id	sid	tid	token	lemma	pos
cana-1220	1	1	communications	communication	NOUN
cana-1220	1	2	on	on	ADP
cana-1220	1	3	applied	apply	VERB
cana-1220	1	4	nonlinear	nonlinear	ADJ
cana-1220	1	5	analysis	analysis	NOUN
cana-1220	1	6	issn	issn	NOUN
cana-1220	1	7	:	:	PUNCT
cana-1220	1	8	1074	1074	NUM
cana-1220	1	9	-	-	PUNCT
cana-1220	1	10	133x	133x	NUM
cana-1220	1	11	vol	vol	NOUN
cana-1220	1	12	31	31	NUM
cana-1220	1	13	no	no	NOUN
cana-1220	1	14	.	.	PUNCT
cana-1220	2	1	6s	6s	NUM
cana-1220	2	2	(	(	PUNCT
cana-1220	2	3	2024	2024	NUM
cana-1220	2	4	)	)	PUNCT
cana-1220	2	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	2	6	259	259	NUM
cana-1220	2	7	prime	prime	ADJ
cana-1220	2	8	graceful	graceful	ADJ
cana-1220	2	9	chromatic	chromatic	ADJ
cana-1220	2	10	number	number	NOUN
cana-1220	2	11	of	of	ADP
cana-1220	2	12	diverse	diverse	ADJ
cana-1220	2	13	graphs	graph	NOUN
cana-1220	2	14	m.	m.	NOUN
cana-1220	2	15	keerthika	keerthika	PROPN
cana-1220	2	16	1	1	NUM
cana-1220	2	17	,	,	PUNCT
cana-1220	2	18	v.	v.	CCONJ
cana-1220	2	19	kowsalya2	kowsalya2	PROPN
cana-1220	2	20	1	1	NUM
cana-1220	2	21	research	research	NOUN
cana-1220	2	22	scholar	scholar	NOUN
cana-1220	2	23	,	,	PUNCT
cana-1220	2	24	pg	pg	PROPN
cana-1220	2	25	&	&	CCONJ
cana-1220	2	26	research	research	PROPN
cana-1220	2	27	department	department	PROPN
cana-1220	2	28	of	of	ADP
cana-1220	2	29	mathematics	mathematic	NOUN
cana-1220	2	30	,	,	PUNCT
cana-1220	2	31	sri	sri	PROPN
cana-1220	2	32	ramakrishna	ramakrishna	PROPN
cana-1220	2	33	college	college	PROPN
cana-1220	2	34	of	of	ADP
cana-1220	2	35	arts	arts	PROPN
cana-1220	2	36	&	&	CCONJ
cana-1220	2	37	science	science	PROPN
cana-1220	2	38	(	(	PUNCT
cana-1220	2	39	autonomous	autonomous	ADJ
cana-1220	2	40	)	)	PUNCT
cana-1220	2	41	,	,	PUNCT
cana-1220	2	42	coimbatore	coimbatore	PROPN
cana-1220	2	43	,	,	PUNCT
cana-1220	2	44	tamil	tamil	PROPN
cana-1220	2	45	nadu	nadu	NOUN
cana-1220	2	46	.	.	PUNCT
cana-1220	3	1	(	(	PUNCT
cana-1220	3	2	e	e	NOUN
cana-1220	3	3	-	-	NOUN
cana-1220	3	4	mail	mail	NOUN
cana-1220	3	5	:	:	PUNCT
cana-1220	3	6	keerthika.m@srcas.ac.in	keerthika.m@srcas.ac.in	NOUN
cana-1220	3	7	)	)	PUNCT
cana-1220	3	8	2associate	2associate	NUM
cana-1220	3	9	professor	professor	NOUN
cana-1220	3	10	,	,	PUNCT
cana-1220	3	11	pg	pg	PROPN
cana-1220	3	12	&	&	CCONJ
cana-1220	3	13	research	research	PROPN
cana-1220	3	14	department	department	PROPN
cana-1220	3	15	of	of	ADP
cana-1220	3	16	mathematics	mathematic	NOUN
cana-1220	3	17	,	,	PUNCT
cana-1220	3	18	sri	sri	PROPN
cana-1220	3	19	ramakrishna	ramakrishna	PROPN
cana-1220	3	20	college	college	PROPN
cana-1220	3	21	of	of	ADP
cana-1220	3	22	arts	arts	PROPN
cana-1220	3	23	&	&	CCONJ
cana-1220	3	24	science	science	PROPN
cana-1220	3	25	(	(	PUNCT
cana-1220	3	26	autonomous	autonomous	ADJ
cana-1220	3	27	)	)	PUNCT
cana-1220	3	28	,	,	PUNCT
cana-1220	3	29	coimbatore	coimbatore	PROPN
cana-1220	3	30	,	,	PUNCT
cana-1220	3	31	tamil	tamil	PROPN
cana-1220	3	32	nadu	nadu	NOUN
cana-1220	3	33	.	.	PUNCT
cana-1220	4	1	(	(	PUNCT
cana-1220	4	2	e	e	NOUN
cana-1220	4	3	-	-	NOUN
cana-1220	4	4	mail	mail	NOUN
cana-1220	4	5	:	:	PUNCT
cana-1220	4	6	kowsalya@srcas.ac.in	kowsalya@srcas.ac.in	NOUN
cana-1220	4	7	)	)	PUNCT
cana-1220	4	8	article	article	NOUN
cana-1220	4	9	history	history	NOUN
cana-1220	4	10	:	:	PUNCT
cana-1220	4	11	received	receive	VERB
cana-1220	4	12	:	:	PUNCT
cana-1220	4	13	03	03	NUM
cana-1220	4	14	-	-	PUNCT
cana-1220	4	15	06	06	NUM
cana-1220	4	16	-	-	PUNCT
cana-1220	4	17	2024	2024	NUM
cana-1220	4	18	revised	revise	VERB
cana-1220	4	19	:	:	PUNCT
cana-1220	4	20	05	05	NUM
cana-1220	4	21	-	-	PUNCT
cana-1220	4	22	07	07	NUM
cana-1220	4	23	-	-	PUNCT
cana-1220	4	24	2024	2024	NUM
cana-1220	4	25	accepted	accept	VERB
cana-1220	4	26	:	:	PUNCT
cana-1220	4	27	25	25	NUM
cana-1220	4	28	-	-	PUNCT
cana-1220	4	29	07	07	NUM
cana-1220	4	30	-	-	PUNCT
cana-1220	4	31	2024	2024	NUM
cana-1220	4	32	abstract	abstract	NOUN
cana-1220	4	33	:	:	PUNCT
cana-1220	4	34	in	in	ADP
cana-1220	4	35	this	this	DET
cana-1220	4	36	paper	paper	NOUN
cana-1220	4	37	,	,	PUNCT
cana-1220	4	38	prime	prime	ADJ
cana-1220	4	39	graceful	graceful	ADJ
cana-1220	4	40	coloring	coloring	NOUN
cana-1220	4	41	is	be	AUX
cana-1220	4	42	introduced	introduce	VERB
cana-1220	4	43	which	which	PRON
cana-1220	4	44	aims	aim	VERB
cana-1220	4	45	to	to	PART
cana-1220	4	46	incorporate	incorporate	VERB
cana-1220	4	47	principles	principle	NOUN
cana-1220	4	48	of	of	ADP
cana-1220	4	49	prime	prime	ADJ
cana-1220	4	50	and	and	CCONJ
cana-1220	4	51	graceful	graceful	ADJ
cana-1220	4	52	coloring	coloring	NOUN
cana-1220	4	53	.	.	PUNCT
cana-1220	5	1	the	the	DET
cana-1220	5	2	prime	prime	ADJ
cana-1220	5	3	graceful	graceful	ADJ
cana-1220	5	4	coloring	coloring	NOUN
cana-1220	5	5	of	of	ADP
cana-1220	5	6	star	star	NOUN
cana-1220	5	7	,	,	PUNCT
cana-1220	5	8	path	path	NOUN
cana-1220	5	9	,	,	PUNCT
cana-1220	5	10	cycle	cycle	NOUN
cana-1220	5	11	,	,	PUNCT
cana-1220	5	12	friendship	friendship	NOUN
cana-1220	5	13	,	,	PUNCT
cana-1220	5	14	pan	pan	NOUN
cana-1220	5	15	and	and	CCONJ
cana-1220	5	16	bistar	bistar	ADJ
cana-1220	5	17	graphs	graph	NOUN
cana-1220	5	18	are	be	AUX
cana-1220	5	19	proposed	propose	VERB
cana-1220	5	20	.	.	PUNCT
cana-1220	6	1	this	this	DET
cana-1220	6	2	new	new	ADJ
cana-1220	6	3	approach	approach	NOUN
cana-1220	6	4	is	be	AUX
cana-1220	6	5	related	relate	VERB
cana-1220	6	6	to	to	ADP
cana-1220	6	7	the	the	DET
cana-1220	6	8	prime	prime	ADJ
cana-1220	6	9	graceful	graceful	ADJ
cana-1220	6	10	chromatic	chromatic	ADJ
cana-1220	6	11	number	number	NOUN
cana-1220	6	12	which	which	PRON
cana-1220	6	13	represent	represent	VERB
cana-1220	6	14	the	the	DET
cana-1220	6	15	fewest	few	ADJ
cana-1220	6	16	colors	color	NOUN
cana-1220	6	17	needed	need	VERB
cana-1220	6	18	to	to	PART
cana-1220	6	19	color	color	VERB
cana-1220	6	20	any	any	DET
cana-1220	6	21	graph	graph	NOUN
cana-1220	6	22	adhering	adhere	VERB
cana-1220	6	23	to	to	ADP
cana-1220	6	24	the	the	DET
cana-1220	6	25	principles	principle	NOUN
cana-1220	6	26	of	of	ADP
cana-1220	6	27	prime	prime	ADJ
cana-1220	6	28	graceful	graceful	ADJ
cana-1220	6	29	coloring	coloring	NOUN
cana-1220	6	30	.	.	PUNCT
cana-1220	7	1	also	also	ADV
cana-1220	7	2	,	,	PUNCT
cana-1220	7	3	the	the	DET
cana-1220	7	4	efficiency	efficiency	NOUN
cana-1220	7	5	of	of	ADP
cana-1220	7	6	new	new	ADJ
cana-1220	7	7	coloring	coloring	NOUN
cana-1220	7	8	are	be	AUX
cana-1220	7	9	analyzed	analyze	VERB
cana-1220	7	10	with	with	ADP
cana-1220	7	11	the	the	DET
cana-1220	7	12	examples	example	NOUN
cana-1220	7	13	.	.	PUNCT
cana-1220	8	1	keywords	keyword	NOUN
cana-1220	8	2	:	:	PUNCT
cana-1220	8	3	prime	prime	ADJ
cana-1220	8	4	coloring	coloring	NOUN
cana-1220	8	5	,	,	PUNCT
cana-1220	8	6	graceful	graceful	ADJ
cana-1220	8	7	coloring	coloring	NOUN
cana-1220	8	8	,	,	PUNCT
cana-1220	8	9	prime	prime	ADJ
cana-1220	8	10	graceful	graceful	ADJ
cana-1220	8	11	labeling	labeling	NOUN
cana-1220	8	12	,	,	PUNCT
cana-1220	8	13	prime	prime	ADJ
cana-1220	8	14	graceful	graceful	ADJ
cana-1220	8	15	coloring	coloring	NOUN
cana-1220	8	16	.	.	PUNCT
cana-1220	9	1	1	1	X
cana-1220	9	2	.	.	X
cana-1220	9	3	introduction	introduction	NOUN
cana-1220	9	4	let	let	VERB
cana-1220	9	5	g	g	PRON
cana-1220	9	6	be	be	AUX
cana-1220	9	7	a	a	DET
cana-1220	9	8	finite	finite	ADJ
cana-1220	9	9	simple	simple	ADJ
cana-1220	9	10	undirected	undirected	ADJ
cana-1220	9	11	graph	graph	NOUN
cana-1220	9	12	.	.	PUNCT
cana-1220	10	1	graph	graph	NOUN
cana-1220	10	2	labeling	labeling	NOUN
cana-1220	10	3	technique	technique	NOUN
cana-1220	10	4	was	be	AUX
cana-1220	10	5	first	first	ADV
cana-1220	10	6	developed	develop	VERB
cana-1220	10	7	by	by	ADP
cana-1220	10	8	rosa	rosa	PROPN
cana-1220	10	9	[	[	X
cana-1220	10	10	5	5	NUM
cana-1220	10	11	]	]	PUNCT
cana-1220	10	12	who	who	PRON
cana-1220	10	13	also	also	ADV
cana-1220	10	14	provided	provide	VERB
cana-1220	10	15	numerous	numerous	ADJ
cana-1220	10	16	graph	graph	NOUN
cana-1220	10	17	labeling	labeling	NOUN
cana-1220	10	18	techniques	technique	NOUN
cana-1220	10	19	and	and	CCONJ
cana-1220	10	20	gallian	gallian	ADJ
cana-1220	10	21	j.a	j.a	PROPN
cana-1220	11	1	[	[	X
cana-1220	11	2	2	2	NUM
cana-1220	11	3	]	]	PUNCT
cana-1220	11	4	conducted	conduct	VERB
cana-1220	11	5	the	the	DET
cana-1220	11	6	survey	survey	NOUN
cana-1220	11	7	on	on	ADP
cana-1220	11	8	graph	graph	NOUN
cana-1220	11	9	labeling	labeling	NOUN
cana-1220	11	10	.	.	PUNCT
cana-1220	12	1	the	the	DET
cana-1220	12	2	proper	proper	ADJ
cana-1220	12	3	coloring	coloring	NOUN
cana-1220	12	4	is	be	AUX
cana-1220	12	5	alluded	allude	VERB
cana-1220	12	6	from	from	ADP
cana-1220	12	7	[	[	X
cana-1220	12	8	10	10	NUM
cana-1220	12	9	]	]	PUNCT
cana-1220	12	10	.	.	PUNCT
cana-1220	13	1	roger	roger	PROPN
cana-1220	13	2	entringer	entringer	PROPN
cana-1220	13	3	defined	define	VERB
cana-1220	13	4	prime	prime	ADJ
cana-1220	13	5	labeling	labeling	NOUN
cana-1220	13	6	and	and	CCONJ
cana-1220	13	7	it	it	PRON
cana-1220	13	8	is	be	AUX
cana-1220	13	9	introduced	introduce	VERB
cana-1220	13	10	by	by	ADP
cana-1220	13	11	tout	tout	NOUN
cana-1220	13	12	et.all	et.all	PROPN
cana-1220	14	1	[	[	X
cana-1220	14	2	9	9	NUM
cana-1220	14	3	]	]	PUNCT
cana-1220	14	4	and	and	CCONJ
cana-1220	14	5	[	[	X
cana-1220	14	6	3	3	X
cana-1220	14	7	]	]	PUNCT
cana-1220	14	8	gave	give	VERB
cana-1220	14	9	the	the	DET
cana-1220	14	10	precise	precise	ADJ
cana-1220	14	11	value	value	NOUN
cana-1220	14	12	of	of	ADP
cana-1220	14	13	prime	prime	ADJ
cana-1220	14	14	chromatic	chromatic	ADJ
cana-1220	14	15	number	number	NOUN
cana-1220	14	16	of	of	ADP
cana-1220	14	17	several	several	ADJ
cana-1220	14	18	graphs	graph	NOUN
cana-1220	14	19	.	.	PUNCT
cana-1220	15	1	zhenming	zhenme	VERB
cana-1220	15	2	bi	bi	ADJ
cana-1220	15	3	et.all	et.all	PROPN
cana-1220	15	4	[	[	X
cana-1220	15	5	1,11	1,11	X
cana-1220	15	6	]	]	PUNCT
cana-1220	15	7	introduced	introduce	VERB
cana-1220	15	8	the	the	DET
cana-1220	15	9	concept	concept	NOUN
cana-1220	15	10	of	of	ADP
cana-1220	15	11	graceful	graceful	ADJ
cana-1220	15	12	colorings	coloring	NOUN
cana-1220	15	13	of	of	ADP
cana-1220	15	14	graphs	graph	NOUN
cana-1220	15	15	and	and	CCONJ
cana-1220	15	16	the	the	DET
cana-1220	15	17	graceful	graceful	ADJ
cana-1220	15	18	chromatic	chromatic	ADJ
cana-1220	15	19	number	number	NOUN
cana-1220	15	20	for	for	ADP
cana-1220	15	21	wheel	wheel	NOUN
cana-1220	15	22	graph	graph	NOUN
cana-1220	15	23	family	family	NOUN
cana-1220	15	24	are	be	AUX
cana-1220	15	25	shown	show	VERB
cana-1220	15	26	to	to	PART
cana-1220	15	27	be	be	AUX
cana-1220	15	28	graceful	graceful	ADJ
cana-1220	16	1	[	[	X
cana-1220	16	2	8	8	NUM
cana-1220	16	3	]	]	PUNCT
cana-1220	16	4	.	.	PUNCT
cana-1220	17	1	in	in	ADP
cana-1220	17	2	2018	2018	NUM
cana-1220	17	3	,	,	PUNCT
cana-1220	17	4	selvarajan.t.m	selvarajan.t.m	NOUN
cana-1220	17	5	,	,	PUNCT
cana-1220	17	6	subramoniam.r	subramoniam.r	VERB
cana-1220	18	1	[	[	X
cana-1220	18	2	7	7	X
cana-1220	18	3	]	]	PUNCT
cana-1220	18	4	combined	combine	VERB
cana-1220	18	5	the	the	DET
cana-1220	18	6	characteristics	characteristic	NOUN
cana-1220	18	7	of	of	ADP
cana-1220	18	8	prime	prime	ADJ
cana-1220	18	9	and	and	CCONJ
cana-1220	18	10	graceful	graceful	ADJ
cana-1220	18	11	labeling	labeling	NOUN
cana-1220	18	12	and	and	CCONJ
cana-1220	18	13	introduced	introduce	VERB
cana-1220	18	14	a	a	DET
cana-1220	18	15	new	new	ADJ
cana-1220	18	16	labeling	labeling	NOUN
cana-1220	18	17	technique	technique	NOUN
cana-1220	18	18	prime	prime	ADJ
cana-1220	18	19	graceful	graceful	ADJ
cana-1220	18	20	labeling	labeling	NOUN
cana-1220	18	21	and	and	CCONJ
cana-1220	18	22	demonstrated	demonstrate	VERB
cana-1220	18	23	the	the	DET
cana-1220	18	24	existence	existence	NOUN
cana-1220	18	25	of	of	ADP
cana-1220	18	26	prime	prime	ADJ
cana-1220	18	27	graceful	graceful	ADJ
cana-1220	18	28	labeling	labeling	NOUN
cana-1220	18	29	in	in	ADP
cana-1220	18	30	some	some	DET
cana-1220	18	31	graphs	graph	NOUN
cana-1220	18	32	.	.	PUNCT
cana-1220	19	1	sayan	sayan	ADJ
cana-1220	19	2	panma	panma	NOUN
cana-1220	19	3	and	and	CCONJ
cana-1220	19	4	penying	penye	VERB
cana-1220	19	5	rochanakul	rochanakul	NOUN
cana-1220	19	6	[	[	X
cana-1220	19	7	6	6	NUM
cana-1220	19	8	]	]	PUNCT
cana-1220	19	9	defined	define	VERB
cana-1220	19	10	prime	prime	ADJ
cana-1220	19	11	-	-	PUNCT
cana-1220	19	12	graceful	graceful	ADJ
cana-1220	19	13	number	number	NOUN
cana-1220	19	14	and	and	CCONJ
cana-1220	19	15	applied	apply	VERB
cana-1220	19	16	prime	prime	ADJ
cana-1220	19	17	graceful	graceful	ADJ
cana-1220	19	18	labeling	labeling	NOUN
cana-1220	19	19	to	to	ADP
cana-1220	19	20	certain	certain	ADJ
cana-1220	19	21	graphs	graph	NOUN
cana-1220	19	22	,	,	PUNCT
cana-1220	19	23	then	then	ADV
cana-1220	19	24	nandhini.s.p	nandhini.s.p	PROPN
cana-1220	19	25	and	and	CCONJ
cana-1220	19	26	pooja	pooja	ADV
cana-1220	19	27	lakshmi.b	lakshmi.b	PUNCT
cana-1220	20	1	[	[	X
cana-1220	20	2	4	4	X
cana-1220	20	3	]	]	PUNCT
cana-1220	20	4	generalized	generalize	VERB
cana-1220	20	5	the	the	DET
cana-1220	20	6	cardinality	cardinality	NOUN
cana-1220	20	7	of	of	ADP
cana-1220	20	8	the	the	DET
cana-1220	20	9	edges	edge	NOUN
cana-1220	20	10	for	for	ADP
cana-1220	20	11	the	the	DET
cana-1220	20	12	triangular	triangular	NOUN
cana-1220	20	13	snake	snake	NOUN
cana-1220	20	14	graph	graph	NOUN
cana-1220	20	15	.	.	PUNCT
cana-1220	21	1	in	in	ADP
cana-1220	21	2	this	this	DET
cana-1220	21	3	paper	paper	NOUN
cana-1220	21	4	,	,	PUNCT
cana-1220	21	5	we	we	PRON
cana-1220	21	6	define	define	VERB
cana-1220	21	7	the	the	DET
cana-1220	21	8	new	new	ADJ
cana-1220	21	9	technique	technique	NOUN
cana-1220	21	10	prime	prime	ADJ
cana-1220	21	11	graceful	graceful	ADJ
cana-1220	21	12	coloring	coloring	NOUN
cana-1220	21	13	and	and	CCONJ
cana-1220	21	14	its	its	PRON
cana-1220	21	15	chromatic	chromatic	ADJ
cana-1220	21	16	number	number	NOUN
cana-1220	21	17	are	be	AUX
cana-1220	21	18	analyzed	analyze	VERB
cana-1220	21	19	for	for	ADP
cana-1220	21	20	some	some	DET
cana-1220	21	21	graphs	graph	NOUN
cana-1220	21	22	.	.	PUNCT
cana-1220	22	1	2	2	X
cana-1220	22	2	.	.	X
cana-1220	22	3	preliminaries	preliminary	NOUN
cana-1220	22	4	definition	definition	NOUN
cana-1220	22	5	:	:	PUNCT
cana-1220	22	6	2.1[3	2.1[3	NUM
cana-1220	22	7	]	]	X
cana-1220	22	8	prime	prime	ADJ
cana-1220	22	9	coloring	coloring	NOUN
cana-1220	22	10	is	be	AUX
cana-1220	22	11	defined	define	VERB
cana-1220	22	12	as	as	SCONJ
cana-1220	22	13	g	g	PROPN
cana-1220	22	14	be	be	AUX
cana-1220	22	15	a	a	DET
cana-1220	22	16	loop	loop	NOUN
cana-1220	22	17	less	less	ADV
cana-1220	22	18	and	and	CCONJ
cana-1220	22	19	without	without	ADP
cana-1220	22	20	multiple	multiple	ADJ
cana-1220	22	21	edges	edge	NOUN
cana-1220	22	22	with	with	ADP
cana-1220	22	23	p	p	X
cana-1220	22	24	distinct	distinct	ADJ
cana-1220	22	25	vertices	vertex	NOUN
cana-1220	22	26	ꞷ	ꞷ	NOUN
cana-1220	22	27	:	:	PUNCT
cana-1220	22	28	v(g)→{1	v(g)→{1	VERB
cana-1220	22	29	,	,	PUNCT
cana-1220	22	30	2	2	NUM
cana-1220	22	31	,	,	PUNCT
cana-1220	22	32	…	…	PUNCT
cana-1220	22	33	p},if	p},if	VERB
cana-1220	22	34	each	each	DET
cana-1220	22	35	edge	edge	NOUN
cana-1220	22	36	e	e	NOUN
cana-1220	23	1	=	=	NOUN
cana-1220	23	2	cicj	cicj	NOUN
cana-1220	23	3	,	,	PUNCT
cana-1220	23	4	i≠j	i≠j	NOUN
cana-1220	23	5	,	,	PUNCT
cana-1220	23	6	gcd	gcd	X
cana-1220	23	7	{	{	PUNCT
cana-1220	23	8	ꞷ	ꞷ	X
cana-1220	23	9	(	(	PUNCT
cana-1220	23	10	vi	vi	NOUN
cana-1220	23	11	)	)	PUNCT
cana-1220	23	12	,	,	PUNCT
cana-1220	23	13	ꞷ	ꞷ	PROPN
cana-1220	23	14	(	(	PUNCT
cana-1220	23	15	vj	vj	PROPN
cana-1220	23	16	)	)	PUNCT
cana-1220	23	17	}	}	PUNCT
cana-1220	23	18	=	=	SYM
cana-1220	24	1	1	1	NUM
cana-1220	24	2	,	,	PUNCT
cana-1220	24	3	ꞷ	ꞷ	PROPN
cana-1220	24	4	(	(	PUNCT
cana-1220	24	5	vi	vi	NOUN
cana-1220	24	6	)	)	PUNCT
cana-1220	24	7	,	,	PUNCT
cana-1220	24	8	ꞷ	ꞷ	PROPN
cana-1220	24	9	(	(	PUNCT
cana-1220	24	10	vj	vj	INTJ
cana-1220	24	11	)	)	PUNCT
cana-1220	24	12	receives	receive	VERB
cana-1220	24	13	distinct	distinct	ADJ
cana-1220	24	14	colors	color	NOUN
cana-1220	24	15	.	.	PUNCT
cana-1220	25	1	it	it	PRON
cana-1220	25	2	is	be	AUX
cana-1220	25	3	denoted	denote	VERB
cana-1220	25	4	by	by	ADP
cana-1220	25	5	η(g	η(g	PROPN
cana-1220	25	6	)	)	PUNCT
cana-1220	25	7	.	.	PUNCT
cana-1220	26	1	definition	definition	NOUN
cana-1220	26	2	:	:	PUNCT
cana-1220	26	3	2.2[8	2.2[8	NUM
cana-1220	26	4	]	]	X
cana-1220	26	5	a	a	DET
cana-1220	26	6	graceful	graceful	ADJ
cana-1220	26	7	k	k	NOUN
cana-1220	26	8	-	-	NOUN
cana-1220	26	9	coloring	coloring	NOUN
cana-1220	26	10	of	of	ADP
cana-1220	26	11	a	a	DET
cana-1220	26	12	non	non	ADJ
cana-1220	26	13	-	-	ADJ
cana-1220	26	14	empty	empty	ADJ
cana-1220	26	15	graph	graph	NOUN
cana-1220	26	16	g=(v	g=(v	NOUN
cana-1220	26	17	,	,	PUNCT
cana-1220	26	18	e	e	NOUN
cana-1220	26	19	)	)	PUNCT
cana-1220	26	20	is	be	AUX
cana-1220	26	21	a	a	DET
cana-1220	26	22	proper	proper	ADJ
cana-1220	26	23	vertex	vertex	NOUN
cana-1220	26	24	coloring	color	VERB
cana-1220	26	25	ꞷ	ꞷ	PRON
cana-1220	26	26	:	:	PUNCT
cana-1220	26	27	v(g)→{1	v(g)→{1	PROPN
cana-1220	26	28	,	,	PUNCT
cana-1220	26	29	2	2	NUM
cana-1220	26	30	,	,	PUNCT
cana-1220	26	31	…	…	PUNCT
cana-1220	26	32	k	k	X
cana-1220	26	33	}	}	PUNCT
cana-1220	26	34	,	,	PUNCT
cana-1220	26	35	k	k	PROPN
cana-1220	26	36	≥2	≥2	X
cana-1220	26	37	which	which	PRON
cana-1220	26	38	induces	induce	VERB
cana-1220	26	39	a	a	DET
cana-1220	26	40	proper	proper	ADJ
cana-1220	26	41	edge	edge	NOUN
cana-1220	26	42	coloring	color	VERB
cana-1220	26	43	ꞷ	ꞷ	PRON
cana-1220	26	44	∗	∗	NOUN
cana-1220	26	45	:	:	PUNCT
cana-1220	26	46	e(g)→{1,2,	e(g)→{1,2,	NOUN
cana-1220	26	47	…	…	SYM
cana-1220	26	48	k-1	k-1	PROPN
cana-1220	26	49	}	}	PUNCT
cana-1220	26	50	defined	define	VERB
cana-1220	26	51	by	by	ADP
cana-1220	26	52	ꞷ	ꞷ	PROPN
cana-1220	26	53	∗(𝑣𝑖	∗(𝑣𝑖	NOUN
cana-1220	26	54	,	,	PUNCT
cana-1220	26	55	𝑣𝑗	𝑣𝑗	ADP
cana-1220	26	56	)	)	PUNCT
cana-1220	26	57	=	=	SYM
cana-1220	26	58	|ꞷ	|ꞷ	PROPN
cana-1220	26	59	(	(	PUNCT
cana-1220	26	60	𝑣𝑖	𝑣𝑖	PROPN
cana-1220	26	61	)	)	PUNCT
cana-1220	26	62	ꞷ	ꞷ	PROPN
cana-1220	26	63	(	(	PUNCT
cana-1220	26	64	𝑣𝑗)|	𝑣𝑗)|	ADP
cana-1220	26	65	where	where	SCONJ
cana-1220	26	66	𝑣𝑖	𝑣𝑖	ADV
cana-1220	26	67	,	,	PUNCT
cana-1220	26	68	𝑣𝑗∈	𝑣𝑗∈	PRON
cana-1220	26	69	v(g	v(g	NOUN
cana-1220	26	70	)	)	PUNCT
cana-1220	26	71	.	.	PUNCT
cana-1220	27	1	the	the	DET
cana-1220	27	2	minimum	minimum	PROPN
cana-1220	27	3	k	k	PROPN
cana-1220	27	4	for	for	ADP
cana-1220	27	5	which	which	PRON
cana-1220	27	6	g	g	NOUN
cana-1220	27	7	has	have	VERB
cana-1220	27	8	a	a	DET
cana-1220	27	9	graceful	graceful	ADJ
cana-1220	27	10	k	k	NOUN
cana-1220	27	11	-	-	ADJ
cana-1220	27	12	coloring	coloring	NOUN
cana-1220	27	13	is	be	AUX
cana-1220	27	14	called	call	VERB
cana-1220	27	15	graceful	graceful	ADJ
cana-1220	27	16	chromatic	chromatic	ADJ
cana-1220	27	17	number	number	NOUN
cana-1220	27	18	χg(g	χg(g	NUM
cana-1220	27	19	)	)	PUNCT
cana-1220	27	20	.	.	PUNCT
cana-1220	28	1	communications	communication	NOUN
cana-1220	28	2	on	on	ADP
cana-1220	28	3	applied	apply	VERB
cana-1220	28	4	nonlinear	nonlinear	ADJ
cana-1220	28	5	analysis	analysis	NOUN
cana-1220	28	6	issn	issn	NOUN
cana-1220	28	7	:	:	PUNCT
cana-1220	28	8	1074	1074	NUM
cana-1220	28	9	-	-	PUNCT
cana-1220	28	10	133x	133x	NUM
cana-1220	28	11	vol	vol	NOUN
cana-1220	28	12	31	31	NUM
cana-1220	28	13	no	no	NOUN
cana-1220	28	14	.	.	PUNCT
cana-1220	29	1	6s	6s	NUM
cana-1220	29	2	(	(	PUNCT
cana-1220	29	3	2024	2024	NUM
cana-1220	29	4	)	)	PUNCT
cana-1220	29	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	29	6	260	260	NUM
cana-1220	29	7	definition	definition	NOUN
cana-1220	29	8	:	:	PUNCT
cana-1220	29	9	2.3	2.3	NUM
cana-1220	30	1	[	[	NOUN
cana-1220	30	2	4	4	X
cana-1220	30	3	]	]	X
cana-1220	30	4	a	a	DET
cana-1220	30	5	graph	graph	NOUN
cana-1220	30	6	g	g	NOUN
cana-1220	30	7	with	with	ADP
cana-1220	30	8	p	p	NOUN
cana-1220	30	9	vertices	vertex	NOUN
cana-1220	30	10	and	and	CCONJ
cana-1220	30	11	q	q	NOUN
cana-1220	30	12	edges	edge	NOUN
cana-1220	30	13	is	be	AUX
cana-1220	30	14	referred	refer	VERB
cana-1220	30	15	to	to	PART
