id	sid	tid	token	lemma	pos
ejpam-5336	1	1	european	european	PROPN
ejpam-5336	1	2	journal	journal	PROPN
ejpam-5336	1	3	of	of	ADP
ejpam-5336	1	4	pure	pure	ADJ
ejpam-5336	1	5	and	and	CCONJ
ejpam-5336	1	6	applied	apply	VERB
ejpam-5336	1	7	mathematics	mathematic	NOUN
ejpam-5336	1	8	vol	vol	NOUN
ejpam-5336	1	9	.	.	PROPN
ejpam-5336	2	1	17	17	NUM
ejpam-5336	2	2	,	,	PUNCT
ejpam-5336	2	3	no	no	INTJ
ejpam-5336	2	4	.	.	NOUN
ejpam-5336	2	5	4	4	NUM
ejpam-5336	2	6	,	,	PUNCT
ejpam-5336	2	7	2024	2024	NUM
ejpam-5336	2	8	,	,	PUNCT
ejpam-5336	2	9	3557	3557	NUM
ejpam-5336	2	10	-	-	SYM
ejpam-5336	2	11	3566	3566	NUM
ejpam-5336	2	12	issn	issn	PROPN
ejpam-5336	2	13	1307	1307	NUM
ejpam-5336	2	14	-	-	SYM
ejpam-5336	2	15	5543	5543	NUM
ejpam-5336	2	16	–	–	PUNCT
ejpam-5336	3	1	ejpam.com	ejpam.com	X
ejpam-5336	3	2	published	publish	VERB
ejpam-5336	3	3	by	by	ADP
ejpam-5336	3	4	new	new	PROPN
ejpam-5336	3	5	york	york	PROPN
ejpam-5336	3	6	business	business	PROPN
ejpam-5336	3	7	global	global	ADJ
ejpam-5336	3	8	prime	prime	ADJ
ejpam-5336	3	9	labeling	labeling	NOUN
ejpam-5336	3	10	of	of	ADP
ejpam-5336	3	11	union	union	NOUN
ejpam-5336	3	12	of	of	ADP
ejpam-5336	3	13	some	some	DET
ejpam-5336	3	14	graphs	graph	NOUN
ejpam-5336	3	15	omar	omar	PROPN
ejpam-5336	3	16	a.	a.	PROPN
ejpam-5336	3	17	abughneim1,∗	abughneim1,∗	PROPN
ejpam-5336	3	18	,	,	PUNCT
ejpam-5336	3	19	baha	baha	X
ejpam-5336	3	20	’	'	PUNCT
ejpam-5336	3	21	abughazaleh2	abughazaleh2	NOUN
ejpam-5336	3	22	1	1	NUM
ejpam-5336	3	23	department	department	NOUN
ejpam-5336	3	24	of	of	ADP
ejpam-5336	3	25	mathematics	mathematic	NOUN
ejpam-5336	3	26	,	,	PUNCT
ejpam-5336	3	27	faculty	faculty	NOUN
ejpam-5336	3	28	of	of	ADP
ejpam-5336	3	29	sciences	science	NOUN
ejpam-5336	3	30	,	,	PUNCT
ejpam-5336	3	31	the	the	DET
ejpam-5336	3	32	university	university	PROPN
ejpam-5336	3	33	of	of	ADP
ejpam-5336	3	34	jordan	jordan	PROPN
ejpam-5336	3	35	,	,	PUNCT
ejpam-5336	3	36	amman	amman	PROPN
ejpam-5336	3	37	,	,	PUNCT
ejpam-5336	3	38	jordan	jordan	PROPN
ejpam-5336	3	39	2	2	NUM
ejpam-5336	3	40	department	department	NOUN
ejpam-5336	3	41	of	of	ADP
ejpam-5336	3	42	mathematics	mathematic	NOUN
ejpam-5336	3	43	,	,	PUNCT
ejpam-5336	3	44	faculty	faculty	NOUN
ejpam-5336	3	45	of	of	ADP
ejpam-5336	3	46	sciences	science	NOUN
ejpam-5336	3	47	,	,	PUNCT
ejpam-5336	3	48	isra	isra	PROPN
ejpam-5336	3	49	university	university	PROPN
ejpam-5336	3	50	,	,	PUNCT
ejpam-5336	3	51	amman	amman	PROPN
ejpam-5336	3	52	,	,	PUNCT
ejpam-5336	3	53	jordan	jordan	PROPN
ejpam-5336	3	54	abstract	abstract	PROPN
ejpam-5336	3	55	.	.	PUNCT
ejpam-5336	4	1	a	a	DET
ejpam-5336	4	2	prime	prime	ADJ
ejpam-5336	4	3	labeling	labeling	NOUN
ejpam-5336	4	4	of	of	ADP
ejpam-5336	4	5	a	a	DET
ejpam-5336	4	6	graph	graph	NOUN
ejpam-5336	4	7	g	g	NOUN
ejpam-5336	4	8	is	be	AUX
ejpam-5336	4	9	a	a	DET
ejpam-5336	4	10	map	map	NOUN
ejpam-5336	4	11	from	from	ADP
ejpam-5336	4	12	the	the	DET
ejpam-5336	4	13	vertex	vertex	NOUN
ejpam-5336	4	14	set	set	NOUN
ejpam-5336	4	15	of	of	ADP
ejpam-5336	4	16	g	g	PROPN
ejpam-5336	4	17	,	,	PUNCT
ejpam-5336	4	18	v	v	NOUN
ejpam-5336	4	19	(	(	PUNCT
ejpam-5336	4	20	g	g	NOUN
ejpam-5336	4	21	)	)	PUNCT
ejpam-5336	4	22	,	,	PUNCT
ejpam-5336	4	23	to	to	ADP
ejpam-5336	4	24	the	the	DET
ejpam-5336	4	25	set	set	NOUN
ejpam-5336	4	26	{	{	PUNCT
ejpam-5336	4	27	1	1	NUM
ejpam-5336	4	28	,	,	PUNCT
ejpam-5336	4	29	2	2	NUM
ejpam-5336	4	30	,	,	PUNCT
ejpam-5336	4	31	...	...	PUNCT
ejpam-5336	4	32	,	,	PUNCT
ejpam-5336	4	33	|v	|v	PROPN
ejpam-5336	4	34	(	(	PUNCT
ejpam-5336	4	35	g)|	g)|	PROPN
ejpam-5336	4	36	}	}	PUNCT
ejpam-5336	4	37	such	such	ADJ
ejpam-5336	4	38	that	that	SCONJ
ejpam-5336	4	39	any	any	DET
ejpam-5336	4	40	two	two	NUM
ejpam-5336	4	41	adjacent	adjacent	ADJ
ejpam-5336	4	42	vertices	vertex	NOUN
ejpam-5336	4	43	in	in	ADP
ejpam-5336	4	44	the	the	DET
ejpam-5336	4	45	graph	graph	NOUN
ejpam-5336	4	46	g	g	NOUN
ejpam-5336	4	47	have	have	AUX
ejpam-5336	4	48	labels	label	NOUN
ejpam-5336	4	49	that	that	PRON
ejpam-5336	4	50	are	be	AUX
ejpam-5336	4	51	relatively	relatively	ADV
ejpam-5336	4	52	prime	prime	ADJ
ejpam-5336	4	53	.	.	PUNCT
ejpam-5336	5	1	in	in	ADP
ejpam-5336	5	2	this	this	DET
ejpam-5336	5	3	paper	paper	NOUN
ejpam-5336	5	4	,	,	PUNCT
ejpam-5336	5	5	we	we	PRON
ejpam-5336	5	6	discuss	discuss	VERB
ejpam-5336	5	7	when	when	SCONJ
ejpam-5336	5	8	the	the	DET
ejpam-5336	5	9	disjoint	disjoint	PROPN
ejpam-5336	5	10	union	union	NOUN
ejpam-5336	5	11	of	of	ADP
ejpam-5336	5	12	some	some	DET
ejpam-5336	5	13	graphs	graph	NOUN
ejpam-5336	5	14	is	be	AUX
ejpam-5336	5	15	a	a	DET
ejpam-5336	5	16	prime	prime	ADJ
ejpam-5336	5	17	graph	graph	NOUN
ejpam-5336	5	18	.	.	PUNCT
ejpam-5336	6	1	2020	2020	NUM
ejpam-5336	6	2	mathematics	mathematic	NOUN
ejpam-5336	6	3	subject	subject	NOUN
ejpam-5336	6	4	classifications	classification	NOUN
ejpam-5336	6	5	:	:	PUNCT
ejpam-5336	6	6	05c78	05c78	NUM
ejpam-5336	6	7	key	key	ADJ
ejpam-5336	6	8	words	word	NOUN
ejpam-5336	6	9	and	and	CCONJ
ejpam-5336	6	10	phrases	phrase	NOUN
ejpam-5336	6	11	:	:	PUNCT
ejpam-5336	6	12	independence	independence	NOUN
ejpam-5336	6	13	number	number	NOUN
ejpam-5336	6	14	,	,	PUNCT
ejpam-5336	6	15	even	even	ADV
ejpam-5336	6	16	cycles	cycle	NOUN
ejpam-5336	6	17	,	,	PUNCT
ejpam-5336	6	18	wheels	wheel	NOUN
ejpam-5336	6	19	,	,	PUNCT
ejpam-5336	6	20	prime	prime	ADJ
ejpam-5336	6	21	labeling	labeling	NOUN
ejpam-5336	6	22	,	,	PUNCT
ejpam-5336	6	23	prime	prime	ADJ
ejpam-5336	6	24	graphs	graph	NOUN
ejpam-5336	6	25	,	,	PUNCT
ejpam-5336	6	26	maximal	maximal	ADJ
ejpam-5336	6	27	prime	prime	ADJ
ejpam-5336	6	28	graphs	graph	NOUN
ejpam-5336	6	29	1	1	NUM
ejpam-5336	6	30	.	.	PUNCT
ejpam-5336	6	31	introduction	introduction	NOUN
ejpam-5336	6	32	a	a	DET
ejpam-5336	6	33	path	path	NOUN
ejpam-5336	6	34	pm	pm	NOUN
ejpam-5336	6	35	in	in	ADP
ejpam-5336	6	36	a	a	DET
ejpam-5336	6	37	graph	graph	NOUN
ejpam-5336	6	38	is	be	AUX
ejpam-5336	6	39	an	an	DET
ejpam-5336	6	40	alternative	alternative	ADJ
ejpam-5336	6	41	sequence	sequence	NOUN
ejpam-5336	6	42	of	of	ADP
ejpam-5336	6	43	vertices	vertex	NOUN
ejpam-5336	6	44	and	and	CCONJ
ejpam-5336	6	45	edges	edge	NOUN
ejpam-5336	6	46	with	with	ADP
ejpam-5336	6	47	no	no	DET
ejpam-5336	6	48	repeated	repeat	VERB
ejpam-5336	6	49	vertices	vertex	NOUN
ejpam-5336	6	50	,	,	PUNCT
ejpam-5336	6	51	a	a	DET
ejpam-5336	6	52	cycle	cycle	NOUN
ejpam-5336	6	53	cm	cm	NOUN
ejpam-5336	6	54	in	in	ADP
ejpam-5336	6	55	a	a	DET
ejpam-5336	6	56	graph	graph	NOUN
ejpam-5336	6	57	is	be	AUX
ejpam-5336	6	58	a	a	DET
ejpam-5336	6	59	path	path	NOUN
ejpam-5336	6	60	that	that	PRON
ejpam-5336	6	61	begins	begin	VERB
ejpam-5336	6	62	and	and	CCONJ
ejpam-5336	6	63	ends	end	VERB
ejpam-5336	6	64	at	at	ADP
ejpam-5336	6	65	the	the	DET
ejpam-5336	6	66	same	same	ADJ
ejpam-5336	6	67	vertex	vertex	NOUN
ejpam-5336	6	68	and	and	CCONJ
ejpam-5336	6	69	a	a	DET
ejpam-5336	6	70	wheel	wheel	NOUN
ejpam-5336	6	71	graph	graph	NOUN
ejpam-5336	6	72	wm	wm	PROPN
ejpam-5336	6	73	is	be	AUX
ejpam-5336	6	74	formed	form	VERB
ejpam-5336	6	75	by	by	ADP
ejpam-5336	6	76	joining	join	VERB
ejpam-5336	6	77	a	a	DET
ejpam-5336	6	78	single	single	ADJ
ejpam-5336	6	79	vertex	vertex	NOUN
ejpam-5336	6	80	,	,	PUNCT
ejpam-5336	6	81	known	know	VERB
ejpam-5336	6	82	as	as	ADP
ejpam-5336	6	83	the	the	DET
ejpam-5336	6	84	apex	apex	NOUN
ejpam-5336	6	85	vertex	vertex	NOUN
ejpam-5336	6	86	,	,	PUNCT
ejpam-5336	6	87	to	to	ADP
ejpam-5336	6	88	all	all	DET
ejpam-5336	6	89	vertices	vertex	NOUN
ejpam-5336	6	90	of	of	ADP
ejpam-5336	6	91	a	a	DET
ejpam-5336	6	92	cycle	cycle	NOUN
ejpam-5336	6	93	cm	cm	NOUN
ejpam-5336	6	94	,	,	PUNCT
ejpam-5336	6	95	these	these	DET
ejpam-5336	6	96	vertices	vertex	NOUN
ejpam-5336	6	97	are	be	AUX
ejpam-5336	6	98	known	know	VERB
ejpam-5336	6	99	as	as	ADP
ejpam-5336	6	100	the	the	DET
ejpam-5336	6	101	rim	rim	NOUN
ejpam-5336	6	102	vertices	vertice	VERB
ejpam-5336	6	103	.	.	PUNCT
ejpam-5336	7	1	a	a	DET
ejpam-5336	7	2	bijective	bijective	ADJ
ejpam-5336	7	3	map	map	NOUN
ejpam-5336	7	4	f	f	PROPN
ejpam-5336	7	5	from	from	ADP
ejpam-5336	7	6	the	the	DET
ejpam-5336	7	7	vertex	vertex	NOUN
ejpam-5336	7	8	set	set	NOUN
ejpam-5336	7	9	of	of	ADP
ejpam-5336	7	10	a	a	DET
ejpam-5336	7	11	graph	graph	NOUN
ejpam-5336	7	12	g	g	NOUN
ejpam-5336	7	13	to	to	ADP
ejpam-5336	7	14	{	{	PUNCT
ejpam-5336	7	15	1	1	NUM
ejpam-5336	7	16	,	,	PUNCT
ejpam-5336	7	17	2	2	NUM
ejpam-5336	7	18	,	,	PUNCT
ejpam-5336	7	19	...	...	PUNCT
ejpam-5336	7	20	,	,	PUNCT
ejpam-5336	7	21	|v	|v	PROPN
ejpam-5336	7	22	(	(	PUNCT
ejpam-5336	7	23	g)|	g)|	PROPN
ejpam-5336	7	24	}	}	PUNCT
ejpam-5336	7	25	such	such	ADJ
ejpam-5336	7	26	that	that	SCONJ
ejpam-5336	7	27	f	f	PROPN
ejpam-5336	7	28	(	(	PUNCT
ejpam-5336	7	29	u	u	NOUN
ejpam-5336	7	30	)	)	PUNCT
ejpam-5336	7	31	and	and	CCONJ
ejpam-5336	7	32	f	f	PROPN
ejpam-5336	7	33	(	(	PUNCT
ejpam-5336	7	34	v	v	NOUN
ejpam-5336	7	35	)	)	PUNCT
ejpam-5336	7	36	are	be	AUX
ejpam-5336	7	37	relatively	relatively	ADV
ejpam-5336	7	38	prime	prime	ADJ
ejpam-5336	7	39	whenever	whenever	SCONJ
ejpam-5336	7	40	u	u	NOUN
ejpam-5336	7	41	and	and	CCONJ
ejpam-5336	7	42	v	v	NOUN
ejpam-5336	7	43	are	be	AUX
ejpam-5336	7	44	adjacent	adjacent	ADJ
ejpam-5336	7	45	in	in	ADP
ejpam-5336	7	46	g	g	PROPN
ejpam-5336	7	47	is	be	AUX
ejpam-5336	7	48	called	call	VERB
ejpam-5336	7	49	a	a	DET
ejpam-5336	7	50	prime	prime	ADJ
ejpam-5336	7	51	labeling	labeling	NOUN
ejpam-5336	7	52	(	(	PUNCT
ejpam-5336	7	53	pl	pl	NOUN
ejpam-5336	7	54	)	)	PUNCT
ejpam-5336	7	55	of	of	ADP
ejpam-5336	7	56	g	g	PROPN
ejpam-5336	7	57	and	and	CCONJ
ejpam-5336	7	58	a	a	DET
ejpam-5336	7	59	graph	graph	NOUN
ejpam-5336	7	60	g	g	NOUN
ejpam-5336	7	61	is	be	AUX
ejpam-5336	7	62	called	call	VERB
ejpam-5336	7	63	a	a	DET
ejpam-5336	7	64	prime	prime	ADJ
ejpam-5336	7	65	graph	graph	NOUN
ejpam-5336	7	66	(	(	PUNCT
ejpam-5336	7	67	pg	pg	INTJ
ejpam-5336	7	68	)	)	PUNCT
ejpam-5336	7	69	if	if	SCONJ
ejpam-5336	7	70	g	g	PROPN
ejpam-5336	7	71	has	have	VERB
ejpam-5336	7	72	a	a	DET
ejpam-5336	7	73	pl	pl	PROPN
ejpam-5336	7	74	.	.	PROPN
ejpam-5336	7	75	entringer	entringer	NOUN
ejpam-5336	7	76	defined	define	VERB
ejpam-5336	7	77	the	the	DET
ejpam-5336	7	78	pl	pl	NOUN
ejpam-5336	7	79	that	that	PRON
ejpam-5336	7	80	was	be	AUX
ejpam-5336	7	81	introduced	introduce	VERB
ejpam-5336	7	82	by	by	ADP
ejpam-5336	7	83	tout	tout	PROPN
ejpam-5336	7	84	et	et	PROPN
ejpam-5336	7	85	.	.	PUNCT
ejpam-5336	8	1	al	al	PROPN
ejpam-5336	8	2	.	.	PUNCT
ejpam-5336	9	1	in	in	ADP
ejpam-5336	9	2	[	[	X
ejpam-5336	9	3	1	1	NUM
ejpam-5336	9	4	]	]	PUNCT
ejpam-5336	9	5	.	.	PUNCT
ejpam-5336	10	1	entringer	entringer	NOUN
ejpam-5336	10	2	conjectured	conjecture	VERB
ejpam-5336	10	3	that	that	SCONJ
ejpam-5336	10	4	all	all	DET
ejpam-5336	10	5	trees	tree	NOUN
ejpam-5336	10	6	could	could	AUX
ejpam-5336	10	7	be	be	AUX
ejpam-5336	10	8	prime	prime	ADJ
ejpam-5336	10	9	labeled	label	VERB
ejpam-5336	10	10	,	,	PUNCT
ejpam-5336	10	11	a	a	DET
ejpam-5336	10	12	hypothesis	hypothesis	NOUN
ejpam-5336	10	13	supported	support	VERB
ejpam-5336	10	14	by	by	ADP
ejpam-5336	10	15	haxell	haxell	PROPN
ejpam-5336	10	16	et	et	PROPN
ejpam-5336	10	17	.	.	PUNCT
ejpam-5336	11	1	al	al	PROPN
ejpam-5336	11	2	.	.	PUNCT
ejpam-5336	12	1	in	in	ADP
ejpam-5336	12	2	[	[	X
ejpam-5336	12	3	8	8	NUM
ejpam-5336	12	4	]	]	PUNCT
ejpam-5336	12	5	proving	prove	VERB
ejpam-5336	12	6	that	that	SCONJ
ejpam-5336	12	7	all	all	DET
ejpam-5336	12	8	sufficiently	sufficiently	ADV
ejpam-5336	12	9	large	large	ADJ
ejpam-5336	12	10	trees	tree	NOUN
ejpam-5336	12	11	have	have	VERB
ejpam-5336	12	12	this	this	DET
ejpam-5336	12	13	property	property	NOUN
ejpam-5336	12	14	.	.	PUNCT
ejpam-5336	13	1	seoud	seoud	PROPN
ejpam-5336	13	2	et	et	PROPN
ejpam-5336	13	3	.	.	PUNCT
ejpam-5336	14	1	al	al	PROPN
ejpam-5336	14	2	.	.	PUNCT
ejpam-5336	15	1	in	in	ADP
ejpam-5336	15	2	[	[	X
ejpam-5336	15	3	7	7	NUM
ejpam-5336	15	4	]	]	PUNCT
ejpam-5336	15	5	further	far	ADV
ejpam-5336	15	6	contributed	contribute	VERB
ejpam-5336	15	7	by	by	ADP
ejpam-5336	15	8	providing	provide	VERB
ejpam-5336	15	9	necessary	necessary	ADJ
ejpam-5336	15	10	and	and	CCONJ
ejpam-5336	15	11	sufficient	sufficient	ADJ
ejpam-5336	15	12	conditions	condition	NOUN
ejpam-5336	15	13	for	for	ADP
ejpam-5336	15	14	a	a	DET
ejpam-5336	15	15	graph	graph	NOUN
ejpam-5336	15	16	to	to	PART
ejpam-5336	15	17	admit	admit	VERB
ejpam-5336	15	18	a	a	DET
ejpam-5336	15	19	prime	prime	ADJ
ejpam-5336	15	20	labeling	labeling	NOUN
ejpam-5336	15	21	.	.	PUNCT
ejpam-5336	16	1	for	for	SCONJ
ejpam-5336	16	2	more	more	ADJ
ejpam-5336	16	3	details	detail	NOUN
ejpam-5336	16	4	about	about	ADP
ejpam-5336	16	5	prime	prime	ADJ
ejpam-5336	16	6	graphs	graph	NOUN
ejpam-5336	16	7	see	see	VERB
ejpam-5336	16	8	for	for	ADP
ejpam-5336	16	9	example	example	NOUN
ejpam-5336	16	10	[	[	X
ejpam-5336	16	11	2	2	NUM
ejpam-5336	16	12	]	]	PUNCT
ejpam-5336	16	13	,	,	PUNCT
ejpam-5336	16	14	[	[	X
ejpam-5336	16	15	5	5	NUM
ejpam-5336	16	16	]	]	PUNCT
ejpam-5336	16	17	,	,	PUNCT
ejpam-5336	16	18	[	[	X
ejpam-5336	16	19	6	6	NUM
ejpam-5336	16	20	]	]	PUNCT
ejpam-5336	16	21	,	,	PUNCT
ejpam-5336	16	22	[	[	X
ejpam-5336	16	23	10	10	NUM
ejpam-5336	16	24	]	]	PUNCT
ejpam-5336	16	25	.	.	PUNCT
ejpam-5336	17	1	in	in	ADP
ejpam-5336	17	2	this	this	DET
ejpam-5336	17	3	paper	paper	NOUN
ejpam-5336	17	4	,	,	PUNCT
ejpam-5336	17	5	we	we	PRON
ejpam-5336	17	6	discuss	discuss	VERB
ejpam-5336	17	7	when	when	SCONJ
ejpam-5336	17	8	the	the	DET
ejpam-5336	17	9	disjoint	disjoint	PROPN
ejpam-5336	17	10	union	union	NOUN
ejpam-5336	17	11	of	of	ADP
ejpam-5336	17	12	some	some	DET
ejpam-5336	17	13	graphs	graph	NOUN
ejpam-5336	17	14	is	be	AUX
ejpam-5336	17	15	a	a	DET
ejpam-5336	17	16	pg	pg	NOUN
ejpam-5336	17	17	.	.	PUNCT
ejpam-5336	18	1	we	we	PRON
ejpam-5336	18	2	prove	prove	VERB
ejpam-5336	18	3	that	that	SCONJ
ejpam-5336	18	4	wm∪pn	wm∪pn	NOUN
ejpam-5336	18	5	is	be	AUX
ejpam-5336	18	6	a	a	DET
ejpam-5336	18	7	pg	pg	NOUN
ejpam-5336	18	8	if	if	SCONJ
ejpam-5336	18	9	and	and	CCONJ
ejpam-5336	18	10	only	only	ADV
ejpam-5336	18	11	if	if	SCONJ
ejpam-5336	18	12	m	m	NOUN
ejpam-5336	18	13	is	be	AUX
ejpam-5336	18	14	even	even	ADV
ejpam-5336	18	15	or	or	CCONJ
ejpam-5336	18	16	n	n	PRON
ejpam-5336	18	17	is	be	AUX
ejpam-5336	18	18	odd	odd	ADJ
ejpam-5336	18	19	.	.	PUNCT
ejpam-5336	19	1	also	also	ADV
ejpam-5336	19	2	,	,	PUNCT
ejpam-5336	19	3	we	we	PRON
ejpam-5336	19	4	show	show	VERB
ejpam-5336	19	5	that	that	SCONJ
ejpam-5336	19	6	c2n∪c2n∪w2	c2n∪c2n∪w2	NOUN
ejpam-5336	19	7	m	m	NOUN
ejpam-5336	19	8	and	and	CCONJ
ejpam-5336	19	9	c2n	c2n	NOUN
ejpam-5336	19	10	∪	∪	ADP
ejpam-5336	19	11	c2n	c2n	NOUN
ejpam-5336	19	12	∪	∪	NOUN
ejpam-5336	19	13	c2n	c2n	NOUN
ejpam-5336	19	14	∪	∪	NOUN
ejpam-5336	19	15	w2	w2	NOUN
ejpam-5336	19	16	m	m	NOUN
ejpam-5336	19	17	are	be	AUX
ejpam-5336	19	18	pgs	pgs	ADJ
ejpam-5336	19	19	.	.	PUNCT
ejpam-5336	20	1	finally	finally	ADV
ejpam-5336	20	2	,	,	PUNCT
ejpam-5336	20	3	we	we	PRON
ejpam-5336	20	4	study	study	VERB
ejpam-5336	20	5	some	some	DET
ejpam-5336	20	6	properties	property	NOUN
ejpam-5336	20	7	of	of	ADP
ejpam-5336	20	8	the	the	DET
ejpam-5336	20	9	disjoint	disjoint	PROPN
ejpam-5336	20	10	union	union	NOUN
ejpam-5336	20	11	between	between	ADP
ejpam-5336	20	12	a	a	DET
ejpam-5336	20	13	complete	complete	ADJ
ejpam-5336	20	14	graph	graph	NOUN
ejpam-5336	20	15	and	and	CCONJ
ejpam-5336	20	16	any	any	DET
ejpam-5336	20	17	graph	graph	NOUN
ejpam-5336	20	18	such	such	ADJ
ejpam-5336	20	19	that	that	SCONJ
ejpam-5336	20	20	this	this	DET
ejpam-5336	20	21	union	union	NOUN
ejpam-5336	20	22	is	be	AUX
ejpam-5336	20	23	a	a	DET
ejpam-5336	20	24	pg	pg	NOUN
ejpam-5336	20	25	.	.	PUNCT
ejpam-5336	21	1	readers	reader	NOUN
ejpam-5336	21	2	are	be	AUX
ejpam-5336	21	3	advised	advise	VERB
ejpam-5336	21	4	to	to	PART
ejpam-5336	21	5	refer	refer	VERB
ejpam-5336	21	6	to	to	ADP
ejpam-5336	21	7	the	the	DET
ejpam-5336	21	8	appropriate	appropriate	ADJ
ejpam-5336	21	9	references	reference	NOUN
ejpam-5336	21	10	or	or	CCONJ
ejpam-5336	21	11	sources	source	NOUN
ejpam-5336	21	12	for	for	ADP
ejpam-5336	21	13	clarification	clarification	NOUN
ejpam-5336	21	14	on	on	ADP
ejpam-5336	21	15	terms	term	NOUN
ejpam-5336	21	16	and	and	CCONJ
ejpam-5336	21	17	concepts	concept	NOUN
ejpam-5336	21	18	that	that	PRON
ejpam-5336	21	19	have	have	AUX
ejpam-5336	21	20	not	not	PART
ejpam-5336	21	21	been	be	AUX
ejpam-5336	21	22	defined	define	VERB
ejpam-5336	21	23	in	in	ADP
ejpam-5336	21	24	the	the	DET
ejpam-5336	21	25	text	text	NOUN
ejpam-5336	21	26	in	in	ADP
ejpam-5336	21	27	[	[	X
ejpam-5336	21	28	3	3	NUM
ejpam-5336	21	29	]	]	PUNCT
ejpam-5336	21	30	and	and	CCONJ
ejpam-5336	21	31	[	[	X
ejpam-5336	21	32	4	4	NUM
ejpam-5336	21	33	]	]	PUNCT
ejpam-5336	21	34	.	.	PUNCT
ejpam-5336	22	1	∗corresponding	∗corresponde	VERB
ejpam-5336	22	2	author	author	NOUN
ejpam-5336	22	3	.	.	PUNCT
ejpam-5336	23	1	doi	doi	NOUN
ejpam-5336	23	2	:	:	PUNCT
ejpam-5336	23	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5336	https://doi.org/10.29020/nybg.ejpam.v17i4.5336	PROPN
ejpam-5336	23	4	email	email	NOUN
ejpam-5336	23	5	addresses	address	NOUN
ejpam-5336	23	6	:	:	PUNCT
ejpam-5336	24	1	o.abughneim@ju.edu.jo	o.abughneim@ju.edu.jo	PROPN
ejpam-5336	24	2	(	(	PUNCT
ejpam-5336	24	3	o.a	o.a	PROPN
ejpam-5336	24	4	.	.	PROPN
ejpam-5336	24	5	abughneim	abughneim	PROPN
ejpam-5336	24	6	)	)	PUNCT
ejpam-5336	24	7	,	,	PUNCT
ejpam-5336	24	8	baha.abughazaleh@iu.edu.jo	baha.abughazaleh@iu.edu.jo	NOUN
ejpam-5336	24	9	(	(	PUNCT
ejpam-5336	24	10	b.	b.	PROPN
ejpam-5336	24	11	abughazaleh	abughazaleh	PROPN
ejpam-5336	24	12	)	)	PUNCT
ejpam-5336	24	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5336	24	14	3557	3557	NUM
ejpam-5336	25	1	copyright	copyright	NOUN
ejpam-5336	25	2	:	:	PUNCT
ejpam-5336	25	3	©	©	PROPN
ejpam-5336	25	4	2024	2024	NUM
ejpam-5336	25	5	the	the	DET
