id	sid	tid	token	lemma	pos
ejpam-4574	1	1	european	european	PROPN
ejpam-4574	1	2	journal	journal	PROPN
ejpam-4574	1	3	of	of	ADP
ejpam-4574	1	4	pure	pure	ADJ
ejpam-4574	1	5	and	and	CCONJ
ejpam-4574	1	6	applied	apply	VERB
ejpam-4574	1	7	mathematics	mathematic	NOUN
ejpam-4574	1	8	vol	vol	NOUN
ejpam-4574	1	9	.	.	PROPN
ejpam-4574	2	1	15	15	NUM
ejpam-4574	2	2	,	,	PUNCT
ejpam-4574	2	3	no	no	INTJ
ejpam-4574	2	4	.	.	NOUN
ejpam-4574	2	5	4	4	NUM
ejpam-4574	2	6	,	,	PUNCT
ejpam-4574	2	7	2022	2022	NUM
ejpam-4574	2	8	,	,	PUNCT
ejpam-4574	2	9	1822	1822	NUM
ejpam-4574	2	10	-	-	SYM
ejpam-4574	2	11	1835	1835	NUM
ejpam-4574	2	12	issn	issn	VERB
ejpam-4574	2	13	1307	1307	NUM
ejpam-4574	2	14	-	-	SYM
ejpam-4574	2	15	5543	5543	NUM
ejpam-4574	2	16	–	–	PUNCT
ejpam-4574	2	17	ejpam.com	ejpam.com	X
ejpam-4574	2	18	published	publish	VERB
ejpam-4574	2	19	by	by	ADP
ejpam-4574	2	20	new	new	PROPN
ejpam-4574	2	21	york	york	PROPN
ejpam-4574	2	22	business	business	PROPN
ejpam-4574	2	23	global	global	ADJ
ejpam-4574	2	24	acyclic	acyclic	PROPN
ejpam-4574	2	25	and	and	CCONJ
ejpam-4574	2	26	star	star	NOUN
ejpam-4574	2	27	coloring	coloring	NOUN
ejpam-4574	2	28	of	of	ADP
ejpam-4574	2	29	powers	power	NOUN
ejpam-4574	2	30	of	of	ADP
ejpam-4574	2	31	paths	path	NOUN
ejpam-4574	2	32	and	and	CCONJ
ejpam-4574	2	33	cycles	cycle	NOUN
ejpam-4574	2	34	ali	ali	PROPN
ejpam-4574	2	35	etawi1,∗	etawi1,∗	PROPN
ejpam-4574	2	36	,	,	PUNCT
ejpam-4574	2	37	manal	manal	ADJ
ejpam-4574	2	38	ghanem1	ghanem1	PROPN
ejpam-4574	2	39	,	,	PUNCT
ejpam-4574	2	40	hasan	hasan	PROPN
ejpam-4574	2	41	al	al	PROPN
ejpam-4574	2	42	-	-	PUNCT
ejpam-4574	2	43	ezeh1	ezeh1	PROPN
ejpam-4574	2	44	1	1	NUM
ejpam-4574	2	45	mathematics	mathematic	NOUN
ejpam-4574	2	46	,	,	PUNCT
ejpam-4574	2	47	the	the	DET
ejpam-4574	2	48	university	university	PROPN
ejpam-4574	2	49	of	of	ADP
ejpam-4574	2	50	jordan	jordan	PROPN
ejpam-4574	2	51	,	,	PUNCT
ejpam-4574	2	52	amman	amman	PROPN
ejpam-4574	2	53	,	,	PUNCT
ejpam-4574	2	54	jordan	jordan	PROPN
ejpam-4574	2	55	abstract	abstract	PROPN
ejpam-4574	2	56	.	.	PUNCT
ejpam-4574	3	1	let	let	VERB
ejpam-4574	3	2	g	g	PROPN
ejpam-4574	3	3	=	=	SYM
ejpam-4574	3	4	(	(	PUNCT
ejpam-4574	3	5	v	v	NOUN
ejpam-4574	3	6	,	,	PUNCT
ejpam-4574	3	7	e	e	NOUN
ejpam-4574	3	8	)	)	PUNCT
ejpam-4574	3	9	be	be	AUX
ejpam-4574	3	10	a	a	DET
ejpam-4574	3	11	graph	graph	NOUN
ejpam-4574	3	12	.	.	PUNCT
ejpam-4574	4	1	the	the	DET
ejpam-4574	4	2	kth−	kth−	PROPN
ejpam-4574	4	3	power	power	NOUN
ejpam-4574	4	4	of	of	ADP
ejpam-4574	4	5	g	g	PROPN
ejpam-4574	4	6	denoted	denote	VERB
ejpam-4574	4	7	by	by	ADP
ejpam-4574	4	8	gk	gk	PROPN
ejpam-4574	4	9	is	be	AUX
ejpam-4574	4	10	the	the	DET
ejpam-4574	4	11	graph	graph	NOUN
ejpam-4574	4	12	whose	whose	DET
ejpam-4574	4	13	vertex	vertex	NOUN
ejpam-4574	4	14	set	set	NOUN
ejpam-4574	4	15	is	be	AUX
ejpam-4574	4	16	v	v	ADJ
ejpam-4574	4	17	and	and	CCONJ
ejpam-4574	4	18	in	in	ADP
ejpam-4574	4	19	which	which	PRON
ejpam-4574	4	20	two	two	NUM
ejpam-4574	4	21	vertices	vertex	NOUN
ejpam-4574	4	22	are	be	AUX
ejpam-4574	4	23	adjacent	adjacent	ADJ
ejpam-4574	4	24	if	if	SCONJ
ejpam-4574	4	25	and	and	CCONJ
ejpam-4574	4	26	only	only	ADV
ejpam-4574	4	27	if	if	SCONJ
ejpam-4574	4	28	their	their	PRON
ejpam-4574	4	29	distance	distance	NOUN
ejpam-4574	4	30	in	in	ADP
ejpam-4574	4	31	g	g	PROPN
ejpam-4574	4	32	is	be	AUX
ejpam-4574	4	33	at	at	ADP
ejpam-4574	4	34	most	most	ADJ
ejpam-4574	4	35	k.	k.	NOUN
ejpam-4574	4	36	a	a	DET
ejpam-4574	4	37	vertex	vertex	NOUN
ejpam-4574	4	38	coloring	coloring	NOUN
ejpam-4574	4	39	of	of	ADP
ejpam-4574	4	40	g	g	PROPN
ejpam-4574	4	41	is	be	AUX
ejpam-4574	4	42	acyclic	acyclic	ADJ
ejpam-4574	4	43	if	if	SCONJ
ejpam-4574	4	44	each	each	DET
ejpam-4574	4	45	bichromatic	bichromatic	ADJ
ejpam-4574	4	46	subgraph	subgraph	NOUN
ejpam-4574	4	47	is	be	AUX
ejpam-4574	4	48	a	a	DET
ejpam-4574	4	49	forest	forest	NOUN
ejpam-4574	4	50	.	.	PUNCT
ejpam-4574	5	1	a	a	DET
ejpam-4574	5	2	star	star	NOUN
ejpam-4574	5	3	coloring	coloring	NOUN
ejpam-4574	5	4	of	of	ADP
ejpam-4574	5	5	g	g	PROPN
ejpam-4574	5	6	is	be	AUX
ejpam-4574	5	7	an	an	DET
ejpam-4574	5	8	acyclic	acyclic	ADJ
ejpam-4574	5	9	coloring	coloring	NOUN
ejpam-4574	5	10	in	in	ADP
ejpam-4574	5	11	which	which	PRON
ejpam-4574	5	12	each	each	DET
ejpam-4574	5	13	bichromatic	bichromatic	ADJ
ejpam-4574	5	14	subgraph	subgraph	NOUN
ejpam-4574	5	15	is	be	AUX
ejpam-4574	5	16	a	a	DET
ejpam-4574	5	17	star	star	NOUN
ejpam-4574	5	18	forest	forest	NOUN
ejpam-4574	5	19	.	.	PUNCT
ejpam-4574	6	1	the	the	DET
ejpam-4574	6	2	minimum	minimum	ADJ
ejpam-4574	6	3	number	number	NOUN
ejpam-4574	6	4	of	of	ADP
ejpam-4574	6	5	colors	color	NOUN
ejpam-4574	6	6	such	such	ADJ
ejpam-4574	6	7	that	that	SCONJ
ejpam-4574	6	8	g	g	PROPN
ejpam-4574	6	9	admits	admit	VERB
ejpam-4574	6	10	an	an	DET
ejpam-4574	6	11	acyclic	acyclic	ADJ
ejpam-4574	6	12	(	(	PUNCT
ejpam-4574	6	13	star	star	NOUN
ejpam-4574	6	14	)	)	PUNCT
ejpam-4574	6	15	coloring	coloring	NOUN
ejpam-4574	6	16	is	be	AUX
ejpam-4574	6	17	called	call	VERB
ejpam-4574	6	18	the	the	DET
ejpam-4574	6	19	acyclic	acyclic	ADJ
ejpam-4574	6	20	(	(	PUNCT
ejpam-4574	6	21	star	star	NOUN
ejpam-4574	6	22	)	)	PUNCT
ejpam-4574	6	23	chromatic	chromatic	ADJ
ejpam-4574	6	24	number	number	NOUN
ejpam-4574	6	25	of	of	ADP
ejpam-4574	6	26	g	g	NOUN
ejpam-4574	6	27	and	and	CCONJ
ejpam-4574	6	28	is	be	AUX
ejpam-4574	6	29	denoted	denote	VERB
ejpam-4574	6	30	by	by	ADP
ejpam-4574	6	31	χa(g)(χs(g	χa(g)(χs(g	NOUN
ejpam-4574	6	32	)	)	PUNCT
ejpam-4574	6	33	)	)	PUNCT
ejpam-4574	6	34	.	.	PUNCT
ejpam-4574	7	1	in	in	ADP
ejpam-4574	7	2	this	this	DET
ejpam-4574	7	3	paper	paper	NOUN
ejpam-4574	7	4	we	we	PRON
ejpam-4574	7	5	prove	prove	VERB
ejpam-4574	7	6	that	that	SCONJ
ejpam-4574	7	7	for	for	ADP
ejpam-4574	7	8	n	n	PRON
ejpam-4574	7	9	≥	≥	NOUN
ejpam-4574	7	10	k	k	NOUN
ejpam-4574	8	1	+	+	CCONJ
ejpam-4574	8	2	1	1	NUM
ejpam-4574	8	3	,	,	PUNCT
ejpam-4574	8	4	χs(p	χs(p	PRON
ejpam-4574	8	5	k	k	PROPN
ejpam-4574	8	6	n	n	PROPN
ejpam-4574	8	7	)	)	PUNCT
ejpam-4574	8	8	=	=	SYM
ejpam-4574	8	9	min{⌊k+n+1	min{⌊k+n+1	PROPN
ejpam-4574	8	10	2	2	NUM
ejpam-4574	8	11	⌋	⌋	NOUN
ejpam-4574	8	12	,	,	PUNCT
ejpam-4574	8	13	2k	2k	NUM
ejpam-4574	8	14	+	+	CCONJ
ejpam-4574	8	15	1	1	NUM
ejpam-4574	8	16	}	}	PUNCT
ejpam-4574	8	17	and	and	CCONJ
ejpam-4574	8	18	χa(p	χa(p	ADJ
ejpam-4574	8	19	k	k	PROPN
ejpam-4574	8	20	n	n	PROPN
ejpam-4574	8	21	)	)	PUNCT
ejpam-4574	9	1	=	=	PUNCT
ejpam-4574	9	2	k	k	PROPN
ejpam-4574	10	1	+	+	NOUN
ejpam-4574	10	2	1	1	X
ejpam-4574	10	3	.	.	PUNCT
ejpam-4574	10	4	further	far	ADV
ejpam-4574	10	5	,	,	PUNCT
ejpam-4574	10	6	we	we	PRON
ejpam-4574	10	7	show	show	VERB
ejpam-4574	10	8	that	that	SCONJ
ejpam-4574	10	9	for	for	ADP
ejpam-4574	10	10	n	n	PRON
ejpam-4574	10	11	≥	≥	NOUN
ejpam-4574	10	12	(	(	PUNCT
ejpam-4574	10	13	k	k	PROPN
ejpam-4574	10	14	+	+	PROPN
ejpam-4574	10	15	1)2	1)2	NUM
ejpam-4574	10	16	,	,	PUNCT
ejpam-4574	10	17	2k+1	2k+1	NOUN
ejpam-4574	10	18	≤	≤	NOUN
ejpam-4574	10	19	χs(c	χs(c	ADV
ejpam-4574	10	20	k	k	PROPN
ejpam-4574	10	21	n	n	CCONJ
ejpam-4574	10	22	)	)	PUNCT
ejpam-4574	10	23	≤	≤	NOUN
ejpam-4574	10	24	2k+2	2k+2	PROPN
ejpam-4574	10	25	and	and	CCONJ
ejpam-4574	10	26	k+2	k+2	PROPN
ejpam-4574	10	27	≤	≤	NOUN
ejpam-4574	10	28	χa(c	χa(c	VERB
ejpam-4574	10	29	k	k	PROPN
ejpam-4574	10	30	n	n	CCONJ
ejpam-4574	10	31	)	)	PUNCT
ejpam-4574	10	32	≤	≤	NOUN
ejpam-4574	11	1	k+3	k+3	PROPN
ejpam-4574	11	2	.	.	PUNCT
ejpam-4574	12	1	finally	finally	ADV
ejpam-4574	12	2	,	,	PUNCT
ejpam-4574	12	3	we	we	PRON
ejpam-4574	12	4	derive	derive	VERB
ejpam-4574	12	5	the	the	DET
ejpam-4574	12	6	formula	formula	NOUN
ejpam-4574	12	7	χa(c	χa(c	VERB
ejpam-4574	12	8	k	k	NOUN
ejpam-4574	12	9	n	n	CCONJ
ejpam-4574	12	10	)	)	PUNCT
ejpam-4574	12	11	=	=	PUNCT
ejpam-4574	12	12	k+2	k+2	PROPN
ejpam-4574	12	13	for	for	ADP
ejpam-4574	12	14	n	n	PRON
ejpam-4574	12	15	≥	≥	NOUN
ejpam-4574	12	16	(	(	PUNCT
ejpam-4574	12	17	k	k	PROPN
ejpam-4574	12	18	+	+	PROPN
ejpam-4574	12	19	1)3	1)3	PROPN
ejpam-4574	12	20	.	.	PUNCT
ejpam-4574	12	21	2020	2020	NUM
ejpam-4574	12	22	mathematics	mathematic	NOUN
ejpam-4574	12	23	subject	subject	NOUN
ejpam-4574	12	24	classifications	classification	NOUN
ejpam-4574	12	25	:	:	PUNCT
ejpam-4574	12	26	05c15	05c15	NUM
ejpam-4574	12	27	,	,	PUNCT
ejpam-4574	12	28	05c38	05c38	NOUN
ejpam-4574	12	29	key	key	ADJ
ejpam-4574	12	30	words	word	NOUN
ejpam-4574	12	31	and	and	CCONJ
ejpam-4574	12	32	phrases	phrase	NOUN
ejpam-4574	12	33	:	:	PUNCT
ejpam-4574	12	34	acyclic	acyclic	ADJ
ejpam-4574	12	35	coloring	coloring	NOUN
ejpam-4574	12	36	,	,	PUNCT
ejpam-4574	12	37	powers	power	NOUN
ejpam-4574	12	38	of	of	ADP
ejpam-4574	12	39	cycles	cycle	NOUN
ejpam-4574	12	40	,	,	PUNCT
ejpam-4574	12	41	powers	power	NOUN
ejpam-4574	12	42	of	of	ADP
ejpam-4574	12	43	paths	path	NOUN
ejpam-4574	12	44	,	,	PUNCT
ejpam-4574	12	45	star	star	NOUN
ejpam-4574	12	46	coloring	color	VERB
ejpam-4574	12	47	1	1	NUM
ejpam-4574	12	48	.	.	PUNCT
ejpam-4574	12	49	introduction	introduction	NOUN
ejpam-4574	12	50	graph	graph	NOUN
ejpam-4574	12	51	theory	theory	NOUN
ejpam-4574	12	52	is	be	AUX
ejpam-4574	12	53	widely	widely	ADV
ejpam-4574	12	54	used	use	VERB
ejpam-4574	12	55	in	in	ADP
ejpam-4574	12	56	many	many	ADJ
ejpam-4574	12	57	areas	area	NOUN
ejpam-4574	12	58	such	such	ADJ
ejpam-4574	12	59	as	as	ADP
ejpam-4574	12	60	the	the	DET
ejpam-4574	12	61	study	study	NOUN
ejpam-4574	12	62	of	of	ADP
ejpam-4574	12	63	molecules	molecule	NOUN
ejpam-4574	12	64	and	and	CCONJ
ejpam-4574	12	65	construction	construction	NOUN
ejpam-4574	12	66	of	of	ADP
ejpam-4574	12	67	bonds	bond	NOUN
ejpam-4574	12	68	in	in	ADP
ejpam-4574	12	69	chemistry	chemistry	NOUN
ejpam-4574	12	70	,	,	PUNCT
ejpam-4574	12	71	operations	operation	NOUN
ejpam-4574	12	72	research	research	NOUN
ejpam-4574	12	73	,	,	PUNCT
ejpam-4574	12	74	modeling	model	VERB
ejpam-4574	12	75	transport	transport	NOUN
ejpam-4574	12	76	networks	network	NOUN
ejpam-4574	12	77	,	,	PUNCT
ejpam-4574	12	78	activity	activity	NOUN
ejpam-4574	12	79	networks	network	NOUN
ejpam-4574	12	80	,	,	PUNCT
ejpam-4574	12	81	computational	computational	ADJ
ejpam-4574	12	82	biochemistry	biochemistry	NOUN
ejpam-4574	12	83	,	,	PUNCT
ejpam-4574	12	84	map	map	VERB
ejpam-4574	12	85	coloring	coloring	NOUN
ejpam-4574	12	86	,	,	PUNCT
ejpam-4574	12	87	and	and	CCONJ
ejpam-4574	12	88	gsm	gsm	PROPN
ejpam-4574	12	89	mobile	mobile	ADJ
ejpam-4574	12	90	phone	phone	NOUN
ejpam-4574	12	91	networks	network	NOUN
ejpam-4574	12	92	,	,	PUNCT
ejpam-4574	12	93	and	and	CCONJ
ejpam-4574	12	94	others	other	NOUN
ejpam-4574	13	1	[	[	X
ejpam-4574	13	2	8	8	NUM
ejpam-4574	13	3	]	]	PUNCT
ejpam-4574	13	4	.	.	PUNCT
ejpam-4574	14	1	graph	graph	NOUN
ejpam-4574	14	2	coloring	coloring	NOUN
ejpam-4574	14	3	is	be	AUX
ejpam-4574	14	4	a	a	DET
ejpam-4574	14	5	branch	branch	NOUN
ejpam-4574	14	6	of	of	ADP
ejpam-4574	14	7	graph	graph	NOUN
ejpam-4574	14	8	theory	theory	NOUN
ejpam-4574	14	9	that	that	SCONJ
ejpam-4574	14	10	deals	deal	NOUN
ejpam-4574	14	11	with	with	ADP
ejpam-4574	14	12	such	such	ADJ
ejpam-4574	14	13	applications	application	NOUN
ejpam-4574	14	14	.	.	PUNCT
ejpam-4574	15	1	coloring	coloring	NOUN
ejpam-4574	15	2	of	of	ADP
ejpam-4574	15	3	a	a	DET
ejpam-4574	15	4	graph	graph	NOUN
ejpam-4574	15	5	is	be	AUX
ejpam-4574	15	6	an	an	DET
ejpam-4574	15	7	assignment	assignment	NOUN
ejpam-4574	15	8	of	of	ADP
ejpam-4574	15	9	colors	color	NOUN
ejpam-4574	15	10	to	to	ADP
ejpam-4574	15	11	the	the	DET
ejpam-4574	15	12	elements	element	NOUN
ejpam-4574	15	13	like	like	ADP
ejpam-4574	15	14	vertices	vertex	NOUN
ejpam-4574	15	15	,	,	PUNCT
ejpam-4574	15	16	edges	edge	NOUN
ejpam-4574	15	17	,	,	PUNCT
ejpam-4574	15	18	or	or	CCONJ
ejpam-4574	15	19	faces	face	NOUN
ejpam-4574	15	20	(	(	PUNCT
ejpam-4574	15	21	regions	region	NOUN
ejpam-4574	15	22	)	)	PUNCT
ejpam-4574	15	23	of	of	ADP
ejpam-4574	15	24	a	a	DET
ejpam-4574	15	25	graph	graph	NOUN
ejpam-4574	15	26	.	.	PUNCT
ejpam-4574	16	1	a	a	DET
ejpam-4574	16	2	coloring	coloring	NOUN
ejpam-4574	16	3	is	be	AUX
ejpam-4574	16	4	called	call	VERB
ejpam-4574	16	5	proper	proper	ADJ
ejpam-4574	16	6	coloring	coloring	NOUN
ejpam-4574	16	7	if	if	SCONJ
ejpam-4574	16	8	no	no	DET
ejpam-4574	16	9	two	two	NUM
ejpam-4574	16	10	adjacent	adjacent	ADJ
ejpam-4574	16	11	elements	element	NOUN
ejpam-4574	16	12	are	be	AUX
ejpam-4574	16	13	assigned	assign	VERB
ejpam-4574	16	14	the	the	DET
ejpam-4574	16	15	same	same	ADJ
ejpam-4574	16	16	color	color	NOUN
ejpam-4574	16	17	.	.	PUNCT
ejpam-4574	17	1	the	the	DET
ejpam-4574	17	2	most	most	ADV
ejpam-4574	17	3	common	common	ADJ
ejpam-4574	17	4	types	type	NOUN
ejpam-4574	17	5	of	of	ADP
ejpam-4574	17	6	graph	graph	NOUN
ejpam-4574	17	7	colorings	coloring	NOUN
ejpam-4574	17	8	are	be	AUX
ejpam-4574	17	9	vertex	vertex	NOUN
ejpam-4574	17	10	coloring	coloring	NOUN
ejpam-4574	17	11	,	,	PUNCT
ejpam-4574	17	12	edge	edge	NOUN
ejpam-4574	17	13	coloring	coloring	NOUN
ejpam-4574	17	14	,	,	PUNCT
ejpam-4574	17	15	and	and	CCONJ
ejpam-4574	17	16	face	face	VERB
ejpam-4574	17	17	coloring	color	VERB
ejpam-4574	17	18	.	.	PUNCT
ejpam-4574	18	1	a	a	DET
ejpam-4574	18	2	k−	k−	PROPN
ejpam-4574	18	3	coloring	coloring	NOUN
ejpam-4574	18	4	of	of	ADP
ejpam-4574	18	5	a	a	DET
ejpam-4574	18	6	graph	graph	NOUN
ejpam-4574	18	7	g	g	NOUN
ejpam-4574	18	8	=	=	PUNCT
ejpam-4574	18	9	(	(	PUNCT
ejpam-4574	18	10	v	v	NOUN
ejpam-4574	18	11	(	(	PUNCT
ejpam-4574	18	12	g	g	NOUN
ejpam-4574	18	13	)	)	PUNCT
ejpam-4574	18	14	,	,	PUNCT
ejpam-4574	18	15	e(g	e(g	PROPN
ejpam-4574	18	16	)	)	PUNCT
ejpam-4574	18	17	)	)	PUNCT
ejpam-4574	18	18	is	be	AUX
ejpam-4574	18	19	a	a	DET
ejpam-4574	18	20	function	function	NOUN
ejpam-4574	18	21	c	c	NOUN
ejpam-4574	18	22	:	:	PUNCT
ejpam-4574	18	23	v	v	NOUN
ejpam-4574	18	24	(	(	PUNCT
ejpam-4574	18	25	g	g	NOUN
ejpam-4574	18	26	)	)	PUNCT
ejpam-4574	18	27	→	→	SYM
ejpam-4574	18	28	{	{	PUNCT
ejpam-4574	18	29	1	1	NUM
ejpam-4574	18	30	,	,	PUNCT
ejpam-4574	18	31	2	2	NUM
ejpam-4574	18	32	,	,	PUNCT
ejpam-4574	18	33	...	...	PUNCT
ejpam-4574	18	34	,	,	PUNCT
ejpam-4574	18	35	k	k	NOUN
ejpam-4574	18	36	}	}	PUNCT
ejpam-4574	18	37	.	.	PUNCT
ejpam-4574	19	1	an	an	DET
ejpam-4574	19	2	acyclic	acyclic	ADJ
ejpam-4574	19	3	coloring	coloring	NOUN
ejpam-4574	19	4	of	of	ADP
ejpam-4574	19	5	a	a	DET
ejpam-4574	19	6	graph	graph	NOUN
ejpam-4574	19	7	g	g	NOUN
ejpam-4574	19	8	is	be	AUX
ejpam-4574	19	9	a	a	DET
ejpam-4574	19	10	proper	proper	ADJ
ejpam-4574	19	11	coloring	coloring	NOUN
ejpam-4574	19	12	such	such	ADJ
ejpam-4574	19	13	that	that	SCONJ
ejpam-4574	19	14	all	all	PRON
ejpam-4574	19	15	induced	induce	VERB
ejpam-4574	19	16	bicolored	bicolore	VERB
ejpam-4574	19	17	subgraphs	subgraph	NOUN
ejpam-4574	19	18	of	of	ADP
ejpam-4574	19	19	g	g	NOUN
ejpam-4574	19	20	contain	contain	VERB
ejpam-4574	19	21	no	no	DET
ejpam-4574	19	22	cycles	cycle	NOUN
ejpam-4574	19	23	,	,	PUNCT
ejpam-4574	19	24	in	in	ADP
ejpam-4574	19	25	other	other	ADJ
ejpam-4574	19	26	words	word	NOUN
ejpam-4574	19	27	,	,	PUNCT
ejpam-4574	19	28	every	every	DET
ejpam-4574	19	29	two	two	NUM
ejpam-4574	19	30	color	color	NOUN
ejpam-4574	19	31	classes	class	NOUN
ejpam-4574	19	32	induce	induce	VERB
ejpam-4574	19	33	a	a	DET
ejpam-4574	19	34	forest	forest	NOUN
ejpam-4574	19	35	.	.	PUNCT
ejpam-4574	20	1	star	star	NOUN
ejpam-4574	20	2	coloring	coloring	NOUN
ejpam-4574	20	3	is	be	AUX
ejpam-4574	20	4	acyclic	acyclic	ADJ
ejpam-4574	20	5	coloring	coloring	NOUN
ejpam-4574	20	6	where	where	SCONJ
ejpam-4574	20	7	every	every	DET
ejpam-4574	20	8	bicolored	bicolore	VERB
ejpam-4574	20	9	subgraph	subgraph	NOUN
ejpam-4574	20	10	induces	induce	VERB
ejpam-4574	20	11	a	a	DET
ejpam-4574	20	12	star	star	NOUN
ejpam-4574	20	13	forest	forest	NOUN
ejpam-4574	20	14	.	.	PUNCT
ejpam-4574	21	1	the	the	DET
ejpam-4574	21	2	chromatic	chromatic	ADJ
ejpam-4574	21	3	number	number	NOUN
ejpam-4574	21	4	of	of	ADP
ejpam-4574	21	5	g	g	NOUN
ejpam-4574	21	6	,	,	PUNCT
ejpam-4574	21	7	denoted	denote	VERB
ejpam-4574	21	8	χ(g	χ(g	PROPN
ejpam-4574	21	9	)	)	PUNCT
ejpam-4574	21	10	,	,	PUNCT
ejpam-4574	21	11	is	be	AUX
ejpam-4574	21	12	the	the	DET
ejpam-4574	21	13	minimum	minimum	NOUN
ejpam-4574	21	14	α	α	NOUN
ejpam-4574	21	15	such	such	ADJ
ejpam-4574	21	16	that	that	SCONJ
ejpam-4574	21	17	g	g	PROPN
ejpam-4574	21	18	admits	admit	VERB
ejpam-4574	21	19	a	a	DET
ejpam-4574	21	20	k−proper	k−proper	NOUN
ejpam-4574	21	21	coloring	coloring	NOUN
ejpam-4574	21	22	;	;	PUNCT
ejpam-4574	21	23	the	the	DET
ejpam-4574	21	24	acyclic	acyclic	ADJ
ejpam-4574	21	25	chromatic	chromatic	ADJ
ejpam-4574	21	26	number	number	NOUN
ejpam-4574	21	27	of	of	ADP
ejpam-4574	21	28	a	a	DET
ejpam-4574	21	29	graph	graph	NOUN
ejpam-4574	21	30	g	g	NOUN
ejpam-4574	21	31	,	,	PUNCT
ejpam-4574	21	32	denoted	denote	VERB
ejpam-4574	21	33	χa(g	χa(g	NOUN
ejpam-4574	21	34	)	)	PUNCT
ejpam-4574	21	35	,	,	PUNCT
ejpam-4574	21	36	is	be	AUX
ejpam-4574	21	37	the	the	DET
ejpam-4574	21	38	minimum	minimum	ADJ
ejpam-4574	21	39	number	number	NOUN
ejpam-4574	21	40	k	k	NOUN
ejpam-4574	21	41	such	such	ADJ
ejpam-4574	21	42	that	that	SCONJ
ejpam-4574	21	43	g	g	PROPN
ejpam-4574	21	44	admits	admit	VERB
ejpam-4574	21	45	a	a	DET
ejpam-4574	21	46	k	k	NOUN
ejpam-4574	21	47	−	−	NOUN
ejpam-4574	21	48	acyclic	acyclic	ADJ
ejpam-4574	21	49	coloring	coloring	NOUN
ejpam-4574	21	50	;	;	PUNCT
ejpam-4574	21	51	∗corresponding	∗corresponde	VERB
ejpam-4574	21	52	author	author	NOUN
ejpam-4574	21	53	.	.	PUNCT
ejpam-4574	22	1	doi	doi	NOUN
ejpam-4574	22	2	:	:	PUNCT
ejpam-4574	22	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4574	https://doi.org/10.29020/nybg.ejpam.v15i4.4574	ADJ
ejpam-4574	22	4	email	email	NOUN
ejpam-4574	22	5	addresses	address	NOUN
ejpam-4574	22	6	:	:	PUNCT
ejpam-4574	22	7	etawi	etawi	PROPN
ejpam-4574	22	8	1412@yahoo.com	1412@yahoo.com	X
ejpam-4574	22	9	(	(	PUNCT
ejpam-4574	22	10	a.	a.	PROPN
ejpam-4574	22	11	etawi	etawi	PROPN
ejpam-4574	22	12	)	)	PUNCT
ejpam-4574	22	13	,	,	PUNCT
ejpam-4574	22	14	m.ghanem@ju.edu.jo	m.ghanem@ju.edu.jo	PROPN
ejpam-4574	22	15	(	(	PUNCT
ejpam-4574	22	16	m.	m.	NOUN
ejpam-4574	22	17	ghanem	ghanem	PROPN
ejpam-4574	22	18	)	)	PUNCT
ejpam-4574	22	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4574	22	20	1822	1822	NUM
ejpam-4574	23	1	©	©	ADP
ejpam-4574	23	2	2022	2022	NUM
ejpam-4574	23	3	ejpam	ejpam	VERB
ejpam-4574	23	4	all	all	DET
ejpam-4574	23	5	rights	right	NOUN
ejpam-4574	23	6	reserved	reserve	VERB
ejpam-4574	23	7	.	.	PUNCT
ejpam-4574	24	1	a.	a.	PROPN
ejpam-4574	24	2	etawi	etawi	PROPN
ejpam-4574	24	3	,	,	PUNCT
ejpam-4574	24	4	m.	m.	PROPN
ejpam-4574	24	5	ghanem	ghanem	PROPN
ejpam-4574	24	6	,	,	PUNCT
ejpam-4574	24	7	h.	h.	PROPN
ejpam-4574	24	8	al	al	PROPN
ejpam-4574	24	9	-	-	PUNCT
ejpam-4574	24	10	ezeh	ezeh	PROPN
ejpam-4574	24	11	/	/	SYM
ejpam-4574	24	12	eur	eur	NOUN
ejpam-4574	24	13	.	.	PUNCT
ejpam-4574	25	1	j.	j.	PROPN
ejpam-4574	25	2	pure	pure	PROPN
ejpam-4574	25	3	appl	appl	PROPN
ejpam-4574	25	4	.	.	PROPN
ejpam-4574	25	5	math	math	PROPN
ejpam-4574	25	6	,	,	PUNCT
ejpam-4574	25	7	15	15	NUM
ejpam-4574	25	8	(	(	PUNCT
ejpam-4574	25	9	4	4	NUM
ejpam-4574	25	10	)	)	PUNCT
ejpam-4574	25	11	(	(	PUNCT
ejpam-4574	25	12	2022	2022	NUM
ejpam-4574	25	13	)	)	PUNCT
ejpam-4574	25	14	,	,	PUNCT
ejpam-4574	25	15	1822	1822	NUM
ejpam-4574	25	16	-	-	SYM
ejpam-4574	25	17	1835	1835	NUM
ejpam-4574	25	18	1823	1823	NUM
ejpam-4574	25	19	and	and	CCONJ
ejpam-4574	25	20	the	the	DET
ejpam-4574	25	21	star	star	NOUN
ejpam-4574	25	22	chromatic	chromatic	ADJ
ejpam-4574	25	23	number	number	NOUN
ejpam-4574	25	24	of	of	ADP
ejpam-4574	25	25	a	a	DET
ejpam-4574	25	26	graph	graph	NOUN
ejpam-4574	25	27	g	g	NOUN
ejpam-4574	25	28	,	,	PUNCT
ejpam-4574	25	29	denoted	denote	VERB
ejpam-4574	25	30	χs(g	χs(g	NOUN
ejpam-4574	25	31	)	)	PUNCT
ejpam-4574	25	32	,	,	PUNCT
ejpam-4574	25	33	is	be	AUX
ejpam-4574	25	34	the	the	DET
ejpam-4574	25	35	minimum	minimum	ADJ
ejpam-4574	25	36	number	number	NOUN
ejpam-4574	25	37	k	k	NOUN
ejpam-4574	25	38	such	such	ADJ
ejpam-4574	25	39	that	that	SCONJ
ejpam-4574	25	40	g	g	PROPN
ejpam-4574	25	41	admits	admit	VERB
ejpam-4574	25	42	a	a	DET
ejpam-4574	25	43	k	k	NOUN
ejpam-4574	25	44	−	−	PROPN
ejpam-4574	25	45	star	star	NOUN
ejpam-4574	25	46	coloring	coloring	NOUN
ejpam-4574	25	47	.	.	PUNCT
ejpam-4574	26	1	all	all	DET
ejpam-4574	26	2	graphs	graph	NOUN
ejpam-4574	26	3	considered	consider	VERB
ejpam-4574	26	4	in	in	ADP
ejpam-4574	26	5	this	this	DET
ejpam-4574	26	6	article	article	NOUN
ejpam-4574	26	7	are	be	AUX
ejpam-4574	26	8	finite	finite	ADJ
ejpam-4574	26	9	undirected	undirected	ADJ
ejpam-4574	26	10	and	and	CCONJ
ejpam-4574	26	11	simple	simple	ADJ
ejpam-4574	26	12	(	(	PUNCT
ejpam-4574	26	13	no	no	DET
ejpam-4574	26	14	loops	loop	NOUN
ejpam-4574	26	15	or	or	CCONJ
ejpam-4574	26	16	multiple	multiple	ADJ
ejpam-4574	26	17	edges	edge	NOUN
ejpam-4574	26	18	)	)	PUNCT
ejpam-4574	26	19	.	.	PUNCT
ejpam-4574	27	1	all	all	DET
ejpam-4574	27	2	coloring	color	VERB
ejpam-4574	27	3	considered	consider	VERB
ejpam-4574	27	4	in	in	ADP
ejpam-4574	27	5	this	this	DET
ejpam-4574	27	6	article	article	NOUN
ejpam-4574	27	7	is	be	AUX
ejpam-4574	27	8	vertex	vertex	NOUN
ejpam-4574	27	9	coloring	coloring	NOUN
ejpam-4574	27	10	.	.	PUNCT
ejpam-4574	28	1	the	the	DET
ejpam-4574	28	2	following	follow	VERB
ejpam-4574	28	3	is	be	AUX
ejpam-4574	28	4	an	an	DET
ejpam-4574	28	5	obvious	obvious	ADJ
ejpam-4574	28	6	observation	observation	NOUN
ejpam-4574	28	7	that	that	SCONJ
ejpam-4574	28	8	we	we	PRON
ejpam-4574	28	9	use	use	VERB
ejpam-4574	28	10	in	in	ADP
ejpam-4574	28	11	our	our	PRON
ejpam-4574	28	12	work	work	NOUN
ejpam-4574	28	13	.	.	PUNCT
ejpam-4574	29	1	observation	observation	NOUN
ejpam-4574	29	2	:	:	PUNCT
ejpam-4574	29	3	for	for	ADP
ejpam-4574	29	4	a	a	DET
ejpam-4574	29	5	proper	proper	ADJ
ejpam-4574	29	6	coloring	coloring	NOUN
ejpam-4574	29	7	c	c	NOUN
ejpam-4574	29	8	of	of	ADP
ejpam-4574	29	9	a	a	DET
ejpam-4574	29	10	graph	graph	NOUN
ejpam-4574	29	11	g	g	ADP
ejpam-4574	29	12	the	the	DET
ejpam-4574	29	13	following	follow	VERB
ejpam-4574	29	14	hold	hold	NOUN
ejpam-4574	29	15	:	:	PUNCT
ejpam-4574	29	16	(	(	PUNCT
ejpam-4574	29	17	1	1	X
ejpam-4574	29	18	)	)	PUNCT
ejpam-4574	29	19	c	c	NOUN
ejpam-4574	29	20	is	be	AUX
ejpam-4574	29	21	an	an	DET
ejpam-4574	29	22	acyclic	acyclic	ADJ
ejpam-4574	29	23	coloring	coloring	NOUN
ejpam-4574	29	24	of	of	ADP
ejpam-4574	29	25	g	g	PROPN
ejpam-4574	29	26	if	if	SCONJ
ejpam-4574	30	1	and	and	CCONJ
ejpam-4574	30	2	only	only	ADV
ejpam-4574	30	3	if	if	SCONJ
ejpam-4574	30	4	every	every	DET
ejpam-4574	30	5	cycle	cycle	NOUN
ejpam-4574	30	6	in	in	ADP
ejpam-4574	30	7	g	g	PROPN
ejpam-4574	30	8	admits	admit	VERB
ejpam-4574	30	9	at	at	ADP
ejpam-4574	30	10	least	least	ADV
ejpam-4574	30	11	three	three	NUM
ejpam-4574	30	12	colors	color	NOUN
ejpam-4574	30	13	.	.	PUNCT
ejpam-4574	31	1	(	(	PUNCT
ejpam-4574	31	2	2	2	X
ejpam-4574	31	3	)	)	PUNCT
ejpam-4574	31	4	c	c	NOUN
ejpam-4574	31	5	is	be	AUX
ejpam-4574	31	6	a	a	DET
ejpam-4574	31	7	star	star	NOUN
ejpam-4574	31	8	coloring	coloring	NOUN
ejpam-4574	31	9	of	of	ADP
ejpam-4574	31	10	g	g	PROPN
ejpam-4574	31	11	if	if	SCONJ
ejpam-4574	32	1	and	and	CCONJ
ejpam-4574	32	2	only	only	ADV
ejpam-4574	32	3	if	if	SCONJ
ejpam-4574	32	4	every	every	DET
ejpam-4574	32	5	path	path	NOUN
ejpam-4574	32	6	on	on	ADP
ejpam-4574	32	7	four	four	NUM
ejpam-4574	32	8	vertices	vertex	NOUN
ejpam-4574	32	9	in	in	ADP
ejpam-4574	32	10	g	g	PROPN
ejpam-4574	32	11	admits	admit	VERB
ejpam-4574	32	12	at	at	ADP
ejpam-4574	32	13	least	least	ADV
ejpam-4574	32	14	three	three	NUM
ejpam-4574	32	15	colors	color	NOUN
ejpam-4574	32	16	.	.	PUNCT
ejpam-4574	33	1	acyclic	acyclic	ADJ
ejpam-4574	33	2	and	and	CCONJ
ejpam-4574	33	3	star	star	NOUN
ejpam-4574	33	4	coloring	coloring	NOUN
ejpam-4574	33	5	were	be	AUX
ejpam-4574	33	6	introduced	introduce	VERB
ejpam-4574	33	7	in	in	ADP
ejpam-4574	33	8	the	the	DET
ejpam-4574	33	9	early	early	ADJ
ejpam-4574	33	10	seventies	seventy	NOUN
ejpam-4574	33	11	by	by	ADP
ejpam-4574	33	12	grünbaum	grünbaum	PROPN
ejpam-4574	34	1	[	[	X
ejpam-4574	34	2	3	3	NUM
ejpam-4574	34	3	]	]	PUNCT
ejpam-4574	34	4	.	.	PUNCT
ejpam-4574	35	1	grünbaum	grünbaum	PROPN
ejpam-4574	35	2	showed	show	VERB
ejpam-4574	35	3	that	that	SCONJ
ejpam-4574	35	4	a	a	DET
ejpam-4574	35	5	graph	graph	NOUN
ejpam-4574	35	6	with	with	ADP
ejpam-4574	35	7	a	a	DET
ejpam-4574	35	8	maximum	maximum	ADJ
ejpam-4574	35	9	degree	degree	NOUN
ejpam-4574	35	10	3	3	NUM
ejpam-4574	35	11	has	have	VERB
ejpam-4574	35	12	4−	4−	NUM
ejpam-4574	35	13	acyclic	acyclic	ADJ
ejpam-4574	35	14	colorings	coloring	NOUN
ejpam-4574	35	15	.	.	PUNCT
ejpam-4574	36	1	burnstein	burnstein	PROPN
ejpam-4574	37	1	[	[	X
ejpam-4574	37	2	2	2	X
ejpam-4574	37	3	]	]	PUNCT
ejpam-4574	37	4	proved	prove	VERB
ejpam-4574	37	5	that	that	SCONJ
ejpam-4574	37	6	a	a	DET
ejpam-4574	37	7	graph	graph	NOUN
ejpam-4574	37	8	with	with	ADP
ejpam-4574	37	9	a	a	DET
ejpam-4574	37	10	maximum	maximum	ADJ
ejpam-4574	37	11	degree	degree	NOUN
ejpam-4574	37	12	4	4	NUM
ejpam-4574	37	13	has	have	VERB
ejpam-4574	37	14	5	5	NUM
ejpam-4574	37	15	−	−	NOUN
ejpam-4574	37	16	acyclic	acyclic	ADJ
ejpam-4574	37	17	colorings	coloring	NOUN
ejpam-4574	37	18	.	.	PUNCT
ejpam-4574	38	1	wood	wood	NOUN
ejpam-4574	39	1	[	[	X
ejpam-4574	39	2	10	10	NUM
ejpam-4574	39	3	]	]	PUNCT
ejpam-4574	39	4	studied	study	VERB
ejpam-4574	39	5	the	the	DET
ejpam-4574	39	6	star	star	NOUN
ejpam-4574	39	7	and	and	CCONJ
ejpam-4574	39	8	acyclic	acyclic	ADJ
ejpam-4574	39	9	chromatic	chromatic	ADJ
ejpam-4574	39	10	numbers	number	NOUN
ejpam-4574	39	11	of	of	ADP
ejpam-4574	39	12	subdivision	subdivision	NOUN
ejpam-4574	39	13	graph	graph	NOUN
ejpam-4574	39	14	g′	g′	NOUN
ejpam-4574	39	15	of	of	ADP
ejpam-4574	39	16	a	a	DET
ejpam-4574	39	17	graph	graph	NOUN
ejpam-4574	39	18	g.	g.	NOUN
ejpam-4574	39	19	a	a	DET
ejpam-4574	39	20	great	great	ADJ
ejpam-4574	39	21	deal	deal	NOUN
ejpam-4574	39	22	of	of	ADP
ejpam-4574	39	23	research	research	NOUN
ejpam-4574	39	24	has	have	AUX
ejpam-4574	39	25	been	be	AUX
ejpam-4574	39	26	conducted	conduct	VERB
ejpam-4574	39	27	since	since	SCONJ
ejpam-4574	39	28	then	then	ADV
ejpam-4574	39	29	.	.	PUNCT
ejpam-4574	40	1	recently	recently	ADV
ejpam-4574	40	2	,	,	PUNCT
ejpam-4574	40	3	wang	wang	PROPN
ejpam-4574	40	4	et	et	PROPN
ejpam-4574	40	5	al	al	PROPN
ejpam-4574	40	6	.	.	PUNCT
ejpam-4574	41	1	[	[	X
ejpam-4574	41	2	9	9	NUM
ejpam-4574	41	3	]	]	PUNCT
ejpam-4574	41	4	studied	study	VERB
ejpam-4574	41	5	the	the	DET
ejpam-4574	41	6	acyclic	acyclic	ADJ
ejpam-4574	41	7	choosability	choosability	NOUN
ejpam-4574	41	8	of	of	ADP
ejpam-4574	41	9	graphs	graph	NOUN
ejpam-4574	41	10	with	with	ADP
ejpam-4574	41	11	bounded	bounded	ADJ
ejpam-4574	41	12	degrees	degree	NOUN
ejpam-4574	41	13	.	.	PUNCT
ejpam-4574	42	1	acyclic	acyclic	ADJ
ejpam-4574	42	2	and	and	CCONJ
ejpam-4574	42	3	star	star	NOUN
ejpam-4574	42	4	coloring	coloring	NOUN
ejpam-4574	42	5	problems	problem	NOUN
ejpam-4574	42	6	are	be	AUX
ejpam-4574	42	7	specialized	specialized	ADJ
ejpam-4574	42	8	vertex	vertex	NOUN
ejpam-4574	42	9	coloring	coloring	NOUN
ejpam-4574	42	10	problems	problem	NOUN
ejpam-4574	42	11	that	that	PRON
ejpam-4574	42	12	arise	arise	VERB
ejpam-4574	42	13	in	in	ADP
ejpam-4574	42	14	the	the	DET
ejpam-4574	42	15	efficient	efficient	ADJ
ejpam-4574	42	16	computation	computation	NOUN
ejpam-4574	42	17	of	of	ADP
ejpam-4574	42	18	hessians	hessian	NOUN
ejpam-4574	42	19	using	use	VERB
ejpam-4574	42	20	automatic	automatic	ADJ
ejpam-4574	42	21	differentiation	differentiation	NOUN
ejpam-4574	42	22	or	or	CCONJ
ejpam-4574	42	23	finite	finite	NOUN
ejpam-4574	42	24	differencing	differencing	NOUN
ejpam-4574	42	25	when	when	SCONJ
ejpam-4574	42	26	both	both	DET
ejpam-4574	42	27	sparsity	sparsity	NOUN
ejpam-4574	42	28	and	and	CCONJ
ejpam-4574	42	29	symmetry	symmetry	NOUN
ejpam-4574	42	30	are	be	AUX
ejpam-4574	42	31	exploited	exploit	VERB
ejpam-4574	42	32	.	.	PUNCT
ejpam-4574	43	1	the	the	DET
ejpam-4574	43	2	kth	kth	PROPN
ejpam-4574	43	3	power	power	NOUN
ejpam-4574	43	4	of	of	ADP
ejpam-4574	43	5	a	a	DET
ejpam-4574	43	6	graph	graph	NOUN
ejpam-4574	43	7	g	g	NOUN
ejpam-4574	43	8	is	be	AUX
ejpam-4574	43	9	defined	define	VERB
ejpam-4574	43	10	on	on	ADP
ejpam-4574	43	11	the	the	DET
ejpam-4574	43	12	same	same	ADJ
ejpam-4574	43	13	set	set	NOUN
ejpam-4574	43	14	of	of	ADP
ejpam-4574	43	15	vertices	vertex	NOUN
ejpam-4574	43	16	as	as	ADP
ejpam-4574	43	17	g	g	NOUN
ejpam-4574	43	18	and	and	CCONJ
ejpam-4574	43	19	has	have	VERB
ejpam-4574	43	20	an	an	DET
ejpam-4574	43	21	edge	edge	NOUN
ejpam-4574	43	22	between	between	ADP
ejpam-4574	43	23	any	any	DET
ejpam-4574	43	24	pair	pair	NOUN
ejpam-4574	43	25	of	of	ADP
ejpam-4574	43	26	vertices	vertex	NOUN
ejpam-4574	43	27	of	of	ADP
ejpam-4574	43	28	distance	distance	NOUN
ejpam-4574	43	29	at	at	ADP
ejpam-4574	43	30	most	most	ADJ
ejpam-4574	43	31	k	k	PROPN
ejpam-4574	43	32	in	in	ADP
ejpam-4574	43	33	g.	g.	PROPN
ejpam-4574	43	34	the	the	DET
ejpam-4574	43	35	problem	problem	NOUN
ejpam-4574	43	36	of	of	ADP
ejpam-4574	43	37	the	the	DET
ejpam-4574	43	38	coloring	coloring	NOUN
ejpam-4574	43	39	of	of	ADP
ejpam-4574	43	40	squares	square	NOUN
ejpam-4574	43	41	of	of	ADP
ejpam-4574	43	42	graphs	graph	NOUN
ejpam-4574	43	43	has	have	VERB
ejpam-4574	43	44	applications	application	NOUN
ejpam-4574	43	45	to	to	PART
ejpam-4574	43	46	frequency	frequency	VERB
ejpam-4574	43	47	allocation	allocation	NOUN
ejpam-4574	43	48	.	.	PUNCT
ejpam-4574	44	1	transceivers	transceiver	NOUN
ejpam-4574	44	2	in	in	ADP
ejpam-4574	44	3	a	a	DET
ejpam-4574	44	4	radio	radio	NOUN
ejpam-4574	44	5	network	network	NOUN
ejpam-4574	44	6	communicate	communicate	VERB
ejpam-4574	44	7	using	use	VERB
ejpam-4574	44	8	channels	channel	NOUN
ejpam-4574	44	9	at	at	ADP
ejpam-4574	44	10	given	give	VERB
ejpam-4574	44	11	radio	radio	NOUN
ejpam-4574	44	12	frequencies	frequency	NOUN
ejpam-4574	44	13	.	.	PUNCT
ejpam-4574	45	1	graph	graph	NOUN
ejpam-4574	45	2	coloring	coloring	NOUN
ejpam-4574	45	3	formalizes	formalize	VERB
ejpam-4574	45	4	this	this	DET
ejpam-4574	45	5	problem	problem	NOUN
ejpam-4574	45	6	.	.	PUNCT
ejpam-4574	46	1	when	when	SCONJ
ejpam-4574	46	2	the	the	DET
ejpam-4574	46	3	constraint	constraint	NOUN
ejpam-4574	46	4	is	be	AUX
ejpam-4574	46	5	that	that	SCONJ
ejpam-4574	46	6	nearby	nearby	ADJ
ejpam-4574	46	7	pairs	pair	NOUN
ejpam-4574	46	8	of	of	ADP
ejpam-4574	46	9	transceivers	transceiver	NOUN
ejpam-4574	46	10	can	can	AUX
ejpam-4574	46	11	not	not	PART
ejpam-4574	46	12	use	use	VERB
ejpam-4574	46	13	the	the	DET
ejpam-4574	46	14	same	same	ADJ
ejpam-4574	46	15	channel	channel	NOUN
ejpam-4574	46	16	due	due	ADP
ejpam-4574	46	17	to	to	ADP
ejpam-4574	46	18	interference	interference	VERB
ejpam-4574	46	19	.	.	PUNCT
ejpam-4574	47	1	however	however	ADV
ejpam-4574	47	2	,	,	PUNCT
ejpam-4574	47	3	if	if	SCONJ
ejpam-4574	47	4	two	two	NUM
ejpam-4574	47	5	transceivers	transceiver	NOUN
ejpam-4574	47	6	are	be	AUX
ejpam-4574	47	7	using	use	VERB
ejpam-4574	47	8	the	the	DET
ejpam-4574	47	9	same	same	ADJ
ejpam-4574	47	10	channel	channel	NOUN
ejpam-4574	47	11	and	and	CCONJ
ejpam-4574	47	12	both	both	PRON
ejpam-4574	47	13	are	be	AUX
ejpam-4574	47	14	adjacent	adjacent	ADJ
ejpam-4574	47	15	to	to	ADP
ejpam-4574	47	16	a	a	DET
ejpam-4574	47	17	third	third	ADJ
ejpam-4574	47	18	station	station	NOUN
ejpam-4574	47	19	,	,	PUNCT
ejpam-4574	47	20	a	a	DET
ejpam-4574	47	21	clashing	clashing	NOUN
ejpam-4574	47	22	of	of	ADP
ejpam-4574	47	23	signals	signal	NOUN
ejpam-4574	47	24	is	be	AUX
ejpam-4574	47	25	experienced	experience	VERB
ejpam-4574	47	26	at	at	ADP
ejpam-4574	47	27	that	that	DET
ejpam-4574	47	28	third	third	ADJ
ejpam-4574	47	29	station	station	NOUN
ejpam-4574	47	30	.	.	PUNCT
ejpam-4574	48	1	this	this	PRON
ejpam-4574	48	2	can	can	AUX
ejpam-4574	48	3	be	be	AUX
ejpam-4574	48	4	avoided	avoid	VERB
ejpam-4574	48	5	by	by	ADP
ejpam-4574	48	6	additionally	additionally	ADV
ejpam-4574	48	7	requiring	require	VERB
ejpam-4574	48	8	all	all	DET
ejpam-4574	48	9	neighbors	neighbor	NOUN
ejpam-4574	48	10	of	of	ADP
ejpam-4574	48	11	a	a	DET
ejpam-4574	48	12	node	node	NOUN
ejpam-4574	48	13	to	to	PART
ejpam-4574	48	14	be	be	AUX
ejpam-4574	48	15	assigned	assign	VERB
ejpam-4574	48	16	different	different	ADJ
ejpam-4574	48	17	colors	color	NOUN
ejpam-4574	48	18	,	,	PUNCT
ejpam-4574	48	19	i.e.	i.e.	X
ejpam-4574	48	20	,	,	PUNCT
ejpam-4574	48	21	that	that	DET
ejpam-4574	48	22	vertices	vertice	VERB
ejpam-4574	48	23	of	of	ADP
ejpam-4574	48	24	distance	distance	NOUN
ejpam-4574	48	25	at	at	ADP
ejpam-4574	48	26	most	most	ADJ
ejpam-4574	48	27	2	2	NUM
ejpam-4574	48	28	receive	receive	VERB
ejpam-4574	48	29	different	different	ADJ
ejpam-4574	48	30	colors	color	NOUN
ejpam-4574	48	31	.	.	PUNCT
ejpam-4574	49	1	this	this	PRON
ejpam-4574	49	2	is	be	AUX
ejpam-4574	49	3	equivalent	equivalent	ADJ
ejpam-4574	49	4	to	to	ADP
ejpam-4574	49	5	coloring	color	VERB
ejpam-4574	49	6	the	the	DET
ejpam-4574	49	7	square	square	NOUN
ejpam-4574	49	8	of	of	ADP
ejpam-4574	49	9	the	the	DET
ejpam-4574	49	10	underlying	underlie	VERB
ejpam-4574	49	11	network	network	NOUN
ejpam-4574	49	12	.	.	PUNCT
ejpam-4574	50	1	we	we	PRON
ejpam-4574	50	2	attempt	attempt	VERB
ejpam-4574	50	3	here	here	ADV
ejpam-4574	50	4	to	to	PART
ejpam-4574	50	5	contribute	contribute	VERB
ejpam-4574	50	6	to	to	ADP
ejpam-4574	50	7	both	both	PRON
ejpam-4574	50	8	of	of	ADP
ejpam-4574	50	9	these	these	DET
ejpam-4574	50	10	perspectives	perspective	NOUN
ejpam-4574	50	11	,	,	PUNCT
ejpam-4574	50	12	graph	graph	NOUN
ejpam-4574	50	13	powers	power	NOUN
ejpam-4574	50	14	and	and	CCONJ
ejpam-4574	50	15	acyclic	acyclic	ADJ
ejpam-4574	50	16	(	(	PUNCT
ejpam-4574	50	17	star	star	NOUN
ejpam-4574	50	18	)	)	PUNCT
ejpam-4574	50	19	colorings	coloring	NOUN
ejpam-4574	50	20	.	.	PUNCT
ejpam-4574	51	1	we	we	PRON
ejpam-4574	51	2	focus	focus	VERB
ejpam-4574	51	3	on	on	ADP
ejpam-4574	51	4	the	the	DET
ejpam-4574	51	5	powers	power	NOUN
ejpam-4574	51	6	of	of	ADP
ejpam-4574	51	7	paths	path	NOUN
ejpam-4574	51	8	and	and	CCONJ
ejpam-4574	51	9	cycles	cycle	NOUN
ejpam-4574	51	10	.	.	PUNCT
ejpam-4574	52	1	as	as	ADP
ejpam-4574	52	2	usual	usual	ADJ
ejpam-4574	52	3	,	,	PUNCT
ejpam-4574	52	4	pn	pn	PROPN
ejpam-4574	52	5	denotes	denote	VERB
ejpam-4574	52	6	the	the	DET
ejpam-4574	52	7	path	path	NOUN
ejpam-4574	52	8	on	on	ADP
ejpam-4574	52	9	n	n	DET
ejpam-4574	52	10	vertices	vertex	NOUN
ejpam-4574	52	11	;	;	PUNCT
ejpam-4574	52	12	and	and	CCONJ
ejpam-4574	52	13	cn	cn	PROPN
ejpam-4574	52	14	denotes	denote	VERB
ejpam-4574	52	15	the	the	DET
ejpam-4574	52	16	cycle	cycle	NOUN
ejpam-4574	52	17	on	on	ADP
ejpam-4574	52	18	n	n	DET
ejpam-4574	52	19	vertices	vertex	NOUN
ejpam-4574	52	20	.	.	PUNCT
ejpam-4574	53	1	acyclic	acyclic	ADJ
