id	sid	tid	token	lemma	pos
ejpam-6188	1	1	european	european	PROPN
ejpam-6188	1	2	journal	journal	PROPN
ejpam-6188	1	3	of	of	ADP
ejpam-6188	1	4	pure	pure	ADJ
ejpam-6188	1	5	and	and	CCONJ
ejpam-6188	1	6	applied	applied	ADJ
ejpam-6188	1	7	mathematics	mathematic	NOUN
ejpam-6188	1	8	2025	2025	NUM
ejpam-6188	1	9	,	,	PUNCT
ejpam-6188	1	10	vol	vol	NOUN
ejpam-6188	1	11	.	.	PROPN
ejpam-6188	1	12	18	18	NUM
ejpam-6188	1	13	,	,	PUNCT
ejpam-6188	1	14	issue	issue	NOUN
ejpam-6188	1	15	3	3	NUM
ejpam-6188	1	16	,	,	PUNCT
ejpam-6188	1	17	article	article	NOUN
ejpam-6188	1	18	number	number	NOUN
ejpam-6188	1	19	6188	6188	NUM
ejpam-6188	1	20	issn	issn	VERB
ejpam-6188	1	21	1307	1307	NUM
ejpam-6188	1	22	-	-	SYM
ejpam-6188	1	23	5543	5543	NUM
ejpam-6188	1	24	–	–	PUNCT
ejpam-6188	1	25	ejpam.com	ejpam.com	X
ejpam-6188	1	26	published	publish	VERB
ejpam-6188	1	27	by	by	ADP
ejpam-6188	1	28	new	new	PROPN
ejpam-6188	1	29	york	york	PROPN
ejpam-6188	1	30	business	business	PROPN
ejpam-6188	1	31	global	global	PROPN
ejpam-6188	1	32	on	on	ADP
ejpam-6188	1	33	a	a	DET
ejpam-6188	1	34	b	b	NOUN
ejpam-6188	1	35	-	-	PUNCT
ejpam-6188	1	36	chromatic	chromatic	ADJ
ejpam-6188	1	37	sum	sum	NOUN
ejpam-6188	1	38	of	of	ADP
ejpam-6188	1	39	a	a	DET
ejpam-6188	1	40	mycielskian	mycielskian	NOUN
ejpam-6188	1	41	of	of	ADP
ejpam-6188	1	42	paths	path	NOUN
ejpam-6188	1	43	panupong	panupong	VERB
ejpam-6188	1	44	vichitkunakorn1,2	vichitkunakorn1,2	PROPN
ejpam-6188	1	45	,	,	PUNCT
ejpam-6188	1	46	wipawee	wipawee	NOUN
ejpam-6188	1	47	tangjai3,∗	tangjai3,∗	PROPN
ejpam-6188	1	48	,	,	PUNCT
ejpam-6188	1	49	sittichai	sittichai	PROPN
ejpam-6188	1	50	chaiyakhot4	chaiyakhot4	PROPN
ejpam-6188	1	51	,	,	PUNCT
ejpam-6188	1	52	rawit	rawit	VERB
ejpam-6188	1	53	sinthuket3	sinthuket3	ADJ
ejpam-6188	1	54	1	1	NUM
ejpam-6188	1	55	division	division	NOUN
ejpam-6188	1	56	of	of	ADP
ejpam-6188	1	57	computational	computational	ADJ
ejpam-6188	1	58	science	science	NOUN
ejpam-6188	1	59	,	,	PUNCT
ejpam-6188	1	60	faculty	faculty	NOUN
ejpam-6188	1	61	of	of	ADP
ejpam-6188	1	62	science	science	NOUN
ejpam-6188	1	63	,	,	PUNCT
ejpam-6188	1	64	prince	prince	NOUN
ejpam-6188	1	65	of	of	ADP
ejpam-6188	1	66	songkla	songkla	PROPN
ejpam-6188	1	67	university	university	PROPN
ejpam-6188	1	68	,	,	PUNCT
ejpam-6188	1	69	songkla	songkla	PROPN
ejpam-6188	1	70	,	,	PUNCT
ejpam-6188	1	71	thailand	thailand	PROPN
ejpam-6188	1	72	2	2	NUM
ejpam-6188	1	73	research	research	NOUN
ejpam-6188	1	74	center	center	NOUN
ejpam-6188	1	75	in	in	ADP
ejpam-6188	1	76	mathematics	mathematic	NOUN
ejpam-6188	1	77	and	and	CCONJ
ejpam-6188	1	78	statistics	statistic	NOUN
ejpam-6188	1	79	with	with	ADP
ejpam-6188	1	80	applications	application	NOUN
ejpam-6188	1	81	,	,	PUNCT
ejpam-6188	1	82	prince	prince	NOUN
ejpam-6188	1	83	of	of	ADP
ejpam-6188	1	84	songkla	songkla	PROPN
ejpam-6188	1	85	university	university	PROPN
ejpam-6188	1	86	,	,	PUNCT
ejpam-6188	1	87	songkhla	songkhla	PROPN
ejpam-6188	1	88	,	,	PUNCT
ejpam-6188	1	89	thailand	thailand	PROPN
ejpam-6188	1	90	3	3	NUM
ejpam-6188	1	91	department	department	NOUN
ejpam-6188	1	92	of	of	ADP
ejpam-6188	1	93	mathematics	mathematic	NOUN
ejpam-6188	1	94	,	,	PUNCT
ejpam-6188	1	95	faculty	faculty	NOUN
ejpam-6188	1	96	of	of	ADP
ejpam-6188	1	97	science	science	NOUN
ejpam-6188	1	98	,	,	PUNCT
ejpam-6188	1	99	mahasarakham	mahasarakham	PROPN
ejpam-6188	1	100	university	university	PROPN
ejpam-6188	1	101	,	,	PUNCT
ejpam-6188	1	102	maha	maha	PROPN
ejpam-6188	1	103	sarakham	sarakham	PROPN
ejpam-6188	1	104	,	,	PUNCT
ejpam-6188	1	105	thailand	thailand	PROPN
ejpam-6188	1	106	4	4	NUM
ejpam-6188	1	107	department	department	NOUN
ejpam-6188	1	108	of	of	ADP
ejpam-6188	1	109	mathematics	mathematic	NOUN
ejpam-6188	1	110	,	,	PUNCT
ejpam-6188	1	111	faculty	faculty	NOUN
ejpam-6188	1	112	of	of	ADP
ejpam-6188	1	113	science	science	NOUN
ejpam-6188	1	114	,	,	PUNCT
ejpam-6188	1	115	khon	khon	PROPN
ejpam-6188	1	116	kaen	kaen	PROPN
ejpam-6188	1	117	university	university	PROPN
ejpam-6188	1	118	,	,	PUNCT
ejpam-6188	1	119	khon	khon	PROPN
ejpam-6188	1	120	kaen	kaen	PROPN
ejpam-6188	1	121	,	,	PUNCT
ejpam-6188	1	122	thailand	thailand	PROPN
ejpam-6188	1	123	abstract	abstract	PROPN
ejpam-6188	1	124	.	.	PUNCT
ejpam-6188	2	1	a	a	DET
ejpam-6188	2	2	b	b	X
ejpam-6188	2	3	-	-	PUNCT
ejpam-6188	2	4	coloring	coloring	NOUN
ejpam-6188	2	5	of	of	ADP
ejpam-6188	2	6	a	a	DET
ejpam-6188	2	7	graph	graph	NOUN
ejpam-6188	2	8	g	g	NOUN
ejpam-6188	2	9	is	be	AUX
ejpam-6188	2	10	a	a	DET
ejpam-6188	2	11	proper	proper	ADJ
ejpam-6188	2	12	coloring	coloring	NOUN
ejpam-6188	2	13	such	such	ADJ
ejpam-6188	2	14	that	that	SCONJ
ejpam-6188	2	15	there	there	PRON
ejpam-6188	2	16	exists	exist	VERB
ejpam-6188	2	17	a	a	DET
ejpam-6188	2	18	vertex	vertex	NOUN
ejpam-6188	2	19	in	in	ADP
ejpam-6188	2	20	each	each	DET
ejpam-6188	2	21	color	color	NOUN
ejpam-6188	2	22	class	class	NOUN
ejpam-6188	2	23	that	that	PRON
ejpam-6188	2	24	is	be	AUX
ejpam-6188	2	25	adjacent	adjacent	ADJ
ejpam-6188	2	26	to	to	ADP
ejpam-6188	2	27	at	at	ADV
ejpam-6188	2	28	least	least	ADV
ejpam-6188	2	29	one	one	NUM
ejpam-6188	2	30	vertex	vertex	NOUN
ejpam-6188	2	31	in	in	ADP
ejpam-6188	2	32	other	other	ADJ
ejpam-6188	2	33	color	color	NOUN
ejpam-6188	2	34	classes	class	NOUN
ejpam-6188	2	35	.	.	PUNCT
ejpam-6188	3	1	the	the	DET
ejpam-6188	3	2	b	b	NOUN
ejpam-6188	3	3	-	-	PUNCT
ejpam-6188	3	4	chromatic	chromatic	ADJ
ejpam-6188	3	5	number	number	NOUN
ejpam-6188	3	6	of	of	ADP
ejpam-6188	3	7	a	a	DET
ejpam-6188	3	8	graph	graph	NOUN
ejpam-6188	3	9	g	g	NOUN
ejpam-6188	3	10	,	,	PUNCT
ejpam-6188	3	11	denoted	denote	VERB
ejpam-6188	3	12	by	by	ADP
ejpam-6188	3	13	φ(g	φ(g	PROPN
ejpam-6188	3	14	)	)	PUNCT
ejpam-6188	3	15	,	,	PUNCT
ejpam-6188	3	16	is	be	AUX
ejpam-6188	3	17	the	the	DET
ejpam-6188	3	18	largest	large	ADJ
ejpam-6188	3	19	integer	integer	NOUN
ejpam-6188	3	20	k	k	PROPN
ejpam-6188	3	21	such	such	ADJ
ejpam-6188	3	22	that	that	SCONJ
ejpam-6188	3	23	g	g	PROPN
ejpam-6188	3	24	has	have	VERB
ejpam-6188	3	25	a	a	DET
ejpam-6188	3	26	b	b	NOUN
ejpam-6188	3	27	-	-	PUNCT
ejpam-6188	3	28	coloring	coloring	NOUN
ejpam-6188	3	29	with	with	ADP
ejpam-6188	3	30	k	k	PROPN
ejpam-6188	3	31	colors	color	NOUN
ejpam-6188	3	32	.	.	PUNCT
ejpam-6188	4	1	the	the	DET
ejpam-6188	4	2	b	b	NOUN
ejpam-6188	4	3	-	-	PUNCT
ejpam-6188	4	4	chromatic	chromatic	ADJ
ejpam-6188	4	5	sum	sum	NOUN
ejpam-6188	4	6	of	of	ADP
ejpam-6188	4	7	a	a	DET
ejpam-6188	4	8	graph	graph	NOUN
ejpam-6188	4	9	g	g	NOUN
ejpam-6188	4	10	,	,	PUNCT
ejpam-6188	4	11	denoted	denote	VERB
ejpam-6188	4	12	by	by	ADP
ejpam-6188	4	13	φ′(g	φ′(g	NOUN
ejpam-6188	4	14	)	)	PUNCT
ejpam-6188	4	15	,	,	PUNCT
ejpam-6188	4	16	is	be	AUX
ejpam-6188	4	17	defined	define	VERB
ejpam-6188	4	18	as	as	ADP
ejpam-6188	4	19	the	the	DET
ejpam-6188	4	20	minimum	minimum	ADJ
ejpam-6188	4	21	sum	sum	NOUN
ejpam-6188	4	22	of	of	ADP
ejpam-6188	4	23	the	the	DET
ejpam-6188	4	24	colors	color	NOUN
ejpam-6188	4	25	c(v	c(v	PROPN
ejpam-6188	4	26	)	)	PUNCT
ejpam-6188	4	27	of	of	ADP
ejpam-6188	4	28	v	v	NOUN
ejpam-6188	4	29	for	for	ADP
ejpam-6188	4	30	all	all	DET
ejpam-6188	4	31	v	v	NOUN
ejpam-6188	4	32	∈	∈	NOUN
ejpam-6188	4	33	v	v	NOUN
ejpam-6188	4	34	where	where	SCONJ
ejpam-6188	4	35	c	c	PROPN
ejpam-6188	4	36	is	be	AUX
ejpam-6188	4	37	a	a	DET
ejpam-6188	4	38	b	b	NOUN
ejpam-6188	4	39	-	-	PUNCT
ejpam-6188	4	40	coloring	coloring	NOUN
ejpam-6188	4	41	using	use	VERB
ejpam-6188	4	42	φ(g	φ(g	NOUN
ejpam-6188	4	43	)	)	PUNCT
ejpam-6188	4	44	colors	color	NOUN
ejpam-6188	4	45	.	.	PUNCT
ejpam-6188	5	1	in	in	ADP
ejpam-6188	5	2	this	this	DET
ejpam-6188	5	3	work	work	NOUN
ejpam-6188	5	4	,	,	PUNCT
ejpam-6188	5	5	we	we	PRON
ejpam-6188	5	6	improve	improve	VERB
ejpam-6188	5	7	the	the	DET
ejpam-6188	5	8	bounds	bound	NOUN
ejpam-6188	5	9	on	on	ADP
ejpam-6188	5	10	the	the	DET
ejpam-6188	5	11	b	b	NOUN
ejpam-6188	5	12	-	-	PUNCT
ejpam-6188	5	13	chromatic	chromatic	ADJ
ejpam-6188	5	14	sum	sum	NOUN
ejpam-6188	5	15	given	give	VERB
ejpam-6188	5	16	by	by	ADP
ejpam-6188	5	17	lisna	lisna	NOUN
ejpam-6188	5	18	and	and	CCONJ
ejpam-6188	5	19	sunitha	sunitha	VERB
ejpam-6188	6	1	[	[	X
ejpam-6188	6	2	1	1	NUM
ejpam-6188	6	3	]	]	PUNCT
ejpam-6188	6	4	.	.	PUNCT
ejpam-6188	7	1	we	we	PRON
ejpam-6188	7	2	give	give	VERB
ejpam-6188	7	3	the	the	DET
ejpam-6188	7	4	b	b	NOUN
ejpam-6188	7	5	-	-	PUNCT
ejpam-6188	7	6	chromatic	chromatic	ADJ
ejpam-6188	7	7	sum	sum	NOUN
ejpam-6188	7	8	of	of	ADP
ejpam-6188	7	9	the	the	DET
ejpam-6188	7	10	mycielskian	mycielskian	NOUN
ejpam-6188	7	11	of	of	ADP
ejpam-6188	7	12	a	a	DET
ejpam-6188	7	13	path	path	NOUN
ejpam-6188	7	14	µ(pn	µ(pn	NOUN
ejpam-6188	7	15	)	)	PUNCT
ejpam-6188	7	16	when	when	SCONJ
ejpam-6188	7	17	n	n	X
ejpam-6188	7	18	=	=	SYM
ejpam-6188	7	19	7	7	NUM
ejpam-6188	7	20	,	,	PUNCT
ejpam-6188	7	21	9	9	NUM
ejpam-6188	7	22	and	and	CCONJ
ejpam-6188	7	23	n	n	PRON
ejpam-6188	7	24	≥	≥	NOUN
ejpam-6188	7	25	16	16	NUM
ejpam-6188	7	26	.	.	PUNCT
ejpam-6188	8	1	for	for	ADP
ejpam-6188	8	2	the	the	DET
ejpam-6188	8	3	case	case	NOUN
ejpam-6188	8	4	10	10	NUM
ejpam-6188	8	5	≤	≤	NUM
ejpam-6188	8	6	n	n	PRON
ejpam-6188	8	7	≤	≤	NUM
ejpam-6188	8	8	15	15	NUM
ejpam-6188	8	9	,	,	PUNCT
ejpam-6188	8	10	we	we	PRON
ejpam-6188	8	11	give	give	VERB
ejpam-6188	8	12	bounds	bound	NOUN
ejpam-6188	8	13	on	on	ADP
ejpam-6188	8	14	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	8	15	)	)	PUNCT
ejpam-6188	8	16	)	)	PUNCT
ejpam-6188	8	17	.	.	PUNCT
ejpam-6188	9	1	2020	2020	NUM
ejpam-6188	9	2	mathematics	mathematic	NOUN
ejpam-6188	9	3	subject	subject	NOUN
ejpam-6188	9	4	classifications	classification	NOUN
ejpam-6188	9	5	:	:	PUNCT
ejpam-6188	9	6	05c15	05c15	NUM
ejpam-6188	9	7	,	,	PUNCT
ejpam-6188	9	8	05c78	05c78	NUM
ejpam-6188	9	9	,	,	PUNCT
ejpam-6188	9	10	05c76	05c76	NUM
ejpam-6188	9	11	,	,	PUNCT
ejpam-6188	9	12	05c38	05c38	NUM
ejpam-6188	9	13	,	,	PUNCT
ejpam-6188	9	14	05c69	05c69	X
ejpam-6188	9	15	key	key	ADJ
ejpam-6188	9	16	words	word	NOUN
ejpam-6188	9	17	and	and	CCONJ
ejpam-6188	9	18	phrases	phrase	NOUN
ejpam-6188	9	19	:	:	PUNCT
ejpam-6188	9	20	b	b	X
ejpam-6188	9	21	-	-	PUNCT
ejpam-6188	9	22	coloring	coloring	ADJ
ejpam-6188	9	23	,	,	PUNCT
ejpam-6188	9	24	b	b	X
ejpam-6188	9	25	-	-	PUNCT
ejpam-6188	9	26	chromatic	chromatic	ADJ
ejpam-6188	9	27	number	number	NOUN
ejpam-6188	9	28	,	,	PUNCT
ejpam-6188	9	29	b	b	X
ejpam-6188	9	30	-	-	PUNCT
ejpam-6188	9	31	dominating	dominating	ADJ
ejpam-6188	9	32	,	,	PUNCT
ejpam-6188	9	33	b	b	NOUN
ejpam-6188	9	34	-	-	PUNCT
ejpam-6188	9	35	chromatic	chromatic	ADJ
ejpam-6188	9	36	sum	sum	NOUN
ejpam-6188	9	37	,	,	PUNCT
ejpam-6188	9	38	path	path	NOUN
ejpam-6188	9	39	,	,	PUNCT
ejpam-6188	9	40	mycielskian	mycielskian	ADJ
ejpam-6188	9	41	1	1	NUM
ejpam-6188	9	42	.	.	PUNCT
ejpam-6188	9	43	introduction	introduction	NOUN
ejpam-6188	9	44	for	for	ADP
ejpam-6188	9	45	a	a	DET
ejpam-6188	9	46	graph	graph	NOUN
ejpam-6188	9	47	g	g	NOUN
ejpam-6188	9	48	=	=	PUNCT
ejpam-6188	9	49	(	(	PUNCT
ejpam-6188	9	50	v	v	NOUN
ejpam-6188	9	51	,	,	PUNCT
ejpam-6188	9	52	e	e	NOUN
ejpam-6188	9	53	)	)	PUNCT
ejpam-6188	9	54	with	with	ADP
ejpam-6188	9	55	a	a	DET
ejpam-6188	9	56	proper	proper	ADJ
ejpam-6188	9	57	coloring	coloring	NOUN
ejpam-6188	9	58	c	c	NOUN
ejpam-6188	9	59	:	:	PUNCT
ejpam-6188	9	60	v	v	X
ejpam-6188	9	61	→	→	SYM
ejpam-6188	9	62	{	{	PUNCT
ejpam-6188	9	63	1	1	NUM
ejpam-6188	9	64	,	,	PUNCT
ejpam-6188	9	65	.	.	PUNCT
ejpam-6188	9	66	.	.	PUNCT
ejpam-6188	9	67	.	.	PUNCT
ejpam-6188	10	1	,	,	PUNCT
ejpam-6188	10	2	k	k	X
ejpam-6188	10	3	}	}	PUNCT
ejpam-6188	10	4	,	,	PUNCT
ejpam-6188	10	5	we	we	PRON
ejpam-6188	10	6	denote	denote	VERB
ejpam-6188	10	7	a	a	DET
ejpam-6188	10	8	color	color	NOUN
ejpam-6188	10	9	class	class	NOUN
ejpam-6188	10	10	of	of	ADP
ejpam-6188	10	11	ci	ci	NOUN
ejpam-6188	10	12	=	=	PUNCT
ejpam-6188	10	13	{	{	PUNCT
ejpam-6188	10	14	v	v	NUM
ejpam-6188	10	15	∈	∈	NOUN
ejpam-6188	10	16	v	v	NOUN
ejpam-6188	10	17	:	:	PUNCT
ejpam-6188	10	18	c(v	c(v	PROPN
ejpam-6188	10	19	)	)	PUNCT
ejpam-6188	11	1	=	=	PUNCT
ejpam-6188	11	2	i	i	PROPN
ejpam-6188	11	3	}	}	PUNCT
ejpam-6188	11	4	for	for	ADP
ejpam-6188	11	5	i	i	PROPN
ejpam-6188	11	6	=	=	NOUN
ejpam-6188	11	7	1	1	NUM
ejpam-6188	11	8	,	,	PUNCT
ejpam-6188	11	9	.	.	PUNCT
ejpam-6188	11	10	.	.	PUNCT
ejpam-6188	11	11	.	.	PUNCT
ejpam-6188	12	1	,	,	PUNCT
ejpam-6188	12	2	k.	k.	PROPN
ejpam-6188	12	3	for	for	ADP
ejpam-6188	12	4	s	s	PROPN
ejpam-6188	12	5	⊂	⊂	PROPN
ejpam-6188	12	6	v	v	PROPN
ejpam-6188	12	7	,	,	PUNCT
ejpam-6188	12	8	we	we	PRON
ejpam-6188	12	9	denote	denote	VERB
ejpam-6188	12	10	c(s	c(	NOUN
ejpam-6188	12	11	)	)	PUNCT
ejpam-6188	12	12	=	=	NOUN
ejpam-6188	12	13	{	{	PUNCT
ejpam-6188	12	14	c(s	c(	NOUN
ejpam-6188	12	15	)	)	PUNCT
ejpam-6188	12	16	:	:	PUNCT
ejpam-6188	12	17	s	s	VERB
ejpam-6188	12	18	∈	∈	PROPN
ejpam-6188	12	19	s	s	PART
ejpam-6188	12	20	}	}	PUNCT
ejpam-6188	12	21	.	.	PUNCT
ejpam-6188	13	1	a	a	DET
ejpam-6188	13	2	vertex	vertex	NOUN
ejpam-6188	13	3	v	v	NOUN
ejpam-6188	13	4	is	be	AUX
ejpam-6188	13	5	a	a	DET
ejpam-6188	13	6	b	b	NOUN
ejpam-6188	13	7	-	-	PUNCT
ejpam-6188	13	8	dominating	dominate	VERB
ejpam-6188	13	9	vertex	vertex	NOUN
ejpam-6188	13	10	of	of	ADP
ejpam-6188	13	11	color	color	NOUN
ejpam-6188	13	12	class	class	NOUN
ejpam-6188	14	1	i	i	PRON
ejpam-6188	14	2	if	if	SCONJ
ejpam-6188	14	3	v	v	NOUN
ejpam-6188	14	4	is	be	AUX
ejpam-6188	14	5	adjacent	adjacent	ADJ
ejpam-6188	14	6	to	to	ADP
ejpam-6188	14	7	a	a	DET
ejpam-6188	14	8	vertex	vertex	NOUN
ejpam-6188	14	9	from	from	ADP
ejpam-6188	14	10	each	each	DET
ejpam-6188	14	11	color	color	NOUN
ejpam-6188	14	12	class	class	NOUN
ejpam-6188	14	13	j	j	PROPN
ejpam-6188	14	14	̸=	̸=	PROPN
ejpam-6188	14	15	i.	i.	NOUN
ejpam-6188	14	16	a	a	DET
ejpam-6188	14	17	b	b	X
ejpam-6188	14	18	-	-	PUNCT
ejpam-6188	14	19	coloring	coloring	NOUN
ejpam-6188	14	20	is	be	AUX
ejpam-6188	14	21	a	a	DET
ejpam-6188	14	22	proper	proper	ADJ
ejpam-6188	14	23	coloring	coloring	NOUN
ejpam-6188	14	24	where	where	SCONJ
ejpam-6188	14	25	each	each	DET
ejpam-6188	14	26	color	color	NOUN
ejpam-6188	14	27	has	have	VERB
ejpam-6188	14	28	a	a	DET
ejpam-6188	14	29	b	b	NOUN
ejpam-6188	14	30	-	-	PUNCT
ejpam-6188	14	31	dominating	dominate	VERB
ejpam-6188	14	32	vertex	vertex	NOUN
ejpam-6188	14	33	.	.	PUNCT
ejpam-6188	15	1	r.	r.	PROPN
ejpam-6188	15	2	w.	w.	PROPN
ejpam-6188	15	3	irving	irving	PROPN
ejpam-6188	15	4	and	and	CCONJ
ejpam-6188	15	5	d.	d.	PROPN
ejpam-6188	15	6	f.	f.	PROPN
ejpam-6188	15	7	manlove	manlove	PROPN
ejpam-6188	16	1	[	[	X
ejpam-6188	16	2	2	2	NUM
ejpam-6188	16	3	]	]	PUNCT
ejpam-6188	16	4	introduced	introduce	VERB
ejpam-6188	16	5	the	the	DET
ejpam-6188	16	6	concept	concept	NOUN
ejpam-6188	16	7	of	of	ADP
ejpam-6188	16	8	the	the	DET
ejpam-6188	16	9	b	b	NOUN
ejpam-6188	16	10	-	-	PUNCT
ejpam-6188	16	11	chromatic	chromatic	ADJ
ejpam-6188	16	12	number	number	NOUN
ejpam-6188	16	13	of	of	ADP
ejpam-6188	16	14	a	a	DET
ejpam-6188	16	15	graph	graph	NOUN
ejpam-6188	16	16	g.	g.	NOUN
ejpam-6188	16	17	the	the	DET
ejpam-6188	16	18	b	b	NOUN
ejpam-6188	16	19	-	-	PUNCT
ejpam-6188	16	20	chromatic	chromatic	ADJ
ejpam-6188	16	21	number	number	NOUN
ejpam-6188	16	22	φ(g	φ(g	PROPN
ejpam-6188	16	23	)	)	PUNCT
ejpam-6188	16	24	of	of	ADP
ejpam-6188	16	25	a	a	DET
ejpam-6188	16	26	graph	graph	NOUN
ejpam-6188	16	27	g	g	NOUN
ejpam-6188	16	28	is	be	AUX
ejpam-6188	16	29	the	the	DET
ejpam-6188	16	30	largest	large	ADJ
ejpam-6188	16	31	positive	positive	ADJ
ejpam-6188	16	32	integer	integer	NOUN
ejpam-6188	16	33	k	k	PROPN
ejpam-6188	16	34	such	such	ADJ
ejpam-6188	16	35	that	that	SCONJ
ejpam-6188	16	36	g	g	PROPN
ejpam-6188	16	37	admits	admit	VERB
ejpam-6188	16	38	a	a	DET
ejpam-6188	16	39	b	b	NOUN
ejpam-6188	16	40	-	-	PUNCT
ejpam-6188	16	41	coloring	coloring	NOUN
ejpam-6188	16	42	.	.	PUNCT
ejpam-6188	17	1	many	many	ADJ
ejpam-6188	17	2	research	research	NOUN
ejpam-6188	17	3	on	on	ADP
ejpam-6188	17	4	the	the	DET
ejpam-6188	17	5	∗corresponding	∗corresponde	VERB
ejpam-6188	17	6	author	author	NOUN
ejpam-6188	17	7	.	.	PUNCT
ejpam-6188	18	1	doi	doi	NOUN
ejpam-6188	18	2	:	:	PUNCT
ejpam-6188	18	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6188	https://doi.org/10.29020/nybg.ejpam.v18i3.6188	NOUN
ejpam-6188	18	4	email	email	NOUN
ejpam-6188	18	5	addresses	address	NOUN
ejpam-6188	18	6	:	:	PUNCT
ejpam-6188	18	7	panupong.v@psu.ac.th	panupong.v@psu.ac.th	ADP
ejpam-6188	18	8	(	(	PUNCT
ejpam-6188	18	9	p.	p.	NOUN
ejpam-6188	18	10	vichitkunakorn	vichitkunakorn	PROPN
ejpam-6188	18	11	)	)	PUNCT
ejpam-6188	18	12	,	,	PUNCT
ejpam-6188	18	13	wipawee.t@msu.ac.th	wipawee.t@msu.ac.th	PROPN
ejpam-6188	18	14	(	(	PUNCT
ejpam-6188	18	15	w.	w.	PROPN
ejpam-6188	18	16	tangjai	tangjai	PROPN
ejpam-6188	18	17	)	)	PUNCT
ejpam-6188	18	18	,	,	PUNCT
ejpam-6188	18	19	63010213002@msu.ac.th	63010213002@msu.ac.th	NOUN
ejpam-6188	18	20	(	(	PUNCT
ejpam-6188	18	21	r.	r.	PROPN
ejpam-6188	18	22	sinthuket	sinthuket	PROPN
ejpam-6188	18	23	)	)	PUNCT
ejpam-6188	18	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6188	18	25	1	1	NUM
ejpam-6188	18	26	copyright	copyright	NOUN
ejpam-6188	18	27	:	:	PUNCT
ejpam-6188	19	1	©	©	PROPN
ejpam-6188	19	2	2025	2025	NUM
ejpam-6188	19	3	the	the	DET
ejpam-6188	19	4	author(s	author(s	NOUN
ejpam-6188	19	5	)	)	PUNCT
ejpam-6188	19	6	.	.	PUNCT
ejpam-6188	20	1	(	(	PUNCT
ejpam-6188	20	2	cc	cc	NOUN
ejpam-6188	20	3	by	by	ADP
ejpam-6188	20	4	-	-	PUNCT
ejpam-6188	20	5	nc	nc	PROPN
ejpam-6188	20	6	4.0	4.0	NUM
ejpam-6188	20	7	)	)	PUNCT
ejpam-6188	21	1	p.	p.	NOUN
ejpam-6188	21	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	21	3	et	et	PROPN
ejpam-6188	21	4	al	al	PROPN
ejpam-6188	21	5	.	.	PUNCT
ejpam-6188	21	6	/	/	SYM
ejpam-6188	21	7	eur	eur	PROPN
ejpam-6188	21	8	.	.	PUNCT
ejpam-6188	22	1	j.	j.	PROPN
ejpam-6188	22	2	pure	pure	PROPN
ejpam-6188	22	3	appl	appl	PROPN
ejpam-6188	22	4	.	.	PROPN
ejpam-6188	22	5	math	math	PROPN
ejpam-6188	22	6	,	,	PUNCT
ejpam-6188	22	7	18	18	NUM
ejpam-6188	22	8	(	(	PUNCT
ejpam-6188	22	9	3	3	NUM
ejpam-6188	22	10	)	)	PUNCT
ejpam-6188	22	11	(	(	PUNCT
ejpam-6188	22	12	2025	2025	NUM
ejpam-6188	22	13	)	)	PUNCT
ejpam-6188	22	14	,	,	PUNCT
ejpam-6188	22	15	6188	6188	NUM
ejpam-6188	22	16	2	2	NUM
ejpam-6188	22	17	of	of	ADP
ejpam-6188	22	18	13	13	NUM
ejpam-6188	22	19	b	b	NOUN
ejpam-6188	22	20	-	-	PUNCT
ejpam-6188	22	21	chromatic	chromatic	ADJ
ejpam-6188	22	22	number	number	NOUN
ejpam-6188	22	23	and	and	CCONJ
ejpam-6188	22	24	its	its	PRON
ejpam-6188	22	25	bounds	bound	NOUN
ejpam-6188	22	26	have	have	AUX
ejpam-6188	22	27	been	be	AUX
ejpam-6188	22	28	studied	study	VERB
ejpam-6188	22	29	[	[	X
ejpam-6188	22	30	3–5	3–5	NOUN
ejpam-6188	22	31	]	]	PUNCT
ejpam-6188	22	32	.	.	PUNCT
ejpam-6188	23	1	a	a	DET
ejpam-6188	23	2	collection	collection	NOUN
ejpam-6188	23	3	of	of	ADP
ejpam-6188	23	4	results	result	NOUN
ejpam-6188	23	5	on	on	ADP
ejpam-6188	23	6	the	the	DET
ejpam-6188	23	7	b	b	NOUN
ejpam-6188	23	8	-	-	PUNCT
ejpam-6188	23	9	chromatic	chromatic	ADJ
ejpam-6188	23	10	number	number	NOUN
ejpam-6188	23	11	appeared	appear	VERB
ejpam-6188	23	12	in	in	ADP
ejpam-6188	23	13	[	[	X
ejpam-6188	23	14	6	6	NUM
ejpam-6188	23	15	]	]	PUNCT
ejpam-6188	23	16	.	.	PUNCT
ejpam-6188	24	1	the	the	DET
ejpam-6188	24	2	application	application	NOUN
ejpam-6188	24	3	of	of	ADP
ejpam-6188	24	4	the	the	DET
ejpam-6188	24	5	b	b	NOUN
ejpam-6188	24	6	-	-	PUNCT
ejpam-6188	24	7	chromatic	chromatic	ADJ
ejpam-6188	24	8	number	number	NOUN
ejpam-6188	24	9	appears	appear	VERB
ejpam-6188	24	10	in	in	ADP
ejpam-6188	24	11	data	datum	NOUN
ejpam-6188	24	12	clustering	cluster	VERB
ejpam-6188	24	13	[	[	X
ejpam-6188	24	14	7	7	NUM
ejpam-6188	24	15	]	]	PUNCT
ejpam-6188	24	16	.	.	PUNCT
ejpam-6188	25	1	in	in	ADP
ejpam-6188	25	2	comparison	comparison	NOUN
ejpam-6188	25	3	to	to	ADP
ejpam-6188	25	4	the	the	DET
ejpam-6188	25	5	well	well	ADV
ejpam-6188	25	6	-	-	PUNCT
ejpam-6188	25	7	known	know	VERB
ejpam-6188	25	8	chromatic	chromatic	ADJ
ejpam-6188	25	9	number	number	NOUN
ejpam-6188	25	10	,	,	PUNCT
ejpam-6188	25	11	the	the	DET
ejpam-6188	25	12	idea	idea	NOUN
ejpam-6188	25	13	of	of	ADP
ejpam-6188	25	14	chromatic	chromatic	ADJ
ejpam-6188	25	15	sum	sum	NOUN
ejpam-6188	25	16	was	be	AUX
ejpam-6188	25	17	originated	originate	VERB
ejpam-6188	25	18	in	in	ADP
ejpam-6188	25	19	1989	1989	NUM
ejpam-6188	25	20	by	by	ADP
ejpam-6188	25	21	kubicka	kubicka	NOUN
ejpam-6188	25	22	and	and	CCONJ
ejpam-6188	25	23	schwenk	schwenk	PROPN
ejpam-6188	26	1	[	[	X
ejpam-6188	26	2	8	8	NUM
ejpam-6188	26	3	]	]	PUNCT
ejpam-6188	26	4	.	.	PUNCT
ejpam-6188	27	1	the	the	DET
ejpam-6188	27	2	chromatic	chromatic	ADJ
ejpam-6188	27	3	sum	sum	NOUN
ejpam-6188	27	4	had	have	AUX
ejpam-6188	27	5	also	also	ADV
ejpam-6188	27	6	gained	gain	VERB
ejpam-6188	27	7	more	more	ADJ
ejpam-6188	27	8	attention	attention	NOUN
ejpam-6188	27	9	from	from	ADP
ejpam-6188	27	10	its	its	PRON
ejpam-6188	27	11	application	application	NOUN
ejpam-6188	27	12	in	in	ADP
ejpam-6188	27	13	the	the	DET
ejpam-6188	27	14	resource	resource	NOUN
ejpam-6188	27	15	allocation	allocation	NOUN
ejpam-6188	27	16	problem	problem	NOUN
ejpam-6188	27	17	[	[	X
ejpam-6188	27	18	9	9	NUM
ejpam-6188	27	19	]	]	PUNCT
ejpam-6188	27	20	.	.	PUNCT
ejpam-6188	28	1	later	later	ADV
ejpam-6188	28	2	in	in	ADP
ejpam-6188	28	3	2015	2015	NUM
ejpam-6188	28	4	,	,	PUNCT
ejpam-6188	28	5	lisna	lisna	NOUN
ejpam-6188	28	6	and	and	CCONJ
ejpam-6188	28	7	sunitha	sunitha	ADJ
ejpam-6188	29	1	[	[	X
ejpam-6188	29	2	10	10	NUM
ejpam-6188	29	3	]	]	PUNCT
ejpam-6188	29	4	introduced	introduce	VERB
ejpam-6188	29	5	the	the	DET
ejpam-6188	29	6	b	b	NOUN
ejpam-6188	29	7	-	-	PUNCT
ejpam-6188	29	8	chromatic	chromatic	ADJ
ejpam-6188	29	9	sum	sum	NOUN
ejpam-6188	29	10	following	follow	VERB
ejpam-6188	29	11	the	the	DET
ejpam-6188	29	12	same	same	ADJ
ejpam-6188	29	13	structure	structure	NOUN
ejpam-6188	29	14	as	as	ADP
ejpam-6188	29	15	the	the	DET
ejpam-6188	29	16	chromatic	chromatic	ADJ
ejpam-6188	29	17	sum	sum	NOUN
ejpam-6188	29	18	.	.	PUNCT
ejpam-6188	30	1	the	the	DET
ejpam-6188	30	2	b	b	NOUN
ejpam-6188	30	3	-	-	PUNCT
ejpam-6188	30	4	chromatic	chromatic	ADJ
ejpam-6188	30	5	sum	sum	NOUN
ejpam-6188	30	6	of	of	ADP
ejpam-6188	30	7	a	a	DET
ejpam-6188	30	8	graph	graph	NOUN
ejpam-6188	30	9	g	g	NOUN
ejpam-6188	30	10	,	,	PUNCT
ejpam-6188	30	11	denoted	denote	VERB
ejpam-6188	30	12	by	by	ADP
ejpam-6188	30	13	φ′(g	φ′(g	NOUN
ejpam-6188	30	14	)	)	PUNCT
ejpam-6188	30	15	,	,	PUNCT
ejpam-6188	30	16	is	be	AUX
ejpam-6188	30	17	the	the	DET
ejpam-6188	30	18	minimum	minimum	NOUN
ejpam-6188	30	19	of	of	ADP
ejpam-6188	30	20	∑	∑	DET
ejpam-6188	30	21	v∈v	v∈v	PROPN
ejpam-6188	30	22	c(v	c(v	PROPN
ejpam-6188	30	23	)	)	PUNCT
ejpam-6188	30	24	over	over	ADP
ejpam-6188	30	25	a	a	DET
ejpam-6188	30	26	b	b	NOUN
ejpam-6188	30	27	-	-	PUNCT
ejpam-6188	30	28	coloring	color	VERB
ejpam-6188	30	29	c	c	NOUN
ejpam-6188	30	30	giving	give	VERB
ejpam-6188	30	31	the	the	DET
ejpam-6188	30	32	b	b	NOUN
ejpam-6188	30	33	-	-	PUNCT
ejpam-6188	30	34	chromatic	chromatic	ADJ
ejpam-6188	30	35	number	number	NOUN
ejpam-6188	30	36	.	.	PUNCT
ejpam-6188	31	1	many	many	ADJ
ejpam-6188	31	2	questions	question	NOUN
ejpam-6188	31	3	related	relate	VERB
ejpam-6188	31	4	to	to	ADP
ejpam-6188	31	5	the	the	DET
ejpam-6188	31	6	b	b	NOUN
ejpam-6188	31	7	-	-	PUNCT
ejpam-6188	31	8	chromatic	chromatic	ADJ
ejpam-6188	31	9	sum	sum	NOUN
ejpam-6188	31	10	are	be	AUX
ejpam-6188	31	11	still	still	ADV
ejpam-6188	31	12	widely	widely	ADV
ejpam-6188	31	13	open	open	ADJ
ejpam-6188	31	14	.	.	PUNCT
ejpam-6188	32	1	the	the	DET
ejpam-6188	32	2	b	b	NOUN
ejpam-6188	32	3	-	-	PUNCT
ejpam-6188	32	4	chromatic	chromatic	ADJ
ejpam-6188	32	5	sum	sum	NOUN
ejpam-6188	32	6	of	of	ADP
ejpam-6188	32	7	only	only	ADV
ejpam-6188	32	8	a	a	DET
ejpam-6188	32	9	few	few	ADJ
ejpam-6188	32	10	classes	class	NOUN
ejpam-6188	32	11	of	of	ADP
ejpam-6188	32	12	graphs	graph	NOUN
ejpam-6188	32	13	had	have	AUX
ejpam-6188	32	14	been	be	AUX
ejpam-6188	32	15	investigated	investigate	VERB
ejpam-6188	32	16	.	.	PUNCT
ejpam-6188	33	1	some	some	DET
ejpam-6188	33	2	examples	example	NOUN
ejpam-6188	33	3	include	include	VERB
ejpam-6188	33	4	paths	path	NOUN
ejpam-6188	33	5	,	,	PUNCT
ejpam-6188	33	6	cycles	cycle	NOUN
ejpam-6188	33	7	,	,	PUNCT
ejpam-6188	33	8	wheel	wheel	NOUN
ejpam-6188	33	9	graphs	graph	NOUN
ejpam-6188	33	10	,	,	PUNCT
ejpam-6188	33	11	complete	complete	ADJ
ejpam-6188	33	12	graphs	graph	NOUN
ejpam-6188	33	13	[	[	X
ejpam-6188	33	14	11	11	NUM
ejpam-6188	33	15	,	,	PUNCT
ejpam-6188	33	16	12	12	NUM
ejpam-6188	33	17	]	]	PUNCT
ejpam-6188	33	18	.	.	PUNCT
ejpam-6188	34	1	for	for	ADP
ejpam-6188	34	2	a	a	DET
ejpam-6188	34	3	graph	graph	NOUN
ejpam-6188	34	4	g	g	NOUN
ejpam-6188	34	5	=	=	PUNCT
ejpam-6188	34	6	(	(	PUNCT
ejpam-6188	34	7	v	v	NOUN
ejpam-6188	34	8	,	,	PUNCT
ejpam-6188	34	9	e	e	NOUN
ejpam-6188	34	10	)	)	PUNCT
ejpam-6188	34	11	where	where	SCONJ
ejpam-6188	34	12	v	v	NOUN
ejpam-6188	34	13	=	=	SYM
ejpam-6188	34	14	{	{	PUNCT
ejpam-6188	34	15	v1	v1	PROPN
ejpam-6188	34	16	,	,	PUNCT
ejpam-6188	34	17	v2	v2	PROPN
ejpam-6188	34	18	,	,	PUNCT
ejpam-6188	34	19	.	.	PUNCT
ejpam-6188	34	20	.	.	PUNCT
ejpam-6188	34	21	.	.	PUNCT
ejpam-6188	35	1	,	,	PUNCT
ejpam-6188	35	2	vn	vn	PROPN
ejpam-6188	35	3	}	}	PUNCT
ejpam-6188	35	4	,	,	PUNCT
ejpam-6188	35	5	the	the	DET
ejpam-6188	35	6	mycielskian	mycielskian	ADJ
ejpam-6188	35	7	or	or	CCONJ
ejpam-6188	35	8	mycielski	mycielski	ADJ
ejpam-6188	35	9	graph	graph	NOUN
ejpam-6188	35	10	µ(g	µ(g	PROPN
ejpam-6188	35	11	)	)	PUNCT
ejpam-6188	35	12	of	of	ADP
ejpam-6188	35	13	g	g	PROPN
ejpam-6188	35	14	is	be	AUX
ejpam-6188	35	15	a	a	DET
ejpam-6188	35	16	graph	graph	NOUN
ejpam-6188	35	17	in	in	ADP
ejpam-6188	35	18	which	which	PRON
ejpam-6188	35	19	the	the	DET
ejpam-6188	35	20	vertex	vertex	NOUN
ejpam-6188	35	21	set	set	VERB
ejpam-6188	35	22	consists	consist	VERB
ejpam-6188	35	23	of	of	ADP
ejpam-6188	35	24	two	two	NUM
ejpam-6188	35	25	copies	copy	NOUN
ejpam-6188	35	26	of	of	ADP
ejpam-6188	35	27	the	the	DET
ejpam-6188	35	28	vertices	vertex	NOUN
ejpam-6188	35	29	in	in	ADP
ejpam-6188	35	30	g	g	PROPN
ejpam-6188	35	31	and	and	CCONJ
ejpam-6188	35	32	a	a	DET
ejpam-6188	35	33	vertex	vertex	NOUN
ejpam-6188	35	34	u	u	NOUN
ejpam-6188	35	35	,	,	PUNCT
ejpam-6188	35	36	i.e.	i.e.	X
ejpam-6188	35	37	,	,	PUNCT
ejpam-6188	35	38	v	v	INTJ
ejpam-6188	35	39	(	(	PUNCT
ejpam-6188	35	40	µ(g	µ(g	NOUN
ejpam-6188	35	41	)	)	PUNCT
ejpam-6188	35	42	)	)	PUNCT
ejpam-6188	36	1	=	=	SYM
ejpam-6188	36	2	v	v	ADP
ejpam-6188	36	3	∪u	∪u	X
ejpam-6188	36	4	∪	∪	X
ejpam-6188	36	5	{	{	PUNCT
ejpam-6188	36	6	u	u	NOUN
ejpam-6188	36	7	}	}	PUNCT
ejpam-6188	36	8	such	such	ADJ
ejpam-6188	36	9	that	that	PRON
ejpam-6188	36	10	v	v	NOUN
ejpam-6188	36	11	=	=	SYM
ejpam-6188	36	12	{	{	PUNCT
ejpam-6188	36	13	v1	v1	NOUN
ejpam-6188	36	14	,	,	PUNCT
ejpam-6188	36	15	.	.	PUNCT
ejpam-6188	36	16	.	.	PUNCT
ejpam-6188	36	17	.	.	PUNCT
ejpam-6188	37	1	,	,	PUNCT
ejpam-6188	37	2	vn	vn	NOUN
ejpam-6188	37	3	}	}	PUNCT
ejpam-6188	37	4	and	and	CCONJ
ejpam-6188	37	5	u	u	X
ejpam-6188	37	6	=	=	NOUN
ejpam-6188	37	7	{	{	PUNCT
ejpam-6188	37	8	u1	u1	NOUN
ejpam-6188	37	9	,	,	PUNCT
ejpam-6188	37	10	.	.	PUNCT
ejpam-6188	37	11	.	.	PUNCT
ejpam-6188	38	1	.	.	PUNCT
ejpam-6188	39	1	,	,	PUNCT
ejpam-6188	39	2	un	un	PROPN
ejpam-6188	39	3	}	}	PUNCT
ejpam-6188	39	4	where	where	SCONJ
