id	sid	tid	token	lemma	pos
ejpam-5973	1	1	european	european	PROPN
ejpam-5973	1	2	journal	journal	PROPN
ejpam-5973	1	3	of	of	ADP
ejpam-5973	1	4	pure	pure	ADJ
ejpam-5973	1	5	and	and	CCONJ
ejpam-5973	1	6	applied	applied	ADJ
ejpam-5973	1	7	mathematics	mathematic	NOUN
ejpam-5973	1	8	2025	2025	NUM
ejpam-5973	1	9	,	,	PUNCT
ejpam-5973	1	10	vol	vol	NOUN
ejpam-5973	1	11	.	.	PROPN
ejpam-5973	1	12	18	18	NUM
ejpam-5973	1	13	,	,	PUNCT
ejpam-5973	1	14	issue	issue	NOUN
ejpam-5973	1	15	2	2	NUM
ejpam-5973	1	16	,	,	PUNCT
ejpam-5973	1	17	article	article	NOUN
ejpam-5973	1	18	number	number	NOUN
ejpam-5973	1	19	5973	5973	NUM
ejpam-5973	1	20	issn	issn	PROPN
ejpam-5973	1	21	1307	1307	NUM
ejpam-5973	1	22	-	-	SYM
ejpam-5973	1	23	5543	5543	NUM
ejpam-5973	1	24	–	–	PUNCT
ejpam-5973	1	25	ejpam.com	ejpam.com	X
ejpam-5973	1	26	published	publish	VERB
ejpam-5973	1	27	by	by	ADP
ejpam-5973	1	28	new	new	PROPN
ejpam-5973	1	29	york	york	PROPN
ejpam-5973	1	30	business	business	PROPN
ejpam-5973	1	31	global	global	PROPN
ejpam-5973	1	32	hop	hop	PROPN
ejpam-5973	1	33	k	k	ADJ
ejpam-5973	1	34	-	-	PUNCT
ejpam-5973	1	35	rainbow	rainbow	NOUN
ejpam-5973	1	36	domination	domination	NOUN
ejpam-5973	1	37	in	in	ADP
ejpam-5973	1	38	graphs	graph	NOUN
ejpam-5973	1	39	jamil	jamil	PROPN
ejpam-5973	1	40	j.	j.	PROPN
ejpam-5973	1	41	hamja1,2	hamja1,2	PROPN
ejpam-5973	1	42	,	,	PUNCT
ejpam-5973	1	43	seyed	seyed	PROPN
ejpam-5973	1	44	mahmoud	mahmoud	PROPN
ejpam-5973	1	45	sheikholeslami3,∗	sheikholeslami3,∗	PROPN
ejpam-5973	1	46	,	,	PUNCT
ejpam-5973	1	47	imelda	imelda	PROPN
ejpam-5973	1	48	s.	s.	PROPN
ejpam-5973	1	49	aniversario2,4	aniversario2,4	PROPN
ejpam-5973	1	50	,	,	PUNCT
ejpam-5973	2	1	lyster	lyster	PROPN
ejpam-5973	2	2	rey	rey	PROPN
ejpam-5973	2	3	b.	b.	PROPN
ejpam-5973	3	1	cabardo2,4	cabardo2,4	PROPN
ejpam-5973	3	2	1	1	NUM
ejpam-5973	3	3	mathematics	mathematic	NOUN
ejpam-5973	3	4	and	and	CCONJ
ejpam-5973	3	5	sciences	sciences	PROPN
ejpam-5973	3	6	department	department	PROPN
ejpam-5973	3	7	,	,	PUNCT
ejpam-5973	3	8	college	college	NOUN
ejpam-5973	3	9	of	of	ADP
ejpam-5973	3	10	arts	art	NOUN
ejpam-5973	3	11	and	and	CCONJ
ejpam-5973	3	12	sciences	science	NOUN
ejpam-5973	3	13	,	,	PUNCT
ejpam-5973	3	14	msu	msu	PROPN
ejpam-5973	3	15	-	-	PUNCT
ejpam-5973	3	16	tawi	tawi	NOUN
ejpam-5973	3	17	-	-	PUNCT
ejpam-5973	3	18	tawi	tawi	NOUN
ejpam-5973	3	19	college	college	PROPN
ejpam-5973	3	20	of	of	ADP
ejpam-5973	3	21	technology	technology	NOUN
ejpam-5973	3	22	and	and	CCONJ
ejpam-5973	3	23	oceanography	oceanography	NOUN
ejpam-5973	3	24	,	,	PUNCT
ejpam-5973	3	25	7500	7500	NUM
ejpam-5973	3	26	tawi	tawi	NOUN
ejpam-5973	3	27	-	-	PUNCT
ejpam-5973	3	28	tawi	tawi	NOUN
ejpam-5973	3	29	,	,	PUNCT
ejpam-5973	3	30	philippines	philippines	PROPN
ejpam-5973	3	31	2	2	NUM
ejpam-5973	3	32	department	department	NOUN
ejpam-5973	3	33	of	of	ADP
ejpam-5973	3	34	mathematics	mathematic	NOUN
ejpam-5973	3	35	and	and	CCONJ
ejpam-5973	3	36	statistics	statistic	NOUN
ejpam-5973	3	37	,	,	PUNCT
ejpam-5973	3	38	college	college	NOUN
ejpam-5973	3	39	of	of	ADP
ejpam-5973	3	40	science	science	NOUN
ejpam-5973	3	41	and	and	CCONJ
ejpam-5973	3	42	mathematics	mathematic	NOUN
ejpam-5973	3	43	,	,	PUNCT
ejpam-5973	3	44	mindanao	mindanao	PROPN
ejpam-5973	3	45	state	state	PROPN
ejpam-5973	3	46	university	university	PROPN
ejpam-5973	3	47	-	-	PUNCT
ejpam-5973	3	48	iligan	iligan	PROPN
ejpam-5973	3	49	institute	institute	PROPN
ejpam-5973	3	50	of	of	ADP
ejpam-5973	3	51	technology	technology	PROPN
ejpam-5973	3	52	,	,	PUNCT
ejpam-5973	3	53	9200	9200	NUM
ejpam-5973	3	54	iligan	iligan	ADJ
ejpam-5973	3	55	city	city	NOUN
ejpam-5973	3	56	,	,	PUNCT
ejpam-5973	3	57	philippines	philippines	PROPN
ejpam-5973	3	58	3	3	NUM
ejpam-5973	3	59	department	department	NOUN
ejpam-5973	3	60	of	of	ADP
ejpam-5973	3	61	mathematics	mathematics	PROPN
ejpam-5973	3	62	,	,	PUNCT
ejpam-5973	3	63	azarbaijan	azarbaijan	NOUN
ejpam-5973	3	64	shahid	shahid	PROPN
ejpam-5973	3	65	madani	madani	PROPN
ejpam-5973	3	66	university	university	PROPN
ejpam-5973	3	67	,	,	PUNCT
ejpam-5973	3	68	tabriz	tabriz	NOUN
ejpam-5973	3	69	,	,	PUNCT
ejpam-5973	3	70	iran	iran	PROPN
ejpam-5973	3	71	4	4	NUM
ejpam-5973	3	72	center	center	NOUN
ejpam-5973	3	73	for	for	ADP
ejpam-5973	3	74	mathematical	mathematical	ADJ
ejpam-5973	3	75	and	and	CCONJ
ejpam-5973	3	76	theoretical	theoretical	ADJ
ejpam-5973	3	77	physical	physical	ADJ
ejpam-5973	3	78	sciences	science	NOUN
ejpam-5973	3	79	,	,	PUNCT
ejpam-5973	3	80	premier	premier	PROPN
ejpam-5973	3	81	research	research	PROPN
ejpam-5973	3	82	institute	institute	PROPN
ejpam-5973	3	83	of	of	ADP
ejpam-5973	3	84	science	science	NOUN
ejpam-5973	3	85	and	and	CCONJ
ejpam-5973	3	86	mathematics	mathematics	PROPN
ejpam-5973	3	87	(	(	PUNCT
ejpam-5973	3	88	prism	prism	NOUN
ejpam-5973	3	89	)	)	PUNCT
ejpam-5973	3	90	,	,	PUNCT
ejpam-5973	3	91	mindanao	mindanao	PROPN
ejpam-5973	3	92	state	state	PROPN
ejpam-5973	3	93	university	university	PROPN
ejpam-5973	3	94	-	-	PUNCT
ejpam-5973	3	95	iligan	iligan	PROPN
ejpam-5973	3	96	institute	institute	PROPN
ejpam-5973	3	97	of	of	ADP
ejpam-5973	3	98	technology	technology	PROPN
ejpam-5973	3	99	,	,	PUNCT
ejpam-5973	3	100	9200	9200	NUM
ejpam-5973	3	101	iligan	iligan	ADJ
ejpam-5973	3	102	city	city	NOUN
ejpam-5973	3	103	,	,	PUNCT
ejpam-5973	3	104	philippines	philippine	NOUN
ejpam-5973	3	105	abstract	abstract	ADJ
ejpam-5973	3	106	.	.	PUNCT
ejpam-5973	4	1	let	let	VERB
ejpam-5973	4	2	g	g	PROPN
ejpam-5973	4	3	=	=	SYM
ejpam-5973	4	4	(	(	PUNCT
ejpam-5973	4	5	v	v	NOUN
ejpam-5973	4	6	(	(	PUNCT
ejpam-5973	4	7	g	g	NOUN
ejpam-5973	4	8	)	)	PUNCT
ejpam-5973	4	9	,	,	PUNCT
ejpam-5973	4	10	e(g	e(g	PROPN
ejpam-5973	4	11	)	)	PUNCT
ejpam-5973	4	12	)	)	PUNCT
ejpam-5973	5	1	be	be	AUX
ejpam-5973	5	2	a	a	DET
ejpam-5973	5	3	graph	graph	NOUN
ejpam-5973	5	4	.	.	PUNCT
ejpam-5973	6	1	a	a	DET
ejpam-5973	6	2	function	function	NOUN
ejpam-5973	6	3	f	f	PROPN
ejpam-5973	6	4	that	that	PRON
ejpam-5973	6	5	assigns	assign	VERB
ejpam-5973	6	6	to	to	ADP
ejpam-5973	6	7	each	each	DET
ejpam-5973	6	8	vertex	vertex	NOUN
ejpam-5973	6	9	of	of	ADP
ejpam-5973	6	10	g	g	PROPN
ejpam-5973	6	11	a	a	DET
ejpam-5973	6	12	subset	subset	NOUN
ejpam-5973	6	13	of	of	ADP
ejpam-5973	6	14	colors	color	NOUN
ejpam-5973	6	15	from	from	ADP
ejpam-5973	6	16	the	the	DET
ejpam-5973	6	17	set	set	NOUN
ejpam-5973	6	18	{	{	PUNCT
ejpam-5973	6	19	1	1	NUM
ejpam-5973	6	20	,	,	PUNCT
ejpam-5973	6	21	2	2	NUM
ejpam-5973	6	22	,	,	PUNCT
ejpam-5973	6	23	.	.	PUNCT
ejpam-5973	6	24	.	.	PUNCT
ejpam-5973	6	25	.	.	PUNCT
ejpam-5973	7	1	,	,	PUNCT
ejpam-5973	7	2	k	k	X
ejpam-5973	7	3	}	}	PUNCT
ejpam-5973	7	4	,	,	PUNCT
ejpam-5973	7	5	i.e.	i.e.	X
ejpam-5973	7	6	,	,	PUNCT
ejpam-5973	7	7	f	f	X
ejpam-5973	7	8	:	:	PUNCT
ejpam-5973	7	9	v	v	X
ejpam-5973	7	10	(	(	PUNCT
ejpam-5973	7	11	g	g	NOUN
ejpam-5973	7	12	)	)	PUNCT
ejpam-5973	7	13	→	→	SYM
ejpam-5973	8	1	p	p	X
ejpam-5973	8	2	(	(	PUNCT
ejpam-5973	8	3	{	{	PUNCT
ejpam-5973	8	4	1	1	NUM
ejpam-5973	8	5	,	,	PUNCT
ejpam-5973	8	6	2	2	NUM
ejpam-5973	8	7	,	,	PUNCT
ejpam-5973	8	8	3	3	NUM
ejpam-5973	8	9	,	,	PUNCT
ejpam-5973	8	10	.	.	PUNCT
ejpam-5973	8	11	.	.	PUNCT
ejpam-5973	8	12	.	.	PUNCT
ejpam-5973	9	1	,	,	PUNCT
ejpam-5973	9	2	k	k	X
ejpam-5973	9	3	}	}	PUNCT
ejpam-5973	9	4	)	)	PUNCT
ejpam-5973	9	5	,	,	PUNCT
ejpam-5973	9	6	is	be	AUX
ejpam-5973	9	7	called	call	VERB
ejpam-5973	9	8	a	a	DET
ejpam-5973	9	9	hop	hop	NOUN
ejpam-5973	9	10	k	k	ADJ
ejpam-5973	9	11	-	-	PUNCT
ejpam-5973	9	12	rainbow	rainbow	NOUN
ejpam-5973	9	13	dominating	dominating	NOUN
ejpam-5973	9	14	function	function	NOUN
ejpam-5973	9	15	(	(	PUNCT
ejpam-5973	9	16	hkrdf	hkrdf	NOUN
ejpam-5973	9	17	)	)	PUNCT
ejpam-5973	9	18	of	of	ADP
ejpam-5973	9	19	g	g	PROPN
ejpam-5973	9	20	if	if	SCONJ
ejpam-5973	9	21	for	for	ADP
ejpam-5973	9	22	every	every	DET
ejpam-5973	9	23	vertex	vertex	NOUN
ejpam-5973	9	24	v	v	ADP
ejpam-5973	9	25	∈	∈	NOUN
ejpam-5973	9	26	v	v	NOUN
ejpam-5973	9	27	(	(	PUNCT
ejpam-5973	9	28	g	g	NOUN
ejpam-5973	9	29	)	)	PUNCT
ejpam-5973	9	30	with	with	ADP
ejpam-5973	9	31	f(v	f(v	NOUN
ejpam-5973	9	32	)	)	PUNCT
ejpam-5973	9	33	=	=	SYM
ejpam-5973	9	34	∅	∅	NOUN
ejpam-5973	9	35	,	,	PUNCT
ejpam-5973	9	36	we	we	PRON
ejpam-5973	9	37	have	have	VERB
ejpam-5973	9	38	⋃	⋃	ADJ
ejpam-5973	9	39	u∈n2	u∈n2	ADJ
ejpam-5973	9	40	g(v	g(v	NOUN
ejpam-5973	9	41	)	)	PUNCT
ejpam-5973	9	42	f(u	f(u	PROPN
ejpam-5973	9	43	)	)	PUNCT
ejpam-5973	9	44	=	=	PRON
ejpam-5973	9	45	{	{	PUNCT
ejpam-5973	9	46	1	1	NUM
ejpam-5973	9	47	,	,	PUNCT
ejpam-5973	9	48	2	2	NUM
ejpam-5973	9	49	,	,	PUNCT
ejpam-5973	9	50	.	.	PUNCT
ejpam-5973	9	51	.	.	PUNCT
ejpam-5973	9	52	.	.	PUNCT
ejpam-5973	10	1	,	,	PUNCT
ejpam-5973	10	2	k	k	X
ejpam-5973	10	3	}	}	PUNCT
ejpam-5973	10	4	where	where	SCONJ
ejpam-5973	10	5	n2	n2	ADJ
ejpam-5973	10	6	g(v	g(v	PROPN
ejpam-5973	10	7	)	)	PUNCT
ejpam-5973	10	8	is	be	AUX
ejpam-5973	10	9	the	the	DET
ejpam-5973	10	10	set	set	NOUN
ejpam-5973	10	11	of	of	ADP
ejpam-5973	10	12	vertices	vertex	NOUN
ejpam-5973	10	13	of	of	ADP
ejpam-5973	10	14	g	g	NOUN
ejpam-5973	10	15	at	at	ADP
ejpam-5973	10	16	distance	distance	NOUN
ejpam-5973	10	17	two	two	NUM
ejpam-5973	10	18	from	from	ADP
ejpam-5973	10	19	v.	v.	ADP
ejpam-5973	10	20	the	the	DET
ejpam-5973	10	21	weight	weight	NOUN
ejpam-5973	10	22	of	of	ADP
ejpam-5973	10	23	f	f	PROPN
ejpam-5973	10	24	,	,	PUNCT
ejpam-5973	10	25	denoted	denote	VERB
ejpam-5973	10	26	ω(f	ω(f	ADJ
ejpam-5973	10	27	)	)	PUNCT
ejpam-5973	10	28	,	,	PUNCT
ejpam-5973	10	29	is	be	AUX
ejpam-5973	10	30	defined	define	VERB
ejpam-5973	10	31	as	as	ADP
ejpam-5973	10	32	ω(f	ω(f	ADJ
ejpam-5973	10	33	)	)	PUNCT
ejpam-5973	10	34	=	=	PUNCT
ejpam-5973	11	1	∑	∑	PUNCT
ejpam-5973	11	2	x∈v	x∈v	PROPN
ejpam-5973	11	3	(	(	PUNCT
ejpam-5973	11	4	g	g	NOUN
ejpam-5973	11	5	)	)	PUNCT
ejpam-5973	11	6	|f(x)|	|f(x)|	NOUN
ejpam-5973	11	7	.	.	PUNCT
ejpam-5973	12	1	the	the	DET
ejpam-5973	12	2	hop	hop	NOUN
ejpam-5973	12	3	k	k	ADJ
ejpam-5973	12	4	-	-	PUNCT
ejpam-5973	12	5	rainbow	rainbow	NOUN
ejpam-5973	12	6	domination	domination	NOUN
ejpam-5973	12	7	number	number	NOUN
ejpam-5973	12	8	of	of	ADP
ejpam-5973	12	9	g	g	NOUN
ejpam-5973	12	10	,	,	PUNCT
ejpam-5973	12	11	denoted	denote	VERB
ejpam-5973	12	12	γhrk(g	γhrk(g	PROPN
ejpam-5973	12	13	)	)	PUNCT
ejpam-5973	12	14	,	,	PUNCT
ejpam-5973	12	15	is	be	AUX
ejpam-5973	12	16	the	the	DET
ejpam-5973	12	17	minimum	minimum	ADJ
ejpam-5973	12	18	weight	weight	NOUN
ejpam-5973	12	19	of	of	ADP
ejpam-5973	12	20	a	a	DET
ejpam-5973	12	21	hop	hop	NOUN
ejpam-5973	12	22	k	k	ADJ
ejpam-5973	12	23	-	-	PUNCT
ejpam-5973	12	24	rainbow	rainbow	NOUN
ejpam-5973	12	25	dominating	dominating	NOUN
ejpam-5973	12	26	function	function	NOUN
ejpam-5973	12	27	of	of	ADP
ejpam-5973	12	28	g.	g.	PROPN
ejpam-5973	12	29	a	a	DET
ejpam-5973	12	30	hop	hop	NOUN
ejpam-5973	12	31	k	k	ADJ
ejpam-5973	12	32	-	-	PUNCT
ejpam-5973	12	33	rainbow	rainbow	NOUN
ejpam-5973	12	34	dominating	dominating	NOUN
ejpam-5973	12	35	function	function	NOUN
ejpam-5973	12	36	of	of	ADP
ejpam-5973	12	37	g	g	NOUN
ejpam-5973	12	38	with	with	ADP
ejpam-5973	12	39	weight	weight	NOUN
ejpam-5973	12	40	γhrk(g	γhrk(g	PROPN
ejpam-5973	12	41	)	)	PUNCT
ejpam-5973	12	42	is	be	AUX
ejpam-5973	12	43	a	a	DET
ejpam-5973	12	44	γhrk	γhrk	NOUN
ejpam-5973	12	45	-	-	PUNCT
ejpam-5973	12	46	function	function	NOUN
ejpam-5973	12	47	of	of	ADP
ejpam-5973	12	48	g.	g.	PROPN
ejpam-5973	12	49	in	in	ADP
ejpam-5973	12	50	this	this	DET
ejpam-5973	12	51	paper	paper	NOUN
ejpam-5973	12	52	,	,	PUNCT
ejpam-5973	12	53	we	we	PRON
ejpam-5973	12	54	initiate	initiate	VERB
ejpam-5973	12	55	the	the	DET
ejpam-5973	12	56	study	study	NOUN
ejpam-5973	12	57	of	of	ADP
ejpam-5973	12	58	hop	hop	PROPN
ejpam-5973	12	59	k	k	ADJ
ejpam-5973	12	60	-	-	PUNCT
ejpam-5973	12	61	rainbow	rainbow	NOUN
ejpam-5973	12	62	domination	domination	NOUN
ejpam-5973	12	63	in	in	ADP
ejpam-5973	12	64	graphs	graph	NOUN
ejpam-5973	12	65	.	.	PUNCT
ejpam-5973	13	1	we	we	PRON
ejpam-5973	13	2	begin	begin	VERB
ejpam-5973	13	3	by	by	ADP
ejpam-5973	13	4	exploring	explore	VERB
ejpam-5973	13	5	fundamental	fundamental	ADJ
ejpam-5973	13	6	properties	property	NOUN
ejpam-5973	13	7	of	of	ADP
ejpam-5973	13	8	this	this	DET
ejpam-5973	13	9	parameter	parameter	NOUN
ejpam-5973	13	10	and	and	CCONJ
ejpam-5973	13	11	then	then	ADV
ejpam-5973	13	12	establish	establish	VERB
ejpam-5973	13	13	various	various	ADJ
ejpam-5973	13	14	bounds	bound	NOUN
ejpam-5973	13	15	on	on	ADP
ejpam-5973	13	16	γhrk(g	γhrk(g	NOUN
ejpam-5973	13	17	)	)	PUNCT
ejpam-5973	13	18	.	.	PUNCT
ejpam-5973	14	1	furthermore	furthermore	ADV
ejpam-5973	14	2	,	,	PUNCT
ejpam-5973	14	3	we	we	PRON
ejpam-5973	14	4	identify	identify	VERB
ejpam-5973	14	5	the	the	DET
ejpam-5973	14	6	graphs	graph	NOUN
ejpam-5973	14	7	for	for	ADP
ejpam-5973	14	8	which	which	PRON
ejpam-5973	14	9	γhrk(g	γhrk(g	PROPN
ejpam-5973	14	10	)	)	PUNCT
ejpam-5973	14	11	=	=	SYM
ejpam-5973	14	12	n	n	NOUN
ejpam-5973	14	13	and	and	CCONJ
ejpam-5973	14	14	determine	determine	VERB
ejpam-5973	14	15	exact	exact	ADJ
ejpam-5973	14	16	values	value	NOUN
ejpam-5973	14	17	for	for	ADP
ejpam-5973	14	18	certain	certain	ADJ
ejpam-5973	14	19	graph	graph	NOUN
ejpam-5973	14	20	classes	class	NOUN
ejpam-5973	14	21	,	,	PUNCT
ejpam-5973	14	22	including	include	VERB
ejpam-5973	14	23	complete	complete	ADJ
ejpam-5973	14	24	graphs	graph	NOUN
ejpam-5973	14	25	,	,	PUNCT
ejpam-5973	14	26	complete	complete	ADJ
ejpam-5973	14	27	bipartite	bipartite	NOUN
ejpam-5973	14	28	graphs	graph	NOUN
ejpam-5973	14	29	,	,	PUNCT
ejpam-5973	14	30	paths	path	NOUN
ejpam-5973	14	31	,	,	PUNCT
ejpam-5973	14	32	and	and	CCONJ
ejpam-5973	14	33	cycles	cycle	NOUN
ejpam-5973	14	34	.	.	PUNCT
ejpam-5973	15	1	additionally	additionally	ADV
ejpam-5973	15	2	,	,	PUNCT
ejpam-5973	15	3	for	for	ADP
ejpam-5973	15	4	any	any	DET
ejpam-5973	15	5	positive	positive	ADJ
ejpam-5973	15	6	integer	integer	NOUN
ejpam-5973	15	7	a	a	PRON
ejpam-5973	15	8	,	,	PUNCT
ejpam-5973	15	9	we	we	PRON
ejpam-5973	15	10	construct	construct	VERB
ejpam-5973	15	11	connected	connected	ADJ
ejpam-5973	15	12	graphs	graph	NOUN
ejpam-5973	15	13	satisfying	satisfy	VERB
ejpam-5973	15	14	γhr2(g	γhr2(g	NUM
ejpam-5973	15	15	)	)	PUNCT
ejpam-5973	15	16	=	=	SYM
ejpam-5973	16	1	γr2(g	γr2(g	PROPN
ejpam-5973	16	2	)	)	PUNCT
ejpam-5973	16	3	=	=	SYM
ejpam-5973	16	4	a.	a.	NOUN
ejpam-5973	16	5	finally	finally	ADV
ejpam-5973	16	6	,	,	PUNCT
ejpam-5973	16	7	we	we	PRON
ejpam-5973	16	8	provide	provide	VERB
ejpam-5973	16	9	a	a	DET
ejpam-5973	16	10	characterization	characterization	NOUN
ejpam-5973	16	11	of	of	ADP
ejpam-5973	16	12	all	all	DET
ejpam-5973	16	13	graphs	graph	NOUN
ejpam-5973	16	14	where	where	SCONJ
ejpam-5973	16	15	γhr2(g	γhr2(g	NUM
ejpam-5973	16	16	)	)	PUNCT
ejpam-5973	16	17	=	=	SYM
ejpam-5973	16	18	n.	n.	NOUN
ejpam-5973	16	19	2020	2020	NUM
ejpam-5973	16	20	mathematics	mathematic	NOUN
ejpam-5973	16	21	subject	subject	NOUN
ejpam-5973	16	22	classifications	classification	NOUN
ejpam-5973	16	23	:	:	PUNCT
ejpam-5973	16	24	05c69	05c69	X
ejpam-5973	16	25	key	key	ADJ
ejpam-5973	16	26	words	word	NOUN
ejpam-5973	16	27	and	and	CCONJ
ejpam-5973	16	28	phrases	phrase	NOUN
ejpam-5973	16	29	:	:	PUNCT
ejpam-5973	16	30	hop	hop	NOUN
ejpam-5973	16	31	domination	domination	NOUN
ejpam-5973	16	32	number	number	NOUN
ejpam-5973	16	33	,	,	PUNCT
ejpam-5973	16	34	k	k	ADJ
ejpam-5973	16	35	-	-	PUNCT
ejpam-5973	16	36	rainbow	rainbow	NOUN
ejpam-5973	16	37	domination	domination	NOUN
ejpam-5973	16	38	number	number	NOUN
ejpam-5973	16	39	,	,	PUNCT
ejpam-5973	16	40	hop	hop	NOUN
ejpam-5973	16	41	krainbow	krainbow	VERB
ejpam-5973	16	42	domination	domination	NOUN
ejpam-5973	16	43	number	number	NOUN
ejpam-5973	16	44	∗corresponding	∗corresponde	VERB
ejpam-5973	16	45	author	author	NOUN
ejpam-5973	16	46	.	.	PUNCT
ejpam-5973	17	1	doi	doi	NOUN
ejpam-5973	17	2	:	:	PUNCT
ejpam-5973	17	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5973	https://doi.org/10.29020/nybg.ejpam.v18i2.5973	NUM
ejpam-5973	17	4	email	email	NOUN
ejpam-5973	17	5	addresses	address	NOUN
ejpam-5973	17	6	:	:	PUNCT
ejpam-5973	17	7	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-5973	17	8	(	(	PUNCT
ejpam-5973	17	9	j.	j.	PROPN
ejpam-5973	17	10	j.	j.	PROPN
ejpam-5973	17	11	hamja	hamja	PROPN
ejpam-5973	17	12	)	)	PUNCT
ejpam-5973	17	13	,	,	PUNCT
ejpam-5973	17	14	s.m.sheikholeslami@azaruniv.ac.ir	s.m.sheikholeslami@azaruniv.ac.ir	PUNCT
ejpam-5973	17	15	(	(	PUNCT
ejpam-5973	17	16	s.	s.	PROPN
ejpam-5973	17	17	m.	m.	PROPN
ejpam-5973	17	18	sheikholeslami	sheikholeslami	PROPN
ejpam-5973	17	19	)	)	PUNCT
ejpam-5973	17	20	,	,	PUNCT
ejpam-5973	17	21	imelda.aniversario@g.msuiit.edu.ph	imelda.aniversario@g.msuiit.edu.ph	PROPN
ejpam-5973	17	22	(	(	PUNCT
ejpam-5973	17	23	i.	i.	PROPN
ejpam-5973	17	24	s.	s.	PROPN
ejpam-5973	17	25	aniversario	aniversario	PROPN
ejpam-5973	17	26	)	)	PUNCT
ejpam-5973	17	27	,	,	PUNCT
ejpam-5973	17	28	lysterrey.cabardo@g.msuiit.edu.ph	lysterrey.cabardo@g.msuiit.edu.ph	PROPN
ejpam-5973	17	29	(	(	PUNCT
ejpam-5973	17	30	l.	l.	PROPN
ejpam-5973	17	31	b.	b.	PROPN
ejpam-5973	17	32	cabardo	cabardo	PROPN
ejpam-5973	17	33	)	)	PUNCT
ejpam-5973	17	34	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5973	17	35	1	1	NUM
ejpam-5973	17	36	copyright	copyright	NOUN
ejpam-5973	17	37	:	:	PUNCT
ejpam-5973	18	1	©	©	PROPN
ejpam-5973	18	2	2025	2025	NUM
ejpam-5973	18	3	the	the	DET
ejpam-5973	18	4	author(s	author(s	NOUN
ejpam-5973	18	5	)	)	PUNCT
ejpam-5973	18	6	.	.	PUNCT
ejpam-5973	19	1	(	(	PUNCT
ejpam-5973	19	2	cc	cc	NOUN
ejpam-5973	19	3	by	by	ADP
ejpam-5973	19	4	-	-	PUNCT
ejpam-5973	19	5	nc	nc	PROPN
ejpam-5973	19	6	4.0	4.0	NUM
ejpam-5973	19	7	)	)	PUNCT
ejpam-5973	19	8	j.	j.	PROPN
ejpam-5973	19	9	j.	j.	PROPN
ejpam-5973	19	10	hamja	hamja	PROPN
ejpam-5973	20	1	et	et	PROPN
ejpam-5973	20	2	al	al	PROPN
ejpam-5973	20	3	.	.	PUNCT
ejpam-5973	20	4	/	/	SYM
ejpam-5973	20	5	eur	eur	PROPN
ejpam-5973	20	6	.	.	PUNCT
ejpam-5973	21	1	j.	j.	PROPN
ejpam-5973	21	2	pure	pure	PROPN
ejpam-5973	21	3	appl	appl	PROPN
ejpam-5973	21	4	.	.	PROPN
ejpam-5973	21	5	math	math	PROPN
ejpam-5973	21	6	,	,	PUNCT
ejpam-5973	21	7	18	18	NUM
ejpam-5973	21	8	(	(	PUNCT
ejpam-5973	21	9	2	2	NUM
ejpam-5973	21	10	)	)	PUNCT
ejpam-5973	21	11	(	(	PUNCT
ejpam-5973	21	12	2025	2025	NUM
ejpam-5973	21	13	)	)	PUNCT
ejpam-5973	21	14	,	,	PUNCT
ejpam-5973	21	15	5973	5973	NUM
ejpam-5973	21	16	2	2	NUM
ejpam-5973	21	17	of	of	ADP
ejpam-5973	21	18	17	17	NUM
ejpam-5973	21	19	1	1	NUM
ejpam-5973	21	20	.	.	PUNCT
ejpam-5973	22	1	introduction	introduction	NOUN
ejpam-5973	22	2	domination	domination	NOUN
ejpam-5973	22	3	in	in	ADP
ejpam-5973	22	4	graphs	graph	NOUN
ejpam-5973	22	5	is	be	AUX
ejpam-5973	22	6	frequently	frequently	ADV
ejpam-5973	22	7	used	use	VERB
ejpam-5973	22	8	as	as	ADP
ejpam-5973	22	9	a	a	DET
ejpam-5973	22	10	model	model	NOUN
ejpam-5973	22	11	for	for	ADP
ejpam-5973	22	12	real	real	ADJ
ejpam-5973	22	13	-	-	PUNCT
ejpam-5973	22	14	world	world	NOUN
ejpam-5973	22	15	applications	application	NOUN
ejpam-5973	22	16	,	,	PUNCT
ejpam-5973	22	17	where	where	SCONJ
ejpam-5973	22	18	the	the	DET
ejpam-5973	22	19	vertices	vertex	NOUN
ejpam-5973	22	20	in	in	ADP
ejpam-5973	22	21	a	a	DET
ejpam-5973	22	22	dominating	dominating	NOUN
ejpam-5973	22	23	set	set	NOUN
ejpam-5973	22	24	provide	provide	VERB
ejpam-5973	22	25	a	a	DET
ejpam-5973	22	26	service	service	NOUN
ejpam-5973	22	27	or	or	CCONJ
ejpam-5973	22	28	product	product	NOUN
ejpam-5973	22	29	that	that	PRON
ejpam-5973	22	30	must	must	AUX
ejpam-5973	22	31	be	be	AUX
ejpam-5973	22	32	accessible	accessible	ADJ
ejpam-5973	22	33	to	to	ADP
ejpam-5973	22	34	every	every	DET
ejpam-5973	22	35	vertex	vertex	NOUN
ejpam-5973	22	36	in	in	ADP
ejpam-5973	22	37	the	the	DET
ejpam-5973	22	38	network	network	NOUN
ejpam-5973	22	39	.	.	PUNCT
ejpam-5973	23	1	in	in	ADP
ejpam-5973	23	2	2020	2020	NUM
ejpam-5973	23	3	,	,	PUNCT
ejpam-5973	23	4	haynes	hayne	NOUN
ejpam-5973	23	5	et	et	NOUN
ejpam-5973	23	6	al	al	PROPN
ejpam-5973	23	7	.	.	PUNCT
ejpam-5973	24	1	(	(	PUNCT
ejpam-5973	24	2	[	[	X
ejpam-5973	24	3	1	1	NUM
ejpam-5973	24	4	]	]	PUNCT
ejpam-5973	24	5	)	)	PUNCT
ejpam-5973	24	6	published	publish	VERB
ejpam-5973	24	7	a	a	DET
ejpam-5973	24	8	comprehensive	comprehensive	ADJ
ejpam-5973	24	9	survey	survey	NOUN
ejpam-5973	24	10	on	on	ADP
ejpam-5973	24	11	domination	domination	NOUN
ejpam-5973	24	12	in	in	ADP
ejpam-5973	24	13	graphs	graph	NOUN
ejpam-5973	24	14	.	.	PUNCT
ejpam-5973	25	1	rainbow	rainbow	PROPN
ejpam-5973	25	2	domination	domination	PROPN
ejpam-5973	25	3	extends	extend	VERB
ejpam-5973	25	4	this	this	DET
ejpam-5973	25	5	concept	concept	NOUN
ejpam-5973	25	6	by	by	ADP
ejpam-5973	25	7	introducing	introduce	VERB
ejpam-5973	25	8	multiple	multiple	ADJ
ejpam-5973	25	9	service	service	NOUN
ejpam-5973	25	10	types	type	NOUN
ejpam-5973	25	11	,	,	PUNCT
ejpam-5973	25	12	represented	represent	VERB
ejpam-5973	25	13	by	by	ADP
ejpam-5973	25	14	different	different	ADJ
ejpam-5973	25	15	colors	color	NOUN
ejpam-5973	25	16	,	,	PUNCT
ejpam-5973	25	17	and	and	CCONJ
ejpam-5973	25	18	ensuring	ensure	VERB
ejpam-5973	25	19	that	that	SCONJ
ejpam-5973	25	20	every	every	DET
ejpam-5973	25	21	vertex	vertex	NOUN
ejpam-5973	25	22	without	without	ADP
ejpam-5973	25	23	direct	direct	ADJ
ejpam-5973	25	24	service	service	NOUN
ejpam-5973	25	25	has	have	VERB
ejpam-5973	25	26	access	access	NOUN
ejpam-5973	25	27	to	to	ADP
ejpam-5973	25	28	all	all	DET
ejpam-5973	25	29	service	service	NOUN
ejpam-5973	25	30	types	type	NOUN
ejpam-5973	25	31	in	in	ADP
ejpam-5973	25	32	its	its	PRON
ejpam-5973	25	33	neighborhood	neighborhood	NOUN
ejpam-5973	25	34	.	.	PUNCT
ejpam-5973	26	1	rainbow	rainbow	PROPN
ejpam-5973	26	2	domination	domination	NOUN
ejpam-5973	26	3	is	be	AUX
ejpam-5973	26	4	being	be	AUX
ejpam-5973	26	5	studied	study	VERB
ejpam-5973	26	6	because	because	SCONJ
ejpam-5973	26	7	it	it	PRON
ejpam-5973	26	8	has	have	VERB
ejpam-5973	26	9	a	a	DET
ejpam-5973	26	10	lot	lot	NOUN
ejpam-5973	26	11	to	to	PART
ejpam-5973	26	12	do	do	VERB
ejpam-5973	26	13	with	with	ADP
ejpam-5973	26	14	domination	domination	NOUN
ejpam-5973	26	15	in	in	ADP
ejpam-5973	26	16	cartesian	cartesian	ADJ
ejpam-5973	26	17	products	product	NOUN
ejpam-5973	26	18	of	of	ADP
ejpam-5973	26	19	graphs	graph	NOUN
ejpam-5973	26	20	and	and	CCONJ
ejpam-5973	26	21	what	what	PRON
ejpam-5973	26	22	that	that	PRON
ejpam-5973	26	23	means	mean	VERB
ejpam-5973	26	24	for	for	ADP
ejpam-5973	26	25	vizing	vize	VERB
ejpam-5973	26	26	’s	’s	PART
ejpam-5973	26	27	conjecture	conjecture	NOUN
ejpam-5973	26	28	,	,	PUNCT
ejpam-5973	26	29	even	even	ADV
ejpam-5973	26	30	though	though	SCONJ
ejpam-5973	26	31	its	its	PRON
ejpam-5973	26	32	practical	practical	ADJ
ejpam-5973	26	33	uses	use	NOUN
ejpam-5973	26	34	are	be	AUX
ejpam-5973	26	35	still	still	ADV
ejpam-5973	26	36	unknown	unknown	ADJ
ejpam-5973	26	37	.	.	PUNCT
ejpam-5973	27	1	the	the	DET
ejpam-5973	27	2	rainbow	rainbow	NOUN
ejpam-5973	27	3	domination	domination	NOUN
ejpam-5973	27	4	number	number	NOUN
ejpam-5973	27	5	was	be	AUX
ejpam-5973	27	6	first	first	ADV
ejpam-5973	27	7	proposed	propose	VERB
ejpam-5973	27	8	by	by	ADP
ejpam-5973	27	9	brešar	brešar	PROPN
ejpam-5973	27	10	et	et	PROPN
ejpam-5973	27	11	al	al	PROPN
ejpam-5973	27	12	.	.	PROPN
ejpam-5973	28	1	in	in	ADP
ejpam-5973	28	2	2008	2008	NUM
ejpam-5973	28	3	[	[	X
ejpam-5973	28	4	2	2	NUM
ejpam-5973	28	5	]	]	PUNCT
ejpam-5973	28	6	.	.	PUNCT
ejpam-5973	29	1	they	they	PRON
ejpam-5973	29	2	showed	show	VERB
ejpam-5973	29	3	how	how	SCONJ
ejpam-5973	29	4	it	it	PRON
ejpam-5973	29	5	works	work	VERB
ejpam-5973	29	6	in	in	ADP
ejpam-5973	29	7	paired	pair	VERB
ejpam-5973	29	8	-	-	PUNCT
ejpam-5973	29	9	domination	domination	NOUN
ejpam-5973	29	10	within	within	ADP
ejpam-5973	29	11	cartesian	cartesian	ADJ
ejpam-5973	29	12	products	product	NOUN
ejpam-5973	29	13	of	of	ADP
ejpam-5973	29	14	graphs	graph	NOUN
ejpam-5973	29	15	and	and	CCONJ
ejpam-5973	29	16	explained	explain	VERB
ejpam-5973	29	17	how	how	SCONJ
ejpam-5973	29	18	it	it	PRON
ejpam-5973	29	19	is	be	AUX
ejpam-5973	29	20	related	relate	VERB
ejpam-5973	29	21	to	to	ADP
ejpam-5973	29	22	standard	standard	ADJ
ejpam-5973	29	23	domination	domination	NOUN
ejpam-5973	29	24	.	.	PUNCT
ejpam-5973	30	1	later	later	ADV
ejpam-5973	30	2	,	,	PUNCT
ejpam-5973	30	3	in	in	ADP
ejpam-5973	30	4	2014	2014	NUM
ejpam-5973	30	5	,	,	PUNCT
ejpam-5973	30	6	z.	z.	PROPN
ejpam-5973	30	7	shao	shao	PROPN
ejpam-5973	30	8	determined	determine	VERB
ejpam-5973	30	9	bounds	bound	NOUN
ejpam-5973	30	10	for	for	ADP
ejpam-5973	30	11	the	the	DET
ejpam-5973	30	12	k	k	ADJ
ejpam-5973	30	13	-	-	PUNCT
ejpam-5973	30	14	rainbow	rainbow	NOUN
ejpam-5973	30	15	domination	domination	NOUN
ejpam-5973	30	16	number	number	NOUN
ejpam-5973	30	17	of	of	ADP
ejpam-5973	30	18	any	any	DET
ejpam-5973	30	19	arbitrary	arbitrary	ADJ
ejpam-5973	30	20	graph	graph	NOUN
ejpam-5973	30	21	for	for	ADP
ejpam-5973	30	22	any	any	DET
ejpam-5973	30	23	positive	positive	ADJ
ejpam-5973	30	24	integer	integer	NOUN
ejpam-5973	30	25	k	k	PROPN
ejpam-5973	31	1	[	[	X
ejpam-5973	31	2	3	3	NUM
ejpam-5973	31	3	]	]	PUNCT
ejpam-5973	31	4	.	.	PUNCT
ejpam-5973	32	1	since	since	SCONJ
ejpam-5973	32	2	then	then	ADV
ejpam-5973	32	3	,	,	PUNCT
ejpam-5973	32	4	this	this	DET
ejpam-5973	32	5	concept	concept	NOUN
ejpam-5973	32	6	has	have	AUX
ejpam-5973	32	7	been	be	AUX
ejpam-5973	32	8	widely	widely	ADV
ejpam-5973	32	9	explored	explore	VERB
ejpam-5973	32	10	(	(	PUNCT
ejpam-5973	32	11	see	see	VERB
ejpam-5973	32	12	,	,	PUNCT
ejpam-5973	32	13	for	for	ADP
ejpam-5973	32	14	example	example	NOUN
ejpam-5973	32	15	,	,	PUNCT
ejpam-5973	32	16	[	[	X
ejpam-5973	32	17	4–9	4–9	X
ejpam-5973	32	18	]	]	X
ejpam-5973	32	19	)	)	PUNCT
ejpam-5973	32	20	.	.	PUNCT
ejpam-5973	33	1	in	in	ADP
ejpam-5973	33	2	this	this	DET
ejpam-5973	33	3	paper	paper	NOUN
ejpam-5973	33	4	,	,	PUNCT
ejpam-5973	33	5	we	we	PRON
ejpam-5973	33	6	introduce	introduce	VERB
ejpam-5973	33	7	the	the	DET
ejpam-5973	33	8	study	study	NOUN
ejpam-5973	33	9	of	of	ADP
ejpam-5973	33	10	hop	hop	PROPN
ejpam-5973	33	11	k	k	ADJ
ejpam-5973	33	12	-	-	PUNCT
ejpam-5973	33	13	rainbow	rainbow	NOUN
ejpam-5973	33	14	domination	domination	NOUN
ejpam-5973	33	15	in	in	ADP
ejpam-5973	33	16	graphs	graph	NOUN
ejpam-5973	33	17	,	,	PUNCT
ejpam-5973	33	18	integrating	integrate	VERB
ejpam-5973	33	19	the	the	DET
ejpam-5973	33	20	ideas	idea	NOUN
ejpam-5973	33	21	of	of	ADP
ejpam-5973	33	22	hop	hop	NOUN
ejpam-5973	33	23	domination	domination	NOUN
ejpam-5973	33	24	and	and	CCONJ
ejpam-5973	33	25	k	k	ADJ
ejpam-5973	33	26	-	-	PUNCT
ejpam-5973	33	27	rainbow	rainbow	NOUN
ejpam-5973	33	28	domination	domination	NOUN
ejpam-5973	33	29	parameters	parameter	NOUN
ejpam-5973	33	30	.	.	PUNCT
ejpam-5973	34	1	the	the	DET
ejpam-5973	34	2	incorporation	incorporation	NOUN
ejpam-5973	34	3	of	of	ADP
ejpam-5973	34	4	hop	hop	NOUN
ejpam-5973	34	5	distance	distance	NOUN
ejpam-5973	34	6	constraints	constraint	NOUN
ejpam-5973	34	7	influences	influence	VERB
ejpam-5973	34	8	the	the	DET
ejpam-5973	34	9	behavior	behavior	NOUN
ejpam-5973	34	10	of	of	ADP
ejpam-5973	34	11	minimum	minimum	NOUN
ejpam-5973	34	12	rainbow	rainbow	NOUN
ejpam-5973	34	13	dominating	dominating	NOUN
ejpam-5973	34	14	functions	function	NOUN
ejpam-5973	34	15	,	,	PUNCT
ejpam-5973	34	16	leading	lead	VERB
ejpam-5973	34	17	to	to	ADP
ejpam-5973	34	18	new	new	ADJ
ejpam-5973	34	19	theoretical	theoretical	ADJ
ejpam-5973	34	20	bounds	bound	NOUN
ejpam-5973	34	21	and	and	CCONJ
ejpam-5973	34	22	extremal	extremal	ADJ
ejpam-5973	34	23	results	result	NOUN
ejpam-5973	34	24	.	.	PUNCT
ejpam-5973	35	1	the	the	DET
ejpam-5973	35	2	concept	concept	NOUN
ejpam-5973	35	3	of	of	ADP
ejpam-5973	35	4	a	a	DET
ejpam-5973	35	5	hop	hop	NOUN
ejpam-5973	35	6	dominating	dominating	NOUN
ejpam-5973	35	7	set	set	NOUN
ejpam-5973	35	8	was	be	AUX
ejpam-5973	35	9	first	first	ADV
ejpam-5973	35	10	proposed	propose	VERB
ejpam-5973	35	11	by	by	ADP
ejpam-5973	35	12	natarajan	natarajan	PROPN
ejpam-5973	35	13	et	et	PROPN
ejpam-5973	35	14	al	al	PROPN
ejpam-5973	35	15	.	.	PROPN
ejpam-5973	36	1	in	in	ADP
ejpam-5973	36	2	2015	2015	NUM
ejpam-5973	36	3	[	[	X
ejpam-5973	36	4	10	10	NUM
ejpam-5973	36	5	]	]	PUNCT
ejpam-5973	36	6	,	,	PUNCT
ejpam-5973	36	7	and	and	CCONJ
ejpam-5973	36	8	it	it	PRON
ejpam-5973	36	9	has	have	AUX
ejpam-5973	36	10	since	since	ADV
ejpam-5973	36	11	been	be	AUX
ejpam-5973	36	12	expanded	expand	VERB
ejpam-5973	36	13	by	by	ADP
ejpam-5973	36	14	many	many	ADJ
ejpam-5973	36	15	researchers	researcher	NOUN
ejpam-5973	36	16	who	who	PRON
ejpam-5973	36	17	have	have	AUX
ejpam-5973	36	18	applied	apply	VERB
ejpam-5973	36	19	it	it	PRON
ejpam-5973	36	20	to	to	ADP
ejpam-5973	36	21	various	various	ADJ
ejpam-5973	36	22	domination	domination	NOUN
ejpam-5973	36	23	variants	variant	NOUN
ejpam-5973	36	24	.	.	PUNCT
ejpam-5973	37	1	for	for	ADP
ejpam-5973	37	2	more	more	ADJ
ejpam-5973	37	3	details	detail	NOUN
ejpam-5973	37	4	on	on	ADP
ejpam-5973	37	5	hop	hop	PROPN
ejpam-5973	37	6	domination	domination	NOUN
ejpam-5973	37	7	,	,	PUNCT
ejpam-5973	37	8	see	see	VERB
ejpam-5973	37	9	for	for	ADP
ejpam-5973	37	10	instance	instance	NOUN
ejpam-5973	37	11	,	,	PUNCT
ejpam-5973	37	12	[	[	X
ejpam-5973	37	13	11–14	11–14	NUM
ejpam-5973	37	14	]	]	PUNCT
ejpam-5973	37	15	.	.	PUNCT
ejpam-5973	38	1	2	2	X
ejpam-5973	38	2	.	.	X
ejpam-5973	38	3	terminology	terminology	NOUN
ejpam-5973	38	4	and	and	CCONJ
ejpam-5973	38	5	notation	notation	NOUN
ejpam-5973	38	6	let	let	VERB
ejpam-5973	38	7	g	g	PRON
ejpam-5973	38	8	be	be	AUX
ejpam-5973	38	9	a	a	DET
ejpam-5973	38	10	graph	graph	NOUN
ejpam-5973	38	11	with	with	ADP
ejpam-5973	38	12	the	the	DET
ejpam-5973	38	13	vertex	vertex	NOUN
ejpam-5973	38	14	set	set	VERB
ejpam-5973	38	15	v	v	NOUN
ejpam-5973	38	16	(	(	PUNCT
ejpam-5973	38	17	g	g	NOUN
ejpam-5973	38	18	)	)	PUNCT
ejpam-5973	38	19	and	and	CCONJ
ejpam-5973	38	20	the	the	DET
ejpam-5973	38	21	edge	edge	NOUN
ejpam-5973	38	22	set	set	VERB
ejpam-5973	38	23	e(g	e(g	PROPN
ejpam-5973	38	24	)	)	PUNCT
ejpam-5973	38	25	,	,	PUNCT
ejpam-5973	38	26	and	and	CCONJ
ejpam-5973	38	27	of	of	ADP
ejpam-5973	38	28	order	order	NOUN
ejpam-5973	38	29	n	n	NOUN
ejpam-5973	38	30	=	=	SYM
ejpam-5973	38	31	|v	|v	X
ejpam-5973	38	32	(	(	PUNCT
ejpam-5973	38	33	g)|	g)|	NOUN
ejpam-5973	38	34	and	and	CCONJ
ejpam-5973	38	35	size	size	NOUN
ejpam-5973	38	36	m	m	PROPN
ejpam-5973	38	37	=	=	SYM
ejpam-5973	38	38	|e(g)|	|e(g)|	NOUN
ejpam-5973	38	39	.	.	PUNCT
ejpam-5973	39	1	the	the	DET
ejpam-5973	39	2	set	set	NOUN
ejpam-5973	39	3	of	of	ADP
ejpam-5973	39	4	neighbors	neighbor	NOUN
ejpam-5973	39	5	of	of	ADP
ejpam-5973	39	6	a	a	DET
ejpam-5973	39	7	vertex	vertex	NOUN
ejpam-5973	39	8	u	u	NOUN
ejpam-5973	39	9	in	in	ADP
ejpam-5973	39	10	g	g	PROPN
ejpam-5973	39	11	is	be	AUX
ejpam-5973	39	12	called	call	VERB
ejpam-5973	39	13	the	the	DET
ejpam-5973	39	14	open	open	ADJ
ejpam-5973	39	15	neighborhood	neighborhood	NOUN
ejpam-5973	39	16	of	of	ADP
ejpam-5973	39	17	u	u	NOUN
ejpam-5973	39	18	in	in	ADP
ejpam-5973	39	19	g	g	NOUN
ejpam-5973	39	20	,	,	PUNCT
ejpam-5973	39	21	denoted	denote	VERB
ejpam-5973	39	22	by	by	ADP
ejpam-5973	39	23	ng(u	ng(u	NOUN
ejpam-5973	39	24	)	)	PUNCT
ejpam-5973	39	25	=	=	PRON
ejpam-5973	39	26	{	{	PUNCT
ejpam-5973	39	27	v	v	NUM
ejpam-5973	39	28	∈	∈	NOUN
ejpam-5973	39	29	v	v	NOUN
ejpam-5973	39	30	(	(	PUNCT
ejpam-5973	39	31	g	g	NOUN
ejpam-5973	39	32	)	)	PUNCT
ejpam-5973	39	33	:	:	PUNCT
ejpam-5973	39	34	uv	uv	PROPN
ejpam-5973	39	35	∈	∈	PROPN
ejpam-5973	39	36	e(g	e(g	PROPN
ejpam-5973	39	37	)	)	PUNCT
ejpam-5973	39	38	}	}	PUNCT
ejpam-5973	39	39	.	.	PUNCT
ejpam-5973	40	1	the	the	DET
ejpam-5973	40	2	closed	closed	ADJ
ejpam-5973	40	3	neighborhood	neighborhood	NOUN
ejpam-5973	40	4	of	of	ADP
ejpam-5973	40	5	u	u	NOUN
ejpam-5973	40	6	in	in	ADP
ejpam-5973	40	7	g	g	PROPN
ejpam-5973	40	8	is	be	AUX
ejpam-5973	40	9	the	the	DET
ejpam-5973	40	10	set	set	NOUN
ejpam-5973	40	11	ng[u	ng[u	PROPN
ejpam-5973	40	12	]	]	X
ejpam-5973	40	13	=	=	PUNCT
ejpam-5973	40	14	ng(u)∪	ng(u)∪	INTJ
ejpam-5973	40	15	{	{	PUNCT
ejpam-5973	40	16	u	u	NOUN
ejpam-5973	40	17	}	}	PUNCT
ejpam-5973	40	18	and	and	CCONJ
ejpam-5973	40	19	the	the	DET
ejpam-5973	40	20	closed	closed	ADJ
ejpam-5973	40	21	neighborhood	neighborhood	NOUN
ejpam-5973	40	22	of	of	ADP
ejpam-5973	40	23	a	a	DET
ejpam-5973	40	24	subset	subset	NOUN
ejpam-5973	40	25	s	s	NOUN
ejpam-5973	40	26	of	of	ADP
ejpam-5973	40	27	v	v	NOUN
ejpam-5973	40	28	(	(	PUNCT
ejpam-5973	40	29	g	g	NOUN
ejpam-5973	40	30	)	)	PUNCT
ejpam-5973	40	31	is	be	AUX
ejpam-5973	40	32	the	the	DET
ejpam-5973	40	33	set	set	VERB
ejpam-5973	40	34	ng[s	ng[	NOUN
ejpam-5973	40	35	]	]	PUNCT
ejpam-5973	40	36	=	=	SYM
ejpam-5973	40	37	ng(s	ng(s	X
ejpam-5973	40	38	)	)	PUNCT
ejpam-5973	40	39	∪	∪	ADP
ejpam-5973	40	40	s.	s.	PROPN
ejpam-5973	40	41	the	the	DET
ejpam-5973	40	42	degree	degree	NOUN
ejpam-5973	40	43	of	of	ADP
ejpam-5973	40	44	a	a	DET
ejpam-5973	40	45	vertex	vertex	NOUN
ejpam-5973	40	46	u	u	NOUN
ejpam-5973	40	47	in	in	ADP
ejpam-5973	40	48	g	g	PROPN
ejpam-5973	40	49	is	be	AUX
ejpam-5973	40	50	the	the	DET
ejpam-5973	40	51	number	number	NOUN
ejpam-5973	40	52	of	of	ADP
ejpam-5973	40	53	neighbors	neighbor	NOUN
ejpam-5973	40	54	u	u	NOUN
ejpam-5973	40	55	in	in	ADP
ejpam-5973	40	56	g	g	NOUN
ejpam-5973	40	57	,	,	PUNCT
ejpam-5973	40	58	denoted	denote	VERB
ejpam-5973	40	59	by	by	ADP
ejpam-5973	40	60	deg(u	deg(u	PROPN
ejpam-5973	40	61	)	)	PUNCT
ejpam-5973	40	62	.	.	PUNCT
ejpam-5973	41	1	the	the	DET
ejpam-5973	41	2	maximum	maximum	ADJ
ejpam-5973	41	3	(	(	PUNCT
ejpam-5973	41	4	minimum	minimum	ADJ
ejpam-5973	41	5	)	)	PUNCT
ejpam-5973	41	6	degree	degree	NOUN
ejpam-5973	41	7	among	among	ADP
ejpam-5973	41	8	the	the	DET
ejpam-5973	41	9	vertices	vertex	NOUN
ejpam-5973	41	10	of	of	ADP
ejpam-5973	41	11	g	g	PROPN
ejpam-5973	41	12	is	be	AUX
ejpam-5973	41	13	denoted	denote	VERB
ejpam-5973	41	14	by	by	ADP
ejpam-5973	41	15	∆(g	∆(g	PROPN
ejpam-5973	41	16	)	)	PUNCT
ejpam-5973	41	17	(	(	PUNCT
ejpam-5973	41	18	δ(g	δ(g	PROPN
ejpam-5973	41	19	)	)	PUNCT
ejpam-5973	41	20	,	,	PUNCT
ejpam-5973	41	21	respectively	respectively	ADV
ejpam-5973	41	22	)	)	PUNCT
ejpam-5973	41	23	.	.	PUNCT
ejpam-5973	42	1	the	the	DET
ejpam-5973	42	2	distance	distance	NOUN
ejpam-5973	42	3	dg(u	dg(u	NOUN
ejpam-5973	42	4	,	,	PUNCT
ejpam-5973	42	5	v	v	NOUN
ejpam-5973	42	6	)	)	PUNCT
ejpam-5973	42	7	between	between	ADP
ejpam-5973	42	8	two	two	NUM
ejpam-5973	42	9	vertices	vertex	NOUN
ejpam-5973	42	10	u	u	NOUN
ejpam-5973	42	11	and	and	CCONJ
ejpam-5973	42	12	v	v	NOUN
ejpam-5973	42	13	in	in	ADP
ejpam-5973	42	14	a	a	DET
ejpam-5973	42	15	connected	connected	ADJ
ejpam-5973	42	16	graph	graph	NOUN
ejpam-5973	42	17	g	g	PROPN
ejpam-5973	42	18	is	be	AUX
ejpam-5973	42	19	the	the	DET
ejpam-5973	42	20	length	length	NOUN
ejpam-5973	42	21	of	of	ADP
ejpam-5973	42	22	a	a	DET
ejpam-5973	42	23	shortest	short	ADJ
ejpam-5973	42	24	u	u	NOUN
ejpam-5973	42	25	-	-	NOUN
ejpam-5973	42	26	v	v	ADJ
ejpam-5973	42	27	path	path	NOUN
ejpam-5973	42	28	in	in	ADP
ejpam-5973	42	29	g	g	PROPN
ejpam-5973	42	30	while	while	SCONJ
ejpam-5973	42	31	the	the	DET
ejpam-5973	42	32	diameter	diameter	NOUN
ejpam-5973	42	33	of	of	ADP
ejpam-5973	42	34	g	g	PROPN
ejpam-5973	42	35	,	,	PUNCT
ejpam-5973	42	36	denoted	denote	VERB
ejpam-5973	42	37	by	by	ADP
ejpam-5973	42	38	diam(g	diam(g	PROPN
ejpam-5973	42	39	)	)	PUNCT
ejpam-5973	42	40	,	,	PUNCT
ejpam-5973	42	41	is	be	AUX
ejpam-5973	42	42	the	the	DET
ejpam-5973	42	43	maximum	maximum	ADJ
ejpam-5973	42	44	distance	distance	NOUN
ejpam-5973	42	45	among	among	ADP
ejpam-5973	42	46	all	all	DET
ejpam-5973	42	47	pairs	pair	NOUN
ejpam-5973	42	48	of	of	ADP
ejpam-5973	42	49	vertices	vertex	NOUN
ejpam-5973	42	50	in	in	ADP
ejpam-5973	42	51	g.	g.	PROPN
ejpam-5973	42	52	a	a	DET
ejpam-5973	42	53	complete	complete	ADJ
ejpam-5973	42	54	graph	graph	NOUN
ejpam-5973	42	55	on	on	ADP
ejpam-5973	42	56	n	n	DET
ejpam-5973	42	57	vertices	vertex	NOUN
ejpam-5973	42	58	is	be	AUX
ejpam-5973	42	59	denoted	denote	VERB
ejpam-5973	42	60	by	by	ADP
ejpam-5973	42	61	kn	kn	PROPN
ejpam-5973	42	62	,	,	PUNCT
ejpam-5973	42	63	while	while	SCONJ
ejpam-5973	42	64	a	a	DET
ejpam-5973	42	65	complete	complete	ADJ
ejpam-5973	42	66	bipartite	bipartite	NOUN
ejpam-5973	42	67	graph	graph	NOUN
ejpam-5973	42	68	with	with	ADP
ejpam-5973	42	69	partite	partite	ADJ
ejpam-5973	42	70	sets	set	NOUN
ejpam-5973	42	71	of	of	ADP
ejpam-5973	42	72	size	size	NOUN
ejpam-5973	42	73	p	p	NOUN
ejpam-5973	42	74	and	and	CCONJ
ejpam-5973	42	75	q	q	NOUN
ejpam-5973	42	76	is	be	AUX
ejpam-5973	42	77	denoted	denote	VERB
ejpam-5973	42	78	by	by	ADP
ejpam-5973	42	79	kp	kp	PROPN
ejpam-5973	42	80	,	,	PUNCT
ejpam-5973	42	81	q.	q.	PROPN
ejpam-5973	42	82	we	we	PRON
ejpam-5973	42	83	write	write	VERB
ejpam-5973	42	84	pn	pn	PROPN
ejpam-5973	42	85	for	for	ADP
ejpam-5973	42	86	the	the	DET
ejpam-5973	42	87	path	path	NOUN
ejpam-5973	42	88	of	of	ADP
ejpam-5973	42	89	order	order	NOUN
ejpam-5973	42	90	n	n	CCONJ
ejpam-5973	42	91	,	,	PUNCT
ejpam-5973	42	92	cn	cn	PROPN
ejpam-5973	42	93	for	for	ADP
ejpam-5973	42	94	the	the	DET
ejpam-5973	42	95	cycle	cycle	NOUN
