id	sid	tid	token	lemma	pos
cana-6080	1	1	communications	communication	NOUN
cana-6080	1	2	on	on	ADP
cana-6080	1	3	applied	apply	VERB
cana-6080	1	4	nonlinear	nonlinear	ADJ
cana-6080	1	5	analysis	analysis	NOUN
cana-6080	1	6	issn	issn	NOUN
cana-6080	1	7	:	:	PUNCT
cana-6080	1	8	1074	1074	NUM
cana-6080	1	9	-	-	PUNCT
cana-6080	1	10	133x	133x	NUM
cana-6080	1	11	vol	vol	NOUN
cana-6080	1	12	31	31	NUM
cana-6080	1	13	no	no	NOUN
cana-6080	1	14	.	.	PUNCT
cana-6080	2	1	8s	8s	PROPN
cana-6080	2	2	(	(	PUNCT
cana-6080	2	3	2024	2024	NUM
cana-6080	2	4	)	)	PUNCT
cana-6080	2	5	1194	1194	NUM
cana-6080	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-6080	3	2	results	result	VERB
cana-6080	3	3	on	on	ADP
cana-6080	3	4	grundy	grundy	PROPN
cana-6080	3	5	number	number	NOUN
cana-6080	3	6	of	of	ADP
cana-6080	3	7	fan	fan	NOUN
cana-6080	3	8	graph	graph	NOUN
cana-6080	3	9	families	family	NOUN
cana-6080	3	10	k.	k.	PROPN
cana-6080	3	11	annathurai1	annathurai1	PROPN
cana-6080	3	12	,	,	PUNCT
cana-6080	3	13	p.	p.	NOUN
cana-6080	3	14	periasamy	periasamy	NOUN
cana-6080	3	15	,	,	PUNCT
cana-6080	3	16	v.	v.	ADP
cana-6080	3	17	sankar	sankar	NOUN
cana-6080	3	18	raj3	raj3	PROPN
cana-6080	3	19	1	1	NUM
cana-6080	3	20	department	department	NOUN
cana-6080	3	21	of	of	ADP
cana-6080	3	22	mathematics	mathematic	NOUN
cana-6080	3	23	,	,	PUNCT
cana-6080	3	24	thiruvalluvar	thiruvalluvar	NOUN
cana-6080	3	25	college	college	NOUN
cana-6080	3	26	,	,	PUNCT
cana-6080	3	27	papanasam	papanasam	PROPN
cana-6080	3	28	627425	627425	NUM
cana-6080	3	29	,	,	PUNCT
cana-6080	3	30	tamil	tamil	PROPN
cana-6080	3	31	nadu	nadu	PROPN
cana-6080	3	32	,	,	PUNCT
cana-6080	4	1	india	india	PROPN
cana-6080	4	2	.	.	PUNCT
cana-6080	4	3	email	email	NOUN
cana-6080	4	4	:	:	PUNCT
cana-6080	4	5	kannathuraitvcmaths@gmail.com	kannathuraitvcmaths@gmail.com	PROPN
cana-6080	4	6	2	2	NUM
cana-6080	4	7	department	department	NOUN
cana-6080	4	8	of	of	ADP
cana-6080	4	9	mathematics	mathematic	NOUN
cana-6080	4	10	,	,	PUNCT
cana-6080	4	11	research	research	NOUN
cana-6080	4	12	scholar	scholar	NOUN
cana-6080	4	13	,	,	PUNCT
cana-6080	4	14	register	register	VERB
cana-6080	4	15	number	number	NOUN
cana-6080	4	16	21124012091019	21124012091019	NUM
cana-6080	4	17	,	,	PUNCT
cana-6080	4	18	manonmaniam	manonmaniam	PROPN
cana-6080	4	19	sundaranar	sundaranar	PROPN
cana-6080	4	20	university	university	PROPN
cana-6080	4	21	,	,	PUNCT
cana-6080	4	22	tirunelveli	tirunelveli	PROPN
cana-6080	4	23	627012	627012	NUM
cana-6080	4	24	,	,	PUNCT
cana-6080	4	25	tamilnadu	tamilnadu	NOUN
cana-6080	4	26	,	,	PUNCT
cana-6080	4	27	india	india	PROPN
cana-6080	4	28	.	.	PUNCT
cana-6080	4	29	email	email	NOUN
cana-6080	4	30	:	:	PUNCT
cana-6080	4	31	ppskmttc2013@gmail.com	ppskmttc2013@gmail.com	PROPN
cana-6080	4	32	3	3	NUM
cana-6080	4	33	department	department	NOUN
cana-6080	4	34	of	of	ADP
cana-6080	4	35	mathematics	mathematics	PROPN
cana-6080	4	36	,	,	PUNCT
cana-6080	4	37	manonmaniam	manonmaniam	PROPN
cana-6080	4	38	sundaranar	sundaranar	PROPN
cana-6080	4	39	university	university	PROPN
cana-6080	4	40	,	,	PUNCT
cana-6080	4	41	tirunelveli	tirunelveli	ADJ
cana-6080	4	42	–	–	PUNCT
cana-6080	4	43	627012	627012	NUM
cana-6080	4	44	,	,	PUNCT
cana-6080	4	45	tamilnadu	tamilnadu	NOUN
cana-6080	4	46	,	,	PUNCT
cana-6080	5	1	india	india	PROPN
cana-6080	5	2	.	.	PUNCT
cana-6080	5	3	email	email	NOUN
cana-6080	5	4	:	:	PUNCT
cana-6080	6	1	sankarrajv@gmail.com	sankarrajv@gmail.com	X
cana-6080	6	2	article	article	NOUN
cana-6080	6	3	history	history	NOUN
cana-6080	6	4	:	:	PUNCT
cana-6080	6	5	received:01/10/2024	received:01/10/2024	NOUN
cana-6080	6	6	revised	revised	ADJ
cana-6080	6	7	:	:	PUNCT
cana-6080	6	8	06/11/2024	06/11/2024	NUM
cana-6080	6	9	accepted	accept	VERB
cana-6080	6	10	:	:	PUNCT
cana-6080	6	11	30/12/2024	30/12/2024	NUM
cana-6080	6	12	1	1	NUM
cana-6080	6	13	introduction	introduction	NOUN
cana-6080	6	14	abstract	abstract	NOUN
cana-6080	6	15	:	:	PUNCT
cana-6080	6	16	a	a	DET
cana-6080	6	17	grundy	grundy	PROPN
cana-6080	6	18	k−coloring	k−coloring	NOUN
cana-6080	6	19	of	of	ADP
cana-6080	6	20	a	a	DET
cana-6080	6	21	graph	graph	NOUN
cana-6080	6	22	g	g	NOUN
cana-6080	6	23	is	be	AUX
cana-6080	6	24	a	a	DET
cana-6080	6	25	proper	proper	ADJ
cana-6080	6	26	k−coloring	k−coloring	NOUN
cana-6080	6	27	of	of	ADP
cana-6080	6	28	vertices	vertex	NOUN
cana-6080	6	29	in	in	ADP
cana-6080	6	30	g	g	NOUN
cana-6080	6	31	using	use	VERB
cana-6080	6	32	colors	color	NOUN
cana-6080	6	33	{	{	PUNCT
cana-6080	6	34	1	1	NUM
cana-6080	6	35	,	,	PUNCT
cana-6080	6	36	2	2	NUM
cana-6080	6	37	,	,	PUNCT
cana-6080	6	38	·	·	PUNCT
cana-6080	6	39	·	·	PUNCT
cana-6080	6	40	·	·	PUNCT
cana-6080	6	41	,	,	PUNCT
cana-6080	6	42	k	k	X
cana-6080	6	43	}	}	PUNCT
cana-6080	6	44	such	such	ADJ
cana-6080	6	45	that	that	SCONJ
cana-6080	6	46	for	for	ADP
cana-6080	6	47	any	any	DET
cana-6080	6	48	two	two	NUM
cana-6080	6	49	colors	color	NOUN
cana-6080	6	50	i	i	PRON
cana-6080	6	51	and	and	CCONJ
cana-6080	6	52	j	j	PROPN
cana-6080	6	53	,	,	PUNCT
cana-6080	6	54	i	i	PRON
cana-6080	6	55	<	<	X
cana-6080	6	56	j	j	PROPN
cana-6080	6	57	,	,	PUNCT
cana-6080	6	58	any	any	DET
cana-6080	6	59	vertex	vertex	NOUN
cana-6080	6	60	colored	color	VERB
cana-6080	6	61	j	j	PROPN
cana-6080	6	62	is	be	AUX
cana-6080	6	63	adjacent	adjacent	ADJ
cana-6080	6	64	to	to	ADP
cana-6080	6	65	some	some	DET
cana-6080	6	66	vertex	vertex	NOUN
cana-6080	6	67	colored	color	VERB
cana-6080	6	68	i.	i.	NOUN
cana-6080	6	69	the	the	DET
cana-6080	6	70	first	first	ADJ
cana-6080	6	71	-	-	PUNCT
cana-6080	6	72	fit	fit	ADJ
cana-6080	6	73	or	or	CCONJ
cana-6080	6	74	grundy	grundy	PROPN
cana-6080	6	75	chromatic	chromatic	ADJ
cana-6080	6	76	number	number	NOUN
cana-6080	6	77	(	(	PUNCT
cana-6080	6	78	or	or	CCONJ
cana-6080	6	79	simply	simply	ADV
cana-6080	6	80	grundy	grundy	PROPN
cana-6080	6	81	number	number	NOUN
cana-6080	6	82	)	)	PUNCT
cana-6080	6	83	of	of	ADP
cana-6080	6	84	a	a	DET
cana-6080	6	85	graph	graph	NOUN
cana-6080	6	86	g	g	NOUN
cana-6080	6	87	,	,	PUNCT
cana-6080	6	88	denoted	denote	VERB
cana-6080	6	89	by	by	ADP
cana-6080	6	90	γ	γ	PROPN
cana-6080	6	91	(	(	PUNCT
cana-6080	6	92	g	g	NOUN
cana-6080	6	93	)	)	PUNCT
cana-6080	6	94	,	,	PUNCT
cana-6080	6	95	is	be	AUX
cana-6080	6	96	the	the	DET
cana-6080	6	97	largest	large	ADJ
cana-6080	6	98	integer	integer	NOUN
cana-6080	6	99	k	k	NOUN
cana-6080	6	100	,	,	PUNCT
cana-6080	6	101	such	such	ADJ
cana-6080	6	102	that	that	SCONJ
cana-6080	6	103	there	there	PRON
cana-6080	6	104	exists	exist	VERB
cana-6080	6	105	a	a	DET
cana-6080	6	106	grundy	grundy	PROPN
cana-6080	6	107	k−coloring	k−coloring	PROPN
cana-6080	6	108	for	for	ADP
cana-6080	6	109	g.	g.	PROPN
cana-6080	6	110	it	it	PRON
cana-6080	6	111	can	can	AUX
cana-6080	6	112	be	be	AUX
cana-6080	6	113	easily	easily	ADV
cana-6080	6	114	seen	see	VERB
cana-6080	6	115	thatγ	thatγ	NOUN
cana-6080	6	116	(	(	PUNCT
cana-6080	6	117	g	g	NOUN
cana-6080	6	118	)	)	PUNCT
cana-6080	6	119	equals	equal	VERB
cana-6080	6	120	to	to	ADP
cana-6080	6	121	the	the	DET
cana-6080	6	122	maximum	maximum	ADJ
cana-6080	6	123	number	number	NOUN
cana-6080	6	124	of	of	ADP
cana-6080	6	125	colors	color	NOUN
cana-6080	6	126	used	use	VERB
cana-6080	6	127	by	by	ADP
cana-6080	6	128	the	the	DET
cana-6080	6	129	greedy	greedy	ADJ
cana-6080	6	130	(	(	PUNCT
cana-6080	6	131	or	or	CCONJ
cana-6080	6	132	first	first	ADJ
cana-6080	6	133	-	-	PUNCT
cana-6080	6	134	fit	fit	NOUN
cana-6080	6	135	)	)	PUNCT
cana-6080	6	136	coloring	coloring	NOUN
cana-6080	6	137	of	of	ADP
cana-6080	6	138	g	g	PROPN
cana-6080	6	139	.	.	PUNCT
cana-6080	7	1	in	in	ADP
cana-6080	7	2	this	this	DET
cana-6080	7	3	paper	paper	NOUN
cana-6080	7	4	,	,	PUNCT
cana-6080	7	5	we	we	PRON
cana-6080	7	6	obtain	obtain	VERB
cana-6080	7	7	the	the	DET
cana-6080	7	8	grundy	grundy	ADJ
cana-6080	7	9	chromatic	chromatic	ADJ
cana-6080	7	10	number	number	NOUN
cana-6080	7	11	of	of	ADP
cana-6080	7	12	fan	fan	NOUN
cana-6080	7	13	graph	graph	NOUN
cana-6080	7	14	,	,	PUNCT
cana-6080	7	15	line	line	NOUN
cana-6080	7	16	graph	graph	NOUN
cana-6080	7	17	of	of	ADP
cana-6080	7	18	fan	fan	NOUN
cana-6080	7	19	graph	graph	NOUN
cana-6080	7	20	,	,	PUNCT
cana-6080	7	21	middle	middle	ADJ
cana-6080	7	22	graph	graph	NOUN
cana-6080	7	23	of	of	ADP
cana-6080	7	24	fan	fan	NOUN
cana-6080	7	25	graph	graph	NOUN
cana-6080	7	26	and	and	CCONJ
cana-6080	7	27	total	total	ADJ
cana-6080	7	28	graph	graph	NOUN
cana-6080	7	29	of	of	ADP
cana-6080	7	30	fan	fan	NOUN
cana-6080	7	31	graph	graph	NOUN
cana-6080	7	32	.	.	PUNCT
cana-6080	8	1	keywords	keyword	NOUN
cana-6080	8	2	:	:	PUNCT
cana-6080	8	3	grundy	grundy	PROPN
cana-6080	8	4	chromatic	chromatic	ADJ
cana-6080	8	5	number	number	NOUN
cana-6080	8	6	,	,	PUNCT
cana-6080	8	7	fan	fan	NOUN
cana-6080	8	8	graph	graph	NOUN
cana-6080	8	9	,	,	PUNCT
cana-6080	8	10	line	line	NOUN
cana-6080	8	11	graph	graph	NOUN
cana-6080	8	12	,	,	PUNCT
cana-6080	8	13	middle	middle	ADJ
cana-6080	8	14	graph	graph	NOUN
cana-6080	8	15	,	,	PUNCT
cana-6080	8	16	total	total	ADJ
cana-6080	8	17	graph	graph	NOUN
cana-6080	8	18	.	.	PUNCT
cana-6080	9	1	the	the	DET
cana-6080	9	2	notion	notion	NOUN
cana-6080	9	3	of	of	ADP
cana-6080	9	4	grundy	grundy	PROPN
cana-6080	9	5	colorings	coloring	NOUN
cana-6080	9	6	was	be	AUX
cana-6080	9	7	introduced	introduce	VERB
cana-6080	9	8	by	by	ADP
cana-6080	9	9	patrick	patrick	PROPN
cana-6080	9	10	michael	michael	PROPN
cana-6080	9	11	grundy	grundy	PROPN
cana-6080	9	12	in	in	ADP
cana-6080	9	13	1939	1939	NUM
cana-6080	9	14	.	.	PUNCT
cana-6080	10	1	author	author	NOUN
cana-6080	10	2	has	have	AUX
cana-6080	10	3	dealing	deal	VERB
cana-6080	10	4	with	with	ADP
cana-6080	10	5	combinatorial	combinatorial	ADJ
cana-6080	10	6	games	game	NOUN
cana-6080	10	7	contained	contain	VERB
cana-6080	10	8	ideas	idea	NOUN
cana-6080	10	9	that	that	PRON
cana-6080	10	10	led	lead	VERB
cana-6080	10	11	to	to	ADP
cana-6080	10	12	the	the	DET
cana-6080	10	13	concept	concept	NOUN
cana-6080	10	14	of	of	ADP
cana-6080	10	15	grundy	grundy	PROPN
cana-6080	10	16	colorings	coloring	NOUN
cana-6080	10	17	of	of	ADP
cana-6080	10	18	graphs	graph	NOUN
cana-6080	10	19	.	.	PUNCT
cana-6080	11	1	a	a	DET
cana-6080	11	2	grundy	grundy	PROPN
cana-6080	11	3	coloring	coloring	NOUN
cana-6080	11	4	[	[	X
cana-6080	11	5	4	4	NUM
cana-6080	11	6	,	,	PUNCT
cana-6080	11	7	6	6	NUM
cana-6080	11	8	,	,	PUNCT
cana-6080	11	9	16	16	NUM
cana-6080	11	10	]	]	PUNCT
cana-6080	11	11	of	of	ADP
cana-6080	11	12	a	a	DET
cana-6080	11	13	graph	graph	NOUN
cana-6080	11	14	g	g	NOUN
cana-6080	11	15	is	be	AUX
cana-6080	11	16	a	a	DET
cana-6080	11	17	proper	proper	ADJ
cana-6080	11	18	vertex	vertex	NOUN
cana-6080	11	19	coloring	coloring	NOUN
cana-6080	11	20	of	of	ADP
cana-6080	11	21	g	g	PROPN
cana-6080	11	22	(	(	PUNCT
cana-6080	11	23	whose	whose	DET
cana-6080	11	24	colors	color	NOUN
cana-6080	11	25	,	,	PUNCT
cana-6080	11	26	as	as	ADP
cana-6080	11	27	usual	usual	ADJ
cana-6080	11	28	,	,	PUNCT
cana-6080	11	29	are	be	AUX
cana-6080	11	30	positive	positive	ADJ
cana-6080	11	31	integers	integer	NOUN
cana-6080	11	32	)	)	PUNCT
cana-6080	11	33	having	have	VERB
cana-6080	11	34	the	the	DET
cana-6080	11	35	property	property	NOUN
cana-6080	11	36	that	that	PRON
cana-6080	11	37	for	for	ADP
cana-6080	11	38	every	every	DET
cana-6080	11	39	two	two	NUM
cana-6080	11	40	colors	color	NOUN
cana-6080	11	41	i	i	PRON
cana-6080	11	42	and	and	CCONJ
cana-6080	11	43	j	j	PROPN
cana-6080	11	44	with	with	ADP
cana-6080	11	45	i	i	PRON
cana-6080	11	46	<	<	X
cana-6080	11	47	j	j	PROPN
cana-6080	11	48	,	,	PUNCT
cana-6080	11	49	every	every	DET
cana-6080	11	50	vertex	vertex	NOUN
cana-6080	11	51	colored	color	VERB
cana-6080	11	52	j	j	PROPN
cana-6080	11	53	has	have	VERB
cana-6080	11	54	a	a	DET
cana-6080	11	55	neighbor	neighbor	NOUN
cana-6080	11	56	colored	color	VERB
cana-6080	11	57	i.	i.	PROPN
cana-6080	11	58	consequently	consequently	ADV
cana-6080	11	59	,	,	PUNCT
cana-6080	11	60	every	every	DET
cana-6080	11	61	grundy	grundy	PROPN
cana-6080	11	62	coloring	coloring	NOUN
cana-6080	11	63	is	be	AUX
cana-6080	11	64	a	a	DET
cana-6080	11	65	complete	complete	ADJ
cana-6080	11	66	coloring	coloring	NOUN
cana-6080	11	67	.	.	PUNCT
cana-6080	12	1	the	the	DET
cana-6080	12	2	4	4	NUM
cana-6080	12	3	-	-	PUNCT
cana-6080	12	4	coloring	coloring	NOUN
cana-6080	12	5	of	of	ADP
cana-6080	12	6	the	the	DET
cana-6080	12	7	tree	tree	NOUN
cana-6080	12	8	t1	t1	NOUN