cana-1220	30	16	have	have	VERB
cana-1220	30	17	prime	prime	ADJ
cana-1220	30	18	graceful	graceful	ADJ
cana-1220	30	19	labeling	labeling	NOUN
cana-1220	30	20	if	if	SCONJ
cana-1220	30	21	the	the	DET
cana-1220	30	22	map	map	NOUN
cana-1220	30	23	depicts	depict	VERB
cana-1220	30	24	a	a	DET
cana-1220	30	25	one	one	NUM
cana-1220	30	26	to	to	ADP
cana-1220	30	27	one	one	NUM
cana-1220	30	28	function	function	NOUN
cana-1220	30	29	ꞷ	ꞷ	NOUN
cana-1220	30	30	:	:	PUNCT
cana-1220	30	31	v(g)→{1	v(g)→{1	PROPN
cana-1220	30	32	,	,	PUNCT
cana-1220	30	33	2	2	NUM
cana-1220	30	34	,	,	PUNCT
cana-1220	30	35	…	…	PUNCT
cana-1220	30	36	k	k	X
cana-1220	30	37	}	}	PUNCT
cana-1220	30	38	.	.	PUNCT
cana-1220	31	1	in	in	ADP
cana-1220	31	2	this	this	DET
cana-1220	31	3	instance	instance	NOUN
cana-1220	31	4	,	,	PUNCT
cana-1220	31	5	k	k	PROPN
cana-1220	31	6	=	=	SYM
cana-1220	31	7	min	min	PROPN
cana-1220	31	8	{	{	PUNCT
cana-1220	31	9	p	p	X
cana-1220	31	10	,	,	PUNCT
cana-1220	31	11	q	q	X
cana-1220	31	12	}	}	PUNCT
cana-1220	31	13	such	such	ADJ
cana-1220	31	14	that	that	DET
cana-1220	31	15	gcd	gcd	NOUN
cana-1220	31	16	(	(	PUNCT
cana-1220	31	17	ꞷ	ꞷ	NOUN
cana-1220	31	18	(	(	PUNCT
cana-1220	31	19	𝑣𝑖	𝑣𝑖	PROPN
cana-1220	31	20	)	)	PUNCT
cana-1220	31	21	,	,	PUNCT
cana-1220	31	22	ꞷ	ꞷ	PROPN
cana-1220	31	23	(	(	PUNCT
cana-1220	31	24	𝑣𝑗	𝑣𝑗	NOUN
cana-1220	31	25	)	)	PUNCT
cana-1220	31	26	)	)	PUNCT
cana-1220	32	1	=	=	SYM
cana-1220	32	2	1	1	NUM
cana-1220	32	3	and	and	CCONJ
cana-1220	32	4	the	the	DET
cana-1220	32	5	map	map	NOUN
cana-1220	32	6	reflect	reflect	VERB
cana-1220	32	7	the	the	DET
cana-1220	32	8	induced	induced	ADJ
cana-1220	32	9	one	one	NUM
cana-1220	32	10	to	to	ADP
cana-1220	32	11	one	one	NUM
cana-1220	32	12	function	function	NOUN
cana-1220	32	13	ꞷ	ꞷ	NOUN
cana-1220	32	14	∗	∗	NOUN
cana-1220	32	15	:	:	PUNCT
cana-1220	32	16	e(g)→{1,2,	e(g)→{1,2,	NOUN
cana-1220	32	17	…	…	SYM
cana-1220	32	18	k-1	k-1	PROPN
cana-1220	32	19	}	}	PUNCT
cana-1220	32	20	defined	define	VERB
cana-1220	32	21	by	by	ADP
cana-1220	32	22	ꞷ	ꞷ	PROPN
cana-1220	32	23	∗(𝑣𝑖𝑣𝑗	∗(𝑣𝑖𝑣𝑗	PUNCT
cana-1220	32	24	)	)	PUNCT
cana-1220	33	1	=	=	SYM
cana-1220	33	2	|ꞷ	|ꞷ	PROPN
cana-1220	33	3	(	(	PUNCT
cana-1220	33	4	𝑣𝑖	𝑣𝑖	NOUN
cana-1220	33	5	)	)	PUNCT
cana-1220	33	6	ꞷ	ꞷ	PROPN
cana-1220	33	7	(	(	PUNCT
cana-1220	33	8	𝑣𝑗)|	𝑣𝑗)|	ADP
cana-1220	33	9	such	such	ADJ
cana-1220	33	10	that	that	SCONJ
cana-1220	33	11	the	the	DET
cana-1220	33	12	corresponding	corresponding	ADJ
cana-1220	33	13	edge	edge	NOUN
cana-1220	33	14	labels	label	NOUN
cana-1220	33	15	are	be	AUX
cana-1220	33	16	distinct	distinct	ADJ
cana-1220	33	17	.	.	PUNCT
cana-1220	34	1	definition	definition	NOUN
cana-1220	34	2	:	:	PUNCT
cana-1220	34	3	2.4	2.4	NUM
cana-1220	34	4	a	a	DET
cana-1220	34	5	prime	prime	ADJ
cana-1220	34	6	graceful	graceful	NOUN
cana-1220	34	7	k	k	NOUN
cana-1220	34	8	-	-	PUNCT
cana-1220	34	9	coloring	coloring	NOUN
cana-1220	34	10	of	of	ADP
cana-1220	34	11	non	non	ADJ
cana-1220	34	12	-	-	ADJ
cana-1220	34	13	empty	empty	ADJ
cana-1220	34	14	simple	simple	ADJ
cana-1220	34	15	graph	graph	NOUN
cana-1220	34	16	g	g	PROPN
cana-1220	34	17	is	be	AUX
cana-1220	34	18	a	a	DET
cana-1220	34	19	proper	proper	ADJ
cana-1220	34	20	vertex	vertex	NOUN
cana-1220	34	21	coloring	color	VERB
cana-1220	34	22	ꞷ	ꞷ	PRON
cana-1220	34	23	:	:	PUNCT
cana-1220	34	24	v(g)→{1	v(g)→{1	PROPN
cana-1220	34	25	,	,	PUNCT
cana-1220	34	26	2,3	2,3	NUM
cana-1220	34	27	…	…	SYM
cana-1220	34	28	…	…	SYM
cana-1220	34	29	k	k	X
cana-1220	34	30	}	}	PUNCT
cana-1220	34	31	such	such	ADJ
cana-1220	34	32	that	that	DET
cana-1220	34	33	gcd	gcd	NOUN
cana-1220	34	34	(	(	PUNCT
cana-1220	34	35	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	34	36	)	)	PUNCT
cana-1220	34	37	,	,	PUNCT
cana-1220	34	38	ꞷ(vj	ꞷ(vj	NOUN
cana-1220	34	39	)	)	PUNCT
cana-1220	34	40	)	)	PUNCT
cana-1220	35	1	=	=	PUNCT
cana-1220	35	2	1	1	NUM
cana-1220	35	3	where	where	SCONJ
cana-1220	35	4	k	k	PROPN
cana-1220	35	5	≥2	≥2	PROPN
cana-1220	35	6	bring	bring	VERB
cana-1220	35	7	about	about	ADP
cana-1220	35	8	a	a	DET
cana-1220	35	9	proper	proper	ADJ
cana-1220	35	10	edge	edge	NOUN
cana-1220	35	11	coloring	color	VERB
cana-1220	35	12	ꞷ	ꞷ	PRON
cana-1220	35	13	∗:e(g)→	∗:e(g)→	X
cana-1220	35	14	{	{	PUNCT
cana-1220	35	15	1,2,3	1,2,3	NUM
cana-1220	35	16	…	…	NUM
cana-1220	35	17	k-1}defined	k-1}define	VERB
cana-1220	35	18	by	by	ADP
cana-1220	35	19	ꞷ	ꞷ	PROPN
cana-1220	35	20	∗(vivj	∗(vivj	X
cana-1220	35	21	)	)	PUNCT
cana-1220	35	22	=	=	SYM
cana-1220	35	23	|ꞷ(vi)ꞷ(vj)|	|ꞷ(vi)ꞷ(vj)|	ADJ
cana-1220	35	24	,	,	PUNCT
cana-1220	35	25	ɐ	ɐ	PROPN
cana-1220	35	26	k∈n	k∈n	PROPN
cana-1220	35	27	the	the	DET
cana-1220	35	28	least	least	ADJ
cana-1220	35	29	k	k	NOUN
cana-1220	35	30	for	for	ADP
cana-1220	35	31	which	which	PRON
cana-1220	35	32	g	g	NOUN
cana-1220	35	33	exhibits	exhibit	VERB
cana-1220	35	34	prime	prime	ADJ
cana-1220	35	35	graceful	graceful	ADJ
cana-1220	35	36	k	k	PROPN
cana-1220	35	37	-	-	ADJ
cana-1220	35	38	coloring	coloring	NOUN
cana-1220	35	39	is	be	AUX
cana-1220	35	40	known	know	VERB
cana-1220	35	41	as	as	ADP
cana-1220	35	42	the	the	DET
cana-1220	35	43	chromatic	chromatic	ADJ
cana-1220	35	44	number	number	NOUN
cana-1220	35	45	of	of	ADP
cana-1220	35	46	prime	prime	ADJ
cana-1220	35	47	graceful	graceful	ADJ
cana-1220	35	48	coloring	coloring	NOUN
cana-1220	35	49	.	.	PUNCT
cana-1220	36	1	it	it	PRON
cana-1220	36	2	is	be	AUX
cana-1220	36	3	denoted	denote	VERB
cana-1220	36	4	by	by	ADP
cana-1220	36	5	χpg(g	χpg(g	PROPN
cana-1220	36	6	)	)	PUNCT
cana-1220	36	7	.	.	PUNCT
cana-1220	37	1	3	3	X
cana-1220	37	2	.	.	X
cana-1220	37	3	main	main	ADJ
cana-1220	37	4	results	result	NOUN
cana-1220	37	5	theorem	theorem	VERB
cana-1220	37	6	:	:	PUNCT
cana-1220	37	7	3.1	3.1	NUM
cana-1220	37	8	let	let	VERB
cana-1220	37	9	k1,n	k1,n	PROPN
cana-1220	37	10	be	be	AUX
cana-1220	37	11	a	a	DET
cana-1220	37	12	star	star	NOUN
cana-1220	37	13	graph	graph	NOUN
cana-1220	37	14	,	,	PUNCT
cana-1220	37	15	then	then	ADV
cana-1220	37	16	χpg(k1,n)=	χpg(k1,n)=	PUNCT
cana-1220	37	17	n+1	n+1	PROPN
cana-1220	37	18	.	.	PUNCT
cana-1220	38	1	proof	proof	NOUN
cana-1220	38	2	:	:	PUNCT
cana-1220	38	3	let	let	VERB
cana-1220	38	4	v	v	X
cana-1220	38	5	(	(	PUNCT
cana-1220	38	6	k1,n	k1,n	PROPN
cana-1220	38	7	)	)	PUNCT
cana-1220	38	8	=	=	PRON
cana-1220	38	9	{	{	PUNCT
cana-1220	38	10	vi	vi	NOUN
cana-1220	38	11	:	:	PUNCT
cana-1220	38	12	1	1	NUM
cana-1220	38	13	≤	≤	NUM
cana-1220	39	1	i	i	X
cana-1220	39	2	≤	≤	NOUN
cana-1220	39	3	n+1	n+1	VERB
cana-1220	39	4	}	}	PUNCT
cana-1220	39	5	and	and	CCONJ
cana-1220	39	6	e	e	PROPN
cana-1220	39	7	(	(	PUNCT
cana-1220	39	8	k1,n	k1,n	PROPN
cana-1220	39	9	)	)	PUNCT
cana-1220	39	10	=	=	PRON
cana-1220	39	11	{	{	PUNCT
cana-1220	39	12	vivi+1	vivi+1	ADJ
cana-1220	39	13	∶	∶	NOUN
cana-1220	39	14	1	1	NUM
cana-1220	39	15	≤	≤	NUM
cana-1220	39	16	i	i	PRON
cana-1220	39	17	≤	≤	NOUN
cana-1220	39	18	n	n	CCONJ
cana-1220	39	19	}	}	PUNCT
cana-1220	39	20	.	.	PUNCT
cana-1220	40	1	let	let	VERB
cana-1220	40	2	ꞷ	ꞷ	PRON
cana-1220	40	3	:	:	PUNCT
cana-1220	40	4	v(k1,n)→{c1	v(k1,n)→{c1	ADJ
cana-1220	40	5	,	,	PUNCT
cana-1220	40	6	c2	c2	PROPN
cana-1220	40	7	,	,	PUNCT
cana-1220	40	8	…	…	PUNCT
cana-1220	40	9	,	,	PUNCT
cana-1220	40	10	cn+1}where	cn+1}where	ADV
cana-1220	40	11	the	the	DET
cana-1220	40	12	vertex	vertex	NOUN
cana-1220	40	13	of	of	ADP
cana-1220	40	14	degree	degree	NOUN
cana-1220	40	15	n	n	NOUN
cana-1220	40	16	is	be	AUX
cana-1220	40	17	designated	designate	VERB
cana-1220	40	18	as	as	ADP
cana-1220	40	19	c1	c1	PROPN
cana-1220	40	20	(	(	PUNCT
cana-1220	40	21	i.e),ꞷ(v1	i.e),ꞷ(v1	PROPN
cana-1220	40	22	)	)	PUNCT
cana-1220	40	23	=	=	SYM
cana-1220	40	24	c1	c1	PROPN
cana-1220	40	25	and	and	CCONJ
cana-1220	40	26	n	n	CCONJ
cana-1220	40	27	-	-	PUNCT
cana-1220	40	28	pendant	pendant	ADJ
cana-1220	40	29	vertices	vertex	NOUN
cana-1220	40	30	are	be	AUX
cana-1220	40	31	designated	designate	VERB
cana-1220	40	32	as	as	ADP
cana-1220	40	33	ꞷ	ꞷ	PROPN
cana-1220	40	34	(	(	PUNCT
cana-1220	40	35	vi)=	vi)=	NOUN
cana-1220	40	36	ci	ci	NOUN
cana-1220	40	37	which	which	PRON
cana-1220	40	38	satisfy	satisfy	NOUN
cana-1220	40	39	gcd	gcd	PROPN
cana-1220	40	40	(	(	PUNCT
cana-1220	40	41	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	40	42	)	)	PUNCT
cana-1220	40	43	,	,	PUNCT
cana-1220	40	44	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	40	45	)	)	PUNCT
cana-1220	40	46	)	)	PUNCT
cana-1220	41	1	=	=	SYM
cana-1220	41	2	1	1	NUM
cana-1220	41	3	for	for	ADP
cana-1220	41	4	2	2	NUM
cana-1220	41	5	≤	≤	NUM
cana-1220	41	6	i	i	PRON
cana-1220	41	7	≤	≤	NOUN
cana-1220	41	8	n	n	PRON
cana-1220	41	9	+1	+1	PROPN
cana-1220	41	10	.	.	PUNCT
cana-1220	42	1	hence	hence	ADV
cana-1220	42	2	,	,	PUNCT
cana-1220	42	3	adjacent	adjacent	ADJ
cana-1220	42	4	vertices	vertex	NOUN
cana-1220	42	5	receive	receive	VERB
cana-1220	42	6	distinct	distinct	ADJ
cana-1220	42	7	colors	color	NOUN
cana-1220	42	8	.	.	PUNCT
cana-1220	43	1	let	let	VERB
cana-1220	43	2	ꞷ*:e(k1,n)→{c1	ꞷ*:e(k1,n)→{c1	NOUN
cana-1220	43	3	,	,	PUNCT
cana-1220	43	4	c2	c2	PROPN
cana-1220	43	5	,	,	PUNCT
cana-1220	43	6	…	…	PUNCT
cana-1220	43	7	,	,	PUNCT
cana-1220	43	8	cn	cn	PROPN
cana-1220	43	9	.	.	PROPN
cana-1220	44	1	for	for	ADP
cana-1220	44	2	2	2	NUM
cana-1220	44	3	≤	≤	NOUN
cana-1220	44	4	i	i	PRON
cana-1220	44	5	≤	≤	NOUN
cana-1220	44	6	n+1	n+1	ADV
cana-1220	44	7	,	,	PUNCT
cana-1220	44	8	ꞷ*(v1vi	ꞷ*(v1vi	NOUN
cana-1220	44	9	)	)	PUNCT
cana-1220	44	10	=	=	PUNCT
cana-1220	44	11	ci-1	ci-1	NOUN
cana-1220	44	12	.	.	PUNCT
cana-1220	45	1	it	it	PRON
cana-1220	45	2	is	be	AUX
cana-1220	45	3	determined	determine	VERB
cana-1220	45	4	by	by	ADP
cana-1220	45	5	ꞷ	ꞷ	PROPN
cana-1220	45	6	∗	∗	NOUN
cana-1220	45	7	(	(	PUNCT
cana-1220	45	8	v1vi)=	v1vi)=	NUM
cana-1220	46	1	|	|	ADV
cana-1220	46	2	ꞷ	ꞷ	X
cana-1220	46	3	(	(	PUNCT
cana-1220	46	4	v1	v1	PROPN
cana-1220	46	5	)	)	PUNCT
cana-1220	46	6	ꞷ	ꞷ	PROPN
cana-1220	47	1	(	(	PUNCT
cana-1220	47	2	𝑣𝑖)|	𝑣𝑖)|	NOUN
cana-1220	47	3	=	=	SYM
cana-1220	47	4	ci-1	ci-1	NOUN
cana-1220	47	5	.	.	PUNCT
cana-1220	48	1	consequently	consequently	ADV
cana-1220	48	2	,	,	PUNCT
cana-1220	48	3	ꞷ*(v1vi	ꞷ*(v1vi	X
cana-1220	48	4	)	)	PUNCT
cana-1220	48	5	receives	receive	VERB
cana-1220	48	6	distinct	distinct	ADJ
cana-1220	48	7	colors	color	NOUN
cana-1220	48	8	.	.	PUNCT
cana-1220	49	1	we	we	PRON
cana-1220	49	2	proved	prove	VERB
cana-1220	49	3	that	that	SCONJ
cana-1220	49	4	χpg(k1,n	χpg(k1,n	NOUN
cana-1220	49	5	)	)	PUNCT
cana-1220	49	6	≤	≤	NOUN
cana-1220	49	7	n	n	CCONJ
cana-1220	49	8	+	+	NOUN
cana-1220	49	9	1	1	X
cana-1220	49	10	.	.	PUNCT
cana-1220	49	11	to	to	PART
cana-1220	49	12	prove	prove	VERB
cana-1220	49	13	χpg(k1,n	χpg(k1,n	NOUN
cana-1220	49	14	)	)	PUNCT
cana-1220	49	15	≥	≥	NOUN
cana-1220	49	16	n	n	PROPN
cana-1220	49	17	+	+	NUM
cana-1220	49	18	1	1	NUM
cana-1220	49	19	,	,	PUNCT
cana-1220	49	20	let	let	VERB
cana-1220	49	21	us	we	PRON
cana-1220	49	22	assume	assume	VERB
cana-1220	49	23	that	that	SCONJ
cana-1220	49	24	χpg(k1,n	χpg(k1,n	NOUN
cana-1220	49	25	)	)	PUNCT
cana-1220	49	26	<	<	X
cana-1220	50	1	n	n	PROPN
cana-1220	50	2	+	+	NUM
cana-1220	50	3	1	1	NUM
cana-1220	50	4	,	,	PUNCT
cana-1220	50	5	say	say	VERB
cana-1220	50	6	n.	n.	NOUN
cana-1220	50	7	we	we	PRON
cana-1220	50	8	define	define	VERB
cana-1220	50	9	proper	proper	ADJ
cana-1220	50	10	vertex	vertex	NOUN
cana-1220	50	11	coloring	coloring	NOUN
cana-1220	50	12	of	of	ADP
cana-1220	50	13	k1,n	k1,n	PROPN
cana-1220	50	14	is	be	AUX
cana-1220	50	15	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	50	16	)	)	PUNCT
cana-1220	51	1	=	=	NOUN
cana-1220	51	2	c1	c1	NOUN
cana-1220	51	3	,	,	PUNCT
cana-1220	51	4	ꞷ	ꞷ	PROPN
cana-1220	51	5	(	(	PUNCT
cana-1220	51	6	vi)=ci-1	vi)=ci-1	NOUN
cana-1220	51	7	ɐ	ɐ	NOUN
cana-1220	51	8	2	2	NUM
cana-1220	51	9	≤	≤	NUM
cana-1220	51	10	i	i	PRON
cana-1220	51	11	≤	≤	NOUN
cana-1220	52	1	n+1	n+1	VERB
cana-1220	52	2	.	.	PUNCT
cana-1220	53	1	then	then	ADV
cana-1220	53	2	,	,	PUNCT
cana-1220	53	3	it	it	PRON
cana-1220	53	4	induces	induce	VERB
cana-1220	53	5	edge	edge	NOUN
cana-1220	53	6	coloring	color	VERB
cana-1220	53	7	ꞷ	ꞷ	PRON
cana-1220	53	8	∗	∗	NOUN
cana-1220	53	9	(	(	PUNCT
cana-1220	53	10	v1v2)=	v1v2)=	NOUN
cana-1220	53	11	𝑐1	𝑐1	NOUN
cana-1220	53	12	,	,	PUNCT
cana-1220	53	13	ꞷ	ꞷ	PROPN
cana-1220	53	14	∗	∗	NOUN
cana-1220	53	15	(	(	PUNCT
cana-1220	53	16	v1v3)=	v1v3)=	ADJ
cana-1220	53	17	𝑐1	𝑐1	NOUN
cana-1220	53	18	,	,	PUNCT
cana-1220	53	19	ꞷ	ꞷ	PROPN
cana-1220	53	20	∗	∗	NOUN
cana-1220	53	21	(	(	PUNCT
cana-1220	53	22	v1v4)=	v1v4)=	PROPN
cana-1220	53	23	c2	c2	PROPN
cana-1220	53	24	,	,	PUNCT
cana-1220	53	25	ꞷ	ꞷ	PROPN
cana-1220	53	26	∗	∗	NOUN
cana-1220	53	27	(	(	PUNCT
cana-1220	53	28	v1v5)=	v1v5)=	NOUN
cana-1220	53	29	c3	c3	PROPN
cana-1220	53	30	and	and	CCONJ
cana-1220	53	31	so	so	ADV
cana-1220	53	32	on	on	ADP
cana-1220	53	33	which	which	PRON
cana-1220	53	34	is	be	AUX
cana-1220	53	35	a	a	DET
cana-1220	53	36	contradiction	contradiction	NOUN
cana-1220	53	37	with	with	ADP
cana-1220	53	38	the	the	DET
cana-1220	53	39	definition	definition	NOUN
cana-1220	53	40	of	of	ADP
cana-1220	53	41	prime	prime	ADJ
cana-1220	53	42	graceful	graceful	ADJ
cana-1220	53	43	coloring	coloring	NOUN
cana-1220	53	44	since	since	SCONJ
cana-1220	53	45	the	the	DET
cana-1220	53	46	color	color	NOUN
cana-1220	53	47	of	of	ADP
cana-1220	53	48	any	any	DET
cana-1220	53	49	two	two	NUM
cana-1220	53	50	adjacent	adjacent	ADJ
cana-1220	53	51	edges	edge	NOUN
cana-1220	53	52	are	be	AUX
cana-1220	53	53	distinct	distinct	ADJ
cana-1220	53	54	.	.	PUNCT
cana-1220	54	1	thus	thus	ADV
cana-1220	54	2	,	,	PUNCT
cana-1220	54	3	χpg(k1,n	χpg(k1,n	PROPN
cana-1220	54	4	)	)	PUNCT
cana-1220	54	5	≥	≥	NOUN
cana-1220	54	6	n	n	NOUN
cana-1220	54	7	+	+	NUM
cana-1220	54	8	1	1	NUM
cana-1220	54	9	.	.	X
cana-1220	55	1	therefore	therefore	ADV
cana-1220	55	2	,	,	PUNCT
cana-1220	55	3	χpg(k1,n)=	χpg(k1,n)=	ADV
cana-1220	55	4	n+1	n+1	PROPN
cana-1220	55	5	.	.	NOUN
cana-1220	55	6	figure	figure	VERB
cana-1220	55	7	1	1	NUM
cana-1220	55	8	analytical	analytical	ADJ
cana-1220	55	9	evaluation	evaluation	NOUN
cana-1220	55	10	of	of	ADP
cana-1220	55	11	the	the	DET
cana-1220	55	12	𝐾1,5	𝐾1,5	NOUN
cana-1220	55	13	communications	communication	NOUN
cana-1220	55	14	on	on	ADP
cana-1220	55	15	applied	apply	VERB
cana-1220	55	16	nonlinear	nonlinear	ADJ
cana-1220	55	17	analysis	analysis	NOUN
cana-1220	55	18	issn	issn	NOUN
cana-1220	55	19	:	:	PUNCT
cana-1220	55	20	1074	1074	NUM
cana-1220	55	21	-	-	PUNCT
cana-1220	55	22	133x	133x	NUM
cana-1220	55	23	vol	vol	NOUN
cana-1220	55	24	31	31	NUM
cana-1220	55	25	no	no	NOUN
cana-1220	55	26	.	.	PUNCT
cana-1220	56	1	6s	6s	NUM
cana-1220	56	2	(	(	PUNCT
cana-1220	56	3	2024	2024	NUM
cana-1220	56	4	)	)	PUNCT
cana-1220	56	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	56	6	261	261	NUM
cana-1220	56	7	theorem	theorem	VERB
cana-1220	56	8	:	:	PUNCT
cana-1220	56	9	3.2	3.2	NUM
cana-1220	56	10	let	let	VERB
cana-1220	56	11	𝑛	𝑛	DET
cana-1220	56	12	≥	≥	PRON
cana-1220	56	13	5	5	NUM
cana-1220	56	14	be	be	AUX
cana-1220	56	15	a	a	DET
cana-1220	56	16	positive	positive	ADJ
cana-1220	56	17	integer	integer	NOUN
cana-1220	56	18	,	,	PUNCT
cana-1220	56	19	then	then	ADV
cana-1220	56	20	χpg(𝑃𝑛)=	χpg(𝑃𝑛)=	NOUN
cana-1220	56	21	4	4	NUM
cana-1220	56	22	.	.	PUNCT
cana-1220	57	1	proof	proof	NOUN
cana-1220	57	2	:	:	PUNCT
cana-1220	57	3	define	define	VERB
cana-1220	57	4	a	a	DET
cana-1220	57	5	proper	proper	ADJ
cana-1220	57	6	vertex	vertex	NOUN
cana-1220	57	7	coloring	color	VERB
cana-1220	57	8	ꞷ	ꞷ	NOUN
cana-1220	57	9	:	:	PUNCT
cana-1220	57	10	v(pn	v(pn	NOUN
cana-1220	57	11	)	)	PUNCT
cana-1220	57	12	→{c1	→{c1	PROPN
cana-1220	57	13	,	,	PUNCT
cana-1220	57	14	c2	c2	PROPN
cana-1220	57	15	,	,	PUNCT
cana-1220	57	16	c3	c3	PROPN
cana-1220	57	17	,	,	PUNCT
cana-1220	57	18	c4}for	c4}for	PROPN
cana-1220	57	19	1≤	1≤	NUM
cana-1220	57	20	i	i	PRON
cana-1220	57	21	≤	≤	PUNCT
cana-1220	57	22	n	n	PRON
cana-1220	57	23	ꞷ(vi	ꞷ(vi	NOUN
cana-1220	57	24	)	)	PUNCT
cana-1220	57	25	=	=	PRON
cana-1220	57	26	{	{	PUNCT
cana-1220	57	27	𝑐1	𝑐1	NOUN
cana-1220	57	28	,	,	PUNCT
cana-1220	57	29	for	for	ADP
cana-1220	57	30	i	i	PROPN
cana-1220	57	31	≡	≡	PROPN
cana-1220	57	32	0	0	PUNCT
cana-1220	57	33	mod	mod	ADJ
cana-1220	57	34	3	3	NUM
cana-1220	57	35	𝑐2	𝑐2	NOUN
cana-1220	57	36	,	,	PUNCT
cana-1220	57	37	for	for	ADP
cana-1220	57	38	𝑣1	𝑣1	NOUN
cana-1220	57	39	𝑐3	𝑐3	NOUN
cana-1220	57	40	,	,	PUNCT
cana-1220	57	41	for	for	ADP
cana-1220	57	42	i	i	PROPN
cana-1220	57	43	≡	≡	PROPN
cana-1220	57	44	2	2	NUM
cana-1220	57	45	mod	mod	NOUN
cana-1220	57	46	3	3	NUM
cana-1220	57	47	𝑐4	𝑐4	NOUN
cana-1220	57	48	,	,	PUNCT
cana-1220	57	49	for	for	ADP
cana-1220	57	50	i	i	PROPN
cana-1220	57	51	≡	≡	PROPN
cana-1220	57	52	1	1	NUM
cana-1220	57	53	mod	mod	NOUN
cana-1220	57	54	3	3	NUM
cana-1220	57	55	,	,	PUNCT
cana-1220	57	56	i	i	PRON
cana-1220	57	57	≠	≠	PROPN
cana-1220	57	58	1	1	NUM
cana-1220	57	59	it	it	PRON
cana-1220	57	60	is	be	AUX
cana-1220	57	61	clear	clear	ADJ
cana-1220	57	62	that	that	SCONJ
cana-1220	57	63	gcd	gcd	PROPN
cana-1220	57	64	(	(	PUNCT
cana-1220	57	65	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	57	66	)	)	PUNCT
cana-1220	57	67	,	,	PUNCT
cana-1220	57	68	ꞷ(vi+1	ꞷ(vi+1	ADJ
cana-1220	57	69	)	)	PUNCT
cana-1220	57	70	)	)	PUNCT
cana-1220	58	1	=	=	PUNCT
cana-1220	58	2	1	1	X
cana-1220	58	3	.	.	X
cana-1220	58	4	it	it	PRON
cana-1220	58	5	induce	induce	VERB
cana-1220	58	6	a	a	DET
cana-1220	58	7	proper	proper	ADJ
cana-1220	58	8	edge	edge	NOUN
cana-1220	58	9	coloring	color	VERB
cana-1220	58	10	ꞷ*:e(pn)→{c1	ꞷ*:e(pn)→{c1	PROPN
cana-1220	58	11	,	,	PUNCT
cana-1220	58	12	c2	c2	PROPN
cana-1220	58	13	,	,	PUNCT
cana-1220	58	14	c3	c3	PROPN
cana-1220	58	15	}	}	PUNCT
cana-1220	58	16	as	as	SCONJ
cana-1220	58	17	follows	follow	VERB
cana-1220	58	18	:	:	PUNCT
cana-1220	58	19	ꞷ	ꞷ	X
cana-1220	58	20	*	*	PUNCT
cana-1220	58	21	(	(	PUNCT
cana-1220	58	22	vivi+1	vivi+1	NOUN
cana-1220	58	23	)	)	PUNCT
cana-1220	58	24	=	=	NOUN
cana-1220	58	25	{	{	PUNCT
cana-1220	58	26	𝑐1	𝑐1	NOUN
cana-1220	58	27	,	,	PUNCT
cana-1220	58	28	for	for	ADP
cana-1220	58	29	i	i	PROPN
cana-1220	58	30	≡	≡	PROPN
cana-1220	58	31	1	1	NUM
cana-1220	58	32	mod	mod	PROPN
cana-1220	58	33	3	3	NUM
cana-1220	58	34	𝑐2	𝑐2	NOUN
cana-1220	58	35	,	,	PUNCT
cana-1220	58	36	for	for	ADP
cana-1220	58	37	i	i	PROPN
cana-1220	58	38	≡	≡	PROPN
cana-1220	58	39	2	2	NUM
cana-1220	58	40	mod	mod	NOUN
cana-1220	58	41	3	3	NUM
cana-1220	58	42	𝑐3	𝑐3	NOUN
cana-1220	58	43	,	,	PUNCT
cana-1220	58	44	for	for	ADP
cana-1220	58	45	i	i	PROPN
cana-1220	58	46	≡	≡	PROPN
cana-1220	58	47	0	0	PUNCT
cana-1220	58	48	mod	mod	NOUN
cana-1220	59	1	3	3	NUM
cana-1220	60	1	it	it	PRON
cana-1220	60	2	satisfy	satisfy	VERB
cana-1220	60	3	ꞷ	ꞷ	PROPN
cana-1220	60	4	∗	∗	NOUN
cana-1220	60	5	(	(	PUNCT
cana-1220	60	6	vivi+1)=	vivi+1)=	NOUN
cana-1220	60	7	|	|	ADV
cana-1220	60	8	ꞷ	ꞷ	X
cana-1220	60	9	(	(	PUNCT
cana-1220	60	10	vi	vi	PROPN
cana-1220	61	1	)	)	PUNCT
cana-1220	61	2	ꞷ	ꞷ	PROPN
cana-1220	61	3	(	(	PUNCT
cana-1220	61	4	vi+1)|	vi+1)|	PROPN
cana-1220	61	5	=	=	PROPN
cana-1220	61	6	c2	c2	PROPN
cana-1220	61	7	−	−	PROPN
cana-1220	62	1	c3	c3	PROPN
cana-1220	62	2	=	=	PROPN
cana-1220	62	3	c3	c3	PROPN
cana-1220	62	4	−	−	PROPN
cana-1220	62	5	c1	c1	PROPN
cana-1220	62	6	=	=	PROPN
cana-1220	62	7	c1	c1	PROPN
cana-1220	62	8	−	−	PROPN
cana-1220	62	9	c4	c4	NOUN
cana-1220	62	10	=	=	SYM
cana-1220	62	11	c4	c4	NOUN
cana-1220	62	12	−	−	PROPN
cana-1220	62	13	c3	c3	PROPN
cana-1220	62	14	=	=	PROPN
cana-1220	62	15	c1	c1	PROPN
cana-1220	62	16	,	,	PUNCT
cana-1220	62	17	c2	c2	PROPN
cana-1220	62	18	,	,	PUNCT
cana-1220	62	19	c3	c3	PROPN
cana-1220	62	20	,	,	PUNCT
cana-1220	62	21	c1	c1	PROPN
cana-1220	62	22	hence	hence	ADV
cana-1220	62	23	,	,	PUNCT
cana-1220	62	24	adjacent	adjacent	ADJ
cana-1220	62	25	vertices	vertex	NOUN
cana-1220	62	26	and	and	CCONJ
cana-1220	62	27	edges	edge	NOUN
cana-1220	62	28	receive	receive	VERB
cana-1220	62	29	distinct	distinct	ADJ
cana-1220	62	30	colors	color	NOUN
cana-1220	62	31	.	.	PUNCT
cana-1220	63	1	thus	thus	ADV
cana-1220	63	2	,	,	PUNCT