ejpam-5336	25	6	author(s	author(s	NOUN
ejpam-5336	25	7	)	)	PUNCT
ejpam-5336	25	8	.	.	PUNCT
ejpam-5336	26	1	(	(	PUNCT
ejpam-5336	26	2	cc	cc	NOUN
ejpam-5336	26	3	by	by	ADP
ejpam-5336	26	4	-	-	PUNCT
ejpam-5336	26	5	nc	nc	PROPN
ejpam-5336	26	6	4.0	4.0	NUM
ejpam-5336	26	7	)	)	PUNCT
ejpam-5336	26	8	o.	o.	NOUN
ejpam-5336	26	9	a.	a.	PROPN
ejpam-5336	26	10	abughneim	abughneim	PROPN
ejpam-5336	26	11	,	,	PUNCT
ejpam-5336	26	12	b.	b.	PROPN
ejpam-5336	26	13	abughazaleh	abughazaleh	PROPN
ejpam-5336	26	14	/	/	SYM
ejpam-5336	26	15	eur	eur	PROPN
ejpam-5336	26	16	.	.	PUNCT
ejpam-5336	27	1	j.	j.	PROPN
ejpam-5336	27	2	pure	pure	PROPN
ejpam-5336	27	3	appl	appl	PROPN
ejpam-5336	27	4	.	.	PROPN
ejpam-5336	27	5	math	math	PROPN
ejpam-5336	27	6	,	,	PUNCT
ejpam-5336	27	7	17	17	NUM
ejpam-5336	27	8	(	(	PUNCT
ejpam-5336	27	9	4	4	NUM
ejpam-5336	27	10	)	)	PUNCT
ejpam-5336	27	11	(	(	PUNCT
ejpam-5336	27	12	2024	2024	NUM
ejpam-5336	27	13	)	)	PUNCT
ejpam-5336	27	14	,	,	PUNCT
ejpam-5336	27	15	3557	3557	NUM
ejpam-5336	27	16	-	-	SYM
ejpam-5336	27	17	3566	3566	NUM
ejpam-5336	27	18	3558	3558	NUM
ejpam-5336	27	19	2	2	NUM
ejpam-5336	27	20	.	.	PUNCT
ejpam-5336	27	21	prime	prime	ADJ
ejpam-5336	27	22	labeling	labeling	NOUN
ejpam-5336	27	23	of	of	ADP
ejpam-5336	27	24	union	union	NOUN
ejpam-5336	27	25	of	of	ADP
ejpam-5336	27	26	some	some	DET
ejpam-5336	27	27	graphs	graph	NOUN
ejpam-5336	27	28	in	in	ADP
ejpam-5336	27	29	this	this	DET
ejpam-5336	27	30	section	section	NOUN
ejpam-5336	27	31	,	,	PUNCT
ejpam-5336	27	32	we	we	PRON
ejpam-5336	27	33	generalize	generalize	VERB
ejpam-5336	27	34	a	a	DET
ejpam-5336	27	35	result	result	NOUN
ejpam-5336	27	36	in	in	ADP
ejpam-5336	27	37	[	[	X
ejpam-5336	27	38	11	11	NUM
ejpam-5336	27	39	]	]	PUNCT
ejpam-5336	27	40	,	,	PUNCT
ejpam-5336	27	41	we	we	PRON
ejpam-5336	27	42	prove	prove	VERB
ejpam-5336	27	43	that	that	SCONJ
ejpam-5336	27	44	wm∪pn	wm∪pn	NOUN
ejpam-5336	27	45	is	be	AUX
ejpam-5336	27	46	a	a	DET
ejpam-5336	27	47	pg	pg	NOUN
ejpam-5336	27	48	if	if	SCONJ
ejpam-5336	27	49	and	and	CCONJ
ejpam-5336	27	50	only	only	ADV
ejpam-5336	27	51	if	if	SCONJ
ejpam-5336	27	52	m	m	NOUN
ejpam-5336	27	53	is	be	AUX
ejpam-5336	27	54	even	even	ADV
ejpam-5336	27	55	or	or	CCONJ
ejpam-5336	27	56	n	n	PRON
ejpam-5336	27	57	is	be	AUX
ejpam-5336	27	58	odd	odd	ADJ
ejpam-5336	27	59	.	.	PUNCT
ejpam-5336	28	1	also	also	ADV
ejpam-5336	28	2	,	,	PUNCT
ejpam-5336	28	3	we	we	PRON
ejpam-5336	28	4	show	show	VERB
ejpam-5336	28	5	that	that	SCONJ
ejpam-5336	28	6	c2n	c2n	NOUN
ejpam-5336	28	7	∪c2n	∪c2n	NOUN
ejpam-5336	28	8	∪w2	∪w2	NOUN
ejpam-5336	28	9	m	m	VERB
ejpam-5336	28	10	and	and	CCONJ
ejpam-5336	28	11	c2n	c2n	PROPN
ejpam-5336	28	12	∪c2n	∪c2n	NOUN
ejpam-5336	28	13	∪c2n	∪c2n	NOUN
ejpam-5336	28	14	∪w2	∪w2	NOUN
ejpam-5336	28	15	m	m	VERB
ejpam-5336	28	16	are	be	AUX
ejpam-5336	28	17	pgs	pgs	ADJ
ejpam-5336	28	18	.	.	PUNCT
ejpam-5336	29	1	the	the	DET
ejpam-5336	29	2	following	follow	VERB
ejpam-5336	29	3	lemma	lemma	PROPN
ejpam-5336	29	4	imposes	impose	VERB
ejpam-5336	29	5	certain	certain	ADJ
ejpam-5336	29	6	restrictions	restriction	NOUN
ejpam-5336	29	7	on	on	ADP
ejpam-5336	29	8	the	the	DET
ejpam-5336	29	9	independence	independence	NOUN
ejpam-5336	29	10	number	number	NOUN
ejpam-5336	29	11	of	of	ADP
ejpam-5336	29	12	pgs	pgs	PROPN
ejpam-5336	29	13	.	.	PUNCT
ejpam-5336	30	1	lemma	lemma	PROPN
ejpam-5336	30	2	1	1	NUM
ejpam-5336	30	3	.	.	PUNCT
ejpam-5336	31	1	[	[	X
ejpam-5336	31	2	14	14	NUM
ejpam-5336	31	3	]	]	PUNCT
ejpam-5336	31	4	“	"	PUNCT
ejpam-5336	31	5	for	for	ADP
ejpam-5336	31	6	any	any	DET
ejpam-5336	31	7	pg	pg	NOUN
ejpam-5336	31	8	g	g	NOUN
ejpam-5336	31	9	,	,	PUNCT
ejpam-5336	31	10	we	we	PRON
ejpam-5336	31	11	have	have	VERB
ejpam-5336	31	12	α(g	α(g	NUM
ejpam-5336	31	13	)	)	PUNCT
ejpam-5336	31	14	≥	≥	NOUN
ejpam-5336	31	15	[	[	PUNCT
ejpam-5336	31	16	|v	|v	X
ejpam-5336	31	17	(	(	PUNCT
ejpam-5336	31	18	g)|	g)|	NOUN
ejpam-5336	31	19	2	2	NUM
ejpam-5336	31	20	]	]	PUNCT
ejpam-5336	31	21	.	.	PUNCT
ejpam-5336	31	22	”	"	PUNCT
ejpam-5336	32	1	the	the	DET
ejpam-5336	32	2	authors	author	NOUN
ejpam-5336	32	3	in	in	ADP
ejpam-5336	32	4	[	[	X
ejpam-5336	32	5	11	11	NUM
ejpam-5336	32	6	]	]	PUNCT
ejpam-5336	32	7	proved	prove	VERB
ejpam-5336	32	8	that	that	SCONJ
ejpam-5336	32	9	“	"	PUNCT
ejpam-5336	32	10	the	the	DET
ejpam-5336	32	11	disjoint	disjoint	PROPN
ejpam-5336	32	12	union	union	NOUN
ejpam-5336	32	13	of	of	ADP
ejpam-5336	32	14	a	a	DET
ejpam-5336	32	15	pg	pg	NOUN
ejpam-5336	32	16	of	of	ADP
ejpam-5336	32	17	even	even	ADV
ejpam-5336	32	18	order	order	NOUN
ejpam-5336	32	19	and	and	CCONJ
ejpam-5336	32	20	a	a	DET
ejpam-5336	32	21	graph	graph	NOUN
ejpam-5336	32	22	of	of	ADP
ejpam-5336	32	23	order	order	NOUN
ejpam-5336	32	24	3	3	NUM
ejpam-5336	32	25	is	be	AUX
ejpam-5336	32	26	a	a	DET
ejpam-5336	32	27	pg	pg	NOUN
ejpam-5336	32	28	.	.	PUNCT
ejpam-5336	32	29	”	"	PUNCT
ejpam-5336	33	1	in	in	ADP
ejpam-5336	33	2	the	the	DET
ejpam-5336	33	3	following	following	NOUN
ejpam-5336	33	4	theorem	theorem	NOUN
ejpam-5336	33	5	,	,	PUNCT
ejpam-5336	33	6	we	we	PRON
ejpam-5336	33	7	generalize	generalize	VERB
ejpam-5336	33	8	this	this	DET
ejpam-5336	33	9	result	result	NOUN
ejpam-5336	33	10	.	.	PUNCT
ejpam-5336	34	1	theorem	theorem	NOUN
ejpam-5336	34	2	1	1	X
ejpam-5336	34	3	.	.	PUNCT
ejpam-5336	35	1	let	let	VERB
ejpam-5336	35	2	g1	g1	PROPN
ejpam-5336	35	3	and	and	CCONJ
ejpam-5336	35	4	g2	g2	PROPN
ejpam-5336	35	5	be	be	AUX
ejpam-5336	35	6	pgs	pgs	NOUN
ejpam-5336	35	7	of	of	ADP
ejpam-5336	35	8	orders	order	NOUN
ejpam-5336	35	9	n	n	PRON
ejpam-5336	35	10	and	and	CCONJ
ejpam-5336	35	11	m	m	VERB
ejpam-5336	35	12	respectively	respectively	ADV
ejpam-5336	35	13	.	.	PUNCT
ejpam-5336	36	1	if	if	SCONJ
ejpam-5336	36	2	for	for	ADP
ejpam-5336	36	3	any	any	DET
ejpam-5336	36	4	prime	prime	NOUN
ejpam-5336	36	5	p	p	NOUN
ejpam-5336	36	6	≤	≤	NOUN
ejpam-5336	36	7	m−	m−	PROPN
ejpam-5336	36	8	1	1	NUM
ejpam-5336	36	9	,	,	PUNCT
ejpam-5336	36	10	we	we	PRON
ejpam-5336	36	11	get	get	VERB
ejpam-5336	36	12	p	p	NOUN
ejpam-5336	36	13	divides	divide	NOUN
ejpam-5336	36	14	n	n	CCONJ
ejpam-5336	36	15	,	,	PUNCT
ejpam-5336	36	16	then	then	ADV
ejpam-5336	36	17	g1	g1	VERB
ejpam-5336	36	18	∪g2	∪g2	PROPN
ejpam-5336	36	19	is	be	AUX
ejpam-5336	36	20	a	a	DET
ejpam-5336	36	21	pg	pg	NOUN
ejpam-5336	36	22	.	.	PUNCT
ejpam-5336	37	1	proof	proof	NOUN
ejpam-5336	37	2	.	.	PUNCT
ejpam-5336	38	1	let	let	VERB
ejpam-5336	38	2	u1	u1	NOUN
ejpam-5336	38	3	,	,	PUNCT
ejpam-5336	38	4	u2	u2	PROPN
ejpam-5336	38	5	,	,	PUNCT
ejpam-5336	38	6	...	...	PUNCT
ejpam-5336	38	7	,	,	PUNCT
ejpam-5336	38	8	un	un	PROPN
ejpam-5336	38	9	be	be	VERB
ejpam-5336	38	10	the	the	DET
ejpam-5336	38	11	vertices	vertex	NOUN
ejpam-5336	38	12	of	of	ADP
ejpam-5336	38	13	g1	g1	NOUN
ejpam-5336	38	14	,	,	PUNCT
ejpam-5336	38	15	v1	v1	NOUN
ejpam-5336	38	16	,	,	PUNCT
ejpam-5336	38	17	v2	v2	PROPN
ejpam-5336	38	18	,	,	PUNCT
ejpam-5336	38	19	...	...	PUNCT
ejpam-5336	38	20	vm	vm	PROPN
ejpam-5336	38	21	be	be	AUX
ejpam-5336	38	22	the	the	DET
ejpam-5336	38	23	vertices	vertex	NOUN
ejpam-5336	38	24	of	of	ADP
ejpam-5336	38	25	g2	g2	PROPN
ejpam-5336	38	26	,	,	PUNCT
ejpam-5336	38	27	f	f	X
ejpam-5336	38	28	:	:	PUNCT
ejpam-5336	38	29	v	v	X
ejpam-5336	38	30	(	(	PUNCT
ejpam-5336	38	31	g1	g1	PROPN
ejpam-5336	38	32	)	)	PUNCT
ejpam-5336	38	33	−→	−→	NOUN
ejpam-5336	38	34	{	{	PUNCT
ejpam-5336	38	35	1	1	NUM
ejpam-5336	38	36	,	,	PUNCT
ejpam-5336	38	37	2	2	NUM
ejpam-5336	38	38	,	,	PUNCT
ejpam-5336	38	39	...	...	PUNCT
ejpam-5336	38	40	,	,	PUNCT
ejpam-5336	38	41	n	n	CCONJ
ejpam-5336	38	42	}	}	PUNCT
ejpam-5336	38	43	be	be	AUX
ejpam-5336	38	44	a	a	DET
ejpam-5336	38	45	pl	pl	NOUN
ejpam-5336	38	46	of	of	ADP
ejpam-5336	38	47	g1	g1	PROPN
ejpam-5336	38	48	and	and	CCONJ
ejpam-5336	38	49	g	g	NOUN
ejpam-5336	38	50	:	:	PUNCT
ejpam-5336	38	51	v	v	PROPN
ejpam-5336	38	52	(	(	PUNCT
ejpam-5336	38	53	g2	g2	PROPN
ejpam-5336	38	54	)	)	PUNCT
ejpam-5336	39	1	−→	−→	NOUN
ejpam-5336	39	2	{	{	PUNCT
ejpam-5336	39	3	1	1	NUM
ejpam-5336	39	4	,	,	PUNCT
ejpam-5336	39	5	2	2	NUM
ejpam-5336	39	6	,	,	PUNCT
ejpam-5336	39	7	...	...	PUNCT
ejpam-5336	39	8	,	,	PUNCT
ejpam-5336	39	9	m	m	AUX
ejpam-5336	39	10	}	}	PUNCT
ejpam-5336	39	11	be	be	AUX
ejpam-5336	39	12	a	a	DET
ejpam-5336	39	13	pl	pl	NOUN
ejpam-5336	39	14	of	of	ADP
ejpam-5336	39	15	g2	g2	PROPN
ejpam-5336	39	16	.	.	PUNCT
ejpam-5336	40	1	define	define	VERB
ejpam-5336	40	2	h	h	NOUN
ejpam-5336	40	3	:	:	PUNCT
ejpam-5336	40	4	v	v	X
ejpam-5336	40	5	(	(	PUNCT
ejpam-5336	40	6	g1	g1	PROPN
ejpam-5336	40	7	∪g2	∪g2	PROPN
ejpam-5336	40	8	)	)	PUNCT
ejpam-5336	40	9	−→	−→	NOUN
ejpam-5336	40	10	{	{	PUNCT
ejpam-5336	40	11	1	1	NUM
ejpam-5336	40	12	,	,	PUNCT
ejpam-5336	40	13	2	2	NUM
ejpam-5336	40	14	,	,	PUNCT
ejpam-5336	40	15	...	...	PUNCT
ejpam-5336	40	16	,	,	PUNCT
ejpam-5336	40	17	n+m	n+m	CCONJ
ejpam-5336	40	18	}	}	PUNCT
ejpam-5336	40	19	by	by	ADP
ejpam-5336	40	20	h	h	PROPN
ejpam-5336	40	21	(	(	PUNCT
ejpam-5336	40	22	ui	ui	PROPN
ejpam-5336	40	23	)	)	PUNCT
ejpam-5336	40	24	=	=	SYM
ejpam-5336	41	1	f	f	PROPN
ejpam-5336	41	2	(	(	PUNCT
ejpam-5336	41	3	ui	ui	PROPN
ejpam-5336	41	4	)	)	PUNCT
ejpam-5336	41	5	for	for	ADP
ejpam-5336	41	6	all	all	DET
ejpam-5336	41	7	1	1	NUM
ejpam-5336	41	8	≤	≤	NUM
ejpam-5336	41	9	i	i	PRON
ejpam-5336	41	10	≤	≤	NOUN
ejpam-5336	41	11	n	n	CCONJ
ejpam-5336	41	12	,	,	PUNCT
ejpam-5336	41	13	and	and	CCONJ
ejpam-5336	41	14	h(vj	h(vj	NOUN
ejpam-5336	41	15	)	)	PUNCT
ejpam-5336	41	16	=	=	PUNCT
ejpam-5336	41	17	n+	n+	NOUN
ejpam-5336	41	18	g(vj	g(vj	PROPN
ejpam-5336	41	19	)	)	PUNCT
ejpam-5336	41	20	for	for	ADP
ejpam-5336	41	21	all	all	PRON
ejpam-5336	41	22	1	1	NUM
ejpam-5336	41	23	≤	≤	NUM
ejpam-5336	41	24	j	j	PROPN
ejpam-5336	41	25	≤	≤	PROPN
ejpam-5336	41	26	m.	m.	NOUN
ejpam-5336	41	27	if	if	SCONJ
ejpam-5336	41	28	ui	ui	PROPN
ejpam-5336	41	29	and	and	CCONJ
ejpam-5336	41	30	uj	uj	PROPN
ejpam-5336	41	31	are	be	AUX
ejpam-5336	41	32	adjacent	adjacent	ADJ
ejpam-5336	41	33	in	in	ADP
ejpam-5336	41	34	g1	g1	PROPN
ejpam-5336	41	35	.	.	PUNCT
ejpam-5336	42	1	then	then	ADV
ejpam-5336	42	2	(	(	PUNCT
ejpam-5336	42	3	h	h	NOUN
ejpam-5336	42	4	(	(	PUNCT
ejpam-5336	42	5	ui	ui	PROPN
ejpam-5336	42	6	)	)	PUNCT
ejpam-5336	42	7	,	,	PUNCT
ejpam-5336	42	8	h	h	PROPN
ejpam-5336	42	9	(	(	PUNCT
ejpam-5336	42	10	uj	uj	PROPN
ejpam-5336	42	11	)	)	PUNCT
ejpam-5336	42	12	)	)	PUNCT
ejpam-5336	43	1	=	=	PRON
ejpam-5336	43	2	(	(	PUNCT
ejpam-5336	43	3	f	f	X
ejpam-5336	43	4	(	(	PUNCT
ejpam-5336	43	5	ui	ui	PROPN
ejpam-5336	43	6	)	)	PUNCT
ejpam-5336	43	7	,	,	PUNCT
ejpam-5336	43	8	f	f	PROPN
ejpam-5336	43	9	(	(	PUNCT
ejpam-5336	43	10	uj	uj	PROPN
ejpam-5336	43	11	)	)	PUNCT
ejpam-5336	43	12	)	)	PUNCT
ejpam-5336	44	1	=	=	PUNCT
ejpam-5336	44	2	1	1	NUM
ejpam-5336	44	3	because	because	SCONJ
ejpam-5336	44	4	f	f	PROPN
ejpam-5336	44	5	is	be	AUX
ejpam-5336	44	6	a	a	DET
ejpam-5336	44	7	pl	pl	PROPN
ejpam-5336	44	8	.	.	PROPN
ejpam-5336	44	9	suppose	suppose	VERB
ejpam-5336	44	10	vi	vi	PROPN
ejpam-5336	45	1	and	and	CCONJ
ejpam-5336	45	2	vj	vj	NOUN
ejpam-5336	45	3	are	be	AUX
ejpam-5336	45	4	adjacent	adjacent	ADJ
ejpam-5336	45	5	in	in	ADP
ejpam-5336	45	6	g2	g2	PROPN
ejpam-5336	45	7	and	and	CCONJ
ejpam-5336	45	8	d	d	NOUN
ejpam-5336	45	9	=	=	PUNCT
ejpam-5336	45	10	(	(	PUNCT
ejpam-5336	45	11	h	h	PROPN
ejpam-5336	45	12	(	(	PUNCT
ejpam-5336	45	13	vi	vi	NOUN
ejpam-5336	45	14	)	)	PUNCT
ejpam-5336	45	15	,	,	PUNCT
ejpam-5336	45	16	h	h	NOUN
ejpam-5336	45	17	(	(	PUNCT
ejpam-5336	45	18	vj	vj	PROPN
ejpam-5336	45	19	)	)	PUNCT
ejpam-5336	45	20	)	)	PUNCT
ejpam-5336	46	1	=	=	PUNCT
ejpam-5336	46	2	(	(	PUNCT
ejpam-5336	46	3	n+	n+	ADP
ejpam-5336	46	4	g	g	PROPN
ejpam-5336	46	5	(	(	PUNCT
ejpam-5336	46	6	ui	ui	PROPN
ejpam-5336	46	7	)	)	PUNCT
ejpam-5336	46	8	,	,	PUNCT
ejpam-5336	46	9	n+	n+	ADP
ejpam-5336	46	10	g	g	PROPN
ejpam-5336	46	11	(	(	PUNCT
ejpam-5336	46	12	uj	uj	PROPN
ejpam-5336	46	13	)	)	PUNCT
ejpam-5336	46	14	)	)	PUNCT
ejpam-5336	46	15	.	.	PUNCT
ejpam-5336	47	1	thus	thus	ADV
ejpam-5336	47	2	d	d	X
ejpam-5336	47	3	divides	divide	VERB
ejpam-5336	47	4	g	g	PROPN
ejpam-5336	47	5	(	(	PUNCT
ejpam-5336	47	6	ui	ui	PROPN
ejpam-5336	47	7	)	)	PUNCT
ejpam-5336	48	1	−	−	PROPN
ejpam-5336	48	2	g	g	PROPN
ejpam-5336	48	3	(	(	PUNCT
ejpam-5336	48	4	uj	uj	PROPN
ejpam-5336	48	5	)	)	PUNCT
ejpam-5336	48	6	and	and	CCONJ
ejpam-5336	48	7	|g	|g	NOUN
ejpam-5336	48	8	(	(	PUNCT
ejpam-5336	48	9	ui)−	ui)−	NUM
ejpam-5336	48	10	g	g	NOUN
ejpam-5336	48	11	(	(	PUNCT
ejpam-5336	48	12	uj)|	uj)|	NOUN
ejpam-5336	48	13	≤	≤	NOUN
ejpam-5336	48	14	m	m	VERB
ejpam-5336	48	15	−	−	NOUN
ejpam-5336	48	16	1	1	NUM
ejpam-5336	48	17	.	.	PUNCT
ejpam-5336	49	1	if	if	SCONJ
ejpam-5336	49	2	d	d	PROPN
ejpam-5336	49	3	>	>	X
ejpam-5336	49	4	1	1	NUM
ejpam-5336	49	5	,	,	PUNCT
ejpam-5336	49	6	then	then	ADV
ejpam-5336	49	7	d	d	PROPN
ejpam-5336	49	8	has	have	VERB
ejpam-5336	49	9	a	a	DET
ejpam-5336	49	10	prime	prime	ADJ
ejpam-5336	49	11	divisor	divisor	NOUN
ejpam-5336	49	12	say	say	VERB
ejpam-5336	49	13	p.	p.	NOUN
ejpam-5336	49	14	therefore	therefore	ADV
ejpam-5336	49	15	,	,	PUNCT
ejpam-5336	50	1	p	p	NOUN
ejpam-5336	50	2	≤	≤	NUM
ejpam-5336	50	3	d	d	NOUN
ejpam-5336	50	4	≤	≤	NUM
ejpam-5336	50	5	m	m	VERB
ejpam-5336	50	6	−	−	NOUN
ejpam-5336	50	7	1	1	NUM
ejpam-5336	50	8	and	and	CCONJ
ejpam-5336	50	9	by	by	ADP
ejpam-5336	50	10	assumption	assumption	NOUN
ejpam-5336	50	11	p	p	NOUN
ejpam-5336	50	12	divides	divide	VERB
ejpam-5336	50	13	n.	n.	PROPN
ejpam-5336	50	14	but	but	CCONJ
ejpam-5336	50	15	p	p	NOUN
ejpam-5336	50	16	divides	divide	VERB
ejpam-5336	50	17	n	n	PROPN
ejpam-5336	50	18	+	+	CCONJ
ejpam-5336	50	19	g	g	PROPN
ejpam-5336	50	20	(	(	PUNCT
ejpam-5336	50	21	ui	ui	NOUN
ejpam-5336	50	22	)	)	PUNCT
ejpam-5336	50	23	and	and	CCONJ
ejpam-5336	50	24	p	p	NOUN
ejpam-5336	50	25	divides	divide	VERB
ejpam-5336	50	26	n	n	PROPN
ejpam-5336	50	27	+	+	CCONJ
ejpam-5336	50	28	g	g	PROPN
ejpam-5336	50	29	(	(	PUNCT
ejpam-5336	50	30	uj	uj	PROPN
ejpam-5336	50	31	)	)	PUNCT
ejpam-5336	50	32	.	.	PUNCT
ejpam-5336	51	1	thus	thus	ADV
ejpam-5336	51	2	p	p	X
ejpam-5336	51	3	divides	divide	VERB
ejpam-5336	51	4	g	g	PROPN
ejpam-5336	51	5	(	(	PUNCT
ejpam-5336	51	6	ui	ui	PROPN
ejpam-5336	51	7	)	)	PUNCT
ejpam-5336	51	8	and	and	CCONJ
ejpam-5336	51	9	p	p	NOUN
ejpam-5336	51	10	divides	divide	VERB
ejpam-5336	51	11	g	g	PROPN
ejpam-5336	51	12	(	(	PUNCT
ejpam-5336	51	13	uj	uj	PROPN
ejpam-5336	51	14	)	)	PUNCT
ejpam-5336	51	15	and	and	CCONJ
ejpam-5336	51	16	hence	hence	ADV
ejpam-5336	51	17	(	(	PUNCT
ejpam-5336	51	18	g	g	PROPN
ejpam-5336	51	19	(	(	PUNCT
ejpam-5336	51	20	ui	ui	PROPN
ejpam-5336	51	21	)	)	PUNCT
ejpam-5336	51	22	,	,	PUNCT
ejpam-5336	51	23	g	g	PROPN
ejpam-5336	51	24	(	(	PUNCT
ejpam-5336	51	25	uj	uj	PROPN
ejpam-5336	51	26	)	)	PUNCT
ejpam-5336	51	27	)	)	PUNCT
ejpam-5336	51	28	≥	≥	PROPN
ejpam-5336	51	29	p	p	X
ejpam-5336	51	30	which	which	PRON
ejpam-5336	51	31	is	be	AUX
ejpam-5336	51	32	a	a	DET
ejpam-5336	51	33	contradiction	contradiction	NOUN
ejpam-5336	51	34	,	,	PUNCT
ejpam-5336	51	35	because	because	SCONJ
ejpam-5336	51	36	g	g	PROPN
ejpam-5336	51	37	is	be	AUX
ejpam-5336	51	38	a	a	DET
ejpam-5336	51	39	pl	pl	NOUN
ejpam-5336	51	40	.	.	PUNCT
ejpam-5336	51	41	therefore	therefore	ADV
ejpam-5336	51	42	,	,	PUNCT
ejpam-5336	51	43	(	(	PUNCT
ejpam-5336	51	44	h	h	NOUN
ejpam-5336	51	45	(	(	PUNCT
ejpam-5336	51	46	vi	vi	NOUN
ejpam-5336	51	47	)	)	PUNCT
ejpam-5336	51	48	,	,	PUNCT
ejpam-5336	51	49	h	h	NOUN
ejpam-5336	51	50	(	(	PUNCT
ejpam-5336	51	51	vj	vj	PROPN
ejpam-5336	51	52	)	)	PUNCT
ejpam-5336	51	53	)	)	PUNCT
ejpam-5336	51	54	=	=	SYM
ejpam-5336	52	1	1	1	NUM
ejpam-5336	53	1	and	and	CCONJ
ejpam-5336	53	2	so	so	ADV
ejpam-5336	53	3	h	h	NOUN
ejpam-5336	53	4	is	be	AUX
ejpam-5336	53	5	a	a	DET
ejpam-5336	53	6	pl	pl	PROPN
ejpam-5336	53	7	of	of	ADP
ejpam-5336	53	8	g1	g1	PROPN
ejpam-5336	53	9	∪g2	∪g2	PROPN
ejpam-5336	53	10	.	.	PUNCT
ejpam-5336	54	1	vaidya	vaidya	PROPN
ejpam-5336	54	2	et	et	PROPN
ejpam-5336	54	3	.	.	PUNCT
ejpam-5336	55	1	al	al	PROPN
ejpam-5336	55	2	.	.	PUNCT
ejpam-5336	56	1	in	in	ADP
ejpam-5336	56	2	[	[	X
ejpam-5336	56	3	13	13	NUM
ejpam-5336	56	4	]	]	PUNCT
ejpam-5336	56	5	proved	prove	VERB
ejpam-5336	56	6	the	the	DET
ejpam-5336	56	7	following	follow	VERB
ejpam-5336	56	8	theorem	theorem	NOUN
ejpam-5336	56	9	theorem	theorem	NOUN
ejpam-5336	56	10	2	2	NUM
ejpam-5336	56	11	.	.	PUNCT
ejpam-5336	57	1	[	[	X
ejpam-5336	57	2	13	13	NUM
ejpam-5336	57	3	]	]	PUNCT
ejpam-5336	57	4	“	"	PUNCT
ejpam-5336	57	5	w2k	w2k	PROPN
ejpam-5336	57	6	∪	∪	ADJ
ejpam-5336	57	7	pm	pm	NOUN
ejpam-5336	57	8	is	be	AUX
ejpam-5336	57	9	a	a	DET
ejpam-5336	57	10	pg	pg	NOUN
ejpam-5336	57	11	.	.	PUNCT
ejpam-5336	57	12	”	"	PUNCT
ejpam-5336	58	1	next	next	ADV
ejpam-5336	58	2	,	,	PUNCT
ejpam-5336	58	3	we	we	PRON
ejpam-5336	58	4	show	show	VERB
ejpam-5336	58	5	when	when	SCONJ
ejpam-5336	58	6	,	,	PUNCT
ejpam-5336	58	7	in	in	ADP
ejpam-5336	58	8	general	general	ADJ
ejpam-5336	58	9	,	,	PUNCT
ejpam-5336	58	10	wm	wm	PROPN
ejpam-5336	58	11	∪	∪	ADP
ejpam-5336	58	12	pn	pn	PROPN
ejpam-5336	58	13	is	be	AUX
ejpam-5336	58	14	a	a	DET
ejpam-5336	58	15	pg	pg	PROPN
ejpam-5336	58	16	.	.	PUNCT
ejpam-5336	58	17	theorem	theorem	NOUN
ejpam-5336	58	18	3	3	NUM
ejpam-5336	58	19	.	.	PUNCT
ejpam-5336	58	20	wm	wm	PROPN
ejpam-5336	58	21	∪	∪	ADP
ejpam-5336	58	22	pn	pn	PROPN
ejpam-5336	58	23	is	be	AUX
ejpam-5336	58	24	a	a	DET
ejpam-5336	58	25	pg	pg	NOUN
ejpam-5336	58	26	if	if	SCONJ
ejpam-5336	59	1	and	and	CCONJ
ejpam-5336	59	2	only	only	ADV
ejpam-5336	59	3	if	if	SCONJ
ejpam-5336	59	4	m	m	NOUN
ejpam-5336	59	5	is	be	AUX
ejpam-5336	59	6	even	even	ADV
ejpam-5336	59	7	or	or	CCONJ
ejpam-5336	59	8	n	n	PRON
ejpam-5336	59	9	is	be	AUX
ejpam-5336	59	10	odd	odd	ADJ
ejpam-5336	59	11	.	.	PUNCT
ejpam-5336	60	1	proof	proof	NOUN
ejpam-5336	60	2	.	.	PUNCT
ejpam-5336	61	1	we	we	PRON