ejpam-6188	39	5	ui	ui	PROPN
ejpam-6188	39	6	is	be	AUX
ejpam-6188	39	7	a	a	DET
ejpam-6188	39	8	copy	copy	NOUN
ejpam-6188	39	9	of	of	ADP
ejpam-6188	39	10	vi	vi	PROPN
ejpam-6188	39	11	.	.	PROPN
ejpam-6188	40	1	for	for	ADP
ejpam-6188	40	2	the	the	DET
ejpam-6188	40	3	edge	edge	NOUN
ejpam-6188	40	4	set	set	NOUN
ejpam-6188	40	5	of	of	ADP
ejpam-6188	40	6	µ(g	µ(g	PROPN
ejpam-6188	40	7	)	)	PUNCT
ejpam-6188	40	8	,	,	PUNCT
ejpam-6188	40	9	we	we	PRON
ejpam-6188	40	10	keep	keep	VERB
ejpam-6188	40	11	the	the	DET
ejpam-6188	40	12	adjacency	adjacency	NOUN
ejpam-6188	40	13	of	of	ADP
ejpam-6188	40	14	the	the	DET
ejpam-6188	40	15	vertices	vertex	NOUN
ejpam-6188	40	16	in	in	ADP
ejpam-6188	40	17	v	v	ADP
ejpam-6188	40	18	⊂	⊂	PROPN
ejpam-6188	40	19	v	v	X
ejpam-6188	40	20	(	(	PUNCT
ejpam-6188	40	21	µ(g	µ(g	PROPN
ejpam-6188	40	22	)	)	PUNCT
ejpam-6188	40	23	)	)	PUNCT
ejpam-6188	40	24	as	as	ADP
ejpam-6188	40	25	in	in	ADP
ejpam-6188	40	26	g	g	NOUN
ejpam-6188	40	27	,	,	PUNCT
ejpam-6188	40	28	then	then	ADV
ejpam-6188	40	29	join	join	VERB
ejpam-6188	40	30	u	u	PRON
ejpam-6188	40	31	to	to	ADP
ejpam-6188	40	32	each	each	DET
ejpam-6188	40	33	vertex	vertex	NOUN
ejpam-6188	40	34	ui	ui	NOUN
ejpam-6188	40	35	,	,	PUNCT
ejpam-6188	40	36	for	for	ADP
ejpam-6188	40	37	1	1	NUM
ejpam-6188	40	38	≤	≤	NUM
ejpam-6188	40	39	i	i	PRON
ejpam-6188	40	40	≤	≤	NOUN
ejpam-6188	40	41	n	n	CCONJ
ejpam-6188	40	42	,	,	PUNCT
ejpam-6188	40	43	and	and	CCONJ
ejpam-6188	40	44	join	join	VERB
ejpam-6188	40	45	ui	ui	PROPN
ejpam-6188	40	46	to	to	ADP
ejpam-6188	40	47	each	each	DET
ejpam-6188	40	48	neighbor	neighbor	NOUN
ejpam-6188	40	49	of	of	ADP
ejpam-6188	40	50	vi	vi	PROPN
ejpam-6188	40	51	in	in	ADP
ejpam-6188	40	52	g.	g.	PROPN
ejpam-6188	40	53	in	in	ADP
ejpam-6188	40	54	2011	2011	NUM
ejpam-6188	40	55	,	,	PUNCT
ejpam-6188	40	56	massimiliano	massimiliano	PROPN
ejpam-6188	40	57	et	et	PROPN
ejpam-6188	40	58	.	.	PUNCT
ejpam-6188	41	1	al	al	PROPN
ejpam-6188	41	2	.	.	PUNCT
ejpam-6188	42	1	[	[	X
ejpam-6188	42	2	13	13	NUM
ejpam-6188	42	3	]	]	PUNCT
ejpam-6188	42	4	discussed	discuss	VERB
ejpam-6188	42	5	the	the	DET
ejpam-6188	42	6	applications	application	NOUN
ejpam-6188	42	7	of	of	ADP
ejpam-6188	42	8	mycielski	mycielski	ADJ
ejpam-6188	42	9	graphs	graph	NOUN
ejpam-6188	42	10	in	in	ADP
ejpam-6188	42	11	the	the	DET
ejpam-6188	42	12	multiprocessor	multiprocessor	NOUN
ejpam-6188	42	13	task	task	NOUN
ejpam-6188	42	14	scheduling	scheduling	NOUN
ejpam-6188	42	15	problem	problem	NOUN
ejpam-6188	42	16	.	.	PUNCT
ejpam-6188	43	1	in	in	ADP
ejpam-6188	43	2	2017	2017	NUM
ejpam-6188	43	3	,	,	PUNCT
ejpam-6188	43	4	lisna	lisna	NOUN
ejpam-6188	43	5	and	and	CCONJ
ejpam-6188	43	6	sunitha	sunitha	VERB
ejpam-6188	43	7	[	[	X
ejpam-6188	43	8	1	1	X
ejpam-6188	43	9	]	]	PUNCT
ejpam-6188	43	10	gave	give	VERB
ejpam-6188	43	11	an	an	DET
ejpam-6188	43	12	upper	upper	ADJ
ejpam-6188	43	13	bound	bind	VERB
ejpam-6188	43	14	on	on	ADP
ejpam-6188	43	15	the	the	DET
ejpam-6188	43	16	b	b	NOUN
ejpam-6188	43	17	-	-	PUNCT
ejpam-6188	43	18	chromatic	chromatic	ADJ
ejpam-6188	43	19	sum	sum	NOUN
ejpam-6188	43	20	of	of	ADP
ejpam-6188	43	21	a	a	DET
ejpam-6188	43	22	mycielskian	mycielskian	ADJ
ejpam-6188	43	23	path	path	NOUN
ejpam-6188	43	24	µ(pn	µ(pn	NOUN
ejpam-6188	43	25	)	)	PUNCT
ejpam-6188	43	26	.	.	PUNCT
ejpam-6188	44	1	they	they	PRON
ejpam-6188	44	2	stated	state	VERB
ejpam-6188	44	3	that	that	SCONJ
ejpam-6188	44	4	such	such	DET
ejpam-6188	44	5	a	a	DET
ejpam-6188	44	6	bound	bind	VERB
ejpam-6188	44	7	is	be	AUX
ejpam-6188	44	8	the	the	DET
ejpam-6188	44	9	b	b	NOUN
ejpam-6188	44	10	-	-	PUNCT
ejpam-6188	44	11	chromatic	chromatic	ADJ
ejpam-6188	44	12	sum	sum	NOUN
ejpam-6188	44	13	of	of	ADP
ejpam-6188	44	14	µ(pn	µ(pn	NOUN
ejpam-6188	44	15	)	)	PUNCT
ejpam-6188	44	16	for	for	ADP
ejpam-6188	44	17	n	n	X
ejpam-6188	44	18	≥	≥	NOUN
ejpam-6188	44	19	2	2	NUM
ejpam-6188	44	20	;	;	PUNCT
ejpam-6188	44	21	however	however	ADV
ejpam-6188	44	22	,	,	PUNCT
ejpam-6188	44	23	we	we	PRON
ejpam-6188	44	24	find	find	VERB
ejpam-6188	44	25	that	that	SCONJ
ejpam-6188	44	26	such	such	ADJ
ejpam-6188	44	27	results	result	NOUN
ejpam-6188	44	28	are	be	AUX
ejpam-6188	44	29	a	a	DET
ejpam-6188	44	30	close	close	ADJ
ejpam-6188	44	31	upper	upper	ADJ
ejpam-6188	44	32	bound	bind	VERB
ejpam-6188	44	33	on	on	ADP
ejpam-6188	44	34	the	the	DET
ejpam-6188	44	35	b	b	NOUN
ejpam-6188	44	36	-	-	PUNCT
ejpam-6188	44	37	chromatic	chromatic	ADJ
ejpam-6188	44	38	sum	sum	NOUN
ejpam-6188	44	39	of	of	ADP
ejpam-6188	44	40	µ(pn	µ(pn	NOUN
ejpam-6188	44	41	)	)	PUNCT
ejpam-6188	44	42	when	when	SCONJ
ejpam-6188	44	43	n	n	X
ejpam-6188	44	44	=	=	SYM
ejpam-6188	44	45	7	7	NUM
ejpam-6188	44	46	or	or	CCONJ
ejpam-6188	44	47	n	n	PRON
ejpam-6188	44	48	≥	≥	NOUN
ejpam-6188	44	49	9	9	NUM
ejpam-6188	44	50	.	.	PUNCT
ejpam-6188	45	1	in	in	ADP
ejpam-6188	45	2	this	this	DET
ejpam-6188	45	3	work	work	NOUN
ejpam-6188	45	4	,	,	PUNCT
ejpam-6188	45	5	we	we	PRON
ejpam-6188	45	6	improve	improve	VERB
ejpam-6188	45	7	such	such	ADJ
ejpam-6188	45	8	results	result	NOUN
ejpam-6188	45	9	and	and	CCONJ
ejpam-6188	45	10	give	give	VERB
ejpam-6188	45	11	the	the	DET
ejpam-6188	45	12	b	b	NOUN
ejpam-6188	45	13	-	-	PUNCT
ejpam-6188	45	14	chromatic	chromatic	ADJ
ejpam-6188	45	15	sum	sum	NOUN
ejpam-6188	45	16	of	of	ADP
ejpam-6188	45	17	µ(pn	µ(pn	NOUN
ejpam-6188	45	18	)	)	PUNCT
ejpam-6188	45	19	when	when	SCONJ
ejpam-6188	45	20	n	n	X
ejpam-6188	45	21	=	=	SYM
ejpam-6188	45	22	7	7	NUM
ejpam-6188	45	23	,	,	PUNCT
ejpam-6188	45	24	9	9	NUM
ejpam-6188	45	25	or	or	CCONJ
ejpam-6188	45	26	n	n	PRON
ejpam-6188	45	27	≥	≥	NOUN
ejpam-6188	45	28	16	16	NUM
ejpam-6188	45	29	and	and	CCONJ
ejpam-6188	45	30	give	give	VERB
ejpam-6188	45	31	a	a	DET
ejpam-6188	45	32	lower	lower	ADV
ejpam-6188	45	33	bound	bind	VERB
ejpam-6188	45	34	and	and	CCONJ
ejpam-6188	45	35	an	an	DET
ejpam-6188	45	36	upper	upper	ADJ
ejpam-6188	45	37	bound	bind	VERB
ejpam-6188	45	38	when	when	SCONJ
ejpam-6188	45	39	10	10	NUM
ejpam-6188	45	40	≤	≤	NUM
ejpam-6188	45	41	n	n	PRON
ejpam-6188	45	42	≤	≤	NUM
ejpam-6188	45	43	15	15	NUM
ejpam-6188	45	44	.	.	NOUN
ejpam-6188	46	1	2	2	NUM
ejpam-6188	46	2	.	.	NUM
ejpam-6188	46	3	preliminaries	preliminary	NOUN
ejpam-6188	46	4	in	in	ADP
ejpam-6188	46	5	this	this	DET
ejpam-6188	46	6	section	section	NOUN
ejpam-6188	46	7	,	,	PUNCT
ejpam-6188	46	8	we	we	PRON
ejpam-6188	46	9	present	present	VERB
ejpam-6188	46	10	definitions	definition	NOUN
ejpam-6188	46	11	and	and	CCONJ
ejpam-6188	46	12	related	relate	VERB
ejpam-6188	46	13	results	result	NOUN
ejpam-6188	46	14	for	for	ADP
ejpam-6188	46	15	the	the	DET
ejpam-6188	46	16	mycielskian	mycielskian	ADJ
ejpam-6188	46	17	graph	graph	NOUN
ejpam-6188	46	18	and	and	CCONJ
ejpam-6188	46	19	the	the	DET
ejpam-6188	46	20	b	b	NOUN
ejpam-6188	46	21	-	-	PUNCT
ejpam-6188	46	22	chromatic	chromatic	ADJ
ejpam-6188	46	23	sum	sum	NOUN
ejpam-6188	46	24	.	.	PUNCT
ejpam-6188	47	1	definition	definition	NOUN
ejpam-6188	47	2	1	1	NUM
ejpam-6188	47	3	.	.	PUNCT
ejpam-6188	48	1	(	(	PUNCT
ejpam-6188	48	2	[	[	X
ejpam-6188	48	3	14	14	NUM
ejpam-6188	48	4	]	]	PUNCT
ejpam-6188	48	5	)	)	PUNCT
ejpam-6188	48	6	.	.	PUNCT
ejpam-6188	49	1	let	let	VERB
ejpam-6188	49	2	g	g	PRON
ejpam-6188	49	3	be	be	AUX
ejpam-6188	49	4	a	a	DET
ejpam-6188	49	5	graph	graph	NOUN
ejpam-6188	49	6	with	with	ADP
ejpam-6188	49	7	n	n	ADP
ejpam-6188	49	8	vertices	vertex	NOUN
ejpam-6188	49	9	,	,	PUNCT
ejpam-6188	49	10	where	where	SCONJ
ejpam-6188	49	11	v	v	X
ejpam-6188	49	12	(	(	PUNCT
ejpam-6188	49	13	g	g	NOUN
ejpam-6188	49	14	)	)	PUNCT
ejpam-6188	49	15	=	=	SYM
ejpam-6188	49	16	{	{	PUNCT
ejpam-6188	49	17	v1	v1	PROPN
ejpam-6188	49	18	,	,	PUNCT
ejpam-6188	49	19	v2	v2	PROPN
ejpam-6188	49	20	,	,	PUNCT
ejpam-6188	49	21	.	.	PUNCT
ejpam-6188	49	22	.	.	PUNCT
ejpam-6188	50	1	.	.	PUNCT
ejpam-6188	51	1	,	,	PUNCT
ejpam-6188	51	2	vn	vn	PROPN
ejpam-6188	51	3	}	}	PUNCT
ejpam-6188	51	4	.	.	PUNCT
ejpam-6188	52	1	the	the	DET
ejpam-6188	52	2	mycielskian	mycielskian	ADJ
ejpam-6188	52	3	or	or	CCONJ
ejpam-6188	52	4	mycielski	mycielski	ADJ
ejpam-6188	52	5	graph	graph	NOUN
ejpam-6188	52	6	µ(g	µ(g	PROPN
ejpam-6188	52	7	)	)	PUNCT
ejpam-6188	52	8	is	be	AUX
ejpam-6188	52	9	a	a	DET
ejpam-6188	52	10	graph	graph	NOUN
ejpam-6188	52	11	where	where	SCONJ
ejpam-6188	52	12	v	v	X
ejpam-6188	52	13	(	(	PUNCT
ejpam-6188	52	14	µ(g	µ(g	NOUN
ejpam-6188	52	15	)	)	PUNCT
ejpam-6188	52	16	)	)	PUNCT
ejpam-6188	53	1	=	=	SYM
ejpam-6188	53	2	v	v	X
ejpam-6188	53	3	(	(	PUNCT
ejpam-6188	53	4	g)∪{u	g)∪{u	PROPN
ejpam-6188	53	5	,	,	PUNCT
ejpam-6188	53	6	u1	u1	NOUN
ejpam-6188	53	7	,	,	PUNCT
ejpam-6188	53	8	.	.	PUNCT
ejpam-6188	53	9	.	.	PUNCT
ejpam-6188	53	10	.	.	PUNCT
ejpam-6188	53	11	,	,	PUNCT
ejpam-6188	53	12	un	un	PROPN
ejpam-6188	53	13	}	}	PUNCT
ejpam-6188	53	14	and	and	CCONJ
ejpam-6188	53	15	e(µ(g	e(µ(g	PROPN
ejpam-6188	53	16	)	)	PUNCT
ejpam-6188	53	17	)	)	PUNCT
ejpam-6188	54	1	=	=	SYM
ejpam-6188	54	2	e(g	e(g	NOUN
ejpam-6188	54	3	)	)	PUNCT
ejpam-6188	54	4	∪	∪	VERB
ejpam-6188	54	5	{	{	PUNCT
ejpam-6188	54	6	uui	uui	NOUN
ejpam-6188	54	7	:	:	PUNCT
ejpam-6188	54	8	1	1	NUM
ejpam-6188	54	9	≤	≤	NUM
ejpam-6188	54	10	i	i	PRON
ejpam-6188	54	11	≤	≤	NOUN
ejpam-6188	54	12	n	n	CCONJ
ejpam-6188	54	13	}	}	PUNCT
ejpam-6188	54	14	∪	∪	ADJ
ejpam-6188	54	15	{	{	PUNCT
ejpam-6188	54	16	uiv	uiv	INTJ
ejpam-6188	54	17	:	:	PUNCT
ejpam-6188	54	18	1	1	NUM
ejpam-6188	54	19	≤	≤	NUM
ejpam-6188	54	20	i	i	PRON
ejpam-6188	54	21	≤	≤	ADJ
ejpam-6188	54	22	n	n	CCONJ
ejpam-6188	54	23	and	and	CCONJ
ejpam-6188	54	24	v	v	ADP
ejpam-6188	54	25	∈	∈	PROPN
ejpam-6188	54	26	ng(vi	ng(vi	NOUN
ejpam-6188	54	27	)	)	PUNCT
ejpam-6188	54	28	}	}	PUNCT
ejpam-6188	54	29	.	.	PUNCT
ejpam-6188	55	1	let	let	VERB
ejpam-6188	55	2	un	un	PROPN
ejpam-6188	55	3	=	=	PRON
ejpam-6188	55	4	{	{	PUNCT
ejpam-6188	55	5	ui	ui	NOUN
ejpam-6188	55	6	:	:	PUNCT
ejpam-6188	55	7	1	1	NUM
ejpam-6188	55	8	≤	≤	NUM
ejpam-6188	55	9	i	i	PRON
ejpam-6188	55	10	≤	≤	NOUN
ejpam-6188	55	11	n	n	CCONJ
ejpam-6188	55	12	}	}	PUNCT
ejpam-6188	55	13	.	.	PUNCT
ejpam-6188	56	1	we	we	PRON
ejpam-6188	56	2	partition	partition	VERB
ejpam-6188	56	3	v	v	ADP
ejpam-6188	56	4	(	(	PUNCT
ejpam-6188	56	5	µ(pn	µ(pn	NOUN
ejpam-6188	56	6	)	)	PUNCT
ejpam-6188	56	7	)	)	PUNCT
ejpam-6188	56	8	according	accord	VERB
ejpam-6188	56	9	to	to	ADP
ejpam-6188	56	10	the	the	DET
ejpam-6188	56	11	definition	definition	NOUN
ejpam-6188	56	12	of	of	ADP
ejpam-6188	56	13	the	the	DET
ejpam-6188	56	14	mycielskian	mycielskian	NOUN
ejpam-6188	56	15	into	into	ADP
ejpam-6188	56	16	{	{	PUNCT
ejpam-6188	56	17	v	v	NOUN
ejpam-6188	56	18	(	(	PUNCT
ejpam-6188	56	19	pn	pn	NOUN
ejpam-6188	56	20	)	)	PUNCT
ejpam-6188	56	21	,	,	PUNCT
ejpam-6188	56	22	un	un	PROPN
ejpam-6188	56	23	,	,	PUNCT
ejpam-6188	56	24	{	{	PUNCT
ejpam-6188	56	25	u	u	NOUN
ejpam-6188	56	26	}	}	PUNCT
ejpam-6188	56	27	}	}	PUNCT
ejpam-6188	56	28	as	as	SCONJ
ejpam-6188	56	29	shown	show	VERB
ejpam-6188	56	30	in	in	ADP
ejpam-6188	56	31	figure	figure	NOUN
ejpam-6188	56	32	1	1	NUM
ejpam-6188	56	33	.	.	PUNCT
ejpam-6188	56	34	theorem	theorem	NOUN
ejpam-6188	56	35	1	1	NUM
ejpam-6188	56	36	provides	provide	VERB
ejpam-6188	56	37	the	the	DET
ejpam-6188	56	38	maximum	maximum	ADJ
ejpam-6188	56	39	number	number	NOUN
ejpam-6188	56	40	of	of	ADP
ejpam-6188	56	41	colors	color	NOUN
ejpam-6188	56	42	that	that	PRON
ejpam-6188	56	43	µ(pn	µ(pn	NOUN
ejpam-6188	56	44	)	)	PUNCT
ejpam-6188	56	45	admit	admit	VERB
ejpam-6188	56	46	a	a	DET
ejpam-6188	56	47	b	b	NOUN
ejpam-6188	56	48	-	-	PUNCT
ejpam-6188	56	49	coloring	coloring	NOUN
ejpam-6188	56	50	.	.	PUNCT
ejpam-6188	57	1	we	we	PRON
ejpam-6188	57	2	note	note	VERB
ejpam-6188	57	3	that	that	SCONJ
ejpam-6188	57	4	for	for	SCONJ
ejpam-6188	57	5	a	a	DET
ejpam-6188	57	6	graph	graph	NOUN
ejpam-6188	57	7	g	g	NOUN
ejpam-6188	57	8	and	and	CCONJ
ejpam-6188	57	9	a	a	DET
ejpam-6188	57	10	coloring	color	VERB
ejpam-6188	57	11	c	c	NOUN
ejpam-6188	57	12	to	to	PART
ejpam-6188	57	13	admit	admit	VERB
ejpam-6188	57	14	a	a	DET
ejpam-6188	57	15	b	b	NOUN
ejpam-6188	57	16	-	-	PUNCT
ejpam-6188	57	17	chromatic	chromatic	ADJ
ejpam-6188	57	18	number	number	NOUN
ejpam-6188	57	19	,	,	PUNCT
ejpam-6188	57	20	there	there	PRON
ejpam-6188	57	21	must	must	AUX
ejpam-6188	57	22	be	be	AUX
ejpam-6188	57	23	at	at	ADP
ejpam-6188	57	24	least	least	ADJ
ejpam-6188	57	25	φ(g	φ(g	ADJ
ejpam-6188	57	26	)	)	PUNCT
ejpam-6188	57	27	b	b	X
ejpam-6188	57	28	-	-	PUNCT
ejpam-6188	57	29	dominating	dominating	NOUN
ejpam-6188	57	30	vertices	vertex	NOUN
ejpam-6188	57	31	.	.	PUNCT
ejpam-6188	58	1	thus	thus	ADV
ejpam-6188	58	2	,	,	PUNCT
ejpam-6188	58	3	g	g	PROPN
ejpam-6188	58	4	has	have	VERB
ejpam-6188	58	5	at	at	ADV
ejpam-6188	58	6	least	least	ADJ
ejpam-6188	58	7	φ(g	φ(g	ADJ
ejpam-6188	58	8	)	)	PUNCT
ejpam-6188	58	9	vertices	vertex	NOUN
ejpam-6188	58	10	with	with	ADP
ejpam-6188	58	11	φ(g	φ(g	PROPN
ejpam-6188	58	12	)	)	PUNCT
ejpam-6188	58	13	−	−	PROPN
ejpam-6188	58	14	1	1	NUM
ejpam-6188	58	15	degree	degree	NOUN
ejpam-6188	58	16	.	.	PUNCT
ejpam-6188	59	1	p.	p.	NOUN
ejpam-6188	59	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	59	3	et	et	PROPN
ejpam-6188	59	4	al	al	PROPN
ejpam-6188	59	5	.	.	PUNCT
ejpam-6188	59	6	/	/	SYM
ejpam-6188	59	7	eur	eur	PROPN
ejpam-6188	59	8	.	.	PUNCT
ejpam-6188	60	1	j.	j.	PROPN
ejpam-6188	60	2	pure	pure	PROPN
ejpam-6188	60	3	appl	appl	PROPN
ejpam-6188	60	4	.	.	PROPN
ejpam-6188	60	5	math	math	PROPN
ejpam-6188	60	6	,	,	PUNCT
ejpam-6188	60	7	18	18	NUM
ejpam-6188	60	8	(	(	PUNCT
ejpam-6188	60	9	3	3	NUM
ejpam-6188	60	10	)	)	PUNCT
ejpam-6188	60	11	(	(	PUNCT
ejpam-6188	60	12	2025	2025	NUM
ejpam-6188	60	13	)	)	PUNCT
ejpam-6188	60	14	,	,	PUNCT
ejpam-6188	60	15	6188	6188	NUM
ejpam-6188	60	16	3	3	NUM
ejpam-6188	60	17	of	of	ADP
ejpam-6188	60	18	13	13	NUM
ejpam-6188	60	19	u	u	NOUN
ejpam-6188	60	20	u1	u1	NOUN
ejpam-6188	60	21	u2	u2	PROPN
ejpam-6188	60	22	u3	u3	PROPN
ejpam-6188	60	23	u4	u4	PROPN
ejpam-6188	60	24	u5	u5	PROPN
ejpam-6188	60	25	.	.	PUNCT
ejpam-6188	60	26	.	.	PUNCT
ejpam-6188	60	27	.	.	PUNCT
ejpam-6188	61	1	un	un	PROPN
ejpam-6188	61	2	v1	v1	PROPN
ejpam-6188	61	3	v2	v2	PROPN
ejpam-6188	61	4	v3	v3	PROPN
ejpam-6188	61	5	v4	v4	PROPN
ejpam-6188	61	6	v5	v5	PROPN
ejpam-6188	61	7	·	·	PUNCT
ejpam-6188	61	8	·	·	PUNCT
ejpam-6188	61	9	·	·	PUNCT
ejpam-6188	62	1	vn	vn	X
ejpam-6188	62	2	figure	figure	NOUN
ejpam-6188	62	3	1	1	NUM
ejpam-6188	62	4	:	:	PUNCT
ejpam-6188	62	5	the	the	DET
ejpam-6188	62	6	graph	graph	NOUN
ejpam-6188	62	7	of	of	ADP
ejpam-6188	62	8	µ(pn	µ(pn	NOUN
ejpam-6188	62	9	)	)	PUNCT
ejpam-6188	62	10	theorem	theorem	NOUN
ejpam-6188	62	11	1	1	NUM
ejpam-6188	62	12	.	.	PUNCT
ejpam-6188	63	1	(	(	PUNCT
ejpam-6188	63	2	[	[	X
ejpam-6188	63	3	15	15	NUM
ejpam-6188	63	4	]	]	NUM
ejpam-6188	63	5	)	)	PUNCT
ejpam-6188	63	6	.	.	PUNCT
ejpam-6188	64	1	the	the	DET
ejpam-6188	64	2	b	b	NOUN
ejpam-6188	64	3	-	-	PUNCT
ejpam-6188	64	4	chromatic	chromatic	ADJ
ejpam-6188	64	5	number	number	NOUN
ejpam-6188	64	6	of	of	ADP
ejpam-6188	64	7	mycielskian	mycielskian	ADJ
ejpam-6188	64	8	path	path	PROPN
ejpam-6188	64	9	µ(pn	µ(pn	NOUN
ejpam-6188	64	10	)	)	PUNCT
ejpam-6188	64	11	is	be	AUX
ejpam-6188	64	12	φ(µ(pn	φ(µ(pn	NOUN
ejpam-6188	64	13	)	)	PUNCT
ejpam-6188	64	14	)	)	PUNCT
ejpam-6188	65	1	=	=	PUNCT
ejpam-6188	65	2			NOUN
ejpam-6188	65	3	3	3	NUM
ejpam-6188	65	4	if	if	SCONJ
ejpam-6188	65	5	2	2	NUM
ejpam-6188	65	6	≤	≤	NOUN
ejpam-6188	65	7	n	n	CCONJ
ejpam-6188	65	8	≤	≤	NOUN
ejpam-6188	65	9	4	4	NUM
ejpam-6188	65	10	,	,	PUNCT
ejpam-6188	65	11	4	4	NUM
ejpam-6188	65	12	if	if	SCONJ
ejpam-6188	65	13	5	5	NUM
ejpam-6188	65	14	≤	≤	NUM
ejpam-6188	65	15	n	n	PRON
ejpam-6188	65	16	≤	≤	NOUN
ejpam-6188	65	17	7	7	NUM
ejpam-6188	65	18	,	,	PUNCT
ejpam-6188	65	19	5	5	NUM
ejpam-6188	65	20	if	if	SCONJ
ejpam-6188	65	21	n	n	PRON
ejpam-6188	65	22	≥	≥	NOUN
ejpam-6188	65	23	8	8	NUM
ejpam-6188	65	24	.	.	PUNCT
ejpam-6188	65	25	lisna	lisna	NOUN
ejpam-6188	65	26	and	and	CCONJ
ejpam-6188	65	27	sunitha	sunitha	VERB
ejpam-6188	66	1	[	[	X
ejpam-6188	66	2	1	1	NUM
ejpam-6188	66	3	]	]	PUNCT
ejpam-6188	66	4	provided	provide	VERB
ejpam-6188	66	5	the	the	DET
ejpam-6188	66	6	following	follow	VERB
ejpam-6188	66	7	results	result	NOUN
ejpam-6188	66	8	.	.	PUNCT
ejpam-6188	67	1	we	we	PRON
ejpam-6188	67	2	note	note	VERB
ejpam-6188	67	3	that	that	SCONJ
ejpam-6188	67	4	theorem	theorem	NOUN
ejpam-6188	67	5	3	3	NUM
ejpam-6188	67	6	was	be	AUX
ejpam-6188	67	7	originally	originally	ADV
ejpam-6188	67	8	stated	state	VERB
ejpam-6188	67	9	as	as	ADP
ejpam-6188	67	10	the	the	DET
ejpam-6188	67	11	exact	exact	ADJ
ejpam-6188	67	12	value	value	NOUN
ejpam-6188	67	13	of	of	ADP
ejpam-6188	67	14	the	the	DET
ejpam-6188	67	15	b	b	NOUN
ejpam-6188	67	16	-	-	PUNCT
ejpam-6188	67	17	chromatic	chromatic	ADJ
ejpam-6188	67	18	sum	sum	NOUN
ejpam-6188	67	19	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	67	20	)	)	PUNCT
ejpam-6188	67	21	)	)	PUNCT
ejpam-6188	67	22	.	.	PUNCT
ejpam-6188	68	1	however	however	ADV
ejpam-6188	68	2	,	,	PUNCT
ejpam-6188	68	3	the	the	DET
ejpam-6188	68	4	proof	proof	NOUN
ejpam-6188	68	5	assumes	assume	VERB
ejpam-6188	68	6	that	that	SCONJ
ejpam-6188	68	7	a	a	DET
ejpam-6188	68	8	b	b	NOUN
ejpam-6188	68	9	-	-	PUNCT
ejpam-6188	68	10	coloring	color	VERB
ejpam-6188	68	11	cn	cn	PROPN
ejpam-6188	68	12	yielding	yield	VERB
ejpam-6188	68	13	the	the	DET
ejpam-6188	68	14	b	b	NOUN
ejpam-6188	68	15	-	-	PUNCT
ejpam-6188	68	16	chromatic	chromatic	ADJ
ejpam-6188	68	17	sum	sum	NOUN
ejpam-6188	68	18	of	of	ADP
ejpam-6188	68	19	µ(pn	µ(pn	NOUN
ejpam-6188	68	20	)	)	PUNCT
ejpam-6188	68	21	must	must	AUX
ejpam-6188	68	22	be	be	AUX
ejpam-6188	68	23	obtained	obtain	VERB
ejpam-6188	68	24	from	from	ADP
ejpam-6188	68	25	a	a	DET
ejpam-6188	68	26	b	b	NOUN
ejpam-6188	68	27	-	-	PUNCT
ejpam-6188	68	28	coloring	color	VERB
ejpam-6188	68	29	cn−1	cn−1	PROPN
ejpam-6188	68	30	yielding	yield	VERB
ejpam-6188	68	31	the	the	DET
ejpam-6188	68	32	b	b	NOUN
ejpam-6188	68	33	-	-	PUNCT
ejpam-6188	68	34	chromatic	chromatic	ADJ
ejpam-6188	68	35	sum	sum	NOUN
ejpam-6188	68	36	of	of	ADP
ejpam-6188	68	37	µ(pn−1	µ(pn−1	NOUN
ejpam-6188	68	38	)	)	PUNCT
ejpam-6188	68	39	for	for	ADP
ejpam-6188	68	40	n	n	PRON
ejpam-6188	68	41	≥	≥	NUM
ejpam-6188	68	42	9	9	NUM
ejpam-6188	68	43	.	.	PUNCT
ejpam-6188	69	1	however	however	ADV
ejpam-6188	69	2	,	,	PUNCT
ejpam-6188	69	3	this	this	DET
ejpam-6188	69	4	assumption	assumption	NOUN
ejpam-6188	69	5	does	do	AUX
ejpam-6188	69	6	not	not	PART
ejpam-6188	69	7	always	always	ADV
ejpam-6188	69	8	hold	hold	VERB
ejpam-6188	69	9	.	.	PUNCT
ejpam-6188	70	1	so	so	ADV
ejpam-6188	70	2	,	,	PUNCT
ejpam-6188	70	3	the	the	DET
ejpam-6188	70	4	b	b	NOUN
ejpam-6188	70	5	-	-	PUNCT
ejpam-6188	70	6	colorings	coloring	NOUN
ejpam-6188	70	7	provided	provide	VERB
ejpam-6188	70	8	in	in	ADP
ejpam-6188	70	9	the	the	DET
ejpam-6188	70	10	proof	proof	NOUN
ejpam-6188	70	11	only	only	ADV
ejpam-6188	70	12	give	give	VERB
ejpam-6188	70	13	an	an	DET
ejpam-6188	70	14	upper	upper	ADJ
ejpam-6188	70	15	bound	bind	VERB
ejpam-6188	70	16	on	on	ADP
ejpam-6188	70	17	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	70	18	)	)	PUNCT
ejpam-6188	70	19	)	)	PUNCT
ejpam-6188	70	20	.	.	PUNCT
ejpam-6188	71	1	the	the	DET
ejpam-6188	71	2	results	result	NOUN
ejpam-6188	71	3	for	for	ADP
ejpam-6188	71	4	small	small	ADJ
ejpam-6188	71	5	values	value	NOUN
ejpam-6188	71	6	of	of	ADP
ejpam-6188	71	7	n	n	CCONJ
ejpam-6188	71	8	,	,	PUNCT
ejpam-6188	71	9	except	except	SCONJ
ejpam-6188	71	10	for	for	ADP
ejpam-6188	71	11	n	n	NOUN
ejpam-6188	71	12	=	=	SYM
ejpam-6188	71	13	7	7	NUM
ejpam-6188	71	14	,	,	PUNCT
ejpam-6188	71	15	are	be	AUX
ejpam-6188	71	16	valid	valid	ADJ
ejpam-6188	71	17	as	as	SCONJ
ejpam-6188	71	18	listed	list	VERB
ejpam-6188	71	19	in	in	ADP
ejpam-6188	71	20	theorem	theorem	NOUN
ejpam-6188	71	21	2	2	NUM
ejpam-6188	71	22	.	.	PUNCT
ejpam-6188	71	23	theorem	theorem	NOUN
ejpam-6188	71	24	2	2	NUM
ejpam-6188	71	25	.	.	PUNCT
ejpam-6188	72	1	(	(	PUNCT
ejpam-6188	72	2	[	[	X
ejpam-6188	72	3	1	1	NUM
ejpam-6188	72	4	]	]	NUM
ejpam-6188	72	5	)	)	PUNCT
ejpam-6188	72	6	.	.	PUNCT
ejpam-6188	73	1	the	the	DET
ejpam-6188	73	2	b	b	NOUN
ejpam-6188	73	3	-	-	PUNCT
ejpam-6188	73	4	chromatic	chromatic	ADJ
ejpam-6188	73	5	sum	sum	NOUN
ejpam-6188	73	6	of	of	ADP
ejpam-6188	73	7	mycielskian	mycielskian	NOUN
ejpam-6188	73	8	of	of	ADP
ejpam-6188	73	9	path	path	PROPN
ejpam-6188	73	10	µ(pn	µ(pn	NOUN
ejpam-6188	73	11	)	)	PUNCT
ejpam-6188	73	12	is	be	AUX
ejpam-6188	73	13	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	73	14	)	)	PUNCT
ejpam-6188	73	15	)	)	PUNCT
ejpam-6188	74	1	=	=	PUNCT
ejpam-6188	74	2			NOUN
ejpam-6188	74	3	4	4	NUM
ejpam-6188	74	4	if	if	SCONJ
ejpam-6188	74	5	n	n	NOUN
ejpam-6188	74	6	=	=	SYM
ejpam-6188	74	7	1	1	NUM
ejpam-6188	74	8	,	,	PUNCT
ejpam-6188	74	9	3	3	NUM
ejpam-6188	74	10	+	+	SYM
ejpam-6188	74	11	2(⌈n2	2(⌈n2	NUM
ejpam-6188	74	12	⌉+	⌉+	NUM
ejpam-6188	74	13	2⌊n2	2⌊n2	NUM
ejpam-6188	74	14	⌋	⌋	NUM
ejpam-6188	74	15	)	)	PUNCT
ejpam-6188	75	1	if	if	SCONJ
ejpam-6188	75	2	n	n	NOUN
ejpam-6188	75	3	=	=	SYM
ejpam-6188	75	4	2	2	NUM
ejpam-6188	75	5	,	,	PUNCT
ejpam-6188	75	6	3	3	NUM
ejpam-6188	75	7	,	,	PUNCT
ejpam-6188	75	8	4	4	NUM
ejpam-6188	75	9	,	,	PUNCT
ejpam-6188	75	10	6	6	NUM
ejpam-6188	75	11	+	+	CCONJ
ejpam-6188	75	12	n+	n+	X
ejpam-6188	75	13	3⌊n−1	3⌊n−1	NUM
ejpam-6188	75	14	2	2	NUM
ejpam-6188	75	15	⌋+	⌋+	PUNCT
ejpam-6188	75	16	2⌈n−1	2⌈n−1	NUM
ejpam-6188	75	17	2	2	NUM
ejpam-6188	75	18	⌉	⌉	X
ejpam-6188	75	19	if	if	SCONJ
ejpam-6188	75	20	n	n	PROPN
ejpam-6188	75	21	=	=	SYM
ejpam-6188	75	22	5	5	NUM
ejpam-6188	75	23	,	,	PUNCT
ejpam-6188	75	24	6	6	NUM
ejpam-6188	75	25	,	,	PUNCT
ejpam-6188	75	26	44	44	NUM
ejpam-6188	75	27	if	if	SCONJ
ejpam-6188	75	28	n	n	NOUN
ejpam-6188	75	29	=	=	SYM
ejpam-6188	75	30	8	8	NUM
ejpam-6188	75	31	.	.	PUNCT
ejpam-6188	75	32	theorem	theorem	NOUN
ejpam-6188	75	33	3	3	NUM
ejpam-6188	75	34	.	.	PUNCT
ejpam-6188	76	1	(	(	PUNCT
ejpam-6188	76	2	[	[	X
ejpam-6188	76	3	1	1	NUM
ejpam-6188	76	4	]	]	NUM
ejpam-6188	76	5	)	)	PUNCT
ejpam-6188	76	6	.	.	PUNCT
ejpam-6188	77	1	the	the	DET
ejpam-6188	77	2	b	b	NOUN
ejpam-6188	77	3	-	-	PUNCT
ejpam-6188	77	4	chromatic	chromatic	ADJ
ejpam-6188	77	5	sum	sum	NOUN
ejpam-6188	77	6	of	of	ADP
ejpam-6188	77	7	mycielskian	mycielskian	NOUN
ejpam-6188	77	8	of	of	ADP
ejpam-6188	77	9	path	path	PROPN
ejpam-6188	77	10	µ(pn	µ(pn	NOUN
ejpam-6188	77	11	)	)	PUNCT
ejpam-6188	77	12	is	be	AUX
ejpam-6188	77	13	as	as	SCONJ
ejpam-6188	77	14	follows	follow	VERB
ejpam-6188	77	15	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	77	16	)	)	PUNCT
ejpam-6188	77	17	)	)	PUNCT
ejpam-6188	77	18	≤	≤	NUM
ejpam-6188	78	1			NUM
ejpam-6188	78	2	28	28	NUM
ejpam-6188	78	3	if	if	SCONJ
ejpam-6188	78	4	n	n	NOUN
ejpam-6188	78	5	=	=	SYM
ejpam-6188	78	6	7	7	NUM
ejpam-6188	78	7	,	,	PUNCT
ejpam-6188	78	8	48	48	NUM
ejpam-6188	78	9	if	if	SCONJ
ejpam-6188	78	10	n	n	NOUN
ejpam-6188	78	11	=	=	SYM
ejpam-6188	78	12	9	9	NUM
ejpam-6188	78	13	,	,	PUNCT
ejpam-6188	78	14	45	45	NUM
ejpam-6188	78	15	+	+	SYM
ejpam-6188	78	16	4⌈n−8	4⌈n−8	NUM
ejpam-6188	78	17	2	2	NUM
ejpam-6188	78	18	⌉+	⌉+	NUM
ejpam-6188	78	19	2⌊n−8	2⌊n−8	NUM
ejpam-6188	78	20	2	2	NUM
ejpam-6188	78	21	⌋	⌋	NOUN
ejpam-6188	78	22	if	if	SCONJ
ejpam-6188	78	23	n	n	PRON
ejpam-6188	78	24	≥	≥	NOUN
ejpam-6188	78	25	10	10	NUM
ejpam-6188	78	26	.	.	PUNCT
ejpam-6188	79	1	for	for	ADP
ejpam-6188	79	2	n	n	X
ejpam-6188	79	3	≥	≥	NUM
ejpam-6188	79	4	10	10	NUM
ejpam-6188	79	5	,	,	PUNCT
ejpam-6188	79	6	theorem	theorem	VERB
ejpam-6188	79	7	3	3	NUM
ejpam-6188	79	8	can	can	AUX
ejpam-6188	79	9	be	be	AUX
ejpam-6188	79	10	restated	restate	VERB
ejpam-6188	79	11	as	as	ADP
ejpam-6188	79	12	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	79	13	)	)	PUNCT
ejpam-6188	79	14	)	)	PUNCT
ejpam-6188	79	15	≤	≤	NOUN
ejpam-6188	79	16	{	{	PUNCT
ejpam-6188	79	17	3n+	3n+	NUM
ejpam-6188	79	18	21	21	NUM
ejpam-6188	79	19	if	if	SCONJ
ejpam-6188	79	20	n	n	PRON
ejpam-6188	79	21	is	be	AUX
ejpam-6188	79	22	even	even	ADV
ejpam-6188	79	23	,	,	PUNCT
ejpam-6188	79	24	3n+	3n+	ADV
ejpam-6188	79	25	22	22	NUM
ejpam-6188	79	26	if	if	SCONJ
ejpam-6188	79	27	n	n	ADJ
ejpam-6188	79	28	is	be	AUX
ejpam-6188	79	29	odd	odd	ADJ
ejpam-6188	79	30	.	.	PUNCT
ejpam-6188	80	1	in	in	ADP
ejpam-6188	80	2	this	this	DET
ejpam-6188	80	3	work	work	NOUN
ejpam-6188	80	4	,	,	PUNCT
ejpam-6188	80	5	we	we	PRON
ejpam-6188	80	6	improve	improve	VERB
ejpam-6188	80	7	theorem	theorem	VERB
ejpam-6188	80	8	3	3	NUM
ejpam-6188	80	9	by	by	ADP
ejpam-6188	80	10	giving	give	VERB
ejpam-6188	80	11	the	the	DET
ejpam-6188	80	12	value	value	NOUN
ejpam-6188	80	13	of	of	ADP
ejpam-6188	80	14	the	the	DET
ejpam-6188	80	15	b	b	NOUN
ejpam-6188	80	16	-	-	PUNCT
ejpam-6188	80	17	chromatic	chromatic	ADJ
ejpam-6188	80	18	sum	sum	NOUN
ejpam-6188	80	19	in	in	ADP
ejpam-6188	80	20	the	the	DET
ejpam-6188	80	21	case	case	NOUN
ejpam-6188	80	22	of	of	ADP
ejpam-6188	80	23	n	n	NOUN
ejpam-6188	80	24	=	=	SYM
ejpam-6188	80	25	7	7	NUM
ejpam-6188	80	26	,	,	PUNCT
ejpam-6188	80	27	9	9	NUM
ejpam-6188	80	28	and	and	CCONJ
ejpam-6188	80	29	n	n	PRON
ejpam-6188	80	30	≥	≥	NOUN
ejpam-6188	80	31	16	16	NUM
ejpam-6188	80	32	and	and	CCONJ
ejpam-6188	80	33	we	we	PRON
ejpam-6188	80	34	lower	lower	VERB
ejpam-6188	80	35	the	the	DET
ejpam-6188	80	36	upper	upper	ADJ
ejpam-6188	80	37	bound	bind	VERB
ejpam-6188	80	38	given	give	VERB
ejpam-6188	80	39	by	by	ADP
ejpam-6188	80	40	lisna	lisna	NOUN
ejpam-6188	80	41	and	and	CCONJ
ejpam-6188	80	42	sunitha	sunitha	VERB
ejpam-6188	80	43	in	in	ADP
ejpam-6188	80	44	the	the	DET
ejpam-6188	80	45	case	case	NOUN
ejpam-6188	80	46	when	when	SCONJ
ejpam-6188	80	47	10	10	NUM
ejpam-6188	80	48	≤	≤	NUM
ejpam-6188	80	49	n	n	PRON
ejpam-6188	80	50	≤	≤	NUM
ejpam-6188	80	51	15	15	NUM
ejpam-6188	80	52	.	.	PUNCT
ejpam-6188	81	1	we	we	PRON
ejpam-6188	81	2	also	also	ADV
ejpam-6188	81	3	give	give	VERB
ejpam-6188	81	4	an	an	DET
ejpam-6188	81	5	explicit	explicit	ADJ
ejpam-6188	81	6	coloring	coloring	NOUN
ejpam-6188	81	7	giving	give	VERB
ejpam-6188	81	8	the	the	DET
ejpam-6188	81	9	sum	sum	NOUN
ejpam-6188	81	10	of	of	ADP
ejpam-6188	81	11	the	the	DET
ejpam-6188	81	12	colors	color	NOUN
ejpam-6188	81	13	.	.	PUNCT
ejpam-6188	82	1	to	to	PART
ejpam-6188	82	2	be	be	AUX
ejpam-6188	82	3	more	more	ADV
ejpam-6188	82	4	precise	precise	ADJ
ejpam-6188	82	5	,	,	PUNCT
ejpam-6188	82	6	we	we	PRON
ejpam-6188	82	7	show	show	VERB
ejpam-6188	82	8	that	that	SCONJ
ejpam-6188	82	9	φ′(µ(p7	φ′(µ(p7	VERB
ejpam-6188	82	10	)	)	PUNCT
ejpam-6188	82	11	)	)	PUNCT
ejpam-6188	83	1	=	=	SYM
ejpam-6188	83	2	27	27	NUM
ejpam-6188	83	3	,	,	PUNCT
ejpam-6188	83	4	φ′(µ(p9	φ′(µ(p9	NUM
ejpam-6188	83	5	)	)	PUNCT
ejpam-6188	83	6	)	)	PUNCT
ejpam-6188	84	1	=	=	SYM
ejpam-6188	84	2	46	46	NUM
ejpam-6188	84	3	,	,	PUNCT
ejpam-6188	84	4	3n+	3n+	NUM
ejpam-6188	84	5	18	18	NUM
ejpam-6188	84	6	≤	≤	NUM
ejpam-6188	84	7	φ′(µ(pn	φ′(µ(pn	NUM
ejpam-6188	84	8	)	)	PUNCT
ejpam-6188	84	9	)	)	PUNCT
ejpam-6188	84	10	≤	≤	NUM
ejpam-6188	85	1	3n+	3n+	NUM
ejpam-6188	85	2	19	19	NUM
ejpam-6188	85	3	for	for	ADP
ejpam-6188	85	4	10	10	NUM
ejpam-6188	85	5	≤	≤	NOUN
ejpam-6188	85	6	n	n	PRON
ejpam-6188	85	7	≤	≤	NUM
ejpam-6188	85	8	15	15	NUM
ejpam-6188	85	9	,	,	PUNCT
ejpam-6188	85	10	and	and	CCONJ
ejpam-6188	85	11	φ′(µ(pn	φ′(µ(pn	NUM
ejpam-6188	85	12	)	)	PUNCT
