id	sid	tid	token	lemma	pos
cana-4446	1	1	communications	communication	NOUN
cana-4446	1	2	on	on	ADP
cana-4446	1	3	applied	apply	VERB
cana-4446	1	4	nonlinear	nonlinear	ADJ
cana-4446	1	5	analysis	analysis	NOUN
cana-4446	1	6	issn	issn	NOUN
cana-4446	1	7	:	:	PUNCT
cana-4446	1	8	1074	1074	NUM
cana-4446	1	9	-	-	PUNCT
cana-4446	1	10	133x	133x	NUM
cana-4446	1	11	vol	vol	NOUN
cana-4446	1	12	32	32	NUM
cana-4446	1	13	no	no	NOUN
cana-4446	1	14	.	.	PUNCT
cana-4446	2	1	9s	9s	NUM
cana-4446	2	2	(	(	PUNCT
cana-4446	2	3	2025	2025	NUM
cana-4446	2	4	)	)	PUNCT
cana-4446	2	5	2044	2044	NUM
cana-4446	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4446	3	2	rainbow	rainbow	VERB
cana-4446	3	3	dynamic	dynamic	ADJ
cana-4446	3	4	coloring	coloring	NOUN
cana-4446	3	5	in	in	ADP
cana-4446	3	6	few	few	ADJ
cana-4446	3	7	brick	brick	NOUN
cana-4446	3	8	product	product	NOUN
cana-4446	3	9	graphs	graph	VERB
cana-4446	3	10	gayathri	gayathri	PROPN
cana-4446	3	11	annasagaram	annasagaram	PROPN
cana-4446	3	12	1	1	NUM
cana-4446	3	13	,	,	PUNCT
cana-4446	3	14	r.	r.	PROPN
cana-4446	3	15	murali	murali	PROPN
cana-4446	3	16	2	2	NUM
cana-4446	3	17	,	,	PUNCT
cana-4446	3	18	and	and	CCONJ
cana-4446	3	19	kulkarni	kulkarni	PROPN
cana-4446	3	20	sunita	sunita	PROPN
cana-4446	3	21	jagannatharao	jagannatharao	PROPN
cana-4446	3	22	3	3	NUM
cana-4446	3	23	dr.ambedkar	dr.ambedkar	NOUN
cana-4446	3	24	institute	institute	NOUN
cana-4446	3	25	of	of	ADP
cana-4446	3	26	technology	technology	PROPN
cana-4446	3	27	,	,	PUNCT
cana-4446	3	28	bangalore	bangalore	PROPN
cana-4446	3	29	,	,	PUNCT
cana-4446	3	30	affiliated	affiliate	VERB
cana-4446	3	31	to	to	AUX
cana-4446	3	32	visvesvaraya	visvesvaraya	VERB
cana-4446	3	33	technological	technological	ADJ
cana-4446	3	34	university	university	NOUN
cana-4446	3	35	,	,	PUNCT
cana-4446	3	36	belagavi	belagavi	VERB
cana-4446	3	37	,	,	PUNCT
cana-4446	3	38	india.1	india.1	PROPN
cana-4446	3	39	dr.ambedkar	dr.ambedkar	NOUN
cana-4446	3	40	institute	institute	NOUN
cana-4446	3	41	of	of	ADP
cana-4446	3	42	technology	technology	PROPN
cana-4446	3	43	,	,	PUNCT
cana-4446	3	44	bangalore	bangalore	PROPN
cana-4446	3	45	,	,	PUNCT
cana-4446	3	46	affiliated	affiliate	VERB
cana-4446	3	47	to	to	AUX
cana-4446	3	48	visvesvaraya	visvesvaraya	VERB
cana-4446	3	49	technological	technological	ADJ
cana-4446	3	50	university	university	NOUN
cana-4446	3	51	,	,	PUNCT
cana-4446	3	52	belagavi	belagavi	VERB
cana-4446	3	53	,	,	PUNCT
cana-4446	3	54	india.2	india.2	ADV
cana-4446	3	55	dr.ambedkar	dr.ambedkar	NOUN
cana-4446	3	56	institute	institute	NOUN
cana-4446	3	57	of	of	ADP
cana-4446	3	58	technology	technology	PROPN
cana-4446	3	59	,	,	PUNCT
cana-4446	3	60	bangalore	bangalore	PROPN
cana-4446	3	61	,	,	PUNCT
cana-4446	3	62	affiliated	affiliate	VERB
cana-4446	3	63	to	to	AUX
cana-4446	3	64	visvesvaraya	visvesvaraya	VERB
cana-4446	3	65	technological	technological	ADJ
cana-4446	3	66	university	university	NOUN
cana-4446	3	67	,	,	PUNCT
cana-4446	3	68	belagavi	belagavi	VERB
cana-4446	3	69	,	,	PUNCT
cana-4446	3	70	india.3	india.3	ADJ
cana-4446	3	71	article	article	NOUN
cana-4446	3	72	history	history	NOUN
cana-4446	3	73	:	:	PUNCT
cana-4446	3	74	received	receive	VERB
cana-4446	3	75	:	:	PUNCT
cana-4446	3	76	12	12	NUM
cana-4446	3	77	-	-	SYM
cana-4446	3	78	01	01	NUM
cana-4446	3	79	-	-	PUNCT
cana-4446	3	80	2025	2025	NUM
cana-4446	3	81	revised	revise	VERB
cana-4446	3	82	:	:	PUNCT
cana-4446	3	83	15	15	NUM
cana-4446	3	84	-	-	NUM
cana-4446	3	85	02	02	NUM
cana-4446	3	86	-	-	PUNCT
cana-4446	3	87	2025	2025	NUM
cana-4446	3	88	accepted	accept	VERB
cana-4446	3	89	:	:	PUNCT
cana-4446	3	90	01	01	NUM
cana-4446	3	91	-	-	SYM
cana-4446	3	92	03	03	NUM
cana-4446	3	93	-	-	PUNCT
cana-4446	3	94	2025	2025	NUM
cana-4446	3	95	abstract	abstract	NOUN
cana-4446	3	96	:	:	PUNCT
cana-4446	3	97	consider	consider	VERB
cana-4446	3	98	a	a	DET
cana-4446	3	99	connected	connected	ADJ
cana-4446	3	100	graph	graph	NOUN
cana-4446	3	101	that	that	PRON
cana-4446	3	102	is	be	AUX
cana-4446	3	103	nontrivial	nontrivial	ADJ
cana-4446	3	104	,	,	PUNCT
cana-4446	3	105	defined	define	VERB
cana-4446	3	106	as	as	ADP
cana-4446	3	107	a	a	DET
cana-4446	3	108	coloring	coloring	NOUN
cana-4446	3	109	c	c	NOUN
cana-4446	3	110	:	:	PUNCT
cana-4446	3	111	v	v	X
cana-4446	3	112	(	(	PUNCT
cana-4446	3	113	g	g	NOUN
cana-4446	3	114	)	)	PUNCT
cana-4446	3	115	→	→	SYM
cana-4446	3	116	{	{	PUNCT
cana-4446	3	117	1	1	NUM
cana-4446	3	118	,	,	PUNCT
cana-4446	3	119	2	2	NUM
cana-4446	3	120	,	,	PUNCT
cana-4446	3	121	.	.	PUNCT
cana-4446	3	122	.	.	PUNCT
cana-4446	3	123	.	.	PUNCT
cana-4446	4	1	.	.	PUNCT
cana-4446	5	1	,	,	PUNCT
cana-4446	5	2	k	k	X
cana-4446	5	3	}	}	PUNCT
cana-4446	5	4	,	,	PUNCT
cana-4446	5	5	k	k	PROPN
cana-4446	5	6	∈	∈	PROPN
cana-4446	5	7	n	n	PRON
cana-4446	5	8	of	of	ADP
cana-4446	5	9	the	the	DET
cana-4446	5	10	vertices	vertex	NOUN
cana-4446	5	11	of	of	ADP
cana-4446	5	12	g.	g.	PROPN
cana-4446	5	13	a	a	DET
cana-4446	5	14	rainbow	rainbow	NOUN
cana-4446	5	15	dynamic	dynamic	ADJ
cana-4446	5	16	coloring	coloring	NOUN
cana-4446	5	17	of	of	ADP
cana-4446	5	18	a	a	DET
cana-4446	5	19	graph	graph	NOUN
cana-4446	5	20	is	be	AUX
cana-4446	5	21	a	a	DET
cana-4446	5	22	dynamic	dynamic	ADJ
cana-4446	5	23	coloring	coloring	NOUN
cana-4446	5	24	,	,	PUNCT
cana-4446	5	25	and	and	CCONJ
cana-4446	5	26	a	a	DET
cana-4446	5	27	minimum	minimum	ADJ
cana-4446	5	28	number	number	NOUN
cana-4446	5	29	of	of	ADP
cana-4446	5	30	colors	color	NOUN
cana-4446	5	31	required	require	VERB
cana-4446	5	32	,	,	PUNCT
cana-4446	5	33	such	such	ADJ
cana-4446	5	34	that	that	SCONJ
cana-4446	5	35	every	every	DET
cana-4446	5	36	pair	pair	NOUN
cana-4446	5	37	of	of	ADP
cana-4446	5	38	vertices	vertex	NOUN
cana-4446	5	39	is	be	AUX
cana-4446	5	40	connected	connect	VERB
cana-4446	5	41	by	by	ADP
cana-4446	5	42	at	at	ADV
cana-4446	5	43	least	least	ADV
cana-4446	5	44	one	one	NUM
cana-4446	5	45	path	path	NOUN
cana-4446	5	46	whose	whose	DET
cana-4446	5	47	within	within	ADP
cana-4446	5	48	vertices	vertex	NOUN
cana-4446	5	49	have	have	VERB
cana-4446	5	50	different	different	ADJ
cana-4446	5	51	colors	color	NOUN
cana-4446	5	52	.	.	PUNCT
cana-4446	6	1	the	the	DET
cana-4446	6	2	minimum	minimum	PROPN
cana-4446	6	3	k	k	PROPN
cana-4446	6	4	for	for	ADP
cana-4446	6	5	which	which	PRON
cana-4446	6	6	k	k	ADJ
cana-4446	6	7	-	-	PUNCT
cana-4446	6	8	vertex	vertex	NOUN
cana-4446	6	9	coloring	coloring	NOUN
cana-4446	6	10	exists	exist	VERB
cana-4446	6	11	is	be	AUX
cana-4446	6	12	called	call	VERB
cana-4446	6	13	the	the	DET
cana-4446	6	14	rainbow	rainbow	NOUN
cana-4446	6	15	dynamic	dynamic	ADJ
cana-4446	6	16	coloring	coloring	NOUN
cana-4446	6	17	of	of	ADP
cana-4446	6	18	g	g	NOUN
cana-4446	6	19	,	,	PUNCT
cana-4446	6	20	denoted	denote	VERB
cana-4446	6	21	by	by	ADP
cana-4446	6	22	rdyc(g	rdyc(g	NOUN
cana-4446	6	23	)	)	PUNCT
cana-4446	6	24	.	.	PUNCT
cana-4446	7	1	in	in	ADP
cana-4446	7	2	this	this	DET
cana-4446	7	3	paper	paper	NOUN
cana-4446	7	4	,	,	PUNCT
cana-4446	7	5	we	we	PRON
cana-4446	7	6	determine	determine	VERB
cana-4446	7	7	the	the	DET
cana-4446	7	8	rdyc	rdyc	NOUN
cana-4446	7	9	of	of	ADP
cana-4446	7	10	some	some	DET
cana-4446	7	11	graphs	graph	NOUN
cana-4446	7	12	of	of	ADP
cana-4446	7	13	brick	brick	NOUN
cana-4446	7	14	products	product	NOUN
cana-4446	7	15	c(2n	c(2n	NOUN
cana-4446	7	16	,	,	PUNCT
cana-4446	7	17	m	m	PRON
cana-4446	7	18	,	,	PUNCT
cana-4446	7	19	r	r	NOUN
cana-4446	7	20	)	)	PUNCT
cana-4446	7	21	associated	associate	VERB
cana-4446	7	22	with	with	ADP
cana-4446	7	23	odd	odd	ADJ
cana-4446	7	24	cycles	cycle	NOUN
cana-4446	7	25	for	for	ADP
cana-4446	7	26	m	m	PROPN
cana-4446	7	27	=	=	SYM
cana-4446	7	28	1	1	X
cana-4446	7	29	.	.	PUNCT
cana-4446	7	30	objectives	objective	NOUN
cana-4446	7	31	:	:	PUNCT
cana-4446	7	32	to	to	PART
cana-4446	7	33	find	find	VERB
cana-4446	7	34	the	the	DET
cana-4446	7	35	rdyc	rdyc	NOUN
cana-4446	7	36	of	of	ADP
cana-4446	7	37	few	few	ADJ
cana-4446	7	38	brick	brick	NOUN
cana-4446	7	39	product	product	NOUN
cana-4446	7	40	graphs	graph	NOUN
cana-4446	7	41	.	.	PUNCT
cana-4446	8	1	conclusions	conclusion	NOUN
cana-4446	8	2	:	:	PUNCT
cana-4446	8	3	in	in	ADP
cana-4446	8	4	this	this	DET
cana-4446	8	5	paper	paper	NOUN
cana-4446	8	6	,	,	PUNCT
cana-4446	8	7	we	we	PRON
cana-4446	8	8	obtain	obtain	VERB
cana-4446	8	9	the	the	DET
cana-4446	8	10	rainbow	rainbow	NOUN
cana-4446	8	11	dynamic	dynamic	ADJ
cana-4446	8	12	coloring	coloring	NOUN
cana-4446	8	13	of	of	ADP
cana-4446	8	14	fewbrick	fewbrick	NOUN
cana-4446	8	15	product	product	NOUN
cana-4446	8	16	graphs	graph	VERB
cana-4446	8	17	c(2n	c(2n	NOUN
cana-4446	8	18	,	,	PUNCT
cana-4446	8	19	m	m	NOUN
cana-4446	8	20	,	,	PUNCT
cana-4446	8	21	r	r	NOUN
cana-4446	8	22	)	)	PUNCT
cana-4446	8	23	for	for	ADP
cana-4446	8	24	m	m	PROPN
cana-4446	8	25	=	=	SYM
cana-4446	8	26	1	1	NUM
cana-4446	8	27	and	and	CCONJ
cana-4446	8	28	r	r	NOUN
cana-4446	8	29	=	=	SYM
cana-4446	8	30	3	3	NUM
cana-4446	8	31	,	,	PUNCT
cana-4446	8	32	5	5	NUM
cana-4446	8	33	,	,	PUNCT
cana-4446	8	34	7	7	NUM
cana-4446	8	35	.	.	PUNCT
cana-4446	9	1	keywords	keyword	NOUN
cana-4446	9	2	:	:	PUNCT
cana-4446	9	3	rainbow	rainbow	VERB
cana-4446	9	4	vertex	vertex	NOUN
cana-4446	9	5	connection	connection	NOUN
cana-4446	9	6	number	number	NOUN
cana-4446	9	7	,	,	PUNCT
cana-4446	9	8	dynamiccoloring	dynamiccoloring	NOUN
cana-4446	9	9	,	,	PUNCT
cana-4446	9	10	brick	brick	NOUN
