id	sid	tid	token	lemma	pos
ejpam-5770	1	1	european	european	PROPN
ejpam-5770	1	2	journal	journal	PROPN
ejpam-5770	1	3	of	of	ADP
ejpam-5770	1	4	pure	pure	ADJ
ejpam-5770	1	5	and	and	CCONJ
ejpam-5770	1	6	applied	applied	ADJ
ejpam-5770	1	7	mathematics	mathematic	NOUN
ejpam-5770	1	8	2025	2025	NUM
ejpam-5770	1	9	,	,	PUNCT
ejpam-5770	1	10	vol	vol	NOUN
ejpam-5770	1	11	.	.	PROPN
ejpam-5770	1	12	18	18	NUM
ejpam-5770	1	13	,	,	PUNCT
ejpam-5770	1	14	issue	issue	NOUN
ejpam-5770	1	15	2	2	NUM
ejpam-5770	1	16	,	,	PUNCT
ejpam-5770	1	17	article	article	NOUN
ejpam-5770	1	18	number	number	NOUN
ejpam-5770	1	19	5770	5770	NUM
ejpam-5770	1	20	issn	issn	PROPN
ejpam-5770	1	21	1307	1307	NUM
ejpam-5770	1	22	-	-	SYM
ejpam-5770	1	23	5543	5543	NUM
ejpam-5770	1	24	–	–	PUNCT
ejpam-5770	1	25	ejpam.com	ejpam.com	X
ejpam-5770	1	26	published	publish	VERB
ejpam-5770	1	27	by	by	ADP
ejpam-5770	1	28	new	new	PROPN
ejpam-5770	1	29	york	york	PROPN
ejpam-5770	1	30	business	business	PROPN
ejpam-5770	1	31	global	global	ADJ
ejpam-5770	1	32	weakly	weakly	ADJ
ejpam-5770	1	33	connected	connected	ADJ
ejpam-5770	1	34	k	k	ADJ
ejpam-5770	1	35	-	-	PUNCT
ejpam-5770	1	36	rainbow	rainbow	NOUN
ejpam-5770	1	37	domination	domination	NOUN
ejpam-5770	1	38	in	in	ADP
ejpam-5770	1	39	graphs	graph	NOUN
ejpam-5770	1	40	jamil	jamil	PROPN
ejpam-5770	1	41	j.	j.	PROPN
ejpam-5770	1	42	hamja1,2	hamja1,2	PROPN
ejpam-5770	1	43	,	,	PUNCT
ejpam-5770	1	44	seyed	seyed	PROPN
ejpam-5770	1	45	mahmoud	mahmoud	PROPN
ejpam-5770	1	46	sheikholeslami3,∗	sheikholeslami3,∗	PROPN
ejpam-5770	1	47	,	,	PUNCT
ejpam-5770	1	48	imelda	imelda	PROPN
ejpam-5770	1	49	s.	s.	PROPN
ejpam-5770	1	50	aniversario2,4	aniversario2,4	PROPN
ejpam-5770	1	51	,	,	PUNCT
ejpam-5770	2	1	lyster	lyster	PROPN
ejpam-5770	2	2	rey	rey	PROPN
ejpam-5770	2	3	b.	b.	PROPN
ejpam-5770	3	1	cabardo2,4	cabardo2,4	PROPN
ejpam-5770	3	2	1	1	NUM
ejpam-5770	3	3	mathematics	mathematic	NOUN
ejpam-5770	3	4	and	and	CCONJ
ejpam-5770	3	5	sciences	sciences	PROPN
ejpam-5770	3	6	department	department	PROPN
ejpam-5770	3	7	,	,	PUNCT
ejpam-5770	3	8	college	college	NOUN
ejpam-5770	3	9	of	of	ADP
ejpam-5770	3	10	arts	art	NOUN
ejpam-5770	3	11	and	and	CCONJ
ejpam-5770	3	12	sciences	science	NOUN
ejpam-5770	3	13	,	,	PUNCT
ejpam-5770	3	14	msu	msu	PROPN
ejpam-5770	3	15	tawi	tawi	PROPN
ejpam-5770	3	16	-	-	PUNCT
ejpam-5770	3	17	tawi	tawi	PROPN
ejpam-5770	3	18	college	college	PROPN
ejpam-5770	3	19	of	of	ADP
ejpam-5770	3	20	technology	technology	NOUN
ejpam-5770	3	21	and	and	CCONJ
ejpam-5770	3	22	oceanography	oceanography	NOUN
ejpam-5770	3	23	,	,	PUNCT
ejpam-5770	3	24	7500	7500	NUM
ejpam-5770	3	25	tawi	tawi	NOUN
ejpam-5770	3	26	-	-	PUNCT
ejpam-5770	3	27	tawi	tawi	NOUN
ejpam-5770	3	28	,	,	PUNCT
ejpam-5770	3	29	philippines	philippines	PROPN
ejpam-5770	3	30	2	2	NUM
ejpam-5770	3	31	department	department	NOUN
ejpam-5770	3	32	of	of	ADP
ejpam-5770	3	33	mathematics	mathematic	NOUN
ejpam-5770	3	34	and	and	CCONJ
ejpam-5770	3	35	statistics	statistic	NOUN
ejpam-5770	3	36	,	,	PUNCT
ejpam-5770	3	37	college	college	NOUN