ejpam-6188	85	13	)	)	PUNCT
ejpam-6188	86	1	=	=	PUNCT
ejpam-6188	86	2	3n+	3n+	NUM
ejpam-6188	86	3	19	19	NUM
ejpam-6188	86	4	for	for	ADP
ejpam-6188	86	5	n	n	X
ejpam-6188	86	6	≥	≥	NOUN
ejpam-6188	86	7	16	16	NUM
ejpam-6188	86	8	.	.	PUNCT
ejpam-6188	87	1	p.	p.	NOUN
ejpam-6188	87	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	87	3	et	et	PROPN
ejpam-6188	87	4	al	al	PROPN
ejpam-6188	87	5	.	.	PUNCT
ejpam-6188	87	6	/	/	SYM
ejpam-6188	87	7	eur	eur	PROPN
ejpam-6188	87	8	.	.	PUNCT
ejpam-6188	88	1	j.	j.	PROPN
ejpam-6188	88	2	pure	pure	PROPN
ejpam-6188	88	3	appl	appl	PROPN
ejpam-6188	88	4	.	.	PROPN
ejpam-6188	88	5	math	math	PROPN
ejpam-6188	88	6	,	,	PUNCT
ejpam-6188	88	7	18	18	NUM
ejpam-6188	88	8	(	(	PUNCT
ejpam-6188	88	9	3	3	NUM
ejpam-6188	88	10	)	)	PUNCT
ejpam-6188	88	11	(	(	PUNCT
ejpam-6188	88	12	2025	2025	NUM
ejpam-6188	88	13	)	)	PUNCT
ejpam-6188	88	14	,	,	PUNCT
ejpam-6188	88	15	6188	6188	NUM
ejpam-6188	88	16	4	4	NUM
ejpam-6188	88	17	of	of	ADP
ejpam-6188	88	18	13	13	NUM
ejpam-6188	88	19	3	3	NUM
ejpam-6188	88	20	.	.	PUNCT
ejpam-6188	88	21	main	main	ADJ
ejpam-6188	88	22	results	result	NOUN
ejpam-6188	88	23	in	in	ADP
ejpam-6188	88	24	[	[	X
ejpam-6188	88	25	1	1	NUM
ejpam-6188	88	26	]	]	PUNCT
ejpam-6188	89	1	,	,	PUNCT
ejpam-6188	89	2	they	they	PRON
ejpam-6188	89	3	gave	give	VERB
ejpam-6188	89	4	the	the	DET
ejpam-6188	89	5	b	b	NOUN
ejpam-6188	89	6	-	-	PUNCT
ejpam-6188	89	7	chromatic	chromatic	ADJ
ejpam-6188	89	8	sum	sum	NOUN
ejpam-6188	89	9	of	of	ADP
ejpam-6188	89	10	µ(p8	µ(p8	NOUN
ejpam-6188	89	11	)	)	PUNCT
ejpam-6188	89	12	by	by	ADP
ejpam-6188	89	13	giving	give	VERB
ejpam-6188	89	14	a	a	DET
ejpam-6188	89	15	b	b	NOUN
ejpam-6188	89	16	-	-	PUNCT
ejpam-6188	89	17	coloring	coloring	NOUN
ejpam-6188	89	18	that	that	PRON
ejpam-6188	89	19	achieves	achieve	VERB
ejpam-6188	89	20	such	such	ADJ
ejpam-6188	89	21	value	value	NOUN
ejpam-6188	89	22	and	and	CCONJ
ejpam-6188	89	23	showed	show	VERB
ejpam-6188	89	24	the	the	DET
ejpam-6188	89	25	minimality	minimality	NOUN
ejpam-6188	89	26	by	by	ADP
ejpam-6188	89	27	counting	count	VERB
ejpam-6188	89	28	the	the	DET
ejpam-6188	89	29	number	number	NOUN
ejpam-6188	89	30	of	of	ADP
ejpam-6188	89	31	vertices	vertex	NOUN
ejpam-6188	89	32	in	in	ADP
ejpam-6188	89	33	each	each	DET
ejpam-6188	89	34	color	color	NOUN
ejpam-6188	89	35	class	class	NOUN
ejpam-6188	89	36	.	.	PUNCT
ejpam-6188	90	1	then	then	ADV
ejpam-6188	90	2	they	they	PRON
ejpam-6188	90	3	adjusted	adjust	VERB
ejpam-6188	90	4	such	such	ADJ
ejpam-6188	90	5	coloring	coloring	NOUN
ejpam-6188	90	6	to	to	PART
ejpam-6188	90	7	be	be	AUX
ejpam-6188	90	8	cn	cn	ADJ
ejpam-6188	90	9	for	for	ADP
ejpam-6188	90	10	n	n	X
ejpam-6188	90	11	≥	≥	NUM
ejpam-6188	90	12	9	9	NUM
ejpam-6188	90	13	.	.	PUNCT
ejpam-6188	91	1	however	however	ADV
ejpam-6188	91	2	,	,	PUNCT
ejpam-6188	91	3	the	the	DET
ejpam-6188	91	4	adjusted	adjusted	ADJ
ejpam-6188	91	5	coloring	coloring	NOUN
ejpam-6188	91	6	does	do	AUX
ejpam-6188	91	7	not	not	PART
ejpam-6188	91	8	yield	yield	VERB
ejpam-6188	91	9	the	the	DET
ejpam-6188	91	10	b	b	NOUN
ejpam-6188	91	11	-	-	PUNCT
ejpam-6188	91	12	chromatic	chromatic	ADJ
ejpam-6188	91	13	sum	sum	NOUN
ejpam-6188	91	14	.	.	PUNCT
ejpam-6188	92	1	in	in	ADP
ejpam-6188	92	2	this	this	DET
ejpam-6188	92	3	section	section	NOUN
ejpam-6188	92	4	,	,	PUNCT
ejpam-6188	92	5	we	we	PRON
ejpam-6188	92	6	improve	improve	VERB
ejpam-6188	92	7	the	the	DET
ejpam-6188	92	8	result	result	NOUN
ejpam-6188	92	9	of	of	ADP
ejpam-6188	92	10	lisna	lisna	NOUN
ejpam-6188	92	11	and	and	CCONJ
ejpam-6188	92	12	sunitha	sunitha	VERB
ejpam-6188	93	1	[	[	X
ejpam-6188	93	2	1	1	NUM
ejpam-6188	93	3	]	]	PUNCT
ejpam-6188	93	4	.	.	PUNCT
ejpam-6188	94	1	we	we	PRON
ejpam-6188	94	2	give	give	VERB
ejpam-6188	94	3	b	b	NOUN
ejpam-6188	94	4	-	-	PUNCT
ejpam-6188	94	5	chromatic	chromatic	ADJ
ejpam-6188	94	6	sum	sum	NOUN
ejpam-6188	94	7	of	of	ADP
ejpam-6188	94	8	µ(pn	µ(pn	NOUN
ejpam-6188	94	9	)	)	PUNCT
ejpam-6188	94	10	when	when	SCONJ
ejpam-6188	94	11	n	n	X
ejpam-6188	94	12	=	=	SYM
ejpam-6188	94	13	7	7	NUM
ejpam-6188	94	14	,	,	PUNCT
ejpam-6188	94	15	9	9	NUM
ejpam-6188	94	16	and	and	CCONJ
ejpam-6188	94	17	n	n	PRON
ejpam-6188	94	18	≥	≥	NOUN
ejpam-6188	94	19	16	16	NUM
ejpam-6188	94	20	,	,	PUNCT
ejpam-6188	94	21	and	and	CCONJ
ejpam-6188	94	22	its	its	PRON
ejpam-6188	94	23	bound	bind	VERB
ejpam-6188	94	24	when	when	SCONJ
ejpam-6188	94	25	10	10	NUM
ejpam-6188	94	26	≤	≤	NUM
ejpam-6188	94	27	n	n	PRON
ejpam-6188	94	28	≤	≤	NUM
ejpam-6188	94	29	15	15	NUM
ejpam-6188	94	30	.	.	PUNCT
ejpam-6188	95	1	we	we	PRON
ejpam-6188	95	2	give	give	VERB
ejpam-6188	95	3	the	the	DET
ejpam-6188	95	4	b	b	NOUN
ejpam-6188	95	5	-	-	PUNCT
ejpam-6188	95	6	chromatic	chromatic	ADJ
ejpam-6188	95	7	sum	sum	NOUN
ejpam-6188	95	8	of	of	ADP
ejpam-6188	95	9	µ(p7	µ(p7	NOUN
ejpam-6188	95	10	)	)	PUNCT
ejpam-6188	95	11	in	in	ADP
ejpam-6188	95	12	theorem	theorem	NOUN
ejpam-6188	95	13	4	4	NUM
ejpam-6188	95	14	.	.	PUNCT
ejpam-6188	96	1	then	then	ADV
ejpam-6188	96	2	we	we	PRON
ejpam-6188	96	3	investigate	investigate	VERB
ejpam-6188	96	4	several	several	ADJ
ejpam-6188	96	5	properties	property	NOUN
ejpam-6188	96	6	of	of	ADP
ejpam-6188	96	7	µ(pn	µ(pn	NOUN
ejpam-6188	96	8	)	)	PUNCT
ejpam-6188	96	9	and	and	CCONJ
ejpam-6188	96	10	its	its	PRON
ejpam-6188	96	11	colorings	coloring	NOUN
ejpam-6188	96	12	that	that	PRON
ejpam-6188	96	13	lead	lead	VERB
ejpam-6188	96	14	to	to	ADP
ejpam-6188	96	15	the	the	DET
ejpam-6188	96	16	bound	bind	VERB
ejpam-6188	96	17	and	and	CCONJ
ejpam-6188	96	18	the	the	DET
ejpam-6188	96	19	exact	exact	ADJ
ejpam-6188	96	20	value	value	NOUN
ejpam-6188	96	21	of	of	ADP
ejpam-6188	96	22	the	the	DET
ejpam-6188	96	23	b	b	NOUN
ejpam-6188	96	24	-	-	PUNCT
ejpam-6188	96	25	chromatic	chromatic	ADJ
ejpam-6188	96	26	sum	sum	NOUN
ejpam-6188	96	27	as	as	SCONJ
ejpam-6188	96	28	mentioned	mention	VERB
ejpam-6188	96	29	.	.	PUNCT
ejpam-6188	97	1	theorem	theorem	VERB
ejpam-6188	97	2	4	4	NUM
ejpam-6188	97	3	.	.	NUM
ejpam-6188	97	4	φ′(µ(p7	φ′(µ(p7	NOUN
ejpam-6188	97	5	)	)	PUNCT
ejpam-6188	97	6	)	)	PUNCT
ejpam-6188	98	1	=	=	SYM
ejpam-6188	98	2	27	27	NUM
ejpam-6188	98	3	.	.	PUNCT
ejpam-6188	99	1	proof	proof	NOUN
ejpam-6188	99	2	.	.	PUNCT
ejpam-6188	100	1	by	by	ADP
ejpam-6188	100	2	theorem	theorem	NOUN
ejpam-6188	100	3	1	1	NUM
ejpam-6188	100	4	,	,	PUNCT
ejpam-6188	100	5	we	we	PRON
ejpam-6188	100	6	have	have	VERB
ejpam-6188	100	7	φ(µ(p7	φ(µ(p7	NOUN
ejpam-6188	100	8	)	)	PUNCT
ejpam-6188	100	9	)	)	PUNCT
ejpam-6188	101	1	=	=	SYM
ejpam-6188	101	2	4	4	X
ejpam-6188	101	3	.	.	PUNCT
ejpam-6188	101	4	let	let	VERB
ejpam-6188	101	5	c	c	NOUN
ejpam-6188	101	6	:	:	PUNCT
ejpam-6188	101	7	v	v	X
ejpam-6188	101	8	(	(	PUNCT
ejpam-6188	101	9	µ(p7	µ(p7	NOUN
ejpam-6188	101	10	)	)	PUNCT
ejpam-6188	101	11	)	)	PUNCT
ejpam-6188	101	12	→	→	PUNCT
ejpam-6188	101	13	{	{	PUNCT
ejpam-6188	101	14	1	1	NUM
ejpam-6188	101	15	,	,	PUNCT
ejpam-6188	101	16	2	2	NUM
ejpam-6188	101	17	,	,	PUNCT
ejpam-6188	101	18	3	3	NUM
ejpam-6188	101	19	,	,	PUNCT
ejpam-6188	101	20	4	4	NUM
ejpam-6188	101	21	}	}	PUNCT
ejpam-6188	101	22	be	be	AUX
ejpam-6188	101	23	a	a	DET
ejpam-6188	101	24	b	b	NOUN
ejpam-6188	101	25	-	-	PUNCT
ejpam-6188	101	26	coloring	coloring	NOUN
ejpam-6188	101	27	giving	give	VERB
ejpam-6188	101	28	a	a	DET
ejpam-6188	101	29	b	b	NOUN
ejpam-6188	101	30	-	-	PUNCT
ejpam-6188	101	31	chromatic	chromatic	ADJ
ejpam-6188	101	32	number	number	NOUN
ejpam-6188	101	33	of	of	ADP
ejpam-6188	101	34	µ(p7	µ(p7	PROPN
ejpam-6188	101	35	)	)	PUNCT
ejpam-6188	101	36	.	.	PUNCT
ejpam-6188	102	1	we	we	PRON
ejpam-6188	102	2	first	first	ADV
ejpam-6188	102	3	show	show	VERB
ejpam-6188	102	4	that	that	SCONJ
ejpam-6188	102	5	there	there	PRON
ejpam-6188	102	6	are	be	VERB
ejpam-6188	102	7	at	at	ADP
ejpam-6188	102	8	most	most	ADJ
ejpam-6188	102	9	7	7	NUM
ejpam-6188	102	10	vertices	vertex	NOUN
ejpam-6188	102	11	with	with	ADP
ejpam-6188	102	12	color	color	NOUN
ejpam-6188	102	13	1	1	NUM
ejpam-6188	102	14	.	.	PUNCT
ejpam-6188	102	15	suppose	suppose	VERB
ejpam-6188	102	16	to	to	ADP
ejpam-6188	102	17	the	the	DET
ejpam-6188	102	18	contrary	contrary	NOUN
ejpam-6188	102	19	that	that	SCONJ
ejpam-6188	102	20	there	there	PRON
ejpam-6188	102	21	exists	exist	VERB
ejpam-6188	102	22	the	the	DET
ejpam-6188	102	23	b	b	NOUN
ejpam-6188	102	24	-	-	PUNCT
ejpam-6188	102	25	coloring	coloring	NOUN
ejpam-6188	102	26	giving	give	VERB
ejpam-6188	102	27	a	a	DET
ejpam-6188	102	28	b	b	NOUN
ejpam-6188	102	29	-	-	PUNCT
ejpam-6188	102	30	chromatic	chromatic	ADJ
ejpam-6188	102	31	number	number	NOUN
ejpam-6188	102	32	with	with	ADP
ejpam-6188	102	33	at	at	ADV
ejpam-6188	102	34	least	least	ADJ
ejpam-6188	102	35	8	8	NUM
ejpam-6188	102	36	vertices	vertex	NOUN
ejpam-6188	102	37	of	of	ADP
ejpam-6188	102	38	color	color	NOUN
ejpam-6188	102	39	1	1	NUM
ejpam-6188	102	40	.	.	PUNCT
ejpam-6188	103	1	there	there	PRON
ejpam-6188	103	2	are	be	VERB
ejpam-6188	103	3	at	at	ADV
ejpam-6188	103	4	least	least	ADJ
ejpam-6188	103	5	three	three	NUM
ejpam-6188	103	6	b	b	X
ejpam-6188	103	7	-	-	PUNCT
ejpam-6188	103	8	dominating	dominating	NOUN
ejpam-6188	103	9	vertices	vertex	NOUN
ejpam-6188	103	10	of	of	ADP
ejpam-6188	103	11	colors	color	NOUN
ejpam-6188	103	12	other	other	ADJ
ejpam-6188	103	13	than	than	ADP
ejpam-6188	103	14	1	1	NUM
ejpam-6188	103	15	.	.	PUNCT
ejpam-6188	104	1	if	if	SCONJ
ejpam-6188	104	2	c(u	c(u	PROPN
ejpam-6188	104	3	)	)	PUNCT
ejpam-6188	105	1	=	=	SYM
ejpam-6188	105	2	1	1	NUM
ejpam-6188	105	3	,	,	PUNCT
ejpam-6188	105	4	then	then	ADV
ejpam-6188	105	5	1	1	NUM
ejpam-6188	105	6	/∈	/∈	NOUN
ejpam-6188	105	7	c(u7	c(u7	PROPN
ejpam-6188	105	8	)	)	PUNCT
ejpam-6188	105	9	.	.	PUNCT
ejpam-6188	106	1	hence	hence	ADV
ejpam-6188	106	2	|c1|	|c1|	VERB
ejpam-6188	106	3	≤	≤	NOUN
ejpam-6188	106	4	⌈	⌈	NUM
ejpam-6188	106	5	|v	|v	X
ejpam-6188	106	6	(	(	PUNCT
ejpam-6188	106	7	p7)|	p7)|	NOUN
ejpam-6188	106	8	2	2	NUM
ejpam-6188	106	9	⌉	⌉	NOUN
ejpam-6188	106	10	+	+	CCONJ
ejpam-6188	106	11	1	1	NUM
ejpam-6188	106	12	=	=	SYM
ejpam-6188	106	13	5	5	NUM
ejpam-6188	106	14	is	be	AUX
ejpam-6188	106	15	a	a	DET
ejpam-6188	106	16	contradiction	contradiction	NOUN
ejpam-6188	106	17	.	.	PUNCT
ejpam-6188	107	1	now	now	ADV
ejpam-6188	107	2	,	,	PUNCT
ejpam-6188	107	3	we	we	PRON
ejpam-6188	107	4	suppose	suppose	VERB
ejpam-6188	107	5	that	that	SCONJ
ejpam-6188	107	6	c(u	c(u	PROPN
ejpam-6188	107	7	)	)	PUNCT
ejpam-6188	107	8	̸=	̸=	PROPN
ejpam-6188	107	9	1	1	NUM
ejpam-6188	107	10	.	.	PUNCT
ejpam-6188	108	1	since	since	SCONJ
ejpam-6188	108	2	there	there	PRON
ejpam-6188	108	3	are	be	VERB
ejpam-6188	108	4	at	at	ADP
ejpam-6188	108	5	most	most	ADJ
ejpam-6188	108	6	4	4	NUM
ejpam-6188	108	7	vertices	vertex	NOUN
ejpam-6188	108	8	of	of	ADP
ejpam-6188	108	9	color	color	NOUN
ejpam-6188	108	10	1	1	NUM
ejpam-6188	108	11	in	in	ADP
ejpam-6188	108	12	v	v	NUM
ejpam-6188	108	13	(	(	PUNCT
ejpam-6188	108	14	p7	p7	PROPN
ejpam-6188	108	15	)	)	PUNCT
ejpam-6188	108	16	,	,	PUNCT
ejpam-6188	108	17	there	there	PRON
ejpam-6188	108	18	are	be	VERB
ejpam-6188	108	19	at	at	ADV
ejpam-6188	108	20	least	least	ADJ
ejpam-6188	108	21	4	4	NUM
ejpam-6188	108	22	vertices	vertex	NOUN
ejpam-6188	108	23	of	of	ADP
ejpam-6188	108	24	color	color	NOUN
ejpam-6188	108	25	1	1	NUM
ejpam-6188	108	26	in	in	ADP
ejpam-6188	108	27	u7	u7	PROPN
ejpam-6188	108	28	.	.	PUNCT
ejpam-6188	109	1	if	if	SCONJ
ejpam-6188	109	2	|c1	|c1	NOUN
ejpam-6188	109	3	∩	∩	ADJ
ejpam-6188	109	4	u7|	u7|	NOUN
ejpam-6188	109	5	=	=	SYM
ejpam-6188	109	6	4	4	NUM
ejpam-6188	109	7	,	,	PUNCT
ejpam-6188	109	8	then	then	ADV
ejpam-6188	109	9	|c1	|c1	VERB
ejpam-6188	109	10	∩	∩	ADJ
ejpam-6188	109	11	v	v	ADP
ejpam-6188	109	12	(	(	PUNCT
ejpam-6188	109	13	p7)|	p7)|	NOUN
ejpam-6188	109	14	=	=	NOUN
ejpam-6188	109	15	4	4	X
ejpam-6188	109	16	.	.	PUNCT
ejpam-6188	110	1	it	it	PRON
ejpam-6188	110	2	follows	follow	VERB
ejpam-6188	110	3	that	that	DET
ejpam-6188	110	4	c1	c1	PROPN
ejpam-6188	110	5	=	=	PROPN
ejpam-6188	110	6	{	{	PUNCT
ejpam-6188	110	7	v1	v1	PROPN
ejpam-6188	110	8	,	,	PUNCT
ejpam-6188	110	9	v3	v3	PROPN
ejpam-6188	110	10	,	,	PUNCT
ejpam-6188	110	11	v5	v5	PROPN
ejpam-6188	110	12	,	,	PUNCT
ejpam-6188	110	13	v7	v7	NUM
ejpam-6188	110	14	,	,	PUNCT
ejpam-6188	110	15	u1	u1	NOUN
ejpam-6188	110	16	,	,	PUNCT
ejpam-6188	110	17	u3	u3	PROPN
ejpam-6188	110	18	,	,	PUNCT
ejpam-6188	110	19	u5	u5	PROPN
ejpam-6188	110	20	,	,	PUNCT
ejpam-6188	110	21	u7	u7	PROPN
ejpam-6188	110	22	}	}	PUNCT
ejpam-6188	110	23	.	.	PUNCT
ejpam-6188	111	1	hence	hence	ADV
ejpam-6188	111	2	,	,	PUNCT
ejpam-6188	111	3	the	the	DET
ejpam-6188	111	4	vertex	vertex	NOUN
ejpam-6188	111	5	u	u	NOUN
ejpam-6188	111	6	is	be	AUX
ejpam-6188	111	7	the	the	DET
ejpam-6188	111	8	only	only	ADJ
ejpam-6188	111	9	possible	possible	ADJ
ejpam-6188	111	10	b	b	X
ejpam-6188	111	11	-	-	PUNCT
ejpam-6188	111	12	dominating	dominate	VERB
ejpam-6188	111	13	vertex	vertex	NOUN
ejpam-6188	111	14	of	of	ADP
ejpam-6188	111	15	color	color	NOUN
ejpam-6188	111	16	other	other	ADJ
ejpam-6188	111	17	than	than	ADP
ejpam-6188	111	18	1	1	NUM
ejpam-6188	111	19	,	,	PUNCT
ejpam-6188	111	20	which	which	PRON
ejpam-6188	111	21	is	be	AUX
ejpam-6188	111	22	not	not	PART
ejpam-6188	111	23	possible	possible	ADJ
ejpam-6188	111	24	.	.	PUNCT
ejpam-6188	112	1	for	for	ADP
ejpam-6188	112	2	j	j	PROPN
ejpam-6188	112	3	=	=	SYM
ejpam-6188	112	4	1	1	NUM
ejpam-6188	112	5	,	,	PUNCT
ejpam-6188	112	6	2	2	NUM
ejpam-6188	112	7	,	,	PUNCT
ejpam-6188	112	8	3	3	NUM
ejpam-6188	112	9	,	,	PUNCT
ejpam-6188	112	10	if	if	SCONJ
ejpam-6188	112	11	|c1	|c1	NOUN
ejpam-6188	112	12	∩	∩	ADJ
ejpam-6188	112	13	u7|	u7|	NOUN
ejpam-6188	112	14	=	=	SYM
ejpam-6188	112	15	4	4	NUM
ejpam-6188	112	16	+	+	SYM
ejpam-6188	112	17	j	j	NOUN
ejpam-6188	112	18	,	,	PUNCT
ejpam-6188	112	19	then	then	ADV
ejpam-6188	112	20	|np7(c1	|np7(c1	NOUN
ejpam-6188	112	21	∩	∩	ADJ
ejpam-6188	112	22	u7)|	u7)|	PROPN
ejpam-6188	112	23	≥	≥	NOUN
ejpam-6188	112	24	|c1	|c1	NOUN
ejpam-6188	112	25	∩	∩	ADJ
ejpam-6188	112	26	u7|	u7|	NOUN
ejpam-6188	112	27	≥	≥	NOUN
ejpam-6188	112	28	4	4	NUM
ejpam-6188	112	29	+	+	CCONJ
ejpam-6188	112	30	j.	j.	PROPN
ejpam-6188	112	31	thus	thus	ADV
ejpam-6188	112	32	,	,	PUNCT
ejpam-6188	112	33	|v	|v	PROPN
ejpam-6188	112	34	(	(	PUNCT
ejpam-6188	112	35	p7	p7	PROPN
ejpam-6188	112	36	)	)	PUNCT
ejpam-6188	112	37	\nµ(p7)(c1	\nµ(p7)(c1	ADP
ejpam-6188	112	38	∩	∩	ADJ
ejpam-6188	112	39	u7)|	u7)|	NOUN
ejpam-6188	112	40	≤	≤	NUM
ejpam-6188	112	41	3	3	NUM
ejpam-6188	112	42	−	−	PROPN
ejpam-6188	112	43	j	j	NOUN
ejpam-6188	112	44	which	which	PRON
ejpam-6188	112	45	is	be	AUX
ejpam-6188	112	46	not	not	PART
ejpam-6188	112	47	enough	enough	ADJ
ejpam-6188	112	48	to	to	PART
ejpam-6188	112	49	complete	complete	VERB
ejpam-6188	112	50	the	the	DET
ejpam-6188	112	51	color	color	NOUN
ejpam-6188	112	52	1	1	NUM
ejpam-6188	112	53	.	.	PUNCT
ejpam-6188	112	54	therefore	therefore	ADV
ejpam-6188	112	55	|c1|	|c1|	VERB
ejpam-6188	112	56	≤	≤	NOUN
ejpam-6188	112	57	7	7	NUM
ejpam-6188	112	58	.	.	PUNCT
ejpam-6188	113	1	next	next	ADV
ejpam-6188	113	2	,	,	PUNCT
ejpam-6188	113	3	we	we	PRON
ejpam-6188	113	4	show	show	VERB
ejpam-6188	113	5	that	that	SCONJ
ejpam-6188	113	6	if	if	SCONJ
ejpam-6188	113	7	|c4|	|c4|	NOUN
ejpam-6188	113	8	=	=	SYM
ejpam-6188	113	9	1	1	NUM
ejpam-6188	113	10	,	,	PUNCT
ejpam-6188	113	11	then	then	ADV
ejpam-6188	113	12	|c3|	|c3|	ADJ
ejpam-6188	113	13	≥	≥	NOUN
ejpam-6188	113	14	2	2	X
ejpam-6188	113	15	.	.	PUNCT
ejpam-6188	113	16	suppose	suppose	VERB
ejpam-6188	113	17	to	to	ADP
ejpam-6188	113	18	the	the	DET
ejpam-6188	113	19	contrary	contrary	NOUN
ejpam-6188	113	20	that	that	PRON
ejpam-6188	113	21	|c3|	|c3|	NOUN
ejpam-6188	113	22	=	=	NOUN
ejpam-6188	113	23	1	1	X
ejpam-6188	113	24	.	.	PUNCT
ejpam-6188	114	1	thus	thus	ADV
ejpam-6188	114	2	,	,	PUNCT
ejpam-6188	114	3	the	the	DET
ejpam-6188	114	4	vertices	vertex	NOUN
ejpam-6188	114	5	with	with	ADP
ejpam-6188	114	6	color	color	NOUN
ejpam-6188	114	7	3	3	NUM
ejpam-6188	114	8	and	and	CCONJ
ejpam-6188	114	9	4	4	NUM
ejpam-6188	114	10	are	be	AUX
ejpam-6188	114	11	both	both	PRON
ejpam-6188	114	12	b	b	X
ejpam-6188	114	13	-	-	PUNCT
ejpam-6188	114	14	dominating	dominating	NOUN
ejpam-6188	114	15	vertices	vertex	NOUN
ejpam-6188	114	16	and	and	CCONJ
ejpam-6188	114	17	are	be	AUX
ejpam-6188	114	18	adjacent	adjacent	ADJ
ejpam-6188	114	19	.	.	PUNCT
ejpam-6188	115	1	since	since	SCONJ
ejpam-6188	115	2	µ(p7	µ(p7	PROPN
ejpam-6188	115	3	)	)	PUNCT
ejpam-6188	115	4	is	be	AUX
ejpam-6188	115	5	triangle	triangle	NOUN
ejpam-6188	115	6	-	-	PUNCT
ejpam-6188	115	7	free	free	ADJ
ejpam-6188	115	8	,	,	PUNCT
ejpam-6188	115	9	there	there	PRON
ejpam-6188	115	10	is	be	VERB
ejpam-6188	115	11	no	no	DET
ejpam-6188	115	12	b	b	NOUN
ejpam-6188	115	13	-	-	PUNCT
ejpam-6188	115	14	dominating	dominating	NOUN
ejpam-6188	115	15	vertices	vertex	NOUN
ejpam-6188	115	16	for	for	ADP
ejpam-6188	115	17	colors	color	NOUN
ejpam-6188	115	18	1	1	NUM
ejpam-6188	115	19	and	and	CCONJ
ejpam-6188	115	20	2	2	NUM
ejpam-6188	115	21	,	,	PUNCT
ejpam-6188	115	22	a	a	DET
ejpam-6188	115	23	contradiction	contradiction	NOUN
ejpam-6188	115	24	.	.	PUNCT
ejpam-6188	116	1	thus	thus	ADV
ejpam-6188	116	2	,	,	PUNCT
ejpam-6188	116	3	if	if	SCONJ
ejpam-6188	116	4	|c4|	|c4|	NOUN
ejpam-6188	116	5	=	=	SYM
ejpam-6188	116	6	1	1	NUM
ejpam-6188	116	7	,	,	PUNCT
ejpam-6188	116	8	then	then	ADV
ejpam-6188	116	9	|c3|	|c3|	ADJ
ejpam-6188	116	10	≥	≥	NOUN
ejpam-6188	116	11	2	2	X
ejpam-6188	116	12	.	.	PUNCT
ejpam-6188	117	1	if	if	SCONJ
ejpam-6188	117	2	|c4|	|c4|	NOUN
ejpam-6188	117	3	=	=	SYM
ejpam-6188	117	4	1	1	NUM
ejpam-6188	117	5	,	,	PUNCT
ejpam-6188	117	6	then	then	ADV
ejpam-6188	117	7	∑	∑	PUNCT
ejpam-6188	117	8	v∈v	v∈v	PROPN
ejpam-6188	117	9	(	(	PUNCT
ejpam-6188	117	10	µ(p7	µ(p7	NOUN
ejpam-6188	117	11	)	)	PUNCT
ejpam-6188	117	12	)	)	PUNCT
ejpam-6188	117	13	c(v	c(v	PROPN
ejpam-6188	117	14	)	)	PUNCT
ejpam-6188	117	15	≥	≥	NOUN
ejpam-6188	117	16	4|c4|+	4|c4|+	NUM
ejpam-6188	117	17	3|c3|+	3|c3|+	NUM
ejpam-6188	117	18	2|c2|+	2|c2|+	PROPN
ejpam-6188	117	19	|c1|	|c1|	PROPN
ejpam-6188	117	20	≥	≥	NUM
ejpam-6188	117	21	4	4	NUM
ejpam-6188	117	22	·	·	SYM
ejpam-6188	117	23	1	1	NUM
ejpam-6188	117	24	+	+	CCONJ
ejpam-6188	117	25	3	3	NUM
ejpam-6188	117	26	·	·	SYM
ejpam-6188	117	27	2	2	NUM
ejpam-6188	117	28	+	+	CCONJ
ejpam-6188	117	29	2	2	NUM
ejpam-6188	117	30	·	·	SYM
ejpam-6188	117	31	5	5	NUM
ejpam-6188	117	32	+	+	CCONJ
ejpam-6188	117	33	1	1	NUM
ejpam-6188	117	34	·	·	SYM
ejpam-6188	117	35	7	7	NUM
ejpam-6188	117	36	=	=	SYM
ejpam-6188	117	37	27	27	NUM
ejpam-6188	117	38	.	.	PUNCT
ejpam-6188	118	1	if	if	SCONJ
ejpam-6188	118	2	|c4|	|c4|	NOUN
ejpam-6188	118	3	≥	≥	NOUN
ejpam-6188	118	4	2	2	NUM
ejpam-6188	118	5	,	,	PUNCT
ejpam-6188	118	6	then	then	ADV
ejpam-6188	118	7	∑	∑	PUNCT
ejpam-6188	118	8	v∈v	v∈v	PROPN
ejpam-6188	118	9	(	(	PUNCT
ejpam-6188	118	10	µ(p7	µ(p7	NOUN
ejpam-6188	118	11	)	)	PUNCT
ejpam-6188	118	12	)	)	PUNCT
ejpam-6188	118	13	c(v	c(v	PROPN
ejpam-6188	118	14	)	)	PUNCT
ejpam-6188	118	15	≥	≥	NOUN
ejpam-6188	118	16	4	4	NUM
ejpam-6188	118	17	·	·	SYM
ejpam-6188	118	18	2	2	NUM
ejpam-6188	119	1	+	+	CCONJ
ejpam-6188	119	2	3	3	NUM
ejpam-6188	119	3	·	·	SYM
ejpam-6188	119	4	1	1	NUM
ejpam-6188	120	1	+	+	CCONJ
ejpam-6188	120	2	2	2	NUM
ejpam-6188	120	3	·	·	SYM
ejpam-6188	120	4	5	5	NUM
ejpam-6188	120	5	+	+	CCONJ
ejpam-6188	120	6	1	1	NUM
ejpam-6188	120	7	·	·	SYM
ejpam-6188	120	8	7	7	NUM
ejpam-6188	120	9	=	=	SYM
ejpam-6188	120	10	28	28	NUM
ejpam-6188	120	11	.	.	PUNCT
ejpam-6188	121	1	thus	thus	ADV
ejpam-6188	121	2	,	,	PUNCT
ejpam-6188	121	3	φ′(µ(p7	φ′(µ(p7	NOUN
ejpam-6188	121	4	)	)	PUNCT
ejpam-6188	121	5	)	)	PUNCT
ejpam-6188	121	6	≥	≥	NOUN
ejpam-6188	121	7	27	27	NUM
ejpam-6188	121	8	.	.	PUNCT
ejpam-6188	122	1	next	next	ADV
ejpam-6188	122	2	,	,	PUNCT
ejpam-6188	122	3	we	we	PRON
ejpam-6188	122	4	give	give	VERB
ejpam-6188	122	5	a	a	DET
ejpam-6188	122	6	b	b	NOUN
ejpam-6188	122	7	-	-	PUNCT
ejpam-6188	122	8	coloring	coloring	NOUN
ejpam-6188	122	9	giving	give	VERB
ejpam-6188	122	10	the	the	DET
ejpam-6188	122	11	b	b	NOUN
ejpam-6188	122	12	-	-	PUNCT
ejpam-6188	122	13	chromatic	chromatic	ADJ
ejpam-6188	122	14	sum	sum	NOUN
ejpam-6188	122	15	of	of	ADP
ejpam-6188	122	16	µ(pn	µ(pn	NOUN
ejpam-6188	122	17	)	)	PUNCT
ejpam-6188	122	18	,	,	PUNCT
ejpam-6188	122	19	as	as	SCONJ
ejpam-6188	122	20	shown	show	VERB
ejpam-6188	122	21	in	in	ADP
ejpam-6188	122	22	figure	figure	NOUN
ejpam-6188	122	23	2	2	NUM
ejpam-6188	122	24	.	.	X
ejpam-6188	123	1	for	for	ADP
ejpam-6188	123	2	the	the	DET
ejpam-6188	123	3	rest	rest	NOUN
ejpam-6188	123	4	of	of	ADP
ejpam-6188	123	5	the	the	DET
ejpam-6188	123	6	paper	paper	NOUN
ejpam-6188	123	7	,	,	PUNCT
ejpam-6188	123	8	each	each	DET
ejpam-6188	123	9	bold	bold	ADJ
ejpam-6188	123	10	vertex	vertex	NOUN
ejpam-6188	123	11	in	in	ADP
ejpam-6188	123	12	a	a	DET
ejpam-6188	123	13	figure	figure	NOUN
ejpam-6188	123	14	represents	represent	VERB
ejpam-6188	123	15	a	a	DET
ejpam-6188	123	16	b	b	NOUN
ejpam-6188	123	17	-	-	PUNCT
ejpam-6188	123	18	dominating	dominate	VERB
ejpam-6188	123	19	vertex	vertex	NOUN
ejpam-6188	123	20	p.	p.	PROPN
ejpam-6188	123	21	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	123	22	et	et	PROPN
ejpam-6188	123	23	al	al	PROPN
ejpam-6188	123	24	.	.	PUNCT
ejpam-6188	123	25	/	/	SYM
ejpam-6188	123	26	eur	eur	PROPN
ejpam-6188	123	27	.	.	PUNCT
ejpam-6188	124	1	j.	j.	PROPN
ejpam-6188	124	2	pure	pure	PROPN
ejpam-6188	124	3	appl	appl	PROPN
ejpam-6188	124	4	.	.	PROPN
ejpam-6188	124	5	math	math	PROPN
ejpam-6188	124	6	,	,	PUNCT
ejpam-6188	124	7	18	18	NUM
ejpam-6188	124	8	(	(	PUNCT
ejpam-6188	124	9	3	3	NUM
ejpam-6188	124	10	)	)	PUNCT
ejpam-6188	124	11	(	(	PUNCT
ejpam-6188	124	12	2025	2025	NUM
ejpam-6188	124	13	)	)	PUNCT
ejpam-6188	124	14	,	,	PUNCT
ejpam-6188	124	15	6188	6188	NUM
ejpam-6188	124	16	5	5	NUM
ejpam-6188	124	17	of	of	ADP
ejpam-6188	124	18	13	13	NUM
ejpam-6188	124	19	and	and	CCONJ
ejpam-6188	124	20	the	the	DET
ejpam-6188	124	21	number	number	NOUN
ejpam-6188	124	22	above	above	ADP
ejpam-6188	124	23	each	each	DET
ejpam-6188	124	24	vertex	vertex	NOUN
ejpam-6188	124	25	represents	represent	VERB
ejpam-6188	124	26	its	its	PRON
ejpam-6188	124	27	assigned	assign	VERB
ejpam-6188	124	28	color	color	NOUN
ejpam-6188	124	29	.	.	PUNCT
ejpam-6188	125	1	define	define	VERB
ejpam-6188	125	2	c7	c7	PROPN
ejpam-6188	125	3	:	:	PUNCT
ejpam-6188	125	4	v	v	PROPN
ejpam-6188	125	5	(	(	PUNCT
ejpam-6188	125	6	µ(p7	µ(p7	NOUN
ejpam-6188	125	7	)	)	PUNCT
ejpam-6188	125	8	)	)	PUNCT
ejpam-6188	126	1	→	→	PUNCT
ejpam-6188	126	2	{	{	PUNCT
ejpam-6188	126	3	1	1	NUM
ejpam-6188	126	4	,	,	PUNCT
ejpam-6188	126	5	2	2	NUM
ejpam-6188	126	6	,	,	PUNCT
ejpam-6188	126	7	3	3	NUM
ejpam-6188	126	8	,	,	PUNCT
ejpam-6188	126	9	4	4	NUM
ejpam-6188	126	10	}	}	PUNCT
ejpam-6188	126	11	by	by	ADP
ejpam-6188	126	12	c7(x	c7(x	NUM
ejpam-6188	126	13	)	)	PUNCT
ejpam-6188	126	14	=	=	PUNCT
ejpam-6188	126	15			NOUN
ejpam-6188	126	16	1	1	NUM
ejpam-6188	126	17	if	if	SCONJ
ejpam-6188	126	18	x	x	X
ejpam-6188	126	19	=	=	SYM
ejpam-6188	126	20	v1	v1	PROPN
ejpam-6188	126	21	,	,	PUNCT
ejpam-6188	126	22	v3	v3	PROPN
ejpam-6188	126	23	,	,	PUNCT
ejpam-6188	126	24	u1	u1	NOUN
ejpam-6188	126	25	,	,	PUNCT
ejpam-6188	126	26	u3	u3	PROPN
ejpam-6188	126	27	,	,	PUNCT
ejpam-6188	126	28	u5	u5	PROPN
ejpam-6188	126	29	,	,	PUNCT
ejpam-6188	126	30	u6	u6	PROPN
ejpam-6188	126	31	,	,	PUNCT
ejpam-6188	126	32	u7	u7	PROPN
ejpam-6188	126	33	,	,	PUNCT
ejpam-6188	126	34	2	2	NUM
ejpam-6188	126	35	if	if	SCONJ
ejpam-6188	126	36	x	x	X
ejpam-6188	126	37	=	=	SYM
ejpam-6188	126	38	v2	v2	PROPN
ejpam-6188	126	39	,	,	PUNCT
ejpam-6188	126	40	v4	v4	NOUN
ejpam-6188	126	41	,	,	PUNCT
ejpam-6188	126	42	v7	v7	NUM
ejpam-6188	126	43	,	,	PUNCT
ejpam-6188	126	44	u2	u2	NOUN
ejpam-6188	126	45	,	,	PUNCT
ejpam-6188	126	46	u4	u4	PROPN
ejpam-6188	126	47	,	,	PUNCT
ejpam-6188	126	48	3	3	NUM
ejpam-6188	126	49	if	if	SCONJ
ejpam-6188	126	50	x	x	PROPN
ejpam-6188	126	51	=	=	SYM
ejpam-6188	126	52	v6	v6	PROPN
ejpam-6188	126	53	,	,	PUNCT
ejpam-6188	126	54	u	u	NOUN
ejpam-6188	126	55	,	,	PUNCT
ejpam-6188	126	56	4	4	NUM
ejpam-6188	126	57	if	if	SCONJ
ejpam-6188	126	58	x	x	X
ejpam-6188	126	59	=	=	SYM
ejpam-6188	126	60	v5	v5	PROPN
ejpam-6188	126	61	.	.	PUNCT
ejpam-6188	127	1	(	(	PUNCT
ejpam-6188	127	2	1	1	X
ejpam-6188	127	3	)	)	PUNCT
ejpam-6188	127	4	we	we	PRON
ejpam-6188	127	5	can	can	AUX
ejpam-6188	127	6	see	see	VERB
ejpam-6188	127	7	that	that	SCONJ
ejpam-6188	127	8	c7	c7	PROPN
ejpam-6188	127	9	is	be	AUX
ejpam-6188	127	10	a	a	DET
ejpam-6188	127	11	b	b	NOUN
ejpam-6188	127	12	-	-	PUNCT
ejpam-6188	127	13	coloring	coloring	NOUN
ejpam-6188	127	14	and	and	CCONJ
ejpam-6188	127	15	φ′(µ(p7	φ′(µ(p7	NOUN
ejpam-6188	127	16	)	)	PUNCT
ejpam-6188	127	17	)	)	PUNCT
ejpam-6188	128	1	=	=	SYM
ejpam-6188	128	2	27	27	NUM
ejpam-6188	128	3	.	.	PUNCT
ejpam-6188	129	1	u	u	NOUN
ejpam-6188	129	2	3	3	NUM
ejpam-6188	129	3	u1	u1	NOUN
ejpam-6188	129	4	1	1	NUM
ejpam-6188	129	5	u2	u2	PROPN
ejpam-6188	129	6	2	2	NUM
ejpam-6188	129	7	u3	u3	NOUN
ejpam-6188	129	8	1	1	NUM
ejpam-6188	129	9	u4	u4	PROPN
ejpam-6188	129	10	2	2	NUM
ejpam-6188	129	11	u5	u5	PROPN
ejpam-6188	129	12	1	1	NUM
ejpam-6188	129	13	u6	u6	PROPN
ejpam-6188	129	14	1	1	NUM
ejpam-6188	129	15	u7	u7	PROPN
ejpam-6188	129	16	1	1	NUM
ejpam-6188	129	17	v1	v1	NOUN
ejpam-6188	129	18	1	1	NUM
ejpam-6188	129	19	v2	v2	PROPN
ejpam-6188	129	20	2	2	NUM
ejpam-6188	129	21	v3	v3	PROPN
ejpam-6188	129	22	1	1	NUM
ejpam-6188	129	23	v4	v4	NOUN
ejpam-6188	129	24	2	2	NUM
ejpam-6188	129	25	v5	v5	PROPN
ejpam-6188	129	26	4	4	NUM
ejpam-6188	129	27	v6	v6	NOUN
ejpam-6188	129	28	3	3	NUM
ejpam-6188	129	29	v7	v7	NOUN
ejpam-6188	129	30	2	2	NUM
ejpam-6188	129	31	figure	figure	NOUN
ejpam-6188	129	32	2	2	NUM
ejpam-6188	129	33	:	:	PUNCT
ejpam-6188	129	34	a	a	DET
ejpam-6188	129	35	b	b	NOUN
ejpam-6188	129	36	-	-	PUNCT
ejpam-6188	129	37	coloring	coloring	NOUN
ejpam-6188	129	38	of	of	ADP
ejpam-6188	129	39	µ(p7	µ(p7	PROPN
ejpam-6188	129	40	)	)	PUNCT
ejpam-6188	129	41	with	with	ADP
ejpam-6188	129	42	4	4	NUM
ejpam-6188	129	43	colors	color	NOUN
ejpam-6188	129	44	lemmas	lemma	VERB
ejpam-6188	129	45	1–5	1–5	PRON
ejpam-6188	129	46	give	give	VERB
ejpam-6188	129	47	a	a	DET
ejpam-6188	129	48	structure	structure	NOUN
ejpam-6188	129	49	of	of	ADP
ejpam-6188	129	50	colors	color	NOUN
ejpam-6188	129	51	in	in	ADP
ejpam-6188	129	52	a	a	DET
ejpam-6188	129	53	b	b	NOUN
ejpam-6188	129	54	-	-	PUNCT
ejpam-6188	129	55	coloring	coloring	NOUN
ejpam-6188	129	56	giving	give	VERB
ejpam-6188	129	57	the	the	DET
ejpam-6188	129	58	b	b	NOUN
ejpam-6188	129	59	-	-	PUNCT
ejpam-6188	129	60	chromatic	chromatic	ADJ
ejpam-6188	129	61	number	number	NOUN
ejpam-6188	129	62	.	.	PUNCT
ejpam-6188	130	1	the	the	DET
ejpam-6188	130	2	structure	structure	NOUN
ejpam-6188	130	3	will	will	AUX
ejpam-6188	130	4	be	be	AUX
ejpam-6188	130	5	used	use	VERB
ejpam-6188	130	6	to	to	PART
ejpam-6188	130	7	determine	determine	VERB
ejpam-6188	130	8	the	the	DET
ejpam-6188	130	9	lower	low	ADJ
ejpam-6188	130	10	bound	bind	VERB
ejpam-6188	130	11	of	of	ADP
ejpam-6188	130	12	the	the	DET
ejpam-6188	130	13	b	b	NOUN
ejpam-6188	130	14	-	-	PUNCT
ejpam-6188	130	15	chromatic	chromatic	ADJ
ejpam-6188	130	16	sum	sum	NOUN
ejpam-6188	130	17	in	in	ADP