cana-4446	9	11	product	product	NOUN
cana-4446	9	12	,	,	PUNCT
cana-4446	9	13	rainbow	rainbow	VERB
cana-4446	9	14	dynamic	dynamic	ADJ
cana-4446	9	15	coloring	coloring	NOUN
cana-4446	9	16	.	.	PUNCT
cana-4446	10	1	1	1	X
cana-4446	10	2	.	.	X
cana-4446	10	3	introduction	introduction	NOUN
cana-4446	10	4	all	all	DET
cana-4446	10	5	graphs	graph	NOUN
cana-4446	10	6	considered	consider	VERB
cana-4446	10	7	in	in	ADP
cana-4446	10	8	this	this	DET
cana-4446	10	9	paper	paper	NOUN
cana-4446	10	10	are	be	AUX
cana-4446	10	11	simple	simple	ADJ
cana-4446	10	12	,	,	PUNCT
cana-4446	10	13	finite	finite	ADJ
cana-4446	10	14	,	,	PUNCT
cana-4446	10	15	and	and	CCONJ
cana-4446	10	16	undirected	undirected	ADJ
cana-4446	10	17	.	.	PUNCT
cana-4446	11	1	let	let	VERB
cana-4446	11	2	g	g	PRON
cana-4446	11	3	be	be	AUX
cana-4446	11	4	non	non	ADJ
cana-4446	11	5	-	-	ADJ
cana-4446	11	6	trivial	trivial	ADJ
cana-4446	11	7	with	with	ADP
cana-4446	11	8	a	a	DET
cana-4446	11	9	vertex	vertex	NOUN
cana-4446	11	10	coloring	coloring	NOUN
cana-4446	11	11	c	c	NOUN
cana-4446	11	12	:	:	PUNCT
cana-4446	11	13	v	v	X
cana-4446	11	14	(	(	PUNCT
cana-4446	11	15	g	g	NOUN
cana-4446	11	16	)	)	PUNCT
cana-4446	11	17	→	→	SYM
cana-4446	11	18	{	{	PUNCT
cana-4446	11	19	1	1	NUM
cana-4446	11	20	,	,	PUNCT
cana-4446	11	21	2	2	NUM
cana-4446	11	22	,	,	PUNCT
cana-4446	11	23	.	.	PUNCT
cana-4446	11	24	.	.	PUNCT
cana-4446	11	25	.	.	PUNCT
cana-4446	12	1	.	.	PUNCT
cana-4446	13	1	,	,	PUNCT
cana-4446	13	2	k	k	X
cana-4446	13	3	}	}	PUNCT
cana-4446	13	4	,	,	PUNCT
cana-4446	13	5	k	k	PROPN
cana-4446	13	6	∈	∈	PROPN
cana-4446	13	7	n.	n.	NOUN
cana-4446	13	8	in	in	ADP
cana-4446	13	9	a	a	DET
cana-4446	13	10	proper	proper	ADJ
cana-4446	13	11	vertex	vertex	NOUN
cana-4446	13	12	-	-	PUNCT
cana-4446	13	13	colored	color	VERB
cana-4446	13	14	graph	graph	NOUN
cana-4446	13	15	g	g	NOUN
cana-4446	13	16	,	,	PUNCT
cana-4446	13	17	a	a	DET
cana-4446	13	18	path	path	NOUN
cana-4446	13	19	p	p	NOUN
cana-4446	13	20	is	be	AUX
cana-4446	13	21	in	in	ADP
cana-4446	13	22	a	a	DET
cana-4446	13	23	rainbow	rainbow	NOUN
cana-4446	13	24	path	path	NOUN
cana-4446	13	25	if	if	SCONJ
cana-4446	13	26	no	no	DET
cana-4446	13	27	two	two	NUM
cana-4446	13	28	vertices	vertex	NOUN
cana-4446	13	29	p	p	NOUN
cana-4446	13	30	are	be	AUX
cana-4446	13	31	of	of	ADP
cana-4446	13	32	the	the	DET
cana-4446	13	33	same	same	ADJ
cana-4446	13	34	color	color	NOUN
cana-4446	13	35	,	,	PUNCT
cana-4446	13	36	except	except	SCONJ
cana-4446	13	37	possibly	possibly	ADV
cana-4446	13	38	the	the	DET
cana-4446	13	39	end	end	NOUN
cana-4446	13	40	vertices	vertex	NOUN
cana-4446	13	41	of	of	ADP
cana-4446	13	42	p.	p.	NOUN
cana-4446	13	43	if	if	SCONJ
cana-4446	13	44	a	a	DET
cana-4446	13	45	rainbow	rainbow	NOUN
cana-4446	13	46	path	path	NOUN
cana-4446	13	47	connects	connect	VERB
cana-4446	13	48	every	every	DET
cana-4446	13	49	two	two	NUM
cana-4446	13	50	vertices	vertex	NOUN
cana-4446	13	51	of	of	ADP
cana-4446	13	52	g	g	NOUN
cana-4446	13	53	,	,	PUNCT
cana-4446	13	54	then	then	ADV
cana-4446	13	55	g	g	PROPN
cana-4446	13	56	is	be	AUX
cana-4446	13	57	a	a	DET
cana-4446	13	58	rainbow	rainbow	NOUN
cana-4446	13	59	connected	connect	VERB
cana-4446	13	60	to	to	ADP
cana-4446	13	61	the	the	DET
cana-4446	13	62	vertex	vertex	NOUN
cana-4446	13	63	.	.	PUNCT
cana-4446	14	1	a	a	DET
cana-4446	14	2	proper	proper	ADJ
cana-4446	14	3	vertex	vertex	NOUN
cana-4446	14	4	coloring	coloring	NOUN
cana-4446	14	5	of	of	ADP
cana-4446	14	6	a	a	DET
cana-4446	14	7	connected	connected	ADJ
cana-4446	14	8	graph	graph	NOUN
cana-4446	14	9	g	g	NOUN
cana-4446	14	10	that	that	PRON
cana-4446	14	11	results	result	VERB
cana-4446	14	12	in	in	ADP
cana-4446	14	13	a	a	DET
cana-4446	14	14	connected	connected	ADJ
cana-4446	14	15	graph	graph	NOUN
cana-4446	14	16	with	with	ADP
cana-4446	14	17	vertex	vertex	NOUN
cana-4446	14	18	rainbow	rainbow	NOUN
cana-4446	14	19	is	be	AUX
cana-4446	14	20	a	a	DET
cana-4446	14	21	rainbow	rainbow	NOUN
cana-4446	14	22	vertex	vertex	NOUN
cana-4446	14	23	coloring	coloring	NOUN
cana-4446	14	24	of	of	ADP
cana-4446	14	25	g.	g.	PROPN
cana-4446	14	26	the	the	DET
cana-4446	14	27	minimum	minimum	ADJ
cana-4446	14	28	number	number	NOUN
cana-4446	14	29	of	of	ADP
cana-4446	14	30	colors	color	NOUN
cana-4446	14	31	needed	need	VERB
cana-4446	14	32	for	for	ADP
cana-4446	14	33	the	the	DET
cana-4446	14	34	vertex	vertex	NOUN
cana-4446	14	35	rainbow	rainbow	NOUN
cana-4446	14	36	coloring	coloring	NOUN
cana-4446	14	37	of	of	ADP
cana-4446	14	38	g	g	PROPN
cana-4446	14	39	is	be	AUX
cana-4446	14	40	the	the	DET
cana-4446	14	41	vertex	vertex	NOUN
cana-4446	14	42	rainbow	rainbow	NOUN
cana-4446	14	43	connection	connection	NOUN
cana-4446	14	44	number	number	NOUN
cana-4446	14	45	rvc(g	rvc(g	PROPN
cana-4446	14	46	)	)	PUNCT
cana-4446	14	47	.	.	PUNCT
cana-4446	15	1	bruce	bruce	PROPN
cana-4446	15	2	montgomery	montgomery	PROPN
cana-4446	15	3	introduced	introduce	VERB
cana-4446	15	4	a	a	DET
cana-4446	15	5	relatively	relatively	ADV
cana-4446	15	6	new	new	ADJ
cana-4446	15	7	concept	concept	NOUN
cana-4446	15	8	in	in	ADP
cana-4446	15	9	vertex	vertex	NOUN
cana-4446	15	10	coloring	coloring	NOUN
cana-4446	15	11	,	,	PUNCT
cana-4446	15	12	called	call	VERB
cana-4446	15	13	dynamic	dynamic	ADJ
cana-4446	15	14	coloring	coloring	NOUN
cana-4446	15	15	,	,	PUNCT
cana-4446	15	16	in	in	ADP
cana-4446	15	17	2001	2001	NUM
cana-4446	15	18	[	[	X
cana-4446	15	19	2	2	NUM
cana-4446	15	20	]	]	PUNCT
cana-4446	15	21	.	.	PUNCT
cana-4446	16	1	a	a	DET
cana-4446	16	2	dynamic	dynamic	ADJ
cana-4446	16	3	graph	graph	NOUN
cana-4446	16	4	coloring	coloring	NOUN
cana-4446	16	5	is	be	AUX
cana-4446	16	6	a	a	DET
cana-4446	16	7	proper	proper	ADJ
cana-4446	16	8	coloring	coloring	NOUN
cana-4446	16	9	of	of	ADP
cana-4446	16	10	the	the	DET
cana-4446	16	11	set	set	NOUN
cana-4446	16	12	of	of	ADP
cana-4446	16	13	vertex	vertex	NOUN
cana-4446	16	14	such	such	ADJ
cana-4446	16	15	that	that	SCONJ
cana-4446	16	16	each	each	DET
cana-4446	16	17	vertex	vertex	NOUN
cana-4446	16	18	of	of	ADP
cana-4446	16	19	degree	degree	NOUN
cana-4446	16	20	at	at	ADV
cana-4446	16	21	least	least	ADJ
cana-4446	16	22	two	two	NUM
cana-4446	16	23	of	of	ADP
cana-4446	16	24	its	its	PRON
cana-4446	16	25	neighbors	neighbor	NOUN
cana-4446	16	26	receives	receive	VERB
cana-4446	16	27	at	at	ADV
cana-4446	16	28	least	least	ADV
cana-4446	16	29	two	two	NUM
cana-4446	16	30	different	different	ADJ
cana-4446	16	31	colors	color	NOUN
cana-4446	16	32	.	.	PUNCT
cana-4446	17	1	krivelevich	krivelevich	PROPN
cana-4446	17	2	and	and	CCONJ
cana-4446	17	3	yuster	yuster	PROPN
cana-4446	17	4	introduced	introduce	VERB
cana-4446	17	5	a	a	DET
cana-4446	17	6	rainbow	rainbow	NOUN
cana-4446	17	7	vertex	vertex	NOUN
cana-4446	17	8	coloring	coloring	NOUN
cana-4446	17	9	concept	concept	NOUN
cana-4446	17	10	in	in	ADP
cana-4446	17	11	2010	2010	NUM
cana-4446	17	12	[	[	X
cana-4446	17	13	3	3	NUM
cana-4446	17	14	]	]	PUNCT
cana-4446	17	15	.	.	PUNCT
cana-4446	18	1	a	a	DET
cana-4446	18	2	rainbow	rainbow	NOUN
cana-4446	18	3	vertex	vertex	NOUN
cana-4446	18	4	connection	connection	NOUN
cana-4446	18	5	number	number	NOUN
cana-4446	18	6	,	,	PUNCT
cana-4446	18	7	rvc(g	rvc(g	PROPN
cana-4446	18	8	)	)	PUNCT
cana-4446	18	9	of	of	ADP
cana-4446	18	10	a	a	DET
cana-4446	18	11	connected	connected	ADJ
cana-4446	18	12	graph	graph	NOUN
cana-4446	18	13	,	,	PUNCT
cana-4446	18	14	is	be	AUX
cana-4446	18	15	the	the	DET
cana-4446	18	16	minimum	minimum	ADJ
cana-4446	18	17	number	number	NOUN
cana-4446	18	18	of	of	ADP
cana-4446	18	19	colors	color	NOUN
cana-4446	18	20	required	require	VERB
cana-4446	18	21	to	to	PART
cana-4446	18	22	color	color	VERB
cana-4446	18	23	its	its	PRON
cana-4446	18	24	vertices	vertex	NOUN
cana-4446	18	25	.	.	PUNCT
cana-4446	19	1	every	every	DET
cana-4446	19	2	pair	pair	NOUN
cana-4446	19	3	of	of	ADP
cana-4446	19	4	vertices	vertex	NOUN
cana-4446	19	5	is	be	AUX
cana-4446	19	6	connected	connect	VERB
cana-4446	19	7	by	by	ADP
cana-4446	19	8	at	at	ADV
cana-4446	19	9	least	least	ADV
cana-4446	19	10	one	one	NUM
cana-4446	19	11	path	path	NOUN
cana-4446	19	12	,	,	PUNCT
cana-4446	19	13	which	which	PRON
cana-4446	19	14	has	have	VERB
cana-4446	19	15	different	different	ADJ
cana-4446	19	16	colors	color	NOUN
cana-4446	19	17	within	within	ADP
cana-4446	19	18	the	the	DET
cana-4446	19	19	vertices	vertex	NOUN
cana-4446	19	20	.	.	PUNCT
cana-4446	20	1	a	a	DET
cana-4446	20	2	rainbow	rainbow	NOUN
cana-4446	20	3	dynamic	dynamic	ADJ
cana-4446	20	4	coloring	coloring	NOUN
cana-4446	20	5	of	of	ADP
cana-4446	20	6	a	a	DET
cana-4446	20	7	graph	graph	NOUN
cana-4446	20	8	is	be	AUX
cana-4446	20	9	not	not	PART
cana-4446	20	10	just	just	ADV
cana-4446	20	11	a	a	DET
cana-4446	20	12	theoretical	theoretical	ADJ
cana-4446	20	13	concept	concept	NOUN
cana-4446	20	14	but	but	CCONJ
cana-4446	20	15	a	a	DET
cana-4446	20	16	practical	practical	ADJ
cana-4446	20	17	one	one	NUM
cana-4446	20	18	.	.	PUNCT
cana-4446	21	1	it	it	PRON
cana-4446	21	2	is	be	AUX
cana-4446	21	3	a	a	DET
cana-4446	21	4	dynamic	dynamic	ADJ
cana-4446	21	5	coloring	coloring	NOUN
cana-4446	21	6	,	,	PUNCT
cana-4446	21	7	and	and	CCONJ
cana-4446	21	8	a	a	DET
cana-4446	21	9	minimum	minimum	ADJ
cana-4446	21	10	number	number	NOUN
cana-4446	21	11	of	of	ADP
cana-4446	21	12	colors	color	NOUN
cana-4446	21	13	is	be	AUX
cana-4446	21	14	required	require	VERB
cana-4446	21	15	such	such	ADJ