ejpam-5770	3	38	of	of	ADP
ejpam-5770	3	39	science	science	NOUN
ejpam-5770	3	40	and	and	CCONJ
ejpam-5770	3	41	mathematics	mathematic	NOUN
ejpam-5770	3	42	,	,	PUNCT
ejpam-5770	3	43	msu	msu	PROPN
ejpam-5770	3	44	iligan	iligan	PROPN
ejpam-5770	3	45	institute	institute	PROPN
ejpam-5770	3	46	of	of	ADP
ejpam-5770	3	47	technology	technology	PROPN
ejpam-5770	3	48	,	,	PUNCT
ejpam-5770	3	49	9200	9200	NUM
ejpam-5770	3	50	iligan	iligan	ADJ
ejpam-5770	3	51	city	city	NOUN
ejpam-5770	3	52	,	,	PUNCT
ejpam-5770	3	53	philippines	philippines	PROPN
ejpam-5770	3	54	3	3	NUM
ejpam-5770	3	55	department	department	NOUN
ejpam-5770	3	56	of	of	ADP
ejpam-5770	3	57	mathematics	mathematics	PROPN
ejpam-5770	3	58	,	,	PUNCT
ejpam-5770	3	59	azarbaijan	azarbaijan	NOUN
ejpam-5770	3	60	shahid	shahid	PROPN
ejpam-5770	3	61	madani	madani	PROPN
ejpam-5770	3	62	university	university	PROPN
ejpam-5770	3	63	,	,	PUNCT
ejpam-5770	3	64	tabriz	tabriz	NOUN
ejpam-5770	3	65	,	,	PUNCT
ejpam-5770	3	66	iran	iran	PROPN
ejpam-5770	3	67	4	4	NUM
ejpam-5770	3	68	center	center	NOUN
ejpam-5770	3	69	for	for	ADP
ejpam-5770	3	70	mathematical	mathematical	ADJ
ejpam-5770	3	71	and	and	CCONJ
ejpam-5770	3	72	theoretical	theoretical	ADJ
ejpam-5770	3	73	physical	physical	ADJ
ejpam-5770	3	74	sciences	science	NOUN
ejpam-5770	3	75	,	,	PUNCT
ejpam-5770	3	76	premier	premier	PROPN
ejpam-5770	3	77	research	research	PROPN
ejpam-5770	3	78	institute	institute	PROPN
ejpam-5770	3	79	of	of	ADP
ejpam-5770	3	80	science	science	NOUN
ejpam-5770	3	81	and	and	CCONJ
ejpam-5770	3	82	mathematics	mathematics	PROPN
ejpam-5770	3	83	(	(	PUNCT
ejpam-5770	3	84	prism	prism	NOUN
ejpam-5770	3	85	)	)	PUNCT
ejpam-5770	3	86	,	,	PUNCT
ejpam-5770	3	87	msu	msu	PROPN
ejpam-5770	3	88	iligan	iligan	PROPN
ejpam-5770	3	89	institute	institute	PROPN
ejpam-5770	3	90	of	of	ADP
ejpam-5770	3	91	technology	technology	PROPN
ejpam-5770	3	92	,	,	PUNCT
ejpam-5770	3	93	9200	9200	NUM
ejpam-5770	3	94	iligan	iligan	ADJ
ejpam-5770	3	95	city	city	NOUN
ejpam-5770	3	96	,	,	PUNCT
ejpam-5770	3	97	philippines	philippine	NOUN
ejpam-5770	3	98	abstract	abstract	ADJ
ejpam-5770	3	99	.	.	PUNCT
ejpam-5770	4	1	let	let	VERB
ejpam-5770	4	2	g	g	PRON
ejpam-5770	4	3	be	be	AUX
ejpam-5770	4	4	a	a	DET
ejpam-5770	4	5	simple	simple	ADJ
ejpam-5770	4	6	and	and	CCONJ
ejpam-5770	4	7	connected	connected	ADJ
ejpam-5770	4	8	graph	graph	NOUN
ejpam-5770	4	9	,	,	PUNCT
ejpam-5770	4	10	and	and	CCONJ
ejpam-5770	4	11	let	let	VERB
ejpam-5770	4	12	f	f	PRON
ejpam-5770	4	13	be	be	AUX
ejpam-5770	4	14	a	a	DET
ejpam-5770	4	15	function	function	NOUN
ejpam-5770	4	16	that	that	PRON
ejpam-5770	4	17	assigns	assign	VERB
ejpam-5770	4	18	to	to	ADP
ejpam-5770	4	19	each	each	DET
ejpam-5770	4	20	vertex	vertex	NOUN
ejpam-5770	4	21	a	a	DET
ejpam-5770	4	22	set	set	NOUN
ejpam-5770	4	23	of	of	ADP
ejpam-5770	4	24	colors	color	NOUN
ejpam-5770	4	25	chosen	choose	VERB
ejpam-5770	4	26	from	from	ADP
ejpam-5770	4	27	the	the	DET
ejpam-5770	4	28	set	set	NOUN
ejpam-5770	4	29	{	{	PUNCT
ejpam-5770	4	30	1	1	NUM