cana-6080	12	9	of	of	ADP
cana-6080	12	10	figure	figure	NOUN
cana-6080	12	11	1	1	NUM
cana-6080	12	12	is	be	AUX
cana-6080	12	13	a	a	DET
cana-6080	12	14	grundy	grundy	PROPN
cana-6080	12	15	4	4	ADV
cana-6080	12	16	-	-	PUNCT
cana-6080	12	17	coloring	coloring	NOUN
cana-6080	12	18	and	and	CCONJ
cana-6080	12	19	is	be	AUX
cana-6080	12	20	therefore	therefore	ADV
cana-6080	12	21	a	a	DET
cana-6080	12	22	complete	complete	ADJ
cana-6080	12	23	4	4	NUM
cana-6080	12	24	-	-	PUNCT
cana-6080	12	25	coloring	coloring	NOUN
cana-6080	12	26	as	as	ADV
cana-6080	12	27	well	well	ADV
cana-6080	12	28	.	.	PUNCT
cana-6080	13	1	however	however	ADV
cana-6080	13	2	,	,	PUNCT
cana-6080	13	3	the	the	DET
cana-6080	13	4	complete	complete	ADJ
cana-6080	13	5	3	3	NUM
cana-6080	13	6	-	-	PUNCT
cana-6080	13	7	coloring	coloring	NOUN
cana-6080	13	8	of	of	ADP
cana-6080	13	9	t2	t2	NOUN
cana-6080	13	10	shown	show	VERB
cana-6080	13	11	in	in	ADP
cana-6080	13	12	figure	figure	NOUN
cana-6080	13	13	1.1	1.1	NUM
cana-6080	13	14	is	be	AUX
cana-6080	13	15	not	not	PART
cana-6080	13	16	a	a	DET
cana-6080	13	17	grundy	grundy	PROPN
cana-6080	13	18	3	3	ADV
cana-6080	13	19	-	-	PUNCT
cana-6080	13	20	coloring	coloring	NOUN
cana-6080	13	21	.	.	PUNCT
cana-6080	14	1	recall	recall	VERB
cana-6080	14	2	that	that	SCONJ
cana-6080	14	3	a	a	DET
cana-6080	14	4	greedy	greedy	ADJ
cana-6080	14	5	coloring	coloring	NOUN
cana-6080	14	6	[	[	X
cana-6080	14	7	1	1	NUM
cana-6080	14	8	]	]	X
cana-6080	14	9	c	c	NOUN
cana-6080	14	10	of	of	ADP
cana-6080	14	11	a	a	DET
cana-6080	14	12	graph	graph	NOUN
cana-6080	14	13	g	g	NOUN
cana-6080	14	14	is	be	AUX
cana-6080	14	15	obtained	obtain	VERB
cana-6080	14	16	from	from	ADP
cana-6080	14	17	an	an	DET
cana-6080	14	18	ordering	ordering	NOUN
cana-6080	14	19	φ	φ	NOUN
cana-6080	14	20	:	:	PUNCT
cana-6080	14	21	v1	v1	NOUN
cana-6080	14	22	,	,	PUNCT
cana-6080	14	23	v2	v2	PROPN
cana-6080	14	24	,	,	PUNCT
cana-6080	14	25	·	·	PUNCT
cana-6080	14	26	·	·	PUNCT
cana-6080	14	27	·	·	PUNCT
cana-6080	14	28	,	,	PUNCT
cana-6080	14	29	vn	vn	INTJ
cana-6080	14	30	of	of	ADP
cana-6080	14	31	the	the	DET
cana-6080	14	32	vertices	vertex	NOUN
cana-6080	14	33	of	of	ADP
cana-6080	14	34	g	g	NOUN
cana-6080	14	35	in	in	ADP
cana-6080	14	36	some	some	DET
cana-6080	14	37	manner	manner	NOUN
cana-6080	14	38	,	,	PUNCT
cana-6080	14	39	by	by	ADP
cana-6080	14	40	defining	define	VERB
cana-6080	14	41	c(v1	c(v1	NOUN
cana-6080	14	42	)	)	PUNCT
cana-6080	15	1	=	=	SYM
cana-6080	15	2	1	1	NUM
cana-6080	15	3	,	,	PUNCT
cana-6080	15	4	and	and	CCONJ
cana-6080	15	5	figure	figure	VERB
cana-6080	15	6	1.1	1.1	NUM
cana-6080	15	7	:	:	PUNCT
cana-6080	16	1	complete	complete	ADJ
cana-6080	16	2	and	and	CCONJ
cana-6080	16	3	grundy	grundy	PROPN
cana-6080	16	4	colorings	coloring	NOUN
cana-6080	16	5	mailto:kannathuraitvcmaths@gmail.com	mailto:kannathuraitvcmaths@gmail.com	PROPN
cana-6080	16	6	mailto:ppskmttc2013@gmail.com	mailto:ppskmttc2013@gmail.com	PROPN
cana-6080	16	7	mailto:sankarrajv@gmail.com	mailto:sankarrajv@gmail.com	PROPN
cana-6080	16	8	communications	communication	NOUN
cana-6080	16	9	on	on	ADP
cana-6080	16	10	applied	apply	VERB
cana-6080	16	11	nonlinear	nonlinear	ADJ
cana-6080	16	12	analysis	analysis	NOUN
cana-6080	16	13	issn	issn	NOUN
cana-6080	16	14	:	:	PUNCT
cana-6080	16	15	1074	1074	NUM
cana-6080	16	16	-	-	PUNCT
cana-6080	16	17	133x	133x	NUM
cana-6080	16	18	vol	vol	NOUN
cana-6080	16	19	31	31	NUM
cana-6080	16	20	no	no	NOUN
cana-6080	16	21	.	.	PUNCT
cana-6080	17	1	8s	8s	PROPN
cana-6080	17	2	(	(	PUNCT
cana-6080	17	3	2024	2024	NUM
cana-6080	17	4	)	)	PUNCT
cana-6080	17	5	1195	1195	NUM
cana-6080	17	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6080	17	7	once	once	SCONJ
cana-6080	17	8	colors	color	NOUN
cana-6080	17	9	have	have	AUX
cana-6080	17	10	been	be	AUX
cana-6080	17	11	assigned	assign	VERB
cana-6080	17	12	to	to	PART
cana-6080	17	13	v1	v1	VERB
cana-6080	17	14	,	,	PUNCT
cana-6080	17	15	v2	v2	PROPN
cana-6080	17	16	,	,	PUNCT
cana-6080	17	17	·	·	PUNCT
cana-6080	17	18	·	·	PUNCT
cana-6080	17	19	·	·	PUNCT
cana-6080	17	20	,	,	PUNCT
cana-6080	17	21	vp	vp	PROPN
cana-6080	17	22	for	for	ADP
cana-6080	17	23	some	some	DET
cana-6080	17	24	integer	integer	NOUN
cana-6080	17	25	p	p	NOUN
cana-6080	17	26	with	with	ADP
cana-6080	17	27	1	1	NUM
cana-6080	17	28	≤	≤	NOUN
cana-6080	17	29	p	p	NOUN
cana-6080	17	30	≤	≤	NUM
cana-6080	17	31	n	n	CCONJ
cana-6080	17	32	,	,	PUNCT
cana-6080	17	33	c(vp+1	c(vp+1	PROPN
cana-6080	17	34	)	)	PUNCT
cana-6080	17	35	is	be	AUX
cana-6080	17	36	defined	define	VERB
cana-6080	17	37	as	as	ADP
cana-6080	17	38	the	the	DET
cana-6080	17	39	smallest	small	ADJ
cana-6080	17	40	color	color	NOUN
cana-6080	17	41	not	not	PART
cana-6080	17	42	assigned	assign	VERB
cana-6080	17	43	to	to	ADP
cana-6080	17	44	any	any	DET
cana-6080	17	45	neighbor	neighbor	NOUN
cana-6080	17	46	of	of	ADP
cana-6080	17	47	vp+1	vp+1	NOUN
cana-6080	17	48	belonging	belong	VERB
cana-6080	17	49	to	to	ADP
cana-6080	17	50	the	the	DET
cana-6080	17	51	set	set	NOUN
cana-6080	17	52	{	{	PUNCT
cana-6080	17	53	v1	v1	NOUN
cana-6080	17	54	,	,	PUNCT
cana-6080	17	55	v2	v2	PROPN
cana-6080	17	56	,	,	PUNCT
cana-6080	17	57	·	·	PUNCT
cana-6080	17	58	·	·	PUNCT
cana-6080	17	59	·	·	PUNCT
cana-6080	17	60	,	,	PUNCT
cana-6080	17	61	vp	vp	PROPN
cana-6080	17	62	}	}	PUNCT
cana-6080	17	63	.	.	PUNCT
cana-6080	18	1	the	the	DET
cana-6080	18	2	coloring	coloring	NOUN
cana-6080	18	3	c	c	PROPN
cana-6080	18	4	so	so	ADV
cana-6080	18	5	produced	produce	VERB
cana-6080	18	6	is	be	AUX
cana-6080	18	7	then	then	ADV
cana-6080	18	8	a	a	DET
cana-6080	18	9	grundy	grundy	PROPN
cana-6080	18	10	coloring	coloring	NOUN
cana-6080	18	11	of	of	ADP
cana-6080	18	12	g.	g.	PROPN
cana-6080	18	13	(	(	PUNCT
cana-6080	18	14	i.e.	i.e.	X
cana-6080	18	15	)	)	PUNCT
cana-6080	18	16	,	,	PUNCT
cana-6080	18	17	every	every	DET
cana-6080	18	18	greedy	greedy	ADJ
cana-6080	18	19	coloring	coloring	NOUN
cana-6080	18	20	is	be	AUX
cana-6080	18	21	a	a	DET
cana-6080	18	22	grundy	grundy	PROPN
cana-6080	18	23	coloring	coloring	NOUN
cana-6080	18	24	.	.	PUNCT
cana-6080	19	1	the	the	DET
cana-6080	19	2	maximum	maximum	ADJ
cana-6080	19	3	positive	positive	ADJ
cana-6080	19	4	integer	integer	NOUN
cana-6080	19	5	k	k	PROPN
cana-6080	19	6	for	for	ADP
cana-6080	19	7	which	which	PRON
cana-6080	19	8	a	a	DET
cana-6080	19	9	graph	graph	NOUN
cana-6080	19	10	g	g	PROPN
cana-6080	19	11	has	have	VERB
cana-6080	19	12	a	a	DET
cana-6080	19	13	grundy	grundy	PROPN
cana-6080	19	14	k	k	X
cana-6080	19	15	-	-	PUNCT
cana-6080	19	16	coloring	coloring	NOUN
cana-6080	19	17	is	be	AUX
cana-6080	19	18	denoted	denote	VERB
cana-6080	19	19	by	by	ADP
cana-6080	19	20	γ(g	γ(g	PROPN
cana-6080	19	21	)	)	PUNCT
cana-6080	19	22	and	and	CCONJ
cana-6080	19	23	is	be	AUX
cana-6080	19	24	called	call	VERB
cana-6080	19	25	the	the	DET
cana-6080	19	26	grundy	grundy	ADJ
cana-6080	19	27	chromatic	chromatic	ADJ
cana-6080	19	28	number	number	NOUN
cana-6080	19	29	of	of	ADP
cana-6080	19	30	g	g	NOUN
cana-6080	19	31	or	or	CCONJ
cana-6080	19	32	more	more	ADJ
cana-6080	19	33	simply	simply	ADV
cana-6080	19	34	the	the	DET
cana-6080	19	35	grundy	grundy	ADJ
cana-6080	19	36	number	number	NOUN
cana-6080	19	37	of	of	ADP
cana-6080	19	38	g.	g.	PROPN
cana-6080	19	39	if	if	SCONJ
cana-6080	19	40	the	the	DET
cana-6080	19	41	grundy	grundy	PROPN
cana-6080	19	42	number	number	NOUN
cana-6080	19	43	of	of	ADP
cana-6080	19	44	a	a	DET
cana-6080	19	45	graph	graph	NOUN
cana-6080	19	46	g	g	PROPN
cana-6080	19	47	is	be	AUX
cana-6080	19	48	k	k	NOUN
cana-6080	19	49	,	,	PUNCT
cana-6080	19	50	then	then	ADV
cana-6080	19	51	in	in	ADP
cana-6080	19	52	any	any	DET
cana-6080	19	53	grundy	grundy	PROPN
cana-6080	19	54	k	k	PROPN
cana-6080	19	55	-	-	PUNCT
cana-6080	19	56	coloring	coloring	NOUN
cana-6080	19	57	of	of	ADP
cana-6080	19	58	g	g	PROPN
cana-6080	19	59	(	(	PUNCT
cana-6080	19	60	using	use	VERB
cana-6080	19	61	the	the	DET
cana-6080	19	62	colors	color	NOUN
cana-6080	19	63	1	1	NUM
cana-6080	19	64	,	,	PUNCT
cana-6080	19	65	2	2	NUM
cana-6080	19	66	,	,	PUNCT
cana-6080	19	67	·	·	PUNCT
cana-6080	19	68	·	·	PUNCT
cana-6080	19	69	·	·	PUNCT
cana-6080	19	70	,	,	PUNCT
cana-6080	19	71	k	k	X
cana-6080	19	72	)	)	PUNCT
cana-6080	19	73	,	,	PUNCT
cana-6080	19	74	every	every	DET
cana-6080	19	75	vertex	vertex	NOUN
cana-6080	19	76	v	v	NOUN
cana-6080	19	77	of	of	ADP
cana-6080	19	78	g	g	NOUN
cana-6080	19	79	colored	colored	ADJ
cana-6080	19	80	k	k	PROPN
cana-6080	19	81	must	must	AUX
cana-6080	19	82	be	be	AUX
cana-6080	19	83	adjacent	adjacent	ADJ
cana-6080	19	84	to	to	ADP
cana-6080	19	85	a	a	DET
cana-6080	19	86	vertex	vertex	NOUN
cana-6080	19	87	colored	color	VERB
cana-6080	19	88	i	i	PRON
cana-6080	19	89	for	for	ADP
cana-6080	19	90	each	each	DET
cana-6080	19	91	integer	integer	NOUN
cana-6080	19	92	i	i	PRON
cana-6080	19	93	with	with	ADP
cana-6080	19	94	1	1	NUM
cana-6080	19	95	≤	≤	NUM
cana-6080	19	96	i	i	PRON
cana-6080	19	97	≤	≤	PROPN
cana-6080	19	98	k.	k.	PROPN
cana-6080	20	1	thus	thus	ADV
cana-6080	20	2	∆(g	∆(g	PROPN
cana-6080	20	3	)	)	PUNCT
cana-6080	20	4	≥	≥	NOUN
cana-6080	20	5	deg(v	deg(v	PROPN
cana-6080	20	6	)	)	PUNCT
cana-6080	20	7	≥	≥	NOUN
cana-6080	20	8	(	(	PUNCT
cana-6080	20	9	k	k	NOUN
cana-6080	20	10	−	−	PROPN
cana-6080	20	11	1	1	NUM
cana-6080	20	12	)	)	PUNCT
cana-6080	20	13	and	and	CCONJ
cana-6080	20	14	so	so	ADV
cana-6080	20	15	γ(g	γ(g	PROPN
cana-6080	20	16	)	)	PUNCT
cana-6080	20	17	≤	≤	PROPN
cana-6080	20	18	∆(g	∆(g	NOUN
cana-6080	20	19	)	)	PUNCT
cana-6080	21	1	+	+	CCONJ
cana-6080	21	2	1	1	NUM
cana-6080	21	3	(	(	PUNCT
cana-6080	21	4	1	1	NUM
cana-6080	21	5	)	)	PUNCT
cana-6080	21	6	for	for	ADP
cana-6080	21	7	every	every	DET
cana-6080	21	8	graph	graph	NOUN
cana-6080	21	9	g.	g.	NOUN
cana-6080	21	10	since	since	SCONJ
cana-6080	21	11	every	every	DET
cana-6080	21	12	grundy	grundy	PROPN
cana-6080	21	13	coloring	coloring	NOUN
cana-6080	21	14	of	of	ADP
cana-6080	21	15	a	a	DET
cana-6080	21	16	graph	graph	NOUN
cana-6080	21	17	g	g	NOUN
cana-6080	21	18	is	be	AUX
cana-6080	21	19	a	a	DET
cana-6080	21	20	proper	proper	ADJ
cana-6080	21	21	coloring	coloring	NOUN
cana-6080	21	22	,	,	PUNCT
cana-6080	21	23	it	it	PRON
cana-6080	21	24	follows	follow	VERB
cana-6080	21	25	that	that	SCONJ
cana-6080	21	26	,	,	PUNCT
cana-6080	21	27	χ(g	χ(g	PROPN
cana-6080	21	28	)	)	PUNCT
cana-6080	21	29	≤	≤	PROPN
cana-6080	21	30	γ(g	γ(g	PROPN
cana-6080	21	31	)	)	PUNCT
cana-6080	21	32	.	.	PUNCT
cana-6080	22	1	the	the	DET
cana-6080	22	2	grundy	grundy	PROPN
cana-6080	22	3	number	number	NOUN
cana-6080	22	4	of	of	ADP
cana-6080	22	5	a	a	DET
cana-6080	22	6	graph	graph	NOUN
cana-6080	22	7	was	be	AUX
cana-6080	22	8	perhaps	perhaps	ADV
cana-6080	22	9	introduced	introduce	VERB
cana-6080	22	10	for	for	ADP
cana-6080	22	11	the	the	DET
cana-6080	22	12	first	first	ADJ
cana-6080	22	13	time	time	NOUN
cana-6080	22	14	by	by	ADP
cana-6080	22	15	christen	christen	NOUN
cana-6080	22	16	and	and	CCONJ
cana-6080	22	17	selkow	selkow	VERB
cana-6080	22	18	[	[	X
cana-6080	22	19	2	2	NUM
cana-6080	22	20	]	]	PUNCT
cana-6080	22	21	.	.	PUNCT
cana-6080	23	1	in	in	ADP
cana-6080	23	2	[	[	X
cana-6080	23	3	3	3	NUM
cana-6080	23	4	]	]	PUNCT
cana-6080	23	5	,	,	PUNCT
cana-6080	23	6	erd¨os	erd¨os	X
cana-6080	23	7	et	et	PROPN
cana-6080	23	8	.	.	PROPN
cana-6080	23	9	,	,	PUNCT
cana-6080	23	10	al	al	PROPN
cana-6080	23	11	.	.	PROPN
cana-6080	23	12	proved	prove	VERB
cana-6080	23	13	that	that	SCONJ
cana-6080	23	14	the	the	DET
cana-6080	23	15	grundy	grundy	PROPN
cana-6080	23	16	number	number	NOUN
cana-6080	23	17	of	of	ADP
cana-6080	23	18	a	a	DET
cana-6080	23	19	graph	graph	NOUN
cana-6080	23	20	is	be	AUX
cana-6080	23	21	in	in	ADP
cana-6080	23	22	fact	fact	NOUN
cana-6080	23	23	the	the	DET
cana-6080	23	24	same	same	ADJ
cana-6080	23	25	as	as	ADP
cana-6080	23	26	a	a	DET
cana-6080	23	27	chromatic	chromatic	ADJ
cana-6080	23	28	number	number	NOUN
cana-6080	23	29	of	of	ADP
cana-6080	23	30	a	a	DET
cana-6080	23	31	graph	graph	NOUN
cana-6080	23	32	which	which	PRON
cana-6080	23	33	was	be	AUX
cana-6080	23	34	defined	define	VERB
cana-6080	23	35	and	and	CCONJ
cana-6080	23	36	studied	study	VERB
cana-6080	23	37	independently	independently	ADV
cana-6080	23	38	by	by	ADP
cana-6080	23	39	simmons	simmon	NOUN
cana-6080	24	1	[	[	X
cana-6080	24	2	12	12	NUM
cana-6080	24	3	]	]	PUNCT
cana-6080	24	4	.	.	PUNCT
cana-6080	25	1	in	in	ADP