ejpam-4574	53	2	colorings	coloring	NOUN
ejpam-4574	53	3	are	be	AUX
ejpam-4574	53	4	hereditary	hereditary	ADJ
ejpam-4574	53	5	in	in	ADP
ejpam-4574	53	6	the	the	DET
ejpam-4574	53	7	sense	sense	NOUN
ejpam-4574	53	8	that	that	SCONJ
ejpam-4574	53	9	the	the	DET
ejpam-4574	53	10	restriction	restriction	NOUN
ejpam-4574	53	11	of	of	ADP
ejpam-4574	53	12	an	an	DET
ejpam-4574	53	13	acyclic	acyclic	ADJ
ejpam-4574	53	14	coloring	coloring	NOUN
ejpam-4574	53	15	to	to	ADP
ejpam-4574	53	16	a	a	DET
ejpam-4574	53	17	subgraph	subgraph	NOUN
ejpam-4574	53	18	is	be	AUX
ejpam-4574	53	19	an	an	DET
ejpam-4574	53	20	acyclic	acyclic	ADJ
ejpam-4574	53	21	coloring	coloring	NOUN
ejpam-4574	53	22	.	.	PUNCT
ejpam-4574	54	1	thus	thus	ADV
ejpam-4574	54	2	,	,	PUNCT
ejpam-4574	54	3	the	the	DET
ejpam-4574	54	4	acyclic	acyclic	ADJ
ejpam-4574	54	5	chromatic	chromatic	ADJ
ejpam-4574	54	6	number	number	NOUN
ejpam-4574	54	7	is	be	AUX
ejpam-4574	54	8	nondecreasing	nondecrease	VERB
ejpam-4574	54	9	from	from	ADP
ejpam-4574	54	10	subgraph	subgraph	NOUN
ejpam-4574	54	11	to	to	PART
ejpam-4574	54	12	supergraph	supergraph	VERB
ejpam-4574	54	13	.	.	PUNCT
ejpam-4574	55	1	the	the	DET
ejpam-4574	55	2	main	main	ADJ
ejpam-4574	55	3	purpose	purpose	NOUN
ejpam-4574	55	4	of	of	ADP
ejpam-4574	55	5	this	this	DET
ejpam-4574	55	6	article	article	NOUN
ejpam-4574	55	7	is	be	AUX
ejpam-4574	55	8	to	to	PART
ejpam-4574	55	9	bound	bind	VERB
ejpam-4574	55	10	and	and	CCONJ
ejpam-4574	55	11	determine	determine	VERB
ejpam-4574	55	12	the	the	DET
ejpam-4574	55	13	star	star	NOUN
ejpam-4574	55	14	and	and	CCONJ
ejpam-4574	55	15	acyclic	acyclic	ADJ
ejpam-4574	55	16	chromatic	chromatic	ADJ
ejpam-4574	55	17	numbers	number	NOUN
ejpam-4574	55	18	of	of	ADP
ejpam-4574	55	19	powers	power	NOUN
ejpam-4574	55	20	of	of	ADP
ejpam-4574	55	21	paths	path	NOUN
ejpam-4574	55	22	and	and	CCONJ
ejpam-4574	55	23	cycles	cycle	NOUN
ejpam-4574	55	24	.	.	PUNCT
ejpam-4574	56	1	we	we	PRON
ejpam-4574	56	2	prove	prove	VERB
ejpam-4574	56	3	that	that	SCONJ
ejpam-4574	56	4	for	for	ADP
ejpam-4574	56	5	large	large	ADJ
ejpam-4574	56	6	graph	graph	NOUN
ejpam-4574	56	7	sizes	size	NOUN
ejpam-4574	56	8	,	,	PUNCT
ejpam-4574	56	9	the	the	DET
ejpam-4574	56	10	star	star	NOUN
ejpam-4574	56	11	and	and	CCONJ
ejpam-4574	56	12	acyclic	acyclic	ADJ
ejpam-4574	56	13	chromatic	chromatic	ADJ
ejpam-4574	56	14	numbers	number	NOUN
ejpam-4574	56	15	of	of	ADP
ejpam-4574	56	16	powers	power	NOUN
ejpam-4574	56	17	of	of	ADP
ejpam-4574	56	18	paths	path	NOUN
ejpam-4574	56	19	and	and	CCONJ
ejpam-4574	56	20	cycles	cycle	NOUN
ejpam-4574	56	21	tend	tend	VERB
ejpam-4574	56	22	to	to	PART
ejpam-4574	56	23	have	have	VERB
ejpam-4574	56	24	exact	exact	ADJ
ejpam-4574	56	25	formulas	formula	NOUN
ejpam-4574	56	26	in	in	ADP
ejpam-4574	56	27	terms	term	NOUN
ejpam-4574	56	28	of	of	ADP
ejpam-4574	56	29	the	the	DET
ejpam-4574	56	30	power	power	NOUN
ejpam-4574	56	31	k.	k.	PROPN
ejpam-4574	56	32	as	as	ADP
ejpam-4574	56	33	a	a	DET
ejpam-4574	56	34	consequence	consequence	NOUN
ejpam-4574	56	35	,	,	PUNCT
ejpam-4574	56	36	we	we	PRON
ejpam-4574	56	37	find	find	VERB
ejpam-4574	56	38	the	the	DET
ejpam-4574	56	39	value	value	NOUN
ejpam-4574	56	40	of	of	ADP
ejpam-4574	56	41	χa(p	χa(p	PROPN
ejpam-4574	56	42	k	k	PROPN
ejpam-4574	56	43	n	n	PROPN
ejpam-4574	56	44	)	)	PUNCT
ejpam-4574	56	45	and	and	CCONJ
ejpam-4574	56	46	a.	a.	PROPN
ejpam-4574	56	47	etawi	etawi	PROPN
ejpam-4574	56	48	,	,	PUNCT
ejpam-4574	56	49	m.	m.	PROPN
ejpam-4574	56	50	ghanem	ghanem	PROPN
ejpam-4574	56	51	,	,	PUNCT
ejpam-4574	56	52	h.	h.	PROPN
ejpam-4574	56	53	al	al	PROPN
ejpam-4574	56	54	-	-	PUNCT
ejpam-4574	56	55	ezeh	ezeh	PROPN
ejpam-4574	56	56	/	/	SYM
ejpam-4574	56	57	eur	eur	NOUN
ejpam-4574	56	58	.	.	PUNCT
ejpam-4574	57	1	j.	j.	PROPN
ejpam-4574	57	2	pure	pure	PROPN
ejpam-4574	57	3	appl	appl	PROPN
ejpam-4574	57	4	.	.	PROPN
ejpam-4574	57	5	math	math	PROPN
ejpam-4574	57	6	,	,	PUNCT
ejpam-4574	57	7	15	15	NUM
ejpam-4574	57	8	(	(	PUNCT
ejpam-4574	57	9	4	4	NUM
ejpam-4574	57	10	)	)	PUNCT
ejpam-4574	57	11	(	(	PUNCT
ejpam-4574	57	12	2022	2022	NUM
ejpam-4574	57	13	)	)	PUNCT
ejpam-4574	57	14	,	,	PUNCT
ejpam-4574	57	15	1822	1822	NUM
ejpam-4574	57	16	-	-	SYM
ejpam-4574	57	17	1835	1835	NUM
ejpam-4574	57	18	1824	1824	NUM
ejpam-4574	57	19	χs(p	χs(p	X
ejpam-4574	57	20	k	k	PROPN
ejpam-4574	57	21	n	n	PROPN
ejpam-4574	57	22	)	)	PUNCT
ejpam-4574	57	23	in	in	ADP
ejpam-4574	57	24	terms	term	NOUN
ejpam-4574	57	25	of	of	ADP
ejpam-4574	57	26	k	k	NOUN
ejpam-4574	57	27	,	,	PUNCT
ejpam-4574	57	28	we	we	PRON
ejpam-4574	57	29	give	give	VERB
ejpam-4574	57	30	an	an	DET
ejpam-4574	57	31	upper	upper	ADJ
ejpam-4574	57	32	bound	bind	VERB
ejpam-4574	57	33	and	and	CCONJ
ejpam-4574	57	34	a	a	DET
ejpam-4574	57	35	sharp	sharp	ADV
ejpam-4574	57	36	lower	low	ADJ
ejpam-4574	57	37	bound	bind	VERB
ejpam-4574	57	38	of	of	ADP
ejpam-4574	57	39	χa(c	χa(c	ADJ
ejpam-4574	57	40	k	k	PROPN
ejpam-4574	57	41	n	n	CCONJ
ejpam-4574	57	42	)	)	PUNCT
ejpam-4574	57	43	in	in	ADP
ejpam-4574	57	44	terms	term	NOUN
ejpam-4574	57	45	of	of	ADP
ejpam-4574	57	46	k	k	X
ejpam-4574	58	1	when	when	SCONJ
ejpam-4574	58	2	(	(	PUNCT
ejpam-4574	58	3	k	k	NOUN
ejpam-4574	58	4	+	+	PROPN
ejpam-4574	58	5	1)2	1)2	NUM
ejpam-4574	58	6	≤	≤	NOUN
ejpam-4574	58	7	n	n	CCONJ
ejpam-4574	58	8	<	<	X
ejpam-4574	58	9	(	(	PUNCT
ejpam-4574	58	10	k	k	PROPN
ejpam-4574	58	11	+	+	PROPN
ejpam-4574	58	12	1)3	1)3	NUM
ejpam-4574	58	13	.	.	PUNCT
ejpam-4574	59	1	we	we	PRON
ejpam-4574	59	2	derived	derive	VERB
ejpam-4574	59	3	the	the	DET
ejpam-4574	59	4	exact	exact	ADJ
ejpam-4574	59	5	value	value	NOUN
ejpam-4574	59	6	of	of	ADP
ejpam-4574	59	7	χa(c	χa(c	ADJ
ejpam-4574	59	8	k	k	PROPN
ejpam-4574	59	9	n	n	CCONJ
ejpam-4574	59	10	)	)	PUNCT
ejpam-4574	59	11	in	in	ADP
ejpam-4574	59	12	terms	term	NOUN
ejpam-4574	59	13	of	of	ADP
ejpam-4574	59	14	k	k	PROPN
ejpam-4574	59	15	for	for	ADP
ejpam-4574	59	16	n	n	PRON
ejpam-4574	59	17	≥	≥	X
ejpam-4574	59	18	(	(	PUNCT
ejpam-4574	59	19	k	k	PROPN
ejpam-4574	59	20	+	+	PROPN
ejpam-4574	59	21	1)3	1)3	PROPN
ejpam-4574	59	22	.	.	PUNCT
ejpam-4574	60	1	additionally	additionally	ADV
ejpam-4574	60	2	,	,	PUNCT
ejpam-4574	60	3	we	we	PRON
ejpam-4574	60	4	give	give	VERB
ejpam-4574	60	5	an	an	DET
ejpam-4574	60	6	upper	upper	ADJ
ejpam-4574	60	7	bound	bind	VERB
ejpam-4574	60	8	and	and	CCONJ
ejpam-4574	60	9	sharp	sharp	ADJ
ejpam-4574	60	10	lower	lower	ADV
ejpam-4574	60	11	bound	bind	VERB
ejpam-4574	60	12	χs(c	χs(c	NOUN
ejpam-4574	60	13	k	k	PROPN
ejpam-4574	60	14	n	n	CCONJ
ejpam-4574	60	15	)	)	PUNCT
ejpam-4574	60	16	in	in	ADP
ejpam-4574	60	17	terms	term	NOUN
ejpam-4574	60	18	of	of	ADP
ejpam-4574	60	19	k	k	PROPN
ejpam-4574	60	20	for	for	ADP
ejpam-4574	60	21	(	(	PUNCT
ejpam-4574	60	22	k	k	PROPN
ejpam-4574	60	23	+	+	PROPN
ejpam-4574	60	24	1)2	1)2	NUM
ejpam-4574	60	25	≤	≤	NUM
ejpam-4574	60	26	n.	n.	NOUN
ejpam-4574	60	27	the	the	DET
ejpam-4574	60	28	underlying	underlie	VERB
ejpam-4574	60	29	common	common	ADJ
ejpam-4574	60	30	technique	technique	NOUN
ejpam-4574	60	31	is	be	AUX
ejpam-4574	60	32	the	the	DET
ejpam-4574	60	33	exploitation	exploitation	NOUN
ejpam-4574	60	34	of	of	ADP
ejpam-4574	60	35	the	the	DET
ejpam-4574	60	36	structure	structure	NOUN
ejpam-4574	60	37	of	of	ADP
ejpam-4574	60	38	bicolored	bicolored	ADJ
ejpam-4574	60	39	induced	induced	ADJ
ejpam-4574	60	40	subgraphs	subgraph	NOUN
ejpam-4574	60	41	,	,	PUNCT
ejpam-4574	60	42	the	the	DET
ejpam-4574	60	43	bounds	bound	NOUN
ejpam-4574	60	44	that	that	PRON
ejpam-4574	60	45	we	we	PRON
ejpam-4574	60	46	reach	reach	VERB
ejpam-4574	60	47	in	in	ADP
ejpam-4574	60	48	this	this	DET
ejpam-4574	60	49	article	article	NOUN
ejpam-4574	60	50	are	be	AUX
ejpam-4574	60	51	tight	tight	ADJ
ejpam-4574	60	52	with	with	ADP
ejpam-4574	60	53	intervals	interval	NOUN
ejpam-4574	60	54	of	of	ADP
ejpam-4574	60	55	two	two	NUM
ejpam-4574	60	56	values	value	NOUN
ejpam-4574	60	57	only	only	ADV
ejpam-4574	60	58	.	.	PUNCT
ejpam-4574	61	1	our	our	PRON
ejpam-4574	61	2	results	result	NOUN
ejpam-4574	61	3	are	be	AUX
ejpam-4574	61	4	summarized	summarize	VERB
ejpam-4574	61	5	below	below	ADP
ejpam-4574	61	6	,	,	PUNCT
ejpam-4574	61	7	[	[	X
ejpam-4574	61	8	bold	bold	ADJ
ejpam-4574	61	9	]	]	X
ejpam-4574	61	10	bounds	bound	NOUN
ejpam-4574	61	11	are	be	AUX
ejpam-4574	61	12	sharp	sharp	ADJ
ejpam-4574	61	13	.	.	PUNCT
ejpam-4574	62	1	graph	graph	NOUN
ejpam-4574	62	2	g	g	PROPN
ejpam-4574	62	3	range	range	NOUN
ejpam-4574	62	4	of	of	ADP
ejpam-4574	62	5	n	n	PRON
ejpam-4574	62	6	χs(g	χs(g	PRON
ejpam-4574	62	7	)	)	PUNCT
ejpam-4574	62	8	χa(g	χa(g	NOUN
ejpam-4574	62	9	)	)	PUNCT
ejpam-4574	63	1	p	p	NOUN
ejpam-4574	63	2	k	k	PROPN
ejpam-4574	63	3	n	n	CCONJ
ejpam-4574	63	4	1	1	NUM
ejpam-4574	63	5	≤	≤	NUM
ejpam-4574	63	6	n	n	PRON
ejpam-4574	63	7	≤	≤	NOUN
ejpam-4574	63	8	k	k	X
ejpam-4574	64	1	+	+	CCONJ
ejpam-4574	64	2	1	1	NUM
ejpam-4574	64	3	n	n	CCONJ
ejpam-4574	64	4	n	n	PRON
ejpam-4574	64	5	k	k	NOUN
ejpam-4574	64	6	+	+	CCONJ
ejpam-4574	64	7	2	2	NUM
ejpam-4574	64	8	≤	≤	NOUN
ejpam-4574	64	9	n	n	PRON
ejpam-4574	64	10	≤	≤	NOUN
ejpam-4574	64	11	3k	3k	NOUN
ejpam-4574	65	1	+	+	CCONJ
ejpam-4574	65	2	1	1	NUM
ejpam-4574	65	3	⌊k+n+1	⌊k+n+1	PROPN
ejpam-4574	65	4	2	2	NUM
ejpam-4574	65	5	⌋	⌋	NOUN
ejpam-4574	65	6	k+1	k+1	X
ejpam-4574	65	7	n	n	PRON
ejpam-4574	65	8	≥	≥	NOUN
ejpam-4574	65	9	3k	3k	X
ejpam-4574	65	10	+	+	CCONJ
ejpam-4574	65	11	1	1	NUM
ejpam-4574	65	12	2k+1	2k+1	NOUN
ejpam-4574	65	13	ck	ck	ADP
ejpam-4574	65	14	n	n	CCONJ
ejpam-4574	65	15	1	1	NUM
ejpam-4574	65	16	≤	≤	NUM
ejpam-4574	65	17	n	n	PRON
ejpam-4574	65	18	≤	≤	NOUN
ejpam-4574	65	19	2k	2k	NOUN
ejpam-4574	65	20	+	+	CCONJ
ejpam-4574	65	21	1	1	NUM
ejpam-4574	65	22	n	n	NUM
ejpam-4574	65	23	n	n	PRON
ejpam-4574	65	24	2k	2k	NOUN
ejpam-4574	65	25	+	+	CCONJ
ejpam-4574	65	26	2	2	NUM
ejpam-4574	65	27	≤	≤	NOUN
ejpam-4574	65	28	n	n	CCONJ
ejpam-4574	65	29	<	<	X
ejpam-4574	65	30	(	(	PUNCT
ejpam-4574	65	31	k	k	PROPN
ejpam-4574	65	32	+	+	PROPN
ejpam-4574	65	33	1)2	1)2	NUM
ejpam-4574	65	34	k+2	k+2	PROPN
ejpam-4574	65	35	≤	≤	NOUN
ejpam-4574	65	36	χs(g	χs(g	NUM
ejpam-4574	65	37	)	)	PUNCT
ejpam-4574	65	38	≤	≤	NOUN
ejpam-4574	65	39	n	n	CCONJ
ejpam-4574	65	40	k	k	NOUN
ejpam-4574	65	41	+	+	CCONJ
ejpam-4574	65	42	2	2	NUM
ejpam-4574	65	43	≤	≤	NUM
ejpam-4574	65	44	χa(g	χa(g	NOUN
ejpam-4574	65	45	)	)	PUNCT
ejpam-4574	65	46	≤	≤	NOUN
ejpam-4574	65	47	n	n	CCONJ
ejpam-4574	65	48	(	(	PUNCT
ejpam-4574	65	49	k	k	PROPN
ejpam-4574	65	50	+	+	PROPN
ejpam-4574	65	51	1)2	1)2	NUM
ejpam-4574	65	52	≤	≤	NOUN
ejpam-4574	65	53	n	n	CCONJ
ejpam-4574	65	54	<	<	X
ejpam-4574	65	55	(	(	PUNCT
ejpam-4574	65	56	k	k	PROPN
ejpam-4574	65	57	+	+	PROPN
ejpam-4574	65	58	1)3	1)3	PROPN
ejpam-4574	65	59	k+2	k+2	PROPN
ejpam-4574	65	60	≤	≤	NOUN
ejpam-4574	65	61	χa(g	χa(g	NOUN
ejpam-4574	65	62	)	)	PUNCT
ejpam-4574	65	63	≤	≤	PUNCT
ejpam-4574	65	64	k	k	NOUN
ejpam-4574	66	1	+	+	CCONJ
ejpam-4574	66	2	3	3	NUM
ejpam-4574	66	3	n	n	NUM
ejpam-4574	66	4	≥	≥	NOUN
ejpam-4574	66	5	(	(	PUNCT
ejpam-4574	66	6	k	k	PROPN
ejpam-4574	66	7	+	+	PROPN
ejpam-4574	66	8	1)3	1)3	PROPN
ejpam-4574	66	9	2k+1	2k+1	PROPN
ejpam-4574	66	10	≤	≤	NOUN
ejpam-4574	66	11	χs(g	χs(g	NOUN
ejpam-4574	66	12	)	)	PUNCT
ejpam-4574	66	13	≤	≤	NOUN
ejpam-4574	66	14	2k	2k	NOUN
ejpam-4574	66	15	+	+	CCONJ
ejpam-4574	66	16	2	2	NUM
ejpam-4574	66	17	k+2	k+2	NUM
ejpam-4574	66	18	2	2	NUM
ejpam-4574	66	19	.	.	PUNCT
ejpam-4574	66	20	acyclic	acyclic	ADJ
ejpam-4574	66	21	coloring	coloring	NOUN
ejpam-4574	66	22	of	of	ADP
ejpam-4574	66	23	p	p	PROPN
ejpam-4574	66	24	k	k	PROPN
ejpam-4574	66	25	n	n	ADV
ejpam-4574	66	26	let	let	VERB
ejpam-4574	66	27	p	p	PRON
ejpam-4574	66	28	k	k	PROPN
ejpam-4574	66	29	n	n	ADV
ejpam-4574	66	30	denote	denote	VERB
ejpam-4574	66	31	the	the	DET
ejpam-4574	66	32	path	path	NOUN
ejpam-4574	66	33	of	of	ADP
ejpam-4574	66	34	order	order	NOUN
ejpam-4574	66	35	n	n	PRON
ejpam-4574	66	36	with	with	ADP
ejpam-4574	66	37	vertex	vertex	NOUN
ejpam-4574	66	38	set	set	VERB
ejpam-4574	66	39	v	v	NOUN
ejpam-4574	66	40	(	(	PUNCT
ejpam-4574	66	41	p	p	X
ejpam-4574	66	42	k	k	PROPN
ejpam-4574	66	43	n	n	PROPN
ejpam-4574	66	44	)	)	PUNCT
ejpam-4574	66	45	=	=	SYM
ejpam-4574	66	46	{	{	PUNCT
ejpam-4574	66	47	v0	v0	NOUN
ejpam-4574	66	48	,	,	PUNCT
ejpam-4574	66	49	v1	v1	NOUN
ejpam-4574	66	50	,	,	PUNCT
ejpam-4574	66	51	...	...	PUNCT
ejpam-4574	66	52	,	,	PUNCT
ejpam-4574	66	53	vn−1	vn−1	ADJ
ejpam-4574	66	54	}	}	PUNCT
ejpam-4574	66	55	and	and	CCONJ
ejpam-4574	66	56	edge	edge	NOUN
ejpam-4574	66	57	set	set	VERB
ejpam-4574	66	58	e(p	e(p	PROPN
ejpam-4574	66	59	k	k	PROPN
ejpam-4574	66	60	n	n	PROPN
ejpam-4574	66	61	)	)	PUNCT
ejpam-4574	66	62	=	=	NOUN
ejpam-4574	66	63	{	{	PUNCT
ejpam-4574	66	64	vivj	vivj	NOUN
ejpam-4574	66	65	:	:	PUNCT
ejpam-4574	66	66	1	1	NUM
ejpam-4574	66	67	≤	≤	NUM
ejpam-4574	66	68	|i−	|i−	NOUN
ejpam-4574	66	69	j|	j|	VERB
ejpam-4574	66	70	≤	≤	ADV
ejpam-4574	66	71	k	k	X
ejpam-4574	66	72	}	}	PUNCT
ejpam-4574	66	73	.	.	PUNCT
ejpam-4574	67	1	clearly	clearly	ADV
ejpam-4574	67	2	,	,	PUNCT
ejpam-4574	67	3	p	p	PROPN
ejpam-4574	67	4	k	k	PROPN
ejpam-4574	67	5	n	n	PROPN
ejpam-4574	67	6	is	be	AUX
ejpam-4574	67	7	a	a	DET
ejpam-4574	67	8	complete	complete	ADJ
ejpam-4574	67	9	graph	graph	NOUN
ejpam-4574	67	10	when	when	SCONJ
ejpam-4574	67	11	n	n	PRON
ejpam-4574	67	12	≤	≤	X
ejpam-4574	67	13	k	k	PROPN
ejpam-4574	68	1	+	+	CCONJ
ejpam-4574	68	2	1	1	NUM
ejpam-4574	68	3	and	and	CCONJ
ejpam-4574	68	4	hence	hence	ADV
ejpam-4574	68	5	χa(p	χa(p	VERB
ejpam-4574	68	6	k	k	PROPN
ejpam-4574	68	7	n	n	PROPN
ejpam-4574	68	8	)	)	PUNCT
ejpam-4574	69	1	=	=	VERB
ejpam-4574	69	2	n.	n.	NOUN
ejpam-4574	69	3	example	example	NOUN
ejpam-4574	70	1	1	1	X
ejpam-4574	70	2	.	.	PUNCT
ejpam-4574	71	1	in	in	ADP
ejpam-4574	71	2	figure	figure	NOUN
ejpam-4574	71	3	1	1	NUM
ejpam-4574	71	4	(	(	PUNCT
ejpam-4574	71	5	a	a	NOUN
ejpam-4574	71	6	)	)	PUNCT
ejpam-4574	71	7	,	,	PUNCT
ejpam-4574	71	8	p	p	NOUN
ejpam-4574	71	9	2	2	NUM
ejpam-4574	71	10	8	8	NUM
ejpam-4574	71	11	admits	admit	VERB
ejpam-4574	71	12	an	an	DET
ejpam-4574	71	13	acyclic	acyclic	ADJ
ejpam-4574	71	14	coloring	coloring	NOUN
ejpam-4574	71	15	as	as	SCONJ
ejpam-4574	71	16	shown	show	VERB
ejpam-4574	71	17	,	,	PUNCT
ejpam-4574	71	18	the	the	DET
ejpam-4574	71	19	induced	induced	ADJ
ejpam-4574	71	20	subgraph	subgraph	NOUN
ejpam-4574	71	21	over	over	ADP
ejpam-4574	71	22	the	the	DET
ejpam-4574	71	23	vertices	vertex	NOUN
ejpam-4574	71	24	colored	color	VERB
ejpam-4574	71	25	by	by	ADP
ejpam-4574	71	26	the	the	DET
ejpam-4574	71	27	color	color	NOUN
ejpam-4574	71	28	classes	class	NOUN
ejpam-4574	71	29	{	{	PUNCT
ejpam-4574	71	30	a	a	DET
ejpam-4574	71	31	,	,	PUNCT
ejpam-4574	71	32	b	b	NOUN
ejpam-4574	71	33	}	}	PUNCT
ejpam-4574	71	34	is	be	AUX
ejpam-4574	71	35	p6	p6	PROPN
ejpam-4574	71	36	,	,	PUNCT
ejpam-4574	71	37	which	which	PRON
ejpam-4574	71	38	is	be	AUX
ejpam-4574	71	39	not	not	PART
ejpam-4574	71	40	a	a	DET
ejpam-4574	71	41	star	star	NOUN
ejpam-4574	71	42	.	.	PUNCT
ejpam-4574	72	1	a	a	DET
ejpam-4574	72	2	v1	v1	PROPN
ejpam-4574	72	3	b	b	PROPN
ejpam-4574	72	4	v2	v2	PROPN
ejpam-4574	72	5	c	c	PROPN
ejpam-4574	72	6	v3	v3	PROPN
ejpam-4574	72	7	a	a	DET
ejpam-4574	72	8	v4	v4	PROPN
ejpam-4574	72	9	b	b	PROPN
ejpam-4574	72	10	v5	v5	PROPN
ejpam-4574	72	11	c	c	PROPN
ejpam-4574	72	12	v6	v6	NOUN
ejpam-4574	72	13	a	a	DET
ejpam-4574	72	14	v7	v7	NOUN
ejpam-4574	72	15	b	b	PROPN
ejpam-4574	72	16	v8	v8	NOUN
ejpam-4574	72	17	(	(	PUNCT
ejpam-4574	72	18	a	a	X
ejpam-4574	72	19	)	)	PUNCT
ejpam-4574	72	20	acyclic	acyclic	ADJ
ejpam-4574	72	21	coloring	coloring	NOUN
ejpam-4574	72	22	of	of	ADP
ejpam-4574	72	23	p	p	NOUN
ejpam-4574	72	24	2	2	NUM
ejpam-4574	72	25	8	8	NUM
ejpam-4574	72	26	1	1	NUM
ejpam-4574	72	27	2	2	NUM
ejpam-4574	72	28	4	4	NUM
ejpam-4574	72	29	5	5	NUM
ejpam-4574	72	30	7	7	NUM
ejpam-4574	72	31	8	8	NUM
ejpam-4574	72	32	(	(	PUNCT
ejpam-4574	72	33	b	b	NOUN
ejpam-4574	72	34	)	)	PUNCT
ejpam-4574	72	35	the	the	DET
ejpam-4574	72	36	induced	induced	ADJ
ejpam-4574	72	37	subgraph	subgraph	NOUN
ejpam-4574	72	38	over	over	ADP
ejpam-4574	72	39	the	the	DET
ejpam-4574	72	40	color	color	NOUN
ejpam-4574	72	41	classes	class	NOUN
ejpam-4574	72	42	{	{	PUNCT
ejpam-4574	72	43	a	a	DET
ejpam-4574	72	44	,	,	PUNCT
ejpam-4574	72	45	b	b	NOUN
ejpam-4574	72	46	}	}	PUNCT
ejpam-4574	72	47	figure	figure	NOUN
ejpam-4574	72	48	1	1	NUM
ejpam-4574	72	49	:	:	PUNCT
ejpam-4574	72	50	acyclic	acyclic	ADJ
ejpam-4574	72	51	coloring	coloring	NOUN
ejpam-4574	72	52	but	but	CCONJ
ejpam-4574	72	53	not	not	PART
ejpam-4574	72	54	star	star	NOUN
ejpam-4574	72	55	coloring	color	VERB
ejpam-4574	72	56	obviously	obviously	ADV
ejpam-4574	72	57	,	,	PUNCT
ejpam-4574	72	58	every	every	DET
ejpam-4574	72	59	star	star	NOUN
ejpam-4574	72	60	coloring	coloring	NOUN
ejpam-4574	72	61	is	be	AUX
ejpam-4574	72	62	acyclic	acyclic	ADJ
ejpam-4574	72	63	coloring	coloring	NOUN
ejpam-4574	72	64	while	while	SCONJ
ejpam-4574	72	65	the	the	DET
ejpam-4574	72	66	converse	converse	NOUN
ejpam-4574	72	67	need	need	AUX
ejpam-4574	72	68	not	not	PART
ejpam-4574	72	69	be	be	AUX
ejpam-4574	72	70	true	true	ADJ
ejpam-4574	72	71	in	in	ADP
ejpam-4574	72	72	general	general	ADJ
ejpam-4574	72	73	.	.	PUNCT
ejpam-4574	73	1	a.	a.	PROPN
ejpam-4574	73	2	etawi	etawi	PROPN
ejpam-4574	73	3	,	,	PUNCT
ejpam-4574	73	4	m.	m.	PROPN
ejpam-4574	73	5	ghanem	ghanem	PROPN
ejpam-4574	73	6	,	,	PUNCT
ejpam-4574	73	7	h.	h.	PROPN
ejpam-4574	73	8	al	al	PROPN
ejpam-4574	73	9	-	-	PUNCT
ejpam-4574	73	10	ezeh	ezeh	PROPN
ejpam-4574	73	11	/	/	SYM
ejpam-4574	73	12	eur	eur	NOUN
ejpam-4574	73	13	.	.	PUNCT
ejpam-4574	74	1	j.	j.	PROPN
ejpam-4574	74	2	pure	pure	PROPN
ejpam-4574	74	3	appl	appl	PROPN
ejpam-4574	74	4	.	.	PROPN
ejpam-4574	74	5	math	math	PROPN
ejpam-4574	74	6	,	,	PUNCT
ejpam-4574	74	7	15	15	NUM
ejpam-4574	74	8	(	(	PUNCT
ejpam-4574	74	9	4	4	NUM
ejpam-4574	74	10	)	)	PUNCT
ejpam-4574	74	11	(	(	PUNCT
ejpam-4574	74	12	2022	2022	NUM
ejpam-4574	74	13	)	)	PUNCT
ejpam-4574	74	14	,	,	PUNCT
ejpam-4574	74	15	1822	1822	NUM
ejpam-4574	74	16	-	-	SYM
ejpam-4574	74	17	1835	1835	NUM
ejpam-4574	74	18	1825	1825	NUM
ejpam-4574	74	19	in	in	ADP
ejpam-4574	74	20	this	this	DET
ejpam-4574	74	21	section	section	NOUN
ejpam-4574	74	22	we	we	PRON
ejpam-4574	74	23	calculate	calculate	VERB
ejpam-4574	74	24	the	the	DET
ejpam-4574	74	25	acyclic	acyclic	ADJ
ejpam-4574	74	26	chromatic	chromatic	ADJ
ejpam-4574	74	27	number	number	NOUN
ejpam-4574	74	28	as	as	ADV
ejpam-4574	74	29	well	well	ADV
ejpam-4574	74	30	as	as	ADP
ejpam-4574	74	31	the	the	DET
ejpam-4574	74	32	star	star	NOUN
ejpam-4574	74	33	chromatic	chromatic	ADJ
ejpam-4574	74	34	number	number	NOUN
ejpam-4574	74	35	of	of	ADP
ejpam-4574	74	36	p	p	PROPN
ejpam-4574	74	37	k	k	PROPN
ejpam-4574	74	38	n	n	PROPN
ejpam-4574	74	39	.	.	PUNCT
ejpam-4574	75	1	definition	definition	NOUN
ejpam-4574	75	2	1	1	NUM
ejpam-4574	75	3	.	.	PUNCT
ejpam-4574	76	1	chordless	chordless	ADJ
ejpam-4574	76	2	cycles	cycle	NOUN
ejpam-4574	76	3	:	:	PUNCT
ejpam-4574	76	4	a	a	DET
ejpam-4574	76	5	chordless	chordless	ADJ
ejpam-4574	76	6	cycle	cycle	NOUN
ejpam-4574	76	7	in	in	ADP
ejpam-4574	76	8	a	a	DET
ejpam-4574	76	9	graph	graph	NOUN
ejpam-4574	76	10	,	,	PUNCT
ejpam-4574	76	11	also	also	ADV
ejpam-4574	76	12	called	call	VERB
ejpam-4574	76	13	a	a	DET
ejpam-4574	76	14	hole	hole	NOUN
ejpam-4574	76	15	or	or	CCONJ
ejpam-4574	76	16	an	an	DET
ejpam-4574	76	17	induced	induced	ADJ
ejpam-4574	76	18	cycle	cycle	NOUN
ejpam-4574	76	19	,	,	PUNCT
ejpam-4574	76	20	is	be	AUX
ejpam-4574	76	21	a	a	DET
ejpam-4574	76	22	cycle	cycle	NOUN
ejpam-4574	76	23	such	such	ADJ
ejpam-4574	76	24	that	that	SCONJ
ejpam-4574	76	25	no	no	DET
ejpam-4574	76	26	two	two	NUM
ejpam-4574	76	27	vertices	vertex	NOUN
ejpam-4574	76	28	of	of	ADP
ejpam-4574	76	29	the	the	DET
ejpam-4574	76	30	cycle	cycle	NOUN
ejpam-4574	76	31	are	be	AUX
ejpam-4574	76	32	connected	connect	VERB
ejpam-4574	76	33	by	by	ADP
ejpam-4574	76	34	an	an	DET
ejpam-4574	76	35	edge	edge	NOUN
ejpam-4574	76	36	that	that	PRON
ejpam-4574	76	37	does	do	AUX
ejpam-4574	76	38	not	not	PART
ejpam-4574	76	39	itself	itself	PRON
ejpam-4574	76	40	belong	belong	VERB
ejpam-4574	76	41	to	to	ADP
ejpam-4574	76	42	the	the	DET
ejpam-4574	76	43	cycle	cycle	NOUN
ejpam-4574	76	44	.	.	PUNCT
ejpam-4574	77	1	definition	definition	NOUN
ejpam-4574	77	2	2	2	NUM
ejpam-4574	77	3	.	.	PUNCT
ejpam-4574	77	4	chordal	chordal	NOUN
ejpam-4574	77	5	graph	graph	NOUN
ejpam-4574	77	6	:	:	PUNCT
ejpam-4574	77	7	a	a	DET
ejpam-4574	77	8	chordal	chordal	ADJ
ejpam-4574	77	9	graph	graph	NOUN
ejpam-4574	77	10	is	be	AUX
ejpam-4574	77	11	a	a	DET
ejpam-4574	77	12	graph	graph	NOUN
ejpam-4574	77	13	in	in	ADP
ejpam-4574	77	14	which	which	PRON
ejpam-4574	77	15	all	all	DET
ejpam-4574	77	16	cycles	cycle	NOUN
ejpam-4574	77	17	of	of	ADP
ejpam-4574	77	18	four	four	NUM
ejpam-4574	77	19	or	or	CCONJ
ejpam-4574	77	20	more	more	ADJ
ejpam-4574	77	21	vertices	vertex	NOUN
ejpam-4574	77	22	have	have	VERB
ejpam-4574	77	23	a	a	DET
ejpam-4574	77	24	chord	chord	NOUN
ejpam-4574	77	25	.	.	PUNCT
ejpam-4574	78	1	proposition	proposition	NOUN
ejpam-4574	78	2	1	1	NUM
ejpam-4574	78	3	.	.	PUNCT
ejpam-4574	79	1	(	(	PUNCT
ejpam-4574	79	2	[	[	X
ejpam-4574	79	3	1	1	NUM
ejpam-4574	79	4	]	]	PUNCT
ejpam-4574	79	5	)	)	PUNCT
ejpam-4574	79	6	.	.	PUNCT
ejpam-4574	80	1	for	for	ADP
ejpam-4574	80	2	every	every	DET
ejpam-4574	80	3	chordal	chordal	NOUN
ejpam-4574	80	4	graph	graph	NOUN
ejpam-4574	80	5	g	g	NOUN
ejpam-4574	80	6	,	,	PUNCT
ejpam-4574	80	7	χa(g	χa(g	NOUN
ejpam-4574	80	8	)	)	PUNCT
ejpam-4574	80	9	=	=	SYM
ejpam-4574	80	10	χ(g	χ(g	PROPN
ejpam-4574	80	11	)	)	PUNCT
ejpam-4574	80	12	.	.	PUNCT
ejpam-4574	81	1	in	in	ADP
ejpam-4574	81	2	the	the	DET
ejpam-4574	81	3	following	following	NOUN
ejpam-4574	81	4	theorem	theorem	NOUN
ejpam-4574	81	5	,	,	PUNCT
ejpam-4574	81	6	we	we	PRON
ejpam-4574	81	7	will	will	AUX
ejpam-4574	81	8	determine	determine	VERB
ejpam-4574	81	9	χa(p	χa(p	PROPN
ejpam-4574	81	10	k	k	PROPN
ejpam-4574	81	11	n	n	PROPN
ejpam-4574	81	12	)	)	PUNCT
ejpam-4574	81	13	for	for	ADP
ejpam-4574	81	14	n	n	DET
ejpam-4574	81	15	≥	≥	NOUN
ejpam-4574	81	16	k	k	NOUN
ejpam-4574	82	1	+	+	CCONJ
ejpam-4574	82	2	2	2	X
ejpam-4574	82	3	.	.	X
ejpam-4574	82	4	theorem	theorem	NOUN
ejpam-4574	82	5	1	1	NUM
ejpam-4574	82	6	.	.	PUNCT
ejpam-4574	82	7	for	for	ADP
ejpam-4574	82	8	n	n	PRON
ejpam-4574	82	9	≥	≥	NOUN
ejpam-4574	82	10	k	k	NOUN
ejpam-4574	83	1	+	+	CCONJ
ejpam-4574	83	2	2	2	NUM
ejpam-4574	83	3	,	,	PUNCT
ejpam-4574	83	4	χa(p	χa(p	NOUN
ejpam-4574	83	5	k	k	PROPN
ejpam-4574	83	6	n	n	PROPN
ejpam-4574	83	7	)	)	PUNCT
ejpam-4574	84	1	=	=	PUNCT
ejpam-4574	84	2	k	k	PROPN
ejpam-4574	85	1	+	+	NOUN
ejpam-4574	85	2	1	1	X
ejpam-4574	85	3	.	.	X
ejpam-4574	85	4	proof	proof	NOUN
ejpam-4574	85	5	.	.	PUNCT
ejpam-4574	86	1	observing	observe	VERB
ejpam-4574	86	2	that	that	PRON
ejpam-4574	86	3	for	for	ADP
ejpam-4574	86	4	k	k	PROPN
ejpam-4574	86	5	≥	≥	PROPN
ejpam-4574	86	6	3	3	NUM
ejpam-4574	86	7	,	,	PUNCT
ejpam-4574	86	8	p	p	PROPN
ejpam-4574	86	9	k	k	PROPN
ejpam-4574	86	10	n	n	PROPN
ejpam-4574	86	11	contains	contain	VERB
ejpam-4574	86	12	no	no	DET
ejpam-4574	86	13	cordless	cordless	NOUN
ejpam-4574	86	14	cycle	cycle	NOUN
ejpam-4574	86	15	,	,	PUNCT
ejpam-4574	86	16	and	and	CCONJ
ejpam-4574	86	17	so	so	ADV
ejpam-4574	86	18	by	by	ADP
ejpam-4574	86	19	proposition	proposition	NOUN
ejpam-4574	86	20	1	1	NUM
ejpam-4574	86	21	we	we	PRON
ejpam-4574	86	22	have	have	AUX
ejpam-4574	86	23	χa(p	χa(p	VERB
ejpam-4574	86	24	k	k	PROPN
ejpam-4574	86	25	n	n	NOUN
ejpam-4574	86	26	)	)	PUNCT
ejpam-4574	86	27	=	=	PUNCT
ejpam-4574	86	28	χ(p	χ(p	PROPN
ejpam-4574	86	29	k	k	PROPN
ejpam-4574	86	30	n	n	PROPN
ejpam-4574	86	31	)	)	PUNCT
ejpam-4574	86	32	=	=	PUNCT
ejpam-4574	87	1	k	k	PROPN
ejpam-4574	88	1	+	+	NOUN
ejpam-4574	88	2	1	1	NUM
ejpam-4574	88	3	.	.	X
ejpam-4574	88	4	3	3	NUM
ejpam-4574	88	5	.	.	X
ejpam-4574	88	6	star	star	NOUN
ejpam-4574	88	7	coloring	coloring	NOUN
ejpam-4574	88	8	of	of	ADP
ejpam-4574	88	9	p	p	PROPN
ejpam-4574	88	10	k	k	PROPN
ejpam-4574	88	11	n	n	PROPN
ejpam-4574	88	12	in	in	ADP
ejpam-4574	88	13	this	this	DET
ejpam-4574	88	14	section	section	NOUN
ejpam-4574	88	15	,	,	PUNCT
ejpam-4574	88	16	we	we	PRON
ejpam-4574	88	17	divide	divide	VERB
ejpam-4574	88	18	the	the	DET
ejpam-4574	88	19	vertices	vertex	NOUN
ejpam-4574	88	20	of	of	ADP
ejpam-4574	88	21	paths	path	NOUN
ejpam-4574	88	22	pn	pn	VERB
ejpam-4574	88	23	into	into	ADP
ejpam-4574	88	24	three	three	NUM
ejpam-4574	88	25	partitions	partition	NOUN
ejpam-4574	88	26	prefix	prefix	NOUN
ejpam-4574	88	27	,	,	PUNCT
ejpam-4574	88	28	main	main	ADJ
ejpam-4574	88	29	and	and	CCONJ
ejpam-4574	88	30	suffix	suffix	NOUN
ejpam-4574	88	31	.	.	PUNCT
ejpam-4574	89	1	by	by	ADP
ejpam-4574	89	2	that	that	PRON
ejpam-4574	89	3	,	,	PUNCT
ejpam-4574	89	4	we	we	PRON
ejpam-4574	89	5	were	be	AUX
ejpam-4574	89	6	able	able	ADJ
ejpam-4574	89	7	to	to	PART
ejpam-4574	89	8	determine	determine	VERB
ejpam-4574	89	9	the	the	DET
ejpam-4574	89	10	value	value	NOUN
ejpam-4574	89	11	of	of	ADP
ejpam-4574	89	12	star	star	ADJ
ejpam-4574	89	13	chromatic	chromatic	ADJ
ejpam-4574	89	14	number	number	NOUN
ejpam-4574	89	15	of	of	ADP
ejpam-4574	89	16	p	p	PROPN
ejpam-4574	89	17	k	k	PROPN
ejpam-4574	89	18	n	n	PROPN
ejpam-4574	89	19	.	.	PUNCT
ejpam-4574	90	1	lemma	lemma	PROPN
ejpam-4574	90	2	1	1	NUM
ejpam-4574	90	3	.	.	PUNCT
ejpam-4574	91	1	for	for	ADP
ejpam-4574	91	2	n	n	PRON
ejpam-4574	91	3	≥	≥	NOUN
ejpam-4574	91	4	k	k	NOUN
ejpam-4574	92	1	+	+	CCONJ
ejpam-4574	92	2	1	1	NUM
ejpam-4574	92	3	,	,	PUNCT
ejpam-4574	92	4	χs(p	χs(p	PUNCT
ejpam-4574	92	5	k	k	PROPN
ejpam-4574	92	6	n	n	PROPN
ejpam-4574	92	7	)	)	PUNCT
ejpam-4574	92	8	≤	≤	NOUN
ejpam-4574	92	9	min{2k	min{2k	PUNCT
ejpam-4574	93	1	+	+	CCONJ
ejpam-4574	93	2	1	1	NUM
ejpam-4574	93	3	,	,	PUNCT
ejpam-4574	93	4	⌊n+k+1	⌊n+k+1	ADP
ejpam-4574	93	5	2	2	NUM
ejpam-4574	93	6	⌋	⌋	NOUN
ejpam-4574	93	7	}	}	PUNCT
ejpam-4574	93	8	.	.	PUNCT
ejpam-4574	94	1	proof	proof	NOUN
ejpam-4574	94	2	.	.	PUNCT
ejpam-4574	95	1	define	define	VERB
ejpam-4574	95	2	c	c	NOUN
ejpam-4574	95	3	:	:	PUNCT
ejpam-4574	95	4	v	v	NOUN
ejpam-4574	95	5	(	(	PUNCT
ejpam-4574	95	6	p	p	X
ejpam-4574	95	7	k	k	PROPN
ejpam-4574	95	8	n	n	PROPN
ejpam-4574	95	9	)	)	PUNCT
ejpam-4574	95	10	→	→	SYM
ejpam-4574	95	11	{	{	PUNCT
ejpam-4574	95	12	c0	c0	NOUN
ejpam-4574	95	13	,	,	PUNCT
ejpam-4574	95	14	c1	c1	PROPN
ejpam-4574	95	15	,	,	PUNCT
ejpam-4574	95	16	...	...	PUNCT
ejpam-4574	95	17	,	,	PUNCT
ejpam-4574	95	18	c2k	c2k	X
ejpam-4574	95	19	}	}	PUNCT
ejpam-4574	95	20	by	by	ADP
ejpam-4574	95	21	c(vj	c(vj	PROPN
ejpam-4574	95	22	)	)	PUNCT
ejpam-4574	95	23	=	=	SYM
ejpam-4574	95	24	cj	cj	NOUN
ejpam-4574	95	25	mod	mod	PROPN
ejpam-4574	95	26	(	(	PUNCT
ejpam-4574	95	27	2k+1	2k+1	NUM
ejpam-4574	95	28	)	)	PUNCT
ejpam-4574	95	29	.	.	PUNCT
ejpam-4574	96	1	if	if	SCONJ
ejpam-4574	96	2	va	va	PROPN
ejpam-4574	96	3	−	−	PROPN
ejpam-4574	96	4	vb	vb	NOUN
ejpam-4574	96	5	−	−	ADP
ejpam-4574	96	6	vc	vc	NOUN
ejpam-4574	96	7	−	−	PROPN
ejpam-4574	96	8	vd	vd	NOUN
ejpam-4574	96	9	is	be	AUX
ejpam-4574	96	10	a	a	DET
ejpam-4574	96	11	bicolored	bicolore	VERB
ejpam-4574	96	12	path	path	NOUN
ejpam-4574	96	13	in	in	ADP
ejpam-4574	96	14	p	p	PROPN
ejpam-4574	96	15	k	k	PROPN
ejpam-4574	96	16	n	n	PROPN
ejpam-4574	96	17	,	,	PUNCT
ejpam-4574	96	18	then	then	ADV
ejpam-4574	96	19	2k	2k	NUM
ejpam-4574	96	20	+	+	CCONJ
ejpam-4574	96	21	1	1	NUM
ejpam-4574	96	22	=	=	SYM
ejpam-4574	96	23	dpn(va	dpn(va	NOUN
ejpam-4574	96	24	,	,	PUNCT
ejpam-4574	96	25	vc	vc	NOUN
ejpam-4574	96	26	)	)	PUNCT
ejpam-4574	96	27	≤	≤	NOUN
ejpam-4574	96	28	dpn(va	dpn(va	NOUN
ejpam-4574	96	29	,	,	PUNCT
ejpam-4574	96	30	vb	vb	NOUN
ejpam-4574	96	31	)	)	PUNCT
ejpam-4574	96	32	+	+	CCONJ
ejpam-4574	96	33	dpn(vb	dpn(vb	NOUN
ejpam-4574	96	34	,	,	PUNCT
ejpam-4574	96	35	vc	vc	NOUN
ejpam-4574	96	36	)	)	PUNCT
ejpam-4574	96	37	≤	≤	NOUN
ejpam-4574	96	38	2k	2k	NUM
ejpam-4574	96	39	,	,	PUNCT
ejpam-4574	96	40	a	a	DET
ejpam-4574	96	41	contradiction	contradiction	NOUN
ejpam-4574	96	42	.	.	PUNCT
ejpam-4574	97	1	so	so	ADV
ejpam-4574	97	2	χs(p	χs(p	PUNCT
ejpam-4574	97	3	k	k	PROPN
ejpam-4574	97	4	n	n	PROPN
ejpam-4574	97	5	)	)	PUNCT
ejpam-4574	97	6	≤	≤	NOUN
ejpam-4574	97	7	2k	2k	NOUN
ejpam-4574	97	8	+	+	CCONJ
ejpam-4574	97	9	1.clearly	1.clearly	NUM
ejpam-4574	97	10	,	,	PUNCT
ejpam-4574	97	11	χs(p	χs(p	PUNCT
ejpam-4574	97	12	k	k	PROPN
ejpam-4574	97	13	n	n	PROPN
ejpam-4574	97	14	)	)	PUNCT
ejpam-4574	97	15	=	=	SYM
ejpam-4574	97	16	n	n	PRON
ejpam-4574	97	17	≤	≤	NOUN
ejpam-4574	97	18	⌊n+k+1	⌊n+k+1	ADP
ejpam-4574	97	19	2	2	NUM
ejpam-4574	97	20	⌋	⌋	NOUN
ejpam-4574	97	21	for	for	ADP
ejpam-4574	97	22	n	n	NOUN
ejpam-4574	97	23	=	=	SYM
ejpam-4574	97	24	k	k	PROPN
ejpam-4574	98	1	+	+	PROPN
ejpam-4574	98	2	1	1	X
ejpam-4574	98	3	.	.	PUNCT
ejpam-4574	99	1	and	and	CCONJ
ejpam-4574	99	2	χs(p	χs(p	PRON
ejpam-4574	99	3	k	k	PROPN
ejpam-4574	99	4	n	n	PROPN
ejpam-4574	99	5	)	)	PUNCT
ejpam-4574	99	6	≤	≤	NUM
ejpam-4574	100	1	2k+1	2k+1	NOUN
ejpam-4574	100	2	≤	≤	NUM
ejpam-4574	100	3	⌊n+k+1	⌊n+k+1	ADP
ejpam-4574	100	4	2	2	NUM
ejpam-4574	100	5	⌋	⌋	NOUN
ejpam-4574	100	6	for	for	ADP
ejpam-4574	100	7	n	n	PRON
ejpam-4574	100	8	≥	≥	NOUN
ejpam-4574	100	9	3k+2	3k+2	NUM
ejpam-4574	100	10	.	.	PUNCT
ejpam-4574	101	1	now	now	ADV
ejpam-4574	101	2	for	for	ADP
ejpam-4574	101	3	k+1	k+1	X
ejpam-4574	101	4	<	<	X
ejpam-4574	101	5	n	n	CCONJ
ejpam-4574	101	6	≤	≤	ADV
ejpam-4574	101	7	3k+1	3k+1	NOUN
ejpam-4574	101	8	we	we	PRON
ejpam-4574	101	9	have	have	VERB
ejpam-4574	101	10	two	two	NUM
ejpam-4574	101	11	cases	case	NOUN
ejpam-4574	101	12	:	:	PUNCT
ejpam-4574	101	13	case	case	NOUN
ejpam-4574	101	14	1	1	NUM
ejpam-4574	101	15	.	.	PUNCT
ejpam-4574	102	1	n	n	NOUN
ejpam-4574	102	2	=	=	SYM
ejpam-4574	102	3	k	k	PROPN
ejpam-4574	103	1	+	+	CCONJ
ejpam-4574	103	2	(	(	PUNCT
ejpam-4574	103	3	2i+	2i+	NUM
ejpam-4574	103	4	1	1	NUM
ejpam-4574	103	5	)	)	PUNCT
ejpam-4574	103	6	for	for	ADP
ejpam-4574	103	7	1	1	NUM
ejpam-4574	103	8	≤	≤	NUM
ejpam-4574	103	9	i	i	PRON
ejpam-4574	103	10	≤	≤	PROPN
ejpam-4574	104	1	k.	k.	PROPN
ejpam-4574	104	2	then	then	ADV
ejpam-4574	104	3	χs(p	χs(p	PUNCT
ejpam-4574	104	4	k	k	PROPN
ejpam-4574	104	5	n	n	PROPN
ejpam-4574	104	6	)	)	PUNCT
ejpam-4574	104	7	≤	≤	PUNCT
ejpam-4574	105	1	k	k	NOUN
ejpam-4574	106	1	+	+	PUNCT
ejpam-4574	106	2	i+	i+	PUNCT
ejpam-4574	106	3	1	1	X
ejpam-4574	106	4	=	=	SYM
ejpam-4574	106	5	⌊k+n+1	⌊k+n+1	PROPN
ejpam-4574	106	6	2	2	NUM
ejpam-4574	106	7	⌋.	⌋.	NOUN
ejpam-4574	106	8	to	to	PART
ejpam-4574	106	9	see	see	VERB
ejpam-4574	106	10	that	that	PRON
ejpam-4574	106	11	,	,	PUNCT
ejpam-4574	106	12	define	define	VERB
ejpam-4574	106	13	the	the	DET
ejpam-4574	106	14	(	(	PUNCT
ejpam-4574	106	15	k	k	NOUN
ejpam-4574	106	16	+	+	CCONJ
ejpam-4574	106	17	i	i	PROPN
ejpam-4574	106	18	+	+	NOUN
ejpam-4574	106	19	1)−	1)−	NUM
ejpam-4574	106	20	proper	proper	ADJ
ejpam-4574	106	21	coloring	coloring	NOUN
ejpam-4574	106	22	c	c	NOUN
ejpam-4574	106	23	:	:	PUNCT
ejpam-4574	106	24	v	v	X
ejpam-4574	106	25	(	(	PUNCT
ejpam-4574	106	26	p	p	X
ejpam-4574	106	27	k	k	PROPN
ejpam-4574	106	28	n	n	PROPN
ejpam-4574	106	29	)	)	PUNCT
ejpam-4574	106	30	→	→	SYM
ejpam-4574	106	31	{	{	PUNCT
ejpam-4574	106	32	c0	c0	NOUN
ejpam-4574	106	33	,	,	PUNCT
ejpam-4574	106	34	c1	c1	PROPN
ejpam-4574	106	35	,	,	PUNCT
ejpam-4574	106	36	...	...	PUNCT
ejpam-4574	106	37	,	,	PUNCT
ejpam-4574	106	38	ck+i	ck+i	PROPN
ejpam-4574	106	39	}	}	PUNCT
ejpam-4574	106	40	by	by	ADP
ejpam-4574	106	41	c(vj	c(vj	PROPN
ejpam-4574	106	42	)	)	PUNCT
ejpam-4574	106	43	=	=	SYM
ejpam-4574	107	1	cj	cj	NOUN
ejpam-4574	107	2	mod(k+i+1	mod(k+i+1	NOUN
ejpam-4574	107	3	)	)	PUNCT
ejpam-4574	107	4	,	,	PUNCT
ejpam-4574	108	1	and	and	CCONJ
ejpam-4574	108	2	suppose	suppose	VERB
ejpam-4574	108	3	that	that	SCONJ
ejpam-4574	108	4	va	va	PROPN
ejpam-4574	108	5	−	−	PROPN
ejpam-4574	108	6	vb	vb	NOUN
ejpam-4574	108	7	−	−	ADP
ejpam-4574	108	8	vc	vc	NOUN
ejpam-4574	108	9	−	−	PROPN
ejpam-4574	108	10	vd	vd	NOUN
ejpam-4574	108	11	is	be	AUX
ejpam-4574	108	12	a	a	DET
ejpam-4574	108	13	bicolored	bicolore	VERB
ejpam-4574	108	14	path	path	NOUN
ejpam-4574	108	15	in	in	ADP
ejpam-4574	108	16	p	p	PROPN
ejpam-4574	108	17	k	k	PROPN
ejpam-4574	108	18	n	n	PROPN
ejpam-4574	108	19	where	where	SCONJ
ejpam-4574	108	20	a	a	DET
ejpam-4574	108	21	<	<	X
ejpam-4574	108	22	b	b	X
ejpam-4574	108	23	<	<	X
ejpam-4574	108	24	c	c	X
ejpam-4574	108	25	<	<	X
ejpam-4574	108	26	d	d	PROPN
ejpam-4574	108	27	,	,	PUNCT
ejpam-4574	108	28	then	then	ADV
ejpam-4574	108	29	dpn	dpn	PROPN
ejpam-4574	108	30	(	(	PUNCT
ejpam-4574	108	31	va	va	NOUN
ejpam-4574	108	32	,	,	PUNCT
ejpam-4574	108	33	vd	vd	NOUN
ejpam-4574	108	34	)	)	PUNCT
ejpam-4574	108	35	=	=	SYM
ejpam-4574	108	36	dpn	dpn	PROPN
ejpam-4574	108	37	(	(	PUNCT
ejpam-4574	108	38	va	va	NOUN
ejpam-4574	108	39	,	,	PUNCT
ejpam-4574	108	40	vc	vc	PROPN
ejpam-4574	108	41	)	)	PUNCT
ejpam-4574	108	42	+	+	NUM
ejpam-4574	109	1	dpn(vb	dpn(vb	NOUN
ejpam-4574	109	2	,	,	PUNCT
ejpam-4574	109	3	vd)−	vd)−	PROPN
ejpam-4574	109	4	dpn(vb	dpn(vb	NOUN
ejpam-4574	109	5	,	,	PUNCT
ejpam-4574	109	6	vc	vc	NOUN
ejpam-4574	109	7	)	)	PUNCT
ejpam-4574	109	8	≤	≤	NOUN
ejpam-4574	110	1	k	k	NOUN
ejpam-4574	111	1	+	+	PUNCT
ejpam-4574	111	2	2i	2i	NUM
ejpam-4574	111	3	=	=	SYM
ejpam-4574	111	4	n−	n−	NOUN
ejpam-4574	111	5	1	1	NUM
ejpam-4574	111	6	.	.	PUNCT
ejpam-4574	112	1	so	so	ADV
ejpam-4574	112	2	,	,	PUNCT
ejpam-4574	112	3	dpn	dpn	PROPN
ejpam-4574	112	4	(	(	PUNCT
ejpam-4574	112	5	vb	vb	NOUN
ejpam-4574	112	6	,	,	PUNCT
ejpam-4574	112	7	vc	vc	NOUN
ejpam-4574	112	8	)	)	PUNCT
ejpam-4574	112	9	≥	≥	NOUN
ejpam-4574	112	10	k	k	NOUN
ejpam-4574	113	1	+	+	CCONJ
ejpam-4574	113	2	2	2	NUM
ejpam-4574	113	3	,	,	PUNCT
ejpam-4574	113	4	a	a	DET
ejpam-4574	113	5	contradiction	contradiction	NOUN
ejpam-4574	113	6	.	.	PUNCT
ejpam-4574	114	1	case	case	NOUN
ejpam-4574	114	2	2	2	NUM
ejpam-4574	114	3	.	.	PUNCT
ejpam-4574	115	1	n	n	NOUN
ejpam-4574	115	2	=	=	SYM
ejpam-4574	115	3	k	k	PROPN
ejpam-4574	116	1	+	+	X
ejpam-4574	116	2	(	(	PUNCT
ejpam-4574	116	3	2i	2i	NUM
ejpam-4574	116	4	)	)	PUNCT
ejpam-4574	116	5	for	for	ADP
ejpam-4574	116	6	1	1	NUM
ejpam-4574	116	7	≤	≤	NUM
ejpam-4574	116	8	i	i	PRON
ejpam-4574	116	9	≤	≤	PROPN
ejpam-4574	117	1	k.	k.	PROPN
ejpam-4574	118	1	then	then	ADV
ejpam-4574	118	2	χs(p	χs(p	PUNCT
ejpam-4574	118	3	k	k	PROPN
ejpam-4574	118	4	n	n	PROPN
ejpam-4574	118	5	)	)	PUNCT
ejpam-4574	118	6	≤	≤	PUNCT
ejpam-4574	119	1	k	k	NOUN
ejpam-4574	120	1	+	+	CCONJ
ejpam-4574	120	2	i	i	NOUN
ejpam-4574	120	3	=	=	SYM
ejpam-4574	120	4	⌊k+n+1	⌊k+n+1	PROPN
ejpam-4574	120	5	2	2	NUM
ejpam-4574	120	6	⌋.	⌋.	NOUN
ejpam-4574	120	7	to	to	PART
ejpam-4574	120	8	see	see	VERB
ejpam-4574	120	9	that	that	PRON