ejpam-5770	4	31	,	,	PUNCT
ejpam-5770	4	32	2	2	NUM
ejpam-5770	4	33	,	,	PUNCT
ejpam-5770	4	34	3	3	NUM
ejpam-5770	4	35	,	,	PUNCT
ejpam-5770	4	36	.	.	PUNCT
ejpam-5770	4	37	.	.	PUNCT
ejpam-5770	5	1	.	.	PUNCT
ejpam-5770	6	1	,	,	PUNCT
ejpam-5770	6	2	k	k	X
ejpam-5770	6	3	}	}	PUNCT
ejpam-5770	6	4	,	,	PUNCT
ejpam-5770	6	5	i.e.	i.e.	X
ejpam-5770	6	6	,	,	PUNCT
ejpam-5770	6	7	f	f	X
ejpam-5770	6	8	:	:	PUNCT
ejpam-5770	6	9	v	v	X
ejpam-5770	6	10	(	(	PUNCT
ejpam-5770	6	11	g	g	NOUN
ejpam-5770	6	12	)	)	PUNCT
ejpam-5770	6	13	→	→	SYM
ejpam-5770	6	14	p({1	p({1	PROPN
ejpam-5770	6	15	,	,	PUNCT
ejpam-5770	6	16	2	2	NUM
ejpam-5770	6	17	,	,	PUNCT
ejpam-5770	6	18	3	3	NUM
ejpam-5770	6	19	,	,	PUNCT
ejpam-5770	6	20	.	.	PUNCT
ejpam-5770	6	21	.	.	PUNCT
ejpam-5770	7	1	.	.	PUNCT
ejpam-5770	8	1	,	,	PUNCT
ejpam-5770	8	2	k	k	X
ejpam-5770	8	3	}	}	PUNCT
ejpam-5770	8	4	)	)	PUNCT
ejpam-5770	8	5	.	.	PUNCT
ejpam-5770	9	1	if	if	SCONJ
ejpam-5770	9	2	for	for	ADP
ejpam-5770	9	3	each	each	DET
ejpam-5770	9	4	vertex	vertex	NOUN
ejpam-5770	9	5	v	v	ADP
ejpam-5770	9	6	∈	∈	PROPN
ejpam-5770	9	7	v	v	NOUN
ejpam-5770	9	8	(	(	PUNCT
ejpam-5770	9	9	g	g	NOUN
ejpam-5770	9	10	)	)	PUNCT
ejpam-5770	9	11	such	such	ADJ
ejpam-5770	9	12	that	that	SCONJ
ejpam-5770	9	13	f(v	f(v	NOUN
ejpam-5770	9	14	)	)	PUNCT
ejpam-5770	9	15	=	=	SYM
ejpam-5770	9	16	∅	∅	NOUN
ejpam-5770	9	17	,	,	PUNCT
ejpam-5770	9	18	we	we	PRON
ejpam-5770	9	19	have	have	VERB
ejpam-5770	9	20	⋃	⋃	PROPN
ejpam-5770	9	21	u∈ng(v	u∈ng(v	PROPN
ejpam-5770	9	22	)	)	PUNCT
ejpam-5770	9	23	f(u	f(u	PROPN
ejpam-5770	9	24	)	)	PUNCT
ejpam-5770	10	1	=	=	PRON
ejpam-5770	10	2	{	{	PUNCT
ejpam-5770	10	3	1	1	NUM
ejpam-5770	10	4	,	,	PUNCT
ejpam-5770	10	5	2	2	NUM
ejpam-5770	10	6	,	,	PUNCT
ejpam-5770	10	7	3	3	NUM
ejpam-5770	10	8	,	,	PUNCT
ejpam-5770	10	9	.	.	PUNCT
ejpam-5770	10	10	.	.	PUNCT
ejpam-5770	11	1	.	.	PUNCT
ejpam-5770	12	1	,	,	PUNCT
ejpam-5770	12	2	k	k	X
ejpam-5770	12	3	}	}	PUNCT
ejpam-5770	12	4	,	,	PUNCT
ejpam-5770	12	5	then	then	ADV
ejpam-5770	12	6	f	f	PROPN
ejpam-5770	12	7	is	be	AUX
ejpam-5770	12	8	called	call	VERB
ejpam-5770	12	9	a	a	DET
ejpam-5770	12	10	k	k	ADJ
ejpam-5770	12	11	-	-	PUNCT
ejpam-5770	12	12	rainbow	rainbow	NOUN
ejpam-5770	12	13	dominating	dominating	NOUN
ejpam-5770	12	14	function	function	NOUN
ejpam-5770	12	15	(	(	PUNCT
ejpam-5770	12	16	krdf	krdf	PROPN
ejpam-5770	12	17	)	)	PUNCT
ejpam-5770	12	18	of	of	ADP
ejpam-5770	12	19	g.	g.	PROPN
ejpam-5770	12	20	a	a	DET
ejpam-5770	12	21	krdf	krdf	NOUN
ejpam-5770	13	1	f	f	X
ejpam-5770	13	2	:	:	PUNCT
ejpam-5770	13	3	v	v	X
ejpam-5770	13	4	(	(	PUNCT
ejpam-5770	13	5	g	g	NOUN
ejpam-5770	13	6	)	)	PUNCT
ejpam-5770	13	7	→	→	SYM
ejpam-5770	13	8	p({1	p({1	PROPN
ejpam-5770	13	9	,	,	PUNCT
ejpam-5770	13	10	2	2	NUM
ejpam-5770	13	11	,	,	PUNCT
ejpam-5770	13	12	.	.	PUNCT
ejpam-5770	13	13	.	.	PUNCT
ejpam-5770	13	14	.	.	PUNCT
ejpam-5770	14	1	,	,	PUNCT
ejpam-5770	14	2	k	k	X
ejpam-5770	14	3	}	}	PUNCT
ejpam-5770	14	4	)	)	PUNCT
ejpam-5770	14	5	is	be	AUX
ejpam-5770	14	6	said	say	VERB
ejpam-5770	14	7	to	to	PART