ejpam-5336	61	2	separate	separate	VERB
ejpam-5336	61	3	the	the	DET
ejpam-5336	61	4	proof	proof	NOUN
ejpam-5336	61	5	in	in	ADP
ejpam-5336	61	6	the	the	DET
ejpam-5336	61	7	following	following	ADJ
ejpam-5336	61	8	cases	case	NOUN
ejpam-5336	61	9	,	,	PUNCT
ejpam-5336	61	10	(	(	PUNCT
ejpam-5336	61	11	i	i	NOUN
ejpam-5336	61	12	)	)	PUNCT
ejpam-5336	61	13	suppose	suppose	VERB
ejpam-5336	61	14	m	m	NOUN
ejpam-5336	61	15	is	be	AUX
ejpam-5336	61	16	odd	odd	ADJ
ejpam-5336	61	17	and	and	CCONJ
ejpam-5336	61	18	n	n	PRON
ejpam-5336	61	19	is	be	AUX
ejpam-5336	61	20	even	even	ADV
ejpam-5336	61	21	.	.	PUNCT
ejpam-5336	62	1	let	let	VERB
ejpam-5336	62	2	m	m	NOUN
ejpam-5336	62	3	=	=	VERB
ejpam-5336	62	4	2k	2k	NUM
ejpam-5336	62	5	+	+	CCONJ
ejpam-5336	62	6	1	1	NUM
ejpam-5336	62	7	and	and	CCONJ
ejpam-5336	62	8	n	n	NOUN
ejpam-5336	62	9	=	=	NUM
ejpam-5336	62	10	2h	2h	NUM
ejpam-5336	62	11	.	.	PUNCT
ejpam-5336	63	1	then	then	ADV
ejpam-5336	63	2	α(wm∪pn	α(wm∪pn	VERB
ejpam-5336	63	3	)	)	PUNCT
ejpam-5336	63	4	=	=	SYM
ejpam-5336	63	5	α(wm)+α	α(wm)+α	NOUN
ejpam-5336	63	6	(	(	PUNCT
ejpam-5336	63	7	pn	pn	NOUN
ejpam-5336	63	8	)	)	PUNCT
ejpam-5336	63	9	=	=	SYM
ejpam-5336	64	1	k+h	k+h	X
ejpam-5336	64	2	<	<	X
ejpam-5336	65	1	[	[	PUNCT
ejpam-5336	65	2	|wm	|wm	X
ejpam-5336	65	3	∪	∪	ADP
ejpam-5336	65	4	pn|	pn|	ADJ
ejpam-5336	65	5	2	2	NUM
ejpam-5336	65	6	]	]	PUNCT
ejpam-5336	65	7	=	=	PUNCT
ejpam-5336	65	8	[	[	PUNCT
ejpam-5336	65	9	2k	2k	NOUN
ejpam-5336	65	10	+	+	CCONJ
ejpam-5336	65	11	2	2	NUM
ejpam-5336	65	12	+	+	NUM
ejpam-5336	65	13	2h	2h	NUM
ejpam-5336	65	14	2	2	NUM
ejpam-5336	65	15	]	]	PUNCT
ejpam-5336	65	16	=	=	PUNCT
ejpam-5336	65	17	k+h+1	k+h+1	PROPN
ejpam-5336	65	18	.	.	PUNCT
ejpam-5336	66	1	by	by	ADP
ejpam-5336	66	2	lemma	lemma	PROPN
ejpam-5336	66	3	1	1	NUM
ejpam-5336	66	4	,	,	PUNCT
ejpam-5336	66	5	we	we	PRON
ejpam-5336	66	6	get	get	VERB
ejpam-5336	66	7	wm	wm	PROPN
ejpam-5336	66	8	∪	∪	PROPN
ejpam-5336	66	9	pn	pn	PROPN
ejpam-5336	66	10	is	be	AUX
ejpam-5336	66	11	not	not	PART
ejpam-5336	66	12	a	a	DET
ejpam-5336	66	13	pg	pg	NOUN
ejpam-5336	66	14	.	.	PUNCT
ejpam-5336	67	1	o.	o.	PROPN
ejpam-5336	67	2	a.	a.	PROPN
ejpam-5336	67	3	abughneim	abughneim	PROPN
ejpam-5336	67	4	,	,	PUNCT
ejpam-5336	67	5	b.	b.	PROPN
ejpam-5336	67	6	abughazaleh	abughazaleh	PROPN
ejpam-5336	67	7	/	/	SYM
ejpam-5336	67	8	eur	eur	PROPN
ejpam-5336	67	9	.	.	PUNCT
ejpam-5336	68	1	j.	j.	PROPN
ejpam-5336	68	2	pure	pure	PROPN
ejpam-5336	68	3	appl	appl	PROPN
ejpam-5336	68	4	.	.	PROPN
ejpam-5336	68	5	math	math	PROPN
ejpam-5336	68	6	,	,	PUNCT
ejpam-5336	68	7	17	17	NUM
ejpam-5336	68	8	(	(	PUNCT
ejpam-5336	68	9	4	4	NUM
ejpam-5336	68	10	)	)	PUNCT
ejpam-5336	68	11	(	(	PUNCT
ejpam-5336	68	12	2024	2024	NUM
ejpam-5336	68	13	)	)	PUNCT
ejpam-5336	68	14	,	,	PUNCT
ejpam-5336	68	15	3557	3557	NUM
ejpam-5336	68	16	-	-	SYM
ejpam-5336	68	17	3566	3566	NUM
ejpam-5336	68	18	3559	3559	NUM
ejpam-5336	68	19	(	(	PUNCT
ejpam-5336	68	20	ii	ii	NOUN
ejpam-5336	68	21	)	)	PUNCT
ejpam-5336	68	22	suppose	suppose	VERB
ejpam-5336	68	23	m	m	NOUN
ejpam-5336	68	24	is	be	AUX
ejpam-5336	68	25	even	even	ADV
ejpam-5336	68	26	.	.	PUNCT
ejpam-5336	69	1	by	by	ADP
ejpam-5336	69	2	theorem	theorem	NOUN
ejpam-5336	69	3	2	2	NUM
ejpam-5336	69	4	,	,	PUNCT
ejpam-5336	69	5	wm	wm	PROPN
ejpam-5336	69	6	∪	∪	PROPN
ejpam-5336	69	7	pn	pn	PROPN
ejpam-5336	69	8	is	be	AUX
ejpam-5336	69	9	a	a	DET
ejpam-5336	69	10	pg	pg	NOUN
ejpam-5336	69	11	.	.	PUNCT
ejpam-5336	70	1	(	(	PUNCT
ejpam-5336	70	2	iii	iii	X
ejpam-5336	70	3	)	)	PUNCT
ejpam-5336	70	4	suppose	suppose	VERB
ejpam-5336	70	5	m	m	VERB
ejpam-5336	70	6	and	and	CCONJ
ejpam-5336	70	7	n	n	PRON
ejpam-5336	70	8	are	be	AUX
ejpam-5336	70	9	odd	odd	ADJ
ejpam-5336	70	10	.	.	PUNCT
ejpam-5336	71	1	let	let	VERB
ejpam-5336	71	2	u0	u0	ADJ
ejpam-5336	71	3	be	be	AUX
ejpam-5336	71	4	the	the	DET
ejpam-5336	71	5	apex	apex	NOUN
ejpam-5336	71	6	vertex	vertex	NOUN
ejpam-5336	71	7	of	of	ADP
ejpam-5336	71	8	wm	wm	PROPN
ejpam-5336	71	9	,	,	PUNCT
ejpam-5336	71	10	u1	u1	PROPN
ejpam-5336	71	11	,	,	PUNCT
ejpam-5336	71	12	u2	u2	NOUN
ejpam-5336	71	13	,	,	PUNCT
ejpam-5336	71	14	...	...	PUNCT
ejpam-5336	71	15	,	,	PUNCT
ejpam-5336	71	16	um	um	INTJ
ejpam-5336	71	17	be	be	AUX
ejpam-5336	71	18	the	the	DET
ejpam-5336	71	19	consecutive	consecutive	ADJ
ejpam-5336	71	20	rim	rim	NOUN
ejpam-5336	71	21	vertices	vertex	NOUN
ejpam-5336	71	22	of	of	ADP
ejpam-5336	71	23	wm	wm	PROPN
ejpam-5336	71	24	and	and	CCONJ
ejpam-5336	71	25	v1v2	v1v2	PROPN
ejpam-5336	71	26	...	...	PROPN
ejpam-5336	71	27	vn	vn	AUX
ejpam-5336	71	28	be	be	AUX
ejpam-5336	71	29	the	the	DET
ejpam-5336	71	30	path	path	NOUN
ejpam-5336	71	31	pn	pn	NOUN
ejpam-5336	71	32	and	and	CCONJ
ejpam-5336	71	33	define	define	VERB
ejpam-5336	71	34	f	f	PROPN
ejpam-5336	71	35	:	:	PUNCT
ejpam-5336	71	36	v	v	PROPN
ejpam-5336	71	37	(	(	PUNCT
ejpam-5336	71	38	wm	wm	PROPN
ejpam-5336	71	39	∪	∪	ADP
ejpam-5336	71	40	pn	pn	PROPN
ejpam-5336	71	41	)	)	PUNCT
ejpam-5336	71	42	−→	−→	NOUN
ejpam-5336	71	43	{	{	PUNCT
ejpam-5336	71	44	1	1	NUM
ejpam-5336	71	45	,	,	PUNCT
ejpam-5336	71	46	2	2	NUM
ejpam-5336	71	47	,	,	PUNCT
ejpam-5336	71	48	...	...	PUNCT
ejpam-5336	71	49	,	,	PUNCT
ejpam-5336	71	50	m+	m+	NUM
ejpam-5336	71	51	n+	n+	PUNCT
ejpam-5336	71	52	1	1	X
ejpam-5336	71	53	}	}	PUNCT
ejpam-5336	71	54	as	as	SCONJ
ejpam-5336	71	55	follows	follow	VERB
ejpam-5336	71	56	:	:	PUNCT
ejpam-5336	71	57	f(ui	f(ui	PROPN
ejpam-5336	71	58	)	)	PUNCT
ejpam-5336	72	1	=	=	PRON
ejpam-5336	72	2	{	{	PUNCT
ejpam-5336	72	3	i+	i+	NOUN
ejpam-5336	72	4	1	1	NUM
ejpam-5336	72	5	,	,	PUNCT
ejpam-5336	72	6	0	0	NUM
ejpam-5336	72	7	≤	≤	NUM
ejpam-5336	73	1	i	i	PRON
ejpam-5336	73	2	≤	≤	ADV
ejpam-5336	73	3	2	2	NUM
ejpam-5336	73	4	i+	i+	NUM
ejpam-5336	73	5	2	2	NUM
ejpam-5336	73	6	,	,	PUNCT
ejpam-5336	73	7	3	3	NUM
ejpam-5336	73	8	≤	≤	NUM
ejpam-5336	73	9	i	i	PRON
ejpam-5336	73	10	≤	≤	NOUN
ejpam-5336	73	11	m	m	VERB
ejpam-5336	73	12	and	and	CCONJ
ejpam-5336	73	13	f(vj	f(vj	PRON
ejpam-5336	73	14	)	)	PUNCT
ejpam-5336	74	1	=	=	PRON
ejpam-5336	74	2	{	{	PUNCT
ejpam-5336	74	3	m+	m+	NUM
ejpam-5336	74	4	j	j	PROPN
ejpam-5336	74	5	+	+	CCONJ
ejpam-5336	74	6	2	2	NUM
ejpam-5336	74	7	,	,	PUNCT
ejpam-5336	74	8	1	1	NUM
ejpam-5336	74	9	≤	≤	NUM
ejpam-5336	74	10	j	j	PROPN
ejpam-5336	74	11	≤	≤	ADJ
ejpam-5336	74	12	n−	n−	PROPN
ejpam-5336	74	13	1	1	NUM
ejpam-5336	74	14	4	4	NUM
ejpam-5336	74	15	,	,	PUNCT
ejpam-5336	74	16	j	j	PROPN
ejpam-5336	74	17	=	=	SYM
ejpam-5336	74	18	n	n	PROPN
ejpam-5336	74	19	.	.	PUNCT
ejpam-5336	75	1	since	since	SCONJ
ejpam-5336	75	2	f(u0	f(u0	ADJ
ejpam-5336	75	3	)	)	PUNCT
ejpam-5336	75	4	=	=	SYM
ejpam-5336	75	5	1	1	NUM
ejpam-5336	75	6	,	,	PUNCT
ejpam-5336	75	7	f(u0	f(u0	NOUN
ejpam-5336	75	8	)	)	PUNCT
ejpam-5336	75	9	is	be	AUX
ejpam-5336	75	10	relatively	relatively	ADV
ejpam-5336	75	11	prime	prime	ADJ
ejpam-5336	75	12	to	to	ADP
ejpam-5336	75	13	f(ui	f(ui	NOUN
ejpam-5336	75	14	)	)	PUNCT
ejpam-5336	75	15	for	for	ADP
ejpam-5336	75	16	all	all	DET
ejpam-5336	75	17	1	1	NUM
ejpam-5336	75	18	≤	≤	NUM
ejpam-5336	75	19	i	i	PRON
ejpam-5336	75	20	≤	≤	NUM
ejpam-5336	75	21	m.	m.	NOUN
ejpam-5336	75	22	also	also	ADV
ejpam-5336	75	23	,	,	PUNCT
ejpam-5336	75	24	(	(	PUNCT
ejpam-5336	75	25	f(u2	f(u2	NOUN
ejpam-5336	75	26	)	)	PUNCT
ejpam-5336	75	27	,	,	PUNCT
ejpam-5336	75	28	f(u3	f(u3	NOUN
ejpam-5336	75	29	)	)	PUNCT
ejpam-5336	75	30	)	)	PUNCT
ejpam-5336	76	1	=	=	PUNCT
ejpam-5336	76	2	(	(	PUNCT
ejpam-5336	76	3	3	3	NUM
ejpam-5336	76	4	,	,	PUNCT
ejpam-5336	76	5	5	5	NUM
ejpam-5336	76	6	)	)	PUNCT
ejpam-5336	76	7	=	=	SYM
ejpam-5336	76	8	1	1	NUM
ejpam-5336	76	9	,	,	PUNCT
ejpam-5336	76	10	(	(	PUNCT
ejpam-5336	76	11	f(u1	f(u1	NOUN
ejpam-5336	76	12	)	)	PUNCT
ejpam-5336	76	13	,	,	PUNCT
ejpam-5336	76	14	f(um	f(um	PROPN
ejpam-5336	76	15	)	)	PUNCT
ejpam-5336	76	16	)	)	PUNCT
ejpam-5336	77	1	=	=	PUNCT
ejpam-5336	77	2	(	(	PUNCT
ejpam-5336	77	3	2	2	NUM
ejpam-5336	77	4	,	,	PUNCT
ejpam-5336	77	5	n+	n+	X
ejpam-5336	77	6	2	2	NUM
ejpam-5336	77	7	)	)	PUNCT
ejpam-5336	77	8	=	=	SYM
ejpam-5336	77	9	1	1	NUM
ejpam-5336	77	10	,	,	PUNCT
ejpam-5336	77	11	because	because	SCONJ
ejpam-5336	77	12	m	m	PROPN
ejpam-5336	77	13	is	be	AUX
ejpam-5336	77	14	odd	odd	ADJ
ejpam-5336	77	15	.	.	PUNCT
ejpam-5336	78	1	now	now	ADV
ejpam-5336	78	2	,	,	PUNCT
ejpam-5336	78	3	(	(	PUNCT
ejpam-5336	78	4	f(vn−1	f(vn−1	PROPN
ejpam-5336	78	5	)	)	PUNCT
ejpam-5336	78	6	,	,	PUNCT
ejpam-5336	78	7	f(vn	f(vn	NOUN
ejpam-5336	78	8	)	)	PUNCT
ejpam-5336	78	9	)	)	PUNCT
ejpam-5336	78	10	=	=	PRON
ejpam-5336	78	11	(	(	PUNCT
ejpam-5336	78	12	m+	m+	NUM
ejpam-5336	78	13	n+	n+	NUM
ejpam-5336	78	14	1	1	NUM
ejpam-5336	78	15	,	,	PUNCT
ejpam-5336	78	16	4	4	NUM
ejpam-5336	78	17	)	)	PUNCT
ejpam-5336	78	18	=	=	SYM
ejpam-5336	78	19	1	1	NUM
ejpam-5336	78	20	,	,	PUNCT
ejpam-5336	78	21	because	because	SCONJ
ejpam-5336	78	22	m+	m+	NUM
ejpam-5336	78	23	n+	n+	SYM
ejpam-5336	78	24	1	1	NUM
ejpam-5336	78	25	is	be	AUX
ejpam-5336	78	26	odd	odd	ADJ
ejpam-5336	78	27	.	.	PUNCT
ejpam-5336	79	1	the	the	DET
ejpam-5336	79	2	labels	label	NOUN
ejpam-5336	79	3	assigned	assign	VERB
ejpam-5336	79	4	to	to	ADP
ejpam-5336	79	5	adjacent	adjacent	ADJ
ejpam-5336	79	6	vertices	vertex	NOUN
ejpam-5336	79	7	within	within	ADP
ejpam-5336	79	8	the	the	DET
ejpam-5336	79	9	graph	graph	NOUN
ejpam-5336	79	10	wm∪pn	wm∪pn	NOUN
ejpam-5336	79	11	exhibit	exhibit	VERB
ejpam-5336	79	12	a	a	DET
ejpam-5336	79	13	property	property	NOUN
ejpam-5336	79	14	of	of	ADP
ejpam-5336	79	15	being	be	AUX
ejpam-5336	79	16	mutually	mutually	ADV
ejpam-5336	79	17	prime	prime	ADJ
ejpam-5336	79	18	because	because	SCONJ
ejpam-5336	79	19	these	these	DET
ejpam-5336	79	20	labels	label	NOUN
ejpam-5336	79	21	are	be	AUX
ejpam-5336	79	22	two	two	NUM
ejpam-5336	79	23	consecutive	consecutive	ADJ
ejpam-5336	79	24	integers	integer	NOUN
ejpam-5336	79	25	.	.	PUNCT
ejpam-5336	80	1	so	so	ADV
ejpam-5336	80	2	f	f	PROPN
ejpam-5336	80	3	is	be	AUX
ejpam-5336	80	4	a	a	DET
ejpam-5336	80	5	pl	pl	PROPN
ejpam-5336	80	6	.	.	PUNCT
ejpam-5336	80	7	theorem	theorem	NOUN
ejpam-5336	80	8	4	4	NUM
ejpam-5336	80	9	.	.	PUNCT
ejpam-5336	81	1	the	the	DET
ejpam-5336	81	2	disjoint	disjoint	PROPN
ejpam-5336	81	3	union	union	NOUN
ejpam-5336	81	4	of	of	ADP
ejpam-5336	81	5	two	two	NUM
ejpam-5336	81	6	wheels	wheel	NOUN
ejpam-5336	81	7	is	be	AUX
ejpam-5336	81	8	not	not	PART
ejpam-5336	81	9	a	a	DET
ejpam-5336	81	10	pg	pg	NOUN
ejpam-5336	81	11	.	.	PUNCT
ejpam-5336	82	1	proof	proof	NOUN
ejpam-5336	82	2	.	.	PUNCT
ejpam-5336	83	1	let	let	VERB
ejpam-5336	83	2	wn	wn	PROPN
ejpam-5336	83	3	and	and	CCONJ
ejpam-5336	83	4	wm	wm	AUX
ejpam-5336	83	5	be	be	AUX
ejpam-5336	83	6	any	any	DET
ejpam-5336	83	7	two	two	NUM
ejpam-5336	83	8	wheels	wheel	NOUN
ejpam-5336	83	9	.	.	PUNCT
ejpam-5336	84	1	then	then	ADV
ejpam-5336	84	2	α(wn	α(wn	X
ejpam-5336	84	3	∪wm	∪wm	PROPN
ejpam-5336	84	4	)	)	PUNCT
ejpam-5336	84	5	=	=	SYM
ejpam-5336	84	6	α(wn	α(wn	NOUN
ejpam-5336	84	7	)	)	PUNCT
ejpam-5336	85	1	+	+	CCONJ
ejpam-5336	85	2	α	α	PROPN
ejpam-5336	85	3	(	(	PUNCT
ejpam-5336	85	4	wm	wm	PROPN
ejpam-5336	85	5	)	)	PUNCT
ejpam-5336	85	6	=	=	PUNCT
ejpam-5336	86	1	[	[	X
ejpam-5336	86	2	n	n	X
ejpam-5336	86	3	2	2	NUM
ejpam-5336	86	4	]	]	PUNCT
ejpam-5336	87	1	+	+	CCONJ
ejpam-5336	87	2	[	[	X
ejpam-5336	87	3	m	m	VERB
ejpam-5336	87	4	2	2	NUM
ejpam-5336	87	5	]	]	PUNCT
ejpam-5336	87	6	≤	≤	X
ejpam-5336	87	7	[	[	PUNCT
ejpam-5336	87	8	n+m	n+m	NUM
ejpam-5336	87	9	2	2	NUM
ejpam-5336	87	10	]	]	PUNCT
ejpam-5336	87	11	<	<	X
ejpam-5336	87	12	[	[	PUNCT
ejpam-5336	87	13	|wn	|wn	NUM
ejpam-5336	87	14	∪wm|	∪wm|	PROPN
ejpam-5336	87	15	2	2	NUM
ejpam-5336	87	16	]	]	PUNCT
ejpam-5336	87	17	=	=	PUNCT
ejpam-5336	88	1	[	[	PUNCT
ejpam-5336	88	2	n+m+	n+m+	X
ejpam-5336	88	3	2	2	NUM
ejpam-5336	88	4	2	2	NUM
ejpam-5336	88	5	]	]	PUNCT
ejpam-5336	88	6	=	=	PUNCT
ejpam-5336	88	7	[	[	PUNCT
ejpam-5336	88	8	n+m	n+m	NUM
ejpam-5336	88	9	2	2	NUM
ejpam-5336	88	10	]	]	PUNCT
ejpam-5336	89	1	+	+	CCONJ
ejpam-5336	89	2	1	1	X
ejpam-5336	89	3	.	.	PUNCT
ejpam-5336	89	4	by	by	ADP
ejpam-5336	89	5	lemma	lemma	PROPN
ejpam-5336	89	6	1	1	NUM
ejpam-5336	89	7	,	,	PUNCT
ejpam-5336	89	8	we	we	PRON
ejpam-5336	89	9	get	get	VERB
ejpam-5336	89	10	wn	wn	PROPN
ejpam-5336	89	11	∪wm	∪wm	PROPN
ejpam-5336	89	12	is	be	AUX
ejpam-5336	89	13	not	not	PART
ejpam-5336	89	14	a	a	DET
ejpam-5336	89	15	pg	pg	NOUN
ejpam-5336	89	16	.	.	PUNCT
ejpam-5336	90	1	patel	patel	PROPN
ejpam-5336	90	2	et	et	PROPN
ejpam-5336	90	3	.	.	PUNCT
ejpam-5336	91	1	al	al	PROPN
ejpam-5336	91	2	.	.	PUNCT
ejpam-5336	92	1	in	in	ADP
ejpam-5336	92	2	[	[	X
ejpam-5336	92	3	9	9	NUM
ejpam-5336	92	4	]	]	PUNCT
ejpam-5336	92	5	proved	prove	VERB
ejpam-5336	92	6	that	that	SCONJ
ejpam-5336	92	7	“	"	PUNCT
ejpam-5336	92	8	the	the	DET
ejpam-5336	92	9	disjoint	disjoint	PROPN
ejpam-5336	92	10	union	union	NOUN
ejpam-5336	92	11	of	of	ADP
ejpam-5336	92	12	an	an	DET
ejpam-5336	92	13	even	even	ADJ
ejpam-5336	92	14	wheel	wheel	NOUN
ejpam-5336	92	15	and	and	CCONJ
ejpam-5336	92	16	an	an	DET
ejpam-5336	92	17	even	even	ADJ
ejpam-5336	92	18	cycle	cycle	NOUN
ejpam-5336	92	19	is	be	AUX
ejpam-5336	92	20	a	a	DET
ejpam-5336	92	21	pg	pg	NOUN
ejpam-5336	92	22	.	.	PUNCT
ejpam-5336	92	23	”	"	PUNCT
ejpam-5336	93	1	in	in	ADP
ejpam-5336	93	2	theorem	theorem	NOUN
ejpam-5336	93	3	5	5	NUM
ejpam-5336	93	4	,	,	PUNCT
ejpam-5336	93	5	we	we	PRON
ejpam-5336	93	6	prove	prove	VERB
ejpam-5336	93	7	that	that	SCONJ
ejpam-5336	93	8	c2n	c2n	NOUN
ejpam-5336	93	9	∪	∪	ADP
ejpam-5336	93	10	c2n	c2n	NOUN
ejpam-5336	93	11	∪w2	∪w2	NOUN
ejpam-5336	93	12	m	m	VERB
ejpam-5336	93	13	and	and	CCONJ
ejpam-5336	93	14	c2n	c2n	NOUN
ejpam-5336	93	15	∪	∪	ADP
ejpam-5336	93	16	c2n	c2n	NOUN
ejpam-5336	93	17	∪	∪	ADP
ejpam-5336	93	18	c2n	c2n	NOUN
ejpam-5336	93	19	∪w2	∪w2	NOUN
ejpam-5336	93	20	m	m	VERB
ejpam-5336	93	21	are	be	AUX
ejpam-5336	93	22	pgs	pgs	ADJ
ejpam-5336	93	23	.	.	PUNCT
ejpam-5336	94	1	theorem	theorem	NOUN
ejpam-5336	94	2	5	5	NUM
ejpam-5336	94	3	.	.	PUNCT
ejpam-5336	95	1	c2n	c2n	NOUN
ejpam-5336	95	2	∪	∪	ADP
ejpam-5336	95	3	c2n	c2n	NOUN
ejpam-5336	95	4	∪w2	∪w2	NOUN
ejpam-5336	95	5	m	m	VERB
ejpam-5336	95	6	and	and	CCONJ
ejpam-5336	95	7	c2n	c2n	NOUN
ejpam-5336	95	8	∪	∪	ADP
ejpam-5336	95	9	c2n	c2n	NOUN
ejpam-5336	95	10	∪	∪	ADP
ejpam-5336	95	11	c2n	c2n	NOUN
ejpam-5336	95	12	∪w2	∪w2	NOUN
ejpam-5336	95	13	m	m	VERB
ejpam-5336	95	14	are	be	AUX
ejpam-5336	95	15	pgs	pgs	ADJ
ejpam-5336	95	16	for	for	ADP
ejpam-5336	95	17	all	all	DET
ejpam-5336	95	18	n	n	CCONJ
ejpam-5336	95	19	,	,	PUNCT
ejpam-5336	95	20	m.	m.	NOUN
ejpam-5336	95	21	proof	proof	NOUN
ejpam-5336	95	22	.	.	PUNCT
ejpam-5336	96	1	let	let	VERB
ejpam-5336	96	2	u1	u1	NOUN
ejpam-5336	96	3	,	,	PUNCT
ejpam-5336	96	4	u2	u2	PROPN
ejpam-5336	96	5	,	,	PUNCT
ejpam-5336	96	6	...	...	PUNCT
ejpam-5336	96	7	,	,	PUNCT
ejpam-5336	96	8	u2n	u2n	PUNCT
ejpam-5336	96	9	be	be	AUX
ejpam-5336	96	10	the	the	DET
ejpam-5336	96	11	vertices	vertex	NOUN
ejpam-5336	96	12	of	of	ADP
ejpam-5336	96	13	the	the	DET
ejpam-5336	96	14	first	first	ADJ
ejpam-5336	96	15	cycle	cycle	NOUN
ejpam-5336	96	16	,	,	PUNCT
ejpam-5336	96	17	u2n+1	u2n+1	PROPN
ejpam-5336	96	18	,	,	PUNCT
ejpam-5336	96	19	u2n+2	u2n+2	PROPN
ejpam-5336	96	20	,	,	PUNCT
ejpam-5336	96	21	...	...	PUNCT
ejpam-5336	96	22	,	,	PUNCT
ejpam-5336	96	23	u4n	u4n	PROPN
ejpam-5336	96	24	be	be	VERB
ejpam-5336	96	25	the	the	DET
ejpam-5336	96	26	vertices	vertex	NOUN
ejpam-5336	96	27	of	of	ADP
ejpam-5336	96	28	the	the	DET
ejpam-5336	96	29	second	second	ADJ
ejpam-5336	96	30	cycle	cycle	NOUN
ejpam-5336	96	31	,	,	PUNCT
ejpam-5336	96	32	u4n+1	u4n+1	ADJ
ejpam-5336	96	33	,	,	PUNCT
ejpam-5336	96	34	u4n+2	u4n+2	PROPN
ejpam-5336	96	35	,	,	PUNCT
ejpam-5336	96	36	...	...	PUNCT
ejpam-5336	96	37	,	,	PUNCT
ejpam-5336	96	38	u6n	u6n	PROPN
ejpam-5336	96	39	be	be	AUX
ejpam-5336	96	40	the	the	DET
ejpam-5336	96	41	vertices	vertex	NOUN
ejpam-5336	96	42	of	of	ADP
ejpam-5336	96	43	the	the	DET
ejpam-5336	96	44	third	third	ADJ
ejpam-5336	96	45	cycle	cycle	NOUN
ejpam-5336	96	46	,	,	PUNCT
ejpam-5336	96	47	v0	v0	NOUN
ejpam-5336	96	48	be	be	AUX
ejpam-5336	96	49	the	the	DET
ejpam-5336	96	50	apex	apex	NOUN
ejpam-5336	96	51	vertex	vertex	NOUN
ejpam-5336	96	52	of	of	ADP
ejpam-5336	96	53	wn	wn	PROPN
ejpam-5336	96	54	and	and	CCONJ
ejpam-5336	96	55	v1	v1	PROPN
ejpam-5336	96	56	,	,	PUNCT
ejpam-5336	96	57	v2	v2	PROPN
ejpam-5336	96	58	,	,	PUNCT
ejpam-5336	96	59	...	...	PUNCT
ejpam-5336	96	60	,	,	PUNCT
ejpam-5336	96	61	v2	v2	PROPN
ejpam-5336	96	62	m	m	VERB
ejpam-5336	96	63	be	be	VERB
ejpam-5336	96	64	the	the	DET
ejpam-5336	96	65	consecutive	consecutive	ADJ
ejpam-5336	96	66	rim	rim	NOUN
ejpam-5336	96	67	vertices	vertex	NOUN
ejpam-5336	96	68	of	of	ADP
ejpam-5336	96	69	w2	w2	NOUN
ejpam-5336	96	70	m.	m.	NOUN
ejpam-5336	96	71	(	(	PUNCT
ejpam-5336	96	72	i	i	NOUN
ejpam-5336	96	73	)	)	PUNCT
ejpam-5336	96	74	to	to	PART
ejpam-5336	96	75	show	show	VERB
ejpam-5336	96	76	that	that	SCONJ
ejpam-5336	96	77	c2n	c2n	NOUN
ejpam-5336	96	78	∪	∪	ADP
ejpam-5336	96	79	c2n	c2n	NOUN
ejpam-5336	96	80	∪w2	∪w2	NOUN
ejpam-5336	96	81	m	m	VERB
ejpam-5336	96	82	is	be	AUX
ejpam-5336	96	83	a	a	DET
ejpam-5336	96	84	pg	pg	NOUN
ejpam-5336	96	85	.	.	PUNCT
ejpam-5336	97	1	we	we	PRON
ejpam-5336	97	2	have	have	VERB
ejpam-5336	97	3	the	the	DET
ejpam-5336	97	4	following	follow	VERB
ejpam-5336	97	5	two	two	NUM
ejpam-5336	97	6	cases	case	NOUN
ejpam-5336	97	7	:	:	PUNCT
ejpam-5336	97	8	(	(	PUNCT
ejpam-5336	97	9	a	a	X
ejpam-5336	97	10	)	)	PUNCT
ejpam-5336	97	11	i.	i.	NOUN
ejpam-5336	97	12	if	if	SCONJ
ejpam-5336	97	13	3	3	NUM
ejpam-5336	97	14	does	do	AUX
ejpam-5336	97	15	not	not	PART
ejpam-5336	97	16	divide	divide	VERB
ejpam-5336	97	17	n+	n+	ADP
ejpam-5336	97	18	1	1	NUM
ejpam-5336	97	19	,	,	PUNCT
ejpam-5336	97	20	define	define	VERB
ejpam-5336	97	21	f	f	X
ejpam-5336	97	22	:	:	PUNCT
ejpam-5336	97	23	v	v	X