cana-6080	25	2	[	[	X
cana-6080	25	3	8	8	NUM
cana-6080	25	4	]	]	PUNCT
cana-6080	25	5	the	the	DET
cana-6080	25	6	authors	author	NOUN
cana-6080	25	7	studied	study	VERB
cana-6080	25	8	the	the	DET
cana-6080	25	9	grundy	grundy	PROPN
cana-6080	25	10	number	number	NOUN
cana-6080	25	11	of	of	ADP
cana-6080	25	12	hypercubes	hypercube	NOUN
cana-6080	25	13	and	and	CCONJ
cana-6080	25	14	determined	determine	VERB
cana-6080	25	15	the	the	DET
cana-6080	25	16	exact	exact	ADJ
cana-6080	25	17	values	value	NOUN
cana-6080	25	18	.	.	PUNCT
cana-6080	26	1	from	from	ADP
cana-6080	26	2	computational	computational	ADJ
cana-6080	26	3	point	point	NOUN
cana-6080	26	4	of	of	ADP
cana-6080	26	5	view	view	NOUN
cana-6080	26	6	,	,	PUNCT
cana-6080	26	7	polynomial	polynomial	ADJ
cana-6080	26	8	time	time	NOUN
cana-6080	26	9	algorithms	algorithm	NOUN
cana-6080	26	10	for	for	ADP
cana-6080	26	11	determining	determine	VERB
cana-6080	26	12	the	the	DET
cana-6080	26	13	grundy	grundy	PROPN
cana-6080	26	14	number	number	NOUN
cana-6080	26	15	have	have	AUX
cana-6080	26	16	been	be	AUX
cana-6080	26	17	given	give	VERB
cana-6080	26	18	for	for	ADP
cana-6080	26	19	trees	tree	NOUN
cana-6080	26	20	in	in	ADP
cana-6080	26	21	[	[	X
cana-6080	26	22	7	7	NUM
cana-6080	26	23	]	]	PUNCT
cana-6080	26	24	and	and	CCONJ
cana-6080	26	25	for	for	ADP
cana-6080	26	26	partial	partial	ADJ
cana-6080	26	27	k	k	NOUN
cana-6080	26	28	-	-	NOUN
cana-6080	26	29	trees	tree	NOUN
cana-6080	26	30	in	in	ADP
cana-6080	26	31	[	[	X
cana-6080	26	32	13	13	NUM
cana-6080	26	33	]	]	PUNCT
cana-6080	26	34	.	.	PUNCT
cana-6080	27	1	in	in	ADP
cana-6080	27	2	a	a	DET
cana-6080	27	3	manuscript	manuscript	NOUN
cana-6080	27	4	[	[	X
cana-6080	27	5	5	5	NUM
cana-6080	27	6	]	]	PUNCT
cana-6080	27	7	the	the	DET
cana-6080	27	8	np	np	NOUN
cana-6080	27	9	-	-	PUNCT
cana-6080	27	10	completeness	completeness	NOUN
cana-6080	27	11	of	of	ADP
cana-6080	27	12	determining	determine	VERB
cana-6080	27	13	the	the	DET
cana-6080	27	14	grundy	grundy	ADJ
cana-6080	27	15	number	number	NOUN
cana-6080	27	16	of	of	ADP
cana-6080	27	17	general	general	ADJ
cana-6080	27	18	graphs	graph	NOUN
cana-6080	27	19	has	have	AUX
cana-6080	27	20	been	be	AUX
cana-6080	27	21	proved	prove	VERB
cana-6080	27	22	.	.	PUNCT
cana-6080	28	1	therefore	therefore	ADV
cana-6080	28	2	,	,	PUNCT
cana-6080	28	3	they	they	PRON
cana-6080	28	4	gave	give	VERB
cana-6080	28	5	an	an	DET
cana-6080	28	6	affirmative	affirmative	ADJ
cana-6080	28	7	answer	answer	NOUN
cana-6080	28	8	to	to	ADP
cana-6080	28	9	the	the	DET
cana-6080	28	10	problem	problem	NOUN
cana-6080	28	11	10.4	10.4	NUM
cana-6080	28	12	posed	pose	VERB
cana-6080	28	13	in	in	ADP
cana-6080	28	14	the	the	DET
cana-6080	28	15	graph	graph	NOUN
cana-6080	28	16	coloring	coloring	NOUN
cana-6080	28	17	problem	problem	NOUN
cana-6080	28	18	book	book	NOUN
cana-6080	28	19	[	[	X
cana-6080	28	20	9	9	NUM
cana-6080	28	21	]	]	PUNCT
cana-6080	28	22	which	which	PRON
cana-6080	28	23	asks	ask	VERB
cana-6080	28	24	whether	whether	SCONJ
cana-6080	28	25	determining	determine	VERB
cana-6080	28	26	the	the	DET
cana-6080	28	27	grundy	grundy	ADJ
cana-6080	28	28	chromatic	chromatic	ADJ
cana-6080	28	29	number	number	NOUN
cana-6080	28	30	of	of	ADP
cana-6080	28	31	graphs	graph	NOUN
cana-6080	28	32	is	be	AUX
cana-6080	28	33	an	an	DET
cana-6080	28	34	np	np	NOUN
cana-6080	28	35	-	-	PUNCT
cana-6080	28	36	complete	complete	ADJ
cana-6080	28	37	problem	problem	NOUN
cana-6080	28	38	.	.	PUNCT
cana-6080	29	1	2	2	NUM
cana-6080	29	2	preliminaries	preliminary	NOUN
cana-6080	29	3	all	all	DET
cana-6080	29	4	graphs	graph	NOUN
cana-6080	29	5	we	we	PRON
cana-6080	29	6	consider	consider	VERB
cana-6080	29	7	are	be	AUX
cana-6080	29	8	simple	simple	ADJ
cana-6080	29	9	and	and	CCONJ
cana-6080	29	10	fnite	fnite	ADJ
cana-6080	29	11	.	.	PUNCT
cana-6080	30	1	definition	definition	NOUN
cana-6080	30	2	2.1	2.1	NUM
cana-6080	30	3	the	the	DET
cana-6080	30	4	generalized	generalized	ADJ
cana-6080	30	5	fan	fan	NOUN
cana-6080	30	6	graph	graph	NOUN
cana-6080	30	7	fn	fn	PROPN
cana-6080	30	8	,	,	PUNCT
cana-6080	30	9	m	m	VERB
cana-6080	30	10	is	be	AUX
cana-6080	30	11	defined	define	VERB
cana-6080	30	12	as	as	ADP
cana-6080	30	13	the	the	DET
cana-6080	30	14	graph	graph	NOUN
cana-6080	30	15	join	join	VERB
cana-6080	30	16	kn+pm	kn+pm	NOUN
cana-6080	30	17	,	,	PUNCT
cana-6080	30	18	where	where	SCONJ
cana-6080	30	19	kn	kn	PROPN
cana-6080	30	20	is	be	AUX
cana-6080	30	21	the	the	DET
cana-6080	30	22	empty	empty	ADJ
cana-6080	30	23	graph	graph	NOUN
cana-6080	30	24	with	with	ADP
cana-6080	30	25	n	n	ADP
cana-6080	30	26	vertices	vertex	NOUN
cana-6080	30	27	and	and	CCONJ
cana-6080	30	28	pm	pm	NOUN
cana-6080	30	29	is	be	AUX
cana-6080	30	30	the	the	DET
cana-6080	30	31	path	path	NOUN
cana-6080	30	32	graph	graph	NOUN
cana-6080	30	33	with	with	ADP
cana-6080	30	34	m	m	PROPN
cana-6080	30	35	vertices	vertex	NOUN
cana-6080	30	36	.	.	PUNCT
cana-6080	31	1	the	the	DET
cana-6080	31	2	case	case	NOUN
cana-6080	31	3	n	n	NOUN
cana-6080	31	4	=	=	SYM
cana-6080	31	5	1	1	NUM
cana-6080	31	6	corresponds	correspond	VERB
cana-6080	31	7	to	to	ADP
cana-6080	31	8	the	the	DET
cana-6080	31	9	usual	usual	ADJ
cana-6080	31	10	fan	fan	NOUN
cana-6080	31	11	graphs	graph	NOUN
cana-6080	31	12	,	,	PUNCT
cana-6080	31	13	while	while	SCONJ
cana-6080	31	14	n	n	NOUN
cana-6080	31	15	=	=	SYM
cana-6080	31	16	2	2	NUM
cana-6080	31	17	corresponds	correspond	NOUN
cana-6080	31	18	to	to	ADP
cana-6080	31	19	the	the	DET
cana-6080	31	20	double	double	ADJ
cana-6080	31	21	fan	fan	NOUN
cana-6080	31	22	graphs	graph	NOUN
cana-6080	31	23	,	,	PUNCT
cana-6080	31	24	etc	etc	X
cana-6080	31	25	.	.	X
cana-6080	31	26	number	number	NOUN
cana-6080	31	27	of	of	ADP
cana-6080	31	28	edges	edge	NOUN
cana-6080	31	29	of	of	ADP
cana-6080	31	30	the	the	DET
cana-6080	31	31	generalized	generalized	ADJ
cana-6080	31	32	fan	fan	NOUN
cana-6080	31	33	graph	graph	NOUN
cana-6080	31	34	fn	fn	PROPN
cana-6080	31	35	,	,	PUNCT
cana-6080	31	36	m	m	VERB
cana-6080	31	37	with	with	ADP
cana-6080	31	38	(	(	PUNCT
cana-6080	31	39	n	n	X
cana-6080	31	40	+	+	NUM
cana-6080	31	41	m	m	NOUN
cana-6080	31	42	)	)	PUNCT
cana-6080	31	43	vertices	vertex	NOUN
cana-6080	31	44	is	be	AUX
cana-6080	31	45	nm	nm	ADJ
cana-6080	31	46	+	+	CCONJ
cana-6080	31	47	(	(	PUNCT
cana-6080	31	48	m	m	VERB
cana-6080	31	49	−	−	NOUN
cana-6080	31	50	1	1	NUM
cana-6080	31	51	)	)	PUNCT
cana-6080	31	52	.	.	PUNCT
cana-6080	32	1	definition	definition	NOUN
cana-6080	32	2	2.2	2.2	NUM
cana-6080	32	3	the	the	DET
cana-6080	32	4	line	line	NOUN
cana-6080	32	5	graph	graph	NOUN
cana-6080	32	6	of	of	ADP
cana-6080	32	7	g	g	PROPN
cana-6080	32	8	denoted	denote	VERB
cana-6080	32	9	by	by	ADP
cana-6080	32	10	l(g	l(g	NOUN
cana-6080	32	11	)	)	PUNCT
cana-6080	32	12	is	be	AUX
cana-6080	32	13	the	the	DET
cana-6080	32	14	graph	graph	NOUN
cana-6080	32	15	with	with	ADP
cana-6080	32	16	vertices	vertex	NOUN
cana-6080	32	17	are	be	AUX
cana-6080	32	18	the	the	DET
cana-6080	32	19	edges	edge	NOUN
cana-6080	32	20	of	of	ADP
cana-6080	32	21	g	g	NOUN
cana-6080	32	22	with	with	ADP
cana-6080	32	23	two	two	NUM
cana-6080	32	24	vertices	vertex	NOUN
cana-6080	32	25	of	of	ADP
cana-6080	32	26	l(g	l(g	NOUN
cana-6080	32	27	)	)	PUNCT
cana-6080	32	28	adjacent	adjacent	ADJ
cana-6080	32	29	whenever	whenever	SCONJ
cana-6080	32	30	the	the	DET
cana-6080	32	31	corresponding	corresponding	ADJ
cana-6080	32	32	edges	edge	NOUN
cana-6080	32	33	of	of	ADP
cana-6080	32	34	g	g	NOUN
cana-6080	32	35	are	be	AUX
cana-6080	32	36	adjacent	adjacent	ADJ
cana-6080	32	37	.	.	PUNCT
cana-6080	33	1	definition	definition	NOUN
cana-6080	33	2	2.3	2.3	NUM
cana-6080	33	3	middle	middle	ADJ
cana-6080	33	4	graph	graph	NOUN
cana-6080	33	5	m(g	m(g	PROPN
cana-6080	33	6	)	)	PUNCT
cana-6080	33	7	of	of	ADP
cana-6080	33	8	a	a	DET
cana-6080	33	9	graph	graph	NOUN
cana-6080	33	10	g	g	NOUN
cana-6080	33	11	is	be	AUX
cana-6080	33	12	defined	define	VERB
cana-6080	33	13	with	with	ADP
cana-6080	33	14	the	the	DET
cana-6080	33	15	vertex	vertex	NOUN
cana-6080	33	16	set	set	VERB
cana-6080	33	17	v	v	NOUN
cana-6080	33	18	(	(	PUNCT
cana-6080	33	19	g	g	NOUN
cana-6080	33	20	)	)	PUNCT
cana-6080	33	21	∪	∪	ADP
cana-6080	33	22	e(g	e(g	PROPN
cana-6080	33	23	)	)	PUNCT
cana-6080	33	24	,	,	PUNCT
cana-6080	33	25	in	in	ADP
cana-6080	33	26	which	which	PRON
cana-6080	33	27	two	two	NUM
cana-6080	33	28	elements	element	NOUN
cana-6080	33	29	are	be	AUX
cana-6080	33	30	adjacent	adjacent	ADJ
cana-6080	33	31	if	if	SCONJ
cana-6080	33	32	and	and	CCONJ
cana-6080	33	33	only	only	ADV
cana-6080	33	34	if	if	SCONJ
cana-6080	33	35	either	either	DET
cana-6080	33	36	both	both	PRON
cana-6080	33	37	are	be	AUX
cana-6080	33	38	adjacent	adjacent	ADJ
cana-6080	33	39	edges	edge	NOUN
cana-6080	33	40	in	in	ADP
cana-6080	33	41	g	g	NOUN
cana-6080	33	42	or	or	CCONJ
cana-6080	33	43	one	one	NUM
cana-6080	33	44	of	of	ADP
cana-6080	33	45	the	the	DET
cana-6080	33	46	elements	element	NOUN
cana-6080	33	47	is	be	AUX
cana-6080	33	48	a	a	DET
cana-6080	33	49	vertex	vertex	NOUN
cana-6080	33	50	and	and	CCONJ
cana-6080	33	51	the	the	DET
cana-6080	33	52	other	other	ADJ
cana-6080	33	53	one	one	NOUN
cana-6080	33	54	is	be	AUX
cana-6080	33	55	an	an	DET
cana-6080	33	56	edge	edge	NOUN
cana-6080	33	57	incident	incident	NOUN
cana-6080	33	58	to	to	ADP
cana-6080	33	59	the	the	DET
cana-6080	33	60	vertex	vertex	NOUN
cana-6080	33	61	in	in	ADP
cana-6080	33	62	g.	g.	PROPN
cana-6080	33	63	definition	definition	NOUN
cana-6080	33	64	2.4	2.4	NUM
cana-6080	33	65	total	total	ADJ
cana-6080	33	66	graph	graph	NOUN
cana-6080	33	67	t(g	t(g	NOUN
cana-6080	33	68	)	)	PUNCT
cana-6080	33	69	of	of	ADP
cana-6080	33	70	a	a	DET
cana-6080	33	71	graph	graph	NOUN
cana-6080	33	72	g	g	NOUN
cana-6080	33	73	defined	define	VERB
cana-6080	33	74	with	with	ADP
cana-6080	33	75	the	the	DET
cana-6080	33	76	vertex	vertex	NOUN
cana-6080	33	77	set	set	VERB
cana-6080	33	78	v	v	NOUN
cana-6080	33	79	(	(	PUNCT
cana-6080	33	80	g	g	NOUN
cana-6080	33	81	)	)	PUNCT
cana-6080	33	82	∪	∪	ADP
cana-6080	33	83	e(g	e(g	PROPN
cana-6080	33	84	)	)	PUNCT
cana-6080	33	85	,	,	PUNCT
cana-6080	33	86	in	in	ADP
cana-6080	33	87	which	which	PRON
cana-6080	33	88	two	two	NUM
cana-6080	33	89	elements	element	NOUN
cana-6080	33	90	are	be	AUX
cana-6080	33	91	adjacent	adjacent	ADJ
cana-6080	33	92	if	if	SCONJ
cana-6080	33	93	and	and	CCONJ
cana-6080	33	94	only	only	ADV
cana-6080	33	95	if	if	SCONJ
cana-6080	33	96	one	one	NUM
cana-6080	33	97	of	of	ADP
cana-6080	33	98	the	the	DET
cana-6080	33	99	following	following	NOUN
cana-6080	33	100	holds	hold	VERB
cana-6080	33	101	true	true	ADJ
cana-6080	33	102	(	(	PUNCT
cana-6080	33	103	i	i	NOUN
cana-6080	33	104	)	)	PUNCT
cana-6080	33	105	both	both	PRON
cana-6080	33	106	are	be	AUX
cana-6080	33	107	adjacent	adjacent	ADJ
cana-6080	33	108	edges	edge	NOUN
cana-6080	33	109	or	or	CCONJ
cana-6080	33	110	vertices	vertex	NOUN
cana-6080	33	111	in	in	ADP
cana-6080	33	112	g	g	PROPN
cana-6080	33	113	(	(	PUNCT
cana-6080	33	114	ii	ii	NOUN
cana-6080	33	115	)	)	PUNCT
cana-6080	33	116	one	one	NOUN
cana-6080	33	117	is	be	AUX
cana-6080	33	118	a	a	DET
cana-6080	33	119	vertex	vertex	NOUN
cana-6080	33	120	and	and	CCONJ
cana-6080	33	121	other	other	ADJ
cana-6080	33	122	is	be	AUX
cana-6080	33	123	an	an	DET
cana-6080	33	124	edge	edge	NOUN
cana-6080	33	125	incident	incident	NOUN
cana-6080	33	126	to	to	ADP
cana-6080	33	127	it	it	PRON
cana-6080	33	128	in	in	ADP
cana-6080	33	129	g.	g.	PROPN
cana-6080	33	130	3	3	NUM
cana-6080	33	131	main	main	ADJ
cana-6080	33	132	results	result	NOUN
cana-6080	33	133	we	we	PRON
cana-6080	33	134	begin	begin	VERB
cana-6080	33	135	by	by	ADP
cana-6080	33	136	presenting	present	VERB
cana-6080	33	137	the	the	DET
cana-6080	33	138	existing	exist	VERB
cana-6080	33	139	results	result	NOUN
cana-6080	33	140	concerning	concern	VERB
cana-6080	33	141	the	the	DET
cana-6080	33	142	grundy	grundy	ADJ
cana-6080	33	143	chromatic	chromatic	ADJ
cana-6080	33	144	number	number	NOUN
cana-6080	33	145	of	of	ADP
cana-6080	33	146	fan	fan	NOUN
cana-6080	33	147	graph	graph	NOUN
cana-6080	33	148	,	,	PUNCT
cana-6080	33	149	line	line	NOUN
cana-6080	33	150	graph	graph	NOUN
cana-6080	33	151	of	of	ADP
cana-6080	33	152	fan	fan	NOUN
cana-6080	33	153	graph	graph	NOUN
cana-6080	33	154	,	,	PUNCT
cana-6080	33	155	central	central	ADJ
cana-6080	33	156	graph	graph	NOUN
cana-6080	33	157	of	of	ADP