ejpam-5973	42	96	of	of	ADP
ejpam-5973	42	97	length	length	NOUN
ejpam-5973	42	98	n	n	PROPN
ejpam-5973	42	99	and	and	CCONJ
ejpam-5973	42	100	kn	kn	PROPN
ejpam-5973	42	101	for	for	ADP
ejpam-5973	42	102	the	the	DET
ejpam-5973	42	103	graph	graph	NOUN
ejpam-5973	42	104	with	with	ADP
ejpam-5973	42	105	n	n	ADP
ejpam-5973	42	106	vertices	vertex	NOUN
ejpam-5973	42	107	and	and	CCONJ
ejpam-5973	42	108	no	no	DET
ejpam-5973	42	109	edges	edge	NOUN
ejpam-5973	42	110	,	,	PUNCT
ejpam-5973	42	111	as	as	SCONJ
ejpam-5973	42	112	defined	define	VERB
ejpam-5973	42	113	by	by	ADP
ejpam-5973	42	114	harary	harary	NOUN
ejpam-5973	42	115	in	in	ADP
ejpam-5973	42	116	[	[	X
ejpam-5973	42	117	15	15	NUM
ejpam-5973	42	118	]	]	PUNCT
ejpam-5973	42	119	.	.	PUNCT
ejpam-5973	43	1	a	a	DET
ejpam-5973	43	2	vertex	vertex	NOUN
ejpam-5973	43	3	v	v	NOUN
ejpam-5973	43	4	in	in	ADP
ejpam-5973	43	5	g	g	PROPN
ejpam-5973	43	6	is	be	AUX
ejpam-5973	43	7	called	call	VERB
ejpam-5973	43	8	an	an	DET
ejpam-5973	43	9	ℓ-neighbor	ℓ-neighbor	NOUN
ejpam-5973	43	10	of	of	ADP
ejpam-5973	43	11	a	a	DET
ejpam-5973	43	12	vertex	vertex	NOUN
ejpam-5973	43	13	u	u	NOUN
ejpam-5973	43	14	in	in	ADP
ejpam-5973	43	15	g	g	PROPN
ejpam-5973	43	16	if	if	SCONJ
ejpam-5973	43	17	dg(u	dg(u	NOUN
ejpam-5973	43	18	,	,	PUNCT
ejpam-5973	43	19	v	v	NOUN
ejpam-5973	43	20	)	)	PUNCT
ejpam-5973	43	21	=	=	VERB
ejpam-5973	43	22	ℓ.	ℓ.	NOUN
ejpam-5973	43	23	the	the	DET
ejpam-5973	43	24	set	set	NOUN
ejpam-5973	43	25	n	n	NOUN
ejpam-5973	43	26	ℓ	ℓ	NOUN
ejpam-5973	43	27	g(u	g(u	PROPN
ejpam-5973	43	28	)	)	PUNCT
ejpam-5973	44	1	=	=	PRON
ejpam-5973	44	2	{	{	PUNCT
ejpam-5973	44	3	v	v	NUM
ejpam-5973	44	4	∈	∈	NOUN
ejpam-5973	44	5	v	v	NOUN
ejpam-5973	44	6	(	(	PUNCT
ejpam-5973	44	7	g	g	NOUN
ejpam-5973	44	8	)	)	PUNCT
ejpam-5973	44	9	:	:	PUNCT
ejpam-5973	44	10	dg(v	dg(v	X
ejpam-5973	44	11	,	,	PUNCT
ejpam-5973	44	12	u	u	NOUN
ejpam-5973	44	13	)	)	PUNCT
ejpam-5973	44	14	=	=	SYM
ejpam-5973	44	15	ℓ	ℓ	X
ejpam-5973	44	16	}	}	PUNCT
ejpam-5973	44	17	is	be	AUX
ejpam-5973	44	18	called	call	VERB
ejpam-5973	44	19	the	the	DET
ejpam-5973	44	20	open	open	ADJ
ejpam-5973	44	21	ℓ-neighborhood	ℓ-neighborhood	NOUN
ejpam-5973	44	22	of	of	ADP
ejpam-5973	44	23	u.	u.	PROPN
ejpam-5973	44	24	the	the	DET
ejpam-5973	44	25	closed	closed	ADJ
ejpam-5973	44	26	ℓ-neighborhood	ℓ-neighborhood	NOUN
ejpam-5973	44	27	of	of	ADP
ejpam-5973	44	28	u	u	PROPN
ejpam-5973	44	29	in	in	ADP
ejpam-5973	44	30	g	g	PROPN
ejpam-5973	44	31	is	be	AUX
ejpam-5973	44	32	given	give	VERB
ejpam-5973	44	33	by	by	ADP
ejpam-5973	44	34	n	n	PROPN
ejpam-5973	44	35	ℓ	ℓ	PROPN
ejpam-5973	44	36	g[u	g[u	X
ejpam-5973	44	37	]	]	X
ejpam-5973	44	38	=	=	SYM
ejpam-5973	44	39	n	n	NUM
ejpam-5973	44	40	ℓ	ℓ	NOUN
ejpam-5973	44	41	g(u	g(u	PROPN
ejpam-5973	44	42	)	)	PUNCT
ejpam-5973	44	43	∪	∪	NOUN
ejpam-5973	44	44	{	{	PUNCT
ejpam-5973	44	45	u	u	NOUN
ejpam-5973	44	46	}	}	PUNCT
ejpam-5973	44	47	.	.	PUNCT
ejpam-5973	45	1	the	the	DET
ejpam-5973	45	2	open	open	ADJ
ejpam-5973	45	3	ℓ-neighborhood	ℓ-neighborhood	NOUN
ejpam-5973	45	4	of	of	ADP
ejpam-5973	45	5	x	x	PROPN
ejpam-5973	45	6	⊆	⊆	NUM
ejpam-5973	45	7	v	v	ADP
ejpam-5973	45	8	(	(	PUNCT
ejpam-5973	45	9	g	g	NOUN
ejpam-5973	45	10	)	)	PUNCT
ejpam-5973	45	11	is	be	AUX
ejpam-5973	45	12	the	the	DET
ejpam-5973	45	13	set	set	NOUN
ejpam-5973	45	14	n	n	NOUN
ejpam-5973	45	15	ℓ	ℓ	NOUN
ejpam-5973	45	16	g(x	g(x	NOUN
ejpam-5973	45	17	)	)	PUNCT
ejpam-5973	46	1	=	=	SYM
ejpam-5973	46	2	⋃	⋃	NOUN
ejpam-5973	46	3	u∈x	u∈x	ADJ
ejpam-5973	46	4	n	n	NOUN
ejpam-5973	46	5	ℓ	ℓ	NOUN
ejpam-5973	46	6	g(u	g(u	PROPN
ejpam-5973	46	7	)	)	PUNCT
ejpam-5973	46	8	.	.	PUNCT
ejpam-5973	47	1	the	the	DET
ejpam-5973	47	2	closed	closed	ADJ
ejpam-5973	47	3	ℓ-neighborhood	ℓ-neighborhood	NOUN
ejpam-5973	47	4	of	of	ADP
ejpam-5973	47	5	x	x	PRON
ejpam-5973	47	6	in	in	ADP
ejpam-5973	47	7	g	g	PROPN
ejpam-5973	47	8	is	be	AUX
ejpam-5973	47	9	the	the	DET
ejpam-5973	47	10	j.	j.	PROPN
ejpam-5973	47	11	j.	j.	PROPN
ejpam-5973	47	12	hamja	hamja	PROPN
ejpam-5973	48	1	et	et	PROPN
ejpam-5973	48	2	al	al	PROPN
ejpam-5973	48	3	.	.	PUNCT
ejpam-5973	48	4	/	/	SYM
ejpam-5973	48	5	eur	eur	PROPN
ejpam-5973	48	6	.	.	PUNCT
ejpam-5973	49	1	j.	j.	PROPN
ejpam-5973	49	2	pure	pure	PROPN
ejpam-5973	49	3	appl	appl	PROPN
ejpam-5973	49	4	.	.	PROPN
ejpam-5973	49	5	math	math	PROPN
ejpam-5973	49	6	,	,	PUNCT
ejpam-5973	49	7	18	18	NUM
ejpam-5973	49	8	(	(	PUNCT
ejpam-5973	49	9	2	2	NUM
ejpam-5973	49	10	)	)	PUNCT
ejpam-5973	49	11	(	(	PUNCT
ejpam-5973	49	12	2025	2025	NUM
ejpam-5973	49	13	)	)	PUNCT
ejpam-5973	49	14	,	,	PUNCT
ejpam-5973	49	15	5973	5973	NUM
ejpam-5973	49	16	3	3	NUM
ejpam-5973	49	17	of	of	ADP
ejpam-5973	49	18	17	17	NUM
ejpam-5973	49	19	set	set	VERB
ejpam-5973	49	20	n	n	PROPN
ejpam-5973	49	21	ℓ	ℓ	X
ejpam-5973	49	22	g[x	g[x	NOUN
ejpam-5973	49	23	]	]	X
ejpam-5973	49	24	=	=	SYM
ejpam-5973	49	25	n	n	PROPN
ejpam-5973	49	26	ℓ	ℓ	NOUN
ejpam-5973	49	27	g(x	g(x	NOUN
ejpam-5973	49	28	)	)	PUNCT
ejpam-5973	49	29	∪x	∪x	NUM
ejpam-5973	49	30	.	.	PUNCT
ejpam-5973	50	1	a	a	DET
ejpam-5973	50	2	set	set	NOUN
ejpam-5973	50	3	s	s	NOUN
ejpam-5973	50	4	⊆	⊆	NUM
ejpam-5973	50	5	v	v	NOUN
ejpam-5973	50	6	(	(	PUNCT
ejpam-5973	50	7	g	g	NOUN
ejpam-5973	50	8	)	)	PUNCT
ejpam-5973	50	9	is	be	AUX
ejpam-5973	50	10	said	say	VERB
ejpam-5973	50	11	to	to	PART
ejpam-5973	50	12	be	be	AUX
ejpam-5973	50	13	a	a	DET
ejpam-5973	50	14	dominating	dominating	NOUN
ejpam-5973	50	15	set	set	NOUN
ejpam-5973	50	16	if	if	SCONJ
ejpam-5973	50	17	ng[s	ng[	NOUN
ejpam-5973	50	18	]	]	PUNCT
ejpam-5973	50	19	=	=	SYM
ejpam-5973	50	20	v	v	NOUN
ejpam-5973	50	21	(	(	PUNCT
ejpam-5973	50	22	g	g	NOUN
ejpam-5973	50	23	)	)	PUNCT
ejpam-5973	50	24	.	.	PUNCT
ejpam-5973	51	1	a	a	DET
ejpam-5973	51	2	dominating	dominating	NOUN
ejpam-5973	51	3	set	set	NOUN
ejpam-5973	51	4	s	s	VERB
ejpam-5973	51	5	is	be	AUX
ejpam-5973	51	6	a	a	DET
ejpam-5973	51	7	minimal	minimal	ADJ
ejpam-5973	51	8	dominating	dominating	NOUN
ejpam-5973	51	9	set	set	NOUN
ejpam-5973	51	10	if	if	SCONJ
ejpam-5973	51	11	no	no	DET
ejpam-5973	51	12	proper	proper	ADJ
ejpam-5973	51	13	subset	subset	NOUN
ejpam-5973	51	14	of	of	ADP
ejpam-5973	51	15	s	s	PROPN
ejpam-5973	51	16	is	be	AUX
ejpam-5973	51	17	a	a	DET
ejpam-5973	51	18	dominating	dominating	NOUN
ejpam-5973	51	19	set	set	NOUN
ejpam-5973	51	20	.	.	PUNCT
ejpam-5973	52	1	the	the	DET
ejpam-5973	52	2	domination	domination	NOUN
ejpam-5973	52	3	number	number	NOUN
ejpam-5973	52	4	of	of	ADP
ejpam-5973	52	5	a	a	DET
ejpam-5973	52	6	graph	graph	NOUN
ejpam-5973	52	7	g	g	NOUN
ejpam-5973	52	8	,	,	PUNCT
ejpam-5973	52	9	denoted	denote	VERB
ejpam-5973	52	10	by	by	ADP
ejpam-5973	52	11	γ(g	γ(g	PROPN
ejpam-5973	52	12	)	)	PUNCT
ejpam-5973	52	13	,	,	PUNCT
ejpam-5973	52	14	is	be	AUX
ejpam-5973	52	15	the	the	DET
ejpam-5973	52	16	minimum	minimum	ADJ
ejpam-5973	52	17	cardinality	cardinality	NOUN
ejpam-5973	52	18	of	of	ADP
ejpam-5973	52	19	a	a	DET
ejpam-5973	52	20	dominating	dominating	NOUN
ejpam-5973	52	21	set	set	NOUN
ejpam-5973	52	22	of	of	ADP
ejpam-5973	52	23	g.	g.	PROPN
ejpam-5973	52	24	a	a	DET
ejpam-5973	52	25	dominating	dominating	NOUN
ejpam-5973	52	26	set	set	NOUN
ejpam-5973	52	27	s	s	NOUN
ejpam-5973	52	28	with	with	ADP
ejpam-5973	52	29	the	the	DET
ejpam-5973	52	30	cardinality	cardinality	NOUN
ejpam-5973	52	31	equal	equal	ADJ
ejpam-5973	52	32	to	to	ADP
ejpam-5973	52	33	γ(g	γ(g	PROPN
ejpam-5973	52	34	)	)	PUNCT
ejpam-5973	52	35	is	be	AUX
ejpam-5973	52	36	said	say	VERB
ejpam-5973	52	37	to	to	PART
ejpam-5973	52	38	be	be	AUX
ejpam-5973	52	39	a	a	DET
ejpam-5973	52	40	γ	γ	NOUN
ejpam-5973	52	41	-	-	PUNCT
ejpam-5973	52	42	set	set	NOUN
ejpam-5973	52	43	of	of	ADP
ejpam-5973	52	44	g	g	NOUN
ejpam-5973	52	45	or	or	CCONJ
ejpam-5973	52	46	γ(g)-set	γ(g)-set	PROPN
ejpam-5973	52	47	.	.	PUNCT
ejpam-5973	53	1	a	a	DET
ejpam-5973	53	2	set	set	NOUN
ejpam-5973	53	3	s	s	NOUN
ejpam-5973	53	4	⊆	⊆	NUM
ejpam-5973	53	5	v	v	NOUN
ejpam-5973	53	6	(	(	PUNCT
ejpam-5973	53	7	g	g	NOUN
ejpam-5973	53	8	)	)	PUNCT
ejpam-5973	53	9	is	be	AUX
ejpam-5973	53	10	a	a	DET
ejpam-5973	53	11	hop	hop	NOUN
ejpam-5973	53	12	dominating	dominating	NOUN
ejpam-5973	53	13	set	set	NOUN
ejpam-5973	53	14	of	of	ADP
ejpam-5973	53	15	g	g	PROPN
ejpam-5973	53	16	if	if	SCONJ
ejpam-5973	53	17	n2	n2	ADJ
ejpam-5973	53	18	g[s	g[s	PROPN
ejpam-5973	53	19	]	]	X
ejpam-5973	53	20	=	=	SYM
ejpam-5973	53	21	v	v	NOUN
ejpam-5973	53	22	(	(	PUNCT
ejpam-5973	53	23	g	g	NOUN
ejpam-5973	53	24	)	)	PUNCT
ejpam-5973	53	25	,	,	PUNCT
ejpam-5973	53	26	that	that	ADV
ejpam-5973	53	27	is	is	ADV
ejpam-5973	53	28	,	,	PUNCT
ejpam-5973	53	29	for	for	ADP
ejpam-5973	53	30	every	every	PRON
ejpam-5973	53	31	v	v	NUM
ejpam-5973	53	32	∈	∈	PROPN
ejpam-5973	53	33	v	v	NOUN
ejpam-5973	53	34	(	(	PUNCT
ejpam-5973	53	35	g	g	NOUN
ejpam-5973	53	36	)	)	PUNCT
ejpam-5973	53	37	\	\	PROPN
ejpam-5973	54	1	s	s	X
ejpam-5973	54	2	,	,	PUNCT
ejpam-5973	54	3	there	there	PRON
ejpam-5973	54	4	exists	exist	VERB
ejpam-5973	54	5	u	u	PROPN
ejpam-5973	54	6	∈	∈	PROPN
ejpam-5973	54	7	s	s	VERB
ejpam-5973	54	8	such	such	ADJ
ejpam-5973	54	9	that	that	DET
ejpam-5973	54	10	dg(u	dg(u	ADJ
ejpam-5973	54	11	,	,	PUNCT
ejpam-5973	54	12	v	v	NOUN
ejpam-5973	54	13	)	)	PUNCT
ejpam-5973	54	14	=	=	SYM
ejpam-5973	54	15	2	2	X
ejpam-5973	54	16	.	.	X
ejpam-5973	54	17	the	the	DET
ejpam-5973	54	18	hop	hop	NOUN
ejpam-5973	54	19	domination	domination	NOUN
ejpam-5973	54	20	number	number	NOUN
ejpam-5973	54	21	of	of	ADP
ejpam-5973	54	22	g	g	NOUN
ejpam-5973	54	23	,	,	PUNCT
ejpam-5973	54	24	denoted	denote	VERB
ejpam-5973	54	25	by	by	ADP
ejpam-5973	54	26	γh(g	γh(g	NOUN
ejpam-5973	54	27	)	)	PUNCT
ejpam-5973	54	28	,	,	PUNCT
ejpam-5973	54	29	is	be	AUX
ejpam-5973	54	30	the	the	DET
ejpam-5973	54	31	minimum	minimum	ADJ
ejpam-5973	54	32	cardinality	cardinality	NOUN
ejpam-5973	54	33	of	of	ADP
ejpam-5973	54	34	a	a	DET
ejpam-5973	54	35	hop	hop	NOUN
ejpam-5973	54	36	dominating	dominating	NOUN
ejpam-5973	54	37	set	set	NOUN
ejpam-5973	54	38	of	of	ADP
ejpam-5973	54	39	g.	g.	PROPN
ejpam-5973	54	40	a	a	DET
ejpam-5973	54	41	hop	hop	NOUN
ejpam-5973	54	42	dominating	dominating	NOUN
ejpam-5973	54	43	set	set	VERB
ejpam-5973	54	44	with	with	ADP
ejpam-5973	54	45	cardinality	cardinality	NOUN
ejpam-5973	54	46	equal	equal	ADJ
ejpam-5973	54	47	to	to	ADP
ejpam-5973	54	48	γh(g	γh(g	NOUN
ejpam-5973	54	49	)	)	PUNCT
ejpam-5973	54	50	is	be	AUX
ejpam-5973	54	51	called	call	VERB
ejpam-5973	54	52	a	a	DET
ejpam-5973	54	53	γh	γh	ADV
ejpam-5973	54	54	-	-	PUNCT
ejpam-5973	54	55	set	set	NOUN
ejpam-5973	54	56	of	of	ADP
ejpam-5973	54	57	g	g	NOUN
ejpam-5973	54	58	,	,	PUNCT
ejpam-5973	54	59	as	as	SCONJ
ejpam-5973	54	60	defined	define	VERB
ejpam-5973	54	61	by	by	ADP
ejpam-5973	54	62	natarajan	natarajan	PROPN
ejpam-5973	54	63	et	et	PROPN
ejpam-5973	54	64	al	al	PROPN
ejpam-5973	54	65	.	.	PUNCT
ejpam-5973	55	1	in	in	ADP
ejpam-5973	55	2	[	[	X
ejpam-5973	55	3	10	10	NUM
ejpam-5973	55	4	]	]	PUNCT
ejpam-5973	55	5	.	.	PUNCT
ejpam-5973	56	1	for	for	ADP
ejpam-5973	56	2	a	a	DET
ejpam-5973	56	3	positive	positive	ADJ
ejpam-5973	56	4	integer	integer	NOUN
ejpam-5973	56	5	ℓ	ℓ	NOUN
ejpam-5973	56	6	,	,	PUNCT
ejpam-5973	56	7	the	the	DET
ejpam-5973	56	8	ℓ-degree	ℓ-degree	NOUN
ejpam-5973	56	9	of	of	ADP
ejpam-5973	56	10	a	a	DET
ejpam-5973	56	11	vertex	vertex	NOUN
ejpam-5973	56	12	v	v	NOUN
ejpam-5973	56	13	in	in	ADP
ejpam-5973	56	14	a	a	DET
ejpam-5973	56	15	graph	graph	NOUN
ejpam-5973	56	16	g	g	NOUN
ejpam-5973	56	17	,	,	PUNCT
ejpam-5973	56	18	denoted	denote	VERB
ejpam-5973	56	19	by	by	ADP
ejpam-5973	56	20	degℓ(v	degℓ(v	PROPN
ejpam-5973	56	21	)	)	PUNCT
ejpam-5973	56	22	,	,	PUNCT
ejpam-5973	56	23	is	be	AUX
ejpam-5973	56	24	defined	define	VERB
ejpam-5973	56	25	as	as	ADP
ejpam-5973	56	26	the	the	DET
ejpam-5973	56	27	number	number	NOUN
ejpam-5973	56	28	of	of	ADP
ejpam-5973	56	29	vertices	vertex	NOUN
ejpam-5973	56	30	at	at	ADP
ejpam-5973	56	31	distance	distance	NOUN
ejpam-5973	56	32	ℓ	ℓ	NOUN
ejpam-5973	56	33	from	from	ADP
ejpam-5973	56	34	v	v	NUM
ejpam-5973	56	35	in	in	ADP
ejpam-5973	56	36	g.	g.	PROPN
ejpam-5973	56	37	the	the	DET
ejpam-5973	56	38	maximum	maximum	ADJ
ejpam-5973	56	39	ℓ-degree	ℓ-degree	NOUN
ejpam-5973	56	40	among	among	ADP
ejpam-5973	56	41	the	the	DET
ejpam-5973	56	42	vertices	vertex	NOUN
ejpam-5973	56	43	of	of	ADP
ejpam-5973	56	44	g	g	PROPN
ejpam-5973	56	45	is	be	AUX
ejpam-5973	56	46	denoted	denote	VERB
ejpam-5973	56	47	by	by	ADP
ejpam-5973	56	48	∆ℓ(g	∆ℓ(g	NOUN
ejpam-5973	56	49	)	)	PUNCT
ejpam-5973	56	50	.	.	PUNCT
ejpam-5973	57	1	in	in	ADP
ejpam-5973	57	2	the	the	DET
ejpam-5973	57	3	special	special	ADJ
ejpam-5973	57	4	case	case	NOUN
ejpam-5973	57	5	ℓ	ℓ	NOUN
ejpam-5973	57	6	=	=	SYM
ejpam-5973	57	7	2	2	NUM
ejpam-5973	57	8	,	,	PUNCT
ejpam-5973	57	9	a	a	DET
ejpam-5973	57	10	2	2	NUM
ejpam-5973	57	11	-	-	PUNCT
ejpam-5973	57	12	neighbor	neighbor	NOUN
ejpam-5973	57	13	is	be	AUX
ejpam-5973	57	14	called	call	VERB
ejpam-5973	57	15	a	a	DET
ejpam-5973	57	16	hop	hop	NOUN
ejpam-5973	57	17	-	-	PUNCT
ejpam-5973	57	18	neighbor	neighbor	NOUN
ejpam-5973	57	19	,	,	PUNCT
ejpam-5973	57	20	denoted	denote	VERB
ejpam-5973	57	21	by	by	ADP
ejpam-5973	57	22	n2	n2	ADJ
ejpam-5973	57	23	g(u	g(u	PROPN
ejpam-5973	57	24	)	)	PUNCT
ejpam-5973	57	25	.	.	PUNCT
ejpam-5973	58	1	the	the	DET
ejpam-5973	58	2	2	2	NUM
ejpam-5973	58	3	-	-	PUNCT
ejpam-5973	58	4	degree	degree	NOUN
ejpam-5973	58	5	is	be	AUX
ejpam-5973	58	6	called	call	VERB
ejpam-5973	58	7	the	the	DET
ejpam-5973	58	8	hop	hop	NOUN
ejpam-5973	58	9	-	-	PUNCT
ejpam-5973	58	10	degree	degree	NOUN
ejpam-5973	58	11	,	,	PUNCT
ejpam-5973	58	12	denoted	denote	VERB
ejpam-5973	58	13	by	by	ADP
ejpam-5973	58	14	deg2(v	deg2(v	NOUN
ejpam-5973	58	15	)	)	PUNCT
ejpam-5973	58	16	.	.	PUNCT
ejpam-5973	59	1	the	the	DET
ejpam-5973	59	2	maximum	maximum	ADJ
ejpam-5973	59	3	hop	hop	NOUN
ejpam-5973	59	4	-	-	PUNCT
ejpam-5973	59	5	degree	degree	NOUN
ejpam-5973	59	6	among	among	ADP
ejpam-5973	59	7	the	the	DET
ejpam-5973	59	8	vertices	vertex	NOUN
ejpam-5973	59	9	of	of	ADP
ejpam-5973	59	10	g	g	PROPN
ejpam-5973	59	11	is	be	AUX
ejpam-5973	59	12	denoted	denote	VERB
ejpam-5973	59	13	by	by	ADP
ejpam-5973	59	14	∆h(g	∆h(g	NOUN
ejpam-5973	59	15	)	)	PUNCT
ejpam-5973	59	16	,	,	PUNCT
ejpam-5973	59	17	as	as	SCONJ
ejpam-5973	59	18	defined	define	VERB
ejpam-5973	59	19	by	by	ADP
ejpam-5973	59	20	shabani	shabani	PROPN
ejpam-5973	59	21	et	et	PROPN
ejpam-5973	59	22	al	al	PROPN
ejpam-5973	59	23	.	.	PUNCT
ejpam-5973	60	1	in	in	ADP
ejpam-5973	60	2	[	[	X
ejpam-5973	60	3	16	16	NUM
ejpam-5973	60	4	]	]	PUNCT
ejpam-5973	60	5	.	.	PUNCT
ejpam-5973	61	1	a	a	DET
ejpam-5973	61	2	function	function	NOUN
ejpam-5973	61	3	f	f	NOUN
ejpam-5973	61	4	:	:	PUNCT
ejpam-5973	61	5	v	v	X
ejpam-5973	61	6	→	→	SYM
ejpam-5973	61	7	{	{	PUNCT
ejpam-5973	61	8	0	0	NUM
ejpam-5973	61	9	,	,	PUNCT
ejpam-5973	61	10	1	1	NUM
ejpam-5973	61	11	,	,	PUNCT
ejpam-5973	61	12	2	2	NUM
ejpam-5973	61	13	}	}	PUNCT
ejpam-5973	61	14	is	be	AUX
ejpam-5973	61	15	a	a	DET
ejpam-5973	61	16	roman	roman	ADJ
ejpam-5973	61	17	dominating	dominating	NOUN
ejpam-5973	61	18	function	function	NOUN
ejpam-5973	61	19	(	(	PUNCT
ejpam-5973	61	20	rd	rd	NOUN
ejpam-5973	61	21	-	-	NOUN
ejpam-5973	61	22	function	function	NOUN
ejpam-5973	61	23	,	,	PUNCT
ejpam-5973	61	24	for	for	ADP
ejpam-5973	61	25	short	short	ADJ
ejpam-5973	61	26	)	)	PUNCT
ejpam-5973	61	27	on	on	ADP
ejpam-5973	61	28	g	g	PROPN
ejpam-5973	61	29	if	if	SCONJ
ejpam-5973	61	30	every	every	DET
ejpam-5973	61	31	vertex	vertex	NOUN
ejpam-5973	61	32	u	u	NOUN
ejpam-5973	61	33	∈	∈	NOUN
ejpam-5973	61	34	v	v	NOUN
ejpam-5973	61	35	for	for	ADP
ejpam-5973	61	36	which	which	PRON
ejpam-5973	61	37	f(u	f(u	PROPN
ejpam-5973	61	38	)	)	PUNCT
ejpam-5973	62	1	=	=	SYM
ejpam-5973	62	2	0	0	NUM
ejpam-5973	62	3	is	be	AUX
ejpam-5973	62	4	adjacent	adjacent	ADJ
ejpam-5973	62	5	to	to	ADP
ejpam-5973	62	6	at	at	ADV
ejpam-5973	62	7	least	least	ADV
ejpam-5973	62	8	one	one	NUM
ejpam-5973	62	9	vertex	vertex	NOUN
ejpam-5973	62	10	v	v	NOUN
ejpam-5973	62	11	for	for	ADP
ejpam-5973	62	12	which	which	PRON
ejpam-5973	62	13	f(v	f(v	NOUN
ejpam-5973	62	14	)	)	PUNCT
ejpam-5973	62	15	=	=	SYM
ejpam-5973	62	16	2	2	NUM
ejpam-5973	62	17	,	,	PUNCT
ejpam-5973	62	18	as	as	SCONJ
ejpam-5973	62	19	defined	define	VERB
ejpam-5973	62	20	by	by	ADP
ejpam-5973	62	21	e.	e.	PROPN
ejpam-5973	62	22	j.	j.	PROPN
ejpam-5973	62	23	cockayne	cockayne	PROPN
ejpam-5973	62	24	in	in	ADP
ejpam-5973	62	25	[	[	X
ejpam-5973	62	26	17	17	NUM
ejpam-5973	62	27	]	]	PUNCT
ejpam-5973	62	28	.	.	PUNCT
ejpam-5973	63	1	a	a	DET
ejpam-5973	63	2	hop	hop	NOUN
ejpam-5973	63	3	roman	roman	ADJ
ejpam-5973	63	4	dominating	dominating	NOUN
ejpam-5973	63	5	function	function	NOUN
ejpam-5973	63	6	(	(	PUNCT
ejpam-5973	63	7	hrdfunction	hrdfunction	NOUN
ejpam-5973	63	8	)	)	PUNCT
ejpam-5973	63	9	of	of	ADP
ejpam-5973	63	10	g	g	PROPN
ejpam-5973	63	11	is	be	AUX
ejpam-5973	63	12	a	a	DET
ejpam-5973	63	13	function	function	NOUN
ejpam-5973	63	14	g	g	NOUN
ejpam-5973	63	15	defined	define	VERB
ejpam-5973	63	16	on	on	ADP
ejpam-5973	63	17	v	v	ADP
ejpam-5973	63	18	(	(	PUNCT
ejpam-5973	63	19	g	g	NOUN
ejpam-5973	63	20	)	)	PUNCT
ejpam-5973	63	21	into	into	ADP
ejpam-5973	63	22	{	{	PUNCT
ejpam-5973	63	23	0	0	NUM
ejpam-5973	63	24	,	,	PUNCT
ejpam-5973	63	25	1	1	NUM
ejpam-5973	63	26	,	,	PUNCT
ejpam-5973	63	27	2	2	NUM
ejpam-5973	63	28	}	}	PUNCT
ejpam-5973	63	29	having	have	VERB
ejpam-5973	63	30	the	the	DET
ejpam-5973	63	31	property	property	NOUN
ejpam-5973	64	1	that	that	PRON
ejpam-5973	64	2	for	for	ADP
ejpam-5973	64	3	every	every	DET
ejpam-5973	64	4	vertex	vertex	NOUN
ejpam-5973	64	5	u	u	NOUN
ejpam-5973	64	6	∈	∈	NOUN
ejpam-5973	64	7	v	v	NOUN
ejpam-5973	64	8	with	with	ADP
ejpam-5973	64	9	g(u	g(u	PROPN
ejpam-5973	64	10	)	)	PUNCT
ejpam-5973	64	11	=	=	SYM
ejpam-5973	64	12	0	0	X
ejpam-5973	65	1	there	there	PRON
ejpam-5973	65	2	is	be	VERB
ejpam-5973	65	3	a	a	DET
ejpam-5973	65	4	vertex	vertex	NOUN
ejpam-5973	65	5	w	w	NOUN
ejpam-5973	65	6	with	with	ADP
ejpam-5973	65	7	g(w	g(w	PROPN
ejpam-5973	65	8	)	)	PUNCT
ejpam-5973	65	9	=	=	SYM
ejpam-5973	65	10	2	2	NUM
ejpam-5973	65	11	and	and	CCONJ
ejpam-5973	65	12	d(u	d(u	PROPN
ejpam-5973	65	13	,	,	PUNCT
ejpam-5973	65	14	w	w	NOUN
ejpam-5973	65	15	)	)	PUNCT
ejpam-5973	65	16	=	=	SYM
ejpam-5973	65	17	2	2	X
ejpam-5973	65	18	.	.	X
ejpam-5973	65	19	the	the	DET
ejpam-5973	65	20	hop	hop	PROPN
ejpam-5973	65	21	roman	roman	ADJ
ejpam-5973	65	22	domination	domination	NOUN
ejpam-5973	65	23	number	number	NOUN
ejpam-5973	65	24	γhr(g	γhr(g	PROPN
ejpam-5973	65	25	)	)	PUNCT
ejpam-5973	65	26	is	be	AUX
ejpam-5973	65	27	equal	equal	ADJ
ejpam-5973	65	28	to	to	ADP
ejpam-5973	65	29	the	the	DET
ejpam-5973	65	30	minimum	minimum	ADJ
ejpam-5973	65	31	weight	weight	NOUN
ejpam-5973	65	32	of	of	ADP
ejpam-5973	65	33	a	a	DET
ejpam-5973	65	34	hrd	hrd	NOUN
ejpam-5973	65	35	-	-	PUNCT
ejpam-5973	65	36	function	function	NOUN
ejpam-5973	65	37	in	in	ADP
ejpam-5973	65	38	g.	g.	NOUN
ejpam-5973	65	39	for	for	ADP
ejpam-5973	65	40	more	more	ADJ
ejpam-5973	65	41	details	detail	NOUN
ejpam-5973	65	42	on	on	ADP
ejpam-5973	65	43	hop	hop	PROPN
ejpam-5973	65	44	roman	roman	ADJ
ejpam-5973	65	45	domination	domination	NOUN
ejpam-5973	65	46	see	see	VERB
ejpam-5973	65	47	for	for	ADP
ejpam-5973	65	48	example	example	NOUN
ejpam-5973	65	49	[	[	X
ejpam-5973	65	50	16	16	NUM
ejpam-5973	65	51	]	]	PUNCT
ejpam-5973	65	52	.	.	PUNCT
ejpam-5973	66	1	let	let	VERB
ejpam-5973	66	2	g	g	PRON
ejpam-5973	66	3	be	be	AUX
ejpam-5973	66	4	a	a	DET
ejpam-5973	66	5	graph	graph	NOUN
ejpam-5973	66	6	and	and	CCONJ
ejpam-5973	66	7	let	let	VERB
ejpam-5973	66	8	f	f	PRON
ejpam-5973	66	9	be	be	AUX
ejpam-5973	66	10	a	a	DET
ejpam-5973	66	11	function	function	NOUN
ejpam-5973	66	12	that	that	PRON
ejpam-5973	66	13	assigns	assign	VERB
ejpam-5973	66	14	to	to	ADP
ejpam-5973	66	15	each	each	DET
ejpam-5973	66	16	vertex	vertex	NOUN
ejpam-5973	66	17	a	a	DET
ejpam-5973	66	18	set	set	NOUN
ejpam-5973	66	19	of	of	ADP
ejpam-5973	66	20	colors	color	NOUN
ejpam-5973	66	21	chosen	choose	VERB
ejpam-5973	66	22	from	from	ADP
ejpam-5973	66	23	the	the	DET
ejpam-5973	66	24	set	set	NOUN
ejpam-5973	66	25	{	{	PUNCT
ejpam-5973	66	26	1	1	NUM
ejpam-5973	66	27	,	,	PUNCT
ejpam-5973	66	28	2	2	NUM
ejpam-5973	66	29	,	,	PUNCT
ejpam-5973	66	30	3	3	NUM
ejpam-5973	66	31	,	,	PUNCT
ejpam-5973	66	32	.	.	PUNCT
ejpam-5973	66	33	.	.	PUNCT
ejpam-5973	67	1	.	.	PUNCT
ejpam-5973	68	1	,	,	PUNCT
ejpam-5973	68	2	k	k	X
ejpam-5973	68	3	}	}	PUNCT
ejpam-5973	68	4	,	,	PUNCT
ejpam-5973	68	5	that	that	ADV
ejpam-5973	68	6	is	is	ADV
ejpam-5973	68	7	,	,	PUNCT
ejpam-5973	68	8	f	f	X
ejpam-5973	68	9	:	:	PUNCT
ejpam-5973	68	10	v	v	X
ejpam-5973	68	11	(	(	PUNCT
ejpam-5973	68	12	g	g	NOUN
ejpam-5973	68	13	)	)	PUNCT
ejpam-5973	68	14	→	→	SYM
ejpam-5973	69	1	p	p	X
ejpam-5973	69	2	(	(	PUNCT
ejpam-5973	69	3	{	{	PUNCT
ejpam-5973	69	4	1	1	NUM
ejpam-5973	69	5	,	,	PUNCT
ejpam-5973	69	6	2	2	NUM
ejpam-5973	69	7	,	,	PUNCT
ejpam-5973	69	8	3	3	NUM
ejpam-5973	69	9	,	,	PUNCT
ejpam-5973	69	10	.	.	PUNCT
ejpam-5973	69	11	.	.	PUNCT
ejpam-5973	69	12	.	.	PUNCT
ejpam-5973	70	1	,	,	PUNCT
ejpam-5973	70	2	k	k	X
ejpam-5973	70	3	}	}	PUNCT
ejpam-5973	70	4	)	)	PUNCT
ejpam-5973	70	5	.	.	PUNCT
ejpam-5973	71	1	if	if	SCONJ
ejpam-5973	71	2	for	for	ADP
ejpam-5973	71	3	each	each	DET
ejpam-5973	71	4	vertex	vertex	NOUN
ejpam-5973	71	5	v	v	ADP
ejpam-5973	71	6	∈	∈	PROPN
ejpam-5973	71	7	v	v	NOUN
ejpam-5973	71	8	(	(	PUNCT
ejpam-5973	71	9	g	g	NOUN
ejpam-5973	71	10	)	)	PUNCT
ejpam-5973	71	11	with	with	ADP
ejpam-5973	71	12	f(v	f(v	NOUN
ejpam-5973	71	13	)	)	PUNCT
ejpam-5973	71	14	=	=	SYM
ejpam-5973	71	15	∅	∅	NOUN
ejpam-5973	71	16	,	,	PUNCT
ejpam-5973	71	17	we	we	PRON
ejpam-5973	71	18	have	have	VERB
ejpam-5973	71	19	⋃	⋃	PROPN
ejpam-5973	71	20	u∈ng(v	u∈ng(v	PROPN
ejpam-5973	71	21	)	)	PUNCT
ejpam-5973	71	22	f(u	f(u	PROPN
ejpam-5973	71	23	)	)	PUNCT
ejpam-5973	72	1	=	=	PRON
ejpam-5973	72	2	{	{	PUNCT
ejpam-5973	72	3	1	1	NUM
ejpam-5973	72	4	,	,	PUNCT
ejpam-5973	72	5	2	2	NUM
ejpam-5973	72	6	,	,	PUNCT
ejpam-5973	72	7	3	3	NUM
ejpam-5973	72	8	,	,	PUNCT
ejpam-5973	72	9	.	.	PUNCT
ejpam-5973	72	10	.	.	PUNCT
ejpam-5973	73	1	.	.	PUNCT
ejpam-5973	74	1	,	,	PUNCT
ejpam-5973	74	2	k	k	X
ejpam-5973	74	3	}	}	PUNCT
ejpam-5973	74	4	,	,	PUNCT
ejpam-5973	74	5	then	then	ADV
ejpam-5973	74	6	f	f	PROPN
ejpam-5973	74	7	is	be	AUX
ejpam-5973	74	8	called	call	VERB
ejpam-5973	74	9	the	the	DET
ejpam-5973	74	10	k	k	ADJ
ejpam-5973	74	11	-	-	PUNCT
ejpam-5973	74	12	rainbow	rainbow	NOUN
ejpam-5973	74	13	dominating	dominating	NOUN
ejpam-5973	74	14	function	function	NOUN
ejpam-5973	74	15	(	(	PUNCT
ejpam-5973	74	16	krdf	krdf	PROPN
ejpam-5973	74	17	)	)	PUNCT
ejpam-5973	74	18	of	of	ADP
ejpam-5973	74	19	g.	g.	PROPN
ejpam-5973	74	20	the	the	DET
ejpam-5973	74	21	weight	weight	NOUN
ejpam-5973	74	22	ω(f	ω(f	PUNCT
ejpam-5973	74	23	)	)	PUNCT
ejpam-5973	74	24	of	of	ADP
ejpam-5973	74	25	f	f	PROPN
ejpam-5973	74	26	is	be	AUX
ejpam-5973	74	27	defined	define	VERB
ejpam-5973	74	28	as	as	ADP
ejpam-5973	74	29	ω(f	ω(f	ADJ
ejpam-5973	74	30	)	)	PUNCT
ejpam-5973	74	31	=	=	SYM
ejpam-5973	75	1	∑	∑	PUNCT
ejpam-5973	75	2	v∈v	v∈v	NOUN
ejpam-5973	75	3	(	(	PUNCT
ejpam-5973	75	4	g)|f(v)|	g)|f(v)|	PROPN
ejpam-5973	75	5	.	.	PUNCT
ejpam-5973	76	1	the	the	DET
ejpam-5973	76	2	k	k	ADJ
ejpam-5973	76	3	-	-	PUNCT
ejpam-5973	76	4	rainbow	rainbow	NOUN
ejpam-5973	76	5	domination	domination	NOUN
ejpam-5973	76	6	number	number	NOUN
ejpam-5973	76	7	of	of	ADP
ejpam-5973	76	8	g	g	NOUN
ejpam-5973	76	9	,	,	PUNCT
ejpam-5973	76	10	denoted	denote	VERB
ejpam-5973	76	11	by	by	ADP
ejpam-5973	76	12	γrk(g	γrk(g	PROPN
ejpam-5973	76	13	)	)	PUNCT
ejpam-5973	76	14	,	,	PUNCT
ejpam-5973	76	15	is	be	AUX
ejpam-5973	76	16	the	the	DET
ejpam-5973	76	17	minimum	minimum	ADJ
ejpam-5973	76	18	weight	weight	NOUN
ejpam-5973	76	19	of	of	ADP
ejpam-5973	76	20	a	a	DET
ejpam-5973	76	21	krdf	krdf	NOUN
ejpam-5973	76	22	.	.	PUNCT
ejpam-5973	77	1	a	a	DET
ejpam-5973	77	2	k	k	ADJ
ejpam-5973	77	3	-	-	PUNCT
ejpam-5973	77	4	rainbow	rainbow	NOUN
ejpam-5973	77	5	dominating	dominating	NOUN
ejpam-5973	77	6	function	function	NOUN
ejpam-5973	77	7	of	of	ADP
ejpam-5973	77	8	g	g	NOUN
ejpam-5973	77	9	with	with	ADP
ejpam-5973	77	10	weight	weight	NOUN
ejpam-5973	77	11	γrk(g	γrk(g	PROPN
ejpam-5973	77	12	)	)	PUNCT
ejpam-5973	77	13	is	be	AUX
ejpam-5973	77	14	a	a	DET
ejpam-5973	77	15	γrk	γrk	NOUN
ejpam-5973	77	16	-	-	PUNCT
ejpam-5973	77	17	function	function	NOUN
ejpam-5973	77	18	of	of	ADP
ejpam-5973	77	19	g	g	NOUN
ejpam-5973	77	20	,	,	PUNCT
ejpam-5973	77	21	as	as	SCONJ
ejpam-5973	77	22	defined	define	VERB
ejpam-5973	77	23	by	by	ADP
ejpam-5973	77	24	b.	b.	PROPN
ejpam-5973	77	25	brešar	brešar	PROPN
ejpam-5973	77	26	in	in	ADP
ejpam-5973	77	27	[	[	X
ejpam-5973	77	28	2	2	NUM
ejpam-5973	77	29	]	]	PUNCT
ejpam-5973	77	30	.	.	PUNCT
ejpam-5973	78	1	a	a	DET
ejpam-5973	78	2	function	function	NOUN
ejpam-5973	78	3	f	f	NOUN
ejpam-5973	78	4	:	:	PUNCT
ejpam-5973	78	5	v	v	X
ejpam-5973	78	6	(	(	PUNCT
ejpam-5973	78	7	g	g	NOUN
ejpam-5973	78	8	)	)	PUNCT
ejpam-5973	78	9	→	→	SYM
ejpam-5973	79	1	p	p	X
ejpam-5973	79	2	(	(	PUNCT
ejpam-5973	79	3	{	{	PUNCT
ejpam-5973	79	4	1	1	NUM
ejpam-5973	79	5	,	,	PUNCT
ejpam-5973	79	6	2	2	NUM
ejpam-5973	79	7	,	,	PUNCT
ejpam-5973	79	8	.	.	PUNCT
ejpam-5973	79	9	.	.	PUNCT
ejpam-5973	79	10	.	.	PUNCT
ejpam-5973	80	1	,	,	PUNCT
ejpam-5973	80	2	k	k	X
ejpam-5973	80	3	}	}	PUNCT
ejpam-5973	80	4	)	)	PUNCT
ejpam-5973	80	5	is	be	AUX
ejpam-5973	80	6	a	a	DET
ejpam-5973	80	7	hop	hop	NOUN
ejpam-5973	80	8	k	k	ADJ
ejpam-5973	80	9	-	-	PUNCT
ejpam-5973	80	10	rainbow	rainbow	NOUN
ejpam-5973	80	11	dominating	dominating	NOUN
ejpam-5973	80	12	function	function	NOUN
ejpam-5973	80	13	(	(	PUNCT
ejpam-5973	80	14	hkrd	hkrd	NOUN
ejpam-5973	80	15	-	-	PUNCT
ejpam-5973	80	16	function	function	NOUN
ejpam-5973	80	17	)	)	PUNCT
ejpam-5973	80	18	if	if	SCONJ
ejpam-5973	80	19	for	for	ADP
ejpam-5973	80	20	every	every	DET
ejpam-5973	80	21	vertex	vertex	NOUN
ejpam-5973	80	22	v	v	ADP
ejpam-5973	80	23	∈	∈	NOUN
ejpam-5973	80	24	v	v	NOUN
ejpam-5973	80	25	(	(	PUNCT
ejpam-5973	80	26	g	g	NOUN
ejpam-5973	80	27	)	)	PUNCT
ejpam-5973	80	28	with	with	ADP
ejpam-5973	80	29	f(v	f(v	NOUN
ejpam-5973	80	30	)	)	PUNCT
ejpam-5973	80	31	=	=	SYM
ejpam-5973	80	32	∅	∅	NOUN
ejpam-5973	80	33	,	,	PUNCT
ejpam-5973	80	34	we	we	PRON
ejpam-5973	80	35	have	have	VERB
ejpam-5973	80	36	⋃	⋃	ADJ
ejpam-5973	80	37	u∈n2	u∈n2	ADJ
ejpam-5973	80	38	g(v	g(v	NOUN
ejpam-5973	80	39	)	)	PUNCT
ejpam-5973	80	40	f(u	f(u	PROPN
ejpam-5973	80	41	)	)	PUNCT
ejpam-5973	80	42	=	=	PRON
ejpam-5973	80	43	{	{	PUNCT
ejpam-5973	80	44	1	1	NUM
ejpam-5973	80	45	,	,	PUNCT
ejpam-5973	80	46	2	2	NUM
ejpam-5973	80	47	,	,	PUNCT
ejpam-5973	80	48	.	.	PUNCT
ejpam-5973	80	49	.	.	PUNCT
ejpam-5973	81	1	.	.	PUNCT
ejpam-5973	82	1	,	,	PUNCT
ejpam-5973	82	2	k	k	X
ejpam-5973	82	3	}	}	PUNCT
ejpam-5973	82	4	.	.	PUNCT
ejpam-5973	83	1	the	the	DET
ejpam-5973	83	2	weight	weight	NOUN
ejpam-5973	83	3	of	of	ADP
ejpam-5973	83	4	a	a	DET
ejpam-5973	83	5	hop	hop	NOUN
ejpam-5973	83	6	k	k	ADJ
ejpam-5973	83	7	-	-	PUNCT
ejpam-5973	83	8	rainbow	rainbow	NOUN
ejpam-5973	83	9	dominating	dominating	NOUN
ejpam-5973	83	10	function	function	NOUN
ejpam-5973	83	11	is	be	AUX
ejpam-5973	83	12	ω(f	ω(f	ADV
ejpam-5973	83	13	)	)	PUNCT
ejpam-5973	83	14	=	=	SYM
ejpam-5973	84	1	∑	∑	PUNCT
ejpam-5973	84	2	v∈v	v∈v	NOUN
ejpam-5973	84	3	(	(	PUNCT
ejpam-5973	84	4	g	g	NOUN
ejpam-5973	84	5	)	)	PUNCT
ejpam-5973	84	6	|f(v)|	|f(v)|	PROPN
ejpam-5973	84	7	.	.	PUNCT
ejpam-5973	85	1	the	the	DET
ejpam-5973	85	2	hop	hop	NOUN
ejpam-5973	85	3	k	k	ADJ
ejpam-5973	85	4	-	-	PUNCT
ejpam-5973	85	5	rainbow	rainbow	NOUN
ejpam-5973	85	6	domination	domination	NOUN
ejpam-5973	85	7	number	number	NOUN
ejpam-5973	85	8	of	of	ADP
ejpam-5973	85	9	g	g	NOUN
ejpam-5973	85	10	,	,	PUNCT
ejpam-5973	85	11	denoted	denote	VERB
ejpam-5973	85	12	γhrk(g	γhrk(g	PROPN
ejpam-5973	85	13	)	)	PUNCT
ejpam-5973	85	14	,	,	PUNCT
ejpam-5973	85	15	is	be	AUX
ejpam-5973	85	16	the	the	DET
ejpam-5973	85	17	minimum	minimum	ADJ
ejpam-5973	85	18	weight	weight	NOUN
ejpam-5973	85	19	of	of	ADP
ejpam-5973	85	20	a	a	DET
ejpam-5973	85	21	hop	hop	NOUN
ejpam-5973	85	22	k	k	ADJ
ejpam-5973	85	23	-	-	PUNCT
ejpam-5973	85	24	rainbow	rainbow	NOUN
ejpam-5973	85	25	dominating	dominating	NOUN
ejpam-5973	85	26	function	function	NOUN
ejpam-5973	85	27	of	of	ADP
ejpam-5973	85	28	g.	g.	PROPN
ejpam-5973	85	29	a	a	DET
ejpam-5973	85	30	hop	hop	NOUN
ejpam-5973	85	31	k	k	ADJ
ejpam-5973	85	32	-	-	PUNCT
ejpam-5973	85	33	rainbow	rainbow	NOUN
ejpam-5973	85	34	dominating	dominating	NOUN
ejpam-5973	85	35	function	function	NOUN
ejpam-5973	85	36	of	of	ADP
ejpam-5973	85	37	g	g	NOUN
ejpam-5973	85	38	with	with	ADP
ejpam-5973	85	39	weight	weight	NOUN
ejpam-5973	85	40	γhrk(g	γhrk(g	PROPN
ejpam-5973	85	41	)	)	PUNCT
ejpam-5973	85	42	is	be	AUX
ejpam-5973	85	43	a	a	DET
ejpam-5973	85	44	γhrk	γhrk	NOUN
ejpam-5973	85	45	-	-	PUNCT
ejpam-5973	85	46	function	function	NOUN
ejpam-5973	85	47	of	of	ADP
ejpam-5973	85	48	g.	g.	PROPN
ejpam-5973	85	49	for	for	ADP
ejpam-5973	85	50	the	the	DET
ejpam-5973	85	51	sake	sake	NOUN
ejpam-5973	85	52	of	of	ADP
ejpam-5973	85	53	simplicity	simplicity	NOUN
ejpam-5973	85	54	,	,	PUNCT
ejpam-5973	85	55	we	we	PRON
ejpam-5973	85	56	will	will	AUX
ejpam-5973	85	57	write	write	VERB
ejpam-5973	85	58	hkrd	hkrd	NOUN
ejpam-5973	85	59	-	-	PUNCT
ejpam-5973	85	60	number	number	NOUN
ejpam-5973	85	61	instead	instead	ADV
ejpam-5973	85	62	of	of	ADP
ejpam-5973	85	63	hop	hop	NOUN
ejpam-5973	85	64	k	k	ADJ
ejpam-5973	85	65	-	-	PUNCT
ejpam-5973	85	66	rainbow	rainbow	NOUN
ejpam-5973	85	67	domination	domination	NOUN
ejpam-5973	85	68	number	number	NOUN
ejpam-5973	85	69	.	.	PUNCT
ejpam-5973	86	1	clearly	clearly	ADV
ejpam-5973	86	2	,	,	PUNCT
ejpam-5973	86	3	when	when	SCONJ
ejpam-5973	86	4	k	k	PROPN
ejpam-5973	86	5	=	=	SYM
ejpam-5973	86	6	1	1	NUM
ejpam-5973	86	7	,	,	PUNCT
ejpam-5973	86	8	γh1r(g	γh1r(g	NOUN
ejpam-5973	86	9	)	)	PUNCT
ejpam-5973	86	10	matches	match	VERB
ejpam-5973	86	11	with	with	ADP
ejpam-5973	86	12	the	the	DET
ejpam-5973	86	13	usual	usual	ADJ
ejpam-5973	86	14	hop	hop	NOUN
ejpam-5973	86	15	domination	domination	NOUN
ejpam-5973	86	16	number	number	NOUN
ejpam-5973	86	17	γh(g	γh(g	PUNCT
ejpam-5973	86	18	)	)	PUNCT
ejpam-5973	86	19	.	.	PUNCT
ejpam-5973	87	1	j.	j.	PROPN
ejpam-5973	87	2	j.	j.	PROPN
ejpam-5973	87	3	hamja	hamja	PROPN
ejpam-5973	87	4	et	et	PROPN
ejpam-5973	87	5	al	al	PROPN
ejpam-5973	87	6	.	.	PUNCT
ejpam-5973	87	7	/	/	SYM
ejpam-5973	87	8	eur	eur	PROPN
ejpam-5973	87	9	.	.	PUNCT
ejpam-5973	88	1	j.	j.	PROPN
ejpam-5973	88	2	pure	pure	PROPN
ejpam-5973	88	3	appl	appl	PROPN
ejpam-5973	88	4	.	.	PROPN
ejpam-5973	88	5	math	math	PROPN
ejpam-5973	88	6	,	,	PUNCT
ejpam-5973	88	7	18	18	NUM
ejpam-5973	88	8	(	(	PUNCT
ejpam-5973	88	9	2	2	NUM
ejpam-5973	88	10	)	)	PUNCT
ejpam-5973	88	11	(	(	PUNCT
ejpam-5973	88	12	2025	2025	NUM
ejpam-5973	88	13	)	)	PUNCT
ejpam-5973	88	14	,	,	PUNCT
ejpam-5973	88	15	5973	5973	NUM
ejpam-5973	88	16	4	4	NUM
ejpam-5973	88	17	of	of	ADP
ejpam-5973	88	18	17	17	NUM
ejpam-5973	88	19	example	example	NOUN
ejpam-5973	88	20	1	1	NUM
ejpam-5973	88	21	.	.	X
ejpam-5973	88	22	consider	consider	VERB
ejpam-5973	88	23	the	the	DET
ejpam-5973	88	24	graph	graph	NOUN
ejpam-5973	88	25	g	g	NOUN
ejpam-5973	88	26	in	in	ADP
ejpam-5973	88	27	figure	figure	NOUN
ejpam-5973	88	28	1	1	NUM
ejpam-5973	88	29	.	.	PUNCT
ejpam-5973	89	1	let	let	VERB
ejpam-5973	89	2	s	s	VERB
ejpam-5973	89	3	=	=	NOUN
ejpam-5973	89	4	{	{	PUNCT
ejpam-5973	89	5	v1	v1	PROPN
ejpam-5973	89	6	,	,	PUNCT
ejpam-5973	89	7	v2	v2	PROPN
ejpam-5973	89	8	,	,	PUNCT
ejpam-5973	89	9	v5	v5	PROPN
ejpam-5973	89	10	,	,	PUNCT
ejpam-5973	89	11	v6	v6	NOUN
ejpam-5973	89	12	,	,	PUNCT
ejpam-5973	89	13	v7	v7	NUM
ejpam-5973	89	14	,	,	PUNCT
ejpam-5973	89	15	v8	v8	PROPN
ejpam-5973	89	16	}	}	PUNCT
ejpam-5973	89	17	and	and	CCONJ
ejpam-5973	89	18	let	let	VERB
ejpam-5973	89	19	f	f	NOUN
ejpam-5973	89	20	:	:	PUNCT
ejpam-5973	89	21	v	v	X
ejpam-5973	89	22	(	(	PUNCT
ejpam-5973	89	23	g	g	NOUN
ejpam-5973	89	24	)	)	PUNCT
ejpam-5973	89	25	→	→	SYM
ejpam-5973	89	26	p	p	X
ejpam-5973	89	27	(	(	PUNCT
ejpam-5973	89	28	{	{	PUNCT
ejpam-5973	89	29	1	1	NUM
ejpam-5973	89	30	,	,	PUNCT
ejpam-5973	89	31	2	2	NUM
ejpam-5973	89	32	}	}	PUNCT
ejpam-5973	89	33	)	)	PUNCT
ejpam-5973	89	34	be	be	AUX
ejpam-5973	89	35	a	a	DET
ejpam-5973	89	36	function	function	NOUN
ejpam-5973	89	37	defined	define	VERB
ejpam-5973	89	38	by	by	ADP
ejpam-5973	89	39	f(v1	f(v1	NOUN
ejpam-5973	89	40	)	)	PUNCT
ejpam-5973	89	41	=	=	SYM
ejpam-5973	89	42	f(v5	f(v5	NOUN
ejpam-5973	89	43	)	)	PUNCT
ejpam-5973	89	44	=	=	SYM
ejpam-5973	89	45	f(v7	f(v7	NOUN
ejpam-5973	89	46	)	)	PUNCT
ejpam-5973	89	47	=	=	PUNCT
ejpam-5973	89	48	f(v8	f(v8	NOUN
ejpam-5973	89	49	)	)	PUNCT
ejpam-5973	90	1	=	=	PRON
ejpam-5973	90	2	{	{	PUNCT
ejpam-5973	90	3	2	2	NUM
ejpam-5973	90	4	}	}	PUNCT
ejpam-5973	90	5	,	,	PUNCT
ejpam-5973	90	6	f(v2	f(v2	NOUN
ejpam-5973	90	7	)	)	PUNCT
ejpam-5973	90	8	=	=	SYM
ejpam-5973	90	9	f(v6	f(v6	NOUN
ejpam-5973	90	10	)	)	PUNCT
ejpam-5973	90	11	=	=	PUNCT
ejpam-5973	90	12	{	{	PUNCT
ejpam-5973	90	13	1	1	NUM
ejpam-5973	90	14	}	}	PUNCT
ejpam-5973	90	15	,	,	PUNCT
ejpam-5973	90	16	and	and	CCONJ
ejpam-5973	90	17	f(v3	f(v3	X
ejpam-5973	90	18	)	)	PUNCT
ejpam-5973	90	19	=	=	SYM
ejpam-5973	90	20	f(v4	f(v4	NOUN
ejpam-5973	90	21	)	)	PUNCT
ejpam-5973	90	22	=	=	VERB
ejpam-5973	90	23	∅.	∅.	AUX
ejpam-5973	90	24	observe	observe	VERB
ejpam-5973	90	25	that	that	DET
ejpam-5973	90	26	dg(v3	dg(v3	NOUN
ejpam-5973	90	27	,	,	PUNCT
ejpam-5973	90	28	v1	v1	NOUN
ejpam-5973	90	29	)	)	PUNCT
ejpam-5973	90	30	=	=	SYM
ejpam-5973	90	31	2	2	NUM
ejpam-5973	90	32	,	,	PUNCT
ejpam-5973	90	33	and	and	CCONJ
ejpam-5973	90	34	dg(v4	dg(v4	NOUN
ejpam-5973	90	35	,	,	PUNCT
ejpam-5973	90	36	v2	v2	PROPN
ejpam-5973	90	37	)	)	PUNCT
ejpam-5973	91	1	=	=	SYM
ejpam-5973	91	2	2	2	X
ejpam-5973	91	3	.	.	PUNCT
ejpam-5973	92	1	thus	thus	ADV
ejpam-5973	92	2	,	,	PUNCT
ejpam-5973	92	3	s	s	VERB
ejpam-5973	92	4	is	be	AUX
ejpam-5973	92	5	a	a	DET
ejpam-5973	92	6	hop	hop	NOUN
ejpam-5973	92	7	dominating	dominating	NOUN
ejpam-5973	92	8	set	set	NOUN
ejpam-5973	92	9	of	of	ADP
ejpam-5973	92	10	g.	g.	PROPN
ejpam-5973	92	11	further	far	ADV
ejpam-5973	92	12	,	,	PUNCT
ejpam-5973	92	13	notice	notice	VERB
ejpam-5973	92	14	that	that	SCONJ
ejpam-5973	92	15	⋃	⋃	ADP
ejpam-5973	92	16	u1∈n2	u1∈n2	ADJ
ejpam-5973	92	17	g(v3	g(v3	NOUN
ejpam-5973	92	18	)	)	PUNCT
ejpam-5973	92	19	f(u1	f(u1	NOUN
ejpam-5973	92	20	)	)	PUNCT
ejpam-5973	92	21	=	=	PRON
ejpam-5973	92	22	{	{	PUNCT
ejpam-5973	92	23	1	1	NUM
ejpam-5973	92	24	,	,	PUNCT
ejpam-5973	92	25	2	2	NUM
ejpam-5973	92	26	}	}	PUNCT
ejpam-5973	92	27	and	and	CCONJ
ejpam-5973	92	28	⋃	⋃	PROPN
ejpam-5973	92	29	u2∈n2	u2∈n2	PROPN
ejpam-5973	92	30	g(v4	g(v4	PROPN
ejpam-5973	92	31	)	)	PUNCT
ejpam-5973	92	32	f(u2	f(u2	PROPN
ejpam-5973	92	33	)	)	PUNCT
ejpam-5973	93	1	=	=	PRON
ejpam-5973	93	2	{	{	PUNCT
ejpam-5973	93	3	1	1	NUM
ejpam-5973	93	4	,	,	PUNCT
ejpam-5973	93	5	2	2	NUM
ejpam-5973	93	6	}	}	PUNCT
ejpam-5973	93	7	.	.	PUNCT
ejpam-5973	94	1	then	then	ADV
ejpam-5973	94	2	f	f	PROPN
ejpam-5973	94	3	is	be	AUX
ejpam-5973	94	4	a	a	DET
ejpam-5973	94	5	hop	hop	NOUN
ejpam-5973	94	6	2	2	NUM
ejpam-5973	94	7	-	-	PUNCT
ejpam-5973	94	8	rainbow	rainbow	NOUN
ejpam-5973	94	9	dominating	dominating	NOUN
ejpam-5973	94	10	function	function	NOUN
ejpam-5973	94	11	of	of	ADP
ejpam-5973	94	12	g.	g.	PROPN
ejpam-5973	94	13	the	the	DET
ejpam-5973	94	14	weight	weight	NOUN
ejpam-5973	94	15	of	of	ADP
ejpam-5973	94	16	f	f	PROPN
ejpam-5973	94	17	is	be	AUX
ejpam-5973	94	18	given	give	VERB
ejpam-5973	94	19	ω(f	ω(f	PUNCT
ejpam-5973	94	20	)	)	PUNCT
ejpam-5973	94	21	=	=	SYM
ejpam-5973	95	1	|f(v1)|+	|f(v1)|+	PROPN
ejpam-5973	95	2	|f(v2)|+	|f(v2)|+	PROPN
ejpam-5973	95	3	|f(v3)|+	|f(v3)|+	PROPN
ejpam-5973	95	4	|f(v4)|+	|f(v4)|+	PROPN
ejpam-5973	95	5	|f(v5)|+	|f(v5)|+	PROPN
ejpam-5973	95	6	|f(v6)|+	|f(v6)|+	PROPN
ejpam-5973	95	7	|f(v7)|+	|f(v7)|+	PROPN
ejpam-5973	95	8	|f(v8)|	|f(v8)|	NUM
ejpam-5973	96	1	=	=	SYM
ejpam-5973	96	2	1	1	NUM
ejpam-5973	96	3	+	+	NUM
ejpam-5973	96	4	1	1	NUM
ejpam-5973	96	5	+	+	SYM
ejpam-5973	96	6	0	0	NUM
ejpam-5973	97	1	+	+	CCONJ
ejpam-5973	97	2	0	0	NUM
ejpam-5973	98	1	+	+	CCONJ
ejpam-5973	98	2	1	1	NUM
ejpam-5973	98	3	+	+	NUM
ejpam-5973	98	4	1	1	NUM
ejpam-5973	98	5	+	+	SYM
ejpam-5973	98	6	1	1	NUM
ejpam-5973	98	7	+	+	SYM
ejpam-5973	98	8	1	1	NUM
ejpam-5973	98	9	=	=	SYM
ejpam-5973	98	10	6	6	NUM
ejpam-5973	98	11	.	.	PUNCT
ejpam-5973	98	12	v1	v1	PROPN
ejpam-5973	98	13	v2	v2	PROPN
ejpam-5973	98	14	v3	v3	PROPN
ejpam-5973	98	15	v4	v4	PROPN
ejpam-5973	98	16	v5	v5	PROPN
ejpam-5973	98	17	v6	v6	NOUN
ejpam-5973	98	18	v7	v7	VERB
ejpam-5973	98	19	v8	v8	PROPN
ejpam-5973	98	20	g	g	PROPN
ejpam-5973	98	21	:	:	PUNCT
ejpam-5973	98	22	figure	figure	NOUN
ejpam-5973	98	23	1	1	NUM
ejpam-5973	98	24	:	:	PUNCT
ejpam-5973	98	25	a	a	DET
ejpam-5973	98	26	graph	graph	NOUN
ejpam-5973	98	27	g	g	NOUN
ejpam-5973	98	28	of	of	ADP
ejpam-5973	98	29	order	order	NOUN
ejpam-5973	98	30	8	8	NUM
ejpam-5973	98	31	with	with	ADP
ejpam-5973	98	32	a	a	DET
