id	sid	tid	token	lemma	pos
cana-4834	1	1	communications	communication	NOUN
cana-4834	1	2	on	on	ADP
cana-4834	1	3	applied	apply	VERB
cana-4834	1	4	nonlinear	nonlinear	ADJ
cana-4834	1	5	analysis	analysis	NOUN
cana-4834	1	6	issn	issn	NOUN
cana-4834	1	7	:	:	PUNCT
cana-4834	1	8	1074	1074	NUM
cana-4834	1	9	-	-	PUNCT
cana-4834	1	10	133x	133x	NUM
cana-4834	1	11	vol	vol	VERB
cana-4834	1	12	32	32	NUM
cana-4834	1	13	no	no	NOUN
cana-4834	1	14	.	.	PUNCT
cana-4834	2	1	10s	10	NOUN
cana-4834	2	2	(	(	PUNCT
cana-4834	2	3	2025	2025	NUM
cana-4834	2	4	)	)	PUNCT
cana-4834	2	5	424	424	NUM
cana-4834	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4834	2	7	on	on	ADP
cana-4834	2	8	irregular	irregular	ADJ
cana-4834	2	9	coloring	coloring	NOUN
cana-4834	2	10	of	of	ADP
cana-4834	2	11	middle	middle	ADJ
cana-4834	2	12	graph	graph	NOUN
cana-4834	2	13	of	of	ADP
cana-4834	2	14	snake	snake	NOUN
cana-4834	2	15	graphs	graph	NOUN
cana-4834	2	16	dr	dr	PROPN
cana-4834	2	17	.	.	PROPN
cana-4834	2	18	g.	g.	PROPN
cana-4834	2	19	jothilakshmi.1	jothilakshmi.1	PROPN
cana-4834	2	20	,	,	PUNCT
cana-4834	2	21	dr	dr	PROPN
cana-4834	2	22	.	.	PROPN
cana-4834	2	23	k.	k.	PROPN
cana-4834	2	24	rajam	rajam	PROPN
cana-4834	2	25	2	2	NUM
cana-4834	2	26	,	,	PUNCT
cana-4834	2	27	dr	dr	PROPN
cana-4834	2	28	.	.	PROPN
cana-4834	2	29	s.	s.	PROPN
cana-4834	2	30	manimekalai	manimekalai	PROPN
cana-4834	2	31	3	3	NUM
cana-4834	2	32	,	,	PUNCT
cana-4834	2	33	dr	dr	PROPN
cana-4834	2	34	.	.	PROPN
cana-4834	2	35	u.	u.	PROPN
cana-4834	2	36	s.	s.	PROPN
cana-4834	2	37	monolisa4	monolisa4	PROPN
cana-4834	2	38	,	,	PUNCT
cana-4834	2	39	k.	k.	PUNCT
cana-4834	2	40	nandhini5	nandhini5	PROPN
cana-4834	3	1	1,5	1,5	NUM
cana-4834	3	2	assistant	assistant	NOUN
cana-4834	3	3	professor	professor	NOUN
cana-4834	3	4	,	,	PUNCT
cana-4834	3	5	department	department	NOUN
cana-4834	3	6	of	of	ADP
cana-4834	3	7	mathematics	mathematic	NOUN
cana-4834	3	8	,	,	PUNCT
cana-4834	3	9	rvs	rvs	ADJ
cana-4834	3	10	college	college	NOUN
cana-4834	3	11	of	of	ADP
cana-4834	3	12	arts	art	NOUN
cana-4834	3	13	and	and	CCONJ
cana-4834	3	14	science	science	NOUN
cana-4834	3	15	(	(	PUNCT
cana-4834	3	16	affiliated	affiliate	VERB
cana-4834	3	17	to	to	ADP
cana-4834	3	18	bharathiar	bharathiar	PROPN
cana-4834	3	19	university	university	PROPN
cana-4834	3	20	)	)	PUNCT
cana-4834	3	21	coimbatore	coimbatore	PROPN
cana-4834	3	22	,	,	PUNCT
cana-4834	3	23	india	india	PROPN
cana-4834	3	24	.	.	PUNCT
cana-4834	3	25	email	email	NOUN
cana-4834	3	26	:	:	PUNCT
cana-4834	3	27	jothikrishna11@gmail.com	jothikrishna11@gmail.com	PROPN
cana-4834	3	28	2	2	NUM
cana-4834	3	29	,	,	PUNCT
cana-4834	3	30	assistant	assistant	NOUN
cana-4834	3	31	professor	professor	NOUN
cana-4834	3	32	(	(	PUNCT
cana-4834	3	33	ss	ss	PROPN
cana-4834	3	34	)	)	PUNCT
cana-4834	3	35	,	,	PUNCT
cana-4834	3	36	department	department	NOUN
cana-4834	3	37	of	of	ADP
cana-4834	3	38	mathematics	mathematics	PROPN
cana-4834	3	39	,	,	PUNCT
cana-4834	3	40	dr	dr	PROPN
cana-4834	3	41	.	.	PROPN
cana-4834	3	42	ngp	ngp	PROPN
cana-4834	3	43	institute	institute	PROPN
cana-4834	3	44	of	of	ADP
cana-4834	3	45	technology	technology	PROPN
cana-4834	3	46	(	(	PUNCT
cana-4834	3	47	affiliated	affiliate	VERB
cana-4834	3	48	to	to	ADP
cana-4834	3	49	anna	anna	PROPN
cana-4834	3	50	university	university	PROPN
cana-4834	3	51	)	)	PUNCT
cana-4834	3	52	coimbatore	coimbatore	PROPN
cana-4834	3	53	,	,	PUNCT
cana-4834	3	54	india	india	PROPN
cana-4834	3	55	,	,	PUNCT
cana-4834	3	56	email	email	NOUN
cana-4834	3	57	:	:	PUNCT
cana-4834	3	58	rajee.mat@gmail.com	rajee.mat@gmail.com	NOUN
cana-4834	3	59	3	3	NUM
cana-4834	3	60	,	,	PUNCT
cana-4834	3	61	associate	associate	ADJ
cana-4834	3	62	professor	professor	NOUN
cana-4834	3	63	,	,	PUNCT
cana-4834	3	64	department	department	NOUN
cana-4834	3	65	of	of	ADP
cana-4834	3	66	mathematics	mathematic	NOUN
cana-4834	3	67	,	,	PUNCT
cana-4834	3	68	dr.n.g.p	dr.n.g.p	NOUN
cana-4834	3	69	arts	art	NOUN
cana-4834	3	70	and	and	CCONJ
cana-4834	3	71	science	science	PROPN
cana-4834	3	72	college	college	PROPN
cana-4834	3	73	(	(	PUNCT
cana-4834	3	74	affiliated	affiliate	VERB
cana-4834	3	75	to	to	ADP
cana-4834	3	76	bharathiar	bharathiar	PROPN
cana-4834	3	77	university	university	PROPN
cana-4834	3	78	)	)	PUNCT
cana-4834	3	79	coimbatore	coimbatore	PROPN
cana-4834	3	80	,	,	PUNCT
cana-4834	3	81	india	india	PROPN
cana-4834	3	82	email:manimekalai@drngpasc.ac.in	email:manimekalai@drngpasc.ac.in	PROPN
cana-4834	3	83	4	4	NUM
cana-4834	3	84	assistant	assistant	NOUN
cana-4834	3	85	professor	professor	NOUN
cana-4834	3	86	,	,	PUNCT
cana-4834	3	87	department	department	NOUN
cana-4834	3	88	of	of	ADP
cana-4834	3	89	mathematics	mathematics	PROPN
cana-4834	3	90	,	,	PUNCT
cana-4834	3	91	dr	dr	PROPN
cana-4834	3	92	.	.	PROPN
cana-4834	3	93	mahalingam	mahalingam	PROPN
cana-4834	3	94	college	college	PROPN
cana-4834	3	95	of	of	ADP
cana-4834	3	96	engineering	engineering	NOUN
cana-4834	3	97	and	and	CCONJ
cana-4834	3	98	technology	technology	NOUN
cana-4834	3	99	(	(	PUNCT
cana-4834	3	100	affiliated	affiliate	VERB
cana-4834	3	101	to	to	ADP
cana-4834	3	102	anna	anna	PROPN
cana-4834	3	103	university	university	PROPN
cana-4834	3	104	)	)	PUNCT
cana-4834	3	105	coimbatore	coimbatore	PROPN
cana-4834	3	106	,	,	PUNCT
cana-4834	3	107	india	india	PROPN
cana-4834	3	108	email	email	NOUN
cana-4834	3	109	:	:	PUNCT
cana-4834	3	110	monolisamat@gmail.com	monolisamat@gmail.com	X
cana-4834	3	111	article	article	NOUN
cana-4834	3	112	history	history	NOUN
cana-4834	3	113	:	:	PUNCT
cana-4834	3	114	received	receive	VERB
cana-4834	3	115	:	:	PUNCT
cana-4834	3	116	12	12	NUM
cana-4834	3	117	-	-	SYM
cana-4834	3	118	01	01	NUM
cana-4834	3	119	-	-	PUNCT
cana-4834	3	120	2025	2025	NUM
cana-4834	3	121	revised	revise	VERB
cana-4834	3	122	:	:	PUNCT
cana-4834	3	123	15	15	NUM
cana-4834	3	124	-	-	NUM
cana-4834	3	125	02	02	NUM
cana-4834	3	126	-	-	PUNCT
cana-4834	3	127	2025	2025	NUM
cana-4834	3	128	accepted	accept	VERB
cana-4834	3	129	:	:	PUNCT
cana-4834	3	130	01	01	NUM
cana-4834	3	131	-	-	SYM
cana-4834	3	132	03	03	NUM
cana-4834	3	133	-	-	PUNCT
cana-4834	3	134	2025	2025	NUM
cana-4834	3	135	abstract	abstract	NOUN
cana-4834	3	136	:	:	PUNCT
cana-4834	3	137	an	an	DET
cana-4834	3	138	irregular	irregular	ADJ
cana-4834	3	139	coloring	coloring	NOUN
cana-4834	3	140	is	be	AUX
cana-4834	3	141	a	a	DET
cana-4834	3	142	proper	proper	ADJ
cana-4834	3	143	coloring	coloring	NOUN
cana-4834	3	144	in	in	ADP
cana-4834	3	145	which	which	PRON
cana-4834	3	146	distinct	distinct	ADJ
cana-4834	3	147	vertices	vertex	NOUN
cana-4834	3	148	have	have	VERB
cana-4834	3	149	different	different	ADJ
cana-4834	3	150	color	color	NOUN
cana-4834	3	151	codes	code	NOUN
cana-4834	3	152	.	.	PUNCT
cana-4834	4	1	in	in	ADP
cana-4834	4	2	this	this	DET
cana-4834	4	3	paper	paper	NOUN
cana-4834	4	4	we	we	PRON
cana-4834	4	5	obtain	obtain	VERB
cana-4834	4	6	the	the	DET
cana-4834	4	7	irregular	irregular	ADJ
cana-4834	4	8	chromatic	chromatic	ADJ