ejpam-5770	14	8	be	be	AUX
ejpam-5770	14	9	a	a	DET
ejpam-5770	14	10	weakly	weakly	ADV
ejpam-5770	14	11	connected	connected	ADJ
ejpam-5770	14	12	k	k	ADJ
ejpam-5770	14	13	-	-	PUNCT
ejpam-5770	14	14	rainbow	rainbow	NOUN
ejpam-5770	14	15	dominating	dominating	NOUN
ejpam-5770	14	16	function	function	NOUN
ejpam-5770	14	17	(	(	PUNCT
ejpam-5770	14	18	wckrdf	wckrdf	PROPN
ejpam-5770	14	19	)	)	PUNCT
ejpam-5770	14	20	if	if	SCONJ
ejpam-5770	14	21	the	the	DET
ejpam-5770	14	22	set	set	NOUN
ejpam-5770	14	23	s	s	AUX
ejpam-5770	14	24	=	=	PUNCT
ejpam-5770	14	25	{	{	PUNCT
ejpam-5770	14	26	v	v	NUM
ejpam-5770	14	27	∈	∈	NOUN
ejpam-5770	14	28	v	v	NOUN
ejpam-5770	14	29	(	(	PUNCT
ejpam-5770	14	30	g	g	NOUN
ejpam-5770	14	31	)	)	PUNCT
ejpam-5770	14	32	:	:	PUNCT
ejpam-5770	14	33	f(v	f(v	NOUN
ejpam-5770	14	34	)	)	PUNCT
ejpam-5770	14	35	̸=	̸=	NOUN
ejpam-5770	14	36	∅	∅	NOUN
ejpam-5770	14	37	}	}	PUNCT
ejpam-5770	14	38	is	be	AUX
ejpam-5770	14	39	weakly	weakly	ADV
ejpam-5770	14	40	connected	connected	ADJ
ejpam-5770	14	41	dominating	dominating	NOUN
ejpam-5770	14	42	.	.	PUNCT
ejpam-5770	15	1	the	the	DET
ejpam-5770	15	2	weight	weight	PROPN
ejpam-5770	15	3	w(f	w(f	PROPN
ejpam-5770	15	4	)	)	PUNCT
ejpam-5770	15	5	of	of	ADP
ejpam-5770	15	6	f	f	PROPN
ejpam-5770	15	7	is	be	AUX
ejpam-5770	15	8	defined	define	VERB
ejpam-5770	15	9	as	as	ADP
ejpam-5770	15	10	ω(f	ω(f	ADJ
ejpam-5770	15	11	)	)	PUNCT
ejpam-5770	16	1	=	=	SYM
ejpam-5770	16	2	∑	∑	PUNCT
ejpam-5770	16	3	v∈v	v∈v	NOUN
ejpam-5770	16	4	(	(	PUNCT
ejpam-5770	16	5	g)|f(v)|	g)|f(v)|	PROPN
ejpam-5770	16	6	.	.	PUNCT
ejpam-5770	17	1	the	the	DET
ejpam-5770	17	2	weakly	weakly	ADV
ejpam-5770	17	3	connected	connected	ADJ
ejpam-5770	17	4	k	k	ADJ
ejpam-5770	17	5	-	-	PUNCT
ejpam-5770	17	6	rainbow	rainbow	NOUN
ejpam-5770	17	7	domination	domination	NOUN
ejpam-5770	17	8	number	number	NOUN
ejpam-5770	17	9	of	of	ADP
ejpam-5770	17	10	g	g	NOUN
ejpam-5770	17	11	,	,	PUNCT
ejpam-5770	17	12	denoted	denote	VERB
ejpam-5770	17	13	by	by	ADP
ejpam-5770	17	14	γwc	γwc	ADJ
ejpam-5770	17	15	rk	rk	PROPN
ejpam-5770	17	16	(	(	PUNCT
ejpam-5770	17	17	g	g	NOUN
ejpam-5770	17	18	)	)	PUNCT
ejpam-5770	17	19	is	be	AUX
ejpam-5770	17	20	the	the	DET
ejpam-5770	17	21	minimum	minimum	ADJ
ejpam-5770	17	22	weight	weight	NOUN
ejpam-5770	17	23	of	of	ADP
ejpam-5770	17	24	wckrdf	wckrdf	NOUN
ejpam-5770	17	25	.	.	PUNCT
ejpam-5770	18	1	a	a	DET
ejpam-5770	18	2	weakly	weakly	ADV
ejpam-5770	18	3	connected	connected	ADJ
ejpam-5770	18	4	k	k	ADJ
ejpam-5770	18	5	-	-	PUNCT
ejpam-5770	18	6	rainbow	rainbow	NOUN
ejpam-5770	18	7	dominating	dominating	NOUN
ejpam-5770	18	8	function	function	NOUN
ejpam-5770	18	9	of	of	ADP
ejpam-5770	18	10	g	g	NOUN
ejpam-5770	18	11	with	with	ADP
ejpam-5770	18	12	weight	weight	NOUN
ejpam-5770	18	13	γwc	γwc	PROPN
ejpam-5770	18	14	rk	rk	PROPN
ejpam-5770	18	15	(	(	PUNCT
ejpam-5770	18	16	g	g	NOUN
ejpam-5770	18	17	)	)	PUNCT
ejpam-5770	18	18	,	,	PUNCT
ejpam-5770	18	19	i.e.	i.e.	X
ejpam-5770	18	20	,	,	PUNCT
ejpam-5770	18	21	ω(f	ω(f	NUM
ejpam-5770	18	22	)	)	PUNCT
ejpam-5770	18	23	=	=	SYM