ejpam-5973	98	33	hop	hop	NOUN
ejpam-5973	98	34	2	2	NUM
ejpam-5973	98	35	-	-	PUNCT
ejpam-5973	98	36	rainbow	rainbow	NOUN
ejpam-5973	98	37	dominating	dominating	NOUN
ejpam-5973	98	38	function	function	NOUN
ejpam-5973	98	39	of	of	ADP
ejpam-5973	98	40	weight	weight	NOUN
ejpam-5973	98	41	6	6	NUM
ejpam-5973	98	42	.	.	NOUN
ejpam-5973	98	43	3	3	NUM
ejpam-5973	98	44	.	.	X
ejpam-5973	98	45	preliminary	preliminary	ADJ
ejpam-5973	98	46	results	result	NOUN
ejpam-5973	98	47	in	in	ADP
ejpam-5973	98	48	this	this	DET
ejpam-5973	98	49	section	section	NOUN
ejpam-5973	98	50	,	,	PUNCT
ejpam-5973	98	51	we	we	PRON
ejpam-5973	98	52	study	study	VERB
ejpam-5973	98	53	the	the	DET
ejpam-5973	98	54	basic	basic	ADJ
ejpam-5973	98	55	properties	property	NOUN
ejpam-5973	98	56	of	of	ADP
ejpam-5973	98	57	hkrd	hkrd	NOUN
ejpam-5973	98	58	-	-	PUNCT
ejpam-5973	98	59	function	function	NOUN
ejpam-5973	98	60	and	and	CCONJ
ejpam-5973	98	61	we	we	PRON
ejpam-5973	98	62	establish	establish	VERB
ejpam-5973	98	63	various	various	ADJ
ejpam-5973	98	64	bounds	bound	NOUN
ejpam-5973	98	65	for	for	ADP
ejpam-5973	98	66	hkrd	hkrd	NOUN
ejpam-5973	98	67	-	-	PUNCT
ejpam-5973	98	68	number	number	NOUN
ejpam-5973	98	69	of	of	ADP
ejpam-5973	98	70	a	a	DET
ejpam-5973	98	71	graph	graph	NOUN
ejpam-5973	98	72	g.	g.	NOUN
ejpam-5973	98	73	theorem	theorem	NOUN
ejpam-5973	98	74	1	1	X
ejpam-5973	98	75	.	.	PUNCT
ejpam-5973	99	1	let	let	VERB
ejpam-5973	99	2	k	k	PRON
ejpam-5973	99	3	be	be	AUX
ejpam-5973	99	4	a	a	DET
ejpam-5973	99	5	positive	positive	ADJ
ejpam-5973	99	6	integer	integer	NOUN
ejpam-5973	99	7	,	,	PUNCT
ejpam-5973	99	8	and	and	CCONJ
ejpam-5973	99	9	let	let	VERB
ejpam-5973	99	10	g	g	PRON
ejpam-5973	99	11	be	be	AUX
ejpam-5973	99	12	a	a	DET
ejpam-5973	99	13	graph	graph	NOUN
ejpam-5973	99	14	of	of	ADP
ejpam-5973	99	15	order	order	NOUN
ejpam-5973	99	16	n	n	PROPN
ejpam-5973	99	17	=	=	SYM
ejpam-5973	99	18	n1+n2	n1+n2	PROPN
ejpam-5973	99	19	+	+	PROPN
ejpam-5973	99	20	·	·	PUNCT
ejpam-5973	99	21	·	·	PUNCT
ejpam-5973	99	22	·	·	PUNCT
ejpam-5973	100	1	+	+	NOUN
ejpam-5973	100	2	np	np	X
ejpam-5973	100	3	with	with	ADP
ejpam-5973	100	4	p	p	PROPN
ejpam-5973	100	5	disjoint	disjoint	NOUN
ejpam-5973	100	6	components	component	NOUN
ejpam-5973	100	7	g1	g1	PROPN
ejpam-5973	100	8	,	,	PUNCT
ejpam-5973	100	9	g2	g2	PROPN
ejpam-5973	100	10	,	,	PUNCT
ejpam-5973	100	11	.	.	PUNCT
ejpam-5973	100	12	.	.	PUNCT
ejpam-5973	100	13	.	.	PUNCT
ejpam-5973	101	1	,	,	PUNCT
ejpam-5973	101	2	gp	gp	NOUN
ejpam-5973	101	3	with	with	ADP
ejpam-5973	101	4	n1	n1	NOUN
ejpam-5973	101	5	,	,	PUNCT
ejpam-5973	101	6	n2	n2	NOUN
ejpam-5973	101	7	,	,	PUNCT
ejpam-5973	101	8	.	.	PUNCT
ejpam-5973	101	9	.	.	PUNCT
ejpam-5973	102	1	.	.	PUNCT
ejpam-5973	103	1	,	,	PUNCT
ejpam-5973	103	2	np	np	NOUN
ejpam-5973	103	3	vertices	vertex	NOUN
ejpam-5973	103	4	,	,	PUNCT
ejpam-5973	103	5	respectively	respectively	ADV
ejpam-5973	103	6	.	.	PUNCT
ejpam-5973	104	1	then	then	ADV
ejpam-5973	104	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	104	3	)	)	PUNCT
ejpam-5973	104	4	=	=	PUNCT
ejpam-5973	105	1	p∑	p∑	X
ejpam-5973	105	2	i=1	i=1	PROPN
ejpam-5973	105	3	γhrk(gi	γhrk(gi	PROPN
ejpam-5973	105	4	)	)	PUNCT
ejpam-5973	105	5	.	.	PUNCT
ejpam-5973	106	1	proof	proof	NOUN
ejpam-5973	106	2	.	.	PUNCT
ejpam-5973	107	1	let	let	VERB
ejpam-5973	107	2	g	g	PRON
ejpam-5973	107	3	be	be	AUX
ejpam-5973	107	4	a	a	DET
ejpam-5973	107	5	graph	graph	NOUN
ejpam-5973	107	6	of	of	ADP
ejpam-5973	107	7	order	order	NOUN
ejpam-5973	107	8	n	n	NOUN
ejpam-5973	107	9	=	=	SYM
ejpam-5973	107	10	n1	n1	PROPN
ejpam-5973	107	11	+	+	CCONJ
ejpam-5973	107	12	n2	n2	ADJ
ejpam-5973	107	13	+	+	X
ejpam-5973	107	14	·	·	PUNCT
ejpam-5973	107	15	·	·	PUNCT
ejpam-5973	107	16	·	·	PUNCT
ejpam-5973	108	1	+	+	CCONJ
ejpam-5973	108	2	np	np	INTJ
ejpam-5973	108	3	consisting	consist	VERB
ejpam-5973	108	4	of	of	ADP
ejpam-5973	108	5	p	p	PROPN
ejpam-5973	108	6	disjoint	disjoint	NOUN
ejpam-5973	108	7	components	component	NOUN
ejpam-5973	108	8	g1	g1	PROPN
ejpam-5973	108	9	,	,	PUNCT
ejpam-5973	108	10	g2	g2	PROPN
ejpam-5973	108	11	,	,	PUNCT
ejpam-5973	108	12	.	.	PUNCT
ejpam-5973	108	13	.	.	PUNCT
ejpam-5973	108	14	.	.	PUNCT
ejpam-5973	108	15	,	,	PUNCT
ejpam-5973	108	16	gp	gp	NOUN
ejpam-5973	108	17	,	,	PUNCT
ejpam-5973	108	18	where	where	SCONJ
ejpam-5973	108	19	each	each	DET
ejpam-5973	108	20	gi	gi	NOUN
ejpam-5973	108	21	has	have	VERB
ejpam-5973	108	22	ni	ni	NOUN
ejpam-5973	108	23	vertices	vertex	NOUN
ejpam-5973	108	24	.	.	PUNCT
ejpam-5973	109	1	first	first	ADV
ejpam-5973	109	2	,	,	PUNCT
ejpam-5973	109	3	for	for	ADP
ejpam-5973	109	4	each	each	DET
ejpam-5973	109	5	i	i	PRON
ejpam-5973	109	6	∈	∈	PROPN
ejpam-5973	109	7	{	{	PUNCT
ejpam-5973	109	8	1	1	NUM
ejpam-5973	109	9	,	,	PUNCT
ejpam-5973	109	10	2	2	NUM
ejpam-5973	109	11	,	,	PUNCT
ejpam-5973	109	12	.	.	PUNCT
ejpam-5973	109	13	.	.	PUNCT
ejpam-5973	109	14	.	.	PUNCT
ejpam-5973	110	1	,	,	PUNCT
ejpam-5973	110	2	p	p	X
ejpam-5973	110	3	}	}	PUNCT
ejpam-5973	110	4	,	,	PUNCT
ejpam-5973	110	5	let	let	VERB
ejpam-5973	110	6	fi	fi	NOUN
ejpam-5973	110	7	be	be	AUX
ejpam-5973	110	8	a	a	DET
ejpam-5973	110	9	γhrk	γhrk	NOUN
ejpam-5973	110	10	-	-	PUNCT
ejpam-5973	110	11	function	function	NOUN
ejpam-5973	110	12	of	of	ADP
ejpam-5973	110	13	gi	gi	NOUN
ejpam-5973	110	14	,	,	PUNCT
ejpam-5973	110	15	meaning	mean	VERB
ejpam-5973	110	16	that	that	SCONJ
ejpam-5973	110	17	fi	fi	NOUN
ejpam-5973	110	18	is	be	AUX
ejpam-5973	110	19	a	a	DET
ejpam-5973	110	20	hop	hop	NOUN
ejpam-5973	110	21	k	k	ADJ
ejpam-5973	110	22	-	-	PUNCT
ejpam-5973	110	23	rainbow	rainbow	NOUN
ejpam-5973	110	24	dominating	dominating	NOUN
ejpam-5973	110	25	function	function	NOUN
ejpam-5973	110	26	of	of	ADP
ejpam-5973	110	27	gi	gi	NOUN
ejpam-5973	110	28	of	of	ADP
ejpam-5973	110	29	weight	weight	NOUN
ejpam-5973	110	30	ω(fi	ω(fi	NOUN
ejpam-5973	110	31	)	)	PUNCT
ejpam-5973	110	32	=	=	SYM
ejpam-5973	110	33	γhrk(gi	γhrk(gi	NOUN
ejpam-5973	110	34	)	)	PUNCT
ejpam-5973	110	35	.	.	PUNCT
ejpam-5973	111	1	define	define	VERB
ejpam-5973	111	2	a	a	DET
ejpam-5973	111	3	function	function	NOUN
ejpam-5973	111	4	f	f	NOUN
ejpam-5973	111	5	:	:	PUNCT
ejpam-5973	111	6	v	v	X
ejpam-5973	111	7	(	(	PUNCT
ejpam-5973	111	8	g	g	NOUN
ejpam-5973	111	9	)	)	PUNCT
ejpam-5973	111	10	→	→	SYM
ejpam-5973	112	1	p({1	p({1	PROPN
ejpam-5973	112	2	,	,	PUNCT
ejpam-5973	112	3	2	2	NUM
ejpam-5973	112	4	,	,	PUNCT
ejpam-5973	112	5	.	.	PUNCT
ejpam-5973	112	6	.	.	PUNCT
ejpam-5973	113	1	.	.	PUNCT
ejpam-5973	114	1	,	,	PUNCT
ejpam-5973	114	2	k	k	X
ejpam-5973	114	3	}	}	PUNCT
ejpam-5973	114	4	)	)	PUNCT
ejpam-5973	114	5	by	by	ADP
ejpam-5973	114	6	f(v	f(v	NOUN
ejpam-5973	114	7	)	)	PUNCT
ejpam-5973	114	8	=	=	SYM
ejpam-5973	114	9	fi(v	fi(v	X
ejpam-5973	114	10	)	)	PUNCT
ejpam-5973	114	11	if	if	SCONJ
ejpam-5973	114	12	v	v	NUM
ejpam-5973	114	13	∈	∈	PROPN
ejpam-5973	114	14	v	v	NOUN
ejpam-5973	114	15	(	(	PUNCT
ejpam-5973	114	16	gi	gi	NOUN
ejpam-5973	114	17	)	)	PUNCT
ejpam-5973	114	18	,	,	PUNCT
ejpam-5973	114	19	for	for	ADP
ejpam-5973	114	20	each	each	DET
ejpam-5973	114	21	i	i	PRON
ejpam-5973	114	22	∈	∈	PROPN
ejpam-5973	114	23	{	{	PUNCT
ejpam-5973	114	24	1	1	NUM
ejpam-5973	114	25	,	,	PUNCT
ejpam-5973	114	26	2	2	NUM
ejpam-5973	114	27	,	,	PUNCT
ejpam-5973	114	28	.	.	PUNCT
ejpam-5973	114	29	.	.	PUNCT
ejpam-5973	115	1	.	.	PUNCT
ejpam-5973	116	1	,	,	PUNCT
ejpam-5973	116	2	p	p	X
ejpam-5973	116	3	}	}	PUNCT
ejpam-5973	116	4	.	.	PUNCT
ejpam-5973	117	1	since	since	SCONJ
ejpam-5973	117	2	g1	g1	PROPN
ejpam-5973	117	3	,	,	PUNCT
ejpam-5973	117	4	g2	g2	PROPN
ejpam-5973	117	5	,	,	PUNCT
ejpam-5973	117	6	.	.	PUNCT
ejpam-5973	117	7	.	.	PUNCT
ejpam-5973	117	8	.	.	PUNCT
ejpam-5973	118	1	,	,	PUNCT
ejpam-5973	118	2	gp	gp	NOUN
ejpam-5973	118	3	are	be	AUX
ejpam-5973	118	4	disjoint	disjoint	ADJ
ejpam-5973	118	5	,	,	PUNCT
ejpam-5973	118	6	it	it	PRON
ejpam-5973	118	7	follows	follow	VERB
ejpam-5973	118	8	that	that	SCONJ
ejpam-5973	118	9	f	f	PROPN
ejpam-5973	118	10	is	be	AUX
ejpam-5973	118	11	a	a	DET
ejpam-5973	118	12	hop	hop	NOUN
ejpam-5973	118	13	k	k	ADJ
ejpam-5973	118	14	-	-	PUNCT
ejpam-5973	118	15	rainbow	rainbow	NOUN
ejpam-5973	118	16	dominating	dominating	NOUN
ejpam-5973	118	17	function	function	NOUN
ejpam-5973	118	18	of	of	ADP
ejpam-5973	118	19	g	g	NOUN
ejpam-5973	118	20	of	of	ADP
ejpam-5973	118	21	weight	weight	NOUN
ejpam-5973	118	22	ω(f	ω(f	PUNCT
ejpam-5973	118	23	)	)	PUNCT
ejpam-5973	118	24	=	=	PUNCT
ejpam-5973	119	1	∑p	∑p	PROPN
ejpam-5973	119	2	i=1	i=1	PROPN
ejpam-5973	119	3	ω(fi	ω(fi	NUM
ejpam-5973	119	4	)	)	PUNCT
ejpam-5973	119	5	.	.	PUNCT
ejpam-5973	120	1	by	by	ADP
ejpam-5973	120	2	the	the	DET
ejpam-5973	120	3	minimality	minimality	NOUN
ejpam-5973	120	4	of	of	ADP
ejpam-5973	120	5	γhrk(g	γhrk(g	PROPN
ejpam-5973	120	6	)	)	PUNCT
ejpam-5973	120	7	,	,	PUNCT
ejpam-5973	120	8	we	we	PRON
ejpam-5973	120	9	obtain	obtain	VERB
ejpam-5973	120	10	γhrk(g	γhrk(g	NOUN
ejpam-5973	120	11	)	)	PUNCT
ejpam-5973	120	12	≤	≤	NOUN
ejpam-5973	120	13	ω(f	ω(f	NUM
ejpam-5973	120	14	)	)	PUNCT
ejpam-5973	121	1	=	=	PUNCT
ejpam-5973	122	1	p∑	p∑	X
ejpam-5973	122	2	i=1	i=1	PROPN
ejpam-5973	122	3	γhrk(gi	γhrk(gi	PROPN
ejpam-5973	122	4	)	)	PUNCT
ejpam-5973	122	5	.	.	PUNCT
ejpam-5973	123	1	j.	j.	PROPN
ejpam-5973	123	2	j.	j.	PROPN
ejpam-5973	123	3	hamja	hamja	PROPN
ejpam-5973	123	4	et	et	PROPN
ejpam-5973	123	5	al	al	PROPN
ejpam-5973	123	6	.	.	PUNCT
ejpam-5973	123	7	/	/	SYM
ejpam-5973	123	8	eur	eur	PROPN
ejpam-5973	123	9	.	.	PUNCT
ejpam-5973	124	1	j.	j.	PROPN
ejpam-5973	124	2	pure	pure	PROPN
ejpam-5973	124	3	appl	appl	PROPN
ejpam-5973	124	4	.	.	PROPN
ejpam-5973	124	5	math	math	PROPN
ejpam-5973	124	6	,	,	PUNCT
ejpam-5973	124	7	18	18	NUM
ejpam-5973	124	8	(	(	PUNCT
ejpam-5973	124	9	2	2	NUM
ejpam-5973	124	10	)	)	PUNCT
ejpam-5973	124	11	(	(	PUNCT
ejpam-5973	124	12	2025	2025	NUM
ejpam-5973	124	13	)	)	PUNCT
ejpam-5973	124	14	,	,	PUNCT
ejpam-5973	124	15	5973	5973	NUM
ejpam-5973	124	16	5	5	NUM
ejpam-5973	124	17	of	of	ADP
ejpam-5973	124	18	17	17	NUM
ejpam-5973	124	19	conversely	conversely	ADV
ejpam-5973	124	20	,	,	PUNCT
ejpam-5973	124	21	suppose	suppose	VERB
ejpam-5973	124	22	f	f	X
ejpam-5973	124	23	′	′	NOUN
ejpam-5973	124	24	is	be	AUX
ejpam-5973	124	25	a	a	DET
ejpam-5973	124	26	γhrk	γhrk	NOUN
ejpam-5973	124	27	-	-	PUNCT
ejpam-5973	124	28	function	function	NOUN
ejpam-5973	124	29	of	of	ADP
ejpam-5973	124	30	g.	g.	PROPN
ejpam-5973	124	31	since	since	SCONJ
ejpam-5973	124	32	the	the	DET
ejpam-5973	124	33	components	component	NOUN
ejpam-5973	124	34	of	of	ADP
ejpam-5973	124	35	g	g	PROPN
ejpam-5973	124	36	are	be	AUX
ejpam-5973	124	37	disjoint	disjoint	ADJ
ejpam-5973	124	38	,	,	PUNCT
ejpam-5973	124	39	we	we	PRON
ejpam-5973	124	40	can	can	AUX
ejpam-5973	124	41	define	define	VERB
ejpam-5973	124	42	f	f	NOUN
ejpam-5973	125	1	′	′	NUM
ejpam-5973	126	1	i	i	PRON
ejpam-5973	126	2	as	as	ADP
ejpam-5973	126	3	the	the	DET
ejpam-5973	126	4	restriction	restriction	NOUN
ejpam-5973	126	5	of	of	ADP
ejpam-5973	126	6	f	f	PROPN
ejpam-5973	126	7	′	′	NOUN
ejpam-5973	126	8	to	to	ADP
ejpam-5973	126	9	gi	gi	PROPN
ejpam-5973	126	10	,	,	PUNCT
ejpam-5973	126	11	i.e.	i.e.	X
ejpam-5973	126	12	,	,	PUNCT
ejpam-5973	126	13	f	f	NOUN
ejpam-5973	126	14	′	′	NUM
ejpam-5973	127	1	i	i	PRON
ejpam-5973	127	2	=	=	SYM
ejpam-5973	127	3	f	f	PROPN
ejpam-5973	127	4	′|v	′|v	PROPN
ejpam-5973	127	5	(	(	PUNCT
ejpam-5973	127	6	gi	gi	NOUN
ejpam-5973	127	7	)	)	PUNCT
ejpam-5973	127	8	.	.	PUNCT
ejpam-5973	128	1	since	since	SCONJ
ejpam-5973	128	2	each	each	DET
ejpam-5973	128	3	f	f	NOUN
ejpam-5973	128	4	′	′	INTJ
ejpam-5973	129	1	i	i	PRON
ejpam-5973	129	2	is	be	AUX
ejpam-5973	129	3	a	a	DET
ejpam-5973	129	4	hop	hop	NOUN
ejpam-5973	129	5	krainbow	krainbow	NOUN
ejpam-5973	129	6	dominating	dominate	VERB
ejpam-5973	129	7	function	function	NOUN
ejpam-5973	129	8	of	of	ADP
ejpam-5973	129	9	gi	gi	NOUN
ejpam-5973	129	10	,	,	PUNCT
ejpam-5973	129	11	we	we	PRON
ejpam-5973	129	12	have	have	VERB
ejpam-5973	129	13	ω(f	ω(f	PROPN
ejpam-5973	129	14	′	′	NUM
ejpam-5973	129	15	i	i	PROPN
ejpam-5973	129	16	)	)	PUNCT
ejpam-5973	129	17	≥	≥	PROPN
ejpam-5973	129	18	γhrk(gi	γhrk(gi	NOUN
ejpam-5973	129	19	)	)	PUNCT
ejpam-5973	129	20	.	.	PUNCT
ejpam-5973	130	1	thus	thus	ADV
ejpam-5973	130	2	,	,	PUNCT
ejpam-5973	130	3	ω(f	ω(f	ADJ
ejpam-5973	130	4	′	′	NUM
ejpam-5973	130	5	)	)	PUNCT
ejpam-5973	130	6	=	=	PUNCT
ejpam-5973	131	1	∑p	∑p	ADJ
ejpam-5973	132	1	i=1	i=1	X
ejpam-5973	132	2	ω(f	ω(f	ADJ
ejpam-5973	132	3	′	′	NUM
ejpam-5973	132	4	i	i	NOUN
ejpam-5973	132	5	)	)	PUNCT
ejpam-5973	132	6	≥∑p	≥∑p	VERB
ejpam-5973	132	7	i=1	i=1	PROPN
ejpam-5973	132	8	γhrk(gi	γhrk(gi	PROPN
ejpam-5973	132	9	)	)	PUNCT
ejpam-5973	132	10	.	.	PUNCT
ejpam-5973	133	1	since	since	SCONJ
ejpam-5973	133	2	f	f	PROPN
ejpam-5973	133	3	′	′	NUM
ejpam-5973	133	4	is	be	AUX
ejpam-5973	133	5	an	an	DET
ejpam-5973	133	6	optimal	optimal	ADJ
ejpam-5973	133	7	function	function	NOUN
ejpam-5973	133	8	for	for	ADP
ejpam-5973	133	9	g	g	NOUN
ejpam-5973	133	10	,	,	PUNCT
ejpam-5973	133	11	it	it	PRON
ejpam-5973	133	12	follows	follow	VERB
ejpam-5973	133	13	that	that	SCONJ
ejpam-5973	133	14	γhrk(g	γhrk(g	NOUN
ejpam-5973	133	15	)	)	PUNCT
ejpam-5973	133	16	=	=	SYM
ejpam-5973	134	1	ω(f	ω(f	PROPN
ejpam-5973	134	2	′	′	NUM
ejpam-5973	134	3	)	)	PUNCT
ejpam-5973	134	4	≥	≥	NOUN
ejpam-5973	135	1	p∑	p∑	NOUN
ejpam-5973	136	1	i=1	i=1	PROPN
ejpam-5973	136	2	γhrk(gi	γhrk(gi	PROPN
ejpam-5973	136	3	)	)	PUNCT
ejpam-5973	136	4	.	.	PUNCT
ejpam-5973	137	1	from	from	ADP
ejpam-5973	137	2	the	the	DET
ejpam-5973	137	3	two	two	NUM
ejpam-5973	137	4	inequalities	inequality	NOUN
ejpam-5973	137	5	,	,	PUNCT
ejpam-5973	137	6	we	we	PRON
ejpam-5973	137	7	conclude	conclude	VERB
ejpam-5973	137	8	that	that	PRON
ejpam-5973	137	9	γhrk(g	γhrk(g	NOUN
ejpam-5973	137	10	)	)	PUNCT
ejpam-5973	137	11	=	=	PUNCT
ejpam-5973	138	1	p∑	p∑	X
ejpam-5973	138	2	i=1	i=1	PROPN
ejpam-5973	138	3	γhrk(gi	γhrk(gi	PROPN
ejpam-5973	138	4	)	)	PUNCT
ejpam-5973	138	5	.	.	PUNCT
ejpam-5973	139	1	this	this	PRON
ejpam-5973	139	2	completes	complete	VERB
ejpam-5973	139	3	the	the	DET
ejpam-5973	139	4	proof	proof	NOUN
ejpam-5973	139	5	.	.	PUNCT
ejpam-5973	140	1	thus	thus	ADV
ejpam-5973	140	2	,	,	PUNCT
ejpam-5973	140	3	in	in	ADP
ejpam-5973	140	4	the	the	DET
ejpam-5973	140	5	following	follow	VERB
ejpam-5973	140	6	discussion	discussion	NOUN
ejpam-5973	140	7	,	,	PUNCT
ejpam-5973	140	8	we	we	PRON
ejpam-5973	140	9	consider	consider	VERB
ejpam-5973	140	10	only	only	ADV
ejpam-5973	140	11	connected	connected	ADJ
ejpam-5973	140	12	graphs	graph	NOUN
ejpam-5973	140	13	.	.	PUNCT
ejpam-5973	141	1	for	for	ADP
ejpam-5973	141	2	any	any	DET
ejpam-5973	141	3	graph	graph	NOUN
ejpam-5973	141	4	g	g	NOUN
ejpam-5973	141	5	and	and	CCONJ
ejpam-5973	141	6	a	a	DET
ejpam-5973	141	7	γhrk	γhrk	NOUN
ejpam-5973	141	8	-	-	PUNCT
ejpam-5973	141	9	function	function	NOUN
ejpam-5973	141	10	f	f	NOUN
ejpam-5973	141	11	of	of	ADP
ejpam-5973	141	12	g	g	PROPN
ejpam-5973	141	13	,	,	PUNCT
ejpam-5973	141	14	define	define	VERB
ejpam-5973	141	15	the	the	DET
ejpam-5973	141	16	set	set	NOUN
ejpam-5973	141	17	v	v	NOUN
ejpam-5973	141	18	f	f	NOUN
ejpam-5973	141	19	i	i	PRON
ejpam-5973	141	20	=	=	PUNCT
ejpam-5973	141	21	{	{	PUNCT
ejpam-5973	141	22	x	x	PROPN
ejpam-5973	141	23	∈	∈	PROPN
ejpam-5973	141	24	v	v	NOUN
ejpam-5973	141	25	(	(	PUNCT
ejpam-5973	141	26	g	g	NOUN
ejpam-5973	141	27	)	)	PUNCT
ejpam-5973	141	28	|	|	ADV
ejpam-5973	141	29	|f(x)|	|f(x)|	PROPN
ejpam-5973	141	30	=	=	SYM
ejpam-5973	141	31	i	i	PROPN
ejpam-5973	141	32	}	}	PUNCT
ejpam-5973	141	33	,	,	PUNCT
ejpam-5973	141	34	for	for	ADP
ejpam-5973	141	35	i	i	PRON
ejpam-5973	141	36	∈	∈	PROPN
ejpam-5973	141	37	{	{	PUNCT
ejpam-5973	141	38	0	0	NUM
ejpam-5973	141	39	,	,	PUNCT
ejpam-5973	141	40	1	1	NUM
ejpam-5973	141	41	,	,	PUNCT
ejpam-5973	141	42	.	.	PUNCT
ejpam-5973	141	43	.	.	PUNCT
ejpam-5973	142	1	.	.	PUNCT
ejpam-5973	143	1	,	,	PUNCT
ejpam-5973	143	2	k	k	X
ejpam-5973	143	3	}	}	PUNCT
ejpam-5973	143	4	.	.	PUNCT
ejpam-5973	143	5	example	example	NOUN
ejpam-5973	144	1	2	2	NUM
ejpam-5973	144	2	.	.	X
ejpam-5973	144	3	consider	consider	VERB
ejpam-5973	144	4	the	the	DET
ejpam-5973	144	5	graph	graph	NOUN
ejpam-5973	144	6	g	g	NOUN
ejpam-5973	144	7	in	in	ADP
ejpam-5973	144	8	figure	figure	NOUN
ejpam-5973	144	9	1	1	NUM
ejpam-5973	144	10	and	and	CCONJ
ejpam-5973	144	11	let	let	VERB
ejpam-5973	144	12	f	f	NOUN
ejpam-5973	144	13	:	:	PUNCT
ejpam-5973	144	14	v	v	X
ejpam-5973	144	15	(	(	PUNCT
ejpam-5973	144	16	g	g	NOUN
ejpam-5973	144	17	)	)	PUNCT
ejpam-5973	144	18	→	→	SYM
ejpam-5973	144	19	p({1	p({1	PROPN
ejpam-5973	144	20	,	,	PUNCT
ejpam-5973	144	21	2	2	NUM
ejpam-5973	144	22	}	}	PUNCT
ejpam-5973	144	23	)	)	PUNCT
ejpam-5973	144	24	be	be	AUX
ejpam-5973	144	25	the	the	DET
ejpam-5973	144	26	hop	hop	ADJ
ejpam-5973	144	27	2	2	NUM
ejpam-5973	144	28	-	-	PUNCT
ejpam-5973	144	29	rainbow	rainbow	NOUN
ejpam-5973	144	30	dominating	dominating	NOUN
ejpam-5973	144	31	function	function	NOUN
ejpam-5973	144	32	of	of	ADP
ejpam-5973	144	33	g	g	PROPN
ejpam-5973	144	34	defined	define	VERB
ejpam-5973	144	35	in	in	ADP
ejpam-5973	144	36	example	example	NOUN
ejpam-5973	145	1	1	1	X
ejpam-5973	145	2	.	.	PUNCT
ejpam-5973	146	1	then	then	ADV
ejpam-5973	146	2	,	,	PUNCT
ejpam-5973	146	3	the	the	DET
ejpam-5973	146	4	sets	set	NOUN
ejpam-5973	146	5	of	of	ADP
ejpam-5973	146	6	vertices	vertex	NOUN
ejpam-5973	146	7	corresponding	correspond	VERB
ejpam-5973	146	8	to	to	ADP
ejpam-5973	146	9	each	each	DET
ejpam-5973	146	10	function	function	NOUN
ejpam-5973	146	11	value	value	NOUN
ejpam-5973	146	12	are	be	AUX
ejpam-5973	146	13	as	as	SCONJ
ejpam-5973	146	14	follows	follow	VERB
ejpam-5973	146	15	:	:	PUNCT
ejpam-5973	146	16	v	v	NUM
ejpam-5973	146	17	f	f	PROPN
ejpam-5973	146	18	0	0	PUNCT
ejpam-5973	147	1	=	=	SYM
ejpam-5973	147	2	{	{	PUNCT
ejpam-5973	147	3	v3	v3	PROPN
ejpam-5973	147	4	,	,	PUNCT
ejpam-5973	147	5	v4	v4	PROPN
ejpam-5973	147	6	}	}	PUNCT
ejpam-5973	147	7	,	,	PUNCT
ejpam-5973	147	8	v	v	NOUN
ejpam-5973	147	9	f	f	PROPN
ejpam-5973	147	10	1	1	NUM
ejpam-5973	147	11	=	=	SYM
ejpam-5973	147	12	{	{	PUNCT
ejpam-5973	147	13	v1	v1	PROPN
ejpam-5973	147	14	,	,	PUNCT
ejpam-5973	147	15	v2	v2	PROPN
ejpam-5973	147	16	,	,	PUNCT
ejpam-5973	147	17	v5	v5	PROPN
ejpam-5973	147	18	,	,	PUNCT
ejpam-5973	147	19	v6	v6	NOUN
ejpam-5973	147	20	,	,	PUNCT
ejpam-5973	147	21	v7	v7	NUM
ejpam-5973	147	22	,	,	PUNCT
ejpam-5973	147	23	v8	v8	PROPN
ejpam-5973	147	24	}	}	PUNCT
ejpam-5973	147	25	,	,	PUNCT
ejpam-5973	147	26	v	v	AUX
ejpam-5973	147	27	f	f	PROPN
ejpam-5973	147	28	2	2	NUM
ejpam-5973	147	29	=	=	PUNCT
ejpam-5973	147	30	∅.	∅.	AUX
ejpam-5973	147	31	let	let	VERB
ejpam-5973	147	32	g	g	NOUN
ejpam-5973	147	33	be	be	AUX
ejpam-5973	147	34	a	a	DET
ejpam-5973	147	35	connected	connected	ADJ
ejpam-5973	147	36	graph	graph	NOUN
ejpam-5973	147	37	and	and	CCONJ
ejpam-5973	147	38	consider	consider	VERB
ejpam-5973	147	39	the	the	DET
ejpam-5973	147	40	graph	graph	NOUN
ejpam-5973	147	41	g2	g2	PROPN
ejpam-5973	147	42	whose	whose	DET
ejpam-5973	147	43	vertex	vertex	NOUN
ejpam-5973	147	44	set	set	NOUN
ejpam-5973	147	45	is	be	AUX
ejpam-5973	147	46	v	v	NOUN
ejpam-5973	147	47	(	(	PUNCT
ejpam-5973	147	48	g	g	NOUN
ejpam-5973	147	49	)	)	PUNCT
ejpam-5973	147	50	,	,	PUNCT
ejpam-5973	147	51	where	where	SCONJ
ejpam-5973	147	52	two	two	NUM
ejpam-5973	147	53	vertices	vertice	VERB
ejpam-5973	147	54	u	u	NOUN
ejpam-5973	147	55	and	and	CCONJ
ejpam-5973	147	56	v	v	NOUN
ejpam-5973	147	57	are	be	AUX
ejpam-5973	147	58	adjacent	adjacent	ADJ
ejpam-5973	147	59	in	in	ADP
ejpam-5973	147	60	g2	g2	PROPN
ejpam-5973	147	61	if	if	SCONJ
ejpam-5973	147	62	dg(u	dg(u	NOUN
ejpam-5973	147	63	,	,	PUNCT
ejpam-5973	147	64	v	v	NOUN
ejpam-5973	147	65	)	)	PUNCT
ejpam-5973	147	66	=	=	SYM
ejpam-5973	147	67	2	2	X
ejpam-5973	147	68	.	.	PUNCT
ejpam-5973	147	69	clearly	clearly	ADV
ejpam-5973	147	70	,	,	PUNCT
ejpam-5973	147	71	any	any	DET
ejpam-5973	147	72	krd	krd	NOUN
ejpam-5973	147	73	-	-	PUNCT
ejpam-5973	147	74	function	function	NOUN
ejpam-5973	147	75	of	of	ADP
ejpam-5973	147	76	g2	g2	PROPN
ejpam-5973	147	77	is	be	AUX
ejpam-5973	147	78	an	an	DET
ejpam-5973	147	79	hkrd	hkrd	NOUN
ejpam-5973	147	80	-	-	PUNCT
ejpam-5973	147	81	function	function	NOUN
ejpam-5973	147	82	of	of	ADP
ejpam-5973	147	83	g	g	NOUN
ejpam-5973	147	84	and	and	CCONJ
ejpam-5973	147	85	vice	vice	ADV
ejpam-5973	147	86	versa	versa	ADV
ejpam-5973	147	87	.	.	PUNCT
ejpam-5973	148	1	thus	thus	ADV
ejpam-5973	148	2	,	,	PUNCT
ejpam-5973	148	3	we	we	PRON
ejpam-5973	148	4	have	have	VERB
ejpam-5973	148	5	the	the	DET
ejpam-5973	148	6	following	following	NOUN
ejpam-5973	148	7	.	.	PUNCT
ejpam-5973	149	1	remark	remark	NOUN
ejpam-5973	149	2	1	1	NUM
ejpam-5973	149	3	.	.	PUNCT
ejpam-5973	150	1	for	for	ADP
ejpam-5973	150	2	any	any	DET
ejpam-5973	150	3	graph	graph	NOUN
ejpam-5973	150	4	g	g	NOUN
ejpam-5973	150	5	,	,	PUNCT
ejpam-5973	150	6	γhrk(g	γhrk(g	PROPN
ejpam-5973	150	7	)	)	PUNCT
ejpam-5973	150	8	=	=	SYM
ejpam-5973	151	1	γrk(g	γrk(g	ADP
ejpam-5973	151	2	2	2	NUM
ejpam-5973	151	3	)	)	PUNCT
ejpam-5973	151	4	.	.	PUNCT
ejpam-5973	152	1	remark	remark	NOUN
ejpam-5973	152	2	2	2	NUM
ejpam-5973	152	3	.	.	PUNCT
ejpam-5973	153	1	if	if	SCONJ
ejpam-5973	153	2	g	g	PROPN
ejpam-5973	153	3	is	be	AUX
ejpam-5973	153	4	a	a	DET
ejpam-5973	153	5	graph	graph	NOUN
ejpam-5973	153	6	of	of	ADP
ejpam-5973	153	7	order	order	NOUN
ejpam-5973	153	8	n	n	PRON
ejpam-5973	153	9	≥	≥	NOUN
ejpam-5973	153	10	1	1	NUM
ejpam-5973	153	11	with	with	ADP
ejpam-5973	153	12	δ(g	δ(g	PROPN
ejpam-5973	153	13	)	)	PUNCT
ejpam-5973	153	14	≥	≥	NOUN
ejpam-5973	153	15	1	1	NUM
ejpam-5973	153	16	.	.	PUNCT
ejpam-5973	154	1	let	let	VERB
ejpam-5973	154	2	f	f	PRON
ejpam-5973	154	3	be	be	AUX
ejpam-5973	154	4	a	a	DET
ejpam-5973	154	5	γhrk	γhrk	NOUN
ejpam-5973	154	6	-	-	PUNCT
ejpam-5973	154	7	function	function	NOUN
ejpam-5973	154	8	of	of	ADP
ejpam-5973	154	9	g	g	NOUN
ejpam-5973	154	10	,	,	PUNCT
ejpam-5973	154	11	then	then	ADV
ejpam-5973	154	12	(	(	PUNCT
ejpam-5973	154	13	i	i	NOUN
ejpam-5973	154	14	)	)	PUNCT
ejpam-5973	154	15	n	n	PROPN
ejpam-5973	155	1	=	=	PRON
ejpam-5973	155	2	∑k	∑k	PROPN
ejpam-5973	155	3	j=0|v	j=0|v	VERB
ejpam-5973	155	4	f	f	PROPN
ejpam-5973	155	5	j	j	PROPN
ejpam-5973	156	1	|	|	ADV
ejpam-5973	156	2	,	,	PUNCT
ejpam-5973	156	3	(	(	PUNCT
ejpam-5973	156	4	ii	ii	NOUN
ejpam-5973	156	5	)	)	PUNCT
ejpam-5973	156	6	γhrk(g	γhrk(g	PROPN
ejpam-5973	156	7	)	)	PUNCT
ejpam-5973	156	8	=	=	SYM
ejpam-5973	157	1	∑k	∑k	PROPN
ejpam-5973	157	2	j=1	j=1	PROPN
ejpam-5973	157	3	j|v	j|v	PUNCT
ejpam-5973	158	1	f	f	PROPN
ejpam-5973	158	2	j	j	PROPN
ejpam-5973	158	3	|	|	ADV
ejpam-5973	158	4	,	,	PUNCT
ejpam-5973	158	5	and	and	CCONJ
ejpam-5973	158	6	(	(	PUNCT
ejpam-5973	158	7	iii	iii	X
ejpam-5973	158	8	)	)	PUNCT
ejpam-5973	158	9	|v	|v	PROPN
ejpam-5973	158	10	f	f	PROPN
ejpam-5973	158	11	0	0	NUM
ejpam-5973	159	1	|	|	ADV
ejpam-5973	159	2	≥	≥	NOUN
ejpam-5973	160	1	∑k	∑k	PROPN
ejpam-5973	160	2	j=2(j	j=2(j	PROPN
ejpam-5973	160	3	−	−	PROPN
ejpam-5973	160	4	1)|v	1)|v	NUM
ejpam-5973	160	5	f	f	PROPN
ejpam-5973	160	6	j	j	PROPN
ejpam-5973	160	7	|	|	ADV
ejpam-5973	160	8	.	.	PUNCT
ejpam-5973	161	1	proof	proof	NOUN
ejpam-5973	161	2	.	.	PUNCT
ejpam-5973	162	1	suppose	suppose	VERB
ejpam-5973	162	2	g	g	PROPN
ejpam-5973	162	3	is	be	AUX
ejpam-5973	162	4	a	a	DET
ejpam-5973	162	5	graph	graph	NOUN
ejpam-5973	162	6	of	of	ADP
ejpam-5973	162	7	order	order	NOUN
ejpam-5973	162	8	n	n	PRON
ejpam-5973	162	9	≥	≥	NOUN
ejpam-5973	162	10	1	1	NUM
ejpam-5973	162	11	with	with	ADP
ejpam-5973	162	12	δ(g	δ(g	PROPN
ejpam-5973	162	13	)	)	PUNCT
ejpam-5973	162	14	≥	≥	NOUN
ejpam-5973	162	15	1	1	NUM
ejpam-5973	162	16	,	,	PUNCT
ejpam-5973	162	17	and	and	CCONJ
ejpam-5973	162	18	let	let	VERB
ejpam-5973	162	19	f	f	PRON
ejpam-5973	162	20	be	be	AUX
ejpam-5973	162	21	a	a	DET
ejpam-5973	162	22	γhrk	γhrk	NOUN
ejpam-5973	162	23	-	-	PUNCT
ejpam-5973	162	24	function	function	NOUN
ejpam-5973	162	25	of	of	ADP
ejpam-5973	162	26	g.	g.	PROPN
ejpam-5973	162	27	by	by	ADP
ejpam-5973	162	28	definition	definition	NOUN
ejpam-5973	162	29	,	,	PUNCT
ejpam-5973	162	30	v	v	NOUN
ejpam-5973	163	1	f	f	X
ejpam-5973	163	2	i	i	PRON
ejpam-5973	163	3	=	=	PUNCT
ejpam-5973	163	4	{	{	PUNCT
ejpam-5973	163	5	v	v	NUM
ejpam-5973	163	6	∈	∈	NOUN
ejpam-5973	163	7	v	v	NOUN
ejpam-5973	163	8	(	(	PUNCT
ejpam-5973	163	9	g	g	NOUN
ejpam-5973	163	10	)	)	PUNCT
ejpam-5973	163	11	:	:	PUNCT
ejpam-5973	164	1	|f(v)|	|f(v)|	PROPN
ejpam-5973	164	2	=	=	SYM
ejpam-5973	164	3	i	i	PROPN
ejpam-5973	164	4	}	}	PUNCT
ejpam-5973	164	5	for	for	ADP
ejpam-5973	164	6	each	each	DET
ejpam-5973	164	7	i	i	PRON
ejpam-5973	164	8	∈	∈	PROPN
ejpam-5973	164	9	{	{	PUNCT
ejpam-5973	164	10	0	0	NUM
ejpam-5973	164	11	,	,	PUNCT
ejpam-5973	164	12	1	1	NUM
ejpam-5973	164	13	,	,	PUNCT
ejpam-5973	164	14	.	.	PUNCT
ejpam-5973	164	15	.	.	PUNCT
ejpam-5973	164	16	.	.	PUNCT
ejpam-5973	165	1	,	,	PUNCT
ejpam-5973	165	2	k	k	X
ejpam-5973	165	3	}	}	PUNCT
ejpam-5973	165	4	.	.	PUNCT
ejpam-5973	166	1	since	since	SCONJ
ejpam-5973	166	2	every	every	DET
ejpam-5973	166	3	vertex	vertex	NOUN
ejpam-5973	166	4	belongs	belong	VERB
ejpam-5973	166	5	to	to	ADP
ejpam-5973	166	6	exactly	exactly	ADV
ejpam-5973	166	7	one	one	NUM
ejpam-5973	166	8	of	of	ADP
ejpam-5973	166	9	these	these	DET
ejpam-5973	166	10	sets	set	NOUN
ejpam-5973	166	11	,	,	PUNCT
ejpam-5973	166	12	we	we	PRON
ejpam-5973	166	13	have	have	VERB
ejpam-5973	166	14	n	n	NOUN
ejpam-5973	167	1	=	=	PUNCT
ejpam-5973	167	2	∑k	∑k	PROPN
ejpam-5973	167	3	j=0|v	j=0|v	VERB
ejpam-5973	167	4	f	f	PROPN
ejpam-5973	167	5	j	j	PROPN
ejpam-5973	168	1	|	|	ADV
ejpam-5973	168	2	.	.	PUNCT
ejpam-5973	169	1	hence	hence	ADV
ejpam-5973	169	2	,	,	PUNCT
ejpam-5973	169	3	(	(	PUNCT
ejpam-5973	169	4	i	i	NOUN
ejpam-5973	169	5	)	)	PUNCT
ejpam-5973	169	6	holds	hold	VERB
ejpam-5973	169	7	.	.	PUNCT
ejpam-5973	170	1	since	since	SCONJ
ejpam-5973	170	2	ω(f	ω(f	NUM
ejpam-5973	170	3	)	)	PUNCT
ejpam-5973	170	4	=	=	SYM
ejpam-5973	170	5	∑	∑	PUNCT
ejpam-5973	170	6	v∈v	v∈v	NOUN
ejpam-5973	170	7	(	(	PUNCT
ejpam-5973	170	8	g	g	NOUN
ejpam-5973	170	9	)	)	PUNCT
ejpam-5973	170	10	|f(v)|	|f(v)|	PROPN
ejpam-5973	170	11	,	,	PUNCT
ejpam-5973	170	12	we	we	PRON
ejpam-5973	170	13	can	can	AUX
ejpam-5973	170	14	rewrite	rewrite	VERB
ejpam-5973	170	15	this	this	DET
ejpam-5973	170	16	sum	sum	NOUN
ejpam-5973	170	17	in	in	ADP
ejpam-5973	170	18	terms	term	NOUN
ejpam-5973	170	19	of	of	ADP
ejpam-5973	170	20	v	v	NOUN
ejpam-5973	170	21	f	f	PROPN
ejpam-5973	170	22	j	j	PROPN
ejpam-5973	170	23	by	by	ADP
ejpam-5973	170	24	γhrk(g	γhrk(g	PROPN
ejpam-5973	170	25	)	)	PUNCT
ejpam-5973	170	26	=	=	PUNCT
ejpam-5973	171	1	∑k	∑k	PROPN
ejpam-5973	171	2	j=1	j=1	PROPN
ejpam-5973	171	3	j|v	j|v	PUNCT
ejpam-5973	172	1	f	f	PROPN
ejpam-5973	172	2	j	j	PROPN
ejpam-5973	172	3	|	|	ADV
ejpam-5973	172	4	.	.	PUNCT
ejpam-5973	173	1	therefore	therefore	ADV
ejpam-5973	173	2	,	,	PUNCT
ejpam-5973	173	3	(	(	PUNCT
ejpam-5973	173	4	ii	ii	NOUN
ejpam-5973	173	5	)	)	PUNCT
ejpam-5973	173	6	holds	hold	VERB
ejpam-5973	173	7	.	.	PUNCT
ejpam-5973	174	1	let	let	VERB
ejpam-5973	174	2	v	v	PART
ejpam-5973	174	3	be	be	AUX
ejpam-5973	174	4	any	any	DET
ejpam-5973	174	5	vertex	vertex	NOUN
ejpam-5973	174	6	in	in	ADP
ejpam-5973	174	7	g	g	NOUN
ejpam-5973	174	8	with	with	ADP
ejpam-5973	174	9	f(v	f(v	NOUN
ejpam-5973	174	10	)	)	PUNCT
ejpam-5973	175	1	=	=	PUNCT
ejpam-5973	175	2	∅.	∅.	NOUN
ejpam-5973	175	3	then	then	ADV
ejpam-5973	175	4	⋃	⋃	NOUN
ejpam-5973	175	5	u∈n2	u∈n2	ADJ
ejpam-5973	175	6	g(v	g(v	PROPN
ejpam-5973	175	7	)	)	PUNCT
ejpam-5973	175	8	f(u	f(u	PROPN
ejpam-5973	175	9	)	)	PUNCT
ejpam-5973	175	10	=	=	PUNCT
ejpam-5973	176	1	j.	j.	PROPN
ejpam-5973	176	2	j.	j.	PROPN
ejpam-5973	176	3	hamja	hamja	PROPN
ejpam-5973	176	4	et	et	PROPN
ejpam-5973	176	5	al	al	PROPN
ejpam-5973	176	6	.	.	PUNCT
ejpam-5973	176	7	/	/	SYM
ejpam-5973	176	8	eur	eur	PROPN
ejpam-5973	176	9	.	.	PUNCT
ejpam-5973	177	1	j.	j.	PROPN
ejpam-5973	177	2	pure	pure	PROPN
ejpam-5973	177	3	appl	appl	PROPN
ejpam-5973	177	4	.	.	PROPN
ejpam-5973	177	5	math	math	PROPN
ejpam-5973	177	6	,	,	PUNCT
ejpam-5973	177	7	18	18	NUM
ejpam-5973	177	8	(	(	PUNCT
ejpam-5973	177	9	2	2	NUM
ejpam-5973	177	10	)	)	PUNCT
ejpam-5973	177	11	(	(	PUNCT
ejpam-5973	177	12	2025	2025	NUM
ejpam-5973	177	13	)	)	PUNCT
ejpam-5973	177	14	,	,	PUNCT
ejpam-5973	177	15	5973	5973	NUM
ejpam-5973	177	16	6	6	NUM
ejpam-5973	177	17	of	of	ADP
ejpam-5973	177	18	17	17	NUM
ejpam-5973	177	19	{	{	PUNCT
ejpam-5973	177	20	1	1	NUM
ejpam-5973	177	21	,	,	PUNCT
ejpam-5973	177	22	2	2	NUM
ejpam-5973	177	23	,	,	PUNCT
ejpam-5973	177	24	.	.	PUNCT
ejpam-5973	177	25	.	.	PUNCT
ejpam-5973	177	26	.	.	PUNCT
ejpam-5973	178	1	,	,	PUNCT
ejpam-5973	178	2	k	k	X
ejpam-5973	178	3	}	}	PUNCT
ejpam-5973	178	4	.	.	PUNCT
ejpam-5973	179	1	this	this	PRON
ejpam-5973	179	2	implies	imply	VERB
ejpam-5973	179	3	that	that	SCONJ
ejpam-5973	179	4	vertices	vertice	VERB
ejpam-5973	179	5	in	in	ADP
ejpam-5973	179	6	v	v	PROPN
ejpam-5973	179	7	f	f	PROPN
ejpam-5973	179	8	0	0	NUM
ejpam-5973	179	9	rely	rely	VERB
ejpam-5973	179	10	on	on	ADP
ejpam-5973	179	11	vertices	vertex	NOUN
ejpam-5973	179	12	in	in	ADP
ejpam-5973	179	13	v	v	PROPN
ejpam-5973	179	14	f	f	PROPN
ejpam-5973	179	15	j	j	PROPN
ejpam-5973	179	16	for	for	ADP
ejpam-5973	179	17	j	j	PROPN
ejpam-5973	179	18	≥	≥	PROPN
ejpam-5973	179	19	2	2	NUM
ejpam-5973	179	20	to	to	PART
ejpam-5973	179	21	contribute	contribute	VERB
ejpam-5973	179	22	enough	enough	ADJ
ejpam-5973	179	23	colors	color	NOUN
ejpam-5973	179	24	.	.	PUNCT
ejpam-5973	180	1	since	since	SCONJ
ejpam-5973	180	2	each	each	DET
ejpam-5973	180	3	vertex	vertex	NOUN
ejpam-5973	180	4	in	in	ADP
ejpam-5973	180	5	v	v	PROPN
ejpam-5973	180	6	f	f	PROPN
ejpam-5973	180	7	j	j	PROPN
ejpam-5973	180	8	contributes	contribute	VERB
ejpam-5973	180	9	j	j	PROPN
ejpam-5973	180	10	colors	color	NOUN
ejpam-5973	180	11	,	,	PUNCT
ejpam-5973	180	12	but	but	CCONJ
ejpam-5973	180	13	at	at	ADV
ejpam-5973	180	14	least	least	ADV
ejpam-5973	180	15	one	one	NUM
ejpam-5973	180	16	color	color	NOUN
ejpam-5973	180	17	is	be	AUX
ejpam-5973	180	18	required	require	VERB
ejpam-5973	180	19	per	per	ADP
ejpam-5973	180	20	vertex	vertex	NOUN
ejpam-5973	180	21	in	in	ADP
ejpam-5973	180	22	v	v	NUM
ejpam-5973	180	23	f	f	NOUN
ejpam-5973	180	24	0	0	NUM
ejpam-5973	180	25	,	,	PUNCT
ejpam-5973	180	26	it	it	PRON
ejpam-5973	180	27	follows	follow	VERB
ejpam-5973	180	28	that	that	SCONJ
ejpam-5973	180	29	|v	|v	PROPN
ejpam-5973	180	30	f	f	PROPN
ejpam-5973	180	31	0	0	NUM
ejpam-5973	181	1	|	|	ADV
ejpam-5973	181	2	≥	≥	NOUN
ejpam-5973	182	1	∑k	∑k	PROPN
ejpam-5973	182	2	j=2(j	j=2(j	PROPN
ejpam-5973	182	3	−	−	PROPN
ejpam-5973	182	4	1)|v	1)|v	NUM
ejpam-5973	182	5	f	f	PROPN
ejpam-5973	182	6	j	j	PROPN
ejpam-5973	182	7	|	|	ADV
ejpam-5973	182	8	.	.	PUNCT
ejpam-5973	183	1	this	this	PRON
ejpam-5973	183	2	proves	prove	VERB
ejpam-5973	183	3	(	(	PUNCT
ejpam-5973	183	4	iii	iii	NOUN
ejpam-5973	183	5	)	)	PUNCT
ejpam-5973	183	6	.	.	PUNCT
ejpam-5973	184	1	the	the	DET
ejpam-5973	184	2	proof	proof	NOUN
ejpam-5973	184	3	is	be	AUX
ejpam-5973	184	4	complete	complete	ADJ
ejpam-5973	184	5	.	.	PUNCT
ejpam-5973	185	1	theorem	theorem	NOUN
ejpam-5973	185	2	2	2	NUM
ejpam-5973	185	3	.	.	PUNCT
ejpam-5973	186	1	let	let	VERB
ejpam-5973	186	2	k	k	PRON
ejpam-5973	186	3	be	be	AUX
ejpam-5973	186	4	a	a	DET
ejpam-5973	186	5	positive	positive	ADJ
ejpam-5973	186	6	integer	integer	NOUN
ejpam-5973	186	7	and	and	CCONJ
ejpam-5973	186	8	g	g	PROPN
ejpam-5973	186	9	be	be	AUX
ejpam-5973	186	10	a	a	DET
ejpam-5973	186	11	graph	graph	NOUN
ejpam-5973	186	12	of	of	ADP
ejpam-5973	186	13	order	order	NOUN
ejpam-5973	186	14	n	n	NOUN
ejpam-5973	186	15	=	=	SYM
ejpam-5973	186	16	n1	n1	PROPN
ejpam-5973	186	17	+	+	NOUN
ejpam-5973	186	18	n2	n2	ADJ
ejpam-5973	186	19	+	+	X
ejpam-5973	186	20	·	·	PUNCT
ejpam-5973	186	21	·	·	PUNCT
ejpam-5973	187	1	·	·	PUNCT
ejpam-5973	187	2	+	+	NOUN
ejpam-5973	187	3	np	np	INTJ
ejpam-5973	187	4	.	.	PUNCT
ejpam-5973	188	1	if	if	SCONJ
ejpam-5973	188	2	g	g	PROPN
ejpam-5973	188	3	is	be	AUX
ejpam-5973	188	4	a	a	DET
ejpam-5973	188	5	disjoint	disjoint	ADJ
ejpam-5973	188	6	union	union	NOUN
ejpam-5973	188	7	of	of	ADP
ejpam-5973	188	8	cliques	clique	NOUN
ejpam-5973	188	9	and	and	CCONJ
ejpam-5973	188	10	isolates	isolate	NOUN
ejpam-5973	188	11	,	,	PUNCT
ejpam-5973	188	12	then	then	ADV
ejpam-5973	188	13	γhrk(g	γhrk(g	PROPN
ejpam-5973	188	14	)	)	PUNCT
ejpam-5973	188	15	=	=	SYM
ejpam-5973	188	16	n.	n.	NOUN
ejpam-5973	188	17	proof	proof	NOUN
ejpam-5973	188	18	.	.	PUNCT
ejpam-5973	189	1	assume	assume	VERB
ejpam-5973	189	2	that	that	SCONJ
ejpam-5973	189	3	g	g	PROPN
ejpam-5973	189	4	is	be	AUX
ejpam-5973	189	5	a	a	DET
ejpam-5973	189	6	disjoint	disjoint	ADJ
ejpam-5973	189	7	union	union	NOUN
ejpam-5973	189	8	of	of	ADP
ejpam-5973	189	9	cliques	clique	NOUN
ejpam-5973	189	10	and	and	CCONJ
ejpam-5973	189	11	isolates	isolate	NOUN
ejpam-5973	189	12	.	.	PUNCT
ejpam-5973	190	1	let	let	VERB
ejpam-5973	190	2	g	g	PROPN
ejpam-5973	190	3	=	=	PROPN
ejpam-5973	190	4	⋃p	⋃p	PROPN
ejpam-5973	190	5	i=1gi	i=1gi	NOUN
ejpam-5973	190	6	,	,	PUNCT
ejpam-5973	190	7	where	where	SCONJ
ejpam-5973	190	8	gi	gi	NOUN
ejpam-5973	190	9	=	=	SYM
ejpam-5973	190	10	kr	kr	PROPN
ejpam-5973	190	11	for	for	ADP
ejpam-5973	190	12	some	some	DET
ejpam-5973	190	13	r	r	NOUN
ejpam-5973	190	14	≥	≥	NOUN
ejpam-5973	190	15	1	1	NUM
ejpam-5973	190	16	.	.	PUNCT
ejpam-5973	190	17	by	by	ADP
ejpam-5973	190	18	theorem	theorem	NOUN
ejpam-5973	190	19	1	1	NUM
ejpam-5973	190	20	,	,	PUNCT
ejpam-5973	190	21	γhrk(g	γhrk(g	PROPN
ejpam-5973	190	22	)	)	PUNCT
ejpam-5973	190	23	=	=	PUNCT
ejpam-5973	191	1	∑p	∑p	PROPN
ejpam-5973	191	2	i=1	i=1	PROPN
ejpam-5973	191	3	γhrk(gi	γhrk(gi	NOUN
ejpam-5973	191	4	)	)	PUNCT
ejpam-5973	191	5	.	.	PUNCT
ejpam-5973	192	1	therefore	therefore	ADV
ejpam-5973	192	2	,	,	PUNCT
ejpam-5973	192	3	we	we	PRON
ejpam-5973	192	4	have	have	VERB
ejpam-5973	192	5	γhrk(g	γhrk(g	NOUN
ejpam-5973	192	6	)	)	PUNCT
ejpam-5973	193	1	=	=	PUNCT
ejpam-5973	194	1	p∑	p∑	X
ejpam-5973	194	2	i=1	i=1	PROPN
ejpam-5973	194	3	γhrk(gi	γhrk(gi	NOUN
ejpam-5973	194	4	)	)	PUNCT
ejpam-5973	194	5	=	=	PUNCT
ejpam-5973	195	1	p∑	p∑	X
ejpam-5973	196	1	i=1	i=1	PROPN
ejpam-5973	196	2	|v	|v	PROPN
ejpam-5973	196	3	(	(	PUNCT
ejpam-5973	196	4	gi)|	gi)|	X
ejpam-5973	196	5	=	=	SYM
ejpam-5973	196	6	n1	n1	PROPN
ejpam-5973	196	7	+	+	CCONJ
ejpam-5973	196	8	n2	n2	ADJ
ejpam-5973	196	9	+	+	X
ejpam-5973	196	10	·	·	PUNCT
ejpam-5973	196	11	·	·	PUNCT
ejpam-5973	196	12	·	·	PUNCT
ejpam-5973	196	13	+	+	NUM
ejpam-5973	196	14	np	np	X
ejpam-5973	196	15	=	=	PUNCT
ejpam-5973	196	16	n.	n.	NOUN
ejpam-5973	196	17	the	the	DET
ejpam-5973	196	18	next	next	ADJ
ejpam-5973	196	19	result	result	NOUN
ejpam-5973	196	20	is	be	AUX
ejpam-5973	196	21	a	a	DET
ejpam-5973	196	22	direct	direct	ADJ
ejpam-5973	196	23	consequence	consequence	NOUN
ejpam-5973	196	24	of	of	ADP
ejpam-5973	196	25	theorem	theorem	ADJ
ejpam-5973	196	26	2	2	NUM
ejpam-5973	196	27	.	.	PUNCT
ejpam-5973	196	28	corollary	corollary	ADJ
ejpam-5973	196	29	1	1	NUM
ejpam-5973	196	30	.	.	PUNCT
ejpam-5973	197	1	let	let	VERB
ejpam-5973	197	2	k	k	NOUN
ejpam-5973	197	3	and	and	CCONJ
ejpam-5973	197	4	n	n	CCONJ
ejpam-5973	197	5	be	be	VERB
ejpam-5973	197	6	positive	positive	ADJ
ejpam-5973	197	7	integers	integer	NOUN
ejpam-5973	197	8	.	.	PUNCT
ejpam-5973	198	1	then	then	ADV
ejpam-5973	198	2	γhrk(kn	γhrk(kn	NOUN
ejpam-5973	198	3	)	)	PUNCT
ejpam-5973	198	4	=	=	SYM
ejpam-5973	198	5	γhrk(kn	γhrk(kn	NOUN
ejpam-5973	198	6	)	)	PUNCT
ejpam-5973	198	7	=	=	SYM
ejpam-5973	199	1	n.	n.	NOUN
ejpam-5973	199	2	theorem	theorem	NOUN
ejpam-5973	199	3	3	3	X
ejpam-5973	199	4	.	.	PUNCT
ejpam-5973	200	1	let	let	VERB
ejpam-5973	200	2	k	k	PROPN
ejpam-5973	200	3	≥	≥	NUM
ejpam-5973	200	4	2	2	NUM
ejpam-5973	200	5	be	be	AUX
ejpam-5973	200	6	an	an	DET
ejpam-5973	200	7	integer	integer	NOUN
ejpam-5973	200	8	and	and	CCONJ
ejpam-5973	200	9	let	let	VERB
ejpam-5973	200	10	g	g	PRON
ejpam-5973	200	11	be	be	AUX
ejpam-5973	200	12	a	a	DET
ejpam-5973	200	13	connected	connected	ADJ
ejpam-5973	200	14	graph	graph	NOUN
ejpam-5973	200	15	of	of	ADP
ejpam-5973	200	16	order	order	NOUN
ejpam-5973	200	17	n	n	PRON
ejpam-5973	200	18	≥	≥	NOUN
ejpam-5973	200	19	k	k	NOUN
ejpam-5973	201	1	+	+	CCONJ
ejpam-5973	201	2	1	1	X
ejpam-5973	201	3	.	.	X
ejpam-5973	201	4	then	then	ADV
ejpam-5973	201	5	γhrk(g	γhrk(g	PROPN
ejpam-5973	201	6	)	)	PUNCT
ejpam-5973	201	7	≥	≥	NOUN
ejpam-5973	201	8	k	k	X
ejpam-5973	202	1	+	+	CCONJ
ejpam-5973	202	2	1	1	X
ejpam-5973	202	3	.	.	PUNCT
ejpam-5973	202	4	moreover	moreover	ADV
ejpam-5973	202	5	,	,	PUNCT
ejpam-5973	202	6	the	the	DET
ejpam-5973	202	7	bound	bind	VERB
ejpam-5973	202	8	is	be	AUX
ejpam-5973	202	9	sharp	sharp	ADJ
ejpam-5973	202	10	for	for	ADP
ejpam-5973	202	11	stars	star	NOUN
ejpam-5973	202	12	k1,m	k1,m	PROPN
ejpam-5973	202	13	(	(	PUNCT
ejpam-5973	202	14	m	m	PROPN
ejpam-5973	202	15	≥	≥	NOUN
ejpam-5973	202	16	k	k	NOUN
ejpam-5973	202	17	)	)	PUNCT
ejpam-5973	202	18	.	.	PUNCT
ejpam-5973	203	1	proof	proof	NOUN
ejpam-5973	203	2	.	.	PUNCT
ejpam-5973	204	1	since	since	SCONJ
ejpam-5973	204	2	g	g	PROPN