cana-4834	4	9	number	number	NOUN
cana-4834	4	10	of	of	ADP
cana-4834	4	11	middle	middle	ADJ
cana-4834	4	12	graph	graph	NOUN
cana-4834	4	13	of	of	ADP
cana-4834	4	14	certain	certain	ADJ
cana-4834	4	15	snake	snake	NOUN
cana-4834	4	16	graphs	graph	NOUN
cana-4834	4	17	of	of	ADP
cana-4834	4	18	snake	snake	NOUN
cana-4834	4	19	graph	graph	NOUN
cana-4834	4	20	families	family	NOUN
cana-4834	4	21	.	.	PUNCT
cana-4834	5	1	keywords	keyword	NOUN
cana-4834	5	2	:	:	PUNCT
cana-4834	5	3	double	double	ADJ
cana-4834	5	4	triangular	triangular	NOUN
cana-4834	5	5	snake	snake	NOUN
cana-4834	5	6	graph	graph	NOUN
cana-4834	5	7	,	,	PUNCT
cana-4834	5	8	diamond	diamond	NOUN
cana-4834	5	9	snake	snake	NOUN
cana-4834	5	10	graph	graph	NOUN
cana-4834	5	11	,	,	PUNCT
cana-4834	5	12	graph	graph	NOUN
cana-4834	5	13	coloring	coloring	NOUN
cana-4834	5	14	,	,	PUNCT
cana-4834	5	15	irregular	irregular	ADJ
cana-4834	5	16	chromatic	chromatic	ADJ
cana-4834	5	17	number	number	NOUN
cana-4834	5	18	,	,	PUNCT
cana-4834	5	19	irregular	irregular	ADJ
cana-4834	5	20	coloring	coloring	NOUN
cana-4834	5	21	,	,	PUNCT
cana-4834	5	22	middle	middle	ADJ
cana-4834	5	23	graph	graph	NOUN
cana-4834	5	24	,	,	PUNCT
cana-4834	5	25	triangular	triangular	NOUN
cana-4834	5	26	snake	snake	NOUN
cana-4834	5	27	graph	graph	NOUN
cana-4834	5	28	classification	classification	NOUN
cana-4834	5	29	number	number	NOUN
cana-4834	5	30	:	:	PUNCT
cana-4834	5	31	05c15	05c15	NOUN
cana-4834	5	32	,	,	PUNCT
cana-4834	5	33	05c76	05c76	NUM
cana-4834	5	34	1	1	X
cana-4834	5	35	.	.	X
cana-4834	5	36	introduction	introduction	NOUN
cana-4834	5	37	let	let	VERB
cana-4834	5	38	g	g	PRON
cana-4834	5	39	be	be	AUX
cana-4834	5	40	a	a	DET
cana-4834	5	41	finite	finite	ADJ
cana-4834	5	42	,	,	PUNCT
cana-4834	5	43	undirected	undirected	ADJ
cana-4834	5	44	graph	graph	NOUN
cana-4834	5	45	[	[	X
cana-4834	5	46	1	1	NUM
cana-4834	5	47	]	]	PUNCT
cana-4834	5	48	with	with	ADP
cana-4834	5	49	no	no	DET
cana-4834	5	50	loops	loop	NOUN
cana-4834	5	51	and	and	CCONJ
cana-4834	5	52	multiple	multiple	ADJ
cana-4834	5	53	edges	edge	NOUN
cana-4834	5	54	.	.	PUNCT
cana-4834	6	1	the	the	DET
cana-4834	6	2	graph	graph	NOUN
cana-4834	6	3	g	g	NOUN
cana-4834	6	4	has	have	VERB
cana-4834	6	5	the	the	DET
cana-4834	6	6	vertex	vertex	NOUN
cana-4834	6	7	set	set	VERB
cana-4834	6	8	v	v	NOUN
cana-4834	6	9	(	(	PUNCT
cana-4834	6	10	g	g	NOUN
cana-4834	6	11	)	)	PUNCT
cana-4834	6	12	and	and	CCONJ
cana-4834	6	13	the	the	DET
cana-4834	6	14	edge	edge	NOUN
cana-4834	6	15	set	set	NOUN
cana-4834	6	16	e	e	X
cana-4834	6	17	(	(	PUNCT
cana-4834	6	18	g	g	NOUN
cana-4834	6	19	)	)	PUNCT
cana-4834	6	20	.	.	PUNCT
cana-4834	7	1	graph	graph	NOUN
cana-4834	7	2	coloring	color	VERB
cana-4834	7	3	[	[	X
cana-4834	7	4	5	5	NUM
cana-4834	7	5	]	]	PUNCT
cana-4834	7	6	is	be	AUX
cana-4834	7	7	coloring	color	VERB
cana-4834	7	8	of	of	ADP
cana-4834	7	9	g	g	PROPN
cana-4834	7	10	such	such	ADJ
cana-4834	7	11	that	that	SCONJ
cana-4834	7	12	no	no	DET
cana-4834	7	13	two	two	NUM
cana-4834	7	14	adjacent	adjacent	ADJ
cana-4834	7	15	vertices	vertex	NOUN
cana-4834	7	16	share	share	VERB
cana-4834	7	17	the	the	DET
cana-4834	7	18	same	same	ADJ
cana-4834	7	19	color	color	NOUN
cana-4834	7	20	.	.	PUNCT
cana-4834	8	1	a	a	DET
cana-4834	8	2	proper	proper	ADJ
cana-4834	8	3	coloring	coloring	NOUN
cana-4834	8	4	c	c	NOUN
cana-4834	8	5	is	be	AUX
cana-4834	8	6	an	an	DET
cana-4834	8	7	irregular	irregular	ADJ
cana-4834	8	8	coloring	coloring	NOUN
cana-4834	8	9	[	[	X
cana-4834	8	10	1	1	X
cana-4834	8	11	]	]	PUNCT
cana-4834	8	12	if	if	SCONJ
cana-4834	8	13	no	no	DET
cana-4834	8	14	two	two	NUM
cana-4834	8	15	like	like	ADJ
cana-4834	8	16	-	-	PUNCT
cana-4834	8	17	colored	colored	ADJ
cana-4834	8	18	vertices	vertex	NOUN
cana-4834	8	19	have	have	VERB
cana-4834	8	20	the	the	DET
cana-4834	8	21	same	same	ADJ
cana-4834	8	22	color	color	NOUN
cana-4834	8	23	code	code	NOUN
cana-4834	8	24	.	.	PUNCT
cana-4834	9	1	i.e.	i.e.	X
cana-4834	9	2	,	,	PUNCT
cana-4834	9	3	for	for	ADP
cana-4834	9	4	every	every	DET
cana-4834	9	5	pair	pair	NOUN
cana-4834	9	6	of	of	ADP
cana-4834	9	7	vertices	vertex	NOUN
cana-4834	9	8	u	u	NOUN
cana-4834	9	9	and	and	CCONJ
cana-4834	9	10	w	w	PROPN
cana-4834	9	11	;	;	PUNCT
cana-4834	9	12	code(u	code(u	PROPN
cana-4834	9	13	)	)	PUNCT
cana-4834	9	14	≠code(w	≠code(w	PROPN
cana-4834	9	15	)	)	PUNCT
cana-4834	9	16	whenever	whenever	SCONJ
cana-4834	9	17	c(u	c(u	PROPN
cana-4834	9	18	)	)	PUNCT
cana-4834	9	19	=	=	SYM
cana-4834	9	20	c(w	c(w	PROPN
cana-4834	9	21	)	)	PUNCT
cana-4834	9	22	.	.	PUNCT
cana-4834	10	1	thus	thus	ADV
cana-4834	10	2	,	,	PUNCT
cana-4834	10	3	an	an	DET
cana-4834	10	4	irregular	irregular	ADJ
cana-4834	10	5	coloring	coloring	NOUN
cana-4834	10	6	distinguishes	distinguish	VERB
cana-4834	10	7	each	each	DET
cana-4834	10	8	vertex	vertex	NOUN
cana-4834	10	9	from	from	ADP
cana-4834	10	10	each	each	PRON
cana-4834	10	11	of	of	ADP
cana-4834	10	12	other	other	ADJ
cana-4834	10	13	vertex	vertex	NOUN
cana-4834	10	14	by	by	ADP
cana-4834	10	15	its	its	PRON
cana-4834	10	16	color	color	NOUN
cana-4834	10	17	or	or	CCONJ
cana-4834	10	18	by	by	ADP
cana-4834	10	19	its	its	PRON
cana-4834	10	20	color	color	NOUN
cana-4834	10	21	code	code	NOUN
cana-4834	10	22	.	.	PUNCT
cana-4834	11	1	mary	mary	PROPN
cana-4834	11	2	radcliffe	radcliffe	PROPN
cana-4834	11	3	and	and	CCONJ
cana-4834	11	4	ping	ping	PROPN
cana-4834	11	5	zhang[8,9	zhang[8,9	PROPN
cana-4834	11	6	]	]	PUNCT
cana-4834	11	7	established	establish	VERB
cana-4834	11	8	sharp	sharp	ADJ
cana-4834	11	9	upper	upper	ADJ
cana-4834	11	10	and	and	CCONJ
cana-4834	11	11	lower	low	ADJ
cana-4834	11	12	bounds	bound	NOUN
cana-4834	11	13	for	for	ADP
cana-4834	11	14	the	the	DET
cana-4834	11	15	irregular	irregular	ADJ
cana-4834	11	16	chromatic	chromatic	ADJ
cana-4834	11	17	number	number	NOUN
cana-4834	11	18	of	of	ADP
cana-4834	11	19	a	a	DET
cana-4834	11	20	disconnected	disconnected	ADJ
cana-4834	11	21	graph	graph	NOUN
cana-4834	11	22	in	in	ADP
cana-4834	11	23	terms	term	NOUN
cana-4834	11	24	of	of	ADP
cana-4834	11	25	the	the	DET
cana-4834	11	26	irregular	irregular	ADJ
cana-4834	11	27	chromatic	chromatic	ADJ
cana-4834	11	28	numbers	number	NOUN
cana-4834	11	29	of	of	ADP
cana-4834	11	30	its	its	PRON
cana-4834	11	31	components	component	NOUN
cana-4834	11	32	.	.	PUNCT
cana-4834	11	33	a.rohini	a.rohini	PROPN
cana-4834	11	34	and	and	CCONJ
cana-4834	11	35	m.venkatachalam	m.venkatachalam	X
cana-4834	12	1	[	[	X
cana-4834	12	2	11	11	NUM
cana-4834	12	3	]	]	PUNCT
cana-4834	12	4	discussed	discuss	VERB
cana-4834	12	5	the	the	DET
cana-4834	12	6	irregular	irregular	ADJ
cana-4834	12	7	coloring	coloring	NOUN
cana-4834	12	8	of	of	ADP
cana-4834	12	9	wheel	wheel	NOUN
cana-4834	12	10	related	relate	VERB
cana-4834	12	11	graphs	graph	NOUN
cana-4834	12	12	.	.	PUNCT
cana-4834	13	1	further	far	ADV
cana-4834	13	2	this	this	DET
cana-4834	13	3	paper	paper	NOUN
cana-4834	13	4	exhibits	exhibit	VERB
cana-4834	13	5	the	the	DET