ejpam-6188	130	18	theorem	theorem	NOUN
ejpam-6188	130	19	5	5	NUM
ejpam-6188	130	20	.	.	PUNCT
ejpam-6188	131	1	the	the	DET
ejpam-6188	131	2	idea	idea	NOUN
ejpam-6188	131	3	of	of	ADP
ejpam-6188	131	4	proof	proof	NOUN
ejpam-6188	131	5	of	of	ADP
ejpam-6188	131	6	lemma	lemma	PROPN
ejpam-6188	131	7	1	1	NUM
ejpam-6188	131	8	is	be	AUX
ejpam-6188	131	9	extended	extend	VERB
ejpam-6188	131	10	from	from	ADP
ejpam-6188	131	11	the	the	DET
ejpam-6188	131	12	calculation	calculation	NOUN
ejpam-6188	131	13	of	of	ADP
ejpam-6188	131	14	φ′(µ(p8	φ′(µ(p8	NOUN
ejpam-6188	131	15	)	)	PUNCT
ejpam-6188	131	16	)	)	PUNCT
ejpam-6188	131	17	in	in	ADP
ejpam-6188	131	18	[	[	X
ejpam-6188	131	19	1	1	NUM
ejpam-6188	131	20	]	]	PUNCT
ejpam-6188	131	21	.	.	PUNCT
ejpam-6188	132	1	lemma	lemma	PROPN
ejpam-6188	132	2	1	1	NUM
ejpam-6188	132	3	.	.	PUNCT
ejpam-6188	133	1	for	for	ADP
ejpam-6188	133	2	a	a	DET
ejpam-6188	133	3	graph	graph	NOUN
ejpam-6188	133	4	µ(pn	µ(pn	NOUN
ejpam-6188	133	5	)	)	PUNCT
ejpam-6188	133	6	where	where	SCONJ
ejpam-6188	133	7	n	n	PRON
ejpam-6188	133	8	≥	≥	VERB
ejpam-6188	133	9	8	8	NUM
ejpam-6188	133	10	with	with	ADP
ejpam-6188	133	11	a	a	DET
ejpam-6188	133	12	coloring	coloring	NOUN
ejpam-6188	133	13	cn	cn	PROPN
ejpam-6188	133	14	giving	give	VERB
ejpam-6188	133	15	b	b	NOUN
ejpam-6188	133	16	-	-	PUNCT
ejpam-6188	133	17	chromatic	chromatic	ADJ
ejpam-6188	133	18	number	number	NOUN
ejpam-6188	133	19	,	,	PUNCT
ejpam-6188	133	20	the	the	DET
ejpam-6188	133	21	following	follow	VERB
ejpam-6188	133	22	properties	property	NOUN
ejpam-6188	133	23	hold	hold	VERB
ejpam-6188	133	24	:	:	PUNCT
ejpam-6188	133	25	(	(	PUNCT
ejpam-6188	133	26	i	i	NOUN
ejpam-6188	133	27	)	)	PUNCT
ejpam-6188	133	28	the	the	DET
ejpam-6188	133	29	b	b	NOUN
ejpam-6188	133	30	-	-	PUNCT
ejpam-6188	133	31	dominating	dominating	NOUN
ejpam-6188	133	32	vertices	vertex	NOUN
ejpam-6188	133	33	are	be	AUX
ejpam-6188	133	34	in	in	ADP
ejpam-6188	133	35	v	v	NOUN
ejpam-6188	133	36	(	(	PUNCT
ejpam-6188	133	37	pn	pn	NOUN
ejpam-6188	133	38	)	)	PUNCT
ejpam-6188	133	39	∪	∪	NOUN
ejpam-6188	133	40	{	{	PUNCT
ejpam-6188	133	41	u	u	NOUN
ejpam-6188	133	42	}	}	PUNCT
ejpam-6188	133	43	,	,	PUNCT
ejpam-6188	133	44	(	(	PUNCT
ejpam-6188	133	45	ii	ii	NOUN
ejpam-6188	133	46	)	)	PUNCT
ejpam-6188	133	47	each	each	DET
ejpam-6188	133	48	color	color	NOUN
ejpam-6188	133	49	appears	appear	VERB
ejpam-6188	133	50	at	at	ADV
ejpam-6188	133	51	least	least	ADJ
ejpam-6188	133	52	twice	twice	ADV
ejpam-6188	133	53	in	in	ADP
ejpam-6188	133	54	v	v	NOUN
ejpam-6188	133	55	(	(	PUNCT
ejpam-6188	133	56	pn	pn	NOUN
ejpam-6188	133	57	)	)	PUNCT
ejpam-6188	133	58	∪	∪	ADJ
ejpam-6188	133	59	un	un	PROPN
ejpam-6188	133	60	.	.	PROPN
ejpam-6188	133	61	proof	proof	NOUN
ejpam-6188	133	62	.	.	PUNCT
ejpam-6188	134	1	(	(	PUNCT
ejpam-6188	134	2	1	1	X
ejpam-6188	134	3	)	)	PUNCT
ejpam-6188	134	4	let	let	VERB
ejpam-6188	134	5	n	n	PRON
ejpam-6188	134	6	≥	≥	NOUN
ejpam-6188	134	7	8	8	NUM
ejpam-6188	134	8	.	.	PUNCT
ejpam-6188	134	9	from	from	ADP
ejpam-6188	134	10	φ(µ(pn	φ(µ(pn	NOUN
ejpam-6188	134	11	)	)	PUNCT
ejpam-6188	134	12	)	)	PUNCT
ejpam-6188	135	1	=	=	SYM
ejpam-6188	135	2	5	5	NUM
ejpam-6188	135	3	,	,	PUNCT
ejpam-6188	135	4	the	the	DET
ejpam-6188	135	5	b	b	NOUN
ejpam-6188	135	6	-	-	PUNCT
ejpam-6188	135	7	vertices	vertex	NOUN
ejpam-6188	135	8	have	have	VERB
ejpam-6188	135	9	degree	degree	NOUN
ejpam-6188	135	10	4	4	NUM
ejpam-6188	135	11	in	in	ADP
ejpam-6188	135	12	µ(pn	µ(pn	NOUN
ejpam-6188	135	13	)	)	PUNCT
ejpam-6188	135	14	.	.	PUNCT
ejpam-6188	136	1	since	since	SCONJ
ejpam-6188	136	2	ui	ui	PROPN
ejpam-6188	136	3	has	have	AUX
ejpam-6188	136	4	degree	degree	NOUN
ejpam-6188	136	5	3	3	NUM
ejpam-6188	136	6	in	in	ADP
ejpam-6188	136	7	µ(pn	µ(pn	NOUN
ejpam-6188	136	8	)	)	PUNCT
ejpam-6188	136	9	for	for	ADP
ejpam-6188	136	10	1	1	NUM
ejpam-6188	136	11	≤	≤	NUM
ejpam-6188	136	12	i	i	PRON
ejpam-6188	136	13	≤	≤	PROPN
ejpam-6188	136	14	n	n	CCONJ
ejpam-6188	136	15	,	,	PUNCT
ejpam-6188	136	16	it	it	PRON
ejpam-6188	136	17	follows	follow	VERB
ejpam-6188	136	18	that	that	SCONJ
ejpam-6188	136	19	the	the	DET
ejpam-6188	136	20	b	b	X
ejpam-6188	136	21	-	-	PUNCT
ejpam-6188	136	22	vertices	vertex	NOUN
ejpam-6188	136	23	must	must	AUX
ejpam-6188	136	24	be	be	AUX
ejpam-6188	136	25	in	in	ADP
ejpam-6188	136	26	v	v	NOUN
ejpam-6188	136	27	(	(	PUNCT
ejpam-6188	136	28	pn	pn	NOUN
ejpam-6188	136	29	)	)	PUNCT
ejpam-6188	136	30	∪	∪	NOUN
ejpam-6188	136	31	{	{	PUNCT
ejpam-6188	136	32	u	u	NOUN
ejpam-6188	136	33	}	}	PUNCT
ejpam-6188	136	34	.	.	PUNCT
ejpam-6188	137	1	(	(	PUNCT
ejpam-6188	137	2	2	2	X
ejpam-6188	137	3	)	)	PUNCT
ejpam-6188	137	4	suppose	suppose	VERB
ejpam-6188	137	5	there	there	PRON
ejpam-6188	137	6	is	be	VERB
ejpam-6188	137	7	only	only	ADV
ejpam-6188	137	8	one	one	NUM
ejpam-6188	137	9	vertex	vertex	NOUN
ejpam-6188	137	10	x	x	SYM
ejpam-6188	137	11	∈	∈	NOUN
ejpam-6188	137	12	v	v	NOUN
ejpam-6188	137	13	(	(	PUNCT
ejpam-6188	137	14	pn)∪un	pn)∪un	NOUN
ejpam-6188	137	15	of	of	ADP
ejpam-6188	137	16	color	color	NOUN
ejpam-6188	137	17	1	1	NUM
ejpam-6188	137	18	.	.	PUNCT
ejpam-6188	138	1	since	since	SCONJ
ejpam-6188	138	2	the	the	DET
ejpam-6188	138	3	b	b	NOUN
ejpam-6188	138	4	-	-	PUNCT
ejpam-6188	138	5	dominating	dominating	NOUN
ejpam-6188	138	6	vertices	vertex	NOUN
ejpam-6188	138	7	of	of	ADP
ejpam-6188	138	8	colors	color	NOUN
ejpam-6188	138	9	2	2	NUM
ejpam-6188	138	10	,	,	PUNCT
ejpam-6188	138	11	3	3	NUM
ejpam-6188	138	12	,	,	PUNCT
ejpam-6188	138	13	4	4	NUM
ejpam-6188	138	14	,	,	PUNCT
ejpam-6188	138	15	5	5	NUM
ejpam-6188	138	16	are	be	AUX
ejpam-6188	138	17	in	in	ADP
ejpam-6188	138	18	v	v	NOUN
ejpam-6188	138	19	(	(	PUNCT
ejpam-6188	138	20	pn	pn	NOUN
ejpam-6188	138	21	)	)	PUNCT
ejpam-6188	138	22	∪	∪	NOUN
ejpam-6188	138	23	{	{	PUNCT
ejpam-6188	138	24	u	u	NOUN
ejpam-6188	138	25	}	}	PUNCT
ejpam-6188	138	26	,	,	PUNCT
ejpam-6188	138	27	at	at	ADP
ejpam-6188	138	28	least	least	ADJ
ejpam-6188	138	29	three	three	NUM
ejpam-6188	138	30	of	of	ADP
ejpam-6188	138	31	them	they	PRON
ejpam-6188	138	32	are	be	AUX
ejpam-6188	138	33	in	in	ADP
ejpam-6188	138	34	v	v	NOUN
ejpam-6188	138	35	(	(	PUNCT
ejpam-6188	138	36	pn	pn	NOUN
ejpam-6188	138	37	)	)	PUNCT
ejpam-6188	138	38	.	.	PUNCT
ejpam-6188	139	1	these	these	DET
ejpam-6188	139	2	three	three	NUM
ejpam-6188	139	3	vertices	vertex	NOUN
ejpam-6188	139	4	must	must	AUX
ejpam-6188	139	5	be	be	AUX
ejpam-6188	139	6	all	all	ADV
ejpam-6188	139	7	adjacent	adjacent	ADJ
ejpam-6188	139	8	to	to	PART
ejpam-6188	139	9	x.	x.	VERB
ejpam-6188	139	10	however	however	ADV
ejpam-6188	139	11	,	,	PUNCT
ejpam-6188	139	12	x	x	PUNCT
ejpam-6188	139	13	has	have	VERB
ejpam-6188	139	14	at	at	ADP
ejpam-6188	139	15	most	most	ADV
ejpam-6188	139	16	two	two	NUM
ejpam-6188	139	17	neighbors	neighbor	NOUN
ejpam-6188	139	18	in	in	ADP
ejpam-6188	139	19	v	v	PROPN
ejpam-6188	139	20	(	(	PUNCT
ejpam-6188	139	21	pn	pn	NOUN
ejpam-6188	139	22	)	)	PUNCT
ejpam-6188	139	23	,	,	PUNCT
ejpam-6188	139	24	a	a	DET
ejpam-6188	139	25	contradiction	contradiction	NOUN
ejpam-6188	139	26	.	.	PUNCT
ejpam-6188	140	1	lemma	lemma	PROPN
ejpam-6188	140	2	2	2	NUM
ejpam-6188	140	3	.	.	X
ejpam-6188	140	4	for	for	ADP
ejpam-6188	140	5	a	a	DET
ejpam-6188	140	6	graph	graph	NOUN
ejpam-6188	140	7	µ(pn	µ(pn	NOUN
ejpam-6188	140	8	)	)	PUNCT
ejpam-6188	140	9	where	where	SCONJ
ejpam-6188	140	10	n	n	PRON
ejpam-6188	140	11	≥	≥	VERB
ejpam-6188	140	12	8	8	NUM
ejpam-6188	140	13	with	with	ADP
ejpam-6188	140	14	a	a	DET
ejpam-6188	140	15	coloring	coloring	NOUN
ejpam-6188	140	16	cn	cn	PROPN
ejpam-6188	140	17	giving	give	VERB
ejpam-6188	140	18	the	the	DET
ejpam-6188	140	19	b	b	NOUN
ejpam-6188	140	20	-	-	PUNCT
ejpam-6188	140	21	chromatic	chromatic	ADJ
ejpam-6188	140	22	number	number	NOUN
ejpam-6188	140	23	,	,	PUNCT
ejpam-6188	140	24	the	the	DET
ejpam-6188	140	25	following	follow	VERB
ejpam-6188	140	26	properties	property	NOUN
ejpam-6188	140	27	hold	hold	VERB
ejpam-6188	140	28	:	:	PUNCT
ejpam-6188	140	29	(	(	PUNCT
ejpam-6188	140	30	i	i	NOUN
ejpam-6188	140	31	)	)	PUNCT
ejpam-6188	140	32	|cn(v	|cn(v	PROPN
ejpam-6188	141	1	(	(	PUNCT
ejpam-6188	141	2	pn))|	pn))|	X
ejpam-6188	141	3	=	=	SYM
ejpam-6188	141	4	5	5	NUM
ejpam-6188	141	5	,	,	PUNCT
ejpam-6188	141	6	(	(	PUNCT
ejpam-6188	141	7	ii	ii	NOUN
ejpam-6188	141	8	)	)	PUNCT
ejpam-6188	141	9	|ccn(u)|	|ccn(u)|	VERB
ejpam-6188	141	10	≤	≤	NUM
ejpam-6188	141	11	⌈	⌈	ADP
ejpam-6188	141	12	n	n	CCONJ
ejpam-6188	141	13	2	2	NUM
ejpam-6188	141	14	⌉	⌉	NOUN
ejpam-6188	141	15	+	+	NOUN
ejpam-6188	141	16	1	1	X
ejpam-6188	141	17	.	.	PUNCT
ejpam-6188	142	1	p.	p.	NOUN
ejpam-6188	142	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	142	3	et	et	PROPN
ejpam-6188	142	4	al	al	PROPN
ejpam-6188	142	5	.	.	PUNCT
ejpam-6188	142	6	/	/	SYM
ejpam-6188	142	7	eur	eur	PROPN
ejpam-6188	142	8	.	.	PUNCT
ejpam-6188	143	1	j.	j.	PROPN
ejpam-6188	143	2	pure	pure	PROPN
ejpam-6188	143	3	appl	appl	PROPN
ejpam-6188	143	4	.	.	PROPN
ejpam-6188	143	5	math	math	PROPN
ejpam-6188	143	6	,	,	PUNCT
ejpam-6188	143	7	18	18	NUM
ejpam-6188	143	8	(	(	PUNCT
ejpam-6188	143	9	3	3	NUM
ejpam-6188	143	10	)	)	PUNCT
ejpam-6188	143	11	(	(	PUNCT
ejpam-6188	143	12	2025	2025	NUM
ejpam-6188	143	13	)	)	PUNCT
ejpam-6188	143	14	,	,	PUNCT
ejpam-6188	143	15	6188	6188	NUM
ejpam-6188	143	16	6	6	NUM
ejpam-6188	143	17	of	of	ADP
ejpam-6188	143	18	13	13	NUM
ejpam-6188	143	19	proof	proof	NOUN
ejpam-6188	143	20	.	.	PUNCT
ejpam-6188	144	1	(	(	PUNCT
ejpam-6188	144	2	1	1	X
ejpam-6188	144	3	)	)	PUNCT
ejpam-6188	144	4	we	we	PRON
ejpam-6188	144	5	note	note	VERB
ejpam-6188	144	6	that	that	SCONJ
ejpam-6188	144	7	cn(u	cn(u	NOUN
ejpam-6188	144	8	)	)	PUNCT
ejpam-6188	144	9	does	do	AUX
ejpam-6188	144	10	not	not	PART
ejpam-6188	144	11	appear	appear	VERB
ejpam-6188	144	12	in	in	ADP
ejpam-6188	144	13	un	un	PROPN
ejpam-6188	144	14	.	.	PROPN
ejpam-6188	144	15	for	for	SCONJ
ejpam-6188	144	16	a	a	DET
ejpam-6188	144	17	vertex	vertex	NOUN
ejpam-6188	144	18	in	in	ADP
ejpam-6188	144	19	v	v	NOUN
ejpam-6188	144	20	(	(	PUNCT
ejpam-6188	144	21	pn	pn	NOUN
ejpam-6188	144	22	)	)	PUNCT
ejpam-6188	144	23	to	to	PART
ejpam-6188	144	24	be	be	AUX
ejpam-6188	144	25	a	a	DET
ejpam-6188	144	26	bdominating	bdominating	NOUN
ejpam-6188	144	27	vertex	vertex	NOUN
ejpam-6188	144	28	,	,	PUNCT
ejpam-6188	144	29	the	the	DET
ejpam-6188	144	30	color	color	NOUN
ejpam-6188	144	31	cn(u	cn(u	NOUN
ejpam-6188	144	32	)	)	PUNCT
ejpam-6188	144	33	must	must	AUX
ejpam-6188	144	34	appear	appear	VERB
ejpam-6188	144	35	in	in	ADP
ejpam-6188	144	36	v	v	PROPN
ejpam-6188	144	37	(	(	PUNCT
ejpam-6188	144	38	pn	pn	NOUN
ejpam-6188	144	39	)	)	PUNCT
ejpam-6188	144	40	.	.	PUNCT
ejpam-6188	145	1	thus	thus	ADV
ejpam-6188	145	2	|cn(v	|cn(v	X
ejpam-6188	145	3	(	(	PUNCT
ejpam-6188	145	4	pn))|	pn))|	X
ejpam-6188	145	5	=	=	SYM
ejpam-6188	145	6	5	5	X
ejpam-6188	145	7	.	.	PUNCT
ejpam-6188	145	8	(	(	PUNCT
ejpam-6188	145	9	2	2	X
ejpam-6188	145	10	)	)	PUNCT
ejpam-6188	145	11	since	since	SCONJ
ejpam-6188	145	12	vi	vi	NOUN
ejpam-6188	145	13	and	and	CCONJ
ejpam-6188	145	14	vi+1	vi+1	ADV
ejpam-6188	145	15	in	in	ADP
ejpam-6188	145	16	v	v	NOUN
ejpam-6188	145	17	(	(	PUNCT
ejpam-6188	145	18	pn	pn	NOUN
ejpam-6188	145	19	)	)	PUNCT
ejpam-6188	145	20	can	can	AUX
ejpam-6188	145	21	not	not	PART
ejpam-6188	145	22	get	get	VERB
ejpam-6188	145	23	the	the	DET
ejpam-6188	145	24	same	same	ADJ
ejpam-6188	145	25	color	color	NOUN
ejpam-6188	145	26	,	,	PUNCT
ejpam-6188	145	27	there	there	PRON
ejpam-6188	145	28	are	be	VERB
ejpam-6188	145	29	at	at	ADP
ejpam-6188	145	30	most	most	ADJ
ejpam-6188	145	31	⌈	⌈	NUM
ejpam-6188	145	32	n	n	PRON
ejpam-6188	145	33	2	2	NUM
ejpam-6188	145	34	⌉	⌉	PRON
ejpam-6188	145	35	vertices	vertice	VERB
ejpam-6188	145	36	in	in	ADP
ejpam-6188	145	37	v	v	NOUN
ejpam-6188	145	38	(	(	PUNCT
ejpam-6188	145	39	pn	pn	NOUN
ejpam-6188	145	40	)	)	PUNCT
ejpam-6188	145	41	with	with	ADP
ejpam-6188	145	42	color	color	NOUN
ejpam-6188	145	43	cn(u	cn(u	NOUN
ejpam-6188	145	44	)	)	PUNCT
ejpam-6188	145	45	.	.	PUNCT
ejpam-6188	146	1	since	since	SCONJ
ejpam-6188	146	2	cn(u	cn(u	NUM
ejpam-6188	146	3	)	)	PUNCT
ejpam-6188	146	4	does	do	AUX
ejpam-6188	146	5	not	not	PART
ejpam-6188	146	6	appear	appear	VERB
ejpam-6188	146	7	in	in	ADP
ejpam-6188	146	8	un	un	PROPN
ejpam-6188	146	9	,	,	PUNCT
ejpam-6188	146	10	we	we	PRON
ejpam-6188	146	11	have	have	VERB
ejpam-6188	146	12	|ccn(u)|	|ccn(u)|	NUM
ejpam-6188	146	13	≤	≤	NUM
ejpam-6188	146	14	⌈	⌈	X
ejpam-6188	146	15	n	n	CCONJ
ejpam-6188	146	16	2	2	NUM
ejpam-6188	146	17	⌉	⌉	X
ejpam-6188	146	18	+1	+1	PROPN
ejpam-6188	146	19	.	.	PUNCT
ejpam-6188	147	1	lemma	lemma	PROPN
ejpam-6188	147	2	3	3	X
ejpam-6188	147	3	.	.	PUNCT
ejpam-6188	147	4	for	for	ADP
ejpam-6188	147	5	a	a	DET
ejpam-6188	147	6	graph	graph	NOUN
ejpam-6188	147	7	µ(pn	µ(pn	NOUN
ejpam-6188	147	8	)	)	PUNCT
ejpam-6188	147	9	where	where	SCONJ
ejpam-6188	147	10	n	n	PRON
ejpam-6188	147	11	≥	≥	VERB
ejpam-6188	147	12	8	8	NUM
ejpam-6188	147	13	with	with	ADP
ejpam-6188	147	14	a	a	DET
ejpam-6188	147	15	coloring	coloring	NOUN
ejpam-6188	147	16	cn	cn	PROPN
ejpam-6188	147	17	giving	give	VERB
ejpam-6188	147	18	the	the	DET
ejpam-6188	147	19	b	b	NOUN
ejpam-6188	147	20	-	-	PUNCT
ejpam-6188	147	21	chromatic	chromatic	ADJ
ejpam-6188	147	22	number	number	NOUN
ejpam-6188	147	23	,	,	PUNCT
ejpam-6188	147	24	if	if	SCONJ
ejpam-6188	147	25	there	there	PRON
ejpam-6188	147	26	exists	exist	VERB
ejpam-6188	147	27	m	m	VERB
ejpam-6188	147	28	∈	∈	NOUN
ejpam-6188	147	29	{	{	PUNCT
ejpam-6188	147	30	1	1	NUM
ejpam-6188	147	31	,	,	PUNCT
ejpam-6188	147	32	2	2	NUM
ejpam-6188	147	33	,	,	PUNCT
ejpam-6188	147	34	3	3	NUM
ejpam-6188	147	35	,	,	PUNCT
ejpam-6188	147	36	4	4	NUM
ejpam-6188	147	37	,	,	PUNCT
ejpam-6188	147	38	5	5	NUM
ejpam-6188	147	39	}	}	PUNCT
ejpam-6188	147	40	where	where	SCONJ
ejpam-6188	147	41	|cm|	|cm|	NOUN
ejpam-6188	147	42	=	=	SYM
ejpam-6188	147	43	2	2	NUM
ejpam-6188	147	44	,	,	PUNCT
ejpam-6188	147	45	then	then	ADV
ejpam-6188	147	46	|ci|	|ci|	PROPN
ejpam-6188	147	47	≥	≥	NUM
ejpam-6188	147	48	3	3	NUM
ejpam-6188	147	49	for	for	ADP
ejpam-6188	147	50	i	i	PRON
ejpam-6188	147	51	̸=	̸=	PROPN
ejpam-6188	147	52	m.	m.	NOUN
ejpam-6188	147	53	proof	proof	NOUN
ejpam-6188	147	54	.	.	PUNCT
ejpam-6188	148	1	by	by	ADP
ejpam-6188	148	2	theorem	theorem	NOUN
ejpam-6188	148	3	1	1	NUM
ejpam-6188	148	4	,	,	PUNCT
ejpam-6188	148	5	we	we	PRON
ejpam-6188	148	6	have	have	VERB
ejpam-6188	148	7	φ(µ(pn	φ(µ(pn	NOUN
ejpam-6188	148	8	)	)	PUNCT
ejpam-6188	148	9	)	)	PUNCT
ejpam-6188	149	1	=	=	SYM
ejpam-6188	149	2	5	5	X
ejpam-6188	149	3	.	.	NOUN
ejpam-6188	149	4	without	without	ADP
ejpam-6188	149	5	loss	loss	NOUN
ejpam-6188	149	6	of	of	ADP
ejpam-6188	149	7	generality	generality	NOUN
ejpam-6188	149	8	,	,	PUNCT
ejpam-6188	149	9	we	we	PRON
ejpam-6188	149	10	suppose	suppose	VERB
ejpam-6188	149	11	|c5|	|c5|	NOUN
ejpam-6188	149	12	=	=	SYM
ejpam-6188	149	13	2	2	X
ejpam-6188	149	14	.	.	PUNCT
ejpam-6188	149	15	by	by	ADP
ejpam-6188	149	16	lemma	lemma	PROPN
ejpam-6188	149	17	1	1	NUM
ejpam-6188	149	18	,	,	PUNCT
ejpam-6188	149	19	we	we	PRON
ejpam-6188	149	20	have	have	VERB
ejpam-6188	149	21	cn(u	cn(u	NOUN
ejpam-6188	149	22	)	)	PUNCT
ejpam-6188	149	23	̸=	̸=	PROPN
ejpam-6188	149	24	5	5	NUM
ejpam-6188	149	25	.	.	PUNCT
ejpam-6188	150	1	hence	hence	ADV
ejpam-6188	150	2	5	5	NUM
ejpam-6188	150	3	̸∈	̸∈	PROPN
ejpam-6188	150	4	{	{	PUNCT
ejpam-6188	150	5	cn(u	cn(u	NOUN
ejpam-6188	150	6	)	)	PUNCT
ejpam-6188	150	7	}	}	PUNCT
ejpam-6188	150	8	∪	∪	X
ejpam-6188	150	9	{	{	PUNCT
ejpam-6188	150	10	cn(ui	cn(ui	PROPN
ejpam-6188	150	11	)	)	PUNCT
ejpam-6188	150	12	,	,	PUNCT
ejpam-6188	150	13	cn(vi	cn(vi	PROPN
ejpam-6188	150	14	)	)	PUNCT
ejpam-6188	150	15	:	:	PUNCT
ejpam-6188	151	1	i	i	NOUN
ejpam-6188	151	2	=	=	NOUN
ejpam-6188	151	3	1	1	NUM
ejpam-6188	151	4	,	,	PUNCT
ejpam-6188	151	5	n	n	CCONJ
ejpam-6188	151	6	}	}	PUNCT
ejpam-6188	151	7	.	.	PUNCT
ejpam-6188	152	1	let	let	VERB
ejpam-6188	152	2	k	k	PROPN
ejpam-6188	152	3	∈	∈	PROPN
ejpam-6188	152	4	{	{	PUNCT
ejpam-6188	152	5	2	2	NUM
ejpam-6188	152	6	,	,	PUNCT
ejpam-6188	152	7	.	.	PUNCT
ejpam-6188	152	8	.	.	PUNCT
ejpam-6188	153	1	.	.	PUNCT
ejpam-6188	154	1	,	,	PUNCT
ejpam-6188	154	2	n	n	CCONJ
ejpam-6188	154	3	−	−	PROPN
ejpam-6188	154	4	1	1	NUM
ejpam-6188	154	5	}	}	PUNCT
ejpam-6188	154	6	be	be	AUX
ejpam-6188	154	7	such	such	ADJ
ejpam-6188	154	8	that	that	PRON
ejpam-6188	154	9	vk	vk	NOUN
ejpam-6188	154	10	is	be	AUX
ejpam-6188	154	11	the	the	DET
ejpam-6188	154	12	b	b	NOUN
ejpam-6188	154	13	-	-	PUNCT
ejpam-6188	154	14	dominating	dominate	VERB
ejpam-6188	154	15	vertex	vertex	NOUN
ejpam-6188	154	16	of	of	ADP
ejpam-6188	154	17	color	color	NOUN
ejpam-6188	154	18	5	5	NUM
ejpam-6188	154	19	and	and	CCONJ
ejpam-6188	154	20	j	j	PROPN
ejpam-6188	154	21	∈	∈	PROPN
ejpam-6188	154	22	{	{	PUNCT
ejpam-6188	154	23	1	1	NUM
ejpam-6188	154	24	,	,	PUNCT
ejpam-6188	154	25	.	.	PUNCT
ejpam-6188	154	26	.	.	PUNCT
ejpam-6188	155	1	.	.	PUNCT
ejpam-6188	156	1	,	,	PUNCT
ejpam-6188	156	2	n	n	CCONJ
ejpam-6188	156	3	}	}	PUNCT
ejpam-6188	156	4	be	be	AUX
ejpam-6188	156	5	such	such	ADJ
ejpam-6188	156	6	that	that	SCONJ
ejpam-6188	156	7	5	5	NUM
ejpam-6188	156	8	∈	∈	PROPN
ejpam-6188	156	9	{	{	PUNCT
ejpam-6188	156	10	cn(uj	cn(uj	NOUN
ejpam-6188	156	11	)	)	PUNCT
ejpam-6188	156	12	,	,	PUNCT
ejpam-6188	156	13	cn(vj	cn(vj	NOUN
ejpam-6188	156	14	)	)	PUNCT
ejpam-6188	156	15	}	}	PUNCT
ejpam-6188	156	16	.	.	PUNCT
ejpam-6188	157	1	at	at	ADV
ejpam-6188	157	2	least	least	ADJ
ejpam-6188	157	3	one	one	NUM
ejpam-6188	157	4	of	of	ADP
ejpam-6188	157	5	vk−1	vk−1	NOUN
ejpam-6188	157	6	and	and	CCONJ
ejpam-6188	157	7	vk+1	vk+1	NOUN
ejpam-6188	157	8	is	be	AUX
ejpam-6188	157	9	a	a	DET
ejpam-6188	157	10	bvertex	bvertex	NOUN
ejpam-6188	157	11	.	.	PUNCT
ejpam-6188	158	1	suppose	suppose	VERB
ejpam-6188	158	2	that	that	SCONJ
ejpam-6188	158	3	vk+1	vk+1	ADJ
ejpam-6188	158	4	is	be	AUX
ejpam-6188	158	5	a	a	DET
ejpam-6188	158	6	b	b	NOUN
ejpam-6188	158	7	-	-	PUNCT
ejpam-6188	158	8	dominating	dominate	VERB
ejpam-6188	158	9	vertex	vertex	NOUN
ejpam-6188	158	10	of	of	ADP
ejpam-6188	158	11	color	color	NOUN
ejpam-6188	158	12	4	4	NUM
ejpam-6188	158	13	,	,	PUNCT
ejpam-6188	158	14	and	and	CCONJ
ejpam-6188	158	15	cn(vk−1	cn(vk−1	NOUN
ejpam-6188	158	16	)	)	PUNCT
ejpam-6188	158	17	=	=	SYM
ejpam-6188	158	18	3	3	NUM
ejpam-6188	158	19	(	(	PUNCT
ejpam-6188	158	20	see	see	VERB
ejpam-6188	158	21	figure	figure	NOUN
ejpam-6188	158	22	3	3	NUM
ejpam-6188	158	23	)	)	PUNCT
ejpam-6188	158	24	.	.	PUNCT
ejpam-6188	159	1	it	it	PRON
ejpam-6188	159	2	follows	follow	VERB
ejpam-6188	159	3	that	that	SCONJ
ejpam-6188	159	4	cn(vk+2	cn(vk+2	NOUN
ejpam-6188	159	5	)	)	PUNCT
ejpam-6188	159	6	=	=	SYM
ejpam-6188	159	7	3	3	NUM
ejpam-6188	159	8	and	and	CCONJ
ejpam-6188	159	9	cn({uk−1	cn({uk−1	PROPN
ejpam-6188	159	10	,	,	PUNCT
ejpam-6188	159	11	uk+1	uk+1	NOUN
ejpam-6188	159	12	}	}	PUNCT
ejpam-6188	159	13	)	)	PUNCT
ejpam-6188	159	14	=	=	SYM
ejpam-6188	159	15	cn({uk	cn({uk	NOUN
ejpam-6188	159	16	,	,	PUNCT
ejpam-6188	159	17	uk+2	uk+2	NOUN
ejpam-6188	159	18	}	}	PUNCT
ejpam-6188	159	19	)	)	PUNCT
ejpam-6188	159	20	=	=	PRON
ejpam-6188	159	21	{	{	PUNCT
ejpam-6188	159	22	1	1	NUM
ejpam-6188	159	23	,	,	PUNCT
ejpam-6188	159	24	2	2	NUM
ejpam-6188	159	25	}	}	PUNCT
ejpam-6188	159	26	.	.	PUNCT
ejpam-6188	160	1	thus	thus	ADV
ejpam-6188	160	2	cn(u	cn(u	X
ejpam-6188	160	3	)	)	PUNCT
ejpam-6188	160	4	∈	∈	PROPN
ejpam-6188	160	5	{	{	PUNCT
ejpam-6188	160	6	3	3	NUM
ejpam-6188	160	7	,	,	PUNCT
ejpam-6188	160	8	4	4	NUM
ejpam-6188	160	9	}	}	PUNCT
ejpam-6188	160	10	,	,	PUNCT
ejpam-6188	160	11	says	say	VERB
ejpam-6188	160	12	cn(u	cn(u	X
ejpam-6188	160	13	)	)	PUNCT
ejpam-6188	160	14	=	=	SYM
ejpam-6188	161	1	3	3	X
ejpam-6188	161	2	.	.	PUNCT
ejpam-6188	162	1	so	so	ADV
ejpam-6188	162	2	|c3|	|c3|	ADJ
ejpam-6188	162	3	≥	≥	NOUN
ejpam-6188	162	4	3	3	NUM
ejpam-6188	162	5	.	.	PUNCT
ejpam-6188	163	1	since	since	SCONJ
ejpam-6188	163	2	the	the	DET
ejpam-6188	163	3	vertices	vertex	NOUN
ejpam-6188	163	4	in	in	ADP
ejpam-6188	163	5	un	un	PROPN
ejpam-6188	163	6	are	be	AUX
ejpam-6188	163	7	not	not	PART
ejpam-6188	163	8	b	b	NOUN
ejpam-6188	163	9	-	-	PUNCT
ejpam-6188	163	10	dominating	dominating	NOUN
ejpam-6188	163	11	vertices	vertex	NOUN
ejpam-6188	163	12	,	,	PUNCT
ejpam-6188	163	13	there	there	PRON
ejpam-6188	163	14	exists	exist	VERB
ejpam-6188	163	15	b	b	X
ejpam-6188	163	16	-	-	PUNCT
ejpam-6188	163	17	dominating	dominating	NOUN
ejpam-6188	163	18	vertices	vertex	NOUN
ejpam-6188	163	19	of	of	ADP
ejpam-6188	163	20	colors	color	NOUN
ejpam-6188	163	21	1	1	NUM
ejpam-6188	163	22	and	and	CCONJ
ejpam-6188	163	23	2	2	NUM
ejpam-6188	163	24	in	in	ADP
ejpam-6188	163	25	v	v	NUM
ejpam-6188	163	26	(	(	PUNCT
ejpam-6188	163	27	pn	pn	NOUN
ejpam-6188	163	28	)	)	PUNCT
ejpam-6188	163	29	.	.	PUNCT
ejpam-6188	164	1	thus	thus	ADV
ejpam-6188	164	2	|c1|	|c1|	VERB
ejpam-6188	164	3	≥	≥	NUM
ejpam-6188	164	4	3	3	NUM
ejpam-6188	164	5	and	and	CCONJ
ejpam-6188	164	6	|c2|	|c2|	VERB
ejpam-6188	164	7	≥	≥	NOUN
ejpam-6188	164	8	3	3	NUM
ejpam-6188	164	9	.	.	PUNCT
ejpam-6188	165	1	there	there	PRON
ejpam-6188	165	2	are	be	VERB
ejpam-6188	165	3	2	2	NUM
ejpam-6188	165	4	vertices	vertex	NOUN
ejpam-6188	165	5	of	of	ADP
ejpam-6188	165	6	color	color	NOUN
ejpam-6188	165	7	5	5	NUM
ejpam-6188	165	8	.	.	PUNCT
ejpam-6188	165	9	since	since	SCONJ
ejpam-6188	165	10	vk−1	vk−1	NOUN
ejpam-6188	165	11	and	and	CCONJ
ejpam-6188	165	12	u	u	NOUN
ejpam-6188	165	13	can	can	AUX
ejpam-6188	165	14	not	not	PART
ejpam-6188	165	15	be	be	AUX
ejpam-6188	165	16	a	a	DET
ejpam-6188	165	17	b	b	NOUN
ejpam-6188	165	18	-	-	PUNCT
ejpam-6188	165	19	dominating	dominate	VERB
ejpam-6188	165	20	vertex	vertex	NOUN
ejpam-6188	165	21	of	of	ADP
ejpam-6188	165	22	color	color	NOUN
ejpam-6188	165	23	1	1	NUM
ejpam-6188	165	24	nor	nor	CCONJ
ejpam-6188	165	25	2	2	NUM
ejpam-6188	165	26	,	,	PUNCT
ejpam-6188	165	27	we	we	PRON
ejpam-6188	165	28	have	have	VERB
ejpam-6188	165	29	j	j	PROPN
ejpam-6188	165	30	∈	∈	PROPN
ejpam-6188	165	31	{	{	PUNCT
ejpam-6188	165	32	2	2	NUM
ejpam-6188	165	33	,	,	PUNCT
ejpam-6188	165	34	3	3	NUM
ejpam-6188	165	35	,	,	PUNCT
ejpam-6188	165	36	.	.	PUNCT
ejpam-6188	165	37	.	.	PUNCT
ejpam-6188	165	38	.	.	PUNCT
ejpam-6188	166	1	,	,	PUNCT
ejpam-6188	166	2	n	n	CCONJ
ejpam-6188	166	3	−	−	PROPN
ejpam-6188	166	4	1	1	NUM
ejpam-6188	166	5	}	}	PUNCT
ejpam-6188	166	6	and	and	CCONJ
ejpam-6188	166	7	the	the	DET
ejpam-6188	166	8	vertices	vertex	NOUN
ejpam-6188	166	9	vj−1	vj−1	PROPN
ejpam-6188	166	10	and	and	CCONJ
ejpam-6188	166	11	vj+1	vj+1	PROPN
ejpam-6188	166	12	are	be	AUX
ejpam-6188	166	13	the	the	DET
ejpam-6188	166	14	b	b	NOUN
ejpam-6188	166	15	-	-	PUNCT
ejpam-6188	166	16	dominating	dominating	NOUN
ejpam-6188	166	17	vertices	vertex	NOUN
ejpam-6188	166	18	of	of	ADP
ejpam-6188	166	19	color	color	NOUN
ejpam-6188	166	20	1	1	NUM
ejpam-6188	166	21	and	and	CCONJ
ejpam-6188	166	22	2	2	NUM
ejpam-6188	166	23	.	.	X
ejpam-6188	167	1	it	it	PRON
ejpam-6188	167	2	remains	remain	VERB
ejpam-6188	167	3	3	3	NUM
ejpam-6188	167	4	b	b	X
ejpam-6188	167	5	-	-	PUNCT
ejpam-6188	167	6	dominating	dominating	NOUN
ejpam-6188	167	7	vertices	vertex	NOUN
ejpam-6188	167	8	that	that	PRON
ejpam-6188	167	9	have	have	VERB
ejpam-6188	167	10	to	to	PART
ejpam-6188	167	11	be	be	AUX
ejpam-6188	167	12	adjacent	adjacent	ADJ
ejpam-6188	167	13	to	to	ADP
ejpam-6188	167	14	a	a	DET
ejpam-6188	167	15	vertex	vertex	NOUN
ejpam-6188	167	16	of	of	ADP
ejpam-6188	167	17	color	color	NOUN
ejpam-6188	167	18	4	4	NUM
ejpam-6188	167	19	and	and	CCONJ
ejpam-6188	167	20	they	they	PRON
ejpam-6188	167	21	have	have	VERB
ejpam-6188	167	22	no	no	DET
ejpam-6188	167	23	common	common	ADJ
ejpam-6188	167	24	neighbor	neighbor	NOUN
ejpam-6188	167	25	.	.	PUNCT
ejpam-6188	168	1	so	so	ADV
ejpam-6188	168	2	,	,	PUNCT
ejpam-6188	168	3	it	it	PRON
ejpam-6188	168	4	requires	require	VERB
ejpam-6188	168	5	at	at	ADV
ejpam-6188	168	6	least	least	ADV
ejpam-6188	168	7	2	2	NUM
ejpam-6188	168	8	more	more	ADJ
ejpam-6188	168	9	vertices	vertex	NOUN
ejpam-6188	168	10	of	of	ADP
ejpam-6188	168	11	color	color	NOUN
ejpam-6188	168	12	4	4	NUM
ejpam-6188	168	13	.	.	PUNCT
ejpam-6188	169	1	now	now	ADV
ejpam-6188	169	2	|c4|	|c4|	VERB
ejpam-6188	169	3	≥	≥	NUM
ejpam-6188	169	4	3	3	NUM
ejpam-6188	169	5	.	.	PUNCT
ejpam-6188	170	1	therefore	therefore	ADV
ejpam-6188	170	2	,	,	PUNCT
ejpam-6188	170	3	|ci|	|ci|	PROPN
ejpam-6188	170	4	≥	≥	NUM
ejpam-6188	170	5	3	3	NUM
ejpam-6188	170	6	for	for	ADP
ejpam-6188	170	7	i	i	PRON
ejpam-6188	170	8	=	=	NOUN
ejpam-6188	170	9	1	1	NUM
ejpam-6188	170	10	,	,	PUNCT
ejpam-6188	170	11	.	.	PUNCT
ejpam-6188	170	12	.	.	PUNCT
ejpam-6188	171	1	.	.	PUNCT
ejpam-6188	172	1	,	,	PUNCT
ejpam-6188	173	1	4	4	X
ejpam-6188	173	2	.	.	PUNCT
ejpam-6188	173	3	this	this	PRON
ejpam-6188	173	4	completes	complete	VERB
ejpam-6188	173	5	the	the	DET
ejpam-6188	173	6	proof	proof	NOUN
ejpam-6188	173	7	.	.	PUNCT
ejpam-6188	174	1	u	u	NOUN
ejpam-6188	174	2	3	3	NUM
ejpam-6188	174	3	uk−2	uk−2	PROPN
ejpam-6188	174	4	uk−1	uk−1	PROPN
ejpam-6188	174	5	1	1	NUM
ejpam-6188	174	6	uk	uk	PROPN
ejpam-6188	174	7	1	1	NUM
ejpam-6188	174	8	uk+1	uk+1	SYM
ejpam-6188	174	9	2	2	NUM
ejpam-6188	174	10	uk+2	uk+2	SYM
ejpam-6188	174	11	2	2	NUM
ejpam-6188	174	12	vk−2	vk−2	NOUN
ejpam-6188	174	13	vk−1	vk−1	NOUN
ejpam-6188	174	14	3	3	NUM
ejpam-6188	174	15	vk	vk	ADP
ejpam-6188	174	16	5	5	NUM
ejpam-6188	174	17	vk+1	vk+1	NUM
ejpam-6188	174	18	4	4	NUM
ejpam-6188	174	19	vk+2	vk+2	NUM
ejpam-6188	174	20	3	3	NUM
ejpam-6188	174	21	figure	figure	NOUN
ejpam-6188	174	22	3	3	NUM
ejpam-6188	174	23	:	:	PUNCT
ejpam-6188	174	24	a	a	DET
ejpam-6188	174	25	coloring	coloring	NOUN
ejpam-6188	174	26	in	in	ADP
ejpam-6188	174	27	lemma	lemma	PROPN
ejpam-6188	174	28	3	3	NUM
ejpam-6188	174	29	for	for	ADP
ejpam-6188	174	30	n	n	X
ejpam-6188	174	31	≥	≥	NOUN
ejpam-6188	174	32	8	8	NUM
ejpam-6188	174	33	,	,	PUNCT
ejpam-6188	174	34	we	we	PRON
ejpam-6188	174	35	consider	consider	VERB
ejpam-6188	174	36	a	a	DET
ejpam-6188	174	37	b	b	NOUN
ejpam-6188	174	38	-	-	PUNCT
ejpam-6188	174	39	coloring	color	VERB
ejpam-6188	174	40	cn	cn	PROPN
ejpam-6188	174	41	giving	give	VERB
ejpam-6188	174	42	the	the	DET
ejpam-6188	174	43	b	b	NOUN
ejpam-6188	174	44	-	-	PUNCT
ejpam-6188	174	45	chromatic	chromatic	ADJ
ejpam-6188	174	46	sum	sum	NOUN
ejpam-6188	174	47	of	of	ADP
ejpam-6188	174	48	µ(pn	µ(pn	NOUN
ejpam-6188	174	49	)	)	PUNCT
ejpam-6188	174	50	.	.	PUNCT
ejpam-6188	175	1	if	if	SCONJ
ejpam-6188	175	2	|c5|	|c5|	NOUN
ejpam-6188	175	3	=	=	SYM
ejpam-6188	175	4	2	2	NUM
ejpam-6188	175	5	,	,	PUNCT
ejpam-6188	175	6	then	then	ADV
ejpam-6188	175	7	∑	∑	PUNCT
ejpam-6188	175	8	v∈v	v∈v	NOUN
ejpam-6188	175	9	(	(	PUNCT
ejpam-6188	175	10	µ(pn	µ(pn	NOUN
ejpam-6188	175	11	)	)	PUNCT
ejpam-6188	175	12	)	)	PUNCT
ejpam-6188	175	13	cn(v	cn(v	VERB
ejpam-6188	175	14	)	)	PUNCT
ejpam-6188	175	15	≥	≥	NOUN
ejpam-6188	175	16	5	5	NUM
ejpam-6188	175	17	·	·	SYM
ejpam-6188	175	18	2	2	NUM
ejpam-6188	175	19	+	+	CCONJ
ejpam-6188	175	20	4	4	NUM
ejpam-6188	175	21	·	·	SYM
ejpam-6188	175	22	3	3	NUM
ejpam-6188	175	23	+	+	CCONJ
ejpam-6188	175	24	3	3	NUM
ejpam-6188	175	25	·	·	SYM
ejpam-6188	175	26	3	3	NUM
ejpam-6188	175	27	+	+	CCONJ
ejpam-6188	175	28	2	2	NUM
ejpam-6188	175	29	·	·	PUNCT
ejpam-6188	175	30	(	(	PUNCT
ejpam-6188	175	31	2n−	2n−	PROPN