cana-4446	21	16	that	that	SCONJ
cana-4446	21	17	every	every	DET
cana-4446	21	18	pair	pair	NOUN
cana-4446	21	19	of	of	ADP
cana-4446	21	20	vertices	vertex	NOUN
cana-4446	21	21	is	be	AUX
cana-4446	21	22	connected	connect	VERB
cana-4446	21	23	by	by	ADP
cana-4446	21	24	at	at	ADV
cana-4446	21	25	least	least	ADV
cana-4446	21	26	one	one	NUM
cana-4446	21	27	path	path	NOUN
cana-4446	21	28	whose	whose	DET
cana-4446	21	29	within	within	ADP
cana-4446	21	30	vertices	vertex	NOUN
cana-4446	21	31	have	have	VERB
cana-4446	21	32	different	different	ADJ
cana-4446	21	33	colors	color	NOUN
cana-4446	21	34	.	.	PUNCT
cana-4446	22	1	the	the	DET
cana-4446	22	2	minimum	minimum	PROPN
cana-4446	22	3	k	k	PROPN
cana-4446	22	4	for	for	ADP
cana-4446	22	5	which	which	PRON
cana-4446	22	6	k	k	ADJ
cana-4446	22	7	-	-	PUNCT
cana-4446	22	8	vertex	vertex	NOUN
cana-4446	22	9	coloring	coloring	NOUN
cana-4446	22	10	exists	exist	VERB
cana-4446	22	11	is	be	AUX
cana-4446	22	12	called	call	VERB
cana-4446	22	13	the	the	DET
cana-4446	22	14	dynamic	dynamic	ADJ
cana-4446	22	15	rainbow	rainbow	NOUN
cana-4446	22	16	coloring	coloring	NOUN
cana-4446	22	17	of	of	ADP
cana-4446	22	18	g	g	NOUN
cana-4446	22	19	,	,	PUNCT
cana-4446	22	20	denoted	denote	VERB
cana-4446	22	21	by	by	ADP
cana-4446	22	22	rdyc(g	rdyc(g	NOUN
cana-4446	22	23	)	)	PUNCT
cana-4446	22	24	.	.	PUNCT
cana-4446	23	1	we	we	PRON
cana-4446	23	2	define	define	VERB
cana-4446	23	3	a	a	DET
cana-4446	23	4	brick	brick	NOUN
cana-4446	23	5	product	product	NOUN
cana-4446	23	6	graph	graph	NOUN
cana-4446	23	7	associated	associate	VERB
cana-4446	23	8	with	with	ADP
cana-4446	23	9	an	an	DET
cana-4446	23	10	odd	odd	ADJ
cana-4446	23	11	cycle	cycle	NOUN
cana-4446	23	12	.	.	PUNCT
cana-4446	24	1	communications	communication	NOUN
cana-4446	24	2	on	on	ADP
cana-4446	24	3	applied	apply	VERB
cana-4446	24	4	nonlinear	nonlinear	ADJ
cana-4446	24	5	analysis	analysis	NOUN
cana-4446	24	6	issn	issn	NOUN
cana-4446	24	7	:	:	PUNCT
cana-4446	24	8	1074	1074	NUM
cana-4446	24	9	-	-	PUNCT
cana-4446	24	10	133x	133x	NUM
cana-4446	24	11	vol	vol	NOUN
cana-4446	24	12	32	32	NUM
cana-4446	24	13	no	no	NOUN
cana-4446	24	14	.	.	PUNCT
cana-4446	25	1	9s	9s	NUM
cana-4446	25	2	(	(	PUNCT
cana-4446	25	3	2025	2025	NUM
cana-4446	25	4	)	)	PUNCT
cana-4446	25	5	2045	2045	NUM
cana-4446	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4446	25	7	2	2	X
cana-4446	25	8	.	.	NOUN
cana-4446	25	9	results	result	NOUN
cana-4446	25	10	definition	definition	NOUN
cana-4446	25	11	let	let	VERB
cana-4446	25	12	m	m	PRON
cana-4446	25	13	,	,	PUNCT
cana-4446	25	14	n	n	PROPN
cana-4446	25	15	and	and	CCONJ
cana-4446	25	16	r	r	NOUN
cana-4446	25	17	be	be	AUX
cana-4446	25	18	positive	positive	ADJ
cana-4446	25	19	integers	integer	NOUN
cana-4446	25	20	.	.	PUNCT
cana-4446	26	1	a	a	DET
cana-4446	26	2	cycle	cycle	NOUN
cana-4446	26	3	of	of	ADP
cana-4446	26	4	order	order	NOUN
cana-4446	26	5	2n	2n	NUM
cana-4446	26	6	is	be	AUX
cana-4446	26	7	indicated	indicate	VERB
cana-4446	26	8	by	by	ADP
cana-4446	26	9	the	the	DET
cana-4446	26	10	notation	notation	NOUN
cana-4446	26	11	𝐶2𝑛	𝐶2𝑛	PROPN
cana-4446	26	12	=	=	SYM
cana-4446	26	13	𝑣1	𝑣1	PROPN
cana-4446	26	14	,	,	PUNCT
cana-4446	26	15	𝑣2	𝑣2	PROPN
cana-4446	26	16	,	,	PUNCT
cana-4446	26	17	.	.	PUNCT
cana-4446	26	18	.	.	PUNCT
cana-4446	26	19	.	.	PUNCT
cana-4446	27	1	.	.	PUNCT
cana-4446	28	1	𝑣2𝑛−1	𝑣2𝑛−1	NOUN
cana-4446	28	2	,	,	PUNCT
cana-4446	28	3	𝑣2𝑛	𝑣2𝑛	PROPN
cana-4446	28	4	=	=	SYM
cana-4446	28	5	𝑣1	𝑣1	PROPN
cana-4446	28	6	.	.	PUNCT
cana-4446	29	1	the	the	DET
cana-4446	29	2	brick	brick	NOUN
cana-4446	29	3	product	product	NOUN
cana-4446	29	4	(	(	PUNCT
cana-4446	29	5	m	m	NOUN
cana-4446	29	6	,	,	PUNCT
cana-4446	29	7	r	r	NOUN
cana-4446	29	8	)	)	PUNCT
cana-4446	29	9	of	of	ADP
cana-4446	29	10	𝐶2𝑛	𝐶2𝑛	PROPN
cana-4446	29	11	represented	represent	VERB
cana-4446	29	12	by	by	ADP
cana-4446	29	13	c(2n	c(2n	NOUN
cana-4446	29	14	,	,	PUNCT
cana-4446	29	15	m	m	PRON
cana-4446	29	16	,	,	PUNCT
cana-4446	29	17	r	r	NOUN
cana-4446	29	18	)	)	PUNCT
cana-4446	29	19	is	be	AUX
cana-4446	29	20	defined	define	VERB
cana-4446	29	21	in	in	ADP
cana-4446	29	22	the	the	DET
cana-4446	29	23	following	following	ADJ
cana-4446	29	24	way	way	NOUN
cana-4446	29	25	in	in	ADP
cana-4446	29	26	two	two	NUM
cana-4446	29	27	cases	case	NOUN
cana-4446	29	28	.	.	PUNCT
cana-4446	30	1	1	1	X
cana-4446	30	2	.	.	X
cana-4446	30	3	we	we	PRON
cana-4446	30	4	need	need	VERB
cana-4446	30	5	r	r	NOUN
cana-4446	30	6	to	to	PART
cana-4446	30	7	be	be	AUX
cana-4446	30	8	odd	odd	ADJ
cana-4446	30	9	and	and	CCONJ
cana-4446	30	10	greater	great	ADJ
cana-4446	30	11	than	than	ADP
cana-4446	30	12	1	1	NUM
cana-4446	30	13	for	for	ADP
cana-4446	30	14	m	m	PROPN
cana-4446	30	15	=	=	NOUN
cana-4446	30	16	1	1	NUM
cana-4446	30	17	.	.	PUNCT
cana-4446	31	1	next	next	ADJ
cana-4446	31	2	,	,	PUNCT
cana-4446	31	3	from𝐶2𝑛	from𝐶2𝑛	PROPN
cana-4446	31	4	,	,	PUNCT
cana-4446	31	5	c(2n	c(2n	NOUN
cana-4446	31	6	,	,	PUNCT
cana-4446	31	7	m	m	NOUN
cana-4446	31	8	,	,	PUNCT
cana-4446	31	9	r	r	NOUN
cana-4446	31	10	)	)	PUNCT
cana-4446	31	11	can	can	AUX
cana-4446	31	12	be	be	AUX
cana-4446	31	13	determined	determine	VERB
cana-4446	31	14	by	by	ADP
cana-4446	31	15	adding	add	VERB
cana-4446	31	16	chords	chord	NOUN
cana-4446	31	17	𝑣2𝑘(𝑣2𝑘+𝑟	𝑣2𝑘(𝑣2𝑘+𝑟	PROPN
cana-4446	31	18	)	)	PUNCT
cana-4446	31	19	,	,	PUNCT
cana-4446	31	20	is	be	AUX
cana-4446	31	21	obtained	obtain	VERB
cana-4446	31	22	by	by	ADP
cana-4446	31	23	adding	add	VERB
cana-4446	31	24	chords𝑣2𝑘(𝑣2𝑘+𝑟	chords𝑣2𝑘(𝑣2𝑘+𝑟	PROPN
cana-4446	31	25	)	)	PUNCT
cana-4446	31	26	,	,	PUNCT
cana-4446	32	1	k	k	X
cana-4446	32	2	=	=	SYM
cana-4446	32	3	1	1	NUM
cana-4446	32	4	,	,	PUNCT
cana-4446	32	5	2	2	NUM
cana-4446	32	6	,	,	PUNCT
cana-4446	32	7	...	...	PUNCT
cana-4446	32	8	,	,	PUNCT
cana-4446	32	9	n	n	CCONJ
cana-4446	32	10	,	,	PUNCT
cana-4446	32	11	where	where	SCONJ
cana-4446	32	12	the	the	DET
cana-4446	32	13	computation	computation	NOUN
cana-4446	32	14	is	be	AUX
cana-4446	32	15	performed	perform	VERB
cana-4446	32	16	modulo	modulo	PROPN
cana-4446	32	17	2n	2n	NUM
cana-4446	32	18	.	.	PUNCT
cana-4446	33	1	2	2	X
cana-4446	33	2	.	.	X
cana-4446	33	3	we	we	PRON
cana-4446	33	4	need	need	VERB
cana-4446	33	5	m	m	VERB
cana-4446	33	6	+	+	NOUN
cana-4446	33	7	r	r	NOUN
cana-4446	33	8	to	to	PART
cana-4446	33	9	be	be	AUX
cana-4446	33	10	even	even	ADV
cana-4446	33	11	for	for	ADP
cana-4446	33	12	m	m	PROPN
cana-4446	33	13	>	>	X
cana-4446	33	14	1	1	NUM
cana-4446	33	15	.	.	PUNCT
cana-4446	34	1	then	then	ADV
cana-4446	34	2	c(2n	c(2n	NOUN
cana-4446	34	3	,	,	PUNCT
cana-4446	34	4	m	m	PRON
cana-4446	34	5	,	,	PUNCT
cana-4446	34	6	r	r	NOUN
cana-4446	34	7	)	)	PUNCT
cana-4446	34	8	is	be	AUX
cana-4446	34	9	obtained	obtain	VERB
cana-4446	34	10	by	by	ADP
cana-4446	34	11	first	first	ADV
cana-4446	34	12	taking	take	VERB
cana-4446	34	13	the	the	DET
cana-4446	34	14	disjoint	disjoint	NOUN
cana-4446	34	15	union	union	NOUN
cana-4446	34	16	of	of	ADP
cana-4446	34	17	m	m	PROPN
cana-4446	34	18	copies	copy	NOUN
cana-4446	34	19	of	of	ADP
cana-4446	34	20	𝐶2𝑛	𝐶2𝑛	PROPN
cana-4446	34	21	,	,	PUNCT
cana-4446	34	22	that	that	ADV
cana-4446	34	23	is	is	ADV
cana-4446	34	24	,	,	PUNCT
cana-4446	34	25	𝐶2𝑛(1	𝐶2𝑛(1	PROPN
cana-4446	34	26	)	)	PUNCT
cana-4446	34	27	,	,	PUNCT
cana-4446	34	28	𝐶2𝑛(2	𝐶2𝑛(2	PROPN
cana-4446	34	29	)	)	PUNCT
cana-4446	34	30	,	,	PUNCT
cana-4446	34	31	....	....	PUNCT
cana-4446	34	32	,	,	PUNCT
cana-4446	34	33	𝐶2𝑛(m	𝐶2𝑛(m	NOUN
cana-4446	34	34	)	)	PUNCT
cana-4446	34	35	where	where	SCONJ
cana-4446	34	36	for	for	ADP
cana-4446	34	37	every	every	DET
cana-4446	34	38	i	i	NOUN
cana-4446	34	39	=	=	NOUN
cana-4446	34	40	1	1	NUM
cana-4446	34	41	,	,	PUNCT
cana-4446	34	42	2	2	NUM
cana-4446	34	43	,	,	PUNCT
cana-4446	34	44	...	...	PUNCT
cana-4446	34	45	,	,	PUNCT
cana-4446	34	46	m	m	PROPN
cana-4446	34	47	,	,	PUNCT
cana-4446	34	48	𝐶2𝑛(i	𝐶2𝑛(i	NOUN
cana-4446	34	49	)	)	PUNCT
cana-4446	35	1	=	=	PUNCT
cana-4446	35	2	𝑣𝑖1	𝑣𝑖1	NOUN
cana-4446	35	3	,	,	PUNCT
cana-4446	35	4	𝑣𝑖1,,𝑣𝑖1	𝑣𝑖1,,𝑣𝑖1	NUM
cana-4446	35	5	.	.	PUNCT
cana-4446	35	6	.	.	PUNCT
cana-4446	35	7	.	.	PUNCT
cana-4446	35	8	.	.	PUNCT
cana-4446	35	9	.	.	PUNCT
cana-4446	35	10	.	.	PUNCT
cana-4446	35	11	.	.	PUNCT
cana-4446	36	1	𝑣𝑖2𝑛	𝑣𝑖2𝑛	ADV
cana-4446	36	2	,	,	PUNCT
cana-4446	36	3	.	.	PUNCT
cana-4446	37	1	next	next	ADV
cana-4446	37	2	,	,	PUNCT
cana-4446	37	3	an	an	DET
cana-4446	37	4	edge	edge	NOUN
cana-4446	37	5	(	(	PUNCT
cana-4446	37	6	also	also	ADV
cana-4446	37	7	known	know	VERB
cana-4446	37	8	as	as	ADP
cana-4446	37	9	a	a	DET
cana-4446	37	10	brick	brick	NOUN
cana-4446	37	11	edge	edge	NOUN
cana-4446	37	12	)	)	PUNCT
cana-4446	37	13	is	be	AUX
cana-4446	37	14	drawn	draw	VERB
cana-4446	37	15	to	to	PART
cana-4446	37	16	join	join	VERB
cana-4446	37	17	(	(	PUNCT
cana-4446	37	18	𝑣𝑖	𝑣𝑖	ADP
cana-4446	37	19	,	,	PUNCT
cana-4446	37	20	,	,	PUNCT
cana-4446	37	21	𝑣𝑘	𝑣𝑘	INTJ
cana-4446	37	22	,	,	PUNCT
cana-4446	37	23	)	)	PUNCT
cana-4446	37	24	to	to	PART
cana-4446	37	25	(	(	PUNCT
cana-4446	37	26	𝑣𝑖+1	𝑣𝑖+1	PROPN
cana-4446	37	27	,	,	PUNCT
cana-4446	37	28	,	,	PUNCT
cana-4446	37	29	𝑣𝑘	𝑣𝑘	INTJ
cana-4446	37	30	,	,	PUNCT
cana-4446	37	31	)	)	PUNCT
cana-4446	37	32	for	for	ADP
cana-4446	37	33	each	each	DET
cana-4446	37	34	odd	odd	ADJ
cana-4446	37	35	i	i	NOUN
cana-4446	37	36	=	=	NOUN
cana-4446	37	37	1	1	NUM
cana-4446	37	38	,	,	PUNCT
cana-4446	37	39	2	2	NUM
cana-4446	37	40	,	,	PUNCT
cana-4446	37	41	....	....	PUNCT