cana-4834	13	6	irregular	irregular	ADJ
cana-4834	13	7	coloring	coloring	NOUN
cana-4834	13	8	of	of	ADP
cana-4834	13	9	snake	snake	NOUN
cana-4834	13	10	graph	graph	NOUN
cana-4834	13	11	families	family	NOUN
cana-4834	13	12	.	.	PUNCT
cana-4834	14	1	the	the	DET
cana-4834	14	2	central	central	ADJ
cana-4834	14	3	graph	graph	NOUN
cana-4834	14	4	c(g	c(g	PROPN
cana-4834	14	5	)	)	PUNCT
cana-4834	14	6	is	be	AUX
cana-4834	14	7	obtained	obtain	VERB
cana-4834	14	8	by	by	ADP
cana-4834	14	9	subdividing	subdivide	VERB
cana-4834	14	10	each	each	DET
cana-4834	14	11	edge	edge	NOUN
cana-4834	14	12	of	of	ADP
cana-4834	14	13	g	g	NOUN
cana-4834	14	14	exactly	exactly	ADV
cana-4834	14	15	once	once	ADV
cana-4834	14	16	and	and	CCONJ
cana-4834	14	17	joining	join	VERB
cana-4834	14	18	all	all	DET
cana-4834	14	19	the	the	DET
cana-4834	14	20	non	non	ADJ
cana-4834	14	21	-	-	ADJ
cana-4834	14	22	adjacent	adjacent	ADJ
cana-4834	14	23	vertices	vertex	NOUN
cana-4834	14	24	of	of	ADP
cana-4834	14	25	g.	g.	PROPN
cana-4834	14	26	the	the	DET
cana-4834	14	27	vertex	vertex	NOUN
cana-4834	14	28	set	set	NOUN
cana-4834	14	29	of	of	ADP
cana-4834	14	30	middle	middle	ADJ
cana-4834	14	31	graph	graph	NOUN
cana-4834	14	32	m(g	m(g	PROPN
cana-4834	14	33	)	)	PUNCT
cana-4834	14	34	is	be	AUX
cana-4834	14	35	v(g)∪e(g	v(g)∪e(g	ADJ
cana-4834	14	36	)	)	PUNCT
cana-4834	14	37	in	in	ADP
cana-4834	14	38	which	which	PRON
cana-4834	14	39	two	two	NUM
cana-4834	14	40	elements	element	NOUN
cana-4834	14	41	are	be	AUX
cana-4834	14	42	adjacent	adjacent	ADJ
cana-4834	14	43	in	in	ADP
cana-4834	14	44	m(g	m(g	PROPN
cana-4834	14	45	)	)	PUNCT
cana-4834	14	46	if	if	SCONJ
cana-4834	14	47	the	the	DET
cana-4834	14	48	following	follow	VERB
cana-4834	14	49	conditions	condition	NOUN
cana-4834	14	50	hold	hold	VERB
cana-4834	14	51	.	.	PUNCT
cana-4834	15	1	(	(	PUNCT
cana-4834	15	2	i	i	NOUN
cana-4834	15	3	)	)	PUNCT
cana-4834	15	4	x	x	NOUN
cana-4834	15	5	,	,	PUNCT
cana-4834	15	6	y	y	PROPN
cana-4834	15	7	∈	∈	PROPN
cana-4834	15	8	e(g	e(g	PROPN
cana-4834	15	9	)	)	PUNCT
cana-4834	15	10	and	and	CCONJ
cana-4834	15	11	x	x	X
cana-4834	15	12	,	,	PUNCT
cana-4834	15	13	y	y	PROPN
cana-4834	15	14	are	be	AUX
cana-4834	15	15	adjacent	adjacent	ADJ
cana-4834	15	16	in	in	ADP
cana-4834	15	17	g.	g.	PROPN
cana-4834	15	18	(	(	PUNCT
cana-4834	15	19	ii	ii	PROPN
cana-4834	15	20	)	)	PUNCT
cana-4834	15	21	x	x	PROPN
cana-4834	16	1	∈v(g	∈v(g	PROPN
cana-4834	16	2	)	)	PUNCT
cana-4834	16	3	,	,	PUNCT
cana-4834	16	4	y	y	PROPN
cana-4834	16	5	∈	∈	PROPN
cana-4834	16	6	e(g	e(g	PROPN
cana-4834	16	7	)	)	PUNCT
cana-4834	16	8	and	and	CCONJ
cana-4834	16	9	y	y	PROPN
cana-4834	16	10	is	be	AUX
cana-4834	16	11	incident	incident	NOUN
cana-4834	16	12	on	on	ADP
cana-4834	16	13	x	x	PUNCT
cana-4834	16	14	in	in	ADP
cana-4834	16	15	g.	g.	PROPN
cana-4834	16	16	communications	communication	NOUN
cana-4834	16	17	on	on	ADP
cana-4834	16	18	applied	apply	VERB
cana-4834	16	19	nonlinear	nonlinear	ADJ
cana-4834	16	20	analysis	analysis	NOUN
cana-4834	16	21	issn	issn	NOUN
cana-4834	16	22	:	:	PUNCT
cana-4834	16	23	1074	1074	NUM
cana-4834	16	24	-	-	PUNCT
cana-4834	16	25	133x	133x	NUM
cana-4834	16	26	vol	vol	VERB
cana-4834	16	27	32	32	NUM
cana-4834	16	28	no	no	NOUN
cana-4834	16	29	.	.	PUNCT
cana-4834	17	1	10s	10	NOUN
cana-4834	17	2	(	(	PUNCT
cana-4834	17	3	2025	2025	NUM
cana-4834	17	4	)	)	PUNCT
cana-4834	17	5	425	425	NUM
cana-4834	17	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4834	17	7	a	a	DET
cana-4834	17	8	triangular	triangular	NOUN
cana-4834	17	9	snake	snake	NOUN
cana-4834	17	10	graph	graph	NOUN
cana-4834	17	11	tn	tn	PROPN
cana-4834	18	1	[	[	X
cana-4834	18	2	3	3	X
cana-4834	18	3	]	]	PUNCT
cana-4834	18	4	is	be	AUX
cana-4834	18	5	obtained	obtain	VERB
cana-4834	18	6	from	from	ADP
cana-4834	18	7	a	a	DET
cana-4834	18	8	path	path	NOUN
cana-4834	18	9	v1	v1	NOUN
cana-4834	18	10	,	,	PUNCT
cana-4834	18	11	v2	v2	PROPN
cana-4834	18	12	,	,	PUNCT
cana-4834	18	13	v3,	v3,	ADV
cana-4834	18	14	…	…	SYM
cana-4834	18	15	…	…	SYM
cana-4834	18	16	.vn+1	.vn+1	NUM
cana-4834	18	17	,	,	PUNCT
cana-4834	18	18	u1	u1	NOUN
cana-4834	18	19	,	,	PUNCT
cana-4834	18	20	u2	u2	PROPN
cana-4834	18	21	…	…	PUNCT
cana-4834	18	22	.	.	PUNCT
cana-4834	19	1	un	un	PROPN
cana-4834	19	2	.it	.it	PROPN
cana-4834	19	3	has	have	VERB
cana-4834	19	4	2n+1	2n+1	PROPN
cana-4834	19	5	vertices	vertex	NOUN
cana-4834	19	6	and	and	CCONJ
cana-4834	19	7	3n	3n	NUM
cana-4834	19	8	edges	edge	NOUN
cana-4834	19	9	,	,	PUNCT
cana-4834	19	10	where	where	SCONJ
cana-4834	19	11	n	n	PRON
cana-4834	19	12	is	be	AUX
cana-4834	19	13	the	the	DET
cana-4834	19	14	number	number	NOUN
cana-4834	19	15	of	of	ADP
cana-4834	19	16	blocks	block	NOUN
cana-4834	19	17	in	in	ADP
cana-4834	19	18	the	the	DET
cana-4834	19	19	triangular	triangular	NOUN
cana-4834	19	20	snake	snake	NOUN
cana-4834	19	21	and	and	CCONJ
cana-4834	19	22	a	a	DET
cana-4834	19	23	double	double	ADJ
cana-4834	19	24	triangular	triangular	NOUN
cana-4834	19	25	snake	snake	NOUN
cana-4834	19	26	graph	graph	NOUN
cana-4834	19	27	d(tn	d(tn	NOUN
cana-4834	19	28	)	)	PUNCT
cana-4834	20	1	[	[	X
cana-4834	20	2	3	3	X
cana-4834	20	3	]	]	PUNCT
cana-4834	20	4	is	be	AUX
cana-4834	20	5	obtained	obtain	VERB
cana-4834	20	6	from	from	ADP
cana-4834	20	7	a	a	DET
cana-4834	20	8	path	path	NOUN
cana-4834	20	9	v1	v1	NOUN
cana-4834	20	10	,	,	PUNCT
cana-4834	20	11	v2	v2	PROPN
cana-4834	20	12	,	,	PUNCT
cana-4834	20	13	v3,	v3,	ADV
cana-4834	20	14	…	…	PUNCT
cana-4834	20	15	…	…	PUNCT
cana-4834	20	16	.vn+1	.vn+1	PUNCT
cana-4834	20	17	by	by	ADP
cana-4834	20	18	joining	join	VERB
cana-4834	20	19	vi	vi	PROPN
cana-4834	20	20	and	and	CCONJ
cana-4834	20	21	vi+1	vi+1	ADV
cana-4834	20	22	to	to	ADP
cana-4834	20	23	a	a	DET
cana-4834	20	24	new	new	ADJ
cana-4834	20	25	vertex	vertex	NOUN
cana-4834	20	26	wi	wi	PROPN
cana-4834	20	27	to	to	ADP
cana-4834	20	28	another	another	DET
cana-4834	20	29	new	new	ADJ
cana-4834	20	30	vertex	vertex	NOUN
cana-4834	20	31	ui	ui	NOUN
cana-4834	20	32	.	.	PUNCT
cana-4834	21	1	a	a	DET
cana-4834	21	2	diamond	diamond	NOUN
cana-4834	21	3	snake	snake	NOUN
cana-4834	21	4	graph	graph	NOUN
cana-4834	21	5	dn	dn	NOUN
cana-4834	22	1	[	[	X
cana-4834	22	2	3	3	X
cana-4834	22	3	]	]	PUNCT
cana-4834	22	4	is	be	AUX
cana-4834	22	5	obtained	obtain	VERB
cana-4834	22	6	from	from	ADP
cana-4834	22	7	a	a	DET
cana-4834	22	8	path	path	NOUN
cana-4834	22	9	v1	v1	NOUN
cana-4834	22	10	,	,	PUNCT
cana-4834	22	11	v2	v2	PROPN
cana-4834	22	12	,	,	PUNCT
cana-4834	22	13	v3,	v3,	ADV
cana-4834	22	14	…	…	PUNCT
cana-4834	22	15	…	…	PUNCT
cana-4834	22	16	.vn+1	.vn+1	PUNCT
cana-4834	22	17	by	by	ADP
cana-4834	22	18	joining	join	VERB
cana-4834	22	19	vi	vi	PROPN
cana-4834	22	20	and	and	CCONJ
cana-4834	22	21	vi+1	vi+1	ADV
cana-4834	22	22	to	to	ADP
cana-4834	22	23	a	a	DET
cana-4834	22	24	new	new	ADJ
cana-4834	22	25	vertex	vertex	NOUN
cana-4834	22	26	wi	wi	PROPN
cana-4834	22	27	to	to	ADP
cana-4834	22	28	another	another	DET
cana-4834	22	29	new	new	ADJ
