-4870	148	41	edges	edge	NOUN
ejpam-4870	148	42	of	of	ADP
ejpam-4870	148	43	graph	graph	NOUN
ejpam-4870	148	44	g.	g.	PROPN
ejpam-4870	148	45	however	however	ADV
ejpam-4870	148	46	,	,	PUNCT
ejpam-4870	148	47	edge	edge	VERB
ejpam-4870	148	48	[	[	X
ejpam-4870	148	49	x2	x2	X
ejpam-4870	148	50	,	,	PUNCT
ejpam-4870	148	51	x3	x3	ADJ
ejpam-4870	148	52	]	]	PUNCT
ejpam-4870	148	53	∈	∈	NOUN
ejpam-4870	148	54	e(h2	e(h2	NOUN
ejpam-4870	148	55	)	)	PUNCT
ejpam-4870	148	56	but	but	CCONJ
ejpam-4870	148	57	edge	edge	NOUN
ejpam-4870	149	1	[	[	X
ejpam-4870	149	2	x2	x2	X
ejpam-4870	149	3	,	,	PUNCT
ejpam-4870	149	4	x3	x3	ADJ
ejpam-4870	149	5	]	]	PUNCT
ejpam-4870	149	6	/∈	/∈	PUNCT
ejpam-4870	149	7	e(g	e(g	PROPN
ejpam-4870	149	8	)	)	PUNCT
ejpam-4870	149	9	.	.	PUNCT
ejpam-4870	150	1	hence	hence	ADV
ejpam-4870	150	2	,	,	PUNCT
ejpam-4870	150	3	h2	h2	PROPN
ejpam-4870	150	4	is	be	AUX
ejpam-4870	150	5	not	not	PART
ejpam-4870	150	6	a	a	DET
ejpam-4870	150	7	subgraph	subgraph	NOUN
ejpam-4870	150	8	of	of	ADP
ejpam-4870	150	9	g.	g.	PROPN
ejpam-4870	150	10	definition	definition	NOUN
ejpam-4870	150	11	11	11	NUM
ejpam-4870	150	12	.	.	PUNCT
ejpam-4870	151	1	a	a	DET
ejpam-4870	151	2	subgraph	subgraph	NOUN
ejpam-4870	151	3	h	h	NOUN
ejpam-4870	151	4	of	of	ADP
ejpam-4870	151	5	a	a	DET
ejpam-4870	151	6	graph	graph	NOUN
ejpam-4870	151	7	g	g	NOUN
ejpam-4870	151	8	is	be	AUX
ejpam-4870	151	9	called	call	VERB
ejpam-4870	151	10	an	an	DET
ejpam-4870	151	11	induced	induced	ADJ
ejpam-4870	151	12	-	-	PUNCT
ejpam-4870	151	13	subgraph	subgraph	NOUN
ejpam-4870	151	14	,	,	PUNCT
ejpam-4870	151	15	written	write	VERB
ejpam-4870	151	16	as	as	ADP
ejpam-4870	151	17	⟨h⟩	⟨h⟩	PROPN
ejpam-4870	151	18	,	,	PUNCT
ejpam-4870	151	19	if	if	SCONJ
ejpam-4870	151	20	whenever	whenever	SCONJ
ejpam-4870	151	21	x	x	X
ejpam-4870	151	22	,	,	PUNCT
ejpam-4870	151	23	y	y	PROPN
ejpam-4870	151	24	∈	∈	PROPN
ejpam-4870	151	25	h	h	NOUN
ejpam-4870	151	26	and	and	CCONJ
ejpam-4870	151	27	[	[	X
ejpam-4870	151	28	x	x	X
ejpam-4870	151	29	,	,	PUNCT
ejpam-4870	151	30	y	y	PROPN
ejpam-4870	151	31	]	]	X
ejpam-4870	151	32	∈	∈	PROPN
ejpam-4870	151	33	e(g	e(g	PROPN
ejpam-4870	151	34	)	)	PUNCT
ejpam-4870	151	35	,	,	PUNCT
ejpam-4870	151	36	then	then	ADV
ejpam-4870	151	37	[	[	X
ejpam-4870	151	38	x	x	X
ejpam-4870	151	39	,	,	PUNCT
ejpam-4870	151	40	y	y	PROPN
ejpam-4870	151	41	]	]	X
ejpam-4870	151	42	is	be	AUX
ejpam-4870	151	43	an	an	DET
ejpam-4870	151	44	edge	edge	NOUN
ejpam-4870	151	45	of	of	ADP
ejpam-4870	151	46	⟨h⟩.	⟨h⟩.	PROPN
ejpam-4870	151	47	in	in	ADP
ejpam-4870	151	48	simple	simple	ADJ
ejpam-4870	151	49	terms	term	NOUN
ejpam-4870	151	50	,	,	PUNCT
ejpam-4870	151	51	an	an	DET
ejpam-4870	151	52	induced	induced	ADJ
ejpam-4870	151	53	-	-	PUNCT
ejpam-4870	151	54	subgraph	subgraph	NOUN
ejpam-4870	151	55	⟨h⟩	⟨h⟩	PROPN
ejpam-4870	151	56	of	of	ADP
ejpam-4870	151	57	a	a	DET
ejpam-4870	151	58	graph	graph	NOUN
ejpam-4870	151	59	g	g	NOUN
ejpam-4870	151	60	has	have	VERB
ejpam-4870	151	61	a	a	DET
ejpam-4870	151	62	vertex	vertex	NOUN
ejpam-4870	151	63	set	set	NOUN
ejpam-4870	151	64	that	that	PRON
ejpam-4870	151	65	is	be	AUX
ejpam-4870	151	66	a	a	DET
ejpam-4870	151	67	subset	subset	NOUN
ejpam-4870	151	68	of	of	ADP
ejpam-4870	151	69	v	v	NOUN
ejpam-4870	151	70	(	(	PUNCT
ejpam-4870	151	71	g	g	NOUN
ejpam-4870	151	72	)	)	PUNCT
ejpam-4870	151	73	together	together	ADV
ejpam-4870	151	74	with	with	ADP
ejpam-4870	151	75	the	the	DET
ejpam-4870	151	76	edges	edge	NOUN
ejpam-4870	151	77	whose	whose	DET
ejpam-4870	151	78	vertices	vertex	NOUN
ejpam-4870	151	79	are	be	AUX
ejpam-4870	151	80	contained	contain	VERB
ejpam-4870	151	81	in	in	ADP
ejpam-4870	151	82	the	the	DET
ejpam-4870	151	83	subset	subset	NOUN
ejpam-4870	151	84	h.	h.	PROPN
ejpam-4870	151	85	example	example	PROPN
ejpam-4870	151	86	7	7	X
ejpam-4870	151	87	.	.	X
ejpam-4870	151	88	consider	consider	VERB
ejpam-4870	151	89	the	the	DET
ejpam-4870	151	90	pictorial	pictorial	ADJ
ejpam-4870	151	91	representation	representation	NOUN
ejpam-4870	151	92	of	of	ADP
ejpam-4870	151	93	graphs	graph	NOUN
ejpam-4870	151	94	g	g	NOUN
ejpam-4870	151	95	and	and	CCONJ
ejpam-4870	151	96	h	h	NOUN
ejpam-4870	151	97	shown	show	VERB
ejpam-4870	151	98	in	in	ADP
ejpam-4870	151	99	figure	figure	NOUN
ejpam-4870	151	100	9	9	NUM
ejpam-4870	151	101	.	.	PUNCT
ejpam-4870	152	1	the	the	DET
ejpam-4870	152	2	graph	graph	NOUN
ejpam-4870	152	3	h	h	NOUN
ejpam-4870	152	4	is	be	AUX
ejpam-4870	152	5	a	a	DET
ejpam-4870	152	6	subgraph	subgraph	NOUN
ejpam-4870	152	7	of	of	ADP
ejpam-4870	152	8	g	g	PROPN
ejpam-4870	152	9	since	since	SCONJ
ejpam-4870	152	10	v	v	PROPN
ejpam-4870	152	11	(	(	PUNCT
ejpam-4870	152	12	h	h	NOUN
ejpam-4870	152	13	)	)	PUNCT
ejpam-4870	152	14	⊆	⊆	NUM
ejpam-4870	152	15	v	v	NOUN
ejpam-4870	152	16	(	(	PUNCT
ejpam-4870	152	17	g	g	NOUN
ejpam-4870	152	18	)	)	PUNCT
ejpam-4870	152	19	and	and	CCONJ
ejpam-4870	152	20	e(h	e(h	NOUN
ejpam-4870	152	21	)	)	PUNCT
ejpam-4870	152	22	⊆	⊆	NUM
ejpam-4870	152	23	e(g	e(g	PROPN
ejpam-4870	152	24	)	)	PUNCT
ejpam-4870	152	25	.	.	PUNCT
ejpam-4870	153	1	now	now	ADV
ejpam-4870	153	2	,	,	PUNCT
ejpam-4870	153	3	consider	consider	VERB
ejpam-4870	153	4	e(h	e(h	PRON
ejpam-4870	153	5	)	)	PUNCT
ejpam-4870	153	6	=	=	PRON
ejpam-4870	153	7	{	{	PUNCT
ejpam-4870	154	1	[	[	X
ejpam-4870	154	2	x1	x1	PROPN
ejpam-4870	154	3	,	,	PUNCT
ejpam-4870	154	4	x2	x2	PROPN
ejpam-4870	154	5	]	]	PUNCT
ejpam-4870	154	6	,	,	PUNCT
ejpam-4870	154	7	[	[	X
ejpam-4870	154	8	x1	x1	PROPN
ejpam-4870	154	9	,	,	PUNCT
ejpam-4870	154	10	x5	x5	PROPN
ejpam-4870	154	11	]	]	X
ejpam-4870	154	12	,	,	PUNCT
ejpam-4870	155	1	[	[	X
ejpam-4870	155	2	x2	x2	X
ejpam-4870	155	3	,	,	PUNCT
ejpam-4870	155	4	x5	x5	PROPN
ejpam-4870	155	5	]	]	X
ejpam-4870	155	6	,	,	PUNCT
ejpam-4870	155	7	[	[	X
ejpam-4870	155	8	x2	x2	X
ejpam-4870	155	9	,	,	PUNCT
ejpam-4870	155	10	x3	x3	ADJ
ejpam-4870	155	11	]	]	PUNCT
ejpam-4870	155	12	}	}	PUNCT
ejpam-4870	155	13	∈	∈	PROPN
ejpam-4870	155	14	e(g	e(g	PROPN
ejpam-4870	155	15	)	)	PUNCT
ejpam-4870	155	16	.	.	PUNCT
ejpam-4870	156	1	it	it	PRON
ejpam-4870	156	2	follows	follow	VERB
ejpam-4870	156	3	that	that	SCONJ
ejpam-4870	156	4	h	h	NOUN
ejpam-4870	156	5	is	be	AUX
ejpam-4870	156	6	an	an	DET
ejpam-4870	156	7	inducedsubgraph	inducedsubgraph	NOUN
ejpam-4870	156	8	of	of	ADP
ejpam-4870	156	9	g.	g.	PROPN
ejpam-4870	156	10	in	in	ADP
ejpam-4870	156	11	some	some	DET
ejpam-4870	156	12	cases	case	NOUN
ejpam-4870	156	13	,	,	PUNCT
ejpam-4870	156	14	a	a	DET
ejpam-4870	156	15	graph	graph	NOUN
ejpam-4870	156	16	h	h	NOUN
ejpam-4870	156	17	is	be	AUX
ejpam-4870	156	18	a	a	DET
ejpam-4870	156	19	subgraph	subgraph	NOUN
ejpam-4870	156	20	of	of	ADP
ejpam-4870	156	21	g	g	PROPN
ejpam-4870	156	22	and	and	CCONJ
ejpam-4870	156	23	the	the	DET
ejpam-4870	156	24	order	order	NOUN
ejpam-4870	156	25	of	of	ADP
ejpam-4870	156	26	h	h	NOUN
ejpam-4870	156	27	and	and	CCONJ
ejpam-4870	156	28	g	g	PROPN
ejpam-4870	156	29	are	be	AUX
ejpam-4870	156	30	equal	equal	ADJ
ejpam-4870	156	31	.	.	PUNCT
ejpam-4870	157	1	this	this	DET
ejpam-4870	157	2	idea	idea	NOUN
ejpam-4870	157	3	gives	give	VERB
ejpam-4870	157	4	the	the	DET
ejpam-4870	157	5	notion	notion	NOUN
ejpam-4870	157	6	of	of	ADP
ejpam-4870	157	7	a	a	DET
ejpam-4870	157	8	spanning	span	VERB
ejpam-4870	157	9	subgraph	subgraph	NOUN
ejpam-4870	157	10	.	.	PUNCT
ejpam-4870	158	1	definition	definition	NOUN
ejpam-4870	158	2	12	12	NUM
ejpam-4870	158	3	.	.	PUNCT
ejpam-4870	159	1	a	a	DET
ejpam-4870	159	2	graph	graph	NOUN
ejpam-4870	159	3	h	h	NOUN
ejpam-4870	159	4	is	be	AUX
ejpam-4870	159	5	a	a	DET
ejpam-4870	159	6	spanning	span	VERB
ejpam-4870	159	7	subgraph	subgraph	NOUN
ejpam-4870	159	8	of	of	ADP
ejpam-4870	159	9	a	a	DET
ejpam-4870	159	10	graph	graph	NOUN
ejpam-4870	159	11	g	g	NOUN
ejpam-4870	159	12	if	if	SCONJ
ejpam-4870	159	13	h	h	NOUN
ejpam-4870	159	14	is	be	AUX
ejpam-4870	159	15	a	a	DET
ejpam-4870	159	16	subgraph	subgraph	NOUN
ejpam-4870	159	17	of	of	ADP
ejpam-4870	159	18	g	g	PROPN
ejpam-4870	159	19	and	and	CCONJ
ejpam-4870	159	20	v	v	NOUN
ejpam-4870	159	21	(	(	PUNCT
ejpam-4870	159	22	g	g	NOUN
ejpam-4870	159	23	)	)	PUNCT
ejpam-4870	160	1	=	=	NOUN
ejpam-4870	160	2	v	v	X
ejpam-4870	160	3	(	(	PUNCT
ejpam-4870	160	4	h	h	NOUN
ejpam-4870	160	5	)	)	PUNCT
ejpam-4870	160	6	.	.	PUNCT
ejpam-4870	161	1	note	note	VERB
ejpam-4870	161	2	that	that	SCONJ
ejpam-4870	161	3	the	the	DET
ejpam-4870	161	4	edge	edge	NOUN
ejpam-4870	161	5	set	set	NOUN
ejpam-4870	161	6	of	of	ADP
ejpam-4870	161	7	graph	graph	NOUN
ejpam-4870	161	8	h	h	NOUN
ejpam-4870	161	9	is	be	AUX
ejpam-4870	161	10	a	a	DET
ejpam-4870	161	11	subset	subset	NOUN
ejpam-4870	161	12	of	of	ADP
ejpam-4870	161	13	the	the	DET
ejpam-4870	161	14	edge	edge	NOUN
ejpam-4870	161	15	set	set	NOUN
ejpam-4870	161	16	of	of	ADP
ejpam-4870	161	17	g.	g.	PROPN
ejpam-4870	161	18	j.c	j.c	PROPN
ejpam-4870	161	19	.	.	PROPN
ejpam-4870	161	20	bonifacio	bonifacio	PROPN
ejpam-4870	161	21	,	,	PUNCT
ejpam-4870	161	22	c.j	c.j	PROPN
ejpam-4870	161	23	.	.	PROPN
ejpam-4870	161	24	andaya	andaya	PROPN
ejpam-4870	161	25	,	,	PUNCT
ejpam-4870	161	26	d.	d.	PROPN
ejpam-4870	161	27	magpantay	magpantay	PROPN
ejpam-4870	161	28	/	/	SYM
ejpam-4870	161	29	eur	eur	PROPN
ejpam-4870	161	30	.	.	PUNCT
ejpam-4870	162	1	j.	j.	PROPN
ejpam-4870	162	2	pure	pure	PROPN
ejpam-4870	162	3	appl	appl	PROPN
ejpam-4870	162	4	.	.	PROPN
ejpam-4870	162	5	math	math	PROPN
ejpam-4870	162	6	,	,	PUNCT
ejpam-4870	162	7	16	16	NUM
ejpam-4870	162	8	(	(	PUNCT
ejpam-4870	162	9	4	4	NUM
ejpam-4870	162	10	)	)	PUNCT
ejpam-4870	162	11	(	(	PUNCT
ejpam-4870	162	12	2023	2023	NUM
ejpam-4870	162	13	)	)	PUNCT
ejpam-4870	162	14	,	,	PUNCT
ejpam-4870	162	15	2476	2476	NUM
ejpam-4870	162	16	-	-	SYM
ejpam-4870	162	17	2498	2498	NUM
ejpam-4870	162	18	2484	2484	NUM
ejpam-4870	162	19	x1	x1	NOUN
ejpam-4870	163	1	x2	x2	PROPN
ejpam-4870	163	2	x5x4	x5x4	PROPN
ejpam-4870	164	1	x3	x3	VERB
ejpam-4870	164	2	x1	x1	PROPN
ejpam-4870	165	1	x2	x2	PROPN
ejpam-4870	165	2	x5	x5	NOUN
ejpam-4870	165	3	x3	x3	NOUN
ejpam-4870	165	4	figure	figure	NOUN
ejpam-4870	165	5	9	9	NUM
ejpam-4870	165	6	:	:	PUNCT
ejpam-4870	165	7	graph	graph	NOUN
ejpam-4870	165	8	g	g	NOUN
ejpam-4870	165	9	and	and	CCONJ
ejpam-4870	165	10	its	its	PRON
ejpam-4870	165	11	subgraph	subgraph	NOUN
ejpam-4870	165	12	h	h	NOUN
ejpam-4870	165	13	3	3	NUM
ejpam-4870	165	14	1	1	NUM
ejpam-4870	165	15	2	2	NUM
ejpam-4870	165	16	figure	figure	NOUN
ejpam-4870	165	17	10	10	NUM
ejpam-4870	165	18	:	:	PUNCT
ejpam-4870	165	19	spanning	span	VERB
ejpam-4870	165	20	subgraphs	subgraph	NOUN
ejpam-4870	165	21	of	of	ADP
ejpam-4870	165	22	c3	c3	PROPN
ejpam-4870	165	23	with	with	ADP
ejpam-4870	165	24	0	0	NUM
ejpam-4870	165	25	edge	edge	NOUN
ejpam-4870	165	26	example	example	NOUN
ejpam-4870	165	27	8	8	NUM
ejpam-4870	165	28	.	.	PUNCT
ejpam-4870	165	29	consider	consider	VERB
ejpam-4870	165	30	cycle	cycle	NOUN
ejpam-4870	165	31	graph	graph	NOUN
ejpam-4870	165	32	c3	c3	PROPN
ejpam-4870	165	33	with	with	ADP
ejpam-4870	165	34	pictorial	pictorial	ADJ
ejpam-4870	165	35	representation	representation	NOUN
ejpam-4870	165	36	shown	show	VERB
ejpam-4870	165	37	in	in	ADP
ejpam-4870	165	38	figure	figure	NOUN
ejpam-4870	165	39	7	7	NUM
ejpam-4870	165	40	.	.	PUNCT
ejpam-4870	165	41	by	by	ADP
ejpam-4870	165	42	definition	definition	NOUN
ejpam-4870	165	43	12	12	NUM
ejpam-4870	165	44	,	,	PUNCT
ejpam-4870	165	45	the	the	DET
ejpam-4870	165	46	spanning	span	VERB
ejpam-4870	165	47	subgraphs	subgraph	NOUN
ejpam-4870	165	48	of	of	ADP
ejpam-4870	165	49	c3	c3	PROPN
ejpam-4870	165	50	have	have	VERB
ejpam-4870	165	51	3	3	NUM
ejpam-4870	165	52	vertices	vertex	NOUN
ejpam-4870	165	53	,	,	PUNCT
ejpam-4870	165	54	and	and	CCONJ
ejpam-4870	165	55	the	the	DET
ejpam-4870	165	56	edges	edge	NOUN
ejpam-4870	165	57	are	be	AUX
ejpam-4870	165	58	subset	subset	VERB
ejpam-4870	165	59	of	of	ADP
ejpam-4870	165	60	c3	c3	PROPN
ejpam-4870	165	61	.	.	PUNCT
ejpam-4870	166	1	to	to	PART
ejpam-4870	166	2	get	get	VERB
ejpam-4870	166	3	the	the	DET
ejpam-4870	166	4	spanning	span	VERB
ejpam-4870	166	5	subgraph	subgraph	NOUN
ejpam-4870	166	6	of	of	ADP
ejpam-4870	166	7	c3	c3	PROPN
ejpam-4870	166	8	,	,	PUNCT
ejpam-4870	166	9	first	first	ADV
ejpam-4870	166	10	we	we	PRON
ejpam-4870	166	11	consider	consider	VERB
ejpam-4870	166	12	the	the	DET
ejpam-4870	166	13	spanning	span	VERB
ejpam-4870	166	14	subgraph	subgraph	NOUN
ejpam-4870	166	15	with	with	ADP
ejpam-4870	166	16	0	0	NUM
ejpam-4870	166	17	edge	edge	NOUN
ejpam-4870	166	18	.	.	PUNCT
ejpam-4870	167	1	refer	refer	VERB
ejpam-4870	167	2	to	to	PART
ejpam-4870	167	3	figure	figure	VERB
ejpam-4870	167	4	10	10	NUM
ejpam-4870	167	5	.	.	PUNCT
ejpam-4870	168	1	next	next	ADV
ejpam-4870	168	2	,	,	PUNCT
ejpam-4870	168	3	consider	consider	VERB
ejpam-4870	168	4	the	the	DET
ejpam-4870	168	5	spanning	span	VERB
ejpam-4870	168	6	subgraphs	subgraph	NOUN
ejpam-4870	168	7	with	with	ADP
ejpam-4870	168	8	1	1	NUM
ejpam-4870	168	9	edge	edge	NOUN
ejpam-4870	168	10	.	.	PUNCT
ejpam-4870	169	1	consider	consider	VERB
ejpam-4870	169	2	the	the	DET
ejpam-4870	169	3	first	first	ADJ
ejpam-4870	169	4	spanning	span	VERB
ejpam-4870	169	5	subgraphs	subgraph	NOUN
ejpam-4870	169	6	with	with	ADP
ejpam-4870	169	7	1	1	NUM
ejpam-4870	169	8	edge	edge	NOUN
ejpam-4870	169	9	.	.	PUNCT
ejpam-4870	170	1	it	it	PRON
ejpam-4870	170	2	can	can	AUX
ejpam-4870	170	3	be	be	AUX
ejpam-4870	170	4	seen	see	VERB
ejpam-4870	170	5	that	that	SCONJ
ejpam-4870	170	6	the	the	DET
ejpam-4870	170	7	edge	edge	NOUN
ejpam-4870	170	8	12	12	NUM
ejpam-4870	170	9	is	be	AUX
ejpam-4870	170	10	present	present	ADJ
ejpam-4870	170	11	hence	hence	ADV
ejpam-4870	170	12	,	,	PUNCT
ejpam-4870	170	13	this	this	DET
ejpam-4870	170	14	spanning	span	VERB
ejpam-4870	170	15	subgraph	subgraph	NOUN
ejpam-4870	170	16	is	be	AUX
ejpam-4870	170	17	denoted	denote	VERB
ejpam-4870	170	18	as	as	ADP
ejpam-4870	170	19	{	{	PUNCT
ejpam-4870	170	20	12	12	NUM
ejpam-4870	170	21	}	}	PUNCT
ejpam-4870	170	22	.	.	PUNCT
ejpam-4870	171	1	similarly	similarly	ADV
ejpam-4870	171	2	,	,	PUNCT
ejpam-4870	171	3	we	we	PRON
ejpam-4870	171	4	get	get	VERB
ejpam-4870	171	5	{	{	PUNCT
ejpam-4870	171	6	23	23	NUM
ejpam-4870	171	7	}	}	PUNCT
ejpam-4870	171	8	and	and	CCONJ
ejpam-4870	171	9	{	{	PUNCT
ejpam-4870	171	10	31	31	NUM
ejpam-4870	171	11	}	}	PUNCT
ejpam-4870	171	12	are	be	AUX
ejpam-4870	171	13	obtained	obtain	VERB
ejpam-4870	171	14	.	.	PUNCT
ejpam-4870	172	1	now	now	ADV
ejpam-4870	172	2	,	,	PUNCT
ejpam-4870	172	3	consider	consider	VERB
ejpam-4870	172	4	the	the	DET
ejpam-4870	172	5	spanning	span	VERB
ejpam-4870	172	6	subgraphs	subgraph	NOUN
ejpam-4870	172	7	with	with	ADP
ejpam-4870	172	8	2	2	NUM
ejpam-4870	172	9	edges	edge	NOUN
ejpam-4870	172	10	.	.	PUNCT
ejpam-4870	173	1	consider	consider	VERB
ejpam-4870	173	2	the	the	DET
ejpam-4870	173	3	first	first	ADJ
ejpam-4870	173	4	spanning	span	VERB
ejpam-4870	173	5	subgraph	subgraph	NOUN
ejpam-4870	173	6	with	with	ADP
ejpam-4870	173	7	2	2	NUM
ejpam-4870	173	8	edges	edge	NOUN
ejpam-4870	173	9	,	,	PUNCT
ejpam-4870	173	10	the	the	DET
ejpam-4870	173	11	edges	edge	NOUN
ejpam-4870	173	12	12	12	NUM
ejpam-4870	173	13	and	and	CCONJ
ejpam-4870	173	14	23	23	NUM
ejpam-4870	173	15	are	be	AUX
ejpam-4870	173	16	present	present	ADJ
ejpam-4870	173	17	so	so	ADV
ejpam-4870	173	18	,	,	PUNCT
ejpam-4870	173	19	this	this	PRON
ejpam-4870	173	20	is	be	AUX
ejpam-4870	173	21	denoted	denote	VERB
ejpam-4870	173	22	by	by	ADP
ejpam-4870	173	23	{	{	PUNCT
ejpam-4870	173	24	12	12	NUM
ejpam-4870	173	25	,	,	PUNCT
ejpam-4870	173	26	23	23	NUM
ejpam-4870	173	27	}	}	PUNCT
ejpam-4870	173	28	.	.	PUNCT
ejpam-4870	174	1	similarly	similarly	ADV
ejpam-4870	174	2	,	,	PUNCT
ejpam-4870	174	3	we	we	PRON
ejpam-4870	174	4	get	get	VERB
ejpam-4870	174	5	{	{	PUNCT
ejpam-4870	174	6	12	12	NUM
ejpam-4870	174	7	,	,	PUNCT
ejpam-4870	174	8	31	31	NUM
ejpam-4870	174	9	}	}	PUNCT
ejpam-4870	174	10	and	and	CCONJ
ejpam-4870	174	11	{	{	PUNCT
ejpam-4870	174	12	23	23	NUM
ejpam-4870	174	13	,	,	PUNCT
ejpam-4870	174	14	31	31	NUM
ejpam-4870	174	15	}	}	PUNCT
ejpam-4870	174	16	.	.	PUNCT
ejpam-4870	175	1	lastly	lastly	ADV
ejpam-4870	175	2	,	,	PUNCT
ejpam-4870	175	3	consider	consider	VERB
ejpam-4870	175	4	the	the	DET
ejpam-4870	175	5	spanning	span	VERB
ejpam-4870	175	6	subgraph	subgraph	NOUN
ejpam-4870	175	7	with	with	ADP
ejpam-4870	175	8	3	3	NUM
ejpam-4870	175	9	edges	edge	NOUN
ejpam-4870	175	10	.	.	PUNCT
ejpam-4870	176	1	it	it	PRON
ejpam-4870	176	2	can	can	AUX
ejpam-4870	176	3	be	be	AUX
ejpam-4870	176	4	observed	observe	VERB
ejpam-4870	176	5	that	that	SCONJ
ejpam-4870	176	6	the	the	DET
ejpam-4870	176	7	edges	edge	NOUN
ejpam-4870	176	8	12	12	NUM
ejpam-4870	176	9	,	,	PUNCT
ejpam-4870	176	10	23	23	NUM
ejpam-4870	176	11	,	,	PUNCT
ejpam-4870	176	12	and	and	CCONJ
ejpam-4870	176	13	31	31	NUM
ejpam-4870	176	14	are	be	AUX
ejpam-4870	176	15	present	present	ADJ
ejpam-4870	176	16	in	in	ADP
ejpam-4870	176	17	the	the	DET
ejpam-4870	176	18	spanning	span	VERB
ejpam-4870	176	19	subgraph	subgraph	NOUN
ejpam-4870	176	20	shown	show	VERB
ejpam-4870	176	21	in	in	ADP
ejpam-4870	176	22	figure	figure	NOUN
ejpam-4870	176	23	13	13	NUM
ejpam-4870	176	24	.	.	PUNCT
ejpam-4870	177	1	hence	hence	ADV
ejpam-4870	177	2	,	,	PUNCT
ejpam-4870	177	3	this	this	PRON
ejpam-4870	177	4	is	be	AUX
ejpam-4870	177	5	denoted	denote	VERB
ejpam-4870	177	6	by	by	ADP
ejpam-4870	177	7	{	{	PUNCT
ejpam-4870	177	8	12	12	NUM
ejpam-4870	177	9	,	,	PUNCT
ejpam-4870	177	10	23	23	NUM
ejpam-4870	177	11	,	,	PUNCT
ejpam-4870	177	12	31	31	NUM
ejpam-4870	177	13	}	}	PUNCT
ejpam-4870	177	14	.	.	PUNCT
ejpam-4870	178	1	note	note	VERB
ejpam-4870	178	2	that	that	SCONJ
ejpam-4870	178	3	{	{	PUNCT
ejpam-4870	178	4	12	12	NUM
ejpam-4870	178	5	}	}	PUNCT
ejpam-4870	178	6	and	and	CCONJ
ejpam-4870	178	7	{	{	PUNCT
ejpam-4870	178	8	21	21	NUM
ejpam-4870	178	9	}	}	PUNCT
ejpam-4870	178	10	are	be	AUX
ejpam-4870	178	11	just	just	ADV
ejpam-4870	178	12	the	the	DET
ejpam-4870	178	13	same	same	ADJ
ejpam-4870	178	14	since	since	SCONJ
ejpam-4870	178	15	undirected	undirected	ADJ
ejpam-4870	178	16	graphs	graph	NOUN
ejpam-4870	178	17	are	be	AUX
ejpam-4870	178	18	considered	consider	VERB
ejpam-4870	178	19	,	,	PUNCT
ejpam-4870	178	20	which	which	PRON
ejpam-4870	178	21	means	mean	VERB
ejpam-4870	178	22	the	the	DET
ejpam-4870	178	23	edges	edge	NOUN
ejpam-4870	178	24	have	have	VERB
ejpam-4870	178	25	no	no	DET
ejpam-4870	178	26	directions	direction	NOUN
ejpam-4870	178	27	.	.	PUNCT
ejpam-4870	179	1	the	the	DET
ejpam-4870	179	2	following	follow	VERB
ejpam-4870	179	3	theorem	theorem	NOUN
ejpam-4870	179	4	determines	determine	VERB
ejpam-4870	179	5	the	the	DET
ejpam-4870	179	6	number	number	NOUN
ejpam-4870	179	7	of	of	ADP
ejpam-4870	179	8	spanning	span	VERB
ejpam-4870	179	9	subgraphs	subgraph	NOUN
ejpam-4870	179	10	with	with	ADP
ejpam-4870	179	11	j	j	PROPN
ejpam-4870	179	12	edges	edge	NOUN
ejpam-4870	179	13	of	of	ADP
ejpam-4870	179	14	a	a	DET
ejpam-4870	179	15	graph	graph	NOUN
ejpam-4870	179	16	.	.	PUNCT
ejpam-4870	180	1	the	the	DET
ejpam-4870	180	2	proof	proof	NOUN
ejpam-4870	180	3	of	of	ADP
ejpam-4870	180	4	theorem	theorem	ADJ
ejpam-4870	180	5	2	2	NUM
ejpam-4870	180	6	can	can	AUX
ejpam-4870	180	7	be	be	AUX
ejpam-4870	180	8	seen	see	VERB
ejpam-4870	180	9	in	in	ADP
ejpam-4870	180	10	[	[	X
ejpam-4870	180	11	6	6	NUM
ejpam-4870	180	12	]	]	PUNCT
ejpam-4870	180	13	.	.	PUNCT
ejpam-4870	181	1	theorem	theorem	NOUN
ejpam-4870	181	2	2	2	NUM
ejpam-4870	181	3	.	.	PUNCT
ejpam-4870	182	1	let	let	VERB
ejpam-4870	182	2	g	g	PRON
ejpam-4870	182	3	be	be	AUX
ejpam-4870	182	4	a	a	DET
ejpam-4870	182	5	graph	graph	NOUN
ejpam-4870	182	6	of	of	ADP
ejpam-4870	182	7	size	size	NOUN
ejpam-4870	182	8	m	m	PROPN
ejpam-4870	182	9	,	,	PUNCT
ejpam-4870	182	10	then	then	ADV
ejpam-4870	182	11	there	there	PRON
ejpam-4870	182	12	are	be	VERB
ejpam-4870	182	13	(	(	PUNCT
ejpam-4870	182	14	m	m	PROPN
ejpam-4870	182	15	j	j	NOUN
ejpam-4870	182	16	)	)	PUNCT
ejpam-4870	182	17	spanning	span	VERB
ejpam-4870	182	18	subgraph	subgraph	NOUN
ejpam-4870	182	19	with	with	ADP
ejpam-4870	182	20	exactly	exactly	ADV
ejpam-4870	182	21	j	j	PROPN
ejpam-4870	182	22	edges	edge	NOUN
ejpam-4870	182	23	where	where	SCONJ
ejpam-4870	182	24	0	0	NUM
ejpam-4870	182	25	≤	≤	NUM
ejpam-4870	182	26	j	j	PROPN
ejpam-4870	182	27	≤	≤	PROPN
ejpam-4870	182	28	m.	m.	NOUN
ejpam-4870	182	29	illustration	illustration	NOUN
ejpam-4870	182	30	1	1	NUM
ejpam-4870	182	31	shows	show	VERB
ejpam-4870	182	32	the	the	DET
ejpam-4870	182	33	spanning	span	VERB
ejpam-4870	182	34	subgraphs	subgraph	NOUN
ejpam-4870	182	35	of	of	ADP
ejpam-4870	182	36	cn	cn	PROPN
ejpam-4870	182	37	when	when	SCONJ
ejpam-4870	182	38	m	m	VERB
ejpam-4870	182	39	=	=	SYM
ejpam-4870	182	40	3	3	NUM
ejpam-4870	182	41	and	and	CCONJ
ejpam-4870	183	1	j	j	NOUN
ejpam-4870	183	2	=	=	NOUN
ejpam-4870	183	3	2	2	X
ejpam-4870	183	4	.	.	X
ejpam-4870	183	5	j.c	j.c	PROPN
ejpam-4870	183	6	.	.	PROPN
ejpam-4870	183	7	bonifacio	bonifacio	PROPN
ejpam-4870	183	8	,	,	PUNCT
ejpam-4870	183	9	c.j	c.j	PROPN
ejpam-4870	183	10	.	.	PROPN
ejpam-4870	183	11	andaya	andaya	PROPN
ejpam-4870	183	12	,	,	PUNCT
ejpam-4870	183	13	d.	d.	PROPN
ejpam-4870	183	14	magpantay	magpantay	PROPN
ejpam-4870	183	15	/	/	SYM
ejpam-4870	183	16	eur	eur	PROPN
ejpam-4870	183	17	.	.	PUNCT
ejpam-4870	184	1	j.	j.	PROPN
ejpam-4870	184	2	pure	pure	PROPN
ejpam-4870	184	3	appl	appl	PROPN
ejpam-4870	184	4	.	.	PROPN
ejpam-4870	184	5	math	math	PROPN
ejpam-4870	184	6	,	,	PUNCT
ejpam-4870	184	7	16	16	NUM
ejpam-4870	184	8	(	(	PUNCT
ejpam-4870	184	9	4	4	NUM
ejpam-4870	184	10	)	)	PUNCT
ejpam-4870	184	11	(	(	PUNCT
ejpam-4870	184	12	2023	2023	NUM
ejpam-4870	184	13	)	)	PUNCT
ejpam-4870	184	14	,	,	PUNCT
ejpam-4870	184	15	2476	2476	NUM
ejpam-4870	184	16	-	-	SYM
ejpam-4870	184	17	2498	2498	NUM
ejpam-4870	184	18	2485	2485	NUM
ejpam-4870	184	19	3	3	NUM
ejpam-4870	184	20	1	1	NUM
ejpam-4870	184	21	2	2	NUM
ejpam-4870	184	22	3	3	NUM
ejpam-4870	184	23	1	1	NUM
ejpam-4870	184	24	2	2	NUM
ejpam-4870	184	25	3	3	NUM
ejpam-4870	184	26	1	1	NUM
ejpam-4870	184	27	2	2	NUM
ejpam-4870	184	28	figure	figure	NOUN
ejpam-4870	184	29	11	11	NUM
ejpam-4870	184	30	:	:	PUNCT
ejpam-4870	184	31	spanning	span	VERB
ejpam-4870	184	32	subgraphs	subgraph	NOUN
ejpam-4870	184	33	of	of	ADP
ejpam-4870	184	34	c3	c3	PROPN
ejpam-4870	184	35	with	with	ADP
ejpam-4870	184	36	1	1	NUM
ejpam-4870	184	37	edge	edge	NOUN
ejpam-4870	184	38	3	3	NUM
ejpam-4870	184	39	1	1	NUM
ejpam-4870	184	40	2	2	NUM
ejpam-4870	184	41	3	3	NUM
ejpam-4870	184	42	1	1	NUM
ejpam-4870	184	43	2	2	NUM
ejpam-4870	184	44	3	3	NUM
ejpam-4870	184	45	1	1	NUM
ejpam-4870	184	46	2	2	NUM
ejpam-4870	184	47	figure	figure	NOUN
ejpam-4870	184	48	12	12	NUM
ejpam-4870	184	49	:	:	PUNCT
ejpam-4870	184	50	spanning	span	VERB
ejpam-4870	184	51	subgraphs	subgraph	NOUN
ejpam-4870	184	52	of	of	ADP
ejpam-4870	184	53	c3	c3	PROPN
ejpam-4870	184	54	with	with	ADP
ejpam-4870	184	55	2	2	NUM
ejpam-4870	184	56	edges	edge	NOUN
ejpam-4870	184	57	illustration	illustration	NOUN
ejpam-4870	184	58	1	1	NUM
ejpam-4870	184	59	.	.	PUNCT
ejpam-4870	184	60	consider	consider	VERB
ejpam-4870	184	61	the	the	DET
ejpam-4870	184	62	spanning	span	VERB
ejpam-4870	184	63	subgraphs	subgraph	NOUN
ejpam-4870	184	64	of	of	ADP
ejpam-4870	184	65	c3	c3	PROPN
ejpam-4870	184	66	shown	show	VERB
ejpam-4870	184	67	in	in	ADP
ejpam-4870	184	68	example	example	NOUN
ejpam-4870	184	69	8	8	NUM
ejpam-4870	184	70	.	.	PUNCT
ejpam-4870	185	1	if	if	SCONJ
ejpam-4870	185	2	j	j	PROPN
ejpam-4870	185	3	=	=	SYM
ejpam-4870	185	4	2	2	NUM
ejpam-4870	185	5	,	,	PUNCT
ejpam-4870	185	6	then	then	ADV
ejpam-4870	185	7	by	by	ADP
ejpam-4870	185	8	counting	count	VERB
ejpam-4870	185	9	,	,	PUNCT
ejpam-4870	185	10	there	there	PRON
ejpam-4870	185	11	are	be	VERB
ejpam-4870	185	12	3	3	NUM
ejpam-4870	185	13	spanning	span	VERB
ejpam-4870	185	14	subgraphs	subgraph	NOUN
ejpam-4870	185	15	with	with	ADP
ejpam-4870	185	16	2	2	NUM
ejpam-4870	185	17	edges	edge	NOUN
ejpam-4870	185	18	.	.	PUNCT
ejpam-4870	186	1	verifying	verify	VERB
ejpam-4870	186	2	this	this	PRON
ejpam-4870	186	3	by	by	ADP
ejpam-4870	186	4	using	use	VERB
ejpam-4870	186	5	theorem	theorem	NOUN
ejpam-4870	186	6	2	2	NUM
ejpam-4870	186	7	,	,	PUNCT
ejpam-4870	186	8	we	we	PRON
ejpam-4870	186	9	have	have	VERB
ejpam-4870	186	10	(	(	PUNCT
ejpam-4870	186	11	m	m	NOUN
ejpam-4870	186	12	j	j	NOUN
ejpam-4870	186	13	)	)	PUNCT
ejpam-4870	187	1	=	=	PUNCT
ejpam-4870	187	2	(	(	PUNCT
ejpam-4870	187	3	3	3	NUM
ejpam-4870	187	4	2	2	NUM
ejpam-4870	187	5	)	)	PUNCT
ejpam-4870	187	6	=	=	SYM
ejpam-4870	188	1	3	3	X
ejpam-4870	188	2	.	.	X
ejpam-4870	188	3	two	two	NUM
ejpam-4870	188	4	graphs	graph	NOUN
ejpam-4870	188	5	g	g	NOUN
ejpam-4870	188	6	and	and	CCONJ
ejpam-4870	188	7	g′	g′	NOUN
ejpam-4870	188	8	are	be	AUX
ejpam-4870	188	9	said	say	VERB
ejpam-4870	188	10	to	to	PART
ejpam-4870	188	11	be	be	AUX
ejpam-4870	188	12	equal	equal	ADJ
ejpam-4870	188	13	if	if	SCONJ
ejpam-4870	188	14	v	v	NOUN
ejpam-4870	188	15	(	(	PUNCT
ejpam-4870	188	16	g	g	NOUN
ejpam-4870	188	17	)	)	PUNCT
ejpam-4870	188	18	=	=	NOUN
ejpam-4870	188	19	v	v	X
ejpam-4870	188	20	(	(	PUNCT
ejpam-4870	188	21	g′	g′	NOUN
ejpam-4870	188	22	)	)	PUNCT
ejpam-4870	188	23	and	and	CCONJ
ejpam-4870	188	24	e(g	e(g	PROPN
ejpam-4870	188	25	)	)	PUNCT
ejpam-4870	189	1	=	=	SYM
ejpam-4870	189	2	e(g′	e(g′	NUM
ejpam-4870	189	3	)	)	PUNCT
ejpam-4870	189	4	.	.	PUNCT
ejpam-4870	190	1	however	however	ADV
ejpam-4870	190	2	,	,	PUNCT
ejpam-4870	190	3	graphs	graph	NOUN
ejpam-4870	190	4	are	be	AUX
ejpam-4870	190	5	possible	possible	ADJ
ejpam-4870	190	6	of	of	ADP
ejpam-4870	190	7	similar	similar	ADJ
ejpam-4870	190	8	form	form	NOUN
ejpam-4870	190	9	even	even	ADV
ejpam-4870	190	10	if	if	SCONJ
ejpam-4870	190	11	they	they	PRON
ejpam-4870	190	12	have	have	VERB
ejpam-4870	190	13	unequal	unequal	ADJ
ejpam-4870	190	14	vertex	vertex	NOUN
ejpam-4870	190	15	and/or	and/or	CCONJ
ejpam-4870	190	16	edge	edge	NOUN
ejpam-4870	190	17	sets	set	NOUN
ejpam-4870	190	18	,	,	PUNCT
ejpam-4870	190	19	that	that	ADV
ejpam-4870	190	20	is	is	ADV
ejpam-4870	190	21	,	,	PUNCT
ejpam-4870	190	22	if	if	SCONJ
ejpam-4870	190	23	there	there	PRON
ejpam-4870	190	24	exists	exist	VERB
ejpam-4870	190	25	an	an	DET
ejpam-4870	190	26	isomorphism	isomorphism	NOUN
ejpam-4870	190	27	between	between	ADP
ejpam-4870	190	28	them	they	PRON
ejpam-4870	190	29	.	.	PUNCT
ejpam-4870	191	1	definition	definition	NOUN
ejpam-4870	191	2	13	13	NUM
ejpam-4870	191	3	.	.	PUNCT
ejpam-4870	192	1	let	let	VERB
ejpam-4870	192	2	g	g	PROPN
ejpam-4870	192	3	=	=	SYM
ejpam-4870	192	4	(	(	PUNCT
ejpam-4870	192	5	v	v	NOUN
ejpam-4870	192	6	(	(	PUNCT
ejpam-4870	192	7	g	g	NOUN
ejpam-4870	192	8	)	)	PUNCT
ejpam-4870	192	9	,	,	PUNCT
ejpam-4870	192	10	e(g	e(g	PROPN
ejpam-4870	192	11	)	)	PUNCT
ejpam-4870	192	12	)	)	PUNCT
ejpam-4870	192	13	and	and	CCONJ
ejpam-4870	192	14	g′	g′	NOUN
ejpam-4870	192	15	=	=	SYM
ejpam-4870	192	16	(	(	PUNCT
ejpam-4870	192	17	v	v	NOUN
ejpam-4870	192	18	(	(	PUNCT
ejpam-4870	192	19	g′	g′	NOUN
ejpam-4870	192	20	)	)	PUNCT
ejpam-4870	192	21	,	,	PUNCT
ejpam-4870	192	22	e(g′	e(g′	NUM
ejpam-4870	192	23	)	)	PUNCT
ejpam-4870	192	24	)	)	PUNCT
ejpam-4870	192	25	be	be	AUX
ejpam-4870	192	26	graphs	graph	NOUN
ejpam-4870	192	27	.	.	PUNCT
ejpam-4870	193	1	a	a	DET
ejpam-4870	193	2	mapping	mapping	NOUN
ejpam-4870	193	3	ϕ	ϕ	NOUN
ejpam-4870	193	4	:	:	PUNCT
ejpam-4870	193	5	v	v	X
ejpam-4870	193	6	(	(	PUNCT
ejpam-4870	193	7	g	g	NOUN
ejpam-4870	193	8	)	)	PUNCT
ejpam-4870	193	9	7−→	7−→	NOUN
ejpam-4870	193	10	v	v	NOUN
ejpam-4870	193	11	(	(	PUNCT
ejpam-4870	193	12	g′	g′	NOUN
ejpam-4870	193	13	)	)	PUNCT
ejpam-4870	193	14	is	be	AUX
ejpam-4870	193	15	called	call	VERB
ejpam-4870	193	16	isomorphism	isomorphism	NOUN
ejpam-4870	193	17	if	if	SCONJ
ejpam-4870	193	18	the	the	DET
ejpam-4870	193	19	following	follow	VERB
ejpam-4870	193	20	conditions	condition	NOUN
ejpam-4870	193	21	are	be	AUX
ejpam-4870	193	22	satisfied	satisfied	ADJ
ejpam-4870	193	23	:	:	PUNCT
ejpam-4870	193	24	(	(	PUNCT
ejpam-4870	193	25	i	i	NOUN
ejpam-4870	193	26	)	)	PUNCT
ejpam-4870	193	27	ϕ	ϕ	PROPN
ejpam-4870	193	28	is	be	AUX
ejpam-4870	193	29	bijective	bijective	ADJ
ejpam-4870	193	30	,	,	PUNCT
ejpam-4870	193	31	that	that	PRON
ejpam-4870	193	32	is	be	AUX
ejpam-4870	193	33	both	both	PRON
ejpam-4870	193	34	one	one	NUM
ejpam-4870	193	35	-	-	PUNCT
ejpam-4870	193	36	to	to	ADP
ejpam-4870	193	37	-	-	PUNCT
ejpam-4870	193	38	one	one	NUM
ejpam-4870	193	39	and	and	CCONJ
ejpam-4870	193	40	onto	onto	ADP
ejpam-4870	193	41	;	;	PUNCT
ejpam-4870	193	42	(	(	PUNCT
ejpam-4870	193	43	ii	ii	NOUN
ejpam-4870	193	44	)	)	PUNCT
ejpam-4870	194	1	[	[	X
ejpam-4870	194	2	a	a	PRON
ejpam-4870	194	3	,	,	PUNCT
ejpam-4870	194	4	b	b	X
ejpam-4870	194	5	]	]	X
ejpam-4870	194	6	∈	∈	PROPN
ejpam-4870	194	7	e(g	e(g	PROPN
ejpam-4870	194	8	)	)	PUNCT
ejpam-4870	194	9	⇒	⇒	VERB
ejpam-4870	194	10	[	[	X
ejpam-4870	194	11	ϕ(a	ϕ(a	NOUN
ejpam-4870	194	12	)	)	PUNCT
ejpam-4870	194	13	,	,	PUNCT
ejpam-4870	194	14	ϕ(b	ϕ(b	PROPN
ejpam-4870	194	15	)	)	PUNCT
ejpam-4870	194	16	]	]	PUNCT
ejpam-4870	195	1	∈	∈	PROPN
ejpam-4870	195	2	e(g′	e(g′	NUM
ejpam-4870	195	3	)	)	PUNCT
ejpam-4870	195	4	;	;	PUNCT
ejpam-4870	195	5	(	(	PUNCT
ejpam-4870	195	6	iii	iii	X
ejpam-4870	195	7	)	)	PUNCT
ejpam-4870	196	1	[	[	X
ejpam-4870	196	2	c	c	X
ejpam-4870	196	3	,	,	PUNCT
ejpam-4870	196	4	d	d	X
ejpam-4870	196	5	]	]	X
ejpam-4870	196	6	∈	∈	PROPN
ejpam-4870	196	7	e(g′	e(g′	NUM
ejpam-4870	196	8	)	)	PUNCT
ejpam-4870	196	9	⇒	⇒	NOUN
ejpam-4870	197	1	[	[	X
ejpam-4870	197	2	ϕ−1(c	ϕ−1(c	PROPN
ejpam-4870	197	3	)	)	PUNCT
ejpam-4870	197	4	,	,	PUNCT
ejpam-4870	197	5	ϕ−1(d)];∈	ϕ−1(d)];∈	PROPN
ejpam-4870	197	6	e(g	e(g	PROPN
ejpam-4870	197	7	)	)	PUNCT
ejpam-4870	197	8	.	.	PUNCT
ejpam-4870	198	1	a	a	DET
ejpam-4870	198	2	function	function	NOUN
ejpam-4870	198	3	mapping	mapping	NOUN
ejpam-4870	198	4	is	be	AUX
ejpam-4870	198	5	one	one	NUM
ejpam-4870	198	6	-	-	PUNCT
ejpam-4870	198	7	to	to	ADP
ejpam-4870	198	8	-	-	PUNCT
ejpam-4870	198	9	one	one	NUM
ejpam-4870	198	10	and	and	CCONJ
ejpam-4870	198	11	onto	onto	ADP
ejpam-4870	198	12	if	if	SCONJ
ejpam-4870	198	13	every	every	DET
ejpam-4870	198	14	element	element	NOUN
ejpam-4870	198	15	of	of	ADP
ejpam-4870	198	16	v	v	NOUN
ejpam-4870	198	17	(	(	PUNCT
ejpam-4870	198	18	g	g	NOUN
ejpam-4870	198	19	)	)	PUNCT
ejpam-4870	198	20	is	be	AUX
ejpam-4870	198	21	mapped	map	VERB
ejpam-4870	198	22	into	into	ADP
ejpam-4870	198	23	exactly	exactly	ADV
ejpam-4870	198	24	one	one	NUM
ejpam-4870	198	25	element	element	NOUN
ejpam-4870	198	26	of	of	ADP
ejpam-4870	198	27	set	set	NOUN
ejpam-4870	198	28	v	v	NOUN
ejpam-4870	198	29	(	(	PUNCT
ejpam-4870	198	30	g′	g′	NOUN
ejpam-4870	198	31	)	)	PUNCT
ejpam-4870	198	32	.	.	PUNCT
ejpam-4870	199	1	example	example	NOUN
ejpam-4870	199	2	9	9	NUM
ejpam-4870	199	3	.	.	X
ejpam-4870	199	4	consider	consider	VERB
ejpam-4870	199	5	graphs	graph	NOUN
ejpam-4870	199	6	g	g	NOUN
ejpam-4870	199	7	and	and	CCONJ
ejpam-4870	199	8	g′	g′	NOUN
ejpam-4870	199	9	in	in	ADP
ejpam-4870	199	10	figure	figure	NOUN
ejpam-4870	199	11	14	14	NUM
ejpam-4870	199	12	.	.	PUNCT
ejpam-4870	200	1	define	define	VERB
ejpam-4870	200	2	the	the	DET
ejpam-4870	200	3	mapping	mapping	NOUN
ejpam-4870	200	4	ϕ	ϕ	NOUN
ejpam-4870	200	5	:	:	PUNCT
ejpam-4870	200	6	v	v	X
ejpam-4870	200	7	(	(	PUNCT
ejpam-4870	200	8	g	g	NOUN
ejpam-4870	200	9	)	)	PUNCT
ejpam-4870	200	10	7−→	7−→	NOUN
ejpam-4870	200	11	v	v	NOUN
ejpam-4870	200	12	(	(	PUNCT
ejpam-4870	200	13	g′	g′	NOUN
ejpam-4870	200	14	)	)	PUNCT
ejpam-4870	200	15	by	by	ADP
ejpam-4870	200	16	ϕ	ϕ	NOUN
ejpam-4870	200	17	:	:	PUNCT
ejpam-4870	200	18	x1	x1	PROPN
ejpam-4870	200	19	7−→	7−→	NOUN
ejpam-4870	200	20	y1	y1	NOUN
ejpam-4870	200	21	x2	x2	NOUN
ejpam-4870	200	22	7−→	7−→	NOUN
ejpam-4870	201	1	y2	y2	NOUN
ejpam-4870	202	1	x3	x3	NOUN
ejpam-4870	202	2	7−→	7−→	PROPN
ejpam-4870	202	3	y4	y4	NOUN
ejpam-4870	202	4	x4	x4	PROPN
ejpam-4870	202	5	7−→	7−→	PROPN
ejpam-4870	202	6	y3	y3	NOUN
ejpam-4870	202	7	by	by	ADP
ejpam-4870	202	8	mapping	map	VERB
ejpam-4870	202	9	ϕ	ϕ	NOUN
ejpam-4870	202	10	,	,	PUNCT
ejpam-4870	202	11	it	it	PRON
ejpam-4870	202	12	can	can	AUX
ejpam-4870	202	13	be	be	AUX
ejpam-4870	202	14	observed	observe	VERB
ejpam-4870	202	15	that	that	SCONJ
ejpam-4870	202	16	ϕ	ϕ	NOUN
ejpam-4870	202	17	is	be	AUX
ejpam-4870	202	18	one	one	NUM
ejpam-4870	202	19	-	-	PUNCT
ejpam-4870	202	20	to	to	ADP
ejpam-4870	202	21	-	-	PUNCT
ejpam-4870	202	22	one	one	NUM
ejpam-4870	202	23	and	and	CCONJ
ejpam-4870	202	24	onto	onto	NOUN
ejpam-4870	202	25	,	,	PUNCT
ejpam-4870	202	26	hence	hence	ADV
ejpam-4870	202	27	condition	condition	NOUN
ejpam-4870	202	28	(	(	PUNCT
ejpam-4870	202	29	i	i	NOUN
ejpam-4870	202	30	)	)	PUNCT
ejpam-4870	202	31	of	of	ADP
ejpam-4870	202	32	the	the	DET
ejpam-4870	202	33	definition	definition	NOUN
ejpam-4870	202	34	is	be	AUX
ejpam-4870	202	35	satisfied	satisfied	ADJ
ejpam-4870	202	36	.	.	PUNCT
ejpam-4870	203	1	to	to	PART
ejpam-4870	203	2	verify	verify	VERB
ejpam-4870	203	3	condition	condition	NOUN
ejpam-4870	203	4	(	(	PUNCT
ejpam-4870	203	5	ii	ii	NOUN
ejpam-4870	203	6	)	)	PUNCT
ejpam-4870	203	7	,	,	PUNCT
ejpam-4870	203	8	we	we	PRON
ejpam-4870	203	9	have	have	VERB
ejpam-4870	203	10	:	:	PUNCT
ejpam-4870	204	1	[	[	X
ejpam-4870	204	2	x1	x1	X
ejpam-4870	204	3	,	,	PUNCT
ejpam-4870	204	4	x2	x2	PROPN
ejpam-4870	204	5	]	]	X
ejpam-4870	204	6	∈	∈	PROPN
ejpam-4870	204	7	e(g	e(g	PROPN
ejpam-4870	204	8	)	)	PUNCT
ejpam-4870	204	9	⇒	⇒	VERB
ejpam-4870	204	10	[	[	X
ejpam-4870	204	11	ϕ(x1	ϕ(x1	ADJ
ejpam-4870	204	12	)	)	PUNCT
ejpam-4870	204	13	,	,	PUNCT
ejpam-4870	204	14	ϕ(x2	ϕ(x2	NOUN
ejpam-4870	204	15	)	)	PUNCT
ejpam-4870	204	16	]	]	PUNCT
ejpam-4870	205	1	=	=	PUNCT
ejpam-4870	206	1	[	[	X
ejpam-4870	206	2	y1	y1	INTJ
ejpam-4870	206	3	,	,	PUNCT
ejpam-4870	206	4	y2	y2	PROPN
ejpam-4870	206	5	]	]	PUNCT
ejpam-4870	206	6	∈	∈	PROPN
ejpam-4870	206	7	e(g′	e(g′	NUM
ejpam-4870	206	8	)	)	PUNCT
ejpam-4870	206	9	j.c	j.c	PROPN
ejpam-4870	206	10	.	.	PROPN
ejpam-4870	206	11	bonifacio	bonifacio	PROPN
ejpam-4870	206	12	,	,	PUNCT
ejpam-4870	206	13	c.j	c.j	PROPN
ejpam-4870	206	14	.	.	PROPN
ejpam-4870	206	15	andaya	andaya	PROPN
ejpam-4870	206	16	,	,	PUNCT
ejpam-4870	206	17	d.	d.	PROPN
ejpam-4870	206	18	magpantay	magpantay	PROPN
ejpam-4870	206	19	/	/	SYM
ejpam-4870	206	20	eur	eur	PROPN
ejpam-4870	206	21	.	.	PUNCT
ejpam-4870	207	1	j.	j.	PROPN
ejpam-4870	207	2	pure	pure	PROPN
ejpam-4870	207	3	appl	appl	PROPN
ejpam-4870	207	4	.	.	PROPN
ejpam-4870	207	5	math	math	PROPN
ejpam-4870	207	6	,	,	PUNCT