cana-4446	37	42	,	,	PUNCT
cana-4446	37	43	m	m	VERB
cana-4446	37	44	−	−	NOUN
cana-4446	37	45	1	1	NUM
cana-4446	37	46	and	and	CCONJ
cana-4446	37	47	each	each	PRON
cana-4446	37	48	even	even	ADV
cana-4446	37	49	k	k	PROPN
cana-4446	37	50	=	=	SYM
cana-4446	37	51	0	0	NUM
cana-4446	37	52	,	,	PUNCT
cana-4446	37	53	1	1	NUM
cana-4446	37	54	,	,	PUNCT
cana-4446	37	55	2	2	NUM
cana-4446	37	56	,	,	PUNCT
cana-4446	37	57	.....	.....	PUNCT
cana-4446	37	58	,	,	PUNCT
cana-4446	37	59	2n	2n	NUM
cana-4446	37	60	−	−	ADP
cana-4446	37	61	2	2	NUM
cana-4446	37	62	,	,	PUNCT
cana-4446	37	63	.	.	PUNCT
cana-4446	38	1	similarly	similarly	ADV
cana-4446	38	2	,	,	PUNCT
cana-4446	38	3	for	for	ADP
cana-4446	38	4	each	each	DET
cana-4446	38	5	even	even	ADV
cana-4446	38	6	i	i	NOUN
cana-4446	38	7	=	=	NOUN
cana-4446	38	8	1	1	NUM
cana-4446	38	9	,	,	PUNCT
cana-4446	38	10	2	2	NUM
cana-4446	38	11	,	,	PUNCT
cana-4446	38	12	....	....	PUNCT
cana-4446	38	13	m	m	VERB
cana-4446	38	14	−	−	NOUN
cana-4446	38	15	1	1	NUM
cana-4446	38	16	and	and	CCONJ
cana-4446	38	17	each	each	DET
cana-4446	38	18	odd	odd	ADJ
cana-4446	38	19	k	k	NOUN
cana-4446	38	20	=	=	SYM
cana-4446	38	21	1	1	NUM
cana-4446	38	22	,	,	PUNCT
cana-4446	38	23	2	2	NUM
cana-4446	38	24	,	,	PUNCT
cana-4446	38	25	...	...	PUNCT
cana-4446	38	26	,	,	PUNCT
cana-4446	38	27	2n	2n	NUM
cana-4446	38	28	−	−	NOUN
cana-4446	38	29	1	1	NUM
cana-4446	38	30	,	,	PUNCT
cana-4446	38	31	an	an	DET
cana-4446	38	32	edge	edge	NOUN
cana-4446	38	33	(	(	PUNCT
cana-4446	38	34	also	also	ADV
cana-4446	38	35	known	know	VERB
cana-4446	38	36	as	as	ADP
cana-4446	38	37	a	a	DET
cana-4446	38	38	brick	brick	NOUN
cana-4446	38	39	edge	edge	NOUN
cana-4446	38	40	)	)	PUNCT
cana-4446	38	41	is	be	AUX
cana-4446	38	42	drawn	draw	VERB
cana-4446	38	43	to	to	PART
cana-4446	38	44	join	join	VERB
cana-4446	38	45	(	(	PUNCT
cana-4446	38	46	𝑣𝑖+1	𝑣𝑖+1	PROPN
cana-4446	38	47	,	,	PUNCT
cana-4446	38	48	,	,	PUNCT
cana-4446	38	49	𝑣𝑘	𝑣𝑘	INTJ
cana-4446	38	50	,	,	PUNCT
cana-4446	38	51	)	)	PUNCT
cana-4446	38	52	.	.	PUNCT
cana-4446	39	1	lastly	lastly	ADV
cana-4446	39	2	,	,	PUNCT
cana-4446	39	3	an	an	DET
cana-4446	39	4	edge	edge	NOUN
cana-4446	39	5	(	(	PUNCT
cana-4446	39	6	referred	refer	VERB
cana-4446	39	7	to	to	ADP
cana-4446	39	8	as	as	ADP
cana-4446	39	9	a	a	DET
cana-4446	39	10	hooking	hooking	NOUN
cana-4446	39	11	edge	edge	NOUN
cana-4446	39	12	)	)	PUNCT
cana-4446	39	13	is	be	AUX
cana-4446	39	14	created	create	VERB
cana-4446	39	15	to	to	PART
cana-4446	39	16	join	join	VERB
cana-4446	39	17	(	(	PUNCT
cana-4446	39	18	𝑣1	𝑣1	PROPN
cana-4446	39	19	,	,	PUNCT
cana-4446	39	20	,	,	PUNCT
cana-4446	39	21	𝑣𝑘	𝑣𝑘	INTJ
cana-4446	39	22	,	,	PUNCT
cana-4446	39	23	)	)	PUNCT
cana-4446	39	24	to	to	PART
cana-4446	39	25	(	(	PUNCT
cana-4446	39	26	𝑣𝑚	𝑣𝑚	ADJ
cana-4446	39	27	,	,	PUNCT
cana-4446	39	28	,	,	PUNCT
cana-4446	39	29	𝑣𝑘+𝑟	𝑣𝑘+𝑟	NUM
cana-4446	39	30	,	,	PUNCT
cana-4446	39	31	)	)	PUNCT
cana-4446	39	32	for	for	ADP
cana-4446	39	33	each	each	DET
cana-4446	39	34	odd	odd	ADJ
cana-4446	39	35	k	k	NOUN
cana-4446	39	36	=	=	SYM
cana-4446	39	37	1	1	NUM
cana-4446	39	38	,	,	PUNCT
cana-4446	39	39	2	2	NUM
cana-4446	39	40	,	,	PUNCT
cana-4446	39	41	...	...	PUNCT
cana-4446	39	42	,	,	PUNCT
cana-4446	39	43	2n	2n	NUM
cana-4446	39	44	−	−	NOUN
cana-4446	39	45	1	1	X
cana-4446	39	46	.	.	PUNCT
cana-4446	40	1	a	a	DET
cana-4446	40	2	flat	flat	ADJ
cana-4446	40	3	edge	edge	NOUN
cana-4446	40	4	is	be	AUX
cana-4446	40	5	an	an	DET
cana-4446	40	6	edge	edge	NOUN
cana-4446	40	7	in	in	ADP
cana-4446	40	8	c(2n	c(2n	NOUN
cana-4446	40	9	,	,	PUNCT
cana-4446	40	10	m	m	NOUN
cana-4446	40	11	,	,	PUNCT
cana-4446	40	12	r	r	NOUN
cana-4446	40	13	)	)	PUNCT
cana-4446	40	14	that	that	PRON
cana-4446	40	15	is	be	AUX
cana-4446	40	16	either	either	CCONJ
cana-4446	40	17	a	a	DET
cana-4446	40	18	brick	brick	NOUN
cana-4446	40	19	or	or	CCONJ
cana-4446	40	20	a	a	DET
cana-4446	40	21	hooked	hooked	ADJ
cana-4446	40	22	edge	edge	NOUN
cana-4446	40	23	.	.	PUNCT
cana-4446	41	1	theorem	theorem	NOUN
cana-4446	41	2	1	1	X
cana-4446	41	3	.	.	X
cana-4446	42	1	consider	consider	VERB
cana-4446	42	2	g	g	NOUN
cana-4446	42	3	=	=	SYM
cana-4446	42	4	(	(	PUNCT
cana-4446	42	5	2n	2n	NUM
cana-4446	42	6	,	,	PUNCT
cana-4446	42	7	m	m	PROPN
cana-4446	42	8	,	,	PUNCT
cana-4446	42	9	r	r	NOUN
cana-4446	42	10	)	)	PUNCT
cana-4446	42	11	.	.	PUNCT
cana-4446	43	1	then	then	ADV
cana-4446	43	2	,	,	PUNCT
cana-4446	43	3	for	for	SCONJ
cana-4446	43	4	r	r	NOUN
cana-4446	43	5	=	=	SYM
cana-4446	43	6	3	3	NUM
cana-4446	43	7	and	and	CCONJ
cana-4446	43	8	m	m	PROPN
cana-4446	43	9	=	=	ADJ
cana-4446	43	10	1	1	NUM
cana-4446	43	11	,	,	PUNCT
cana-4446	43	12	and	and	CCONJ
cana-4446	43	13	n	n	PRON
cana-4446	43	14	≥	≥	NOUN
cana-4446	43	15	3	3	NUM
cana-4446	43	16	,	,	PUNCT
cana-4446	43	17	𝑟𝑑𝑦𝑐(𝐺	𝑟𝑑𝑦𝑐(𝐺	NUM
cana-4446	43	18	)	)	PUNCT
cana-4446	43	19	=	=	NOUN
cana-4446	43	20	{	{	PUNCT
cana-4446	43	21	4	4	NUM
cana-4446	43	22	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4446	43	23	3	3	NUM
cana-4446	43	24	≤	≤	NUM
cana-4446	43	25	𝑛	𝑛	DET
cana-4446	43	26	≤	≤	NUM
cana-4446	43	27	7	7	NUM
cana-4446	43	28	5	5	NUM
cana-4446	43	29	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4446	43	30	8	8	NUM
cana-4446	43	31	≤	≤	NUM
cana-4446	43	32	𝑛	𝑛	DET
cana-4446	43	33	≤	≤	NUM
cana-4446	43	34	10	10	NUM
cana-4446	44	1	⌊	⌊	PROPN
cana-4446	44	2	2𝑛	2𝑛	PROPN
cana-4446	44	3	3	3	NUM
cana-4446	44	4	⌋	⌋	NOUN
cana-4446	44	5	−	−	PROPN
cana-4446	44	6	1	1	NUM
cana-4446	44	7	,	,	PUNCT
cana-4446	44	8	2𝑛	2𝑛	PROPN
cana-4446	44	9	≅	≅	PROPN
cana-4446	44	10	1(𝑚𝑜𝑑3	1(𝑚𝑜𝑑3	NUM
cana-4446	44	11	)	)	PUNCT
cana-4446	45	1	⌊	⌊	ADP
cana-4446	45	2	2𝑛	2𝑛	PROPN
cana-4446	45	3	3	3	NUM
cana-4446	45	4	⌋	⌋	NOUN
cana-4446	45	5	−	−	PROPN
cana-4446	45	6	2	2	NUM
cana-4446	45	7	,	,	PUNCT
cana-4446	45	8	2𝑛	2𝑛	PROPN
cana-4446	45	9	≅	≅	PROPN
cana-4446	45	10	0(𝑚𝑜𝑑3	0(𝑚𝑜𝑑3	PROPN
cana-4446	45	11	)	)	PUNCT
cana-4446	46	1	⌊	⌊	ADP
cana-4446	46	2	2𝑛	2𝑛	PROPN
cana-4446	46	3	3	3	NUM
cana-4446	46	4	⌋	⌋	NOUN
cana-4446	46	5	−	−	PROPN
cana-4446	46	6	1	1	NUM
cana-4446	46	7	,	,	PUNCT
cana-4446	46	8	2𝑛	2𝑛	PROPN
cana-4446	46	9	≅	≅	PROPN
cana-4446	46	10	2(𝑚𝑜𝑑3	2(𝑚𝑜𝑑3	NUM
cana-4446	46	11	the	the	DET
cana-4446	46	12	vertex	vertex	NOUN
cana-4446	46	13	set	set	NOUN
cana-4446	46	14	of	of	ADP
cana-4446	46	15	g	g	PROPN
cana-4446	46	16	is	be	AUX
cana-4446	46	17	defined	define	VERB
cana-4446	46	18	as	as	ADP
cana-4446	46	19	v(g	v(g	ADJ
cana-4446	46	20	)	)	PUNCT
cana-4446	46	21	=	=	SYM
cana-4446	46	22	{	{	PUNCT
cana-4446	46	23	v1	v1	PROPN
cana-4446	46	24	,	,	PUNCT
cana-4446	46	25	v2	v2	PROPN
cana-4446	46	26	,	,	PUNCT
cana-4446	46	27	.	.	PUNCT
cana-4446	46	28	.	.	PUNCT
cana-4446	46	29	.	.	PUNCT
cana-4446	46	30	.	.	PUNCT
cana-4446	47	1	.	.	PUNCT
cana-4446	48	1	.	.	PUNCT
cana-4446	49	1	,	,	PUNCT
cana-4446	49	2	v2n−1	v2n−1	PROPN
cana-4446	49	3	,	,	PUNCT
cana-4446	49	4	v2n	v2n	NOUN
cana-4446	49	5	=	=	SYM
cana-4446	49	6	v1	v1	PROPN
cana-4446	49	7	}	}	PUNCT
cana-4446	49	8	and	and	CCONJ
cana-4446	49	9	the	the	DET
cana-4446	49	10	edge	edge	NOUN
cana-4446	49	11	set	set	NOUN
cana-4446	49	12	of	of	ADP
cana-4446	49	13	g	g	PROPN
cana-4446	49	14	is	be	AUX
cana-4446	49	15	defined	define	VERB
cana-4446	49	16	as	as	ADP
cana-4446	49	17	e(g	e(g	NOUN
cana-4446	49	18	)	)	PUNCT
cana-4446	50	1	=	=	PRON
cana-4446	50	2	{	{	PUNCT
cana-4446	50	3	ei	ei	X
cana-4446	50	4	:	:	PUNCT
cana-4446	50	5	1	1	NUM
cana-4446	50	6	≤	≤	NUM
cana-4446	50	7	i	i	PRON
cana-4446	50	8	≤	≤	NUM
cana-4446	50	9	2n	2n	NUM
cana-4446	50	10	}	}	PUNCT
cana-4446	50	11	∪	∪	X
cana-4446	50	12	{	{	PUNCT
cana-4446	50	13	ei′	ei′	NOUN
cana-4446	50	14	:	:	SYM
cana-4446	50	15	1	1	NUM
cana-4446	50	16	≤	≤	NUM
cana-4446	50	17	i	i	PRON
cana-4446	50	18	≤	≤	NOUN
cana-4446	50	19	n	n	CCONJ
cana-4446	50	20	}	}	PUNCT
cana-4446	50	21	where	where	SCONJ
cana-4446	50	22	𝑒𝑖	𝑒𝑖	NOUN
cana-4446	50	23	represents	represent	VERB
cana-4446	50	24	the	the	DET
cana-4446	50	25	cycle	cycle	NOUN
cana-4446	50	26	edge	edge	NOUN
cana-4446	50	27	(	(	PUNCT
cana-4446	50	28	𝑣𝑖−1	𝑣𝑖−1	NOUN
cana-4446	50	29	,	,	PUNCT
cana-4446	50	30	𝑣𝑖	𝑣𝑖	NOUN
cana-4446	50	31	)	)	PUNCT
cana-4446	50	32	and	and	CCONJ
cana-4446	50	33	𝑒𝑖′	𝑒𝑖′	ADJ
cana-4446	50	34	represents	represent	VERB
cana-4446	50	35	the	the	DET
cana-4446	50	36	brick	brick	NOUN
cana-4446	50	37	edge	edge	NOUN
cana-4446	50	38	(	(	PUNCT
cana-4446	50	39	v2k	v2k	NOUN
cana-4446	50	40	,	,	PUNCT
cana-4446	50	41	𝑣2𝑘+𝑟	𝑣2𝑘+𝑟	PROPN
cana-4446	50	42	)	)	PUNCT
cana-4446	50	43	,	,	PUNCT
cana-4446	50	44	𝑘	𝑘	X
cana-4446	50	45	=	=	NOUN
cana-4446	50	46	0,1,2	0,1,2	NUM
cana-4446	50	47	,	,	PUNCT
cana-4446	50	48	.	.	PUNCT
cana-4446	50	49	.	.	PUNCT
cana-4446	50	50	.	.	PUNCT
cana-4446	50	51	.	.	PUNCT
cana-4446	51	1	.	.	PUNCT
cana-4446	52	1	,	,	PUNCT
cana-4446	52	2	𝑛.	𝑛.	NOUN
cana-4446	52	3	in	in	ADP
cana-4446	52	4	this	this	DET
cana-4446	52	5	case	case	NOUN
cana-4446	52	6	,	,	PUNCT
cana-4446	52	7	2k	2k	NUM
cana-4446	52	8	+	+	CCONJ
cana-4446	52	9	r	r	NOUN
cana-4446	52	10	is	be	AUX
cana-4446	52	11	computed	compute	VERB
cana-4446	52	12	modulo	modulo	PROPN
cana-4446	52	13	2n	2n	NUM
cana-4446	52	14	.	.	PUNCT
cana-4446	53	1	let	let	VERB
cana-4446	53	2	’s	’s	PRON
cana-4446	53	3	define	define	VERB
cana-4446	53	4	coloring	color	VERB
cana-4446	53	5	to	to	ADP
cana-4446	53	6	g	g	PROPN
cana-4446	53	7	’s	’s	PART
cana-4446	53	8	vertices	vertex	NOUN
cana-4446	53	9	in	in	ADP
cana-4446	53	10	the	the	DET