cana-1220	63	3	χpg(pn	χpg(pn	ADJ
cana-1220	63	4	)	)	PUNCT
cana-1220	63	5	≤4	≤4	NOUN
cana-1220	63	6	,	,	PUNCT
cana-1220	63	7	for	for	ADP
cana-1220	63	8	n	n	X
cana-1220	63	9	≥	≥	NUM
cana-1220	63	10	5	5	NUM
cana-1220	63	11	.	.	PUNCT
cana-1220	64	1	to	to	PART
cana-1220	64	2	prove	prove	VERB
cana-1220	64	3	χpg(pn)≥	χpg(pn)≥	NOUN
cana-1220	64	4	4	4	NUM
cana-1220	64	5	,	,	PUNCT
cana-1220	64	6	let	let	VERB
cana-1220	64	7	us	we	PRON
cana-1220	64	8	assume	assume	VERB
cana-1220	64	9	that	that	SCONJ
cana-1220	64	10	χpg(pn	χpg(pn	X
cana-1220	64	11	)	)	PUNCT
cana-1220	64	12	<	<	X
cana-1220	64	13	4	4	NUM
cana-1220	64	14	,	,	PUNCT
cana-1220	64	15	say	say	VERB
cana-1220	64	16	3	3	X
cana-1220	64	17	.	.	X
cana-1220	65	1	we	we	PRON
cana-1220	65	2	define	define	VERB
cana-1220	65	3	vertex	vertex	NOUN
cana-1220	65	4	coloring	coloring	NOUN
cana-1220	65	5	of	of	ADP
cana-1220	65	6	p5	p5	PROPN
cana-1220	65	7	is	be	AUX
cana-1220	65	8	c2	c2	PROPN
cana-1220	65	9	,	,	PUNCT
cana-1220	65	10	c1	c1	PROPN
cana-1220	65	11	,	,	PUNCT
cana-1220	65	12	c3	c3	PROPN
cana-1220	65	13	,	,	PUNCT
cana-1220	65	14	c2	c2	PROPN
cana-1220	65	15	,	,	PUNCT
cana-1220	65	16	c1	c1	PROPN
cana-1220	65	17	.	.	PUNCT
cana-1220	66	1	then	then	ADV
cana-1220	66	2	,	,	PUNCT
cana-1220	66	3	ꞷ	ꞷ	PROPN
cana-1220	66	4	∗	∗	NOUN
cana-1220	66	5	(	(	PUNCT
cana-1220	66	6	v1v2	v1v2	X
cana-1220	66	7	)	)	PUNCT
cana-1220	66	8	=	=	NOUN
cana-1220	67	1	𝑐1	𝑐1	NOUN
cana-1220	67	2	,	,	PUNCT
cana-1220	67	3	ꞷ	ꞷ	PROPN
cana-1220	67	4	∗	∗	NOUN
cana-1220	67	5	(	(	PUNCT
cana-1220	67	6	v2v3	v2v3	NOUN
cana-1220	67	7	)	)	PUNCT
cana-1220	67	8	=	=	SYM
cana-1220	67	9	𝑐2	𝑐2	NOUN
cana-1220	67	10	,	,	PUNCT
cana-1220	67	11	ꞷ	ꞷ	PROPN
cana-1220	67	12	∗	∗	X
cana-1220	67	13	(	(	PUNCT
cana-1220	67	14	v3v4	v3v4	NOUN
cana-1220	67	15	)	)	PUNCT
cana-1220	68	1	=	=	NOUN
cana-1220	68	2	𝑐1	𝑐1	NOUN
cana-1220	68	3	,	,	PUNCT
cana-1220	68	4	ꞷ	ꞷ	PROPN
cana-1220	68	5	∗	∗	NOUN
cana-1220	68	6	(	(	PUNCT
cana-1220	68	7	v4v5	v4v5	NOUN
cana-1220	68	8	)	)	PUNCT
cana-1220	68	9	=	=	NOUN
cana-1220	69	1	𝑐1	𝑐1	NOUN
cana-1220	69	2	which	which	PRON
cana-1220	69	3	is	be	AUX
cana-1220	69	4	a	a	DET
cana-1220	69	5	contradiction	contradiction	NOUN
cana-1220	69	6	with	with	ADP
cana-1220	69	7	the	the	DET
cana-1220	69	8	definition	definition	NOUN
cana-1220	69	9	of	of	ADP
cana-1220	69	10	prime	prime	ADJ
cana-1220	69	11	graceful	graceful	ADJ
cana-1220	69	12	coloring	coloring	NOUN
cana-1220	69	13	since	since	SCONJ
cana-1220	69	14	the	the	DET
cana-1220	69	15	color	color	NOUN
cana-1220	69	16	of	of	ADP
cana-1220	69	17	any	any	DET
cana-1220	69	18	two	two	NUM
cana-1220	69	19	adjacent	adjacent	ADJ
cana-1220	69	20	edges	edge	NOUN
cana-1220	69	21	are	be	AUX
cana-1220	69	22	distinct	distinct	ADJ
cana-1220	69	23	.	.	PUNCT
cana-1220	70	1	thus	thus	ADV
cana-1220	70	2	,	,	PUNCT
cana-1220	70	3	χpg(pn)≥	χpg(pn)≥	VERB
cana-1220	70	4	4	4	NUM
cana-1220	70	5	.	.	PUNCT
cana-1220	71	1	therefore	therefore	ADV
cana-1220	71	2	,	,	PUNCT
cana-1220	71	3	χpg(pn)=	χpg(pn)=	ADV
cana-1220	71	4	4	4	NUM
cana-1220	71	5	,	,	PUNCT
cana-1220	71	6	if	if	SCONJ
cana-1220	71	7	n	n	PRON
cana-1220	71	8	≥	≥	NOUN
cana-1220	71	9	5	5	NUM
cana-1220	71	10	.	.	PUNCT
cana-1220	71	11	corollary	corollary	ADJ
cana-1220	71	12	:	:	PUNCT
cana-1220	71	13	3.3	3.3	NUM
cana-1220	71	14	for	for	ADP
cana-1220	71	15	any	any	DET
cana-1220	71	16	path	path	NOUN
cana-1220	71	17	,	,	PUNCT
cana-1220	71	18	χpg(pn)=	χpg(pn)=	ADV
cana-1220	71	19	2	2	NUM
cana-1220	71	20	,	,	PUNCT
cana-1220	71	21	if	if	SCONJ
cana-1220	71	22	n	n	NOUN
cana-1220	71	23	=	=	SYM
cana-1220	71	24	2	2	X
cana-1220	71	25	.	.	X
cana-1220	71	26	corollary	corollary	ADJ
cana-1220	71	27	:	:	PUNCT
cana-1220	71	28	3.4	3.4	NUM
cana-1220	71	29	for	for	ADP
cana-1220	71	30	any	any	DET
cana-1220	71	31	path	path	NOUN
cana-1220	71	32	,	,	PUNCT
cana-1220	71	33	χpg(pn)=	χpg(pn)=	ADV
cana-1220	71	34	3	3	NUM
cana-1220	71	35	,	,	PUNCT
cana-1220	71	36	if	if	SCONJ
cana-1220	71	37	n	n	NOUN
cana-1220	71	38	=	=	SYM
cana-1220	71	39	3	3	NUM
cana-1220	71	40	and	and	CCONJ
cana-1220	71	41	4	4	NUM
cana-1220	71	42	.	.	X
cana-1220	71	43	figure	figure	VERB
cana-1220	71	44	2	2	NUM
cana-1220	71	45	analytical	analytical	ADJ
cana-1220	71	46	evaluation	evaluation	NOUN
cana-1220	71	47	of	of	ADP
cana-1220	71	48	the	the	DET
cana-1220	71	49	𝑃7	𝑃7	NOUN
cana-1220	71	50	theorem	theorem	VERB
cana-1220	71	51	:	:	PUNCT
cana-1220	71	52	3.5	3.5	NUM
cana-1220	71	53	let	let	VERB
cana-1220	71	54	n	n	PRON
cana-1220	71	55	≥	≥	NUM
cana-1220	71	56	5	5	NUM
cana-1220	71	57	be	be	AUX
cana-1220	71	58	a	a	DET
cana-1220	71	59	positive	positive	ADJ
cana-1220	71	60	integer	integer	NOUN
cana-1220	71	61	,	,	PUNCT
cana-1220	71	62	then	then	ADV
cana-1220	71	63	χpg(cn)=	χpg(cn)=	ADP
cana-1220	71	64	5	5	NUM
cana-1220	71	65	.	.	X
cana-1220	72	1	proof	proof	NOUN
cana-1220	72	2	:	:	PUNCT
cana-1220	72	3	the	the	DET
cana-1220	72	4	vertices	vertex	NOUN
cana-1220	72	5	and	and	CCONJ
cana-1220	72	6	edges	edge	NOUN
cana-1220	72	7	of	of	ADP
cana-1220	72	8	cn	cn	PROPN
cana-1220	72	9	are	be	AUX
cana-1220	72	10	v	v	X
cana-1220	72	11	(	(	PUNCT
cana-1220	72	12	cn	cn	NOUN
cana-1220	72	13	)	)	PUNCT
cana-1220	72	14	=	=	NOUN
cana-1220	72	15	{	{	PUNCT
cana-1220	72	16	vi	vi	PROPN
cana-1220	72	17	:1	:1	PUNCT
cana-1220	72	18	≤	≤	NUM
cana-1220	72	19	i	i	PRON
cana-1220	72	20	≤	≤	NOUN
cana-1220	72	21	n	n	CCONJ
cana-1220	72	22	}	}	PUNCT
cana-1220	72	23	and	and	CCONJ
cana-1220	72	24	e(cn)={vivi+1	e(cn)={vivi+1	NOUN
cana-1220	72	25	:	:	PUNCT
cana-1220	72	26	1	1	NUM
cana-1220	72	27	≤	≤	NUM
cana-1220	72	28	i	i	PRON
cana-1220	72	29	≤	≤	NOUN
cana-1220	72	30	n	n	CCONJ
cana-1220	72	31	}	}	PUNCT
cana-1220	72	32	.	.	PUNCT
cana-1220	73	1	case	case	NOUN
cana-1220	73	2	1	1	NUM
cana-1220	73	3	:	:	PUNCT
cana-1220	73	4	n	n	NUM
cana-1220	73	5	≠	≠	PROPN
cana-1220	73	6	3m+3	3m+3	PROPN
cana-1220	73	7	,	,	PUNCT
cana-1220	73	8	m∈n	m∈n	NOUN
cana-1220	73	9	define	define	VERB
cana-1220	73	10	a	a	DET
cana-1220	73	11	proper	proper	ADJ
cana-1220	73	12	vertex	vertex	NOUN
cana-1220	73	13	coloring	color	VERB
cana-1220	73	14	ꞷ	ꞷ	NOUN
cana-1220	73	15	:	:	PUNCT
cana-1220	73	16	v(cn	v(cn	NUM
cana-1220	73	17	)	)	PUNCT
cana-1220	73	18	→{c1	→{c1	PROPN
cana-1220	73	19	,	,	PUNCT
cana-1220	73	20	c2	c2	PROPN
cana-1220	73	21	,	,	PUNCT
cana-1220	73	22	c3	c3	PROPN
cana-1220	73	23	,	,	PUNCT
cana-1220	73	24	c4	c4	NOUN
cana-1220	73	25	,	,	PUNCT
cana-1220	73	26	c5}as	c5}as	PRON
cana-1220	73	27	follows	follow	VERB
cana-1220	73	28	:	:	PUNCT
cana-1220	73	29	for	for	ADP
cana-1220	73	30	1≤	1≤	NUM
cana-1220	74	1	i	i	NOUN
cana-1220	74	2	≤	≤	PUNCT
cana-1220	74	3	n	n	PRON
cana-1220	74	4	ꞷ(vi	ꞷ(vi	NOUN
cana-1220	74	5	)	)	PUNCT
cana-1220	74	6	=	=	PRON
cana-1220	74	7	{	{	PUNCT
cana-1220	74	8	𝑐1	𝑐1	NOUN
cana-1220	74	9	,	,	PUNCT
cana-1220	74	10	for	for	ADP
cana-1220	74	11	𝑣2	𝑣2	NOUN
cana-1220	74	12	𝑐2	𝑐2	NOUN
cana-1220	74	13	,	,	PUNCT
cana-1220	74	14	for	for	ADP
cana-1220	74	15	i	i	PROPN
cana-1220	74	16	≡	≡	PROPN
cana-1220	74	17	0	0	PUNCT
cana-1220	74	18	mod	mod	ADJ
cana-1220	74	19	3	3	NUM
cana-1220	74	20	𝑐3	𝑐3	NOUN
cana-1220	74	21	,	,	PUNCT
cana-1220	74	22	for	for	ADP
cana-1220	74	23	i	i	PROPN
cana-1220	74	24	≡	≡	PROPN
cana-1220	74	25	2	2	NUM
cana-1220	74	26	mod	mod	NOUN
cana-1220	74	27	3	3	NUM
cana-1220	75	1	and	and	CCONJ
cana-1220	75	2	i	i	PRON
cana-1220	75	3	≠	≠	ADJ
cana-1220	75	4	2	2	NUM
cana-1220	75	5	𝑐4	𝑐4	NOUN
cana-1220	75	6	,	,	PUNCT
cana-1220	75	7	for	for	ADP
cana-1220	75	8	𝑣1	𝑣1	NOUN
cana-1220	75	9	𝑐5	𝑐5	NOUN
cana-1220	75	10	,	,	PUNCT
cana-1220	75	11	for	for	ADP
cana-1220	75	12	i	i	PRON
cana-1220	75	13	≡	≡	PROPN
cana-1220	75	14	1	1	NUM
cana-1220	75	15	mod	mod	NOUN
cana-1220	75	16	3	3	NUM
cana-1220	75	17	and	and	CCONJ
cana-1220	75	18	i	i	PRON
cana-1220	75	19	≠	≠	PROPN
cana-1220	75	20	1	1	NUM
cana-1220	75	21	such	such	ADJ
cana-1220	75	22	that	that	DET
cana-1220	75	23	gcd	gcd	NOUN
cana-1220	75	24	(	(	PUNCT
cana-1220	75	25	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	75	26	)	)	PUNCT
cana-1220	75	27	,	,	PUNCT
cana-1220	75	28	ꞷ(vi+1	ꞷ(vi+1	ADJ
cana-1220	75	29	)	)	PUNCT
cana-1220	75	30	)	)	PUNCT
cana-1220	76	1	=	=	PUNCT
cana-1220	76	2	1	1	X
cana-1220	76	3	.	.	X
cana-1220	76	4	it	it	PRON
cana-1220	76	5	induce	induce	VERB
cana-1220	76	6	a	a	DET
cana-1220	76	7	proper	proper	ADJ
cana-1220	76	8	edge	edge	NOUN
cana-1220	76	9	coloring	color	VERB
cana-1220	76	10	ꞷ*:e(cn)→{c1	ꞷ*:e(cn)→{c1	PROPN
cana-1220	76	11	,	,	PUNCT
cana-1220	76	12	c2	c2	PROPN
cana-1220	76	13	,	,	PUNCT
cana-1220	76	14	c3	c3	PROPN
cana-1220	76	15	}	}	PUNCT
cana-1220	76	16	as	as	SCONJ
cana-1220	76	17	follows	follow	VERB
cana-1220	76	18	:	:	PUNCT
cana-1220	76	19	communications	communication	NOUN
cana-1220	76	20	on	on	ADP
cana-1220	76	21	applied	apply	VERB
cana-1220	76	22	nonlinear	nonlinear	ADJ
cana-1220	76	23	analysis	analysis	NOUN
cana-1220	76	24	issn	issn	NOUN
cana-1220	76	25	:	:	PUNCT
cana-1220	76	26	1074	1074	NUM
cana-1220	76	27	-	-	PUNCT
cana-1220	76	28	133x	133x	NUM
cana-1220	76	29	vol	vol	NOUN
cana-1220	76	30	31	31	NUM
cana-1220	76	31	no	no	NOUN
cana-1220	76	32	.	.	PUNCT
cana-1220	77	1	6s	6s	NUM
cana-1220	77	2	(	(	PUNCT
cana-1220	77	3	2024	2024	NUM
cana-1220	77	4	)	)	PUNCT
cana-1220	77	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	77	6	262	262	NUM
cana-1220	77	7	ꞷ	ꞷ	X
cana-1220	77	8	*	*	PUNCT
cana-1220	77	9	(	(	PUNCT
cana-1220	77	10	vivi+1	vivi+1	NOUN
cana-1220	77	11	)	)	PUNCT
cana-1220	77	12	=	=	NOUN
cana-1220	77	13	{	{	PUNCT
cana-1220	77	14	𝑐3	𝑐3	NOUN
cana-1220	77	15	,	,	PUNCT
cana-1220	77	16	for	for	ADP
cana-1220	77	17	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1220	77	18	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1220	78	1	i	i	PROPN
cana-1220	78	2	≡	≡	PROPN
cana-1220	78	3	0	0	NUM
cana-1220	78	4	mod	mod	ADJ
cana-1220	78	5	3	3	NUM
cana-1220	78	6	𝑐2	𝑐2	NOUN
cana-1220	78	7	,	,	PUNCT
cana-1220	78	8	for	for	ADP
cana-1220	78	9	i	i	PRON
cana-1220	78	10	≡	≡	PROPN
cana-1220	78	11	1	1	NUM
cana-1220	78	12	mod	mod	NOUN
cana-1220	78	13	3	3	NUM
cana-1220	78	14	and	and	CCONJ
cana-1220	78	15	i	i	PRON
cana-1220	78	16	≠	≠	PROPN
cana-1220	78	17	1	1	NUM
cana-1220	78	18	𝑐1	𝑐1	NOUN
cana-1220	78	19	,	,	PUNCT
cana-1220	78	20	for	for	ADP
cana-1220	78	21	𝑣𝑛𝑣1	𝑣𝑛𝑣1	NOUN
cana-1220	78	22	and	and	CCONJ
cana-1220	78	23	i	i	PRON
cana-1220	78	24	≡	≡	PROPN
cana-1220	78	25	2	2	NUM
cana-1220	78	26	mod	mod	NOUN
cana-1220	78	27	3	3	NUM
cana-1220	78	28	it	it	PRON
cana-1220	78	29	satisfy	satisfy	VERB
cana-1220	78	30	ꞷ	ꞷ	PROPN
cana-1220	78	31	∗	∗	NOUN
cana-1220	78	32	(	(	PUNCT
cana-1220	78	33	v1v2)=	v1v2)=	NOUN
cana-1220	78	34	|	|	INTJ
cana-1220	78	35	ꞷ	ꞷ	NOUN
cana-1220	78	36	(	(	PUNCT
cana-1220	78	37	v1	v1	PROPN
cana-1220	78	38	)	)	PUNCT
cana-1220	78	39	ꞷ	ꞷ	PROPN
cana-1220	79	1	(	(	PUNCT
cana-1220	79	2	𝑣2)|	𝑣2)|	NOUN
cana-1220	79	3	=	=	SYM
cana-1220	79	4	𝑐4	𝑐4	PROPN
cana-1220	79	5	−	−	NOUN
cana-1220	79	6	𝑐1	𝑐1	NOUN
cana-1220	79	7	=	=	SYM
cana-1220	79	8	𝑐3	𝑐3	NOUN
cana-1220	79	9	ꞷ	ꞷ	PROPN
cana-1220	79	10	∗	∗	NOUN
cana-1220	79	11	(	(	PUNCT
cana-1220	79	12	v2v3)=	v2v3)=	NOUN
cana-1220	80	1	|	|	ADV
cana-1220	81	1	ꞷ	ꞷ	PROPN
cana-1220	82	1	(	(	PUNCT
cana-1220	82	2	v2	v2	PROPN
cana-1220	82	3	)	)	PUNCT
cana-1220	82	4	ꞷ	ꞷ	PROPN
cana-1220	83	1	(	(	PUNCT
cana-1220	83	2	𝑣3)|	𝑣3)|	NUM
cana-1220	83	3	=	=	NOUN
cana-1220	83	4	𝑐1	𝑐1	NOUN
cana-1220	83	5	−	−	NOUN
cana-1220	83	6	𝑐2	𝑐2	NOUN
cana-1220	83	7	=	=	NOUN
cana-1220	84	1	𝑐1	𝑐1	NOUN
cana-1220	84	2	ꞷ	ꞷ	X
cana-1220	84	3	∗	∗	NOUN
cana-1220	84	4	(	(	PUNCT
cana-1220	84	5	v3v4)=	v3v4)=	X
cana-1220	85	1	|	|	NOUN
cana-1220	85	2	ꞷ	ꞷ	PROPN
cana-1220	85	3	(	(	PUNCT
cana-1220	85	4	v3	v3	PROPN
cana-1220	85	5	)	)	PUNCT
cana-1220	85	6	ꞷ	ꞷ	PROPN
cana-1220	86	1	(	(	PUNCT
cana-1220	86	2	𝑣4)|	𝑣4)|	NOUN
cana-1220	86	3	=	=	PUNCT
cana-1220	86	4	𝑐2	𝑐2	NOUN
cana-1220	86	5	−	−	NOUN
cana-1220	86	6	𝑐5	𝑐5	NOUN
cana-1220	86	7	=	=	SYM
cana-1220	86	8	𝑐3	𝑐3	NOUN
cana-1220	86	9	ꞷ	ꞷ	PROPN
cana-1220	86	10	∗	∗	NOUN
cana-1220	86	11	(	(	PUNCT
cana-1220	86	12	v4v5)=	v4v5)=	PROPN
cana-1220	87	1	|	|	NOUN
cana-1220	87	2	ꞷ	ꞷ	PROPN
cana-1220	87	3	(	(	PUNCT
cana-1220	87	4	v4	v4	PROPN
cana-1220	87	5	)	)	PUNCT
cana-1220	87	6	ꞷ	ꞷ	PROPN
cana-1220	87	7	(	(	PUNCT
cana-1220	87	8	𝑣5)|	𝑣5)|	PROPN
cana-1220	87	9	=	=	SYM
cana-1220	87	10	𝑐5	𝑐5	PROPN
cana-1220	87	11	−	−	PROPN
cana-1220	87	12	𝑐3	𝑐3	NOUN
cana-1220	88	1	=	=	NOUN
cana-1220	88	2	𝑐2	𝑐2	NOUN
cana-1220	88	3	hence	hence	ADV
cana-1220	88	4	,	,	PUNCT
cana-1220	88	5	adjacent	adjacent	ADJ
cana-1220	88	6	vertices	vertex	NOUN
cana-1220	88	7	and	and	CCONJ
cana-1220	88	8	edges	edge	NOUN
cana-1220	88	9	receive	receive	VERB
cana-1220	88	10	distinct	distinct	ADJ
cana-1220	88	11	colors	color	NOUN
cana-1220	88	12	.	.	PUNCT
cana-1220	89	1	thus	thus	ADV
cana-1220	89	2	,	,	PUNCT
cana-1220	89	3	χpg(cn)≤5	χpg(cn)≤5	PROPN
cana-1220	89	4	.	.	PUNCT
cana-1220	90	1	to	to	PART
cana-1220	90	2	prove	prove	VERB
cana-1220	90	3	χpg(cn)≥	χpg(cn)≥	NOUN
cana-1220	90	4	5	5	NUM
cana-1220	90	5	,	,	PUNCT
cana-1220	90	6	let	let	VERB
cana-1220	90	7	us	we	PRON
cana-1220	90	8	assume	assume	VERB
cana-1220	90	9	that	that	SCONJ
cana-1220	90	10	χpg(cn	χpg(cn	AUX
cana-1220	90	11	)	)	PUNCT
cana-1220	90	12	<	<	X
cana-1220	90	13	5	5	NUM
cana-1220	90	14	,	,	PUNCT
cana-1220	90	15	say	say	VERB
cana-1220	90	16	4	4	X
cana-1220	90	17	.	.	X
cana-1220	91	1	we	we	PRON
cana-1220	91	2	define	define	VERB
cana-1220	91	3	vertex	vertex	NOUN
cana-1220	91	4	coloring	coloring	NOUN
cana-1220	91	5	of	of	ADP
cana-1220	91	6	c5	c5	PROPN
cana-1220	91	7	is	be	AUX
cana-1220	91	8	c4	c4	NOUN
cana-1220	91	9	,	,	PUNCT
cana-1220	91	10	c1	c1	PROPN
cana-1220	91	11	,	,	PUNCT
cana-1220	91	12	c2	c2	PROPN
cana-1220	91	13	,	,	PUNCT
cana-1220	91	14	c3	c3	PROPN
cana-1220	91	15	,	,	PUNCT
cana-1220	91	16	c1	c1	PROPN
cana-1220	91	17	.	.	PUNCT
cana-1220	92	1	then	then	ADV
cana-1220	92	2	,	,	PUNCT
cana-1220	92	3	ꞷ	ꞷ	PROPN
cana-1220	92	4	∗	∗	NOUN
cana-1220	92	5	(	(	PUNCT
cana-1220	92	6	v1v2)=	v1v2)=	NOUN
cana-1220	92	7	𝑐3	𝑐3	NOUN
cana-1220	92	8	,	,	PUNCT
cana-1220	92	9	ꞷ	ꞷ	NOUN
cana-1220	92	10	∗	∗	NOUN
cana-1220	92	11	(	(	PUNCT
cana-1220	92	12	v2v3)=	v2v3)=	ADP
cana-1220	92	13	𝑐1	𝑐1	NOUN
cana-1220	92	14	,	,	PUNCT
cana-1220	92	15	ꞷ	ꞷ	PROPN
cana-1220	92	16	∗	∗	NOUN
cana-1220	92	17	(	(	PUNCT
cana-1220	92	18	v3v4)=	v3v4)=	ADJ
cana-1220	92	19	𝑐1	𝑐1	NOUN
cana-1220	92	20	,	,	PUNCT
cana-1220	92	21	ꞷ	ꞷ	PROPN
cana-1220	92	22	∗	∗	NOUN
cana-1220	92	23	(	(	PUNCT
cana-1220	92	24	v4v5)=	v4v5)=	NOUN
cana-1220	92	25	𝑐2	𝑐2	PROPN
cana-1220	92	26	,	,	PUNCT
cana-1220	92	27	ꞷ	ꞷ	PROPN
cana-1220	92	28	∗	∗	NOUN
cana-1220	92	29	(	(	PUNCT
cana-1220	92	30	v5v1	v5v1	NOUN
cana-1220	92	31	)	)	PUNCT
cana-1220	92	32	=	=	NOUN
cana-1220	92	33	𝑐3	𝑐3	NOUN
cana-1220	92	34	.	.	PUNCT
cana-1220	93	1	since	since	SCONJ
cana-1220	93	2	ꞷ	ꞷ	PROPN
cana-1220	93	3	∗	∗	X
cana-1220	93	4	(	(	PUNCT
cana-1220	93	5	v2v3	v2v3	NOUN
cana-1220	93	6	)	)	PUNCT
cana-1220	93	7	=	=	NOUN
cana-1220	93	8	𝑐1	𝑐1	NOUN
cana-1220	93	9	and	and	CCONJ
cana-1220	93	10	ꞷ	ꞷ	PRON
cana-1220	93	11	∗	∗	NOUN
cana-1220	93	12	(	(	PUNCT
cana-1220	93	13	v3v4	v3v4	NOUN
cana-1220	93	14	)	)	PUNCT
cana-1220	93	15	=	=	NOUN
cana-1220	94	1	𝑐1	𝑐1	NOUN
cana-1220	94	2	receives	receive	VERB
cana-1220	94	3	same	same	ADJ
cana-1220	94	4	color	color	NOUN
cana-1220	94	5	which	which	PRON
cana-1220	94	6	is	be	AUX
cana-1220	94	7	a	a	DET
cana-1220	94	8	contradiction	contradiction	NOUN
cana-1220	94	9	with	with	ADP
cana-1220	94	10	the	the	DET
cana-1220	94	11	definition	definition	NOUN
cana-1220	94	12	of	of	ADP
cana-1220	94	13	prime	prime	ADJ
cana-1220	94	14	graceful	graceful	ADJ
cana-1220	94	15	coloring	coloring	NOUN
cana-1220	94	16	since	since	SCONJ
cana-1220	94	17	the	the	DET
cana-1220	94	18	color	color	NOUN
cana-1220	94	19	of	of	ADP
cana-1220	94	20	any	any	DET
cana-1220	94	21	two	two	NUM
cana-1220	94	22	adjacent	adjacent	ADJ
cana-1220	94	23	edges	edge	NOUN
cana-1220	94	24	are	be	AUX
cana-1220	94	25	distinct	distinct	ADJ
cana-1220	94	26	.	.	PUNCT
cana-1220	95	1	thus	thus	ADV
cana-1220	95	2	,	,	PUNCT
cana-1220	95	3	χpg(cn)≥	χpg(cn)≥	NOUN
cana-1220	95	4	5	5	NUM
cana-1220	95	5	.	.	PUNCT
cana-1220	96	1	therefore	therefore	ADV
cana-1220	96	2	,	,	PUNCT
cana-1220	96	3	χpg(cn)=	χpg(cn)=	ADV
cana-1220	96	4	5	5	NUM
cana-1220	96	5	for	for	ADP
cana-1220	96	6	n	n	DET
cana-1220	96	7	≠3m+3	≠3m+3	NOUN
cana-1220	96	8	,	,	PUNCT
cana-1220	96	9	m∈n	m∈n	PROPN
cana-1220	96	10	.	.	PUNCT
cana-1220	97	1	case	case	NOUN
cana-1220	97	2	2	2	NUM
cana-1220	97	3	:	:	SYM
cana-1220	97	4	n	n	PROPN
cana-1220	97	5	=	=	SYM
cana-1220	97	6	3	3	NUM
cana-1220	97	7	m	m	NOUN
cana-1220	97	8	+	+	NOUN
cana-1220	97	9	3	3	NUM
cana-1220	97	10	define	define	VERB
cana-1220	97	11	a	a	DET
cana-1220	97	12	proper	proper	ADJ
cana-1220	97	13	vertex	vertex	NOUN
cana-1220	97	14	coloring	color	VERB
cana-1220	97	15	ꞷ	ꞷ	NOUN
cana-1220	97	16	:	:	PUNCT
cana-1220	97	17	v(cn	v(cn	NUM
cana-1220	97	18	)	)	PUNCT
cana-1220	97	19	→{c1	→{c1	PROPN
cana-1220	97	20	,	,	PUNCT
cana-1220	97	21	c2	c2	PROPN
cana-1220	97	22	,	,	PUNCT
cana-1220	97	23	c3	c3	PROPN
cana-1220	97	24	,	,	PUNCT
cana-1220	97	25	c4	c4	NOUN
cana-1220	97	26	,	,	PUNCT
cana-1220	97	27	c5}as	c5}as	PRON
cana-1220	97	28	follows	follow	VERB
cana-1220	97	29	:	:	PUNCT
cana-1220	97	30	for	for	ADP
cana-1220	97	31	1≤	1≤	NUM
cana-1220	98	1	i	i	NOUN
cana-1220	98	2	≤	≤	PUNCT
cana-1220	98	3	n	n	PRON
cana-1220	98	4	ꞷ(vi	ꞷ(vi	NOUN
cana-1220	98	5	)	)	PUNCT
cana-1220	98	6	=	=	PRON
cana-1220	98	7	{	{	PUNCT
cana-1220	98	8	𝑐1	𝑐1	NOUN
cana-1220	98	9	,	,	PUNCT
cana-1220	98	10	for	for	ADP
cana-1220	98	11	𝑣2	𝑣2	NOUN
cana-1220	98	12	𝑐2	𝑐2	NOUN
cana-1220	98	13	,	,	PUNCT
cana-1220	98	14	for	for	ADP
cana-1220	98	15	i	i	PRON
cana-1220	98	16	≡	≡	PROPN
cana-1220	98	17	1	1	NUM
cana-1220	98	18	mod	mod	NOUN
cana-1220	98	19	3	3	NUM
cana-1220	99	1	and	and	CCONJ
cana-1220	99	2	i	i	PRON
cana-1220	99	3	≠	≠	ADJ
cana-1220	99	4	1	1	NUM
cana-1220	99	5	𝑐3	𝑐3	NOUN
cana-1220	99	6	,	,	PUNCT
cana-1220	99	7	for	for	ADP
cana-1220	99	8	i	i	PROPN
cana-1220	99	9	≡	≡	PROPN
cana-1220	99	10	2	2	NUM
cana-1220	99	11	mod	mod	NOUN
cana-1220	99	12	3	3	NUM
cana-1220	99	13	and	and	CCONJ
cana-1220	99	14	i	i	PRON
cana-1220	99	15	≠	≠	ADJ
cana-1220	99	16	2	2	NUM
cana-1220	99	17	𝑐4	𝑐4	NOUN
cana-1220	99	18	,	,	PUNCT
cana-1220	99	19	for	for	ADP
cana-1220	99	20	𝑣1	𝑣1	NOUN
cana-1220	99	21	𝑐5	𝑐5	NOUN
cana-1220	99	22	,	,	PUNCT
cana-1220	99	23	for	for	ADP
cana-1220	99	24	i	i	PROPN
cana-1220	99	25	≡	≡	PROPN
cana-1220	99	26	0	0	PUNCT
cana-1220	99	27	mod	mod	PROPN
cana-1220	99	28	3	3	NUM
cana-1220	99	29	such	such	ADJ
cana-1220	99	30	that	that	DET
cana-1220	99	31	gcd	gcd	NOUN
cana-1220	99	32	(	(	PUNCT
cana-1220	99	33	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	99	34	)	)	PUNCT
cana-1220	99	35	,	,	PUNCT
cana-1220	99	36	ꞷ(vi+1	ꞷ(vi+1	ADJ
cana-1220	99	37	)	)	PUNCT
cana-1220	99	38	)	)	PUNCT
cana-1220	100	1	=	=	PUNCT
cana-1220	100	2	1	1	X
cana-1220	100	3	.	.	X
cana-1220	100	4	it	it	PRON
cana-1220	100	5	induce	induce	VERB
cana-1220	100	6	a	a	DET
cana-1220	100	7	proper	proper	ADJ
cana-1220	100	8	edge	edge	NOUN
cana-1220	100	9	coloring	color	VERB
cana-1220	100	10	ꞷ*:e(cn)→{c1	ꞷ*:e(cn)→{c1	PROPN
cana-1220	100	11	,	,	PUNCT
cana-1220	100	12	c2	c2	PROPN
cana-1220	100	13	,	,	PUNCT
cana-1220	100	14	c3	c3	PROPN
cana-1220	100	15	}	}	PUNCT
cana-1220	100	16	as	as	SCONJ
cana-1220	100	17	follows	follow	VERB
cana-1220	100	18	:	:	PUNCT
cana-1220	100	19	ꞷ	ꞷ	X
cana-1220	100	20	*	*	PUNCT
cana-1220	100	21	(	(	PUNCT
cana-1220	100	22	vivi+1	vivi+1	NOUN
cana-1220	100	23	)	)	PUNCT
cana-1220	100	24	=	=	PRON
cana-1220	100	25	{	{	PUNCT
cana-1220	100	26	𝑐1	𝑐1	NOUN
cana-1220	100	27	,	,	PUNCT
cana-1220	100	28	for	for	ADP
cana-1220	100	29	𝑣𝑛𝑣1	𝑣𝑛𝑣1	PROPN
cana-1220	100	30	,	,	PUNCT
cana-1220	100	31	i	i	PRON
cana-1220	100	32	≡	≡	PROPN
cana-1220	100	33	1	1	NUM
cana-1220	100	34	mod	mod	NOUN
cana-1220	100	35	3	3	NUM
cana-1220	101	1	and	and	CCONJ
cana-1220	101	2	i	i	PRON
cana-1220	101	3	≠	≠	PROPN
cana-1220	101	4	1	1	NUM
cana-1220	101	5	𝑐2	𝑐2	NOUN