ejpam-5770	18	24	γwc	γwc	ADJ
ejpam-5770	18	25	rk	rk	PROPN
ejpam-5770	18	26	(	(	PUNCT
ejpam-5770	18	27	g	g	NOUN
ejpam-5770	18	28	)	)	PUNCT
ejpam-5770	18	29	is	be	AUX
ejpam-5770	18	30	referred	refer	VERB
ejpam-5770	18	31	to	to	ADP
ejpam-5770	18	32	as	as	ADP
ejpam-5770	18	33	a	a	DET
ejpam-5770	18	34	γwc	γwc	ADJ
ejpam-5770	18	35	rk	rk	NOUN
ejpam-5770	18	36	-function	-function	NOUN
ejpam-5770	18	37	of	of	ADP
ejpam-5770	18	38	g.	g.	NOUN
ejpam-5770	18	39	in	in	ADP
ejpam-5770	18	40	this	this	DET
ejpam-5770	18	41	paper	paper	NOUN
ejpam-5770	18	42	,	,	PUNCT
ejpam-5770	18	43	we	we	PRON
ejpam-5770	18	44	initiate	initiate	VERB
ejpam-5770	18	45	the	the	DET
ejpam-5770	18	46	study	study	NOUN
ejpam-5770	18	47	of	of	ADP
ejpam-5770	18	48	the	the	DET
ejpam-5770	18	49	weakly	weakly	ADJ
ejpam-5770	18	50	connected	connected	ADJ
ejpam-5770	18	51	k	k	ADJ
ejpam-5770	18	52	-	-	PUNCT
ejpam-5770	18	53	rainbow	rainbow	NOUN
ejpam-5770	18	54	domination	domination	NOUN
ejpam-5770	18	55	parameter	parameter	NOUN
ejpam-5770	18	56	.	.	PUNCT
ejpam-5770	19	1	first	first	ADV
ejpam-5770	19	2	,	,	PUNCT
ejpam-5770	19	3	we	we	PRON
ejpam-5770	19	4	establish	establish	VERB
ejpam-5770	19	5	fundamental	fundamental	ADJ
ejpam-5770	19	6	properties	property	NOUN
ejpam-5770	19	7	and	and	CCONJ
ejpam-5770	19	8	bounds	bound	NOUN
ejpam-5770	19	9	for	for	ADP
ejpam-5770	19	10	weakly	weakly	ADV
ejpam-5770	19	11	connected	connected	ADJ
ejpam-5770	19	12	k	k	ADJ
ejpam-5770	19	13	-	-	PUNCT
ejpam-5770	19	14	rainbow	rainbow	NOUN
ejpam-5770	19	15	domination	domination	NOUN
ejpam-5770	19	16	.	.	PUNCT
ejpam-5770	20	1	then	then	ADV
ejpam-5770	20	2	,	,	PUNCT
ejpam-5770	20	3	we	we	PRON
ejpam-5770	20	4	determine	determine	VERB
ejpam-5770	20	5	the	the	DET
ejpam-5770	20	6	weakly	weakly	ADJ
ejpam-5770	20	7	connected	connected	ADJ
ejpam-5770	20	8	k	k	ADJ
ejpam-5770	20	9	-	-	PUNCT
ejpam-5770	20	10	rainbow	rainbow	NOUN
ejpam-5770	20	11	domination	domination	NOUN
ejpam-5770	20	12	number	number	NOUN
ejpam-5770	20	13	for	for	ADP
ejpam-5770	20	14	various	various	ADJ
ejpam-5770	20	15	classes	class	NOUN
ejpam-5770	20	16	of	of	ADP
ejpam-5770	20	17	graphs	graph	NOUN
ejpam-5770	20	18	.	.	PUNCT
ejpam-5770	21	1	furthermore	furthermore	ADV
ejpam-5770	21	2	,	,	PUNCT
ejpam-5770	21	3	we	we	PRON
ejpam-5770	21	4	characterize	characterize	VERB
ejpam-5770	21	5	the	the	DET
ejpam-5770	21	6	weakly	weakly	ADV
ejpam-5770	21	7	connected	connected	ADJ
ejpam-5770	21	8	k	k	ADJ
ejpam-5770	21	9	-	-	PUNCT
ejpam-5770	21	10	rainbow	rainbow	NOUN
ejpam-5770	21	11	dominating	dominating	NOUN
ejpam-5770	21	12	function	function	NOUN
ejpam-5770	21	13	under	under	ADP
ejpam-5770	21	14	the	the	DET
ejpam-5770	21	15	join	join	NOUN
ejpam-5770	21	16	of	of	ADP
ejpam-5770	21	17	graphs	graph	NOUN
ejpam-5770	21	18	and	and	CCONJ
ejpam-5770	21	19	determine	determine	VERB
ejpam-5770	21	20	the	the	DET
ejpam-5770	21	21	weakly	weakly	ADJ
ejpam-5770	21	22	connected	connected	ADJ
ejpam-5770	21	23	k	k	ADJ
ejpam-5770	21	24	-	-	PUNCT
ejpam-5770	21	25	rainbow	rainbow	NOUN