ejpam-4574	120	10	,	,	PUNCT
ejpam-4574	120	11	define	define	VERB
ejpam-4574	120	12	the	the	DET
ejpam-4574	120	13	(	(	PUNCT
ejpam-4574	120	14	k	k	PROPN
ejpam-4574	120	15	+	+	PROPN
ejpam-4574	120	16	i)−	i)−	ADJ
ejpam-4574	120	17	proper	proper	ADJ
ejpam-4574	120	18	coloring	coloring	NOUN
ejpam-4574	120	19	c	c	NOUN
ejpam-4574	120	20	:	:	PUNCT
ejpam-4574	120	21	v	v	X
ejpam-4574	120	22	(	(	PUNCT
ejpam-4574	120	23	p	p	X
ejpam-4574	120	24	k	k	PROPN
ejpam-4574	120	25	n	n	PROPN
ejpam-4574	120	26	)	)	PUNCT
ejpam-4574	120	27	→	→	SYM
ejpam-4574	120	28	{	{	PUNCT
ejpam-4574	120	29	c0	c0	NOUN
ejpam-4574	120	30	,	,	PUNCT
ejpam-4574	120	31	c1	c1	PROPN
ejpam-4574	120	32	,	,	PUNCT
ejpam-4574	120	33	...	...	PUNCT
ejpam-4574	120	34	,	,	PUNCT
ejpam-4574	120	35	ck+i−1	ck+i−1	PROPN
ejpam-4574	120	36	}	}	PUNCT
ejpam-4574	120	37	by	by	ADP
ejpam-4574	120	38	c(vj	c(vj	PROPN
ejpam-4574	120	39	)	)	PUNCT
ejpam-4574	120	40	=	=	PUNCT
ejpam-4574	120	41	cj	cj	NOUN
ejpam-4574	120	42	mod(k+i	mod(k+i	NOUN
ejpam-4574	120	43	)	)	PUNCT
ejpam-4574	120	44	,	,	PUNCT
ejpam-4574	120	45	and	and	CCONJ
ejpam-4574	120	46	suppose	suppose	VERB
ejpam-4574	120	47	that	that	SCONJ
ejpam-4574	120	48	va	va	PROPN
ejpam-4574	120	49	−	−	PROPN
ejpam-4574	120	50	vb	vb	NOUN
ejpam-4574	120	51	−	−	ADP
ejpam-4574	120	52	vc	vc	NOUN
ejpam-4574	120	53	−	−	PROPN
ejpam-4574	120	54	vd	vd	NOUN
ejpam-4574	120	55	is	be	AUX
ejpam-4574	120	56	a	a	DET
ejpam-4574	120	57	bicolored	bicolore	VERB
ejpam-4574	120	58	path	path	NOUN
ejpam-4574	120	59	in	in	ADP
ejpam-4574	120	60	p	p	PROPN
ejpam-4574	120	61	k	k	PROPN
ejpam-4574	120	62	n	n	PROPN
ejpam-4574	120	63	.	.	PUNCT
ejpam-4574	121	1	since	since	SCONJ
ejpam-4574	121	2	dpn(va	dpn(va	NOUN
ejpam-4574	121	3	,	,	PUNCT
ejpam-4574	121	4	vd	vd	NOUN
ejpam-4574	121	5	)	)	PUNCT
ejpam-4574	121	6	=	=	SYM
ejpam-4574	121	7	dpn(va	dpn(va	NOUN
ejpam-4574	121	8	,	,	PUNCT
ejpam-4574	121	9	vc)+dpn(vb	vc)+dpn(vb	PROPN
ejpam-4574	121	10	,	,	PUNCT
ejpam-4574	121	11	vd)−dpn(vb	vd)−dpn(vb	PROPN
ejpam-4574	121	12	,	,	PUNCT
ejpam-4574	121	13	vc	vc	NOUN
ejpam-4574	121	14	)	)	PUNCT
ejpam-4574	121	15	≤	≤	PROPN
ejpam-4574	121	16	k+2i−1	k+2i−1	PROPN
ejpam-4574	121	17	,	,	PUNCT
ejpam-4574	121	18	we	we	PRON
ejpam-4574	121	19	have	have	VERB
ejpam-4574	121	20	dpn(vb	dpn(vb	NOUN
ejpam-4574	121	21	,	,	PUNCT
ejpam-4574	121	22	vc	vc	NOUN
ejpam-4574	121	23	)	)	PUNCT
ejpam-4574	121	24	≥	≥	NOUN
ejpam-4574	121	25	k+1	k+1	NOUN
ejpam-4574	121	26	,	,	PUNCT
ejpam-4574	121	27	a	a	DET
ejpam-4574	121	28	contradiction	contradiction	NOUN
ejpam-4574	121	29	.	.	PUNCT
ejpam-4574	122	1	a.	a.	PROPN
ejpam-4574	122	2	etawi	etawi	PROPN
ejpam-4574	122	3	,	,	PUNCT
ejpam-4574	122	4	m.	m.	PROPN
ejpam-4574	122	5	ghanem	ghanem	PROPN
ejpam-4574	122	6	,	,	PUNCT
ejpam-4574	122	7	h.	h.	PROPN
ejpam-4574	122	8	al	al	PROPN
ejpam-4574	122	9	-	-	PUNCT
ejpam-4574	122	10	ezeh	ezeh	PROPN
ejpam-4574	122	11	/	/	SYM
ejpam-4574	122	12	eur	eur	NOUN
ejpam-4574	122	13	.	.	PUNCT
ejpam-4574	123	1	j.	j.	PROPN
ejpam-4574	123	2	pure	pure	PROPN
ejpam-4574	123	3	appl	appl	PROPN
ejpam-4574	123	4	.	.	PROPN
ejpam-4574	123	5	math	math	PROPN
ejpam-4574	123	6	,	,	PUNCT
ejpam-4574	123	7	15	15	NUM
ejpam-4574	123	8	(	(	PUNCT
ejpam-4574	123	9	4	4	NUM
ejpam-4574	123	10	)	)	PUNCT
ejpam-4574	123	11	(	(	PUNCT
ejpam-4574	123	12	2022	2022	NUM
ejpam-4574	123	13	)	)	PUNCT
ejpam-4574	123	14	,	,	PUNCT
ejpam-4574	123	15	1822	1822	NUM
ejpam-4574	123	16	-	-	SYM
ejpam-4574	123	17	1835	1835	NUM
ejpam-4574	123	18	1826	1826	NUM
ejpam-4574	123	19	lemma	lemma	PROPN
ejpam-4574	123	20	2	2	NUM
ejpam-4574	123	21	.	.	X
ejpam-4574	123	22	for	for	ADP
ejpam-4574	123	23	n	n	NOUN
ejpam-4574	123	24	=	=	SYM
ejpam-4574	123	25	k	k	PROPN
ejpam-4574	124	1	+	+	CCONJ
ejpam-4574	124	2	2i+	2i+	NUM
ejpam-4574	124	3	1	1	NUM
ejpam-4574	124	4	where	where	SCONJ
ejpam-4574	124	5	0	0	NUM
ejpam-4574	124	6	≤	≤	PUNCT
ejpam-4574	124	7	i	i	NOUN
ejpam-4574	124	8	≤	≤	PUNCT
ejpam-4574	125	1	k	k	X
ejpam-4574	125	2	,	,	PUNCT
ejpam-4574	125	3	χs(p	χs(p	PUNCT
ejpam-4574	125	4	k	k	PROPN
ejpam-4574	125	5	n	n	PROPN
ejpam-4574	125	6	)	)	PUNCT
ejpam-4574	125	7	≥	≥	NOUN
ejpam-4574	125	8	k	k	NOUN
ejpam-4574	126	1	+	+	NOUN
ejpam-4574	126	2	i+	i+	NOUN
ejpam-4574	126	3	1	1	NUM
ejpam-4574	126	4	.	.	PUNCT
ejpam-4574	126	5	proof	proof	NOUN
ejpam-4574	126	6	.	.	PUNCT
ejpam-4574	127	1	let	let	VERB
ejpam-4574	127	2	pn	pn	PROPN
ejpam-4574	127	3	denote	denote	VERB
ejpam-4574	127	4	the	the	DET
ejpam-4574	127	5	path	path	NOUN
ejpam-4574	127	6	of	of	ADP
ejpam-4574	127	7	order	order	NOUN
ejpam-4574	127	8	n	n	PRON
ejpam-4574	127	9	with	with	ADP
ejpam-4574	127	10	v	v	PROPN
ejpam-4574	127	11	(	(	PUNCT
ejpam-4574	127	12	pn	pn	NOUN
ejpam-4574	127	13	)	)	PUNCT
ejpam-4574	127	14	=	=	SYM
ejpam-4574	127	15	{	{	PUNCT
ejpam-4574	127	16	pi	pi	NOUN
ejpam-4574	127	17	,	,	PUNCT
ejpam-4574	127	18	pi−1	pi−1	PROPN
ejpam-4574	127	19	,	,	PUNCT
ejpam-4574	127	20	...	...	PUNCT
ejpam-4574	127	21	,	,	PUNCT
ejpam-4574	127	22	p2	p2	NOUN
ejpam-4574	127	23	,	,	PUNCT
ejpam-4574	127	24	p1,v1,v2	p1,v1,v2	NOUN
ejpam-4574	127	25	,	,	PUNCT
ejpam-4574	127	26	...	...	PUNCT
ejpam-4574	127	27	,	,	PUNCT
ejpam-4574	127	28	vk+1	vk+1	ADJ
ejpam-4574	127	29	,	,	PUNCT
ejpam-4574	127	30	s1	s1	NOUN
ejpam-4574	127	31	,	,	PUNCT
ejpam-4574	127	32	s2	s2	PROPN
ejpam-4574	127	33	,	,	PUNCT
ejpam-4574	127	34	...	...	PUNCT
ejpam-4574	127	35	,	,	PUNCT
ejpam-4574	127	36	si	si	X
ejpam-4574	127	37	}	}	PUNCT
ejpam-4574	127	38	,	,	PUNCT
ejpam-4574	127	39	and	and	CCONJ
ejpam-4574	127	40	edge	edge	VERB
ejpam-4574	127	41	set	set	VERB
ejpam-4574	127	42	e(pn	e(pn	NOUN
ejpam-4574	127	43	)	)	PUNCT
ejpam-4574	127	44	=	=	PRON
ejpam-4574	127	45	{	{	PUNCT
ejpam-4574	127	46	pxpx+1	pxpx+1	NOUN
ejpam-4574	127	47	:	:	PUNCT
ejpam-4574	127	48	x	x	SYM
ejpam-4574	127	49	=	=	SYM
ejpam-4574	127	50	1	1	NUM
ejpam-4574	127	51	,	,	PUNCT
ejpam-4574	127	52	2	2	NUM
ejpam-4574	127	53	,	,	PUNCT
ejpam-4574	127	54	...	...	PUNCT
ejpam-4574	127	55	,	,	PUNCT
ejpam-4574	127	56	i−	i−	PROPN
ejpam-4574	127	57	1}∪{vxvx+1	1}∪{vxvx+1	NOUN
ejpam-4574	127	58	:	:	PUNCT
ejpam-4574	127	59	x	x	SYM
ejpam-4574	127	60	=	=	SYM
ejpam-4574	127	61	1	1	NUM
ejpam-4574	127	62	,	,	PUNCT
ejpam-4574	127	63	2	2	NUM
ejpam-4574	127	64	,	,	PUNCT
ejpam-4574	127	65	...	...	PUNCT
ejpam-4574	127	66	,	,	PUNCT
ejpam-4574	127	67	k}∪{sxsx+1	k}∪{sxsx+1	PROPN
ejpam-4574	127	68	:	:	PUNCT
ejpam-4574	127	69	x	x	SYM
ejpam-4574	127	70	=	=	SYM
ejpam-4574	127	71	1	1	NUM
ejpam-4574	127	72	,	,	PUNCT
ejpam-4574	127	73	2	2	NUM
ejpam-4574	127	74	,	,	PUNCT
ejpam-4574	127	75	...	...	PUNCT
ejpam-4574	127	76	,	,	PUNCT
ejpam-4574	127	77	i−	i−	PROPN
ejpam-4574	127	78	1}∪{p1v1	1}∪{p1v1	NOUN
ejpam-4574	127	79	,	,	PUNCT
ejpam-4574	127	80	vk+1s1	vk+1s1	NOUN
ejpam-4574	127	81	}	}	PUNCT
ejpam-4574	127	82	.	.	PUNCT
ejpam-4574	128	1	define	define	VERB
ejpam-4574	128	2	three	three	NUM
ejpam-4574	128	3	induced	induced	ADJ
ejpam-4574	128	4	cliques	clique	NOUN
ejpam-4574	128	5	of	of	ADP
ejpam-4574	128	6	p	p	PROPN
ejpam-4574	128	7	k	k	PROPN
ejpam-4574	128	8	n	n	PROPN
ejpam-4574	128	9	,	,	PUNCT
ejpam-4574	128	10	prefix	prefix	X
ejpam-4574	128	11	(	(	PUNCT
ejpam-4574	128	12	pr(p	pr(p	NOUN
ejpam-4574	128	13	k	k	PROPN
ejpam-4574	128	14	n	n	PROPN
ejpam-4574	128	15	)	)	PUNCT
ejpam-4574	128	16	)	)	PUNCT
ejpam-4574	128	17	,	,	PUNCT
ejpam-4574	128	18	main	main	ADJ
ejpam-4574	128	19	(	(	PUNCT
ejpam-4574	128	20	ma(p	ma(p	NOUN
ejpam-4574	128	21	k	k	PROPN
ejpam-4574	128	22	n	n	PROPN
ejpam-4574	128	23	)	)	PUNCT
ejpam-4574	128	24	)	)	PUNCT
ejpam-4574	128	25	and	and	CCONJ
ejpam-4574	128	26	suffix	suffix	PROPN
ejpam-4574	128	27	(	(	PUNCT
ejpam-4574	128	28	su(p	su(p	X
ejpam-4574	128	29	k	k	PROPN
ejpam-4574	128	30	n	n	PROPN
ejpam-4574	128	31	)	)	PUNCT
ejpam-4574	128	32	)	)	PUNCT
ejpam-4574	128	33	with	with	ADP
ejpam-4574	128	34	vertex	vertex	NOUN
ejpam-4574	128	35	sets	set	VERB
ejpam-4574	128	36	v	v	NOUN
ejpam-4574	128	37	(	(	PUNCT
ejpam-4574	128	38	pr(p	pr(p	NOUN
ejpam-4574	128	39	k	k	PROPN
ejpam-4574	128	40	n	n	PROPN
ejpam-4574	128	41	)	)	PUNCT
ejpam-4574	128	42	)	)	PUNCT
ejpam-4574	129	1	=	=	PRON
ejpam-4574	129	2	{	{	PUNCT
ejpam-4574	129	3	p1	p1	NOUN
ejpam-4574	129	4	,	,	PUNCT
ejpam-4574	129	5	p2	p2	NOUN
ejpam-4574	129	6	,	,	PUNCT
ejpam-4574	129	7	...	...	PUNCT
ejpam-4574	129	8	,	,	PUNCT
ejpam-4574	129	9	pi	pi	NOUN
ejpam-4574	129	10	}	}	PUNCT
ejpam-4574	129	11	,	,	PUNCT
ejpam-4574	129	12	v	v	INTJ
ejpam-4574	129	13	(	(	PUNCT
ejpam-4574	129	14	ma(p	ma(p	NOUN
ejpam-4574	129	15	k	k	PROPN
ejpam-4574	129	16	n	n	PROPN
ejpam-4574	129	17	)	)	PUNCT
ejpam-4574	129	18	)	)	PUNCT
ejpam-4574	130	1	=	=	PRON
ejpam-4574	130	2	{	{	PUNCT
ejpam-4574	130	3	v1	v1	PROPN
ejpam-4574	130	4	,	,	PUNCT
ejpam-4574	130	5	v2	v2	PROPN
ejpam-4574	130	6	,	,	PUNCT
ejpam-4574	130	7	...	...	PUNCT
ejpam-4574	130	8	,	,	PUNCT
ejpam-4574	130	9	vk+1	vk+1	X
ejpam-4574	130	10	}	}	PUNCT
ejpam-4574	130	11	and	and	CCONJ
ejpam-4574	130	12	v	v	X
ejpam-4574	130	13	(	(	PUNCT
ejpam-4574	130	14	su(p	su(p	NOUN
ejpam-4574	130	15	k	k	NOUN
ejpam-4574	130	16	n	n	PROPN
ejpam-4574	130	17	)	)	PUNCT
ejpam-4574	130	18	)	)	PUNCT
ejpam-4574	131	1	=	=	PRON
ejpam-4574	131	2	{	{	PUNCT
ejpam-4574	131	3	s1	s1	NOUN
ejpam-4574	131	4	,	,	PUNCT
ejpam-4574	131	5	s2	s2	PROPN
ejpam-4574	131	6	,	,	PUNCT
ejpam-4574	131	7	...	...	PUNCT
ejpam-4574	131	8	,	,	PUNCT
ejpam-4574	131	9	si	si	X
ejpam-4574	131	10	}	}	PUNCT
ejpam-4574	131	11	.	.	PUNCT
ejpam-4574	132	1	let	let	VERB
ejpam-4574	132	2	m	m	VERB
ejpam-4574	132	3	=	=	PRON
ejpam-4574	132	4	{	{	PUNCT
ejpam-4574	132	5	m1,m2	m1,m2	PROPN
ejpam-4574	132	6	,	,	PUNCT
ejpam-4574	132	7	...	...	PUNCT
ejpam-4574	132	8	,	,	PUNCT
ejpam-4574	132	9	mk+1	mk+1	NOUN
ejpam-4574	132	10	}	}	PUNCT
ejpam-4574	132	11	and	and	CCONJ
ejpam-4574	132	12	n	n	CCONJ
ejpam-4574	132	13	=	=	CCONJ
ejpam-4574	132	14	{	{	PUNCT
ejpam-4574	132	15	n1	n1	PROPN
ejpam-4574	132	16	,	,	PUNCT
ejpam-4574	132	17	n2	n2	NOUN
ejpam-4574	132	18	,	,	PUNCT
ejpam-4574	132	19	...	...	PUNCT
ejpam-4574	132	20	,	,	PUNCT
ejpam-4574	132	21	ni−1	ni−1	PROPN
ejpam-4574	132	22	}	}	PUNCT
ejpam-4574	132	23	be	be	VERB
ejpam-4574	132	24	two	two	NUM
ejpam-4574	132	25	disjoint	disjoint	NOUN
ejpam-4574	132	26	sets	set	NOUN
ejpam-4574	132	27	of	of	ADP
ejpam-4574	132	28	colors	color	NOUN
ejpam-4574	132	29	,	,	PUNCT
ejpam-4574	132	30	and	and	CCONJ
ejpam-4574	132	31	let	let	VERB
ejpam-4574	132	32	c	c	NOUN
ejpam-4574	132	33	:	:	PUNCT
ejpam-4574	132	34	v	v	X
ejpam-4574	132	35	(	(	PUNCT
ejpam-4574	132	36	p	p	X
ejpam-4574	132	37	k	k	PROPN
ejpam-4574	132	38	n	n	PROPN
ejpam-4574	132	39	)	)	PUNCT
ejpam-4574	133	1	→	→	SYM
ejpam-4574	133	2	m	m	AUX
ejpam-4574	133	3	∪n	∪n	AUX
ejpam-4574	133	4	be	be	AUX
ejpam-4574	133	5	a	a	DET
ejpam-4574	133	6	(	(	PUNCT
ejpam-4574	133	7	k	k	NOUN
ejpam-4574	133	8	+	+	CCONJ
ejpam-4574	133	9	i	i	NOUN
ejpam-4574	133	10	)	)	PUNCT
ejpam-4574	133	11	−	−	VERB
ejpam-4574	133	12	proper	proper	ADJ
ejpam-4574	133	13	coloring	coloring	NOUN
ejpam-4574	133	14	with	with	ADP
ejpam-4574	133	15	c(vα	c(vα	PROPN
ejpam-4574	133	16	)	)	PUNCT
ejpam-4574	133	17	=	=	SYM
ejpam-4574	133	18	mα	mα	PROPN
ejpam-4574	133	19	.	.	PUNCT
ejpam-4574	134	1	then	then	ADV
ejpam-4574	134	2	c(pr(p	c(pr(p	VERB
ejpam-4574	134	3	k	k	PROPN
ejpam-4574	134	4	n	n	PROPN
ejpam-4574	134	5	)	)	PUNCT
ejpam-4574	134	6	)	)	PUNCT
ejpam-4574	135	1	∩m	∩m	PROPN
ejpam-4574	135	2	̸=	̸=	PROPN
ejpam-4574	135	3	ϕ	ϕ	PROPN
ejpam-4574	135	4	and	and	CCONJ
ejpam-4574	135	5	c(su(p	c(su(p	PROPN
ejpam-4574	135	6	k	k	PROPN
ejpam-4574	135	7	n	n	PROPN
ejpam-4574	135	8	)	)	PUNCT
ejpam-4574	135	9	)	)	PUNCT
ejpam-4574	136	1	∩m	∩m	PROPN
ejpam-4574	136	2	̸=	̸=	PROPN
ejpam-4574	136	3	ϕ.	ϕ.	NOUN
ejpam-4574	136	4	let	let	VERB
ejpam-4574	136	5	j	j	PROPN
ejpam-4574	136	6	and	and	CCONJ
ejpam-4574	136	7	h	h	PROPN
ejpam-4574	136	8	be	be	VERB
ejpam-4574	136	9	the	the	DET
ejpam-4574	136	10	least	least	ADJ
ejpam-4574	136	11	indices	index	NOUN
ejpam-4574	136	12	where	where	SCONJ
ejpam-4574	136	13	c(pj	c(pj	VERB
ejpam-4574	136	14	)	)	PUNCT
ejpam-4574	136	15	=	=	SYM
ejpam-4574	136	16	mx	mx	PROPN
ejpam-4574	136	17	and	and	CCONJ
ejpam-4574	136	18	c(sh	c(sh	PROPN
ejpam-4574	136	19	)	)	PUNCT
ejpam-4574	137	1	=	=	PUNCT
ejpam-4574	137	2	my	my	PRON
ejpam-4574	137	3	for	for	ADP
ejpam-4574	137	4	some	some	DET
ejpam-4574	137	5	x	x	NOUN
ejpam-4574	137	6	,	,	PUNCT
ejpam-4574	137	7	y	y	PROPN
ejpam-4574	137	8	∈	∈	PROPN
ejpam-4574	137	9	{	{	PUNCT
ejpam-4574	137	10	1	1	NUM
ejpam-4574	137	11	,	,	PUNCT
ejpam-4574	137	12	2	2	NUM
ejpam-4574	137	13	,	,	PUNCT
ejpam-4574	137	14	...	...	PUNCT
ejpam-4574	137	15	,	,	PUNCT
ejpam-4574	137	16	k	k	PROPN
ejpam-4574	137	17	+	+	PROPN
ejpam-4574	137	18	1	1	NUM
ejpam-4574	137	19	}	}	PUNCT
ejpam-4574	137	20	.	.	PUNCT
ejpam-4574	138	1	then	then	ADV
ejpam-4574	138	2	dpn(pj	dpn(pj	VERB
ejpam-4574	138	3	,	,	PUNCT
ejpam-4574	138	4	vx	vx	PROPN
ejpam-4574	138	5	)	)	PUNCT
ejpam-4574	138	6	,	,	PUNCT
ejpam-4574	138	7	dpn(vy	dpn(vy	PROPN
ejpam-4574	138	8	,	,	PUNCT
ejpam-4574	138	9	sh	sh	PROPN
ejpam-4574	138	10	)	)	PUNCT
ejpam-4574	138	11	≥	≥	NOUN
ejpam-4574	138	12	k	k	NOUN
ejpam-4574	139	1	+	+	CCONJ
ejpam-4574	139	2	1	1	NUM
ejpam-4574	139	3	,	,	PUNCT
ejpam-4574	139	4	and	and	CCONJ
ejpam-4574	139	5	hence	hence	ADV
ejpam-4574	139	6	x	x	PUNCT
ejpam-4574	139	7	≥	≥	PROPN
ejpam-4574	140	1	k	k	NOUN
ejpam-4574	140	2	−	−	PROPN
ejpam-4574	141	1	j	j	PROPN
ejpam-4574	142	1	+	+	CCONJ
ejpam-4574	142	2	2	2	NUM
ejpam-4574	142	3	,	,	PUNCT
ejpam-4574	142	4	y	y	PROPN
ejpam-4574	142	5	≤	≤	PROPN
ejpam-4574	142	6	h.	h.	NOUN
ejpam-4574	142	7	claim	claim	NOUN
ejpam-4574	142	8	(	(	PUNCT
ejpam-4574	142	9	1	1	X
ejpam-4574	142	10	)	)	PUNCT
ejpam-4574	142	11	h	h	NOUN
ejpam-4574	142	12	>	>	X
ejpam-4574	142	13	i−	i−	PROPN
ejpam-4574	142	14	j	j	PROPN
ejpam-4574	142	15	+	+	CCONJ
ejpam-4574	142	16	1	1	X
ejpam-4574	142	17	.	.	PUNCT
ejpam-4574	143	1	let	let	VERB
ejpam-4574	143	2	x	x	PRON
ejpam-4574	143	3	,	,	PUNCT
ejpam-4574	143	4	y	y	PROPN
ejpam-4574	143	5	∈	∈	PROPN
ejpam-4574	143	6	{	{	PUNCT
ejpam-4574	143	7	1	1	NUM
ejpam-4574	143	8	,	,	PUNCT
ejpam-4574	143	9	2	2	NUM
ejpam-4574	143	10	,	,	PUNCT
ejpam-4574	143	11	...	...	PUNCT
ejpam-4574	143	12	,	,	PUNCT
ejpam-4574	143	13	k	k	PROPN
ejpam-4574	144	1	+	+	PROPN
ejpam-4574	144	2	1	1	X
ejpam-4574	144	3	}	}	PUNCT
ejpam-4574	144	4	where	where	SCONJ
ejpam-4574	144	5	c(pj	c(pj	X
ejpam-4574	144	6	)	)	PUNCT
ejpam-4574	144	7	=	=	SYM
ejpam-4574	144	8	mx	mx	AUX
ejpam-4574	144	9	.	.	PROPN
ejpam-4574	144	10	suppose	suppose	VERB
ejpam-4574	144	11	that	that	SCONJ
ejpam-4574	144	12	there	there	PRON
ejpam-4574	144	13	exists	exist	VERB
ejpam-4574	144	14	l	l	NOUN
ejpam-4574	144	15	≤	≤	PUNCT
ejpam-4574	145	1	i	i	PRON
ejpam-4574	145	2	−	−	PUNCT
ejpam-4574	146	1	j	j	NOUN
ejpam-4574	147	1	+	+	CCONJ
ejpam-4574	147	2	1	1	NUM
ejpam-4574	147	3	such	such	ADJ
ejpam-4574	147	4	that	that	SCONJ
ejpam-4574	147	5	c(sl	c(sl	NOUN
ejpam-4574	147	6	)	)	PUNCT
ejpam-4574	147	7	=	=	PUNCT
ejpam-4574	148	1	my	my	PRON
ejpam-4574	148	2	,	,	PUNCT
ejpam-4574	148	3	then	then	ADV
ejpam-4574	148	4	k	k	PROPN
ejpam-4574	148	5	+	+	CCONJ
ejpam-4574	148	6	1	1	NUM
ejpam-4574	148	7	≤	≤	NOUN
ejpam-4574	148	8	dpn(vy	dpn(vy	NOUN
ejpam-4574	148	9	,	,	PUNCT
ejpam-4574	148	10	sl	sl	NUM
ejpam-4574	148	11	)	)	PUNCT
ejpam-4574	148	12	=	=	PUNCT
ejpam-4574	148	13	k	k	PROPN
ejpam-4574	149	1	+	+	CCONJ
ejpam-4574	149	2	1	1	NUM
ejpam-4574	149	3	−	−	PROPN
ejpam-4574	149	4	y	y	PROPN
ejpam-4574	149	5	+	+	CCONJ
ejpam-4574	149	6	l	l	NOUN
ejpam-4574	149	7	and	and	CCONJ
ejpam-4574	149	8	hence	hence	ADV
ejpam-4574	149	9	y	y	PROPN
ejpam-4574	149	10	≤	≤	X
ejpam-4574	149	11	l.	l.	PROPN
ejpam-4574	149	12	but	but	CCONJ
ejpam-4574	149	13	dpn(pj	dpn(pj	NOUN
ejpam-4574	149	14	,	,	PUNCT
ejpam-4574	149	15	vy	vy	PROPN
ejpam-4574	149	16	)	)	PUNCT
ejpam-4574	149	17	=	=	PUNCT
ejpam-4574	150	1	j+y−1	j+y−1	ADJ
ejpam-4574	150	2	≤	≤	NOUN
ejpam-4574	150	3	j+	j+	NUM
ejpam-4574	150	4	l−1	l−1	PROPN
ejpam-4574	150	5	≤	≤	ADV
ejpam-4574	150	6	i	i	PROPN
ejpam-4574	150	7	and	and	CCONJ
ejpam-4574	150	8	dpn(vx	dpn(vx	NOUN
ejpam-4574	150	9	,	,	PUNCT
ejpam-4574	150	10	sl	sl	NUM
ejpam-4574	150	11	)	)	PUNCT
ejpam-4574	150	12	=	=	SYM
ejpam-4574	151	1	k+1−x+	k+1−x+	PROPN
ejpam-4574	151	2	l	l	NOUN
ejpam-4574	151	3	≤	≤	PROPN
ejpam-4574	151	4	i	i	PRON
ejpam-4574	151	5	,	,	PUNCT
ejpam-4574	151	6	so	so	ADV
ejpam-4574	151	7	pj−vy−vx−sl	pj−vy−vx−sl	NOUN
ejpam-4574	151	8	is	be	AUX
ejpam-4574	151	9	a	a	DET
ejpam-4574	151	10	bicolored	bicolored	ADJ
ejpam-4574	151	11	path	path	NOUN
ejpam-4574	151	12	in	in	ADP
ejpam-4574	151	13	p	p	PROPN
ejpam-4574	151	14	k	k	PROPN
ejpam-4574	151	15	n	n	PROPN
ejpam-4574	151	16	.	.	PUNCT
ejpam-4574	152	1	hence	hence	ADV
ejpam-4574	152	2	c(sl	c(sl	NUM
ejpam-4574	152	3	)	)	PUNCT
ejpam-4574	152	4	∈	∈	PROPN
ejpam-4574	152	5	n	n	X
ejpam-4574	152	6	for	for	ADP
ejpam-4574	152	7	all	all	DET
ejpam-4574	152	8	l	l	NOUN
ejpam-4574	152	9	=	=	SYM
ejpam-4574	152	10	1	1	NUM
ejpam-4574	152	11	,	,	PUNCT
ejpam-4574	152	12	2	2	NUM
ejpam-4574	152	13	,	,	PUNCT
ejpam-4574	152	14	...	...	PUNCT
ejpam-4574	152	15	,	,	PUNCT
ejpam-4574	152	16	i−	i−	PROPN
ejpam-4574	152	17	j	j	PROPN
ejpam-4574	152	18	+	+	CCONJ
ejpam-4574	152	19	1	1	X
ejpam-4574	152	20	.	.	X
ejpam-4574	153	1	claim	claim	NOUN
ejpam-4574	153	2	(	(	PUNCT
ejpam-4574	153	3	2	2	X
ejpam-4574	153	4	)	)	PUNCT
ejpam-4574	153	5	if	if	SCONJ
ejpam-4574	153	6	a	a	PRON
ejpam-4574	153	7	=	=	X
ejpam-4574	153	8	{	{	PUNCT
ejpam-4574	153	9	c(p1	c(p1	NOUN
ejpam-4574	153	10	)	)	PUNCT
ejpam-4574	153	11	,	,	PUNCT
ejpam-4574	153	12	c(p2	c(p2	NOUN
ejpam-4574	153	13	)	)	PUNCT
ejpam-4574	153	14	,	,	PUNCT
ejpam-4574	153	15	...	...	PUNCT
ejpam-4574	153	16	,	,	PUNCT
ejpam-4574	153	17	c(pj−1	c(pj−1	NOUN
ejpam-4574	153	18	)	)	PUNCT
ejpam-4574	153	19	}	}	PUNCT
ejpam-4574	153	20	and	and	CCONJ
ejpam-4574	153	21	b	b	X
ejpam-4574	153	22	=	=	SYM
ejpam-4574	153	23	{	{	PUNCT
ejpam-4574	153	24	c(s1	c(s1	NOUN
ejpam-4574	153	25	)	)	PUNCT
ejpam-4574	153	26	,	,	PUNCT
ejpam-4574	153	27	c(s2	c(s2	PROPN
ejpam-4574	153	28	)	)	PUNCT
ejpam-4574	153	29	,	,	PUNCT
ejpam-4574	153	30	...	...	PUNCT
ejpam-4574	153	31	,	,	PUNCT
ejpam-4574	153	32	c(sh−1	c(sh−1	NOUN
ejpam-4574	153	33	)	)	PUNCT
ejpam-4574	153	34	}	}	PUNCT
ejpam-4574	153	35	,	,	PUNCT
ejpam-4574	153	36	then	then	ADV
ejpam-4574	153	37	a	a	DET
ejpam-4574	153	38	∩b	∩b	NOUN
ejpam-4574	153	39	has	have	VERB
ejpam-4574	153	40	at	at	ADP
ejpam-4574	153	41	most	most	ADJ
ejpam-4574	153	42	h−	h−	ADJ
ejpam-4574	153	43	i+	i+	PRON
ejpam-4574	154	1	j	j	NOUN
ejpam-4574	154	2	−	−	PROPN
ejpam-4574	154	3	2	2	NUM
ejpam-4574	154	4	distinct	distinct	ADJ
ejpam-4574	154	5	colors	color	NOUN
ejpam-4574	154	6	.	.	PUNCT
ejpam-4574	155	1	let	let	VERB
ejpam-4574	155	2	t	t	NOUN
ejpam-4574	155	3	=	=	SYM
ejpam-4574	155	4	h−	h−	PROPN
ejpam-4574	155	5	(	(	PUNCT
ejpam-4574	155	6	i−	i−	PROPN
ejpam-4574	155	7	j	j	PROPN
ejpam-4574	155	8	+	+	PROPN
ejpam-4574	155	9	1	1	NUM
ejpam-4574	155	10	)	)	PUNCT
ejpam-4574	155	11	.	.	PUNCT
ejpam-4574	156	1	if	if	SCONJ
ejpam-4574	156	2	i+	i+	NOUN
ejpam-4574	156	3	t	t	X
ejpam-4574	156	4	<	<	X
ejpam-4574	156	5	k	k	X
ejpam-4574	156	6	,	,	PUNCT
ejpam-4574	156	7	then	then	ADV
ejpam-4574	156	8	h+j−1	h+j−1	PROPN
ejpam-4574	156	9	≤	≤	PUNCT
ejpam-4574	156	10	k	k	NOUN
ejpam-4574	156	11	,	,	PUNCT
ejpam-4574	156	12	so	so	ADV
ejpam-4574	156	13	y+j−1	y+j−1	NOUN
ejpam-4574	156	14	≤	≤	NOUN
ejpam-4574	156	15	k.	k.	INTJ
ejpam-4574	157	1	since	since	SCONJ
ejpam-4574	157	2	dpn(pj	dpn(pj	NOUN
ejpam-4574	157	3	,	,	PUNCT
ejpam-4574	157	4	vy	vy	PROPN
ejpam-4574	157	5	)	)	PUNCT
ejpam-4574	157	6	=	=	PUNCT
ejpam-4574	158	1	j+y−1	j+y−1	ADJ
ejpam-4574	158	2	≤	≤	PUNCT
ejpam-4574	158	3	j+h−1	j+h−1	PROPN
ejpam-4574	158	4	≤	≤	NOUN
ejpam-4574	159	1	j+(k−j+1)−1	j+(k−j+1)−1	NOUN
ejpam-4574	160	1	=	=	SYM
ejpam-4574	160	2	k	k	PROPN
ejpam-4574	160	3	and	and	CCONJ
ejpam-4574	160	4	dpn(vx	dpn(vx	PROPN
ejpam-4574	160	5	,	,	PUNCT
ejpam-4574	160	6	sh	sh	PROPN
ejpam-4574	160	7	)	)	PUNCT
ejpam-4574	160	8	=	=	SYM
ejpam-4574	161	1	(	(	PUNCT
ejpam-4574	161	2	k	k	PROPN
ejpam-4574	161	3	+	+	SYM
ejpam-4574	161	4	1−	1−	NUM
ejpam-4574	161	5	x	x	NOUN
ejpam-4574	161	6	)	)	PUNCT
ejpam-4574	162	1	+	+	CCONJ
ejpam-4574	162	2	h	h	NOUN
ejpam-4574	162	3	≤	≤	NOUN
ejpam-4574	162	4	(	(	PUNCT
ejpam-4574	162	5	k	k	PROPN
ejpam-4574	162	6	+	+	PROPN
ejpam-4574	162	7	1−	1−	NUM
ejpam-4574	162	8	(	(	PUNCT
ejpam-4574	162	9	k	k	NOUN
ejpam-4574	162	10	−	−	PROPN
ejpam-4574	162	11	j	j	PROPN
ejpam-4574	162	12	+	+	CCONJ
ejpam-4574	162	13	2	2	NUM
ejpam-4574	162	14	)	)	PUNCT
ejpam-4574	162	15	)	)	PUNCT
ejpam-4574	163	1	+	+	CCONJ
ejpam-4574	163	2	(	(	PUNCT
ejpam-4574	163	3	t+	t+	NOUN
ejpam-4574	163	4	i−	i−	PROPN
ejpam-4574	163	5	j	j	PROPN
ejpam-4574	163	6	+	+	CCONJ
ejpam-4574	163	7	1	1	X
ejpam-4574	163	8	)	)	PUNCT
ejpam-4574	163	9	=	=	VERB
ejpam-4574	163	10	t+	t+	PUNCT
ejpam-4574	164	1	i	i	PRON
ejpam-4574	164	2	<	<	X
ejpam-4574	164	3	k	k	X
ejpam-4574	164	4	,	,	PUNCT
ejpam-4574	164	5	we	we	PRON
ejpam-4574	164	6	have	have	VERB
ejpam-4574	164	7	a	a	DET
ejpam-4574	164	8	bicolored	bicolore	VERB
ejpam-4574	164	9	path	path	NOUN
ejpam-4574	164	10	pj	pj	PROPN
ejpam-4574	164	11	−	−	PROPN
ejpam-4574	165	1	vy	vy	PROPN
ejpam-4574	165	2	−	−	PROPN
ejpam-4574	165	3	vx	vx	PROPN
ejpam-4574	165	4	−	−	PROPN
ejpam-4574	165	5	sh	sh	PROPN
ejpam-4574	165	6	in	in	ADP
ejpam-4574	165	7	p	p	PROPN
ejpam-4574	165	8	k	k	PROPN
ejpam-4574	165	9	n	n	PROPN
ejpam-4574	165	10	.	.	PUNCT
ejpam-4574	166	1	so	so	ADV
ejpam-4574	166	2	i+	i+	NOUN
ejpam-4574	166	3	t	t	PROPN
ejpam-4574	166	4	≥	≥	PROPN
ejpam-4574	166	5	k.	k.	PROPN
ejpam-4574	167	1	now	now	ADV
ejpam-4574	167	2	suppose	suppose	VERB
ejpam-4574	167	3	that	that	SCONJ
ejpam-4574	167	4	x	x	PRON
ejpam-4574	167	5	≥	≥	VERB
ejpam-4574	167	6	i	i	NOUN
ejpam-4574	167	7	+	+	NOUN
ejpam-4574	167	8	t	t	NOUN
ejpam-4574	167	9	+	+	CCONJ
ejpam-4574	167	10	2	2	NUM
ejpam-4574	167	11	−	−	PROPN
ejpam-4574	167	12	j	j	PROPN
ejpam-4574	167	13	and	and	CCONJ
ejpam-4574	167	14	y	y	PROPN
ejpam-4574	167	15	≤	≤	PROPN
ejpam-4574	168	1	k	k	PROPN
ejpam-4574	168	2	+	+	CCONJ
ejpam-4574	168	3	1	1	NUM
ejpam-4574	168	4	−	−	PROPN
ejpam-4574	168	5	j	j	NOUN
ejpam-4574	168	6	,	,	PUNCT
ejpam-4574	168	7	then	then	ADV
ejpam-4574	168	8	pj	pj	PROPN
ejpam-4574	168	9	−	−	PROPN
ejpam-4574	169	1	vy	vy	PROPN
ejpam-4574	169	2	−	−	PROPN
ejpam-4574	170	1	vx	vx	PROPN
ejpam-4574	170	2	−	−	PROPN
ejpam-4574	170	3	sh	sh	PROPN
ejpam-4574	170	4	is	be	AUX
ejpam-4574	170	5	a	a	DET
ejpam-4574	170	6	bicolored	bicolored	ADJ
ejpam-4574	170	7	path	path	NOUN
ejpam-4574	170	8	in	in	ADP
ejpam-4574	170	9	p	p	PROPN
ejpam-4574	170	10	k	k	PROPN
ejpam-4574	170	11	n	n	PROPN
ejpam-4574	170	12	.	.	PUNCT
ejpam-4574	171	1	since	since	SCONJ
ejpam-4574	171	2	x	x	X
ejpam-4574	171	3	≥	≥	PROPN
ejpam-4574	172	1	k	k	NOUN
ejpam-4574	172	2	−	−	PROPN
ejpam-4574	173	1	j	j	PROPN
ejpam-4574	173	2	+	+	CCONJ
ejpam-4574	173	3	2	2	NUM
ejpam-4574	173	4	and	and	CCONJ
ejpam-4574	173	5	y	y	PROPN
ejpam-4574	173	6	≤	≤	PROPN
ejpam-4574	173	7	h	h	NOUN
ejpam-4574	173	8	=	=	SYM
ejpam-4574	173	9	t	t	PROPN
ejpam-4574	174	1	+	+	CCONJ
ejpam-4574	174	2	i	i	PRON
ejpam-4574	174	3	−	−	VERB
ejpam-4574	175	1	j	j	PROPN
ejpam-4574	176	1	+	+	CCONJ
ejpam-4574	176	2	1	1	NUM
ejpam-4574	176	3	,	,	PUNCT
ejpam-4574	176	4	either	either	CCONJ
ejpam-4574	176	5	x	x	SYM
ejpam-4574	176	6	or	or	CCONJ
ejpam-4574	176	7	y	y	PROPN
ejpam-4574	176	8	∈	∈	PROPN
ejpam-4574	176	9	{	{	PUNCT
ejpam-4574	177	1	k	k	PROPN
ejpam-4574	177	2	+	+	NUM
ejpam-4574	177	3	2−	2−	NUM
ejpam-4574	177	4	j	j	NOUN
ejpam-4574	177	5	,	,	PUNCT
ejpam-4574	177	6	k	k	PROPN
ejpam-4574	178	1	+	+	CCONJ
ejpam-4574	178	2	3−	3−	PROPN
ejpam-4574	178	3	j	j	PROPN
ejpam-4574	178	4	,	,	PUNCT
ejpam-4574	178	5	...	...	PUNCT
ejpam-4574	178	6	,	,	PUNCT
ejpam-4574	178	7	i+	i+	NUM
ejpam-4574	178	8	t+	t+	NOUN
ejpam-4574	178	9	1−	1−	NUM
ejpam-4574	178	10	j	j	NOUN
ejpam-4574	178	11	}	}	PUNCT
ejpam-4574	178	12	.	.	PUNCT
ejpam-4574	179	1	case	case	NOUN
ejpam-4574	179	2	1	1	NUM
ejpam-4574	179	3	.	.	PUNCT
ejpam-4574	180	1	k	k	PROPN
ejpam-4574	181	1	+	+	NUM
ejpam-4574	181	2	2−	2−	NUM
ejpam-4574	181	3	j	j	PROPN
ejpam-4574	181	4	≤	≤	PROPN
ejpam-4574	181	5	y	y	PROPN
ejpam-4574	181	6	≤	≤	PROPN
ejpam-4574	181	7	i+	i+	NUM
ejpam-4574	181	8	t+	t+	NUM
ejpam-4574	181	9	1−	1−	NUM
ejpam-4574	181	10	j.	j.	PROPN
ejpam-4574	181	11	let	let	VERB
ejpam-4574	181	12	y	y	PROPN
ejpam-4574	181	13	=	=	PUNCT
ejpam-4574	182	1	k	k	PROPN
ejpam-4574	183	1	+	+	CCONJ
ejpam-4574	183	2	2	2	NUM
ejpam-4574	183	3	−	−	PROPN
ejpam-4574	183	4	j	j	PROPN
ejpam-4574	184	1	+	+	CCONJ
ejpam-4574	184	2	w	w	NOUN
ejpam-4574	184	3	for	for	ADP
ejpam-4574	184	4	0	0	NUM
ejpam-4574	184	5	≤	≤	NOUN
ejpam-4574	184	6	w	w	NOUN
ejpam-4574	184	7	≤	≤	PUNCT
ejpam-4574	185	1	i	i	PRON
ejpam-4574	185	2	+	+	PROPN
ejpam-4574	185	3	t	t	PROPN
ejpam-4574	185	4	−	−	PROPN
ejpam-4574	185	5	k	k	NOUN
ejpam-4574	185	6	−	−	PROPN
ejpam-4574	185	7	1	1	NUM
ejpam-4574	185	8	and	and	CCONJ
ejpam-4574	185	9	let	let	VERB
ejpam-4574	185	10	w1	w1	PROPN
ejpam-4574	185	11	∈	∈	PROPN
ejpam-4574	185	12	{	{	PUNCT
ejpam-4574	185	13	1	1	NUM
ejpam-4574	185	14	,	,	PUNCT
ejpam-4574	185	15	2	2	NUM
ejpam-4574	185	16	,	,	PUNCT
ejpam-4574	185	17	...	...	PUNCT
ejpam-4574	185	18	,	,	PUNCT
ejpam-4574	185	19	j	j	PROPN
ejpam-4574	185	20	−	−	PROPN
ejpam-4574	185	21	1	1	NUM
ejpam-4574	185	22	}	}	PUNCT
ejpam-4574	185	23	,	,	PUNCT
ejpam-4574	185	24	w2	w2	NOUN
ejpam-4574	185	25	∈	∈	PROPN
ejpam-4574	185	26	{	{	PUNCT
ejpam-4574	185	27	1	1	NUM
ejpam-4574	185	28	,	,	PUNCT
ejpam-4574	185	29	2	2	NUM
ejpam-4574	185	30	,	,	PUNCT
ejpam-4574	185	31	...	...	PUNCT
ejpam-4574	185	32	,	,	PUNCT
ejpam-4574	185	33	h	h	NOUN
ejpam-4574	185	34	−	−	NOUN
ejpam-4574	185	35	1	1	NUM
ejpam-4574	185	36	}	}	PUNCT
ejpam-4574	185	37	.	.	PUNCT
ejpam-4574	186	1	clearly	clearly	ADV
ejpam-4574	186	2	,	,	PUNCT
ejpam-4574	186	3	c(pw1	c(pw1	VERB
ejpam-4574	186	4	)	)	PUNCT
ejpam-4574	186	5	,	,	PUNCT
ejpam-4574	186	6	c(sw2	c(sw2	NOUN
ejpam-4574	186	7	)	)	PUNCT
ejpam-4574	186	8	∈	∈	PROPN
ejpam-4574	186	9	n.	n.	NOUN
ejpam-4574	186	10	if	if	SCONJ
ejpam-4574	186	11	there	there	PRON
ejpam-4574	186	12	exist	exist	VERB
ejpam-4574	186	13	pw1	pw1	NOUN
ejpam-4574	186	14	,	,	PUNCT
ejpam-4574	186	15	sw2	sw2	PROPN
ejpam-4574	186	16	adjacent	adjacent	ADJ
ejpam-4574	186	17	to	to	AUX
ejpam-4574	186	18	vy	vy	VERB
ejpam-4574	186	19	such	such	ADJ
ejpam-4574	186	20	that	that	DET
ejpam-4574	186	21	c(pw1	c(pw1	VERB
ejpam-4574	186	22	)	)	PUNCT
ejpam-4574	186	23	=	=	PUNCT
ejpam-4574	186	24	c(sw2	c(sw2	NOUN
ejpam-4574	186	25	)	)	PUNCT
ejpam-4574	186	26	,	,	PUNCT
ejpam-4574	186	27	then	then	ADV
ejpam-4574	186	28	pw1	pw1	VERB
ejpam-4574	186	29	−	−	PROPN
ejpam-4574	186	30	vy	vy	PROPN
ejpam-4574	186	31	−	−	PROPN
ejpam-4574	186	32	sw2	sw2	PROPN
ejpam-4574	186	33	−	−	PROPN
ejpam-4574	186	34	sh	sh	PROPN
ejpam-4574	186	35	is	be	AUX
ejpam-4574	186	36	a	a	DET
ejpam-4574	186	37	bicolored	bicolored	ADJ
ejpam-4574	186	38	path	path	NOUN
ejpam-4574	186	39	in	in	ADP
ejpam-4574	186	40	p	p	PROPN
ejpam-4574	186	41	k	k	PROPN
ejpam-4574	186	42	n	n	PROPN
ejpam-4574	186	43	.	.	PUNCT
ejpam-4574	187	1	therefore	therefore	ADV
ejpam-4574	187	2	,	,	PUNCT
ejpam-4574	187	3	c(pw1	c(pw1	VERB
ejpam-4574	187	4	)	)	PUNCT
ejpam-4574	187	5	̸=	̸=	PROPN
ejpam-4574	187	6	c(sw2	c(sw2	NOUN
ejpam-4574	187	7	)	)	PUNCT
ejpam-4574	187	8	when	when	SCONJ
ejpam-4574	187	9	pw1	pw1	PROPN
ejpam-4574	187	10	and	and	CCONJ
ejpam-4574	187	11	sw2	sw2	PROPN
ejpam-4574	187	12	are	be	AUX
ejpam-4574	187	13	adjacent	adjacent	ADJ
ejpam-4574	187	14	to	to	PART
ejpam-4574	187	15	vy	vy	VERB
ejpam-4574	187	16	.	.	PUNCT
ejpam-4574	188	1	let	let	VERB
ejpam-4574	188	2	|a	|a	NOUN
ejpam-4574	188	3	∩	∩	NOUN
ejpam-4574	188	4	b|	b|	PROPN
ejpam-4574	188	5	=	=	SYM
ejpam-4574	188	6	t	t	PROPN
ejpam-4574	188	7	+	+	NUM
ejpam-4574	188	8	α	α	NOUN
ejpam-4574	188	9	for	for	ADP
ejpam-4574	188	10	some	some	DET
ejpam-4574	188	11	integer	integer	NOUN
ejpam-4574	188	12	α	α	NOUN
ejpam-4574	188	13	.	.	PUNCT
ejpam-4574	189	1	if	if	SCONJ
ejpam-4574	189	2	a	a	PRON
ejpam-4574	189	3	is	be	AUX
ejpam-4574	189	4	the	the	DET
ejpam-4574	189	5	number	number	NOUN
ejpam-4574	189	6	of	of	ADP
ejpam-4574	189	7	vertices	vertex	NOUN
ejpam-4574	189	8	in	in	ADP
ejpam-4574	189	9	pr(p	pr(p	NOUN
ejpam-4574	189	10	k	k	PROPN
ejpam-4574	189	11	n	n	PROPN
ejpam-4574	189	12	)	)	PUNCT
ejpam-4574	189	13	that	that	PRON
ejpam-4574	189	14	are	be	AUX
ejpam-4574	189	15	adjacent	adjacent	ADJ
ejpam-4574	189	16	to	to	PART
ejpam-4574	189	17	vy	vy	VERB
ejpam-4574	189	18	,	,	PUNCT
ejpam-4574	189	19	then	then	ADV
ejpam-4574	189	20	dpn(vy	dpn(vy	NUM
ejpam-4574	189	21	,	,	PUNCT
ejpam-4574	189	22	pa	pa	PROPN
ejpam-4574	189	23	)	)	PUNCT
ejpam-4574	189	24	=	=	PUNCT
ejpam-4574	190	1	a	a	PRON
ejpam-4574	190	2	+	+	NOUN
ejpam-4574	190	3	y	y	NOUN
ejpam-4574	190	4	−	−	PROPN
ejpam-4574	190	5	1	1	NUM
ejpam-4574	190	6	=	=	SYM
ejpam-4574	190	7	a	a	PRON
ejpam-4574	191	1	+	+	X
ejpam-4574	191	2	(	(	PUNCT
ejpam-4574	191	3	k	k	NOUN
ejpam-4574	191	4	+	+	PROPN
ejpam-4574	191	5	2	2	NUM
ejpam-4574	191	6	−	−	PROPN
ejpam-4574	191	7	j	j	PROPN
ejpam-4574	191	8	+	+	CCONJ
ejpam-4574	191	9	w	w	PROPN
ejpam-4574	191	10	)	)	PUNCT
ejpam-4574	191	11	−	−	PROPN
ejpam-4574	191	12	1	1	NUM
ejpam-4574	191	13	=	=	SYM
ejpam-4574	191	14	k	k	NOUN
ejpam-4574	191	15	,	,	PUNCT
ejpam-4574	191	16	and	and	CCONJ
ejpam-4574	191	17	thus	thus	ADV
ejpam-4574	191	18	a	a	DET
ejpam-4574	191	19	=	=	SYM
ejpam-4574	191	20	j−w−	j−w−	ADP
ejpam-4574	191	21	1	1	NUM
ejpam-4574	191	22	.	.	PUNCT
ejpam-4574	192	1	so	so	ADV
ejpam-4574	192	2	there	there	PRON
ejpam-4574	192	3	exist	exist	VERB
ejpam-4574	192	4	w	w	NOUN
ejpam-4574	192	5	=	=	SYM
ejpam-4574	192	6	j−	j−	PROPN
ejpam-4574	192	7	1−a	1−a	NUM
ejpam-4574	192	8	vertices	vertex	NOUN
ejpam-4574	192	9	that	that	PRON
ejpam-4574	192	10	are	be	AUX
ejpam-4574	192	11	colored	color	VERB
ejpam-4574	192	12	from	from	ADP
ejpam-4574	192	13	the	the	DET
ejpam-4574	192	14	set	set	NOUN
ejpam-4574	192	15	a	a	PRON
ejpam-4574	192	16	and	and	CCONJ
ejpam-4574	192	17	are	be	AUX
ejpam-4574	192	18	not	not	PART
ejpam-4574	192	19	adjacent	adjacent	ADJ
ejpam-4574	192	20	to	to	PART
ejpam-4574	192	21	vy	vy	PROPN
ejpam-4574	192	22	.	.	PUNCT
ejpam-4574	193	1	therefore	therefore	ADV
ejpam-4574	193	2	,	,	PUNCT
ejpam-4574	193	3	there	there	PRON
ejpam-4574	193	4	exist	exist	VERB
ejpam-4574	193	5	at	at	ADV
ejpam-4574	193	6	least	least	ADJ
ejpam-4574	193	7	(	(	PUNCT
ejpam-4574	193	8	t	t	NOUN
ejpam-4574	193	9	+	+	NUM
ejpam-4574	193	10	α	α	X
ejpam-4574	193	11	)	)	PUNCT
ejpam-4574	194	1	−	−	PROPN
ejpam-4574	195	1	w	w	NOUN
ejpam-4574	195	2	vertices	vertex	NOUN
ejpam-4574	195	3	from	from	ADP
ejpam-4574	195	4	pr(p	pr(p	NOUN
ejpam-4574	195	5	k	k	PROPN
ejpam-4574	195	6	n	n	PROPN
ejpam-4574	195	7	)	)	PUNCT
ejpam-4574	195	8	that	that	PRON
ejpam-4574	195	9	are	be	AUX
ejpam-4574	195	10	adjacent	adjacent	ADJ
ejpam-4574	195	11	to	to	PART
ejpam-4574	195	12	vy	vy	VERB
ejpam-4574	195	13	and	and	CCONJ
ejpam-4574	195	14	colored	colored	ADJ
ejpam-4574	195	15	using	use	VERB
ejpam-4574	195	16	colors	color	NOUN
ejpam-4574	195	17	from	from	ADP
ejpam-4574	195	18	a	a	DET
ejpam-4574	195	19	∩b	∩b	NOUN
ejpam-4574	195	20	.	.	PUNCT
ejpam-4574	196	1	also	also	ADV
ejpam-4574	196	2	,	,	PUNCT
ejpam-4574	196	3	if	if	SCONJ
ejpam-4574	196	4	b	b	PROPN
ejpam-4574	196	5	is	be	AUX
ejpam-4574	196	6	the	the	DET
ejpam-4574	196	7	number	number	NOUN
ejpam-4574	196	8	of	of	ADP
ejpam-4574	196	9	vertices	vertex	NOUN
ejpam-4574	196	10	in	in	ADP
ejpam-4574	196	11	su(p	su(p	NOUN
ejpam-4574	196	12	k	k	PROPN
ejpam-4574	196	13	n	n	PROPN
ejpam-4574	196	14	)	)	PUNCT
ejpam-4574	196	15	that	that	PRON
ejpam-4574	196	16	are	be	AUX
ejpam-4574	196	17	adjacent	adjacent	ADJ
ejpam-4574	196	18	to	to	PART
ejpam-4574	196	19	vy	vy	VERB
ejpam-4574	196	20	,	,	PUNCT
ejpam-4574	196	21	then	then	ADV
ejpam-4574	196	22	k	k	PROPN
ejpam-4574	196	23	=	=	SYM
ejpam-4574	196	24	dpn(vy	dpn(vy	PROPN
ejpam-4574	196	25	,	,	PUNCT
ejpam-4574	196	26	sb	sb	NOUN
ejpam-4574	196	27	)	)	PUNCT
ejpam-4574	196	28	.	.	PUNCT
ejpam-4574	197	1	so	so	ADV
ejpam-4574	197	2	,	,	PUNCT
ejpam-4574	197	3	b	b	X
ejpam-4574	198	1	=	=	SYM
ejpam-4574	198	2	k	k	PROPN
ejpam-4574	199	1	−	−	PROPN
ejpam-4574	200	1	j	j	PROPN
ejpam-4574	201	1	+	+	CCONJ
ejpam-4574	201	2	1	1	NUM
ejpam-4574	201	3	+	+	CCONJ
ejpam-4574	201	4	w	w	NOUN
ejpam-4574	201	5	,	,	PUNCT
ejpam-4574	201	6	and	and	CCONJ
ejpam-4574	201	7	thus	thus	ADV
ejpam-4574	201	8	the	the	DET
ejpam-4574	201	9	number	number	NOUN
ejpam-4574	201	10	of	of	ADP
ejpam-4574	201	11	vertices	vertex	NOUN
ejpam-4574	201	12	in	in	ADP
ejpam-4574	201	13	su(p	su(p	NOUN
ejpam-4574	201	14	k	k	PROPN
ejpam-4574	201	15	n	n	CCONJ
ejpam-4574	201	16	)	)	PUNCT
ejpam-4574	201	17	colored	color	VERB
ejpam-4574	201	18	from	from	ADP
ejpam-4574	201	19	the	the	DET
ejpam-4574	201	20	set	set	PROPN
ejpam-4574	201	21	b	b	PROPN
ejpam-4574	201	22	and	and	CCONJ
ejpam-4574	201	23	are	be	AUX
ejpam-4574	201	24	not	not	PART
ejpam-4574	201	25	adjacent	adjacent	ADJ
ejpam-4574	201	26	to	to	ADP
ejpam-4574	201	27	vy	vy	VERB
ejpam-4574	201	28	is	be	AUX
ejpam-4574	201	29	(	(	PUNCT
ejpam-4574	201	30	h	h	NOUN
ejpam-4574	201	31	−	−	NOUN
ejpam-4574	201	32	1	1	NUM
ejpam-4574	201	33	)	)	PUNCT
ejpam-4574	202	1	−	−	PROPN
ejpam-4574	202	2	b	b	X
ejpam-4574	202	3	=	=	SYM
ejpam-4574	202	4	t	t	PROPN
ejpam-4574	203	1	+	+	CCONJ
ejpam-4574	203	2	i	i	PRON
ejpam-4574	203	3	−	−	VERB
ejpam-4574	204	1	k	k	NOUN
ejpam-4574	205	1	−	−	PROPN
ejpam-4574	205	2	1	1	NUM
ejpam-4574	205	3	−	−	PROPN
ejpam-4574	205	4	w.	w.	NOUN
ejpam-4574	205	5	then	then	ADV
ejpam-4574	205	6	there	there	PRON
ejpam-4574	205	7	exist	exist	VERB
ejpam-4574	205	8	at	at	ADV
ejpam-4574	205	9	least	least	ADJ
ejpam-4574	205	10	(	(	PUNCT
ejpam-4574	205	11	t	t	NOUN
ejpam-4574	205	12	+	+	NUM
ejpam-4574	205	13	α	α	NOUN
ejpam-4574	205	14	)	)	PUNCT
ejpam-4574	205	15	−	−	PROPN
ejpam-4574	206	1	(	(	PUNCT
ejpam-4574	206	2	t	t	PROPN
ejpam-4574	206	3	+	+	CCONJ
ejpam-4574	207	1	i	i	PRON
ejpam-4574	207	2	−	−	VERB
ejpam-4574	208	1	k	k	INTJ
ejpam-4574	209	1	−	−	PROPN
ejpam-4574	209	2	w	w	NOUN
ejpam-4574	209	3	−	−	NOUN
ejpam-4574	209	4	1	1	NUM
ejpam-4574	209	5	)	)	PUNCT
ejpam-4574	210	1	=	=	SYM
ejpam-4574	210	2	α	α	PROPN
ejpam-4574	211	1	+	+	X
ejpam-4574	211	2	k	k	PROPN
ejpam-4574	212	1	+	+	CCONJ
ejpam-4574	212	2	w	w	PROPN
ejpam-4574	213	1	+	+	ADP
ejpam-4574	213	2	1	1	NUM
ejpam-4574	213	3	−	−	NOUN
ejpam-4574	213	4	i	i	PRON
ejpam-4574	213	5	vertices	vertice	VERB
ejpam-4574	213	6	from	from	ADP
ejpam-4574	213	7	su(p	su(p	NOUN
ejpam-4574	213	8	k	k	PROPN