cana-1220	101	6	,	,	PUNCT
cana-1220	101	7	for	for	ADP
cana-1220	101	8	i	i	PROPN
cana-1220	101	9	≡	≡	PROPN
cana-1220	101	10	2	2	NUM
cana-1220	101	11	mod	mod	NOUN
cana-1220	101	12	3	3	NUM
cana-1220	101	13	and	and	CCONJ
cana-1220	101	14	i	i	PRON
cana-1220	101	15	≠	≠	ADJ
cana-1220	101	16	2	2	NUM
cana-1220	101	17	𝑐3	𝑐3	NOUN
cana-1220	101	18	,	,	PUNCT
cana-1220	101	19	for	for	ADP
cana-1220	101	20	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1220	101	21	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1220	101	22	i	i	PROPN
cana-1220	101	23	≡	≡	PROPN
cana-1220	101	24	0	0	NUM
cana-1220	101	25	mod	mod	ADJ
cana-1220	101	26	3	3	NUM
cana-1220	101	27	𝑐4	𝑐4	NOUN
cana-1220	101	28	,	,	PUNCT
cana-1220	101	29	for	for	ADP
cana-1220	101	30	𝑣2𝑣3	𝑣2𝑣3	PRON
cana-1220	101	31	it	it	PRON
cana-1220	101	32	satisfy	satisfy	VERB
cana-1220	101	33	ꞷ	ꞷ	PROPN
cana-1220	101	34	∗	∗	NOUN
cana-1220	101	35	(	(	PUNCT
cana-1220	101	36	v1v2)=	v1v2)=	NOUN
cana-1220	102	1	|	|	INTJ
cana-1220	102	2	ꞷ	ꞷ	NOUN
cana-1220	102	3	(	(	PUNCT
cana-1220	102	4	v1	v1	PROPN
cana-1220	102	5	)	)	PUNCT
cana-1220	102	6	ꞷ	ꞷ	PROPN
cana-1220	103	1	(	(	PUNCT
cana-1220	103	2	𝑣2)|	𝑣2)|	NOUN
cana-1220	103	3	=	=	SYM
cana-1220	103	4	𝑐4	𝑐4	PROPN
cana-1220	103	5	−	−	NOUN
cana-1220	103	6	𝑐1	𝑐1	NOUN
cana-1220	103	7	=	=	SYM
cana-1220	103	8	𝑐3	𝑐3	NOUN
cana-1220	103	9	ꞷ	ꞷ	PROPN
cana-1220	103	10	∗	∗	NOUN
cana-1220	103	11	(	(	PUNCT
cana-1220	103	12	v2v3)=	v2v3)=	NOUN
cana-1220	104	1	|	|	ADV
cana-1220	105	1	ꞷ	ꞷ	PROPN
cana-1220	106	1	(	(	PUNCT
cana-1220	106	2	v2	v2	PROPN
cana-1220	106	3	)	)	PUNCT
cana-1220	106	4	ꞷ	ꞷ	PROPN
cana-1220	107	1	(	(	PUNCT
cana-1220	107	2	𝑣3)|	𝑣3)|	NUM
cana-1220	107	3	=	=	NOUN
cana-1220	107	4	𝑐1	𝑐1	NOUN
cana-1220	107	5	−	−	NOUN
cana-1220	107	6	𝑐5	𝑐5	NOUN
cana-1220	107	7	=	=	SYM
cana-1220	107	8	𝑐4	𝑐4	PROPN
cana-1220	107	9	ꞷ	ꞷ	NOUN
cana-1220	107	10	∗	∗	NOUN
cana-1220	107	11	(	(	PUNCT
cana-1220	107	12	v3v4)=	v3v4)=	X
cana-1220	108	1	|	|	NOUN
cana-1220	108	2	ꞷ	ꞷ	PROPN
cana-1220	108	3	(	(	PUNCT
cana-1220	108	4	v3	v3	PROPN
cana-1220	108	5	)	)	PUNCT
cana-1220	108	6	ꞷ	ꞷ	PROPN
cana-1220	109	1	(	(	PUNCT
cana-1220	109	2	𝑣4)|	𝑣4)|	NOUN
cana-1220	109	3	=	=	PUNCT
cana-1220	109	4	𝑐5	𝑐5	ADJ
cana-1220	109	5	−	−	NOUN
cana-1220	109	6	𝑐2	𝑐2	NOUN
cana-1220	109	7	=	=	PRON
cana-1220	109	8	𝑐3	𝑐3	NOUN
cana-1220	109	9	ꞷ	ꞷ	PROPN
cana-1220	109	10	∗	∗	NOUN
cana-1220	109	11	(	(	PUNCT
cana-1220	109	12	v4v5)=	v4v5)=	PROPN
cana-1220	110	1	|	|	NOUN
cana-1220	111	1	ꞷ	ꞷ	PROPN
cana-1220	112	1	(	(	PUNCT
cana-1220	112	2	v4	v4	PROPN
cana-1220	112	3	)	)	PUNCT
cana-1220	112	4	ꞷ	ꞷ	PROPN
cana-1220	113	1	(	(	PUNCT
cana-1220	113	2	𝑣5)|	𝑣5)|	PROPN
cana-1220	113	3	=	=	PUNCT
cana-1220	113	4	𝑐2	𝑐2	NOUN
cana-1220	113	5	−	−	PROPN
cana-1220	113	6	𝑐3	𝑐3	NOUN
cana-1220	113	7	=	=	NOUN
cana-1220	113	8	𝑐1	𝑐1	NOUN
cana-1220	113	9	hence	hence	ADV
cana-1220	113	10	,	,	PUNCT
cana-1220	113	11	adjacent	adjacent	ADJ
cana-1220	113	12	vertices	vertex	NOUN
cana-1220	113	13	and	and	CCONJ
cana-1220	113	14	edges	edge	NOUN
cana-1220	113	15	receive	receive	VERB
cana-1220	113	16	distinct	distinct	ADJ
cana-1220	113	17	colors	color	NOUN
cana-1220	113	18	.	.	PUNCT
cana-1220	114	1	thus	thus	ADV
cana-1220	114	2	,	,	PUNCT
cana-1220	114	3	χpg(cn)≤5	χpg(cn)≤5	PROPN
cana-1220	114	4	.	.	PUNCT
cana-1220	115	1	to	to	PART
cana-1220	115	2	prove	prove	VERB
cana-1220	115	3	χpg(cn)≥	χpg(cn)≥	NOUN
cana-1220	115	4	5	5	NUM
cana-1220	115	5	,	,	PUNCT
cana-1220	115	6	let	let	VERB
cana-1220	115	7	us	we	PRON
cana-1220	115	8	assume	assume	VERB
cana-1220	115	9	that	that	SCONJ
cana-1220	115	10	χpg(cn	χpg(cn	AUX
cana-1220	115	11	)	)	PUNCT
cana-1220	115	12	<	<	X
cana-1220	115	13	5	5	NUM
cana-1220	115	14	,	,	PUNCT
cana-1220	115	15	say	say	VERB
cana-1220	115	16	4	4	X
cana-1220	115	17	.	.	X
cana-1220	116	1	we	we	PRON
cana-1220	116	2	define	define	VERB
cana-1220	116	3	vertex	vertex	NOUN
cana-1220	116	4	coloring	coloring	NOUN
cana-1220	116	5	of	of	ADP
cana-1220	116	6	c6	c6	PROPN
cana-1220	116	7	is	be	AUX
cana-1220	116	8	c4	c4	NOUN
cana-1220	116	9	,	,	PUNCT
cana-1220	116	10	c1	c1	PROPN
cana-1220	116	11	,	,	PUNCT
cana-1220	116	12	c2	c2	PROPN
cana-1220	116	13	,	,	PUNCT
cana-1220	116	14	c4	c4	NOUN
cana-1220	116	15	,	,	PUNCT
cana-1220	116	16	c3	c3	PROPN
cana-1220	116	17	,	,	PUNCT
cana-1220	116	18	c2	c2	PROPN
cana-1220	116	19	.	.	PUNCT
cana-1220	117	1	then	then	ADV
cana-1220	117	2	,	,	PUNCT
cana-1220	117	3	ꞷ	ꞷ	PROPN
cana-1220	117	4	∗	∗	NOUN
cana-1220	117	5	(	(	PUNCT
cana-1220	117	6	v1v2	v1v2	NOUN
cana-1220	117	7	)	)	PUNCT
cana-1220	117	8	=	=	NOUN
cana-1220	117	9	𝑐3	𝑐3	NOUN
cana-1220	117	10	,	,	PUNCT
cana-1220	117	11	ꞷ	ꞷ	PROPN
cana-1220	117	12	∗	∗	NOUN
cana-1220	117	13	(	(	PUNCT
cana-1220	117	14	v2v3	v2v3	NOUN
cana-1220	117	15	)	)	PUNCT
cana-1220	117	16	=	=	NOUN
cana-1220	117	17	𝑐1	𝑐1	NOUN
cana-1220	117	18	,	,	PUNCT
cana-1220	117	19	ꞷ	ꞷ	PROPN
cana-1220	117	20	∗	∗	X
cana-1220	117	21	(	(	PUNCT
cana-1220	117	22	v3v4	v3v4	NOUN
cana-1220	117	23	)	)	PUNCT
cana-1220	117	24	=	=	SYM
cana-1220	118	1	𝑐2	𝑐2	NOUN
cana-1220	118	2	,	,	PUNCT
cana-1220	118	3	ꞷ	ꞷ	PROPN
cana-1220	118	4	∗	∗	NOUN
cana-1220	118	5	(	(	PUNCT
cana-1220	118	6	v4v5	v4v5	NOUN
cana-1220	118	7	)	)	PUNCT
cana-1220	119	1	=	=	NOUN
cana-1220	119	2	𝑐1	𝑐1	NOUN
cana-1220	119	3	,	,	PUNCT
cana-1220	119	4	ꞷ	ꞷ	PROPN
cana-1220	119	5	∗	∗	NOUN
cana-1220	119	6	(	(	PUNCT
cana-1220	119	7	v5v6	v5v6	NOUN
cana-1220	119	8	)	)	PUNCT
cana-1220	119	9	=	=	SYM
cana-1220	119	10	𝑐1	𝑐1	NOUN
cana-1220	119	11	and	and	CCONJ
cana-1220	119	12	ꞷ	ꞷ	NOUN
cana-1220	119	13	∗	∗	NOUN
cana-1220	119	14	(	(	PUNCT
cana-1220	119	15	v6v1	v6v1	NOUN
cana-1220	119	16	)	)	PUNCT
cana-1220	119	17	=	=	SYM
cana-1220	119	18	𝑐2	𝑐2	NOUN
cana-1220	119	19	.	.	PUNCT
cana-1220	120	1	since	since	SCONJ
cana-1220	120	2	ꞷ	ꞷ	PROPN
cana-1220	120	3	∗	∗	X
cana-1220	120	4	(	(	PUNCT
cana-1220	120	5	v4v5	v4v5	NOUN
cana-1220	120	6	)	)	PUNCT
cana-1220	120	7	=	=	NOUN
cana-1220	120	8	𝑐1	𝑐1	NOUN
cana-1220	120	9	and	and	CCONJ
cana-1220	120	10	ꞷ	ꞷ	PRON
cana-1220	120	11	∗	∗	NOUN
cana-1220	120	12	(	(	PUNCT
cana-1220	120	13	v5v6	v5v6	NOUN
cana-1220	120	14	)	)	PUNCT
cana-1220	120	15	=	=	NOUN
cana-1220	121	1	𝑐1	𝑐1	NOUN
cana-1220	121	2	receives	receive	VERB
cana-1220	121	3	same	same	ADJ
cana-1220	121	4	color	color	NOUN
cana-1220	121	5	which	which	PRON
cana-1220	121	6	is	be	AUX
cana-1220	121	7	a	a	DET
cana-1220	121	8	contradiction	contradiction	NOUN
cana-1220	121	9	with	with	ADP
cana-1220	121	10	the	the	DET
cana-1220	121	11	definition	definition	NOUN
cana-1220	121	12	of	of	ADP
cana-1220	121	13	prime	prime	ADJ
cana-1220	121	14	graceful	graceful	ADJ
cana-1220	121	15	coloring	coloring	NOUN
cana-1220	121	16	since	since	SCONJ
cana-1220	121	17	the	the	DET
cana-1220	121	18	color	color	NOUN
cana-1220	121	19	of	of	ADP
cana-1220	121	20	any	any	DET
cana-1220	121	21	two	two	NUM
cana-1220	121	22	adjacent	adjacent	ADJ
cana-1220	121	23	edges	edge	NOUN
cana-1220	121	24	are	be	AUX
cana-1220	121	25	distinct	distinct	ADJ
cana-1220	121	26	.	.	PUNCT
cana-1220	122	1	thus	thus	ADV
cana-1220	122	2	,	,	PUNCT
cana-1220	122	3	χpg(cn)≥	χpg(cn)≥	NOUN
cana-1220	122	4	5	5	NUM
cana-1220	122	5	.	.	PUNCT
cana-1220	123	1	therefore	therefore	ADV
cana-1220	123	2	,	,	PUNCT
cana-1220	123	3	χpg(cn)=	χpg(cn)=	ADV
cana-1220	123	4	5	5	NUM
cana-1220	123	5	for	for	ADP
cana-1220	123	6	n	n	DET
cana-1220	123	7	=	=	SYM
cana-1220	123	8	3m+3	3m+3	NUM
cana-1220	123	9	,	,	PUNCT
cana-1220	123	10	m∈n	m∈n	NOUN
cana-1220	123	11	.	.	PUNCT
cana-1220	124	1	communications	communication	NOUN
cana-1220	124	2	on	on	ADP
cana-1220	124	3	applied	apply	VERB
cana-1220	124	4	nonlinear	nonlinear	ADJ
cana-1220	124	5	analysis	analysis	NOUN
cana-1220	124	6	issn	issn	NOUN
cana-1220	124	7	:	:	PUNCT
cana-1220	124	8	1074	1074	NUM
cana-1220	124	9	-	-	PUNCT
cana-1220	124	10	133x	133x	NUM
cana-1220	124	11	vol	vol	NOUN
cana-1220	124	12	31	31	NUM
cana-1220	124	13	no	no	NOUN
cana-1220	124	14	.	.	PUNCT
cana-1220	125	1	6s	6s	NUM
cana-1220	125	2	(	(	PUNCT
cana-1220	125	3	2024	2024	NUM
cana-1220	125	4	)	)	PUNCT
cana-1220	125	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	125	6	263	263	NUM
cana-1220	125	7	figure	figure	NOUN
cana-1220	125	8	3	3	NUM
cana-1220	125	9	analytical	analytical	ADJ
cana-1220	125	10	evaluation	evaluation	NOUN
cana-1220	125	11	of	of	ADP
cana-1220	125	12	the	the	DET
cana-1220	125	13	𝐶6	𝐶6	NOUN
cana-1220	125	14	theorem	theorem	NOUN
cana-1220	125	15	:	:	PUNCT
cana-1220	125	16	3.6	3.6	NUM
cana-1220	125	17	let	let	VERB
cana-1220	125	18	fn	fn	PRON
cana-1220	125	19	be	be	AUX
cana-1220	125	20	a	a	DET
cana-1220	125	21	friendship	friendship	NOUN
cana-1220	125	22	graph	graph	NOUN
cana-1220	125	23	,	,	PUNCT
cana-1220	125	24	then	then	ADV
cana-1220	125	25	for	for	ADP
cana-1220	125	26	n	n	PROPN
cana-1220	125	27	≥	≥	NUM
cana-1220	125	28	2	2	NUM
cana-1220	125	29	χpg(fn)=	χpg(fn)=	NOUN
cana-1220	125	30	2n+1	2n+1	NOUN
cana-1220	125	31	.	.	PUNCT
cana-1220	126	1	proof	proof	NOUN
cana-1220	126	2	:	:	PUNCT
cana-1220	126	3	the	the	DET
cana-1220	126	4	vertices	vertex	NOUN
cana-1220	126	5	and	and	CCONJ
cana-1220	126	6	edges	edge	NOUN
cana-1220	126	7	of	of	ADP
cana-1220	126	8	fn	fn	NOUN
cana-1220	126	9	are	be	AUX
cana-1220	126	10	v	v	PRON
cana-1220	126	11	(	(	PUNCT
cana-1220	126	12	fn	fn	NOUN
cana-1220	126	13	)	)	PUNCT
cana-1220	126	14	=	=	PRON
cana-1220	126	15	{	{	PUNCT
cana-1220	126	16	vi	vi	NOUN
cana-1220	126	17	:	:	PUNCT
cana-1220	126	18	1	1	NUM
cana-1220	126	19	≤	≤	NUM
cana-1220	126	20	i	i	PRON
cana-1220	126	21	≤	≤	PROPN
cana-1220	126	22	2n+1	2n+1	PROPN
cana-1220	126	23	}	}	PUNCT
cana-1220	126	24	and	and	CCONJ
cana-1220	126	25	e	e	NOUN
cana-1220	126	26	(	(	PUNCT
cana-1220	126	27	fn	fn	NOUN
cana-1220	126	28	)	)	PUNCT
cana-1220	126	29	=	=	PRON
cana-1220	126	30	{	{	PUNCT
cana-1220	126	31	vivi+1	vivi+1	ADJ
cana-1220	126	32	∶	∶	NOUN
cana-1220	126	33	1	1	NUM
cana-1220	126	34	≤	≤	NUM
cana-1220	126	35	i	i	PRON
cana-1220	126	36	≤	≤	ADJ
cana-1220	126	37	3n	3n	NOUN
cana-1220	126	38	}	}	PUNCT
cana-1220	126	39	.	.	PUNCT
cana-1220	127	1	let	let	VERB
cana-1220	127	2	ꞷ	ꞷ	PRON
cana-1220	127	3	:	:	PUNCT
cana-1220	127	4	v(fn)→{c1	v(fn)→{c1	PROPN
cana-1220	127	5	,	,	PUNCT
cana-1220	127	6	c2	c2	PROPN
cana-1220	127	7	,	,	PUNCT
cana-1220	127	8	…	…	PUNCT
cana-1220	127	9	,	,	PUNCT
cana-1220	127	10	c2n+1}be	c2n+1}be	X
cana-1220	127	11	a	a	DET
cana-1220	127	12	proper	proper	ADJ
cana-1220	127	13	vertex	vertex	NOUN
cana-1220	127	14	coloring	coloring	NOUN
cana-1220	127	15	of	of	ADP
cana-1220	127	16	fn	fn	NOUN
cana-1220	127	17	.	.	PUNCT
cana-1220	128	1	it	it	PRON
cana-1220	128	2	is	be	AUX
cana-1220	128	3	defined	define	VERB
cana-1220	128	4	as	as	ADP
cana-1220	128	5	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	128	6	)	)	PUNCT
cana-1220	129	1	=	=	NOUN
cana-1220	129	2	c1	c1	NOUN
cana-1220	129	3	(	(	PUNCT
cana-1220	129	4	i.e	i.e	PROPN
cana-1220	129	5	)	)	PUNCT
cana-1220	129	6	,	,	PUNCT
cana-1220	129	7	vertex	vertex	NOUN
cana-1220	129	8	of	of	ADP
cana-1220	129	9	degree	degree	NOUN
cana-1220	129	10	2n	2n	NUM
cana-1220	129	11	,	,	PUNCT
cana-1220	129	12	ꞷ	ꞷ	PROPN
cana-1220	129	13	(	(	PUNCT
cana-1220	129	14	v2i)=	v2i)=	X
cana-1220	129	15	c2i	c2i	ADJ
cana-1220	129	16	∀	∀	NUM
cana-1220	129	17	1	1	NUM
cana-1220	129	18	≤	≤	NUM
cana-1220	129	19	i	i	PRON
cana-1220	129	20	≤	≤	ADJ
cana-1220	129	21	n.	n.	NOUN
cana-1220	129	22	for	for	ADP
cana-1220	129	23	i=1	i=1	PROPN
cana-1220	129	24	,	,	PUNCT
cana-1220	129	25	ꞷ	ꞷ	PROPN
cana-1220	129	26	(	(	PUNCT
cana-1220	129	27	v2i+1)=	v2i+1)=	PROPN
cana-1220	129	28	c2n+1	c2n+1	PROPN
cana-1220	129	29	and	and	CCONJ
cana-1220	129	30	ꞷ	ꞷ	PROPN
cana-1220	129	31	(	(	PUNCT
cana-1220	129	32	v2i+1)=	v2i+1)=	NOUN
cana-1220	129	33	c2i-1	c2i-1	X
cana-1220	129	34	∀	∀	X
cana-1220	129	35	2	2	NUM
cana-1220	129	36	≤	≤	NUM
cana-1220	129	37	i	i	PRON
cana-1220	129	38	≤	≤	PROPN
cana-1220	129	39	n.	n.	NOUN
cana-1220	129	40	since	since	SCONJ
cana-1220	129	41	these	these	DET
cana-1220	129	42	colors	color	NOUN
cana-1220	129	43	are	be	AUX
cana-1220	129	44	consecutive	consecutive	ADJ
cana-1220	129	45	colors	color	NOUN
cana-1220	129	46	,	,	PUNCT
cana-1220	129	47	gcd(ꞷ(v1),ꞷ(v2i	gcd(ꞷ(v1),ꞷ(v2i	NOUN
cana-1220	129	48	)	)	PUNCT
cana-1220	129	49	)	)	PUNCT
cana-1220	130	1	=	=	SYM
cana-1220	130	2	1	1	NUM
cana-1220	130	3	,	,	PUNCT
cana-1220	130	4	gcd	gcd	NOUN
cana-1220	130	5	(	(	PUNCT
cana-1220	130	6	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	130	7	)	)	PUNCT
cana-1220	130	8	,	,	PUNCT
cana-1220	130	9	ꞷ(v2i+1	ꞷ(v2i+1	NUM
cana-1220	130	10	)	)	PUNCT
cana-1220	130	11	)	)	PUNCT
cana-1220	130	12	=	=	SYM
cana-1220	130	13	1	1	NUM
cana-1220	130	14	and	and	CCONJ
cana-1220	130	15	gcd	gcd	VERB
cana-1220	130	16	(	(	PUNCT
cana-1220	130	17	ꞷ(v2i	ꞷ(v2i	NOUN
cana-1220	130	18	)	)	PUNCT
cana-1220	130	19	,	,	PUNCT
cana-1220	130	20	ꞷ(v2i+1	ꞷ(v2i+1	NUM
cana-1220	130	21	)	)	PUNCT
cana-1220	130	22	)	)	PUNCT
cana-1220	131	1	=	=	SYM
cana-1220	131	2	1	1	NUM
cana-1220	131	3	let	let	VERB
cana-1220	131	4	ꞷ*:e(fn)→{c1	ꞷ*:e(fn)→{c1	PROPN
cana-1220	131	5	,	,	PUNCT
cana-1220	131	6	c2	c2	PROPN
cana-1220	131	7	,	,	PUNCT
cana-1220	131	8	…	…	PUNCT
cana-1220	131	9	,	,	PUNCT
cana-1220	131	10	c2n	c2n	NOUN
cana-1220	131	11	}	}	PUNCT
cana-1220	131	12	be	be	AUX
cana-1220	131	13	a	a	DET
cana-1220	131	14	proper	proper	ADJ
cana-1220	131	15	edge	edge	NOUN
cana-1220	131	16	coloring	coloring	NOUN
cana-1220	131	17	of	of	ADP
cana-1220	131	18	fn	fn	PROPN
cana-1220	131	19	.	.	PUNCT
cana-1220	132	1	it	it	PRON
cana-1220	132	2	is	be	AUX
cana-1220	132	3	defined	define	VERB
cana-1220	132	4	as	as	ADP
cana-1220	132	5	,	,	PUNCT
cana-1220	132	6	ꞷ	ꞷ	PROPN
cana-1220	132	7	∗	∗	NOUN
cana-1220	132	8	(	(	PUNCT
cana-1220	132	9	vivi+1)=	vivi+1)=	NOUN
cana-1220	132	10	|	|	ADV
cana-1220	133	1	ꞷ	ꞷ	X
cana-1220	133	2	(	(	PUNCT
cana-1220	133	3	vi	vi	PROPN
cana-1220	133	4	)	)	PUNCT
cana-1220	133	5	ꞷ	ꞷ	PROPN
cana-1220	134	1	(	(	PUNCT
cana-1220	134	2	vi+1)|	vi+1)|	NOUN
cana-1220	134	3	∀	∀	NOUN
cana-1220	134	4	1	1	NUM
cana-1220	134	5	≤	≤	NUM
cana-1220	134	6	i	i	PRON
cana-1220	134	7	≤	≤	ADJ
cana-1220	134	8	n.	n.	NOUN
cana-1220	134	9	ꞷ*(v1v2i	ꞷ*(v1v2i	NOUN
cana-1220	134	10	)	)	PUNCT
cana-1220	134	11	=	=	SYM
cana-1220	135	1	c2i-1	c2i-1	NOUN
cana-1220	135	2	,	,	PUNCT
cana-1220	135	3	ꞷ*(v1v2i+1	ꞷ*(v1v2i+1	NOUN
cana-1220	135	4	)	)	PUNCT
cana-1220	135	5	=	=	PUNCT
cana-1220	136	1	c2i	c2i	X
cana-1220	136	2	for	for	ADP
cana-1220	136	3	i=1	i=1	X
cana-1220	136	4	,	,	PUNCT
cana-1220	136	5	ꞷ*(v2iv2i+1	ꞷ*(v2iv2i+1	PUNCT
cana-1220	136	6	)	)	PUNCT
cana-1220	137	1	=	=	SYM
cana-1220	137	2	c2n-1	c2n-1	NOUN
cana-1220	137	3	and	and	CCONJ
cana-1220	137	4	for	for	ADP
cana-1220	137	5	i≠1	i≠1	ADJ
cana-1220	137	6	,	,	PUNCT
cana-1220	137	7	ꞷ*(v2iv2i+1	ꞷ*(v2iv2i+1	PUNCT
cana-1220	137	8	)	)	PUNCT
cana-1220	137	9	=	=	SYM
cana-1220	137	10	c1	c1	PROPN
cana-1220	137	11	.	.	PUNCT
cana-1220	138	1	consequently	consequently	ADV
cana-1220	138	2	,	,	PUNCT
cana-1220	138	3	ꞷ*(vivi+1	ꞷ*(vivi+1	NOUN
cana-1220	138	4	)	)	PUNCT
cana-1220	138	5	receives	receive	VERB
cana-1220	138	6	distinct	distinct	ADJ
cana-1220	138	7	colors	color	NOUN
cana-1220	138	8	.	.	PUNCT
cana-1220	139	1	thus	thus	ADV
cana-1220	139	2	,	,	PUNCT
cana-1220	139	3	χpg(fn)≤2n	χpg(fn)≤2n	PROPN
cana-1220	139	4	+	+	NOUN
cana-1220	139	5	1	1	X
cana-1220	139	6	.	.	PUNCT
cana-1220	139	7	to	to	PART
cana-1220	139	8	prove	prove	VERB
cana-1220	139	9	χpg(fn)≥	χpg(fn)≥	NOUN
cana-1220	139	10	2n	2n	NUM
cana-1220	139	11	+	+	CCONJ
cana-1220	139	12	1	1	NUM
cana-1220	139	13	,	,	PUNCT
cana-1220	139	14	let	let	VERB
cana-1220	139	15	us	we	PRON
cana-1220	139	16	assume	assume	VERB
cana-1220	139	17	that	that	SCONJ
cana-1220	139	18	χpg(fn	χpg(fn	NOUN
cana-1220	139	19	)	)	PUNCT
cana-1220	139	20	<	<	X
cana-1220	139	21	2n	2n	NUM
cana-1220	140	1	+	+	CCONJ
cana-1220	140	2	1	1	NUM
cana-1220	140	3	,	,	PUNCT
cana-1220	140	4	say	say	VERB
cana-1220	140	5	2n	2n	NUM
cana-1220	140	6	.	.	PUNCT
cana-1220	141	1	we	we	PRON
cana-1220	141	2	must	must	AUX
cana-1220	141	3	assign	assign	VERB
cana-1220	141	4	2n	2n	NUM
cana-1220	141	5	colors	color	NOUN
cana-1220	141	6	for	for	ADP
cana-1220	141	7	{	{	PUNCT
cana-1220	141	8	v2i	v2i	NOUN
cana-1220	141	9	,	,	PUNCT
cana-1220	141	10	v2i+1	v2i+1	NOUN
cana-1220	141	11	:	:	PUNCT
cana-1220	141	12	1	1	NUM
cana-1220	141	13	≤	≤	NUM
cana-1220	141	14	i	i	PRON
cana-1220	141	15	≤	≤	NOUN
cana-1220	141	16	n	n	CCONJ
cana-1220	141	17	}	}	PUNCT
cana-1220	141	18	for	for	ADP
cana-1220	141	19	proper	proper	ADJ
cana-1220	141	20	vertex	vertex	NOUN
cana-1220	141	21	coloring	coloring	NOUN
cana-1220	141	22	.	.	PUNCT
cana-1220	142	1	since	since	SCONJ
cana-1220	142	2	one	one	NUM
cana-1220	142	3	vertex	vertex	NOUN
cana-1220	142	4	is	be	AUX
cana-1220	142	5	adjacent	adjacent	ADJ
cana-1220	142	6	to	to	ADP
cana-1220	142	7	remaining	remain	VERB
cana-1220	142	8	vertices	vertex	NOUN
cana-1220	142	9	,	,	PUNCT
cana-1220	142	10	we	we	PRON
cana-1220	142	11	can	can	AUX
cana-1220	142	12	not	not	PART
cana-1220	142	13	assign	assign	VERB
cana-1220	142	14	the	the	DET
cana-1220	142	15	same	same	ADJ
cana-1220	142	16	color	color	NOUN
cana-1220	142	17	which	which	PRON
cana-1220	142	18	is	be	AUX
cana-1220	142	19	a	a	DET
cana-1220	142	20	contradiction	contradiction	NOUN
cana-1220	142	21	with	with	ADP
cana-1220	142	22	the	the	DET
cana-1220	142	23	definition	definition	NOUN
cana-1220	142	24	of	of	ADP
cana-1220	142	25	prime	prime	ADJ
cana-1220	142	26	graceful	graceful	ADJ
cana-1220	142	27	coloring	coloring	NOUN
cana-1220	142	28	since	since	SCONJ
cana-1220	142	29	the	the	DET
cana-1220	142	30	color	color	NOUN
cana-1220	142	31	of	of	ADP
cana-1220	142	32	any	any	DET
cana-1220	142	33	two	two	NUM
cana-1220	142	34	adjacent	adjacent	ADJ
cana-1220	142	35	vertices	vertex	NOUN
cana-1220	142	36	are	be	AUX
cana-1220	142	37	distinct	distinct	ADJ
cana-1220	142	38	.	.	PUNCT
cana-1220	143	1	thus	thus	ADV
cana-1220	143	2	,	,	PUNCT
cana-1220	143	3	χpg(fn)≥	χpg(fn)≥	VERB
cana-1220	143	4	2n	2n	NUM
cana-1220	143	5	+	+	CCONJ
cana-1220	143	6	1	1	X
cana-1220	143	7	.	.	X
cana-1220	143	8	therefore	therefore	ADV
cana-1220	143	9	,	,	PUNCT
cana-1220	143	10	χpg(fn)=	χpg(fn)=	PROPN
cana-1220	143	11	2n+1	2n+1	PROPN
cana-1220	143	12	.	.	PUNCT
cana-1220	144	1	communications	communication	NOUN
cana-1220	144	2	on	on	ADP
cana-1220	144	3	applied	apply	VERB
cana-1220	144	4	nonlinear	nonlinear	ADJ
cana-1220	144	5	analysis	analysis	NOUN
cana-1220	144	6	issn	issn	NOUN
cana-1220	144	7	:	:	PUNCT
cana-1220	144	8	1074	1074	NUM
cana-1220	144	9	-	-	PUNCT
cana-1220	144	10	133x	133x	NUM
cana-1220	144	11	vol	vol	NOUN
cana-1220	144	12	31	31	NUM
cana-1220	144	13	no	no	NOUN
cana-1220	144	14	.	.	PUNCT
cana-1220	145	1	6s	6s	NUM
cana-1220	145	2	(	(	PUNCT
cana-1220	145	3	2024	2024	NUM
cana-1220	145	4	)	)	PUNCT
cana-1220	145	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	145	6	264	264	NUM
cana-1220	145	7	figure	figure	NOUN
cana-1220	145	8	4	4	NUM
cana-1220	145	9	analytical	analytical	ADJ
cana-1220	145	10	evaluation	evaluation	NOUN
cana-1220	145	11	of	of	ADP
cana-1220	145	12	the	the	DET
cana-1220	145	13	𝐹4	𝐹4	NOUN
cana-1220	145	14	theorem	theorem	PROPN
cana-1220	145	15	:	:	PUNCT
cana-1220	145	16	3.7	3.7	NUM
cana-1220	145	17	let	let	VERB
cana-1220	145	18	n	n	PRON
cana-1220	145	19	≥	≥	X
cana-1220	145	20	3	3	NUM
cana-1220	145	21	be	be	AUX
cana-1220	145	22	a	a	DET
cana-1220	145	23	positive	positive	ADJ
cana-1220	145	24	integer	integer	NOUN
cana-1220	145	25	,	,	PUNCT
cana-1220	145	26	then	then	ADV
cana-1220	145	27	χpg(n	χpg(n	PROPN
cana-1220	145	28	-	-	PUNCT
cana-1220	145	29	pan	pan	NOUN
cana-1220	145	30	graph)=	graph)=	NOUN
cana-1220	145	31	{	{	PUNCT
cana-1220	145	32	4	4	NUM
cana-1220	145	33	,	,	PUNCT
cana-1220	145	34	if	if	SCONJ
cana-1220	145	35	n	n	NOUN
cana-1220	145	36	=	=	SYM
cana-1220	145	37	3	3	NUM
cana-1220	145	38	5	5	NUM
cana-1220	145	39	,	,	PUNCT
cana-1220	145	40	if	if	SCONJ
cana-1220	145	41	n	n	PRON
cana-1220	145	42	≠	≠	ADJ
cana-1220	145	43	3	3	NUM
cana-1220	145	44	proof	proof	NOUN
cana-1220	145	45	:	:	PUNCT
