id	sid	tid	token	lemma	pos
cana-6025	1	1	communications	communication	NOUN
cana-6025	1	2	on	on	ADP
cana-6025	1	3	applied	apply	VERB
cana-6025	1	4	nonlinear	nonlinear	ADJ
cana-6025	1	5	analysis	analysis	NOUN
cana-6025	1	6	issn	issn	NOUN
cana-6025	1	7	:	:	PUNCT
cana-6025	1	8	1074	1074	NUM
cana-6025	1	9	-	-	PUNCT
cana-6025	1	10	133x	133x	NUM
cana-6025	1	11	vol	vol	NOUN
cana-6025	1	12	31	31	NUM
cana-6025	1	13	no	no	NOUN
cana-6025	1	14	.	.	NOUN
cana-6025	1	15	2	2	NUM
cana-6025	1	16	(	(	PUNCT
cana-6025	1	17	2024	2024	NUM
cana-6025	1	18	)	)	PUNCT
cana-6025	1	19	489	489	NUM
cana-6025	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	1	21	on	on	ADP
cana-6025	1	22	the	the	DET
cana-6025	1	23	chromatic	chromatic	ADJ
cana-6025	1	24	restrained	restrain	VERB
cana-6025	1	25	domination	domination	NOUN
cana-6025	1	26	number	number	NOUN
cana-6025	1	27	of	of	ADP
cana-6025	1	28	strong	strong	ADJ
cana-6025	1	29	product	product	NOUN
cana-6025	1	30	of	of	ADP
cana-6025	1	31	graphs	graph	NOUN
cana-6025	1	32	r.	r.	PROPN
cana-6025	1	33	divinelin	divinelin	PROPN
cana-6025	1	34	kumari	kumari	PROPN
cana-6025	1	35	𝟏	𝟏	NUM
cana-6025	1	36	,	,	PUNCT
cana-6025	1	37	m.	m.	NOUN
cana-6025	1	38	k.	k.	PROPN
cana-6025	1	39	angel	angel	PROPN
cana-6025	2	1	jebitha	jebitha	PROPN
cana-6025	2	2	𝟐∗	𝟐∗	PROPN
cana-6025	2	3	and	and	CCONJ
cana-6025	2	4	s.	s.	PROPN
cana-6025	2	5	sujitha	sujitha	VERB
cana-6025	2	6	3	3	NUM
cana-6025	2	7	1	1	NUM
cana-6025	2	8	research	research	NOUN
cana-6025	2	9	scholar	scholar	NOUN
cana-6025	2	10	(	(	PUNCT
cana-6025	2	11	reg	reg	NOUN
cana-6025	2	12	.	.	PUNCT
cana-6025	3	1	no	no	INTJ
cana-6025	3	2	.	.	NOUN
cana-6025	3	3	21213042092002	21213042092002	NUM
cana-6025	3	4	)	)	PUNCT
cana-6025	3	5	,	,	PUNCT
cana-6025	3	6	2,3	2,3	NUM
cana-6025	3	7	assistant	assistant	NOUN
cana-6025	3	8	professor	professor	NOUN
cana-6025	3	9	,	,	PUNCT
cana-6025	3	10	1,2,3	1,2,3	NUM
cana-6025	3	11	pg	pg	NOUN
cana-6025	3	12	&	&	CCONJ
cana-6025	3	13	research	research	PROPN
cana-6025	3	14	department	department	PROPN
cana-6025	3	15	of	of	ADP
cana-6025	3	16	mathematics	mathematics	PROPN
cana-6025	3	17	,	,	PUNCT
cana-6025	3	18	holy	holy	PROPN
cana-6025	3	19	cross	cross	PROPN
cana-6025	3	20	college	college	PROPN
cana-6025	3	21	(	(	PUNCT
cana-6025	3	22	autonomous	autonomous	ADJ
cana-6025	3	23	)	)	PUNCT
cana-6025	3	24	,	,	PUNCT
cana-6025	3	25	nagercoil	nagercoil	NOUN
cana-6025	3	26	,	,	PUNCT
cana-6025	3	27	tamil	tamil	PROPN
cana-6025	3	28	nadu	nadu	PROPN
cana-6025	3	29	,	,	PUNCT
cana-6025	3	30	india	india	PROPN
cana-6025	3	31	.	.	PUNCT
cana-6025	4	1	(	(	PUNCT
cana-6025	4	2	affiliated	affiliate	VERB
cana-6025	4	3	to	to	ADP
cana-6025	4	4	manonmaniam	manonmaniam	PROPN
cana-6025	4	5	sundaranar	sundaranar	PROPN
cana-6025	4	6	university	university	PROPN
cana-6025	4	7	,	,	PUNCT
cana-6025	4	8	abishekapatti	abishekapatti	ADJ
cana-6025	4	9	,	,	PUNCT
cana-6025	4	10	tirunelveli-627012	tirunelveli-627012	NOUN
cana-6025	4	11	,	,	PUNCT
cana-6025	4	12	tamil	tamil	PROPN
cana-6025	4	13	nadu	nadu	PROPN
cana-6025	4	14	,	,	PUNCT
cana-6025	4	15	india	india	PROPN
cana-6025	4	16	)	)	PUNCT
cana-6025	4	17	email	email	NOUN
cana-6025	4	18	:	:	PUNCT
cana-6025	4	19	1divinelinr@gmail.com	1divinelinr@gmail.com	NUM
cana-6025	4	20	,	,	PUNCT
cana-6025	4	21	2angeljebitha@holycrossngl.edu.in	2angeljebitha@holycrossngl.edu.in	NUM
cana-6025	4	22	,	,	PUNCT
cana-6025	4	23	3sujitha.s@holycrossngl.edu.in	3sujitha.s@holycrossngl.edu.in	NUM
cana-6025	4	24	article	article	NOUN
cana-6025	4	25	history	history	NOUN
cana-6025	4	26	:	:	PUNCT
cana-6025	4	27	received	receive	VERB
cana-6025	4	28	:	:	PUNCT
cana-6025	4	29	12	12	NUM
cana-6025	4	30	-	-	SYM
cana-6025	4	31	01	01	NUM
cana-6025	4	32	-	-	PUNCT
cana-6025	4	33	2024	2024	NUM
cana-6025	4	34	revised	revise	VERB
cana-6025	4	35	:	:	PUNCT
cana-6025	4	36	15	15	NUM
cana-6025	4	37	-	-	NUM
cana-6025	4	38	02	02	NUM
cana-6025	4	39	-	-	PUNCT
cana-6025	4	40	2024	2024	NUM
cana-6025	4	41	accepted	accept	VERB
cana-6025	4	42	:	:	PUNCT
cana-6025	4	43	01	01	NUM
cana-6025	4	44	-	-	PUNCT
cana-6025	4	45	03	03	NUM
cana-6025	4	46	-	-	PUNCT
cana-6025	4	47	2024	2024	NUM
cana-6025	4	48	abstract	abstract	NOUN
cana-6025	4	49	:	:	PUNCT
cana-6025	4	50	let	let	VERB
cana-6025	4	51	𝐺	𝐺	PROPN
cana-6025	4	52	=	=	SYM
cana-6025	4	53	(	(	PUNCT
cana-6025	4	54	𝑉	𝑉	PROPN
cana-6025	4	55	,	,	PUNCT
cana-6025	4	56	𝐸	𝐸	PROPN
cana-6025	4	57	)	)	PUNCT
cana-6025	4	58	be	be	VERB
cana-6025	4	59	a	a	DET
cana-6025	4	60	graph	graph	NOUN
cana-6025	4	61	.	.	PUNCT
cana-6025	5	1	a	a	DET
cana-6025	5	2	subset	subset	NOUN
cana-6025	5	3	𝐷	𝐷	NOUN
cana-6025	5	4	of	of	ADP
cana-6025	5	5	𝑉	𝑉	PROPN
cana-6025	5	6	is	be	AUX
cana-6025	5	7	said	say	VERB
cana-6025	5	8	to	to	PART
cana-6025	5	9	be	be	AUX
cana-6025	5	10	a	a	DET
cana-6025	5	11	chromatic	chromatic	ADJ
cana-6025	5	12	restrained	restrain	VERB
cana-6025	5	13	dominating	dominating	NOUN
cana-6025	5	14	set	set	NOUN
cana-6025	5	15	(	(	PUNCT
cana-6025	5	16	or	or	CCONJ
cana-6025	5	17	crd	crd	NOUN
cana-6025	5	18	-	-	PUNCT
cana-6025	5	19	set	set	NOUN
cana-6025	5	20	)	)	PUNCT
cana-6025	5	21	if	if	SCONJ
cana-6025	5	22	𝐷	𝐷	PROPN
cana-6025	5	23	is	be	AUX
cana-6025	5	24	a	a	DET
cana-6025	5	25	restrained	restrained	ADJ
cana-6025	5	26	dominating	dominating	NOUN
cana-6025	5	27	set	set	NOUN
cana-6025	5	28	and	and	CCONJ
cana-6025	5	29	𝜒	𝜒	X
cana-6025	5	30	(	(	PUNCT
cana-6025	5	31	<	<	X
cana-6025	5	32	𝐷	𝐷	PROPN
cana-6025	5	33	>	>	PUNCT
cana-6025	5	34	)	)	PUNCT
cana-6025	5	35	=	=	SYM
cana-6025	5	36	𝜒(𝐺	𝜒(𝐺	NOUN
cana-6025	5	37	)	)	PUNCT
cana-6025	5	38	.	.	PUNCT
cana-6025	6	1	the	the	DET
cana-6025	6	2	minimum	minimum	ADJ
cana-6025	6	3	cardinality	cardinality	NOUN
cana-6025	6	4	taken	take	VERB
cana-6025	6	5	over	over	ADP
cana-6025	6	6	all	all	DET
cana-6025	6	7	minimal	minimal	ADJ
cana-6025	6	8	chromatic	chromatic	ADJ
cana-6025	6	9	restrained	restrain	VERB
cana-6025	6	10	dominating	dominating	NOUN
cana-6025	6	11	sets	set	NOUN
cana-6025	6	12	is	be	AUX
cana-6025	6	13	called	call	VERB
cana-6025	6	14	the	the	DET
cana-6025	6	15	chromatic	chromatic	ADJ
cana-6025	6	16	restrained	restrain	VERB
cana-6025	6	17	domination	domination	NOUN
cana-6025	6	18	number	number	NOUN
cana-6025	6	19	of	of	ADP
cana-6025	6	20	𝐺	𝐺	PROPN
cana-6025	6	21	and	and	CCONJ
cana-6025	6	22	is	be	AUX
cana-6025	6	23	denoted	denote	VERB
cana-6025	6	24	by	by	ADP
cana-6025	6	25	𝛾𝑟	𝛾𝑟	ADP
cana-6025	6	26	𝑐(𝐺	𝑐(𝐺	NUM
cana-6025	6	27	)	)	PUNCT
cana-6025	6	28	.	.	PUNCT
cana-6025	7	1	in	in	ADP
cana-6025	7	2	this	this	DET
cana-6025	7	3	paper	paper	NOUN
cana-6025	7	4	,	,	PUNCT
cana-6025	7	5	we	we	PRON
cana-6025	7	6	obtain	obtain	VERB
cana-6025	7	7	the	the	DET
cana-6025	7	8	chromatic	chromatic	ADJ
cana-6025	7	9	restrained	restrain	VERB
cana-6025	7	10	domination	domination	NOUN
cana-6025	7	11	number	number	NOUN
cana-6025	7	12	for	for	ADP
cana-6025	7	13	the	the	DET
cana-6025	7	14	strong	strong	ADJ
cana-6025	7	15	product	product	NOUN
cana-6025	7	16	of	of	ADP
cana-6025	7	17	some	some	DET
cana-6025	7	18	standard	standard	ADJ
cana-6025	7	19	graphs	graph	NOUN
cana-6025	7	20	.	.	PUNCT
cana-6025	8	1	keywords	keyword	NOUN
cana-6025	8	2	:	:	PUNCT
cana-6025	8	3	domination	domination	NOUN
cana-6025	8	4	,	,	PUNCT
cana-6025	8	5	restrained	restrained	ADJ
cana-6025	8	6	domination	domination	NOUN
cana-6025	8	7	,	,	PUNCT
cana-6025	8	8	chromatic	chromatic	ADJ
cana-6025	8	9	number	number	NOUN
cana-6025	8	10	,	,	PUNCT
cana-6025	8	11	strong	strong	ADJ
cana-6025	8	12	product	product	NOUN
cana-6025	8	13	.	.	PUNCT
cana-6025	9	1	ams	am	NOUN
cana-6025	9	2	subject	subject	ADJ
cana-6025	9	3	classification	classification	NOUN
cana-6025	9	4	:	:	PUNCT
cana-6025	9	5	05c15	05c15	NOUN
cana-6025	9	6	,	,	PUNCT
cana-6025	9	7	05c69	05c69	NOUN
cana-6025	9	8	1	1	NUM
cana-6025	9	9	.	.	X
cana-6025	9	10	introduction	introduction	NOUN
cana-6025	9	11	all	all	DET
cana-6025	9	12	the	the	DET
cana-6025	9	13	graphs	graph	NOUN
cana-6025	9	14	𝐺	𝐺	NOUN
cana-6025	9	15	=	=	SYM
cana-6025	9	16	(	(	PUNCT
cana-6025	9	17	𝑉	𝑉	PROPN
cana-6025	9	18	,	,	PUNCT
cana-6025	9	19	𝐸	𝐸	PROPN
cana-6025	9	20	)	)	PUNCT
cana-6025	9	21	=	=	PUNCT
cana-6025	9	22	(	(	PUNCT
cana-6025	9	23	𝑛	𝑛	PROPN
cana-6025	9	24	,	,	PUNCT
cana-6025	9	25	𝑚	𝑚	NOUN
cana-6025	9	26	)	)	PUNCT
cana-6025	9	27	considered	consider	VERB
cana-6025	9	28	here	here	ADV
cana-6025	9	29	are	be	AUX
cana-6025	9	30	simple	simple	ADJ
cana-6025	9	31	,	,	PUNCT
cana-6025	9	32	finite	finite	ADJ
cana-6025	9	33	and	and	CCONJ
cana-6025	9	34	undirected	undirected	ADJ
cana-6025	9	35	,	,	PUNCT
cana-6025	9	36	with	with	ADP
cana-6025	9	37	neither	neither	CCONJ
cana-6025	9	38	loops	loop	NOUN
cana-6025	9	39	nor	nor	CCONJ
cana-6025	9	40	multiple	multiple	ADJ
cana-6025	9	41	edges	edge	NOUN
cana-6025	9	42	.	.	PUNCT
cana-6025	10	1	for	for	ADP
cana-6025	10	2	𝐷	𝐷	PROPN
cana-6025	10	3	⊆	⊆	PROPN
cana-6025	10	4	𝑉	𝑉	PROPN
cana-6025	10	5	,	,	PUNCT
cana-6025	10	6	the	the	DET
cana-6025	10	7	subgraph	subgraph	NOUN
cana-6025	10	8	induced	induce	VERB
cana-6025	10	9	by	by	ADP
cana-6025	10	10	𝐷	𝐷	PROPN
cana-6025	10	11	is	be	AUX
cana-6025	10	12	denoted	denote	VERB
cana-6025	10	13	by	by	ADP
cana-6025	10	14	⟨𝐷⟩.	⟨𝐷⟩.	DET
cana-6025	10	15	a	a	DET
cana-6025	10	16	k	k	NOUN
cana-6025	10	17	-	-	ADJ
cana-6025	10	18	vertexcoloring	vertexcoloring	NOUN
cana-6025	10	19	of	of	ADP
cana-6025	10	20	a	a	DET
cana-6025	10	21	graph	graph	NOUN
cana-6025	10	22	,	,	PUNCT
cana-6025	10	23	or	or	CCONJ
cana-6025	10	24	simply	simply	ADV
cana-6025	10	25	a	a	DET
cana-6025	10	26	k	k	NOUN
cana-6025	10	27	-	-	ADJ
cana-6025	10	28	coloring	coloring	NOUN
cana-6025	10	29	,	,	PUNCT
cana-6025	10	30	is	be	AUX
cana-6025	10	31	an	an	DET
cana-6025	10	32	assignment	assignment	NOUN
cana-6025	10	33	of	of	ADP
cana-6025	10	34	k	k	NOUN
cana-6025	10	35	-	-	NOUN
cana-6025	10	36	colors	color	NOUN
cana-6025	10	37	to	to	ADP
cana-6025	10	38	its	its	PRON
cana-6025	10	39	vertices	vertex	NOUN
cana-6025	10	40	.	.	PUNCT
cana-6025	11	1	the	the	DET
cana-6025	11	2	coloring	coloring	NOUN
cana-6025	11	3	is	be	AUX
cana-6025	11	4	proper	proper	ADJ
cana-6025	11	5	if	if	SCONJ
cana-6025	11	6	no	no	DET
cana-6025	11	7	two	two	NUM
cana-6025	11	8	adjacent	adjacent	ADJ
cana-6025	11	9	vertices	vertex	NOUN
cana-6025	11	10	are	be	AUX
cana-6025	11	11	assigned	assign	VERB
cana-6025	11	12	the	the	DET
cana-6025	11	13	same	same	ADJ
cana-6025	11	14	color	color	NOUN
cana-6025	11	15	.	.	PUNCT
cana-6025	12	1	a	a	DET
cana-6025	12	2	coloring	coloring	NOUN
cana-6025	12	3	in	in	ADP
cana-6025	12	4	which	which	PRON
cana-6025	12	5	k	k	NOUN
cana-6025	12	6	-	-	PUNCT
cana-6025	12	7	colors	color	NOUN
cana-6025	12	8	are	be	AUX
cana-6025	12	9	used	use	VERB
cana-6025	12	10	is	be	AUX
cana-6025	12	11	a	a	DET
cana-6025	12	12	k	k	NOUN
cana-6025	12	13	-	-	NOUN
cana-6025	12	14	coloring	coloring	NOUN
cana-6025	12	15	.	.	PUNCT
cana-6025	13	1	a	a	DET
cana-6025	13	2	graph	graph	NOUN
cana-6025	13	3	is	be	AUX
cana-6025	13	4	k	k	ADJ
cana-6025	13	5	-	-	ADJ
cana-6025	13	6	colorable	colorable	ADJ
cana-6025	13	7	if	if	SCONJ
cana-6025	13	8	it	it	PRON
cana-6025	13	9	has	have	VERB
cana-6025	13	10	a	a	DET
cana-6025	13	11	proper	proper	ADJ
cana-6025	13	12	k	k	NOUN
cana-6025	13	13	-	-	NOUN
cana-6025	13	14	coloring	coloring	NOUN
cana-6025	13	15	.	.	PUNCT
cana-6025	14	1	the	the	DET
cana-6025	14	2	minimum	minimum	NOUN
cana-6025	14	3	𝑘	𝑘	X
cana-6025	14	4	for	for	ADP
cana-6025	14	5	which	which	PRON
cana-6025	14	6	a	a	DET
cana-6025	14	7	graph	graph	NOUN
cana-6025	14	8	mailto:divinelinr@gmail.com	mailto:divinelinr@gmail.com	NOUN
cana-6025	15	1	mailto:angeljebitha@holycrossngl.edu.in	mailto:angeljebitha@holycrossngl.edu.in	NOUN
cana-6025	15	2	mailto:sujitha.s@holycrossngl.edu.in	mailto:sujitha.s@holycrossngl.edu.in	NOUN
cana-6025	15	3	communications	communication	NOUN
cana-6025	15	4	on	on	ADP
cana-6025	15	5	applied	apply	VERB
cana-6025	15	6	nonlinear	nonlinear	ADJ
cana-6025	15	7	analysis	analysis	NOUN
cana-6025	15	8	issn	issn	NOUN
cana-6025	15	9	:	:	PUNCT
cana-6025	15	10	1074	1074	NUM
cana-6025	15	11	-	-	PUNCT
cana-6025	15	12	133x	133x	NUM
cana-6025	15	13	vol	vol	NOUN
cana-6025	15	14	31	31	NUM
cana-6025	15	15	no	no	NOUN
cana-6025	15	16	.	.	NOUN
cana-6025	15	17	2	2	NUM
cana-6025	15	18	(	(	PUNCT
cana-6025	15	19	2024	2024	NUM
cana-6025	15	20	)	)	PUNCT
cana-6025	15	21	490	490	NUM
cana-6025	15	22	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	15	23	𝐺	𝐺	PROPN
cana-6025	15	24	is	be	AUX
cana-6025	15	25	k	k	ADJ
cana-6025	15	26	-	-	ADJ
cana-6025	15	27	colorable	colorable	ADJ
cana-6025	15	28	is	be	AUX
cana-6025	15	29	called	call	VERB
cana-6025	15	30	its	its	PRON
cana-6025	15	31	chromatic	chromatic	ADJ
cana-6025	15	32	number	number	NOUN
cana-6025	15	33	,	,	PUNCT
cana-6025	15	34	and	and	CCONJ
cana-6025	15	35	denoted	denote	VERB
cana-6025	15	36	by	by	ADP
cana-6025	15	37	𝜒(𝐺	𝜒(𝐺	NOUN
cana-6025	15	38	)	)	PUNCT
cana-6025	15	39	.	.	PUNCT
cana-6025	16	1	graph	graph	NOUN
cana-6025	16	2	theory	theory	NOUN
cana-6025	16	3	terminologies	terminology	NOUN
cana-6025	16	4	which	which	PRON
cana-6025	16	5	are	be	AUX
cana-6025	16	6	not	not	PART
cana-6025	16	7	defined	define	VERB
cana-6025	16	8	here	here	ADV
cana-6025	16	9	can	can	AUX
cana-6025	16	10	be	be	AUX
cana-6025	16	11	seen	see	VERB
cana-6025	16	12	in	in	ADP
cana-6025	16	13	[	[	X
cana-6025	16	14	1	1	NUM
cana-6025	16	15	]	]	PUNCT
cana-6025	16	16	and	and	CCONJ
cana-6025	16	17	[	[	X
cana-6025	16	18	2	2	NUM
cana-6025	16	19	]	]	PUNCT
cana-6025	16	20	.	.	PUNCT
cana-6025	17	1	a	a	DET
cana-6025	17	2	set	set	VERB
cana-6025	17	3	𝐷	𝐷	PROPN
cana-6025	17	4	⊆	⊆	NUM
cana-6025	17	5	𝑉	𝑉	PROPN
cana-6025	17	6	of	of	ADP
cana-6025	17	7	vertices	vertex	NOUN
cana-6025	17	8	in	in	ADP
cana-6025	17	9	a	a	DET
cana-6025	17	10	graph	graph	NOUN
cana-6025	17	11	𝐺	𝐺	NOUN
cana-6025	17	12	is	be	AUX
cana-6025	17	13	called	call	VERB
cana-6025	17	14	a	a	DET
cana-6025	17	15	dominating	dominating	NOUN
cana-6025	17	16	set	set	NOUN
cana-6025	17	17	if	if	SCONJ
cana-6025	17	18	every	every	DET
cana-6025	17	19	vertex	vertex	NOUN
cana-6025	17	20	𝑢	𝑢	ADP
cana-6025	17	21	∈	∈	PROPN
cana-6025	17	22	𝑉	𝑉	PROPN
cana-6025	17	23	is	be	AUX
cana-6025	17	24	either	either	CCONJ
cana-6025	17	25	an	an	DET
cana-6025	17	26	element	element	NOUN
cana-6025	17	27	of	of	ADP
cana-6025	17	28	𝐷	𝐷	NOUN
cana-6025	17	29	or	or	CCONJ
cana-6025	17	30	is	be	AUX
cana-6025	17	31	adjacent	adjacent	ADJ
cana-6025	17	32	to	to	ADP
cana-6025	17	33	an	an	DET
cana-6025	17	34	element	element	NOUN
cana-6025	17	35	of	of	ADP
cana-6025	17	36	𝐷.	𝐷.	PROPN
cana-6025	17	37	the	the	DET
cana-6025	17	38	minimum	minimum	ADJ
cana-6025	17	39	cardinality	cardinality	NOUN
cana-6025	17	40	taken	take	VERB
cana-6025	17	41	over	over	ADP
cana-6025	17	42	all	all	DET
cana-6025	17	43	minimal	minimal	ADJ
cana-6025	17	44	dominating	dominating	NOUN
cana-6025	17	45	sets	set	NOUN
cana-6025	17	46	is	be	AUX
cana-6025	17	47	called	call	VERB
cana-6025	17	48	the	the	DET
cana-6025	17	49	domination	domination	NOUN
cana-6025	17	50	number	number	NOUN
cana-6025	17	51	of	of	ADP
cana-6025	17	52	𝐺	𝐺	PROPN
cana-6025	17	53	and	and	CCONJ
cana-6025	17	54	is	be	AUX
cana-6025	17	55	denoted	denote	VERB
cana-6025	17	56	by	by	ADP
cana-6025	17	57	𝛾(𝐺	𝛾(𝐺	PROPN
cana-6025	17	58	)	)	PUNCT
cana-6025	17	59	.	.	PUNCT
cana-6025	18	1	a	a	DET
cana-6025	18	2	set	set	VERB
cana-6025	18	3	𝐷	𝐷	NOUN
cana-6025	18	4	⊆	⊆	NUM
cana-6025	18	5	𝑉	𝑉	PROPN
cana-6025	18	6	is	be	AUX
cana-6025	18	7	a	a	DET
cana-6025	18	8	restrained	restrained	ADJ
cana-6025	18	9	dominating	dominating	NOUN
cana-6025	18	10	set	set	VERB
cana-6025	18	11	if	if	SCONJ
cana-6025	18	12	every	every	DET
cana-6025	18	13	vertex	vertex	NOUN
cana-6025	18	14	in	in	ADP
cana-6025	18	15	𝑉	𝑉	PROPN
cana-6025	18	16	−	−	PROPN
cana-6025	18	17	𝐷	𝐷	PROPN
cana-6025	18	18	is	be	AUX
cana-6025	18	19	adjacent	adjacent	ADJ
cana-6025	18	20	to	to	ADP
cana-6025	18	21	a	a	DET
cana-6025	18	22	vertex	vertex	NOUN
cana-6025	18	23	in	in	ADP
cana-6025	18	24	𝐷	𝐷	NOUN
cana-6025	18	25	and	and	CCONJ
cana-6025	18	26	another	another	DET
cana-6025	18	27	vertex	vertex	NOUN
cana-6025	18	28	in	in	ADP
cana-6025	18	29	𝑉	𝑉	PROPN
cana-6025	18	30	−	−	PROPN
cana-6025	18	31	𝐷	𝐷	PROPN
cana-6025	18	32	[	[	NOUN
cana-6025	18	33	3	3	NUM
cana-6025	18	34	]	]	PUNCT
cana-6025	18	35	.	.	PUNCT
cana-6025	19	1	the	the	DET
cana-6025	19	2	minimum	minimum	ADJ
cana-6025	19	3	cardinality	cardinality	NOUN
cana-6025	19	4	taken	take	VERB
cana-6025	19	5	over	over	ADP
cana-6025	19	6	all	all	DET
cana-6025	19	7	minimal	minimal	ADJ
cana-6025	19	8	restrained	restrained	ADJ
cana-6025	19	9	dominating	dominating	NOUN
cana-6025	19	10	sets	set	NOUN
cana-6025	19	11	is	be	AUX
cana-6025	19	12	called	call	VERB
cana-6025	19	13	the	the	DET
cana-6025	19	14	restrained	restrained	ADJ
cana-6025	19	15	domination	domination	NOUN
cana-6025	19	16	number	number	NOUN
cana-6025	19	17	of	of	ADP
cana-6025	19	18	𝐺	𝐺	PROPN
cana-6025	19	19	and	and	CCONJ
cana-6025	19	20	is	be	AUX
cana-6025	19	21	denoted	denote	VERB
cana-6025	19	22	by	by	ADP
cana-6025	19	23	𝛾𝑟(𝐺	𝛾𝑟(𝐺	PROPN
cana-6025	19	24	)	)	PUNCT
cana-6025	19	25	.	.	PUNCT
cana-6025	20	1	a	a	DET
cana-6025	20	2	set	set	VERB
cana-6025	20	3	𝐷	𝐷	NOUN
cana-6025	20	4	is	be	AUX
cana-6025	20	5	a	a	DET
cana-6025	20	6	𝛾𝑟	𝛾𝑟	NOUN
cana-6025	20	7	set	set	NOUN
cana-6025	20	8	if	if	SCONJ
cana-6025	20	9	𝐷	𝐷	PROPN
cana-6025	20	10	is	be	AUX
cana-6025	20	11	a	a	DET
cana-6025	20	12	restrained	restrained	ADJ
cana-6025	20	13	dominating	dominating	NOUN
cana-6025	20	14	set	set	NOUN
cana-6025	20	15	of	of	ADP
cana-6025	20	16	cardinality	cardinality	PROPN
cana-6025	20	17	𝛾𝑟(𝐺	𝛾𝑟(𝐺	PROPN
cana-6025	20	18	)	)	PUNCT
cana-6025	20	19	.	.	PUNCT
cana-6025	21	1	strong	strong	ADJ
cana-6025	21	2	product	product	NOUN
cana-6025	21	3	of	of	ADP
cana-6025	21	4	two	two	NUM
cana-6025	21	5	graphs	graph	NOUN
cana-6025	21	6	𝐺	𝐺	PROPN
cana-6025	21	7	and	and	CCONJ
cana-6025	21	8	𝐻	𝐻	PROPN
cana-6025	21	9	is	be	AUX
cana-6025	21	10	the	the	DET
cana-6025	21	11	graph	graph	NOUN
cana-6025	21	12	𝐺	𝐺	PROPN
cana-6025	21	13	⊠	⊠	PROPN
cana-6025	21	14	𝐻	𝐻	PROPN
cana-6025	21	15	whose	whose	DET
cana-6025	21	16	vertex	vertex	NOUN
cana-6025	21	17	set	set	NOUN
cana-6025	21	18	is	be	AUX
cana-6025	21	19	𝑉(𝐺	𝑉(𝐺	NOUN
cana-6025	21	20	)	)	PUNCT
cana-6025	21	21	×	×	NOUN
cana-6025	21	22	𝑉(𝐻	𝑉(𝐻	NOUN
cana-6025	21	23	)	)	PUNCT
cana-6025	21	24	,	,	PUNCT
cana-6025	21	25	vertices	vertice	VERB
cana-6025	21	26	(	(	PUNCT
cana-6025	21	27	𝑢	𝑢	X
cana-6025	21	28	,	,	PUNCT
cana-6025	21	29	𝑥	𝑥	NOUN
cana-6025	21	30	)	)	PUNCT
cana-6025	21	31	and	and	CCONJ
cana-6025	21	32	(	(	PUNCT
cana-6025	21	33	𝑣	𝑣	NOUN
cana-6025	21	34	,	,	PUNCT
cana-6025	21	35	𝑦	𝑦	NOUN
cana-6025	21	36	)	)	PUNCT
cana-6025	21	37	being	be	AUX
cana-6025	21	38	adjacent	adjacent	ADJ
cana-6025	21	39	if	if	SCONJ
cana-6025	21	40	and	and	CCONJ
cana-6025	21	41	only	only	ADV
cana-6025	21	42	if	if	SCONJ
cana-6025	21	43	𝑢𝑣	𝑢𝑣	PROPN
cana-6025	21	44	∈	∈	PROPN
cana-6025	21	45	𝐸(𝐺	𝐸(𝐺	NOUN
cana-6025	21	46	)	)	PUNCT
cana-6025	21	47	and	and	CCONJ
cana-6025	21	48	𝑥	𝑥	X
cana-6025	21	49	=	=	SYM
cana-6025	21	50	𝑦	𝑦	NOUN
cana-6025	21	51	(	(	PUNCT
cana-6025	21	52	or	or	CCONJ
cana-6025	21	53	)	)	PUNCT
cana-6025	21	54	𝑢	𝑢	PROPN
cana-6025	21	55	=	=	SYM
cana-6025	21	56	𝑣	𝑣	NOUN
cana-6025	21	57	and	and	CCONJ
cana-6025	21	58	𝑥𝑦	𝑥𝑦	PROPN
cana-6025	21	59	∈	∈	PROPN
cana-6025	21	60	𝐸(𝐻	𝐸(𝐻	NOUN
cana-6025	21	61	)	)	PUNCT
cana-6025	21	62	or	or	CCONJ
cana-6025	21	63	𝑢𝑣	𝑢𝑣	PROPN
cana-6025	21	64	∈	∈	PROPN
cana-6025	21	65	𝐸(𝐺	𝐸(𝐺	NOUN
cana-6025	21	66	)	)	PUNCT
cana-6025	21	67	and	and	CCONJ
cana-6025	21	68	𝑥𝑦	𝑥𝑦	PROPN
cana-6025	21	69	∈	∈	PROPN
cana-6025	21	70	𝐸(𝐻	𝐸(𝐻	NOUN
cana-6025	21	71	)	)	PUNCT
cana-6025	22	1	[	[	X
cana-6025	22	2	4	4	NUM
cana-6025	22	3	]	]	PUNCT
cana-6025	22	4	.	.	PUNCT
cana-6025	23	1	t.	t.	PROPN
cana-6025	23	2	n.	n.	PROPN
cana-6025	23	3	janakiraman	janakiraman	PROPN
cana-6025	23	4	and	and	CCONJ
cana-6025	23	5	m.	m.	NOUN
cana-6025	23	6	poobalaranjani	poobalaranjani	NOUN
cana-6025	23	7	introduced	introduce	VERB
cana-6025	23	8	the	the	DET
cana-6025	23	9	concept	concept	NOUN
cana-6025	23	10	of	of	ADP
cana-6025	23	11	chromatic	chromatic	ADJ
cana-6025	23	12	preserving	preserve	VERB
cana-6025	23	13	set	set	NOUN
cana-6025	23	14	.	.	PUNCT
cana-6025	24	1	a	a	DET
cana-6025	24	2	set	set	VERB
cana-6025	24	3	𝐷	𝐷	NOUN
cana-6025	24	4	⊆	⊆	NUM
cana-6025	24	5	𝑉	𝑉	PROPN
cana-6025	24	6	is	be	AUX
cana-6025	24	7	a	a	DET
cana-6025	24	8	chromatic	chromatic	ADJ
cana-6025	24	9	preserving	preserve	VERB
cana-6025	24	10	set	set	NOUN
cana-6025	24	11	or	or	CCONJ
cana-6025	24	12	a	a	DET
cana-6025	24	13	cp	cp	NOUN
cana-6025	24	14	-	-	PUNCT
cana-6025	24	15	set	set	VERB
cana-6025	24	16	if	if	SCONJ
cana-6025	24	17	𝜒	𝜒	X
cana-6025	24	18	(	(	PUNCT
cana-6025	24	19	<	<	X
cana-6025	24	20	𝐷	𝐷	PROPN
cana-6025	24	21	>	>	PUNCT
cana-6025	24	22	)	)	PUNCT
cana-6025	24	23	=	=	SYM
cana-6025	24	24	𝜒(𝐺	𝜒(𝐺	NOUN
cana-6025	24	25	)	)	PUNCT
cana-6025	24	26	and	and	CCONJ
cana-6025	24	27	the	the	DET
cana-6025	24	28	minimum	minimum	ADJ
cana-6025	24	29	cardinality	cardinality	NOUN
cana-6025	24	30	taken	take	VERB
cana-6025	24	31	over	over	ADP
cana-6025	24	32	all	all	DET
cana-6025	24	33	cp	cp	NOUN
cana-6025	24	34	-	-	NOUN
cana-6025	24	35	sets	set	NOUN
cana-6025	24	36	in	in	ADP
cana-6025	24	37	𝐺	𝐺	PROPN
cana-6025	24	38	is	be	AUX
cana-6025	24	39	called	call	VERB
cana-6025	24	40	the	the	DET
cana-6025	24	41	chromatic	chromatic	ADJ
cana-6025	24	42	preserving	preserve	VERB
cana-6025	24	43	number	number	NOUN
cana-6025	24	44	or	or	CCONJ
cana-6025	24	45	cp	cp	NOUN
cana-6025	24	46	-	-	PUNCT
cana-6025	24	47	number	number	NOUN
cana-6025	24	48	of	of	ADP
cana-6025	24	49	𝐺	𝐺	PROPN
cana-6025	24	50	and	and	CCONJ
cana-6025	24	51	is	be	AUX
cana-6025	24	52	denoted	denote	VERB
cana-6025	24	53	by	by	ADP
cana-6025	24	54	𝑐𝑝𝑛(𝐺	𝑐𝑝𝑛(𝐺	NOUN
cana-6025	24	55	)	)	PUNCT
cana-6025	25	1	[	[	X
cana-6025	25	2	5	5	NUM
cana-6025	25	3	]	]	PUNCT
cana-6025	25	4	.	.	PUNCT
cana-6025	26	1	a	a	DET
cana-6025	26	2	subset	subset	NOUN
cana-6025	26	3	𝐷	𝐷	NOUN
cana-6025	26	4	of	of	ADP
cana-6025	26	5	𝑉	𝑉	PROPN
cana-6025	26	6	is	be	AUX
cana-6025	26	7	said	say	VERB
cana-6025	26	8	to	to	PART
cana-6025	26	9	be	be	AUX
cana-6025	26	10	a	a	DET
cana-6025	26	11	dom	dom	NOUN
cana-6025	26	12	-	-	PUNCT
cana-6025	26	13	chromatic	chromatic	ADJ
cana-6025	26	14	set	set	NOUN
cana-6025	26	15	(	(	PUNCT
cana-6025	26	16	or	or	CCONJ
cana-6025	26	17	dc	dc	PROPN
cana-6025	26	18	-	-	PUNCT
cana-6025	26	19	set)if	set)if	PROPN
cana-6025	26	20	𝐷	𝐷	PROPN
cana-6025	26	21	is	be	AUX
cana-6025	26	22	a	a	DET
cana-6025	26	23	dominating	dominating	NOUN
cana-6025	26	24	set	set	NOUN
cana-6025	26	25	and	and	CCONJ
cana-6025	26	26	𝜒	𝜒	X
cana-6025	26	27	(	(	PUNCT
cana-6025	26	28	<	<	X
cana-6025	26	29	𝐷	𝐷	PROPN
cana-6025	26	30	>	>	PUNCT
cana-6025	26	31	)	)	PUNCT
cana-6025	26	32	=	=	SYM
cana-6025	26	33	𝜒(𝐺	𝜒(𝐺	NOUN
cana-6025	26	34	)	)	PUNCT
cana-6025	26	35	.	.	PUNCT
cana-6025	27	1	the	the	DET
cana-6025	27	2	minimum	minimum	ADJ
cana-6025	27	3	cardinality	cardinality	NOUN
cana-6025	27	4	taken	take	VERB
cana-6025	27	5	over	over	ADP
cana-6025	27	6	all	all	DET
cana-6025	27	7	minimal	minimal	ADJ
cana-6025	27	8	domchromatic	domchromatic	ADJ
cana-6025	27	9	sets	set	NOUN
cana-6025	27	10	in	in	ADP
cana-6025	27	11	𝐺	𝐺	PROPN
cana-6025	27	12	is	be	AUX
cana-6025	27	13	called	call	VERB
cana-6025	27	14	the	the	DET
cana-6025	27	15	dom	dom	NOUN
cana-6025	27	16	-	-	PUNCT
cana-6025	27	17	chromatic	chromatic	ADJ
cana-6025	27	18	number	number	NOUN
cana-6025	27	19	and	and	CCONJ
cana-6025	27	20	is	be	AUX
cana-6025	27	21	denoted	denote	VERB
cana-6025	27	22	by	by	ADP
cana-6025	27	23	𝛾𝑐ℎ(𝐺)[6	𝛾𝑐ℎ(𝐺)[6	NOUN
cana-6025	27	24	]	]	PUNCT
cana-6025	27	25	.	.	PUNCT
cana-6025	28	1	in	in	ADP
cana-6025	28	2	this	this	DET
cana-6025	28	3	paper	paper	NOUN
cana-6025	28	4	,	,	PUNCT
cana-6025	28	5	the	the	DET
cana-6025	28	6	chromatic	chromatic	ADJ
cana-6025	28	7	restrained	restrained	ADJ
cana-6025	28	8	domination	domination	NOUN
cana-6025	28	9	number	number	NOUN
cana-6025	28	10	on	on	ADP
cana-6025	28	11	the	the	DET
cana-6025	28	12	strong	strong	ADJ
cana-6025	28	13	product	product	NOUN
cana-6025	28	14	of	of	ADP
cana-6025	28	15	some	some	DET
cana-6025	28	16	standard	standard	ADJ
cana-6025	28	17	graphs	graph	NOUN
cana-6025	28	18	are	be	AUX
cana-6025	28	19	obtained	obtain	VERB
cana-6025	28	20	.	.	PUNCT
cana-6025	28	21	2	2	NUM
cana-6025	28	22	main	main	ADJ
cana-6025	28	23	results	result	NOUN
cana-6025	28	24	in	in	ADP
cana-6025	28	25	this	this	DET
cana-6025	28	26	section	section	NOUN
cana-6025	28	27	,	,	PUNCT
cana-6025	28	28	we	we	PRON
cana-6025	28	29	obtain	obtain	VERB
cana-6025	28	30	the	the	DET
cana-6025	28	31	chromatic	chromatic	ADJ
cana-6025	28	32	restrained	restrain	VERB
cana-6025	28	33	domination	domination	NOUN
cana-6025	28	34	number	number	NOUN
cana-6025	28	35	for	for	ADP
cana-6025	28	36	the	the	DET
cana-6025	28	37	strong	strong	ADJ
cana-6025	28	38	product	product	NOUN
cana-6025	28	39	of	of	ADP