cana-4834	22	30	vertex	vertex	NOUN
cana-4834	22	31	ui	ui	NOUN
cana-4834	22	32	2	2	NUM
cana-4834	22	33	.	.	PUNCT
cana-4834	22	34	structural	structural	ADJ
cana-4834	22	35	properties	property	NOUN
cana-4834	22	36	of	of	ADP
cana-4834	22	37	middle	middle	ADJ
cana-4834	22	38	graph	graph	NOUN
cana-4834	22	39	of	of	ADP
cana-4834	22	40	m	m	PROPN
cana-4834	22	41	(	(	PUNCT
cana-4834	22	42	𝐓𝐧	𝐓𝐧	NOUN
cana-4834	22	43	)	)	PUNCT
cana-4834	22	44	,	,	PUNCT
cana-4834	22	45	m	m	VERB
cana-4834	22	46	[	[	X
cana-4834	22	47	d(tn	d(tn	X
cana-4834	22	48	)	)	PUNCT
cana-4834	22	49	]	]	PUNCT
cana-4834	22	50	and	and	CCONJ
cana-4834	22	51	m[dn	m[dn	NOUN
cana-4834	22	52	]	]	X
cana-4834	22	53	•	•	NUM
cana-4834	22	54	number	number	NOUN
cana-4834	22	55	of	of	ADP
cana-4834	22	56	vertices	vertex	NOUN
cana-4834	22	57	in	in	ADP
cana-4834	22	58	m	m	PROPN
cana-4834	22	59	(	(	PUNCT
cana-4834	22	60	tn	tn	PROPN
cana-4834	22	61	)	)	PUNCT
cana-4834	22	62	,	,	PUNCT
cana-4834	23	1	p=	p=	PROPN
cana-4834	23	2	5n+1	5n+1	NOUN
cana-4834	23	3	•	•	NOUN
cana-4834	23	4	number	number	NOUN
cana-4834	23	5	of	of	ADP
cana-4834	23	6	vertices	vertex	NOUN
cana-4834	23	7	in	in	ADP
cana-4834	23	8	m[d(tn	m[d(tn	PROPN
cana-4834	23	9	)	)	PUNCT
cana-4834	23	10	]	]	PUNCT
cana-4834	23	11	,	,	PUNCT
cana-4834	23	12	p=8n+1	p=8n+1	VERB
cana-4834	23	13	•	•	NUM
cana-4834	23	14	number	number	NOUN
cana-4834	23	15	of	of	ADP
cana-4834	23	16	vertices	vertex	NOUN
cana-4834	23	17	in	in	ADP
cana-4834	23	18	m[dn	m[dn	NOUN
cana-4834	23	19	]	]	PUNCT
cana-4834	23	20	,	,	PUNCT
cana-4834	23	21	p=7n+1	p=7n+1	NOUN
cana-4834	23	22	•	•	PRON
cana-4834	23	23	maximum	maximum	NOUN
cana-4834	23	24	degree	degree	NOUN
cana-4834	23	25	in	in	ADP
cana-4834	23	26	m	m	PROPN
cana-4834	23	27	(	(	PUNCT
cana-4834	23	28	tn	tn	NOUN
cana-4834	23	29	)	)	PUNCT
cana-4834	23	30	and	and	CCONJ
cana-4834	23	31	m[dn	m[dn	NOUN
cana-4834	23	32	]	]	X
cana-4834	23	33	,	,	PUNCT
cana-4834	23	34	∆=6	∆=6	NOUN
cana-4834	23	35	•	•	NUM
cana-4834	23	36	maximum	maximum	ADJ
cana-4834	23	37	degree	degree	NOUN
cana-4834	23	38	in	in	ADP
cana-4834	23	39	m[d(tn	m[d(tn	PROPN
cana-4834	23	40	)	)	PUNCT
cana-4834	23	41	]	]	PUNCT
cana-4834	23	42	,	,	PUNCT
cana-4834	23	43	∆=7	∆=7	X
cana-4834	23	44	•	•	ADP
cana-4834	23	45	minimum	minimum	NOUN
cana-4834	23	46	degree	degree	NOUN
cana-4834	23	47	in	in	ADP
cana-4834	23	48	m(tn),m[dn	m(tn),m[dn	NOUN
cana-4834	23	49	]	]	PUNCT
cana-4834	23	50	,	,	PUNCT
cana-4834	23	51	δ	δ	PROPN
cana-4834	23	52	=	=	NOUN
cana-4834	23	53	2	2	NUM
cana-4834	23	54	•	•	NUM
cana-4834	23	55	minimum	minimum	NOUN
cana-4834	23	56	degree	degree	NOUN
cana-4834	23	57	in	in	ADP
cana-4834	23	58	m[d(tn	m[d(tn	PROPN
cana-4834	23	59	)	)	PUNCT
cana-4834	23	60	]	]	PUNCT
cana-4834	23	61	,	,	PUNCT
cana-4834	23	62	δ	δ	X
cana-4834	23	63	=	=	SYM
cana-4834	23	64	3	3	NUM
cana-4834	23	65	3	3	NUM
cana-4834	23	66	.	.	PUNCT
cana-4834	23	67	irregular	irregular	ADJ
cana-4834	23	68	coloring	coloring	NOUN
cana-4834	23	69	of	of	ADP
cana-4834	23	70	m	m	PROPN
cana-4834	23	71	(	(	PUNCT
cana-4834	23	72	𝐓𝐧	𝐓𝐧	NOUN
cana-4834	23	73	)	)	PUNCT
cana-4834	23	74	,	,	PUNCT
cana-4834	23	75	m	m	VERB
cana-4834	23	76	[	[	X
cana-4834	23	77	d(tn	d(tn	X
cana-4834	23	78	)	)	PUNCT
cana-4834	23	79	]	]	PUNCT
cana-4834	23	80	and	and	CCONJ
cana-4834	23	81	m[dn	m[dn	NOUN
cana-4834	23	82	]	]	PUNCT
cana-4834	23	83	theorem	theorem	VERB
cana-4834	23	84	3.1	3.1	NUM
cana-4834	23	85	:	:	PUNCT
cana-4834	23	86	for	for	ADP
cana-4834	23	87	m(𝑇n	m(𝑇n	NOUN
cana-4834	23	88	)	)	PUNCT
cana-4834	23	89	,	,	PUNCT
cana-4834	23	90	𝜒𝑖𝑟	𝜒𝑖𝑟	NOUN
cana-4834	23	91	[	[	X
cana-4834	23	92	[	[	X
cana-4834	23	93	m	m	X
cana-4834	23	94	(	(	PUNCT
cana-4834	23	95	tn	tn	NOUN
cana-4834	23	96	)	)	PUNCT
cana-4834	23	97	]	]	PUNCT
cana-4834	24	1	=	=	SYM
cana-4834	24	2	2𝑛	2𝑛	PROPN
cana-4834	24	3	+	+	CCONJ
cana-4834	24	4	2	2	NUM
cana-4834	24	5	,	,	PUNCT
cana-4834	24	6	n≥2	n≥2	NOUN
cana-4834	24	7	proof	proof	NOUN
cana-4834	24	8	:	:	PUNCT
cana-4834	24	9	now	now	ADV
cana-4834	24	10	by	by	ADP
cana-4834	24	11	definition	definition	NOUN
cana-4834	24	12	of	of	ADP
cana-4834	24	13	middle	middle	ADJ
cana-4834	24	14	graph	graph	NOUN
cana-4834	24	15	,	,	PUNCT
cana-4834	24	16	each	each	DET
cana-4834	24	17	edge	edge	NOUN
cana-4834	24	18	of	of	ADP
cana-4834	24	19	the	the	DET
cana-4834	24	20	triangular	triangular	NOUN
cana-4834	24	21	snake	snake	NOUN
cana-4834	24	22	graph	graph	NOUN
cana-4834	24	23	tn	tn	PROPN
cana-4834	24	24	is	be	AUX
cana-4834	24	25	sub	sub	ADJ
cana-4834	24	26	-	-	ADJ
cana-4834	24	27	divided	divide	VERB
cana-4834	24	28	by	by	ADP
cana-4834	24	29	a	a	DET
cana-4834	24	30	new	new	ADJ
cana-4834	24	31	vertex	vertex	NOUN
cana-4834	24	32	.	.	PUNCT
cana-4834	25	1	assume	assume	VERB
cana-4834	25	2	that	that	SCONJ
cana-4834	25	3	each	each	DET
cana-4834	25	4	edge	edge	NOUN
cana-4834	25	5	(	(	PUNCT
cana-4834	25	6	vi	vi	NOUN
cana-4834	25	7	,	,	PUNCT
cana-4834	25	8	vi+1	vi+1	NOUN
cana-4834	25	9	)	)	PUNCT
cana-4834	25	10	and	and	CCONJ
cana-4834	25	11	the	the	DET
cana-4834	25	12	line	line	NOUN
cana-4834	25	13	joining	join	VERB
cana-4834	25	14	vi	vi	PROPN
cana-4834	25	15	and	and	CCONJ
cana-4834	25	16	vi+1	vi+1	ADV
cana-4834	25	17	to	to	ADP
cana-4834	25	18	a	a	DET
cana-4834	25	19	vertex	vertex	NOUN
cana-4834	25	20	ui	ui	NOUN
cana-4834	25	21	,	,	PUNCT
cana-4834	25	22	i=1,2,3,	i=1,2,3,	NOUN
cana-4834	25	23	…	…	NUM
cana-4834	25	24	.n	.n	ADJ
cana-4834	25	25	are	be	AUX
cana-4834	25	26	subdivided	subdivide	VERB
cana-4834	25	27	by	by	ADP
cana-4834	25	28	the	the	DET
cana-4834	25	29	vertices	vertex	NOUN
cana-4834	25	30	wi	wi	PROPN
cana-4834	25	31	,	,	PUNCT
cana-4834	25	32	ejj	ejj	ADJ
cana-4834	25	33	and	and	CCONJ
cana-4834	25	34	ejj+1	ejj+1	PROPN
cana-4834	25	35	,	,	PUNCT
cana-4834	25	36	j=1,2,3,	j=1,2,3,	NOUN
cana-4834	25	37	…	…	NUM
cana-4834	25	38	.n	.n	ADJ
cana-4834	25	39	respectively	respectively	ADV
cana-4834	25	40	.	.	PUNCT
cana-4834	26	1	assign	assign	VERB
cana-4834	26	2	the	the	DET
cana-4834	26	3	color	color	NOUN
cana-4834	26	4	c1	c1	NOUN
cana-4834	26	5	to	to	ADP
cana-4834	26	6	vi	vi	PROPN
cana-4834	26	7	,	,	PUNCT
cana-4834	26	8	ui	ui	PROPN
cana-4834	26	9	and	and	CCONJ
cana-4834	26	10	c2	c2	PROPN
cana-4834	26	11	,	,	PUNCT
cana-4834	26	12	c3	c3	PROPN
cana-4834	26	13	…	…	PUNCT
cana-4834	26	14	.c2n+1	.c2n+1	NOUN
cana-4834	26	15	to	to	ADP
cana-4834	26	16	e1,e	e1,e	PROPN
cana-4834	26	17	2	2	NUM
cana-4834	26	18	…	…	PUNCT
cana-4834	26	19	…	…	PUNCT
cana-4834	26	20	e2n	e2n	X
cana-4834	26	21	assign	assign	VERB
cana-4834	26	22	the	the	DET
cana-4834	26	23	color	color	NOUN
cana-4834	26	24	cn+3	cn+3	PROPN
cana-4834	26	25	,	,	PUNCT
cana-4834	26	26	cn+4	cn+4	NOUN
cana-4834	26	27	…	…	PUNCT
cana-4834	26	28	..	..	PUNCT
cana-4834	26	29	c2n+2	c2n+2	VERB
cana-4834	26	30	to	to	ADP
cana-4834	26	31	e2n+1,e2n+2	e2n+1,e2n+2	PROPN
cana-4834	26	32	…	…	PUNCT
cana-4834	26	33	…	…	PUNCT
cana-4834	26	34	e3n	e3n	VERB
cana-4834	26	35	to	to	PART
cana-4834	26	36	prove	prove	VERB