ejpam-4870	207	7	16	16	NUM
ejpam-4870	207	8	(	(	PUNCT
ejpam-4870	207	9	4	4	NUM
ejpam-4870	207	10	)	)	PUNCT
ejpam-4870	207	11	(	(	PUNCT
ejpam-4870	207	12	2023	2023	NUM
ejpam-4870	207	13	)	)	PUNCT
ejpam-4870	207	14	,	,	PUNCT
ejpam-4870	207	15	2476	2476	NUM
ejpam-4870	207	16	-	-	SYM
ejpam-4870	207	17	2498	2498	NUM
ejpam-4870	207	18	2486	2486	NUM
ejpam-4870	207	19	3	3	NUM
ejpam-4870	207	20	1	1	NUM
ejpam-4870	207	21	2	2	NUM
ejpam-4870	207	22	figure	figure	NOUN
ejpam-4870	207	23	13	13	NUM
ejpam-4870	207	24	:	:	PUNCT
ejpam-4870	207	25	spanning	span	VERB
ejpam-4870	207	26	subgraphs	subgraph	NOUN
ejpam-4870	207	27	of	of	ADP
ejpam-4870	207	28	c3	c3	PROPN
ejpam-4870	207	29	with	with	ADP
ejpam-4870	207	30	3	3	NUM
ejpam-4870	207	31	edge	edge	NOUN
ejpam-4870	207	32	x1x4	x1x4	X
ejpam-4870	207	33	x2x3	x2x3	X
ejpam-4870	207	34	y1y4	y1y4	X
ejpam-4870	208	1	y2y3	y2y3	NOUN
ejpam-4870	208	2	figure	figure	NOUN
ejpam-4870	208	3	14	14	NUM
ejpam-4870	208	4	:	:	PUNCT
ejpam-4870	208	5	graph	graph	VERB
ejpam-4870	208	6	g	g	NOUN
ejpam-4870	208	7	and	and	CCONJ
ejpam-4870	208	8	g′	g′	NOUN
ejpam-4870	209	1	[	[	X
ejpam-4870	209	2	x3	x3	ADJ
ejpam-4870	209	3	,	,	PUNCT
ejpam-4870	209	4	x4	x4	PROPN
ejpam-4870	209	5	]	]	X
ejpam-4870	209	6	∈	∈	PROPN
ejpam-4870	209	7	e(g	e(g	PROPN
ejpam-4870	209	8	)	)	PUNCT
ejpam-4870	209	9	⇒	⇒	VERB
ejpam-4870	209	10	[	[	X
ejpam-4870	209	11	ϕ(x3	ϕ(x3	NOUN
ejpam-4870	209	12	)	)	PUNCT
ejpam-4870	209	13	,	,	PUNCT
ejpam-4870	209	14	ϕ(x4	ϕ(x4	NOUN
ejpam-4870	209	15	)	)	PUNCT
ejpam-4870	209	16	]	]	PUNCT
ejpam-4870	210	1	=	=	PUNCT
ejpam-4870	211	1	[	[	X
ejpam-4870	211	2	y4	y4	X
ejpam-4870	211	3	,	,	PUNCT
ejpam-4870	211	4	y3	y3	PROPN
ejpam-4870	211	5	]	]	PUNCT
ejpam-4870	211	6	∈	∈	PROPN
ejpam-4870	211	7	e(g′	e(g′	NUM
ejpam-4870	211	8	)	)	PUNCT
ejpam-4870	212	1	[	[	X
ejpam-4870	212	2	x2	x2	X
ejpam-4870	212	3	,	,	PUNCT
ejpam-4870	212	4	x4	x4	PROPN
ejpam-4870	212	5	]	]	X
ejpam-4870	212	6	∈	∈	PROPN
ejpam-4870	212	7	e(g	e(g	PROPN
ejpam-4870	212	8	)	)	PUNCT
ejpam-4870	212	9	⇒	⇒	VERB
ejpam-4870	212	10	[	[	X
ejpam-4870	212	11	ϕ(x2	ϕ(x2	NOUN
ejpam-4870	212	12	)	)	PUNCT
ejpam-4870	212	13	,	,	PUNCT
ejpam-4870	212	14	ϕ(x4	ϕ(x4	NOUN
ejpam-4870	212	15	)	)	PUNCT
ejpam-4870	212	16	]	]	PUNCT
ejpam-4870	213	1	=	=	PUNCT
ejpam-4870	214	1	[	[	X
ejpam-4870	214	2	y2	y2	PROPN
ejpam-4870	214	3	,	,	PUNCT
ejpam-4870	214	4	y3	y3	PROPN
ejpam-4870	214	5	]	]	PUNCT
ejpam-4870	214	6	∈	∈	PROPN
ejpam-4870	214	7	e(g′	e(g′	NUM
ejpam-4870	214	8	)	)	PUNCT
ejpam-4870	215	1	[	[	X
ejpam-4870	215	2	x1	x1	X
ejpam-4870	215	3	,	,	PUNCT
ejpam-4870	215	4	x3	x3	ADJ
ejpam-4870	215	5	]	]	X
ejpam-4870	215	6	∈	∈	PROPN
ejpam-4870	215	7	e(g	e(g	PROPN
ejpam-4870	215	8	)	)	PUNCT
ejpam-4870	215	9	⇒	⇒	VERB
ejpam-4870	215	10	[	[	X
ejpam-4870	215	11	ϕ(x1	ϕ(x1	ADJ
ejpam-4870	215	12	)	)	PUNCT
ejpam-4870	215	13	,	,	PUNCT
ejpam-4870	215	14	ϕ(x3	ϕ(x3	NOUN
ejpam-4870	215	15	)	)	PUNCT
ejpam-4870	215	16	]	]	PUNCT
ejpam-4870	216	1	=	=	PUNCT
ejpam-4870	217	1	[	[	X
ejpam-4870	217	2	y1	y1	X
ejpam-4870	217	3	,	,	PUNCT
ejpam-4870	217	4	y4	y4	X
ejpam-4870	217	5	]	]	X
ejpam-4870	217	6	∈	∈	PROPN
ejpam-4870	217	7	e(g′	e(g′	NUM
ejpam-4870	217	8	)	)	PUNCT
ejpam-4870	217	9	hence	hence	ADV
ejpam-4870	217	10	,	,	PUNCT
ejpam-4870	217	11	the	the	DET
ejpam-4870	217	12	condition	condition	NOUN
ejpam-4870	217	13	(	(	PUNCT
ejpam-4870	217	14	ii	ii	NOUN
ejpam-4870	217	15	)	)	PUNCT
ejpam-4870	217	16	of	of	ADP
ejpam-4870	217	17	the	the	DET
ejpam-4870	217	18	definition	definition	NOUN
ejpam-4870	217	19	13	13	NUM
ejpam-4870	217	20	is	be	AUX
ejpam-4870	217	21	satisfied	satisfied	ADJ
ejpam-4870	217	22	.	.	PUNCT
ejpam-4870	218	1	lastly	lastly	ADV
ejpam-4870	218	2	,	,	PUNCT
ejpam-4870	218	3	to	to	PART
ejpam-4870	218	4	verify	verify	VERB
ejpam-4870	218	5	condition	condition	NOUN
ejpam-4870	218	6	(	(	PUNCT
ejpam-4870	218	7	iii	iii	X
ejpam-4870	218	8	)	)	PUNCT
ejpam-4870	218	9	we	we	PRON
ejpam-4870	218	10	have	have	VERB
ejpam-4870	218	11	:	:	PUNCT
ejpam-4870	219	1	[	[	X
ejpam-4870	219	2	y1	y1	X
ejpam-4870	219	3	,	,	PUNCT
ejpam-4870	219	4	y2	y2	PROPN
ejpam-4870	219	5	]	]	PUNCT
ejpam-4870	219	6	∈	∈	PROPN
ejpam-4870	219	7	e(g′	e(g′	NUM
ejpam-4870	219	8	)	)	PUNCT
ejpam-4870	219	9	⇒	⇒	NOUN
ejpam-4870	219	10	[	[	X
ejpam-4870	219	11	ϕ−1(x1	ϕ−1(x1	NOUN
ejpam-4870	219	12	)	)	PUNCT
ejpam-4870	219	13	,	,	PUNCT
ejpam-4870	219	14	ϕ	ϕ	X
ejpam-4870	219	15	−1(x2	−1(x2	NOUN
ejpam-4870	219	16	)	)	PUNCT
ejpam-4870	219	17	]	]	PUNCT
ejpam-4870	220	1	=	=	PUNCT
ejpam-4870	221	1	[	[	X
ejpam-4870	221	2	x1	x1	X
ejpam-4870	221	3	,	,	PUNCT
ejpam-4870	221	4	x2	x2	PROPN
ejpam-4870	221	5	]	]	X
ejpam-4870	221	6	∈	∈	PROPN
ejpam-4870	221	7	e(g	e(g	PROPN
ejpam-4870	221	8	)	)	PUNCT
ejpam-4870	222	1	[	[	X
ejpam-4870	222	2	y4	y4	X
ejpam-4870	222	3	,	,	PUNCT
ejpam-4870	222	4	y3	y3	PROPN
ejpam-4870	222	5	]	]	PUNCT
ejpam-4870	222	6	∈	∈	PROPN
ejpam-4870	222	7	e(g′	e(g′	NUM
ejpam-4870	222	8	)	)	PUNCT
ejpam-4870	222	9	⇒	⇒	NOUN
ejpam-4870	223	1	[	[	X
ejpam-4870	224	1	ϕ−1(x1	ϕ−1(x1	NOUN
ejpam-4870	224	2	)	)	PUNCT
ejpam-4870	224	3	,	,	PUNCT
ejpam-4870	224	4	ϕ	ϕ	X
ejpam-4870	224	5	−1(x2	−1(x2	NOUN
ejpam-4870	224	6	)	)	PUNCT
ejpam-4870	224	7	]	]	PUNCT
ejpam-4870	225	1	=	=	PUNCT
ejpam-4870	226	1	[	[	X
ejpam-4870	226	2	x3	x3	ADJ
ejpam-4870	226	3	,	,	PUNCT
ejpam-4870	226	4	x4	x4	PROPN
ejpam-4870	226	5	]	]	X
ejpam-4870	226	6	∈	∈	PROPN
ejpam-4870	226	7	e(g	e(g	PROPN
ejpam-4870	226	8	)	)	PUNCT
ejpam-4870	227	1	[	[	X
ejpam-4870	227	2	y2	y2	PROPN
ejpam-4870	227	3	,	,	PUNCT
ejpam-4870	227	4	y3	y3	PROPN
ejpam-4870	227	5	]	]	PUNCT
ejpam-4870	227	6	∈	∈	PROPN
ejpam-4870	227	7	e(g′	e(g′	NUM
ejpam-4870	227	8	)	)	PUNCT
ejpam-4870	227	9	⇒	⇒	NOUN
ejpam-4870	228	1	[	[	X
ejpam-4870	229	1	ϕ−1(x1	ϕ−1(x1	NOUN
ejpam-4870	229	2	)	)	PUNCT
ejpam-4870	229	3	,	,	PUNCT
ejpam-4870	229	4	ϕ	ϕ	X
ejpam-4870	229	5	−1(x2	−1(x2	NOUN
ejpam-4870	229	6	)	)	PUNCT
ejpam-4870	229	7	]	]	PUNCT
ejpam-4870	230	1	=	=	PUNCT
ejpam-4870	231	1	[	[	X
ejpam-4870	231	2	x2	x2	PROPN
ejpam-4870	231	3	,	,	PUNCT
ejpam-4870	231	4	x4	x4	PROPN
ejpam-4870	231	5	]	]	X
ejpam-4870	231	6	∈	∈	PROPN
ejpam-4870	231	7	e(g	e(g	PROPN
ejpam-4870	231	8	)	)	PUNCT
ejpam-4870	232	1	[	[	X
ejpam-4870	232	2	y1	y1	X
ejpam-4870	232	3	,	,	PUNCT
ejpam-4870	232	4	y4	y4	X
ejpam-4870	232	5	]	]	X
ejpam-4870	232	6	∈	∈	PROPN
ejpam-4870	232	7	e(g′	e(g′	NUM
ejpam-4870	232	8	)	)	PUNCT
ejpam-4870	232	9	⇒	⇒	NOUN
ejpam-4870	232	10	[	[	X
ejpam-4870	232	11	ϕ−1(x1	ϕ−1(x1	NOUN
ejpam-4870	232	12	)	)	PUNCT
ejpam-4870	232	13	,	,	PUNCT
ejpam-4870	232	14	ϕ	ϕ	X
ejpam-4870	232	15	−1(x2	−1(x2	NOUN
ejpam-4870	232	16	)	)	PUNCT
ejpam-4870	232	17	]	]	PUNCT
ejpam-4870	233	1	=	=	PUNCT
ejpam-4870	234	1	[	[	X
ejpam-4870	234	2	x1	x1	X
ejpam-4870	234	3	,	,	PUNCT
ejpam-4870	234	4	x3	x3	ADJ
ejpam-4870	234	5	]	]	X
ejpam-4870	234	6	∈	∈	PROPN
ejpam-4870	234	7	e(g	e(g	PROPN
ejpam-4870	234	8	)	)	PUNCT
ejpam-4870	234	9	thus	thus	ADV
ejpam-4870	234	10	,	,	PUNCT
ejpam-4870	234	11	condition	condition	NOUN
ejpam-4870	234	12	(	(	PUNCT
ejpam-4870	234	13	iii	iii	NOUN
ejpam-4870	234	14	)	)	PUNCT
ejpam-4870	234	15	of	of	ADP
ejpam-4870	234	16	the	the	DET
ejpam-4870	234	17	definition	definition	NOUN
ejpam-4870	234	18	13	13	NUM
ejpam-4870	234	19	is	be	AUX
ejpam-4870	234	20	satisfied	satisfied	ADJ
ejpam-4870	234	21	.	.	PUNCT
ejpam-4870	235	1	therefore	therefore	ADV
ejpam-4870	235	2	,	,	PUNCT
ejpam-4870	235	3	ϕ	ϕ	PROPN
ejpam-4870	235	4	is	be	AUX
ejpam-4870	235	5	an	an	DET
ejpam-4870	235	6	isomorphism	isomorphism	NOUN
ejpam-4870	235	7	.	.	PUNCT
ejpam-4870	236	1	definition	definition	NOUN
ejpam-4870	236	2	14	14	NUM
ejpam-4870	236	3	.	.	PUNCT
ejpam-4870	237	1	let	let	VERB
ejpam-4870	237	2	g	g	NOUN
ejpam-4870	237	3	and	and	CCONJ
ejpam-4870	237	4	g′	g′	NOUN
ejpam-4870	237	5	be	be	AUX
ejpam-4870	237	6	graphs	graph	NOUN
ejpam-4870	237	7	.	.	PUNCT
ejpam-4870	238	1	a	a	DET
ejpam-4870	238	2	graph	graph	NOUN
ejpam-4870	238	3	g	g	NOUN
ejpam-4870	238	4	is	be	AUX
ejpam-4870	238	5	isomorphic	isomorphic	ADJ
ejpam-4870	238	6	to	to	ADP
ejpam-4870	238	7	g′	g′	NOUN
ejpam-4870	238	8	,	,	PUNCT
ejpam-4870	238	9	denoted	denote	VERB
ejpam-4870	238	10	by	by	ADP
ejpam-4870	238	11	g	g	PROPN
ejpam-4870	238	12	≃	≃	PROPN
ejpam-4870	238	13	g′	g′	NOUN
ejpam-4870	238	14	,	,	PUNCT
ejpam-4870	238	15	if	if	SCONJ
ejpam-4870	238	16	there	there	PRON
ejpam-4870	238	17	exists	exist	VERB
ejpam-4870	238	18	an	an	DET
ejpam-4870	238	19	isomorphism	isomorphism	NOUN
ejpam-4870	238	20	ϕ	ϕ	NOUN
ejpam-4870	238	21	:	:	PUNCT
ejpam-4870	238	22	v	v	NOUN
ejpam-4870	238	23	(	(	PUNCT
ejpam-4870	238	24	g	g	NOUN
ejpam-4870	238	25	)	)	PUNCT
ejpam-4870	238	26	7−→	7−→	NOUN
ejpam-4870	238	27	v	v	NOUN
ejpam-4870	238	28	(	(	PUNCT
ejpam-4870	238	29	g′	g′	NOUN
ejpam-4870	238	30	)	)	PUNCT
ejpam-4870	238	31	.	.	PUNCT
ejpam-4870	239	1	note	note	VERB
ejpam-4870	239	2	that	that	SCONJ
ejpam-4870	239	3	if	if	SCONJ
ejpam-4870	239	4	a	a	DET
ejpam-4870	239	5	graph	graph	NOUN
ejpam-4870	239	6	g	g	NOUN
ejpam-4870	239	7	has	have	VERB
ejpam-4870	239	8	cycle	cycle	NOUN
ejpam-4870	239	9	,	,	PUNCT
ejpam-4870	239	10	the	the	DET
ejpam-4870	239	11	isomorphic	isomorphic	ADJ
ejpam-4870	239	12	graph	graph	NOUN
ejpam-4870	239	13	g′	g′	NOUN
ejpam-4870	239	14	should	should	AUX
ejpam-4870	239	15	also	also	ADV
ejpam-4870	239	16	preserves	preserve	VERB
ejpam-4870	239	17	the	the	DET
ejpam-4870	239	18	cycle	cycle	NOUN
ejpam-4870	239	19	.	.	PUNCT
ejpam-4870	240	1	also	also	ADV
ejpam-4870	240	2	,	,	PUNCT
ejpam-4870	240	3	it	it	PRON
ejpam-4870	240	4	preserves	preserve	VERB
ejpam-4870	240	5	the	the	DET
ejpam-4870	240	6	degree	degree	NOUN
ejpam-4870	240	7	sequence	sequence	NOUN
ejpam-4870	240	8	of	of	ADP
ejpam-4870	240	9	the	the	DET
ejpam-4870	240	10	graph	graph	NOUN
ejpam-4870	240	11	which	which	PRON
ejpam-4870	240	12	is	be	AUX
ejpam-4870	240	13	just	just	ADV
ejpam-4870	240	14	the	the	DET
ejpam-4870	240	15	list	list	NOUN
ejpam-4870	240	16	of	of	ADP
ejpam-4870	240	17	degrees	degree	NOUN
ejpam-4870	240	18	of	of	ADP
ejpam-4870	240	19	each	each	DET
ejpam-4870	240	20	vertex	vertex	NOUN
ejpam-4870	240	21	in	in	ADP
ejpam-4870	240	22	a	a	DET
ejpam-4870	240	23	particular	particular	ADJ
ejpam-4870	240	24	graph	graph	NOUN
ejpam-4870	240	25	.	.	PUNCT
ejpam-4870	240	26	example	example	NOUN
ejpam-4870	240	27	10	10	NUM
ejpam-4870	240	28	.	.	PUNCT
ejpam-4870	241	1	in	in	ADP
ejpam-4870	241	2	example	example	NOUN
ejpam-4870	241	3	9	9	NUM
ejpam-4870	241	4	,	,	PUNCT
ejpam-4870	241	5	since	since	SCONJ
ejpam-4870	241	6	there	there	PRON
ejpam-4870	241	7	exists	exist	VERB
ejpam-4870	241	8	a	a	DET
ejpam-4870	241	9	function	function	NOUN
ejpam-4870	241	10	mapping	mapping	NOUN
ejpam-4870	241	11	ϕ	ϕ	NOUN
ejpam-4870	241	12	:	:	PUNCT
ejpam-4870	241	13	v	v	X
ejpam-4870	241	14	(	(	PUNCT
ejpam-4870	241	15	g	g	NOUN
ejpam-4870	241	16	)	)	PUNCT
ejpam-4870	241	17	7−→	7−→	NOUN
ejpam-4870	241	18	v	v	NOUN
ejpam-4870	241	19	(	(	PUNCT
ejpam-4870	241	20	h	h	NOUN
ejpam-4870	241	21	)	)	PUNCT
ejpam-4870	241	22	which	which	PRON
ejpam-4870	241	23	is	be	AUX
ejpam-4870	241	24	an	an	DET
ejpam-4870	241	25	isomorphism	isomorphism	NOUN
ejpam-4870	241	26	,	,	PUNCT
ejpam-4870	241	27	it	it	PRON
ejpam-4870	241	28	follows	follow	VERB
ejpam-4870	241	29	that	that	SCONJ
ejpam-4870	241	30	g	g	PROPN
ejpam-4870	241	31	is	be	AUX
ejpam-4870	241	32	isomorphic	isomorphic	ADJ
ejpam-4870	241	33	to	to	AUX
ejpam-4870	241	34	h.	h.	PROPN
ejpam-4870	241	35	recall	recall	VERB
ejpam-4870	241	36	that	that	SCONJ
ejpam-4870	241	37	a	a	DET
ejpam-4870	241	38	cycle	cycle	NOUN
ejpam-4870	241	39	graph	graph	NOUN
ejpam-4870	241	40	of	of	ADP
ejpam-4870	241	41	order	order	NOUN
ejpam-4870	241	42	3	3	NUM
ejpam-4870	241	43	and	and	CCONJ
ejpam-4870	241	44	size	size	NOUN
ejpam-4870	241	45	3	3	NUM
ejpam-4870	241	46	is	be	AUX
ejpam-4870	241	47	a	a	DET
ejpam-4870	241	48	2	2	NUM
ejpam-4870	241	49	-	-	PUNCT
ejpam-4870	241	50	regular	regular	ADJ
ejpam-4870	241	51	graph	graph	NOUN
ejpam-4870	241	52	.	.	PUNCT
ejpam-4870	242	1	moreover	moreover	ADV
ejpam-4870	242	2	,	,	PUNCT
ejpam-4870	242	3	it	it	PRON
ejpam-4870	242	4	can	can	AUX
ejpam-4870	242	5	be	be	AUX
ejpam-4870	242	6	observed	observe	VERB
ejpam-4870	242	7	that	that	SCONJ
ejpam-4870	242	8	a	a	DET
ejpam-4870	242	9	complete	complete	ADJ
ejpam-4870	242	10	graph	graph	NOUN
ejpam-4870	242	11	of	of	ADP
ejpam-4870	242	12	order	order	NOUN
ejpam-4870	242	13	3	3	NUM
ejpam-4870	242	14	is	be	AUX
ejpam-4870	242	15	also	also	ADV
ejpam-4870	242	16	a	a	DET
ejpam-4870	242	17	2	2	NUM
ejpam-4870	242	18	-	-	PUNCT
ejpam-4870	242	19	regular	regular	ADJ
ejpam-4870	242	20	graph	graph	NOUN
ejpam-4870	242	21	which	which	PRON
ejpam-4870	242	22	has	have	VERB
ejpam-4870	242	23	also	also	ADV
ejpam-4870	242	24	a	a	DET
ejpam-4870	242	25	j.c	j.c	PROPN
ejpam-4870	242	26	.	.	PROPN
ejpam-4870	242	27	bonifacio	bonifacio	PROPN
ejpam-4870	242	28	,	,	PUNCT
ejpam-4870	242	29	c.j	c.j	PROPN
ejpam-4870	242	30	.	.	PROPN
ejpam-4870	242	31	andaya	andaya	PROPN
ejpam-4870	242	32	,	,	PUNCT
ejpam-4870	242	33	d.	d.	PROPN
ejpam-4870	242	34	magpantay	magpantay	PROPN
ejpam-4870	242	35	/	/	SYM
ejpam-4870	242	36	eur	eur	PROPN
ejpam-4870	242	37	.	.	PUNCT
ejpam-4870	243	1	j.	j.	PROPN
ejpam-4870	243	2	pure	pure	PROPN
ejpam-4870	243	3	appl	appl	PROPN
ejpam-4870	243	4	.	.	PROPN
ejpam-4870	243	5	math	math	PROPN
ejpam-4870	243	6	,	,	PUNCT
ejpam-4870	243	7	16	16	NUM
ejpam-4870	243	8	(	(	PUNCT
ejpam-4870	243	9	4	4	NUM
ejpam-4870	243	10	)	)	PUNCT
ejpam-4870	243	11	(	(	PUNCT
ejpam-4870	243	12	2023	2023	NUM
ejpam-4870	243	13	)	)	PUNCT
ejpam-4870	243	14	,	,	PUNCT
ejpam-4870	243	15	2476	2476	NUM
ejpam-4870	243	16	-	-	SYM
ejpam-4870	243	17	2498	2498	NUM
ejpam-4870	243	18	2487	2487	NUM
ejpam-4870	243	19	size	size	NOUN
ejpam-4870	243	20	of	of	ADP
ejpam-4870	243	21	3	3	NUM
ejpam-4870	243	22	.	.	PUNCT
ejpam-4870	244	1	now	now	ADV
ejpam-4870	244	2	,	,	PUNCT
ejpam-4870	244	3	since	since	SCONJ
ejpam-4870	244	4	c3	c3	PROPN
ejpam-4870	244	5	have	have	VERB
ejpam-4870	244	6	the	the	DET
ejpam-4870	244	7	same	same	ADJ
ejpam-4870	244	8	order	order	NOUN
ejpam-4870	244	9	,	,	PUNCT
ejpam-4870	244	10	size	size	NOUN
ejpam-4870	244	11	,	,	PUNCT
ejpam-4870	244	12	and	and	CCONJ
ejpam-4870	244	13	the	the	DET
ejpam-4870	244	14	degree	degree	NOUN
ejpam-4870	244	15	of	of	ADP
ejpam-4870	244	16	every	every	DET
ejpam-4870	244	17	vertex	vertex	NOUN
ejpam-4870	244	18	as	as	ADP
ejpam-4870	244	19	that	that	PRON
ejpam-4870	244	20	of	of	ADP
ejpam-4870	244	21	k3	k3	PROPN
ejpam-4870	244	22	,	,	PUNCT
ejpam-4870	244	23	it	it	PRON
ejpam-4870	244	24	can	can	AUX
ejpam-4870	244	25	be	be	AUX
ejpam-4870	244	26	verified	verify	VERB
ejpam-4870	244	27	that	that	SCONJ
ejpam-4870	244	28	there	there	PRON
ejpam-4870	244	29	is	be	VERB
ejpam-4870	244	30	an	an	DET
ejpam-4870	244	31	isomorphism	isomorphism	NOUN
ejpam-4870	244	32	between	between	ADP
ejpam-4870	244	33	the	the	DET
ejpam-4870	244	34	two	two	NUM
ejpam-4870	244	35	graphs	graph	NOUN
ejpam-4870	244	36	.	.	PUNCT
ejpam-4870	245	1	refer	refer	VERB
ejpam-4870	245	2	to	to	PART
ejpam-4870	245	3	figure	figure	VERB
ejpam-4870	245	4	15	15	NUM
ejpam-4870	245	5	.	.	NOUN
ejpam-4870	245	6	3	3	NUM
ejpam-4870	245	7	1	1	NUM
ejpam-4870	245	8	2	2	NUM
ejpam-4870	245	9	x3	x3	NOUN
ejpam-4870	245	10	x1	x1	NOUN
ejpam-4870	246	1	x2	x2	PROPN
ejpam-4870	246	2	figure	figure	NOUN
ejpam-4870	246	3	15	15	NUM
ejpam-4870	246	4	:	:	PUNCT
ejpam-4870	246	5	pictorial	pictorial	ADJ
ejpam-4870	246	6	representations	representation	NOUN
ejpam-4870	246	7	of	of	ADP
ejpam-4870	246	8	c3	c3	PROPN
ejpam-4870	246	9	and	and	CCONJ
ejpam-4870	246	10	k3	k3	VERB
ejpam-4870	246	11	remark	remark	NOUN
ejpam-4870	246	12	1	1	NUM
ejpam-4870	246	13	.	.	PUNCT
ejpam-4870	247	1	let	let	VERB
ejpam-4870	247	2	c3	c3	NOUN
ejpam-4870	247	3	be	be	AUX
ejpam-4870	247	4	a	a	DET
ejpam-4870	247	5	cycle	cycle	NOUN
ejpam-4870	247	6	graph	graph	NOUN
ejpam-4870	247	7	of	of	ADP
ejpam-4870	247	8	order	order	NOUN
ejpam-4870	247	9	3	3	NUM
ejpam-4870	247	10	and	and	CCONJ
ejpam-4870	247	11	let	let	VERB
ejpam-4870	247	12	k3	k3	NOUN
ejpam-4870	247	13	be	be	AUX
ejpam-4870	247	14	a	a	DET
ejpam-4870	247	15	complete	complete	ADJ
ejpam-4870	247	16	graph	graph	NOUN
ejpam-4870	247	17	of	of	ADP
ejpam-4870	247	18	order	order	NOUN
ejpam-4870	247	19	3	3	X
ejpam-4870	247	20	.	.	PUNCT
ejpam-4870	248	1	then	then	ADV
ejpam-4870	248	2	c3	c3	PROPN
ejpam-4870	248	3	≃	≃	PROPN
ejpam-4870	248	4	k3	k3	PROPN
ejpam-4870	248	5	.	.	PUNCT
ejpam-4870	249	1	this	this	DET
ejpam-4870	249	2	section	section	NOUN
ejpam-4870	249	3	discusses	discuss	VERB
ejpam-4870	249	4	some	some	PRON
ejpam-4870	249	5	of	of	ADP
ejpam-4870	249	6	the	the	DET
ejpam-4870	249	7	parameters	parameter	NOUN
ejpam-4870	249	8	that	that	PRON
ejpam-4870	249	9	will	will	AUX
ejpam-4870	249	10	be	be	AUX
ejpam-4870	249	11	helpful	helpful	ADJ
ejpam-4870	249	12	in	in	ADP
ejpam-4870	249	13	determining	determine	VERB
ejpam-4870	249	14	a	a	DET
ejpam-4870	249	15	graph	graph	NOUN
ejpam-4870	249	16	.	.	PUNCT
ejpam-4870	250	1	definition	definition	NOUN
ejpam-4870	250	2	15	15	NUM
ejpam-4870	250	3	.	.	PUNCT
ejpam-4870	251	1	let	let	VERB
ejpam-4870	251	2	g	g	PRON
ejpam-4870	251	3	be	be	AUX
ejpam-4870	251	4	a	a	DET
ejpam-4870	251	5	graph	graph	NOUN
ejpam-4870	251	6	.	.	PUNCT
ejpam-4870	252	1	the	the	DET
ejpam-4870	252	2	nonempty	nonempty	ADV
ejpam-4870	252	3	set	set	VERB
ejpam-4870	252	4	i	i	PRON
ejpam-4870	252	5	⊆	⊆	NUM
ejpam-4870	252	6	v	v	ADP
ejpam-4870	252	7	(	(	PUNCT
ejpam-4870	252	8	g	g	NOUN
ejpam-4870	252	9	)	)	PUNCT
ejpam-4870	252	10	is	be	AUX
ejpam-4870	252	11	called	call	VERB
ejpam-4870	252	12	an	an	DET
ejpam-4870	252	13	independent	independent	ADJ
ejpam-4870	252	14	set	set	NOUN
ejpam-4870	252	15	in	in	ADP
ejpam-4870	252	16	a	a	DET
ejpam-4870	252	17	graph	graph	NOUN
ejpam-4870	252	18	g	g	NOUN
ejpam-4870	252	19	if	if	SCONJ
ejpam-4870	252	20	for	for	ADP
ejpam-4870	252	21	every	every	DET
ejpam-4870	252	22	x	x	NOUN
ejpam-4870	252	23	,	,	PUNCT
ejpam-4870	252	24	y	y	PROPN
ejpam-4870	252	25	∈	∈	PROPN
ejpam-4870	253	1	i	i	PRON
ejpam-4870	253	2	,	,	PUNCT
ejpam-4870	253	3	then	then	ADV
ejpam-4870	253	4	[	[	X
ejpam-4870	253	5	x	x	X
ejpam-4870	253	6	,	,	PUNCT
ejpam-4870	253	7	y	y	PROPN
ejpam-4870	253	8	]	]	PUNCT
ejpam-4870	253	9	/∈	/∈	PUNCT
ejpam-4870	253	10	e(g	e(g	PROPN
ejpam-4870	253	11	)	)	PUNCT
ejpam-4870	253	12	.	.	PUNCT
ejpam-4870	254	1	the	the	DET
ejpam-4870	254	2	independence	independence	NOUN
ejpam-4870	254	3	number	number	NOUN
ejpam-4870	254	4	of	of	ADP
ejpam-4870	254	5	a	a	DET
ejpam-4870	254	6	graph	graph	NOUN
ejpam-4870	254	7	g	g	NOUN
ejpam-4870	254	8	,	,	PUNCT
ejpam-4870	254	9	denoted	denote	VERB
ejpam-4870	254	10	by	by	ADP
ejpam-4870	254	11	α(g	α(g	NOUN
ejpam-4870	254	12	)	)	PUNCT
ejpam-4870	254	13	,	,	PUNCT
ejpam-4870	254	14	is	be	AUX
ejpam-4870	254	15	the	the	DET
ejpam-4870	254	16	cardinality	cardinality	NOUN
ejpam-4870	254	17	of	of	ADP
ejpam-4870	254	18	the	the	DET
ejpam-4870	254	19	largest	large	ADJ
ejpam-4870	254	20	independent	independent	ADJ
ejpam-4870	254	21	set	set	NOUN
ejpam-4870	254	22	of	of	ADP
ejpam-4870	254	23	g.	g.	PROPN
ejpam-4870	254	24	note	note	VERB
ejpam-4870	254	25	that	that	SCONJ
ejpam-4870	254	26	if	if	SCONJ
ejpam-4870	254	27	there	there	PRON
ejpam-4870	254	28	exists	exist	VERB
ejpam-4870	254	29	a	a	DET
ejpam-4870	254	30	set	set	NOUN
ejpam-4870	254	31	i	i	PRON
ejpam-4870	254	32	⊆	⊆	NUM
ejpam-4870	254	33	v	v	NOUN
ejpam-4870	254	34	(	(	PUNCT
ejpam-4870	254	35	g	g	NOUN
ejpam-4870	254	36	)	)	PUNCT
ejpam-4870	254	37	that	that	PRON
ejpam-4870	254	38	is	be	AUX
ejpam-4870	254	39	an	an	DET
ejpam-4870	254	40	independent	independent	ADJ
ejpam-4870	254	41	set	set	NOUN
ejpam-4870	254	42	of	of	ADP
ejpam-4870	254	43	g	g	NOUN
ejpam-4870	254	44	,	,	PUNCT
ejpam-4870	254	45	then	then	ADV
ejpam-4870	254	46	it	it	PRON
ejpam-4870	254	47	can	can	AUX
ejpam-4870	254	48	be	be	AUX
ejpam-4870	254	49	verified	verify	VERB
ejpam-4870	254	50	that	that	SCONJ
ejpam-4870	254	51	α(g	α(g	PROPN
ejpam-4870	254	52	)	)	PUNCT
ejpam-4870	254	53	≥	≥	NOUN
ejpam-4870	254	54	|i|	|i|	PROPN
ejpam-4870	254	55	.	.	PUNCT
ejpam-4870	254	56	definition	definition	NOUN
ejpam-4870	254	57	16	16	NUM
ejpam-4870	254	58	.	.	PUNCT
ejpam-4870	255	1	let	let	VERB
ejpam-4870	255	2	g	g	PROPN
ejpam-4870	255	3	=	=	SYM
ejpam-4870	255	4	(	(	PUNCT
ejpam-4870	255	5	v	v	NOUN
ejpam-4870	255	6	(	(	PUNCT
ejpam-4870	255	7	g	g	NOUN
ejpam-4870	255	8	)	)	PUNCT
ejpam-4870	255	9	,	,	PUNCT
ejpam-4870	255	10	e(g	e(g	PROPN
ejpam-4870	255	11	)	)	PUNCT
ejpam-4870	255	12	)	)	PUNCT
ejpam-4870	256	1	be	be	AUX
ejpam-4870	256	2	a	a	DET
ejpam-4870	256	3	graph	graph	NOUN
ejpam-4870	256	4	.	.	PUNCT
ejpam-4870	257	1	a	a	DET
ejpam-4870	257	2	nonempty	nonempty	NOUN
ejpam-4870	257	3	subset	subset	VERB
ejpam-4870	257	4	i	i	PRON
ejpam-4870	257	5	of	of	ADP
ejpam-4870	257	6	v	v	NOUN
ejpam-4870	257	7	(	(	PUNCT
ejpam-4870	257	8	g	g	NOUN
ejpam-4870	257	9	)	)	PUNCT
ejpam-4870	257	10	is	be	AUX
ejpam-4870	257	11	called	call	VERB
ejpam-4870	257	12	dominating	dominate	VERB
ejpam-4870	257	13	set	set	NOUN
ejpam-4870	257	14	of	of	ADP
ejpam-4870	257	15	g	g	PROPN
ejpam-4870	257	16	if	if	SCONJ
ejpam-4870	257	17	every	every	DET
ejpam-4870	257	18	element	element	NOUN
ejpam-4870	257	19	of	of	ADP
ejpam-4870	257	20	v	v	NOUN
ejpam-4870	257	21	(	(	PUNCT
ejpam-4870	257	22	g)\i	g)\i	NOUN
ejpam-4870	257	23	is	be	AUX
ejpam-4870	257	24	adjacent	adjacent	ADJ
ejpam-4870	257	25	to	to	ADP
ejpam-4870	257	26	some	some	DET
ejpam-4870	257	27	element	element	NOUN
ejpam-4870	257	28	of	of	ADP
ejpam-4870	257	29	i.	i.	PROPN
ejpam-4870	257	30	moreover	moreover	ADV
ejpam-4870	257	31	,	,	PUNCT
ejpam-4870	257	32	domination	domination	NOUN
ejpam-4870	257	33	number	number	NOUN
ejpam-4870	257	34	,	,	PUNCT
ejpam-4870	257	35	written	write	VERB
ejpam-4870	257	36	as	as	ADP
ejpam-4870	257	37	γ(g	γ(g	PROPN
ejpam-4870	257	38	)	)	PUNCT
ejpam-4870	257	39	,	,	PUNCT
ejpam-4870	257	40	of	of	ADP
ejpam-4870	257	41	a	a	DET
ejpam-4870	257	42	graph	graph	NOUN
ejpam-4870	257	43	g	g	NOUN
ejpam-4870	257	44	is	be	AUX
ejpam-4870	257	45	the	the	DET
ejpam-4870	257	46	minimum	minimum	ADJ
ejpam-4870	257	47	cardinality	cardinality	NOUN
ejpam-4870	257	48	among	among	ADP
ejpam-4870	257	49	all	all	DET
ejpam-4870	257	50	the	the	DET
ejpam-4870	257	51	dominating	dominating	NOUN
ejpam-4870	257	52	set	set	NOUN
ejpam-4870	257	53	of	of	ADP
ejpam-4870	257	54	g.	g.	PROPN
ejpam-4870	257	55	the	the	DET
ejpam-4870	257	56	statement	statement	NOUN
ejpam-4870	257	57	that	that	SCONJ
ejpam-4870	257	58	if	if	SCONJ
ejpam-4870	257	59	i	i	PRON
ejpam-4870	257	60	=	=	SYM
ejpam-4870	257	61	v	v	X
ejpam-4870	257	62	(	(	PUNCT
ejpam-4870	257	63	g	g	NOUN
ejpam-4870	257	64	)	)	PUNCT
ejpam-4870	257	65	then	then	ADV
ejpam-4870	257	66	i	i	PRON
ejpam-4870	257	67	is	be	AUX
ejpam-4870	257	68	a	a	DET
ejpam-4870	257	69	dominating	dominating	NOUN
ejpam-4870	257	70	set	set	NOUN
ejpam-4870	257	71	is	be	AUX
ejpam-4870	257	72	vacuously	vacuously	ADV
ejpam-4870	257	73	true	true	ADJ
ejpam-4870	257	74	since	since	SCONJ
ejpam-4870	257	75	v	v	NOUN
ejpam-4870	257	76	(	(	PUNCT
ejpam-4870	257	77	g)\i	g)\i	NOUN
ejpam-4870	257	78	=	=	SYM
ejpam-4870	257	79	∅	∅	NOUN
ejpam-4870	257	80	so	so	CCONJ
ejpam-4870	257	81	there	there	PRON
ejpam-4870	257	82	are	be	VERB
ejpam-4870	257	83	no	no	DET
ejpam-4870	257	84	elements	element	NOUN
ejpam-4870	257	85	to	to	PART
ejpam-4870	257	86	be	be	AUX
ejpam-4870	257	87	considered	consider	VERB
ejpam-4870	257	88	.	.	PUNCT
ejpam-4870	258	1	in	in	ADP
ejpam-4870	258	2	mathematics	mathematic	NOUN
ejpam-4870	258	3	,	,	PUNCT
ejpam-4870	258	4	a	a	DET
ejpam-4870	258	5	vacuous	vacuous	ADJ
ejpam-4870	258	6	truth	truth	NOUN
ejpam-4870	258	7	is	be	AUX
ejpam-4870	258	8	a	a	DET
ejpam-4870	258	9	universal	universal	ADJ
ejpam-4870	258	10	or	or	CCONJ
ejpam-4870	258	11	conditional	conditional	ADJ
ejpam-4870	258	12	statement	statement	NOUN
ejpam-4870	258	13	that	that	PRON
ejpam-4870	258	14	is	be	AUX
ejpam-4870	258	15	deemed	deem	VERB
ejpam-4870	258	16	to	to	PART
ejpam-4870	258	17	be	be	AUX
ejpam-4870	258	18	true	true	ADJ
ejpam-4870	258	19	.	.	PUNCT
ejpam-4870	259	1	also	also	ADV
ejpam-4870	259	2	,	,	PUNCT
ejpam-4870	259	3	it	it	PRON
ejpam-4870	259	4	can	can	AUX
ejpam-4870	259	5	be	be	AUX
ejpam-4870	259	6	observed	observe	VERB
ejpam-4870	259	7	that	that	SCONJ
ejpam-4870	259	8	if	if	SCONJ
ejpam-4870	259	9	i	i	PRON
ejpam-4870	259	10	⊆	⊆	NUM
ejpam-4870	259	11	v	v	ADP
ejpam-4870	259	12	(	(	PUNCT
ejpam-4870	259	13	g	g	NOUN
ejpam-4870	259	14	)	)	PUNCT
ejpam-4870	259	15	is	be	AUX
ejpam-4870	259	16	dominating	dominate	VERB
ejpam-4870	259	17	set	set	VERB
ejpam-4870	259	18	in	in	ADP
ejpam-4870	259	19	g	g	NOUN
ejpam-4870	259	20	,	,	PUNCT
ejpam-4870	259	21	thus	thus	ADV
ejpam-4870	259	22	,	,	PUNCT
ejpam-4870	259	23	γ(g	γ(g	PROPN
ejpam-4870	259	24	)	)	PUNCT
ejpam-4870	259	25	≤	≤	NOUN
ejpam-4870	259	26	|i|	|i|	PROPN
ejpam-4870	259	27	.	.	PROPN
ejpam-4870	260	1	3	3	NUM
ejpam-4870	260	2	.	.	X
ejpam-4870	260	3	j	j	NOUN
ejpam-4870	260	4	-	-	PUNCT
ejpam-4870	260	5	edge	edge	NOUN
ejpam-4870	260	6	intersection	intersection	NOUN
ejpam-4870	260	7	graph	graph	NOUN
ejpam-4870	260	8	of	of	ADP
ejpam-4870	260	9	cycle	cycle	NOUN
ejpam-4870	260	10	graph	graph	NOUN
ejpam-4870	260	11	and	and	CCONJ
ejpam-4870	260	12	its	its	PRON
ejpam-4870	260	13	basic	basic	ADJ
ejpam-4870	260	14	parameters	parameter	NOUN
ejpam-4870	260	15	recall	recall	VERB
ejpam-4870	260	16	that	that	PRON
ejpam-4870	260	17	for	for	ADP
ejpam-4870	260	18	an	an	DET
ejpam-4870	260	19	arbitrary	arbitrary	ADJ
ejpam-4870	260	20	edge	edge	NOUN
ejpam-4870	260	21	of	of	ADP
ejpam-4870	260	22	the	the	DET
ejpam-4870	260	23	cycle	cycle	NOUN
ejpam-4870	260	24	graph	graph	NOUN
ejpam-4870	260	25	cn	cn	PROPN
ejpam-4870	260	26	,	,	PUNCT
ejpam-4870	260	27	say	say	VERB
ejpam-4870	260	28	[	[	X
ejpam-4870	260	29	1	1	NUM
ejpam-4870	260	30	,	,	PUNCT
ejpam-4870	260	31	2	2	NUM
ejpam-4870	260	32	]	]	PUNCT
ejpam-4870	260	33	is	be	AUX
ejpam-4870	260	34	relabeled	relabele	VERB
ejpam-4870	260	35	as	as	ADP
ejpam-4870	260	36	12	12	NUM
ejpam-4870	260	37	.	.	PUNCT
ejpam-4870	261	1	moreover	moreover	ADV
ejpam-4870	261	2	,	,	PUNCT
ejpam-4870	261	3	the	the	DET
ejpam-4870	261	4	variable	variable	ADJ
ejpam-4870	261	5	j	j	PROPN
ejpam-4870	261	6	is	be	AUX
ejpam-4870	261	7	used	use	VERB
ejpam-4870	261	8	as	as	ADP
ejpam-4870	261	9	the	the	DET
ejpam-4870	261	10	number	number	NOUN
ejpam-4870	261	11	of	of	ADP
ejpam-4870	261	12	edges	edge	NOUN
ejpam-4870	261	13	of	of	ADP
ejpam-4870	261	14	the	the	DET
ejpam-4870	261	15	spanning	span	VERB
ejpam-4870	261	16	subgraph	subgraph	NOUN
ejpam-4870	261	17	of	of	ADP
ejpam-4870	261	18	cn	cn	PROPN
ejpam-4870	261	19	and	and	CCONJ
ejpam-4870	261	20	the	the	DET
ejpam-4870	261	21	researchers	researcher	NOUN
ejpam-4870	261	22	are	be	AUX
ejpam-4870	261	23	focusing	focus	VERB
ejpam-4870	261	24	only	only	ADV
ejpam-4870	261	25	on	on	ADP
ejpam-4870	261	26	the	the	DET
ejpam-4870	261	27	spanning	span	VERB
ejpam-4870	261	28	subgraph	subgraph	NOUN
ejpam-4870	261	29	when	when	SCONJ
ejpam-4870	261	30	1	1	NUM
ejpam-4870	261	31	≤	≤	NUM
ejpam-4870	261	32	j	j	PROPN
ejpam-4870	261	33	≤	≤	PROPN
ejpam-4870	261	34	n.	n.	PROPN
ejpam-4870	261	35	j.c	j.c	PROPN
ejpam-4870	261	36	.	.	PROPN
ejpam-4870	261	37	bonifacio	bonifacio	PROPN
ejpam-4870	261	38	,	,	PUNCT
ejpam-4870	261	39	c.j	c.j	PROPN
ejpam-4870	261	40	.	.	PROPN
ejpam-4870	261	41	andaya	andaya	PROPN
ejpam-4870	261	42	,	,	PUNCT
ejpam-4870	261	43	d.	d.	PROPN
ejpam-4870	261	44	magpantay	magpantay	PROPN
ejpam-4870	261	45	/	/	SYM
ejpam-4870	261	46	eur	eur	PROPN
ejpam-4870	261	47	.	.	PUNCT
ejpam-4870	262	1	j.	j.	PROPN
ejpam-4870	262	2	pure	pure	PROPN
ejpam-4870	262	3	appl	appl	PROPN
ejpam-4870	262	4	.	.	PROPN
ejpam-4870	262	5	math	math	PROPN
ejpam-4870	262	6	,	,	PUNCT
ejpam-4870	262	7	16	16	NUM
ejpam-4870	262	8	(	(	PUNCT
ejpam-4870	262	9	4	4	NUM
ejpam-4870	262	10	)	)	PUNCT
ejpam-4870	262	11	(	(	PUNCT
ejpam-4870	262	12	2023	2023	NUM
ejpam-4870	262	13	)	)	PUNCT
ejpam-4870	262	14	,	,	PUNCT
ejpam-4870	262	15	2476	2476	NUM
ejpam-4870	262	16	-	-	SYM
ejpam-4870	262	17	2498	2498	NUM
ejpam-4870	262	18	2488	2488	NUM
ejpam-4870	262	19	definition	definition	NOUN
ejpam-4870	262	20	17	17	NUM
ejpam-4870	262	21	.	.	PUNCT
ejpam-4870	263	1	let	let	VERB
ejpam-4870	263	2	cn	cn	PROPN
ejpam-4870	263	3	be	be	AUX
ejpam-4870	263	4	a	a	DET
ejpam-4870	263	5	cycle	cycle	NOUN
ejpam-4870	263	6	graph	graph	NOUN
ejpam-4870	263	7	of	of	ADP
ejpam-4870	263	8	order	order	NOUN
ejpam-4870	263	9	n	n	PRON
ejpam-4870	263	10	where	where	SCONJ
ejpam-4870	263	11	n	n	PRON
ejpam-4870	263	12	≥	≥	NOUN
ejpam-4870	263	13	3	3	NUM
ejpam-4870	263	14	.	.	PUNCT
ejpam-4870	264	1	for	for	ADP
ejpam-4870	264	2	1	1	NUM
ejpam-4870	264	3	≤	≤	NUM
ejpam-4870	264	4	j	j	PROPN
ejpam-4870	264	5	≤	≤	NUM
ejpam-4870	264	6	n	n	CCONJ
ejpam-4870	264	7	,	,	PUNCT
ejpam-4870	264	8	a	a	DET
ejpam-4870	264	9	j	j	NOUN
ejpam-4870	264	10	-	-	PUNCT
ejpam-4870	264	11	edge	edge	NOUN
ejpam-4870	264	12	intersection	intersection	NOUN
ejpam-4870	264	13	graph	graph	NOUN
ejpam-4870	264	14	of	of	ADP
ejpam-4870	264	15	cn	cn	PROPN
ejpam-4870	264	16	,	,	PUNCT
ejpam-4870	264	17	denoted	denote	VERB
ejpam-4870	264	18	by	by	ADP
ejpam-4870	264	19	ec(n	ec(n	PROPN
ejpam-4870	264	20	,	,	PUNCT
ejpam-4870	264	21	j	j	PROPN
ejpam-4870	264	22	)	)	PUNCT
ejpam-4870	264	23	,	,	PUNCT
ejpam-4870	264	24	is	be	AUX
ejpam-4870	264	25	the	the	DET
ejpam-4870	264	26	graph	graph	NOUN
ejpam-4870	264	27	whose	whose	DET
ejpam-4870	264	28	vertex	vertex	NOUN
ejpam-4870	264	29	set	set	NOUN
ejpam-4870	264	30	is	be	AUX
ejpam-4870	264	31	v	v	NOUN
ejpam-4870	264	32	(	(	PUNCT
ejpam-4870	264	33	ec(n	ec(n	NUM
ejpam-4870	264	34	,	,	PUNCT
ejpam-4870	264	35	j	j	NOUN
ejpam-4870	264	36	)	)	PUNCT
ejpam-4870	264	37	)	)	PUNCT
ejpam-4870	265	1	=	=	PRON
ejpam-4870	265	2	{	{	PUNCT
ejpam-4870	265	3	{	{	PUNCT
ejpam-4870	265	4	e1	e1	PROPN
ejpam-4870	265	5	,	,	PUNCT
ejpam-4870	265	6	e2	e2	PROPN
ejpam-4870	265	7	,	,	PUNCT
ejpam-4870	265	8	·	·	PUNCT
ejpam-4870	265	9	·	·	PUNCT
ejpam-4870	265	10	·	·	PUNCT
ejpam-4870	265	11	,	,	PUNCT
ejpam-4870	265	12	ej}|ei	ej}|ei	PROPN
ejpam-4870	265	13	is	be	AUX
ejpam-4870	265	14	an	an	DET
ejpam-4870	265	15	edge	edge	NOUN
ejpam-4870	265	16	in	in	ADP
ejpam-4870	265	17	e(cn	e(cn	NOUN
ejpam-4870	265	18	)	)	PUNCT
ejpam-4870	265	19	,	,	PUNCT
ejpam-4870	265	20	1	1	NUM
ejpam-4870	265	21	≤	≤	NUM
ejpam-4870	265	22	i	i	X
ejpam-4870	265	23	≤	≤	PROPN
ejpam-4870	265	24	j	j	NOUN
ejpam-4870	265	25	}	}	PUNCT
ejpam-4870	265	26	.	.	PUNCT
ejpam-4870	266	1	moreover	moreover	ADV
ejpam-4870	266	2	,	,	PUNCT
ejpam-4870	266	3	two	two	NUM
ejpam-4870	266	4	distinct	distinct	ADJ
ejpam-4870	266	5	vertices	vertice	VERB
ejpam-4870	266	6	a	a	DET
ejpam-4870	266	7	,	,	PUNCT
ejpam-4870	266	8	b	b	PROPN
ejpam-4870	266	9	∈	∈	PROPN
ejpam-4870	266	10	v	v	NOUN
ejpam-4870	266	11	(	(	PUNCT
ejpam-4870	266	12	ec(n	ec(n	PROPN
ejpam-4870	266	13	,	,	PUNCT
ejpam-4870	266	14	j	j	NOUN
ejpam-4870	266	15	)	)	PUNCT
ejpam-4870	266	16	)	)	PUNCT
ejpam-4870	266	17	are	be	AUX
ejpam-4870	266	18	adjacent	adjacent	ADJ
ejpam-4870	266	19	whenever	whenever	SCONJ
ejpam-4870	266	20	|a	|a	X
ejpam-4870	266	21	∩b|	∩b|	PROPN
ejpam-4870	266	22	=	=	SYM
ejpam-4870	266	23	1	1	X
ejpam-4870	266	24	.	.	X
ejpam-4870	266	25	note	note	VERB
ejpam-4870	266	26	that	that	SCONJ
ejpam-4870	266	27	the	the	DET
ejpam-4870	266	28	spanning	span	VERB
ejpam-4870	266	29	subgraphs	subgraph	NOUN
ejpam-4870	266	30	of	of	ADP
ejpam-4870	266	31	cn	cn	PROPN
ejpam-4870	266	32	can	can	AUX
ejpam-4870	266	33	be	be	AUX
ejpam-4870	266	34	uniquely	uniquely	ADV
ejpam-4870	266	35	determined	determine	VERB
ejpam-4870	266	36	by	by	ADP
ejpam-4870	266	37	the	the	DET
ejpam-4870	266	38	vertices	vertex	NOUN
ejpam-4870	266	39	of	of	ADP
ejpam-4870	266	40	j	j	NOUN
ejpam-4870	266	41	-	-	PUNCT
ejpam-4870	266	42	edge	edge	NOUN
ejpam-4870	266	43	intersection	intersection	NOUN
ejpam-4870	266	44	graph	graph	NOUN
ejpam-4870	266	45	of	of	ADP
ejpam-4870	266	46	cn	cn	PROPN
ejpam-4870	266	47	.	.	PUNCT
ejpam-4870	267	1	now	now	ADV
ejpam-4870	267	2	,	,	PUNCT
ejpam-4870	267	3	the	the	DET
ejpam-4870	267	4	elements	element	NOUN
ejpam-4870	267	5	of	of	ADP
ejpam-4870	267	6	v	v	NOUN
ejpam-4870	267	7	(	(	PUNCT
ejpam-4870	267	8	ec(n	ec(n	NUM
ejpam-4870	267	9	,	,	PUNCT
ejpam-4870	267	10	j	j	NOUN
ejpam-4870	267	11	)	)	PUNCT
ejpam-4870	267	12	)	)	PUNCT
ejpam-4870	267	13	are	be	AUX
ejpam-4870	267	14	the	the	DET
ejpam-4870	267	15	spanning	span	VERB
ejpam-4870	267	16	subgraphs	subgraph	NOUN
ejpam-4870	267	17	of	of	ADP
ejpam-4870	267	18	cn	cn	PROPN
ejpam-4870	267	19	with	with	ADP
ejpam-4870	267	20	j	j	PROPN
ejpam-4870	267	21	edges	edge	NOUN
ejpam-4870	267	22	where	where	SCONJ
ejpam-4870	267	23	1	1	NUM
ejpam-4870	267	24	≤	≤	NUM
ejpam-4870	267	25	j	j	PROPN
ejpam-4870	267	26	≤	≤	PROPN
ejpam-4870	267	27	n.	n.	NOUN
ejpam-4870	267	28	moreover	moreover	ADV
ejpam-4870	267	29	,	,	PUNCT
ejpam-4870	267	30	distinct	distinct	ADJ
ejpam-4870	267	31	pairs	pair	NOUN
ejpam-4870	267	32	of	of	ADP
ejpam-4870	267	33	vertices	vertex	NOUN
ejpam-4870	267	34	are	be	AUX
ejpam-4870	267	35	elements	element	NOUN
ejpam-4870	267	36	of	of	ADP
ejpam-4870	267	37	e(ec(n	e(ec(n	PROPN
ejpam-4870	267	38	,	,	PUNCT
ejpam-4870	267	39	j	j	PROPN
ejpam-4870	267	40	)	)	PUNCT
ejpam-4870	267	41	)	)	PUNCT
ejpam-4870	268	1	if	if	SCONJ
ejpam-4870	268	2	they	they	PRON
ejpam-4870	268	3	share	share	VERB
ejpam-4870	268	4	exactly	exactly	ADV
ejpam-4870	268	5	one	one	NUM
ejpam-4870	268	6	edge	edge	NOUN
ejpam-4870	268	7	.	.	PUNCT
ejpam-4870	269	1	to	to	PART
ejpam-4870	269	2	understand	understand	VERB
ejpam-4870	269	3	this	this	PRON
ejpam-4870	269	4	,	,	PUNCT
ejpam-4870	269	5	given	give	VERB
ejpam-4870	269	6	in	in	ADP
ejpam-4870	269	7	example	example	NOUN
ejpam-4870	269	8	11	11	NUM
ejpam-4870	269	9	is	be	AUX
ejpam-4870	269	10	an	an	DET
ejpam-4870	269	11	illustration	illustration	NOUN
ejpam-4870	269	12	of	of	ADP
ejpam-4870	269	13	ec(n	ec(n	PROPN
ejpam-4870	269	14	,	,	PUNCT
ejpam-4870	269	15	j	j	PROPN
ejpam-4870	269	16	)	)	PUNCT
ejpam-4870	270	1	where	where	SCONJ
ejpam-4870	270	2	n	n	NOUN
ejpam-4870	270	3	=	=	SYM
ejpam-4870	270	4	4	4	NUM
ejpam-4870	270	5	and	and	CCONJ
ejpam-4870	270	6	j	j	PROPN
ejpam-4870	270	7	=	=	SYM
ejpam-4870	270	8	2	2	NUM
ejpam-4870	270	9	.	.	NOUN
ejpam-4870	270	10	example	example	NOUN
ejpam-4870	270	11	11	11	NUM
ejpam-4870	270	12	.	.	PUNCT