ejpam-5336	97	24	(	(	PUNCT
ejpam-5336	97	25	c2n	c2n	NOUN
ejpam-5336	97	26	∪	∪	ADP
ejpam-5336	97	27	c2n	c2n	NOUN
ejpam-5336	97	28	∪w2	∪w2	NOUN
ejpam-5336	97	29	m	m	NOUN
ejpam-5336	97	30	)	)	PUNCT
ejpam-5336	97	31	−→	−→	ADV
ejpam-5336	97	32	{	{	PUNCT
ejpam-5336	97	33	1	1	NUM
ejpam-5336	97	34	,	,	PUNCT
ejpam-5336	97	35	2	2	NUM
ejpam-5336	97	36	,	,	PUNCT
ejpam-5336	97	37	...	...	PUNCT
ejpam-5336	97	38	,	,	PUNCT
ejpam-5336	97	39	4n+	4n+	NUM
ejpam-5336	97	40	2m+	2m+	NUM
ejpam-5336	97	41	1	1	NUM
ejpam-5336	97	42	}	}	PUNCT
ejpam-5336	97	43	as	as	SCONJ
ejpam-5336	97	44	follows	follow	VERB
ejpam-5336	97	45	:	:	PUNCT
ejpam-5336	97	46	f	f	PROPN
ejpam-5336	97	47	(	(	PUNCT
ejpam-5336	97	48	ui	ui	PROPN
ejpam-5336	97	49	)	)	PUNCT
ejpam-5336	97	50	=	=	NOUN
ejpam-5336	98	1	i+	i+	NUM
ejpam-5336	98	2	2	2	NUM
ejpam-5336	98	3	,	,	PUNCT
ejpam-5336	98	4	for	for	ADP
ejpam-5336	98	5	all	all	DET
ejpam-5336	98	6	1	1	NUM
ejpam-5336	98	7	≤	≤	NUM
ejpam-5336	98	8	i	i	PRON
ejpam-5336	98	9	≤	≤	NOUN
ejpam-5336	98	10	4n	4n	NOUN
ejpam-5336	98	11	,	,	PUNCT
ejpam-5336	98	12	o.	o.	PROPN
ejpam-5336	98	13	a.	a.	PROPN
ejpam-5336	98	14	abughneim	abughneim	PROPN
ejpam-5336	98	15	,	,	PUNCT
ejpam-5336	98	16	b.	b.	PROPN
ejpam-5336	98	17	abughazaleh	abughazaleh	PROPN
ejpam-5336	98	18	/	/	SYM
ejpam-5336	98	19	eur	eur	PROPN
ejpam-5336	98	20	.	.	PUNCT
ejpam-5336	99	1	j.	j.	PROPN
ejpam-5336	99	2	pure	pure	PROPN
ejpam-5336	99	3	appl	appl	PROPN
ejpam-5336	99	4	.	.	PROPN
ejpam-5336	99	5	math	math	PROPN
ejpam-5336	99	6	,	,	PUNCT
ejpam-5336	99	7	17	17	NUM
ejpam-5336	99	8	(	(	PUNCT
ejpam-5336	99	9	4	4	NUM
ejpam-5336	99	10	)	)	PUNCT
ejpam-5336	99	11	(	(	PUNCT
ejpam-5336	99	12	2024	2024	NUM
ejpam-5336	99	13	)	)	PUNCT
ejpam-5336	99	14	,	,	PUNCT
ejpam-5336	99	15	3557	3557	NUM
ejpam-5336	99	16	-	-	SYM
ejpam-5336	99	17	3566	3566	NUM
ejpam-5336	99	18	3560	3560	NUM
ejpam-5336	99	19	f	f	PROPN
ejpam-5336	99	20	(	(	PUNCT
ejpam-5336	99	21	vj	vj	PROPN
ejpam-5336	99	22	)	)	PUNCT
ejpam-5336	99	23	=	=	SYM
ejpam-5336	99	24	j	j	PROPN
ejpam-5336	100	1	+	+	CCONJ
ejpam-5336	100	2	1	1	NUM
ejpam-5336	100	3	for	for	ADP
ejpam-5336	100	4	j	j	PROPN
ejpam-5336	100	5	=	=	SYM
ejpam-5336	100	6	0	0	NUM
ejpam-5336	100	7	and	and	CCONJ
ejpam-5336	100	8	1	1	NUM
ejpam-5336	100	9	,	,	PUNCT
ejpam-5336	100	10	f	f	PROPN
ejpam-5336	100	11	(	(	PUNCT
ejpam-5336	100	12	vj	vj	PROPN
ejpam-5336	100	13	)	)	PUNCT
ejpam-5336	100	14	=	=	SYM
ejpam-5336	101	1	4n+	4n+	NUM
ejpam-5336	101	2	j	j	NOUN
ejpam-5336	102	1	+	+	NOUN
ejpam-5336	102	2	1	1	NUM
ejpam-5336	102	3	,	,	PUNCT
ejpam-5336	102	4	for	for	ADP
ejpam-5336	102	5	all	all	DET
ejpam-5336	102	6	2	2	NUM
ejpam-5336	102	7	≤	≤	NUM
ejpam-5336	102	8	j	j	PROPN
ejpam-5336	102	9	≤	≤	ADV
ejpam-5336	102	10	2	2	NUM
ejpam-5336	102	11	m.	m.	NOUN
ejpam-5336	102	12	we	we	PRON
ejpam-5336	102	13	get	get	VERB
ejpam-5336	102	14	(	(	PUNCT
ejpam-5336	102	15	f(u1	f(u1	NOUN
ejpam-5336	102	16	)	)	PUNCT
ejpam-5336	102	17	,	,	PUNCT
ejpam-5336	102	18	f(u2n	f(u2n	NOUN
ejpam-5336	102	19	)	)	PUNCT
ejpam-5336	102	20	)	)	PUNCT
ejpam-5336	103	1	=	=	PUNCT
ejpam-5336	103	2	(	(	PUNCT
ejpam-5336	103	3	3	3	NUM
ejpam-5336	103	4	,	,	PUNCT
ejpam-5336	103	5	2n+	2n+	NUM
ejpam-5336	103	6	2	2	NUM
ejpam-5336	103	7	)	)	PUNCT
ejpam-5336	103	8	=	=	SYM
ejpam-5336	103	9	1	1	NUM
ejpam-5336	103	10	,	,	PUNCT
ejpam-5336	103	11	because	because	SCONJ
ejpam-5336	103	12	3	3	NUM
ejpam-5336	103	13	does	do	AUX
ejpam-5336	103	14	not	not	PART
ejpam-5336	103	15	divide	divide	VERB
ejpam-5336	103	16	n+	n+	ADP
ejpam-5336	103	17	1	1	X
ejpam-5336	103	18	.	.	PUNCT
ejpam-5336	103	19	also	also	ADV
ejpam-5336	103	20	,	,	PUNCT
ejpam-5336	103	21	(	(	PUNCT
ejpam-5336	103	22	f(u2n+1	f(u2n+1	NOUN
ejpam-5336	103	23	)	)	PUNCT
ejpam-5336	103	24	,	,	PUNCT
ejpam-5336	103	25	f(u4n	f(u4n	NOUN
ejpam-5336	103	26	)	)	PUNCT
ejpam-5336	103	27	)	)	PUNCT
ejpam-5336	103	28	=	=	PUNCT
ejpam-5336	104	1	(	(	PUNCT
ejpam-5336	104	2	2n+	2n+	NUM
ejpam-5336	104	3	3	3	NUM
ejpam-5336	104	4	,	,	PUNCT
ejpam-5336	104	5	4n+	4n+	NUM
ejpam-5336	104	6	2	2	NUM
ejpam-5336	104	7	)	)	PUNCT
ejpam-5336	104	8	=	=	SYM
ejpam-5336	104	9	1	1	NUM
ejpam-5336	104	10	because	because	SCONJ
ejpam-5336	104	11	if	if	SCONJ
ejpam-5336	104	12	d	d	X
ejpam-5336	104	13	=	=	SYM
ejpam-5336	104	14	(	(	PUNCT
ejpam-5336	104	15	f(u2n+1	f(u2n+1	PROPN
ejpam-5336	104	16	)	)	PUNCT
ejpam-5336	104	17	,	,	PUNCT
ejpam-5336	104	18	f(u4n	f(u4n	NOUN
ejpam-5336	104	19	)	)	PUNCT
ejpam-5336	104	20	)	)	PUNCT
ejpam-5336	104	21	,	,	PUNCT
ejpam-5336	104	22	then	then	ADV
ejpam-5336	104	23	d	d	X
ejpam-5336	104	24	divids	divid	VERB
ejpam-5336	104	25	2n+3	2n+3	PROPN
ejpam-5336	104	26	and	and	CCONJ
ejpam-5336	104	27	hence	hence	ADV
ejpam-5336	104	28	d	d	PRON
ejpam-5336	104	29	is	be	AUX
ejpam-5336	104	30	odd	odd	ADJ
ejpam-5336	104	31	and	and	CCONJ
ejpam-5336	104	32	d	d	NOUN
ejpam-5336	104	33	divids	divids	ADJ
ejpam-5336	105	1	2(2n+3)−(4n+2	2(2n+3)−(4n+2	NUM
ejpam-5336	105	2	)	)	PUNCT
ejpam-5336	105	3	=	=	SYM
ejpam-5336	105	4	4	4	X
ejpam-5336	105	5	.	.	PUNCT
ejpam-5336	105	6	thus	thus	ADV
ejpam-5336	105	7	d	d	X
ejpam-5336	105	8	=	=	SYM
ejpam-5336	105	9	1	1	X
ejpam-5336	105	10	.	.	PUNCT
ejpam-5336	105	11	clearly	clearly	ADV
ejpam-5336	105	12	,	,	PUNCT
ejpam-5336	105	13	any	any	DET
ejpam-5336	105	14	other	other	ADJ
ejpam-5336	105	15	adjacent	adjacent	ADJ
ejpam-5336	105	16	vertices	vertex	NOUN
ejpam-5336	105	17	have	have	VERB
ejpam-5336	105	18	relatively	relatively	ADV
ejpam-5336	105	19	prime	prime	ADJ
ejpam-5336	105	20	labels	label	NOUN
ejpam-5336	105	21	.	.	PUNCT
ejpam-5336	106	1	so	so	ADV
ejpam-5336	106	2	,	,	PUNCT
ejpam-5336	106	3	f	f	PROPN
ejpam-5336	106	4	is	be	AUX
ejpam-5336	106	5	a	a	DET
ejpam-5336	106	6	pl	pl	PROPN
ejpam-5336	106	7	.	.	PROPN
ejpam-5336	106	8	ii	ii	PROPN
ejpam-5336	106	9	.	.	PUNCT
ejpam-5336	107	1	if	if	SCONJ
ejpam-5336	107	2	3	3	NUM
ejpam-5336	107	3	divides	divide	NOUN
ejpam-5336	107	4	n+	n+	PUNCT
ejpam-5336	107	5	1	1	NUM
ejpam-5336	107	6	,	,	PUNCT
ejpam-5336	107	7	define	define	VERB
ejpam-5336	107	8	f	f	X
ejpam-5336	107	9	:	:	PUNCT
ejpam-5336	107	10	v	v	X
ejpam-5336	107	11	(	(	PUNCT
ejpam-5336	107	12	c2n	c2n	NOUN
ejpam-5336	107	13	∪	∪	ADP
ejpam-5336	107	14	c2n	c2n	NOUN
ejpam-5336	107	15	∪w2	∪w2	NOUN
ejpam-5336	107	16	m	m	NOUN
ejpam-5336	107	17	)	)	PUNCT
ejpam-5336	107	18	−→	−→	ADV
ejpam-5336	107	19	{	{	PUNCT
ejpam-5336	107	20	1	1	NUM
ejpam-5336	107	21	,	,	PUNCT
ejpam-5336	107	22	2	2	NUM
ejpam-5336	107	23	,	,	PUNCT
ejpam-5336	107	24	...	...	PUNCT
ejpam-5336	107	25	,	,	PUNCT
ejpam-5336	107	26	4n+	4n+	NUM
ejpam-5336	107	27	2m+	2m+	NUM
ejpam-5336	107	28	1	1	NUM
ejpam-5336	107	29	}	}	PUNCT
ejpam-5336	107	30	as	as	SCONJ
ejpam-5336	107	31	follows	follow	VERB
ejpam-5336	107	32	:	:	PUNCT
ejpam-5336	108	1	f	f	PROPN
ejpam-5336	108	2	(	(	PUNCT
ejpam-5336	108	3	ui	ui	PROPN
ejpam-5336	108	4	)	)	PUNCT
ejpam-5336	108	5	=	=	NOUN
ejpam-5336	108	6	i+	i+	NUM
ejpam-5336	108	7	3	3	NUM
ejpam-5336	108	8	,	,	PUNCT
ejpam-5336	108	9	for	for	ADP
ejpam-5336	108	10	all	all	DET
ejpam-5336	108	11	1	1	NUM
ejpam-5336	108	12	≤	≤	NUM
ejpam-5336	108	13	i	i	PRON
ejpam-5336	108	14	≤	≤	NOUN
ejpam-5336	109	1	4n−	4n−	NUM
ejpam-5336	109	2	1	1	NUM
ejpam-5336	109	3	,	,	PUNCT
ejpam-5336	109	4	f	f	PROPN
ejpam-5336	109	5	(	(	PUNCT
ejpam-5336	109	6	u4n	u4n	PROPN
ejpam-5336	109	7	)	)	PUNCT
ejpam-5336	109	8	=	=	SYM
ejpam-5336	109	9	3	3	NUM
ejpam-5336	109	10	,	,	PUNCT
ejpam-5336	109	11	f	f	PROPN
ejpam-5336	109	12	(	(	PUNCT
ejpam-5336	109	13	vj	vj	PROPN
ejpam-5336	109	14	)	)	PUNCT
ejpam-5336	109	15	=	=	SYM
ejpam-5336	109	16	j	j	PROPN
ejpam-5336	110	1	+	+	CCONJ
ejpam-5336	110	2	1	1	NUM
ejpam-5336	110	3	for	for	ADP
ejpam-5336	110	4	j	j	PROPN
ejpam-5336	110	5	=	=	SYM
ejpam-5336	110	6	0	0	NUM
ejpam-5336	110	7	and	and	CCONJ
ejpam-5336	110	8	1	1	NUM
ejpam-5336	110	9	,	,	PUNCT
ejpam-5336	110	10	f	f	PROPN
ejpam-5336	110	11	(	(	PUNCT
ejpam-5336	110	12	vj	vj	PROPN
ejpam-5336	110	13	)	)	PUNCT
ejpam-5336	110	14	=	=	SYM
ejpam-5336	111	1	4n+	4n+	NUM
ejpam-5336	111	2	j	j	NOUN
ejpam-5336	112	1	+	+	NOUN
ejpam-5336	112	2	1	1	NUM
ejpam-5336	112	3	,	,	PUNCT
ejpam-5336	112	4	for	for	ADP
ejpam-5336	112	5	all	all	DET
ejpam-5336	112	6	2	2	NUM
ejpam-5336	112	7	≤	≤	NUM
ejpam-5336	112	8	j	j	PROPN
ejpam-5336	112	9	≤	≤	ADV
ejpam-5336	112	10	2	2	NUM
ejpam-5336	112	11	m.	m.	NOUN
ejpam-5336	112	12	since	since	SCONJ
ejpam-5336	112	13	3	3	NUM
ejpam-5336	112	14	divides	divide	NOUN
ejpam-5336	112	15	n+1	n+1	PROPN
ejpam-5336	112	16	,	,	PUNCT
ejpam-5336	112	17	3	3	NUM
ejpam-5336	112	18	does	do	AUX
ejpam-5336	112	19	not	not	PART
ejpam-5336	112	20	divide	divide	VERB
ejpam-5336	112	21	2n+4	2n+4	PROPN
ejpam-5336	112	22	and	and	CCONJ
ejpam-5336	112	23	4n+2	4n+2	PROPN
ejpam-5336	112	24	.	.	PUNCT
ejpam-5336	113	1	so	so	ADV
ejpam-5336	113	2	,	,	PUNCT
ejpam-5336	113	3	(	(	PUNCT
ejpam-5336	113	4	f(u2n+1	f(u2n+1	NOUN
ejpam-5336	113	5	)	)	PUNCT
ejpam-5336	113	6	,	,	PUNCT
ejpam-5336	113	7	f(u4n	f(u4n	NOUN
ejpam-5336	113	8	)	)	PUNCT
ejpam-5336	113	9	)	)	PUNCT
ejpam-5336	114	1	=	=	PUNCT
ejpam-5336	114	2	(	(	PUNCT
ejpam-5336	114	3	2n+	2n+	NUM
ejpam-5336	114	4	4	4	NUM
ejpam-5336	114	5	,	,	PUNCT
ejpam-5336	114	6	3	3	X
ejpam-5336	114	7	)	)	PUNCT
ejpam-5336	114	8	=	=	SYM
ejpam-5336	114	9	1	1	NUM
ejpam-5336	114	10	and	and	CCONJ
ejpam-5336	114	11	(	(	PUNCT
ejpam-5336	114	12	f(u4n−1	f(u4n−1	PROPN
ejpam-5336	114	13	)	)	PUNCT
ejpam-5336	114	14	,	,	PUNCT
ejpam-5336	114	15	f(u4n	f(u4n	NOUN
ejpam-5336	114	16	)	)	PUNCT
ejpam-5336	114	17	)	)	PUNCT
ejpam-5336	115	1	=	=	PRON
ejpam-5336	115	2	(	(	PUNCT
ejpam-5336	115	3	4n+	4n+	NUM
ejpam-5336	115	4	2	2	NUM
ejpam-5336	115	5	,	,	PUNCT
ejpam-5336	115	6	3	3	NUM
ejpam-5336	115	7	)	)	PUNCT
ejpam-5336	115	8	=	=	SYM
ejpam-5336	115	9	1	1	X
ejpam-5336	115	10	.	.	PUNCT
ejpam-5336	116	1	it	it	PRON
ejpam-5336	116	2	is	be	AUX
ejpam-5336	116	3	clear	clear	ADJ
ejpam-5336	116	4	that	that	SCONJ
ejpam-5336	116	5	all	all	DET
ejpam-5336	116	6	other	other	ADJ
ejpam-5336	116	7	adjacent	adjacent	ADJ
ejpam-5336	116	8	vertices	vertex	NOUN
ejpam-5336	116	9	have	have	VERB
ejpam-5336	116	10	relatively	relatively	ADV
ejpam-5336	116	11	prime	prime	ADJ
ejpam-5336	116	12	labels	label	NOUN
ejpam-5336	116	13	.	.	PUNCT
ejpam-5336	117	1	therefore	therefore	ADV
ejpam-5336	117	2	,	,	PUNCT
ejpam-5336	117	3	f	f	PROPN
ejpam-5336	117	4	is	be	AUX
ejpam-5336	117	5	a	a	DET
ejpam-5336	117	6	pl	pl	NOUN
ejpam-5336	117	7	.	.	PUNCT
ejpam-5336	117	8	(	(	PUNCT
ejpam-5336	117	9	ii	ii	NOUN
ejpam-5336	117	10	)	)	PUNCT
ejpam-5336	117	11	to	to	PART
ejpam-5336	117	12	show	show	VERB
ejpam-5336	117	13	that	that	SCONJ
ejpam-5336	117	14	c2n	c2n	NOUN
ejpam-5336	117	15	∪	∪	VERB
ejpam-5336	117	16	c2n	c2n	NOUN
ejpam-5336	117	17	∪	∪	ADP
ejpam-5336	117	18	c2n	c2n	NOUN
ejpam-5336	117	19	∪w2	∪w2	NOUN
ejpam-5336	117	20	m	m	VERB
ejpam-5336	117	21	is	be	AUX
ejpam-5336	117	22	a	a	DET
ejpam-5336	117	23	pg	pg	NOUN
ejpam-5336	117	24	.	.	PUNCT
ejpam-5336	118	1	we	we	PRON
ejpam-5336	118	2	have	have	VERB
ejpam-5336	118	3	the	the	DET
ejpam-5336	118	4	following	follow	VERB
ejpam-5336	118	5	two	two	NUM
ejpam-5336	118	6	cases	case	NOUN
ejpam-5336	118	7	:	:	PUNCT
ejpam-5336	118	8	(	(	PUNCT
ejpam-5336	118	9	a	a	X
ejpam-5336	118	10	)	)	PUNCT
ejpam-5336	118	11	if	if	SCONJ
ejpam-5336	118	12	3	3	NUM
ejpam-5336	118	13	does	do	AUX
ejpam-5336	118	14	not	not	PART
ejpam-5336	118	15	divide	divide	VERB
ejpam-5336	118	16	4n+	4n+	NUM
ejpam-5336	118	17	1	1	NUM
ejpam-5336	118	18	,	,	PUNCT
ejpam-5336	118	19	define	define	VERB
ejpam-5336	118	20	f	f	X
ejpam-5336	118	21	:	:	PUNCT
ejpam-5336	118	22	v	v	X
ejpam-5336	118	23	(	(	PUNCT
ejpam-5336	118	24	c2n	c2n	NOUN
ejpam-5336	118	25	∪	∪	NOUN
ejpam-5336	118	26	c2n	c2n	NOUN
ejpam-5336	118	27	∪	∪	ADP
ejpam-5336	118	28	c2n	c2n	NOUN
ejpam-5336	118	29	∪w2	∪w2	NOUN
ejpam-5336	118	30	m	m	NOUN
ejpam-5336	118	31	)	)	PUNCT
ejpam-5336	118	32	−→	−→	ADV
ejpam-5336	118	33	{	{	PUNCT
ejpam-5336	118	34	1	1	NUM
ejpam-5336	118	35	,	,	PUNCT
ejpam-5336	118	36	2	2	NUM
ejpam-5336	118	37	,	,	PUNCT
ejpam-5336	118	38	...	...	PUNCT
ejpam-5336	118	39	,	,	PUNCT
ejpam-5336	118	40	6n+	6n+	NUM
ejpam-5336	118	41	2m+	2m+	NUM
ejpam-5336	118	42	1	1	NUM
ejpam-5336	118	43	}	}	PUNCT
ejpam-5336	118	44	as	as	SCONJ
ejpam-5336	118	45	follows	follow	VERB
ejpam-5336	118	46	f	f	PROPN
ejpam-5336	118	47	(	(	PUNCT
ejpam-5336	118	48	ui	ui	PROPN
ejpam-5336	118	49	)	)	PUNCT
ejpam-5336	118	50	=	=	SYM
ejpam-5336	119	1	6n+	6n+	NUM
ejpam-5336	119	2	i	i	PRON
ejpam-5336	119	3	for	for	ADP
ejpam-5336	119	4	i	i	PRON
ejpam-5336	119	5	=	=	NOUN
ejpam-5336	119	6	1	1	NUM
ejpam-5336	119	7	,	,	PUNCT
ejpam-5336	119	8	2	2	NUM
ejpam-5336	119	9	,	,	PUNCT
ejpam-5336	119	10	f	f	PROPN
ejpam-5336	119	11	(	(	PUNCT
ejpam-5336	119	12	ui	ui	PROPN
ejpam-5336	119	13	)	)	PUNCT
ejpam-5336	119	14	=	=	VERB
ejpam-5336	120	1	i	i	PRON
ejpam-5336	120	2	for	for	ADP
ejpam-5336	120	3	all	all	DET
ejpam-5336	120	4	3	3	NUM
ejpam-5336	120	5	≤	≤	NUM
ejpam-5336	120	6	i	i	PRON
ejpam-5336	120	7	≤	≤	ADJ
ejpam-5336	120	8	6n	6n	PROPN
ejpam-5336	120	9	,	,	PUNCT
ejpam-5336	120	10	f	f	PROPN
ejpam-5336	120	11	(	(	PUNCT
ejpam-5336	120	12	vj	vj	PROPN
ejpam-5336	120	13	)	)	PUNCT
ejpam-5336	120	14	=	=	SYM
ejpam-5336	120	15	j	j	PROPN
ejpam-5336	120	16	+	+	CCONJ
ejpam-5336	120	17	1	1	NUM
ejpam-5336	120	18	for	for	ADP
ejpam-5336	120	19	j	j	PROPN
ejpam-5336	120	20	=	=	SYM
ejpam-5336	120	21	0	0	NUM
ejpam-5336	120	22	and	and	CCONJ
ejpam-5336	120	23	1	1	NUM
ejpam-5336	120	24	,	,	PUNCT
ejpam-5336	120	25	f	f	PROPN
ejpam-5336	120	26	(	(	PUNCT
ejpam-5336	120	27	vj	vj	PROPN
ejpam-5336	120	28	)	)	PUNCT
ejpam-5336	120	29	=	=	SYM
ejpam-5336	121	1	6n+	6n+	NUM
ejpam-5336	121	2	j	j	NOUN
ejpam-5336	121	3	+	+	CCONJ
ejpam-5336	121	4	1	1	NUM
ejpam-5336	121	5	for	for	ADP
ejpam-5336	121	6	all	all	DET
ejpam-5336	121	7	2	2	NUM
ejpam-5336	121	8	≤	≤	NUM
ejpam-5336	121	9	j	j	PROPN
ejpam-5336	121	10	≤	≤	ADV
ejpam-5336	121	11	2	2	NUM
ejpam-5336	121	12	m.	m.	NOUN
ejpam-5336	121	13	we	we	PRON
ejpam-5336	121	14	have	have	VERB
ejpam-5336	121	15	(	(	PUNCT
ejpam-5336	121	16	f(u2	f(u2	NOUN
ejpam-5336	121	17	)	)	PUNCT
ejpam-5336	121	18	,	,	PUNCT
ejpam-5336	121	19	f(u3	f(u3	NOUN
ejpam-5336	121	20	)	)	PUNCT
ejpam-5336	121	21	)	)	PUNCT
ejpam-5336	122	1	=	=	PUNCT
ejpam-5336	122	2	(	(	PUNCT
ejpam-5336	122	3	6n+	6n+	NUM
ejpam-5336	122	4	2	2	NUM
ejpam-5336	122	5	,	,	PUNCT
ejpam-5336	122	6	3	3	NUM
ejpam-5336	122	7	)	)	PUNCT
ejpam-5336	122	8	=	=	SYM
ejpam-5336	122	9	1	1	NUM
ejpam-5336	122	10	,	,	PUNCT
ejpam-5336	122	11	because	because	SCONJ
ejpam-5336	122	12	3	3	NUM
ejpam-5336	122	13	does	do	AUX
ejpam-5336	122	14	not	not	PART
ejpam-5336	122	15	divide	divide	VERB
ejpam-5336	122	16	6n+	6n+	NUM
ejpam-5336	122	17	2	2	NUM
ejpam-5336	122	18	,	,	PUNCT
ejpam-5336	122	19	(	(	PUNCT
ejpam-5336	122	20	f(u1	f(u1	NOUN
ejpam-5336	122	21	)	)	PUNCT
ejpam-5336	122	22	,	,	PUNCT
ejpam-5336	122	23	f(u2n	f(u2n	NOUN
ejpam-5336	122	24	)	)	PUNCT
ejpam-5336	122	25	)	)	PUNCT
ejpam-5336	123	1	=	=	PUNCT
ejpam-5336	124	1	(	(	PUNCT
ejpam-5336	124	2	6n+	6n+	NUM
ejpam-5336	124	3	1	1	NUM
ejpam-5336	124	4	,	,	PUNCT
ejpam-5336	124	5	2n	2n	NUM
ejpam-5336	124	6	)	)	PUNCT
ejpam-5336	124	7	=	=	SYM
ejpam-5336	125	1	1	1	NUM
ejpam-5336	125	2	,	,	PUNCT
ejpam-5336	125	3	because	because	SCONJ
ejpam-5336	125	4	1	1	NUM
ejpam-5336	125	5	=	=	SYM
ejpam-5336	125	6	(	(	PUNCT
ejpam-5336	125	7	6n+	6n+	NUM
ejpam-5336	125	8	1)−	1)−	NUM
ejpam-5336	125	9	3	3	NUM
ejpam-5336	125	10	(	(	PUNCT
ejpam-5336	125	11	2n	2n	NUM
ejpam-5336	125	12	)	)	PUNCT
ejpam-5336	125	13	and	and	CCONJ
ejpam-5336	125	14	(	(	PUNCT
ejpam-5336	125	15	f(u2n+1	f(u2n+1	PROPN
ejpam-5336	125	16	)	)	PUNCT
ejpam-5336	125	17	,	,	PUNCT
ejpam-5336	125	18	f(u4n	f(u4n	NOUN
ejpam-5336	125	19	)	)	PUNCT
ejpam-5336	125	20	)	)	PUNCT
ejpam-5336	125	21	=	=	PUNCT
ejpam-5336	126	1	(	(	PUNCT
ejpam-5336	126	2	2n+	2n+	NUM
ejpam-5336	126	3	1	1	NUM
ejpam-5336	126	4	,	,	PUNCT
ejpam-5336	126	5	4n	4n	NOUN
ejpam-5336	126	6	)	)	PUNCT
ejpam-5336	126	7	=	=	SYM
ejpam-5336	126	8	1	1	NUM
ejpam-5336	126	9	,	,	PUNCT
ejpam-5336	126	10	because	because	SCONJ
ejpam-5336	126	11	2	2	NUM
ejpam-5336	126	12	=	=	SYM
ejpam-5336	126	13	2(2n+	2(2n+	NUM
ejpam-5336	126	14	1)−	1)−	NUM
ejpam-5336	126	15	4n	4n	NOUN
ejpam-5336	126	16	and	and	CCONJ
ejpam-5336	126	17	2	2	NUM
ejpam-5336	126	18	does	do	AUX
ejpam-5336	126	19	not	not	PART
ejpam-5336	126	20	divide	divide	VERB
ejpam-5336	126	21	2n+	2n+	NUM
ejpam-5336	126	22	1	1	NUM
ejpam-5336	126	23	.	.	PUNCT
ejpam-5336	127	1	also	also	ADV
ejpam-5336	127	2	,	,	PUNCT
ejpam-5336	127	3	(	(	PUNCT
ejpam-5336	127	4	f(u4n+1	f(u4n+1	ADJ
ejpam-5336	127	5	)	)	PUNCT
ejpam-5336	127	6	,	,	PUNCT
ejpam-5336	127	7	f(u6n	f(u6n	NOUN
ejpam-5336	127	8	)	)	PUNCT
ejpam-5336	127	9	)	)	PUNCT
ejpam-5336	128	1	=	=	PRON
ejpam-5336	128	2	(	(	PUNCT
ejpam-5336	128	3	4n+	4n+	NUM
ejpam-5336	128	4	1	1	NUM
ejpam-5336	128	5	,	,	PUNCT
ejpam-5336	128	6	6n	6n	NUM
ejpam-5336	128	7	)	)	PUNCT
ejpam-5336	128	8	=	=	SYM
ejpam-5336	129	1	1	1	NUM
ejpam-5336	129	2	,	,	PUNCT
ejpam-5336	129	3	because	because	SCONJ
ejpam-5336	129	4	3	3	NUM
ejpam-5336	129	5	=	=	SYM
ejpam-5336	129	6	3(4n+	3(4n+	NUM
ejpam-5336	129	7	1)−	1)−	NUM
ejpam-5336	129	8	2	2	NUM
ejpam-5336	129	9	(	(	PUNCT
ejpam-5336	129	10	6n	6n	NOUN
ejpam-5336	129	11	)	)	PUNCT
ejpam-5336	129	12	and	and	CCONJ
ejpam-5336	129	13	3	3	NUM
ejpam-5336	129	14	does	do	AUX
ejpam-5336	129	15	not	not	PART
ejpam-5336	129	16	divide	divide	VERB
ejpam-5336	129	17	4n+	4n+	NOUN
ejpam-5336	129	18	1	1	NUM
ejpam-5336	129	19	.	.	PUNCT
ejpam-5336	130	1	thus	thus	ADV
ejpam-5336	130	2	f	f	PROPN
ejpam-5336	130	3	is	be	AUX
ejpam-5336	130	4	a	a	DET
ejpam-5336	130	5	pl	pl	NOUN
ejpam-5336	130	6	.	.	PUNCT
ejpam-5336	130	7	o.	o.	PROPN
ejpam-5336	130	8	a.	a.	PROPN