cana-4834	26	37	(	(	PUNCT
cana-4834	26	38	2n+2)coloring	2n+2)coloring	NUM
cana-4834	26	39	is	be	AUX
cana-4834	26	40	an	an	DET
cana-4834	26	41	irregular	irregular	ADJ
cana-4834	26	42	coloring	coloring	NOUN
cana-4834	26	43	of	of	ADP
cana-4834	26	44	m	m	PROPN
cana-4834	26	45	(	(	PUNCT
cana-4834	26	46	tn	tn	PROPN
cana-4834	26	47	)	)	PUNCT
cana-4834	26	48	,	,	PUNCT
cana-4834	26	49	since	since	SCONJ
cana-4834	26	50	deg(ui	deg(ui	PRON
cana-4834	26	51	)	)	PUNCT
cana-4834	26	52	≠	≠	PROPN
cana-4834	26	53	deg(ei),it	deg(ei),it	NOUN
cana-4834	26	54	shows	show	VERB
cana-4834	26	55	that	that	SCONJ
cana-4834	26	56	code(ui	code(ui	NOUN
cana-4834	26	57	)	)	PUNCT
cana-4834	26	58	≠	≠	PROPN
cana-4834	26	59	code	code	NOUN
cana-4834	26	60	(	(	PUNCT
cana-4834	26	61	ei	ei	NOUN
cana-4834	26	62	)	)	PUNCT
cana-4834	26	63	since	since	SCONJ
cana-4834	26	64	each	each	DET
cana-4834	26	65	ui	ui	PROPN
cana-4834	26	66	’s	’	VERB
cana-4834	26	67	are	be	AUX
cana-4834	26	68	adjacent	adjacent	ADJ
cana-4834	26	69	to	to	ADP
cana-4834	26	70	ei	ei	VERB
cana-4834	26	71	’s	’s	X
cana-4834	27	1	but	but	CCONJ
cana-4834	27	2	vi	vi	PROPN
cana-4834	27	3	’s	’	VERB
cana-4834	27	4	are	be	AUX
cana-4834	27	5	not	not	PART
cana-4834	27	6	adjacent	adjacent	ADJ
cana-4834	27	7	to	to	ADP
cana-4834	27	8	ui	ui	PROPN
cana-4834	27	9	’s	’s	NOUN
cana-4834	27	10	.	.	PUNCT
cana-4834	28	1	hence	hence	ADV
cana-4834	28	2	code	code	NOUN
cana-4834	28	3	(	(	PUNCT
cana-4834	28	4	ei	ei	NOUN
cana-4834	28	5	)	)	PUNCT
cana-4834	28	6	≠code(vi	≠code(vi	NOUN
cana-4834	28	7	)	)	PUNCT
cana-4834	28	8	.thus	.thus	ADV
cana-4834	28	9	𝜒𝑖𝑟[m	𝜒𝑖𝑟[m	PROPN
cana-4834	28	10	(	(	PUNCT
cana-4834	28	11	tn	tn	NOUN
cana-4834	28	12	)	)	PUNCT
cana-4834	28	13	]	]	PUNCT
cana-4834	29	1	≤	≤	NUM
cana-4834	29	2	2𝑛	2𝑛	NOUN
cana-4834	29	3	+	+	CCONJ
cana-4834	29	4	2	2	NUM
cana-4834	29	5	by	by	ADP
cana-4834	29	6	the	the	DET
cana-4834	29	7	definition	definition	NOUN
cana-4834	29	8	of	of	ADP
cana-4834	29	9	middle	middle	PROPN
cana-4834	29	10	graph{v	graph{v	PROPN
cana-4834	29	11	,	,	PUNCT
cana-4834	29	12	ei}induces	ei}induce	VERB
cana-4834	29	13	a	a	DET
cana-4834	29	14	clique	clique	NOUN
cana-4834	29	15	of	of	ADP
cana-4834	29	16	order	order	NOUN
cana-4834	29	17	2n+2	2n+2	NUM
cana-4834	29	18	in	in	ADP
cana-4834	29	19	m(tn	m(tn	NOUN
cana-4834	29	20	)	)	PUNCT
cana-4834	30	1	[	[	X
cana-4834	30	2	9	9	NUM
cana-4834	30	3	]	]	PUNCT
cana-4834	30	4	it	it	PRON
cana-4834	30	5	follows	follow	VERB
cana-4834	30	6	that,𝜒𝑖𝑟[m	that,𝜒𝑖𝑟[m	PROPN
cana-4834	30	7	(	(	PUNCT
cana-4834	30	8	tn	tn	NOUN
cana-4834	30	9	)	)	PUNCT
cana-4834	30	10	]	]	PUNCT
cana-4834	30	11	≥	≥	PRON
cana-4834	30	12	𝜒[m	𝜒[m	PRON
cana-4834	30	13	(	(	PUNCT
cana-4834	30	14	tn	tn	NOUN
cana-4834	30	15	)	)	PUNCT
cana-4834	30	16	]	]	PUNCT
cana-4834	31	1	=	=	SYM
cana-4834	31	2	2𝑛	2𝑛	PROPN
cana-4834	31	3	+	+	CCONJ
cana-4834	32	1	2	2	X
cana-4834	32	2	.	.	X
cana-4834	32	3	hence	hence	ADV
cana-4834	32	4	𝜒𝑖𝑟	𝜒𝑖𝑟	NOUN
cana-4834	32	5	[	[	X
cana-4834	32	6	[	[	X
cana-4834	32	7	m	m	PROPN
cana-4834	32	8	(	(	PUNCT
cana-4834	32	9	tn	tn	NOUN
cana-4834	32	10	)	)	PUNCT
cana-4834	32	11	]	]	PUNCT
cana-4834	33	1	=	=	SYM
cana-4834	33	2	2𝑛	2𝑛	PROPN
cana-4834	33	3	+	+	CCONJ
cana-4834	33	4	2	2	NUM
cana-4834	33	5	,	,	PUNCT
cana-4834	33	6	n≥2	n≥2	NOUN
cana-4834	33	7	by	by	ADP
cana-4834	33	8	the	the	DET
cana-4834	33	9	definition	definition	NOUN
cana-4834	33	10	of	of	ADP
cana-4834	33	11	middle	middle	PROPN
cana-4834	33	12	graph{v	graph{v	PROPN
cana-4834	33	13	,	,	PUNCT
cana-4834	33	14	ei}induces	ei}induce	VERB
cana-4834	33	15	a	a	DET
cana-4834	33	16	clique	clique	NOUN
cana-4834	33	17	of	of	ADP
cana-4834	33	18	order	order	NOUN
cana-4834	33	19	2n+2	2n+2	NUM
cana-4834	33	20	in	in	ADP
cana-4834	33	21	m(tn	m(tn	NOUN
cana-4834	33	22	)	)	PUNCT
cana-4834	34	1	[	[	X
cana-4834	34	2	9	9	NUM
cana-4834	34	3	]	]	PUNCT
cana-4834	34	4	it	it	PRON
cana-4834	34	5	follows	follow	VERB
cana-4834	34	6	that,𝜒𝑖𝑟[m	that,𝜒𝑖𝑟[m	PROPN
cana-4834	34	7	(	(	PUNCT
cana-4834	34	8	tn	tn	NOUN
cana-4834	34	9	)	)	PUNCT
cana-4834	34	10	]	]	PUNCT
cana-4834	34	11	≥	≥	PRON
cana-4834	34	12	𝜒[m	𝜒[m	PRON
cana-4834	34	13	(	(	PUNCT
cana-4834	34	14	tn	tn	NOUN
cana-4834	34	15	)	)	PUNCT
cana-4834	34	16	]	]	PUNCT
cana-4834	35	1	=	=	SYM
cana-4834	35	2	2𝑛	2𝑛	PROPN
cana-4834	35	3	+	+	CCONJ
cana-4834	36	1	2	2	X
cana-4834	36	2	.	.	X
cana-4834	36	3	hence	hence	ADV
cana-4834	36	4	𝜒𝑖𝑟	𝜒𝑖𝑟	NOUN
cana-4834	36	5	[	[	X
cana-4834	36	6	[	[	X
cana-4834	36	7	m	m	PROPN
cana-4834	36	8	(	(	PUNCT
cana-4834	36	9	tn	tn	NOUN
cana-4834	36	10	)	)	PUNCT
cana-4834	36	11	]	]	PUNCT
cana-4834	37	1	=	=	SYM
cana-4834	37	2	2𝑛	2𝑛	PROPN
cana-4834	37	3	+	+	CCONJ
cana-4834	37	4	2	2	NUM
cana-4834	37	5	,	,	PUNCT
cana-4834	37	6	n≥2	n≥2	NOUN
cana-4834	37	7	communications	communication	NOUN
cana-4834	37	8	on	on	ADP
cana-4834	37	9	applied	apply	VERB
cana-4834	37	10	nonlinear	nonlinear	ADJ
cana-4834	37	11	analysis	analysis	NOUN
cana-4834	37	12	issn	issn	NOUN
cana-4834	37	13	:	:	PUNCT
cana-4834	37	14	1074	1074	NUM
cana-4834	37	15	-	-	PUNCT
cana-4834	37	16	133x	133x	NUM
cana-4834	37	17	vol	vol	VERB
cana-4834	37	18	32	32	NUM
cana-4834	37	19	no	no	NOUN
cana-4834	37	20	.	.	PUNCT
cana-4834	38	1	10s	10	NOUN
cana-4834	38	2	(	(	PUNCT
cana-4834	38	3	2025	2025	NUM
cana-4834	38	4	)	)	PUNCT
cana-4834	38	5	426	426	NUM
cana-4834	38	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4834	38	7	figure	figure	NOUN
cana-4834	38	8	1	1	NUM
cana-4834	38	9	:	:	PUNCT
cana-4834	38	10	middle	middle	ADJ
cana-4834	38	11	graph	graph	NOUN
cana-4834	38	12	of	of	ADP
cana-4834	38	13	triangular	triangular	NOUN
cana-4834	38	14	snake	snake	NOUN
cana-4834	38	15	graph	graph	NOUN
cana-4834	38	16	m(t2	m(t2	PROPN
cana-4834	38	17	)	)	PUNCT
cana-4834	38	18	theorem	theorem	VERB
cana-4834	38	19	3.2	3.2	NUM
cana-4834	38	20	:	:	PUNCT
cana-4834	38	21	for	for	ADP
cana-4834	38	22	[	[	PUNCT
cana-4834	38	23	m(d	m(d	PROPN
cana-4834	38	24	(	(	PUNCT
cana-4834	38	25	tn	tn	NOUN
cana-4834	38	26	)	)	PUNCT
cana-4834	38	27	)	)	PUNCT
cana-4834	38	28	]	]	PUNCT
cana-4834	39	1	,	,	PUNCT
cana-4834	39	2	the	the	DET
cana-4834	39	3	irregular	irregular	ADJ
cana-4834	39	4	chromatic	chromatic	ADJ
cana-4834	39	5	number	number	NOUN
cana-4834	39	6	is	be	AUX
cana-4834	39	7	6	6	NUM
cana-4834	39	8	.	.	PUNCT
cana-4834	40	1	(	(	PUNCT
cana-4834	40	2	𝑖.	𝑖.	NOUN
cana-4834	40	3	𝑒	𝑒	X
cana-4834	40	4	)	)	PUNCT
cana-4834	40	5	𝜒𝑖𝑟	𝜒𝑖𝑟	NOUN
cana-4834	41	1	[	[	X
cana-4834	41	2	m(d	m(d	PROPN
cana-4834	41	3	(	(	PUNCT
cana-4834	41	4	tn	tn	NOUN
cana-4834	41	5	)	)	PUNCT
cana-4834	41	6	)	)	PUNCT
cana-4834	41	7	]	]	PUNCT
cana-4834	42	1	=	=	SYM
cana-4834	42	2	3𝑛	3𝑛	NUM
cana-4834	42	3	+	+	CCONJ
cana-4834	42	4	3	3	NUM
cana-4834	42	5	n	n	NOUN
cana-4834	42	6	≥	≥	NOUN
cana-4834	42	7	2	2	NUM
cana-4834	42	8	proof	proof	NOUN
cana-4834	42	9	:	:	PUNCT
cana-4834	42	10	let	let	VERB
cana-4834	42	11	{	{	PUNCT
cana-4834	42	12	v1,v2	v1,v2	PROPN
cana-4834	42	13	…	…	SYM
cana-4834	42	14	…	…	PUNCT