cana-4446	53	11	following	following	ADJ
cana-4446	53	12	manner	manner	NOUN
cana-4446	53	13	.	.	PUNCT
cana-4446	54	1	case	case	NOUN
cana-4446	54	2	1	1	NUM
cana-4446	54	3	.	.	SYM
cana-4446	54	4	3	3	NUM
cana-4446	54	5	≤	≤	NOUN
cana-4446	54	6	n	n	PRON
cana-4446	54	7	≤	≤	NUM
cana-4446	54	8	7	7	NUM
cana-4446	54	9	coloring	coloring	NOUN
cana-4446	54	10	is	be	AUX
cana-4446	54	11	defined	define	VERB
cana-4446	54	12	by	by	ADP
cana-4446	54	13	𝑣𝑖+4𝑘	𝑣𝑖+4𝑘	NOUN
cana-4446	54	14	=	=	SYM
cana-4446	54	15	𝑖	𝑖	SYM
cana-4446	54	16	,	,	PUNCT
cana-4446	54	17	1	1	NUM
cana-4446	54	18	≤	≤	NUM
cana-4446	54	19	𝑖	𝑖	SYM
cana-4446	54	20	≤	≤	NUM
cana-4446	54	21	4,0	4,0	NUM
cana-4446	54	22	≤	≤	NOUN
cana-4446	54	23	𝑘	𝑘	DET
cana-4446	54	24	≤	≤	NUM
cana-4446	54	25	⌈	⌈	NOUN
cana-4446	54	26	𝑛	𝑛	ADP
cana-4446	54	27	2	2	NUM
cana-4446	54	28	⌉	⌉	ADP
cana-4446	54	29	−	−	PROPN
cana-4446	54	30	1	1	NUM
cana-4446	54	31	case	case	NOUN
cana-4446	54	32	2	2	NUM
cana-4446	54	33	.	.	NOUN
cana-4446	54	34	8	8	NUM
cana-4446	54	35	≤	≤	NOUN
cana-4446	54	36	n	n	PRON
cana-4446	54	37	≤	≤	NOUN
cana-4446	54	38	10	10	NUM
cana-4446	54	39	for	for	ADP
cana-4446	54	40	n=8	n=8	ADP
cana-4446	54	41	coloring	coloring	NOUN
cana-4446	54	42	is	be	AUX
cana-4446	54	43	defined	define	VERB
cana-4446	54	44	by	by	ADP
cana-4446	54	45	𝑣𝑖+5𝑘	𝑣𝑖+5𝑘	PROPN
cana-4446	54	46	=	=	SYM
cana-4446	54	47	𝑖	𝑖	SYM
cana-4446	54	48	,	,	PUNCT
cana-4446	54	49	1	1	NUM
cana-4446	54	50	≤	≤	NUM
cana-4446	54	51	i	i	X
cana-4446	54	52	≤	≤	NOUN
cana-4446	54	53	5	5	NUM
cana-4446	54	54	,	,	PUNCT
cana-4446	54	55	0	0	NUM
cana-4446	54	56	≤	≤	NUM
cana-4446	55	1	k	k	X
cana-4446	55	2	≤	≤	NUM
cana-4446	55	3	2	2	NUM
cana-4446	55	4	,	,	PUNCT
cana-4446	55	5	𝑣2𝑛	𝑣2𝑛	ADJ
cana-4446	55	6	=	=	SYM
cana-4446	55	7	2	2	NUM
cana-4446	55	8	for	for	ADP
cana-4446	55	9	n	n	NOUN
cana-4446	55	10	=	=	SYM
cana-4446	55	11	9	9	NUM
cana-4446	55	12	,	,	PUNCT
cana-4446	55	13	10	10	NUM
cana-4446	55	14	coloring	coloring	NOUN
cana-4446	55	15	is	be	AUX
cana-4446	55	16	defined	define	VERB
cana-4446	55	17	by	by	ADP
cana-4446	55	18	𝑣𝑖+5𝑘	𝑣𝑖+5𝑘	PROPN
cana-4446	55	19	=	=	SYM
cana-4446	55	20	𝑖	𝑖	SYM
cana-4446	55	21	,	,	PUNCT
cana-4446	55	22	1	1	NUM
cana-4446	55	23	≤	≤	NUM
cana-4446	55	24	i	i	X
cana-4446	55	25	≤	≤	NOUN
cana-4446	55	26	5	5	NUM
cana-4446	55	27	,	,	PUNCT
cana-4446	55	28	0	0	NUM
cana-4446	55	29	≤	≤	NUM
cana-4446	55	30	k	k	X
cana-4446	55	31	≤	≤	ADJ
cana-4446	55	32	3	3	NUM
cana-4446	55	33	case	case	NOUN
cana-4446	55	34	3	3	NUM
cana-4446	55	35	.	.	X
cana-4446	55	36	2n	2n	NUM
cana-4446	55	37	≡	≡	PROPN
cana-4446	55	38	1	1	NUM
cana-4446	55	39	(	(	PUNCT
cana-4446	55	40	mod	mod	NOUN
cana-4446	55	41	3	3	X
cana-4446	55	42	)	)	PUNCT
cana-4446	55	43	coloring	coloring	NOUN
cana-4446	55	44	is	be	AUX
cana-4446	55	45	defined	define	VERB
cana-4446	55	46	by	by	ADP
cana-4446	55	47	𝑣𝑖	𝑣𝑖	ADV
cana-4446	55	48	+	+	CCONJ
cana-4446	55	49	⌈	⌈	NOUN
cana-4446	55	50	𝑛	𝑛	ADP
cana-4446	55	51	2	2	NUM
cana-4446	55	52	⌉	⌉	X
cana-4446	55	53	𝑘	𝑘	X
cana-4446	55	54	=	=	SYM
cana-4446	55	55	𝑖	𝑖	NUM
cana-4446	55	56	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4446	55	57	1	1	NUM
cana-4446	55	58	≤	≤	NOUN
cana-4446	55	59	𝑖	𝑖	SYM
cana-4446	55	60	≤	≤	NOUN
cana-4446	55	61	⌈	⌈	NOUN
cana-4446	55	62	𝑛	𝑛	ADP
cana-4446	55	63	2	2	NUM
cana-4446	55	64	⌉	⌉	NOUN
cana-4446	55	65	,	,	PUNCT
cana-4446	55	66	0	0	NUM
cana-4446	55	67	≤	≤	NOUN
cana-4446	55	68	𝑘	𝑘	DET
cana-4446	55	69	≤	≤	ADJ
cana-4446	55	70	3	3	NUM
cana-4446	55	71	.	.	PUNCT
cana-4446	56	1	communications	communication	NOUN
cana-4446	56	2	on	on	ADP
cana-4446	56	3	applied	apply	VERB
cana-4446	56	4	nonlinear	nonlinear	ADJ
cana-4446	56	5	analysis	analysis	NOUN
cana-4446	56	6	issn	issn	NOUN
cana-4446	56	7	:	:	PUNCT
cana-4446	56	8	1074	1074	NUM
cana-4446	56	9	-	-	PUNCT
cana-4446	56	10	133x	133x	NUM
cana-4446	56	11	vol	vol	NOUN
cana-4446	56	12	32	32	NUM
cana-4446	56	13	no	no	NOUN
cana-4446	56	14	.	.	PUNCT
cana-4446	57	1	9s	9s	NUM
cana-4446	57	2	(	(	PUNCT
cana-4446	57	3	2025	2025	NUM
cana-4446	57	4	)	)	PUNCT
cana-4446	57	5	2046	2046	NUM
cana-4446	57	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4446	58	1	it	it	PRON
cana-4446	58	2	is	be	AUX
cana-4446	58	3	clear	clear	ADJ
cana-4446	58	4	from	from	ADP
cana-4446	58	5	this	this	DET
cana-4446	58	6	color	color	NOUN
cana-4446	58	7	pattern	pattern	NOUN
cana-4446	58	8	that	that	PRON
cana-4446	58	9	rdyc(g	rdyc(g	VERB
cana-4446	58	10	)	)	PUNCT
cana-4446	58	11	=	=	SYM
cana-4446	58	12	rdyc(g	rdyc(g	NOUN
cana-4446	58	13	)	)	PUNCT
cana-4446	58	14	=	=	PUNCT
cana-4446	59	1	⌊	⌊	ADP
cana-4446	59	2	2𝑛	2𝑛	NUM
cana-4446	59	3	3	3	NUM
cana-4446	59	4	⌋	⌋	NOUN
cana-4446	59	5	−	−	NOUN
cana-4446	59	6	1	1	X
cana-4446	59	7	.	.	PUNCT
cana-4446	59	8	case	case	NOUN
cana-4446	59	9	4	4	NUM
cana-4446	59	10	.	.	X
cana-4446	59	11	2n	2n	NUM
cana-4446	59	12	≡	≡	PROPN
cana-4446	59	13	0	0	PUNCT
cana-4446	60	1	(	(	PUNCT
cana-4446	60	2	mod	mod	NOUN
cana-4446	60	3	3	3	X
cana-4446	60	4	)	)	PUNCT
cana-4446	60	5	coloring	coloring	NOUN
cana-4446	60	6	is	be	AUX
cana-4446	60	7	defined	define	VERB
cana-4446	60	8	by	by	ADP
cana-4446	60	9	𝑣	𝑣	PROPN
cana-4446	60	10	𝑖+(⌈	𝑖+(⌈	PUNCT
cana-4446	60	11	2𝑛	2𝑛	PROPN
cana-4446	60	12	3	3	NUM
cana-4446	60	13	⌉−2)𝑘=𝑖	⌉−2)𝑘=𝑖	PROPN
cana-4446	60	14	for	for	ADP
cana-4446	60	15	1	1	NUM
cana-4446	60	16	≤	≤	NUM
cana-4446	60	17	i	i	PRON
cana-4446	60	18	≤	≤	NOUN
cana-4446	60	19	⌈	⌈	NUM
cana-4446	60	20	2n	2n	NUM
cana-4446	60	21	3	3	NUM
cana-4446	60	22	⌉	⌉	SCONJ
cana-4446	60	23	−	−	PROPN
cana-4446	60	24	2	2	NUM
cana-4446	60	25	,	,	PUNCT
cana-4446	60	26	0	0	NUM
cana-4446	60	27	≤	≤	NUM
cana-4446	60	28	k	k	X
cana-4446	60	29	≤	≤	NUM
cana-4446	60	30	3	3	NUM
cana-4446	60	31	.	.	PUNCT
cana-4446	61	1	it	it	PRON
cana-4446	61	2	is	be	AUX
cana-4446	61	3	clear	clear	ADJ
cana-4446	61	4	from	from	ADP
cana-4446	61	5	this	this	DET
cana-4446	61	6	color	color	NOUN
cana-4446	61	7	pattern	pattern	NOUN
cana-4446	61	8	that	that	PRON
cana-4446	61	9	rdyc(g	rdyc(g	VERB
cana-4446	61	10	)	)	PUNCT
cana-4446	61	11	=	=	PUNCT
cana-4446	62	1	⌊	⌊	VERB
cana-4446	62	2	2n	2n	NUM
cana-4446	62	3	3	3	NUM
cana-4446	62	4	⌋	⌋	NOUN
cana-4446	62	5	−	−	NOUN
cana-4446	62	6	2	2	NUM
cana-4446	62	7	case	case	NOUN
cana-4446	62	8	5	5	NUM
cana-4446	62	9	.	.	X
cana-4446	62	10	2n	2n	NUM
cana-4446	62	11	≡	≡	PROPN
cana-4446	62	12	2	2	NUM
cana-4446	62	13	(	(	PUNCT
cana-4446	62	14	mod	mod	NOUN
cana-4446	62	15	3	3	X
cana-4446	62	16	)	)	PUNCT
cana-4446	62	17	coloring	coloring	NOUN
cana-4446	62	18	is	be	AUX
cana-4446	62	19	defined	define	VERB
cana-4446	62	20	by	by	ADP
cana-4446	62	21	𝑣	𝑣	PROPN
cana-4446	62	22	𝑖+(⌈	𝑖+(⌈	SYM
cana-4446	62	23	2𝑛	2𝑛	PROPN
cana-4446	62	24	3	3	NUM
cana-4446	62	25	⌉−1)𝑘=𝑖	⌉−1)𝑘=𝑖	NOUN
cana-4446	62	26	for	for	ADP
cana-4446	62	27	1	1	NUM
cana-4446	62	28	≤	≤	NUM
cana-4446	63	1	i	i	PRON
cana-4446	63	2	≤	≤	NOUN
cana-4446	63	3	⌈	⌈	NUM
cana-4446	63	4	2n	2n	NUM
cana-4446	63	5	3	3	NUM
cana-4446	63	6	⌉	⌉	SCONJ
cana-4446	63	7	−	−	PROPN
cana-4446	63	8	1	1	NUM
cana-4446	63	9	,	,	PUNCT
cana-4446	63	10	0	0	NUM
cana-4446	63	11	≤	≤	NUM
cana-4446	63	12	k	k	X
cana-4446	63	13	≤	≤	NUM
cana-4446	63	14	3	3	NUM
cana-4446	63	15	.	.	PUNCT
cana-4446	64	1	it	it	PRON
cana-4446	64	2	is	be	AUX
cana-4446	64	3	clear	clear	ADJ
cana-4446	64	4	from	from	ADP
cana-4446	64	5	this	this	DET
cana-4446	64	6	color	color	NOUN
cana-4446	64	7	pattern	pattern	NOUN
cana-4446	64	8	that	that	PRON
cana-4446	64	9	rdyc(g	rdyc(g	VERB
cana-4446	64	10	)	)	PUNCT
cana-4446	64	11	=	=	PUNCT
cana-4446	65	1	⌊	⌊	AUX
cana-4446	65	2	2n	2n	NUM
cana-4446	65	3	3	3	NUM
cana-4446	65	4	⌋	⌋	NOUN
cana-4446	65	5	−	−	NOUN
cana-4446	65	6	1	1	X
cana-4446	65	7	.	.	PUNCT
cana-4446	65	8	figure	figure	NOUN
cana-4446	65	9	1	1	NUM
cana-4446	65	10	:	:	PUNCT
cana-4446	65	11	a	a	DET
cana-4446	65	12	graph	graph	NOUN
cana-4446	65	13	illustrating	illustrate	VERB
cana-4446	65	14	the	the	DET
cana-4446	65	15	way	way	NOUN
cana-4446	65	16	colors	color	NOUN
cana-4446	65	17	are	be	AUX
cana-4446	65	18	assigned	assign	VERB
cana-4446	65	19	to	to	ADP
cana-4446	65	20	brick	brick	NOUN
cana-4446	65	21	products	product	NOUN
cana-4446	65	22	in	in	ADP
cana-4446	65	23	c(14,1,3	c(14,1,3	PROPN
cana-4446	65	24	)	)	PUNCT
cana-4446	65	25	theorem	theorem	VERB
cana-4446	65	26	2	2	NUM
cana-4446	65	27	.	.	X
cana-4446	66	1	consider	consider	VERB
cana-4446	66	2	g	g	NOUN
cana-4446	66	3	=	=	SYM
cana-4446	66	4	(	(	PUNCT
cana-4446	66	5	2n	2n	NUM
cana-4446	66	6	,	,	PUNCT
cana-4446	66	7	m	m	PROPN
cana-4446	66	8	,	,	PUNCT
cana-4446	66	9	r	r	NOUN
cana-4446	66	10	)	)	PUNCT
cana-4446	66	11	.	.	PUNCT
cana-4446	67	1	then	then	ADV
cana-4446	67	2	,	,	PUNCT
cana-4446	67	3	for	for	ADP
cana-4446	67	4	r	r	NOUN
cana-4446	67	5	=	=	SYM
cana-4446	67	6	7	7	NUM
cana-4446	67	7	and	and	CCONJ
cana-4446	67	8	m	m	VERB
cana-4446	67	9	=	=	ADJ
cana-4446	67	10	1	1	NUM
cana-4446	67	11	,	,	PUNCT
cana-4446	67	12	and	and	CCONJ
cana-4446	67	13	n	n	PRON
cana-4446	67	14	≥	≥	NOUN
cana-4446	67	15	5	5	NUM
cana-4446	67	16	,	,	PUNCT
cana-4446	67	17	𝑟𝑑𝑦𝑐(𝐺	𝑟𝑑𝑦𝑐(𝐺	NUM
cana-4446	67	18	)	)	PUNCT
cana-4446	67	19	=	=	NOUN
cana-4446	67	20	{	{	PUNCT
cana-4446	67	21	4	4	NUM
cana-4446	67	22	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-4446	67	23	𝑛	𝑛	X
cana-4446	67	24	=	=	SYM
cana-4446	67	25	5	5	NUM
cana-4446	67	26	,	,	PUNCT
cana-4446	67	27	7	7	NUM
cana-4446	67	28	≤	≤	NUM
cana-4446	67	29	𝑛	𝑛	DET
cana-4446	67	30	≤	≤	NUM
cana-4446	67	31	12	12	NUM