cana-6025	28	40	some	some	DET
cana-6025	28	41	standard	standard	ADJ
cana-6025	28	42	graphs	graph	NOUN
cana-6025	28	43	.	.	PUNCT
cana-6025	29	1	definition	definition	NOUN
cana-6025	29	2	2.1	2.1	NUM
cana-6025	29	3	let	let	VERB
cana-6025	29	4	𝐺	𝐺	PROPN
cana-6025	29	5	=	=	SYM
cana-6025	29	6	(	(	PUNCT
cana-6025	29	7	𝑉	𝑉	PROPN
cana-6025	29	8	,	,	PUNCT
cana-6025	29	9	𝐸	𝐸	PROPN
cana-6025	29	10	)	)	PUNCT
cana-6025	29	11	be	be	VERB
cana-6025	29	12	a	a	DET
cana-6025	29	13	graph	graph	NOUN
cana-6025	29	14	.	.	PUNCT
cana-6025	30	1	a	a	DET
cana-6025	30	2	subset	subset	NOUN
cana-6025	30	3	𝐷	𝐷	NOUN
cana-6025	30	4	of	of	ADP
cana-6025	30	5	𝑉	𝑉	PROPN
cana-6025	30	6	is	be	AUX
cana-6025	30	7	said	say	VERB
cana-6025	30	8	to	to	PART
cana-6025	30	9	be	be	AUX
cana-6025	30	10	a	a	DET
cana-6025	30	11	chromatic	chromatic	ADJ
cana-6025	30	12	restrained	restrain	VERB
cana-6025	30	13	dominating	dominating	NOUN
cana-6025	30	14	set	set	NOUN
cana-6025	30	15	(	(	PUNCT
cana-6025	30	16	or	or	CCONJ
cana-6025	30	17	crd	crd	NOUN
cana-6025	30	18	-	-	PUNCT
cana-6025	30	19	set	set	NOUN
cana-6025	30	20	)	)	PUNCT
cana-6025	30	21	if	if	SCONJ
cana-6025	30	22	𝐷	𝐷	PROPN
cana-6025	30	23	is	be	AUX
cana-6025	30	24	a	a	DET
cana-6025	30	25	restrained	restrained	ADJ
cana-6025	30	26	dominating	dominating	NOUN
cana-6025	30	27	set	set	NOUN
cana-6025	30	28	and	and	CCONJ
cana-6025	30	29	𝜒	𝜒	X
cana-6025	30	30	(	(	PUNCT
cana-6025	30	31	<	<	X
cana-6025	30	32	𝐷	𝐷	PROPN
cana-6025	30	33	>	>	PUNCT
cana-6025	30	34	)	)	PUNCT
cana-6025	30	35	=	=	SYM
cana-6025	30	36	𝜒(𝐺	𝜒(𝐺	NOUN
cana-6025	30	37	)	)	PUNCT
cana-6025	30	38	.	.	PUNCT
cana-6025	31	1	the	the	DET
cana-6025	31	2	minimum	minimum	ADJ
cana-6025	31	3	cardinality	cardinality	NOUN
cana-6025	31	4	taken	take	VERB
cana-6025	31	5	over	over	ADP
cana-6025	31	6	all	all	DET
cana-6025	31	7	minimal	minimal	ADJ
cana-6025	31	8	chromatic	chromatic	ADJ
cana-6025	31	9	restrained	restrain	VERB
cana-6025	31	10	dominating	dominating	NOUN
cana-6025	31	11	sets	set	NOUN
cana-6025	31	12	is	be	AUX
cana-6025	31	13	called	call	VERB
cana-6025	31	14	the	the	DET
cana-6025	31	15	chromatic	chromatic	ADJ
cana-6025	31	16	restrained	restrain	VERB
cana-6025	31	17	domination	domination	NOUN
cana-6025	31	18	number	number	NOUN
cana-6025	31	19	and	and	CCONJ
cana-6025	31	20	is	be	AUX
cana-6025	31	21	denoted	denote	VERB
cana-6025	31	22	by	by	ADP
cana-6025	31	23	𝛾𝑟	𝛾𝑟	ADP
cana-6025	31	24	𝑐(𝐺	𝑐(𝐺	NUM
cana-6025	31	25	)	)	PUNCT
cana-6025	31	26	.	.	PUNCT
cana-6025	32	1	throughout	throughout	ADP
cana-6025	32	2	this	this	DET
cana-6025	32	3	paper	paper	NOUN
cana-6025	32	4	,	,	PUNCT
cana-6025	32	5	we	we	PRON
cana-6025	32	6	denote	denote	VERB
cana-6025	32	7	the	the	DET
cana-6025	32	8	chromatic	chromatic	ADJ
cana-6025	32	9	restrained	restrained	ADJ
cana-6025	32	10	domination	domination	NOUN
cana-6025	32	11	number	number	NOUN
cana-6025	32	12	on	on	ADP
cana-6025	32	13	the	the	DET
cana-6025	32	14	strong	strong	ADJ
cana-6025	32	15	product	product	NOUN
cana-6025	32	16	of	of	ADP
cana-6025	32	17	two	two	NUM
cana-6025	32	18	graphs	graph	NOUN
cana-6025	32	19	𝐺	𝐺	NOUN
cana-6025	32	20	and	and	CCONJ
cana-6025	32	21	𝐻	𝐻	PROPN
cana-6025	32	22	by	by	ADP
cana-6025	32	23	𝛾𝑟	𝛾𝑟	ADP
cana-6025	32	24	𝑐(𝐺	𝑐(𝐺	NUM
cana-6025	32	25	⊠	⊠	PROPN
cana-6025	32	26	𝐻	𝐻	PROPN
cana-6025	32	27	)	)	PUNCT
cana-6025	32	28	.	.	PUNCT
cana-6025	33	1	theorem	theorem	VERB
cana-6025	33	2	2.2	2.2	NUM
cana-6025	33	3	for	for	ADP
cana-6025	33	4	𝑟	𝑟	NOUN
cana-6025	33	5	,	,	PUNCT
cana-6025	33	6	𝑠	𝑠	PRON
cana-6025	33	7	≥	≥	NUM
cana-6025	33	8	2	2	NUM
cana-6025	33	9	,	,	PUNCT
cana-6025	33	10	𝛾𝑟	𝛾𝑟	ADP
cana-6025	33	11	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	33	12	⊠	⊠	PROPN
cana-6025	33	13	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	33	14	)	)	PUNCT
cana-6025	33	15	=	=	SYM
cana-6025	33	16	𝑟𝑠.	𝑟𝑠.	NOUN
cana-6025	33	17	communications	communication	NOUN
cana-6025	33	18	on	on	ADP
cana-6025	33	19	applied	apply	VERB
cana-6025	33	20	nonlinear	nonlinear	ADJ
cana-6025	33	21	analysis	analysis	NOUN
cana-6025	33	22	issn	issn	NOUN
cana-6025	33	23	:	:	PUNCT
cana-6025	33	24	1074	1074	NUM
cana-6025	33	25	-	-	PUNCT
cana-6025	33	26	133x	133x	NUM
cana-6025	33	27	vol	vol	NOUN
cana-6025	33	28	31	31	NUM
cana-6025	33	29	no	no	NOUN
cana-6025	33	30	.	.	NOUN
cana-6025	33	31	2	2	NUM
cana-6025	33	32	(	(	PUNCT
cana-6025	33	33	2024	2024	NUM
cana-6025	33	34	)	)	PUNCT
cana-6025	33	35	491	491	NUM
cana-6025	33	36	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	33	37	proof	proof	NOUN
cana-6025	33	38	.	.	PUNCT
cana-6025	34	1	let	let	VERB
cana-6025	34	2	𝑉(𝐾𝑟	𝑉(𝐾𝑟	NUM
cana-6025	34	3	)	)	PUNCT
cana-6025	35	1	=	=	PRON
cana-6025	35	2	{	{	PUNCT
cana-6025	35	3	𝑢1	𝑢1	PROPN
cana-6025	35	4	,	,	PUNCT
cana-6025	35	5	𝑢2	𝑢2	PROPN
cana-6025	35	6	,	,	PUNCT
cana-6025	35	7	𝑢3	𝑢3	PROPN
cana-6025	35	8	,	,	PUNCT
cana-6025	35	9	.	.	PUNCT
cana-6025	35	10	.	.	PUNCT
cana-6025	36	1	.	.	PUNCT
cana-6025	37	1	,	,	PUNCT
cana-6025	37	2	𝑢𝑟	𝑢𝑟	ADV
cana-6025	37	3	}	}	PUNCT
cana-6025	37	4	and	and	CCONJ
cana-6025	37	5	𝑉(𝐾𝑠	𝑉(𝐾𝑠	NOUN
cana-6025	37	6	)	)	PUNCT
cana-6025	37	7	=	=	PRON
cana-6025	37	8	{	{	PUNCT
cana-6025	37	9	𝑣1	𝑣1	PROPN
cana-6025	37	10	,	,	PUNCT
cana-6025	37	11	𝑣2	𝑣2	PROPN
cana-6025	37	12	,	,	PUNCT
cana-6025	37	13	𝑣3	𝑣3	ADJ
cana-6025	37	14	,	,	PUNCT
cana-6025	37	15	.	.	PUNCT
cana-6025	37	16	.	.	PUNCT
cana-6025	38	1	.	.	PUNCT
cana-6025	39	1	,	,	PUNCT
cana-6025	39	2	𝑣𝑠	𝑣𝑠	ADP
cana-6025	39	3	}	}	PUNCT
cana-6025	39	4	.	.	PUNCT
cana-6025	40	1	then	then	ADV
cana-6025	40	2	,	,	PUNCT
cana-6025	40	3	𝑉(𝐾𝑟	𝑉(𝐾𝑟	ADJ
cana-6025	40	4	⊠	⊠	PROPN
cana-6025	40	5	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	40	6	)	)	PUNCT
cana-6025	40	7	=	=	PRON
cana-6025	40	8	{	{	PUNCT
cana-6025	40	9	(	(	PUNCT
cana-6025	40	10	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	40	11	,	,	PUNCT
cana-6025	40	12	𝑣𝑗)/1	𝑣𝑗)/1	PROPN
cana-6025	40	13	≤	≤	PROPN
cana-6025	40	14	𝑖	𝑖	SYM
cana-6025	40	15	≤	≤	NOUN
cana-6025	40	16	𝑟	𝑟	NOUN
cana-6025	40	17	,	,	PUNCT
cana-6025	40	18	1	1	NUM
cana-6025	40	19	≤	≤	NUM
cana-6025	40	20	𝑗	𝑗	PRON
cana-6025	40	21	≤	≤	NUM
cana-6025	40	22	𝑠	𝑠	NOUN
cana-6025	40	23	}	}	PUNCT
cana-6025	40	24	where	where	SCONJ
cana-6025	40	25	each	each	DET
cana-6025	40	26	vertex	vertex	NOUN
cana-6025	40	27	in	in	ADP
cana-6025	40	28	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	40	29	⊠	⊠	PROPN
cana-6025	40	30	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	40	31	is	be	AUX
cana-6025	40	32	a	a	DET
cana-6025	40	33	full	full	ADJ
cana-6025	40	34	degree	degree	NOUN
cana-6025	40	35	vertex	vertex	NOUN
cana-6025	40	36	and	and	CCONJ
cana-6025	40	37	|𝑉(𝐾𝑟	|𝑉(𝐾𝑟	ADJ
cana-6025	40	38	⊠	⊠	PROPN
cana-6025	40	39	𝐾𝑠)|	𝐾𝑠)|	NOUN
cana-6025	40	40	=	=	PUNCT
cana-6025	40	41	𝑟𝑠.	𝑟𝑠.	NOUN
cana-6025	40	42	also	also	ADV
cana-6025	40	43	,	,	PUNCT
cana-6025	40	44	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	40	45	⊠	⊠	PROPN
cana-6025	40	46	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	40	47	)	)	PUNCT
cana-6025	40	48	=	=	SYM
cana-6025	40	49	𝑟𝑠	𝑟𝑠	PROPN
cana-6025	40	50	as	as	SCONJ
cana-6025	40	51	each	each	DET
cana-6025	40	52	vertex	vertex	NOUN
cana-6025	40	53	can	can	AUX
cana-6025	40	54	be	be	AUX
cana-6025	40	55	given	give	VERB
cana-6025	40	56	different	different	ADJ
cana-6025	40	57	colors	color	NOUN
cana-6025	40	58	.	.	PUNCT
cana-6025	41	1	let	let	VERB
cana-6025	41	2	𝐷	𝐷	NOUN
cana-6025	41	3	be	be	AUX
cana-6025	41	4	a	a	DET
cana-6025	41	5	𝛾𝑟	𝛾𝑟	ADP
cana-6025	41	6	𝑐	𝑐	PROPN
cana-6025	41	7	−	−	PROPN
cana-6025	41	8	set	set	NOUN
cana-6025	41	9	of	of	ADP
cana-6025	41	10	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	41	11	⊠	⊠	PROPN
cana-6025	41	12	𝐾𝑠.	𝐾𝑠.	PROPN
cana-6025	41	13	for	for	ADP
cana-6025	41	14	any	any	DET
cana-6025	41	15	proper	proper	ADJ
cana-6025	41	16	subset	subset	NOUN
cana-6025	41	17	𝑆	𝑆	PROPN
cana-6025	41	18	of	of	ADP
cana-6025	41	19	𝑉(𝐾𝑟	𝑉(𝐾𝑟	PROPN
cana-6025	41	20	⊠	⊠	PROPN
cana-6025	41	21	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	41	22	)	)	PUNCT
cana-6025	41	23	,	,	PUNCT
cana-6025	41	24	𝜒(⟨𝑆⟩	𝜒(⟨𝑆⟩	NOUN
cana-6025	41	25	)	)	PUNCT
cana-6025	41	26	<	<	X
cana-6025	42	1	𝑟𝑠.	𝑟𝑠.	PROPN
cana-6025	42	2	thus	thus	ADV
cana-6025	42	3	,	,	PUNCT
cana-6025	42	4	𝐷	𝐷	PROPN
cana-6025	42	5	=	=	PUNCT
cana-6025	42	6	𝑉(𝐾𝑟	𝑉(𝐾𝑟	VERB
cana-6025	42	7	⊠	⊠	PROPN
cana-6025	42	8	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	42	9	)	)	PUNCT
cana-6025	42	10	is	be	AUX
cana-6025	42	11	the	the	DET
cana-6025	42	12	only	only	ADJ
cana-6025	42	13	chromatic	chromatic	ADJ
cana-6025	42	14	restrained	restrain	VERB
cana-6025	42	15	dominating	dominating	NOUN
cana-6025	42	16	set	set	NOUN
cana-6025	42	17	of	of	ADP
cana-6025	42	18	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	42	19	⊠	⊠	PROPN
cana-6025	42	20	𝐾𝑠.	𝐾𝑠.	PROPN
cana-6025	42	21	therefore	therefore	ADV
cana-6025	42	22	,	,	PUNCT
cana-6025	42	23	𝛾𝑟	𝛾𝑟	ADP
cana-6025	42	24	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	42	25	⊠	⊠	PROPN
cana-6025	42	26	𝐾𝑠	𝐾𝑠	PROPN
cana-6025	42	27	)	)	PUNCT
cana-6025	42	28	=	=	SYM
cana-6025	42	29	|𝐷|	|𝐷|	NOUN
cana-6025	42	30	=	=	SYM
cana-6025	42	31	𝑟𝑠.	𝑟𝑠.	NOUN
cana-6025	42	32	theorem	theorem	VERB
cana-6025	42	33	2.3	2.3	NUM
cana-6025	42	34	for	for	ADP
cana-6025	42	35	𝑠	𝑠	PROPN
cana-6025	42	36	≥	≥	NUM
cana-6025	42	37	4	4	NUM
cana-6025	42	38	,	,	PUNCT
cana-6025	42	39	𝛾𝑟	𝛾𝑟	ADP
cana-6025	42	40	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	42	41	⊠	⊠	PROPN
cana-6025	42	42	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	42	43	)	)	PUNCT
cana-6025	42	44	=	=	PUNCT
cana-6025	42	45	⌈	⌈	PROPN
cana-6025	43	1	𝑠−4	𝑠−4	ADP
cana-6025	43	2	3	3	NUM
cana-6025	43	3	⌉	⌉	NOUN
cana-6025	43	4	+	+	CCONJ
cana-6025	43	5	2𝑟.	2𝑟.	NUM
cana-6025	43	6	proof	proof	NOUN
cana-6025	43	7	.	.	PUNCT
cana-6025	44	1	let	let	VERB
cana-6025	44	2	𝑉(𝐾𝑟	𝑉(𝐾𝑟	NUM
cana-6025	44	3	)	)	PUNCT
cana-6025	45	1	=	=	PRON
cana-6025	45	2	{	{	PUNCT
cana-6025	45	3	𝑢1	𝑢1	PROPN
cana-6025	45	4	,	,	PUNCT
cana-6025	45	5	𝑢2	𝑢2	PROPN
cana-6025	45	6	,	,	PUNCT
cana-6025	45	7	𝑢3	𝑢3	PROPN
cana-6025	45	8	,	,	PUNCT
cana-6025	45	9	.	.	PUNCT
cana-6025	45	10	.	.	PUNCT
cana-6025	46	1	.	.	PUNCT
cana-6025	47	1	,	,	PUNCT
cana-6025	47	2	𝑢𝑟	𝑢𝑟	ADV
cana-6025	47	3	}	}	PUNCT
cana-6025	47	4	and	and	CCONJ
cana-6025	47	5	𝑉(𝑃𝑠	𝑉(𝑃𝑠	NUM
cana-6025	47	6	)	)	PUNCT
cana-6025	47	7	=	=	PRON
cana-6025	47	8	{	{	PUNCT
cana-6025	47	9	𝑣1	𝑣1	PROPN
cana-6025	47	10	,	,	PUNCT
cana-6025	47	11	𝑣2	𝑣2	PROPN
cana-6025	47	12	,	,	PUNCT
cana-6025	47	13	𝑣3	𝑣3	ADJ
cana-6025	47	14	,	,	PUNCT
cana-6025	47	15	.	.	PUNCT
cana-6025	47	16	.	.	PUNCT
cana-6025	48	1	.	.	PUNCT
cana-6025	49	1	,	,	PUNCT
cana-6025	49	2	𝑣𝑠	𝑣𝑠	ADP
cana-6025	49	3	}	}	PUNCT
cana-6025	49	4	.	.	PUNCT
cana-6025	50	1	then	then	ADV
cana-6025	50	2	𝑉(𝐾𝑟	𝑉(𝐾𝑟	PROPN
cana-6025	50	3	⊠	⊠	PROPN
cana-6025	50	4	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	50	5	)	)	PUNCT
cana-6025	50	6	=	=	PRON
cana-6025	50	7	{	{	PUNCT
cana-6025	50	8	(	(	PUNCT
cana-6025	50	9	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	50	10	,	,	PUNCT
cana-6025	50	11	𝑣𝑗)/1	𝑣𝑗)/1	PROPN
cana-6025	50	12	≤	≤	PROPN
cana-6025	50	13	𝑖	𝑖	SYM
cana-6025	50	14	≤	≤	NOUN
cana-6025	50	15	𝑟	𝑟	NOUN
cana-6025	50	16	,	,	PUNCT
cana-6025	50	17	1	1	NUM
cana-6025	50	18	≤	≤	NUM
cana-6025	50	19	𝑗	𝑗	PRON
cana-6025	50	20	≤	≤	NUM
cana-6025	50	21	𝑠	𝑠	NOUN
cana-6025	50	22	}	}	PUNCT
cana-6025	50	23	with	with	ADP
cana-6025	50	24	cardinality	cardinality	NOUN
cana-6025	50	25	𝑟𝑠.	𝑟𝑠.	PROPN
cana-6025	50	26	also	also	ADV
cana-6025	50	27	,	,	PUNCT
cana-6025	50	28	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	50	29	⊠	⊠	PROPN
cana-6025	50	30	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	50	31	contains	contain	VERB
cana-6025	50	32	𝑟	𝑟	NOUN
cana-6025	50	33	rows	row	NOUN
cana-6025	50	34	and	and	CCONJ
cana-6025	50	35	𝑠	𝑠	PRON
cana-6025	50	36	columns	column	NOUN
cana-6025	50	37	𝑉1	𝑉1	PROPN
cana-6025	50	38	,	,	PUNCT
cana-6025	50	39	𝑉2	𝑉2	NOUN
cana-6025	50	40	,	,	PUNCT
cana-6025	50	41	𝑉3	𝑉3	NOUN
cana-6025	50	42	,	,	PUNCT
cana-6025	50	43	.	.	PUNCT
cana-6025	50	44	.	.	PUNCT
cana-6025	51	1	.	.	PUNCT
cana-6025	52	1	,	,	PUNCT
cana-6025	52	2	𝑉𝑠.	𝑉𝑠.	PROPN
cana-6025	52	3	since	since	SCONJ
cana-6025	52	4	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	52	5	)	)	PUNCT
cana-6025	52	6	=	=	SYM
cana-6025	52	7	𝑟	𝑟	NOUN
cana-6025	52	8	and	and	CCONJ
cana-6025	52	9	𝜒(𝑃𝑠	𝜒(𝑃𝑠	NOUN
cana-6025	52	10	)	)	PUNCT
cana-6025	52	11	=	=	SYM
cana-6025	52	12	2	2	NUM
cana-6025	52	13	,	,	PUNCT
cana-6025	52	14	each	each	DET
cana-6025	52	15	column	column	NOUN
cana-6025	52	16	𝑉𝑖	𝑉𝑖	PROPN
cana-6025	52	17	,	,	PUNCT
cana-6025	52	18	𝑖	𝑖	VERB
cana-6025	52	19	is	be	AUX
cana-6025	52	20	odd	odd	ADJ
cana-6025	52	21	can	can	AUX
cana-6025	52	22	be	be	AUX
cana-6025	52	23	colored	color	VERB
cana-6025	52	24	with	with	ADP
cana-6025	52	25	𝑟	𝑟	NOUN
cana-6025	52	26	colors	color	NOUN
cana-6025	52	27	and	and	CCONJ
cana-6025	52	28	each	each	DET
cana-6025	52	29	𝑉𝑗	𝑉𝑗	PROPN
cana-6025	52	30	,	,	PUNCT
cana-6025	52	31	𝑗	𝑗	X
cana-6025	52	32	is	be	AUX
cana-6025	52	33	even	even	ADV
cana-6025	52	34	can	can	AUX
cana-6025	52	35	be	be	AUX
cana-6025	52	36	colored	color	VERB
cana-6025	52	37	with	with	ADP
cana-6025	52	38	another	another	DET
cana-6025	52	39	𝑟	𝑟	NOUN
cana-6025	52	40	colors	color	NOUN
cana-6025	52	41	.	.	PUNCT
cana-6025	53	1	thus	thus	ADV
cana-6025	53	2	,	,	PUNCT
cana-6025	53	3	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	53	4	⊠	⊠	PROPN
cana-6025	53	5	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	53	6	)	)	PUNCT
cana-6025	53	7	=	=	NUM
cana-6025	53	8	2𝑟.	2𝑟.	NUM
cana-6025	53	9	case	case	NOUN
cana-6025	53	10	(	(	PUNCT
cana-6025	53	11	i	i	NOUN
cana-6025	53	12	):	):	PUNCT
cana-6025	53	13	𝑠	𝑠	PROPN
cana-6025	53	14	≡	≡	PROPN
cana-6025	53	15	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	NOUN
cana-6025	53	16	3	3	X
cana-6025	53	17	)	)	PUNCT
cana-6025	53	18	let	let	VERB
cana-6025	53	19	𝐷	𝐷	NOUN
cana-6025	53	20	=	=	SYM
cana-6025	53	21	{	{	PUNCT
cana-6025	53	22	(	(	PUNCT
cana-6025	53	23	𝑢1	𝑢1	PROPN
cana-6025	53	24	,	,	PUNCT
cana-6025	53	25	𝑣3𝑘−1)/1	𝑣3𝑘−1)/1	PROPN
cana-6025	53	26	≤	≤	NOUN
cana-6025	53	27	𝑘	𝑘	DET
cana-6025	53	28	≤	≤	NUM
cana-6025	53	29	𝑠	𝑠	PRON
cana-6025	53	30	3	3	X
cana-6025	53	31	}	}	PUNCT
cana-6025	53	32	where	where	SCONJ
cana-6025	53	33	|𝐷|	|𝐷|	X
cana-6025	53	34	=	=	PUNCT
cana-6025	53	35	𝑠	𝑠	PROPN
cana-6025	53	36	3	3	NUM
cana-6025	53	37	.	.	PUNCT
cana-6025	54	1	then	then	ADV
cana-6025	54	2	,	,	PUNCT
cana-6025	54	3	𝐷	𝐷	PROPN
cana-6025	54	4	is	be	AUX
cana-6025	54	5	a	a	DET
cana-6025	54	6	dominating	dominating	NOUN
cana-6025	54	7	set	set	NOUN
cana-6025	54	8	and	and	CCONJ
cana-6025	54	9	there	there	PRON
cana-6025	54	10	does	do	AUX
cana-6025	54	11	not	not	PART
cana-6025	54	12	exists	exist	VERB
cana-6025	54	13	any	any	DET
cana-6025	54	14	isolated	isolated	ADJ
cana-6025	54	15	vertex	vertex	NOUN
cana-6025	54	16	in	in	ADP
cana-6025	54	17	⟨𝑉	⟨𝑉	PROPN
cana-6025	54	18	−	−	PROPN
cana-6025	54	19	𝐷⟩.	𝐷⟩.	PROPN
cana-6025	54	20	thus	thus	ADV
cana-6025	54	21	,	,	PUNCT
cana-6025	54	22	𝐷	𝐷	PROPN
cana-6025	54	23	is	be	AUX
cana-6025	54	24	a	a	DET
cana-6025	54	25	restrained	restrained	ADJ
cana-6025	54	26	dominating	dominating	NOUN
cana-6025	54	27	set	set	NOUN
cana-6025	54	28	and	and	CCONJ
cana-6025	54	29	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	54	30	⊠	⊠	PROPN
cana-6025	54	31	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	54	32	)	)	PUNCT
cana-6025	54	33	≤	≤	NUM
cana-6025	54	34	|𝐷|	|𝐷|	NOUN
cana-6025	54	35	=	=	SYM
cana-6025	54	36	𝑠	𝑠	PROPN
cana-6025	54	37	3	3	NUM
cana-6025	54	38	.	.	PUNCT
cana-6025	55	1	since	since	SCONJ
cana-6025	55	2	,	,	PUNCT
cana-6025	55	3	𝛾(𝑃𝑠	𝛾(𝑃𝑠	PROPN
cana-6025	55	4	)	)	PUNCT
cana-6025	55	5	=	=	SYM
cana-6025	55	6	𝑠	𝑠	ADP
cana-6025	55	7	3	3	NUM
cana-6025	55	8	and	and	CCONJ
cana-6025	55	9	each	each	DET
cana-6025	55	10	vertex	vertex	NOUN
cana-6025	55	11	in	in	ADP
cana-6025	55	12	column	column	NOUN
cana-6025	55	13	𝑉𝑗	𝑉𝑗	PROPN
cana-6025	55	14	is	be	AUX
cana-6025	55	15	adjacent	adjacent	ADJ
cana-6025	55	16	to	to	ADP
cana-6025	55	17	all	all	DET
cana-6025	55	18	the	the	DET
cana-6025	55	19	vertices	vertex	NOUN
cana-6025	55	20	in	in	ADP
cana-6025	55	21	𝑉𝑗−1	𝑉𝑗−1	PROPN
cana-6025	55	22	,	,	PUNCT
cana-6025	55	23	𝑉𝑗	𝑉𝑗	PROPN
cana-6025	55	24	and	and	CCONJ
cana-6025	55	25	𝑉𝑗+1	𝑉𝑗+1	VERB
cana-6025	55	26	,	,	PUNCT
cana-6025	55	27	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	55	28	⊠	⊠	PROPN
cana-6025	55	29	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	55	30	)	)	PUNCT
cana-6025	55	31	≥	≥	NOUN
cana-6025	55	32	𝑠	𝑠	PROPN
cana-6025	55	33	3	3	NUM
cana-6025	55	34	.	.	PUNCT
cana-6025	56	1	thus	thus	ADV
cana-6025	56	2	,	,	PUNCT
cana-6025	56	3	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	56	4	⊠	⊠	PROPN
cana-6025	56	5	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	56	6	)	)	PUNCT
cana-6025	56	7	=	=	SYM
cana-6025	56	8	𝑠	𝑠	DET
cana-6025	56	9	3	3	NUM
cana-6025	56	10	.	.	PUNCT
cana-6025	57	1	but	but	CCONJ
cana-6025	57	2	,	,	PUNCT
cana-6025	57	3	every	every	DET
cana-6025	57	4	minimum	minimum	NOUN
cana-6025	57	5	restrained	restrain	VERB
cana-6025	57	6	dominating	dominating	NOUN
cana-6025	57	7	set	set	NOUN
cana-6025	57	8	is	be	AUX
cana-6025	57	9	independent	independent	ADJ
cana-6025	57	10	and	and	CCONJ
cana-6025	57	11	so	so	ADV
cana-6025	57	12	,	,	PUNCT
cana-6025	57	13	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	57	14	)	)	PUNCT
cana-6025	57	15	=	=	SYM
cana-6025	57	16	1	1	NUM
cana-6025	57	17	≠	≠	PROPN
cana-6025	57	18	𝜒(𝐾𝑟	𝜒(𝐾𝑟	VERB
cana-6025	57	19	⊠	⊠	PROPN
cana-6025	57	20	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	57	21	)	)	PUNCT
cana-6025	57	22	.	.	PUNCT
cana-6025	58	1	thus	thus	ADV
cana-6025	58	2	,	,	PUNCT
cana-6025	58	3	𝐷	𝐷	PROPN
cana-6025	58	4	is	be	AUX
cana-6025	58	5	not	not	PART
cana-6025	58	6	a	a	DET
cana-6025	58	7	chromatic	chromatic	ADJ
cana-6025	58	8	restrained	restrain	VERB
cana-6025	58	9	dominating	dominating	NOUN
cana-6025	58	10	set	set	NOUN
cana-6025	58	11	of	of	ADP
cana-6025	58	12	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	58	13	⊠	⊠	PROPN
cana-6025	58	14	𝑃𝑠.	𝑃𝑠.	PROPN
cana-6025	58	15	consider	consider	VERB
cana-6025	58	16	𝐷1	𝐷1	NOUN
cana-6025	58	17	=	=	PROPN
cana-6025	58	18	𝐷	𝐷	PROPN
cana-6025	58	19	∪	∪	X
cana-6025	58	20	{	{	PUNCT
cana-6025	58	21	(	(	PUNCT
cana-6025	58	22	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	58	23	,	,	PUNCT
cana-6025	58	24	𝑣2	𝑣2	PROPN
cana-6025	58	25	)	)	PUNCT
cana-6025	58	26	,	,	PUNCT
cana-6025	58	27	(	(	PUNCT
cana-6025	58	28	𝑢𝑗	𝑢𝑗	NOUN
cana-6025	58	29	,	,	PUNCT
cana-6025	58	30	𝑣3)/2	𝑣3)/2	NOUN
cana-6025	58	31	≤	≤	NOUN
cana-6025	58	32	𝑖	𝑖	SYM
cana-6025	58	33	≤	≤	NOUN
cana-6025	58	34	𝑟	𝑟	NOUN
cana-6025	58	35	,	,	PUNCT
cana-6025	58	36	1	1	NUM
cana-6025	58	37	≤	≤	NUM
cana-6025	58	38	𝑗	𝑗	PRON
cana-6025	58	39	≤	≤	ADJ
cana-6025	58	40	𝑟	𝑟	NOUN
cana-6025	58	41	}	}	PUNCT
cana-6025	58	42	.	.	PUNCT
cana-6025	59	1	since	since	SCONJ
cana-6025	59	2	(	(	PUNCT
cana-6025	59	3	𝑢1	𝑢1	PROPN
cana-6025	59	4	,	,	PUNCT
cana-6025	59	5	𝑣2	𝑣2	NOUN
cana-6025	59	6	)	)	PUNCT
cana-6025	59	7	∈	∈	PROPN
cana-6025	59	8	𝐷	𝐷	PROPN
cana-6025	59	9	,	,	PUNCT
cana-6025	59	10	⟨{(𝑢𝑖	⟨{(𝑢𝑖	PROPN
cana-6025	59	11	,	,	PUNCT
cana-6025	59	12	𝑣2	𝑣2	PROPN
cana-6025	59	13	)	)	PUNCT
cana-6025	59	14	,	,	PUNCT
cana-6025	59	15	(	(	PUNCT
cana-6025	59	16	𝑢𝑗	𝑢𝑗	NOUN
cana-6025	59	17	,	,	PUNCT
cana-6025	59	18	𝑣3)/1	𝑣3)/1	NOUN
cana-6025	59	19	≤	≤	NUM
cana-6025	59	20	𝑖	𝑖	ADP
cana-6025	59	21	,	,	PUNCT
cana-6025	59	22	𝑗	𝑗	PROPN
cana-6025	59	23	≤	≤	PRON
cana-6025	59	24	𝑟}⟩	𝑟}⟩	PROPN
cana-6025	59	25	is	be	AUX
cana-6025	59	26	a	a	DET
cana-6025	59	27	complete	complete	ADJ
cana-6025	59	28	subgraph	subgraph	NOUN
cana-6025	59	29	on	on	ADP
cana-6025	59	30	2𝑟	2𝑟	NUM
cana-6025	59	31	vertices	vertex	NOUN
cana-6025	59	32	and	and	CCONJ
cana-6025	59	33	so	so	ADV
cana-6025	59	34	,	,	PUNCT
cana-6025	59	35	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	59	36	)	)	PUNCT
cana-6025	59	37	=	=	SYM
cana-6025	59	38	2𝑟	2𝑟	NUM
cana-6025	59	39	=	=	SYM
cana-6025	59	40	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	59	41	⊠	⊠	PROPN
cana-6025	59	42	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	59	43	)	)	PUNCT
cana-6025	59	44	.	.	PUNCT
cana-6025	60	1	clearly	clearly	ADV
cana-6025	60	2	,	,	PUNCT
cana-6025	60	3	𝐷1	𝐷1	PROPN
cana-6025	60	4	is	be	AUX
cana-6025	60	5	a	a	DET
cana-6025	60	6	restrained	restrain	VERB
cana-6025	60	7	dominating	dominating	NOUN
cana-6025	60	8	set	set	NOUN
cana-6025	60	9	.	.	PUNCT
cana-6025	61	1	thus	thus	ADV
cana-6025	61	2	,	,	PUNCT
cana-6025	61	3	𝐷1	𝐷1	PROPN
cana-6025	61	4	is	be	AUX
cana-6025	61	5	a	a	DET
cana-6025	61	6	chromatic	chromatic	ADJ
cana-6025	61	7	restrained	restrain	VERB
cana-6025	61	8	dominating	dominating	NOUN
cana-6025	61	9	set	set	NOUN
cana-6025	61	10	of	of	ADP
cana-6025	61	11	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	61	12	⊠	⊠	PROPN
cana-6025	61	13	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	61	14	and	and	CCONJ
cana-6025	61	15	𝛾𝑟	𝛾𝑟	ADP
cana-6025	61	16	𝑐(𝐾𝑟	𝑐(𝐾𝑟	X
cana-6025	61	17	⊠	⊠	PROPN
cana-6025	61	18	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	61	19	)	)	PUNCT
cana-6025	61	20	≤	≤	NUM
cana-6025	61	21	|𝐷1|	|𝐷1|	NOUN
cana-6025	61	22	=	=	SYM
cana-6025	61	23	𝑠	𝑠	ADP
cana-6025	61	24	3	3	NUM
cana-6025	61	25	+	+	CCONJ
cana-6025	61	26	2𝑟	2𝑟	NUM
cana-6025	61	27	−	−	NOUN
cana-6025	61	28	1	1	NUM
cana-6025	62	1	=	=	SYM
cana-6025	62	2	𝑠−3	𝑠−3	PROPN
cana-6025	62	3	3	3	NUM
cana-6025	62	4	+	+	NUM
cana-6025	62	5	2𝑟	2𝑟	NUM
cana-6025	62	6	=	=	SYM
cana-6025	62	7	⌈	⌈	SYM
cana-6025	62	8	𝑠−4	𝑠−4	ADP
cana-6025	62	9	3	3	NUM
cana-6025	62	10	⌉	⌉	NOUN
cana-6025	62	11	+	+	NUM
cana-6025	62	12	2𝑟.	2𝑟.	NUM
cana-6025	62	13	then	then	ADV
cana-6025	62	14	,	,	PUNCT
cana-6025	62	15	it	it	PRON
cana-6025	62	16	remains	remain	VERB
cana-6025	62	17	to	to	PART
cana-6025	62	18	show	show	VERB
cana-6025	62	19	that	that	SCONJ
cana-6025	62	20	,	,	PUNCT
cana-6025	62	21	𝛾𝑟	𝛾𝑟	ADP
cana-6025	62	22	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	62	23	⊠	⊠	PROPN
cana-6025	62	24	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	62	25	)	)	PUNCT
cana-6025	62	26	≥	≥	NOUN
cana-6025	62	27	⌈	⌈	NOUN
cana-6025	62	28	𝑠−4	𝑠−4	ADP
cana-6025	62	29	3	3	NUM
cana-6025	62	30	⌉	⌉	NOUN
cana-6025	62	31	+	+	CCONJ
cana-6025	62	32	2𝑟.	2𝑟.	NUM
cana-6025	62	33	since	since	SCONJ
cana-6025	62	34	,	,	PUNCT
cana-6025	62	35	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	62	36	⊠	⊠	PROPN
cana-6025	62	37	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	62	38	)	)	PUNCT
cana-6025	62	39	=	=	SYM
cana-6025	62	40	2𝑟	2𝑟	NUM
cana-6025	62	41	and	and	CCONJ
cana-6025	62	42	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	62	43	⊠	⊠	PROPN
cana-6025	62	44	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	62	45	contains	contain	VERB
cana-6025	62	46	induced	induced	ADJ
cana-6025	62	47	subgraph	subgraph	NOUN
cana-6025	62	48	which	which	PRON
cana-6025	62	49	is	be	AUX
cana-6025	62	50	complete	complete	ADJ
cana-6025	62	51	on	on	ADP
cana-6025	62	52	2𝑟	2𝑟	NUM
cana-6025	62	53	vertices	vertex	NOUN
cana-6025	62	54	,	,	PUNCT
cana-6025	62	55	any	any	DET
cana-6025	62	56	minimum	minimum	ADJ
cana-6025	62	57	chromatic	chromatic	ADJ
cana-6025	62	58	restrained	restrain	VERB
cana-6025	62	59	dominating	dominating	NOUN
cana-6025	62	60	set	set	NOUN
cana-6025	62	61	must	must	AUX
cana-6025	62	62	contain	contain	VERB
cana-6025	62	63	those	those	DET
cana-6025	62	64	2𝑟	2𝑟	NUM
cana-6025	62	65	vertices	vertex	NOUN
cana-6025	62	66	which	which	PRON
cana-6025	62	67	are	be	AUX
cana-6025	62	68	the	the	DET
cana-6025	62	69	vertices	vertex	NOUN
cana-6025	62	70	of	of	ADP
cana-6025	62	71	two	two	NUM
cana-6025	62	72	adjacent	adjacent	ADJ
cana-6025	62	73	columns	column	NOUN
cana-6025	62	74	.	.	PUNCT
cana-6025	63	1	let	let	VERB
cana-6025	63	2	them	they	PRON
cana-6025	63	3	be	be	AUX
cana-6025	63	4	𝑉2	𝑉2	NOUN
cana-6025	63	5	and	and	CCONJ
cana-6025	63	6	𝑉3	𝑉3	NOUN
cana-6025	63	7	,	,	PUNCT
cana-6025	63	8	so	so	SCONJ
cana-6025	63	9	that	that	SCONJ
cana-6025	63	10	,	,	PUNCT
cana-6025	63	11	all	all	DET
cana-6025	63	12	the	the	DET
cana-6025	63	13	vertices	vertex	NOUN
cana-6025	63	14	of	of	ADP
cana-6025	63	15	𝑉1	𝑉1	NOUN
cana-6025	63	16	and	and	CCONJ
cana-6025	63	17	𝑉4	𝑉4	NOUN
cana-6025	63	18	are	be	AUX
cana-6025	63	19	adjacent	adjacent	ADJ
cana-6025	63	20	to	to	ADP
cana-6025	63	21	the	the	DET
cana-6025	63	22	vertices	vertex	NOUN
cana-6025	63	23	of	of	ADP
cana-6025	63	24	𝑉2	𝑉2	NOUN
cana-6025	63	25	and	and	CCONJ
cana-6025	63	26	𝑉3	𝑉3	NOUN
cana-6025	63	27	.	.	PUNCT
cana-6025	64	1	from	from	ADP
cana-6025	64	2	the	the	DET
cana-6025	64	3	remaining	remain	VERB
cana-6025	64	4	𝑠	𝑠	PROPN
cana-6025	64	5	−	−	PROPN
cana-6025	64	6	4	4	NUM
cana-6025	64	7	columns	column	NOUN
cana-6025	64	8	,	,	PUNCT
cana-6025	64	9	choose	choose	VERB
cana-6025	64	10	a	a	DET
cana-6025	64	11	vertex	vertex	NOUN
cana-6025	64	12	of	of	ADP
cana-6025	64	13	each	each	DET