ejpam-4870	271	1	consider	consider	VERB
ejpam-4870	271	2	the	the	DET
ejpam-4870	271	3	cycle	cycle	NOUN
ejpam-4870	271	4	graph	graph	NOUN
ejpam-4870	271	5	c4	c4	NOUN
ejpam-4870	271	6	where	where	SCONJ
ejpam-4870	271	7	v	v	NOUN
ejpam-4870	271	8	(	(	PUNCT
ejpam-4870	271	9	c4	c4	NOUN
ejpam-4870	271	10	)	)	PUNCT
ejpam-4870	271	11	=	=	PUNCT
ejpam-4870	271	12	{	{	PUNCT
ejpam-4870	271	13	1	1	NUM
ejpam-4870	271	14	,	,	PUNCT
ejpam-4870	271	15	2	2	NUM
ejpam-4870	271	16	,	,	PUNCT
ejpam-4870	271	17	3	3	NUM
ejpam-4870	271	18	,	,	PUNCT
ejpam-4870	271	19	4	4	NUM
ejpam-4870	271	20	}	}	PUNCT
ejpam-4870	271	21	,	,	PUNCT
ejpam-4870	271	22	e(c4	e(c4	NOUN
ejpam-4870	271	23	)	)	PUNCT
ejpam-4870	272	1	=	=	PRON
ejpam-4870	272	2	{	{	PUNCT
ejpam-4870	272	3	12	12	NUM
ejpam-4870	272	4	,	,	PUNCT
ejpam-4870	272	5	23	23	NUM
ejpam-4870	272	6	,	,	PUNCT
ejpam-4870	272	7	34	34	NUM
ejpam-4870	272	8	,	,	PUNCT
ejpam-4870	272	9	41	41	NUM
ejpam-4870	272	10	}	}	PUNCT
ejpam-4870	272	11	and	and	CCONJ
ejpam-4870	272	12	let	let	VERB
ejpam-4870	272	13	j	j	PROPN
ejpam-4870	272	14	=	=	NOUN
ejpam-4870	272	15	2	2	X
ejpam-4870	272	16	.	.	PUNCT
ejpam-4870	273	1	the	the	DET
ejpam-4870	273	2	vertex	vertex	NOUN
ejpam-4870	273	3	set	set	NOUN
ejpam-4870	273	4	of	of	ADP
ejpam-4870	273	5	ec(4,2	ec(4,2	NOUN
ejpam-4870	273	6	)	)	PUNCT
ejpam-4870	273	7	is	be	AUX
ejpam-4870	273	8	given	give	VERB
ejpam-4870	273	9	by	by	ADP
ejpam-4870	273	10	v	v	NOUN
ejpam-4870	273	11	(	(	PUNCT
ejpam-4870	273	12	ec(4,2	ec(4,2	NOUN
ejpam-4870	273	13	)	)	PUNCT
ejpam-4870	273	14	)	)	PUNCT
ejpam-4870	274	1	=	=	PRON
ejpam-4870	274	2	{	{	PUNCT
ejpam-4870	274	3	{	{	PUNCT
ejpam-4870	274	4	12	12	NUM
ejpam-4870	274	5	,	,	PUNCT
ejpam-4870	274	6	23	23	NUM
ejpam-4870	274	7	}	}	PUNCT
ejpam-4870	274	8	,	,	PUNCT
ejpam-4870	274	9	{	{	PUNCT
ejpam-4870	274	10	12	12	NUM
ejpam-4870	274	11	,	,	PUNCT
ejpam-4870	274	12	34	34	NUM
ejpam-4870	274	13	}	}	PUNCT
ejpam-4870	274	14	,	,	PUNCT
ejpam-4870	274	15	{	{	PUNCT
ejpam-4870	274	16	12	12	NUM
ejpam-4870	274	17	,	,	PUNCT
ejpam-4870	274	18	41	41	NUM
ejpam-4870	274	19	}	}	PUNCT
ejpam-4870	274	20	,	,	PUNCT
ejpam-4870	274	21	{	{	PUNCT
ejpam-4870	274	22	23	23	NUM
ejpam-4870	274	23	,	,	PUNCT
ejpam-4870	274	24	34	34	NUM
ejpam-4870	274	25	}	}	PUNCT
ejpam-4870	274	26	,	,	PUNCT
ejpam-4870	274	27	{	{	PUNCT
ejpam-4870	274	28	23	23	NUM
ejpam-4870	274	29	,	,	PUNCT
ejpam-4870	274	30	41	41	NUM
ejpam-4870	274	31	}	}	PUNCT
ejpam-4870	274	32	,	,	PUNCT
ejpam-4870	274	33	{	{	PUNCT
ejpam-4870	274	34	34	34	NUM
ejpam-4870	274	35	,	,	PUNCT
ejpam-4870	274	36	41	41	NUM
ejpam-4870	274	37	}	}	PUNCT
ejpam-4870	274	38	}	}	PUNCT
ejpam-4870	274	39	.	.	PUNCT
ejpam-4870	275	1	now	now	ADV
ejpam-4870	275	2	,	,	PUNCT
ejpam-4870	275	3	since	since	SCONJ
ejpam-4870	275	4	{	{	PUNCT
ejpam-4870	275	5	12	12	NUM
ejpam-4870	275	6	,	,	PUNCT
ejpam-4870	275	7	23	23	NUM
ejpam-4870	275	8	}	}	PUNCT
ejpam-4870	275	9	∩	∩	NOUN
ejpam-4870	275	10	{	{	PUNCT
ejpam-4870	275	11	12	12	NUM
ejpam-4870	275	12	,	,	PUNCT
ejpam-4870	275	13	34	34	NUM
ejpam-4870	275	14	}	}	PUNCT
ejpam-4870	275	15	=	=	PUNCT
ejpam-4870	275	16	{	{	PUNCT
ejpam-4870	275	17	12	12	NUM
ejpam-4870	275	18	}	}	PUNCT
ejpam-4870	275	19	,	,	PUNCT
ejpam-4870	275	20	it	it	PRON
ejpam-4870	275	21	follows	follow	VERB
ejpam-4870	275	22	that	that	SCONJ
ejpam-4870	275	23	[	[	X
ejpam-4870	275	24	{	{	PUNCT
ejpam-4870	275	25	12	12	NUM
ejpam-4870	275	26	,	,	PUNCT
ejpam-4870	275	27	23	23	NUM
ejpam-4870	275	28	}	}	PUNCT
ejpam-4870	275	29	,	,	PUNCT
ejpam-4870	275	30	{	{	PUNCT
ejpam-4870	275	31	12	12	NUM
ejpam-4870	275	32	,	,	PUNCT
ejpam-4870	275	33	34	34	NUM
ejpam-4870	275	34	}	}	PUNCT
ejpam-4870	275	35	]	]	PUNCT
ejpam-4870	275	36	∈	∈	PROPN
ejpam-4870	275	37	e(ec(4,2	e(ec(4,2	NOUN
ejpam-4870	275	38	)	)	PUNCT
ejpam-4870	275	39	)	)	PUNCT
ejpam-4870	275	40	.	.	PUNCT
ejpam-4870	276	1	similarly	similarly	ADV
ejpam-4870	276	2	,	,	PUNCT
ejpam-4870	276	3	{	{	PUNCT
ejpam-4870	276	4	23	23	NUM
ejpam-4870	276	5	,	,	PUNCT
ejpam-4870	276	6	41	41	NUM
ejpam-4870	276	7	}	}	PUNCT
ejpam-4870	276	8	∩	∩	NOUN
ejpam-4870	276	9	{	{	PUNCT
ejpam-4870	276	10	34	34	NUM
ejpam-4870	276	11	,	,	PUNCT
ejpam-4870	276	12	41	41	NUM
ejpam-4870	276	13	}	}	PUNCT
ejpam-4870	276	14	=	=	PUNCT
ejpam-4870	276	15	{	{	PUNCT
ejpam-4870	276	16	41	41	NUM
ejpam-4870	276	17	}	}	PUNCT
ejpam-4870	276	18	,	,	PUNCT
ejpam-4870	276	19	so	so	CCONJ
ejpam-4870	276	20	[	[	X
ejpam-4870	276	21	{	{	PUNCT
ejpam-4870	276	22	23	23	NUM
ejpam-4870	276	23	,	,	PUNCT
ejpam-4870	276	24	41	41	NUM
ejpam-4870	276	25	}	}	PUNCT
ejpam-4870	276	26	,	,	PUNCT
ejpam-4870	276	27	{	{	PUNCT
ejpam-4870	276	28	34	34	NUM
ejpam-4870	276	29	,	,	PUNCT
ejpam-4870	276	30	41	41	NUM
ejpam-4870	276	31	}	}	PUNCT
ejpam-4870	276	32	]	]	PUNCT
ejpam-4870	276	33	is	be	AUX
ejpam-4870	276	34	also	also	ADV
ejpam-4870	276	35	in	in	ADP
ejpam-4870	276	36	e(ec(4,2	e(ec(4,2	NOUN
ejpam-4870	276	37	)	)	PUNCT
ejpam-4870	276	38	)	)	PUNCT
ejpam-4870	276	39	.	.	PUNCT
ejpam-4870	277	1	however	however	ADV
ejpam-4870	277	2	,	,	PUNCT
ejpam-4870	277	3	vertices	vertice	VERB
ejpam-4870	277	4	{	{	PUNCT
ejpam-4870	277	5	12	12	NUM
ejpam-4870	277	6	,	,	PUNCT
ejpam-4870	277	7	23	23	NUM
ejpam-4870	277	8	}	}	PUNCT
ejpam-4870	277	9	and	and	CCONJ
ejpam-4870	277	10	{	{	PUNCT
ejpam-4870	277	11	34	34	NUM
ejpam-4870	277	12	,	,	PUNCT
ejpam-4870	277	13	41	41	NUM
ejpam-4870	277	14	}	}	PUNCT
ejpam-4870	277	15	are	be	AUX
ejpam-4870	277	16	not	not	PART
ejpam-4870	277	17	adjacent	adjacent	ADJ
ejpam-4870	277	18	since	since	SCONJ
ejpam-4870	277	19	{	{	PUNCT
ejpam-4870	277	20	12	12	NUM
ejpam-4870	277	21	,	,	PUNCT
ejpam-4870	277	22	23	23	NUM
ejpam-4870	277	23	}	}	PUNCT
ejpam-4870	277	24	∩	∩	NOUN
ejpam-4870	277	25	{	{	PUNCT
ejpam-4870	277	26	34	34	NUM
ejpam-4870	277	27	,	,	PUNCT
ejpam-4870	277	28	41	41	NUM
ejpam-4870	277	29	}	}	PUNCT
ejpam-4870	277	30	=	=	PUNCT
ejpam-4870	277	31	∅.	∅.	AUX
ejpam-4870	277	32	doing	do	VERB
ejpam-4870	277	33	the	the	DET
ejpam-4870	277	34	same	same	ADJ
ejpam-4870	277	35	process	process	NOUN
ejpam-4870	277	36	for	for	ADP
ejpam-4870	277	37	any	any	DET
ejpam-4870	277	38	two	two	NUM
ejpam-4870	277	39	distinct	distinct	ADJ
ejpam-4870	277	40	vertices	vertex	NOUN
ejpam-4870	277	41	in	in	ADP
ejpam-4870	277	42	v	v	NOUN
ejpam-4870	277	43	(	(	PUNCT
ejpam-4870	277	44	ec(4,2	ec(4,2	NOUN
ejpam-4870	277	45	)	)	PUNCT
ejpam-4870	277	46	)	)	PUNCT
ejpam-4870	277	47	,	,	PUNCT
ejpam-4870	277	48	we	we	PRON
ejpam-4870	277	49	have	have	VERB
ejpam-4870	277	50	e(ec(4,2	e(ec(4,2	NOUN
ejpam-4870	277	51	)	)	PUNCT
ejpam-4870	277	52	)	)	PUNCT
ejpam-4870	278	1	=	=	X
ejpam-4870	278	2	{	{	PUNCT
ejpam-4870	278	3	[	[	X
ejpam-4870	278	4	{	{	PUNCT
ejpam-4870	278	5	12	12	NUM
ejpam-4870	278	6	,	,	PUNCT
ejpam-4870	278	7	23	23	NUM
ejpam-4870	278	8	}	}	PUNCT
ejpam-4870	278	9	,	,	PUNCT
ejpam-4870	278	10	{	{	PUNCT
ejpam-4870	278	11	12	12	NUM
ejpam-4870	278	12	,	,	PUNCT
ejpam-4870	278	13	34	34	NUM
ejpam-4870	278	14	}	}	PUNCT
ejpam-4870	278	15	]	]	PUNCT
ejpam-4870	278	16	,	,	PUNCT
ejpam-4870	278	17	[	[	X
ejpam-4870	278	18	{	{	PUNCT
ejpam-4870	278	19	12	12	NUM
ejpam-4870	278	20	,	,	PUNCT
ejpam-4870	278	21	23	23	NUM
ejpam-4870	278	22	}	}	PUNCT
ejpam-4870	278	23	,	,	PUNCT
ejpam-4870	278	24	{	{	PUNCT
ejpam-4870	278	25	12	12	NUM
ejpam-4870	278	26	,	,	PUNCT
ejpam-4870	278	27	41	41	NUM
ejpam-4870	278	28	}	}	PUNCT
ejpam-4870	278	29	]	]	PUNCT
ejpam-4870	278	30	,	,	PUNCT
ejpam-4870	278	31	[	[	X
ejpam-4870	278	32	{	{	PUNCT
ejpam-4870	278	33	12	12	NUM
ejpam-4870	278	34	,	,	PUNCT
ejpam-4870	278	35	23	23	NUM
ejpam-4870	278	36	}	}	PUNCT
ejpam-4870	278	37	,	,	PUNCT
ejpam-4870	278	38	{	{	PUNCT
ejpam-4870	278	39	23	23	NUM
ejpam-4870	278	40	,	,	PUNCT
ejpam-4870	278	41	34	34	NUM
ejpam-4870	278	42	}	}	PUNCT
ejpam-4870	278	43	]	]	PUNCT
ejpam-4870	278	44	,	,	PUNCT
ejpam-4870	278	45	[	[	X
ejpam-4870	278	46	{	{	PUNCT
ejpam-4870	278	47	12	12	NUM
ejpam-4870	278	48	,	,	PUNCT
ejpam-4870	278	49	23	23	NUM
ejpam-4870	278	50	}	}	PUNCT
ejpam-4870	278	51	,	,	PUNCT
ejpam-4870	278	52	{	{	PUNCT
ejpam-4870	278	53	23	23	NUM
ejpam-4870	278	54	,	,	PUNCT
ejpam-4870	278	55	41	41	NUM
ejpam-4870	278	56	}	}	PUNCT
ejpam-4870	278	57	]	]	PUNCT
ejpam-4870	278	58	,	,	PUNCT
ejpam-4870	278	59	[	[	X
ejpam-4870	278	60	{	{	PUNCT
ejpam-4870	278	61	12	12	NUM
ejpam-4870	278	62	,	,	PUNCT
ejpam-4870	278	63	34	34	NUM
ejpam-4870	278	64	}	}	PUNCT
ejpam-4870	278	65	,	,	PUNCT
ejpam-4870	278	66	{	{	PUNCT
ejpam-4870	278	67	12	12	NUM
ejpam-4870	278	68	,	,	PUNCT
ejpam-4870	278	69	41	41	NUM
ejpam-4870	278	70	}	}	PUNCT
ejpam-4870	278	71	]	]	PUNCT
ejpam-4870	278	72	,	,	PUNCT
ejpam-4870	278	73	[	[	X
ejpam-4870	278	74	{	{	PUNCT
ejpam-4870	278	75	12	12	NUM
ejpam-4870	278	76	,	,	PUNCT
ejpam-4870	278	77	34	34	NUM
ejpam-4870	278	78	}	}	PUNCT
ejpam-4870	278	79	,	,	PUNCT
ejpam-4870	278	80	{	{	PUNCT
ejpam-4870	278	81	23	23	NUM
ejpam-4870	278	82	,	,	PUNCT
ejpam-4870	278	83	34	34	NUM
ejpam-4870	278	84	}	}	PUNCT
ejpam-4870	278	85	]	]	PUNCT
ejpam-4870	278	86	,	,	PUNCT
ejpam-4870	278	87	[	[	X
ejpam-4870	278	88	{	{	PUNCT
ejpam-4870	278	89	12	12	NUM
ejpam-4870	278	90	,	,	PUNCT
ejpam-4870	278	91	34	34	NUM
ejpam-4870	278	92	}	}	PUNCT
ejpam-4870	278	93	,	,	PUNCT
ejpam-4870	278	94	{	{	PUNCT
ejpam-4870	278	95	34	34	NUM
ejpam-4870	278	96	,	,	PUNCT
ejpam-4870	278	97	41	41	NUM
ejpam-4870	278	98	}	}	PUNCT
ejpam-4870	278	99	]	]	PUNCT
ejpam-4870	278	100	,	,	PUNCT
ejpam-4870	278	101	[	[	X
ejpam-4870	278	102	{	{	PUNCT
ejpam-4870	278	103	12	12	NUM
ejpam-4870	278	104	,	,	PUNCT
ejpam-4870	278	105	41	41	NUM
ejpam-4870	278	106	}	}	PUNCT
ejpam-4870	278	107	,	,	PUNCT
ejpam-4870	278	108	{	{	PUNCT
ejpam-4870	278	109	23	23	NUM
ejpam-4870	278	110	,	,	PUNCT
ejpam-4870	278	111	41	41	NUM
ejpam-4870	278	112	}	}	PUNCT
ejpam-4870	278	113	]	]	PUNCT
ejpam-4870	278	114	,	,	PUNCT
ejpam-4870	278	115	[	[	X
ejpam-4870	278	116	{	{	PUNCT
ejpam-4870	278	117	12	12	NUM
ejpam-4870	278	118	,	,	PUNCT
ejpam-4870	278	119	41	41	NUM
ejpam-4870	278	120	}	}	PUNCT
ejpam-4870	278	121	,	,	PUNCT
ejpam-4870	278	122	{	{	PUNCT
ejpam-4870	278	123	34	34	NUM
ejpam-4870	278	124	,	,	PUNCT
ejpam-4870	278	125	41	41	NUM
ejpam-4870	278	126	}	}	PUNCT
ejpam-4870	278	127	]	]	PUNCT
ejpam-4870	278	128	,	,	PUNCT
ejpam-4870	278	129	[	[	X
ejpam-4870	278	130	{	{	PUNCT
ejpam-4870	278	131	23	23	NUM
ejpam-4870	278	132	,	,	PUNCT
ejpam-4870	278	133	34	34	NUM
ejpam-4870	278	134	}	}	PUNCT
ejpam-4870	278	135	,	,	PUNCT
ejpam-4870	278	136	{	{	PUNCT
ejpam-4870	278	137	23	23	NUM
ejpam-4870	278	138	,	,	PUNCT
ejpam-4870	278	139	41	41	NUM
ejpam-4870	278	140	}	}	PUNCT
ejpam-4870	278	141	]	]	PUNCT
ejpam-4870	278	142	,	,	PUNCT
ejpam-4870	278	143	[	[	X
ejpam-4870	278	144	{	{	PUNCT
ejpam-4870	278	145	23	23	NUM
ejpam-4870	278	146	,	,	PUNCT
ejpam-4870	278	147	34	34	NUM
ejpam-4870	278	148	}	}	PUNCT
ejpam-4870	278	149	,	,	PUNCT
ejpam-4870	278	150	{	{	PUNCT
ejpam-4870	278	151	34	34	NUM
ejpam-4870	278	152	,	,	PUNCT
ejpam-4870	278	153	41	41	NUM
ejpam-4870	278	154	}	}	PUNCT
ejpam-4870	278	155	]	]	PUNCT
ejpam-4870	278	156	,	,	PUNCT
ejpam-4870	278	157	[	[	X
ejpam-4870	278	158	{	{	PUNCT
ejpam-4870	278	159	23	23	NUM
ejpam-4870	278	160	,	,	PUNCT
ejpam-4870	278	161	41	41	NUM
ejpam-4870	278	162	}	}	PUNCT
ejpam-4870	278	163	,	,	PUNCT
ejpam-4870	278	164	{	{	PUNCT
ejpam-4870	278	165	34	34	NUM
ejpam-4870	278	166	,	,	PUNCT
ejpam-4870	278	167	41	41	NUM
ejpam-4870	278	168	}	}	PUNCT
ejpam-4870	278	169	]	]	PUNCT
ejpam-4870	278	170	}	}	PUNCT
ejpam-4870	278	171	.	.	PUNCT
ejpam-4870	279	1	it	it	PRON
ejpam-4870	279	2	can	can	AUX
ejpam-4870	279	3	be	be	AUX
ejpam-4870	279	4	noted	note	VERB
ejpam-4870	279	5	that	that	SCONJ
ejpam-4870	279	6	the	the	DET
ejpam-4870	279	7	order	order	NOUN
ejpam-4870	279	8	of	of	ADP
ejpam-4870	279	9	ec(4,2	ec(4,2	NOUN
ejpam-4870	279	10	)	)	PUNCT
ejpam-4870	279	11	is	be	AUX
ejpam-4870	279	12	6	6	NUM
ejpam-4870	279	13	and	and	CCONJ
ejpam-4870	279	14	its	its	PRON
ejpam-4870	279	15	size	size	NOUN
ejpam-4870	279	16	is	be	AUX
ejpam-4870	279	17	12	12	NUM
ejpam-4870	279	18	.	.	PUNCT
ejpam-4870	280	1	a	a	DET
ejpam-4870	280	2	pictorial	pictorial	ADJ
ejpam-4870	280	3	representation	representation	NOUN
ejpam-4870	280	4	of	of	ADP
ejpam-4870	280	5	ec(4,2	ec(4,2	NOUN
ejpam-4870	280	6	)	)	PUNCT
ejpam-4870	280	7	is	be	AUX
ejpam-4870	280	8	given	give	VERB
ejpam-4870	280	9	in	in	ADP
ejpam-4870	280	10	figure	figure	NOUN
ejpam-4870	280	11	16	16	NUM
ejpam-4870	280	12	.	.	PUNCT
ejpam-4870	281	1	{	{	PUNCT
ejpam-4870	281	2	34	34	NUM
ejpam-4870	281	3	,	,	PUNCT
ejpam-4870	281	4	41	41	NUM
ejpam-4870	281	5	}	}	PUNCT
ejpam-4870	281	6	{	{	PUNCT
ejpam-4870	281	7	12	12	NUM
ejpam-4870	281	8	,	,	PUNCT
ejpam-4870	281	9	23	23	NUM
ejpam-4870	281	10	}	}	PUNCT
ejpam-4870	281	11	{	{	PUNCT
ejpam-4870	281	12	12	12	NUM
ejpam-4870	281	13	,	,	PUNCT
ejpam-4870	281	14	34	34	NUM
ejpam-4870	281	15	}	}	PUNCT
ejpam-4870	281	16	{	{	PUNCT
ejpam-4870	281	17	12	12	NUM
ejpam-4870	281	18	,	,	PUNCT
ejpam-4870	281	19	41	41	NUM
ejpam-4870	281	20	}	}	PUNCT
ejpam-4870	281	21	{	{	PUNCT
ejpam-4870	281	22	23	23	NUM
ejpam-4870	281	23	,	,	PUNCT
ejpam-4870	281	24	34}{23	34}{23	NUM
ejpam-4870	281	25	,	,	PUNCT
ejpam-4870	281	26	41	41	NUM
ejpam-4870	281	27	}	}	PUNCT
ejpam-4870	281	28	figure	figure	NOUN
ejpam-4870	281	29	16	16	NUM
ejpam-4870	281	30	:	:	PUNCT
ejpam-4870	281	31	2	2	NUM
ejpam-4870	281	32	-	-	PUNCT
ejpam-4870	281	33	edge	edge	NOUN
ejpam-4870	281	34	intersection	intersection	NOUN
ejpam-4870	281	35	graph	graph	NOUN
ejpam-4870	281	36	of	of	ADP
ejpam-4870	281	37	c4	c4	NOUN
ejpam-4870	281	38	it	it	PRON
ejpam-4870	281	39	can	can	AUX
ejpam-4870	281	40	be	be	AUX
ejpam-4870	281	41	observed	observe	VERB
ejpam-4870	281	42	that	that	SCONJ
ejpam-4870	281	43	ec(n	ec(n	PROPN
ejpam-4870	281	44	,	,	PUNCT
ejpam-4870	281	45	j	j	NOUN
ejpam-4870	281	46	)	)	PUNCT
ejpam-4870	281	47	does	do	AUX
ejpam-4870	281	48	not	not	PART
ejpam-4870	281	49	contain	contain	VERB
ejpam-4870	281	50	any	any	DET
ejpam-4870	281	51	loop	loop	NOUN
ejpam-4870	281	52	.	.	PUNCT
ejpam-4870	282	1	moreover	moreover	ADV
ejpam-4870	282	2	,	,	PUNCT
ejpam-4870	282	3	since	since	SCONJ
ejpam-4870	282	4	v	v	NOUN
ejpam-4870	282	5	(	(	PUNCT
ejpam-4870	282	6	ec(n	ec(n	NUM
ejpam-4870	282	7	,	,	PUNCT
ejpam-4870	282	8	j	j	NOUN
ejpam-4870	282	9	)	)	PUNCT
ejpam-4870	282	10	)	)	PUNCT
ejpam-4870	283	1	j.c	j.c	PROPN
ejpam-4870	283	2	.	.	PROPN
ejpam-4870	283	3	bonifacio	bonifacio	PROPN
ejpam-4870	283	4	,	,	PUNCT
ejpam-4870	283	5	c.j	c.j	PROPN
ejpam-4870	283	6	.	.	PROPN
ejpam-4870	283	7	andaya	andaya	PROPN
ejpam-4870	283	8	,	,	PUNCT
ejpam-4870	283	9	d.	d.	PROPN
ejpam-4870	283	10	magpantay	magpantay	PROPN
ejpam-4870	283	11	/	/	SYM
ejpam-4870	283	12	eur	eur	PROPN
ejpam-4870	283	13	.	.	PUNCT
ejpam-4870	284	1	j.	j.	PROPN
ejpam-4870	284	2	pure	pure	PROPN
ejpam-4870	284	3	appl	appl	PROPN
ejpam-4870	284	4	.	.	PROPN
ejpam-4870	284	5	math	math	PROPN
ejpam-4870	284	6	,	,	PUNCT
ejpam-4870	284	7	16	16	NUM
ejpam-4870	284	8	(	(	PUNCT
ejpam-4870	284	9	4	4	NUM
ejpam-4870	284	10	)	)	PUNCT
ejpam-4870	284	11	(	(	PUNCT
ejpam-4870	284	12	2023	2023	NUM
ejpam-4870	284	13	)	)	PUNCT
ejpam-4870	284	14	,	,	PUNCT
ejpam-4870	284	15	2476	2476	NUM
ejpam-4870	284	16	-	-	SYM
ejpam-4870	284	17	2498	2498	NUM
ejpam-4870	284	18	2489	2489	NUM
ejpam-4870	284	19	is	be	AUX
ejpam-4870	284	20	the	the	DET
ejpam-4870	284	21	collection	collection	NOUN
ejpam-4870	284	22	of	of	ADP
ejpam-4870	284	23	all	all	DET
ejpam-4870	284	24	distinct	distinct	ADJ
ejpam-4870	284	25	spanning	span	VERB
ejpam-4870	284	26	subgraphs	subgraph	NOUN
ejpam-4870	284	27	of	of	ADP
ejpam-4870	284	28	cn	cn	PROPN
ejpam-4870	284	29	with	with	ADP
ejpam-4870	284	30	j	j	PROPN
ejpam-4870	284	31	edges	edge	NOUN
ejpam-4870	284	32	,	,	PUNCT
ejpam-4870	284	33	it	it	PRON
ejpam-4870	284	34	follows	follow	VERB
ejpam-4870	284	35	that	that	SCONJ
ejpam-4870	284	36	e(ec(n	e(ec(n	PROPN
ejpam-4870	284	37	,	,	PUNCT
ejpam-4870	284	38	j	j	PROPN
ejpam-4870	284	39	)	)	PUNCT
ejpam-4870	284	40	)	)	PUNCT
ejpam-4870	284	41	does	do	AUX
ejpam-4870	284	42	not	not	PART
ejpam-4870	284	43	have	have	VERB
ejpam-4870	284	44	the	the	DET
ejpam-4870	284	45	same	same	ADJ
ejpam-4870	284	46	pair	pair	NOUN
ejpam-4870	284	47	of	of	ADP
ejpam-4870	284	48	vertices	vertex	NOUN
ejpam-4870	284	49	which	which	PRON
ejpam-4870	284	50	means	mean	VERB
ejpam-4870	284	51	that	that	SCONJ
ejpam-4870	284	52	ec(n	ec(n	PROPN
ejpam-4870	284	53	,	,	PUNCT
ejpam-4870	284	54	j	j	PROPN
ejpam-4870	284	55	)	)	PUNCT
ejpam-4870	284	56	has	have	VERB
ejpam-4870	284	57	no	no	DET
ejpam-4870	284	58	multiple	multiple	ADJ
ejpam-4870	284	59	edges	edge	NOUN
ejpam-4870	284	60	.	.	PUNCT
ejpam-4870	285	1	equivalently	equivalently	ADV
ejpam-4870	285	2	,	,	PUNCT
ejpam-4870	285	3	the	the	DET
ejpam-4870	285	4	following	follow	VERB
ejpam-4870	285	5	remark	remark	NOUN
ejpam-4870	285	6	is	be	AUX
ejpam-4870	285	7	given	give	VERB
ejpam-4870	285	8	.	.	PUNCT
ejpam-4870	285	9	remark	remark	NOUN
ejpam-4870	285	10	2	2	NUM
ejpam-4870	285	11	.	.	PUNCT
ejpam-4870	286	1	a	a	DET
ejpam-4870	286	2	j	j	NOUN
ejpam-4870	286	3	-	-	PUNCT
ejpam-4870	286	4	edge	edge	NOUN
ejpam-4870	286	5	graph	graph	NOUN
ejpam-4870	286	6	of	of	ADP
ejpam-4870	286	7	cycle	cycle	NOUN
ejpam-4870	286	8	graph	graph	NOUN
ejpam-4870	286	9	ec(n	ec(n	PROPN
ejpam-4870	286	10	,	,	PUNCT
ejpam-4870	286	11	j	j	PROPN
ejpam-4870	286	12	)	)	PUNCT
ejpam-4870	286	13	is	be	AUX
ejpam-4870	286	14	a	a	DET
ejpam-4870	286	15	simple	simple	ADJ
ejpam-4870	286	16	graph	graph	NOUN
ejpam-4870	286	17	.	.	PUNCT
ejpam-4870	287	1	the	the	DET
ejpam-4870	287	2	first	first	ADJ
ejpam-4870	287	3	theorem	theorem	NOUN
ejpam-4870	287	4	determines	determine	VERB
ejpam-4870	287	5	the	the	DET
ejpam-4870	287	6	order	order	NOUN
ejpam-4870	287	7	of	of	ADP
ejpam-4870	287	8	ec(n	ec(n	PROPN
ejpam-4870	287	9	,	,	PUNCT
ejpam-4870	287	10	j	j	PROPN
ejpam-4870	287	11	)	)	PUNCT
ejpam-4870	287	12	.	.	PUNCT
ejpam-4870	288	1	theorem	theorem	NOUN
ejpam-4870	288	2	3	3	X
ejpam-4870	288	3	.	.	PUNCT
ejpam-4870	289	1	let	let	VERB
ejpam-4870	289	2	n	n	PRON
ejpam-4870	289	3	≥	≥	X
ejpam-4870	289	4	3	3	NUM
ejpam-4870	289	5	and	and	CCONJ
ejpam-4870	289	6	1	1	NUM
ejpam-4870	289	7	≤	≤	NUM
ejpam-4870	289	8	j	j	PROPN
ejpam-4870	289	9	≤	≤	PROPN
ejpam-4870	289	10	n.	n.	NOUN
ejpam-4870	289	11	if	if	SCONJ
ejpam-4870	289	12	ec(n	ec(n	PROPN
ejpam-4870	289	13	,	,	PUNCT
ejpam-4870	289	14	j	j	PROPN
ejpam-4870	289	15	)	)	PUNCT
ejpam-4870	289	16	is	be	AUX
ejpam-4870	289	17	the	the	DET
ejpam-4870	289	18	j	j	NOUN
ejpam-4870	289	19	-	-	PUNCT
ejpam-4870	289	20	edge	edge	NOUN
ejpam-4870	289	21	intersection	intersection	NOUN
ejpam-4870	289	22	graph	graph	NOUN
ejpam-4870	289	23	of	of	ADP
ejpam-4870	289	24	cn	cn	PROPN
ejpam-4870	289	25	.	.	PUNCT
ejpam-4870	290	1	then	then	ADV
ejpam-4870	290	2	the	the	DET
ejpam-4870	290	3	order	order	NOUN
ejpam-4870	290	4	of	of	ADP
ejpam-4870	290	5	ec(n	ec(n	PROPN
ejpam-4870	290	6	,	,	PUNCT
ejpam-4870	290	7	j	j	PROPN
ejpam-4870	290	8	)	)	PUNCT
ejpam-4870	290	9	is	be	AUX
ejpam-4870	290	10	(	(	PUNCT
ejpam-4870	290	11	n	n	X
ejpam-4870	290	12	j	j	PROPN
ejpam-4870	290	13	)	)	PUNCT
ejpam-4870	290	14	.	.	PUNCT
ejpam-4870	291	1	proof	proof	NOUN
ejpam-4870	291	2	.	.	PUNCT
ejpam-4870	292	1	by	by	ADP
ejpam-4870	292	2	definition	definition	NOUN
ejpam-4870	292	3	17	17	NUM
ejpam-4870	292	4	,	,	PUNCT
ejpam-4870	292	5	v	v	NUM
ejpam-4870	292	6	(	(	PUNCT
ejpam-4870	292	7	ec(n	ec(n	NUM
ejpam-4870	292	8	,	,	PUNCT
ejpam-4870	292	9	j	j	NOUN
ejpam-4870	292	10	)	)	PUNCT
ejpam-4870	292	11	)	)	PUNCT
ejpam-4870	292	12	contains	contain	VERB
ejpam-4870	292	13	the	the	DET
ejpam-4870	292	14	spanning	span	VERB
ejpam-4870	292	15	subgraphs	subgraph	NOUN
ejpam-4870	292	16	of	of	ADP
ejpam-4870	292	17	cn	cn	PROPN
ejpam-4870	292	18	with	with	ADP
ejpam-4870	292	19	exactly	exactly	ADV
ejpam-4870	292	20	j	j	PROPN
ejpam-4870	292	21	edges	edge	NOUN
ejpam-4870	292	22	.	.	PUNCT
ejpam-4870	293	1	by	by	ADP
ejpam-4870	293	2	theorem	theorem	NOUN
ejpam-4870	293	3	2	2	NUM
ejpam-4870	293	4	,	,	PUNCT
ejpam-4870	293	5	for	for	ADP
ejpam-4870	293	6	any	any	DET
ejpam-4870	293	7	graph	graph	NOUN
ejpam-4870	293	8	g	g	NOUN
ejpam-4870	293	9	of	of	ADP
ejpam-4870	293	10	size	size	NOUN
ejpam-4870	293	11	m	m	PROPN
ejpam-4870	293	12	,	,	PUNCT
ejpam-4870	293	13	there	there	PRON
ejpam-4870	293	14	are	be	VERB
ejpam-4870	293	15	(	(	PUNCT
ejpam-4870	293	16	m	m	PROPN
ejpam-4870	293	17	j	j	NOUN
ejpam-4870	293	18	)	)	PUNCT
ejpam-4870	293	19	spanning	span	VERB
ejpam-4870	293	20	subgraphs	subgraph	NOUN
ejpam-4870	293	21	with	with	ADP
ejpam-4870	293	22	exactly	exactly	ADV
ejpam-4870	293	23	j	j	PROPN
ejpam-4870	293	24	edges	edge	NOUN
ejpam-4870	293	25	.	.	PUNCT
ejpam-4870	294	1	since	since	SCONJ
ejpam-4870	294	2	the	the	DET
ejpam-4870	294	3	size	size	NOUN
ejpam-4870	294	4	of	of	ADP
ejpam-4870	294	5	cn	cn	PROPN
ejpam-4870	294	6	is	be	AUX
ejpam-4870	294	7	n	n	CCONJ
ejpam-4870	294	8	,	,	PUNCT
ejpam-4870	294	9	it	it	PRON
ejpam-4870	294	10	follows	follow	VERB
ejpam-4870	294	11	that	that	SCONJ
ejpam-4870	294	12	there	there	PRON
ejpam-4870	294	13	are	be	VERB
ejpam-4870	294	14	(	(	PUNCT
ejpam-4870	294	15	n	n	CCONJ
ejpam-4870	294	16	j	j	NOUN
ejpam-4870	294	17	)	)	PUNCT
ejpam-4870	294	18	spanning	span	VERB
ejpam-4870	294	19	subgraph	subgraph	NOUN
ejpam-4870	294	20	of	of	ADP
ejpam-4870	294	21	cn	cn	PROPN
ejpam-4870	294	22	.	.	PUNCT
ejpam-4870	295	1	therefore	therefore	ADV
ejpam-4870	295	2	,	,	PUNCT
ejpam-4870	295	3	|v	|v	PROPN
ejpam-4870	295	4	(	(	PUNCT
ejpam-4870	295	5	ec(n	ec(n	PROPN
ejpam-4870	295	6	,	,	PUNCT
ejpam-4870	295	7	j	j	NOUN
ejpam-4870	295	8	)	)	PUNCT
ejpam-4870	295	9	)	)	PUNCT
ejpam-4870	296	1	|	|	ADV
ejpam-4870	296	2	=	=	SYM
ejpam-4870	296	3	(	(	PUNCT
ejpam-4870	296	4	n	n	X
ejpam-4870	296	5	j	j	PROPN
ejpam-4870	296	6	)	)	PUNCT
ejpam-4870	296	7	.	.	PUNCT
ejpam-4870	297	1	note	note	VERB
ejpam-4870	297	2	that	that	SCONJ
ejpam-4870	297	3	(	(	PUNCT
ejpam-4870	297	4	n	n	X
ejpam-4870	297	5	j	j	PROPN
ejpam-4870	297	6	)	)	PUNCT
ejpam-4870	297	7	is	be	AUX
ejpam-4870	297	8	a	a	DET
ejpam-4870	297	9	positive	positive	ADJ
ejpam-4870	297	10	integer	integer	NOUN
ejpam-4870	297	11	.	.	PUNCT
ejpam-4870	298	1	this	this	PRON
ejpam-4870	298	2	means	mean	VERB
ejpam-4870	298	3	that	that	SCONJ
ejpam-4870	298	4	v	v	NOUN
ejpam-4870	298	5	(	(	PUNCT
ejpam-4870	298	6	ec(n	ec(n	PROPN
ejpam-4870	298	7	,	,	PUNCT
ejpam-4870	298	8	j	j	NOUN
ejpam-4870	298	9	)	)	PUNCT
ejpam-4870	298	10	)	)	PUNCT
ejpam-4870	298	11	is	be	AUX
ejpam-4870	298	12	always	always	ADV
ejpam-4870	298	13	nonempty	nonempty	ADJ
ejpam-4870	298	14	for	for	ADP
ejpam-4870	298	15	values	value	NOUN
ejpam-4870	298	16	of	of	ADP
ejpam-4870	298	17	n	n	PRON
ejpam-4870	298	18	≥	≥	NOUN
ejpam-4870	298	19	3	3	NUM
ejpam-4870	298	20	and	and	CCONJ
ejpam-4870	298	21	1	1	NUM
ejpam-4870	298	22	≤	≤	NUM
ejpam-4870	298	23	j	j	PROPN
ejpam-4870	298	24	≤	≤	PROPN
ejpam-4870	298	25	n.	n.	NOUN
ejpam-4870	298	26	illustration	illustration	NOUN
ejpam-4870	298	27	2	2	NUM
ejpam-4870	298	28	.	.	PUNCT
ejpam-4870	298	29	consider	consider	VERB
ejpam-4870	298	30	ec(4,2	ec(4,2	NOUN
ejpam-4870	298	31	)	)	PUNCT
ejpam-4870	298	32	with	with	ADP
ejpam-4870	298	33	the	the	DET
ejpam-4870	298	34	pictorial	pictorial	ADJ
ejpam-4870	298	35	illustration	illustration	NOUN
ejpam-4870	298	36	in	in	ADP
ejpam-4870	298	37	figure	figure	NOUN
ejpam-4870	298	38	16	16	NUM
ejpam-4870	298	39	.	.	PUNCT
ejpam-4870	299	1	it	it	PRON
ejpam-4870	299	2	can	can	AUX
ejpam-4870	299	3	be	be	AUX
ejpam-4870	299	4	noted	note	VERB
ejpam-4870	299	5	that	that	SCONJ
ejpam-4870	299	6	the	the	DET
ejpam-4870	299	7	order	order	NOUN
ejpam-4870	299	8	of	of	ADP
ejpam-4870	299	9	ec(4,2	ec(4,2	NOUN
ejpam-4870	299	10	)	)	PUNCT
ejpam-4870	299	11	is	be	AUX
ejpam-4870	299	12	6	6	NUM
ejpam-4870	299	13	.	.	PUNCT
ejpam-4870	299	14	using	use	VERB
ejpam-4870	299	15	theorem	theorem	NOUN
ejpam-4870	299	16	3	3	NUM
ejpam-4870	299	17	,	,	PUNCT
ejpam-4870	299	18	with	with	ADP
ejpam-4870	299	19	n	n	NOUN
ejpam-4870	299	20	=	=	SYM
ejpam-4870	299	21	4	4	NUM
ejpam-4870	299	22	and	and	CCONJ
ejpam-4870	299	23	j	j	PROPN
ejpam-4870	299	24	=	=	SYM
ejpam-4870	299	25	2	2	NUM
ejpam-4870	299	26	,	,	PUNCT
ejpam-4870	299	27	we	we	PRON
ejpam-4870	299	28	have	have	VERB
ejpam-4870	299	29	|v	|v	NOUN
ejpam-4870	299	30	(	(	PUNCT
ejpam-4870	299	31	ec(4,2	ec(4,2	NOUN
ejpam-4870	299	32	)	)	PUNCT
ejpam-4870	299	33	)	)	PUNCT
ejpam-4870	300	1	|	|	ADV
ejpam-4870	300	2	=	=	SYM
ejpam-4870	300	3	(	(	PUNCT
ejpam-4870	300	4	4	4	NUM
ejpam-4870	300	5	2	2	NUM
ejpam-4870	300	6	)	)	PUNCT
ejpam-4870	300	7	=	=	SYM
ejpam-4870	301	1	6	6	X
ejpam-4870	301	2	.	.	X
ejpam-4870	302	1	there	there	PRON
ejpam-4870	302	2	are	be	VERB
ejpam-4870	302	3	times	time	NOUN
ejpam-4870	302	4	that	that	SCONJ
ejpam-4870	302	5	ec(n	ec(n	PROPN
ejpam-4870	302	6	,	,	PUNCT
ejpam-4870	302	7	j	j	PROPN
ejpam-4870	302	8	)	)	PUNCT
ejpam-4870	302	9	contains	contain	VERB
ejpam-4870	302	10	only	only	ADV
ejpam-4870	302	11	one	one	NUM
ejpam-4870	302	12	vertex	vertex	NOUN
ejpam-4870	302	13	.	.	PUNCT
ejpam-4870	303	1	since	since	SCONJ
ejpam-4870	303	2	j	j	PROPN
ejpam-4870	303	3	≥	≥	NUM
ejpam-4870	303	4	1	1	NUM
ejpam-4870	303	5	and	and	CCONJ
ejpam-4870	303	6	using	use	VERB
ejpam-4870	303	7	theorem	theorem	NOUN
ejpam-4870	303	8	3	3	NUM
ejpam-4870	303	9	,	,	PUNCT
ejpam-4870	303	10	(	(	PUNCT
ejpam-4870	303	11	n	n	X
ejpam-4870	303	12	j	j	NOUN
ejpam-4870	303	13	)	)	PUNCT
ejpam-4870	304	1	=	=	PUNCT
ejpam-4870	304	2	1	1	NUM
ejpam-4870	304	3	if	if	SCONJ
ejpam-4870	304	4	and	and	CCONJ
ejpam-4870	304	5	only	only	ADV
ejpam-4870	304	6	if	if	SCONJ
ejpam-4870	304	7	j	j	PROPN
ejpam-4870	304	8	=	=	SYM
ejpam-4870	304	9	n.	n.	PROPN
ejpam-4870	304	10	theorem	theorem	VERB
ejpam-4870	304	11	4	4	NUM
ejpam-4870	304	12	discusses	discuss	VERB
ejpam-4870	304	13	a	a	DET
ejpam-4870	304	14	property	property	NOUN
ejpam-4870	304	15	of	of	ADP
ejpam-4870	304	16	ec(n	ec(n	PROPN
ejpam-4870	304	17	,	,	PUNCT
ejpam-4870	304	18	j	j	PROPN
ejpam-4870	304	19	)	)	PUNCT
ejpam-4870	304	20	when	when	SCONJ
ejpam-4870	304	21	j	j	PROPN
ejpam-4870	304	22	=	=	PROPN
ejpam-4870	304	23	n.	n.	PROPN
ejpam-4870	304	24	theorem	theorem	VERB
ejpam-4870	304	25	4	4	NUM
ejpam-4870	304	26	.	.	PUNCT
ejpam-4870	305	1	let	let	VERB
ejpam-4870	305	2	n	n	PRON
ejpam-4870	305	3	≥	≥	X
ejpam-4870	305	4	3	3	NUM
ejpam-4870	305	5	and	and	CCONJ
ejpam-4870	305	6	1	1	NUM
ejpam-4870	305	7	≤	≤	NUM
ejpam-4870	305	8	j	j	PROPN
ejpam-4870	305	9	≤	≤	PROPN
ejpam-4870	305	10	n.	n.	NOUN
ejpam-4870	305	11	if	if	SCONJ
ejpam-4870	305	12	ec(n	ec(n	PROPN
ejpam-4870	305	13	,	,	PUNCT
ejpam-4870	305	14	j	j	PROPN
ejpam-4870	305	15	)	)	PUNCT
ejpam-4870	305	16	is	be	AUX
ejpam-4870	305	17	the	the	DET
ejpam-4870	305	18	j	j	NOUN
ejpam-4870	305	19	-	-	PUNCT
ejpam-4870	305	20	edge	edge	NOUN
ejpam-4870	305	21	intersection	intersection	NOUN
ejpam-4870	305	22	graph	graph	NOUN
ejpam-4870	305	23	of	of	ADP
ejpam-4870	305	24	cn	cn	PROPN
ejpam-4870	305	25	.	.	PUNCT
ejpam-4870	306	1	then	then	ADV
ejpam-4870	306	2	ec(n	ec(n	NUM
ejpam-4870	306	3	,	,	PUNCT
ejpam-4870	306	4	j	j	PROPN
ejpam-4870	306	5	)	)	PUNCT
ejpam-4870	306	6	is	be	AUX
ejpam-4870	306	7	a	a	DET
ejpam-4870	306	8	trivial	trivial	ADJ
ejpam-4870	306	9	graph	graph	NOUN
ejpam-4870	306	10	if	if	SCONJ
ejpam-4870	307	1	and	and	CCONJ
ejpam-4870	307	2	only	only	ADV
ejpam-4870	307	3	if	if	SCONJ
ejpam-4870	307	4	j	j	PROPN
ejpam-4870	307	5	=	=	SYM
ejpam-4870	307	6	n.	n.	PROPN
ejpam-4870	307	7	proof	proof	NOUN
ejpam-4870	307	8	.	.	PUNCT
ejpam-4870	308	1	assume	assume	VERB
ejpam-4870	308	2	ec(n	ec(n	PROPN
ejpam-4870	308	3	,	,	PUNCT
ejpam-4870	308	4	j	j	PROPN
ejpam-4870	308	5	)	)	PUNCT
ejpam-4870	308	6	is	be	AUX
ejpam-4870	308	7	a	a	DET
ejpam-4870	308	8	trivial	trivial	ADJ
ejpam-4870	308	9	graph	graph	NOUN
ejpam-4870	308	10	.	.	PUNCT
ejpam-4870	309	1	by	by	ADP
ejpam-4870	309	2	definition	definition	NOUN
ejpam-4870	309	3	4	4	NUM
ejpam-4870	309	4	,	,	PUNCT
ejpam-4870	309	5	|v	|v	X
ejpam-4870	309	6	(	(	PUNCT
ejpam-4870	309	7	ec(n	ec(n	PROPN
ejpam-4870	309	8	,	,	PUNCT
ejpam-4870	309	9	j	j	NOUN
ejpam-4870	309	10	)	)	PUNCT
ejpam-4870	309	11	)	)	PUNCT
ejpam-4870	309	12	|	|	ADV
ejpam-4870	309	13	=	=	SYM
ejpam-4870	309	14	1	1	X
ejpam-4870	309	15	.	.	PUNCT
ejpam-4870	310	1	since	since	SCONJ
ejpam-4870	310	2	the	the	DET
ejpam-4870	310	3	order	order	NOUN
ejpam-4870	310	4	of	of	ADP
ejpam-4870	310	5	ec(n	ec(n	PROPN
ejpam-4870	310	6	,	,	PUNCT
ejpam-4870	310	7	j	j	NOUN
ejpam-4870	310	8	)	)	PUNCT
ejpam-4870	310	9	is	be	AUX
ejpam-4870	310	10	equal	equal	ADJ
ejpam-4870	310	11	to	to	ADP
ejpam-4870	310	12	(	(	PUNCT
ejpam-4870	310	13	n	n	X
ejpam-4870	310	14	j	j	PROPN
ejpam-4870	310	15	)	)	PUNCT
ejpam-4870	310	16	,	,	PUNCT
ejpam-4870	310	17	we	we	PRON
ejpam-4870	310	18	have	have	VERB
ejpam-4870	310	19	(	(	PUNCT
ejpam-4870	310	20	n	n	X
ejpam-4870	310	21	j	j	NOUN
ejpam-4870	310	22	)	)	PUNCT
ejpam-4870	311	1	=	=	PUNCT
ejpam-4870	311	2	1	1	X
ejpam-4870	311	3	.	.	PUNCT
ejpam-4870	311	4	hence	hence	ADV
ejpam-4870	311	5	,	,	PUNCT
ejpam-4870	311	6	by	by	ADP
ejpam-4870	311	7	theorem	theorem	NOUN
ejpam-4870	311	8	3	3	NUM
ejpam-4870	311	9	,	,	PUNCT
ejpam-4870	311	10	j	j	PROPN
ejpam-4870	311	11	is	be	AUX
ejpam-4870	311	12	equal	equal	ADJ
ejpam-4870	311	13	to	to	PART
ejpam-4870	311	14	n.	n.	VERB
ejpam-4870	311	15	conversely	conversely	ADV
ejpam-4870	311	16	,	,	PUNCT
ejpam-4870	311	17	assume	assume	VERB
ejpam-4870	311	18	that	that	SCONJ
ejpam-4870	311	19	j	j	PROPN
ejpam-4870	311	20	=	=	PUNCT
ejpam-4870	311	21	n.	n.	PROPN
ejpam-4870	311	22	if	if	SCONJ
ejpam-4870	311	23	j	j	PROPN
ejpam-4870	311	24	=	=	SYM
ejpam-4870	311	25	n	n	CCONJ
ejpam-4870	311	26	,	,	PUNCT
ejpam-4870	311	27	then	then	ADV
ejpam-4870	311	28	v	v	X
ejpam-4870	311	29	(	(	PUNCT
ejpam-4870	311	30	ec(n	ec(n	NUM
ejpam-4870	311	31	,	,	PUNCT
ejpam-4870	311	32	n	n	CCONJ
ejpam-4870	311	33	)	)	PUNCT
ejpam-4870	311	34	)	)	PUNCT
ejpam-4870	312	1	only	only	ADV
ejpam-4870	312	2	contains	contain	VERB
ejpam-4870	312	3	{	{	PUNCT
ejpam-4870	312	4	e1	e1	NOUN
ejpam-4870	312	5	,	,	PUNCT
ejpam-4870	312	6	e2	e2	PROPN
ejpam-4870	312	7	,	,	PUNCT
ejpam-4870	312	8	·	·	PUNCT
ejpam-4870	312	9	·	·	PUNCT
ejpam-4870	312	10	·	·	PUNCT
ejpam-4870	312	11	,	,	PUNCT
ejpam-4870	312	12	en	en	ADP
ejpam-4870	312	13	}	}	PUNCT
ejpam-4870	312	14	.	.	PUNCT
ejpam-4870	313	1	since	since	SCONJ
ejpam-4870	313	2	there	there	PRON
ejpam-4870	313	3	is	be	VERB
ejpam-4870	313	4	only	only	ADV
ejpam-4870	313	5	one	one	NUM
ejpam-4870	313	6	spanning	span	VERB
ejpam-4870	313	7	subgraph	subgraph	NOUN
ejpam-4870	313	8	of	of	ADP
ejpam-4870	313	9	cn	cn	PROPN
ejpam-4870	313	10	with	with	ADP
ejpam-4870	313	11	n	n	NOUN
ejpam-4870	313	12	edges	edge	NOUN
ejpam-4870	313	13	,	,	PUNCT
ejpam-4870	313	14	it	it	PRON
ejpam-4870	313	15	follows	follow	VERB
ejpam-4870	313	16	that	that	SCONJ
ejpam-4870	313	17	e(ec(n	e(ec(n	PROPN
ejpam-4870	313	18	,	,	PUNCT
ejpam-4870	313	19	n	n	CCONJ
ejpam-4870	313	20	)	)	PUNCT
ejpam-4870	313	21	)	)	PUNCT
ejpam-4870	313	22	is	be	AUX
ejpam-4870	313	23	empty	empty	ADJ
ejpam-4870	313	24	.	.	PUNCT
ejpam-4870	314	1	by	by	ADP
ejpam-4870	314	2	remark	remark	NOUN
ejpam-4870	314	3	2	2	NUM
ejpam-4870	314	4	,	,	PUNCT
ejpam-4870	314	5	[	[	X
ejpam-4870	314	6	{	{	PUNCT
ejpam-4870	314	7	e1	e1	PROPN
ejpam-4870	314	8	,	,	PUNCT
ejpam-4870	314	9	e2	e2	PROPN
ejpam-4870	314	10	,	,	PUNCT
ejpam-4870	314	11	·	·	PUNCT
ejpam-4870	314	12	·	·	PUNCT
ejpam-4870	314	13	·	·	PUNCT
ejpam-4870	314	14	,	,	PUNCT
ejpam-4870	314	15	en	en	X
ejpam-4870	314	16	}	}	PUNCT
ejpam-4870	314	17	,	,	PUNCT
ejpam-4870	314	18	{	{	PUNCT
ejpam-4870	314	19	e1	e1	PROPN
ejpam-4870	314	20	,	,	PUNCT
ejpam-4870	314	21	e2	e2	PROPN
ejpam-4870	314	22	,	,	PUNCT
ejpam-4870	314	23	·	·	PUNCT
ejpam-4870	314	24	·	·	PUNCT
ejpam-4870	314	25	·	·	PUNCT
ejpam-4870	314	26	,	,	PUNCT
ejpam-4870	314	27	en	en	X
ejpam-4870	314	28	}	}	PUNCT
ejpam-4870	314	29	]	]	PUNCT
ejpam-4870	314	30	/∈	/∈	PUNCT
ejpam-4870	315	1	e(ec(n	e(ec(n	PROPN
ejpam-4870	315	2	,	,	PUNCT
ejpam-4870	315	3	n	n	CCONJ
ejpam-4870	315	4	)	)	PUNCT
ejpam-4870	315	5	)	)	PUNCT
ejpam-4870	316	1	it	it	PRON
ejpam-4870	316	2	implies	imply	VERB
ejpam-4870	316	3	that	that	SCONJ
ejpam-4870	316	4	e(ec(n	e(ec(n	PROPN
ejpam-4870	316	5	,	,	PUNCT
ejpam-4870	316	6	n	n	CCONJ
ejpam-4870	316	7	)	)	PUNCT
ejpam-4870	316	8	)	)	PUNCT
ejpam-4870	317	1	=	=	NOUN
ejpam-4870	317	2	∅	∅	NOUN
ejpam-4870	317	3	.	.	PUNCT
ejpam-4870	318	1	therefore	therefore	ADV
ejpam-4870	318	2	,	,	PUNCT
ejpam-4870	318	3	ec(n	ec(n	NUM
ejpam-4870	318	4	,	,	PUNCT
ejpam-4870	318	5	n	n	CCONJ
ejpam-4870	318	6	)	)	PUNCT
ejpam-4870	318	7	is	be	AUX
ejpam-4870	318	8	a	a	DET
ejpam-4870	318	9	trivial	trivial	ADJ
ejpam-4870	318	10	graph	graph	NOUN
ejpam-4870	318	11	.	.	PUNCT
ejpam-4870	319	1	the	the	DET
ejpam-4870	319	2	illustration	illustration	NOUN
ejpam-4870	319	3	below	below	ADV
ejpam-4870	319	4	provides	provide	VERB
ejpam-4870	319	5	an	an	DET
ejpam-4870	319	6	example	example	NOUN
ejpam-4870	319	7	given	give	VERB
ejpam-4870	319	8	that	that	DET
ejpam-4870	319	9	n	n	NOUN
ejpam-4870	319	10	=	=	SYM
ejpam-4870	319	11	4	4	NUM
ejpam-4870	319	12	and	and	CCONJ
ejpam-4870	319	13	j	j	NOUN
ejpam-4870	319	14	=	=	NOUN
ejpam-4870	319	15	4	4	X
ejpam-4870	319	16	.	.	X
ejpam-4870	319	17	illustration	illustration	NOUN
ejpam-4870	319	18	3	3	NUM
ejpam-4870	319	19	.	.	X
ejpam-4870	319	20	consider	consider	VERB
ejpam-4870	319	21	cycle	cycle	NOUN
ejpam-4870	319	22	graph	graph	NOUN
ejpam-4870	319	23	c4	c4	NOUN
ejpam-4870	319	24	where	where	SCONJ
ejpam-4870	319	25	e(c4	e(c4	NOUN
ejpam-4870	319	26	)	)	PUNCT
ejpam-4870	320	1	=	=	PRON
ejpam-4870	320	2	{	{	PUNCT
ejpam-4870	320	3	12	12	NUM
ejpam-4870	320	4	,	,	PUNCT
ejpam-4870	320	5	23	23	NUM
ejpam-4870	320	6	,	,	PUNCT
ejpam-4870	320	7	34	34	NUM
ejpam-4870	320	8	,	,	PUNCT
ejpam-4870	320	9	41	41	NUM
ejpam-4870	320	10	}	}	PUNCT
ejpam-4870	320	11	.	.	PUNCT
ejpam-4870	321	1	the	the	DET
ejpam-4870	321	2	number	number	NOUN
ejpam-4870	321	3	of	of	ADP
ejpam-4870	321	4	spanning	span	VERB
ejpam-4870	321	5	subgraphs	subgraph	NOUN
ejpam-4870	321	6	of	of	ADP
ejpam-4870	321	7	c4	c4	NOUN
ejpam-4870	321	8	with	with	ADP
ejpam-4870	321	9	4	4	NUM
ejpam-4870	321	10	edges	edge	NOUN
ejpam-4870	321	11	is	be	AUX
ejpam-4870	321	12	(	(	PUNCT
ejpam-4870	321	13	4	4	NUM
ejpam-4870	321	14	4	4	NUM
ejpam-4870	321	15	)	)	PUNCT
ejpam-4870	321	16	=	=	SYM
ejpam-4870	321	17	1	1	X
ejpam-4870	321	18	,	,	PUNCT
ejpam-4870	321	19	by	by	ADP
ejpam-4870	321	20	theorem	theorem	NOUN