ejpam-4574	213	9	n	n	PROPN
ejpam-4574	213	10	)	)	PUNCT
ejpam-4574	213	11	that	that	PRON
ejpam-4574	213	12	are	be	AUX
ejpam-4574	213	13	adjacent	adjacent	ADJ
ejpam-4574	213	14	to	to	PART
ejpam-4574	213	15	vy	vy	VERB
ejpam-4574	213	16	and	and	CCONJ
ejpam-4574	213	17	colored	color	VERB
ejpam-4574	213	18	from	from	ADP
ejpam-4574	213	19	a	a	DET
ejpam-4574	213	20	∩	∩	ADJ
ejpam-4574	213	21	b.	b.	NOUN
ejpam-4574	213	22	hence	hence	ADV
ejpam-4574	213	23	the	the	DET
ejpam-4574	213	24	total	total	ADJ
ejpam-4574	213	25	number	number	NOUN
ejpam-4574	213	26	of	of	ADP
ejpam-4574	213	27	distinct	distinct	ADJ
ejpam-4574	213	28	colors	color	NOUN
ejpam-4574	213	29	is	be	AUX
ejpam-4574	213	30	greater	great	ADJ
ejpam-4574	213	31	than	than	ADP
ejpam-4574	213	32	or	or	CCONJ
ejpam-4574	213	33	equal	equal	ADJ
ejpam-4574	213	34	(	(	PUNCT
ejpam-4574	213	35	α	α	NOUN
ejpam-4574	214	1	+	+	X
ejpam-4574	214	2	k	k	X
ejpam-4574	214	3	+	+	CCONJ
ejpam-4574	214	4	1	1	NUM
ejpam-4574	215	1	+	+	CCONJ
ejpam-4574	215	2	w	w	PROPN
ejpam-4574	215	3	−	−	PROPN
ejpam-4574	215	4	i	i	NOUN
ejpam-4574	215	5	)	)	PUNCT
ejpam-4574	216	1	+	+	CCONJ
ejpam-4574	216	2	(	(	PUNCT
ejpam-4574	216	3	t	t	X
ejpam-4574	216	4	+	+	CCONJ
ejpam-4574	216	5	α	α	PROPN
ejpam-4574	216	6	−	−	NOUN
ejpam-4574	216	7	w	w	NOUN
ejpam-4574	216	8	)	)	PUNCT
ejpam-4574	216	9	=	=	SYM
ejpam-4574	216	10	t	t	NOUN
ejpam-4574	217	1	+	+	CCONJ
ejpam-4574	217	2	2α	2α	NOUN
ejpam-4574	217	3	+	+	CCONJ
ejpam-4574	217	4	(	(	PUNCT
ejpam-4574	217	5	k	k	NOUN
ejpam-4574	217	6	−	−	PROPN
ejpam-4574	217	7	i	i	PROPN
ejpam-4574	217	8	)	)	PUNCT
ejpam-4574	218	1	+	+	CCONJ
ejpam-4574	218	2	1	1	NUM
ejpam-4574	218	3	≥	≥	NOUN
ejpam-4574	218	4	1	1	NUM
ejpam-4574	218	5	+	+	NUM
ejpam-4574	218	6	t	t	NOUN
ejpam-4574	218	7	+	+	CCONJ
ejpam-4574	218	8	2α	2α	NOUN
ejpam-4574	218	9	,	,	PUNCT
ejpam-4574	218	10	but	but	CCONJ
ejpam-4574	218	11	,	,	PUNCT
ejpam-4574	218	12	1	1	NUM
ejpam-4574	218	13	+	+	ADJ
ejpam-4574	218	14	t+	t+	NOUN
ejpam-4574	218	15	2α	2α	NOUN
ejpam-4574	218	16	≤	≤	PRON
ejpam-4574	218	17	|(a	|(a	PROPN
ejpam-4574	218	18	∩b)|	∩b)|	PUNCT
ejpam-4574	218	19	=	=	SYM
ejpam-4574	218	20	t+	t+	PUNCT
ejpam-4574	218	21	α	α	NOUN
ejpam-4574	218	22	yields	yield	NOUN
ejpam-4574	218	23	to	to	ADP
ejpam-4574	218	24	α	α	PRON
ejpam-4574	218	25	≤	≤	NUM
ejpam-4574	218	26	−1	−1	NOUN
ejpam-4574	218	27	.	.	PUNCT
ejpam-4574	219	1	case	case	NOUN
ejpam-4574	219	2	2	2	NUM
ejpam-4574	219	3	.	.	NUM
ejpam-4574	219	4	x	x	SYM
ejpam-4574	219	5	≤	≤	ADJ
ejpam-4574	219	6	i+	i+	NUM
ejpam-4574	219	7	t+	t+	NUM
ejpam-4574	219	8	1−	1−	NUM
ejpam-4574	219	9	j	j	PROPN
ejpam-4574	219	10	a.	a.	PROPN
ejpam-4574	219	11	etawi	etawi	PROPN
ejpam-4574	219	12	,	,	PUNCT
ejpam-4574	219	13	m.	m.	PROPN
ejpam-4574	219	14	ghanem	ghanem	PROPN
ejpam-4574	219	15	,	,	PUNCT
ejpam-4574	219	16	h.	h.	PROPN
ejpam-4574	219	17	al	al	PROPN
ejpam-4574	219	18	-	-	PUNCT
ejpam-4574	219	19	ezeh	ezeh	PROPN
ejpam-4574	219	20	/	/	SYM
ejpam-4574	219	21	eur	eur	NOUN
ejpam-4574	219	22	.	.	PUNCT
ejpam-4574	220	1	j.	j.	PROPN
ejpam-4574	220	2	pure	pure	PROPN
ejpam-4574	220	3	appl	appl	PROPN
ejpam-4574	220	4	.	.	PROPN
ejpam-4574	220	5	math	math	PROPN
ejpam-4574	220	6	,	,	PUNCT
ejpam-4574	220	7	15	15	NUM
ejpam-4574	220	8	(	(	PUNCT
ejpam-4574	220	9	4	4	NUM
ejpam-4574	220	10	)	)	PUNCT
ejpam-4574	220	11	(	(	PUNCT
ejpam-4574	220	12	2022	2022	NUM
ejpam-4574	220	13	)	)	PUNCT
ejpam-4574	220	14	,	,	PUNCT
ejpam-4574	220	15	1822	1822	NUM
ejpam-4574	220	16	-	-	SYM
ejpam-4574	220	17	1835	1835	NUM
ejpam-4574	220	18	1827	1827	NUM
ejpam-4574	220	19	if	if	SCONJ
ejpam-4574	220	20	we	we	PRON
ejpam-4574	220	21	mimic	mimic	VERB
ejpam-4574	220	22	the	the	DET
ejpam-4574	220	23	proof	proof	NOUN
ejpam-4574	220	24	of	of	ADP
ejpam-4574	220	25	case	case	NOUN
ejpam-4574	220	26	(	(	PUNCT
ejpam-4574	220	27	1	1	NUM
ejpam-4574	220	28	)	)	PUNCT
ejpam-4574	220	29	,	,	PUNCT
ejpam-4574	220	30	we	we	PRON
ejpam-4574	220	31	get	get	VERB
ejpam-4574	220	32	|a∩b|	|a∩b|	NOUN
ejpam-4574	220	33	≤	≤	NOUN
ejpam-4574	220	34	t−	t−	ADP
ejpam-4574	220	35	1.so	1.so	NUM
ejpam-4574	220	36	the	the	DET
ejpam-4574	220	37	number	number	NOUN
ejpam-4574	220	38	of	of	ADP
ejpam-4574	220	39	distinct	distinct	ADJ
ejpam-4574	220	40	colors	color	NOUN
ejpam-4574	220	41	to	to	PART
ejpam-4574	220	42	color	color	VERB
ejpam-4574	220	43	su(p	su(p	NOUN
ejpam-4574	220	44	k	k	PROPN
ejpam-4574	220	45	n	n	PROPN
ejpam-4574	220	46	)	)	PUNCT
ejpam-4574	220	47	and	and	CCONJ
ejpam-4574	220	48	pr(p	pr(p	VERB
ejpam-4574	220	49	k	k	PROPN
ejpam-4574	220	50	n	n	ADJ
ejpam-4574	220	51	)	)	PUNCT
ejpam-4574	220	52	from	from	ADP
ejpam-4574	220	53	n	n	PROPN
ejpam-4574	220	54	is	be	AUX
ejpam-4574	220	55	|a|+	|a|+	VERB
ejpam-4574	220	56	|b|	|b|	PRON
ejpam-4574	220	57	−	−	PROPN
ejpam-4574	220	58	|a∩b|	|a∩b|	NOUN
ejpam-4574	220	59	≥	≥	NOUN
ejpam-4574	220	60	(	(	PUNCT
ejpam-4574	220	61	j−	j−	PROPN
ejpam-4574	220	62	1)+	1)+	NUM
ejpam-4574	220	63	(	(	PUNCT
ejpam-4574	220	64	h−	h−	PROPN
ejpam-4574	220	65	1)−	1)−	PROPN
ejpam-4574	220	66	(	(	PUNCT
ejpam-4574	220	67	t−	t−	PROPN
ejpam-4574	220	68	1	1	NUM
ejpam-4574	220	69	)	)	PUNCT
ejpam-4574	220	70	=	=	NOUN
ejpam-4574	221	1	i	i	PRON
ejpam-4574	221	2	colors	color	VERB
ejpam-4574	221	3	.	.	PUNCT
ejpam-4574	222	1	therefore	therefore	ADV
ejpam-4574	222	2	,	,	PUNCT
ejpam-4574	222	3	for	for	ADP
ejpam-4574	222	4	n	n	NOUN
ejpam-4574	222	5	=	=	SYM
ejpam-4574	222	6	k	k	PROPN
ejpam-4574	222	7	+	+	CCONJ
ejpam-4574	222	8	2i+	2i+	NUM
ejpam-4574	222	9	1	1	NUM
ejpam-4574	222	10	,	,	PUNCT
ejpam-4574	222	11	χs(p	χs(p	PRON
ejpam-4574	222	12	k	k	PROPN
ejpam-4574	222	13	n	n	PROPN
ejpam-4574	222	14	)	)	PUNCT
ejpam-4574	222	15	≥	≥	NOUN
ejpam-4574	222	16	k	k	NOUN
ejpam-4574	223	1	+	+	NOUN
ejpam-4574	223	2	i+	i+	NUM
ejpam-4574	223	3	1	1	NUM
ejpam-4574	223	4	.	.	PUNCT
ejpam-4574	223	5	example	example	NOUN
ejpam-4574	223	6	2	2	NUM
ejpam-4574	223	7	.	.	X
ejpam-4574	223	8	figure	figure	NOUN
ejpam-4574	223	9	2	2	NUM
ejpam-4574	223	10	shows	show	VERB
ejpam-4574	223	11	an	an	DET
ejpam-4574	223	12	example	example	NOUN
ejpam-4574	223	13	of	of	ADP
ejpam-4574	223	14	the	the	DET
ejpam-4574	223	15	cases	case	NOUN
ejpam-4574	223	16	in	in	ADP
ejpam-4574	223	17	lemma	lemma	PROPN
ejpam-4574	223	18	2	2	NUM
ejpam-4574	223	19	.	.	PUNCT
ejpam-4574	224	1	let	let	VERB
ejpam-4574	224	2	m	m	VERB
ejpam-4574	224	3	=	=	SYM
ejpam-4574	224	4	{	{	PUNCT
ejpam-4574	224	5	k+2−	k+2−	PROPN
ejpam-4574	224	6	j	j	PROPN
ejpam-4574	224	7	,	,	PUNCT
ejpam-4574	224	8	i+	i+	NUM
ejpam-4574	224	9	t+	t+	NUM
ejpam-4574	224	10	1−	1−	NUM
ejpam-4574	224	11	j	j	NOUN
ejpam-4574	224	12	}	}	PUNCT
ejpam-4574	224	13	.	.	PUNCT
ejpam-4574	225	1	p3	p3	PROPN
ejpam-4574	225	2	p2	p2	PROPN
ejpam-4574	225	3	p1	p1	PROPN
ejpam-4574	225	4	1	1	NUM
ejpam-4574	225	5	v1	v1	NOUN
ejpam-4574	225	6	2	2	NUM
ejpam-4574	225	7	v2	v2	PROPN
ejpam-4574	225	8	3	3	NUM
ejpam-4574	225	9	v3	v3	PROPN
ejpam-4574	225	10	4	4	NUM
ejpam-4574	225	11	v4	v4	NOUN
ejpam-4574	225	12	s1	s1	PROPN
ejpam-4574	225	13	s2	s2	PROPN
ejpam-4574	225	14	s3	s3	PROPN
ejpam-4574	225	15	case	case	NOUN
ejpam-4574	225	16	j	j	PROPN
ejpam-4574	225	17	h	h	NOUN
ejpam-4574	225	18	a	a	DET
ejpam-4574	225	19	b	b	PROPN
ejpam-4574	225	20	path	path	NOUN
ejpam-4574	225	21	p3	p3	PROPN
ejpam-4574	225	22	p2	p2	PROPN
ejpam-4574	225	23	p1	p1	PROPN
ejpam-4574	225	24	1	1	NUM
ejpam-4574	225	25	v1	v1	NOUN
ejpam-4574	225	26	2	2	NUM
ejpam-4574	225	27	v2	v2	PROPN
ejpam-4574	225	28	3	3	NUM
ejpam-4574	225	29	v3	v3	PROPN
ejpam-4574	225	30	4	4	NUM
ejpam-4574	225	31	v4	v4	NOUN
ejpam-4574	225	32	s1	s1	PROPN
ejpam-4574	225	33	s2	s2	PROPN
ejpam-4574	225	34	s3	s3	PROPN
ejpam-4574	225	35	case	case	NOUN
ejpam-4574	225	36	j	j	PROPN
ejpam-4574	225	37	h	h	NOUN
ejpam-4574	225	38	a	a	DET
ejpam-4574	225	39	b	b	PROPN
ejpam-4574	225	40	path	path	NOUN
ejpam-4574	225	41	p3	p3	PROPN
ejpam-4574	225	42	p2	p2	PROPN
ejpam-4574	225	43	p1	p1	PROPN
ejpam-4574	225	44	1	1	NUM
ejpam-4574	225	45	v1	v1	NOUN
ejpam-4574	225	46	2	2	NUM
ejpam-4574	225	47	v2	v2	PROPN
ejpam-4574	225	48	3	3	NUM
ejpam-4574	225	49	v3	v3	PROPN
ejpam-4574	225	50	4	4	NUM
ejpam-4574	225	51	v4	v4	NOUN
ejpam-4574	225	52	s1	s1	PROPN
ejpam-4574	225	53	s2	s2	PROPN
ejpam-4574	225	54	s3	s3	PROPN
ejpam-4574	225	55	case	case	NOUN
ejpam-4574	225	56	j	j	PROPN
ejpam-4574	225	57	h	h	NOUN
ejpam-4574	225	58	a	a	DET
ejpam-4574	225	59	b	b	PROPN
ejpam-4574	225	60	path	path	NOUN
ejpam-4574	225	61	p3	p3	PROPN
ejpam-4574	225	62	p2	p2	PROPN
ejpam-4574	225	63	p1	p1	PROPN
ejpam-4574	225	64	1	1	NUM
ejpam-4574	225	65	v1	v1	NOUN
ejpam-4574	225	66	2	2	NUM
ejpam-4574	225	67	v2	v2	PROPN
ejpam-4574	225	68	3	3	NUM
ejpam-4574	225	69	v3	v3	PROPN
ejpam-4574	225	70	4	4	NUM
ejpam-4574	225	71	v4	v4	NOUN
ejpam-4574	225	72	s1	s1	PROPN
ejpam-4574	225	73	s2	s2	PROPN
ejpam-4574	225	74	s3	s3	PROPN
ejpam-4574	225	75	case	case	NOUN
ejpam-4574	225	76	j	j	PROPN
ejpam-4574	225	77	h	h	NOUN
ejpam-4574	225	78	a	a	DET
ejpam-4574	225	79	b	b	PROPN
ejpam-4574	225	80	path	path	NOUN
ejpam-4574	225	81	prefix	prefix	VERB
ejpam-4574	225	82	main	main	ADJ
ejpam-4574	225	83	suffix	suffix	NOUN
ejpam-4574	225	84	graph	graph	NOUN
ejpam-4574	225	85	parameters	parameter	NOUN
ejpam-4574	225	86	:	:	PUNCT
ejpam-4574	225	87	n	n	PROPN
ejpam-4574	225	88	=	=	SYM
ejpam-4574	225	89	10	10	NUM
ejpam-4574	225	90	k	k	NOUN
ejpam-4574	225	91	=	=	SYM
ejpam-4574	225	92	3	3	NUM
ejpam-4574	226	1	i	i	NOUN
ejpam-4574	226	2	=	=	NOUN
ejpam-4574	226	3	3	3	NUM
ejpam-4574	226	4	t	t	NOUN
ejpam-4574	226	5	=	=	SYM
ejpam-4574	226	6	h−	h−	PROPN
ejpam-4574	226	7	i+	i+	PUNCT
ejpam-4574	227	1	j	j	NOUN
ejpam-4574	227	2	−	−	PROPN
ejpam-4574	228	1	1	1	NUM
ejpam-4574	228	2	j	j	PROPN
ejpam-4574	228	3	is	be	AUX
ejpam-4574	228	4	the	the	DET
ejpam-4574	228	5	smallest	small	ADJ
ejpam-4574	228	6	index	index	NOUN
ejpam-4574	228	7	in	in	ADP
ejpam-4574	228	8	prefix	prefix	NOUN
ejpam-4574	228	9	such	such	ADJ
ejpam-4574	228	10	that	that	SCONJ
ejpam-4574	228	11	v(pj	v(pj	NOUN
ejpam-4574	228	12	)	)	PUNCT
ejpam-4574	228	13	∈	∈	PROPN
ejpam-4574	228	14	m	m	PROPN
ejpam-4574	228	15	,	,	PUNCT
ejpam-4574	228	16	c(pj	c(pj	X
ejpam-4574	228	17	)	)	PUNCT
ejpam-4574	228	18	=	=	SYM
ejpam-4574	228	19	c(vx	c(vx	NOUN
ejpam-4574	228	20	)	)	PUNCT
ejpam-4574	229	1	a	a	PRON
ejpam-4574	229	2	=	=	PUNCT
ejpam-4574	229	3	{	{	PUNCT
ejpam-4574	229	4	c(p1	c(p1	NOUN
ejpam-4574	229	5	)	)	PUNCT
ejpam-4574	229	6	,	,	PUNCT
ejpam-4574	229	7	c(p2	c(p2	NOUN
ejpam-4574	229	8	)	)	PUNCT
ejpam-4574	229	9	,	,	PUNCT
ejpam-4574	229	10	...	...	PUNCT
ejpam-4574	229	11	,	,	PUNCT
ejpam-4574	229	12	c(pj−1	c(pj−1	NOUN
ejpam-4574	229	13	)	)	PUNCT
ejpam-4574	229	14	}	}	PUNCT
ejpam-4574	229	15	h	h	NOUN
ejpam-4574	229	16	is	be	AUX
ejpam-4574	229	17	the	the	DET
ejpam-4574	229	18	smallest	small	ADJ
ejpam-4574	229	19	index	index	NOUN
ejpam-4574	229	20	in	in	ADP
ejpam-4574	229	21	suffix	suffix	NOUN
ejpam-4574	229	22	such	such	ADJ
ejpam-4574	229	23	that	that	SCONJ
ejpam-4574	229	24	v(sh	v(sh	PROPN
ejpam-4574	229	25	)	)	PUNCT
ejpam-4574	229	26	∈	∈	PROPN
ejpam-4574	229	27	m	m	PROPN
ejpam-4574	229	28	,	,	PUNCT
ejpam-4574	229	29	c(sh	c(sh	PROPN
ejpam-4574	229	30	)	)	PUNCT
ejpam-4574	229	31	=	=	SYM
ejpam-4574	229	32	c(vy	c(vy	NOUN
ejpam-4574	229	33	)	)	PUNCT
ejpam-4574	229	34	b	b	NOUN
ejpam-4574	229	35	=	=	SYM
ejpam-4574	229	36	{	{	PUNCT
ejpam-4574	229	37	c(s1	c(s1	NOUN
ejpam-4574	229	38	)	)	PUNCT
ejpam-4574	229	39	,	,	PUNCT
ejpam-4574	229	40	c(s2	c(s2	PROPN
ejpam-4574	229	41	)	)	PUNCT
ejpam-4574	229	42	,	,	PUNCT
ejpam-4574	229	43	...	...	PUNCT
ejpam-4574	229	44	,	,	PUNCT
ejpam-4574	229	45	c(sh−1	c(sh−1	NOUN
ejpam-4574	229	46	)	)	PUNCT
ejpam-4574	229	47	}	}	PUNCT
ejpam-4574	229	48	(	(	PUNCT
ejpam-4574	229	49	a	a	X
ejpam-4574	229	50	)	)	PUNCT
ejpam-4574	229	51	h	h	NOUN
ejpam-4574	229	52	≤	≤	NOUN
ejpam-4574	230	1	i−	i−	PROPN
ejpam-4574	230	2	j	j	PROPN
ejpam-4574	230	3	+	+	CCONJ
ejpam-4574	230	4	1	1	NUM
ejpam-4574	230	5	=	=	SYM
ejpam-4574	230	6	2	2	NUM
ejpam-4574	230	7	2	2	NUM
ejpam-4574	230	8	2	2	NUM
ejpam-4574	230	9	p1	p1	NOUN
ejpam-4574	230	10	s1	s1	NOUN
ejpam-4574	230	11	p2	p2	PROPN
ejpam-4574	230	12	−	−	PROPN
ejpam-4574	230	13	v1	v1	PROPN
ejpam-4574	230	14	−	−	PROPN
ejpam-4574	230	15	v3	v3	PROPN
ejpam-4574	230	16	−	−	PROPN
ejpam-4574	230	17	s2	s2	PROPN
ejpam-4574	230	18	3	3	NUM
ejpam-4574	230	19	1	1	NUM
ejpam-4574	230	20	(	(	PUNCT
ejpam-4574	230	21	b)x	b)x	X
ejpam-4574	230	22	>	>	X
ejpam-4574	230	23	i+	i+	NUM
ejpam-4574	230	24	t+	t+	NOUN
ejpam-4574	230	25	1−	1−	NUM
ejpam-4574	230	26	j	j	X
ejpam-4574	230	27	=	=	SYM
ejpam-4574	230	28	3	3	NUM
ejpam-4574	230	29	y	y	NOUN
ejpam-4574	230	30	<	<	X
ejpam-4574	230	31	k	k	PROPN
ejpam-4574	231	1	+	+	NUM
ejpam-4574	231	2	2−	2−	NUM
ejpam-4574	231	3	j	j	NOUN
ejpam-4574	231	4	=	=	SYM
ejpam-4574	231	5	3	3	NUM
ejpam-4574	231	6	2	2	NUM
ejpam-4574	231	7	3	3	NUM
ejpam-4574	231	8	p1	p1	PROPN
ejpam-4574	231	9	s1,s2	s1,s2	PROPN
ejpam-4574	231	10	p2	p2	PROPN
ejpam-4574	231	11	−	−	PROPN
ejpam-4574	231	12	v2	v2	PROPN
ejpam-4574	231	13	−	−	PROPN
ejpam-4574	231	14	v4	v4	NOUN
ejpam-4574	231	15	−	−	PROPN
ejpam-4574	231	16	s3	s3	PROPN
ejpam-4574	231	17	4	4	NUM
ejpam-4574	231	18	2	2	NUM
ejpam-4574	231	19	(	(	PUNCT
ejpam-4574	231	20	c	c	NOUN
ejpam-4574	231	21	)	)	PUNCT
ejpam-4574	231	22	|a	|a	VERB
ejpam-4574	232	1	∩	∩	NOUN
ejpam-4574	232	2	b|	b|	PROPN
ejpam-4574	232	3	>	>	X
ejpam-4574	232	4	h	h	NOUN
ejpam-4574	233	1	−	−	PROPN
ejpam-4574	233	2	i	i	PRON
ejpam-4574	233	3	+	+	NUM
ejpam-4574	234	1	j	j	PROPN
ejpam-4574	235	1	−	−	PROPN
ejpam-4574	235	2	2	2	NUM
ejpam-4574	235	3	|a	|a	VERB
ejpam-4574	235	4	∩	∩	NOUN
ejpam-4574	235	5	b|	b|	PROPN
ejpam-4574	235	6	>	>	X
ejpam-4574	235	7	1	1	NUM
ejpam-4574	235	8	and	and	CCONJ
ejpam-4574	235	9	y	y	NOUN
ejpam-4574	235	10	=	=	NOUN
ejpam-4574	235	11	3	3	NUM
ejpam-4574	235	12	3	3	NUM
ejpam-4574	235	13	3	3	NUM
ejpam-4574	235	14	p1,p2	p1,p2	PROPN
ejpam-4574	235	15	s1,s2	s1,s2	PROPN
ejpam-4574	235	16	p1	p1	PROPN
ejpam-4574	235	17	−	−	PROPN
ejpam-4574	235	18	v3	v3	PROPN
ejpam-4574	235	19	−	−	PROPN
ejpam-4574	235	20	s2	s2	PROPN
ejpam-4574	235	21	−	−	PROPN
ejpam-4574	235	22	s3	s3	PROPN
ejpam-4574	235	23	4	4	NUM
ejpam-4574	235	24	6	6	NUM
ejpam-4574	235	25	5	5	NUM
ejpam-4574	235	26	6	6	NUM
ejpam-4574	235	27	5	5	NUM
ejpam-4574	235	28	3	3	NUM
ejpam-4574	235	29	(	(	PUNCT
ejpam-4574	235	30	d	d	NOUN
ejpam-4574	235	31	)	)	PUNCT
ejpam-4574	236	1	|a	|a	VERB
ejpam-4574	236	2	∩	∩	NOUN
ejpam-4574	236	3	b|	b|	PROPN
ejpam-4574	236	4	>	>	X
ejpam-4574	236	5	h	h	NOUN
ejpam-4574	237	1	−	−	PROPN
ejpam-4574	237	2	i	i	PRON
ejpam-4574	237	3	+	+	NUM
ejpam-4574	238	1	j	j	PROPN
ejpam-4574	239	1	−	−	PROPN
ejpam-4574	239	2	2	2	NUM
ejpam-4574	239	3	|a	|a	VERB
ejpam-4574	239	4	∩	∩	NOUN
ejpam-4574	239	5	b|	b|	PROPN
ejpam-4574	239	6	>	>	X
ejpam-4574	239	7	1	1	NUM
ejpam-4574	239	8	and	and	CCONJ
ejpam-4574	239	9	x	x	X
ejpam-4574	239	10	=	=	SYM
ejpam-4574	239	11	3	3	NUM
ejpam-4574	239	12	3	3	NUM
ejpam-4574	239	13	3	3	NUM
ejpam-4574	239	14	p1,p2	p1,p2	PROPN
ejpam-4574	239	15	s1,s2	s1,s2	PROPN
ejpam-4574	239	16	p3	p3	PROPN
ejpam-4574	239	17	−	−	PROPN
ejpam-4574	239	18	p1	p1	PROPN
ejpam-4574	239	19	−	−	PROPN
ejpam-4574	239	20	v3	v3	PROPN
ejpam-4574	239	21	−	−	PROPN
ejpam-4574	239	22	s3	s3	PROPN
ejpam-4574	239	23	3	3	NUM
ejpam-4574	239	24	6	6	NUM
ejpam-4574	239	25	5	5	NUM
ejpam-4574	239	26	6	6	NUM
ejpam-4574	239	27	5	5	NUM
ejpam-4574	239	28	1	1	NUM
ejpam-4574	239	29	figure	figure	NOUN
ejpam-4574	239	30	2	2	NUM
ejpam-4574	239	31	:	:	PUNCT
ejpam-4574	239	32	cases	case	NOUN
ejpam-4574	239	33	of	of	ADP
ejpam-4574	239	34	star	star	NOUN
ejpam-4574	239	35	coloring	color	VERB
ejpam-4574	239	36	p	p	PROPN
ejpam-4574	239	37	3	3	NUM
ejpam-4574	239	38	10	10	NUM
ejpam-4574	239	39	lemma	lemma	PROPN
ejpam-4574	239	40	3	3	NUM
ejpam-4574	239	41	.	.	X
ejpam-4574	239	42	for	for	ADP
ejpam-4574	239	43	n	n	NOUN
ejpam-4574	239	44	=	=	SYM
ejpam-4574	239	45	k	k	PROPN
ejpam-4574	240	1	+	+	NOUN
ejpam-4574	240	2	2i	2i	NUM
ejpam-4574	240	3	where	where	SCONJ
ejpam-4574	240	4	1	1	NUM
ejpam-4574	240	5	≤	≤	NUM
ejpam-4574	240	6	i	i	NOUN
ejpam-4574	240	7	≤	≤	PUNCT
ejpam-4574	241	1	k	k	X
ejpam-4574	241	2	,	,	PUNCT
ejpam-4574	241	3	χs(p	χs(p	PUNCT
ejpam-4574	241	4	k	k	PROPN
ejpam-4574	241	5	n	n	PROPN
ejpam-4574	241	6	)	)	PUNCT
ejpam-4574	241	7	≥	≥	PROPN
ejpam-4574	241	8	k	k	PROPN
ejpam-4574	242	1	+	+	CCONJ
ejpam-4574	242	2	i.	i.	NOUN
ejpam-4574	242	3	proof	proof	NOUN
ejpam-4574	242	4	.	.	PUNCT
ejpam-4574	243	1	let	let	VERB
ejpam-4574	243	2	n1	n1	PROPN
ejpam-4574	243	3	=	=	SYM
ejpam-4574	243	4	n	n	CCONJ
ejpam-4574	243	5	−	−	PROPN
ejpam-4574	243	6	1	1	NUM
ejpam-4574	243	7	.	.	PUNCT
ejpam-4574	244	1	then	then	ADV
ejpam-4574	244	2	χs(p	χs(p	PUNCT
ejpam-4574	244	3	k	k	PROPN
ejpam-4574	244	4	n1	n1	PROPN
ejpam-4574	244	5	)	)	PUNCT
ejpam-4574	244	6	≤	≤	NOUN
ejpam-4574	244	7	χs(p	χs(p	X
ejpam-4574	244	8	k	k	PROPN
ejpam-4574	244	9	n	n	PROPN
ejpam-4574	244	10	)	)	PUNCT
ejpam-4574	244	11	since	since	SCONJ
ejpam-4574	244	12	p	p	PROPN
ejpam-4574	244	13	k	k	PROPN
ejpam-4574	244	14	n1	n1	PROPN
ejpam-4574	244	15	is	be	AUX
ejpam-4574	244	16	a	a	DET
ejpam-4574	244	17	subgraph	subgraph	NOUN
ejpam-4574	244	18	of	of	ADP
ejpam-4574	244	19	p	p	PROPN
ejpam-4574	244	20	k	k	PROPN
ejpam-4574	244	21	n	n	PROPN
ejpam-4574	244	22	.	.	PUNCT
ejpam-4574	245	1	but	but	CCONJ
ejpam-4574	245	2	n1	n1	PROPN
ejpam-4574	245	3	=	=	SYM
ejpam-4574	245	4	k+2(i−	k+2(i−	NOUN
ejpam-4574	246	1	1)+	1)+	NUM
ejpam-4574	246	2	1	1	NUM
ejpam-4574	246	3	,	,	PUNCT
ejpam-4574	246	4	so	so	CCONJ
ejpam-4574	246	5	by	by	ADP
ejpam-4574	246	6	using	use	VERB
ejpam-4574	246	7	lemma	lemma	PROPN
ejpam-4574	246	8	2	2	NUM
ejpam-4574	246	9	we	we	PRON
ejpam-4574	246	10	get	get	VERB
ejpam-4574	246	11	χs(p	χs(p	PRON
ejpam-4574	246	12	k	k	PROPN
ejpam-4574	246	13	n1	n1	PROPN
ejpam-4574	246	14	)	)	PUNCT
ejpam-4574	246	15	≥	≥	NOUN
ejpam-4574	246	16	k+	k+	X
ejpam-4574	247	1	(	(	PUNCT
ejpam-4574	247	2	i−	i−	PROPN
ejpam-4574	247	3	1)+	1)+	NUM
ejpam-4574	247	4	1	1	NUM
ejpam-4574	247	5	=	=	SYM
ejpam-4574	247	6	k+	k+	NOUN
ejpam-4574	247	7	i.	i.	NOUN
ejpam-4574	247	8	as	as	ADP
ejpam-4574	247	9	a	a	DET
ejpam-4574	247	10	consequence	consequence	NOUN
ejpam-4574	247	11	of	of	ADP
ejpam-4574	247	12	lemmas	lemmas	PROPN
ejpam-4574	247	13	12	12	NUM
ejpam-4574	247	14	we	we	PRON
ejpam-4574	247	15	have	have	VERB
ejpam-4574	247	16	χs(p	χs(p	NOUN
ejpam-4574	247	17	k	k	PROPN
ejpam-4574	247	18	n	n	PROPN
ejpam-4574	247	19	)	)	PUNCT
ejpam-4574	247	20	≤	≤	PUNCT
ejpam-4574	247	21	min{⌊n+k+1	min{⌊n+k+1	PROPN
ejpam-4574	247	22	2	2	NUM
ejpam-4574	247	23	⌋	⌋	NOUN
ejpam-4574	247	24	,	,	PUNCT
ejpam-4574	247	25	2k	2k	NUM
ejpam-4574	247	26	+	+	CCONJ
ejpam-4574	247	27	1	1	NUM
ejpam-4574	247	28	}	}	PUNCT
ejpam-4574	247	29	for	for	ADP
ejpam-4574	247	30	n	n	DET
ejpam-4574	247	31	≥	≥	NOUN
ejpam-4574	247	32	k	k	NOUN
ejpam-4574	248	1	+	+	CCONJ
ejpam-4574	248	2	1	1	NUM
ejpam-4574	248	3	,	,	PUNCT
ejpam-4574	248	4	and	and	CCONJ
ejpam-4574	248	5	χs(p	χs(p	ADP
ejpam-4574	248	6	k	k	PROPN
ejpam-4574	248	7	n	n	PROPN
ejpam-4574	248	8	)	)	PUNCT
ejpam-4574	248	9	≥	≥	PROPN
ejpam-4574	248	10	⌊n+k+1	⌊n+k+1	ADP
ejpam-4574	248	11	2	2	NUM
ejpam-4574	248	12	⌋	⌋	NOUN
ejpam-4574	248	13	for	for	ADP
ejpam-4574	248	14	n	n	DET
ejpam-4574	248	15	∈	∈	PROPN
ejpam-4574	248	16	{	{	PUNCT
ejpam-4574	248	17	k	k	NOUN
ejpam-4574	248	18	+	+	NUM
ejpam-4574	248	19	2i	2i	NUM
ejpam-4574	248	20	,	,	PUNCT
ejpam-4574	249	1	k	k	PROPN
ejpam-4574	250	1	+	+	PUNCT
ejpam-4574	250	2	2i	2i	NUM
ejpam-4574	250	3	+	+	CCONJ
ejpam-4574	250	4	1	1	NUM
ejpam-4574	250	5	}	}	PUNCT
ejpam-4574	250	6	,	,	PUNCT
ejpam-4574	250	7	where	where	SCONJ
ejpam-4574	250	8	0	0	NUM
ejpam-4574	250	9	≤	≤	NUM
ejpam-4574	250	10	i	i	PROPN
ejpam-4574	250	11	≤	≤	PROPN
ejpam-4574	251	1	k.	k.	PROPN
ejpam-4574	252	1	moreover	moreover	ADV
ejpam-4574	252	2	,	,	PUNCT
ejpam-4574	252	3	2k	2k	NUM
ejpam-4574	252	4	+	+	CCONJ
ejpam-4574	252	5	1	1	NUM
ejpam-4574	252	6	≤	≤	NOUN
ejpam-4574	252	7	χs(p	χs(p	X
ejpam-4574	252	8	k	k	PROPN
ejpam-4574	252	9	n	n	PROPN
ejpam-4574	252	10	)	)	PUNCT
ejpam-4574	252	11	≤	≤	NOUN
ejpam-4574	252	12	χs(p	χs(p	X
ejpam-4574	252	13	k	k	PROPN
ejpam-4574	252	14	n′	n′	PROPN
ejpam-4574	252	15	)	)	PUNCT
ejpam-4574	252	16	for	for	ADP
ejpam-4574	252	17	all	all	DET
ejpam-4574	252	18	n′	n′	PROPN
ejpam-4574	252	19	≥	≥	NOUN
ejpam-4574	252	20	n.	n.	VERB
ejpam-4574	252	21	so	so	ADV
ejpam-4574	252	22	,	,	PUNCT
ejpam-4574	252	23	we	we	PRON
ejpam-4574	252	24	can	can	AUX
ejpam-4574	252	25	conclude	conclude	VERB
ejpam-4574	252	26	the	the	DET
ejpam-4574	252	27	following	follow	VERB
ejpam-4574	252	28	theorem	theorem	PROPN
ejpam-4574	252	29	.	.	PUNCT
ejpam-4574	252	30	theorem	theorem	NOUN
ejpam-4574	252	31	2	2	NUM
ejpam-4574	252	32	.	.	X
ejpam-4574	252	33	for	for	ADP
ejpam-4574	252	34	n	n	PRON
ejpam-4574	252	35	≥	≥	NOUN
ejpam-4574	252	36	k	k	NOUN
ejpam-4574	253	1	+	+	CCONJ
ejpam-4574	253	2	1	1	NUM
ejpam-4574	253	3	,	,	PUNCT
ejpam-4574	253	4	χs(p	χs(p	PRON
ejpam-4574	253	5	k	k	PROPN
ejpam-4574	253	6	n	n	PROPN
ejpam-4574	253	7	)	)	PUNCT
ejpam-4574	253	8	=	=	PUNCT
ejpam-4574	253	9	min{⌊n+k+1	min{⌊n+k+1	PROPN
ejpam-4574	253	10	2	2	NUM
ejpam-4574	253	11	⌋	⌋	NOUN
ejpam-4574	253	12	,	,	PUNCT
ejpam-4574	253	13	2k	2k	NUM
ejpam-4574	253	14	+	+	CCONJ
ejpam-4574	253	15	1	1	NUM
ejpam-4574	253	16	}	}	PUNCT
ejpam-4574	253	17	.	.	PUNCT
ejpam-4574	254	1	example	example	NOUN
ejpam-4574	254	2	3	3	NUM
ejpam-4574	254	3	.	.	NOUN
ejpam-4574	254	4	χs(p	χs(p	PUNCT
ejpam-4574	254	5	2	2	NUM
ejpam-4574	254	6	10	10	NUM
ejpam-4574	254	7	)	)	PUNCT
ejpam-4574	254	8	=	=	SYM
ejpam-4574	255	1	min{⌊10	min{⌊10	PROPN
ejpam-4574	255	2	+	+	PROPN
ejpam-4574	255	3	2	2	NUM
ejpam-4574	255	4	+	+	ADJ
ejpam-4574	255	5	1	1	NUM
ejpam-4574	255	6	2	2	NUM
ejpam-4574	255	7	⌋	⌋	NOUN
ejpam-4574	255	8	,	,	PUNCT
ejpam-4574	255	9	2(2	2(2	NUM
ejpam-4574	255	10	)	)	PUNCT
ejpam-4574	256	1	+	+	CCONJ
ejpam-4574	256	2	1	1	X
ejpam-4574	256	3	}	}	PUNCT
ejpam-4574	256	4	=	=	SYM
ejpam-4574	256	5	min{6	min{6	NOUN
ejpam-4574	256	6	,	,	PUNCT
ejpam-4574	256	7	5	5	NUM
ejpam-4574	256	8	}	}	PUNCT
ejpam-4574	256	9	=	=	SYM
ejpam-4574	256	10	5	5	NUM
ejpam-4574	256	11	example	example	NOUN
ejpam-4574	256	12	4	4	NUM
ejpam-4574	256	13	.	.	NOUN
ejpam-4574	256	14	χs(p	χs(p	PUNCT
ejpam-4574	256	15	3	3	NUM
ejpam-4574	256	16	8	8	NUM
ejpam-4574	256	17	)	)	PUNCT
ejpam-4574	256	18	=	=	PUNCT
ejpam-4574	257	1	min{⌊8	min{⌊8	PROPN
ejpam-4574	257	2	+	+	PROPN
ejpam-4574	257	3	3	3	NUM
ejpam-4574	257	4	+	+	ADJ
ejpam-4574	257	5	1	1	NUM
ejpam-4574	257	6	2	2	NUM
ejpam-4574	257	7	⌋	⌋	NOUN
ejpam-4574	257	8	,	,	PUNCT
ejpam-4574	257	9	2(3	2(3	NUM
ejpam-4574	257	10	)	)	PUNCT
ejpam-4574	257	11	+	+	CCONJ
ejpam-4574	257	12	1	1	NUM
ejpam-4574	257	13	}	}	PUNCT
ejpam-4574	257	14	=	=	SYM
ejpam-4574	257	15	min{6	min{6	NOUN
ejpam-4574	257	16	,	,	PUNCT
ejpam-4574	257	17	7	7	NUM
ejpam-4574	257	18	}	}	PUNCT
ejpam-4574	257	19	=	=	SYM
ejpam-4574	257	20	6	6	NUM
ejpam-4574	257	21	a.	a.	NOUN
ejpam-4574	257	22	etawi	etawi	PROPN
ejpam-4574	257	23	,	,	PUNCT
ejpam-4574	257	24	m.	m.	PROPN
ejpam-4574	257	25	ghanem	ghanem	PROPN
ejpam-4574	257	26	,	,	PUNCT
ejpam-4574	257	27	h.	h.	PROPN
ejpam-4574	257	28	al	al	PROPN
ejpam-4574	257	29	-	-	PUNCT
ejpam-4574	257	30	ezeh	ezeh	PROPN
ejpam-4574	257	31	/	/	SYM
ejpam-4574	257	32	eur	eur	NOUN
ejpam-4574	257	33	.	.	PUNCT
ejpam-4574	258	1	j.	j.	PROPN
ejpam-4574	258	2	pure	pure	PROPN
ejpam-4574	258	3	appl	appl	PROPN
ejpam-4574	258	4	.	.	PROPN
ejpam-4574	258	5	math	math	PROPN
ejpam-4574	258	6	,	,	PUNCT
ejpam-4574	258	7	15	15	NUM
ejpam-4574	258	8	(	(	PUNCT
ejpam-4574	258	9	4	4	NUM
ejpam-4574	258	10	)	)	PUNCT
ejpam-4574	258	11	(	(	PUNCT
ejpam-4574	258	12	2022	2022	NUM
ejpam-4574	258	13	)	)	PUNCT
ejpam-4574	258	14	,	,	PUNCT
ejpam-4574	258	15	1822	1822	NUM
ejpam-4574	258	16	-	-	SYM
ejpam-4574	258	17	1835	1835	NUM
ejpam-4574	258	18	1828	1828	NUM
ejpam-4574	258	19	1	1	NUM
ejpam-4574	258	20	v1	v1	NOUN
ejpam-4574	258	21	2	2	NUM
ejpam-4574	258	22	v2	v2	PROPN
ejpam-4574	258	23	3	3	NUM
ejpam-4574	258	24	v3	v3	PROPN
ejpam-4574	258	25	4	4	NUM
ejpam-4574	258	26	v4	v4	NOUN
ejpam-4574	258	27	5	5	NUM
ejpam-4574	258	28	v5	v5	PROPN
ejpam-4574	258	29	1	1	NUM
ejpam-4574	258	30	v6	v6	NOUN
ejpam-4574	258	31	2	2	NUM
ejpam-4574	258	32	v7	v7	NOUN
ejpam-4574	258	33	3	3	NUM
ejpam-4574	258	34	v8	v8	PROPN
ejpam-4574	258	35	4	4	NUM
ejpam-4574	258	36	v9	v9	PROPN
ejpam-4574	258	37	5	5	NUM
ejpam-4574	258	38	v10	v10	NOUN
ejpam-4574	258	39	figure	figure	NOUN
ejpam-4574	258	40	3	3	NUM
ejpam-4574	258	41	:	:	PUNCT
ejpam-4574	258	42	star	star	NOUN
ejpam-4574	258	43	coloring	coloring	NOUN
ejpam-4574	258	44	of	of	ADP
ejpam-4574	258	45	p	p	PROPN
ejpam-4574	258	46	2	2	NUM
ejpam-4574	258	47	10	10	NUM
ejpam-4574	258	48	1	1	NUM
ejpam-4574	258	49	v1	v1	NOUN
ejpam-4574	258	50	2	2	NUM
ejpam-4574	258	51	v2	v2	PROPN
ejpam-4574	258	52	3	3	NUM
ejpam-4574	258	53	v3	v3	PROPN
ejpam-4574	258	54	4	4	NUM
ejpam-4574	258	55	v4	v4	NOUN
ejpam-4574	258	56	5	5	NUM
ejpam-4574	258	57	v5	v5	PROPN
ejpam-4574	258	58	6	6	NUM
ejpam-4574	258	59	v6	v6	NOUN
ejpam-4574	258	60	1	1	NUM
ejpam-4574	258	61	v7	v7	NOUN
ejpam-4574	258	62	2	2	NUM
ejpam-4574	258	63	v8	v8	NOUN
ejpam-4574	258	64	figure	figure	NOUN
ejpam-4574	258	65	4	4	NUM
ejpam-4574	258	66	:	:	PUNCT
ejpam-4574	258	67	star	star	NOUN
ejpam-4574	258	68	coloring	coloring	NOUN
ejpam-4574	258	69	of	of	ADP
ejpam-4574	258	70	p	p	PROPN
ejpam-4574	258	71	3	3	NUM
ejpam-4574	258	72	8	8	NUM
ejpam-4574	258	73	4	4	NUM
ejpam-4574	258	74	.	.	PUNCT
ejpam-4574	258	75	acyclic	acyclic	ADJ
ejpam-4574	258	76	coloring	coloring	NOUN
ejpam-4574	258	77	of	of	ADP
ejpam-4574	258	78	ck	ck	PROPN
ejpam-4574	258	79	n	n	PRON
ejpam-4574	258	80	the	the	DET
ejpam-4574	258	81	technique	technique	NOUN
ejpam-4574	258	82	that	that	PRON
ejpam-4574	258	83	we	we	PRON
ejpam-4574	258	84	followed	follow	VERB
ejpam-4574	258	85	in	in	ADP
ejpam-4574	258	86	this	this	DET
ejpam-4574	258	87	section	section	NOUN
ejpam-4574	258	88	was	be	AUX
ejpam-4574	258	89	to	to	PART
ejpam-4574	258	90	squeeze	squeeze	VERB
ejpam-4574	258	91	χa(c	χa(c	ADV
ejpam-4574	258	92	k	k	PROPN
ejpam-4574	258	93	n	n	CCONJ
ejpam-4574	258	94	)	)	PUNCT
ejpam-4574	258	95	between	between	ADP
ejpam-4574	258	96	upper	upper	ADJ
ejpam-4574	258	97	and	and	CCONJ
ejpam-4574	258	98	lower	low	ADJ
ejpam-4574	258	99	bounds	bound	NOUN
ejpam-4574	258	100	until	until	SCONJ
ejpam-4574	258	101	we	we	PRON
ejpam-4574	258	102	reached	reach	VERB
ejpam-4574	258	103	the	the	DET
ejpam-4574	258	104	exact	exact	ADJ
ejpam-4574	258	105	value	value	NOUN
ejpam-4574	258	106	of	of	ADP
ejpam-4574	258	107	χa(c	χa(c	ADJ
ejpam-4574	258	108	k	k	PROPN
ejpam-4574	258	109	n	n	CCONJ
ejpam-4574	258	110	)	)	PUNCT
ejpam-4574	258	111	for	for	ADP
ejpam-4574	258	112	a	a	DET
ejpam-4574	258	113	wide	wide	ADJ
ejpam-4574	258	114	range	range	NOUN
ejpam-4574	258	115	of	of	ADP
ejpam-4574	258	116	cases	case	NOUN
ejpam-4574	258	117	,	,	PUNCT
ejpam-4574	258	118	and	and	CCONJ
ejpam-4574	258	119	to	to	PART
ejpam-4574	258	120	find	find	VERB
ejpam-4574	258	121	an	an	DET
ejpam-4574	258	122	upper	upper	ADJ
ejpam-4574	258	123	bound	bind	VERB
ejpam-4574	258	124	and	and	CCONJ
ejpam-4574	258	125	a	a	DET
ejpam-4574	258	126	sharp	sharp	ADV
ejpam-4574	258	127	lower	lower	ADV
ejpam-4574	258	128	bound	bind	VERB
ejpam-4574	258	129	for	for	ADP
ejpam-4574	258	130	χa(c	χa(c	VERB
ejpam-4574	258	131	k	k	PROPN
ejpam-4574	258	132	n	n	CCONJ
ejpam-4574	258	133	)	)	PUNCT
ejpam-4574	258	134	for	for	ADP
ejpam-4574	258	135	other	other	ADJ
ejpam-4574	258	136	cases	case	NOUN
ejpam-4574	258	137	.	.	PUNCT
ejpam-4574	259	1	then	then	ADV
ejpam-4574	259	2	we	we	PRON
ejpam-4574	259	3	built	build	VERB
ejpam-4574	259	4	on	on	ADP
ejpam-4574	259	5	previous	previous	ADJ
ejpam-4574	259	6	studies	study	NOUN
ejpam-4574	259	7	on	on	ADP
ejpam-4574	259	8	the	the	DET
ejpam-4574	259	9	proper	proper	ADJ
ejpam-4574	259	10	coloring	coloring	NOUN
ejpam-4574	259	11	of	of	ADP
ejpam-4574	259	12	ck	ck	PROPN
ejpam-4574	259	13	n	n	CCONJ
ejpam-4574	259	14	,	,	PUNCT
ejpam-4574	259	15	and	and	CCONJ
ejpam-4574	259	16	added	add	VERB
ejpam-4574	259	17	some	some	DET
ejpam-4574	259	18	colors	color	NOUN
ejpam-4574	259	19	to	to	PART
ejpam-4574	259	20	break	break	VERB
ejpam-4574	259	21	one	one	NUM
ejpam-4574	259	22	of	of	ADP
ejpam-4574	259	23	these	these	DET
ejpam-4574	259	24	conditions	condition	NOUN
ejpam-4574	259	25	,	,	PUNCT
ejpam-4574	259	26	by	by	ADP
ejpam-4574	259	27	which	which	PRON
ejpam-4574	259	28	we	we	PRON
ejpam-4574	259	29	were	be	AUX
ejpam-4574	259	30	able	able	ADJ
ejpam-4574	259	31	to	to	PART
ejpam-4574	259	32	determine	determine	VERB
ejpam-4574	259	33	upper	upper	ADJ
ejpam-4574	259	34	bounds	bound	NOUN
ejpam-4574	259	35	for	for	ADP
ejpam-4574	259	36	χa(c	χa(c	ADJ
ejpam-4574	259	37	k	k	PROPN
ejpam-4574	259	38	n	n	CCONJ
ejpam-4574	259	39	)	)	PUNCT
ejpam-4574	259	40	.	.	PUNCT
ejpam-4574	260	1	let	let	VERB
ejpam-4574	260	2	ck	ck	PRON
ejpam-4574	260	3	n	n	ADV
ejpam-4574	260	4	denote	denote	VERB
ejpam-4574	260	5	the	the	DET
ejpam-4574	260	6	cycle	cycle	NOUN
ejpam-4574	260	7	of	of	ADP
ejpam-4574	260	8	order	order	NOUN
ejpam-4574	260	9	n	n	CCONJ
ejpam-4574	260	10	with	with	ADP
ejpam-4574	260	11	vertex	vertex	NOUN
ejpam-4574	260	12	set	set	VERB
ejpam-4574	260	13	v	v	NOUN
ejpam-4574	260	14	(	(	PUNCT
ejpam-4574	260	15	ck	ck	NOUN
ejpam-4574	260	16	n	n	CCONJ
ejpam-4574	260	17	)	)	PUNCT
ejpam-4574	260	18	=	=	SYM
ejpam-4574	260	19	{	{	PUNCT
ejpam-4574	260	20	v0	v0	NOUN
ejpam-4574	260	21	,	,	PUNCT
ejpam-4574	260	22	v1	v1	NOUN
ejpam-4574	260	23	,	,	PUNCT
ejpam-4574	260	24	...	...	PUNCT
ejpam-4574	260	25	,	,	PUNCT
ejpam-4574	260	26	vn−1	vn−1	ADJ
ejpam-4574	260	27	}	}	PUNCT
ejpam-4574	260	28	and	and	CCONJ
ejpam-4574	260	29	edge	edge	NOUN
ejpam-4574	260	30	set	set	VERB
ejpam-4574	260	31	e(ck	e(ck	PROPN
ejpam-4574	260	32	n)={vivj	n)={vivj	NOUN
ejpam-4574	260	33	:	:	PUNCT
ejpam-4574	260	34	1	1	NUM
ejpam-4574	260	35	≤	≤	NUM
ejpam-4574	260	36	|i−	|i−	NOUN
ejpam-4574	260	37	j|	j|	PROPN
ejpam-4574	260	38	,	,	PUNCT
ejpam-4574	260	39	n−	n−	PROPN
ejpam-4574	260	40	|i−	|i−	NOUN
ejpam-4574	260	41	j|	j|	VERB
ejpam-4574	260	42	≤	≤	ADV
ejpam-4574	261	1	k	k	X
ejpam-4574	261	2	}	}	PUNCT
ejpam-4574	261	3	.	.	PUNCT
ejpam-4574	262	1	clearly	clearly	ADV
ejpam-4574	262	2	,	,	PUNCT
ejpam-4574	262	3	ck	ck	PROPN
ejpam-4574	262	4	n	n	PRON
ejpam-4574	262	5	is	be	AUX
ejpam-4574	262	6	a	a	DET
ejpam-4574	262	7	complete	complete	ADJ
ejpam-4574	262	8	graph	graph	NOUN
ejpam-4574	262	9	when	when	SCONJ
ejpam-4574	262	10	n	n	PRON
ejpam-4574	262	11	≤	≤	NOUN
ejpam-4574	262	12	2k	2k	NOUN
ejpam-4574	262	13	+	+	CCONJ
ejpam-4574	262	14	1	1	NUM
ejpam-4574	262	15	and	and	CCONJ
ejpam-4574	262	16	hence	hence	ADV
ejpam-4574	262	17	χa(c	χa(c	VERB
ejpam-4574	262	18	k	k	PROPN
ejpam-4574	262	19	n	n	CCONJ
ejpam-4574	262	20	)	)	PUNCT
ejpam-4574	263	1	=	=	VERB
ejpam-4574	263	2	n.	n.	NOUN
ejpam-4574	263	3	we	we	PRON
ejpam-4574	263	4	will	will	AUX
ejpam-4574	263	5	start	start	VERB
ejpam-4574	263	6	by	by	ADP
ejpam-4574	263	7	determining	determine	VERB
ejpam-4574	263	8	a	a	DET
ejpam-4574	263	9	lower	low	ADJ
ejpam-4574	263	10	bound	bind	VERB
ejpam-4574	263	11	for	for	ADP
ejpam-4574	263	12	χa(c	χa(c	VERB
ejpam-4574	263	13	k	k	PROPN
ejpam-4574	263	14	n	n	CCONJ
ejpam-4574	263	15	)	)	PUNCT
ejpam-4574	263	16	when	when	SCONJ
ejpam-4574	263	17	n	n	X
ejpam-4574	263	18	>	>	X
ejpam-4574	263	19	2k	2k	NUM
ejpam-4574	263	20	+	+	CCONJ
ejpam-4574	263	21	1	1	X
ejpam-4574	263	22	.	.	X
ejpam-4574	263	23	theorem	theorem	NOUN
ejpam-4574	263	24	3	3	NUM
ejpam-4574	263	25	.	.	X
ejpam-4574	263	26	for	for	ADP
ejpam-4574	263	27	n	n	NOUN
ejpam-4574	263	28	>	>	X
ejpam-4574	263	29	2k	2k	PROPN
ejpam-4574	264	1	+	+	CCONJ
ejpam-4574	264	2	1	1	NUM
ejpam-4574	264	3	,	,	PUNCT
ejpam-4574	264	4	χa(c	χa(c	VERB
ejpam-4574	264	5	k	k	PROPN
ejpam-4574	264	6	n	n	CCONJ
ejpam-4574	264	7	)	)	PUNCT
ejpam-4574	264	8	≥	≥	NOUN
ejpam-4574	264	9	k	k	NOUN
ejpam-4574	265	1	+	+	CCONJ
ejpam-4574	265	2	2	2	X
ejpam-4574	265	3	.	.	X
ejpam-4574	265	4	proof	proof	NOUN
ejpam-4574	265	5	.	.	PUNCT
ejpam-4574	266	1	let	let	VERB
ejpam-4574	266	2	c	c	NOUN
ejpam-4574	266	3	:	:	PUNCT
ejpam-4574	266	4	v	v	X
ejpam-4574	266	5	(	(	PUNCT
ejpam-4574	266	6	ck	ck	NOUN
ejpam-4574	266	7	n	n	CCONJ
ejpam-4574	266	8	)	)	PUNCT
ejpam-4574	266	9	→	→	NOUN
ejpam-4574	266	10	{	{	PUNCT
ejpam-4574	266	11	c0	c0	NOUN
ejpam-4574	266	12	,	,	PUNCT
ejpam-4574	266	13	c1	c1	PROPN
ejpam-4574	266	14	,	,	PUNCT
ejpam-4574	266	15	...	...	PUNCT
ejpam-4574	266	16	,	,	PUNCT
ejpam-4574	266	17	ck	ck	X
ejpam-4574	266	18	}	}	PUNCT
ejpam-4574	266	19	be	be	AUX
ejpam-4574	266	20	a	a	DET
ejpam-4574	266	21	(	(	PUNCT
ejpam-4574	266	22	k	k	PROPN
ejpam-4574	266	23	+	+	PROPN
ejpam-4574	266	24	1)−	1)−	PROPN
ejpam-4574	266	25	coloring	coloring	NOUN
ejpam-4574	266	26	of	of	ADP
ejpam-4574	266	27	ck	ck	PROPN
ejpam-4574	266	28	n.	n.	PROPN
ejpam-4574	266	29	since	since	SCONJ
ejpam-4574	266	30	{	{	PUNCT
ejpam-4574	266	31	v0	v0	PROPN
ejpam-4574	266	32	,	,	PUNCT
ejpam-4574	266	33	v1	v1	NOUN
ejpam-4574	266	34	,	,	PUNCT
ejpam-4574	266	35	...	...	PUNCT
ejpam-4574	266	36	,	,	PUNCT
ejpam-4574	266	37	vk	vk	NOUN
ejpam-4574	266	38	}	}	PUNCT
ejpam-4574	266	39	induces	induce	VERB
ejpam-4574	266	40	a	a	DET
ejpam-4574	266	41	(	(	PUNCT
ejpam-4574	266	42	k+1)−	k+1)−	PROPN
ejpam-4574	266	43	clique	clique	NOUN
ejpam-4574	266	44	in	in	ADP
ejpam-4574	266	45	ck	ck	PROPN
ejpam-4574	266	46	n	n	CCONJ
ejpam-4574	266	47	,	,	PUNCT
ejpam-4574	266	48	without	without	ADP
ejpam-4574	266	49	loss	loss	NOUN
ejpam-4574	266	50	of	of	ADP
ejpam-4574	266	51	generality	generality	NOUN
ejpam-4574	266	52	,	,	PUNCT
ejpam-4574	266	53	define	define	VERB
ejpam-4574	266	54	c	c	NOUN
ejpam-4574	266	55	by	by	ADP
ejpam-4574	266	56	c(vj	c(vj	PROPN
ejpam-4574	266	57	)	)	PUNCT
ejpam-4574	267	1	=	=	SYM
ejpam-4574	267	2	cj	cj	NOUN
ejpam-4574	267	3	mod	mod	PROPN
ejpam-4574	267	4	(	(	PUNCT
ejpam-4574	267	5	k+1	k+1	NOUN
ejpam-4574	267	6	)	)	PUNCT
ejpam-4574	267	7	for	for	ADP
ejpam-4574	267	8	j	j	PROPN
ejpam-4574	267	9	=	=	SYM
ejpam-4574	267	10	0	0	PROPN
ejpam-4574	267	11	,	,	PUNCT
ejpam-4574	267	12	1	1	NUM
ejpam-4574	267	13	,	,	PUNCT
ejpam-4574	267	14	...	...	PUNCT
ejpam-4574	267	15	,	,	PUNCT
ejpam-4574	267	16	n−	n−	NOUN
ejpam-4574	267	17	1	1	NUM
ejpam-4574	267	18	.	.	PUNCT
ejpam-4574	268	1	let	let	VERB
ejpam-4574	268	2	n	n	NOUN
ejpam-4574	268	3	=	=	SYM
ejpam-4574	268	4	q(k+1)+	q(k+1)+	PROPN
ejpam-4574	268	5	r	r	NOUN
ejpam-4574	268	6	where	where	SCONJ
ejpam-4574	268	7	q	q	PROPN
ejpam-4574	268	8	≥	≥	NUM
ejpam-4574	268	9	2	2	NUM
ejpam-4574	268	10	and	and	CCONJ