ejpam-5973	204	3	is	be	AUX
ejpam-5973	204	4	a	a	DET
ejpam-5973	204	5	connected	connected	ADJ
ejpam-5973	204	6	graph	graph	NOUN
ejpam-5973	204	7	of	of	ADP
ejpam-5973	204	8	order	order	NOUN
ejpam-5973	204	9	n	n	PRON
ejpam-5973	204	10	≥	≥	NOUN
ejpam-5973	204	11	k	k	NOUN
ejpam-5973	205	1	+	+	CCONJ
ejpam-5973	205	2	1	1	NUM
ejpam-5973	205	3	,	,	PUNCT
ejpam-5973	205	4	we	we	PRON
ejpam-5973	205	5	assume	assume	VERB
ejpam-5973	205	6	γhrk(g	γhrk(g	ADP
ejpam-5973	205	7	)	)	PUNCT
ejpam-5973	206	1	=	=	VERB
ejpam-5973	206	2	k.	k.	PROPN
ejpam-5973	206	3	let	let	VERB
ejpam-5973	206	4	s	s	NOUN
ejpam-5973	206	5	=	=	NOUN
ejpam-5973	206	6	v	v	ADJ
ejpam-5973	206	7	(	(	PUNCT
ejpam-5973	206	8	g	g	NOUN
ejpam-5973	206	9	)	)	PUNCT
ejpam-5973	206	10	\	\	PROPN
ejpam-5973	206	11	v	v	ADP
ejpam-5973	206	12	g	g	NOUN
ejpam-5973	206	13	0	0	NUM
ejpam-5973	206	14	,	,	PUNCT
ejpam-5973	206	15	that	that	PRON
ejpam-5973	206	16	is	be	AUX
ejpam-5973	206	17	the	the	DET
ejpam-5973	206	18	set	set	NOUN
ejpam-5973	206	19	of	of	ADP
ejpam-5973	206	20	vertices	vertex	NOUN
ejpam-5973	206	21	assigned	assign	VERB
ejpam-5973	206	22	at	at	ADP
ejpam-5973	206	23	least	least	ADV
ejpam-5973	206	24	one	one	NUM
ejpam-5973	206	25	color	color	NOUN
ejpam-5973	206	26	.	.	PUNCT
ejpam-5973	207	1	given	give	VERB
ejpam-5973	207	2	n	n	PRON
ejpam-5973	207	3	≥	≥	NOUN
ejpam-5973	207	4	k	k	NOUN
ejpam-5973	208	1	+	+	CCONJ
ejpam-5973	208	2	1	1	NUM
ejpam-5973	208	3	,	,	PUNCT
ejpam-5973	208	4	it	it	PRON
ejpam-5973	208	5	follows	follow	VERB
ejpam-5973	208	6	that	that	SCONJ
ejpam-5973	208	7	v	v	ADP
ejpam-5973	208	8	g	g	NOUN
ejpam-5973	208	9	0	0	NUM
ejpam-5973	208	10	̸=	̸=	PROPN
ejpam-5973	208	11	∅.	∅.	ADV
ejpam-5973	208	12	now	now	ADV
ejpam-5973	208	13	,	,	PUNCT
ejpam-5973	208	14	consider	consider	VERB
ejpam-5973	208	15	any	any	DET
ejpam-5973	208	16	x	x	SYM
ejpam-5973	208	17	∈	∈	PROPN
ejpam-5973	208	18	v	v	ADP
ejpam-5973	208	19	g	g	NOUN
ejpam-5973	208	20	0	0	NUM
ejpam-5973	208	21	and	and	CCONJ
ejpam-5973	208	22	any	any	DET
ejpam-5973	208	23	y	y	PROPN
ejpam-5973	208	24	∈	∈	PROPN
ejpam-5973	208	25	s.	s.	PROPN
ejpam-5973	208	26	since	since	SCONJ
ejpam-5973	208	27	x	x	PRON
ejpam-5973	208	28	is	be	AUX
ejpam-5973	208	29	not	not	PART
ejpam-5973	208	30	assigned	assign	VERB
ejpam-5973	208	31	a	a	DET
ejpam-5973	208	32	color	color	NOUN
ejpam-5973	208	33	,	,	PUNCT
ejpam-5973	208	34	there	there	PRON
ejpam-5973	208	35	exist	exist	VERB
ejpam-5973	208	36	a	a	DET
ejpam-5973	208	37	vertex	vertex	NOUN
ejpam-5973	208	38	z	z	NOUN
ejpam-5973	208	39	such	such	ADJ
ejpam-5973	208	40	that	that	SCONJ
ejpam-5973	208	41	z	z	PROPN
ejpam-5973	208	42	∈	∈	PROPN
ejpam-5973	208	43	ng(x)∩ng(y	ng(x)∩ng(y	PROPN
ejpam-5973	208	44	)	)	PUNCT
ejpam-5973	208	45	,	,	PUNCT
ejpam-5973	208	46	ensuring	ensure	VERB
ejpam-5973	208	47	x	x	PUNCT
ejpam-5973	208	48	is	be	AUX
ejpam-5973	208	49	at	at	ADP
ejpam-5973	208	50	distance	distance	NOUN
ejpam-5973	208	51	2	2	NUM
ejpam-5973	208	52	from	from	ADP
ejpam-5973	208	53	y.	y.	PROPN
ejpam-5973	208	54	however	however	ADV
ejpam-5973	208	55	,	,	PUNCT
ejpam-5973	208	56	since	since	SCONJ
ejpam-5973	208	57	x	x	PRON
ejpam-5973	208	58	is	be	AUX
ejpam-5973	208	59	at	at	ADP
ejpam-5973	208	60	distance	distance	NOUN
ejpam-5973	208	61	2	2	NUM
ejpam-5973	208	62	from	from	ADP
ejpam-5973	208	63	every	every	DET
ejpam-5973	208	64	vertex	vertex	NOUN
ejpam-5973	208	65	in	in	ADP
ejpam-5973	208	66	s	s	PROPN
ejpam-5973	208	67	,	,	PUNCT
ejpam-5973	208	68	it	it	PRON
ejpam-5973	208	69	follows	follow	VERB
ejpam-5973	208	70	that	that	PRON
ejpam-5973	208	71	z	z	NOUN
ejpam-5973	208	72	/∈	/∈	PUNCT
ejpam-5973	209	1	s	s	X
ejpam-5973	209	2	,	,	PUNCT
ejpam-5973	209	3	meaning	mean	VERB
ejpam-5973	209	4	z	z	PROPN
ejpam-5973	209	5	∈	∈	PROPN
ejpam-5973	209	6	v	v	ADP
ejpam-5973	209	7	g	g	PROPN
ejpam-5973	209	8	0	0	NUM
ejpam-5973	209	9	.	.	PUNCT
ejpam-5973	210	1	but	but	CCONJ
ejpam-5973	210	2	then	then	ADV
ejpam-5973	210	3	,	,	PUNCT
ejpam-5973	210	4	z	z	PROPN
ejpam-5973	210	5	would	would	AUX
ejpam-5973	210	6	also	also	ADV
ejpam-5973	210	7	need	need	VERB
ejpam-5973	210	8	to	to	PART
ejpam-5973	210	9	be	be	AUX
ejpam-5973	210	10	assigned	assign	VERB
ejpam-5973	210	11	the	the	DET
ejpam-5973	210	12	set	set	NOUN
ejpam-5973	210	13	of	of	ADP
ejpam-5973	210	14	colors	color	NOUN
ejpam-5973	210	15	from	from	ADP
ejpam-5973	210	16	some	some	DET
ejpam-5973	210	17	vertex	vertex	NOUN
ejpam-5973	210	18	in	in	ADP
ejpam-5973	210	19	s	s	PROPN
ejpam-5973	210	20	,	,	PUNCT
ejpam-5973	210	21	contradicting	contradict	VERB
ejpam-5973	210	22	the	the	DET
ejpam-5973	210	23	assumption	assumption	NOUN
ejpam-5973	210	24	that	that	SCONJ
ejpam-5973	210	25	γhrk(g	γhrk(g	NOUN
ejpam-5973	210	26	)	)	PUNCT
ejpam-5973	210	27	=	=	SYM
ejpam-5973	211	1	k.	k.	PROPN
ejpam-5973	212	1	therefore	therefore	ADV
ejpam-5973	212	2	,	,	PUNCT
ejpam-5973	212	3	γhrk(g	γhrk(g	PROPN
ejpam-5973	212	4	)	)	PUNCT
ejpam-5973	212	5	≥	≥	NOUN
ejpam-5973	212	6	k	k	X
ejpam-5973	213	1	+	+	CCONJ
ejpam-5973	213	2	1	1	X
ejpam-5973	213	3	.	.	X
ejpam-5973	213	4	theorem	theorem	NOUN
ejpam-5973	213	5	4	4	NUM
ejpam-5973	213	6	.	.	PUNCT
ejpam-5973	214	1	let	let	VERB
ejpam-5973	214	2	k	k	PROPN
ejpam-5973	214	3	≥	≥	NUM
ejpam-5973	214	4	1	1	NUM
ejpam-5973	214	5	be	be	AUX
ejpam-5973	214	6	an	an	DET
ejpam-5973	214	7	integer	integer	NOUN
ejpam-5973	214	8	and	and	CCONJ
ejpam-5973	214	9	g	g	PROPN
ejpam-5973	214	10	be	be	AUX
ejpam-5973	214	11	a	a	DET
ejpam-5973	214	12	graph	graph	NOUN
ejpam-5973	214	13	of	of	ADP
ejpam-5973	214	14	order	order	NOUN
ejpam-5973	214	15	n	n	PRON
ejpam-5973	214	16	≥	≥	NOUN
ejpam-5973	214	17	1	1	NUM
ejpam-5973	214	18	.	.	PUNCT
ejpam-5973	215	1	then	then	ADV
ejpam-5973	215	2	min{n	min{n	PROPN
ejpam-5973	215	3	,	,	PUNCT
ejpam-5973	215	4	k	k	PROPN
ejpam-5973	215	5	+	+	PROPN
ejpam-5973	215	6	1	1	NUM
ejpam-5973	215	7	}	}	PUNCT
ejpam-5973	215	8	≤	≤	NUM
ejpam-5973	215	9	γhrk(g	γhrk(g	PROPN
ejpam-5973	215	10	)	)	PUNCT
ejpam-5973	215	11	≤	≤	NOUN
ejpam-5973	215	12	n.	n.	NOUN
ejpam-5973	215	13	in	in	ADP
ejpam-5973	215	14	particular	particular	ADJ
ejpam-5973	215	15	,	,	PUNCT
ejpam-5973	215	16	if	if	SCONJ
ejpam-5973	215	17	1	1	NUM
ejpam-5973	215	18	≤	≤	NUM
ejpam-5973	215	19	n	n	PRON
ejpam-5973	215	20	≤	≤	NOUN
ejpam-5973	215	21	k	k	PROPN
ejpam-5973	216	1	+	+	CCONJ
ejpam-5973	216	2	1	1	NUM
ejpam-5973	216	3	,	,	PUNCT
ejpam-5973	216	4	then	then	ADV
ejpam-5973	216	5	γhrk(g	γhrk(g	PROPN
ejpam-5973	216	6	)	)	PUNCT
ejpam-5973	216	7	=	=	SYM
ejpam-5973	217	1	n.	n.	NOUN
ejpam-5973	217	2	proof	proof	NOUN
ejpam-5973	217	3	.	.	PUNCT
ejpam-5973	218	1	if	if	SCONJ
ejpam-5973	218	2	n	n	NUM
ejpam-5973	218	3	≥	≥	X
ejpam-5973	218	4	k	k	NOUN
ejpam-5973	219	1	+	+	CCONJ
ejpam-5973	219	2	1	1	NUM
ejpam-5973	219	3	,	,	PUNCT
ejpam-5973	219	4	then	then	ADV
ejpam-5973	219	5	it	it	PRON
ejpam-5973	219	6	follows	follow	VERB
ejpam-5973	219	7	from	from	ADP
ejpam-5973	219	8	theorem	theorem	ADJ
ejpam-5973	219	9	3	3	NUM
ejpam-5973	219	10	that	that	PRON
ejpam-5973	219	11	γhrk(g	γhrk(g	PROPN
ejpam-5973	219	12	)	)	PUNCT
ejpam-5973	219	13	≥	≥	NOUN
ejpam-5973	219	14	k	k	X
ejpam-5973	220	1	+	+	CCONJ
ejpam-5973	220	2	1	1	X
ejpam-5973	220	3	.	.	PUNCT
ejpam-5973	220	4	let	let	VERB
ejpam-5973	220	5	n	n	PRON
ejpam-5973	220	6	≤	≤	NOUN
ejpam-5973	220	7	k	k	PROPN
ejpam-5973	220	8	and	and	CCONJ
ejpam-5973	220	9	let	let	VERB
ejpam-5973	220	10	g	g	NOUN
ejpam-5973	220	11	be	be	AUX
ejpam-5973	220	12	γhrk	γhrk	NOUN
ejpam-5973	220	13	-	-	PUNCT
ejpam-5973	220	14	function	function	NOUN
ejpam-5973	220	15	of	of	ADP
ejpam-5973	220	16	g.	g.	PROPN
ejpam-5973	221	1	if	if	SCONJ
ejpam-5973	221	2	v	v	NUM
ejpam-5973	221	3	g	g	NOUN
ejpam-5973	221	4	0	0	NUM
ejpam-5973	221	5	̸=	̸=	PROPN
ejpam-5973	221	6	∅	∅	NOUN
ejpam-5973	221	7	and	and	CCONJ
ejpam-5973	221	8	v	v	ADP
ejpam-5973	221	9	∈	∈	PROPN
ejpam-5973	221	10	v	v	ADP
ejpam-5973	221	11	g	g	NOUN
ejpam-5973	221	12	0	0	NUM
ejpam-5973	221	13	,	,	PUNCT
ejpam-5973	221	14	then	then	ADV
ejpam-5973	221	15	by	by	ADP
ejpam-5973	221	16	the	the	DET
ejpam-5973	221	17	definition	definition	NOUN
ejpam-5973	221	18	we	we	PRON
ejpam-5973	221	19	have	have	VERB
ejpam-5973	221	20	⋃	⋃	NOUN
ejpam-5973	221	21	u∈n2	u∈n2	ADJ
ejpam-5973	221	22	g(v	g(v	NOUN
ejpam-5973	221	23	)	)	PUNCT
ejpam-5973	221	24	g(u	g(u	PROPN
ejpam-5973	221	25	)	)	PUNCT
ejpam-5973	222	1	=	=	PRON
ejpam-5973	222	2	{	{	PUNCT
ejpam-5973	222	3	1	1	NUM
ejpam-5973	222	4	,	,	PUNCT
ejpam-5973	222	5	2	2	NUM
ejpam-5973	222	6	,	,	PUNCT
ejpam-5973	222	7	.	.	PUNCT
ejpam-5973	222	8	.	.	PUNCT
ejpam-5973	222	9	.	.	PUNCT
ejpam-5973	223	1	,	,	PUNCT
ejpam-5973	223	2	k	k	X
ejpam-5973	223	3	}	}	PUNCT
ejpam-5973	223	4	,	,	PUNCT
ejpam-5973	223	5	and	and	CCONJ
ejpam-5973	223	6	so	so	ADV
ejpam-5973	223	7	γhrk(g	γhrk(g	PROPN
ejpam-5973	223	8	)	)	PUNCT
ejpam-5973	223	9	≥	≥	NOUN
ejpam-5973	223	10	∑	∑	ADV
ejpam-5973	223	11	u∈n2	u∈n2	ADJ
ejpam-5973	223	12	g(v	g(v	NOUN
ejpam-5973	223	13	)	)	PUNCT
ejpam-5973	223	14	|g(u)|	|g(u)|	PROPN
ejpam-5973	223	15	≥	≥	NUM
ejpam-5973	223	16	k	k	X
ejpam-5973	223	17	≥	≥	PROPN
ejpam-5973	223	18	n.	n.	PROPN
ejpam-5973	223	19	assume	assume	PROPN
ejpam-5973	223	20	j.	j.	PROPN
ejpam-5973	223	21	j.	j.	PROPN
ejpam-5973	223	22	hamja	hamja	PROPN
ejpam-5973	224	1	et	et	PROPN
ejpam-5973	224	2	al	al	PROPN
ejpam-5973	224	3	.	.	PUNCT
ejpam-5973	224	4	/	/	SYM
ejpam-5973	224	5	eur	eur	PROPN
ejpam-5973	224	6	.	.	PUNCT
ejpam-5973	225	1	j.	j.	PROPN
ejpam-5973	225	2	pure	pure	PROPN
ejpam-5973	225	3	appl	appl	PROPN
ejpam-5973	225	4	.	.	PROPN
ejpam-5973	225	5	math	math	PROPN
ejpam-5973	225	6	,	,	PUNCT
ejpam-5973	225	7	18	18	NUM
ejpam-5973	225	8	(	(	PUNCT
ejpam-5973	225	9	2	2	NUM
ejpam-5973	225	10	)	)	PUNCT
ejpam-5973	225	11	(	(	PUNCT
ejpam-5973	225	12	2025	2025	NUM
ejpam-5973	225	13	)	)	PUNCT
ejpam-5973	225	14	,	,	PUNCT
ejpam-5973	225	15	5973	5973	NUM
ejpam-5973	225	16	7	7	NUM
ejpam-5973	225	17	of	of	ADP
ejpam-5973	225	18	17	17	NUM
ejpam-5973	226	1	that	that	PRON
ejpam-5973	226	2	v	v	ADP
ejpam-5973	226	3	g	g	NOUN
ejpam-5973	226	4	0	0	NUM
ejpam-5973	226	5	=	=	SYM
ejpam-5973	226	6	∅.	∅.	NOUN
ejpam-5973	226	7	then	then	ADV
ejpam-5973	226	8	by	by	ADP
ejpam-5973	226	9	remark	remark	NOUN
ejpam-5973	226	10	2	2	NUM
ejpam-5973	226	11	,	,	PUNCT
ejpam-5973	226	12	we	we	PRON
ejpam-5973	226	13	have	have	VERB
ejpam-5973	226	14	γhrk(g	γhrk(g	NOUN
ejpam-5973	226	15	)	)	PUNCT
ejpam-5973	227	1	=	=	SYM
ejpam-5973	228	1	∑k	∑k	PROPN
ejpam-5973	228	2	j=1	j=1	PROPN
ejpam-5973	228	3	j|v	j|v	PUNCT
ejpam-5973	229	1	f	f	PROPN
ejpam-5973	229	2	j	j	PROPN
ejpam-5973	230	1	|	|	ADV
ejpam-5973	230	2	≥	≥	NOUN
ejpam-5973	230	3	∑k	∑k	PROPN
ejpam-5973	230	4	j=1	j=1	PROPN
ejpam-5973	230	5	|v	|v	PROPN
ejpam-5973	230	6	f	f	PROPN
ejpam-5973	230	7	j	j	PROPN
ejpam-5973	231	1	|	|	ADV
ejpam-5973	231	2	=	=	PROPN
ejpam-5973	231	3	n.	n.	PROPN
ejpam-5973	231	4	consequently	consequently	ADV
ejpam-5973	231	5	,	,	PUNCT
ejpam-5973	231	6	we	we	PRON
ejpam-5973	231	7	have	have	VERB
ejpam-5973	231	8	γhrk(g	γhrk(g	PROPN
ejpam-5973	231	9	)	)	PUNCT
ejpam-5973	231	10	≥	≥	NOUN
ejpam-5973	231	11	min{n	min{n	NOUN
ejpam-5973	231	12	,	,	PUNCT
ejpam-5973	231	13	k	k	PROPN
ejpam-5973	231	14	+	+	PROPN
ejpam-5973	231	15	1	1	NUM
ejpam-5973	231	16	}	}	PUNCT
ejpam-5973	231	17	.	.	PUNCT
ejpam-5973	232	1	for	for	ADP
ejpam-5973	232	2	the	the	DET
ejpam-5973	232	3	upper	upper	ADJ
ejpam-5973	232	4	bound	bound	NOUN
ejpam-5973	232	5	,	,	PUNCT
ejpam-5973	232	6	consider	consider	VERB
ejpam-5973	232	7	the	the	DET
ejpam-5973	232	8	function	function	NOUN
ejpam-5973	232	9	h	h	NOUN
ejpam-5973	232	10	:	:	PUNCT
ejpam-5973	232	11	v	v	X
ejpam-5973	232	12	(	(	PUNCT
ejpam-5973	232	13	g	g	NOUN
ejpam-5973	232	14	)	)	PUNCT
ejpam-5973	232	15	→	→	SYM
ejpam-5973	232	16	p({1	p({1	PROPN
ejpam-5973	232	17	,	,	PUNCT
ejpam-5973	232	18	2	2	NUM
ejpam-5973	232	19	,	,	PUNCT
ejpam-5973	232	20	.	.	PUNCT
ejpam-5973	232	21	.	.	PUNCT
ejpam-5973	233	1	.	.	PUNCT
ejpam-5973	234	1	,	,	PUNCT
ejpam-5973	234	2	k	k	X
ejpam-5973	234	3	}	}	PUNCT
ejpam-5973	234	4	)	)	PUNCT
ejpam-5973	234	5	defined	define	VERB
ejpam-5973	234	6	by	by	ADP
ejpam-5973	234	7	h(v	h(v	PROPN
ejpam-5973	234	8	)	)	PUNCT
ejpam-5973	235	1	=	=	PRON
ejpam-5973	235	2	{	{	PUNCT
ejpam-5973	235	3	1	1	NUM
ejpam-5973	235	4	}	}	PUNCT
ejpam-5973	235	5	for	for	ADP
ejpam-5973	235	6	all	all	PRON
ejpam-5973	235	7	v	v	ADP
ejpam-5973	235	8	∈	∈	NOUN
ejpam-5973	235	9	v	v	NOUN
ejpam-5973	235	10	(	(	PUNCT
ejpam-5973	235	11	g	g	NOUN
ejpam-5973	235	12	)	)	PUNCT
ejpam-5973	235	13	.	.	PUNCT
ejpam-5973	236	1	clearly	clearly	ADV
ejpam-5973	236	2	,	,	PUNCT
ejpam-5973	236	3	h	h	NOUN
ejpam-5973	236	4	is	be	AUX
ejpam-5973	236	5	an	an	DET
ejpam-5973	236	6	hkrd	hkrd	NOUN
ejpam-5973	236	7	-	-	PUNCT
ejpam-5973	236	8	function	function	NOUN
ejpam-5973	236	9	of	of	ADP
ejpam-5973	236	10	g	g	NOUN
ejpam-5973	236	11	of	of	ADP
ejpam-5973	236	12	weight	weight	NOUN
ejpam-5973	236	13	n	n	VERB
ejpam-5973	236	14	leading	lead	VERB
ejpam-5973	236	15	to	to	ADP
ejpam-5973	236	16	γhrk(g	γhrk(g	PROPN
ejpam-5973	236	17	)	)	PUNCT
ejpam-5973	236	18	≤	≤	PROPN
ejpam-5973	236	19	n.	n.	NOUN
ejpam-5973	236	20	therefore	therefore	ADV
ejpam-5973	236	21	,	,	PUNCT
ejpam-5973	236	22	we	we	PRON
ejpam-5973	236	23	have	have	AUX
ejpam-5973	236	24	established	establish	VERB
ejpam-5973	236	25	that	that	DET
ejpam-5973	236	26	min{n	min{n	NOUN
ejpam-5973	236	27	,	,	PUNCT
ejpam-5973	236	28	k	k	PROPN
ejpam-5973	237	1	+	+	PROPN
ejpam-5973	237	2	1	1	NUM
ejpam-5973	237	3	}	}	PUNCT
ejpam-5973	237	4	≤	≤	NUM
ejpam-5973	237	5	γhrk(g	γhrk(g	PROPN
ejpam-5973	237	6	)	)	PUNCT
ejpam-5973	237	7	≤	≤	NOUN
ejpam-5973	237	8	n.	n.	NOUN
ejpam-5973	237	9	in	in	ADP
ejpam-5973	237	10	particular	particular	ADJ
ejpam-5973	237	11	,	,	PUNCT
ejpam-5973	237	12	if	if	SCONJ
ejpam-5973	237	13	n	n	PRON
ejpam-5973	237	14	≤	≤	X
ejpam-5973	237	15	k+1	k+1	NOUN
ejpam-5973	237	16	,	,	PUNCT
ejpam-5973	237	17	then	then	ADV
ejpam-5973	237	18	the	the	DET
ejpam-5973	237	19	lower	lower	ADV
ejpam-5973	237	20	bound	bind	VERB
ejpam-5973	237	21	is	be	AUX
ejpam-5973	237	22	the	the	DET
ejpam-5973	237	23	min{n	min{n	NOUN
ejpam-5973	237	24	,	,	PUNCT
ejpam-5973	237	25	k+1	k+1	NOUN
ejpam-5973	237	26	}	}	PUNCT
ejpam-5973	237	27	=	=	SYM
ejpam-5973	237	28	n	n	CCONJ
ejpam-5973	237	29	,	,	PUNCT
ejpam-5973	237	30	implying	imply	VERB
ejpam-5973	237	31	that	that	SCONJ
ejpam-5973	237	32	γhrk(g	γhrk(g	NOUN
ejpam-5973	237	33	)	)	PUNCT
ejpam-5973	237	34	=	=	PUNCT
ejpam-5973	237	35	n.	n.	NOUN
ejpam-5973	237	36	proposition	proposition	NOUN
ejpam-5973	237	37	1	1	X
ejpam-5973	237	38	.	.	PUNCT
ejpam-5973	238	1	let	let	VERB
ejpam-5973	238	2	k	k	PROPN
ejpam-5973	238	3	≥	≥	NUM
ejpam-5973	238	4	2	2	NUM
ejpam-5973	238	5	be	be	AUX
ejpam-5973	238	6	an	an	DET
ejpam-5973	238	7	integer	integer	NOUN
ejpam-5973	238	8	and	and	CCONJ
ejpam-5973	238	9	g	g	PROPN
ejpam-5973	238	10	be	be	AUX
ejpam-5973	238	11	a	a	DET
ejpam-5973	238	12	graph	graph	NOUN
ejpam-5973	238	13	of	of	ADP
ejpam-5973	238	14	order	order	NOUN
ejpam-5973	238	15	n	n	PRON
ejpam-5973	238	16	≥	≥	NOUN
ejpam-5973	238	17	k	k	X
ejpam-5973	238	18	with	with	ADP
ejpam-5973	238	19	δ(g	δ(g	PROPN
ejpam-5973	238	20	)	)	PUNCT
ejpam-5973	238	21	≥	≥	NOUN
ejpam-5973	238	22	n−2	n−2	PROPN
ejpam-5973	238	23	.	.	PUNCT
ejpam-5973	239	1	then	then	ADV
ejpam-5973	239	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	239	3	)	)	PUNCT
ejpam-5973	239	4	=	=	SYM
ejpam-5973	240	1	n.	n.	NOUN
ejpam-5973	240	2	proof	proof	NOUN
ejpam-5973	240	3	.	.	PUNCT
ejpam-5973	241	1	let	let	VERB
ejpam-5973	241	2	f	f	PRON
ejpam-5973	241	3	be	be	AUX
ejpam-5973	241	4	a	a	DET
ejpam-5973	241	5	γhrk	γhrk	NOUN
ejpam-5973	241	6	-	-	PUNCT
ejpam-5973	241	7	function	function	NOUN
ejpam-5973	241	8	of	of	ADP
ejpam-5973	241	9	g	g	NOUN
ejpam-5973	241	10	such	such	ADJ
ejpam-5973	241	11	that	that	SCONJ
ejpam-5973	241	12	|v	|v	PROPN
ejpam-5973	241	13	f	f	NOUN
ejpam-5973	241	14	0	0	NUM
ejpam-5973	242	1	|	|	ADV
ejpam-5973	242	2	is	be	AUX
ejpam-5973	242	3	minimized	minimize	VERB
ejpam-5973	242	4	.	.	PUNCT
ejpam-5973	243	1	if	if	SCONJ
ejpam-5973	243	2	v	v	NUM
ejpam-5973	243	3	f	f	NOUN
ejpam-5973	243	4	0	0	NUM
ejpam-5973	244	1	=	=	NOUN
ejpam-5973	244	2	∅	∅	NOUN
ejpam-5973	244	3	,	,	PUNCT
ejpam-5973	244	4	then	then	ADV
ejpam-5973	244	5	we	we	PRON
ejpam-5973	244	6	are	be	AUX
ejpam-5973	244	7	done	do	VERB
ejpam-5973	244	8	.	.	PUNCT
ejpam-5973	245	1	assume	assume	VERB
ejpam-5973	245	2	,	,	PUNCT
ejpam-5973	245	3	for	for	ADP
ejpam-5973	245	4	contradiction	contradiction	NOUN
ejpam-5973	245	5	,	,	PUNCT
ejpam-5973	245	6	that	that	SCONJ
ejpam-5973	245	7	v	v	X
ejpam-5973	245	8	f	f	PROPN
ejpam-5973	245	9	0	0	NUM
ejpam-5973	245	10	̸=	̸=	PROPN
ejpam-5973	245	11	∅	∅	NOUN
ejpam-5973	245	12	and	and	CCONJ
ejpam-5973	245	13	let	let	VERB
ejpam-5973	245	14	x	x	PUNCT
ejpam-5973	245	15	∈	∈	PROPN
ejpam-5973	245	16	v	v	PROPN
ejpam-5973	245	17	f	f	PROPN
ejpam-5973	245	18	0	0	PROPN
ejpam-5973	245	19	.	.	PUNCT
ejpam-5973	246	1	hence	hence	ADV
ejpam-5973	246	2	,	,	PUNCT
ejpam-5973	246	3	we	we	PRON
ejpam-5973	246	4	must	must	AUX
ejpam-5973	246	5	have⋃	have⋃	VERB
ejpam-5973	246	6	u∈n2	u∈n2	ADJ
ejpam-5973	246	7	g(x	g(x	NOUN
ejpam-5973	246	8	)	)	PUNCT
ejpam-5973	246	9	f(u	f(u	PROPN
ejpam-5973	246	10	)	)	PUNCT
ejpam-5973	246	11	=	=	PRON
ejpam-5973	247	1	{	{	PUNCT
ejpam-5973	247	2	1	1	NUM
ejpam-5973	247	3	,	,	PUNCT
ejpam-5973	247	4	2	2	NUM
ejpam-5973	247	5	,	,	PUNCT
ejpam-5973	247	6	.	.	PUNCT
ejpam-5973	247	7	.	.	PUNCT
ejpam-5973	248	1	.	.	PUNCT
ejpam-5973	249	1	,	,	PUNCT
ejpam-5973	249	2	k	k	X
ejpam-5973	249	3	}	}	PUNCT
ejpam-5973	249	4	.	.	PUNCT
ejpam-5973	250	1	it	it	PRON
ejpam-5973	250	2	follows	follow	VERB
ejpam-5973	250	3	from	from	ADP
ejpam-5973	250	4	δ(g	δ(g	PROPN
ejpam-5973	250	5	)	)	PUNCT
ejpam-5973	250	6	≥	≥	NOUN
ejpam-5973	250	7	n	n	CCONJ
ejpam-5973	250	8	−	−	PROPN
ejpam-5973	250	9	2	2	NUM
ejpam-5973	250	10	that	that	PRON
ejpam-5973	250	11	|n2	|n2	PRON
ejpam-5973	250	12	g(x)|	g(x)|	NOUN
ejpam-5973	250	13	=	=	NOUN
ejpam-5973	250	14	1	1	X
ejpam-5973	250	15	.	.	X
ejpam-5973	250	16	assume	assume	VERB
ejpam-5973	250	17	that	that	SCONJ
ejpam-5973	250	18	y	y	PROPN
ejpam-5973	250	19	∈	∈	PROPN
ejpam-5973	250	20	n2	n2	NOUN
ejpam-5973	250	21	g(x	g(x	PROPN
ejpam-5973	250	22	)	)	PUNCT
ejpam-5973	250	23	.	.	PUNCT
ejpam-5973	251	1	then	then	ADV
ejpam-5973	251	2	f(y	f(y	NOUN
ejpam-5973	251	3	)	)	PUNCT
ejpam-5973	251	4	=	=	PRON
ejpam-5973	251	5	{	{	PUNCT
ejpam-5973	251	6	1	1	NUM
ejpam-5973	251	7	,	,	PUNCT
ejpam-5973	251	8	2	2	NUM
ejpam-5973	251	9	,	,	PUNCT
ejpam-5973	251	10	.	.	PUNCT
ejpam-5973	251	11	.	.	PUNCT
ejpam-5973	251	12	.	.	PUNCT
ejpam-5973	252	1	,	,	PUNCT
ejpam-5973	252	2	k	k	X
ejpam-5973	252	3	}	}	PUNCT
ejpam-5973	252	4	.	.	PUNCT
ejpam-5973	253	1	since	since	SCONJ
ejpam-5973	253	2	y	y	PROPN
ejpam-5973	253	3	is	be	AUX
ejpam-5973	253	4	adjacent	adjacent	ADJ
ejpam-5973	253	5	to	to	ADP
ejpam-5973	253	6	all	all	DET
ejpam-5973	253	7	vertices	vertex	NOUN
ejpam-5973	253	8	other	other	ADJ
ejpam-5973	253	9	than	than	ADP
ejpam-5973	253	10	x	x	PRON
ejpam-5973	253	11	,	,	PUNCT
ejpam-5973	253	12	the	the	DET
ejpam-5973	253	13	function	function	NOUN
ejpam-5973	253	14	g	g	PROPN
ejpam-5973	253	15	defined	define	VERB
ejpam-5973	253	16	on	on	ADP
ejpam-5973	253	17	g(x	g(x	NOUN
ejpam-5973	253	18	)	)	PUNCT
ejpam-5973	254	1	=	=	PUNCT
ejpam-5973	254	2	g(y	g(y	NOUN
ejpam-5973	254	3	)	)	PUNCT
ejpam-5973	254	4	=	=	PUNCT
ejpam-5973	254	5	{	{	PUNCT
ejpam-5973	254	6	1	1	NUM
ejpam-5973	254	7	}	}	PUNCT
ejpam-5973	254	8	and	and	CCONJ
ejpam-5973	254	9	g(z	g(z	ADJ
ejpam-5973	254	10	)	)	PUNCT
ejpam-5973	254	11	=	=	SYM
ejpam-5973	254	12	f(z	f(z	PROPN
ejpam-5973	254	13	)	)	PUNCT
ejpam-5973	254	14	for	for	ADP
ejpam-5973	254	15	other	other	ADJ
ejpam-5973	254	16	vertices	vertex	NOUN
ejpam-5973	254	17	is	be	AUX
ejpam-5973	254	18	an	an	DET
ejpam-5973	254	19	hkrd	hkrd	NOUN
ejpam-5973	254	20	-	-	PUNCT
ejpam-5973	254	21	function	function	NOUN
ejpam-5973	254	22	of	of	ADP
ejpam-5973	254	23	g	g	NOUN
ejpam-5973	254	24	of	of	ADP
ejpam-5973	254	25	weight	weight	NOUN
ejpam-5973	254	26	at	at	ADP
ejpam-5973	254	27	most	most	ADV
ejpam-5973	254	28	ω(f	ω(f	NUM
ejpam-5973	254	29	)	)	PUNCT
ejpam-5973	254	30	with	with	ADP
ejpam-5973	254	31	|v	|v	PROPN
ejpam-5973	254	32	g	g	PROPN
ejpam-5973	254	33	0	0	NUM
ejpam-5973	255	1	|	|	ADV
ejpam-5973	255	2	<	<	X
ejpam-5973	255	3	|v	|v	PROPN
ejpam-5973	255	4	f	f	PROPN
ejpam-5973	255	5	0	0	NUM
ejpam-5973	256	1	|	|	ADV
ejpam-5973	256	2	,	,	PUNCT
ejpam-5973	256	3	contradicting	contradict	VERB
ejpam-5973	256	4	the	the	DET
ejpam-5973	256	5	choice	choice	NOUN
ejpam-5973	256	6	of	of	ADP
ejpam-5973	256	7	f	f	PROPN
ejpam-5973	256	8	.	.	PUNCT
ejpam-5973	257	1	therefore	therefore	ADV
ejpam-5973	257	2	,	,	PUNCT
ejpam-5973	257	3	v	v	PROPN
ejpam-5973	257	4	f	f	NOUN
ejpam-5973	257	5	0	0	NUM
ejpam-5973	257	6	=	=	SYM
ejpam-5973	257	7	∅	∅	NOUN
ejpam-5973	257	8	,	,	PUNCT
ejpam-5973	257	9	and	and	CCONJ
ejpam-5973	257	10	it	it	PRON
ejpam-5973	257	11	follows	follow	VERB
ejpam-5973	257	12	that	that	SCONJ
ejpam-5973	257	13	γhrk(g	γhrk(g	NOUN
ejpam-5973	257	14	)	)	PUNCT
ejpam-5973	257	15	=	=	PUNCT
ejpam-5973	257	16	ω(f	ω(f	X
ejpam-5973	257	17	)	)	PUNCT
ejpam-5973	257	18	=	=	VERB
ejpam-5973	258	1	n.	n.	NOUN
ejpam-5973	258	2	in	in	ADP
ejpam-5973	258	3	the	the	DET
ejpam-5973	258	4	next	next	ADJ
ejpam-5973	258	5	theorem	theorem	NOUN
ejpam-5973	258	6	we	we	PRON
ejpam-5973	258	7	provide	provide	VERB
ejpam-5973	258	8	a	a	DET
ejpam-5973	258	9	sufficient	sufficient	ADJ
ejpam-5973	258	10	condition	condition	NOUN
ejpam-5973	258	11	for	for	ADP
ejpam-5973	258	12	a	a	DET
ejpam-5973	258	13	graph	graph	NOUN
ejpam-5973	258	14	g	g	NOUN
ejpam-5973	258	15	to	to	PART
ejpam-5973	258	16	have	have	VERB
ejpam-5973	258	17	γhrk(g	γhrk(g	NOUN
ejpam-5973	258	18	)	)	PUNCT
ejpam-5973	258	19	=	=	SYM
ejpam-5973	258	20	n.	n.	NOUN
ejpam-5973	258	21	theorem	theorem	VERB
ejpam-5973	258	22	5	5	NUM
ejpam-5973	258	23	.	.	PUNCT
ejpam-5973	259	1	for	for	ADP
ejpam-5973	259	2	positive	positive	ADJ
ejpam-5973	259	3	integers	integer	NOUN
ejpam-5973	259	4	n	n	PRON
ejpam-5973	259	5	and	and	CCONJ
ejpam-5973	259	6	k	k	PROPN
ejpam-5973	259	7	≥	≥	NUM
ejpam-5973	259	8	2	2	NUM
ejpam-5973	259	9	,	,	PUNCT
ejpam-5973	259	10	let	let	VERB
ejpam-5973	259	11	g	g	PRON
ejpam-5973	259	12	be	be	AUX
ejpam-5973	259	13	a	a	DET
ejpam-5973	259	14	graph	graph	NOUN
ejpam-5973	259	15	of	of	ADP
ejpam-5973	259	16	order	order	NOUN
ejpam-5973	259	17	n	n	PRON
ejpam-5973	259	18	≥	≥	NOUN
ejpam-5973	259	19	k	k	X
ejpam-5973	259	20	with	with	ADP
ejpam-5973	259	21	k	k	PROPN
ejpam-5973	259	22	>	>	X
ejpam-5973	259	23	∆h(g	∆h(g	PROPN
ejpam-5973	259	24	2	2	NUM
ejpam-5973	259	25	)	)	PUNCT
ejpam-5973	259	26	.	.	PUNCT
ejpam-5973	260	1	then	then	ADV
ejpam-5973	260	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	260	3	)	)	PUNCT
ejpam-5973	260	4	=	=	SYM
ejpam-5973	260	5	n.	n.	NOUN
ejpam-5973	260	6	proof	proof	NOUN
ejpam-5973	260	7	.	.	PUNCT
ejpam-5973	261	1	suppose	suppose	VERB
ejpam-5973	261	2	,	,	PUNCT
ejpam-5973	261	3	for	for	ADP
ejpam-5973	261	4	the	the	DET
ejpam-5973	261	5	sake	sake	NOUN
ejpam-5973	261	6	of	of	ADP
ejpam-5973	261	7	contradiction	contradiction	NOUN
ejpam-5973	261	8	,	,	PUNCT
ejpam-5973	261	9	that	that	SCONJ
ejpam-5973	261	10	γhrk(g	γhrk(g	NOUN
ejpam-5973	261	11	)	)	PUNCT
ejpam-5973	261	12	<	<	X
ejpam-5973	261	13	n.	n.	PROPN
ejpam-5973	261	14	let	let	VERB
ejpam-5973	261	15	f	f	PRON
ejpam-5973	261	16	be	be	AUX
ejpam-5973	261	17	a	a	DET
ejpam-5973	261	18	γhrkfunction	γhrkfunction	NOUN
ejpam-5973	261	19	of	of	ADP
ejpam-5973	261	20	g	g	PROPN
ejpam-5973	261	21	such	such	ADJ
ejpam-5973	261	22	that	that	SCONJ
ejpam-5973	261	23	|v	|v	PROPN
ejpam-5973	261	24	f	f	NOUN
ejpam-5973	261	25	0	0	NUM
ejpam-5973	262	1	|	|	ADV
ejpam-5973	262	2	is	be	AUX
ejpam-5973	262	3	as	as	ADV
ejpam-5973	262	4	large	large	ADJ
ejpam-5973	262	5	as	as	ADP
ejpam-5973	262	6	possible	possible	ADJ
ejpam-5973	262	7	,	,	PUNCT
ejpam-5973	262	8	where	where	SCONJ
ejpam-5973	262	9	v	v	ADP
ejpam-5973	262	10	f	f	NOUN
ejpam-5973	262	11	0	0	NUM
ejpam-5973	263	1	=	=	SYM
ejpam-5973	263	2	{	{	PUNCT
ejpam-5973	263	3	v	v	NUM
ejpam-5973	263	4	∈	∈	NOUN
ejpam-5973	263	5	v	v	NOUN
ejpam-5973	263	6	(	(	PUNCT
ejpam-5973	263	7	g	g	NOUN
ejpam-5973	263	8	)	)	PUNCT
ejpam-5973	263	9	:	:	PUNCT
ejpam-5973	263	10	f(v	f(v	NOUN
ejpam-5973	263	11	)	)	PUNCT
ejpam-5973	263	12	=	=	SYM
ejpam-5973	263	13	∅	∅	NOUN
ejpam-5973	263	14	}	}	PUNCT
ejpam-5973	263	15	.	.	PUNCT
ejpam-5973	264	1	since	since	SCONJ
ejpam-5973	264	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	264	3	)	)	PUNCT
ejpam-5973	264	4	<	<	X
ejpam-5973	264	5	n	n	CCONJ
ejpam-5973	264	6	,	,	PUNCT
ejpam-5973	264	7	there	there	PRON
ejpam-5973	264	8	is	be	VERB
ejpam-5973	264	9	a	a	DET
ejpam-5973	264	10	vertex	vertex	NOUN
ejpam-5973	264	11	v	v	ADP
ejpam-5973	264	12	∈	∈	PROPN
ejpam-5973	264	13	v	v	ADP
ejpam-5973	264	14	f	f	PROPN
ejpam-5973	264	15	0	0	NUM
ejpam-5973	264	16	and	and	CCONJ
ejpam-5973	264	17	by	by	ADP
ejpam-5973	264	18	the	the	DET
ejpam-5973	264	19	definition	definition	NOUN
ejpam-5973	264	20	we	we	PRON
ejpam-5973	264	21	have	have	VERB
ejpam-5973	264	22	⋃	⋃	NOUN
ejpam-5973	264	23	x∈n2	x∈n2	ADJ
ejpam-5973	264	24	g(v	g(v	NOUN
ejpam-5973	264	25	)	)	PUNCT
ejpam-5973	264	26	f(x	f(x	PROPN
ejpam-5973	264	27	)	)	PUNCT
ejpam-5973	264	28	=	=	PRON
ejpam-5973	264	29	{	{	PUNCT
ejpam-5973	264	30	1	1	NUM
ejpam-5973	264	31	,	,	PUNCT
ejpam-5973	264	32	2	2	NUM
ejpam-5973	264	33	,	,	PUNCT
ejpam-5973	264	34	.	.	PUNCT
ejpam-5973	264	35	.	.	PUNCT
ejpam-5973	265	1	.	.	PUNCT
ejpam-5973	266	1	,	,	PUNCT
ejpam-5973	266	2	k	k	X
ejpam-5973	266	3	}	}	PUNCT
ejpam-5973	266	4	.	.	PUNCT
ejpam-5973	267	1	since	since	SCONJ
ejpam-5973	267	2	k	k	PROPN
ejpam-5973	267	3	>	>	X
ejpam-5973	267	4	∆h(g	∆h(g	PROPN
ejpam-5973	267	5	2	2	NUM
ejpam-5973	267	6	)	)	PUNCT
ejpam-5973	267	7	,	,	PUNCT
ejpam-5973	267	8	there	there	PRON
ejpam-5973	267	9	exists	exist	VERB
ejpam-5973	267	10	a	a	DET
ejpam-5973	267	11	vertex	vertex	NOUN
ejpam-5973	267	12	x	x	SYM
ejpam-5973	267	13	∈	∈	NOUN
ejpam-5973	267	14	n2	n2	ADJ
ejpam-5973	267	15	g(v	g(v	PROPN
ejpam-5973	267	16	)	)	PUNCT
ejpam-5973	267	17	such	such	ADJ
ejpam-5973	267	18	that	that	SCONJ
ejpam-5973	267	19	|f(x)|	|f(x)|	NOUN
ejpam-5973	267	20	≥	≥	NOUN
ejpam-5973	267	21	∆h(g	∆h(g	NOUN
ejpam-5973	267	22	)	)	PUNCT
ejpam-5973	268	1	+	+	CCONJ
ejpam-5973	268	2	1	1	X
ejpam-5973	268	3	.	.	X
ejpam-5973	268	4	assume	assume	VERB
ejpam-5973	268	5	,	,	PUNCT
ejpam-5973	268	6	without	without	ADP
ejpam-5973	268	7	loss	loss	NOUN
ejpam-5973	268	8	of	of	ADP
ejpam-5973	268	9	generality	generality	NOUN
ejpam-5973	268	10	,	,	PUNCT
ejpam-5973	268	11	that	that	SCONJ
ejpam-5973	268	12	{	{	PUNCT
ejpam-5973	268	13	1	1	NUM
ejpam-5973	268	14	,	,	PUNCT
ejpam-5973	268	15	2	2	NUM
ejpam-5973	268	16	,	,	PUNCT
ejpam-5973	268	17	.	.	PUNCT
ejpam-5973	268	18	.	.	PUNCT
ejpam-5973	268	19	.	.	PUNCT
ejpam-5973	269	1	,	,	PUNCT
ejpam-5973	269	2	∆h(g	∆h(g	NOUN
ejpam-5973	269	3	)	)	PUNCT
ejpam-5973	270	1	+	+	CCONJ
ejpam-5973	270	2	1	1	NUM
ejpam-5973	270	3	}	}	SYM
ejpam-5973	270	4	⊆	⊆	NUM
ejpam-5973	270	5	f(x	f(x	PROPN
ejpam-5973	270	6	)	)	PUNCT
ejpam-5973	270	7	.	.	PUNCT
ejpam-5973	271	1	let	let	VERB
ejpam-5973	271	2	x1	x1	NUM
ejpam-5973	271	3	,	,	PUNCT
ejpam-5973	271	4	x2	x2	PROPN
ejpam-5973	271	5	,	,	PUNCT
ejpam-5973	271	6	.	.	PUNCT
ejpam-5973	271	7	.	.	PUNCT
ejpam-5973	272	1	.	.	PUNCT
ejpam-5973	273	1	,	,	PUNCT
ejpam-5973	273	2	xdeg2(x	xdeg2(x	X
ejpam-5973	273	3	)	)	PUNCT
ejpam-5973	273	4	be	be	VERB
ejpam-5973	273	5	the	the	DET
ejpam-5973	273	6	vertices	vertex	NOUN
ejpam-5973	273	7	of	of	ADP
ejpam-5973	273	8	g	g	NOUN
ejpam-5973	273	9	at	at	ADP
ejpam-5973	273	10	distance	distance	NOUN
ejpam-5973	273	11	2	2	NUM
ejpam-5973	273	12	from	from	ADP
ejpam-5973	273	13	x	x	PUNCT
ejpam-5973	273	14	and	and	CCONJ
ejpam-5973	273	15	define	define	VERB
ejpam-5973	273	16	the	the	DET
ejpam-5973	273	17	function	function	NOUN
ejpam-5973	273	18	g	g	NOUN
ejpam-5973	273	19	:	:	PUNCT
ejpam-5973	273	20	v	v	NOUN
ejpam-5973	273	21	(	(	PUNCT
ejpam-5973	273	22	g	g	NOUN
ejpam-5973	273	23	)	)	PUNCT
ejpam-5973	273	24	→	→	SYM
ejpam-5973	273	25	p({1	p({1	PROPN
ejpam-5973	273	26	,	,	PUNCT
ejpam-5973	273	27	2	2	NUM
ejpam-5973	273	28	,	,	PUNCT
ejpam-5973	273	29	.	.	PUNCT
ejpam-5973	273	30	.	.	PUNCT
ejpam-5973	274	1	.	.	PUNCT
ejpam-5973	275	1	,	,	PUNCT
ejpam-5973	275	2	k	k	X
ejpam-5973	275	3	}	}	PUNCT
ejpam-5973	275	4	)	)	PUNCT
ejpam-5973	275	5	by	by	ADP
ejpam-5973	275	6	g(xi	g(xi	PROPN
ejpam-5973	275	7	)	)	PUNCT
ejpam-5973	275	8	=	=	SYM
ejpam-5973	275	9	f(xi	f(xi	PROPN
ejpam-5973	275	10	)	)	PUNCT
ejpam-5973	275	11	∪	∪	NOUN
ejpam-5973	275	12	{	{	PUNCT
ejpam-5973	275	13	i	i	NOUN
ejpam-5973	275	14	}	}	PUNCT
ejpam-5973	275	15	for	for	ADP
ejpam-5973	275	16	each	each	DET
ejpam-5973	275	17	i	i	PRON
ejpam-5973	275	18	∈	∈	PROPN
ejpam-5973	275	19	{	{	PUNCT
ejpam-5973	275	20	1	1	NUM
ejpam-5973	275	21	,	,	PUNCT
ejpam-5973	275	22	2	2	NUM
ejpam-5973	275	23	,	,	PUNCT
ejpam-5973	275	24	.	.	PUNCT
ejpam-5973	275	25	.	.	PUNCT
ejpam-5973	276	1	.	.	PUNCT
ejpam-5973	277	1	,	,	PUNCT
ejpam-5973	278	1	deg2(x	deg2(x	NUM
ejpam-5973	278	2	)	)	PUNCT
ejpam-5973	278	3	}	}	PUNCT
ejpam-5973	278	4	,	,	PUNCT
ejpam-5973	278	5	g(x	g(x	NOUN
ejpam-5973	278	6	)	)	PUNCT
ejpam-5973	278	7	=	=	SYM
ejpam-5973	278	8	f(x	f(x	PROPN
ejpam-5973	278	9	)	)	PUNCT
ejpam-5973	279	1	−	−	PROPN
ejpam-5973	279	2	{	{	PUNCT
ejpam-5973	279	3	1	1	NUM
ejpam-5973	279	4	,	,	PUNCT
ejpam-5973	279	5	.	.	PUNCT
ejpam-5973	279	6	.	.	PUNCT
ejpam-5973	280	1	.	.	PUNCT
ejpam-5973	281	1	,	,	PUNCT
ejpam-5973	281	2	deg2(x	deg2(x	NUM
ejpam-5973	281	3	)	)	PUNCT
ejpam-5973	281	4	}	}	PUNCT
ejpam-5973	281	5	and	and	CCONJ
ejpam-5973	281	6	g(y	g(y	NOUN
ejpam-5973	281	7	)	)	PUNCT
ejpam-5973	281	8	=	=	SYM
ejpam-5973	281	9	f(y	f(y	NOUN
ejpam-5973	281	10	)	)	PUNCT
ejpam-5973	281	11	for	for	ADP
ejpam-5973	281	12	other	other	ADJ
ejpam-5973	281	13	vertices	vertex	NOUN
ejpam-5973	281	14	.	.	PUNCT
ejpam-5973	282	1	then	then	ADV
ejpam-5973	282	2	g	g	PROPN
ejpam-5973	282	3	is	be	AUX
ejpam-5973	282	4	an	an	DET
ejpam-5973	282	5	hkrdfunction	hkrdfunction	NOUN
ejpam-5973	282	6	of	of	ADP
ejpam-5973	282	7	g	g	NOUN
ejpam-5973	282	8	of	of	ADP
ejpam-5973	282	9	weight	weight	NOUN
ejpam-5973	282	10	ω(f	ω(f	ADV
ejpam-5973	282	11	)	)	PUNCT
ejpam-5973	282	12	such	such	ADJ
ejpam-5973	282	13	that	that	SCONJ
ejpam-5973	282	14	|v	|v	PROPN
ejpam-5973	282	15	g	g	PROPN
ejpam-5973	282	16	0	0	NUM
ejpam-5973	283	1	|	|	ADV
ejpam-5973	283	2	<	<	X
ejpam-5973	283	3	|v	|v	PROPN
ejpam-5973	283	4	f	f	NOUN
ejpam-5973	283	5	0	0	NUM
ejpam-5973	284	1	|	|	ADV
ejpam-5973	284	2	a	a	DET
ejpam-5973	284	3	contradiction	contradiction	NOUN
ejpam-5973	284	4	with	with	ADP
ejpam-5973	284	5	the	the	DET
ejpam-5973	284	6	choice	choice	NOUN
ejpam-5973	284	7	of	of	ADP
ejpam-5973	284	8	f	f	PROPN
ejpam-5973	284	9	.	.	PUNCT
ejpam-5973	285	1	consequently	consequently	ADV
ejpam-5973	285	2	,	,	PUNCT
ejpam-5973	285	3	γhrk(g	γhrk(g	PROPN
ejpam-5973	285	4	)	)	PUNCT
ejpam-5973	285	5	=	=	VERB
ejpam-5973	286	1	n.	n.	NOUN
ejpam-5973	286	2	the	the	DET
ejpam-5973	286	3	next	next	ADJ
ejpam-5973	286	4	result	result	NOUN
ejpam-5973	286	5	is	be	AUX
ejpam-5973	286	6	immediate	immediate	ADJ
ejpam-5973	286	7	from	from	ADP
ejpam-5973	286	8	theorem	theorem	ADJ
ejpam-5973	286	9	5	5	NUM
ejpam-5973	286	10	.	.	PUNCT
ejpam-5973	286	11	corollary	corollary	ADJ
ejpam-5973	286	12	2	2	NUM
ejpam-5973	286	13	.	.	PUNCT
ejpam-5973	286	14	for	for	ADP
ejpam-5973	286	15	positive	positive	ADJ
ejpam-5973	286	16	integers	integer	NOUN
ejpam-5973	286	17	n	n	PRON
ejpam-5973	286	18	and	and	CCONJ
ejpam-5973	286	19	k	k	X
ejpam-5973	286	20	with	with	ADP
ejpam-5973	286	21	n	n	PRON
ejpam-5973	286	22	≥	≥	NOUN
ejpam-5973	286	23	k	k	X
ejpam-5973	286	24	≥	≥	NUM
ejpam-5973	286	25	5	5	NUM
ejpam-5973	286	26	,	,	PUNCT
ejpam-5973	286	27	γhrk(pn	γhrk(pn	NOUN
ejpam-5973	286	28	)	)	PUNCT
ejpam-5973	286	29	=	=	SYM
ejpam-5973	286	30	γhrk(cn	γhrk(cn	PROPN
ejpam-5973	286	31	)	)	PUNCT
ejpam-5973	286	32	=	=	SYM
ejpam-5973	286	33	n.	n.	NOUN
ejpam-5973	286	34	theorem	theorem	VERB
ejpam-5973	286	35	6	6	NUM
ejpam-5973	286	36	.	.	PUNCT
ejpam-5973	287	1	let	let	VERB
ejpam-5973	287	2	k	k	PROPN
ejpam-5973	287	3	≥	≥	NUM
ejpam-5973	287	4	1	1	NUM
ejpam-5973	287	5	be	be	AUX
ejpam-5973	287	6	an	an	DET
ejpam-5973	287	7	integer	integer	NOUN
ejpam-5973	287	8	and	and	CCONJ
ejpam-5973	287	9	g	g	PROPN
ejpam-5973	287	10	be	be	AUX
ejpam-5973	287	11	a	a	DET
ejpam-5973	287	12	connected	connected	ADJ
ejpam-5973	287	13	graph	graph	NOUN
ejpam-5973	287	14	of	of	ADP
ejpam-5973	287	15	order	order	NOUN
ejpam-5973	287	16	n	n	PRON
ejpam-5973	287	17	≥	≥	NOUN
ejpam-5973	287	18	k	k	X
ejpam-5973	287	19	with	with	ADP
ejpam-5973	287	20	hop	hop	PROPN
ejpam-5973	287	21	maximum	maximum	PROPN
ejpam-5973	287	22	degree	degree	NOUN
ejpam-5973	287	23	∆h	∆h	PROPN
ejpam-5973	287	24	.	.	PUNCT
ejpam-5973	288	1	then	then	ADV
ejpam-5973	288	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	288	3	)	)	PUNCT
ejpam-5973	288	4	≥	≥	NOUN
ejpam-5973	288	5	nk/(∆h	nk/(∆h	NOUN
ejpam-5973	288	6	+	+	CCONJ
ejpam-5973	288	7	k	k	NOUN
ejpam-5973	288	8	)	)	PUNCT
ejpam-5973	288	9	.	.	PUNCT
ejpam-5973	289	1	j.	j.	PROPN
ejpam-5973	289	2	j.	j.	PROPN
ejpam-5973	289	3	hamja	hamja	PROPN
ejpam-5973	289	4	et	et	PROPN
ejpam-5973	289	5	al	al	PROPN
ejpam-5973	289	6	.	.	PUNCT
ejpam-5973	289	7	/	/	SYM
ejpam-5973	289	8	eur	eur	PROPN
ejpam-5973	289	9	.	.	PUNCT
ejpam-5973	290	1	j.	j.	PROPN
ejpam-5973	290	2	pure	pure	PROPN
ejpam-5973	290	3	appl	appl	PROPN
ejpam-5973	290	4	.	.	PROPN
ejpam-5973	290	5	math	math	PROPN
ejpam-5973	290	6	,	,	PUNCT
ejpam-5973	290	7	18	18	NUM
ejpam-5973	290	8	(	(	PUNCT
ejpam-5973	290	9	2	2	NUM
ejpam-5973	290	10	)	)	PUNCT
ejpam-5973	290	11	(	(	PUNCT
ejpam-5973	290	12	2025	2025	NUM
ejpam-5973	290	13	)	)	PUNCT
ejpam-5973	290	14	,	,	PUNCT
ejpam-5973	290	15	5973	5973	NUM
ejpam-5973	290	16	8	8	NUM
ejpam-5973	290	17	of	of	ADP
ejpam-5973	290	18	17	17	NUM
ejpam-5973	290	19	proof	proof	NOUN
ejpam-5973	290	20	.	.	PUNCT
ejpam-5973	291	1	let	let	VERB
ejpam-5973	291	2	g	g	PRON
ejpam-5973	291	3	be	be	AUX
ejpam-5973	291	4	a	a	DET
ejpam-5973	291	5	γhrk	γhrk	NOUN
ejpam-5973	291	6	-	-	PUNCT
ejpam-5973	291	7	function	function	NOUN
ejpam-5973	291	8	of	of	ADP
ejpam-5973	291	9	g	g	PROPN
ejpam-5973	291	10	,	,	PUNCT
ejpam-5973	291	11	i.e	i.e	PROPN
ejpam-5973	291	12	ω(g	ω(g	NOUN
ejpam-5973	291	13	)	)	PUNCT
ejpam-5973	292	1	=	=	SYM
ejpam-5973	292	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	292	3	)	)	PUNCT
ejpam-5973	292	4	,	,	PUNCT
ejpam-5973	292	5	and	and	CCONJ
ejpam-5973	292	6	let	let	VERB
ejpam-5973	292	7	v	v	NOUN
ejpam-5973	292	8	g	g	NOUN
ejpam-5973	292	9	0	0	NUM
ejpam-5973	292	10	be	be	AUX
ejpam-5973	292	11	the	the	DET
ejpam-5973	292	12	set	set	NOUN
ejpam-5973	292	13	of	of	ADP
ejpam-5973	292	14	vertices	vertex	NOUN
ejpam-5973	292	15	assigned	assign	VERB
ejpam-5973	292	16	∅	∅	NOUN
ejpam-5973	292	17	under	under	ADP
ejpam-5973	292	18	g.	g.	PROPN
ejpam-5973	292	19	clearly	clearly	ADV
ejpam-5973	292	20	,	,	PUNCT
ejpam-5973	292	21	γhrk(g	γhrk(g	PROPN
ejpam-5973	292	22	)	)	PUNCT
ejpam-5973	292	23	≥	≥	NOUN
ejpam-5973	292	24	n	n	CCONJ
ejpam-5973	292	25	−	−	PROPN
ejpam-5973	292	26	|v	|v	PROPN
ejpam-5973	292	27	g	g	PROPN
ejpam-5973	292	28	0	0	NUM
ejpam-5973	293	1	|	|	NOUN
ejpam-5973	293	2	.	.	PUNCT
ejpam-5973	294	1	on	on	ADP
ejpam-5973	294	2	the	the	DET
ejpam-5973	294	3	other	other	ADJ
ejpam-5973	294	4	hand	hand	NOUN
ejpam-5973	294	5	,	,	PUNCT
ejpam-5973	294	6	sine	sine	VERB
ejpam-5973	294	7	each	each	DET
ejpam-5973	294	8	vertex	vertex	NOUN
ejpam-5973	294	9	x	x	PUNCT
ejpam-5973	294	10	in	in	ADP
ejpam-5973	294	11	v	v	NUM
ejpam-5973	294	12	g	g	NOUN
ejpam-5973	294	13	0	0	NUM
ejpam-5973	294	14	must	must	AUX
ejpam-5973	294	15	have	have	VERB
ejpam-5973	294	16	each	each	PRON
ejpam-5973	294	17	of	of	ADP
ejpam-5973	294	18	k	k	PROPN
ejpam-5973	294	19	colors	color	NOUN
ejpam-5973	294	20	in	in	ADP
ejpam-5973	294	21	n2	n2	ADJ
ejpam-5973	294	22	g(x	g(x	PROPN
ejpam-5973	294	23	)	)	PUNCT
ejpam-5973	294	24	,	,	PUNCT
ejpam-5973	294	25	we	we	PRON
ejpam-5973	294	26	have	have	VERB
ejpam-5973	294	27	∆hγhrk(g	∆hγhrk(g	NOUN
ejpam-5973	294	28	)	)	PUNCT
ejpam-5973	294	29	≥	≥	PROPN
ejpam-5973	294	30	k|v	k|v	NOUN
ejpam-5973	295	1	g	g	NOUN
ejpam-5973	295	2	0	0	NUM
ejpam-5973	296	1	|	|	INTJ
ejpam-5973	296	2	.	.	PUNCT
ejpam-5973	297	1	since	since	SCONJ
ejpam-5973	297	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	297	3	)	)	PUNCT
ejpam-5973	297	4	≥	≥	NOUN
ejpam-5973	297	5	n	n	CCONJ
ejpam-5973	297	6	−	−	PROPN
ejpam-5973	297	7	|v	|v	PROPN
ejpam-5973	297	8	g	g	PROPN
ejpam-5973	297	9	0	0	NUM
ejpam-5973	297	10	|	|	ADV
ejpam-5973	297	11	,	,	PUNCT
ejpam-5973	297	12	it	it	PRON
ejpam-5973	297	13	would	would	AUX
ejpam-5973	297	14	imply	imply	VERB
ejpam-5973	297	15	that	that	DET
ejpam-5973	297	16	kγhrk(g	kγhrk(g	PROPN
ejpam-5973	297	17	)	)	PUNCT