cana-6080	33	158	fan	fan	NOUN
cana-6080	33	159	graph	graph	NOUN
cana-6080	33	160	,	,	PUNCT
cana-6080	33	161	middle	middle	ADJ
cana-6080	33	162	graph	graph	NOUN
cana-6080	33	163	of	of	ADP
cana-6080	33	164	fan	fan	NOUN
cana-6080	33	165	graph	graph	NOUN
cana-6080	33	166	and	and	CCONJ
cana-6080	33	167	total	total	ADJ
cana-6080	33	168	graph	graph	NOUN
cana-6080	33	169	of	of	ADP
cana-6080	33	170	fan	fan	NOUN
cana-6080	33	171	graph	graph	NOUN
cana-6080	33	172	.	.	PUNCT
cana-6080	34	1	communications	communication	NOUN
cana-6080	34	2	on	on	ADP
cana-6080	34	3	applied	apply	VERB
cana-6080	34	4	nonlinear	nonlinear	ADJ
cana-6080	34	5	analysis	analysis	NOUN
cana-6080	34	6	issn	issn	NOUN
cana-6080	34	7	:	:	PUNCT
cana-6080	34	8	1074	1074	NUM
cana-6080	34	9	-	-	PUNCT
cana-6080	34	10	133x	133x	NUM
cana-6080	34	11	vol	vol	NOUN
cana-6080	34	12	31	31	NUM
cana-6080	34	13	no	no	NOUN
cana-6080	34	14	.	.	PUNCT
cana-6080	35	1	8s	8s	PROPN
cana-6080	35	2	(	(	PUNCT
cana-6080	35	3	2024	2024	NUM
cana-6080	35	4	)	)	PUNCT
cana-6080	35	5	1196	1196	NUM
cana-6080	35	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6080	35	7	theorem	theorem	VERB
cana-6080	35	8	3.1	3.1	NUM
cana-6080	35	9	for	for	ADP
cana-6080	35	10	any	any	DET
cana-6080	35	11	positive	positive	ADJ
cana-6080	35	12	integer	integer	NOUN
cana-6080	35	13	m	m	NOUN
cana-6080	35	14	and	and	CCONJ
cana-6080	35	15	n	n	PRON
cana-6080	35	16	≥	≥	NUM
cana-6080	35	17	2	2	NUM
cana-6080	35	18	,	,	PUNCT
cana-6080	35	19	then	then	ADV
cana-6080	35	20	γ(fm	γ(fm	PROPN
cana-6080	35	21	,	,	PUNCT
cana-6080	35	22	n	n	CCONJ
cana-6080	35	23	)	)	PUNCT
cana-6080	35	24	is	be	AUX
cana-6080	35	25	3	3	NUM
cana-6080	35	26	,	,	PUNCT
cana-6080	35	27	if	if	SCONJ
cana-6080	35	28	n	n	NOUN
cana-6080	35	29	=	=	SYM
cana-6080	35	30	2	2	NUM
cana-6080	35	31	,	,	PUNCT
cana-6080	35	32	3	3	NUM
cana-6080	35	33	and	and	CCONJ
cana-6080	35	34	is	be	AUX
cana-6080	35	35	4	4	NUM
cana-6080	35	36	,	,	PUNCT
cana-6080	35	37	if	if	SCONJ
cana-6080	35	38	n	n	PRON
cana-6080	35	39	≥	≥	NOUN
cana-6080	35	40	4	4	NUM
cana-6080	35	41	.	.	PUNCT
cana-6080	36	1	proof	proof	NOUN
cana-6080	36	2	:	:	PUNCT
cana-6080	36	3	let	let	VERB
cana-6080	36	4	us	we	PRON
cana-6080	36	5	consider	consider	VERB
cana-6080	36	6	the	the	DET
cana-6080	36	7	fan	fan	NOUN
cana-6080	36	8	graph	graph	NOUN
cana-6080	36	9	of	of	ADP
cana-6080	36	10	the	the	DET
cana-6080	36	11	form	form	NOUN
cana-6080	36	12	fm	fm	PROPN
cana-6080	36	13	,	,	PUNCT
cana-6080	36	14	n	n	CCONJ
cana-6080	36	15	,	,	PUNCT
cana-6080	36	16	where	where	SCONJ
cana-6080	36	17	m	m	PROPN
cana-6080	36	18	≥	≥	VERB
cana-6080	36	19	1	1	NUM
cana-6080	36	20	and	and	CCONJ
cana-6080	36	21	n	n	PRON
cana-6080	36	22	≥	≥	NOUN
cana-6080	36	23	2	2	NUM
cana-6080	36	24	.	.	PUNCT
cana-6080	37	1	the	the	DET
cana-6080	37	2	grundy	grundy	PROPN
cana-6080	37	3	chromatic	chromatic	ADJ
cana-6080	37	4	number	number	NOUN
cana-6080	37	5	of	of	ADP
cana-6080	37	6	fm	fm	PROPN
cana-6080	37	7	,	,	PUNCT
cana-6080	37	8	n	n	PRON
cana-6080	37	9	be	be	AUX
cana-6080	37	10	denoted	denote	VERB
cana-6080	37	11	as	as	ADP
cana-6080	37	12	γ(fm	γ(fm	PROPN
cana-6080	37	13	,	,	PUNCT
cana-6080	37	14	n	n	CCONJ
cana-6080	37	15	)	)	PUNCT
cana-6080	37	16	.	.	PUNCT
cana-6080	38	1	let	let	VERB
cana-6080	38	2	v(fm	v(fm	PROPN
cana-6080	38	3	,	,	PUNCT
cana-6080	38	4	n	n	CCONJ
cana-6080	38	5	)	)	PUNCT
cana-6080	38	6	represent	represent	VERB
cana-6080	38	7	the	the	DET
cana-6080	38	8	vertices	vertex	NOUN
cana-6080	38	9	of	of	ADP
cana-6080	38	10	fm	fm	PROPN
cana-6080	38	11	,	,	PUNCT
cana-6080	38	12	n.	n.	PROPN
cana-6080	38	13	that	that	PRON
cana-6080	38	14	is	be	AUX
cana-6080	38	15	,	,	PUNCT
cana-6080	38	16	v(fm	v(fm	PROPN
cana-6080	38	17	,	,	PUNCT
cana-6080	38	18	n	n	CCONJ
cana-6080	38	19	)	)	PUNCT
cana-6080	38	20	is	be	AUX
cana-6080	38	21	ui	ui	NOUN
cana-6080	38	22	(	(	PUNCT
cana-6080	38	23	1	1	NUM
cana-6080	38	24	≤	≤	NUM
cana-6080	38	25	i	i	X
cana-6080	38	26	≤	≤	NOUN
cana-6080	38	27	m	m	VERB
cana-6080	38	28	)	)	PUNCT
cana-6080	38	29	and	and	CCONJ
cana-6080	38	30	vj	vj	INTJ
cana-6080	38	31	(	(	PUNCT
cana-6080	38	32	1	1	NUM
cana-6080	38	33	≤	≤	NUM
cana-6080	38	34	j	j	PROPN
cana-6080	38	35	≤	≤	NUM
cana-6080	38	36	n	n	CCONJ
cana-6080	38	37	)	)	PUNCT
cana-6080	38	38	.	.	PUNCT
cana-6080	39	1	we	we	PRON
cana-6080	39	2	define	define	VERB
cana-6080	39	3	a	a	DET
cana-6080	39	4	mapping	mapping	NOUN
cana-6080	39	5	,	,	PUNCT
cana-6080	39	6	σ	σ	NOUN
cana-6080	39	7	:	:	PUNCT
cana-6080	39	8	v	v	X
cana-6080	39	9	(	(	PUNCT
cana-6080	39	10	fm	fm	PROPN
cana-6080	39	11	,	,	PUNCT
cana-6080	39	12	n	n	CCONJ
cana-6080	39	13	)	)	PUNCT
cana-6080	39	14	→	→	PUNCT
cana-6080	39	15	n	n	CCONJ
cana-6080	39	16	as	as	SCONJ
cana-6080	39	17	follows	follow	VERB
cana-6080	39	18	:	:	PUNCT
cana-6080	39	19	case	case	NOUN
cana-6080	39	20	(	(	PUNCT
cana-6080	39	21	i	i	NOUN
cana-6080	39	22	):	):	PUNCT
cana-6080	39	23	for	for	ADP
cana-6080	39	24	n	n	NOUN
cana-6080	39	25	=	=	SYM
cana-6080	39	26	2	2	NUM
cana-6080	39	27	,	,	PUNCT
cana-6080	39	28	3	3	NUM
cana-6080	39	29	.	.	X
cana-6080	39	30	for	for	ADP
cana-6080	39	31	1	1	NUM
cana-6080	39	32	≤	≤	NUM
cana-6080	39	33	i	i	PRON
cana-6080	39	34	≤	≤	NOUN
cana-6080	39	35	m	m	ADP
cana-6080	39	36	,	,	PUNCT
cana-6080	39	37	1	1	NUM
cana-6080	39	38	≤	≤	NUM
cana-6080	39	39	j	j	PROPN
cana-6080	39	40	≤	≤	ADV
cana-6080	39	41	3	3	NUM
cana-6080	39	42	,	,	PUNCT
cana-6080	39	43	σ	σ	PROPN
cana-6080	39	44	(	(	PUNCT
cana-6080	39	45	ui	ui	PROPN
cana-6080	39	46	)	)	PUNCT
cana-6080	39	47	=	=	SYM
cana-6080	40	1	3	3	X
cana-6080	40	2	.	.	X
cana-6080	41	1	σ	σ	NOUN
cana-6080	41	2	(	(	PUNCT
cana-6080	41	3	vj	vj	INTJ
cana-6080	41	4	)	)	PUNCT
cana-6080	41	5	=	=	SYM
cana-6080	42	1	1	1	X
cana-6080	42	2	,	,	PUNCT
cana-6080	42	3	if	if	SCONJ
cana-6080	42	4	j	j	PROPN
cana-6080	42	5	≡	≡	PROPN
cana-6080	42	6	1	1	NUM
cana-6080	42	7	mod	mod	NOUN
cana-6080	42	8	2	2	NUM
cana-6080	42	9	and	and	CCONJ
cana-6080	42	10	2	2	NUM
cana-6080	42	11	,	,	PUNCT
cana-6080	42	12	if	if	SCONJ
cana-6080	42	13	j	j	PROPN
cana-6080	42	14	≡	≡	PROPN
cana-6080	42	15	0	0	NUM
cana-6080	42	16	mod	mod	PROPN
cana-6080	42	17	2	2	NUM
cana-6080	42	18	since	since	SCONJ
cana-6080	42	19	γ(fn	γ(fn	NUM
cana-6080	42	20	,	,	PUNCT
cana-6080	42	21	m	m	NOUN
cana-6080	42	22	)	)	PUNCT
cana-6080	42	23	≤	≤	NUM
cana-6080	42	24	3	3	NUM
cana-6080	42	25	.	.	X
cana-6080	42	26	assume	assume	VERB
cana-6080	42	27	that	that	SCONJ
cana-6080	42	28	γ(fn	γ(fn	NUM
cana-6080	42	29	,	,	PUNCT
cana-6080	42	30	m	m	NOUN
cana-6080	42	31	)	)	PUNCT
cana-6080	42	32	>	>	X
cana-6080	43	1	3	3	X
cana-6080	43	2	.	.	X
cana-6080	43	3	for	for	ADP
cana-6080	43	4	n	n	NOUN
cana-6080	43	5	=	=	SYM
cana-6080	43	6	2	2	NUM
cana-6080	43	7	,	,	PUNCT
cana-6080	43	8	there	there	PRON
cana-6080	43	9	exist	exist	VERB
cana-6080	43	10	a	a	DET
cana-6080	43	11	clique	clique	NOUN
cana-6080	43	12	of	of	ADP
cana-6080	43	13	order	order	NOUN
cana-6080	43	14	3	3	X
cana-6080	43	15	.	.	PUNCT
cana-6080	44	1	it	it	PRON
cana-6080	44	2	is	be	AUX
cana-6080	44	3	obviously	obviously	ADV
cana-6080	44	4	the	the	DET
cana-6080	44	5	color	color	NOUN
cana-6080	44	6	is	be	AUX
cana-6080	44	7	3	3	NUM
cana-6080	44	8	.	.	NOUN
cana-6080	44	9	for	for	ADP
cana-6080	44	10	n	n	NOUN
cana-6080	44	11	=	=	SYM
cana-6080	44	12	3	3	NUM
cana-6080	44	13	,	,	PUNCT
cana-6080	44	14	suppose	suppose	VERB
cana-6080	44	15	,	,	PUNCT
cana-6080	44	16	the	the	DET
cana-6080	44	17	vertices	vertex	NOUN
cana-6080	44	18	ui	ui	NOUN
cana-6080	44	19	(	(	PUNCT
cana-6080	44	20	1	1	NUM
cana-6080	44	21	≤	≤	NUM
cana-6080	44	22	i	i	X
cana-6080	44	23	≤	≤	PROPN
cana-6080	44	24	m	m	VERB
cana-6080	44	25	)	)	PUNCT
cana-6080	44	26	received	receive	VERB
cana-6080	44	27	any	any	DET
cana-6080	44	28	one	one	NUM
cana-6080	44	29	of	of	ADP
cana-6080	44	30	the	the	DET
cana-6080	44	31	color	color	NOUN
cana-6080	44	32	1	1	NUM
cana-6080	44	33	to	to	ADP
cana-6080	44	34	4	4	NUM
cana-6080	44	35	and	and	CCONJ
cana-6080	44	36	remaining	remain	VERB
cana-6080	44	37	three	three	NUM
cana-6080	44	38	color	color	NOUN
cana-6080	44	39	are	be	AUX
cana-6080	44	40	assigned	assign	VERB
cana-6080	44	41	to	to	ADP
cana-6080	44	42	the	the	DET
cana-6080	44	43	vertices	vertex	NOUN
cana-6080	44	44	vj	vj	INTJ
cana-6080	44	45	(	(	PUNCT
cana-6080	44	46	1	1	NUM
cana-6080	44	47	≤	≤	NUM
cana-6080	44	48	j	j	PROPN
cana-6080	44	49	≤	≤	NOUN
cana-6080	44	50	3	3	NUM
cana-6080	44	51	)	)	PUNCT
cana-6080	44	52	.	.	PUNCT
cana-6080	45	1	this	this	PRON
cana-6080	45	2	arises	arise	VERB
cana-6080	45	3	a	a	DET
cana-6080	45	4	contradiction	contradiction	NOUN
cana-6080	45	5	of	of	ADP
cana-6080	45	6	grundy	grundy	PROPN
cana-6080	45	7	chromatic	chromatic	ADJ
cana-6080	45	8	number	number	NOUN
cana-6080	45	9	definition	definition	NOUN
cana-6080	45	10	.	.	PUNCT
cana-6080	46	1	hence	hence	ADV
cana-6080	46	2	γ(fm	γ(fm	PROPN
cana-6080	46	3	,	,	PUNCT
cana-6080	46	4	n	n	CCONJ
cana-6080	46	5	)	)	PUNCT
cana-6080	46	6	≤	≤	NUM
cana-6080	46	7	3	3	NUM
cana-6080	46	8	.	.	PUNCT
cana-6080	47	1	therefore	therefore	ADV
cana-6080	47	2	,	,	PUNCT
cana-6080	47	3	γ(fm	γ(fm	PROPN
cana-6080	47	4	,	,	PUNCT
cana-6080	47	5	n	n	CCONJ
cana-6080	47	6	)	)	PUNCT
cana-6080	47	7	=	=	SYM
cana-6080	47	8	3	3	NUM
cana-6080	47	9	,	,	PUNCT
cana-6080	47	10	(	(	PUNCT
cana-6080	47	11	m	m	NOUN
cana-6080	47	12	≥	≥	NOUN
cana-6080	47	13	1	1	NUM
cana-6080	47	14	;	;	PUNCT
cana-6080	47	15	2	2	NUM
cana-6080	47	16	≤	≤	NOUN
cana-6080	47	17	n	n	CCONJ
cana-6080	47	18	≤	≤	NOUN
cana-6080	47	19	3	3	NUM
cana-6080	47	20	)	)	PUNCT
cana-6080	47	21	.	.	PUNCT
cana-6080	48	1	case	case	NOUN
cana-6080	48	2	(	(	PUNCT
cana-6080	48	3	ii	ii	NUM
cana-6080	48	4	):	):	PUNCT
cana-6080	48	5	for	for	ADP
cana-6080	48	6	n	n	X
cana-6080	48	7	≥	≥	NOUN
cana-6080	48	8	4	4	NUM
cana-6080	48	9	.	.	PUNCT
cana-6080	49	1	for	for	ADP
cana-6080	49	2	1	1	NUM
cana-6080	49	3	≤	≤	NUM
cana-6080	49	4	i	i	PRON
cana-6080	49	5	≤	≤	NOUN
cana-6080	49	6	m	m	PROPN
cana-6080	49	7	,	,	PUNCT
cana-6080	49	8	σ	σ	PROPN
cana-6080	49	9	(	(	PUNCT
cana-6080	49	10	ui	ui	PROPN
cana-6080	49	11	)	)	PUNCT
cana-6080	49	12	=	=	SYM
cana-6080	49	13	4	4	X
cana-6080	49	14	.	.	NOUN
cana-6080	49	15	subcase	subcase	PROPN
cana-6080	49	16	(	(	PUNCT
cana-6080	49	17	i	i	NOUN
cana-6080	49	18	):	):	PUNCT
cana-6080	49	19	for	for	ADP
cana-6080	49	20	n	n	PRON
cana-6080	49	21	≡	≡	PROPN
cana-6080	49	22	0	0	NUM
cana-6080	49	23	mod	mod	PROPN
cana-6080	50	1	3	3	X
cana-6080	50	2	.	.	PUNCT
cana-6080	50	3	σ	σ	PROPN
cana-6080	50	4	(	(	PUNCT
cana-6080	50	5	vj	vj	INTJ
cana-6080	50	6	)	)	PUNCT
cana-6080	50	7	=	=	SYM
cana-6080	50	8	(	(	PUNCT
cana-6080	50	9	for	for	ADP
cana-6080	50	10	1	1	NUM
cana-6080	50	11	≤	≤	NUM
cana-6080	50	12	j	j	PROPN
cana-6080	50	13	≤	≤	PROPN
cana-6080	50	14	n	n	CCONJ
cana-6080	50	15	−	−	PROPN
cana-6080	50	16	2	2	NUM
cana-6080	50	17	)	)	PUNCT
cana-6080	50	18	1	1	NUM
cana-6080	50	19	,	,	PUNCT
cana-6080	50	20	if	if	SCONJ
cana-6080	50	21	j	j	PROPN
cana-6080	50	22	≡	≡	PROPN
cana-6080	50	23	1	1	NUM
cana-6080	50	24	mod	mod	PROPN
cana-6080	50	25	3	3	NUM
cana-6080	50	26	3	3	NUM
cana-6080	50	27	,	,	PUNCT
cana-6080	50	28	if	if	SCONJ
cana-6080	50	29	j	j	PROPN
cana-6080	50	30	≡	≡	PROPN
cana-6080	50	31	2	2	NUM
cana-6080	50	32	mod	mod	NOUN
cana-6080	50	33	3	3	NUM
cana-6080	50	34	and	and	CCONJ
cana-6080	50	35	2	2	NUM
cana-6080	50	36	,	,	PUNCT
cana-6080	50	37	if	if	SCONJ
cana-6080	50	38	j	j	PROPN
cana-6080	50	39	≡	≡	PROPN
cana-6080	50	40	0	0	NUM
cana-6080	50	41	mod	mod	PROPN
cana-6080	50	42	3	3	NUM
cana-6080	50	43	σ	σ	PROPN
cana-6080	50	44	(	(	PUNCT
cana-6080	50	45	vn−1	vn−1	PROPN
cana-6080	50	46	)	)	PUNCT
cana-6080	50	47	=	=	SYM
cana-6080	50	48	2	2	NUM
cana-6080	50	49	;	;	PUNCT
cana-6080	50	50	σ	σ	PROPN
cana-6080	50	51	(	(	PUNCT
cana-6080	50	52	vn	vn	NOUN
cana-6080	50	53	)	)	PUNCT
cana-6080	50	54	=	=	SYM
cana-6080	50	55	1	1	X
cana-6080	50	56	.	.	X
cana-6080	50	57	subcase	subcase	PROPN
cana-6080	50	58	(	(	PUNCT
cana-6080	50	59	ii	ii	PROPN
cana-6080	50	60	):	):	PUNCT
cana-6080	50	61	for	for	ADP
cana-6080	50	62	n	n	PRON
cana-6080	50	63	≡	≡	PROPN
cana-6080	50	64	2	2	NUM
cana-6080	50	65	mod	mod	NOUN
cana-6080	50	66	3	3	NUM
cana-6080	50	67	.	.	PUNCT
cana-6080	51	1	σ	σ	PROPN
cana-6080	51	2	(	(	PUNCT
cana-6080	51	3	vj	vj	INTJ
cana-6080	51	4	)	)	PUNCT
cana-6080	51	5	=	=	SYM
cana-6080	51	6	(	(	PUNCT
cana-6080	51	7	for	for	ADP
cana-6080	51	8	1	1	NUM