ejpam-4574	268	11	0	0	NUM
ejpam-4574	268	12	≤	≤	NUM
ejpam-4574	268	13	r	r	NOUN
ejpam-4574	268	14	≤	≤	PUNCT
ejpam-4574	268	15	k.	k.	NOUN
ejpam-4574	269	1	then	then	ADV
ejpam-4574	269	2	we	we	PRON
ejpam-4574	269	3	have	have	VERB
ejpam-4574	269	4	two	two	NUM
ejpam-4574	269	5	cases	case	NOUN
ejpam-4574	269	6	:	:	PUNCT
ejpam-4574	269	7	case	case	NOUN
ejpam-4574	269	8	1	1	NUM
ejpam-4574	269	9	.	.	PUNCT
ejpam-4574	270	1	r	r	NOUN
ejpam-4574	270	2	>	>	X
ejpam-4574	270	3	0	0	NUM
ejpam-4574	270	4	.	.	PUNCT
ejpam-4574	271	1	then	then	ADV
ejpam-4574	271	2	c(vr−1	c(vr−1	PUNCT
ejpam-4574	271	3	)	)	PUNCT
ejpam-4574	271	4	=	=	SYM
ejpam-4574	271	5	c(vn−1	c(vn−1	X
ejpam-4574	271	6	)	)	PUNCT
ejpam-4574	271	7	and	and	CCONJ
ejpam-4574	271	8	dcn(vn−1	dcn(vn−1	PROPN
ejpam-4574	271	9	,	,	PUNCT
ejpam-4574	271	10	vr−1	vr−1	PROPN
ejpam-4574	271	11	)	)	PUNCT
ejpam-4574	271	12	=	=	SYM
ejpam-4574	271	13	r	r	NOUN
ejpam-4574	271	14	≤	≤	NUM
ejpam-4574	271	15	k	k	NOUN
ejpam-4574	271	16	,	,	PUNCT
ejpam-4574	271	17	a	a	DET
ejpam-4574	271	18	contradiction	contradiction	NOUN
ejpam-4574	271	19	.	.	PUNCT
ejpam-4574	272	1	case	case	NOUN
ejpam-4574	272	2	2	2	NUM
ejpam-4574	272	3	.	.	PUNCT
ejpam-4574	273	1	r	r	NOUN
ejpam-4574	273	2	=	=	SYM
ejpam-4574	273	3	0	0	NUM
ejpam-4574	273	4	.	.	PUNCT
ejpam-4574	274	1	for	for	ADP
ejpam-4574	274	2	1	1	NUM
ejpam-4574	274	3	≤	≤	NUM
ejpam-4574	275	1	i	i	NOUN
ejpam-4574	275	2	≤	≤	PROPN
ejpam-4574	276	1	k	k	X
ejpam-4574	276	2	,	,	PUNCT
ejpam-4574	276	3	the	the	DET
ejpam-4574	276	4	induced	induced	ADJ
ejpam-4574	276	5	subgraph	subgraph	NOUN
ejpam-4574	276	6	ci	ci	PROPN
ejpam-4574	276	7	of	of	ADP
ejpam-4574	276	8	c	c	PROPN
ejpam-4574	276	9	k	k	PROPN
ejpam-4574	276	10	n	n	PROPN
ejpam-4574	276	11	where	where	SCONJ
ejpam-4574	276	12	v	v	X
ejpam-4574	276	13	(	(	PUNCT
ejpam-4574	276	14	ci	ci	NOUN
ejpam-4574	276	15	)	)	PUNCT
ejpam-4574	276	16	=	=	SYM
ejpam-4574	276	17	{	{	PUNCT
ejpam-4574	276	18	v0	v0	PROPN
ejpam-4574	276	19	,	,	PUNCT
ejpam-4574	276	20	vi	vi	PROPN
ejpam-4574	276	21	,	,	PUNCT
ejpam-4574	276	22	v(k+1	v(k+1	NOUN
ejpam-4574	276	23	)	)	PUNCT
ejpam-4574	276	24	,	,	PUNCT
ejpam-4574	276	25	v(k+1)+i	v(k+1)+i	PROPN
ejpam-4574	276	26	,	,	PUNCT
ejpam-4574	276	27	...	...	PUNCT
ejpam-4574	276	28	,	,	PUNCT
ejpam-4574	276	29	v(q−1)(k+1	v(q−1)(k+1	NOUN
ejpam-4574	276	30	)	)	PUNCT
ejpam-4574	276	31	,	,	PUNCT
ejpam-4574	276	32	v(q−1)(k+1)+i	v(q−1)(k+1)+i	PROPN
ejpam-4574	276	33	}	}	PUNCT
ejpam-4574	276	34	,	,	PUNCT
ejpam-4574	276	35	is	be	AUX
ejpam-4574	276	36	a	a	DET
ejpam-4574	276	37	bicolored	bicolore	VERB
ejpam-4574	276	38	cycle	cycle	NOUN
ejpam-4574	276	39	of	of	ADP
ejpam-4574	276	40	ck	ck	PROPN
ejpam-4574	276	41	n	n	NOUN
ejpam-4574	276	42	using	use	VERB
ejpam-4574	276	43	two	two	NUM
ejpam-4574	276	44	colors	color	NOUN
ejpam-4574	276	45	c0	c0	NOUN
ejpam-4574	276	46	and	and	CCONJ
ejpam-4574	276	47	ci	ci	PROPN
ejpam-4574	276	48	,	,	PUNCT
ejpam-4574	276	49	a	a	DET
ejpam-4574	276	50	contradiction	contradiction	NOUN
ejpam-4574	276	51	.	.	PUNCT
ejpam-4574	277	1	hence	hence	ADV
ejpam-4574	277	2	χa(c	χa(c	VERB
ejpam-4574	277	3	k	k	PROPN
ejpam-4574	277	4	n	n	CCONJ
ejpam-4574	277	5	)	)	PUNCT
ejpam-4574	277	6	≥	≥	NOUN
ejpam-4574	277	7	k	k	NOUN
ejpam-4574	278	1	+	+	CCONJ
ejpam-4574	278	2	2	2	X
ejpam-4574	278	3	.	.	X
ejpam-4574	278	4	lemma	lemma	PROPN
ejpam-4574	278	5	4	4	X
ejpam-4574	278	6	.	.	PUNCT
ejpam-4574	278	7	let	let	VERB
ejpam-4574	278	8	c	c	NOUN
ejpam-4574	278	9	=	=	NOUN
ejpam-4574	278	10	{	{	PUNCT
ejpam-4574	278	11	c1	c1	PROPN
ejpam-4574	278	12	,	,	PUNCT
ejpam-4574	278	13	c2	c2	PROPN
ejpam-4574	278	14	,	,	PUNCT
ejpam-4574	278	15	...	...	PUNCT
ejpam-4574	278	16	,	,	PUNCT
ejpam-4574	278	17	cr	cr	PART
ejpam-4574	278	18	}	}	PUNCT
ejpam-4574	278	19	be	be	AUX
ejpam-4574	278	20	a	a	DET
ejpam-4574	278	21	proper	proper	ADJ
ejpam-4574	278	22	coloring	coloring	NOUN
ejpam-4574	278	23	of	of	ADP
ejpam-4574	278	24	ck	ck	PROPN
ejpam-4574	278	25	n	n	PROPN
ejpam-4574	278	26	and	and	CCONJ
ejpam-4574	278	27	p	p	NOUN
ejpam-4574	278	28	=	=	PUNCT
ejpam-4574	278	29	{	{	PUNCT
ejpam-4574	278	30	ti	ti	NOUN
ejpam-4574	278	31	:	:	PUNCT
ejpam-4574	278	32	i	i	NOUN
ejpam-4574	278	33	=	=	NOUN
ejpam-4574	278	34	1	1	NUM
ejpam-4574	278	35	,	,	PUNCT
ejpam-4574	278	36	2	2	NUM
ejpam-4574	278	37	,	,	PUNCT
ejpam-4574	278	38	...	...	PUNCT
ejpam-4574	278	39	,	,	PUNCT
ejpam-4574	278	40	r	r	AUX
ejpam-4574	278	41	}	}	PUNCT
ejpam-4574	278	42	be	be	AUX
ejpam-4574	278	43	the	the	DET
ejpam-4574	278	44	color	color	NOUN
ejpam-4574	278	45	classes	class	NOUN
ejpam-4574	278	46	of	of	ADP
ejpam-4574	278	47	c.	c.	NOUN
ejpam-4574	278	48	if	if	SCONJ
ejpam-4574	278	49	cl	cl	NOUN
ejpam-4574	278	50	is	be	AUX
ejpam-4574	278	51	a	a	DET
ejpam-4574	278	52	bicolored	bicolore	VERB
ejpam-4574	278	53	cycle	cycle	NOUN
ejpam-4574	278	54	in	in	ADP
ejpam-4574	278	55	ck	ck	PROPN
ejpam-4574	278	56	n	n	CCONJ
ejpam-4574	278	57	,	,	PUNCT
ejpam-4574	278	58	then	then	ADV
ejpam-4574	278	59	the	the	DET
ejpam-4574	278	60	following	follow	VERB
ejpam-4574	278	61	holds	hold	VERB
ejpam-4574	278	62	.	.	PUNCT
ejpam-4574	279	1	if	if	SCONJ
ejpam-4574	279	2	ti	ti	PROPN
ejpam-4574	279	3	∩	∩	X
ejpam-4574	279	4	v	v	ADP
ejpam-4574	279	5	(	(	PUNCT
ejpam-4574	279	6	cl	cl	NOUN
ejpam-4574	279	7	)	)	PUNCT
ejpam-4574	279	8	̸=	̸=	PROPN
ejpam-4574	279	9	ϕ	ϕ	NOUN
ejpam-4574	279	10	then	then	ADV
ejpam-4574	279	11	ti	ti	PROPN
ejpam-4574	279	12	⊆	⊆	NUM
ejpam-4574	279	13	v	v	NOUN
ejpam-4574	279	14	(	(	PUNCT
ejpam-4574	279	15	cl	cl	NOUN
ejpam-4574	279	16	)	)	PUNCT
ejpam-4574	279	17	.	.	PUNCT
ejpam-4574	280	1	proof	proof	NOUN
ejpam-4574	280	2	.	.	PUNCT
ejpam-4574	281	1	clearly	clearly	ADV
ejpam-4574	281	2	,	,	PUNCT
ejpam-4574	281	3	l	l	NOUN
ejpam-4574	281	4	is	be	AUX
ejpam-4574	281	5	even	even	ADV
ejpam-4574	281	6	,	,	PUNCT
ejpam-4574	281	7	l	l	NOUN
ejpam-4574	281	8	≥	≥	NUM
ejpam-4574	281	9	4	4	NUM
ejpam-4574	281	10	.	.	PUNCT
ejpam-4574	282	1	obviously	obviously	ADV
ejpam-4574	282	2	,	,	PUNCT
ejpam-4574	282	3	when	when	SCONJ
ejpam-4574	282	4	|ti|	|ti|	ADJ
ejpam-4574	282	5	=	=	NOUN
ejpam-4574	282	6	2	2	NUM
ejpam-4574	282	7	ti	ti	NOUN
ejpam-4574	282	8	⊆	⊆	NUM
ejpam-4574	282	9	v	v	NOUN
ejpam-4574	282	10	(	(	PUNCT
ejpam-4574	282	11	cl	cl	NOUN
ejpam-4574	282	12	)	)	PUNCT
ejpam-4574	282	13	.	.	PUNCT
ejpam-4574	283	1	now	now	ADV
ejpam-4574	283	2	let	let	VERB
ejpam-4574	283	3	|ti|	|ti|	PROPN
ejpam-4574	283	4	≥	≥	NOUN
ejpam-4574	283	5	3	3	NUM
ejpam-4574	283	6	and	and	CCONJ
ejpam-4574	283	7	cl	cl	NOUN
ejpam-4574	283	8	be	be	AUX
ejpam-4574	283	9	a	a	DET
ejpam-4574	283	10	bicolored	bicolore	VERB
ejpam-4574	283	11	cycle	cycle	NOUN
ejpam-4574	283	12	with	with	ADP
ejpam-4574	283	13	v	v	NOUN
ejpam-4574	283	14	(	(	PUNCT
ejpam-4574	283	15	cl	cl	NOUN
ejpam-4574	283	16	)	)	PUNCT
ejpam-4574	283	17	=	=	SYM
ejpam-4574	283	18	{	{	PUNCT
ejpam-4574	283	19	vx1	vx1	NOUN
ejpam-4574	283	20	,	,	PUNCT
ejpam-4574	283	21	vy1	vy1	NOUN
ejpam-4574	283	22	,	,	PUNCT
ejpam-4574	283	23	vx2	vx2	INTJ
ejpam-4574	283	24	,	,	PUNCT
ejpam-4574	283	25	vy2	vy2	NOUN
ejpam-4574	283	26	,	,	PUNCT
ejpam-4574	283	27	...	...	PUNCT
ejpam-4574	283	28	,	,	PUNCT
ejpam-4574	283	29	vxl	vxl	PROPN
ejpam-4574	283	30	2	2	NUM
ejpam-4574	283	31	,	,	PUNCT
ejpam-4574	283	32	vyl	vyl	NOUN
ejpam-4574	283	33	2	2	NUM
ejpam-4574	283	34	}	}	PUNCT
ejpam-4574	283	35	and	and	CCONJ
ejpam-4574	283	36	a.	a.	PROPN
ejpam-4574	283	37	etawi	etawi	PROPN
ejpam-4574	283	38	,	,	PUNCT
ejpam-4574	283	39	m.	m.	PROPN
ejpam-4574	283	40	ghanem	ghanem	PROPN
ejpam-4574	283	41	,	,	PUNCT
ejpam-4574	283	42	h.	h.	PROPN
ejpam-4574	283	43	al	al	PROPN
ejpam-4574	283	44	-	-	PUNCT
ejpam-4574	283	45	ezeh	ezeh	PROPN
ejpam-4574	283	46	/	/	SYM
ejpam-4574	283	47	eur	eur	NOUN
ejpam-4574	283	48	.	.	PUNCT
ejpam-4574	284	1	j.	j.	PROPN
ejpam-4574	284	2	pure	pure	PROPN
ejpam-4574	284	3	appl	appl	PROPN
ejpam-4574	284	4	.	.	PROPN
ejpam-4574	284	5	math	math	PROPN
ejpam-4574	284	6	,	,	PUNCT
ejpam-4574	284	7	15	15	NUM
ejpam-4574	284	8	(	(	PUNCT
ejpam-4574	284	9	4	4	NUM
ejpam-4574	284	10	)	)	PUNCT
ejpam-4574	284	11	(	(	PUNCT
ejpam-4574	284	12	2022	2022	NUM
ejpam-4574	284	13	)	)	PUNCT
ejpam-4574	284	14	,	,	PUNCT
ejpam-4574	284	15	1822	1822	NUM
ejpam-4574	284	16	-	-	SYM
ejpam-4574	284	17	1835	1835	NUM
ejpam-4574	284	18	1829	1829	NUM
ejpam-4574	284	19	e(cl	e(cl	NOUN
ejpam-4574	284	20	)	)	PUNCT
ejpam-4574	284	21	=	=	PRON
ejpam-4574	284	22	{	{	PUNCT
ejpam-4574	284	23	vxivyi	vxivyi	NOUN
ejpam-4574	284	24	:	:	PUNCT
ejpam-4574	285	1	i	i	NOUN
ejpam-4574	285	2	=	=	NOUN
ejpam-4574	285	3	1	1	NUM
ejpam-4574	285	4	,	,	PUNCT
ejpam-4574	285	5	2	2	NUM
ejpam-4574	285	6	,	,	PUNCT
ejpam-4574	285	7	...	...	PUNCT
ejpam-4574	285	8	,	,	PUNCT
ejpam-4574	285	9	l2	l2	NOUN
ejpam-4574	285	10	}	}	PUNCT
ejpam-4574	285	11	∪	∪	NOUN
ejpam-4574	285	12	{	{	PUNCT
ejpam-4574	285	13	vyivxi+1	vyivxi+1	NOUN
ejpam-4574	285	14	:	:	PUNCT
ejpam-4574	285	15	i	i	NOUN
ejpam-4574	285	16	=	=	NOUN
ejpam-4574	285	17	1	1	NUM
ejpam-4574	285	18	,	,	PUNCT
ejpam-4574	285	19	2	2	NUM
ejpam-4574	285	20	,	,	PUNCT
ejpam-4574	285	21	...	...	PUNCT
ejpam-4574	285	22	,	,	PUNCT
ejpam-4574	285	23	l2	l2	VERB
ejpam-4574	285	24	−	−	NOUN
ejpam-4574	285	25	1	1	NUM
ejpam-4574	285	26	}	}	PUNCT
ejpam-4574	285	27	∪	∪	NOUN
ejpam-4574	285	28	{	{	PUNCT
ejpam-4574	285	29	vyl	vyl	NOUN
ejpam-4574	285	30	2	2	NUM
ejpam-4574	285	31	vx1	vx1	NOUN
ejpam-4574	285	32	}	}	PUNCT
ejpam-4574	285	33	.	.	PUNCT
ejpam-4574	286	1	assume	assume	VERB
ejpam-4574	286	2	that	that	SCONJ
ejpam-4574	286	3	there	there	PRON
ejpam-4574	286	4	exists	exist	VERB
ejpam-4574	286	5	vh	vh	PROPN
ejpam-4574	286	6	∈	∈	PROPN
ejpam-4574	286	7	ti	ti	PROPN
ejpam-4574	286	8	−	−	PROPN
ejpam-4574	286	9	v	v	PROPN
ejpam-4574	286	10	(	(	PUNCT
ejpam-4574	286	11	cl	cl	NOUN
ejpam-4574	286	12	)	)	PUNCT
ejpam-4574	286	13	.	.	PUNCT
ejpam-4574	287	1	if	if	SCONJ
ejpam-4574	287	2	vh	vh	PROPN
ejpam-4574	287	3	lies	lie	VERB
ejpam-4574	287	4	between	between	ADP
ejpam-4574	287	5	vxi	vxi	NOUN
ejpam-4574	287	6	and	and	CCONJ
ejpam-4574	287	7	vxi+1	vxi+1	VERB
ejpam-4574	287	8	for	for	ADP
ejpam-4574	287	9	some	some	DET
ejpam-4574	287	10	i	i	NOUN
ejpam-4574	287	11	=	=	NOUN
ejpam-4574	287	12	1	1	NUM
ejpam-4574	287	13	,	,	PUNCT
ejpam-4574	287	14	2	2	NUM
ejpam-4574	287	15	,	,	PUNCT
ejpam-4574	287	16	...	...	PUNCT
ejpam-4574	287	17	,	,	PUNCT
ejpam-4574	287	18	l2	l2	VERB
ejpam-4574	287	19	−	−	NOUN
ejpam-4574	287	20	1	1	NUM
ejpam-4574	287	21	,	,	PUNCT
ejpam-4574	287	22	then	then	ADV
ejpam-4574	287	23	dcn(vxi	dcn(vxi	NOUN
ejpam-4574	287	24	,	,	PUNCT
ejpam-4574	287	25	vh	vh	PROPN
ejpam-4574	287	26	)	)	PUNCT
ejpam-4574	287	27	<	<	X
ejpam-4574	287	28	dcn(vxi	dcn(vxi	X
ejpam-4574	287	29	,	,	PUNCT
ejpam-4574	287	30	vyi	vyi	PROPN
ejpam-4574	287	31	)	)	PUNCT
ejpam-4574	287	32	≤	≤	PUNCT
ejpam-4574	288	1	k	k	NOUN
ejpam-4574	288	2	or	or	CCONJ
ejpam-4574	288	3	dcn(vh	dcn(vh	ADJ
ejpam-4574	288	4	,	,	PUNCT
ejpam-4574	288	5	vxi+1	vxi+1	ADJ
ejpam-4574	288	6	)	)	PUNCT
ejpam-4574	288	7	<	<	X
ejpam-4574	288	8	dcn(vyi	dcn(vyi	X
ejpam-4574	288	9	,	,	PUNCT
ejpam-4574	288	10	vxi+1	vxi+1	PROPN
ejpam-4574	288	11	)	)	PUNCT
ejpam-4574	288	12	≤	≤	NOUN
ejpam-4574	289	1	k	k	PROPN
ejpam-4574	289	2	,	,	PUNCT
ejpam-4574	289	3	a	a	DET
ejpam-4574	289	4	contradiction	contradiction	NOUN
ejpam-4574	289	5	.	.	PUNCT
ejpam-4574	290	1	and	and	CCONJ
ejpam-4574	290	2	if	if	SCONJ
ejpam-4574	290	3	vh	vh	PROPN
ejpam-4574	290	4	lies	lie	VERB
ejpam-4574	290	5	between	between	ADP
ejpam-4574	290	6	vxl	vxl	PROPN
ejpam-4574	290	7	2	2	NUM
ejpam-4574	290	8	and	and	CCONJ
ejpam-4574	290	9	vx1	vx1	NOUN
ejpam-4574	290	10	,	,	PUNCT
ejpam-4574	290	11	then	then	ADV
ejpam-4574	290	12	dcn(vxl	dcn(vxl	NOUN
ejpam-4574	290	13	2	2	NUM
ejpam-4574	290	14	,	,	PUNCT
ejpam-4574	290	15	v	v	NOUN
ejpam-4574	290	16	h	h	NOUN
ejpam-4574	290	17	)	)	PUNCT
ejpam-4574	290	18	<	<	X
ejpam-4574	290	19	dcn(vxl	dcn(vxl	NOUN
ejpam-4574	290	20	2	2	NUM
ejpam-4574	290	21	,	,	PUNCT
ejpam-4574	290	22	vyi	vyi	NOUN
ejpam-4574	290	23	)	)	PUNCT
ejpam-4574	290	24	≤	≤	PUNCT
ejpam-4574	290	25	k	k	NOUN
ejpam-4574	290	26	or	or	CCONJ
ejpam-4574	290	27	dcn(vh	dcn(vh	ADJ
ejpam-4574	290	28	,	,	PUNCT
ejpam-4574	290	29	vx1	vx1	NOUN
ejpam-4574	290	30	)	)	PUNCT
ejpam-4574	290	31	<	<	X
ejpam-4574	290	32	dcn(vyi	dcn(vyi	X
ejpam-4574	290	33	,	,	PUNCT
ejpam-4574	290	34	vx1	vx1	NOUN
ejpam-4574	290	35	)	)	PUNCT
ejpam-4574	290	36	≤	≤	PUNCT
ejpam-4574	291	1	k	k	PROPN
ejpam-4574	291	2	,	,	PUNCT
ejpam-4574	291	3	a	a	DET
ejpam-4574	291	4	contradiction	contradiction	NOUN
ejpam-4574	291	5	.	.	PUNCT
ejpam-4574	292	1	thus	thus	ADV
ejpam-4574	292	2	ti	ti	X
ejpam-4574	292	3	⊆	⊆	NUM
ejpam-4574	292	4	v	v	NOUN
ejpam-4574	292	5	(	(	PUNCT
ejpam-4574	292	6	cl	cl	NOUN
ejpam-4574	292	7	)	)	PUNCT
ejpam-4574	292	8	.	.	PUNCT
ejpam-4574	293	1	since	since	SCONJ
ejpam-4574	293	2	cl	cl	NOUN
ejpam-4574	293	3	is	be	AUX
ejpam-4574	293	4	an	an	DET
ejpam-4574	293	5	induced	induce	VERB
ejpam-4574	293	6	bicolored	bicolore	VERB
ejpam-4574	293	7	cycle	cycle	NOUN
ejpam-4574	293	8	of	of	ADP
ejpam-4574	293	9	ck	ck	PROPN
ejpam-4574	293	10	n	n	CCONJ
ejpam-4574	293	11	,	,	PUNCT
ejpam-4574	293	12	we	we	PRON
ejpam-4574	293	13	have	have	VERB
ejpam-4574	293	14	k	k	X
ejpam-4574	293	15	<	<	X
ejpam-4574	293	16	dcn(vxi	dcn(vxi	X
ejpam-4574	293	17	,	,	PUNCT
ejpam-4574	293	18	vxi+1	vxi+1	PROPN
ejpam-4574	293	19	)	)	PUNCT
ejpam-4574	293	20	,	,	PUNCT
ejpam-4574	293	21	dcn(vyi	dcn(vyi	X
ejpam-4574	293	22	,	,	PUNCT
ejpam-4574	293	23	vyi+1	vyi+1	NOUN
ejpam-4574	293	24	)	)	PUNCT
ejpam-4574	293	25	≤	≤	NOUN
ejpam-4574	293	26	2k	2k	NOUN
ejpam-4574	293	27	for	for	ADP
ejpam-4574	293	28	i	i	PRON
ejpam-4574	293	29	=	=	NOUN
ejpam-4574	293	30	1	1	NUM
ejpam-4574	293	31	,	,	PUNCT
ejpam-4574	293	32	2	2	NUM
ejpam-4574	293	33	,	,	PUNCT
ejpam-4574	293	34	...	...	PUNCT
ejpam-4574	293	35	,	,	PUNCT
ejpam-4574	293	36	l2	l2	VERB
ejpam-4574	293	37	−	−	NOUN
ejpam-4574	293	38	1	1	NUM
ejpam-4574	293	39	,	,	PUNCT
ejpam-4574	293	40	and	and	CCONJ
ejpam-4574	293	41	k	k	X
ejpam-4574	293	42	<	<	X
ejpam-4574	293	43	dcn(vxl	dcn(vxl	NOUN
ejpam-4574	293	44	2	2	NUM
ejpam-4574	293	45	,	,	PUNCT
ejpam-4574	293	46	vx1	vx1	NOUN
ejpam-4574	293	47	)	)	PUNCT
ejpam-4574	293	48	,	,	PUNCT
ejpam-4574	293	49	dcn(vyl	dcn(vyl	VERB
ejpam-4574	293	50	2	2	NUM
ejpam-4574	293	51	,	,	PUNCT
ejpam-4574	293	52	vy1	vy1	NOUN
ejpam-4574	293	53	)	)	PUNCT
ejpam-4574	293	54	≤	≤	NOUN
ejpam-4574	293	55	2k	2k	NUM
ejpam-4574	293	56	.	.	PUNCT
ejpam-4574	294	1	using	use	VERB
ejpam-4574	294	2	4	4	NUM
ejpam-4574	294	3	to	to	PART
ejpam-4574	294	4	get	get	VERB
ejpam-4574	294	5	ti	ti	NOUN
ejpam-4574	294	6	,	,	PUNCT
ejpam-4574	294	7	tj	tj	PROPN
ejpam-4574	294	8	⊆	⊆	NUM
ejpam-4574	294	9	v	v	ADP
ejpam-4574	294	10	(	(	PUNCT
ejpam-4574	294	11	cl	cl	NOUN
ejpam-4574	294	12	)	)	PUNCT
ejpam-4574	294	13	are	be	AUX
ejpam-4574	294	14	induced	induce	VERB
ejpam-4574	294	15	cycles	cycle	NOUN
ejpam-4574	294	16	of	of	ADP
ejpam-4574	294	17	c2k	c2k	NOUN
ejpam-4574	294	18	n	n	NOUN
ejpam-4574	294	19	,	,	PUNCT
ejpam-4574	294	20	and	and	CCONJ
ejpam-4574	294	21	|ti|	|ti|	NOUN
ejpam-4574	294	22	=	=	PUNCT
ejpam-4574	295	1	|tj	|tj	NUM
ejpam-4574	296	1	|	|	ADV
ejpam-4574	296	2	≥	≥	VERB
ejpam-4574	296	3	n	n	PRON
ejpam-4574	296	4	2k	2k	NOUN
ejpam-4574	296	5	for	for	ADP
ejpam-4574	296	6	some	some	DET
ejpam-4574	296	7	i	i	PROPN
ejpam-4574	296	8	and	and	CCONJ
ejpam-4574	296	9	j.	j.	PROPN
ejpam-4574	296	10	definition	definition	NOUN
ejpam-4574	296	11	3	3	NUM
ejpam-4574	296	12	.	.	PUNCT
ejpam-4574	297	1	a	a	DET
ejpam-4574	297	2	clique	clique	NOUN
ejpam-4574	297	3	of	of	ADP
ejpam-4574	297	4	ck	ck	PROPN
ejpam-4574	297	5	n	n	PROPN
ejpam-4574	297	6	is	be	AUX
ejpam-4574	297	7	called	call	VERB
ejpam-4574	297	8	consecutive	consecutive	ADJ
ejpam-4574	297	9	if	if	SCONJ
ejpam-4574	297	10	it	it	PRON
ejpam-4574	297	11	is	be	AUX
ejpam-4574	297	12	composed	compose	VERB
ejpam-4574	297	13	of	of	ADP
ejpam-4574	297	14	vertices	vertex	NOUN
ejpam-4574	297	15	of	of	ADP
ejpam-4574	297	16	consecutive	consecutive	ADJ
ejpam-4574	297	17	integer	integer	NOUN
ejpam-4574	297	18	indices	index	NOUN
ejpam-4574	297	19	(	(	PUNCT
ejpam-4574	297	20	module	module	NOUN
ejpam-4574	297	21	n	n	CCONJ
ejpam-4574	297	22	)	)	PUNCT
ejpam-4574	297	23	.	.	PUNCT
ejpam-4574	298	1	lemma	lemma	PROPN
ejpam-4574	298	2	5	5	NUM
ejpam-4574	298	3	.	.	PUNCT
ejpam-4574	299	1	[	[	X
ejpam-4574	299	2	7	7	X
ejpam-4574	299	3	]	]	X
ejpam-4574	299	4	let	let	VERB
ejpam-4574	299	5	n	n	PRON
ejpam-4574	299	6	≥	≥	PRON
ejpam-4574	299	7	max{3	max{3	NOUN
ejpam-4574	299	8	,	,	PUNCT
ejpam-4574	299	9	k+	k+	NOUN
ejpam-4574	299	10	1	1	NUM
ejpam-4574	299	11	}	}	PUNCT
ejpam-4574	299	12	,	,	PUNCT
ejpam-4574	299	13	write	write	VERB
ejpam-4574	299	14	n	n	NOUN
ejpam-4574	299	15	=	=	PUNCT
ejpam-4574	299	16	q(k+	q(k+	PROPN
ejpam-4574	299	17	1	1	NUM
ejpam-4574	299	18	)	)	PUNCT
ejpam-4574	300	1	+	+	CCONJ
ejpam-4574	300	2	r	r	NOUN
ejpam-4574	300	3	where	where	SCONJ
ejpam-4574	300	4	q	q	PROPN
ejpam-4574	300	5	≥	≥	NUM
ejpam-4574	300	6	1	1	NUM
ejpam-4574	300	7	and	and	CCONJ
ejpam-4574	300	8	0	0	NUM
ejpam-4574	300	9	≤	≤	NUM
ejpam-4574	300	10	r	r	NOUN
ejpam-4574	300	11	≤	≤	NUM
ejpam-4574	300	12	k.	k.	NOUN
ejpam-4574	300	13	then	then	ADV
ejpam-4574	300	14	χ(ck	χ(ck	PROPN
ejpam-4574	300	15	n	n	CCONJ
ejpam-4574	300	16	)	)	PUNCT
ejpam-4574	300	17	=	=	SYM
ejpam-4574	301	1	k	k	PROPN
ejpam-4574	302	1	+	+	CCONJ
ejpam-4574	302	2	1	1	NUM
ejpam-4574	302	3	+	+	CCONJ
ejpam-4574	302	4	⌈	⌈	NUM
ejpam-4574	302	5	rq	rq	VERB
ejpam-4574	302	6	⌉.	⌉.	ADV
ejpam-4574	302	7	lemma	lemma	PROPN
ejpam-4574	302	8	6	6	NUM
ejpam-4574	302	9	.	.	PUNCT
ejpam-4574	303	1	[	[	X
ejpam-4574	303	2	6	6	NUM
ejpam-4574	303	3	]	]	PUNCT
ejpam-4574	303	4	for	for	ADP
ejpam-4574	303	5	any	any	DET
ejpam-4574	303	6	two	two	NUM
ejpam-4574	303	7	integers	integer	NOUN
ejpam-4574	303	8	n	n	CCONJ
ejpam-4574	303	9	,	,	PUNCT
ejpam-4574	303	10	k	k	PROPN
ejpam-4574	303	11	if	if	SCONJ
ejpam-4574	303	12	α|n	α|n	X
ejpam-4574	303	13	and	and	CCONJ
ejpam-4574	303	14	k	k	X
ejpam-4574	303	15	<	<	X
ejpam-4574	303	16	h	h	NOUN
ejpam-4574	303	17	then	then	ADV
ejpam-4574	303	18	ck	ck	INTJ
ejpam-4574	303	19	n	n	ADV
ejpam-4574	303	20	is	be	AUX
ejpam-4574	303	21	α−colorable	α−colorable	ADJ
ejpam-4574	303	22	if	if	SCONJ
ejpam-4574	304	1	and	and	CCONJ
ejpam-4574	304	2	only	only	ADV
ejpam-4574	305	1	if	if	SCONJ
ejpam-4574	305	2	ck	ck	PROPN
ejpam-4574	305	3	n	n	ADV
ejpam-4574	305	4	contains	contain	VERB
ejpam-4574	305	5	no	no	DET
ejpam-4574	305	6	α+	α+	NOUN
ejpam-4574	305	7	1	1	NUM
ejpam-4574	305	8	consecutive	consecutive	ADJ
ejpam-4574	305	9	clique	clique	NOUN
ejpam-4574	305	10	..	..	PUNCT
ejpam-4574	305	11	lemma	lemma	PROPN
ejpam-4574	305	12	7	7	X
ejpam-4574	305	13	.	.	PUNCT
ejpam-4574	306	1	[	[	X
ejpam-4574	306	2	5	5	NUM
ejpam-4574	306	3	]	]	PUNCT
ejpam-4574	306	4	let	let	VERB
ejpam-4574	306	5	n	n	X
ejpam-4574	306	6	=	=	PUNCT
ejpam-4574	306	7	q(k	q(k	PROPN
ejpam-4574	306	8	+	+	CCONJ
ejpam-4574	306	9	1	1	X
ejpam-4574	306	10	)	)	PUNCT
ejpam-4574	307	1	+	+	NOUN
ejpam-4574	307	2	r	r	NOUN
ejpam-4574	308	1	where	where	SCONJ
ejpam-4574	308	2	q	q	PROPN
ejpam-4574	308	3	≥	≥	NUM
ejpam-4574	308	4	1	1	NUM
ejpam-4574	308	5	and	and	CCONJ
ejpam-4574	308	6	0	0	NUM
ejpam-4574	308	7	≤	≤	NUM
ejpam-4574	308	8	r	r	NOUN
ejpam-4574	308	9	≤	≤	PUNCT
ejpam-4574	308	10	k.	k.	NOUN
ejpam-4574	308	11	then	then	ADV
ejpam-4574	308	12	(	(	PUNCT
ejpam-4574	308	13	1	1	X
ejpam-4574	308	14	)	)	PUNCT
ejpam-4574	308	15	r	r	NOUN
ejpam-4574	308	16	=	=	SYM
ejpam-4574	308	17	0	0	NUM
ejpam-4574	308	18	implies	imply	VERB
ejpam-4574	308	19	that	that	PRON
ejpam-4574	308	20	c′1	c′1	NOUN
ejpam-4574	308	21	defined	define	VERB
ejpam-4574	308	22	by	by	ADP
ejpam-4574	308	23	c′1(vi	c′1(vi	NOUN
ejpam-4574	308	24	)	)	PUNCT
ejpam-4574	308	25	=	=	VERB
ejpam-4574	309	1	i	i	PRON
ejpam-4574	309	2	mod	mod	NOUN
ejpam-4574	309	3	(	(	PUNCT
ejpam-4574	309	4	k+1	k+1	NOUN
ejpam-4574	309	5	)	)	PUNCT
ejpam-4574	309	6	is	be	AUX
ejpam-4574	309	7	a	a	DET
ejpam-4574	309	8	(	(	PUNCT
ejpam-4574	309	9	k+1)−proper	k+1)−proper	X
ejpam-4574	309	10	coloring	coloring	NOUN
ejpam-4574	309	11	of	of	ADP
ejpam-4574	309	12	ck	ck	PROPN
ejpam-4574	309	13	n.	n.	NOUN
ejpam-4574	309	14	(	(	PUNCT
ejpam-4574	309	15	2	2	NUM
ejpam-4574	309	16	)	)	PUNCT
ejpam-4574	309	17	r	r	NOUN
ejpam-4574	309	18	̸=	̸=	PROPN
ejpam-4574	309	19	0	0	NUM
ejpam-4574	309	20	,	,	PUNCT
ejpam-4574	309	21	k1	k1	NOUN
ejpam-4574	309	22	=	=	SYM
ejpam-4574	309	23	⌈	⌈	NOUN
ejpam-4574	309	24	rq	rq	X
ejpam-4574	309	25	⌉	⌉	NOUN
ejpam-4574	309	26	,	,	PUNCT
ejpam-4574	309	27	and	and	CCONJ
ejpam-4574	309	28	t	t	X
ejpam-4574	309	29	=	=	PUNCT
ejpam-4574	309	30	⌊	⌊	VERB
ejpam-4574	309	31	r	r	NOUN
ejpam-4574	309	32	k1	k1	NOUN
ejpam-4574	309	33	⌋	⌋	NOUN
ejpam-4574	309	34	,	,	PUNCT
ejpam-4574	310	1	w	w	PROPN
ejpam-4574	310	2	=	=	SYM
ejpam-4574	310	3	k	k	PROPN
ejpam-4574	311	1	+	+	PROPN
ejpam-4574	311	2	1	1	NUM
ejpam-4574	311	3	+	+	NUM
ejpam-4574	311	4	r	r	NOUN
ejpam-4574	311	5	−	−	PROPN
ejpam-4574	311	6	k1	k1	PROPN
ejpam-4574	311	7	t	t	PROPN
ejpam-4574	311	8	,	,	PUNCT
ejpam-4574	311	9	and	and	CCONJ
ejpam-4574	311	10	α	α	X
ejpam-4574	311	11	=	=	PUNCT
ejpam-4574	311	12	k	k	X
ejpam-4574	312	1	+	+	PROPN
ejpam-4574	312	2	1	1	NUM
ejpam-4574	312	3	+	+	NUM
ejpam-4574	312	4	k1	k1	NOUN
ejpam-4574	312	5	imply	imply	VERB
ejpam-4574	312	6	that	that	SCONJ
ejpam-4574	312	7	c′2	c′2	NOUN
ejpam-4574	312	8	defined	define	VERB
ejpam-4574	312	9	by	by	ADP
ejpam-4574	312	10	c′2(vi	c′2(vi	NOUN
ejpam-4574	312	11	)	)	PUNCT
ejpam-4574	312	12	=	=	NOUN
ejpam-4574	312	13	{	{	PUNCT
ejpam-4574	312	14	ci	ci	PROPN
ejpam-4574	312	15	mod	mod	PROPN
ejpam-4574	312	16	α	α	PROPN
ejpam-4574	312	17	if	if	SCONJ
ejpam-4574	312	18	i	i	PRON
ejpam-4574	312	19	∈	∈	PROPN
ejpam-4574	312	20	{	{	PUNCT
ejpam-4574	312	21	0	0	NUM
ejpam-4574	312	22	,	,	PUNCT
ejpam-4574	312	23	1	1	NUM
ejpam-4574	312	24	,	,	PUNCT
ejpam-4574	312	25	2	2	NUM
ejpam-4574	312	26	,	,	PUNCT
ejpam-4574	312	27	...	...	PUNCT
ejpam-4574	312	28	,	,	PUNCT
ejpam-4574	312	29	tα+	tα+	ADV
ejpam-4574	313	1	w	w	NOUN
ejpam-4574	313	2	−	−	PROPN
ejpam-4574	313	3	1	1	NUM
ejpam-4574	313	4	}	}	PUNCT
ejpam-4574	313	5	c(i−(tα+w	c(i−(tα+w	NOUN
ejpam-4574	313	6	)	)	PUNCT
ejpam-4574	313	7	)	)	PUNCT
ejpam-4574	314	1	mod	mod	NOUN
ejpam-4574	314	2	(	(	PUNCT
ejpam-4574	314	3	k+1	k+1	NOUN
ejpam-4574	314	4	)	)	PUNCT
ejpam-4574	314	5	if	if	SCONJ
ejpam-4574	314	6	i	i	PRON
ejpam-4574	314	7	∈	∈	PROPN
ejpam-4574	314	8	{	{	PUNCT
ejpam-4574	314	9	tα+	tα+	NOUN
ejpam-4574	314	10	w	w	PROPN
ejpam-4574	314	11	,	,	PUNCT
ejpam-4574	314	12	...	...	PUNCT
ejpam-4574	314	13	,	,	PUNCT
ejpam-4574	314	14	n−	n−	NOUN
ejpam-4574	314	15	1	1	NUM
ejpam-4574	314	16	}	}	PUNCT
ejpam-4574	314	17	is	be	AUX
ejpam-4574	314	18	a	a	DET
ejpam-4574	314	19	proper	proper	ADJ
ejpam-4574	314	20	-	-	PUNCT
ejpam-4574	314	21	coloring	coloring	NOUN
ejpam-4574	314	22	of	of	ADP
ejpam-4574	314	23	ck	ck	NOUN
ejpam-4574	314	24	n	n	NOUN
ejpam-4574	314	25	using	use	VERB
ejpam-4574	314	26	h	h	NOUN
ejpam-4574	314	27	colors	color	NOUN
ejpam-4574	314	28	only	only	ADV
ejpam-4574	314	29	.	.	PUNCT
ejpam-4574	315	1	proof	proof	NOUN
ejpam-4574	315	2	.	.	PUNCT
ejpam-4574	316	1	(	(	PUNCT
ejpam-4574	316	2	1	1	X
ejpam-4574	316	3	)	)	PUNCT
ejpam-4574	316	4	r	r	NOUN
ejpam-4574	316	5	=	=	SYM
ejpam-4574	316	6	0	0	NUM
ejpam-4574	316	7	implies	imply	VERB
ejpam-4574	316	8	that	that	SCONJ
ejpam-4574	316	9	α	α	NOUN
ejpam-4574	317	1	=	=	PUNCT
ejpam-4574	317	2	k	k	X
ejpam-4574	317	3	+	+	CCONJ
ejpam-4574	317	4	1	1	NUM
ejpam-4574	317	5	divides	divide	NOUN
ejpam-4574	317	6	n	n	ADV
ejpam-4574	317	7	and	and	CCONJ
ejpam-4574	317	8	by	by	ADP
ejpam-4574	317	9	lemma	lemma	PROPN
ejpam-4574	317	10	6	6	NUM
ejpam-4574	317	11	,	,	PUNCT
ejpam-4574	317	12	ck	ck	PRON
ejpam-4574	317	13	n	n	PRON
ejpam-4574	317	14	can	can	AUX
ejpam-4574	317	15	be	be	AUX
ejpam-4574	317	16	colored	color	VERB
ejpam-4574	317	17	using	use	VERB
ejpam-4574	317	18	k	k	PROPN
ejpam-4574	317	19	+	+	CCONJ
ejpam-4574	317	20	1	1	NUM
ejpam-4574	317	21	colors	color	NOUN
ejpam-4574	317	22	.	.	PUNCT
ejpam-4574	318	1	(	(	PUNCT
ejpam-4574	318	2	2	2	X
ejpam-4574	318	3	)	)	PUNCT
ejpam-4574	318	4	let	let	VERB
ejpam-4574	318	5	r	r	PRON
ejpam-4574	318	6	̸=	̸=	PROPN
ejpam-4574	318	7	0	0	NUM
ejpam-4574	318	8	,	,	PUNCT
ejpam-4574	318	9	k1	k1	NOUN
ejpam-4574	318	10	=	=	SYM
ejpam-4574	318	11	⌈	⌈	NOUN
ejpam-4574	318	12	rq	rq	X
ejpam-4574	318	13	⌉	⌉	NOUN
ejpam-4574	318	14	,	,	PUNCT
ejpam-4574	318	15	t	t	X
ejpam-4574	318	16	=	=	SYM
ejpam-4574	318	17	⌊	⌊	VERB
ejpam-4574	318	18	r	r	NOUN
ejpam-4574	318	19	k1	k1	NOUN
ejpam-4574	318	20	⌋	⌋	NOUN
ejpam-4574	318	21	,	,	PUNCT
ejpam-4574	318	22	and	and	CCONJ
ejpam-4574	319	1	α	α	X
ejpam-4574	319	2	=	=	PUNCT
ejpam-4574	320	1	k	k	X
ejpam-4574	321	1	+	+	CCONJ
ejpam-4574	321	2	1	1	NUM
ejpam-4574	321	3	+	+	NUM
ejpam-4574	321	4	k1	k1	NOUN
ejpam-4574	321	5	.	.	PUNCT
ejpam-4574	322	1	then	then	ADV
ejpam-4574	322	2	we	we	PRON
ejpam-4574	322	3	have	have	VERB
ejpam-4574	322	4	two	two	NUM
ejpam-4574	322	5	cases	case	NOUN
ejpam-4574	322	6	,	,	PUNCT
ejpam-4574	322	7	case	case	NOUN
ejpam-4574	322	8	1	1	NUM
ejpam-4574	322	9	.	.	PUNCT
ejpam-4574	322	10	q	q	PUNCT
ejpam-4574	323	1	=	=	PUNCT
ejpam-4574	323	2	t.	t.	NOUN
ejpam-4574	323	3	in	in	ADP
ejpam-4574	323	4	this	this	DET
ejpam-4574	323	5	case	case	NOUN
ejpam-4574	323	6	we	we	PRON
ejpam-4574	323	7	have	have	VERB
ejpam-4574	323	8	q	q	NOUN
ejpam-4574	323	9	=	=	SYM
ejpam-4574	323	10	t	t	NOUN
ejpam-4574	323	11	≤	≤	NUM
ejpam-4574	323	12	r	r	NOUN
ejpam-4574	323	13	k1	k1	NOUN
ejpam-4574	323	14	≤	≤	NUM
ejpam-4574	323	15	r	r	NOUN
ejpam-4574	323	16	r	r	NOUN
ejpam-4574	323	17	q	q	NOUN
ejpam-4574	324	1	=	=	SYM
ejpam-4574	324	2	q	q	NOUN
ejpam-4574	324	3	,	,	PUNCT
ejpam-4574	324	4	and	and	CCONJ
ejpam-4574	324	5	thus	thus	ADV
ejpam-4574	324	6	k1	k1	X
ejpam-4574	324	7	=	=	SYM
ejpam-4574	324	8	r	r	NOUN
ejpam-4574	324	9	q	q	NOUN
ejpam-4574	324	10	,	,	PUNCT
ejpam-4574	324	11	which	which	PRON
ejpam-4574	324	12	implies	imply	VERB
ejpam-4574	324	13	that	that	SCONJ
ejpam-4574	324	14	α	α	NOUN
ejpam-4574	325	1	=	=	PUNCT
ejpam-4574	325	2	k	k	PROPN
ejpam-4574	326	1	+	+	CCONJ
ejpam-4574	326	2	1	1	NUM
ejpam-4574	326	3	+	+	NUM
ejpam-4574	326	4	r	r	NOUN
ejpam-4574	326	5	q	q	NOUN
ejpam-4574	327	1	and	and	CCONJ
ejpam-4574	327	2	so	so	ADV
ejpam-4574	327	3	,	,	PUNCT
ejpam-4574	327	4	qα	qα	PROPN
ejpam-4574	328	1	=	=	SYM
ejpam-4574	328	2	q(k	q(k	PROPN
ejpam-4574	328	3	+	+	CCONJ
ejpam-4574	328	4	1	1	X
ejpam-4574	328	5	)	)	PUNCT
ejpam-4574	328	6	+	+	CCONJ
ejpam-4574	328	7	r	r	NOUN
ejpam-4574	328	8	=	=	SYM
ejpam-4574	328	9	n.	n.	NOUN
ejpam-4574	328	10	therefore	therefore	ADV
ejpam-4574	328	11	,	,	PUNCT
ejpam-4574	328	12	α	α	PROPN
ejpam-4574	328	13	divides	divide	VERB
ejpam-4574	328	14	n	n	ADV
ejpam-4574	328	15	and	and	CCONJ
ejpam-4574	328	16	by	by	ADP
ejpam-4574	328	17	lemma	lemma	PROPN
ejpam-4574	328	18	6	6	NUM
ejpam-4574	328	19	,	,	PUNCT
ejpam-4574	328	20	c′2	c′2	NOUN
ejpam-4574	328	21	is	be	AUX
ejpam-4574	328	22	a	a	PRON
ejpam-4574	328	23	α−	α−	ADP
ejpam-4574	328	24	proper	proper	ADJ
ejpam-4574	328	25	coloring	coloring	NOUN
ejpam-4574	328	26	of	of	ADP
ejpam-4574	328	27	ck	ck	PROPN
ejpam-4574	328	28	n.	n.	PROPN
ejpam-4574	328	29	case	case	NOUN
ejpam-4574	328	30	2	2	X
ejpam-4574	328	31	.	.	PUNCT
ejpam-4574	329	1	q	q	PROPN
ejpam-4574	329	2	̸=	̸=	PROPN
ejpam-4574	329	3	t.	t.	NOUN
ejpam-4574	329	4	let	let	VERB
ejpam-4574	329	5	w	w	NOUN
ejpam-4574	329	6	=	=	PUNCT
ejpam-4574	329	7	k	k	PROPN
ejpam-4574	330	1	+	+	CCONJ
ejpam-4574	330	2	1	1	NUM
ejpam-4574	330	3	+	+	NUM
ejpam-4574	330	4	r	r	NOUN
ejpam-4574	330	5	−	−	PROPN
ejpam-4574	330	6	k1	k1	PROPN
ejpam-4574	330	7	t	t	PROPN
ejpam-4574	330	8	,	,	PUNCT
ejpam-4574	330	9	and	and	CCONJ
ejpam-4574	330	10	color	color	NOUN
ejpam-4574	330	11	ck	ck	PRON
ejpam-4574	331	1	n	n	CCONJ
ejpam-4574	331	2	using	use	VERB
ejpam-4574	331	3	c′2	c′2	NOUN
ejpam-4574	331	4	.	.	PUNCT
ejpam-4574	332	1	notice	notice	VERB
ejpam-4574	332	2	that	that	SCONJ
ejpam-4574	332	3	r	r	NOUN
ejpam-4574	332	4	≥	≥	NOUN
ejpam-4574	332	5	k1⌊	k1⌊	PROPN
ejpam-4574	332	6	r	r	NOUN
ejpam-4574	332	7	k1	k1	NOUN
ejpam-4574	332	8	⌋	⌋	NOUN
ejpam-4574	332	9	=	=	SYM
ejpam-4574	332	10	k1	k1	PROPN
ejpam-4574	332	11	t	t	PROPN
ejpam-4574	332	12	and	and	CCONJ
ejpam-4574	332	13	r	r	NOUN
ejpam-4574	332	14	−	−	PROPN
ejpam-4574	332	15	k1	k1	NOUN
ejpam-4574	332	16	=	=	SYM
ejpam-4574	332	17	k1	k1	PROPN
ejpam-4574	332	18	(	(	PUNCT
ejpam-4574	332	19	r	r	NOUN
ejpam-4574	332	20	k1	k1	NOUN
ejpam-4574	332	21	−	−	PROPN
ejpam-4574	332	22	1	1	NUM
ejpam-4574	332	23	)	)	PUNCT
ejpam-4574	332	24	≤	≤	NOUN
ejpam-4574	332	25	k1⌊	k1⌊	PROPN
ejpam-4574	332	26	r	r	NOUN
ejpam-4574	332	27	k1	k1	NOUN
ejpam-4574	332	28	⌋	⌋	NOUN
ejpam-4574	332	29	=	=	SYM
ejpam-4574	332	30	k1	k1	PROPN
ejpam-4574	332	31	t	t	PROPN
ejpam-4574	332	32	which	which	PRON
ejpam-4574	332	33	leads	lead	VERB
ejpam-4574	332	34	to	to	ADP
ejpam-4574	332	35	0	0	NUM
ejpam-4574	332	36	≤	≤	NOUN
ejpam-4574	332	37	r	r	NOUN
ejpam-4574	332	38	−	−	NOUN
ejpam-4574	332	39	k1	k1	PROPN
ejpam-4574	332	40	t	t	NOUN
ejpam-4574	332	41	≤	≤	NUM
ejpam-4574	332	42	k1	k1	NOUN
ejpam-4574	332	43	.	.	PUNCT
ejpam-4574	333	1	thus	thus	ADV
ejpam-4574	333	2	k	k	X
ejpam-4574	333	3	+	+	CCONJ
ejpam-4574	333	4	1	1	NUM
ejpam-4574	333	5	≤	≤	NOUN
ejpam-4574	333	6	w	w	NOUN
ejpam-4574	333	7	≤	≤	NUM
ejpam-4574	333	8	α	α	NOUN
ejpam-4574	333	9	.	.	PUNCT
ejpam-4574	334	1	moreover	moreover	ADV
ejpam-4574	334	2	,	,	PUNCT
ejpam-4574	334	3	the	the	DET
ejpam-4574	334	4	cardinality	cardinality	NOUN
ejpam-4574	334	5	of	of	ADP
ejpam-4574	334	6	the	the	DET
ejpam-4574	334	7	subset	subset	NOUN
ejpam-4574	334	8	of	of	ADP
ejpam-4574	334	9	vertices	vertex	NOUN
ejpam-4574	334	10	{	{	PUNCT
ejpam-4574	334	11	vtα+w	vtα+w	NUM
ejpam-4574	334	12	,	,	PUNCT
ejpam-4574	334	13	...	...	PUNCT
ejpam-4574	334	14	,	,	PUNCT
ejpam-4574	334	15	vn−1	vn−1	PROPN
ejpam-4574	334	16	}	}	PUNCT
ejpam-4574	334	17	is	be	AUX
ejpam-4574	334	18	a	a	DET
ejpam-4574	334	19	multiple	multiple	NOUN
ejpam-4574	334	20	of	of	ADP
ejpam-4574	334	21	(	(	PUNCT
ejpam-4574	334	22	k	k	PROPN
ejpam-4574	334	23	+	+	PROPN
ejpam-4574	334	24	1	1	X
ejpam-4574	334	25	)	)	PUNCT
ejpam-4574	334	26	since	since	SCONJ
ejpam-4574	334	27	n−	n−	NOUN
ejpam-4574	334	28	tα−	tα−	PUNCT
ejpam-4574	334	29	w	w	NOUN
ejpam-4574	334	30	=	=	SYM
ejpam-4574	334	31	(	(	PUNCT
ejpam-4574	334	32	q	q	NOUN
ejpam-4574	334	33	−	−	PROPN
ejpam-4574	334	34	t−	t−	PROPN
ejpam-4574	334	35	1)(k	1)(k	NUM
ejpam-4574	334	36	+	+	CCONJ
ejpam-4574	334	37	1	1	NUM
ejpam-4574	334	38	)	)	PUNCT
ejpam-4574	334	39	and	and	CCONJ
ejpam-4574	334	40	t	t	X
ejpam-4574	334	41	≤	≤	NUM
ejpam-4574	334	42	q	q	NOUN
ejpam-4574	334	43	−	−	PROPN
ejpam-4574	334	44	1	1	NUM
ejpam-4574	334	45	.	.	PUNCT
ejpam-4574	335	1	finally	finally	ADV
ejpam-4574	335	2	,	,	PUNCT
ejpam-4574	335	3	note	note	VERB
ejpam-4574	335	4	that	that	SCONJ
ejpam-4574	335	5	c′2(vtα+w−1	c′2(vtα+w−1	NOUN
ejpam-4574	335	6	)	)	PUNCT
ejpam-4574	336	1	=	=	PUNCT
ejpam-4574	337	1	w	w	ADP
ejpam-4574	337	2	−	−	NOUN
ejpam-4574	337	3	1	1	NUM
ejpam-4574	337	4	,	,	PUNCT
ejpam-4574	337	5	and	and	CCONJ
ejpam-4574	337	6	so	so	ADV
ejpam-4574	337	7	k	k	PROPN
ejpam-4574	337	8	≤	≤	PROPN
ejpam-4574	337	9	c′2(vtα+w−1	c′2(vtα+w−1	NOUN
ejpam-4574	337	10	)	)	PUNCT
ejpam-4574	337	11	≤	≤	NOUN
ejpam-4574	338	1	α	α	PRON
ejpam-4574	338	2	−	−	NOUN
ejpam-4574	338	3	1	1	NUM
ejpam-4574	338	4	.	.	PUNCT
ejpam-4574	338	5	also	also	ADV
ejpam-4574	338	6	c′2(vn−1	c′2(vn−1	ADJ
ejpam-4574	338	7	)	)	PUNCT
ejpam-4574	338	8	=	=	VERB
ejpam-4574	339	1	k.	k.	PROPN
ejpam-4574	339	2	hence	hence	ADV
ejpam-4574	339	3	c′2	c′2	NOUN
ejpam-4574	339	4	is	be	AUX
ejpam-4574	339	5	a	a	DET
ejpam-4574	339	6	proper	proper	ADJ
ejpam-4574	339	7	coloring	coloring	NOUN
ejpam-4574	339	8	of	of	ADP
ejpam-4574	339	9	ck	ck	PROPN
ejpam-4574	339	10	n	n	PROPN
ejpam-4574	339	11	that	that	PRON
ejpam-4574	339	12	uses	use	VERB
ejpam-4574	339	13	at	at	ADP
ejpam-4574	339	14	most	most	ADJ
ejpam-4574	339	15	α	α	NUM
ejpam-4574	339	16	colors	color	NOUN
ejpam-4574	339	17	.	.	PUNCT
ejpam-4574	340	1	in	in	ADP
ejpam-4574	340	2	general	general	ADJ
ejpam-4574	340	3	when	when	SCONJ
ejpam-4574	340	4	n	n	X
ejpam-4574	340	5	<	<	X
ejpam-4574	340	6	(	(	PUNCT
ejpam-4574	340	7	k+1)2	k+1)2	PROPN
ejpam-4574	340	8	,	,	PUNCT
ejpam-4574	340	9	χa(c	χa(c	X
ejpam-4574	340	10	k	k	PROPN
ejpam-4574	340	11	n	n	CCONJ
ejpam-4574	340	12	)	)	PUNCT
ejpam-4574	340	13	does	do	AUX
ejpam-4574	340	14	not	not	PART
ejpam-4574	340	15	have	have	VERB
ejpam-4574	340	16	a	a	DET
ejpam-4574	340	17	lower	low	ADJ
ejpam-4574	340	18	bound	bind	VERB
ejpam-4574	340	19	in	in	ADP
ejpam-4574	340	20	terms	term	NOUN
ejpam-4574	340	21	of	of	ADP
ejpam-4574	340	22	k	k	PRON
ejpam-4574	340	23	since	since	SCONJ
ejpam-4574	340	24	the	the	DET
ejpam-4574	340	25	value	value	NOUN
ejpam-4574	340	26	of	of	ADP
ejpam-4574	340	27	χ(ck	χ(ck	ADJ
ejpam-4574	340	28	n	n	CCONJ
ejpam-4574	340	29	)	)	PUNCT
ejpam-4574	340	30	changes	change	NOUN
ejpam-4574	340	31	as	as	ADP
ejpam-4574	340	32	the	the	DET
ejpam-4574	340	33	ratio	ratio	NOUN
ejpam-4574	340	34	r	r	NOUN
ejpam-4574	340	35	q	q	NOUN
ejpam-4574	340	36	change	change	NOUN
ejpam-4574	340	37	,	,	PUNCT
ejpam-4574	340	38	to	to	PART
ejpam-4574	340	39	illustrate	illustrate	VERB
ejpam-4574	340	40	this	this	PRON
ejpam-4574	340	41	consider	consider	VERB
ejpam-4574	340	42	the	the	DET
ejpam-4574	340	43	following	follow	VERB
ejpam-4574	340	44	example	example	NOUN
ejpam-4574	340	45	.	.	PUNCT
ejpam-4574	341	1	a.	a.	PROPN
ejpam-4574	341	2	etawi	etawi	PROPN
ejpam-4574	341	3	,	,	PUNCT
ejpam-4574	341	4	m.	m.	PROPN
ejpam-4574	341	5	ghanem	ghanem	PROPN
ejpam-4574	341	6	,	,	PUNCT
ejpam-4574	341	7	h.	h.	PROPN
ejpam-4574	341	8	al	al	PROPN
ejpam-4574	341	9	-	-	PUNCT
ejpam-4574	341	10	ezeh	ezeh	PROPN
ejpam-4574	341	11	/	/	SYM
ejpam-4574	341	12	eur	eur	NOUN
ejpam-4574	341	13	.	.	PUNCT
ejpam-4574	342	1	j.	j.	PROPN
ejpam-4574	342	2	pure	pure	PROPN
ejpam-4574	342	3	appl	appl	PROPN
ejpam-4574	342	4	.	.	PROPN
ejpam-4574	342	5	math	math	PROPN
ejpam-4574	342	6	,	,	PUNCT
ejpam-4574	342	7	15	15	NUM
ejpam-4574	342	8	(	(	PUNCT
ejpam-4574	342	9	4	4	NUM
ejpam-4574	342	10	)	)	PUNCT
ejpam-4574	342	11	(	(	PUNCT
ejpam-4574	342	12	2022	2022	NUM
ejpam-4574	342	13	)	)	PUNCT
ejpam-4574	342	14	,	,	PUNCT
ejpam-4574	342	15	1822	1822	NUM
ejpam-4574	342	16	-	-	SYM
ejpam-4574	342	17	1835	1835	NUM
ejpam-4574	342	18	1830	1830	NUM
ejpam-4574	342	19	example	example	NOUN
ejpam-4574	343	1	5	5	NUM
ejpam-4574	343	2	.	.	PUNCT
ejpam-4574	343	3	let	let	VERB
ejpam-4574	343	4	k	k	NOUN