ejpam-5973	297	18	≥	≥	NOUN
ejpam-5973	297	19	kn	kn	PROPN
ejpam-5973	297	20	−	−	PROPN
ejpam-5973	297	21	k|v	k|v	NOUN
ejpam-5973	298	1	g	g	NOUN
ejpam-5973	298	2	0	0	NUM
ejpam-5973	298	3	|	|	NOUN
ejpam-5973	298	4	.	.	PUNCT
ejpam-5973	299	1	moreover	moreover	ADV
ejpam-5973	299	2	,	,	PUNCT
ejpam-5973	299	3	since	since	SCONJ
ejpam-5973	299	4	k|v	k|v	PROPN
ejpam-5973	299	5	g	g	PROPN
ejpam-5973	299	6	0	0	NUM
ejpam-5973	299	7	|	|	ADV
ejpam-5973	299	8	≤	≤	NUM
ejpam-5973	299	9	∆hγhrk(g	∆hγhrk(g	NOUN
ejpam-5973	299	10	)	)	PUNCT
ejpam-5973	299	11	,	,	PUNCT
ejpam-5973	299	12	it	it	PRON
ejpam-5973	299	13	follows	follow	VERB
ejpam-5973	299	14	that	that	SCONJ
ejpam-5973	299	15	kγhrk(g)+∆hγhrk(g	kγhrk(g)+∆hγhrk(g	NOUN
ejpam-5973	299	16	)	)	PUNCT
ejpam-5973	299	17	≥	≥	NOUN
ejpam-5973	299	18	kn	kn	PROPN
ejpam-5973	299	19	.	.	PUNCT
ejpam-5973	300	1	hence	hence	ADV
ejpam-5973	300	2	,	,	PUNCT
ejpam-5973	300	3	(	(	PUNCT
ejpam-5973	300	4	k+∆h)γhrk(g	k+∆h)γhrk(g	X
ejpam-5973	300	5	)	)	PUNCT
ejpam-5973	300	6	≥	≥	NOUN
ejpam-5973	301	1	kn	kn	PROPN
ejpam-5973	301	2	.	.	PUNCT
ejpam-5973	302	1	therefore	therefore	ADV
ejpam-5973	302	2	,	,	PUNCT
ejpam-5973	302	3	γhrk(g	γhrk(g	PROPN
ejpam-5973	302	4	)	)	PUNCT
ejpam-5973	302	5	≥	≥	NOUN
ejpam-5973	302	6	nk/(∆h	nk/(∆h	NOUN
ejpam-5973	302	7	+	+	CCONJ
ejpam-5973	302	8	k	k	NOUN
ejpam-5973	302	9	)	)	PUNCT
ejpam-5973	302	10	as	as	SCONJ
ejpam-5973	302	11	desired	desire	VERB
ejpam-5973	302	12	.	.	PUNCT
ejpam-5973	303	1	the	the	DET
ejpam-5973	303	2	next	next	ADJ
ejpam-5973	303	3	result	result	NOUN
ejpam-5973	303	4	will	will	AUX
ejpam-5973	303	5	be	be	AUX
ejpam-5973	303	6	used	use	VERB
ejpam-5973	303	7	in	in	ADP
ejpam-5973	303	8	the	the	DET
ejpam-5973	303	9	subsequent	subsequent	ADJ
ejpam-5973	303	10	discussion	discussion	NOUN
ejpam-5973	303	11	.	.	PUNCT
ejpam-5973	304	1	proposition	proposition	NOUN
ejpam-5973	304	2	2	2	NUM
ejpam-5973	304	3	.	.	PUNCT
ejpam-5973	305	1	[	[	X
ejpam-5973	305	2	18	18	NUM
ejpam-5973	305	3	]	]	PUNCT
ejpam-5973	305	4	for	for	ADP
ejpam-5973	305	5	any	any	DET
ejpam-5973	305	6	positive	positive	ADJ
ejpam-5973	305	7	integer	integer	NOUN
ejpam-5973	305	8	n	n	CCONJ
ejpam-5973	305	9	,	,	PUNCT
ejpam-5973	305	10	we	we	PRON
ejpam-5973	305	11	have	have	VERB
ejpam-5973	305	12	:	:	PUNCT
ejpam-5973	305	13	γrk(kn	γrk(kn	NUM
ejpam-5973	305	14	)	)	PUNCT
ejpam-5973	306	1	=	=	PRON
ejpam-5973	306	2	{	{	PUNCT
ejpam-5973	306	3	k	k	NOUN
ejpam-5973	306	4	,	,	PUNCT
ejpam-5973	306	5	if	if	SCONJ
ejpam-5973	306	6	n	n	NUM
ejpam-5973	306	7	≥	≥	X
ejpam-5973	306	8	k	k	NOUN
ejpam-5973	306	9	,	,	PUNCT
ejpam-5973	306	10	n	n	CCONJ
ejpam-5973	306	11	,	,	PUNCT
ejpam-5973	306	12	if	if	SCONJ
ejpam-5973	306	13	n	n	PROPN
ejpam-5973	306	14	<	<	X
ejpam-5973	306	15	k.	k.	PROPN
ejpam-5973	306	16	proposition	proposition	PROPN
ejpam-5973	306	17	3	3	X
ejpam-5973	306	18	.	.	PUNCT
ejpam-5973	307	1	let	let	VERB
ejpam-5973	307	2	m	m	PRON
ejpam-5973	307	3	,	,	PUNCT
ejpam-5973	307	4	n	n	CCONJ
ejpam-5973	307	5	,	,	PUNCT
ejpam-5973	307	6	and	and	CCONJ
ejpam-5973	307	7	k	k	PROPN
ejpam-5973	307	8	be	be	AUX
ejpam-5973	307	9	positive	positive	ADJ
ejpam-5973	307	10	integers	integer	NOUN
ejpam-5973	307	11	with	with	ADP
ejpam-5973	307	12	k	k	PROPN
ejpam-5973	307	13	≥	≥	NUM
ejpam-5973	307	14	1	1	NUM
ejpam-5973	307	15	and	and	CCONJ
ejpam-5973	307	16	m	m	PROPN
ejpam-5973	307	17	≤	≤	NOUN
ejpam-5973	307	18	n.	n.	NOUN
ejpam-5973	307	19	then	then	ADV
ejpam-5973	307	20	γhrk(km	γhrk(km	NOUN
ejpam-5973	307	21	,	,	PUNCT
ejpam-5973	307	22	n	n	CCONJ
ejpam-5973	307	23	)	)	PUNCT
ejpam-5973	308	1	=	=	SYM
ejpam-5973	309	1			NOUN
ejpam-5973	309	2	2k	2k	NOUN
ejpam-5973	309	3	if	if	SCONJ
ejpam-5973	309	4	m	m	PROPN
ejpam-5973	309	5	≥	≥	VERB
ejpam-5973	309	6	k	k	NOUN
ejpam-5973	309	7	,	,	PUNCT
ejpam-5973	309	8	k	k	PROPN
ejpam-5973	310	1	+	+	PROPN
ejpam-5973	310	2	m	m	VERB
ejpam-5973	310	3	if	if	SCONJ
ejpam-5973	310	4	m	m	VERB
ejpam-5973	310	5	<	<	X
ejpam-5973	310	6	k	k	PROPN
ejpam-5973	310	7	and	and	CCONJ
ejpam-5973	310	8	n	n	PRON
ejpam-5973	310	9	≥	≥	NOUN
ejpam-5973	310	10	k	k	NOUN
ejpam-5973	310	11	,	,	PUNCT
ejpam-5973	310	12	m+	m+	NUM
ejpam-5973	310	13	n	n	NOUN
ejpam-5973	310	14	if	if	SCONJ
ejpam-5973	310	15	n	n	CCONJ
ejpam-5973	310	16	<	<	X
ejpam-5973	310	17	k.	k.	PROPN
ejpam-5973	310	18	proof	proof	NOUN
ejpam-5973	310	19	.	.	PUNCT
ejpam-5973	311	1	it	it	PRON
ejpam-5973	311	2	is	be	AUX
ejpam-5973	311	3	easy	easy	ADJ
ejpam-5973	311	4	to	to	PART
ejpam-5973	311	5	see	see	VERB
ejpam-5973	311	6	that	that	DET
ejpam-5973	311	7	k2	k2	PROPN
ejpam-5973	311	8	m	m	PROPN
ejpam-5973	311	9	,	,	PUNCT
ejpam-5973	311	10	n	n	NOUN
ejpam-5973	311	11	=	=	NUM
ejpam-5973	311	12	km	km	PROPN
ejpam-5973	311	13	∪	∪	PROPN
ejpam-5973	311	14	kn	kn	PROPN
ejpam-5973	311	15	.	.	PUNCT
ejpam-5973	312	1	by	by	ADP
ejpam-5973	312	2	proposition	proposition	NOUN
ejpam-5973	312	3	2	2	NUM
ejpam-5973	312	4	,	,	PUNCT
ejpam-5973	312	5	for	for	ADP
ejpam-5973	312	6	any	any	DET
ejpam-5973	312	7	positive	positive	ADJ
ejpam-5973	312	8	integer	integer	NOUN
ejpam-5973	312	9	s	s	NOUN
ejpam-5973	312	10	,	,	PUNCT
ejpam-5973	312	11	we	we	PRON
ejpam-5973	312	12	have	have	VERB
ejpam-5973	312	13	γrk(ks	γrk(ks	NOUN
ejpam-5973	312	14	)	)	PUNCT
ejpam-5973	313	1	=	=	SYM
ejpam-5973	313	2	min{k	min{k	X
ejpam-5973	313	3	,	,	PUNCT
ejpam-5973	313	4	s	s	NOUN
ejpam-5973	313	5	}	}	PUNCT
ejpam-5973	313	6	.	.	PUNCT
ejpam-5973	314	1	so	so	ADV
ejpam-5973	314	2	,	,	PUNCT
ejpam-5973	314	3	applying	apply	VERB
ejpam-5973	314	4	remark	remark	NOUN
ejpam-5973	314	5	1	1	NUM
ejpam-5973	314	6	yields	yield	NOUN
ejpam-5973	314	7	γhrk(km	γhrk(km	ADJ
ejpam-5973	314	8	,	,	PUNCT
ejpam-5973	314	9	n	n	CCONJ
ejpam-5973	314	10	)	)	PUNCT
ejpam-5973	314	11	=	=	SYM
ejpam-5973	315	1	γrk(k	γrk(k	PROPN
ejpam-5973	315	2	2	2	NUM
ejpam-5973	315	3	m	m	NOUN
ejpam-5973	315	4	,	,	PUNCT
ejpam-5973	315	5	n	n	CCONJ
ejpam-5973	315	6	)	)	PUNCT
ejpam-5973	315	7	=	=	SYM
ejpam-5973	315	8	γrk(km	γrk(km	NOUN
ejpam-5973	315	9	)	)	PUNCT
ejpam-5973	315	10	+	+	CCONJ
ejpam-5973	315	11	γrk(kn	γrk(kn	NUM
ejpam-5973	315	12	)	)	PUNCT
ejpam-5973	315	13	=	=	SYM
ejpam-5973	315	14	min{k	min{k	NOUN
ejpam-5973	315	15	,	,	PUNCT
ejpam-5973	315	16	m}+min{k	m}+min{k	PROPN
ejpam-5973	315	17	,	,	PUNCT
ejpam-5973	315	18	n	n	CCONJ
ejpam-5973	315	19	}	}	PUNCT
ejpam-5973	315	20	,	,	PUNCT
ejpam-5973	315	21	as	as	SCONJ
ejpam-5973	315	22	desired	desire	VERB
ejpam-5973	315	23	.	.	PUNCT
ejpam-5973	316	1	in	in	ADP
ejpam-5973	316	2	the	the	DET
ejpam-5973	316	3	next	next	ADJ
ejpam-5973	316	4	theorem	theorem	NOUN
ejpam-5973	316	5	we	we	PRON
ejpam-5973	316	6	provide	provide	VERB
ejpam-5973	316	7	an	an	DET
ejpam-5973	316	8	upper	upper	NOUN
ejpam-5973	316	9	and	and	CCONJ
ejpam-5973	316	10	a	a	DET
ejpam-5973	316	11	lower	lower	ADV
ejpam-5973	316	12	bound	bind	VERB
ejpam-5973	316	13	on	on	ADP
ejpam-5973	316	14	the	the	DET
ejpam-5973	316	15	hop	hop	NOUN
ejpam-5973	316	16	k	k	ADJ
ejpam-5973	316	17	-	-	PUNCT
ejpam-5973	316	18	rainbow	rainbow	NOUN
ejpam-5973	316	19	domination	domination	NOUN
ejpam-5973	316	20	number	number	NOUN
ejpam-5973	316	21	in	in	ADP
ejpam-5973	316	22	terms	term	NOUN
ejpam-5973	316	23	of	of	ADP
ejpam-5973	316	24	hop	hop	NOUN
ejpam-5973	316	25	domination	domination	NOUN
ejpam-5973	316	26	number	number	NOUN
ejpam-5973	316	27	.	.	PUNCT
ejpam-5973	317	1	we	we	PRON
ejpam-5973	317	2	recall	recall	VERB
ejpam-5973	317	3	that	that	DET
ejpam-5973	317	4	γh(kn	γh(kn	PROPN
ejpam-5973	317	5	)	)	PUNCT
ejpam-5973	318	1	=	=	SYM
ejpam-5973	318	2	n	n	PROPN
ejpam-5973	318	3	and	and	CCONJ
ejpam-5973	318	4	γh(km	γh(km	NOUN
ejpam-5973	318	5	,	,	PUNCT
ejpam-5973	318	6	n	n	CCONJ
ejpam-5973	318	7	)	)	PUNCT
ejpam-5973	318	8	=	=	SYM
ejpam-5973	318	9	2	2	NUM
ejpam-5973	318	10	(	(	PUNCT
ejpam-5973	318	11	see	see	VERB
ejpam-5973	318	12	[	[	X
ejpam-5973	318	13	10	10	NUM
ejpam-5973	318	14	]	]	NUM
ejpam-5973	318	15	)	)	PUNCT
ejpam-5973	318	16	.	.	PUNCT
ejpam-5973	319	1	theorem	theorem	ADJ
ejpam-5973	319	2	7	7	NUM
ejpam-5973	319	3	.	.	PUNCT
ejpam-5973	320	1	let	let	VERB
ejpam-5973	320	2	k	k	PROPN
ejpam-5973	320	3	≥	≥	NUM
ejpam-5973	320	4	2	2	NUM
ejpam-5973	320	5	be	be	AUX
ejpam-5973	320	6	an	an	DET
ejpam-5973	320	7	integer	integer	NOUN
ejpam-5973	320	8	and	and	CCONJ
ejpam-5973	320	9	let	let	VERB
ejpam-5973	320	10	g	g	PRON
ejpam-5973	320	11	be	be	AUX
ejpam-5973	320	12	a	a	DET
ejpam-5973	320	13	graph	graph	NOUN
ejpam-5973	320	14	of	of	ADP
ejpam-5973	320	15	order	order	NOUN
ejpam-5973	320	16	n	n	PRON
ejpam-5973	320	17	≥	≥	NOUN
ejpam-5973	320	18	k.	k.	PROPN
ejpam-5973	321	1	then	then	ADV
ejpam-5973	321	2	γh(g	γh(g	PUNCT
ejpam-5973	321	3	)	)	PUNCT
ejpam-5973	321	4	≤	≤	NUM
ejpam-5973	321	5	γhrk(g	γhrk(g	PROPN
ejpam-5973	321	6	)	)	PUNCT
ejpam-5973	321	7	≤	≤	NUM
ejpam-5973	321	8	kγh(g	kγh(g	NOUN
ejpam-5973	321	9	)	)	PUNCT
ejpam-5973	321	10	.	.	PUNCT
ejpam-5973	322	1	moreover	moreover	ADV
ejpam-5973	322	2	,	,	PUNCT
ejpam-5973	322	3	these	these	DET
ejpam-5973	322	4	bounds	bound	NOUN
ejpam-5973	322	5	are	be	AUX
ejpam-5973	322	6	sharp	sharp	ADJ
ejpam-5973	322	7	.	.	PUNCT
ejpam-5973	323	1	proof	proof	NOUN
ejpam-5973	323	2	.	.	PUNCT
ejpam-5973	324	1	we	we	PRON
ejpam-5973	324	2	first	first	ADV
ejpam-5973	324	3	show	show	VERB
ejpam-5973	324	4	that	that	SCONJ
ejpam-5973	324	5	the	the	DET
ejpam-5973	324	6	left	left	ADJ
ejpam-5973	324	7	inequality	inequality	NOUN
ejpam-5973	324	8	holds	hold	VERB
ejpam-5973	324	9	.	.	PUNCT
ejpam-5973	325	1	let	let	VERB
ejpam-5973	325	2	f	f	PRON
ejpam-5973	325	3	be	be	AUX
ejpam-5973	325	4	a	a	DET
ejpam-5973	325	5	γhrk	γhrk	NOUN
ejpam-5973	325	6	-	-	PUNCT
ejpam-5973	325	7	function	function	NOUN
ejpam-5973	325	8	of	of	ADP
ejpam-5973	325	9	g.	g.	PROPN
ejpam-5973	325	10	then	then	ADV
ejpam-5973	325	11	clearly	clearly	ADV
ejpam-5973	325	12	the	the	DET
ejpam-5973	325	13	set	set	NOUN
ejpam-5973	325	14	s	s	PART
ejpam-5973	325	15	=	=	PUNCT
ejpam-5973	325	16	{	{	PUNCT
ejpam-5973	325	17	v	v	NUM
ejpam-5973	325	18	∈	∈	NOUN
ejpam-5973	325	19	v	v	NOUN
ejpam-5973	325	20	(	(	PUNCT
ejpam-5973	325	21	g	g	NOUN
ejpam-5973	325	22	)	)	PUNCT
ejpam-5973	325	23	:	:	PUNCT
ejpam-5973	325	24	f(v	f(v	NOUN
ejpam-5973	325	25	)	)	PUNCT
ejpam-5973	325	26	̸=	̸=	NOUN
ejpam-5973	325	27	∅	∅	NOUN
ejpam-5973	325	28	}	}	PUNCT
ejpam-5973	325	29	is	be	AUX
ejpam-5973	325	30	a	a	DET
ejpam-5973	325	31	hop	hop	NOUN
ejpam-5973	325	32	dominating	dominating	NOUN
ejpam-5973	325	33	set	set	NOUN
ejpam-5973	325	34	of	of	ADP
ejpam-5973	325	35	g.	g.	PROPN
ejpam-5973	325	36	therefore	therefore	ADV
ejpam-5973	325	37	,	,	PUNCT
ejpam-5973	325	38	|s|	|s|	VERB
ejpam-5973	325	39	≥	≥	NOUN
ejpam-5973	325	40	γh(g	γh(g	NOUN
ejpam-5973	325	41	)	)	PUNCT
ejpam-5973	325	42	.	.	PUNCT
ejpam-5973	326	1	since	since	SCONJ
ejpam-5973	326	2	the	the	DET
ejpam-5973	326	3	weight	weight	NOUN
ejpam-5973	326	4	ω(f	ω(f	PUNCT
ejpam-5973	326	5	)	)	PUNCT
ejpam-5973	326	6	=	=	SYM
ejpam-5973	326	7	∑	∑	PUNCT
ejpam-5973	326	8	v∈v	v∈v	NOUN
ejpam-5973	326	9	(	(	PUNCT
ejpam-5973	326	10	g	g	NOUN
ejpam-5973	326	11	)	)	PUNCT
ejpam-5973	326	12	|f(v)|	|f(v)|	PROPN
ejpam-5973	326	13	and	and	CCONJ
ejpam-5973	326	14	|f(v)|	|f(v)|	ADP
ejpam-5973	326	15	≥	≥	NUM
ejpam-5973	326	16	1	1	NUM
ejpam-5973	326	17	for	for	ADP
ejpam-5973	326	18	each	each	DET
ejpam-5973	326	19	v	v	NUM
ejpam-5973	326	20	∈	∈	PROPN
ejpam-5973	326	21	s	s	NOUN
ejpam-5973	326	22	,	,	PUNCT
ejpam-5973	326	23	it	it	PRON
ejpam-5973	326	24	follows	follow	VERB
ejpam-5973	326	25	that	that	SCONJ
ejpam-5973	326	26	ω(f	ω(f	ADJ
ejpam-5973	326	27	)	)	PUNCT
ejpam-5973	326	28	≥	≥	NOUN
ejpam-5973	326	29	|s|	|s|	VERB
ejpam-5973	326	30	≥	≥	NOUN
ejpam-5973	326	31	γh(g	γh(g	NOUN
ejpam-5973	326	32	)	)	PUNCT
ejpam-5973	326	33	.	.	PUNCT
ejpam-5973	327	1	this	this	PRON
ejpam-5973	327	2	implies	imply	VERB
ejpam-5973	327	3	that	that	SCONJ
ejpam-5973	327	4	γh(g	γh(g	NOUN
ejpam-5973	327	5	)	)	PUNCT
ejpam-5973	327	6	≤	≤	NUM
ejpam-5973	327	7	γhrk(g	γhrk(g	PROPN
ejpam-5973	327	8	)	)	PUNCT
ejpam-5973	327	9	.	.	PUNCT
ejpam-5973	328	1	next	next	ADV
ejpam-5973	328	2	,	,	PUNCT
ejpam-5973	328	3	we	we	PRON
ejpam-5973	328	4	will	will	AUX
ejpam-5973	328	5	show	show	VERB
ejpam-5973	328	6	that	that	PRON
ejpam-5973	328	7	γhrk(g	γhrk(g	NOUN
ejpam-5973	328	8	)	)	PUNCT
ejpam-5973	328	9	≤	≤	NUM
ejpam-5973	328	10	kγh(g	kγh(g	NOUN
ejpam-5973	328	11	)	)	PUNCT
ejpam-5973	328	12	.	.	PUNCT
ejpam-5973	329	1	consider	consider	VERB
ejpam-5973	329	2	a	a	DET
ejpam-5973	329	3	minimum	minimum	ADJ
ejpam-5973	329	4	hop	hop	NOUN
ejpam-5973	329	5	dominating	dominating	NOUN
ejpam-5973	329	6	set	set	NOUN
ejpam-5973	329	7	s	s	NOUN
ejpam-5973	329	8	of	of	ADP
ejpam-5973	329	9	g	g	NOUN
ejpam-5973	329	10	with	with	ADP
ejpam-5973	329	11	|s|	|s|	NOUN
ejpam-5973	329	12	=	=	NOUN
ejpam-5973	329	13	γh(g	γh(g	NOUN
ejpam-5973	329	14	)	)	PUNCT
ejpam-5973	329	15	.	.	PUNCT
ejpam-5973	330	1	define	define	VERB
ejpam-5973	330	2	a	a	DET
ejpam-5973	330	3	function	function	NOUN
ejpam-5973	330	4	f	f	NOUN
ejpam-5973	330	5	:	:	PUNCT
ejpam-5973	330	6	v	v	X
ejpam-5973	330	7	(	(	PUNCT
ejpam-5973	330	8	g	g	NOUN
ejpam-5973	330	9	)	)	PUNCT
ejpam-5973	330	10	→	→	SYM
ejpam-5973	331	1	p({1	p({1	PROPN
ejpam-5973	331	2	,	,	PUNCT
ejpam-5973	331	3	2	2	NUM
ejpam-5973	331	4	,	,	PUNCT
ejpam-5973	331	5	.	.	PUNCT
ejpam-5973	331	6	.	.	PUNCT
ejpam-5973	332	1	.	.	PUNCT
ejpam-5973	333	1	,	,	PUNCT
ejpam-5973	333	2	k	k	X
ejpam-5973	333	3	}	}	PUNCT
ejpam-5973	333	4	)	)	PUNCT
ejpam-5973	333	5	by	by	ADP
ejpam-5973	333	6	f(v	f(v	NOUN
ejpam-5973	333	7	)	)	PUNCT
ejpam-5973	334	1	=	=	PRON
ejpam-5973	334	2	{	{	PUNCT
ejpam-5973	334	3	1	1	NUM
ejpam-5973	334	4	,	,	PUNCT
ejpam-5973	334	5	2	2	NUM
ejpam-5973	334	6	,	,	PUNCT
ejpam-5973	334	7	.	.	PUNCT
ejpam-5973	334	8	.	.	PUNCT
ejpam-5973	334	9	.	.	PUNCT
ejpam-5973	335	1	,	,	PUNCT
ejpam-5973	335	2	k	k	X
ejpam-5973	335	3	}	}	PUNCT
ejpam-5973	335	4	for	for	ADP
ejpam-5973	335	5	each	each	DET
ejpam-5973	335	6	vertex	vertex	NOUN
ejpam-5973	335	7	v	v	ADP
ejpam-5973	335	8	∈	∈	PROPN
ejpam-5973	335	9	s	s	NOUN
ejpam-5973	335	10	and	and	CCONJ
ejpam-5973	335	11	f(v	f(v	NOUN
ejpam-5973	335	12	)	)	PUNCT
ejpam-5973	335	13	=	=	NOUN
ejpam-5973	335	14	∅	∅	NOUN
ejpam-5973	335	15	for	for	ADP
ejpam-5973	335	16	every	every	PRON
ejpam-5973	335	17	v	v	NUM
ejpam-5973	335	18	∈	∈	PROPN
ejpam-5973	335	19	v	v	NOUN
ejpam-5973	335	20	(	(	PUNCT
ejpam-5973	335	21	g	g	NOUN
ejpam-5973	335	22	)	)	PUNCT
ejpam-5973	335	23	\	\	NOUN
ejpam-5973	336	1	s.	s.	PROPN
ejpam-5973	336	2	for	for	ADP
ejpam-5973	336	3	each	each	DET
ejpam-5973	336	4	v	v	NUM
ejpam-5973	336	5	∈	∈	NOUN
ejpam-5973	336	6	v	v	NOUN
ejpam-5973	336	7	(	(	PUNCT
ejpam-5973	336	8	g)\s	g)\s	NOUN
ejpam-5973	336	9	,	,	PUNCT
ejpam-5973	336	10	there	there	PRON
ejpam-5973	336	11	exists	exist	VERB
ejpam-5973	336	12	a	a	DET
ejpam-5973	336	13	vertex	vertex	NOUN
ejpam-5973	336	14	u	u	NOUN
ejpam-5973	336	15	∈	∈	NOUN
ejpam-5973	336	16	s	s	VERB
ejpam-5973	336	17	such	such	ADJ
ejpam-5973	336	18	that	that	SCONJ
ejpam-5973	336	19	d(u	d(u	PROPN
ejpam-5973	336	20	,	,	PUNCT
ejpam-5973	336	21	v	v	NOUN
ejpam-5973	336	22	)	)	PUNCT
ejpam-5973	336	23	=	=	SYM
ejpam-5973	336	24	2	2	NUM
ejpam-5973	336	25	since	since	SCONJ
ejpam-5973	336	26	s	s	NOUN
ejpam-5973	336	27	is	be	AUX
ejpam-5973	336	28	a	a	DET
ejpam-5973	336	29	hop	hop	NOUN
ejpam-5973	336	30	dominating	dominating	NOUN
ejpam-5973	336	31	set	set	NOUN
ejpam-5973	336	32	.	.	PUNCT
ejpam-5973	337	1	since	since	SCONJ
ejpam-5973	337	2	f(u	f(u	PROPN
ejpam-5973	337	3	)	)	PUNCT
ejpam-5973	337	4	=	=	PRON
ejpam-5973	337	5	{	{	PUNCT
ejpam-5973	337	6	1	1	NUM
ejpam-5973	337	7	,	,	PUNCT
ejpam-5973	337	8	2	2	NUM
ejpam-5973	337	9	,	,	PUNCT
ejpam-5973	337	10	.	.	PUNCT
ejpam-5973	337	11	.	.	PUNCT
ejpam-5973	337	12	.	.	PUNCT
ejpam-5973	338	1	,	,	PUNCT
ejpam-5973	338	2	k	k	X
ejpam-5973	338	3	}	}	PUNCT
ejpam-5973	338	4	for	for	ADP
ejpam-5973	338	5	each	each	DET
ejpam-5973	338	6	u	u	PROPN
ejpam-5973	338	7	∈	∈	PROPN
ejpam-5973	338	8	s	s	PART
ejpam-5973	338	9	,	,	PUNCT
ejpam-5973	338	10	it	it	PRON
ejpam-5973	338	11	follows	follow	VERB
ejpam-5973	338	12	that	that	SCONJ
ejpam-5973	338	13	⋃	⋃	ADJ
ejpam-5973	338	14	u∈n2	u∈n2	ADJ
ejpam-5973	338	15	g(v	g(v	NOUN
ejpam-5973	338	16	)	)	PUNCT
ejpam-5973	338	17	f(u	f(u	PROPN
ejpam-5973	338	18	)	)	PUNCT
ejpam-5973	338	19	=	=	PRON
ejpam-5973	338	20	{	{	PUNCT
ejpam-5973	338	21	1	1	NUM
ejpam-5973	338	22	,	,	PUNCT
ejpam-5973	338	23	2	2	NUM
ejpam-5973	338	24	,	,	PUNCT
ejpam-5973	338	25	.	.	PUNCT
ejpam-5973	338	26	.	.	PUNCT
ejpam-5973	339	1	.	.	PUNCT
ejpam-5973	340	1	,	,	PUNCT
ejpam-5973	340	2	k	k	X
ejpam-5973	340	3	}	}	PUNCT
ejpam-5973	340	4	j.	j.	PROPN
ejpam-5973	340	5	j.	j.	PROPN
ejpam-5973	340	6	hamja	hamja	PROPN
ejpam-5973	341	1	et	et	PROPN
ejpam-5973	341	2	al	al	PROPN
ejpam-5973	341	3	.	.	PUNCT
ejpam-5973	341	4	/	/	SYM
ejpam-5973	341	5	eur	eur	PROPN
ejpam-5973	341	6	.	.	PUNCT
ejpam-5973	342	1	j.	j.	PROPN
ejpam-5973	342	2	pure	pure	PROPN
ejpam-5973	342	3	appl	appl	PROPN
ejpam-5973	342	4	.	.	PROPN
ejpam-5973	342	5	math	math	PROPN
ejpam-5973	342	6	,	,	PUNCT
ejpam-5973	342	7	18	18	NUM
ejpam-5973	342	8	(	(	PUNCT
ejpam-5973	342	9	2	2	NUM
ejpam-5973	342	10	)	)	PUNCT
ejpam-5973	342	11	(	(	PUNCT
ejpam-5973	342	12	2025	2025	NUM
ejpam-5973	342	13	)	)	PUNCT
ejpam-5973	342	14	,	,	PUNCT
ejpam-5973	342	15	5973	5973	NUM
ejpam-5973	342	16	9	9	NUM
ejpam-5973	342	17	of	of	ADP
ejpam-5973	342	18	17	17	NUM
ejpam-5973	342	19	for	for	ADP
ejpam-5973	342	20	each	each	PRON
ejpam-5973	342	21	v	v	NUM
ejpam-5973	342	22	∈	∈	PROPN
ejpam-5973	342	23	v	v	NOUN
ejpam-5973	342	24	(	(	PUNCT
ejpam-5973	342	25	g	g	NOUN
ejpam-5973	342	26	)	)	PUNCT
ejpam-5973	342	27	\	\	NOUN
ejpam-5973	343	1	s.	s.	PROPN
ejpam-5973	343	2	thus	thus	ADV
ejpam-5973	343	3	,	,	PUNCT
ejpam-5973	343	4	f	f	PROPN
ejpam-5973	343	5	is	be	AUX
ejpam-5973	343	6	a	a	DET
ejpam-5973	343	7	hop	hop	NOUN
ejpam-5973	343	8	k	k	ADJ
ejpam-5973	343	9	-	-	PUNCT
ejpam-5973	343	10	rainbow	rainbow	NOUN
ejpam-5973	343	11	dominating	dominating	NOUN
ejpam-5973	343	12	function	function	NOUN
ejpam-5973	343	13	of	of	ADP
ejpam-5973	343	14	g	g	NOUN
ejpam-5973	343	15	of	of	ADP
ejpam-5973	343	16	weight	weight	NOUN
ejpam-5973	343	17	ω(f	ω(f	PUNCT
ejpam-5973	343	18	)	)	PUNCT
ejpam-5973	343	19	=	=	SYM
ejpam-5973	343	20	kγh(g	kγh(g	NOUN
ejpam-5973	343	21	)	)	PUNCT
ejpam-5973	343	22	.	.	PUNCT
ejpam-5973	344	1	hence	hence	ADV
ejpam-5973	344	2	,	,	PUNCT
ejpam-5973	344	3	we	we	PRON
ejpam-5973	344	4	have	have	VERB
ejpam-5973	344	5	γhrk(g	γhrk(g	NOUN
ejpam-5973	344	6	)	)	PUNCT
ejpam-5973	344	7	≤	≤	NUM
ejpam-5973	344	8	kγh(g	kγh(g	NOUN
ejpam-5973	344	9	)	)	PUNCT
ejpam-5973	344	10	.	.	PUNCT
ejpam-5973	345	1	for	for	ADP
ejpam-5973	345	2	the	the	DET
ejpam-5973	345	3	sharpness	sharpness	NOUN
ejpam-5973	345	4	of	of	ADP
ejpam-5973	345	5	the	the	DET
ejpam-5973	345	6	lower	lower	ADV
ejpam-5973	345	7	bound	bind	VERB
ejpam-5973	345	8	,	,	PUNCT
ejpam-5973	345	9	let	let	VERB
ejpam-5973	345	10	g	g	PROPN
ejpam-5973	345	11	=	=	PROPN
ejpam-5973	345	12	kn	kn	PROPN
ejpam-5973	345	13	.	.	PUNCT
ejpam-5973	346	1	then	then	ADV
ejpam-5973	346	2	by	by	ADP
ejpam-5973	346	3	corollary	corollary	ADJ
ejpam-5973	346	4	1	1	NUM
ejpam-5973	346	5	,	,	PUNCT
ejpam-5973	346	6	we	we	PRON
ejpam-5973	346	7	have	have	VERB
ejpam-5973	346	8	γh(g	γh(g	NOUN
ejpam-5973	346	9	)	)	PUNCT
ejpam-5973	347	1	=	=	SYM
ejpam-5973	347	2	γhrk(g	γhrk(g	PROPN
ejpam-5973	347	3	)	)	PUNCT
ejpam-5973	347	4	.	.	PUNCT
ejpam-5973	348	1	for	for	ADP
ejpam-5973	348	2	the	the	DET
ejpam-5973	348	3	upper	upper	ADJ
ejpam-5973	348	4	bound	bound	NOUN
ejpam-5973	348	5	,	,	PUNCT
ejpam-5973	348	6	let	let	VERB
ejpam-5973	348	7	g	g	NOUN
ejpam-5973	348	8	=	=	PROPN
ejpam-5973	348	9	km	km	PROPN
ejpam-5973	348	10	,	,	PUNCT
ejpam-5973	348	11	n	n	CCONJ
ejpam-5973	348	12	,	,	PUNCT
ejpam-5973	348	13	where	where	SCONJ
ejpam-5973	348	14	n	n	PRON
ejpam-5973	348	15	≥	≥	NOUN
ejpam-5973	348	16	m	m	VERB
ejpam-5973	348	17	≥	≥	PROPN
ejpam-5973	348	18	k.	k.	PROPN
ejpam-5973	348	19	then	then	ADV
ejpam-5973	348	20	by	by	ADP
ejpam-5973	348	21	proposition	proposition	NOUN
ejpam-5973	348	22	3	3	NUM
ejpam-5973	348	23	,	,	PUNCT
ejpam-5973	348	24	we	we	PRON
ejpam-5973	348	25	have	have	VERB
ejpam-5973	348	26	γhrk(km	γhrk(km	ADJ
ejpam-5973	348	27	,	,	PUNCT
ejpam-5973	348	28	n	n	CCONJ
ejpam-5973	348	29	)	)	PUNCT
ejpam-5973	349	1	=	=	SYM
ejpam-5973	349	2	2k	2k	NUM
ejpam-5973	349	3	=	=	SYM
ejpam-5973	349	4	kγh(km	kγh(km	X
ejpam-5973	349	5	,	,	PUNCT
ejpam-5973	349	6	n	n	CCONJ
ejpam-5973	349	7	)	)	PUNCT
ejpam-5973	349	8	.	.	PUNCT
ejpam-5973	350	1	4	4	X
ejpam-5973	350	2	.	.	X
ejpam-5973	350	3	exact	exact	ADJ
ejpam-5973	350	4	values	value	NOUN
ejpam-5973	350	5	the	the	DET
ejpam-5973	350	6	exact	exact	ADJ
ejpam-5973	350	7	values	value	NOUN
ejpam-5973	350	8	of	of	ADP
ejpam-5973	350	9	the	the	DET
ejpam-5973	350	10	k	k	ADJ
ejpam-5973	350	11	-	-	PUNCT
ejpam-5973	350	12	rainbow	rainbow	NOUN
ejpam-5973	350	13	domination	domination	NOUN
ejpam-5973	350	14	number	number	NOUN
ejpam-5973	350	15	for	for	ADP
ejpam-5973	350	16	k	k	PROPN
ejpam-5973	350	17	∈	∈	PROPN
ejpam-5973	350	18	{	{	PUNCT
ejpam-5973	350	19	2	2	NUM
ejpam-5973	350	20	,	,	PUNCT
ejpam-5973	350	21	3	3	NUM
ejpam-5973	350	22	}	}	PUNCT
ejpam-5973	350	23	of	of	ADP
ejpam-5973	350	24	paths	path	NOUN
ejpam-5973	350	25	and	and	CCONJ
ejpam-5973	350	26	cycles	cycle	NOUN
ejpam-5973	350	27	are	be	AUX
ejpam-5973	350	28	determined	determine	VERB
ejpam-5973	350	29	as	as	SCONJ
ejpam-5973	350	30	follows	follow	VERB
ejpam-5973	350	31	.	.	PUNCT
ejpam-5973	351	1	theorem	theorem	ADJ
ejpam-5973	351	2	8	8	NUM
ejpam-5973	351	3	.	.	PUNCT
ejpam-5973	352	1	[	[	X
ejpam-5973	352	2	3	3	NUM
ejpam-5973	352	3	,	,	PUNCT
ejpam-5973	352	4	6	6	NUM
ejpam-5973	352	5	]	]	PUNCT
ejpam-5973	352	6	let	let	VERB
ejpam-5973	352	7	n	n	PRON
ejpam-5973	352	8	be	be	AUX
ejpam-5973	352	9	a	a	DET
ejpam-5973	352	10	positive	positive	ADJ
ejpam-5973	352	11	integer	integer	NOUN
ejpam-5973	352	12	.	.	PUNCT
ejpam-5973	353	1	then	then	ADV
ejpam-5973	353	2	(	(	PUNCT
ejpam-5973	353	3	i	i	NOUN
ejpam-5973	353	4	)	)	PUNCT
ejpam-5973	353	5	γr2(pn	γr2(pn	PROPN
ejpam-5973	353	6	)	)	PUNCT
ejpam-5973	353	7	=	=	PUNCT
ejpam-5973	353	8	⌊	⌊	VERB
ejpam-5973	353	9	n	n	ADV
ejpam-5973	353	10	2	2	NUM
ejpam-5973	353	11	⌋	⌋	NOUN
ejpam-5973	353	12	+	+	CCONJ
ejpam-5973	353	13	1	1	X
ejpam-5973	353	14	.	.	X
ejpam-5973	353	15	(	(	PUNCT
ejpam-5973	353	16	ii	ii	NOUN
ejpam-5973	353	17	)	)	PUNCT
ejpam-5973	353	18	for	for	ADP
ejpam-5973	353	19	n	n	X
ejpam-5973	353	20	≥	≥	NUM
ejpam-5973	353	21	3	3	NUM
ejpam-5973	353	22	,	,	PUNCT
ejpam-5973	353	23	γr2(cn	γr2(cn	NOUN
ejpam-5973	353	24	)	)	PUNCT
ejpam-5973	353	25	=	=	SYM
ejpam-5973	354	1	⌊	⌊	VERB
ejpam-5973	354	2	n	n	ADV
ejpam-5973	354	3	2	2	NUM
ejpam-5973	354	4	⌋	⌋	NOUN
ejpam-5973	354	5	+	+	CCONJ
ejpam-5973	354	6	⌈	⌈	SYM
ejpam-5973	354	7	n	n	PRON
ejpam-5973	354	8	4	4	NUM
ejpam-5973	354	9	⌉	⌉	PART
ejpam-5973	354	10	−	−	ADP
ejpam-5973	354	11	⌊	⌊	PROPN
ejpam-5973	354	12	n	n	ADV
ejpam-5973	354	13	4	4	NUM
ejpam-5973	354	14	⌋	⌋	NOUN
ejpam-5973	354	15	.	.	PUNCT
ejpam-5973	355	1	(	(	PUNCT
ejpam-5973	355	2	iii	iii	NOUN
ejpam-5973	355	3	)	)	PUNCT
ejpam-5973	355	4	for	for	ADP
ejpam-5973	355	5	n	n	X
ejpam-5973	355	6	≥	≥	NUM
ejpam-5973	355	7	5	5	NUM
ejpam-5973	355	8	,	,	PUNCT
ejpam-5973	355	9	γr3(pn	γr3(pn	NOUN
ejpam-5973	355	10	)	)	PUNCT
ejpam-5973	355	11	=	=	PUNCT
ejpam-5973	355	12			PUNCT
ejpam-5973	356	1	⌈	⌈	NOUN
ejpam-5973	356	2	3n	3n	NUM
ejpam-5973	356	3	4	4	NUM
ejpam-5973	356	4	⌉	⌉	NOUN
ejpam-5973	356	5	+	+	CCONJ
ejpam-5973	356	6	1	1	NUM
ejpam-5973	356	7	if	if	SCONJ
ejpam-5973	356	8	n	n	PRON
ejpam-5973	356	9	≡	≡	PROPN
ejpam-5973	356	10	0	0	PUNCT
ejpam-5973	356	11	(	(	PUNCT
ejpam-5973	356	12	mod	mod	PROPN
ejpam-5973	356	13	4),⌈	4),⌈	NUM
ejpam-5973	356	14	3n	3n	NUM
ejpam-5973	356	15	4	4	NUM
ejpam-5973	356	16	⌉	⌉	NOUN
ejpam-5973	356	17	if	if	SCONJ
ejpam-5973	356	18	n	n	PRON
ejpam-5973	356	19	≡	≡	PROPN
ejpam-5973	356	20	1	1	NUM
ejpam-5973	356	21	,	,	PUNCT
ejpam-5973	356	22	2	2	NUM
ejpam-5973	356	23	,	,	PUNCT
ejpam-5973	356	24	3	3	NUM
ejpam-5973	356	25	(	(	PUNCT
ejpam-5973	356	26	mod	mod	NOUN
ejpam-5973	356	27	4	4	NUM
ejpam-5973	356	28	)	)	PUNCT
ejpam-5973	356	29	.	.	PUNCT
ejpam-5973	357	1	(	(	PUNCT
ejpam-5973	357	2	iv	iv	X
ejpam-5973	357	3	)	)	PUNCT
ejpam-5973	357	4	for	for	ADP
ejpam-5973	357	5	n	n	X
ejpam-5973	357	6	≥	≥	NUM
ejpam-5973	357	7	5	5	NUM
ejpam-5973	357	8	,	,	PUNCT
ejpam-5973	357	9	γr3(cn	γr3(cn	NUM
ejpam-5973	357	10	)	)	PUNCT
ejpam-5973	358	1	=	=	PUNCT
ejpam-5973	359	1	⌈	⌈	NOUN
ejpam-5973	359	2	3n	3n	NUM
ejpam-5973	359	3	4	4	NUM
ejpam-5973	359	4	⌉	⌉	NOUN
ejpam-5973	359	5	.	.	PUNCT
ejpam-5973	360	1	using	use	VERB
ejpam-5973	360	2	theorem	theorem	ADJ
ejpam-5973	360	3	8	8	NUM
ejpam-5973	360	4	and	and	CCONJ
ejpam-5973	360	5	proposition	proposition	NOUN
ejpam-5973	360	6	1	1	NUM
ejpam-5973	360	7	,	,	PUNCT
ejpam-5973	360	8	we	we	PRON
ejpam-5973	360	9	determine	determine	VERB
ejpam-5973	360	10	the	the	DET
ejpam-5973	360	11	exact	exact	ADJ
ejpam-5973	360	12	value	value	NOUN
ejpam-5973	360	13	of	of	ADP
ejpam-5973	360	14	hop	hop	NOUN
ejpam-5973	360	15	k	k	ADJ
ejpam-5973	360	16	-	-	PUNCT
ejpam-5973	360	17	rainbow	rainbow	NOUN
ejpam-5973	360	18	domination	domination	NOUN
ejpam-5973	360	19	number	number	NOUN
ejpam-5973	360	20	for	for	ADP
ejpam-5973	360	21	k	k	PROPN
ejpam-5973	360	22	∈	∈	PROPN
ejpam-5973	360	23	{	{	PUNCT
ejpam-5973	360	24	2	2	NUM
ejpam-5973	360	25	,	,	PUNCT
ejpam-5973	360	26	3	3	NUM
ejpam-5973	360	27	}	}	PUNCT
ejpam-5973	360	28	of	of	ADP
ejpam-5973	360	29	paths	path	NOUN
ejpam-5973	360	30	and	and	CCONJ
ejpam-5973	360	31	cycles	cycle	NOUN
ejpam-5973	360	32	in	in	ADP
ejpam-5973	360	33	the	the	DET
ejpam-5973	360	34	next	next	ADJ
ejpam-5973	360	35	theorem	theorem	PROPN
ejpam-5973	360	36	.	.	PUNCT
ejpam-5973	360	37	theorem	theorem	NOUN
ejpam-5973	360	38	9	9	NUM
ejpam-5973	360	39	.	.	PUNCT
ejpam-5973	361	1	let	let	VERB
ejpam-5973	361	2	n	n	PRON
ejpam-5973	361	3	be	be	AUX
ejpam-5973	361	4	a	a	DET
ejpam-5973	361	5	positive	positive	ADJ
ejpam-5973	361	6	integer	integer	NOUN
ejpam-5973	361	7	.	.	PUNCT
ejpam-5973	362	1	then	then	ADV
ejpam-5973	362	2	(	(	PUNCT
ejpam-5973	362	3	i	i	NOUN
ejpam-5973	362	4	)	)	PUNCT
ejpam-5973	362	5	γhr2(pn	γhr2(pn	PROPN
ejpam-5973	362	6	)	)	PUNCT
ejpam-5973	362	7	=	=	PUNCT
ejpam-5973	362	8			PUNCT
ejpam-5973	362	9	⌊	⌊	VERB
ejpam-5973	362	10	n+1	n+1	NUM
ejpam-5973	362	11	4	4	NUM
ejpam-5973	362	12	⌋	⌋	NOUN
ejpam-5973	363	1	+	+	CCONJ
ejpam-5973	363	2	⌊	⌊	PROPN
ejpam-5973	363	3	n−1	n−1	PROPN
ejpam-5973	363	4	4	4	NUM
ejpam-5973	363	5	⌋	⌋	NOUN
ejpam-5973	364	1	+	+	CCONJ
ejpam-5973	364	2	2	2	NUM
ejpam-5973	364	3	if	if	SCONJ
ejpam-5973	364	4	n	n	PRON
ejpam-5973	364	5	≡	≡	PROPN
ejpam-5973	364	6	1	1	NUM
ejpam-5973	364	7	,	,	PUNCT
ejpam-5973	364	8	3	3	NUM
ejpam-5973	364	9	(	(	PUNCT
ejpam-5973	364	10	mod	mod	NOUN
ejpam-5973	364	11	4	4	NUM
ejpam-5973	364	12	)	)	PUNCT
ejpam-5973	364	13	,	,	PUNCT
ejpam-5973	364	14	2	2	NUM
ejpam-5973	364	15	⌊	⌊	PROPN
ejpam-5973	364	16	n	n	ADV
ejpam-5973	364	17	4	4	NUM
ejpam-5973	364	18	⌋	⌋	ADJ
ejpam-5973	364	19	+	+	CCONJ
ejpam-5973	364	20	2	2	NUM
ejpam-5973	364	21	if	if	SCONJ
ejpam-5973	364	22	n	n	PRON
ejpam-5973	364	23	≡	≡	PROPN
ejpam-5973	364	24	0	0	NUM
ejpam-5973	364	25	,	,	PUNCT
ejpam-5973	364	26	2	2	NUM
ejpam-5973	364	27	(	(	PUNCT
ejpam-5973	364	28	mod	mod	NOUN
ejpam-5973	364	29	4	4	NUM
ejpam-5973	364	30	)	)	PUNCT
ejpam-5973	364	31	.	.	PUNCT
ejpam-5973	365	1	(	(	PUNCT
ejpam-5973	365	2	ii	ii	NOUN
ejpam-5973	365	3	)	)	PUNCT
ejpam-5973	365	4	for	for	ADP
ejpam-5973	365	5	n	n	X
ejpam-5973	365	6	≥	≥	NUM
ejpam-5973	365	7	2	2	NUM
ejpam-5973	365	8	,	,	PUNCT
ejpam-5973	365	9	γhr3(pn	γhr3(pn	NOUN
ejpam-5973	365	10	)	)	PUNCT
ejpam-5973	365	11	=	=	SYM
ejpam-5973	366	1			NUM
ejpam-5973	366	2	2	2	NUM
ejpam-5973	366	3	⌈	⌈	NUM
ejpam-5973	366	4	3n	3n	NUM
ejpam-5973	366	5	8	8	NUM
ejpam-5973	366	6	⌉	⌉	NOUN
ejpam-5973	366	7	+	+	CCONJ
ejpam-5973	366	8	2	2	NUM
ejpam-5973	366	9	if	if	SCONJ
ejpam-5973	366	10	n	n	PRON
ejpam-5973	366	11	≡	≡	PROPN
ejpam-5973	366	12	0	0	PUNCT
ejpam-5973	367	1	(	(	PUNCT
ejpam-5973	367	2	mod	mod	PROPN
ejpam-5973	367	3	8),⌈	8),⌈	NUM
ejpam-5973	367	4	3(n+1	3(n+1	NUM
ejpam-5973	367	5	)	)	PUNCT
ejpam-5973	367	6	8	8	NUM
ejpam-5973	367	7	⌉	⌉	NOUN
ejpam-5973	367	8	+	+	CCONJ
ejpam-5973	367	9	⌈	⌈	SYM
ejpam-5973	367	10	3(n−1	3(n−1	ADJ
ejpam-5973	367	11	)	)	PUNCT
ejpam-5973	367	12	8	8	NUM
ejpam-5973	367	13	⌉	⌉	NOUN
ejpam-5973	367	14	+	+	CCONJ
ejpam-5973	367	15	1	1	NUM
ejpam-5973	367	16	if	if	SCONJ
ejpam-5973	367	17	n	n	PRON
ejpam-5973	367	18	≡	≡	PROPN
ejpam-5973	367	19	1	1	NUM
ejpam-5973	367	20	,	,	PUNCT
ejpam-5973	367	21	7	7	NUM
ejpam-5973	367	22	(	(	PUNCT
ejpam-5973	367	23	mod	mod	NOUN
ejpam-5973	367	24	8)	8)	NUM
ejpam-5973	367	25	,	,	PUNCT
ejpam-5973	367	26	2	2	NUM
ejpam-5973	367	27	⌈	⌈	NUM
ejpam-5973	367	28	3n	3n	NUM
ejpam-5973	367	29	8	8	NUM
ejpam-5973	367	30	⌉	⌉	NOUN
ejpam-5973	367	31	if	if	SCONJ
ejpam-5973	367	32	n	n	PRON
ejpam-5973	367	33	≡	≡	PROPN
ejpam-5973	367	34	2	2	NUM
ejpam-5973	367	35	,	,	PUNCT
ejpam-5973	367	36	4	4	NUM
ejpam-5973	367	37	,	,	PUNCT
ejpam-5973	367	38	6	6	NUM
ejpam-5973	367	39	(	(	PUNCT
ejpam-5973	367	40	mod	mod	PROPN
ejpam-5973	367	41	8),⌈	8),⌈	NUM
ejpam-5973	367	42	3(n+1	3(n+1	NUM
ejpam-5973	367	43	)	)	PUNCT
ejpam-5973	367	44	8	8	NUM
ejpam-5973	367	45	⌉	⌉	NOUN
ejpam-5973	367	46	+	+	CCONJ
ejpam-5973	367	47	⌈	⌈	SYM
ejpam-5973	367	48	3(n−1	3(n−1	ADJ
ejpam-5973	367	49	)	)	PUNCT
ejpam-5973	367	50	8	8	NUM
ejpam-5973	367	51	⌉	⌉	PRON
ejpam-5973	367	52	if	if	SCONJ
ejpam-5973	367	53	n	n	PRON
ejpam-5973	367	54	≡	≡	PROPN
ejpam-5973	367	55	3	3	NUM
ejpam-5973	367	56	,	,	PUNCT
ejpam-5973	367	57	5	5	NUM
ejpam-5973	367	58	(	(	PUNCT
ejpam-5973	367	59	mod	mod	PROPN
ejpam-5973	367	60	8)	8)	NUM
ejpam-5973	367	61	.	.	PUNCT
ejpam-5973	368	1	(	(	PUNCT
ejpam-5973	368	2	iii	iii	NOUN
ejpam-5973	368	3	)	)	PUNCT
ejpam-5973	368	4	for	for	ADP
ejpam-5973	368	5	n	n	X
ejpam-5973	368	6	≥	≥	NOUN
ejpam-5973	368	7	4	4	NUM
ejpam-5973	368	8	,	,	PUNCT
ejpam-5973	368	9	γhr2(cn	γhr2(cn	NUM
ejpam-5973	368	10	)	)	PUNCT
ejpam-5973	369	1	=	=	PUNCT
ejpam-5973	369	2			PUNCT
ejpam-5973	369	3	⌊	⌊	PROPN
ejpam-5973	369	4	n	n	PRON
ejpam-5973	369	5	2	2	NUM
ejpam-5973	369	6	⌋	⌋	NOUN
ejpam-5973	369	7	+	+	CCONJ
ejpam-5973	369	8	⌈	⌈	SYM
ejpam-5973	369	9	n	n	PRON
ejpam-5973	369	10	4	4	NUM
ejpam-5973	369	11	⌉	⌉	PART
ejpam-5973	369	12	−	−	ADP
ejpam-5973	369	13	⌊	⌊	PROPN
ejpam-5973	369	14	n	n	ADV
ejpam-5973	369	15	4	4	NUM
ejpam-5973	369	16	⌋	⌋	NOUN
ejpam-5973	369	17	if	if	SCONJ
ejpam-5973	369	18	n	n	PRON
ejpam-5973	369	19	≡	≡	PROPN
ejpam-5973	369	20	1	1	NUM
ejpam-5973	369	21	(	(	PUNCT
ejpam-5973	369	22	mod	mod	NOUN
ejpam-5973	369	23	2	2	NUM
ejpam-5973	369	24	)	)	PUNCT
ejpam-5973	369	25	,	,	PUNCT
ejpam-5973	369	26	2	2	NUM
ejpam-5973	369	27	(	(	PUNCT
ejpam-5973	369	28	⌊	⌊	VERB
ejpam-5973	369	29	n	n	PRON
ejpam-5973	369	30	4	4	NUM
ejpam-5973	369	31	⌋	⌋	ADJ
ejpam-5973	369	32	+	+	CCONJ
ejpam-5973	369	33	⌈	⌈	NUM
ejpam-5973	369	34	n	n	CCONJ
ejpam-5973	369	35	8	8	NUM
ejpam-5973	369	36	⌉	⌉	PRON
ejpam-5973	369	37	−	−	ADP
ejpam-5973	369	38	⌊	⌊	PROPN
ejpam-5973	369	39	n	n	ADV
ejpam-5973	369	40	8	8	NUM
ejpam-5973	369	41	⌋	⌋	NOUN
ejpam-5973	369	42	)	)	PUNCT
ejpam-5973	369	43	otherwise	otherwise	ADV
ejpam-5973	369	44	.	.	PUNCT
ejpam-5973	370	1	(	(	PUNCT
ejpam-5973	370	2	iv	iv	X
ejpam-5973	370	3	)	)	PUNCT
ejpam-5973	370	4	for	for	ADP
ejpam-5973	370	5	n	n	X
ejpam-5973	370	6	≥	≥	NUM
ejpam-5973	370	7	5	5	NUM
ejpam-5973	370	8	,	,	PUNCT
ejpam-5973	370	9	γhr3(cn	γhr3(cn	PROPN
ejpam-5973	370	10	)	)	PUNCT
ejpam-5973	370	11	=	=	PUNCT
ejpam-5973	371	1			PUNCT
ejpam-5973	371	2	⌈	⌈	NOUN
ejpam-5973	371	3	3n	3n	NUM
ejpam-5973	371	4	4	4	NUM
ejpam-5973	371	5	⌉	⌉	NOUN
ejpam-5973	371	6	if	if	SCONJ
ejpam-5973	371	7	n	n	PRON
ejpam-5973	371	8	≡	≡	PROPN
ejpam-5973	371	9	1	1	NUM
ejpam-5973	371	10	(	(	PUNCT
ejpam-5973	371	11	mod	mod	NOUN
ejpam-5973	371	12	2	2	NUM
ejpam-5973	371	13	)	)	PUNCT
ejpam-5973	371	14	,	,	PUNCT
ejpam-5973	371	15	2	2	NUM
ejpam-5973	371	16	⌈	⌈	NUM
ejpam-5973	371	17	3n	3n	NUM
ejpam-5973	371	18	8	8	NUM
ejpam-5973	371	19	⌉	⌉	SCONJ
ejpam-5973	371	20	otherwise	otherwise	ADV
ejpam-5973	371	21	.	.	PUNCT
ejpam-5973	372	1	j.	j.	PROPN
ejpam-5973	372	2	j.	j.	PROPN
ejpam-5973	372	3	hamja	hamja	PROPN
ejpam-5973	372	4	et	et	PROPN
ejpam-5973	372	5	al	al	PROPN
ejpam-5973	372	6	.	.	PUNCT
ejpam-5973	372	7	/	/	SYM
ejpam-5973	372	8	eur	eur	PROPN
ejpam-5973	372	9	.	.	PUNCT
ejpam-5973	373	1	j.	j.	PROPN
ejpam-5973	373	2	pure	pure	PROPN
ejpam-5973	373	3	appl	appl	PROPN
ejpam-5973	373	4	.	.	PROPN
ejpam-5973	373	5	math	math	PROPN
ejpam-5973	373	6	,	,	PUNCT
ejpam-5973	373	7	18	18	NUM
ejpam-5973	373	8	(	(	PUNCT
ejpam-5973	373	9	2	2	NUM
ejpam-5973	373	10	)	)	PUNCT
ejpam-5973	373	11	(	(	PUNCT
ejpam-5973	373	12	2025	2025	NUM
ejpam-5973	373	13	)	)	PUNCT
ejpam-5973	373	14	,	,	PUNCT
ejpam-5973	373	15	5973	5973	NUM
ejpam-5973	373	16	10	10	NUM
ejpam-5973	373	17	of	of	ADP
ejpam-5973	373	18	17	17	NUM
ejpam-5973	373	19	proof	proof	NOUN
ejpam-5973	373	20	.	.	PUNCT
ejpam-5973	374	1	let	let	VERB
ejpam-5973	374	2	pn	pn	VERB
ejpam-5973	374	3	=	=	PUNCT
ejpam-5973	375	1	[	[	X
ejpam-5973	375	2	v1	v1	NOUN
ejpam-5973	375	3	,	,	PUNCT
ejpam-5973	375	4	v2	v2	NOUN
ejpam-5973	375	5	,	,	PUNCT
ejpam-5973	375	6	.	.	PUNCT
ejpam-5973	375	7	.	.	PUNCT
ejpam-5973	375	8	.	.	PUNCT
ejpam-5973	376	1	vn	vn	AUX
ejpam-5973	376	2	]	]	X
ejpam-5973	376	3	be	be	AUX
ejpam-5973	376	4	a	a	DET
ejpam-5973	376	5	path	path	NOUN
ejpam-5973	376	6	on	on	ADP
ejpam-5973	376	7	n	n	DET
ejpam-5973	376	8	vertices	vertex	NOUN
ejpam-5973	376	9	.	.	PUNCT
ejpam-5973	377	1	by	by	ADP
ejpam-5973	377	2	theorem	theorem	NOUN
ejpam-5973	377	3	4	4	NUM
ejpam-5973	377	4	and	and	CCONJ
ejpam-5973	377	5	proposition	proposition	NOUN
ejpam-5973	377	6	1	1	NUM
ejpam-5973	377	7	,	,	PUNCT
ejpam-5973	377	8	we	we	PRON
ejpam-5973	377	9	have	have	VERB
ejpam-5973	377	10	γhr2(pn	γhr2(pn	NOUN
ejpam-5973	377	11	)	)	PUNCT
ejpam-5973	377	12	=	=	SYM
ejpam-5973	377	13	γhr3(pn	γhr3(pn	PROPN
ejpam-5973	377	14	)	)	PUNCT
ejpam-5973	377	15	=	=	SYM
ejpam-5973	377	16	n	n	NOUN
ejpam-5973	377	17	for	for	ADP
ejpam-5973	377	18	n	n	PRON
ejpam-5973	377	19	∈	∈	NOUN
ejpam-5973	377	20	{	{	PUNCT
ejpam-5973	377	21	1	1	NUM
ejpam-5973	377	22	,	,	PUNCT
ejpam-5973	377	23	2	2	NUM
ejpam-5973	377	24	,	,	PUNCT
ejpam-5973	377	25	3	3	NUM
ejpam-5973	377	26	,	,	PUNCT
ejpam-5973	377	27	4	4	NUM
ejpam-5973	377	28	}	}	PUNCT
ejpam-5973	377	29	.	.	PUNCT
ejpam-5973	378	1	assume	assume	VERB
ejpam-5973	378	2	that	that	SCONJ
ejpam-5973	378	3	n	n	NUM
ejpam-5973	378	4	≥	≥	NUM
ejpam-5973	378	5	5	5	NUM
ejpam-5973	378	6	.	.	PUNCT
ejpam-5973	379	1	if	if	SCONJ
ejpam-5973	379	2	n	n	NOUN
ejpam-5973	379	3	is	be	AUX
ejpam-5973	379	4	even	even	ADV
ejpam-5973	379	5	,	,	PUNCT
ejpam-5973	379	6	then	then	ADV
ejpam-5973	379	7	p	p	X
ejpam-5973	379	8	2	2	NUM
ejpam-5973	379	9	n	n	NOUN
ejpam-5973	379	10	is	be	AUX
ejpam-5973	379	11	the	the	DET
ejpam-5973	379	12	union	union	NOUN
ejpam-5973	379	13	of	of	ADP
ejpam-5973	379	14	two	two	NUM
ejpam-5973	379	15	paths	path	NOUN
ejpam-5973	379	16	:	:	PUNCT
ejpam-5973	379	17	p	p	X
ejpam-5973	379	18	=	=	PUNCT
ejpam-5973	380	1	[	[	X
ejpam-5973	380	2	v1	v1	NOUN
ejpam-5973	380	3	,	,	PUNCT
ejpam-5973	380	4	v3	v3	PROPN
ejpam-5973	380	5	,	,	PUNCT
ejpam-5973	380	6	.	.	PUNCT
ejpam-5973	380	7	.	.	PUNCT
ejpam-5973	381	1	.	.	PUNCT
ejpam-5973	382	1	,	,	PUNCT
ejpam-5973	382	2	vn−1	vn−1	PROPN
ejpam-5973	382	3	]	]	PUNCT
ejpam-5973	382	4	and	and	CCONJ
ejpam-5973	382	5	q	q	NOUN
ejpam-5973	382	6	=	=	PUNCT
ejpam-5973	383	1	[	[	X
ejpam-5973	383	2	v2	v2	PROPN
ejpam-5973	383	3	,	,	PUNCT
ejpam-5973	383	4	v4	v4	NOUN
ejpam-5973	383	5	,	,	PUNCT
ejpam-5973	383	6	.	.	PUNCT
ejpam-5973	383	7	.	.	PUNCT
ejpam-5973	384	1	.	.	PUNCT
ejpam-5973	385	1	,	,	PUNCT
ejpam-5973	385	2	vn	vn	X
ejpam-5973	385	3	]	]	PUNCT
ejpam-5973	385	4	.	.	PUNCT
ejpam-5973	386	1	if	if	SCONJ
ejpam-5973	386	2	n	n	PROPN
ejpam-5973	386	3	is	be	AUX
ejpam-5973	386	4	odd	odd	ADJ
ejpam-5973	386	5	,	,	PUNCT
ejpam-5973	386	6	then	then	ADV
ejpam-5973	386	7	p	p	X
ejpam-5973	386	8	2	2	NUM
ejpam-5973	386	9	n	n	NOUN
ejpam-5973	386	10	is	be	AUX
ejpam-5973	386	11	the	the	DET
ejpam-5973	386	12	union	union	NOUN
ejpam-5973	386	13	of	of	ADP
ejpam-5973	386	14	two	two	NUM
ejpam-5973	386	15	paths	path	NOUN
ejpam-5973	386	16	:	:	PUNCT
ejpam-5973	386	17	p	p	NOUN
ejpam-5973	386	18	′	′	NOUN
ejpam-5973	386	19	=	=	PUNCT
ejpam-5973	387	1	[	[	X
ejpam-5973	387	2	v1	v1	NOUN
ejpam-5973	387	3	,	,	PUNCT
ejpam-5973	387	4	v3	v3	PROPN
ejpam-5973	387	5	,	,	PUNCT
ejpam-5973	387	6	.	.	PUNCT
ejpam-5973	387	7	.	.	PUNCT
ejpam-5973	387	8	.	.	PUNCT
ejpam-5973	388	1	,	,	PUNCT
ejpam-5973	388	2	vn	vn	X
ejpam-5973	388	3	]	]	PUNCT
ejpam-5973	388	4	and	and	CCONJ
ejpam-5973	388	5	q′	q′	NOUN
ejpam-5973	388	6	=	=	PUNCT
ejpam-5973	389	1	[	[	X
ejpam-5973	389	2	v2	v2	PROPN
ejpam-5973	389	3	,	,	PUNCT
ejpam-5973	389	4	v4	v4	NOUN