cana-4446	67	32	⌊	⌊	PROPN
cana-4446	67	33	𝑛	𝑛	DET
cana-4446	67	34	2	2	NUM
cana-4446	67	35	⌋	⌋	NOUN
cana-4446	67	36	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4446	67	37	𝑛	𝑛	X
cana-4446	67	38	=	=	SYM
cana-4446	67	39	6	6	NUM
cana-4446	67	40	,	,	PUNCT
cana-4446	67	41	13	13	NUM
cana-4446	67	42	⌊	⌊	NOUN
cana-4446	67	43	𝑛	𝑛	DET
cana-4446	67	44	2	2	NUM
cana-4446	67	45	⌋	⌋	NOUN
cana-4446	67	46	−	−	NOUN
cana-4446	67	47	1	1	NUM
cana-4446	67	48	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-4446	67	49	𝑛	𝑛	DET
cana-4446	67	50	≥	≥	NOUN
cana-4446	67	51	14	14	NUM
cana-4446	67	52	let	let	VERB
cana-4446	67	53	us	we	PRON
cana-4446	67	54	define	define	VERB
cana-4446	67	55	the	the	DET
cana-4446	67	56	vertex	vertex	NOUN
cana-4446	67	57	set	set	NOUN
cana-4446	67	58	and	and	CCONJ
cana-4446	67	59	edge	edge	NOUN
cana-4446	67	60	set	set	VERB
cana-4446	67	61	as	as	ADP
cana-4446	67	62	in	in	ADP
cana-4446	67	63	theorem1	theorem1	PROPN
cana-4446	67	64	.	.	PUNCT
cana-4446	68	1	let	let	VERB
cana-4446	68	2	’s	’s	PRON
cana-4446	68	3	define	define	VERB
cana-4446	68	4	coloring	color	VERB
cana-4446	68	5	to	to	ADP
cana-4446	68	6	g	g	PROPN
cana-4446	68	7	’s	’s	PART
cana-4446	68	8	vertices	vertex	NOUN
cana-4446	68	9	in	in	ADP
cana-4446	68	10	the	the	DET
cana-4446	68	11	following	following	ADJ
cana-4446	68	12	manner	manner	NOUN
cana-4446	68	13	.	.	PUNCT
cana-4446	69	1	case	case	NOUN
cana-4446	69	2	1	1	NUM
cana-4446	69	3	.	.	X
cana-4446	70	1	for	for	ADP
cana-4446	70	2	n	n	NOUN
cana-4446	70	3	=	=	SYM
cana-4446	70	4	5	5	NUM
cana-4446	70	5	,	,	PUNCT
cana-4446	70	6	7	7	NUM
cana-4446	70	7	,	,	PUNCT
cana-4446	70	8	9	9	NUM
cana-4446	70	9	,	,	PUNCT
cana-4446	70	10	11	11	NUM
cana-4446	70	11	coloring	coloring	NOUN
cana-4446	70	12	is	be	AUX
cana-4446	70	13	defined	define	VERB
cana-4446	70	14	by	by	ADP
cana-4446	70	15	𝑣𝑖+4𝑘	𝑣𝑖+4𝑘	NOUN
cana-4446	70	16	=	=	SYM
cana-4446	70	17	𝑖	𝑖	SYM
cana-4446	70	18	,	,	PUNCT
cana-4446	70	19	1	1	NUM
cana-4446	70	20	≤	≤	NUM
cana-4446	70	21	i	i	X
cana-4446	70	22	≤	≤	ADJ
cana-4446	70	23	4	4	NUM
cana-4446	70	24	,	,	PUNCT
cana-4446	70	25	0	0	NUM
cana-4446	70	26	≤	≤	NUM
cana-4446	70	27	k	k	X
cana-4446	70	28	≤	≤	NUM
cana-4446	70	29	⌊	⌊	VERB
cana-4446	70	30	n	n	ADV
cana-4446	70	31	2	2	NUM
cana-4446	70	32	⌋	⌋	NOUN
cana-4446	70	33	−	−	NOUN
cana-4446	70	34	1	1	NUM
cana-4446	70	35	;	;	PUNCT
cana-4446	70	36	vi+4k	vi+4k	NOUN
cana-4446	71	1	=	=	SYM
cana-4446	71	2	i	i	PRON
cana-4446	71	3	+	+	NOUN
cana-4446	71	4	1	1	NUM
cana-4446	71	5	,	,	PUNCT
cana-4446	71	6	1	1	NUM
cana-4446	71	7	≤	≤	NUM
cana-4446	71	8	i	i	X
cana-4446	71	9	≤	≤	ADV
cana-4446	71	10	2	2	NUM
cana-4446	71	11	,	,	PUNCT
cana-4446	71	12	k	k	X
cana-4446	72	1	=	=	SYM
cana-4446	72	2	⌊	⌊	PROPN
cana-4446	72	3	n	n	ADV
cana-4446	72	4	2	2	NUM
cana-4446	72	5	⌋	⌋	NOUN
cana-4446	72	6	for	for	ADP
cana-4446	72	7	n	n	NOUN
cana-4446	72	8	=	=	SYM
cana-4446	72	9	8	8	NUM
cana-4446	72	10	,	,	PUNCT
cana-4446	72	11	10	10	NUM
cana-4446	72	12	,	,	PUNCT
cana-4446	72	13	12	12	NUM
cana-4446	72	14	coloring	coloring	NOUN
cana-4446	72	15	is	be	AUX
cana-4446	72	16	defined	define	VERB
cana-4446	72	17	by	by	ADP
cana-4446	72	18	𝑣𝑖+4𝑘	𝑣𝑖+4𝑘	NOUN
cana-4446	72	19	=	=	SYM
cana-4446	72	20	𝑖	𝑖	SYM
cana-4446	72	21	,	,	PUNCT
cana-4446	72	22	1	1	NUM
cana-4446	72	23	≤	≤	NUM
cana-4446	72	24	i	i	X
cana-4446	72	25	≤	≤	ADJ
cana-4446	72	26	4	4	NUM
cana-4446	72	27	,	,	PUNCT
cana-4446	72	28	0	0	NUM
cana-4446	72	29	≤	≤	NUM
cana-4446	72	30	k	k	NOUN
cana-4446	72	31	≤	≤	NUM
cana-4446	72	32	n	n	CCONJ
cana-4446	72	33	2	2	NUM
cana-4446	72	34	−	−	NUM
cana-4446	72	35	1	1	NUM
cana-4446	72	36	.	.	PUNCT
cana-4446	72	37	case	case	NOUN
cana-4446	72	38	2	2	NUM
cana-4446	72	39	.	.	PUNCT
cana-4446	73	1	n=6,13	n=6,13	PROPN
cana-4446	73	2	communications	communication	NOUN
cana-4446	73	3	on	on	ADP
cana-4446	73	4	applied	apply	VERB
cana-4446	73	5	nonlinear	nonlinear	ADJ
cana-4446	73	6	analysis	analysis	NOUN
cana-4446	73	7	issn	issn	NOUN
cana-4446	73	8	:	:	PUNCT
cana-4446	73	9	1074	1074	NUM
cana-4446	73	10	-	-	PUNCT
cana-4446	73	11	133x	133x	NUM
cana-4446	73	12	vol	vol	NOUN
cana-4446	73	13	32	32	NUM
cana-4446	73	14	no	no	NOUN
cana-4446	73	15	.	.	PUNCT
cana-4446	74	1	9s	9s	NUM
cana-4446	74	2	(	(	PUNCT
cana-4446	74	3	2025	2025	NUM
cana-4446	74	4	)	)	PUNCT
cana-4446	74	5	2047	2047	NUM
cana-4446	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4446	74	7	coloring	coloring	NOUN
cana-4446	74	8	is	be	AUX
cana-4446	74	9	defined	define	VERB
cana-4446	74	10	by	by	ADP
cana-4446	74	11	𝑣	𝑣	PROPN
cana-4446	74	12	𝑖+	𝑖+	PRON
cana-4446	74	13	(	(	PUNCT
cana-4446	74	14	𝑛	𝑛	DET
cana-4446	74	15	2	2	NUM
cana-4446	74	16	)	)	PUNCT
cana-4446	74	17	𝑘=𝑖	𝑘=𝑖	NOUN
cana-4446	74	18	for	for	ADP
cana-4446	74	19	1	1	NUM
cana-4446	74	20	≤	≤	NUM
cana-4446	74	21	i	i	PRON
cana-4446	74	22	≤	≤	PUNCT
cana-4446	74	23	⌊	⌊	VERB
cana-4446	74	24	n	n	DET
cana-4446	74	25	2	2	NUM
cana-4446	74	26	⌋	⌋	NOUN
cana-4446	74	27	,	,	PUNCT
cana-4446	74	28	0	0	NUM
cana-4446	74	29	≤	≤	NUM
cana-4446	74	30	k	k	X
cana-4446	74	31	≤	≤	NUM
cana-4446	74	32	⌊	⌊	VERB
cana-4446	74	33	n	n	PRON
cana-4446	74	34	3	3	NUM
cana-4446	74	35	⌋	⌋	NOUN
cana-4446	74	36	+	+	CCONJ
cana-4446	75	1	1	1	X
cana-4446	75	2	.	.	X
cana-4446	75	3	it	it	PRON
cana-4446	75	4	is	be	AUX
cana-4446	75	5	clear	clear	ADJ
cana-4446	75	6	from	from	ADP
cana-4446	75	7	this	this	DET
cana-4446	75	8	color	color	NOUN
cana-4446	75	9	pattern	pattern	NOUN
cana-4446	75	10	that	that	PRON
cana-4446	75	11	rdyc(g	rdyc(g	VERB
cana-4446	75	12	)	)	PUNCT
cana-4446	75	13	=	=	PUNCT
cana-4446	76	1	⌊	⌊	VERB
cana-4446	76	2	n	n	PRON
cana-4446	76	3	2	2	NUM
cana-4446	76	4	⌋	⌋	NOUN
cana-4446	76	5	case	case	NOUN
cana-4446	76	6	3	3	NUM
cana-4446	76	7	.	.	X
cana-4446	76	8	n	n	PRON
cana-4446	76	9	≥	≥	NOUN
cana-4446	76	10	14	14	NUM
cana-4446	76	11	coloring	coloring	NOUN
cana-4446	76	12	is	be	AUX
cana-4446	76	13	defined	define	VERB
cana-4446	76	14	by	by	ADP
cana-4446	76	15	𝑣	𝑣	PROPN
cana-4446	76	16	(	(	PUNCT
cana-4446	76	17	⌊	⌊	VERB
cana-4446	76	18	𝑛	𝑛	DET
cana-4446	76	19	2	2	NUM
cana-4446	76	20	⌋−1)𝑘	⌋−1)𝑘	NOUN
cana-4446	76	21	=	=	SYM
cana-4446	76	22	𝑖	𝑖	PROPN
cana-4446	76	23	for	for	ADP
cana-4446	76	24	1	1	NUM
cana-4446	76	25	≤	≤	NUM
cana-4446	77	1	i	i	PRON
cana-4446	77	2	≤	≤	PUNCT
cana-4446	77	3	⌊	⌊	VERB
cana-4446	77	4	n	n	ADV
cana-4446	77	5	2	2	NUM
cana-4446	77	6	⌋	⌋	NOUN
cana-4446	77	7	−	−	NOUN
cana-4446	77	8	1	1	NUM
cana-4446	77	9	,	,	PUNCT
cana-4446	77	10	0	0	NUM
cana-4446	77	11	≤	≤	NUM
cana-4446	77	12	k	k	X
cana-4446	77	13	≤	≤	NUM
cana-4446	77	14	4	4	NUM
cana-4446	77	15	.	.	PUNCT
cana-4446	78	1	it	it	PRON
cana-4446	78	2	is	be	AUX
cana-4446	78	3	clear	clear	ADJ
cana-4446	78	4	from	from	ADP
cana-4446	78	5	this	this	DET
cana-4446	78	6	color	color	NOUN
cana-4446	78	7	pattern	pattern	NOUN
cana-4446	78	8	that	that	PRON
cana-4446	78	9	rdyc(g	rdyc(g	VERB
cana-4446	78	10	)	)	PUNCT
cana-4446	78	11	=	=	PUNCT
cana-4446	79	1	⌊	⌊	AUX
cana-4446	79	2	n	n	ADV
cana-4446	79	3	2	2	NUM
cana-4446	79	4	⌋	⌋	NOUN
cana-4446	79	5	−	−	NOUN
cana-4446	79	6	1	1	X
cana-4446	79	7	.	.	PUNCT
cana-4446	79	8	figure	figure	NOUN
cana-4446	79	9	2	2	NUM
cana-4446	79	10	:	:	PUNCT
cana-4446	79	11	a	a	DET
cana-4446	79	12	graph	graph	NOUN
cana-4446	79	13	illustrating	illustrate	VERB
cana-4446	79	14	the	the	DET
cana-4446	79	15	way	way	NOUN
cana-4446	79	16	colors	color	NOUN
cana-4446	79	17	are	be	AUX
cana-4446	79	18	assigned	assign	VERB
cana-4446	79	19	to	to	ADP
cana-4446	79	20	brick	brick	NOUN
cana-4446	79	21	products	product	NOUN
cana-4446	79	22	in	in	ADP
cana-4446	79	23	c(16,1,5	c(16,1,5	NOUN
cana-4446	79	24	)	)	PUNCT
cana-4446	79	25	theorem	theorem	NOUN
cana-4446	79	26	3	3	X
cana-4446	79	27	.	.	X
cana-4446	79	28	consider	consider	VERB
cana-4446	79	29	g	g	NOUN
cana-4446	79	30	=	=	SYM
cana-4446	79	31	(	(	PUNCT
cana-4446	79	32	2n	2n	NUM
cana-4446	79	33	,	,	PUNCT
cana-4446	79	34	m	m	PROPN
cana-4446	79	35	,	,	PUNCT
cana-4446	79	36	r	r	NOUN
cana-4446	79	37	)	)	PUNCT
cana-4446	79	38	.	.	PUNCT
cana-4446	80	1	then	then	ADV
cana-4446	80	2	,	,	PUNCT
cana-4446	80	3	for	for	ADP
cana-4446	80	4	r	r	NOUN
cana-4446	80	5	=	=	SYM
cana-4446	80	6	7	7	NUM
cana-4446	80	7	and	and	CCONJ
cana-4446	80	8	m	m	VERB
cana-4446	80	9	=	=	ADJ
cana-4446	80	10	1	1	NUM
cana-4446	80	11	,	,	PUNCT
cana-4446	80	12	and	and	CCONJ
cana-4446	80	13	n	n	PRON
cana-4446	80	14	≥	≥	NOUN
cana-4446	80	15	7	7	NUM
cana-4446	80	16	,	,	PUNCT
cana-4446	80	17	𝑟𝑑𝑦𝑐(𝐺	𝑟𝑑𝑦𝑐(𝐺	NUM
cana-4446	80	18	)	)	PUNCT
cana-4446	80	19	=	=	NOUN
cana-4446	80	20	{	{	PUNCT
cana-4446	80	21	3	3	NUM
cana-4446	80	22	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-4446	80	23	𝑛	𝑛	X
cana-4446	80	24	=	=	SYM
cana-4446	80	25	7	7	NUM
cana-4446	80	26	4	4	NUM
cana-4446	80	27	for	for	ADP
cana-4446	80	28	n	n	PRON
cana-4446	80	29	=	=	SYM
cana-4446	80	30	7,8,10	7,8,10	NUM
cana-4446	80	31	≤	≤	NOUN
cana-4446	80	32	n	n	DET
cana-4446	80	33	≤	≤	NUM
cana-4446	80	34	14	14	NUM
cana-4446	80	35	5	5	NUM
cana-4446	80	36	for	for	ADP
cana-4446	80	37	n	n	NOUN
cana-4446	80	38	=	=	SYM
cana-4446	80	39	15,17,16	15,17,16	NUM
cana-4446	80	40	6	6	NUM
cana-4446	80	41	for	for	ADP
cana-4446	80	42	n	n	NOUN
cana-4446	80	43	=	=	SYM
cana-4446	80	44	16,18	16,18	NOUN
cana-4446	80	45	⌊	⌊	NOUN
cana-4446	80	46	𝑛	𝑛	DET
cana-4446	80	47	3	3	NUM
cana-4446	80	48	⌋	⌋	NOUN
cana-4446	80	49	+	+	CCONJ
cana-4446	80	50	2	2	NUM
cana-4446	80	51	,	,	PUNCT
cana-4446	80	52	2𝑛	2𝑛	PROPN