cana-1220	145	46	the	the	DET
cana-1220	145	47	vertices	vertex	NOUN
cana-1220	145	48	and	and	CCONJ
cana-1220	145	49	edges	edge	NOUN
cana-1220	145	50	of	of	ADP
cana-1220	145	51	pan	pan	NOUN
cana-1220	145	52	graph	graph	NOUN
cana-1220	145	53	are	be	AUX
cana-1220	145	54	v	v	ADP
cana-1220	145	55	(	(	PUNCT
cana-1220	145	56	g	g	NOUN
cana-1220	145	57	)	)	PUNCT
cana-1220	145	58	=	=	PRON
cana-1220	145	59	{	{	PUNCT
cana-1220	145	60	vi	vi	X
cana-1220	145	61	:1	:1	PUNCT
cana-1220	145	62	≤	≤	NUM
cana-1220	146	1	i	i	PRON
cana-1220	146	2	≤	≤	ADJ
cana-1220	146	3	n	n	CCONJ
cana-1220	146	4	+	+	CCONJ
cana-1220	146	5	1	1	NUM
cana-1220	146	6	}	}	PUNCT
cana-1220	146	7	and	and	CCONJ
cana-1220	146	8	e(g)={vivi+1	e(g)={vivi+1	VERB
cana-1220	146	9	:	:	PUNCT
cana-1220	146	10	1	1	NUM
cana-1220	146	11	≤	≤	NUM
cana-1220	146	12	i	i	PRON
cana-1220	146	13	≤	≤	ADJ
cana-1220	146	14	n	n	CCONJ
cana-1220	146	15	+	+	NOUN
cana-1220	146	16	1	1	NUM
cana-1220	146	17	}	}	PUNCT
cana-1220	146	18	.	.	PUNCT
cana-1220	147	1	case	case	NOUN
cana-1220	147	2	1	1	NUM
cana-1220	147	3	:	:	PUNCT
cana-1220	147	4	n	n	CCONJ
cana-1220	147	5	=3	=3	VERB
cana-1220	147	6	define	define	VERB
cana-1220	147	7	a	a	DET
cana-1220	147	8	proper	proper	ADJ
cana-1220	147	9	vertex	vertex	NOUN
cana-1220	147	10	coloring	color	VERB
cana-1220	147	11	ꞷ	ꞷ	PRON
cana-1220	147	12	:	:	PUNCT
cana-1220	147	13	v(g	v(g	ADJ
cana-1220	147	14	)	)	PUNCT
cana-1220	148	1	→{c1	→{c1	PROPN
cana-1220	148	2	,	,	PUNCT
cana-1220	148	3	c2	c2	PROPN
cana-1220	148	4	,	,	PUNCT
cana-1220	148	5	c3	c3	PROPN
cana-1220	148	6	,	,	PUNCT
cana-1220	148	7	c4}as	c4}as	X
cana-1220	148	8	follows	follow	VERB
cana-1220	148	9	:	:	PUNCT
cana-1220	148	10	for	for	ADP
cana-1220	148	11	1≤	1≤	NUM
cana-1220	148	12	i	i	PRON
cana-1220	148	13	≤	≤	PROPN
cana-1220	148	14	n+1	n+1	PROPN
cana-1220	148	15	,	,	PUNCT
cana-1220	148	16	such	such	ADJ
cana-1220	148	17	that	that	DET
cana-1220	148	18	gcd	gcd	NOUN
cana-1220	148	19	(	(	PUNCT
cana-1220	148	20	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	148	21	)	)	PUNCT
cana-1220	148	22	,	,	PUNCT
cana-1220	148	23	ꞷ(vi+1	ꞷ(vi+1	ADJ
cana-1220	148	24	)	)	PUNCT
cana-1220	148	25	)	)	PUNCT
cana-1220	149	1	=	=	PUNCT
cana-1220	149	2	1	1	X
cana-1220	149	3	.	.	X
cana-1220	149	4	it	it	PRON
cana-1220	149	5	induce	induce	VERB
cana-1220	149	6	a	a	DET
cana-1220	149	7	proper	proper	ADJ
cana-1220	149	8	edge	edge	NOUN
cana-1220	149	9	coloring	coloring	NOUN
cana-1220	149	10	ꞷ*:e(g)→{c1	ꞷ*:e(g)→{c1	NOUN
cana-1220	149	11	,	,	PUNCT
cana-1220	149	12	c2	c2	PROPN
cana-1220	149	13	,	,	PUNCT
cana-1220	149	14	c3	c3	PROPN
cana-1220	149	15	}	}	PUNCT
cana-1220	149	16	as	as	SCONJ
cana-1220	149	17	follows	follow	VERB
cana-1220	149	18	:	:	PUNCT
cana-1220	149	19	ꞷ(vi	ꞷ(vi	X
cana-1220	149	20	)	)	PUNCT
cana-1220	149	21	=	=	PRON
cana-1220	149	22	{	{	PUNCT
cana-1220	149	23	𝑐1	𝑐1	NOUN
cana-1220	149	24	,	,	PUNCT
cana-1220	149	25	for	for	ADP
cana-1220	149	26	𝑣2	𝑣2	NOUN
cana-1220	149	27	𝑐2	𝑐2	NOUN
cana-1220	149	28	,	,	PUNCT
cana-1220	149	29	for	for	ADP
cana-1220	149	30	𝑣4	𝑣4	NOUN
cana-1220	149	31	𝑐3	𝑐3	NOUN
cana-1220	149	32	,	,	PUNCT
cana-1220	149	33	for	for	ADP
cana-1220	149	34	𝑣1	𝑣1	NOUN
cana-1220	149	35	𝑐4	𝑐4	NOUN
cana-1220	149	36	,	,	PUNCT
cana-1220	149	37	for	for	ADP
cana-1220	149	38	𝑣3	𝑣3	PROPN
cana-1220	149	39	ꞷ	ꞷ	PROPN
cana-1220	149	40	*	*	X
cana-1220	149	41	(	(	PUNCT
cana-1220	149	42	vivi+1	vivi+1	NOUN
cana-1220	149	43	)	)	PUNCT
cana-1220	149	44	=	=	NOUN
cana-1220	149	45	{	{	PUNCT
cana-1220	149	46	c2	c2	PROPN
cana-1220	149	47	,	,	PUNCT
cana-1220	149	48	for	for	ADP
cana-1220	149	49	v1v2	v1v2	SYM
cana-1220	149	50	c3	c3	PROPN
cana-1220	149	51	,	,	PUNCT
cana-1220	149	52	for	for	ADP
cana-1220	149	53	v2v3	v2v3	PROPN
cana-1220	149	54	c1	c1	PROPN
cana-1220	149	55	,	,	PUNCT
cana-1220	149	56	for	for	ADP
cana-1220	149	57	v3v1	v3v1	PUNCT
cana-1220	149	58	and	and	CCONJ
cana-1220	149	59	v1v4	v1v4	ADJ
cana-1220	149	60	hence	hence	ADV
cana-1220	149	61	,	,	PUNCT
cana-1220	149	62	adjacent	adjacent	ADJ
cana-1220	149	63	vertices	vertex	NOUN
cana-1220	149	64	and	and	CCONJ
cana-1220	149	65	edges	edge	NOUN
cana-1220	149	66	receive	receive	VERB
cana-1220	149	67	distinct	distinct	ADJ
cana-1220	149	68	colors	color	NOUN
cana-1220	149	69	.	.	PUNCT
cana-1220	150	1	therefore	therefore	ADV
cana-1220	150	2	,	,	PUNCT
cana-1220	150	3	χpg(n	χpg(n	PROPN
cana-1220	150	4	-	-	PUNCT
cana-1220	150	5	pan	pan	NOUN
cana-1220	150	6	graph)=	graph)=	NOUN
cana-1220	150	7	4	4	NUM
cana-1220	150	8	for	for	ADP
cana-1220	150	9	n=3	n=3	DET
cana-1220	150	10	case	case	NOUN
cana-1220	150	11	2	2	NUM
cana-1220	150	12	:	:	PUNCT
cana-1220	150	13	n	n	NUM
cana-1220	150	14	≠	≠	PROPN
cana-1220	150	15	3m+3	3m+3	PROPN
cana-1220	150	16	,	,	PUNCT
cana-1220	150	17	m∈n	m∈n	NOUN
cana-1220	150	18	define	define	VERB
cana-1220	150	19	a	a	DET
cana-1220	150	20	proper	proper	ADJ
cana-1220	150	21	vertex	vertex	NOUN
cana-1220	150	22	coloring	color	VERB
cana-1220	150	23	ꞷ	ꞷ	PRON
cana-1220	150	24	:	:	PUNCT
cana-1220	150	25	v(g	v(g	ADJ
cana-1220	150	26	)	)	PUNCT
cana-1220	151	1	→{c1	→{c1	PROPN
cana-1220	151	2	,	,	PUNCT
cana-1220	151	3	c2	c2	PROPN
cana-1220	151	4	,	,	PUNCT
cana-1220	151	5	c3	c3	PROPN
cana-1220	151	6	,	,	PUNCT
cana-1220	151	7	c4	c4	NOUN
cana-1220	151	8	,	,	PUNCT
cana-1220	151	9	c5}as	c5}as	PRON
cana-1220	151	10	follows	follow	VERB
cana-1220	151	11	:	:	PUNCT
cana-1220	151	12	for	for	ADP
cana-1220	151	13	1≤	1≤	NUM
cana-1220	151	14	i	i	PRON
cana-1220	151	15	≤	≤	VERB
cana-1220	151	16	n+1	n+1	NUM
cana-1220	151	17	ꞷ(vi	ꞷ(vi	NOUN
cana-1220	151	18	)	)	PUNCT
cana-1220	151	19	=	=	PRON
cana-1220	151	20	{	{	PUNCT
cana-1220	151	21	𝑐1	𝑐1	NOUN
cana-1220	151	22	,	,	PUNCT
cana-1220	151	23	for	for	ADP
cana-1220	151	24	𝑣2	𝑣2	NOUN
cana-1220	151	25	𝑐2	𝑐2	NOUN
cana-1220	151	26	,	,	PUNCT
cana-1220	151	27	for	for	ADP
cana-1220	151	28	i	i	PROPN
cana-1220	151	29	≡	≡	PROPN
cana-1220	151	30	0	0	PUNCT
cana-1220	151	31	mod	mod	ADJ
cana-1220	151	32	3	3	NUM
cana-1220	151	33	𝑐3	𝑐3	NOUN
cana-1220	151	34	,	,	PUNCT
cana-1220	151	35	for	for	ADP
cana-1220	151	36	i	i	PROPN
cana-1220	151	37	≡	≡	PROPN
cana-1220	151	38	2	2	NUM
cana-1220	151	39	mod	mod	NOUN
cana-1220	151	40	3	3	NUM
cana-1220	151	41	,	,	PUNCT
cana-1220	151	42	vn+1	vn+1	PROPN
cana-1220	151	43	and	and	CCONJ
cana-1220	151	44	i	i	PRON
cana-1220	151	45	≠	≠	PROPN
cana-1220	151	46	2	2	NUM
cana-1220	151	47	𝑐4	𝑐4	NOUN
cana-1220	151	48	,	,	PUNCT
cana-1220	151	49	for	for	ADP
cana-1220	151	50	𝑣1	𝑣1	NOUN
cana-1220	151	51	𝑐5	𝑐5	NOUN
cana-1220	151	52	,	,	PUNCT
cana-1220	151	53	for	for	ADP
cana-1220	151	54	i	i	PRON
cana-1220	151	55	≡	≡	PROPN
cana-1220	151	56	1	1	NUM
cana-1220	151	57	mod	mod	NOUN
cana-1220	151	58	3	3	NUM
cana-1220	152	1	and	and	CCONJ
cana-1220	152	2	i	i	PRON
cana-1220	152	3	≠	≠	PROPN
cana-1220	152	4	1	1	NUM
cana-1220	152	5	such	such	ADJ
cana-1220	152	6	that	that	DET
cana-1220	152	7	gcd	gcd	NOUN
cana-1220	152	8	(	(	PUNCT
cana-1220	152	9	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	152	10	)	)	PUNCT
cana-1220	152	11	,	,	PUNCT
cana-1220	152	12	ꞷ(vi+1	ꞷ(vi+1	ADJ
cana-1220	152	13	)	)	PUNCT
cana-1220	152	14	)	)	PUNCT
cana-1220	153	1	=	=	PUNCT
cana-1220	153	2	1	1	X
cana-1220	153	3	.	.	X
cana-1220	153	4	it	it	PRON
cana-1220	153	5	induce	induce	VERB
cana-1220	153	6	a	a	DET
cana-1220	153	7	proper	proper	ADJ
cana-1220	153	8	edge	edge	NOUN
cana-1220	153	9	coloring	coloring	NOUN
cana-1220	153	10	ꞷ*:e(g)→{c1	ꞷ*:e(g)→{c1	NOUN
cana-1220	153	11	,	,	PUNCT
cana-1220	153	12	c2	c2	PROPN
cana-1220	153	13	,	,	PUNCT
cana-1220	153	14	c3	c3	PROPN
cana-1220	153	15	}	}	PUNCT
cana-1220	153	16	as	as	SCONJ
cana-1220	153	17	follows	follow	VERB
cana-1220	153	18	:	:	PUNCT
cana-1220	153	19	ꞷ	ꞷ	X
cana-1220	153	20	*	*	PUNCT
cana-1220	153	21	(	(	PUNCT
cana-1220	153	22	vivi+1	vivi+1	NOUN
cana-1220	153	23	)	)	PUNCT
cana-1220	153	24	=	=	PRON
cana-1220	153	25	{	{	PUNCT
cana-1220	153	26	𝑐1	𝑐1	NOUN
cana-1220	153	27	,	,	PUNCT
cana-1220	153	28	for	for	ADP
cana-1220	153	29	i	i	PROPN
cana-1220	153	30	≡	≡	PROPN
cana-1220	153	31	2	2	NUM
cana-1220	153	32	mod	mod	NOUN
cana-1220	153	33	3	3	NUM
cana-1220	153	34	and	and	CCONJ
cana-1220	153	35	vnv1	vnv1	NOUN
cana-1220	153	36	𝑐2	𝑐2	NOUN
cana-1220	153	37	,	,	PUNCT
cana-1220	153	38	for	for	ADP
cana-1220	153	39	i	i	PRON
cana-1220	153	40	≡	≡	PROPN
cana-1220	153	41	1	1	NUM
cana-1220	153	42	mod	mod	PROPN
cana-1220	153	43	3	3	NUM
cana-1220	153	44	,	,	PUNCT
cana-1220	153	45	v2vn+1	v2vn+1	NOUN
cana-1220	153	46	and	and	CCONJ
cana-1220	153	47	i	i	PRON
cana-1220	153	48	≠	≠	ADJ
cana-1220	153	49	1	1	NUM
cana-1220	153	50	𝑐3	𝑐3	NOUN
cana-1220	153	51	,	,	PUNCT
cana-1220	153	52	for	for	ADP
cana-1220	153	53	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1220	153	54	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1220	154	1	i	i	PROPN
cana-1220	154	2	≡	≡	PROPN
cana-1220	154	3	0	0	NUM
cana-1220	154	4	mod	mod	ADJ
cana-1220	154	5	3	3	NUM
cana-1220	154	6	communications	communication	NOUN
cana-1220	154	7	on	on	ADP
cana-1220	154	8	applied	apply	VERB
cana-1220	154	9	nonlinear	nonlinear	ADJ
cana-1220	154	10	analysis	analysis	NOUN
cana-1220	154	11	issn	issn	NOUN
cana-1220	154	12	:	:	PUNCT
cana-1220	154	13	1074	1074	NUM
cana-1220	154	14	-	-	PUNCT
cana-1220	154	15	133x	133x	NUM
cana-1220	154	16	vol	vol	NOUN
cana-1220	154	17	31	31	NUM
cana-1220	154	18	no	no	NOUN
cana-1220	154	19	.	.	PUNCT
cana-1220	155	1	6s	6s	NUM
cana-1220	155	2	(	(	PUNCT
cana-1220	155	3	2024	2024	NUM
cana-1220	155	4	)	)	PUNCT
cana-1220	155	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	155	6	265	265	NUM
cana-1220	155	7	it	it	PRON
cana-1220	155	8	satisfy	satisfy	VERB
cana-1220	155	9	ꞷ	ꞷ	PROPN
cana-1220	155	10	∗	∗	NOUN
cana-1220	155	11	(	(	PUNCT
cana-1220	155	12	vivi+1)=	vivi+1)=	NOUN
cana-1220	156	1	|	|	ADV
cana-1220	156	2	ꞷ	ꞷ	X
cana-1220	156	3	(	(	PUNCT
cana-1220	156	4	vi	vi	PROPN
cana-1220	156	5	)	)	PUNCT
cana-1220	156	6	ꞷ	ꞷ	PROPN
cana-1220	156	7	(	(	PUNCT
cana-1220	156	8	𝑣𝑖+1)|	𝑣𝑖+1)|	PRON
cana-1220	156	9	hence	hence	ADV
cana-1220	156	10	,	,	PUNCT
cana-1220	156	11	adjacent	adjacent	ADJ
cana-1220	156	12	vertices	vertex	NOUN
cana-1220	156	13	and	and	CCONJ
cana-1220	156	14	edges	edge	NOUN
cana-1220	156	15	receive	receive	VERB
cana-1220	156	16	distinct	distinct	ADJ
cana-1220	156	17	colors	color	NOUN
cana-1220	156	18	.	.	PUNCT
cana-1220	157	1	thus	thus	ADV
cana-1220	157	2	,	,	PUNCT
cana-1220	157	3	χpg(n	χpg(n	PROPN
cana-1220	157	4	-	-	PUNCT
cana-1220	157	5	pan	pan	NOUN
cana-1220	157	6	graph)≤5	graph)≤5	NOUN
cana-1220	157	7	.	.	PUNCT
cana-1220	158	1	to	to	PART
cana-1220	158	2	prove	prove	VERB
cana-1220	158	3	χpg(n	χpg(n	PROPN
cana-1220	158	4	-	-	PUNCT
cana-1220	158	5	pan	pan	NOUN
cana-1220	158	6	graph)≥	graph)≥	PROPN
cana-1220	158	7	5	5	NUM
cana-1220	158	8	,	,	PUNCT
cana-1220	158	9	let	let	VERB
cana-1220	158	10	us	we	PRON
cana-1220	158	11	assume	assume	VERB
cana-1220	158	12	that	that	SCONJ
cana-1220	158	13	χpg(cn	χpg(cn	NOUN
cana-1220	158	14	)	)	PUNCT
cana-1220	158	15	<	<	X
cana-1220	158	16	5	5	NUM
cana-1220	158	17	,	,	PUNCT
cana-1220	158	18	say	say	VERB
cana-1220	158	19	4	4	NUM
cana-1220	158	20	.	.	PUNCT
cana-1220	159	1	we	we	PRON
cana-1220	159	2	define	define	VERB
cana-1220	159	3	vertex	vertex	NOUN
cana-1220	159	4	coloring	coloring	NOUN
cana-1220	159	5	as	as	ADP
cana-1220	159	6	c4	c4	NOUN
cana-1220	159	7	,	,	PUNCT
cana-1220	159	8	c1	c1	PROPN
cana-1220	159	9	,	,	PUNCT
cana-1220	159	10	c2	c2	PROPN
cana-1220	159	11	,	,	PUNCT
cana-1220	159	12	c3	c3	PROPN
cana-1220	159	13	,	,	PUNCT
cana-1220	159	14	c1	c1	PROPN
cana-1220	159	15	for	for	ADP
cana-1220	159	16	5	5	NUM
cana-1220	159	17	-	-	PUNCT
cana-1220	159	18	pan	pan	NOUN
cana-1220	159	19	graph	graph	NOUN
cana-1220	159	20	and	and	CCONJ
cana-1220	159	21	the	the	DET
cana-1220	159	22	vertex	vertex	NOUN
cana-1220	159	23	with	with	ADP
cana-1220	159	24	degree	degree	NOUN
cana-1220	159	25	1	1	NUM
cana-1220	159	26	is	be	AUX
cana-1220	159	27	assigned	assign	VERB
cana-1220	159	28	with	with	ADP
cana-1220	159	29	the	the	DET
cana-1220	159	30	color	color	NOUN
cana-1220	159	31	c3	c3	PROPN
cana-1220	159	32	.	.	PUNCT
cana-1220	160	1	then	then	ADV
cana-1220	160	2	,	,	PUNCT
cana-1220	160	3	ꞷ	ꞷ	PROPN
cana-1220	160	4	∗	∗	NOUN
cana-1220	160	5	(	(	PUNCT
cana-1220	160	6	v1v2	v1v2	NOUN
cana-1220	160	7	)	)	PUNCT
cana-1220	160	8	=	=	NOUN
cana-1220	160	9	𝑐3	𝑐3	NOUN
cana-1220	160	10	,	,	PUNCT
cana-1220	160	11	ꞷ	ꞷ	PROPN
cana-1220	160	12	∗	∗	NOUN
cana-1220	160	13	(	(	PUNCT
cana-1220	160	14	v2v3	v2v3	NOUN
cana-1220	160	15	)	)	PUNCT
cana-1220	160	16	=	=	NOUN
cana-1220	160	17	𝑐1	𝑐1	NOUN
cana-1220	160	18	,	,	PUNCT
cana-1220	160	19	ꞷ	ꞷ	PROPN
cana-1220	160	20	∗	∗	NOUN
cana-1220	160	21	(	(	PUNCT
cana-1220	160	22	v2v6	v2v6	NOUN
cana-1220	160	23	)	)	PUNCT
cana-1220	160	24	=	=	SYM
cana-1220	160	25	𝑐2	𝑐2	PROPN
cana-1220	160	26	,	,	PUNCT
cana-1220	160	27	ꞷ	ꞷ	PROPN
cana-1220	160	28	∗	∗	NOUN
cana-1220	160	29	(	(	PUNCT
cana-1220	160	30	v3v4	v3v4	NOUN
cana-1220	160	31	)	)	PUNCT
cana-1220	160	32	=	=	NOUN
cana-1220	161	1	𝑐1	𝑐1	NOUN
cana-1220	161	2	,	,	PUNCT
cana-1220	161	3	ꞷ	ꞷ	PROPN
cana-1220	161	4	∗	∗	NOUN
cana-1220	161	5	(	(	PUNCT
cana-1220	161	6	v4v5	v4v5	NOUN
cana-1220	161	7	)	)	PUNCT
cana-1220	161	8	=	=	SYM
cana-1220	162	1	𝑐2	𝑐2	NOUN
cana-1220	162	2	,	,	PUNCT
cana-1220	162	3	ꞷ	ꞷ	PROPN
cana-1220	162	4	∗	∗	NOUN
cana-1220	162	5	(	(	PUNCT
cana-1220	162	6	v5v1	v5v1	NOUN
cana-1220	162	7	)	)	PUNCT
cana-1220	162	8	=	=	NOUN
cana-1220	162	9	𝑐3	𝑐3	NOUN
cana-1220	162	10	.	.	PUNCT
cana-1220	163	1	since	since	SCONJ
cana-1220	163	2	ꞷ	ꞷ	PROPN
cana-1220	163	3	∗	∗	X
cana-1220	163	4	(	(	PUNCT
cana-1220	163	5	v2v3	v2v3	NOUN
cana-1220	163	6	)	)	PUNCT
cana-1220	163	7	=	=	NOUN
cana-1220	163	8	𝑐1	𝑐1	NOUN
cana-1220	163	9	and	and	CCONJ
cana-1220	163	10	ꞷ	ꞷ	PRON
cana-1220	163	11	∗	∗	NOUN
cana-1220	163	12	(	(	PUNCT
cana-1220	163	13	v3v4	v3v4	NOUN
cana-1220	163	14	)	)	PUNCT
cana-1220	163	15	=	=	NOUN
cana-1220	164	1	𝑐1	𝑐1	NOUN
cana-1220	164	2	receives	receive	VERB
cana-1220	164	3	same	same	ADJ
cana-1220	164	4	color	color	NOUN
cana-1220	164	5	which	which	PRON
cana-1220	164	6	is	be	AUX
cana-1220	164	7	a	a	DET
cana-1220	164	8	contradiction	contradiction	NOUN
cana-1220	164	9	with	with	ADP
cana-1220	164	10	the	the	DET
cana-1220	164	11	definition	definition	NOUN
cana-1220	164	12	of	of	ADP
cana-1220	164	13	prime	prime	ADJ
cana-1220	164	14	graceful	graceful	ADJ
cana-1220	164	15	coloring	coloring	NOUN
cana-1220	164	16	since	since	SCONJ
cana-1220	164	17	the	the	DET
cana-1220	164	18	color	color	NOUN
cana-1220	164	19	of	of	ADP
cana-1220	164	20	any	any	DET
cana-1220	164	21	two	two	NUM
cana-1220	164	22	adjacent	adjacent	ADJ
cana-1220	164	23	edges	edge	NOUN
cana-1220	164	24	are	be	AUX
cana-1220	164	25	distinct	distinct	ADJ
cana-1220	164	26	.	.	PUNCT
cana-1220	165	1	thus	thus	ADV
cana-1220	165	2	,	,	PUNCT
cana-1220	165	3	χpg(n	χpg(n	PROPN
cana-1220	165	4	-	-	PUNCT
cana-1220	165	5	pan	pan	NOUN
cana-1220	165	6	graph)≥	graph)≥	PROPN
cana-1220	165	7	5	5	NUM
cana-1220	165	8	.	.	PUNCT
cana-1220	166	1	therefore	therefore	ADV
cana-1220	166	2	,	,	PUNCT
cana-1220	166	3	χpg(n	χpg(n	PROPN
cana-1220	166	4	-	-	PUNCT
cana-1220	166	5	pan	pan	NOUN
cana-1220	166	6	graph)=	graph)=	NOUN
cana-1220	166	7	5	5	NUM
cana-1220	166	8	for	for	ADP
cana-1220	166	9	n	n	DET
cana-1220	166	10	≠3m+3	≠3m+3	NOUN
cana-1220	166	11	,	,	PUNCT
cana-1220	166	12	m∈n	m∈n	PROPN
cana-1220	166	13	.	.	PUNCT
cana-1220	166	14	case	case	NOUN
cana-1220	166	15	3	3	NUM
cana-1220	166	16	:	:	PUNCT
cana-1220	166	17	n	n	PROPN
cana-1220	166	18	=	=	SYM
cana-1220	166	19	3	3	NUM
cana-1220	166	20	m	m	NOUN
cana-1220	166	21	+	+	NOUN
cana-1220	166	22	3	3	NUM
cana-1220	166	23	define	define	VERB
cana-1220	166	24	a	a	DET
cana-1220	166	25	proper	proper	ADJ
cana-1220	166	26	vertex	vertex	NOUN
cana-1220	166	27	coloring	color	VERB
cana-1220	166	28	ꞷ	ꞷ	PRON
cana-1220	166	29	:	:	PUNCT
cana-1220	166	30	v(g	v(g	ADJ
cana-1220	166	31	)	)	PUNCT
cana-1220	167	1	→{c1	→{c1	PROPN
cana-1220	167	2	,	,	PUNCT
cana-1220	167	3	c2	c2	PROPN
cana-1220	167	4	,	,	PUNCT
cana-1220	167	5	c3	c3	PROPN
cana-1220	167	6	,	,	PUNCT
cana-1220	167	7	c4	c4	NOUN
cana-1220	167	8	,	,	PUNCT
cana-1220	167	9	c5}as	c5}as	PRON
cana-1220	167	10	follows	follow	VERB
cana-1220	167	11	:	:	PUNCT
cana-1220	167	12	for	for	ADP
cana-1220	167	13	1≤	1≤	NUM
cana-1220	167	14	i	i	PRON
cana-1220	167	15	≤	≤	PROPN
cana-1220	167	16	n+1	n+1	PROPN
cana-1220	167	17	,	,	PUNCT
cana-1220	167	18	such	such	ADJ
cana-1220	167	19	that	that	DET
cana-1220	167	20	gcd	gcd	NOUN
cana-1220	167	21	(	(	PUNCT
cana-1220	167	22	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	167	23	)	)	PUNCT
cana-1220	167	24	,	,	PUNCT
cana-1220	167	25	ꞷ(vi+1	ꞷ(vi+1	ADJ
cana-1220	167	26	)	)	PUNCT
cana-1220	167	27	)	)	PUNCT
cana-1220	168	1	=	=	PUNCT
cana-1220	168	2	1	1	X
cana-1220	168	3	.	.	X
cana-1220	168	4	it	it	PRON
cana-1220	168	5	induce	induce	VERB
cana-1220	168	6	a	a	DET
cana-1220	168	7	proper	proper	ADJ
cana-1220	168	8	edge	edge	NOUN
cana-1220	168	9	coloring	coloring	NOUN
cana-1220	168	10	ꞷ*:e(g)→{c1	ꞷ*:e(g)→{c1	NOUN
cana-1220	168	11	,	,	PUNCT
cana-1220	168	12	c2	c2	PROPN
cana-1220	168	13	,	,	PUNCT
cana-1220	168	14	c3	c3	PROPN
cana-1220	168	15	}	}	PUNCT
cana-1220	168	16	as	as	SCONJ
cana-1220	168	17	follows	follow	VERB
cana-1220	168	18	:	:	PUNCT
cana-1220	168	19	ꞷ(vi	ꞷ(vi	X
cana-1220	168	20	)	)	PUNCT
cana-1220	168	21	=	=	PRON
cana-1220	168	22	{	{	PUNCT
cana-1220	168	23	𝑐1	𝑐1	NOUN
cana-1220	168	24	,	,	PUNCT
cana-1220	168	25	for	for	ADP
cana-1220	168	26	𝑣2	𝑣2	NOUN
cana-1220	168	27	𝑐2	𝑐2	NOUN
cana-1220	168	28	,	,	PUNCT
cana-1220	168	29	for	for	ADP
cana-1220	168	30	i	i	PRON
cana-1220	168	31	≡	≡	PROPN
cana-1220	168	32	1	1	NUM
cana-1220	168	33	mod	mod	NOUN
cana-1220	168	34	3	3	NUM
cana-1220	168	35	and	and	CCONJ
cana-1220	168	36	i	i	PRON
cana-1220	168	37	≠	≠	ADJ
cana-1220	168	38	1	1	NUM
cana-1220	168	39	𝑐3	𝑐3	NOUN
cana-1220	168	40	,	,	PUNCT
cana-1220	168	41	for	for	ADP
cana-1220	168	42	i	i	PROPN
cana-1220	168	43	≡	≡	PROPN
cana-1220	168	44	2	2	NUM
cana-1220	168	45	mod	mod	NOUN
cana-1220	168	46	3	3	NUM
cana-1220	168	47	,	,	PUNCT
cana-1220	168	48	vn+1	vn+1	PROPN
cana-1220	168	49	and	and	CCONJ
cana-1220	168	50	i	i	PRON
cana-1220	168	51	≠	≠	PROPN
cana-1220	168	52	2	2	NUM
cana-1220	168	53	𝑐4	𝑐4	NOUN
cana-1220	168	54	,	,	PUNCT
cana-1220	168	55	for	for	ADP
cana-1220	168	56	𝑣1	𝑣1	NOUN
cana-1220	168	57	𝑐5	𝑐5	NOUN
cana-1220	168	58	,	,	PUNCT
cana-1220	168	59	for	for	ADP
cana-1220	168	60	i	i	PROPN
cana-1220	168	61	≡	≡	PROPN
cana-1220	168	62	0	0	PUNCT
cana-1220	168	63	mod	mod	PROPN
cana-1220	168	64	3	3	NUM
cana-1220	168	65	ꞷ	ꞷ	SYM
cana-1220	168	66	*	*	PUNCT
cana-1220	168	67	(	(	PUNCT
cana-1220	168	68	vivi+1	vivi+1	NOUN
cana-1220	168	69	)	)	PUNCT
cana-1220	168	70	=	=	SYM
cana-1220	168	71	{	{	PUNCT
cana-1220	168	72	𝑐4	𝑐4	PROPN
cana-1220	168	73	,	,	PUNCT
cana-1220	168	74	for	for	ADP
cana-1220	168	75	𝑣2𝑣3	𝑣2𝑣3	PROPN
cana-1220	168	76	𝑐3	𝑐3	NOUN
cana-1220	168	77	,	,	PUNCT
cana-1220	168	78	for	for	ADP
cana-1220	168	79	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1220	168	80	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1220	168	81	i	i	PROPN
cana-1220	168	82	≡	≡	PROPN
cana-1220	168	83	0	0	NUM
cana-1220	169	1	mod	mod	ADJ
cana-1220	169	2	3	3	NUM
cana-1220	169	3	𝑐2	𝑐2	NOUN
cana-1220	169	4	,	,	PUNCT
cana-1220	169	5	for	for	ADP
cana-1220	169	6	i	i	PROPN
cana-1220	169	7	≡	≡	PROPN
cana-1220	169	8	2	2	NUM
cana-1220	169	9	mod	mod	NOUN
cana-1220	169	10	3	3	NUM
cana-1220	169	11	,	,	PUNCT
cana-1220	169	12	v2vn+1	v2vn+1	NOUN
cana-1220	169	13	and	and	CCONJ
cana-1220	169	14	i	i	PRON
cana-1220	169	15	≠	≠	PROPN
cana-1220	169	16	2	2	NUM
cana-1220	169	17	𝑐1	𝑐1	NOUN
cana-1220	169	18	,	,	PUNCT
cana-1220	169	19	for	for	ADP
cana-1220	169	20	vnv1	vnv1	NOUN
cana-1220	169	21	,	,	PUNCT
cana-1220	169	22	i	i	PRON
cana-1220	169	23	≡	≡	PROPN
cana-1220	169	24	1	1	NUM
cana-1220	169	25	mod	mod	NOUN
cana-1220	169	26	3	3	NUM
cana-1220	170	1	and	and	CCONJ
cana-1220	170	2	i	i	PRON