ejpam-5973	389	5	,	,	PUNCT
ejpam-5973	389	6	.	.	PUNCT
ejpam-5973	389	7	.	.	PUNCT
ejpam-5973	390	1	.	.	PUNCT
ejpam-5973	391	1	,	,	PUNCT
ejpam-5973	391	2	vn−1	vn−1	PROPN
ejpam-5973	391	3	]	]	PUNCT
ejpam-5973	391	4	.	.	PUNCT
ejpam-5973	392	1	(	(	PUNCT
ejpam-5973	392	2	i	i	NOUN
ejpam-5973	392	3	)	)	PUNCT
ejpam-5973	392	4	since	since	SCONJ
ejpam-5973	392	5	both	both	CCONJ
ejpam-5973	392	6	p	p	NOUN
ejpam-5973	392	7	and	and	CCONJ
ejpam-5973	392	8	q	q	NOUN
ejpam-5973	392	9	have	have	VERB
ejpam-5973	392	10	n/2	n/2	NOUN
ejpam-5973	392	11	vertices	vertex	NOUN
ejpam-5973	392	12	,	,	PUNCT
ejpam-5973	392	13	we	we	PRON
ejpam-5973	392	14	can	can	AUX
ejpam-5973	392	15	use	use	VERB
ejpam-5973	392	16	theorem	theorem	NOUN
ejpam-5973	392	17	8-(i	8-(i	NUM
ejpam-5973	392	18	)	)	PUNCT
ejpam-5973	392	19	to	to	PART
ejpam-5973	392	20	find	find	VERB
ejpam-5973	392	21	that	that	SCONJ
ejpam-5973	392	22	γr2(p	γr2(p	PROPN
ejpam-5973	392	23	)	)	PUNCT
ejpam-5973	393	1	=	=	PUNCT
ejpam-5973	393	2	⌊	⌊	VERB
ejpam-5973	393	3	n/2	n/2	NUM
ejpam-5973	393	4	2	2	NUM
ejpam-5973	393	5	⌋	⌋	NOUN
ejpam-5973	393	6	+	+	CCONJ
ejpam-5973	393	7	1	1	NUM
ejpam-5973	393	8	=	=	SYM
ejpam-5973	393	9	⌊n/4⌋+	⌊n/4⌋+	PROPN
ejpam-5973	393	10	1	1	NUM
ejpam-5973	393	11	,	,	PUNCT
ejpam-5973	393	12	and	and	CCONJ
ejpam-5973	393	13	similarly	similarly	ADV
ejpam-5973	393	14	,	,	PUNCT
ejpam-5973	393	15	γr2(q	γr2(q	PROPN
ejpam-5973	393	16	)	)	PUNCT
ejpam-5973	394	1	=	=	PUNCT
ejpam-5973	395	1	⌊n/4⌋+	⌊n/4⌋+	PROPN
ejpam-5973	395	2	1	1	X
ejpam-5973	395	3	.	.	PUNCT
ejpam-5973	396	1	if	if	SCONJ
ejpam-5973	396	2	n	n	NOUN
ejpam-5973	396	3	is	be	AUX
ejpam-5973	396	4	even	even	ADV
ejpam-5973	396	5	,	,	PUNCT
ejpam-5973	396	6	then	then	ADV
ejpam-5973	396	7	by	by	ADP
ejpam-5973	396	8	remark	remark	NOUN
ejpam-5973	396	9	1	1	NUM
ejpam-5973	396	10	,	,	PUNCT
ejpam-5973	396	11	we	we	PRON
ejpam-5973	396	12	obtain	obtain	VERB
ejpam-5973	396	13	γhr2(pn	γhr2(pn	NOUN
ejpam-5973	396	14	)	)	PUNCT
ejpam-5973	396	15	=	=	SYM
ejpam-5973	396	16	γr2(p	γr2(p	PROPN
ejpam-5973	396	17	2	2	NUM
ejpam-5973	396	18	n	n	CCONJ
ejpam-5973	396	19	)	)	PUNCT
ejpam-5973	396	20	=	=	SYM
ejpam-5973	396	21	γr2(p	γr2(p	PROPN
ejpam-5973	396	22	)	)	PUNCT
ejpam-5973	397	1	+	+	NUM
ejpam-5973	397	2	γr2(q	γr2(q	PROPN
ejpam-5973	397	3	)	)	PUNCT
ejpam-5973	398	1	=	=	SYM
ejpam-5973	398	2	2	2	NUM
ejpam-5973	398	3	(	(	PUNCT
ejpam-5973	398	4	⌊n	⌊n	X
ejpam-5973	398	5	4	4	NUM
ejpam-5973	398	6	⌋	⌋	NOUN
ejpam-5973	398	7	+	+	CCONJ
ejpam-5973	398	8	1	1	X
ejpam-5973	398	9	)	)	PUNCT
ejpam-5973	398	10	=	=	SYM
ejpam-5973	398	11	2	2	NUM
ejpam-5973	398	12	⌊n	⌊n	NOUN
ejpam-5973	398	13	4	4	NUM
ejpam-5973	398	14	⌋	⌋	NOUN
ejpam-5973	398	15	+	+	CCONJ
ejpam-5973	398	16	2	2	NUM
ejpam-5973	398	17	,	,	PUNCT
ejpam-5973	398	18	as	as	SCONJ
ejpam-5973	398	19	desired	desire	VERB
ejpam-5973	398	20	.	.	PUNCT
ejpam-5973	399	1	assume	assume	VERB
ejpam-5973	399	2	that	that	SCONJ
ejpam-5973	399	3	n	n	PRON
ejpam-5973	399	4	is	be	AUX
ejpam-5973	399	5	odd	odd	ADJ
ejpam-5973	399	6	.	.	PUNCT
ejpam-5973	400	1	then	then	ADV
ejpam-5973	400	2	p	p	PROPN
ejpam-5973	400	3	and	and	CCONJ
ejpam-5973	400	4	q	q	NOUN
ejpam-5973	400	5	have	have	VERB
ejpam-5973	400	6	lengths	length	NOUN
ejpam-5973	400	7	n+1	n+1	PROPN
ejpam-5973	400	8	2	2	NUM
ejpam-5973	400	9	and	and	CCONJ
ejpam-5973	400	10	n−1	n−1	PROPN
ejpam-5973	400	11	2	2	NUM
ejpam-5973	400	12	,	,	PUNCT
ejpam-5973	400	13	respectively	respectively	ADV
ejpam-5973	400	14	.	.	PUNCT
ejpam-5973	401	1	applying	apply	VERB
ejpam-5973	401	2	theorem	theorem	NOUN
ejpam-5973	401	3	8-(i	8-(i	NUM
ejpam-5973	401	4	)	)	PUNCT
ejpam-5973	401	5	again	again	ADV
ejpam-5973	401	6	,	,	PUNCT
ejpam-5973	401	7	we	we	PRON
ejpam-5973	401	8	obtain	obtain	VERB
ejpam-5973	401	9	γr2(p	γr2(p	PROPN
ejpam-5973	401	10	)	)	PUNCT
ejpam-5973	402	1	=	=	PUNCT
ejpam-5973	402	2	⌊	⌊	PROPN
ejpam-5973	402	3	(	(	PUNCT
ejpam-5973	402	4	n+1)/2	n+1)/2	NOUN
ejpam-5973	402	5	2	2	NUM
ejpam-5973	402	6	⌋	⌋	NOUN
ejpam-5973	402	7	+	+	CCONJ
ejpam-5973	402	8	1	1	X
ejpam-5973	402	9	=	=	SYM
ejpam-5973	402	10	⌊	⌊	VERB
ejpam-5973	402	11	n+1	n+1	NUM
ejpam-5973	402	12	4	4	NUM
ejpam-5973	402	13	⌋	⌋	NOUN
ejpam-5973	402	14	+	+	CCONJ
ejpam-5973	402	15	1	1	NUM
ejpam-5973	402	16	and	and	CCONJ
ejpam-5973	402	17	γr2(q	γr2(q	PROPN
ejpam-5973	402	18	)	)	PUNCT
ejpam-5973	403	1	=	=	SYM
ejpam-5973	403	2	⌊	⌊	PROPN
ejpam-5973	403	3	(	(	PUNCT
ejpam-5973	403	4	n−1)/2	n−1)/2	NOUN
ejpam-5973	403	5	2	2	NUM
ejpam-5973	403	6	⌋	⌋	NOUN
ejpam-5973	403	7	+	+	CCONJ
ejpam-5973	403	8	1	1	X
ejpam-5973	403	9	=	=	SYM
ejpam-5973	403	10	⌊	⌊	VERB
ejpam-5973	403	11	n−1	n−1	PROPN
ejpam-5973	403	12	4	4	NUM
ejpam-5973	403	13	⌋	⌋	NOUN
ejpam-5973	403	14	+	+	CCONJ
ejpam-5973	403	15	1	1	X
ejpam-5973	403	16	.	.	X
ejpam-5973	403	17	therefore	therefore	ADV
ejpam-5973	403	18	,	,	PUNCT
ejpam-5973	403	19	by	by	ADP
ejpam-5973	403	20	remark	remark	NOUN
ejpam-5973	403	21	1	1	NUM
ejpam-5973	403	22	,	,	PUNCT
ejpam-5973	403	23	we	we	PRON
ejpam-5973	403	24	have	have	VERB
ejpam-5973	403	25	γhr2(pn	γhr2(pn	NOUN
ejpam-5973	403	26	)	)	PUNCT
ejpam-5973	403	27	=	=	SYM
ejpam-5973	403	28	γr2(p	γr2(p	PROPN
ejpam-5973	403	29	)	)	PUNCT
ejpam-5973	404	1	+	+	NUM
ejpam-5973	404	2	γr2(q	γr2(q	PROPN
ejpam-5973	404	3	)	)	PUNCT
ejpam-5973	405	1	=	=	PRON
ejpam-5973	405	2	(	(	PUNCT
ejpam-5973	405	3	⌊	⌊	VERB
ejpam-5973	405	4	n+	n+	NUM
ejpam-5973	405	5	1	1	NUM
ejpam-5973	405	6	4	4	NUM
ejpam-5973	405	7	⌋	⌋	ADJ
ejpam-5973	405	8	+	+	CCONJ
ejpam-5973	405	9	1	1	NUM
ejpam-5973	405	10	)	)	PUNCT
ejpam-5973	405	11	+	+	CCONJ
ejpam-5973	405	12	(	(	PUNCT
ejpam-5973	405	13	⌊	⌊	VERB
ejpam-5973	405	14	n−	n−	NOUN
ejpam-5973	405	15	1	1	NUM
ejpam-5973	405	16	4	4	NUM
ejpam-5973	405	17	⌋	⌋	ADJ
ejpam-5973	405	18	+	+	CCONJ
ejpam-5973	405	19	1	1	NUM
ejpam-5973	405	20	)	)	PUNCT
ejpam-5973	405	21	.	.	PUNCT
ejpam-5973	406	1	simplifying	simplify	VERB
ejpam-5973	406	2	,	,	PUNCT
ejpam-5973	406	3	we	we	PRON
ejpam-5973	406	4	get	get	VERB
ejpam-5973	406	5	γhr2(pn	γhr2(pn	NOUN
ejpam-5973	406	6	)	)	PUNCT
ejpam-5973	406	7	=	=	PUNCT
ejpam-5973	407	1	⌊	⌊	VERB
ejpam-5973	407	2	n+	n+	NUM
ejpam-5973	407	3	1	1	NUM
ejpam-5973	407	4	4	4	NUM
ejpam-5973	407	5	⌋	⌋	AUX
ejpam-5973	407	6	+	+	CCONJ
ejpam-5973	407	7	⌊	⌊	PROPN
ejpam-5973	407	8	n−	n−	NOUN
ejpam-5973	407	9	1	1	NUM
ejpam-5973	407	10	4	4	NUM
ejpam-5973	407	11	⌋	⌋	NOUN
ejpam-5973	407	12	+	+	CCONJ
ejpam-5973	407	13	2	2	NUM
ejpam-5973	407	14	=	=	SYM
ejpam-5973	407	15	⌊n	⌊n	NOUN
ejpam-5973	407	16	4	4	NUM
ejpam-5973	407	17	⌋	⌋	NOUN
ejpam-5973	407	18	+	+	CCONJ
ejpam-5973	407	19	⌈n	⌈n	NOUN
ejpam-5973	407	20	4	4	NUM
ejpam-5973	407	21	⌉	⌉	NOUN
ejpam-5973	407	22	+	+	ADJ
ejpam-5973	407	23	2	2	NUM
ejpam-5973	407	24	,	,	PUNCT
ejpam-5973	407	25	as	as	SCONJ
ejpam-5973	407	26	desired	desire	VERB
ejpam-5973	407	27	.	.	PUNCT
ejpam-5973	408	1	(	(	PUNCT
ejpam-5973	408	2	ii	ii	NOUN
ejpam-5973	408	3	)	)	PUNCT
ejpam-5973	408	4	if	if	SCONJ
ejpam-5973	408	5	n	n	PRON
ejpam-5973	408	6	≡	≡	PROPN
ejpam-5973	408	7	0	0	PUNCT
ejpam-5973	408	8	(	(	PUNCT
ejpam-5973	408	9	mod	mod	PROPN
ejpam-5973	408	10	8)	8)	NUM
ejpam-5973	408	11	,	,	PUNCT
ejpam-5973	408	12	then	then	ADV
ejpam-5973	408	13	|v	|v	PROPN
ejpam-5973	408	14	(	(	PUNCT
ejpam-5973	408	15	p	p	NOUN
ejpam-5973	408	16	)	)	PUNCT
ejpam-5973	409	1	|	|	NOUN
ejpam-5973	409	2	=	=	SYM
ejpam-5973	409	3	n	n	PRON
ejpam-5973	409	4	2	2	NUM
ejpam-5973	409	5	≡	≡	PROPN
ejpam-5973	409	6	0	0	PUNCT
ejpam-5973	409	7	(	(	PUNCT
ejpam-5973	409	8	mod	mod	PROPN
ejpam-5973	409	9	4	4	NUM
ejpam-5973	409	10	)	)	PUNCT
ejpam-5973	409	11	,	,	PUNCT
ejpam-5973	409	12	|v	|v	PROPN
ejpam-5973	409	13	(	(	PUNCT
ejpam-5973	409	14	q)|	q)|	NOUN
ejpam-5973	409	15	=	=	SYM
ejpam-5973	409	16	n	n	CCONJ
ejpam-5973	409	17	2	2	NUM
ejpam-5973	409	18	≡	≡	PROPN
ejpam-5973	409	19	0	0	PUNCT
ejpam-5973	409	20	(	(	PUNCT
ejpam-5973	409	21	mod	mod	NOUN
ejpam-5973	409	22	4	4	NUM
ejpam-5973	409	23	)	)	PUNCT
ejpam-5973	409	24	and	and	CCONJ
ejpam-5973	409	25	by	by	ADP
ejpam-5973	409	26	remark	remark	NOUN
ejpam-5973	409	27	1	1	NUM
ejpam-5973	409	28	and	and	CCONJ
ejpam-5973	409	29	theorem	theorem	VERB
ejpam-5973	409	30	8-(iii	8-(iii	NOUN
ejpam-5973	409	31	)	)	PUNCT
ejpam-5973	409	32	,	,	PUNCT
ejpam-5973	409	33	we	we	PRON
ejpam-5973	409	34	have	have	VERB
ejpam-5973	409	35	γhr3(pn	γhr3(pn	NOUN
ejpam-5973	409	36	)	)	PUNCT
ejpam-5973	409	37	=	=	PUNCT
ejpam-5973	409	38	γr3(p	γr3(p	NOUN
ejpam-5973	409	39	2	2	NUM
ejpam-5973	409	40	n	n	CCONJ
ejpam-5973	409	41	)	)	PUNCT
ejpam-5973	409	42	=	=	SYM
ejpam-5973	409	43	γr3(p	γr3(p	NOUN
ejpam-5973	409	44	)	)	PUNCT
ejpam-5973	410	1	+	+	CCONJ
ejpam-5973	410	2	γr3(q	γr3(q	PROPN
ejpam-5973	410	3	)	)	PUNCT
ejpam-5973	410	4	=	=	SYM
ejpam-5973	410	5	2	2	NUM
ejpam-5973	410	6	(	(	PUNCT
ejpam-5973	410	7	⌈	⌈	NUM
ejpam-5973	410	8	3n	3n	NUM
ejpam-5973	410	9	2	2	NUM
ejpam-5973	410	10	4	4	NUM
ejpam-5973	410	11	⌉	⌉	NOUN
ejpam-5973	410	12	+	+	CCONJ
ejpam-5973	410	13	1	1	X
ejpam-5973	410	14	)	)	PUNCT
ejpam-5973	410	15	=	=	SYM
ejpam-5973	410	16	2	2	NUM
ejpam-5973	410	17	(	(	PUNCT
ejpam-5973	410	18	⌈	⌈	NUM
ejpam-5973	410	19	3n	3n	NUM
ejpam-5973	410	20	8	8	NUM
ejpam-5973	410	21	⌉	⌉	NOUN
ejpam-5973	410	22	+	+	X
ejpam-5973	410	23	1	1	NUM
ejpam-5973	410	24	)	)	PUNCT
ejpam-5973	410	25	,	,	PUNCT
ejpam-5973	410	26	as	as	SCONJ
ejpam-5973	410	27	desired	desire	VERB
ejpam-5973	410	28	.	.	PUNCT
ejpam-5973	411	1	if	if	SCONJ
ejpam-5973	411	2	n	n	NUM
ejpam-5973	411	3	≡	≡	PROPN
ejpam-5973	411	4	1	1	NUM
ejpam-5973	411	5	(	(	PUNCT
ejpam-5973	411	6	mod	mod	PROPN
ejpam-5973	411	7	8)	8)	NUM
ejpam-5973	411	8	(	(	PUNCT
ejpam-5973	411	9	the	the	DET
ejpam-5973	411	10	case	case	NOUN
ejpam-5973	411	11	n	n	X
ejpam-5973	411	12	≡	≡	PROPN
ejpam-5973	411	13	7	7	NUM
ejpam-5973	411	14	(	(	PUNCT
ejpam-5973	411	15	mod	mod	PROPN
ejpam-5973	411	16	8)	8)	NUM
ejpam-5973	411	17	is	be	AUX
ejpam-5973	411	18	similar	similar	ADJ
ejpam-5973	411	19	)	)	PUNCT
ejpam-5973	411	20	,	,	PUNCT
ejpam-5973	411	21	then	then	ADV
ejpam-5973	411	22	|v	|v	PROPN
ejpam-5973	411	23	(	(	PUNCT
ejpam-5973	411	24	p	p	NOUN
ejpam-5973	411	25	)	)	PUNCT
ejpam-5973	411	26	|	|	ADV
ejpam-5973	411	27	=	=	SYM
ejpam-5973	411	28	n+1	n+1	PROPN
ejpam-5973	411	29	2	2	NUM
ejpam-5973	411	30	≡	≡	PROPN
ejpam-5973	411	31	1	1	NUM
ejpam-5973	411	32	(	(	PUNCT
ejpam-5973	411	33	mod	mod	NOUN
ejpam-5973	411	34	4	4	NUM
ejpam-5973	411	35	)	)	PUNCT
ejpam-5973	411	36	and	and	CCONJ
ejpam-5973	411	37	|v	|v	PROPN
ejpam-5973	411	38	(	(	PUNCT
ejpam-5973	411	39	q)|	q)|	NOUN
ejpam-5973	411	40	=	=	SYM
ejpam-5973	411	41	n−1	n−1	PROPN
ejpam-5973	411	42	2	2	NUM
ejpam-5973	411	43	≡	≡	PROPN
ejpam-5973	411	44	0	0	PUNCT
ejpam-5973	412	1	(	(	PUNCT
ejpam-5973	412	2	mod	mod	PROPN
ejpam-5973	412	3	4	4	NUM
ejpam-5973	412	4	)	)	PUNCT
ejpam-5973	412	5	.	.	PUNCT
ejpam-5973	413	1	applying	apply	VERB
ejpam-5973	413	2	remark	remark	NOUN
ejpam-5973	413	3	1	1	NUM
ejpam-5973	413	4	and	and	CCONJ
ejpam-5973	413	5	theorem	theorem	VERB
ejpam-5973	413	6	8-(iii	8-(iii	NOUN
ejpam-5973	413	7	)	)	PUNCT
ejpam-5973	413	8	,	,	PUNCT
ejpam-5973	413	9	it	it	PRON
ejpam-5973	413	10	follows	follow	VERB
ejpam-5973	413	11	that	that	PRON
ejpam-5973	413	12	γhr3(pn	γhr3(pn	NOUN
ejpam-5973	413	13	)	)	PUNCT
ejpam-5973	413	14	=	=	PUNCT
ejpam-5973	413	15	γr3(p	γr3(p	NOUN
ejpam-5973	413	16	)	)	PUNCT
ejpam-5973	414	1	+	+	CCONJ
ejpam-5973	414	2	γr3(q	γr3(q	PROPN
ejpam-5973	414	3	)	)	PUNCT
ejpam-5973	414	4	=	=	PUNCT
ejpam-5973	415	1	⌈	⌈	X
ejpam-5973	415	2	3n+1	3n+1	NUM
ejpam-5973	415	3	2	2	NUM
ejpam-5973	415	4	4	4	NUM
ejpam-5973	415	5	⌉	⌉	NOUN
ejpam-5973	415	6	+	+	CCONJ
ejpam-5973	415	7	⌈	⌈	NOUN
ejpam-5973	415	8	3n−1	3n−1	NUM
ejpam-5973	415	9	2	2	NUM
ejpam-5973	415	10	4	4	NUM
ejpam-5973	415	11	⌉	⌉	NOUN
ejpam-5973	415	12	+	+	SYM
ejpam-5973	415	13	1	1	NUM
ejpam-5973	415	14	j.	j.	PROPN
ejpam-5973	415	15	j.	j.	PROPN
ejpam-5973	415	16	hamja	hamja	PROPN
ejpam-5973	415	17	et	et	PROPN
ejpam-5973	415	18	al	al	PROPN
ejpam-5973	415	19	.	.	PUNCT
ejpam-5973	415	20	/	/	SYM
ejpam-5973	415	21	eur	eur	PROPN
ejpam-5973	415	22	.	.	PUNCT
ejpam-5973	416	1	j.	j.	PROPN
ejpam-5973	416	2	pure	pure	PROPN
ejpam-5973	416	3	appl	appl	PROPN
ejpam-5973	416	4	.	.	PROPN
ejpam-5973	416	5	math	math	PROPN
ejpam-5973	416	6	,	,	PUNCT
ejpam-5973	416	7	18	18	NUM
ejpam-5973	416	8	(	(	PUNCT
ejpam-5973	416	9	2	2	NUM
ejpam-5973	416	10	)	)	PUNCT
ejpam-5973	416	11	(	(	PUNCT
ejpam-5973	416	12	2025	2025	NUM
ejpam-5973	416	13	)	)	PUNCT
ejpam-5973	416	14	,	,	PUNCT
ejpam-5973	416	15	5973	5973	NUM
ejpam-5973	416	16	11	11	NUM
ejpam-5973	416	17	of	of	ADP
ejpam-5973	416	18	17	17	NUM
ejpam-5973	416	19	=	=	SYM
ejpam-5973	416	20	⌈	⌈	NOUN
ejpam-5973	416	21	3(n+	3(n+	NUM
ejpam-5973	416	22	1	1	NUM
ejpam-5973	416	23	)	)	PUNCT
ejpam-5973	416	24	8	8	NUM
ejpam-5973	416	25	⌉	⌉	NOUN
ejpam-5973	416	26	+	+	PUNCT
ejpam-5973	416	27	⌈	⌈	NOUN
ejpam-5973	416	28	3(n−	3(n−	NUM
ejpam-5973	416	29	1	1	NUM
ejpam-5973	416	30	)	)	PUNCT
ejpam-5973	416	31	8	8	NUM
ejpam-5973	416	32	⌉	⌉	NOUN
ejpam-5973	416	33	+	+	NOUN
ejpam-5973	416	34	1	1	X
ejpam-5973	416	35	.	.	X
ejpam-5973	417	1	if	if	SCONJ
ejpam-5973	417	2	n	n	NUM
ejpam-5973	417	3	≡	≡	PROPN
ejpam-5973	417	4	3	3	NUM
ejpam-5973	417	5	(	(	PUNCT
ejpam-5973	417	6	mod	mod	PROPN
ejpam-5973	417	7	8)	8)	NUM
ejpam-5973	417	8	(	(	PUNCT
ejpam-5973	417	9	the	the	DET
ejpam-5973	417	10	case	case	NOUN
ejpam-5973	417	11	n	n	NUM
ejpam-5973	417	12	≡	≡	PROPN
ejpam-5973	417	13	5	5	NUM
ejpam-5973	417	14	(	(	PUNCT
ejpam-5973	417	15	mod	mod	PROPN
ejpam-5973	417	16	8)	8)	NUM
ejpam-5973	417	17	is	be	AUX
ejpam-5973	417	18	similar	similar	ADJ
ejpam-5973	417	19	)	)	PUNCT
ejpam-5973	417	20	,	,	PUNCT
ejpam-5973	417	21	then	then	ADV
ejpam-5973	417	22	|v	|v	PROPN
ejpam-5973	417	23	(	(	PUNCT
ejpam-5973	417	24	p	p	NOUN
ejpam-5973	417	25	)	)	PUNCT
ejpam-5973	418	1	|	|	ADV
ejpam-5973	418	2	=	=	SYM
ejpam-5973	418	3	n+1	n+1	PROPN
ejpam-5973	418	4	2	2	NUM
ejpam-5973	418	5	≡	≡	PROPN
ejpam-5973	418	6	2	2	NUM
ejpam-5973	418	7	(	(	PUNCT
ejpam-5973	418	8	mod	mod	NOUN
ejpam-5973	418	9	4	4	NUM
ejpam-5973	418	10	)	)	PUNCT
ejpam-5973	418	11	and	and	CCONJ
ejpam-5973	418	12	|v	|v	PROPN
ejpam-5973	418	13	(	(	PUNCT
ejpam-5973	418	14	q)|	q)|	NOUN
ejpam-5973	418	15	=	=	SYM
ejpam-5973	418	16	n−1	n−1	PROPN
ejpam-5973	418	17	2	2	NUM
ejpam-5973	418	18	≡	≡	PROPN
ejpam-5973	418	19	1	1	NUM
ejpam-5973	418	20	(	(	PUNCT
ejpam-5973	418	21	mod	mod	NOUN
ejpam-5973	418	22	4	4	NUM
ejpam-5973	418	23	)	)	PUNCT
ejpam-5973	418	24	.	.	PUNCT
ejpam-5973	419	1	applying	apply	VERB
ejpam-5973	419	2	remark	remark	NOUN
ejpam-5973	419	3	1	1	NUM
ejpam-5973	419	4	and	and	CCONJ
ejpam-5973	419	5	theorem	theorem	VERB
ejpam-5973	419	6	8-(iii	8-(iii	NOUN
ejpam-5973	419	7	)	)	PUNCT
ejpam-5973	419	8	,	,	PUNCT
ejpam-5973	419	9	it	it	PRON
ejpam-5973	419	10	follows	follow	VERB
ejpam-5973	419	11	that	that	PRON
ejpam-5973	419	12	γhr3(pn	γhr3(pn	NOUN
ejpam-5973	419	13	)	)	PUNCT
ejpam-5973	419	14	=	=	PUNCT
ejpam-5973	419	15	γr3(p	γr3(p	NOUN
ejpam-5973	419	16	)	)	PUNCT
ejpam-5973	420	1	+	+	CCONJ
ejpam-5973	420	2	γr3(q	γr3(q	PROPN
ejpam-5973	420	3	)	)	PUNCT
ejpam-5973	420	4	=	=	PUNCT
ejpam-5973	421	1	⌈	⌈	X
ejpam-5973	421	2	3n+1	3n+1	NUM
ejpam-5973	421	3	2	2	NUM
ejpam-5973	421	4	4	4	NUM
ejpam-5973	421	5	⌉	⌉	NOUN
ejpam-5973	421	6	+	+	CCONJ
ejpam-5973	421	7	⌈	⌈	NOUN
ejpam-5973	421	8	3n−1	3n−1	NUM
ejpam-5973	421	9	2	2	NUM
ejpam-5973	421	10	4	4	NUM
ejpam-5973	421	11	⌉	⌉	NOUN
ejpam-5973	421	12	=	=	SYM
ejpam-5973	421	13	⌈	⌈	NOUN
ejpam-5973	421	14	3(n+	3(n+	NUM
ejpam-5973	421	15	1	1	NUM
ejpam-5973	421	16	)	)	PUNCT
ejpam-5973	421	17	8	8	NUM
ejpam-5973	421	18	⌉	⌉	NOUN
ejpam-5973	421	19	+	+	PUNCT
ejpam-5973	421	20	⌈	⌈	NOUN
ejpam-5973	421	21	3(n−	3(n−	NUM
ejpam-5973	421	22	1	1	NUM
ejpam-5973	421	23	)	)	PUNCT
ejpam-5973	421	24	8	8	NUM
ejpam-5973	421	25	⌉	⌉	NOUN
ejpam-5973	421	26	.	.	PUNCT
ejpam-5973	422	1	if	if	SCONJ
ejpam-5973	422	2	n	n	PRON
ejpam-5973	422	3	≡	≡	PROPN
ejpam-5973	422	4	2	2	NUM
ejpam-5973	422	5	,	,	PUNCT
ejpam-5973	422	6	4	4	NUM
ejpam-5973	422	7	,	,	PUNCT
ejpam-5973	422	8	6	6	NUM
ejpam-5973	422	9	(	(	PUNCT
ejpam-5973	422	10	mod	mod	NOUN
ejpam-5973	422	11	8)	8)	NUM
ejpam-5973	422	12	,	,	PUNCT
ejpam-5973	422	13	then	then	ADV
ejpam-5973	422	14	|v	|v	PROPN
ejpam-5973	422	15	(	(	PUNCT
ejpam-5973	422	16	p	p	NOUN
ejpam-5973	422	17	)	)	PUNCT
ejpam-5973	422	18	|	|	ADV
ejpam-5973	422	19	=	=	SYM
ejpam-5973	422	20	|v	|v	PROPN
ejpam-5973	422	21	(	(	PUNCT
ejpam-5973	422	22	q)|	q)|	NOUN
ejpam-5973	422	23	=	=	SYM
ejpam-5973	422	24	n	n	PRON
ejpam-5973	422	25	2	2	NUM
ejpam-5973	422	26	̸≡	̸≡	NOUN
ejpam-5973	422	27	0	0	NUM
ejpam-5973	423	1	(	(	PUNCT
ejpam-5973	423	2	mod	mod	PROPN
ejpam-5973	423	3	4	4	NUM
ejpam-5973	423	4	)	)	PUNCT
ejpam-5973	423	5	.	.	PUNCT
ejpam-5973	424	1	applying	apply	VERB
ejpam-5973	424	2	theorem	theorem	ADJ
ejpam-5973	424	3	8-(iii	8-(iii	NOUN
ejpam-5973	424	4	)	)	PUNCT
ejpam-5973	424	5	,	,	PUNCT
ejpam-5973	424	6	it	it	PRON
ejpam-5973	424	7	follows	follow	VERB
ejpam-5973	424	8	that	that	DET
ejpam-5973	424	9	γhr3(pn	γhr3(pn	NOUN
ejpam-5973	424	10	)	)	PUNCT
ejpam-5973	424	11	=	=	PUNCT
ejpam-5973	425	1	γr3(p	γr3(p	NOUN
ejpam-5973	425	2	2	2	NUM
ejpam-5973	425	3	n	n	CCONJ
ejpam-5973	425	4	)	)	PUNCT
ejpam-5973	425	5	=	=	SYM
ejpam-5973	425	6	γr3(p	γr3(p	NOUN
ejpam-5973	425	7	)	)	PUNCT
ejpam-5973	426	1	+	+	CCONJ
ejpam-5973	426	2	γr3(q	γr3(q	PROPN
ejpam-5973	426	3	)	)	PUNCT
ejpam-5973	426	4	=	=	SYM
ejpam-5973	427	1	2	2	NUM
ejpam-5973	427	2	⌈	⌈	NUM
ejpam-5973	427	3	3n	3n	NUM
ejpam-5973	427	4	2	2	NUM
ejpam-5973	427	5	4	4	NUM
ejpam-5973	427	6	⌉	⌉	NOUN
ejpam-5973	427	7	=	=	SYM
ejpam-5973	427	8	2	2	NUM
ejpam-5973	427	9	⌈	⌈	NUM
ejpam-5973	427	10	3n	3n	NUM
ejpam-5973	427	11	8	8	NUM
ejpam-5973	427	12	⌉	⌉	X
ejpam-5973	427	13	.	.	PUNCT
ejpam-5973	428	1	(	(	PUNCT
ejpam-5973	428	2	iii	iii	X
ejpam-5973	428	3	)	)	PUNCT
ejpam-5973	428	4	let	let	VERB
ejpam-5973	428	5	cn	cn	PROPN
ejpam-5973	428	6	=	=	PUNCT
ejpam-5973	429	1	[	[	X
ejpam-5973	429	2	v1	v1	NOUN
ejpam-5973	429	3	,	,	PUNCT
ejpam-5973	429	4	v2	v2	NOUN
ejpam-5973	429	5	,	,	PUNCT
ejpam-5973	429	6	.	.	PUNCT
ejpam-5973	429	7	.	.	PUNCT
ejpam-5973	429	8	.	.	PUNCT
ejpam-5973	430	1	,	,	PUNCT
ejpam-5973	430	2	vn	vn	X
ejpam-5973	430	3	,	,	PUNCT
ejpam-5973	430	4	v1	v1	PROPN
ejpam-5973	430	5	]	]	PUNCT
ejpam-5973	430	6	be	be	VERB
ejpam-5973	430	7	a	a	DET
ejpam-5973	430	8	cycle	cycle	NOUN
ejpam-5973	430	9	on	on	ADP
ejpam-5973	430	10	n	n	DET
ejpam-5973	430	11	vertices	vertex	NOUN
ejpam-5973	430	12	.	.	PUNCT
ejpam-5973	431	1	by	by	ADP
ejpam-5973	431	2	theorem	theorem	NOUN
ejpam-5973	431	3	4	4	NUM
ejpam-5973	431	4	and	and	CCONJ
ejpam-5973	431	5	remark	remark	NOUN
ejpam-5973	431	6	1	1	NUM
ejpam-5973	431	7	,	,	PUNCT
ejpam-5973	431	8	we	we	PRON
ejpam-5973	431	9	have	have	VERB
ejpam-5973	431	10	γhr2(c3	γhr2(c3	PROPN
ejpam-5973	431	11	)	)	PUNCT
ejpam-5973	432	1	=	=	SYM
ejpam-5973	432	2	3	3	NUM
ejpam-5973	432	3	and	and	CCONJ
ejpam-5973	432	4	γhr2(c4	γhr2(c4	PROPN
ejpam-5973	432	5	)	)	PUNCT
ejpam-5973	432	6	=	=	SYM
ejpam-5973	433	1	4	4	X
ejpam-5973	433	2	.	.	X
ejpam-5973	433	3	assume	assume	VERB
ejpam-5973	433	4	that	that	SCONJ
ejpam-5973	433	5	n	n	NUM
ejpam-5973	433	6	≥	≥	NUM
ejpam-5973	433	7	5	5	NUM
ejpam-5973	433	8	.	.	PUNCT
ejpam-5973	434	1	if	if	SCONJ
ejpam-5973	434	2	n	n	NOUN
ejpam-5973	434	3	is	be	AUX
ejpam-5973	434	4	even	even	ADV
ejpam-5973	434	5	,	,	PUNCT
ejpam-5973	434	6	then	then	ADV
ejpam-5973	434	7	c2	c2	PROPN
ejpam-5973	434	8	n	n	PART
ejpam-5973	434	9	is	be	AUX
ejpam-5973	434	10	the	the	DET
ejpam-5973	434	11	union	union	NOUN
ejpam-5973	434	12	of	of	ADP
ejpam-5973	434	13	two	two	NUM
ejpam-5973	434	14	cycles	cycle	NOUN
ejpam-5973	434	15	:	:	PUNCT
ejpam-5973	434	16	c	c	X
ejpam-5973	434	17	=	=	PUNCT
ejpam-5973	435	1	[	[	X
ejpam-5973	435	2	v1	v1	NOUN
ejpam-5973	435	3	,	,	PUNCT
ejpam-5973	435	4	v3	v3	PROPN
ejpam-5973	435	5	,	,	PUNCT
ejpam-5973	435	6	.	.	PUNCT
ejpam-5973	435	7	.	.	PUNCT
ejpam-5973	435	8	.	.	PUNCT
ejpam-5973	436	1	,	,	PUNCT
ejpam-5973	436	2	vn−1	vn−1	ADJ
ejpam-5973	436	3	,	,	PUNCT
ejpam-5973	436	4	v1	v1	NOUN
ejpam-5973	436	5	]	]	PUNCT
ejpam-5973	436	6	and	and	CCONJ
ejpam-5973	436	7	c	c	NOUN
ejpam-5973	436	8	′	′	NOUN
ejpam-5973	437	1	=	=	PUNCT
ejpam-5973	438	1	[	[	X
ejpam-5973	438	2	v2	v2	PROPN
ejpam-5973	438	3	,	,	PUNCT
ejpam-5973	438	4	v4	v4	NOUN
ejpam-5973	438	5	,	,	PUNCT
ejpam-5973	438	6	.	.	PUNCT
ejpam-5973	438	7	.	.	PUNCT
ejpam-5973	439	1	.	.	PUNCT
ejpam-5973	440	1	,	,	PUNCT
ejpam-5973	440	2	vn	vn	INTJ
ejpam-5973	440	3	,	,	PUNCT
ejpam-5973	440	4	v2	v2	PROPN
ejpam-5973	440	5	]	]	PUNCT
ejpam-5973	440	6	,	,	PUNCT
ejpam-5973	440	7	of	of	ADP
ejpam-5973	440	8	order	order	NOUN
ejpam-5973	440	9	n	n	PRON
ejpam-5973	440	10	2	2	NUM
ejpam-5973	440	11	.	.	PUNCT
ejpam-5973	441	1	by	by	ADP
ejpam-5973	441	2	remark	remark	NOUN
ejpam-5973	441	3	1	1	NUM
ejpam-5973	441	4	and	and	CCONJ
ejpam-5973	441	5	theorem	theorem	VERB
ejpam-5973	441	6	8-(ii	8-(ii	NUM
ejpam-5973	441	7	)	)	PUNCT
ejpam-5973	441	8	,	,	PUNCT
ejpam-5973	441	9	we	we	PRON
ejpam-5973	441	10	get	get	VERB
ejpam-5973	441	11	γhr2(cn	γhr2(cn	NUM
ejpam-5973	441	12	)	)	PUNCT
ejpam-5973	442	1	=	=	SYM
ejpam-5973	442	2	γr2(c	γr2(c	PROPN
ejpam-5973	442	3	2	2	NUM
ejpam-5973	442	4	n	n	CCONJ
ejpam-5973	442	5	)	)	PUNCT
ejpam-5973	442	6	=	=	SYM
ejpam-5973	442	7	γr2(c	γr2(c	PROPN
ejpam-5973	442	8	)	)	PUNCT
ejpam-5973	443	1	+	+	CCONJ
ejpam-5973	443	2	γr2(c	γr2(c	PROPN
ejpam-5973	443	3	′	′	NUM
ejpam-5973	443	4	)	)	PUNCT
ejpam-5973	443	5	=	=	SYM
ejpam-5973	443	6	2	2	NUM
ejpam-5973	443	7	(	(	PUNCT
ejpam-5973	443	8	⌊	⌊	VERB
ejpam-5973	443	9	n	n	PRON
ejpam-5973	443	10	2	2	NUM
ejpam-5973	443	11	2	2	NUM
ejpam-5973	443	12	⌋	⌋	NOUN
ejpam-5973	443	13	+	+	CCONJ
ejpam-5973	443	14	⌈	⌈	SYM
ejpam-5973	443	15	n	n	CCONJ
ejpam-5973	443	16	2	2	NUM
ejpam-5973	443	17	4	4	NUM
ejpam-5973	443	18	⌉	⌉	ADP
ejpam-5973	443	19	−	−	NOUN
ejpam-5973	443	20	⌊	⌊	PROPN
ejpam-5973	443	21	n	n	ADV
ejpam-5973	443	22	2	2	NUM
ejpam-5973	443	23	4	4	NUM
ejpam-5973	443	24	⌋	⌋	NOUN
ejpam-5973	443	25	)	)	PUNCT
ejpam-5973	444	1	=	=	SYM
ejpam-5973	444	2	2	2	NUM
ejpam-5973	444	3	(	(	PUNCT
ejpam-5973	444	4	⌊n	⌊n	X
ejpam-5973	444	5	4	4	NUM
ejpam-5973	444	6	⌋	⌋	NOUN
ejpam-5973	444	7	+	+	CCONJ
ejpam-5973	444	8	⌈n	⌈n	NOUN
ejpam-5973	444	9	8	8	NUM
ejpam-5973	444	10	⌉	⌉	ADP
ejpam-5973	444	11	−	−	NOUN
ejpam-5973	444	12	⌊n	⌊n	ADJ
ejpam-5973	444	13	8	8	NUM
ejpam-5973	444	14	⌋	⌋	NOUN
ejpam-5973	444	15	)	)	PUNCT
ejpam-5973	444	16	,	,	PUNCT
ejpam-5973	444	17	as	as	SCONJ
ejpam-5973	444	18	desired	desire	VERB
ejpam-5973	444	19	.	.	PUNCT
ejpam-5973	445	1	assume	assume	VERB
ejpam-5973	445	2	that	that	SCONJ
ejpam-5973	445	3	n	n	PRON
ejpam-5973	445	4	is	be	AUX
ejpam-5973	445	5	odd	odd	ADJ
ejpam-5973	445	6	.	.	PUNCT
ejpam-5973	446	1	then	then	ADV
ejpam-5973	446	2	,	,	PUNCT
ejpam-5973	446	3	clearly	clearly	ADV
ejpam-5973	446	4	,	,	PUNCT
ejpam-5973	446	5	c2	c2	PROPN
ejpam-5973	446	6	n	n	PROPN
ejpam-5973	446	7	=	=	SYM
ejpam-5973	446	8	cn	cn	PROPN
ejpam-5973	446	9	,	,	PUNCT
ejpam-5973	446	10	and	and	CCONJ
ejpam-5973	446	11	by	by	ADP
ejpam-5973	446	12	theorem	theorem	PROPN
ejpam-5973	446	13	8-(ii	8-(ii	NUM
ejpam-5973	446	14	)	)	PUNCT
ejpam-5973	446	15	,	,	PUNCT
ejpam-5973	446	16	we	we	PRON
ejpam-5973	446	17	have	have	VERB
ejpam-5973	446	18	γhr2(cn	γhr2(cn	NUM
ejpam-5973	446	19	)	)	PUNCT
ejpam-5973	447	1	=	=	SYM
ejpam-5973	447	2	γr2(c	γr2(c	PROPN
ejpam-5973	447	3	2	2	NUM
ejpam-5973	447	4	n	n	CCONJ
ejpam-5973	447	5	)	)	PUNCT
ejpam-5973	447	6	=	=	SYM
ejpam-5973	447	7	γr2(cn	γr2(cn	NOUN
ejpam-5973	447	8	)	)	PUNCT
ejpam-5973	447	9	=	=	PUNCT
ejpam-5973	447	10	⌊n	⌊n	NUM
ejpam-5973	447	11	2	2	NUM
ejpam-5973	447	12	⌋	⌋	NOUN
ejpam-5973	447	13	+	+	CCONJ
ejpam-5973	447	14	⌈n	⌈n	NOUN
ejpam-5973	447	15	4	4	NUM
ejpam-5973	447	16	⌉	⌉	ADP
ejpam-5973	447	17	−	−	NOUN
ejpam-5973	447	18	⌊n	⌊n	ADJ
ejpam-5973	447	19	4	4	NUM
ejpam-5973	447	20	⌋	⌋	NOUN
ejpam-5973	447	21	.	.	PUNCT
ejpam-5973	448	1	(	(	PUNCT
ejpam-5973	448	2	iv	iv	X
ejpam-5973	448	3	)	)	PUNCT
ejpam-5973	448	4	the	the	DET
ejpam-5973	448	5	proof	proof	NOUN
ejpam-5973	448	6	is	be	AUX
ejpam-5973	448	7	similar	similar	ADJ
ejpam-5973	448	8	to	to	ADP
ejpam-5973	448	9	that	that	PRON
ejpam-5973	448	10	of	of	ADP
ejpam-5973	448	11	item	item	NOUN
ejpam-5973	448	12	(	(	PUNCT
ejpam-5973	448	13	iii	iii	NOUN
ejpam-5973	448	14	)	)	PUNCT
ejpam-5973	448	15	.	.	PUNCT
ejpam-5973	449	1	let	let	VERB
ejpam-5973	449	2	cn	cn	PROPN
ejpam-5973	449	3	=	=	PUNCT
ejpam-5973	450	1	[	[	X
ejpam-5973	450	2	v1	v1	NOUN
ejpam-5973	450	3	,	,	PUNCT
ejpam-5973	450	4	v2	v2	NOUN
ejpam-5973	450	5	,	,	PUNCT
ejpam-5973	450	6	.	.	PUNCT
ejpam-5973	450	7	.	.	PUNCT
ejpam-5973	450	8	.	.	PUNCT
ejpam-5973	451	1	,	,	PUNCT
ejpam-5973	451	2	vn	vn	X
ejpam-5973	451	3	,	,	PUNCT
ejpam-5973	451	4	v1	v1	PROPN
ejpam-5973	451	5	]	]	PUNCT
ejpam-5973	451	6	be	be	VERB
ejpam-5973	451	7	a	a	DET
ejpam-5973	451	8	cycle	cycle	NOUN
ejpam-5973	451	9	of	of	ADP
ejpam-5973	451	10	order	order	NOUN
ejpam-5973	452	1	n.	n.	NOUN
ejpam-5973	452	2	we	we	PRON
ejpam-5973	452	3	know	know	VERB
ejpam-5973	452	4	from	from	ADP
ejpam-5973	452	5	theorem	theorem	ADJ
ejpam-5973	452	6	4	4	NUM
ejpam-5973	452	7	and	and	CCONJ
ejpam-5973	452	8	remark	remark	NOUN
ejpam-5973	452	9	1	1	NUM
ejpam-5973	452	10	that	that	DET
ejpam-5973	452	11	γhr3(cn	γhr3(cn	NOUN
ejpam-5973	452	12	)	)	PUNCT
ejpam-5973	452	13	=	=	PUNCT
ejpam-5973	453	1	γr3(c	γr3(c	ADP
ejpam-5973	453	2	2	2	NUM
ejpam-5973	453	3	n	n	CCONJ
ejpam-5973	453	4	)	)	PUNCT
ejpam-5973	453	5	.	.	PUNCT
ejpam-5973	454	1	assume	assume	VERB
ejpam-5973	454	2	that	that	SCONJ
ejpam-5973	454	3	n	n	NUM
ejpam-5973	454	4	≥	≥	NUM
ejpam-5973	454	5	5	5	NUM
ejpam-5973	454	6	.	.	PUNCT
ejpam-5973	455	1	if	if	SCONJ
ejpam-5973	455	2	n	n	NOUN
ejpam-5973	455	3	is	be	AUX
ejpam-5973	455	4	odd	odd	ADJ
ejpam-5973	455	5	.	.	PUNCT
ejpam-5973	456	1	then	then	ADV
ejpam-5973	456	2	clearly	clearly	ADV
ejpam-5973	456	3	c2	c2	VERB
ejpam-5973	456	4	n	n	PROPN
ejpam-5973	456	5	=	=	SYM
ejpam-5973	456	6	cn	cn	PROPN
ejpam-5973	456	7	.	.	PUNCT
ejpam-5973	457	1	hence	hence	ADV
ejpam-5973	457	2	,	,	PUNCT
ejpam-5973	457	3	γhr3(cn	γhr3(cn	PROPN
ejpam-5973	457	4	)	)	PUNCT
ejpam-5973	457	5	=	=	PUNCT
ejpam-5973	458	1	γr3(c	γr3(c	ADP
ejpam-5973	458	2	2	2	NUM
ejpam-5973	458	3	n	n	CCONJ
ejpam-5973	458	4	)	)	PUNCT
ejpam-5973	458	5	=	=	SYM
ejpam-5973	458	6	γr3(cn	γr3(cn	PROPN
ejpam-5973	458	7	)	)	PUNCT
ejpam-5973	458	8	.	.	PUNCT
ejpam-5973	459	1	applying	apply	VERB
ejpam-5973	459	2	theorem	theorem	ADJ
ejpam-5973	459	3	8(iv	8(iv	NUM
ejpam-5973	459	4	)	)	PUNCT
ejpam-5973	459	5	,	,	PUNCT
ejpam-5973	459	6	we	we	PRON
ejpam-5973	459	7	get	get	VERB
ejpam-5973	459	8	γhr3(cn	γhr3(cn	NOUN
ejpam-5973	459	9	)	)	PUNCT
ejpam-5973	460	1	=	=	PUNCT
ejpam-5973	461	1	⌈	⌈	NOUN
ejpam-5973	461	2	3n	3n	NUM
ejpam-5973	461	3	4	4	NUM
ejpam-5973	461	4	⌉	⌉	NOUN
ejpam-5973	461	5	,	,	PUNCT
ejpam-5973	461	6	j.	j.	PROPN
ejpam-5973	461	7	j.	j.	PROPN
ejpam-5973	461	8	hamja	hamja	PROPN
ejpam-5973	461	9	et	et	PROPN
ejpam-5973	461	10	al	al	PROPN
ejpam-5973	461	11	.	.	PUNCT
ejpam-5973	461	12	/	/	SYM
ejpam-5973	461	13	eur	eur	PROPN
ejpam-5973	461	14	.	.	PUNCT
ejpam-5973	462	1	j.	j.	PROPN
ejpam-5973	462	2	pure	pure	PROPN
ejpam-5973	462	3	appl	appl	PROPN
ejpam-5973	462	4	.	.	PROPN
ejpam-5973	462	5	math	math	PROPN
ejpam-5973	462	6	,	,	PUNCT
ejpam-5973	462	7	18	18	NUM
ejpam-5973	462	8	(	(	PUNCT
ejpam-5973	462	9	2	2	NUM
ejpam-5973	462	10	)	)	PUNCT
ejpam-5973	462	11	(	(	PUNCT
ejpam-5973	462	12	2025	2025	NUM
ejpam-5973	462	13	)	)	PUNCT
ejpam-5973	462	14	,	,	PUNCT
ejpam-5973	462	15	5973	5973	NUM
ejpam-5973	462	16	12	12	NUM
ejpam-5973	462	17	of	of	ADP
ejpam-5973	462	18	17	17	NUM
ejpam-5973	462	19	as	as	SCONJ
ejpam-5973	462	20	desired	desire	VERB
ejpam-5973	462	21	.	.	PUNCT
ejpam-5973	463	1	if	if	SCONJ
ejpam-5973	463	2	n	n	PRON
ejpam-5973	463	3	is	be	AUX
ejpam-5973	463	4	even	even	ADV
ejpam-5973	463	5	.	.	PUNCT
ejpam-5973	464	1	then	then	ADV
ejpam-5973	464	2	c2	c2	PROPN
ejpam-5973	464	3	n	n	PART
ejpam-5973	464	4	is	be	AUX
ejpam-5973	464	5	the	the	DET
ejpam-5973	464	6	union	union	NOUN
ejpam-5973	464	7	of	of	ADP
ejpam-5973	464	8	two	two	NUM
ejpam-5973	464	9	cycles	cycle	NOUN
ejpam-5973	464	10	,	,	PUNCT
ejpam-5973	464	11	each	each	PRON
ejpam-5973	464	12	of	of	ADP
ejpam-5973	464	13	length	length	NOUN
ejpam-5973	464	14	n	n	ADP
ejpam-5973	464	15	2	2	NUM
ejpam-5973	464	16	:	:	PUNCT
ejpam-5973	464	17	c	c	X
ejpam-5973	464	18	=	=	PUNCT
ejpam-5973	465	1	[	[	X
ejpam-5973	465	2	v1	v1	NOUN
ejpam-5973	465	3	,	,	PUNCT
ejpam-5973	465	4	v3	v3	PROPN
ejpam-5973	465	5	,	,	PUNCT
ejpam-5973	465	6	.	.	PUNCT
ejpam-5973	465	7	.	.	PUNCT
ejpam-5973	465	8	.	.	PUNCT
ejpam-5973	466	1	,	,	PUNCT
ejpam-5973	466	2	vn−1	vn−1	ADJ
ejpam-5973	466	3	,	,	PUNCT
ejpam-5973	466	4	v1	v1	NOUN
ejpam-5973	466	5	]	]	PUNCT
ejpam-5973	466	6	and	and	CCONJ
ejpam-5973	466	7	c	c	NOUN
ejpam-5973	466	8	′	′	NOUN
ejpam-5973	467	1	=	=	PUNCT
ejpam-5973	468	1	[	[	X
ejpam-5973	468	2	v2	v2	PROPN
ejpam-5973	468	3	,	,	PUNCT
ejpam-5973	468	4	v4	v4	NOUN
ejpam-5973	468	5	,	,	PUNCT
ejpam-5973	468	6	.	.	PUNCT
ejpam-5973	468	7	.	.	PUNCT
ejpam-5973	469	1	.	.	PUNCT
ejpam-5973	470	1	,	,	PUNCT
ejpam-5973	470	2	vn	vn	INTJ
ejpam-5973	470	3	,	,	PUNCT
ejpam-5973	470	4	v2	v2	PROPN
ejpam-5973	470	5	]	]	PUNCT
ejpam-5973	470	6	.	.	PUNCT
ejpam-5973	471	1	thus	thus	ADV
ejpam-5973	471	2	,	,	PUNCT
ejpam-5973	471	3	γhr3(cn	γhr3(cn	PROPN
ejpam-5973	471	4	)	)	PUNCT
ejpam-5973	471	5	=	=	PUNCT
ejpam-5973	472	1	γr3(c	γr3(c	ADP
ejpam-5973	472	2	2	2	NUM
ejpam-5973	472	3	n	n	CCONJ
ejpam-5973	472	4	)	)	PUNCT
ejpam-5973	472	5	=	=	SYM
ejpam-5973	472	6	γr3(c	γr3(c	X
ejpam-5973	472	7	)	)	PUNCT
ejpam-5973	472	8	+	+	CCONJ
ejpam-5973	472	9	γr3(c	γr3(c	PROPN
ejpam-5973	472	10	′	′	NUM
ejpam-5973	472	11	)	)	PUNCT
ejpam-5973	472	12	.	.	PUNCT
ejpam-5973	473	1	since	since	SCONJ
ejpam-5973	473	2	both	both	DET
ejpam-5973	473	3	c	c	PROPN
ejpam-5973	473	4	and	and	CCONJ
ejpam-5973	473	5	c	c	PROPN
ejpam-5973	473	6	′	′	NOUN
ejpam-5973	473	7	are	be	AUX
ejpam-5973	473	8	cycles	cycle	NOUN
ejpam-5973	473	9	of	of	ADP
ejpam-5973	473	10	order	order	NOUN
ejpam-5973	473	11	n	n	PRON
ejpam-5973	473	12	2	2	NUM
ejpam-5973	473	13	,	,	PUNCT
ejpam-5973	473	14	we	we	PRON
ejpam-5973	473	15	can	can	AUX
ejpam-5973	473	16	apply	apply	VERB
ejpam-5973	473	17	theorem	theorem	ADJ
ejpam-5973	473	18	8-(iv	8-(iv	NOUN
ejpam-5973	473	19	)	)	PUNCT
ejpam-5973	473	20	again	again	ADV
ejpam-5973	473	21	,	,	PUNCT
ejpam-5973	473	22	γr3(c	γr3(c	ADJ
ejpam-5973	473	23	)	)	PUNCT
ejpam-5973	473	24	=	=	PUNCT
ejpam-5973	474	1	⌈	⌈	NUM
ejpam-5973	474	2	3n	3n	NUM
ejpam-5973	474	3	2	2	NUM
ejpam-5973	474	4	4	4	NUM
ejpam-5973	474	5	⌉	⌉	NOUN
ejpam-5973	474	6	=	=	PUNCT
ejpam-5973	474	7	⌈	⌈	NUM
ejpam-5973	474	8	3n	3n	NUM
ejpam-5973	474	9	8	8	NUM
ejpam-5973	474	10	⌉	⌉	NOUN
ejpam-5973	474	11	,	,	PUNCT
ejpam-5973	474	12	and	and	CCONJ
ejpam-5973	474	13	similarly	similarly	ADV
ejpam-5973	474	14	,	,	PUNCT
ejpam-5973	474	15	γr3(c	γr3(c	PROPN
ejpam-5973	474	16	′	′	NOUN
ejpam-5973	474	17	)	)	PUNCT
ejpam-5973	475	1	=	=	PUNCT
ejpam-5973	476	1	⌈	⌈	NOUN
ejpam-5973	476	2	3n	3n	NUM
ejpam-5973	476	3	8	8	NUM
ejpam-5973	476	4	⌉	⌉	NOUN
ejpam-5973	476	5	.	.	PUNCT
ejpam-5973	477	1	hence	hence	ADV
ejpam-5973	477	2	,	,	PUNCT
ejpam-5973	477	3	γhr3(cn	γhr3(cn	PROPN
ejpam-5973	477	4	)	)	PUNCT
ejpam-5973	477	5	=	=	PUNCT
ejpam-5973	478	1	⌈	⌈	NUM
ejpam-5973	478	2	3n	3n	NUM
ejpam-5973	478	3	8	8	NUM
ejpam-5973	478	4	⌉	⌉	NOUN
ejpam-5973	478	5	+	+	PUNCT
ejpam-5973	478	6	⌈	⌈	NUM
ejpam-5973	478	7	3n	3n	NUM
ejpam-5973	478	8	8	8	NUM
ejpam-5973	478	9	⌉	⌉	NOUN
ejpam-5973	478	10	=	=	SYM
ejpam-5973	478	11	2	2	NUM
ejpam-5973	478	12	⌈	⌈	NUM
ejpam-5973	478	13	3n	3n	NUM
ejpam-5973	478	14	8	8	NUM
ejpam-5973	478	15	⌉	⌉	NOUN
ejpam-5973	478	16	,	,	PUNCT
ejpam-5973	478	17	this	this	PRON
ejpam-5973	478	18	completes	complete	VERB
ejpam-5973	478	19	the	the	DET
ejpam-5973	478	20	proof	proof	NOUN
ejpam-5973	478	21	.	.	PUNCT
ejpam-5973	479	1	applying	apply	VERB
ejpam-5973	479	2	theorem	theorem	NOUN
ejpam-5973	479	3	9	9	NUM
ejpam-5973	479	4	and	and	CCONJ
ejpam-5973	479	5	a	a	DET
ejpam-5973	479	6	similar	similar	ADJ
ejpam-5973	479	7	method	method	NOUN
ejpam-5973	479	8	used	use	VERB
ejpam-5973	479	9	in	in	ADP
ejpam-5973	479	10	[	[	X
ejpam-5973	479	11	3	3	NUM
ejpam-5973	479	12	,	,	PUNCT
ejpam-5973	479	13	9	9	NUM
ejpam-5973	479	14	]	]	PUNCT
ejpam-5973	479	15	,	,	PUNCT
ejpam-5973	479	16	we	we	PRON
ejpam-5973	479	17	obtain	obtain	VERB
ejpam-5973	479	18	the	the	DET
ejpam-5973	479	19	following	follow	VERB
ejpam-5973	479	20	result	result	NOUN
ejpam-5973	479	21	.	.	PUNCT
ejpam-5973	480	1	corollary	corollary	ADJ
ejpam-5973	480	2	3	3	NUM
ejpam-5973	480	3	.	.	PUNCT
ejpam-5973	481	1	for	for	ADP
ejpam-5973	481	2	any	any	DET
ejpam-5973	481	3	connected	connected	ADJ
ejpam-5973	481	4	graph	graph	NOUN
ejpam-5973	481	5	g	g	NOUN
ejpam-5973	481	6	of	of	ADP
ejpam-5973	481	7	order	order	NOUN
ejpam-5973	481	8	n	n	CCONJ
ejpam-5973	481	9	,	,	PUNCT
ejpam-5973	481	10	(	(	PUNCT
ejpam-5973	481	11	i	i	NOUN
ejpam-5973	481	12	)	)	PUNCT
ejpam-5973	481	13	γhr2(g	γhr2(g	NUM
ejpam-5973	481	14	)	)	PUNCT
ejpam-5973	481	15	≤	≤	NUM
ejpam-5973	481	16			NOUN
ejpam-5973	481	17	n+	n+	NUM
ejpam-5973	482	1	2−	2−	NUM
ejpam-5973	482	2	diam(g	diam(g	NOUN
ejpam-5973	482	3	)	)	PUNCT
ejpam-5973	482	4	2	2	NUM
ejpam-5973	482	5	,	,	PUNCT
ejpam-5973	482	6	if	if	SCONJ
ejpam-5973	482	7	diam(g	diam(g	NOUN
ejpam-5973	482	8	)	)	PUNCT
ejpam-5973	482	9	≡	≡	PROPN
ejpam-5973	482	10	0	0	PUNCT
ejpam-5973	483	1	(	(	PUNCT
ejpam-5973	483	2	mod	mod	PROPN
ejpam-5973	483	3	4	4	NUM
ejpam-5973	483	4	)	)	PUNCT
ejpam-5973	483	5	,	,	PUNCT
ejpam-5973	483	6	n+	n+	NUM
ejpam-5973	483	7	1−	1−	NUM
ejpam-5973	483	8	diam(g	diam(g	NOUN
ejpam-5973	483	9	)	)	PUNCT
ejpam-5973	483	10	2	2	NUM
ejpam-5973	483	11	,	,	PUNCT
ejpam-5973	483	12	if	if	SCONJ
ejpam-5973	483	13	diam(g	diam(g	NOUN
ejpam-5973	483	14	)	)	PUNCT
ejpam-5973	483	15	≡	≡	PROPN
ejpam-5973	483	16	2	2	NUM
ejpam-5973	483	17	(	(	PUNCT
ejpam-5973	483	18	mod	mod	NOUN
ejpam-5973	483	19	4	4	NUM
ejpam-5973	483	20	)	)	PUNCT
ejpam-5973	483	21	,	,	PUNCT
ejpam-5973	483	22	n+	n+	X
ejpam-5973	483	23	1−	1−	NUM
ejpam-5973	483	24	diam(g)+1	diam(g)+1	NOUN
ejpam-5973	483	25	2	2	NUM
ejpam-5973	483	26	,	,	PUNCT
ejpam-5973	483	27	if	if	SCONJ
ejpam-5973	483	28	diam(g	diam(g	NOUN
ejpam-5973	483	29	)	)	PUNCT
ejpam-5973	483	30	≡	≡	PROPN
ejpam-5973	483	31	1	1	NUM
ejpam-5973	483	32	(	(	PUNCT
ejpam-5973	483	33	mod	mod	NOUN
ejpam-5973	483	34	4	4	NUM
ejpam-5973	483	35	)	)	PUNCT
ejpam-5973	483	36	,	,	PUNCT
ejpam-5973	483	37	n+	n+	ADP
ejpam-5973	483	38	1	1	NUM
ejpam-5973	483	39	+	+	NUM
ejpam-5973	483	40	1−diam(g	1−diam(g	ADJ
ejpam-5973	483	41	)	)	PUNCT
ejpam-5973	483	42	2	2	NUM
ejpam-5973	483	43	,	,	PUNCT
ejpam-5973	483	44	if	if	SCONJ
ejpam-5973	483	45	diam(g	diam(g	NOUN
ejpam-5973	483	46	)	)	PUNCT
ejpam-5973	483	47	≡	≡	PROPN
ejpam-5973	483	48	3	3	NUM
ejpam-5973	483	49	(	(	PUNCT
ejpam-5973	483	50	mod	mod	NOUN
ejpam-5973	483	51	4	4	NUM
ejpam-5973	483	52	)	)	PUNCT
ejpam-5973	483	53	.	.	PUNCT
ejpam-5973	484	1	(	(	PUNCT
ejpam-5973	484	2	ii	ii	NOUN
ejpam-5973	484	3	)	)	PUNCT
ejpam-5973	484	4	γhr3(g	γhr3(g	NUM
ejpam-5973	484	5	)	)	PUNCT
ejpam-5973	484	6	≤	≤	NUM
ejpam-5973	484	7			NUM
ejpam-5973	484	8	n−	n−	NOUN
ejpam-5973	484	9	diam(g	diam(g	NOUN
ejpam-5973	484	10	)	)	PUNCT
ejpam-5973	485	1	+	+	NUM
ejpam-5973	485	2	⌈	⌈	NUM
ejpam-5973	485	3	3diam(g)+6	3diam(g)+6	NOUN
ejpam-5973	485	4	8	8	NUM
ejpam-5973	485	5	⌉	⌉	NOUN
ejpam-5973	485	6	+	+	CCONJ
ejpam-5973	485	7	⌈	⌈	NOUN
ejpam-5973	485	8	3diam(g	3diam(g	NUM
ejpam-5973	485	9	)	)	PUNCT
ejpam-5973	485	10	8	8	NUM
ejpam-5973	485	11	⌉	⌉	NOUN
ejpam-5973	485	12	,	,	PUNCT
ejpam-5973	485	13	if	if	SCONJ
ejpam-5973	485	14	diam(g	diam(g	NOUN
ejpam-5973	485	15	)	)	PUNCT
ejpam-5973	485	16	≡	≡	PROPN
ejpam-5973	485	17	0	0	NUM
ejpam-5973	485	18	,	,	PUNCT
ejpam-5973	485	19	6	6	NUM
ejpam-5973	485	20	(	(	PUNCT
ejpam-5973	485	21	mod	mod	PROPN
ejpam-5973	485	22	8)	8)	NUM
ejpam-5973	485	23	,	,	PUNCT
ejpam-5973	485	24	n−	n−	PROPN
ejpam-5973	485	25	1−	1−	NUM
ejpam-5973	485	26	diam(g	diam(g	NOUN
ejpam-5973	485	27	)	)	PUNCT
ejpam-5973	486	1	+	+	CCONJ
ejpam-5973	486	2	2	2	NUM
ejpam-5973	486	3	⌈	⌈	NOUN
ejpam-5973	486	4	3diam(g)+3	3diam(g)+3	NUM
ejpam-5973	486	5	8	8	NUM
ejpam-5973	486	6	⌉	⌉	NOUN
ejpam-5973	486	7	,	,	PUNCT
ejpam-5973	486	8	if	if	SCONJ
ejpam-5973	486	9	diam(g	diam(g	NOUN
ejpam-5973	486	10	)	)	PUNCT
ejpam-5973	486	11	≡	≡	PROPN
ejpam-5973	486	12	1	1	NUM
ejpam-5973	486	13	,	,	PUNCT
ejpam-5973	486	14	3	3	NUM
ejpam-5973	486	15	,	,	PUNCT
ejpam-5973	486	16	5	5	NUM
ejpam-5973	486	17	(	(	PUNCT
ejpam-5973	486	18	mod	mod	NOUN
ejpam-5973	486	19	8)	8)	NUM
ejpam-5973	486	20	,	,	PUNCT
ejpam-5973	486	21	n−	n−	PROPN
ejpam-5973	486	22	1−	1−	NUM
ejpam-5973	486	23	diam(g	diam(g	NOUN
ejpam-5973	486	24	)	)	PUNCT
ejpam-5973	487	1	+	+	NUM
ejpam-5973	487	2	⌈	⌈	NUM
ejpam-5973	487	3	3diam(g)+6	3diam(g)+6	NOUN
ejpam-5973	487	4	8	8	NUM
ejpam-5973	487	5	⌉	⌉	NOUN