cana-6025	64	14	column	column	NOUN
cana-6025	64	15	𝑉3(𝑘+1	𝑉3(𝑘+1	PROPN
cana-6025	64	16	)	)	PUNCT
cana-6025	64	17	,	,	PUNCT
cana-6025	64	18	1	1	NUM
cana-6025	64	19	≤	≤	NOUN
cana-6025	64	20	𝑘	𝑘	PRON
cana-6025	64	21	≤	≤	NUM
cana-6025	64	22	⌈	⌈	NOUN
cana-6025	64	23	𝑠−4	𝑠−4	ADP
cana-6025	64	24	3	3	NUM
cana-6025	64	25	⌉	⌉	NOUN
cana-6025	64	26	which	which	PRON
cana-6025	64	27	is	be	AUX
cana-6025	64	28	adjacent	adjacent	ADJ
cana-6025	64	29	to	to	ADP
cana-6025	64	30	all	all	DET
cana-6025	64	31	the	the	DET
cana-6025	64	32	vertices	vertex	NOUN
cana-6025	64	33	of	of	ADP
cana-6025	64	34	𝑉3(𝑘+1	𝑉3(𝑘+1	NOUN
cana-6025	64	35	)	)	PUNCT
cana-6025	65	1	−	−	PROPN
cana-6025	65	2	1	1	NUM
cana-6025	65	3	,	,	PUNCT
cana-6025	65	4	𝑉3(𝑘+1	𝑉3(𝑘+1	NOUN
cana-6025	65	5	)	)	PUNCT
cana-6025	65	6	and	and	CCONJ
cana-6025	65	7	𝑉3(𝑘+1	𝑉3(𝑘+1	NOUN
cana-6025	65	8	)	)	PUNCT
cana-6025	66	1	+	+	CCONJ
cana-6025	66	2	1	1	X
cana-6025	66	3	.	.	PUNCT
cana-6025	66	4	thus	thus	ADV
cana-6025	66	5	,	,	PUNCT
cana-6025	66	6	𝛾𝑟	𝛾𝑟	ADP
cana-6025	66	7	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	66	8	⊠	⊠	PROPN
cana-6025	66	9	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	66	10	)	)	PUNCT
cana-6025	66	11	≥	≥	NOUN
cana-6025	66	12	⌈	⌈	NOUN
cana-6025	66	13	𝑠−4	𝑠−4	ADP
cana-6025	66	14	3	3	NUM
cana-6025	66	15	⌉	⌉	NOUN
cana-6025	66	16	+	+	CCONJ
cana-6025	66	17	2𝑟.	2𝑟.	NUM
cana-6025	66	18	therefore	therefore	ADV
cana-6025	66	19	,	,	PUNCT
cana-6025	66	20	𝛾𝑟	𝛾𝑟	ADP
cana-6025	66	21	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	66	22	⊠	⊠	PROPN
cana-6025	66	23	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	66	24	)	)	PUNCT
cana-6025	66	25	=	=	PUNCT
cana-6025	66	26	⌈	⌈	PROPN
cana-6025	66	27	𝑠−4	𝑠−4	ADP
cana-6025	66	28	3	3	NUM
cana-6025	66	29	⌉	⌉	NOUN
cana-6025	66	30	+	+	NUM
cana-6025	66	31	2𝑟.	2𝑟.	NUM
cana-6025	66	32	communications	communication	NOUN
cana-6025	66	33	on	on	ADP
cana-6025	66	34	applied	apply	VERB
cana-6025	66	35	nonlinear	nonlinear	ADJ
cana-6025	66	36	analysis	analysis	NOUN
cana-6025	66	37	issn	issn	NOUN
cana-6025	66	38	:	:	PUNCT
cana-6025	66	39	1074	1074	NUM
cana-6025	66	40	-	-	PUNCT
cana-6025	66	41	133x	133x	NUM
cana-6025	66	42	vol	vol	NOUN
cana-6025	66	43	31	31	NUM
cana-6025	66	44	no	no	NOUN
cana-6025	66	45	.	.	NOUN
cana-6025	66	46	2	2	NUM
cana-6025	66	47	(	(	PUNCT
cana-6025	66	48	2024	2024	NUM
cana-6025	66	49	)	)	PUNCT
cana-6025	66	50	492	492	NUM
cana-6025	66	51	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	66	52	case	case	NOUN
cana-6025	66	53	(	(	PUNCT
cana-6025	66	54	ii	ii	NUM
cana-6025	66	55	):	):	PUNCT
cana-6025	66	56	𝑠	𝑠	PROPN
cana-6025	66	57	≡	≡	PROPN
cana-6025	66	58	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	NOUN
cana-6025	66	59	3	3	X
cana-6025	66	60	)	)	PUNCT
cana-6025	66	61	let	let	VERB
cana-6025	66	62	𝐷2	𝐷2	NOUN
cana-6025	66	63	=	=	PRON
cana-6025	66	64	{	{	PUNCT
cana-6025	66	65	(	(	PUNCT
cana-6025	66	66	𝑢1	𝑢1	PROPN
cana-6025	66	67	,	,	PUNCT
cana-6025	66	68	𝑣3𝑘−1)/1	𝑣3𝑘−1)/1	PROPN
cana-6025	66	69	≤	≤	NOUN
cana-6025	66	70	𝑘	𝑘	DET
cana-6025	66	71	≤	≤	NOUN
cana-6025	67	1	⌊	⌊	VERB
cana-6025	67	2	𝑠	𝑠	PRON
cana-6025	67	3	3	3	NUM
cana-6025	67	4	⌋	⌋	NOUN
cana-6025	67	5	}	}	PUNCT
cana-6025	67	6	∪	∪	X
cana-6025	67	7	{	{	PUNCT
cana-6025	67	8	(	(	PUNCT
cana-6025	67	9	𝑢1	𝑢1	PROPN
cana-6025	67	10	,	,	PUNCT
cana-6025	67	11	𝑣𝑠	𝑣𝑠	NOUN
cana-6025	67	12	)	)	PUNCT
cana-6025	67	13	}	}	PUNCT
cana-6025	67	14	where	where	SCONJ
cana-6025	67	15	|𝐷2|	|𝐷2|	X
cana-6025	67	16	=	=	SYM
cana-6025	67	17	⌊	⌊	PROPN
cana-6025	67	18	𝑠	𝑠	NUM
cana-6025	67	19	3	3	NUM
cana-6025	67	20	⌋	⌋	NOUN
cana-6025	67	21	+	+	CCONJ
cana-6025	67	22	1	1	NUM
cana-6025	67	23	=	=	SYM
cana-6025	67	24	⌈	⌈	NOUN
cana-6025	67	25	𝑠	𝑠	ADP
cana-6025	67	26	3	3	NUM
cana-6025	67	27	⌉.	⌉.	ADV
cana-6025	67	28	then	then	ADV
cana-6025	67	29	,	,	PUNCT
cana-6025	67	30	every	every	DET
cana-6025	67	31	vertex	vertex	NOUN
cana-6025	67	32	of	of	ADP
cana-6025	67	33	𝑉	𝑉	PROPN
cana-6025	67	34	−	−	PROPN
cana-6025	67	35	𝐷2	𝐷2	NOUN
cana-6025	67	36	is	be	AUX
cana-6025	67	37	adjacent	adjacent	ADJ
cana-6025	67	38	to	to	ADP
cana-6025	67	39	at	at	ADV
cana-6025	67	40	least	least	ADV
cana-6025	67	41	one	one	NUM
cana-6025	67	42	vertex	vertex	NOUN
cana-6025	67	43	of	of	ADP
cana-6025	67	44	𝐷2	𝐷2	NOUN
cana-6025	67	45	and	and	CCONJ
cana-6025	67	46	at	at	ADP
cana-6025	67	47	least	least	ADJ
cana-6025	67	48	one	one	NUM
cana-6025	67	49	another	another	DET
cana-6025	67	50	vertex	vertex	NOUN
cana-6025	67	51	in	in	ADP
cana-6025	67	52	𝑉	𝑉	PROPN
cana-6025	67	53	−	−	PROPN
cana-6025	67	54	𝐷2	𝐷2	NOUN
cana-6025	67	55	.	.	PUNCT
cana-6025	68	1	thus	thus	ADV
cana-6025	68	2	,	,	PUNCT
cana-6025	68	3	𝐷2	𝐷2	PROPN
cana-6025	68	4	is	be	AUX
cana-6025	68	5	a	a	DET
cana-6025	68	6	restrained	restrain	VERB
cana-6025	68	7	dominating	dominating	NOUN
cana-6025	68	8	set	set	NOUN
cana-6025	68	9	and	and	CCONJ
cana-6025	68	10	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	68	11	⊠	⊠	PROPN
cana-6025	68	12	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	68	13	)	)	PUNCT
cana-6025	68	14	≤	≤	NOUN
cana-6025	69	1	|𝐷2|	|𝐷2|	X
cana-6025	69	2	=	=	SYM
cana-6025	69	3	⌈	⌈	NOUN
cana-6025	69	4	𝑠	𝑠	ADP
cana-6025	69	5	3	3	NUM
cana-6025	69	6	⌉.	⌉.	ADV
cana-6025	69	7	since	since	SCONJ
cana-6025	69	8	,	,	PUNCT
cana-6025	69	9	𝛾(𝑃𝑠	𝛾(𝑃𝑠	PROPN
cana-6025	69	10	)	)	PUNCT
cana-6025	69	11	=	=	PUNCT
cana-6025	70	1	⌈	⌈	NOUN
cana-6025	70	2	𝑠	𝑠	ADP
cana-6025	70	3	3	3	NUM
cana-6025	70	4	⌉	⌉	NOUN
cana-6025	70	5	and	and	CCONJ
cana-6025	70	6	each	each	DET
cana-6025	70	7	vertex	vertex	NOUN
cana-6025	70	8	in	in	ADP
cana-6025	70	9	column	column	NOUN
cana-6025	70	10	𝑉𝑗	𝑉𝑗	PROPN
cana-6025	70	11	is	be	AUX
cana-6025	70	12	adjacent	adjacent	ADJ
cana-6025	70	13	to	to	ADP
cana-6025	70	14	all	all	DET
cana-6025	70	15	the	the	DET
cana-6025	70	16	vertices	vertex	NOUN
cana-6025	70	17	in	in	ADP
cana-6025	70	18	𝑉𝑗−1	𝑉𝑗−1	PROPN
cana-6025	70	19	,	,	PUNCT
cana-6025	70	20	𝑉𝑗	𝑉𝑗	PROPN
cana-6025	70	21	and	and	CCONJ
cana-6025	70	22	𝑉𝑗+1	𝑉𝑗+1	VERB
cana-6025	70	23	,	,	PUNCT
cana-6025	70	24	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	70	25	⊠	⊠	PROPN
cana-6025	70	26	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	70	27	)	)	PUNCT
cana-6025	70	28	≥	≥	NOUN
cana-6025	70	29	⌈	⌈	NOUN
cana-6025	70	30	𝑠	𝑠	ADP
cana-6025	70	31	3	3	NUM
cana-6025	70	32	⌉.	⌉.	ADV
cana-6025	70	33	therefore	therefore	ADV
cana-6025	70	34	,	,	PUNCT
cana-6025	70	35	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	70	36	⊠	⊠	PROPN
cana-6025	70	37	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	70	38	)	)	PUNCT
cana-6025	70	39	=	=	PUNCT
cana-6025	71	1	⌈	⌈	NOUN
cana-6025	71	2	𝑠	𝑠	ADP
cana-6025	71	3	3	3	NUM
cana-6025	71	4	⌉.	⌉.	ADV
cana-6025	71	5	consider	consider	VERB
cana-6025	71	6	𝐷3	𝐷3	NOUN
cana-6025	71	7	=	=	PROPN
cana-6025	71	8	𝑉2	𝑉2	PROPN
cana-6025	71	9	∪	∪	VERB
cana-6025	71	10	𝑉3	𝑉3	NOUN
cana-6025	71	11	∪	∪	X
cana-6025	71	12	{	{	PUNCT
cana-6025	71	13	(	(	PUNCT
cana-6025	71	14	𝑢1	𝑢1	PROPN
cana-6025	71	15	,	,	PUNCT
cana-6025	71	16	𝑣3(𝑘+1))/1	𝑣3(𝑘+1))/1	NOUN
cana-6025	71	17	≤	≤	NOUN
cana-6025	71	18	𝑘	𝑘	DET
cana-6025	71	19	≤	≤	ADJ
cana-6025	71	20	𝑠−4	𝑠−4	ADP
cana-6025	71	21	3	3	NUM
cana-6025	71	22	}	}	PUNCT
cana-6025	71	23	.	.	PUNCT
cana-6025	72	1	clearly	clearly	ADV
cana-6025	72	2	,	,	PUNCT
cana-6025	72	3	𝐷3	𝐷3	PROPN
cana-6025	72	4	is	be	AUX
cana-6025	72	5	a	a	DET
cana-6025	72	6	restrained	restrain	VERB
cana-6025	72	7	dominating	dominating	NOUN
cana-6025	72	8	set	set	NOUN
cana-6025	72	9	and	and	CCONJ
cana-6025	72	10	⟨𝑉2	⟨𝑉2	NOUN
cana-6025	72	11	∪	∪	ADJ
cana-6025	72	12	𝑉3⟩	𝑉3⟩	NOUN
cana-6025	72	13	is	be	AUX
cana-6025	72	14	a	a	DET
cana-6025	72	15	complete	complete	ADJ
cana-6025	72	16	graph	graph	NOUN
cana-6025	72	17	on	on	ADP
cana-6025	72	18	2𝑟	2𝑟	NUM
cana-6025	72	19	vertices	vertex	NOUN
cana-6025	72	20	.	.	PUNCT
cana-6025	73	1	thus	thus	ADV
cana-6025	73	2	,	,	PUNCT
cana-6025	73	3	𝜒(⟨𝐷3⟩	𝜒(⟨𝐷3⟩	ADJ
cana-6025	73	4	)	)	PUNCT
cana-6025	73	5	=	=	SYM
cana-6025	73	6	2𝑟	2𝑟	NUM
cana-6025	73	7	=	=	SYM
cana-6025	73	8	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	73	9	⊠	⊠	PROPN
cana-6025	73	10	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	73	11	)	)	PUNCT
cana-6025	73	12	and	and	CCONJ
cana-6025	73	13	so	so	ADV
cana-6025	73	14	,	,	PUNCT
cana-6025	73	15	𝐷3	𝐷3	PROPN
cana-6025	73	16	is	be	AUX
cana-6025	73	17	a	a	DET
cana-6025	73	18	chromatic	chromatic	ADJ
cana-6025	73	19	restrained	restrain	VERB
cana-6025	73	20	dominating	dominating	NOUN
cana-6025	73	21	set	set	NOUN
cana-6025	73	22	of	of	ADP
cana-6025	73	23	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	73	24	⊠	⊠	PROPN
cana-6025	73	25	𝑃𝑠.	𝑃𝑠.	PROPN
cana-6025	73	26	then	then	ADV
cana-6025	73	27	,	,	PUNCT
cana-6025	73	28	𝛾𝑟	𝛾𝑟	ADP
cana-6025	73	29	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	73	30	⊠	⊠	PROPN
cana-6025	73	31	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	73	32	)	)	PUNCT
cana-6025	73	33	≤	≤	PUNCT
cana-6025	74	1	|𝐷3|	|𝐷3|	X
cana-6025	74	2	=	=	PUNCT
cana-6025	74	3	⌈	⌈	PROPN
cana-6025	74	4	𝑠−4	𝑠−4	ADP
cana-6025	74	5	3	3	NUM
cana-6025	74	6	⌉	⌉	NOUN
cana-6025	74	7	+	+	CCONJ
cana-6025	74	8	2𝑟.	2𝑟.	NUM
cana-6025	74	9	since	since	SCONJ
cana-6025	74	10	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	74	11	⊠	⊠	PROPN
cana-6025	74	12	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	74	13	)	)	PUNCT
cana-6025	74	14	=	=	SYM
cana-6025	74	15	2𝑟	2𝑟	NUM
cana-6025	74	16	,	,	PUNCT
cana-6025	74	17	any	any	DET
cana-6025	74	18	minimum	minimum	ADJ
cana-6025	74	19	chromatic	chromatic	ADJ
cana-6025	74	20	restrained	restrain	VERB
cana-6025	74	21	dominating	dominating	NOUN
cana-6025	74	22	set	set	NOUN
cana-6025	74	23	must	must	AUX
cana-6025	74	24	contain	contain	VERB
cana-6025	74	25	all	all	DET
cana-6025	74	26	the	the	DET
cana-6025	74	27	𝑟	𝑟	NOUN
cana-6025	74	28	vertices	vertex	NOUN
cana-6025	74	29	of	of	ADP
cana-6025	74	30	two	two	NUM
cana-6025	74	31	adjacent	adjacent	ADJ
cana-6025	74	32	columns	column	NOUN
cana-6025	74	33	.	.	PUNCT
cana-6025	75	1	let	let	VERB
cana-6025	75	2	them	they	PRON
cana-6025	75	3	be	be	AUX
cana-6025	75	4	𝑉2	𝑉2	NOUN
cana-6025	75	5	and	and	CCONJ
cana-6025	75	6	𝑉3	𝑉3	NOUN
cana-6025	75	7	which	which	PRON
cana-6025	75	8	is	be	AUX
cana-6025	75	9	adjacent	adjacent	ADJ
cana-6025	75	10	to	to	ADP
cana-6025	75	11	all	all	DET
cana-6025	75	12	the	the	DET
cana-6025	75	13	vertices	vertex	NOUN
cana-6025	75	14	of	of	ADP
cana-6025	75	15	𝑉1	𝑉1	NOUN
cana-6025	75	16	and	and	CCONJ
cana-6025	75	17	𝑉4	𝑉4	NOUN
cana-6025	75	18	.	.	PUNCT
cana-6025	76	1	again	again	ADV
cana-6025	76	2	from	from	ADP
cana-6025	76	3	the	the	DET
cana-6025	76	4	remaining	remain	VERB
cana-6025	76	5	𝑠	𝑠	PROPN
cana-6025	76	6	−	−	PROPN
cana-6025	76	7	4	4	NUM
cana-6025	76	8	columns	column	NOUN
cana-6025	76	9	,	,	PUNCT
cana-6025	76	10	choose	choose	VERB
cana-6025	76	11	a	a	DET
cana-6025	76	12	vertex	vertex	NOUN
cana-6025	76	13	from	from	ADP
cana-6025	76	14	columns	column	NOUN
cana-6025	76	15	𝑉3(𝑘+1	𝑉3(𝑘+1	NOUN
cana-6025	76	16	)	)	PUNCT
cana-6025	76	17	,	,	PUNCT
cana-6025	76	18	1	1	NUM
cana-6025	76	19	≤	≤	NOUN
cana-6025	76	20	𝑘	𝑘	DET
cana-6025	76	21	≤	≤	ADJ
cana-6025	76	22	𝑠−4	𝑠−4	ADP
cana-6025	76	23	3	3	NUM
cana-6025	76	24	.	.	PUNCT
cana-6025	77	1	thus	thus	ADV
cana-6025	77	2	,	,	PUNCT
cana-6025	77	3	we	we	PRON
cana-6025	77	4	get	get	VERB
cana-6025	77	5	a	a	DET
cana-6025	77	6	minimum	minimum	ADJ
cana-6025	77	7	chromatic	chromatic	ADJ
cana-6025	77	8	restrained	restrain	VERB
cana-6025	77	9	dominating	dominating	NOUN
cana-6025	77	10	set	set	NOUN
cana-6025	77	11	of	of	ADP
cana-6025	77	12	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	77	13	⊠	⊠	PROPN
cana-6025	77	14	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	77	15	and	and	CCONJ
cana-6025	77	16	𝛾𝑟	𝛾𝑟	ADP
cana-6025	77	17	𝑐(𝐾𝑟	𝑐(𝐾𝑟	X
cana-6025	77	18	⊠	⊠	PROPN
cana-6025	77	19	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	77	20	)	)	PUNCT
cana-6025	77	21	≥	≥	NOUN
cana-6025	77	22	⌈	⌈	NOUN
cana-6025	77	23	𝑠−4	𝑠−4	ADP
cana-6025	77	24	3	3	NUM
cana-6025	77	25	⌉	⌉	NOUN
cana-6025	77	26	+	+	CCONJ
cana-6025	77	27	2𝑟.	2𝑟.	NUM
cana-6025	77	28	therefore	therefore	ADV
cana-6025	77	29	,	,	PUNCT
cana-6025	77	30	𝛾𝑟	𝛾𝑟	ADP
cana-6025	77	31	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	77	32	⊠	⊠	PROPN
cana-6025	77	33	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	77	34	)	)	PUNCT
cana-6025	77	35	=	=	PUNCT
cana-6025	78	1	⌈	⌈	PROPN
cana-6025	78	2	𝑠−4	𝑠−4	ADP
cana-6025	78	3	3	3	NUM
cana-6025	78	4	⌉	⌉	NOUN
cana-6025	78	5	+	+	NUM
cana-6025	78	6	2𝑟.	2𝑟.	NUM
cana-6025	78	7	case	case	NOUN
cana-6025	78	8	(	(	PUNCT
cana-6025	78	9	iii	iii	NOUN
cana-6025	78	10	):	):	PUNCT
cana-6025	78	11	𝑠	𝑠	PROPN
cana-6025	78	12	≡	≡	PROPN
cana-6025	78	13	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-6025	78	14	3	3	NUM
cana-6025	78	15	)	)	PUNCT
cana-6025	78	16	clearly	clearly	ADV
cana-6025	78	17	,	,	PUNCT
cana-6025	78	18	𝐷2	𝐷2	PROPN
cana-6025	78	19	is	be	AUX
cana-6025	78	20	a	a	DET
cana-6025	78	21	restrained	restrained	ADJ
cana-6025	78	22	dominating	dominating	NOUN
cana-6025	78	23	set	set	NOUN
cana-6025	78	24	of	of	ADP
cana-6025	78	25	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	78	26	⊠	⊠	PROPN
cana-6025	78	27	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	78	28	and	and	CCONJ
cana-6025	78	29	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	78	30	⊠	⊠	PROPN
cana-6025	78	31	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	78	32	)	)	PUNCT
cana-6025	78	33	=	=	PUNCT
cana-6025	79	1	⌈	⌈	NOUN
cana-6025	79	2	𝑠	𝑠	NUM
cana-6025	79	3	3	3	NUM
cana-6025	79	4	⌉.	⌉.	ADV
cana-6025	79	5	also	also	ADV
cana-6025	79	6	,	,	PUNCT
cana-6025	79	7	𝐷1	𝐷1	NOUN
cana-6025	79	8	=	=	PROPN
cana-6025	79	9	𝐷2	𝐷2	PROPN
cana-6025	79	10	∪	∪	X
cana-6025	79	11	{	{	PUNCT
cana-6025	79	12	(	(	PUNCT
cana-6025	79	13	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	79	14	,	,	PUNCT
cana-6025	79	15	𝑣2	𝑣2	PROPN
cana-6025	79	16	)	)	PUNCT
cana-6025	79	17	,	,	PUNCT
cana-6025	79	18	(	(	PUNCT
cana-6025	79	19	𝑢𝑗	𝑢𝑗	NOUN
cana-6025	79	20	,	,	PUNCT
cana-6025	79	21	𝑣3)/2	𝑣3)/2	NOUN
cana-6025	79	22	≤	≤	NOUN
cana-6025	79	23	𝑖	𝑖	SYM
cana-6025	79	24	≤	≤	NOUN
cana-6025	79	25	𝑟	𝑟	NOUN
cana-6025	79	26	,	,	PUNCT
cana-6025	79	27	1	1	NUM
cana-6025	79	28	≤	≤	NUM
cana-6025	79	29	𝑗	𝑗	PRON
cana-6025	79	30	≤	≤	ADJ
cana-6025	79	31	𝑟	𝑟	NOUN
cana-6025	79	32	}	}	PUNCT
cana-6025	79	33	is	be	AUX
cana-6025	79	34	a	a	DET
cana-6025	79	35	chromatic	chromatic	ADJ
cana-6025	79	36	restrained	restrain	VERB
cana-6025	79	37	dominating	dominating	NOUN
cana-6025	79	38	set	set	NOUN
cana-6025	79	39	of	of	ADP
cana-6025	79	40	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	79	41	⊠	⊠	PROPN
cana-6025	79	42	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	79	43	and	and	CCONJ
cana-6025	79	44	so	so	ADV
cana-6025	79	45	,	,	PUNCT
cana-6025	79	46	𝛾𝑟	𝛾𝑟	ADP
cana-6025	79	47	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	79	48	⊠	⊠	PROPN
cana-6025	79	49	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	79	50	)	)	PUNCT
cana-6025	79	51	=	=	PUNCT
cana-6025	80	1	⌈	⌈	PROPN
cana-6025	80	2	𝑠−4	𝑠−4	ADP
cana-6025	80	3	3	3	NUM
cana-6025	80	4	⌉	⌉	NOUN
cana-6025	80	5	+	+	ADJ
cana-6025	80	6	2𝑟.	2𝑟.	NUM
cana-6025	80	7	theorem	theorem	VERB
cana-6025	80	8	2.4	2.4	NUM
cana-6025	80	9	for	for	ADP
cana-6025	80	10	𝑟	𝑟	NOUN
cana-6025	80	11	,	,	PUNCT
cana-6025	80	12	𝑠	𝑠	PRON
cana-6025	80	13	≥	≥	NUM
cana-6025	80	14	2	2	NUM
cana-6025	80	15	,	,	PUNCT
cana-6025	80	16	𝛾𝑟	𝛾𝑟	ADP
cana-6025	80	17	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	80	18	⊠	⊠	PROPN
cana-6025	80	19	𝐾1,𝑠	𝐾1,𝑠	PUNCT
cana-6025	80	20	)	)	PUNCT
cana-6025	81	1	=	=	SYM
cana-6025	82	1	2𝑟.	2𝑟.	NUM
cana-6025	82	2	proof	proof	NOUN
cana-6025	82	3	.	.	PUNCT
cana-6025	83	1	let	let	VERB
cana-6025	83	2	𝑉(𝐾𝑟	𝑉(𝐾𝑟	NUM
cana-6025	83	3	)	)	PUNCT
cana-6025	84	1	=	=	PRON
cana-6025	84	2	{	{	PUNCT
cana-6025	84	3	𝑢1	𝑢1	PROPN
cana-6025	84	4	,	,	PUNCT
cana-6025	84	5	𝑢2	𝑢2	PROPN
cana-6025	84	6	,	,	PUNCT
cana-6025	84	7	𝑢3	𝑢3	PROPN
cana-6025	84	8	,	,	PUNCT
cana-6025	84	9	.	.	PUNCT
cana-6025	84	10	.	.	PUNCT
cana-6025	85	1	.	.	PUNCT
cana-6025	86	1	,	,	PUNCT
cana-6025	86	2	𝑢𝑟	𝑢𝑟	ADV
cana-6025	86	3	}	}	PUNCT
cana-6025	86	4	and	and	CCONJ
cana-6025	86	5	𝑉(𝐾1,𝑠	𝑉(𝐾1,𝑠	NOUN
cana-6025	86	6	)	)	PUNCT
cana-6025	86	7	=	=	PRON
cana-6025	86	8	{	{	PUNCT
cana-6025	86	9	𝑣0	𝑣0	PROPN
cana-6025	86	10	,	,	PUNCT
cana-6025	86	11	𝑣1	𝑣1	PROPN
cana-6025	86	12	,	,	PUNCT
cana-6025	86	13	𝑣2	𝑣2	PROPN
cana-6025	86	14	,	,	PUNCT
cana-6025	86	15	𝑣3	𝑣3	ADJ
cana-6025	86	16	,	,	PUNCT
cana-6025	86	17	.	.	PUNCT
cana-6025	86	18	.	.	PUNCT
cana-6025	87	1	.	.	PUNCT
cana-6025	88	1	,	,	PUNCT
cana-6025	88	2	𝑣𝑠	𝑣𝑠	ADP
cana-6025	88	3	}	}	PUNCT
cana-6025	88	4	where	where	SCONJ
cana-6025	88	5	𝑣0	𝑣0	PROPN
cana-6025	88	6	is	be	AUX
cana-6025	88	7	the	the	DET
cana-6025	88	8	full	full	ADJ
cana-6025	88	9	degree	degree	NOUN
cana-6025	88	10	vertex	vertex	NOUN
cana-6025	88	11	of	of	ADP
cana-6025	88	12	𝐾1,𝑠.	𝐾1,𝑠.	PUNCT
cana-6025	88	13	then	then	ADV
cana-6025	88	14	,	,	PUNCT
cana-6025	88	15	𝑉(𝐾𝑟	𝑉(𝐾𝑟	PROPN
cana-6025	88	16	⊠	⊠	PROPN
cana-6025	88	17	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	88	18	)	)	PUNCT
cana-6025	89	1	=	=	PRON
cana-6025	89	2	{	{	PUNCT
cana-6025	89	3	(	(	PUNCT
cana-6025	89	4	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	89	5	,	,	PUNCT
cana-6025	89	6	𝑣𝑗)/1	𝑣𝑗)/1	PROPN
cana-6025	89	7	≤	≤	PROPN
cana-6025	89	8	𝑖	𝑖	SYM
cana-6025	89	9	≤	≤	NOUN
cana-6025	89	10	𝑟	𝑟	NOUN
cana-6025	89	11	,	,	PUNCT
cana-6025	89	12	0	0	NUM
cana-6025	89	13	≤	≤	NUM
cana-6025	89	14	𝑗	𝑗	PRON
cana-6025	89	15	≤	≤	NUM
cana-6025	89	16	𝑠	𝑠	ADP
cana-6025	89	17	}	}	PUNCT
cana-6025	89	18	and	and	CCONJ
cana-6025	89	19	|𝑉(𝐾𝑟	|𝑉(𝐾𝑟	ADJ
cana-6025	89	20	⊠	⊠	PROPN
cana-6025	89	21	𝐾1,𝑠)|	𝐾1,𝑠)|	NOUN
cana-6025	89	22	=	=	SYM
cana-6025	89	23	(	(	PUNCT
cana-6025	89	24	𝑠	𝑠	PROPN
cana-6025	89	25	+	+	CCONJ
cana-6025	89	26	1)𝑟.	1)𝑟.	NUM
cana-6025	89	27	clearly	clearly	ADV
cana-6025	89	28	,	,	PUNCT
cana-6025	89	29	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	89	30	⊠	⊠	PROPN
cana-6025	89	31	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	89	32	contains	contain	VERB
cana-6025	89	33	𝑟	𝑟	NOUN
cana-6025	89	34	rows	row	NOUN
cana-6025	89	35	and	and	CCONJ
cana-6025	89	36	𝑠	𝑠	INTJ
cana-6025	89	37	+	+	CCONJ
cana-6025	89	38	1	1	NUM
cana-6025	89	39	columns	column	NOUN
cana-6025	89	40	(	(	PUNCT
cana-6025	89	41	𝑉1	𝑉1	PROPN
cana-6025	89	42	,	,	PUNCT
cana-6025	89	43	𝑉2	𝑉2	NOUN
cana-6025	89	44	,	,	PUNCT
cana-6025	89	45	𝑉3	𝑉3	NOUN
cana-6025	89	46	,	,	PUNCT
cana-6025	89	47	.	.	PUNCT
cana-6025	89	48	.	.	PUNCT
cana-6025	89	49	.	.	PUNCT
cana-6025	90	1	,	,	PUNCT
cana-6025	90	2	𝑉𝑠+1	𝑉𝑠+1	NOUN
cana-6025	90	3	)	)	PUNCT
cana-6025	90	4	where	where	SCONJ
cana-6025	90	5	the	the	DET
cana-6025	90	6	induced	induced	ADJ
cana-6025	90	7	subgraph	subgraph	NOUN
cana-6025	90	8	of	of	ADP
cana-6025	90	9	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	90	10	⊠	⊠	PROPN
cana-6025	90	11	𝐾1,𝑠	𝐾1,𝑠	ADV
cana-6025	90	12	formed	form	VERB
cana-6025	90	13	from	from	ADP
cana-6025	90	14	all	all	DET
cana-6025	90	15	the	the	DET
cana-6025	90	16	vertices	vertex	NOUN
cana-6025	90	17	of	of	ADP
cana-6025	90	18	two	two	NUM
cana-6025	90	19	columns	column	NOUN
cana-6025	90	20	𝑉1	𝑉1	NOUN
cana-6025	90	21	and	and	CCONJ
cana-6025	90	22	𝑉𝑖	𝑉𝑖	PROPN
cana-6025	90	23	,	,	PUNCT
cana-6025	90	24	𝑖	𝑖	PRON
cana-6025	90	25	≠	≠	NOUN
cana-6025	90	26	1	1	NUM
cana-6025	90	27	is	be	AUX
cana-6025	90	28	a	a	DET
cana-6025	90	29	complete	complete	ADJ
cana-6025	90	30	subgraph	subgraph	NOUN
cana-6025	90	31	on	on	ADP
cana-6025	90	32	2𝑟	2𝑟	NUM
cana-6025	90	33	vertices	vertex	NOUN
cana-6025	90	34	.	.	PUNCT
cana-6025	91	1	then	then	ADV
cana-6025	91	2	,	,	PUNCT
cana-6025	91	3	𝜒(⟨𝑉1	𝜒(⟨𝑉1	VERB
cana-6025	91	4	∪	∪	ADJ
cana-6025	91	5	𝑉2⟩	𝑉2⟩	NOUN
cana-6025	91	6	)	)	PUNCT
cana-6025	91	7	=	=	SYM
cana-6025	91	8	2𝑟	2𝑟	NUM
cana-6025	91	9	and	and	CCONJ
cana-6025	91	10	all	all	DET
cana-6025	91	11	the	the	DET
cana-6025	91	12	remaining	remain	VERB
cana-6025	91	13	vertices	vertex	NOUN
cana-6025	91	14	can	can	AUX
cana-6025	91	15	be	be	AUX
cana-6025	91	16	colored	color	VERB
cana-6025	91	17	using	use	VERB
cana-6025	91	18	𝑟	𝑟	NOUN
cana-6025	91	19	colors	color	NOUN
cana-6025	91	20	used	use	VERB
cana-6025	91	21	for	for	ADP
cana-6025	91	22	coloring	color	VERB
cana-6025	91	23	the	the	DET
cana-6025	91	24	column	column	NOUN
cana-6025	91	25	𝑉2	𝑉2	NOUN
cana-6025	91	26	,	,	PUNCT
cana-6025	91	27	since	since	SCONJ
cana-6025	91	28	there	there	PRON
cana-6025	91	29	does	do	AUX
cana-6025	91	30	not	not	PART
cana-6025	91	31	exists	exist	VERB
cana-6025	91	32	adjacency	adjacency	NOUN
cana-6025	91	33	between	between	ADP
cana-6025	91	34	columns	column	NOUN
cana-6025	91	35	𝑉2	𝑉2	PROPN
cana-6025	91	36	,	,	PUNCT
cana-6025	91	37	𝑉3	𝑉3	NOUN
cana-6025	91	38	,	,	PUNCT
cana-6025	91	39	.	.	PUNCT
cana-6025	91	40	.	.	PUNCT
cana-6025	92	1	.	.	PUNCT
cana-6025	93	1	,	,	PUNCT
cana-6025	93	2	𝑉𝑠+1	𝑉𝑠+1	NOUN
cana-6025	93	3	.	.	PUNCT
cana-6025	93	4	thus	thus	ADV
cana-6025	93	5	,	,	PUNCT
cana-6025	93	6	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	93	7	⊠	⊠	PROPN
cana-6025	93	8	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	93	9	)	)	PUNCT
cana-6025	94	1	=	=	PRON
cana-6025	94	2	2𝑟.	2𝑟.	NUM
cana-6025	94	3	clearly	clearly	ADV
cana-6025	94	4	,	,	PUNCT
cana-6025	94	5	𝐷	𝐷	PROPN
cana-6025	94	6	=	=	SYM
cana-6025	94	7	{	{	PUNCT
cana-6025	94	8	(	(	PUNCT
cana-6025	94	9	𝑢1	𝑢1	PROPN
cana-6025	94	10	,	,	PUNCT
cana-6025	94	11	𝑣0	𝑣0	NOUN
cana-6025	94	12	)	)	PUNCT
cana-6025	94	13	}	}	PUNCT
cana-6025	94	14	is	be	AUX
cana-6025	94	15	a	a	DET
cana-6025	94	16	restrained	restrain	VERB
cana-6025	94	17	dominating	dominating	NOUN
cana-6025	94	18	set	set	NOUN
cana-6025	94	19	,	,	PUNCT
cana-6025	94	20	as	as	ADP
cana-6025	94	21	(	(	PUNCT
cana-6025	94	22	𝑢1	𝑢1	PROPN
cana-6025	94	23	,	,	PUNCT
cana-6025	94	24	𝑣0	𝑣0	NOUN
cana-6025	94	25	)	)	PUNCT
cana-6025	94	26	is	be	AUX
cana-6025	94	27	a	a	DET
cana-6025	94	28	full	full	ADJ
cana-6025	94	29	degree	degree	NOUN
cana-6025	94	30	vertex	vertex	NOUN
cana-6025	94	31	of	of	ADP
cana-6025	94	32	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	94	33	⊠	⊠	PROPN
cana-6025	94	34	𝐾1,𝑠.	𝐾1,𝑠.	X
cana-6025	94	35	therefore	therefore	ADV
cana-6025	94	36	,	,	PUNCT
cana-6025	94	37	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	94	38	⊠	⊠	PROPN
cana-6025	94	39	𝐾1,𝑠	𝐾1,𝑠	PUNCT
cana-6025	94	40	)	)	PUNCT
cana-6025	95	1	=	=	PUNCT
cana-6025	95	2	1	1	X
cana-6025	95	3	.	.	PUNCT
cana-6025	96	1	but	but	CCONJ
cana-6025	96	2	,	,	PUNCT
cana-6025	96	3	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	96	4	)	)	PUNCT
cana-6025	96	5	=	=	SYM
cana-6025	96	6	1	1	NUM
cana-6025	96	7	≠	≠	PROPN
cana-6025	96	8	communications	communication	NOUN
cana-6025	96	9	on	on	ADP
cana-6025	96	10	applied	apply	VERB
cana-6025	96	11	nonlinear	nonlinear	ADJ
cana-6025	96	12	analysis	analysis	NOUN
cana-6025	96	13	issn	issn	NOUN
cana-6025	96	14	:	:	PUNCT
cana-6025	96	15	1074	1074	NUM
cana-6025	96	16	-	-	PUNCT
cana-6025	96	17	133x	133x	NUM
cana-6025	96	18	vol	vol	NOUN
cana-6025	96	19	31	31	NUM
cana-6025	96	20	no	no	NOUN
cana-6025	96	21	.	.	NOUN
cana-6025	96	22	2	2	NUM
cana-6025	96	23	(	(	PUNCT
cana-6025	96	24	2024	2024	NUM
cana-6025	96	25	)	)	PUNCT
cana-6025	96	26	493	493	NUM
cana-6025	96	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	96	28	𝜒(𝐾𝑟	𝜒(𝐾𝑟	NOUN
cana-6025	96	29	⊠	⊠	PROPN
cana-6025	96	30	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	96	31	)	)	PUNCT
cana-6025	96	32	,	,	PUNCT
cana-6025	96	33	and	and	CCONJ
cana-6025	96	34	so	so	ADV
cana-6025	96	35	𝐷	𝐷	PROPN
cana-6025	96	36	is	be	AUX
cana-6025	96	37	not	not	PART
cana-6025	96	38	a	a	DET
cana-6025	96	39	chromatic	chromatic	ADJ
cana-6025	96	40	restrained	restrain	VERB
cana-6025	96	41	dominating	dominating	NOUN
cana-6025	96	42	set	set	NOUN
cana-6025	96	43	.	.	PUNCT
cana-6025	97	1	consider	consider	VERB
cana-6025	97	2	𝐷1	𝐷1	NOUN
cana-6025	97	3	=	=	VERB
cana-6025	97	4	𝑉1	𝑉1	PROPN
cana-6025	97	5	∪	∪	PROPN
cana-6025	97	6	𝑉2	𝑉2	NOUN
cana-6025	97	7	,	,	PUNCT
cana-6025	97	8	where	where	SCONJ
cana-6025	97	9	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	VERB
cana-6025	97	10	)	)	PUNCT
cana-6025	97	11	=	=	SYM
cana-6025	97	12	2𝑟	2𝑟	NUM
cana-6025	97	13	=	=	SYM
cana-6025	97	14	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	97	15	⊠	⊠	PROPN
cana-6025	97	16	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	97	17	)	)	PUNCT
cana-6025	97	18	.	.	PUNCT
cana-6025	98	1	since	since	SCONJ
cana-6025	98	2	(	(	PUNCT
cana-6025	98	3	𝑢1	𝑢1	PROPN
cana-6025	98	4	,	,	PUNCT
cana-6025	98	5	𝑣0	𝑣0	NOUN
cana-6025	98	6	)	)	PUNCT
cana-6025	98	7	∈	∈	PROPN
cana-6025	98	8	𝐷1	𝐷1	NOUN
cana-6025	98	9	,	,	PUNCT
cana-6025	98	10	𝐷1	𝐷1	PROPN
cana-6025	98	11	is	be	AUX
cana-6025	98	12	also	also	ADV
cana-6025	98	13	a	a	DET
cana-6025	98	14	restrained	restrained	ADJ
cana-6025	98	15	dominating	dominating	NOUN
cana-6025	98	16	set	set	NOUN
cana-6025	98	17	.	.	PUNCT