ejpam-5336	130	9	abughneim	abughneim	PROPN
ejpam-5336	130	10	,	,	PUNCT
ejpam-5336	130	11	b.	b.	PROPN
ejpam-5336	130	12	abughazaleh	abughazaleh	PROPN
ejpam-5336	130	13	/	/	SYM
ejpam-5336	130	14	eur	eur	PROPN
ejpam-5336	130	15	.	.	PUNCT
ejpam-5336	131	1	j.	j.	PROPN
ejpam-5336	131	2	pure	pure	PROPN
ejpam-5336	131	3	appl	appl	PROPN
ejpam-5336	131	4	.	.	PROPN
ejpam-5336	131	5	math	math	PROPN
ejpam-5336	131	6	,	,	PUNCT
ejpam-5336	131	7	17	17	NUM
ejpam-5336	131	8	(	(	PUNCT
ejpam-5336	131	9	4	4	NUM
ejpam-5336	131	10	)	)	PUNCT
ejpam-5336	131	11	(	(	PUNCT
ejpam-5336	131	12	2024	2024	NUM
ejpam-5336	131	13	)	)	PUNCT
ejpam-5336	131	14	,	,	PUNCT
ejpam-5336	131	15	3557	3557	NUM
ejpam-5336	131	16	-	-	SYM
ejpam-5336	131	17	3566	3566	NUM
ejpam-5336	131	18	3561	3561	NUM
ejpam-5336	131	19	(	(	PUNCT
ejpam-5336	131	20	b	b	X
ejpam-5336	131	21	)	)	PUNCT
ejpam-5336	131	22	if	if	SCONJ
ejpam-5336	131	23	3	3	NUM
ejpam-5336	131	24	divides	divide	VERB
ejpam-5336	131	25	4n+	4n+	NUM
ejpam-5336	131	26	1	1	NUM
ejpam-5336	131	27	,	,	PUNCT
ejpam-5336	131	28	define	define	VERB
ejpam-5336	131	29	f	f	X
ejpam-5336	131	30	:	:	PUNCT
ejpam-5336	131	31	v	v	X
ejpam-5336	131	32	(	(	PUNCT
ejpam-5336	131	33	c2n	c2n	NOUN
ejpam-5336	131	34	∪	∪	NOUN
ejpam-5336	131	35	c2n	c2n	NOUN
ejpam-5336	131	36	∪	∪	ADP
ejpam-5336	131	37	c2n	c2n	NOUN
ejpam-5336	131	38	∪w2	∪w2	NOUN
ejpam-5336	131	39	m	m	NOUN
ejpam-5336	131	40	)	)	PUNCT
ejpam-5336	131	41	−→	−→	ADV
ejpam-5336	131	42	{	{	PUNCT
ejpam-5336	131	43	1	1	NUM
ejpam-5336	131	44	,	,	PUNCT
ejpam-5336	131	45	2	2	NUM
ejpam-5336	131	46	,	,	PUNCT
ejpam-5336	131	47	...	...	PUNCT
ejpam-5336	131	48	,	,	PUNCT
ejpam-5336	131	49	6n+	6n+	NUM
ejpam-5336	131	50	2m+	2m+	NUM
ejpam-5336	131	51	1	1	NUM
ejpam-5336	131	52	}	}	PUNCT
ejpam-5336	131	53	as	as	SCONJ
ejpam-5336	131	54	follows	follow	VERB
ejpam-5336	131	55	f	f	PROPN
ejpam-5336	131	56	(	(	PUNCT
ejpam-5336	131	57	ui	ui	PROPN
ejpam-5336	131	58	)	)	PUNCT
ejpam-5336	131	59	=	=	SYM
ejpam-5336	132	1	6n+	6n+	NUM
ejpam-5336	132	2	i	i	PRON
ejpam-5336	132	3	for	for	ADP
ejpam-5336	132	4	i	i	PRON
ejpam-5336	132	5	=	=	SYM
ejpam-5336	132	6	1	1	NUM
ejpam-5336	132	7	and	and	CCONJ
ejpam-5336	132	8	2	2	NUM
ejpam-5336	132	9	,	,	PUNCT
ejpam-5336	132	10	f	f	PROPN
ejpam-5336	132	11	(	(	PUNCT
ejpam-5336	132	12	u2n	u2n	PROPN
ejpam-5336	132	13	)	)	PUNCT
ejpam-5336	132	14	=	=	SYM
ejpam-5336	133	1	4n	4n	NOUN
ejpam-5336	133	2	,	,	PUNCT
ejpam-5336	133	3	f	f	PROPN
ejpam-5336	133	4	(	(	PUNCT
ejpam-5336	133	5	u4n	u4n	PROPN
ejpam-5336	133	6	)	)	PUNCT
ejpam-5336	133	7	=	=	SYM
ejpam-5336	133	8	6n	6n	PROPN
ejpam-5336	133	9	,	,	PUNCT
ejpam-5336	133	10	f	f	PROPN
ejpam-5336	133	11	(	(	PUNCT
ejpam-5336	133	12	u6n	u6n	PROPN
ejpam-5336	133	13	)	)	PUNCT
ejpam-5336	133	14	=	=	SYM
ejpam-5336	133	15	2n	2n	NUM
ejpam-5336	133	16	,	,	PUNCT
ejpam-5336	133	17	f	f	PROPN
ejpam-5336	133	18	(	(	PUNCT
ejpam-5336	133	19	ui	ui	PROPN
ejpam-5336	133	20	)	)	PUNCT
ejpam-5336	133	21	=	=	VERB
ejpam-5336	134	1	i	i	PRON
ejpam-5336	134	2	for	for	ADP
ejpam-5336	134	3	all	all	PRON
ejpam-5336	134	4	i	i	PRON
ejpam-5336	134	5	̸=	̸=	PROPN
ejpam-5336	134	6	1	1	NUM
ejpam-5336	134	7	,	,	PUNCT
ejpam-5336	134	8	2	2	NUM
ejpam-5336	134	9	,	,	PUNCT
ejpam-5336	134	10	2n	2n	NUM
ejpam-5336	134	11	,	,	PUNCT
ejpam-5336	134	12	4n	4n	NOUN
ejpam-5336	134	13	and	and	CCONJ
ejpam-5336	134	14	6n	6n	NUM
ejpam-5336	134	15	,	,	PUNCT
ejpam-5336	134	16	f	f	PROPN
ejpam-5336	134	17	(	(	PUNCT
ejpam-5336	134	18	vj	vj	PROPN
ejpam-5336	134	19	)	)	PUNCT
ejpam-5336	134	20	=	=	SYM
ejpam-5336	134	21	j	j	PROPN
ejpam-5336	134	22	+	+	CCONJ
ejpam-5336	134	23	1	1	NUM
ejpam-5336	134	24	for	for	ADP
ejpam-5336	134	25	j	j	PROPN
ejpam-5336	134	26	=	=	SYM
ejpam-5336	134	27	0	0	NUM
ejpam-5336	134	28	and	and	CCONJ
ejpam-5336	134	29	1	1	NUM
ejpam-5336	134	30	,	,	PUNCT
ejpam-5336	134	31	f	f	PROPN
ejpam-5336	134	32	(	(	PUNCT
ejpam-5336	134	33	vj	vj	PROPN
ejpam-5336	134	34	)	)	PUNCT
ejpam-5336	134	35	=	=	SYM
ejpam-5336	135	1	6n+	6n+	NUM
ejpam-5336	135	2	j	j	NOUN
ejpam-5336	135	3	+	+	CCONJ
ejpam-5336	135	4	1	1	NUM
ejpam-5336	135	5	for	for	ADP
ejpam-5336	135	6	all	all	DET
ejpam-5336	135	7	2	2	NUM
ejpam-5336	135	8	≤	≤	NUM
ejpam-5336	135	9	j	j	PROPN
ejpam-5336	135	10	≤	≤	ADV
ejpam-5336	135	11	2	2	NUM
ejpam-5336	135	12	m.	m.	NOUN
ejpam-5336	135	13	then	then	ADV
ejpam-5336	135	14	,	,	PUNCT
ejpam-5336	135	15	(	(	PUNCT
ejpam-5336	135	16	f(u1	f(u1	NOUN
ejpam-5336	135	17	)	)	PUNCT
ejpam-5336	135	18	,	,	PUNCT
ejpam-5336	135	19	f(u2n	f(u2n	NOUN
ejpam-5336	135	20	)	)	PUNCT
ejpam-5336	135	21	)	)	PUNCT
ejpam-5336	136	1	=	=	PUNCT
ejpam-5336	137	1	(	(	PUNCT
ejpam-5336	137	2	6n+	6n+	NUM
ejpam-5336	137	3	1	1	NUM
ejpam-5336	137	4	,	,	PUNCT
ejpam-5336	137	5	4n	4n	NOUN
ejpam-5336	137	6	)	)	PUNCT
ejpam-5336	137	7	=	=	SYM
ejpam-5336	137	8	1	1	NUM
ejpam-5336	137	9	,	,	PUNCT
ejpam-5336	137	10	because	because	SCONJ
ejpam-5336	137	11	2	2	NUM
ejpam-5336	137	12	=	=	SYM
ejpam-5336	137	13	2(6n+	2(6n+	NUM
ejpam-5336	137	14	1)−	1)−	NUM
ejpam-5336	137	15	3	3	NUM
ejpam-5336	137	16	(	(	PUNCT
ejpam-5336	137	17	4n	4n	X
ejpam-5336	137	18	)	)	PUNCT
ejpam-5336	137	19	and	and	CCONJ
ejpam-5336	137	20	6n+	6n+	NUM
ejpam-5336	137	21	1	1	NUM
ejpam-5336	137	22	is	be	AUX
ejpam-5336	137	23	odd	odd	ADJ
ejpam-5336	137	24	,	,	PUNCT
ejpam-5336	137	25	(	(	PUNCT
ejpam-5336	137	26	f(u2n−1	f(u2n−1	PROPN
ejpam-5336	137	27	)	)	PUNCT
ejpam-5336	137	28	,	,	PUNCT
ejpam-5336	137	29	f(u2n	f(u2n	NOUN
ejpam-5336	137	30	)	)	PUNCT
ejpam-5336	137	31	)	)	PUNCT
ejpam-5336	138	1	=	=	PUNCT
ejpam-5336	138	2	(	(	PUNCT
ejpam-5336	138	3	2n−	2n−	PROPN
ejpam-5336	138	4	1	1	NUM
ejpam-5336	138	5	,	,	PUNCT
ejpam-5336	138	6	4n	4n	NOUN
ejpam-5336	138	7	)	)	PUNCT
ejpam-5336	138	8	=	=	SYM
ejpam-5336	139	1	(	(	PUNCT
ejpam-5336	139	2	2n−	2n−	PROPN
ejpam-5336	139	3	1	1	NUM
ejpam-5336	139	4	,	,	PUNCT
ejpam-5336	139	5	2n	2n	NUM
ejpam-5336	139	6	)	)	PUNCT
ejpam-5336	140	1	=	=	SYM
ejpam-5336	140	2	1	1	NUM
ejpam-5336	140	3	,	,	PUNCT
ejpam-5336	140	4	(	(	PUNCT
ejpam-5336	140	5	f(u2	f(u2	NOUN
ejpam-5336	140	6	)	)	PUNCT
ejpam-5336	140	7	,	,	PUNCT
ejpam-5336	140	8	f(u3	f(u3	NOUN
ejpam-5336	140	9	)	)	PUNCT
ejpam-5336	140	10	)	)	PUNCT
ejpam-5336	141	1	=	=	PUNCT
ejpam-5336	141	2	(	(	PUNCT
ejpam-5336	141	3	6n+	6n+	NUM
ejpam-5336	141	4	2	2	NUM
ejpam-5336	141	5	,	,	PUNCT
ejpam-5336	141	6	3	3	NUM
ejpam-5336	141	7	)	)	PUNCT
ejpam-5336	141	8	=	=	SYM
ejpam-5336	141	9	1	1	NUM
ejpam-5336	141	10	,	,	PUNCT
ejpam-5336	141	11	because	because	SCONJ
ejpam-5336	141	12	3	3	NUM
ejpam-5336	141	13	does	do	AUX
ejpam-5336	141	14	not	not	PART
ejpam-5336	141	15	divide	divide	VERB
ejpam-5336	141	16	6n+	6n+	NUM
ejpam-5336	141	17	2	2	NUM
ejpam-5336	141	18	.	.	PUNCT
ejpam-5336	141	19	(	(	PUNCT
ejpam-5336	141	20	f(u4n+1	f(u4n+1	NUM
ejpam-5336	141	21	)	)	PUNCT
ejpam-5336	141	22	,	,	PUNCT
ejpam-5336	141	23	f(u6n	f(u6n	NOUN
ejpam-5336	141	24	)	)	PUNCT
ejpam-5336	141	25	)	)	PUNCT
ejpam-5336	142	1	=	=	PRON
ejpam-5336	143	1	(	(	PUNCT
ejpam-5336	143	2	4n+	4n+	NUM
ejpam-5336	143	3	1	1	NUM
ejpam-5336	143	4	,	,	PUNCT
ejpam-5336	143	5	2n	2n	NUM
ejpam-5336	143	6	)	)	PUNCT
ejpam-5336	143	7	=	=	SYM
ejpam-5336	143	8	1	1	NUM
ejpam-5336	143	9	,	,	PUNCT
ejpam-5336	143	10	because	because	SCONJ
ejpam-5336	143	11	1	1	NUM
ejpam-5336	143	12	=	=	SYM
ejpam-5336	143	13	(	(	PUNCT
ejpam-5336	143	14	4n+	4n+	NUM
ejpam-5336	143	15	1)−	1)−	NUM
ejpam-5336	143	16	2(2n	2(2n	NUM
ejpam-5336	143	17	)	)	PUNCT
ejpam-5336	143	18	.	.	PUNCT
ejpam-5336	144	1	(	(	PUNCT
ejpam-5336	144	2	f(u6n−1	f(u6n−1	PROPN
ejpam-5336	144	3	)	)	PUNCT
ejpam-5336	144	4	,	,	PUNCT
ejpam-5336	144	5	f(u6n	f(u6n	NOUN
ejpam-5336	144	6	)	)	PUNCT
ejpam-5336	144	7	)	)	PUNCT
ejpam-5336	145	1	=	=	PUNCT
ejpam-5336	145	2	(	(	PUNCT
ejpam-5336	145	3	6n−	6n−	PROPN
ejpam-5336	145	4	1	1	NUM
ejpam-5336	145	5	,	,	PUNCT
ejpam-5336	145	6	2n	2n	NUM
ejpam-5336	145	7	)	)	PUNCT
ejpam-5336	146	1	=	=	SYM
ejpam-5336	146	2	1	1	NUM
ejpam-5336	146	3	,	,	PUNCT
ejpam-5336	146	4	because	because	SCONJ
ejpam-5336	146	5	1	1	NUM
ejpam-5336	146	6	=	=	SYM
ejpam-5336	146	7	3	3	NUM
ejpam-5336	146	8	(	(	PUNCT
ejpam-5336	146	9	2n)−	2n)−	NUM
ejpam-5336	146	10	(	(	PUNCT
ejpam-5336	146	11	6n−	6n−	PROPN
ejpam-5336	146	12	1	1	NUM
ejpam-5336	146	13	)	)	PUNCT
ejpam-5336	146	14	.	.	PUNCT
ejpam-5336	147	1	now	now	ADV
ejpam-5336	147	2	,	,	PUNCT
ejpam-5336	147	3	since	since	SCONJ
ejpam-5336	147	4	1	1	NUM
ejpam-5336	147	5	=	=	SYM
ejpam-5336	147	6	2(2n	2(2n	NUM
ejpam-5336	147	7	+	+	CCONJ
ejpam-5336	147	8	1	1	NUM
ejpam-5336	147	9	)	)	PUNCT
ejpam-5336	147	10	−	−	PROPN
ejpam-5336	147	11	(	(	PUNCT
ejpam-5336	147	12	4n	4n	X
ejpam-5336	147	13	+	+	NOUN
ejpam-5336	147	14	1	1	NUM
ejpam-5336	147	15	)	)	PUNCT
ejpam-5336	147	16	and	and	CCONJ
ejpam-5336	147	17	3	3	NUM
ejpam-5336	147	18	divides	divide	VERB
ejpam-5336	147	19	4n	4n	NOUN
ejpam-5336	147	20	+	+	X
ejpam-5336	147	21	1	1	NUM
ejpam-5336	147	22	,	,	PUNCT
ejpam-5336	147	23	3	3	NUM
ejpam-5336	147	24	does	do	AUX
ejpam-5336	147	25	not	not	PART
ejpam-5336	147	26	divide	divide	VERB
ejpam-5336	147	27	2n+	2n+	NUM
ejpam-5336	147	28	1	1	NUM
ejpam-5336	147	29	.	.	PUNCT
ejpam-5336	148	1	therefore	therefore	ADV
ejpam-5336	148	2	,	,	PUNCT
ejpam-5336	148	3	(	(	PUNCT
ejpam-5336	148	4	f(u2n+1	f(u2n+1	NOUN
ejpam-5336	148	5	)	)	PUNCT
ejpam-5336	148	6	,	,	PUNCT
ejpam-5336	148	7	f(u4n	f(u4n	NOUN
ejpam-5336	148	8	)	)	PUNCT
ejpam-5336	148	9	)	)	PUNCT
ejpam-5336	149	1	=	=	PUNCT
ejpam-5336	149	2	(	(	PUNCT
ejpam-5336	149	3	2n+	2n+	NUM
ejpam-5336	149	4	1	1	NUM
ejpam-5336	149	5	,	,	PUNCT
ejpam-5336	149	6	6n	6n	NUM
ejpam-5336	149	7	)	)	PUNCT
ejpam-5336	149	8	=	=	PUNCT
ejpam-5336	150	1	(	(	PUNCT
ejpam-5336	150	2	2n+	2n+	NUM
ejpam-5336	150	3	1	1	NUM
ejpam-5336	150	4	,	,	PUNCT
ejpam-5336	150	5	2n	2n	NUM
ejpam-5336	150	6	)	)	PUNCT
ejpam-5336	150	7	=	=	SYM
ejpam-5336	151	1	1	1	X
ejpam-5336	151	2	.	.	PUNCT
ejpam-5336	151	3	also	also	ADV
ejpam-5336	151	4	,	,	PUNCT
ejpam-5336	151	5	3	3	NUM
ejpam-5336	151	6	does	do	AUX
ejpam-5336	151	7	not	not	PART
ejpam-5336	151	8	divide	divide	VERB
ejpam-5336	151	9	4n−	4n−	PROPN
ejpam-5336	151	10	1	1	NUM
ejpam-5336	151	11	because	because	SCONJ
ejpam-5336	151	12	3	3	NUM
ejpam-5336	151	13	divides	divide	VERB
ejpam-5336	151	14	4n+	4n+	NOUN
ejpam-5336	151	15	1	1	NUM
ejpam-5336	151	16	.	.	PUNCT
ejpam-5336	152	1	thus	thus	ADV
ejpam-5336	152	2	(	(	PUNCT
ejpam-5336	152	3	f(u4n−1	f(u4n−1	ADJ
ejpam-5336	152	4	)	)	PUNCT
ejpam-5336	152	5	,	,	PUNCT
ejpam-5336	152	6	f(u4n	f(u4n	NOUN
ejpam-5336	152	7	)	)	PUNCT
ejpam-5336	152	8	)	)	PUNCT
ejpam-5336	153	1	=	=	PUNCT
ejpam-5336	153	2	(	(	PUNCT
ejpam-5336	153	3	4n−	4n−	NUM
ejpam-5336	153	4	1	1	NUM
ejpam-5336	153	5	,	,	PUNCT
ejpam-5336	153	6	6n	6n	NOUN
ejpam-5336	153	7	)	)	PUNCT
ejpam-5336	153	8	=	=	PUNCT
ejpam-5336	154	1	(	(	PUNCT
ejpam-5336	154	2	4n−	4n−	NUM
ejpam-5336	154	3	1	1	NUM
ejpam-5336	154	4	,	,	PUNCT
ejpam-5336	154	5	2n	2n	NUM
ejpam-5336	154	6	)	)	PUNCT
ejpam-5336	154	7	=	=	SYM
ejpam-5336	155	1	1	1	X
ejpam-5336	155	2	.	.	X
ejpam-5336	155	3	therefore	therefore	ADV
ejpam-5336	155	4	f	f	PROPN
ejpam-5336	155	5	is	be	AUX
ejpam-5336	155	6	a	a	DET
ejpam-5336	155	7	pl	pl	NOUN
ejpam-5336	155	8	.	.	PUNCT
ejpam-5336	155	9	o.	o.	PROPN
ejpam-5336	155	10	a.	a.	PROPN
ejpam-5336	155	11	abughneim	abughneim	PROPN
ejpam-5336	155	12	,	,	PUNCT
ejpam-5336	155	13	b.	b.	PROPN
ejpam-5336	155	14	abughazaleh	abughazaleh	PROPN
ejpam-5336	155	15	/	/	SYM
ejpam-5336	155	16	eur	eur	PROPN
ejpam-5336	155	17	.	.	PUNCT
ejpam-5336	156	1	j.	j.	PROPN
ejpam-5336	156	2	pure	pure	PROPN
ejpam-5336	156	3	appl	appl	PROPN
ejpam-5336	156	4	.	.	PROPN
ejpam-5336	156	5	math	math	PROPN
ejpam-5336	156	6	,	,	PUNCT
ejpam-5336	156	7	17	17	NUM
ejpam-5336	156	8	(	(	PUNCT
ejpam-5336	156	9	4	4	NUM
ejpam-5336	156	10	)	)	PUNCT
ejpam-5336	156	11	(	(	PUNCT
ejpam-5336	156	12	2024	2024	NUM
ejpam-5336	156	13	)	)	PUNCT
ejpam-5336	156	14	,	,	PUNCT
ejpam-5336	156	15	3557	3557	NUM
ejpam-5336	156	16	-	-	SYM
ejpam-5336	156	17	3566	3566	NUM
ejpam-5336	156	18	3562	3562	NUM
ejpam-5336	156	19	3	3	NUM
ejpam-5336	156	20	.	.	PUNCT
ejpam-5336	156	21	prime	prime	ADJ
ejpam-5336	156	22	labeling	labeling	NOUN
ejpam-5336	156	23	of	of	ADP
ejpam-5336	156	24	union	union	NOUN
ejpam-5336	156	25	of	of	ADP
ejpam-5336	156	26	complete	complete	ADJ
ejpam-5336	156	27	graphs	graph	NOUN
ejpam-5336	156	28	and	and	CCONJ
ejpam-5336	156	29	graphs	graph	NOUN
ejpam-5336	156	30	with	with	ADP
ejpam-5336	156	31	maximal	maximal	ADJ
ejpam-5336	156	32	size	size	NOUN
ejpam-5336	156	33	in	in	ADP
ejpam-5336	156	34	this	this	DET
ejpam-5336	156	35	section	section	NOUN
ejpam-5336	157	1	,	,	PUNCT
ejpam-5336	157	2	we	we	PRON
ejpam-5336	157	3	will	will	AUX
ejpam-5336	157	4	study	study	VERB
ejpam-5336	157	5	some	some	DET
ejpam-5336	157	6	properties	property	NOUN
ejpam-5336	157	7	of	of	ADP
ejpam-5336	157	8	the	the	DET
ejpam-5336	157	9	disjoint	disjoint	PROPN
ejpam-5336	157	10	union	union	NOUN
ejpam-5336	157	11	between	between	ADP
ejpam-5336	157	12	a	a	DET
ejpam-5336	157	13	complete	complete	ADJ
ejpam-5336	157	14	graph	graph	NOUN
ejpam-5336	157	15	and	and	CCONJ
ejpam-5336	157	16	any	any	DET
ejpam-5336	157	17	graph	graph	NOUN
ejpam-5336	157	18	such	such	ADJ
ejpam-5336	157	19	that	that	SCONJ
ejpam-5336	157	20	this	this	DET
ejpam-5336	157	21	union	union	NOUN
ejpam-5336	157	22	is	be	AUX
ejpam-5336	157	23	a	a	DET
ejpam-5336	157	24	pg	pg	NOUN
ejpam-5336	157	25	.	.	PUNCT
ejpam-5336	157	26	seoud	seoud	PROPN
ejpam-5336	157	27	et	et	PROPN
ejpam-5336	157	28	.	.	PUNCT
ejpam-5336	158	1	al	al	PROPN
ejpam-5336	158	2	.	.	PUNCT
ejpam-5336	159	1	in	in	ADP
ejpam-5336	159	2	[	[	X
ejpam-5336	159	3	12	12	NUM
ejpam-5336	159	4	]	]	PUNCT
ejpam-5336	159	5	define	define	VERB
ejpam-5336	159	6	a	a	DET
ejpam-5336	159	7	maximal	maximal	ADJ
ejpam-5336	159	8	pg	pg	NOUN
ejpam-5336	159	9	as	as	SCONJ
ejpam-5336	159	10	follows	follow	VERB
ejpam-5336	159	11	:	:	PUNCT
ejpam-5336	159	12	definition	definition	NOUN
ejpam-5336	159	13	1	1	NUM
ejpam-5336	159	14	.	.	PUNCT
ejpam-5336	160	1	[	[	X
ejpam-5336	160	2	12	12	NUM
ejpam-5336	160	3	]	]	PUNCT
ejpam-5336	160	4	“	"	PUNCT
ejpam-5336	160	5	a	a	DET
ejpam-5336	160	6	maximal	maximal	ADJ
ejpam-5336	160	7	pg	pg	NOUN
ejpam-5336	160	8	is	be	AUX
ejpam-5336	160	9	a	a	DET
ejpam-5336	160	10	pg	pg	NOUN
ejpam-5336	160	11	of	of	ADP
ejpam-5336	160	12	n	n	PRON
ejpam-5336	160	13	vertices	vertice	VERB
ejpam-5336	160	14	such	such	ADJ
ejpam-5336	160	15	that	that	SCONJ
ejpam-5336	160	16	adding	add	VERB
ejpam-5336	160	17	any	any	DET
ejpam-5336	160	18	new	new	ADJ
ejpam-5336	160	19	edge	edge	NOUN
ejpam-5336	160	20	yields	yield	VERB
ejpam-5336	160	21	a	a	DET
ejpam-5336	160	22	non	non	NOUN
ejpam-5336	160	23	-	-	NOUN
ejpam-5336	160	24	pg	pg	NOUN
ejpam-5336	160	25	.	.	PUNCT
ejpam-5336	161	1	usually	usually	ADV
ejpam-5336	161	2	this	this	DET
ejpam-5336	161	3	graph	graph	NOUN
ejpam-5336	161	4	is	be	AUX
ejpam-5336	161	5	denoted	denote	VERB
ejpam-5336	161	6	by	by	ADP
ejpam-5336	161	7	r(n	r(n	NOUN
ejpam-5336	161	8	)	)	PUNCT
ejpam-5336	161	9	.	.	PUNCT
ejpam-5336	161	10	”	"	PUNCT
ejpam-5336	161	11	theorem	theorem	VERB
ejpam-5336	161	12	6	6	NUM
ejpam-5336	161	13	.	.	PUNCT
ejpam-5336	162	1	[	[	X
ejpam-5336	162	2	14	14	NUM
ejpam-5336	162	3	]	]	PUNCT
ejpam-5336	162	4	“	"	PUNCT
ejpam-5336	162	5	the	the	DET
ejpam-5336	162	6	largest	large	ADJ
ejpam-5336	162	7	complete	complete	ADJ
ejpam-5336	162	8	subgraph	subgraph	NOUN
ejpam-5336	162	9	in	in	ADP
ejpam-5336	162	10	the	the	DET
ejpam-5336	162	11	maximal	maximal	ADJ
ejpam-5336	162	12	pg	pg	NOUN
ejpam-5336	162	13	of	of	ADP
ejpam-5336	162	14	n	n	PRON
ejpam-5336	162	15	vertices	vertex	NOUN
ejpam-5336	162	16	is	be	AUX
ejpam-5336	162	17	of	of	ADP
ejpam-5336	162	18	order	order	NOUN
ejpam-5336	162	19	π(n	π(n	NUM
ejpam-5336	162	20	)	)	PUNCT
ejpam-5336	163	1	+	+	CCONJ
ejpam-5336	163	2	1	1	NUM
ejpam-5336	163	3	,	,	PUNCT
ejpam-5336	163	4	where	where	SCONJ
ejpam-5336	163	5	π(n	π(n	NOUN
ejpam-5336	163	6	)	)	PUNCT
ejpam-5336	163	7	is	be	AUX
ejpam-5336	163	8	the	the	DET
ejpam-5336	163	9	number	number	NOUN
ejpam-5336	163	10	of	of	ADP
ejpam-5336	163	11	primes	prime	NOUN
ejpam-5336	163	12	less	less	ADJ
ejpam-5336	163	13	than	than	ADP
ejpam-5336	163	14	or	or	CCONJ
ejpam-5336	163	15	equal	equal	ADJ
ejpam-5336	163	16	to	to	PART
ejpam-5336	163	17	n.	n.	NOUN
ejpam-5336	163	18	”	"	PUNCT
ejpam-5336	163	19	remark	remark	NOUN
ejpam-5336	163	20	1	1	NUM
ejpam-5336	163	21	.	.	PUNCT
ejpam-5336	164	1	let	let	VERB
ejpam-5336	164	2	h	h	NOUN
ejpam-5336	164	3	be	be	AUX
ejpam-5336	164	4	the	the	DET
ejpam-5336	164	5	largest	large	ADJ
ejpam-5336	164	6	complete	complete	ADJ
ejpam-5336	164	7	subgraph	subgraph	NOUN
ejpam-5336	164	8	in	in	ADP
ejpam-5336	164	9	the	the	DET
ejpam-5336	164	10	maximal	maximal	ADJ
ejpam-5336	164	11	pg	pg	NOUN
ejpam-5336	164	12	of	of	ADP
ejpam-5336	164	13	n	n	PRON
ejpam-5336	164	14	vertices	vertex	NOUN
ejpam-5336	164	15	.	.	PUNCT
ejpam-5336	165	1	then	then	ADV
ejpam-5336	165	2	we	we	PRON
ejpam-5336	165	3	can	can	AUX
ejpam-5336	165	4	label	label	VERB
ejpam-5336	165	5	the	the	DET
ejpam-5336	165	6	vertices	vertex	NOUN
ejpam-5336	165	7	of	of	ADP
ejpam-5336	165	8	h	h	NOUN
ejpam-5336	165	9	by	by	ADP
ejpam-5336	165	10	the	the	DET
ejpam-5336	165	11	primes	prime	NOUN
ejpam-5336	165	12	less	less	ADJ
ejpam-5336	165	13	than	than	ADP
ejpam-5336	165	14	or	or	CCONJ
ejpam-5336	165	15	equal	equal	ADJ
ejpam-5336	165	16	to	to	ADP
ejpam-5336	165	17	n	n	PRON
ejpam-5336	165	18	together	together	ADV
ejpam-5336	165	19	with	with	ADP
ejpam-5336	165	20	1	1	NUM
ejpam-5336	165	21	namely	namely	ADV
ejpam-5336	165	22	,	,	PUNCT
ejpam-5336	165	23	1	1	NUM
ejpam-5336	165	24	,	,	PUNCT
ejpam-5336	165	25	p1	p1	NOUN
ejpam-5336	165	26	,	,	PUNCT
ejpam-5336	165	27	p2	p2	NOUN
ejpam-5336	165	28	,	,	PUNCT