ejpam-5973	487	6	+	+	CCONJ
ejpam-5973	487	7	⌈	⌈	NOUN
ejpam-5973	487	8	3diam(g	3diam(g	NUM
ejpam-5973	487	9	)	)	PUNCT
ejpam-5973	487	10	8	8	NUM
ejpam-5973	487	11	⌉	⌉	NOUN
ejpam-5973	487	12	,	,	PUNCT
ejpam-5973	487	13	if	if	SCONJ
ejpam-5973	487	14	diam(g	diam(g	NOUN
ejpam-5973	487	15	)	)	PUNCT
ejpam-5973	487	16	≡	≡	PROPN
ejpam-5973	487	17	2	2	NUM
ejpam-5973	487	18	,	,	PUNCT
ejpam-5973	487	19	4	4	NUM
ejpam-5973	487	20	(	(	PUNCT
ejpam-5973	487	21	mod	mod	NOUN
ejpam-5973	487	22	8)	8)	NUM
ejpam-5973	487	23	,	,	PUNCT
ejpam-5973	487	24	n+	n+	ADP
ejpam-5973	487	25	1	1	NUM
ejpam-5973	487	26	+	+	CCONJ
ejpam-5973	487	27	3−diam(g	3−diam(g	ADJ
ejpam-5973	487	28	)	)	PUNCT
ejpam-5973	487	29	4	4	NUM
ejpam-5973	487	30	,	,	PUNCT
ejpam-5973	487	31	if	if	SCONJ
ejpam-5973	487	32	diam(g	diam(g	NOUN
ejpam-5973	487	33	)	)	PUNCT
ejpam-5973	487	34	≡	≡	PROPN
ejpam-5973	487	35	7	7	NUM
ejpam-5973	487	36	(	(	PUNCT
ejpam-5973	487	37	mod	mod	NOUN
ejpam-5973	487	38	8)	8)	NUM
ejpam-5973	487	39	.	.	PUNCT
ejpam-5973	488	1	proof	proof	NOUN
ejpam-5973	488	2	.	.	PUNCT
ejpam-5973	489	1	let	let	VERB
ejpam-5973	489	2	d	d	NOUN
ejpam-5973	489	3	=	=	SYM
ejpam-5973	489	4	diam(g	diam(g	PROPN
ejpam-5973	489	5	)	)	PUNCT
ejpam-5973	490	1	+	+	CCONJ
ejpam-5973	490	2	1	1	NUM
ejpam-5973	490	3	and	and	CCONJ
ejpam-5973	490	4	p	p	NOUN
ejpam-5973	490	5	=	=	PUNCT
ejpam-5973	491	1	[	[	X
ejpam-5973	491	2	x1	x1	PROPN
ejpam-5973	491	3	,	,	PUNCT
ejpam-5973	491	4	x2	x2	PROPN
ejpam-5973	491	5	,	,	PUNCT
ejpam-5973	491	6	.	.	PUNCT
ejpam-5973	491	7	.	.	PUNCT
ejpam-5973	492	1	.	.	PUNCT
ejpam-5973	493	1	,	,	PUNCT
ejpam-5973	493	2	xd	xd	ADP
ejpam-5973	493	3	]	]	PUNCT
ejpam-5973	493	4	be	be	AUX
ejpam-5973	493	5	a	a	DET
ejpam-5973	493	6	diametral	diametral	ADJ
ejpam-5973	493	7	path	path	NOUN
ejpam-5973	493	8	in	in	ADP
ejpam-5973	493	9	g.	g.	PROPN
ejpam-5973	493	10	let	let	VERB
ejpam-5973	493	11	f	f	PRON
ejpam-5973	493	12	be	be	AUX
ejpam-5973	493	13	a	a	DET
ejpam-5973	493	14	γhrs	γhrs	NOUN
ejpam-5973	493	15	-	-	PUNCT
ejpam-5973	493	16	function	function	NOUN
ejpam-5973	493	17	of	of	ADP
ejpam-5973	493	18	p	p	NOUN
ejpam-5973	493	19	where	where	SCONJ
ejpam-5973	493	20	s	s	VERB
ejpam-5973	493	21	∈	∈	PROPN
ejpam-5973	493	22	{	{	PUNCT
ejpam-5973	493	23	2	2	NUM
ejpam-5973	493	24	,	,	PUNCT
ejpam-5973	493	25	3	3	NUM
ejpam-5973	493	26	}	}	PUNCT
ejpam-5973	493	27	,	,	PUNCT
ejpam-5973	493	28	and	and	CCONJ
ejpam-5973	493	29	define	define	VERB
ejpam-5973	493	30	the	the	DET
ejpam-5973	493	31	function	function	NOUN
ejpam-5973	493	32	g	g	NOUN
ejpam-5973	493	33	:	:	PUNCT
ejpam-5973	493	34	v	v	NOUN
ejpam-5973	493	35	(	(	PUNCT
ejpam-5973	493	36	g	g	NOUN
ejpam-5973	493	37	)	)	PUNCT
ejpam-5973	493	38	→	→	SYM
ejpam-5973	493	39	p({1	p({1	PROPN
ejpam-5973	493	40	,	,	PUNCT
ejpam-5973	493	41	.	.	PUNCT
ejpam-5973	493	42	.	.	PUNCT
ejpam-5973	494	1	.	.	PUNCT
ejpam-5973	495	1	,	,	PUNCT
ejpam-5973	495	2	s	s	X
ejpam-5973	495	3	}	}	PUNCT
ejpam-5973	495	4	)	)	PUNCT
ejpam-5973	495	5	by	by	ADP
ejpam-5973	495	6	g(x	g(x	NOUN
ejpam-5973	495	7	)	)	PUNCT
ejpam-5973	495	8	=	=	SYM
ejpam-5973	495	9	f(x	f(x	PROPN
ejpam-5973	495	10	)	)	PUNCT
ejpam-5973	495	11	for	for	ADP
ejpam-5973	495	12	x	x	PROPN
ejpam-5973	495	13	∈	∈	PROPN
ejpam-5973	495	14	v	v	ADP
ejpam-5973	495	15	(	(	PUNCT
ejpam-5973	495	16	p	p	NOUN
ejpam-5973	495	17	)	)	PUNCT
ejpam-5973	495	18	and	and	CCONJ
ejpam-5973	495	19	g(x	g(x	NOUN
ejpam-5973	495	20	)	)	PUNCT
ejpam-5973	495	21	=	=	SYM
ejpam-5973	496	1	1	1	NUM
ejpam-5973	496	2	for	for	ADP
ejpam-5973	496	3	x	x	PROPN
ejpam-5973	496	4	∈	∈	PROPN
ejpam-5973	496	5	v	v	ADP
ejpam-5973	496	6	(	(	PUNCT
ejpam-5973	496	7	g	g	NOUN
ejpam-5973	496	8	)	)	PUNCT
ejpam-5973	496	9	\	\	PROPN
ejpam-5973	497	1	v	v	X
ejpam-5973	497	2	(	(	PUNCT
ejpam-5973	497	3	p	p	NOUN
ejpam-5973	497	4	)	)	PUNCT
ejpam-5973	497	5	.	.	PUNCT
ejpam-5973	498	1	clearly	clearly	ADV
ejpam-5973	498	2	,	,	PUNCT
ejpam-5973	498	3	g	g	PROPN
ejpam-5973	498	4	is	be	AUX
ejpam-5973	498	5	an	an	DET
ejpam-5973	498	6	hsrdfunction	hsrdfunction	NOUN
ejpam-5973	498	7	of	of	ADP
ejpam-5973	498	8	g	g	NOUN
ejpam-5973	498	9	of	of	ADP
ejpam-5973	498	10	weight	weight	NOUN
ejpam-5973	498	11	ω(f	ω(f	PUNCT
ejpam-5973	498	12	)	)	PUNCT
ejpam-5973	499	1	+	+	CCONJ
ejpam-5973	500	1	n	n	CCONJ
ejpam-5973	500	2	−	−	NOUN
ejpam-5973	501	1	d	d	NOUN
ejpam-5973	501	2	−	−	PROPN
ejpam-5973	502	1	1	1	NUM
ejpam-5973	502	2	.	.	PUNCT
ejpam-5973	503	1	this	this	PRON
ejpam-5973	503	2	implies	imply	VERB
ejpam-5973	503	3	that	that	PRON
ejpam-5973	503	4	γhrs(g	γhrs(g	VERB
ejpam-5973	503	5	)	)	PUNCT
ejpam-5973	503	6	≤	≤	NOUN
ejpam-5973	503	7	ω(f	ω(f	NUM
ejpam-5973	503	8	)	)	PUNCT
ejpam-5973	504	1	+	+	CCONJ
ejpam-5973	505	1	n	n	CCONJ
ejpam-5973	505	2	−	−	NOUN
ejpam-5973	506	1	d	d	NOUN
ejpam-5973	506	2	−	−	PROPN
ejpam-5973	506	3	1	1	NUM
ejpam-5973	506	4	.	.	PUNCT
ejpam-5973	507	1	thus	thus	ADV
ejpam-5973	507	2	,	,	PUNCT
ejpam-5973	507	3	the	the	DET
ejpam-5973	507	4	corollary	corollary	NOUN
ejpam-5973	507	5	follows	follow	VERB
ejpam-5973	507	6	by	by	ADP
ejpam-5973	507	7	theorem	theorem	NOUN
ejpam-5973	507	8	2	2	NUM
ejpam-5973	507	9	.	.	PUNCT
ejpam-5973	507	10	using	use	VERB
ejpam-5973	507	11	theorem	theorem	NOUN
ejpam-5973	507	12	9	9	NUM
ejpam-5973	507	13	we	we	PRON
ejpam-5973	507	14	obtain	obtain	VERB
ejpam-5973	507	15	the	the	DET
ejpam-5973	507	16	next	next	ADJ
ejpam-5973	507	17	result	result	NOUN
ejpam-5973	507	18	.	.	PUNCT
ejpam-5973	508	1	corollary	corollary	ADJ
ejpam-5973	508	2	4	4	NUM
ejpam-5973	508	3	.	.	PUNCT
ejpam-5973	509	1	for	for	ADP
ejpam-5973	509	2	every	every	DET
ejpam-5973	509	3	positive	positive	ADJ
ejpam-5973	509	4	integer	integer	NOUN
ejpam-5973	509	5	a	a	PRON
ejpam-5973	509	6	,	,	PUNCT
ejpam-5973	509	7	there	there	PRON
ejpam-5973	509	8	exists	exist	VERB
ejpam-5973	509	9	a	a	DET
ejpam-5973	509	10	connected	connected	ADJ
ejpam-5973	509	11	graph	graph	NOUN
ejpam-5973	509	12	g	g	ADP
ejpam-5973	509	13	such	such	ADJ
ejpam-5973	509	14	that	that	SCONJ
ejpam-5973	509	15	γhr2(g	γhr2(g	NUM
ejpam-5973	509	16	)	)	PUNCT
ejpam-5973	509	17	=	=	SYM
ejpam-5973	509	18	γr2(g	γr2(g	PROPN
ejpam-5973	509	19	)	)	PUNCT
ejpam-5973	509	20	=	=	SYM
ejpam-5973	509	21	a.	a.	NOUN
ejpam-5973	509	22	proof	proof	NOUN
ejpam-5973	509	23	.	.	PUNCT
ejpam-5973	510	1	clearly	clearly	ADV
ejpam-5973	510	2	,	,	PUNCT
ejpam-5973	510	3	γhr2(ka	γhr2(ka	PROPN
ejpam-5973	510	4	)	)	PUNCT
ejpam-5973	510	5	=	=	SYM
ejpam-5973	510	6	γr2(ka	γr2(ka	NOUN
ejpam-5973	510	7	)	)	PUNCT
ejpam-5973	510	8	=	=	PUNCT
ejpam-5973	511	1	a	a	PRON
ejpam-5973	511	2	for	for	ADP
ejpam-5973	511	3	a	a	DET
ejpam-5973	511	4	∈	∈	PROPN
ejpam-5973	511	5	{	{	PUNCT
ejpam-5973	511	6	1	1	NUM
ejpam-5973	511	7	,	,	PUNCT
ejpam-5973	511	8	2	2	NUM
ejpam-5973	511	9	}	}	PUNCT
ejpam-5973	511	10	.	.	PUNCT
ejpam-5973	512	1	assume	assume	VERB
ejpam-5973	512	2	that	that	SCONJ
ejpam-5973	512	3	a	a	DET
ejpam-5973	512	4	≥	≥	NOUN
ejpam-5973	512	5	3	3	NUM
ejpam-5973	512	6	.	.	PUNCT
ejpam-5973	513	1	if	if	SCONJ
ejpam-5973	513	2	a	a	PRON
ejpam-5973	513	3	is	be	AUX
ejpam-5973	513	4	odd	odd	ADJ
ejpam-5973	513	5	and	and	CCONJ
ejpam-5973	513	6	a	a	DET
ejpam-5973	513	7	=	=	ADJ
ejpam-5973	513	8	2	2	NUM
ejpam-5973	513	9	m	m	NOUN
ejpam-5973	513	10	+	+	NOUN
ejpam-5973	513	11	1	1	NUM
ejpam-5973	513	12	,	,	PUNCT
ejpam-5973	513	13	then	then	ADV
ejpam-5973	513	14	by	by	ADP
ejpam-5973	513	15	theorems	theorem	NOUN
ejpam-5973	513	16	8	8	NUM
ejpam-5973	513	17	and	and	CCONJ
ejpam-5973	513	18	9	9	NUM
ejpam-5973	513	19	we	we	PRON
ejpam-5973	513	20	have	have	VERB
ejpam-5973	513	21	that	that	PRON
ejpam-5973	513	22	γhr2(c4m+1	γhr2(c4m+1	PROPN
ejpam-5973	513	23	)	)	PUNCT
ejpam-5973	513	24	=	=	SYM
ejpam-5973	513	25	γr2(c4m+1	γr2(c4m+1	NUM
ejpam-5973	513	26	)	)	PUNCT
ejpam-5973	513	27	=	=	SYM
ejpam-5973	513	28	2	2	NUM
ejpam-5973	513	29	m	m	NOUN
ejpam-5973	513	30	+	+	NOUN
ejpam-5973	513	31	1	1	NUM
ejpam-5973	513	32	=	=	NOUN
ejpam-5973	513	33	a.	a.	NOUN
ejpam-5973	513	34	assume	assume	VERB
ejpam-5973	513	35	that	that	SCONJ
ejpam-5973	513	36	a	a	PRON
ejpam-5973	513	37	is	be	AUX
ejpam-5973	513	38	even	even	ADV
ejpam-5973	513	39	and	and	CCONJ
ejpam-5973	513	40	let	let	VERB
ejpam-5973	513	41	a	a	DET
ejpam-5973	513	42	=	=	SYM
ejpam-5973	513	43	2	2	NUM
ejpam-5973	513	44	m	m	NOUN
ejpam-5973	513	45	for	for	ADP
ejpam-5973	513	46	some	some	DET
ejpam-5973	513	47	m	m	NOUN
ejpam-5973	513	48	≥	≥	NOUN
ejpam-5973	513	49	2	2	NUM
ejpam-5973	513	50	.	.	PUNCT
ejpam-5973	514	1	then	then	ADV
ejpam-5973	514	2	again	again	ADV
ejpam-5973	514	3	by	by	ADP
ejpam-5973	514	4	theorems	theorem	NOUN
ejpam-5973	514	5	8	8	NUM
ejpam-5973	514	6	and	and	CCONJ
ejpam-5973	514	7	9	9	NUM
ejpam-5973	514	8	we	we	PRON
ejpam-5973	514	9	have	have	VERB
ejpam-5973	514	10	γhr2(c4(m−1)+3	γhr2(c4(m−1)+3	PUNCT
ejpam-5973	514	11	)	)	PUNCT
ejpam-5973	515	1	=	=	SYM
ejpam-5973	515	2	γr2(c4(m−1)+3	γr2(c4(m−1)+3	NOUN
ejpam-5973	515	3	)	)	PUNCT
ejpam-5973	515	4	=	=	SYM
ejpam-5973	515	5	2	2	NUM
ejpam-5973	515	6	m	m	NOUN
ejpam-5973	515	7	=	=	PUNCT
ejpam-5973	515	8	a.	a.	PROPN
ejpam-5973	515	9	j.	j.	PROPN
ejpam-5973	515	10	j.	j.	PROPN
ejpam-5973	516	1	hamja	hamja	PROPN
ejpam-5973	516	2	et	et	PROPN
ejpam-5973	516	3	al	al	PROPN
ejpam-5973	516	4	.	.	PUNCT
ejpam-5973	516	5	/	/	SYM
ejpam-5973	516	6	eur	eur	PROPN
ejpam-5973	516	7	.	.	PUNCT
ejpam-5973	517	1	j.	j.	PROPN
ejpam-5973	517	2	pure	pure	PROPN
ejpam-5973	517	3	appl	appl	PROPN
ejpam-5973	517	4	.	.	PROPN
ejpam-5973	517	5	math	math	PROPN
ejpam-5973	517	6	,	,	PUNCT
ejpam-5973	517	7	18	18	NUM
ejpam-5973	517	8	(	(	PUNCT
ejpam-5973	517	9	2	2	NUM
ejpam-5973	517	10	)	)	PUNCT
ejpam-5973	517	11	(	(	PUNCT
ejpam-5973	517	12	2025	2025	NUM
ejpam-5973	517	13	)	)	PUNCT
ejpam-5973	517	14	,	,	PUNCT
ejpam-5973	517	15	5973	5973	NUM
ejpam-5973	517	16	13	13	NUM
ejpam-5973	517	17	of	of	ADP
ejpam-5973	517	18	17	17	NUM
ejpam-5973	517	19	proposition	proposition	NOUN
ejpam-5973	517	20	4	4	NUM
ejpam-5973	517	21	.	.	X
ejpam-5973	518	1	for	for	ADP
ejpam-5973	518	2	any	any	DET
ejpam-5973	518	3	non	non	ADJ
ejpam-5973	518	4	-	-	ADJ
ejpam-5973	518	5	negative	negative	ADJ
ejpam-5973	518	6	integer	integer	NOUN
ejpam-5973	518	7	a	a	PRON
ejpam-5973	518	8	,	,	PUNCT
ejpam-5973	518	9	there	there	PRON
ejpam-5973	518	10	is	be	VERB
ejpam-5973	518	11	a	a	DET
ejpam-5973	518	12	connected	connected	ADJ
ejpam-5973	518	13	graph	graph	NOUN
ejpam-5973	518	14	g	g	ADP
ejpam-5973	518	15	such	such	ADJ
ejpam-5973	518	16	that	that	SCONJ
ejpam-5973	518	17	γr2(g)−	γr2(g)−	NOUN
ejpam-5973	518	18	γhr2(g	γhr2(g	NUM
ejpam-5973	518	19	)	)	PUNCT
ejpam-5973	518	20	=	=	SYM
ejpam-5973	518	21	a.	a.	NOUN
ejpam-5973	518	22	proof	proof	NOUN
ejpam-5973	518	23	.	.	PUNCT
ejpam-5973	519	1	let	let	VERB
ejpam-5973	519	2	g	g	NOUN
ejpam-5973	519	3	be	be	AUX
ejpam-5973	519	4	the	the	DET
ejpam-5973	519	5	graph	graph	NOUN
ejpam-5973	519	6	obtained	obtain	VERB
ejpam-5973	519	7	from	from	ADP
ejpam-5973	519	8	a	a	DET
ejpam-5973	519	9	star	star	NOUN
ejpam-5973	519	10	k1,a+4	k1,a+4	NOUN
ejpam-5973	519	11	by	by	ADP
ejpam-5973	519	12	subdividing	subdivide	VERB
ejpam-5973	519	13	each	each	DET
ejpam-5973	519	14	edge	edge	NOUN
ejpam-5973	519	15	twice	twice	ADV
ejpam-5973	519	16	.	.	PUNCT
ejpam-5973	520	1	it	it	PRON
ejpam-5973	520	2	is	be	AUX
ejpam-5973	520	3	not	not	PART
ejpam-5973	520	4	hard	hard	ADJ
ejpam-5973	520	5	to	to	PART
ejpam-5973	520	6	see	see	VERB
ejpam-5973	520	7	that	that	SCONJ
ejpam-5973	520	8	γr2(g	γr2(g	NOUN
ejpam-5973	520	9	)	)	PUNCT
ejpam-5973	520	10	=	=	SYM
ejpam-5973	520	11	2a	2a	NUM
ejpam-5973	520	12	+	+	CCONJ
ejpam-5973	520	13	8	8	NUM
ejpam-5973	520	14	and	and	CCONJ
ejpam-5973	520	15	γhr2(g	γhr2(g	NUM
ejpam-5973	520	16	)	)	PUNCT
ejpam-5973	520	17	=	=	PUNCT
ejpam-5973	520	18	a	a	DET
ejpam-5973	520	19	+	+	NUM
ejpam-5973	520	20	8	8	NUM
ejpam-5973	520	21	.	.	PUNCT
ejpam-5973	521	1	therefore	therefore	ADV
ejpam-5973	521	2	,	,	PUNCT
ejpam-5973	521	3	γr2(g)−	γr2(g)−	NOUN
ejpam-5973	521	4	γhr2(g	γhr2(g	NUM
ejpam-5973	521	5	)	)	PUNCT
ejpam-5973	521	6	=	=	SYM
ejpam-5973	521	7	a.	a.	NOUN
ejpam-5973	521	8	proposition	proposition	NOUN
ejpam-5973	521	9	5	5	NUM
ejpam-5973	521	10	.	.	PUNCT
ejpam-5973	522	1	for	for	ADP
ejpam-5973	522	2	any	any	DET
ejpam-5973	522	3	positive	positive	ADJ
ejpam-5973	522	4	integer	integer	NOUN
ejpam-5973	522	5	a	a	PRON
ejpam-5973	522	6	,	,	PUNCT
ejpam-5973	522	7	there	there	PRON
ejpam-5973	522	8	is	be	VERB
ejpam-5973	522	9	a	a	DET
ejpam-5973	522	10	connected	connected	ADJ
ejpam-5973	522	11	graph	graph	NOUN
ejpam-5973	522	12	g	g	ADP
ejpam-5973	522	13	such	such	ADJ
ejpam-5973	522	14	that	that	SCONJ
ejpam-5973	522	15	γhr2(g)−	γhr2(g)−	NOUN
ejpam-5973	522	16	γr2(g	γr2(g	X
ejpam-5973	522	17	)	)	PUNCT
ejpam-5973	522	18	=	=	NOUN
ejpam-5973	522	19	a.	a.	NOUN
ejpam-5973	522	20	proof	proof	NOUN
ejpam-5973	522	21	.	.	PUNCT
ejpam-5973	523	1	let	let	VERB
ejpam-5973	523	2	g	g	NOUN
ejpam-5973	523	3	be	be	AUX
ejpam-5973	523	4	the	the	DET
ejpam-5973	523	5	graph	graph	NOUN
ejpam-5973	523	6	obtained	obtain	VERB
ejpam-5973	523	7	from	from	ADP
ejpam-5973	523	8	a	a	DET
ejpam-5973	523	9	star	star	NOUN
ejpam-5973	523	10	k1,a+1	k1,a+1	PROPN
ejpam-5973	523	11	by	by	ADP
ejpam-5973	523	12	first	first	ADV
ejpam-5973	523	13	subdividing	subdivide	VERB
ejpam-5973	523	14	each	each	DET
ejpam-5973	523	15	edge	edge	NOUN
ejpam-5973	523	16	twice	twice	ADV
ejpam-5973	523	17	and	and	CCONJ
ejpam-5973	523	18	then	then	ADV
ejpam-5973	523	19	adding	add	VERB
ejpam-5973	523	20	a	a	DET
ejpam-5973	523	21	new	new	ADJ
ejpam-5973	523	22	pendant	pendant	ADJ
ejpam-5973	523	23	edge	edge	NOUN
ejpam-5973	523	24	at	at	ADP
ejpam-5973	523	25	each	each	DET
ejpam-5973	523	26	support	support	NOUN
ejpam-5973	523	27	vertex	vertex	NOUN
ejpam-5973	523	28	.	.	PUNCT
ejpam-5973	524	1	it	it	PRON
ejpam-5973	524	2	is	be	AUX
ejpam-5973	524	3	easy	easy	ADJ
ejpam-5973	524	4	to	to	PART
ejpam-5973	524	5	verify	verify	VERB
ejpam-5973	524	6	that	that	PRON
ejpam-5973	524	7	γhr2(g	γhr2(g	NUM
ejpam-5973	524	8	)	)	PUNCT
ejpam-5973	524	9	=	=	SYM
ejpam-5973	524	10	2(a+	2(a+	NUM
ejpam-5973	524	11	1	1	NUM
ejpam-5973	524	12	)	)	PUNCT
ejpam-5973	524	13	+	+	CCONJ
ejpam-5973	524	14	2	2	NUM
ejpam-5973	524	15	and	and	CCONJ
ejpam-5973	524	16	γr2(g	γr2(g	PROPN
ejpam-5973	524	17	)	)	PUNCT
ejpam-5973	524	18	=	=	PRON
ejpam-5973	524	19	a+	a+	PUNCT
ejpam-5973	524	20	4	4	X
ejpam-5973	524	21	.	.	X
ejpam-5973	525	1	therfore	therfore	ADV
ejpam-5973	525	2	,	,	PUNCT
ejpam-5973	525	3	γhr2(g)−	γhr2(g)−	NOUN
ejpam-5973	525	4	γr2(g	γr2(g	X
ejpam-5973	525	5	)	)	PUNCT
ejpam-5973	525	6	=	=	NOUN
ejpam-5973	525	7	a.	a.	NOUN
ejpam-5973	525	8	remark	remark	NOUN
ejpam-5973	525	9	3	3	NUM
ejpam-5973	525	10	.	.	PUNCT
ejpam-5973	526	1	let	let	VERB
ejpam-5973	526	2	g	g	PRON
ejpam-5973	526	3	be	be	AUX
ejpam-5973	526	4	a	a	DET
ejpam-5973	526	5	connected	connected	ADJ
ejpam-5973	526	6	graph	graph	NOUN
ejpam-5973	526	7	.	.	PUNCT
ejpam-5973	527	1	then	then	ADV
ejpam-5973	527	2	the	the	DET
ejpam-5973	527	3	hop	hop	NOUN
ejpam-5973	527	4	k	k	ADJ
ejpam-5973	527	5	-	-	PUNCT
ejpam-5973	527	6	rainbow	rainbow	NOUN
ejpam-5973	527	7	domination	domination	NOUN
ejpam-5973	527	8	and	and	CCONJ
ejpam-5973	527	9	the	the	DET
ejpam-5973	527	10	k	k	ADJ
ejpam-5973	527	11	-	-	PUNCT
ejpam-5973	527	12	rainbow	rainbow	NOUN
ejpam-5973	527	13	domination	domination	NOUN
ejpam-5973	527	14	parameters	parameter	NOUN
ejpam-5973	527	15	are	be	AUX
ejpam-5973	527	16	incomparable	incomparable	ADJ
ejpam-5973	527	17	.	.	PUNCT
ejpam-5973	528	1	proof	proof	NOUN
ejpam-5973	528	2	.	.	PUNCT
ejpam-5973	529	1	by	by	ADP
ejpam-5973	529	2	corollary	corollary	ADJ
ejpam-5973	529	3	4	4	NUM
ejpam-5973	529	4	,	,	PUNCT
ejpam-5973	529	5	there	there	PRON
ejpam-5973	529	6	exists	exist	VERB
ejpam-5973	529	7	a	a	DET
ejpam-5973	529	8	connected	connected	ADJ
ejpam-5973	529	9	graph	graph	NOUN
ejpam-5973	529	10	g	g	ADP
ejpam-5973	529	11	such	such	ADJ
ejpam-5973	529	12	that	that	SCONJ
ejpam-5973	529	13	γhr2(g	γhr2(g	NUM
ejpam-5973	529	14	)	)	PUNCT
ejpam-5973	529	15	=	=	SYM
ejpam-5973	529	16	γr2(g	γr2(g	PROPN
ejpam-5973	529	17	)	)	PUNCT
ejpam-5973	529	18	.	.	PUNCT
ejpam-5973	530	1	by	by	ADP
ejpam-5973	530	2	proposition	proposition	NOUN
ejpam-5973	530	3	4	4	NUM
ejpam-5973	530	4	,	,	PUNCT
ejpam-5973	530	5	there	there	PRON
ejpam-5973	530	6	exists	exist	VERB
ejpam-5973	530	7	a	a	DET
ejpam-5973	530	8	connected	connected	ADJ
ejpam-5973	530	9	graph	graph	NOUN
ejpam-5973	530	10	g′	g′	NOUN
ejpam-5973	530	11	such	such	ADJ
ejpam-5973	530	12	that	that	SCONJ
ejpam-5973	530	13	γr2(g	γr2(g	PROPN
ejpam-5973	530	14	′	′	NOUN
ejpam-5973	530	15	)	)	PUNCT
ejpam-5973	530	16	>	>	X
ejpam-5973	530	17	γhr2(g	γhr2(g	NUM
ejpam-5973	530	18	′	′	NUM
ejpam-5973	530	19	)	)	PUNCT
ejpam-5973	530	20	.	.	PUNCT
ejpam-5973	531	1	similarly	similarly	ADV
ejpam-5973	531	2	,	,	PUNCT
ejpam-5973	531	3	by	by	ADP
ejpam-5973	531	4	proposition	proposition	NOUN
ejpam-5973	531	5	5	5	NUM
ejpam-5973	531	6	,	,	PUNCT
ejpam-5973	531	7	there	there	PRON
ejpam-5973	531	8	exists	exist	VERB
ejpam-5973	531	9	a	a	DET
ejpam-5973	531	10	connected	connected	ADJ
ejpam-5973	531	11	graph	graph	NOUN
ejpam-5973	531	12	g′′	g′′	PROPN
ejpam-5973	531	13	such	such	ADJ
ejpam-5973	531	14	that	that	SCONJ
ejpam-5973	531	15	γhr2(g	γhr2(g	NUM
ejpam-5973	531	16	′′	′′	PROPN
ejpam-5973	531	17	)	)	PUNCT
ejpam-5973	531	18	>	>	PUNCT
ejpam-5973	531	19	γr2(g	γr2(g	PROPN
ejpam-5973	531	20	′′	′′	PROPN
ejpam-5973	531	21	)	)	PUNCT
ejpam-5973	531	22	.	.	PUNCT
ejpam-5973	532	1	these	these	DET
ejpam-5973	532	2	results	result	NOUN
ejpam-5973	532	3	demonstrate	demonstrate	VERB
ejpam-5973	532	4	that	that	SCONJ
ejpam-5973	532	5	neither	neither	CCONJ
ejpam-5973	532	6	γr2(g	γr2(g	PROPN
ejpam-5973	532	7	)	)	PUNCT
ejpam-5973	532	8	≤	≤	NOUN
ejpam-5973	532	9	γhr2(g	γhr2(g	NUM
ejpam-5973	532	10	)	)	PUNCT
ejpam-5973	532	11	nor	nor	CCONJ
ejpam-5973	532	12	γhr2(g	γhr2(g	NUM
ejpam-5973	532	13	)	)	PUNCT
ejpam-5973	532	14	≤	≤	NUM
ejpam-5973	532	15	γr2(g	γr2(g	PROPN
ejpam-5973	532	16	)	)	PUNCT
ejpam-5973	532	17	hold	hold	VERB
ejpam-5973	532	18	universally	universally	ADV
ejpam-5973	532	19	.	.	PUNCT
ejpam-5973	533	1	thus	thus	ADV
ejpam-5973	533	2	,	,	PUNCT
ejpam-5973	533	3	γr2(g	γr2(g	PROPN
ejpam-5973	533	4	)	)	PUNCT
ejpam-5973	533	5	and	and	CCONJ
ejpam-5973	533	6	γhr2(g	γhr2(g	NUM
ejpam-5973	533	7	)	)	PUNCT
ejpam-5973	533	8	are	be	AUX
ejpam-5973	533	9	incomparable	incomparable	ADJ
ejpam-5973	533	10	.	.	PUNCT
ejpam-5973	534	1	5	5	X
ejpam-5973	534	2	.	.	X
ejpam-5973	534	3	graphs	graph	NOUN
ejpam-5973	534	4	with	with	ADP
ejpam-5973	534	5	γhr2(g	γhr2(g	NUM
ejpam-5973	534	6	)	)	PUNCT
ejpam-5973	534	7	=	=	SYM
ejpam-5973	535	1	n	n	CCONJ
ejpam-5973	535	2	in	in	ADP
ejpam-5973	535	3	the	the	DET
ejpam-5973	535	4	next	next	ADJ
ejpam-5973	535	5	theorem	theorem	NOUN
ejpam-5973	535	6	we	we	PRON
ejpam-5973	535	7	characterize	characterize	VERB
ejpam-5973	535	8	all	all	DET
ejpam-5973	535	9	graphs	graph	NOUN
ejpam-5973	535	10	g	g	NOUN
ejpam-5973	535	11	with	with	ADP
ejpam-5973	535	12	γhr2(g	γhr2(g	NUM
ejpam-5973	535	13	)	)	PUNCT
ejpam-5973	535	14	=	=	VERB
ejpam-5973	536	1	n.	n.	NOUN
ejpam-5973	536	2	for	for	ADP
ejpam-5973	536	3	this	this	DET
ejpam-5973	536	4	purpose	purpose	NOUN
ejpam-5973	536	5	,	,	PUNCT
ejpam-5973	536	6	consider	consider	VERB
ejpam-5973	536	7	a	a	DET
ejpam-5973	536	8	family	family	NOUN
ejpam-5973	536	9	of	of	ADP
ejpam-5973	536	10	graphs	graph	NOUN
ejpam-5973	536	11	defined	define	VERB
ejpam-5973	536	12	as	as	ADP
ejpam-5973	536	13	follows	follow	VERB
ejpam-5973	536	14	.	.	PUNCT
ejpam-5973	537	1	define	define	VERB
ejpam-5973	537	2	f	f	PROPN
ejpam-5973	537	3	be	be	AUX
ejpam-5973	537	4	the	the	DET
ejpam-5973	537	5	family	family	NOUN
ejpam-5973	537	6	of	of	ADP
ejpam-5973	537	7	graphs	graph	NOUN
ejpam-5973	537	8	g	g	ADP
ejpam-5973	537	9	such	such	ADJ
ejpam-5973	537	10	that	that	SCONJ
ejpam-5973	537	11	g	g	PROPN
ejpam-5973	537	12	can	can	AUX
ejpam-5973	537	13	be	be	AUX
ejpam-5973	537	14	constructed	construct	VERB
ejpam-5973	537	15	from	from	ADP
ejpam-5973	537	16	a	a	DET
ejpam-5973	537	17	sequence	sequence	NOUN
ejpam-5973	537	18	h0	h0	NOUN
ejpam-5973	537	19	,	,	PUNCT
ejpam-5973	537	20	h1	h1	PROPN
ejpam-5973	537	21	,	,	PUNCT
ejpam-5973	537	22	.	.	PUNCT
ejpam-5973	537	23	.	.	PUNCT
ejpam-5973	538	1	.	.	PUNCT
ejpam-5973	539	1	,	,	PUNCT
ejpam-5973	539	2	ht	ht	INTJ
ejpam-5973	539	3	,	,	PUNCT
ejpam-5973	539	4	(	(	PUNCT
ejpam-5973	539	5	t	t	PROPN
ejpam-5973	539	6	≥	≥	NUM
ejpam-5973	539	7	1	1	NUM
ejpam-5973	539	8	)	)	PUNCT
ejpam-5973	539	9	,	,	PUNCT
ejpam-5973	539	10	of	of	ADP
ejpam-5973	539	11	graphs	graph	NOUN
ejpam-5973	539	12	,	,	PUNCT
ejpam-5973	539	13	where	where	SCONJ
ejpam-5973	539	14	h0	h0	PROPN
ejpam-5973	539	15	is	be	AUX
ejpam-5973	539	16	a	a	DET
ejpam-5973	539	17	complete	complete	ADJ
ejpam-5973	539	18	graph	graph	NOUN
ejpam-5973	539	19	as	as	SCONJ
ejpam-5973	539	20	demonstrated	demonstrate	VERB
ejpam-5973	539	21	in	in	ADP
ejpam-5973	539	22	figure	figure	NOUN
ejpam-5973	539	23	2	2	NUM
ejpam-5973	539	24	or	or	CCONJ
ejpam-5973	539	25	h0	h0	NOUN
ejpam-5973	539	26	=	=	PROPN
ejpam-5973	539	27	k2	k2	PROPN
ejpam-5973	539	28	as	as	SCONJ
ejpam-5973	539	29	demonstrated	demonstrate	VERB
ejpam-5973	539	30	in	in	ADP
ejpam-5973	539	31	figure	figure	NOUN
ejpam-5973	539	32	3	3	NUM
ejpam-5973	539	33	,	,	PUNCT
ejpam-5973	539	34	g	g	PROPN
ejpam-5973	539	35	=	=	SYM
ejpam-5973	539	36	ht	ht	PROPN
ejpam-5973	540	1	and	and	CCONJ
ejpam-5973	540	2	,	,	PUNCT
ejpam-5973	540	3	if	if	SCONJ
ejpam-5973	540	4	t	t	PROPN
ejpam-5973	540	5	≥	≥	NUM
ejpam-5973	540	6	1	1	NUM
ejpam-5973	540	7	,	,	PUNCT
ejpam-5973	540	8	then	then	ADV
ejpam-5973	540	9	hi+1	hi+1	NOUN
ejpam-5973	540	10	can	can	AUX
ejpam-5973	540	11	be	be	AUX
ejpam-5973	540	12	obtained	obtain	VERB
ejpam-5973	540	13	recursively	recursively	ADV
ejpam-5973	540	14	from	from	ADP
ejpam-5973	540	15	hi	hi	INTJ
ejpam-5973	540	16	by	by	ADP
ejpam-5973	540	17	adding	add	VERB
ejpam-5973	540	18	2	2	NUM
ejpam-5973	540	19	new	new	ADJ
ejpam-5973	540	20	vertices	vertex	NOUN
ejpam-5973	540	21	and	and	CCONJ
ejpam-5973	540	22	joining	join	VERB
ejpam-5973	540	23	each	each	PRON
ejpam-5973	540	24	of	of	ADP
ejpam-5973	540	25	the	the	DET
ejpam-5973	540	26	new	new	ADJ
ejpam-5973	540	27	vertices	vertex	NOUN
ejpam-5973	540	28	to	to	ADP
ejpam-5973	540	29	all	all	DET
ejpam-5973	540	30	vertices	vertex	NOUN
ejpam-5973	540	31	in	in	ADP
ejpam-5973	540	32	hi	hi	PROPN
ejpam-5973	540	33	.	.	PUNCT
ejpam-5973	541	1	j.	j.	PROPN
ejpam-5973	541	2	j.	j.	PROPN
ejpam-5973	541	3	hamja	hamja	PROPN
ejpam-5973	541	4	et	et	PROPN
ejpam-5973	541	5	al	al	PROPN
ejpam-5973	541	6	.	.	PUNCT
ejpam-5973	541	7	/	/	SYM
ejpam-5973	541	8	eur	eur	PROPN
ejpam-5973	541	9	.	.	PUNCT
ejpam-5973	542	1	j.	j.	PROPN
ejpam-5973	542	2	pure	pure	PROPN
ejpam-5973	542	3	appl	appl	PROPN
ejpam-5973	542	4	.	.	PROPN
ejpam-5973	542	5	math	math	PROPN
ejpam-5973	542	6	,	,	PUNCT
ejpam-5973	542	7	18	18	NUM
ejpam-5973	542	8	(	(	PUNCT
ejpam-5973	542	9	2	2	NUM
ejpam-5973	542	10	)	)	PUNCT
ejpam-5973	542	11	(	(	PUNCT
ejpam-5973	542	12	2025	2025	NUM
ejpam-5973	542	13	)	)	PUNCT
ejpam-5973	542	14	,	,	PUNCT
ejpam-5973	542	15	5973	5973	NUM
ejpam-5973	542	16	14	14	NUM
ejpam-5973	542	17	of	of	ADP
ejpam-5973	542	18	17	17	NUM
ejpam-5973	542	19	h0	h0	PROPN
ejpam-5973	542	20	h1	h1	PROPN
ejpam-5973	542	21	h2	h2	PROPN
ejpam-5973	542	22	figure	figure	NOUN
ejpam-5973	542	23	2	2	NUM
ejpam-5973	542	24	:	:	PUNCT
ejpam-5973	542	25	example	example	NOUN
ejpam-5973	542	26	of	of	ADP
ejpam-5973	542	27	graphs	graph	NOUN
ejpam-5973	542	28	in	in	ADP
ejpam-5973	542	29	the	the	DET
ejpam-5973	542	30	family	family	NOUN
ejpam-5973	542	31	f	f	NOUN
ejpam-5973	543	1	if	if	SCONJ
ejpam-5973	543	2	h0	h0	PROPN
ejpam-5973	543	3	=	=	PROPN
ejpam-5973	543	4	k2	k2	PROPN
ejpam-5973	543	5	h1	h1	NOUN
ejpam-5973	543	6	h2h0	h2h0	PRON
ejpam-5973	543	7	figure	figure	VERB
ejpam-5973	543	8	3	3	NUM
ejpam-5973	543	9	:	:	PUNCT
ejpam-5973	543	10	example	example	NOUN
ejpam-5973	543	11	of	of	ADP
ejpam-5973	543	12	graphs	graph	NOUN
ejpam-5973	543	13	in	in	ADP
ejpam-5973	543	14	the	the	DET
ejpam-5973	543	15	family	family	NOUN
ejpam-5973	543	16	f	f	NOUN
ejpam-5973	544	1	if	if	SCONJ
ejpam-5973	544	2	h0	h0	PROPN
ejpam-5973	544	3	=	=	PROPN
ejpam-5973	544	4	k2	k2	PROPN
ejpam-5973	544	5	this	this	DET
ejpam-5973	544	6	family	family	NOUN
ejpam-5973	544	7	of	of	ADP
ejpam-5973	544	8	graphs	graph	NOUN
ejpam-5973	544	9	was	be	AUX
ejpam-5973	544	10	introduced	introduce	VERB
ejpam-5973	544	11	by	by	ADP
ejpam-5973	544	12	shabani	shabani	PROPN
ejpam-5973	544	13	et	et	PROPN
ejpam-5973	544	14	al	al	PROPN
ejpam-5973	544	15	.	.	PUNCT
ejpam-5973	545	1	in	in	ADP
ejpam-5973	545	2	[	[	X
ejpam-5973	545	3	16	16	NUM
ejpam-5973	545	4	]	]	PUNCT
ejpam-5973	545	5	for	for	ADP
ejpam-5973	545	6	characterizing	characterize	VERB
ejpam-5973	545	7	the	the	DET
ejpam-5973	545	8	graphs	graph	NOUN
ejpam-5973	545	9	g	g	ADP
ejpam-5973	545	10	of	of	ADP
ejpam-5973	545	11	order	order	NOUN
ejpam-5973	545	12	n	n	X
ejpam-5973	545	13	with	with	ADP
ejpam-5973	545	14	γhr(g	γhr(g	PROPN
ejpam-5973	545	15	)	)	PUNCT
ejpam-5973	545	16	=	=	SYM
ejpam-5973	545	17	n	n	CCONJ
ejpam-5973	545	18	(	(	PUNCT
ejpam-5973	545	19	see	see	VERB
ejpam-5973	545	20	proposition	proposition	NOUN
ejpam-5973	545	21	6	6	NUM
ejpam-5973	545	22	)	)	PUNCT
ejpam-5973	545	23	.	.	PUNCT
ejpam-5973	546	1	in	in	ADP
ejpam-5973	546	2	what	what	PRON
ejpam-5973	546	3	follows	follow	VERB
ejpam-5973	546	4	is	be	AUX
ejpam-5973	546	5	a	a	DET
ejpam-5973	546	6	useful	useful	ADJ
ejpam-5973	546	7	result	result	NOUN
ejpam-5973	546	8	:	:	PUNCT
ejpam-5973	546	9	proposition	proposition	NOUN
ejpam-5973	546	10	6	6	NUM
ejpam-5973	546	11	.	.	PUNCT
ejpam-5973	547	1	[	[	X
ejpam-5973	547	2	16	16	NUM
ejpam-5973	547	3	]	]	X
ejpam-5973	547	4	if	if	SCONJ
ejpam-5973	547	5	g	g	PROPN
ejpam-5973	547	6	is	be	AUX
ejpam-5973	547	7	a	a	DET
ejpam-5973	547	8	connected	connected	ADJ
ejpam-5973	547	9	graph	graph	NOUN
ejpam-5973	547	10	of	of	ADP
ejpam-5973	547	11	order	order	NOUN
ejpam-5973	547	12	n	n	CCONJ
ejpam-5973	547	13	,	,	PUNCT
ejpam-5973	547	14	then	then	ADV
ejpam-5973	547	15	γhr(g	γhr(g	NUM
ejpam-5973	547	16	)	)	PUNCT
ejpam-5973	547	17	=	=	SYM
ejpam-5973	548	1	n	n	NOUN
ejpam-5973	548	2	if	if	SCONJ
ejpam-5973	549	1	and	and	CCONJ
ejpam-5973	549	2	only	only	ADV
ejpam-5973	549	3	if	if	SCONJ
ejpam-5973	549	4	g	g	PROPN
ejpam-5973	549	5	∈	∈	PROPN
ejpam-5973	549	6	f	f	PROPN
ejpam-5973	549	7	∪	∪	X
ejpam-5973	549	8	{	{	PUNCT
ejpam-5973	549	9	p4	p4	ADJ
ejpam-5973	549	10	}	}	PUNCT
ejpam-5973	549	11	.	.	PUNCT
ejpam-5973	550	1	theorem	theorem	NOUN
ejpam-5973	550	2	10	10	NUM
ejpam-5973	550	3	.	.	PUNCT
ejpam-5973	551	1	let	let	VERB
ejpam-5973	551	2	k	k	PROPN
ejpam-5973	551	3	≤	≤	ADJ
ejpam-5973	551	4	2	2	NUM
ejpam-5973	551	5	be	be	AUX
ejpam-5973	551	6	an	an	DET
ejpam-5973	551	7	integer	integer	NOUN
ejpam-5973	551	8	and	and	CCONJ
ejpam-5973	551	9	g	g	PROPN
ejpam-5973	551	10	be	be	AUX
ejpam-5973	551	11	a	a	DET
ejpam-5973	551	12	connected	connected	ADJ
ejpam-5973	551	13	graph	graph	NOUN
ejpam-5973	551	14	of	of	ADP
ejpam-5973	551	15	order	order	NOUN
ejpam-5973	551	16	n	n	PRON
ejpam-5973	551	17	≥	≥	NOUN
ejpam-5973	551	18	1	1	NUM
ejpam-5973	551	19	.	.	PUNCT
ejpam-5973	552	1	then	then	ADV
ejpam-5973	552	2	γhr2(g	γhr2(g	NUM
ejpam-5973	552	3	)	)	PUNCT
ejpam-5973	553	1	=	=	SYM
ejpam-5973	553	2	n	n	NOUN
ejpam-5973	553	3	if	if	SCONJ
ejpam-5973	553	4	and	and	CCONJ
ejpam-5973	553	5	only	only	ADV
ejpam-5973	553	6	if	if	SCONJ
ejpam-5973	553	7	g	g	PROPN
ejpam-5973	553	8	∈	∈	PROPN
ejpam-5973	553	9	f	f	PROPN
ejpam-5973	553	10	∪	∪	X
ejpam-5973	553	11	{	{	PUNCT
ejpam-5973	553	12	p4	p4	ADJ
ejpam-5973	553	13	}	}	PUNCT
ejpam-5973	553	14	.	.	PUNCT
ejpam-5973	554	1	proof	proof	NOUN
ejpam-5973	554	2	.	.	PUNCT
ejpam-5973	555	1	let	let	VERB
ejpam-5973	555	2	g	g	PRON
ejpam-5973	555	3	be	be	AUX
ejpam-5973	555	4	a	a	DET
ejpam-5973	555	5	graph	graph	NOUN
ejpam-5973	555	6	of	of	ADP
ejpam-5973	555	7	order	order	NOUN
ejpam-5973	555	8	n	n	PRON
ejpam-5973	555	9	≥	≥	NOUN
ejpam-5973	555	10	1	1	NUM
ejpam-5973	555	11	with	with	ADP
ejpam-5973	555	12	γhr2(g	γhr2(g	NUM
ejpam-5973	555	13	)	)	PUNCT
ejpam-5973	555	14	=	=	VERB
ejpam-5973	556	1	n.	n.	NOUN
ejpam-5973	556	2	we	we	PRON
ejpam-5973	556	3	deduce	deduce	VERB
ejpam-5973	556	4	from	from	ADP
ejpam-5973	556	5	n	n	NOUN
ejpam-5973	556	6	=	=	SYM
ejpam-5973	556	7	γhr2(g	γhr2(g	NUM
ejpam-5973	556	8	)	)	PUNCT
ejpam-5973	556	9	≤	≤	NOUN
ejpam-5973	556	10	γhr(g	γhr(g	NUM
ejpam-5973	556	11	)	)	PUNCT
ejpam-5973	556	12	≤	≤	NOUN
ejpam-5973	557	1	n	n	CCONJ
ejpam-5973	557	2	that	that	SCONJ
ejpam-5973	557	3	γhr(g	γhr(g	ADV
ejpam-5973	557	4	)	)	PUNCT
ejpam-5973	557	5	=	=	SYM
ejpam-5973	558	1	n	n	NOUN
ejpam-5973	558	2	and	and	CCONJ
ejpam-5973	558	3	proposition	proposition	NOUN
ejpam-5973	558	4	6	6	NUM
ejpam-5973	558	5	leads	lead	VERB
ejpam-5973	558	6	to	to	ADP
ejpam-5973	558	7	g	g	PROPN
ejpam-5973	558	8	∈	∈	PROPN
ejpam-5973	558	9	f	f	PROPN
ejpam-5973	558	10	∪	∪	X
ejpam-5973	558	11	{	{	PUNCT
ejpam-5973	558	12	p4	p4	ADJ
ejpam-5973	558	13	}	}	PUNCT
ejpam-5973	558	14	.	.	PUNCT
ejpam-5973	559	1	conversely	conversely	ADV
ejpam-5973	559	2	,	,	PUNCT
ejpam-5973	559	3	assume	assume	VERB
ejpam-5973	559	4	that	that	SCONJ
ejpam-5973	559	5	g	g	PROPN
ejpam-5973	559	6	∈	∈	PROPN
ejpam-5973	559	7	f	f	PROPN
ejpam-5973	559	8	∪	∪	X
ejpam-5973	559	9	{	{	PUNCT
ejpam-5973	559	10	p4	p4	ADJ
ejpam-5973	559	11	}	}	PUNCT
ejpam-5973	559	12	.	.	PUNCT
ejpam-5973	560	1	if	if	SCONJ
ejpam-5973	560	2	g	g	NOUN
ejpam-5973	560	3	=	=	SYM
ejpam-5973	560	4	p4	p4	ADJ
ejpam-5973	560	5	,	,	PUNCT
ejpam-5973	560	6	then	then	ADV
ejpam-5973	560	7	clearly	clearly	ADV
ejpam-5973	560	8	γhr2(g	γhr2(g	NUM
ejpam-5973	560	9	)	)	PUNCT
ejpam-5973	560	10	=	=	SYM
ejpam-5973	560	11	4	4	X
ejpam-5973	560	12	.	.	X
ejpam-5973	560	13	assume	assume	VERB
ejpam-5973	560	14	that	that	SCONJ
ejpam-5973	560	15	g	g	PROPN
ejpam-5973	560	16	∈	∈	PROPN
ejpam-5973	560	17	f	f	X
ejpam-5973	560	18	.	.	PUNCT
ejpam-5973	561	1	by	by	ADP
ejpam-5973	561	2	the	the	DET
ejpam-5973	561	3	construction	construction	NOUN
ejpam-5973	561	4	of	of	ADP
ejpam-5973	561	5	g	g	NOUN
ejpam-5973	561	6	we	we	PRON
ejpam-5973	561	7	have	have	AUX
ejpam-5973	561	8	δ(g	δ(g	VERB
ejpam-5973	561	9	)	)	PUNCT
ejpam-5973	561	10	≥	≥	NOUN
ejpam-5973	561	11	n−	n−	NOUN
ejpam-5973	561	12	2	2	NUM
ejpam-5973	561	13	.	.	PUNCT
ejpam-5973	561	14	then	then	ADV
ejpam-5973	561	15	by	by	ADP
ejpam-5973	561	16	proposition	proposition	NOUN
ejpam-5973	561	17	4	4	NUM
ejpam-5973	561	18	,	,	PUNCT
ejpam-5973	561	19	we	we	PRON
ejpam-5973	561	20	have	have	VERB
ejpam-5973	561	21	γhr2(g	γhr2(g	NUM
ejpam-5973	561	22	)	)	PUNCT
ejpam-5973	561	23	=	=	SYM
ejpam-5973	561	24	n.	n.	NOUN
ejpam-5973	561	25	6	6	NUM
ejpam-5973	561	26	.	.	PUNCT
ejpam-5973	562	1	open	open	ADJ
ejpam-5973	562	2	questions	question	NOUN
ejpam-5973	562	3	and	and	CCONJ
ejpam-5973	562	4	problems	problem	NOUN
ejpam-5973	562	5	we	we	PRON
ejpam-5973	562	6	conclude	conclude	VERB
ejpam-5973	562	7	this	this	DET
ejpam-5973	562	8	paper	paper	NOUN
ejpam-5973	562	9	by	by	ADP
ejpam-5973	562	10	mentioning	mention	VERB
ejpam-5973	562	11	some	some	DET
ejpam-5973	562	12	questions	question	NOUN
ejpam-5973	562	13	and	and	CCONJ
ejpam-5973	562	14	problems	problem	NOUN
ejpam-5973	562	15	suggested	suggest	VERB
ejpam-5973	562	16	by	by	ADP
ejpam-5973	562	17	this	this	DET
ejpam-5973	562	18	research	research	NOUN
ejpam-5973	562	19	.	.	PUNCT
ejpam-5973	563	1	using	use	VERB
ejpam-5973	563	2	the	the	DET
ejpam-5973	563	3	construction	construction	NOUN
ejpam-5973	563	4	introduced	introduce	VERB
ejpam-5973	563	5	by	by	ADP
ejpam-5973	563	6	shabani	shabani	PROPN
ejpam-5973	563	7	et	et	PROPN
ejpam-5973	563	8	al	al	PROPN
ejpam-5973	563	9	.	.	PUNCT
ejpam-5973	564	1	in	in	ADP
ejpam-5973	564	2	[	[	X
ejpam-5973	564	3	16	16	NUM
ejpam-5973	564	4	]	]	PUNCT
ejpam-5973	564	5	,	,	PUNCT
ejpam-5973	564	6	we	we	PRON
ejpam-5973	564	7	characterize	characterize	VERB
ejpam-5973	564	8	all	all	DET
ejpam-5973	564	9	connected	connected	ADJ
ejpam-5973	564	10	graph	graph	NOUN
ejpam-5973	564	11	g	g	NOUN
ejpam-5973	564	12	of	of	ADP
ejpam-5973	564	13	order	order	NOUN
ejpam-5973	564	14	n	n	CCONJ
ejpam-5973	564	15	with	with	ADP
ejpam-5973	564	16	γhr2(g	γhr2(g	NUM
ejpam-5973	564	17	)	)	PUNCT
ejpam-5973	564	18	=	=	VERB
ejpam-5973	564	19	n.	n.	NOUN
ejpam-5973	564	20	shabani	shabani	PROPN
ejpam-5973	564	21	et	et	PROPN
ejpam-5973	564	22	al	al	PROPN
ejpam-5973	564	23	.	.	PUNCT
ejpam-5973	565	1	in	in	ADP
ejpam-5973	565	2	[	[	X
ejpam-5973	565	3	16	16	NUM
ejpam-5973	565	4	]	]	PUNCT
ejpam-5973	565	5	,	,	PUNCT
ejpam-5973	565	6	also	also	ADV
ejpam-5973	565	7	characterize	characterize	VERB
ejpam-5973	565	8	all	all	DET
ejpam-5973	565	9	connected	connected	ADJ
ejpam-5973	565	10	graph	graph	NOUN
ejpam-5973	565	11	g	g	NOUN
ejpam-5973	565	12	of	of	ADP
ejpam-5973	565	13	order	order	NOUN
ejpam-5973	565	14	n	n	PRON
ejpam-5973	565	15	with	with	ADP
ejpam-5973	565	16	γhr(g	γhr(g	PROPN
ejpam-5973	565	17	)	)	PUNCT
ejpam-5973	565	18	=	=	SYM
ejpam-5973	566	1	n−1	n−1	PROPN
ejpam-5973	566	2	.	.	PUNCT
ejpam-5973	567	1	we	we	PRON
ejpam-5973	567	2	think	think	VERB
ejpam-5973	567	3	that	that	SCONJ
ejpam-5973	567	4	using	use	VERB
ejpam-5973	567	5	theirs	theirs	PRON
ejpam-5973	567	6	construction	construction	NOUN
ejpam-5973	567	7	one	one	PRON
ejpam-5973	567	8	can	can	AUX
ejpam-5973	567	9	characterize	characterize	VERB
ejpam-5973	567	10	all	all	DET
ejpam-5973	567	11	connected	connected	ADJ
ejpam-5973	567	12	graph	graph	NOUN
ejpam-5973	567	13	g	g	NOUN
ejpam-5973	567	14	of	of	ADP
ejpam-5973	567	15	order	order	NOUN
ejpam-5973	567	16	n	n	CCONJ
ejpam-5973	567	17	with	with	ADP
ejpam-5973	567	18	γhr2(g	γhr2(g	NUM
ejpam-5973	567	19	)	)	PUNCT
ejpam-5973	567	20	=	=	SYM
ejpam-5973	567	21	n−	n−	NOUN
ejpam-5973	567	22	1	1	NUM
ejpam-5973	567	23	.	.	PUNCT
ejpam-5973	568	1	j.	j.	PROPN
ejpam-5973	568	2	j.	j.	PROPN
ejpam-5973	568	3	hamja	hamja	PROPN
ejpam-5973	568	4	et	et	PROPN
ejpam-5973	568	5	al	al	PROPN
ejpam-5973	568	6	.	.	PUNCT
ejpam-5973	568	7	/	/	SYM
ejpam-5973	568	8	eur	eur	PROPN
ejpam-5973	568	9	.	.	PUNCT
ejpam-5973	569	1	j.	j.	PROPN
ejpam-5973	569	2	pure	pure	PROPN
ejpam-5973	569	3	appl	appl	PROPN
ejpam-5973	569	4	.	.	PROPN
ejpam-5973	569	5	math	math	PROPN
ejpam-5973	569	6	,	,	PUNCT
ejpam-5973	569	7	18	18	NUM
ejpam-5973	569	8	(	(	PUNCT
ejpam-5973	569	9	2	2	NUM
ejpam-5973	569	10	)	)	PUNCT
ejpam-5973	569	11	(	(	PUNCT
ejpam-5973	569	12	2025	2025	NUM
ejpam-5973	569	13	)	)	PUNCT
ejpam-5973	569	14	,	,	PUNCT
ejpam-5973	569	15	5973	5973	NUM
ejpam-5973	569	16	15	15	NUM
ejpam-5973	569	17	of	of	ADP
ejpam-5973	569	18	17	17	NUM
ejpam-5973	569	19	problem	problem	NOUN
ejpam-5973	569	20	1	1	NUM
ejpam-5973	569	21	.	.	PUNCT
ejpam-5973	569	22	characterize	characterize	VERB
ejpam-5973	569	23	all	all	DET
ejpam-5973	569	24	connected	connected	ADJ
ejpam-5973	569	25	graphs	graph	NOUN
ejpam-5973	569	26	g	g	ADP
ejpam-5973	569	27	of	of	ADP
ejpam-5973	569	28	order	order	NOUN
ejpam-5973	569	29	n	n	PRON
ejpam-5973	569	30	such	such	ADJ
ejpam-5973	569	31	that	that	SCONJ
ejpam-5973	569	32	γhr2(g	γhr2(g	NUM
ejpam-5973	569	33	)	)	PUNCT
ejpam-5973	569	34	=	=	SYM
ejpam-5973	569	35	n−	n−	NOUN
ejpam-5973	569	36	1	1	NUM
ejpam-5973	569	37	.	.	PUNCT
ejpam-5973	569	38	problem	problem	NOUN
ejpam-5973	569	39	2	2	NUM
ejpam-5973	569	40	.	.	X
ejpam-5973	569	41	for	for	ADP
ejpam-5973	569	42	positive	positive	ADJ
ejpam-5973	569	43	integer	integer	NOUN
ejpam-5973	569	44	k	k	PROPN
ejpam-5973	569	45	≥	≥	NUM
ejpam-5973	569	46	3	3	NUM
ejpam-5973	569	47	,	,	PUNCT
ejpam-5973	569	48	characterize	characterize	VERB
ejpam-5973	569	49	the	the	DET
ejpam-5973	569	50	graphs	graph	NOUN
ejpam-5973	569	51	g	g	ADP
ejpam-5973	569	52	of	of	ADP
ejpam-5973	569	53	order	order	NOUN
ejpam-5973	569	54	n	n	PRON
ejpam-5973	569	55	such	such	ADJ
ejpam-5973	569	56	that	that	DET
ejpam-5973	569	57	γhrk(g	γhrk(g	NOUN
ejpam-5973	569	58	)	)	PUNCT
ejpam-5973	569	59	=	=	VERB
ejpam-5973	570	1	n.	n.	NOUN
ejpam-5973	570	2	in	in	ADP
ejpam-5973	570	3	theorem	theorem	NOUN
ejpam-5973	570	4	3	3	NUM
ejpam-5973	570	5	,	,	PUNCT
ejpam-5973	570	6	we	we	PRON
ejpam-5973	570	7	observed	observe	VERB
ejpam-5973	570	8	that	that	SCONJ
ejpam-5973	570	9	for	for	ADP
ejpam-5973	570	10	any	any	DET
ejpam-5973	570	11	positive	positive	ADJ
ejpam-5973	570	12	integer	integer	NOUN
ejpam-5973	570	13	k	k	PROPN
ejpam-5973	570	14	≥	≥	NUM
ejpam-5973	570	15	2	2	NUM
ejpam-5973	570	16	and	and	CCONJ
ejpam-5973	570	17	every	every	DET
ejpam-5973	570	18	graph	graph	NOUN
ejpam-5973	570	19	g	g	NOUN
ejpam-5973	570	20	of	of	ADP
ejpam-5973	570	21	order	order	NOUN
ejpam-5973	570	22	n	n	PRON
ejpam-5973	570	23	≥	≥	NOUN
ejpam-5973	570	24	k+2	k+2	NOUN
ejpam-5973	570	25	we	we	PRON
ejpam-5973	570	26	have	have	VERB
ejpam-5973	570	27	γhrk(g	γhrk(g	PROPN
ejpam-5973	570	28	)	)	PUNCT
ejpam-5973	570	29	≥	≥	NOUN
ejpam-5973	570	30	k+1	k+1	NOUN
ejpam-5973	570	31	.	.	NOUN
ejpam-5973	570	32	to	to	PART