ejpam-5770	21	26	domination	domination	NOUN
ejpam-5770	21	27	number	number	NOUN
ejpam-5770	21	28	for	for	ADP
ejpam-5770	21	29	this	this	DET
ejpam-5770	21	30	binary	binary	ADJ
ejpam-5770	21	31	operation	operation	NOUN
ejpam-5770	21	32	.	.	PUNCT
ejpam-5770	22	1	2020	2020	NUM
ejpam-5770	22	2	mathematics	mathematic	NOUN
ejpam-5770	22	3	subject	subject	NOUN
ejpam-5770	22	4	classifications	classification	NOUN
ejpam-5770	22	5	:	:	PUNCT
ejpam-5770	22	6	05c69	05c69	X
ejpam-5770	22	7	key	key	ADJ
ejpam-5770	22	8	words	word	NOUN
ejpam-5770	22	9	and	and	CCONJ
ejpam-5770	22	10	phrases	phrase	NOUN
ejpam-5770	22	11	:	:	PUNCT
ejpam-5770	22	12	weakly	weakly	ADJ
ejpam-5770	22	13	connected	connected	ADJ
ejpam-5770	22	14	dominating	dominating	NOUN
ejpam-5770	22	15	set	set	NOUN
ejpam-5770	22	16	,	,	PUNCT
ejpam-5770	22	17	k	k	ADJ
ejpam-5770	22	18	-	-	PUNCT
ejpam-5770	22	19	rainbow	rainbow	NOUN
ejpam-5770	22	20	dominating	dominating	NOUN
ejpam-5770	22	21	function	function	NOUN
ejpam-5770	22	22	,	,	PUNCT
ejpam-5770	22	23	weakly	weakly	ADV
ejpam-5770	22	24	connected	connected	ADJ
ejpam-5770	22	25	k	k	ADJ
ejpam-5770	22	26	-	-	PUNCT
ejpam-5770	22	27	rainbow	rainbow	NOUN
ejpam-5770	22	28	dominating	dominating	NOUN
ejpam-5770	22	29	function	function	NOUN
ejpam-5770	22	30	∗corresponding	∗corresponde	VERB
ejpam-5770	22	31	author	author	NOUN
ejpam-5770	22	32	.	.	PUNCT
ejpam-5770	23	1	doi	doi	NOUN
ejpam-5770	23	2	:	:	PUNCT
ejpam-5770	23	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5770	https://doi.org/10.29020/nybg.ejpam.v18i2.5770	NOUN
ejpam-5770	23	4	email	email	NOUN
ejpam-5770	23	5	addresses	address	NOUN
ejpam-5770	23	6	:	:	PUNCT
ejpam-5770	23	7	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-5770	23	8	(	(	PUNCT
ejpam-5770	23	9	j.	j.	PROPN
ejpam-5770	23	10	j.	j.	PROPN
ejpam-5770	23	11	hamja	hamja	PROPN
ejpam-5770	23	12	)	)	PUNCT
ejpam-5770	23	13	,	,	PUNCT
ejpam-5770	23	14	s.m.sheikholeslami@azaruniv.ac.ir	s.m.sheikholeslami@azaruniv.ac.ir	PUNCT
ejpam-5770	23	15	(	(	PUNCT
ejpam-5770	23	16	s.	s.	PROPN
ejpam-5770	23	17	m.	m.	PROPN
ejpam-5770	23	18	sheikholeslami	sheikholeslami	PROPN
ejpam-5770	23	19	)	)	PUNCT
ejpam-5770	23	20	,	,	PUNCT
ejpam-5770	23	21	imelda.aniversario@g.msuiit.edu.ph	imelda.aniversario@g.msuiit.edu.ph	PROPN
ejpam-5770	23	22	(	(	PUNCT
ejpam-5770	23	23	i.	i.	PROPN
ejpam-5770	23	24	s.	s.	PROPN
ejpam-5770	23	25	aniversario	aniversario	PROPN
ejpam-5770	23	26	)	)	PUNCT
ejpam-5770	23	27	,	,	PUNCT
ejpam-5770	23	28	lysterrey.cabardo@g.msuiit.edu.ph	lysterrey.cabardo@g.msuiit.edu.ph	PROPN
ejpam-5770	23	29	(	(	PUNCT
ejpam-5770	23	30	l.	l.	PROPN
ejpam-5770	23	31	b.	b.	PROPN
ejpam-5770	23	32	cabardo	cabardo	PROPN
ejpam-5770	23	33	)	)	PUNCT
ejpam-5770	23	34	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5770	24	1	1	1	NUM
ejpam-5770	24	2	copyright	copyright	NOUN
ejpam-5770	24	3	:	:	PUNCT
ejpam-5770	24	4	©	©	PROPN
ejpam-5770	24	5	2025	2025	NUM
ejpam-5770	24	6	the	the	DET
ejpam-5770	24	7	author(s	author(s	NOUN
ejpam-5770	24	8	)	)	PUNCT