ejpam-5336	165	29	....	....	PUNCT
ejpam-5336	165	30	,	,	PUNCT
ejpam-5336	165	31	pπ(n	pπ(n	NUM
ejpam-5336	165	32	)	)	PUNCT
ejpam-5336	165	33	.	.	PUNCT
ejpam-5336	166	1	also	also	ADV
ejpam-5336	166	2	,	,	PUNCT
ejpam-5336	166	3	we	we	PRON
ejpam-5336	166	4	can	can	AUX
ejpam-5336	166	5	replace	replace	VERB
ejpam-5336	166	6	the	the	DET
ejpam-5336	166	7	label	label	NOUN
ejpam-5336	166	8	pi	pi	NOUN
ejpam-5336	166	9	by	by	ADP
ejpam-5336	166	10	pki	pki	PROPN
ejpam-5336	166	11	for	for	ADP
ejpam-5336	166	12	some	some	DET
ejpam-5336	166	13	k	k	PROPN
ejpam-5336	166	14	≥	≥	NUM
ejpam-5336	166	15	2	2	NUM
ejpam-5336	166	16	and	and	CCONJ
ejpam-5336	166	17	pki	pki	PROPN
ejpam-5336	166	18	≤	≤	NOUN
ejpam-5336	166	19	n	n	VERB
ejpam-5336	166	20	because	because	SCONJ
ejpam-5336	166	21	for	for	ADP
ejpam-5336	166	22	any	any	DET
ejpam-5336	166	23	a	a	DET
ejpam-5336	166	24	∈	∈	NOUN
ejpam-5336	166	25	z+	z+	NUM
ejpam-5336	166	26	,	,	PUNCT
ejpam-5336	166	27	(	(	PUNCT
ejpam-5336	166	28	a	a	DET
ejpam-5336	166	29	,	,	PUNCT
ejpam-5336	166	30	pi	pi	NOUN
ejpam-5336	166	31	)	)	PUNCT
ejpam-5336	166	32	=	=	SYM
ejpam-5336	166	33	1	1	NUM
ejpam-5336	167	1	if	if	SCONJ
ejpam-5336	167	2	and	and	CCONJ
ejpam-5336	167	3	only	only	ADV
ejpam-5336	167	4	if	if	SCONJ
ejpam-5336	167	5	(	(	PUNCT
ejpam-5336	167	6	a	a	X
ejpam-5336	167	7	,	,	PUNCT
ejpam-5336	167	8	pki	pki	PROPN
ejpam-5336	167	9	)	)	PUNCT
ejpam-5336	167	10	=	=	SYM
ejpam-5336	167	11	1	1	X
ejpam-5336	167	12	.	.	X
ejpam-5336	167	13	theorem	theorem	NOUN
ejpam-5336	167	14	7	7	NUM
ejpam-5336	167	15	.	.	PUNCT
ejpam-5336	167	16	suppose	suppose	VERB
ejpam-5336	167	17	kn	kn	PROPN
ejpam-5336	167	18	is	be	AUX
ejpam-5336	167	19	the	the	DET
ejpam-5336	167	20	complete	complete	ADJ
ejpam-5336	167	21	graph	graph	NOUN
ejpam-5336	167	22	of	of	ADP
ejpam-5336	167	23	order	order	NOUN
ejpam-5336	167	24	n	n	NOUN
ejpam-5336	167	25	and	and	CCONJ
ejpam-5336	167	26	gm	gm	PROPN
ejpam-5336	167	27	is	be	AUX
ejpam-5336	167	28	any	any	DET
ejpam-5336	167	29	graph	graph	NOUN
ejpam-5336	167	30	of	of	ADP
ejpam-5336	167	31	order	order	NOUN
ejpam-5336	167	32	m	m	VERB
ejpam-5336	167	33	such	such	ADJ
ejpam-5336	167	34	that	that	SCONJ
ejpam-5336	167	35	kn	kn	PROPN
ejpam-5336	167	36	∪gm	∪gm	PROPN
ejpam-5336	167	37	is	be	AUX
ejpam-5336	167	38	a	a	DET
ejpam-5336	167	39	pg	pg	NOUN
ejpam-5336	167	40	.	.	PUNCT
ejpam-5336	168	1	then	then	ADV
ejpam-5336	168	2	(	(	PUNCT
ejpam-5336	168	3	i	i	NOUN
ejpam-5336	168	4	)	)	PUNCT
ejpam-5336	168	5	π	π	PROPN
ejpam-5336	168	6	(	(	PUNCT
ejpam-5336	168	7	n+m	n+m	NUM
ejpam-5336	168	8	)	)	PUNCT
ejpam-5336	168	9	≥	≥	NOUN
ejpam-5336	168	10	n−	n−	NOUN
ejpam-5336	168	11	1	1	NUM
ejpam-5336	168	12	.	.	PUNCT
ejpam-5336	168	13	(	(	PUNCT
ejpam-5336	168	14	ii	ii	NOUN
ejpam-5336	168	15	)	)	PUNCT
ejpam-5336	168	16	α	α	PROPN
ejpam-5336	168	17	(	(	PUNCT
ejpam-5336	168	18	gm	gm	PROPN
ejpam-5336	168	19	)	)	PUNCT
ejpam-5336	168	20	≥	≥	NOUN
ejpam-5336	168	21	[	[	PUNCT
ejpam-5336	168	22	n+m	n+m	NUM
ejpam-5336	168	23	2	2	NUM
ejpam-5336	168	24	]	]	PUNCT
ejpam-5336	168	25	−	−	PROPN
ejpam-5336	168	26	1	1	X
ejpam-5336	168	27	.	.	PUNCT
ejpam-5336	169	1	proof	proof	NOUN
ejpam-5336	169	2	.	.	PUNCT
ejpam-5336	170	1	(	(	PUNCT
ejpam-5336	170	2	i	i	NOUN
ejpam-5336	170	3	)	)	PUNCT
ejpam-5336	170	4	by	by	ADP
ejpam-5336	170	5	theorem	theorem	NOUN
ejpam-5336	170	6	6	6	NUM
ejpam-5336	170	7	,	,	PUNCT
ejpam-5336	170	8	n	n	NOUN
ejpam-5336	170	9	=	=	PRON
ejpam-5336	170	10	|v	|v	X
ejpam-5336	170	11	(	(	PUNCT
ejpam-5336	170	12	kn)|	kn)|	PROPN
ejpam-5336	170	13	≤	≤	PROPN
ejpam-5336	170	14	π	π	PROPN
ejpam-5336	170	15	(	(	PUNCT
ejpam-5336	170	16	n+m	n+m	NUM
ejpam-5336	170	17	)	)	PUNCT
ejpam-5336	171	1	+	+	CCONJ
ejpam-5336	171	2	1	1	X
ejpam-5336	171	3	.	.	PUNCT
ejpam-5336	172	1	so	so	ADV
ejpam-5336	172	2	,	,	PUNCT
ejpam-5336	172	3	π	π	X
ejpam-5336	172	4	(	(	PUNCT
ejpam-5336	172	5	n+m	n+m	NUM
ejpam-5336	172	6	)	)	PUNCT
ejpam-5336	172	7	≥	≥	NOUN
ejpam-5336	172	8	n−	n−	NOUN
ejpam-5336	172	9	1	1	NUM
ejpam-5336	172	10	.	.	PUNCT
ejpam-5336	172	11	(	(	PUNCT
ejpam-5336	172	12	ii	ii	NOUN
ejpam-5336	172	13	)	)	PUNCT
ejpam-5336	172	14	since	since	SCONJ
ejpam-5336	172	15	at	at	ADP
ejpam-5336	172	16	most	most	ADJ
ejpam-5336	172	17	one	one	NUM
ejpam-5336	172	18	of	of	ADP
ejpam-5336	172	19	the	the	DET
ejpam-5336	172	20	vertices	vertex	NOUN
ejpam-5336	172	21	of	of	ADP
ejpam-5336	172	22	kn	kn	PROPN
ejpam-5336	172	23	has	have	AUX
ejpam-5336	172	24	even	even	ADV
ejpam-5336	172	25	label	label	VERB
ejpam-5336	172	26	,	,	PUNCT
ejpam-5336	172	27	the	the	DET
ejpam-5336	172	28	set	set	NOUN
ejpam-5336	172	29	s	s	PART
ejpam-5336	172	30	=	=	PUNCT
ejpam-5336	172	31	{	{	PUNCT
ejpam-5336	172	32	u	u	NOUN
ejpam-5336	172	33	∈	∈	PROPN
ejpam-5336	172	34	v	v	NOUN
ejpam-5336	172	35	(	(	PUNCT
ejpam-5336	172	36	gm	gm	PROPN
ejpam-5336	172	37	)	)	PUNCT
ejpam-5336	172	38	:	:	PUNCT
ejpam-5336	173	1	the	the	DET
ejpam-5336	173	2	label	label	NOUN
ejpam-5336	173	3	of	of	ADP
ejpam-5336	173	4	u	u	NOUN
ejpam-5336	173	5	is	be	AUX
ejpam-5336	173	6	even	even	ADV
ejpam-5336	173	7	}	}	PUNCT
ejpam-5336	173	8	is	be	AUX
ejpam-5336	173	9	an	an	DET
ejpam-5336	173	10	independent	independent	ADJ
ejpam-5336	173	11	set	set	NOUN
ejpam-5336	173	12	of	of	ADP
ejpam-5336	173	13	gm	gm	PROPN
ejpam-5336	173	14	with	with	ADP
ejpam-5336	173	15	cardinality	cardinality	NOUN
ejpam-5336	173	16	at	at	ADP
ejpam-5336	173	17	least	least	ADJ
ejpam-5336	173	18	[	[	PUNCT
ejpam-5336	173	19	n+m	n+m	NUM
ejpam-5336	173	20	2	2	NUM
ejpam-5336	173	21	]	]	PUNCT
ejpam-5336	173	22	−	−	PROPN
ejpam-5336	174	1	1	1	X
ejpam-5336	174	2	.	.	PUNCT
ejpam-5336	175	1	so	so	ADV
ejpam-5336	175	2	,	,	PUNCT
ejpam-5336	175	3	α(gm	α(gm	PROPN
ejpam-5336	175	4	)	)	PUNCT
ejpam-5336	175	5	≥	≥	NOUN
ejpam-5336	175	6	[	[	PUNCT
ejpam-5336	175	7	n+m	n+m	NUM
ejpam-5336	175	8	2	2	NUM
ejpam-5336	175	9	]	]	PUNCT
ejpam-5336	175	10	−	−	PROPN
ejpam-5336	176	1	1	1	X
ejpam-5336	176	2	.	.	PUNCT
ejpam-5336	176	3	let	let	VERB
ejpam-5336	176	4	gm	gm	PROPN
ejpam-5336	176	5	be	be	AUX
ejpam-5336	176	6	a	a	DET
ejpam-5336	176	7	graph	graph	NOUN
ejpam-5336	176	8	with	with	ADP
ejpam-5336	176	9	maximum	maximum	ADJ
ejpam-5336	176	10	size	size	NOUN
ejpam-5336	176	11	such	such	ADJ
ejpam-5336	176	12	that	that	SCONJ
ejpam-5336	176	13	kn	kn	PROPN
ejpam-5336	176	14	∪gm	∪gm	PROPN
ejpam-5336	176	15	be	be	AUX
ejpam-5336	176	16	a	a	DET
ejpam-5336	176	17	pg	pg	NOUN
ejpam-5336	176	18	.	.	PUNCT
ejpam-5336	177	1	we	we	PRON
ejpam-5336	177	2	will	will	AUX
ejpam-5336	177	3	examine	examine	VERB
ejpam-5336	177	4	when	when	SCONJ
ejpam-5336	177	5	gm	gm	PROPN
ejpam-5336	177	6	is	be	AUX
ejpam-5336	177	7	connected	connect	VERB
ejpam-5336	177	8	.	.	PUNCT
ejpam-5336	178	1	firstly	firstly	ADV
ejpam-5336	178	2	,	,	PUNCT
ejpam-5336	178	3	we	we	PRON
ejpam-5336	178	4	need	need	VERB
ejpam-5336	178	5	the	the	DET
ejpam-5336	178	6	following	follow	VERB
ejpam-5336	178	7	lemma	lemma	PROPN
ejpam-5336	178	8	and	and	CCONJ
ejpam-5336	178	9	corollary	corollary	ADJ
ejpam-5336	178	10	.	.	PUNCT
ejpam-5336	179	1	lemma	lemma	PROPN
ejpam-5336	179	2	2	2	NUM
ejpam-5336	179	3	.	.	PUNCT
ejpam-5336	180	1	[	[	X
ejpam-5336	180	2	4]“(bonse	4]“(bonse	NOUN
ejpam-5336	180	3	’s	’s	PART
ejpam-5336	180	4	inequality	inequality	NOUN
ejpam-5336	180	5	)	)	PUNCT
ejpam-5336	180	6	let	let	VERB
ejpam-5336	180	7	k	k	PROPN
ejpam-5336	180	8	≥	≥	NUM
ejpam-5336	180	9	5	5	NUM
ejpam-5336	180	10	and	and	CCONJ
ejpam-5336	180	11	p1	p1	NOUN
ejpam-5336	180	12	,	,	PUNCT
ejpam-5336	180	13	p2	p2	NOUN
ejpam-5336	180	14	,	,	PUNCT
ejpam-5336	180	15	...	...	PUNCT
ejpam-5336	180	16	,	,	PUNCT
ejpam-5336	180	17	pk	pk	NOUN
ejpam-5336	180	18	be	be	AUX
ejpam-5336	180	19	the	the	DET
ejpam-5336	180	20	first	first	ADJ
ejpam-5336	180	21	k	k	PROPN
ejpam-5336	180	22	primes	prime	NOUN
ejpam-5336	180	23	.	.	PUNCT
ejpam-5336	181	1	then	then	ADV
ejpam-5336	181	2	p2k+1	p2k+1	VERB
ejpam-5336	181	3	<	<	X
ejpam-5336	181	4	k∏	k∏	X
ejpam-5336	181	5	i=1	i=1	X
ejpam-5336	181	6	pi	pi	NOUN
ejpam-5336	181	7	where	where	SCONJ
ejpam-5336	181	8	pk+1	pk+1	NOUN
ejpam-5336	181	9	is	be	AUX
ejpam-5336	181	10	the	the	DET
ejpam-5336	181	11	prime	prime	NOUN
ejpam-5336	181	12	next	next	ADJ
ejpam-5336	181	13	to	to	ADP
ejpam-5336	181	14	pk	pk	PROPN
ejpam-5336	181	15	.	.	PUNCT
ejpam-5336	181	16	”	"	PUNCT
ejpam-5336	182	1	also	also	ADV
ejpam-5336	182	2	,	,	PUNCT
ejpam-5336	182	3	if	if	SCONJ
ejpam-5336	182	4	k	k	PROPN
ejpam-5336	182	5	=	=	SYM
ejpam-5336	182	6	4	4	NUM
ejpam-5336	182	7	,	,	PUNCT
ejpam-5336	182	8	then	then	ADV
ejpam-5336	182	9	pk	pk	NOUN
ejpam-5336	182	10	=	=	PROPN
ejpam-5336	182	11	7	7	NUM
ejpam-5336	182	12	and	and	CCONJ
ejpam-5336	182	13	pk+1	pk+1	VERB
ejpam-5336	182	14	=	=	SYM
ejpam-5336	182	15	11	11	NUM
ejpam-5336	182	16	and	and	CCONJ
ejpam-5336	182	17	its	its	PRON
ejpam-5336	182	18	clear	clear	ADJ
ejpam-5336	182	19	112	112	NUM
ejpam-5336	182	20	<	<	X
ejpam-5336	182	21	(	(	PUNCT
ejpam-5336	182	22	2)(3)(5)(7	2)(3)(5)(7	NUM
ejpam-5336	182	23	)	)	PUNCT
ejpam-5336	182	24	.	.	PUNCT
ejpam-5336	183	1	so	so	ADV
ejpam-5336	183	2	,	,	PUNCT
ejpam-5336	183	3	we	we	PRON
ejpam-5336	183	4	have	have	VERB
ejpam-5336	183	5	the	the	DET
ejpam-5336	183	6	following	follow	VERB
ejpam-5336	183	7	corollary	corollary	NOUN
ejpam-5336	183	8	.	.	PUNCT
ejpam-5336	184	1	corollary	corollary	ADJ
ejpam-5336	184	2	1	1	NUM
ejpam-5336	184	3	.	.	PUNCT
ejpam-5336	185	1	let	let	VERB
ejpam-5336	185	2	k	k	PROPN
ejpam-5336	185	3	≥	≥	NUM
ejpam-5336	185	4	4	4	NUM
ejpam-5336	185	5	and	and	CCONJ
ejpam-5336	185	6	p1	p1	NOUN
ejpam-5336	185	7	,	,	PUNCT
ejpam-5336	185	8	p2	p2	NOUN
ejpam-5336	185	9	,	,	PUNCT
ejpam-5336	185	10	...	...	PUNCT
ejpam-5336	185	11	,	,	PUNCT
ejpam-5336	185	12	pk	pk	NOUN
ejpam-5336	185	13	be	be	AUX
ejpam-5336	185	14	the	the	DET
ejpam-5336	185	15	first	first	ADJ
ejpam-5336	185	16	k	k	PROPN
ejpam-5336	185	17	primes	prime	NOUN
ejpam-5336	185	18	.	.	PUNCT
ejpam-5336	186	1	then	then	ADV
ejpam-5336	186	2	p2k+1	p2k+1	VERB
ejpam-5336	186	3	<	<	X
ejpam-5336	186	4	k∏	k∏	X
ejpam-5336	186	5	i=1	i=1	X
ejpam-5336	186	6	pi	pi	NOUN
ejpam-5336	186	7	where	where	SCONJ
ejpam-5336	186	8	pk+1	pk+1	NOUN
ejpam-5336	186	9	is	be	AUX
ejpam-5336	186	10	the	the	DET
ejpam-5336	186	11	prime	prime	NOUN
ejpam-5336	186	12	next	next	ADJ
ejpam-5336	186	13	to	to	ADP
ejpam-5336	186	14	pk	pk	PROPN
ejpam-5336	186	15	.	.	PUNCT
ejpam-5336	186	16	o.	o.	PROPN
ejpam-5336	186	17	a.	a.	PROPN
ejpam-5336	186	18	abughneim	abughneim	PROPN
ejpam-5336	186	19	,	,	PUNCT
ejpam-5336	186	20	b.	b.	PROPN
ejpam-5336	186	21	abughazaleh	abughazaleh	PROPN
ejpam-5336	186	22	/	/	SYM
ejpam-5336	186	23	eur	eur	PROPN
ejpam-5336	186	24	.	.	PUNCT
ejpam-5336	187	1	j.	j.	PROPN
ejpam-5336	187	2	pure	pure	PROPN
ejpam-5336	187	3	appl	appl	PROPN
ejpam-5336	187	4	.	.	PROPN
ejpam-5336	187	5	math	math	PROPN
ejpam-5336	187	6	,	,	PUNCT
ejpam-5336	187	7	17	17	NUM
ejpam-5336	187	8	(	(	PUNCT
ejpam-5336	187	9	4	4	NUM
ejpam-5336	187	10	)	)	PUNCT
ejpam-5336	187	11	(	(	PUNCT
ejpam-5336	187	12	2024	2024	NUM
ejpam-5336	187	13	)	)	PUNCT
ejpam-5336	187	14	,	,	PUNCT
ejpam-5336	187	15	3557	3557	NUM
ejpam-5336	187	16	-	-	SYM
ejpam-5336	187	17	3566	3566	NUM
ejpam-5336	187	18	3563	3563	NUM
ejpam-5336	187	19	theorem	theorem	VERB
ejpam-5336	187	20	8	8	NUM
ejpam-5336	187	21	.	.	PUNCT
ejpam-5336	188	1	let	let	VERB
ejpam-5336	188	2	gm	gm	PROPN
ejpam-5336	188	3	be	be	AUX
ejpam-5336	188	4	a	a	DET
ejpam-5336	188	5	graph	graph	NOUN
ejpam-5336	188	6	with	with	ADP
ejpam-5336	188	7	maximum	maximum	ADJ
ejpam-5336	188	8	size	size	NOUN
ejpam-5336	188	9	such	such	ADJ
ejpam-5336	188	10	that	that	SCONJ
ejpam-5336	188	11	kn	kn	PROPN
ejpam-5336	188	12	∪	∪	ADP
ejpam-5336	188	13	gm	gm	PROPN
ejpam-5336	188	14	be	be	AUX
ejpam-5336	188	15	a	a	DET
ejpam-5336	188	16	pg	pg	NOUN
ejpam-5336	188	17	and	and	CCONJ
ejpam-5336	188	18	π	π	PROPN
ejpam-5336	188	19	(	(	PUNCT
ejpam-5336	188	20	n+m	n+m	NUM
ejpam-5336	188	21	)	)	PUNCT
ejpam-5336	188	22	≥	≥	PROPN
ejpam-5336	188	23	n.	n.	NOUN
ejpam-5336	188	24	then	then	ADV
ejpam-5336	188	25	gm	gm	PROPN
ejpam-5336	188	26	is	be	AUX
ejpam-5336	188	27	connected	connect	VERB
ejpam-5336	188	28	.	.	PUNCT
ejpam-5336	189	1	proof	proof	NOUN
ejpam-5336	189	2	.	.	PUNCT
ejpam-5336	190	1	since	since	SCONJ
ejpam-5336	190	2	π	π	PROPN
ejpam-5336	190	3	(	(	PUNCT
ejpam-5336	190	4	n+m	n+m	NUM
ejpam-5336	190	5	)	)	PUNCT
ejpam-5336	190	6	≥	≥	NOUN
ejpam-5336	190	7	n	n	CCONJ
ejpam-5336	190	8	,	,	PUNCT
ejpam-5336	190	9	then	then	ADV
ejpam-5336	190	10	the	the	DET
ejpam-5336	190	11	number	number	NOUN
ejpam-5336	190	12	of	of	ADP
ejpam-5336	190	13	primes	prime	NOUN
ejpam-5336	190	14	less	less	ADJ
ejpam-5336	190	15	than	than	ADP
ejpam-5336	190	16	or	or	CCONJ
ejpam-5336	190	17	equal	equal	ADJ
ejpam-5336	190	18	to	to	ADP
ejpam-5336	190	19	n	n	PROPN
ejpam-5336	190	20	+	+	CCONJ
ejpam-5336	190	21	m	m	VERB
ejpam-5336	190	22	is	be	AUX
ejpam-5336	190	23	greater	great	ADJ
ejpam-5336	190	24	than	than	ADP
ejpam-5336	190	25	or	or	CCONJ
ejpam-5336	190	26	equal	equal	ADJ
ejpam-5336	190	27	to	to	ADP
ejpam-5336	190	28	the	the	DET
ejpam-5336	190	29	number	number	NOUN
ejpam-5336	190	30	of	of	ADP
ejpam-5336	190	31	vertices	vertex	NOUN
ejpam-5336	190	32	of	of	ADP
ejpam-5336	190	33	kn	kn	PROPN
ejpam-5336	190	34	and	and	CCONJ
ejpam-5336	190	35	these	these	DET
ejpam-5336	190	36	primes	prime	NOUN
ejpam-5336	190	37	are	be	AUX
ejpam-5336	190	38	mutually	mutually	ADV
ejpam-5336	190	39	relatively	relatively	ADV
ejpam-5336	190	40	prime	prime	ADJ
ejpam-5336	190	41	.	.	PUNCT
ejpam-5336	191	1	so	so	ADV
ejpam-5336	191	2	,	,	PUNCT
ejpam-5336	191	3	we	we	PRON
ejpam-5336	191	4	can	can	AUX
ejpam-5336	191	5	use	use	VERB
ejpam-5336	191	6	a	a	DET
ejpam-5336	191	7	subset	subset	NOUN
ejpam-5336	191	8	of	of	ADP
ejpam-5336	191	9	these	these	DET
ejpam-5336	191	10	primes	prime	NOUN
ejpam-5336	191	11	to	to	PART
ejpam-5336	191	12	label	label	VERB
ejpam-5336	191	13	the	the	DET
ejpam-5336	191	14	vertices	vertex	NOUN
ejpam-5336	191	15	of	of	ADP
ejpam-5336	191	16	kn	kn	PROPN
ejpam-5336	191	17	and	and	CCONJ
ejpam-5336	191	18	hence	hence	ADV
ejpam-5336	191	19	one	one	NUM
ejpam-5336	191	20	of	of	ADP
ejpam-5336	191	21	the	the	DET
ejpam-5336	191	22	vertices	vertex	NOUN
ejpam-5336	191	23	of	of	ADP
ejpam-5336	191	24	gm	gm	PROPN
ejpam-5336	191	25	will	will	AUX
ejpam-5336	191	26	be	be	AUX
ejpam-5336	191	27	labeled	label	VERB
ejpam-5336	191	28	by	by	ADP
ejpam-5336	191	29	1	1	NUM
ejpam-5336	191	30	.	.	PUNCT
ejpam-5336	192	1	this	this	DET
ejpam-5336	192	2	vertex	vertex	NOUN
ejpam-5336	192	3	is	be	AUX
ejpam-5336	192	4	adjacent	adjacent	ADJ
ejpam-5336	192	5	to	to	ADP
ejpam-5336	192	6	all	all	DET
ejpam-5336	192	7	other	other	ADJ
ejpam-5336	192	8	vertices	vertex	NOUN
ejpam-5336	192	9	of	of	ADP
ejpam-5336	192	10	gm	gm	PROPN
ejpam-5336	192	11	,	,	PUNCT
ejpam-5336	192	12	because	because	SCONJ
ejpam-5336	192	13	gm	gm	PROPN
ejpam-5336	192	14	is	be	AUX
ejpam-5336	192	15	a	a	DET
ejpam-5336	192	16	graph	graph	NOUN
ejpam-5336	192	17	with	with	ADP
ejpam-5336	192	18	maximum	maximum	ADJ
ejpam-5336	192	19	size	size	NOUN
ejpam-5336	192	20	such	such	ADJ
ejpam-5336	192	21	that	that	SCONJ
ejpam-5336	192	22	kn	kn	PROPN
ejpam-5336	192	23	∪	∪	ADP
ejpam-5336	192	24	gm	gm	PROPN
ejpam-5336	192	25	is	be	AUX
ejpam-5336	192	26	a	a	DET
ejpam-5336	192	27	pg	pg	NOUN
ejpam-5336	192	28	.	.	PUNCT
ejpam-5336	193	1	thus	thus	ADV
ejpam-5336	193	2	,	,	PUNCT
ejpam-5336	193	3	gm	gm	PROPN
ejpam-5336	193	4	is	be	AUX
ejpam-5336	193	5	connected	connect	VERB
ejpam-5336	193	6	.	.	PUNCT
ejpam-5336	194	1	theorem	theorem	ADJ
ejpam-5336	194	2	9	9	NUM
ejpam-5336	194	3	.	.	PUNCT
ejpam-5336	195	1	let	let	VERB
ejpam-5336	195	2	gm	gm	PROPN
ejpam-5336	195	3	be	be	AUX
ejpam-5336	195	4	a	a	DET
ejpam-5336	195	5	graph	graph	NOUN
ejpam-5336	195	6	with	with	ADP
ejpam-5336	195	7	maximum	maximum	ADJ
ejpam-5336	195	8	size	size	NOUN
ejpam-5336	195	9	such	such	ADJ
ejpam-5336	195	10	that	that	SCONJ
ejpam-5336	195	11	kn	kn	PROPN
ejpam-5336	195	12	∪	∪	ADP
ejpam-5336	195	13	gm	gm	PROPN
ejpam-5336	195	14	be	be	AUX
ejpam-5336	195	15	a	a	DET
ejpam-5336	195	16	pg	pg	NOUN
ejpam-5336	195	17	and	and	CCONJ
ejpam-5336	195	18	π	π	PROPN
ejpam-5336	195	19	(	(	PUNCT
ejpam-5336	195	20	n+m	n+m	NUM
ejpam-5336	195	21	)	)	PUNCT
ejpam-5336	196	1	=	=	PUNCT
ejpam-5336	196	2	n−	n−	NOUN
ejpam-5336	196	3	1	1	NUM
ejpam-5336	196	4	.	.	PUNCT
ejpam-5336	197	1	then	then	ADV
ejpam-5336	197	2	(	(	PUNCT
ejpam-5336	197	3	i	i	NOUN
ejpam-5336	197	4	)	)	PUNCT
ejpam-5336	197	5	gm	gm	PROPN
ejpam-5336	197	6	is	be	AUX
ejpam-5336	197	7	the	the	DET
ejpam-5336	197	8	trivial	trivial	ADJ
ejpam-5336	197	9	graph	graph	NOUN
ejpam-5336	197	10	(	(	PUNCT
ejpam-5336	197	11	m	m	NOUN
ejpam-5336	197	12	=	=	NOUN
ejpam-5336	197	13	1	1	X
ejpam-5336	197	14	)	)	PUNCT
ejpam-5336	197	15	whenever	whenever	SCONJ
ejpam-5336	197	16	n+m	n+m	NUM
ejpam-5336	197	17	=	=	SYM
ejpam-5336	197	18	4	4	NUM
ejpam-5336	197	19	or	or	CCONJ
ejpam-5336	197	20	5	5	NUM
ejpam-5336	197	21	.	.	PUNCT
ejpam-5336	197	22	(	(	PUNCT
ejpam-5336	197	23	ii	ii	NOUN
ejpam-5336	197	24	)	)	PUNCT
ejpam-5336	197	25	gm	gm	PROPN
ejpam-5336	197	26	is	be	AUX
ejpam-5336	197	27	disconnected	disconnect	VERB
ejpam-5336	197	28	whenever	whenever	SCONJ
ejpam-5336	197	29	6	6	NUM
ejpam-5336	197	30	≤	≤	NOUN
ejpam-5336	197	31	n+m	n+m	PROPN
ejpam-5336	197	32	<	<	X
ejpam-5336	197	33	25	25	NUM
ejpam-5336	197	34	or	or	CCONJ
ejpam-5336	197	35	30	30	NUM
ejpam-5336	197	36	≤	≤	NOUN
ejpam-5336	197	37	n+m	n+m	PROPN
ejpam-5336	197	38	<	<	X
ejpam-5336	197	39	49	49	NUM
ejpam-5336	197	40	.	.	PUNCT
ejpam-5336	198	1	(	(	PUNCT
ejpam-5336	198	2	iii	iii	X
ejpam-5336	198	3	)	)	PUNCT
ejpam-5336	198	4	gm	gm	PROPN
ejpam-5336	198	5	is	be	AUX
ejpam-5336	198	6	connected	connect	VERB