ejpam-5973	570	33	see	see	VERB
ejpam-5973	570	34	the	the	DET
ejpam-5973	570	35	sharpness	sharpness	NOUN
ejpam-5973	570	36	,	,	PUNCT
ejpam-5973	570	37	let	let	VERB
ejpam-5973	570	38	g	g	PRON
ejpam-5973	570	39	be	be	AUX
ejpam-5973	570	40	a	a	DET
ejpam-5973	570	41	graph	graph	NOUN
ejpam-5973	570	42	obtained	obtain	VERB
ejpam-5973	570	43	from	from	ADP
ejpam-5973	570	44	complete	complete	ADJ
ejpam-5973	570	45	graph	graph	NOUN
ejpam-5973	570	46	kk+1	kk+1	VERB
ejpam-5973	570	47	with	with	ADP
ejpam-5973	570	48	vertex	vertex	NOUN
ejpam-5973	570	49	set	set	NOUN
ejpam-5973	570	50	v1	v1	NOUN
ejpam-5973	570	51	,	,	PUNCT
ejpam-5973	570	52	.	.	PUNCT
ejpam-5973	570	53	.	.	PUNCT
ejpam-5973	571	1	.	.	PUNCT
ejpam-5973	572	1	,	,	PUNCT
ejpam-5973	572	2	vk+1	vk+1	VERB
ejpam-5973	572	3	by	by	ADP
ejpam-5973	572	4	first	first	ADV
ejpam-5973	572	5	adding	add	VERB
ejpam-5973	572	6	r	r	PRON
ejpam-5973	572	7	≥	≥	NOUN
ejpam-5973	572	8	0	0	NUM
ejpam-5973	572	9	new	new	ADJ
ejpam-5973	572	10	vertices	vertex	NOUN
ejpam-5973	572	11	and	and	CCONJ
ejpam-5973	572	12	connecting	connect	VERB
ejpam-5973	572	13	them	they	PRON
ejpam-5973	572	14	to	to	PART
ejpam-5973	572	15	v1	v1	VERB
ejpam-5973	572	16	and	and	CCONJ
ejpam-5973	572	17	then	then	ADV
ejpam-5973	572	18	adding	add	VERB
ejpam-5973	572	19	s	s	PRON
ejpam-5973	572	20	≥	≥	NOUN
ejpam-5973	572	21	0	0	NUM
ejpam-5973	572	22	new	new	ADJ
ejpam-5973	572	23	vertices	vertex	NOUN
ejpam-5973	572	24	and	and	CCONJ
ejpam-5973	572	25	connecting	connect	VERB
ejpam-5973	572	26	them	they	PRON
ejpam-5973	572	27	to	to	PART
ejpam-5973	572	28	vk+1	vk+1	ADJ
ejpam-5973	572	29	.	.	PUNCT
ejpam-5973	573	1	clearly	clearly	ADV
ejpam-5973	573	2	assigning	assign	VERB
ejpam-5973	573	3	{	{	PUNCT
ejpam-5973	573	4	1	1	NUM
ejpam-5973	573	5	}	}	PUNCT
ejpam-5973	573	6	to	to	PART
ejpam-5973	573	7	v1	v1	VERB
ejpam-5973	573	8	,	,	PUNCT
ejpam-5973	573	9	vk+1	vk+1	VERB
ejpam-5973	573	10	,	,	PUNCT
ejpam-5973	573	11	{	{	PUNCT
ejpam-5973	573	12	i	i	NOUN
ejpam-5973	573	13	}	}	PUNCT
ejpam-5973	573	14	to	to	ADP
ejpam-5973	573	15	vertex	vertex	PROPN
ejpam-5973	573	16	vi	vi	PROPN
ejpam-5973	573	17	for	for	ADP
ejpam-5973	573	18	i	i	PRON
ejpam-5973	573	19	∈	∈	PROPN
ejpam-5973	573	20	{	{	PUNCT
ejpam-5973	573	21	2	2	NUM
ejpam-5973	573	22	,	,	PUNCT
ejpam-5973	573	23	.	.	PUNCT
ejpam-5973	573	24	.	.	PUNCT
ejpam-5973	573	25	.	.	PUNCT
ejpam-5973	574	1	,	,	PUNCT
ejpam-5973	574	2	k	k	X
ejpam-5973	574	3	}	}	PUNCT
ejpam-5973	574	4	and	and	CCONJ
ejpam-5973	574	5	∅	∅	NOUN
ejpam-5973	574	6	to	to	ADP
ejpam-5973	574	7	the	the	DET
ejpam-5973	574	8	remaining	remain	VERB
ejpam-5973	574	9	vertices	vertex	NOUN
ejpam-5973	574	10	,	,	PUNCT
ejpam-5973	574	11	if	if	SCONJ
ejpam-5973	574	12	any	any	PRON
ejpam-5973	574	13	,	,	PUNCT
ejpam-5973	574	14	provides	provide	VERB
ejpam-5973	574	15	a	a	DET
ejpam-5973	574	16	hop	hop	NOUN
ejpam-5973	574	17	k	k	ADJ
ejpam-5973	574	18	-	-	PUNCT
ejpam-5973	574	19	rainbow	rainbow	NOUN
ejpam-5973	574	20	dominating	dominating	NOUN
ejpam-5973	574	21	function	function	NOUN
ejpam-5973	574	22	on	on	ADP
ejpam-5973	574	23	g	g	NOUN
ejpam-5973	574	24	of	of	ADP
ejpam-5973	574	25	weight	weight	NOUN
ejpam-5973	574	26	k+1	k+1	NOUN
ejpam-5973	574	27	and	and	CCONJ
ejpam-5973	574	28	so	so	ADV
ejpam-5973	574	29	γhrk(g	γhrk(g	PROPN
ejpam-5973	574	30	)	)	PUNCT
ejpam-5973	574	31	≥	≥	NOUN
ejpam-5973	574	32	k+1	k+1	X
ejpam-5973	574	33	.	.	PUNCT
ejpam-5973	575	1	thus	thus	ADV
ejpam-5973	575	2	,	,	PUNCT
ejpam-5973	575	3	γhrk(g	γhrk(g	PROPN
ejpam-5973	575	4	)	)	PUNCT
ejpam-5973	575	5	=	=	SYM
ejpam-5973	576	1	k+1	k+1	X
ejpam-5973	576	2	.	.	PUNCT
ejpam-5973	577	1	this	this	DET
ejpam-5973	577	2	example	example	NOUN
ejpam-5973	577	3	demonstrate	demonstrate	VERB
ejpam-5973	577	4	that	that	SCONJ
ejpam-5973	577	5	the	the	DET
ejpam-5973	577	6	bound	bound	NOUN
ejpam-5973	577	7	of	of	ADP
ejpam-5973	577	8	theorem	theorem	NOUN
ejpam-5973	577	9	4	4	NUM
ejpam-5973	577	10	is	be	AUX
ejpam-5973	577	11	sharp	sharp	ADJ
ejpam-5973	577	12	.	.	PUNCT
ejpam-5973	578	1	hence	hence	ADV
ejpam-5973	578	2	,	,	PUNCT
ejpam-5973	578	3	we	we	PRON
ejpam-5973	578	4	pose	pose	VERB
ejpam-5973	578	5	the	the	DET
ejpam-5973	578	6	following	following	ADJ
ejpam-5973	578	7	problem	problem	NOUN
ejpam-5973	578	8	.	.	PUNCT
ejpam-5973	579	1	problem	problem	NOUN
ejpam-5973	579	2	3	3	NUM
ejpam-5973	579	3	.	.	X
ejpam-5973	580	1	for	for	ADP
ejpam-5973	580	2	positive	positive	ADJ
ejpam-5973	580	3	integer	integer	NOUN
ejpam-5973	580	4	k	k	PROPN
ejpam-5973	580	5	≥	≥	NUM
ejpam-5973	580	6	2	2	NUM
ejpam-5973	580	7	,	,	PUNCT
ejpam-5973	580	8	characterize	characterize	VERB
ejpam-5973	580	9	the	the	DET
ejpam-5973	580	10	graphs	graph	NOUN
ejpam-5973	580	11	g	g	ADP
ejpam-5973	580	12	of	of	ADP
ejpam-5973	580	13	order	order	NOUN
ejpam-5973	580	14	n	n	PRON
ejpam-5973	580	15	such	such	ADJ
ejpam-5973	580	16	that	that	DET
ejpam-5973	580	17	γhrk(g	γhrk(g	NOUN
ejpam-5973	580	18	)	)	PUNCT
ejpam-5973	581	1	=	=	SYM
ejpam-5973	581	2	k	k	PROPN
ejpam-5973	582	1	+	+	NOUN
ejpam-5973	582	2	1	1	X
ejpam-5973	582	3	.	.	AUX
ejpam-5973	582	4	applying	apply	VERB
ejpam-5973	582	5	the	the	DET
ejpam-5973	582	6	bounds	bound	NOUN
ejpam-5973	582	7	presented	present	VERB
ejpam-5973	582	8	in	in	ADP
ejpam-5973	582	9	theorem	theorem	ADJ
ejpam-5973	582	10	4	4	NUM
ejpam-5973	582	11	,	,	PUNCT
ejpam-5973	582	12	for	for	ADP
ejpam-5973	582	13	any	any	DET
ejpam-5973	582	14	positive	positive	ADJ
ejpam-5973	582	15	integer	integer	NOUN
ejpam-5973	582	16	k	k	PROPN
ejpam-5973	582	17	≥	≥	NUM
ejpam-5973	582	18	1	1	NUM
ejpam-5973	582	19	and	and	CCONJ
ejpam-5973	582	20	every	every	DET
ejpam-5973	582	21	graph	graph	NOUN
ejpam-5973	582	22	g	g	NOUN
ejpam-5973	582	23	of	of	ADP
ejpam-5973	582	24	order	order	NOUN
ejpam-5973	582	25	n	n	CCONJ
ejpam-5973	582	26	,	,	PUNCT
ejpam-5973	582	27	we	we	PRON
ejpam-5973	582	28	have	have	VERB
ejpam-5973	582	29	that	that	DET
ejpam-5973	582	30	2min{n	2min{n	NUM
ejpam-5973	582	31	,	,	PUNCT
ejpam-5973	582	32	k	k	PROPN
ejpam-5973	583	1	+	+	PROPN
ejpam-5973	583	2	1	1	NUM
ejpam-5973	583	3	}	}	PUNCT
ejpam-5973	583	4	≤	≤	NUM
ejpam-5973	583	5	γhrk(g	γhrk(g	PROPN
ejpam-5973	583	6	)	)	PUNCT
ejpam-5973	584	1	+	+	NUM
ejpam-5973	584	2	γhrk(g	γhrk(g	NOUN
ejpam-5973	584	3	)	)	PUNCT
ejpam-5973	584	4	≤	≤	NUM
ejpam-5973	584	5	2n	2n	NUM
ejpam-5973	584	6	.	.	PUNCT
ejpam-5973	585	1	in	in	ADP
ejpam-5973	585	2	particular	particular	ADJ
ejpam-5973	585	3	,	,	PUNCT
ejpam-5973	585	4	if	if	SCONJ
ejpam-5973	585	5	n	n	PRON
ejpam-5973	585	6	≤	≤	X
ejpam-5973	585	7	k	k	X
ejpam-5973	585	8	+	+	NOUN
ejpam-5973	585	9	1	1	NUM
ejpam-5973	585	10	,	,	PUNCT
ejpam-5973	585	11	then	then	ADV
ejpam-5973	585	12	we	we	PRON
ejpam-5973	585	13	have	have	VERB
ejpam-5973	585	14	γhrk(g	γhrk(g	PROPN
ejpam-5973	585	15	)	)	PUNCT
ejpam-5973	586	1	+	+	NUM
ejpam-5973	586	2	γhrk(g	γhrk(g	NOUN
ejpam-5973	586	3	)	)	PUNCT
ejpam-5973	586	4	≤	≤	NUM
ejpam-5973	586	5	2n	2n	NUM
ejpam-5973	586	6	.	.	PUNCT
ejpam-5973	587	1	so	so	ADV
ejpam-5973	587	2	,	,	PUNCT
ejpam-5973	587	3	finding	find	VERB
ejpam-5973	587	4	nordhausgaddum	nordhausgaddum	ADJ
ejpam-5973	587	5	type	type	NOUN
ejpam-5973	587	6	results	result	NOUN
ejpam-5973	587	7	for	for	ADP
ejpam-5973	587	8	graphs	graph	NOUN
ejpam-5973	587	9	g	g	ADP
ejpam-5973	587	10	of	of	ADP
ejpam-5973	587	11	order	order	NOUN
ejpam-5973	587	12	n	n	PRON
ejpam-5973	587	13	≥	≥	NOUN
ejpam-5973	587	14	k	k	NOUN
ejpam-5973	588	1	+	+	CCONJ
ejpam-5973	588	2	2	2	NUM
ejpam-5973	588	3	is	be	AUX
ejpam-5973	588	4	of	of	ADP
ejpam-5973	588	5	interest	interest	NOUN
ejpam-5973	588	6	.	.	PUNCT
ejpam-5973	589	1	problem	problem	NOUN
ejpam-5973	589	2	4	4	NUM
ejpam-5973	589	3	.	.	PUNCT
ejpam-5973	590	1	for	for	ADP
ejpam-5973	590	2	graphs	graph	NOUN
ejpam-5973	590	3	g	g	ADP
ejpam-5973	590	4	of	of	ADP
ejpam-5973	590	5	order	order	NOUN
ejpam-5973	590	6	n	n	PRON
ejpam-5973	590	7	≥	≥	NOUN
ejpam-5973	590	8	k	k	NOUN
ejpam-5973	591	1	+	+	CCONJ
ejpam-5973	591	2	2	2	NUM
ejpam-5973	591	3	,	,	PUNCT
ejpam-5973	591	4	determine	determine	VERB
ejpam-5973	591	5	nordhaus	nordhaus	NOUN
ejpam-5973	591	6	-	-	PUNCT
ejpam-5973	591	7	gaddum	gaddum	NOUN
ejpam-5973	591	8	type	type	NOUN
ejpam-5973	591	9	results	result	NOUN
ejpam-5973	591	10	for	for	ADP
ejpam-5973	591	11	γhrk(g	γhrk(g	NOUN
ejpam-5973	591	12	)	)	PUNCT
ejpam-5973	591	13	.	.	PUNCT
ejpam-5973	592	1	problem	problem	NOUN
ejpam-5973	592	2	5	5	NUM
ejpam-5973	592	3	.	.	PUNCT
ejpam-5973	592	4	design	design	VERB
ejpam-5973	592	5	an	an	DET
ejpam-5973	592	6	algorithm	algorithm	NOUN
ejpam-5973	592	7	for	for	ADP
ejpam-5973	592	8	computing	compute	VERB
ejpam-5973	592	9	the	the	DET
ejpam-5973	592	10	value	value	NOUN
ejpam-5973	592	11	of	of	ADP
ejpam-5973	592	12	γhrk(g	γhrk(g	PROPN
ejpam-5973	592	13	)	)	PUNCT
ejpam-5973	592	14	for	for	ADP
ejpam-5973	592	15	any	any	DET
ejpam-5973	592	16	tree	tree	NOUN
ejpam-5973	592	17	t	t	PROPN
ejpam-5973	592	18	and	and	CCONJ
ejpam-5973	592	19	k	k	PROPN
ejpam-5973	592	20	≥	≥	NUM
ejpam-5973	592	21	3	3	NUM
ejpam-5973	592	22	.	.	NOUN
ejpam-5973	592	23	7	7	NUM
ejpam-5973	592	24	.	.	X
ejpam-5973	592	25	conclusion	conclusion	NOUN
ejpam-5973	592	26	in	in	ADP
ejpam-5973	592	27	this	this	DET
ejpam-5973	592	28	paper	paper	NOUN
ejpam-5973	592	29	,	,	PUNCT
ejpam-5973	592	30	we	we	PRON
ejpam-5973	592	31	have	have	AUX
ejpam-5973	592	32	introduced	introduce	VERB
ejpam-5973	592	33	and	and	CCONJ
ejpam-5973	592	34	analyzed	analyze	VERB
ejpam-5973	592	35	the	the	DET
ejpam-5973	592	36	concept	concept	NOUN
ejpam-5973	592	37	of	of	ADP
ejpam-5973	592	38	hop	hop	PROPN
ejpam-5973	592	39	k	k	ADJ
ejpam-5973	592	40	-	-	PUNCT
ejpam-5973	592	41	rainbow	rainbow	NOUN
ejpam-5973	592	42	domination	domination	NOUN
ejpam-5973	592	43	in	in	ADP
ejpam-5973	592	44	graphs	graph	NOUN
ejpam-5973	592	45	.	.	PUNCT
ejpam-5973	593	1	we	we	PRON
ejpam-5973	593	2	established	establish	VERB
ejpam-5973	593	3	fundamental	fundamental	ADJ
ejpam-5973	593	4	properties	property	NOUN
ejpam-5973	593	5	,	,	PUNCT
ejpam-5973	593	6	derived	derive	VERB
ejpam-5973	593	7	bounds	bound	NOUN
ejpam-5973	593	8	,	,	PUNCT
ejpam-5973	593	9	and	and	CCONJ
ejpam-5973	593	10	determined	determine	VERB
ejpam-5973	593	11	exact	exact	ADJ
ejpam-5973	593	12	values	value	NOUN
ejpam-5973	593	13	of	of	ADP
ejpam-5973	593	14	γhrk(g	γhrk(g	NOUN
ejpam-5973	593	15	)	)	PUNCT
ejpam-5973	593	16	for	for	ADP
ejpam-5973	593	17	several	several	ADJ
ejpam-5973	593	18	graph	graph	NOUN
ejpam-5973	593	19	classes	class	NOUN
ejpam-5973	593	20	.	.	PUNCT
ejpam-5973	594	1	additionally	additionally	ADV
ejpam-5973	594	2	,	,	PUNCT
ejpam-5973	594	3	we	we	PRON
ejpam-5973	594	4	identified	identify	VERB
ejpam-5973	594	5	graphs	graph	NOUN
ejpam-5973	594	6	where	where	SCONJ
ejpam-5973	594	7	γhrk(g	γhrk(g	NOUN
ejpam-5973	594	8	)	)	PUNCT
ejpam-5973	594	9	=	=	SYM
ejpam-5973	594	10	n	n	NOUN
ejpam-5973	594	11	and	and	CCONJ
ejpam-5973	594	12	showed	show	VERB
ejpam-5973	594	13	that	that	SCONJ
ejpam-5973	594	14	the	the	DET
ejpam-5973	594	15	hop	hop	NOUN
ejpam-5973	594	16	k	k	ADJ
ejpam-5973	594	17	-	-	PUNCT
ejpam-5973	594	18	rainbow	rainbow	NOUN
ejpam-5973	594	19	domination	domination	NOUN
ejpam-5973	594	20	and	and	CCONJ
ejpam-5973	594	21	the	the	DET
ejpam-5973	594	22	k	k	ADJ
ejpam-5973	594	23	-	-	PUNCT
ejpam-5973	594	24	rainbow	rainbow	NOUN
ejpam-5973	594	25	domination	domination	NOUN
ejpam-5973	594	26	parameters	parameter	NOUN
ejpam-5973	594	27	are	be	AUX
ejpam-5973	594	28	incomparable	incomparable	ADJ
ejpam-5973	594	29	.	.	PUNCT
ejpam-5973	595	1	these	these	DET
ejpam-5973	595	2	results	result	NOUN
ejpam-5973	595	3	lay	lie	VERB
ejpam-5973	595	4	the	the	DET
ejpam-5973	595	5	groundwork	groundwork	NOUN
ejpam-5973	595	6	for	for	ADP
ejpam-5973	595	7	further	further	ADJ
ejpam-5973	595	8	exploration	exploration	NOUN
ejpam-5973	595	9	of	of	ADP
ejpam-5973	595	10	hop	hop	PROPN
ejpam-5973	595	11	k	k	ADJ
ejpam-5973	595	12	-	-	PUNCT
ejpam-5973	595	13	rainbow	rainbow	NOUN
ejpam-5973	595	14	domination	domination	NOUN
ejpam-5973	595	15	and	and	CCONJ
ejpam-5973	595	16	its	its	PRON
ejpam-5973	595	17	applications	application	NOUN
ejpam-5973	595	18	in	in	ADP
ejpam-5973	595	19	graph	graph	NOUN
ejpam-5973	595	20	theory	theory	NOUN
ejpam-5973	595	21	.	.	PUNCT
ejpam-5973	596	1	j.	j.	PROPN
ejpam-5973	596	2	j.	j.	PROPN
ejpam-5973	596	3	hamja	hamja	PROPN
ejpam-5973	596	4	et	et	PROPN
ejpam-5973	596	5	al	al	PROPN
ejpam-5973	596	6	.	.	PUNCT
ejpam-5973	596	7	/	/	SYM
ejpam-5973	596	8	eur	eur	PROPN
ejpam-5973	596	9	.	.	PUNCT
ejpam-5973	597	1	j.	j.	PROPN
ejpam-5973	597	2	pure	pure	PROPN
ejpam-5973	597	3	appl	appl	PROPN
ejpam-5973	597	4	.	.	PROPN
ejpam-5973	597	5	math	math	PROPN
ejpam-5973	597	6	,	,	PUNCT
ejpam-5973	597	7	18	18	NUM
ejpam-5973	597	8	(	(	PUNCT
ejpam-5973	597	9	2	2	NUM
ejpam-5973	597	10	)	)	PUNCT
ejpam-5973	597	11	(	(	PUNCT
ejpam-5973	597	12	2025	2025	NUM
ejpam-5973	597	13	)	)	PUNCT
ejpam-5973	597	14	,	,	PUNCT
ejpam-5973	597	15	5973	5973	NUM
ejpam-5973	597	16	16	16	NUM
ejpam-5973	597	17	of	of	ADP
ejpam-5973	597	18	17	17	NUM
ejpam-5973	597	19	acknowledgements	acknowledgement	NOUN
ejpam-5973	597	20	the	the	DET
ejpam-5973	597	21	authors	author	NOUN
ejpam-5973	597	22	extend	extend	VERB
ejpam-5973	597	23	their	their	PRON
ejpam-5973	597	24	heartfelt	heartfelt	ADJ
ejpam-5973	597	25	appreciation	appreciation	NOUN
ejpam-5973	597	26	to	to	ADP
ejpam-5973	597	27	the	the	DET
ejpam-5973	597	28	reviewers	reviewer	NOUN
ejpam-5973	597	29	for	for	ADP
ejpam-5973	597	30	their	their	PRON
ejpam-5973	597	31	valuable	valuable	ADJ
ejpam-5973	597	32	comments	comment	NOUN
ejpam-5973	597	33	and	and	CCONJ
ejpam-5973	597	34	suggestions	suggestion	NOUN
ejpam-5973	597	35	,	,	PUNCT
ejpam-5973	597	36	which	which	PRON
ejpam-5973	597	37	have	have	AUX
ejpam-5973	597	38	significantly	significantly	ADV
ejpam-5973	597	39	improved	improve	VERB
ejpam-5973	597	40	the	the	DET
ejpam-5973	597	41	quality	quality	NOUN
ejpam-5973	597	42	of	of	ADP
ejpam-5973	597	43	this	this	DET
ejpam-5973	597	44	paper	paper	NOUN
ejpam-5973	597	45	.	.	PUNCT
ejpam-5973	598	1	they	they	PRON
ejpam-5973	598	2	are	be	AUX
ejpam-5973	598	3	also	also	ADV
ejpam-5973	598	4	deeply	deeply	ADV
ejpam-5973	598	5	grateful	grateful	ADJ
ejpam-5973	598	6	for	for	ADP
ejpam-5973	598	7	the	the	DET
ejpam-5973	598	8	financial	financial	ADJ
ejpam-5973	598	9	support	support	NOUN
ejpam-5973	598	10	provided	provide	VERB
ejpam-5973	598	11	by	by	ADP
ejpam-5973	598	12	the	the	DET
ejpam-5973	598	13	department	department	PROPN
ejpam-5973	598	14	of	of	ADP
ejpam-5973	598	15	science	science	NOUN
ejpam-5973	598	16	and	and	CCONJ
ejpam-5973	598	17	technology	technology	NOUN
ejpam-5973	598	18	–	–	PUNCT
ejpam-5973	598	19	accelerated	accelerate	VERB
ejpam-5973	598	20	science	science	NOUN
ejpam-5973	598	21	and	and	CCONJ
ejpam-5973	598	22	technology	technology	NOUN
ejpam-5973	598	23	human	human	ADJ
ejpam-5973	598	24	resource	resource	NOUN
ejpam-5973	598	25	development	development	NOUN
ejpam-5973	598	26	program	program	NOUN
ejpam-5973	598	27	(	(	PUNCT
ejpam-5973	598	28	dost	dost	NOUN
ejpam-5973	598	29	-	-	PUNCT
ejpam-5973	598	30	asthrdp	asthrdp	NOUN
ejpam-5973	598	31	)	)	PUNCT
ejpam-5973	598	32	,	,	PUNCT
ejpam-5973	598	33	mindanao	mindanao	PROPN
ejpam-5973	598	34	state	state	PROPN
ejpam-5973	598	35	university	university	PROPN
ejpam-5973	598	36	–	–	PUNCT
ejpam-5973	598	37	tawi	tawi	NOUN
ejpam-5973	598	38	-	-	PUNCT
ejpam-5973	598	39	tawi	tawi	NOUN
ejpam-5973	598	40	college	college	PROPN
ejpam-5973	598	41	of	of	ADP
ejpam-5973	598	42	technology	technology	NOUN
ejpam-5973	598	43	and	and	CCONJ
ejpam-5973	598	44	oceanography	oceanography	NOUN
ejpam-5973	598	45	(	(	PUNCT
ejpam-5973	598	46	msu	msu	PROPN
ejpam-5973	598	47	-	-	PUNCT
ejpam-5973	598	48	tcto	tcto	NOUN
ejpam-5973	598	49	)	)	PUNCT
ejpam-5973	598	50	,	,	PUNCT
ejpam-5973	598	51	and	and	CCONJ
ejpam-5973	598	52	mindanao	mindanao	PROPN
ejpam-5973	598	53	state	state	PROPN
ejpam-5973	598	54	university	university	PROPN
ejpam-5973	598	55	–	–	PUNCT
ejpam-5973	598	56	iligan	iligan	PROPN
ejpam-5973	598	57	institute	institute	PROPN
ejpam-5973	598	58	of	of	ADP
ejpam-5973	598	59	technology	technology	PROPN
ejpam-5973	598	60	(	(	PUNCT
ejpam-5973	598	61	msu	msu	PROPN
ejpam-5973	598	62	-	-	PUNCT
ejpam-5973	598	63	iit	iit	NOUN
ejpam-5973	598	64	)	)	PUNCT
ejpam-5973	598	65	,	,	PUNCT
ejpam-5973	598	66	which	which	PRON
ejpam-5973	598	67	made	make	VERB
ejpam-5973	598	68	the	the	DET
ejpam-5973	598	69	publication	publication	NOUN
ejpam-5973	598	70	of	of	ADP
ejpam-5973	598	71	this	this	DET
ejpam-5973	598	72	work	work	NOUN
ejpam-5973	598	73	possible	possible	ADJ
ejpam-5973	598	74	.	.	PUNCT
ejpam-5973	599	1	references	reference	NOUN
ejpam-5973	599	2	[	[	X
ejpam-5973	599	3	1	1	X
ejpam-5973	599	4	]	]	PUNCT
ejpam-5973	599	5	t.	t.	PROPN
ejpam-5973	599	6	w.	w.	PROPN
ejpam-5973	599	7	haynes	haynes	PROPN
ejpam-5973	599	8	,	,	PUNCT
ejpam-5973	599	9	s.	s.	PROPN
ejpam-5973	599	10	t.	t.	PROPN
ejpam-5973	599	11	hedetniemi	hedetniemi	PROPN
ejpam-5973	599	12	,	,	PUNCT
ejpam-5973	599	13	and	and	CCONJ
ejpam-5973	599	14	m.	m.	PROPN
ejpam-5973	599	15	a.	a.	PROPN
ejpam-5973	599	16	henning	henning	PROPN
ejpam-5973	599	17	.	.	PUNCT
ejpam-5973	600	1	topics	topic	NOUN
ejpam-5973	600	2	in	in	ADP
ejpam-5973	600	3	domination	domination	NOUN
ejpam-5973	600	4	in	in	ADP
ejpam-5973	600	5	graphs	graph	NOUN
ejpam-5973	600	6	,	,	PUNCT
ejpam-5973	600	7	volume	volume	NOUN
ejpam-5973	600	8	64	64	NUM
ejpam-5973	600	9	of	of	ADP
ejpam-5973	600	10	developments	development	NOUN
ejpam-5973	600	11	in	in	ADP
ejpam-5973	600	12	mathematics	mathematic	NOUN
ejpam-5973	600	13	.	.	PUNCT
ejpam-5973	601	1	springer	springer	PROPN
ejpam-5973	601	2	,	,	PUNCT
ejpam-5973	601	3	cham	cham	PROPN
ejpam-5973	601	4	,	,	PUNCT
ejpam-5973	601	5	2020	2020	NUM
ejpam-5973	601	6	.	.	PUNCT
ejpam-5973	602	1	[	[	X
ejpam-5973	602	2	2	2	NUM
ejpam-5973	602	3	]	]	PUNCT
ejpam-5973	602	4	b.	b.	PROPN
ejpam-5973	602	5	brešar	brešar	PROPN
ejpam-5973	602	6	,	,	PUNCT
ejpam-5973	602	7	m.	m.	NOUN
ejpam-5973	602	8	a.	a.	PROPN
ejpam-5973	602	9	henning	henning	PROPN
ejpam-5973	602	10	,	,	PUNCT
ejpam-5973	602	11	and	and	CCONJ
ejpam-5973	602	12	d.	d.	PROPN
ejpam-5973	602	13	f.	f.	PROPN
ejpam-5973	602	14	rall	rall	PROPN
ejpam-5973	602	15	.	.	PUNCT
ejpam-5973	603	1	rainbow	rainbow	PROPN
ejpam-5973	603	2	domination	domination	NOUN
ejpam-5973	603	3	in	in	ADP
ejpam-5973	603	4	graphs	graph	NOUN
ejpam-5973	603	5	.	.	PUNCT
ejpam-5973	604	1	taiwanese	taiwanese	ADJ
ejpam-5973	604	2	journal	journal	NOUN
ejpam-5973	604	3	of	of	ADP
ejpam-5973	604	4	mathematics	mathematic	NOUN
ejpam-5973	604	5	,	,	PUNCT
ejpam-5973	604	6	12(1):213–225	12(1):213–225	PROPN
ejpam-5973	604	7	,	,	PUNCT
ejpam-5973	604	8	2008	2008	NUM
ejpam-5973	604	9	.	.	PUNCT
ejpam-5973	605	1	[	[	X
ejpam-5973	605	2	3	3	X
ejpam-5973	605	3	]	]	PUNCT
ejpam-5973	605	4	z.	z.	PROPN
ejpam-5973	605	5	shao	shao	PROPN
ejpam-5973	605	6	,	,	PUNCT
ejpam-5973	605	7	m.	m.	PROPN
ejpam-5973	605	8	liang	liang	PROPN
ejpam-5973	605	9	,	,	PUNCT
ejpam-5973	605	10	c.	c.	PROPN
ejpam-5973	605	11	yin	yin	PROPN
ejpam-5973	605	12	,	,	PUNCT
ejpam-5973	605	13	x.	x.	PROPN
ejpam-5973	605	14	xu	xu	PROPN
ejpam-5973	605	15	,	,	PUNCT
ejpam-5973	605	16	p.	p.	NOUN
ejpam-5973	605	17	pavlič	pavlič	PROPN
ejpam-5973	605	18	,	,	PUNCT
ejpam-5973	605	19	and	and	CCONJ
ejpam-5973	605	20	j.	j.	PROPN
ejpam-5973	605	21	žerovnik	žerovnik	VERB
ejpam-5973	605	22	.	.	PUNCT
ejpam-5973	606	1	on	on	ADP
ejpam-5973	606	2	rainbow	rainbow	PROPN
ejpam-5973	606	3	domination	domination	NOUN
ejpam-5973	606	4	numbers	number	NOUN
ejpam-5973	606	5	of	of	ADP
ejpam-5973	606	6	graphs	graph	NOUN
ejpam-5973	606	7	.	.	PUNCT
ejpam-5973	607	1	information	information	NOUN
ejpam-5973	607	2	sciences	sciences	PROPN
ejpam-5973	607	3	,	,	PUNCT
ejpam-5973	607	4	254:225–234	254:225–234	NUM
ejpam-5973	607	5	,	,	PUNCT
ejpam-5973	607	6	2014	2014	NUM
ejpam-5973	607	7	.	.	PUNCT
ejpam-5973	608	1	[	[	X
ejpam-5973	608	2	4	4	X
ejpam-5973	608	3	]	]	X
ejpam-5973	608	4	h.	h.	NOUN
ejpam-5973	608	5	abdollahzadeh	abdollahzadeh	PROPN
ejpam-5973	608	6	ahangar	ahangar	NOUN
ejpam-5973	608	7	,	,	PUNCT
ejpam-5973	608	8	j.	j.	PROPN
ejpam-5973	608	9	amjadi	amjadi	PROPN
ejpam-5973	608	10	,	,	PUNCT
ejpam-5973	608	11	n.	n.	PROPN
ejpam-5973	608	12	jafari	jafari	PROPN
ejpam-5973	608	13	rad	rad	PROPN
ejpam-5973	608	14	,	,	PUNCT
ejpam-5973	608	15	and	and	CCONJ
ejpam-5973	608	16	v.	v.	ADP
ejpam-5973	608	17	d.	d.	PROPN
ejpam-5973	608	18	samodivkin	samodivkin	PROPN
ejpam-5973	608	19	.	.	PUNCT
ejpam-5973	609	1	total	total	ADJ
ejpam-5973	609	2	k	k	ADJ
ejpam-5973	609	3	-	-	PUNCT
ejpam-5973	609	4	rainbow	rainbow	NOUN
ejpam-5973	609	5	domination	domination	NOUN
ejpam-5973	609	6	numbers	number	NOUN
ejpam-5973	609	7	in	in	ADP
ejpam-5973	609	8	graphs	graph	NOUN
ejpam-5973	609	9	.	.	PUNCT
ejpam-5973	610	1	communications	communication	NOUN
ejpam-5973	610	2	in	in	ADP
ejpam-5973	610	3	combinatorics	combinatoric	NOUN
ejpam-5973	610	4	and	and	CCONJ
ejpam-5973	610	5	optimization	optimization	NOUN
ejpam-5973	610	6	,	,	PUNCT
ejpam-5973	610	7	3(1):37–50	3(1):37–50	NUM
ejpam-5973	610	8	,	,	PUNCT
ejpam-5973	610	9	2018	2018	NUM
ejpam-5973	610	10	.	.	PUNCT
ejpam-5973	611	1	[	[	X
ejpam-5973	611	2	5	5	X
ejpam-5973	611	3	]	]	PUNCT
ejpam-5973	611	4	b.	b.	PROPN
ejpam-5973	611	5	brešar	brešar	PROPN
ejpam-5973	611	6	and	and	CCONJ
ejpam-5973	611	7	t.	t.	PROPN
ejpam-5973	611	8	k.	k.	PROPN
ejpam-5973	611	9	šumenjak	šumenjak	PROPN
ejpam-5973	611	10	.	.	PUNCT
ejpam-5973	612	1	on	on	ADP
ejpam-5973	612	2	the	the	DET
ejpam-5973	612	3	2	2	NUM
ejpam-5973	612	4	-	-	PUNCT
ejpam-5973	612	5	rainbow	rainbow	NOUN
ejpam-5973	612	6	domination	domination	NOUN
ejpam-5973	612	7	in	in	ADP
ejpam-5973	612	8	graphs	graph	NOUN
ejpam-5973	612	9	.	.	PUNCT
ejpam-5973	613	1	discrete	discrete	ADJ
ejpam-5973	613	2	applied	applied	ADJ
ejpam-5973	613	3	mathematics	mathematic	NOUN
ejpam-5973	613	4	,	,	PUNCT
ejpam-5973	613	5	155(17):2394–2400	155(17):2394–2400	NUM
ejpam-5973	613	6	,	,	PUNCT
ejpam-5973	613	7	2007	2007	NUM
ejpam-5973	613	8	.	.	PUNCT
ejpam-5973	614	1	[	[	X
ejpam-5973	614	2	6	6	NUM
ejpam-5973	614	3	]	]	PUNCT
ejpam-5973	614	4	b.	b.	PROPN
ejpam-5973	614	5	brešar	brešar	PROPN
ejpam-5973	614	6	and	and	CCONJ
ejpam-5973	614	7	t.k	t.k	PROPN
ejpam-5973	614	8	.	.	PROPN
ejpam-5973	614	9	šumenjak	šumenjak	PROPN
ejpam-5973	614	10	.	.	PUNCT
ejpam-5973	615	1	on	on	ADP
ejpam-5973	615	2	the	the	DET
ejpam-5973	615	3	2	2	NUM
ejpam-5973	615	4	-	-	PUNCT
ejpam-5973	615	5	rainbow	rainbow	NOUN
ejpam-5973	615	6	domination	domination	NOUN
ejpam-5973	615	7	in	in	ADP
ejpam-5973	615	8	graphs	graph	NOUN
ejpam-5973	615	9	.	.	PUNCT
ejpam-5973	616	1	discrete	discrete	ADJ
ejpam-5973	616	2	applied	apply	VERB
ejpam-5973	616	3	mathematics	mathematic	NOUN
ejpam-5973	616	4	,	,	PUNCT
ejpam-5973	616	5	155:2394–2400	155:2394–2400	NUM
ejpam-5973	616	6	,	,	PUNCT
ejpam-5973	616	7	2007	2007	NUM
ejpam-5973	616	8	.	.	PUNCT
ejpam-5973	617	1	[	[	X
ejpam-5973	617	2	7	7	X
ejpam-5973	617	3	]	]	PUNCT
ejpam-5973	617	4	a.	a.	NOUN
ejpam-5973	617	5	mahmoodi	mahmoodi	NOUN
ejpam-5973	617	6	and	and	CCONJ
ejpam-5973	617	7	l.	l.	PROPN
ejpam-5973	617	8	volkmann	volkmann	PROPN
ejpam-5973	617	9	.	.	PUNCT
ejpam-5973	618	1	outer	outer	ADJ
ejpam-5973	618	2	-	-	PUNCT
ejpam-5973	618	3	independent	independent	ADJ
ejpam-5973	618	4	total	total	ADJ
ejpam-5973	618	5	2	2	NUM
ejpam-5973	618	6	-	-	PUNCT
ejpam-5973	618	7	rainbow	rainbow	NOUN
ejpam-5973	618	8	dominating	dominating	NOUN
ejpam-5973	618	9	functions	function	NOUN
ejpam-5973	618	10	in	in	ADP
ejpam-5973	618	11	graphs	graph	NOUN
ejpam-5973	618	12	.	.	PUNCT
ejpam-5973	619	1	communications	communication	NOUN
ejpam-5973	619	2	in	in	ADP
ejpam-5973	619	3	combinatorics	combinatoric	NOUN
ejpam-5973	619	4	and	and	CCONJ
ejpam-5973	619	5	optimization	optimization	NOUN
ejpam-5973	619	6	,	,	PUNCT
ejpam-5973	619	7	8(2):431–444	8(2):431–444	NUM
ejpam-5973	619	8	,	,	PUNCT
ejpam-5973	619	9	2023	2023	NUM
ejpam-5973	619	10	.	.	PUNCT
ejpam-5973	620	1	[	[	X
ejpam-5973	620	2	8	8	NUM
ejpam-5973	620	3	]	]	X
ejpam-5973	620	4	r.	r.	PROPN
ejpam-5973	620	5	y.	y.	PROPN
ejpam-5973	620	6	salkhori	salkhori	PROPN
ejpam-5973	620	7	,	,	PUNCT
ejpam-5973	620	8	e.	e.	PROPN
ejpam-5973	620	9	vatandoost	vatandoost	PROPN
ejpam-5973	620	10	,	,	PUNCT
ejpam-5973	620	11	and	and	CCONJ
ejpam-5973	620	12	a.	a.	NOUN
ejpam-5973	620	13	behtoei	behtoei	PROPN
ejpam-5973	620	14	.	.	PUNCT
ejpam-5973	621	1	2	2	NUM
ejpam-5973	621	2	-	-	PUNCT
ejpam-5973	621	3	rainbow	rainbow	NOUN
ejpam-5973	621	4	domination	domination	NOUN
ejpam-5973	621	5	number	number	NOUN
ejpam-5973	621	6	of	of	ADP
ejpam-5973	621	7	the	the	DET
ejpam-5973	621	8	subdivision	subdivision	NOUN
ejpam-5973	621	9	of	of	ADP
ejpam-5973	621	10	graphs	graph	NOUN
ejpam-5973	621	11	.	.	PUNCT
ejpam-5973	622	1	communications	communication	NOUN
ejpam-5973	622	2	in	in	ADP
ejpam-5973	622	3	combinatorics	combinatoric	NOUN
ejpam-5973	622	4	and	and	CCONJ
ejpam-5973	622	5	optimization	optimization	NOUN
ejpam-5973	622	6	,	,	PUNCT
ejpam-5973	622	7	2023	2023	NUM
ejpam-5973	622	8	.	.	PUNCT
ejpam-5973	623	1	in	in	ADP
ejpam-5973	623	2	press	press	NOUN
ejpam-5973	623	3	.	.	PUNCT
ejpam-5973	624	1	[	[	X
ejpam-5973	624	2	9	9	X
ejpam-5973	624	3	]	]	X
ejpam-5973	624	4	y.	y.	PROPN
ejpam-5973	624	5	wu	wu	PROPN
ejpam-5973	624	6	and	and	CCONJ
ejpam-5973	624	7	n.	n.	PROPN
ejpam-5973	624	8	jafari	jafari	PROPN
ejpam-5973	624	9	rad	rad	PROPN
ejpam-5973	624	10	.	.	PROPN
ejpam-5973	624	11	bounds	bound	NOUN
ejpam-5973	624	12	on	on	ADP
ejpam-5973	624	13	the	the	DET
ejpam-5973	624	14	2	2	NUM
ejpam-5973	624	15	-	-	PUNCT
ejpam-5973	624	16	rainbow	rainbow	NOUN
ejpam-5973	624	17	domination	domination	NOUN
ejpam-5973	624	18	number	number	NOUN
ejpam-5973	624	19	of	of	ADP
ejpam-5973	624	20	graphs	graph	NOUN
ejpam-5973	624	21	.	.	PUNCT
ejpam-5973	625	1	graphs	graph	NOUN
ejpam-5973	625	2	and	and	CCONJ
ejpam-5973	625	3	combinatorics	combinatoric	NOUN
ejpam-5973	625	4	,	,	PUNCT
ejpam-5973	625	5	29(4):1125–1133	29(4):1125–1133	NUM
ejpam-5973	625	6	,	,	PUNCT
ejpam-5973	625	7	2013	2013	NUM
ejpam-5973	625	8	.	.	PUNCT
ejpam-5973	626	1	[	[	X
ejpam-5973	626	2	10	10	NUM
ejpam-5973	626	3	]	]	X
ejpam-5973	626	4	c.	c.	PROPN
ejpam-5973	626	5	natarajan	natarajan	PROPN
ejpam-5973	626	6	and	and	CCONJ
ejpam-5973	626	7	s.	s.	PROPN
ejpam-5973	626	8	k.	k.	PROPN
ejpam-5973	626	9	ayyaswamy	ayyaswamy	PROPN
ejpam-5973	626	10	.	.	PUNCT
ejpam-5973	627	1	hop	hop	PROPN
ejpam-5973	627	2	domination	domination	NOUN
ejpam-5973	627	3	in	in	ADP
ejpam-5973	627	4	graphs	graphs	PROPN
ejpam-5973	627	5	ii	ii	PROPN
ejpam-5973	627	6	.	.	PUNCT
ejpam-5973	628	1	analele	analele	ADP
ejpam-5973	628	2	ştiinţifice	ştiinţifice	PROPN
ejpam-5973	628	3	ale	ale	NOUN
ejpam-5973	628	4	universităţii	universităţii	PROPN
ejpam-5973	628	5	ovidius	ovidius	NOUN
ejpam-5973	628	6	constanţa	constanţa	NOUN
ejpam-5973	628	7	,	,	PUNCT
ejpam-5973	628	8	seria	seria	PROPN
ejpam-5973	628	9	matematică	matematică	PROPN
ejpam-5973	628	10	,	,	PUNCT
ejpam-5973	628	11	23(2):187–199	23(2):187–199	PROPN
ejpam-5973	628	12	,	,	PUNCT
ejpam-5973	628	13	2015	2015	NUM
ejpam-5973	628	14	.	.	PUNCT
ejpam-5973	629	1	[	[	X
ejpam-5973	629	2	11	11	NUM
ejpam-5973	629	3	]	]	X
ejpam-5973	629	4	d.	d.	PROPN
ejpam-5973	629	5	anusha	anusha	PROPN
ejpam-5973	629	6	,	,	PUNCT
ejpam-5973	629	7	j.	j.	PROPN
ejpam-5973	629	8	john	john	PROPN
ejpam-5973	629	9	,	,	PUNCT
ejpam-5973	629	10	and	and	CCONJ
ejpam-5973	629	11	s.	s.	PROPN
ejpam-5973	629	12	joseph	joseph	PROPN
ejpam-5973	629	13	robin	robin	PROPN
ejpam-5973	629	14	.	.	PUNCT
ejpam-5973	630	1	graphs	graph	NOUN
ejpam-5973	630	2	with	with	ADP
ejpam-5973	630	3	small	small	ADJ
ejpam-5973	630	4	and	and	CCONJ
ejpam-5973	630	5	large	large	ADJ
ejpam-5973	630	6	hop	hop	NOUN
ejpam-5973	630	7	domination	domination	NOUN
ejpam-5973	630	8	numbers	number	NOUN
ejpam-5973	630	9	.	.	PUNCT
ejpam-5973	631	1	bulletin	bulletin	NOUN
ejpam-5973	631	2	of	of	ADP
ejpam-5973	631	3	the	the	DET
ejpam-5973	631	4	international	international	ADJ
ejpam-5973	631	5	mathematical	mathematical	ADJ
ejpam-5973	631	6	virtual	virtual	PROPN
ejpam-5973	631	7	institute	institute	PROPN
ejpam-5973	631	8	,	,	PUNCT
ejpam-5973	631	9	11(3):483–489	11(3):483–489	PROPN
ejpam-5973	631	10	,	,	PUNCT
ejpam-5973	631	11	2021	2021	NUM
ejpam-5973	631	12	.	.	PUNCT
ejpam-5973	632	1	[	[	X
ejpam-5973	632	2	12	12	NUM
ejpam-5973	632	3	]	]	X
ejpam-5973	632	4	d.	d.	PROPN
ejpam-5973	632	5	anusha	anusha	PROPN
ejpam-5973	632	6	,	,	PUNCT
ejpam-5973	632	7	s.	s.	PROPN
ejpam-5973	632	8	joseph	joseph	PROPN
ejpam-5973	632	9	robin	robin	PROPN
ejpam-5973	632	10	,	,	PUNCT
ejpam-5973	632	11	and	and	CCONJ
ejpam-5973	632	12	j.	j.	PROPN
ejpam-5973	632	13	john	john	PROPN
ejpam-5973	632	14	.	.	PUNCT
ejpam-5973	633	1	further	further	PROPN
ejpam-5973	633	2	results	result	NOUN
ejpam-5973	633	3	on	on	ADP
ejpam-5973	633	4	the	the	DET
ejpam-5973	633	5	hop	hop	NOUN
ejpam-5973	633	6	domination	domination	NOUN
ejpam-5973	633	7	number	number	NOUN
ejpam-5973	633	8	of	of	ADP
ejpam-5973	633	9	a	a	DET
ejpam-5973	633	10	graph	graph	NOUN
ejpam-5973	633	11	.	.	PUNCT
ejpam-5973	634	1	boletim	boletim	PROPN
ejpam-5973	634	2	da	da	PROPN
ejpam-5973	634	3	sociedade	sociedade	PROPN
ejpam-5973	634	4	paranaense	paranaense	PROPN
ejpam-5973	634	5	de	de	PROPN
ejpam-5973	634	6	matemática	matemática	PROPN
ejpam-5973	634	7	,	,	PUNCT
ejpam-5973	634	8	42:1–12	42:1–12	NUM
ejpam-5973	634	9	,	,	PUNCT
ejpam-5973	634	10	2024	2024	NUM
ejpam-5973	634	11	.	.	PUNCT
ejpam-5973	635	1	[	[	X
ejpam-5973	635	2	13	13	NUM
ejpam-5973	635	3	]	]	PUNCT
ejpam-5973	635	4	s.	s.	PROPN
ejpam-5973	635	5	k.	k.	PROPN
ejpam-5973	635	6	ayyaswamy	ayyaswamy	PROPN
ejpam-5973	635	7	,	,	PUNCT
ejpam-5973	635	8	c.	c.	PROPN
ejpam-5973	635	9	natarajan	natarajan	PROPN
ejpam-5973	635	10	,	,	PUNCT
ejpam-5973	635	11	and	and	CCONJ
ejpam-5973	635	12	g.	g.	PROPN
ejpam-5973	635	13	sathiamoorthy	sathiamoorthy	PROPN
ejpam-5973	635	14	.	.	PUNCT
ejpam-5973	636	1	a	a	DET
ejpam-5973	636	2	note	note	NOUN
ejpam-5973	636	3	on	on	ADP
ejpam-5973	636	4	hop	hop	NOUN
ejpam-5973	636	5	domination	domination	NOUN
ejpam-5973	636	6	number	number	NOUN
ejpam-5973	636	7	of	of	ADP
ejpam-5973	636	8	some	some	DET
ejpam-5973	636	9	special	special	ADJ
ejpam-5973	636	10	families	family	NOUN
ejpam-5973	636	11	of	of	ADP
ejpam-5973	636	12	graphs	graph	NOUN
ejpam-5973	636	13	.	.	PUNCT
ejpam-5973	637	1	international	international	ADJ
ejpam-5973	637	2	journal	journal	NOUN
ejpam-5973	637	3	of	of	ADP
ejpam-5973	637	4	pure	pure	ADJ
ejpam-5973	637	5	and	and	CCONJ
ejpam-5973	637	6	applied	applied	ADJ
ejpam-5973	637	7	mathematics	mathematic	NOUN
ejpam-5973	637	8	,	,	PUNCT
ejpam-5973	637	9	119(12):11465–14171	119(12):11465–14171	NUM
ejpam-5973	637	10	,	,	PUNCT
ejpam-5973	637	11	2018	2018	NUM
ejpam-5973	637	12	.	.	PUNCT
ejpam-5973	638	1	j.	j.	PROPN
ejpam-5973	638	2	j.	j.	PROPN
ejpam-5973	638	3	hamja	hamja	PROPN
ejpam-5973	638	4	et	et	PROPN
ejpam-5973	638	5	al	al	PROPN
ejpam-5973	638	6	.	.	PUNCT
ejpam-5973	638	7	/	/	SYM
ejpam-5973	638	8	eur	eur	PROPN
ejpam-5973	638	9	.	.	PUNCT
ejpam-5973	639	1	j.	j.	PROPN
ejpam-5973	639	2	pure	pure	PROPN
ejpam-5973	639	3	appl	appl	PROPN
ejpam-5973	639	4	.	.	PROPN
ejpam-5973	639	5	math	math	PROPN
ejpam-5973	639	6	,	,	PUNCT
ejpam-5973	639	7	18	18	NUM
ejpam-5973	639	8	(	(	PUNCT
ejpam-5973	639	9	2	2	NUM
ejpam-5973	639	10	)	)	PUNCT
ejpam-5973	639	11	(	(	PUNCT
ejpam-5973	639	12	2025	2025	NUM
ejpam-5973	639	13	)	)	PUNCT
ejpam-5973	639	14	,	,	PUNCT
ejpam-5973	639	15	5973	5973	NUM
ejpam-5973	639	16	17	17	NUM
ejpam-5973	639	17	of	of	ADP
ejpam-5973	639	18	17	17	NUM
ejpam-5973	639	19	[	[	SYM
ejpam-5973	639	20	14	14	NUM
ejpam-5973	639	21	]	]	PUNCT
ejpam-5973	639	22	a.	a.	PROPN
ejpam-5973	639	23	e.	e.	PROPN
ejpam-5973	639	24	hassan	hassan	PROPN
ejpam-5973	639	25	,	,	PUNCT
ejpam-5973	639	26	a.	a.	PROPN
ejpam-5973	639	27	e.	e.	PROPN
ejpam-5973	639	28	gamorez	gamorez	PROPN
ejpam-5973	639	29	,	,	PUNCT
ejpam-5973	639	30	e.	e.	PROPN
ejpam-5973	639	31	c.	c.	PROPN
ejpam-5973	639	32	ahmad	ahmad	PROPN
ejpam-5973	639	33	,	,	PUNCT
ejpam-5973	639	34	and	and	CCONJ
ejpam-5973	639	35	j.	j.	PROPN
ejpam-5973	639	36	j.	j.	PROPN
ejpam-5973	639	37	hamja	hamja	PROPN
ejpam-5973	639	38	.	.	PUNCT
ejpam-5973	640	1	certified	certify	VERB
ejpam-5973	640	2	hop	hop	NOUN
ejpam-5973	640	3	domination	domination	NOUN
ejpam-5973	640	4	in	in	ADP
ejpam-5973	640	5	graphs	graph	NOUN
ejpam-5973	640	6	.	.	PUNCT
ejpam-5973	641	1	international	international	ADJ
ejpam-5973	641	2	journal	journal	NOUN
ejpam-5973	641	3	of	of	ADP
ejpam-5973	641	4	mathematics	mathematic	NOUN
ejpam-5973	641	5	and	and	CCONJ
ejpam-5973	641	6	computer	computer	NOUN
ejpam-5973	641	7	science	science	NOUN
ejpam-5973	641	8	,	,	PUNCT
ejpam-5973	641	9	19(4):1105	19(4):1105	NUM
ejpam-5973	641	10	–	–	PUNCT
ejpam-5973	641	11	1110	1110	NUM
ejpam-5973	641	12	,	,	PUNCT
ejpam-5973	641	13	2024	2024	NUM
ejpam-5973	641	14	.	.	PUNCT
ejpam-5973	642	1	[	[	X
ejpam-5973	642	2	15	15	NUM
ejpam-5973	642	3	]	]	X
ejpam-5973	642	4	f.	f.	PROPN
ejpam-5973	642	5	harary	harary	PROPN
ejpam-5973	642	6	.	.	PUNCT
ejpam-5973	643	1	graph	graph	NOUN
ejpam-5973	643	2	theory	theory	NOUN
ejpam-5973	643	3	.	.	PUNCT
ejpam-5973	644	1	addison	addison	PROPN
ejpam-5973	644	2	-	-	PUNCT
ejpam-5973	644	3	wesley	wesley	PROPN
ejpam-5973	644	4	publishing	publishing	PROPN
ejpam-5973	644	5	company	company	NOUN
ejpam-5973	644	6	,	,	PUNCT
ejpam-5973	644	7	massachusetts	massachusetts	PROPN
ejpam-5973	644	8	,	,	PUNCT
ejpam-5973	644	9	1969	1969	NUM
ejpam-5973	644	10	.	.	PUNCT
ejpam-5973	645	1	[	[	X
ejpam-5973	645	2	16	16	NUM
ejpam-5973	645	3	]	]	X
ejpam-5973	645	4	e.	e.	PROPN
ejpam-5973	645	5	shabani	shabani	PROPN
ejpam-5973	645	6	,	,	PUNCT
ejpam-5973	645	7	n.	n.	PROPN
ejpam-5973	645	8	jafari	jafari	PROPN
ejpam-5973	645	9	rad	rad	PROPN
ejpam-5973	645	10	,	,	PUNCT
ejpam-5973	645	11	and	and	CCONJ
ejpam-5973	645	12	a.	a.	NOUN
ejpam-5973	645	13	poureidi	poureidi	PROPN
ejpam-5973	645	14	.	.	PUNCT
ejpam-5973	646	1	graphs	graph	NOUN
ejpam-5973	646	2	with	with	ADP
ejpam-5973	646	3	large	large	ADJ
ejpam-5973	646	4	hop	hop	NOUN
ejpam-5973	646	5	roman	roman	ADJ
ejpam-5973	646	6	domination	domination	NOUN
ejpam-5973	646	7	number	number	NOUN
ejpam-5973	646	8	.	.	PUNCT
ejpam-5973	647	1	computer	computer	NOUN
ejpam-5973	647	2	science	science	PROPN
ejpam-5973	647	3	journal	journal	PROPN
ejpam-5973	647	4	of	of	ADP
ejpam-5973	647	5	moldova	moldova	PROPN
ejpam-5973	647	6	,	,	PUNCT
ejpam-5973	647	7	27(1):3–22	27(1):3–22	NUM
ejpam-5973	647	8	,	,	PUNCT
ejpam-5973	647	9	2019	2019	NUM
ejpam-5973	647	10	.	.	PUNCT
ejpam-5973	648	1	[	[	X
ejpam-5973	648	2	17	17	NUM
ejpam-5973	648	3	]	]	X
ejpam-5973	648	4	e.	e.	PROPN
ejpam-5973	648	5	j.	j.	PROPN
ejpam-5973	648	6	cockayne	cockayne	PROPN
ejpam-5973	648	7	,	,	PUNCT
ejpam-5973	648	8	p.	p.	NOUN
ejpam-5973	648	9	m.	m.	NOUN
ejpam-5973	648	10	dreyer	dreyer	PROPN
ejpam-5973	648	11	jr	jr	PROPN
ejpam-5973	648	12	.	.	PROPN
ejpam-5973	648	13	,	,	PUNCT
ejpam-5973	648	14	s.	s.	PROPN
ejpam-5973	648	15	m.	m.	PROPN
ejpam-5973	648	16	hedetniemi	hedetniemi	ADV
ejpam-5973	648	17	,	,	PUNCT
ejpam-5973	648	18	and	and	CCONJ
ejpam-5973	648	19	s.	s.	PROPN
ejpam-5973	648	20	t.	t.	PROPN
ejpam-5973	648	21	hedetniemi	hedetniemi	PROPN
ejpam-5973	648	22	.	.	PUNCT
ejpam-5973	649	1	on	on	ADP
ejpam-5973	649	2	roman	roman	ADJ
ejpam-5973	649	3	domination	domination	NOUN
ejpam-5973	649	4	in	in	ADP
ejpam-5973	649	5	graphs	graph	NOUN
ejpam-5973	649	6	.	.	PUNCT
ejpam-5973	650	1	discrete	discrete	ADJ
ejpam-5973	650	2	mathematics	mathematic	NOUN
ejpam-5973	650	3	,	,	PUNCT
ejpam-5973	650	4	278(1	278(1	NUM
ejpam-5973	650	5	-	-	SYM
ejpam-5973	650	6	3):11–22	3):11–22	NUM
ejpam-5973	650	7	,	,	PUNCT
ejpam-5973	650	8	2004	2004	NUM
ejpam-5973	650	9	.	.	PUNCT
ejpam-5973	651	1	[	[	X
ejpam-5973	651	2	18	18	NUM
ejpam-5973	651	3	]	]	X
ejpam-5973	651	4	d.	d.	PROPN
ejpam-5973	651	5	a.	a.	NOUN
ejpam-5973	651	6	mojdeh	mojdeh	PROPN
ejpam-5973	651	7	and	and	CCONJ
ejpam-5973	651	8	z.	z.	PROPN
ejpam-5973	651	9	mansouri	mansouri	PROPN
ejpam-5973	651	10	.	.	PUNCT
ejpam-5973	652	1	rainbow	rainbow	PROPN
ejpam-5973	652	2	domination	domination	NOUN
ejpam-5973	652	3	of	of	ADP
ejpam-5973	652	4	graphs	graph	NOUN
ejpam-5973	652	5	.	.	PUNCT
ejpam-5973	653	1	transactions	transaction	NOUN
ejpam-5973	653	2	on	on	ADP
ejpam-5973	653	3	combinatorics	combinatoric	NOUN
ejpam-5973	653	4	,	,	PUNCT
ejpam-5973	653	5	7(4):99–106	7(4):99–106	NUM
ejpam-5973	653	6	,	,	PUNCT
ejpam-5973	653	7	2018	2018	NUM
ejpam-5973	653	8	.	.	PUNCT
ejpam-5973	653	9	presented	present	VERB
ejpam-5973	653	10	at	at	ADP
ejpam-5973	653	11	the	the	DET
ejpam-5973	653	12	3rd	3rd	ADJ
ejpam-5973	653	13	international	international	ADJ
ejpam-5973	653	14	conference	conference	NOUN
ejpam-5973	653	15	on	on	ADP
ejpam-5973	653	16	combinatorics	combinatoric	NOUN
ejpam-5973	653	17	.	.	PUNCT