ejpam-4870	321	21	3	3	NUM
ejpam-4870	321	22	.	.	PUNCT
ejpam-4870	322	1	thus	thus	ADV
ejpam-4870	322	2	,	,	PUNCT
ejpam-4870	322	3	the	the	DET
ejpam-4870	322	4	order	order	NOUN
ejpam-4870	322	5	of	of	ADP
ejpam-4870	322	6	ec(4,4	ec(4,4	PROPN
ejpam-4870	322	7	)	)	PUNCT
ejpam-4870	322	8	is	be	AUX
ejpam-4870	322	9	1	1	NUM
ejpam-4870	322	10	.	.	PUNCT
ejpam-4870	322	11	figure	figure	NOUN
ejpam-4870	322	12	17	17	NUM
ejpam-4870	322	13	shows	show	VERB
ejpam-4870	322	14	the	the	DET
ejpam-4870	322	15	pictorial	pictorial	ADJ
ejpam-4870	322	16	representation	representation	NOUN
ejpam-4870	322	17	of	of	ADP
ejpam-4870	322	18	ec(4,4	ec(4,4	PROPN
ejpam-4870	322	19	)	)	PUNCT
ejpam-4870	322	20	.	.	PUNCT
ejpam-4870	323	1	{	{	PUNCT
ejpam-4870	323	2	12	12	NUM
ejpam-4870	323	3	,	,	PUNCT
ejpam-4870	323	4	23	23	NUM
ejpam-4870	323	5	,	,	PUNCT
ejpam-4870	323	6	34	34	NUM
ejpam-4870	323	7	,	,	PUNCT
ejpam-4870	323	8	41	41	NUM
ejpam-4870	323	9	}	}	PUNCT
ejpam-4870	323	10	figure	figure	NOUN
ejpam-4870	323	11	17	17	NUM
ejpam-4870	323	12	:	:	PUNCT
ejpam-4870	323	13	a	a	DET
ejpam-4870	323	14	pictorial	pictorial	ADJ
ejpam-4870	323	15	representation	representation	NOUN
ejpam-4870	323	16	of	of	ADP
ejpam-4870	323	17	ec(4,4	ec(4,4	PROPN
ejpam-4870	323	18	)	)	PUNCT
ejpam-4870	324	1	the	the	DET
ejpam-4870	324	2	next	next	ADJ
ejpam-4870	324	3	proposition	proposition	NOUN
ejpam-4870	324	4	discusses	discuss	VERB
ejpam-4870	324	5	when	when	SCONJ
ejpam-4870	324	6	ec(n	ec(n	NUM
ejpam-4870	324	7	,	,	PUNCT
ejpam-4870	324	8	j	j	PROPN
ejpam-4870	324	9	)	)	PUNCT
ejpam-4870	324	10	contains	contain	VERB
ejpam-4870	324	11	no	no	DET
ejpam-4870	324	12	edge	edge	NOUN
ejpam-4870	324	13	,	,	PUNCT
ejpam-4870	324	14	that	that	PRON
ejpam-4870	324	15	is	be	AUX
ejpam-4870	324	16	when	when	SCONJ
ejpam-4870	324	17	j	j	PROPN
ejpam-4870	324	18	=	=	PROPN
ejpam-4870	324	19	1	1	X
ejpam-4870	324	20	.	.	PUNCT
ejpam-4870	325	1	j.c	j.c	PROPN
ejpam-4870	325	2	.	.	PROPN
ejpam-4870	325	3	bonifacio	bonifacio	PROPN
ejpam-4870	325	4	,	,	PUNCT
ejpam-4870	325	5	c.j	c.j	PROPN
ejpam-4870	325	6	.	.	PROPN
ejpam-4870	325	7	andaya	andaya	PROPN
ejpam-4870	325	8	,	,	PUNCT
ejpam-4870	325	9	d.	d.	PROPN
ejpam-4870	325	10	magpantay	magpantay	PROPN
ejpam-4870	325	11	/	/	SYM
ejpam-4870	325	12	eur	eur	PROPN
ejpam-4870	325	13	.	.	PUNCT
ejpam-4870	326	1	j.	j.	PROPN
ejpam-4870	326	2	pure	pure	PROPN
ejpam-4870	326	3	appl	appl	PROPN
ejpam-4870	326	4	.	.	PROPN
ejpam-4870	326	5	math	math	PROPN
ejpam-4870	326	6	,	,	PUNCT
ejpam-4870	326	7	16	16	NUM
ejpam-4870	326	8	(	(	PUNCT
ejpam-4870	326	9	4	4	NUM
ejpam-4870	326	10	)	)	PUNCT
ejpam-4870	326	11	(	(	PUNCT
ejpam-4870	326	12	2023	2023	NUM
ejpam-4870	326	13	)	)	PUNCT
ejpam-4870	326	14	,	,	PUNCT
ejpam-4870	326	15	2476	2476	NUM
ejpam-4870	326	16	-	-	SYM
ejpam-4870	326	17	2498	2498	NUM
ejpam-4870	326	18	2490	2490	NUM
ejpam-4870	326	19	proposition	proposition	NOUN
ejpam-4870	326	20	1	1	NUM
ejpam-4870	326	21	.	.	PUNCT
ejpam-4870	327	1	let	let	VERB
ejpam-4870	327	2	n	n	PRON
ejpam-4870	327	3	≥	≥	X
ejpam-4870	327	4	3	3	NUM
ejpam-4870	327	5	and	and	CCONJ
ejpam-4870	327	6	1	1	NUM
ejpam-4870	327	7	≤	≤	NUM
ejpam-4870	327	8	j	j	PROPN
ejpam-4870	327	9	≤	≤	NOUN
ejpam-4870	327	10	n	n	NOUN
ejpam-4870	327	11	are	be	AUX
ejpam-4870	327	12	integers	integer	NOUN
ejpam-4870	327	13	.	.	PUNCT
ejpam-4870	328	1	if	if	SCONJ
ejpam-4870	328	2	ec(n	ec(n	NUM
ejpam-4870	328	3	,	,	PUNCT
ejpam-4870	328	4	j	j	PROPN
ejpam-4870	328	5	)	)	PUNCT
ejpam-4870	328	6	is	be	AUX
ejpam-4870	328	7	the	the	DET
ejpam-4870	328	8	j	j	NOUN
ejpam-4870	328	9	-	-	PUNCT
ejpam-4870	328	10	edge	edge	NOUN
ejpam-4870	328	11	intersection	intersection	NOUN
ejpam-4870	328	12	graph	graph	NOUN
ejpam-4870	328	13	of	of	ADP
ejpam-4870	328	14	cn	cn	PROPN
ejpam-4870	328	15	and	and	CCONJ
ejpam-4870	328	16	j	j	PROPN
ejpam-4870	328	17	=	=	SYM
ejpam-4870	328	18	1	1	NUM
ejpam-4870	328	19	,	,	PUNCT
ejpam-4870	328	20	then	then	ADV
ejpam-4870	328	21	ec(n	ec(n	NUM
ejpam-4870	328	22	,	,	PUNCT
ejpam-4870	328	23	j	j	PROPN
ejpam-4870	328	24	)	)	PUNCT
ejpam-4870	328	25	is	be	AUX
ejpam-4870	328	26	an	an	DET
ejpam-4870	328	27	empty	empty	ADJ
ejpam-4870	328	28	graph	graph	NOUN
ejpam-4870	328	29	of	of	ADP
ejpam-4870	328	30	order	order	NOUN
ejpam-4870	328	31	n.	n.	NOUN
ejpam-4870	328	32	proof	proof	NOUN
ejpam-4870	328	33	.	.	PUNCT
ejpam-4870	329	1	assume	assume	VERB
ejpam-4870	329	2	that	that	SCONJ
ejpam-4870	329	3	j	j	PROPN
ejpam-4870	329	4	=	=	NOUN
ejpam-4870	329	5	1	1	X
ejpam-4870	329	6	.	.	PUNCT
ejpam-4870	329	7	then	then	ADV
ejpam-4870	329	8	v	v	X
ejpam-4870	329	9	(	(	PUNCT
ejpam-4870	329	10	ec(n,1	ec(n,1	PROPN
ejpam-4870	329	11	)	)	PUNCT
ejpam-4870	329	12	)	)	PUNCT
ejpam-4870	330	1	=	=	PRON
ejpam-4870	330	2	{	{	PUNCT
ejpam-4870	330	3	{	{	PUNCT
ejpam-4870	330	4	e1	e1	PROPN
ejpam-4870	330	5	}	}	PUNCT
ejpam-4870	330	6	,	,	PUNCT
ejpam-4870	330	7	{	{	PUNCT
ejpam-4870	330	8	e2	e2	PROPN
ejpam-4870	330	9	}	}	PUNCT
ejpam-4870	330	10	,	,	PUNCT
ejpam-4870	330	11	·	·	PUNCT
ejpam-4870	330	12	·	·	PUNCT
ejpam-4870	330	13	·	·	PUNCT
ejpam-4870	330	14	,	,	PUNCT
ejpam-4870	330	15	{	{	PUNCT
ejpam-4870	330	16	en	en	X
ejpam-4870	330	17	}	}	PUNCT
ejpam-4870	330	18	}	}	PUNCT
ejpam-4870	330	19	.	.	PUNCT
ejpam-4870	331	1	now	now	ADV
ejpam-4870	331	2	,	,	PUNCT
ejpam-4870	331	3	observe	observe	VERB
ejpam-4870	331	4	that	that	SCONJ
ejpam-4870	331	5	ea∩eb	ea∩eb	PROPN
ejpam-4870	331	6	=	=	SYM
ejpam-4870	331	7	∅	∅	NOUN
ejpam-4870	331	8	for	for	ADP
ejpam-4870	331	9	all	all	DET
ejpam-4870	331	10	1	1	NUM
ejpam-4870	331	11	≤	≤	NOUN
ejpam-4870	331	12	a	a	DET
ejpam-4870	331	13	,	,	PUNCT
ejpam-4870	331	14	b	b	PROPN
ejpam-4870	331	15	≤	≤	PROPN
ejpam-4870	331	16	n.	n.	NOUN
ejpam-4870	331	17	it	it	PRON
ejpam-4870	331	18	means	mean	VERB
ejpam-4870	331	19	that	that	SCONJ
ejpam-4870	331	20	there	there	PRON
ejpam-4870	331	21	are	be	VERB
ejpam-4870	331	22	no	no	DET
ejpam-4870	331	23	adjacent	adjacent	ADJ
ejpam-4870	331	24	vertices	vertex	NOUN
ejpam-4870	331	25	in	in	ADP
ejpam-4870	331	26	ec(n,1	ec(n,1	NOUN
ejpam-4870	331	27	)	)	PUNCT
ejpam-4870	331	28	.	.	PUNCT
ejpam-4870	332	1	since	since	SCONJ
ejpam-4870	332	2	|v	|v	PROPN
ejpam-4870	332	3	(	(	PUNCT
ejpam-4870	332	4	ec(n,1	ec(n,1	PROPN
ejpam-4870	332	5	)	)	PUNCT
ejpam-4870	332	6	)	)	PUNCT
ejpam-4870	333	1	|	|	ADV
ejpam-4870	333	2	=	=	SYM
ejpam-4870	333	3	n	n	CCONJ
ejpam-4870	333	4	,	,	PUNCT
ejpam-4870	333	5	it	it	PRON
ejpam-4870	333	6	follows	follow	VERB
ejpam-4870	333	7	that	that	SCONJ
ejpam-4870	333	8	ec(n,1	ec(n,1	PROPN
ejpam-4870	333	9	)	)	PUNCT
ejpam-4870	333	10	is	be	AUX
ejpam-4870	333	11	an	an	DET
ejpam-4870	333	12	empty	empty	ADJ
ejpam-4870	333	13	graph	graph	NOUN
ejpam-4870	333	14	of	of	ADP
ejpam-4870	333	15	order	order	NOUN
ejpam-4870	333	16	n.	n.	NOUN
ejpam-4870	333	17	illustration	illustration	NOUN
ejpam-4870	333	18	4	4	NUM
ejpam-4870	333	19	shows	show	VERB
ejpam-4870	333	20	that	that	SCONJ
ejpam-4870	333	21	ec(n,1	ec(n,1	PROPN
ejpam-4870	333	22	)	)	PUNCT
ejpam-4870	333	23	is	be	AUX
ejpam-4870	333	24	an	an	DET
ejpam-4870	333	25	empty	empty	ADJ
ejpam-4870	333	26	graph	graph	NOUN
ejpam-4870	333	27	when	when	SCONJ
ejpam-4870	333	28	j	j	PROPN
ejpam-4870	333	29	=	=	NOUN
ejpam-4870	333	30	1	1	PROPN
ejpam-4870	333	31	.	.	X
ejpam-4870	333	32	illustration	illustration	NOUN
ejpam-4870	333	33	4	4	NUM
ejpam-4870	333	34	.	.	PUNCT
ejpam-4870	333	35	consider	consider	VERB
ejpam-4870	333	36	the	the	DET
ejpam-4870	333	37	cycle	cycle	NOUN
ejpam-4870	333	38	graph	graph	NOUN
ejpam-4870	333	39	c3	c3	PROPN
ejpam-4870	333	40	where	where	SCONJ
ejpam-4870	333	41	e(c3	e(c3	NOUN
ejpam-4870	333	42	)	)	PUNCT
ejpam-4870	333	43	=	=	PRON
ejpam-4870	333	44	{	{	PUNCT
ejpam-4870	333	45	12	12	NUM
ejpam-4870	333	46	,	,	PUNCT
ejpam-4870	333	47	23	23	NUM
ejpam-4870	333	48	,	,	PUNCT
ejpam-4870	333	49	31	31	NUM
ejpam-4870	333	50	}	}	PUNCT
ejpam-4870	333	51	and	and	CCONJ
ejpam-4870	333	52	let	let	VERB
ejpam-4870	333	53	j	j	PROPN
ejpam-4870	333	54	=	=	NOUN
ejpam-4870	333	55	1	1	X
ejpam-4870	333	56	.	.	PUNCT
ejpam-4870	334	1	then	then	ADV
ejpam-4870	334	2	we	we	PRON
ejpam-4870	334	3	have	have	VERB
ejpam-4870	334	4	v	v	NOUN
ejpam-4870	334	5	(	(	PUNCT
ejpam-4870	334	6	ec(3,1	ec(3,1	PROPN
ejpam-4870	334	7	)	)	PUNCT
ejpam-4870	334	8	)	)	PUNCT
ejpam-4870	335	1	=	=	PRON
ejpam-4870	335	2	{	{	PUNCT
ejpam-4870	335	3	{	{	PUNCT
ejpam-4870	335	4	12	12	NUM
ejpam-4870	335	5	}	}	PUNCT
ejpam-4870	335	6	,	,	PUNCT
ejpam-4870	335	7	{	{	PUNCT
ejpam-4870	335	8	23	23	NUM
ejpam-4870	335	9	}	}	PUNCT
ejpam-4870	335	10	,	,	PUNCT
ejpam-4870	335	11	{	{	PUNCT
ejpam-4870	335	12	31	31	NUM
ejpam-4870	335	13	}	}	PUNCT
ejpam-4870	335	14	}	}	PUNCT
ejpam-4870	335	15	.	.	PUNCT
ejpam-4870	336	1	it	it	PRON
ejpam-4870	336	2	can	can	AUX
ejpam-4870	336	3	be	be	AUX
ejpam-4870	336	4	observed	observe	VERB
ejpam-4870	336	5	that	that	SCONJ
ejpam-4870	336	6	the	the	DET
ejpam-4870	336	7	spanning	span	VERB
ejpam-4870	336	8	subgraphs	subgraph	NOUN
ejpam-4870	336	9	have	have	VERB
ejpam-4870	336	10	no	no	DET
ejpam-4870	336	11	common	common	ADJ
ejpam-4870	336	12	edge	edge	NOUN
ejpam-4870	336	13	.	.	PUNCT
ejpam-4870	337	1	thus	thus	ADV
ejpam-4870	337	2	,	,	PUNCT
ejpam-4870	337	3	e(ec(3,1	e(ec(3,1	NOUN
ejpam-4870	337	4	)	)	PUNCT
ejpam-4870	337	5	)	)	PUNCT
ejpam-4870	338	1	=	=	PUNCT
ejpam-4870	338	2	∅.	∅.	NOUN
ejpam-4870	338	3	it	it	PRON
ejpam-4870	338	4	follows	follow	VERB
ejpam-4870	338	5	that	that	SCONJ
ejpam-4870	338	6	ec(3,1	ec(3,1	NOUN
ejpam-4870	338	7	)	)	PUNCT
ejpam-4870	338	8	is	be	AUX
ejpam-4870	338	9	an	an	DET
ejpam-4870	338	10	empty	empty	ADJ
ejpam-4870	338	11	graph	graph	NOUN
ejpam-4870	338	12	of	of	ADP
ejpam-4870	338	13	order	order	NOUN
ejpam-4870	338	14	3	3	NUM
ejpam-4870	338	15	shown	show	VERB
ejpam-4870	338	16	in	in	ADP
ejpam-4870	338	17	figure	figure	NOUN
ejpam-4870	338	18	18	18	NUM
ejpam-4870	338	19	.	.	PUNCT
ejpam-4870	339	1	{	{	PUNCT
ejpam-4870	339	2	31	31	NUM
ejpam-4870	339	3	}	}	PUNCT
ejpam-4870	339	4	{	{	PUNCT
ejpam-4870	339	5	12	12	NUM
ejpam-4870	339	6	}	}	PUNCT
ejpam-4870	339	7	{	{	PUNCT
ejpam-4870	339	8	23	23	NUM
ejpam-4870	339	9	}	}	PUNCT
ejpam-4870	339	10	figure	figure	NOUN
ejpam-4870	339	11	18	18	NUM
ejpam-4870	339	12	:	:	PUNCT
ejpam-4870	339	13	an	an	DET
ejpam-4870	339	14	empty	empty	ADJ
ejpam-4870	339	15	graph	graph	NOUN
ejpam-4870	339	16	ec(3,1	ec(3,1	NOUN
ejpam-4870	339	17	)	)	PUNCT
ejpam-4870	339	18	in	in	ADP
ejpam-4870	339	19	figure	figure	NOUN
ejpam-4870	339	20	18	18	NUM
ejpam-4870	339	21	,	,	PUNCT
ejpam-4870	339	22	it	it	PRON
ejpam-4870	339	23	can	can	AUX
ejpam-4870	339	24	be	be	AUX
ejpam-4870	339	25	observed	observe	VERB
ejpam-4870	339	26	that	that	SCONJ
ejpam-4870	339	27	the	the	DET
ejpam-4870	339	28	vertices	vertex	NOUN
ejpam-4870	339	29	are	be	AUX
ejpam-4870	339	30	not	not	PART
ejpam-4870	339	31	adjacent	adjacent	ADJ
ejpam-4870	339	32	to	to	ADP
ejpam-4870	339	33	each	each	DET
ejpam-4870	339	34	other	other	ADJ
ejpam-4870	339	35	,	,	PUNCT
ejpam-4870	339	36	thus	thus	ADV
ejpam-4870	339	37	the	the	DET
ejpam-4870	339	38	degree	degree	NOUN
ejpam-4870	339	39	of	of	ADP
ejpam-4870	339	40	every	every	DET
ejpam-4870	339	41	vertex	vertex	NOUN
ejpam-4870	339	42	in	in	ADP
ejpam-4870	339	43	ec(n	ec(n	PROPN
ejpam-4870	339	44	,	,	PUNCT
ejpam-4870	339	45	j	j	PROPN
ejpam-4870	339	46	)	)	PUNCT
ejpam-4870	339	47	when	when	SCONJ
ejpam-4870	339	48	j	j	PROPN
ejpam-4870	339	49	=	=	NOUN
ejpam-4870	339	50	1	1	NUM
ejpam-4870	339	51	is	be	AUX
ejpam-4870	339	52	equal	equal	ADJ
ejpam-4870	339	53	to	to	ADP
ejpam-4870	339	54	zero	zero	NUM
ejpam-4870	339	55	.	.	PUNCT
ejpam-4870	340	1	note	note	VERB
ejpam-4870	340	2	that	that	SCONJ
ejpam-4870	340	3	the	the	DET
ejpam-4870	340	4	degree	degree	NOUN
ejpam-4870	340	5	of	of	ADP
ejpam-4870	340	6	every	every	DET
ejpam-4870	340	7	vertex	vertex	NOUN
ejpam-4870	340	8	in	in	ADP
ejpam-4870	340	9	ec(n	ec(n	PROPN
ejpam-4870	340	10	,	,	PUNCT
ejpam-4870	340	11	j	j	PROPN
ejpam-4870	340	12	)	)	PUNCT
ejpam-4870	340	13	depends	depend	VERB
ejpam-4870	340	14	on	on	ADP
ejpam-4870	340	15	the	the	DET
ejpam-4870	340	16	value	value	NOUN
ejpam-4870	340	17	of	of	ADP
ejpam-4870	340	18	the	the	DET
ejpam-4870	340	19	nonnegative	nonnegative	ADJ
ejpam-4870	340	20	integer	integer	NOUN
ejpam-4870	340	21	j.	j.	PROPN
ejpam-4870	340	22	lemma	lemma	PROPN
ejpam-4870	340	23	1	1	NUM
ejpam-4870	340	24	shows	show	VERB
ejpam-4870	340	25	that	that	SCONJ
ejpam-4870	340	26	for	for	ADP
ejpam-4870	340	27	any	any	DET
ejpam-4870	340	28	two	two	NUM
ejpam-4870	340	29	distinct	distinct	ADJ
ejpam-4870	340	30	spanning	span	VERB
ejpam-4870	340	31	subgraphs	subgraph	NOUN
ejpam-4870	340	32	of	of	ADP
ejpam-4870	340	33	cn	cn	PROPN
ejpam-4870	340	34	with	with	ADP
ejpam-4870	340	35	j	j	PROPN
ejpam-4870	340	36	edges	edge	NOUN
ejpam-4870	340	37	,	,	PUNCT
ejpam-4870	340	38	their	their	PRON
ejpam-4870	340	39	common	common	ADJ
ejpam-4870	340	40	edge	edge	NOUN
ejpam-4870	340	41	is	be	AUX
ejpam-4870	340	42	always	always	ADV
ejpam-4870	340	43	greater	great	ADJ
ejpam-4870	340	44	than	than	ADP
ejpam-4870	340	45	1	1	NUM
ejpam-4870	340	46	that	that	PRON
ejpam-4870	340	47	is	be	AUX
ejpam-4870	340	48	when	when	SCONJ
ejpam-4870	340	49	⌈n2	⌈n2	NOUN
ejpam-4870	340	50	⌉	⌉	ADP
ejpam-4870	340	51	<	<	X
ejpam-4870	340	52	j	j	PROPN
ejpam-4870	340	53	≤	≤	PROPN
ejpam-4870	340	54	n.	n.	PROPN
ejpam-4870	340	55	lemma	lemma	PROPN
ejpam-4870	341	1	1	1	X
ejpam-4870	341	2	.	.	PUNCT
ejpam-4870	342	1	if	if	SCONJ
ejpam-4870	342	2	⌈n2	⌈n2	NOUN
ejpam-4870	342	3	⌉	⌉	ADP
ejpam-4870	342	4	<	<	X
ejpam-4870	342	5	j	j	PROPN
ejpam-4870	342	6	≤	≤	NUM
ejpam-4870	342	7	n	n	CCONJ
ejpam-4870	342	8	,	,	PUNCT
ejpam-4870	342	9	then	then	ADV
ejpam-4870	342	10	for	for	ADP
ejpam-4870	342	11	all	all	DET
ejpam-4870	342	12	a	a	DET
ejpam-4870	342	13	,	,	PUNCT
ejpam-4870	342	14	b	b	PROPN
ejpam-4870	342	15	∈	∈	PROPN
ejpam-4870	342	16	v	v	NOUN
ejpam-4870	342	17	(	(	PUNCT
ejpam-4870	342	18	ec(n	ec(n	PROPN
ejpam-4870	342	19	,	,	PUNCT
ejpam-4870	342	20	j	j	NOUN
ejpam-4870	342	21	)	)	PUNCT
ejpam-4870	342	22	)	)	PUNCT
ejpam-4870	342	23	,	,	PUNCT
ejpam-4870	342	24	|a	|a	PRON
ejpam-4870	342	25	∩b|	∩b|	PROPN
ejpam-4870	342	26	>	>	X
ejpam-4870	342	27	1	1	NUM
ejpam-4870	342	28	.	.	PUNCT
ejpam-4870	342	29	proof	proof	NOUN
ejpam-4870	342	30	.	.	PUNCT
ejpam-4870	343	1	suppose	suppose	VERB
ejpam-4870	343	2	|a	|a	PRON
ejpam-4870	343	3	∩b|	∩b|	PROPN
ejpam-4870	343	4	≯	≯	NOUN
ejpam-4870	343	5	1	1	NUM
ejpam-4870	343	6	.	.	PUNCT
ejpam-4870	343	7	then	then	ADV
ejpam-4870	343	8	|a	|a	PRON
ejpam-4870	343	9	∩b|	∩b|	PROPN
ejpam-4870	343	10	is	be	AUX
ejpam-4870	343	11	either	either	CCONJ
ejpam-4870	343	12	0	0	NUM
ejpam-4870	343	13	or	or	CCONJ
ejpam-4870	343	14	1	1	NUM
ejpam-4870	343	15	.	.	X
ejpam-4870	343	16	consider	consider	VERB
ejpam-4870	343	17	two	two	NUM
ejpam-4870	343	18	cases	case	NOUN
ejpam-4870	343	19	:	:	PUNCT
ejpam-4870	343	20	case	case	NOUN
ejpam-4870	343	21	1	1	NUM
ejpam-4870	343	22	:	:	PUNCT
ejpam-4870	343	23	if	if	SCONJ
ejpam-4870	343	24	|a	|a	ADP
ejpam-4870	343	25	∩	∩	NOUN
ejpam-4870	343	26	b|	b|	PROPN
ejpam-4870	343	27	=	=	SYM
ejpam-4870	343	28	0	0	NUM
ejpam-4870	343	29	,	,	PUNCT
ejpam-4870	343	30	then	then	ADV
ejpam-4870	343	31	a	a	PRON
ejpam-4870	343	32	and	and	CCONJ
ejpam-4870	343	33	b	b	NOUN
ejpam-4870	343	34	are	be	AUX
ejpam-4870	343	35	disjoint	disjoint	ADJ
ejpam-4870	343	36	.	.	PUNCT
ejpam-4870	344	1	now	now	ADV
ejpam-4870	344	2	,	,	PUNCT
ejpam-4870	344	3	if	if	SCONJ
ejpam-4870	344	4	n	n	PRON
ejpam-4870	344	5	is	be	AUX
ejpam-4870	344	6	even	even	ADV
ejpam-4870	344	7	,	,	PUNCT
ejpam-4870	344	8	by	by	ADP
ejpam-4870	344	9	the	the	DET
ejpam-4870	344	10	definition	definition	NOUN
ejpam-4870	344	11	of	of	ADP
ejpam-4870	344	12	ceiling	ceiling	NOUN
ejpam-4870	344	13	function	function	NOUN
ejpam-4870	344	14	,	,	PUNCT
ejpam-4870	344	15	let	let	VERB
ejpam-4870	344	16	j	j	PROPN
ejpam-4870	344	17	=	=	VERB
ejpam-4870	344	18	⌈n2	⌈n2	NOUN
ejpam-4870	344	19	⌉+	⌉+	NOUN
ejpam-4870	344	20	1	1	NUM
ejpam-4870	344	21	=	=	SYM
ejpam-4870	344	22	n	n	PRON
ejpam-4870	344	23	2	2	NUM
ejpam-4870	345	1	+	+	SYM
ejpam-4870	345	2	1	1	NUM
ejpam-4870	345	3	where	where	SCONJ
ejpam-4870	345	4	it	it	PRON
ejpam-4870	345	5	is	be	AUX
ejpam-4870	345	6	the	the	DET
ejpam-4870	345	7	minimum	minimum	ADJ
ejpam-4870	345	8	value	value	NOUN
ejpam-4870	345	9	of	of	ADP
ejpam-4870	345	10	j.	j.	PROPN
ejpam-4870	345	11	then	then	ADV
ejpam-4870	345	12	|a	|a	VERB
ejpam-4870	345	13	∪b|	∪b|	PROPN
ejpam-4870	345	14	=	=	SYM
ejpam-4870	345	15	(	(	PUNCT
ejpam-4870	345	16	n	n	ADV
ejpam-4870	345	17	2	2	NUM
ejpam-4870	345	18	+	+	NUM
ejpam-4870	345	19	1	1	NUM
ejpam-4870	345	20	)	)	PUNCT
ejpam-4870	345	21	+	+	CCONJ
ejpam-4870	345	22	(	(	PUNCT
ejpam-4870	345	23	n	n	ADV
ejpam-4870	345	24	2	2	NUM
ejpam-4870	345	25	+	+	NUM
ejpam-4870	345	26	1	1	NUM
ejpam-4870	345	27	)	)	PUNCT
ejpam-4870	345	28	=	=	PUNCT
ejpam-4870	345	29	n+	n+	PUNCT
ejpam-4870	345	30	2	2	X
ejpam-4870	345	31	.	.	X
ejpam-4870	346	1	this	this	PRON
ejpam-4870	346	2	is	be	AUX
ejpam-4870	346	3	a	a	DET
ejpam-4870	346	4	contradiction	contradiction	NOUN
ejpam-4870	346	5	to	to	ADP
ejpam-4870	346	6	the	the	DET
ejpam-4870	346	7	fact	fact	NOUN
ejpam-4870	346	8	that	that	SCONJ
ejpam-4870	346	9	j	j	PROPN
ejpam-4870	346	10	≤	≤	PROPN
ejpam-4870	346	11	n.	n.	NOUN
ejpam-4870	346	12	moreover	moreover	ADV
ejpam-4870	346	13	,	,	PUNCT
ejpam-4870	346	14	if	if	SCONJ
ejpam-4870	346	15	n	n	PRON
ejpam-4870	346	16	is	be	AUX
ejpam-4870	346	17	odd	odd	ADJ
ejpam-4870	346	18	,	,	PUNCT
ejpam-4870	346	19	let	let	VERB
ejpam-4870	346	20	j	j	PROPN
ejpam-4870	346	21	=	=	PUNCT
ejpam-4870	346	22	⌈n2	⌈n2	NOUN
ejpam-4870	346	23	⌉	⌉	ADJ
ejpam-4870	346	24	+	+	CCONJ
ejpam-4870	346	25	1	1	NUM
ejpam-4870	346	26	=	=	SYM
ejpam-4870	346	27	n+1	n+1	NUM
ejpam-4870	346	28	2	2	NUM
ejpam-4870	346	29	+	+	NUM
ejpam-4870	346	30	1	1	NUM
ejpam-4870	346	31	.	.	PUNCT
ejpam-4870	346	32	then	then	ADV
ejpam-4870	346	33	|a	|a	VERB
ejpam-4870	346	34	∪b|	∪b|	PROPN
ejpam-4870	346	35	=	=	PRON
ejpam-4870	346	36	(	(	PUNCT
ejpam-4870	346	37	n+	n+	NUM
ejpam-4870	346	38	1	1	NUM
ejpam-4870	346	39	2	2	NUM
ejpam-4870	346	40	+	+	NUM
ejpam-4870	346	41	1	1	NUM
ejpam-4870	346	42	)	)	PUNCT
ejpam-4870	346	43	+	+	CCONJ
ejpam-4870	346	44	(	(	PUNCT
ejpam-4870	346	45	n+	n+	NUM
ejpam-4870	346	46	1	1	NUM
ejpam-4870	346	47	2	2	NUM
ejpam-4870	346	48	+	+	NUM
ejpam-4870	346	49	1	1	NUM
ejpam-4870	346	50	)	)	PUNCT
ejpam-4870	346	51	=	=	PUNCT
ejpam-4870	346	52	n+	n+	PUNCT
ejpam-4870	347	1	3	3	X
ejpam-4870	347	2	.	.	PUNCT
ejpam-4870	348	1	this	this	PRON
ejpam-4870	348	2	is	be	AUX
ejpam-4870	348	3	also	also	ADV
ejpam-4870	348	4	a	a	DET
ejpam-4870	348	5	contradiction	contradiction	NOUN
ejpam-4870	348	6	to	to	ADP
ejpam-4870	348	7	the	the	DET
ejpam-4870	348	8	fact	fact	NOUN
ejpam-4870	348	9	that	that	SCONJ
ejpam-4870	348	10	j	j	PROPN
ejpam-4870	348	11	≤	≤	PROPN
ejpam-4870	348	12	n.	n.	PROPN
ejpam-4870	348	13	j.c	j.c	PROPN
ejpam-4870	348	14	.	.	PROPN
ejpam-4870	348	15	bonifacio	bonifacio	PROPN
ejpam-4870	348	16	,	,	PUNCT
ejpam-4870	348	17	c.j	c.j	PROPN
ejpam-4870	348	18	.	.	PROPN
ejpam-4870	348	19	andaya	andaya	PROPN
ejpam-4870	348	20	,	,	PUNCT
ejpam-4870	348	21	d.	d.	PROPN
ejpam-4870	348	22	magpantay	magpantay	PROPN
ejpam-4870	348	23	/	/	SYM
ejpam-4870	348	24	eur	eur	PROPN
ejpam-4870	348	25	.	.	PUNCT
ejpam-4870	349	1	j.	j.	PROPN
ejpam-4870	349	2	pure	pure	PROPN
ejpam-4870	349	3	appl	appl	PROPN
ejpam-4870	349	4	.	.	PROPN
ejpam-4870	349	5	math	math	PROPN
ejpam-4870	349	6	,	,	PUNCT
ejpam-4870	349	7	16	16	NUM
ejpam-4870	349	8	(	(	PUNCT
ejpam-4870	349	9	4	4	NUM
ejpam-4870	349	10	)	)	PUNCT
ejpam-4870	349	11	(	(	PUNCT
ejpam-4870	349	12	2023	2023	NUM
ejpam-4870	349	13	)	)	PUNCT
ejpam-4870	349	14	,	,	PUNCT
ejpam-4870	349	15	2476	2476	NUM
ejpam-4870	349	16	-	-	SYM
ejpam-4870	349	17	2498	2498	NUM
ejpam-4870	349	18	2491	2491	NUM
ejpam-4870	349	19	case	case	NOUN
ejpam-4870	349	20	2	2	NUM
ejpam-4870	349	21	:	:	PUNCT
ejpam-4870	349	22	if	if	SCONJ
ejpam-4870	349	23	|a	|a	VERB
ejpam-4870	349	24	∩	∩	NOUN
ejpam-4870	349	25	b|	b|	PROPN
ejpam-4870	349	26	=	=	SYM
ejpam-4870	349	27	1	1	NUM
ejpam-4870	349	28	,	,	PUNCT
ejpam-4870	349	29	then	then	ADV
ejpam-4870	349	30	there	there	PRON
ejpam-4870	349	31	is	be	VERB
ejpam-4870	349	32	exactly	exactly	ADV
ejpam-4870	349	33	one	one	NUM
ejpam-4870	349	34	common	common	ADJ
ejpam-4870	349	35	element	element	NOUN
ejpam-4870	349	36	.	.	PUNCT
ejpam-4870	350	1	now	now	ADV
ejpam-4870	350	2	,	,	PUNCT
ejpam-4870	350	3	if	if	SCONJ
ejpam-4870	350	4	n	n	PRON
ejpam-4870	350	5	is	be	AUX
ejpam-4870	350	6	even	even	ADV
ejpam-4870	350	7	,	,	PUNCT
ejpam-4870	350	8	by	by	ADP
ejpam-4870	350	9	definition	definition	NOUN
ejpam-4870	350	10	?	?	PUNCT
ejpam-4870	350	11	?	?	PUNCT
ejpam-4870	350	12	,	,	PUNCT
ejpam-4870	350	13	let	let	VERB
ejpam-4870	350	14	j	j	PROPN
ejpam-4870	350	15	=	=	VERB
ejpam-4870	350	16	⌈n2	⌈n2	NOUN
ejpam-4870	350	17	⌉+	⌉+	NOUN
ejpam-4870	350	18	1	1	NUM
ejpam-4870	350	19	=	=	SYM
ejpam-4870	350	20	n	n	PRON
ejpam-4870	350	21	2	2	NUM
ejpam-4870	350	22	+	+	NUM
ejpam-4870	350	23	1	1	NUM
ejpam-4870	350	24	.	.	PUNCT
ejpam-4870	350	25	then	then	ADV
ejpam-4870	350	26	|a	|a	VERB
ejpam-4870	350	27	∪b|	∪b|	PROPN
ejpam-4870	350	28	=	=	SYM
ejpam-4870	350	29	(	(	PUNCT
ejpam-4870	350	30	n	n	ADV
ejpam-4870	350	31	2	2	NUM
ejpam-4870	350	32	+	+	CCONJ
ejpam-4870	350	33	1−	1−	NUM
ejpam-4870	350	34	1	1	NUM
ejpam-4870	350	35	)	)	PUNCT
ejpam-4870	350	36	+	+	CCONJ
ejpam-4870	350	37	(	(	PUNCT
ejpam-4870	350	38	n	n	CCONJ
ejpam-4870	350	39	2	2	NUM
ejpam-4870	350	40	+	+	CCONJ
ejpam-4870	350	41	1−	1−	NUM
ejpam-4870	350	42	1	1	NUM
ejpam-4870	350	43	)	)	PUNCT
ejpam-4870	350	44	+	+	CCONJ
ejpam-4870	350	45	1	1	NUM
ejpam-4870	350	46	=	=	SYM
ejpam-4870	350	47	n+	n+	PRON
ejpam-4870	350	48	1	1	X
ejpam-4870	350	49	.	.	PUNCT
ejpam-4870	351	1	this	this	PRON
ejpam-4870	351	2	is	be	AUX
ejpam-4870	351	3	a	a	DET
ejpam-4870	351	4	contradiction	contradiction	NOUN
ejpam-4870	351	5	to	to	ADP
ejpam-4870	351	6	the	the	DET
ejpam-4870	351	7	fact	fact	NOUN
ejpam-4870	351	8	that	that	SCONJ
ejpam-4870	351	9	j	j	PROPN
ejpam-4870	351	10	≤	≤	PROPN
ejpam-4870	351	11	n.	n.	NOUN
ejpam-4870	351	12	moreover	moreover	ADV
ejpam-4870	351	13	,	,	PUNCT
ejpam-4870	351	14	if	if	SCONJ
ejpam-4870	351	15	n	n	PRON
ejpam-4870	351	16	is	be	AUX
ejpam-4870	351	17	odd	odd	ADJ
ejpam-4870	351	18	,	,	PUNCT
ejpam-4870	351	19	let	let	VERB
ejpam-4870	351	20	j	j	PROPN
ejpam-4870	351	21	=	=	PUNCT
ejpam-4870	351	22	⌈n2	⌈n2	NOUN
ejpam-4870	351	23	⌉	⌉	ADJ
ejpam-4870	351	24	+	+	CCONJ
ejpam-4870	351	25	1	1	NUM
ejpam-4870	351	26	=	=	SYM
ejpam-4870	351	27	n+1	n+1	NUM
ejpam-4870	351	28	2	2	NUM
ejpam-4870	351	29	+	+	NUM
ejpam-4870	351	30	1	1	NUM
ejpam-4870	351	31	.	.	PUNCT
ejpam-4870	351	32	then	then	ADV
ejpam-4870	351	33	|a	|a	VERB
ejpam-4870	351	34	∪b|	∪b|	PROPN
ejpam-4870	351	35	=	=	PRON
ejpam-4870	351	36	(	(	PUNCT
ejpam-4870	351	37	n+	n+	NUM
ejpam-4870	351	38	1	1	NUM
ejpam-4870	351	39	2	2	NUM
ejpam-4870	351	40	+	+	CCONJ
ejpam-4870	351	41	1−	1−	NUM
ejpam-4870	351	42	1	1	NUM
ejpam-4870	351	43	)	)	PUNCT
ejpam-4870	351	44	+	+	CCONJ
ejpam-4870	351	45	(	(	PUNCT
ejpam-4870	351	46	n+	n+	NUM
ejpam-4870	351	47	1	1	NUM
ejpam-4870	351	48	2	2	NUM
ejpam-4870	351	49	+	+	CCONJ
ejpam-4870	351	50	1−	1−	NUM
ejpam-4870	351	51	1	1	NUM
ejpam-4870	351	52	)	)	PUNCT
ejpam-4870	351	53	+	+	CCONJ
ejpam-4870	351	54	1	1	NUM
ejpam-4870	351	55	=	=	SYM
ejpam-4870	351	56	n+	n+	ADP
ejpam-4870	351	57	2	2	X
ejpam-4870	351	58	.	.	X
ejpam-4870	352	1	this	this	PRON
ejpam-4870	352	2	is	be	AUX
ejpam-4870	352	3	also	also	ADV
ejpam-4870	352	4	a	a	DET
ejpam-4870	352	5	contradiction	contradiction	NOUN
ejpam-4870	352	6	to	to	ADP
ejpam-4870	352	7	the	the	DET
ejpam-4870	352	8	fact	fact	NOUN
ejpam-4870	352	9	that	that	SCONJ
ejpam-4870	352	10	j	j	PROPN
ejpam-4870	352	11	≤	≤	PROPN
ejpam-4870	352	12	n.	n.	NOUN
ejpam-4870	352	13	therefore	therefore	ADV
ejpam-4870	352	14	,	,	PUNCT
ejpam-4870	352	15	if	if	SCONJ
ejpam-4870	352	16	⌈n2	⌈n2	NOUN
ejpam-4870	352	17	⌉	⌉	ADP
ejpam-4870	352	18	<	<	X
ejpam-4870	352	19	j	j	PROPN
ejpam-4870	352	20	≤	≤	NUM
ejpam-4870	352	21	n	n	CCONJ
ejpam-4870	352	22	,	,	PUNCT
ejpam-4870	352	23	then	then	ADV
ejpam-4870	352	24	|a	|a	VERB
ejpam-4870	352	25	∩b|	∩b|	PROPN
ejpam-4870	352	26	>	>	X
ejpam-4870	352	27	1	1	NUM
ejpam-4870	352	28	.	.	X
ejpam-4870	352	29	illustration	illustration	NOUN
ejpam-4870	352	30	5	5	NUM
ejpam-4870	352	31	shows	show	VERB
ejpam-4870	352	32	that	that	SCONJ
ejpam-4870	352	33	for	for	ADP
ejpam-4870	352	34	two	two	NUM
ejpam-4870	352	35	distinct	distinct	ADJ
ejpam-4870	352	36	spanning	span	VERB
ejpam-4870	352	37	subgraphs	subgraph	NOUN
ejpam-4870	352	38	of	of	ADP
ejpam-4870	352	39	cn	cn	PROPN
ejpam-4870	352	40	with	with	ADP
ejpam-4870	352	41	j	j	PROPN
ejpam-4870	352	42	edges	edge	NOUN
ejpam-4870	352	43	,	,	PUNCT
ejpam-4870	352	44	their	their	PRON
ejpam-4870	352	45	common	common	ADJ
ejpam-4870	352	46	edge	edge	NOUN
ejpam-4870	352	47	is	be	AUX
ejpam-4870	352	48	always	always	ADV
ejpam-4870	352	49	greater	great	ADJ
ejpam-4870	352	50	than	than	ADP
ejpam-4870	352	51	1	1	NUM
ejpam-4870	352	52	that	that	PRON
ejpam-4870	352	53	is	be	AUX
ejpam-4870	352	54	when	when	SCONJ
ejpam-4870	352	55	⌈n2	⌈n2	NOUN
ejpam-4870	352	56	⌉	⌉	ADP
ejpam-4870	352	57	<	<	X
ejpam-4870	352	58	j	j	PROPN
ejpam-4870	352	59	≤	≤	PROPN
ejpam-4870	352	60	n.	n.	NOUN
ejpam-4870	352	61	setting	set	VERB
ejpam-4870	352	62	n	n	NOUN
ejpam-4870	352	63	=	=	SYM
ejpam-4870	352	64	4	4	NUM
ejpam-4870	352	65	and	and	CCONJ
ejpam-4870	352	66	j	j	PROPN
ejpam-4870	352	67	=	=	SYM
ejpam-4870	352	68	3	3	X
ejpam-4870	352	69	.	.	X
ejpam-4870	352	70	illustration	illustration	NOUN
ejpam-4870	352	71	5	5	NUM
ejpam-4870	352	72	.	.	PUNCT
ejpam-4870	352	73	consider	consider	VERB
ejpam-4870	352	74	c4	c4	NOUN
ejpam-4870	352	75	with	with	ADP
ejpam-4870	352	76	e(c4	e(c4	PROPN
ejpam-4870	352	77	)	)	PUNCT
ejpam-4870	352	78	=	=	PRON
ejpam-4870	352	79	{	{	PUNCT
ejpam-4870	352	80	12	12	NUM
ejpam-4870	352	81	,	,	PUNCT
ejpam-4870	352	82	32	32	NUM
ejpam-4870	352	83	,	,	PUNCT
ejpam-4870	352	84	34	34	NUM
ejpam-4870	352	85	,	,	PUNCT
ejpam-4870	352	86	41	41	NUM
ejpam-4870	352	87	}	}	PUNCT
ejpam-4870	352	88	and	and	CCONJ
ejpam-4870	352	89	let	let	VERB
ejpam-4870	352	90	j	j	PROPN
ejpam-4870	352	91	=	=	NOUN
ejpam-4870	352	92	3	3	X
ejpam-4870	352	93	.	.	PUNCT
ejpam-4870	353	1	it	it	PRON
ejpam-4870	353	2	can	can	AUX
ejpam-4870	353	3	be	be	AUX
ejpam-4870	353	4	observed	observe	VERB
ejpam-4870	353	5	that	that	SCONJ
ejpam-4870	353	6	j	j	PROPN
ejpam-4870	353	7	=	=	SYM
ejpam-4870	353	8	3	3	NUM
ejpam-4870	353	9	>	>	SYM
ejpam-4870	353	10	⌈42⌉.	⌈42⌉.	PROPN
ejpam-4870	353	11	the	the	DET
ejpam-4870	353	12	vertex	vertex	NOUN
ejpam-4870	353	13	set	set	NOUN
ejpam-4870	353	14	of	of	ADP
ejpam-4870	353	15	ec(4,3	ec(4,3	PROPN
ejpam-4870	353	16	)	)	PUNCT
ejpam-4870	353	17	is	be	AUX
ejpam-4870	353	18	given	give	VERB
ejpam-4870	353	19	by	by	ADP
ejpam-4870	353	20	v	v	PROPN
ejpam-4870	353	21	(	(	PUNCT
ejpam-4870	353	22	ec(4,3	ec(4,3	PROPN
ejpam-4870	353	23	)	)	PUNCT
ejpam-4870	353	24	)	)	PUNCT
ejpam-4870	354	1	=	=	PRON
ejpam-4870	354	2	{	{	PUNCT
ejpam-4870	354	3	{	{	PUNCT
ejpam-4870	354	4	12	12	NUM
ejpam-4870	354	5	,	,	PUNCT
ejpam-4870	354	6	23	23	NUM
ejpam-4870	354	7	,	,	PUNCT
ejpam-4870	354	8	34	34	NUM
ejpam-4870	354	9	}	}	PUNCT
ejpam-4870	354	10	,	,	PUNCT
ejpam-4870	354	11	{	{	PUNCT
ejpam-4870	354	12	12	12	NUM
ejpam-4870	354	13	,	,	PUNCT
ejpam-4870	354	14	23	23	NUM
ejpam-4870	354	15	,	,	PUNCT
ejpam-4870	354	16	41	41	NUM
ejpam-4870	354	17	}	}	PUNCT
ejpam-4870	354	18	,	,	PUNCT
ejpam-4870	354	19	{	{	PUNCT
ejpam-4870	354	20	12	12	NUM
ejpam-4870	354	21	,	,	PUNCT
ejpam-4870	354	22	34	34	NUM
ejpam-4870	354	23	,	,	PUNCT
ejpam-4870	354	24	41	41	NUM
ejpam-4870	354	25	}	}	PUNCT
ejpam-4870	354	26	,	,	PUNCT
ejpam-4870	354	27	{	{	PUNCT
ejpam-4870	354	28	23	23	NUM
ejpam-4870	354	29	,	,	PUNCT
ejpam-4870	354	30	34	34	NUM
ejpam-4870	354	31	,	,	PUNCT
ejpam-4870	354	32	41	41	NUM
ejpam-4870	354	33	}	}	PUNCT
ejpam-4870	354	34	}	}	PUNCT
ejpam-4870	354	35	.	.	PUNCT
ejpam-4870	355	1	now	now	ADV
ejpam-4870	355	2	,	,	PUNCT
ejpam-4870	355	3	take	take	VERB
ejpam-4870	355	4	two	two	NUM
ejpam-4870	355	5	arbitrary	arbitrary	ADJ
ejpam-4870	355	6	elements	element	NOUN
ejpam-4870	355	7	in	in	ADP
ejpam-4870	355	8	v	v	NUM
ejpam-4870	355	9	(	(	PUNCT
ejpam-4870	355	10	ec(4,3	ec(4,3	PROPN
ejpam-4870	355	11	)	)	PUNCT
ejpam-4870	355	12	)	)	PUNCT
ejpam-4870	355	13	,	,	PUNCT
ejpam-4870	355	14	say	say	VERB
ejpam-4870	355	15	{	{	PUNCT
ejpam-4870	355	16	12	12	NUM
ejpam-4870	355	17	,	,	PUNCT
ejpam-4870	355	18	23	23	NUM
ejpam-4870	355	19	,	,	PUNCT
ejpam-4870	355	20	34	34	NUM
ejpam-4870	355	21	}	}	PUNCT
ejpam-4870	355	22	and	and	CCONJ
ejpam-4870	355	23	{	{	PUNCT
ejpam-4870	355	24	12	12	NUM
ejpam-4870	355	25	,	,	PUNCT
ejpam-4870	355	26	23	23	NUM
ejpam-4870	355	27	,	,	PUNCT
ejpam-4870	355	28	41	41	NUM
ejpam-4870	355	29	}	}	PUNCT
ejpam-4870	355	30	,	,	PUNCT
ejpam-4870	355	31	{	{	PUNCT
ejpam-4870	355	32	12	12	NUM
ejpam-4870	355	33	,	,	PUNCT
ejpam-4870	355	34	23	23	NUM
ejpam-4870	355	35	,	,	PUNCT
ejpam-4870	355	36	34}∩	34}∩	NUM
ejpam-4870	355	37	{	{	PUNCT
ejpam-4870	355	38	12	12	NUM
ejpam-4870	355	39	,	,	PUNCT
ejpam-4870	355	40	23	23	NUM
ejpam-4870	355	41	,	,	PUNCT
ejpam-4870	355	42	41	41	NUM
ejpam-4870	355	43	}	}	PUNCT
ejpam-4870	355	44	=	=	SYM
ejpam-4870	355	45	{	{	PUNCT
ejpam-4870	355	46	12	12	NUM
ejpam-4870	355	47	,	,	PUNCT
ejpam-4870	355	48	23	23	NUM
ejpam-4870	355	49	}	}	PUNCT
ejpam-4870	355	50	with	with	ADP
ejpam-4870	355	51	cardinality	cardinality	NOUN
ejpam-4870	355	52	equal	equal	ADJ
ejpam-4870	355	53	to	to	ADP
ejpam-4870	355	54	2	2	NUM
ejpam-4870	355	55	.	.	PUNCT
ejpam-4870	356	1	it	it	PRON
ejpam-4870	356	2	can	can	AUX
ejpam-4870	356	3	be	be	AUX
ejpam-4870	356	4	noted	note	VERB
ejpam-4870	356	5	that	that	SCONJ
ejpam-4870	356	6	when	when	SCONJ
ejpam-4870	356	7	two	two	NUM
ejpam-4870	356	8	distinct	distinct	ADJ
ejpam-4870	356	9	spanning	span	VERB
ejpam-4870	356	10	subgraphs	subgraph	NOUN
ejpam-4870	356	11	of	of	ADP
ejpam-4870	356	12	cn	cn	PROPN
ejpam-4870	356	13	with	with	ADP
ejpam-4870	356	14	j	j	PROPN
ejpam-4870	356	15	edges	edge	NOUN
ejpam-4870	356	16	share	share	VERB
ejpam-4870	356	17	more	more	ADJ
ejpam-4870	356	18	than	than	ADP
ejpam-4870	356	19	1	1	NUM
ejpam-4870	356	20	edge	edge	NOUN
ejpam-4870	356	21	,	,	PUNCT
ejpam-4870	356	22	the	the	DET
ejpam-4870	356	23	degree	degree	NOUN
ejpam-4870	356	24	of	of	ADP
ejpam-4870	356	25	every	every	DET
ejpam-4870	356	26	vertex	vertex	NOUN
ejpam-4870	356	27	in	in	ADP
ejpam-4870	356	28	ec(n	ec(n	PROPN
ejpam-4870	356	29	,	,	PUNCT
ejpam-4870	356	30	j	j	NOUN
ejpam-4870	356	31	)	)	PUNCT
ejpam-4870	356	32	is	be	AUX
ejpam-4870	356	33	equal	equal	ADJ
ejpam-4870	356	34	to	to	ADP
ejpam-4870	356	35	0	0	NUM
ejpam-4870	356	36	.	.	PUNCT
ejpam-4870	356	37	theorem	theorem	NOUN
ejpam-4870	356	38	5	5	NUM
ejpam-4870	356	39	.	.	PUNCT
ejpam-4870	356	40	if	if	SCONJ
ejpam-4870	356	41	⌈n2	⌈n2	NOUN
ejpam-4870	356	42	⌉	⌉	ADP
ejpam-4870	356	43	<	<	X
ejpam-4870	356	44	j	j	PROPN
ejpam-4870	356	45	≤	≤	NUM
ejpam-4870	356	46	n	n	CCONJ
ejpam-4870	356	47	,	,	PUNCT
ejpam-4870	356	48	then	then	ADV
ejpam-4870	356	49	for	for	ADP
ejpam-4870	356	50	all	all	DET
ejpam-4870	356	51	a	a	DET
ejpam-4870	356	52	∈	∈	PROPN
ejpam-4870	356	53	v	v	NOUN
ejpam-4870	356	54	(	(	PUNCT
ejpam-4870	356	55	ec(n	ec(n	PROPN
ejpam-4870	356	56	,	,	PUNCT
ejpam-4870	356	57	j	j	NOUN
ejpam-4870	356	58	)	)	PUNCT
ejpam-4870	356	59	)	)	PUNCT
ejpam-4870	356	60	,	,	PUNCT
ejpam-4870	356	61	the	the	DET
ejpam-4870	356	62	deg(a	deg(a	PROPN
ejpam-4870	356	63	)	)	PUNCT
ejpam-4870	356	64	=	=	NOUN
ejpam-4870	356	65	0	0	X
ejpam-4870	356	66	.	.	PUNCT
ejpam-4870	357	1	proof	proof	NOUN
ejpam-4870	357	2	.	.	PUNCT
ejpam-4870	358	1	by	by	ADP
ejpam-4870	358	2	lemma	lemma	PROPN
ejpam-4870	358	3	1	1	NUM
ejpam-4870	358	4	,	,	PUNCT
ejpam-4870	358	5	if	if	SCONJ
ejpam-4870	358	6	a	a	DET
ejpam-4870	358	7	,	,	PUNCT
ejpam-4870	358	8	b	b	PROPN
ejpam-4870	358	9	∈	∈	PROPN
ejpam-4870	358	10	v	v	NOUN
ejpam-4870	358	11	(	(	PUNCT
ejpam-4870	358	12	ec(n	ec(n	PROPN
ejpam-4870	358	13	,	,	PUNCT
ejpam-4870	358	14	j	j	NOUN
ejpam-4870	358	15	)	)	PUNCT
ejpam-4870	358	16	)	)	PUNCT
ejpam-4870	358	17	,	,	PUNCT
ejpam-4870	358	18	then	then	ADV
ejpam-4870	358	19	|a	|a	PRON
ejpam-4870	358	20	∩	∩	PROPN
ejpam-4870	358	21	b|	b|	PROPN
ejpam-4870	358	22	>	>	X
ejpam-4870	358	23	1	1	NUM
ejpam-4870	358	24	when	when	SCONJ
ejpam-4870	358	25	⌈n2	⌈n2	NOUN
ejpam-4870	358	26	⌉	⌉	ADP
ejpam-4870	358	27	<	<	X
ejpam-4870	358	28	j	j	PROPN
ejpam-4870	358	29	≤	≤	X
ejpam-4870	358	30	n.	n.	NOUN
ejpam-4870	358	31	by	by	ADP
ejpam-4870	358	32	definition	definition	NOUN
ejpam-4870	358	33	17	17	NUM
ejpam-4870	358	34	,	,	PUNCT
ejpam-4870	358	35	two	two	NUM
ejpam-4870	358	36	distinct	distinct	ADJ
ejpam-4870	358	37	vertices	vertex	NOUN
ejpam-4870	358	38	are	be	AUX
ejpam-4870	358	39	adjacent	adjacent	ADJ
ejpam-4870	358	40	if	if	SCONJ
ejpam-4870	358	41	they	they	PRON
ejpam-4870	358	42	share	share	VERB
ejpam-4870	358	43	exactly	exactly	ADV
ejpam-4870	358	44	one	one	NUM
ejpam-4870	358	45	edge	edge	NOUN
ejpam-4870	358	46	.	.	PUNCT
ejpam-4870	359	1	so	so	ADV
ejpam-4870	359	2	,	,	PUNCT
ejpam-4870	359	3	if	if	SCONJ
ejpam-4870	359	4	|a	|a	VERB
ejpam-4870	359	5	∩	∩	NOUN
ejpam-4870	359	6	b|	b|	PROPN
ejpam-4870	359	7	>	>	X
ejpam-4870	359	8	1	1	NUM
ejpam-4870	359	9	,	,	PUNCT
ejpam-4870	359	10	then	then	ADV
ejpam-4870	359	11	[	[	X
ejpam-4870	359	12	a	a	DET
ejpam-4870	359	13	,	,	PUNCT
ejpam-4870	359	14	b	b	NOUN
ejpam-4870	359	15	]	]	X
ejpam-4870	359	16	/∈	/∈	PUNCT
ejpam-4870	360	1	e(ec(n	e(ec(n	PROPN
ejpam-4870	360	2	,	,	PUNCT
ejpam-4870	360	3	j	j	PROPN
ejpam-4870	360	4	)	)	PUNCT
ejpam-4870	360	5	)	)	PUNCT
ejpam-4870	361	1	for	for	ADP
ejpam-4870	361	2	all	all	DET
ejpam-4870	361	3	a	a	DET
ejpam-4870	361	4	∈	∈	PROPN
ejpam-4870	361	5	v	v	NOUN
ejpam-4870	361	6	(	(	PUNCT
ejpam-4870	361	7	ec(n	ec(n	PROPN
ejpam-4870	361	8	,	,	PUNCT
ejpam-4870	361	9	j	j	NOUN
ejpam-4870	361	10	)	)	PUNCT
ejpam-4870	361	11	)	)	PUNCT
ejpam-4870	361	12	.	.	PUNCT
ejpam-4870	362	1	thus	thus	ADV
ejpam-4870	362	2	,	,	PUNCT
ejpam-4870	362	3	when	when	SCONJ
ejpam-4870	362	4	⌈n2	⌈n2	NOUN
ejpam-4870	362	5	⌉	⌉	ADP
ejpam-4870	362	6	<	<	X
ejpam-4870	362	7	j	j	PROPN
ejpam-4870	362	8	≤	≤	PROPN
ejpam-4870	362	9	n	n	CCONJ
ejpam-4870	362	10	,	,	PUNCT