ejpam-4574	343	5	=	=	SYM
ejpam-4574	343	6	5	5	X
ejpam-4574	343	7	.	.	PUNCT
ejpam-4574	344	1	then	then	ADV
ejpam-4574	344	2	:	:	PUNCT
ejpam-4574	344	3	if	if	SCONJ
ejpam-4574	344	4	n	n	ADV
ejpam-4574	344	5	=	=	SYM
ejpam-4574	344	6	2(k	2(k	NUM
ejpam-4574	344	7	+	+	CCONJ
ejpam-4574	344	8	1	1	X
ejpam-4574	344	9	)	)	PUNCT
ejpam-4574	344	10	+	+	NUM
ejpam-4574	344	11	1	1	NUM
ejpam-4574	344	12	,	,	PUNCT
ejpam-4574	344	13	then	then	ADV
ejpam-4574	344	14	χ(ck	χ(ck	ADJ
ejpam-4574	344	15	n	n	CCONJ
ejpam-4574	344	16	)	)	PUNCT
ejpam-4574	344	17	=	=	SYM
ejpam-4574	345	1	5	5	NUM
ejpam-4574	345	2	+	+	CCONJ
ejpam-4574	345	3	1	1	NUM
ejpam-4574	345	4	+	+	CCONJ
ejpam-4574	345	5	⌈	⌈	SYM
ejpam-4574	345	6	1	1	NUM
ejpam-4574	345	7	2	2	NUM
ejpam-4574	345	8	⌉	⌉	NOUN
ejpam-4574	345	9	=	=	SYM
ejpam-4574	345	10	k	k	PROPN
ejpam-4574	346	1	+	+	PROPN
ejpam-4574	346	2	2	2	NUM
ejpam-4574	346	3	,	,	PUNCT
ejpam-4574	346	4	and	and	CCONJ
ejpam-4574	346	5	χa(c	χa(c	VERB
ejpam-4574	346	6	k	k	PROPN
ejpam-4574	346	7	n	n	CCONJ
ejpam-4574	346	8	)	)	PUNCT
ejpam-4574	346	9	≥	≥	NOUN
ejpam-4574	346	10	k	k	NOUN
ejpam-4574	347	1	+	+	CCONJ
ejpam-4574	347	2	2	2	X
ejpam-4574	347	3	.	.	X
ejpam-4574	348	1	if	if	SCONJ
ejpam-4574	348	2	n	n	NOUN
ejpam-4574	348	3	=	=	SYM
ejpam-4574	348	4	2(k	2(k	NUM
ejpam-4574	349	1	+	+	CCONJ
ejpam-4574	349	2	1	1	X
ejpam-4574	349	3	)	)	PUNCT
ejpam-4574	349	4	+	+	NUM
ejpam-4574	349	5	5	5	NUM
ejpam-4574	349	6	,	,	PUNCT
ejpam-4574	349	7	then	then	ADV
ejpam-4574	349	8	χ(ck	χ(ck	ADJ
ejpam-4574	349	9	n	n	CCONJ
ejpam-4574	349	10	)	)	PUNCT
ejpam-4574	350	1	=	=	SYM
ejpam-4574	350	2	5	5	NUM
ejpam-4574	350	3	+	+	CCONJ
ejpam-4574	350	4	1	1	NUM
ejpam-4574	350	5	+	+	CCONJ
ejpam-4574	350	6	⌈	⌈	SYM
ejpam-4574	350	7	5	5	NUM
ejpam-4574	350	8	2	2	NUM
ejpam-4574	350	9	⌉	⌉	NOUN
ejpam-4574	350	10	=	=	SYM
ejpam-4574	350	11	k	k	PROPN
ejpam-4574	350	12	+	+	NOUN
ejpam-4574	350	13	4	4	NUM
ejpam-4574	350	14	,	,	PUNCT
ejpam-4574	350	15	and	and	CCONJ
ejpam-4574	350	16	χa(c	χa(c	VERB
ejpam-4574	350	17	k	k	PROPN
ejpam-4574	350	18	n	n	CCONJ
ejpam-4574	350	19	)	)	PUNCT
ejpam-4574	350	20	≥	≥	NOUN
ejpam-4574	350	21	k	k	NOUN
ejpam-4574	351	1	+	+	CCONJ
ejpam-4574	351	2	4	4	X
ejpam-4574	351	3	.	.	X
ejpam-4574	351	4	remark	remark	NOUN
ejpam-4574	351	5	1	1	NUM
ejpam-4574	351	6	.	.	PUNCT
ejpam-4574	352	1	:	:	PUNCT
ejpam-4574	352	2	let	let	VERB
ejpam-4574	352	3	n	n	X
ejpam-4574	352	4	=	=	SYM
ejpam-4574	352	5	q(k+1)+	q(k+1)+	PROPN
ejpam-4574	352	6	r	r	NOUN
ejpam-4574	352	7	where	where	SCONJ
ejpam-4574	352	8	q	q	PROPN
ejpam-4574	352	9	≥	≥	X
ejpam-4574	352	10	k+1	k+1	X
ejpam-4574	352	11	and	and	CCONJ
ejpam-4574	352	12	0	0	NUM
ejpam-4574	352	13	≤	≤	NOUN
ejpam-4574	352	14	r	r	NOUN
ejpam-4574	352	15	≤	≤	PROPN
ejpam-4574	352	16	k.	k.	NOUN
ejpam-4574	352	17	apply	apply	VERB
ejpam-4574	352	18	lemma	lemma	PROPN
ejpam-4574	352	19	7	7	NUM
ejpam-4574	352	20	to	to	PART
ejpam-4574	352	21	get	get	VERB
ejpam-4574	352	22	k1	k1	NOUN
ejpam-4574	352	23	=	=	SYM
ejpam-4574	352	24	1	1	NUM
ejpam-4574	352	25	,	,	PUNCT
ejpam-4574	352	26	t	t	NOUN
ejpam-4574	352	27	=	=	SYM
ejpam-4574	352	28	r	r	NOUN
ejpam-4574	352	29	,	,	PUNCT
ejpam-4574	352	30	w	w	NOUN
ejpam-4574	352	31	=	=	SYM
ejpam-4574	352	32	k	k	PROPN
ejpam-4574	353	1	+	+	NOUN
ejpam-4574	353	2	1	1	NUM
ejpam-4574	353	3	,	,	PUNCT
ejpam-4574	353	4	t	t	PROPN
ejpam-4574	353	5	̸=	̸=	PROPN
ejpam-4574	353	6	q	q	PROPN
ejpam-4574	353	7	,	,	PUNCT
ejpam-4574	353	8	α	α	X
ejpam-4574	353	9	=	=	PUNCT
ejpam-4574	353	10	k	k	PROPN
ejpam-4574	354	1	+	+	PROPN
ejpam-4574	354	2	2	2	NUM
ejpam-4574	354	3	,	,	PUNCT
ejpam-4574	354	4	and	and	CCONJ
ejpam-4574	354	5	c′2(vi	c′2(vi	NUM
ejpam-4574	354	6	)	)	PUNCT
ejpam-4574	354	7	=	=	NOUN
ejpam-4574	354	8	{	{	PUNCT
ejpam-4574	354	9	ci	ci	PROPN
ejpam-4574	354	10	mod	mod	PROPN
ejpam-4574	354	11	(	(	PUNCT
ejpam-4574	354	12	k+2	k+2	NOUN
ejpam-4574	354	13	)	)	PUNCT
ejpam-4574	354	14	if	if	SCONJ
ejpam-4574	354	15	i	i	PRON
ejpam-4574	354	16	∈	∈	PROPN
ejpam-4574	354	17	{	{	PUNCT
ejpam-4574	354	18	0	0	NUM
ejpam-4574	354	19	,	,	PUNCT
ejpam-4574	354	20	1	1	NUM
ejpam-4574	354	21	,	,	PUNCT
ejpam-4574	354	22	...	...	PUNCT
ejpam-4574	354	23	,	,	PUNCT
ejpam-4574	355	1	r(k	r(k	PROPN
ejpam-4574	355	2	+	+	ADJ
ejpam-4574	355	3	2	2	NUM
ejpam-4574	355	4	)	)	PUNCT
ejpam-4574	355	5	+	+	CCONJ
ejpam-4574	355	6	k	k	ADJ
ejpam-4574	355	7	}	}	PUNCT
ejpam-4574	355	8	c{i−r(k+2)+k+1	c{i−r(k+2)+k+1	NOUN
ejpam-4574	355	9	}	}	PUNCT
ejpam-4574	355	10	mod	mod	NOUN
ejpam-4574	355	11	(	(	PUNCT
ejpam-4574	355	12	k+1	k+1	NOUN
ejpam-4574	355	13	)	)	PUNCT
ejpam-4574	355	14	if	if	SCONJ
ejpam-4574	355	15	i	i	PRON
ejpam-4574	355	16	∈	∈	PROPN
ejpam-4574	355	17	{	{	PUNCT
ejpam-4574	355	18	r(k	r(k	PROPN
ejpam-4574	355	19	+	+	PROPN
ejpam-4574	355	20	2	2	NUM
ejpam-4574	355	21	)	)	PUNCT
ejpam-4574	355	22	+	+	NOUN
ejpam-4574	355	23	k	k	PROPN
ejpam-4574	355	24	+	+	NOUN
ejpam-4574	355	25	1	1	NUM
ejpam-4574	355	26	,	,	PUNCT
ejpam-4574	355	27	...	...	PUNCT
ejpam-4574	355	28	,	,	PUNCT
ejpam-4574	355	29	n−	n−	NOUN
ejpam-4574	355	30	1	1	NUM
ejpam-4574	355	31	}	}	PUNCT
ejpam-4574	355	32	is	be	AUX
ejpam-4574	355	33	a	a	DET
ejpam-4574	355	34	(	(	PUNCT
ejpam-4574	355	35	k	k	NOUN
ejpam-4574	355	36	+	+	PROPN
ejpam-4574	355	37	2)−proper	2)−proper	NUM
ejpam-4574	355	38	coloring	coloring	NOUN
ejpam-4574	355	39	of	of	ADP
ejpam-4574	355	40	ck	ck	PROPN
ejpam-4574	355	41	n.	n.	NOUN
ejpam-4574	355	42	in	in	ADP
ejpam-4574	355	43	the	the	DET
ejpam-4574	355	44	following	follow	VERB
ejpam-4574	355	45	lemmas	lemmas	PROPN
ejpam-4574	355	46	,	,	PUNCT
ejpam-4574	355	47	we	we	PRON
ejpam-4574	355	48	will	will	AUX
ejpam-4574	355	49	consider	consider	VERB
ejpam-4574	355	50	n	n	PRON
ejpam-4574	355	51	≥	≥	X
ejpam-4574	355	52	(	(	PUNCT
ejpam-4574	355	53	k	k	PROPN
ejpam-4574	355	54	+	+	PROPN
ejpam-4574	355	55	1)2	1)2	NUM
ejpam-4574	355	56	.	.	PUNCT
ejpam-4574	356	1	lemma	lemma	PROPN
ejpam-4574	356	2	8	8	NUM
ejpam-4574	356	3	.	.	PUNCT
ejpam-4574	357	1	let	let	VERB
ejpam-4574	357	2	n	n	NOUN
ejpam-4574	357	3	=	=	PUNCT
ejpam-4574	357	4	q(k	q(k	PROPN
ejpam-4574	357	5	+	+	NOUN
ejpam-4574	357	6	1	1	X
ejpam-4574	357	7	)	)	PUNCT
ejpam-4574	358	1	where	where	SCONJ
ejpam-4574	358	2	q	q	PRON
ejpam-4574	358	3	≥	≥	X
ejpam-4574	358	4	k	k	X
ejpam-4574	358	5	+	+	CCONJ
ejpam-4574	358	6	1	1	NUM
ejpam-4574	358	7	,	,	PUNCT
ejpam-4574	358	8	and	and	CCONJ
ejpam-4574	358	9	c′3(vi	c′3(vi	NUM
ejpam-4574	358	10	)	)	PUNCT
ejpam-4574	358	11	=	=	NOUN
ejpam-4574	358	12	{	{	PUNCT
ejpam-4574	358	13	ck+1	ck+1	VERB
ejpam-4574	358	14	if	if	SCONJ
ejpam-4574	358	15	i	i	PRON
ejpam-4574	358	16	=	=	PUNCT
ejpam-4574	358	17	h(k	h(k	PROPN
ejpam-4574	358	18	+	+	CCONJ
ejpam-4574	358	19	2	2	X
ejpam-4574	358	20	)	)	PUNCT
ejpam-4574	358	21	and	and	CCONJ
ejpam-4574	358	22	h	h	NOUN
ejpam-4574	358	23	=	=	SYM
ejpam-4574	358	24	0	0	NUM
ejpam-4574	358	25	,	,	PUNCT
ejpam-4574	358	26	1	1	NUM
ejpam-4574	358	27	,	,	PUNCT
ejpam-4574	358	28	...	...	PUNCT
ejpam-4574	358	29	,	,	PUNCT
ejpam-4574	358	30	k	k	PROPN
ejpam-4574	358	31	−	−	PROPN
ejpam-4574	358	32	1	1	NUM
ejpam-4574	358	33	c′1(vi	c′1(vi	SYM
ejpam-4574	358	34	)	)	PUNCT
ejpam-4574	358	35	otherwise	otherwise	ADV
ejpam-4574	358	36	.	.	PUNCT
ejpam-4574	359	1	then	then	ADV
ejpam-4574	359	2	:	:	PUNCT
ejpam-4574	359	3	(	(	PUNCT
ejpam-4574	359	4	1	1	X
ejpam-4574	359	5	)	)	PUNCT
ejpam-4574	359	6	c′3	c′3	NOUN
ejpam-4574	359	7	is	be	AUX
ejpam-4574	359	8	a	a	DET
ejpam-4574	359	9	(	(	PUNCT
ejpam-4574	359	10	k	k	X
ejpam-4574	359	11	+	+	PROPN
ejpam-4574	359	12	2)−	2)−	NUM
ejpam-4574	359	13	proper	proper	ADJ
ejpam-4574	359	14	coloring	coloring	NOUN
ejpam-4574	359	15	of	of	ADP
ejpam-4574	359	16	ck	ck	PROPN
ejpam-4574	359	17	n.	n.	NOUN
ejpam-4574	359	18	(	(	PUNCT
ejpam-4574	359	19	2	2	X
ejpam-4574	359	20	)	)	PUNCT
ejpam-4574	359	21	ti	ti	NOUN
ejpam-4574	359	22	does	do	AUX
ejpam-4574	359	23	not	not	PART
ejpam-4574	359	24	induce	induce	VERB
ejpam-4574	359	25	a	a	DET
ejpam-4574	359	26	cycle	cycle	NOUN
ejpam-4574	359	27	in	in	ADP
ejpam-4574	359	28	c2k	c2k	PROPN
ejpam-4574	359	29	n	n	NOUN
ejpam-4574	359	30	for	for	ADP
ejpam-4574	359	31	0	0	NUM
ejpam-4574	359	32	≤	≤	NUM
ejpam-4574	360	1	i	i	PRON
ejpam-4574	360	2	≤	≤	NOUN
ejpam-4574	361	1	k	k	PRON
ejpam-4574	362	1	−	−	NOUN
ejpam-4574	362	2	1	1	X
ejpam-4574	362	3	.	.	PUNCT
ejpam-4574	362	4	(	(	PUNCT
ejpam-4574	362	5	3	3	X
ejpam-4574	362	6	)	)	PUNCT
ejpam-4574	362	7	c′3	c′3	NOUN
ejpam-4574	362	8	is	be	AUX
ejpam-4574	362	9	a	a	DET
ejpam-4574	362	10	(	(	PUNCT
ejpam-4574	362	11	k	k	PROPN
ejpam-4574	362	12	+	+	CCONJ
ejpam-4574	362	13	2)−	2)−	NUM
ejpam-4574	362	14	acyclic	acyclic	ADJ
ejpam-4574	362	15	coloring	coloring	NOUN
ejpam-4574	362	16	of	of	ADP
ejpam-4574	362	17	ck	ck	PROPN
ejpam-4574	362	18	n.	n.	NOUN
ejpam-4574	362	19	proof	proof	NOUN
ejpam-4574	362	20	.	.	PUNCT
ejpam-4574	363	1	(	(	PUNCT
ejpam-4574	363	2	1	1	X
ejpam-4574	363	3	)	)	PUNCT
ejpam-4574	363	4	since	since	SCONJ
ejpam-4574	363	5	c′1	c′1	NOUN
ejpam-4574	363	6	is	be	AUX
ejpam-4574	363	7	a	a	DET
ejpam-4574	363	8	proper	proper	ADJ
ejpam-4574	363	9	coloring	coloring	NOUN
ejpam-4574	363	10	it	it	PRON
ejpam-4574	363	11	is	be	AUX
ejpam-4574	363	12	enough	enough	ADJ
ejpam-4574	363	13	to	to	PART
ejpam-4574	363	14	check	check	VERB
ejpam-4574	363	15	the	the	DET
ejpam-4574	363	16	vertices	vertex	NOUN
ejpam-4574	363	17	with	with	ADP
ejpam-4574	363	18	color	color	NOUN
ejpam-4574	363	19	ck+1	ck+1	NOUN
ejpam-4574	363	20	.	.	PUNCT
ejpam-4574	364	1	since	since	SCONJ
ejpam-4574	364	2	dcn(vh(k+2	dcn(vh(k+2	PROPN
ejpam-4574	364	3	)	)	PUNCT
ejpam-4574	364	4	,	,	PUNCT
ejpam-4574	364	5	v(h+1)(k+2	v(h+1)(k+2	NOUN
ejpam-4574	364	6	)	)	PUNCT
ejpam-4574	364	7	)	)	PUNCT
ejpam-4574	365	1	=	=	PUNCT
ejpam-4574	365	2	k+2	k+2	PROPN
ejpam-4574	365	3	for	for	ADP
ejpam-4574	365	4	0	0	NUM
ejpam-4574	365	5	≤	≤	NUM
ejpam-4574	365	6	h	h	NOUN
ejpam-4574	365	7	≤	≤	PROPN
ejpam-4574	365	8	k−1	k−1	PROPN
ejpam-4574	365	9	,	,	PUNCT
ejpam-4574	365	10	and	and	CCONJ
ejpam-4574	365	11	dcn(v(k−1)(k+2	dcn(v(k−1)(k+2	NOUN
ejpam-4574	365	12	)	)	PUNCT
ejpam-4574	365	13	,	,	PUNCT
ejpam-4574	365	14	v0	v0	PROPN
ejpam-4574	365	15	)	)	PUNCT
ejpam-4574	365	16	≥	≥	NOUN
ejpam-4574	365	17	k+2	k+2	NUM
ejpam-4574	365	18	,	,	PUNCT
ejpam-4574	365	19	we	we	PRON
ejpam-4574	365	20	have	have	AUX
ejpam-4574	365	21	c′3	c′3	NOUN
ejpam-4574	365	22	is	be	AUX
ejpam-4574	365	23	a	a	DET
ejpam-4574	365	24	proper	proper	ADJ
ejpam-4574	365	25	coloring	coloring	NOUN
ejpam-4574	365	26	of	of	ADP
ejpam-4574	365	27	ck	ck	PROPN
ejpam-4574	365	28	n.	n.	NOUN
ejpam-4574	365	29	(	(	PUNCT
ejpam-4574	365	30	2	2	NUM
ejpam-4574	365	31	)	)	PUNCT
ejpam-4574	365	32	if	if	SCONJ
ejpam-4574	365	33	q	q	PROPN
ejpam-4574	365	34	≤	≤	NUM
ejpam-4574	365	35	3	3	NUM
ejpam-4574	365	36	,	,	PUNCT
ejpam-4574	365	37	then	then	ADV
ejpam-4574	365	38	|t0|	|t0|	X
ejpam-4574	365	39	=	=	PUNCT
ejpam-4574	365	40	q	q	PROPN
ejpam-4574	365	41	−	−	PROPN
ejpam-4574	365	42	1	1	NUM
ejpam-4574	365	43	<	<	X
ejpam-4574	365	44	q(k+1	q(k+1	PROPN
ejpam-4574	365	45	)	)	PUNCT
ejpam-4574	365	46	2k	2k	NOUN
ejpam-4574	365	47	.	.	PUNCT
ejpam-4574	366	1	now	now	ADV
ejpam-4574	366	2	,	,	PUNCT
ejpam-4574	366	3	consider	consider	VERB
ejpam-4574	366	4	q	q	NOUN
ejpam-4574	366	5	>	>	X
ejpam-4574	366	6	3	3	NUM
ejpam-4574	366	7	,	,	PUNCT
ejpam-4574	366	8	then	then	ADV
ejpam-4574	366	9	t0	t0	PROPN
ejpam-4574	366	10	does	do	AUX
ejpam-4574	366	11	not	not	PART
ejpam-4574	366	12	induce	induce	VERB
ejpam-4574	366	13	a	a	DET
ejpam-4574	366	14	cycle	cycle	NOUN
ejpam-4574	366	15	in	in	ADP
ejpam-4574	366	16	c2k	c2k	PROPN
ejpam-4574	366	17	n	n	CCONJ
ejpam-4574	366	18	since	since	SCONJ
ejpam-4574	366	19	dcn(v(q−1)(k+1	dcn(v(q−1)(k+1	NOUN
ejpam-4574	366	20	)	)	PUNCT
ejpam-4574	366	21	,	,	PUNCT
ejpam-4574	366	22	v(k+1	v(k+1	NOUN
ejpam-4574	366	23	)	)	PUNCT
ejpam-4574	366	24	)	)	PUNCT
ejpam-4574	366	25	≥	≥	NOUN
ejpam-4574	366	26	2(k	2(k	NUM
ejpam-4574	367	1	+	+	CCONJ
ejpam-4574	367	2	1	1	NUM
ejpam-4574	367	3	)	)	PUNCT
ejpam-4574	367	4	.	.	PUNCT
ejpam-4574	368	1	also	also	ADV
ejpam-4574	368	2	ti	ti	X
ejpam-4574	368	3	where	where	SCONJ
ejpam-4574	368	4	1	1	NUM
ejpam-4574	368	5	≤	≤	NUM
ejpam-4574	368	6	i	i	NOUN
ejpam-4574	368	7	≤	≤	PUNCT
ejpam-4574	369	1	k	k	ADP
ejpam-4574	370	1	−	−	NOUN
ejpam-4574	370	2	1	1	NUM
ejpam-4574	370	3	does	do	AUX
ejpam-4574	370	4	n’t	not	PART
ejpam-4574	370	5	induce	induce	VERB
ejpam-4574	370	6	a	a	DET
ejpam-4574	370	7	cycle	cycle	NOUN
ejpam-4574	370	8	in	in	ADP
ejpam-4574	370	9	c2k	c2k	PROPN
ejpam-4574	370	10	n	n	CCONJ
ejpam-4574	370	11	since	since	SCONJ
ejpam-4574	370	12	vi(k+1)+i	vi(k+1)+i	ADV
ejpam-4574	370	13	/∈	/∈	PUNCT
ejpam-4574	370	14	ti	ti	X
ejpam-4574	370	15	and	and	CCONJ
ejpam-4574	370	16	dcn(v(i−1)(k+1)+i	dcn(v(i−1)(k+1)+i	PROPN
ejpam-4574	370	17	,	,	PUNCT
ejpam-4574	370	18	v(i+1)(k+1)+i	v(i+1)(k+1)+i	ADJ
ejpam-4574	370	19	)	)	PUNCT
ejpam-4574	370	20	=	=	SYM
ejpam-4574	371	1	2(k	2(k	NUM
ejpam-4574	371	2	+	+	CCONJ
ejpam-4574	371	3	1	1	NUM
ejpam-4574	371	4	)	)	PUNCT
ejpam-4574	371	5	.	.	PUNCT
ejpam-4574	372	1	note	note	VERB
ejpam-4574	372	2	that	that	SCONJ
ejpam-4574	372	3	the	the	DET
ejpam-4574	372	4	coloring	coloring	NOUN
ejpam-4574	372	5	c′1	c′1	NOUN
ejpam-4574	372	6	colors	color	VERB
ejpam-4574	372	7	the	the	DET
ejpam-4574	372	8	vertices	vertex	NOUN
ejpam-4574	372	9	of	of	ADP
ejpam-4574	372	10	ck	ck	INTJ
ejpam-4574	372	11	n	n	X
ejpam-4574	372	12	by	by	ADP
ejpam-4574	372	13	repeating	repeat	VERB
ejpam-4574	372	14	c1	c1	PROPN
ejpam-4574	372	15	,	,	PUNCT
ejpam-4574	372	16	c2	c2	PROPN
ejpam-4574	372	17	,	,	PUNCT
ejpam-4574	372	18	...	...	PUNCT
ejpam-4574	372	19	,	,	PUNCT
ejpam-4574	372	20	ck	ck	INTJ
ejpam-4574	372	21	,	,	PUNCT
ejpam-4574	372	22	which	which	PRON
ejpam-4574	372	23	makes	make	VERB
ejpam-4574	372	24	the	the	DET
ejpam-4574	372	25	distance	distance	NOUN
ejpam-4574	372	26	between	between	ADP
ejpam-4574	372	27	any	any	DET
ejpam-4574	372	28	two	two	NUM
ejpam-4574	372	29	vertices	vertex	NOUN
ejpam-4574	372	30	having	have	VERB
ejpam-4574	372	31	the	the	DET
ejpam-4574	372	32	same	same	ADJ
ejpam-4574	372	33	color	color	NOUN
ejpam-4574	372	34	be	be	AUX
ejpam-4574	372	35	(	(	PUNCT
ejpam-4574	372	36	k	k	PROPN
ejpam-4574	372	37	+	+	PROPN
ejpam-4574	372	38	1	1	NUM
ejpam-4574	372	39	)	)	PUNCT
ejpam-4574	372	40	.	.	PUNCT
ejpam-4574	373	1	adding	add	VERB
ejpam-4574	373	2	ck+1	ck+1	ADV
ejpam-4574	373	3	in	in	ADP
ejpam-4574	373	4	c′3	c′3	NOUN
ejpam-4574	373	5	denies	deny	VERB
ejpam-4574	373	6	one	one	NUM
ejpam-4574	373	7	occurrence	occurrence	NOUN
ejpam-4574	373	8	of	of	ADP
ejpam-4574	373	9	each	each	DET
ejpam-4574	373	10	color	color	NOUN
ejpam-4574	373	11	of	of	ADP
ejpam-4574	373	12	c1	c1	PROPN
ejpam-4574	373	13	,	,	PUNCT
ejpam-4574	373	14	c2	c2	PROPN
ejpam-4574	373	15	,	,	PUNCT
ejpam-4574	373	16	...	...	PUNCT
ejpam-4574	373	17	,	,	PUNCT
ejpam-4574	373	18	ck−1	ck−1	NOUN
ejpam-4574	373	19	which	which	PRON
ejpam-4574	373	20	makes	make	VERB
ejpam-4574	373	21	a	a	DET
ejpam-4574	373	22	distance	distance	NOUN
ejpam-4574	373	23	between	between	ADP
ejpam-4574	373	24	two	two	NUM
ejpam-4574	373	25	vertices	vertex	NOUN
ejpam-4574	373	26	having	have	VERB
ejpam-4574	373	27	the	the	DET
ejpam-4574	373	28	same	same	ADJ
ejpam-4574	373	29	color	color	NOUN
ejpam-4574	373	30	become	become	VERB
ejpam-4574	373	31	2(k	2(k	NUM
ejpam-4574	374	1	+	+	CCONJ
ejpam-4574	375	1	1	1	NUM
ejpam-4574	375	2	)	)	PUNCT
ejpam-4574	375	3	.	.	PUNCT
ejpam-4574	376	1	accordingly	accordingly	ADV
ejpam-4574	376	2	,	,	PUNCT
ejpam-4574	376	3	the	the	DET
ejpam-4574	376	4	color	color	NOUN
ejpam-4574	376	5	classes	class	NOUN
ejpam-4574	376	6	t1	t1	NOUN
ejpam-4574	376	7	,	,	PUNCT
ejpam-4574	376	8	t2	t2	NOUN
ejpam-4574	376	9	,	,	PUNCT
ejpam-4574	376	10	...	...	PUNCT
ejpam-4574	376	11	,	,	PUNCT
ejpam-4574	376	12	tk−1	tk−1	PROPN
ejpam-4574	376	13	will	will	AUX
ejpam-4574	376	14	not	not	PART
ejpam-4574	376	15	induce	induce	VERB
ejpam-4574	376	16	a	a	DET
ejpam-4574	376	17	cycle	cycle	NOUN
ejpam-4574	376	18	in	in	ADP
ejpam-4574	376	19	c2k	c2k	PROPN
ejpam-4574	376	20	n	n	CCONJ
ejpam-4574	376	21	as	as	SCONJ
ejpam-4574	376	22	shown	show	VERB
ejpam-4574	376	23	in	in	ADP
ejpam-4574	376	24	the	the	DET
ejpam-4574	376	25	below	below	ADJ
ejpam-4574	376	26	table	table	NOUN
ejpam-4574	376	27	:	:	PUNCT
ejpam-4574	376	28	rule	rule	NOUN
ejpam-4574	376	29	i	i	PRON
ejpam-4574	376	30	mod(k	mod(k	PROPN
ejpam-4574	377	1	+	+	CCONJ
ejpam-4574	377	2	1	1	X
ejpam-4574	377	3	)	)	PUNCT
ejpam-4574	377	4	i	i	NOUN
ejpam-4574	377	5	0	0	NUM
ejpam-4574	377	6	1	1	NUM
ejpam-4574	377	7	...	...	PUNCT
ejpam-4574	378	1	k	k	X
ejpam-4574	379	1	+	+	CCONJ
ejpam-4574	379	2	1	1	NUM
ejpam-4574	379	3	k	k	NOUN
ejpam-4574	379	4	+	+	CCONJ
ejpam-4574	379	5	2	2	NUM
ejpam-4574	379	6	k	k	NOUN
ejpam-4574	379	7	+	+	ADP
ejpam-4574	379	8	3	3	NUM
ejpam-4574	379	9	...	...	PUNCT
ejpam-4574	379	10	2k	2k	NOUN
ejpam-4574	379	11	+	+	CCONJ
ejpam-4574	379	12	3	3	NUM
ejpam-4574	379	13	2	2	NUM
ejpam-4574	379	14	(	(	PUNCT
ejpam-4574	379	15	k	k	NOUN
ejpam-4574	379	16	+	+	PROPN
ejpam-4574	379	17	2	2	X
ejpam-4574	379	18	)	)	PUNCT
ejpam-4574	379	19	2k	2k	NOUN
ejpam-4574	379	20	+	+	CCONJ
ejpam-4574	379	21	5	5	NUM
ejpam-4574	379	22	...	...	PUNCT
ejpam-4574	379	23	3k	3k	X
ejpam-4574	380	1	+	+	CCONJ
ejpam-4574	380	2	5	5	NUM
ejpam-4574	380	3	3	3	NUM
ejpam-4574	380	4	(	(	PUNCT
ejpam-4574	380	5	k	k	NOUN
ejpam-4574	380	6	+	+	PROPN
ejpam-4574	380	7	2	2	X
ejpam-4574	380	8	)	)	PUNCT
ejpam-4574	380	9	3k	3k	NOUN
ejpam-4574	381	1	+	+	CCONJ
ejpam-4574	381	2	7	7	NUM
ejpam-4574	381	3	...	...	PUNCT
ejpam-4574	381	4	(	(	PUNCT
ejpam-4574	381	5	k	k	X
ejpam-4574	381	6	−	−	PROPN
ejpam-4574	381	7	1	1	NUM
ejpam-4574	381	8	)	)	PUNCT
ejpam-4574	381	9	(	(	PUNCT
ejpam-4574	382	1	k	k	X
ejpam-4574	382	2	+	+	CCONJ
ejpam-4574	382	3	2	2	X
ejpam-4574	382	4	)	)	PUNCT
ejpam-4574	382	5	−	−	NOUN
ejpam-4574	383	1	1	1	NUM
ejpam-4574	383	2	(	(	PUNCT
ejpam-4574	383	3	k	k	NOUN
ejpam-4574	383	4	−	−	PROPN
ejpam-4574	383	5	1	1	NUM
ejpam-4574	383	6	)	)	PUNCT
ejpam-4574	383	7	(	(	PUNCT
ejpam-4574	383	8	k	k	X
ejpam-4574	384	1	+	+	PROPN
ejpam-4574	384	2	2	2	X
ejpam-4574	384	3	)	)	PUNCT
ejpam-4574	384	4	(	(	PUNCT
ejpam-4574	384	5	k	k	NOUN
ejpam-4574	384	6	−	−	PROPN
ejpam-4574	384	7	1	1	NUM
ejpam-4574	384	8	)	)	PUNCT
ejpam-4574	384	9	(	(	PUNCT
ejpam-4574	384	10	k	k	NOUN
ejpam-4574	384	11	+	+	CCONJ
ejpam-4574	384	12	2	2	X
ejpam-4574	384	13	)	)	PUNCT
ejpam-4574	385	1	+	+	CCONJ
ejpam-4574	385	2	1	1	NUM
ejpam-4574	385	3	...	...	PUNCT
ejpam-4574	385	4	q	q	X
ejpam-4574	385	5	(	(	PUNCT
ejpam-4574	385	6	k	k	NOUN
ejpam-4574	385	7	+	+	PROPN
ejpam-4574	385	8	1	1	X
ejpam-4574	385	9	)	)	PUNCT
ejpam-4574	385	10	−	−	PROPN
ejpam-4574	385	11	2	2	NUM
ejpam-4574	385	12	n	n	NOUN
ejpam-4574	385	13	−	−	PROPN
ejpam-4574	385	14	1	1	NUM
ejpam-4574	385	15	c1(vi	c1(vi	PROPN
ejpam-4574	385	16	)	)	PUNCT
ejpam-4574	385	17	c0	c0	PROPN
ejpam-4574	385	18	c1	c1	PROPN
ejpam-4574	385	19	...	...	PUNCT
ejpam-4574	386	1	c0	c0	PROPN
ejpam-4574	386	2	c1	c1	PROPN
ejpam-4574	386	3	c2	c2	PROPN
ejpam-4574	386	4	...	...	PUNCT
ejpam-4574	387	1	c1	c1	PROPN
ejpam-4574	387	2	c2	c2	PROPN
ejpam-4574	387	3	c3	c3	PROPN
ejpam-4574	387	4	...	...	PUNCT
ejpam-4574	388	1	c2	c2	PROPN
ejpam-4574	388	2	c3	c3	PROPN
ejpam-4574	388	3	c4	c4	PROPN
ejpam-4574	388	4	...	...	PUNCT
ejpam-4574	389	1	ck−2	ck−2	PROPN
ejpam-4574	389	2	ck−1	ck−1	NOUN
ejpam-4574	389	3	ck	ck	INTJ
ejpam-4574	389	4	...	...	PUNCT
ejpam-4574	390	1	ck−1	ck−1	NOUN
ejpam-4574	390	2	ck	ck	NOUN
ejpam-4574	390	3	c3(vi	c3(vi	PROPN
ejpam-4574	390	4	)	)	PUNCT
ejpam-4574	390	5	ck+1	ck+1	NUM
ejpam-4574	390	6	c1	c1	PROPN
ejpam-4574	390	7	...	...	PUNCT
ejpam-4574	391	1	c0	c0	PROPN
ejpam-4574	391	2	ck+1	ck+1	AUX
ejpam-4574	391	3	c2	c2	PROPN
ejpam-4574	391	4	...	...	PUNCT
ejpam-4574	392	1	c1	c1	PROPN
ejpam-4574	392	2	ck+1	ck+1	AUX
ejpam-4574	392	3	c3	c3	PROPN
ejpam-4574	392	4	...	...	PUNCT
ejpam-4574	393	1	c2	c2	PROPN
ejpam-4574	393	2	ck+1	ck+1	AUX
ejpam-4574	393	3	c4	c4	PROPN
ejpam-4574	393	4	...	...	PUNCT
ejpam-4574	394	1	ck−2	ck−2	INTJ
ejpam-4574	394	2	ck+1	ck+1	AUX
ejpam-4574	394	3	ck	ck	INTJ
ejpam-4574	394	4	...	...	PUNCT
ejpam-4574	395	1	ck−1	ck−1	NOUN
ejpam-4574	395	2	ck	ck	NOUN
ejpam-4574	395	3	⌞	⌞	X
ejpam-4574	395	4	2(k	2(k	NUM
ejpam-4574	396	1	+	+	CCONJ
ejpam-4574	396	2	1	1	X
ejpam-4574	396	3	)	)	PUNCT
ejpam-4574	396	4	⌟	⌟	VERB
ejpam-4574	396	5	⌞	⌞	PUNCT
ejpam-4574	396	6	2(k	2(k	NUM
ejpam-4574	397	1	+	+	CCONJ
ejpam-4574	397	2	1	1	X
ejpam-4574	397	3	)	)	PUNCT
ejpam-4574	397	4	⌟	⌟	VERB
ejpam-4574	397	5	a.	a.	NOUN
ejpam-4574	397	6	etawi	etawi	PROPN
ejpam-4574	397	7	,	,	PUNCT
ejpam-4574	397	8	m.	m.	PROPN
ejpam-4574	397	9	ghanem	ghanem	PROPN
ejpam-4574	397	10	,	,	PUNCT
ejpam-4574	397	11	h.	h.	PROPN
ejpam-4574	397	12	al	al	PROPN
ejpam-4574	397	13	-	-	PUNCT
ejpam-4574	397	14	ezeh	ezeh	PROPN
ejpam-4574	397	15	/	/	SYM
ejpam-4574	397	16	eur	eur	NOUN
ejpam-4574	397	17	.	.	PUNCT
ejpam-4574	398	1	j.	j.	PROPN
ejpam-4574	398	2	pure	pure	PROPN
ejpam-4574	398	3	appl	appl	PROPN
ejpam-4574	398	4	.	.	PROPN
ejpam-4574	398	5	math	math	PROPN
ejpam-4574	398	6	,	,	PUNCT
ejpam-4574	398	7	15	15	NUM
ejpam-4574	398	8	(	(	PUNCT
ejpam-4574	398	9	4	4	NUM
ejpam-4574	398	10	)	)	PUNCT
ejpam-4574	398	11	(	(	PUNCT
ejpam-4574	398	12	2022	2022	NUM
ejpam-4574	398	13	)	)	PUNCT
ejpam-4574	398	14	,	,	PUNCT
ejpam-4574	398	15	1822	1822	NUM
ejpam-4574	398	16	-	-	SYM
ejpam-4574	398	17	1835	1835	NUM
ejpam-4574	398	18	1831	1831	NUM
ejpam-4574	398	19	(	(	PUNCT
ejpam-4574	398	20	3	3	X
ejpam-4574	398	21	)	)	PUNCT
ejpam-4574	398	22	suppose	suppose	VERB
ejpam-4574	398	23	that	that	SCONJ
ejpam-4574	398	24	cl	cl	NOUN
ejpam-4574	398	25	is	be	AUX
ejpam-4574	398	26	a	a	DET
ejpam-4574	398	27	bicolored	bicolore	VERB
ejpam-4574	398	28	cycle	cycle	NOUN
ejpam-4574	398	29	in	in	ADP
ejpam-4574	398	30	ck	ck	PROPN
ejpam-4574	398	31	n	n	CCONJ
ejpam-4574	398	32	,	,	PUNCT
ejpam-4574	398	33	then	then	ADV
ejpam-4574	398	34	v	v	X
ejpam-4574	398	35	(	(	PUNCT
ejpam-4574	398	36	cl	cl	NOUN
ejpam-4574	398	37	)	)	PUNCT
ejpam-4574	398	38	=	=	SYM
ejpam-4574	398	39	ti	ti	NOUN
ejpam-4574	398	40	∪	∪	NOUN
ejpam-4574	398	41	tj	tj	NOUN
ejpam-4574	398	42	for	for	ADP
ejpam-4574	398	43	some	some	DET
ejpam-4574	398	44	i	i	PROPN
ejpam-4574	398	45	and	and	CCONJ
ejpam-4574	398	46	j.	j.	PROPN
ejpam-4574	398	47	note	note	VERB
ejpam-4574	398	48	that	that	SCONJ
ejpam-4574	398	49	k	k	PROPN
ejpam-4574	399	1	=	=	PUNCT
ejpam-4574	399	2	|tk+1|	|tk+1|	NOUN
ejpam-4574	399	3	=	=	X
ejpam-4574	399	4	̸	̸	NUM
ejpam-4574	399	5	|tk|	|tk|	PROPN
ejpam-4574	399	6	≥	≥	NOUN
ejpam-4574	399	7	k	k	NOUN
ejpam-4574	400	1	+	+	CCONJ
ejpam-4574	400	2	1	1	X
ejpam-4574	400	3	.	.	PUNCT
ejpam-4574	400	4	so	so	ADV
ejpam-4574	400	5	v	v	ADJ
ejpam-4574	400	6	(	(	PUNCT
ejpam-4574	400	7	cl	cl	NOUN
ejpam-4574	400	8	)	)	PUNCT
ejpam-4574	400	9	̸=	̸=	PROPN
ejpam-4574	400	10	tk+1	tk+1	NUM
ejpam-4574	400	11	∪	∪	VERB
ejpam-4574	400	12	tk	tk	NOUN
ejpam-4574	400	13	and	and	CCONJ
ejpam-4574	400	14	hence	hence	ADV
ejpam-4574	400	15	i	i	PRON
ejpam-4574	400	16	or	or	CCONJ
ejpam-4574	400	17	j	j	PROPN
ejpam-4574	400	18	≤	≤	PROPN
ejpam-4574	401	1	k	k	INTJ
ejpam-4574	402	1	−	−	PROPN
ejpam-4574	402	2	1	1	NUM
ejpam-4574	402	3	.	.	PUNCT
ejpam-4574	402	4	from	from	ADP
ejpam-4574	402	5	part	part	NOUN
ejpam-4574	402	6	(	(	PUNCT
ejpam-4574	402	7	2	2	NUM
ejpam-4574	402	8	)	)	PUNCT
ejpam-4574	402	9	and	and	CCONJ
ejpam-4574	402	10	lemma	lemma	PROPN
ejpam-4574	402	11	4	4	NUM
ejpam-4574	402	12	,	,	PUNCT
ejpam-4574	402	13	we	we	PRON
ejpam-4574	402	14	get	get	VERB
ejpam-4574	402	15	a	a	DET
ejpam-4574	402	16	contradiction	contradiction	NOUN
ejpam-4574	402	17	.	.	PUNCT
ejpam-4574	403	1	therefore	therefore	ADV
ejpam-4574	403	2	,	,	PUNCT
ejpam-4574	403	3	c′3	c′3	PROPN
ejpam-4574	403	4	is	be	AUX
ejpam-4574	403	5	a	a	DET
ejpam-4574	403	6	(	(	PUNCT
ejpam-4574	403	7	k	k	PROPN
ejpam-4574	403	8	+	+	CCONJ
ejpam-4574	403	9	2)−	2)−	NUM
ejpam-4574	403	10	acyclic	acyclic	ADJ
ejpam-4574	403	11	coloring	coloring	NOUN
ejpam-4574	403	12	of	of	ADP
ejpam-4574	403	13	ck	ck	PROPN
ejpam-4574	403	14	n.	n.	PROPN
ejpam-4574	403	15	lemma	lemma	PROPN
ejpam-4574	403	16	9	9	X
ejpam-4574	403	17	.	.	PUNCT
ejpam-4574	404	1	let	let	VERB
ejpam-4574	404	2	n	n	NOUN
ejpam-4574	404	3	=	=	SYM
ejpam-4574	404	4	(	(	PUNCT
ejpam-4574	404	5	k	k	PROPN
ejpam-4574	404	6	+	+	CCONJ
ejpam-4574	404	7	1)2	1)2	NUM
ejpam-4574	404	8	+	+	CCONJ
ejpam-4574	404	9	k	k	ADJ
ejpam-4574	404	10	,	,	PUNCT
ejpam-4574	404	11	and	and	CCONJ
ejpam-4574	404	12	c′4(vi	c′4(vi	PRON
ejpam-4574	404	13	)	)	PUNCT
ejpam-4574	404	14	=	=	PUNCT
ejpam-4574	404	15			X
ejpam-4574	404	16	ck+2	ck+2	NOUN
ejpam-4574	404	17	if	if	SCONJ
ejpam-4574	404	18	i	i	PRON
ejpam-4574	404	19	=	=	PUNCT
ejpam-4574	404	20	j(k	j(k	X
ejpam-4574	404	21	+	+	CCONJ
ejpam-4574	404	22	3	3	X
ejpam-4574	404	23	)	)	PUNCT
ejpam-4574	404	24	for	for	ADP
ejpam-4574	404	25	j	j	PROPN
ejpam-4574	404	26	∈	∈	PROPN
ejpam-4574	404	27	{	{	PUNCT
ejpam-4574	404	28	0	0	NUM
ejpam-4574	404	29	,	,	PUNCT
ejpam-4574	404	30	1	1	NUM
ejpam-4574	404	31	,	,	PUNCT
ejpam-4574	404	32	...	...	PUNCT
ejpam-4574	404	33	,	,	PUNCT
ejpam-4574	404	34	k	k	PROPN
ejpam-4574	405	1	−	−	PROPN
ejpam-4574	405	2	1	1	NUM
ejpam-4574	405	3	}	}	PUNCT
ejpam-4574	405	4	ck+1	ck+1	VERB
ejpam-4574	405	5	if	if	SCONJ
ejpam-4574	405	6	i	i	PRON
ejpam-4574	405	7	=	=	SYM
ejpam-4574	406	1	k(k	k(k	PROPN
ejpam-4574	406	2	+	+	CCONJ
ejpam-4574	406	3	3	3	X
ejpam-4574	406	4	)	)	PUNCT
ejpam-4574	406	5	c′2(vi	c′2(vi	NOUN
ejpam-4574	406	6	)	)	PUNCT
ejpam-4574	406	7	otherwise	otherwise	ADV
ejpam-4574	406	8	.	.	PUNCT
ejpam-4574	407	1	then	then	ADV
ejpam-4574	407	2	(	(	PUNCT
ejpam-4574	407	3	1	1	X
ejpam-4574	407	4	)	)	PUNCT
ejpam-4574	407	5	c′4	c′4	X
ejpam-4574	407	6	is	be	AUX
ejpam-4574	407	7	a	a	DET
ejpam-4574	407	8	(	(	PUNCT
ejpam-4574	407	9	k	k	PROPN
ejpam-4574	407	10	+	+	CCONJ
ejpam-4574	407	11	3)−proper	3)−proper	PROPN
ejpam-4574	407	12	coloring	coloring	NOUN
ejpam-4574	407	13	of	of	ADP
ejpam-4574	407	14	ck	ck	PROPN
ejpam-4574	407	15	n.	n.	NOUN
ejpam-4574	407	16	(	(	PUNCT
ejpam-4574	407	17	2	2	NUM
ejpam-4574	407	18	)	)	PUNCT
ejpam-4574	407	19	for	for	ADP
ejpam-4574	407	20	0	0	NUM
ejpam-4574	407	21	≤	≤	NOUN
ejpam-4574	408	1	i	i	NOUN
ejpam-4574	408	2	≤	≤	PROPN
ejpam-4574	409	1	k	k	PUNCT
ejpam-4574	409	2	,	,	PUNCT
ejpam-4574	409	3	ti	ti	PROPN
ejpam-4574	409	4	is	be	AUX
ejpam-4574	409	5	not	not	PART
ejpam-4574	409	6	an	an	DET
ejpam-4574	409	7	induced	induced	ADJ
ejpam-4574	409	8	cycle	cycle	NOUN
ejpam-4574	409	9	of	of	ADP
ejpam-4574	409	10	c2k	c2k	NOUN
ejpam-4574	409	11	n	n	NOUN
ejpam-4574	409	12	.	.	PUNCT
ejpam-4574	410	1	(	(	PUNCT
ejpam-4574	410	2	3	3	X
ejpam-4574	410	3	)	)	PUNCT
ejpam-4574	410	4	c′4	c′4	X
ejpam-4574	410	5	is	be	AUX
ejpam-4574	410	6	a	a	DET
ejpam-4574	410	7	(	(	PUNCT
ejpam-4574	410	8	k	k	X
ejpam-4574	410	9	+	+	CCONJ
ejpam-4574	410	10	3)−	3)−	NUM
ejpam-4574	410	11	acyclic	acyclic	ADJ
ejpam-4574	410	12	coloring	coloring	NOUN
ejpam-4574	410	13	of	of	ADP
ejpam-4574	410	14	ck	ck	PROPN
ejpam-4574	410	15	n.	n.	NOUN
ejpam-4574	410	16	proof	proof	NOUN
ejpam-4574	410	17	.	.	PUNCT
ejpam-4574	411	1	(	(	PUNCT
ejpam-4574	411	2	1	1	X
ejpam-4574	411	3	)	)	PUNCT
ejpam-4574	411	4	since	since	SCONJ
ejpam-4574	411	5	c′2	c′2	NOUN
ejpam-4574	411	6	is	be	AUX
ejpam-4574	411	7	a	a	DET
ejpam-4574	411	8	proper	proper	ADJ
ejpam-4574	411	9	coloring	coloring	NOUN
ejpam-4574	411	10	,	,	PUNCT
ejpam-4574	411	11	dcn(vh(k+3	dcn(vh(k+3	PROPN
ejpam-4574	411	12	)	)	PUNCT
ejpam-4574	411	13	,	,	PUNCT
ejpam-4574	411	14	v(h+1)(k+3	v(h+1)(k+3	PROPN
ejpam-4574	411	15	)	)	PUNCT
ejpam-4574	411	16	)	)	PUNCT
ejpam-4574	412	1	=	=	PUNCT
ejpam-4574	413	1	k	k	PROPN
ejpam-4574	414	1	+	+	CCONJ
ejpam-4574	414	2	3	3	NUM
ejpam-4574	414	3	for	for	ADP
ejpam-4574	414	4	0	0	NUM
ejpam-4574	414	5	≤	≤	NUM
ejpam-4574	414	6	h	h	NOUN
ejpam-4574	414	7	≤	≤	NUM
ejpam-4574	414	8	k	k	PRON
ejpam-4574	415	1	−	−	PROPN
ejpam-4574	415	2	1	1	NUM
ejpam-4574	415	3	,	,	PUNCT
ejpam-4574	415	4	and	and	CCONJ
ejpam-4574	415	5	dcn(v(k−1)(k+3	dcn(v(k−1)(k+3	PROPN
ejpam-4574	415	6	)	)	PUNCT
ejpam-4574	415	7	,	,	PUNCT
ejpam-4574	415	8	v0	v0	NOUN
ejpam-4574	415	9	)	)	PUNCT
ejpam-4574	415	10	=	=	PUNCT
ejpam-4574	416	1	k	k	PROPN
ejpam-4574	417	1	+	+	NOUN
ejpam-4574	417	2	4	4	NUM
ejpam-4574	417	3	,	,	PUNCT
ejpam-4574	417	4	we	we	PRON
ejpam-4574	417	5	have	have	AUX
ejpam-4574	417	6	c′4	c′4	X
ejpam-4574	417	7	is	be	AUX
ejpam-4574	417	8	a	a	DET
ejpam-4574	417	9	(	(	PUNCT
ejpam-4574	417	10	k	k	PROPN
ejpam-4574	417	11	+	+	CCONJ
ejpam-4574	417	12	3)−proper	3)−proper	PROPN
ejpam-4574	417	13	coloring	coloring	NOUN
ejpam-4574	417	14	of	of	ADP
ejpam-4574	417	15	ck	ck	PROPN
ejpam-4574	417	16	n.	n.	PROPN
ejpam-4574	417	17	moreover	moreover	ADV
ejpam-4574	417	18	,	,	PUNCT
ejpam-4574	417	19	to	to	PART
ejpam-4574	417	20	keep	keep	VERB
ejpam-4574	417	21	c′4	c′4	ADJ
ejpam-4574	417	22	proper	proper	ADJ
ejpam-4574	417	23	coloring	coloring	NOUN
ejpam-4574	417	24	,	,	PUNCT
ejpam-4574	417	25	ck+1	ck+1	PRON
ejpam-4574	417	26	was	be	AUX
ejpam-4574	417	27	assigned	assign	VERB
ejpam-4574	417	28	to	to	PART
ejpam-4574	417	29	vn−1	vn−1	VERB
ejpam-4574	417	30	instead	instead	ADV
ejpam-4574	417	31	ck+2	ck+2	NUM
ejpam-4574	417	32	since	since	SCONJ
ejpam-4574	417	33	dcn(v0	dcn(v0	PROPN
ejpam-4574	417	34	,	,	PUNCT
ejpam-4574	417	35	vn−1	vn−1	ADJ
ejpam-4574	417	36	)	)	PUNCT
ejpam-4574	417	37	=	=	SYM
ejpam-4574	418	1	1	1	X
ejpam-4574	418	2	.	.	PUNCT
ejpam-4574	418	3	(	(	PUNCT
ejpam-4574	418	4	2	2	X
ejpam-4574	418	5	)	)	PUNCT
ejpam-4574	418	6	if	if	SCONJ
ejpam-4574	418	7	k	k	PROPN
ejpam-4574	418	8	≤	≤	PROPN
ejpam-4574	418	9	2	2	NUM
ejpam-4574	418	10	,	,	PUNCT
ejpam-4574	418	11	then	then	ADV
ejpam-4574	418	12	|t0|	|t0|	X
ejpam-4574	418	13	=	=	SYM
ejpam-4574	418	14	k	k	X
ejpam-4574	418	15	<	<	X
ejpam-4574	418	16	(	(	PUNCT
ejpam-4574	418	17	k+1)2+k	k+1)2+k	PROPN
ejpam-4574	418	18	2k	2k	PROPN
ejpam-4574	418	19	.	.	PUNCT
ejpam-4574	419	1	if	if	SCONJ
ejpam-4574	419	2	k	k	PROPN
ejpam-4574	419	3	>	>	X
ejpam-4574	419	4	2	2	NUM
ejpam-4574	419	5	,	,	PUNCT
ejpam-4574	419	6	then	then	ADV
ejpam-4574	419	7	dcn(vk(k+2	dcn(vk(k+2	PROPN
ejpam-4574	419	8	)	)	PUNCT
ejpam-4574	419	9	,	,	PUNCT
ejpam-4574	419	10	v(k+2	v(k+2	NUM
ejpam-4574	419	11	)	)	PUNCT
ejpam-4574	419	12	)	)	PUNCT
ejpam-4574	420	1	≥	≥	NOUN
ejpam-4574	420	2	2(k	2(k	NUM
ejpam-4574	421	1	+	+	CCONJ
ejpam-4574	421	2	1	1	NUM
ejpam-4574	421	3	)	)	PUNCT
ejpam-4574	421	4	.	.	PUNCT
ejpam-4574	422	1	if	if	SCONJ
ejpam-4574	422	2	0	0	NUM
ejpam-4574	422	3	<	<	X
ejpam-4574	422	4	i	i	NOUN
ejpam-4574	422	5	≤	≤	PUNCT
ejpam-4574	423	1	k	k	X
ejpam-4574	423	2	,	,	PUNCT
ejpam-4574	423	3	then	then	ADV
ejpam-4574	423	4	vi(k+2)+i	vi(k+2)+i	NOUN
ejpam-4574	423	5	/∈	/∈	PUNCT
ejpam-4574	423	6	ti	ti	NOUN
ejpam-4574	423	7	and	and	CCONJ
ejpam-4574	423	8	thus	thus	ADV
ejpam-4574	423	9	dcn(v(i−1)(k+2)+i	dcn(v(i−1)(k+2)+i	PROPN
ejpam-4574	423	10	,	,	PUNCT
ejpam-4574	423	11	v(i+1)(k+2)+i	v(i+1)(k+2)+i	NOUN
ejpam-4574	423	12	)	)	PUNCT
ejpam-4574	423	13	=	=	PUNCT
ejpam-4574	424	1	2(k	2(k	NUM
ejpam-4574	424	2	+	+	CCONJ
ejpam-4574	424	3	2	2	NUM
ejpam-4574	424	4	)	)	PUNCT
ejpam-4574	424	5	.	.	PUNCT
ejpam-4574	425	1	adding	add	VERB
ejpam-4574	425	2	ck+2	ck+2	NOUN
ejpam-4574	425	3	in	in	ADP
ejpam-4574	425	4	c′4	c′4	PROPN
ejpam-4574	425	5	denies	deny	VERB
ejpam-4574	425	6	one	one	NUM
ejpam-4574	425	7	occurrence	occurrence	NOUN
ejpam-4574	425	8	of	of	ADP
ejpam-4574	425	9	each	each	DET
ejpam-4574	425	10	color	color	NOUN
ejpam-4574	425	11	of	of	ADP
ejpam-4574	425	12	c1	c1	PROPN
ejpam-4574	425	13	,	,	PUNCT
ejpam-4574	425	14	c2	c2	PROPN
ejpam-4574	425	15	,	,	PUNCT
ejpam-4574	425	16	...	...	PUNCT
ejpam-4574	425	17	,	,	PUNCT
ejpam-4574	425	18	ck	ck	INTJ
ejpam-4574	425	19	which	which	PRON
ejpam-4574	425	20	makes	make	VERB
ejpam-4574	425	21	a	a	DET
ejpam-4574	425	22	distance	distance	NOUN
ejpam-4574	425	23	between	between	ADP
ejpam-4574	425	24	two	two	NUM
ejpam-4574	425	25	vertices	vertex	NOUN
ejpam-4574	425	26	having	have	VERB
ejpam-4574	425	27	the	the	DET
ejpam-4574	425	28	same	same	ADJ
ejpam-4574	425	29	color	color	NOUN
ejpam-4574	425	30	become	become	VERB
ejpam-4574	425	31	2(k+1	2(k+1	NOUN
ejpam-4574	425	32	)	)	PUNCT
ejpam-4574	425	33	.	.	PUNCT
ejpam-4574	426	1	accordingly	accordingly	ADV
ejpam-4574	426	2	,	,	PUNCT
ejpam-4574	426	3	the	the	DET
ejpam-4574	426	4	color	color	NOUN
ejpam-4574	426	5	classes	class	NOUN
ejpam-4574	426	6	t1	t1	NOUN
ejpam-4574	426	7	,	,	PUNCT
ejpam-4574	426	8	t2	t2	NOUN
ejpam-4574	426	9	,	,	PUNCT
ejpam-4574	426	10	...	...	PUNCT
ejpam-4574	426	11	,	,	PUNCT
ejpam-4574	426	12	tk	tk	PROPN
ejpam-4574	426	13	will	will	AUX
ejpam-4574	426	14	not	not	PART
ejpam-4574	426	15	induce	induce	VERB
ejpam-4574	426	16	a	a	DET
ejpam-4574	426	17	cycle	cycle	NOUN
ejpam-4574	426	18	in	in	ADP
ejpam-4574	426	19	c2k	c2k	PROPN
ejpam-4574	426	20	n	n	CCONJ
ejpam-4574	426	21	as	as	SCONJ
ejpam-4574	426	22	shown	show	VERB
ejpam-4574	426	23	in	in	ADP
ejpam-4574	426	24	the	the	DET
ejpam-4574	426	25	below	below	ADJ
ejpam-4574	426	26	table	table	NOUN
ejpam-4574	426	27	:	:	PUNCT
ejpam-4574	426	28	rule	rule	NOUN
ejpam-4574	426	29	i	i	PRON
ejpam-4574	426	30	mod(k	mod(k	PROPN
ejpam-4574	427	1	+	+	CCONJ
ejpam-4574	427	2	2	2	X
ejpam-4574	427	3	)	)	PUNCT
ejpam-4574	427	4	i	i	NOUN
ejpam-4574	427	5	0	0	NUM
ejpam-4574	427	6	1	1	NUM
ejpam-4574	427	7	...	...	PUNCT
ejpam-4574	428	1	k	k	X
ejpam-4574	429	1	+	+	CCONJ
ejpam-4574	429	2	2	2	NUM
ejpam-4574	429	3	k	k	NOUN
ejpam-4574	429	4	+	+	NOUN
ejpam-4574	429	5	3	3	NUM
ejpam-4574	429	6	k	k	NOUN
ejpam-4574	429	7	+	+	ADP
ejpam-4574	429	8	4	4	NUM
ejpam-4574	429	9	...	...	PUNCT
ejpam-4574	429	10	2k	2k	NOUN
ejpam-4574	429	11	+	+	CCONJ
ejpam-4574	429	12	5	5	NUM
ejpam-4574	429	13	2	2	NUM
ejpam-4574	429	14	(	(	PUNCT
ejpam-4574	429	15	k	k	NOUN
ejpam-4574	429	16	+	+	PROPN
ejpam-4574	429	17	2	2	X
ejpam-4574	429	18	)	)	PUNCT
ejpam-4574	429	19	+	+	CCONJ
ejpam-4574	429	20	2	2	NUM
ejpam-4574	429	21	2k	2k	NUM
ejpam-4574	429	22	+	+	CCONJ
ejpam-4574	429	23	7	7	NUM
ejpam-4574	429	24	...	...	PUNCT
ejpam-4574	429	25	3k	3k	X
ejpam-4574	430	1	+	+	CCONJ
ejpam-4574	430	2	8	8	NUM
ejpam-4574	430	3	3	3	NUM
ejpam-4574	430	4	(	(	PUNCT
ejpam-4574	430	5	k	k	NOUN
ejpam-4574	430	6	+	+	PROPN
ejpam-4574	430	7	2	2	X
ejpam-4574	430	8	)	)	PUNCT
ejpam-4574	430	9	+	+	CCONJ
ejpam-4574	430	10	3	3	NUM
ejpam-4574	430	11	...	...	PUNCT
ejpam-4574	431	1	(	(	PUNCT
ejpam-4574	431	2	k	k	X
ejpam-4574	431	3	−	−	PROPN
ejpam-4574	431	4	1	1	NUM
ejpam-4574	431	5	)	)	PUNCT
ejpam-4574	431	6	(	(	PUNCT
ejpam-4574	431	7	k	k	NOUN
ejpam-4574	431	8	+	+	CCONJ