cana-6025	99	1	thus	thus	ADV
cana-6025	99	2	,	,	PUNCT
cana-6025	99	3	𝐷1	𝐷1	PROPN
cana-6025	99	4	is	be	AUX
cana-6025	99	5	a	a	DET
cana-6025	99	6	chromatic	chromatic	ADJ
cana-6025	99	7	restrained	restrain	VERB
cana-6025	99	8	dominating	dominating	NOUN
cana-6025	99	9	set	set	NOUN
cana-6025	99	10	of	of	ADP
cana-6025	99	11	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	99	12	⊠	⊠	PROPN
cana-6025	100	1	𝐾1,𝑠	𝐾1,𝑠	ADV
cana-6025	100	2	and	and	CCONJ
cana-6025	100	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	100	4	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	100	5	⊠	⊠	PROPN
cana-6025	100	6	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	101	1	)	)	PUNCT
cana-6025	101	2	≤	≤	NUM
cana-6025	101	3	|𝐷1|	|𝐷1|	NOUN
cana-6025	101	4	=	=	SYM
cana-6025	101	5	|𝑉1|	|𝑉1|	PROPN
cana-6025	101	6	+	+	NUM
cana-6025	101	7	|𝑉2|	|𝑉2|	NOUN
cana-6025	101	8	=	=	SYM
cana-6025	101	9	2𝑟.	2𝑟.	NUM
cana-6025	101	10	since	since	SCONJ
cana-6025	101	11	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	101	12	⊠	⊠	PROPN
cana-6025	101	13	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	101	14	)	)	PUNCT
cana-6025	102	1	=	=	SYM
cana-6025	102	2	2𝑟	2𝑟	NUM
cana-6025	102	3	,	,	PUNCT
cana-6025	102	4	any	any	DET
cana-6025	102	5	minimum	minimum	ADJ
cana-6025	102	6	chromatic	chromatic	ADJ
cana-6025	102	7	restrained	restrain	VERB
cana-6025	102	8	dominating	dominating	NOUN
cana-6025	102	9	set	set	NOUN
cana-6025	102	10	must	must	AUX
cana-6025	102	11	contain	contain	VERB
cana-6025	102	12	a	a	DET
cana-6025	102	13	minimum	minimum	NOUN
cana-6025	102	14	of	of	ADP
cana-6025	102	15	2𝑟	2𝑟	NUM
cana-6025	102	16	vertices	vertex	NOUN
cana-6025	102	17	.	.	PUNCT
cana-6025	103	1	therefore	therefore	ADV
cana-6025	103	2	,	,	PUNCT
cana-6025	103	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	103	4	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	103	5	⊠	⊠	PROPN
cana-6025	103	6	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	103	7	)	)	PUNCT
cana-6025	103	8	≥	≥	NOUN
cana-6025	103	9	2𝑟.	2𝑟.	NUM
cana-6025	103	10	hence	hence	ADV
cana-6025	103	11	,	,	PUNCT
cana-6025	103	12	𝛾𝑟	𝛾𝑟	ADP
cana-6025	103	13	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	103	14	⊠	⊠	PROPN
cana-6025	103	15	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	103	16	)	)	PUNCT
cana-6025	104	1	=	=	SYM
cana-6025	104	2	2𝑟.	2𝑟.	NUM
cana-6025	104	3	theorem	theorem	VERB
cana-6025	104	4	2.5	2.5	NUM
cana-6025	104	5	for	for	ADP
cana-6025	104	6	𝑟	𝑟	NOUN
cana-6025	104	7	,	,	PUNCT
cana-6025	104	8	𝑠	𝑠	PROPN
cana-6025	104	9	,	,	PUNCT
cana-6025	104	10	𝑚	𝑚	X
cana-6025	104	11	≥	≥	NUM
cana-6025	104	12	2	2	NUM
cana-6025	104	13	,	,	PUNCT
cana-6025	104	14	𝛾𝑟	𝛾𝑟	ADP
cana-6025	104	15	𝑐(𝐾𝑚	𝑐(𝐾𝑚	NOUN
cana-6025	104	16	⊠	⊠	PROPN
cana-6025	104	17	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	104	18	)	)	PUNCT
cana-6025	104	19	=	=	SYM
cana-6025	104	20	2𝑚.	2𝑚.	NUM
cana-6025	104	21	proof	proof	NOUN
cana-6025	104	22	.	.	PUNCT
cana-6025	105	1	let	let	VERB
cana-6025	105	2	𝑉(𝐾𝑚	𝑉(𝐾𝑚	NUM
cana-6025	105	3	)	)	PUNCT
cana-6025	105	4	=	=	PRON
cana-6025	105	5	{	{	PUNCT
cana-6025	105	6	𝑢1	𝑢1	PROPN
cana-6025	105	7	,	,	PUNCT
cana-6025	105	8	𝑢2	𝑢2	PROPN
cana-6025	105	9	,	,	PUNCT
cana-6025	105	10	𝑢3	𝑢3	PROPN
cana-6025	105	11	,	,	PUNCT
cana-6025	105	12	…	…	PUNCT
cana-6025	105	13	,	,	PUNCT
cana-6025	105	14	𝑢𝑚	𝑢𝑚	NOUN
cana-6025	105	15	}	}	PUNCT
cana-6025	105	16	and	and	CCONJ
cana-6025	105	17	𝑉(𝐾𝑟,𝑠	𝑉(𝐾𝑟,𝑠	ADJ
cana-6025	105	18	)	)	PUNCT
cana-6025	105	19	=	=	PRON
cana-6025	105	20	{	{	PUNCT
cana-6025	105	21	𝑣1	𝑣1	PROPN
cana-6025	105	22	,	,	PUNCT
cana-6025	105	23	𝑣2	𝑣2	PROPN
cana-6025	105	24	,	,	PUNCT
cana-6025	105	25	𝑣3	𝑣3	ADJ
cana-6025	105	26	,	,	PUNCT
cana-6025	105	27	…	…	PUNCT
cana-6025	105	28	,	,	PUNCT
cana-6025	105	29	𝑣𝑟	𝑣𝑟	INTJ
cana-6025	105	30	,	,	PUNCT
cana-6025	105	31	𝑣𝑟+1	𝑣𝑟+1	ADV
cana-6025	105	32	,	,	PUNCT
cana-6025	105	33	𝑣𝑟+2	𝑣𝑟+2	NUM
cana-6025	105	34	,	,	PUNCT
cana-6025	105	35	…	…	PUNCT
cana-6025	105	36	,	,	PUNCT
cana-6025	105	37	𝑣𝑟+𝑠	𝑣𝑟+𝑠	NOUN
cana-6025	105	38	}	}	PUNCT
cana-6025	105	39	.	.	PUNCT
cana-6025	106	1	then	then	ADV
cana-6025	106	2	,	,	PUNCT
cana-6025	106	3	𝑉(𝐾𝑚	𝑉(𝐾𝑚	VERB
cana-6025	106	4	⊠	⊠	PROPN
cana-6025	106	5	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	106	6	)	)	PUNCT
cana-6025	107	1	=	=	PRON
cana-6025	107	2	{	{	PUNCT
cana-6025	107	3	(	(	PUNCT
cana-6025	107	4	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	107	5	,	,	PUNCT
cana-6025	107	6	𝑣𝑗)/1	𝑣𝑗)/1	PROPN
cana-6025	107	7	≤	≤	PROPN
cana-6025	107	8	𝑖	𝑖	SYM
cana-6025	107	9	≤	≤	NUM
cana-6025	107	10	𝑚	𝑚	ADP
cana-6025	107	11	,	,	PUNCT
cana-6025	107	12	1	1	NUM
cana-6025	107	13	≤	≤	NUM
cana-6025	107	14	𝑗	𝑗	PRON
cana-6025	107	15	≤	≤	ADJ
cana-6025	107	16	𝑟	𝑟	NOUN
cana-6025	107	17	+	+	X
cana-6025	107	18	𝑠	𝑠	NOUN
cana-6025	107	19	}	}	PUNCT
cana-6025	107	20	.	.	PUNCT
cana-6025	108	1	clearly	clearly	ADV
cana-6025	108	2	,	,	PUNCT
cana-6025	108	3	𝐾𝑚	𝐾𝑚	PROPN
cana-6025	108	4	⊠	⊠	PROPN
cana-6025	108	5	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	108	6	contains	contain	VERB
cana-6025	108	7	𝑚	𝑚	ADP
cana-6025	108	8	rows	row	NOUN
cana-6025	108	9	and	and	CCONJ
cana-6025	108	10	𝑟	𝑟	PRON
cana-6025	108	11	+	+	CCONJ
cana-6025	108	12	𝑠	𝑠	ADP
cana-6025	108	13	columns	column	NOUN
cana-6025	108	14	where	where	SCONJ
cana-6025	108	15	𝑉1	𝑉1	NOUN
cana-6025	108	16	,	,	PUNCT
cana-6025	108	17	𝑉2	𝑉2	NOUN
cana-6025	108	18	,	,	PUNCT
cana-6025	108	19	.	.	PUNCT
cana-6025	108	20	.	.	PUNCT
cana-6025	109	1	.	.	PUNCT
cana-6025	110	1	,	,	PUNCT
cana-6025	110	2	𝑉𝑟+𝑠	𝑉𝑟+𝑠	PROPN
cana-6025	110	3	denotes	denote	VERB
cana-6025	110	4	the	the	DET
cana-6025	110	5	columns	column	NOUN
cana-6025	110	6	.	.	PUNCT
cana-6025	111	1	also	also	ADV
cana-6025	111	2	,	,	PUNCT
cana-6025	111	3	the	the	DET
cana-6025	111	4	induced	induced	ADJ
cana-6025	111	5	subgraph	subgraph	NOUN
cana-6025	111	6	of	of	ADP
cana-6025	111	7	𝐾𝑚	𝐾𝑚	PROPN
cana-6025	111	8	⊠	⊠	PROPN
cana-6025	111	9	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	111	10	formed	form	VERB
cana-6025	111	11	from	from	ADP
cana-6025	111	12	all	all	DET
cana-6025	111	13	the	the	DET
cana-6025	111	14	vertices	vertex	NOUN
cana-6025	111	15	of	of	ADP
cana-6025	111	16	two	two	NUM
cana-6025	111	17	columns	column	NOUN
cana-6025	111	18	,	,	PUNCT
cana-6025	111	19	one	one	NUM
cana-6025	111	20	among	among	ADP
cana-6025	111	21	the	the	DET
cana-6025	111	22	columns	column	NOUN
cana-6025	111	23	𝑉1	𝑉1	PROPN
cana-6025	111	24	,	,	PUNCT
cana-6025	111	25	𝑉2	𝑉2	NOUN
cana-6025	111	26	,	,	PUNCT
cana-6025	111	27	.	.	PUNCT
cana-6025	111	28	.	.	PUNCT
cana-6025	112	1	.	.	PUNCT
cana-6025	113	1	,	,	PUNCT
cana-6025	113	2	𝑉𝑟	𝑉𝑟	PROPN
cana-6025	113	3	and	and	CCONJ
cana-6025	113	4	another	another	PRON
cana-6025	113	5	among	among	ADP
cana-6025	113	6	the	the	DET
cana-6025	113	7	columns	column	NOUN
cana-6025	113	8	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	113	9	,	,	PUNCT
cana-6025	113	10	𝑉𝑟+2	𝑉𝑟+2	NOUN
cana-6025	113	11	,	,	PUNCT
cana-6025	113	12	.	.	PUNCT
cana-6025	113	13	.	.	PUNCT
cana-6025	114	1	.	.	PUNCT
cana-6025	115	1	,	,	PUNCT
cana-6025	115	2	𝑉𝑟+𝑠	𝑉𝑟+𝑠	PROPN
cana-6025	115	3	is	be	AUX
cana-6025	115	4	a	a	DET
cana-6025	115	5	complete	complete	ADJ
cana-6025	115	6	subgraph	subgraph	NOUN
cana-6025	115	7	on	on	ADP
cana-6025	115	8	2𝑚	2𝑚	NUM
cana-6025	115	9	vertices	vertex	NOUN
cana-6025	115	10	.	.	PUNCT
cana-6025	116	1	clearly	clearly	ADV
cana-6025	116	2	,	,	PUNCT
cana-6025	116	3	the	the	DET
cana-6025	116	4	columns	column	NOUN
cana-6025	116	5	𝑉1	𝑉1	PROPN
cana-6025	116	6	,	,	PUNCT
cana-6025	116	7	𝑉2	𝑉2	NOUN
cana-6025	116	8	,	,	PUNCT
cana-6025	116	9	.	.	PUNCT
cana-6025	116	10	.	.	PUNCT
cana-6025	116	11	.	.	PUNCT
cana-6025	117	1	,	,	PUNCT
cana-6025	117	2	𝑉𝑟	𝑉𝑟	PROPN
cana-6025	117	3	can	can	AUX
cana-6025	117	4	be	be	AUX
cana-6025	117	5	colored	color	VERB
cana-6025	117	6	with	with	ADP
cana-6025	117	7	𝑚	𝑚	ADP
cana-6025	117	8	colors	color	NOUN
cana-6025	117	9	and	and	CCONJ
cana-6025	117	10	the	the	DET
cana-6025	117	11	remaining	remain	VERB
cana-6025	117	12	columns	column	NOUN
cana-6025	117	13	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	117	14	,	,	PUNCT
cana-6025	117	15	𝑉𝑟+2	𝑉𝑟+2	NOUN
cana-6025	117	16	,	,	PUNCT
cana-6025	117	17	.	.	PUNCT
cana-6025	117	18	.	.	PUNCT
cana-6025	117	19	.	.	PUNCT
cana-6025	118	1	.	.	PUNCT
cana-6025	119	1	,	,	PUNCT
cana-6025	119	2	𝑉𝑟+𝑠	𝑉𝑟+𝑠	PROPN
cana-6025	119	3	can	can	AUX
cana-6025	119	4	be	be	AUX
cana-6025	119	5	colored	color	VERB
cana-6025	119	6	with	with	ADP
cana-6025	119	7	another	another	DET
cana-6025	119	8	𝑚	𝑚	PROPN
cana-6025	119	9	colors	color	NOUN
cana-6025	119	10	.	.	PUNCT
cana-6025	120	1	thus	thus	ADV
cana-6025	120	2	,	,	PUNCT
cana-6025	120	3	𝜒(𝐾𝑚	𝜒(𝐾𝑚	ADP
cana-6025	120	4	⊠	⊠	PROPN
cana-6025	120	5	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	120	6	)	)	PUNCT
cana-6025	120	7	=	=	SYM
cana-6025	120	8	2𝑚.	2𝑚.	NOUN
cana-6025	120	9	consider	consider	VERB
cana-6025	120	10	a	a	DET
cana-6025	120	11	vertex	vertex	NOUN
cana-6025	120	12	from	from	ADP
cana-6025	120	13	one	one	NUM
cana-6025	120	14	of	of	ADP
cana-6025	120	15	the	the	DET
cana-6025	120	16	columns	column	NOUN
cana-6025	120	17	𝑉1	𝑉1	PROPN
cana-6025	120	18	,	,	PUNCT
cana-6025	120	19	𝑉2	𝑉2	NOUN
cana-6025	120	20	,	,	PUNCT
cana-6025	120	21	.	.	PUNCT
cana-6025	120	22	.	.	PUNCT
cana-6025	121	1	.	.	PUNCT
cana-6025	122	1	,	,	PUNCT
cana-6025	122	2	𝑉𝑟	𝑉𝑟	PROPN
cana-6025	122	3	and	and	CCONJ
cana-6025	122	4	another	another	DET
cana-6025	122	5	vertex	vertex	NOUN
cana-6025	122	6	from	from	ADP
cana-6025	122	7	one	one	NUM
cana-6025	122	8	of	of	ADP
cana-6025	122	9	the	the	DET
cana-6025	122	10	columns	column	NOUN
cana-6025	122	11	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	122	12	,	,	PUNCT
cana-6025	122	13	𝑉𝑟+2	𝑉𝑟+2	NOUN
cana-6025	122	14	,	,	PUNCT
cana-6025	122	15	.	.	PUNCT
cana-6025	122	16	.	.	PUNCT
cana-6025	122	17	.	.	PUNCT
cana-6025	123	1	,	,	PUNCT
cana-6025	123	2	𝑉𝑟+𝑠.	𝑉𝑟+𝑠.	PROPN
cana-6025	123	3	so	so	ADV
cana-6025	123	4	,	,	PUNCT
cana-6025	123	5	let	let	VERB
cana-6025	123	6	𝐷	𝐷	NOUN
cana-6025	123	7	=	=	SYM
cana-6025	123	8	{	{	PUNCT
cana-6025	123	9	(	(	PUNCT
cana-6025	123	10	𝑢1	𝑢1	PROPN
cana-6025	123	11	,	,	PUNCT
cana-6025	123	12	𝑣𝑟	𝑣𝑟	INTJ
cana-6025	123	13	)	)	PUNCT
cana-6025	123	14	,	,	PUNCT
cana-6025	123	15	(	(	PUNCT
cana-6025	123	16	𝑢1	𝑢1	PROPN
cana-6025	123	17	,	,	PUNCT
cana-6025	123	18	𝑣𝑟+1	𝑣𝑟+1	NOUN
cana-6025	123	19	)	)	PUNCT
cana-6025	123	20	}	}	PUNCT
cana-6025	123	21	.	.	PUNCT
cana-6025	124	1	clearly	clearly	ADV
cana-6025	124	2	,	,	PUNCT
cana-6025	124	3	𝐷	𝐷	PROPN
cana-6025	124	4	is	be	AUX
cana-6025	124	5	a	a	DET
cana-6025	124	6	restrained	restrained	ADJ
cana-6025	124	7	dominating	dominating	NOUN
cana-6025	124	8	set	set	NOUN
cana-6025	124	9	and	and	CCONJ
cana-6025	124	10	𝛾𝑟(𝐾𝑚	𝛾𝑟(𝐾𝑚	PROPN
cana-6025	124	11	⊠	⊠	PROPN
cana-6025	124	12	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	124	13	)	)	PUNCT
cana-6025	124	14	=	=	SYM
cana-6025	124	15	2	2	X
cana-6025	124	16	.	.	PUNCT
cana-6025	125	1	but	but	CCONJ
cana-6025	125	2	,	,	PUNCT
cana-6025	125	3	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	125	4	)	)	PUNCT
cana-6025	125	5	=	=	SYM
cana-6025	125	6	2	2	NUM
cana-6025	125	7	≠	≠	PROPN
cana-6025	125	8	𝜒(𝐾𝑚	𝜒(𝐾𝑚	NOUN
cana-6025	125	9	⊠	⊠	PROPN
cana-6025	125	10	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	125	11	)	)	PUNCT
cana-6025	125	12	and	and	CCONJ
cana-6025	125	13	so	so	ADV
cana-6025	125	14	,	,	PUNCT
cana-6025	125	15	𝐷	𝐷	PROPN
cana-6025	125	16	is	be	AUX
cana-6025	125	17	not	not	PART
cana-6025	125	18	a	a	DET
cana-6025	125	19	chromatic	chromatic	ADJ
cana-6025	125	20	restrained	restrain	VERB
cana-6025	125	21	dominating	dominating	NOUN
cana-6025	125	22	set	set	NOUN
cana-6025	125	23	of	of	ADP
cana-6025	125	24	𝐾𝑚	𝐾𝑚	PROPN
cana-6025	125	25	⊠	⊠	PROPN
cana-6025	125	26	𝐾𝑟,𝑠.	𝐾𝑟,𝑠.	NOUN
cana-6025	125	27	consider	consider	VERB
cana-6025	125	28	𝐷1	𝐷1	NOUN
cana-6025	125	29	=	=	PUNCT
cana-6025	125	30	𝑉𝑟	𝑉𝑟	PROPN
cana-6025	125	31	∪	∪	ADJ
cana-6025	125	32	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	125	33	.	.	PUNCT
cana-6025	126	1	then	then	ADV
cana-6025	126	2	,	,	PUNCT
cana-6025	126	3	𝐷1	𝐷1	PROPN
cana-6025	126	4	is	be	AUX
cana-6025	126	5	a	a	DET
cana-6025	126	6	restrained	restrain	VERB
cana-6025	126	7	dominating	dominating	NOUN
cana-6025	126	8	set	set	NOUN
cana-6025	126	9	and	and	CCONJ
cana-6025	126	10	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	126	11	)	)	PUNCT
cana-6025	126	12	=	=	SYM
cana-6025	126	13	|𝑉𝑟|	|𝑉𝑟|	NOUN
cana-6025	126	14	+	+	NUM
cana-6025	126	15	|𝑉𝑟+1|	|𝑉𝑟+1|	NOUN
cana-6025	126	16	=	=	SYM
cana-6025	126	17	2𝑚	2𝑚	NUM
cana-6025	126	18	=	=	SYM
cana-6025	126	19	𝜒(𝐾𝑚	𝜒(𝐾𝑚	NOUN
cana-6025	126	20	⊠	⊠	PROPN
cana-6025	126	21	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	126	22	)	)	PUNCT
cana-6025	126	23	.	.	PUNCT
cana-6025	127	1	thus	thus	ADV
cana-6025	127	2	,	,	PUNCT
cana-6025	127	3	𝐷1	𝐷1	PROPN
cana-6025	127	4	is	be	AUX
cana-6025	127	5	a	a	DET
cana-6025	127	6	chromatic	chromatic	ADJ
cana-6025	127	7	restrained	restrain	VERB
cana-6025	127	8	dominating	dominating	NOUN
cana-6025	127	9	set	set	NOUN
cana-6025	127	10	of	of	ADP
cana-6025	127	11	𝐾𝑚	𝐾𝑚	PROPN
cana-6025	127	12	⊠	⊠	PROPN
cana-6025	127	13	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	127	14	and	and	CCONJ
cana-6025	127	15	𝛾𝑟	𝛾𝑟	ADP
cana-6025	127	16	𝑐(𝐾𝑚	𝑐(𝐾𝑚	X
cana-6025	127	17	⊠	⊠	PROPN
cana-6025	127	18	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	127	19	)	)	PUNCT
cana-6025	127	20	≤	≤	NUM
cana-6025	127	21	|𝐷1|	|𝐷1|	NOUN
cana-6025	127	22	=	=	SYM
cana-6025	127	23	2𝑚.	2𝑚.	NUM
cana-6025	127	24	since	since	SCONJ
cana-6025	127	25	𝜒(𝐾𝑚	𝜒(𝐾𝑚	X
cana-6025	127	26	⊠	⊠	PROPN
cana-6025	127	27	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	127	28	)	)	PUNCT
cana-6025	127	29	=	=	SYM
cana-6025	127	30	2𝑚	2𝑚	NOUN
cana-6025	127	31	,	,	PUNCT
cana-6025	127	32	any	any	DET
cana-6025	127	33	minimum	minimum	ADJ
cana-6025	127	34	chromatic	chromatic	ADJ
cana-6025	127	35	restrained	restrain	VERB
cana-6025	127	36	dominating	dominating	NOUN
cana-6025	127	37	set	set	NOUN
cana-6025	127	38	must	must	AUX
cana-6025	127	39	contain	contain	VERB
cana-6025	127	40	at	at	ADV
cana-6025	127	41	least	least	ADJ
cana-6025	127	42	2𝑚	2𝑚	NUM
cana-6025	127	43	vertices	vertex	NOUN
cana-6025	127	44	.	.	PUNCT
cana-6025	128	1	therefore	therefore	ADV
cana-6025	128	2	,	,	PUNCT
cana-6025	128	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	128	4	𝑐(𝐾𝑚	𝑐(𝐾𝑚	NOUN
cana-6025	128	5	⊠	⊠	PROPN
cana-6025	128	6	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	128	7	)	)	PUNCT
cana-6025	128	8	≥	≥	NOUN
cana-6025	128	9	2𝑚.	2𝑚.	NUM
cana-6025	128	10	hence	hence	ADV
cana-6025	128	11	,	,	PUNCT
cana-6025	128	12	𝛾𝑟	𝛾𝑟	ADP
cana-6025	128	13	𝑐(𝐾𝑚	𝑐(𝐾𝑚	NOUN
cana-6025	128	14	⊠	⊠	PROPN
cana-6025	128	15	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	128	16	)	)	PUNCT
cana-6025	128	17	=	=	SYM
cana-6025	129	1	2𝑚.	2𝑚.	NUM
cana-6025	129	2	theorem	theorem	VERB
cana-6025	129	3	2.6	2.6	NUM
cana-6025	129	4	for	for	ADP
cana-6025	129	5	𝑠	𝑠	PROPN
cana-6025	129	6	≥	≥	NUM
cana-6025	129	7	5	5	NUM
cana-6025	129	8	and	and	CCONJ
cana-6025	129	9	𝑠	𝑠	PROPN
cana-6025	129	10	is	be	AUX
cana-6025	129	11	odd	odd	ADJ
cana-6025	129	12	,	,	PUNCT
cana-6025	129	13	𝛾𝑟	𝛾𝑟	ADP
cana-6025	129	14	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	129	15	⊠	⊠	PROPN
cana-6025	129	16	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	129	17	)	)	PUNCT
cana-6025	129	18	=	=	SYM
cana-6025	129	19	3𝑟.	3𝑟.	NUM
cana-6025	129	20	proof	proof	NOUN
cana-6025	129	21	.	.	PUNCT
cana-6025	130	1	let	let	VERB
cana-6025	130	2	𝑉(𝐾𝑟	𝑉(𝐾𝑟	NUM
cana-6025	130	3	)	)	PUNCT
cana-6025	131	1	=	=	PRON
cana-6025	131	2	{	{	PUNCT
cana-6025	131	3	𝑢1	𝑢1	PROPN
cana-6025	131	4	,	,	PUNCT
cana-6025	131	5	𝑢2	𝑢2	PROPN
cana-6025	131	6	,	,	PUNCT
cana-6025	131	7	𝑢3	𝑢3	PROPN
cana-6025	131	8	,	,	PUNCT
cana-6025	131	9	…	…	PUNCT
cana-6025	131	10	,	,	PUNCT
cana-6025	131	11	𝑢𝑟	𝑢𝑟	ADP
cana-6025	131	12	}	}	PUNCT
cana-6025	131	13	and	and	CCONJ
cana-6025	131	14	𝑉(𝑊𝑠	𝑉(𝑊𝑠	NUM
cana-6025	131	15	)	)	PUNCT
cana-6025	131	16	=	=	SYM
cana-6025	131	17	{	{	PUNCT
cana-6025	131	18	𝑣0	𝑣0	PROPN
cana-6025	131	19	,	,	PUNCT
cana-6025	131	20	𝑣1	𝑣1	PROPN
cana-6025	131	21	,	,	PUNCT
cana-6025	131	22	𝑣2	𝑣2	PROPN
cana-6025	131	23	,	,	PUNCT
cana-6025	131	24	…	…	PUNCT
cana-6025	131	25	,	,	PUNCT
cana-6025	131	26	𝑣𝑠−1	𝑣𝑠−1	PROPN
cana-6025	131	27	}	}	PUNCT
cana-6025	131	28	.	.	PUNCT
cana-6025	132	1	then	then	ADV
cana-6025	132	2	,	,	PUNCT
cana-6025	132	3	𝑉(𝐾𝑟	𝑉(𝐾𝑟	PROPN
cana-6025	132	4	⊠	⊠	PROPN
cana-6025	132	5	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	132	6	)	)	PUNCT
cana-6025	132	7	=	=	PRON
cana-6025	132	8	{	{	PUNCT
cana-6025	132	9	(	(	PUNCT
cana-6025	132	10	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	132	11	,	,	PUNCT
cana-6025	132	12	𝑣𝑗)/1	𝑣𝑗)/1	PROPN
cana-6025	132	13	≤	≤	PROPN
cana-6025	132	14	𝑖	𝑖	SYM
cana-6025	132	15	≤	≤	NOUN
cana-6025	132	16	𝑟	𝑟	NOUN
cana-6025	132	17	,	,	PUNCT
cana-6025	132	18	0	0	NUM
cana-6025	132	19	≤	≤	NUM
cana-6025	133	1	𝑗	𝑗	PRON
cana-6025	133	2	≤	≤	NUM
cana-6025	133	3	𝑠	𝑠	INTJ
cana-6025	133	4	−	−	PROPN
cana-6025	133	5	1	1	NUM
cana-6025	133	6	}	}	PUNCT
cana-6025	133	7	where	where	SCONJ
cana-6025	133	8	(	(	PUNCT
cana-6025	133	9	𝑢1	𝑢1	PROPN
cana-6025	133	10	,	,	PUNCT
cana-6025	133	11	𝑣0	𝑣0	NOUN
cana-6025	133	12	)	)	PUNCT
cana-6025	133	13	is	be	AUX
cana-6025	133	14	the	the	DET
cana-6025	133	15	full	full	ADJ
cana-6025	133	16	degree	degree	NOUN
cana-6025	133	17	vertex	vertex	NOUN
cana-6025	133	18	in	in	ADP
cana-6025	133	19	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	133	20	⊠	⊠	PROPN
cana-6025	133	21	𝑊𝑠.	𝑊𝑠.	PROPN
cana-6025	133	22	also	also	ADV
cana-6025	133	23	,	,	PUNCT
cana-6025	133	24	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	133	25	⊠	⊠	PROPN
cana-6025	133	26	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	133	27	consists	consist	VERB
cana-6025	133	28	of	of	ADP
cana-6025	133	29	𝑟	𝑟	NOUN
cana-6025	133	30	rows	row	NOUN
cana-6025	133	31	and	and	CCONJ
cana-6025	133	32	𝑠	𝑠	DET
cana-6025	133	33	columns	column	NOUN
cana-6025	133	34	,	,	PUNCT
cana-6025	133	35	where	where	SCONJ
cana-6025	133	36	𝑉1	𝑉1	NOUN
cana-6025	133	37	,	,	PUNCT
cana-6025	133	38	𝑉2	𝑉2	NOUN
cana-6025	133	39	,	,	PUNCT
cana-6025	133	40	…	…	PUNCT
cana-6025	133	41	,	,	PUNCT
cana-6025	133	42	𝑉𝑠	𝑉𝑠	PROPN
cana-6025	133	43	denotes	denote	VERB
cana-6025	133	44	the	the	DET
cana-6025	133	45	columns	column	NOUN
cana-6025	133	46	.	.	PUNCT
cana-6025	134	1	clearly	clearly	ADV
cana-6025	134	2	,	,	PUNCT
cana-6025	134	3	𝑉1	𝑉1	PROPN
cana-6025	134	4	can	can	AUX
cana-6025	134	5	be	be	AUX
cana-6025	134	6	colored	color	VERB
cana-6025	134	7	with	with	ADP
cana-6025	134	8	𝑟	𝑟	NOUN
cana-6025	134	9	colors	color	NOUN
cana-6025	134	10	,	,	PUNCT
cana-6025	134	11	𝑉2	𝑉2	PROPN
cana-6025	134	12	can	can	AUX
cana-6025	134	13	be	be	AUX
cana-6025	134	14	colored	color	VERB
cana-6025	134	15	with	with	ADP
cana-6025	134	16	another	another	DET
cana-6025	134	17	𝑟	𝑟	PRON
cana-6025	134	18	colors	color	NOUN
cana-6025	134	19	,	,	PUNCT
cana-6025	134	20	𝑉3	𝑉3	NOUN
cana-6025	134	21	can	can	AUX
cana-6025	134	22	be	be	AUX
cana-6025	134	23	colored	color	VERB
cana-6025	134	24	with	with	ADP
cana-6025	134	25	another	another	DET
cana-6025	134	26	𝑟	𝑟	NOUN
cana-6025	134	27	colors	color	NOUN
cana-6025	134	28	and	and	CCONJ
cana-6025	134	29	the	the	DET
cana-6025	134	30	remaining	remain	VERB
cana-6025	134	31	vertices	vertex	NOUN
cana-6025	134	32	can	can	AUX
cana-6025	134	33	be	be	AUX
cana-6025	134	34	colored	color	VERB
cana-6025	134	35	with	with	ADP
cana-6025	134	36	one	one	NUM
cana-6025	134	37	among	among	ADP
cana-6025	134	38	those	those	DET
cana-6025	134	39	3𝑟	3𝑟	NUM
cana-6025	134	40	colors	color	NOUN
cana-6025	134	41	.	.	PUNCT
cana-6025	135	1	then	then	ADV
cana-6025	135	2	,	,	PUNCT
cana-6025	135	3	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	135	4	⊠	⊠	PROPN
cana-6025	135	5	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	135	6	)	)	PUNCT
cana-6025	135	7	=	=	PUNCT
cana-6025	136	1	3𝑟.	3𝑟.	NOUN
cana-6025	136	2	clearly	clearly	ADV
cana-6025	136	3	,	,	PUNCT
cana-6025	136	4	𝐷	𝐷	PROPN
cana-6025	136	5	=	=	SYM
cana-6025	136	6	{	{	PUNCT
cana-6025	136	7	(	(	PUNCT
cana-6025	136	8	𝑢1	𝑢1	PROPN
cana-6025	136	9	,	,	PUNCT
cana-6025	136	10	𝑣0	𝑣0	NOUN
cana-6025	136	11	)	)	PUNCT
cana-6025	136	12	}	}	PUNCT
cana-6025	136	13	is	be	AUX
cana-6025	136	14	a	a	DET
cana-6025	136	15	restrained	restrained	ADJ
cana-6025	136	16	dominating	dominating	NOUN
cana-6025	136	17	set	set	NOUN
cana-6025	136	18	of	of	ADP
cana-6025	136	19	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	136	20	⊠	⊠	PROPN
cana-6025	136	21	𝑊𝑠.	𝑊𝑠.	PROPN
cana-6025	136	22	then	then	ADV
cana-6025	136	23	,	,	PUNCT
cana-6025	136	24	𝛾𝑟(𝐾𝑟	𝛾𝑟(𝐾𝑟	PROPN
cana-6025	136	25	⊠	⊠	PROPN
cana-6025	136	26	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	136	27	)	)	PUNCT
cana-6025	136	28	=	=	SYM
cana-6025	137	1	1	1	X
cana-6025	137	2	.	.	PUNCT
cana-6025	138	1	but	but	CCONJ
cana-6025	138	2	,	,	PUNCT
cana-6025	138	3	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	138	4	)	)	PUNCT
cana-6025	138	5	=	=	SYM
cana-6025	138	6	1	1	NUM
cana-6025	138	7	≠	≠	PROPN
cana-6025	138	8	𝜒(𝐾𝑟	𝜒(𝐾𝑟	NOUN
cana-6025	138	9	⊠	⊠	PROPN
cana-6025	138	10	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	138	11	)	)	PUNCT
cana-6025	138	12	.	.	PUNCT
cana-6025	139	1	this	this	PRON
cana-6025	139	2	implies	imply	VERB
cana-6025	139	3	that	that	SCONJ
cana-6025	139	4	,	,	PUNCT
cana-6025	139	5	𝐷	𝐷	PROPN
cana-6025	139	6	is	be	AUX
cana-6025	139	7	not	not	PART
cana-6025	139	8	a	a	DET
cana-6025	139	9	chromatic	chromatic	ADJ
cana-6025	139	10	restrained	restrain	VERB
cana-6025	139	11	dominating	dominating	NOUN
cana-6025	139	12	set	set	NOUN
cana-6025	139	13	of	of	ADP
cana-6025	139	14	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	139	15	⊠	⊠	PROPN
cana-6025	139	16	𝑊𝑠.	𝑊𝑠.	PROPN
cana-6025	139	17	let	let	VERB
cana-6025	139	18	𝐷1	𝐷1	NOUN
cana-6025	139	19	=	=	VERB
cana-6025	139	20	𝑉1	𝑉1	PROPN
cana-6025	139	21	∪	∪	ADP
cana-6025	139	22	𝑉2	𝑉2	PROPN
cana-6025	139	23	∪	∪	ADP
cana-6025	139	24	𝑉3	𝑉3	NOUN
cana-6025	139	25	=	=	SYM
cana-6025	139	26	{	{	PUNCT
cana-6025	139	27	(	(	PUNCT
cana-6025	139	28	𝑢1	𝑢1	PROPN
cana-6025	139	29	,	,	PUNCT
cana-6025	139	30	communications	communication	NOUN
cana-6025	139	31	on	on	ADP
cana-6025	139	32	applied	apply	VERB
cana-6025	139	33	nonlinear	nonlinear	ADJ
cana-6025	139	34	analysis	analysis	NOUN
cana-6025	139	35	issn	issn	NOUN
cana-6025	139	36	:	:	PUNCT
cana-6025	139	37	1074	1074	NUM
cana-6025	139	38	-	-	PUNCT
cana-6025	139	39	133x	133x	NUM
cana-6025	139	40	vol	vol	NOUN
cana-6025	139	41	31	31	NUM
cana-6025	139	42	no	no	NOUN
cana-6025	139	43	.	.	NOUN
cana-6025	139	44	2	2	NUM
cana-6025	139	45	(	(	PUNCT
cana-6025	139	46	2024	2024	NUM
cana-6025	139	47	)	)	PUNCT
cana-6025	139	48	494	494	NUM
cana-6025	139	49	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	139	50	𝑣0	𝑣0	PROPN
cana-6025	139	51	)	)	PUNCT
cana-6025	139	52	,	,	PUNCT
cana-6025	139	53	(	(	PUNCT
cana-6025	139	54	𝑢2	𝑢2	PROPN
cana-6025	139	55	,	,	PUNCT
cana-6025	139	56	𝑣0	𝑣0	PROPN
cana-6025	139	57	)	)	PUNCT
cana-6025	139	58	,	,	PUNCT
cana-6025	139	59	…	…	PUNCT
cana-6025	139	60	,	,	PUNCT
cana-6025	139	61	(	(	PUNCT
cana-6025	139	62	𝑢𝑟	𝑢𝑟	ADV
cana-6025	139	63	,	,	PUNCT
cana-6025	139	64	𝑣0	𝑣0	PROPN
cana-6025	139	65	)	)	PUNCT
cana-6025	139	66	,	,	PUNCT
cana-6025	139	67	(	(	PUNCT
cana-6025	139	68	𝑢1	𝑢1	PROPN
cana-6025	139	69	,	,	PUNCT
cana-6025	139	70	𝑣1	𝑣1	PROPN
cana-6025	139	71	)	)	PUNCT
cana-6025	139	72	,	,	PUNCT
cana-6025	139	73	(	(	PUNCT
cana-6025	139	74	𝑢2	𝑢2	PROPN
cana-6025	139	75	,	,	PUNCT
cana-6025	139	76	𝑣1	𝑣1	PROPN
cana-6025	139	77	)	)	PUNCT
cana-6025	139	78	,	,	PUNCT
cana-6025	139	79	.	.	PUNCT
cana-6025	139	80	.	.	PUNCT
cana-6025	139	81	.	.	PUNCT
cana-6025	140	1	,	,	PUNCT
cana-6025	140	2	(	(	PUNCT
cana-6025	140	3	𝑢𝑟	𝑢𝑟	ADV
cana-6025	140	4	,	,	PUNCT
cana-6025	140	5	𝑣1	𝑣1	PROPN
cana-6025	140	6	)	)	PUNCT
cana-6025	140	7	,	,	PUNCT
cana-6025	140	8	(	(	PUNCT
cana-6025	140	9	𝑢1	𝑢1	PROPN
cana-6025	140	10	,	,	PUNCT
cana-6025	140	11	𝑣2	𝑣2	PROPN
cana-6025	140	12	)	)	PUNCT
cana-6025	140	13	,	,	PUNCT
cana-6025	140	14	(	(	PUNCT
cana-6025	140	15	𝑢2	𝑢2	PROPN
cana-6025	140	16	,	,	PUNCT
cana-6025	140	17	𝑣2	𝑣2	PROPN
cana-6025	140	18	)	)	PUNCT
cana-6025	140	19	,	,	PUNCT
cana-6025	140	20	.	.	PUNCT
cana-6025	140	21	.	.	PUNCT
cana-6025	141	1	.	.	PUNCT
cana-6025	142	1	,	,	PUNCT
cana-6025	142	2	(	(	PUNCT
cana-6025	142	3	𝑢𝑟	𝑢𝑟	ADV
cana-6025	142	4	,	,	PUNCT
cana-6025	142	5	𝑣2	𝑣2	PROPN
cana-6025	142	6	)	)	PUNCT
cana-6025	142	7	}	}	PUNCT
cana-6025	142	8	.	.	PUNCT
cana-6025	143	1	since	since	SCONJ
cana-6025	143	2	the	the	DET
cana-6025	143	3	induced	induced	ADJ
cana-6025	143	4	subgraph	subgraph	NOUN
cana-6025	143	5	formed	form	VERB
cana-6025	143	6	from	from	ADP
cana-6025	143	7	all	all	DET
cana-6025	143	8	the	the	DET
cana-6025	143	9	vertices	vertex	NOUN
cana-6025	143	10	of	of	ADP