ejpam-4870	362	11	deg(a	deg(a	PROPN
ejpam-4870	362	12	)	)	PUNCT
ejpam-4870	362	13	=	=	NOUN
ejpam-4870	362	14	0	0	X
ejpam-4870	362	15	.	.	X
ejpam-4870	362	16	illustration	illustration	NOUN
ejpam-4870	362	17	6	6	NUM
ejpam-4870	362	18	shows	show	VERB
ejpam-4870	362	19	that	that	SCONJ
ejpam-4870	362	20	when	when	SCONJ
ejpam-4870	362	21	⌈n2	⌈n2	NOUN
ejpam-4870	362	22	⌉	⌉	ADP
ejpam-4870	362	23	<	<	X
ejpam-4870	362	24	j	j	PROPN
ejpam-4870	362	25	≤	≤	PROPN
ejpam-4870	362	26	n	n	CCONJ
ejpam-4870	362	27	the	the	DET
ejpam-4870	362	28	degree	degree	NOUN
ejpam-4870	362	29	of	of	ADP
ejpam-4870	362	30	every	every	DET
ejpam-4870	362	31	vertex	vertex	NOUN
ejpam-4870	362	32	in	in	ADP
ejpam-4870	362	33	ec(n	ec(n	PROPN
ejpam-4870	362	34	,	,	PUNCT
ejpam-4870	362	35	j	j	NOUN
ejpam-4870	362	36	)	)	PUNCT
ejpam-4870	362	37	is	be	AUX
ejpam-4870	362	38	equal	equal	ADJ
ejpam-4870	362	39	to	to	ADP
ejpam-4870	362	40	0	0	NUM
ejpam-4870	362	41	.	.	PUNCT
ejpam-4870	362	42	illustration	illustration	NOUN
ejpam-4870	362	43	6	6	NUM
ejpam-4870	362	44	.	.	PUNCT
ejpam-4870	363	1	let	let	VERB
ejpam-4870	363	2	c4	c4	NOUN
ejpam-4870	363	3	be	be	AUX
ejpam-4870	363	4	a	a	DET
ejpam-4870	363	5	cycle	cycle	NOUN
ejpam-4870	363	6	graph	graph	NOUN
ejpam-4870	363	7	of	of	ADP
ejpam-4870	363	8	order	order	NOUN
ejpam-4870	363	9	4	4	NUM
ejpam-4870	363	10	with	with	ADP
ejpam-4870	363	11	e(c4	e(c4	NOUN
ejpam-4870	363	12	)	)	PUNCT
ejpam-4870	364	1	=	=	PRON
ejpam-4870	364	2	{	{	PUNCT
ejpam-4870	364	3	12	12	NUM
ejpam-4870	364	4	,	,	PUNCT
ejpam-4870	364	5	23	23	NUM
ejpam-4870	364	6	,	,	PUNCT
ejpam-4870	364	7	34	34	NUM
ejpam-4870	364	8	,	,	PUNCT
ejpam-4870	364	9	41	41	NUM
ejpam-4870	364	10	}	}	PUNCT
ejpam-4870	365	1	and	and	CCONJ
ejpam-4870	365	2	let	let	VERB
ejpam-4870	365	3	j	j	PROPN
ejpam-4870	365	4	=	=	SYM
ejpam-4870	365	5	3	3	X
ejpam-4870	365	6	.	.	X
ejpam-4870	365	7	observe	observe	VERB
ejpam-4870	365	8	that	that	PRON
ejpam-4870	365	9	j	j	PROPN
ejpam-4870	365	10	=	=	SYM
ejpam-4870	365	11	3	3	NUM
ejpam-4870	365	12	>	>	SYM
ejpam-4870	365	13	⌈42⌉.	⌈42⌉.	PROPN
ejpam-4870	365	14	the	the	DET
ejpam-4870	365	15	vertex	vertex	NOUN
ejpam-4870	365	16	set	set	NOUN
ejpam-4870	365	17	of	of	ADP
ejpam-4870	365	18	ec(4,3	ec(4,3	PROPN
ejpam-4870	365	19	)	)	PUNCT
ejpam-4870	365	20	is	be	AUX
ejpam-4870	365	21	given	give	VERB
ejpam-4870	365	22	by	by	ADP
ejpam-4870	365	23	v	v	PROPN
ejpam-4870	365	24	(	(	PUNCT
ejpam-4870	365	25	ec(4,3	ec(4,3	PROPN
ejpam-4870	365	26	)	)	PUNCT
ejpam-4870	365	27	)	)	PUNCT
ejpam-4870	366	1	=	=	PRON
ejpam-4870	366	2	{	{	PUNCT
ejpam-4870	366	3	{	{	PUNCT
ejpam-4870	366	4	12	12	NUM
ejpam-4870	366	5	,	,	PUNCT
ejpam-4870	366	6	23	23	NUM
ejpam-4870	366	7	,	,	PUNCT
ejpam-4870	366	8	34	34	NUM
ejpam-4870	366	9	}	}	PUNCT
ejpam-4870	366	10	,	,	PUNCT
ejpam-4870	366	11	{	{	PUNCT
ejpam-4870	366	12	12	12	NUM
ejpam-4870	366	13	,	,	PUNCT
ejpam-4870	366	14	23	23	NUM
ejpam-4870	366	15	,	,	PUNCT
ejpam-4870	366	16	41	41	NUM
ejpam-4870	366	17	}	}	PUNCT
ejpam-4870	366	18	,	,	PUNCT
ejpam-4870	366	19	{	{	PUNCT
ejpam-4870	366	20	12	12	NUM
ejpam-4870	366	21	,	,	PUNCT
ejpam-4870	366	22	34	34	NUM
ejpam-4870	366	23	,	,	PUNCT
ejpam-4870	366	24	41	41	NUM
ejpam-4870	366	25	}	}	PUNCT
ejpam-4870	366	26	,	,	PUNCT
ejpam-4870	366	27	{	{	PUNCT
ejpam-4870	366	28	23	23	NUM
ejpam-4870	366	29	,	,	PUNCT
ejpam-4870	366	30	34	34	NUM
ejpam-4870	366	31	,	,	PUNCT
ejpam-4870	366	32	41	41	NUM
ejpam-4870	366	33	}	}	PUNCT
ejpam-4870	366	34	}	}	PUNCT
ejpam-4870	366	35	now	now	ADV
ejpam-4870	366	36	,	,	PUNCT
ejpam-4870	366	37	observe	observe	VERB
ejpam-4870	366	38	that	that	SCONJ
ejpam-4870	366	39	{	{	PUNCT
ejpam-4870	366	40	12	12	NUM
ejpam-4870	366	41	,	,	PUNCT
ejpam-4870	366	42	23	23	NUM
ejpam-4870	366	43	,	,	PUNCT
ejpam-4870	366	44	34}∩{12	34}∩{12	NUM
ejpam-4870	366	45	,	,	PUNCT
ejpam-4870	366	46	23	23	NUM
ejpam-4870	366	47	,	,	PUNCT
ejpam-4870	366	48	41	41	NUM
ejpam-4870	366	49	}	}	PUNCT
ejpam-4870	366	50	=	=	SYM
ejpam-4870	366	51	{	{	PUNCT
ejpam-4870	366	52	12	12	NUM
ejpam-4870	366	53	,	,	PUNCT
ejpam-4870	366	54	23	23	NUM
ejpam-4870	366	55	}	}	PUNCT
ejpam-4870	366	56	with	with	ADP
ejpam-4870	366	57	cardinality	cardinality	NOUN
ejpam-4870	366	58	equal	equal	ADJ
ejpam-4870	366	59	to	to	ADP
ejpam-4870	366	60	2	2	NUM
ejpam-4870	366	61	,	,	PUNCT
ejpam-4870	366	62	by	by	ADP
ejpam-4870	366	63	definition	definition	NOUN
ejpam-4870	366	64	17	17	NUM
ejpam-4870	366	65	,	,	PUNCT
ejpam-4870	366	66	{	{	PUNCT
ejpam-4870	366	67	12	12	NUM
ejpam-4870	366	68	,	,	PUNCT
ejpam-4870	366	69	23	23	NUM
ejpam-4870	366	70	,	,	PUNCT
ejpam-4870	366	71	34	34	NUM
ejpam-4870	366	72	}	}	PUNCT
ejpam-4870	366	73	and	and	CCONJ
ejpam-4870	366	74	{	{	PUNCT
ejpam-4870	366	75	12	12	NUM
ejpam-4870	366	76	,	,	PUNCT
ejpam-4870	366	77	23	23	NUM
ejpam-4870	366	78	,	,	PUNCT
ejpam-4870	366	79	41	41	NUM
ejpam-4870	366	80	}	}	PUNCT
ejpam-4870	366	81	in	in	ADP
ejpam-4870	366	82	v	v	NUM
ejpam-4870	366	83	(	(	PUNCT
ejpam-4870	366	84	ec(4,3	ec(4,3	PROPN
ejpam-4870	366	85	)	)	PUNCT
ejpam-4870	366	86	)	)	PUNCT
ejpam-4870	366	87	are	be	AUX
ejpam-4870	366	88	not	not	PART
ejpam-4870	366	89	adjacent	adjacent	ADJ
ejpam-4870	366	90	.	.	PUNCT
ejpam-4870	367	1	similarly	similarly	ADV
ejpam-4870	367	2	,	,	PUNCT
ejpam-4870	367	3	{	{	PUNCT
ejpam-4870	367	4	12	12	NUM
ejpam-4870	367	5	,	,	PUNCT
ejpam-4870	367	6	34	34	NUM
ejpam-4870	367	7	,	,	PUNCT
ejpam-4870	367	8	41	41	NUM
ejpam-4870	367	9	}	}	PUNCT
ejpam-4870	367	10	and	and	CCONJ
ejpam-4870	367	11	{	{	PUNCT
ejpam-4870	367	12	23	23	NUM
ejpam-4870	367	13	,	,	PUNCT
ejpam-4870	367	14	34	34	NUM
ejpam-4870	367	15	,	,	PUNCT
ejpam-4870	367	16	41	41	NUM
ejpam-4870	367	17	}	}	PUNCT
ejpam-4870	367	18	in	in	ADP
ejpam-4870	367	19	v	v	NUM
ejpam-4870	367	20	(	(	PUNCT
ejpam-4870	367	21	ec(4,3	ec(4,3	PROPN
ejpam-4870	367	22	)	)	PUNCT
ejpam-4870	367	23	)	)	PUNCT
ejpam-4870	367	24	are	be	AUX
ejpam-4870	367	25	also	also	ADV
ejpam-4870	367	26	not	not	PART
ejpam-4870	367	27	adjacent	adjacent	ADJ
ejpam-4870	367	28	since	since	SCONJ
ejpam-4870	367	29	|{12	|{12	NOUN
ejpam-4870	367	30	,	,	PUNCT
ejpam-4870	367	31	34	34	NUM
ejpam-4870	367	32	,	,	PUNCT
ejpam-4870	367	33	41	41	NUM
ejpam-4870	367	34	}	}	PUNCT
ejpam-4870	367	35	∩	∩	NOUN
ejpam-4870	367	36	{	{	PUNCT
ejpam-4870	367	37	23	23	NUM
ejpam-4870	367	38	,	,	PUNCT
ejpam-4870	367	39	34	34	NUM
ejpam-4870	367	40	,	,	PUNCT
ejpam-4870	367	41	41}|	41}|	PROPN
ejpam-4870	367	42	=	=	SYM
ejpam-4870	367	43	2	2	NUM
ejpam-4870	367	44	.	.	PUNCT
ejpam-4870	368	1	hence	hence	ADV
ejpam-4870	368	2	,	,	PUNCT
ejpam-4870	368	3	the	the	DET
ejpam-4870	368	4	degree	degree	NOUN
ejpam-4870	368	5	of	of	ADP
ejpam-4870	368	6	any	any	DET
ejpam-4870	368	7	vertex	vertex	NOUN
ejpam-4870	368	8	in	in	ADP
ejpam-4870	368	9	ec(4,3	ec(4,3	PROPN
ejpam-4870	368	10	)	)	PUNCT
ejpam-4870	368	11	is	be	AUX
ejpam-4870	368	12	equal	equal	ADJ
ejpam-4870	368	13	to	to	ADP
ejpam-4870	368	14	0	0	NUM
ejpam-4870	368	15	.	.	PUNCT
ejpam-4870	369	1	j.c	j.c	PROPN
ejpam-4870	369	2	.	.	PROPN
ejpam-4870	369	3	bonifacio	bonifacio	PROPN
ejpam-4870	369	4	,	,	PUNCT
ejpam-4870	369	5	c.j	c.j	PROPN
ejpam-4870	369	6	.	.	PROPN
ejpam-4870	369	7	andaya	andaya	PROPN
ejpam-4870	369	8	,	,	PUNCT
ejpam-4870	369	9	d.	d.	PROPN
ejpam-4870	369	10	magpantay	magpantay	PROPN
ejpam-4870	369	11	/	/	SYM
ejpam-4870	369	12	eur	eur	PROPN
ejpam-4870	369	13	.	.	PUNCT
ejpam-4870	370	1	j.	j.	PROPN
ejpam-4870	370	2	pure	pure	PROPN
ejpam-4870	370	3	appl	appl	PROPN
ejpam-4870	370	4	.	.	PROPN
ejpam-4870	370	5	math	math	PROPN
ejpam-4870	370	6	,	,	PUNCT
ejpam-4870	370	7	16	16	NUM
ejpam-4870	370	8	(	(	PUNCT
ejpam-4870	370	9	4	4	NUM
ejpam-4870	370	10	)	)	PUNCT
ejpam-4870	370	11	(	(	PUNCT
ejpam-4870	370	12	2023	2023	NUM
ejpam-4870	370	13	)	)	PUNCT
ejpam-4870	370	14	,	,	PUNCT
ejpam-4870	370	15	2476	2476	NUM
ejpam-4870	370	16	-	-	SYM
ejpam-4870	370	17	2498	2498	NUM
ejpam-4870	370	18	2492	2492	NUM
ejpam-4870	370	19	it	it	PRON
ejpam-4870	370	20	can	can	AUX
ejpam-4870	370	21	be	be	AUX
ejpam-4870	370	22	noted	note	VERB
ejpam-4870	370	23	that	that	SCONJ
ejpam-4870	370	24	when	when	SCONJ
ejpam-4870	370	25	⌈n2	⌈n2	NOUN
ejpam-4870	370	26	⌉	⌉	ADP
ejpam-4870	370	27	<	<	X
ejpam-4870	370	28	j	j	PROPN
ejpam-4870	370	29	≤	≤	PROPN
ejpam-4870	370	30	n	n	CCONJ
ejpam-4870	370	31	,	,	PUNCT
ejpam-4870	370	32	ec(n	ec(n	PROPN
ejpam-4870	370	33	,	,	PUNCT
ejpam-4870	370	34	j	j	NOUN
ejpam-4870	370	35	)	)	PUNCT
ejpam-4870	370	36	is	be	AUX
ejpam-4870	370	37	a	a	DET
ejpam-4870	370	38	empty	empty	ADJ
ejpam-4870	370	39	graph	graph	NOUN
ejpam-4870	370	40	of	of	ADP
ejpam-4870	370	41	order	order	NOUN
ejpam-4870	370	42	(	(	PUNCT
ejpam-4870	370	43	n	n	X
ejpam-4870	370	44	j	j	PROPN
ejpam-4870	370	45	)	)	PUNCT
ejpam-4870	370	46	since	since	SCONJ
ejpam-4870	370	47	the	the	DET
ejpam-4870	370	48	degree	degree	NOUN
ejpam-4870	370	49	of	of	ADP
ejpam-4870	370	50	every	every	DET
ejpam-4870	370	51	vertex	vertex	NOUN
ejpam-4870	370	52	in	in	ADP
ejpam-4870	370	53	ec(n	ec(n	PROPN
ejpam-4870	370	54	,	,	PUNCT
ejpam-4870	370	55	j	j	NOUN
ejpam-4870	370	56	)	)	PUNCT
ejpam-4870	370	57	is	be	AUX
ejpam-4870	370	58	equal	equal	ADJ
ejpam-4870	370	59	to	to	ADP
ejpam-4870	370	60	0	0	NUM
ejpam-4870	370	61	.	.	PUNCT
ejpam-4870	371	1	it	it	PRON
ejpam-4870	371	2	means	mean	VERB
ejpam-4870	371	3	that	that	SCONJ
ejpam-4870	371	4	there	there	PRON
ejpam-4870	371	5	are	be	VERB
ejpam-4870	371	6	no	no	DET
ejpam-4870	371	7	adjacent	adjacent	ADJ
ejpam-4870	371	8	vertices	vertex	NOUN
ejpam-4870	371	9	in	in	ADP
ejpam-4870	371	10	ec(n	ec(n	PROPN
ejpam-4870	371	11	,	,	PUNCT
ejpam-4870	371	12	j	j	PROPN
ejpam-4870	371	13	)	)	PUNCT
ejpam-4870	371	14	.	.	PUNCT
ejpam-4870	372	1	since	since	SCONJ
ejpam-4870	372	2	we	we	PRON
ejpam-4870	372	3	have	have	AUX
ejpam-4870	372	4	already	already	ADV
ejpam-4870	372	5	explored	explore	VERB
ejpam-4870	372	6	the	the	DET
ejpam-4870	372	7	case	case	NOUN
ejpam-4870	372	8	when	when	SCONJ
ejpam-4870	372	9	j	j	PROPN
ejpam-4870	372	10	=	=	NOUN
ejpam-4870	372	11	1	1	NUM
ejpam-4870	372	12	in	in	ADP
ejpam-4870	372	13	theorem	theorem	NOUN
ejpam-4870	372	14	1	1	NUM
ejpam-4870	372	15	,	,	PUNCT
ejpam-4870	372	16	we	we	PRON
ejpam-4870	372	17	will	will	AUX
ejpam-4870	372	18	proceed	proceed	VERB
ejpam-4870	372	19	to	to	ADP
ejpam-4870	372	20	the	the	DET
ejpam-4870	372	21	case	case	NOUN
ejpam-4870	372	22	when	when	SCONJ
ejpam-4870	372	23	2	2	NUM
ejpam-4870	372	24	≤	≤	NUM
ejpam-4870	372	25	j	j	PROPN
ejpam-4870	372	26	≤	≤	PROPN
ejpam-4870	372	27	⌈n2	⌈n2	NOUN
ejpam-4870	372	28	⌉.	⌉.	ADV
ejpam-4870	372	29	theorem	theorem	ADJ
ejpam-4870	372	30	6	6	NUM
ejpam-4870	372	31	determines	determine	VERB
ejpam-4870	372	32	the	the	DET
ejpam-4870	372	33	degree	degree	NOUN
ejpam-4870	372	34	of	of	ADP
ejpam-4870	372	35	every	every	DET
ejpam-4870	372	36	vertex	vertex	NOUN
ejpam-4870	372	37	in	in	ADP
ejpam-4870	372	38	ec(n	ec(n	PROPN
ejpam-4870	372	39	,	,	PUNCT
ejpam-4870	372	40	j	j	NOUN
ejpam-4870	372	41	)	)	PUNCT
ejpam-4870	372	42	when	when	SCONJ
ejpam-4870	372	43	2	2	NUM
ejpam-4870	372	44	≤	≤	NUM
ejpam-4870	372	45	j	j	PROPN
ejpam-4870	372	46	≤	≤	PROPN
ejpam-4870	372	47	⌈n2	⌈n2	NOUN
ejpam-4870	372	48	⌉.	⌉.	ADV
ejpam-4870	372	49	theorem	theorem	ADJ
ejpam-4870	372	50	6	6	NUM
ejpam-4870	372	51	.	.	PUNCT
ejpam-4870	372	52	for	for	ADP
ejpam-4870	372	53	any	any	DET
ejpam-4870	372	54	arbitrary	arbitrary	ADJ
ejpam-4870	372	55	vertex	vertex	NOUN
ejpam-4870	372	56	a	a	DET
ejpam-4870	372	57	∈	∈	PROPN
ejpam-4870	372	58	v	v	NOUN
ejpam-4870	372	59	(	(	PUNCT
ejpam-4870	372	60	ec(n	ec(n	PROPN
ejpam-4870	372	61	,	,	PUNCT
ejpam-4870	372	62	j	j	NOUN
ejpam-4870	372	63	)	)	PUNCT
ejpam-4870	372	64	)	)	PUNCT
ejpam-4870	372	65	where	where	SCONJ
ejpam-4870	372	66	2	2	NUM
ejpam-4870	372	67	≤	≤	NUM
ejpam-4870	372	68	j	j	PROPN
ejpam-4870	372	69	≤	≤	PROPN
ejpam-4870	372	70	⌈n2	⌈n2	NOUN
ejpam-4870	372	71	⌉	⌉	NOUN
ejpam-4870	372	72	,	,	PUNCT
ejpam-4870	372	73	deg(a	deg(a	PROPN
ejpam-4870	372	74	)	)	PUNCT
ejpam-4870	372	75	=	=	SYM
ejpam-4870	372	76	j	j	PROPN
ejpam-4870	372	77	(	(	PUNCT
ejpam-4870	372	78	n−j	n−j	ADV
ejpam-4870	372	79	j−1	j−1	PROPN
ejpam-4870	372	80	)	)	PUNCT
ejpam-4870	372	81	.	.	PUNCT
ejpam-4870	373	1	proof	proof	NOUN
ejpam-4870	373	2	.	.	PUNCT
ejpam-4870	374	1	let	let	VERB
ejpam-4870	374	2	a	a	DET
ejpam-4870	374	3	∈	∈	PROPN
ejpam-4870	374	4	v	v	NOUN
ejpam-4870	374	5	(	(	PUNCT
ejpam-4870	374	6	ec(n	ec(n	PROPN
ejpam-4870	374	7	,	,	PUNCT
ejpam-4870	374	8	j	j	NOUN
ejpam-4870	374	9	)	)	PUNCT
ejpam-4870	374	10	)	)	PUNCT
ejpam-4870	374	11	.	.	PUNCT
ejpam-4870	375	1	the	the	DET
ejpam-4870	375	2	vertices	vertex	NOUN
ejpam-4870	375	3	adjacent	adjacent	ADJ
ejpam-4870	375	4	to	to	ADP
ejpam-4870	375	5	a	a	PRON
ejpam-4870	375	6	are	be	AUX
ejpam-4870	375	7	the	the	DET
ejpam-4870	375	8	spanning	span	VERB
ejpam-4870	375	9	subgraphs	subgraph	NOUN
ejpam-4870	375	10	of	of	ADP
ejpam-4870	375	11	cn	cn	PROPN
ejpam-4870	375	12	with	with	ADP
ejpam-4870	375	13	j	j	PROPN
ejpam-4870	375	14	edges	edge	NOUN
ejpam-4870	375	15	and	and	CCONJ
ejpam-4870	375	16	having	have	VERB
ejpam-4870	375	17	exactly	exactly	ADV
ejpam-4870	375	18	one	one	NUM
ejpam-4870	375	19	common	common	ADJ
ejpam-4870	375	20	edge	edge	NOUN
ejpam-4870	375	21	.	.	PUNCT
ejpam-4870	376	1	without	without	ADP
ejpam-4870	376	2	loss	loss	NOUN
ejpam-4870	376	3	of	of	ADP
ejpam-4870	376	4	generality	generality	NOUN
ejpam-4870	376	5	,	,	PUNCT
ejpam-4870	376	6	fix	fix	NOUN
ejpam-4870	376	7	e1	e1	NOUN
ejpam-4870	376	8	as	as	ADP
ejpam-4870	376	9	the	the	DET
ejpam-4870	376	10	common	common	ADJ
ejpam-4870	376	11	edge	edge	NOUN
ejpam-4870	376	12	.	.	PUNCT
ejpam-4870	377	1	hence	hence	ADV
ejpam-4870	377	2	,	,	PUNCT
ejpam-4870	377	3	there	there	PRON
ejpam-4870	377	4	are	be	VERB
ejpam-4870	377	5	j−1	j−1	PROPN
ejpam-4870	377	6	edges	edge	VERB
ejpam-4870	377	7	different	different	ADJ
ejpam-4870	377	8	from	from	ADP
ejpam-4870	377	9	{	{	PUNCT
ejpam-4870	377	10	e2	e2	PROPN
ejpam-4870	377	11	,	,	PUNCT
ejpam-4870	377	12	·	·	PUNCT
ejpam-4870	377	13	·	·	PUNCT
ejpam-4870	377	14	·	·	PUNCT
ejpam-4870	377	15	,	,	PUNCT
ejpam-4870	377	16	ej	ej	PROPN
ejpam-4870	377	17	}	}	PUNCT
ejpam-4870	377	18	.	.	PUNCT
ejpam-4870	378	1	these	these	DET
ejpam-4870	378	2	edges	edge	NOUN
ejpam-4870	378	3	must	must	AUX
ejpam-4870	378	4	be	be	AUX
ejpam-4870	378	5	chosen	choose	VERB
ejpam-4870	378	6	from	from	ADP
ejpam-4870	378	7	the	the	DET
ejpam-4870	378	8	other	other	ADJ
ejpam-4870	378	9	n−	n−	NOUN
ejpam-4870	378	10	j	j	PROPN
ejpam-4870	378	11	edges	edge	NOUN
ejpam-4870	378	12	of	of	ADP
ejpam-4870	378	13	cn	cn	PROPN
ejpam-4870	378	14	.	.	PUNCT
ejpam-4870	379	1	thus	thus	ADV
ejpam-4870	379	2	,	,	PUNCT
ejpam-4870	379	3	these	these	PRON
ejpam-4870	379	4	are	be	AUX
ejpam-4870	379	5	(	(	PUNCT
ejpam-4870	379	6	n−j	n−j	ADV
ejpam-4870	379	7	j−1	j−1	PROPN
ejpam-4870	379	8	)	)	PUNCT
ejpam-4870	379	9	ways	way	NOUN
ejpam-4870	379	10	to	to	PART
ejpam-4870	379	11	do	do	VERB
ejpam-4870	379	12	this	this	PRON
ejpam-4870	379	13	.	.	PUNCT
ejpam-4870	380	1	since	since	SCONJ
ejpam-4870	380	2	there	there	PRON
ejpam-4870	380	3	are	be	VERB
ejpam-4870	380	4	j	j	PROPN
ejpam-4870	380	5	edges	edge	NOUN
ejpam-4870	380	6	contained	contain	VERB
ejpam-4870	380	7	in	in	ADP
ejpam-4870	380	8	each	each	DET
ejpam-4870	380	9	vertex	vertex	NOUN
ejpam-4870	380	10	,	,	PUNCT
ejpam-4870	380	11	it	it	PRON
ejpam-4870	380	12	follows	follow	VERB
ejpam-4870	380	13	that	that	SCONJ
ejpam-4870	380	14	there	there	PRON
ejpam-4870	380	15	are	be	VERB
ejpam-4870	380	16	j	j	PROPN
ejpam-4870	380	17	(	(	PUNCT
ejpam-4870	380	18	n−j	n−j	X
ejpam-4870	380	19	j−1	j−1	PROPN
ejpam-4870	380	20	)	)	PUNCT
ejpam-4870	380	21	vertices	vertice	VERB
ejpam-4870	380	22	adjacent	adjacent	ADJ
ejpam-4870	380	23	to	to	PART
ejpam-4870	380	24	a.	a.	VERB
ejpam-4870	380	25	in	in	ADP
ejpam-4870	380	26	the	the	DET
ejpam-4870	380	27	succeeding	succeed	VERB
ejpam-4870	380	28	discussions	discussion	NOUN
ejpam-4870	380	29	,	,	PUNCT
ejpam-4870	380	30	we	we	PRON
ejpam-4870	380	31	just	just	ADV
ejpam-4870	380	32	focus	focus	VERB
ejpam-4870	380	33	on	on	ADP
ejpam-4870	380	34	ec(n	ec(n	PROPN
ejpam-4870	380	35	,	,	PUNCT
ejpam-4870	380	36	j	j	NOUN
ejpam-4870	380	37	)	)	PUNCT
ejpam-4870	380	38	when	when	SCONJ
ejpam-4870	380	39	2	2	NUM
ejpam-4870	380	40	≤	≤	NUM
ejpam-4870	380	41	j	j	PROPN
ejpam-4870	380	42	≤	≤	NUM
ejpam-4870	380	43	⌈n2	⌈n2	NOUN
ejpam-4870	380	44	⌉	⌉	PUNCT
ejpam-4870	380	45	since	since	SCONJ
ejpam-4870	380	46	when	when	SCONJ
ejpam-4870	380	47	j	j	PROPN
ejpam-4870	380	48	=	=	SYM
ejpam-4870	380	49	1	1	NUM
ejpam-4870	380	50	and	and	CCONJ
ejpam-4870	380	51	⌈n2	⌈n2	NOUN
ejpam-4870	380	52	⌉	⌉	ADP
ejpam-4870	380	53	<	<	X
ejpam-4870	380	54	j	j	PROPN
ejpam-4870	380	55	≤	≤	PUNCT
ejpam-4870	380	56	n	n	CCONJ
ejpam-4870	380	57	we	we	PRON
ejpam-4870	380	58	produce	produce	VERB
ejpam-4870	380	59	an	an	DET
ejpam-4870	380	60	empty	empty	ADJ
ejpam-4870	380	61	graph	graph	NOUN
ejpam-4870	380	62	.	.	PUNCT
ejpam-4870	381	1	presented	present	VERB
ejpam-4870	381	2	in	in	ADP
ejpam-4870	381	3	illustration	illustration	NOUN
ejpam-4870	381	4	7	7	NUM
ejpam-4870	381	5	is	be	AUX
ejpam-4870	381	6	an	an	DET
ejpam-4870	381	7	example	example	NOUN
ejpam-4870	381	8	for	for	ADP
ejpam-4870	381	9	the	the	DET
ejpam-4870	381	10	degree	degree	NOUN
ejpam-4870	381	11	of	of	ADP
ejpam-4870	381	12	every	every	DET
ejpam-4870	381	13	vertex	vertex	NOUN
ejpam-4870	381	14	of	of	ADP
ejpam-4870	381	15	a	a	DET
ejpam-4870	381	16	ec(n	ec(n	PROPN
ejpam-4870	381	17	,	,	PUNCT
ejpam-4870	381	18	j	j	NOUN
ejpam-4870	381	19	)	)	PUNCT
ejpam-4870	381	20	when	when	SCONJ
ejpam-4870	381	21	2	2	NUM
ejpam-4870	381	22	≤	≤	NUM
ejpam-4870	381	23	j	j	PROPN
ejpam-4870	381	24	≤	≤	NUM
ejpam-4870	381	25	⌈n2	⌈n2	NOUN
ejpam-4870	381	26	⌉	⌉	SCONJ
ejpam-4870	381	27	where	where	SCONJ
ejpam-4870	381	28	n	n	X
ejpam-4870	381	29	=	=	SYM
ejpam-4870	381	30	5	5	NUM
ejpam-4870	381	31	and	and	CCONJ
ejpam-4870	381	32	j	j	NOUN
ejpam-4870	382	1	=	=	SYM
ejpam-4870	382	2	3	3	X
ejpam-4870	382	3	.	.	X
ejpam-4870	382	4	illustration	illustration	NOUN
ejpam-4870	382	5	7	7	NUM
ejpam-4870	382	6	.	.	PUNCT
ejpam-4870	382	7	consider	consider	VERB
ejpam-4870	382	8	the	the	DET
ejpam-4870	382	9	cycle	cycle	NOUN
ejpam-4870	382	10	graph	graph	NOUN
ejpam-4870	382	11	c5	c5	PROPN
ejpam-4870	382	12	where	where	SCONJ
ejpam-4870	382	13	e(c5	e(c5	ADJ
ejpam-4870	382	14	)	)	PUNCT
ejpam-4870	382	15	=	=	PRON
ejpam-4870	382	16	{	{	PUNCT
ejpam-4870	382	17	12	12	NUM
ejpam-4870	382	18	,	,	PUNCT
ejpam-4870	382	19	23	23	NUM
ejpam-4870	382	20	,	,	PUNCT
ejpam-4870	382	21	34	34	NUM
ejpam-4870	382	22	,	,	PUNCT
ejpam-4870	382	23	45	45	NUM
ejpam-4870	382	24	,	,	PUNCT
ejpam-4870	382	25	51	51	NUM
ejpam-4870	382	26	}	}	PUNCT
ejpam-4870	382	27	and	and	CCONJ
ejpam-4870	383	1	j	j	PROPN
ejpam-4870	383	2	=	=	SYM
ejpam-4870	383	3	3	3	X
ejpam-4870	383	4	.	.	PUNCT
ejpam-4870	384	1	the	the	DET
ejpam-4870	384	2	vertex	vertex	NOUN
ejpam-4870	384	3	set	set	NOUN
ejpam-4870	384	4	of	of	ADP
ejpam-4870	384	5	ec(5,3	ec(5,3	PROPN
ejpam-4870	384	6	)	)	PUNCT
ejpam-4870	384	7	is	be	AUX
ejpam-4870	384	8	given	give	VERB
ejpam-4870	384	9	by	by	ADP
ejpam-4870	384	10	v	v	PROPN
ejpam-4870	384	11	(	(	PUNCT
ejpam-4870	384	12	ec(5,3	ec(5,3	PROPN
ejpam-4870	384	13	)	)	PUNCT
ejpam-4870	384	14	)	)	PUNCT
ejpam-4870	385	1	=	=	PRON
ejpam-4870	385	2	{	{	PUNCT
ejpam-4870	385	3	{	{	PUNCT
ejpam-4870	385	4	12	12	NUM
ejpam-4870	385	5	,	,	PUNCT
ejpam-4870	385	6	23	23	NUM
ejpam-4870	385	7	,	,	PUNCT
ejpam-4870	385	8	34	34	NUM
ejpam-4870	385	9	}	}	PUNCT
ejpam-4870	385	10	,	,	PUNCT
ejpam-4870	385	11	{	{	PUNCT
ejpam-4870	385	12	12	12	NUM
ejpam-4870	385	13	,	,	PUNCT
ejpam-4870	385	14	23	23	NUM
ejpam-4870	385	15	,	,	PUNCT
ejpam-4870	385	16	45	45	NUM
ejpam-4870	385	17	}	}	PUNCT
ejpam-4870	385	18	,	,	PUNCT
ejpam-4870	385	19	{	{	PUNCT
ejpam-4870	385	20	12	12	NUM
ejpam-4870	385	21	,	,	PUNCT
ejpam-4870	385	22	23	23	NUM
ejpam-4870	385	23	,	,	PUNCT
ejpam-4870	385	24	51	51	NUM
ejpam-4870	385	25	}	}	PUNCT
ejpam-4870	385	26	,	,	PUNCT
ejpam-4870	385	27	{	{	PUNCT
ejpam-4870	385	28	12	12	NUM
ejpam-4870	385	29	,	,	PUNCT
ejpam-4870	385	30	34	34	NUM
ejpam-4870	385	31	,	,	PUNCT
ejpam-4870	385	32	45	45	NUM
ejpam-4870	385	33	}	}	PUNCT
ejpam-4870	385	34	,	,	PUNCT
ejpam-4870	385	35	{	{	PUNCT
ejpam-4870	385	36	12	12	NUM
ejpam-4870	385	37	,	,	PUNCT
ejpam-4870	385	38	34	34	NUM
ejpam-4870	385	39	,	,	PUNCT
ejpam-4870	385	40	51	51	NUM
ejpam-4870	385	41	}	}	PUNCT
ejpam-4870	385	42	,	,	PUNCT
ejpam-4870	385	43	{	{	PUNCT
ejpam-4870	385	44	12	12	NUM
ejpam-4870	385	45	,	,	PUNCT
ejpam-4870	385	46	45	45	NUM
ejpam-4870	385	47	,	,	PUNCT
ejpam-4870	385	48	51	51	NUM
ejpam-4870	385	49	}	}	PUNCT
ejpam-4870	385	50	,	,	PUNCT
ejpam-4870	385	51	{	{	PUNCT
ejpam-4870	385	52	23	23	NUM
ejpam-4870	385	53	,	,	PUNCT
ejpam-4870	385	54	34	34	NUM
ejpam-4870	385	55	,	,	PUNCT
ejpam-4870	385	56	45	45	NUM
ejpam-4870	385	57	}	}	PUNCT
ejpam-4870	385	58	,	,	PUNCT
ejpam-4870	385	59	{	{	PUNCT
ejpam-4870	385	60	23	23	NUM
ejpam-4870	385	61	,	,	PUNCT
ejpam-4870	385	62	34	34	NUM
ejpam-4870	385	63	,	,	PUNCT
ejpam-4870	385	64	51	51	NUM
ejpam-4870	385	65	}	}	PUNCT
ejpam-4870	385	66	,	,	PUNCT
ejpam-4870	385	67	{	{	PUNCT
ejpam-4870	385	68	23	23	NUM
ejpam-4870	385	69	,	,	PUNCT
ejpam-4870	385	70	45	45	NUM
ejpam-4870	385	71	,	,	PUNCT
ejpam-4870	385	72	51	51	NUM
ejpam-4870	385	73	}	}	PUNCT
ejpam-4870	385	74	,	,	PUNCT
ejpam-4870	385	75	{	{	PUNCT
ejpam-4870	385	76	34	34	NUM
ejpam-4870	385	77	,	,	PUNCT
ejpam-4870	385	78	45	45	NUM
ejpam-4870	385	79	,	,	PUNCT
ejpam-4870	385	80	51	51	NUM
ejpam-4870	385	81	}	}	PUNCT
ejpam-4870	385	82	}	}	PUNCT
ejpam-4870	385	83	figure	figure	NOUN
ejpam-4870	385	84	19	19	NUM
ejpam-4870	385	85	is	be	AUX
ejpam-4870	385	86	a	a	DET
ejpam-4870	385	87	pictorial	pictorial	ADJ
ejpam-4870	385	88	representation	representation	NOUN
ejpam-4870	385	89	of	of	ADP
ejpam-4870	385	90	sc(5,3	sc(5,3	PROPN
ejpam-4870	385	91	)	)	PUNCT
ejpam-4870	385	92	.	.	PUNCT
ejpam-4870	386	1	observe	observe	VERB
ejpam-4870	386	2	that	that	SCONJ
ejpam-4870	386	3	the	the	DET
ejpam-4870	386	4	degree	degree	NOUN
ejpam-4870	386	5	of	of	ADP
ejpam-4870	386	6	each	each	DET
ejpam-4870	386	7	vertex	vertex	NOUN
ejpam-4870	386	8	of	of	ADP
ejpam-4870	386	9	ec(5,3	ec(5,3	PROPN
ejpam-4870	386	10	)	)	PUNCT
ejpam-4870	386	11	is	be	AUX
ejpam-4870	386	12	3	3	NUM
ejpam-4870	386	13	.	.	PUNCT
ejpam-4870	387	1	now	now	ADV
ejpam-4870	387	2	,	,	PUNCT
ejpam-4870	387	3	to	to	PART
ejpam-4870	387	4	verify	verify	VERB
ejpam-4870	387	5	this	this	DET
ejpam-4870	387	6	using	use	VERB
ejpam-4870	387	7	theorem	theorem	NOUN
ejpam-4870	387	8	6	6	NUM
ejpam-4870	387	9	,	,	PUNCT
ejpam-4870	387	10	the	the	DET
ejpam-4870	387	11	degree	degree	NOUN
ejpam-4870	387	12	of	of	ADP
ejpam-4870	387	13	every	every	DET
ejpam-4870	387	14	vertex	vertex	NOUN
ejpam-4870	387	15	a	a	PRON
ejpam-4870	387	16	of	of	ADP
ejpam-4870	387	17	ec(5,3	ec(5,3	PROPN
ejpam-4870	387	18	)	)	PUNCT
ejpam-4870	387	19	is	be	AUX
ejpam-4870	387	20	given	give	VERB
ejpam-4870	387	21	by	by	ADP
ejpam-4870	387	22	deg(a	deg(a	PROPN
ejpam-4870	387	23	)	)	PUNCT
ejpam-4870	388	1	=	=	NOUN
ejpam-4870	388	2	j	j	PROPN
ejpam-4870	388	3	(	(	PUNCT
ejpam-4870	388	4	n−	n−	NOUN
ejpam-4870	388	5	j	j	PROPN
ejpam-4870	388	6	j	j	PROPN
ejpam-4870	388	7	−	−	PROPN
ejpam-4870	388	8	1	1	NUM
ejpam-4870	388	9	)	)	PUNCT
ejpam-4870	388	10	=3	=3	PROPN
ejpam-4870	388	11	(	(	PUNCT
ejpam-4870	388	12	5−	5−	NUM
ejpam-4870	388	13	3	3	NUM
ejpam-4870	388	14	3−	3−	NUM
ejpam-4870	388	15	1	1	NUM
ejpam-4870	388	16	)	)	PUNCT
ejpam-4870	388	17	=3	=3	NOUN
ejpam-4870	388	18	(	(	PUNCT
ejpam-4870	388	19	2	2	NUM
ejpam-4870	388	20	2	2	NUM
ejpam-4870	388	21	)	)	PUNCT
ejpam-4870	388	22	=	=	NOUN
ejpam-4870	388	23	3(1	3(1	NUM
ejpam-4870	388	24	)	)	PUNCT
ejpam-4870	388	25	=3	=3	VERB
ejpam-4870	388	26	.	.	PUNCT
ejpam-4870	389	1	it	it	PRON
ejpam-4870	389	2	can	can	AUX
ejpam-4870	389	3	be	be	AUX
ejpam-4870	389	4	observed	observe	VERB
ejpam-4870	389	5	that	that	SCONJ
ejpam-4870	389	6	the	the	DET
ejpam-4870	389	7	vertices	vertex	NOUN
ejpam-4870	389	8	of	of	ADP
ejpam-4870	389	9	ec(n	ec(n	PROPN
ejpam-4870	389	10	,	,	PUNCT
ejpam-4870	389	11	j	j	NOUN
ejpam-4870	389	12	)	)	PUNCT
ejpam-4870	389	13	have	have	VERB
ejpam-4870	389	14	the	the	DET
ejpam-4870	389	15	same	same	ADJ
ejpam-4870	389	16	degree	degree	NOUN
ejpam-4870	389	17	which	which	PRON
ejpam-4870	389	18	means	mean	VERB
ejpam-4870	389	19	that	that	SCONJ
ejpam-4870	389	20	ec(n	ec(n	PROPN
ejpam-4870	389	21	,	,	PUNCT
ejpam-4870	389	22	j	j	NOUN
ejpam-4870	389	23	)	)	PUNCT
ejpam-4870	389	24	is	be	AUX
ejpam-4870	389	25	a	a	DET
ejpam-4870	389	26	regular	regular	ADJ
ejpam-4870	389	27	graph	graph	NOUN
ejpam-4870	389	28	.	.	PUNCT
ejpam-4870	390	1	j.c	j.c	PROPN
ejpam-4870	390	2	.	.	PROPN
ejpam-4870	390	3	bonifacio	bonifacio	PROPN
ejpam-4870	390	4	,	,	PUNCT
ejpam-4870	390	5	c.j	c.j	PROPN
ejpam-4870	390	6	.	.	PROPN
ejpam-4870	390	7	andaya	andaya	PROPN
ejpam-4870	390	8	,	,	PUNCT
ejpam-4870	390	9	d.	d.	PROPN
ejpam-4870	390	10	magpantay	magpantay	PROPN
ejpam-4870	390	11	/	/	SYM
ejpam-4870	390	12	eur	eur	PROPN
ejpam-4870	390	13	.	.	PUNCT
ejpam-4870	391	1	j.	j.	PROPN
ejpam-4870	391	2	pure	pure	PROPN
ejpam-4870	391	3	appl	appl	PROPN
ejpam-4870	391	4	.	.	PROPN
ejpam-4870	391	5	math	math	PROPN
ejpam-4870	391	6	,	,	PUNCT
ejpam-4870	391	7	16	16	NUM
ejpam-4870	391	8	(	(	PUNCT
ejpam-4870	391	9	4	4	NUM
ejpam-4870	391	10	)	)	PUNCT
ejpam-4870	391	11	(	(	PUNCT
ejpam-4870	391	12	2023	2023	NUM
ejpam-4870	391	13	)	)	PUNCT
ejpam-4870	391	14	,	,	PUNCT
ejpam-4870	391	15	2476	2476	NUM
ejpam-4870	391	16	-	-	SYM
ejpam-4870	391	17	2498	2498	NUM
ejpam-4870	391	18	2493	2493	NUM
ejpam-4870	391	19	{	{	PUNCT
ejpam-4870	391	20	12	12	NUM
ejpam-4870	391	21	,	,	PUNCT
ejpam-4870	391	22	23	23	NUM
ejpam-4870	391	23	,	,	PUNCT
ejpam-4870	391	24	34	34	NUM
ejpam-4870	391	25	}	}	PUNCT
ejpam-4870	391	26	{	{	PUNCT
ejpam-4870	391	27	12	12	NUM
ejpam-4870	391	28	,	,	PUNCT
ejpam-4870	391	29	23	23	NUM
ejpam-4870	391	30	,	,	PUNCT
ejpam-4870	391	31	45	45	NUM
ejpam-4870	391	32	}	}	PUNCT
ejpam-4870	391	33	{	{	PUNCT
ejpam-4870	391	34	12	12	NUM
ejpam-4870	391	35	,	,	PUNCT
ejpam-4870	391	36	23	23	NUM
ejpam-4870	391	37	,	,	PUNCT
ejpam-4870	391	38	51	51	NUM
ejpam-4870	391	39	}	}	PUNCT
ejpam-4870	391	40	{	{	PUNCT
ejpam-4870	391	41	12	12	NUM
ejpam-4870	391	42	,	,	PUNCT
ejpam-4870	391	43	34	34	NUM
ejpam-4870	391	44	,	,	PUNCT
ejpam-4870	391	45	45	45	NUM
ejpam-4870	391	46	}	}	PUNCT
ejpam-4870	391	47	{	{	PUNCT
ejpam-4870	391	48	12	12	NUM
ejpam-4870	391	49	,	,	PUNCT
ejpam-4870	391	50	34	34	NUM
ejpam-4870	391	51	,	,	PUNCT
ejpam-4870	391	52	51	51	NUM
ejpam-4870	391	53	}	}	PUNCT
ejpam-4870	391	54	{	{	PUNCT
ejpam-4870	391	55	12	12	NUM
ejpam-4870	391	56	,	,	PUNCT
ejpam-4870	391	57	45	45	NUM
ejpam-4870	391	58	,	,	PUNCT
ejpam-4870	391	59	51	51	NUM
ejpam-4870	391	60	}	}	PUNCT
ejpam-4870	391	61	{	{	PUNCT
ejpam-4870	391	62	23	23	NUM
ejpam-4870	391	63	,	,	PUNCT
ejpam-4870	391	64	34	34	NUM
ejpam-4870	391	65	,	,	PUNCT
ejpam-4870	391	66	45	45	NUM
ejpam-4870	391	67	}	}	PUNCT
ejpam-4870	391	68	{	{	PUNCT
ejpam-4870	391	69	23	23	NUM
ejpam-4870	391	70	,	,	PUNCT
ejpam-4870	391	71	34	34	NUM
ejpam-4870	391	72	,	,	PUNCT
ejpam-4870	391	73	51	51	NUM
ejpam-4870	391	74	}	}	PUNCT
ejpam-4870	391	75	{	{	PUNCT
ejpam-4870	391	76	23	23	NUM
ejpam-4870	391	77	,	,	PUNCT
ejpam-4870	391	78	45	45	NUM
ejpam-4870	391	79	,	,	PUNCT
ejpam-4870	391	80	51	51	NUM
ejpam-4870	391	81	}	}	PUNCT
ejpam-4870	391	82	{	{	PUNCT
ejpam-4870	391	83	34	34	NUM
ejpam-4870	391	84	,	,	PUNCT
ejpam-4870	391	85	45	45	NUM
ejpam-4870	391	86	,	,	PUNCT
ejpam-4870	391	87	51	51	NUM
ejpam-4870	391	88	}	}	PUNCT
ejpam-4870	391	89	figure	figure	NOUN
ejpam-4870	391	90	19	19	NUM
ejpam-4870	391	91	:	:	PUNCT
ejpam-4870	391	92	a	a	DET
ejpam-4870	391	93	pictorial	pictorial	ADJ
ejpam-4870	391	94	representation	representation	NOUN
ejpam-4870	391	95	of	of	ADP
ejpam-4870	391	96	ec(5,3	ec(5,3	PROPN
ejpam-4870	391	97	)	)	PUNCT
ejpam-4870	391	98	corollary	corollary	ADJ
ejpam-4870	391	99	1	1	NUM
ejpam-4870	391	100	.	.	PUNCT
ejpam-4870	392	1	let	let	AUX
ejpam-4870	392	2	ec(n	ec(n	PROPN
ejpam-4870	392	3	,	,	PUNCT
ejpam-4870	392	4	j	j	PROPN
ejpam-4870	392	5	)	)	PUNCT
ejpam-4870	392	6	be	be	VERB
ejpam-4870	392	7	a	a	DET
ejpam-4870	392	8	j	j	NOUN
ejpam-4870	392	9	-	-	PUNCT
ejpam-4870	392	10	edge	edge	NOUN
ejpam-4870	392	11	intersection	intersection	NOUN
ejpam-4870	392	12	graph	graph	NOUN
ejpam-4870	392	13	of	of	ADP
ejpam-4870	392	14	cn	cn	PROPN
ejpam-4870	392	15	.	.	PUNCT
ejpam-4870	393	1	then	then	ADV
ejpam-4870	393	2	ec(n	ec(n	NUM
ejpam-4870	393	3	,	,	PUNCT
ejpam-4870	393	4	j	j	PROPN
ejpam-4870	393	5	)	)	PUNCT
ejpam-4870	393	6	is	be	AUX
ejpam-4870	393	7	an	an	DET
ejpam-4870	393	8	r	r	NOUN
ejpam-4870	393	9	-	-	PUNCT
ejpam-4870	393	10	regular	regular	ADJ
ejpam-4870	393	11	graph	graph	NOUN
ejpam-4870	393	12	where	where	SCONJ
ejpam-4870	393	13	r	r	NOUN
ejpam-4870	393	14	=	=	SYM
ejpam-4870	393	15	j	j	PROPN
ejpam-4870	393	16	(	(	PUNCT
ejpam-4870	393	17	n−j	n−j	ADV
ejpam-4870	393	18	j−1	j−1	PROPN
ejpam-4870	393	19	)	)	PUNCT
ejpam-4870	393	20	if	if	SCONJ
ejpam-4870	393	21	2	2	NUM
ejpam-4870	393	22	≤	≤	NUM
ejpam-4870	393	23	j	j	PROPN
ejpam-4870	393	24	≤	≤	PROPN
ejpam-4870	393	25	⌈n2	⌈n2	VERB
ejpam-4870	393	26	⌉.	⌉.	ADJ
ejpam-4870	393	27	proof	proof	NOUN
ejpam-4870	393	28	.	.	PUNCT
ejpam-4870	394	1	this	this	PRON
ejpam-4870	394	2	is	be	AUX
ejpam-4870	394	3	the	the	DET
ejpam-4870	394	4	direct	direct	ADJ
ejpam-4870	394	5	consequence	consequence	NOUN
ejpam-4870	394	6	of	of	ADP
ejpam-4870	394	7	theorem	theorem	ADJ
ejpam-4870	394	8	6	6	NUM
ejpam-4870	394	9	.	.	PUNCT
ejpam-4870	394	10	illustration	illustration	NOUN
ejpam-4870	394	11	8	8	NUM
ejpam-4870	394	12	.	.	PUNCT
ejpam-4870	395	1	consider	consider	VERB
ejpam-4870	395	2	the	the	DET
ejpam-4870	395	3	pictorial	pictorial	ADJ
ejpam-4870	395	4	representation	representation	NOUN
ejpam-4870	395	5	of	of	ADP
ejpam-4870	395	6	ec(5,3	ec(5,3	PROPN
ejpam-4870	395	7	)	)	PUNCT
ejpam-4870	395	8	in	in	ADP
ejpam-4870	395	9	figure	figure	NOUN
ejpam-4870	395	10	19	19	NUM
ejpam-4870	395	11	.	.	PUNCT
ejpam-4870	396	1	since	since	SCONJ
ejpam-4870	396	2	deg(a	deg(a	PROPN
ejpam-4870	396	3	)	)	PUNCT
ejpam-4870	396	4	=	=	SYM
ejpam-4870	396	5	3	3	NUM
ejpam-4870	396	6	for	for	ADP
ejpam-4870	396	7	all	all	DET
ejpam-4870	396	8	a	a	DET
ejpam-4870	396	9	∈	∈	PROPN
ejpam-4870	396	10	ec(5,3	ec(5,3	PROPN
ejpam-4870	396	11	)	)	PUNCT
ejpam-4870	396	12	,	,	PUNCT
ejpam-4870	396	13	it	it	PRON
ejpam-4870	396	14	follows	follow	VERB
ejpam-4870	396	15	that	that	SCONJ
ejpam-4870	396	16	ec(5,3	ec(5,3	PROPN
ejpam-4870	396	17	)	)	PUNCT
ejpam-4870	396	18	is	be	AUX
ejpam-4870	396	19	a	a	DET
ejpam-4870	396	20	3	3	NUM
ejpam-4870	396	21	-	-	PUNCT
ejpam-4870	396	22	regular	regular	ADJ
ejpam-4870	396	23	graph	graph	NOUN
ejpam-4870	396	24	.	.	PUNCT
ejpam-4870	397	1	in	in	ADP
ejpam-4870	397	2	describing	describe	VERB
ejpam-4870	397	3	a	a	DET
ejpam-4870	397	4	graph	graph	NOUN
ejpam-4870	397	5	,	,	PUNCT
ejpam-4870	397	6	the	the	DET
ejpam-4870	397	7	size	size	NOUN
ejpam-4870	397	8	of	of	ADP
ejpam-4870	397	9	the	the	DET
ejpam-4870	397	10	graph	graph	NOUN
ejpam-4870	397	11	is	be	AUX
ejpam-4870	397	12	one	one	NUM
ejpam-4870	397	13	important	important	ADJ
ejpam-4870	397	14	characteristic	characteristic	ADJ
ejpam-4870	397	15	to	to	PART
ejpam-4870	397	16	consider	consider	VERB
ejpam-4870	397	17	.	.	PUNCT
ejpam-4870	398	1	it	it	PRON
ejpam-4870	398	2	can	can	AUX
ejpam-4870	398	3	be	be	AUX
ejpam-4870	398	4	noted	note	VERB
ejpam-4870	398	5	that	that	SCONJ
ejpam-4870	398	6	ec(n	ec(n	PROPN
ejpam-4870	398	7	,	,	PUNCT
ejpam-4870	398	8	j	j	NOUN
ejpam-4870	398	9	)	)	PUNCT
ejpam-4870	398	10	is	be	AUX
ejpam-4870	398	11	a	a	DET
ejpam-4870	398	12	regular	regular	ADJ
ejpam-4870	398	13	graph	graph	NOUN
ejpam-4870	398	14	;	;	PUNCT
ejpam-4870	398	15	thus	thus	ADV
ejpam-4870	398	16	,	,	PUNCT
ejpam-4870	398	17	corollary	corollary	ADJ
ejpam-4870	398	18	1	1	NUM
ejpam-4870	398	19	and	and	CCONJ
ejpam-4870	398	20	equation	equation	NOUN
ejpam-4870	398	21	1	1	NUM
ejpam-4870	398	22	can	can	AUX
ejpam-4870	398	23	be	be	AUX
ejpam-4870	398	24	used	use	VERB
ejpam-4870	398	25	to	to	PART
ejpam-4870	398	26	determine	determine	VERB
ejpam-4870	398	27	the	the	DET
ejpam-4870	398	28	size	size	NOUN
ejpam-4870	398	29	of	of	ADP
ejpam-4870	398	30	ec(n	ec(n	PROPN
ejpam-4870	398	31	,	,	PUNCT
ejpam-4870	398	32	j	j	PROPN
ejpam-4870	398	33	)	)	PUNCT
ejpam-4870	398	34	.	.	PUNCT
ejpam-4870	399	1	theorem	theorem	VERB
ejpam-4870	399	2	7	7	NUM
ejpam-4870	399	3	.	.	PUNCT
ejpam-4870	400	1	let	let	AUX
ejpam-4870	400	2	ec(n	ec(n	PROPN
ejpam-4870	400	3	,	,	PUNCT
ejpam-4870	400	4	j	j	PROPN
ejpam-4870	400	5	)	)	PUNCT
ejpam-4870	400	6	be	be	VERB
ejpam-4870	400	7	a	a	DET
ejpam-4870	400	8	j	j	NOUN
ejpam-4870	400	9	-	-	PUNCT
ejpam-4870	400	10	edge	edge	NOUN
ejpam-4870	400	11	intersection	intersection	NOUN
ejpam-4870	400	12	graph	graph	NOUN
ejpam-4870	400	13	of	of	ADP
ejpam-4870	400	14	cn	cn	PROPN
ejpam-4870	400	15	.	.	PUNCT
ejpam-4870	401	1	if	if	SCONJ
ejpam-4870	401	2	2	2	NUM
ejpam-4870	401	3	≤	≤	NUM
ejpam-4870	401	4	j	j	PROPN
ejpam-4870	401	5	≤	≤	PROPN
ejpam-4870	401	6	⌈n2	⌈n2	NOUN
ejpam-4870	401	7	⌉	⌉	NOUN
ejpam-4870	401	8	,	,	PUNCT
ejpam-4870	401	9	then	then	ADV
ejpam-4870	401	10	the	the	DET
ejpam-4870	401	11	size	size	NOUN
ejpam-4870	401	12	of	of	ADP
ejpam-4870	401	13	ec(n	ec(n	PROPN
ejpam-4870	401	14	,	,	PUNCT
ejpam-4870	401	15	j	j	NOUN
ejpam-4870	401	16	)	)	PUNCT
ejpam-4870	401	17	is	be	AUX
ejpam-4870	401	18	given	give	VERB
ejpam-4870	401	19	by	by	ADP
ejpam-4870	401	20	|e(ec(n	|e(ec(n	NOUN
ejpam-4870	401	21	,	,	PUNCT
ejpam-4870	401	22	j	j	NOUN
ejpam-4870	401	23	)	)	PUNCT
ejpam-4870	401	24	)	)	PUNCT
ejpam-4870	402	1	|	|	ADV
ejpam-4870	402	2	=	=	SYM
ejpam-4870	402	3	j(n−j	j(n−j	NUM
ejpam-4870	402	4	j−1	j−1	PROPN
ejpam-4870	402	5	)	)	PUNCT
ejpam-4870	402	6	(	(	PUNCT
ejpam-4870	402	7	n	n	X
ejpam-4870	402	8	j	j	NOUN
ejpam-4870	402	9	)	)	PUNCT
ejpam-4870	402	10	2	2	NUM
ejpam-4870	402	11	.	.	PUNCT
ejpam-4870	403	1	proof	proof	NOUN
ejpam-4870	403	2	.	.	PUNCT
ejpam-4870	404	1	by	by	ADP
ejpam-4870	404	2	theorem	theorem	NOUN
ejpam-4870	404	3	11	11	NUM
ejpam-4870	404	4	,	,	PUNCT
ejpam-4870	404	5	the	the	DET
ejpam-4870	404	6	order	order	NOUN
ejpam-4870	404	7	of	of	ADP
ejpam-4870	404	8	ec(n	ec(n	PROPN
ejpam-4870	404	9	,	,	PUNCT
ejpam-4870	404	10	j	j	PROPN