cana-1220	170	3	≠	≠	PROPN
cana-1220	170	4	1	1	NUM
cana-1220	170	5	it	it	PRON
cana-1220	170	6	satisfy	satisfy	VERB
cana-1220	170	7	ꞷ	ꞷ	PROPN
cana-1220	170	8	∗	∗	NOUN
cana-1220	170	9	(	(	PUNCT
cana-1220	170	10	vivi+1)=	vivi+1)=	NOUN
cana-1220	171	1	|	|	ADV
cana-1220	171	2	ꞷ	ꞷ	X
cana-1220	171	3	(	(	PUNCT
cana-1220	171	4	vi	vi	PROPN
cana-1220	171	5	)	)	PUNCT
cana-1220	171	6	ꞷ	ꞷ	PROPN
cana-1220	171	7	(	(	PUNCT
cana-1220	171	8	𝑣𝑖+1)|	𝑣𝑖+1)|	PRON
cana-1220	171	9	hence	hence	ADV
cana-1220	171	10	,	,	PUNCT
cana-1220	171	11	adjacent	adjacent	ADJ
cana-1220	171	12	vertices	vertex	NOUN
cana-1220	171	13	and	and	CCONJ
cana-1220	171	14	edges	edge	NOUN
cana-1220	171	15	receive	receive	VERB
cana-1220	171	16	distinct	distinct	ADJ
cana-1220	171	17	colors	color	NOUN
cana-1220	171	18	.	.	PUNCT
cana-1220	172	1	thus	thus	ADV
cana-1220	172	2	,	,	PUNCT
cana-1220	172	3	χpg(n	χpg(n	PROPN
cana-1220	172	4	-	-	PUNCT
cana-1220	172	5	pan	pan	NOUN
cana-1220	172	6	graph)≤5	graph)≤5	NOUN
cana-1220	172	7	.	.	PUNCT
cana-1220	173	1	to	to	PART
cana-1220	173	2	prove	prove	VERB
cana-1220	173	3	χpg(npan	χpg(npan	NOUN
cana-1220	173	4	graph)≥	graph)≥	PROPN
cana-1220	173	5	5	5	NUM
cana-1220	173	6	,	,	PUNCT
cana-1220	173	7	let	let	VERB
cana-1220	173	8	us	we	PRON
cana-1220	173	9	assume	assume	VERB
cana-1220	173	10	that	that	SCONJ
cana-1220	173	11	χpg(n	χpg(n	PROPN
cana-1220	173	12	-	-	PUNCT
cana-1220	173	13	pan	pan	NOUN
cana-1220	173	14	graph	graph	NOUN
cana-1220	173	15	)	)	PUNCT
cana-1220	173	16	<	<	X
cana-1220	173	17	5	5	NUM
cana-1220	173	18	,	,	PUNCT
cana-1220	173	19	say	say	VERB
cana-1220	173	20	4	4	X
cana-1220	173	21	.	.	X
cana-1220	174	1	we	we	PRON
cana-1220	174	2	define	define	VERB
cana-1220	174	3	vertex	vertex	NOUN
cana-1220	174	4	coloring	coloring	NOUN
cana-1220	174	5	as	as	ADP
cana-1220	174	6	c4	c4	NOUN
cana-1220	174	7	,	,	PUNCT
cana-1220	174	8	c1	c1	PROPN
cana-1220	174	9	,	,	PUNCT
cana-1220	174	10	c2	c2	PROPN
cana-1220	174	11	,	,	PUNCT
cana-1220	174	12	c3	c3	PROPN
cana-1220	174	13	,	,	PUNCT
cana-1220	174	14	c1	c1	PROPN
cana-1220	174	15	,	,	PUNCT
cana-1220	174	16	c3	c3	NOUN
cana-1220	174	17	for	for	ADP
cana-1220	174	18	6	6	NUM
cana-1220	174	19	-	-	PUNCT
cana-1220	174	20	pan	pan	NOUN
cana-1220	174	21	graph	graph	NOUN
cana-1220	174	22	and	and	CCONJ
cana-1220	174	23	the	the	DET
cana-1220	174	24	vertex	vertex	NOUN
cana-1220	174	25	with	with	ADP
cana-1220	174	26	degree	degree	NOUN
cana-1220	174	27	1	1	NUM
cana-1220	174	28	is	be	AUX
cana-1220	174	29	assigned	assign	VERB
cana-1220	174	30	with	with	ADP
cana-1220	174	31	the	the	DET
cana-1220	174	32	color	color	NOUN
cana-1220	174	33	c3	c3	PROPN
cana-1220	174	34	.	.	PUNCT
cana-1220	175	1	then	then	ADV
cana-1220	175	2	,	,	PUNCT
cana-1220	175	3	ꞷ	ꞷ	PROPN
cana-1220	175	4	∗	∗	NOUN
cana-1220	175	5	(	(	PUNCT
cana-1220	175	6	v1v2)=	v1v2)=	ADJ
cana-1220	175	7	𝑐3	𝑐3	NOUN
cana-1220	175	8	,	,	PUNCT
cana-1220	175	9	ꞷ	ꞷ	PROPN
cana-1220	175	10	∗	∗	NOUN
cana-1220	175	11	(	(	PUNCT
cana-1220	175	12	v2v3	v2v3	NOUN
cana-1220	175	13	)	)	PUNCT
cana-1220	175	14	=	=	NOUN
cana-1220	175	15	𝑐1	𝑐1	NOUN
cana-1220	175	16	,	,	PUNCT
cana-1220	175	17	ꞷ	ꞷ	PROPN
cana-1220	175	18	∗	∗	NOUN
cana-1220	175	19	(	(	PUNCT
cana-1220	175	20	v2v6	v2v6	NOUN
cana-1220	175	21	)	)	PUNCT
cana-1220	175	22	=	=	SYM
cana-1220	175	23	𝑐2	𝑐2	PROPN
cana-1220	175	24	,	,	PUNCT
cana-1220	175	25	ꞷ	ꞷ	PROPN
cana-1220	175	26	∗	∗	NOUN
cana-1220	175	27	(	(	PUNCT
cana-1220	175	28	v3v4	v3v4	NOUN
cana-1220	175	29	)	)	PUNCT
cana-1220	175	30	=	=	NOUN
cana-1220	176	1	𝑐1	𝑐1	NOUN
cana-1220	176	2	,	,	PUNCT
cana-1220	176	3	ꞷ	ꞷ	PROPN
cana-1220	176	4	∗	∗	NOUN
cana-1220	176	5	(	(	PUNCT
cana-1220	176	6	v4v5	v4v5	NOUN
cana-1220	176	7	)	)	PUNCT
cana-1220	176	8	=	=	SYM
cana-1220	177	1	𝑐2	𝑐2	NOUN
cana-1220	177	2	,	,	PUNCT
cana-1220	177	3	ꞷ	ꞷ	PROPN
cana-1220	177	4	∗	∗	NOUN
cana-1220	177	5	(	(	PUNCT
cana-1220	177	6	v5v6	v5v6	NOUN
cana-1220	177	7	)	)	PUNCT
cana-1220	177	8	=	=	SYM
cana-1220	178	1	𝑐2,ꞷ	𝑐2,ꞷ	NOUN
cana-1220	178	2	∗	∗	NOUN
cana-1220	178	3	(	(	PUNCT
cana-1220	178	4	v6v1	v6v1	NOUN
cana-1220	178	5	)	)	PUNCT
cana-1220	178	6	=	=	NOUN
cana-1220	178	7	𝑐1	𝑐1	NOUN
cana-1220	178	8	.	.	PUNCT
cana-1220	179	1	since	since	SCONJ
cana-1220	179	2	ꞷ	ꞷ	PROPN
cana-1220	179	3	∗	∗	X
cana-1220	179	4	(	(	PUNCT
cana-1220	179	5	v2v3	v2v3	NOUN
cana-1220	179	6	)	)	PUNCT
cana-1220	179	7	=	=	NOUN
cana-1220	179	8	𝑐1	𝑐1	NOUN
cana-1220	179	9	and	and	CCONJ
cana-1220	179	10	ꞷ	ꞷ	PRON
cana-1220	179	11	∗	∗	NOUN
cana-1220	179	12	(	(	PUNCT
cana-1220	179	13	v3v4	v3v4	NOUN
cana-1220	179	14	)	)	PUNCT
cana-1220	180	1	=	=	NOUN
cana-1220	180	2	𝑐1	𝑐1	NOUN
cana-1220	180	3	receive	receive	VERB
cana-1220	180	4	same	same	ADJ
cana-1220	180	5	color	color	NOUN
cana-1220	180	6	which	which	PRON
cana-1220	180	7	is	be	AUX
cana-1220	180	8	a	a	DET
cana-1220	180	9	contradiction	contradiction	NOUN
cana-1220	180	10	with	with	ADP
cana-1220	180	11	the	the	DET
cana-1220	180	12	definition	definition	NOUN
cana-1220	180	13	of	of	ADP
cana-1220	180	14	prime	prime	ADJ
cana-1220	180	15	graceful	graceful	ADJ
cana-1220	180	16	coloring	coloring	NOUN
cana-1220	180	17	since	since	SCONJ
cana-1220	180	18	the	the	DET
cana-1220	180	19	color	color	NOUN
cana-1220	180	20	of	of	ADP
cana-1220	180	21	any	any	DET
cana-1220	180	22	two	two	NUM
cana-1220	180	23	adjacent	adjacent	ADJ
cana-1220	180	24	edges	edge	NOUN
cana-1220	180	25	are	be	AUX
cana-1220	180	26	distinct	distinct	ADJ
cana-1220	180	27	.	.	PUNCT
cana-1220	181	1	thus	thus	ADV
cana-1220	181	2	,	,	PUNCT
cana-1220	181	3	χpg(npan	χpg(npan	X
cana-1220	181	4	graph)≥	graph)≥	PROPN
cana-1220	181	5	5	5	NUM
cana-1220	181	6	.	.	PUNCT
cana-1220	182	1	therefore	therefore	ADV
cana-1220	182	2	,	,	PUNCT
cana-1220	182	3	χpg(n	χpg(n	PROPN
cana-1220	182	4	-	-	PUNCT
cana-1220	182	5	pan	pan	NOUN
cana-1220	182	6	graph)=	graph)=	NOUN
cana-1220	182	7	5	5	NUM
cana-1220	182	8	for	for	ADP
cana-1220	182	9	n	n	DET
cana-1220	182	10	=	=	SYM
cana-1220	182	11	3m+3	3m+3	PROPN
cana-1220	182	12	,	,	PUNCT
cana-1220	182	13	m∈n	m∈n	PROPN
cana-1220	182	14	.	.	PUNCT
cana-1220	182	15	figure	figure	VERB
cana-1220	182	16	5	5	NUM
cana-1220	182	17	analytical	analytical	ADJ
cana-1220	182	18	evaluation	evaluation	NOUN
cana-1220	182	19	of	of	ADP
cana-1220	182	20	the	the	DET
cana-1220	182	21	5	5	NUM
cana-1220	182	22	pan	pan	NOUN
cana-1220	182	23	graph	graph	NOUN
cana-1220	182	24	theorem	theorem	VERB
cana-1220	182	25	:	:	PUNCT
cana-1220	182	26	3.8	3.8	NUM
cana-1220	182	27	let	let	VERB
cana-1220	182	28	bn	bn	ADV
cana-1220	182	29	,	,	PUNCT
cana-1220	182	30	n	n	PRON
cana-1220	182	31	be	be	VERB
cana-1220	182	32	a	a	DET
cana-1220	182	33	bistar	bistar	NOUN
cana-1220	182	34	graph	graph	NOUN
cana-1220	182	35	,	,	PUNCT
cana-1220	182	36	then	then	ADV
cana-1220	182	37	χpg(bn	χpg(bn	NUM
cana-1220	182	38	,	,	PUNCT
cana-1220	182	39	n	n	NOUN
cana-1220	182	40	)	)	PUNCT
cana-1220	183	1	=	=	NOUN
cana-1220	183	2	{	{	PUNCT
cana-1220	183	3	n	n	PROPN
cana-1220	183	4	+	+	NUM
cana-1220	183	5	2	2	NUM
cana-1220	183	6	,	,	PUNCT
cana-1220	183	7	when	when	SCONJ
cana-1220	183	8	n	n	PRON
cana-1220	183	9	is	be	AUX
cana-1220	183	10	odd	odd	ADJ
cana-1220	183	11	n	n	NOUN
cana-1220	183	12	+	+	NUM
cana-1220	183	13	3	3	NUM
cana-1220	183	14	,	,	PUNCT
cana-1220	183	15	when	when	SCONJ
cana-1220	183	16	n	n	PRON
cana-1220	183	17	is	be	AUX
cana-1220	183	18	even	even	ADV
cana-1220	183	19	n	n	PRON
cana-1220	183	20	+	+	NOUN
cana-1220	183	21	4	4	NUM
cana-1220	183	22	,	,	PUNCT
cana-1220	183	23	when	when	SCONJ
cana-1220	183	24	n	n	X
cana-1220	183	25	≡	≡	PROPN
cana-1220	183	26	1	1	NUM
cana-1220	183	27	mod	mod	NOUN
cana-1220	183	28	6	6	NUM
cana-1220	183	29	and	and	CCONJ
cana-1220	183	30	n	n	CCONJ
cana-1220	183	31	≠	≠	PROPN
cana-1220	183	32	1	1	NUM
cana-1220	183	33	n	n	NOUN
cana-1220	183	34	+	+	NUM
cana-1220	183	35	5	5	NUM
cana-1220	183	36	,	,	PUNCT
cana-1220	183	37	when	when	SCONJ
cana-1220	183	38	n	n	X
cana-1220	183	39	≡	≡	PROPN
cana-1220	183	40	0	0	NUM
cana-1220	183	41	mod	mod	ADJ
cana-1220	183	42	6	6	NUM
cana-1220	183	43	communications	communication	NOUN
cana-1220	183	44	on	on	ADP
cana-1220	183	45	applied	apply	VERB
cana-1220	183	46	nonlinear	nonlinear	ADJ
cana-1220	183	47	analysis	analysis	NOUN
cana-1220	183	48	issn	issn	NOUN
cana-1220	183	49	:	:	PUNCT
cana-1220	183	50	1074	1074	NUM
cana-1220	183	51	-	-	PUNCT
cana-1220	183	52	133x	133x	NUM
cana-1220	183	53	vol	vol	NOUN
cana-1220	183	54	31	31	NUM
cana-1220	183	55	no	no	NOUN
cana-1220	183	56	.	.	PUNCT
cana-1220	184	1	6s	6s	NUM
cana-1220	184	2	(	(	PUNCT
cana-1220	184	3	2024	2024	NUM
cana-1220	184	4	)	)	PUNCT
cana-1220	184	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	184	6	266	266	NUM
cana-1220	184	7	proof	proof	NOUN
cana-1220	184	8	:	:	PUNCT
cana-1220	184	9	the	the	DET
cana-1220	184	10	bistar	bistar	NOUN
cana-1220	184	11	graph	graph	NOUN
cana-1220	184	12	has	have	VERB
cana-1220	184	13	2n+2	2n+2	PROPN
cana-1220	184	14	vertices	vertex	NOUN
cana-1220	184	15	and	and	CCONJ
cana-1220	184	16	2n+1	2n+1	NOUN
cana-1220	184	17	edges	edge	NOUN
cana-1220	184	18	.	.	PUNCT
cana-1220	185	1	case	case	NOUN
cana-1220	185	2	1	1	NUM
cana-1220	185	3	:	:	PUNCT
cana-1220	185	4	n	n	PRON
cana-1220	185	5	is	be	AUX
cana-1220	185	6	odd	odd	ADJ
cana-1220	185	7	let	let	VERB
cana-1220	185	8	v	v	NOUN
cana-1220	185	9	(	(	PUNCT
cana-1220	185	10	bn	bn	NOUN
cana-1220	185	11	,	,	PUNCT
cana-1220	185	12	n	n	CCONJ
cana-1220	185	13	)	)	PUNCT
cana-1220	185	14	=	=	PRON
cana-1220	185	15	{	{	PUNCT
cana-1220	186	1	ui	ui	NOUN
cana-1220	186	2	:1	:1	PUNCT
cana-1220	186	3	≤	≤	NUM
cana-1220	186	4	i	i	PRON
cana-1220	186	5	≤	≤	ADJ
cana-1220	186	6	n	n	CCONJ
cana-1220	186	7	+	+	CCONJ
cana-1220	186	8	1	1	NUM
cana-1220	186	9	}	}	PUNCT
cana-1220	186	10	∪	∪	ADJ
cana-1220	186	11	{	{	PUNCT
cana-1220	186	12	vi	vi	NOUN
cana-1220	186	13	:1	:1	PUNCT
cana-1220	186	14	≤	≤	NUM
cana-1220	186	15	i	i	PRON
cana-1220	186	16	≤	≤	ADJ
cana-1220	186	17	n	n	CCONJ
cana-1220	186	18	+	+	NOUN
cana-1220	186	19	1	1	NUM
cana-1220	186	20	}	}	PUNCT
cana-1220	186	21	.	.	PUNCT
cana-1220	187	1	define	define	VERB
cana-1220	187	2	a	a	DET
cana-1220	187	3	proper	proper	ADJ
cana-1220	187	4	vertex	vertex	NOUN
cana-1220	187	5	coloring	color	VERB
cana-1220	187	6	ꞷ	ꞷ	NOUN
cana-1220	187	7	:	:	PUNCT
cana-1220	187	8	v(bn	v(bn	VERB
cana-1220	187	9	,	,	PUNCT
cana-1220	187	10	n	n	NOUN
cana-1220	187	11	)	)	PUNCT
cana-1220	187	12	→{c1	→{c1	PROPN
cana-1220	187	13	,	,	PUNCT
cana-1220	187	14	c2	c2	PROPN
cana-1220	187	15	,	,	PUNCT
cana-1220	187	16	c3	c3	PROPN
cana-1220	187	17	…	…	PUNCT
cana-1220	187	18	…	…	PUNCT
cana-1220	187	19	cn+2	cn+2	X
cana-1220	187	20	}	}	PUNCT
cana-1220	187	21	.	.	PUNCT
cana-1220	188	1	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	188	2	)	)	PUNCT
cana-1220	189	1	=	=	SYM
cana-1220	189	2	c1	c1	PROPN
cana-1220	189	3	and	and	CCONJ
cana-1220	189	4	ꞷ	ꞷ	PROPN
cana-1220	189	5	(	(	PUNCT
cana-1220	189	6	v1)=	v1)=	ADP
cana-1220	189	7	cn+2	cn+2	X
cana-1220	189	8	and	and	CCONJ
cana-1220	189	9	ꞷ(ui	ꞷ(ui	NUM
cana-1220	189	10	)	)	PUNCT
cana-1220	189	11	=	=	SYM
cana-1220	189	12	ci	ci	PROPN
cana-1220	189	13	,	,	PUNCT
cana-1220	189	14	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	189	15	)	)	PUNCT
cana-1220	189	16	=	=	SYM
cana-1220	189	17	ci	ci	NOUN
cana-1220	189	18	ɐ	ɐ	PROPN
cana-1220	189	19	2	2	NUM
cana-1220	189	20	≤	≤	NUM
cana-1220	189	21	i	i	PRON
cana-1220	189	22	≤	≤	ADJ
cana-1220	189	23	n	n	CCONJ
cana-1220	189	24	+	+	NOUN
cana-1220	189	25	1	1	NUM
cana-1220	189	26	.	.	X
cana-1220	189	27	such	such	ADJ
cana-1220	189	28	that	that	DET
cana-1220	189	29	gcd	gcd	NOUN
cana-1220	189	30	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	189	31	,	,	PUNCT
cana-1220	189	32	ui	ui	NOUN
cana-1220	189	33	)	)	PUNCT
cana-1220	189	34	=	=	SYM
cana-1220	189	35	1	1	NUM
cana-1220	189	36	,	,	PUNCT
cana-1220	189	37	gcd	gcd	NOUN
cana-1220	189	38	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	189	39	,	,	PUNCT
cana-1220	189	40	vi	vi	NOUN
cana-1220	189	41	)	)	PUNCT
cana-1220	189	42	=	=	SYM
cana-1220	189	43	1	1	NUM
cana-1220	189	44	and	and	CCONJ
cana-1220	189	45	gcd	gcd	VERB
cana-1220	189	46	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	189	47	,	,	PUNCT
cana-1220	189	48	v1	v1	NOUN
cana-1220	189	49	)	)	PUNCT
cana-1220	189	50	=	=	SYM
cana-1220	190	1	1	1	X
cana-1220	190	2	.	.	X
cana-1220	190	3	define	define	VERB
cana-1220	190	4	a	a	DET
cana-1220	190	5	proper	proper	ADJ
cana-1220	190	6	edge	edge	NOUN
cana-1220	190	7	coloring	color	VERB
cana-1220	190	8	ꞷ*:e(bn	ꞷ*:e(bn	PROPN
cana-1220	190	9	,	,	PUNCT
cana-1220	190	10	n)→	n)→	PROPN
cana-1220	190	11	{	{	PUNCT
cana-1220	190	12	c1	c1	NOUN
cana-1220	190	13	,	,	PUNCT
cana-1220	190	14	c2	c2	PROPN
cana-1220	190	15	,	,	PUNCT
cana-1220	190	16	c3	c3	PROPN
cana-1220	190	17	…	…	PUNCT
cana-1220	190	18	…	…	PUNCT
cana-1220	190	19	cn+1	cn+1	VERB
cana-1220	190	20	}	}	PUNCT
cana-1220	190	21	ꞷ	ꞷ	NOUN
cana-1220	190	22	∗	∗	NOUN
cana-1220	190	23	(	(	PUNCT
cana-1220	190	24	u1v1)=	u1v1)=	PROPN
cana-1220	190	25	c1	c1	PROPN
cana-1220	190	26	−	−	PROPN
cana-1220	190	27	cn+2	cn+2	PROPN
cana-1220	191	1	=	=	SYM
cana-1220	191	2	cn+1	cn+1	PROPN
cana-1220	191	3	,	,	PUNCT
cana-1220	191	4	ꞷ	ꞷ	PROPN
cana-1220	191	5	∗	∗	NOUN
cana-1220	191	6	(	(	PUNCT
cana-1220	191	7	u1ui)=	u1ui)=	NUM
cana-1220	191	8	c1	c1	NOUN
cana-1220	191	9	−	−	PROPN
cana-1220	191	10	ci	ci	PROPN
cana-1220	191	11	=	=	PROPN
cana-1220	191	12	ci−1	ci−1	PROPN
cana-1220	191	13	,	,	PUNCT
cana-1220	191	14	ꞷ	ꞷ	PROPN
cana-1220	191	15	∗	∗	NOUN
cana-1220	191	16	(	(	PUNCT
cana-1220	191	17	v1vi)=	v1vi)=	NUM
cana-1220	191	18	cn+2	cn+2	NUM
cana-1220	191	19	−	−	PROPN
cana-1220	191	20	ci	ci	NOUN
cana-1220	191	21	=	=	NOUN
cana-1220	191	22	cn+2−i	cn+2−i	NOUN
cana-1220	191	23	hence	hence	ADV
cana-1220	191	24	,	,	PUNCT
cana-1220	191	25	adjacent	adjacent	ADJ
cana-1220	191	26	vertices	vertex	NOUN
cana-1220	191	27	and	and	CCONJ
cana-1220	191	28	edges	edge	NOUN
cana-1220	191	29	receive	receive	VERB
cana-1220	191	30	distinct	distinct	ADJ
cana-1220	191	31	colors	color	NOUN
cana-1220	191	32	.	.	PUNCT
cana-1220	192	1	we	we	PRON
cana-1220	192	2	proved	prove	VERB
cana-1220	192	3	that	that	SCONJ
cana-1220	192	4	χpg(bn	χpg(bn	PROPN
cana-1220	192	5	,	,	PUNCT
cana-1220	192	6	n	n	CCONJ
cana-1220	192	7	)	)	PUNCT
cana-1220	192	8	≤	≤	NOUN
cana-1220	192	9	n	n	PRON
cana-1220	192	10	+	+	NOUN
cana-1220	192	11	2	2	X
cana-1220	192	12	.	.	PUNCT
cana-1220	192	13	to	to	PART
cana-1220	192	14	prove	prove	VERB
cana-1220	192	15	χpg(bn	χpg(bn	NOUN
cana-1220	192	16	,	,	PUNCT
cana-1220	192	17	n	n	CCONJ
cana-1220	192	18	)	)	PUNCT
cana-1220	192	19	≥	≥	NOUN
cana-1220	192	20	n	n	NOUN
cana-1220	192	21	+	+	NUM
cana-1220	192	22	2	2	NUM
cana-1220	192	23	,	,	PUNCT
cana-1220	192	24	let	let	VERB
cana-1220	192	25	us	we	PRON
cana-1220	192	26	assume	assume	VERB
cana-1220	192	27	that	that	SCONJ
cana-1220	192	28	χpg(bn	χpg(bn	PROPN
cana-1220	192	29	,	,	PUNCT
cana-1220	192	30	n	n	CCONJ
cana-1220	192	31	)	)	PUNCT
cana-1220	192	32	<	<	X
cana-1220	193	1	n	n	PROPN
cana-1220	193	2	+	+	CCONJ
cana-1220	193	3	2	2	NUM
cana-1220	193	4	say	say	VERB
cana-1220	193	5	n	n	NOUN
cana-1220	193	6	+	+	NOUN
cana-1220	193	7	1	1	X
cana-1220	193	8	.	.	X
cana-1220	194	1	we	we	PRON
cana-1220	194	2	define	define	VERB
cana-1220	194	3	proper	proper	ADJ
cana-1220	194	4	vertex	vertex	NOUN
cana-1220	194	5	coloring	coloring	NOUN
cana-1220	194	6	of	of	ADP
cana-1220	194	7	bn	bn	PROPN
cana-1220	194	8	,	,	PUNCT
cana-1220	194	9	n	n	X
cana-1220	194	10	is	be	AUX
cana-1220	194	11	ꞷ(ui	ꞷ(ui	X
cana-1220	194	12	)	)	PUNCT
cana-1220	195	1	=	=	SYM
cana-1220	195	2	ci	ci	PROPN
cana-1220	195	3	,	,	PUNCT
cana-1220	195	4	ꞷ	ꞷ	PROPN
cana-1220	195	5	(	(	PUNCT
cana-1220	195	6	vi)=ci	vi)=ci	NOUN
cana-1220	195	7	ɐ	ɐ	NOUN
cana-1220	195	8	2	2	NUM
cana-1220	195	9	≤	≤	NUM
cana-1220	195	10	i	i	PRON
cana-1220	195	11	≤	≤	NOUN
cana-1220	195	12	n+1	n+1	ADV
cana-1220	195	13	since	since	SCONJ
cana-1220	195	14	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	195	15	)	)	PUNCT
cana-1220	195	16	and	and	CCONJ
cana-1220	195	17	ꞷ	ꞷ	PROPN
cana-1220	195	18	(	(	PUNCT
cana-1220	195	19	v1	v1	NOUN
cana-1220	195	20	)	)	PUNCT
cana-1220	195	21	are	be	AUX
cana-1220	195	22	adjacent	adjacent	ADJ
cana-1220	195	23	vertices	vertex	NOUN
cana-1220	195	24	we	we	PRON
cana-1220	195	25	can	can	AUX
cana-1220	195	26	not	not	PART
cana-1220	195	27	assign	assign	VERB
cana-1220	195	28	the	the	DET
cana-1220	195	29	same	same	ADJ
cana-1220	195	30	color	color	NOUN
cana-1220	195	31	c1	c1	NOUN
cana-1220	195	32	which	which	PRON
cana-1220	195	33	is	be	AUX
cana-1220	195	34	a	a	DET
cana-1220	195	35	contradiction	contradiction	NOUN
cana-1220	195	36	with	with	ADP
cana-1220	195	37	the	the	DET
cana-1220	195	38	definition	definition	NOUN
cana-1220	195	39	of	of	ADP
cana-1220	195	40	prime	prime	ADJ
cana-1220	195	41	graceful	graceful	ADJ
cana-1220	195	42	coloring	coloring	NOUN
cana-1220	195	43	since	since	SCONJ
cana-1220	195	44	the	the	DET
cana-1220	195	45	color	color	NOUN
cana-1220	195	46	of	of	ADP
cana-1220	195	47	any	any	DET
cana-1220	195	48	two	two	NUM
cana-1220	195	49	adjacent	adjacent	ADJ
cana-1220	195	50	vertices	vertex	NOUN
cana-1220	195	51	are	be	AUX
cana-1220	195	52	distinct	distinct	ADJ
cana-1220	195	53	.	.	PUNCT
cana-1220	196	1	thus	thus	ADV
cana-1220	196	2	,	,	PUNCT
cana-1220	196	3	χpg(bn	χpg(bn	PROPN
cana-1220	196	4	,	,	PUNCT
cana-1220	196	5	n	n	CCONJ
cana-1220	196	6	)	)	PUNCT
cana-1220	196	7	≥	≥	NOUN
cana-1220	196	8	n	n	NOUN
cana-1220	196	9	+	+	NUM
cana-1220	196	10	2	2	NUM
cana-1220	196	11	.	.	X
cana-1220	197	1	therefore	therefore	ADV
cana-1220	197	2	,	,	PUNCT
cana-1220	197	3	χpg(bn	χpg(bn	PROPN
cana-1220	197	4	,	,	PUNCT
cana-1220	197	5	n)=	n)=	NOUN
cana-1220	197	6	n+	n+	PUNCT
cana-1220	197	7	2	2	NUM
cana-1220	197	8	when	when	SCONJ
cana-1220	197	9	n	n	X
cana-1220	197	10	is	be	AUX
cana-1220	197	11	odd	odd	ADJ
cana-1220	197	12	.	.	PUNCT
cana-1220	198	1	case	case	NOUN
cana-1220	198	2	2	2	NUM
cana-1220	198	3	:	:	PUNCT
cana-1220	198	4	n	n	PRON
cana-1220	198	5	is	be	AUX
cana-1220	198	6	even	even	ADV
cana-1220	198	7	let	let	VERB
cana-1220	198	8	v	v	X
cana-1220	198	9	(	(	PUNCT
cana-1220	198	10	bn	bn	NOUN
cana-1220	198	11	,	,	PUNCT
cana-1220	198	12	n	n	CCONJ
cana-1220	198	13	)	)	PUNCT
cana-1220	198	14	=	=	PRON
cana-1220	198	15	{	{	PUNCT
cana-1220	199	1	ui	ui	NOUN
cana-1220	199	2	:1	:1	PUNCT
cana-1220	199	3	≤	≤	NUM
cana-1220	199	4	i	i	PRON
cana-1220	199	5	≤	≤	ADJ
cana-1220	199	6	n	n	CCONJ
cana-1220	199	7	+	+	CCONJ
cana-1220	199	8	1	1	NUM
cana-1220	199	9	}	}	PUNCT
cana-1220	199	10	∪	∪	ADJ
cana-1220	199	11	{	{	PUNCT
cana-1220	199	12	vi	vi	NOUN
cana-1220	199	13	:1	:1	PUNCT
cana-1220	199	14	≤	≤	NUM
cana-1220	199	15	i	i	PRON
cana-1220	199	16	≤	≤	ADJ
cana-1220	199	17	n	n	CCONJ
cana-1220	199	18	+	+	NOUN
cana-1220	199	19	1	1	NUM
cana-1220	199	20	}	}	PUNCT
cana-1220	199	21	.	.	PUNCT
cana-1220	200	1	define	define	VERB
cana-1220	200	2	a	a	DET
cana-1220	200	3	proper	proper	ADJ
cana-1220	200	4	vertex	vertex	NOUN
cana-1220	200	5	coloring	color	VERB
cana-1220	200	6	ꞷ	ꞷ	NOUN
cana-1220	200	7	:	:	PUNCT
cana-1220	200	8	v(bn	v(bn	VERB
cana-1220	200	9	,	,	PUNCT
cana-1220	200	10	n	n	NOUN
cana-1220	200	11	)	)	PUNCT
cana-1220	200	12	→{c1	→{c1	PROPN
cana-1220	200	13	,	,	PUNCT
cana-1220	200	14	c2	c2	PROPN
cana-1220	200	15	,	,	PUNCT
cana-1220	200	16	c3	c3	PROPN
cana-1220	200	17	…	…	PUNCT
cana-1220	200	18	…	…	PUNCT
cana-1220	200	19	cn+3	cn+3	NUM
cana-1220	200	20	}	}	PUNCT
cana-1220	200	21	.	.	PUNCT
cana-1220	201	1	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	201	2	)	)	PUNCT
cana-1220	202	1	=	=	SYM
cana-1220	202	2	c1	c1	PROPN
cana-1220	202	3	and	and	CCONJ
cana-1220	202	4	ꞷ	ꞷ	PROPN
cana-1220	202	5	(	(	PUNCT
cana-1220	202	6	v1)=	v1)=	ADP
cana-1220	202	7	cn+3	cn+3	PROPN
cana-1220	202	8	and	and	CCONJ
cana-1220	202	9	ꞷ(ui	ꞷ(ui	NUM
cana-1220	202	10	)	)	PUNCT
cana-1220	202	11	=	=	SYM
cana-1220	202	12	ci	ci	PROPN
cana-1220	202	13	,	,	PUNCT
cana-1220	202	14	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	202	15	)	)	PUNCT