cana-6025	143	11	𝑉1	𝑉1	PROPN
cana-6025	143	12	,	,	PUNCT
cana-6025	143	13	𝑉2	𝑉2	NOUN
cana-6025	143	14	and	and	CCONJ
cana-6025	143	15	𝑉3	𝑉3	NOUN
cana-6025	143	16	is	be	AUX
cana-6025	143	17	a	a	DET
cana-6025	143	18	complete	complete	ADJ
cana-6025	143	19	graph	graph	NOUN
cana-6025	143	20	on	on	ADP
cana-6025	143	21	3𝑟	3𝑟	ADJ
cana-6025	143	22	vertices	vertex	NOUN
cana-6025	143	23	,	,	PUNCT
cana-6025	143	24	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	143	25	)	)	PUNCT
cana-6025	143	26	=	=	SYM
cana-6025	143	27	3𝑟	3𝑟	NOUN
cana-6025	143	28	=	=	SYM
cana-6025	143	29	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	143	30	⊠	⊠	PROPN
cana-6025	143	31	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	143	32	)	)	PUNCT
cana-6025	143	33	.	.	PUNCT
cana-6025	144	1	also	also	ADV
cana-6025	144	2	,	,	PUNCT
cana-6025	144	3	𝐷1	𝐷1	PROPN
cana-6025	144	4	is	be	AUX
cana-6025	144	5	a	a	DET
cana-6025	144	6	restrained	restrain	VERB
cana-6025	144	7	dominating	dominating	NOUN
cana-6025	144	8	set	set	NOUN
cana-6025	144	9	.	.	PUNCT
cana-6025	145	1	therefore	therefore	ADV
cana-6025	145	2	,	,	PUNCT
cana-6025	145	3	𝐷1	𝐷1	PROPN
cana-6025	145	4	is	be	AUX
cana-6025	145	5	a	a	DET
cana-6025	145	6	chromatic	chromatic	ADJ
cana-6025	145	7	restrained	restrain	VERB
cana-6025	145	8	dominating	dominating	NOUN
cana-6025	145	9	set	set	NOUN
cana-6025	145	10	of	of	ADP
cana-6025	145	11	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	145	12	⊠	⊠	PROPN
cana-6025	145	13	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	145	14	and	and	CCONJ
cana-6025	145	15	𝛾𝑟	𝛾𝑟	ADP
cana-6025	145	16	𝑐(𝐾𝑟	𝑐(𝐾𝑟	X
cana-6025	145	17	⊠	⊠	PROPN
cana-6025	145	18	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	145	19	)	)	PUNCT
cana-6025	145	20	≤	≤	NUM
cana-6025	145	21	|𝐷1|	|𝐷1|	NOUN
cana-6025	145	22	=	=	SYM
cana-6025	145	23	3𝑟.	3𝑟.	NOUN
cana-6025	145	24	since	since	SCONJ
cana-6025	145	25	𝜒(𝐾𝑟	𝜒(𝐾𝑟	PROPN
cana-6025	145	26	⊠	⊠	PROPN
cana-6025	145	27	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	145	28	)	)	PUNCT
cana-6025	145	29	=	=	SYM
cana-6025	145	30	3𝑟	3𝑟	NOUN
cana-6025	145	31	,	,	PUNCT
cana-6025	145	32	any	any	DET
cana-6025	145	33	minimum	minimum	ADJ
cana-6025	145	34	chromatic	chromatic	ADJ
cana-6025	145	35	restrained	restrain	VERB
cana-6025	145	36	dominating	dominating	NOUN
cana-6025	145	37	set	set	NOUN
cana-6025	145	38	of	of	ADP
cana-6025	145	39	𝐾𝑟	𝐾𝑟	PROPN
cana-6025	145	40	⊠	⊠	PROPN
cana-6025	145	41	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	145	42	must	must	AUX
cana-6025	145	43	contain	contain	VERB
cana-6025	145	44	a	a	DET
cana-6025	145	45	minimum	minimum	NOUN
cana-6025	145	46	of	of	ADP
cana-6025	145	47	3𝑟	3𝑟	ADJ
cana-6025	145	48	vertices	vertex	NOUN
cana-6025	145	49	and	and	CCONJ
cana-6025	145	50	so	so	ADV
cana-6025	145	51	𝛾𝑟	𝛾𝑟	ADP
cana-6025	145	52	𝑐(𝐾𝑟	𝑐(𝐾𝑟	X
cana-6025	145	53	⊠	⊠	PROPN
cana-6025	145	54	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	145	55	)	)	PUNCT
cana-6025	145	56	≥	≥	NOUN
cana-6025	145	57	3𝑟.	3𝑟.	ADV
cana-6025	145	58	hence	hence	ADV
cana-6025	145	59	,	,	PUNCT
cana-6025	145	60	𝛾𝑟	𝛾𝑟	ADP
cana-6025	145	61	𝑐(𝐾𝑟	𝑐(𝐾𝑟	NOUN
cana-6025	145	62	⊠	⊠	PROPN
cana-6025	145	63	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	145	64	)	)	PUNCT
cana-6025	145	65	=	=	SYM
cana-6025	146	1	3𝑟.	3𝑟.	NOUN
cana-6025	146	2	theorem	theorem	VERB
cana-6025	146	3	2.7	2.7	NUM
cana-6025	146	4	for	for	ADP
cana-6025	146	5	any	any	DET
cana-6025	146	6	𝑟	𝑟	NOUN
cana-6025	146	7	,	,	PUNCT
cana-6025	146	8	𝑠	𝑠	PRON
cana-6025	146	9	≥	≥	NUM
cana-6025	146	10	2	2	NUM
cana-6025	146	11	,	,	PUNCT
cana-6025	146	12	𝛾𝑟	𝛾𝑟	ADP
cana-6025	146	13	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	146	14	⊠	⊠	PROPN
cana-6025	146	15	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	146	16	)	)	PUNCT
cana-6025	146	17	=	=	PUNCT
cana-6025	147	1	⌈	⌈	PROPN
cana-6025	147	2	𝑠−4	𝑠−4	ADP
cana-6025	147	3	3	3	NUM
cana-6025	147	4	⌉	⌉	NOUN
cana-6025	147	5	+	+	NOUN
cana-6025	147	6	4	4	X
cana-6025	147	7	.	.	X
cana-6025	147	8	proof	proof	NOUN
cana-6025	147	9	.	.	PUNCT
cana-6025	148	1	let	let	VERB
cana-6025	148	2	𝑉(𝐾1,𝑟	𝑉(𝐾1,𝑟	NOUN
cana-6025	148	3	)	)	PUNCT
cana-6025	149	1	=	=	PRON
cana-6025	149	2	{	{	PUNCT
cana-6025	149	3	𝑢0	𝑢0	PROPN
cana-6025	149	4	,	,	PUNCT
cana-6025	149	5	𝑢1	𝑢1	NOUN
cana-6025	149	6	,	,	PUNCT
cana-6025	149	7	𝑢2	𝑢2	PROPN
cana-6025	149	8	,	,	PUNCT
cana-6025	149	9	𝑢3	𝑢3	PROPN
cana-6025	149	10	,	,	PUNCT
cana-6025	149	11	.	.	PUNCT
cana-6025	149	12	.	.	PUNCT
cana-6025	150	1	.	.	PUNCT
cana-6025	151	1	,	,	PUNCT
cana-6025	151	2	𝑢𝑟	𝑢𝑟	ADP
cana-6025	151	3	}	}	PUNCT
cana-6025	151	4	where	where	SCONJ
cana-6025	151	5	𝑢0	𝑢0	PROPN
cana-6025	151	6	is	be	AUX
cana-6025	151	7	the	the	DET
cana-6025	151	8	full	full	ADJ
cana-6025	151	9	degree	degree	NOUN
cana-6025	151	10	vertex	vertex	NOUN
cana-6025	151	11	of	of	ADP
cana-6025	151	12	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	151	13	and	and	CCONJ
cana-6025	151	14	𝑉(𝑃𝑠	𝑉(𝑃𝑠	NUM
cana-6025	151	15	)	)	PUNCT
cana-6025	152	1	=	=	PRON
cana-6025	152	2	{	{	PUNCT
cana-6025	152	3	𝑣1	𝑣1	PROPN
cana-6025	152	4	,	,	PUNCT
cana-6025	152	5	𝑣2	𝑣2	PROPN
cana-6025	152	6	,	,	PUNCT
cana-6025	152	7	𝑣3	𝑣3	ADJ
cana-6025	152	8	,	,	PUNCT
cana-6025	152	9	.	.	PUNCT
cana-6025	152	10	.	.	PUNCT
cana-6025	152	11	.	.	PUNCT
cana-6025	153	1	,	,	PUNCT
cana-6025	153	2	𝑣𝑠	𝑣𝑠	ADP
cana-6025	153	3	}	}	PUNCT
cana-6025	153	4	.	.	PUNCT
cana-6025	154	1	then	then	ADV
cana-6025	154	2	,	,	PUNCT
cana-6025	154	3	𝑉(𝐾1,𝑟	𝑉(𝐾1,𝑟	PROPN
cana-6025	154	4	⊠	⊠	PROPN
cana-6025	154	5	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	154	6	)	)	PUNCT
cana-6025	154	7	=	=	PRON
cana-6025	154	8	{	{	PUNCT
cana-6025	154	9	(	(	PUNCT
cana-6025	154	10	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	154	11	,	,	PUNCT
cana-6025	154	12	𝑣𝑗)/0	𝑣𝑗)/0	NOUN
cana-6025	154	13	≤	≤	NOUN
cana-6025	154	14	𝑖	𝑖	SYM
cana-6025	154	15	≤	≤	NOUN
cana-6025	154	16	𝑟	𝑟	NOUN
cana-6025	154	17	,	,	PUNCT
cana-6025	154	18	1	1	NUM
cana-6025	154	19	≤	≤	NUM
cana-6025	154	20	𝑗	𝑗	PRON
cana-6025	154	21	≤	≤	NUM
cana-6025	154	22	𝑠	𝑠	PROPN
cana-6025	154	23	}	}	PUNCT
cana-6025	154	24	and	and	CCONJ
cana-6025	154	25	|𝑉(𝐾1,𝑟	|𝑉(𝐾1,𝑟	VERB
cana-6025	154	26	⊠	⊠	PROPN
cana-6025	154	27	𝑃𝑠)|	𝑃𝑠)|	NOUN
cana-6025	154	28	=	=	PUNCT
cana-6025	154	29	(	(	PUNCT
cana-6025	154	30	𝑟	𝑟	NOUN
cana-6025	154	31	+	+	X
cana-6025	154	32	1)𝑠.	1)𝑠.	PROPN
cana-6025	154	33	clearly	clearly	ADV
cana-6025	154	34	,	,	PUNCT
cana-6025	154	35	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	154	36	⊠	⊠	PROPN
cana-6025	154	37	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	154	38	consists	consist	VERB
cana-6025	154	39	of	of	ADP
cana-6025	154	40	𝑟	𝑟	NOUN
cana-6025	154	41	+	+	CCONJ
cana-6025	154	42	1	1	NUM
cana-6025	154	43	rows	row	NOUN
cana-6025	154	44	and	and	CCONJ
cana-6025	154	45	𝑠	𝑠	PRON
cana-6025	154	46	columns	column	NOUN
cana-6025	154	47	denoted	denote	VERB
cana-6025	154	48	as	as	ADP
cana-6025	154	49	𝑉1	𝑉1	PROPN
cana-6025	154	50	,	,	PUNCT
cana-6025	154	51	𝑉2	𝑉2	NOUN
cana-6025	154	52	,	,	PUNCT
cana-6025	154	53	𝑉3	𝑉3	NOUN
cana-6025	154	54	,	,	PUNCT
cana-6025	154	55	.	.	PUNCT
cana-6025	154	56	.	.	PUNCT
cana-6025	155	1	.	.	PUNCT
cana-6025	156	1	,	,	PUNCT
cana-6025	156	2	𝑉𝑠.	𝑉𝑠.	NOUN
cana-6025	156	3	now	now	ADV
cana-6025	156	4	,	,	PUNCT
cana-6025	156	5	the	the	DET
cana-6025	156	6	first	first	ADJ
cana-6025	156	7	row	row	NOUN
cana-6025	156	8	can	can	AUX
cana-6025	156	9	be	be	AUX
cana-6025	156	10	colored	color	VERB
cana-6025	156	11	with	with	ADP
cana-6025	156	12	two	two	NUM
cana-6025	156	13	colors	color	NOUN
cana-6025	156	14	and	and	CCONJ
cana-6025	156	15	the	the	DET
cana-6025	156	16	remaining	remain	VERB
cana-6025	156	17	𝑟	𝑟	NOUN
cana-6025	156	18	rows	row	NOUN
cana-6025	156	19	can	can	AUX
cana-6025	156	20	be	be	AUX
cana-6025	156	21	colored	color	VERB
cana-6025	156	22	with	with	ADP
cana-6025	156	23	extra	extra	ADJ
cana-6025	156	24	two	two	NUM
cana-6025	156	25	colors	color	NOUN
cana-6025	156	26	since	since	SCONJ
cana-6025	156	27	no	no	DET
cana-6025	156	28	two	two	NUM
cana-6025	156	29	vertices	vertex	NOUN
cana-6025	156	30	belonging	belong	VERB
cana-6025	156	31	to	to	ADP
cana-6025	156	32	different	different	ADJ
cana-6025	156	33	rows	row	NOUN
cana-6025	156	34	(	(	PUNCT
cana-6025	156	35	among	among	ADP
cana-6025	156	36	those	those	DET
cana-6025	156	37	𝑟	𝑟	NOUN
cana-6025	156	38	rows	row	NOUN
cana-6025	156	39	)	)	PUNCT
cana-6025	156	40	are	be	AUX
cana-6025	156	41	adjacent	adjacent	ADJ
cana-6025	156	42	.	.	PUNCT
cana-6025	157	1	this	this	PRON
cana-6025	157	2	implies	imply	VERB
cana-6025	157	3	that	that	SCONJ
cana-6025	157	4	,	,	PUNCT
cana-6025	157	5	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	PRON
cana-6025	157	6	⊠	⊠	PROPN
cana-6025	157	7	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	157	8	)	)	PUNCT
cana-6025	157	9	=	=	SYM
cana-6025	157	10	4	4	X
cana-6025	157	11	.	.	X
cana-6025	157	12	case	case	NOUN
cana-6025	157	13	(	(	PUNCT
cana-6025	157	14	i	i	NOUN
cana-6025	157	15	):	):	PUNCT
cana-6025	157	16	𝑠	𝑠	PROPN
cana-6025	157	17	≡	≡	PROPN
cana-6025	157	18	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	NOUN
cana-6025	158	1	3	3	X
cana-6025	158	2	)	)	PUNCT
cana-6025	158	3	let	let	VERB
cana-6025	158	4	𝐷1	𝐷1	NOUN
cana-6025	158	5	=	=	PRON
cana-6025	158	6	{	{	PUNCT
cana-6025	158	7	(	(	PUNCT
cana-6025	158	8	𝑢0	𝑢0	PROPN
cana-6025	158	9	,	,	PUNCT
cana-6025	158	10	𝑣3𝑘−1)/1	𝑣3𝑘−1)/1	PROPN
cana-6025	158	11	≤	≤	NOUN
cana-6025	158	12	𝑘	𝑘	DET
cana-6025	158	13	≤	≤	NUM
cana-6025	158	14	𝑠	𝑠	PRON
cana-6025	158	15	3	3	NUM
cana-6025	158	16	}	}	PUNCT
cana-6025	158	17	where	where	SCONJ
cana-6025	158	18	|𝐷1|	|𝐷1|	NOUN
cana-6025	158	19	=	=	SYM
cana-6025	158	20	𝑠	𝑠	ADP
cana-6025	158	21	3	3	NUM
cana-6025	158	22	.	.	PUNCT
cana-6025	159	1	then	then	ADV
cana-6025	159	2	,	,	PUNCT
cana-6025	159	3	𝐷1	𝐷1	PROPN
cana-6025	159	4	is	be	AUX
cana-6025	159	5	a	a	DET
cana-6025	159	6	restrained	restrained	ADJ
cana-6025	159	7	dominating	dominating	NOUN
cana-6025	159	8	set	set	VERB
cana-6025	159	9	since	since	SCONJ
cana-6025	159	10	𝐷1	𝐷1	PROPN
cana-6025	159	11	is	be	AUX
cana-6025	159	12	a	a	DET
cana-6025	159	13	dominating	dominating	NOUN
cana-6025	159	14	set	set	NOUN
cana-6025	159	15	and	and	CCONJ
cana-6025	159	16	⟨𝑉	⟨𝑉	PROPN
cana-6025	159	17	−	−	PROPN
cana-6025	159	18	𝐷1⟩	𝐷1⟩	NOUN
cana-6025	159	19	has	have	VERB
cana-6025	159	20	no	no	DET
cana-6025	159	21	vertices	vertex	NOUN
cana-6025	159	22	of	of	ADP
cana-6025	159	23	degree	degree	NOUN
cana-6025	159	24	one	one	NUM
cana-6025	159	25	.	.	PUNCT
cana-6025	160	1	thus	thus	ADV
cana-6025	160	2	,	,	PUNCT
cana-6025	160	3	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	160	4	⊠	⊠	PROPN
cana-6025	160	5	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	160	6	)	)	PUNCT
cana-6025	160	7	≤	≤	NUM
cana-6025	160	8	|𝐷1|	|𝐷1|	NOUN
cana-6025	160	9	=	=	SYM
cana-6025	160	10	𝑠	𝑠	ADP
cana-6025	160	11	3	3	NUM
cana-6025	160	12	.	.	PUNCT
cana-6025	161	1	since	since	SCONJ
cana-6025	161	2	𝛾(𝐾1,𝑟	𝛾(𝐾1,𝑟	VERB
cana-6025	161	3	⊠	⊠	PROPN
cana-6025	161	4	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	161	5	)	)	PUNCT
cana-6025	161	6	=	=	SYM
cana-6025	161	7	𝑠	𝑠	NUM
cana-6025	161	8	3	3	NUM
cana-6025	161	9	,	,	PUNCT
cana-6025	161	10	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	161	11	⊠	⊠	PROPN
cana-6025	161	12	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	161	13	)	)	PUNCT
cana-6025	161	14	≥	≥	NOUN
cana-6025	161	15	𝑠	𝑠	PROPN
cana-6025	161	16	3	3	NUM
cana-6025	161	17	.	.	PUNCT
cana-6025	162	1	therefore	therefore	ADV
cana-6025	162	2	,	,	PUNCT
cana-6025	162	3	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	162	4	⊠	⊠	PROPN
cana-6025	162	5	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	162	6	)	)	PUNCT
cana-6025	162	7	=	=	SYM
cana-6025	162	8	𝑠	𝑠	PRON
cana-6025	162	9	3	3	NUM
cana-6025	162	10	.	.	PUNCT
cana-6025	163	1	since	since	SCONJ
cana-6025	163	2	𝐷1	𝐷1	PROPN
cana-6025	163	3	is	be	AUX
cana-6025	163	4	an	an	DET
cana-6025	163	5	independent	independent	ADJ
cana-6025	163	6	set	set	NOUN
cana-6025	163	7	,	,	PUNCT
cana-6025	163	8	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	163	9	)	)	PUNCT
cana-6025	163	10	=	=	SYM
cana-6025	163	11	1	1	NUM
cana-6025	163	12	≠	≠	PROPN
cana-6025	163	13	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	VERB
cana-6025	163	14	⊠	⊠	PROPN
cana-6025	163	15	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	163	16	)	)	PUNCT
cana-6025	163	17	and	and	CCONJ
cana-6025	163	18	so	so	ADV
cana-6025	163	19	,	,	PUNCT
cana-6025	163	20	𝐷1	𝐷1	PROPN
cana-6025	163	21	is	be	AUX
cana-6025	163	22	not	not	PART
cana-6025	163	23	a	a	DET
cana-6025	163	24	chromatic	chromatic	ADJ
cana-6025	163	25	restrained	restrain	VERB
cana-6025	163	26	dominating	dominating	NOUN
cana-6025	163	27	set	set	NOUN
cana-6025	163	28	.	.	PUNCT
cana-6025	164	1	consider	consider	VERB
cana-6025	164	2	𝐷2	𝐷2	NOUN
cana-6025	164	3	=	=	PUNCT
cana-6025	164	4	𝐷1	𝐷1	PROPN
cana-6025	164	5	∪	∪	X
cana-6025	164	6	{	{	PUNCT
cana-6025	164	7	(	(	PUNCT
cana-6025	164	8	𝑢0	𝑢0	PROPN
cana-6025	164	9	,	,	PUNCT
cana-6025	164	10	𝑣3	𝑣3	ADJ
cana-6025	164	11	)	)	PUNCT
cana-6025	164	12	,	,	PUNCT
cana-6025	164	13	(	(	PUNCT
cana-6025	164	14	𝑢1	𝑢1	PROPN
cana-6025	164	15	,	,	PUNCT
cana-6025	164	16	𝑣2	𝑣2	PROPN
cana-6025	164	17	)	)	PUNCT
cana-6025	164	18	,	,	PUNCT
cana-6025	164	19	(	(	PUNCT
cana-6025	164	20	𝑢1	𝑢1	NOUN
cana-6025	164	21	,	,	PUNCT
cana-6025	164	22	𝑣3	𝑣3	ADJ
cana-6025	164	23	)	)	PUNCT
cana-6025	164	24	}	}	PUNCT
cana-6025	164	25	.	.	PUNCT
cana-6025	165	1	then	then	ADV
cana-6025	165	2	,	,	PUNCT
cana-6025	165	3	⟨𝐷2⟩	⟨𝐷2⟩	PROPN
cana-6025	165	4	contains	contain	VERB
cana-6025	165	5	a	a	DET
cana-6025	165	6	complete	complete	ADJ
cana-6025	165	7	subgraph	subgraph	NOUN
cana-6025	165	8	on	on	ADP
cana-6025	165	9	four	four	NUM
cana-6025	165	10	vertices	vertex	NOUN
cana-6025	165	11	(	(	PUNCT
cana-6025	165	12	𝑢0	𝑢0	PROPN
cana-6025	165	13	,	,	PUNCT
cana-6025	165	14	𝑣2	𝑣2	PROPN
cana-6025	165	15	)	)	PUNCT
cana-6025	165	16	,	,	PUNCT
cana-6025	165	17	(	(	PUNCT
cana-6025	165	18	𝑢0	𝑢0	NOUN
cana-6025	165	19	,	,	PUNCT
cana-6025	165	20	𝑣3	𝑣3	ADJ
cana-6025	165	21	)	)	PUNCT
cana-6025	165	22	,	,	PUNCT
cana-6025	165	23	(	(	PUNCT
cana-6025	165	24	𝑢1	𝑢1	PROPN
cana-6025	165	25	,	,	PUNCT
cana-6025	165	26	𝑣2	𝑣2	NUM
cana-6025	165	27	)	)	PUNCT
cana-6025	165	28	and	and	CCONJ
cana-6025	165	29	(	(	PUNCT
cana-6025	165	30	𝑢1	𝑢1	PROPN
cana-6025	165	31	,	,	PUNCT
cana-6025	165	32	𝑣3	𝑣3	ADJ
cana-6025	165	33	)	)	PUNCT
cana-6025	165	34	.	.	PUNCT
cana-6025	166	1	thus	thus	ADV
cana-6025	166	2	,	,	PUNCT
cana-6025	166	3	𝜒(⟨𝐷2⟩	𝜒(⟨𝐷2⟩	PROPN
cana-6025	166	4	)	)	PUNCT
cana-6025	166	5	=	=	SYM
cana-6025	166	6	4	4	NUM
cana-6025	166	7	=	=	SYM
cana-6025	166	8	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	NUM
cana-6025	166	9	⊠	⊠	PROPN
cana-6025	166	10	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	166	11	)	)	PUNCT
cana-6025	166	12	.	.	PUNCT
cana-6025	167	1	also	also	ADV
cana-6025	167	2	,	,	PUNCT
cana-6025	167	3	𝐷2	𝐷2	PROPN
cana-6025	167	4	is	be	AUX
cana-6025	167	5	a	a	DET
cana-6025	167	6	restrained	restrained	ADJ
cana-6025	167	7	dominating	dominating	NOUN
cana-6025	167	8	set	set	NOUN
cana-6025	167	9	.	.	PUNCT
cana-6025	168	1	this	this	PRON
cana-6025	168	2	implies	imply	VERB
cana-6025	168	3	that	that	SCONJ
cana-6025	168	4	,	,	PUNCT
cana-6025	168	5	𝐷2	𝐷2	PROPN
cana-6025	168	6	is	be	AUX
cana-6025	168	7	a	a	DET
cana-6025	168	8	chromatic	chromatic	ADJ
cana-6025	168	9	restrained	restrain	VERB
cana-6025	168	10	dominating	dominating	NOUN
cana-6025	168	11	set	set	NOUN
cana-6025	168	12	and	and	CCONJ
cana-6025	168	13	𝛾𝑟	𝛾𝑟	ADP
cana-6025	168	14	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	168	15	⊠	⊠	PROPN
cana-6025	168	16	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	168	17	)	)	PUNCT
cana-6025	168	18	≤	≤	NOUN
cana-6025	168	19	|𝐷2|	|𝐷2|	X
cana-6025	168	20	=	=	SYM
cana-6025	168	21	|𝐷1|	|𝐷1|	PROPN
cana-6025	168	22	+	+	CCONJ
cana-6025	168	23	3	3	NUM
cana-6025	168	24	=	=	SYM
cana-6025	168	25	𝑠	𝑠	NUM
cana-6025	168	26	3	3	NUM
cana-6025	168	27	+	+	CCONJ
cana-6025	168	28	3	3	X
cana-6025	168	29	.	.	PUNCT
cana-6025	168	30	suppose	suppose	VERB
cana-6025	168	31	there	there	PRON
cana-6025	168	32	exists	exist	VERB
cana-6025	168	33	a	a	DET
cana-6025	168	34	chromatic	chromatic	ADJ
cana-6025	168	35	restrained	restrain	VERB
cana-6025	168	36	dominating	dominating	NOUN
cana-6025	168	37	set	set	VERB
cana-6025	168	38	𝑆1	𝑆1	NOUN
cana-6025	168	39	such	such	ADJ
cana-6025	168	40	that	that	SCONJ
cana-6025	168	41	|𝑆1|	|𝑆1|	PROPN
cana-6025	168	42	<	<	X
cana-6025	168	43	𝑠	𝑠	PROPN
cana-6025	168	44	3	3	NUM
cana-6025	168	45	+	+	CCONJ
cana-6025	168	46	3	3	X
cana-6025	168	47	.	.	PUNCT
cana-6025	168	48	then	then	ADV
cana-6025	168	49	|𝐷1|	|𝐷1|	VERB
cana-6025	168	50	<	<	X
cana-6025	168	51	|𝑆1|	|𝑆1|	PUNCT
cana-6025	168	52	<	<	X
cana-6025	168	53	𝑠	𝑠	PROPN
cana-6025	168	54	3	3	NUM
cana-6025	168	55	+	+	CCONJ
cana-6025	168	56	3	3	NUM
cana-6025	168	57	=	=	NOUN
cana-6025	168	58	|𝐷1|	|𝐷1|	NOUN
cana-6025	168	59	+	+	CCONJ
cana-6025	168	60	3	3	NUM
cana-6025	168	61	and	and	CCONJ
cana-6025	168	62	the	the	DET
cana-6025	168	63	only	only	ADJ
cana-6025	168	64	possible	possible	ADJ
cana-6025	168	65	case	case	NOUN
cana-6025	168	66	for	for	ADP
cana-6025	168	67	cardinality	cardinality	NOUN
cana-6025	168	68	of	of	ADP
cana-6025	168	69	𝑆1	𝑆1	NOUN
cana-6025	168	70	is	be	AUX
cana-6025	168	71	either	either	CCONJ
cana-6025	168	72	𝑠	𝑠	PROPN
cana-6025	168	73	3	3	NUM
cana-6025	168	74	+	+	CCONJ
cana-6025	168	75	1	1	NUM
cana-6025	168	76	or	or	CCONJ
cana-6025	168	77	𝑠	𝑠	NUM
cana-6025	168	78	3	3	NUM
cana-6025	168	79	+	+	CCONJ
cana-6025	168	80	2	2	NUM
cana-6025	168	81	.	.	PUNCT
cana-6025	169	1	but	but	CCONJ
cana-6025	169	2	,	,	PUNCT
cana-6025	169	3	there	there	PRON
cana-6025	169	4	does	do	AUX
cana-6025	169	5	not	not	PART
cana-6025	169	6	exists	exist	VERB
cana-6025	169	7	a	a	DET
cana-6025	169	8	chromatic	chromatic	ADJ
cana-6025	169	9	restrained	restrain	VERB
cana-6025	169	10	dominating	dominating	NOUN
cana-6025	169	11	set	set	VERB
cana-6025	169	12	with	with	ADP
cana-6025	169	13	cardinality	cardinality	NOUN
cana-6025	169	14	𝑠	𝑠	PROPN
cana-6025	169	15	3	3	NUM
cana-6025	169	16	+	+	CCONJ
cana-6025	169	17	1	1	NUM
cana-6025	169	18	or	or	CCONJ
cana-6025	169	19	𝑠	𝑠	NUM
cana-6025	169	20	3	3	NUM
cana-6025	169	21	+	+	CCONJ
cana-6025	169	22	2	2	X
cana-6025	169	23	.	.	X
cana-6025	170	1	therefore	therefore	ADV
cana-6025	170	2	,	,	PUNCT
cana-6025	170	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	170	4	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	170	5	⊠	⊠	PROPN
cana-6025	170	6	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	170	7	)	)	PUNCT
cana-6025	170	8	=	=	SYM
cana-6025	170	9	𝑠	𝑠	ADP
cana-6025	170	10	3	3	NUM
cana-6025	170	11	+	+	CCONJ
cana-6025	170	12	3	3	NUM
cana-6025	170	13	=	=	SYM
cana-6025	170	14	⌈	⌈	NOUN
cana-6025	170	15	𝑠−4	𝑠−4	ADP
cana-6025	170	16	3	3	NUM
cana-6025	170	17	⌉	⌉	NOUN
cana-6025	170	18	+	+	NOUN
cana-6025	170	19	4	4	NUM
cana-6025	170	20	.	.	X
cana-6025	170	21	case	case	NOUN
cana-6025	170	22	(	(	PUNCT
cana-6025	170	23	ii	ii	NUM
cana-6025	170	24	):	):	PUNCT
cana-6025	170	25	𝑠	𝑠	PROPN
cana-6025	170	26	≡	≡	PROPN
cana-6025	170	27	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-6025	170	28	3	3	X
cana-6025	170	29	)	)	PUNCT
cana-6025	170	30	communications	communication	NOUN
cana-6025	170	31	on	on	ADP
cana-6025	170	32	applied	apply	VERB
cana-6025	170	33	nonlinear	nonlinear	ADJ
cana-6025	170	34	analysis	analysis	NOUN
cana-6025	170	35	issn	issn	NOUN
cana-6025	170	36	:	:	PUNCT
cana-6025	170	37	1074	1074	NUM
cana-6025	170	38	-	-	PUNCT
cana-6025	170	39	133x	133x	NUM
cana-6025	170	40	vol	vol	NOUN
cana-6025	170	41	31	31	NUM
cana-6025	170	42	no	no	NOUN
cana-6025	170	43	.	.	NOUN
cana-6025	170	44	2	2	NUM
cana-6025	170	45	(	(	PUNCT
cana-6025	170	46	2024	2024	NUM
cana-6025	170	47	)	)	PUNCT
cana-6025	170	48	495	495	NUM
cana-6025	170	49	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	170	50	let	let	VERB
cana-6025	170	51	𝐷3	𝐷3	PROPN
cana-6025	170	52	=	=	PRON
cana-6025	170	53	{	{	PUNCT
cana-6025	170	54	(	(	PUNCT
cana-6025	170	55	𝑢0	𝑢0	PROPN
cana-6025	170	56	,	,	PUNCT
cana-6025	170	57	𝑣3𝑘−1)/1	𝑣3𝑘−1)/1	PROPN
cana-6025	170	58	≤	≤	NOUN
cana-6025	170	59	𝑘	𝑘	DET
cana-6025	170	60	≤	≤	ADJ
cana-6025	170	61	𝑠−1	𝑠−1	PROPN
cana-6025	170	62	3	3	NUM
cana-6025	170	63	}	}	PUNCT
cana-6025	170	64	∪	∪	X
cana-6025	170	65	{	{	PUNCT
cana-6025	170	66	(	(	PUNCT
cana-6025	170	67	𝑢0	𝑢0	PROPN
cana-6025	170	68	,	,	PUNCT
cana-6025	170	69	𝑣𝑠	𝑣𝑠	NOUN
cana-6025	170	70	)	)	PUNCT
cana-6025	170	71	}	}	PUNCT
cana-6025	170	72	where	where	SCONJ
cana-6025	170	73	|𝐷3|	|𝐷3|	PROPN
cana-6025	170	74	=	=	SYM
cana-6025	170	75	𝑠−1	𝑠−1	PROPN
cana-6025	170	76	3	3	NUM
cana-6025	170	77	+	+	SYM
cana-6025	170	78	1	1	NUM
cana-6025	170	79	=	=	SYM
cana-6025	170	80	⌈	⌈	NOUN
cana-6025	170	81	𝑠	𝑠	ADP
cana-6025	170	82	3	3	NUM
cana-6025	170	83	⌉.	⌉.	ADV
cana-6025	170	84	then	then	ADV
cana-6025	170	85	,	,	PUNCT
cana-6025	170	86	𝐷3	𝐷3	PROPN
cana-6025	170	87	is	be	AUX
cana-6025	170	88	a	a	DET
cana-6025	170	89	dominating	dominating	NOUN
cana-6025	170	90	set	set	NOUN
cana-6025	170	91	and	and	CCONJ
cana-6025	170	92	⟨𝑉	⟨𝑉	NOUN
cana-6025	170	93	−	−	PROPN
cana-6025	170	94	𝐷3⟩	𝐷3⟩	NOUN
cana-6025	170	95	has	have	VERB
cana-6025	170	96	no	no	DET
cana-6025	170	97	isolated	isolated	ADJ
cana-6025	170	98	vertices	vertex	NOUN
cana-6025	170	99	.	.	PUNCT
cana-6025	171	1	thus	thus	ADV
cana-6025	171	2	,	,	PUNCT
cana-6025	171	3	𝐷3	𝐷3	PROPN
cana-6025	171	4	is	be	AUX
cana-6025	171	5	a	a	DET
cana-6025	171	6	restrained	restrain	VERB
cana-6025	171	7	dominating	dominating	NOUN
cana-6025	171	8	set	set	NOUN
cana-6025	171	9	and	and	CCONJ
cana-6025	171	10	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	NOUN
cana-6025	171	11	⊠	⊠	PROPN
cana-6025	171	12	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	171	13	)	)	PUNCT
cana-6025	171	14	≤	≤	PUNCT
cana-6025	172	1	|𝐷3|	|𝐷3|	X
cana-6025	172	2	=	=	PUNCT
cana-6025	172	3	⌈	⌈	PROPN
cana-6025	172	4	𝑠	𝑠	ADP
cana-6025	172	5	3	3	NUM
cana-6025	172	6	⌉.	⌉.	ADV
cana-6025	172	7	since	since	SCONJ
cana-6025	172	8	any	any	DET
cana-6025	172	9	minimum	minimum	ADJ
cana-6025	172	10	dominating	dominating	NOUN
cana-6025	172	11	set	set	NOUN
cana-6025	172	12	must	must	AUX
cana-6025	172	13	contain	contain	VERB
cana-6025	172	14	the	the	DET
cana-6025	172	15	first	first	ADJ
cana-6025	172	16	vertex	vertex	NOUN
cana-6025	172	17	of	of	ADP
cana-6025	172	18	each	each	DET
cana-6025	172	19	column	column	NOUN
cana-6025	172	20	𝑉3𝑗−1	𝑉3𝑗−1	PROPN
cana-6025	172	21	,	,	PUNCT
cana-6025	172	22	1	1	NUM
cana-6025	172	23	≤	≤	NUM
cana-6025	172	24	𝑗	𝑗	PRON
cana-6025	172	25	≤	≤	ADJ
cana-6025	172	26	𝑠−1	𝑠−1	PROPN
cana-6025	172	27	3	3	NUM
cana-6025	172	28	together	together	ADV
cana-6025	172	29	with	with	ADP
cana-6025	172	30	the	the	DET
cana-6025	172	31	first	first	ADJ
cana-6025	172	32	vertex	vertex	NOUN
cana-6025	172	33	of	of	ADP
cana-6025	172	34	column	column	NOUN
cana-6025	172	35	𝑉3	𝑉3	NOUN
cana-6025	172	36	or	or	CCONJ
cana-6025	172	37	the	the	DET
cana-6025	172	38	first	first	ADJ
cana-6025	172	39	vertex	vertex	NOUN
cana-6025	172	40	of	of	ADP
cana-6025	172	41	each	each	DET
cana-6025	172	42	column	column	NOUN
cana-6025	172	43	𝑉3𝑗+1	𝑉3𝑗+1	NOUN
cana-6025	172	44	,	,	PUNCT
cana-6025	172	45	1	1	NUM
cana-6025	172	46	≤	≤	NUM
cana-6025	172	47	𝑗	𝑗	PRON
cana-6025	172	48	≤	≤	ADJ
cana-6025	172	49	𝑠−1	𝑠−1	PROPN
cana-6025	172	50	3	3	NUM
cana-6025	172	51	together	together	ADV
cana-6025	172	52	with	with	ADP
cana-6025	172	53	the	the	DET
cana-6025	172	54	first	first	ADJ
cana-6025	172	55	vertex	vertex	NOUN
cana-6025	172	56	of	of	ADP
cana-6025	172	57	column	column	NOUN
cana-6025	172	58	𝑉1	𝑉1	NOUN
cana-6025	172	59	or	or	CCONJ
cana-6025	172	60	the	the	DET
cana-6025	172	61	first	first	ADJ
cana-6025	172	62	vertex	vertex	NOUN
cana-6025	172	63	of	of	ADP
cana-6025	172	64	each	each	DET
cana-6025	172	65	column	column	NOUN
cana-6025	172	66	𝑉3𝑗	𝑉3𝑗	PROPN
cana-6025	172	67	,	,	PUNCT
cana-6025	172	68	1	1	NUM
cana-6025	172	69	≤	≤	NUM
cana-6025	172	70	𝑗	𝑗	PRON
cana-6025	172	71	≤	≤	ADJ
cana-6025	172	72	𝑠−1	𝑠−1	PROPN
cana-6025	172	73	3	3	NUM
cana-6025	172	74	together	together	ADV
cana-6025	172	75	with	with	ADP
cana-6025	172	76	the	the	DET
cana-6025	172	77	first	first	ADJ
cana-6025	172	78	vertex	vertex	NOUN
cana-6025	172	79	of	of	ADP
cana-6025	172	80	column	column	NOUN
cana-6025	172	81	𝑉1	𝑉1	NOUN
cana-6025	172	82	and	and	CCONJ
cana-6025	172	83	so	so	ADV
cana-6025	172	84	on	on	ADV
cana-6025	172	85	,	,	PUNCT
cana-6025	172	86	𝛾(𝐾1,𝑟	𝛾(𝐾1,𝑟	PROPN
cana-6025	172	87	⊠	⊠	PROPN
cana-6025	172	88	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	172	89	)	)	PUNCT
cana-6025	172	90	=	=	SYM
cana-6025	173	1	𝑠−1	𝑠−1	PROPN
cana-6025	173	2	3	3	NUM
cana-6025	173	3	+	+	SYM
cana-6025	173	4	1	1	NUM
cana-6025	173	5	=	=	SYM
cana-6025	173	6	⌈	⌈	NOUN
cana-6025	173	7	𝑠	𝑠	ADP
cana-6025	173	8	3	3	NUM
cana-6025	173	9	⌉.	⌉.	ADV