ejpam-6188	175	32	7−	7−	NUM
ejpam-6188	175	33	|c1|	|c1|	NOUN
ejpam-6188	175	34	)	)	PUNCT
ejpam-6188	175	35	+	+	CCONJ
ejpam-6188	175	36	|c1|	|c1|	NOUN
ejpam-6188	175	37	=	=	SYM
ejpam-6188	175	38	4n−	4n−	PROPN
ejpam-6188	175	39	|c1|+	|c1|+	NOUN
ejpam-6188	175	40	17	17	NUM
ejpam-6188	175	41	.	.	PUNCT
ejpam-6188	176	1	if	if	SCONJ
ejpam-6188	176	2	|c5|	|c5|	NOUN
ejpam-6188	176	3	≥	≥	NOUN
ejpam-6188	176	4	3	3	NUM
ejpam-6188	176	5	,	,	PUNCT
ejpam-6188	176	6	then∑	then∑	NOUN
ejpam-6188	176	7	v∈v	v∈v	NOUN
ejpam-6188	176	8	(	(	PUNCT
ejpam-6188	176	9	µ(pn	µ(pn	NOUN
ejpam-6188	176	10	)	)	PUNCT
ejpam-6188	176	11	)	)	PUNCT
ejpam-6188	176	12	cn(v	cn(v	VERB
ejpam-6188	176	13	)	)	PUNCT
ejpam-6188	176	14	≥	≥	NOUN
ejpam-6188	176	15	5	5	NUM
ejpam-6188	176	16	·	·	SYM
ejpam-6188	176	17	3	3	NUM
ejpam-6188	177	1	+	+	CCONJ
ejpam-6188	177	2	4	4	NUM
ejpam-6188	177	3	·	·	SYM
ejpam-6188	177	4	2	2	NUM
ejpam-6188	177	5	+	+	CCONJ
ejpam-6188	177	6	3	3	NUM
ejpam-6188	177	7	·	·	SYM
ejpam-6188	177	8	3	3	NUM
ejpam-6188	177	9	+	+	CCONJ
ejpam-6188	177	10	2	2	NUM
ejpam-6188	177	11	·	·	PUNCT
ejpam-6188	177	12	(	(	PUNCT
ejpam-6188	177	13	2n−	2n−	PROPN
ejpam-6188	177	14	7−	7−	NUM
ejpam-6188	177	15	|c1|	|c1|	NOUN
ejpam-6188	177	16	)	)	PUNCT
ejpam-6188	178	1	+	+	CCONJ
ejpam-6188	178	2	|c1|	|c1|	NOUN
ejpam-6188	178	3	=	=	SYM
ejpam-6188	178	4	4n−	4n−	PROPN
ejpam-6188	178	5	|c1|+	|c1|+	NOUN
ejpam-6188	178	6	18	18	NUM
ejpam-6188	178	7	.	.	PUNCT
ejpam-6188	179	1	p.	p.	NOUN
ejpam-6188	179	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	179	3	et	et	PROPN
ejpam-6188	179	4	al	al	PROPN
ejpam-6188	179	5	.	.	PUNCT
ejpam-6188	179	6	/	/	SYM
ejpam-6188	179	7	eur	eur	PROPN
ejpam-6188	179	8	.	.	PUNCT
ejpam-6188	180	1	j.	j.	PROPN
ejpam-6188	180	2	pure	pure	PROPN
ejpam-6188	180	3	appl	appl	PROPN
ejpam-6188	180	4	.	.	PROPN
ejpam-6188	180	5	math	math	PROPN
ejpam-6188	180	6	,	,	PUNCT
ejpam-6188	180	7	18	18	NUM
ejpam-6188	180	8	(	(	PUNCT
ejpam-6188	180	9	3	3	NUM
ejpam-6188	180	10	)	)	PUNCT
ejpam-6188	180	11	(	(	PUNCT
ejpam-6188	180	12	2025	2025	NUM
ejpam-6188	180	13	)	)	PUNCT
ejpam-6188	180	14	,	,	PUNCT
ejpam-6188	180	15	6188	6188	NUM
ejpam-6188	180	16	7	7	NUM
ejpam-6188	180	17	of	of	ADP
ejpam-6188	180	18	13	13	NUM
ejpam-6188	180	19	lemma	lemma	PROPN
ejpam-6188	180	20	4	4	NUM
ejpam-6188	180	21	.	.	PUNCT
ejpam-6188	180	22	for	for	ADP
ejpam-6188	180	23	n	n	NUM
ejpam-6188	180	24	≥	≥	NOUN
ejpam-6188	180	25	8	8	NUM
ejpam-6188	180	26	,	,	PUNCT
ejpam-6188	180	27	let	let	VERB
ejpam-6188	180	28	cn	cn	PROPN
ejpam-6188	180	29	be	be	AUX
ejpam-6188	180	30	a	a	DET
ejpam-6188	180	31	coloring	coloring	NOUN
ejpam-6188	180	32	giving	give	VERB
ejpam-6188	180	33	the	the	DET
ejpam-6188	180	34	b	b	NOUN
ejpam-6188	180	35	-	-	PUNCT
ejpam-6188	180	36	chromatic	chromatic	ADJ
ejpam-6188	180	37	number	number	NOUN
ejpam-6188	180	38	of	of	ADP
ejpam-6188	180	39	µ(pn	µ(pn	NOUN
ejpam-6188	180	40	)	)	PUNCT
ejpam-6188	180	41	.	.	PUNCT
ejpam-6188	181	1	if	if	SCONJ
ejpam-6188	181	2	there	there	PRON
ejpam-6188	181	3	exists	exist	VERB
ejpam-6188	181	4	m	m	VERB
ejpam-6188	181	5	such	such	ADJ
ejpam-6188	181	6	that	that	SCONJ
ejpam-6188	181	7	|cm|	|cm|	NOUN
ejpam-6188	181	8	=	=	SYM
ejpam-6188	181	9	2	2	NUM
ejpam-6188	181	10	,	,	PUNCT
ejpam-6188	181	11	then	then	ADV
ejpam-6188	181	12	c(u	c(u	PROPN
ejpam-6188	181	13	)	)	PUNCT
ejpam-6188	181	14	̸=	̸=	PROPN
ejpam-6188	181	15	m	m	NOUN
ejpam-6188	181	16	and	and	CCONJ
ejpam-6188	181	17	|cc(u)|	|cc(u)|	NUM
ejpam-6188	181	18	≥	≥	NOUN
ejpam-6188	181	19	4	4	NUM
ejpam-6188	181	20	.	.	PUNCT
ejpam-6188	181	21	proof	proof	NOUN
ejpam-6188	181	22	.	.	PUNCT
ejpam-6188	182	1	let	let	VERB
ejpam-6188	182	2	m	m	PRON
ejpam-6188	182	3	be	be	AUX
ejpam-6188	182	4	such	such	ADJ
ejpam-6188	182	5	that	that	SCONJ
ejpam-6188	182	6	|cm|	|cm|	NOUN
ejpam-6188	182	7	=	=	SYM
ejpam-6188	182	8	2	2	X
ejpam-6188	182	9	.	.	X
ejpam-6188	182	10	from	from	ADP
ejpam-6188	182	11	lemma	lemma	PROPN
ejpam-6188	182	12	1	1	NUM
ejpam-6188	182	13	,	,	PUNCT
ejpam-6188	182	14	we	we	PRON
ejpam-6188	182	15	have	have	VERB
ejpam-6188	182	16	c(u	c(u	PROPN
ejpam-6188	182	17	)	)	PUNCT
ejpam-6188	183	1	̸=	̸=	PROPN
ejpam-6188	183	2	m.	m.	NOUN
ejpam-6188	183	3	without	without	ADP
ejpam-6188	183	4	loss	loss	NOUN
ejpam-6188	183	5	of	of	ADP
ejpam-6188	183	6	generality	generality	NOUN
ejpam-6188	183	7	,	,	PUNCT
ejpam-6188	183	8	we	we	PRON
ejpam-6188	183	9	suppose	suppose	VERB
ejpam-6188	183	10	|c5|	|c5|	NOUN
ejpam-6188	183	11	=	=	SYM
ejpam-6188	183	12	2	2	NUM
ejpam-6188	183	13	and	and	CCONJ
ejpam-6188	183	14	c(u	c(u	PROPN
ejpam-6188	183	15	)	)	PUNCT
ejpam-6188	183	16	=	=	SYM
ejpam-6188	184	1	3	3	X
ejpam-6188	184	2	.	.	PUNCT
ejpam-6188	184	3	by	by	ADP
ejpam-6188	184	4	lemma	lemma	PROPN
ejpam-6188	184	5	3	3	NUM
ejpam-6188	184	6	,	,	PUNCT
ejpam-6188	184	7	we	we	PRON
ejpam-6188	184	8	have	have	VERB
ejpam-6188	184	9	|ck|	|ck|	PROPN
ejpam-6188	184	10	≥	≥	NUM
ejpam-6188	184	11	3	3	NUM
ejpam-6188	184	12	for	for	ADP
ejpam-6188	184	13	k	k	PROPN
ejpam-6188	184	14	̸=	̸=	PROPN
ejpam-6188	184	15	5	5	NUM
ejpam-6188	184	16	.	.	PUNCT
ejpam-6188	185	1	next	next	ADV
ejpam-6188	185	2	,	,	PUNCT
ejpam-6188	185	3	we	we	PRON
ejpam-6188	185	4	suppose	suppose	VERB
ejpam-6188	185	5	to	to	ADP
ejpam-6188	185	6	the	the	DET
ejpam-6188	185	7	contrary	contrary	NOUN
ejpam-6188	185	8	that	that	DET
ejpam-6188	185	9	|c3|	|c3|	NOUN
ejpam-6188	185	10	=	=	NOUN
ejpam-6188	186	1	3	3	X
ejpam-6188	186	2	.	.	X
ejpam-6188	187	1	we	we	PRON
ejpam-6188	187	2	see	see	VERB
ejpam-6188	187	3	that	that	SCONJ
ejpam-6188	187	4	3	3	NUM
ejpam-6188	187	5	̸∈	̸∈	PROPN
ejpam-6188	187	6	cn(un	cn(un	PROPN
ejpam-6188	187	7	)	)	PUNCT
ejpam-6188	187	8	.	.	PUNCT
ejpam-6188	188	1	let	let	VERB
ejpam-6188	188	2	vi	vi	NOUN
ejpam-6188	188	3	,	,	PUNCT
ejpam-6188	188	4	vj	vj	INTJ
ejpam-6188	188	5	be	be	AUX
ejpam-6188	188	6	such	such	ADJ
ejpam-6188	188	7	that	that	SCONJ
ejpam-6188	188	8	cn(vi	cn(vi	PROPN
ejpam-6188	188	9	)	)	PUNCT
ejpam-6188	188	10	=	=	SYM
ejpam-6188	188	11	cn(vj	cn(vj	NOUN
ejpam-6188	188	12	)	)	PUNCT
ejpam-6188	189	1	=	=	SYM
ejpam-6188	190	1	3	3	X
ejpam-6188	190	2	.	.	X
ejpam-6188	190	3	we	we	PRON
ejpam-6188	190	4	have	have	VERB
ejpam-6188	190	5	that	that	PRON
ejpam-6188	190	6	npn({vi	npn({vi	NOUN
ejpam-6188	190	7	,	,	PUNCT
ejpam-6188	190	8	vj	vj	INTJ
ejpam-6188	190	9	}	}	PUNCT
ejpam-6188	190	10	)	)	PUNCT
ejpam-6188	191	1	=	=	PRON
ejpam-6188	191	2	{	{	PUNCT
ejpam-6188	191	3	vi−1	vi−1	PROPN
ejpam-6188	191	4	,	,	PUNCT
ejpam-6188	191	5	vi+1	vi+1	NOUN
ejpam-6188	191	6	,	,	PUNCT
ejpam-6188	191	7	vj−1	vj−1	PROPN
ejpam-6188	191	8	,	,	PUNCT
ejpam-6188	191	9	vj+1	vj+1	PROPN
ejpam-6188	191	10	}	}	PUNCT
ejpam-6188	191	11	consists	consist	VERB
ejpam-6188	191	12	of	of	ADP
ejpam-6188	191	13	b	b	NOUN
ejpam-6188	191	14	-	-	PUNCT
ejpam-6188	191	15	dominating	dominating	NOUN
ejpam-6188	191	16	vertices	vertex	NOUN
ejpam-6188	191	17	of	of	ADP
ejpam-6188	191	18	colors	color	NOUN
ejpam-6188	191	19	1,2,4	1,2,4	NUM
ejpam-6188	191	20	and	and	CCONJ
ejpam-6188	191	21	5	5	NUM
ejpam-6188	191	22	,	,	PUNCT
ejpam-6188	191	23	says	say	VERB
ejpam-6188	191	24	c(vj−1	c(vj−1	NOUN
ejpam-6188	191	25	)	)	PUNCT
ejpam-6188	191	26	=	=	SYM
ejpam-6188	192	1	5	5	X
ejpam-6188	192	2	.	.	PUNCT
ejpam-6188	192	3	since	since	SCONJ
ejpam-6188	192	4	|c5|	|c5|	NOUN
ejpam-6188	192	5	=	=	SYM
ejpam-6188	192	6	2	2	NUM
ejpam-6188	192	7	,	,	PUNCT
ejpam-6188	192	8	either	either	CCONJ
ejpam-6188	192	9	cn(u	cn(u	NOUN
ejpam-6188	192	10	)	)	PUNCT
ejpam-6188	192	11	=	=	SYM
ejpam-6188	192	12	5	5	NUM
ejpam-6188	192	13	or	or	CCONJ
ejpam-6188	192	14	vi+1	vi+1	NOUN
ejpam-6188	192	15	and	and	CCONJ
ejpam-6188	192	16	vj−1	vj−1	PROPN
ejpam-6188	192	17	are	be	AUX
ejpam-6188	192	18	adjacent	adjacent	ADJ
ejpam-6188	192	19	.	.	PUNCT
ejpam-6188	193	1	in	in	ADP
ejpam-6188	193	2	either	either	DET
ejpam-6188	193	3	case	case	NOUN
ejpam-6188	193	4	,	,	PUNCT
ejpam-6188	193	5	we	we	PRON
ejpam-6188	193	6	need	need	VERB
ejpam-6188	193	7	3	3	NUM
ejpam-6188	193	8	vertices	vertex	NOUN
ejpam-6188	193	9	of	of	ADP
ejpam-6188	193	10	color	color	NOUN
ejpam-6188	193	11	5	5	NUM
ejpam-6188	193	12	to	to	PART
ejpam-6188	193	13	complete	complete	VERB
ejpam-6188	193	14	the	the	DET
ejpam-6188	193	15	neighbors	neighbor	NOUN
ejpam-6188	193	16	of	of	ADP
ejpam-6188	193	17	the	the	DET
ejpam-6188	193	18	b	b	NOUN
ejpam-6188	193	19	-	-	PUNCT
ejpam-6188	193	20	vertices	vertex	NOUN
ejpam-6188	193	21	,	,	PUNCT
ejpam-6188	193	22	a	a	DET
ejpam-6188	193	23	contradiction	contradiction	NOUN
ejpam-6188	193	24	.	.	PUNCT
ejpam-6188	194	1	we	we	PRON
ejpam-6188	194	2	note	note	VERB
ejpam-6188	194	3	that	that	SCONJ
ejpam-6188	194	4	,	,	PUNCT
ejpam-6188	194	5	by	by	ADP
ejpam-6188	194	6	using	use	VERB
ejpam-6188	194	7	lemmas	lemmas	PROPN
ejpam-6188	194	8	1–4	1–4	PROPN
ejpam-6188	194	9	,	,	PUNCT
ejpam-6188	194	10	a	a	DET
ejpam-6188	194	11	lower	lower	ADV
ejpam-6188	194	12	bound	bind	VERB
ejpam-6188	194	13	of	of	ADP
ejpam-6188	194	14	φ′(µ(p8	φ′(µ(p8	NOUN
ejpam-6188	194	15	)	)	PUNCT
ejpam-6188	194	16	)	)	PUNCT
ejpam-6188	194	17	can	can	AUX
ejpam-6188	194	18	be	be	AUX
ejpam-6188	194	19	obtained	obtain	VERB
ejpam-6188	194	20	from	from	ADP
ejpam-6188	194	21	a	a	DET
ejpam-6188	194	22	configuration	configuration	NOUN
ejpam-6188	194	23	where	where	SCONJ
ejpam-6188	194	24	|c5|	|c5|	NOUN
ejpam-6188	194	25	=	=	SYM
ejpam-6188	194	26	2	2	NUM
ejpam-6188	194	27	,	,	PUNCT
ejpam-6188	195	1	|c4|	|c4|	NOUN
ejpam-6188	195	2	=	=	PUNCT
ejpam-6188	195	3	|c3|	|c3|	NOUN
ejpam-6188	195	4	=	=	SYM
ejpam-6188	195	5	3	3	X
ejpam-6188	195	6	.	.	PUNCT
ejpam-6188	195	7	from	from	ADP
ejpam-6188	195	8	lemmas	lemmas	PROPN
ejpam-6188	195	9	2	2	NUM
ejpam-6188	195	10	and	and	CCONJ
ejpam-6188	195	11	4	4	NUM
ejpam-6188	195	12	,	,	PUNCT
ejpam-6188	195	13	c8(u	c8(u	NUM
ejpam-6188	195	14	)	)	PUNCT
ejpam-6188	195	15	∈	∈	NOUN
ejpam-6188	195	16	{	{	PUNCT
ejpam-6188	195	17	1	1	NUM
ejpam-6188	195	18	,	,	PUNCT
ejpam-6188	195	19	2	2	NUM
ejpam-6188	195	20	}	}	PUNCT
ejpam-6188	195	21	and	and	CCONJ
ejpam-6188	195	22	4	4	NUM
ejpam-6188	195	23	≤	≤	NOUN
ejpam-6188	195	24	|cc8(u)|	|cc8(u)|	VERB
ejpam-6188	195	25	≤	≤	NUM
ejpam-6188	195	26	⌈	⌈	X
ejpam-6188	195	27	n	n	CCONJ
ejpam-6188	195	28	2	2	NUM
ejpam-6188	195	29	⌉	⌉	NOUN
ejpam-6188	195	30	+	+	CCONJ
ejpam-6188	195	31	1	1	NUM
ejpam-6188	195	32	=	=	SYM
ejpam-6188	195	33	5	5	NUM
ejpam-6188	195	34	.	.	PUNCT
ejpam-6188	196	1	so	so	ADV
ejpam-6188	196	2	,	,	PUNCT
ejpam-6188	196	3	|c2|	|c2|	NOUN
ejpam-6188	196	4	=	=	PUNCT
ejpam-6188	196	5	4	4	NUM
ejpam-6188	196	6	and	and	CCONJ
ejpam-6188	196	7	|c1|	|c1|	PROPN
ejpam-6188	196	8	=	=	SYM
ejpam-6188	196	9	5	5	NUM
ejpam-6188	196	10	give	give	VERB
ejpam-6188	196	11	the	the	DET
ejpam-6188	196	12	lowest	low	ADJ
ejpam-6188	196	13	sum	sum	NOUN
ejpam-6188	196	14	.	.	PUNCT
ejpam-6188	197	1	hence,∑	hence,∑	NOUN
ejpam-6188	197	2	v∈v	v∈v	NOUN
ejpam-6188	197	3	(	(	PUNCT
ejpam-6188	197	4	µ(p8	µ(p8	NOUN
ejpam-6188	197	5	)	)	PUNCT
ejpam-6188	197	6	)	)	PUNCT
ejpam-6188	198	1	c8(v	c8(v	NOUN
ejpam-6188	198	2	)	)	PUNCT
ejpam-6188	198	3	≥	≥	NOUN
ejpam-6188	198	4	5	5	NUM
ejpam-6188	198	5	·	·	SYM
ejpam-6188	198	6	2	2	NUM
ejpam-6188	199	1	+	+	CCONJ
ejpam-6188	199	2	4	4	NUM
ejpam-6188	199	3	·	·	SYM
ejpam-6188	199	4	3	3	NUM
ejpam-6188	199	5	+	+	CCONJ
ejpam-6188	199	6	3	3	NUM
ejpam-6188	199	7	·	·	SYM
ejpam-6188	199	8	3	3	NUM
ejpam-6188	199	9	+	+	CCONJ
ejpam-6188	199	10	2	2	NUM
ejpam-6188	199	11	·	·	SYM
ejpam-6188	199	12	4	4	NUM
ejpam-6188	199	13	+	+	SYM
ejpam-6188	199	14	1	1	NUM
ejpam-6188	199	15	·	·	SYM
ejpam-6188	199	16	5	5	NUM
ejpam-6188	199	17	=	=	SYM
ejpam-6188	199	18	44	44	NUM
ejpam-6188	199	19	.	.	PUNCT
ejpam-6188	199	20	combined	combine	VERB
ejpam-6188	199	21	with	with	ADP
ejpam-6188	199	22	the	the	DET
ejpam-6188	199	23	b	b	NOUN
ejpam-6188	199	24	-	-	PUNCT
ejpam-6188	199	25	coloring	coloring	NOUN
ejpam-6188	199	26	provided	provide	VERB
ejpam-6188	199	27	by	by	ADP
ejpam-6188	199	28	[	[	X
ejpam-6188	199	29	1	1	NUM
ejpam-6188	199	30	]	]	PUNCT
ejpam-6188	200	1	,	,	PUNCT
ejpam-6188	200	2	it	it	PRON
ejpam-6188	200	3	follows	follow	VERB
ejpam-6188	200	4	that	that	PRON
ejpam-6188	200	5	φ′(µ(p8	φ′(µ(p8	NOUN
ejpam-6188	200	6	)	)	PUNCT
ejpam-6188	200	7	)	)	PUNCT
ejpam-6188	201	1	=	=	SYM
ejpam-6188	201	2	44	44	NUM
ejpam-6188	201	3	,	,	PUNCT
ejpam-6188	201	4	which	which	PRON
ejpam-6188	201	5	aligns	align	VERB
ejpam-6188	201	6	with	with	ADP
ejpam-6188	201	7	the	the	DET
ejpam-6188	201	8	result	result	NOUN
ejpam-6188	201	9	in	in	ADP
ejpam-6188	201	10	[	[	X
ejpam-6188	201	11	1	1	NUM
ejpam-6188	201	12	]	]	PUNCT
ejpam-6188	201	13	.	.	PUNCT
ejpam-6188	202	1	for	for	ADP
ejpam-6188	202	2	n	n	PRON
ejpam-6188	202	3	≥	≥	NOUN
ejpam-6188	202	4	9	9	NUM
ejpam-6188	202	5	,	,	PUNCT
ejpam-6188	202	6	lemma	lemma	PROPN
ejpam-6188	202	7	5	5	NUM
ejpam-6188	202	8	will	will	AUX
ejpam-6188	202	9	give	give	VERB
ejpam-6188	202	10	a	a	PRON
ejpam-6188	202	11	better	well	ADV
ejpam-6188	202	12	lower	low	ADJ
ejpam-6188	202	13	bound	bind	VERB
ejpam-6188	202	14	on	on	ADP
ejpam-6188	202	15	the	the	DET
ejpam-6188	202	16	b	b	NOUN
ejpam-6188	202	17	-	-	PUNCT
ejpam-6188	202	18	chromatic	chromatic	ADJ
ejpam-6188	202	19	sum	sum	NOUN
ejpam-6188	202	20	(	(	PUNCT
ejpam-6188	202	21	if	if	SCONJ
ejpam-6188	202	22	not	not	PART
ejpam-6188	202	23	exact	exact	ADJ
ejpam-6188	202	24	)	)	PUNCT
ejpam-6188	202	25	than	than	ADP
ejpam-6188	202	26	using	use	VERB
ejpam-6188	202	27	only	only	ADV
ejpam-6188	202	28	lemmas	lemmas	PROPN
ejpam-6188	202	29	1–4	1–4	PROPN
ejpam-6188	202	30	.	.	PUNCT
ejpam-6188	202	31	lemma	lemma	PROPN
ejpam-6188	202	32	5	5	NUM
ejpam-6188	202	33	.	.	PUNCT
ejpam-6188	202	34	for	for	ADP
ejpam-6188	202	35	n	n	PRON
ejpam-6188	202	36	≥	≥	NOUN
ejpam-6188	202	37	9	9	NUM
ejpam-6188	202	38	,	,	PUNCT
ejpam-6188	202	39	let	let	VERB
ejpam-6188	202	40	cn	cn	PROPN
ejpam-6188	202	41	be	be	AUX
ejpam-6188	202	42	a	a	DET
ejpam-6188	202	43	coloring	coloring	NOUN
ejpam-6188	202	44	giving	give	VERB
ejpam-6188	202	45	the	the	DET
ejpam-6188	202	46	b	b	NOUN
ejpam-6188	202	47	-	-	PUNCT
ejpam-6188	202	48	chromatic	chromatic	ADJ
ejpam-6188	202	49	number	number	NOUN
ejpam-6188	202	50	of	of	ADP
ejpam-6188	202	51	µ(pn	µ(pn	NOUN
ejpam-6188	202	52	)	)	PUNCT
ejpam-6188	202	53	.	.	PUNCT
ejpam-6188	203	1	there	there	PRON
ejpam-6188	203	2	are	be	VERB
ejpam-6188	203	3	at	at	ADP
ejpam-6188	203	4	most	most	ADJ
ejpam-6188	203	5	n−	n−	NOUN
ejpam-6188	203	6	1	1	NUM
ejpam-6188	203	7	vertices	vertex	NOUN
ejpam-6188	203	8	of	of	ADP
ejpam-6188	203	9	color	color	NOUN
ejpam-6188	203	10	1	1	NUM
ejpam-6188	203	11	.	.	PUNCT
ejpam-6188	203	12	proof	proof	NOUN
ejpam-6188	203	13	.	.	PUNCT
ejpam-6188	204	1	if	if	SCONJ
ejpam-6188	204	2	u	u	PROPN
ejpam-6188	204	3	∈	∈	PROPN
ejpam-6188	204	4	c1	c1	NOUN
ejpam-6188	204	5	,	,	PUNCT
ejpam-6188	204	6	then	then	ADV
ejpam-6188	204	7	|c1|	|c1|	VERB
ejpam-6188	204	8	≤	≤	NOUN
ejpam-6188	204	9	1	1	NUM
ejpam-6188	204	10	+	+	CCONJ
ejpam-6188	204	11	⌈	⌈	NUM
ejpam-6188	204	12	n	n	PRON
ejpam-6188	204	13	2	2	NUM
ejpam-6188	204	14	⌉	⌉	PRON
ejpam-6188	204	15	≤	≤	NUM
ejpam-6188	204	16	n−	n−	NOUN
ejpam-6188	204	17	1	1	NUM
ejpam-6188	204	18	.	.	PUNCT
ejpam-6188	204	19	without	without	ADP
ejpam-6188	204	20	loss	loss	NOUN
ejpam-6188	204	21	of	of	ADP
ejpam-6188	204	22	generality	generality	NOUN
ejpam-6188	204	23	,	,	PUNCT
ejpam-6188	204	24	we	we	PRON
ejpam-6188	204	25	assume	assume	VERB
ejpam-6188	204	26	that	that	SCONJ
ejpam-6188	204	27	u	u	PROPN
ejpam-6188	204	28	∈	∈	PROPN
ejpam-6188	204	29	c2	c2	PROPN
ejpam-6188	204	30	.	.	PUNCT
ejpam-6188	205	1	let	let	VERB
ejpam-6188	205	2	vk	vk	PART
ejpam-6188	205	3	be	be	AUX
ejpam-6188	205	4	a	a	DET
ejpam-6188	205	5	b	b	NOUN
ejpam-6188	205	6	-	-	PUNCT
ejpam-6188	205	7	dominating	dominate	VERB
ejpam-6188	205	8	vertex	vertex	NOUN
ejpam-6188	205	9	of	of	ADP
ejpam-6188	205	10	color	color	NOUN
ejpam-6188	205	11	3	3	NUM
ejpam-6188	205	12	where	where	SCONJ
ejpam-6188	205	13	2	2	NUM
ejpam-6188	205	14	≤	≤	NOUN
ejpam-6188	205	15	k	k	NOUN
ejpam-6188	205	16	≤	≤	PROPN
ejpam-6188	205	17	n	n	CCONJ
ejpam-6188	205	18	−	−	PROPN
ejpam-6188	205	19	1	1	NUM
ejpam-6188	205	20	.	.	PUNCT
ejpam-6188	206	1	there	there	PRON
ejpam-6188	206	2	is	be	VERB
ejpam-6188	206	3	only	only	ADV
ejpam-6188	206	4	one	one	NUM
ejpam-6188	206	5	vertex	vertex	NOUN
ejpam-6188	206	6	in	in	ADP
ejpam-6188	206	7	nµ(pn)(vk	nµ(pn)(vk	PROPN
ejpam-6188	206	8	)	)	PUNCT
ejpam-6188	207	1	=	=	PRON
ejpam-6188	207	2	{	{	PUNCT
ejpam-6188	207	3	vk−1	vk−1	NOUN
ejpam-6188	207	4	,	,	PUNCT
ejpam-6188	207	5	vk+1	vk+1	NOUN
ejpam-6188	207	6	,	,	PUNCT
ejpam-6188	207	7	uk−1	uk−1	ADJ
ejpam-6188	207	8	,	,	PUNCT
ejpam-6188	207	9	uk+1	uk+1	VERB
ejpam-6188	207	10	}	}	PUNCT
ejpam-6188	207	11	with	with	ADP
ejpam-6188	207	12	color	color	NOUN
ejpam-6188	207	13	1	1	NUM
ejpam-6188	207	14	.	.	PUNCT
ejpam-6188	208	1	we	we	PRON
ejpam-6188	208	2	consider	consider	VERB
ejpam-6188	208	3	two	two	NUM
ejpam-6188	208	4	cases	case	NOUN
ejpam-6188	208	5	of	of	ADP
ejpam-6188	208	6	positions	position	NOUN
ejpam-6188	208	7	of	of	ADP
ejpam-6188	208	8	1	1	NUM
ejpam-6188	208	9	’s	’s	NOUN
ejpam-6188	208	10	up	up	ADV
ejpam-6188	208	11	to	to	ADP
ejpam-6188	208	12	left	leave	VERB
ejpam-6188	208	13	-	-	PUNCT
ejpam-6188	208	14	right	right	NOUN
ejpam-6188	208	15	reflection	reflection	NOUN
ejpam-6188	208	16	as	as	ADP
ejpam-6188	208	17	in	in	ADP
ejpam-6188	208	18	figure	figure	NOUN
ejpam-6188	208	19	4	4	NUM
ejpam-6188	208	20	.	.	PUNCT
ejpam-6188	209	1	the	the	DET
ejpam-6188	209	2	x	x	PUNCT
ejpam-6188	209	3	marked	mark	VERB
ejpam-6188	209	4	in	in	ADP
ejpam-6188	209	5	all	all	DET
ejpam-6188	209	6	figures	figure	NOUN
ejpam-6188	209	7	mentioned	mention	VERB
ejpam-6188	209	8	in	in	ADP
ejpam-6188	209	9	this	this	DET
ejpam-6188	209	10	proof	proof	NOUN
ejpam-6188	209	11	means	mean	VERB
ejpam-6188	209	12	that	that	SCONJ
ejpam-6188	209	13	the	the	DET
ejpam-6188	209	14	color	color	NOUN
ejpam-6188	209	15	of	of	ADP
ejpam-6188	209	16	such	such	ADJ
ejpam-6188	209	17	vertex	vertex	NOUN
ejpam-6188	209	18	can	can	AUX
ejpam-6188	209	19	not	not	PART
ejpam-6188	209	20	be	be	AUX
ejpam-6188	209	21	1	1	NUM
ejpam-6188	209	22	.	.	PUNCT
ejpam-6188	210	1	let	let	VERB
ejpam-6188	210	2	l	l	NOUN
ejpam-6188	210	3	=	=	PUNCT
ejpam-6188	211	1	k	k	NOUN
ejpam-6188	212	1	−	−	PROPN
ejpam-6188	212	2	2	2	NUM
ejpam-6188	212	3	and	and	CCONJ
ejpam-6188	212	4	r	r	NOUN
ejpam-6188	212	5	=	=	SYM
ejpam-6188	212	6	n	n	PROPN
ejpam-6188	212	7	−	−	PROPN
ejpam-6188	213	1	k.	k.	NOUN
ejpam-6188	214	1	we	we	PRON
ejpam-6188	214	2	have	have	VERB
ejpam-6188	214	3	l	l	NOUN
ejpam-6188	214	4	and	and	CCONJ
ejpam-6188	214	5	r	r	NOUN
ejpam-6188	214	6	columns	column	NOUN
ejpam-6188	214	7	that	that	PRON
ejpam-6188	214	8	possibly	possibly	ADV
ejpam-6188	214	9	contain	contain	VERB
ejpam-6188	214	10	1	1	NUM
ejpam-6188	214	11	on	on	ADP
ejpam-6188	214	12	the	the	DET
ejpam-6188	214	13	left	left	NOUN
ejpam-6188	214	14	and	and	CCONJ
ejpam-6188	214	15	right	right	NOUN
ejpam-6188	214	16	of	of	ADP
ejpam-6188	214	17	vk	vk	PROPN
ejpam-6188	214	18	,	,	PUNCT
ejpam-6188	214	19	respectively	respectively	ADV
ejpam-6188	214	20	.	.	PUNCT
ejpam-6188	215	1	case	case	NOUN
ejpam-6188	215	2	1	1	NUM
ejpam-6188	215	3	.	.	PUNCT
ejpam-6188	215	4	cn(vk−1	cn(vk−1	NOUN
ejpam-6188	215	5	)	)	PUNCT
ejpam-6188	216	1	=	=	SYM
ejpam-6188	216	2	1	1	NUM
ejpam-6188	216	3	or	or	CCONJ
ejpam-6188	216	4	cn(vk+1	cn(vk+1	VERB
ejpam-6188	216	5	)	)	PUNCT
ejpam-6188	216	6	=	=	SYM
ejpam-6188	216	7	1	1	X
ejpam-6188	216	8	.	.	NOUN
ejpam-6188	216	9	without	without	ADP
ejpam-6188	216	10	loss	loss	NOUN
ejpam-6188	216	11	of	of	ADP
ejpam-6188	216	12	generality	generality	NOUN
ejpam-6188	216	13	,	,	PUNCT
ejpam-6188	216	14	we	we	PRON
ejpam-6188	216	15	suppose	suppose	VERB
ejpam-6188	216	16	that	that	SCONJ
ejpam-6188	216	17	cn(vk+1	cn(vk+1	VERB
ejpam-6188	216	18	)	)	PUNCT
ejpam-6188	216	19	=	=	SYM
ejpam-6188	216	20	1	1	NUM
ejpam-6188	216	21	as	as	SCONJ
ejpam-6188	216	22	shown	show	VERB
ejpam-6188	216	23	in	in	ADP
ejpam-6188	216	24	figure	figure	NOUN
ejpam-6188	216	25	4	4	NUM
ejpam-6188	216	26	(	(	PUNCT
ejpam-6188	216	27	the	the	DET
ejpam-6188	216	28	left	left	ADJ
ejpam-6188	216	29	figure	figure	NOUN
ejpam-6188	216	30	)	)	PUNCT
ejpam-6188	216	31	.	.	PUNCT
ejpam-6188	217	1	the	the	DET
ejpam-6188	217	2	left	left	ADJ
ejpam-6188	217	3	side	side	NOUN
ejpam-6188	217	4	of	of	ADP
ejpam-6188	217	5	vk	vk	NOUN
ejpam-6188	217	6	contains	contain	VERB
ejpam-6188	217	7	at	at	ADP
ejpam-6188	217	8	most	most	ADJ
ejpam-6188	217	9	l	l	NOUN
ejpam-6188	217	10	of	of	ADP
ejpam-6188	217	11	1	1	NUM
ejpam-6188	217	12	’s	’s	NOUN
ejpam-6188	217	13	when	when	SCONJ
ejpam-6188	217	14	l	l	NOUN
ejpam-6188	217	15	is	be	AUX
ejpam-6188	217	16	even	even	ADV
ejpam-6188	217	17	,	,	PUNCT
ejpam-6188	217	18	and	and	CCONJ
ejpam-6188	217	19	l+1	l+1	PROPN
ejpam-6188	217	20	of	of	ADP
ejpam-6188	217	21	1	1	NUM
ejpam-6188	217	22	’s	’s	NOUN
ejpam-6188	217	23	when	when	SCONJ
ejpam-6188	217	24	l	l	NOUN
ejpam-6188	217	25	is	be	AUX
ejpam-6188	217	26	odd	odd	ADJ
ejpam-6188	217	27	.	.	PUNCT
ejpam-6188	218	1	the	the	DET
ejpam-6188	218	2	right	right	ADJ
ejpam-6188	218	3	side	side	NOUN
ejpam-6188	218	4	of	of	ADP
ejpam-6188	218	5	vk	vk	NOUN
ejpam-6188	218	6	contains	contain	VERB
ejpam-6188	218	7	at	at	ADP
ejpam-6188	218	8	most	most	ADJ
ejpam-6188	218	9	r	r	NOUN
ejpam-6188	218	10	of	of	ADP
ejpam-6188	218	11	1	1	NUM
ejpam-6188	218	12	’s	’s	NOUN
ejpam-6188	218	13	.	.	PUNCT
ejpam-6188	219	1	hence	hence	ADV
ejpam-6188	219	2	|c1|	|c1|	VERB
ejpam-6188	219	3	≤	≤	NOUN
ejpam-6188	219	4	(	(	PUNCT
ejpam-6188	219	5	l	l	NOUN
ejpam-6188	219	6	+	+	NOUN
ejpam-6188	219	7	1	1	NUM
ejpam-6188	219	8	)	)	PUNCT
ejpam-6188	219	9	+	+	CCONJ
ejpam-6188	219	10	r	r	NOUN
ejpam-6188	219	11	=	=	SYM
ejpam-6188	219	12	n−	n−	NOUN
ejpam-6188	219	13	1	1	NUM
ejpam-6188	219	14	.	.	PUNCT
ejpam-6188	219	15	case	case	NOUN
ejpam-6188	219	16	2	2	NUM
ejpam-6188	219	17	.	.	X
ejpam-6188	219	18	cn(uk−1	cn(uk−1	X
ejpam-6188	219	19	)	)	PUNCT
ejpam-6188	219	20	=	=	SYM
ejpam-6188	219	21	1	1	NUM
ejpam-6188	219	22	or	or	CCONJ
ejpam-6188	219	23	cn(uk+1	cn(uk+1	NOUN
ejpam-6188	219	24	)	)	PUNCT
ejpam-6188	219	25	=	=	SYM
ejpam-6188	219	26	1	1	X
ejpam-6188	219	27	.	.	NOUN
ejpam-6188	219	28	without	without	ADP
ejpam-6188	219	29	loss	loss	NOUN
ejpam-6188	219	30	of	of	ADP
ejpam-6188	219	31	generality	generality	NOUN
ejpam-6188	219	32	,	,	PUNCT
ejpam-6188	219	33	we	we	PRON
ejpam-6188	219	34	suppose	suppose	VERB
ejpam-6188	219	35	that	that	SCONJ
ejpam-6188	219	36	cn(uk+1	cn(uk+1	VERB
ejpam-6188	219	37	)	)	PUNCT
ejpam-6188	219	38	=	=	SYM
ejpam-6188	219	39	1	1	NUM
ejpam-6188	219	40	as	as	SCONJ
ejpam-6188	219	41	shown	show	VERB
ejpam-6188	219	42	in	in	ADP
ejpam-6188	219	43	figure	figure	NOUN
ejpam-6188	219	44	4	4	NUM
ejpam-6188	219	45	(	(	PUNCT
ejpam-6188	219	46	the	the	DET
ejpam-6188	219	47	right	right	ADJ
ejpam-6188	219	48	figure	figure	NOUN
ejpam-6188	219	49	)	)	PUNCT
ejpam-6188	219	50	.	.	PUNCT
ejpam-6188	220	1	there	there	PRON
ejpam-6188	220	2	are	be	VERB
ejpam-6188	220	3	at	at	ADP
ejpam-6188	220	4	most	most	ADJ
ejpam-6188	220	5	l+	l+	NOUN
ejpam-6188	220	6	1	1	NUM
ejpam-6188	220	7	and	and	CCONJ
ejpam-6188	220	8	r	r	NOUN
ejpam-6188	220	9	of	of	ADP
ejpam-6188	220	10	1	1	NUM
ejpam-6188	220	11	’s	’s	NOUN
ejpam-6188	220	12	on	on	ADP
ejpam-6188	220	13	the	the	DET
ejpam-6188	220	14	left	left	NOUN
ejpam-6188	220	15	and	and	CCONJ
ejpam-6188	220	16	right	right	NOUN
ejpam-6188	220	17	of	of	ADP
ejpam-6188	220	18	vk	vk	PROPN
ejpam-6188	220	19	,	,	PUNCT
ejpam-6188	220	20	respectively	respectively	ADV
ejpam-6188	220	21	.	.	PUNCT
ejpam-6188	221	1	in	in	ADP
ejpam-6188	221	2	this	this	DET
ejpam-6188	221	3	case	case	NOUN
ejpam-6188	221	4	,	,	PUNCT
ejpam-6188	221	5	it	it	PRON
ejpam-6188	221	6	is	be	AUX
ejpam-6188	221	7	possible	possible	ADJ
ejpam-6188	221	8	that	that	SCONJ
ejpam-6188	221	9	cn(uk	cn(uk	ADJ
ejpam-6188	221	10	)	)	PUNCT
ejpam-6188	222	1	=	=	SYM
ejpam-6188	222	2	1	1	X
ejpam-6188	222	3	.	.	PUNCT
ejpam-6188	222	4	hence	hence	ADV
ejpam-6188	222	5	|c1|	|c1|	VERB
ejpam-6188	222	6	≤	≤	NOUN
ejpam-6188	222	7	(	(	PUNCT
ejpam-6188	222	8	l+1)+	l+1)+	PROPN
ejpam-6188	223	1	r+1	r+1	PROPN
ejpam-6188	223	2	=	=	PUNCT
ejpam-6188	223	3	n.	n.	PROPN
ejpam-6188	223	4	suppose	suppose	VERB
ejpam-6188	223	5	to	to	ADP
ejpam-6188	223	6	a	a	DET
ejpam-6188	223	7	contrary	contrary	NOUN
ejpam-6188	223	8	that	that	SCONJ
ejpam-6188	223	9	|c1|	|c1|	PROPN
ejpam-6188	223	10	=	=	SYM
ejpam-6188	223	11	n.	n.	NOUN
ejpam-6188	223	12	it	it	PRON
ejpam-6188	223	13	implies	imply	VERB
ejpam-6188	223	14	that	that	SCONJ
ejpam-6188	223	15	there	there	PRON
ejpam-6188	223	16	are	be	VERB
ejpam-6188	223	17	exactly	exactly	ADV
ejpam-6188	223	18	l+1	l+1	ADJ
ejpam-6188	223	19	of	of	ADP
ejpam-6188	223	20	1	1	NUM
ejpam-6188	223	21	’s	’s	NOUN
ejpam-6188	223	22	on	on	ADP
ejpam-6188	223	23	the	the	DET
ejpam-6188	223	24	left	left	NOUN
ejpam-6188	223	25	of	of	ADP
ejpam-6188	223	26	vk	vk	NOUN
ejpam-6188	223	27	.	.	PUNCT
ejpam-6188	224	1	thus	thus	ADV
ejpam-6188	224	2	l	l	NOUN
ejpam-6188	224	3	is	be	AUX
ejpam-6188	224	4	odd	odd	ADJ
ejpam-6188	224	5	and	and	CCONJ
ejpam-6188	224	6	the	the	DET
ejpam-6188	224	7	positions	position	NOUN
ejpam-6188	224	8	of	of	ADP
ejpam-6188	224	9	1	1	NUM
ejpam-6188	224	10	’s	’s	PART
ejpam-6188	224	11	must	must	AUX
ejpam-6188	224	12	appear	appear	VERB
ejpam-6188	224	13	as	as	ADP
ejpam-6188	224	14	in	in	ADP
ejpam-6188	224	15	figure	figure	NOUN
ejpam-6188	224	16	5	5	NUM
ejpam-6188	224	17	.	.	PUNCT
ejpam-6188	225	1	it	it	PRON
ejpam-6188	225	2	follows	follow	VERB
ejpam-6188	225	3	that	that	SCONJ
ejpam-6188	225	4	there	there	PRON
ejpam-6188	225	5	are	be	VERB
ejpam-6188	225	6	no	no	DET
ejpam-6188	225	7	b	b	NOUN
ejpam-6188	225	8	-	-	PUNCT
ejpam-6188	225	9	dominating	dominate	VERB
ejpam-6188	225	10	vertex	vertex	NOUN
ejpam-6188	225	11	of	of	ADP
ejpam-6188	225	12	color	color	NOUN
ejpam-6188	225	13	4	4	NUM
ejpam-6188	225	14	and	and	CCONJ
ejpam-6188	225	15	5	5	NUM
ejpam-6188	225	16	in	in	ADP
ejpam-6188	225	17	these	these	DET
ejpam-6188	225	18	columns	column	NOUN
ejpam-6188	225	19	.	.	PUNCT
ejpam-6188	226	1	thus	thus	ADV
ejpam-6188	226	2	,	,	PUNCT
ejpam-6188	226	3	the	the	DET
ejpam-6188	226	4	b	b	NOUN
ejpam-6188	226	5	-	-	PUNCT
ejpam-6188	226	6	dominating	dominating	NOUN
ejpam-6188	226	7	vertices	vertex	NOUN
ejpam-6188	226	8	of	of	ADP
ejpam-6188	226	9	4	4	NUM
ejpam-6188	226	10	and	and	CCONJ
ejpam-6188	226	11	5	5	NUM
ejpam-6188	226	12	are	be	AUX
ejpam-6188	226	13	on	on	ADP
ejpam-6188	226	14	the	the	DET
ejpam-6188	226	15	right	right	ADJ
ejpam-6188	226	16	side	side	NOUN
ejpam-6188	226	17	of	of	ADP
ejpam-6188	226	18	vk	vk	PROPN
ejpam-6188	226	19	.	.	PUNCT
ejpam-6188	227	1	since	since	SCONJ
ejpam-6188	227	2	there	there	PRON
ejpam-6188	227	3	are	be	VERB
ejpam-6188	227	4	exactly	exactly	ADV
ejpam-6188	227	5	r	r	NOUN
ejpam-6188	227	6	vertices	vertex	NOUN
ejpam-6188	227	7	of	of	ADP
ejpam-6188	227	8	color	color	NOUN
ejpam-6188	227	9	1	1	NUM
ejpam-6188	227	10	on	on	ADP
ejpam-6188	227	11	the	the	DET
ejpam-6188	227	12	p.	p.	NOUN
ejpam-6188	227	13	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	227	14	et	et	PROPN