cana-6080	51	9	≤	≤	NUM
cana-6080	51	10	j	j	PROPN
cana-6080	51	11	≤	≤	PROPN
cana-6080	51	12	n	n	CCONJ
cana-6080	51	13	–	–	PUNCT
cana-6080	51	14	1	1	NUM
cana-6080	51	15	)	)	PUNCT
cana-6080	51	16	1	1	NUM
cana-6080	51	17	,	,	PUNCT
cana-6080	51	18	if	if	SCONJ
cana-6080	51	19	j	j	PROPN
cana-6080	51	20	≡	≡	PROPN
cana-6080	51	21	1	1	NUM
cana-6080	51	22	mod	mod	PROPN
cana-6080	51	23	3	3	NUM
cana-6080	51	24	3	3	NUM
cana-6080	51	25	,	,	PUNCT
cana-6080	51	26	if	if	SCONJ
cana-6080	51	27	j	j	PROPN
cana-6080	51	28	≡	≡	PROPN
cana-6080	51	29	2	2	NUM
cana-6080	51	30	mod	mod	NOUN
cana-6080	51	31	3	3	NUM
cana-6080	51	32	and	and	CCONJ
cana-6080	51	33	2	2	NUM
cana-6080	51	34	,	,	PUNCT
cana-6080	51	35	if	if	SCONJ
cana-6080	51	36	j	j	PROPN
cana-6080	51	37	≡	≡	PROPN
cana-6080	51	38	0	0	NUM
cana-6080	51	39	mod	mod	PROPN
cana-6080	51	40	3	3	NUM
cana-6080	51	41	σ	σ	PROPN
cana-6080	51	42	(	(	PUNCT
cana-6080	51	43	vn	vn	NOUN
cana-6080	51	44	)	)	PUNCT
cana-6080	51	45	=	=	SYM
cana-6080	51	46	2	2	X
cana-6080	51	47	.	.	X
cana-6080	51	48	subcase	subcase	PROPN
cana-6080	51	49	(	(	PUNCT
cana-6080	51	50	iii	iii	NOUN
cana-6080	51	51	):	):	PUNCT
cana-6080	51	52	for	for	ADP
cana-6080	51	53	n	n	PRON
cana-6080	51	54	≡	≡	PROPN
cana-6080	51	55	1	1	NUM
cana-6080	51	56	mod	mod	NOUN
cana-6080	51	57	3	3	X
cana-6080	51	58	.	.	PUNCT
cana-6080	52	1	σ	σ	PROPN
cana-6080	52	2	(	(	PUNCT
cana-6080	52	3	vj	vj	INTJ
cana-6080	52	4	)	)	PUNCT
cana-6080	52	5	=	=	SYM
cana-6080	52	6	(	(	PUNCT
cana-6080	52	7	for	for	ADP
cana-6080	52	8	1	1	NUM
cana-6080	52	9	≤	≤	NUM
cana-6080	52	10	j	j	PROPN
cana-6080	52	11	≤	≤	NUM
cana-6080	52	12	n	n	CCONJ
cana-6080	52	13	)	)	PUNCT
cana-6080	52	14	1	1	NUM
cana-6080	52	15	,	,	PUNCT
cana-6080	52	16	if	if	SCONJ
cana-6080	52	17	j	j	PROPN
cana-6080	52	18	≡	≡	PROPN
cana-6080	52	19	1	1	NUM
cana-6080	52	20	mod	mod	PROPN
cana-6080	52	21	3	3	NUM
cana-6080	52	22	3	3	NUM
cana-6080	52	23	,	,	PUNCT
cana-6080	52	24	if	if	SCONJ
cana-6080	52	25	j	j	PROPN
cana-6080	52	26	≡	≡	PROPN
cana-6080	52	27	2	2	NUM
cana-6080	52	28	mod	mod	NOUN
cana-6080	52	29	3	3	NUM
cana-6080	52	30	and	and	CCONJ
cana-6080	52	31	2	2	NUM
cana-6080	52	32	,	,	PUNCT
cana-6080	52	33	if	if	SCONJ
cana-6080	52	34	j	j	PROPN
cana-6080	52	35	≡	≡	PROPN
cana-6080	52	36	0	0	NUM
cana-6080	52	37	mod	mod	ADJ
cana-6080	52	38	3	3	NUM
cana-6080	52	39	communications	communication	NOUN
cana-6080	52	40	on	on	ADP
cana-6080	52	41	applied	apply	VERB
cana-6080	52	42	nonlinear	nonlinear	ADJ
cana-6080	52	43	analysis	analysis	NOUN
cana-6080	52	44	issn	issn	NOUN
cana-6080	52	45	:	:	PUNCT
cana-6080	52	46	1074	1074	NUM
cana-6080	52	47	-	-	PUNCT
cana-6080	52	48	133x	133x	NUM
cana-6080	52	49	vol	vol	NOUN
cana-6080	52	50	31	31	NUM
cana-6080	52	51	no	no	NOUN
cana-6080	52	52	.	.	PUNCT
cana-6080	53	1	8s	8s	PROPN
cana-6080	53	2	(	(	PUNCT
cana-6080	53	3	2024	2024	NUM
cana-6080	53	4	)	)	PUNCT
cana-6080	53	5	1197	1197	NUM
cana-6080	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6080	53	7	since	since	SCONJ
cana-6080	53	8	γ(fn	γ(fn	NUM
cana-6080	53	9	,	,	PUNCT
cana-6080	53	10	m	m	NOUN
cana-6080	53	11	)	)	PUNCT
cana-6080	53	12	≤	≤	NUM
cana-6080	53	13	4	4	NUM
cana-6080	53	14	.	.	PUNCT
cana-6080	53	15	assume	assume	VERB
cana-6080	53	16	that	that	SCONJ
cana-6080	53	17	γ(fn	γ(fn	NUM
cana-6080	53	18	,	,	PUNCT
cana-6080	53	19	m	m	NOUN
cana-6080	53	20	)	)	PUNCT
cana-6080	53	21	>	>	X
cana-6080	53	22	4	4	NUM
cana-6080	53	23	,	,	PUNCT
cana-6080	53	24	suppose	suppose	VERB
cana-6080	53	25	,	,	PUNCT
cana-6080	53	26	the	the	DET
cana-6080	53	27	vertices	vertex	NOUN
cana-6080	53	28	ui	ui	NOUN
cana-6080	53	29	(	(	PUNCT
cana-6080	53	30	1	1	NUM
cana-6080	53	31	≤	≤	NUM
cana-6080	53	32	i	i	X
cana-6080	53	33	≤	≤	PROPN
cana-6080	53	34	m	m	VERB
cana-6080	53	35	)	)	PUNCT
cana-6080	53	36	received	receive	VERB
cana-6080	53	37	any	any	DET
cana-6080	53	38	one	one	NUM
cana-6080	53	39	of	of	ADP
cana-6080	53	40	the	the	DET
cana-6080	53	41	color	color	NOUN
cana-6080	53	42	1	1	NUM
cana-6080	53	43	to	to	ADP
cana-6080	53	44	5	5	NUM
cana-6080	53	45	and	and	CCONJ
cana-6080	53	46	remaining	remain	VERB
cana-6080	53	47	four	four	NUM
cana-6080	53	48	color	color	NOUN
cana-6080	53	49	are	be	AUX
cana-6080	53	50	assigned	assign	VERB
cana-6080	53	51	to	to	ADP
cana-6080	53	52	the	the	DET
cana-6080	53	53	vertices	vertex	NOUN
cana-6080	53	54	vj	vj	INTJ
cana-6080	53	55	(	(	PUNCT
cana-6080	53	56	1	1	NUM
cana-6080	53	57	≤	≤	NUM
cana-6080	53	58	j	j	PROPN
cana-6080	53	59	≤	≤	NUM
cana-6080	53	60	n	n	CCONJ
cana-6080	53	61	)	)	PUNCT
cana-6080	53	62	.	.	PUNCT
cana-6080	54	1	this	this	PRON
cana-6080	54	2	arises	arise	VERB
cana-6080	54	3	a	a	DET
cana-6080	54	4	contradiction	contradiction	NOUN
cana-6080	54	5	of	of	ADP
cana-6080	54	6	grundy	grundy	PROPN
cana-6080	54	7	chromatic	chromatic	ADJ
cana-6080	54	8	number	number	NOUN
cana-6080	54	9	definition	definition	NOUN
cana-6080	54	10	.	.	PUNCT
cana-6080	55	1	hence	hence	ADV
cana-6080	55	2	γ(fm	γ(fm	PROPN
cana-6080	55	3	,	,	PUNCT
cana-6080	55	4	n	n	CCONJ
cana-6080	55	5	)	)	PUNCT
cana-6080	55	6	≤	≤	NUM
cana-6080	55	7	4	4	NUM
cana-6080	55	8	.	.	PUNCT
cana-6080	56	1	therefore	therefore	ADV
cana-6080	56	2	,	,	PUNCT
cana-6080	56	3	γ(fm	γ(fm	PROPN
cana-6080	56	4	,	,	PUNCT
cana-6080	56	5	n	n	CCONJ
cana-6080	56	6	)	)	PUNCT
cana-6080	56	7	=	=	SYM
cana-6080	56	8	4	4	NUM
cana-6080	56	9	,	,	PUNCT
cana-6080	56	10	n	n	PRON
cana-6080	56	11	≥	≥	NOUN
cana-6080	56	12	4	4	NUM
cana-6080	56	13	.	.	PUNCT
cana-6080	56	14	theorem	theorem	VERB
cana-6080	56	15	3.2	3.2	NUM
cana-6080	56	16	for	for	ADP
cana-6080	56	17	any	any	DET
cana-6080	56	18	positive	positive	ADJ
cana-6080	56	19	integer	integer	NOUN
cana-6080	56	20	m	m	NOUN
cana-6080	56	21	and	and	CCONJ
cana-6080	56	22	n	n	PRON
cana-6080	56	23	≥	≥	NOUN
cana-6080	56	24	3	3	NUM
cana-6080	56	25	,	,	PUNCT
cana-6080	56	26	then	then	ADV
cana-6080	56	27	γ	γ	X
cana-6080	56	28	[	[	X
cana-6080	56	29	l(fm	l(fm	PROPN
cana-6080	56	30	,	,	PUNCT
cana-6080	56	31	n	n	CCONJ
cana-6080	56	32	)	)	PUNCT
cana-6080	56	33	]	]	PUNCT
cana-6080	57	1	=	=	PUNCT
cana-6080	57	2	n	n	PROPN
cana-6080	57	3	+	+	NUM
cana-6080	57	4	1	1	NUM
cana-6080	57	5	,	,	PUNCT
cana-6080	57	6	n	n	PRON
cana-6080	57	7	≥	≥	NOUN
cana-6080	57	8	m.	m.	NOUN
cana-6080	57	9	proof	proof	NOUN
cana-6080	57	10	:	:	PUNCT
cana-6080	57	11	let	let	VERB
cana-6080	57	12	us	we	PRON
cana-6080	57	13	consider	consider	VERB
cana-6080	57	14	the	the	DET
cana-6080	57	15	fan	fan	NOUN
cana-6080	57	16	graph	graph	NOUN
cana-6080	57	17	of	of	ADP
cana-6080	57	18	the	the	DET
cana-6080	57	19	form	form	NOUN
cana-6080	57	20	fm	fm	PROPN
cana-6080	57	21	,	,	PUNCT
cana-6080	57	22	n	n	CCONJ
cana-6080	57	23	,	,	PUNCT
cana-6080	57	24	where	where	SCONJ
cana-6080	57	25	m	m	PROPN
cana-6080	57	26	≥	≥	VERB
cana-6080	57	27	1	1	NUM
cana-6080	57	28	and	and	CCONJ
cana-6080	57	29	n	n	PRON
cana-6080	57	30	≥	≥	NOUN
cana-6080	57	31	3	3	X
cana-6080	57	32	.	.	PUNCT
cana-6080	58	1	let	let	VERB
cana-6080	58	2	v	v	X
cana-6080	58	3	(	(	PUNCT
cana-6080	58	4	fm	fm	PROPN
cana-6080	58	5	,	,	PUNCT
cana-6080	58	6	n	n	CCONJ
cana-6080	58	7	)	)	PUNCT
cana-6080	58	8	=	=	PRON
cana-6080	59	1	{	{	PUNCT
cana-6080	59	2	ui	ui	NOUN
cana-6080	59	3	:	:	PUNCT
cana-6080	59	4	1	1	NUM
cana-6080	59	5	≤	≤	NUM
cana-6080	59	6	i	i	X
cana-6080	59	7	≤	≤	NOUN
cana-6080	59	8	m	m	VERB
cana-6080	59	9	}	}	PUNCT
cana-6080	59	10	∪	∪	ADJ
cana-6080	59	11	{	{	PUNCT
cana-6080	59	12	vj	vj	NOUN
cana-6080	59	13	:	:	PUNCT
cana-6080	59	14	1	1	NUM
cana-6080	59	15	≤	≤	NUM
cana-6080	59	16	j	j	PROPN
cana-6080	59	17	≤	≤	PROPN
cana-6080	59	18	n	n	CCONJ
cana-6080	59	19	}	}	PUNCT
cana-6080	59	20	represent	represent	VERB
cana-6080	59	21	the	the	DET
cana-6080	59	22	vertices	vertex	NOUN
cana-6080	59	23	of	of	ADP
cana-6080	59	24	fm	fm	NOUN
cana-6080	59	25	,	,	PUNCT
cana-6080	59	26	n.	n.	VERB
cana-6080	59	27	by	by	ADP
cana-6080	59	28	the	the	DET
cana-6080	59	29	definition	definition	NOUN
cana-6080	59	30	of	of	ADP
cana-6080	59	31	line	line	NOUN
cana-6080	59	32	graph	graph	NOUN
cana-6080	59	33	,	,	PUNCT
cana-6080	59	34	let	let	VERB
cana-6080	59	35	v	v	ADP
cana-6080	59	36	[	[	X
cana-6080	59	37	l(fm	l(fm	PROPN
cana-6080	59	38	,	,	PUNCT
cana-6080	59	39	n	n	CCONJ
cana-6080	59	40	)	)	PUNCT
cana-6080	59	41	]	]	PUNCT
cana-6080	60	1	=	=	PUNCT
cana-6080	60	2	sum	sum	NOUN
cana-6080	60	3	i=1	i=1	PROPN
cana-6080	60	4	to	to	ADP
cana-6080	60	5	m	m	PROPN
cana-6080	60	6	{	{	PUNCT
cana-6080	60	7	ei	ei	PROPN
cana-6080	60	8	,	,	PUNCT
cana-6080	60	9	j	j	NOUN
cana-6080	60	10	:	:	PUNCT
cana-6080	60	11	1	1	NUM
cana-6080	60	12	≤	≤	NUM
cana-6080	60	13	j	j	PROPN
cana-6080	60	14	≤	≤	PROPN
cana-6080	60	15	n	n	CCONJ
cana-6080	60	16	}	}	PUNCT
cana-6080	60	17	∪	∪	X
cana-6080	60	18	{	{	PUNCT
cana-6080	60	19	xj	xj	NOUN
cana-6080	60	20	,	,	PUNCT
cana-6080	60	21	j+1	j+1	NUM
cana-6080	60	22	:	:	PUNCT
cana-6080	60	23	1	1	NUM
cana-6080	60	24	≤	≤	NUM
cana-6080	60	25	j	j	PROPN
cana-6080	60	26	≤	≤	PROPN
cana-6080	60	27	n	n	CCONJ
cana-6080	60	28	−	−	PROPN
cana-6080	60	29	1	1	NUM
cana-6080	60	30	}	}	PUNCT
cana-6080	60	31	.	.	PUNCT
cana-6080	61	1	we	we	PRON
cana-6080	61	2	define	define	VERB
cana-6080	61	3	a	a	DET
cana-6080	61	4	mapping	mapping	NOUN
cana-6080	61	5	,	,	PUNCT
cana-6080	61	6	σ	σ	NOUN
cana-6080	61	7	:	:	PUNCT
cana-6080	61	8	v	v	PART
cana-6080	61	9	[	[	X
cana-6080	61	10	l(fm	l(fm	PROPN
cana-6080	61	11	,	,	PUNCT
cana-6080	61	12	n	n	CCONJ
cana-6080	61	13	)	)	PUNCT
cana-6080	61	14	]	]	PUNCT
cana-6080	62	1	→	→	PUNCT
cana-6080	62	2	n	n	CCONJ
cana-6080	62	3	as	as	SCONJ
cana-6080	62	4	follows	follow	VERB
cana-6080	62	5	:	:	PUNCT
cana-6080	62	6	for	for	ADP
cana-6080	62	7	1	1	NUM
cana-6080	62	8	≤	≤	NUM
cana-6080	62	9	i	i	PRON
cana-6080	62	10	≤	≤	NOUN
cana-6080	62	11	m	m	ADP
cana-6080	62	12	,	,	PUNCT
cana-6080	62	13	1	1	NUM
cana-6080	62	14	≤	≤	NUM
cana-6080	62	15	j	j	PROPN
cana-6080	62	16	≤	≤	PROPN
cana-6080	62	17	n	n	CCONJ
cana-6080	62	18	,	,	PUNCT
cana-6080	62	19	σ	σ	PROPN
cana-6080	62	20	(	(	PUNCT
cana-6080	62	21	ei	ei	PROPN
cana-6080	62	22	,	,	PUNCT
cana-6080	62	23	j	j	PROPN
cana-6080	62	24	)	)	PUNCT
cana-6080	63	1	=	=	PUNCT
cana-6080	64	1	i	i	PRON
cana-6080	64	2	+	+	NUM
cana-6080	64	3	j	j	PROPN
cana-6080	64	4	,	,	PUNCT
cana-6080	64	5	i	i	PRON
cana-6080	64	6	≡	≡	PROPN
cana-6080	65	1	i	i	PRON
cana-6080	65	2	mod	mod	PROPN
cana-6080	65	3	m	m	PROPN
cana-6080	65	4	;	;	PUNCT
cana-6080	65	5	j	j	PROPN
cana-6080	65	6	≡	≡	PROPN
cana-6080	65	7	j	j	PROPN
cana-6080	65	8	mod	mod	PROPN
cana-6080	65	9	n	n	PROPN
cana-6080	65	10	σ	σ	PROPN
cana-6080	65	11	(	(	PUNCT
cana-6080	65	12	ej	ej	PROPN
cana-6080	65	13	,	,	PUNCT
cana-6080	65	14	j+1	j+1	PROPN
cana-6080	65	15	)	)	PUNCT
cana-6080	65	16	=	=	SYM
cana-6080	65	17	1	1	NUM
cana-6080	65	18	,	,	PUNCT
cana-6080	65	19	1	1	NUM
cana-6080	65	20	≤	≤	NUM
cana-6080	65	21	j	j	PROPN
cana-6080	65	22	≤	≤	PROPN
cana-6080	65	23	n	n	CCONJ
cana-6080	65	24	−	−	PROPN
cana-6080	65	25	1	1	NUM
cana-6080	65	26	.	.	PUNCT
cana-6080	66	1	since	since	SCONJ
cana-6080	66	2	above	above	ADP
cana-6080	66	3	color	color	NOUN
cana-6080	66	4	partition	partition	NOUN
cana-6080	66	5	γ(fn	γ(fn	PROPN
cana-6080	66	6	,	,	PUNCT
cana-6080	66	7	m	m	NOUN
cana-6080	66	8	)	)	PUNCT
cana-6080	66	9	≤	≤	NOUN
cana-6080	66	10	n	n	PRON
cana-6080	66	11	+	+	NOUN
cana-6080	66	12	1	1	X
cana-6080	66	13	.	.	X
cana-6080	66	14	assume	assume	VERB
cana-6080	66	15	that	that	SCONJ
cana-6080	66	16	γ(fn	γ(fn	NUM
cana-6080	66	17	,	,	PUNCT
cana-6080	66	18	m	m	NOUN
cana-6080	66	19	)	)	PUNCT
cana-6080	66	20	>	>	X
cana-6080	66	21	n	n	PROPN
cana-6080	67	1	+	+	NOUN
cana-6080	67	2	1	1	X
cana-6080	67	3	.	.	PUNCT
cana-6080	67	4	since	since	SCONJ
cana-6080	67	5	there	there	PRON
cana-6080	67	6	exist	exist	VERB
cana-6080	67	7	a	a	DET
cana-6080	67	8	clique	clique	NOUN
cana-6080	67	9	of	of	ADP
cana-6080	67	10	order	order	NOUN
cana-6080	67	11	n.	n.	NOUN
cana-6080	67	12	by	by	ADP
cana-6080	67	13	the	the	DET
cana-6080	67	14	above	above	ADJ
cana-6080	67	15	equation	equation	NOUN
cana-6080	67	16	γ	γ	X
cana-6080	67	17	[	[	X
cana-6080	67	18	l(fm	l(fm	PROPN
cana-6080	67	19	,	,	PUNCT
cana-6080	67	20	n	n	CCONJ
cana-6080	67	21	)	)	PUNCT
cana-6080	67	22	]	]	PUNCT
cana-6080	67	23	≤	≤	NUM
cana-6080	67	24	n	n	CCONJ
cana-6080	67	25	+	+	NOUN
cana-6080	67	26	1	1	X
cana-6080	67	27	.	.	PUNCT
cana-6080	67	28	this	this	PRON
cana-6080	67	29	arises	arise	VERB
cana-6080	67	30	a	a	DET