ejpam-5770	24	9	.	.	PUNCT
ejpam-5770	25	1	(	(	PUNCT
ejpam-5770	25	2	cc	cc	NOUN
ejpam-5770	25	3	by	by	ADP
ejpam-5770	25	4	-	-	PUNCT
ejpam-5770	25	5	nc	nc	PROPN
ejpam-5770	25	6	4.0	4.0	NUM
ejpam-5770	25	7	)	)	PUNCT
ejpam-5770	25	8	j.	j.	PROPN
ejpam-5770	25	9	j.	j.	PROPN
ejpam-5770	25	10	hamja	hamja	PROPN
ejpam-5770	26	1	et	et	PROPN
ejpam-5770	26	2	al	al	PROPN
ejpam-5770	26	3	.	.	PUNCT
ejpam-5770	26	4	/	/	SYM
ejpam-5770	26	5	eur	eur	PROPN
ejpam-5770	26	6	.	.	PUNCT
ejpam-5770	27	1	j.	j.	PROPN
ejpam-5770	27	2	pure	pure	PROPN
ejpam-5770	27	3	appl	appl	PROPN
ejpam-5770	27	4	.	.	PROPN
ejpam-5770	27	5	math	math	PROPN
ejpam-5770	27	6	,	,	PUNCT
ejpam-5770	27	7	18	18	NUM
ejpam-5770	27	8	(	(	PUNCT
ejpam-5770	27	9	2	2	NUM
ejpam-5770	27	10	)	)	PUNCT
ejpam-5770	27	11	(	(	PUNCT
ejpam-5770	27	12	2025	2025	NUM
ejpam-5770	27	13	)	)	PUNCT
ejpam-5770	27	14	,	,	PUNCT
ejpam-5770	27	15	5770	5770	NUM
ejpam-5770	27	16	2	2	NUM
ejpam-5770	27	17	of	of	ADP
ejpam-5770	27	18	11	11	NUM
ejpam-5770	27	19	1	1	NUM
ejpam-5770	27	20	.	.	PUNCT
ejpam-5770	28	1	introduction	introduction	NOUN
ejpam-5770	28	2	graph	graph	NOUN
ejpam-5770	28	3	domination	domination	NOUN
ejpam-5770	28	4	provides	provide	VERB
ejpam-5770	28	5	a	a	DET
ejpam-5770	28	6	foundational	foundational	ADJ
ejpam-5770	28	7	framework	framework	NOUN
ejpam-5770	28	8	for	for	ADP
ejpam-5770	28	9	various	various	ADJ
ejpam-5770	28	10	applications	application	NOUN
ejpam-5770	28	11	,	,	PUNCT
ejpam-5770	28	12	where	where	SCONJ
ejpam-5770	28	13	the	the	DET
ejpam-5770	28	14	vertices	vertex	NOUN
ejpam-5770	28	15	in	in	ADP
ejpam-5770	28	16	a	a	DET
ejpam-5770	28	17	dominating	dominating	NOUN
ejpam-5770	28	18	set	set	NOUN
ejpam-5770	28	19	represent	represent	VERB
ejpam-5770	28	20	service	service	NOUN
ejpam-5770	28	21	providers	provider	NOUN
ejpam-5770	28	22	or	or	CCONJ
ejpam-5770	28	23	essential	essential	ADJ
ejpam-5770	28	24	resources	resource	NOUN
ejpam-5770	28	25	that	that	PRON
ejpam-5770	28	26	ensure	ensure	VERB
ejpam-5770	28	27	accessibility	accessibility	NOUN
ejpam-5770	28	28	to	to	ADP
ejpam-5770	28	29	every	every	DET
ejpam-5770	28	30	vertex	vertex	NOUN
ejpam-5770	28	31	in	in	ADP
ejpam-5770	28	32	the	the	DET
ejpam-5770	28	33	network	network	NOUN
ejpam-5770	28	34	,	,	PUNCT
ejpam-5770	28	35	as	as	SCONJ
ejpam-5770	28	36	described	describe	VERB
ejpam-5770	28	37	by	by	ADP
ejpam-5770	28	38	t.	t.	PROPN
ejpam-5770	28	39	w.	w.	PROPN
ejpam-5770	28	40	haynes	haynes	PROPN
ejpam-5770	28	41	et	et	PROPN
ejpam-5770	28	42	al	al	PROPN
ejpam-5770	28	43	.	.	PUNCT
ejpam-5770	29	1	[	[	X
ejpam-5770	29	2	1	1	NUM
ejpam-5770	29	3	]	]	PUNCT
ejpam-5770	29	4	.	.	PUNCT
ejpam-5770	30	1	in	in	ADP
ejpam-5770	30	2	1987	1987	NUM
ejpam-5770	30	3	,	,	PUNCT
ejpam-5770	30	4	hedetniemi	hedetniemi	ADP
ejpam-5770	31	1	[	[	X
ejpam-5770	31	2	2	2	X
ejpam-5770	31	3	]	]	PUNCT
ejpam-5770	31	4	introduced	introduce	VERB
ejpam-5770	31	5	the	the	DET
ejpam-5770	31	6	concept	concept	NOUN
ejpam-5770	31	7	of	of	ADP
ejpam-5770	31	8	dominating	dominating	NOUN