ejpam-6188	227	15	al	al	PROPN
ejpam-6188	227	16	.	.	PUNCT
ejpam-6188	227	17	/	/	SYM
ejpam-6188	227	18	eur	eur	PROPN
ejpam-6188	227	19	.	.	PUNCT
ejpam-6188	228	1	j.	j.	PROPN
ejpam-6188	228	2	pure	pure	PROPN
ejpam-6188	228	3	appl	appl	PROPN
ejpam-6188	228	4	.	.	PROPN
ejpam-6188	228	5	math	math	PROPN
ejpam-6188	228	6	,	,	PUNCT
ejpam-6188	228	7	18	18	NUM
ejpam-6188	228	8	(	(	PUNCT
ejpam-6188	228	9	3	3	NUM
ejpam-6188	228	10	)	)	PUNCT
ejpam-6188	228	11	(	(	PUNCT
ejpam-6188	228	12	2025	2025	NUM
ejpam-6188	228	13	)	)	PUNCT
ejpam-6188	228	14	,	,	PUNCT
ejpam-6188	228	15	6188	6188	NUM
ejpam-6188	228	16	8	8	NUM
ejpam-6188	228	17	of	of	ADP
ejpam-6188	228	18	13	13	NUM
ejpam-6188	228	19	right	right	NOUN
ejpam-6188	228	20	of	of	ADP
ejpam-6188	228	21	vk	vk	NOUN
ejpam-6188	228	22	,	,	PUNCT
ejpam-6188	228	23	there	there	PRON
ejpam-6188	228	24	is	be	VERB
ejpam-6188	228	25	only	only	ADV
ejpam-6188	228	26	one	one	NUM
ejpam-6188	228	27	vertex	vertex	NOUN
ejpam-6188	228	28	that	that	PRON
ejpam-6188	228	29	can	can	AUX
ejpam-6188	228	30	possibly	possibly	ADV
ejpam-6188	228	31	be	be	AUX
ejpam-6188	228	32	the	the	DET
ejpam-6188	228	33	b	b	NOUN
ejpam-6188	228	34	-	-	PUNCT
ejpam-6188	228	35	dominating	dominate	VERB
ejpam-6188	228	36	vertex	vertex	NOUN
ejpam-6188	228	37	of	of	ADP
ejpam-6188	228	38	both	both	DET
ejpam-6188	228	39	colors	color	NOUN
ejpam-6188	228	40	4	4	NUM
ejpam-6188	228	41	and	and	CCONJ
ejpam-6188	228	42	5	5	NUM
ejpam-6188	228	43	,	,	PUNCT
ejpam-6188	228	44	as	as	SCONJ
ejpam-6188	228	45	shown	show	VERB
ejpam-6188	228	46	in	in	ADP
ejpam-6188	228	47	figure	figure	NOUN
ejpam-6188	228	48	6	6	NUM
ejpam-6188	228	49	.	.	PUNCT
ejpam-6188	229	1	this	this	PRON
ejpam-6188	229	2	leads	lead	VERB
ejpam-6188	229	3	to	to	ADP
ejpam-6188	229	4	a	a	DET
ejpam-6188	229	5	contradiction	contradiction	NOUN
ejpam-6188	229	6	.	.	PUNCT
ejpam-6188	230	1	therefore	therefore	ADV
ejpam-6188	230	2	|c1|	|c1|	VERB
ejpam-6188	230	3	≤	≤	ADJ
ejpam-6188	230	4	n−	n−	PROPN
ejpam-6188	230	5	1	1	NUM
ejpam-6188	230	6	.	.	PUNCT
ejpam-6188	231	1	uk−1	uk−1	ADJ
ejpam-6188	231	2	x	x	SYM
ejpam-6188	231	3	uk	uk	PROPN
ejpam-6188	231	4	x	x	SYM
ejpam-6188	231	5	uk+1	uk+1	X
ejpam-6188	231	6	x	x	SYM
ejpam-6188	231	7	uk+2	uk+2	PROPN
ejpam-6188	231	8	x	x	PART
ejpam-6188	231	9	vk−1	vk−1	VERB
ejpam-6188	231	10	x	x	SYM
ejpam-6188	231	11	vk	vk	ADP
ejpam-6188	231	12	3	3	NUM
ejpam-6188	231	13	vk+1	vk+1	NOUN
ejpam-6188	231	14	1	1	NUM
ejpam-6188	231	15	vk+2	vk+2	NUM
ejpam-6188	231	16	x	x	NOUN
ejpam-6188	231	17	uk−1	uk−1	PROPN
ejpam-6188	231	18	x	x	VERB
ejpam-6188	231	19	uk	uk	PROPN
ejpam-6188	231	20	uk+1	uk+1	ADP
ejpam-6188	231	21	1	1	NUM
ejpam-6188	231	22	uk+2	uk+2	NOUN
ejpam-6188	231	23	vk−1	vk−1	NOUN
ejpam-6188	231	24	x	x	SYM
ejpam-6188	231	25	vk	vk	ADP
ejpam-6188	231	26	3	3	NUM
ejpam-6188	231	27	vk+1	vk+1	NOUN
ejpam-6188	231	28	x	x	SYM
ejpam-6188	231	29	vk+2	vk+2	NUM
ejpam-6188	231	30	x	x	NOUN
ejpam-6188	231	31	figure	figure	NOUN
ejpam-6188	231	32	4	4	NUM
ejpam-6188	231	33	:	:	PUNCT
ejpam-6188	231	34	positions	position	NOUN
ejpam-6188	231	35	of	of	ADP
ejpam-6188	231	36	1	1	NUM
ejpam-6188	231	37	’s	’s	NOUN
ejpam-6188	231	38	in	in	ADP
ejpam-6188	231	39	each	each	DET
ejpam-6188	231	40	type	type	NOUN
ejpam-6188	231	41	of	of	ADP
ejpam-6188	231	42	b	b	NOUN
ejpam-6188	231	43	-	-	PUNCT
ejpam-6188	231	44	dominating	dominate	VERB
ejpam-6188	231	45	vertex	vertex	NOUN
ejpam-6188	231	46	positioning	positioning	NOUN
ejpam-6188	231	47	u1	u1	NOUN
ejpam-6188	231	48	1	1	NUM
ejpam-6188	231	49	u2	u2	PROPN
ejpam-6188	231	50	x	x	SYM
ejpam-6188	231	51	u3	u3	PROPN
ejpam-6188	231	52	1	1	NUM
ejpam-6188	231	53	u4	u4	PROPN
ejpam-6188	231	54	x	x	PROPN
ejpam-6188	231	55	·	·	PUNCT
ejpam-6188	231	56	·	·	PUNCT
ejpam-6188	231	57	·	·	PUNCT
ejpam-6188	232	1	uk−2	uk−2	NOUN
ejpam-6188	232	2	1	1	NUM
ejpam-6188	232	3	v1	v1	NOUN
ejpam-6188	232	4	1	1	NUM
ejpam-6188	232	5	v2	v2	PROPN
ejpam-6188	232	6	x	x	SYM
ejpam-6188	232	7	v3	v3	PROPN
ejpam-6188	232	8	1	1	NUM
ejpam-6188	232	9	v4	v4	PROPN
ejpam-6188	232	10	x	x	X
ejpam-6188	232	11	·	·	PUNCT
ejpam-6188	232	12	·	·	PUNCT
ejpam-6188	232	13	·	·	PUNCT
ejpam-6188	233	1	vk−2	vk−2	NOUN
ejpam-6188	233	2	1	1	NUM
ejpam-6188	233	3	odd	odd	ADJ
ejpam-6188	233	4	ℓ	ℓ	NOUN
ejpam-6188	233	5	figure	figure	NOUN
ejpam-6188	233	6	5	5	NUM
ejpam-6188	233	7	:	:	PUNCT
ejpam-6188	233	8	positions	position	NOUN
ejpam-6188	233	9	of	of	ADP
ejpam-6188	233	10	1	1	NUM
ejpam-6188	233	11	’s	’s	NOUN
ejpam-6188	233	12	on	on	ADP
ejpam-6188	233	13	the	the	DET
ejpam-6188	233	14	left	left	ADJ
ejpam-6188	233	15	side	side	NOUN
ejpam-6188	233	16	of	of	ADP
ejpam-6188	233	17	a	a	DET
ejpam-6188	233	18	b	b	NOUN
ejpam-6188	233	19	-	-	PUNCT
ejpam-6188	233	20	dominating	dominate	VERB
ejpam-6188	233	21	vertex	vertex	NOUN
ejpam-6188	233	22	vk	vk	ADP
ejpam-6188	233	23	1	1	NUM
ejpam-6188	233	24	·	·	PUNCT
ejpam-6188	233	25	·	·	PUNCT
ejpam-6188	233	26	·	·	PUNCT
ejpam-6188	234	1	1	1	NUM
ejpam-6188	234	2	1	1	NUM
ejpam-6188	234	3	x	x	SYM
ejpam-6188	234	4	1	1	NUM
ejpam-6188	234	5	·	·	PUNCT
ejpam-6188	234	6	·	·	PUNCT
ejpam-6188	234	7	·	·	PUNCT
ejpam-6188	234	8	x	x	SYM
ejpam-6188	234	9	1	1	NUM
ejpam-6188	234	10	x	x	SYM
ejpam-6188	234	11	·	·	PUNCT
ejpam-6188	234	12	·	·	PUNCT
ejpam-6188	234	13	·	·	PUNCT
ejpam-6188	234	14	x	x	PUNCT
ejpam-6188	234	15	x	x	PUNCT
ejpam-6188	234	16	x	x	SYM
ejpam-6188	234	17	1	1	NUM
ejpam-6188	234	18	·	·	PUNCT
ejpam-6188	234	19	·	·	PUNCT
ejpam-6188	234	20	·	·	PUNCT
ejpam-6188	234	21	x	x	SYM
ejpam-6188	234	22	1	1	NUM
ejpam-6188	234	23	r	r	NOUN
ejpam-6188	234	24	figure	figure	NOUN
ejpam-6188	234	25	6	6	NUM
ejpam-6188	234	26	:	:	PUNCT
ejpam-6188	234	27	positions	position	NOUN
ejpam-6188	234	28	of	of	ADP
ejpam-6188	234	29	1	1	NUM
ejpam-6188	234	30	’s	’s	NOUN
ejpam-6188	234	31	on	on	ADP
ejpam-6188	234	32	the	the	DET
ejpam-6188	234	33	right	right	ADJ
ejpam-6188	234	34	side	side	NOUN
ejpam-6188	234	35	of	of	ADP
ejpam-6188	234	36	a	a	DET
ejpam-6188	234	37	b	b	NOUN
ejpam-6188	234	38	-	-	PUNCT
ejpam-6188	234	39	dominating	dominate	VERB
ejpam-6188	234	40	vertex	vertex	NOUN
ejpam-6188	234	41	the	the	DET
ejpam-6188	234	42	following	follow	VERB
ejpam-6188	234	43	theorem	theorem	NOUN
ejpam-6188	234	44	gives	give	VERB
ejpam-6188	234	45	a	a	DET
ejpam-6188	234	46	lower	low	ADJ
ejpam-6188	234	47	bound	bind	VERB
ejpam-6188	234	48	on	on	ADP
ejpam-6188	234	49	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	234	50	)	)	PUNCT
ejpam-6188	234	51	)	)	PUNCT
ejpam-6188	234	52	for	for	ADP
ejpam-6188	234	53	n	n	PRON
ejpam-6188	234	54	≥	≥	NUM
ejpam-6188	234	55	9	9	NUM
ejpam-6188	234	56	.	.	PUNCT
ejpam-6188	234	57	theorem	theorem	NOUN
ejpam-6188	234	58	5	5	NUM
ejpam-6188	234	59	.	.	PUNCT
ejpam-6188	234	60	for	for	ADP
ejpam-6188	234	61	n	n	PRON
ejpam-6188	234	62	≥	≥	NOUN
ejpam-6188	234	63	9	9	NUM
ejpam-6188	234	64	,	,	PUNCT
ejpam-6188	234	65	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	234	66	)	)	PUNCT
ejpam-6188	234	67	)	)	PUNCT
ejpam-6188	234	68	≥	≥	NOUN
ejpam-6188	234	69	{	{	PUNCT
ejpam-6188	234	70	3n+	3n+	NUM
ejpam-6188	234	71	18	18	NUM
ejpam-6188	234	72	for	for	ADP
ejpam-6188	234	73	10	10	NUM
ejpam-6188	234	74	≤	≤	NOUN
ejpam-6188	234	75	n	n	PRON
ejpam-6188	234	76	≤	≤	NUM
ejpam-6188	234	77	15	15	NUM
ejpam-6188	234	78	,	,	PUNCT
ejpam-6188	234	79	3n+	3n+	NUM
ejpam-6188	234	80	19	19	NUM
ejpam-6188	234	81	for	for	ADP
ejpam-6188	234	82	n	n	NOUN
ejpam-6188	234	83	=	=	SYM
ejpam-6188	234	84	9	9	NUM
ejpam-6188	234	85	or	or	CCONJ
ejpam-6188	234	86	n	n	PRON
ejpam-6188	234	87	≥	≥	NOUN
ejpam-6188	234	88	16	16	NUM
ejpam-6188	234	89	.	.	PUNCT
ejpam-6188	235	1	proof	proof	NOUN
ejpam-6188	235	2	.	.	PUNCT
ejpam-6188	236	1	let	let	VERB
ejpam-6188	236	2	cn	cn	PROPN
ejpam-6188	236	3	:	:	PUNCT
ejpam-6188	236	4	v	v	NOUN
ejpam-6188	236	5	(	(	PUNCT
ejpam-6188	236	6	µ(pn	µ(pn	NOUN
ejpam-6188	236	7	)	)	PUNCT
ejpam-6188	236	8	)	)	PUNCT
ejpam-6188	236	9	→	→	PUNCT
ejpam-6188	236	10	{	{	PUNCT
ejpam-6188	236	11	1	1	NUM
ejpam-6188	236	12	,	,	PUNCT
ejpam-6188	236	13	2	2	NUM
ejpam-6188	236	14	,	,	PUNCT
ejpam-6188	236	15	3	3	NUM
ejpam-6188	236	16	,	,	PUNCT
ejpam-6188	236	17	4	4	NUM
ejpam-6188	236	18	,	,	PUNCT
ejpam-6188	236	19	5	5	NUM
ejpam-6188	236	20	}	}	PUNCT
ejpam-6188	236	21	be	be	AUX
ejpam-6188	236	22	a	a	DET
ejpam-6188	236	23	b	b	NOUN
ejpam-6188	236	24	-	-	PUNCT
ejpam-6188	236	25	coloring	coloring	NOUN
ejpam-6188	236	26	of	of	ADP
ejpam-6188	236	27	µ(pn	µ(pn	NOUN
ejpam-6188	236	28	)	)	PUNCT
ejpam-6188	236	29	for	for	ADP
ejpam-6188	236	30	n	n	PRON
ejpam-6188	236	31	≥	≥	NOUN
ejpam-6188	236	32	9	9	NUM
ejpam-6188	236	33	.	.	PUNCT
ejpam-6188	237	1	in	in	ADP
ejpam-6188	237	2	this	this	DET
ejpam-6188	237	3	proof	proof	NOUN
ejpam-6188	237	4	,	,	PUNCT
ejpam-6188	237	5	we	we	PRON
ejpam-6188	237	6	assume	assume	VERB
ejpam-6188	237	7	the	the	DET
ejpam-6188	237	8	minimality	minimality	NOUN
ejpam-6188	237	9	of	of	ADP
ejpam-6188	237	10	the	the	DET
ejpam-6188	237	11	sum	sum	NOUN
ejpam-6188	237	12	of	of	ADP
ejpam-6188	237	13	the	the	DET
ejpam-6188	237	14	colors	color	NOUN
ejpam-6188	237	15	by	by	ADP
ejpam-6188	237	16	maximizing	maximize	VERB
ejpam-6188	237	17	the	the	DET
ejpam-6188	237	18	number	number	NOUN
ejpam-6188	237	19	of	of	ADP
ejpam-6188	237	20	vertices	vertex	NOUN
ejpam-6188	237	21	with	with	ADP
ejpam-6188	237	22	smaller	small	ADJ
ejpam-6188	237	23	colors	color	NOUN
ejpam-6188	237	24	and	and	CCONJ
ejpam-6188	237	25	minimizing	minimize	VERB
ejpam-6188	237	26	the	the	DET
ejpam-6188	237	27	number	number	NOUN
ejpam-6188	237	28	of	of	ADP
ejpam-6188	237	29	vertices	vertex	NOUN
ejpam-6188	237	30	with	with	ADP
ejpam-6188	237	31	larger	large	ADJ
ejpam-6188	237	32	colors	color	NOUN
ejpam-6188	237	33	.	.	PUNCT
ejpam-6188	238	1	case	case	NOUN
ejpam-6188	238	2	1	1	NUM
ejpam-6188	238	3	.	.	PUNCT
ejpam-6188	238	4	cn(u	cn(u	X
ejpam-6188	238	5	)	)	PUNCT
ejpam-6188	238	6	∈	∈	PROPN
ejpam-6188	238	7	{	{	PUNCT
ejpam-6188	238	8	3	3	NUM
ejpam-6188	238	9	,	,	PUNCT
ejpam-6188	238	10	4	4	NUM
ejpam-6188	238	11	,	,	PUNCT
ejpam-6188	238	12	5	5	NUM
ejpam-6188	238	13	}	}	PUNCT
ejpam-6188	238	14	.	.	PUNCT
ejpam-6188	239	1	by	by	ADP
ejpam-6188	239	2	lemmas	lemmas	PROPN
ejpam-6188	239	3	1	1	NUM
ejpam-6188	239	4	,	,	PUNCT
ejpam-6188	239	5	3	3	NUM
ejpam-6188	239	6	and	and	CCONJ
ejpam-6188	239	7	4	4	NUM
ejpam-6188	239	8	,	,	PUNCT
ejpam-6188	239	9	we	we	PRON
ejpam-6188	239	10	have	have	VERB
ejpam-6188	239	11	that	that	DET
ejpam-6188	239	12	|c3|	|c3|	ADJ
ejpam-6188	239	13	≥	≥	NOUN
ejpam-6188	239	14	3	3	NUM
ejpam-6188	239	15	,	,	PUNCT
ejpam-6188	239	16	|c4|	|c4|	NOUN
ejpam-6188	239	17	≥	≥	NUM
ejpam-6188	239	18	3	3	NUM
ejpam-6188	239	19	,	,	PUNCT
ejpam-6188	239	20	|c5|	|c5|	NOUN
ejpam-6188	239	21	≥	≥	NOUN
ejpam-6188	239	22	3	3	NUM
ejpam-6188	239	23	,	,	PUNCT
ejpam-6188	239	24	or	or	CCONJ
ejpam-6188	239	25	|ci|	|ci|	PROPN
ejpam-6188	239	26	≥	≥	NUM
ejpam-6188	239	27	2	2	NUM
ejpam-6188	239	28	,	,	PUNCT
ejpam-6188	239	29	|cj	|cj	PROPN
ejpam-6188	239	30	|	|	ADV
ejpam-6188	239	31	≥	≥	NOUN
ejpam-6188	239	32	3	3	NUM
ejpam-6188	239	33	,	,	PUNCT
ejpam-6188	239	34	|ck|	|ck|	PROPN
ejpam-6188	239	35	≥	≥	NOUN
ejpam-6188	239	36	4	4	NUM
ejpam-6188	239	37	where	where	SCONJ
ejpam-6188	239	38	{	{	PUNCT
ejpam-6188	239	39	i	i	NOUN
ejpam-6188	239	40	,	,	PUNCT
ejpam-6188	239	41	j	j	PROPN
ejpam-6188	239	42	,	,	PUNCT
ejpam-6188	239	43	k	k	NOUN
ejpam-6188	239	44	}	}	PUNCT
ejpam-6188	239	45	=	=	SYM
ejpam-6188	239	46	{	{	PUNCT
ejpam-6188	239	47	3	3	NUM
ejpam-6188	239	48	,	,	PUNCT
ejpam-6188	239	49	4	4	NUM
ejpam-6188	239	50	,	,	PUNCT
ejpam-6188	239	51	5	5	NUM
ejpam-6188	239	52	}	}	PUNCT
ejpam-6188	239	53	.	.	PUNCT
ejpam-6188	240	1	hence	hence	ADV
ejpam-6188	240	2	,	,	PUNCT
ejpam-6188	240	3	the	the	DET
ejpam-6188	240	4	minimum	minimum	NOUN
ejpam-6188	240	5	of	of	ADP
ejpam-6188	240	6	5|c5|	5|c5|	NUM
ejpam-6188	240	7	+	+	SYM
ejpam-6188	240	8	4|c4|	4|c4|	NOUN
ejpam-6188	241	1	+	+	CCONJ
ejpam-6188	241	2	3|c3|	3|c3|	NUM
ejpam-6188	241	3	is	be	AUX
ejpam-6188	241	4	p.	p.	PROPN
ejpam-6188	241	5	vichitkunakorn	vichitkunakorn	ADJ
ejpam-6188	241	6	et	et	PROPN
ejpam-6188	241	7	al	al	PROPN
ejpam-6188	241	8	.	.	PUNCT
ejpam-6188	241	9	/	/	SYM
ejpam-6188	241	10	eur	eur	PROPN
ejpam-6188	241	11	.	.	PUNCT
ejpam-6188	242	1	j.	j.	PROPN
ejpam-6188	242	2	pure	pure	PROPN
ejpam-6188	242	3	appl	appl	PROPN
ejpam-6188	242	4	.	.	PROPN
ejpam-6188	242	5	math	math	PROPN
ejpam-6188	242	6	,	,	PUNCT
ejpam-6188	242	7	18	18	NUM
ejpam-6188	242	8	(	(	PUNCT
ejpam-6188	242	9	3	3	NUM
ejpam-6188	242	10	)	)	PUNCT
ejpam-6188	242	11	(	(	PUNCT
ejpam-6188	242	12	2025	2025	NUM
ejpam-6188	242	13	)	)	PUNCT
ejpam-6188	242	14	,	,	PUNCT
ejpam-6188	242	15	6188	6188	NUM
ejpam-6188	242	16	9	9	NUM
ejpam-6188	242	17	of	of	ADP
ejpam-6188	242	18	13	13	NUM
ejpam-6188	242	19	5	5	NUM
ejpam-6188	242	20	·	·	SYM
ejpam-6188	242	21	2	2	NUM
ejpam-6188	243	1	+	+	CCONJ
ejpam-6188	243	2	4	4	NUM
ejpam-6188	243	3	·	·	SYM
ejpam-6188	243	4	3	3	NUM
ejpam-6188	243	5	+	+	CCONJ
ejpam-6188	243	6	3	3	NUM
ejpam-6188	243	7	·	·	SYM
ejpam-6188	243	8	4	4	NUM
ejpam-6188	243	9	=	=	SYM
ejpam-6188	243	10	34	34	NUM
ejpam-6188	243	11	.	.	PUNCT
ejpam-6188	244	1	by	by	ADP
ejpam-6188	244	2	lemma	lemma	PROPN
ejpam-6188	244	3	5	5	NUM
ejpam-6188	244	4	,	,	PUNCT
ejpam-6188	244	5	to	to	PART
ejpam-6188	244	6	achieve	achieve	VERB
ejpam-6188	244	7	the	the	DET
ejpam-6188	244	8	minimum	minimum	ADJ
ejpam-6188	244	9	sum	sum	NOUN
ejpam-6188	244	10	,	,	PUNCT
ejpam-6188	244	11	we	we	PRON
ejpam-6188	244	12	assume	assume	VERB
ejpam-6188	244	13	the	the	DET
ejpam-6188	244	14	maximality	maximality	NOUN
ejpam-6188	244	15	of	of	ADP
ejpam-6188	244	16	|c1|	|c1|	PROPN
ejpam-6188	244	17	=	=	SYM
ejpam-6188	244	18	n−	n−	NOUN
ejpam-6188	244	19	1	1	NUM
ejpam-6188	244	20	.	.	PUNCT
ejpam-6188	245	1	then	then	ADV
ejpam-6188	245	2	,	,	PUNCT
ejpam-6188	245	3	we	we	PRON
ejpam-6188	245	4	have	have	AUX
ejpam-6188	245	5	|c2|	|c2|	NOUN
ejpam-6188	245	6	=	=	PUNCT
ejpam-6188	245	7	n−	n−	NOUN
ejpam-6188	245	8	7	7	NUM
ejpam-6188	245	9	.	.	PUNCT
ejpam-6188	246	1	thus,∑	thus,∑	NOUN
ejpam-6188	246	2	w∈v	w∈v	PROPN
ejpam-6188	246	3	(	(	PUNCT
ejpam-6188	246	4	µ(pn	µ(pn	NOUN
ejpam-6188	246	5	)	)	PUNCT
ejpam-6188	246	6	)	)	PUNCT
ejpam-6188	246	7	cn(w	cn(w	PUNCT
ejpam-6188	246	8	)	)	PUNCT
ejpam-6188	246	9	≥	≥	NOUN
ejpam-6188	246	10	34	34	NUM
ejpam-6188	246	11	+	+	CCONJ
ejpam-6188	246	12	2(n−	2(n−	NUM
ejpam-6188	246	13	7	7	NUM
ejpam-6188	246	14	)	)	PUNCT
ejpam-6188	246	15	+	+	CCONJ
ejpam-6188	246	16	(	(	PUNCT
ejpam-6188	246	17	n−	n−	NOUN
ejpam-6188	246	18	1	1	NUM
ejpam-6188	246	19	)	)	PUNCT
ejpam-6188	246	20	=	=	NOUN
ejpam-6188	246	21	3n+	3n+	NUM
ejpam-6188	246	22	19	19	NUM
ejpam-6188	246	23	.	.	PUNCT
ejpam-6188	246	24	case	case	NOUN
ejpam-6188	246	25	2	2	NUM
ejpam-6188	246	26	.	.	PUNCT
ejpam-6188	246	27	cn(u	cn(u	X
ejpam-6188	246	28	)	)	PUNCT
ejpam-6188	246	29	=	=	SYM
ejpam-6188	247	1	1	1	X
ejpam-6188	247	2	.	.	PUNCT
ejpam-6188	247	3	by	by	ADP
ejpam-6188	247	4	lemma	lemma	PROPN
ejpam-6188	247	5	2	2	NUM
ejpam-6188	247	6	,	,	PUNCT
ejpam-6188	247	7	we	we	PRON
ejpam-6188	247	8	have	have	AUX
ejpam-6188	247	9	|c1|	|c1|	VERB
ejpam-6188	247	10	≤	≤	NOUN
ejpam-6188	247	11	⌈	⌈	NUM
ejpam-6188	247	12	n	n	CCONJ
ejpam-6188	247	13	2	2	NUM
ejpam-6188	247	14	⌉	⌉	NOUN
ejpam-6188	247	15	+	+	NOUN
ejpam-6188	247	16	1	1	X
ejpam-6188	247	17	.	.	X
ejpam-6188	248	1	we	we	PRON
ejpam-6188	248	2	assume	assume	VERB
ejpam-6188	248	3	the	the	DET
ejpam-6188	248	4	maximum	maximum	ADJ
ejpam-6188	248	5	value	value	NOUN
ejpam-6188	248	6	of	of	ADP
ejpam-6188	248	7	|c1|	|c1|	NOUN
ejpam-6188	248	8	and	and	CCONJ
ejpam-6188	248	9	minimum	minimum	NOUN
ejpam-6188	248	10	values	value	NOUN
ejpam-6188	248	11	of	of	ADP
ejpam-6188	248	12	|c3|	|c3|	NOUN
ejpam-6188	248	13	,	,	PUNCT
ejpam-6188	248	14	|c4|	|c4|	NOUN
ejpam-6188	248	15	and	and	CCONJ
ejpam-6188	248	16	|c5|	|c5|	NOUN
ejpam-6188	248	17	.	.	PUNCT
ejpam-6188	249	1	we	we	PRON
ejpam-6188	249	2	have	have	VERB
ejpam-6188	249	3	that	that	DET
ejpam-6188	249	4	|c1|	|c1|	NOUN
ejpam-6188	249	5	=	=	SYM
ejpam-6188	249	6	⌈	⌈	NOUN
ejpam-6188	249	7	n	n	PRON
ejpam-6188	249	8	2	2	NUM
ejpam-6188	249	9	⌉	⌉	NOUN
ejpam-6188	249	10	+	+	CCONJ
ejpam-6188	249	11	1	1	NUM
ejpam-6188	249	12	,	,	PUNCT
ejpam-6188	249	13	|c3|	|c3|	NOUN
ejpam-6188	249	14	=	=	PUNCT
ejpam-6188	250	1	|c4|	|c4|	NOUN
ejpam-6188	250	2	=	=	SYM
ejpam-6188	250	3	3	3	NUM
ejpam-6188	250	4	and	and	CCONJ
ejpam-6188	250	5	|c5|	|c5|	NOUN
ejpam-6188	250	6	=	=	SYM
ejpam-6188	250	7	2	2	NUM
ejpam-6188	250	8	.	.	PUNCT
ejpam-6188	251	1	hence	hence	ADV
ejpam-6188	251	2	,	,	PUNCT
ejpam-6188	251	3	|c2|	|c2|	NOUN
ejpam-6188	251	4	=	=	PUNCT
ejpam-6188	251	5	n+	n+	NUM
ejpam-6188	251	6	⌊	⌊	PROPN
ejpam-6188	251	7	n	n	DET
ejpam-6188	251	8	2	2	NUM
ejpam-6188	251	9	⌋	⌋	NOUN
ejpam-6188	251	10	−	−	NOUN
ejpam-6188	251	11	8	8	NUM
ejpam-6188	251	12	.	.	PUNCT
ejpam-6188	252	1	it	it	PRON
ejpam-6188	252	2	follows	follow	VERB
ejpam-6188	252	3	that∑	that∑	NOUN
ejpam-6188	252	4	w∈v	w∈v	PROPN
ejpam-6188	252	5	(	(	PUNCT
ejpam-6188	252	6	µ(pn	µ(pn	NOUN
ejpam-6188	252	7	)	)	PUNCT
ejpam-6188	252	8	)	)	PUNCT
ejpam-6188	252	9	cn(w	cn(w	PUNCT
ejpam-6188	252	10	)	)	PUNCT
ejpam-6188	252	11	≥	≥	NOUN
ejpam-6188	252	12	5	5	NUM
ejpam-6188	252	13	·	·	SYM
ejpam-6188	252	14	2	2	NUM
ejpam-6188	253	1	+	+	CCONJ
ejpam-6188	253	2	4	4	NUM
ejpam-6188	253	3	·	·	SYM
ejpam-6188	253	4	3	3	NUM
ejpam-6188	253	5	+	+	CCONJ
ejpam-6188	253	6	3	3	NUM
ejpam-6188	253	7	·	·	SYM
ejpam-6188	253	8	3	3	NUM
ejpam-6188	254	1	+	+	SYM
ejpam-6188	254	2	2	2	NUM
ejpam-6188	254	3	(	(	PUNCT
ejpam-6188	254	4	n+	n+	X
ejpam-6188	254	5	⌊n	⌊n	NOUN
ejpam-6188	254	6	2	2	NUM
ejpam-6188	254	7	⌋	⌋	NOUN
ejpam-6188	254	8	−	−	NOUN
ejpam-6188	254	9	8	8	NUM
ejpam-6188	254	10	)	)	PUNCT
ejpam-6188	255	1	+	+	CCONJ
ejpam-6188	255	2	(	(	PUNCT
ejpam-6188	255	3	⌈n	⌈n	NOUN
ejpam-6188	255	4	2	2	NUM
ejpam-6188	255	5	⌉	⌉	NOUN
ejpam-6188	255	6	+	+	CCONJ
ejpam-6188	255	7	1	1	X
ejpam-6188	255	8	)	)	PUNCT
ejpam-6188	255	9	=	=	NOUN
ejpam-6188	255	10	3n+	3n+	NUM
ejpam-6188	255	11	⌊n	⌊n	NOUN
ejpam-6188	255	12	2	2	NUM
ejpam-6188	255	13	⌋	⌋	NOUN
ejpam-6188	255	14	+	+	CCONJ
ejpam-6188	255	15	16	16	NUM
ejpam-6188	255	16	>	>	SYM
ejpam-6188	255	17	3n+	3n+	NUM
ejpam-6188	255	18	19	19	NUM
ejpam-6188	255	19	.	.	PUNCT
ejpam-6188	256	1	case	case	NOUN
ejpam-6188	256	2	3	3	NUM
ejpam-6188	256	3	.	.	PUNCT
ejpam-6188	256	4	cn(u	cn(u	X
ejpam-6188	256	5	)	)	PUNCT
ejpam-6188	257	1	=	=	SYM
ejpam-6188	258	1	2	2	X
ejpam-6188	258	2	.	.	X
ejpam-6188	258	3	we	we	PRON
ejpam-6188	258	4	first	first	ADV
ejpam-6188	258	5	assume	assume	VERB
ejpam-6188	258	6	the	the	DET
ejpam-6188	258	7	maximum	maximum	ADJ
ejpam-6188	258	8	value	value	NOUN
ejpam-6188	258	9	of	of	ADP
ejpam-6188	258	10	|c1|	|c1|	NOUN
ejpam-6188	258	11	and	and	CCONJ
ejpam-6188	258	12	minimum	minimum	NOUN
ejpam-6188	258	13	values	value	NOUN
ejpam-6188	258	14	of	of	ADP
ejpam-6188	258	15	|c3|	|c3|	NOUN
ejpam-6188	258	16	,	,	PUNCT
ejpam-6188	258	17	|c4|	|c4|	NOUN
ejpam-6188	258	18	and	and	CCONJ
ejpam-6188	258	19	|c5|	|c5|	NOUN
ejpam-6188	258	20	.	.	PUNCT
ejpam-6188	259	1	we	we	PRON
ejpam-6188	259	2	have	have	VERB
ejpam-6188	259	3	that	that	DET
ejpam-6188	259	4	|c1|	|c1|	NOUN
ejpam-6188	259	5	=	=	SYM
ejpam-6188	259	6	n	n	CCONJ
ejpam-6188	259	7	−	−	PROPN
ejpam-6188	259	8	1	1	NUM
ejpam-6188	259	9	,	,	PUNCT
ejpam-6188	259	10	|c3|	|c3|	NOUN
ejpam-6188	259	11	=	=	PUNCT
ejpam-6188	260	1	|c4|	|c4|	NOUN
ejpam-6188	260	2	=	=	SYM
ejpam-6188	260	3	3	3	NUM
ejpam-6188	260	4	and	and	CCONJ
ejpam-6188	260	5	|c5|	|c5|	NOUN
ejpam-6188	260	6	=	=	SYM
ejpam-6188	260	7	2	2	X
ejpam-6188	260	8	.	.	PUNCT
ejpam-6188	261	1	it	it	PRON
ejpam-6188	261	2	follows	follow	VERB
ejpam-6188	261	3	that	that	SCONJ
ejpam-6188	261	4	|c2|	|c2|	NOUN
ejpam-6188	261	5	=	=	SYM
ejpam-6188	261	6	n	n	CCONJ
ejpam-6188	261	7	−	−	PROPN
ejpam-6188	261	8	6	6	NUM
ejpam-6188	261	9	.	.	PUNCT
ejpam-6188	262	1	hence	hence	ADV
ejpam-6188	262	2	,	,	PUNCT
ejpam-6188	262	3	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	262	4	)	)	PUNCT
ejpam-6188	262	5	)	)	PUNCT
ejpam-6188	262	6	≥	≥	NOUN
ejpam-6188	262	7	5	5	NUM
ejpam-6188	262	8	·	·	SYM
ejpam-6188	262	9	2	2	NUM
ejpam-6188	263	1	+	+	CCONJ
ejpam-6188	263	2	4	4	NUM
ejpam-6188	263	3	·	·	SYM
ejpam-6188	263	4	3	3	NUM
ejpam-6188	263	5	+	+	CCONJ
ejpam-6188	263	6	3	3	NUM
ejpam-6188	263	7	·	·	SYM
ejpam-6188	263	8	3	3	NUM
ejpam-6188	264	1	+	+	CCONJ
ejpam-6188	264	2	2(n−	2(n−	NUM
ejpam-6188	264	3	6	6	NUM
ejpam-6188	264	4	)	)	PUNCT
ejpam-6188	264	5	+	+	CCONJ
ejpam-6188	264	6	(	(	PUNCT
ejpam-6188	264	7	n−	n−	NOUN
ejpam-6188	264	8	1	1	NUM
ejpam-6188	264	9	)	)	PUNCT
ejpam-6188	264	10	=	=	NOUN
ejpam-6188	264	11	3n+	3n+	NUM
ejpam-6188	264	12	18	18	NUM
ejpam-6188	264	13	.	.	PUNCT
ejpam-6188	265	1	next	next	ADV
ejpam-6188	265	2	,	,	PUNCT
ejpam-6188	265	3	we	we	PRON
ejpam-6188	265	4	consider	consider	VERB
ejpam-6188	265	5	two	two	NUM
ejpam-6188	265	6	special	special	ADJ
ejpam-6188	265	7	subcases	subcase	NOUN
ejpam-6188	265	8	.	.	PUNCT
ejpam-6188	266	1	•	•	NUM
ejpam-6188	266	2	when	when	SCONJ
ejpam-6188	266	3	n	n	X
ejpam-6188	266	4	=	=	SYM
ejpam-6188	266	5	9	9	NUM
ejpam-6188	266	6	,	,	PUNCT
ejpam-6188	266	7	it	it	PRON
ejpam-6188	266	8	is	be	AUX
ejpam-6188	266	9	not	not	PART
ejpam-6188	266	10	possible	possible	ADJ
ejpam-6188	266	11	that	that	SCONJ
ejpam-6188	266	12	|c1|	|c1|	NOUN
ejpam-6188	266	13	=	=	SYM
ejpam-6188	266	14	8	8	NUM
ejpam-6188	266	15	since	since	SCONJ
ejpam-6188	266	16	lemma	lemma	PROPN
ejpam-6188	266	17	4	4	NUM
ejpam-6188	266	18	gives	give	VERB
ejpam-6188	266	19	an	an	DET
ejpam-6188	266	20	extra	extra	ADJ
ejpam-6188	266	21	condition	condition	NOUN
ejpam-6188	266	22	that	that	PRON
ejpam-6188	266	23	|c2|	|c2|	VERB
ejpam-6188	266	24	≥	≥	PRON
ejpam-6188	266	25	4	4	NUM
ejpam-6188	266	26	,	,	PUNCT
ejpam-6188	266	27	|c3|	|c3|	NOUN
ejpam-6188	266	28	=	=	PUNCT
ejpam-6188	266	29	|c4|	|c4|	NOUN
ejpam-6188	266	30	=	=	SYM
ejpam-6188	266	31	3	3	NUM
ejpam-6188	266	32	and	and	CCONJ
ejpam-6188	266	33	|c5|	|c5|	NOUN
ejpam-6188	266	34	=	=	SYM
ejpam-6188	266	35	2	2	NUM
ejpam-6188	266	36	are	be	AUX
ejpam-6188	266	37	the	the	DET
ejpam-6188	266	38	lowest	low	ADJ
ejpam-6188	266	39	possible	possible	ADJ
ejpam-6188	266	40	values	value	NOUN
ejpam-6188	266	41	.	.	PUNCT
ejpam-6188	267	1	thus	thus	ADV
ejpam-6188	267	2	|c1|	|c1|	VERB
ejpam-6188	267	3	=	=	SYM
ejpam-6188	267	4	7	7	NUM
ejpam-6188	267	5	and	and	CCONJ
ejpam-6188	267	6	|c2|	|c2|	NOUN
ejpam-6188	267	7	=	=	PUNCT
ejpam-6188	267	8	4	4	X
ejpam-6188	267	9	.	.	PUNCT
ejpam-6188	267	10	hence	hence	ADV
ejpam-6188	267	11	,	,	PUNCT
ejpam-6188	267	12	φ′(µ(p9	φ′(µ(p9	ADV
ejpam-6188	267	13	)	)	PUNCT
ejpam-6188	267	14	)	)	PUNCT
ejpam-6188	267	15	≥	≥	NOUN
ejpam-6188	267	16	5	5	NUM
ejpam-6188	267	17	·	·	SYM
ejpam-6188	267	18	2	2	NUM
ejpam-6188	268	1	+	+	CCONJ
ejpam-6188	268	2	4	4	NUM
ejpam-6188	268	3	·	·	SYM
ejpam-6188	268	4	3	3	NUM
ejpam-6188	268	5	+	+	CCONJ
ejpam-6188	268	6	3	3	NUM
ejpam-6188	268	7	·	·	SYM
ejpam-6188	268	8	3	3	NUM
ejpam-6188	268	9	+	+	CCONJ
ejpam-6188	268	10	2	2	NUM
ejpam-6188	268	11	·	·	SYM
ejpam-6188	268	12	4	4	NUM
ejpam-6188	268	13	+	+	SYM
ejpam-6188	268	14	1	1	NUM
ejpam-6188	268	15	·	·	SYM
ejpam-6188	268	16	7	7	NUM
ejpam-6188	268	17	=	=	SYM
ejpam-6188	268	18	46	46	NUM
ejpam-6188	268	19	=	=	SYM
ejpam-6188	268	20	3n+	3n+	NUM
ejpam-6188	268	21	19	19	NUM
ejpam-6188	268	22	.	.	NOUN
ejpam-6188	268	23	•	•	NUM
ejpam-6188	268	24	when	when	SCONJ
ejpam-6188	268	25	n	n	X
ejpam-6188	268	26	≥	≥	NOUN
ejpam-6188	268	27	16	16	NUM
ejpam-6188	268	28	,	,	PUNCT
ejpam-6188	268	29	we	we	PRON
ejpam-6188	268	30	assume	assume	VERB
ejpam-6188	268	31	|c1|	|c1|	PROPN
ejpam-6188	268	32	=	=	SYM
ejpam-6188	268	33	n−	n−	NOUN
ejpam-6188	268	34	1	1	NUM
ejpam-6188	268	35	and	and	CCONJ
ejpam-6188	268	36	|c4|	|c4|	NOUN
ejpam-6188	268	37	=	=	SYM
ejpam-6188	268	38	3	3	NUM
ejpam-6188	268	39	and	and	CCONJ
ejpam-6188	268	40	|c5|	|c5|	NOUN
ejpam-6188	268	41	=	=	SYM
ejpam-6188	268	42	2	2	X
ejpam-6188	268	43	.	.	X
ejpam-6188	268	44	from	from	ADP
ejpam-6188	268	45	lemma	lemma	PROPN
ejpam-6188	268	46	2	2	NUM
ejpam-6188	268	47	,	,	PUNCT
ejpam-6188	268	48	we	we	PRON
ejpam-6188	268	49	assume	assume	VERB
ejpam-6188	268	50	|c2|	|c2|	NOUN
ejpam-6188	268	51	=	=	SYM
ejpam-6188	268	52	⌈	⌈	NOUN
ejpam-6188	268	53	n	n	CCONJ
ejpam-6188	268	54	2	2	NUM
ejpam-6188	268	55	⌉	⌉	NOUN
ejpam-6188	268	56	+	+	NOUN
ejpam-6188	268	57	1	1	X
ejpam-6188	268	58	.	.	PUNCT
ejpam-6188	269	1	it	it	PRON
ejpam-6188	269	2	follows	follow	VERB
ejpam-6188	269	3	that	that	DET
ejpam-6188	269	4	|c3|	|c3|	NOUN
ejpam-6188	269	5	=	=	SYM
ejpam-6188	269	6	(	(	PUNCT
ejpam-6188	269	7	2n+	2n+	NUM
ejpam-6188	269	8	1)−	1)−	NUM
ejpam-6188	269	9	2−	2−	NUM
ejpam-6188	269	10	3−	3−	NUM
ejpam-6188	269	11	(	(	PUNCT
ejpam-6188	269	12	⌈n	⌈n	NOUN
ejpam-6188	269	13	2	2	NUM
ejpam-6188	269	14	⌉	⌉	NOUN
ejpam-6188	269	15	+	+	CCONJ
ejpam-6188	269	16	1	1	NUM
ejpam-6188	269	17	)	)	PUNCT
ejpam-6188	269	18	−	−	PROPN
ejpam-6188	270	1	(	(	PUNCT
ejpam-6188	270	2	n−	n−	NOUN
ejpam-6188	270	3	1	1	NUM
ejpam-6188	270	4	)	)	PUNCT
ejpam-6188	270	5	=	=	PUNCT
ejpam-6188	270	6	⌊n	⌊n	X
ejpam-6188	270	7	2	2	NUM
ejpam-6188	270	8	⌋	⌋	NOUN
ejpam-6188	270	9	−	−	NOUN
ejpam-6188	270	10	4	4	X
ejpam-6188	270	11	.	.	PUNCT
ejpam-6188	271	1	we	we	PRON
ejpam-6188	271	2	have	have	VERB
ejpam-6188	271	3	∑	∑	PROPN
ejpam-6188	271	4	w∈v	w∈v	PROPN
ejpam-6188	271	5	(	(	PUNCT
ejpam-6188	271	6	µ(pn	µ(pn	NOUN
ejpam-6188	271	7	)	)	PUNCT
ejpam-6188	271	8	)	)	PUNCT
ejpam-6188	271	9	cn(w	cn(w	PUNCT
ejpam-6188	271	10	)	)	PUNCT
ejpam-6188	271	11	≥	≥	NOUN
ejpam-6188	271	12	5	5	NUM
ejpam-6188	271	13	·	·	SYM
ejpam-6188	271	14	2	2	NUM
ejpam-6188	272	1	+	+	CCONJ
ejpam-6188	272	2	4	4	NUM
ejpam-6188	272	3	·	·	SYM
ejpam-6188	272	4	3	3	NUM
ejpam-6188	272	5	+	+	NUM
ejpam-6188	272	6	3	3	NUM
ejpam-6188	272	7	(	(	PUNCT
ejpam-6188	272	8	⌊n	⌊n	NOUN
ejpam-6188	272	9	2	2	NUM
ejpam-6188	272	10	⌋	⌋	NOUN
ejpam-6188	272	11	−	−	PROPN
ejpam-6188	272	12	4	4	NUM
ejpam-6188	272	13	)	)	PUNCT
ejpam-6188	272	14	+	+	CCONJ
ejpam-6188	272	15	2	2	NUM
ejpam-6188	272	16	(	(	PUNCT
ejpam-6188	272	17	⌈n	⌈n	NOUN
ejpam-6188	272	18	2	2	NUM
ejpam-6188	272	19	⌉	⌉	NOUN
ejpam-6188	272	20	+	+	CCONJ
ejpam-6188	272	21	1	1	NUM
ejpam-6188	272	22	)	)	PUNCT
ejpam-6188	273	1	+	+	CCONJ
ejpam-6188	273	2	(	(	PUNCT
ejpam-6188	273	3	n−	n−	NOUN
ejpam-6188	273	4	1	1	NUM
ejpam-6188	273	5	)	)	PUNCT
ejpam-6188	273	6	≥	≥	NOUN
ejpam-6188	273	7	3n+	3n+	NUM
ejpam-6188	273	8	⌊n	⌊n	NOUN
ejpam-6188	273	9	2	2	NUM
ejpam-6188	273	10	⌋	⌋	NOUN
ejpam-6188	273	11	+	+	CCONJ
ejpam-6188	273	12	11	11	NUM
ejpam-6188	273	13	≥	≥	NOUN
ejpam-6188	273	14	3n+	3n+	NUM
ejpam-6188	273	15	19	19	NUM
ejpam-6188	273	16	.	.	PUNCT
ejpam-6188	273	17	from	from	ADP
ejpam-6188	273	18	the	the	DET
ejpam-6188	273	19	three	three	NUM
ejpam-6188	273	20	cases	case	NOUN
ejpam-6188	273	21	,	,	PUNCT
ejpam-6188	273	22	we	we	PRON
ejpam-6188	273	23	can	can	AUX
ejpam-6188	273	24	conclude	conclude	VERB
ejpam-6188	273	25	that	that	PRON
ejpam-6188	273	26	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	273	27	)	)	PUNCT
ejpam-6188	273	28	)	)	PUNCT
ejpam-6188	273	29	≥	≥	NOUN