cana-6080	67	31	contradiction	contradiction	NOUN
cana-6080	67	32	.	.	PUNCT
cana-6080	68	1	therefore	therefore	ADV
cana-6080	68	2	,	,	PUNCT
cana-6080	68	3	γ	γ	X
cana-6080	68	4	[	[	X
cana-6080	68	5	l(fm	l(fm	PROPN
cana-6080	68	6	,	,	PUNCT
cana-6080	68	7	n	n	CCONJ
cana-6080	68	8	)	)	PUNCT
cana-6080	68	9	]	]	PUNCT
cana-6080	69	1	=	=	PUNCT
cana-6080	69	2	n	n	PROPN
cana-6080	69	3	+	+	NOUN
cana-6080	69	4	1	1	NUM
cana-6080	69	5	.	.	X
cana-6080	69	6	theorem	theorem	VERB
cana-6080	69	7	3.3	3.3	NUM
cana-6080	69	8	for	for	ADP
cana-6080	69	9	any	any	DET
cana-6080	69	10	positive	positive	ADJ
cana-6080	69	11	integer	integer	NOUN
cana-6080	69	12	m	m	NOUN
cana-6080	69	13	and	and	CCONJ
cana-6080	69	14	n	n	PRON
cana-6080	69	15	≥	≥	NOUN
cana-6080	69	16	3	3	NUM
cana-6080	69	17	,	,	PUNCT
cana-6080	69	18	then	then	ADV
cana-6080	69	19	γ	γ	X
cana-6080	69	20	[	[	X
cana-6080	69	21	m	m	X
cana-6080	69	22	(	(	PUNCT
cana-6080	69	23	fm	fm	PROPN
cana-6080	69	24	,	,	PUNCT
cana-6080	69	25	n	n	CCONJ
cana-6080	69	26	)	)	PUNCT
cana-6080	69	27	]	]	PUNCT
cana-6080	70	1	=	=	PUNCT
cana-6080	70	2	n	n	PROPN
cana-6080	70	3	+	+	NUM
cana-6080	70	4	2	2	NUM
cana-6080	70	5	,	,	PUNCT
cana-6080	70	6	n	n	PRON
cana-6080	70	7	≥	≥	NOUN
cana-6080	70	8	m.	m.	NOUN
cana-6080	70	9	proof	proof	NOUN
cana-6080	70	10	:	:	PUNCT
cana-6080	70	11	let	let	VERB
cana-6080	70	12	us	we	PRON
cana-6080	70	13	consider	consider	VERB
cana-6080	70	14	the	the	DET
cana-6080	70	15	fan	fan	NOUN
cana-6080	70	16	graph	graph	NOUN
cana-6080	70	17	of	of	ADP
cana-6080	70	18	the	the	DET
cana-6080	70	19	form	form	NOUN
cana-6080	70	20	fm	fm	PROPN
cana-6080	70	21	,	,	PUNCT
cana-6080	70	22	n	n	CCONJ
cana-6080	70	23	,	,	PUNCT
cana-6080	70	24	where	where	SCONJ
cana-6080	70	25	m	m	PROPN
cana-6080	70	26	≥	≥	VERB
cana-6080	70	27	1	1	NUM
cana-6080	70	28	and	and	CCONJ
cana-6080	70	29	n	n	PRON
cana-6080	70	30	≥	≥	NOUN
cana-6080	70	31	3	3	X
cana-6080	70	32	.	.	PUNCT
cana-6080	71	1	let	let	VERB
cana-6080	71	2	v	v	X
cana-6080	71	3	(	(	PUNCT
cana-6080	71	4	fm	fm	PROPN
cana-6080	71	5	,	,	PUNCT
cana-6080	71	6	n	n	CCONJ
cana-6080	71	7	)	)	PUNCT
cana-6080	71	8	=	=	PRON
cana-6080	72	1	{	{	PUNCT
cana-6080	72	2	ui	ui	NOUN
cana-6080	72	3	:	:	PUNCT
cana-6080	72	4	1	1	NUM
cana-6080	72	5	≤	≤	NUM
cana-6080	72	6	i	i	X
cana-6080	72	7	≤	≤	NOUN
cana-6080	72	8	m	m	VERB
cana-6080	72	9	}	}	PUNCT
cana-6080	72	10	∪	∪	ADJ
cana-6080	72	11	{	{	PUNCT
cana-6080	72	12	vj	vj	NOUN
cana-6080	72	13	:	:	PUNCT
cana-6080	72	14	1	1	NUM
cana-6080	72	15	≤	≤	NUM
cana-6080	72	16	j	j	PROPN
cana-6080	72	17	≤	≤	PROPN
cana-6080	72	18	n	n	CCONJ
cana-6080	72	19	}	}	PUNCT
cana-6080	72	20	represent	represent	VERB
cana-6080	72	21	the	the	DET
cana-6080	72	22	vertices	vertex	NOUN
cana-6080	72	23	of	of	ADP
cana-6080	72	24	fm	fm	NOUN
cana-6080	72	25	,	,	PUNCT
cana-6080	72	26	n.	n.	VERB
cana-6080	72	27	by	by	ADP
cana-6080	72	28	the	the	DET
cana-6080	72	29	definition	definition	NOUN
cana-6080	72	30	of	of	ADP
cana-6080	72	31	middle	middle	ADJ
cana-6080	72	32	graph	graph	NOUN
cana-6080	72	33	,	,	PUNCT
cana-6080	72	34	let	let	VERB
cana-6080	72	35	v	v	ADP
cana-6080	72	36	[	[	X
cana-6080	72	37	m	m	X
cana-6080	72	38	(	(	PUNCT
cana-6080	72	39	fm	fm	PROPN
cana-6080	72	40	,	,	PUNCT
cana-6080	72	41	n	n	CCONJ
cana-6080	72	42	)	)	PUNCT
cana-6080	72	43	]	]	PUNCT
cana-6080	73	1	=	=	SYM
cana-6080	73	2	v	v	X
cana-6080	73	3	(	(	PUNCT
cana-6080	73	4	fm	fm	PROPN
cana-6080	73	5	,	,	PUNCT
cana-6080	73	6	n	n	CCONJ
cana-6080	73	7	)	)	PUNCT
cana-6080	73	8	∪	∪	ADP
cana-6080	73	9	sum	sum	NOUN
cana-6080	73	10	i=1	i=1	PROPN
cana-6080	73	11	to	to	ADP
cana-6080	73	12	m	m	PROPN
cana-6080	73	13	{	{	PUNCT
cana-6080	73	14	ei	ei	PROPN
cana-6080	73	15	,	,	PUNCT
cana-6080	73	16	j	j	NOUN
cana-6080	73	17	:	:	PUNCT
cana-6080	73	18	1	1	NUM
cana-6080	73	19	≤	≤	NUM
cana-6080	73	20	j	j	PROPN
cana-6080	73	21	≤	≤	PROPN
cana-6080	73	22	n	n	CCONJ
cana-6080	73	23	}	}	PUNCT
cana-6080	73	24	∪	∪	X
cana-6080	73	25	{	{	PUNCT
cana-6080	73	26	xj	xj	NOUN
cana-6080	73	27	,	,	PUNCT
cana-6080	73	28	j+1	j+1	NUM
cana-6080	73	29	:	:	PUNCT
cana-6080	73	30	1	1	NUM
cana-6080	73	31	≤	≤	NUM
cana-6080	73	32	j	j	PROPN
cana-6080	73	33	≤	≤	PROPN
cana-6080	73	34	n−1	n−1	PROPN
cana-6080	73	35	}	}	PUNCT
cana-6080	73	36	.	.	PUNCT
cana-6080	74	1	we	we	PRON
cana-6080	74	2	define	define	VERB
cana-6080	74	3	a	a	DET
cana-6080	74	4	mapping	mapping	NOUN
cana-6080	74	5	,	,	PUNCT
cana-6080	74	6	σ	σ	NOUN
cana-6080	74	7	:	:	PUNCT
cana-6080	74	8	v	v	ADP
cana-6080	74	9	[	[	X
cana-6080	74	10	m	m	X
cana-6080	74	11	(	(	PUNCT
cana-6080	74	12	fm	fm	PROPN
cana-6080	74	13	,	,	PUNCT
cana-6080	74	14	n	n	CCONJ
cana-6080	74	15	)	)	PUNCT
cana-6080	74	16	]	]	PUNCT
cana-6080	75	1	→	→	PUNCT
cana-6080	75	2	n	n	CCONJ
cana-6080	75	3	as	as	SCONJ
cana-6080	75	4	follows	follow	VERB
cana-6080	75	5	:	:	PUNCT
cana-6080	75	6	for	for	ADP
cana-6080	75	7	1	1	NUM
cana-6080	75	8	≤	≤	NUM
cana-6080	75	9	i	i	PRON
cana-6080	75	10	≤	≤	NOUN
cana-6080	75	11	m	m	ADP
cana-6080	75	12	,	,	PUNCT
cana-6080	75	13	1	1	NUM
cana-6080	75	14	≤	≤	NUM
cana-6080	75	15	j	j	PROPN
cana-6080	75	16	≤	≤	PROPN
cana-6080	75	17	n	n	CCONJ
cana-6080	75	18	,	,	PUNCT
cana-6080	75	19	σ	σ	PROPN
cana-6080	75	20	(	(	PUNCT
cana-6080	75	21	ei	ei	PROPN
cana-6080	75	22	,	,	PUNCT
cana-6080	75	23	j	j	PROPN
cana-6080	75	24	)	)	PUNCT
cana-6080	76	1	=	=	PUNCT
cana-6080	77	1	i	i	PRON
cana-6080	77	2	+	+	NUM
cana-6080	77	3	j	j	PROPN
cana-6080	78	1	+	+	NOUN
cana-6080	78	2	1	1	NUM
cana-6080	78	3	,	,	PUNCT
cana-6080	78	4	i	i	PRON
cana-6080	78	5	≡	≡	PROPN
cana-6080	78	6	i	i	PRON
cana-6080	78	7	mod	mod	PROPN
cana-6080	78	8	m	m	PROPN
cana-6080	78	9	;	;	PUNCT
cana-6080	78	10	j	j	PROPN
cana-6080	78	11	≡	≡	PROPN
cana-6080	78	12	j	j	PROPN
cana-6080	78	13	mod	mod	PROPN
cana-6080	78	14	n	n	PROPN
cana-6080	78	15	σ	σ	PROPN
cana-6080	78	16	(	(	PUNCT
cana-6080	78	17	ui	ui	PROPN
cana-6080	78	18	)	)	PUNCT
cana-6080	78	19	=	=	SYM
cana-6080	78	20	1	1	NUM
cana-6080	78	21	;	;	PUNCT
cana-6080	78	22	communications	communication	NOUN
cana-6080	78	23	on	on	ADP
cana-6080	78	24	applied	apply	VERB
cana-6080	78	25	nonlinear	nonlinear	ADJ
cana-6080	78	26	analysis	analysis	NOUN
cana-6080	78	27	issn	issn	NOUN
cana-6080	78	28	:	:	PUNCT
cana-6080	78	29	1074	1074	NUM
cana-6080	78	30	-	-	PUNCT
cana-6080	78	31	133x	133x	NUM
cana-6080	78	32	vol	vol	NOUN
cana-6080	78	33	31	31	NUM
cana-6080	78	34	no	no	NOUN
cana-6080	78	35	.	.	PUNCT
cana-6080	79	1	8s	8s	PROPN
cana-6080	79	2	(	(	PUNCT
cana-6080	79	3	2024	2024	NUM
cana-6080	79	4	)	)	PUNCT
cana-6080	79	5	1198	1198	NUM
cana-6080	79	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6080	80	1	σ	σ	PROPN
cana-6080	80	2	(	(	PUNCT
cana-6080	80	3	vj	vj	INTJ
cana-6080	80	4	)	)	PUNCT
cana-6080	80	5	=	=	SYM
cana-6080	80	6	2	2	NUM
cana-6080	80	7	;	;	PUNCT
cana-6080	80	8	σ	σ	PROPN
cana-6080	80	9	(	(	PUNCT
cana-6080	80	10	ej	ej	PROPN
cana-6080	80	11	,	,	PUNCT
cana-6080	80	12	j+1	j+1	PROPN
cana-6080	80	13	)	)	PUNCT
cana-6080	80	14	=	=	SYM
cana-6080	80	15	1	1	NUM
cana-6080	80	16	,	,	PUNCT
cana-6080	80	17	1	1	NUM
cana-6080	80	18	≤	≤	NUM
cana-6080	80	19	j	j	PROPN
cana-6080	80	20	≤	≤	PROPN
cana-6080	80	21	n	n	CCONJ
cana-6080	80	22	−	−	PROPN
cana-6080	80	23	1	1	NUM
cana-6080	80	24	.	.	PUNCT
cana-6080	81	1	since	since	SCONJ
cana-6080	81	2	above	above	ADP
cana-6080	81	3	color	color	NOUN
cana-6080	81	4	partition	partition	NOUN
cana-6080	81	5	γ(fn	γ(fn	PROPN
cana-6080	81	6	,	,	PUNCT
cana-6080	81	7	m	m	NOUN
cana-6080	81	8	)	)	PUNCT
cana-6080	81	9	≤	≤	NOUN
cana-6080	81	10	n	n	PRON
cana-6080	81	11	+	+	NOUN
cana-6080	81	12	2	2	X
cana-6080	81	13	.	.	X
cana-6080	81	14	assume	assume	VERB
cana-6080	81	15	that	that	SCONJ
cana-6080	81	16	γ(fn	γ(fn	NUM
cana-6080	81	17	,	,	PUNCT
cana-6080	81	18	m	m	NOUN
cana-6080	81	19	)	)	PUNCT
cana-6080	81	20	>	>	X
cana-6080	81	21	n	n	PROPN
cana-6080	82	1	+	+	NOUN
cana-6080	82	2	2	2	X
cana-6080	82	3	.	.	PUNCT
cana-6080	82	4	since	since	SCONJ
cana-6080	82	5	there	there	PRON
cana-6080	82	6	exist	exist	VERB
cana-6080	82	7	a	a	DET
cana-6080	82	8	clique	clique	NOUN
cana-6080	82	9	of	of	ADP
cana-6080	82	10	order	order	NOUN
cana-6080	82	11	n	n	X
cana-6080	82	12	+	+	NOUN
cana-6080	82	13	1	1	X
cana-6080	82	14	.	.	PUNCT
cana-6080	82	15	by	by	ADP
cana-6080	82	16	equation	equation	NOUN
cana-6080	82	17	(	(	PUNCT
cana-6080	82	18	1	1	NUM
cana-6080	82	19	)	)	PUNCT
cana-6080	82	20	,	,	PUNCT
cana-6080	82	21	γ	γ	X
cana-6080	82	22	[	[	X
cana-6080	82	23	m	m	X
cana-6080	82	24	(	(	PUNCT
cana-6080	82	25	fm	fm	PROPN
cana-6080	82	26	,	,	PUNCT
cana-6080	82	27	n	n	CCONJ
cana-6080	82	28	)	)	PUNCT
cana-6080	82	29	]	]	PUNCT
cana-6080	82	30	≤	≤	NUM
cana-6080	82	31	n	n	CCONJ
cana-6080	82	32	+	+	NOUN
cana-6080	82	33	2	2	X
cana-6080	82	34	.	.	X
cana-6080	82	35	this	this	PRON
cana-6080	82	36	arises	arise	VERB
cana-6080	82	37	a	a	DET
cana-6080	82	38	contradiction	contradiction	NOUN
cana-6080	82	39	.	.	PUNCT
cana-6080	83	1	therefore	therefore	ADV
cana-6080	83	2	,	,	PUNCT
cana-6080	83	3	γ	γ	X
cana-6080	83	4	[	[	X
cana-6080	83	5	m	m	X
cana-6080	83	6	(	(	PUNCT
cana-6080	83	7	fm	fm	PROPN
cana-6080	83	8	,	,	PUNCT
cana-6080	83	9	n	n	CCONJ
cana-6080	83	10	)	)	PUNCT
cana-6080	83	11	]	]	PUNCT
cana-6080	84	1	=	=	PUNCT
cana-6080	84	2	n	n	PROPN
cana-6080	84	3	+	+	NOUN
cana-6080	84	4	2	2	NUM
cana-6080	84	5	.	.	X
cana-6080	84	6	theorem	theorem	VERB
cana-6080	84	7	3.4	3.4	NUM
cana-6080	84	8	for	for	ADP
cana-6080	84	9	any	any	DET
cana-6080	84	10	positive	positive	ADJ
cana-6080	84	11	integer	integer	NOUN
cana-6080	84	12	m	m	NOUN
cana-6080	84	13	and	and	CCONJ
cana-6080	84	14	n	n	PRON
cana-6080	84	15	≥	≥	NOUN
cana-6080	84	16	3	3	NUM
cana-6080	84	17	,	,	PUNCT
cana-6080	84	18	then	then	ADV
cana-6080	84	19	γ	γ	PROPN
cana-6080	84	20	[	[	X
cana-6080	84	21	t	t	X
cana-6080	84	22	(	(	PUNCT
cana-6080	84	23	fm	fm	PROPN
cana-6080	84	24	,	,	PUNCT
cana-6080	84	25	n	n	CCONJ
cana-6080	84	26	)	)	PUNCT
cana-6080	84	27	]	]	PUNCT
cana-6080	85	1	=	=	PUNCT
cana-6080	85	2	n	n	PROPN
cana-6080	85	3	+	+	NUM
cana-6080	85	4	2	2	NUM
cana-6080	85	5	,	,	PUNCT
cana-6080	85	6	n	n	PRON
cana-6080	85	7	≥	≥	NOUN
cana-6080	85	8	m.	m.	NOUN
cana-6080	85	9	proof	proof	NOUN
cana-6080	85	10	:	:	PUNCT
cana-6080	85	11	let	let	VERB
cana-6080	85	12	us	we	PRON
cana-6080	85	13	consider	consider	VERB
cana-6080	85	14	the	the	DET
cana-6080	85	15	fan	fan	NOUN
cana-6080	85	16	graph	graph	NOUN
cana-6080	85	17	of	of	ADP
cana-6080	85	18	the	the	DET
cana-6080	85	19	form	form	NOUN
cana-6080	85	20	fm	fm	PROPN
cana-6080	85	21	,	,	PUNCT
cana-6080	85	22	n	n	CCONJ
cana-6080	85	23	,	,	PUNCT
cana-6080	85	24	where	where	SCONJ
cana-6080	85	25	m	m	PROPN
cana-6080	85	26	≥	≥	VERB
cana-6080	85	27	1	1	NUM
cana-6080	85	28	and	and	CCONJ
cana-6080	85	29	n	n	PRON
cana-6080	85	30	≥	≥	NOUN
cana-6080	85	31	3	3	X
cana-6080	85	32	.	.	PUNCT
cana-6080	86	1	let	let	VERB
cana-6080	86	2	v	v	X
cana-6080	86	3	(	(	PUNCT
cana-6080	86	4	fm	fm	PROPN
cana-6080	86	5	,	,	PUNCT
cana-6080	86	6	n	n	CCONJ
cana-6080	86	7	)	)	PUNCT
cana-6080	86	8	=	=	PRON
cana-6080	87	1	{	{	PUNCT
cana-6080	87	2	ui	ui	NOUN
cana-6080	87	3	:	:	PUNCT
cana-6080	87	4	1	1	NUM
cana-6080	87	5	≤	≤	NUM
cana-6080	87	6	i	i	X
cana-6080	87	7	≤	≤	NOUN
cana-6080	87	8	m	m	VERB
cana-6080	87	9	}	}	PUNCT
cana-6080	87	10	∪	∪	ADJ
cana-6080	87	11	{	{	PUNCT
cana-6080	87	12	vj	vj	NOUN
cana-6080	87	13	:	:	PUNCT
cana-6080	87	14	1	1	NUM
cana-6080	87	15	≤	≤	NUM
cana-6080	87	16	j	j	PROPN
cana-6080	87	17	≤	≤	PROPN
cana-6080	87	18	n	n	CCONJ
cana-6080	87	19	}	}	PUNCT
cana-6080	87	20	represent	represent	VERB
cana-6080	87	21	the	the	DET
cana-6080	87	22	vertices	vertex	NOUN
cana-6080	87	23	of	of	ADP
cana-6080	87	24	fm	fm	NOUN
cana-6080	87	25	,	,	PUNCT
cana-6080	87	26	n.	n.	VERB
cana-6080	87	27	by	by	ADP
cana-6080	87	28	the	the	DET
cana-6080	87	29	definition	definition	NOUN
cana-6080	87	30	of	of	ADP
cana-6080	87	31	total	total	ADJ
cana-6080	87	32	graph	graph	NOUN
cana-6080	87	33	,	,	PUNCT
cana-6080	87	34	let	let	VERB
cana-6080	87	35	v	v	ADP
cana-6080	87	36	[	[	X
cana-6080	87	37	t	t	X
cana-6080	87	38	(	(	PUNCT
cana-6080	87	39	fm	fm	PROPN
cana-6080	87	40	,	,	PUNCT
cana-6080	87	41	n	n	CCONJ
cana-6080	87	42	)	)	PUNCT
cana-6080	87	43	]	]	PUNCT
cana-6080	88	1	=	=	SYM
cana-6080	88	2	v	v	X
cana-6080	88	3	(	(	PUNCT
cana-6080	88	4	fm	fm	PROPN
cana-6080	88	5	,	,	PUNCT
cana-6080	88	6	n	n	CCONJ
cana-6080	88	7	)	)	PUNCT