cana-1220	202	16	=	=	SYM
cana-1220	203	1	ci+1	ci+1	NUM
cana-1220	203	2	ɐ	ɐ	NOUN
cana-1220	203	3	2	2	NUM
cana-1220	203	4	≤	≤	NUM
cana-1220	203	5	i	i	PRON
cana-1220	203	6	≤	≤	ADJ
cana-1220	204	1	n	n	CCONJ
cana-1220	205	1	+	+	NOUN
cana-1220	206	1	1	1	NUM
cana-1220	206	2	.	.	X
cana-1220	206	3	such	such	ADJ
cana-1220	206	4	that	that	DET
cana-1220	206	5	gcd	gcd	NOUN
cana-1220	206	6	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	206	7	,	,	PUNCT
cana-1220	206	8	ui	ui	NOUN
cana-1220	206	9	)	)	PUNCT
cana-1220	206	10	=	=	SYM
cana-1220	206	11	1	1	NUM
cana-1220	206	12	,	,	PUNCT
cana-1220	206	13	gcd	gcd	NOUN
cana-1220	206	14	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	206	15	,	,	PUNCT
cana-1220	206	16	vi	vi	NOUN
cana-1220	206	17	)	)	PUNCT
cana-1220	206	18	=	=	SYM
cana-1220	206	19	1	1	NUM
cana-1220	206	20	and	and	CCONJ
cana-1220	206	21	gcd	gcd	VERB
cana-1220	206	22	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	206	23	,	,	PUNCT
cana-1220	206	24	v1	v1	NOUN
cana-1220	206	25	)	)	PUNCT
cana-1220	206	26	=	=	SYM
cana-1220	206	27	1	1	X
cana-1220	206	28	.	.	X
cana-1220	206	29	define	define	VERB
cana-1220	206	30	a	a	DET
cana-1220	206	31	proper	proper	ADJ
cana-1220	206	32	edge	edge	NOUN
cana-1220	206	33	coloring	color	VERB
cana-1220	206	34	ꞷ*:e(bn	ꞷ*:e(bn	PROPN
cana-1220	206	35	,	,	PUNCT
cana-1220	206	36	n)→	n)→	PROPN
cana-1220	206	37	{	{	PUNCT
cana-1220	206	38	c1	c1	NOUN
cana-1220	206	39	,	,	PUNCT
cana-1220	206	40	c2	c2	PROPN
cana-1220	206	41	,	,	PUNCT
cana-1220	206	42	c3	c3	PROPN
cana-1220	206	43	…	…	PUNCT
cana-1220	206	44	…	…	PUNCT
cana-1220	206	45	cn+2	cn+2	SYM
cana-1220	206	46	}	}	PUNCT
cana-1220	206	47	ꞷ	ꞷ	NOUN
cana-1220	206	48	∗	∗	NOUN
cana-1220	206	49	(	(	PUNCT
cana-1220	206	50	u1v1)=	u1v1)=	PROPN
cana-1220	206	51	c1	c1	PROPN
cana-1220	206	52	−	−	PROPN
cana-1220	206	53	cn+3	cn+3	PROPN
cana-1220	206	54	=	=	SYM
cana-1220	206	55	cn+2	cn+2	PROPN
cana-1220	206	56	,	,	PUNCT
cana-1220	206	57	ꞷ	ꞷ	PROPN
cana-1220	206	58	∗	∗	NOUN
cana-1220	206	59	(	(	PUNCT
cana-1220	206	60	u1ui)=	u1ui)=	NUM
cana-1220	206	61	c1	c1	NOUN
cana-1220	206	62	−	−	PROPN
cana-1220	206	63	ci	ci	PROPN
cana-1220	206	64	=	=	PROPN
cana-1220	206	65	ci−1	ci−1	PROPN
cana-1220	206	66	,	,	PUNCT
cana-1220	206	67	ꞷ	ꞷ	PROPN
cana-1220	206	68	∗	∗	NOUN
cana-1220	206	69	(	(	PUNCT
cana-1220	206	70	v1vi)=	v1vi)=	NUM
cana-1220	206	71	cn+3	cn+3	NOUN
cana-1220	206	72	−	−	PROPN
cana-1220	206	73	ci+1	ci+1	PROPN
cana-1220	206	74	=	=	NOUN
cana-1220	206	75	cn+2−i	cn+2−i	NOUN
cana-1220	206	76	hence	hence	ADV
cana-1220	206	77	,	,	PUNCT
cana-1220	206	78	adjacent	adjacent	ADJ
cana-1220	206	79	vertices	vertex	NOUN
cana-1220	206	80	and	and	CCONJ
cana-1220	206	81	edges	edge	NOUN
cana-1220	206	82	receive	receive	VERB
cana-1220	206	83	distinct	distinct	ADJ
cana-1220	206	84	colors	color	NOUN
cana-1220	206	85	.	.	PUNCT
cana-1220	207	1	we	we	PRON
cana-1220	207	2	proved	prove	VERB
cana-1220	207	3	that	that	SCONJ
cana-1220	207	4	χpg(bn	χpg(bn	PROPN
cana-1220	207	5	,	,	PUNCT
cana-1220	207	6	n	n	CCONJ
cana-1220	207	7	)	)	PUNCT
cana-1220	207	8	≤	≤	NOUN
cana-1220	207	9	n	n	PRON
cana-1220	207	10	+	+	NOUN
cana-1220	207	11	3	3	X
cana-1220	207	12	.	.	PUNCT
cana-1220	207	13	to	to	PART
cana-1220	207	14	prove	prove	VERB
cana-1220	207	15	χpg(bn	χpg(bn	NOUN
cana-1220	207	16	,	,	PUNCT
cana-1220	207	17	n	n	CCONJ
cana-1220	207	18	)	)	PUNCT
cana-1220	207	19	≥	≥	NOUN
cana-1220	207	20	n	n	PROPN
cana-1220	207	21	+	+	NUM
cana-1220	207	22	3	3	NUM
cana-1220	207	23	,	,	PUNCT
cana-1220	207	24	let	let	VERB
cana-1220	207	25	us	we	PRON
cana-1220	207	26	assume	assume	VERB
cana-1220	207	27	that	that	SCONJ
cana-1220	207	28	χpg(bn	χpg(bn	PROPN
cana-1220	207	29	,	,	PUNCT
cana-1220	207	30	n	n	CCONJ
cana-1220	207	31	)	)	PUNCT
cana-1220	207	32	<	<	X
cana-1220	208	1	n	n	PROPN
cana-1220	208	2	+	+	CCONJ
cana-1220	208	3	3	3	NUM
cana-1220	208	4	say	say	VERB
cana-1220	208	5	n	n	X
cana-1220	208	6	+	+	NOUN
cana-1220	208	7	2	2	X
cana-1220	208	8	.	.	X
cana-1220	208	9	we	we	PRON
cana-1220	208	10	define	define	VERB
cana-1220	208	11	proper	proper	ADJ
cana-1220	208	12	vertex	vertex	NOUN
cana-1220	208	13	coloring	coloring	NOUN
cana-1220	208	14	of	of	ADP
cana-1220	208	15	bn	bn	PROPN
cana-1220	208	16	,	,	PUNCT
cana-1220	208	17	n	n	X
cana-1220	208	18	is	be	AUX
cana-1220	208	19	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	208	20	)	)	PUNCT
cana-1220	208	21	=	=	SYM
cana-1220	208	22	c1	c1	NOUN
cana-1220	208	23	,	,	PUNCT
cana-1220	208	24	ꞷ(ui	ꞷ(ui	X
cana-1220	208	25	)	)	PUNCT
cana-1220	209	1	=	=	SYM
cana-1220	209	2	ci	ci	PROPN
cana-1220	209	3	,	,	PUNCT
cana-1220	209	4	ꞷ	ꞷ	PROPN
cana-1220	209	5	(	(	PUNCT
cana-1220	209	6	vi)=ci	vi)=ci	NOUN
cana-1220	209	7	ɐ	ɐ	NOUN
cana-1220	209	8	2	2	NUM
cana-1220	209	9	≤	≤	NUM
cana-1220	209	10	i	i	PRON
cana-1220	209	11	≤	≤	NOUN
cana-1220	209	12	n+1	n+1	PROPN
cana-1220	209	13	and	and	CCONJ
cana-1220	209	14	ꞷ	ꞷ	PROPN
cana-1220	209	15	(	(	PUNCT
cana-1220	209	16	v1)=	v1)=	ADP
cana-1220	209	17	cn+2	cn+2	X
cana-1220	209	18	.	.	PUNCT
cana-1220	210	1	since	since	SCONJ
cana-1220	210	2	v1	v1	NOUN
cana-1220	210	3	is	be	AUX
cana-1220	210	4	adjacent	adjacent	ADJ
cana-1220	210	5	to	to	ADP
cana-1220	210	6	vi	vi	PROPN
cana-1220	210	7	ɐ	ɐ	PROPN
cana-1220	210	8	2	2	NUM
cana-1220	210	9	≤	≤	NUM
cana-1220	210	10	i	i	PRON
cana-1220	210	11	≤	≤	NOUN
cana-1220	210	12	n+1	n+1	ADV
cana-1220	210	13	for	for	ADP
cana-1220	210	14	each	each	DET
cana-1220	210	15	edge	edge	NOUN
cana-1220	210	16	vivj	vivj	NOUN
cana-1220	210	17	gcd	gcd	PROPN
cana-1220	210	18	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	210	19	,	,	PUNCT
cana-1220	210	20	vi	vi	NOUN
cana-1220	210	21	)	)	PUNCT
cana-1220	210	22	≠	≠	PROPN
cana-1220	210	23	1	1	NUM
cana-1220	210	24	.	.	PUNCT
cana-1220	211	1	thus	thus	ADV
cana-1220	211	2	,	,	PUNCT
cana-1220	211	3	χpg(bn	χpg(bn	PROPN
cana-1220	211	4	,	,	PUNCT
cana-1220	211	5	n	n	CCONJ
cana-1220	211	6	)	)	PUNCT
cana-1220	211	7	≥	≥	NOUN
cana-1220	211	8	n	n	PROPN
cana-1220	211	9	+	+	NUM
cana-1220	211	10	3	3	X
cana-1220	211	11	.	.	X
cana-1220	211	12	therefore	therefore	ADV
cana-1220	211	13	,	,	PUNCT
cana-1220	211	14	χpg(bn	χpg(bn	PROPN
cana-1220	211	15	,	,	PUNCT
cana-1220	211	16	n)=	n)=	NOUN
cana-1220	211	17	n+	n+	PUNCT
cana-1220	211	18	3	3	NUM
cana-1220	211	19	when	when	SCONJ
cana-1220	211	20	n	n	X
cana-1220	211	21	is	be	AUX
cana-1220	211	22	even	even	ADV
cana-1220	211	23	.	.	PUNCT
cana-1220	212	1	case	case	NOUN
cana-1220	212	2	3	3	NUM
cana-1220	212	3	:	:	PUNCT
cana-1220	212	4	n	n	NUM
cana-1220	212	5	≡	≡	PROPN
cana-1220	212	6	1	1	NUM
cana-1220	212	7	mod	mod	NOUN
cana-1220	212	8	6	6	NUM
cana-1220	212	9	and	and	CCONJ
cana-1220	212	10	n	n	PRON
cana-1220	212	11	≠	≠	PROPN
cana-1220	212	12	1	1	NUM
cana-1220	212	13	let	let	VERB
cana-1220	212	14	v	v	NOUN
cana-1220	212	15	(	(	PUNCT
cana-1220	212	16	bn	bn	NOUN
cana-1220	212	17	,	,	PUNCT
cana-1220	212	18	n	n	CCONJ
cana-1220	212	19	)	)	PUNCT
cana-1220	212	20	=	=	PRON
cana-1220	212	21	{	{	PUNCT
cana-1220	213	1	ui	ui	NOUN
cana-1220	213	2	:1	:1	PUNCT
cana-1220	213	3	≤	≤	NUM
cana-1220	213	4	i	i	PRON
cana-1220	213	5	≤	≤	ADJ
cana-1220	213	6	n	n	CCONJ
cana-1220	213	7	+	+	CCONJ
cana-1220	213	8	1	1	NUM
cana-1220	213	9	}	}	PUNCT
cana-1220	213	10	∪	∪	ADJ
cana-1220	213	11	{	{	PUNCT
cana-1220	213	12	vi	vi	NOUN
cana-1220	213	13	:1	:1	PUNCT
cana-1220	213	14	≤	≤	NUM
cana-1220	213	15	i	i	PRON
cana-1220	213	16	≤	≤	ADJ
cana-1220	213	17	n	n	CCONJ
cana-1220	213	18	+	+	NOUN
cana-1220	213	19	1	1	NUM
cana-1220	213	20	}	}	PUNCT
cana-1220	213	21	.	.	PUNCT
cana-1220	214	1	define	define	VERB
cana-1220	214	2	a	a	DET
cana-1220	214	3	proper	proper	ADJ
cana-1220	214	4	vertex	vertex	NOUN
cana-1220	214	5	coloring	color	VERB
cana-1220	214	6	ꞷ	ꞷ	NOUN
cana-1220	214	7	:	:	PUNCT
cana-1220	214	8	v(bn	v(bn	VERB
cana-1220	214	9	,	,	PUNCT
cana-1220	214	10	n	n	NOUN
cana-1220	214	11	)	)	PUNCT
cana-1220	214	12	→{c1	→{c1	PROPN
cana-1220	214	13	,	,	PUNCT
cana-1220	214	14	c2	c2	PROPN
cana-1220	214	15	,	,	PUNCT
cana-1220	214	16	……	……	NOUN
cana-1220	214	17	cn+4	cn+4	NOUN
cana-1220	214	18	}	}	PUNCT
cana-1220	214	19	.	.	PUNCT
cana-1220	215	1	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	215	2	)	)	PUNCT
cana-1220	216	1	=	=	SYM
cana-1220	216	2	c1	c1	PROPN
cana-1220	216	3	and	and	CCONJ
cana-1220	216	4	ꞷ	ꞷ	PROPN
cana-1220	216	5	(	(	PUNCT
cana-1220	216	6	v1)=	v1)=	ADP
cana-1220	216	7	cn+4	cn+4	NUM
cana-1220	216	8	ꞷ(ui	ꞷ(ui	NUM
cana-1220	216	9	)	)	PUNCT
cana-1220	216	10	=	=	SYM
cana-1220	216	11	ci	ci	PROPN
cana-1220	216	12	,	,	PUNCT
cana-1220	216	13	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	216	14	)	)	PUNCT
cana-1220	216	15	=	=	SYM
cana-1220	217	1	ci+2	ci+2	NUM
cana-1220	217	2	ɐ	ɐ	NOUN
cana-1220	217	3	2	2	NUM
cana-1220	217	4	≤	≤	NUM
cana-1220	217	5	i	i	PRON
cana-1220	217	6	≤	≤	ADJ
cana-1220	217	7	n	n	CCONJ
cana-1220	217	8	+	+	NOUN
cana-1220	217	9	1	1	NUM
cana-1220	217	10	.	.	X
cana-1220	217	11	such	such	ADJ
cana-1220	217	12	that	that	DET
cana-1220	217	13	gcd	gcd	NOUN
cana-1220	217	14	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	217	15	,	,	PUNCT
cana-1220	217	16	ui	ui	NOUN
cana-1220	217	17	)	)	PUNCT
cana-1220	217	18	=	=	SYM
cana-1220	217	19	1	1	NUM
cana-1220	217	20	,	,	PUNCT
cana-1220	217	21	gcd	gcd	NOUN
cana-1220	217	22	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	217	23	,	,	PUNCT
cana-1220	217	24	vi	vi	NOUN
cana-1220	217	25	)	)	PUNCT
cana-1220	217	26	=	=	SYM
cana-1220	217	27	1	1	NUM
cana-1220	217	28	and	and	CCONJ
cana-1220	217	29	gcd	gcd	VERB
cana-1220	217	30	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	217	31	,	,	PUNCT
cana-1220	217	32	v1	v1	NOUN
cana-1220	217	33	)	)	PUNCT
cana-1220	217	34	=	=	SYM
cana-1220	217	35	1	1	X
cana-1220	217	36	.	.	X
cana-1220	217	37	define	define	VERB
cana-1220	217	38	a	a	DET
cana-1220	217	39	proper	proper	ADJ
cana-1220	217	40	edge	edge	NOUN
cana-1220	217	41	coloring	color	VERB
cana-1220	217	42	ꞷ*:e(bn	ꞷ*:e(bn	PROPN
cana-1220	217	43	,	,	PUNCT
cana-1220	217	44	n)→	n)→	PROPN
cana-1220	217	45	{	{	PUNCT
cana-1220	217	46	c1	c1	NOUN
cana-1220	217	47	,	,	PUNCT
cana-1220	217	48	c2	c2	PROPN
cana-1220	217	49	,	,	PUNCT
cana-1220	217	50	c3	c3	PROPN
cana-1220	217	51	…	…	PUNCT
cana-1220	217	52	…	…	SYM
cana-1220	217	53	cn+3	cn+3	NUM
cana-1220	217	54	}	}	PUNCT
cana-1220	217	55	ꞷ	ꞷ	NOUN
cana-1220	217	56	∗	∗	NOUN
cana-1220	217	57	(	(	PUNCT
cana-1220	217	58	u1v1)=	u1v1)=	PROPN
cana-1220	217	59	c1	c1	PROPN
cana-1220	217	60	−	−	PROPN
cana-1220	217	61	cn+4	cn+4	PROPN
cana-1220	217	62	=	=	SYM
cana-1220	217	63	cn+3	cn+3	PROPN
cana-1220	217	64	,	,	PUNCT
cana-1220	217	65	ꞷ	ꞷ	PROPN
cana-1220	217	66	∗	∗	NOUN
cana-1220	217	67	(	(	PUNCT
cana-1220	217	68	u1ui)=	u1ui)=	NUM
cana-1220	217	69	c1	c1	NOUN
cana-1220	217	70	−	−	PROPN
cana-1220	217	71	ci	ci	PROPN
cana-1220	217	72	=	=	PROPN
cana-1220	217	73	ci−1	ci−1	PROPN
cana-1220	217	74	,	,	PUNCT
cana-1220	217	75	ꞷ	ꞷ	PROPN
cana-1220	217	76	∗	∗	NOUN
cana-1220	217	77	(	(	PUNCT
cana-1220	217	78	v1vi)=	v1vi)=	NUM
cana-1220	217	79	cn+4	cn+4	NOUN
cana-1220	217	80	−	−	PROPN
cana-1220	217	81	ci	ci	NOUN
cana-1220	217	82	+2	+2	NOUN
cana-1220	217	83	=	=	NOUN
cana-1220	217	84	cn+2−i	cn+2−i	NOUN
cana-1220	217	85	hence	hence	ADV
cana-1220	217	86	,	,	PUNCT
cana-1220	217	87	adjacent	adjacent	ADJ
cana-1220	217	88	vertices	vertex	NOUN
cana-1220	217	89	and	and	CCONJ
cana-1220	217	90	edges	edge	NOUN
cana-1220	217	91	receive	receive	VERB
cana-1220	217	92	distinct	distinct	ADJ
cana-1220	217	93	colors	color	NOUN
cana-1220	217	94	.	.	PUNCT
cana-1220	218	1	we	we	PRON
cana-1220	218	2	proved	prove	VERB
cana-1220	218	3	that	that	SCONJ
cana-1220	218	4	χpg(bn	χpg(bn	PROPN
cana-1220	218	5	,	,	PUNCT
cana-1220	218	6	n	n	CCONJ
cana-1220	218	7	)	)	PUNCT
cana-1220	218	8	≤	≤	NOUN
cana-1220	218	9	n	n	PRON
cana-1220	218	10	+	+	NOUN
cana-1220	218	11	4	4	NUM
cana-1220	218	12	.	.	PUNCT
cana-1220	218	13	to	to	PART
cana-1220	218	14	prove	prove	VERB
cana-1220	218	15	χpg(bn	χpg(bn	NOUN
cana-1220	218	16	,	,	PUNCT
cana-1220	218	17	n	n	CCONJ
cana-1220	218	18	)	)	PUNCT
cana-1220	218	19	≥	≥	NOUN
cana-1220	218	20	n	n	PROPN
cana-1220	218	21	+	+	NUM
cana-1220	218	22	4	4	NUM
cana-1220	218	23	,	,	PUNCT
cana-1220	218	24	let	let	VERB
cana-1220	218	25	us	we	PRON
cana-1220	218	26	assume	assume	VERB
cana-1220	218	27	that	that	SCONJ
cana-1220	218	28	χpg(bn	χpg(bn	PROPN
cana-1220	218	29	,	,	PUNCT
cana-1220	218	30	n	n	CCONJ
cana-1220	218	31	)	)	PUNCT
cana-1220	218	32	<	<	X
cana-1220	219	1	n	n	PROPN
cana-1220	219	2	+	+	CCONJ
cana-1220	219	3	4	4	NUM
cana-1220	219	4	say	say	VERB
cana-1220	219	5	n	n	PRON
cana-1220	219	6	+	+	NOUN
cana-1220	219	7	3	3	X
cana-1220	219	8	.	.	X
cana-1220	220	1	we	we	PRON
cana-1220	220	2	define	define	VERB
cana-1220	220	3	proper	proper	ADJ
cana-1220	220	4	vertex	vertex	NOUN
cana-1220	220	5	coloring	coloring	NOUN
cana-1220	220	6	of	of	ADP
cana-1220	220	7	bn	bn	PROPN
cana-1220	220	8	,	,	PUNCT
cana-1220	220	9	n	n	X
cana-1220	220	10	is	be	AUX
cana-1220	220	11	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	220	12	)	)	PUNCT
cana-1220	221	1	=	=	SYM
cana-1220	221	2	c1	c1	NOUN
cana-1220	221	3	,	,	PUNCT
cana-1220	221	4	ꞷ(ui	ꞷ(ui	X
cana-1220	221	5	)	)	PUNCT
cana-1220	222	1	=	=	SYM
cana-1220	222	2	ci	ci	PROPN
cana-1220	222	3	,	,	PUNCT
cana-1220	222	4	ꞷ	ꞷ	PROPN
cana-1220	222	5	(	(	PUNCT
cana-1220	222	6	vi)=ci	vi)=ci	NOUN
cana-1220	222	7	ɐ	ɐ	NOUN
cana-1220	222	8	2	2	NUM
cana-1220	222	9	≤	≤	NUM
cana-1220	222	10	i	i	PRON
cana-1220	222	11	≤	≤	NOUN
cana-1220	222	12	n+1	n+1	PROPN
cana-1220	222	13	and	and	CCONJ
cana-1220	222	14	ꞷ	ꞷ	PROPN
cana-1220	222	15	(	(	PUNCT
cana-1220	222	16	v1)=	v1)=	ADP
cana-1220	222	17	cn+3	cn+3	PROPN
cana-1220	222	18	.	.	PUNCT
cana-1220	223	1	since	since	SCONJ
cana-1220	223	2	v1	v1	NOUN
cana-1220	223	3	is	be	AUX
cana-1220	223	4	adjacent	adjacent	ADJ
cana-1220	223	5	to	to	ADP
cana-1220	223	6	vi	vi	PROPN
cana-1220	223	7	ɐ	ɐ	PROPN
cana-1220	223	8	2	2	NUM
cana-1220	223	9	≤	≤	NUM
cana-1220	223	10	i	i	PRON
cana-1220	223	11	≤	≤	NOUN
cana-1220	223	12	n+1	n+1	ADV
cana-1220	223	13	for	for	ADP
cana-1220	223	14	each	each	DET
cana-1220	223	15	edge	edge	NOUN
cana-1220	223	16	vivj	vivj	NOUN
cana-1220	223	17	gcd	gcd	PROPN
cana-1220	223	18	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	223	19	,	,	PUNCT
cana-1220	223	20	vi	vi	NOUN
cana-1220	223	21	)	)	PUNCT
cana-1220	223	22	≠	≠	PROPN
cana-1220	223	23	1	1	NUM
cana-1220	223	24	.	.	PUNCT
cana-1220	224	1	thus	thus	ADV
cana-1220	224	2	,	,	PUNCT
cana-1220	224	3	χpg(bn	χpg(bn	PROPN
cana-1220	224	4	,	,	PUNCT
cana-1220	224	5	n	n	CCONJ
cana-1220	224	6	)	)	PUNCT
cana-1220	224	7	≥	≥	NOUN
cana-1220	224	8	n	n	NOUN
cana-1220	224	9	+	+	NUM
cana-1220	224	10	4	4	NUM
cana-1220	224	11	.	.	PUNCT
cana-1220	225	1	therefore	therefore	ADV
cana-1220	225	2	,	,	PUNCT
cana-1220	225	3	χpg(bn	χpg(bn	PROPN
cana-1220	225	4	,	,	PUNCT
cana-1220	225	5	n)=	n)=	NOUN
cana-1220	225	6	n+	n+	PUNCT
cana-1220	225	7	4	4	NUM
cana-1220	225	8	when	when	SCONJ
cana-1220	225	9	n	n	X
cana-1220	225	10	≡	≡	PROPN
cana-1220	225	11	1	1	NUM
cana-1220	225	12	mod	mod	NOUN
cana-1220	225	13	6	6	NUM
cana-1220	225	14	and	and	CCONJ
cana-1220	225	15	n	n	CCONJ
cana-1220	225	16	≠	≠	PROPN
cana-1220	225	17	1	1	NUM
cana-1220	225	18	.	.	PUNCT
cana-1220	225	19	case	case	NOUN
cana-1220	225	20	4	4	NUM
cana-1220	225	21	:	:	PUNCT
cana-1220	225	22	n	n	NUM
cana-1220	225	23	≡	≡	PROPN
cana-1220	225	24	0	0	PUNCT
cana-1220	226	1	mod	mod	PROPN
cana-1220	227	1	6	6	NUM
cana-1220	227	2	let	let	VERB
cana-1220	227	3	v	v	NOUN
cana-1220	227	4	(	(	PUNCT
cana-1220	227	5	bn	bn	NOUN
cana-1220	227	6	,	,	PUNCT
cana-1220	227	7	n	n	CCONJ
cana-1220	227	8	)	)	PUNCT
cana-1220	227	9	=	=	PRON
cana-1220	227	10	{	{	PUNCT
cana-1220	227	11	ui	ui	NOUN
cana-1220	228	1	:1	:1	PUNCT
cana-1220	228	2	≤	≤	NUM
cana-1220	228	3	i	i	PRON
cana-1220	228	4	≤	≤	ADJ
cana-1220	228	5	n	n	CCONJ
cana-1220	228	6	+	+	CCONJ
cana-1220	228	7	1	1	NUM
cana-1220	228	8	}	}	PUNCT
cana-1220	228	9	∪	∪	ADJ
cana-1220	228	10	{	{	PUNCT
cana-1220	228	11	vi	vi	NOUN
cana-1220	228	12	:1	:1	PUNCT
cana-1220	228	13	≤	≤	NUM
cana-1220	228	14	i	i	PRON
cana-1220	228	15	≤	≤	ADJ
cana-1220	228	16	n	n	CCONJ
cana-1220	228	17	+	+	NOUN
cana-1220	228	18	1	1	NUM
cana-1220	228	19	}	}	PUNCT
cana-1220	228	20	.	.	PUNCT
cana-1220	229	1	define	define	VERB
cana-1220	229	2	a	a	DET
cana-1220	229	3	proper	proper	ADJ
cana-1220	229	4	vertex	vertex	NOUN
cana-1220	229	5	coloring	color	VERB
cana-1220	229	6	ꞷ	ꞷ	NOUN
cana-1220	229	7	:	:	PUNCT
cana-1220	229	8	v(bn	v(bn	VERB
cana-1220	229	9	,	,	PUNCT
cana-1220	229	10	n	n	NOUN
cana-1220	229	11	)	)	PUNCT
cana-1220	229	12	→{c1	→{c1	PROPN
cana-1220	229	13	,	,	PUNCT
cana-1220	229	14	c2	c2	PROPN
cana-1220	229	15	,	,	PUNCT
cana-1220	229	16	c3	c3	PROPN
cana-1220	229	17	…	…	PUNCT
cana-1220	229	18	…	…	PUNCT
cana-1220	229	19	cn+5	cn+5	NOUN
cana-1220	229	20	}	}	PUNCT
cana-1220	229	21	.	.	PUNCT
cana-1220	230	1	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	230	2	)	)	PUNCT
cana-1220	231	1	=	=	SYM
cana-1220	231	2	c1	c1	PROPN
cana-1220	231	3	and	and	CCONJ
cana-1220	231	4	ꞷ	ꞷ	PROPN
cana-1220	231	5	(	(	PUNCT
cana-1220	231	6	v1)=	v1)=	ADP
cana-1220	231	7	cn+5	cn+5	PROPN
cana-1220	231	8	and	and	CCONJ
cana-1220	231	9	ꞷ(ui	ꞷ(ui	NUM
cana-1220	231	10	)	)	PUNCT
cana-1220	231	11	=	=	SYM
cana-1220	231	12	ci	ci	PROPN
cana-1220	231	13	,	,	PUNCT
cana-1220	231	14	ꞷ(vi	ꞷ(vi	PROPN
cana-1220	231	15	)	)	PUNCT
cana-1220	231	16	=	=	SYM
cana-1220	231	17	ci+3	ci+3	NOUN
cana-1220	231	18	ɐ	ɐ	NOUN
cana-1220	231	19	2	2	NUM
cana-1220	231	20	≤	≤	NUM
cana-1220	231	21	i	i	PRON
cana-1220	231	22	≤	≤	ADJ
cana-1220	231	23	n	n	CCONJ
cana-1220	231	24	+	+	NOUN
cana-1220	231	25	1	1	NUM
cana-1220	231	26	.	.	X
cana-1220	231	27	such	such	ADJ
cana-1220	231	28	that	that	DET
cana-1220	231	29	gcd	gcd	NOUN
cana-1220	231	30	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	231	31	,	,	PUNCT
cana-1220	231	32	ui	ui	NOUN
cana-1220	231	33	)	)	PUNCT
cana-1220	231	34	=	=	SYM
cana-1220	231	35	1	1	NUM
cana-1220	231	36	,	,	PUNCT
cana-1220	231	37	gcd	gcd	NOUN
cana-1220	231	38	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	231	39	,	,	PUNCT
cana-1220	231	40	vi	vi	NOUN
cana-1220	231	41	)	)	PUNCT
cana-1220	231	42	=	=	SYM
cana-1220	231	43	1	1	NUM
cana-1220	231	44	and	and	CCONJ
cana-1220	231	45	gcd	gcd	VERB
cana-1220	231	46	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	231	47	,	,	PUNCT
cana-1220	231	48	v1	v1	NOUN
cana-1220	231	49	)	)	PUNCT
cana-1220	231	50	=	=	SYM
cana-1220	232	1	1	1	X
cana-1220	232	2	.	.	X
cana-1220	232	3	define	define	VERB
cana-1220	232	4	a	a	DET
cana-1220	232	5	proper	proper	ADJ
cana-1220	232	6	edge	edge	NOUN
cana-1220	232	7	coloring	color	VERB
cana-1220	232	8	ꞷ*:e(bn	ꞷ*:e(bn	PROPN
cana-1220	232	9	,	,	PUNCT
cana-1220	232	10	n)→	n)→	PROPN
cana-1220	232	11	{	{	PUNCT
cana-1220	232	12	c1	c1	NOUN
cana-1220	232	13	,	,	PUNCT
cana-1220	232	14	c2	c2	PROPN
cana-1220	232	15	,	,	PUNCT
cana-1220	232	16	c3	c3	PROPN
cana-1220	232	17	…	…	PUNCT
cana-1220	232	18	…	…	SYM
cana-1220	232	19	cn+4	cn+4	X
cana-1220	232	20	}	}	PUNCT
cana-1220	232	21	ꞷ	ꞷ	NOUN
cana-1220	232	22	∗	∗	NOUN
cana-1220	232	23	(	(	PUNCT
cana-1220	232	24	u1v1)=	u1v1)=	PROPN
cana-1220	232	25	c1	c1	PROPN
cana-1220	232	26	−	−	PROPN
cana-1220	232	27	cn+5	cn+5	PROPN
cana-1220	233	1	=	=	SYM
cana-1220	233	2	cn+4	cn+4	PROPN
cana-1220	233	3	,	,	PUNCT
cana-1220	233	4	ꞷ	ꞷ	PRON
cana-1220	233	5	∗	∗	NOUN
cana-1220	233	6	(	(	PUNCT
cana-1220	233	7	u1ui)=	u1ui)=	NUM
cana-1220	233	8	c1	c1	NOUN
cana-1220	233	9	−	−	PROPN
cana-1220	233	10	ci	ci	PROPN
cana-1220	233	11	=	=	PROPN
cana-1220	233	12	ci−1	ci−1	PROPN
cana-1220	233	13	,	,	PUNCT
cana-1220	233	14	ꞷ	ꞷ	PROPN
cana-1220	233	15	∗	∗	NOUN
cana-1220	233	16	(	(	PUNCT
cana-1220	233	17	v1vi)=	v1vi)=	NUM
cana-1220	233	18	cn+5	cn+5	NOUN
cana-1220	233	19	−	−	PROPN
cana-1220	233	20	ci	ci	PROPN
cana-1220	233	21	+3	+3	PROPN
cana-1220	233	22	=	=	SYM
cana-1220	233	23	cn+2−i	cn+2−i	NOUN
cana-1220	233	24	hence	hence	ADV
cana-1220	233	25	,	,	PUNCT