cana-6025	173	10	thus	thus	ADV
cana-6025	173	11	,	,	PUNCT
cana-6025	173	12	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	173	13	⊠	⊠	PROPN
cana-6025	173	14	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	173	15	)	)	PUNCT
cana-6025	173	16	≥	≥	NOUN
cana-6025	173	17	⌈	⌈	NOUN
cana-6025	173	18	𝑠	𝑠	ADP
cana-6025	173	19	3	3	NUM
cana-6025	173	20	⌉	⌉	PUNCT
cana-6025	173	21	and	and	CCONJ
cana-6025	173	22	so	so	ADV
cana-6025	173	23	,	,	PUNCT
cana-6025	173	24	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	173	25	⊠	⊠	PROPN
cana-6025	173	26	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	173	27	)	)	PUNCT
cana-6025	173	28	=	=	PUNCT
cana-6025	174	1	⌈	⌈	NOUN
cana-6025	174	2	𝑠	𝑠	ADP
cana-6025	174	3	3	3	NUM
cana-6025	174	4	⌉.	⌉.	ADV
cana-6025	174	5	since	since	SCONJ
cana-6025	174	6	𝐷3	𝐷3	PROPN
cana-6025	174	7	is	be	AUX
cana-6025	174	8	independent	independent	ADJ
cana-6025	174	9	,	,	PUNCT
cana-6025	174	10	𝜒(⟨𝐷3⟩	𝜒(⟨𝐷3⟩	ADJ
cana-6025	174	11	)	)	PUNCT
cana-6025	174	12	=	=	SYM
cana-6025	174	13	1	1	NUM
cana-6025	174	14	≠	≠	PROPN
cana-6025	174	15	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	VERB
cana-6025	174	16	⊠	⊠	PROPN
cana-6025	174	17	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	174	18	)	)	PUNCT
cana-6025	174	19	.	.	PUNCT
cana-6025	175	1	this	this	PRON
cana-6025	175	2	implies	imply	VERB
cana-6025	175	3	that	that	SCONJ
cana-6025	175	4	,	,	PUNCT
cana-6025	175	5	𝐷3	𝐷3	PROPN
cana-6025	175	6	is	be	AUX
cana-6025	175	7	not	not	PART
cana-6025	175	8	a	a	DET
cana-6025	175	9	chromatic	chromatic	ADJ
cana-6025	175	10	restrained	restrain	VERB
cana-6025	175	11	dominating	dominating	NOUN
cana-6025	175	12	set	set	NOUN
cana-6025	175	13	.	.	PUNCT
cana-6025	176	1	consider	consider	VERB
cana-6025	176	2	𝐷4	𝐷4	NOUN
cana-6025	176	3	=	=	SYM
cana-6025	176	4	{	{	PUNCT
cana-6025	176	5	(	(	PUNCT
cana-6025	176	6	𝑢0	𝑢0	PROPN
cana-6025	176	7	,	,	PUNCT
cana-6025	176	8	𝑣2	𝑣2	PROPN
cana-6025	176	9	)	)	PUNCT
cana-6025	176	10	,	,	PUNCT
cana-6025	176	11	(	(	PUNCT
cana-6025	176	12	𝑢0	𝑢0	NOUN
cana-6025	176	13	,	,	PUNCT
cana-6025	176	14	𝑣3	𝑣3	ADJ
cana-6025	176	15	)	)	PUNCT
cana-6025	176	16	,	,	PUNCT
cana-6025	176	17	(	(	PUNCT
cana-6025	176	18	𝑢1	𝑢1	PROPN
cana-6025	176	19	,	,	PUNCT
cana-6025	176	20	𝑣2	𝑣2	PROPN
cana-6025	176	21	)	)	PUNCT
cana-6025	176	22	,	,	PUNCT
cana-6025	176	23	(	(	PUNCT
cana-6025	176	24	𝑢1	𝑢1	NOUN
cana-6025	176	25	,	,	PUNCT
cana-6025	176	26	𝑣3	𝑣3	ADJ
cana-6025	176	27	)	)	PUNCT
cana-6025	176	28	,	,	PUNCT
cana-6025	176	29	(	(	PUNCT
cana-6025	176	30	𝑢0	𝑢0	PROPN
cana-6025	176	31	,	,	PUNCT
cana-6025	176	32	𝑣3𝑘)/2	𝑣3𝑘)/2	NOUN
cana-6025	176	33	≤	≤	NOUN
cana-6025	176	34	𝑘	𝑘	DET
cana-6025	176	35	≤	≤	ADJ
cana-6025	176	36	𝑠−1	𝑠−1	PROPN
cana-6025	176	37	3	3	NUM
cana-6025	176	38	}	}	PUNCT
cana-6025	176	39	.	.	PUNCT
cana-6025	177	1	then	then	ADV
cana-6025	177	2	,	,	PUNCT
cana-6025	177	3	𝐷4	𝐷4	PROPN
cana-6025	177	4	is	be	AUX
cana-6025	177	5	a	a	DET
cana-6025	177	6	restrained	restrained	ADJ
cana-6025	177	7	dominating	dominating	NOUN
cana-6025	177	8	set	set	VERB
cana-6025	177	9	since	since	SCONJ
cana-6025	177	10	the	the	DET
cana-6025	177	11	columns	column	NOUN
cana-6025	177	12	𝑉1	𝑉1	PROPN
cana-6025	177	13	,	,	PUNCT
cana-6025	177	14	𝑉2	𝑉2	NOUN
cana-6025	177	15	,	,	PUNCT
cana-6025	177	16	𝑉3	𝑉3	NOUN
cana-6025	177	17	and	and	CCONJ
cana-6025	177	18	𝑉4	𝑉4	NOUN
cana-6025	177	19	are	be	AUX
cana-6025	177	20	dominated	dominate	VERB
cana-6025	177	21	by	by	ADP
cana-6025	177	22	the	the	DET
cana-6025	177	23	vertices	vertex	NOUN
cana-6025	177	24	(	(	PUNCT
cana-6025	177	25	𝑢0	𝑢0	PROPN
cana-6025	177	26	,	,	PUNCT
cana-6025	177	27	𝑣2	𝑣2	PROPN
cana-6025	177	28	)	)	PUNCT
cana-6025	177	29	,	,	PUNCT
cana-6025	177	30	(	(	PUNCT
cana-6025	177	31	𝑢0	𝑢0	NOUN
cana-6025	177	32	,	,	PUNCT
cana-6025	177	33	𝑣3	𝑣3	ADJ
cana-6025	177	34	)	)	PUNCT
cana-6025	177	35	,	,	PUNCT
cana-6025	177	36	(	(	PUNCT
cana-6025	177	37	𝑢1	𝑢1	PROPN
cana-6025	177	38	,	,	PUNCT
cana-6025	177	39	𝑣2	𝑣2	PROPN
cana-6025	177	40	)	)	PUNCT
cana-6025	177	41	,	,	PUNCT
cana-6025	177	42	(	(	PUNCT
cana-6025	177	43	𝑢1	𝑢1	NOUN
cana-6025	177	44	,	,	PUNCT
cana-6025	177	45	𝑣3	𝑣3	ADJ
cana-6025	177	46	)	)	PUNCT
cana-6025	177	47	and	and	CCONJ
cana-6025	177	48	all	all	DET
cana-6025	177	49	the	the	DET
cana-6025	177	50	remaining	remain	VERB
cana-6025	177	51	vertices	vertex	NOUN
cana-6025	177	52	are	be	AUX
cana-6025	177	53	adjacent	adjacent	ADJ
cana-6025	177	54	to	to	ADP
cana-6025	177	55	one	one	NUM
cana-6025	177	56	of	of	ADP
cana-6025	177	57	the	the	DET
cana-6025	177	58	vertex	vertex	NOUN
cana-6025	177	59	in	in	ADP
cana-6025	177	60	{	{	PUNCT
cana-6025	177	61	(	(	PUNCT
cana-6025	177	62	𝑢0	𝑢0	PROPN
cana-6025	177	63	,	,	PUNCT
cana-6025	177	64	𝑣3𝑘)/2	𝑣3𝑘)/2	NOUN
cana-6025	177	65	≤	≤	NOUN
cana-6025	177	66	𝑘	𝑘	DET
cana-6025	177	67	≤	≤	ADJ
cana-6025	177	68	𝑠−1	𝑠−1	PROPN
cana-6025	177	69	3	3	NUM
cana-6025	177	70	}	}	PUNCT
cana-6025	177	71	.	.	PUNCT
cana-6025	178	1	also	also	ADV
cana-6025	178	2	,	,	PUNCT
cana-6025	178	3	⟨𝐷4⟩	⟨𝐷4⟩	PROPN
cana-6025	178	4	contains	contain	VERB
cana-6025	178	5	𝐾4	𝐾4	NOUN
cana-6025	178	6	as	as	ADP
cana-6025	178	7	an	an	DET
cana-6025	178	8	induced	induced	ADJ
cana-6025	178	9	subgraph	subgraph	NOUN
cana-6025	178	10	and	and	CCONJ
cana-6025	178	11	so	so	ADV
cana-6025	178	12	,	,	PUNCT
cana-6025	178	13	𝜒(⟨𝐷4⟩	𝜒(⟨𝐷4⟩	PROPN
cana-6025	178	14	)	)	PUNCT
cana-6025	178	15	=	=	PUNCT
cana-6025	179	1	4	4	X
cana-6025	179	2	.	.	PUNCT
cana-6025	179	3	therefore	therefore	ADV
cana-6025	179	4	,	,	PUNCT
cana-6025	179	5	𝐷4	𝐷4	PROPN
cana-6025	179	6	is	be	AUX
cana-6025	179	7	a	a	DET
cana-6025	179	8	chromatic	chromatic	ADJ
cana-6025	179	9	restrained	restrain	VERB
cana-6025	179	10	dominating	dominating	NOUN
cana-6025	179	11	set	set	NOUN
cana-6025	179	12	of	of	ADP
cana-6025	179	13	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	179	14	⊠	⊠	PROPN
cana-6025	179	15	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	179	16	and	and	CCONJ
cana-6025	179	17	𝛾𝑟	𝛾𝑟	ADP
cana-6025	179	18	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	179	19	⊠	⊠	PROPN
cana-6025	179	20	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	179	21	)	)	PUNCT
cana-6025	179	22	≤	≤	NOUN
cana-6025	179	23	|𝐷4|	|𝐷4|	ADP
cana-6025	179	24	=	=	SYM
cana-6025	179	25	𝑠−1	𝑠−1	PROPN
cana-6025	179	26	3	3	NUM
cana-6025	179	27	+	+	SYM
cana-6025	179	28	3	3	NUM
cana-6025	179	29	=	=	SYM
cana-6025	179	30	⌈	⌈	NOUN
cana-6025	179	31	𝑠−4	𝑠−4	ADP
cana-6025	179	32	3	3	NUM
cana-6025	179	33	⌉	⌉	NOUN
cana-6025	179	34	+	+	NOUN
cana-6025	179	35	4	4	X
cana-6025	179	36	.	.	PUNCT
cana-6025	179	37	suppose	suppose	VERB
cana-6025	179	38	there	there	PRON
cana-6025	179	39	exists	exist	VERB
cana-6025	179	40	a	a	DET
cana-6025	179	41	chromatic	chromatic	ADJ
cana-6025	179	42	restrained	restrain	VERB
cana-6025	179	43	dominating	dominating	NOUN
cana-6025	179	44	set	set	VERB
cana-6025	179	45	𝑆2	𝑆2	ADV
cana-6025	179	46	such	such	ADJ
cana-6025	179	47	that	that	PRON
cana-6025	179	48	|𝑆2|	|𝑆2|	ADP
cana-6025	179	49	<	<	X
cana-6025	179	50	𝑠−1	𝑠−1	PROPN
cana-6025	179	51	3	3	NUM
cana-6025	179	52	+	+	SYM
cana-6025	179	53	3	3	X
cana-6025	179	54	.	.	PUNCT
cana-6025	179	55	then	then	ADV
cana-6025	179	56	|𝐷3|	|𝐷3|	X
cana-6025	179	57	<	<	X
cana-6025	179	58	|𝑆2|	|𝑆2|	ADP
cana-6025	179	59	<	<	X
cana-6025	179	60	𝑠−1	𝑠−1	PROPN
cana-6025	179	61	3	3	NUM
cana-6025	179	62	+	+	SYM
cana-6025	179	63	3	3	NUM
cana-6025	179	64	=	=	SYM
cana-6025	179	65	|𝐷3|	|𝐷3|	X
cana-6025	180	1	+	+	CCONJ
cana-6025	180	2	2	2	NUM
cana-6025	180	3	and	and	CCONJ
cana-6025	180	4	the	the	DET
cana-6025	180	5	only	only	ADJ
cana-6025	180	6	possible	possible	ADJ
cana-6025	180	7	cardinality	cardinality	NOUN
cana-6025	180	8	of	of	ADP
cana-6025	180	9	𝑆2	𝑆2	PROPN
cana-6025	180	10	is	be	AUX
cana-6025	180	11	𝑠−1	𝑠−1	PROPN
cana-6025	180	12	3	3	NUM
cana-6025	180	13	+	+	SYM
cana-6025	180	14	2	2	NUM
cana-6025	180	15	.	.	PUNCT
cana-6025	181	1	but	but	CCONJ
cana-6025	181	2	there	there	PRON
cana-6025	181	3	does	do	AUX
cana-6025	181	4	not	not	PART
cana-6025	181	5	exists	exist	VERB
cana-6025	181	6	a	a	DET
cana-6025	181	7	chromatic	chromatic	ADJ
cana-6025	181	8	restrained	restrain	VERB
cana-6025	181	9	dominating	dominating	NOUN
cana-6025	181	10	set	set	VERB
cana-6025	181	11	with	with	ADP
cana-6025	181	12	cardinality	cardinality	NOUN
cana-6025	181	13	𝑠−1	𝑠−1	PROPN
cana-6025	181	14	3	3	NUM
cana-6025	181	15	+	+	SYM
cana-6025	181	16	2	2	NUM
cana-6025	181	17	.	.	X
cana-6025	182	1	therefore	therefore	ADV
cana-6025	182	2	,	,	PUNCT
cana-6025	182	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	182	4	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	182	5	⊠	⊠	PROPN
cana-6025	182	6	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	182	7	)	)	PUNCT
cana-6025	182	8	≥	≥	NOUN
cana-6025	182	9	𝑠−1	𝑠−1	PROPN
cana-6025	182	10	3	3	NUM
cana-6025	182	11	+	+	SYM
cana-6025	182	12	3	3	NUM
cana-6025	182	13	.	.	X
cana-6025	182	14	hence	hence	ADV
cana-6025	182	15	,	,	PUNCT
cana-6025	182	16	𝛾𝑟	𝛾𝑟	ADP
cana-6025	182	17	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	182	18	⊠	⊠	PROPN
cana-6025	182	19	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	182	20	)	)	PUNCT
cana-6025	182	21	=	=	SYM
cana-6025	182	22	𝑠−1	𝑠−1	PROPN
cana-6025	182	23	3	3	NUM
cana-6025	182	24	+	+	SYM
cana-6025	182	25	3	3	NUM
cana-6025	182	26	=	=	SYM
cana-6025	182	27	⌈	⌈	NOUN
cana-6025	182	28	𝑠−4	𝑠−4	ADP
cana-6025	182	29	3	3	NUM
cana-6025	182	30	⌉	⌉	NOUN
cana-6025	182	31	+	+	NOUN
cana-6025	182	32	4	4	NUM
cana-6025	182	33	.	.	X
cana-6025	182	34	case	case	NOUN
cana-6025	182	35	(	(	PUNCT
cana-6025	182	36	iii	iii	NOUN
cana-6025	182	37	):	):	PUNCT
cana-6025	182	38	𝑠	𝑠	PROPN
cana-6025	182	39	≡	≡	PROPN
cana-6025	182	40	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-6025	182	41	3	3	X
cana-6025	182	42	)	)	PUNCT
cana-6025	182	43	let	let	VERB
cana-6025	182	44	𝐷5	𝐷5	PROPN
cana-6025	182	45	=	=	PRON
cana-6025	182	46	{	{	PUNCT
cana-6025	182	47	(	(	PUNCT
cana-6025	182	48	𝑢0	𝑢0	PROPN
cana-6025	182	49	,	,	PUNCT
cana-6025	182	50	𝑣3𝑘−1)/1	𝑣3𝑘−1)/1	PROPN
cana-6025	182	51	≤	≤	NOUN
cana-6025	182	52	𝑘	𝑘	DET
cana-6025	182	53	≤	≤	NUM
cana-6025	182	54	⌈	⌈	NOUN
cana-6025	182	55	𝑠	𝑠	ADP
cana-6025	182	56	3	3	NUM
cana-6025	182	57	⌉	⌉	NOUN
cana-6025	182	58	}	}	PUNCT
cana-6025	182	59	with	with	ADP
cana-6025	182	60	cardinality	cardinality	PROPN
cana-6025	182	61	⌈	⌈	PROPN
cana-6025	182	62	𝑠	𝑠	ADP
cana-6025	182	63	3	3	NUM
cana-6025	182	64	⌉.	⌉.	ADV
cana-6025	182	65	since	since	SCONJ
cana-6025	182	66	𝐷5	𝐷5	PROPN
cana-6025	182	67	is	be	AUX
cana-6025	182	68	a	a	DET
cana-6025	182	69	dominating	dominating	NOUN
cana-6025	182	70	set	set	NOUN
cana-6025	182	71	and	and	CCONJ
cana-6025	182	72	⟨𝑉	⟨𝑉	PROPN
cana-6025	182	73	−	−	PROPN
cana-6025	182	74	𝐷5⟩	𝐷5⟩	PRON
cana-6025	182	75	has	have	VERB
cana-6025	182	76	no	no	DET
cana-6025	182	77	isolated	isolated	ADJ
cana-6025	182	78	vertices	vertex	NOUN
cana-6025	182	79	,	,	PUNCT
cana-6025	182	80	𝐷5	𝐷5	PROPN
cana-6025	182	81	is	be	AUX
cana-6025	182	82	a	a	DET
cana-6025	182	83	restrained	restrained	ADJ
cana-6025	182	84	dominating	dominating	NOUN
cana-6025	182	85	set	set	NOUN
cana-6025	182	86	.	.	PUNCT
cana-6025	183	1	thus	thus	ADV
cana-6025	183	2	,	,	PUNCT
cana-6025	183	3	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	183	4	⊠	⊠	PROPN
cana-6025	183	5	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	183	6	)	)	PUNCT
cana-6025	183	7	≤	≤	NOUN
cana-6025	184	1	|𝐷5|	|𝐷5|	NOUN
cana-6025	184	2	=	=	PUNCT
cana-6025	184	3	⌈	⌈	NOUN
cana-6025	184	4	𝑠	𝑠	ADP
cana-6025	184	5	3	3	NUM
cana-6025	184	6	⌉.	⌉.	ADV
cana-6025	184	7	since	since	SCONJ
cana-6025	184	8	𝛾(𝐾1,𝑟	𝛾(𝐾1,𝑟	VERB
cana-6025	184	9	⊠	⊠	PROPN
cana-6025	184	10	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	184	11	)	)	PUNCT
cana-6025	184	12	=	=	PUNCT
cana-6025	185	1	⌈	⌈	NOUN
cana-6025	185	2	𝑠	𝑠	ADP
cana-6025	185	3	3	3	NUM
cana-6025	185	4	⌉	⌉	NOUN
cana-6025	185	5	,	,	PUNCT
cana-6025	185	6	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	185	7	⊠	⊠	PROPN
cana-6025	185	8	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	185	9	)	)	PUNCT
cana-6025	185	10	≥	≥	NOUN
cana-6025	185	11	⌈	⌈	NOUN
cana-6025	185	12	𝑠	𝑠	ADP
cana-6025	185	13	3	3	NUM
cana-6025	185	14	⌉.	⌉.	ADV
cana-6025	185	15	therefore	therefore	ADV
cana-6025	185	16	,	,	PUNCT
cana-6025	185	17	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	PROPN
cana-6025	185	18	⊠	⊠	PROPN
cana-6025	185	19	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	185	20	)	)	PUNCT
cana-6025	185	21	=	=	PUNCT
cana-6025	186	1	⌈	⌈	NOUN
cana-6025	186	2	𝑠	𝑠	ADP
cana-6025	186	3	3	3	NUM
cana-6025	186	4	⌉.	⌉.	ADV
cana-6025	186	5	but	but	CCONJ
cana-6025	186	6	every	every	DET
cana-6025	186	7	minimum	minimum	NOUN
cana-6025	186	8	restrained	restrain	VERB
cana-6025	186	9	dominating	dominating	NOUN
cana-6025	186	10	set	set	NOUN
cana-6025	186	11	is	be	AUX
cana-6025	186	12	independent	independent	ADJ
cana-6025	186	13	and	and	CCONJ
cana-6025	186	14	so	so	ADV
cana-6025	186	15	,	,	PUNCT
cana-6025	186	16	𝜒(⟨𝐷5⟩	𝜒(⟨𝐷5⟩	NUM
cana-6025	186	17	)	)	PUNCT
cana-6025	186	18	=	=	SYM
cana-6025	186	19	1	1	NUM
cana-6025	186	20	≠	≠	PROPN
cana-6025	186	21	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	VERB
cana-6025	186	22	⊠	⊠	PROPN
cana-6025	186	23	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	186	24	)	)	PUNCT
cana-6025	186	25	.	.	PUNCT
cana-6025	187	1	this	this	PRON
cana-6025	187	2	indicates	indicate	VERB
cana-6025	187	3	that	that	SCONJ
cana-6025	187	4	,	,	PUNCT
cana-6025	187	5	𝐷5	𝐷5	PROPN
cana-6025	187	6	is	be	AUX
cana-6025	187	7	not	not	PART
cana-6025	187	8	a	a	DET
cana-6025	187	9	chromatic	chromatic	ADJ
cana-6025	187	10	restrained	restrain	VERB
cana-6025	187	11	dominating	dominating	NOUN
cana-6025	187	12	set	set	NOUN
cana-6025	187	13	.	.	PUNCT
cana-6025	188	1	consider	consider	VERB
cana-6025	188	2	𝐷6	𝐷6	NOUN
cana-6025	188	3	=	=	SYM
cana-6025	188	4	𝐷5	𝐷5	PROPN
cana-6025	188	5	∪	∪	X
cana-6025	188	6	{	{	PUNCT
cana-6025	188	7	(	(	PUNCT
cana-6025	188	8	𝑢0	𝑢0	PROPN
cana-6025	188	9	,	,	PUNCT
cana-6025	188	10	𝑣3	𝑣3	ADJ
cana-6025	188	11	)	)	PUNCT
cana-6025	188	12	,	,	PUNCT
cana-6025	188	13	(	(	PUNCT
cana-6025	188	14	𝑢1	𝑢1	PROPN
cana-6025	188	15	,	,	PUNCT
cana-6025	188	16	𝑣2	𝑣2	PROPN
cana-6025	188	17	)	)	PUNCT
cana-6025	188	18	,	,	PUNCT
cana-6025	188	19	(	(	PUNCT
cana-6025	188	20	𝑢1	𝑢1	NOUN
cana-6025	188	21	,	,	PUNCT
cana-6025	188	22	𝑣3	𝑣3	ADJ
cana-6025	188	23	)	)	PUNCT
cana-6025	188	24	}	}	PUNCT
cana-6025	188	25	.	.	PUNCT
cana-6025	189	1	then	then	ADV
cana-6025	189	2	⟨𝐷6⟩	⟨𝐷6⟩	PROPN
cana-6025	189	3	contains	contain	VERB
cana-6025	189	4	𝐾4	𝐾4	NOUN
cana-6025	189	5	as	as	ADP
cana-6025	189	6	an	an	DET
cana-6025	189	7	induced	induced	ADJ
cana-6025	189	8	subgraph	subgraph	NOUN
cana-6025	189	9	and	and	CCONJ
cana-6025	189	10	so	so	ADV
cana-6025	189	11	,	,	PUNCT
cana-6025	189	12	𝜒(⟨𝐷6⟩	𝜒(⟨𝐷6⟩	PROPN
cana-6025	189	13	)	)	PUNCT
cana-6025	189	14	=	=	SYM
cana-6025	189	15	4	4	NUM
cana-6025	189	16	=	=	SYM
cana-6025	189	17	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	NUM
cana-6025	189	18	⊠	⊠	PROPN
cana-6025	189	19	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	189	20	)	)	PUNCT
cana-6025	189	21	.	.	PUNCT
cana-6025	190	1	also	also	ADV
cana-6025	190	2	,	,	PUNCT
cana-6025	190	3	𝐷6	𝐷6	NOUN
cana-6025	190	4	is	be	AUX
cana-6025	190	5	a	a	DET
cana-6025	190	6	restrained	restrain	VERB
cana-6025	190	7	dominating	dominating	NOUN
cana-6025	190	8	set	set	NOUN
cana-6025	190	9	.	.	PUNCT
cana-6025	191	1	therefore	therefore	ADV
cana-6025	191	2	,	,	PUNCT
cana-6025	191	3	𝐷6	𝐷6	NOUN
cana-6025	191	4	is	be	AUX
cana-6025	191	5	a	a	DET
cana-6025	191	6	chromatic	chromatic	ADJ
cana-6025	191	7	restrained	restrain	VERB
cana-6025	191	8	dominating	dominating	NOUN
cana-6025	191	9	set	set	NOUN
cana-6025	191	10	of	of	ADP
cana-6025	191	11	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	191	12	⊠	⊠	PROPN
cana-6025	191	13	𝑃𝑠.	𝑃𝑠.	PROPN
cana-6025	191	14	thus	thus	ADV
cana-6025	191	15	,	,	PUNCT
cana-6025	191	16	𝛾𝑟	𝛾𝑟	SCONJ
cana-6025	191	17	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	PROPN
cana-6025	191	18	⊠	⊠	PROPN
cana-6025	191	19	communications	communication	NOUN
cana-6025	191	20	on	on	ADP
cana-6025	191	21	applied	apply	VERB
cana-6025	191	22	nonlinear	nonlinear	ADJ
cana-6025	191	23	analysis	analysis	NOUN
cana-6025	191	24	issn	issn	NOUN
cana-6025	191	25	:	:	PUNCT
cana-6025	191	26	1074	1074	NUM
cana-6025	191	27	-	-	PUNCT
cana-6025	191	28	133x	133x	NUM
cana-6025	191	29	vol	vol	NOUN
cana-6025	191	30	31	31	NUM
cana-6025	191	31	no	no	NOUN
cana-6025	191	32	.	.	NOUN
cana-6025	191	33	2	2	NUM
cana-6025	191	34	(	(	PUNCT
cana-6025	191	35	2024	2024	NUM
cana-6025	191	36	)	)	PUNCT
cana-6025	191	37	496	496	NUM
cana-6025	191	38	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	191	39	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	191	40	)	)	PUNCT
cana-6025	191	41	≤	≤	NOUN
cana-6025	191	42	|𝐷6|	|𝐷6|	X
cana-6025	192	1	=	=	SYM
cana-6025	192	2	⌈	⌈	NOUN
cana-6025	192	3	𝑠	𝑠	ADP
cana-6025	192	4	3	3	NUM
cana-6025	192	5	⌉	⌉	NOUN
cana-6025	192	6	+	+	CCONJ
cana-6025	192	7	3	3	NUM
cana-6025	192	8	=	=	SYM
cana-6025	192	9	⌈	⌈	NOUN
cana-6025	192	10	𝑠−4	𝑠−4	ADP
cana-6025	192	11	3	3	NUM
cana-6025	192	12	⌉	⌉	NOUN
cana-6025	192	13	+	+	NOUN
cana-6025	192	14	4	4	X
cana-6025	192	15	.	.	PUNCT
cana-6025	192	16	since	since	SCONJ
cana-6025	192	17	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	PROPN
cana-6025	192	18	⊠	⊠	PROPN
cana-6025	192	19	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	192	20	)	)	PUNCT
cana-6025	192	21	=	=	SYM
cana-6025	192	22	4	4	NUM
cana-6025	192	23	and	and	CCONJ
cana-6025	192	24	any	any	DET
cana-6025	192	25	𝐾1,𝑟	𝐾1,𝑟	NUM
cana-6025	192	26	⊠	⊠	PROPN
cana-6025	192	27	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	192	28	contains	contain	VERB
cana-6025	192	29	𝐾4	𝐾4	NOUN
cana-6025	192	30	as	as	ADP
cana-6025	192	31	an	an	DET
cana-6025	192	32	induced	induced	ADJ
cana-6025	192	33	subgraph	subgraph	NOUN
cana-6025	192	34	,	,	PUNCT
cana-6025	192	35	any	any	DET
cana-6025	192	36	chromatic	chromatic	ADJ
cana-6025	192	37	restrained	restrain	VERB
cana-6025	192	38	dominating	dominating	NOUN
cana-6025	192	39	set	set	NOUN
cana-6025	192	40	must	must	AUX
cana-6025	192	41	contain	contain	VERB
cana-6025	192	42	all	all	DET
cana-6025	192	43	the	the	DET
cana-6025	192	44	four	four	NUM
cana-6025	192	45	vertices	vertex	NOUN
cana-6025	192	46	of	of	ADP
cana-6025	192	47	𝐾4	𝐾4	NOUN
cana-6025	192	48	so	so	SCONJ
cana-6025	192	49	that	that	SCONJ
cana-6025	192	50	,	,	PUNCT
cana-6025	192	51	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	PRON
cana-6025	192	52	⊠	⊠	PROPN
cana-6025	192	53	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	192	54	)	)	PUNCT
cana-6025	192	55	=	=	SYM
cana-6025	192	56	4	4	X
cana-6025	192	57	.	.	PUNCT
cana-6025	192	58	let	let	VERB
cana-6025	192	59	it	it	PRON
cana-6025	192	60	be	be	AUX
cana-6025	192	61	(	(	PUNCT
cana-6025	192	62	𝑢0	𝑢0	PROPN
cana-6025	192	63	,	,	PUNCT
cana-6025	192	64	𝑣2	𝑣2	PROPN
cana-6025	192	65	)	)	PUNCT
cana-6025	192	66	,	,	PUNCT
cana-6025	192	67	(	(	PUNCT
cana-6025	192	68	𝑢0	𝑢0	NOUN
cana-6025	192	69	,	,	PUNCT
cana-6025	192	70	𝑣3	𝑣3	ADJ
cana-6025	192	71	)	)	PUNCT
cana-6025	192	72	,	,	PUNCT
cana-6025	192	73	(	(	PUNCT
cana-6025	192	74	𝑢1	𝑢1	PROPN
cana-6025	192	75	,	,	PUNCT
cana-6025	192	76	𝑣2	𝑣2	PROPN
cana-6025	192	77	)	)	PUNCT
cana-6025	192	78	,	,	PUNCT
cana-6025	192	79	(	(	PUNCT
cana-6025	192	80	𝑢1	𝑢1	NOUN
cana-6025	192	81	,	,	PUNCT
cana-6025	192	82	𝑣3	𝑣3	ADJ
cana-6025	192	83	)	)	PUNCT
cana-6025	192	84	which	which	PRON
cana-6025	192	85	dominates	dominate	VERB
cana-6025	192	86	all	all	DET
cana-6025	192	87	the	the	DET
cana-6025	192	88	vertices	vertex	NOUN
cana-6025	192	89	in	in	ADP
cana-6025	192	90	columns	column	NOUN
cana-6025	192	91	𝑉1	𝑉1	PROPN
cana-6025	192	92	,	,	PUNCT
cana-6025	192	93	𝑉2	𝑉2	NOUN
cana-6025	192	94	,	,	PUNCT
cana-6025	192	95	𝑉3	𝑉3	NOUN
cana-6025	192	96	,	,	PUNCT
cana-6025	192	97	𝑉4	𝑉4	NOUN
cana-6025	192	98	.	.	PUNCT
cana-6025	193	1	from	from	ADP
cana-6025	193	2	the	the	DET
cana-6025	193	3	remaining	remain	VERB
cana-6025	193	4	𝑠	𝑠	PROPN
cana-6025	193	5	−	−	PROPN
cana-6025	193	6	4	4	NUM
cana-6025	193	7	columns	column	NOUN
cana-6025	193	8	,	,	PUNCT
cana-6025	193	9	choosing	choose	VERB
cana-6025	193	10	the	the	DET
cana-6025	193	11	first	first	ADJ
cana-6025	193	12	vertex	vertex	NOUN
cana-6025	193	13	of	of	ADP
cana-6025	193	14	each	each	DET
cana-6025	193	15	column	column	NOUN
cana-6025	193	16	𝑉3𝑘+2	𝑉3𝑘+2	PROPN
cana-6025	193	17	,	,	PUNCT
cana-6025	193	18	1	1	NUM
cana-6025	193	19	≤	≤	NOUN
cana-6025	193	20	𝑘	𝑘	PRON
cana-6025	193	21	≤	≤	NUM
cana-6025	193	22	⌈	⌈	NOUN
cana-6025	193	23	𝑠−4	𝑠−4	ADP
cana-6025	193	24	3	3	NUM
cana-6025	193	25	⌉	⌉	NOUN
cana-6025	193	26	,	,	PUNCT
cana-6025	193	27	we	we	PRON
cana-6025	193	28	get	get	VERB
cana-6025	193	29	a	a	DET
cana-6025	193	30	minimum	minimum	ADJ
cana-6025	193	31	chromatic	chromatic	ADJ
cana-6025	193	32	restrained	restrain	VERB
cana-6025	193	33	dominating	dominating	NOUN
cana-6025	193	34	set	set	NOUN
cana-6025	193	35	.	.	PUNCT
cana-6025	194	1	thus	thus	ADV
cana-6025	194	2	,	,	PUNCT
cana-6025	194	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	194	4	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	194	5	⊠	⊠	PROPN
cana-6025	194	6	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	194	7	)	)	PUNCT
cana-6025	194	8	≥	≥	NOUN
cana-6025	194	9	⌈	⌈	NOUN
cana-6025	194	10	𝑠−4	𝑠−4	ADP
cana-6025	194	11	3	3	NUM
cana-6025	194	12	⌉	⌉	NOUN
cana-6025	194	13	+	+	NOUN
cana-6025	194	14	4	4	X
cana-6025	194	15	.	.	X
cana-6025	194	16	therefore	therefore	ADV
cana-6025	194	17	,	,	PUNCT
cana-6025	194	18	𝛾𝑟	𝛾𝑟	ADP
cana-6025	194	19	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	194	20	⊠	⊠	PROPN
cana-6025	194	21	𝑃𝑠	𝑃𝑠	PROPN
cana-6025	194	22	)	)	PUNCT
cana-6025	194	23	=	=	PUNCT
cana-6025	194	24	⌈	⌈	PROPN
cana-6025	194	25	𝑠−4	𝑠−4	ADP
cana-6025	194	26	3	3	NUM
cana-6025	194	27	⌉	⌉	NOUN
cana-6025	194	28	+	+	NOUN
cana-6025	194	29	4	4	X
cana-6025	194	30	.	.	X
cana-6025	194	31	theorem	theorem	VERB
cana-6025	194	32	2.8	2.8	NUM
cana-6025	194	33	let	let	VERB
cana-6025	194	34	𝑟	𝑟	NOUN
cana-6025	194	35	,	,	PUNCT
cana-6025	194	36	𝑠	𝑠	PRON
cana-6025	194	37	≥	≥	NUM
cana-6025	194	38	2	2	NUM
cana-6025	194	39	.	.	PUNCT
cana-6025	194	40	then	then	ADV
cana-6025	194	41	𝛾𝑟	𝛾𝑟	ADP
cana-6025	194	42	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	194	43	⊠	⊠	PROPN
cana-6025	194	44	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	194	45	)	)	PUNCT
cana-6025	195	1	=	=	SYM
cana-6025	195	2	4	4	X
cana-6025	195	3	.	.	X
cana-6025	195	4	proof	proof	NOUN
cana-6025	195	5	.	.	PUNCT
cana-6025	196	1	let	let	VERB
cana-6025	196	2	𝑉(𝐾1,𝑟	𝑉(𝐾1,𝑟	NOUN
cana-6025	196	3	)	)	PUNCT
cana-6025	197	1	=	=	PRON
cana-6025	197	2	{	{	PUNCT
cana-6025	197	3	𝑢0	𝑢0	PROPN
cana-6025	197	4	,	,	PUNCT
cana-6025	197	5	𝑢1	𝑢1	NOUN
cana-6025	197	6	,	,	PUNCT
cana-6025	197	7	𝑢2	𝑢2	PROPN
cana-6025	197	8	,	,	PUNCT
cana-6025	197	9	…	…	PUNCT
cana-6025	197	10	,	,	PUNCT
cana-6025	197	11	𝑢𝑟	𝑢𝑟	ADP
cana-6025	197	12	}	}	PUNCT
cana-6025	197	13	and	and	CCONJ
cana-6025	197	14	𝑉(𝐾1,𝑠	𝑉(𝐾1,𝑠	NOUN
cana-6025	197	15	)	)	PUNCT
cana-6025	198	1	=	=	PRON
cana-6025	198	2	{	{	PUNCT
cana-6025	198	3	𝑣0	𝑣0	PROPN
cana-6025	198	4	,	,	PUNCT
cana-6025	198	5	𝑣1	𝑣1	PROPN
cana-6025	198	6	,	,	PUNCT
cana-6025	198	7	𝑣2	𝑣2	PROPN
cana-6025	198	8	,	,	PUNCT
cana-6025	198	9	…	…	PUNCT
cana-6025	198	10	,	,	PUNCT
cana-6025	198	11	𝑣𝑠	𝑣𝑠	ADP
cana-6025	198	12	}	}	PUNCT
cana-6025	198	13	.	.	PUNCT
cana-6025	199	1	then	then	ADV
cana-6025	199	2	,	,	PUNCT
cana-6025	199	3	𝑉(𝐾1,𝑟	𝑉(𝐾1,𝑟	PROPN
cana-6025	199	4	⊠	⊠	PROPN
cana-6025	199	5	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	199	6	)	)	PUNCT
cana-6025	200	1	=	=	PRON
cana-6025	200	2	{	{	PUNCT
cana-6025	200	3	(	(	PUNCT
cana-6025	200	4	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	200	5	,	,	PUNCT
cana-6025	200	6	𝑣𝑗)/0	𝑣𝑗)/0	NOUN
cana-6025	200	7	≤	≤	NOUN
cana-6025	200	8	𝑖	𝑖	SYM
cana-6025	200	9	≤	≤	NOUN
cana-6025	200	10	𝑟	𝑟	NOUN
cana-6025	200	11	,	,	PUNCT
cana-6025	200	12	0	0	NUM
cana-6025	200	13	≤	≤	NUM
cana-6025	201	1	𝑗	𝑗	PRON
cana-6025	201	2	≤	≤	NUM
cana-6025	201	3	𝑠	𝑠	NOUN
cana-6025	201	4	}	}	PUNCT
cana-6025	201	5	where	where	SCONJ
cana-6025	201	6	(	(	PUNCT
cana-6025	201	7	𝑢0	𝑢0	PROPN
cana-6025	201	8	,	,	PUNCT
cana-6025	201	9	𝑣0	𝑣0	NOUN
cana-6025	201	10	)	)	PUNCT
cana-6025	201	11	is	be	AUX
cana-6025	201	12	the	the	DET
cana-6025	201	13	full	full	ADJ
cana-6025	201	14	degree	degree	NOUN
cana-6025	201	15	vertex	vertex	NOUN
cana-6025	201	16	of	of	ADP
cana-6025	201	17	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	201	18	⊠	⊠	PROPN
cana-6025	201	19	𝐾1,𝑠.	𝐾1,𝑠.	VERB
cana-6025	201	20	clearly	clearly	ADV
cana-6025	201	21	,	,	PUNCT
cana-6025	201	22	𝐷	𝐷	PROPN
cana-6025	201	23	=	=	SYM
cana-6025	201	24	{	{	PUNCT
cana-6025	201	25	(	(	PUNCT
cana-6025	201	26	𝑢0	𝑢0	PROPN
cana-6025	201	27	,	,	PUNCT
cana-6025	201	28	𝑣0	𝑣0	NOUN
cana-6025	201	29	)	)	PUNCT