ejpam-4870	404	11	)	)	PUNCT
ejpam-4870	404	12	is	be	AUX
ejpam-4870	404	13	(	(	PUNCT
ejpam-4870	404	14	n	n	X
ejpam-4870	404	15	j	j	PROPN
ejpam-4870	404	16	)	)	PUNCT
ejpam-4870	404	17	and	and	CCONJ
ejpam-4870	404	18	by	by	ADP
ejpam-4870	404	19	theorem	theorem	NOUN
ejpam-4870	404	20	1	1	NUM
ejpam-4870	404	21	,	,	PUNCT
ejpam-4870	404	22	ec(n	ec(n	NUM
ejpam-4870	404	23	,	,	PUNCT
ejpam-4870	404	24	j	j	NOUN
ejpam-4870	404	25	)	)	PUNCT
ejpam-4870	404	26	is	be	AUX
ejpam-4870	404	27	a	a	DET
ejpam-4870	404	28	regular	regular	ADJ
ejpam-4870	404	29	graph	graph	NOUN
ejpam-4870	404	30	.	.	PUNCT
ejpam-4870	405	1	using	use	VERB
ejpam-4870	405	2	equation	equation	NOUN
ejpam-4870	405	3	1	1	NUM
ejpam-4870	405	4	,	,	PUNCT
ejpam-4870	405	5	the	the	DET
ejpam-4870	405	6	size	size	NOUN
ejpam-4870	405	7	of	of	ADP
ejpam-4870	405	8	ec(n	ec(n	PROPN
ejpam-4870	405	9	,	,	PUNCT
ejpam-4870	405	10	j	j	PROPN
ejpam-4870	405	11	)	)	PUNCT
ejpam-4870	405	12	is	be	AUX
ejpam-4870	405	13	(	(	PUNCT
ejpam-4870	405	14	nj)(r	nj)(r	NOUN
ejpam-4870	405	15	)	)	PUNCT
ejpam-4870	405	16	2	2	NUM
ejpam-4870	405	17	where	where	SCONJ
ejpam-4870	405	18	r	r	NOUN
ejpam-4870	405	19	is	be	AUX
ejpam-4870	405	20	the	the	DET
ejpam-4870	405	21	degree	degree	NOUN
ejpam-4870	405	22	of	of	ADP
ejpam-4870	405	23	every	every	DET
ejpam-4870	405	24	vertex	vertex	NOUN
ejpam-4870	405	25	in	in	ADP
ejpam-4870	405	26	ec(n	ec(n	PROPN
ejpam-4870	405	27	,	,	PUNCT
ejpam-4870	405	28	j	j	PROPN
ejpam-4870	405	29	)	)	PUNCT
ejpam-4870	405	30	.	.	PUNCT
ejpam-4870	406	1	now	now	ADV
ejpam-4870	406	2	,	,	PUNCT
ejpam-4870	406	3	if	if	SCONJ
ejpam-4870	406	4	2	2	NUM
ejpam-4870	406	5	≤	≤	NUM
ejpam-4870	406	6	j	j	PROPN
ejpam-4870	406	7	≤	≤	PROPN
ejpam-4870	406	8	⌈n2	⌈n2	NOUN
ejpam-4870	406	9	⌉	⌉	NOUN
ejpam-4870	406	10	,	,	PUNCT
ejpam-4870	406	11	s(cn	s(cn	NUM
ejpam-4870	406	12	,	,	PUNCT
ejpam-4870	406	13	j	j	NOUN
ejpam-4870	406	14	)	)	PUNCT
ejpam-4870	406	15	is	be	AUX
ejpam-4870	406	16	a	a	DET
ejpam-4870	406	17	{	{	PUNCT
ejpam-4870	406	18	j	j	PROPN
ejpam-4870	406	19	(	(	PUNCT
ejpam-4870	406	20	n−j	n−j	X
ejpam-4870	406	21	j−1	j−1	X
ejpam-4870	406	22	)	)	PUNCT
ejpam-4870	406	23	}	}	PUNCT
ejpam-4870	406	24	-regular	-regular	ADJ
ejpam-4870	406	25	graph	graph	NOUN
ejpam-4870	406	26	.	.	PUNCT
ejpam-4870	407	1	hence	hence	ADV
ejpam-4870	407	2	,	,	PUNCT
ejpam-4870	407	3	|e(ec(n	|e(ec(n	PROPN
ejpam-4870	407	4	,	,	PUNCT
ejpam-4870	407	5	j	j	NOUN
ejpam-4870	407	6	)	)	PUNCT
ejpam-4870	407	7	)	)	PUNCT
ejpam-4870	408	1	|	|	ADV
ejpam-4870	408	2	=	=	SYM
ejpam-4870	408	3	j(n−j	j(n−j	NUM
ejpam-4870	408	4	j−1	j−1	PROPN
ejpam-4870	408	5	)	)	PUNCT
ejpam-4870	408	6	(	(	PUNCT
ejpam-4870	408	7	n	n	X
ejpam-4870	408	8	j	j	NOUN
ejpam-4870	408	9	)	)	PUNCT
ejpam-4870	408	10	2	2	NUM
ejpam-4870	408	11	.	.	PUNCT
ejpam-4870	409	1	the	the	DET
ejpam-4870	409	2	next	next	ADJ
ejpam-4870	409	3	illustration	illustration	NOUN
ejpam-4870	409	4	shows	show	VERB
ejpam-4870	409	5	the	the	DET
ejpam-4870	409	6	size	size	NOUN
ejpam-4870	409	7	of	of	ADP
ejpam-4870	409	8	ec(n	ec(n	PROPN
ejpam-4870	409	9	,	,	PUNCT
ejpam-4870	409	10	j	j	NOUN
ejpam-4870	409	11	)	)	PUNCT
ejpam-4870	409	12	given	give	VERB
ejpam-4870	409	13	that	that	SCONJ
ejpam-4870	409	14	2	2	NUM
ejpam-4870	409	15	≤	≤	NUM
ejpam-4870	409	16	j	j	PROPN
ejpam-4870	409	17	≤	≤	PROPN
ejpam-4870	409	18	⌈n2	⌈n2	NOUN
ejpam-4870	409	19	⌉.	⌉.	ADV
ejpam-4870	409	20	illustration	illustration	NOUN
ejpam-4870	409	21	9	9	NUM
ejpam-4870	409	22	.	.	PUNCT
ejpam-4870	409	23	given	give	VERB
ejpam-4870	409	24	ec(5,3	ec(5,3	PROPN
ejpam-4870	409	25	)	)	PUNCT
ejpam-4870	409	26	shown	show	VERB
ejpam-4870	409	27	in	in	ADP
ejpam-4870	409	28	figure	figure	NOUN
ejpam-4870	409	29	19	19	NUM
ejpam-4870	409	30	.	.	PUNCT
ejpam-4870	410	1	we	we	PRON
ejpam-4870	410	2	know	know	VERB
ejpam-4870	410	3	that	that	PRON
ejpam-4870	410	4	j	j	PROPN
ejpam-4870	411	1	=	=	PUNCT
ejpam-4870	411	2	3	3	NUM
ejpam-4870	411	3	which	which	PRON
ejpam-4870	411	4	means	mean	VERB
ejpam-4870	411	5	that	that	SCONJ
ejpam-4870	411	6	⌈52⌉	⌈52⌉	NOUN
ejpam-4870	411	7	=	=	NOUN
ejpam-4870	411	8	3	3	X
ejpam-4870	411	9	=	=	SYM
ejpam-4870	411	10	j.	j.	PROPN
ejpam-4870	411	11	since	since	SCONJ
ejpam-4870	411	12	ec(5,3	ec(5,3	PROPN
ejpam-4870	411	13	)	)	PUNCT
ejpam-4870	411	14	is	be	AUX
ejpam-4870	411	15	a	a	DET
ejpam-4870	411	16	graph	graph	NOUN
ejpam-4870	411	17	of	of	ADP
ejpam-4870	411	18	order	order	NOUN
ejpam-4870	411	19	10	10	NUM
ejpam-4870	411	20	and	and	CCONJ
ejpam-4870	411	21	a	a	DET
ejpam-4870	411	22	3	3	NUM
ejpam-4870	411	23	-	-	PUNCT
ejpam-4870	411	24	regular	regular	ADJ
ejpam-4870	411	25	graph	graph	NOUN
ejpam-4870	411	26	,	,	PUNCT
ejpam-4870	411	27	using	use	VERB
ejpam-4870	411	28	theorem	theorem	NOUN
ejpam-4870	411	29	7	7	NUM
ejpam-4870	411	30	,	,	PUNCT
ejpam-4870	411	31	|e(ec(5,3	|e(ec(5,3	PROPN
ejpam-4870	411	32	)	)	PUNCT
ejpam-4870	411	33	)	)	PUNCT
ejpam-4870	412	1	|	|	ADV
ejpam-4870	412	2	=	=	SYM
ejpam-4870	412	3	10(3	10(3	X
ejpam-4870	412	4	)	)	PUNCT
ejpam-4870	412	5	2	2	NUM
ejpam-4870	412	6	=	=	SYM
ejpam-4870	412	7	15	15	NUM
ejpam-4870	412	8	.	.	PUNCT
ejpam-4870	413	1	j.c	j.c	PROPN
ejpam-4870	413	2	.	.	PROPN
ejpam-4870	413	3	bonifacio	bonifacio	PROPN
ejpam-4870	413	4	,	,	PUNCT
ejpam-4870	413	5	c.j	c.j	PROPN
ejpam-4870	413	6	.	.	PROPN
ejpam-4870	413	7	andaya	andaya	PROPN
ejpam-4870	413	8	,	,	PUNCT
ejpam-4870	413	9	d.	d.	PROPN
ejpam-4870	413	10	magpantay	magpantay	PROPN
ejpam-4870	413	11	/	/	SYM
ejpam-4870	413	12	eur	eur	PROPN
ejpam-4870	413	13	.	.	PUNCT
ejpam-4870	414	1	j.	j.	PROPN
ejpam-4870	414	2	pure	pure	PROPN
ejpam-4870	414	3	appl	appl	PROPN
ejpam-4870	414	4	.	.	PROPN
ejpam-4870	414	5	math	math	PROPN
ejpam-4870	414	6	,	,	PUNCT
ejpam-4870	414	7	16	16	NUM
ejpam-4870	414	8	(	(	PUNCT
ejpam-4870	414	9	4	4	NUM
ejpam-4870	414	10	)	)	PUNCT
ejpam-4870	414	11	(	(	PUNCT
ejpam-4870	414	12	2023	2023	NUM
ejpam-4870	414	13	)	)	PUNCT
ejpam-4870	414	14	,	,	PUNCT
ejpam-4870	414	15	2476	2476	NUM
ejpam-4870	414	16	-	-	SYM
ejpam-4870	414	17	2498	2498	NUM
ejpam-4870	414	18	2494	2494	NUM
ejpam-4870	414	19	there	there	PRON
ejpam-4870	414	20	are	be	VERB
ejpam-4870	414	21	times	time	NOUN
ejpam-4870	414	22	that	that	SCONJ
ejpam-4870	414	23	a	a	DET
ejpam-4870	414	24	ec(n	ec(n	PROPN
ejpam-4870	414	25	,	,	PUNCT
ejpam-4870	414	26	j	j	NOUN
ejpam-4870	414	27	)	)	PUNCT
ejpam-4870	414	28	is	be	AUX
ejpam-4870	414	29	isomorphic	isomorphic	ADJ
ejpam-4870	414	30	to	to	ADP
ejpam-4870	414	31	some	some	DET
ejpam-4870	414	32	special	special	ADJ
ejpam-4870	414	33	classes	class	NOUN
ejpam-4870	414	34	of	of	ADP
ejpam-4870	414	35	a	a	DET
ejpam-4870	414	36	graph	graph	NOUN
ejpam-4870	414	37	.	.	PUNCT
ejpam-4870	415	1	the	the	DET
ejpam-4870	415	2	next	next	ADJ
ejpam-4870	415	3	theorem	theorem	NOUN
ejpam-4870	415	4	provides	provide	VERB
ejpam-4870	415	5	necessary	necessary	ADJ
ejpam-4870	415	6	and	and	CCONJ
ejpam-4870	415	7	sufficient	sufficient	ADJ
ejpam-4870	415	8	conditions	condition	NOUN
ejpam-4870	415	9	when	when	SCONJ
ejpam-4870	415	10	ec(n	ec(n	NUM
ejpam-4870	415	11	,	,	PUNCT
ejpam-4870	415	12	j	j	PROPN
ejpam-4870	415	13	)	)	PUNCT
ejpam-4870	415	14	is	be	AUX
ejpam-4870	415	15	a	a	DET
ejpam-4870	415	16	cycle	cycle	NOUN
ejpam-4870	415	17	graph	graph	NOUN
ejpam-4870	415	18	of	of	ADP
ejpam-4870	415	19	order	order	NOUN
ejpam-4870	415	20	3	3	NUM
ejpam-4870	415	21	.	.	PUNCT
ejpam-4870	415	22	theorem	theorem	VERB
ejpam-4870	415	23	8	8	NUM
ejpam-4870	415	24	.	.	PUNCT
ejpam-4870	416	1	a	a	DET
ejpam-4870	416	2	j	j	NOUN
ejpam-4870	416	3	-	-	PUNCT
ejpam-4870	416	4	edge	edge	NOUN
ejpam-4870	416	5	intersection	intersection	NOUN
ejpam-4870	416	6	graph	graph	NOUN
ejpam-4870	416	7	of	of	ADP
ejpam-4870	416	8	cn	cn	PROPN
ejpam-4870	416	9	ec(n	ec(n	PROPN
ejpam-4870	416	10	,	,	PUNCT
ejpam-4870	416	11	j	j	PROPN
ejpam-4870	416	12	)	)	PUNCT
ejpam-4870	416	13	is	be	AUX
ejpam-4870	416	14	a	a	DET
ejpam-4870	416	15	cycle	cycle	NOUN
ejpam-4870	416	16	graph	graph	NOUN
ejpam-4870	416	17	of	of	ADP
ejpam-4870	416	18	order	order	NOUN
ejpam-4870	416	19	3	3	NUM
ejpam-4870	416	20	if	if	SCONJ
ejpam-4870	416	21	and	and	CCONJ
ejpam-4870	416	22	only	only	ADV
ejpam-4870	416	23	if	if	SCONJ
ejpam-4870	416	24	n	n	PROPN
ejpam-4870	416	25	=	=	SYM
ejpam-4870	416	26	3	3	NUM
ejpam-4870	416	27	and	and	CCONJ
ejpam-4870	416	28	j	j	NOUN
ejpam-4870	416	29	=	=	NOUN
ejpam-4870	416	30	2	2	X
ejpam-4870	416	31	.	.	PUNCT
ejpam-4870	416	32	proof	proof	NOUN
ejpam-4870	416	33	.	.	PUNCT
ejpam-4870	417	1	assume	assume	VERB
ejpam-4870	417	2	that	that	SCONJ
ejpam-4870	417	3	ec(n	ec(n	PROPN
ejpam-4870	417	4	,	,	PUNCT
ejpam-4870	417	5	j	j	NOUN
ejpam-4870	417	6	)	)	PUNCT
ejpam-4870	417	7	is	be	AUX
ejpam-4870	417	8	a	a	DET
ejpam-4870	417	9	cycle	cycle	NOUN
ejpam-4870	417	10	graph	graph	NOUN
ejpam-4870	417	11	of	of	ADP
ejpam-4870	417	12	order	order	NOUN
ejpam-4870	417	13	3	3	X
ejpam-4870	417	14	.	.	PUNCT
ejpam-4870	418	1	we	we	PRON
ejpam-4870	418	2	know	know	VERB
ejpam-4870	418	3	that	that	SCONJ
ejpam-4870	418	4	every	every	DET
ejpam-4870	418	5	cycle	cycle	NOUN
ejpam-4870	418	6	graph	graph	NOUN
ejpam-4870	418	7	is	be	AUX
ejpam-4870	418	8	a	a	DET
ejpam-4870	418	9	2	2	NUM
ejpam-4870	418	10	-	-	PUNCT
ejpam-4870	418	11	regular	regular	ADJ
ejpam-4870	418	12	graph	graph	NOUN
ejpam-4870	418	13	.	.	PUNCT
ejpam-4870	419	1	suppose	suppose	VERB
ejpam-4870	419	2	that	that	SCONJ
ejpam-4870	419	3	n	n	PROPN
ejpam-4870	419	4	̸=	̸=	PROPN
ejpam-4870	419	5	3	3	NUM
ejpam-4870	419	6	or	or	CCONJ
ejpam-4870	419	7	j	j	PROPN
ejpam-4870	419	8	̸=	̸=	PROPN
ejpam-4870	419	9	2	2	NUM
ejpam-4870	419	10	.	.	PUNCT
ejpam-4870	420	1	now	now	ADV
ejpam-4870	420	2	,	,	PUNCT
ejpam-4870	420	3	if	if	SCONJ
ejpam-4870	420	4	n	n	PROPN
ejpam-4870	420	5	>	>	X
ejpam-4870	420	6	3	3	NUM
ejpam-4870	420	7	,	,	PUNCT
ejpam-4870	420	8	then	then	ADV
ejpam-4870	420	9	(	(	PUNCT
ejpam-4870	420	10	n	n	X
ejpam-4870	420	11	j	j	NOUN
ejpam-4870	420	12	)	)	PUNCT
ejpam-4870	421	1	=	=	SYM
ejpam-4870	421	2	1	1	NUM
ejpam-4870	421	3	or	or	CCONJ
ejpam-4870	421	4	(	(	PUNCT
ejpam-4870	421	5	n	n	CCONJ
ejpam-4870	421	6	j	j	PROPN
ejpam-4870	421	7	)	)	PUNCT
ejpam-4870	421	8	≥	≥	PROPN
ejpam-4870	421	9	4	4	NUM
ejpam-4870	421	10	.	.	PUNCT
ejpam-4870	422	1	this	this	PRON
ejpam-4870	422	2	contradicts	contradict	VERB
ejpam-4870	422	3	the	the	DET
ejpam-4870	422	4	fact	fact	NOUN
ejpam-4870	422	5	that	that	SCONJ
ejpam-4870	422	6	the	the	DET
ejpam-4870	422	7	order	order	NOUN
ejpam-4870	422	8	of	of	ADP
ejpam-4870	422	9	ec(n	ec(n	PROPN
ejpam-4870	422	10	,	,	PUNCT
ejpam-4870	422	11	j	j	PROPN
ejpam-4870	422	12	)	)	PUNCT
ejpam-4870	422	13	is	be	AUX
ejpam-4870	422	14	3	3	NUM
ejpam-4870	422	15	.	.	PUNCT
ejpam-4870	423	1	on	on	ADP
ejpam-4870	423	2	the	the	DET
ejpam-4870	423	3	other	other	ADJ
ejpam-4870	423	4	hand	hand	NOUN
ejpam-4870	423	5	,	,	PUNCT
ejpam-4870	423	6	if	if	SCONJ
ejpam-4870	423	7	j	j	PROPN
ejpam-4870	423	8	<	<	X
ejpam-4870	423	9	2	2	NUM
ejpam-4870	423	10	,	,	PUNCT
ejpam-4870	423	11	then	then	ADV
ejpam-4870	423	12	j	j	PROPN
ejpam-4870	423	13	is	be	AUX
ejpam-4870	423	14	1	1	NUM
ejpam-4870	423	15	.	.	PUNCT
ejpam-4870	424	1	by	by	ADP
ejpam-4870	424	2	proposition	proposition	NOUN
ejpam-4870	424	3	4	4	NUM
ejpam-4870	424	4	,	,	PUNCT
ejpam-4870	424	5	ec(n	ec(n	NUM
ejpam-4870	424	6	,	,	PUNCT
ejpam-4870	424	7	j	j	NOUN
ejpam-4870	424	8	)	)	PUNCT
ejpam-4870	424	9	is	be	AUX
ejpam-4870	424	10	a	a	DET
ejpam-4870	424	11	trivial	trivial	ADJ
ejpam-4870	424	12	graph	graph	NOUN
ejpam-4870	424	13	,	,	PUNCT
ejpam-4870	424	14	this	this	PRON
ejpam-4870	424	15	is	be	AUX
ejpam-4870	424	16	a	a	DET
ejpam-4870	424	17	contradiction	contradiction	NOUN
ejpam-4870	424	18	to	to	ADP
ejpam-4870	424	19	the	the	DET
ejpam-4870	424	20	assumption	assumption	NOUN
ejpam-4870	424	21	that	that	SCONJ
ejpam-4870	424	22	ec(n	ec(n	PROPN
ejpam-4870	424	23	,	,	PUNCT
ejpam-4870	424	24	j	j	PROPN
ejpam-4870	424	25	)	)	PUNCT
ejpam-4870	424	26	is	be	AUX
ejpam-4870	424	27	a	a	DET
ejpam-4870	424	28	cycle	cycle	NOUN
ejpam-4870	424	29	graph	graph	NOUN
ejpam-4870	424	30	of	of	ADP
ejpam-4870	424	31	order	order	NOUN
ejpam-4870	424	32	3	3	X
ejpam-4870	424	33	.	.	PUNCT
ejpam-4870	425	1	furthermore	furthermore	ADV
ejpam-4870	425	2	,	,	PUNCT
ejpam-4870	425	3	if	if	SCONJ
ejpam-4870	425	4	j	j	PROPN
ejpam-4870	425	5	>	>	X
ejpam-4870	425	6	2	2	NUM
ejpam-4870	425	7	,	,	PUNCT
ejpam-4870	425	8	then	then	ADV
ejpam-4870	425	9	(	(	PUNCT
ejpam-4870	425	10	n	n	X
ejpam-4870	425	11	j	j	PROPN
ejpam-4870	425	12	)	)	PUNCT
ejpam-4870	425	13	is	be	AUX
ejpam-4870	425	14	equal	equal	ADJ
ejpam-4870	425	15	to	to	ADP
ejpam-4870	425	16	1	1	NUM
ejpam-4870	425	17	or	or	CCONJ
ejpam-4870	425	18	greater	great	ADJ
ejpam-4870	425	19	than	than	ADP
ejpam-4870	425	20	or	or	CCONJ
ejpam-4870	425	21	equal	equal	ADJ
ejpam-4870	425	22	to	to	ADP
ejpam-4870	425	23	4	4	NUM
ejpam-4870	425	24	which	which	PRON
ejpam-4870	425	25	is	be	AUX
ejpam-4870	425	26	a	a	DET
ejpam-4870	425	27	contradiction	contradiction	NOUN
ejpam-4870	425	28	that	that	PRON
ejpam-4870	425	29	ec(n	ec(n	PROPN
ejpam-4870	425	30	,	,	PUNCT
ejpam-4870	425	31	j	j	NOUN
ejpam-4870	425	32	)	)	PUNCT
ejpam-4870	425	33	is	be	AUX
ejpam-4870	425	34	a	a	DET
ejpam-4870	425	35	cycle	cycle	NOUN
ejpam-4870	425	36	graph	graph	NOUN
ejpam-4870	425	37	of	of	ADP
ejpam-4870	425	38	order	order	NOUN
ejpam-4870	425	39	3	3	X
ejpam-4870	425	40	.	.	PUNCT
ejpam-4870	426	1	therefore	therefore	ADV
ejpam-4870	426	2	,	,	PUNCT
ejpam-4870	426	3	n	n	PROPN
ejpam-4870	426	4	=	=	SYM
ejpam-4870	426	5	3	3	NUM
ejpam-4870	426	6	and	and	CCONJ
ejpam-4870	426	7	j	j	NOUN
ejpam-4870	426	8	=	=	NOUN
ejpam-4870	426	9	2	2	X
ejpam-4870	426	10	.	.	PUNCT
ejpam-4870	426	11	conversely	conversely	ADV
ejpam-4870	426	12	,	,	PUNCT
ejpam-4870	426	13	assume	assume	VERB
ejpam-4870	426	14	that	that	SCONJ
ejpam-4870	426	15	n	n	NOUN
ejpam-4870	426	16	=	=	SYM
ejpam-4870	426	17	3	3	NUM
ejpam-4870	426	18	and	and	CCONJ
ejpam-4870	426	19	j	j	PROPN
ejpam-4870	426	20	=	=	SYM
ejpam-4870	426	21	2	2	X
ejpam-4870	426	22	.	.	X
ejpam-4870	426	23	observe	observe	VERB
ejpam-4870	426	24	that	that	SCONJ
ejpam-4870	426	25	⌈32⌉	⌈32⌉	NOUN
ejpam-4870	426	26	=	=	SYM
ejpam-4870	426	27	2	2	X
ejpam-4870	426	28	.	.	PUNCT
ejpam-4870	426	29	by	by	ADP
ejpam-4870	426	30	lemma	lemma	PROPN
ejpam-4870	426	31	6	6	NUM
ejpam-4870	426	32	,	,	PUNCT
ejpam-4870	426	33	the	the	DET
ejpam-4870	426	34	degree	degree	NOUN
ejpam-4870	426	35	of	of	ADP
ejpam-4870	426	36	every	every	DET
ejpam-4870	426	37	vertex	vertex	NOUN
ejpam-4870	426	38	in	in	ADP
ejpam-4870	426	39	ec(3,2	ec(3,2	NOUN
ejpam-4870	426	40	)	)	PUNCT
ejpam-4870	426	41	is	be	AUX
ejpam-4870	426	42	equal	equal	ADJ
ejpam-4870	426	43	to	to	ADP
ejpam-4870	426	44	2	2	NUM
ejpam-4870	426	45	.	.	PUNCT
ejpam-4870	426	46	by	by	ADP
ejpam-4870	426	47	theorem	theorem	NOUN
ejpam-4870	426	48	3	3	NUM
ejpam-4870	426	49	,	,	PUNCT
ejpam-4870	426	50	(	(	PUNCT
ejpam-4870	426	51	3	3	NUM
ejpam-4870	426	52	2	2	NUM
ejpam-4870	426	53	)	)	PUNCT
ejpam-4870	426	54	=	=	SYM
ejpam-4870	427	1	3	3	X
ejpam-4870	427	2	.	.	PUNCT
ejpam-4870	427	3	thus	thus	ADV
ejpam-4870	427	4	,	,	PUNCT
ejpam-4870	427	5	every	every	DET
ejpam-4870	427	6	vertex	vertex	NOUN
ejpam-4870	427	7	in	in	ADP
ejpam-4870	427	8	ec(3,2	ec(3,2	NOUN
ejpam-4870	427	9	)	)	PUNCT
ejpam-4870	427	10	is	be	AUX
ejpam-4870	427	11	adjacent	adjacent	ADJ
ejpam-4870	427	12	to	to	ADP
ejpam-4870	427	13	each	each	DET
ejpam-4870	427	14	other	other	ADJ
ejpam-4870	427	15	.	.	PUNCT
ejpam-4870	428	1	therefore	therefore	ADV
ejpam-4870	428	2	,	,	PUNCT
ejpam-4870	428	3	ec(3,2	ec(3,2	NOUN
ejpam-4870	428	4	)	)	PUNCT
ejpam-4870	428	5	is	be	AUX
ejpam-4870	428	6	a	a	DET
ejpam-4870	428	7	cycle	cycle	NOUN
ejpam-4870	428	8	graph	graph	NOUN
ejpam-4870	428	9	of	of	ADP
ejpam-4870	428	10	order	order	NOUN
ejpam-4870	428	11	3	3	X
ejpam-4870	428	12	.	.	PUNCT
ejpam-4870	429	1	the	the	DET
ejpam-4870	429	2	next	next	ADJ
ejpam-4870	429	3	illustration	illustration	NOUN
ejpam-4870	429	4	shows	show	VERB
ejpam-4870	429	5	that	that	SCONJ
ejpam-4870	429	6	a	a	DET
ejpam-4870	429	7	ec(n	ec(n	PROPN
ejpam-4870	429	8	,	,	PUNCT
ejpam-4870	429	9	j	j	NOUN
ejpam-4870	429	10	)	)	PUNCT
ejpam-4870	429	11	is	be	AUX
ejpam-4870	429	12	isomorphic	isomorphic	ADJ
ejpam-4870	429	13	to	to	ADP
ejpam-4870	429	14	the	the	DET
ejpam-4870	429	15	cycle	cycle	NOUN
ejpam-4870	429	16	graph	graph	NOUN
ejpam-4870	429	17	when	when	SCONJ
ejpam-4870	429	18	n	n	PROPN
ejpam-4870	429	19	=	=	SYM
ejpam-4870	429	20	3	3	NUM
ejpam-4870	429	21	and	and	CCONJ
ejpam-4870	429	22	j	j	PROPN
ejpam-4870	430	1	=	=	NOUN
ejpam-4870	430	2	2	2	X
ejpam-4870	430	3	.	.	X
ejpam-4870	430	4	illustration	illustration	NOUN
ejpam-4870	430	5	10	10	NUM
ejpam-4870	430	6	.	.	PUNCT
ejpam-4870	431	1	consider	consider	VERB
ejpam-4870	431	2	cycle	cycle	NOUN
ejpam-4870	431	3	graph	graph	NOUN
ejpam-4870	431	4	c3	c3	PROPN
ejpam-4870	431	5	and	and	CCONJ
ejpam-4870	431	6	let	let	VERB
ejpam-4870	431	7	j	j	PROPN
ejpam-4870	431	8	=	=	NOUN
ejpam-4870	431	9	2	2	X
ejpam-4870	431	10	.	.	PUNCT
ejpam-4870	432	1	the	the	DET
ejpam-4870	432	2	vertex	vertex	NOUN
ejpam-4870	432	3	set	set	NOUN
ejpam-4870	432	4	of	of	ADP
ejpam-4870	432	5	ec(3,2	ec(3,2	NOUN
ejpam-4870	432	6	)	)	PUNCT
ejpam-4870	432	7	is	be	AUX
ejpam-4870	432	8	given	give	VERB
ejpam-4870	432	9	by	by	ADP
ejpam-4870	432	10	v	v	NOUN
ejpam-4870	432	11	(	(	PUNCT
ejpam-4870	432	12	ec(3,2	ec(3,2	NOUN
ejpam-4870	432	13	)	)	PUNCT
ejpam-4870	432	14	)	)	PUNCT
ejpam-4870	433	1	=	=	PRON
ejpam-4870	433	2	{	{	PUNCT
ejpam-4870	433	3	{	{	PUNCT
ejpam-4870	433	4	12	12	NUM
ejpam-4870	433	5	,	,	PUNCT
ejpam-4870	433	6	23	23	NUM
ejpam-4870	433	7	}	}	PUNCT
ejpam-4870	433	8	,	,	PUNCT
ejpam-4870	433	9	{	{	PUNCT
ejpam-4870	433	10	12	12	NUM
ejpam-4870	433	11	,	,	PUNCT
ejpam-4870	433	12	31	31	NUM
ejpam-4870	433	13	}	}	PUNCT
ejpam-4870	433	14	,	,	PUNCT
ejpam-4870	433	15	{	{	PUNCT
ejpam-4870	433	16	23	23	NUM
ejpam-4870	433	17	,	,	PUNCT
ejpam-4870	433	18	31	31	NUM
ejpam-4870	433	19	}	}	PUNCT
ejpam-4870	433	20	}	}	PUNCT
ejpam-4870	433	21	.	.	PUNCT
ejpam-4870	434	1	now	now	ADV
ejpam-4870	434	2	,	,	PUNCT
ejpam-4870	434	3	since	since	SCONJ
ejpam-4870	434	4	{	{	PUNCT
ejpam-4870	434	5	12	12	NUM
ejpam-4870	434	6	,	,	PUNCT
ejpam-4870	434	7	23	23	NUM
ejpam-4870	434	8	}	}	PUNCT
ejpam-4870	434	9	∩	∩	NOUN
ejpam-4870	434	10	{	{	PUNCT
ejpam-4870	434	11	12	12	NUM
ejpam-4870	434	12	,	,	PUNCT
ejpam-4870	434	13	31	31	NUM
ejpam-4870	434	14	}	}	PUNCT
ejpam-4870	434	15	=	=	PUNCT
ejpam-4870	434	16	{	{	PUNCT
ejpam-4870	434	17	12	12	NUM
ejpam-4870	434	18	}	}	PUNCT
ejpam-4870	434	19	,	,	PUNCT
ejpam-4870	434	20	it	it	PRON
ejpam-4870	434	21	follows	follow	VERB
ejpam-4870	434	22	that	that	SCONJ
ejpam-4870	434	23	[	[	X
ejpam-4870	434	24	{	{	PUNCT
ejpam-4870	434	25	12	12	NUM
ejpam-4870	434	26	,	,	PUNCT
ejpam-4870	434	27	23	23	NUM
ejpam-4870	434	28	}	}	PUNCT
ejpam-4870	434	29	,	,	PUNCT
ejpam-4870	434	30	{	{	PUNCT
ejpam-4870	434	31	12	12	NUM
ejpam-4870	434	32	,	,	PUNCT
ejpam-4870	434	33	31	31	NUM
ejpam-4870	434	34	}	}	PUNCT
ejpam-4870	434	35	]	]	PUNCT
ejpam-4870	434	36	∈	∈	PROPN
ejpam-4870	434	37	e(ec(3,2	e(ec(3,2	NOUN
ejpam-4870	434	38	)	)	PUNCT
ejpam-4870	434	39	)	)	PUNCT
ejpam-4870	434	40	.	.	PUNCT
ejpam-4870	435	1	similarly	similarly	ADV
ejpam-4870	435	2	,	,	PUNCT
ejpam-4870	435	3	[	[	X
ejpam-4870	435	4	{	{	PUNCT
ejpam-4870	435	5	12	12	NUM
ejpam-4870	435	6	,	,	PUNCT
ejpam-4870	435	7	23	23	NUM
ejpam-4870	435	8	}	}	PUNCT
ejpam-4870	435	9	,	,	PUNCT
ejpam-4870	435	10	{	{	PUNCT
ejpam-4870	435	11	23	23	NUM
ejpam-4870	435	12	,	,	PUNCT
ejpam-4870	435	13	31	31	NUM
ejpam-4870	435	14	}	}	PUNCT
ejpam-4870	435	15	]	]	PUNCT
ejpam-4870	435	16	and	and	CCONJ
ejpam-4870	435	17	[	[	X
ejpam-4870	435	18	{	{	PUNCT
ejpam-4870	435	19	12	12	NUM
ejpam-4870	435	20	,	,	PUNCT
ejpam-4870	435	21	31	31	NUM
ejpam-4870	435	22	}	}	PUNCT
ejpam-4870	435	23	,	,	PUNCT
ejpam-4870	435	24	{	{	PUNCT
ejpam-4870	435	25	23	23	NUM
ejpam-4870	435	26	,	,	PUNCT
ejpam-4870	435	27	31	31	NUM
ejpam-4870	435	28	}	}	PUNCT
ejpam-4870	435	29	]	]	PUNCT
ejpam-4870	435	30	are	be	AUX
ejpam-4870	435	31	also	also	ADV
ejpam-4870	435	32	elements	element	NOUN
ejpam-4870	435	33	of	of	ADP
ejpam-4870	435	34	e(ec(3,2	e(ec(3,2	NOUN
ejpam-4870	435	35	)	)	PUNCT
ejpam-4870	435	36	)	)	PUNCT
ejpam-4870	435	37	.	.	PUNCT
ejpam-4870	436	1	thus	thus	ADV
ejpam-4870	436	2	,	,	PUNCT
ejpam-4870	436	3	ec(3,2	ec(3,2	NOUN
ejpam-4870	436	4	)	)	PUNCT
ejpam-4870	436	5	is	be	AUX
ejpam-4870	436	6	a	a	DET
ejpam-4870	436	7	cycle	cycle	NOUN
ejpam-4870	436	8	graph	graph	NOUN
ejpam-4870	436	9	of	of	ADP
ejpam-4870	436	10	order	order	NOUN
ejpam-4870	436	11	3	3	X
ejpam-4870	436	12	.	.	PUNCT
ejpam-4870	436	13	figure	figure	NOUN
ejpam-4870	436	14	20	20	NUM
ejpam-4870	436	15	is	be	AUX
ejpam-4870	436	16	a	a	DET
ejpam-4870	436	17	pictorial	pictorial	ADJ
ejpam-4870	436	18	representation	representation	NOUN
ejpam-4870	436	19	of	of	ADP
ejpam-4870	436	20	ec(3,2	ec(3,2	NOUN
ejpam-4870	436	21	)	)	PUNCT
ejpam-4870	436	22	.	.	PUNCT
ejpam-4870	437	1	{	{	PUNCT
ejpam-4870	437	2	23	23	NUM
ejpam-4870	437	3	,	,	PUNCT
ejpam-4870	437	4	31	31	NUM
ejpam-4870	437	5	}	}	PUNCT
ejpam-4870	437	6	{	{	PUNCT
ejpam-4870	437	7	12	12	NUM
ejpam-4870	437	8	,	,	PUNCT
ejpam-4870	437	9	23	23	NUM
ejpam-4870	437	10	}	}	PUNCT
ejpam-4870	437	11	{	{	PUNCT
ejpam-4870	437	12	12	12	NUM
ejpam-4870	437	13	,	,	PUNCT
ejpam-4870	437	14	31	31	NUM
ejpam-4870	437	15	}	}	PUNCT
ejpam-4870	437	16	figure	figure	NOUN
ejpam-4870	437	17	20	20	NUM
ejpam-4870	437	18	:	:	PUNCT
ejpam-4870	437	19	pictorial	pictorial	ADJ
ejpam-4870	437	20	representation	representation	NOUN
ejpam-4870	437	21	of	of	ADP
ejpam-4870	437	22	ec(3,2	ec(3,2	NOUN
ejpam-4870	437	23	)	)	PUNCT
ejpam-4870	437	24	4	4	NUM
ejpam-4870	437	25	.	.	PUNCT
ejpam-4870	437	26	additional	additional	ADJ
ejpam-4870	437	27	parameters	parameter	NOUN
ejpam-4870	437	28	of	of	ADP
ejpam-4870	437	29	ec(n	ec(n	PROPN
ejpam-4870	437	30	,	,	PUNCT
ejpam-4870	437	31	j	j	NOUN
ejpam-4870	437	32	)	)	PUNCT
ejpam-4870	437	33	in	in	ADP
ejpam-4870	437	34	this	this	DET
ejpam-4870	437	35	section	section	NOUN
ejpam-4870	437	36	,	,	PUNCT
ejpam-4870	437	37	other	other	ADJ
ejpam-4870	437	38	parameters	parameter	NOUN
ejpam-4870	437	39	of	of	ADP
ejpam-4870	437	40	a	a	DET
ejpam-4870	437	41	graph	graph	NOUN
ejpam-4870	437	42	such	such	ADJ
ejpam-4870	437	43	as	as	ADP
ejpam-4870	437	44	independence	independence	NOUN
ejpam-4870	437	45	number	number	NOUN
ejpam-4870	437	46	,	,	PUNCT
ejpam-4870	437	47	and	and	CCONJ
ejpam-4870	437	48	domination	domination	NOUN
ejpam-4870	437	49	number	number	NOUN
ejpam-4870	437	50	are	be	AUX
ejpam-4870	437	51	discussed	discuss	VERB
ejpam-4870	437	52	.	.	PUNCT
ejpam-4870	438	1	j.c	j.c	PROPN
ejpam-4870	438	2	.	.	PROPN
ejpam-4870	438	3	bonifacio	bonifacio	PROPN
ejpam-4870	438	4	,	,	PUNCT
ejpam-4870	438	5	c.j	c.j	PROPN
ejpam-4870	438	6	.	.	PROPN
ejpam-4870	438	7	andaya	andaya	PROPN
ejpam-4870	438	8	,	,	PUNCT
ejpam-4870	438	9	d.	d.	PROPN
ejpam-4870	438	10	magpantay	magpantay	PROPN
ejpam-4870	438	11	/	/	SYM
ejpam-4870	438	12	eur	eur	PROPN
ejpam-4870	438	13	.	.	PUNCT
ejpam-4870	439	1	j.	j.	PROPN
ejpam-4870	439	2	pure	pure	PROPN
ejpam-4870	439	3	appl	appl	PROPN
ejpam-4870	439	4	.	.	PROPN
ejpam-4870	439	5	math	math	PROPN
ejpam-4870	439	6	,	,	PUNCT
ejpam-4870	439	7	16	16	NUM
ejpam-4870	439	8	(	(	PUNCT
ejpam-4870	439	9	4	4	NUM
ejpam-4870	439	10	)	)	PUNCT
ejpam-4870	439	11	(	(	PUNCT
ejpam-4870	439	12	2023	2023	NUM
ejpam-4870	439	13	)	)	PUNCT
ejpam-4870	439	14	,	,	PUNCT
ejpam-4870	439	15	2476	2476	NUM
ejpam-4870	439	16	-	-	SYM
ejpam-4870	439	17	2498	2498	NUM
ejpam-4870	439	18	2495	2495	NUM
ejpam-4870	439	19	4.1	4.1	NUM
ejpam-4870	439	20	.	.	PUNCT
ejpam-4870	440	1	independence	independence	NOUN
ejpam-4870	440	2	number	number	NOUN
ejpam-4870	440	3	of	of	ADP
ejpam-4870	440	4	ec(n,2	ec(n,2	PROPN
ejpam-4870	440	5	)	)	PUNCT
ejpam-4870	440	6	this	this	DET
ejpam-4870	440	7	subsection	subsection	NOUN
ejpam-4870	440	8	determines	determine	VERB
ejpam-4870	440	9	a	a	DET
ejpam-4870	440	10	lower	lower	ADV
ejpam-4870	440	11	bound	bind	VERB
ejpam-4870	440	12	for	for	ADP
ejpam-4870	440	13	the	the	DET
ejpam-4870	440	14	independence	independence	NOUN
ejpam-4870	440	15	number	number	NOUN
ejpam-4870	440	16	of	of	ADP
ejpam-4870	440	17	ec(n	ec(n	PROPN
ejpam-4870	440	18	,	,	PUNCT
ejpam-4870	440	19	j	j	PROPN
ejpam-4870	440	20	)	)	PUNCT
ejpam-4870	440	21	when	when	SCONJ
ejpam-4870	440	22	j	j	PROPN
ejpam-4870	440	23	=	=	SYM
ejpam-4870	440	24	2	2	X
ejpam-4870	440	25	.	.	PUNCT
ejpam-4870	440	26	theorem	theorem	NOUN
ejpam-4870	440	27	9	9	NUM
ejpam-4870	440	28	determines	determine	VERB
ejpam-4870	440	29	the	the	DET
ejpam-4870	440	30	existence	existence	NOUN
ejpam-4870	440	31	of	of	ADP
ejpam-4870	440	32	independent	independent	ADJ
ejpam-4870	440	33	set	set	NOUN
ejpam-4870	440	34	of	of	ADP
ejpam-4870	440	35	ec(n,2	ec(n,2	PROPN
ejpam-4870	440	36	)	)	PUNCT
ejpam-4870	440	37	.	.	PUNCT
ejpam-4870	441	1	theorem	theorem	VERB
ejpam-4870	441	2	9	9	NUM
ejpam-4870	441	3	.	.	PUNCT
ejpam-4870	442	1	let	let	AUX
ejpam-4870	442	2	ec(n,2	ec(n,2	PROPN
ejpam-4870	442	3	)	)	PUNCT
ejpam-4870	442	4	be	be	AUX
ejpam-4870	442	5	a	a	DET
ejpam-4870	442	6	2	2	NUM
ejpam-4870	442	7	-	-	PUNCT
ejpam-4870	442	8	edge	edge	NOUN
ejpam-4870	442	9	intersection	intersection	NOUN
ejpam-4870	442	10	graph	graph	NOUN
ejpam-4870	442	11	of	of	ADP
ejpam-4870	442	12	cn	cn	PROPN
ejpam-4870	442	13	.	.	PUNCT
ejpam-4870	443	1	then	then	ADV
ejpam-4870	443	2	α(ec(n,2	α(ec(n,2	NUM
ejpam-4870	443	3	)	)	PUNCT
ejpam-4870	443	4	)	)	PUNCT
ejpam-4870	444	1	≥	≥	X
ejpam-4870	444	2	⌊n2	⌊n2	PUNCT
ejpam-4870	444	3	⌋.	⌋.	PUNCT
ejpam-4870	445	1	proof	proof	NOUN
ejpam-4870	445	2	.	.	PUNCT
ejpam-4870	446	1	let	let	VERB
ejpam-4870	446	2	i	i	PRON
ejpam-4870	446	3	=	=	PRON
ejpam-4870	446	4	{	{	PUNCT
ejpam-4870	446	5	{	{	PUNCT
ejpam-4870	446	6	e1	e1	PROPN
ejpam-4870	446	7	,	,	PUNCT
ejpam-4870	446	8	e2	e2	PROPN
ejpam-4870	446	9	}	}	PUNCT
ejpam-4870	446	10	,	,	PUNCT
ejpam-4870	446	11	{	{	PUNCT
ejpam-4870	446	12	e3	e3	NOUN
ejpam-4870	446	13	,	,	PUNCT
ejpam-4870	446	14	e4	e4	PROPN
ejpam-4870	446	15	}	}	PUNCT
ejpam-4870	446	16	,	,	PUNCT
ejpam-4870	446	17	·	·	PUNCT
ejpam-4870	446	18	·	·	PUNCT
ejpam-4870	446	19	·	·	PUNCT
ejpam-4870	446	20	,	,	PUNCT
ejpam-4870	446	21	{	{	PUNCT
ejpam-4870	446	22	e2⌊n	e2⌊n	SYM
ejpam-4870	446	23	2	2	NUM
ejpam-4870	446	24	⌋−1	⌋−1	NOUN
ejpam-4870	446	25	,	,	PUNCT
ejpam-4870	446	26	e2⌊n	e2⌊n	NOUN
ejpam-4870	446	27	2	2	NUM
ejpam-4870	446	28	⌋	⌋	NOUN
ejpam-4870	446	29	}	}	PUNCT
ejpam-4870	446	30	}	}	PUNCT
ejpam-4870	446	31	.	.	PUNCT
ejpam-4870	447	1	for	for	ADP
ejpam-4870	447	2	every	every	DET
ejpam-4870	447	3	a	a	DET
ejpam-4870	447	4	∈	∈	PROPN
ejpam-4870	447	5	i	i	PRON
ejpam-4870	447	6	,	,	PUNCT
ejpam-4870	447	7	the	the	DET
ejpam-4870	447	8	cardinality	cardinality	NOUN
ejpam-4870	447	9	of	of	ADP
ejpam-4870	447	10	a	a	PRON
ejpam-4870	447	11	is	be	AUX
ejpam-4870	447	12	equal	equal	ADJ
ejpam-4870	447	13	to	to	ADP
ejpam-4870	447	14	2	2	NUM
ejpam-4870	447	15	.	.	PUNCT
ejpam-4870	448	1	this	this	PRON
ejpam-4870	448	2	means	mean	VERB
ejpam-4870	448	3	that	that	SCONJ
ejpam-4870	448	4	i	i	PRON
ejpam-4870	448	5	⊆	⊆	NUM
ejpam-4870	448	6	v	v	NOUN
ejpam-4870	448	7	(	(	PUNCT
ejpam-4870	448	8	ec(n,2	ec(n,2	PROPN
ejpam-4870	448	9	)	)	PUNCT
ejpam-4870	448	10	)	)	PUNCT
ejpam-4870	448	11	.	.	PUNCT
ejpam-4870	449	1	for	for	ADP
ejpam-4870	449	2	all	all	DET
ejpam-4870	449	3	two	two	NUM
ejpam-4870	449	4	distinct	distinct	ADJ
ejpam-4870	449	5	elements	element	NOUN
ejpam-4870	449	6	in	in	ADP
ejpam-4870	449	7	i	i	PRON
ejpam-4870	449	8	,	,	PUNCT
ejpam-4870	449	9	say	say	VERB
ejpam-4870	449	10	a	a	DET
ejpam-4870	449	11	and	and	CCONJ
ejpam-4870	449	12	b	b	NOUN
ejpam-4870	449	13	,	,	PUNCT
ejpam-4870	449	14	a∩b	a∩b	NOUN
ejpam-4870	449	15	=	=	SYM
ejpam-4870	449	16	∅	∅	NOUN
ejpam-4870	449	17	,	,	PUNCT
ejpam-4870	449	18	thus	thus	ADV
ejpam-4870	449	19	[	[	X
ejpam-4870	449	20	a	a	X
ejpam-4870	449	21	,	,	PUNCT
ejpam-4870	449	22	b	b	NOUN
ejpam-4870	449	23	]	]	X
ejpam-4870	449	24	/∈	/∈	PUNCT
ejpam-4870	449	25	e(ec(n,2	e(ec(n,2	ADJ
ejpam-4870	449	26	)	)	PUNCT
ejpam-4870	449	27	)	)	PUNCT
ejpam-4870	449	28	.	.	PUNCT
ejpam-4870	450	1	therefore	therefore	ADV
ejpam-4870	450	2	,	,	PUNCT
ejpam-4870	450	3	i	i	PRON
ejpam-4870	450	4	is	be	AUX
ejpam-4870	450	5	an	an	DET
ejpam-4870	450	6	independent	independent	ADJ
ejpam-4870	450	7	set	set	NOUN
ejpam-4870	450	8	in	in	ADP
ejpam-4870	450	9	ec(n,2	ec(n,2	PROPN
ejpam-4870	450	10	)	)	PUNCT
ejpam-4870	450	11	and	and	CCONJ
ejpam-4870	450	12	α(ec(n,2	α(ec(n,2	NUM
ejpam-4870	450	13	)	)	PUNCT
ejpam-4870	450	14	)	)	PUNCT
ejpam-4870	450	15	≥	≥	PROPN
ejpam-4870	450	16	|i|	|i|	PROPN
ejpam-4870	450	17	.	.	PUNCT
ejpam-4870	450	18	to	to	PART
ejpam-4870	450	19	determine	determine	VERB
ejpam-4870	450	20	|i|	|i|	PROPN
ejpam-4870	450	21	,	,	PUNCT
ejpam-4870	450	22	two	two	NUM
ejpam-4870	450	23	cases	case	NOUN
ejpam-4870	450	24	are	be	AUX
ejpam-4870	450	25	considered	consider	VERB
ejpam-4870	450	26	.	.	PUNCT
ejpam-4870	451	1	if	if	SCONJ
ejpam-4870	451	2	n	n	PRON
ejpam-4870	451	3	is	be	AUX
ejpam-4870	451	4	even	even	ADV
ejpam-4870	451	5	,	,	PUNCT
ejpam-4870	451	6	then	then	ADV
ejpam-4870	451	7	⌊n2	⌊n2	PUNCT
ejpam-4870	451	8	⌋	⌋	NOUN
ejpam-4870	451	9	=	=	PUNCT
ejpam-4870	451	10	n	n	PRON
ejpam-4870	451	11	2	2	NUM
ejpam-4870	451	12	which	which	PRON
ejpam-4870	451	13	implies	imply	VERB
ejpam-4870	451	14	that	that	SCONJ
ejpam-4870	451	15	i	i	PRON
ejpam-4870	451	16	=	=	PRON
ejpam-4870	451	17	{	{	PUNCT
ejpam-4870	451	18	{	{	PUNCT
ejpam-4870	451	19	e1	e1	PROPN
ejpam-4870	451	20	,	,	PUNCT
ejpam-4870	451	21	e2	e2	PROPN
ejpam-4870	451	22	}	}	PUNCT
ejpam-4870	451	23	,	,	PUNCT
ejpam-4870	451	24	{	{	PUNCT
ejpam-4870	451	25	e3	e3	NOUN
ejpam-4870	451	26	,	,	PUNCT
ejpam-4870	451	27	e4	e4	PROPN
ejpam-4870	451	28	}	}	PUNCT
ejpam-4870	451	29	,	,	PUNCT
ejpam-4870	451	30	·	·	PUNCT
ejpam-4870	451	31	·	·	PUNCT
ejpam-4870	451	32	·	·	PUNCT
ejpam-4870	451	33	,	,	PUNCT
ejpam-4870	451	34	{	{	PUNCT
ejpam-4870	451	35	en−1	en−1	PROPN
ejpam-4870	451	36	,	,	PUNCT
ejpam-4870	451	37	en	en	ADP
ejpam-4870	451	38	}	}	PUNCT
ejpam-4870	451	39	}	}	PUNCT
ejpam-4870	451	40	.	.	PUNCT
ejpam-4870	452	1	in	in	ADP
ejpam-4870	452	2	this	this	DET
ejpam-4870	452	3	case	case	NOUN
ejpam-4870	452	4	,	,	PUNCT
ejpam-4870	452	5	|i|	|i|	PROPN
ejpam-4870	452	6	=	=	SYM
ejpam-4870	452	7	n	n	PRON
ejpam-4870	452	8	2	2	NUM
ejpam-4870	452	9	.	.	PUNCT
ejpam-4870	453	1	if	if	SCONJ
ejpam-4870	453	2	n	n	NOUN
ejpam-4870	453	3	is	be	AUX
ejpam-4870	453	4	odd	odd	ADJ
ejpam-4870	453	5	,	,	PUNCT
ejpam-4870	453	6	then	then	ADV
ejpam-4870	453	7	⌊n2	⌊n2	PUNCT
ejpam-4870	453	8	⌋	⌋	PROPN
ejpam-4870	454	1	=	=	SYM
ejpam-4870	454	2	n−1	n−1	PROPN
ejpam-4870	454	3	2	2	NUM
ejpam-4870	454	4	which	which	PRON
ejpam-4870	454	5	menas	mena	VERB
ejpam-4870	455	1	that	that	SCONJ
ejpam-4870	455	2	i	i	PRON
ejpam-4870	455	3	=	=	PRON
ejpam-4870	455	4	{	{	PUNCT
ejpam-4870	455	5	{	{	PUNCT
ejpam-4870	455	6	e1	e1	PROPN
ejpam-4870	455	7	,	,	PUNCT
ejpam-4870	455	8	e2	e2	PROPN
ejpam-4870	455	9	}	}	PUNCT
ejpam-4870	455	10	,	,	PUNCT
ejpam-4870	455	11	{	{	PUNCT
ejpam-4870	455	12	e3	e3	NOUN
ejpam-4870	455	13	,	,	PUNCT
ejpam-4870	455	14	e4	e4	PROPN
ejpam-4870	455	15	}	}	PUNCT
ejpam-4870	455	16	,	,	PUNCT
ejpam-4870	455	17	·	·	PUNCT
ejpam-4870	455	18	·	·	PUNCT
ejpam-4870	455	19	·	·	PUNCT
ejpam-4870	455	20	,	,	PUNCT
ejpam-4870	455	21	{	{	PUNCT
ejpam-4870	455	22	en−2	en−2	PROPN
ejpam-4870	455	23	,	,	PUNCT
ejpam-4870	455	24	en−1	en−1	PROPN
ejpam-4870	455	25	}	}	PUNCT
ejpam-4870	455	26	}	}	PUNCT
ejpam-4870	455	27	.	.	PUNCT
ejpam-4870	456	1	this	this	PRON
ejpam-4870	456	2	indicates	indicate	VERB
ejpam-4870	456	3	that	that	SCONJ
ejpam-4870	456	4	|i|	|i|	VERB
ejpam-4870	456	5	=	=	SYM
ejpam-4870	456	6	n−1	n−1	PROPN
ejpam-4870	456	7	2	2	NUM
ejpam-4870	456	8	.	.	PUNCT
ejpam-4870	457	1	thus	thus	ADV
ejpam-4870	457	2	,	,	PUNCT
ejpam-4870	457	3	α(ec(n,2	α(ec(n,2	NUM
ejpam-4870	457	4	)	)	PUNCT
ejpam-4870	457	5	)	)	PUNCT
ejpam-4870	457	6	≥	≥	X
ejpam-4870	457	7	⌊n2	⌊n2	PUNCT
ejpam-4870	457	8	⌋.	⌋.	PRON
ejpam-4870	458	1	illustration	illustration	NOUN
ejpam-4870	458	2	11	11	NUM
ejpam-4870	458	3	,	,	PUNCT
ejpam-4870	458	4	shows	show	VERB
ejpam-4870	458	5	a	a	DET
ejpam-4870	458	6	lower	lower	ADV
ejpam-4870	458	7	bound	bind	VERB
ejpam-4870	458	8	for	for	ADP
ejpam-4870	458	9	the	the	DET
ejpam-4870	458	10	independence	independence	NOUN
ejpam-4870	458	11	number	number	NOUN
ejpam-4870	458	12	of	of	ADP
ejpam-4870	458	13	ec(n,2	ec(n,2	PROPN
ejpam-4870	458	14	)	)	PUNCT
ejpam-4870	458	15	when	when	SCONJ
ejpam-4870	458	16	n	n	PROPN
ejpam-4870	458	17	=	=	SYM
ejpam-4870	458	18	6	6	NUM
ejpam-4870	458	19	.	.	PUNCT
ejpam-4870	458	20	illustration	illustration	NOUN
ejpam-4870	458	21	11	11	NUM
ejpam-4870	458	22	.	.	PUNCT
ejpam-4870	459	1	consider	consider	VERB
ejpam-4870	459	2	c6	c6	PROPN
ejpam-4870	459	3	,	,	PUNCT
ejpam-4870	459	4	and	and	CCONJ
ejpam-4870	459	5	let	let	VERB
ejpam-4870	459	6	j	j	PROPN
ejpam-4870	459	7	=	=	NOUN
ejpam-4870	459	8	2	2	X
ejpam-4870	459	9	.	.	PUNCT
ejpam-4870	460	1	the	the	DET
ejpam-4870	460	2	vertex	vertex	NOUN
ejpam-4870	460	3	set	set	NOUN
ejpam-4870	460	4	of	of	ADP
ejpam-4870	460	5	ec(6,2	ec(6,2	PROPN
ejpam-4870	460	6	)	)	PUNCT
ejpam-4870	460	7	is	be	AUX
ejpam-4870	460	8	given	give	VERB
ejpam-4870	460	9	by	by	ADP
ejpam-4870	460	10	v	v	PROPN
ejpam-4870	460	11	(	(	PUNCT
ejpam-4870	460	12	ec(6,2	ec(6,2	PROPN
ejpam-4870	460	13	)	)	PUNCT
ejpam-4870	460	14	)	)	PUNCT
ejpam-4870	461	1	=	=	PRON
ejpam-4870	461	2	{	{	PUNCT
ejpam-4870	461	3	{	{	PUNCT
ejpam-4870	461	4	12	12	NUM
ejpam-4870	461	5	,	,	PUNCT
ejpam-4870	461	6	23	23	NUM
ejpam-4870	461	7	}	}	PUNCT
ejpam-4870	461	8	,	,	PUNCT
ejpam-4870	461	9	{	{	PUNCT
ejpam-4870	461	10	12	12	NUM
ejpam-4870	461	11	,	,	PUNCT
ejpam-4870	461	12	34	34	NUM
ejpam-4870	461	13	}	}	PUNCT
ejpam-4870	461	14	,	,	PUNCT
ejpam-4870	461	15	{	{	PUNCT
ejpam-4870	461	16	12	12	NUM
ejpam-4870	461	17	,	,	PUNCT
ejpam-4870	461	18	45	45	NUM
ejpam-4870	461	19	}	}	PUNCT
ejpam-4870	461	20	,	,	PUNCT
ejpam-4870	461	21	{	{	PUNCT
ejpam-4870	461	22	12	12	NUM
ejpam-4870	461	23	,	,	PUNCT
ejpam-4870	461	24	56	56	NUM
ejpam-4870	461	25	}	}	PUNCT
ejpam-4870	461	26	,	,	PUNCT
ejpam-4870	461	27	{	{	PUNCT
ejpam-4870	461	28	12	12	NUM
ejpam-4870	461	29	,	,	PUNCT
ejpam-4870	461	30	61	61	NUM
ejpam-4870	461	31	}	}	PUNCT
ejpam-4870	461	32	,	,	PUNCT