cana-4834	42	15	…	…	X
cana-4834	42	16	vn+1	vn+1	X
cana-4834	42	17	,	,	PUNCT
cana-4834	42	18	u1,u2	u1,u2	PROPN
cana-4834	42	19	…	…	PUNCT
cana-4834	42	20	…	…	PUNCT
cana-4834	42	21	…	…	SYM
cana-4834	42	22	un	un	PROPN
cana-4834	42	23	,	,	PUNCT
cana-4834	42	24	w1,w2	w1,w2	PROPN
cana-4834	42	25	…	…	SYM
cana-4834	42	26	…	…	SYM
cana-4834	42	27	wn	wn	NOUN
cana-4834	42	28	}	}	PUNCT
cana-4834	42	29	be	be	VERB
cana-4834	42	30	the	the	DET
cana-4834	42	31	vertices	vertex	NOUN
cana-4834	42	32	of	of	ADP
cana-4834	42	33	the	the	DET
cana-4834	42	34	double	double	ADJ
cana-4834	42	35	triangular	triangular	NOUN
cana-4834	42	36	snake	snake	NOUN
cana-4834	42	37	graph	graph	NOUN
cana-4834	42	38	d(tn	d(tn	NOUN
cana-4834	42	39	)	)	PUNCT
cana-4834	42	40	.	.	PUNCT
cana-4834	43	1	assume	assume	VERB
cana-4834	43	2	that	that	SCONJ
cana-4834	43	3	each	each	DET
cana-4834	43	4	edge	edge	NOUN
cana-4834	43	5	(	(	PUNCT
cana-4834	43	6	vi	vi	NOUN
cana-4834	43	7	,	,	PUNCT
cana-4834	43	8	vi+1	vi+1	NOUN
cana-4834	43	9	)	)	PUNCT
cana-4834	43	10	(	(	PUNCT
cana-4834	43	11	ui	ui	INTJ
cana-4834	43	12	,	,	PUNCT
cana-4834	43	13	vj	vj	INTJ
cana-4834	43	14	)	)	PUNCT
cana-4834	43	15	and	and	CCONJ
cana-4834	43	16	(	(	PUNCT
cana-4834	43	17	vj	vj	INTJ
cana-4834	43	18	,	,	PUNCT
cana-4834	43	19	wi	wi	PROPN
cana-4834	43	20	)	)	PUNCT
cana-4834	43	21	,	,	PUNCT
cana-4834	43	22	i=1,2,3,	i=1,2,3,	NOUN
cana-4834	43	23	…	…	NUM
cana-4834	43	24	.n	.n	ADJ
cana-4834	43	25	and	and	CCONJ
cana-4834	43	26	j=1,2,3	j=1,2,3	NUM
cana-4834	43	27	…	…	PUNCT
cana-4834	43	28	n+1	n+1	PROPN
cana-4834	43	29	is	be	AUX
cana-4834	43	30	sub	sub	ADJ
cana-4834	43	31	-	-	ADJ
cana-4834	43	32	divided	divide	VERB
cana-4834	43	33	by	by	ADP
cana-4834	43	34	the	the	DET
cana-4834	43	35	vertices	vertex	NOUN
cana-4834	43	36	xi	xi	PROPN
cana-4834	43	37	,	,	PUNCT
cana-4834	43	38	eij	eij	PROPN
cana-4834	43	39	and	and	CCONJ
cana-4834	43	40	fij	fij	PROPN
cana-4834	43	41	for	for	ADP
cana-4834	43	42	,	,	PUNCT
cana-4834	43	43	i=1,2,3,	i=1,2,3,	NOUN
cana-4834	43	44	…	…	NUM
cana-4834	43	45	.n	.n	ADJ
cana-4834	43	46	and	and	CCONJ
cana-4834	43	47	j=1,2,3	j=1,2,3	NUM
cana-4834	43	48	…	…	SYM
cana-4834	43	49	n+1	n+1	NUM
cana-4834	43	50	respectively.[9	respectively.[9	NOUN
cana-4834	43	51	]	]	PUNCT
cana-4834	43	52	color	color	VERB
cana-4834	43	53	the	the	DET
cana-4834	43	54	vertices	vertex	NOUN
cana-4834	43	55	v1,v2	v1,v2	PROPN
cana-4834	43	56	…	…	SYM
cana-4834	43	57	…	…	PUNCT
cana-4834	43	58	…	…	X
cana-4834	43	59	vn+1	vn+1	X
cana-4834	43	60	,	,	PUNCT
cana-4834	43	61	u1,u2	u1,u2	PROPN
cana-4834	43	62	…	…	SYM
cana-4834	43	63	…	…	PUNCT
cana-4834	43	64	…	…	SYM
cana-4834	43	65	un	un	PROPN
cana-4834	43	66	and	and	CCONJ
cana-4834	43	67	w1,w2	w1,w2	PROPN
cana-4834	43	68	…	…	PUNCT
cana-4834	43	69	…	…	SYM
cana-4834	43	70	wn	wn	PROPN
cana-4834	43	71	with	with	ADP
cana-4834	43	72	c1	c1	PROPN
cana-4834	43	73	.	.	PUNCT
cana-4834	44	1	color	color	VERB
cana-4834	44	2	the	the	DET
cana-4834	44	3	elements	element	NOUN
cana-4834	44	4	e1,e	e1,e	PROPN
cana-4834	44	5	2	2	NUM
cana-4834	44	6	…	…	PUNCT
cana-4834	44	7	…	…	PUNCT
cana-4834	44	8	e2n	e2n	X
cana-4834	44	9	with	with	ADP
cana-4834	44	10	c2	c2	PROPN
cana-4834	44	11	,	,	PUNCT
cana-4834	44	12	c3	c3	PROPN
cana-4834	44	13	…	…	PUNCT
cana-4834	44	14	.	.	PUNCT
cana-4834	45	1	c2n+1	c2n+1	NOUN
cana-4834	45	2	and	and	CCONJ
cana-4834	45	3	e2n+1,e2n+2	e2n+1,e2n+2	PROPN
cana-4834	45	4	…	…	PUNCT
cana-4834	45	5	…	…	PUNCT
cana-4834	45	6	e4n	e4n	NOUN
cana-4834	45	7	with	with	ADP
cana-4834	45	8	c5	c5	PROPN
cana-4834	45	9	,	,	PUNCT
cana-4834	45	10	c6	c6	PROPN
cana-4834	45	11	…	…	PUNCT
cana-4834	45	12	..	..	PUNCT
cana-4834	45	13	c2n+4	c2n+4	NOUN
cana-4834	45	14	atlast	atlast	NOUN
cana-4834	45	15	,	,	PUNCT
cana-4834	45	16	color	color	VERB
cana-4834	45	17	the	the	DET
cana-4834	45	18	sub	sub	ADJ
cana-4834	45	19	-	-	ADJ
cana-4834	45	20	divided	divided	ADJ
cana-4834	45	21	vertices	vertex	NOUN
cana-4834	45	22	vi	vi	PROPN
cana-4834	45	23	,	,	PUNCT
cana-4834	45	24	i+1,vi+1,i+2	i+1,vi+1,i+2	NUM
cana-4834	45	25	…	…	SYM
cana-4834	45	26	.vn	.vn	PROPN
cana-4834	45	27	,	,	PUNCT
cana-4834	45	28	i+n	i+n	NUM
cana-4834	45	29	with	with	ADP
cana-4834	45	30	c2n+4	c2n+4	NOUN
cana-4834	45	31	,	,	PUNCT
cana-4834	45	32	c2n+5	c2n+5	PROPN
cana-4834	45	33	…	…	PROPN
cana-4834	45	34	.	.	PUNCT
cana-4834	46	1	c3n+3	c3n+3	PROPN
cana-4834	46	2	hence	hence	ADV
cana-4834	46	3	𝜒𝑖𝑟	𝜒𝑖𝑟	VERB
cana-4834	47	1	[	[	X
cana-4834	47	2	m	m	X
cana-4834	47	3	(	(	PUNCT
cana-4834	47	4	d(tn	d(tn	NOUN
cana-4834	47	5	)	)	PUNCT
cana-4834	47	6	]	]	PUNCT
cana-4834	47	7	=	=	SYM
cana-4834	47	8	3𝑛	3𝑛	NUM
cana-4834	47	9	+	+	CCONJ
cana-4834	47	10	3	3	NUM
cana-4834	47	11	,	,	PUNCT
cana-4834	47	12	n≥2	n≥2	NOUN
cana-4834	47	13	figure	figure	NOUN
cana-4834	47	14	2	2	NUM
cana-4834	47	15	:	:	PUNCT
cana-4834	47	16	middle	middle	ADJ
cana-4834	47	17	graph	graph	NOUN
cana-4834	47	18	of	of	ADP
cana-4834	47	19	double	double	ADJ
cana-4834	47	20	triangular	triangular	NOUN
cana-4834	47	21	snake	snake	NOUN
cana-4834	47	22	graph	graph	NOUN
cana-4834	47	23	m(d(t2	m(d(t2	NOUN
cana-4834	47	24	)	)	PUNCT
cana-4834	47	25	)	)	PUNCT
cana-4834	47	26	theorem	theorem	VERB
cana-4834	47	27	3.3	3.3	NUM
cana-4834	47	28	:	:	PUNCT
cana-4834	47	29	for	for	ADP
cana-4834	47	30	[	[	X
cana-4834	47	31	m	m	X
cana-4834	47	32	(	(	PUNCT
cana-4834	47	33	𝐷n	𝐷n	PROPN
cana-4834	47	34	)	)	PUNCT
cana-4834	47	35	]	]	PUNCT
cana-4834	47	36	,	,	PUNCT
cana-4834	47	37	the	the	DET
cana-4834	47	38	irregular	irregular	ADJ
cana-4834	47	39	-chromatic	-chromatic	ADJ
cana-4834	47	40	number	number	NOUN
cana-4834	47	41	is	be	AUX
cana-4834	47	42	2n+3	2n+3	NOUN
cana-4834	47	43	(	(	PUNCT
cana-4834	47	44	𝑖.	𝑖.	NOUN
cana-4834	47	45	𝑒	𝑒	X
cana-4834	47	46	)	)	PUNCT
cana-4834	47	47	𝜒𝑖𝑟	𝜒𝑖𝑟	NOUN
cana-4834	48	1	[	[	X
cana-4834	48	2	m	m	X
cana-4834	48	3	(	(	PUNCT
cana-4834	48	4	𝐷n	𝐷n	PROPN
cana-4834	48	5	)	)	PUNCT
cana-4834	48	6	]	]	PUNCT
cana-4834	49	1	=	=	SYM
cana-4834	49	2	2n+3	2n+3	PROPN
cana-4834	49	3	,	,	PUNCT
cana-4834	49	4	n≥2	n≥2	NOUN
cana-4834	49	5	proof	proof	NOUN
cana-4834	49	6	:	:	PUNCT
cana-4834	49	7	color	color	VERB
cana-4834	49	8	the	the	DET
cana-4834	49	9	vertices	vertex	NOUN
cana-4834	49	10	{	{	PUNCT
cana-4834	49	11	v1,v2	v1,v2	PROPN
cana-4834	49	12	…	…	SYM
cana-4834	49	13	…	…	PUNCT
cana-4834	49	14	…	…	X
cana-4834	49	15	vn+1	vn+1	X
cana-4834	49	16	,	,	PUNCT
cana-4834	49	17	u1,u2	u1,u2	PROPN
cana-4834	49	18	…	…	PUNCT
cana-4834	49	19	…	…	PUNCT
cana-4834	49	20	…	…	SYM
cana-4834	49	21	un	un	PROPN
cana-4834	49	22	,	,	PUNCT
cana-4834	49	23	w1,w2,	w1,w2,	PROPN
cana-4834	49	24	…	…	PUNCT
cana-4834	49	25	.wn	.wn	PUNCT
cana-4834	49	26	}	}	PUNCT
cana-4834	49	27	with	with	ADP
cana-4834	49	28	c1	c1	PROPN
cana-4834	49	29	.	.	PUNCT
cana-4834	50	1	color	color	VERB
cana-4834	50	2	the	the	DET
cana-4834	50	3	elements	element	NOUN
cana-4834	50	4	e11,e	e11,e	NOUN
cana-4834	50	5	12	12	NUM
cana-4834	50	6	…	…	SYM
cana-4834	50	7	…	…	PUNCT
cana-4834	50	8	e2n	e2n	X