ejpam-4574	431	9	2	2	NUM
ejpam-4574	431	10	)	)	PUNCT
ejpam-4574	432	1	+	+	CCONJ
ejpam-4574	433	1	k	k	X
ejpam-4574	433	2	−	−	PROPN
ejpam-4574	433	3	2	2	NUM
ejpam-4574	433	4	(	(	PUNCT
ejpam-4574	433	5	k	k	NOUN
ejpam-4574	433	6	−	−	PROPN
ejpam-4574	433	7	1	1	NUM
ejpam-4574	433	8	)	)	PUNCT
ejpam-4574	433	9	(	(	PUNCT
ejpam-4574	433	10	k	k	NOUN
ejpam-4574	433	11	+	+	CCONJ
ejpam-4574	433	12	2	2	NUM
ejpam-4574	433	13	)	)	PUNCT
ejpam-4574	434	1	+	+	CCONJ
ejpam-4574	435	1	k	k	X
ejpam-4574	435	2	−	−	PROPN
ejpam-4574	435	3	1	1	NUM
ejpam-4574	435	4	(	(	PUNCT
ejpam-4574	435	5	k	k	NOUN
ejpam-4574	435	6	−	−	PROPN
ejpam-4574	435	7	1	1	NUM
ejpam-4574	435	8	)	)	PUNCT
ejpam-4574	435	9	(	(	PUNCT
ejpam-4574	435	10	k	k	NOUN
ejpam-4574	435	11	+	+	CCONJ
ejpam-4574	435	12	2	2	NUM
ejpam-4574	435	13	)	)	PUNCT
ejpam-4574	436	1	+	+	CCONJ
ejpam-4574	437	1	k	k	PROPN
ejpam-4574	437	2	k	k	X
ejpam-4574	437	3	(	(	PUNCT
ejpam-4574	437	4	k	k	PROPN
ejpam-4574	437	5	+	+	PROPN
ejpam-4574	437	6	2	2	X
ejpam-4574	437	7	)	)	PUNCT
ejpam-4574	437	8	−	−	PROPN
ejpam-4574	437	9	1	1	NUM
ejpam-4574	437	10	...	...	PUNCT
ejpam-4574	438	1	k	k	X
ejpam-4574	438	2	(	(	PUNCT
ejpam-4574	438	3	k	k	PROPN
ejpam-4574	438	4	+	+	PROPN
ejpam-4574	438	5	2	2	NUM
ejpam-4574	438	6	)	)	PUNCT
ejpam-4574	438	7	+	+	CCONJ
ejpam-4574	439	1	k	k	PROPN
ejpam-4574	439	2	−	−	PROPN
ejpam-4574	439	3	1	1	NUM
ejpam-4574	439	4	n	n	CCONJ
ejpam-4574	439	5	−	−	PROPN
ejpam-4574	439	6	1	1	NUM
ejpam-4574	439	7	c2(vi	c2(vi	PROPN
ejpam-4574	439	8	)	)	PUNCT
ejpam-4574	439	9	c0	c0	PROPN
ejpam-4574	439	10	c1	c1	PROPN
ejpam-4574	439	11	...	...	PUNCT
ejpam-4574	440	1	c0	c0	PROPN
ejpam-4574	440	2	c1	c1	PROPN
ejpam-4574	440	3	c2	c2	PROPN
ejpam-4574	440	4	...	...	PUNCT
ejpam-4574	441	1	c1	c1	PROPN
ejpam-4574	441	2	c2	c2	PROPN
ejpam-4574	441	3	c3	c3	PROPN
ejpam-4574	441	4	...	...	PUNCT
ejpam-4574	442	1	c2	c2	PROPN
ejpam-4574	442	2	c3	c3	PROPN
ejpam-4574	442	3	...	...	PUNCT
ejpam-4574	443	1	ck−2	ck−2	PROPN
ejpam-4574	443	2	ck−1	ck−1	NOUN
ejpam-4574	443	3	ck	ck	INTJ
ejpam-4574	443	4	ck+1	ck+1	NOUN
ejpam-4574	443	5	...	...	PUNCT
ejpam-4574	444	1	ck−1	ck−1	NOUN
ejpam-4574	444	2	ck	ck	NOUN
ejpam-4574	444	3	c4(vi	c4(vi	PROPN
ejpam-4574	444	4	)	)	PUNCT
ejpam-4574	444	5	ck+2	ck+2	NOUN
ejpam-4574	444	6	c1	c1	NOUN
ejpam-4574	444	7	...	...	PUNCT
ejpam-4574	445	1	c0	c0	PROPN
ejpam-4574	445	2	ck+2	ck+2	PROPN
ejpam-4574	445	3	c2	c2	PROPN
ejpam-4574	445	4	...	...	PUNCT
ejpam-4574	446	1	c1	c1	PROPN
ejpam-4574	446	2	ck+2	ck+2	NOUN
ejpam-4574	446	3	c3	c3	PROPN
ejpam-4574	446	4	...	...	PUNCT
ejpam-4574	447	1	c2	c2	PROPN
ejpam-4574	447	2	ck+2	ck+2	NOUN
ejpam-4574	447	3	...	...	PUNCT
ejpam-4574	448	1	ck−2	ck−2	INTJ
ejpam-4574	448	2	ck+2	ck+2	NOUN
ejpam-4574	448	3	ck	ck	INTJ
ejpam-4574	448	4	ck+1	ck+1	NOUN
ejpam-4574	448	5	...	...	PUNCT
ejpam-4574	449	1	ck−1	ck−1	NOUN
ejpam-4574	449	2	ck+1	ck+1	X
ejpam-4574	449	3	⌞	⌞	X
ejpam-4574	449	4	2(k	2(k	NUM
ejpam-4574	449	5	+	+	CCONJ
ejpam-4574	449	6	2	2	X
ejpam-4574	449	7	)	)	PUNCT
ejpam-4574	449	8	⌟	⌟	VERB
ejpam-4574	449	9	⌞	⌞	PUNCT
ejpam-4574	449	10	2(k	2(k	NUM
ejpam-4574	449	11	+	+	CCONJ
ejpam-4574	449	12	2	2	X
ejpam-4574	449	13	)	)	PUNCT
ejpam-4574	449	14	⌟	⌟	NOUN
ejpam-4574	449	15	⌞	⌞	PUNCT
ejpam-4574	450	1	k	k	NOUN
ejpam-4574	450	2	+	+	PROPN
ejpam-4574	450	3	1	1	NUM
ejpam-4574	450	4	⌟	⌟	NOUN
ejpam-4574	450	5	(	(	PUNCT
ejpam-4574	450	6	3	3	X
ejpam-4574	450	7	)	)	PUNCT
ejpam-4574	450	8	assume	assume	VERB
ejpam-4574	450	9	that	that	SCONJ
ejpam-4574	450	10	cl	cl	NOUN
ejpam-4574	450	11	is	be	AUX
ejpam-4574	450	12	a	a	DET
ejpam-4574	450	13	bicolored	bicolore	VERB
ejpam-4574	450	14	cycle	cycle	NOUN
ejpam-4574	450	15	in	in	ADP
ejpam-4574	450	16	ck	ck	PROPN
ejpam-4574	450	17	n	n	CCONJ
ejpam-4574	450	18	,	,	PUNCT
ejpam-4574	450	19	then	then	ADV
ejpam-4574	450	20	v	v	X
ejpam-4574	450	21	(	(	PUNCT
ejpam-4574	450	22	cl	cl	NOUN
ejpam-4574	450	23	)	)	PUNCT
ejpam-4574	450	24	=	=	SYM
ejpam-4574	450	25	ti∪tj	ti∪tj	PROPN
ejpam-4574	450	26	for	for	ADP
ejpam-4574	450	27	some	some	DET
ejpam-4574	450	28	i	i	PROPN
ejpam-4574	450	29	,	,	PUNCT
ejpam-4574	450	30	j.	j.	PROPN
ejpam-4574	450	31	clearly	clearly	ADV
ejpam-4574	450	32	k+1	k+1	X
ejpam-4574	450	33	=	=	PUNCT
ejpam-4574	450	34	|tk+1|	|tk+1|	NOUN
ejpam-4574	450	35	=	=	NOUN
ejpam-4574	450	36	̸	̸	NOUN
ejpam-4574	450	37	|tk+2|	|tk+2|	NOUN
ejpam-4574	451	1	=	=	SYM
ejpam-4574	451	2	k	k	PROPN
ejpam-4574	451	3	,	,	PUNCT
ejpam-4574	452	1	so	so	SCONJ
ejpam-4574	452	2	i	i	PRON
ejpam-4574	452	3	or	or	CCONJ
ejpam-4574	452	4	j	j	PROPN
ejpam-4574	452	5	≤	≤	PROPN
ejpam-4574	453	1	k	k	NOUN
ejpam-4574	453	2	say	say	VERB
ejpam-4574	453	3	i	i	PRON
ejpam-4574	453	4	,	,	PUNCT
ejpam-4574	453	5	then	then	ADV
ejpam-4574	453	6	ti	ti	PROPN
ejpam-4574	453	7	is	be	AUX
ejpam-4574	453	8	not	not	PART
ejpam-4574	453	9	an	an	DET
ejpam-4574	453	10	induced	induced	ADJ
ejpam-4574	453	11	cycle	cycle	NOUN
ejpam-4574	453	12	of	of	ADP
ejpam-4574	453	13	c2k	c2k	PROPN
ejpam-4574	453	14	n	n	NOUN
ejpam-4574	453	15	.	.	PUNCT
ejpam-4574	454	1	lemma	lemma	PROPN
ejpam-4574	454	2	10	10	NUM
ejpam-4574	454	3	.	.	PUNCT
ejpam-4574	455	1	let	let	VERB
ejpam-4574	455	2	n	n	NOUN
ejpam-4574	455	3	=	=	PUNCT
ejpam-4574	455	4	q(k	q(k	PROPN
ejpam-4574	455	5	+	+	CCONJ
ejpam-4574	455	6	1	1	X
ejpam-4574	455	7	)	)	PUNCT
ejpam-4574	456	1	+	+	CCONJ
ejpam-4574	456	2	r	r	X
ejpam-4574	456	3	,	,	PUNCT
ejpam-4574	456	4	0	0	NUM
ejpam-4574	456	5	<	<	X
ejpam-4574	456	6	r	r	X
ejpam-4574	456	7	<	<	X
ejpam-4574	456	8	k	k	X
ejpam-4574	457	1	+	+	PROPN
ejpam-4574	457	2	1	1	NUM
ejpam-4574	457	3	,	,	PUNCT
ejpam-4574	457	4	q	q	X
ejpam-4574	457	5	≥	≥	X
ejpam-4574	457	6	k	k	NOUN
ejpam-4574	458	1	+	+	CCONJ
ejpam-4574	458	2	1	1	NUM
ejpam-4574	458	3	and	and	CCONJ
ejpam-4574	458	4	n	n	PRON
ejpam-4574	458	5	̸=	̸=	PROPN
ejpam-4574	458	6	(	(	PUNCT
ejpam-4574	458	7	k	k	PROPN
ejpam-4574	458	8	+	+	PROPN
ejpam-4574	458	9	1)2	1)2	NUM
ejpam-4574	459	1	+	+	CCONJ
ejpam-4574	459	2	k.	k.	PROPN
ejpam-4574	459	3	let	let	VERB
ejpam-4574	459	4	c′5(vi	c′5(vi	PRON
ejpam-4574	459	5	)	)	PUNCT
ejpam-4574	459	6	=	=	PUNCT
ejpam-4574	459	7			X
ejpam-4574	459	8	ck+2	ck+2	NOUN
ejpam-4574	459	9	if	if	SCONJ
ejpam-4574	459	10	i	i	PRON
ejpam-4574	459	11	=	=	PUNCT
ejpam-4574	459	12	j(k	j(k	X
ejpam-4574	459	13	+	+	CCONJ
ejpam-4574	459	14	3	3	X
ejpam-4574	459	15	)	)	PUNCT
ejpam-4574	459	16	for	for	ADP
ejpam-4574	459	17	j	j	PROPN
ejpam-4574	459	18	∈	∈	PROPN
ejpam-4574	459	19	{	{	PUNCT
ejpam-4574	459	20	0	0	NUM
ejpam-4574	459	21	,	,	PUNCT
ejpam-4574	459	22	1	1	NUM
ejpam-4574	459	23	,	,	PUNCT
ejpam-4574	459	24	...	...	PUNCT
ejpam-4574	459	25	,	,	PUNCT
ejpam-4574	459	26	r	r	X
ejpam-4574	459	27	}	}	PUNCT
ejpam-4574	459	28	ck+1	ck+1	PUNCT
ejpam-4574	459	29	if	if	SCONJ
ejpam-4574	459	30	i	i	PRON
ejpam-4574	459	31	=	=	PUNCT
ejpam-4574	460	1	r(k	r(k	PROPN
ejpam-4574	460	2	+	+	NOUN
ejpam-4574	460	3	3	3	X
ejpam-4574	460	4	)	)	PUNCT
ejpam-4574	460	5	+	+	X
ejpam-4574	460	6	j(k	j(k	X
ejpam-4574	460	7	+	+	CCONJ
ejpam-4574	460	8	2	2	X
ejpam-4574	460	9	)	)	PUNCT
ejpam-4574	460	10	for	for	ADP
ejpam-4574	460	11	j	j	PROPN
ejpam-4574	460	12	∈	∈	PROPN
ejpam-4574	460	13	{	{	PUNCT
ejpam-4574	460	14	1	1	NUM
ejpam-4574	460	15	,	,	PUNCT
ejpam-4574	460	16	2	2	NUM
ejpam-4574	460	17	,	,	PUNCT
ejpam-4574	460	18	...	...	PUNCT
ejpam-4574	460	19	,	,	PUNCT
ejpam-4574	460	20	k	k	PROPN
ejpam-4574	460	21	−	−	NOUN
ejpam-4574	460	22	r	r	X
ejpam-4574	460	23	}	}	PUNCT
ejpam-4574	460	24	c′2(vi	c′2(vi	NOUN
ejpam-4574	460	25	)	)	PUNCT
ejpam-4574	460	26	otherwise	otherwise	ADV
ejpam-4574	460	27	.	.	PUNCT
ejpam-4574	461	1	then	then	ADV
ejpam-4574	461	2	:	:	PUNCT
ejpam-4574	461	3	a.	a.	PROPN
ejpam-4574	461	4	etawi	etawi	PROPN
ejpam-4574	461	5	,	,	PUNCT
ejpam-4574	461	6	m.	m.	PROPN
ejpam-4574	461	7	ghanem	ghanem	PROPN
ejpam-4574	461	8	,	,	PUNCT
ejpam-4574	461	9	h.	h.	PROPN
ejpam-4574	461	10	al	al	PROPN
ejpam-4574	461	11	-	-	PUNCT
ejpam-4574	461	12	ezeh	ezeh	PROPN
ejpam-4574	461	13	/	/	SYM
ejpam-4574	461	14	eur	eur	NOUN
ejpam-4574	461	15	.	.	PUNCT
ejpam-4574	462	1	j.	j.	PROPN
ejpam-4574	462	2	pure	pure	PROPN
ejpam-4574	462	3	appl	appl	PROPN
ejpam-4574	462	4	.	.	PROPN
ejpam-4574	462	5	math	math	PROPN
ejpam-4574	462	6	,	,	PUNCT
ejpam-4574	462	7	15	15	NUM
ejpam-4574	462	8	(	(	PUNCT
ejpam-4574	462	9	4	4	NUM
ejpam-4574	462	10	)	)	PUNCT
ejpam-4574	462	11	(	(	PUNCT
ejpam-4574	462	12	2022	2022	NUM
ejpam-4574	462	13	)	)	PUNCT
ejpam-4574	462	14	,	,	PUNCT
ejpam-4574	462	15	1822	1822	NUM
ejpam-4574	462	16	-	-	SYM
ejpam-4574	462	17	1835	1835	NUM
ejpam-4574	462	18	1832	1832	NUM
ejpam-4574	462	19	(	(	PUNCT
ejpam-4574	462	20	1	1	X
ejpam-4574	462	21	)	)	PUNCT
ejpam-4574	462	22	c′5	c′5	NOUN
ejpam-4574	462	23	is	be	AUX
ejpam-4574	462	24	a	a	DET
ejpam-4574	462	25	(	(	PUNCT
ejpam-4574	462	26	k	k	PROPN
ejpam-4574	462	27	+	+	CCONJ
ejpam-4574	462	28	3)−proper	3)−proper	PROPN
ejpam-4574	462	29	coloring	coloring	NOUN
ejpam-4574	462	30	of	of	ADP
ejpam-4574	462	31	ck	ck	PROPN
ejpam-4574	462	32	n.	n.	NOUN
ejpam-4574	462	33	(	(	PUNCT
ejpam-4574	462	34	2	2	NUM
ejpam-4574	462	35	)	)	PUNCT
ejpam-4574	462	36	for	for	ADP
ejpam-4574	462	37	i	i	PROPN
ejpam-4574	462	38	∈	∈	PROPN
ejpam-4574	462	39	{	{	PUNCT
ejpam-4574	462	40	0	0	NUM
ejpam-4574	462	41	,	,	PUNCT
ejpam-4574	462	42	...	...	PUNCT
ejpam-4574	462	43	,	,	PUNCT
ejpam-4574	462	44	k	k	X
ejpam-4574	462	45	}	}	PUNCT
ejpam-4574	462	46	,	,	PUNCT
ejpam-4574	462	47	ti	ti	NOUN
ejpam-4574	462	48	is	be	AUX
ejpam-4574	462	49	not	not	PART
ejpam-4574	462	50	an	an	DET
ejpam-4574	462	51	induced	induced	ADJ
ejpam-4574	462	52	cycle	cycle	NOUN
ejpam-4574	462	53	of	of	ADP
ejpam-4574	462	54	c2k	c2k	NOUN
ejpam-4574	462	55	n	n	NOUN
ejpam-4574	462	56	.	.	PUNCT
ejpam-4574	463	1	(	(	PUNCT
ejpam-4574	463	2	3	3	X
ejpam-4574	463	3	)	)	PUNCT
ejpam-4574	463	4	c′5	c′5	NOUN
ejpam-4574	463	5	is	be	AUX
ejpam-4574	463	6	a	a	DET
ejpam-4574	463	7	(	(	PUNCT
ejpam-4574	463	8	k	k	X
ejpam-4574	463	9	+	+	CCONJ
ejpam-4574	463	10	3)−	3)−	NUM
ejpam-4574	463	11	acyclic	acyclic	ADJ
ejpam-4574	463	12	coloring	coloring	NOUN
ejpam-4574	463	13	of	of	ADP
ejpam-4574	463	14	ck	ck	PROPN
ejpam-4574	463	15	n.	n.	NOUN
ejpam-4574	463	16	define	define	NOUN
ejpam-4574	463	17	p1	p1	PROPN
ejpam-4574	463	18	=	=	SYM
ejpam-4574	463	19	{	{	PUNCT
ejpam-4574	463	20	0	0	NUM
ejpam-4574	463	21	,	,	PUNCT
ejpam-4574	463	22	k+3	k+3	PROPN
ejpam-4574	463	23	,	,	PUNCT
ejpam-4574	463	24	2(k+3	2(k+3	NUM
ejpam-4574	463	25	)	)	PUNCT
ejpam-4574	463	26	,	,	PUNCT
ejpam-4574	463	27	...	...	PUNCT
ejpam-4574	463	28	,	,	PUNCT
ejpam-4574	463	29	r(k+3	r(k+3	NUM
ejpam-4574	463	30	)	)	PUNCT
ejpam-4574	463	31	}	}	PUNCT
ejpam-4574	463	32	and	and	CCONJ
ejpam-4574	463	33	p2	p2	PROPN
ejpam-4574	463	34	=	=	SYM
ejpam-4574	463	35	{	{	PUNCT
ejpam-4574	463	36	r(k+3)+	r(k+3)+	X
ejpam-4574	463	37	(	(	PUNCT
ejpam-4574	463	38	k+2	k+2	NOUN
ejpam-4574	463	39	)	)	PUNCT
ejpam-4574	463	40	,	,	PUNCT
ejpam-4574	463	41	r(k+3)+	r(k+3)+	ADJ
ejpam-4574	463	42	2(k+	2(k+	NUM
ejpam-4574	463	43	2	2	NUM
ejpam-4574	463	44	)	)	PUNCT
ejpam-4574	463	45	,	,	PUNCT
ejpam-4574	463	46	...	...	PUNCT
ejpam-4574	463	47	,	,	PUNCT
ejpam-4574	464	1	r(k	r(k	PROPN
ejpam-4574	464	2	+	+	ADJ
ejpam-4574	464	3	3	3	X
ejpam-4574	464	4	)	)	PUNCT
ejpam-4574	464	5	+	+	CCONJ
ejpam-4574	464	6	(	(	PUNCT
ejpam-4574	464	7	k	k	PROPN
ejpam-4574	464	8	−	−	PROPN
ejpam-4574	464	9	r)(k	r)(k	NOUN
ejpam-4574	464	10	+	+	CCONJ
ejpam-4574	464	11	2	2	NUM
ejpam-4574	464	12	)	)	PUNCT
ejpam-4574	464	13	}	}	PUNCT
ejpam-4574	464	14	.	.	PUNCT
ejpam-4574	465	1	proof	proof	NOUN
ejpam-4574	465	2	.	.	PUNCT
ejpam-4574	466	1	(	(	PUNCT
ejpam-4574	466	2	1	1	X
ejpam-4574	466	3	)	)	PUNCT
ejpam-4574	466	4	since	since	SCONJ
ejpam-4574	466	5	dcn(vh(k+3	dcn(vh(k+3	PROPN
ejpam-4574	466	6	)	)	PUNCT
ejpam-4574	466	7	,	,	PUNCT
ejpam-4574	466	8	v(h+1)(k+3	v(h+1)(k+3	PROPN
ejpam-4574	466	9	)	)	PUNCT
ejpam-4574	466	10	)	)	PUNCT
ejpam-4574	467	1	=	=	SYM
ejpam-4574	467	2	k+3	k+3	PROPN
ejpam-4574	467	3	for	for	ADP
ejpam-4574	467	4	0	0	NUM
ejpam-4574	467	5	≤	≤	NUM
ejpam-4574	467	6	h	h	NOUN
ejpam-4574	467	7	≤	≤	NUM
ejpam-4574	467	8	r−	r−	PROPN
ejpam-4574	467	9	1	1	NUM
ejpam-4574	467	10	,	,	PUNCT
ejpam-4574	467	11	dcn(vr(k+3	dcn(vr(k+3	PROPN
ejpam-4574	467	12	)	)	PUNCT
ejpam-4574	467	13	,	,	PUNCT
ejpam-4574	467	14	v0	v0	PROPN
ejpam-4574	467	15	)	)	PUNCT
ejpam-4574	467	16	≥	≥	NOUN
ejpam-4574	467	17	k	k	X
ejpam-4574	468	1	+	+	CCONJ
ejpam-4574	468	2	3	3	NUM
ejpam-4574	468	3	,	,	PUNCT
ejpam-4574	468	4	dcn(vr(k+3)+h(k+2	dcn(vr(k+3)+h(k+2	PROPN
ejpam-4574	468	5	)	)	PUNCT
ejpam-4574	468	6	,	,	PUNCT
ejpam-4574	468	7	vr(k+3)+(h+1)(k+2	vr(k+3)+(h+1)(k+2	NOUN
ejpam-4574	468	8	)	)	PUNCT
ejpam-4574	468	9	)	)	PUNCT
ejpam-4574	469	1	=	=	PUNCT
ejpam-4574	470	1	k	k	PROPN
ejpam-4574	471	1	+	+	CCONJ
ejpam-4574	471	2	2	2	NUM
ejpam-4574	471	3	for	for	ADP
ejpam-4574	471	4	1	1	NUM
ejpam-4574	471	5	≤	≤	NUM
ejpam-4574	471	6	h	h	NOUN
ejpam-4574	471	7	≤	≤	NUM
ejpam-4574	471	8	k	k	NOUN
ejpam-4574	472	1	−	−	NOUN
ejpam-4574	472	2	r	r	NOUN
ejpam-4574	472	3	−	−	NOUN
ejpam-4574	472	4	1	1	NUM
ejpam-4574	472	5	,	,	PUNCT
ejpam-4574	472	6	dcn(vn−((k−r)(k+2)−q	dcn(vn−((k−r)(k+2)−q	NOUN
ejpam-4574	472	7	)	)	PUNCT
ejpam-4574	472	8	,	,	PUNCT
ejpam-4574	472	9	vk+1	vk+1	X
ejpam-4574	472	10	)	)	PUNCT
ejpam-4574	472	11	≥	≥	NOUN
ejpam-4574	472	12	k	k	NOUN
ejpam-4574	473	1	+	+	CCONJ
ejpam-4574	473	2	2	2	NUM
ejpam-4574	473	3	and	and	CCONJ
ejpam-4574	473	4	c′2	c′2	NOUN
ejpam-4574	473	5	is	be	AUX
ejpam-4574	473	6	a	a	DET
ejpam-4574	473	7	proper	proper	ADJ
ejpam-4574	473	8	coloring	coloring	NOUN
ejpam-4574	473	9	,	,	PUNCT
ejpam-4574	473	10	we	we	PRON
ejpam-4574	473	11	have	have	AUX
ejpam-4574	473	12	c′5	c′5	NOUN
ejpam-4574	473	13	is	be	AUX
ejpam-4574	473	14	a	a	DET
ejpam-4574	473	15	proper	proper	ADJ
ejpam-4574	473	16	coloring	coloring	NOUN
ejpam-4574	473	17	of	of	ADP
ejpam-4574	473	18	ck	ck	PROPN
ejpam-4574	473	19	n.	n.	NOUN
ejpam-4574	473	20	(	(	PUNCT
ejpam-4574	473	21	2	2	NUM
ejpam-4574	473	22	)	)	PUNCT
ejpam-4574	473	23	to	to	PART
ejpam-4574	473	24	show	show	VERB
ejpam-4574	473	25	that	that	SCONJ
ejpam-4574	473	26	ti	ti	NOUN
ejpam-4574	473	27	is	be	AUX
ejpam-4574	473	28	not	not	PART
ejpam-4574	473	29	an	an	DET
ejpam-4574	473	30	induced	induced	ADJ
ejpam-4574	473	31	cycle	cycle	NOUN
ejpam-4574	473	32	of	of	ADP
ejpam-4574	473	33	c2k	c2k	X
ejpam-4574	473	34	n	n	NOUN
ejpam-4574	473	35	for	for	ADP
ejpam-4574	473	36	i	i	PRON
ejpam-4574	473	37	∈	∈	PROPN
ejpam-4574	473	38	{	{	PUNCT
ejpam-4574	473	39	0	0	NUM
ejpam-4574	473	40	,	,	PUNCT
ejpam-4574	473	41	...	...	PUNCT
ejpam-4574	473	42	,	,	PUNCT
ejpam-4574	473	43	k	k	X
ejpam-4574	473	44	}	}	PUNCT
ejpam-4574	473	45	,	,	PUNCT
ejpam-4574	473	46	consider	consider	VERB
ejpam-4574	473	47	the	the	DET
ejpam-4574	473	48	following	follow	VERB
ejpam-4574	473	49	cases	case	NOUN
ejpam-4574	473	50	case	case	NOUN
ejpam-4574	473	51	1	1	NUM
ejpam-4574	473	52	.	.	PUNCT
ejpam-4574	474	1	i	i	PRON
ejpam-4574	474	2	=	=	NOUN
ejpam-4574	475	1	0	0	X
ejpam-4574	475	2	.	.	PUNCT
ejpam-4574	476	1	then	then	ADV
ejpam-4574	476	2	v0	v0	PROPN
ejpam-4574	476	3	/∈	/∈	PUNCT
ejpam-4574	477	1	t0	t0	PROPN
ejpam-4574	477	2	and	and	CCONJ
ejpam-4574	477	3	dcn(vn−k	dcn(vn−k	PROPN
ejpam-4574	477	4	,	,	PUNCT
ejpam-4574	477	5	vk+2	vk+2	NOUN
ejpam-4574	477	6	)	)	PUNCT
ejpam-4574	477	7	=	=	SYM
ejpam-4574	477	8	2k	2k	NOUN
ejpam-4574	477	9	+	+	CCONJ
ejpam-4574	477	10	2	2	X
ejpam-4574	477	11	.	.	X
ejpam-4574	477	12	case	case	NOUN
ejpam-4574	477	13	2	2	NUM
ejpam-4574	477	14	.	.	NOUN
ejpam-4574	477	15	0	0	PUNCT
ejpam-4574	478	1	<	<	X
ejpam-4574	478	2	i	i	X
ejpam-4574	478	3	<	<	X
ejpam-4574	478	4	r.	r.	PROPN
ejpam-4574	478	5	then	then	ADV
ejpam-4574	478	6	vi(k+2)+i	vi(k+2)+i	PROPN
ejpam-4574	478	7	/∈	/∈	PUNCT
ejpam-4574	478	8	ti	ti	PROPN
ejpam-4574	478	9	and	and	CCONJ
ejpam-4574	478	10	dcn(v(i−1)(k+2)+i	dcn(v(i−1)(k+2)+i	PROPN
ejpam-4574	478	11	,	,	PUNCT
ejpam-4574	478	12	v(i+1)(k+2)+i	v(i+1)(k+2)+i	NOUN
ejpam-4574	478	13	)	)	PUNCT
ejpam-4574	479	1	=	=	PUNCT
ejpam-4574	480	1	2(k	2(k	NUM
ejpam-4574	480	2	+	+	CCONJ
ejpam-4574	480	3	2	2	NUM
ejpam-4574	480	4	)	)	PUNCT
ejpam-4574	480	5	.	.	PUNCT
ejpam-4574	481	1	case	case	NOUN
ejpam-4574	482	1	3	3	X
ejpam-4574	482	2	.	.	PUNCT
ejpam-4574	483	1	i	i	PRON
ejpam-4574	483	2	=	=	SYM
ejpam-4574	483	3	r.	r.	PROPN
ejpam-4574	483	4	then	then	ADV
ejpam-4574	483	5	vr(k+2)+i	vr(k+2)+i	PROPN
ejpam-4574	483	6	/∈	/∈	PUNCT
ejpam-4574	484	1	ti	ti	NOUN
ejpam-4574	484	2	and	and	CCONJ
ejpam-4574	484	3	dcn(v(r−1)(k+2)+i	dcn(v(r−1)(k+2)+i	PROPN
ejpam-4574	484	4	,	,	PUNCT
ejpam-4574	484	5	vr(k+2)+(k+1)+i	vr(k+2)+(k+1)+i	NOUN
ejpam-4574	484	6	)	)	PUNCT
ejpam-4574	485	1	=	=	SYM
ejpam-4574	485	2	2k	2k	NUM
ejpam-4574	485	3	+	+	CCONJ
ejpam-4574	485	4	3	3	X
ejpam-4574	485	5	.	.	X
ejpam-4574	485	6	case	case	NOUN
ejpam-4574	485	7	4	4	NUM
ejpam-4574	485	8	.	.	PUNCT
ejpam-4574	486	1	r	r	NOUN
ejpam-4574	486	2	<	<	X
ejpam-4574	486	3	i	i	X
ejpam-4574	486	4	<	<	X
ejpam-4574	486	5	k.	k.	PROPN
ejpam-4574	486	6	then	then	ADV
ejpam-4574	486	7	vr(k+3)+(i−r)(k+2	vr(k+3)+(i−r)(k+2	PROPN
ejpam-4574	486	8	)	)	PUNCT
ejpam-4574	486	9	/∈	/∈	PUNCT
ejpam-4574	487	1	ti	ti	NOUN
ejpam-4574	487	2	and	and	CCONJ
ejpam-4574	487	3	dcn(vr(k+3)+(i−r−1)(k+2	dcn(vr(k+3)+(i−r−1)(k+2	NUM
ejpam-4574	487	4	)	)	PUNCT
ejpam-4574	487	5	,	,	PUNCT
ejpam-4574	487	6	vr(k+3)+(i−r+1)(k+2	vr(k+3)+(i−r+1)(k+2	ADJ
ejpam-4574	487	7	)	)	PUNCT
ejpam-4574	487	8	)	)	PUNCT
ejpam-4574	488	1	=	=	PUNCT
ejpam-4574	489	1	2(k	2(k	NUM
ejpam-4574	489	2	+	+	CCONJ
ejpam-4574	489	3	2	2	NUM
ejpam-4574	489	4	)	)	PUNCT
ejpam-4574	489	5	.	.	PUNCT
ejpam-4574	490	1	case	case	NOUN
ejpam-4574	490	2	5	5	NUM
ejpam-4574	490	3	.	.	PUNCT
ejpam-4574	491	1	i	i	PRON
ejpam-4574	491	2	=	=	SYM
ejpam-4574	491	3	k.	k.	PROPN
ejpam-4574	491	4	then	then	ADV
ejpam-4574	491	5	vr(k+3)+(k−r)(k+2	vr(k+3)+(k−r)(k+2	ADJ
ejpam-4574	491	6	)	)	PUNCT
ejpam-4574	491	7	/∈	/∈	PUNCT
ejpam-4574	492	1	tk	tk	PROPN
ejpam-4574	492	2	and	and	CCONJ
ejpam-4574	492	3	dcn(vr(k+3)+(k−r−1)(k+2	dcn(vr(k+3)+(k−r−1)(k+2	NOUN
ejpam-4574	492	4	)	)	PUNCT
ejpam-4574	492	5	,	,	PUNCT
ejpam-4574	492	6	vk	vk	PROPN
ejpam-4574	492	7	)	)	PUNCT
ejpam-4574	492	8	≥	≥	NOUN
ejpam-4574	492	9	2k	2k	NOUN
ejpam-4574	492	10	+	+	CCONJ
ejpam-4574	492	11	3	3	NUM
ejpam-4574	493	1	when	when	SCONJ
ejpam-4574	493	2	q	q	PROPN
ejpam-4574	493	3	≤	≤	X
ejpam-4574	493	4	k	k	X
ejpam-4574	494	1	+	+	NOUN
ejpam-4574	494	2	2	2	X
ejpam-4574	494	3	.	.	PUNCT
ejpam-4574	494	4	while	while	SCONJ
ejpam-4574	494	5	dcn(vr(k+3)+(k−r−1)(k+2	dcn(vr(k+3)+(k−r−1)(k+2	NOUN
ejpam-4574	494	6	)	)	PUNCT
ejpam-4574	494	7	,	,	PUNCT
ejpam-4574	494	8	vr(k+3)+(k−r+1)(k+2	vr(k+3)+(k−r+1)(k+2	NOUN
ejpam-4574	494	9	)	)	PUNCT
ejpam-4574	494	10	)	)	PUNCT
ejpam-4574	495	1	=	=	PUNCT
ejpam-4574	496	1	2(k	2(k	NUM
ejpam-4574	496	2	+	+	CCONJ
ejpam-4574	496	3	2	2	X
ejpam-4574	496	4	)	)	PUNCT
ejpam-4574	496	5	when	when	SCONJ
ejpam-4574	496	6	q	q	X
ejpam-4574	496	7	≥	≥	AUX
ejpam-4574	496	8	k	k	X
ejpam-4574	497	1	+	+	CCONJ
ejpam-4574	497	2	3	3	X
ejpam-4574	497	3	.	.	X
ejpam-4574	497	4	therefore	therefore	ADV
ejpam-4574	497	5	,	,	PUNCT
ejpam-4574	497	6	tk	tk	PROPN
ejpam-4574	497	7	is	be	AUX
ejpam-4574	497	8	not	not	PART
ejpam-4574	497	9	an	an	DET
ejpam-4574	497	10	induced	induced	ADJ
ejpam-4574	497	11	cycle	cycle	NOUN
ejpam-4574	497	12	of	of	ADP
ejpam-4574	497	13	c2k	c2k	PROPN
ejpam-4574	497	14	n	n	NOUN
ejpam-4574	497	15	.	.	PUNCT
ejpam-4574	498	1	adding	add	VERB
ejpam-4574	498	2	ck+2	ck+2	NOUN
ejpam-4574	498	3	to	to	ADP
ejpam-4574	498	4	p1	p1	PROPN
ejpam-4574	498	5	denies	deny	VERB
ejpam-4574	498	6	a	a	DET
ejpam-4574	498	7	turn	turn	NOUN
ejpam-4574	498	8	of	of	ADP
ejpam-4574	498	9	each	each	DET
ejpam-4574	498	10	color	color	NOUN
ejpam-4574	498	11	of	of	ADP
ejpam-4574	498	12	c1	c1	PROPN
ejpam-4574	498	13	,	,	PUNCT
ejpam-4574	498	14	c2	c2	PROPN
ejpam-4574	498	15	,	,	PUNCT
ejpam-4574	498	16	...	...	PUNCT
ejpam-4574	498	17	,	,	PUNCT
ejpam-4574	498	18	cr	cr	NOUN
ejpam-4574	498	19	,	,	PUNCT
ejpam-4574	498	20	and	and	CCONJ
ejpam-4574	498	21	adding	add	VERB
ejpam-4574	498	22	ck+1	ck+1	ADV
ejpam-4574	498	23	to	to	ADP
ejpam-4574	498	24	p2	p2	PROPN
ejpam-4574	498	25	does	do	VERB
ejpam-4574	498	26	the	the	DET
ejpam-4574	498	27	same	same	ADJ
ejpam-4574	498	28	for	for	ADP
ejpam-4574	498	29	colors	color	NOUN
ejpam-4574	498	30	cr+1	cr+1	PROPN
ejpam-4574	498	31	,	,	PUNCT
ejpam-4574	498	32	cr+2	cr+2	X
ejpam-4574	498	33	,	,	PUNCT
ejpam-4574	498	34	...	...	PUNCT
ejpam-4574	498	35	,	,	PUNCT
ejpam-4574	498	36	ck	ck	PROPN
ejpam-4574	498	37	.	.	PUNCT
ejpam-4574	499	1	also	also	ADV
ejpam-4574	499	2	the	the	DET
ejpam-4574	499	3	distance	distance	NOUN
ejpam-4574	499	4	between	between	ADP
ejpam-4574	499	5	the	the	DET
ejpam-4574	499	6	last	last	ADJ
ejpam-4574	499	7	vertex	vertex	NOUN
ejpam-4574	499	8	in	in	ADP
ejpam-4574	499	9	p1	p1	PROPN
ejpam-4574	499	10	that	that	PRON
ejpam-4574	499	11	is	be	AUX
ejpam-4574	499	12	colored	color	VERB
ejpam-4574	499	13	with	with	ADP
ejpam-4574	499	14	ck+1	ck+1	NOUN
ejpam-4574	499	15	and	and	CCONJ
ejpam-4574	499	16	the	the	DET
ejpam-4574	499	17	first	first	ADJ
ejpam-4574	499	18	vertex	vertex	NOUN
ejpam-4574	499	19	in	in	ADP
ejpam-4574	499	20	p2	p2	PROPN
ejpam-4574	499	21	that	that	PRON
ejpam-4574	499	22	is	be	AUX
ejpam-4574	499	23	colored	color	VERB
ejpam-4574	499	24	with	with	ADP
ejpam-4574	499	25	ck+1	ck+1	ADJ
ejpam-4574	499	26	is	be	AUX
ejpam-4574	499	27	(	(	PUNCT
ejpam-4574	499	28	k	k	X
ejpam-4574	499	29	+	+	PROPN
ejpam-4574	499	30	1	1	X
ejpam-4574	499	31	)	)	PUNCT
ejpam-4574	499	32	which	which	PRON
ejpam-4574	499	33	keeps	keep	VERB
ejpam-4574	499	34	the	the	DET
ejpam-4574	499	35	coloring	coloring	NOUN
ejpam-4574	499	36	proper	proper	ADJ
ejpam-4574	499	37	as	as	SCONJ
ejpam-4574	499	38	shown	show	VERB
ejpam-4574	499	39	in	in	ADP
ejpam-4574	499	40	the	the	DET
ejpam-4574	499	41	below	below	ADJ
ejpam-4574	499	42	table	table	NOUN
ejpam-4574	499	43	:	:	PUNCT
ejpam-4574	499	44	rule	rule	NOUN
ejpam-4574	499	45	i	i	PRON
ejpam-4574	499	46	mod(k	mod(k	PROPN
ejpam-4574	500	1	+	+	CCONJ
ejpam-4574	500	2	2	2	NUM
ejpam-4574	500	3	)	)	PUNCT
ejpam-4574	500	4	(	(	PUNCT
ejpam-4574	500	5	i−	i−	PROPN
ejpam-4574	500	6	x)mod(k	x)mod(k	PROPN
ejpam-4574	500	7	+	+	CCONJ
ejpam-4574	501	1	1	1	X
ejpam-4574	501	2	)	)	PUNCT
ejpam-4574	501	3	i	i	NOUN
ejpam-4574	501	4	0	0	NUM
ejpam-4574	501	5	1	1	NUM
ejpam-4574	501	6	...	...	PUNCT
ejpam-4574	502	1	k	k	X
ejpam-4574	503	1	+	+	CCONJ
ejpam-4574	503	2	2	2	NUM
ejpam-4574	503	3	k	k	NOUN
ejpam-4574	503	4	+	+	NOUN
ejpam-4574	503	5	3	3	NUM
ejpam-4574	503	6	k	k	NOUN
ejpam-4574	503	7	+	+	ADP
ejpam-4574	503	8	4	4	NUM
ejpam-4574	503	9	...	...	SYM
ejpam-4574	503	10	2	2	NUM
ejpam-4574	503	11	k	k	NOUN
ejpam-4574	503	12	+	+	CCONJ
ejpam-4574	503	13	5	5	NUM
ejpam-4574	503	14	2	2	NUM
ejpam-4574	503	15	(	(	PUNCT
ejpam-4574	503	16	k	k	NOUN
ejpam-4574	503	17	+	+	PROPN
ejpam-4574	503	18	3	3	X
ejpam-4574	503	19	)	)	PUNCT
ejpam-4574	503	20	2	2	NUM
ejpam-4574	504	1	k	k	NOUN
ejpam-4574	505	1	+	+	CCONJ
ejpam-4574	505	2	7	7	NUM
ejpam-4574	505	3	...	...	SYM
ejpam-4574	505	4	3	3	NUM
ejpam-4574	505	5	k	k	NOUN
ejpam-4574	505	6	+	+	CCONJ
ejpam-4574	505	7	8	8	NUM
ejpam-4574	505	8	3	3	NUM
ejpam-4574	505	9	(	(	PUNCT
ejpam-4574	505	10	k	k	X
ejpam-4574	505	11	+	+	PROPN
ejpam-4574	505	12	3	3	NUM
ejpam-4574	505	13	)	)	PUNCT
ejpam-4574	505	14	...	...	PUNCT
ejpam-4574	506	1	(	(	PUNCT
ejpam-4574	506	2	r	r	NOUN
ejpam-4574	506	3	−	−	PROPN
ejpam-4574	506	4	1	1	NUM
ejpam-4574	506	5	)	)	PUNCT
ejpam-4574	506	6	(	(	PUNCT
ejpam-4574	506	7	k	k	NOUN
ejpam-4574	506	8	+	+	CCONJ
ejpam-4574	506	9	2	2	X
ejpam-4574	506	10	)	)	PUNCT
ejpam-4574	507	1	+	+	CCONJ
ejpam-4574	507	2	r	r	NOUN
ejpam-4574	507	3	...	...	PUNCT
ejpam-4574	508	1	r	r	X
ejpam-4574	508	2	(	(	PUNCT
ejpam-4574	508	3	k	k	NOUN
ejpam-4574	508	4	+	+	PROPN
ejpam-4574	508	5	3	3	X
ejpam-4574	508	6	)	)	PUNCT
ejpam-4574	508	7	r	r	NOUN
ejpam-4574	508	8	(	(	PUNCT
ejpam-4574	508	9	k	k	NOUN
ejpam-4574	508	10	+	+	PROPN
ejpam-4574	508	11	3	3	NUM
ejpam-4574	508	12	)	)	PUNCT
ejpam-4574	508	13	+	+	CCONJ
ejpam-4574	508	14	1	1	NUM
ejpam-4574	508	15	...	...	PUNCT
ejpam-4574	508	16	x	x	X
ejpam-4574	509	1	−	−	PROPN
ejpam-4574	509	2	1	1	NUM
ejpam-4574	509	3	x	x	NOUN
ejpam-4574	509	4	...	...	PUNCT
ejpam-4574	509	5	x	x	PUNCT
ejpam-4574	510	1	+	+	PUNCT
ejpam-4574	511	1	r	r	NOUN
ejpam-4574	511	2	x	x	SYM
ejpam-4574	512	1	+	+	CCONJ
ejpam-4574	512	2	r	r	NOUN
ejpam-4574	512	3	+	+	NOUN
ejpam-4574	512	4	1	1	NUM
ejpam-4574	512	5	...	...	PUNCT
ejpam-4574	513	1	r	r	NOUN
ejpam-4574	513	2	(	(	PUNCT
ejpam-4574	513	3	k	k	NOUN
ejpam-4574	513	4	+	+	PROPN
ejpam-4574	513	5	3	3	NUM
ejpam-4574	513	6	)	)	PUNCT
ejpam-4574	514	1	+	+	CCONJ
ejpam-4574	515	1	k	k	X
ejpam-4574	516	1	+	+	CCONJ
ejpam-4574	516	2	1	1	NUM
ejpam-4574	516	3	r	r	NOUN
ejpam-4574	516	4	(	(	PUNCT
ejpam-4574	516	5	k	k	NOUN
ejpam-4574	516	6	+	+	PROPN
ejpam-4574	516	7	3	3	NUM
ejpam-4574	516	8	)	)	PUNCT
ejpam-4574	517	1	+	+	CCONJ
ejpam-4574	517	2	k	k	X
ejpam-4574	518	1	+	+	CCONJ
ejpam-4574	518	2	2	2	NUM
ejpam-4574	518	3	...	...	SYM
ejpam-4574	518	4	n	n	CCONJ
ejpam-4574	518	5	-1	-1	ADJ
ejpam-4574	518	6	c2(vi	c2(vi	PROPN
ejpam-4574	518	7	)	)	PUNCT
ejpam-4574	518	8	c0	c0	PROPN
ejpam-4574	518	9	c1	c1	PROPN
ejpam-4574	518	10	...	...	PUNCT
ejpam-4574	519	1	c0	c0	PROPN
ejpam-4574	519	2	c1	c1	PROPN
ejpam-4574	519	3	c2	c2	PROPN
ejpam-4574	519	4	...	...	PUNCT
ejpam-4574	520	1	c1	c1	PROPN
ejpam-4574	520	2	c2	c2	PROPN
ejpam-4574	520	3	c3	c3	PROPN
ejpam-4574	520	4	...	...	PUNCT
ejpam-4574	521	1	c2	c2	PROPN
ejpam-4574	521	2	c3	c3	PROPN
ejpam-4574	521	3	...	...	PUNCT
ejpam-4574	522	1	cr	cr	X
ejpam-4574	522	2	...	...	PUNCT
ejpam-4574	523	1	cr	cr	PROPN
ejpam-4574	523	2	cr+1	cr+1	PROPN
ejpam-4574	523	3	...	...	PUNCT
ejpam-4574	524	1	ck	ck	PROPN
ejpam-4574	524	2	c0	c0	PROPN
ejpam-4574	524	3	...	...	PUNCT
ejpam-4574	525	1	cr	cr	PROPN
ejpam-4574	525	2	cr+1	cr+1	PROPN
ejpam-4574	525	3	...	...	PUNCT
ejpam-4574	526	1	cr	cr	X
ejpam-4574	526	2	cr+1	cr+1	PROPN
ejpam-4574	526	3	...	...	PUNCT
ejpam-4574	526	4	ck+1	ck+1	NUM
ejpam-4574	526	5	c5(vi	c5(vi	NUM
ejpam-4574	526	6	)	)	PUNCT
ejpam-4574	526	7	ck+2	ck+2	NOUN
ejpam-4574	526	8	c1	c1	NOUN
ejpam-4574	526	9	...	...	PUNCT
ejpam-4574	527	1	c0	c0	PROPN
ejpam-4574	527	2	ck+2	ck+2	PROPN
ejpam-4574	527	3	c2	c2	PROPN
ejpam-4574	527	4	...	...	PUNCT
ejpam-4574	528	1	c1	c1	PROPN
ejpam-4574	528	2	ck+2	ck+2	NOUN
ejpam-4574	528	3	c3	c3	PROPN
ejpam-4574	528	4	...	...	PUNCT
ejpam-4574	529	1	c2	c2	PROPN
ejpam-4574	529	2	ck+2	ck+2	NOUN
ejpam-4574	529	3	...	...	PUNCT
ejpam-4574	529	4	cr	cr	X
ejpam-4574	529	5	...	...	PUNCT
ejpam-4574	530	1	ck+2	ck+2	PROPN
ejpam-4574	530	2	cr+1	cr+1	NOUN
ejpam-4574	530	3	...	...	PUNCT
ejpam-4574	530	4	ck	ck	PROPN
ejpam-4574	530	5	c0	c0	PROPN
ejpam-4574	530	6	...	...	PUNCT
ejpam-4574	531	1	cr	cr	PROPN
ejpam-4574	531	2	cr+1	cr+1	PROPN
ejpam-4574	531	3	...	...	PUNCT
ejpam-4574	532	1	cr	cr	NOUN
ejpam-4574	532	2	ck+1	ck+1	VERB
ejpam-4574	532	3	...	...	PUNCT
ejpam-4574	532	4	ck+1	ck+1	VERB
ejpam-4574	532	5	⌞	⌞	NUM
ejpam-4574	532	6	2(k	2(k	NUM
ejpam-4574	532	7	+	+	CCONJ
ejpam-4574	532	8	2	2	X
ejpam-4574	532	9	)	)	PUNCT
ejpam-4574	532	10	⌟	⌟	VERB
ejpam-4574	532	11	⌞	⌞	PUNCT
ejpam-4574	532	12	2(k	2(k	NUM
ejpam-4574	532	13	+	+	CCONJ
ejpam-4574	532	14	1	1	X
ejpam-4574	532	15	)	)	PUNCT
ejpam-4574	532	16	+	+	CCONJ
ejpam-4574	532	17	1	1	NUM
ejpam-4574	532	18	⌟	⌟	X
ejpam-4574	532	19	⌞	⌞	X
ejpam-4574	532	20	2(k	2(k	NUM
ejpam-4574	532	21	+	+	CCONJ
ejpam-4574	532	22	2	2	X
ejpam-4574	532	23	)	)	PUNCT
ejpam-4574	532	24	⌟	⌟	NOUN
ejpam-4574	532	25	⌞	⌞	PUNCT
ejpam-4574	533	1	k	k	NOUN
ejpam-4574	533	2	+	+	PROPN
ejpam-4574	533	3	1	1	NUM
ejpam-4574	533	4	⌟	⌟	NOUN
ejpam-4574	533	5	x	x	X
ejpam-4574	533	6	=	=	SYM
ejpam-4574	534	1	r(k	r(k	PROPN
ejpam-4574	534	2	+	+	PROPN
ejpam-4574	534	3	2	2	NUM
ejpam-4574	534	4	)	)	PUNCT
ejpam-4574	534	5	+	+	NOUN
ejpam-4574	534	6	k	k	X
ejpam-4574	535	1	+	+	CCONJ
ejpam-4574	535	2	1	1	NUM
ejpam-4574	535	3	(	(	PUNCT
ejpam-4574	535	4	3	3	NUM
ejpam-4574	535	5	)	)	PUNCT
ejpam-4574	535	6	since	since	SCONJ
ejpam-4574	535	7	dcn(v0	dcn(v0	VERB
ejpam-4574	535	8	,	,	PUNCT
ejpam-4574	535	9	vk+1	vk+1	X
ejpam-4574	535	10	)	)	PUNCT
ejpam-4574	535	11	=	=	SYM
ejpam-4574	536	1	k	k	PROPN
ejpam-4574	537	1	+	+	CCONJ
ejpam-4574	537	2	1	1	NUM
ejpam-4574	537	3	and	and	CCONJ
ejpam-4574	537	4	ti	ti	X
ejpam-4574	537	5	not	not	PART
ejpam-4574	537	6	an	an	DET
ejpam-4574	537	7	induced	induced	ADJ
ejpam-4574	537	8	cycle	cycle	NOUN
ejpam-4574	537	9	of	of	ADP
ejpam-4574	537	10	c2k	c2k	X
ejpam-4574	537	11	n	n	NOUN
ejpam-4574	537	12	for	for	ADP
ejpam-4574	537	13	i	i	PRON
ejpam-4574	537	14	≤	≤	NOUN
ejpam-4574	538	1	k	k	X
ejpam-4574	538	2	,	,	PUNCT
ejpam-4574	538	3	c′5	c′5	NOUN
ejpam-4574	538	4	is	be	AUX
ejpam-4574	538	5	a	a	DET
ejpam-4574	538	6	(	(	PUNCT
ejpam-4574	538	7	k	k	X
ejpam-4574	538	8	+	+	CCONJ
ejpam-4574	538	9	3)−	3)−	NUM
ejpam-4574	538	10	acyclic	acyclic	ADJ
ejpam-4574	538	11	coloring	coloring	NOUN
ejpam-4574	538	12	for	for	ADP
ejpam-4574	538	13	ck	ck	PROPN
ejpam-4574	538	14	n.	n.	PROPN
ejpam-4574	538	15	lemma	lemma	PROPN
ejpam-4574	538	16	11	11	NUM
ejpam-4574	538	17	.	.	PUNCT
ejpam-4574	539	1	let	let	VERB
ejpam-4574	539	2	n	n	NOUN
ejpam-4574	539	3	=	=	PUNCT
ejpam-4574	539	4	q(k	q(k	PROPN
ejpam-4574	539	5	+	+	CCONJ
ejpam-4574	539	6	1	1	X
ejpam-4574	539	7	)	)	PUNCT
ejpam-4574	540	1	+	+	CCONJ
ejpam-4574	540	2	r	r	X
ejpam-4574	540	3	,	,	PUNCT
ejpam-4574	540	4	0	0	NUM
ejpam-4574	540	5	<	<	X
ejpam-4574	540	6	r	r	NOUN
ejpam-4574	540	7	≤	≤	PUNCT
ejpam-4574	540	8	k	k	NOUN
ejpam-4574	540	9	,	,	PUNCT
ejpam-4574	540	10	q	q	X
ejpam-4574	540	11	≥	≥	X
ejpam-4574	540	12	k	k	X
ejpam-4574	541	1	+	+	CCONJ
ejpam-4574	541	2	1	1	NUM
ejpam-4574	541	3	,	,	PUNCT
ejpam-4574	541	4	q	q	NOUN
ejpam-4574	542	1	−	−	PROPN
ejpam-4574	542	2	r	r	NOUN
ejpam-4574	542	3	≥	≥	NOUN
ejpam-4574	542	4	k	k	NOUN
ejpam-4574	543	1	+	+	CCONJ
ejpam-4574	543	2	1	1	NUM
ejpam-4574	543	3	and	and	CCONJ
ejpam-4574	543	4	c′6(vi	c′6(vi	PUNCT
ejpam-4574	543	5	)	)	PUNCT
ejpam-4574	544	1	=	=	PRON
ejpam-4574	544	2	{	{	PUNCT
ejpam-4574	544	3	ck+1	ck+1	VERB
ejpam-4574	544	4	if	if	SCONJ
ejpam-4574	544	5	i	i	PRON
ejpam-4574	544	6	=	=	PUNCT
ejpam-4574	544	7	(	(	PUNCT
ejpam-4574	544	8	r	r	NOUN
ejpam-4574	544	9	+	+	NUM
ejpam-4574	544	10	j)(k	j)(k	PROPN
ejpam-4574	544	11	+	+	CCONJ
ejpam-4574	544	12	2	2	NUM
ejpam-4574	544	13	)	)	PUNCT
ejpam-4574	544	14	+	+	CCONJ
ejpam-4574	544	15	(	(	PUNCT
ejpam-4574	544	16	k	k	X
ejpam-4574	544	17	+	+	PROPN
ejpam-4574	544	18	1	1	NUM
ejpam-4574	544	19	)	)	PUNCT
ejpam-4574	544	20	for	for	ADP
ejpam-4574	544	21	j	j	PROPN
ejpam-4574	544	22	∈	∈	PROPN
ejpam-4574	544	23	{	{	PUNCT
ejpam-4574	544	24	0	0	NUM
ejpam-4574	544	25	,	,	PUNCT
ejpam-4574	544	26	1	1	NUM
ejpam-4574	544	27	,	,	PUNCT
ejpam-4574	544	28	...	...	PUNCT
ejpam-4574	544	29	,	,	PUNCT
ejpam-4574	544	30	k	k	PROPN
ejpam-4574	544	31	−	−	PROPN
ejpam-4574	544	32	1	1	NUM
ejpam-4574	544	33	}	}	PUNCT
ejpam-4574	544	34	c′2(vi	c′2(vi	NOUN
ejpam-4574	544	35	)	)	PUNCT
ejpam-4574	544	36	otherwise	otherwise	ADV
ejpam-4574	544	37	.	.	PUNCT
ejpam-4574	545	1	a.	a.	PROPN
ejpam-4574	545	2	etawi	etawi	PROPN
ejpam-4574	545	3	,	,	PUNCT
ejpam-4574	545	4	m.	m.	PROPN
ejpam-4574	545	5	ghanem	ghanem	PROPN
ejpam-4574	545	6	,	,	PUNCT
ejpam-4574	545	7	h.	h.	PROPN
ejpam-4574	545	8	al	al	PROPN
ejpam-4574	545	9	-	-	PUNCT
ejpam-4574	545	10	ezeh	ezeh	PROPN
ejpam-4574	545	11	/	/	SYM
ejpam-4574	545	12	eur	eur	NOUN
ejpam-4574	545	13	.	.	PUNCT
ejpam-4574	546	1	j.	j.	PROPN
ejpam-4574	546	2	pure	pure	PROPN
ejpam-4574	546	3	appl	appl	PROPN
ejpam-4574	546	4	.	.	PROPN
ejpam-4574	546	5	math	math	PROPN
ejpam-4574	546	6	,	,	PUNCT
ejpam-4574	546	7	15	15	NUM
ejpam-4574	546	8	(	(	PUNCT
ejpam-4574	546	9	4	4	NUM
ejpam-4574	546	10	)	)	PUNCT
ejpam-4574	546	11	(	(	PUNCT
ejpam-4574	546	12	2022	2022	NUM
ejpam-4574	546	13	)	)	PUNCT
ejpam-4574	546	14	,	,	PUNCT
ejpam-4574	546	15	1822	1822	NUM
ejpam-4574	546	16	-	-	SYM
ejpam-4574	546	17	1835	1835	NUM
ejpam-4574	546	18	1833	1833	NUM
ejpam-4574	546	19	then	then	ADV
ejpam-4574	546	20	:	:	PUNCT
ejpam-4574	546	21	(	(	PUNCT
ejpam-4574	546	22	1	1	X
ejpam-4574	546	23	)	)	PUNCT
ejpam-4574	546	24	c′6	c′6	PROPN
ejpam-4574	546	25	is	be	AUX
ejpam-4574	546	26	a	a	DET
ejpam-4574	546	27	(	(	PUNCT
ejpam-4574	546	28	k	k	NOUN
ejpam-4574	546	29	+	+	PROPN
ejpam-4574	546	30	2)−proper	2)−proper	NUM
ejpam-4574	546	31	coloring	coloring	NOUN
ejpam-4574	546	32	of	of	ADP
ejpam-4574	546	33	ck	ck	PROPN
ejpam-4574	546	34	n.	n.	NOUN
ejpam-4574	546	35	(	(	PUNCT
ejpam-4574	546	36	2	2	NUM
ejpam-4574	546	37	)	)	PUNCT
ejpam-4574	546	38	for	for	ADP
ejpam-4574	546	39	i	i	PROPN
ejpam-4574	546	40	∈	∈	PROPN
ejpam-4574	546	41	{	{	PUNCT
ejpam-4574	546	42	0	0	NUM
ejpam-4574	546	43	,	,	PUNCT
ejpam-4574	546	44	...	...	PUNCT
ejpam-4574	546	45	,	,	PUNCT
ejpam-4574	546	46	k	k	PROPN
ejpam-4574	547	1	−	−	PROPN
ejpam-4574	547	2	1	1	NUM
ejpam-4574	547	3	}	}	PUNCT
ejpam-4574	547	4	,	,	PUNCT
ejpam-4574	547	5	ti	ti	PROPN
ejpam-4574	547	6	is	be	AUX
ejpam-4574	547	7	not	not	PART
ejpam-4574	547	8	an	an	DET
ejpam-4574	547	9	induced	induced	ADJ
ejpam-4574	547	10	cycle	cycle	NOUN
ejpam-4574	547	11	of	of	ADP
ejpam-4574	547	12	c2k	c2k	NOUN
ejpam-4574	547	13	n	n	NOUN
ejpam-4574	547	14	.	.	PUNCT
ejpam-4574	548	1	(	(	PUNCT
ejpam-4574	548	2	3	3	X
ejpam-4574	548	3	)	)	PUNCT
ejpam-4574	548	4	c′6	c′6	PROPN
ejpam-4574	548	5	is	be	AUX
ejpam-4574	548	6	a	a	DET
ejpam-4574	548	7	(	(	PUNCT
ejpam-4574	548	8	k	k	PROPN
ejpam-4574	548	9	+	+	CCONJ
ejpam-4574	548	10	2)−	2)−	NUM
ejpam-4574	548	11	acyclic	acyclic	ADJ
ejpam-4574	548	12	coloring	coloring	NOUN
ejpam-4574	548	13	of	of	ADP
ejpam-4574	548	14	ck	ck	PROPN
ejpam-4574	548	15	n.	n.	NOUN
ejpam-4574	548	16	proof	proof	NOUN
ejpam-4574	548	17	.	.	PUNCT
ejpam-4574	549	1	(	(	PUNCT
ejpam-4574	549	2	1	1	X
ejpam-4574	549	3	)	)	PUNCT
ejpam-4574	549	4	note	note	NOUN
ejpam-4574	549	5	that	that	SCONJ
ejpam-4574	549	6	dcn(v(r+h)(k+2)+k+1	dcn(v(r+h)(k+2)+k+1	ADV
ejpam-4574	549	7	,	,	PUNCT
ejpam-4574	549	8	v(r+h+1)(k+2)+k+1	v(r+h+1)(k+2)+k+1	X
ejpam-4574	549	9	)	)	PUNCT
ejpam-4574	549	10	=	=	SYM
ejpam-4574	550	1	k+2	k+2	PROPN
ejpam-4574	550	2	for	for	ADP
ejpam-4574	550	3	0	0	NUM
ejpam-4574	550	4	≤	≤	NUM
ejpam-4574	550	5	h	h	NOUN
ejpam-4574	550	6	≤	≤	PROPN
ejpam-4574	550	7	k−2	k−2	PROPN
ejpam-4574	550	8	,	,	PUNCT
ejpam-4574	550	9	dcn(v(r+k−1)(k+2)+k+1	dcn(v(r+k−1)(k+2)+k+1	NOUN
ejpam-4574	550	10	,	,	PUNCT
ejpam-4574	550	11	vk+1	vk+1	X
ejpam-4574	550	12	)	)	PUNCT
ejpam-4574	550	13	≥	≥	NOUN
ejpam-4574	550	14	k+3	k+3	PROPN
ejpam-4574	550	15	,	,	PUNCT