cana-6025	201	30	}	}	PUNCT
cana-6025	201	31	is	be	AUX
cana-6025	201	32	a	a	DET
cana-6025	201	33	restrained	restrain	VERB
cana-6025	201	34	dominating	dominating	NOUN
cana-6025	201	35	set	set	NOUN
cana-6025	201	36	and	and	CCONJ
cana-6025	201	37	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	NOUN
cana-6025	201	38	⊠	⊠	PROPN
cana-6025	201	39	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	201	40	)	)	PUNCT
cana-6025	202	1	=	=	SYM
cana-6025	202	2	1	1	X
cana-6025	202	3	.	.	PUNCT
cana-6025	203	1	let	let	VERB
cana-6025	203	2	𝑉1	𝑉1	NOUN
cana-6025	203	3	,	,	PUNCT
cana-6025	203	4	𝑉2	𝑉2	NOUN
cana-6025	203	5	,	,	PUNCT
cana-6025	203	6	𝑉3	𝑉3	NOUN
cana-6025	203	7	,	,	PUNCT
cana-6025	203	8	.	.	PUNCT
cana-6025	203	9	.	.	PUNCT
cana-6025	204	1	.	.	PUNCT
cana-6025	205	1	,	,	PUNCT
cana-6025	205	2	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	205	3	denotes	denote	VERB
cana-6025	205	4	the	the	DET
cana-6025	205	5	𝑟	𝑟	NOUN
cana-6025	205	6	+	+	CCONJ
cana-6025	205	7	1	1	NUM
cana-6025	205	8	rows	row	NOUN
cana-6025	205	9	of	of	ADP
cana-6025	205	10	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	205	11	⊠	⊠	PROPN
cana-6025	205	12	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	205	13	where	where	SCONJ
cana-6025	205	14	𝑉1	𝑉1	PROPN
cana-6025	205	15	can	can	AUX
cana-6025	205	16	be	be	AUX
cana-6025	205	17	colored	color	VERB
cana-6025	205	18	with	with	ADP
cana-6025	205	19	two	two	NUM
cana-6025	205	20	colors	color	NOUN
cana-6025	205	21	and	and	CCONJ
cana-6025	205	22	the	the	DET
cana-6025	205	23	remaining	remain	VERB
cana-6025	205	24	𝑟	𝑟	NOUN
cana-6025	205	25	rows	row	NOUN
cana-6025	205	26	can	can	AUX
cana-6025	205	27	be	be	AUX
cana-6025	205	28	colored	color	VERB
cana-6025	205	29	with	with	ADP
cana-6025	205	30	another	another	DET
cana-6025	205	31	two	two	NUM
cana-6025	205	32	colors	color	NOUN
cana-6025	205	33	.	.	PUNCT
cana-6025	206	1	thus	thus	ADV
cana-6025	206	2	,	,	PUNCT
cana-6025	206	3	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	PROPN
cana-6025	206	4	⊠	⊠	PROPN
cana-6025	206	5	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	206	6	)	)	PUNCT
cana-6025	207	1	=	=	PUNCT
cana-6025	207	2	4	4	X
cana-6025	207	3	.	.	PUNCT
cana-6025	207	4	but	but	CCONJ
cana-6025	207	5	,	,	PUNCT
cana-6025	207	6	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	207	7	)	)	PUNCT
cana-6025	207	8	=	=	SYM
cana-6025	207	9	1	1	NUM
cana-6025	207	10	≠	≠	PROPN
cana-6025	207	11	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	VERB
cana-6025	207	12	⊠	⊠	PROPN
cana-6025	207	13	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	207	14	)	)	PUNCT
cana-6025	207	15	and	and	CCONJ
cana-6025	207	16	so	so	ADV
cana-6025	207	17	,	,	PUNCT
cana-6025	207	18	𝐷	𝐷	PROPN
cana-6025	207	19	is	be	AUX
cana-6025	207	20	not	not	PART
cana-6025	207	21	a	a	DET
cana-6025	207	22	chromatic	chromatic	ADJ
cana-6025	207	23	restrained	restrain	VERB
cana-6025	207	24	dominating	dominating	NOUN
cana-6025	207	25	set	set	NOUN
cana-6025	207	26	.	.	PUNCT
cana-6025	208	1	let	let	VERB
cana-6025	208	2	𝐷1	𝐷1	NOUN
cana-6025	208	3	=	=	PRON
cana-6025	208	4	{	{	PUNCT
cana-6025	208	5	(	(	PUNCT
cana-6025	208	6	𝑢0	𝑢0	PROPN
cana-6025	208	7	,	,	PUNCT
cana-6025	208	8	𝑣0	𝑣0	PROPN
cana-6025	208	9	)	)	PUNCT
cana-6025	208	10	,	,	PUNCT
cana-6025	208	11	(	(	PUNCT
cana-6025	208	12	𝑢0	𝑢0	PROPN
cana-6025	208	13	,	,	PUNCT
cana-6025	208	14	𝑣1	𝑣1	PROPN
cana-6025	208	15	)	)	PUNCT
cana-6025	208	16	,	,	PUNCT
cana-6025	208	17	(	(	PUNCT
cana-6025	208	18	𝑢1	𝑢1	PROPN
cana-6025	208	19	,	,	PUNCT
cana-6025	208	20	𝑣0	𝑣0	NOUN
cana-6025	208	21	)	)	PUNCT
cana-6025	208	22	,	,	PUNCT
cana-6025	208	23	(	(	PUNCT
cana-6025	208	24	𝑢1	𝑢1	PROPN
cana-6025	208	25	,	,	PUNCT
cana-6025	208	26	𝑣1	𝑣1	NOUN
cana-6025	208	27	)	)	PUNCT
cana-6025	208	28	}	}	PUNCT
cana-6025	208	29	where	where	SCONJ
cana-6025	208	30	⟨𝐷1⟩	⟨𝐷1⟩	NOUN
cana-6025	208	31	=	=	SYM
cana-6025	208	32	𝐾4	𝐾4	NOUN
cana-6025	208	33	.	.	PUNCT
cana-6025	209	1	this	this	PRON
cana-6025	209	2	implies	imply	VERB
cana-6025	209	3	that	that	SCONJ
cana-6025	209	4	,	,	PUNCT
cana-6025	209	5	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	209	6	)	)	PUNCT
cana-6025	209	7	=	=	SYM
cana-6025	209	8	4	4	NUM
cana-6025	209	9	and	and	CCONJ
cana-6025	209	10	𝐷1	𝐷1	NOUN
cana-6025	209	11	is	be	AUX
cana-6025	209	12	also	also	ADV
cana-6025	209	13	a	a	DET
cana-6025	209	14	restrained	restrained	ADJ
cana-6025	209	15	dominating	dominating	NOUN
cana-6025	209	16	set	set	NOUN
cana-6025	209	17	.	.	PUNCT
cana-6025	210	1	thus	thus	ADV
cana-6025	210	2	,	,	PUNCT
cana-6025	210	3	𝐷1	𝐷1	PROPN
cana-6025	210	4	is	be	AUX
cana-6025	210	5	a	a	DET
cana-6025	210	6	chromatic	chromatic	ADJ
cana-6025	210	7	restrained	restrain	VERB
cana-6025	210	8	dominating	dominating	NOUN
cana-6025	210	9	set	set	NOUN
cana-6025	210	10	of	of	ADP
cana-6025	210	11	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	210	12	⊠	⊠	PROPN
cana-6025	210	13	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	210	14	and	and	CCONJ
cana-6025	210	15	𝛾𝑟	𝛾𝑟	ADP
cana-6025	210	16	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	210	17	⊠	⊠	PROPN
cana-6025	210	18	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	210	19	)	)	PUNCT
cana-6025	210	20	≤	≤	NUM
cana-6025	210	21	|𝐷1|	|𝐷1|	NUM
cana-6025	210	22	=	=	SYM
cana-6025	210	23	4	4	X
cana-6025	210	24	.	.	PUNCT
cana-6025	211	1	since	since	SCONJ
cana-6025	211	2	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	PROPN
cana-6025	211	3	⊠	⊠	PROPN
cana-6025	211	4	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	211	5	)	)	PUNCT
cana-6025	211	6	=	=	SYM
cana-6025	211	7	4	4	NUM
cana-6025	211	8	,	,	PUNCT
cana-6025	211	9	any	any	DET
cana-6025	211	10	minimum	minimum	ADJ
cana-6025	211	11	chromatic	chromatic	ADJ
cana-6025	211	12	restrained	restrain	VERB
cana-6025	211	13	dominating	dominating	NOUN
cana-6025	211	14	set	set	NOUN
cana-6025	211	15	must	must	AUX
cana-6025	211	16	contain	contain	VERB
cana-6025	211	17	at	at	ADV
cana-6025	211	18	least	least	ADJ
cana-6025	211	19	four	four	NUM
cana-6025	211	20	vertices	vertex	NOUN
cana-6025	211	21	and	and	CCONJ
cana-6025	211	22	so	so	ADV
cana-6025	211	23	,	,	PUNCT
cana-6025	211	24	𝛾𝑟	𝛾𝑟	ADP
cana-6025	211	25	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	211	26	⊠	⊠	PROPN
cana-6025	211	27	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	211	28	)	)	PUNCT
cana-6025	211	29	≥	≥	NOUN
cana-6025	211	30	4	4	NUM
cana-6025	211	31	.	.	PUNCT
cana-6025	212	1	therefore	therefore	ADV
cana-6025	212	2	,	,	PUNCT
cana-6025	212	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	212	4	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	212	5	⊠	⊠	PROPN
cana-6025	212	6	𝐾1,𝑠	𝐾1,𝑠	PROPN
cana-6025	212	7	)	)	PUNCT
cana-6025	212	8	=	=	SYM
cana-6025	212	9	4	4	X
cana-6025	212	10	.	.	PUNCT
cana-6025	212	11	theorem	theorem	VERB
cana-6025	212	12	2.9	2.9	NUM
cana-6025	212	13	for	for	ADP
cana-6025	212	14	any	any	DET
cana-6025	212	15	𝑚	𝑚	NOUN
cana-6025	212	16	,	,	PUNCT
cana-6025	212	17	𝑟	𝑟	NOUN
cana-6025	212	18	,	,	PUNCT
cana-6025	212	19	𝑠	𝑠	PRON
cana-6025	212	20	≥	≥	NUM
cana-6025	212	21	2	2	NUM
cana-6025	212	22	,	,	PUNCT
cana-6025	212	23	𝛾𝑟	𝛾𝑟	ADP
cana-6025	212	24	𝑐(𝐾1,𝑚	𝑐(𝐾1,𝑚	PROPN
cana-6025	212	25	⊠	⊠	PROPN
cana-6025	212	26	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	212	27	)	)	PUNCT
cana-6025	213	1	=	=	SYM
cana-6025	213	2	4	4	X
cana-6025	213	3	.	.	X
cana-6025	213	4	proof	proof	NOUN
cana-6025	213	5	.	.	PUNCT
cana-6025	214	1	let	let	VERB
cana-6025	214	2	𝑉(𝐾1,𝑚	𝑉(𝐾1,𝑚	NOUN
cana-6025	214	3	)	)	PUNCT
cana-6025	214	4	=	=	PRON
cana-6025	214	5	{	{	PUNCT
cana-6025	214	6	𝑢0	𝑢0	PROPN
cana-6025	214	7	,	,	PUNCT
cana-6025	214	8	𝑢1	𝑢1	NOUN
cana-6025	214	9	,	,	PUNCT
cana-6025	214	10	𝑢2	𝑢2	PROPN
cana-6025	214	11	,	,	PUNCT
cana-6025	214	12	…	…	PUNCT
cana-6025	214	13	,	,	PUNCT
cana-6025	214	14	𝑢𝑚	𝑢𝑚	NOUN
cana-6025	214	15	}	}	PUNCT
cana-6025	214	16	and	and	CCONJ
cana-6025	214	17	𝑉(𝐾𝑟,𝑠	𝑉(𝐾𝑟,𝑠	ADJ
cana-6025	214	18	)	)	PUNCT
cana-6025	215	1	=	=	PRON
cana-6025	215	2	{	{	PUNCT
cana-6025	215	3	𝑣1	𝑣1	PROPN
cana-6025	215	4	,	,	PUNCT
cana-6025	215	5	𝑣2	𝑣2	PROPN
cana-6025	215	6	,	,	PUNCT
cana-6025	215	7	𝑣3	𝑣3	ADJ
cana-6025	215	8	,	,	PUNCT
cana-6025	215	9	…	…	PUNCT
cana-6025	215	10	,	,	PUNCT
cana-6025	215	11	𝑣𝑟	𝑣𝑟	INTJ
cana-6025	215	12	,	,	PUNCT
cana-6025	215	13	𝑣𝑟+1	𝑣𝑟+1	ADV
cana-6025	215	14	,	,	PUNCT
cana-6025	215	15	𝑣𝑟+2	𝑣𝑟+2	NUM
cana-6025	215	16	,	,	PUNCT
cana-6025	215	17	…	…	PUNCT
cana-6025	215	18	,	,	PUNCT
cana-6025	215	19	𝑣𝑟+𝑠	𝑣𝑟+𝑠	NOUN
cana-6025	215	20	}	}	PUNCT
cana-6025	215	21	where	where	SCONJ
cana-6025	215	22	𝑢0	𝑢0	PROPN
cana-6025	215	23	is	be	AUX
cana-6025	215	24	the	the	DET
cana-6025	215	25	full	full	ADJ
cana-6025	215	26	degree	degree	NOUN
cana-6025	215	27	vertex	vertex	NOUN
cana-6025	215	28	of	of	ADP
cana-6025	215	29	𝐾1,𝑚.	𝐾1,𝑚.	PUNCT
cana-6025	215	30	now	now	ADV
cana-6025	215	31	,	,	PUNCT
cana-6025	215	32	𝑉(𝐾1,𝑚	𝑉(𝐾1,𝑚	VERB
cana-6025	215	33	⊠	⊠	PROPN
cana-6025	215	34	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	215	35	)	)	PUNCT
cana-6025	216	1	=	=	PRON
cana-6025	216	2	{	{	PUNCT
cana-6025	216	3	(	(	PUNCT
cana-6025	216	4	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	216	5	,	,	PUNCT
cana-6025	216	6	𝑣𝑗)/0	𝑣𝑗)/0	NOUN
cana-6025	216	7	≤	≤	NOUN
cana-6025	216	8	𝑖	𝑖	SYM
cana-6025	216	9	≤	≤	NOUN
cana-6025	216	10	𝑚	𝑚	ADP
cana-6025	216	11	,	,	PUNCT
cana-6025	216	12	1	1	NUM
cana-6025	216	13	≤	≤	NUM
cana-6025	216	14	𝑗	𝑗	PRON
cana-6025	216	15	≤	≤	ADJ
cana-6025	216	16	𝑟	𝑟	NOUN
cana-6025	216	17	+	+	CCONJ
cana-6025	216	18	𝑠	𝑠	NOUN
cana-6025	216	19	}	}	PUNCT
cana-6025	216	20	and	and	CCONJ
cana-6025	216	21	|𝑉(𝐾1,𝑚	|𝑉(𝐾1,𝑚	PROPN
cana-6025	216	22	⊠	⊠	PROPN
cana-6025	216	23	𝐾𝑟,𝑠)|	𝐾𝑟,𝑠)|	NOUN
cana-6025	216	24	=	=	PUNCT
cana-6025	216	25	(	(	PUNCT
cana-6025	216	26	𝑚	𝑚	PROPN
cana-6025	216	27	+	+	NOUN
cana-6025	216	28	1)(𝑟	1)(𝑟	NUM
cana-6025	216	29	+	+	CCONJ
cana-6025	216	30	𝑠	𝑠	NOUN
cana-6025	216	31	)	)	PUNCT
cana-6025	216	32	.	.	PUNCT
cana-6025	217	1	also	also	ADV
cana-6025	217	2	,	,	PUNCT
cana-6025	217	3	there	there	PRON
cana-6025	217	4	exists	exist	VERB
cana-6025	217	5	𝑚	𝑚	X
cana-6025	217	6	+	+	CCONJ
cana-6025	217	7	1	1	NUM
cana-6025	217	8	rows	row	NOUN
cana-6025	217	9	and	and	CCONJ
cana-6025	217	10	𝑟	𝑟	PRON
cana-6025	218	1	+	+	CCONJ
cana-6025	218	2	𝑠	𝑠	PROPN
cana-6025	218	3	columns	column	NOUN
cana-6025	218	4	in	in	ADP
cana-6025	218	5	𝐾1,𝑚	𝐾1,𝑚	DET
cana-6025	218	6	⊠	⊠	PROPN
cana-6025	218	7	𝐾𝑟,𝑠.	𝐾𝑟,𝑠.	PROPN
cana-6025	218	8	clearly	clearly	ADV
cana-6025	218	9	,	,	PUNCT
cana-6025	218	10	the	the	DET
cana-6025	218	11	first	first	ADJ
cana-6025	218	12	row	row	NOUN
cana-6025	218	13	can	can	AUX
cana-6025	218	14	be	be	AUX
cana-6025	218	15	colored	color	VERB
cana-6025	218	16	with	with	ADP
cana-6025	218	17	two	two	NUM
cana-6025	218	18	colors	color	NOUN
cana-6025	218	19	and	and	CCONJ
cana-6025	218	20	the	the	DET
cana-6025	218	21	remaining	remain	VERB
cana-6025	218	22	𝑚	𝑚	PROPN
cana-6025	218	23	rows	row	NOUN
cana-6025	218	24	can	can	AUX
cana-6025	218	25	be	be	AUX
cana-6025	218	26	colored	color	VERB
cana-6025	218	27	with	with	ADP
cana-6025	218	28	another	another	DET
cana-6025	218	29	two	two	NUM
cana-6025	218	30	colors	color	NOUN
cana-6025	218	31	since	since	SCONJ
cana-6025	218	32	there	there	PRON
cana-6025	218	33	does	do	AUX
cana-6025	218	34	not	not	PART
cana-6025	218	35	exists	exist	VERB
cana-6025	218	36	adjacency	adjacency	NOUN
cana-6025	218	37	between	between	ADP
cana-6025	218	38	any	any	DET
cana-6025	218	39	two	two	NUM
cana-6025	218	40	vertices	vertex	NOUN
cana-6025	218	41	belonging	belong	VERB
cana-6025	218	42	to	to	ADP
cana-6025	218	43	those	those	PRON
cana-6025	218	44	𝑚	𝑚	ADP
cana-6025	218	45	different	different	ADJ
cana-6025	218	46	rows	row	NOUN
cana-6025	218	47	.	.	PUNCT
cana-6025	219	1	thus	thus	ADV
cana-6025	219	2	,	,	PUNCT
cana-6025	219	3	𝜒(𝐾1,𝑚	𝜒(𝐾1,𝑚	X
cana-6025	219	4	⊠	⊠	PROPN
cana-6025	219	5	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	219	6	)	)	PUNCT
cana-6025	219	7	=	=	SYM
cana-6025	219	8	4	4	X
cana-6025	219	9	.	.	X
cana-6025	219	10	let	let	VERB
cana-6025	219	11	𝐷	𝐷	NOUN
cana-6025	219	12	=	=	SYM
cana-6025	219	13	{	{	PUNCT
cana-6025	219	14	(	(	PUNCT
cana-6025	219	15	𝑢0	𝑢0	PROPN
cana-6025	219	16	,	,	PUNCT
cana-6025	219	17	𝑣𝑟	𝑣𝑟	NOUN
cana-6025	219	18	)	)	PUNCT
cana-6025	219	19	,	,	PUNCT
cana-6025	219	20	(	(	PUNCT
cana-6025	219	21	𝑢0	𝑢0	PROPN
cana-6025	219	22	,	,	PUNCT
cana-6025	219	23	𝑣𝑟+1	𝑣𝑟+1	NOUN
cana-6025	219	24	)	)	PUNCT
cana-6025	219	25	}	}	PUNCT
cana-6025	219	26	.	.	PUNCT
cana-6025	220	1	then	then	ADV
cana-6025	220	2	𝐷	𝐷	PROPN
cana-6025	220	3	is	be	AUX
cana-6025	220	4	a	a	DET
cana-6025	220	5	restrained	restrained	ADJ
cana-6025	220	6	dominating	dominating	NOUN
cana-6025	220	7	set	set	NOUN
cana-6025	220	8	as	as	ADP
cana-6025	220	9	𝐷	𝐷	PROPN
cana-6025	220	10	is	be	AUX
cana-6025	220	11	a	a	DET
cana-6025	220	12	dominating	dominating	NOUN
cana-6025	220	13	set	set	NOUN
cana-6025	220	14	and	and	CCONJ
cana-6025	220	15	every	every	DET
cana-6025	220	16	vertex	vertex	NOUN
cana-6025	220	17	in	in	ADP
cana-6025	220	18	𝑉	𝑉	PROPN
cana-6025	220	19	−	−	PROPN
cana-6025	220	20	𝐷	𝐷	PROPN
cana-6025	220	21	is	be	AUX
cana-6025	220	22	adjacent	adjacent	ADJ
cana-6025	220	23	to	to	ADP
cana-6025	220	24	at	at	ADV
cana-6025	220	25	least	least	ADV
cana-6025	220	26	one	one	NUM
cana-6025	220	27	another	another	DET
cana-6025	220	28	vertex	vertex	NOUN
cana-6025	220	29	in	in	ADP
cana-6025	220	30	𝑉	𝑉	PROPN
cana-6025	221	1	−	−	PROPN
cana-6025	221	2	𝐷.	𝐷.	PROPN
cana-6025	221	3	thus	thus	ADV
cana-6025	221	4	,	,	PUNCT
cana-6025	221	5	𝛾𝑟(𝐾1,𝑚	𝛾𝑟(𝐾1,𝑚	PROPN
cana-6025	221	6	⊠	⊠	PROPN
cana-6025	221	7	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	221	8	)	)	PUNCT
cana-6025	221	9	≤	≤	NUM
cana-6025	221	10	2	2	NUM
cana-6025	221	11	.	.	PUNCT
cana-6025	222	1	since	since	SCONJ
cana-6025	222	2	there	there	PRON
cana-6025	222	3	does	do	AUX
cana-6025	222	4	not	not	PART
cana-6025	222	5	exists	exist	VERB
cana-6025	222	6	a	a	DET
cana-6025	222	7	full	full	ADJ
cana-6025	222	8	degree	degree	NOUN
cana-6025	222	9	vertex	vertex	NOUN
cana-6025	222	10	in	in	ADP
cana-6025	222	11	𝐾1,𝑚	𝐾1,𝑚	DET
cana-6025	222	12	⊠	⊠	PROPN
cana-6025	222	13	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	222	14	,	,	PUNCT
cana-6025	222	15	𝛾𝑟(𝐾1,𝑚	𝛾𝑟(𝐾1,𝑚	ADJ
cana-6025	222	16	⊠	⊠	PROPN
cana-6025	222	17	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	222	18	)	)	PUNCT
cana-6025	222	19	<	<	X
cana-6025	222	20	2	2	NUM
cana-6025	222	21	is	be	AUX
cana-6025	222	22	impossible	impossible	ADJ
cana-6025	222	23	.	.	PUNCT
cana-6025	223	1	therefore	therefore	ADV
cana-6025	223	2	,	,	PUNCT
cana-6025	223	3	𝛾𝑟(𝐾1,𝑚	𝛾𝑟(𝐾1,𝑚	PROPN
cana-6025	223	4	⊠	⊠	PROPN
cana-6025	223	5	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	223	6	)	)	PUNCT
cana-6025	223	7	=	=	SYM
cana-6025	224	1	2	2	X
cana-6025	224	2	.	.	PUNCT
cana-6025	224	3	but	but	CCONJ
cana-6025	224	4	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	224	5	)	)	PUNCT
cana-6025	224	6	=	=	SYM
cana-6025	224	7	2	2	NUM
cana-6025	224	8	and	and	CCONJ
cana-6025	224	9	so	so	ADV
cana-6025	224	10	,	,	PUNCT
cana-6025	224	11	𝐷	𝐷	PROPN
cana-6025	224	12	is	be	AUX
cana-6025	224	13	not	not	PART
cana-6025	224	14	a	a	DET
cana-6025	224	15	chromatic	chromatic	ADJ
cana-6025	224	16	restrained	restrain	VERB
cana-6025	224	17	dominating	dominating	NOUN
cana-6025	224	18	set	set	NOUN
cana-6025	224	19	of	of	ADP
cana-6025	224	20	𝐾1,𝑚	𝐾1,𝑚	DET
cana-6025	224	21	⊠	⊠	PROPN
cana-6025	224	22	𝐾𝑟,𝑠.	𝐾𝑟,𝑠.	PROPN
cana-6025	224	23	consider	consider	VERB
cana-6025	224	24	𝐷1	𝐷1	NOUN
cana-6025	224	25	=	=	PUNCT
cana-6025	224	26	𝐷	𝐷	PROPN
cana-6025	224	27	∪	∪	NOUN
cana-6025	224	28	communications	communication	NOUN
cana-6025	224	29	on	on	ADP
cana-6025	224	30	applied	apply	VERB
cana-6025	224	31	nonlinear	nonlinear	ADJ
cana-6025	224	32	analysis	analysis	NOUN
cana-6025	224	33	issn	issn	NOUN
cana-6025	224	34	:	:	PUNCT
cana-6025	224	35	1074	1074	NUM
cana-6025	224	36	-	-	PUNCT
cana-6025	224	37	133x	133x	NUM
cana-6025	224	38	vol	vol	NOUN
cana-6025	224	39	31	31	NUM
cana-6025	224	40	no	no	NOUN
cana-6025	224	41	.	.	NOUN
cana-6025	224	42	2	2	NUM
cana-6025	224	43	(	(	PUNCT
cana-6025	224	44	2024	2024	NUM
cana-6025	224	45	)	)	PUNCT
cana-6025	224	46	497	497	NUM
cana-6025	224	47	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	224	48	{	{	PUNCT
cana-6025	224	49	(	(	PUNCT
cana-6025	224	50	𝑢1	𝑢1	PROPN
cana-6025	224	51	,	,	PUNCT
cana-6025	224	52	𝑣𝑟	𝑣𝑟	INTJ
cana-6025	224	53	)	)	PUNCT
cana-6025	224	54	,	,	PUNCT
cana-6025	224	55	(	(	PUNCT
cana-6025	224	56	𝑢1	𝑢1	PROPN
cana-6025	224	57	,	,	PUNCT
cana-6025	224	58	𝑣𝑟+1	𝑣𝑟+1	NOUN
cana-6025	224	59	)	)	PUNCT
cana-6025	224	60	}	}	PUNCT
cana-6025	224	61	.	.	PUNCT
cana-6025	225	1	then	then	ADV
cana-6025	225	2	,	,	PUNCT
cana-6025	225	3	⟨𝐷1⟩	⟨𝐷1⟩	NOUN
cana-6025	225	4	is	be	AUX
cana-6025	225	5	a	a	DET
cana-6025	225	6	complete	complete	ADJ
cana-6025	225	7	graph	graph	NOUN
cana-6025	225	8	on	on	ADP
cana-6025	225	9	four	four	NUM
cana-6025	225	10	vertices	vertex	NOUN
cana-6025	225	11	and	and	CCONJ
cana-6025	225	12	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	225	13	)	)	PUNCT
cana-6025	225	14	=	=	SYM
cana-6025	225	15	4	4	NUM
cana-6025	225	16	=	=	SYM
cana-6025	225	17	𝜒(𝐾1,𝑚	𝜒(𝐾1,𝑚	X
cana-6025	225	18	⊠	⊠	PROPN
cana-6025	225	19	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	225	20	)	)	PUNCT
cana-6025	225	21	.	.	PUNCT
cana-6025	226	1	also	also	ADV
cana-6025	226	2	,	,	PUNCT
cana-6025	226	3	𝐷1	𝐷1	PROPN
cana-6025	226	4	is	be	AUX
cana-6025	226	5	a	a	DET
cana-6025	226	6	restrained	restrain	VERB
cana-6025	226	7	dominating	dominating	NOUN
cana-6025	226	8	set	set	NOUN
cana-6025	226	9	.	.	PUNCT
cana-6025	227	1	thus	thus	ADV
cana-6025	227	2	,	,	PUNCT
cana-6025	227	3	𝐷1	𝐷1	PROPN
cana-6025	227	4	is	be	AUX
cana-6025	227	5	a	a	DET
cana-6025	227	6	chromatic	chromatic	ADJ
cana-6025	227	7	restrained	restrain	VERB
cana-6025	227	8	dominating	dominating	NOUN
cana-6025	227	9	set	set	NOUN
cana-6025	227	10	and	and	CCONJ
cana-6025	227	11	𝛾𝑟	𝛾𝑟	ADP
cana-6025	227	12	𝑐(𝐾1,𝑚	𝑐(𝐾1,𝑚	PROPN
cana-6025	227	13	⊠	⊠	PROPN
cana-6025	227	14	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	227	15	)	)	PUNCT
cana-6025	227	16	≤	≤	NUM
cana-6025	227	17	|𝐷1|	|𝐷1|	NOUN
cana-6025	227	18	=	=	SYM
cana-6025	227	19	4	4	X
cana-6025	227	20	.	.	PUNCT
cana-6025	227	21	since	since	SCONJ
cana-6025	227	22	𝜒(𝐾1,𝑚	𝜒(𝐾1,𝑚	X
cana-6025	227	23	⊠	⊠	PROPN
cana-6025	227	24	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	227	25	)	)	PUNCT
cana-6025	227	26	=	=	SYM
cana-6025	227	27	4	4	NUM
cana-6025	227	28	,	,	PUNCT
cana-6025	227	29	𝛾𝑟	𝛾𝑟	ADP
cana-6025	227	30	𝑐(𝐾1,𝑚	𝑐(𝐾1,𝑚	PROPN
cana-6025	227	31	⊠	⊠	PROPN
cana-6025	227	32	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	227	33	)	)	PUNCT
cana-6025	227	34	≥	≥	NOUN
cana-6025	227	35	4	4	NUM
cana-6025	227	36	.	.	PUNCT
cana-6025	228	1	therefore	therefore	ADV
cana-6025	228	2	,	,	PUNCT
cana-6025	228	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	228	4	𝑐(𝐾1,𝑚	𝑐(𝐾1,𝑚	PROPN
cana-6025	228	5	⊠	⊠	PROPN
cana-6025	228	6	𝐾𝑟,𝑠	𝐾𝑟,𝑠	PROPN
cana-6025	228	7	)	)	PUNCT
cana-6025	228	8	=	=	SYM
cana-6025	228	9	4	4	X
cana-6025	228	10	.	.	PUNCT
cana-6025	228	11	theorem	theorem	VERB
cana-6025	228	12	2.10	2.10	NUM
cana-6025	228	13	for	for	ADP
cana-6025	228	14	any	any	DET
cana-6025	228	15	𝑟	𝑟	NOUN
cana-6025	228	16	≥	≥	NOUN
cana-6025	228	17	3	3	NUM
cana-6025	228	18	,	,	PUNCT
cana-6025	228	19	𝑠	𝑠	PRON
cana-6025	228	20	≥	≥	NUM
cana-6025	228	21	4	4	NUM
cana-6025	228	22	,	,	PUNCT
cana-6025	228	23	𝛾𝑟	𝛾𝑟	ADP
cana-6025	228	24	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	228	25	⊠	⊠	PROPN
cana-6025	228	26	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	228	27	)	)	PUNCT
cana-6025	228	28	=	=	PRON
cana-6025	228	29	{	{	PUNCT
cana-6025	228	30	6	6	NUM
cana-6025	228	31	𝑖𝑓	𝑖𝑓	NOUN
cana-6025	228	32	𝑠	𝑠	PART
cana-6025	228	33	𝑖𝑠	𝑖𝑠	CCONJ
cana-6025	228	34	𝑜𝑑𝑑	𝑜𝑑𝑑	PROPN
cana-6025	228	35	2𝑠	2𝑠	NOUN
cana-6025	229	1	𝑖𝑓	𝑖𝑓	INTJ
cana-6025	230	1	𝑠	𝑠	INTJ
cana-6025	230	2	𝑖𝑠	𝑖𝑠	INTJ
cana-6025	230	3	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	INTJ
cana-6025	230	4	.	.	PUNCT
cana-6025	231	1	proof	proof	NOUN
cana-6025	231	2	.	.	PUNCT
cana-6025	232	1	let	let	VERB
cana-6025	232	2	𝑉(𝐾1,𝑟	𝑉(𝐾1,𝑟	NOUN
cana-6025	232	3	)	)	PUNCT
cana-6025	233	1	=	=	PRON
cana-6025	233	2	{	{	PUNCT
cana-6025	233	3	𝑢0	𝑢0	PROPN
cana-6025	233	4	,	,	PUNCT
cana-6025	233	5	𝑢1	𝑢1	NOUN
cana-6025	233	6	,	,	PUNCT
cana-6025	233	7	𝑢2	𝑢2	PROPN
cana-6025	233	8	,	,	PUNCT
cana-6025	233	9	.	.	PUNCT
cana-6025	233	10	.	.	PUNCT
cana-6025	234	1	.	.	PUNCT
cana-6025	235	1	,	,	PUNCT
cana-6025	235	2	𝑢𝑟	𝑢𝑟	ADV
cana-6025	235	3	}	}	PUNCT
cana-6025	235	4	and	and	CCONJ
cana-6025	235	5	𝑉(𝑊𝑠	𝑉(𝑊𝑠	NUM
cana-6025	235	6	)	)	PUNCT
cana-6025	235	7	=	=	SYM
cana-6025	235	8	{	{	PUNCT
cana-6025	235	9	𝑣0	𝑣0	PROPN
cana-6025	235	10	,	,	PUNCT
cana-6025	235	11	𝑣1	𝑣1	PROPN
cana-6025	235	12	,	,	PUNCT
cana-6025	235	13	𝑣2	𝑣2	PROPN
cana-6025	235	14	,	,	PUNCT
cana-6025	235	15	.	.	PUNCT
cana-6025	235	16	.	.	PUNCT
cana-6025	236	1	.	.	PUNCT
cana-6025	237	1	,	,	PUNCT
cana-6025	237	2	𝑣𝑠−1	𝑣𝑠−1	NOUN
cana-6025	237	3	}	}	PUNCT
cana-6025	237	4	where	where	SCONJ
cana-6025	237	5	𝑢0	𝑢0	PROPN
cana-6025	237	6	and	and	CCONJ
cana-6025	237	7	𝑣0	𝑣0	PROPN
cana-6025	237	8	are	be	AUX
cana-6025	237	9	the	the	DET
cana-6025	237	10	full	full	ADJ
cana-6025	237	11	degree	degree	NOUN
cana-6025	237	12	vertices	vertex	NOUN
cana-6025	237	13	of	of	ADP
cana-6025	237	14	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	237	15	and	and	CCONJ
cana-6025	237	16	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	237	17	respectively	respectively	ADV
cana-6025	237	18	.	.	PUNCT
cana-6025	238	1	then	then	ADV
cana-6025	238	2	,	,	PUNCT
cana-6025	238	3	𝑉(𝐾1,𝑟	𝑉(𝐾1,𝑟	PROPN
cana-6025	238	4	⊠	⊠	PROPN
cana-6025	238	5	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	238	6	)	)	PUNCT
cana-6025	238	7	=	=	PRON
cana-6025	238	8	{	{	PUNCT
cana-6025	238	9	(	(	PUNCT
cana-6025	238	10	𝑢𝑖	𝑢𝑖	INTJ
cana-6025	238	11	,	,	PUNCT
cana-6025	238	12	𝑣𝑗)/0	𝑣𝑗)/0	NOUN
cana-6025	238	13	≤	≤	NOUN
cana-6025	238	14	𝑖	𝑖	SYM
cana-6025	238	15	≤	≤	NOUN
cana-6025	238	16	𝑟	𝑟	NOUN
cana-6025	238	17	,	,	PUNCT
cana-6025	238	18	0	0	NUM
cana-6025	238	19	≤	≤	NUM
cana-6025	239	1	𝑗	𝑗	PRON
cana-6025	239	2	≤	≤	NUM
cana-6025	239	3	𝑠	𝑠	INTJ
cana-6025	239	4	−	−	PROPN
cana-6025	239	5	1	1	NUM
cana-6025	239	6	}	}	PUNCT
cana-6025	239	7	where	where	SCONJ
cana-6025	239	8	|𝑉(𝐾1,𝑟	|𝑉(𝐾1,𝑟	NOUN
cana-6025	239	9	⊠	⊠	PROPN
cana-6025	239	10	𝑊𝑠)|	𝑊𝑠)|	NOUN
cana-6025	239	11	=	=	PUNCT
cana-6025	239	12	(	(	PUNCT
cana-6025	239	13	𝑟	𝑟	NOUN
cana-6025	239	14	+	+	X
cana-6025	239	15	1)𝑠.	1)𝑠.	PROPN
cana-6025	239	16	also	also	ADV
cana-6025	239	17	,	,	PUNCT
cana-6025	239	18	𝑑𝑒𝑔(𝑢0	𝑑𝑒𝑔(𝑢0	ADJ
cana-6025	239	19	,	,	PUNCT
cana-6025	239	20	𝑣0	𝑣0	NOUN
cana-6025	239	21	)	)	PUNCT
cana-6025	239	22	=	=	SYM
cana-6025	239	23	(	(	PUNCT
cana-6025	239	24	𝑟	𝑟	NOUN
cana-6025	239	25	+	+	SYM
cana-6025	239	26	1)𝑠	1)𝑠	NUM
cana-6025	239	27	−	−	PROPN
cana-6025	239	28	1	1	X
cana-6025	239	29	.	.	PUNCT
cana-6025	239	30	clearly	clearly	ADV
cana-6025	239	31	,	,	PUNCT
cana-6025	239	32	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	239	33	⊠	⊠	PROPN
cana-6025	239	34	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	239	35	contains	contain	VERB
cana-6025	239	36	𝑟	𝑟	NOUN
cana-6025	239	37	+	+	SYM
cana-6025	239	38	1	1	NUM
cana-6025	239	39	rows	row	NOUN
cana-6025	239	40	(	(	PUNCT
cana-6025	239	41	𝑉1	𝑉1	PROPN
cana-6025	239	42	,	,	PUNCT
cana-6025	239	43	𝑉2	𝑉2	NOUN
cana-6025	239	44	,	,	PUNCT
cana-6025	239	45	𝑉3	𝑉3	NOUN
cana-6025	239	46	,	,	PUNCT
cana-6025	239	47	.	.	PUNCT
cana-6025	239	48	.	.	PUNCT
cana-6025	240	1	.	.	PUNCT
cana-6025	241	1	,	,	PUNCT
cana-6025	241	2	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	241	3	)	)	PUNCT
cana-6025	241	4	and	and	CCONJ
cana-6025	241	5	𝑠	𝑠	PROPN
cana-6025	241	6	columns	column	NOUN
cana-6025	241	7	.	.	PUNCT
cana-6025	242	1	case	case	NOUN
cana-6025	242	2	(	(	PUNCT
cana-6025	242	3	i	i	NOUN
cana-6025	242	4	):	):	PUNCT
cana-6025	242	5	𝑠	𝑠	PROPN
cana-6025	242	6	is	be	AUX
cana-6025	242	7	odd	odd	ADJ
cana-6025	242	8	then	then	ADV
cana-6025	242	9	the	the	DET
cana-6025	242	10	first	first	ADJ
cana-6025	242	11	row	row	NOUN
cana-6025	242	12	of	of	ADP
cana-6025	242	13	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	242	14	⊠	⊠	PROPN
cana-6025	242	15	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	242	16	can	can	AUX
cana-6025	242	17	be	be	AUX
cana-6025	242	18	colored	color	VERB
cana-6025	242	19	with	with	ADP
cana-6025	242	20	three	three	NUM
cana-6025	242	21	colors	color	NOUN
cana-6025	242	22	and	and	CCONJ
cana-6025	242	23	the	the	DET
cana-6025	242	24	second	second	ADJ
cana-6025	242	25	row	row	NOUN
cana-6025	242	26	can	can	AUX
cana-6025	242	27	be	be	AUX
cana-6025	242	28	colored	color	VERB
cana-6025	242	29	with	with	ADP
cana-6025	242	30	another	another	DET
cana-6025	242	31	three	three	NUM
cana-6025	242	32	colors	color	NOUN
cana-6025	242	33	.	.	PUNCT