cana-1220	233	26	adjacent	adjacent	ADJ
cana-1220	233	27	vertices	vertex	NOUN
cana-1220	233	28	and	and	CCONJ
cana-1220	233	29	edges	edge	NOUN
cana-1220	233	30	receive	receive	VERB
cana-1220	233	31	distinct	distinct	ADJ
cana-1220	233	32	colors	color	NOUN
cana-1220	233	33	.	.	PUNCT
cana-1220	234	1	we	we	PRON
cana-1220	234	2	proved	prove	VERB
cana-1220	234	3	that	that	SCONJ
cana-1220	234	4	χpg(bn	χpg(bn	PROPN
cana-1220	234	5	,	,	PUNCT
cana-1220	234	6	n	n	CCONJ
cana-1220	234	7	)	)	PUNCT
cana-1220	234	8	≤	≤	NOUN
cana-1220	234	9	n	n	PRON
cana-1220	234	10	+	+	NUM
cana-1220	234	11	5	5	NUM
cana-1220	234	12	.	.	PUNCT
cana-1220	234	13	to	to	PART
cana-1220	234	14	prove	prove	VERB
cana-1220	234	15	χpg(bn	χpg(bn	NOUN
cana-1220	234	16	,	,	PUNCT
cana-1220	234	17	n	n	CCONJ
cana-1220	234	18	)	)	PUNCT
cana-1220	234	19	≥	≥	NOUN
cana-1220	234	20	n	n	PROPN
cana-1220	234	21	+	+	NUM
cana-1220	234	22	5	5	NUM
cana-1220	234	23	,	,	PUNCT
cana-1220	234	24	let	let	VERB
cana-1220	234	25	us	we	PRON
cana-1220	234	26	assume	assume	VERB
cana-1220	234	27	that	that	SCONJ
cana-1220	234	28	χpg(bn	χpg(bn	PROPN
cana-1220	234	29	,	,	PUNCT
cana-1220	234	30	n	n	CCONJ
cana-1220	234	31	)	)	PUNCT
cana-1220	234	32	<	<	X
cana-1220	235	1	n	n	PROPN
cana-1220	235	2	+	+	SYM
cana-1220	235	3	5	5	NUM
cana-1220	235	4	say	say	VERB
cana-1220	235	5	n	n	PRON
cana-1220	235	6	+	+	NOUN
cana-1220	235	7	4	4	X
cana-1220	235	8	.	.	X
cana-1220	236	1	we	we	PRON
cana-1220	236	2	define	define	VERB
cana-1220	236	3	proper	proper	ADJ
cana-1220	236	4	vertex	vertex	NOUN
cana-1220	236	5	communications	communication	NOUN
cana-1220	236	6	on	on	ADP
cana-1220	236	7	applied	apply	VERB
cana-1220	236	8	nonlinear	nonlinear	ADJ
cana-1220	236	9	analysis	analysis	NOUN
cana-1220	236	10	issn	issn	NOUN
cana-1220	236	11	:	:	PUNCT
cana-1220	236	12	1074	1074	NUM
cana-1220	236	13	-	-	PUNCT
cana-1220	236	14	133x	133x	NUM
cana-1220	236	15	vol	vol	NOUN
cana-1220	236	16	31	31	NUM
cana-1220	236	17	no	no	NOUN
cana-1220	236	18	.	.	PUNCT
cana-1220	237	1	6s	6s	NUM
cana-1220	237	2	(	(	PUNCT
cana-1220	237	3	2024	2024	NUM
cana-1220	237	4	)	)	PUNCT
cana-1220	237	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-1220	237	6	267	267	NUM
cana-1220	237	7	coloring	coloring	NOUN
cana-1220	237	8	of	of	ADP
cana-1220	237	9	bn	bn	NOUN
cana-1220	237	10	,	,	PUNCT
cana-1220	237	11	n	n	X
cana-1220	237	12	is	be	AUX
cana-1220	237	13	ꞷ(u1	ꞷ(u1	NOUN
cana-1220	237	14	)	)	PUNCT
cana-1220	238	1	=	=	SYM
cana-1220	238	2	c1	c1	NOUN
cana-1220	238	3	,	,	PUNCT
cana-1220	238	4	ꞷ(ui	ꞷ(ui	X
cana-1220	238	5	)	)	PUNCT
cana-1220	239	1	=	=	SYM
cana-1220	239	2	ci	ci	PROPN
cana-1220	239	3	,	,	PUNCT
cana-1220	239	4	ꞷ	ꞷ	PROPN
cana-1220	239	5	(	(	PUNCT
cana-1220	239	6	vi)=ci	vi)=ci	NOUN
cana-1220	239	7	ɐ	ɐ	NOUN
cana-1220	239	8	2	2	NUM
cana-1220	239	9	≤	≤	NUM
cana-1220	239	10	i	i	PRON
cana-1220	239	11	≤	≤	NOUN
cana-1220	239	12	n+1	n+1	PROPN
cana-1220	239	13	and	and	CCONJ
cana-1220	239	14	ꞷ	ꞷ	PROPN
cana-1220	239	15	(	(	PUNCT
cana-1220	239	16	v1)=	v1)=	ADP
cana-1220	239	17	cn+4	cn+4	NOUN
cana-1220	239	18	.	.	PUNCT
cana-1220	240	1	since	since	SCONJ
cana-1220	240	2	v1	v1	NOUN
cana-1220	240	3	is	be	AUX
cana-1220	240	4	adjacent	adjacent	ADJ
cana-1220	240	5	to	to	ADP
cana-1220	240	6	vi	vi	PROPN
cana-1220	240	7	ɐ	ɐ	PROPN
cana-1220	240	8	2	2	NUM
cana-1220	240	9	≤	≤	NUM
cana-1220	240	10	i	i	PRON
cana-1220	240	11	≤	≤	NOUN
cana-1220	240	12	n+1	n+1	ADV
cana-1220	240	13	for	for	ADP
cana-1220	240	14	each	each	DET
cana-1220	240	15	edge	edge	NOUN
cana-1220	240	16	vivj	vivj	NOUN
cana-1220	240	17	gcd	gcd	PROPN
cana-1220	240	18	ꞷ(v1	ꞷ(v1	NOUN
cana-1220	240	19	,	,	PUNCT
cana-1220	240	20	vi	vi	NOUN
cana-1220	240	21	)	)	PUNCT
cana-1220	240	22	≠	≠	PROPN
cana-1220	240	23	1	1	NUM
cana-1220	240	24	.	.	PUNCT
cana-1220	241	1	thus	thus	ADV
cana-1220	241	2	,	,	PUNCT
cana-1220	241	3	χpg(bn	χpg(bn	PROPN
cana-1220	241	4	,	,	PUNCT
cana-1220	241	5	n	n	CCONJ
cana-1220	241	6	)	)	PUNCT
cana-1220	241	7	≥	≥	NOUN
cana-1220	241	8	n	n	NOUN
cana-1220	241	9	+	+	NUM
cana-1220	241	10	5	5	NUM
cana-1220	241	11	.	.	X
cana-1220	242	1	therefore	therefore	ADV
cana-1220	242	2	,	,	PUNCT
cana-1220	242	3	χpg(bn	χpg(bn	PROPN
cana-1220	242	4	,	,	PUNCT
cana-1220	242	5	n)=	n)=	NOUN
cana-1220	242	6	n+	n+	PUNCT
cana-1220	242	7	5	5	NUM
cana-1220	242	8	when	when	SCONJ
cana-1220	242	9	n	n	X
cana-1220	242	10	≡	≡	PROPN
cana-1220	242	11	0	0	NUM
cana-1220	242	12	mod	mod	PROPN
cana-1220	242	13	6	6	NUM
cana-1220	242	14	.	.	PUNCT
cana-1220	242	15	figure	figure	VERB
cana-1220	242	16	6	6	NUM
cana-1220	242	17	analytical	analytical	ADJ
cana-1220	242	18	evaluation	evaluation	NOUN
cana-1220	242	19	of	of	ADP
cana-1220	242	20	the	the	DET
cana-1220	242	21	𝐵5,5	𝐵5,5	PROPN
cana-1220	242	22	4	4	NUM
cana-1220	242	23	.	.	PUNCT
cana-1220	243	1	conclusion	conclusion	NOUN
cana-1220	243	2	this	this	DET
cana-1220	243	3	paper	paper	NOUN
cana-1220	243	4	demonstrates	demonstrate	VERB
cana-1220	243	5	that	that	SCONJ
cana-1220	243	6	several	several	ADJ
cana-1220	243	7	graph	graph	NOUN
cana-1220	243	8	classes	class	NOUN
cana-1220	243	9	admits	admit	VERB
cana-1220	243	10	prime	prime	ADJ
cana-1220	243	11	graceful	graceful	ADJ
cana-1220	243	12	coloring	coloring	NOUN
cana-1220	243	13	.	.	PUNCT
cana-1220	244	1	prime	prime	PROPN
cana-1220	244	2	graceful	graceful	ADJ
cana-1220	244	3	coloring	coloring	NOUN
cana-1220	244	4	offer	offer	VERB
cana-1220	244	5	a	a	DET
cana-1220	244	6	unique	unique	ADJ
cana-1220	244	7	structure	structure	NOUN
cana-1220	244	8	where	where	SCONJ
cana-1220	244	9	vertex	vertex	NOUN
cana-1220	244	10	colors	color	NOUN
cana-1220	244	11	are	be	AUX
cana-1220	244	12	relatively	relatively	ADV
cana-1220	244	13	prime	prime	ADJ
cana-1220	244	14	and	and	CCONJ
cana-1220	244	15	induce	induce	VERB
cana-1220	244	16	a	a	DET
cana-1220	244	17	proper	proper	ADJ
cana-1220	244	18	edge	edge	NOUN
cana-1220	244	19	coloring	coloring	NOUN
cana-1220	244	20	which	which	PRON
cana-1220	244	21	potentially	potentially	ADV
cana-1220	244	22	leads	lead	VERB
cana-1220	244	23	to	to	ADP
cana-1220	244	24	application	application	NOUN
cana-1220	244	25	in	in	ADP
cana-1220	244	26	areas	area	NOUN
cana-1220	244	27	yet	yet	ADV
cana-1220	244	28	to	to	PART
cana-1220	244	29	be	be	AUX
cana-1220	244	30	explored	explore	VERB
cana-1220	244	31	.	.	PUNCT
cana-1220	245	1	further	further	ADJ
cana-1220	245	2	research	research	NOUN
cana-1220	245	3	can	can	AUX
cana-1220	245	4	investigate	investigate	VERB
cana-1220	245	5	prime	prime	ADJ
cana-1220	245	6	graceful	graceful	ADJ
cana-1220	245	7	coloring	coloring	NOUN
cana-1220	245	8	in	in	ADP
cana-1220	245	9	more	more	ADJ
cana-1220	245	10	complex	complex	ADJ
cana-1220	245	11	graph	graph	NOUN
cana-1220	245	12	families	family	NOUN
cana-1220	245	13	and	and	CCONJ
cana-1220	245	14	explore	explore	VERB
cana-1220	245	15	their	their	PRON
cana-1220	245	16	potential	potential	ADJ
cana-1220	245	17	uses	use	NOUN
cana-1220	245	18	in	in	ADP
cana-1220	245	19	areas	area	NOUN
cana-1220	245	20	where	where	SCONJ
cana-1220	245	21	efficient	efficient	ADJ
cana-1220	245	22	and	and	CCONJ
cana-1220	245	23	unique	unique	ADJ
cana-1220	245	24	coloring	coloring	NOUN
cana-1220	245	25	schemes	scheme	NOUN
cana-1220	245	26	are	be	AUX
cana-1220	245	27	crucial	crucial	ADJ
cana-1220	245	28	.	.	PUNCT
cana-1220	246	1	references	reference	NOUN
cana-1220	246	2	[	[	X
cana-1220	246	3	1	1	NUM
cana-1220	246	4	]	]	X
cana-1220	246	5	english	english	PROPN
cana-1220	246	6	,	,	PUNCT
cana-1220	246	7	s.	s.	PROPN
cana-1220	246	8	;	;	PUNCT
cana-1220	246	9	and	and	CCONJ
cana-1220	246	10	zhang	zhang	PROPN
cana-1220	246	11	,	,	PUNCT
cana-1220	246	12	p.	p.	NOUN
cana-1220	246	13	:	:	PUNCT
cana-1220	246	14	on	on	ADP
cana-1220	246	15	graceful	graceful	ADJ
cana-1220	246	16	colorings	coloring	NOUN
cana-1220	246	17	of	of	ADP
cana-1220	246	18	trees	tree	NOUN
cana-1220	246	19	,	,	PUNCT
cana-1220	246	20	mathematica	mathematica	PROPN
cana-1220	246	21	bohemica	bohemica	PROPN
cana-1220	246	22	,	,	PUNCT
cana-1220	246	23	vol.142	vol.142	INTJ
cana-1220	246	24	,	,	PUNCT
cana-1220	246	25	pp	pp	ADJ
cana-1220	246	26	.	.	PUNCT
cana-1220	247	1	57	57	NUM
cana-1220	247	2	-	-	SYM
cana-1220	247	3	73	73	NUM
cana-1220	247	4	,	,	PUNCT
cana-1220	247	5	(	(	PUNCT
cana-1220	247	6	2017	2017	NUM
cana-1220	247	7	)	)	PUNCT
cana-1220	247	8	.	.	PUNCT
cana-1220	248	1	[	[	X
cana-1220	248	2	2	2	NUM
cana-1220	248	3	]	]	PUNCT
cana-1220	248	4	gallian	gallian	ADJ
cana-1220	248	5	,	,	PUNCT
cana-1220	248	6	j.a	j.a	PROPN
cana-1220	248	7	.	.	PROPN
cana-1220	248	8	:	:	PUNCT
cana-1220	249	1	a	a	DET
cana-1220	249	2	dynamic	dynamic	ADJ
cana-1220	249	3	survey	survey	NOUN
cana-1220	249	4	of	of	ADP
cana-1220	249	5	graph	graph	NOUN
cana-1220	249	6	labeling	labeling	NOUN
cana-1220	249	7	,	,	PUNCT
cana-1220	249	8	the	the	DET
cana-1220	249	9	electronic	electronic	ADJ
cana-1220	249	10	journal	journal	NOUN
cana-1220	249	11	of	of	ADP
cana-1220	249	12	combinatorics,18	combinatorics,18	NOUN
cana-1220	249	13	,	,	PUNCT
cana-1220	249	14	(	(	PUNCT
cana-1220	249	15	2015	2015	NUM
cana-1220	249	16	)	)	PUNCT
cana-1220	249	17	.	.	PUNCT
cana-1220	250	1	[	[	X
cana-1220	250	2	3	3	NUM
cana-1220	250	3	]	]	X
cana-1220	250	4	murugarajan	murugarajan	NOUN
cana-1220	250	5	,	,	PUNCT
cana-1220	250	6	p.	p.	NOUN
cana-1220	250	7	;	;	PUNCT
cana-1220	250	8	aruldoss	aruldoss	PROPN
cana-1220	250	9	,	,	PUNCT
cana-1220	250	10	r.	r.	PROPN
cana-1220	250	11	:	:	PUNCT
cana-1220	250	12	prime	prime	ADJ
cana-1220	250	13	coloring	coloring	NOUN
cana-1220	250	14	of	of	ADP
cana-1220	250	15	some	some	DET
cana-1220	250	16	graphs	graph	NOUN
cana-1220	250	17	,	,	PUNCT
cana-1220	250	18	international	international	ADJ
cana-1220	250	19	journal	journal	NOUN
cana-1220	250	20	of	of	ADP
cana-1220	250	21	scientific	scientific	PROPN
cana-1220	250	22	&	&	CCONJ
cana-1220	250	23	technology	technology	PROPN
cana-1220	250	24	research	research	NOUN
cana-1220	250	25	,	,	PUNCT
cana-1220	250	26	volume	volume	NOUN
cana-1220	250	27	8	8	NUM
cana-1220	250	28	,	,	PUNCT
cana-1220	250	29	issue	issue	NOUN
cana-1220	250	30	08	08	NUM
cana-1220	250	31	,	,	PUNCT
cana-1220	250	32	august	august	PROPN
cana-1220	250	33	(	(	PUNCT
cana-1220	250	34	2019	2019	NUM
cana-1220	250	35	)	)	PUNCT
cana-1220	250	36	.	.	PUNCT
cana-1220	251	1	[	[	X
cana-1220	251	2	4	4	NUM
cana-1220	251	3	]	]	SYM
cana-1220	251	4	nandhini	nandhini	PROPN
cana-1220	251	5	,	,	PUNCT
cana-1220	251	6	s.p	s.p	PROPN
cana-1220	251	7	.	.	PROPN
cana-1220	251	8	;	;	PUNCT
cana-1220	251	9	pooja	pooja	PROPN
cana-1220	251	10	lakshmi	lakshmi	PROPN
cana-1220	251	11	,	,	PUNCT
cana-1220	251	12	b.	b.	PROPN
cana-1220	251	13	:	:	PUNCT
cana-1220	251	14	study	study	NOUN
cana-1220	251	15	on	on	ADP
cana-1220	251	16	prime	prime	ADJ
cana-1220	251	17	graceful	graceful	ADJ
cana-1220	251	18	labeling	labeling	NOUN
cana-1220	251	19	for	for	ADP
cana-1220	251	20	some	some	DET
cana-1220	251	21	special	special	ADJ
cana-1220	251	22	graphs	graph	NOUN
cana-1220	251	23	,	,	PUNCT
cana-1220	251	24	nveo	nveo	NOUN
cana-1220	251	25	,	,	PUNCT
cana-1220	251	26	1316113171	1316113171	NUM
cana-1220	251	27	,	,	PUNCT
cana-1220	251	28	(	(	PUNCT
cana-1220	251	29	2021	2021	NUM
cana-1220	251	30	)	)	PUNCT
cana-1220	251	31	.	.	PUNCT
cana-1220	252	1	[	[	X
cana-1220	252	2	5	5	X
cana-1220	252	3	]	]	X
cana-1220	252	4	rosa	rosa	PROPN
cana-1220	252	5	,	,	PUNCT
cana-1220	252	6	a.	a.	NOUN
cana-1220	252	7	:	:	PUNCT
cana-1220	252	8	on	on	ADP
cana-1220	252	9	certain	certain	ADJ
cana-1220	252	10	valuations	valuation	NOUN
cana-1220	252	11	of	of	ADP
cana-1220	252	12	the	the	DET
cana-1220	252	13	vertices	vertex	NOUN
cana-1220	252	14	of	of	ADP
cana-1220	252	15	a	a	DET
cana-1220	252	16	graph	graph	NOUN
cana-1220	252	17	,	,	PUNCT
cana-1220	252	18	theory	theory	NOUN
cana-1220	252	19	of	of	ADP
cana-1220	252	20	graphs	graph	NOUN
cana-1220	252	21	,	,	PUNCT
cana-1220	252	22	(	(	PUNCT
cana-1220	252	23	international)symposium	international)symposium	NOUN
cana-1220	252	24	,	,	PUNCT
cana-1220	252	25	rome	rome	PROPN
cana-1220	252	26	,	,	PUNCT
cana-1220	252	27	july	july	PROPN
cana-1220	252	28	1966	1966	NUM
cana-1220	252	29	)	)	PUNCT
cana-1220	252	30	,	,	PUNCT
cana-1220	252	31	gordon	gordon	PROPN
cana-1220	252	32	and	and	CCONJ
cana-1220	252	33	breach	breach	PROPN
cana-1220	252	34	,	,	PUNCT
cana-1220	252	35	n.y	n.y	PROPN
cana-1220	252	36	.	.	PROPN
cana-1220	252	37	and	and	CCONJ
cana-1220	252	38	dunod	dunod	PROPN
cana-1220	252	39	paris	paris	PROPN
cana-1220	252	40	,	,	PUNCT
cana-1220	252	41	355	355	NUM
cana-1220	252	42	,	,	PUNCT
cana-1220	252	43	(	(	PUNCT
cana-1220	252	44	1967	1967	NUM
cana-1220	252	45	)	)	PUNCT
cana-1220	252	46	.	.	PUNCT
cana-1220	253	1	[	[	X
cana-1220	253	2	6	6	NUM
cana-1220	253	3	]	]	PUNCT
cana-1220	253	4	sayan	sayan	ADJ
cana-1220	253	5	panma	panma	PROPN
cana-1220	253	6	.	.	PUNCT
cana-1220	253	7	;	;	PUNCT
cana-1220	253	8	and	and	CCONJ
cana-1220	254	1	penying	penye	VERB
cana-1220	254	2	rochanakul	rochanakul	NOUN
cana-1220	254	3	.	.	PUNCT
cana-1220	255	1	:	:	PUNCT
cana-1220	255	2	prime	prime	ADJ
cana-1220	255	3	-	-	PUNCT
cana-1220	255	4	graceful	graceful	NOUN
cana-1220	255	5	graphs	graph	NOUN
cana-1220	255	6	,	,	PUNCT
cana-1220	255	7	thai	thai	PROPN
cana-1220	255	8	journal	journal	PROPN
cana-1220	255	9	of	of	ADP
cana-1220	255	10	mathematics	mathematic	NOUN
cana-1220	255	11	,	,	PUNCT
cana-1220	255	12	volume	volume	NOUN
cana-1220	255	13	19	19	NUM
cana-1220	255	14	number	number	NOUN
cana-1220	255	15	4	4	NUM
cana-1220	255	16	,	,	PUNCT
cana-1220	255	17	(	(	PUNCT
cana-1220	255	18	2021	2021	NUM
cana-1220	255	19	)	)	PUNCT
cana-1220	255	20	.	.	PUNCT
cana-1220	256	1	[	[	X
cana-1220	256	2	7	7	NUM
cana-1220	256	3	]	]	X
cana-1220	256	4	selvarajan	selvarajan	PROPN
cana-1220	256	5	,	,	PUNCT
cana-1220	256	6	t.m	t.m	PROPN
cana-1220	256	7	.	.	PROPN
cana-1220	256	8	;	;	PUNCT
cana-1220	256	9	subramoniam	subramoniam	PROPN
cana-1220	256	10	,	,	PUNCT
cana-1220	256	11	r.	r.	PROPN
cana-1220	256	12	:	:	PUNCT
cana-1220	256	13	prime	prime	ADJ
cana-1220	256	14	graceful	graceful	ADJ
cana-1220	256	15	labeling	labeling	NOUN
cana-1220	256	16	,	,	PUNCT
cana-1220	256	17	ijet	ijet	NOUN
cana-1220	256	18	,	,	PUNCT
cana-1220	256	19	750	750	NUM
cana-1220	256	20	-	-	SYM
cana-1220	256	21	752	752	NUM
cana-1220	256	22	,	,	PUNCT
cana-1220	256	23	(	(	PUNCT
cana-1220	256	24	2018	2018	NUM
cana-1220	256	25	)	)	PUNCT
cana-1220	256	26	.	.	PUNCT
cana-1220	257	1	[	[	X
cana-1220	257	2	8	8	NUM
cana-1220	257	3	]	]	PUNCT
cana-1220	257	4	siti	siti	NOUN
cana-1220	257	5	khoirunnisa	khoirunnisa	NOUN
cana-1220	257	6	.	.	PUNCT
cana-1220	257	7	;	;	PUNCT
cana-1220	257	8	dafik	dafik	PROPN
cana-1220	257	9	.	.	PUNCT
cana-1220	257	10	;	;	PUNCT
cana-1220	257	11	arika	arika	PROPN
cana-1220	257	12	indah	indah	PROPN
cana-1220	257	13	kristiana	kristiana	PROPN
cana-1220	257	14	.	.	PROPN
cana-1220	257	15	;	;	PUNCT
cana-1220	257	16	ridho	ridho	PROPN
cana-1220	257	17	alfarisi	alfarisi	PROPN
cana-1220	257	18	.	.	PUNCT
cana-1220	257	19	;	;	PUNCT
cana-1220	257	20	graceful	graceful	ADJ
cana-1220	257	21	coloring	coloring	NOUN
cana-1220	257	22	of	of	ADP
cana-1220	257	23	wheel	wheel	NOUN
cana-1220	257	24	graph	graph	NOUN
cana-1220	257	25	family	family	NOUN
cana-1220	257	26	,	,	PUNCT
cana-1220	257	27	international	international	ADJ
cana-1220	257	28	journal	journal	NOUN
cana-1220	257	29	of	of	ADP
cana-1220	257	30	academic	academic	ADJ
cana-1220	257	31	and	and	CCONJ
cana-1220	257	32	applied	applied	ADJ
cana-1220	257	33	research	research	NOUN
cana-1220	257	34	,	,	PUNCT
cana-1220	257	35	vol	vol	NOUN
cana-1220	257	36	.	.	NOUN
cana-1220	257	37	5	5	NUM
cana-1220	257	38	issue	issue	NOUN
cana-1220	257	39	4	4	NUM
cana-1220	257	40	,	,	PUNCT
cana-1220	257	41	pages	page	NOUN
cana-1220	257	42	:	:	PUNCT
cana-1220	257	43	68	68	NUM
cana-1220	257	44	-	-	SYM
cana-1220	257	45	78	78	NUM
cana-1220	257	46	,	,	PUNCT
cana-1220	257	47	(	(	PUNCT
cana-1220	257	48	2021	2021	NUM
cana-1220	257	49	)	)	PUNCT
cana-1220	257	50	.	.	PUNCT
cana-1220	258	1	[	[	X
cana-1220	258	2	9	9	NUM
cana-1220	258	3	]	]	SYM
cana-1220	258	4	tout	tout	PROPN
cana-1220	258	5	,	,	PUNCT
cana-1220	258	6	a.	a.	NOUN
cana-1220	258	7	;	;	PUNCT
cana-1220	258	8	dabboucy	dabboucy	PROPN
cana-1220	258	9	,	,	PUNCT
cana-1220	258	10	a.n	a.n	PROPN
cana-1220	258	11	.	.	PROPN
cana-1220	258	12	;	;	PUNCT
cana-1220	258	13	and	and	CCONJ
cana-1220	258	14	howalla	howalla	NOUN
cana-1220	258	15	,	,	PUNCT
cana-1220	258	16	k.	k.	PROPN
cana-1220	258	17	:	:	PUNCT
cana-1220	258	18	prime	prime	ADJ
cana-1220	258	19	labeling	labeling	NOUN
cana-1220	258	20	of	of	ADP
cana-1220	258	21	graphs	graph	NOUN
cana-1220	258	22	,	,	PUNCT
cana-1220	258	23	national	national	ADJ
cana-1220	258	24	academyscience	academyscience	NOUN
cana-1220	258	25	letters	letter	NOUN
cana-1220	258	26	,	,	PUNCT
cana-1220	258	27	vol	vol	NOUN
cana-1220	258	28	.	.	PROPN
cana-1220	258	29	11	11	NUM
cana-1220	258	30	,	,	PUNCT
cana-1220	258	31	pp	pp	ADJ
cana-1220	258	32	.	.	PUNCT
cana-1220	259	1	365	365	NUM
cana-1220	259	2	-	-	SYM
cana-1220	259	3	368	368	NUM
cana-1220	259	4	,	,	PUNCT
cana-1220	259	5	(	(	PUNCT
cana-1220	259	6	1982	1982	NUM
cana-1220	259	7	)	)	PUNCT
cana-1220	259	8	.	.	PUNCT
cana-1220	260	1	[	[	X
cana-1220	260	2	10	10	NUM
cana-1220	260	3	]	]	X
cana-1220	260	4	vernold	vernold	ADJ
cana-1220	260	5	vivin	vivin	PROPN
cana-1220	260	6	,	,	PUNCT
cana-1220	260	7	j.	j.	PROPN
cana-1220	260	8	;	;	PUNCT
cana-1220	260	9	kowsalya	kowsalya	PROPN
cana-1220	260	10	,	,	PUNCT
cana-1220	260	11	v.	v.	ADV
cana-1220	260	12	;	;	PUNCT
cana-1220	260	13	vimal	vimal	PROPN
cana-1220	260	14	kumar	kumar	PROPN
cana-1220	260	15	,	,	PUNCT
cana-1220	260	16	s.	s.	PROPN
cana-1220	260	17	:	:	PUNCT
cana-1220	260	18	on	on	ADP
cana-1220	260	19	star	star	NOUN
cana-1220	260	20	chromatic	chromatic	ADJ
cana-1220	260	21	number	number	NOUN
cana-1220	260	22	of	of	ADP
cana-1220	260	23	prism	prism	NOUN
cana-1220	260	24	graph	graph	NOUN
cana-1220	260	25	families	family	NOUN
cana-1220	260	26	,	,	PUNCT
cana-1220	260	27	twms.j.app.eng.math	twms.j.app.eng.math	NOUN
cana-1220	260	28	.	.	PUNCT
cana-1220	260	29	,	,	PUNCT
cana-1220	260	30	9	9	NUM
cana-1220	260	31	,	,	PUNCT
cana-1220	260	32	687	687	NUM
cana-1220	260	33	-	-	SYM
cana-1220	260	34	692	692	NUM
cana-1220	260	35	,	,	PUNCT
cana-1220	260	36	(	(	PUNCT
cana-1220	260	37	2019	2019	NUM
cana-1220	260	38	)	)	PUNCT
cana-1220	260	39	.	.	PUNCT
cana-1220	261	1	[	[	X
cana-1220	261	2	11	11	NUM
cana-1220	261	3	]	]	PUNCT
cana-1220	261	4	zhenming	zhenme	VERB
cana-1220	261	5	bi	bi	NOUN
cana-1220	261	6	.	.	PROPN
cana-1220	261	7	;	;	PUNCT
cana-1220	261	8	alexis	alexis	PROPN
cana-1220	261	9	byers	byers	PROPN
cana-1220	261	10	.	.	PUNCT
cana-1220	261	11	;	;	PUNCT
cana-1220	262	1	sean	sean	PROPN
cana-1220	262	2	english	english	PROPN
cana-1220	262	3	.	.	PUNCT
cana-1220	262	4	;	;	PUNCT
cana-1220	262	5	elliot	elliot	PROPN
cana-1220	262	6	laforge	laforge	PROPN
cana-1220	262	7	.	.	PUNCT
cana-1220	262	8	;	;	PUNCT
cana-1220	262	9	ping	ping	PROPN
cana-1220	262	10	zhang	zhang	PROPN
cana-1220	262	11	.	.	PUNCT
cana-1220	262	12	;	;	PUNCT
cana-1220	262	13	graceful	graceful	ADJ
cana-1220	262	14	colorings	coloring	NOUN
cana-1220	262	15	of	of	ADP
cana-1220	262	16	graph	graph	NOUN
cana-1220	262	17	,	,	PUNCT
cana-1220	262	18	the	the	DET
cana-1220	262	19	journal	journal	NOUN
cana-1220	262	20	of	of	ADP
cana-1220	262	21	combinatorial	combinatorial	ADJ
cana-1220	262	22	mathematics	mathematic	NOUN
cana-1220	262	23	and	and	CCONJ
cana-1220	262	24	combinatorial	combinatorial	ADJ
cana-1220	262	25	computing	computing	NOUN
cana-1220	262	26	,	,	PUNCT
cana-1220	262	27	volume	volume	NOUN
cana-1220	262	28	–	–	PUNCT
cana-1220	262	29	101	101	NUM
cana-1220	262	30	,	,	PUNCT
cana-1220	262	31	pages	page	NOUN
cana-1220	262	32	101	101	NUM
cana-1220	262	33	-	-	SYM
cana-1220	262	34	119	119	NUM
cana-1220	262	35	,	,	PUNCT
cana-1220	262	36	(	(	PUNCT
cana-1220	262	37	2017	2017	NUM
cana-1220	262	38	)	)	PUNCT
cana-1220	262	39	.	.	PUNCT