ejpam-6188	273	30	3n+	3n+	NUM
ejpam-6188	273	31	18	18	NUM
ejpam-6188	273	32	for	for	ADP
ejpam-6188	273	33	10	10	NUM
ejpam-6188	273	34	≤	≤	NOUN
ejpam-6188	273	35	n	n	PRON
ejpam-6188	273	36	≤	≤	NUM
ejpam-6188	273	37	15	15	NUM
ejpam-6188	273	38	,	,	PUNCT
ejpam-6188	273	39	and	and	CCONJ
ejpam-6188	273	40	φ′(µ(pn	φ′(µ(pn	NUM
ejpam-6188	273	41	)	)	PUNCT
ejpam-6188	273	42	)	)	PUNCT
ejpam-6188	273	43	≥	≥	NOUN
ejpam-6188	274	1	3n+	3n+	NUM
ejpam-6188	274	2	19	19	NUM
ejpam-6188	274	3	for	for	ADP
ejpam-6188	274	4	n	n	NOUN
ejpam-6188	274	5	=	=	SYM
ejpam-6188	274	6	9	9	NUM
ejpam-6188	274	7	or	or	CCONJ
ejpam-6188	274	8	n	n	PRON
ejpam-6188	274	9	≥	≥	NOUN
ejpam-6188	274	10	16	16	NUM
ejpam-6188	274	11	.	.	PUNCT
ejpam-6188	275	1	the	the	DET
ejpam-6188	275	2	following	follow	VERB
ejpam-6188	275	3	remark	remark	NOUN
ejpam-6188	275	4	is	be	AUX
ejpam-6188	275	5	obtained	obtain	VERB
ejpam-6188	275	6	from	from	ADP
ejpam-6188	275	7	the	the	DET
ejpam-6188	275	8	proof	proof	NOUN
ejpam-6188	275	9	of	of	ADP
ejpam-6188	275	10	theorem	theorem	NOUN
ejpam-6188	275	11	5	5	NUM
ejpam-6188	275	12	.	.	PUNCT
ejpam-6188	276	1	p.	p.	NOUN
ejpam-6188	276	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	276	3	et	et	PROPN
ejpam-6188	276	4	al	al	PROPN
ejpam-6188	276	5	.	.	PUNCT
ejpam-6188	276	6	/	/	SYM
ejpam-6188	276	7	eur	eur	PROPN
ejpam-6188	276	8	.	.	PUNCT
ejpam-6188	277	1	j.	j.	PROPN
ejpam-6188	277	2	pure	pure	PROPN
ejpam-6188	277	3	appl	appl	PROPN
ejpam-6188	277	4	.	.	PROPN
ejpam-6188	277	5	math	math	PROPN
ejpam-6188	277	6	,	,	PUNCT
ejpam-6188	277	7	18	18	NUM
ejpam-6188	277	8	(	(	PUNCT
ejpam-6188	277	9	3	3	NUM
ejpam-6188	277	10	)	)	PUNCT
ejpam-6188	277	11	(	(	PUNCT
ejpam-6188	277	12	2025	2025	NUM
ejpam-6188	277	13	)	)	PUNCT
ejpam-6188	277	14	,	,	PUNCT
ejpam-6188	277	15	6188	6188	NUM
ejpam-6188	277	16	10	10	NUM
ejpam-6188	277	17	of	of	ADP
ejpam-6188	277	18	13	13	NUM
ejpam-6188	277	19	u	u	NOUN
ejpam-6188	277	20	2	2	NUM
ejpam-6188	277	21	u1	u1	NOUN
ejpam-6188	277	22	1	1	NUM
ejpam-6188	277	23	u2	u2	PROPN
ejpam-6188	277	24	1	1	NUM
ejpam-6188	277	25	u3	u3	NOUN
ejpam-6188	277	26	5	5	NUM
ejpam-6188	277	27	u4	u4	PROPN
ejpam-6188	277	28	1	1	NUM
ejpam-6188	277	29	u5	u5	PROPN
ejpam-6188	277	30	3	3	NUM
ejpam-6188	277	31	u6	u6	PROPN
ejpam-6188	277	32	1	1	NUM
ejpam-6188	277	33	u7	u7	PROPN
ejpam-6188	277	34	1	1	NUM
ejpam-6188	277	35	u8	u8	PROPN
ejpam-6188	277	36	3	3	NUM
ejpam-6188	277	37	u9	u9	PROPN
ejpam-6188	277	38	4	4	NUM
ejpam-6188	277	39	v1	v1	NOUN
ejpam-6188	277	40	2	2	NUM
ejpam-6188	277	41	v2	v2	PROPN
ejpam-6188	277	42	3	3	NUM
ejpam-6188	277	43	v3	v3	PROPN
ejpam-6188	277	44	4	4	NUM
ejpam-6188	277	45	v4	v4	NOUN
ejpam-6188	277	46	1	1	NUM
ejpam-6188	277	47	v5	v5	PROPN
ejpam-6188	277	48	2	2	NUM
ejpam-6188	277	49	v6	v6	NOUN
ejpam-6188	277	50	4	4	NUM
ejpam-6188	277	51	v7	v7	NOUN
ejpam-6188	277	52	5	5	NUM
ejpam-6188	277	53	v8	v8	PROPN
ejpam-6188	277	54	2	2	NUM
ejpam-6188	277	55	v9	v9	NOUN
ejpam-6188	277	56	1	1	NUM
ejpam-6188	277	57	figure	figure	NOUN
ejpam-6188	277	58	7	7	NUM
ejpam-6188	277	59	:	:	PUNCT
ejpam-6188	277	60	a	a	DET
ejpam-6188	277	61	b	b	NOUN
ejpam-6188	277	62	-	-	PUNCT
ejpam-6188	277	63	coloring	coloring	NOUN
ejpam-6188	277	64	of	of	ADP
ejpam-6188	277	65	µ(p9	µ(p9	NOUN
ejpam-6188	277	66	)	)	PUNCT
ejpam-6188	277	67	u	u	NOUN
ejpam-6188	277	68	2	2	NUM
ejpam-6188	277	69	u1	u1	NOUN
ejpam-6188	277	70	1	1	NUM
ejpam-6188	277	71	u2	u2	PROPN
ejpam-6188	277	72	1	1	NUM
ejpam-6188	277	73	u3	u3	NOUN
ejpam-6188	277	74	5	5	NUM
ejpam-6188	277	75	u4	u4	PROPN
ejpam-6188	277	76	1	1	NUM
ejpam-6188	277	77	u5	u5	PROPN
ejpam-6188	277	78	3	3	NUM
ejpam-6188	277	79	u6	u6	PROPN
ejpam-6188	277	80	1	1	NUM
ejpam-6188	277	81	u7	u7	PROPN
ejpam-6188	277	82	1	1	NUM
ejpam-6188	277	83	u8	u8	PROPN
ejpam-6188	277	84	3	3	NUM
ejpam-6188	277	85	u9	u9	PROPN
ejpam-6188	277	86	1	1	NUM
ejpam-6188	277	87	u10	u10	PROPN
ejpam-6188	277	88	4	4	NUM
ejpam-6188	277	89	v1	v1	NOUN
ejpam-6188	277	90	2	2	NUM
ejpam-6188	277	91	v2	v2	PROPN
ejpam-6188	277	92	3	3	NUM
ejpam-6188	277	93	v3	v3	PROPN
ejpam-6188	277	94	4	4	NUM
ejpam-6188	277	95	v4	v4	NOUN
ejpam-6188	277	96	1	1	NUM
ejpam-6188	277	97	v5	v5	PROPN
ejpam-6188	277	98	2	2	NUM
ejpam-6188	277	99	v6	v6	NOUN
ejpam-6188	277	100	5	5	NUM
ejpam-6188	277	101	v7	v7	NOUN
ejpam-6188	277	102	4	4	NUM
ejpam-6188	277	103	v8	v8	PROPN
ejpam-6188	277	104	2	2	NUM
ejpam-6188	277	105	v9	v9	NOUN
ejpam-6188	277	106	1	1	NUM
ejpam-6188	277	107	v10	v10	NOUN
ejpam-6188	277	108	2	2	NUM
ejpam-6188	277	109	figure	figure	NOUN
ejpam-6188	277	110	8	8	NUM
ejpam-6188	277	111	:	:	PUNCT
ejpam-6188	277	112	a	a	DET
ejpam-6188	277	113	b	b	NOUN
ejpam-6188	277	114	-	-	PUNCT
ejpam-6188	277	115	coloring	coloring	NOUN
ejpam-6188	277	116	of	of	ADP
ejpam-6188	277	117	µ(p10	µ(p10	ADJ
ejpam-6188	277	118	)	)	PUNCT
ejpam-6188	277	119	remark	remark	NOUN
ejpam-6188	277	120	1	1	NUM
ejpam-6188	277	121	.	.	PUNCT
ejpam-6188	278	1	for	for	ADP
ejpam-6188	278	2	10	10	NUM
ejpam-6188	278	3	≤	≤	NOUN
ejpam-6188	278	4	n	n	DET
ejpam-6188	278	5	≤	≤	NUM
ejpam-6188	278	6	15	15	NUM
ejpam-6188	278	7	,	,	PUNCT
ejpam-6188	278	8	if	if	SCONJ
ejpam-6188	278	9	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	278	10	)	)	PUNCT
ejpam-6188	278	11	)	)	PUNCT
ejpam-6188	278	12	=	=	SYM
ejpam-6188	278	13	3n+18	3n+18	NUM
ejpam-6188	278	14	,	,	PUNCT
ejpam-6188	278	15	then	then	ADV
ejpam-6188	278	16	cn(u	cn(u	NOUN
ejpam-6188	278	17	)	)	PUNCT
ejpam-6188	278	18	=	=	SYM
ejpam-6188	278	19	2	2	NUM
ejpam-6188	278	20	,	,	PUNCT
ejpam-6188	278	21	|c1|	|c1|	NOUN
ejpam-6188	278	22	=	=	SYM
ejpam-6188	278	23	n−1	n−1	PROPN
ejpam-6188	278	24	,	,	PUNCT
ejpam-6188	278	25	|c2|	|c2|	NOUN
ejpam-6188	278	26	=	=	PUNCT
ejpam-6188	278	27	n−	n−	NOUN
ejpam-6188	278	28	6	6	NUM
ejpam-6188	278	29	,	,	PUNCT
ejpam-6188	278	30	|c3|	|c3|	NOUN
ejpam-6188	278	31	=	=	PUNCT
ejpam-6188	278	32	|c4|	|c4|	NOUN
ejpam-6188	278	33	=	=	SYM
ejpam-6188	278	34	3	3	NUM
ejpam-6188	278	35	and	and	CCONJ
ejpam-6188	278	36	|c5|	|c5|	NOUN
ejpam-6188	278	37	=	=	SYM
ejpam-6188	278	38	2	2	X
ejpam-6188	278	39	.	.	X
ejpam-6188	278	40	theorem	theorem	VERB
ejpam-6188	278	41	6	6	NUM
ejpam-6188	278	42	.	.	PUNCT
ejpam-6188	278	43	for	for	ADP
ejpam-6188	278	44	n	n	PRON
ejpam-6188	278	45	≥	≥	NOUN
ejpam-6188	278	46	9	9	NUM
ejpam-6188	278	47	,	,	PUNCT
ejpam-6188	278	48	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	278	49	)	)	PUNCT
ejpam-6188	278	50	)	)	PUNCT
ejpam-6188	278	51	≤	≤	ADV
ejpam-6188	278	52	3n+	3n+	NUM
ejpam-6188	278	53	19	19	NUM
ejpam-6188	278	54	.	.	PUNCT
ejpam-6188	279	1	proof	proof	NOUN
ejpam-6188	279	2	.	.	PUNCT
ejpam-6188	280	1	we	we	PRON
ejpam-6188	280	2	define	define	VERB
ejpam-6188	280	3	b	b	NUM
ejpam-6188	280	4	-	-	PUNCT
ejpam-6188	280	5	colorings	coloring	NOUN
ejpam-6188	280	6	c9	c9	NOUN
ejpam-6188	280	7	,	,	PUNCT
ejpam-6188	280	8	c10	c10	VERB
ejpam-6188	280	9	and	and	CCONJ
ejpam-6188	280	10	c11	c11	NOUN
ejpam-6188	280	11	of	of	ADP
ejpam-6188	280	12	µ(p9	µ(p9	PRON
ejpam-6188	280	13	)	)	PUNCT
ejpam-6188	280	14	,	,	PUNCT
ejpam-6188	280	15	µ(p10	µ(p10	ADV
ejpam-6188	280	16	)	)	PUNCT
ejpam-6188	280	17	and	and	CCONJ
ejpam-6188	280	18	µ(p11	µ(p11	NUM
ejpam-6188	280	19	)	)	PUNCT
ejpam-6188	280	20	as	as	ADP
ejpam-6188	280	21	in	in	ADP
ejpam-6188	280	22	figures	figure	NOUN
ejpam-6188	280	23	7	7	NUM
ejpam-6188	280	24	,	,	PUNCT
ejpam-6188	280	25	8	8	NUM
ejpam-6188	280	26	and	and	CCONJ
ejpam-6188	280	27	9	9	NUM
ejpam-6188	280	28	,	,	PUNCT
ejpam-6188	280	29	respectively	respectively	ADV
ejpam-6188	280	30	.	.	PUNCT
ejpam-6188	281	1	we	we	PRON
ejpam-6188	281	2	see	see	VERB
ejpam-6188	281	3	that	that	SCONJ
ejpam-6188	281	4	∑	∑	PUNCT
ejpam-6188	281	5	v∈v	v∈v	NOUN
ejpam-6188	281	6	(	(	PUNCT
ejpam-6188	281	7	µ(pn	µ(pn	NOUN
ejpam-6188	281	8	)	)	PUNCT
ejpam-6188	281	9	)	)	PUNCT
ejpam-6188	281	10	cn(v	cn(v	VERB
ejpam-6188	281	11	)	)	PUNCT
ejpam-6188	281	12	=	=	SYM
ejpam-6188	281	13	3n	3n	NOUN
ejpam-6188	281	14	+	+	SYM
ejpam-6188	281	15	19	19	NUM
ejpam-6188	281	16	for	for	ADP
ejpam-6188	281	17	9	9	NUM
ejpam-6188	281	18	≤	≤	NOUN
ejpam-6188	281	19	n	n	PRON
ejpam-6188	281	20	≤	≤	NOUN
ejpam-6188	281	21	11	11	NUM
ejpam-6188	281	22	.	.	PUNCT
ejpam-6188	282	1	thus	thus	ADV
ejpam-6188	282	2	φ′(µ(pn	φ′(µ(pn	NUM
ejpam-6188	282	3	)	)	PUNCT
ejpam-6188	282	4	)	)	PUNCT
ejpam-6188	283	1	≤	≤	NUM
ejpam-6188	283	2	3n+	3n+	NUM
ejpam-6188	283	3	19	19	NUM
ejpam-6188	283	4	for	for	ADP
ejpam-6188	283	5	9	9	NUM
ejpam-6188	283	6	≤	≤	NOUN
ejpam-6188	283	7	n	n	PRON
ejpam-6188	283	8	≤	≤	NOUN
ejpam-6188	283	9	11	11	NUM
ejpam-6188	283	10	.	.	PUNCT
ejpam-6188	284	1	for	for	ADP
ejpam-6188	284	2	an	an	DET
ejpam-6188	284	3	even	even	ADV
ejpam-6188	284	4	n	n	PRON
ejpam-6188	284	5	≥	≥	NOUN
ejpam-6188	284	6	12	12	NUM
ejpam-6188	284	7	,	,	PUNCT
ejpam-6188	284	8	we	we	PRON
ejpam-6188	284	9	define	define	VERB
ejpam-6188	284	10	the	the	DET
ejpam-6188	284	11	proper	proper	ADJ
ejpam-6188	284	12	coloring	coloring	NOUN
ejpam-6188	284	13	cn	cn	NOUN
ejpam-6188	284	14	:	:	PUNCT
ejpam-6188	284	15	v	v	NOUN
ejpam-6188	284	16	(	(	PUNCT
ejpam-6188	284	17	µ(pn	µ(pn	NOUN
ejpam-6188	284	18	)	)	PUNCT
ejpam-6188	284	19	)	)	PUNCT
ejpam-6188	284	20	→	→	PUNCT
ejpam-6188	284	21	{	{	PUNCT
ejpam-6188	284	22	1	1	NUM
ejpam-6188	284	23	,	,	PUNCT
ejpam-6188	284	24	2	2	NUM
ejpam-6188	284	25	,	,	PUNCT
ejpam-6188	284	26	3	3	NUM
ejpam-6188	284	27	,	,	PUNCT
ejpam-6188	284	28	4	4	NUM
ejpam-6188	284	29	,	,	PUNCT
ejpam-6188	284	30	5	5	NUM
ejpam-6188	284	31	}	}	PUNCT
ejpam-6188	284	32	by	by	ADP
ejpam-6188	284	33	cn(xi	cn(xi	PROPN
ejpam-6188	284	34	)	)	PUNCT
ejpam-6188	285	1	=	=	PUNCT
ejpam-6188	286	1			PRON
ejpam-6188	286	2	c11(xi−1	c11(xi−1	VERB
ejpam-6188	286	3	)	)	PUNCT
ejpam-6188	286	4	if	if	SCONJ
ejpam-6188	286	5	xi	xi	PROPN
ejpam-6188	286	6	∈	∈	PROPN
ejpam-6188	286	7	{	{	PUNCT
ejpam-6188	286	8	ui	ui	PROPN
ejpam-6188	286	9	,	,	PUNCT
ejpam-6188	286	10	vi	vi	PROPN
ejpam-6188	286	11	}	}	PUNCT
ejpam-6188	286	12	for	for	ADP
ejpam-6188	286	13	i	i	PROPN
ejpam-6188	286	14	=	=	NOUN
ejpam-6188	286	15	2	2	NUM
ejpam-6188	286	16	,	,	PUNCT
ejpam-6188	286	17	.	.	PUNCT
ejpam-6188	286	18	.	.	PUNCT
ejpam-6188	286	19	.	.	PUNCT
ejpam-6188	287	1	,	,	PUNCT
ejpam-6188	287	2	12	12	NUM
ejpam-6188	287	3	,	,	PUNCT
ejpam-6188	287	4	1	1	NUM
ejpam-6188	287	5	if	if	SCONJ
ejpam-6188	287	6	xi	xi	NOUN
ejpam-6188	287	7	=	=	NOUN
ejpam-6188	287	8	u1	u1	PROPN
ejpam-6188	287	9	or	or	CCONJ
ejpam-6188	287	10	xi	xi	ADP
ejpam-6188	287	11	∈	∈	PROPN
ejpam-6188	287	12	{	{	PUNCT
ejpam-6188	287	13	ui	ui	PROPN
ejpam-6188	287	14	,	,	PUNCT
ejpam-6188	287	15	vi	vi	PROPN
ejpam-6188	287	16	}	}	PUNCT
ejpam-6188	287	17	for	for	ADP
ejpam-6188	287	18	even	even	ADV
ejpam-6188	287	19	i	i	PRON
ejpam-6188	287	20	≥	≥	NOUN
ejpam-6188	287	21	14	14	NUM
ejpam-6188	287	22	,	,	PUNCT
ejpam-6188	287	23	2	2	NUM
ejpam-6188	288	1	if	if	SCONJ
ejpam-6188	288	2	xi	xi	X
ejpam-6188	288	3	=	=	SYM
ejpam-6188	288	4	v1	v1	PROPN
ejpam-6188	288	5	or	or	CCONJ
ejpam-6188	288	6	xi	xi	ADP
ejpam-6188	288	7	∈	∈	PROPN
ejpam-6188	288	8	{	{	PUNCT
ejpam-6188	288	9	ui	ui	PROPN
ejpam-6188	288	10	,	,	PUNCT
ejpam-6188	288	11	vi	vi	PROPN
ejpam-6188	288	12	}	}	PUNCT
ejpam-6188	288	13	for	for	ADP
ejpam-6188	288	14	odd	odd	ADJ
ejpam-6188	288	15	i	i	PRON
ejpam-6188	288	16	≥	≥	NUM
ejpam-6188	288	17	13	13	NUM
ejpam-6188	288	18	.	.	PUNCT
ejpam-6188	289	1	for	for	ADP
ejpam-6188	289	2	an	an	DET
ejpam-6188	289	3	odd	odd	ADJ
ejpam-6188	289	4	n	n	PRON
ejpam-6188	289	5	≥	≥	NOUN
ejpam-6188	289	6	13	13	NUM
ejpam-6188	289	7	,	,	PUNCT
ejpam-6188	289	8	we	we	PRON
ejpam-6188	289	9	define	define	VERB
ejpam-6188	289	10	the	the	DET
ejpam-6188	289	11	proper	proper	ADJ
ejpam-6188	289	12	coloring	coloring	NOUN
ejpam-6188	289	13	cn	cn	NOUN
ejpam-6188	289	14	:	:	PUNCT
ejpam-6188	289	15	v	v	NOUN
ejpam-6188	289	16	(	(	PUNCT
ejpam-6188	289	17	µ(pn	µ(pn	NOUN
ejpam-6188	289	18	)	)	PUNCT
ejpam-6188	289	19	)	)	PUNCT
ejpam-6188	289	20	→	→	PUNCT
ejpam-6188	289	21	{	{	PUNCT
ejpam-6188	289	22	1	1	NUM
ejpam-6188	289	23	,	,	PUNCT
ejpam-6188	289	24	2	2	NUM
ejpam-6188	289	25	,	,	PUNCT
ejpam-6188	289	26	3	3	NUM
ejpam-6188	289	27	,	,	PUNCT
ejpam-6188	289	28	4	4	NUM
ejpam-6188	289	29	,	,	PUNCT
ejpam-6188	289	30	5	5	NUM
ejpam-6188	289	31	}	}	PUNCT
ejpam-6188	289	32	by	by	ADP
ejpam-6188	289	33	cn(xi	cn(xi	PROPN
ejpam-6188	289	34	)	)	PUNCT
ejpam-6188	290	1	=	=	PUNCT
ejpam-6188	291	1			PROPN
ejpam-6188	291	2	c11(xi	c11(xi	PROPN
ejpam-6188	291	3	)	)	PUNCT
ejpam-6188	291	4	if	if	SCONJ
ejpam-6188	291	5	xi	xi	PROPN
ejpam-6188	291	6	∈	∈	PROPN
ejpam-6188	291	7	{	{	PUNCT
ejpam-6188	291	8	ui	ui	PROPN
ejpam-6188	291	9	,	,	PUNCT
ejpam-6188	291	10	vi	vi	PROPN
ejpam-6188	291	11	}	}	PUNCT
ejpam-6188	291	12	for	for	ADP
ejpam-6188	291	13	i	i	PROPN
ejpam-6188	291	14	=	=	NOUN
ejpam-6188	291	15	1	1	NUM
ejpam-6188	291	16	,	,	PUNCT
ejpam-6188	291	17	.	.	PUNCT
ejpam-6188	291	18	.	.	PUNCT
ejpam-6188	291	19	.	.	PUNCT
ejpam-6188	292	1	,	,	PUNCT
ejpam-6188	292	2	11	11	NUM
ejpam-6188	292	3	,	,	PUNCT
ejpam-6188	292	4	2	2	NUM
ejpam-6188	292	5	if	if	SCONJ
ejpam-6188	292	6	xi	xi	PROPN
ejpam-6188	292	7	∈	∈	PROPN
ejpam-6188	292	8	{	{	PUNCT
ejpam-6188	292	9	ui	ui	PROPN
ejpam-6188	292	10	,	,	PUNCT
ejpam-6188	292	11	vi	vi	PROPN
ejpam-6188	292	12	}	}	PUNCT
ejpam-6188	292	13	for	for	ADP
ejpam-6188	292	14	even	even	ADV
ejpam-6188	292	15	i	i	PRON
ejpam-6188	292	16	≥	≥	NOUN
ejpam-6188	292	17	12	12	NUM
ejpam-6188	292	18	,	,	PUNCT
ejpam-6188	292	19	1	1	NUM
ejpam-6188	292	20	if	if	SCONJ
ejpam-6188	292	21	xi	xi	PROPN
ejpam-6188	292	22	∈	∈	PROPN
ejpam-6188	292	23	{	{	PUNCT
ejpam-6188	292	24	ui	ui	PROPN
ejpam-6188	292	25	,	,	PUNCT
ejpam-6188	292	26	vi	vi	PROPN
ejpam-6188	292	27	}	}	PUNCT
ejpam-6188	292	28	for	for	ADP
ejpam-6188	292	29	odd	odd	ADJ
ejpam-6188	292	30	i	i	PRON
ejpam-6188	292	31	≥	≥	NUM
ejpam-6188	292	32	13	13	NUM
ejpam-6188	292	33	.	.	PUNCT
ejpam-6188	293	1	it	it	PRON
ejpam-6188	293	2	is	be	AUX
ejpam-6188	293	3	clear	clear	ADJ
ejpam-6188	293	4	that	that	SCONJ
ejpam-6188	293	5	cn	cn	PROPN
ejpam-6188	293	6	is	be	AUX
ejpam-6188	293	7	a	a	DET
ejpam-6188	293	8	b	b	NOUN
ejpam-6188	293	9	-	-	PUNCT
ejpam-6188	293	10	coloring	coloring	NOUN
ejpam-6188	293	11	and	and	CCONJ
ejpam-6188	294	1	that	that	SCONJ
ejpam-6188	294	2	∑	∑	PUNCT
ejpam-6188	294	3	x∈v	x∈v	PROPN
ejpam-6188	294	4	(	(	PUNCT
ejpam-6188	294	5	µ(pn	µ(pn	NOUN
ejpam-6188	294	6	)	)	PUNCT
ejpam-6188	294	7	)	)	PUNCT
ejpam-6188	294	8	cn(x	cn(x	NUM
ejpam-6188	294	9	)	)	PUNCT
ejpam-6188	295	1	=	=	PUNCT
ejpam-6188	295	2	3n+19	3n+19	NOUN
ejpam-6188	295	3	for	for	ADP
ejpam-6188	295	4	n	n	PRON
ejpam-6188	295	5	≥	≥	NUM
ejpam-6188	295	6	12	12	NUM
ejpam-6188	295	7	.	.	PUNCT
ejpam-6188	296	1	hence	hence	ADV
ejpam-6188	296	2	,	,	PUNCT
ejpam-6188	296	3	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	296	4	)	)	PUNCT
ejpam-6188	296	5	)	)	PUNCT
ejpam-6188	297	1	≤	≤	NUM
ejpam-6188	297	2	3n+	3n+	NUM
ejpam-6188	297	3	19	19	NUM
ejpam-6188	297	4	for	for	ADP
ejpam-6188	297	5	n	n	X
ejpam-6188	297	6	≥	≥	NOUN
ejpam-6188	297	7	12	12	NUM
ejpam-6188	297	8	.	.	PUNCT
ejpam-6188	298	1	therefore	therefore	ADV
ejpam-6188	298	2	,	,	PUNCT
ejpam-6188	298	3	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	298	4	)	)	PUNCT
ejpam-6188	298	5	)	)	PUNCT
ejpam-6188	298	6	≤	≤	NUM
ejpam-6188	298	7	3n+	3n+	NUM
ejpam-6188	298	8	19	19	NUM
ejpam-6188	298	9	for	for	ADP
ejpam-6188	298	10	n	n	X
ejpam-6188	298	11	≥	≥	NUM
ejpam-6188	298	12	9	9	NUM
ejpam-6188	298	13	.	.	PUNCT
ejpam-6188	299	1	the	the	DET
ejpam-6188	299	2	following	follow	VERB
ejpam-6188	299	3	theorem	theorem	NOUN
ejpam-6188	299	4	is	be	AUX
ejpam-6188	299	5	a	a	DET
ejpam-6188	299	6	direct	direct	ADJ
ejpam-6188	299	7	result	result	NOUN
ejpam-6188	299	8	from	from	ADP
ejpam-6188	299	9	theorems	theorem	NOUN
ejpam-6188	299	10	4	4	NUM
ejpam-6188	299	11	,	,	PUNCT
ejpam-6188	299	12	5	5	NUM
ejpam-6188	299	13	and	and	CCONJ
ejpam-6188	299	14	6	6	NUM
ejpam-6188	299	15	.	.	PUNCT
ejpam-6188	300	1	p.	p.	NOUN
ejpam-6188	300	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	300	3	et	et	PROPN
ejpam-6188	300	4	al	al	PROPN
ejpam-6188	300	5	.	.	PUNCT
ejpam-6188	300	6	/	/	SYM
ejpam-6188	300	7	eur	eur	PROPN
ejpam-6188	300	8	.	.	PUNCT
ejpam-6188	301	1	j.	j.	PROPN
ejpam-6188	301	2	pure	pure	PROPN
ejpam-6188	301	3	appl	appl	PROPN
ejpam-6188	301	4	.	.	PROPN
ejpam-6188	301	5	math	math	PROPN
ejpam-6188	301	6	,	,	PUNCT
ejpam-6188	301	7	18	18	NUM
ejpam-6188	301	8	(	(	PUNCT
ejpam-6188	301	9	3	3	NUM
ejpam-6188	301	10	)	)	PUNCT
ejpam-6188	301	11	(	(	PUNCT
ejpam-6188	301	12	2025	2025	NUM
ejpam-6188	301	13	)	)	PUNCT
ejpam-6188	301	14	,	,	PUNCT
ejpam-6188	301	15	6188	6188	NUM
ejpam-6188	301	16	11	11	NUM
ejpam-6188	301	17	of	of	ADP
ejpam-6188	301	18	13	13	NUM
ejpam-6188	301	19	u	u	NOUN
ejpam-6188	301	20	3	3	NUM
ejpam-6188	301	21	u1	u1	NOUN
ejpam-6188	301	22	1	1	NUM
ejpam-6188	301	23	u2	u2	PROPN
ejpam-6188	301	24	1	1	NUM
ejpam-6188	301	25	u3	u3	NOUN
ejpam-6188	301	26	5	5	NUM
ejpam-6188	301	27	u4	u4	PROPN
ejpam-6188	301	28	1	1	NUM
ejpam-6188	301	29	u5	u5	PROPN
ejpam-6188	301	30	2	2	NUM
ejpam-6188	301	31	u6	u6	NOUN
ejpam-6188	301	32	1	1	NUM
ejpam-6188	301	33	u7	u7	PROPN
ejpam-6188	301	34	1	1	NUM
ejpam-6188	301	35	u8	u8	PROPN
ejpam-6188	301	36	2	2	NUM
ejpam-6188	301	37	u9	u9	PROPN
ejpam-6188	301	38	1	1	NUM
ejpam-6188	301	39	u10	u10	PROPN
ejpam-6188	301	40	4	4	NUM
ejpam-6188	301	41	u11	u11	ADJ
ejpam-6188	301	42	1	1	NUM
ejpam-6188	301	43	v1	v1	NOUN
ejpam-6188	301	44	3	3	NUM
ejpam-6188	301	45	v2	v2	PROPN
ejpam-6188	301	46	2	2	NUM
ejpam-6188	301	47	v3	v3	PROPN
ejpam-6188	301	48	4	4	NUM
ejpam-6188	301	49	v4	v4	NOUN
ejpam-6188	301	50	1	1	NUM
ejpam-6188	301	51	v5	v5	PROPN
ejpam-6188	301	52	3	3	NUM
ejpam-6188	301	53	v6	v6	NOUN
ejpam-6188	301	54	5	5	NUM
ejpam-6188	301	55	v7	v7	NOUN
ejpam-6188	301	56	4	4	NUM
ejpam-6188	301	57	v8	v8	PROPN
ejpam-6188	301	58	3	3	NUM
ejpam-6188	301	59	v9	v9	PROPN
ejpam-6188	301	60	1	1	NUM
ejpam-6188	301	61	v10	v10	NOUN
ejpam-6188	301	62	2	2	NUM
ejpam-6188	301	63	v11	v11	NOUN
ejpam-6188	301	64	1	1	NUM
ejpam-6188	301	65	figure	figure	NOUN
ejpam-6188	301	66	9	9	NUM
ejpam-6188	301	67	:	:	PUNCT
ejpam-6188	301	68	a	a	DET
ejpam-6188	301	69	b	b	NOUN
ejpam-6188	301	70	-	-	PUNCT
ejpam-6188	301	71	coloring	coloring	NOUN
ejpam-6188	301	72	of	of	ADP
ejpam-6188	301	73	µ(p11	µ(p11	NOUN
ejpam-6188	301	74	)	)	PUNCT
ejpam-6188	301	75	theorem	theorem	VERB
ejpam-6188	301	76	7	7	NUM
ejpam-6188	301	77	.	.	PUNCT
ejpam-6188	302	1	the	the	DET
ejpam-6188	302	2	followings	following	NOUN
ejpam-6188	302	3	hold	hold	VERB
ejpam-6188	302	4	:	:	PUNCT
ejpam-6188	302	5	•	•	NUM
ejpam-6188	302	6	φ′(µ(p7	φ′(µ(p7	NUM
ejpam-6188	302	7	)	)	PUNCT
ejpam-6188	302	8	=	=	SYM
ejpam-6188	303	1	27	27	NUM
ejpam-6188	303	2	,	,	PUNCT
ejpam-6188	303	3	•	•	NOUN
ejpam-6188	303	4	3n+	3n+	NUM
ejpam-6188	303	5	18	18	NUM
ejpam-6188	303	6	≤	≤	NUM
ejpam-6188	303	7	φ′(µ(pn	φ′(µ(pn	NUM
ejpam-6188	303	8	)	)	PUNCT
ejpam-6188	303	9	)	)	PUNCT
ejpam-6188	304	1	≤	≤	NUM
ejpam-6188	304	2	3n+	3n+	NUM
ejpam-6188	304	3	19	19	NUM
ejpam-6188	304	4	for	for	ADP
ejpam-6188	304	5	10	10	NUM
ejpam-6188	304	6	≤	≤	NOUN
ejpam-6188	304	7	n	n	PRON
ejpam-6188	304	8	≤	≤	NUM
ejpam-6188	304	9	15	15	NUM
ejpam-6188	304	10	,	,	PUNCT
ejpam-6188	304	11	•	•	NUM
ejpam-6188	304	12	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	304	13	)	)	PUNCT
ejpam-6188	304	14	)	)	PUNCT
ejpam-6188	305	1	=	=	PUNCT
ejpam-6188	305	2	3n+	3n+	NUM
ejpam-6188	305	3	19	19	NUM
ejpam-6188	305	4	for	for	ADP
ejpam-6188	305	5	n	n	NOUN
ejpam-6188	305	6	=	=	SYM
ejpam-6188	305	7	9	9	NUM
ejpam-6188	305	8	or	or	CCONJ
ejpam-6188	305	9	n	n	PRON
ejpam-6188	305	10	≥	≥	NOUN
ejpam-6188	305	11	16	16	NUM
ejpam-6188	305	12	.	.	PUNCT
ejpam-6188	306	1	4	4	X
ejpam-6188	306	2	.	.	NOUN
ejpam-6188	306	3	conclusion	conclusion	NOUN
ejpam-6188	306	4	and	and	CCONJ
ejpam-6188	306	5	discussion	discussion	NOUN
ejpam-6188	306	6	in	in	ADP
ejpam-6188	306	7	the	the	DET
ejpam-6188	306	8	work	work	NOUN
ejpam-6188	306	9	of	of	ADP
ejpam-6188	306	10	lisna	lisna	NOUN
ejpam-6188	306	11	and	and	CCONJ
ejpam-6188	306	12	sunitha	sunitha	VERB
ejpam-6188	307	1	[	[	X
ejpam-6188	307	2	1	1	NUM
ejpam-6188	307	3	]	]	PUNCT
ejpam-6188	307	4	,	,	PUNCT
ejpam-6188	307	5	they	they	PRON
ejpam-6188	307	6	gave	give	VERB
ejpam-6188	307	7	the	the	DET
ejpam-6188	307	8	b	b	NOUN
ejpam-6188	307	9	-	-	PUNCT
ejpam-6188	307	10	coloring	coloring	NOUN
ejpam-6188	307	11	giving	give	VERB
ejpam-6188	307	12	a	a	DET
ejpam-6188	307	13	b	b	NOUN
ejpam-6188	307	14	-	-	PUNCT
ejpam-6188	307	15	chromatic	chromatic	ADJ
ejpam-6188	307	16	sum	sum	NOUN
ejpam-6188	307	17	of	of	ADP
ejpam-6188	307	18	µ(p8	µ(p8	NOUN
ejpam-6188	307	19	)	)	PUNCT
ejpam-6188	307	20	,	,	PUNCT
ejpam-6188	307	21	i.e.	i.e.	X
ejpam-6188	307	22	φ′(µ(p8	φ′(µ(p8	NOUN
ejpam-6188	307	23	)	)	PUNCT
ejpam-6188	307	24	)	)	PUNCT
ejpam-6188	308	1	=	=	SYM
ejpam-6188	308	2	44	44	NUM
ejpam-6188	308	3	.	.	PUNCT
ejpam-6188	309	1	then	then	ADV
ejpam-6188	309	2	they	they	PRON
ejpam-6188	309	3	extended	extend	VERB
ejpam-6188	309	4	and	and	CCONJ
ejpam-6188	309	5	adjusted	adjust	VERB
ejpam-6188	309	6	such	such	ADJ
ejpam-6188	309	7	coloring	coloring	NOUN
ejpam-6188	309	8	to	to	ADP
ejpam-6188	309	9	a	a	DET
ejpam-6188	309	10	b	b	NOUN
ejpam-6188	309	11	-	-	PUNCT
ejpam-6188	309	12	coloring	coloring	NOUN
ejpam-6188	309	13	of	of	ADP
ejpam-6188	309	14	µ(pn	µ(pn	NOUN
ejpam-6188	309	15	)	)	PUNCT
ejpam-6188	309	16	for	for	ADP
ejpam-6188	309	17	n	n	PRON
ejpam-6188	309	18	≥	≥	NUM
ejpam-6188	309	19	9	9	NUM
ejpam-6188	309	20	.	.	PUNCT
ejpam-6188	310	1	even	even	ADV
ejpam-6188	310	2	though	though	SCONJ
ejpam-6188	310	3	this	this	DET
ejpam-6188	310	4	extended	extended	ADJ
ejpam-6188	310	5	coloring	coloring	NOUN
ejpam-6188	310	6	is	be	AUX
ejpam-6188	310	7	a	a	DET
ejpam-6188	310	8	b	b	NOUN
ejpam-6188	310	9	-	-	PUNCT
ejpam-6188	310	10	coloring	coloring	NOUN
ejpam-6188	310	11	,	,	PUNCT
ejpam-6188	310	12	the	the	DET
ejpam-6188	310	13	sum	sum	NOUN
ejpam-6188	310	14	of	of	ADP
ejpam-6188	310	15	the	the	DET
ejpam-6188	310	16	colors	color	NOUN
ejpam-6188	310	17	may	may	AUX
ejpam-6188	310	18	not	not	PART
ejpam-6188	310	19	be	be	AUX
ejpam-6188	310	20	minimum	minimum	ADJ
ejpam-6188	310	21	.	.	PUNCT
ejpam-6188	311	1	one	one	NUM
ejpam-6188	311	2	of	of	ADP
ejpam-6188	311	3	the	the	DET
ejpam-6188	311	4	main	main	ADJ
ejpam-6188	311	5	reason	reason	NOUN
ejpam-6188	311	6	is	be	AUX
ejpam-6188	311	7	because	because	SCONJ
ejpam-6188	311	8	the	the	DET
ejpam-6188	311	9	number	number	NOUN
ejpam-6188	311	10	of	of	ADP
ejpam-6188	311	11	color	color	NOUN
ejpam-6188	311	12	1	1	NUM
ejpam-6188	311	13	in	in	ADP
ejpam-6188	311	14	the	the	DET
ejpam-6188	311	15	extended	extended	ADJ
ejpam-6188	311	16	coloring	coloring	NOUN
ejpam-6188	311	17	is	be	AUX
ejpam-6188	311	18	less	less	ADJ
ejpam-6188	311	19	than	than	ADP
ejpam-6188	311	20	the	the	DET
ejpam-6188	311	21	maximum	maximum	ADJ
ejpam-6188	311	22	number	number	NOUN
ejpam-6188	311	23	of	of	ADP
ejpam-6188	311	24	color	color	NOUN
ejpam-6188	311	25	1	1	NUM
ejpam-6188	311	26	as	as	SCONJ
ejpam-6188	311	27	shown	show	VERB
ejpam-6188	311	28	in	in	ADP
ejpam-6188	311	29	lemma	lemma	PROPN
ejpam-6188	311	30	5	5	NUM
ejpam-6188	311	31	.	.	PUNCT
ejpam-6188	312	1	in	in	ADP
ejpam-6188	312	2	this	this	DET
ejpam-6188	312	3	work	work	NOUN
ejpam-6188	312	4	,	,	PUNCT
ejpam-6188	312	5	we	we	PRON
ejpam-6188	312	6	give	give	VERB
ejpam-6188	312	7	a	a	DET
ejpam-6188	312	8	b	b	NOUN
ejpam-6188	312	9	-	-	PUNCT
ejpam-6188	312	10	coloring	coloring	NOUN
ejpam-6188	312	11	giving	give	VERB
ejpam-6188	312	12	a	a	DET
ejpam-6188	312	13	lower	low	ADJ
ejpam-6188	312	14	b	b	NOUN
ejpam-6188	312	15	-	-	PUNCT
ejpam-6188	312	16	chromatic	chromatic	ADJ
ejpam-6188	312	17	sum	sum	NOUN
ejpam-6188	312	18	.	.	PUNCT
ejpam-6188	313	1	we	we	PRON
ejpam-6188	313	2	also	also	ADV
ejpam-6188	313	3	analyze	analyze	VERB
ejpam-6188	313	4	a	a	DET
ejpam-6188	313	5	lower	lower	ADV
ejpam-6188	313	6	bound	bind	VERB
ejpam-6188	313	7	.	.	PUNCT
ejpam-6188	314	1	our	our	PRON
ejpam-6188	314	2	method	method	NOUN
ejpam-6188	314	3	relies	rely	VERB
ejpam-6188	314	4	heavily	heavily	ADV
ejpam-6188	314	5	on	on	ADP
ejpam-6188	314	6	lemma	lemma	PROPN
ejpam-6188	314	7	5	5	NUM
ejpam-6188	314	8	by	by	ADP
ejpam-6188	314	9	setting	set	VERB
ejpam-6188	314	10	|c1|	|c1|	NOUN
ejpam-6188	314	11	=	=	SYM
ejpam-6188	314	12	n	n	CCONJ
ejpam-6188	314	13	−	−	NOUN
ejpam-6188	314	14	1	1	NUM
ejpam-6188	314	15	when	when	SCONJ
ejpam-6188	314	16	computing	compute	VERB
ejpam-6188	314	17	the	the	DET
ejpam-6188	314	18	lower	lower	ADV
ejpam-6188	314	19	bound	bind	VERB
ejpam-6188	314	20	.	.	PUNCT
ejpam-6188	315	1	the	the	DET
ejpam-6188	315	2	proof	proof	NOUN
ejpam-6188	315	3	of	of	ADP
ejpam-6188	315	4	lemma	lemma	PROPN
ejpam-6188	315	5	5	5	NUM
ejpam-6188	315	6	also	also	ADV
ejpam-6188	315	7	describes	describe	VERB
ejpam-6188	315	8	valid	valid	ADJ
ejpam-6188	315	9	configurations	configuration	NOUN
ejpam-6188	315	10	of	of	ADP
ejpam-6188	315	11	color	color	NOUN
ejpam-6188	315	12	1	1	NUM
ejpam-6188	315	13	.	.	PUNCT
ejpam-6188	316	1	as	as	ADP
ejpam-6188	316	2	a	a	DET
ejpam-6188	316	3	result	result	NOUN
ejpam-6188	316	4	,	,	PUNCT
ejpam-6188	316	5	we	we	PRON
ejpam-6188	316	6	give	give	VERB
ejpam-6188	316	7	lower	low	ADJ
ejpam-6188	316	8	and	and	CCONJ
ejpam-6188	316	9	upper	upper	ADJ
ejpam-6188	316	10	bounds	bound	NOUN
ejpam-6188	316	11	on	on	ADP
ejpam-6188	316	12	the	the	DET
ejpam-6188	316	13	b	b	NOUN
ejpam-6188	316	14	-	-	PUNCT
ejpam-6188	316	15	chromatic	chromatic	ADJ
ejpam-6188	316	16	sum	sum	NOUN
ejpam-6188	316	17	of	of	ADP
ejpam-6188	316	18	a	a	DET
ejpam-6188	316	19	mycielskian	mycielskian	ADJ
ejpam-6188	316	20	path	path	NOUN
ejpam-6188	316	21	µ(pn	µ(pn	NOUN
ejpam-6188	316	22	)	)	PUNCT
ejpam-6188	316	23	for	for	ADP
ejpam-6188	316	24	10	10	NUM
ejpam-6188	316	25	≤	≤	NOUN
ejpam-6188	316	26	n	n	PRON
ejpam-6188	316	27	≤	≤	NUM
ejpam-6188	316	28	15	15	NUM
ejpam-6188	316	29	.	.	PUNCT
ejpam-6188	317	1	we	we	PRON
ejpam-6188	317	2	get	get	VERB
ejpam-6188	317	3	the	the	DET
ejpam-6188	317	4	exact	exact	ADJ
ejpam-6188	317	5	value	value	NOUN
ejpam-6188	317	6	of	of	ADP
ejpam-6188	317	7	the	the	DET
ejpam-6188	317	8	b	b	NOUN
ejpam-6188	317	9	-	-	PUNCT
ejpam-6188	317	10	chromatic	chromatic	ADJ
ejpam-6188	317	11	sum	sum	NOUN
ejpam-6188	317	12	φ′(µ(p7	φ′(µ(p7	NOUN
ejpam-6188	317	13	)	)	PUNCT
ejpam-6188	317	14	)	)	PUNCT
ejpam-6188	318	1	=	=	SYM
ejpam-6188	318	2	27	27	NUM
ejpam-6188	318	3	and	and	CCONJ
ejpam-6188	318	4	φ′(µ(pn	φ′(µ(pn	NUM
ejpam-6188	318	5	)	)	PUNCT
ejpam-6188	318	6	)	)	PUNCT
ejpam-6188	319	1	=	=	SYM
ejpam-6188	319	2	3n	3n	NOUN
ejpam-6188	320	1	+	+	SYM
ejpam-6188	320	2	19	19	NUM
ejpam-6188	321	1	when	when	SCONJ
ejpam-6188	321	2	n	n	X
ejpam-6188	321	3	=	=	SYM
ejpam-6188	321	4	9	9	NUM
ejpam-6188	321	5	or	or	CCONJ
ejpam-6188	321	6	n	n	PRON
ejpam-6188	321	7	≥	≥	NOUN
ejpam-6188	321	8	16	16	NUM
ejpam-6188	321	9	.	.	PUNCT
ejpam-6188	321	10	table	table	NOUN
ejpam-6188	321	11	1	1	NUM
ejpam-6188	321	12	compares	compare	VERB
ejpam-6188	321	13	the	the	DET
ejpam-6188	321	14	results	result	NOUN
ejpam-6188	321	15	given	give	VERB
ejpam-6188	321	16	by	by	ADP
ejpam-6188	321	17	lisna	lisna	NOUN
ejpam-6188	321	18	and	and	CCONJ