ejpam-5336	198	7	whenever	whenever	SCONJ
ejpam-5336	198	8	25	25	NUM
ejpam-5336	198	9	≤	≤	NOUN
ejpam-5336	198	10	n+m	n+m	NUM
ejpam-5336	198	11	<	<	X
ejpam-5336	198	12	30	30	NUM
ejpam-5336	198	13	or	or	CCONJ
ejpam-5336	198	14	n+m	n+m	NUM
ejpam-5336	198	15	≥	≥	NOUN
ejpam-5336	198	16	49	49	NUM
ejpam-5336	198	17	.	.	PUNCT
ejpam-5336	199	1	proof	proof	NOUN
ejpam-5336	199	2	.	.	PUNCT
ejpam-5336	200	1	by	by	ADP
ejpam-5336	200	2	remark	remark	NOUN
ejpam-5336	200	3	1	1	NUM
ejpam-5336	200	4	,	,	PUNCT
ejpam-5336	200	5	label	label	VERB
ejpam-5336	200	6	the	the	DET
ejpam-5336	200	7	vertices	vertex	NOUN
ejpam-5336	200	8	of	of	ADP
ejpam-5336	200	9	kn	kn	PROPN
ejpam-5336	200	10	by	by	ADP
ejpam-5336	200	11	the	the	DET
ejpam-5336	200	12	primes	prime	NOUN
ejpam-5336	200	13	less	less	ADJ
ejpam-5336	200	14	than	than	ADP
ejpam-5336	200	15	or	or	CCONJ
ejpam-5336	200	16	equal	equal	ADJ
ejpam-5336	200	17	to	to	ADP
ejpam-5336	200	18	n+m	n+m	PROPN
ejpam-5336	200	19	together	together	ADV
ejpam-5336	200	20	with	with	ADP
ejpam-5336	200	21	1	1	NUM
ejpam-5336	200	22	and	and	CCONJ
ejpam-5336	200	23	label	label	VERB
ejpam-5336	200	24	the	the	DET
ejpam-5336	200	25	vertices	vertex	NOUN
ejpam-5336	200	26	of	of	ADP
ejpam-5336	200	27	gm	gm	PROPN
ejpam-5336	200	28	by	by	ADP
ejpam-5336	200	29	the	the	DET
ejpam-5336	200	30	composite	composite	ADJ
ejpam-5336	200	31	numbers	number	NOUN
ejpam-5336	200	32	less	less	ADJ
ejpam-5336	200	33	than	than	ADP
ejpam-5336	200	34	or	or	CCONJ
ejpam-5336	200	35	equal	equal	ADJ
ejpam-5336	200	36	to	to	ADP
ejpam-5336	200	37	n+m	n+m	NUM
ejpam-5336	200	38	.	.	PUNCT
ejpam-5336	201	1	(	(	PUNCT
ejpam-5336	201	2	i	i	NOUN
ejpam-5336	201	3	)	)	PUNCT
ejpam-5336	201	4	if	if	SCONJ
ejpam-5336	201	5	n+m	n+m	NUM
ejpam-5336	201	6	=	=	SYM
ejpam-5336	201	7	4	4	NUM
ejpam-5336	201	8	,	,	PUNCT
ejpam-5336	201	9	then	then	ADV
ejpam-5336	201	10	π	π	X
ejpam-5336	201	11	(	(	PUNCT
ejpam-5336	201	12	n+m	n+m	NUM
ejpam-5336	201	13	)	)	PUNCT
ejpam-5336	201	14	=	=	SYM
ejpam-5336	202	1	2	2	X
ejpam-5336	202	2	.	.	PUNCT
ejpam-5336	203	1	so	so	ADV
ejpam-5336	203	2	n	n	NOUN
ejpam-5336	203	3	=	=	SYM
ejpam-5336	203	4	π	π	PROPN
ejpam-5336	203	5	(	(	PUNCT
ejpam-5336	203	6	n+m)+1	n+m)+1	NOUN
ejpam-5336	203	7	=	=	SYM
ejpam-5336	203	8	3	3	X
ejpam-5336	203	9	.	.	PUNCT
ejpam-5336	204	1	thus	thus	ADV
ejpam-5336	204	2	m	m	VERB
ejpam-5336	204	3	=	=	ADJ
ejpam-5336	204	4	1	1	X
ejpam-5336	204	5	.	.	PUNCT
ejpam-5336	204	6	similarly	similarly	ADV
ejpam-5336	204	7	,	,	PUNCT
ejpam-5336	204	8	if	if	SCONJ
ejpam-5336	204	9	n+m	n+m	NUM
ejpam-5336	204	10	=	=	SYM
ejpam-5336	204	11	5	5	X
ejpam-5336	204	12	.	.	PUNCT
ejpam-5336	204	13	(	(	PUNCT
ejpam-5336	204	14	ii	ii	NOUN
ejpam-5336	204	15	)	)	PUNCT
ejpam-5336	204	16	if	if	SCONJ
ejpam-5336	204	17	6	6	NUM
ejpam-5336	204	18	≤	≤	NOUN
ejpam-5336	204	19	n+m	n+m	NUM
ejpam-5336	204	20	<	<	X
ejpam-5336	204	21	25	25	NUM
ejpam-5336	204	22	,	,	PUNCT
ejpam-5336	204	23	then	then	ADV
ejpam-5336	204	24	the	the	DET
ejpam-5336	204	25	vertex	vertex	NOUN
ejpam-5336	204	26	of	of	ADP
ejpam-5336	204	27	gm	gm	PROPN
ejpam-5336	204	28	whose	whose	DET
ejpam-5336	204	29	label	label	NOUN
ejpam-5336	204	30	is	be	AUX
ejpam-5336	204	31	6	6	NUM
ejpam-5336	204	32	must	must	AUX
ejpam-5336	204	33	be	be	AUX
ejpam-5336	204	34	an	an	DET
ejpam-5336	204	35	isolated	isolated	ADJ
ejpam-5336	204	36	vertex	vertex	NOUN
ejpam-5336	204	37	in	in	ADP
ejpam-5336	204	38	gm	gm	PROPN
ejpam-5336	204	39	because	because	SCONJ
ejpam-5336	204	40	any	any	DET
ejpam-5336	204	41	composite	composite	ADJ
ejpam-5336	204	42	number	number	NOUN
ejpam-5336	204	43	less	less	ADJ
ejpam-5336	204	44	than	than	ADP
ejpam-5336	204	45	25	25	NUM
ejpam-5336	204	46	is	be	AUX
ejpam-5336	204	47	not	not	PART
ejpam-5336	204	48	relatively	relatively	ADV
ejpam-5336	204	49	prime	prime	ADJ
ejpam-5336	204	50	to	to	ADP
ejpam-5336	204	51	6	6	NUM
ejpam-5336	204	52	.	.	PUNCT
ejpam-5336	205	1	thus	thus	ADV
ejpam-5336	205	2	gm	gm	PROPN
ejpam-5336	205	3	is	be	AUX
ejpam-5336	205	4	disconnected	disconnect	VERB
ejpam-5336	205	5	.	.	PUNCT
ejpam-5336	206	1	if	if	SCONJ
ejpam-5336	206	2	30	30	NUM
ejpam-5336	206	3	≤	≤	NOUN
ejpam-5336	206	4	n+m	n+m	PROPN
ejpam-5336	206	5	<	<	X
ejpam-5336	206	6	49	49	NUM
ejpam-5336	206	7	,	,	PUNCT
ejpam-5336	206	8	then	then	ADV
ejpam-5336	206	9	any	any	DET
ejpam-5336	206	10	composite	composite	ADJ
ejpam-5336	206	11	number	number	NOUN
ejpam-5336	206	12	less	less	ADJ
ejpam-5336	206	13	than	than	ADP
ejpam-5336	206	14	49	49	NUM
ejpam-5336	206	15	is	be	AUX
ejpam-5336	206	16	not	not	PART
ejpam-5336	206	17	relatively	relatively	ADV
ejpam-5336	206	18	prime	prime	ADJ
ejpam-5336	206	19	to	to	ADP
ejpam-5336	206	20	30	30	NUM
ejpam-5336	206	21	so	so	ADV
ejpam-5336	206	22	,	,	PUNCT
ejpam-5336	206	23	30	30	NUM
ejpam-5336	206	24	is	be	AUX
ejpam-5336	206	25	isolated	isolate	VERB
ejpam-5336	206	26	and	and	CCONJ
ejpam-5336	206	27	thus	thus	ADV
ejpam-5336	206	28	gm	gm	PROPN
ejpam-5336	206	29	is	be	AUX
ejpam-5336	206	30	disconnected	disconnect	VERB
ejpam-5336	206	31	.	.	PUNCT
ejpam-5336	207	1	(	(	PUNCT
ejpam-5336	207	2	iii	iii	NOUN
ejpam-5336	207	3	)	)	PUNCT
ejpam-5336	207	4	let	let	VERB
ejpam-5336	207	5	p1	p1	NOUN
ejpam-5336	207	6	,	,	PUNCT
ejpam-5336	207	7	p2	p2	NOUN
ejpam-5336	207	8	,	,	PUNCT
ejpam-5336	207	9	...	...	PUNCT
ejpam-5336	207	10	,	,	PUNCT
ejpam-5336	207	11	pk	pk	NOUN
ejpam-5336	207	12	be	be	AUX
ejpam-5336	207	13	the	the	DET
ejpam-5336	207	14	primes	prime	NOUN
ejpam-5336	207	15	less	less	ADJ
ejpam-5336	207	16	than	than	ADP
ejpam-5336	207	17	or	or	CCONJ
ejpam-5336	207	18	equal	equal	ADJ
ejpam-5336	207	19	√	√	ADJ
ejpam-5336	207	20	n	n	NOUN
ejpam-5336	207	21	in	in	ADP
ejpam-5336	207	22	ascending	ascend	VERB
ejpam-5336	207	23	order	order	NOUN
ejpam-5336	207	24	.	.	PUNCT
ejpam-5336	208	1	we	we	PRON
ejpam-5336	208	2	refer	refer	VERB
ejpam-5336	208	3	to	to	ADP
ejpam-5336	208	4	the	the	DET
ejpam-5336	208	5	vertices	vertex	NOUN
ejpam-5336	208	6	of	of	ADP
ejpam-5336	208	7	gm	gm	PROPN
ejpam-5336	208	8	by	by	ADP
ejpam-5336	208	9	their	their	PRON
ejpam-5336	208	10	labels	label	NOUN
ejpam-5336	208	11	.	.	PUNCT
ejpam-5336	209	1	we	we	PRON
ejpam-5336	209	2	partition	partition	VERB
ejpam-5336	209	3	the	the	DET
ejpam-5336	209	4	vertices	vertex	NOUN
ejpam-5336	209	5	of	of	ADP
ejpam-5336	209	6	gm	gm	PROPN
ejpam-5336	209	7	into	into	ADP
ejpam-5336	209	8	the	the	DET
ejpam-5336	209	9	following	follow	VERB
ejpam-5336	209	10	sets	set	VERB
ejpam-5336	209	11	a0	a0	NOUN
ejpam-5336	209	12	=	=	SYM
ejpam-5336	209	13	{	{	PUNCT
ejpam-5336	209	14	p21	p21	NOUN
ejpam-5336	209	15	,	,	PUNCT
ejpam-5336	209	16	p22	p22	NOUN
ejpam-5336	209	17	,	,	PUNCT
ejpam-5336	209	18	...	...	PUNCT
ejpam-5336	209	19	,	,	PUNCT
ejpam-5336	209	20	p2k	p2k	PROPN
ejpam-5336	209	21	}	}	PUNCT
ejpam-5336	209	22	and	and	CCONJ
ejpam-5336	209	23	ai	ai	VERB
ejpam-5336	209	24	=	=	PUNCT
ejpam-5336	209	25	{	{	PUNCT
ejpam-5336	209	26	s	s	X
ejpam-5336	209	27	:	:	PUNCT
ejpam-5336	209	28	pi	pi	NOUN
ejpam-5336	209	29	does	do	AUX
ejpam-5336	209	30	not	not	PART
ejpam-5336	209	31	divide	divide	VERB
ejpam-5336	209	32	s	s	PRON
ejpam-5336	209	33	}	}	PUNCT
ejpam-5336	209	34	−	−	PROPN
ejpam-5336	209	35	j	j	PROPN
ejpam-5336	209	36	=	=	PROPN
ejpam-5336	209	37	i−1⋃	i−1⋃	PROPN
ejpam-5336	209	38	j=0	j=0	PROPN
ejpam-5336	209	39	aj	aj	PROPN
ejpam-5336	209	40	for	for	ADP
ejpam-5336	209	41	all	all	DET
ejpam-5336	209	42	i	i	PRON
ejpam-5336	209	43	=	=	NOUN
ejpam-5336	209	44	1	1	NUM
ejpam-5336	209	45	,	,	PUNCT
ejpam-5336	209	46	2	2	NUM
ejpam-5336	209	47	,	,	PUNCT
ejpam-5336	209	48	...	...	PUNCT
ejpam-5336	209	49	k.	k.	PROPN
ejpam-5336	209	50	notice	notice	VERB
ejpam-5336	209	51	that	that	SCONJ
ejpam-5336	209	52	a0	a0	NOUN
ejpam-5336	209	53	,	,	PUNCT
ejpam-5336	209	54	a1	a1	PROPN
ejpam-5336	209	55	,	,	PUNCT
ejpam-5336	209	56	a2	a2	PROPN
ejpam-5336	209	57	,	,	PUNCT
ejpam-5336	209	58	...	...	PUNCT
ejpam-5336	209	59	,	,	PUNCT
ejpam-5336	209	60	ak	ak	PROPN
ejpam-5336	209	61	are	be	AUX
ejpam-5336	209	62	mutually	mutually	ADV
ejpam-5336	209	63	disjoint	disjoint	NOUN
ejpam-5336	209	64	sets	set	NOUN
ejpam-5336	209	65	.	.	PUNCT
ejpam-5336	210	1	we	we	PRON
ejpam-5336	210	2	want	want	VERB
ejpam-5336	210	3	to	to	PART
ejpam-5336	210	4	show	show	VERB
ejpam-5336	210	5	that	that	SCONJ
ejpam-5336	210	6	gm	gm	PROPN
ejpam-5336	211	1	=	=	PUNCT
ejpam-5336	211	2	i	i	PROPN
ejpam-5336	211	3	=	=	VERB
ejpam-5336	211	4	k⋃	k⋃	X
ejpam-5336	211	5	i=0	i=0	PROPN
ejpam-5336	211	6	ai	ai	AUX
ejpam-5336	211	7	.	.	PUNCT
ejpam-5336	211	8	suppose	suppose	VERB
ejpam-5336	211	9	there	there	PRON
ejpam-5336	211	10	is	be	VERB
ejpam-5336	211	11	a	a	DET
ejpam-5336	211	12	composite	composite	ADJ
ejpam-5336	211	13	number	number	NOUN
ejpam-5336	211	14	t	t	NOUN
ejpam-5336	211	15	less	less	ADJ
ejpam-5336	211	16	than	than	ADP
ejpam-5336	211	17	or	or	CCONJ
ejpam-5336	211	18	equal	equal	ADJ
ejpam-5336	211	19	to	to	ADP
ejpam-5336	211	20	n+m	n+m	PROPN
ejpam-5336	211	21	such	such	ADJ
ejpam-5336	211	22	that	that	DET
ejpam-5336	211	23	pi	pi	NOUN
ejpam-5336	211	24	divides	divide	VERB
ejpam-5336	211	25	t	t	PROPN
ejpam-5336	211	26	for	for	ADP
ejpam-5336	211	27	all	all	DET
ejpam-5336	211	28	i	i	PRON
ejpam-5336	211	29	=	=	NOUN
ejpam-5336	211	30	1	1	NUM
ejpam-5336	211	31	,	,	PUNCT
ejpam-5336	211	32	2	2	NUM
ejpam-5336	211	33	,	,	PUNCT
ejpam-5336	211	34	...	...	PUNCT
ejpam-5336	212	1	k.	k.	PROPN
ejpam-5336	213	1	if	if	SCONJ
ejpam-5336	213	2	25	25	NUM
ejpam-5336	213	3	≤	≤	NOUN
ejpam-5336	213	4	n+m	n+m	NUM
ejpam-5336	213	5	<	<	X
ejpam-5336	213	6	30	30	NUM
ejpam-5336	213	7	,	,	PUNCT
ejpam-5336	213	8	then	then	ADV
ejpam-5336	213	9	2	2	NUM
ejpam-5336	213	10	divides	divide	VERB
ejpam-5336	213	11	t	t	PROPN
ejpam-5336	213	12	,	,	PUNCT
ejpam-5336	213	13	3	3	NUM
ejpam-5336	213	14	divides	divide	VERB
ejpam-5336	213	15	t	t	NOUN
ejpam-5336	213	16	and	and	CCONJ
ejpam-5336	213	17	5	5	NUM
ejpam-5336	213	18	divides	divide	NOUN
ejpam-5336	213	19	t.	t.	PROPN
ejpam-5336	213	20	so	so	ADV
ejpam-5336	213	21	,	,	PUNCT
ejpam-5336	213	22	t	t	PROPN
ejpam-5336	213	23	≥	≥	NUM
ejpam-5336	213	24	30	30	NUM
ejpam-5336	213	25	which	which	PRON
ejpam-5336	213	26	is	be	AUX
ejpam-5336	213	27	a	a	DET
ejpam-5336	213	28	contradiction	contradiction	NOUN
ejpam-5336	213	29	.	.	PUNCT
ejpam-5336	214	1	if	if	SCONJ
ejpam-5336	214	2	n+m	n+m	NUM
ejpam-5336	214	3	≥	≥	NUM
ejpam-5336	214	4	49	49	NUM
ejpam-5336	214	5	,	,	PUNCT
ejpam-5336	214	6	then	then	ADV
ejpam-5336	214	7	by	by	ADP
ejpam-5336	214	8	corollary	corollary	ADJ
ejpam-5336	214	9	1	1	NUM
ejpam-5336	214	10	we	we	PRON
ejpam-5336	214	11	o.	o.	NOUN
ejpam-5336	214	12	a.	a.	PROPN
ejpam-5336	214	13	abughneim	abughneim	PROPN
ejpam-5336	214	14	,	,	PUNCT
ejpam-5336	214	15	b.	b.	PROPN
ejpam-5336	214	16	abughazaleh	abughazaleh	PROPN
ejpam-5336	214	17	/	/	SYM
ejpam-5336	214	18	eur	eur	PROPN
ejpam-5336	214	19	.	.	PUNCT
ejpam-5336	215	1	j.	j.	PROPN
ejpam-5336	215	2	pure	pure	PROPN
ejpam-5336	215	3	appl	appl	PROPN
ejpam-5336	215	4	.	.	PROPN
ejpam-5336	215	5	math	math	PROPN
ejpam-5336	215	6	,	,	PUNCT
ejpam-5336	215	7	17	17	NUM
ejpam-5336	215	8	(	(	PUNCT
ejpam-5336	215	9	4	4	NUM
ejpam-5336	215	10	)	)	PUNCT
ejpam-5336	215	11	(	(	PUNCT
ejpam-5336	215	12	2024	2024	NUM
ejpam-5336	215	13	)	)	PUNCT
ejpam-5336	215	14	,	,	PUNCT
ejpam-5336	215	15	3557	3557	NUM
ejpam-5336	215	16	-	-	SYM
ejpam-5336	215	17	3566	3566	NUM
ejpam-5336	215	18	3564	3564	NUM
ejpam-5336	215	19	get	get	VERB
ejpam-5336	215	20	p2k+1	p2k+1	NOUN
ejpam-5336	215	21	<	<	X
ejpam-5336	215	22	k∏	k∏	PROPN
ejpam-5336	215	23	i=1	i=1	X
ejpam-5336	215	24	pi	pi	NOUN
ejpam-5336	215	25	where	where	SCONJ
ejpam-5336	215	26	pk+1	pk+1	NOUN
ejpam-5336	215	27	is	be	AUX
ejpam-5336	215	28	the	the	DET
ejpam-5336	215	29	prime	prime	NOUN
ejpam-5336	215	30	next	next	ADJ
ejpam-5336	215	31	to	to	ADP
ejpam-5336	215	32	pk	pk	PROPN
ejpam-5336	215	33	.	.	PUNCT
ejpam-5336	216	1	so	so	ADV
ejpam-5336	216	2	,	,	PUNCT
ejpam-5336	216	3	n+m	n+m	AUX
ejpam-5336	216	4	<	<	X
ejpam-5336	216	5	p2k+1	p2k+1	X
ejpam-5336	216	6	<	<	X
ejpam-5336	216	7	k∏	k∏	PROPN
ejpam-5336	216	8	i=1	i=1	PROPN
ejpam-5336	216	9	pi	pi	NOUN
ejpam-5336	216	10	<	<	X
ejpam-5336	216	11	t	t	PROPN
ejpam-5336	216	12	because	because	SCONJ
ejpam-5336	216	13	pi	pi	NOUN
ejpam-5336	216	14	divides	divide	VERB
ejpam-5336	216	15	t	t	PROPN
ejpam-5336	216	16	for	for	ADP
ejpam-5336	216	17	all	all	DET
ejpam-5336	216	18	i	i	PRON
ejpam-5336	217	1	=	=	NOUN
ejpam-5336	217	2	1	1	NUM
ejpam-5336	217	3	,	,	PUNCT
ejpam-5336	217	4	2	2	NUM
ejpam-5336	217	5	,	,	PUNCT
ejpam-5336	217	6	...	...	PUNCT
ejpam-5336	217	7	,	,	PUNCT
ejpam-5336	217	8	k	k	X
ejpam-5336	217	9	which	which	PRON
ejpam-5336	217	10	is	be	AUX
ejpam-5336	217	11	a	a	DET
ejpam-5336	217	12	contradiction	contradiction	NOUN
ejpam-5336	217	13	.	.	PUNCT
ejpam-5336	218	1	now	now	ADV
ejpam-5336	218	2	,	,	PUNCT
ejpam-5336	218	3	let	let	VERB
ejpam-5336	218	4	u	u	NOUN
ejpam-5336	218	5	,	,	PUNCT
ejpam-5336	218	6	v	v	PROPN
ejpam-5336	218	7	∈	∈	PROPN
ejpam-5336	218	8	gm	gm	PROPN
ejpam-5336	218	9	.	.	PUNCT
ejpam-5336	219	1	we	we	PRON
ejpam-5336	219	2	want	want	VERB
ejpam-5336	219	3	to	to	PART
ejpam-5336	219	4	find	find	VERB
ejpam-5336	219	5	a	a	DET
ejpam-5336	219	6	path	path	NOUN
ejpam-5336	219	7	between	between	ADP
ejpam-5336	219	8	u	u	NOUN
ejpam-5336	219	9	and	and	CCONJ
ejpam-5336	219	10	v	v	NOUN
ejpam-5336	219	11	and	and	CCONJ
ejpam-5336	219	12	this	this	PRON
ejpam-5336	219	13	shows	show	VERB
ejpam-5336	219	14	that	that	SCONJ
ejpam-5336	219	15	gm	gm	PROPN
ejpam-5336	219	16	is	be	AUX
ejpam-5336	219	17	connected	connect	VERB
ejpam-5336	219	18	.	.	PUNCT
ejpam-5336	220	1	we	we	PRON
ejpam-5336	220	2	have	have	VERB
ejpam-5336	220	3	the	the	DET
ejpam-5336	220	4	following	follow	VERB
ejpam-5336	220	5	cases	case	NOUN
ejpam-5336	220	6	:	:	PUNCT
ejpam-5336	220	7	(	(	PUNCT
ejpam-5336	220	8	a	a	X
ejpam-5336	220	9	)	)	PUNCT
ejpam-5336	220	10	if	if	SCONJ
ejpam-5336	220	11	u	u	NOUN
ejpam-5336	220	12	,	,	PUNCT
ejpam-5336	220	13	v	v	PROPN
ejpam-5336	220	14	∈	∈	PROPN
ejpam-5336	220	15	a0	a0	NOUN
ejpam-5336	220	16	,	,	PUNCT
ejpam-5336	220	17	then	then	ADV
ejpam-5336	220	18	u−	u−	PROPN
ejpam-5336	220	19	v	v	NOUN
ejpam-5336	220	20	is	be	AUX
ejpam-5336	220	21	a	a	DET
ejpam-5336	220	22	path	path	NOUN
ejpam-5336	220	23	in	in	ADP
ejpam-5336	220	24	gm	gm	PROPN
ejpam-5336	220	25	.	.	PUNCT
ejpam-5336	221	1	(	(	PUNCT
ejpam-5336	221	2	b	b	X
ejpam-5336	221	3	)	)	PUNCT
ejpam-5336	221	4	if	if	SCONJ
ejpam-5336	221	5	u	u	NOUN
ejpam-5336	221	6	,	,	PUNCT
ejpam-5336	221	7	v	v	PROPN
ejpam-5336	221	8	∈	∈	NOUN
ejpam-5336	221	9	ai	ai	VERB
ejpam-5336	221	10	for	for	ADP
ejpam-5336	221	11	some	some	DET
ejpam-5336	221	12	i	i	NOUN
ejpam-5336	221	13	=	=	NOUN
ejpam-5336	221	14	1	1	NUM
ejpam-5336	221	15	,	,	PUNCT
ejpam-5336	221	16	2	2	NUM
ejpam-5336	221	17	,	,	PUNCT
ejpam-5336	221	18	...	...	PUNCT
ejpam-5336	222	1	k	k	X
ejpam-5336	222	2	,	,	PUNCT
ejpam-5336	222	3	then	then	ADV
ejpam-5336	222	4	u−	u−	PROPN
ejpam-5336	222	5	p2i	p2i	ADV
ejpam-5336	222	6	−	−	PROPN
ejpam-5336	222	7	v	v	NOUN
ejpam-5336	222	8	is	be	AUX
ejpam-5336	222	9	a	a	DET
ejpam-5336	222	10	path	path	NOUN
ejpam-5336	222	11	in	in	ADP
ejpam-5336	222	12	gm	gm	PROPN
ejpam-5336	222	13	.	.	PUNCT
ejpam-5336	223	1	(	(	PUNCT
ejpam-5336	223	2	c	c	X
ejpam-5336	223	3	)	)	PUNCT
ejpam-5336	223	4	if	if	SCONJ
ejpam-5336	223	5	u	u	PROPN
ejpam-5336	223	6	∈	∈	PROPN
ejpam-5336	223	7	ai	ai	VERB
ejpam-5336	223	8	for	for	ADP
ejpam-5336	223	9	some	some	DET
ejpam-5336	223	10	i	i	NOUN
ejpam-5336	223	11	=	=	NOUN
ejpam-5336	223	12	1	1	NUM
ejpam-5336	223	13	,	,	PUNCT
ejpam-5336	223	14	2	2	NUM
ejpam-5336	223	15	,	,	PUNCT
ejpam-5336	223	16	...	...	PUNCT
ejpam-5336	224	1	k	k	PROPN
ejpam-5336	224	2	and	and	CCONJ
ejpam-5336	224	3	v	v	ADP
ejpam-5336	224	4	∈	∈	PROPN
ejpam-5336	224	5	aj	aj	PROPN
ejpam-5336	224	6	for	for	ADP
ejpam-5336	224	7	some	some	PRON
ejpam-5336	224	8	j	j	PROPN
ejpam-5336	224	9	=	=	SYM
ejpam-5336	224	10	1	1	NUM
ejpam-5336	224	11	,	,	PUNCT
ejpam-5336	224	12	2	2	NUM
ejpam-5336	224	13	,	,	PUNCT
ejpam-5336	224	14	...	...	PUNCT
ejpam-5336	225	1	k	k	X
ejpam-5336	225	2	such	such	ADJ
ejpam-5336	225	3	that	that	SCONJ
ejpam-5336	225	4	i	i	PRON
ejpam-5336	225	5	̸=	̸=	PROPN
ejpam-5336	225	6	j	j	PROPN
ejpam-5336	225	7	,	,	PUNCT
ejpam-5336	225	8	then	then	ADV
ejpam-5336	225	9	u−	u−	PROPN
ejpam-5336	225	10	p2i	p2i	ADV
ejpam-5336	225	11	−	−	PROPN
ejpam-5336	225	12	p2j	p2j	PROPN
ejpam-5336	225	13	−	−	PROPN
ejpam-5336	225	14	v	v	NOUN
ejpam-5336	225	15	is	be	AUX
ejpam-5336	225	16	a	a	DET
ejpam-5336	225	17	path	path	NOUN
ejpam-5336	225	18	in	in	ADP
ejpam-5336	225	19	gm	gm	PROPN
ejpam-5336	225	20	.	.	PUNCT
ejpam-5336	226	1	(	(	PUNCT
ejpam-5336	226	2	d	d	X
ejpam-5336	226	3	)	)	PUNCT
ejpam-5336	226	4	if	if	SCONJ
ejpam-5336	226	5	u	u	PROPN
ejpam-5336	226	6	∈	∈	PROPN
ejpam-5336	226	7	a0	a0	NOUN
ejpam-5336	226	8	and	and	CCONJ
ejpam-5336	226	9	v	v	ADP
ejpam-5336	226	10	∈	∈	PROPN
ejpam-5336	226	11	aj	aj	PROPN
ejpam-5336	226	12	for	for	ADP
ejpam-5336	226	13	some	some	PRON
ejpam-5336	226	14	j	j	PROPN
ejpam-5336	226	15	=	=	SYM
ejpam-5336	226	16	1	1	NUM
ejpam-5336	226	17	,	,	PUNCT
ejpam-5336	226	18	2	2	NUM
ejpam-5336	226	19	,	,	PUNCT
ejpam-5336	226	20	...	...	PUNCT
ejpam-5336	227	1	k	k	X
ejpam-5336	227	2	,	,	PUNCT
ejpam-5336	227	3	then	then	ADV
ejpam-5336	227	4	u	u	NOUN
ejpam-5336	227	5	−	−	PROPN
ejpam-5336	227	6	p2j	p2j	PROPN
ejpam-5336	227	7	−	−	PROPN
ejpam-5336	227	8	v	v	NOUN
ejpam-5336	227	9	is	be	AUX
ejpam-5336	227	10	a	a	DET
ejpam-5336	227	11	path	path	NOUN
ejpam-5336	227	12	in	in	ADP
ejpam-5336	227	13	gm	gm	PROPN
ejpam-5336	227	14	whenever	whenever	SCONJ
ejpam-5336	227	15	u	u	PROPN
ejpam-5336	227	16	̸=	̸=	PROPN
ejpam-5336	227	17	p2j	p2j	PROPN