ejpam-4574	550	16	and	and	CCONJ
ejpam-4574	550	17	dcn(vr(k+2)−1	dcn(vr(k+2)−1	NOUN
ejpam-4574	550	18	,	,	PUNCT
ejpam-4574	550	19	vr(k+2)+k+1	vr(k+2)+k+1	ADJ
ejpam-4574	550	20	)	)	PUNCT
ejpam-4574	550	21	=	=	SYM
ejpam-4574	550	22	k+2	k+2	PROPN
ejpam-4574	550	23	.	.	X
ejpam-4574	550	24	therefore	therefore	ADV
ejpam-4574	550	25	,	,	PUNCT
ejpam-4574	550	26	c′6	c′6	PROPN
ejpam-4574	550	27	is	be	AUX
ejpam-4574	550	28	a	a	DET
ejpam-4574	550	29	proper	proper	ADJ
ejpam-4574	550	30	coloring	coloring	NOUN
ejpam-4574	550	31	of	of	ADP
ejpam-4574	550	32	ck	ck	PROPN
ejpam-4574	550	33	n.	n.	NOUN
ejpam-4574	550	34	(	(	PUNCT
ejpam-4574	550	35	2	2	NUM
ejpam-4574	550	36	)	)	PUNCT
ejpam-4574	550	37	for	for	ADP
ejpam-4574	550	38	0	0	NUM
ejpam-4574	550	39	≤	≤	NOUN
ejpam-4574	551	1	i	i	PRON
ejpam-4574	551	2	<	<	X
ejpam-4574	551	3	k	k	X
ejpam-4574	552	1	−	−	PROPN
ejpam-4574	552	2	1	1	NUM
ejpam-4574	552	3	,	,	PUNCT
ejpam-4574	552	4	v(r+i)(k+2)+(k+1	v(r+i)(k+2)+(k+1	NUM
ejpam-4574	552	5	)	)	PUNCT
ejpam-4574	552	6	/∈	/∈	PUNCT
ejpam-4574	553	1	ti	ti	NOUN
ejpam-4574	553	2	and	and	CCONJ
ejpam-4574	553	3	dcn(v(r+i)(k+2	dcn(v(r+i)(k+2	PROPN
ejpam-4574	553	4	)	)	PUNCT
ejpam-4574	553	5	,	,	PUNCT
ejpam-4574	553	6	v(r+i)(k+2)+2(k+1	v(r+i)(k+2)+2(k+1	NOUN
ejpam-4574	553	7	)	)	PUNCT
ejpam-4574	553	8	)	)	PUNCT
ejpam-4574	554	1	=	=	PUNCT
ejpam-4574	555	1	2(k	2(k	NUM
ejpam-4574	555	2	+	+	CCONJ
ejpam-4574	555	3	1	1	NUM
ejpam-4574	555	4	)	)	PUNCT
ejpam-4574	555	5	.	.	PUNCT
ejpam-4574	556	1	for	for	ADP
ejpam-4574	556	2	i	i	PRON
ejpam-4574	556	3	=	=	SYM
ejpam-4574	556	4	k	k	PROPN
ejpam-4574	557	1	−	−	PROPN
ejpam-4574	557	2	1	1	NUM
ejpam-4574	557	3	we	we	PRON
ejpam-4574	557	4	have	have	VERB
ejpam-4574	557	5	the	the	DET
ejpam-4574	557	6	following	follow	VERB
ejpam-4574	557	7	two	two	NUM
ejpam-4574	557	8	cases	case	NOUN
ejpam-4574	557	9	:	:	PUNCT
ejpam-4574	557	10	case	case	NOUN
ejpam-4574	557	11	1	1	NUM
ejpam-4574	557	12	.	.	PUNCT
ejpam-4574	557	13	q	q	NOUN
ejpam-4574	558	1	−	−	NOUN
ejpam-4574	558	2	r	r	NOUN
ejpam-4574	558	3	>	>	X
ejpam-4574	558	4	k	k	PROPN
ejpam-4574	559	1	+	+	NOUN
ejpam-4574	559	2	1	1	X
ejpam-4574	559	3	.	.	PUNCT
ejpam-4574	559	4	then	then	ADV
ejpam-4574	559	5	v(r+k−1)(k+2)+(k+1	v(r+k−1)(k+2)+(k+1	NUM
ejpam-4574	559	6	)	)	PUNCT
ejpam-4574	559	7	/∈	/∈	PUNCT
ejpam-4574	560	1	tk−1	tk−1	PROPN
ejpam-4574	560	2	and	and	CCONJ
ejpam-4574	560	3	dcn(v(r+k−1)(k+2	dcn(v(r+k−1)(k+2	PROPN
ejpam-4574	560	4	)	)	PUNCT
ejpam-4574	560	5	,	,	PUNCT
ejpam-4574	560	6	v(r+k−1)(k+2)+2(k+1	v(r+k−1)(k+2)+2(k+1	PROPN
ejpam-4574	560	7	)	)	PUNCT
ejpam-4574	560	8	)	)	PUNCT
ejpam-4574	561	1	=	=	PUNCT
ejpam-4574	562	1	2(k	2(k	NUM
ejpam-4574	562	2	+	+	CCONJ
ejpam-4574	562	3	1	1	NUM
ejpam-4574	562	4	)	)	PUNCT
ejpam-4574	562	5	.	.	PUNCT
ejpam-4574	563	1	case	case	NOUN
ejpam-4574	563	2	2	2	NUM
ejpam-4574	563	3	.	.	PUNCT
ejpam-4574	563	4	:	:	PUNCT
ejpam-4574	563	5	q	q	X
ejpam-4574	564	1	−	−	NOUN
ejpam-4574	564	2	r	r	NOUN
ejpam-4574	564	3	=	=	SYM
ejpam-4574	564	4	k	k	PROPN
ejpam-4574	565	1	+	+	NOUN
ejpam-4574	565	2	1	1	X
ejpam-4574	565	3	.	.	PUNCT
ejpam-4574	565	4	then	then	ADV
ejpam-4574	565	5	v(r+k−1)(k+2)+(k+1	v(r+k−1)(k+2)+(k+1	NUM
ejpam-4574	565	6	)	)	PUNCT
ejpam-4574	565	7	/∈	/∈	PUNCT
ejpam-4574	566	1	tk−1	tk−1	PROPN
ejpam-4574	566	2	and	and	CCONJ
ejpam-4574	566	3	dcn(v(r+k−1)(k+2	dcn(v(r+k−1)(k+2	PROPN
ejpam-4574	566	4	)	)	PUNCT
ejpam-4574	566	5	,	,	PUNCT
ejpam-4574	566	6	vk−1	vk−1	NOUN
ejpam-4574	566	7	)	)	PUNCT
ejpam-4574	566	8	=	=	PUNCT
ejpam-4574	567	1	2(k	2(k	NUM
ejpam-4574	567	2	+	+	CCONJ
ejpam-4574	567	3	1	1	NUM
ejpam-4574	567	4	)	)	PUNCT
ejpam-4574	567	5	.	.	PUNCT
ejpam-4574	568	1	hence	hence	ADV
ejpam-4574	568	2	ti	ti	PROPN
ejpam-4574	568	3	is	be	AUX
ejpam-4574	568	4	not	not	PART
ejpam-4574	568	5	an	an	DET
ejpam-4574	568	6	induced	induced	ADJ
ejpam-4574	568	7	cycle	cycle	NOUN
ejpam-4574	568	8	of	of	ADP
ejpam-4574	568	9	c2k	c2k	X
ejpam-4574	568	10	n	n	NOUN
ejpam-4574	568	11	for	for	ADP
ejpam-4574	568	12	i	i	PRON
ejpam-4574	568	13	∈	∈	PROPN
ejpam-4574	568	14	{	{	PUNCT
ejpam-4574	568	15	0	0	NUM
ejpam-4574	568	16	,	,	PUNCT
ejpam-4574	568	17	...	...	PUNCT
ejpam-4574	568	18	,	,	PUNCT
ejpam-4574	568	19	k	k	PROPN
ejpam-4574	568	20	−	−	PROPN
ejpam-4574	568	21	1	1	NUM
ejpam-4574	568	22	}	}	PUNCT
ejpam-4574	568	23	.	.	PUNCT
ejpam-4574	569	1	(	(	PUNCT
ejpam-4574	569	2	3	3	X
ejpam-4574	569	3	)	)	PUNCT
ejpam-4574	569	4	note	note	NOUN
ejpam-4574	569	5	that	that	SCONJ
ejpam-4574	569	6	r	r	NOUN
ejpam-4574	569	7	+	+	NOUN
ejpam-4574	569	8	k	k	NOUN
ejpam-4574	569	9	=	=	SYM
ejpam-4574	569	10	|tk+1|	|tk+1|	NUM
ejpam-4574	569	11	̸=	̸=	PROPN
ejpam-4574	569	12	|tk|	|tk|	PROPN
ejpam-4574	569	13	≥	≥	NUM
ejpam-4574	569	14	r	r	NOUN
ejpam-4574	569	15	+	+	NOUN
ejpam-4574	569	16	k	k	PROPN
ejpam-4574	570	1	+	+	NOUN
ejpam-4574	570	2	1	1	X
ejpam-4574	570	3	.	.	PUNCT
ejpam-4574	570	4	moreover	moreover	ADV
ejpam-4574	570	5	,	,	PUNCT
ejpam-4574	570	6	ti	ti	PROPN
ejpam-4574	570	7	is	be	AUX
ejpam-4574	570	8	not	not	PART
ejpam-4574	570	9	an	an	DET
ejpam-4574	570	10	induced	induced	ADJ
ejpam-4574	570	11	cycle	cycle	NOUN
ejpam-4574	570	12	in	in	ADP
ejpam-4574	570	13	c2k	c2k	PROPN
ejpam-4574	570	14	n	n	NOUN
ejpam-4574	570	15	for	for	ADP
ejpam-4574	570	16	i	i	PRON
ejpam-4574	570	17	≤	≤	NOUN
ejpam-4574	571	1	k	k	PRON
ejpam-4574	572	1	−	−	PROPN
ejpam-4574	572	2	1	1	NUM
ejpam-4574	572	3	,	,	PUNCT
ejpam-4574	572	4	so	so	ADV
ejpam-4574	572	5	c′6	c′6	PROPN
ejpam-4574	572	6	is	be	AUX
ejpam-4574	572	7	a	a	DET
ejpam-4574	572	8	(	(	PUNCT
ejpam-4574	572	9	k	k	PROPN
ejpam-4574	572	10	+	+	PROPN
ejpam-4574	572	11	2)−	2)−	NUM
ejpam-4574	572	12	acyclic	acyclic	ADJ
ejpam-4574	572	13	coloring	coloring	NOUN
ejpam-4574	572	14	for	for	ADP
ejpam-4574	572	15	ck	ck	PROPN
ejpam-4574	572	16	n.	n.	PROPN
ejpam-4574	572	17	as	as	ADP
ejpam-4574	572	18	a	a	DET
ejpam-4574	572	19	consequence	consequence	NOUN
ejpam-4574	572	20	of	of	ADP
ejpam-4574	572	21	lemmas	lemmas	ADJ
ejpam-4574	572	22	4	4	NUM
ejpam-4574	572	23	-	-	SYM
ejpam-4574	572	24	11	11	NUM
ejpam-4574	572	25	we	we	PRON
ejpam-4574	572	26	get	get	VERB
ejpam-4574	572	27	the	the	DET
ejpam-4574	572	28	following	follow	VERB
ejpam-4574	572	29	theorem	theorem	VERB
ejpam-4574	572	30	.	.	PUNCT
ejpam-4574	573	1	theorem	theorem	NOUN
ejpam-4574	573	2	4	4	NUM
ejpam-4574	573	3	.	.	PUNCT
ejpam-4574	574	1	let	let	VERB
ejpam-4574	574	2	ck	ck	PRON
ejpam-4574	574	3	n	n	PART
ejpam-4574	574	4	be	be	AUX
ejpam-4574	574	5	the	the	DET
ejpam-4574	574	6	kth	kth	NOUN
ejpam-4574	574	7	-	-	PUNCT
ejpam-4574	574	8	power	power	NOUN
ejpam-4574	574	9	of	of	ADP
ejpam-4574	574	10	a	a	DET
ejpam-4574	574	11	cycle	cycle	NOUN
ejpam-4574	574	12	of	of	ADP
ejpam-4574	574	13	order	order	NOUN
ejpam-4574	575	1	n.	n.	NOUN
ejpam-4574	575	2	then	then	ADV
ejpam-4574	575	3	(	(	PUNCT
ejpam-4574	575	4	1	1	X
ejpam-4574	575	5	)	)	PUNCT
ejpam-4574	575	6	k	k	NOUN
ejpam-4574	576	1	+	+	CCONJ
ejpam-4574	576	2	2	2	NUM
ejpam-4574	576	3	≤	≤	NUM
ejpam-4574	576	4	χa(c	χa(c	VERB
ejpam-4574	576	5	k	k	PROPN
ejpam-4574	576	6	n	n	CCONJ
ejpam-4574	576	7	)	)	PUNCT
ejpam-4574	576	8	≤	≤	PUNCT
ejpam-4574	577	1	k	k	NOUN
ejpam-4574	578	1	+	+	CCONJ
ejpam-4574	578	2	3	3	NUM
ejpam-4574	578	3	if	if	SCONJ
ejpam-4574	578	4	n	n	PRON
ejpam-4574	578	5	≥	≥	X
ejpam-4574	578	6	(	(	PUNCT
ejpam-4574	578	7	k	k	PROPN
ejpam-4574	578	8	+	+	PROPN
ejpam-4574	578	9	1)2	1)2	NUM
ejpam-4574	578	10	.	.	PUNCT
ejpam-4574	579	1	(	(	PUNCT
ejpam-4574	579	2	2	2	X
ejpam-4574	579	3	)	)	PUNCT
ejpam-4574	579	4	χa(c	χa(c	VERB
ejpam-4574	579	5	k	k	PROPN
ejpam-4574	579	6	n	n	CCONJ
ejpam-4574	579	7	)	)	PUNCT
ejpam-4574	580	1	=	=	SYM
ejpam-4574	580	2	k	k	PROPN
ejpam-4574	581	1	+	+	CCONJ
ejpam-4574	581	2	2	2	NUM
ejpam-4574	581	3	if	if	SCONJ
ejpam-4574	581	4	n	n	PRON
ejpam-4574	581	5	=	=	PUNCT
ejpam-4574	581	6	q(k	q(k	PROPN
ejpam-4574	581	7	+	+	CCONJ
ejpam-4574	581	8	1	1	X
ejpam-4574	581	9	)	)	PUNCT
ejpam-4574	581	10	+	+	CCONJ
ejpam-4574	581	11	r	r	NOUN
ejpam-4574	581	12	and	and	CCONJ
ejpam-4574	581	13	q	q	NOUN
ejpam-4574	582	1	−	−	PROPN
ejpam-4574	582	2	r	r	NOUN
ejpam-4574	582	3	≥	≥	NOUN
ejpam-4574	582	4	k	k	NOUN
ejpam-4574	583	1	+	+	NOUN
ejpam-4574	583	2	1	1	X
ejpam-4574	583	3	.	.	PUNCT
ejpam-4574	583	4	(	(	PUNCT
ejpam-4574	583	5	3	3	X
ejpam-4574	583	6	)	)	PUNCT
ejpam-4574	583	7	χa(c	χa(c	VERB
ejpam-4574	583	8	k	k	PROPN
ejpam-4574	583	9	n	n	CCONJ
ejpam-4574	583	10	)	)	PUNCT
ejpam-4574	584	1	=	=	SYM
ejpam-4574	584	2	k	k	PROPN
ejpam-4574	585	1	+	+	CCONJ
ejpam-4574	585	2	2	2	NUM
ejpam-4574	585	3	if	if	SCONJ
ejpam-4574	585	4	n	n	PRON
ejpam-4574	585	5	≥	≥	X
ejpam-4574	585	6	(	(	PUNCT
ejpam-4574	585	7	k	k	PROPN
ejpam-4574	585	8	+	+	PROPN
ejpam-4574	585	9	1)3	1)3	PROPN
ejpam-4574	585	10	.	.	PUNCT
ejpam-4574	586	1	according	accord	VERB
ejpam-4574	586	2	to	to	ADP
ejpam-4574	586	3	theorem	theorem	NOUN
ejpam-4574	586	4	4	4	NUM
ejpam-4574	586	5	when	when	SCONJ
ejpam-4574	586	6	n	n	X
ejpam-4574	586	7	is	be	AUX
ejpam-4574	586	8	between	between	ADP
ejpam-4574	586	9	(	(	PUNCT
ejpam-4574	586	10	k+1)2	k+1)2	PROPN
ejpam-4574	586	11	and	and	CCONJ
ejpam-4574	586	12	(	(	PUNCT
ejpam-4574	586	13	k+1)3	k+1)3	PROPN
ejpam-4574	586	14	,	,	PUNCT
ejpam-4574	586	15	χa(c	χa(c	X
ejpam-4574	586	16	k	k	PROPN
ejpam-4574	586	17	n	n	CCONJ
ejpam-4574	586	18	)	)	PUNCT
ejpam-4574	586	19	varies	vary	VERB
ejpam-4574	586	20	between	between	ADP
ejpam-4574	586	21	k	k	PROPN
ejpam-4574	586	22	+	+	CCONJ
ejpam-4574	586	23	2	2	NUM
ejpam-4574	586	24	and	and	CCONJ
ejpam-4574	586	25	k	k	PROPN
ejpam-4574	586	26	+	+	NOUN
ejpam-4574	586	27	3	3	NUM
ejpam-4574	586	28	,	,	PUNCT
ejpam-4574	586	29	while	while	SCONJ
ejpam-4574	586	30	for	for	ADP
ejpam-4574	586	31	n	n	PRON
ejpam-4574	586	32	≥	≥	X
ejpam-4574	586	33	(	(	PUNCT
ejpam-4574	586	34	k	k	PROPN
ejpam-4574	586	35	+	+	PROPN
ejpam-4574	586	36	1)3	1)3	PROPN
ejpam-4574	586	37	,	,	PUNCT
ejpam-4574	586	38	χa(c	χa(c	PUNCT
ejpam-4574	586	39	k	k	PROPN
ejpam-4574	586	40	n	n	CCONJ
ejpam-4574	586	41	)	)	PUNCT
ejpam-4574	586	42	=	=	SYM
ejpam-4574	586	43	k	k	PROPN
ejpam-4574	587	1	+	+	NOUN
ejpam-4574	587	2	2	2	X
ejpam-4574	587	3	.	.	X
ejpam-4574	587	4	the	the	DET
ejpam-4574	587	5	following	follow	VERB
ejpam-4574	587	6	example	example	NOUN
ejpam-4574	587	7	shows	show	VERB
ejpam-4574	587	8	that	that	SCONJ
ejpam-4574	588	1	k	k	PROPN
ejpam-4574	589	1	+	+	CCONJ
ejpam-4574	589	2	2	2	NUM
ejpam-4574	589	3	is	be	AUX
ejpam-4574	589	4	a	a	DET
ejpam-4574	589	5	sharp	sharp	ADV
ejpam-4574	589	6	lower	lower	ADV
ejpam-4574	589	7	bound	bind	VERB
ejpam-4574	589	8	for	for	ADP
ejpam-4574	589	9	χa(c	χa(c	VERB
ejpam-4574	589	10	k	k	PROPN
ejpam-4574	589	11	n	n	CCONJ
ejpam-4574	589	12	)	)	PUNCT
ejpam-4574	589	13	when	when	SCONJ
ejpam-4574	589	14	n	n	X
ejpam-4574	589	15	=	=	SYM
ejpam-4574	589	16	(	(	PUNCT
ejpam-4574	589	17	k	k	PROPN
ejpam-4574	589	18	+	+	PROPN
ejpam-4574	589	19	1)2	1)2	NUM
ejpam-4574	589	20	.	.	PUNCT
ejpam-4574	589	21	example	example	NOUN
ejpam-4574	590	1	6	6	NUM
ejpam-4574	590	2	.	.	PUNCT
ejpam-4574	590	3	let	let	VERB
ejpam-4574	590	4	k	k	NOUN
ejpam-4574	590	5	=	=	SYM
ejpam-4574	590	6	2	2	NUM
ejpam-4574	590	7	and	and	CCONJ
ejpam-4574	590	8	n	n	NOUN
ejpam-4574	590	9	=	=	SYM
ejpam-4574	590	10	(	(	PUNCT
ejpam-4574	590	11	k	k	PROPN
ejpam-4574	590	12	+	+	PROPN
ejpam-4574	590	13	1)2	1)2	NUM
ejpam-4574	590	14	,	,	PUNCT
ejpam-4574	590	15	then	then	ADV
ejpam-4574	590	16	c′2(c	c′2(c	PROPN
ejpam-4574	590	17	k	k	PROPN
ejpam-4574	590	18	n	n	CCONJ
ejpam-4574	590	19	)	)	PUNCT
ejpam-4574	590	20	uses	use	VERB
ejpam-4574	590	21	only	only	ADV
ejpam-4574	590	22	4	4	NUM
ejpam-4574	590	23	colors	color	NOUN
ejpam-4574	590	24	to	to	PART
ejpam-4574	590	25	acyclic	acyclic	VERB
ejpam-4574	590	26	color	color	NOUN
ejpam-4574	590	27	ck	ck	PROPN
ejpam-4574	590	28	n	n	CCONJ
ejpam-4574	590	29	,	,	PUNCT
ejpam-4574	590	30	χa(c	χa(c	VERB
ejpam-4574	590	31	k	k	PROPN
ejpam-4574	590	32	n	n	CCONJ
ejpam-4574	590	33	)	)	PUNCT
ejpam-4574	590	34	=	=	SYM
ejpam-4574	591	1	k	k	PROPN
ejpam-4574	592	1	+	+	NOUN
ejpam-4574	592	2	2	2	X
ejpam-4574	592	3	.	.	X
ejpam-4574	592	4	the	the	DET
ejpam-4574	592	5	union	union	NOUN
ejpam-4574	592	6	of	of	ADP
ejpam-4574	592	7	any	any	DET
ejpam-4574	592	8	two	two	NUM
ejpam-4574	592	9	color	color	NOUN
ejpam-4574	592	10	classes	class	NOUN
ejpam-4574	592	11	induces	induce	VERB
ejpam-4574	592	12	a	a	DET
ejpam-4574	592	13	disjoint	disjoint	ADJ
ejpam-4574	592	14	collection	collection	NOUN
ejpam-4574	592	15	of	of	ADP
ejpam-4574	592	16	trees	tree	NOUN
ejpam-4574	592	17	.	.	PUNCT
ejpam-4574	593	1	1	1	NUM
ejpam-4574	593	2	v1	v1	NOUN
ejpam-4574	593	3	2	2	NUM
ejpam-4574	593	4	v2	v2	NOUN
ejpam-4574	593	5	3	3	NUM
ejpam-4574	593	6	v3	v3	PROPN
ejpam-4574	593	7	1	1	NUM
ejpam-4574	593	8	v4	v4	NOUN
ejpam-4574	593	9	2	2	NUM
ejpam-4574	593	10	v5	v5	PROPN
ejpam-4574	593	11	4	4	NUM
ejpam-4574	593	12	v6	v6	NOUN
ejpam-4574	593	13	1	1	NUM
ejpam-4574	593	14	v7	v7	NOUN
ejpam-4574	593	15	3	3	NUM
ejpam-4574	593	16	v8	v8	PROPN
ejpam-4574	593	17	4	4	NUM
ejpam-4574	593	18	v9	v9	NOUN
ejpam-4574	593	19	t1,t2	t1,t2	PUNCT
ejpam-4574	593	20	t1,t3	t1,t3	PROPN
ejpam-4574	593	21	t1,t4	t1,t4	PROPN
ejpam-4574	593	22	t2,t3	t2,t3	PROPN
ejpam-4574	593	23	t2,t4	t2,t4	PROPN
ejpam-4574	593	24	t3,t4	t3,t4	PROPN
ejpam-4574	593	25	v1	v1	PROPN
ejpam-4574	593	26	v2	v2	NOUN
ejpam-4574	593	27	v4	v4	NOUN
ejpam-4574	593	28	v5	v5	NOUN
ejpam-4574	593	29	v7	v7	VERB
ejpam-4574	593	30	v1	v1	NOUN
ejpam-4574	593	31	v3	v3	PROPN
ejpam-4574	593	32	v4	v4	NOUN
ejpam-4574	593	33	v7	v7	VERB
ejpam-4574	593	34	v8	v8	PROPN
ejpam-4574	593	35	v4	v4	PROPN
ejpam-4574	593	36	v6	v6	NOUN
ejpam-4574	593	37	v7	v7	VERB
ejpam-4574	593	38	v9	v9	PROPN
ejpam-4574	593	39	v1	v1	PROPN
ejpam-4574	593	40	v2	v2	PROPN
ejpam-4574	593	41	v3	v3	PROPN
ejpam-4574	593	42	v5	v5	PROPN
ejpam-4574	593	43	v8	v8	PROPN
ejpam-4574	593	44	v9	v9	PROPN
ejpam-4574	593	45	v2	v2	PROPN
ejpam-4574	593	46	v5	v5	PROPN
ejpam-4574	593	47	v6	v6	PROPN
ejpam-4574	593	48	v6	v6	PROPN
ejpam-4574	593	49	v8	v8	PROPN
ejpam-4574	593	50	v9	v9	PROPN
ejpam-4574	593	51	v3	v3	PROPN
ejpam-4574	593	52	references	reference	NOUN
ejpam-4574	593	53	1834	1834	NUM
ejpam-4574	593	54	5	5	NUM
ejpam-4574	593	55	.	.	PUNCT
ejpam-4574	593	56	star	star	NOUN
ejpam-4574	593	57	coloring	coloring	PROPN
ejpam-4574	593	58	of	of	ADP
ejpam-4574	593	59	ck	ck	PROPN
ejpam-4574	593	60	n	n	PROPN
ejpam-4574	593	61	in	in	ADP
ejpam-4574	593	62	this	this	DET
ejpam-4574	593	63	section	section	NOUN
ejpam-4574	593	64	we	we	PRON
ejpam-4574	593	65	bound	bind	VERB
ejpam-4574	593	66	χs(c	χs(c	PROPN
ejpam-4574	593	67	k	k	PROPN
ejpam-4574	593	68	n	n	CCONJ
ejpam-4574	593	69	)	)	PUNCT
ejpam-4574	593	70	between	between	ADP
ejpam-4574	593	71	two	two	NUM
ejpam-4574	593	72	values	value	NOUN
ejpam-4574	593	73	by	by	ADP
ejpam-4574	593	74	combining	combine	VERB
ejpam-4574	593	75	some	some	DET
ejpam-4574	593	76	results	result	NOUN
ejpam-4574	593	77	from	from	ADP
ejpam-4574	593	78	previous	previous	ADJ
ejpam-4574	593	79	sections	section	NOUN
ejpam-4574	593	80	with	with	ADP
ejpam-4574	593	81	the	the	DET
ejpam-4574	593	82	relation	relation	NOUN
ejpam-4574	593	83	between	between	ADP
ejpam-4574	593	84	χs(g	χs(g	PROPN
ejpam-4574	593	85	)	)	PUNCT
ejpam-4574	593	86	and	and	CCONJ
ejpam-4574	593	87	χ(g2	χ(g2	NUM
ejpam-4574	593	88	)	)	PUNCT
ejpam-4574	593	89	.	.	PUNCT
ejpam-4574	594	1	lemma	lemma	PROPN
ejpam-4574	594	2	12	12	NUM
ejpam-4574	594	3	.	.	PUNCT
ejpam-4574	595	1	[	[	X
ejpam-4574	595	2	4	4	X
ejpam-4574	595	3	]	]	PUNCT
ejpam-4574	595	4	let	let	VERB
ejpam-4574	595	5	g	g	PRON
ejpam-4574	595	6	be	be	AUX
ejpam-4574	595	7	a	a	DET
ejpam-4574	595	8	graph	graph	NOUN
ejpam-4574	595	9	of	of	ADP
ejpam-4574	595	10	order	order	NOUN
ejpam-4574	595	11	n	n	NOUN
ejpam-4574	595	12	and	and	CCONJ
ejpam-4574	595	13	g2	g2	PROPN
ejpam-4574	595	14	be	be	VERB
ejpam-4574	595	15	the	the	DET
ejpam-4574	595	16	square	square	ADJ
ejpam-4574	595	17	graph	graph	NOUN
ejpam-4574	595	18	of	of	ADP
ejpam-4574	595	19	g.	g.	PROPN
ejpam-4574	595	20	then	then	ADV
ejpam-4574	595	21	,	,	PUNCT
ejpam-4574	595	22	χs(g	χs(g	X
ejpam-4574	595	23	)	)	PUNCT
ejpam-4574	595	24	≤	≤	NOUN
ejpam-4574	595	25	χ(g2	χ(g2	NOUN
ejpam-4574	595	26	)	)	PUNCT
ejpam-4574	595	27	,	,	PUNCT
ejpam-4574	595	28	where	where	SCONJ
ejpam-4574	595	29	χ(g	χ(g	PROPN
ejpam-4574	595	30	)	)	PUNCT
ejpam-4574	595	31	denotes	denote	VERB
ejpam-4574	595	32	the	the	DET
ejpam-4574	595	33	(	(	PUNCT
ejpam-4574	595	34	proper	proper	ADJ
ejpam-4574	595	35	)	)	PUNCT
ejpam-4574	595	36	chromatic	chromatic	ADJ
ejpam-4574	595	37	number	number	NOUN
ejpam-4574	595	38	of	of	ADP
ejpam-4574	595	39	g.	g.	PROPN
ejpam-4574	595	40	theorem	theorem	VERB
ejpam-4574	595	41	5	5	NUM
ejpam-4574	595	42	.	.	PUNCT
ejpam-4574	596	1	for	for	ADP
ejpam-4574	596	2	n	n	NUM
ejpam-4574	596	3	≥	≥	X
ejpam-4574	596	4	(	(	PUNCT
ejpam-4574	596	5	k	k	PROPN
ejpam-4574	596	6	+	+	PROPN
ejpam-4574	596	7	1)2	1)2	NUM
ejpam-4574	596	8	,	,	PUNCT
ejpam-4574	596	9	2k	2k	NUM
ejpam-4574	596	10	+	+	CCONJ
ejpam-4574	596	11	1	1	NUM
ejpam-4574	596	12	≤	≤	NOUN
ejpam-4574	596	13	χs(c	χs(c	ADP
ejpam-4574	596	14	k	k	PROPN
ejpam-4574	596	15	n	n	CCONJ
ejpam-4574	596	16	)	)	PUNCT
ejpam-4574	596	17	≤	≤	NOUN
ejpam-4574	596	18	2k	2k	NOUN
ejpam-4574	596	19	+	+	CCONJ
ejpam-4574	596	20	2	2	X
ejpam-4574	596	21	.	.	X
ejpam-4574	596	22	proof	proof	NOUN
ejpam-4574	596	23	.	.	PUNCT
ejpam-4574	597	1	let	let	VERB
ejpam-4574	597	2	n	n	NOUN
ejpam-4574	597	3	=	=	PUNCT
ejpam-4574	597	4	q(k+1)+r	q(k+1)+r	NOUN
ejpam-4574	597	5	.	.	PUNCT
ejpam-4574	598	1	using	use	VERB
ejpam-4574	598	2	lemmas	lemmas	PROPN
ejpam-4574	598	3	5	5	NUM
ejpam-4574	598	4	and	and	CCONJ
ejpam-4574	598	5	12	12	NUM
ejpam-4574	598	6	to	to	PART
ejpam-4574	598	7	get	get	VERB
ejpam-4574	598	8	χ(c2k	χ(c2k	DET
ejpam-4574	598	9	n	n	PROPN
ejpam-4574	598	10	)	)	PUNCT
ejpam-4574	599	1	=	=	SYM
ejpam-4574	599	2	2k+1	2k+1	NUM
ejpam-4574	599	3	+	+	NUM
ejpam-4574	599	4	⌈	⌈	NOUN
ejpam-4574	599	5	r	r	NOUN
ejpam-4574	599	6	q	q	X
ejpam-4574	599	7	⌉	⌉	X
ejpam-4574	599	8	=	=	SYM
ejpam-4574	599	9	2k+2	2k+2	PROPN
ejpam-4574	599	10	and	and	CCONJ
ejpam-4574	599	11	χs(c	χs(c	PROPN
ejpam-4574	599	12	k	k	PROPN
ejpam-4574	599	13	n	n	CCONJ
ejpam-4574	599	14	)	)	PUNCT
ejpam-4574	599	15	≤	≤	NOUN
ejpam-4574	599	16	2k	2k	NOUN
ejpam-4574	599	17	+	+	CCONJ
ejpam-4574	599	18	2	2	X
ejpam-4574	599	19	.	.	PUNCT
ejpam-4574	600	1	moreover	moreover	ADV
ejpam-4574	600	2	,	,	PUNCT
ejpam-4574	600	3	p	p	PROPN
ejpam-4574	600	4	k	k	PROPN
ejpam-4574	600	5	n	n	PROPN
ejpam-4574	600	6	is	be	AUX
ejpam-4574	600	7	a	a	DET
ejpam-4574	600	8	subgraph	subgraph	NOUN
ejpam-4574	600	9	of	of	ADP
ejpam-4574	600	10	ck	ck	PROPN
ejpam-4574	600	11	n	n	CCONJ
ejpam-4574	600	12	,	,	PUNCT
ejpam-4574	600	13	so	so	SCONJ
ejpam-4574	600	14	χs(p	χs(p	X
ejpam-4574	600	15	k	k	PROPN
ejpam-4574	600	16	n	n	PROPN
ejpam-4574	600	17	)	)	PUNCT
ejpam-4574	600	18	≤	≤	NOUN
ejpam-4574	600	19	χs(c	χs(c	ADP
ejpam-4574	600	20	k	k	PROPN
ejpam-4574	600	21	n	n	CCONJ
ejpam-4574	600	22	)	)	PUNCT
ejpam-4574	600	23	.	.	PUNCT
ejpam-4574	601	1	according	accord	VERB
ejpam-4574	601	2	to	to	ADP
ejpam-4574	601	3	lemma	lemma	PROPN
ejpam-4574	601	4	1	1	NUM
ejpam-4574	601	5	χs(p	χs(p	ADP
ejpam-4574	601	6	k	k	PROPN
ejpam-4574	601	7	n	n	PROPN
ejpam-4574	601	8	)	)	PUNCT
ejpam-4574	602	1	=	=	SYM
ejpam-4574	602	2	2k	2k	NUM
ejpam-4574	602	3	+	+	CCONJ
ejpam-4574	602	4	1	1	NUM
ejpam-4574	602	5	,	,	PUNCT
ejpam-4574	602	6	so	so	ADV
ejpam-4574	602	7	2k	2k	NUM
ejpam-4574	602	8	+	+	CCONJ
ejpam-4574	602	9	1	1	NUM
ejpam-4574	602	10	≤	≤	NOUN
ejpam-4574	602	11	χs(c	χs(c	ADP
ejpam-4574	602	12	k	k	PROPN
ejpam-4574	602	13	n	n	CCONJ
ejpam-4574	602	14	)	)	PUNCT
ejpam-4574	602	15	.	.	PUNCT
ejpam-4574	603	1	the	the	DET
ejpam-4574	603	2	following	follow	VERB
ejpam-4574	603	3	example	example	NOUN
ejpam-4574	603	4	shows	show	VERB
ejpam-4574	603	5	that	that	SCONJ
ejpam-4574	604	1	k	k	PROPN
ejpam-4574	605	1	+	+	CCONJ
ejpam-4574	605	2	2	2	NUM
ejpam-4574	605	3	is	be	AUX
ejpam-4574	605	4	a	a	DET
ejpam-4574	605	5	sharp	sharp	ADV
ejpam-4574	605	6	lower	lower	ADV
ejpam-4574	605	7	bound	bind	VERB
ejpam-4574	605	8	for	for	ADP
ejpam-4574	605	9	χa(c	χa(c	VERB
ejpam-4574	605	10	k	k	PROPN
ejpam-4574	605	11	n	n	CCONJ
ejpam-4574	605	12	)	)	PUNCT
ejpam-4574	605	13	.	.	PUNCT
ejpam-4574	606	1	example	example	NOUN
ejpam-4574	607	1	7	7	NUM
ejpam-4574	607	2	.	.	PUNCT
ejpam-4574	607	3	let	let	VERB
ejpam-4574	607	4	k	k	NOUN
ejpam-4574	607	5	=	=	SYM
ejpam-4574	607	6	2	2	NUM
ejpam-4574	607	7	and	and	CCONJ
ejpam-4574	607	8	n	n	NOUN
ejpam-4574	607	9	=	=	SYM
ejpam-4574	607	10	(	(	PUNCT
ejpam-4574	607	11	k	k	PROPN
ejpam-4574	608	1	+	+	CCONJ
ejpam-4574	608	2	1)2	1)2	NUM
ejpam-4574	608	3	+	+	CCONJ
ejpam-4574	608	4	1	1	NUM
ejpam-4574	608	5	,	,	PUNCT
ejpam-4574	608	6	then	then	ADV
ejpam-4574	608	7	c(ck	c(ck	NUM
ejpam-4574	608	8	n	n	CCONJ
ejpam-4574	608	9	)	)	PUNCT
ejpam-4574	608	10	uses	use	VERB
ejpam-4574	608	11	only	only	ADV
ejpam-4574	608	12	5	5	NUM
ejpam-4574	608	13	colors	color	NOUN
ejpam-4574	608	14	to	to	PART
ejpam-4574	608	15	star	star	VERB
ejpam-4574	608	16	color	color	PROPN
ejpam-4574	608	17	c2	c2	PROPN
ejpam-4574	608	18	10	10	NUM
ejpam-4574	608	19	,	,	PUNCT
ejpam-4574	608	20	χs(c	χs(c	ADV
ejpam-4574	608	21	k	k	PROPN
ejpam-4574	608	22	n	n	CCONJ
ejpam-4574	608	23	)	)	PUNCT
ejpam-4574	608	24	=	=	SYM
ejpam-4574	608	25	2k	2k	NUM
ejpam-4574	609	1	+	+	CCONJ
ejpam-4574	609	2	1	1	X
ejpam-4574	609	3	.	.	X
ejpam-4574	609	4	the	the	DET
ejpam-4574	609	5	union	union	NOUN
ejpam-4574	609	6	of	of	ADP
ejpam-4574	609	7	any	any	DET
ejpam-4574	609	8	two	two	NUM
ejpam-4574	609	9	color	color	NOUN
ejpam-4574	609	10	classes	class	NOUN
ejpam-4574	609	11	induces	induce	VERB
ejpam-4574	609	12	a	a	DET
ejpam-4574	609	13	disjoint	disjoint	ADJ
ejpam-4574	609	14	collection	collection	NOUN
ejpam-4574	609	15	of	of	ADP
ejpam-4574	609	16	stars	star	NOUN
ejpam-4574	609	17	.	.	PUNCT
ejpam-4574	610	1	1	1	NUM
ejpam-4574	610	2	v1	v1	NOUN
ejpam-4574	610	3	2	2	NUM
ejpam-4574	610	4	v2	v2	PROPN
ejpam-4574	610	5	3	3	NUM
ejpam-4574	610	6	v3	v3	PROPN
ejpam-4574	610	7	4	4	NUM
ejpam-4574	610	8	v4	v4	PROPN
ejpam-4574	610	9	5	5	NUM
ejpam-4574	610	10	v51	v51	NOUN
ejpam-4574	610	11	v6	v6	NOUN
ejpam-4574	610	12	4	4	NUM
ejpam-4574	610	13	v7	v7	NOUN
ejpam-4574	610	14	2	2	NUM
ejpam-4574	610	15	v8	v8	PROPN
ejpam-4574	610	16	3	3	NUM
ejpam-4574	610	17	v9	v9	PROPN
ejpam-4574	610	18	5	5	NUM
ejpam-4574	610	19	v10t1,t2	v10t1,t2	PROPN
ejpam-4574	610	20	t1,t3	t1,t3	PROPN
ejpam-4574	610	21	t1,t4	t1,t4	PROPN
ejpam-4574	610	22	t1,t5	t1,t5	PROPN
ejpam-4574	610	23	t2,t3	t2,t3	PROPN
ejpam-4574	610	24	t2,t4	t2,t4	PROPN
ejpam-4574	610	25	t2,t5	t2,t5	PROPN
ejpam-4574	610	26	t3,t4	t3,t4	PROPN
ejpam-4574	610	27	t3,t5	t3,t5	PROPN
ejpam-4574	611	1	t4,t5	t4,t5	PROPN
ejpam-4574	611	2	v1	v1	PROPN
ejpam-4574	611	3	v2	v2	PROPN
ejpam-4574	611	4	v6	v6	NOUN
ejpam-4574	611	5	v7	v7	VERB
ejpam-4574	611	6	v9	v9	PROPN
ejpam-4574	611	7	v1	v1	PROPN
ejpam-4574	611	8	v3	v3	PROPN
ejpam-4574	611	9	v6	v6	PROPN
ejpam-4574	611	10	v4	v4	PROPN
ejpam-4574	611	11	v6	v6	NOUN
ejpam-4574	611	12	v7	v7	NOUN
ejpam-4574	611	13	v1	v1	NOUN
ejpam-4574	611	14	v10	v10	NOUN
ejpam-4574	611	15	v1	v1	NOUN
ejpam-4574	611	16	v5	v5	PROPN
ejpam-4574	611	17	v6	v6	PROPN
ejpam-4574	611	18	v2	v2	PROPN
ejpam-4574	611	19	v3	v3	PROPN
ejpam-4574	611	20	v8	v8	PROPN
ejpam-4574	611	21	v9	v9	PROPN
ejpam-4574	611	22	v2	v2	PROPN
ejpam-4574	611	23	v4	v4	NOUN
ejpam-4574	611	24	v7	v7	VERB
ejpam-4574	611	25	v8	v8	PROPN
ejpam-4574	611	26	v8	v8	PROPN
ejpam-4574	611	27	v10	v10	PROPN
ejpam-4574	611	28	v2	v2	PROPN
ejpam-4574	611	29	v5	v5	PROPN
ejpam-4574	611	30	v3	v3	PROPN
ejpam-4574	611	31	v4	v4	PROPN
ejpam-4574	611	32	v7	v7	VERB
ejpam-4574	611	33	v9	v9	PROPN
ejpam-4574	611	34	v3	v3	PROPN
ejpam-4574	611	35	v5	v5	PROPN
ejpam-4574	611	36	v9	v9	PROPN
ejpam-4574	611	37	v10	v10	PROPN
ejpam-4574	611	38	v4	v4	NOUN
ejpam-4574	611	39	v5	v5	PROPN
ejpam-4574	611	40	v7	v7	VERB
ejpam-4574	611	41	v10	v10	NOUN
ejpam-4574	611	42	references	reference	NOUN
ejpam-4574	611	43	[	[	X
ejpam-4574	611	44	1	1	NUM
ejpam-4574	611	45	]	]	PUNCT
ejpam-4574	611	46	h.	h.	PROPN
ejpam-4574	611	47	l.	l.	PROPN
ejpam-4574	611	48	bodlaender	bodlaender	PROPN
ejpam-4574	611	49	,	,	PUNCT
ejpam-4574	611	50	m.	m.	PROPN
ejpam-4574	611	51	r.	r.	PROPN
ejpam-4574	611	52	fellows	fellows	PROPN
ejpam-4574	611	53	,	,	PUNCT
ejpam-4574	611	54	m.	m.	NOUN
ejpam-4574	611	55	t.	t.	PROPN
ejpam-4574	611	56	hallett	hallett	PROPN
ejpam-4574	611	57	,	,	PUNCT
ejpam-4574	611	58	h.	h.	PROPN
ejpam-4574	611	59	t.	t.	PROPN
ejpam-4574	611	60	wareham	wareham	PROPN
ejpam-4574	611	61	,	,	PUNCT
ejpam-4574	611	62	and	and	CCONJ
ejpam-4574	611	63	t.	t.	PROPN
ejpam-4574	611	64	j.	j.	PROPN
ejpam-4574	611	65	warnow	warnow	PROPN
ejpam-4574	611	66	.	.	PUNCT
ejpam-4574	612	1	the	the	DET
ejpam-4574	612	2	hardness	hardness	NOUN
ejpam-4574	612	3	of	of	ADP
ejpam-4574	612	4	perfect	perfect	ADJ
ejpam-4574	612	5	phylogeny	phylogeny	NOUN
ejpam-4574	612	6	,	,	PUNCT
ejpam-4574	612	7	feasible	feasible	ADJ
ejpam-4574	612	8	register	register	NOUN
ejpam-4574	612	9	assignment	assignment	NOUN
ejpam-4574	612	10	and	and	CCONJ
ejpam-4574	612	11	other	other	ADJ
ejpam-4574	612	12	problems	problem	NOUN
ejpam-4574	612	13	on	on	ADP
ejpam-4574	612	14	thin	thin	ADJ
ejpam-4574	612	15	colored	colored	ADJ
ejpam-4574	612	16	graphs	graph	NOUN
ejpam-4574	612	17	.	.	PUNCT
ejpam-4574	613	1	theoretical	theoretical	ADJ
ejpam-4574	613	2	computer	computer	NOUN
ejpam-4574	613	3	science	science	NOUN
ejpam-4574	613	4	,	,	PUNCT
ejpam-4574	613	5	244:167–188	244:167–188	NUM
ejpam-4574	613	6	,	,	PUNCT
ejpam-4574	613	7	2000	2000	NUM
ejpam-4574	613	8	.	.	PUNCT
ejpam-4574	614	1	[	[	X
ejpam-4574	614	2	2	2	NUM
ejpam-4574	614	3	]	]	X
ejpam-4574	614	4	m.i	m.i	PROPN
ejpam-4574	614	5	.	.	PROPN
ejpam-4574	614	6	burstein	burstein	PROPN
ejpam-4574	614	7	.	.	PUNCT
ejpam-4574	615	1	every	every	DET
ejpam-4574	615	2	4	4	NUM
ejpam-4574	615	3	-	-	PUNCT
ejpam-4574	615	4	valent	valent	NOUN
ejpam-4574	615	5	graph	graph	NOUN
ejpam-4574	615	6	has	have	VERB
ejpam-4574	615	7	an	an	DET
ejpam-4574	615	8	acyclic	acyclic	ADJ
ejpam-4574	615	9	5	5	NUM
ejpam-4574	615	10	-	-	PUNCT
ejpam-4574	615	11	coloring	coloring	NOUN
ejpam-4574	615	12	.	.	PUNCT
ejpam-4574	616	1	soobshch	soobshch	PROPN
ejpam-4574	616	2	.	.	PUNCT
ejpam-4574	617	1	akad	akad	PROPN
ejpam-4574	617	2	.	.	PUNCT
ejpam-4574	618	1	nauk	nauk	PROPN
ejpam-4574	618	2	gruzin	gruzin	PROPN
ejpam-4574	618	3	ssr	ssr	PROPN
ejpam-4574	618	4	,	,	PUNCT
ejpam-4574	618	5	93:21–24	93:21–24	NUM
ejpam-4574	618	6	,	,	PUNCT
ejpam-4574	618	7	1979	1979	NUM
ejpam-4574	618	8	.	.	PUNCT
ejpam-4574	619	1	[	[	X
ejpam-4574	619	2	3	3	X
ejpam-4574	619	3	]	]	X
ejpam-4574	619	4	b.	b.	PROPN
ejpam-4574	619	5	grunbaum	grunbaum	PROPN
ejpam-4574	619	6	.	.	PUNCT
ejpam-4574	620	1	acyclic	acyclic	ADJ
ejpam-4574	620	2	colorings	coloring	NOUN
ejpam-4574	620	3	of	of	ADP
ejpam-4574	620	4	planar	planar	ADJ
ejpam-4574	620	5	graphs	graph	NOUN
ejpam-4574	620	6	.	.	PUNCT
ejpam-4574	621	1	isr.j	isr.j	X
ejpam-4574	621	2	.	.	PUNCT
ejpam-4574	621	3	math	math	PROPN
ejpam-4574	621	4	,	,	PUNCT
ejpam-4574	621	5	19:390–412	19:390–412	NUM
ejpam-4574	621	6	,	,	PUNCT
ejpam-4574	621	7	1973	1973	NUM
ejpam-4574	621	8	.	.	PUNCT
ejpam-4574	622	1	[	[	X
ejpam-4574	622	2	4	4	NUM
ejpam-4574	622	3	]	]	X
ejpam-4574	622	4	f.	f.	PROPN
ejpam-4574	622	5	guillaume	guillaume	PROPN
ejpam-4574	622	6	,	,	PUNCT
ejpam-4574	622	7	a.	a.	NOUN
ejpam-4574	622	8	raspaud	raspaud	PROPN
ejpam-4574	622	9	,	,	PUNCT
ejpam-4574	622	10	and	and	CCONJ
ejpam-4574	622	11	b.	b.	PROPN
ejpam-4574	622	12	reed	reed	PROPN
ejpam-4574	622	13	.	.	PUNCT
ejpam-4574	623	1	star	star	PROPN
ejpam-4574	623	2	coloring	coloring	NOUN
ejpam-4574	623	3	of	of	ADP
ejpam-4574	623	4	graphs	graph	NOUN
ejpam-4574	623	5	.	.	PUNCT
ejpam-4574	624	1	journal	journal	NOUN
ejpam-4574	624	2	of	of	ADP
ejpam-4574	624	3	graph	graph	NOUN
ejpam-4574	624	4	theory	theory	NOUN
ejpam-4574	624	5	wiley	wiley	NOUN
ejpam-4574	624	6	,	,	PUNCT
ejpam-4574	624	7	47(3):163–182	47(3):163–182	PROPN
ejpam-4574	624	8	,	,	PUNCT
ejpam-4574	624	9	2004	2004	NUM
ejpam-4574	624	10	.	.	PUNCT
ejpam-4574	625	1	[	[	X
ejpam-4574	625	2	5	5	NUM
ejpam-4574	625	3	]	]	PUNCT
ejpam-4574	625	4	m.	m.	PROPN
ejpam-4574	625	5	f.	f.	PROPN
ejpam-4574	625	6	jimenez	jimenez	PROPN
ejpam-4574	625	7	and	and	CCONJ
ejpam-4574	625	8	m.v	m.v	PROPN
ejpam-4574	625	9	.	.	PROPN
ejpam-4574	625	10	pabon	pabon	PROPN
ejpam-4574	625	11	.	.	PUNCT
ejpam-4574	626	1	a	a	DET
ejpam-4574	626	2	note	note	NOUN
ejpam-4574	626	3	on	on	ADP
ejpam-4574	626	4	coloring	color	VERB
ejpam-4574	626	5	powers	power	NOUN
ejpam-4574	626	6	of	of	ADP
ejpam-4574	626	7	cycles	cycle	NOUN
ejpam-4574	626	8	.	.	PUNCT
ejpam-4574	627	1	research	research	NOUN
ejpam-4574	627	2	report	report	NOUN
ejpam-4574	627	3	18a	18a	NOUN
ejpam-4574	627	4	70	70	NUM
ejpam-4574	627	5	,	,	PUNCT
ejpam-4574	627	6	facultad	facultad	PROPN
ejpam-4574	627	7	de	de	PROPN
ejpam-4574	627	8	ciencias	ciencias	PROPN
ejpam-4574	627	9	de	de	PROPN
ejpam-4574	627	10	la	la	X
ejpam-4574	627	11	universidad	universidad	PROPN
ejpam-4574	627	12	de	de	PROPN
ejpam-4574	627	13	los	los	PROPN
ejpam-4574	627	14	andes	andes	PROPN
ejpam-4574	627	15	,	,	PUNCT
ejpam-4574	627	16	bogota	bogota	PROPN
ejpam-4574	627	17	,	,	PUNCT
ejpam-4574	627	18	colombia	colombia	PROPN
ejpam-4574	627	19	,	,	PUNCT
ejpam-4574	627	20	2005	2005	NUM
ejpam-4574	627	21	.	.	PUNCT
ejpam-4574	628	1	references	reference	NOUN
ejpam-4574	628	2	1835	1835	NUM
ejpam-4574	628	3	[	[	X
ejpam-4574	628	4	6	6	NUM
ejpam-4574	628	5	]	]	PUNCT
ejpam-4574	628	6	j.	j.	PROPN
ejpam-4574	628	7	b.	b.	PROPN
ejpam-4574	628	8	orlin	orlin	PROPN
ejpam-4574	628	9	,	,	PUNCT
ejpam-4574	628	10	m.	m.	NOUN
ejpam-4574	628	11	a.	a.	PROPN
ejpam-4574	628	12	bonuccelli	bonuccelli	PROPN
ejpam-4574	628	13	,	,	PUNCT
ejpam-4574	628	14	and	and	CCONJ
ejpam-4574	628	15	d.	d.	PROPN
ejpam-4574	628	16	p.	p.	PROPN
ejpam-4574	628	17	bovet	bovet	PROPN
ejpam-4574	628	18	.	.	PUNCT
ejpam-4574	629	1	an	an	DET
ejpam-4574	629	2	o(n2	o(n2	ADJ
ejpam-4574	629	3	)	)	PUNCT
ejpam-4574	629	4	algorithm	algorithm	NOUN
ejpam-4574	629	5	for	for	ADP
ejpam-4574	629	6	coloring	color	VERB
ejpam-4574	629	7	proper	proper	ADJ
ejpam-4574	629	8	circular	circular	ADJ
ejpam-4574	629	9	arc	arc	NOUN
ejpam-4574	629	10	graphs	graph	NOUN
ejpam-4574	629	11	.	.	PUNCT
ejpam-4574	630	1	siam	siam	PROPN
ejpam-4574	630	2	j.	j.	PROPN
ejpam-4574	630	3	alg	alg	PROPN
ejpam-4574	630	4	.	.	PUNCT
ejpam-4574	631	1	disc	disc	PROPN
ejpam-4574	631	2	.	.	PUNCT
ejpam-4574	632	1	meth	meth	NOUN
ejpam-4574	632	2	.	.	PUNCT
ejpam-4574	632	3	,	,	PUNCT
ejpam-4574	632	4	2(2):88–93	2(2):88–93	NUM
ejpam-4574	632	5	,	,	PUNCT
ejpam-4574	632	6	1981	1981	NUM
ejpam-4574	632	7	.	.	PUNCT
ejpam-4574	633	1	[	[	X
ejpam-4574	633	2	7	7	NUM
ejpam-4574	633	3	]	]	PUNCT
ejpam-4574	633	4	a.	a.	NOUN
ejpam-4574	633	5	prowse	prowse	PROPN
ejpam-4574	633	6	and	and	CCONJ
ejpam-4574	633	7	d.	d.	PROPN
ejpam-4574	633	8	r.	r.	PROPN
ejpam-4574	633	9	woodall	woodall	PROPN
ejpam-4574	633	10	.	.	PUNCT
ejpam-4574	634	1	choosability	choosability	NOUN
ejpam-4574	634	2	of	of	ADP
ejpam-4574	634	3	powers	power	NOUN
ejpam-4574	634	4	of	of	ADP
ejpam-4574	634	5	circuits	circuit	NOUN
ejpam-4574	634	6	.	.	PUNCT
ejpam-4574	635	1	graphs	graph	NOUN
ejpam-4574	635	2	and	and	CCONJ
ejpam-4574	635	3	combinatorics	combinatoric	NOUN
ejpam-4574	635	4	,	,	PUNCT
ejpam-4574	635	5	19:137–144	19:137–144	NUM
ejpam-4574	635	6	,	,	PUNCT
ejpam-4574	635	7	2003	2003	NUM
ejpam-4574	635	8	.	.	PUNCT
ejpam-4574	636	1	[	[	X
ejpam-4574	636	2	8	8	NUM
ejpam-4574	636	3	]	]	X
ejpam-4574	636	4	n.	n.	NOUN
ejpam-4574	636	5	vedavathi	vedavathi	PROPN
ejpam-4574	636	6	and	and	CCONJ
ejpam-4574	636	7	d.	d.	PROPN
ejpam-4574	636	8	gurram	gurram	PROPN
ejpam-4574	636	9	.	.	PUNCT
ejpam-4574	637	1	applications	application	NOUN
ejpam-4574	637	2	on	on	ADP
ejpam-4574	637	3	graph	graph	NOUN
ejpam-4574	637	4	theory	theory	NOUN
ejpam-4574	637	5	.	.	PUNCT
ejpam-4574	638	1	international	international	ADJ
ejpam-4574	638	2	journal	journal	PROPN
ejpam-4574	638	3	of	of	ADP
ejpam-4574	638	4	engineering	engineering	NOUN
ejpam-4574	638	5	research	research	NOUN
ejpam-4574	638	6	and	and	CCONJ
ejpam-4574	638	7	technology	technology	NOUN
ejpam-4574	638	8	,	,	PUNCT
ejpam-4574	638	9	2(1):1–4	2(1):1–4	NOUN
ejpam-4574	638	10	,	,	PUNCT
ejpam-4574	638	11	2013	2013	NUM
ejpam-4574	638	12	.	.	PUNCT
ejpam-4574	639	1	[	[	X
ejpam-4574	639	2	9	9	NUM
ejpam-4574	639	3	]	]	PUNCT
ejpam-4574	639	4	j.	j.	PROPN
ejpam-4574	639	5	wang	wang	PROPN
ejpam-4574	639	6	,	,	PUNCT
ejpam-4574	639	7	l.	l.	PROPN
ejpam-4574	639	8	y.	y.	PROPN
ejpam-4574	639	9	miao	miao	PROPN
ejpam-4574	639	10	,	,	PUNCT
ejpam-4574	639	11	j.	j.	PROPN
ejpam-4574	639	12	b.	b.	PROPN
ejpam-4574	639	13	li	li	PROPN
ejpam-4574	639	14	,	,	PUNCT
ejpam-4574	639	15	and	and	CCONJ
ejpam-4574	639	16	y.	y.	PROPN
ejpam-4574	639	17	l.	l.	PROPN
ejpam-4574	639	18	liu	liu	PROPN
ejpam-4574	639	19	.	.	PUNCT
ejpam-4574	640	1	acyclic	acyclic	ADJ
ejpam-4574	640	2	choosability	choosability	NOUN
ejpam-4574	640	3	of	of	ADP
ejpam-4574	640	4	graphs	graph	NOUN
ejpam-4574	640	5	with	with	ADP
ejpam-4574	640	6	bounded	bounded	ADJ
ejpam-4574	640	7	degree	degree	NOUN
ejpam-4574	640	8	.	.	PUNCT
ejpam-4574	641	1	acta	acta	PROPN
ejpam-4574	641	2	mathematica	mathematica	PROPN
ejpam-4574	641	3	sinica	sinica	PROPN
ejpam-4574	641	4	english	english	PROPN
ejpam-4574	641	5	series	series	PROPN
ejpam-4574	641	6	,	,	PUNCT
ejpam-4574	641	7	38:560–570	38:560–570	NUM
ejpam-4574	641	8	,	,	PUNCT
ejpam-4574	641	9	2022	2022	NUM
ejpam-4574	641	10	.	.	PUNCT
ejpam-4574	642	1	[	[	X
ejpam-4574	642	2	10	10	NUM
ejpam-4574	642	3	]	]	X
ejpam-4574	642	4	d.	d.	PROPN
ejpam-4574	642	5	r.	r.	PROPN
ejpam-4574	642	6	wood	wood	PROPN
ejpam-4574	642	7	.	.	PUNCT
ejpam-4574	643	1	acyclic	acyclic	ADJ
ejpam-4574	643	2	,	,	PUNCT
ejpam-4574	643	3	star	star	NOUN
ejpam-4574	643	4	and	and	CCONJ
ejpam-4574	643	5	oriented	orient	VERB
ejpam-4574	643	6	colourings	colouring	NOUN
ejpam-4574	643	7	of	of	ADP
ejpam-4574	643	8	graph	graph	NOUN
ejpam-4574	643	9	subdivisions	subdivision	NOUN
ejpam-4574	643	10	.	.	PUNCT
ejpam-4574	644	1	discrete	discrete	ADJ
ejpam-4574	644	2	mathematics	mathematic	NOUN
ejpam-4574	644	3	and	and	CCONJ
ejpam-4574	644	4	theoretical	theoretical	ADJ
ejpam-4574	644	5	computer	computer	NOUN
ejpam-4574	644	6	science	science	NOUN
ejpam-4574	644	7	,	,	PUNCT
ejpam-4574	644	8	7:37–50	7:37–50	NOUN
ejpam-4574	644	9	,	,	PUNCT
ejpam-4574	644	10	2005	2005	NUM
ejpam-4574	644	11	.	.	PUNCT