cana-4446	80	53	≅	≅	PROPN
cana-4446	80	54	2(𝑚𝑜𝑑3	2(𝑚𝑜𝑑3	NUM
cana-4446	80	55	)	)	PUNCT
cana-4446	80	56	⌊	⌊	VERB
cana-4446	80	57	𝑛	𝑛	DET
cana-4446	80	58	3	3	NUM
cana-4446	80	59	⌋	⌋	NOUN
cana-4446	80	60	+	+	CCONJ
cana-4446	80	61	1	1	NUM
cana-4446	80	62	,	,	PUNCT
cana-4446	80	63	2𝑛	2𝑛	PROPN
cana-4446	80	64	≅	≅	PROPN
cana-4446	80	65	1(𝑚𝑜𝑑3	1(𝑚𝑜𝑑3	NUM
cana-4446	80	66	)	)	PUNCT
cana-4446	80	67	𝑛	𝑛	PRON
cana-4446	80	68	3	3	NUM
cana-4446	80	69	+	+	SYM
cana-4446	80	70	1	1	NUM
cana-4446	80	71	,	,	PUNCT
cana-4446	80	72	2𝑛	2𝑛	PROPN
cana-4446	80	73	≅	≅	PROPN
cana-4446	80	74	0(𝑚𝑜𝑑3	0(𝑚𝑜𝑑3	PROPN
cana-4446	80	75	let	let	VERB
cana-4446	80	76	us	we	PRON
cana-4446	80	77	define	define	VERB
cana-4446	80	78	the	the	DET
cana-4446	80	79	vertex	vertex	NOUN
cana-4446	80	80	set	set	NOUN
cana-4446	80	81	and	and	CCONJ
cana-4446	80	82	edge	edge	NOUN
cana-4446	80	83	set	set	VERB
cana-4446	80	84	as	as	ADP
cana-4446	80	85	in	in	ADP
cana-4446	80	86	theorem1	theorem1	PROPN
cana-4446	80	87	.	.	PUNCT
cana-4446	81	1	let	let	VERB
cana-4446	81	2	’s	’s	PRON
cana-4446	81	3	define	define	VERB
cana-4446	81	4	coloring	color	VERB
cana-4446	81	5	to	to	ADP
cana-4446	81	6	g	g	PROPN
cana-4446	81	7	’s	’s	PART
cana-4446	81	8	vertices	vertex	NOUN
cana-4446	81	9	in	in	ADP
cana-4446	81	10	the	the	DET
cana-4446	81	11	following	following	ADJ
cana-4446	81	12	manner	manner	NOUN
cana-4446	81	13	.	.	PUNCT
cana-4446	82	1	case	case	NOUN
cana-4446	82	2	1	1	NUM
cana-4446	82	3	.	.	PUNCT
cana-4446	83	1	n	n	NOUN
cana-4446	83	2	=	=	SYM
cana-4446	83	3	9	9	NUM
cana-4446	83	4	coloring	coloring	NOUN
cana-4446	83	5	is	be	AUX
cana-4446	83	6	defined	define	VERB
cana-4446	83	7	by	by	ADP
cana-4446	83	8	𝑣𝑖+3𝑘	𝑣𝑖+3𝑘	NOUN
cana-4446	83	9	=	=	SYM
cana-4446	83	10	𝑖	𝑖	SYM
cana-4446	83	11	,	,	PUNCT
cana-4446	83	12	1	1	NUM
cana-4446	83	13	≤	≤	NUM
cana-4446	83	14	𝑖	𝑖	SYM
cana-4446	83	15	≤	≤	NOUN
cana-4446	83	16	3	3	NUM
cana-4446	83	17	,	,	PUNCT
cana-4446	83	18	0	0	NUM
cana-4446	83	19	≤	≤	NOUN
cana-4446	83	20	𝑘	𝑘	DET
cana-4446	83	21	≤	≤	ADJ
cana-4446	83	22	5	5	NUM
cana-4446	83	23	.	.	PUNCT
cana-4446	83	24	case	case	NOUN
cana-4446	83	25	2	2	NUM
cana-4446	83	26	.	.	PUNCT
cana-4446	84	1	n	n	NOUN
cana-4446	84	2	=	=	SYM
cana-4446	84	3	7	7	NUM
cana-4446	84	4	,	,	PUNCT
cana-4446	84	5	8	8	NUM
cana-4446	84	6	,	,	PUNCT
cana-4446	84	7	10	10	NUM
cana-4446	84	8	≤	≤	NUM
cana-4446	84	9	n	n	PRON
cana-4446	84	10	≤	≤	NUM
cana-4446	84	11	14	14	NUM
cana-4446	84	12	coloring	coloring	NOUN
cana-4446	84	13	is	be	AUX
cana-4446	84	14	defined	define	VERB
cana-4446	84	15	by	by	ADP
cana-4446	84	16	𝑣𝑖+4𝑘	𝑣𝑖+4𝑘	NOUN
cana-4446	84	17	=	=	SYM
cana-4446	84	18	𝑖	𝑖	SYM
cana-4446	84	19	,	,	PUNCT
cana-4446	84	20	1	1	NUM
cana-4446	84	21	≤	≤	NUM
cana-4446	84	22	𝑖	𝑖	SYM
cana-4446	84	23	≤	≤	NOUN
cana-4446	84	24	4	4	NUM
cana-4446	84	25	,	,	PUNCT
cana-4446	84	26	0	0	NUM
cana-4446	84	27	≤	≤	NOUN
cana-4446	84	28	𝑘	𝑘	DET
cana-4446	84	29	≤	≤	NUM
cana-4446	84	30	⌈	⌈	NOUN
cana-4446	84	31	𝑛	𝑛	ADP
cana-4446	84	32	2	2	NUM
cana-4446	84	33	⌉	⌉	ADP
cana-4446	84	34	−	−	PROPN
cana-4446	84	35	1	1	NUM
cana-4446	84	36	.	.	PUNCT
cana-4446	85	1	communications	communication	NOUN
cana-4446	85	2	on	on	ADP
cana-4446	85	3	applied	apply	VERB
cana-4446	85	4	nonlinear	nonlinear	ADJ
cana-4446	85	5	analysis	analysis	NOUN
cana-4446	85	6	issn	issn	NOUN
cana-4446	85	7	:	:	PUNCT
cana-4446	85	8	1074	1074	NUM
cana-4446	85	9	-	-	PUNCT
cana-4446	85	10	133x	133x	NUM
cana-4446	85	11	vol	vol	NOUN
cana-4446	85	12	32	32	NUM
cana-4446	85	13	no	no	NOUN
cana-4446	85	14	.	.	PUNCT
cana-4446	86	1	9s	9s	NUM
cana-4446	86	2	(	(	PUNCT
cana-4446	86	3	2025	2025	NUM
cana-4446	86	4	)	)	PUNCT
cana-4446	86	5	2048	2048	NUM
cana-4446	86	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4446	86	7	case	case	NOUN
cana-4446	86	8	3	3	X
cana-4446	86	9	.	.	PUNCT
cana-4446	87	1	n	n	NOUN
cana-4446	87	2	=	=	NUM
cana-4446	87	3	15	15	NUM
cana-4446	87	4	,	,	PUNCT
cana-4446	87	5	17	17	NUM
cana-4446	87	6	coloring	coloring	NOUN
cana-4446	87	7	is	be	AUX
cana-4446	87	8	defined	define	VERB
cana-4446	87	9	by	by	ADP
cana-4446	87	10	𝑣𝑖+4𝑘	𝑣𝑖+4𝑘	NOUN
cana-4446	87	11	=	=	SYM
cana-4446	87	12	𝑖	𝑖	SYM
cana-4446	87	13	,	,	PUNCT
cana-4446	87	14	1	1	NUM
cana-4446	87	15	≤	≤	NUM
cana-4446	87	16	𝑖	𝑖	SYM
cana-4446	87	17	≤	≤	NOUN
cana-4446	87	18	4	4	NUM
cana-4446	87	19	,	,	PUNCT
cana-4446	87	20	0	0	NUM
cana-4446	87	21	≤	≤	NOUN
cana-4446	87	22	𝑘	𝑘	DET
cana-4446	87	23	≤	≤	NUM
cana-4446	87	24	⌈	⌈	NOUN
cana-4446	87	25	𝑛	𝑛	ADP
cana-4446	87	26	2	2	NUM
cana-4446	87	27	⌉	⌉	NOUN
cana-4446	87	28	−	−	PROPN
cana-4446	87	29	1	1	NUM
cana-4446	87	30	.	.	PUNCT
cana-4446	87	31	case	case	NOUN
cana-4446	87	32	4	4	NUM
cana-4446	87	33	.	.	PUNCT
cana-4446	88	1	n	n	NOUN
cana-4446	88	2	=	=	SYM
cana-4446	88	3	16	16	NUM
cana-4446	88	4	,	,	PUNCT
cana-4446	88	5	18	18	NUM
cana-4446	88	6	for	for	ADP
cana-4446	88	7	n	n	NOUN
cana-4446	88	8	=	=	SYM
cana-4446	88	9	16	16	NUM
cana-4446	88	10	coloring	coloring	NOUN
cana-4446	88	11	is	be	AUX
cana-4446	88	12	defined	define	VERB
cana-4446	88	13	by	by	ADP
cana-4446	88	14	𝑣𝑖+6𝑘	𝑣𝑖+6𝑘	PROPN
cana-4446	88	15	=	=	SYM
cana-4446	88	16	i	i	PROPN
cana-4446	88	17	,	,	PUNCT
cana-4446	88	18	1	1	NUM
cana-4446	88	19	≤	≤	NUM
cana-4446	88	20	i	i	X
cana-4446	88	21	≤	≤	NOUN
cana-4446	88	22	6	6	NUM
cana-4446	88	23	,	,	PUNCT
cana-4446	88	24	0	0	NUM
cana-4446	88	25	≤	≤	NUM
cana-4446	88	26	k	k	X
cana-4446	88	27	≤	≤	NUM
cana-4446	88	28	4	4	NUM
cana-4446	88	29	;	;	PUNCT
cana-4446	88	30	𝑣𝑖+6𝑘	𝑣𝑖+6𝑘	PROPN
cana-4446	89	1	=	=	PUNCT
cana-4446	89	2	i	i	PRON
cana-4446	89	3	+	+	NOUN
cana-4446	89	4	1	1	NUM
cana-4446	89	5	,	,	PUNCT
cana-4446	89	6	1	1	NUM
cana-4446	89	7	≤	≤	NUM
cana-4446	89	8	i	i	X
cana-4446	89	9	≤	≤	ADV
cana-4446	89	10	2	2	NUM
cana-4446	89	11	,	,	PUNCT
cana-4446	89	12	k	k	X
cana-4446	90	1	=	=	PUNCT
cana-4446	90	2	⌊	⌊	VERB
cana-4446	90	3	𝑛	𝑛	DET
cana-4446	90	4	3	3	NUM
cana-4446	90	5	⌋.	⌋.	PRON
cana-4446	90	6	for	for	ADP
cana-4446	90	7	n	n	PROPN
cana-4446	90	8	=	=	SYM
cana-4446	90	9	18	18	NUM
cana-4446	90	10	,	,	PUNCT
cana-4446	90	11	𝑣𝑖+6𝑘	𝑣𝑖+6𝑘	NOUN
cana-4446	90	12	=	=	SYM
cana-4446	90	13	i	i	PROPN
cana-4446	90	14	,	,	PUNCT
cana-4446	90	15	1	1	NUM
cana-4446	90	16	≤	≤	NUM
cana-4446	90	17	i	i	X
cana-4446	90	18	≤	≤	NOUN
cana-4446	90	19	6	6	NUM
cana-4446	90	20	,	,	PUNCT
cana-4446	90	21	0	0	NUM
cana-4446	90	22	≤	≤	NUM
cana-4446	90	23	k	k	X
cana-4446	90	24	≤	≤	NUM
cana-4446	90	25	5	5	NUM
cana-4446	90	26	case	case	NOUN
cana-4446	90	27	5	5	NUM
cana-4446	90	28	.	.	X
cana-4446	90	29	2n	2n	NUM
cana-4446	90	30	≡	≡	PROPN
cana-4446	90	31	2	2	NUM
cana-4446	90	32	(	(	PUNCT
cana-4446	90	33	mod	mod	NOUN
cana-4446	90	34	3	3	X
cana-4446	90	35	)	)	PUNCT
cana-4446	90	36	coloring	coloring	NOUN
cana-4446	90	37	is	be	AUX
cana-4446	90	38	defined	define	VERB
cana-4446	90	39	by	by	ADP
cana-4446	90	40	𝑣	𝑣	DET
cana-4446	90	41	𝑖+(⌊	𝑖+(⌊	NOUN
cana-4446	90	42	𝑛	𝑛	ADP
cana-4446	90	43	3	3	NUM
cana-4446	90	44	⌋+2)𝑘	⌋+2)𝑘	ADP
cana-4446	90	45	=	=	PROPN
cana-4446	90	46	i	i	PROPN
cana-4446	90	47	,	,	PUNCT
cana-4446	90	48	1	1	NUM
cana-4446	90	49	≤	≤	NUM
cana-4446	91	1	i	i	PRON
cana-4446	91	2	≤	≤	PUNCT
cana-4446	91	3	⌊	⌊	VERB
cana-4446	91	4	𝑛	𝑛	DET
cana-4446	91	5	3	3	NUM
cana-4446	91	6	+	+	NUM
cana-4446	91	7	2⌋	2⌋	NUM
cana-4446	91	8	,	,	PUNCT
cana-4446	91	9	0	0	NUM
cana-4446	91	10	≤	≤	NUM
cana-4446	92	1	k	k	X
cana-4446	92	2	≤	≤	NUM
cana-4446	92	3	4	4	NUM
cana-4446	92	4	for	for	ADP
cana-4446	92	5	2n	2n	NUM
cana-4446	92	6	≤	≤	NUM
cana-4446	92	7	50	50	NUM
cana-4446	92	8	,	,	PUNCT
cana-4446	92	9	0	0	NUM
cana-4446	92	10	≤	≤	NUM
cana-4446	92	11	k	k	X
cana-4446	92	12	≤	≤	NUM
cana-4446	92	13	5	5	NUM
cana-4446	92	14	for	for	ADP
cana-4446	92	15	2n	2n	NUM
cana-4446	92	16	>	>	SYM
cana-4446	92	17	50	50	NUM
cana-4446	92	18	it	it	PRON
cana-4446	92	19	is	be	AUX
cana-4446	92	20	clear	clear	ADJ
cana-4446	92	21	from	from	ADP
cana-4446	92	22	this	this	DET
cana-4446	92	23	color	color	NOUN
cana-4446	92	24	pattern	pattern	NOUN
cana-4446	92	25	that	that	PRON
cana-4446	92	26	rdyc(g	rdyc(g	VERB
cana-4446	92	27	)	)	PUNCT
cana-4446	93	1	=	=	SYM
cana-4446	93	2	⌊	⌊	VERB
cana-4446	93	3	𝑛	𝑛	DET
cana-4446	93	4	3	3	NUM
cana-4446	93	5	⌋	⌋	NOUN
cana-4446	93	6	+	+	CCONJ
cana-4446	93	7	2	2	X
cana-4446	93	8	.	.	X
cana-4446	93	9	case	case	NOUN
cana-4446	93	10	6	6	NUM
cana-4446	93	11	.	.	X
cana-4446	93	12	2n	2n	NUM
cana-4446	93	13	≡	≡	PROPN
cana-4446	93	14	1	1	NUM
cana-4446	93	15	(	(	PUNCT
cana-4446	93	16	mod	mod	NOUN
cana-4446	93	17	3	3	X
cana-4446	93	18	)	)	PUNCT
cana-4446	93	19	coloring	coloring	NOUN
cana-4446	93	20	is	be	AUX
cana-4446	93	21	defined	define	VERB
cana-4446	93	22	by	by	ADP
cana-4446	93	23	𝑣	𝑣	DET
cana-4446	93	24	𝑖+(⌊	𝑖+(⌊	NOUN
cana-4446	93	25	𝑛	𝑛	PRON
cana-4446	93	26	3	3	NUM
cana-4446	93	27	⌋+1)𝑘	⌋+1)𝑘	NOUN
cana-4446	94	1	=	=	ADJ
cana-4446	94	2	i	i	NOUN
cana-4446	94	3	,	,	PUNCT
cana-4446	94	4	1	1	NUM
cana-4446	94	5	≤	≤	NUM
cana-4446	94	6	i	i	PRON
cana-4446	94	7	≤	≤	PUNCT
cana-4446	94	8	⌊	⌊	VERB
cana-4446	94	9	𝑛	𝑛	DET
cana-4446	94	10	3	3	NUM
cana-4446	94	11	⌋	⌋	NOUN
cana-4446	94	12	+	+	CCONJ
cana-4446	94	13	1	1	NUM
cana-4446	94	14	,	,	PUNCT
cana-4446	94	15	0	0	NUM
cana-4446	94	16	≤	≤	NUM
cana-4446	95	1	k	k	X
cana-4446	95	2	≤	≤	NUM
cana-4446	95	3	5	5	NUM
cana-4446	95	4	it	it	PRON
cana-4446	95	5	is	be	AUX
cana-4446	95	6	clear	clear	ADJ
cana-4446	95	7	from	from	ADP
cana-4446	95	8	this	this	DET
cana-4446	95	9	color	color	NOUN