cana-6025	243	1	since	since	SCONJ
cana-6025	243	2	there	there	PRON
cana-6025	243	3	does	do	AUX
cana-6025	243	4	not	not	PART
cana-6025	243	5	exists	exist	VERB
cana-6025	243	6	adjacency	adjacency	NOUN
cana-6025	243	7	between	between	ADP
cana-6025	243	8	any	any	DET
cana-6025	243	9	two	two	NUM
cana-6025	243	10	vertices	vertex	NOUN
cana-6025	243	11	belonging	belong	VERB
cana-6025	243	12	to	to	ADP
cana-6025	243	13	different	different	ADJ
cana-6025	243	14	rows	row	NOUN
cana-6025	243	15	of	of	ADP
cana-6025	243	16	𝑉2	𝑉2	PROPN
cana-6025	243	17	,	,	PUNCT
cana-6025	243	18	𝑉3	𝑉3	NOUN
cana-6025	243	19	,	,	PUNCT
cana-6025	243	20	…	…	PUNCT
cana-6025	243	21	,	,	PUNCT
cana-6025	243	22	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	243	23	,	,	PUNCT
cana-6025	243	24	all	all	DET
cana-6025	243	25	the	the	DET
cana-6025	243	26	𝑟	𝑟	NOUN
cana-6025	243	27	rows	row	NOUN
cana-6025	243	28	can	can	AUX
cana-6025	243	29	be	be	AUX
cana-6025	243	30	colored	color	VERB
cana-6025	243	31	with	with	ADP
cana-6025	243	32	three	three	NUM
cana-6025	243	33	colors	color	NOUN
cana-6025	243	34	.	.	PUNCT
cana-6025	244	1	then	then	ADV
cana-6025	244	2	,	,	PUNCT
cana-6025	244	3	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	VERB
cana-6025	244	4	⊠	⊠	PROPN
cana-6025	244	5	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	244	6	)	)	PUNCT
cana-6025	244	7	=	=	PUNCT
cana-6025	245	1	6	6	X
cana-6025	245	2	.	.	PUNCT
cana-6025	245	3	since	since	SCONJ
cana-6025	245	4	(	(	PUNCT
cana-6025	245	5	𝑢0	𝑢0	PROPN
cana-6025	245	6	,	,	PUNCT
cana-6025	245	7	𝑣0	𝑣0	NOUN
cana-6025	245	8	)	)	PUNCT
cana-6025	245	9	is	be	AUX
cana-6025	245	10	the	the	DET
cana-6025	245	11	full	full	ADJ
cana-6025	245	12	degree	degree	NOUN
cana-6025	245	13	vertex	vertex	NOUN
cana-6025	245	14	,	,	PUNCT
cana-6025	245	15	𝐷	𝐷	NOUN
cana-6025	245	16	=	=	SYM
cana-6025	245	17	{	{	PUNCT
cana-6025	245	18	(	(	PUNCT
cana-6025	245	19	𝑢0	𝑢0	PROPN
cana-6025	245	20	,	,	PUNCT
cana-6025	245	21	𝑣0	𝑣0	NOUN
cana-6025	245	22	)	)	PUNCT
cana-6025	245	23	}	}	PUNCT
cana-6025	245	24	is	be	AUX
cana-6025	245	25	a	a	DET
cana-6025	245	26	restrained	restrained	ADJ
cana-6025	245	27	dominating	dominating	NOUN
cana-6025	245	28	set	set	NOUN
cana-6025	245	29	of	of	ADP
cana-6025	245	30	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	245	31	⊠	⊠	PROPN
cana-6025	245	32	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	245	33	and	and	CCONJ
cana-6025	245	34	𝛾𝑟(𝐾1,𝑟	𝛾𝑟(𝐾1,𝑟	NOUN
cana-6025	245	35	⊠	⊠	PROPN
cana-6025	245	36	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	245	37	)	)	PUNCT
cana-6025	245	38	=	=	SYM
cana-6025	245	39	1	1	X
cana-6025	245	40	.	.	PUNCT
cana-6025	245	41	but	but	CCONJ
cana-6025	245	42	𝜒(⟨𝐷⟩	𝜒(⟨𝐷⟩	PROPN
cana-6025	245	43	)	)	PUNCT
cana-6025	245	44	=	=	SYM
cana-6025	245	45	1	1	NUM
cana-6025	245	46	and	and	CCONJ
cana-6025	245	47	so	so	ADV
cana-6025	245	48	,	,	PUNCT
cana-6025	245	49	𝐷	𝐷	PROPN
cana-6025	245	50	is	be	AUX
cana-6025	245	51	not	not	PART
cana-6025	245	52	a	a	DET
cana-6025	245	53	chromatic	chromatic	ADJ
cana-6025	245	54	restrained	restrain	VERB
cana-6025	245	55	dominating	dominating	NOUN
cana-6025	245	56	set	set	NOUN
cana-6025	245	57	.	.	PUNCT
cana-6025	246	1	let	let	VERB
cana-6025	246	2	𝐷1	𝐷1	NOUN
cana-6025	246	3	=	=	PRON
cana-6025	246	4	{	{	PUNCT
cana-6025	246	5	(	(	PUNCT
cana-6025	246	6	𝑢0	𝑢0	PROPN
cana-6025	246	7	,	,	PUNCT
cana-6025	246	8	𝑣0	𝑣0	PROPN
cana-6025	246	9	)	)	PUNCT
cana-6025	246	10	,	,	PUNCT
cana-6025	246	11	(	(	PUNCT
cana-6025	246	12	𝑢0	𝑢0	PROPN
cana-6025	246	13	,	,	PUNCT
cana-6025	246	14	𝑣1	𝑣1	PROPN
cana-6025	246	15	)	)	PUNCT
cana-6025	246	16	,	,	PUNCT
cana-6025	246	17	(	(	PUNCT
cana-6025	246	18	𝑢0	𝑢0	PROPN
cana-6025	246	19	,	,	PUNCT
cana-6025	246	20	𝑣2	𝑣2	PROPN
cana-6025	246	21	)	)	PUNCT
cana-6025	246	22	,	,	PUNCT
cana-6025	246	23	(	(	PUNCT
cana-6025	246	24	𝑢1	𝑢1	PROPN
cana-6025	246	25	,	,	PUNCT
cana-6025	246	26	𝑣0	𝑣0	NOUN
cana-6025	246	27	)	)	PUNCT
cana-6025	246	28	,	,	PUNCT
cana-6025	246	29	(	(	PUNCT
cana-6025	246	30	𝑢1	𝑢1	PROPN
cana-6025	246	31	,	,	PUNCT
cana-6025	246	32	𝑣1	𝑣1	PROPN
cana-6025	246	33	)	)	PUNCT
cana-6025	246	34	,	,	PUNCT
cana-6025	246	35	(	(	PUNCT
cana-6025	246	36	𝑢1	𝑢1	PROPN
cana-6025	246	37	,	,	PUNCT
cana-6025	246	38	𝑣2	𝑣2	NOUN
cana-6025	246	39	)	)	PUNCT
cana-6025	246	40	}	}	PUNCT
cana-6025	246	41	where	where	SCONJ
cana-6025	246	42	⟨𝐷1⟩	⟨𝐷1⟩	NOUN
cana-6025	246	43	=	=	SYM
cana-6025	246	44	𝐾6	𝐾6	NOUN
cana-6025	246	45	and	and	CCONJ
cana-6025	246	46	|𝐷1|	|𝐷1|	NUM
cana-6025	246	47	=	=	SYM
cana-6025	246	48	6	6	NUM
cana-6025	246	49	.	.	PUNCT
cana-6025	247	1	then	then	ADV
cana-6025	247	2	,	,	PUNCT
cana-6025	247	3	𝜒(⟨𝐷1⟩	𝜒(⟨𝐷1⟩	PROPN
cana-6025	247	4	)	)	PUNCT
cana-6025	247	5	=	=	SYM
cana-6025	247	6	6	6	NUM
cana-6025	247	7	=	=	SYM
cana-6025	247	8	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	NUM
cana-6025	247	9	⊠	⊠	PROPN
cana-6025	247	10	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	247	11	)	)	PUNCT
cana-6025	247	12	and	and	CCONJ
cana-6025	247	13	𝐷1	𝐷1	NOUN
cana-6025	247	14	is	be	AUX
cana-6025	247	15	a	a	DET
cana-6025	247	16	restrained	restrain	VERB
cana-6025	247	17	dominating	dominating	NOUN
cana-6025	247	18	set	set	NOUN
cana-6025	247	19	.	.	PUNCT
cana-6025	248	1	therefore	therefore	ADV
cana-6025	248	2	,	,	PUNCT
cana-6025	248	3	𝐷1	𝐷1	PROPN
cana-6025	248	4	is	be	AUX
cana-6025	248	5	a	a	DET
cana-6025	248	6	chromatic	chromatic	ADJ
cana-6025	248	7	restrained	restrain	VERB
cana-6025	248	8	dominating	dominating	NOUN
cana-6025	248	9	set	set	NOUN
cana-6025	248	10	and	and	CCONJ
cana-6025	248	11	𝛾𝑟	𝛾𝑟	ADP
cana-6025	248	12	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	248	13	⊠	⊠	PROPN
cana-6025	248	14	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	248	15	)	)	PUNCT
cana-6025	248	16	≤	≤	NUM
cana-6025	248	17	|𝐷1|	|𝐷1|	NOUN
cana-6025	249	1	=	=	SYM
cana-6025	250	1	6	6	NUM
cana-6025	250	2	.	.	PUNCT
cana-6025	251	1	since	since	SCONJ
cana-6025	251	2	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	PROPN
cana-6025	251	3	⊠	⊠	PROPN
cana-6025	251	4	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	251	5	)	)	PUNCT
cana-6025	251	6	=	=	SYM
cana-6025	251	7	6	6	NUM
cana-6025	251	8	,	,	PUNCT
cana-6025	251	9	any	any	DET
cana-6025	251	10	chromatic	chromatic	ADJ
cana-6025	251	11	restrained	restrain	VERB
cana-6025	251	12	dominating	dominating	NOUN
cana-6025	251	13	set	set	NOUN
cana-6025	251	14	must	must	AUX
cana-6025	251	15	contain	contain	VERB
cana-6025	251	16	at	at	ADV
cana-6025	251	17	least	least	ADJ
cana-6025	251	18	six	six	NUM
cana-6025	251	19	vertices	vertex	NOUN
cana-6025	251	20	.	.	PUNCT
cana-6025	252	1	thus	thus	ADV
cana-6025	252	2	,	,	PUNCT
cana-6025	252	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	252	4	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	252	5	⊠	⊠	PROPN
cana-6025	252	6	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	252	7	)	)	PUNCT
cana-6025	252	8	≥	≥	NOUN
cana-6025	252	9	6	6	NUM
cana-6025	252	10	.	.	PUNCT
cana-6025	253	1	therefore	therefore	ADV
cana-6025	253	2	,	,	PUNCT
cana-6025	253	3	𝛾𝑟	𝛾𝑟	ADP
cana-6025	253	4	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	253	5	⊠	⊠	PROPN
cana-6025	253	6	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	253	7	)	)	PUNCT
cana-6025	253	8	=	=	SYM
cana-6025	253	9	6	6	NUM
cana-6025	253	10	.	.	PUNCT
cana-6025	253	11	case	case	NOUN
cana-6025	253	12	(	(	PUNCT
cana-6025	253	13	ii	ii	NUM
cana-6025	253	14	):	):	PUNCT
cana-6025	253	15	𝑠	𝑠	PROPN
cana-6025	253	16	is	be	AUX
cana-6025	253	17	even	even	ADV
cana-6025	253	18	then	then	ADV
cana-6025	253	19	the	the	DET
cana-6025	253	20	first	first	ADJ
cana-6025	253	21	row	row	NOUN
cana-6025	253	22	𝑉1	𝑉1	NOUN
cana-6025	253	23	can	can	AUX
cana-6025	253	24	be	be	AUX
cana-6025	253	25	colored	color	VERB
cana-6025	253	26	with	with	ADP
cana-6025	253	27	four	four	NUM
cana-6025	253	28	colors	color	NOUN
cana-6025	253	29	.	.	PUNCT
cana-6025	254	1	since	since	SCONJ
cana-6025	254	2	,	,	PUNCT
cana-6025	254	3	some	some	PRON
cana-6025	254	4	of	of	ADP
cana-6025	254	5	the	the	DET
cana-6025	254	6	vertices	vertex	NOUN
cana-6025	254	7	in	in	ADP
cana-6025	254	8	𝑉1	𝑉1	NOUN
cana-6025	254	9	and	and	CCONJ
cana-6025	254	10	𝑉2	𝑉2	NOUN
cana-6025	254	11	are	be	AUX
cana-6025	254	12	adjacent	adjacent	ADJ
cana-6025	254	13	,	,	PUNCT
cana-6025	254	14	the	the	DET
cana-6025	254	15	second	second	ADJ
cana-6025	254	16	row	row	NOUN
cana-6025	254	17	𝑉2	𝑉2	NOUN
cana-6025	254	18	can	can	AUX
cana-6025	254	19	be	be	AUX
cana-6025	254	20	colored	color	VERB
cana-6025	254	21	by	by	ADP
cana-6025	254	22	introducing	introduce	VERB
cana-6025	254	23	three	three	NUM
cana-6025	254	24	more	more	ADJ
cana-6025	254	25	colors	color	NOUN
cana-6025	254	26	.	.	PUNCT
cana-6025	255	1	also	also	ADV
cana-6025	255	2	,	,	PUNCT
cana-6025	255	3	the	the	DET
cana-6025	255	4	remaining	remain	VERB
cana-6025	255	5	rows	row	NOUN
cana-6025	255	6	𝑉3	𝑉3	NOUN
cana-6025	255	7	,	,	PUNCT
cana-6025	255	8	𝑉4	𝑉4	NOUN
cana-6025	255	9	,	,	PUNCT
cana-6025	255	10	…	…	PUNCT
cana-6025	255	11	,	,	PUNCT
cana-6025	255	12	𝑉𝑟+1	𝑉𝑟+1	NOUN
cana-6025	255	13	can	can	AUX
cana-6025	255	14	be	be	AUX
cana-6025	255	15	colored	color	VERB
cana-6025	255	16	by	by	ADP
cana-6025	255	17	assigning	assign	VERB
cana-6025	255	18	the	the	DET
cana-6025	255	19	same	same	ADJ
cana-6025	255	20	colors	color	NOUN
cana-6025	255	21	as	as	ADP
cana-6025	255	22	in	in	ADP
cana-6025	255	23	𝑉2	𝑉2	NOUN
cana-6025	255	24	.	.	PUNCT
cana-6025	256	1	thus	thus	ADV
cana-6025	256	2	,	,	PUNCT
cana-6025	256	3	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	NUM
cana-6025	256	4	⊠	⊠	PROPN
cana-6025	256	5	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	256	6	)	)	PUNCT
cana-6025	256	7	=	=	SYM
cana-6025	256	8	7	7	X
cana-6025	256	9	.	.	PUNCT
cana-6025	257	1	let	let	VERB
cana-6025	257	2	𝐷2	𝐷2	PROPN
cana-6025	257	3	=	=	SYM
cana-6025	257	4	{	{	PUNCT
cana-6025	257	5	(	(	PUNCT
cana-6025	257	6	𝑢0	𝑢0	PROPN
cana-6025	257	7	,	,	PUNCT
cana-6025	257	8	𝑣0	𝑣0	PROPN
cana-6025	257	9	)	)	PUNCT
cana-6025	257	10	,	,	PUNCT
cana-6025	257	11	(	(	PUNCT
cana-6025	257	12	𝑢0	𝑢0	PROPN
cana-6025	257	13	,	,	PUNCT
cana-6025	257	14	𝑣1	𝑣1	PROPN
cana-6025	257	15	)	)	PUNCT
cana-6025	257	16	,	,	PUNCT
cana-6025	257	17	(	(	PUNCT
cana-6025	257	18	𝑢0	𝑢0	PROPN
cana-6025	257	19	,	,	PUNCT
cana-6025	257	20	𝑣2	𝑣2	PROPN
cana-6025	257	21	)	)	PUNCT
cana-6025	257	22	,	,	PUNCT
cana-6025	257	23	…	…	PUNCT
cana-6025	257	24	,	,	PUNCT
cana-6025	257	25	(	(	PUNCT
cana-6025	257	26	𝑢0	𝑢0	PROPN
cana-6025	257	27	,	,	PUNCT
cana-6025	257	28	𝑣𝑠−1	𝑣𝑠−1	PROPN
cana-6025	257	29	)	)	PUNCT
cana-6025	257	30	,	,	PUNCT
cana-6025	257	31	(	(	PUNCT
cana-6025	257	32	𝑢1	𝑢1	PROPN
cana-6025	257	33	,	,	PUNCT
cana-6025	257	34	𝑣0	𝑣0	NOUN
cana-6025	257	35	)	)	PUNCT
cana-6025	257	36	,	,	PUNCT
cana-6025	257	37	(	(	PUNCT
cana-6025	257	38	𝑢1	𝑢1	PROPN
cana-6025	257	39	,	,	PUNCT
cana-6025	257	40	𝑣1	𝑣1	PROPN
cana-6025	257	41	)	)	PUNCT
cana-6025	257	42	,	,	PUNCT
cana-6025	257	43	(	(	PUNCT
cana-6025	257	44	𝑢1	𝑢1	PROPN
cana-6025	257	45	,	,	PUNCT
cana-6025	257	46	𝑣2	𝑣2	PROPN
cana-6025	257	47	)	)	PUNCT
cana-6025	257	48	,	,	PUNCT
cana-6025	257	49	.	.	PUNCT
cana-6025	257	50	.	.	PUNCT
cana-6025	258	1	.	.	PUNCT
cana-6025	259	1	,	,	PUNCT
cana-6025	259	2	(	(	PUNCT
cana-6025	259	3	𝑢1	𝑢1	NOUN
cana-6025	259	4	,	,	PUNCT
cana-6025	259	5	𝑣𝑠−2	𝑣𝑠−2	NOUN
cana-6025	259	6	)	)	PUNCT
cana-6025	259	7	}	}	PUNCT
cana-6025	259	8	.	.	PUNCT
cana-6025	260	1	then	then	ADV
cana-6025	260	2	𝜒(⟨𝐷2⟩	𝜒(⟨𝐷2⟩	NUM
cana-6025	260	3	)	)	PUNCT
cana-6025	260	4	=	=	SYM
cana-6025	260	5	7	7	NUM
cana-6025	260	6	=	=	SYM
cana-6025	260	7	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	NUM
cana-6025	260	8	⊠	⊠	PROPN
cana-6025	260	9	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	260	10	)	)	PUNCT
cana-6025	260	11	.	.	PUNCT
cana-6025	261	1	but	but	CCONJ
cana-6025	261	2	𝐷2	𝐷2	PROPN
cana-6025	261	3	is	be	AUX
cana-6025	261	4	not	not	PART
cana-6025	261	5	a	a	DET
cana-6025	261	6	restrained	restrained	ADJ
cana-6025	261	7	dominating	dominating	NOUN
cana-6025	261	8	set	set	VERB
cana-6025	261	9	since	since	SCONJ
cana-6025	261	10	(	(	PUNCT
cana-6025	261	11	𝑢1	𝑢1	PROPN
cana-6025	261	12	,	,	PUNCT
cana-6025	261	13	𝑣𝑠−1	𝑣𝑠−1	PROPN
cana-6025	261	14	)	)	PUNCT
cana-6025	261	15	∈	∈	PROPN
cana-6025	261	16	𝑉	𝑉	PROPN
cana-6025	261	17	−	−	PROPN
cana-6025	261	18	𝐷2	𝐷2	PROPN
cana-6025	261	19	has	have	VERB
cana-6025	261	20	no	no	DET
cana-6025	261	21	adjacent	adjacent	ADJ
cana-6025	261	22	vertex	vertex	NOUN
cana-6025	261	23	in	in	ADP
cana-6025	261	24	𝑉	𝑉	PROPN
cana-6025	261	25	−	−	PROPN
cana-6025	261	26	𝐷2	𝐷2	NOUN
cana-6025	261	27	.	.	PUNCT
cana-6025	262	1	so	so	ADV
cana-6025	262	2	,	,	PUNCT
cana-6025	262	3	consider	consider	VERB
cana-6025	262	4	𝐷3	𝐷3	NOUN
cana-6025	262	5	=	=	NOUN
cana-6025	262	6	𝐷2	𝐷2	PROPN
cana-6025	262	7	∪	∪	X
cana-6025	262	8	{	{	PUNCT
cana-6025	262	9	(	(	PUNCT
cana-6025	262	10	𝑢1	𝑢1	PROPN
cana-6025	262	11	,	,	PUNCT
cana-6025	262	12	𝑣𝑠−1	𝑣𝑠−1	PROPN
cana-6025	262	13	)	)	PUNCT
cana-6025	262	14	}	}	PUNCT
cana-6025	262	15	.	.	PUNCT
cana-6025	263	1	then	then	ADV
cana-6025	263	2	,	,	PUNCT
cana-6025	263	3	𝜒(⟨𝐷3⟩	𝜒(⟨𝐷3⟩	ADJ
cana-6025	263	4	)	)	PUNCT
cana-6025	263	5	=	=	SYM
cana-6025	263	6	7	7	NUM
cana-6025	263	7	and	and	CCONJ
cana-6025	263	8	𝐷3	𝐷3	PROPN
cana-6025	263	9	is	be	AUX
cana-6025	263	10	a	a	DET
cana-6025	263	11	restrained	restrain	VERB
cana-6025	263	12	dominating	dominating	NOUN
cana-6025	263	13	set	set	NOUN
cana-6025	263	14	.	.	PUNCT
cana-6025	264	1	thus	thus	ADV
cana-6025	264	2	,	,	PUNCT
cana-6025	264	3	𝐷3	𝐷3	PROPN
cana-6025	264	4	is	be	AUX
cana-6025	264	5	a	a	DET
cana-6025	264	6	chromatic	chromatic	ADJ
cana-6025	264	7	restrained	restrain	VERB
cana-6025	264	8	dominating	dominating	NOUN
cana-6025	264	9	set	set	NOUN
cana-6025	264	10	of	of	ADP
cana-6025	264	11	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	264	12	⊠	⊠	PROPN
cana-6025	264	13	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	264	14	and	and	CCONJ
cana-6025	264	15	communications	communication	NOUN
cana-6025	264	16	on	on	ADP
cana-6025	264	17	applied	apply	VERB
cana-6025	264	18	nonlinear	nonlinear	ADJ
cana-6025	264	19	analysis	analysis	NOUN
cana-6025	264	20	issn	issn	NOUN
cana-6025	264	21	:	:	PUNCT
cana-6025	264	22	1074	1074	NUM
cana-6025	264	23	-	-	PUNCT
cana-6025	264	24	133x	133x	NUM
cana-6025	264	25	vol	vol	NOUN
cana-6025	264	26	31	31	NUM
cana-6025	264	27	no	no	NOUN
cana-6025	264	28	.	.	NOUN
cana-6025	264	29	2	2	NUM
cana-6025	264	30	(	(	PUNCT
cana-6025	264	31	2024	2024	NUM
cana-6025	264	32	)	)	PUNCT
cana-6025	264	33	498	498	NUM
cana-6025	264	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-6025	264	35	𝛾𝑟	𝛾𝑟	ADP
cana-6025	264	36	𝑐𝐾1,𝑟	𝑐𝐾1,𝑟	PROPN
cana-6025	264	37	⊠	⊠	PROPN
cana-6025	264	38	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	264	39	)	)	PUNCT
cana-6025	264	40	≤	≤	PUNCT
cana-6025	264	41	|𝐷3|	|𝐷3|	X
cana-6025	265	1	=	=	PUNCT
cana-6025	266	1	|𝐷2|	|𝐷2|	X
cana-6025	267	1	+	+	NOUN
cana-6025	267	2	1	1	NUM
cana-6025	267	3	=	=	SYM
cana-6025	267	4	2𝑠.	2𝑠.	NUM
cana-6025	267	5	since	since	SCONJ
cana-6025	267	6	𝑠	𝑠	PROPN
cana-6025	267	7	is	be	AUX
cana-6025	267	8	even	even	ADV
cana-6025	267	9	and	and	CCONJ
cana-6025	267	10	𝜒(𝐾1,𝑟	𝜒(𝐾1,𝑟	NUM
cana-6025	267	11	⊠	⊠	PROPN
cana-6025	267	12	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	267	13	)	)	PUNCT
cana-6025	267	14	=	=	SYM
cana-6025	267	15	7	7	NUM
cana-6025	267	16	,	,	PUNCT
cana-6025	267	17	any	any	DET
cana-6025	267	18	minimum	minimum	ADJ
cana-6025	267	19	chromatic	chromatic	ADJ
cana-6025	267	20	restrained	restrain	VERB
cana-6025	267	21	dominating	dominating	NOUN
cana-6025	267	22	set	set	NOUN
cana-6025	267	23	of	of	ADP
cana-6025	267	24	𝐾1,𝑟	𝐾1,𝑟	PROPN
cana-6025	267	25	⊠	⊠	PROPN
cana-6025	267	26	𝑊𝑠must	𝑊𝑠must	PROPN
cana-6025	267	27	contain	contain	VERB
cana-6025	267	28	at	at	ADP
cana-6025	267	29	least	least	ADJ
cana-6025	267	30	2𝑠	2𝑠	NUM
cana-6025	267	31	vertices	vertex	NOUN
cana-6025	267	32	and	and	CCONJ
cana-6025	267	33	so	so	ADV
cana-6025	267	34	,	,	PUNCT
cana-6025	267	35	𝛾𝑟	𝛾𝑟	ADP
cana-6025	267	36	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	267	37	⊠	⊠	PROPN
cana-6025	267	38	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	267	39	)	)	PUNCT
cana-6025	267	40	≥	≥	NOUN
cana-6025	267	41	2𝑠.	2𝑠.	NUM
cana-6025	267	42	therefore	therefore	ADV
cana-6025	267	43	,	,	PUNCT
cana-6025	267	44	𝛾𝑟	𝛾𝑟	ADP
cana-6025	267	45	𝑐(𝐾1,𝑟	𝑐(𝐾1,𝑟	NOUN
cana-6025	267	46	⊠	⊠	PROPN
cana-6025	267	47	𝑊𝑠	𝑊𝑠	PROPN
cana-6025	267	48	)	)	PUNCT
cana-6025	267	49	=	=	PROPN
cana-6025	267	50	2𝑠.	2𝑠.	NUM
cana-6025	267	51	3	3	NUM
cana-6025	267	52	.	.	PUNCT
cana-6025	267	53	conclusion	conclusion	NOUN
cana-6025	267	54	in	in	ADP
cana-6025	267	55	this	this	DET
cana-6025	267	56	article	article	NOUN
cana-6025	267	57	,	,	PUNCT
cana-6025	267	58	the	the	DET
cana-6025	267	59	chromatic	chromatic	ADJ
cana-6025	267	60	restrained	restrained	ADJ
cana-6025	267	61	domination	domination	NOUN
cana-6025	267	62	number	number	NOUN
cana-6025	267	63	on	on	ADP
cana-6025	267	64	the	the	DET
cana-6025	267	65	strong	strong	ADJ
cana-6025	267	66	product	product	NOUN
cana-6025	267	67	of	of	ADP
cana-6025	267	68	certain	certain	ADJ
cana-6025	267	69	standard	standard	ADJ
cana-6025	267	70	graphs	graph	NOUN
cana-6025	267	71	are	be	AUX
cana-6025	267	72	obtained	obtain	VERB
cana-6025	267	73	.	.	PUNCT
cana-6025	268	1	a	a	DET
cana-6025	268	2	promising	promising	ADJ
cana-6025	268	3	avenue	avenue	NOUN
cana-6025	268	4	for	for	ADP
cana-6025	268	5	future	future	ADJ
cana-6025	268	6	research	research	NOUN
cana-6025	268	7	is	be	AUX
cana-6025	268	8	to	to	PART
cana-6025	268	9	investigate	investigate	VERB
cana-6025	268	10	the	the	DET
cana-6025	268	11	bounds	bound	NOUN
cana-6025	268	12	on	on	ADP
cana-6025	268	13	the	the	DET
cana-6025	268	14	strong	strong	ADJ
cana-6025	268	15	product	product	NOUN
cana-6025	268	16	of	of	ADP
cana-6025	268	17	graphs	graph	NOUN
cana-6025	268	18	and	and	CCONJ
cana-6025	268	19	identify	identify	VERB
cana-6025	268	20	the	the	DET
cana-6025	268	21	extremal	extremal	ADJ
cana-6025	268	22	graphs	graph	NOUN
cana-6025	268	23	that	that	PRON
cana-6025	268	24	define	define	VERB
cana-6025	268	25	the	the	DET
cana-6025	268	26	upper	upper	ADJ
cana-6025	268	27	and	and	CCONJ
cana-6025	268	28	lower	low	ADJ
cana-6025	268	29	limits	limit	NOUN
cana-6025	268	30	of	of	ADP
cana-6025	268	31	the	the	DET
cana-6025	268	32	chromatic	chromatic	ADJ
cana-6025	268	33	restrained	restrain	VERB
cana-6025	268	34	domination	domination	NOUN
cana-6025	268	35	number	number	NOUN
cana-6025	268	36	in	in	ADP
cana-6025	268	37	such	such	ADJ
cana-6025	268	38	products	product	NOUN
cana-6025	268	39	.	.	PUNCT
cana-6025	269	1	references	reference	NOUN
cana-6025	269	2	[	[	X
cana-6025	269	3	1	1	NUM
cana-6025	269	4	]	]	X
cana-6025	269	5	frank	frank	PROPN
cana-6025	269	6	harary	harary	PROPN
cana-6025	269	7	,	,	PUNCT
cana-6025	269	8	graph	graph	NOUN
cana-6025	269	9	theory	theory	NOUN
cana-6025	269	10	,	,	PUNCT
cana-6025	269	11	addison	addison	PROPN
cana-6025	269	12	wesley	wesley	PROPN
cana-6025	269	13	publishing	publishing	PROPN
cana-6025	269	14	company	company	NOUN
cana-6025	269	15	,	,	PUNCT
cana-6025	269	16	1969	1969	NUM
cana-6025	269	17	.	.	PUNCT
cana-6025	270	1	[	[	X
cana-6025	270	2	2	2	NUM
cana-6025	270	3	]	]	X
cana-6025	270	4	teresa	teresa	PROPN
cana-6025	270	5	w.	w.	PROPN
cana-6025	270	6	haynes	haynes	PROPN
cana-6025	270	7	,	,	PUNCT
cana-6025	270	8	stephen	stephen	PROPN
cana-6025	270	9	t.	t.	PROPN
cana-6025	270	10	hedetniemi	hedetniemi	PROPN
cana-6025	270	11	,	,	PUNCT
cana-6025	270	12	peter	peter	PROPN
cana-6025	270	13	j.	j.	PROPN
cana-6025	270	14	slater	slater	PROPN
cana-6025	270	15	,	,	PUNCT
cana-6025	270	16	fundamentals	fundamental	NOUN
cana-6025	270	17	of	of	ADP
cana-6025	270	18	domination	domination	NOUN
cana-6025	270	19	in	in	ADP
cana-6025	270	20	graphs	graph	NOUN
cana-6025	270	21	,	,	PUNCT
cana-6025	270	22	marcel	marcel	PROPN
cana-6025	270	23	dekker	dekker	PROPN
cana-6025	270	24	,	,	PUNCT
cana-6025	270	25	1998	1998	NUM
cana-6025	270	26	.	.	PUNCT
cana-6025	271	1	[	[	X
cana-6025	271	2	3	3	NUM
cana-6025	271	3	]	]	X
cana-6025	271	4	bondy	bondy	PROPN
cana-6025	271	5	.	.	PUNCT
cana-6025	272	1	j.	j.	PROPN
cana-6025	272	2	a	a	PROPN
cana-6025	272	3	and	and	CCONJ
cana-6025	272	4	murty	murty	NOUN
cana-6025	272	5	.	.	PUNCT
cana-6025	273	1	u.	u.	PROPN
cana-6025	273	2	s.	s.	PROPN
cana-6025	273	3	r	r	PROPN
cana-6025	273	4	,	,	PUNCT
cana-6025	273	5	graph	graph	NOUN
cana-6025	273	6	theory	theory	NOUN
cana-6025	273	7	with	with	ADP
cana-6025	273	8	applications	application	NOUN
cana-6025	273	9	,	,	PUNCT
cana-6025	273	10	springer	springer	NOUN
cana-6025	273	11	,	,	PUNCT
cana-6025	273	12	2008	2008	NUM
cana-6025	273	13	.	.	PUNCT
cana-6025	274	1	[	[	X
cana-6025	274	2	4	4	NUM
cana-6025	274	3	]	]	X
cana-6025	274	4	gayla	gayla	ADJ
cana-6025	274	5	s.	s.	PROPN
cana-6025	274	6	domke	domke	PROPN
cana-6025	274	7	,	,	PUNCT
cana-6025	274	8	johannes	johannes	PROPN
cana-6025	274	9	h.	h.	PROPN
cana-6025	274	10	hattingh	hattingh	PROPN
cana-6025	274	11	,	,	PUNCT
cana-6025	274	12	stephen	stephen	PROPN
cana-6025	274	13	t.	t.	PROPN
cana-6025	274	14	hedetniemi	hedetniemi	PROPN
cana-6025	274	15	,	,	PUNCT
cana-6025	274	16	renu	renu	PROPN
cana-6025	274	17	c.	c.	PROPN
cana-6025	274	18	laskar	laskar	PROPN
cana-6025	274	19	,	,	PUNCT
cana-6025	274	20	lisa	lisa	PROPN
cana-6025	274	21	r.	r.	PROPN
cana-6025	274	22	markus	markus	PROPN
cana-6025	274	23	,	,	PUNCT
cana-6025	274	24	restrained	restrained	ADJ
cana-6025	274	25	domination	domination	NOUN
cana-6025	274	26	in	in	ADP
cana-6025	274	27	graphs	graph	NOUN
cana-6025	274	28	,	,	PUNCT
cana-6025	274	29	discrete	discrete	ADJ
cana-6025	274	30	mathematics	mathematic	NOUN
cana-6025	274	31	,	,	PUNCT
cana-6025	274	32	203	203	NUM
cana-6025	274	33	(	(	PUNCT
cana-6025	274	34	1999	1999	NUM
cana-6025	274	35	)	)	PUNCT
cana-6025	274	36	61	61	NUM
cana-6025	274	37	69	69	NUM
cana-6025	274	38	.	.	PUNCT
cana-6025	275	1	[	[	X
cana-6025	275	2	5	5	NUM
cana-6025	275	3	]	]	PUNCT
cana-6025	275	4	janakiraman	janakiraman	NOUN
cana-6025	275	5	.	.	PUNCT
cana-6025	276	1	t.	t.	PROPN
cana-6025	276	2	n	n	PROPN
cana-6025	276	3	and	and	CCONJ
cana-6025	276	4	poobalaranjani	poobalaranjani	NOUN
cana-6025	276	5	.	.	PUNCT
cana-6025	277	1	m	m	PROPN
cana-6025	277	2	,	,	PUNCT
cana-6025	277	3	on	on	ADP
cana-6025	277	4	the	the	DET
cana-6025	277	5	chromatic	chromatic	ADJ
cana-6025	277	6	preserving	preserve	VERB
cana-6025	277	7	sets	set	NOUN
cana-6025	277	8	,	,	PUNCT
cana-6025	277	9	international	international	ADJ
cana-6025	277	10	journal	journal	NOUN
cana-6025	277	11	of	of	ADP
cana-6025	277	12	engineering	engineering	NOUN
cana-6025	277	13	science	science	NOUN
cana-6025	277	14	,	,	PUNCT
cana-6025	277	15	advanced	advanced	ADJ
cana-6025	277	16	computing	computing	NOUN
cana-6025	277	17	and	and	CCONJ
cana-6025	277	18	bio	bio	PROPN
cana-6025	277	19	technology	technology	PROPN
cana-6025	277	20	,	,	PUNCT
cana-6025	277	21	vol	vol	NOUN
cana-6025	277	22	.	.	PROPN
cana-6025	277	23	1	1	NUM
cana-6025	277	24	,	,	PUNCT
cana-6025	277	25	no	no	INTJ
cana-6025	277	26	.	.	NOUN
cana-6025	277	27	1	1	NUM
cana-6025	277	28	,	,	PUNCT
cana-6025	277	29	january	january	PROPN
cana-6025	277	30	march	march	PROPN
cana-6025	277	31	2010	2010	NUM
cana-6025	277	32	,	,	PUNCT
cana-6025	277	33	pp	pp	ADJ
cana-6025	277	34	.	.	PUNCT
cana-6025	278	1	29	29	NUM
cana-6025	278	2	42	42	NUM
cana-6025	278	3	.	.	PUNCT
cana-6025	279	1	[	[	X
cana-6025	279	2	6	6	NUM
cana-6025	279	3	]	]	PUNCT
cana-6025	279	4	janakiraman	janakiraman	NOUN
cana-6025	279	5	.	.	PUNCT
cana-6025	280	1	t.	t.	PROPN
cana-6025	280	2	n	n	PROPN
cana-6025	280	3	and	and	CCONJ
cana-6025	280	4	poobalaranjani	poobalaranjani	NOUN
cana-6025	280	5	.	.	PUNCT
cana-6025	281	1	m	m	PROPN
cana-6025	281	2	,	,	PUNCT
cana-6025	281	3	dom	dom	NOUN
cana-6025	281	4	-	-	PUNCT
cana-6025	281	5	chromatic	chromatic	ADJ
cana-6025	281	6	sets	set	NOUN
cana-6025	281	7	of	of	ADP
cana-6025	281	8	graphs	graph	NOUN
cana-6025	281	9	,	,	PUNCT
cana-6025	281	10	international	international	ADJ
cana-6025	281	11	journal	journal	NOUN
cana-6025	281	12	of	of	ADP
cana-6025	281	13	engineering	engineering	NOUN
cana-6025	281	14	science	science	NOUN
cana-6025	281	15	,	,	PUNCT
cana-6025	281	16	advanced	advanced	ADJ
cana-6025	281	17	computing	computing	NOUN
cana-6025	281	18	and	and	CCONJ
cana-6025	281	19	bio	bio	PROPN
cana-6025	281	20	technology	technology	PROPN
cana-6025	281	21	,	,	PUNCT
cana-6025	281	22	vol	vol	NOUN
cana-6025	281	23	.	.	PROPN
cana-6025	281	24	2	2	NUM
cana-6025	281	25	,	,	PUNCT
cana-6025	281	26	no	no	INTJ
cana-6025	281	27	.	.	NOUN
cana-6025	281	28	2	2	NUM
cana-6025	281	29	,	,	PUNCT
cana-6025	281	30	april	april	PROPN
cana-6025	281	31	june	june	PROPN
cana-6025	281	32	2011	2011	NUM
cana-6025	281	33	,	,	PUNCT
cana-6025	281	34	pp	pp	ADV
cana-6025	281	35	.	.	PUNCT
cana-6025	282	1	88	88	NUM
cana-6025	282	2	103	103	NUM
cana-6025	282	3	.	.	PUNCT