ejpam-4870	461	33	{	{	PUNCT
ejpam-4870	461	34	23	23	NUM
ejpam-4870	461	35	,	,	PUNCT
ejpam-4870	461	36	34	34	NUM
ejpam-4870	461	37	}	}	PUNCT
ejpam-4870	461	38	,	,	PUNCT
ejpam-4870	461	39	{	{	PUNCT
ejpam-4870	461	40	23	23	NUM
ejpam-4870	461	41	,	,	PUNCT
ejpam-4870	461	42	45	45	NUM
ejpam-4870	461	43	}	}	PUNCT
ejpam-4870	461	44	,	,	PUNCT
ejpam-4870	461	45	{	{	PUNCT
ejpam-4870	461	46	23	23	NUM
ejpam-4870	461	47	,	,	PUNCT
ejpam-4870	461	48	56	56	NUM
ejpam-4870	461	49	}	}	PUNCT
ejpam-4870	461	50	,	,	PUNCT
ejpam-4870	461	51	{	{	PUNCT
ejpam-4870	461	52	23	23	NUM
ejpam-4870	461	53	,	,	PUNCT
ejpam-4870	461	54	61	61	NUM
ejpam-4870	461	55	}	}	PUNCT
ejpam-4870	461	56	,	,	PUNCT
ejpam-4870	461	57	{	{	PUNCT
ejpam-4870	461	58	34	34	NUM
ejpam-4870	461	59	,	,	PUNCT
ejpam-4870	461	60	45	45	NUM
ejpam-4870	461	61	}	}	PUNCT
ejpam-4870	461	62	,	,	PUNCT
ejpam-4870	461	63	{	{	PUNCT
ejpam-4870	461	64	34	34	NUM
ejpam-4870	461	65	,	,	PUNCT
ejpam-4870	461	66	56	56	NUM
ejpam-4870	461	67	}	}	PUNCT
ejpam-4870	461	68	,	,	PUNCT
ejpam-4870	461	69	{	{	PUNCT
ejpam-4870	461	70	34	34	NUM
ejpam-4870	461	71	,	,	PUNCT
ejpam-4870	461	72	61	61	NUM
ejpam-4870	461	73	}	}	PUNCT
ejpam-4870	461	74	,	,	PUNCT
ejpam-4870	461	75	{	{	PUNCT
ejpam-4870	461	76	45	45	NUM
ejpam-4870	461	77	,	,	PUNCT
ejpam-4870	461	78	56	56	NUM
ejpam-4870	461	79	}	}	PUNCT
ejpam-4870	461	80	,	,	PUNCT
ejpam-4870	461	81	{	{	PUNCT
ejpam-4870	461	82	45	45	NUM
ejpam-4870	461	83	,	,	PUNCT
ejpam-4870	461	84	61	61	NUM
ejpam-4870	461	85	}	}	PUNCT
ejpam-4870	461	86	,	,	PUNCT
ejpam-4870	461	87	{	{	PUNCT
ejpam-4870	461	88	56	56	NUM
ejpam-4870	461	89	,	,	PUNCT
ejpam-4870	461	90	61	61	NUM
ejpam-4870	461	91	}	}	PUNCT
ejpam-4870	461	92	}	}	PUNCT
ejpam-4870	461	93	.	.	PUNCT
ejpam-4870	462	1	the	the	DET
ejpam-4870	462	2	pictorial	pictorial	ADJ
ejpam-4870	462	3	representation	representation	NOUN
ejpam-4870	462	4	in	in	ADP
ejpam-4870	462	5	figure	figure	NOUN
ejpam-4870	462	6	21	21	NUM
ejpam-4870	462	7	shows	show	NOUN
ejpam-4870	462	8	ec(6,2	ec(6,2	PROPN
ejpam-4870	462	9	)	)	PUNCT
ejpam-4870	462	10	.	.	PUNCT
ejpam-4870	463	1	let	let	VERB
ejpam-4870	463	2	ia	ia	PROPN
ejpam-4870	463	3	⊆	⊆	NUM
ejpam-4870	463	4	v	v	NOUN
ejpam-4870	463	5	(	(	PUNCT
ejpam-4870	463	6	ec(6,2	ec(6,2	PROPN
ejpam-4870	463	7	)	)	PUNCT
ejpam-4870	463	8	)	)	PUNCT
ejpam-4870	463	9	for	for	ADP
ejpam-4870	463	10	1	1	NUM
ejpam-4870	463	11	≤	≤	NOUN
ejpam-4870	463	12	a	a	DET
ejpam-4870	463	13	≤	≤	ADJ
ejpam-4870	463	14	3	3	NUM
ejpam-4870	463	15	.	.	PUNCT
ejpam-4870	464	1	it	it	PRON
ejpam-4870	464	2	can	can	AUX
ejpam-4870	464	3	be	be	AUX
ejpam-4870	464	4	noticed	notice	VERB
ejpam-4870	464	5	that	that	SCONJ
ejpam-4870	464	6	i1	i1	PROPN
ejpam-4870	464	7	=	=	PRON
ejpam-4870	464	8	{	{	PUNCT
ejpam-4870	464	9	{	{	PUNCT
ejpam-4870	464	10	ei	ei	NOUN
ejpam-4870	464	11	}	}	PUNCT
ejpam-4870	464	12	}	}	PUNCT
ejpam-4870	464	13	for	for	ADP
ejpam-4870	464	14	all	all	PRON
ejpam-4870	464	15	i	i	PRON
ejpam-4870	464	16	element	element	NOUN
ejpam-4870	464	17	of	of	ADP
ejpam-4870	464	18	the	the	DET
ejpam-4870	464	19	spanning	span	VERB
ejpam-4870	464	20	subgraph	subgraph	NOUN
ejpam-4870	464	21	with	with	ADP
ejpam-4870	464	22	2	2	NUM
ejpam-4870	464	23	edges	edge	NOUN
ejpam-4870	464	24	is	be	AUX
ejpam-4870	464	25	an	an	DET
ejpam-4870	464	26	independent	independent	ADJ
ejpam-4870	464	27	set	set	NOUN
ejpam-4870	464	28	since	since	SCONJ
ejpam-4870	464	29	ec(6,2	ec(6,2	PROPN
ejpam-4870	464	30	)	)	PUNCT
ejpam-4870	464	31	is	be	AUX
ejpam-4870	464	32	a	a	DET
ejpam-4870	464	33	simple	simple	ADJ
ejpam-4870	464	34	graph	graph	NOUN
ejpam-4870	464	35	.	.	PUNCT
ejpam-4870	465	1	moreover	moreover	ADV
ejpam-4870	465	2	,	,	PUNCT
ejpam-4870	465	3	the	the	DET
ejpam-4870	465	4	set	set	NOUN
ejpam-4870	465	5	i2	i2	PROPN
ejpam-4870	465	6	=	=	PRON
ejpam-4870	465	7	{	{	PUNCT
ejpam-4870	465	8	{	{	PUNCT
ejpam-4870	465	9	12	12	NUM
ejpam-4870	465	10	,	,	PUNCT
ejpam-4870	465	11	23	23	NUM
ejpam-4870	465	12	}	}	PUNCT
ejpam-4870	465	13	,	,	PUNCT
ejpam-4870	465	14	{	{	PUNCT
ejpam-4870	465	15	34	34	NUM
ejpam-4870	465	16	,	,	PUNCT
ejpam-4870	465	17	45	45	NUM
ejpam-4870	465	18	}	}	PUNCT
ejpam-4870	465	19	}	}	PUNCT
ejpam-4870	465	20	is	be	AUX
ejpam-4870	465	21	an	an	DET
ejpam-4870	465	22	independent	independent	ADJ
ejpam-4870	465	23	set	set	NOUN
ejpam-4870	465	24	since	since	SCONJ
ejpam-4870	465	25	{	{	PUNCT
ejpam-4870	465	26	12	12	NUM
ejpam-4870	465	27	,	,	PUNCT
ejpam-4870	465	28	23	23	NUM
ejpam-4870	465	29	}	}	PUNCT
ejpam-4870	465	30	∩	∩	NOUN
ejpam-4870	465	31	{	{	PUNCT
ejpam-4870	465	32	34	34	NUM
ejpam-4870	465	33	,	,	PUNCT
ejpam-4870	465	34	45	45	NUM
ejpam-4870	465	35	}	}	PUNCT
ejpam-4870	465	36	=	=	NOUN
ejpam-4870	465	37	∅	∅	NOUN
ejpam-4870	465	38	so	so	CCONJ
ejpam-4870	465	39	there	there	PRON
ejpam-4870	465	40	is	be	VERB
ejpam-4870	465	41	no	no	DET
ejpam-4870	465	42	edge	edge	NOUN
ejpam-4870	465	43	connecting	connect	VERB
ejpam-4870	465	44	the	the	DET
ejpam-4870	465	45	vertices	vertex	NOUN
ejpam-4870	465	46	{	{	PUNCT
ejpam-4870	465	47	12	12	NUM
ejpam-4870	465	48	,	,	PUNCT
ejpam-4870	465	49	23	23	NUM
ejpam-4870	465	50	}	}	PUNCT
ejpam-4870	465	51	and	and	CCONJ
ejpam-4870	465	52	{	{	PUNCT
ejpam-4870	465	53	34	34	NUM
ejpam-4870	465	54	,	,	PUNCT
ejpam-4870	465	55	45	45	NUM
ejpam-4870	465	56	}	}	PUNCT
ejpam-4870	465	57	.	.	PUNCT
ejpam-4870	466	1	it	it	PRON
ejpam-4870	466	2	means	mean	VERB
ejpam-4870	466	3	that	that	SCONJ
ejpam-4870	466	4	there	there	PRON
ejpam-4870	466	5	exists	exist	VERB
ejpam-4870	466	6	an	an	DET
ejpam-4870	466	7	independent	independent	ADJ
ejpam-4870	466	8	set	set	NOUN
ejpam-4870	466	9	with	with	ADP
ejpam-4870	466	10	cardinality	cardinality	NOUN
ejpam-4870	466	11	2	2	NUM
ejpam-4870	466	12	.	.	PUNCT
ejpam-4870	467	1	furthermore	furthermore	ADV
ejpam-4870	467	2	,	,	PUNCT
ejpam-4870	467	3	i3	i3	NOUN
ejpam-4870	467	4	=	=	SYM
ejpam-4870	467	5	{	{	PUNCT
ejpam-4870	467	6	{	{	PUNCT
ejpam-4870	467	7	12	12	NUM
ejpam-4870	467	8	,	,	PUNCT
ejpam-4870	467	9	23	23	NUM
ejpam-4870	467	10	}	}	PUNCT
ejpam-4870	467	11	,	,	PUNCT
ejpam-4870	467	12	{	{	PUNCT
ejpam-4870	467	13	34	34	NUM
ejpam-4870	467	14	,	,	PUNCT
ejpam-4870	467	15	45	45	NUM
ejpam-4870	467	16	}	}	PUNCT
ejpam-4870	467	17	,	,	PUNCT
ejpam-4870	467	18	{	{	PUNCT
ejpam-4870	467	19	56	56	NUM
ejpam-4870	467	20	,	,	PUNCT
ejpam-4870	467	21	61	61	NUM
ejpam-4870	467	22	}	}	PUNCT
ejpam-4870	467	23	}	}	PUNCT
ejpam-4870	467	24	is	be	AUX
ejpam-4870	467	25	also	also	ADV
ejpam-4870	467	26	an	an	DET
ejpam-4870	467	27	independent	independent	ADJ
ejpam-4870	467	28	set	set	NOUN
ejpam-4870	467	29	since	since	SCONJ
ejpam-4870	467	30	there	there	PRON
ejpam-4870	467	31	is	be	VERB
ejpam-4870	467	32	no	no	DET
ejpam-4870	467	33	edge	edge	NOUN
ejpam-4870	467	34	incident	incident	NOUN
ejpam-4870	467	35	to	to	ADP
ejpam-4870	467	36	themselves	themselves	PRON
ejpam-4870	467	37	.	.	PUNCT
ejpam-4870	468	1	it	it	PRON
ejpam-4870	468	2	can	can	AUX
ejpam-4870	468	3	be	be	AUX
ejpam-4870	468	4	observed	observe	VERB
ejpam-4870	468	5	that	that	SCONJ
ejpam-4870	468	6	⌊62⌋	⌊62⌋	NOUN
ejpam-4870	469	1	=	=	PUNCT
ejpam-4870	469	2	3	3	NUM
ejpam-4870	469	3	and	and	CCONJ
ejpam-4870	469	4	|i3|	|i3|	PRON
ejpam-4870	469	5	=	=	SYM
ejpam-4870	469	6	3	3	X
ejpam-4870	469	7	.	.	PUNCT
ejpam-4870	470	1	thus	thus	ADV
ejpam-4870	470	2	,	,	PUNCT
ejpam-4870	470	3	there	there	PRON
ejpam-4870	470	4	exists	exist	VERB
ejpam-4870	470	5	an	an	DET
ejpam-4870	470	6	independent	independent	ADJ
ejpam-4870	470	7	set	set	NOUN
ejpam-4870	470	8	of	of	ADP
ejpam-4870	470	9	ec(6,2	ec(6,2	PROPN
ejpam-4870	470	10	)	)	PUNCT
ejpam-4870	470	11	with	with	ADP
ejpam-4870	470	12	cardinality	cardinality	NOUN
ejpam-4870	470	13	3	3	NUM
ejpam-4870	470	14	.	.	PUNCT
ejpam-4870	471	1	moreover	moreover	ADV
ejpam-4870	471	2	,	,	PUNCT
ejpam-4870	471	3	there	there	PRON
ejpam-4870	471	4	are	be	VERB
ejpam-4870	471	5	no	no	DET
ejpam-4870	471	6	independent	independent	ADJ
ejpam-4870	471	7	sets	set	NOUN
ejpam-4870	471	8	in	in	ADP
ejpam-4870	471	9	ec(6,2	ec(6,2	PROPN
ejpam-4870	471	10	)	)	PUNCT
ejpam-4870	471	11	of	of	ADP
ejpam-4870	471	12	cardinality	cardinality	NOUN
ejpam-4870	471	13	greater	great	ADJ
ejpam-4870	471	14	than	than	ADP
ejpam-4870	471	15	3	3	NUM
ejpam-4870	471	16	.	.	PUNCT
ejpam-4870	471	17	therefore	therefore	ADV
ejpam-4870	471	18	,	,	PUNCT
ejpam-4870	471	19	α(ec(6,2	α(ec(6,2	X
ejpam-4870	471	20	)	)	PUNCT
ejpam-4870	471	21	)	)	PUNCT
ejpam-4870	471	22	≥	≥	NOUN
ejpam-4870	471	23	3	3	NUM
ejpam-4870	471	24	to	to	PART
ejpam-4870	471	25	verify	verify	VERB
ejpam-4870	471	26	this	this	PRON
ejpam-4870	471	27	,	,	PUNCT
ejpam-4870	471	28	we	we	PRON
ejpam-4870	471	29	will	will	AUX
ejpam-4870	471	30	be	be	AUX
ejpam-4870	471	31	using	use	VERB
ejpam-4870	471	32	theorem	theorem	ADJ
ejpam-4870	471	33	9	9	NUM
ejpam-4870	471	34	,	,	PUNCT
ejpam-4870	471	35	setting	set	VERB
ejpam-4870	471	36	n	n	X
ejpam-4870	471	37	=	=	SYM
ejpam-4870	471	38	6	6	NUM
ejpam-4870	471	39	and	and	CCONJ
ejpam-4870	471	40	j	j	NOUN
ejpam-4870	471	41	=	=	SYM
ejpam-4870	471	42	2	2	NUM
ejpam-4870	471	43	,	,	PUNCT
ejpam-4870	471	44	we	we	PRON
ejpam-4870	471	45	have	have	VERB
ejpam-4870	471	46	α(ec(6,2	α(ec(6,2	X
ejpam-4870	471	47	)	)	PUNCT
ejpam-4870	471	48	)	)	PUNCT
ejpam-4870	472	1	≥⌊6	≥⌊6	NOUN
ejpam-4870	472	2	2	2	NUM
ejpam-4870	472	3	⌋	⌋	NOUN
ejpam-4870	472	4	=3	=3	VERB
ejpam-4870	472	5	.	.	PUNCT
ejpam-4870	473	1	4.2	4.2	NUM
ejpam-4870	473	2	.	.	PUNCT
ejpam-4870	473	3	domination	domination	NOUN
ejpam-4870	473	4	number	number	NOUN
ejpam-4870	473	5	of	of	ADP
ejpam-4870	473	6	ec(n,2	ec(n,2	PROPN
ejpam-4870	473	7	)	)	PUNCT
ejpam-4870	473	8	this	this	DET
ejpam-4870	473	9	subsection	subsection	NOUN
ejpam-4870	473	10	determines	determine	VERB
ejpam-4870	473	11	an	an	DET
ejpam-4870	473	12	upper	upper	ADJ
ejpam-4870	473	13	bound	bind	VERB
ejpam-4870	473	14	for	for	ADP
ejpam-4870	473	15	the	the	DET
ejpam-4870	473	16	domination	domination	NOUN
ejpam-4870	473	17	number	number	NOUN
ejpam-4870	473	18	of	of	ADP
ejpam-4870	473	19	ec(n	ec(n	PROPN
ejpam-4870	473	20	,	,	PUNCT
ejpam-4870	473	21	j	j	PROPN
ejpam-4870	473	22	)	)	PUNCT
ejpam-4870	473	23	when	when	SCONJ
ejpam-4870	473	24	j	j	PROPN
ejpam-4870	473	25	=	=	SYM
ejpam-4870	473	26	2	2	X
ejpam-4870	473	27	.	.	PUNCT
ejpam-4870	473	28	theorem	theorem	VERB
ejpam-4870	473	29	10	10	NUM
ejpam-4870	473	30	determines	determine	VERB
ejpam-4870	473	31	the	the	DET
ejpam-4870	473	32	existence	existence	NOUN
ejpam-4870	473	33	of	of	ADP
ejpam-4870	473	34	dominating	dominate	VERB
ejpam-4870	473	35	set	set	NOUN
ejpam-4870	473	36	of	of	ADP
ejpam-4870	473	37	ec(n,2	ec(n,2	PROPN
ejpam-4870	473	38	)	)	PUNCT
ejpam-4870	473	39	.	.	PUNCT
ejpam-4870	474	1	theorem	theorem	ADJ
ejpam-4870	474	2	10	10	NUM
ejpam-4870	474	3	.	.	PUNCT
ejpam-4870	475	1	let	let	VERB
ejpam-4870	475	2	ec(n,2	ec(n,2	PROPN
ejpam-4870	475	3	)	)	PUNCT
ejpam-4870	475	4	be	be	AUX
ejpam-4870	475	5	a	a	DET
ejpam-4870	475	6	2	2	NUM
ejpam-4870	475	7	-	-	PUNCT
ejpam-4870	475	8	edge	edge	NOUN
ejpam-4870	475	9	intersection	intersection	NOUN
ejpam-4870	475	10	graph	graph	NOUN
ejpam-4870	475	11	of	of	ADP
ejpam-4870	475	12	cn	cn	PROPN
ejpam-4870	475	13	.	.	PUNCT
ejpam-4870	476	1	then	then	ADV
ejpam-4870	476	2	γ(ec(n,2	γ(ec(n,2	PROPN
ejpam-4870	476	3	)	)	PUNCT
ejpam-4870	476	4	)	)	PUNCT
ejpam-4870	477	1	≤	≤	NOUN
ejpam-4870	477	2	⌊n2	⌊n2	PUNCT
ejpam-4870	477	3	⌋.	⌋.	PROPN
ejpam-4870	477	4	j.c	j.c	PROPN
ejpam-4870	477	5	.	.	PROPN
ejpam-4870	477	6	bonifacio	bonifacio	PROPN
ejpam-4870	477	7	,	,	PUNCT
ejpam-4870	477	8	c.j	c.j	PROPN
ejpam-4870	477	9	.	.	PROPN
ejpam-4870	477	10	andaya	andaya	PROPN
ejpam-4870	477	11	,	,	PUNCT
ejpam-4870	477	12	d.	d.	PROPN
ejpam-4870	477	13	magpantay	magpantay	PROPN
ejpam-4870	477	14	/	/	SYM
ejpam-4870	477	15	eur	eur	PROPN
ejpam-4870	477	16	.	.	PUNCT
ejpam-4870	478	1	j.	j.	PROPN
ejpam-4870	478	2	pure	pure	PROPN
ejpam-4870	478	3	appl	appl	PROPN
ejpam-4870	478	4	.	.	PROPN
ejpam-4870	478	5	math	math	PROPN
ejpam-4870	478	6	,	,	PUNCT
ejpam-4870	478	7	16	16	NUM
ejpam-4870	478	8	(	(	PUNCT
ejpam-4870	478	9	4	4	NUM
ejpam-4870	478	10	)	)	PUNCT
ejpam-4870	478	11	(	(	PUNCT
ejpam-4870	478	12	2023	2023	NUM
ejpam-4870	478	13	)	)	PUNCT
ejpam-4870	478	14	,	,	PUNCT
ejpam-4870	478	15	2476	2476	NUM
ejpam-4870	478	16	-	-	SYM
ejpam-4870	478	17	2498	2498	NUM
ejpam-4870	478	18	2496	2496	NUM
ejpam-4870	478	19	{	{	PUNCT
ejpam-4870	478	20	12	12	NUM
ejpam-4870	478	21	,	,	PUNCT
ejpam-4870	478	22	23	23	NUM
ejpam-4870	478	23	}	}	PUNCT
ejpam-4870	478	24	{	{	PUNCT
ejpam-4870	478	25	12	12	NUM
ejpam-4870	478	26	,	,	PUNCT
ejpam-4870	478	27	34	34	NUM
ejpam-4870	478	28	}	}	PUNCT
ejpam-4870	478	29	{	{	PUNCT
ejpam-4870	478	30	12	12	NUM
ejpam-4870	478	31	,	,	PUNCT
ejpam-4870	478	32	45	45	NUM
ejpam-4870	478	33	}	}	PUNCT
ejpam-4870	478	34	{	{	PUNCT
ejpam-4870	478	35	12	12	NUM
ejpam-4870	478	36	,	,	PUNCT
ejpam-4870	478	37	56	56	NUM
ejpam-4870	478	38	}	}	PUNCT
ejpam-4870	478	39	{	{	PUNCT
ejpam-4870	478	40	12	12	NUM
ejpam-4870	478	41	,	,	PUNCT
ejpam-4870	478	42	61	61	NUM
ejpam-4870	478	43	}	}	PUNCT
ejpam-4870	478	44	{	{	PUNCT
ejpam-4870	478	45	23	23	NUM
ejpam-4870	478	46	,	,	PUNCT
ejpam-4870	478	47	34	34	NUM
ejpam-4870	478	48	}	}	PUNCT
ejpam-4870	478	49	{	{	PUNCT
ejpam-4870	478	50	23	23	NUM
ejpam-4870	478	51	,	,	PUNCT
ejpam-4870	478	52	45	45	NUM
ejpam-4870	478	53	}	}	PUNCT
ejpam-4870	478	54	{	{	PUNCT
ejpam-4870	478	55	23	23	NUM
ejpam-4870	478	56	,	,	PUNCT
ejpam-4870	478	57	56	56	NUM
ejpam-4870	478	58	}	}	PUNCT
ejpam-4870	478	59	{	{	PUNCT
ejpam-4870	478	60	23	23	NUM
ejpam-4870	478	61	,	,	PUNCT
ejpam-4870	478	62	61	61	NUM
ejpam-4870	478	63	}	}	PUNCT
ejpam-4870	478	64	{	{	PUNCT
ejpam-4870	478	65	34	34	NUM
ejpam-4870	478	66	,	,	PUNCT
ejpam-4870	478	67	45	45	NUM
ejpam-4870	478	68	}	}	PUNCT
ejpam-4870	478	69	{	{	PUNCT
ejpam-4870	478	70	34	34	NUM
ejpam-4870	478	71	,	,	PUNCT
ejpam-4870	478	72	56	56	NUM
ejpam-4870	478	73	}	}	PUNCT
ejpam-4870	478	74	{	{	PUNCT
ejpam-4870	478	75	34	34	NUM
ejpam-4870	478	76	,	,	PUNCT
ejpam-4870	478	77	61	61	NUM
ejpam-4870	478	78	}	}	PUNCT
ejpam-4870	478	79	{	{	PUNCT
ejpam-4870	478	80	45	45	NUM
ejpam-4870	478	81	,	,	PUNCT
ejpam-4870	478	82	56	56	NUM
ejpam-4870	478	83	}	}	PUNCT
ejpam-4870	478	84	{	{	PUNCT
ejpam-4870	478	85	45	45	NUM
ejpam-4870	478	86	,	,	PUNCT
ejpam-4870	478	87	61	61	NUM
ejpam-4870	478	88	}	}	PUNCT
ejpam-4870	478	89	{	{	PUNCT
ejpam-4870	478	90	56	56	NUM
ejpam-4870	478	91	,	,	PUNCT
ejpam-4870	478	92	61	61	NUM
ejpam-4870	478	93	}	}	PUNCT
ejpam-4870	478	94	figure	figure	NOUN
ejpam-4870	478	95	21	21	NUM
ejpam-4870	478	96	:	:	PUNCT
ejpam-4870	478	97	a	a	DET
ejpam-4870	478	98	pictorial	pictorial	ADJ
ejpam-4870	478	99	representation	representation	NOUN
ejpam-4870	478	100	of	of	ADP
ejpam-4870	478	101	ec(6,2	ec(6,2	NOUN
ejpam-4870	478	102	)	)	PUNCT
ejpam-4870	478	103	proof	proof	NOUN
ejpam-4870	478	104	.	.	PUNCT
ejpam-4870	479	1	let	let	VERB
ejpam-4870	479	2	i	i	PRON
ejpam-4870	479	3	=	=	PRON
ejpam-4870	479	4	{	{	PUNCT
ejpam-4870	479	5	{	{	PUNCT
ejpam-4870	479	6	e1	e1	PROPN
ejpam-4870	479	7	,	,	PUNCT
ejpam-4870	479	8	e2	e2	PROPN
ejpam-4870	479	9	}	}	PUNCT
ejpam-4870	479	10	,	,	PUNCT
ejpam-4870	479	11	{	{	PUNCT
ejpam-4870	479	12	e3	e3	NOUN
ejpam-4870	479	13	,	,	PUNCT
ejpam-4870	479	14	e4	e4	PROPN
ejpam-4870	479	15	}	}	PUNCT
ejpam-4870	479	16	,	,	PUNCT
ejpam-4870	479	17	·	·	PUNCT
ejpam-4870	479	18	·	·	PUNCT
ejpam-4870	479	19	·	·	PUNCT
ejpam-4870	479	20	,	,	PUNCT
ejpam-4870	479	21	{	{	PUNCT
ejpam-4870	479	22	e2⌊n	e2⌊n	SYM
ejpam-4870	479	23	2	2	NUM
ejpam-4870	479	24	⌋−1	⌋−1	NOUN
ejpam-4870	479	25	,	,	PUNCT
ejpam-4870	479	26	e2⌊n	e2⌊n	NOUN
ejpam-4870	479	27	2	2	NUM
ejpam-4870	479	28	⌋	⌋	NOUN
ejpam-4870	479	29	}	}	PUNCT
ejpam-4870	479	30	}	}	PUNCT
ejpam-4870	479	31	.	.	PUNCT
ejpam-4870	480	1	if	if	SCONJ
ejpam-4870	480	2	n	n	PRON
ejpam-4870	480	3	is	be	AUX
ejpam-4870	480	4	even	even	ADV
ejpam-4870	480	5	,	,	PUNCT
ejpam-4870	480	6	then	then	ADV
ejpam-4870	480	7	⌊n2	⌊n2	PUNCT
ejpam-4870	480	8	⌋	⌋	NOUN
ejpam-4870	480	9	=	=	PUNCT
ejpam-4870	480	10	n	n	PRON
ejpam-4870	480	11	2	2	NUM
ejpam-4870	480	12	.	.	PUNCT
ejpam-4870	481	1	in	in	ADP
ejpam-4870	481	2	this	this	DET
ejpam-4870	481	3	case	case	NOUN
ejpam-4870	481	4	,	,	PUNCT
ejpam-4870	481	5	i	i	PRON
ejpam-4870	481	6	=	=	PUNCT
ejpam-4870	481	7	{	{	PUNCT
ejpam-4870	481	8	{	{	PUNCT
ejpam-4870	481	9	e1	e1	PROPN
ejpam-4870	481	10	,	,	PUNCT
ejpam-4870	481	11	e2	e2	PROPN
ejpam-4870	481	12	}	}	PUNCT
ejpam-4870	481	13	,	,	PUNCT
ejpam-4870	481	14	{	{	PUNCT
ejpam-4870	481	15	e3	e3	NOUN
ejpam-4870	481	16	,	,	PUNCT
ejpam-4870	481	17	e4	e4	PROPN
ejpam-4870	481	18	}	}	PUNCT
ejpam-4870	481	19	,	,	PUNCT
ejpam-4870	481	20	·	·	PUNCT
ejpam-4870	481	21	·	·	PUNCT
ejpam-4870	481	22	·	·	PUNCT
ejpam-4870	481	23	,	,	PUNCT
ejpam-4870	481	24	{	{	PUNCT
ejpam-4870	481	25	en−1	en−1	PROPN
ejpam-4870	481	26	,	,	PUNCT
ejpam-4870	481	27	en	en	ADJ
ejpam-4870	481	28	}	}	PUNCT
ejpam-4870	481	29	}	}	PUNCT
ejpam-4870	481	30	and	and	CCONJ
ejpam-4870	481	31	|i|	|i|	PROPN
ejpam-4870	481	32	=	=	SYM
ejpam-4870	481	33	n	n	PRON
ejpam-4870	481	34	2	2	NUM
ejpam-4870	481	35	.	.	PUNCT
ejpam-4870	482	1	since	since	SCONJ
ejpam-4870	482	2	each	each	DET
ejpam-4870	482	3	ei	ei	NOUN
ejpam-4870	482	4	,	,	PUNCT
ejpam-4870	482	5	1	1	NUM
ejpam-4870	482	6	<	<	X
ejpam-4870	482	7	i	i	PRON
ejpam-4870	482	8	<	<	X
ejpam-4870	482	9	n	n	CCONJ
ejpam-4870	482	10	,	,	PUNCT
ejpam-4870	482	11	is	be	AUX
ejpam-4870	482	12	incident	incident	NOUN
ejpam-4870	482	13	with	with	ADP
ejpam-4870	482	14	two	two	NUM
ejpam-4870	482	15	other	other	ADJ
ejpam-4870	482	16	edges	edge	NOUN
ejpam-4870	482	17	,	,	PUNCT
ejpam-4870	482	18	it	it	PRON
ejpam-4870	482	19	follows	follow	VERB
ejpam-4870	482	20	that	that	SCONJ
ejpam-4870	482	21	if	if	SCONJ
ejpam-4870	482	22	{	{	PUNCT
ejpam-4870	482	23	ei−1	ei−1	PROPN
ejpam-4870	482	24	,	,	PUNCT
ejpam-4870	482	25	ei	ei	NOUN
ejpam-4870	482	26	}	}	PUNCT
ejpam-4870	482	27	∈	∈	PROPN
ejpam-4870	482	28	v	v	NOUN
ejpam-4870	482	29	(	(	PUNCT
ejpam-4870	482	30	ec(n,2	ec(n,2	PROPN
ejpam-4870	482	31	)	)	PUNCT
ejpam-4870	482	32	)	)	PUNCT
ejpam-4870	483	1	\i	\i	PROPN
ejpam-4870	484	1	then	then	ADV
ejpam-4870	484	2	we	we	PRON
ejpam-4870	484	3	have	have	VERB
ejpam-4870	484	4	{	{	PUNCT
ejpam-4870	484	5	ei	ei	NOUN
ejpam-4870	484	6	,	,	PUNCT
ejpam-4870	484	7	ei+1	ei+1	NOUN
ejpam-4870	484	8	}	}	PUNCT
ejpam-4870	484	9	∈	∈	PROPN
ejpam-4870	485	1	i	i	PRON
ejpam-4870	485	2	such	such	ADJ
ejpam-4870	485	3	that	that	SCONJ
ejpam-4870	485	4	{	{	PUNCT
ejpam-4870	485	5	ei−1	ei−1	PROPN
ejpam-4870	485	6	,	,	PUNCT
ejpam-4870	485	7	ei	ei	ADJ
ejpam-4870	485	8	}	}	PUNCT
ejpam-4870	485	9	is	be	AUX
ejpam-4870	485	10	adjacent	adjacent	ADJ
ejpam-4870	485	11	to	to	PART
ejpam-4870	485	12	{	{	PUNCT
ejpam-4870	485	13	ei	ei	VERB
ejpam-4870	485	14	,	,	PUNCT
ejpam-4870	485	15	ei+1	ei+1	VERB
ejpam-4870	485	16	}	}	PUNCT
ejpam-4870	485	17	since	since	SCONJ
ejpam-4870	485	18	|{ei−1	|{ei−1	NOUN
ejpam-4870	485	19	,	,	PUNCT
ejpam-4870	485	20	ei	ei	NOUN
ejpam-4870	485	21	}	}	PUNCT
ejpam-4870	485	22	∩	∩	NOUN
ejpam-4870	485	23	{	{	PUNCT
ejpam-4870	485	24	ei	ei	NOUN
ejpam-4870	485	25	,	,	PUNCT
ejpam-4870	485	26	ei+1}|	ei+1}|	NOUN
ejpam-4870	485	27	=	=	SYM
ejpam-4870	485	28	1	1	X
ejpam-4870	485	29	.	.	PUNCT
ejpam-4870	486	1	moreover	moreover	ADV
ejpam-4870	486	2	,	,	PUNCT
ejpam-4870	486	3	since	since	SCONJ
ejpam-4870	486	4	e1	e1	NOUN
ejpam-4870	486	5	is	be	AUX
ejpam-4870	486	6	incident	incident	NOUN
ejpam-4870	486	7	to	to	ADP
ejpam-4870	486	8	e2	e2	PROPN
ejpam-4870	486	9	and	and	CCONJ
ejpam-4870	486	10	en	en	X
ejpam-4870	486	11	,	,	PUNCT
ejpam-4870	486	12	if	if	SCONJ
ejpam-4870	486	13	{	{	PUNCT
ejpam-4870	486	14	en	en	X
ejpam-4870	486	15	,	,	PUNCT
ejpam-4870	486	16	e1	e1	PROPN
ejpam-4870	486	17	}	}	PUNCT
ejpam-4870	486	18	∈	∈	PROPN
ejpam-4870	486	19	v	v	NOUN
ejpam-4870	486	20	(	(	PUNCT
ejpam-4870	486	21	ec(n,2	ec(n,2	PROPN
ejpam-4870	486	22	)	)	PUNCT
ejpam-4870	486	23	)	)	PUNCT
ejpam-4870	487	1	\i	\i	PROPN
ejpam-4870	487	2	then	then	ADV
ejpam-4870	487	3	we	we	PRON
ejpam-4870	487	4	have	have	VERB
ejpam-4870	487	5	{	{	PUNCT
ejpam-4870	487	6	e1	e1	NOUN
ejpam-4870	487	7	,	,	PUNCT
ejpam-4870	487	8	e2	e2	NOUN
ejpam-4870	487	9	}	}	PUNCT
ejpam-4870	487	10	∈	∈	PROPN
ejpam-4870	488	1	i	i	PRON
ejpam-4870	488	2	such	such	ADJ
ejpam-4870	488	3	that	that	SCONJ
ejpam-4870	488	4	{	{	PUNCT
ejpam-4870	488	5	en	en	X
ejpam-4870	488	6	,	,	PUNCT
ejpam-4870	488	7	e1	e1	PROPN
ejpam-4870	488	8	}	}	PUNCT
ejpam-4870	488	9	and	and	CCONJ
ejpam-4870	488	10	{	{	PUNCT
ejpam-4870	488	11	e1	e1	PROPN
ejpam-4870	488	12	,	,	PUNCT
ejpam-4870	488	13	e2	e2	PROPN
ejpam-4870	488	14	}	}	PUNCT
ejpam-4870	488	15	are	be	AUX
ejpam-4870	488	16	adjacent	adjacent	ADJ
ejpam-4870	488	17	.	.	PUNCT
ejpam-4870	489	1	similarly	similarly	ADV
ejpam-4870	489	2	,	,	PUNCT
ejpam-4870	489	3	since	since	SCONJ
ejpam-4870	489	4	en	en	X
ejpam-4870	489	5	is	be	VERB
ejpam-4870	489	6	incident	incident	NOUN
ejpam-4870	489	7	to	to	ADP
ejpam-4870	489	8	en−1	en−1	PROPN
ejpam-4870	489	9	and	and	CCONJ
ejpam-4870	489	10	e1	e1	NOUN
ejpam-4870	489	11	,	,	PUNCT
ejpam-4870	489	12	if	if	SCONJ
ejpam-4870	489	13	{	{	PUNCT
ejpam-4870	489	14	en	en	X
ejpam-4870	489	15	,	,	PUNCT
ejpam-4870	489	16	e1	e1	PROPN
ejpam-4870	489	17	}	}	PUNCT
ejpam-4870	489	18	∈	∈	PROPN
ejpam-4870	489	19	v	v	NOUN
ejpam-4870	489	20	(	(	PUNCT
ejpam-4870	489	21	ec(n,2	ec(n,2	PROPN
ejpam-4870	489	22	)	)	PUNCT
ejpam-4870	489	23	)	)	PUNCT
ejpam-4870	490	1	\i	\i	PROPN
ejpam-4870	490	2	then	then	ADV
ejpam-4870	490	3	we	we	PRON
ejpam-4870	490	4	have	have	VERB
ejpam-4870	490	5	{	{	PUNCT
ejpam-4870	490	6	en−1	en−1	PROPN
ejpam-4870	490	7	,	,	PUNCT
ejpam-4870	490	8	en	en	ADP
ejpam-4870	490	9	}	}	PUNCT
ejpam-4870	490	10	∈	∈	NOUN
ejpam-4870	490	11	i	i	PRON
ejpam-4870	490	12	such	such	ADJ
ejpam-4870	490	13	that	that	SCONJ
ejpam-4870	490	14	{	{	PUNCT
ejpam-4870	490	15	en	en	X
ejpam-4870	490	16	,	,	PUNCT
ejpam-4870	490	17	e1	e1	PROPN
ejpam-4870	490	18	}	}	PUNCT
ejpam-4870	490	19	and	and	CCONJ
ejpam-4870	490	20	{	{	PUNCT
ejpam-4870	490	21	en−1	en−1	PROPN
ejpam-4870	490	22	,	,	PUNCT
ejpam-4870	490	23	en	en	ADP
ejpam-4870	490	24	}	}	PUNCT
ejpam-4870	490	25	are	be	AUX
ejpam-4870	490	26	adjacent	adjacent	ADJ
ejpam-4870	490	27	.	.	PUNCT
ejpam-4870	491	1	thus	thus	ADV
ejpam-4870	491	2	,	,	PUNCT
ejpam-4870	491	3	for	for	ADP
ejpam-4870	491	4	every	every	DET
ejpam-4870	491	5	a	a	DET
ejpam-4870	491	6	∈	∈	PROPN
ejpam-4870	491	7	v	v	NOUN
ejpam-4870	491	8	(	(	PUNCT
ejpam-4870	491	9	ec(n,2	ec(n,2	PROPN
ejpam-4870	491	10	)	)	PUNCT
ejpam-4870	491	11	)	)	PUNCT
ejpam-4870	491	12	\i	\i	ADV
ejpam-4870	491	13	,	,	PUNCT
ejpam-4870	491	14	there	there	PRON
ejpam-4870	491	15	exist	exist	VERB
ejpam-4870	491	16	b	b	PRON
ejpam-4870	491	17	∈	∈	NOUN
ejpam-4870	491	18	i	i	PRON
ejpam-4870	491	19	such	such	ADJ
ejpam-4870	491	20	that	that	SCONJ
ejpam-4870	491	21	|a	|a	VERB
ejpam-4870	491	22	∩	∩	NOUN
ejpam-4870	491	23	b|	b|	PROPN
ejpam-4870	491	24	=	=	SYM
ejpam-4870	491	25	1	1	NUM
ejpam-4870	491	26	.	.	PUNCT
ejpam-4870	492	1	hence	hence	ADV
ejpam-4870	492	2	,	,	PUNCT
ejpam-4870	492	3	[	[	X
ejpam-4870	492	4	a	a	PRON
ejpam-4870	492	5	,	,	PUNCT
ejpam-4870	492	6	b	b	NOUN
ejpam-4870	492	7	]	]	X
ejpam-4870	492	8	∈	∈	NOUN
ejpam-4870	492	9	e(ec(n,2	e(ec(n,2	ADJ
ejpam-4870	492	10	)	)	PUNCT
ejpam-4870	492	11	)	)	PUNCT
ejpam-4870	492	12	.	.	PUNCT
ejpam-4870	493	1	therefore	therefore	ADV
ejpam-4870	493	2	,	,	PUNCT
ejpam-4870	493	3	i	i	PRON
ejpam-4870	493	4	is	be	AUX
ejpam-4870	493	5	a	a	DET
ejpam-4870	493	6	dominating	dominating	NOUN
ejpam-4870	493	7	set	set	VERB
ejpam-4870	493	8	in	in	ADP
ejpam-4870	493	9	ec(n,2	ec(n,2	PROPN
ejpam-4870	493	10	)	)	PUNCT
ejpam-4870	493	11	.	.	PUNCT
ejpam-4870	494	1	on	on	ADP
ejpam-4870	494	2	the	the	DET
ejpam-4870	494	3	other	other	ADJ
ejpam-4870	494	4	hand	hand	NOUN
ejpam-4870	494	5	,	,	PUNCT
ejpam-4870	494	6	if	if	SCONJ
ejpam-4870	494	7	n	n	PRON
ejpam-4870	494	8	is	be	AUX
ejpam-4870	494	9	odd	odd	ADJ
ejpam-4870	494	10	,	,	PUNCT
ejpam-4870	494	11	then	then	ADV
ejpam-4870	494	12	⌊n2	⌊n2	PUNCT
ejpam-4870	494	13	⌋	⌋	PROPN
ejpam-4870	495	1	=	=	SYM
ejpam-4870	495	2	n−1	n−1	PROPN
ejpam-4870	495	3	2	2	NUM
ejpam-4870	495	4	.	.	PUNCT
ejpam-4870	496	1	hence	hence	ADV
ejpam-4870	496	2	,	,	PUNCT
ejpam-4870	496	3	i	i	PRON
ejpam-4870	496	4	=	=	PRON
ejpam-4870	496	5	{	{	PUNCT
ejpam-4870	496	6	{	{	PUNCT
ejpam-4870	496	7	e1	e1	PROPN
ejpam-4870	496	8	,	,	PUNCT
ejpam-4870	496	9	e2	e2	PROPN
ejpam-4870	496	10	}	}	PUNCT
ejpam-4870	496	11	,	,	PUNCT
ejpam-4870	496	12	{	{	PUNCT
ejpam-4870	496	13	e3	e3	NOUN
ejpam-4870	496	14	,	,	PUNCT
ejpam-4870	496	15	e4	e4	PROPN
ejpam-4870	496	16	}	}	PUNCT
ejpam-4870	496	17	,	,	PUNCT
ejpam-4870	496	18	·	·	PUNCT
ejpam-4870	496	19	·	·	PUNCT
ejpam-4870	496	20	·	·	PUNCT
ejpam-4870	496	21	,	,	PUNCT
ejpam-4870	496	22	{	{	PUNCT
ejpam-4870	496	23	en−2	en−2	PROPN
ejpam-4870	496	24	,	,	PUNCT
ejpam-4870	496	25	en−1	en−1	PROPN
ejpam-4870	496	26	}	}	PUNCT
ejpam-4870	496	27	}	}	PUNCT
ejpam-4870	496	28	and	and	CCONJ
ejpam-4870	496	29	i	i	PRON
ejpam-4870	496	30	=	=	SYM
ejpam-4870	496	31	n−1	n−1	PROPN
ejpam-4870	496	32	2	2	NUM
ejpam-4870	496	33	.	.	PUNCT
ejpam-4870	497	1	similarly	similarly	ADV
ejpam-4870	497	2	,	,	PUNCT
ejpam-4870	497	3	for	for	ADP
ejpam-4870	497	4	each	each	DET
ejpam-4870	497	5	ei	ei	NOUN
ejpam-4870	497	6	where	where	SCONJ
ejpam-4870	497	7	1	1	NUM
ejpam-4870	497	8	<	<	X
ejpam-4870	497	9	i	i	PROPN
ejpam-4870	497	10	≤	≤	NOUN
ejpam-4870	497	11	n	n	CCONJ
ejpam-4870	497	12	,	,	PUNCT
ejpam-4870	497	13	if	if	SCONJ
ejpam-4870	497	14	{	{	PUNCT
ejpam-4870	497	15	ei−1	ei−1	PROPN
ejpam-4870	497	16	,	,	PUNCT
ejpam-4870	497	17	ei	ei	NOUN
ejpam-4870	497	18	}	}	PUNCT
ejpam-4870	497	19	∈	∈	PROPN
ejpam-4870	497	20	v	v	NOUN
ejpam-4870	497	21	(	(	PUNCT
ejpam-4870	497	22	ec(n,2	ec(n,2	PROPN
ejpam-4870	497	23	)	)	PUNCT
ejpam-4870	497	24	)	)	PUNCT
ejpam-4870	497	25	\i	\i	PROPN
ejpam-4870	497	26	then	then	ADV
ejpam-4870	497	27	we	we	PRON
ejpam-4870	497	28	have	have	VERB
ejpam-4870	497	29	{	{	PUNCT
ejpam-4870	497	30	ei	ei	NOUN
ejpam-4870	497	31	,	,	PUNCT
ejpam-4870	497	32	ei+1	ei+1	NOUN
ejpam-4870	497	33	}	}	PUNCT
ejpam-4870	497	34	∈	∈	PROPN
ejpam-4870	498	1	i	i	PRON
ejpam-4870	498	2	such	such	ADJ
ejpam-4870	498	3	that	that	SCONJ
ejpam-4870	498	4	{	{	PUNCT
ejpam-4870	498	5	ei−1	ei−1	PROPN
ejpam-4870	498	6	,	,	PUNCT
ejpam-4870	498	7	ei	ei	NOUN
ejpam-4870	498	8	}	}	PUNCT
ejpam-4870	498	9	and	and	CCONJ
ejpam-4870	498	10	{	{	PUNCT
ejpam-4870	498	11	ei	ei	NOUN
ejpam-4870	498	12	,	,	PUNCT
ejpam-4870	498	13	ei+1	ei+1	AUX
ejpam-4870	498	14	}	}	PUNCT
ejpam-4870	498	15	are	be	AUX
ejpam-4870	498	16	adjacent	adjacent	ADJ
ejpam-4870	498	17	.	.	PUNCT
ejpam-4870	499	1	moreover	moreover	ADV
ejpam-4870	499	2	,	,	PUNCT
ejpam-4870	499	3	if	if	SCONJ
ejpam-4870	499	4	{	{	PUNCT
ejpam-4870	499	5	en	en	X
ejpam-4870	499	6	,	,	PUNCT
ejpam-4870	499	7	e1	e1	PROPN
ejpam-4870	499	8	}	}	PUNCT
ejpam-4870	499	9	∈	∈	PROPN
ejpam-4870	499	10	v	v	NOUN
ejpam-4870	499	11	(	(	PUNCT
ejpam-4870	499	12	ec(n,2	ec(n,2	PROPN
ejpam-4870	499	13	)	)	PUNCT
ejpam-4870	499	14	)	)	PUNCT
ejpam-4870	499	15	\i	\i	PROPN
ejpam-4870	499	16	then	then	ADV
ejpam-4870	499	17	we	we	PRON
ejpam-4870	499	18	have	have	VERB
ejpam-4870	499	19	{	{	PUNCT
ejpam-4870	499	20	e1	e1	NOUN
ejpam-4870	499	21	,	,	PUNCT
ejpam-4870	499	22	e2	e2	NOUN
ejpam-4870	499	23	}	}	PUNCT
ejpam-4870	499	24	∈	∈	PROPN
ejpam-4870	500	1	i	i	PRON
ejpam-4870	500	2	such	such	ADJ
ejpam-4870	500	3	that	that	SCONJ
ejpam-4870	500	4	{	{	PUNCT
ejpam-4870	500	5	en	en	X
ejpam-4870	500	6	,	,	PUNCT
ejpam-4870	500	7	e1	e1	PROPN
ejpam-4870	500	8	}	}	PUNCT
ejpam-4870	500	9	and	and	CCONJ
ejpam-4870	500	10	{	{	PUNCT
ejpam-4870	500	11	e1	e1	PROPN
ejpam-4870	500	12	,	,	PUNCT
ejpam-4870	500	13	e2	e2	PROPN
ejpam-4870	500	14	}	}	PUNCT
ejpam-4870	500	15	are	be	AUX
ejpam-4870	500	16	adjacent	adjacent	ADJ
ejpam-4870	500	17	.	.	PUNCT
ejpam-4870	501	1	thus	thus	ADV
ejpam-4870	501	2	,	,	PUNCT
ejpam-4870	501	3	for	for	ADP
ejpam-4870	501	4	all	all	DET
ejpam-4870	501	5	a	a	DET
ejpam-4870	501	6	∈	∈	PROPN
ejpam-4870	501	7	v	v	NOUN
ejpam-4870	501	8	(	(	PUNCT
ejpam-4870	501	9	ec(n,2	ec(n,2	PROPN
ejpam-4870	501	10	)	)	PUNCT
ejpam-4870	501	11	)	)	PUNCT
ejpam-4870	501	12	\i	\i	ADV
ejpam-4870	501	13	,	,	PUNCT
ejpam-4870	501	14	there	there	PRON
ejpam-4870	501	15	exist	exist	VERB
ejpam-4870	501	16	b	b	PRON
ejpam-4870	501	17	∈	∈	PROPN
ejpam-4870	501	18	i	i	PRON
ejpam-4870	501	19	where	where	SCONJ
ejpam-4870	501	20	|a	|a	VERB
ejpam-4870	501	21	∩b|	∩b|	PROPN
ejpam-4870	501	22	=	=	SYM
ejpam-4870	501	23	1	1	X
ejpam-4870	501	24	.	.	PUNCT
ejpam-4870	502	1	this	this	PRON
ejpam-4870	502	2	implies	imply	VERB
ejpam-4870	502	3	that	that	SCONJ
ejpam-4870	502	4	[	[	X
ejpam-4870	502	5	a	a	X
ejpam-4870	502	6	,	,	PUNCT
ejpam-4870	502	7	b	b	NOUN
ejpam-4870	502	8	]	]	X
ejpam-4870	502	9	∈	∈	NOUN
ejpam-4870	502	10	e(ec(n,2	e(ec(n,2	ADJ
ejpam-4870	502	11	)	)	PUNCT
ejpam-4870	502	12	)	)	PUNCT
ejpam-4870	502	13	.	.	PUNCT
ejpam-4870	503	1	thus	thus	ADV
ejpam-4870	503	2	,	,	PUNCT
ejpam-4870	503	3	i	i	PRON
ejpam-4870	503	4	is	be	AUX
ejpam-4870	503	5	a	a	DET
ejpam-4870	503	6	dominating	dominating	NOUN
ejpam-4870	503	7	set	set	VERB
ejpam-4870	503	8	in	in	ADP
ejpam-4870	503	9	ec(n,2	ec(n,2	PROPN
ejpam-4870	503	10	)	)	PUNCT
ejpam-4870	503	11	and	and	CCONJ
ejpam-4870	503	12	γ(ec(n,2	γ(ec(n,2	NUM
ejpam-4870	503	13	)	)	PUNCT
ejpam-4870	503	14	)	)	PUNCT
ejpam-4870	504	1	≤	≤	NUM
ejpam-4870	504	2	|i|	|i|	PROPN
ejpam-4870	504	3	.	.	PUNCT
ejpam-4870	504	4	therefore	therefore	ADV
ejpam-4870	504	5	,	,	PUNCT
ejpam-4870	504	6	γ(ec(n,2	γ(ec(n,2	NUM
ejpam-4870	504	7	)	)	PUNCT
ejpam-4870	504	8	)	)	PUNCT
ejpam-4870	505	1	≤	≤	NOUN
ejpam-4870	505	2	⌊n2	⌊n2	PUNCT
ejpam-4870	505	3	⌋.	⌋.	PRON
ejpam-4870	505	4	illustration	illustration	NOUN
ejpam-4870	505	5	12	12	NUM
ejpam-4870	505	6	shows	show	VERB
ejpam-4870	505	7	an	an	DET
ejpam-4870	505	8	upper	upper	ADJ
ejpam-4870	505	9	bound	bind	VERB
ejpam-4870	505	10	for	for	ADP
ejpam-4870	505	11	domination	domination	NOUN
ejpam-4870	505	12	number	number	NOUN
ejpam-4870	505	13	of	of	ADP
ejpam-4870	505	14	ec(6,2	ec(6,2	PROPN
ejpam-4870	505	15	)	)	PUNCT
ejpam-4870	505	16	.	.	PUNCT
ejpam-4870	506	1	illustration	illustration	NOUN
ejpam-4870	506	2	12	12	NUM
ejpam-4870	506	3	.	.	PUNCT
ejpam-4870	507	1	consider	consider	VERB
ejpam-4870	507	2	ec(6,2	ec(6,2	PROPN
ejpam-4870	507	3	)	)	PUNCT
ejpam-4870	507	4	with	with	ADP
ejpam-4870	507	5	pictorial	pictorial	ADJ
ejpam-4870	507	6	representation	representation	NOUN
ejpam-4870	507	7	shown	show	VERB
ejpam-4870	507	8	in	in	ADP
ejpam-4870	507	9	figure	figure	NOUN
ejpam-4870	507	10	21	21	NUM
ejpam-4870	507	11	.	.	PUNCT
ejpam-4870	508	1	let	let	VERB
ejpam-4870	508	2	i	i	PRON
ejpam-4870	508	3	=	=	PRON
ejpam-4870	508	4	{	{	PUNCT
ejpam-4870	508	5	{	{	PUNCT
ejpam-4870	508	6	12	12	NUM
ejpam-4870	508	7	,	,	PUNCT
ejpam-4870	508	8	23	23	NUM
ejpam-4870	508	9	}	}	PUNCT
ejpam-4870	508	10	,	,	PUNCT
ejpam-4870	508	11	{	{	PUNCT
ejpam-4870	508	12	34	34	NUM
ejpam-4870	508	13	,	,	PUNCT
ejpam-4870	508	14	45	45	NUM
ejpam-4870	508	15	}	}	PUNCT
ejpam-4870	508	16	,	,	PUNCT
ejpam-4870	508	17	{	{	PUNCT
ejpam-4870	508	18	56	56	NUM
ejpam-4870	508	19	,	,	PUNCT
ejpam-4870	508	20	61	61	NUM
ejpam-4870	508	21	}	}	PUNCT
ejpam-4870	508	22	}	}	PUNCT
ejpam-4870	508	23	where	where	SCONJ
ejpam-4870	508	24	i	i	PRON
ejpam-4870	508	25	⊆	⊆	NUM
ejpam-4870	508	26	v	v	NOUN
ejpam-4870	508	27	(	(	PUNCT
ejpam-4870	508	28	ec(6,2	ec(6,2	PROPN
ejpam-4870	508	29	)	)	PUNCT
ejpam-4870	508	30	)	)	PUNCT
ejpam-4870	508	31	.	.	PUNCT
ejpam-4870	509	1	now	now	ADV
ejpam-4870	509	2	,	,	PUNCT
ejpam-4870	509	3	the	the	DET
ejpam-4870	509	4	set	set	NOUN
ejpam-4870	509	5	v	v	NOUN
ejpam-4870	509	6	(	(	PUNCT
ejpam-4870	509	7	ec(6,2	ec(6,2	PROPN
ejpam-4870	509	8	)	)	PUNCT
ejpam-4870	509	9	)	)	PUNCT
ejpam-4870	509	10	\i	\i	PROPN
ejpam-4870	509	11	is	be	AUX
ejpam-4870	509	12	given	give	VERB
ejpam-4870	509	13	j.c	j.c	PROPN
ejpam-4870	509	14	.	.	PROPN
ejpam-4870	509	15	bonifacio	bonifacio	PROPN
ejpam-4870	509	16	,	,	PUNCT
ejpam-4870	509	17	c.j	c.j	PROPN
ejpam-4870	509	18	.	.	PROPN
ejpam-4870	509	19	andaya	andaya	PROPN
ejpam-4870	509	20	,	,	PUNCT
ejpam-4870	509	21	d.	d.	PROPN
ejpam-4870	509	22	magpantay	magpantay	PROPN
ejpam-4870	509	23	/	/	SYM
ejpam-4870	509	24	eur	eur	PROPN
ejpam-4870	509	25	.	.	PUNCT
ejpam-4870	510	1	j.	j.	PROPN
ejpam-4870	510	2	pure	pure	PROPN
ejpam-4870	510	3	appl	appl	PROPN
ejpam-4870	510	4	.	.	PROPN
ejpam-4870	510	5	math	math	PROPN
ejpam-4870	510	6	,	,	PUNCT
ejpam-4870	510	7	16	16	NUM
ejpam-4870	510	8	(	(	PUNCT
ejpam-4870	510	9	4	4	NUM
ejpam-4870	510	10	)	)	PUNCT
ejpam-4870	510	11	(	(	PUNCT
ejpam-4870	510	12	2023	2023	NUM
ejpam-4870	510	13	)	)	PUNCT
ejpam-4870	510	14	,	,	PUNCT
ejpam-4870	510	15	2476	2476	NUM
ejpam-4870	510	16	-	-	SYM
ejpam-4870	510	17	2498	2498	NUM
ejpam-4870	510	18	2497	2497	NUM
ejpam-4870	510	19	by	by	ADP
ejpam-4870	510	20	v	v	NOUN
ejpam-4870	510	21	(	(	PUNCT
ejpam-4870	510	22	ec(6,2	ec(6,2	PROPN
ejpam-4870	510	23	)	)	PUNCT
ejpam-4870	510	24	)	)	PUNCT
ejpam-4870	511	1	\i	\i	NOUN
ejpam-4870	512	1	=	=	NOUN
ejpam-4870	512	2	{	{	PUNCT
ejpam-4870	512	3	{	{	PUNCT
ejpam-4870	512	4	12	12	NUM
ejpam-4870	512	5	,	,	PUNCT
ejpam-4870	512	6	34	34	NUM
ejpam-4870	512	7	}	}	PUNCT
ejpam-4870	512	8	,	,	PUNCT
ejpam-4870	512	9	{	{	PUNCT
ejpam-4870	512	10	12	12	NUM
ejpam-4870	512	11	,	,	PUNCT
ejpam-4870	512	12	45	45	NUM
ejpam-4870	512	13	}	}	PUNCT
ejpam-4870	512	14	,	,	PUNCT
ejpam-4870	512	15	{	{	PUNCT
ejpam-4870	512	16	12	12	NUM
ejpam-4870	512	17	,	,	PUNCT
ejpam-4870	512	18	56	56	NUM
ejpam-4870	512	19	}	}	PUNCT
ejpam-4870	512	20	,	,	PUNCT
ejpam-4870	512	21	{	{	PUNCT
ejpam-4870	512	22	12	12	NUM
ejpam-4870	512	23	,	,	PUNCT
ejpam-4870	512	24	61	61	NUM
ejpam-4870	512	25	}	}	PUNCT
ejpam-4870	512	26	,	,	PUNCT
ejpam-4870	512	27	{	{	PUNCT
ejpam-4870	512	28	23	23	NUM
ejpam-4870	512	29	,	,	PUNCT
ejpam-4870	512	30	34	34	NUM
ejpam-4870	512	31	}	}	PUNCT
ejpam-4870	512	32	,	,	PUNCT
ejpam-4870	512	33	{	{	PUNCT
ejpam-4870	512	34	23	23	NUM
ejpam-4870	512	35	,	,	PUNCT
ejpam-4870	512	36	45	45	NUM
ejpam-4870	512	37	}	}	PUNCT
ejpam-4870	512	38	,	,	PUNCT
ejpam-4870	512	39	{	{	PUNCT
ejpam-4870	512	40	23	23	NUM
ejpam-4870	512	41	,	,	PUNCT
ejpam-4870	512	42	56	56	NUM
ejpam-4870	512	43	}	}	PUNCT
ejpam-4870	512	44	,	,	PUNCT
ejpam-4870	512	45	{	{	PUNCT