cana-4834	50	9	with	with	ADP
cana-4834	50	10	c2	c2	PROPN
cana-4834	50	11	,	,	PUNCT
cana-4834	50	12	c3	c3	PROPN
cana-4834	50	13	.	.	PUNCT
cana-4834	50	14	…	…	PUNCT
cana-4834	50	15	.	.	PUNCT
cana-4834	51	1	c2n+1	c2n+1	VERB
cana-4834	51	2	communications	communication	NOUN
cana-4834	51	3	on	on	ADP
cana-4834	51	4	applied	apply	VERB
cana-4834	51	5	nonlinear	nonlinear	ADJ
cana-4834	51	6	analysis	analysis	NOUN
cana-4834	51	7	issn	issn	NOUN
cana-4834	51	8	:	:	PUNCT
cana-4834	51	9	1074	1074	NUM
cana-4834	51	10	-	-	PUNCT
cana-4834	51	11	133x	133x	NUM
cana-4834	51	12	vol	vol	VERB
cana-4834	51	13	32	32	NUM
cana-4834	51	14	no	no	NOUN
cana-4834	51	15	.	.	PUNCT
cana-4834	52	1	10s	10	NOUN
cana-4834	52	2	(	(	PUNCT
cana-4834	52	3	2025	2025	NUM
cana-4834	52	4	)	)	PUNCT
cana-4834	52	5	427	427	NUM
cana-4834	52	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4834	52	7	color	color	VERB
cana-4834	52	8	the	the	DET
cana-4834	52	9	elements	element	NOUN
cana-4834	52	10	e2n+1,e2n+2	e2n+1,e2n+2	PROPN
cana-4834	52	11	…	…	PUNCT
cana-4834	52	12	…	…	PUNCT
cana-4834	52	13	e4n	e4n	NOUN
cana-4834	52	14	with	with	ADP
cana-4834	52	15	c4	c4	NOUN
cana-4834	52	16	,	,	PUNCT
cana-4834	52	17	c5	c5	PROPN
cana-4834	52	18	…	…	NUM
cana-4834	52	19	..	..	PUNCT
cana-4834	52	20	c2n+3	c2n+3	PROPN
cana-4834	52	21	.	.	PUNCT
cana-4834	53	1	to	to	PART
cana-4834	53	2	prove	prove	VERB
cana-4834	53	3	(	(	PUNCT
cana-4834	53	4	2n+3)coloring	2n+3)coloring	NOUN
cana-4834	53	5	is	be	AUX
cana-4834	53	6	an	an	DET
cana-4834	53	7	irregular	irregular	ADJ
cana-4834	53	8	coloring	coloring	NOUN
cana-4834	53	9	of	of	ADP
cana-4834	53	10	m	m	PROPN
cana-4834	53	11	(	(	PUNCT
cana-4834	53	12	dn	dn	PROPN
cana-4834	53	13	)	)	PUNCT
cana-4834	53	14	,	,	PUNCT
cana-4834	53	15	since	since	SCONJ
cana-4834	53	16	deg(ui	deg(ui	PRON
cana-4834	53	17	)	)	PUNCT
cana-4834	53	18	≠	≠	PROPN
cana-4834	53	19	deg(ei),it	deg(ei),it	NOUN
cana-4834	53	20	shows	show	VERB
cana-4834	53	21	that	that	SCONJ
cana-4834	53	22	code(ui	code(ui	NOUN
cana-4834	53	23	)	)	PUNCT
cana-4834	53	24	≠	≠	PROPN
cana-4834	53	25	code	code	NOUN
cana-4834	53	26	(	(	PUNCT
cana-4834	53	27	ei	ei	NOUN
cana-4834	53	28	)	)	PUNCT
cana-4834	53	29	since	since	SCONJ
cana-4834	53	30	each	each	DET
cana-4834	53	31	ui	ui	PROPN
cana-4834	53	32	’s	’	VERB
cana-4834	53	33	are	be	AUX
cana-4834	53	34	adjacent	adjacent	ADJ
cana-4834	53	35	to	to	ADP
cana-4834	53	36	ei	ei	VERB
cana-4834	53	37	’s	’s	X
cana-4834	54	1	but	but	CCONJ
cana-4834	54	2	vi	vi	PROPN
cana-4834	54	3	’s	’	VERB
cana-4834	54	4	are	be	AUX
cana-4834	54	5	not	not	PART
cana-4834	54	6	adjacent	adjacent	ADJ
cana-4834	54	7	to	to	ADP
cana-4834	54	8	ui	ui	PROPN
cana-4834	54	9	’s	’s	NOUN
cana-4834	54	10	.	.	PUNCT
cana-4834	55	1	hence	hence	ADV
cana-4834	55	2	code	code	NOUN
cana-4834	55	3	(	(	PUNCT
cana-4834	55	4	ei	ei	NOUN
cana-4834	55	5	)	)	PUNCT
cana-4834	55	6	≠code(vi	≠code(vi	NOUN
cana-4834	55	7	)	)	PUNCT
cana-4834	55	8	.	.	PUNCT
cana-4834	56	1	thus	thus	ADV
cana-4834	56	2	𝜒𝑖𝑟[m	𝜒𝑖𝑟[m	NOUN
cana-4834	56	3	(	(	PUNCT
cana-4834	56	4	dn	dn	NOUN
cana-4834	56	5	)	)	PUNCT
cana-4834	56	6	]	]	PUNCT
cana-4834	56	7	≤	≤	NUM
cana-4834	56	8	2𝑛	2𝑛	NOUN
cana-4834	56	9	+	+	CCONJ
cana-4834	56	10	3	3	NUM
cana-4834	56	11	by	by	ADP
cana-4834	56	12	the	the	DET
cana-4834	56	13	definition	definition	NOUN
cana-4834	56	14	of	of	ADP
cana-4834	56	15	middle	middle	PROPN
cana-4834	56	16	graph,{v	graph,{v	PROPN
cana-4834	56	17	,	,	PUNCT
cana-4834	56	18	ei	ei	NOUN
cana-4834	56	19	}	}	PUNCT
cana-4834	56	20	induces	induce	VERB
cana-4834	56	21	a	a	DET
cana-4834	56	22	clique	clique	NOUN
cana-4834	56	23	of	of	ADP
cana-4834	56	24	order	order	NOUN
cana-4834	56	25	2n+3	2n+3	PROPN
cana-4834	56	26	in	in	ADP
cana-4834	56	27	m(dn)[9	m(dn)[9	PROPN
cana-4834	56	28	]	]	X
cana-4834	56	29	it	it	PRON
cana-4834	56	30	follows	follow	VERB
cana-4834	56	31	that,𝜒𝑖𝑟[m	that,𝜒𝑖𝑟[m	PROPN
cana-4834	56	32	(	(	PUNCT
cana-4834	56	33	dn	dn	NOUN
cana-4834	56	34	)	)	PUNCT
cana-4834	56	35	]	]	PUNCT
cana-4834	56	36	≥	≥	PRON
cana-4834	56	37	𝜒[m	𝜒[m	PRON
cana-4834	56	38	(	(	PUNCT
cana-4834	56	39	dn	dn	NOUN
cana-4834	56	40	)	)	PUNCT
cana-4834	56	41	]	]	PUNCT
cana-4834	57	1	=	=	SYM
cana-4834	57	2	2𝑛	2𝑛	PROPN
cana-4834	57	3	+	+	CCONJ
cana-4834	57	4	3	3	NUM
cana-4834	57	5	figure	figure	NOUN
cana-4834	57	6	3	3	NUM
cana-4834	57	7	:	:	PUNCT
cana-4834	57	8	middle	middle	ADJ
cana-4834	57	9	graph	graph	NOUN
cana-4834	57	10	of	of	ADP
cana-4834	57	11	double	double	ADJ
cana-4834	57	12	triangular	triangular	NOUN
cana-4834	57	13	snake	snake	NOUN
cana-4834	57	14	graph	graph	NOUN
cana-4834	57	15	m(d2	m(d2	ADJ
cana-4834	57	16	)	)	PUNCT
cana-4834	57	17	.	.	PUNCT
cana-4834	58	1	5	5	X
cana-4834	58	2	.	.	X
cana-4834	58	3	conclusion	conclusion	NOUN
cana-4834	58	4	an	an	DET
cana-4834	58	5	irregular	irregular	ADJ
cana-4834	58	6	coloring	coloring	NOUN
cana-4834	58	7	play	play	VERB
cana-4834	58	8	an	an	DET
cana-4834	58	9	important	important	ADJ
cana-4834	58	10	role	role	NOUN
cana-4834	58	11	in	in	ADP
cana-4834	58	12	clustering	cluster	VERB
cana-4834	58	13	,	,	PUNCT
cana-4834	58	14	automatic	automatic	ADJ
cana-4834	58	15	reading	reading	NOUN
cana-4834	58	16	system	system	NOUN
cana-4834	58	17	and	and	CCONJ
cana-4834	58	18	distributed	distribute	VERB
cana-4834	58	19	system.the	system.the	DET
cana-4834	58	20	investigation	investigation	NOUN
cana-4834	58	21	of	of	ADP
cana-4834	58	22	similar	similar	ADJ
cana-4834	58	23	results	result	NOUN
cana-4834	58	24	for	for	ADP
cana-4834	58	25	different	different	ADJ
cana-4834	58	26	graphs	graph	NOUN
cana-4834	58	27	as	as	ADV
cana-4834	58	28	well	well	ADV
cana-4834	58	29	in	in	ADP
cana-4834	58	30	the	the	DET
cana-4834	58	31	context	context	NOUN
cana-4834	58	32	of	of	ADP
cana-4834	58	33	various	various	ADJ
cana-4834	58	34	graph	graph	NOUN
cana-4834	58	35	coloring	coloring	NOUN
cana-4834	58	36	problems	problem	NOUN
cana-4834	58	37	is	be	AUX
cana-4834	58	38	an	an	DET
cana-4834	58	39	open	open	ADJ
cana-4834	58	40	area	area	NOUN
cana-4834	58	41	of	of	ADP
cana-4834	58	42	research	research	NOUN
cana-4834	58	43	.	.	PUNCT
cana-4834	59	1	references	reference	NOUN
cana-4834	59	2	[	[	X
cana-4834	59	3	1	1	X
cana-4834	59	4	]	]	PUNCT
cana-4834	59	5	j.	j.	PROPN
cana-4834	59	6	a.	a.	PROPN
cana-4834	59	7	bondy	bondy	PROPN
cana-4834	59	8	and	and	CCONJ
cana-4834	59	9	u.	u.	PROPN
cana-4834	59	10	s.	s.	PROPN
cana-4834	59	11	r.	r.	PROPN
cana-4834	59	12	murty	murty	PROPN
cana-4834	59	13	,	,	PUNCT
cana-4834	59	14	graph	graph	NOUN
cana-4834	59	15	theory	theory	NOUN
cana-4834	59	16	with	with	ADP
cana-4834	59	17	applications	application	NOUN
cana-4834	59	18	,	,	PUNCT
cana-4834	59	19	london	london	PROPN
cana-4834	59	20	macmillan	macmillan	PROPN
cana-4834	60	1	[	[	X
cana-4834	60	2	2	2	NUM
cana-4834	60	3	]	]	X
cana-4834	60	4	frank	frank	PROPN
cana-4834	60	5	harary	harary	PROPN
cana-4834	60	6	,	,	PUNCT
cana-4834	60	7	narosa	narosa	PROPN
cana-4834	60	8	publishing	publishing	PROPN