cana-6080	88	8	∪	∪	ADP
cana-6080	88	9	sum	sum	NOUN
cana-6080	88	10	i=1	i=1	PROPN
cana-6080	88	11	to	to	ADP
cana-6080	88	12	m	m	PROPN
cana-6080	88	13	{	{	PUNCT
cana-6080	88	14	ei	ei	PROPN
cana-6080	88	15	,	,	PUNCT
cana-6080	88	16	j	j	NOUN
cana-6080	88	17	:	:	PUNCT
cana-6080	88	18	1	1	NUM
cana-6080	88	19	≤	≤	NUM
cana-6080	88	20	j	j	PROPN
cana-6080	88	21	≤	≤	PROPN
cana-6080	88	22	n	n	CCONJ
cana-6080	88	23	}	}	PUNCT
cana-6080	88	24	∪	∪	X
cana-6080	88	25	{	{	PUNCT
cana-6080	88	26	xj	xj	NOUN
cana-6080	88	27	,	,	PUNCT
cana-6080	88	28	j+1	j+1	NUM
cana-6080	88	29	:	:	PUNCT
cana-6080	88	30	1	1	NUM
cana-6080	88	31	≤	≤	NUM
cana-6080	88	32	j	j	PROPN
cana-6080	88	33	≤	≤	PROPN
cana-6080	88	34	n	n	CCONJ
cana-6080	88	35	−	−	PROPN
cana-6080	88	36	1	1	NUM
cana-6080	88	37	}	}	PUNCT
cana-6080	88	38	.	.	PUNCT
cana-6080	89	1	we	we	PRON
cana-6080	89	2	define	define	VERB
cana-6080	89	3	a	a	DET
cana-6080	89	4	mapping	mapping	NOUN
cana-6080	89	5	,	,	PUNCT
cana-6080	89	6	σ	σ	NOUN
cana-6080	89	7	:	:	PUNCT
cana-6080	89	8	v	v	ADP
cana-6080	89	9	[	[	X
cana-6080	89	10	t	t	X
cana-6080	89	11	(	(	PUNCT
cana-6080	89	12	fm	fm	PROPN
cana-6080	89	13	,	,	PUNCT
cana-6080	89	14	n	n	CCONJ
cana-6080	89	15	)	)	PUNCT
cana-6080	89	16	]	]	PUNCT
cana-6080	90	1	→	→	PUNCT
cana-6080	90	2	n	n	CCONJ
cana-6080	90	3	as	as	SCONJ
cana-6080	90	4	follows	follow	VERB
cana-6080	90	5	:	:	PUNCT
cana-6080	90	6	for	for	ADP
cana-6080	90	7	1	1	NUM
cana-6080	90	8	≤	≤	NUM
cana-6080	90	9	i	i	PRON
cana-6080	90	10	≤	≤	NOUN
cana-6080	90	11	m	m	ADP
cana-6080	90	12	,	,	PUNCT
cana-6080	90	13	1	1	NUM
cana-6080	90	14	≤	≤	NUM
cana-6080	90	15	j	j	PROPN
cana-6080	90	16	≤	≤	PROPN
cana-6080	90	17	n	n	CCONJ
cana-6080	90	18	,	,	PUNCT
cana-6080	90	19	σ	σ	PROPN
cana-6080	90	20	(	(	PUNCT
cana-6080	90	21	ei	ei	PROPN
cana-6080	90	22	,	,	PUNCT
cana-6080	90	23	j	j	PROPN
cana-6080	90	24	)	)	PUNCT
cana-6080	91	1	=	=	PUNCT
cana-6080	92	1	i	i	PRON
cana-6080	92	2	+	+	NUM
cana-6080	92	3	j	j	PROPN
cana-6080	93	1	+	+	NOUN
cana-6080	93	2	1	1	NUM
cana-6080	93	3	,	,	PUNCT
cana-6080	93	4	i	i	PRON
cana-6080	93	5	≡	≡	PROPN
cana-6080	93	6	i	i	PRON
cana-6080	93	7	mod	mod	PROPN
cana-6080	93	8	m	m	PROPN
cana-6080	93	9	;	;	PUNCT
cana-6080	93	10	j	j	PROPN
cana-6080	93	11	≡	≡	PROPN
cana-6080	93	12	j	j	PROPN
cana-6080	93	13	mod	mod	PROPN
cana-6080	93	14	n	n	PROPN
cana-6080	93	15	σ	σ	PROPN
cana-6080	93	16	(	(	PUNCT
cana-6080	93	17	ui	ui	PROPN
cana-6080	93	18	)	)	PUNCT
cana-6080	93	19	=	=	SYM
cana-6080	93	20	1	1	NUM
cana-6080	93	21	;	;	PUNCT
cana-6080	93	22	σ	σ	PROPN
cana-6080	93	23	(	(	PUNCT
cana-6080	93	24	vj	vj	INTJ
cana-6080	93	25	)	)	PUNCT
cana-6080	94	1	=	=	SYM
cana-6080	94	2	2	2	NUM
cana-6080	94	3	;	;	PUNCT
cana-6080	94	4	σ	σ	PROPN
cana-6080	94	5	(	(	PUNCT
cana-6080	94	6	ej	ej	PROPN
cana-6080	94	7	,	,	PUNCT
cana-6080	94	8	j+1	j+1	PROPN
cana-6080	94	9	)	)	PUNCT
cana-6080	94	10	=	=	SYM
cana-6080	94	11	1	1	NUM
cana-6080	94	12	,	,	PUNCT
cana-6080	94	13	1	1	NUM
cana-6080	94	14	≤	≤	NUM
cana-6080	94	15	j	j	PROPN
cana-6080	94	16	≤	≤	PROPN
cana-6080	94	17	n	n	CCONJ
cana-6080	94	18	−	−	PROPN
cana-6080	94	19	1	1	NUM
cana-6080	94	20	.	.	PUNCT
cana-6080	95	1	since	since	SCONJ
cana-6080	95	2	the	the	DET
cana-6080	95	3	above	above	ADJ
cana-6080	95	4	color	color	NOUN
cana-6080	95	5	partition	partition	NOUN
cana-6080	95	6	γ(fn	γ(fn	PROPN
cana-6080	95	7	,	,	PUNCT
cana-6080	95	8	m	m	NOUN
cana-6080	95	9	)	)	PUNCT
cana-6080	95	10	≤	≤	NOUN
cana-6080	95	11	n+2	n+2	PRON
cana-6080	95	12	.	.	PUNCT
cana-6080	96	1	assume	assume	VERB
cana-6080	96	2	that	that	SCONJ
cana-6080	96	3	γ(fn	γ(fn	NUM
cana-6080	96	4	,	,	PUNCT
cana-6080	96	5	m	m	NOUN
cana-6080	96	6	)	)	PUNCT
cana-6080	96	7	>	>	X
cana-6080	97	1	n+2	n+2	X
cana-6080	97	2	.	.	PUNCT
cana-6080	98	1	since	since	SCONJ
cana-6080	98	2	there	there	PRON
cana-6080	98	3	exist	exist	VERB
cana-6080	98	4	a	a	DET
cana-6080	98	5	clique	clique	NOUN
cana-6080	98	6	of	of	ADP
cana-6080	98	7	order	order	NOUN
cana-6080	98	8	n	n	X
cana-6080	98	9	+	+	NOUN
cana-6080	98	10	1	1	NUM
cana-6080	98	11	.	.	X
cana-6080	99	1	γ	γ	PROPN
cana-6080	99	2	[	[	X
cana-6080	99	3	t	t	X
cana-6080	99	4	(	(	PUNCT
cana-6080	99	5	fm	fm	PROPN
cana-6080	99	6	,	,	PUNCT
cana-6080	99	7	n	n	CCONJ
cana-6080	99	8	)	)	PUNCT
cana-6080	99	9	]	]	PUNCT
cana-6080	99	10	≤	≤	NUM
cana-6080	99	11	n	n	CCONJ
cana-6080	99	12	+	+	NOUN
cana-6080	99	13	2	2	X
cana-6080	99	14	.	.	X
cana-6080	100	1	this	this	PRON
cana-6080	100	2	arises	arise	VERB
cana-6080	100	3	a	a	DET
cana-6080	100	4	contradiction	contradiction	NOUN
cana-6080	100	5	.	.	PUNCT
cana-6080	101	1	therefore	therefore	ADV
cana-6080	101	2	,	,	PUNCT
cana-6080	101	3	γ	γ	X
cana-6080	101	4	[	[	X
cana-6080	101	5	t	t	X
cana-6080	101	6	(	(	PUNCT
cana-6080	101	7	fm	fm	PROPN
cana-6080	101	8	,	,	PUNCT
cana-6080	101	9	n	n	CCONJ
cana-6080	101	10	)	)	PUNCT
cana-6080	101	11	]	]	PUNCT
cana-6080	102	1	=	=	PUNCT
cana-6080	102	2	n	n	PROPN
cana-6080	102	3	+	+	NOUN
cana-6080	102	4	2	2	NUM
cana-6080	102	5	.	.	X
cana-6080	102	6	conclusion	conclusion	NOUN
cana-6080	102	7	in	in	ADP
cana-6080	102	8	this	this	DET
cana-6080	102	9	paper	paper	NOUN
cana-6080	102	10	we	we	PRON
cana-6080	102	11	discussed	discuss	VERB
cana-6080	102	12	about	about	ADP
cana-6080	102	13	the	the	DET
cana-6080	102	14	grundy	grundy	PROPN
cana-6080	102	15	chromatic	chromatic	ADJ
cana-6080	102	16	number	number	NOUN
cana-6080	102	17	of	of	ADP
cana-6080	102	18	fan	fan	NOUN
cana-6080	102	19	graph	graph	NOUN
cana-6080	102	20	,	,	PUNCT
cana-6080	102	21	line	line	NOUN
cana-6080	102	22	graph	graph	NOUN
cana-6080	102	23	of	of	ADP
cana-6080	102	24	fan	fan	NOUN
cana-6080	102	25	graph	graph	NOUN
cana-6080	102	26	,	,	PUNCT
cana-6080	102	27	middle	middle	ADJ
cana-6080	102	28	graph	graph	NOUN
cana-6080	102	29	of	of	ADP
cana-6080	102	30	fan	fan	NOUN
cana-6080	102	31	graph	graph	NOUN
cana-6080	102	32	and	and	CCONJ
cana-6080	102	33	total	total	ADJ
cana-6080	102	34	graph	graph	NOUN
cana-6080	102	35	of	of	ADP
cana-6080	102	36	fan	fan	NOUN
cana-6080	102	37	graph	graph	NOUN
cana-6080	102	38	.	.	PUNCT
cana-6080	103	1	this	this	DET
cana-6080	103	2	paper	paper	NOUN
cana-6080	103	3	motivates	motivate	VERB
cana-6080	103	4	the	the	DET
cana-6080	103	5	reader	reader	NOUN
cana-6080	103	6	to	to	PART
cana-6080	103	7	do	do	VERB
cana-6080	103	8	further	further	ADJ
cana-6080	103	9	research	research	NOUN
cana-6080	103	10	in	in	ADP
cana-6080	103	11	grundy	grundy	PROPN
cana-6080	103	12	coloring	coloring	NOUN
cana-6080	103	13	of	of	ADP
cana-6080	103	14	some	some	DET
cana-6080	103	15	other	other	ADJ
cana-6080	103	16	graphs	graph	NOUN
cana-6080	103	17	and	and	CCONJ
cana-6080	103	18	also	also	ADV
cana-6080	103	19	other	other	ADJ
cana-6080	103	20	colorings	coloring	NOUN
cana-6080	103	21	of	of	ADP
cana-6080	103	22	different	different	ADJ
cana-6080	103	23	types	type	NOUN
cana-6080	103	24	of	of	ADP
cana-6080	103	25	graphs	graph	NOUN
cana-6080	103	26	.	.	PUNCT
cana-6080	104	1	references	reference	NOUN
cana-6080	104	2	[	[	X
cana-6080	104	3	1	1	NUM
cana-6080	104	4	]	]	X
cana-6080	104	5	y.	y.	PROPN
cana-6080	104	6	caro	caro	PROPN
cana-6080	104	7	,	,	PUNCT
cana-6080	104	8	a.	a.	PROPN
cana-6080	104	9	seb¨o	seb¨o	PROPN
cana-6080	104	10	,	,	PUNCT
cana-6080	104	11	m.	m.	NOUN
cana-6080	104	12	tarsi	tarsi	NOUN
cana-6080	104	13	,	,	PUNCT
cana-6080	104	14	recognizing	recognize	VERB
cana-6080	104	15	greedy	greedy	ADJ
cana-6080	104	16	structures	structure	NOUN
cana-6080	104	17	,	,	PUNCT
cana-6080	104	18	j.	j.	PROPN
cana-6080	104	19	algorithms	algorithms	PROPN
cana-6080	104	20	20	20	NUM
cana-6080	104	21	(	(	PUNCT
cana-6080	104	22	1996	1996	NUM
cana-6080	104	23	)	)	PUNCT
cana-6080	104	24	137	137	NUM
cana-6080	104	25	156	156	NUM
cana-6080	104	26	.	.	PUNCT
cana-6080	105	1	[	[	X
cana-6080	105	2	2	2	NUM
cana-6080	105	3	]	]	X
cana-6080	105	4	c.a	c.a	PROPN
cana-6080	105	5	.	.	PROPN
cana-6080	105	6	christen	christen	PROPN
cana-6080	105	7	,	,	PUNCT
cana-6080	105	8	s.m	s.m	PROPN
cana-6080	105	9	.	.	PROPN
cana-6080	105	10	selkow	selkow	PROPN
cana-6080	105	11	,	,	PUNCT
cana-6080	105	12	some	some	DET
cana-6080	105	13	perfect	perfect	ADJ
cana-6080	105	14	coloring	coloring	NOUN
cana-6080	105	15	properties	property	NOUN
cana-6080	105	16	of	of	ADP
cana-6080	105	17	graphs	graph	NOUN
cana-6080	105	18	,	,	PUNCT
cana-6080	105	19	j.	j.	PROPN
cana-6080	105	20	combin	combin	PROPN
cana-6080	105	21	.	.	PUNCT
cana-6080	105	22	theory	theory	NOUN
cana-6080	105	23	ser	ser	PROPN
cana-6080	105	24	.	.	PUNCT
cana-6080	106	1	b	b	ADP
cana-6080	106	2	27	27	NUM
cana-6080	106	3	(	(	PUNCT
cana-6080	106	4	1979	1979	NUM
cana-6080	106	5	)	)	PUNCT
cana-6080	106	6	49	49	NUM
cana-6080	106	7	-	-	SYM
cana-6080	106	8	59	59	NUM
cana-6080	106	9	.	.	PUNCT
cana-6080	107	1	communications	communication	NOUN
cana-6080	107	2	on	on	ADP
cana-6080	107	3	applied	apply	VERB
cana-6080	107	4	nonlinear	nonlinear	ADJ
cana-6080	107	5	analysis	analysis	NOUN
cana-6080	107	6	issn	issn	NOUN
cana-6080	107	7	:	:	PUNCT
cana-6080	107	8	1074	1074	NUM
cana-6080	107	9	-	-	PUNCT
cana-6080	107	10	133x	133x	NUM
cana-6080	107	11	vol	vol	NOUN
cana-6080	107	12	31	31	NUM
cana-6080	107	13	no	no	NOUN
cana-6080	107	14	.	.	PUNCT
cana-6080	108	1	8s	8s	PROPN
cana-6080	108	2	(	(	PUNCT
cana-6080	108	3	2024	2024	NUM
cana-6080	108	4	)	)	PUNCT
cana-6080	108	5	1199	1199	NUM
cana-6080	108	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6080	109	1	[	[	X
cana-6080	109	2	3	3	NUM
cana-6080	109	3	]	]	X
cana-6080	109	4	p.	p.	NOUN
cana-6080	109	5	erd¨os	erd¨os	PROPN
cana-6080	109	6	,	,	PUNCT
cana-6080	109	7	w.r	w.r	PROPN
cana-6080	109	8	.	.	PROPN
cana-6080	109	9	hare	hare	PROPN
cana-6080	109	10	,	,	PUNCT
cana-6080	109	11	s.t	s.t	PROPN
cana-6080	109	12	.	.	PROPN
cana-6080	109	13	hedetniemi	hedetniemi	PROPN
cana-6080	109	14	,	,	PUNCT
cana-6080	109	15	r.	r.	PROPN
cana-6080	109	16	laskar	laskar	PROPN
cana-6080	109	17	,	,	PUNCT
cana-6080	109	18	on	on	ADP
cana-6080	109	19	the	the	DET
cana-6080	109	20	equality	equality	NOUN
cana-6080	109	21	of	of	ADP
cana-6080	109	22	grundy	grundy	PROPN
cana-6080	109	23	and	and	CCONJ
cana-6080	109	24	ochromatic	ochromatic	ADJ
cana-6080	109	25	number	number	NOUN
cana-6080	109	26	of	of	ADP
cana-6080	109	27	graphs	graph	NOUN
cana-6080	109	28	,	,	PUNCT
cana-6080	109	29	j.	j.	PROPN
cana-6080	109	30	graph	graph	NOUN
cana-6080	109	31	theory	theory	NOUN
cana-6080	109	32	11	11	NUM
cana-6080	109	33	(	(	PUNCT
cana-6080	109	34	1987	1987	NUM
cana-6080	109	35	)	)	PUNCT
cana-6080	109	36	,	,	PUNCT
cana-6080	109	37	157	157	NUM
cana-6080	109	38	-	-	SYM
cana-6080	109	39	159	159	NUM
cana-6080	109	40	.	.	PUNCT
cana-6080	110	1	[	[	X
cana-6080	110	2	4	4	X
cana-6080	110	3	]	]	PUNCT
cana-6080	110	4	z.	z.	PROPN
cana-6080	110	5	f¨uredi	f¨uredi	PROPN
cana-6080	110	6	,	,	PUNCT
cana-6080	110	7	a.	a.	NOUN
cana-6080	110	8	gy´arf´as	gy´arf´as	PROPN
cana-6080	110	9	,	,	PUNCT
cana-6080	110	10	g.	g.	PROPN
cana-6080	110	11	s´ark¨ozy	s´ark¨ozy	PROPN
cana-6080	110	12	,	,	PUNCT
cana-6080	110	13	s.	s.	PROPN
cana-6080	110	14	selkow	selkow	PROPN
cana-6080	110	15	,	,	PUNCT
cana-6080	110	16	inequalities	inequality	NOUN
cana-6080	110	17	for	for	ADP
cana-6080	110	18	the	the	DET
cana-6080	110	19	first	first	ADJ
cana-6080	110	20	-	-	PUNCT
cana-6080	110	21	fit	fit	ADJ
cana-6080	110	22	chromatic	chromatic	ADJ
cana-6080	110	23	number	number	NOUN
cana-6080	110	24	,	,	PUNCT
cana-6080	110	25	59(1	59(1	NUM
cana-6080	110	26	)	)	PUNCT
cana-6080	110	27	,	,	PUNCT
cana-6080	110	28	(	(	PUNCT
cana-6080	110	29	2008	2008	NUM
cana-6080	110	30	)	)	PUNCT
cana-6080	110	31	,	,	PUNCT
cana-6080	110	32	75	75	NUM
cana-6080	110	33	-	-	SYM
cana-6080	110	34	88	88	NUM
cana-6080	110	35	.	.	PUNCT
cana-6080	111	1	[	[	X
cana-6080	111	2	5	5	NUM
cana-6080	111	3	]	]	X
cana-6080	111	4	n.	n.	PROPN
cana-6080	111	5	goyal	goyal	PROPN
cana-6080	111	6	,	,	PUNCT
cana-6080	111	7	s.	s.	PROPN
cana-6080	111	8	vishvanathan	vishvanathan	PROPN
cana-6080	111	9	,	,	PUNCT
cana-6080	111	10	np	np	NOUN
cana-6080	111	11	-	-	PUNCT
cana-6080	111	12	completeness	completeness	NOUN
cana-6080	111	13	of	of	ADP
cana-6080	111	14	undirected	undirected	ADJ
cana-6080	111	15	grundy	grundy	PROPN
cana-6080	111	16	numbering	numbering	NOUN
cana-6080	111	17	and	and	CCONJ
cana-6080	111	18	related	related	ADJ
cana-6080	111	19	problems	problem	NOUN
cana-6080	111	20	,	,	PUNCT