ejpam-5336	227	18	and	and	CCONJ
ejpam-5336	227	19	u−	u−	PROPN
ejpam-5336	227	20	v	v	NOUN
ejpam-5336	227	21	is	be	AUX
ejpam-5336	227	22	a	a	DET
ejpam-5336	227	23	path	path	NOUN
ejpam-5336	227	24	in	in	ADP
ejpam-5336	227	25	gmwhenever	gmwhenever	NOUN
ejpam-5336	227	26	u	u	NOUN
ejpam-5336	227	27	=	=	PROPN
ejpam-5336	227	28	p2j	p2j	PROPN
ejpam-5336	227	29	.	.	PUNCT
ejpam-5336	228	1	therefore	therefore	ADV
ejpam-5336	228	2	,	,	PUNCT
ejpam-5336	228	3	gm	gm	PROPN
ejpam-5336	228	4	is	be	AUX
ejpam-5336	228	5	connected	connect	VERB
ejpam-5336	228	6	.	.	PUNCT
ejpam-5336	229	1	example	example	NOUN
ejpam-5336	230	1	1	1	NUM
ejpam-5336	230	2	.	.	PUNCT
ejpam-5336	231	1	(	(	PUNCT
ejpam-5336	231	2	i	i	NOUN
ejpam-5336	231	3	)	)	PUNCT
ejpam-5336	231	4	consider	consider	VERB
ejpam-5336	231	5	the	the	DET
ejpam-5336	231	6	complete	complete	ADJ
ejpam-5336	231	7	graph	graph	NOUN
ejpam-5336	231	8	k5	k5	PROPN
ejpam-5336	231	9	and	and	CCONJ
ejpam-5336	231	10	let	let	VERB
ejpam-5336	231	11	g4	g4	NOUN
ejpam-5336	231	12	be	be	AUX
ejpam-5336	231	13	a	a	DET
ejpam-5336	231	14	graph	graph	NOUN
ejpam-5336	231	15	with	with	ADP
ejpam-5336	231	16	maximum	maximum	ADJ
ejpam-5336	231	17	size	size	NOUN
ejpam-5336	231	18	such	such	ADJ
ejpam-5336	231	19	that	that	SCONJ
ejpam-5336	231	20	k5	k5	PROPN
ejpam-5336	231	21	∪g4	∪g4	NOUN
ejpam-5336	231	22	is	be	AUX
ejpam-5336	231	23	a	a	DET
ejpam-5336	231	24	pg	pg	NOUN
ejpam-5336	231	25	.	.	PUNCT
ejpam-5336	232	1	then	then	ADV
ejpam-5336	232	2	,	,	PUNCT
ejpam-5336	232	3	π(9	π(9	PROPN
ejpam-5336	232	4	)	)	PUNCT
ejpam-5336	232	5	=	=	SYM
ejpam-5336	232	6	4	4	NUM
ejpam-5336	232	7	=	=	SYM
ejpam-5336	232	8	5−	5−	NUM
ejpam-5336	232	9	1	1	NUM
ejpam-5336	232	10	and	and	CCONJ
ejpam-5336	232	11	since	since	SCONJ
ejpam-5336	232	12	k5	k5	PROPN
ejpam-5336	232	13	∪g4	∪g4	NOUN
ejpam-5336	232	14	is	be	AUX
ejpam-5336	232	15	a	a	DET
ejpam-5336	232	16	pg	pg	NOUN
ejpam-5336	232	17	,	,	PUNCT
ejpam-5336	232	18	we	we	PRON
ejpam-5336	232	19	can	can	AUX
ejpam-5336	232	20	label	label	VERB
ejpam-5336	232	21	the	the	DET
ejpam-5336	232	22	vertices	vertex	NOUN
ejpam-5336	232	23	of	of	ADP
ejpam-5336	232	24	k5	k5	PROPN
ejpam-5336	232	25	by	by	ADP
ejpam-5336	232	26	the	the	DET
ejpam-5336	232	27	numbers	number	NOUN
ejpam-5336	232	28	1	1	NUM
ejpam-5336	232	29	,	,	PUNCT
ejpam-5336	232	30	2	2	NUM
ejpam-5336	232	31	,	,	PUNCT
ejpam-5336	232	32	3	3	NUM
ejpam-5336	232	33	,	,	PUNCT
ejpam-5336	232	34	5	5	NUM
ejpam-5336	232	35	,	,	PUNCT
ejpam-5336	232	36	7	7	NUM
ejpam-5336	232	37	and	and	CCONJ
ejpam-5336	232	38	hence	hence	ADV
ejpam-5336	232	39	g4	g4	NOUN
ejpam-5336	232	40	is	be	AUX
ejpam-5336	232	41	the	the	DET
ejpam-5336	232	42	following	follow	VERB
ejpam-5336	232	43	graph	graph	NOUN
ejpam-5336	232	44	4	4	NUM
ejpam-5336	232	45	86	86	NUM
ejpam-5336	232	46	9	9	NUM
ejpam-5336	232	47	g4	g4	NOUN
ejpam-5336	232	48	so	so	ADV
ejpam-5336	232	49	,	,	PUNCT
ejpam-5336	232	50	g4	g4	NOUN
ejpam-5336	232	51	is	be	AUX
ejpam-5336	232	52	disconnected	disconnect	VERB
ejpam-5336	232	53	.	.	PUNCT
ejpam-5336	233	1	(	(	PUNCT
ejpam-5336	233	2	ii	ii	NOUN
ejpam-5336	233	3	)	)	PUNCT
ejpam-5336	233	4	consider	consider	VERB
ejpam-5336	233	5	the	the	DET
ejpam-5336	233	6	complete	complete	ADJ
ejpam-5336	233	7	graph	graph	NOUN
ejpam-5336	233	8	k10	k10	NOUN
ejpam-5336	233	9	and	and	CCONJ
ejpam-5336	233	10	let	let	VERB
ejpam-5336	233	11	g15	g15	PROPN
ejpam-5336	233	12	be	be	AUX
ejpam-5336	233	13	a	a	DET
ejpam-5336	233	14	graph	graph	NOUN
ejpam-5336	233	15	with	with	ADP
ejpam-5336	233	16	maximum	maximum	ADJ
ejpam-5336	233	17	size	size	NOUN
ejpam-5336	233	18	such	such	ADJ
ejpam-5336	233	19	that	that	SCONJ
ejpam-5336	233	20	k10	k10	PROPN
ejpam-5336	233	21	∪g15	∪g15	PROPN
ejpam-5336	233	22	is	be	AUX
ejpam-5336	233	23	a	a	DET
ejpam-5336	233	24	pg	pg	NOUN
ejpam-5336	233	25	.	.	PUNCT
ejpam-5336	234	1	then	then	ADV
ejpam-5336	234	2	,	,	PUNCT
ejpam-5336	234	3	π(25	π(25	PROPN
ejpam-5336	234	4	)	)	PUNCT
ejpam-5336	235	1	=	=	SYM
ejpam-5336	235	2	9	9	NUM
ejpam-5336	235	3	=	=	SYM
ejpam-5336	235	4	10−	10−	NOUN
ejpam-5336	235	5	1	1	NUM
ejpam-5336	235	6	and	and	CCONJ
ejpam-5336	235	7	since	since	SCONJ
ejpam-5336	235	8	k10	k10	PROPN
ejpam-5336	235	9	∪g15	∪g15	PROPN
ejpam-5336	235	10	is	be	AUX
ejpam-5336	235	11	a	a	DET
ejpam-5336	235	12	pg	pg	NOUN
ejpam-5336	235	13	,	,	PUNCT
ejpam-5336	235	14	we	we	PRON
ejpam-5336	235	15	can	can	AUX
ejpam-5336	235	16	label	label	VERB
ejpam-5336	235	17	the	the	DET
ejpam-5336	235	18	vertices	vertex	NOUN
ejpam-5336	235	19	of	of	ADP
ejpam-5336	235	20	k10	k10	NOUN
ejpam-5336	235	21	by	by	ADP
ejpam-5336	235	22	the	the	DET
ejpam-5336	235	23	numbers	number	NOUN
ejpam-5336	235	24	1	1	NUM
ejpam-5336	235	25	,	,	PUNCT
ejpam-5336	235	26	2	2	NUM
ejpam-5336	235	27	,	,	PUNCT
ejpam-5336	235	28	3	3	NUM
ejpam-5336	235	29	,	,	PUNCT
ejpam-5336	235	30	5	5	NUM
ejpam-5336	235	31	,	,	PUNCT
ejpam-5336	235	32	7	7	NUM
ejpam-5336	235	33	,	,	PUNCT
ejpam-5336	235	34	11	11	NUM
ejpam-5336	235	35	,	,	PUNCT
ejpam-5336	235	36	13	13	NUM
ejpam-5336	235	37	,	,	PUNCT
ejpam-5336	235	38	17	17	NUM
ejpam-5336	235	39	,	,	PUNCT
ejpam-5336	235	40	19	19	NUM
ejpam-5336	235	41	,	,	PUNCT
ejpam-5336	235	42	23	23	NUM
ejpam-5336	235	43	and	and	CCONJ
ejpam-5336	235	44	hence	hence	ADV
ejpam-5336	235	45	g15	g15	PROPN
ejpam-5336	235	46	is	be	AUX
ejpam-5336	235	47	the	the	DET
ejpam-5336	235	48	following	follow	VERB
ejpam-5336	235	49	graph	graph	NOUN
ejpam-5336	235	50	references	reference	NOUN
ejpam-5336	235	51	3565	3565	NUM
ejpam-5336	235	52	6	6	NUM
ejpam-5336	235	53	4	4	NUM
ejpam-5336	235	54	8	8	NUM
ejpam-5336	235	55	22	22	NUM
ejpam-5336	235	56	10	10	NUM
ejpam-5336	235	57	12	12	NUM
ejpam-5336	235	58	25	25	NUM
ejpam-5336	235	59	15	15	NUM
ejpam-5336	235	60	2018	2018	NUM
ejpam-5336	235	61	16	16	NUM
ejpam-5336	235	62	21	21	NUM
ejpam-5336	235	63	9	9	NUM
ejpam-5336	235	64	24	24	NUM
ejpam-5336	235	65	14	14	NUM
ejpam-5336	235	66	g15	g15	PROPN
ejpam-5336	235	67	so	so	ADV
ejpam-5336	235	68	,	,	PUNCT
ejpam-5336	235	69	g15	g15	PROPN
ejpam-5336	235	70	is	be	AUX
ejpam-5336	235	71	connected	connect	VERB
ejpam-5336	235	72	.	.	PUNCT
ejpam-5336	236	1	references	reference	NOUN
ejpam-5336	236	2	[	[	X
ejpam-5336	236	3	1	1	X
ejpam-5336	236	4	]	]	PUNCT
ejpam-5336	236	5	an	an	DET
ejpam-5336	236	6	dabboucy	dabboucy	NOUN
ejpam-5336	236	7	a	a	DET
ejpam-5336	236	8	tout	tout	NOUN
ejpam-5336	236	9	and	and	CCONJ
ejpam-5336	236	10	k	k	PROPN
ejpam-5336	236	11	howalla	howalla	NOUN
ejpam-5336	236	12	.	.	PUNCT
ejpam-5336	237	1	prime	prime	ADJ
ejpam-5336	237	2	labeling	labeling	NOUN
ejpam-5336	237	3	of	of	ADP
ejpam-5336	237	4	graphs	graph	NOUN
ejpam-5336	237	5	.	.	PUNCT
ejpam-5336	238	1	nat	nat	PROPN
ejpam-5336	238	2	.	.	PUNCT
ejpam-5336	239	1	acad	acad	PROPN
ejpam-5336	239	2	.	.	PUNCT
ejpam-5336	240	1	sci	sci	PROPN
ejpam-5336	240	2	.	.	PROPN
ejpam-5336	240	3	lett	lett	PROPN
ejpam-5336	240	4	.	.	PROPN
ejpam-5336	240	5	,	,	PUNCT
ejpam-5336	240	6	11:365–368	11:365–368	NUM
ejpam-5336	240	7	,	,	PUNCT
ejpam-5336	240	8	1982	1982	NUM
ejpam-5336	240	9	.	.	PUNCT
ejpam-5336	241	1	[	[	X
ejpam-5336	241	2	2	2	NUM
ejpam-5336	241	3	]	]	SYM
ejpam-5336	241	4	b	b	X
ejpam-5336	241	5	abughazaleh	abughazaleh	NOUN
ejpam-5336	241	6	and	and	CCONJ
ejpam-5336	241	7	oa	oa	PROPN
ejpam-5336	241	8	abughneim	abughneim	PROPN
ejpam-5336	241	9	.	.	PUNCT
ejpam-5336	242	1	prime	prime	ADJ
ejpam-5336	242	2	labeling	labeling	NOUN
ejpam-5336	242	3	of	of	ADP
ejpam-5336	242	4	graphs	graph	NOUN
ejpam-5336	242	5	constructed	construct	VERB
ejpam-5336	242	6	from	from	ADP
ejpam-5336	242	7	wheel	wheel	NOUN
ejpam-5336	242	8	graph	graph	NOUN
ejpam-5336	242	9	.	.	PUNCT
ejpam-5336	243	1	heliyon	heliyon	NOUN
ejpam-5336	243	2	,	,	PUNCT
ejpam-5336	243	3	10(2):e23979	10(2):e23979	NUM
ejpam-5336	243	4	,	,	PUNCT
ejpam-5336	243	5	2024	2024	NUM
ejpam-5336	243	6	.	.	PUNCT
ejpam-5336	244	1	[	[	X
ejpam-5336	244	2	3	3	X
ejpam-5336	244	3	]	]	PUNCT
ejpam-5336	244	4	g	g	NOUN
ejpam-5336	244	5	agnarsson	agnarsson	NOUN
ejpam-5336	244	6	and	and	CCONJ
ejpam-5336	244	7	r	r	NOUN
ejpam-5336	244	8	greenlaw	greenlaw	NOUN
ejpam-5336	244	9	.	.	PUNCT
ejpam-5336	245	1	graph	graph	NOUN
ejpam-5336	245	2	theory	theory	NOUN
ejpam-5336	245	3	:	:	PUNCT
ejpam-5336	245	4	modeling	modeling	NOUN
ejpam-5336	245	5	,	,	PUNCT
ejpam-5336	245	6	applications	application	NOUN
ejpam-5336	245	7	,	,	PUNCT
ejpam-5336	245	8	and	and	CCONJ
ejpam-5336	245	9	algorithms	algorithm	NOUN
ejpam-5336	245	10	,	,	PUNCT
ejpam-5336	245	11	1st	1st	ADJ
ejpam-5336	245	12	ed	ed	NOUN
ejpam-5336	245	13	..	..	PUNCT
ejpam-5336	245	14	pearson	pearson	PROPN
ejpam-5336	245	15	education	education	PROPN
ejpam-5336	245	16	,	,	PUNCT
ejpam-5336	245	17	ann	ann	PROPN
ejpam-5336	245	18	arbor	arbor	PROPN
ejpam-5336	245	19	,	,	PUNCT
ejpam-5336	245	20	michigan	michigan	PROPN
ejpam-5336	245	21	,	,	PUNCT
ejpam-5336	245	22	2007	2007	NUM
ejpam-5336	245	23	.	.	PUNCT
ejpam-5336	246	1	[	[	X
ejpam-5336	246	2	4	4	X
ejpam-5336	246	3	]	]	X
ejpam-5336	246	4	david	david	PROPN
ejpam-5336	246	5	m.	m.	PROPN
ejpam-5336	246	6	burton	burton	PROPN
ejpam-5336	246	7	.	.	PUNCT
ejpam-5336	247	1	elementary	elementary	ADJ
ejpam-5336	247	2	number	number	NOUN
ejpam-5336	247	3	theory	theory	NOUN
ejpam-5336	247	4	.	.	PUNCT
ejpam-5336	248	1	tata	tata	PROPN
ejpam-5336	248	2	mcgraw	mcgraw	PROPN
ejpam-5336	248	3	hill	hill	PROPN
ejpam-5336	248	4	education	education	PROPN
ejpam-5336	248	5	,	,	PUNCT
ejpam-5336	248	6	new	new	PROPN
ejpam-5336	248	7	york	york	PROPN
ejpam-5336	248	8	,	,	PUNCT
ejpam-5336	248	9	7th	7th	ADJ
ejpam-5336	248	10	edition	edition	NOUN
ejpam-5336	248	11	,	,	PUNCT
ejpam-5336	248	12	2009	2009	NUM
ejpam-5336	248	13	.	.	PUNCT
ejpam-5336	249	1	[	[	X
ejpam-5336	249	2	5	5	NUM
ejpam-5336	249	3	]	]	PUNCT
ejpam-5336	249	4	hl	hl	NOUN
ejpam-5336	249	5	fu	fu	NOUN
ejpam-5336	249	6	and	and	CCONJ
ejpam-5336	249	7	kc	kc	PROPN
ejpam-5336	249	8	huang	huang	PROPN
ejpam-5336	249	9	.	.	PROPN
ejpam-5336	250	1	on	on	ADP
ejpam-5336	250	2	prime	prime	ADJ
ejpam-5336	250	3	labellings	labelling	NOUN
ejpam-5336	250	4	.	.	PUNCT
ejpam-5336	251	1	discrete	discrete	ADJ
ejpam-5336	251	2	math	math	NOUN
ejpam-5336	251	3	,	,	PUNCT
ejpam-5336	251	4	127:181–186	127:181–186	NUM
ejpam-5336	251	5	,	,	PUNCT
ejpam-5336	251	6	1994	1994	NUM
ejpam-5336	251	7	.	.	PUNCT
ejpam-5336	252	1	[	[	X
ejpam-5336	252	2	6	6	NUM
ejpam-5336	252	3	]	]	PUNCT
ejpam-5336	252	4	ja	ja	PROPN
ejpam-5336	252	5	gallian	gallian	PROPN
ejpam-5336	252	6	.	.	PUNCT
ejpam-5336	253	1	a	a	DET
ejpam-5336	253	2	dynamic	dynamic	ADJ
ejpam-5336	253	3	survey	survey	NOUN
ejpam-5336	253	4	of	of	ADP
ejpam-5336	253	5	graph	graph	NOUN
ejpam-5336	253	6	labeling	labeling	NOUN
ejpam-5336	253	7	.	.	PUNCT
ejpam-5336	254	1	electron	electron	PROPN
ejpam-5336	254	2	.	.	PUNCT
ejpam-5336	255	1	j.	j.	PROPN
ejpam-5336	255	2	comb	comb	PROPN
ejpam-5336	255	3	.	.	PUNCT
ejpam-5336	255	4	,	,	PUNCT
ejpam-5336	255	5	6(25):4–623	6(25):4–623	NUM
ejpam-5336	255	6	,	,	PUNCT
ejpam-5336	255	7	2022	2022	NUM
ejpam-5336	255	8	.	.	PUNCT
ejpam-5336	256	1	[	[	X
ejpam-5336	256	2	7	7	X
ejpam-5336	256	3	]	]	X
ejpam-5336	256	4	a	a	DET
ejpam-5336	256	5	el	el	PROPN
ejpam-5336	256	6	sonbaty	sonbaty	PROPN
ejpam-5336	256	7	ma	ma	PROPN
ejpam-5336	256	8	seoud	seoud	PROPN
ejpam-5336	256	9	and	and	CCONJ
ejpam-5336	256	10	aea	aea	PROPN
ejpam-5336	256	11	mahran	mahran	PROPN
ejpam-5336	256	12	.	.	PUNCT
ejpam-5336	257	1	on	on	ADP
ejpam-5336	257	2	prime	prime	ADJ
ejpam-5336	257	3	graphs	graph	NOUN
ejpam-5336	257	4	.	.	PUNCT
ejpam-5336	258	1	ars	ars	PROPN
ejpam-5336	258	2	comb	comb	PROPN
ejpam-5336	258	3	.	.	PUNCT
ejpam-5336	259	1	,	,	PUNCT
ejpam-5336	259	2	104:241	104:241	NOUN
ejpam-5336	259	3	–	–	PUNCT
ejpam-5336	259	4	260	260	NUM
ejpam-5336	259	5	,	,	PUNCT
ejpam-5336	259	6	2012	2012	NUM
ejpam-5336	259	7	.	.	PUNCT
ejpam-5336	260	1	[	[	X
ejpam-5336	260	2	8	8	NUM
ejpam-5336	260	3	]	]	X
ejpam-5336	260	4	o	o	NOUN
ejpam-5336	260	5	pikhurko	pikhurko	NOUN
ejpam-5336	260	6	p	p	NOUN
ejpam-5336	260	7	haxell	haxell	NOUN
ejpam-5336	260	8	and	and	CCONJ
ejpam-5336	260	9	a	a	DET
ejpam-5336	260	10	taraz	taraz	NOUN
ejpam-5336	260	11	.	.	PUNCT
ejpam-5336	261	1	primality	primality	NOUN
ejpam-5336	261	2	of	of	ADP
ejpam-5336	261	3	trees	tree	NOUN
ejpam-5336	261	4	.	.	PUNCT
ejpam-5336	262	1	j.	j.	PROPN
ejpam-5336	262	2	combinatorics	combinatorics	PROPN
ejpam-5336	262	3	,	,	PUNCT
ejpam-5336	262	4	2:481–500	2:481–500	NUM
ejpam-5336	262	5	,	,	PUNCT
ejpam-5336	262	6	2011	2011	NUM
ejpam-5336	262	7	.	.	PUNCT
ejpam-5336	263	1	[	[	X
ejpam-5336	263	2	9	9	NUM
ejpam-5336	263	3	]	]	PUNCT
ejpam-5336	263	4	sk	sk	ADP
ejpam-5336	263	5	patel	patel	NOUN
ejpam-5336	263	6	and	and	CCONJ
ejpam-5336	263	7	jb	jb	PROPN
ejpam-5336	263	8	vasava	vasava	PROPN
ejpam-5336	263	9	.	.	PUNCT
ejpam-5336	264	1	on	on	ADP
ejpam-5336	264	2	prime	prime	ADJ
ejpam-5336	264	3	labeling	labeling	NOUN
ejpam-5336	264	4	of	of	ADP
ejpam-5336	264	5	some	some	DET
ejpam-5336	264	6	union	union	NOUN
ejpam-5336	264	7	graphs	graph	NOUN
ejpam-5336	264	8	and	and	CCONJ
ejpam-5336	264	9	circulant	circulant	ADJ
ejpam-5336	264	10	graphs	graph	NOUN
ejpam-5336	264	11	.	.	PUNCT
ejpam-5336	265	1	inter	inter	PROPN
ejpam-5336	265	2	.	.	PUNCT
ejpam-5336	266	1	j.	j.	PROPN
ejpam-5336	266	2	sci	sci	PROPN
ejpam-5336	266	3	.	.	PUNCT
ejpam-5336	267	1	res	res	PROPN
ejpam-5336	267	2	.	.	PROPN
ejpam-5336	267	3	math	math	PROPN
ejpam-5336	267	4	.	.	PUNCT
ejpam-5336	268	1	stat	stat	PROPN
ejpam-5336	268	2	.	.	PUNCT
ejpam-5336	269	1	sci	sci	PROPN
ejpam-5336	269	2	.	.	PROPN
ejpam-5336	269	3	,	,	PUNCT
ejpam-5336	269	4	5(6):248–254	5(6):248–254	NUM
ejpam-5336	269	5	,	,	PUNCT
ejpam-5336	269	6	2018	2018	NUM
ejpam-5336	269	7	.	.	PUNCT
ejpam-5336	270	1	[	[	X
ejpam-5336	270	2	10	10	NUM
ejpam-5336	270	3	]	]	X
ejpam-5336	270	4	o	o	X
ejpam-5336	270	5	pikhurko	pikhurko	NOUN
ejpam-5336	270	6	.	.	PUNCT
ejpam-5336	271	1	trees	tree	NOUN
ejpam-5336	271	2	are	be	AUX
ejpam-5336	271	3	almost	almost	ADV
ejpam-5336	271	4	prime	prime	ADJ
ejpam-5336	271	5	.	.	PUNCT
ejpam-5336	272	1	discrete	discrete	ADJ
ejpam-5336	272	2	math	math	NOUN
ejpam-5336	272	3	,	,	PUNCT
ejpam-5336	272	4	307:1455–1462	307:1455–1462	NUM
ejpam-5336	272	5	,	,	PUNCT
ejpam-5336	272	6	2007	2007	NUM
ejpam-5336	272	7	.	.	PUNCT
ejpam-5336	273	1	[	[	X
ejpam-5336	273	2	11	11	NUM
ejpam-5336	273	3	]	]	X
ejpam-5336	273	4	um	um	INTJ
ejpam-5336	273	5	prajapati	prajapati	PROPN
ejpam-5336	273	6	and	and	CCONJ
ejpam-5336	273	7	sj	sj	PROPN
ejpam-5336	273	8	gajjar	gajjar	NOUN
ejpam-5336	273	9	.	.	PUNCT
ejpam-5336	274	1	some	some	DET
ejpam-5336	274	2	results	result	NOUN
ejpam-5336	274	3	on	on	ADP
ejpam-5336	274	4	prime	prime	ADJ
ejpam-5336	274	5	labeling	labeling	NOUN
ejpam-5336	274	6	.	.	PUNCT
ejpam-5336	275	1	open	open	ADJ
ejpam-5336	275	2	j.	j.	PROPN
ejpam-5336	275	3	discrete	discrete	PROPN
ejpam-5336	275	4	math	math	NOUN
ejpam-5336	275	5	,	,	PUNCT
ejpam-5336	275	6	4:60–66	4:60–66	NUM
ejpam-5336	275	7	,	,	PUNCT
ejpam-5336	275	8	2014	2014	NUM
ejpam-5336	275	9	.	.	PUNCT
ejpam-5336	276	1	[	[	X
ejpam-5336	276	2	12	12	NUM
ejpam-5336	276	3	]	]	X
ejpam-5336	276	4	ma	ma	PROPN
ejpam-5336	276	5	seoud	seoud	PROPN
ejpam-5336	276	6	and	and	CCONJ
ejpam-5336	276	7	mz	mz	PROPN
ejpam-5336	276	8	youssef	youssef	PROPN
ejpam-5336	276	9	.	.	PUNCT
ejpam-5336	277	1	on	on	ADP
ejpam-5336	277	2	prime	prime	ADJ
ejpam-5336	277	3	labelings	labeling	NOUN
ejpam-5336	277	4	of	of	ADP
ejpam-5336	277	5	graphs	graph	NOUN
ejpam-5336	277	6	.	.	PUNCT
ejpam-5336	278	1	congr	congr	NOUN
ejpam-5336	278	2	.	.	PUNCT
ejpam-5336	279	1	numer	numer	PROPN
ejpam-5336	279	2	.	.	PROPN
ejpam-5336	279	3	,	,	PUNCT
ejpam-5336	279	4	141:203	141:203	NUM
ejpam-5336	279	5	–	–	PUNCT
ejpam-5336	279	6	215	215	NUM
ejpam-5336	279	7	,	,	PUNCT
ejpam-5336	279	8	1999	1999	NUM
ejpam-5336	279	9	.	.	PUNCT
ejpam-5336	280	1	references	reference	NOUN
ejpam-5336	280	2	3566	3566	NUM
ejpam-5336	280	3	[	[	X
ejpam-5336	280	4	13	13	NUM
ejpam-5336	280	5	]	]	PUNCT
ejpam-5336	280	6	sk	sk	PROPN
ejpam-5336	280	7	vaidya	vaidya	PROPN
ejpam-5336	280	8	and	and	CCONJ
ejpam-5336	280	9	um	um	INTJ
ejpam-5336	280	10	prajapati	prajapati	PROPN
ejpam-5336	280	11	.	.	PUNCT
ejpam-5336	281	1	some	some	DET
ejpam-5336	281	2	results	result	NOUN
ejpam-5336	281	3	on	on	ADP
ejpam-5336	281	4	prime	prime	ADJ
ejpam-5336	281	5	and	and	CCONJ
ejpam-5336	281	6	k	k	ADJ
ejpam-5336	281	7	-	-	ADJ
ejpam-5336	281	8	prime	prime	ADJ
ejpam-5336	281	9	labeling	labeling	NOUN
ejpam-5336	281	10	.	.	PUNCT
ejpam-5336	282	1	j.	j.	PROPN
ejpam-5336	282	2	math	math	PROPN
ejpam-5336	282	3	.	.	PUNCT
ejpam-5336	283	1	res	re	NOUN
ejpam-5336	283	2	.	.	PROPN
ejpam-5336	283	3	,	,	PUNCT
ejpam-5336	283	4	3(1):248–254	3(1):248–254	NUM
ejpam-5336	283	5	,	,	PUNCT
ejpam-5336	283	6	2011	2011	NUM
ejpam-5336	283	7	.	.	PUNCT
ejpam-5336	284	1	[	[	X
ejpam-5336	284	2	14	14	NUM
ejpam-5336	284	3	]	]	X
ejpam-5336	284	4	mz	mz	PROPN
ejpam-5336	284	5	youssef	youssef	PROPN
ejpam-5336	284	6	.	.	PUNCT
ejpam-5336	285	1	on	on	ADP
ejpam-5336	285	2	graceful	graceful	ADJ
ejpam-5336	285	3	,	,	PUNCT
ejpam-5336	285	4	harmonious	harmonious	ADJ
ejpam-5336	285	5	and	and	CCONJ
ejpam-5336	285	6	prime	prime	ADJ
ejpam-5336	285	7	labelings	labeling	NOUN
ejpam-5336	285	8	of	of	ADP
ejpam-5336	285	9	graphs	graph	NOUN
ejpam-5336	285	10	.	.	PUNCT
ejpam-5336	286	1	phd	phd	NOUN
ejpam-5336	286	2	thesis	thesis	PROPN
ejpam-5336	286	3	,	,	PUNCT
ejpam-5336	286	4	department	department	NOUN
ejpam-5336	286	5	of	of	ADP
ejpam-5336	286	6	mathematics	mathematics	PROPN
ejpam-5336	286	7	,	,	PUNCT
ejpam-5336	286	8	ain	ain	PROPN
ejpam-5336	286	9	shams	sham	NOUN
ejpam-5336	286	10	university	university	NOUN
ejpam-5336	286	11	,	,	PUNCT
ejpam-5336	286	12	2000	2000	NUM
ejpam-5336	286	13	.	.	PUNCT