cana-4834	60	9	home	home	NOUN
cana-4834	60	10	,	,	PUNCT
cana-4834	60	11	(	(	PUNCT
cana-4834	60	12	2001	2001	NUM
cana-4834	60	13	)	)	PUNCT
cana-4834	60	14	.	.	PUNCT
cana-4834	61	1	[	[	X
cana-4834	61	2	3	3	X
cana-4834	61	3	]	]	X
cana-4834	61	4	m.s.franklin	m.s.franklin	NOUN
cana-4834	61	5	thamilselvi	thamilselvi	ADJ
cana-4834	61	6	,	,	PUNCT
cana-4834	61	7	harmonious	harmonious	ADJ
cana-4834	61	8	coloring	coloring	NOUN
cana-4834	61	9	of	of	ADP
cana-4834	61	10	central	central	ADJ
cana-4834	61	11	graph	graph	NOUN
cana-4834	61	12	of	of	ADP
cana-4834	61	13	certain	certain	ADJ
cana-4834	61	14	graphs	graph	NOUN
cana-4834	61	15	,	,	PUNCT
cana-4834	61	16	applied	apply	VERB
cana-4834	61	17	mathematical	mathematical	ADJ
cana-4834	61	18	sciences,5	sciences,5	NOUN
cana-4834	61	19	pp.569	pp.569	PROPN
cana-4834	61	20	-	-	PUNCT
cana-4834	61	21	578(2015	578(2015	NUM
cana-4834	61	22	)	)	PUNCT
cana-4834	62	1	[	[	X
cana-4834	62	2	4	4	X
cana-4834	62	3	]	]	PUNCT
cana-4834	62	4	g.jothilakshmi	g.jothilakshmi	NOUN
cana-4834	62	5	,	,	PUNCT
cana-4834	62	6	on	on	ADP
cana-4834	62	7	harmonious	harmonious	ADJ
cana-4834	62	8	coloring	coloring	NOUN
cana-4834	62	9	of	of	ADP
cana-4834	62	10	c(hn	c(hn	NOUN
cana-4834	62	11	)	)	PUNCT
cana-4834	62	12	and	and	CCONJ
cana-4834	62	13	c(gn	c(gn	NOUN
cana-4834	62	14	)	)	PUNCT
cana-4834	62	15	,	,	PUNCT
cana-4834	62	16	journal	journal	NOUN
cana-4834	62	17	of	of	ADP
cana-4834	62	18	global	global	ADJ
cana-4834	62	19	research	research	NOUN
cana-4834	62	20	in	in	ADP
cana-4834	62	21	mathematical	mathematical	ADJ
cana-4834	62	22	archives	archive	NOUN
cana-4834	62	23	,	,	PUNCT
cana-4834	62	24	volume	volume	NOUN
cana-4834	62	25	2	2	NUM
cana-4834	62	26	,	,	PUNCT
cana-4834	62	27	no	no	INTJ
cana-4834	62	28	.	.	NOUN
cana-4834	62	29	5	5	NUM
cana-4834	62	30	,	,	PUNCT
cana-4834	62	31	(	(	PUNCT
cana-4834	62	32	may	may	NOUN
cana-4834	62	33	2014	2014	NUM
cana-4834	62	34	)	)	PUNCT
cana-4834	63	1	[	[	X
cana-4834	63	2	5	5	X
cana-4834	63	3	]	]	PUNCT
cana-4834	63	4	g.jothilakshmi	g.jothilakshmi	NOUN
cana-4834	63	5	k.rajam	k.rajam	NOUN
cana-4834	63	6	and	and	CCONJ
cana-4834	63	7	a.sathyakala	a.sathyakala	ADV
cana-4834	63	8	,	,	PUNCT
cana-4834	63	9	a	a	DET
cana-4834	63	10	study	study	NOUN
cana-4834	63	11	on	on	ADP
cana-4834	63	12	an	an	DET
cana-4834	63	13	irregular	irregular	ADJ
cana-4834	63	14	colouring	colouring	NOUN
cana-4834	63	15	of	of	ADP
cana-4834	63	16	sunlet	sunlet	NOUN
cana-4834	63	17	and	and	CCONJ
cana-4834	63	18	pangraph	pangraph	NOUN
cana-4834	63	19	families	family	NOUN
cana-4834	63	20	,	,	PUNCT
cana-4834	63	21	indian	indian	ADJ
cana-4834	63	22	journal	journal	NOUN
cana-4834	63	23	of	of	ADP
cana-4834	63	24	natural	natural	ADJ
cana-4834	63	25	sciences	science	NOUN
cana-4834	63	26	,	,	PUNCT
cana-4834	63	27	vol.15	vol.15	NOUN
cana-4834	63	28	,	,	PUNCT
cana-4834	63	29	issue	issue	NOUN
cana-4834	63	30	87	87	NUM
cana-4834	63	31	,	,	PUNCT
cana-4834	63	32	(	(	PUNCT
cana-4834	63	33	dec	dec	PROPN
cana-4834	63	34	2024	2024	NUM
cana-4834	63	35	)	)	PUNCT
cana-4834	64	1	[	[	X
cana-4834	64	2	6	6	NUM
cana-4834	64	3	]	]	SYM
cana-4834	64	4	mary.u	mary.u	PROPN
cana-4834	64	5	,	,	PUNCT
cana-4834	64	6	g.jothilakshmi	g.jothilakshmi	NOUN
cana-4834	64	7	,	,	PUNCT
cana-4834	64	8	on	on	ADP
cana-4834	64	9	harmonious	harmonious	ADJ
cana-4834	64	10	coloring	coloring	NOUN
cana-4834	64	11	of	of	ADP
cana-4834	64	12	m(sn	m(sn	NOUN
cana-4834	64	13	)	)	PUNCT
cana-4834	64	14	and	and	CCONJ
cana-4834	64	15	m(d3	m(d3	NOUN
cana-4834	64	16	m	m	PROPN
cana-4834	64	17	)	)	PUNCT
cana-4834	64	18	,	,	PUNCT
cana-4834	64	19	international	international	ADJ
cana-4834	64	20	journal	journal	NOUN
cana-4834	64	21	of	of	ADP
cana-4834	64	22	computer	computer	NOUN
cana-4834	64	23	applications	application	NOUN
cana-4834	64	24	,	,	PUNCT
cana-4834	64	25	vol	vol	NOUN
cana-4834	64	26	4,pp.239	4,pp.239	NUM
cana-4834	64	27	-	-	PUNCT
cana-4834	64	28	244(2014	244(2014	NUM
cana-4834	64	29	)	)	PUNCT
cana-4834	65	1	[	[	X
cana-4834	65	2	7	7	X
cana-4834	65	3	]	]	SYM
cana-4834	65	4	mary.u	mary.u	NOUN
cana-4834	65	5	and	and	CCONJ
cana-4834	65	6	g.jothilakshmi	g.jothilakshmi	NOUN
cana-4834	65	7	,	,	PUNCT
cana-4834	65	8	a	a	DET
cana-4834	65	9	study	study	NOUN
cana-4834	65	10	on	on	ADP
cana-4834	65	11	harmonious	harmonious	ADJ
cana-4834	65	12	coloring	coloring	NOUN
cana-4834	65	13	of	of	ADP
cana-4834	65	14	middle	middle	ADJ
cana-4834	65	15	graph	graph	NOUN
cana-4834	65	16	of	of	ADP
cana-4834	65	17	certain	certain	ADJ
cana-4834	65	18	snake	snake	NOUN
cana-4834	65	19	graphs	graph	NOUN
cana-4834	65	20	,	,	PUNCT
cana-4834	65	21	journal	journal	NOUN
cana-4834	65	22	of	of	ADP
cana-4834	65	23	applied	apply	VERB
cana-4834	65	24	science	science	NOUN
cana-4834	65	25	and	and	CCONJ
cana-4834	65	26	computations	computation	NOUN
cana-4834	65	27	,	,	PUNCT
cana-4834	65	28	vol	vol	NOUN
cana-4834	65	29	6,pp.30	6,pp.30	NUM
cana-4834	65	30	-	-	PUNCT
cana-4834	65	31	37(2019	37(2019	NUM
cana-4834	65	32	)	)	PUNCT
cana-4834	66	1	[	[	X
cana-4834	66	2	8	8	NUM
cana-4834	66	3	]	]	X
cana-4834	66	4	radcliffe	radcliffe	PROPN
cana-4834	66	5	,	,	PUNCT
cana-4834	66	6	m.	m.	NOUN
cana-4834	66	7	and	and	CCONJ
cana-4834	66	8	zhang	zhang	PROPN
cana-4834	66	9	,	,	PUNCT
cana-4834	66	10	p.	p.	PROPN
cana-4834	66	11	,	,	PUNCT
cana-4834	66	12	irregular	irregular	ADJ
cana-4834	66	13	coloring	coloring	NOUN
cana-4834	66	14	of	of	ADP
cana-4834	66	15	graphs	graph	NOUN
cana-4834	66	16	,	,	PUNCT
cana-4834	66	17	bull	bull	NOUN
cana-4834	66	18	.	.	PUNCT
cana-4834	66	19	inst	inst	PROPN
cana-4834	66	20	.	.	PUNCT
cana-4834	67	1	combin	combin	NOUN
cana-4834	67	2	.	.	PUNCT
cana-4834	68	1	appl	appl	PROPN
cana-4834	68	2	.	.	PROPN
cana-4834	69	1	,49	,49	PROPN
cana-4834	69	2	,	,	PUNCT
cana-4834	69	3	(	(	PUNCT
cana-4834	69	4	2007	2007	NUM
cana-4834	69	5	)	)	PUNCT
cana-4834	69	6	,	,	PUNCT
cana-4834	69	7	41	41	NUM
cana-4834	69	8	-	-	SYM
cana-4834	69	9	59	59	NUM
cana-4834	69	10	.	.	PUNCT
cana-4834	70	1	[	[	X
cana-4834	70	2	9	9	NUM
cana-4834	70	3	]	]	PUNCT
cana-4834	70	4	a.rohini	a.rohini	NOUN
cana-4834	70	5	and	and	CCONJ
cana-4834	70	6	m.venkatachalam	m.venkatachalam	PROPN
cana-4834	70	7	,	,	PUNCT
cana-4834	70	8	on	on	ADP
cana-4834	70	9	irregular	irregular	ADJ
cana-4834	70	10	coloring	coloring	NOUN
cana-4834	70	11	of	of	ADP
cana-4834	70	12	wheel	wheel	NOUN
cana-4834	70	13	related	relate	VERB
cana-4834	70	14	graphs	graph	NOUN
cana-4834	70	15	,	,	PUNCT
cana-4834	70	16	international	international	ADJ
cana-4834	70	17	conference	conference	NOUN
cana-4834	70	18	on	on	ADP
cana-4834	70	19	current	current	ADJ
cana-4834	70	20	scenario	scenario	NOUN
cana-4834	70	21	in	in	ADP
cana-4834	70	22	pure	pure	ADJ
cana-4834	70	23	and	and	CCONJ
cana-4834	70	24	applied	applied	ADJ
cana-4834	70	25	mathematics	mathematic	NOUN
cana-4834	70	26	,	,	PUNCT
cana-4834	70	27	1462	1462	NUM
cana-4834	70	28	1472(2019	1472(2019	NUM
cana-4834	70	29	)	)	PUNCT
cana-4834	70	30	.	.	PUNCT