cana-6080	111	21	manuscript	manuscript	NOUN
cana-6080	111	22	,	,	PUNCT
cana-6080	111	23	bombay	bombay	PROPN
cana-6080	111	24	,	,	PUNCT
cana-6080	111	25	1997	1997	NUM
cana-6080	111	26	.	.	PUNCT
cana-6080	112	1	[	[	X
cana-6080	112	2	6	6	NUM
cana-6080	112	3	]	]	PUNCT
cana-6080	112	4	a.	a.	NOUN
cana-6080	112	5	gy´arf´as	gy´arf´as	PROPN
cana-6080	112	6	,	,	PUNCT
cana-6080	112	7	j.	j.	PROPN
cana-6080	112	8	lehel	lehel	PROPN
cana-6080	112	9	,	,	PUNCT
cana-6080	112	10	on	on	ADP
cana-6080	112	11	-	-	PUNCT
cana-6080	112	12	line	line	NOUN
cana-6080	112	13	and	and	CCONJ
cana-6080	112	14	first	first	ADJ
cana-6080	112	15	-	-	PUNCT
cana-6080	112	16	fit	fit	ADJ
cana-6080	112	17	coloring	coloring	NOUN
cana-6080	112	18	of	of	ADP
cana-6080	112	19	graphs	graph	NOUN
cana-6080	112	20	,	,	PUNCT
cana-6080	112	21	j.	j.	PROPN
cana-6080	112	22	graph	graph	NOUN
cana-6080	112	23	theory	theory	NOUN
cana-6080	112	24	12	12	NUM
cana-6080	112	25	(	(	PUNCT
cana-6080	112	26	1988	1988	NUM
cana-6080	112	27	)	)	PUNCT
cana-6080	112	28	217	217	NUM
cana-6080	112	29	-	-	SYM
cana-6080	112	30	227	227	NUM
cana-6080	112	31	.	.	PUNCT
cana-6080	113	1	[	[	X
cana-6080	113	2	7	7	NUM
cana-6080	113	3	]	]	X
cana-6080	113	4	s.m	s.m	PROPN
cana-6080	113	5	.	.	PROPN
cana-6080	113	6	hedetniemi	hedetniemi	PROPN
cana-6080	113	7	,	,	PUNCT
cana-6080	113	8	s.t	s.t	PROPN
cana-6080	113	9	.	.	PROPN
cana-6080	113	10	hedetniemi	hedetniemi	PROPN
cana-6080	113	11	,	,	PUNCT
cana-6080	113	12	a.	a.	PROPN
cana-6080	113	13	beyer	beyer	PROPN
cana-6080	113	14	,	,	PUNCT
cana-6080	113	15	a	a	DET
cana-6080	113	16	linear	linear	ADJ
cana-6080	113	17	algorithm	algorithm	NOUN
cana-6080	113	18	for	for	ADP
cana-6080	113	19	the	the	DET
cana-6080	113	20	grundy	grundy	PROPN
cana-6080	113	21	(	(	PUNCT
cana-6080	113	22	coloring	coloring	NOUN
cana-6080	113	23	)	)	PUNCT
cana-6080	113	24	number	number	NOUN
cana-6080	113	25	of	of	ADP
cana-6080	113	26	a	a	DET
cana-6080	113	27	tree	tree	NOUN
cana-6080	113	28	,	,	PUNCT
cana-6080	113	29	congr	congr	NOUN
cana-6080	113	30	.	.	PUNCT
cana-6080	114	1	numer	numer	PROPN
cana-6080	114	2	.	.	PROPN
cana-6080	115	1	36	36	NUM
cana-6080	115	2	(	(	PUNCT
cana-6080	115	3	1982	1982	NUM
cana-6080	115	4	)	)	PUNCT
cana-6080	115	5	,	,	PUNCT
cana-6080	115	6	351	351	NUM
cana-6080	115	7	-	-	SYM
cana-6080	115	8	363	363	NUM
cana-6080	115	9	.	.	PUNCT
cana-6080	116	1	[	[	X
cana-6080	116	2	8	8	NUM
cana-6080	116	3	]	]	X
cana-6080	116	4	d.g	d.g	PROPN
cana-6080	116	5	.	.	PROPN
cana-6080	116	6	hoffman	hoffman	PROPN
cana-6080	116	7	,	,	PUNCT
cana-6080	116	8	p.d	p.d	PROPN
cana-6080	116	9	.	.	PROPN
cana-6080	116	10	johnson	johnson	PROPN
cana-6080	116	11	jr	jr	PROPN
cana-6080	116	12	.	.	PROPN
cana-6080	116	13	,	,	PUNCT
cana-6080	116	14	greedy	greedy	ADJ
cana-6080	116	15	colorings	coloring	NOUN
cana-6080	116	16	and	and	CCONJ
cana-6080	116	17	the	the	DET
cana-6080	116	18	grundy	grundy	ADJ
cana-6080	116	19	chromatic	chromatic	ADJ
cana-6080	116	20	number	number	NOUN
cana-6080	116	21	of	of	ADP
cana-6080	116	22	the	the	DET
cana-6080	116	23	n	n	CCONJ
cana-6080	116	24	-	-	PUNCT
cana-6080	116	25	cube	cube	NOUN
cana-6080	116	26	,	,	PUNCT
cana-6080	116	27	bull	bull	NOUN
cana-6080	116	28	.	.	PUNCT
cana-6080	117	1	ica	ica	PROPN
cana-6080	117	2	26	26	NUM
cana-6080	117	3	(	(	PUNCT
cana-6080	117	4	1999	1999	NUM
cana-6080	117	5	)	)	PUNCT
cana-6080	117	6	49	49	NUM
cana-6080	117	7	-	-	SYM
cana-6080	117	8	57	57	NUM
cana-6080	117	9	.	.	PUNCT
cana-6080	118	1	[	[	X
cana-6080	118	2	9	9	NUM
cana-6080	118	3	]	]	SYM
cana-6080	118	4	t.r	t.r	PROPN
cana-6080	118	5	.	.	PROPN
cana-6080	118	6	jensen	jensen	PROPN
cana-6080	118	7	,	,	PUNCT
cana-6080	118	8	b.	b.	PROPN
cana-6080	118	9	toft	toft	PROPN
cana-6080	118	10	,	,	PUNCT
cana-6080	118	11	graph	graph	NOUN
cana-6080	118	12	coloring	coloring	NOUN
cana-6080	118	13	problems	problem	NOUN
cana-6080	118	14	,	,	PUNCT
cana-6080	118	15	wiley	wiley	NOUN
cana-6080	118	16	,	,	PUNCT
cana-6080	118	17	newyork	newyork	PROPN
cana-6080	118	18	,	,	PUNCT
cana-6080	118	19	1995	1995	NUM
cana-6080	118	20	.	.	PUNCT
cana-6080	119	1	[	[	X
cana-6080	119	2	10	10	NUM
cana-6080	119	3	]	]	X
cana-6080	119	4	k	k	PROPN
cana-6080	119	5	kaliraj	kaliraj	PROPN
cana-6080	119	6	,	,	PUNCT
cana-6080	119	7	vj	vj	PROPN
cana-6080	119	8	vernold	vernold	PROPN
cana-6080	119	9	,	,	PUNCT
cana-6080	119	10	on	on	ADP
cana-6080	119	11	equitable	equitable	ADJ
cana-6080	119	12	coloring	coloring	NOUN
cana-6080	119	13	of	of	ADP
cana-6080	119	14	helm	helm	NOUN
cana-6080	119	15	graph	graph	NOUN
cana-6080	119	16	and	and	CCONJ
cana-6080	119	17	gear	gear	NOUN
cana-6080	119	18	graph	graph	NOUN
cana-6080	119	19	,	,	PUNCT
cana-6080	119	20	international	international	ADJ
cana-6080	119	21	journal	journal	NOUN
cana-6080	119	22	of	of	ADP
cana-6080	119	23	mathematical	mathematical	ADJ
cana-6080	119	24	combinatorics	combinatoric	NOUN
cana-6080	119	25	,	,	PUNCT
cana-6080	119	26	vol.4	vol.4	PROPN
cana-6080	119	27	(	(	PUNCT
cana-6080	119	28	2010	2010	NUM
cana-6080	119	29	)	)	PUNCT
cana-6080	119	30	,	,	PUNCT
cana-6080	119	31	32	32	NUM
cana-6080	119	32	-	-	SYM
cana-6080	119	33	37	37	NUM
cana-6080	119	34	.	.	PUNCT
cana-6080	120	1	[	[	X
cana-6080	120	2	11	11	NUM
cana-6080	120	3	]	]	PUNCT
cana-6080	120	4	k.	k.	PROPN
cana-6080	120	5	kaliraj	kaliraj	PROPN
cana-6080	120	6	,	,	PUNCT
cana-6080	120	7	j.	j.	PROPN
cana-6080	120	8	vernold	vernold	PROPN
cana-6080	120	9	vivin	vivin	PROPN
cana-6080	120	10	and	and	CCONJ
cana-6080	120	11	m.m	m.m	PROPN
cana-6080	120	12	.	.	PROPN
cana-6080	120	13	akbar	akbar	PROPN
cana-6080	120	14	ali	ali	PROPN
cana-6080	120	15	,	,	PUNCT
cana-6080	120	16	on	on	ADP
cana-6080	120	17	equitable	equitable	ADJ
cana-6080	120	18	coloring	coloring	NOUN
cana-6080	120	19	of	of	ADP
cana-6080	120	20	sun	sun	NOUN
cana-6080	120	21	let	let	VERB
cana-6080	120	22	graph	graph	NOUN
cana-6080	120	23	families	family	NOUN
cana-6080	120	24	,	,	PUNCT
cana-6080	120	25	vol.ciii	vol.ciii	X
cana-6080	120	26	(	(	PUNCT
cana-6080	120	27	2012	2012	NUM
cana-6080	120	28	)	)	PUNCT
cana-6080	120	29	,	,	PUNCT
cana-6080	120	30	497	497	NUM
cana-6080	120	31	-	-	SYM
cana-6080	120	32	504	504	NUM
cana-6080	120	33	.	.	PUNCT
cana-6080	121	1	[	[	X
cana-6080	121	2	12	12	NUM
cana-6080	121	3	]	]	X
cana-6080	121	4	g.j	g.j	PROPN
cana-6080	121	5	.	.	PROPN
cana-6080	121	6	simmons	simmon	NOUN
cana-6080	121	7	,	,	PUNCT
cana-6080	121	8	on	on	ADP
cana-6080	121	9	the	the	DET
cana-6080	121	10	ochromatic	ochromatic	ADJ
cana-6080	121	11	number	number	NOUN
cana-6080	121	12	of	of	ADP
cana-6080	121	13	a	a	DET
cana-6080	121	14	graph	graph	NOUN
cana-6080	121	15	,	,	PUNCT
cana-6080	121	16	congr	congr	NOUN
cana-6080	121	17	.	.	PUNCT
cana-6080	122	1	numer	numer	PROPN
cana-6080	122	2	.	.	PROPN
cana-6080	123	1	40	40	NUM
cana-6080	123	2	(	(	PUNCT
cana-6080	123	3	1983	1983	NUM
cana-6080	123	4	)	)	PUNCT
cana-6080	123	5	339	339	NUM
cana-6080	123	6	-	-	SYM
cana-6080	123	7	366	366	NUM
cana-6080	123	8	.	.	PUNCT
cana-6080	124	1	[	[	X
cana-6080	124	2	13	13	NUM
cana-6080	124	3	]	]	X
cana-6080	124	4	j.a	j.a	PROPN
cana-6080	124	5	.	.	PROPN
cana-6080	124	6	telle	telle	PROPN
cana-6080	124	7	,	,	PUNCT
cana-6080	124	8	a.	a.	NOUN
cana-6080	124	9	proskurowski	proskurowski	NOUN
cana-6080	124	10	,	,	PUNCT
cana-6080	124	11	algorithms	algorithm	NOUN
cana-6080	124	12	for	for	ADP
cana-6080	124	13	vertex	vertex	NOUN
cana-6080	124	14	partitioning	partitioning	NOUN
cana-6080	124	15	problems	problem	NOUN
cana-6080	124	16	on	on	ADP
cana-6080	124	17	partial	partial	ADJ
cana-6080	124	18	k	k	NOUN
cana-6080	124	19	-	-	PUNCT
cana-6080	124	20	trees	tree	NOUN
cana-6080	124	21	,	,	PUNCT
cana-6080	124	22	siam	siam	ADJ
cana-6080	124	23	j.	j.	PROPN
cana-6080	124	24	discrete	discrete	PROPN
cana-6080	124	25	math	math	NOUN
cana-6080	124	26	.	.	PUNCT
cana-6080	125	1	10	10	NUM
cana-6080	125	2	(	(	PUNCT
cana-6080	125	3	1997	1997	NUM
cana-6080	125	4	)	)	PUNCT
cana-6080	125	5	529	529	NUM
cana-6080	125	6	-	-	SYM
cana-6080	125	7	550	550	NUM
cana-6080	125	8	.	.	PUNCT
cana-6080	126	1	[	[	X
cana-6080	126	2	14	14	NUM
cana-6080	126	3	]	]	PUNCT
cana-6080	126	4	m.	m.	NOUN
cana-6080	126	5	venkatachalam	venkatachalam	PROPN
cana-6080	126	6	,	,	PUNCT
cana-6080	126	7	j.	j.	PROPN
cana-6080	126	8	vernold	vernold	PROPN
cana-6080	126	9	vivin	vivin	PROPN
cana-6080	126	10	and	and	CCONJ
cana-6080	126	11	k.	k.	PROPN
cana-6080	126	12	kaliraj	kaliraj	PROPN
cana-6080	126	13	,	,	PUNCT
cana-6080	126	14	harmonious	harmonious	ADJ
cana-6080	126	15	coloring	coloring	NOUN
cana-6080	126	16	on	on	ADP
cana-6080	126	17	double	double	ADJ
cana-6080	126	18	star	star	NOUN
cana-6080	126	19	graph	graph	NOUN
cana-6080	126	20	families	family	NOUN
cana-6080	126	21	,	,	PUNCT
cana-6080	126	22	43(2	43(2	NUM
cana-6080	126	23	)	)	PUNCT
cana-6080	126	24	(	(	PUNCT
cana-6080	126	25	2012	2012	NUM
cana-6080	126	26	)	)	PUNCT
cana-6080	126	27	,	,	PUNCT
cana-6080	126	28	153	153	NUM
cana-6080	126	29	-	-	SYM
cana-6080	126	30	158	158	NUM
cana-6080	126	31	.	.	PUNCT
cana-6080	126	32	doi:10.5556	doi:10.5556	NOUN
cana-6080	126	33	/	/	SYM
cana-6080	126	34	j.tkjm.43.2012.153	j.tkjm.43.2012.153	PROPN
cana-6080	126	35	-	-	PUNCT
cana-6080	126	36	158	158	NUM
cana-6080	126	37	.	.	PUNCT
cana-6080	127	1	[	[	X
cana-6080	127	2	15	15	NUM
cana-6080	127	3	]	]	X
cana-6080	127	4	j.	j.	PROPN
cana-6080	127	5	v.	v.	PROPN
cana-6080	127	6	vivin	vivin	PROPN
cana-6080	127	7	,	,	PUNCT
cana-6080	127	8	k	k	PROPN
cana-6080	127	9	kaliraj	kaliraj	PROPN
cana-6080	127	10	,	,	PUNCT
cana-6080	127	11	on	on	ADP
cana-6080	127	12	equitable	equitable	ADJ
cana-6080	127	13	coloring	coloring	NOUN
cana-6080	127	14	of	of	ADP
cana-6080	127	15	corona	corona	NOUN
cana-6080	127	16	of	of	ADP
cana-6080	127	17	wheels	wheel	NOUN
cana-6080	127	18	,	,	PUNCT
cana-6080	127	19	electronic	electronic	ADJ
cana-6080	127	20	journal	journal	NOUN
cana-6080	127	21	of	of	ADP
cana-6080	127	22	graph	graph	NOUN
cana-6080	127	23	theory	theory	NOUN
cana-6080	127	24	and	and	CCONJ
cana-6080	127	25	applications	application	NOUN
cana-6080	127	26	,	,	PUNCT
cana-6080	127	27	4	4	NUM
cana-6080	127	28	(	(	PUNCT
cana-6080	127	29	2	2	NUM
cana-6080	127	30	)	)	PUNCT
cana-6080	127	31	,	,	PUNCT
cana-6080	127	32	206	206	NUM
cana-6080	127	33	-	-	SYM
cana-6080	127	34	222	222	NUM
cana-6080	127	35	.	.	PUNCT
cana-6080	128	1	[	[	X
cana-6080	128	2	16	16	NUM
cana-6080	128	3	]	]	PUNCT
cana-6080	128	4	m.	m.	NOUN
cana-6080	128	5	zaker	zaker	PROPN
cana-6080	128	6	,	,	PUNCT
cana-6080	128	7	grundy	grundy	PROPN
cana-6080	128	8	chromatic	chromatic	ADJ
cana-6080	128	9	number	number	NOUN
cana-6080	128	10	of	of	ADP
cana-6080	128	11	the	the	DET
cana-6080	128	12	complement	complement	NOUN
cana-6080	128	13	of	of	ADP
cana-6080	128	14	bipartite	bipartite	NOUN
cana-6080	128	15	graphs	graph	NOUN
cana-6080	128	16	,	,	PUNCT
cana-6080	128	17	australas	australa	NOUN
cana-6080	128	18	.	.	PUNCT
cana-6080	129	1	j.	j.	PROPN
cana-6080	129	2	combin	combin	PROPN
cana-6080	129	3	.	.	PROPN
cana-6080	130	1	31	31	NUM
cana-6080	130	2	(	(	PUNCT
cana-6080	130	3	2005	2005	NUM
cana-6080	130	4	)	)	PUNCT
cana-6080	130	5	325	325	NUM
cana-6080	130	6	-	-	SYM
cana-6080	130	7	329	329	NUM
cana-6080	130	8	.	.	PUNCT
cana-6080	131	1	[	[	X
cana-6080	131	2	17	17	NUM
cana-6080	131	3	]	]	X
cana-6080	131	4	harishchandra	harishchandra	PROPN
cana-6080	131	5	patel	patel	PROPN
cana-6080	131	6	,	,	PUNCT
cana-6080	131	7	“	"	PUNCT
cana-6080	131	8	impedance	impedance	NOUN
cana-6080	131	9	control	control	NOUN
cana-6080	131	10	in	in	ADP
cana-6080	131	11	hdi	hdi	PROPN
cana-6080	131	12	and	and	CCONJ
cana-6080	131	13	substrate	substrate	NOUN
cana-6080	131	14	-	-	PUNCT
cana-6080	131	15	like	like	ADJ
cana-6080	131	16	pcbs	pcb	NOUN
cana-6080	131	17	for	for	ADP
cana-6080	131	18	ai	ai	PROPN
cana-6080	131	19	hardware	hardware	NOUN
cana-6080	131	20	applications	application	NOUN
cana-6080	131	21	”	"	PUNCT
cana-6080	131	22	(	(	PUNCT
cana-6080	131	23	2024	2024	NUM
cana-6080	131	24	)	)	PUNCT
cana-6080	131	25	.	.	PUNCT
cana-6080	132	1	journal	journal	NOUN
cana-6080	132	2	of	of	ADP
cana-6080	132	3	electrical	electrical	ADJ
cana-6080	132	4	systems	system	NOUN
cana-6080	132	5	,	,	PUNCT
cana-6080	132	6	20(11s	20(11s	NUM
cana-6080	132	7	)	)	PUNCT
cana-6080	132	8	,	,	PUNCT
cana-6080	132	9	5109	5109	NUM
cana-6080	132	10	-	-	SYM
cana-6080	132	11	5115	5115	NUM
cana-6080	132	12	.	.	PUNCT