cana-4446	95	10	pattern	pattern	NOUN
cana-4446	95	11	that	that	PRON
cana-4446	95	12	rdyc(g	rdyc(g	VERB
cana-4446	95	13	)	)	PUNCT
cana-4446	96	1	=	=	SYM
cana-4446	96	2	⌊	⌊	VERB
cana-4446	96	3	𝑛	𝑛	DET
cana-4446	96	4	3	3	NUM
cana-4446	96	5	⌋	⌋	NOUN
cana-4446	96	6	+	+	CCONJ
cana-4446	96	7	1	1	NUM
cana-4446	96	8	case	case	NOUN
cana-4446	96	9	7	7	NUM
cana-4446	96	10	.	.	NOUN
cana-4446	96	11	2n	2n	NUM
cana-4446	96	12	≡	≡	PROPN
cana-4446	96	13	0	0	PUNCT
cana-4446	97	1	(	(	PUNCT
cana-4446	97	2	mod	mod	NOUN
cana-4446	97	3	3	3	X
cana-4446	97	4	)	)	PUNCT
cana-4446	97	5	coloring	coloring	NOUN
cana-4446	97	6	is	be	AUX
cana-4446	97	7	defined	define	VERB
cana-4446	97	8	by	by	ADP
cana-4446	97	9	𝑣	𝑣	PROPN
cana-4446	97	10	𝑖+	𝑖+	PRON
cana-4446	97	11	(	(	PUNCT
cana-4446	97	12	𝑛	𝑛	PROPN
cana-4446	97	13	3	3	NUM
cana-4446	97	14	+1)𝑘	+1)𝑘	NOUN
cana-4446	97	15	=	=	SYM
cana-4446	97	16	𝑖	𝑖	PROPN
cana-4446	97	17	,	,	PUNCT
cana-4446	97	18	for	for	ADP
cana-4446	97	19	1	1	NUM
cana-4446	97	20	≤	≤	NUM
cana-4446	97	21	i	i	PRON
cana-4446	97	22	≤	≤	NOUN
cana-4446	97	23	n	n	PRON
cana-4446	97	24	3	3	NUM
cana-4446	97	25	+	+	NUM
cana-4446	97	26	1	1	NUM
cana-4446	97	27	,	,	PUNCT
cana-4446	97	28	0	0	NUM
cana-4446	97	29	≤	≤	NUM
cana-4446	97	30	k	k	X
cana-4446	97	31	≤	≤	NUM
cana-4446	97	32	5	5	NUM
cana-4446	97	33	.	.	PUNCT
cana-4446	98	1	it	it	PRON
cana-4446	98	2	is	be	AUX
cana-4446	98	3	clear	clear	ADJ
cana-4446	98	4	from	from	ADP
cana-4446	98	5	this	this	DET
cana-4446	98	6	color	color	NOUN
cana-4446	98	7	pattern	pattern	NOUN
cana-4446	98	8	that	that	PRON
cana-4446	98	9	rdyc(g	rdyc(g	VERB
cana-4446	98	10	)	)	PUNCT
cana-4446	99	1	=	=	SYM
cana-4446	99	2	⌊	⌊	VERB
cana-4446	99	3	𝑛	𝑛	DET
cana-4446	99	4	3	3	NUM
cana-4446	99	5	⌋	⌋	NOUN
cana-4446	99	6	+	+	CCONJ
cana-4446	99	7	1	1	X
cana-4446	99	8	.	.	X
cana-4446	99	9	figure	figure	VERB
cana-4446	99	10	3	3	NUM
cana-4446	99	11	:	:	PUNCT
cana-4446	99	12	a	a	DET
cana-4446	99	13	graph	graph	NOUN
cana-4446	99	14	illustrating	illustrate	VERB
cana-4446	99	15	the	the	DET
cana-4446	99	16	way	way	NOUN
cana-4446	99	17	colors	color	NOUN
cana-4446	99	18	are	be	AUX
cana-4446	99	19	assigned	assign	VERB
cana-4446	99	20	to	to	ADP
cana-4446	99	21	brick	brick	NOUN
cana-4446	99	22	products	product	NOUN
cana-4446	99	23	in	in	ADP
cana-4446	99	24	c(18,1,7	c(18,1,7	PROPN
cana-4446	99	25	)	)	PUNCT
cana-4446	99	26	refrences	refrence	VERB
cana-4446	100	1	[	[	X
cana-4446	100	2	1	1	X
cana-4446	100	3	]	]	PUNCT
cana-4446	100	4	alspach	alspach	NOUN
cana-4446	100	5	,	,	PUNCT
cana-4446	100	6	c.c.chen	c.c.chen	PROPN
cana-4446	100	7	&	&	CCONJ
cana-4446	100	8	and	and	CCONJ
cana-4446	100	9	kevin	kevin	PROPN
cana-4446	100	10	mcavancy	mcavancy	PROPN
cana-4446	100	11	.	.	PUNCT
cana-4446	101	1	(	(	PUNCT
cana-4446	101	2	1996	1996	NUM
cana-4446	101	3	)	)	PUNCT
cana-4446	101	4	.	.	PUNCT
cana-4446	102	1	on	on	ADP
cana-4446	102	2	a	a	DET
cana-4446	102	3	class	class	NOUN
cana-4446	102	4	of	of	ADP
cana-4446	102	5	hamiltonian	hamiltonian	ADJ
cana-4446	102	6	laceable	laceable	ADJ
cana-4446	102	7	3	3	NUM
cana-4446	102	8	-	-	PUNCT
cana-4446	102	9	regular	regular	ADJ
cana-4446	102	10	graphs	graph	NOUN
cana-4446	102	11	,	,	PUNCT
cana-4446	102	12	discrete	discrete	ADJ
cana-4446	102	13	math	math	NOUN
cana-4446	102	14	.	.	PUNCT
cana-4446	103	1	,151	,151	PROPN
cana-4446	103	2	,	,	PUNCT
cana-4446	103	3	19	19	NUM
cana-4446	103	4	-	-	SYM
cana-4446	103	5	38	38	NUM
cana-4446	103	6	.	.	PUNCT
cana-4446	104	1	[	[	X
cana-4446	104	2	2	2	NUM
cana-4446	104	3	]	]	PUNCT
cana-4446	104	4	g.chatrand	g.chatrand	NOUN
cana-4446	104	5	,	,	PUNCT
cana-4446	104	6	g.l.johns	g.l.johns	NOUN
cana-4446	104	7	,	,	PUNCT
cana-4446	104	8	k.a.mckeon	k.a.mckeon	PROPN
cana-4446	104	9	&	&	CCONJ
cana-4446	104	10	p.zang.(2008	p.zang.(2008	PROPN
cana-4446	104	11	)	)	PUNCT
cana-4446	104	12	.	.	PUNCT
cana-4446	105	1	rainbow	rainbow	NOUN
cana-4446	105	2	connection	connection	NOUN
cana-4446	105	3	in	in	ADP
cana-4446	105	4	graphs	graph	NOUN
cana-4446	105	5	,	,	PUNCT
cana-4446	105	6	math	math	NOUN
cana-4446	105	7	.	.	PUNCT
cana-4446	106	1	bohem	bohem	PROPN
cana-4446	106	2	133	133	NUM
cana-4446	106	3	,	,	PUNCT
cana-4446	106	4	85	85	NUM
cana-4446	106	5	-	-	SYM
cana-4446	106	6	98	98	NUM
cana-4446	106	7	.	.	PUNCT
cana-4446	107	1	communications	communication	NOUN
cana-4446	107	2	on	on	ADP
cana-4446	107	3	applied	apply	VERB
cana-4446	107	4	nonlinear	nonlinear	ADJ
cana-4446	107	5	analysis	analysis	NOUN
cana-4446	107	6	issn	issn	NOUN
cana-4446	107	7	:	:	PUNCT
cana-4446	107	8	1074	1074	NUM
cana-4446	107	9	-	-	PUNCT
cana-4446	107	10	133x	133x	NUM
cana-4446	107	11	vol	vol	NOUN
cana-4446	107	12	32	32	NUM
cana-4446	107	13	no	no	NOUN
cana-4446	107	14	.	.	PUNCT
cana-4446	108	1	9s	9s	NUM
cana-4446	108	2	(	(	PUNCT
cana-4446	108	3	2025	2025	NUM
cana-4446	108	4	)	)	PUNCT
cana-4446	108	5	2049	2049	NUM
cana-4446	108	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4446	109	1	[	[	X
cana-4446	109	2	3	3	X
cana-4446	109	3	]	]	PUNCT
cana-4446	109	4	m.krivelevich	m.krivelevich	NOUN
cana-4446	109	5	,	,	PUNCT
cana-4446	109	6	&	&	CCONJ
cana-4446	109	7	and	and	CCONJ
cana-4446	109	8	r.yuster.(2010	r.yuster.(2010	NOUN
cana-4446	109	9	)	)	PUNCT
cana-4446	109	10	.	.	PUNCT
cana-4446	110	1	the	the	DET
cana-4446	110	2	rainbow	rainbow	NOUN
cana-4446	110	3	connection	connection	NOUN
cana-4446	110	4	of	of	ADP
cana-4446	110	5	a	a	DET
cana-4446	110	6	graph	graph	NOUN
cana-4446	110	7	is	be	AUX
cana-4446	110	8	(	(	PUNCT
cana-4446	110	9	at	at	ADP
cana-4446	110	10	most	most	ADJ
cana-4446	110	11	)	)	PUNCT
cana-4446	110	12	is	be	AUX
cana-4446	110	13	reciprocal	reciprocal	ADJ
cana-4446	110	14	to	to	ADP
cana-4446	110	15	its	its	PRON
cana-4446	110	16	minimum	minimum	NOUN
cana-4446	110	17	degree	degree	NOUN
cana-4446	110	18	,	,	PUNCT
cana-4446	110	19	j.graph	j.graph	NOUN
cana-4446	110	20	theory	theory	NOUN
cana-4446	110	21	,	,	PUNCT
cana-4446	110	22	63	63	NUM
cana-4446	110	23	,	,	PUNCT
cana-4446	110	24	185	185	NUM
cana-4446	110	25	-	-	SYM
cana-4446	110	26	191	191	NUM
cana-4446	110	27	.	.	PUNCT
cana-4446	111	1	[	[	X
cana-4446	111	2	4	4	X
cana-4446	111	3	]	]	PUNCT
cana-4446	111	4	kulkarni	kulkarni	PROPN
cana-4446	111	5	sunita	sunita	PROPN
cana-4446	111	6	jagannatharao	jagannatharao	PROPN
cana-4446	111	7	&	&	CCONJ
cana-4446	111	8	and	and	CCONJ
cana-4446	111	9	r.	r.	PROPN
cana-4446	111	10	murali	murali	PROPN
cana-4446	111	11	.	.	PUNCT
cana-4446	112	1	(	(	PUNCT
cana-4446	112	2	2017	2017	NUM
cana-4446	112	3	)	)	PUNCT
cana-4446	112	4	.	.	PUNCT
cana-4446	113	1	rainbow	rainbow	NOUN
cana-4446	113	2	connection	connection	NOUN
cana-4446	113	3	in	in	ADP
cana-4446	113	4	some	some	DET
cana-4446	113	5	brick	brick	NOUN
cana-4446	113	6	product	product	NOUN
cana-4446	113	7	graphs	graph	NOUN
cana-4446	113	8	,	,	PUNCT
cana-4446	113	9	int	int	NOUN
cana-4446	113	10	.	.	PUNCT
cana-4446	114	1	j.	j.	PROPN
cana-4446	114	2	of	of	ADP
cana-4446	114	3	math	math	PROPN
cana-4446	114	4	.	.	PUNCT
cana-4446	115	1	and	and	CCONJ
cana-4446	115	2	its	its	PRON
cana-4446	115	3	appl	appl	NOUN
cana-4446	115	4	,	,	PUNCT
cana-4446	115	5	4e(5	4e(5	NUM
cana-4446	115	6	)	)	PUNCT
cana-4446	115	7	,	,	PUNCT
cana-4446	115	8	769	769	NUM
cana-4446	115	9	-	-	SYM
cana-4446	115	10	77	77	NUM
cana-4446	116	1	[	[	X
cana-4446	116	2	5	5	NUM
cana-4446	116	3	]	]	PUNCT
cana-4446	116	4	kulkarni	kulkarni	PROPN
cana-4446	116	5	sunita	sunita	PROPN
cana-4446	116	6	jagannatharao	jagannatharao	PROPN
cana-4446	116	7	&	&	CCONJ
cana-4446	116	8	and	and	CCONJ
cana-4446	116	9	r.	r.	PROPN
cana-4446	116	10	murali	murali	PROPN
cana-4446	116	11	.	.	PUNCT
cana-4446	117	1	(	(	PUNCT
cana-4446	117	2	2019	2019	NUM
cana-4446	117	3	)	)	PUNCT
cana-4446	117	4	.	.	PUNCT
cana-4446	118	1	rainbow	rainbow	PROPN
cana-4446	118	2	connection	connection	NOUN
cana-4446	118	3	insome	insome	PROPN
cana-4446	118	4	corona	corona	NOUN
cana-4446	118	5	product	product	NOUN
cana-4446	118	6	graphs	graph	NOUN
cana-4446	118	7	,	,	PUNCT
cana-4446	118	8	malaya	malaya	PROPN
cana-4446	118	9	journal	journal	PROPN
cana-4446	118	10	of	of	ADP
cana-4446	118	11	matematik	matematik	PROPN
cana-4446	118	12	,	,	PUNCT
cana-4446	118	13	7(1	7(1	NUM
cana-4446	118	14	)	)	PUNCT
cana-4446	118	15	,	,	PUNCT
cana-4446	118	16	127	127	NUM
cana-4446	118	17	-	-	SYM
cana-4446	118	18	131	131	NUM
cana-4446	118	19	.	.	PUNCT
cana-4446	119	1	[	[	X
cana-4446	119	2	6	6	NUM
cana-4446	119	3	]	]	PUNCT
cana-4446	119	4	kulkarni	kulkarni	PROPN
cana-4446	119	5	sunita	sunita	PROPN
cana-4446	119	6	jagannatharao	jagannatharao	PROPN
cana-4446	119	7	and	and	CCONJ
cana-4446	119	8	r.	r.	PROPN
cana-4446	119	9	murali	murali	PROPN
cana-4446	119	10	,	,	PUNCT
cana-4446	119	11	”	"	PUNCT
cana-4446	119	12	star	star	NOUN
cana-4446	119	13	rainbow	rainbow	NOUN
cana-4446	119	14	coloring	color	VERB
cana-4446	119	15	in	in	ADP
cana-4446	119	16	brick	brick	NOUN
cana-4446	119	17	product	product	NOUN
cana-4446	119	18	graphs	graph	NOUN
cana-4446	119	19	,	,	PUNCT
cana-4446	119	20	”	"	PUNCT
cana-4446	119	21	journal	journal	NOUN
cana-4446	119	22	of	of	ADP
cana-4446	119	23	physics	physics	NOUN
cana-4446	119	24	conferences	conference	NOUN
cana-4446	119	25	,	,	PUNCT
cana-4446	119	26	vol.1597	vol.1597	NOUN
cana-4446	119	27	,	,	PUNCT
cana-4446	119	28	no.1,012056	no.1,012056	PROPN
cana-4446	119	29	,	,	PUNCT
cana-4446	119	30	2020	2020	NUM
cana-4446	119	31	.	.	PUNCT