ejpam-4870	512	46	23	23	NUM
ejpam-4870	512	47	,	,	PUNCT
ejpam-4870	512	48	61	61	NUM
ejpam-4870	512	49	}	}	PUNCT
ejpam-4870	512	50	,	,	PUNCT
ejpam-4870	512	51	{	{	PUNCT
ejpam-4870	512	52	34	34	NUM
ejpam-4870	512	53	,	,	PUNCT
ejpam-4870	512	54	56	56	NUM
ejpam-4870	512	55	}	}	PUNCT
ejpam-4870	512	56	,	,	PUNCT
ejpam-4870	512	57	{	{	PUNCT
ejpam-4870	512	58	34	34	NUM
ejpam-4870	512	59	,	,	PUNCT
ejpam-4870	512	60	61	61	NUM
ejpam-4870	512	61	}	}	PUNCT
ejpam-4870	512	62	,	,	PUNCT
ejpam-4870	512	63	{	{	PUNCT
ejpam-4870	512	64	45	45	NUM
ejpam-4870	512	65	,	,	PUNCT
ejpam-4870	512	66	56	56	NUM
ejpam-4870	512	67	}	}	PUNCT
ejpam-4870	512	68	,	,	PUNCT
ejpam-4870	512	69	{	{	PUNCT
ejpam-4870	512	70	45	45	NUM
ejpam-4870	512	71	,	,	PUNCT
ejpam-4870	512	72	61	61	NUM
ejpam-4870	512	73	}	}	PUNCT
ejpam-4870	512	74	}	}	PUNCT
ejpam-4870	512	75	.	.	PUNCT
ejpam-4870	513	1	it	it	PRON
ejpam-4870	513	2	can	can	AUX
ejpam-4870	513	3	be	be	AUX
ejpam-4870	513	4	noted	note	VERB
ejpam-4870	513	5	that	that	SCONJ
ejpam-4870	513	6	the	the	DET
ejpam-4870	513	7	vertices	vertex	NOUN
ejpam-4870	513	8	{	{	PUNCT
ejpam-4870	513	9	12	12	NUM
ejpam-4870	513	10	,	,	PUNCT
ejpam-4870	513	11	34	34	NUM
ejpam-4870	513	12	}	}	PUNCT
ejpam-4870	513	13	,	,	PUNCT
ejpam-4870	513	14	{	{	PUNCT
ejpam-4870	513	15	12	12	NUM
ejpam-4870	513	16	,	,	PUNCT
ejpam-4870	513	17	45	45	NUM
ejpam-4870	513	18	}	}	PUNCT
ejpam-4870	513	19	,	,	PUNCT
ejpam-4870	513	20	{	{	PUNCT
ejpam-4870	513	21	12	12	NUM
ejpam-4870	513	22	,	,	PUNCT
ejpam-4870	513	23	56	56	NUM
ejpam-4870	513	24	}	}	PUNCT
ejpam-4870	513	25	,	,	PUNCT
ejpam-4870	513	26	{	{	PUNCT
ejpam-4870	513	27	12	12	NUM
ejpam-4870	513	28	,	,	PUNCT
ejpam-4870	513	29	61	61	NUM
ejpam-4870	513	30	}	}	PUNCT
ejpam-4870	513	31	,	,	PUNCT
ejpam-4870	513	32	{	{	PUNCT
ejpam-4870	513	33	23	23	NUM
ejpam-4870	513	34	,	,	PUNCT
ejpam-4870	513	35	34	34	NUM
ejpam-4870	513	36	}	}	PUNCT
ejpam-4870	513	37	,	,	PUNCT
ejpam-4870	513	38	{	{	PUNCT
ejpam-4870	513	39	23	23	NUM
ejpam-4870	513	40	,	,	PUNCT
ejpam-4870	513	41	45	45	NUM
ejpam-4870	513	42	}	}	PUNCT
ejpam-4870	513	43	,	,	PUNCT
ejpam-4870	513	44	{	{	PUNCT
ejpam-4870	513	45	23	23	NUM
ejpam-4870	513	46	,	,	PUNCT
ejpam-4870	513	47	56	56	NUM
ejpam-4870	513	48	}	}	PUNCT
ejpam-4870	513	49	,	,	PUNCT
ejpam-4870	513	50	{	{	PUNCT
ejpam-4870	513	51	23	23	NUM
ejpam-4870	513	52	,	,	PUNCT
ejpam-4870	513	53	61	61	NUM
ejpam-4870	513	54	}	}	PUNCT
ejpam-4870	513	55	of	of	ADP
ejpam-4870	513	56	v	v	NOUN
ejpam-4870	513	57	(	(	PUNCT
ejpam-4870	513	58	ec(6,2	ec(6,2	PROPN
ejpam-4870	513	59	)	)	PUNCT
ejpam-4870	513	60	)	)	PUNCT
ejpam-4870	513	61	\i	\i	ADJ
ejpam-4870	513	62	are	be	AUX
ejpam-4870	513	63	adjacent	adjacent	ADJ
ejpam-4870	513	64	to	to	ADP
ejpam-4870	513	65	{	{	PUNCT
ejpam-4870	513	66	12	12	NUM
ejpam-4870	513	67	,	,	PUNCT
ejpam-4870	513	68	23	23	NUM
ejpam-4870	513	69	}	}	PUNCT
ejpam-4870	513	70	.	.	PUNCT
ejpam-4870	514	1	also	also	ADV
ejpam-4870	514	2	,	,	PUNCT
ejpam-4870	514	3	{	{	PUNCT
ejpam-4870	514	4	34	34	NUM
ejpam-4870	514	5	,	,	PUNCT
ejpam-4870	514	6	56	56	NUM
ejpam-4870	514	7	}	}	PUNCT
ejpam-4870	514	8	,	,	PUNCT
ejpam-4870	514	9	{	{	PUNCT
ejpam-4870	514	10	34	34	NUM
ejpam-4870	514	11	,	,	PUNCT
ejpam-4870	514	12	61	61	NUM
ejpam-4870	514	13	}	}	PUNCT
ejpam-4870	514	14	,	,	PUNCT
ejpam-4870	514	15	{	{	PUNCT
ejpam-4870	514	16	45	45	NUM
ejpam-4870	514	17	,	,	PUNCT
ejpam-4870	514	18	56	56	NUM
ejpam-4870	514	19	}	}	PUNCT
ejpam-4870	514	20	,	,	PUNCT
ejpam-4870	514	21	{	{	PUNCT
ejpam-4870	514	22	45	45	NUM
ejpam-4870	514	23	,	,	PUNCT
ejpam-4870	514	24	61	61	NUM
ejpam-4870	514	25	}	}	PUNCT
ejpam-4870	514	26	are	be	AUX
ejpam-4870	514	27	vertices	vertex	NOUN
ejpam-4870	514	28	adjacent	adjacent	ADJ
ejpam-4870	514	29	to	to	ADP
ejpam-4870	514	30	{	{	PUNCT
ejpam-4870	514	31	34	34	NUM
ejpam-4870	514	32	,	,	PUNCT
ejpam-4870	514	33	45	45	NUM
ejpam-4870	514	34	}	}	PUNCT
ejpam-4870	514	35	.	.	PUNCT
ejpam-4870	515	1	now	now	ADV
ejpam-4870	515	2	,	,	PUNCT
ejpam-4870	515	3	all	all	PRON
ejpam-4870	515	4	of	of	ADP
ejpam-4870	515	5	the	the	DET
ejpam-4870	515	6	elements	element	NOUN
ejpam-4870	515	7	of	of	ADP
ejpam-4870	515	8	v	v	NOUN
ejpam-4870	515	9	(	(	PUNCT
ejpam-4870	515	10	ec(6,2	ec(6,2	PROPN
ejpam-4870	515	11	)	)	PUNCT
ejpam-4870	515	12	)	)	PUNCT
ejpam-4870	515	13	\i	\i	ADJ
ejpam-4870	515	14	are	be	AUX
ejpam-4870	515	15	adjacent	adjacent	ADJ
ejpam-4870	515	16	to	to	ADP
ejpam-4870	515	17	either	either	CCONJ
ejpam-4870	515	18	{	{	PUNCT
ejpam-4870	515	19	12	12	NUM
ejpam-4870	515	20	,	,	PUNCT
ejpam-4870	515	21	23	23	NUM
ejpam-4870	515	22	}	}	PUNCT
ejpam-4870	515	23	or	or	CCONJ
ejpam-4870	515	24	{	{	PUNCT
ejpam-4870	515	25	34	34	NUM
ejpam-4870	515	26	,	,	PUNCT
ejpam-4870	515	27	45	45	NUM
ejpam-4870	515	28	}	}	PUNCT
ejpam-4870	515	29	.	.	PUNCT
ejpam-4870	516	1	hence	hence	ADV
ejpam-4870	516	2	,	,	PUNCT
ejpam-4870	516	3	i	i	PRON
ejpam-4870	516	4	is	be	AUX
ejpam-4870	516	5	a	a	DET
ejpam-4870	516	6	dominating	dominating	NOUN
ejpam-4870	516	7	set	set	VERB
ejpam-4870	516	8	with	with	ADP
ejpam-4870	516	9	cardinality	cardinality	NOUN
ejpam-4870	516	10	equal	equal	ADJ
ejpam-4870	516	11	to	to	ADP
ejpam-4870	516	12	3	3	NUM
ejpam-4870	516	13	which	which	PRON
ejpam-4870	516	14	is	be	AUX
ejpam-4870	516	15	also	also	ADV
ejpam-4870	516	16	equal	equal	ADJ
ejpam-4870	516	17	to	to	ADP
ejpam-4870	516	18	⌊62⌋.	⌊62⌋.	NOUN
ejpam-4870	516	19	therefore	therefore	ADV
ejpam-4870	516	20	,	,	PUNCT
ejpam-4870	516	21	there	there	PRON
ejpam-4870	516	22	exists	exist	VERB
ejpam-4870	516	23	a	a	DET
ejpam-4870	516	24	dominating	dominating	NOUN
ejpam-4870	516	25	set	set	VERB
ejpam-4870	516	26	in	in	ADP
ejpam-4870	516	27	ec(6,2	ec(6,2	PROPN
ejpam-4870	516	28	)	)	PUNCT
ejpam-4870	516	29	with	with	ADP
ejpam-4870	516	30	cardinality	cardinality	NOUN
ejpam-4870	516	31	equal	equal	ADJ
ejpam-4870	516	32	to	to	ADP
ejpam-4870	516	33	⌊n2	⌊n2	PROPN
ejpam-4870	516	34	⌋.	⌋.	ADV
ejpam-4870	517	1	moreover	moreover	ADV
ejpam-4870	517	2	,	,	PUNCT
ejpam-4870	517	3	there	there	PRON
ejpam-4870	517	4	is	be	VERB
ejpam-4870	517	5	no	no	DET
ejpam-4870	517	6	dominating	dominating	NOUN
ejpam-4870	517	7	set	set	VERB
ejpam-4870	517	8	in	in	ADP
ejpam-4870	517	9	ec(6,2	ec(6,2	PROPN
ejpam-4870	517	10	)	)	PUNCT
ejpam-4870	517	11	with	with	ADP
ejpam-4870	517	12	cardinality	cardinality	NOUN
ejpam-4870	517	13	less	less	ADJ
ejpam-4870	517	14	than	than	ADP
ejpam-4870	517	15	3	3	NUM
ejpam-4870	517	16	.	.	PUNCT
ejpam-4870	517	17	therefore	therefore	ADV
ejpam-4870	517	18	,	,	PUNCT
ejpam-4870	517	19	γ(ec(6,2	γ(ec(6,2	NOUN
ejpam-4870	517	20	)	)	PUNCT
ejpam-4870	517	21	)	)	PUNCT
ejpam-4870	518	1	≤	≤	NUM
ejpam-4870	518	2	3	3	NUM
ejpam-4870	518	3	.	.	PUNCT
ejpam-4870	518	4	to	to	PART
ejpam-4870	518	5	verify	verify	VERB
ejpam-4870	518	6	this	this	PRON
ejpam-4870	518	7	,	,	PUNCT
ejpam-4870	518	8	using	use	VERB
ejpam-4870	518	9	theorem	theorem	NOUN
ejpam-4870	518	10	10	10	NUM
ejpam-4870	518	11	,	,	PUNCT
ejpam-4870	518	12	setting	set	VERB
ejpam-4870	518	13	n	n	X
ejpam-4870	518	14	=	=	SYM
ejpam-4870	518	15	6	6	NUM
ejpam-4870	518	16	and	and	CCONJ
ejpam-4870	518	17	j	j	NOUN
ejpam-4870	518	18	=	=	SYM
ejpam-4870	518	19	2	2	NUM
ejpam-4870	518	20	,	,	PUNCT
ejpam-4870	518	21	we	we	PRON
ejpam-4870	518	22	have	have	VERB
ejpam-4870	518	23	γ(ec(6,2	γ(ec(6,2	VERB
ejpam-4870	518	24	)	)	PUNCT
ejpam-4870	518	25	)	)	PUNCT
ejpam-4870	519	1	≤	≤	NUM
ejpam-4870	519	2	⌊	⌊	VERB
ejpam-4870	519	3	6	6	NUM
ejpam-4870	519	4	2	2	NUM
ejpam-4870	519	5	⌋	⌋	NOUN
ejpam-4870	519	6	=3	=3	VERB
ejpam-4870	519	7	.	.	PUNCT
ejpam-4870	520	1	5	5	X
ejpam-4870	520	2	.	.	X
ejpam-4870	520	3	conclusion	conclusion	NOUN
ejpam-4870	520	4	and	and	CCONJ
ejpam-4870	520	5	recommendations	recommendation	NOUN
ejpam-4870	520	6	this	this	DET
ejpam-4870	520	7	study	study	NOUN
ejpam-4870	520	8	explores	explore	NOUN
ejpam-4870	520	9	and	and	CCONJ
ejpam-4870	520	10	defines	define	VERB
ejpam-4870	520	11	a	a	DET
ejpam-4870	520	12	new	new	ADJ
ejpam-4870	520	13	graph	graph	NOUN
ejpam-4870	520	14	which	which	PRON
ejpam-4870	520	15	is	be	AUX
ejpam-4870	520	16	called	call	VERB
ejpam-4870	520	17	a	a	DET
ejpam-4870	520	18	j	j	NOUN
ejpam-4870	520	19	-	-	PUNCT
ejpam-4870	520	20	edge	edge	NOUN
ejpam-4870	520	21	graph	graph	NOUN
ejpam-4870	520	22	of	of	ADP
ejpam-4870	520	23	cn	cn	PROPN
ejpam-4870	520	24	,	,	PUNCT
ejpam-4870	520	25	as	as	ADV
ejpam-4870	520	26	well	well	ADV
ejpam-4870	520	27	as	as	ADP
ejpam-4870	520	28	some	some	PRON
ejpam-4870	520	29	of	of	ADP
ejpam-4870	520	30	its	its	PRON
ejpam-4870	520	31	parameters	parameter	NOUN
ejpam-4870	520	32	and	and	CCONJ
ejpam-4870	520	33	characteristics	characteristic	NOUN
ejpam-4870	520	34	.	.	PUNCT
ejpam-4870	521	1	a	a	DET
ejpam-4870	521	2	j	j	NOUN
ejpam-4870	521	3	-	-	PUNCT
ejpam-4870	521	4	edge	edge	NOUN
ejpam-4870	521	5	graph	graph	NOUN
ejpam-4870	521	6	of	of	ADP
ejpam-4870	521	7	cn	cn	PROPN
ejpam-4870	521	8	,	,	PUNCT
ejpam-4870	521	9	denoted	denote	VERB
ejpam-4870	521	10	by	by	ADP
ejpam-4870	521	11	ec(n	ec(n	PROPN
ejpam-4870	521	12	,	,	PUNCT
ejpam-4870	521	13	j	j	PROPN
ejpam-4870	521	14	)	)	PUNCT
ejpam-4870	521	15	,	,	PUNCT
ejpam-4870	521	16	is	be	AUX
ejpam-4870	521	17	a	a	DET
ejpam-4870	521	18	graph	graph	NOUN
ejpam-4870	521	19	whose	whose	DET
ejpam-4870	521	20	vertex	vertex	NOUN
ejpam-4870	521	21	set	set	NOUN
ejpam-4870	521	22	contains	contain	VERB
ejpam-4870	521	23	the	the	DET
ejpam-4870	521	24	spanning	span	VERB
ejpam-4870	521	25	subgraphs	subgraph	NOUN
ejpam-4870	521	26	of	of	ADP
ejpam-4870	521	27	cn	cn	PROPN
ejpam-4870	521	28	with	with	ADP
ejpam-4870	521	29	j	j	PROPN
ejpam-4870	521	30	edges	edge	NOUN
ejpam-4870	521	31	.	.	PUNCT
ejpam-4870	522	1	moreover	moreover	ADV
ejpam-4870	522	2	,	,	PUNCT
ejpam-4870	522	3	two	two	NUM
ejpam-4870	522	4	distinct	distinct	ADJ
ejpam-4870	522	5	vertices	vertex	NOUN
ejpam-4870	522	6	are	be	AUX
ejpam-4870	522	7	adjacent	adjacent	ADJ
ejpam-4870	522	8	whenever	whenever	SCONJ
ejpam-4870	522	9	they	they	PRON
ejpam-4870	522	10	share	share	VERB
ejpam-4870	522	11	exactly	exactly	ADV
ejpam-4870	522	12	one	one	NUM
ejpam-4870	522	13	common	common	ADJ
ejpam-4870	522	14	edge	edge	NOUN
ejpam-4870	522	15	.	.	PUNCT
ejpam-4870	523	1	a	a	DET
ejpam-4870	523	2	j	j	NOUN
ejpam-4870	523	3	-	-	PUNCT
ejpam-4870	523	4	edge	edge	NOUN
ejpam-4870	523	5	graph	graph	NOUN
ejpam-4870	523	6	of	of	ADP
ejpam-4870	523	7	cn	cn	PROPN
ejpam-4870	523	8	does	do	AUX
ejpam-4870	523	9	not	not	PART
ejpam-4870	523	10	contain	contain	VERB
ejpam-4870	523	11	any	any	DET
ejpam-4870	523	12	loop	loop	NOUN
ejpam-4870	523	13	.	.	PUNCT
ejpam-4870	524	1	since	since	SCONJ
ejpam-4870	524	2	v	v	NUM
ejpam-4870	524	3	(	(	PUNCT
ejpam-4870	524	4	ec(n	ec(n	NUM
ejpam-4870	524	5	,	,	PUNCT
ejpam-4870	524	6	j	j	NOUN
ejpam-4870	524	7	)	)	PUNCT
ejpam-4870	524	8	)	)	PUNCT
ejpam-4870	524	9	is	be	AUX
ejpam-4870	524	10	the	the	DET
ejpam-4870	524	11	collection	collection	NOUN
ejpam-4870	524	12	of	of	ADP
ejpam-4870	524	13	all	all	DET
ejpam-4870	524	14	distinct	distinct	ADJ
ejpam-4870	524	15	spanning	span	VERB
ejpam-4870	524	16	subgraphs	subgraph	NOUN
ejpam-4870	524	17	of	of	ADP
ejpam-4870	524	18	cn	cn	PROPN
ejpam-4870	524	19	with	with	ADP
ejpam-4870	524	20	j	j	PROPN
ejpam-4870	524	21	edges	edge	NOUN
ejpam-4870	524	22	,	,	PUNCT
ejpam-4870	524	23	it	it	PRON
ejpam-4870	524	24	follows	follow	VERB
ejpam-4870	524	25	that	that	SCONJ
ejpam-4870	524	26	e(ec(n	e(ec(n	PROPN
ejpam-4870	524	27	,	,	PUNCT
ejpam-4870	524	28	j	j	PROPN
ejpam-4870	524	29	)	)	PUNCT
ejpam-4870	524	30	)	)	PUNCT
ejpam-4870	524	31	does	do	AUX
ejpam-4870	524	32	not	not	PART
ejpam-4870	524	33	have	have	VERB
ejpam-4870	524	34	the	the	DET
ejpam-4870	524	35	same	same	ADJ
ejpam-4870	524	36	pair	pair	NOUN
ejpam-4870	524	37	of	of	ADP
ejpam-4870	524	38	vertices	vertex	NOUN
ejpam-4870	524	39	which	which	PRON
ejpam-4870	524	40	means	mean	VERB
ejpam-4870	524	41	that	that	SCONJ
ejpam-4870	524	42	ec(n	ec(n	PROPN
ejpam-4870	524	43	,	,	PUNCT
ejpam-4870	524	44	j	j	PROPN
ejpam-4870	524	45	)	)	PUNCT
ejpam-4870	524	46	has	have	VERB
ejpam-4870	524	47	no	no	DET
ejpam-4870	524	48	multiple	multiple	ADJ
ejpam-4870	524	49	edges	edge	NOUN
ejpam-4870	524	50	and	and	CCONJ
ejpam-4870	524	51	it	it	PRON
ejpam-4870	524	52	is	be	AUX
ejpam-4870	524	53	a	a	DET
ejpam-4870	524	54	simple	simple	ADJ
ejpam-4870	524	55	graph	graph	NOUN
ejpam-4870	524	56	.	.	PUNCT
ejpam-4870	525	1	the	the	DET
ejpam-4870	525	2	researchers	researcher	NOUN
ejpam-4870	525	3	discovered	discover	VERB
ejpam-4870	525	4	that	that	SCONJ
ejpam-4870	525	5	the	the	DET
ejpam-4870	525	6	order	order	NOUN
ejpam-4870	525	7	of	of	ADP
ejpam-4870	525	8	ec(n	ec(n	PROPN
ejpam-4870	525	9	,	,	PUNCT
ejpam-4870	525	10	j	j	NOUN
ejpam-4870	525	11	)	)	PUNCT
ejpam-4870	525	12	is	be	AUX
ejpam-4870	525	13	equal	equal	ADJ
ejpam-4870	525	14	to	to	ADP
ejpam-4870	525	15	(	(	PUNCT
ejpam-4870	525	16	n	n	PRON
ejpam-4870	525	17	j	j	PROPN
ejpam-4870	525	18	)	)	PUNCT
ejpam-4870	525	19	.	.	PUNCT
ejpam-4870	526	1	from	from	ADP
ejpam-4870	526	2	this	this	PRON
ejpam-4870	526	3	,	,	PUNCT
ejpam-4870	526	4	it	it	PRON
ejpam-4870	526	5	was	be	AUX
ejpam-4870	526	6	determined	determine	VERB
ejpam-4870	526	7	that	that	SCONJ
ejpam-4870	526	8	ec(n	ec(n	PROPN
ejpam-4870	526	9	,	,	PUNCT
ejpam-4870	526	10	j	j	NOUN
ejpam-4870	526	11	)	)	PUNCT
ejpam-4870	526	12	is	be	AUX
ejpam-4870	526	13	a	a	DET
ejpam-4870	526	14	trivial	trivial	ADJ
ejpam-4870	526	15	graph	graph	NOUN
ejpam-4870	527	1	if	if	SCONJ
ejpam-4870	527	2	j	j	PROPN
ejpam-4870	527	3	=	=	PROPN
ejpam-4870	527	4	n.	n.	PROPN
ejpam-4870	527	5	moreover	moreover	ADV
ejpam-4870	527	6	,	,	PUNCT
ejpam-4870	527	7	the	the	DET
ejpam-4870	527	8	degree	degree	NOUN
ejpam-4870	527	9	of	of	ADP
ejpam-4870	527	10	every	every	DET
ejpam-4870	527	11	vertex	vertex	NOUN
ejpam-4870	527	12	in	in	ADP
ejpam-4870	527	13	ec(n	ec(n	PROPN
ejpam-4870	527	14	,	,	PUNCT
ejpam-4870	527	15	j	j	PROPN
ejpam-4870	527	16	)	)	PUNCT
ejpam-4870	527	17	when	when	SCONJ
ejpam-4870	527	18	j	j	PROPN
ejpam-4870	527	19	=	=	SYM
ejpam-4870	527	20	1	1	NUM
ejpam-4870	527	21	and	and	CCONJ
ejpam-4870	527	22	⌈n2	⌈n2	NOUN
ejpam-4870	527	23	⌉	⌉	ADP
ejpam-4870	527	24	<	<	X
ejpam-4870	527	25	j	j	PROPN
ejpam-4870	527	26	≤	≤	PUNCT
ejpam-4870	527	27	n	n	CCONJ
ejpam-4870	527	28	is	be	AUX
ejpam-4870	527	29	both	both	ADV
ejpam-4870	527	30	equal	equal	ADJ
ejpam-4870	527	31	to	to	ADP
ejpam-4870	527	32	0	0	NUM
ejpam-4870	527	33	which	which	PRON
ejpam-4870	527	34	will	will	AUX
ejpam-4870	527	35	both	both	PRON
ejpam-4870	527	36	produce	produce	VERB
ejpam-4870	527	37	an	an	DET
ejpam-4870	527	38	empty	empty	ADJ
ejpam-4870	527	39	graph	graph	NOUN
ejpam-4870	527	40	of	of	ADP
ejpam-4870	527	41	order	order	NOUN
ejpam-4870	527	42	(	(	PUNCT
ejpam-4870	527	43	n	n	NOUN
ejpam-4870	527	44	2	2	NUM
ejpam-4870	527	45	)	)	PUNCT
ejpam-4870	527	46	.	.	PUNCT
ejpam-4870	528	1	with	with	ADP
ejpam-4870	528	2	these	these	PRON
ejpam-4870	528	3	,	,	PUNCT
ejpam-4870	528	4	the	the	DET
ejpam-4870	528	5	proponents	proponent	NOUN
ejpam-4870	528	6	focused	focus	VERB
ejpam-4870	528	7	only	only	ADV
ejpam-4870	528	8	on	on	ADP
ejpam-4870	528	9	ec(n	ec(n	PROPN
ejpam-4870	528	10	,	,	PUNCT
ejpam-4870	528	11	j	j	NOUN
ejpam-4870	528	12	)	)	PUNCT
ejpam-4870	528	13	when	when	SCONJ
ejpam-4870	528	14	2	2	NUM
ejpam-4870	528	15	≤	≤	NUM
ejpam-4870	528	16	j	j	PROPN
ejpam-4870	528	17	≤	≤	PROPN
ejpam-4870	528	18	⌈n2	⌈n2	NOUN
ejpam-4870	528	19	⌉.	⌉.	ADV
ejpam-4870	528	20	for	for	ADP
ejpam-4870	528	21	all	all	DET
ejpam-4870	528	22	a	a	DET
ejpam-4870	528	23	∈	∈	PROPN
ejpam-4870	528	24	v	v	NOUN
ejpam-4870	528	25	(	(	PUNCT
ejpam-4870	528	26	ec(n	ec(n	PROPN
ejpam-4870	528	27	,	,	PUNCT
ejpam-4870	528	28	j	j	NOUN
ejpam-4870	528	29	)	)	PUNCT
ejpam-4870	528	30	)	)	PUNCT
ejpam-4870	528	31	where	where	SCONJ
ejpam-4870	528	32	2	2	NUM
ejpam-4870	528	33	≤	≤	NUM
ejpam-4870	528	34	j	j	PROPN
ejpam-4870	528	35	≤	≤	PROPN
ejpam-4870	528	36	⌈n2	⌈n2	NOUN
ejpam-4870	528	37	⌉	⌉	NOUN
ejpam-4870	528	38	,	,	PUNCT
ejpam-4870	528	39	deg(a	deg(a	PROPN
ejpam-4870	528	40	)	)	PUNCT
ejpam-4870	529	1	=	=	SYM
ejpam-4870	529	2	j	j	PROPN
ejpam-4870	529	3	(	(	PUNCT
ejpam-4870	529	4	n−j	n−j	ADV
ejpam-4870	529	5	j−1	j−1	PROPN
ejpam-4870	529	6	)	)	PUNCT
ejpam-4870	529	7	.	.	PUNCT
ejpam-4870	530	1	the	the	DET
ejpam-4870	530	2	size	size	NOUN
ejpam-4870	530	3	of	of	ADP
ejpam-4870	530	4	ec(n	ec(n	PROPN
ejpam-4870	530	5	,	,	PUNCT
ejpam-4870	530	6	j	j	NOUN
ejpam-4870	530	7	)	)	PUNCT
ejpam-4870	530	8	is	be	AUX
ejpam-4870	530	9	equal	equal	ADJ
ejpam-4870	530	10	to	to	ADP
ejpam-4870	530	11	|e(ec(n	|e(ec(n	NOUN
ejpam-4870	530	12	,	,	PUNCT
ejpam-4870	530	13	j	j	NOUN
ejpam-4870	530	14	)	)	PUNCT
ejpam-4870	530	15	)	)	PUNCT
ejpam-4870	531	1	|	|	ADV
ejpam-4870	531	2	=	=	SYM
ejpam-4870	531	3	j(n−j	j(n−j	NUM
ejpam-4870	531	4	j−1	j−1	PROPN
ejpam-4870	531	5	)	)	PUNCT
ejpam-4870	531	6	(	(	PUNCT
ejpam-4870	531	7	n	n	X
ejpam-4870	531	8	j	j	NOUN
ejpam-4870	531	9	)	)	PUNCT
ejpam-4870	531	10	2	2	NUM
ejpam-4870	531	11	if	if	SCONJ
ejpam-4870	531	12	2	2	NUM
ejpam-4870	531	13	≤	≤	NUM
ejpam-4870	531	14	j	j	PROPN
ejpam-4870	531	15	≤	≤	PROPN
ejpam-4870	531	16	⌈n2	⌈n2	NOUN
ejpam-4870	531	17	⌉.	⌉.	ADV
ejpam-4870	531	18	furthermore	furthermore	ADV
ejpam-4870	531	19	,	,	PUNCT
ejpam-4870	531	20	this	this	DET
ejpam-4870	531	21	study	study	NOUN
ejpam-4870	531	22	showed	show	VERB
ejpam-4870	531	23	that	that	SCONJ
ejpam-4870	531	24	ec(n	ec(n	PROPN
ejpam-4870	531	25	,	,	PUNCT
ejpam-4870	531	26	j	j	NOUN
ejpam-4870	531	27	)	)	PUNCT
ejpam-4870	531	28	is	be	AUX
ejpam-4870	531	29	a	a	DET
ejpam-4870	531	30	cycle	cycle	NOUN
ejpam-4870	531	31	graph	graph	NOUN
ejpam-4870	531	32	of	of	ADP
ejpam-4870	531	33	order	order	NOUN
ejpam-4870	531	34	3	3	NUM
ejpam-4870	531	35	if	if	SCONJ
ejpam-4870	531	36	and	and	CCONJ
ejpam-4870	531	37	only	only	ADV
ejpam-4870	531	38	if	if	SCONJ
ejpam-4870	531	39	n	n	PROPN
ejpam-4870	531	40	=	=	SYM
ejpam-4870	531	41	3	3	NUM
ejpam-4870	531	42	and	and	CCONJ
ejpam-4870	531	43	j	j	NOUN
ejpam-4870	532	1	=	=	NOUN
ejpam-4870	532	2	2	2	X
ejpam-4870	532	3	.	.	PUNCT
ejpam-4870	533	1	finally	finally	ADV
ejpam-4870	533	2	,	,	PUNCT
ejpam-4870	533	3	this	this	DET
ejpam-4870	533	4	study	study	NOUN
ejpam-4870	533	5	specified	specify	VERB
ejpam-4870	533	6	other	other	ADJ
ejpam-4870	533	7	parameters	parameter	NOUN
ejpam-4870	533	8	of	of	ADP
ejpam-4870	533	9	ec(n	ec(n	PROPN
ejpam-4870	533	10	,	,	PUNCT
ejpam-4870	533	11	j	j	NOUN
ejpam-4870	533	12	)	)	PUNCT
ejpam-4870	533	13	such	such	ADJ
ejpam-4870	533	14	as	as	ADP
ejpam-4870	533	15	independence	independence	NOUN
ejpam-4870	533	16	number	number	NOUN
ejpam-4870	533	17	,	,	PUNCT
ejpam-4870	533	18	and	and	CCONJ
ejpam-4870	533	19	domination	domination	NOUN
ejpam-4870	533	20	number	number	NOUN
ejpam-4870	533	21	.	.	PUNCT
ejpam-4870	534	1	the	the	DET
ejpam-4870	534	2	proponents	proponent	NOUN
ejpam-4870	534	3	focused	focus	VERB
ejpam-4870	534	4	on	on	ADP
ejpam-4870	534	5	ec(n	ec(n	PROPN
ejpam-4870	534	6	,	,	PUNCT
ejpam-4870	534	7	j	j	PROPN
ejpam-4870	534	8	)	)	PUNCT
ejpam-4870	535	1	when	when	SCONJ
ejpam-4870	535	2	j	j	PROPN
ejpam-4870	535	3	=	=	NOUN
ejpam-4870	535	4	2	2	NUM
ejpam-4870	535	5	on	on	ADP
ejpam-4870	535	6	getting	get	VERB
ejpam-4870	535	7	independence	independence	NOUN
ejpam-4870	535	8	number	number	NOUN
ejpam-4870	535	9	,	,	PUNCT
ejpam-4870	535	10	and	and	CCONJ
ejpam-4870	535	11	domination	domination	NOUN
ejpam-4870	535	12	number	number	NOUN
ejpam-4870	535	13	since	since	SCONJ
ejpam-4870	535	14	ec(n,2	ec(n,2	PROPN
ejpam-4870	535	15	)	)	PUNCT
ejpam-4870	535	16	is	be	AUX
ejpam-4870	535	17	defined	define	VERB
ejpam-4870	535	18	for	for	ADP
ejpam-4870	535	19	all	all	DET
ejpam-4870	535	20	values	value	NOUN
ejpam-4870	535	21	of	of	ADP
ejpam-4870	535	22	n.	n.	NOUN
ejpam-4870	535	23	the	the	DET
ejpam-4870	535	24	researchers	researcher	NOUN
ejpam-4870	535	25	found	find	VERB
ejpam-4870	535	26	a	a	DET
ejpam-4870	535	27	lower	low	ADJ
ejpam-4870	535	28	bound	bind	VERB
ejpam-4870	535	29	of	of	ADP
ejpam-4870	535	30	the	the	DET
ejpam-4870	535	31	independence	independence	NOUN
ejpam-4870	535	32	number	number	NOUN
ejpam-4870	535	33	of	of	ADP
ejpam-4870	535	34	ec(n,2	ec(n,2	PROPN
ejpam-4870	535	35	)	)	PUNCT
ejpam-4870	535	36	which	which	PRON
ejpam-4870	535	37	is	be	AUX
ejpam-4870	535	38	greater	great	ADJ
ejpam-4870	535	39	than	than	ADP
ejpam-4870	535	40	or	or	CCONJ
ejpam-4870	535	41	equal	equal	ADJ
ejpam-4870	535	42	to	to	ADP
ejpam-4870	535	43	⌊n2	⌊n2	PUNCT
ejpam-4870	535	44	⌋.	⌋.	ADV
ejpam-4870	535	45	in	in	ADP
ejpam-4870	535	46	addition	addition	NOUN
ejpam-4870	535	47	,	,	PUNCT
ejpam-4870	535	48	it	it	PRON
ejpam-4870	535	49	was	be	AUX
ejpam-4870	535	50	discovered	discover	VERB
ejpam-4870	535	51	that	that	SCONJ
ejpam-4870	535	52	an	an	DET
ejpam-4870	535	53	upper	upper	ADJ
ejpam-4870	535	54	bound	bound	NOUN
ejpam-4870	535	55	of	of	ADP
ejpam-4870	535	56	references	reference	NOUN
ejpam-4870	535	57	2498	2498	NUM
ejpam-4870	535	58	γ(ec(n,2	γ(ec(n,2	PUNCT
ejpam-4870	535	59	)	)	PUNCT
ejpam-4870	535	60	)	)	PUNCT
ejpam-4870	535	61	≤	≤	NOUN
ejpam-4870	535	62	⌊n2	⌊n2	PUNCT
ejpam-4870	535	63	⌋.	⌋.	PUNCT
ejpam-4870	536	1	the	the	DET
ejpam-4870	536	2	researcher	researcher	NOUN
ejpam-4870	536	3	believed	believe	VERB
ejpam-4870	536	4	that	that	SCONJ
ejpam-4870	536	5	by	by	ADP
ejpam-4870	536	6	focusing	focus	VERB
ejpam-4870	536	7	on	on	ADP
ejpam-4870	536	8	the	the	DET
ejpam-4870	536	9	ec(n	ec(n	PROPN
ejpam-4870	536	10	,	,	PUNCT
ejpam-4870	536	11	j	j	PROPN
ejpam-4870	536	12	)	)	PUNCT
ejpam-4870	536	13	,	,	PUNCT
ejpam-4870	536	14	the	the	DET
ejpam-4870	536	15	parallel	parallel	ADJ
ejpam-4870	536	16	research	research	NOUN
ejpam-4870	536	17	study	study	NOUN
ejpam-4870	536	18	could	could	AUX
ejpam-4870	536	19	be	be	AUX
ejpam-4870	536	20	accomplished	accomplish	VERB
ejpam-4870	536	21	.	.	PUNCT
ejpam-4870	537	1	the	the	DET
ejpam-4870	537	2	researchers	researcher	NOUN
ejpam-4870	537	3	have	have	AUX
ejpam-4870	537	4	made	make	VERB
ejpam-4870	537	5	the	the	DET
ejpam-4870	537	6	following	follow	VERB
ejpam-4870	537	7	recommendations	recommendation	NOUN
ejpam-4870	537	8	in	in	ADP
ejpam-4870	537	9	particular	particular	ADJ
ejpam-4870	537	10	:	:	PUNCT
ejpam-4870	537	11	(	(	PUNCT
ejpam-4870	537	12	i	i	NOUN
ejpam-4870	537	13	)	)	PUNCT
ejpam-4870	537	14	it	it	PRON
ejpam-4870	537	15	is	be	AUX
ejpam-4870	537	16	recommended	recommend	VERB
ejpam-4870	537	17	that	that	SCONJ
ejpam-4870	537	18	future	future	ADJ
ejpam-4870	537	19	studies	study	NOUN
ejpam-4870	537	20	explore	explore	VERB
ejpam-4870	537	21	more	more	ADJ
ejpam-4870	537	22	in	in	ADP
ejpam-4870	537	23	finding	find	VERB
ejpam-4870	537	24	the	the	DET
ejpam-4870	537	25	independence	independence	NOUN
ejpam-4870	537	26	number	number	NOUN
ejpam-4870	537	27	,	,	PUNCT
ejpam-4870	537	28	and	and	CCONJ
ejpam-4870	537	29	domination	domination	NOUN
ejpam-4870	537	30	number	number	NOUN
ejpam-4870	537	31	for	for	ADP
ejpam-4870	537	32	all	all	DET
ejpam-4870	537	33	values	value	NOUN
ejpam-4870	537	34	of	of	ADP
ejpam-4870	537	35	j.	j.	PROPN
ejpam-4870	537	36	also	also	ADV
ejpam-4870	537	37	,	,	PUNCT
ejpam-4870	537	38	finding	find	VERB
ejpam-4870	537	39	other	other	ADJ
ejpam-4870	537	40	parameters	parameter	NOUN
ejpam-4870	537	41	of	of	ADP
ejpam-4870	537	42	ec(n	ec(n	PROPN
ejpam-4870	537	43	,	,	PUNCT
ejpam-4870	537	44	j	j	NOUN
ejpam-4870	537	45	)	)	PUNCT
ejpam-4870	537	46	such	such	ADJ
ejpam-4870	537	47	as	as	ADP
ejpam-4870	537	48	its	its	PRON
ejpam-4870	537	49	distance	distance	NOUN
ejpam-4870	537	50	,	,	PUNCT
ejpam-4870	537	51	adjacency	adjacency	NOUN
ejpam-4870	537	52	matrix	matrix	NOUN
ejpam-4870	537	53	,	,	PUNCT
ejpam-4870	537	54	complement	complement	NOUN
ejpam-4870	537	55	,	,	PUNCT
ejpam-4870	537	56	chromatic	chromatic	ADJ
ejpam-4870	537	57	number	number	NOUN
ejpam-4870	537	58	,	,	PUNCT
ejpam-4870	537	59	and	and	CCONJ
ejpam-4870	537	60	isolate	isolate	VERB
ejpam-4870	537	61	domination	domination	NOUN
ejpam-4870	537	62	number	number	NOUN
ejpam-4870	537	63	can	can	AUX
ejpam-4870	537	64	help	help	VERB
ejpam-4870	537	65	in	in	ADP
ejpam-4870	537	66	determining	determine	VERB
ejpam-4870	537	67	the	the	DET
ejpam-4870	537	68	graph	graph	NOUN
ejpam-4870	537	69	;	;	PUNCT
ejpam-4870	537	70	(	(	PUNCT
ejpam-4870	537	71	ii	ii	X
ejpam-4870	537	72	)	)	PUNCT
ejpam-4870	537	73	the	the	DET
ejpam-4870	537	74	researchers	researcher	NOUN
ejpam-4870	537	75	suggest	suggest	VERB
ejpam-4870	537	76	exploring	explore	VERB
ejpam-4870	537	77	the	the	DET
ejpam-4870	537	78	j	j	NOUN
ejpam-4870	537	79	-	-	PUNCT
ejpam-4870	537	80	edge	edge	NOUN
ejpam-4870	537	81	intersection	intersection	NOUN
ejpam-4870	537	82	graph	graph	NOUN
ejpam-4870	537	83	of	of	ADP
ejpam-4870	537	84	other	other	ADJ
ejpam-4870	537	85	special	special	ADJ
ejpam-4870	537	86	classes	class	NOUN
ejpam-4870	537	87	of	of	ADP
ejpam-4870	537	88	a	a	DET
ejpam-4870	537	89	graph	graph	NOUN
ejpam-4870	537	90	such	such	ADJ
ejpam-4870	537	91	as	as	ADP
ejpam-4870	537	92	a	a	DET
ejpam-4870	537	93	path	path	NOUN
ejpam-4870	537	94	and	and	CCONJ
ejpam-4870	537	95	complete	complete	ADJ
ejpam-4870	537	96	graph	graph	NOUN
ejpam-4870	537	97	.	.	PUNCT
ejpam-4870	538	1	also	also	ADV
ejpam-4870	538	2	,	,	PUNCT
ejpam-4870	538	3	the	the	DET
ejpam-4870	538	4	proponents	proponent	NOUN
ejpam-4870	538	5	suggest	suggest	VERB
ejpam-4870	538	6	the	the	DET
ejpam-4870	538	7	notion	notion	NOUN
ejpam-4870	538	8	of	of	ADP
ejpam-4870	538	9	an	an	DET
ejpam-4870	538	10	edge	edge	NOUN
ejpam-4870	538	11	-	-	PUNCT
ejpam-4870	538	12	induced	induce	VERB
ejpam-4870	538	13	subgraph	subgraph	NOUN
ejpam-4870	538	14	instead	instead	ADV
ejpam-4870	538	15	of	of	ADP
ejpam-4870	538	16	a	a	DET
ejpam-4870	538	17	spanning	span	VERB
ejpam-4870	538	18	subgraph	subgraph	NOUN
ejpam-4870	538	19	.	.	PUNCT
ejpam-4870	539	1	(	(	PUNCT
ejpam-4870	539	2	iii	iii	X
ejpam-4870	539	3	)	)	PUNCT
ejpam-4870	539	4	the	the	DET
ejpam-4870	539	5	researchers	researcher	NOUN
ejpam-4870	539	6	recommend	recommend	VERB
ejpam-4870	539	7	exploring	explore	VERB
ejpam-4870	539	8	the	the	DET
ejpam-4870	539	9	use	use	NOUN
ejpam-4870	539	10	of	of	ADP
ejpam-4870	539	11	ec(n	ec(n	PROPN
ejpam-4870	539	12	,	,	PUNCT
ejpam-4870	539	13	j	j	NOUN
ejpam-4870	539	14	)	)	PUNCT
ejpam-4870	539	15	in	in	ADP
ejpam-4870	539	16	solving	solve	VERB
ejpam-4870	539	17	real	real	ADJ
ejpam-4870	539	18	-	-	PUNCT
ejpam-4870	539	19	life	life	NOUN
ejpam-4870	539	20	problems	problem	NOUN
ejpam-4870	539	21	since	since	SCONJ
ejpam-4870	539	22	many	many	ADJ
ejpam-4870	539	23	of	of	ADP
ejpam-4870	539	24	the	the	DET
ejpam-4870	539	25	results	result	NOUN
ejpam-4870	539	26	in	in	ADP
ejpam-4870	539	27	this	this	DET
ejpam-4870	539	28	study	study	NOUN
ejpam-4870	539	29	are	be	AUX
ejpam-4870	539	30	based	base	VERB
ejpam-4870	539	31	on	on	ADP
ejpam-4870	539	32	combination	combination	NOUN
ejpam-4870	539	33	formula	formula	NOUN
ejpam-4870	539	34	which	which	PRON
ejpam-4870	539	35	has	have	VERB
ejpam-4870	539	36	much	much	ADJ
ejpam-4870	539	37	application	application	NOUN
ejpam-4870	539	38	in	in	ADP
ejpam-4870	539	39	solving	solve	VERB
ejpam-4870	539	40	real	real	ADJ
ejpam-4870	539	41	-	-	PUNCT
ejpam-4870	539	42	life	life	NOUN
ejpam-4870	539	43	problems	problem	NOUN
ejpam-4870	539	44	.	.	PUNCT
ejpam-4870	540	1	references	reference	NOUN
ejpam-4870	540	2	[	[	X
ejpam-4870	540	3	1	1	NUM
ejpam-4870	540	4	]	]	PUNCT
ejpam-4870	540	5	a.	a.	NOUN
ejpam-4870	540	6	asinowski	asinowski	PROPN
ejpam-4870	540	7	,	,	PUNCT
ejpam-4870	540	8	a.	a.	NOUN
ejpam-4870	540	9	suk	suk	NOUN
ejpam-4870	540	10	.	.	PUNCT
ejpam-4870	540	11	’	'	PUNCT
ejpam-4870	540	12	edge	edge	NOUN
ejpam-4870	540	13	intersection	intersection	NOUN
ejpam-4870	540	14	graphs	graph	NOUN
ejpam-4870	540	15	of	of	ADP
ejpam-4870	540	16	systems	system	NOUN
ejpam-4870	540	17	of	of	ADP
ejpam-4870	540	18	paths	path	NOUN
ejpam-4870	540	19	on	on	ADP
ejpam-4870	540	20	a	a	DET
ejpam-4870	540	21	grid	grid	NOUN
ejpam-4870	540	22	with	with	ADP
ejpam-4870	540	23	a	a	DET
ejpam-4870	540	24	bounded	bounded	ADJ
ejpam-4870	540	25	number	number	NOUN
ejpam-4870	540	26	of	of	ADP
ejpam-4870	540	27	bends	bend	NOUN
ejpam-4870	540	28	.	.	PUNCT
ejpam-4870	541	1	discrete	discrete	ADJ
ejpam-4870	541	2	applied	apply	VERB
ejpam-4870	541	3	mathematics	mathematic	NOUN
ejpam-4870	541	4	.	.	PUNCT
ejpam-4870	542	1	2009	2009	NUM
ejpam-4870	543	1	[	[	X
ejpam-4870	543	2	2	2	X
ejpam-4870	543	3	]	]	PUNCT
ejpam-4870	543	4	t.	t.	PROPN
ejpam-4870	543	5	biedl	biedl	PROPN
ejpam-4870	543	6	,	,	PUNCT
ejpam-4870	543	7	m.	m.	NOUN
ejpam-4870	543	8	stern	stern	NOUN
ejpam-4870	543	9	.	.	PUNCT
ejpam-4870	544	1	on	on	ADP
ejpam-4870	544	2	edge	edge	NOUN
ejpam-4870	544	3	-	-	PUNCT
ejpam-4870	544	4	intersection	intersection	NOUN
ejpam-4870	544	5	graphs	graph	NOUN
ejpam-4870	544	6	of	of	ADP
ejpam-4870	544	7	k	k	NOUN
ejpam-4870	544	8	-	-	PUNCT
ejpam-4870	544	9	bends	bend	NOUN
ejpam-4870	544	10	paths	path	NOUN
ejpam-4870	544	11	grids	grid	NOUN
ejpam-4870	544	12	.	.	PUNCT
ejpam-4870	545	1	discrete	discrete	ADJ
ejpam-4870	545	2	mathematics	mathematic	NOUN
ejpam-4870	545	3	and	and	CCONJ
ejpam-4870	545	4	theoretical	theoretical	ADJ
ejpam-4870	545	5	cmputer	cmputer	NOUN
ejpam-4870	545	6	science.2010	science.2010	PROPN
ejpam-4870	545	7	.	.	PUNCT
ejpam-4870	546	1	[	[	X
ejpam-4870	546	2	3	3	X
ejpam-4870	546	3	]	]	X
ejpam-4870	546	4	d.	d.	NOUN
ejpam-4870	546	5	guichard	guichard	PROPN
ejpam-4870	546	6	.	.	PUNCT
ejpam-4870	547	1	graph	graph	NOUN
ejpam-4870	547	2	coloring	coloring	NOUN
ejpam-4870	547	3	and	and	CCONJ
ejpam-4870	547	4	independence	independence	NOUN
ejpam-4870	547	5	number	number	NOUN
ejpam-4870	547	6	of	of	ADP
ejpam-4870	547	7	a	a	DET
ejpam-4870	547	8	complete	complete	ADJ
ejpam-4870	547	9	graph	graph	NOUN
ejpam-4870	547	10	.	.	PUNCT
ejpam-4870	548	1	withman	withman	NOUN
ejpam-4870	548	2	college	college	NOUN
ejpam-4870	548	3	.	.	PUNCT
ejpam-4870	549	1	[	[	X
ejpam-4870	549	2	4	4	NUM
ejpam-4870	549	3	]	]	X
ejpam-4870	549	4	m.c	m.c	PROPN
ejpam-4870	549	5	.	.	PROPN
ejpam-4870	549	6	golumbic	golumbic	PROPN
ejpam-4870	549	7	,	,	PUNCT
ejpam-4870	549	8	r.	r.	PROPN
ejpam-4870	549	9	jamison	jamison	PROPN
ejpam-4870	549	10	.	.	PUNCT
ejpam-4870	550	1	the	the	DET
ejpam-4870	550	2	k	k	ADJ
ejpam-4870	550	3	-	-	PUNCT
ejpam-4870	550	4	edge	edge	NOUN
ejpam-4870	550	5	intersection	intersection	NOUN
ejpam-4870	550	6	graphs	graph	NOUN
ejpam-4870	550	7	of	of	ADP
ejpam-4870	550	8	paths	path	NOUN
ejpam-4870	550	9	in	in	ADP
ejpam-4870	550	10	a	a	DET
ejpam-4870	550	11	tree	tree	NOUN
ejpam-4870	550	12	.	.	PUNCT
ejpam-4870	551	1	journal	journal	NOUN
ejpam-4870	551	2	of	of	ADP
ejpam-4870	551	3	combinatorial	combinatorial	ADJ
ejpam-4870	551	4	theory	theory	NOUN
ejpam-4870	551	5	,	,	PUNCT
ejpam-4870	551	6	.	.	PUNCT
ejpam-4870	552	1	1985	1985	NUM
ejpam-4870	553	1	[	[	X
ejpam-4870	553	2	5	5	NUM
ejpam-4870	553	3	]	]	X
ejpam-4870	553	4	j.m	j.m	PROPN
ejpam-4870	553	5	.	.	PROPN
ejpam-4870	553	6	harris	harris	PROPN
ejpam-4870	553	7	et	et	PROPN
ejpam-4870	553	8	.	.	PUNCT
ejpam-4870	554	1	al	al	PROPN
ejpam-4870	554	2	.	.	PROPN
ejpam-4870	554	3	combinatorics	combinatoric	NOUN
ejpam-4870	554	4	and	and	CCONJ
ejpam-4870	554	5	graph	graph	NOUN
ejpam-4870	554	6	theory	theory	NOUN
ejpam-4870	554	7	.	.	PUNCT
ejpam-4870	555	1	springer	springer	PROPN
ejpam-4870	555	2	new	new	PROPN
ejpam-4870	555	3	york	york	PROPN
ejpam-4870	555	4	,	,	PUNCT
ejpam-4870	555	5	ny.2008	ny.2008	PROPN
ejpam-4870	555	6	.	.	PUNCT
ejpam-4870	556	1	[	[	X
ejpam-4870	556	2	6	6	NUM
ejpam-4870	556	3	]	]	X
ejpam-4870	556	4	b.e	b.e	PROPN
ejpam-4870	556	5	.	.	PROPN
ejpam-4870	556	6	sagan	sagan	PROPN
ejpam-4870	556	7	,	,	PUNCT
ejpam-4870	556	8	combinatorics	combinatoric	NOUN
ejpam-4870	556	9	:	:	PUNCT
ejpam-4870	556	10	the	the	DET
ejpam-4870	556	11	art	art	NOUN
ejpam-4870	556	12	of	of	ADP
ejpam-4870	556	13	counting	counting	NOUN
ejpam-4870	556	14	,	,	PUNCT
ejpam-4870	556	15	american	american	PROPN
ejpam-4870	556	16	mathematical	mathematical	ADJ
ejpam-4870	556	17	society	society	NOUN
ejpam-4870	556	18	.	.	PUNCT
ejpam-4870	557	1	2020	2020	NUM
ejpam-4870	557	2	.	.	PUNCT
ejpam-4870	558	1	[	[	X
ejpam-4870	558	2	7	7	X
ejpam-4870	558	3	]	]	ADJ
ejpam-4870	558	4	i.	i.	PROPN
ejpam-4870	558	5	sahul	sahul	PROPN
ejpam-4870	558	6	hamid	hamid	PROPN
ejpam-4870	558	7	,	,	PUNCT
ejpam-4870	558	8	s.	s.	PROPN
ejpam-4870	558	9	balamurgan	balamurgan	PROPN
ejpam-4870	558	10	.	.	PUNCT
ejpam-4870	558	11	isolate	isolate	VERB
ejpam-4870	558	12	domination	domination	NOUN
ejpam-4870	558	13	in	in	ADP
ejpam-4870	558	14	graphs	graph	NOUN
ejpam-4870	558	15	.	.	PUNCT
ejpam-4870	559	1	arab	arab	PROPN
ejpam-4870	559	2	journal	journal	PROPN
ejpam-4870	559	3	of	of	ADP
ejpam-4870	559	4	mathematical	mathematical	ADJ
ejpam-4870	559	5	sciences	science	NOUN
ejpam-4870	559	6	.	.	PUNCT
ejpam-4870	560	1	2015	2015	NUM
ejpam-4870	560	2	.	.	PUNCT
ejpam-4870	561	1	[	[	X
ejpam-4870	561	2	8	8	NUM
ejpam-4870	561	3	]	]	PUNCT
ejpam-4870	561	4	s.	s.	PROPN
ejpam-4870	561	5	selkow	selkow	PROPN
ejpam-4870	561	6	,	,	PUNCT
ejpam-4870	561	7	the	the	DET
ejpam-4870	561	8	independence	independence	NOUN
ejpam-4870	561	9	number	number	NOUN
ejpam-4870	561	10	of	of	ADP
ejpam-4870	561	11	graphs	graph	NOUN
ejpam-4870	561	12	in	in	ADP
ejpam-4870	561	13	terms	term	NOUN
ejpam-4870	561	14	of	of	ADP
ejpam-4870	561	15	degrees	degree	NOUN
ejpam-4870	561	16	.	.	PUNCT
ejpam-4870	562	1	discrete	discrete	ADJ
ejpam-4870	562	2	mathematics	mathematic	NOUN
ejpam-4870	562	3	.	.	PUNCT
ejpam-4870	563	1	1993	1993	NUM
ejpam-4870	563	2	.	.	PUNCT
ejpam-4870	564	1	[	[	X
ejpam-4870	564	2	9	9	NUM
ejpam-4870	564	3	]	]	PUNCT
ejpam-4870	564	4	a.	a.	NOUN
ejpam-4870	564	5	sugumaran	sugumaran	NOUN
ejpam-4870	564	6	,	,	PUNCT
ejpam-4870	564	7	e.	e.	PROPN
ejpam-4870	564	8	jayachadran	jayachadran	PROPN
ejpam-4870	564	9	.	.	PUNCT
ejpam-4870	565	1	domination	domination	NOUN
ejpam-4870	565	2	of	of	ADP
ejpam-4870	565	3	number	number	NOUN
ejpam-4870	565	4	of	of	ADP
ejpam-4870	565	5	some	some	DET
ejpam-4870	565	6	graph	graph	NOUN
ejpam-4870	565	7	.	.	PUNCT
ejpam-4870	566	1	international	international	ADJ
ejpam-4870	566	2	journal	journal	PROPN
ejpam-4870	566	3	of	of	ADP
ejpam-4870	566	4	scientific	scientific	ADJ
ejpam-4870	566	5	development	development	NOUN
ejpam-4870	566	6	and	and	CCONJ
ejpam-4870	566	7	research	research	NOUN
ejpam-4870	566	8	.	.	PUNCT
ejpam-4870	567	1	2018	2018	NUM
ejpam-4870	567	2	.	.	PUNCT