ejpam-6188	321	19	sunitha	sunitha	ADJ
ejpam-6188	322	1	[	[	X
ejpam-6188	322	2	1	1	X
ejpam-6188	322	3	]	]	PUNCT
ejpam-6188	322	4	and	and	CCONJ
ejpam-6188	322	5	the	the	DET
ejpam-6188	322	6	results	result	NOUN
ejpam-6188	322	7	in	in	ADP
ejpam-6188	322	8	this	this	DET
ejpam-6188	322	9	paper	paper	NOUN
ejpam-6188	322	10	.	.	PUNCT
ejpam-6188	323	1	from	from	ADP
ejpam-6188	323	2	the	the	DET
ejpam-6188	323	3	table	table	NOUN
ejpam-6188	323	4	,	,	PUNCT
ejpam-6188	323	5	we	we	PRON
ejpam-6188	323	6	propose	propose	VERB
ejpam-6188	323	7	the	the	DET
ejpam-6188	323	8	following	following	ADJ
ejpam-6188	323	9	conjecture	conjecture	NOUN
ejpam-6188	323	10	.	.	PUNCT
ejpam-6188	324	1	conjecture	conjecture	NOUN
ejpam-6188	324	2	1	1	NUM
ejpam-6188	324	3	.	.	PUNCT
ejpam-6188	324	4	for	for	ADP
ejpam-6188	324	5	n	n	PRON
ejpam-6188	324	6	≥	≥	NOUN
ejpam-6188	324	7	9	9	NUM
ejpam-6188	324	8	,	,	PUNCT
ejpam-6188	324	9	we	we	PRON
ejpam-6188	324	10	have	have	VERB
ejpam-6188	324	11	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	324	12	)	)	PUNCT
ejpam-6188	324	13	)	)	PUNCT
ejpam-6188	325	1	=	=	PUNCT
ejpam-6188	325	2	3n+	3n+	NUM
ejpam-6188	325	3	19	19	NUM
ejpam-6188	325	4	.	.	PUNCT
ejpam-6188	326	1	a	a	DET
ejpam-6188	326	2	vast	vast	ADJ
ejpam-6188	326	3	area	area	NOUN
ejpam-6188	326	4	of	of	ADP
ejpam-6188	326	5	research	research	NOUN
ejpam-6188	326	6	related	relate	VERB
ejpam-6188	326	7	to	to	ADP
ejpam-6188	326	8	the	the	DET
ejpam-6188	326	9	b	b	NOUN
ejpam-6188	326	10	-	-	PUNCT
ejpam-6188	326	11	chromatic	chromatic	ADJ
ejpam-6188	326	12	sum	sum	NOUN
ejpam-6188	326	13	still	still	ADV
ejpam-6188	326	14	remains	remain	VERB
ejpam-6188	326	15	open	open	ADJ
ejpam-6188	326	16	.	.	PUNCT
ejpam-6188	327	1	the	the	DET
ejpam-6188	327	2	bchromatic	bchromatic	ADJ
ejpam-6188	327	3	sum	sum	NOUN
ejpam-6188	327	4	of	of	ADP
ejpam-6188	327	5	only	only	ADV
ejpam-6188	327	6	a	a	DET
ejpam-6188	327	7	few	few	ADJ
ejpam-6188	327	8	specific	specific	ADJ
ejpam-6188	327	9	classes	class	NOUN
ejpam-6188	327	10	of	of	ADP
ejpam-6188	327	11	graphs	graph	NOUN
ejpam-6188	327	12	such	such	ADJ
ejpam-6188	327	13	as	as	ADP
ejpam-6188	327	14	paths	path	NOUN
ejpam-6188	327	15	,	,	PUNCT
ejpam-6188	327	16	cycles	cycle	NOUN
ejpam-6188	327	17	,	,	PUNCT
ejpam-6188	327	18	stars	star	NOUN
ejpam-6188	327	19	and	and	CCONJ
ejpam-6188	327	20	certain	certain	ADJ
ejpam-6188	327	21	classes	class	NOUN
ejpam-6188	327	22	of	of	ADP
ejpam-6188	327	23	mycielskian	mycielskian	ADJ
ejpam-6188	327	24	graphs	graph	NOUN
ejpam-6188	327	25	had	have	AUX
ejpam-6188	327	26	been	be	AUX
ejpam-6188	327	27	studied	study	VERB
ejpam-6188	327	28	.	.	PUNCT
ejpam-6188	328	1	one	one	NUM
ejpam-6188	328	2	of	of	ADP
ejpam-6188	328	3	the	the	DET
ejpam-6188	328	4	intuitive	intuitive	ADJ
ejpam-6188	328	5	problems	problem	NOUN
ejpam-6188	328	6	is	be	AUX
ejpam-6188	328	7	to	to	PART
ejpam-6188	328	8	find	find	VERB
ejpam-6188	328	9	a	a	DET
ejpam-6188	328	10	bound	bind	VERB
ejpam-6188	328	11	on	on	ADP
ejpam-6188	328	12	the	the	DET
ejpam-6188	328	13	b	b	NOUN
ejpam-6188	328	14	-	-	PUNCT
ejpam-6188	328	15	chromatic	chromatic	ADJ
ejpam-6188	328	16	sum	sum	NOUN
ejpam-6188	328	17	for	for	ADP
ejpam-6188	328	18	more	more	ADJ
ejpam-6188	328	19	generalized	generalized	ADJ
ejpam-6188	328	20	classes	class	NOUN
ejpam-6188	328	21	of	of	ADP
ejpam-6188	328	22	graphs	graph	NOUN
ejpam-6188	328	23	,	,	PUNCT
ejpam-6188	328	24	for	for	ADP
ejpam-6188	328	25	example	example	NOUN
ejpam-6188	328	26	,	,	PUNCT
ejpam-6188	328	27	regular	regular	ADJ
ejpam-6188	328	28	graphs	graph	NOUN
ejpam-6188	328	29	and	and	CCONJ
ejpam-6188	328	30	connected	connected	ADJ
ejpam-6188	328	31	graphs	graph	NOUN
ejpam-6188	328	32	.	.	PUNCT
ejpam-6188	329	1	another	another	DET
ejpam-6188	329	2	possible	possible	ADJ
ejpam-6188	329	3	question	question	NOUN
ejpam-6188	329	4	is	be	AUX
ejpam-6188	329	5	to	to	PART
ejpam-6188	329	6	find	find	VERB
ejpam-6188	329	7	an	an	DET
ejpam-6188	329	8	efficient	efficient	ADJ
ejpam-6188	329	9	algorithm	algorithm	NOUN
ejpam-6188	329	10	to	to	PART
ejpam-6188	329	11	compute	compute	VERB
ejpam-6188	329	12	the	the	DET
ejpam-6188	329	13	b	b	NOUN
ejpam-6188	329	14	-	-	PUNCT
ejpam-6188	329	15	chromatic	chromatic	ADJ
ejpam-6188	329	16	sum	sum	NOUN
ejpam-6188	329	17	of	of	ADP
ejpam-6188	329	18	a	a	DET
ejpam-6188	329	19	graph	graph	NOUN
ejpam-6188	329	20	.	.	PUNCT
ejpam-6188	330	1	p.	p.	NOUN
ejpam-6188	330	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	330	3	et	et	PROPN
ejpam-6188	330	4	al	al	PROPN
ejpam-6188	330	5	.	.	PUNCT
ejpam-6188	330	6	/	/	SYM
ejpam-6188	330	7	eur	eur	PROPN
ejpam-6188	330	8	.	.	PUNCT
ejpam-6188	331	1	j.	j.	PROPN
ejpam-6188	331	2	pure	pure	PROPN
ejpam-6188	331	3	appl	appl	PROPN
ejpam-6188	331	4	.	.	PROPN
ejpam-6188	331	5	math	math	PROPN
ejpam-6188	331	6	,	,	PUNCT
ejpam-6188	331	7	18	18	NUM
ejpam-6188	331	8	(	(	PUNCT
ejpam-6188	331	9	3	3	NUM
ejpam-6188	331	10	)	)	PUNCT
ejpam-6188	331	11	(	(	PUNCT
ejpam-6188	331	12	2025	2025	NUM
ejpam-6188	331	13	)	)	PUNCT
ejpam-6188	331	14	,	,	PUNCT
ejpam-6188	331	15	6188	6188	NUM
ejpam-6188	331	16	12	12	NUM
ejpam-6188	331	17	of	of	ADP
ejpam-6188	331	18	13	13	NUM
ejpam-6188	331	19	n	n	NOUN
ejpam-6188	331	20	our	our	PRON
ejpam-6188	331	21	results	result	NOUN
ejpam-6188	331	22	[	[	X
ejpam-6188	331	23	1	1	X
ejpam-6188	331	24	]	]	X
ejpam-6188	331	25	lower	lower	ADV
ejpam-6188	331	26	bound	bind	VERB
ejpam-6188	331	27	upper	upper	ADJ
ejpam-6188	331	28	bound	bind	VERB
ejpam-6188	331	29	upper	upper	ADJ
ejpam-6188	331	30	bound	bind	VERB
ejpam-6188	331	31	7	7	NUM
ejpam-6188	331	32	27	27	NUM
ejpam-6188	331	33	27	27	NUM
ejpam-6188	331	34	28	28	NUM
ejpam-6188	331	35	8	8	NUM
ejpam-6188	331	36	44	44	NUM
ejpam-6188	331	37	44	44	NUM
ejpam-6188	331	38	44	44	NUM
ejpam-6188	331	39	9	9	NUM
ejpam-6188	331	40	3n+	3n+	NUM
ejpam-6188	331	41	19	19	NUM
ejpam-6188	331	42	3n+	3n+	NUM
ejpam-6188	331	43	19	19	NUM
ejpam-6188	331	44	3n+	3n+	NUM
ejpam-6188	331	45	21	21	NUM
ejpam-6188	331	46	10	10	NUM
ejpam-6188	331	47	,	,	PUNCT
ejpam-6188	331	48	12	12	NUM
ejpam-6188	331	49	,	,	PUNCT
ejpam-6188	331	50	14	14	NUM
ejpam-6188	331	51	3n+	3n+	NUM
ejpam-6188	331	52	18	18	NUM
ejpam-6188	331	53	3n+	3n+	NUM
ejpam-6188	331	54	19	19	NUM
ejpam-6188	331	55	3n+	3n+	NUM
ejpam-6188	331	56	21	21	NUM
ejpam-6188	331	57	11	11	NUM
ejpam-6188	331	58	,	,	PUNCT
ejpam-6188	331	59	13	13	NUM
ejpam-6188	331	60	,	,	PUNCT
ejpam-6188	331	61	15	15	NUM
ejpam-6188	331	62	3n+	3n+	NUM
ejpam-6188	331	63	18	18	NUM
ejpam-6188	331	64	3n+	3n+	NUM
ejpam-6188	331	65	19	19	NUM
ejpam-6188	331	66	3n+	3n+	NUM
ejpam-6188	331	67	22	22	NUM
ejpam-6188	331	68	even	even	ADV
ejpam-6188	331	69	n	n	PRON
ejpam-6188	331	70	≥	≥	NOUN
ejpam-6188	331	71	16	16	NUM
ejpam-6188	331	72	3n+	3n+	NUM
ejpam-6188	331	73	19	19	NUM
ejpam-6188	331	74	3n+	3n+	NUM
ejpam-6188	331	75	19	19	NUM
ejpam-6188	331	76	3n+	3n+	NUM
ejpam-6188	331	77	21	21	NUM
ejpam-6188	331	78	odd	odd	ADJ
ejpam-6188	331	79	n	n	PRON
ejpam-6188	331	80	≥	≥	NUM
ejpam-6188	331	81	17	17	NUM
ejpam-6188	331	82	3n+	3n+	NUM
ejpam-6188	331	83	19	19	NUM
ejpam-6188	331	84	3n+	3n+	NUM
ejpam-6188	331	85	19	19	NUM
ejpam-6188	331	86	3n+	3n+	NUM
ejpam-6188	331	87	22	22	NUM
ejpam-6188	331	88	table	table	NOUN
ejpam-6188	331	89	1	1	NUM
ejpam-6188	331	90	:	:	PUNCT
ejpam-6188	331	91	bounds	bound	VERB
ejpam-6188	331	92	on	on	ADP
ejpam-6188	331	93	φ′(µ(pn	φ′(µ(pn	NOUN
ejpam-6188	331	94	)	)	PUNCT
ejpam-6188	331	95	)	)	PUNCT
ejpam-6188	331	96	for	for	ADP
ejpam-6188	331	97	n	n	X
ejpam-6188	331	98	≥	≥	NUM
ejpam-6188	331	99	7	7	NUM
ejpam-6188	331	100	acknowledgements	acknowledgement	NOUN
ejpam-6188	331	101	we	we	PRON
ejpam-6188	331	102	dedicate	dedicate	VERB
ejpam-6188	331	103	this	this	DET
ejpam-6188	331	104	work	work	NOUN
ejpam-6188	331	105	to	to	ADP
ejpam-6188	331	106	the	the	DET
ejpam-6188	331	107	memory	memory	NOUN
ejpam-6188	331	108	of	of	ADP
ejpam-6188	331	109	sittichai	sittichai	PROPN
ejpam-6188	331	110	chaiyakhot	chaiyakhot	PROPN
ejpam-6188	331	111	,	,	PUNCT
ejpam-6188	331	112	a	a	DET
ejpam-6188	331	113	young	young	ADJ
ejpam-6188	331	114	mathematician	mathematician	NOUN
ejpam-6188	331	115	whose	whose	DET
ejpam-6188	331	116	passion	passion	NOUN
ejpam-6188	331	117	,	,	PUNCT
ejpam-6188	331	118	curiosity	curiosity	NOUN
ejpam-6188	331	119	,	,	PUNCT
ejpam-6188	331	120	and	and	CCONJ
ejpam-6188	331	121	dedication	dedication	NOUN
ejpam-6188	331	122	laid	lay	VERB
ejpam-6188	331	123	the	the	DET
ejpam-6188	331	124	foundation	foundation	NOUN
ejpam-6188	331	125	for	for	ADP
ejpam-6188	331	126	this	this	DET
ejpam-6188	331	127	project	project	NOUN
ejpam-6188	331	128	.	.	PUNCT
ejpam-6188	332	1	his	his	PRON
ejpam-6188	332	2	commitment	commitment	NOUN
ejpam-6188	332	3	to	to	ADP
ejpam-6188	332	4	discovery	discovery	PROPN
ejpam-6188	332	5	continues	continue	VERB
ejpam-6188	332	6	to	to	PART
ejpam-6188	332	7	inspire	inspire	VERB
ejpam-6188	332	8	us	we	PRON
ejpam-6188	332	9	.	.	PUNCT
ejpam-6188	333	1	though	though	SCONJ
ejpam-6188	333	2	he	he	PRON
ejpam-6188	333	3	is	be	AUX
ejpam-6188	333	4	no	no	ADV
ejpam-6188	333	5	longer	long	ADV
ejpam-6188	333	6	with	with	ADP
ejpam-6188	333	7	us	we	PRON
ejpam-6188	333	8	,	,	PUNCT
ejpam-6188	333	9	his	his	PRON
ejpam-6188	333	10	contribution	contribution	NOUN
ejpam-6188	333	11	remains	remain	VERB
ejpam-6188	333	12	at	at	ADP
ejpam-6188	333	13	the	the	DET
ejpam-6188	333	14	heart	heart	NOUN
ejpam-6188	333	15	of	of	ADP
ejpam-6188	333	16	this	this	DET
ejpam-6188	333	17	research	research	NOUN
ejpam-6188	333	18	.	.	PUNCT
ejpam-6188	334	1	we	we	PRON
ejpam-6188	334	2	are	be	AUX
ejpam-6188	334	3	deeply	deeply	ADV
ejpam-6188	334	4	grateful	grateful	ADJ
ejpam-6188	334	5	for	for	ADP
ejpam-6188	334	6	his	his	PRON
ejpam-6188	334	7	work	work	NOUN
ejpam-6188	334	8	and	and	CCONJ
ejpam-6188	334	9	honor	honor	VERB
ejpam-6188	334	10	his	his	PRON
ejpam-6188	334	11	memory	memory	NOUN
ejpam-6188	334	12	by	by	ADP
ejpam-6188	334	13	delivering	deliver	VERB
ejpam-6188	334	14	his	his	PRON
ejpam-6188	334	15	discovery	discovery	NOUN
ejpam-6188	334	16	.	.	PUNCT
ejpam-6188	335	1	this	this	DET
ejpam-6188	335	2	research	research	NOUN
ejpam-6188	335	3	project	project	NOUN
ejpam-6188	335	4	was	be	AUX
ejpam-6188	335	5	financially	financially	ADV
ejpam-6188	335	6	supported	support	VERB
ejpam-6188	335	7	by	by	ADP
ejpam-6188	335	8	mahasarakham	mahasarakham	PROPN
ejpam-6188	335	9	university	university	PROPN
ejpam-6188	335	10	.	.	PUNCT
ejpam-6188	336	1	references	reference	NOUN
ejpam-6188	336	2	[	[	X
ejpam-6188	336	3	1	1	NUM
ejpam-6188	336	4	]	]	PUNCT
ejpam-6188	336	5	p.	p.	NOUN
ejpam-6188	336	6	c.	c.	PROPN
ejpam-6188	336	7	lisna	lisna	PROPN
ejpam-6188	336	8	and	and	CCONJ
ejpam-6188	336	9	m.	m.	PROPN
ejpam-6188	336	10	s.	s.	PROPN
ejpam-6188	336	11	sunitha	sunitha	VERB
ejpam-6188	336	12	.	.	PUNCT
ejpam-6188	337	1	b	b	X
ejpam-6188	337	2	-	-	PUNCT
ejpam-6188	337	3	chromatic	chromatic	ADJ
ejpam-6188	337	4	sum	sum	NOUN
ejpam-6188	337	5	of	of	ADP
ejpam-6188	337	6	mycielskian	mycielskian	NOUN
ejpam-6188	337	7	of	of	ADP
ejpam-6188	337	8	paths	path	NOUN
ejpam-6188	337	9	.	.	PUNCT
ejpam-6188	338	1	electronic	electronic	ADJ
ejpam-6188	338	2	notes	note	NOUN
ejpam-6188	338	3	in	in	ADP
ejpam-6188	338	4	discrete	discrete	ADJ
ejpam-6188	338	5	mathematics	mathematic	NOUN
ejpam-6188	338	6	,	,	PUNCT
ejpam-6188	338	7	63:407–414	63:407–414	NOUN
ejpam-6188	338	8	,	,	PUNCT
ejpam-6188	338	9	2017	2017	NUM
ejpam-6188	338	10	.	.	PUNCT
ejpam-6188	339	1	[	[	X
ejpam-6188	339	2	2	2	NUM
ejpam-6188	339	3	]	]	PUNCT
ejpam-6188	339	4	r.	r.	PROPN
ejpam-6188	339	5	w.	w.	PROPN
ejpam-6188	339	6	irving	irving	PROPN
ejpam-6188	339	7	and	and	CCONJ
ejpam-6188	339	8	d.	d.	PROPN
ejpam-6188	339	9	f.	f.	PROPN
ejpam-6188	339	10	manlove	manlove	PROPN
ejpam-6188	339	11	.	.	PUNCT
ejpam-6188	340	1	the	the	DET
ejpam-6188	340	2	b	b	NOUN
ejpam-6188	340	3	-	-	PUNCT
ejpam-6188	340	4	chromatic	chromatic	ADJ
ejpam-6188	340	5	number	number	NOUN
ejpam-6188	340	6	of	of	ADP
ejpam-6188	340	7	a	a	DET
ejpam-6188	340	8	graph	graph	NOUN
ejpam-6188	340	9	.	.	PUNCT
ejpam-6188	341	1	discrete	discrete	ADJ
ejpam-6188	341	2	applied	apply	VERB
ejpam-6188	341	3	mathematics	mathematic	NOUN
ejpam-6188	341	4	,	,	PUNCT
ejpam-6188	341	5	91(1	91(1	NOUN
ejpam-6188	341	6	-	-	PUNCT
ejpam-6188	341	7	3):127–141	3):127–141	NUM
ejpam-6188	341	8	,	,	PUNCT
ejpam-6188	341	9	1999	1999	NUM
ejpam-6188	341	10	.	.	PUNCT
ejpam-6188	342	1	[	[	X
ejpam-6188	342	2	3	3	NUM
ejpam-6188	342	3	]	]	PUNCT
ejpam-6188	342	4	m.	m.	NOUN
ejpam-6188	342	5	alkhateeb	alkhateeb	NOUN
ejpam-6188	342	6	and	and	CCONJ
ejpam-6188	342	7	a.	a.	NOUN
ejpam-6188	342	8	kohl	kohl	NOUN
ejpam-6188	342	9	.	.	PUNCT
ejpam-6188	343	1	upper	upper	ADJ
ejpam-6188	343	2	bounds	bound	NOUN
ejpam-6188	343	3	on	on	ADP
ejpam-6188	343	4	the	the	DET
ejpam-6188	343	5	b	b	NOUN
ejpam-6188	343	6	-	-	PUNCT
ejpam-6188	343	7	chromatic	chromatic	ADJ
ejpam-6188	343	8	number	number	NOUN
ejpam-6188	343	9	and	and	CCONJ
ejpam-6188	343	10	results	result	NOUN
ejpam-6188	343	11	for	for	ADP
ejpam-6188	343	12	restricted	restrict	VERB
ejpam-6188	343	13	graph	graph	NOUN
ejpam-6188	343	14	classes	class	NOUN
ejpam-6188	343	15	.	.	PUNCT
ejpam-6188	344	1	discuss	discuss	PROPN
ejpam-6188	344	2	.	.	PUNCT
ejpam-6188	345	1	math	math	NOUN
ejpam-6188	345	2	.	.	PUNCT
ejpam-6188	346	1	graph	graph	NOUN
ejpam-6188	346	2	theory	theory	NOUN
ejpam-6188	346	3	,	,	PUNCT
ejpam-6188	346	4	31:709–735	31:709–735	NUM
ejpam-6188	346	5	,	,	PUNCT
ejpam-6188	346	6	2011	2011	NUM
ejpam-6188	346	7	.	.	PUNCT
ejpam-6188	347	1	[	[	X
ejpam-6188	347	2	4	4	NUM
ejpam-6188	347	3	]	]	X
ejpam-6188	347	4	c.	c.	PROPN
ejpam-6188	347	5	guo	guo	PROPN
ejpam-6188	347	6	and	and	CCONJ
ejpam-6188	347	7	m.	m.	PROPN
ejpam-6188	347	8	newman	newman	PROPN
ejpam-6188	347	9	.	.	PUNCT
ejpam-6188	348	1	on	on	ADP
ejpam-6188	348	2	the	the	DET
ejpam-6188	348	3	b	b	NOUN
ejpam-6188	348	4	-	-	PUNCT
ejpam-6188	348	5	chromatic	chromatic	ADJ
ejpam-6188	348	6	number	number	NOUN
ejpam-6188	348	7	of	of	ADP
ejpam-6188	348	8	cartesian	cartesian	ADJ
ejpam-6188	348	9	products	product	NOUN
ejpam-6188	348	10	.	.	PUNCT
ejpam-6188	349	1	discrete	discrete	ADJ
ejpam-6188	349	2	applied	apply	VERB
ejpam-6188	349	3	mathematics	mathematic	NOUN
ejpam-6188	349	4	,	,	PUNCT
ejpam-6188	349	5	239:82–93	239:82–93	NUM
ejpam-6188	349	6	,	,	PUNCT
ejpam-6188	349	7	2018	2018	NUM
ejpam-6188	349	8	.	.	PUNCT
ejpam-6188	350	1	[	[	X
ejpam-6188	350	2	5	5	X
ejpam-6188	350	3	]	]	PUNCT
ejpam-6188	350	4	k.	k.	PROPN
ejpam-6188	350	5	kouider	kouider	PROPN
ejpam-6188	350	6	and	and	CCONJ
ejpam-6188	350	7	m.	m.	NOUN
ejpam-6188	350	8	zaker	zaker	PROPN
ejpam-6188	350	9	.	.	PUNCT
ejpam-6188	351	1	bounds	bound	VERB
ejpam-6188	351	2	for	for	ADP
ejpam-6188	351	3	the	the	DET
ejpam-6188	351	4	b	b	NOUN
ejpam-6188	351	5	-	-	PUNCT
ejpam-6188	351	6	chromatic	chromatic	ADJ
ejpam-6188	351	7	number	number	NOUN
ejpam-6188	351	8	of	of	ADP
ejpam-6188	351	9	some	some	DET
ejpam-6188	351	10	families	family	NOUN
ejpam-6188	351	11	of	of	ADP
ejpam-6188	351	12	graphs	graph	NOUN
ejpam-6188	351	13	.	.	PUNCT
ejpam-6188	352	1	discrete	discrete	ADJ
ejpam-6188	352	2	mathematics	mathematic	NOUN
ejpam-6188	352	3	,	,	PUNCT
ejpam-6188	352	4	306:617–623	306:617–623	NUM
ejpam-6188	352	5	,	,	PUNCT
ejpam-6188	352	6	2006	2006	NUM
ejpam-6188	352	7	.	.	PUNCT
ejpam-6188	353	1	[	[	X
ejpam-6188	353	2	6	6	NUM
ejpam-6188	353	3	]	]	PUNCT
ejpam-6188	353	4	m.	m.	NOUN
ejpam-6188	353	5	jakovac	jakovac	PROPN
ejpam-6188	353	6	and	and	CCONJ
ejpam-6188	353	7	i.	i.	PROPN
ejpam-6188	353	8	peterin	peterin	NOUN
ejpam-6188	353	9	.	.	PUNCT
ejpam-6188	354	1	the	the	DET
ejpam-6188	354	2	b	b	NOUN
ejpam-6188	354	3	-	-	PUNCT
ejpam-6188	354	4	chromatic	chromatic	ADJ
ejpam-6188	354	5	number	number	NOUN
ejpam-6188	354	6	and	and	CCONJ
ejpam-6188	354	7	related	related	ADJ
ejpam-6188	354	8	topics	topic	NOUN
ejpam-6188	354	9	—	—	PUNCT
ejpam-6188	354	10	a	a	DET
ejpam-6188	354	11	survey	survey	NOUN
ejpam-6188	354	12	.	.	PUNCT
ejpam-6188	355	1	discrete	discrete	ADJ
ejpam-6188	355	2	applied	apply	VERB
ejpam-6188	355	3	mathematics	mathematic	NOUN
ejpam-6188	355	4	,	,	PUNCT
ejpam-6188	355	5	235:184–201	235:184–201	NUM
ejpam-6188	355	6	,	,	PUNCT
ejpam-6188	355	7	2018	2018	NUM
ejpam-6188	355	8	.	.	PUNCT
ejpam-6188	356	1	[	[	X
ejpam-6188	356	2	7	7	X
ejpam-6188	356	3	]	]	X
ejpam-6188	356	4	h.	h.	PROPN
ejpam-6188	356	5	elghazel	elghazel	PROPN
ejpam-6188	356	6	,	,	PUNCT
ejpam-6188	356	7	h.	h.	PROPN
ejpam-6188	356	8	kheddouci	kheddouci	PROPN
ejpam-6188	356	9	,	,	PUNCT
ejpam-6188	356	10	v.	v.	ADP
ejpam-6188	356	11	deslandres	deslandre	NOUN
ejpam-6188	356	12	,	,	PUNCT
ejpam-6188	356	13	and	and	CCONJ
ejpam-6188	356	14	a.	a.	NOUN
ejpam-6188	356	15	dussauchoy	dussauchoy	PROPN
ejpam-6188	356	16	.	.	PUNCT
ejpam-6188	357	1	a	a	DET
ejpam-6188	357	2	graph	graph	NOUN
ejpam-6188	357	3	b	b	NOUN
ejpam-6188	357	4	-	-	PUNCT
ejpam-6188	357	5	coloring	color	VERB
ejpam-6188	357	6	framework	framework	NOUN
ejpam-6188	357	7	for	for	ADP
ejpam-6188	357	8	data	datum	NOUN
ejpam-6188	357	9	clustering	clustering	NOUN
ejpam-6188	357	10	.	.	PUNCT
ejpam-6188	358	1	journal	journal	NOUN
ejpam-6188	358	2	of	of	ADP
ejpam-6188	358	3	mathematical	mathematical	ADJ
ejpam-6188	358	4	modelling	modelling	NOUN
ejpam-6188	358	5	and	and	CCONJ
ejpam-6188	358	6	algorithms	algorithm	NOUN
ejpam-6188	358	7	,	,	PUNCT
ejpam-6188	358	8	7:389–423	7:389–423	NUM
ejpam-6188	358	9	,	,	PUNCT
ejpam-6188	358	10	2008	2008	NUM
ejpam-6188	358	11	.	.	PUNCT
ejpam-6188	359	1	[	[	X
ejpam-6188	359	2	8	8	NUM
ejpam-6188	359	3	]	]	X
ejpam-6188	359	4	e.	e.	PROPN
ejpam-6188	359	5	kubicka	kubicka	PROPN
ejpam-6188	359	6	and	and	CCONJ
ejpam-6188	359	7	a.	a.	PROPN
ejpam-6188	359	8	j.	j.	PROPN
ejpam-6188	359	9	schwenk	schwenk	PROPN
ejpam-6188	359	10	.	.	PUNCT
ejpam-6188	360	1	an	an	DET
ejpam-6188	360	2	introduction	introduction	NOUN
ejpam-6188	360	3	to	to	ADP
ejpam-6188	360	4	chromatic	chromatic	ADJ
ejpam-6188	360	5	sums	sum	NOUN
ejpam-6188	360	6	.	.	PUNCT
ejpam-6188	361	1	in	in	ADP
ejpam-6188	361	2	proceedings	proceeding	NOUN
ejpam-6188	361	3	of	of	ADP
ejpam-6188	361	4	the	the	DET
ejpam-6188	361	5	17th	17th	ADJ
ejpam-6188	361	6	conference	conference	NOUN
ejpam-6188	361	7	on	on	ADP
ejpam-6188	361	8	acm	acm	PROPN
ejpam-6188	361	9	annual	annual	ADJ
ejpam-6188	361	10	computer	computer	NOUN
ejpam-6188	361	11	science	science	NOUN
ejpam-6188	361	12	conference	conference	NOUN
ejpam-6188	361	13	,	,	PUNCT
ejpam-6188	361	14	csc	csc	PROPN
ejpam-6188	361	15	’	'	PUNCT
ejpam-6188	361	16	89	89	NUM
ejpam-6188	361	17	,	,	PUNCT
ejpam-6188	361	18	page	page	NOUN
ejpam-6188	361	19	39–45	39–45	NUM
ejpam-6188	361	20	,	,	PUNCT
ejpam-6188	361	21	new	new	PROPN
ejpam-6188	361	22	york	york	PROPN
ejpam-6188	361	23	,	,	PUNCT
ejpam-6188	361	24	ny	ny	PROPN
ejpam-6188	361	25	,	,	PUNCT
ejpam-6188	361	26	usa	usa	PROPN
ejpam-6188	361	27	,	,	PUNCT
ejpam-6188	361	28	1989	1989	NUM
ejpam-6188	361	29	.	.	PUNCT
ejpam-6188	362	1	association	association	NOUN
ejpam-6188	362	2	for	for	ADP
ejpam-6188	362	3	computing	compute	VERB
ejpam-6188	362	4	machinery	machinery	NOUN
ejpam-6188	362	5	.	.	PUNCT
ejpam-6188	363	1	[	[	X
ejpam-6188	363	2	9	9	NUM
ejpam-6188	363	3	]	]	PUNCT
ejpam-6188	363	4	a.	a.	NOUN
ejpam-6188	363	5	bar	bar	NOUN
ejpam-6188	363	6	-	-	PUNCT
ejpam-6188	363	7	noy	noy	NOUN
ejpam-6188	363	8	,	,	PUNCT
ejpam-6188	363	9	m.	m.	NOUN
ejpam-6188	363	10	bellare	bellare	NOUN
ejpam-6188	363	11	,	,	PUNCT
ejpam-6188	363	12	m.	m.	NOUN
ejpam-6188	363	13	m.	m.	NOUN
ejpam-6188	363	14	halldórsson	halldórsson	PROPN
ejpam-6188	363	15	,	,	PUNCT
ejpam-6188	363	16	h.	h.	PROPN
ejpam-6188	363	17	shachnai	shachnai	PROPN
ejpam-6188	363	18	,	,	PUNCT
ejpam-6188	363	19	and	and	CCONJ
ejpam-6188	363	20	t.	t.	PROPN
ejpam-6188	363	21	tamir	tamir	PROPN
ejpam-6188	363	22	.	.	PROPN
ejpam-6188	364	1	on	on	ADP
ejpam-6188	364	2	chromatic	chromatic	ADJ
ejpam-6188	364	3	sums	sum	NOUN
ejpam-6188	364	4	and	and	CCONJ
ejpam-6188	364	5	distributed	distribute	VERB
ejpam-6188	364	6	resource	resource	NOUN
ejpam-6188	364	7	allocation	allocation	NOUN
ejpam-6188	364	8	.	.	PUNCT
ejpam-6188	365	1	information	information	NOUN
ejpam-6188	365	2	and	and	CCONJ
ejpam-6188	365	3	computation	computation	NOUN
ejpam-6188	365	4	,	,	PUNCT
ejpam-6188	365	5	140(2):183	140(2):183	NUM
ejpam-6188	365	6	–	–	PUNCT
ejpam-6188	365	7	202	202	NUM
ejpam-6188	365	8	,	,	PUNCT
ejpam-6188	365	9	1998	1998	NUM
ejpam-6188	365	10	.	.	PUNCT
ejpam-6188	366	1	p.	p.	NOUN
ejpam-6188	366	2	vichitkunakorn	vichitkunakorn	NOUN
ejpam-6188	366	3	et	et	PROPN
ejpam-6188	366	4	al	al	PROPN
ejpam-6188	366	5	.	.	PUNCT
ejpam-6188	366	6	/	/	SYM
ejpam-6188	366	7	eur	eur	PROPN
ejpam-6188	366	8	.	.	PUNCT
ejpam-6188	367	1	j.	j.	PROPN
ejpam-6188	367	2	pure	pure	PROPN
ejpam-6188	367	3	appl	appl	PROPN
ejpam-6188	367	4	.	.	PROPN
ejpam-6188	367	5	math	math	PROPN
ejpam-6188	367	6	,	,	PUNCT
ejpam-6188	367	7	18	18	NUM
ejpam-6188	367	8	(	(	PUNCT
ejpam-6188	367	9	3	3	NUM
ejpam-6188	367	10	)	)	PUNCT
ejpam-6188	367	11	(	(	PUNCT
ejpam-6188	367	12	2025	2025	NUM
ejpam-6188	367	13	)	)	PUNCT
ejpam-6188	367	14	,	,	PUNCT
ejpam-6188	367	15	6188	6188	NUM
ejpam-6188	367	16	13	13	NUM
ejpam-6188	367	17	of	of	ADP
ejpam-6188	367	18	13	13	NUM
ejpam-6188	367	19	[	[	SYM
ejpam-6188	367	20	10	10	NUM
ejpam-6188	367	21	]	]	PUNCT
ejpam-6188	367	22	p.	p.	NOUN
ejpam-6188	367	23	c.	c.	PROPN
ejpam-6188	367	24	lisna	lisna	PROPN
ejpam-6188	367	25	and	and	CCONJ
ejpam-6188	367	26	m.	m.	PROPN
ejpam-6188	367	27	s.	s.	PROPN
ejpam-6188	367	28	sunitha	sunitha	VERB
ejpam-6188	367	29	.	.	PUNCT
ejpam-6188	368	1	b	b	X
ejpam-6188	368	2	-	-	PUNCT
ejpam-6188	368	3	chromatic	chromatic	ADJ
ejpam-6188	368	4	sum	sum	NOUN
ejpam-6188	368	5	of	of	ADP
ejpam-6188	368	6	a	a	DET
ejpam-6188	368	7	graph	graph	NOUN
ejpam-6188	368	8	.	.	PUNCT
ejpam-6188	369	1	discrete	discrete	ADJ
ejpam-6188	369	2	mathematics	mathematic	NOUN
ejpam-6188	369	3	,	,	PUNCT
ejpam-6188	369	4	algorithms	algorithm	NOUN
ejpam-6188	369	5	and	and	CCONJ
ejpam-6188	369	6	applications	application	NOUN
ejpam-6188	369	7	,	,	PUNCT
ejpam-6188	369	8	7(3):1550040	7(3):1550040	NUM
ejpam-6188	369	9	,	,	PUNCT
ejpam-6188	369	10	2015	2015	NUM
ejpam-6188	369	11	.	.	PUNCT
ejpam-6188	370	1	[	[	X
ejpam-6188	370	2	11	11	NUM
ejpam-6188	370	3	]	]	PUNCT
ejpam-6188	370	4	p.	p.	PROPN
ejpam-6188	370	5	c.	c.	PROPN
ejpam-6188	370	6	lisna	lisna	PROPN
ejpam-6188	370	7	and	and	CCONJ
ejpam-6188	370	8	m.	m.	PROPN
ejpam-6188	370	9	s.	s.	PROPN
ejpam-6188	370	10	sunitha	sunitha	VERB
ejpam-6188	370	11	.	.	PUNCT
ejpam-6188	371	1	b	b	X
ejpam-6188	371	2	-	-	PUNCT
ejpam-6188	371	3	chromatic	chromatic	ADJ
ejpam-6188	371	4	sum	sum	NOUN
ejpam-6188	371	5	of	of	ADP
ejpam-6188	371	6	a	a	DET
ejpam-6188	371	7	graph	graph	NOUN
ejpam-6188	371	8	.	.	PUNCT
ejpam-6188	372	1	discrete	discrete	ADJ
ejpam-6188	372	2	mathematics	mathematic	NOUN
ejpam-6188	372	3	,	,	PUNCT
ejpam-6188	372	4	algorithms	algorithm	NOUN
ejpam-6188	372	5	and	and	CCONJ
ejpam-6188	372	6	applications	application	NOUN
ejpam-6188	372	7	,	,	PUNCT
ejpam-6188	372	8	07(04):1550040	07(04):1550040	NUM
ejpam-6188	372	9	,	,	PUNCT
ejpam-6188	372	10	2015	2015	NUM
ejpam-6188	372	11	.	.	PUNCT
ejpam-6188	373	1	[	[	X
ejpam-6188	373	2	12	12	NUM
ejpam-6188	373	3	]	]	PUNCT
ejpam-6188	373	4	p.	p.	NOUN
ejpam-6188	373	5	c.	c.	PROPN
ejpam-6188	373	6	lisna	lisna	PROPN
ejpam-6188	373	7	and	and	CCONJ
ejpam-6188	373	8	m.	m.	PROPN
ejpam-6188	373	9	s.	s.	PROPN
ejpam-6188	373	10	sunitha	sunitha	PROPN
ejpam-6188	373	11	.	.	PUNCT
ejpam-6188	374	1	on	on	ADP
ejpam-6188	374	2	the	the	DET
ejpam-6188	374	3	b	b	NOUN
ejpam-6188	374	4	-	-	PUNCT
ejpam-6188	374	5	chromatic	chromatic	ADJ
ejpam-6188	374	6	sum	sum	NOUN
ejpam-6188	374	7	of	of	ADP
ejpam-6188	374	8	mycielskian	mycielskian	NOUN
ejpam-6188	374	9	of	of	ADP
ejpam-6188	374	10	km	km	PROPN
ejpam-6188	374	11	,	,	PUNCT
ejpam-6188	374	12	n	n	CCONJ
ejpam-6188	374	13	,	,	PUNCT
ejpam-6188	374	14	kn	kn	PROPN
ejpam-6188	374	15	and	and	CCONJ
ejpam-6188	374	16	cn	cn	PROPN
ejpam-6188	374	17	.	.	PROPN
ejpam-6188	374	18	journal	journal	PROPN
ejpam-6188	374	19	of	of	ADP
ejpam-6188	374	20	interconnection	interconnection	NOUN
ejpam-6188	374	21	networks	network	NOUN
ejpam-6188	374	22	,	,	PUNCT
ejpam-6188	374	23	20(02):2050007	20(02):2050007	NUM
ejpam-6188	374	24	,	,	PUNCT
ejpam-6188	374	25	2020	2020	NUM
ejpam-6188	374	26	.	.	PUNCT
ejpam-6188	375	1	[	[	X
ejpam-6188	375	2	13	13	NUM
ejpam-6188	375	3	]	]	PUNCT
ejpam-6188	375	4	m.	m.	NOUN
ejpam-6188	375	5	caramia	caramia	NOUN
ejpam-6188	375	6	and	and	CCONJ
ejpam-6188	375	7	p.	p.	NOUN
ejpam-6188	375	8	dell’olmo	dell’olmo	NOUN
ejpam-6188	375	9	.	.	PUNCT
ejpam-6188	376	1	a	a	DET
ejpam-6188	376	2	lower	lower	ADV
ejpam-6188	376	3	bound	bind	VERB
ejpam-6188	376	4	on	on	ADP
ejpam-6188	376	5	the	the	DET
ejpam-6188	376	6	chromatic	chromatic	ADJ
ejpam-6188	376	7	number	number	NOUN
ejpam-6188	376	8	of	of	ADP
ejpam-6188	376	9	mycielski	mycielski	ADJ
ejpam-6188	376	10	graphs	graph	NOUN
ejpam-6188	376	11	.	.	PUNCT
ejpam-6188	377	1	discrete	discrete	ADJ
ejpam-6188	377	2	mathematics	mathematic	NOUN
ejpam-6188	377	3	,	,	PUNCT
ejpam-6188	377	4	235(1	235(1	NUM
ejpam-6188	377	5	-	-	SYM
ejpam-6188	377	6	3):79–86	3):79–86	NUM
ejpam-6188	377	7	,	,	PUNCT
ejpam-6188	377	8	2001	2001	NUM
ejpam-6188	377	9	.	.	PUNCT
ejpam-6188	378	1	[	[	X
ejpam-6188	378	2	14	14	NUM
ejpam-6188	378	3	]	]	X
ejpam-6188	378	4	j.	j.	PROPN
ejpam-6188	378	5	mycielski	mycielski	PROPN
ejpam-6188	378	6	.	.	PUNCT
ejpam-6188	379	1	sur	sur	PROPN
ejpam-6188	379	2	le	le	PROPN
ejpam-6188	379	3	coloriage	coloriage	PROPN
ejpam-6188	379	4	des	des	PROPN
ejpam-6188	379	5	graphs	graph	NOUN
ejpam-6188	379	6	.	.	PUNCT
ejpam-6188	380	1	in	in	ADP
ejpam-6188	380	2	colloquium	colloquium	NOUN
ejpam-6188	380	3	mathematicae	mathematicae	PROPN
ejpam-6188	380	4	,	,	PUNCT
ejpam-6188	380	5	volume	volume	NOUN
ejpam-6188	380	6	3	3	NUM
ejpam-6188	380	7	,	,	PUNCT
ejpam-6188	380	8	pages	page	NOUN
ejpam-6188	380	9	161–162	161–162	NUM
ejpam-6188	380	10	,	,	PUNCT
ejpam-6188	380	11	1955	1955	NUM
ejpam-6188	380	12	.	.	PUNCT
ejpam-6188	381	1	[	[	X
ejpam-6188	381	2	15	15	NUM
ejpam-6188	381	3	]	]	X
ejpam-6188	381	4	p.	p.	NOUN
ejpam-6188	381	5	c.	c.	PROPN
ejpam-6188	381	6	lisna	lisna	PROPN
ejpam-6188	381	7	and	and	CCONJ
ejpam-6188	381	8	m.	m.	PROPN
ejpam-6188	381	9	s.	s.	PROPN
ejpam-6188	381	10	sunitha	sunitha	PROPN
ejpam-6188	381	11	.	.	PUNCT
ejpam-6188	382	1	the	the	DET
ejpam-6188	382	2	b	b	NOUN
ejpam-6188	382	3	-	-	PUNCT
ejpam-6188	382	4	chromatic	chromatic	ADJ
ejpam-6188	382	5	number	number	NOUN
ejpam-6188	382	6	of	of	ADP
ejpam-6188	382	7	mycielskian	mycielskian	NOUN
ejpam-6188	382	8	of	of	ADP
ejpam-6188	382	9	some	some	DET
ejpam-6188	382	10	graphs	graph	NOUN
ejpam-6188	382	11	.	.	PUNCT
ejpam-6188	383	1	international	international	ADJ
ejpam-6188	383	2	journal	journal	NOUN
ejpam-6188	383	3	of	of	ADP
ejpam-6188	383	4	convergence	convergence	NOUN
ejpam-6188	383	5	computing	computing	NOUN
ejpam-6188	383	6	,	,	PUNCT
ejpam-6188	383	7	2(1):23–40	2(1):23–40	NUM
ejpam-6188	383	8	,	,	PUNCT
ejpam-6188	383	9	2016	2016	NUM
ejpam-6188	383	10	.	.	PUNCT