ejpam-5770	31	9	functions	function	NOUN
ejpam-5770	31	10	,	,	PUNCT
ejpam-5770	31	11	offering	offer	VERB
ejpam-5770	31	12	an	an	DET
ejpam-5770	31	13	analytical	analytical	ADJ
ejpam-5770	31	14	approach	approach	NOUN
ejpam-5770	31	15	to	to	ADP
ejpam-5770	31	16	studying	study	VERB
ejpam-5770	31	17	this	this	DET
ejpam-5770	31	18	discrete	discrete	ADJ
ejpam-5770	31	19	structure	structure	NOUN
ejpam-5770	31	20	.	.	PUNCT
ejpam-5770	32	1	this	this	DET
ejpam-5770	32	2	framework	framework	NOUN
ejpam-5770	32	3	established	establish	VERB
ejpam-5770	32	4	connections	connection	NOUN
ejpam-5770	32	5	between	between	ADP
ejpam-5770	32	6	domination	domination	NOUN
ejpam-5770	32	7	,	,	PUNCT
ejpam-5770	32	8	graph	graph	NOUN
ejpam-5770	32	9	labelings	labeling	NOUN
ejpam-5770	32	10	,	,	PUNCT
ejpam-5770	32	11	and	and	CCONJ
ejpam-5770	32	12	colorings	coloring	NOUN
ejpam-5770	32	13	,	,	PUNCT
ejpam-5770	32	14	leading	lead	VERB
ejpam-5770	32	15	to	to	ADP
ejpam-5770	32	16	the	the	DET
ejpam-5770	32	17	development	development	NOUN
ejpam-5770	32	18	of	of	ADP
ejpam-5770	32	19	new	new	ADJ
ejpam-5770	32	20	domination	domination	NOUN
ejpam-5770	32	21	function	function	NOUN
ejpam-5770	32	22	parameters	parameter	NOUN
ejpam-5770	32	23	.	.	PUNCT
ejpam-5770	33	1	two	two	NUM
ejpam-5770	33	2	decades	decade	NOUN
ejpam-5770	33	3	later	later	ADV
ejpam-5770	33	4	,	,	PUNCT
ejpam-5770	33	5	in	in	ADP
ejpam-5770	33	6	2008	2008	NUM
ejpam-5770	33	7	,	,	PUNCT
ejpam-5770	33	8	brešar	brešar	VERB
ejpam-5770	33	9	et	et	PROPN
ejpam-5770	33	10	al	al	PROPN
ejpam-5770	33	11	.	.	PUNCT
ejpam-5770	34	1	[	[	X
ejpam-5770	34	2	3	3	X
ejpam-5770	34	3	]	]	PUNCT
ejpam-5770	34	4	introduced	introduce	VERB
ejpam-5770	34	5	the	the	DET
ejpam-5770	34	6	concept	concept	NOUN
ejpam-5770	34	7	of	of	ADP
ejpam-5770	34	8	rainbow	rainbow	NOUN
ejpam-5770	34	9	domination	domination	NOUN
ejpam-5770	34	10	,	,	PUNCT
ejpam-5770	34	11	which	which	PRON
ejpam-5770	34	12	extends	extend	VERB
ejpam-5770	34	13	the	the	DET
ejpam-5770	34	14	notion	notion	NOUN
ejpam-5770	34	15	of	of	ADP
ejpam-5770	34	16	domination	domination	NOUN
ejpam-5770	34	17	by	by	ADP
ejpam-5770	34	18	incorporating	incorporate	VERB
ejpam-5770	34	19	multiple	multiple	ADJ
ejpam-5770	34	20	service	service	NOUN
ejpam-5770	34	21	types	type	NOUN
ejpam-5770	34	22	,	,	PUNCT
ejpam-5770	34	23	each	each	PRON
ejpam-5770	34	24	represented	represent	VERB
ejpam-5770	34	25	by	by	ADP
ejpam-5770	34	26	a	a	DET
ejpam-5770	34	27	distinct	distinct	ADJ
ejpam-5770	34	28	color	color	NOUN
ejpam-5770	34	29	.	.	PUNCT
ejpam-5770	35	1	the	the	DET
ejpam-5770	35	2	goal	goal	NOUN
ejpam-5770	35	3	of	of	ADP
ejpam-5770	35	4	rainbow	rainbow	NOUN
ejpam-5770	35	5	domination	domination	NOUN
ejpam-5770	35	6	is	be	AUX
ejpam-5770	35	7	to	to	PART
ejpam-5770	35	8	assign	assign	VERB
ejpam-5770	35	9	services	service	NOUN
ejpam-5770	35	10	so	so	SCONJ
ejpam-5770	35	11	that	that	SCONJ
ejpam-5770	35	12	any	any	DET
ejpam-5770	35	13	vertex	vertex	NOUN
ejpam-5770	35	14	not	not	PART
ejpam-5770	35	15	directly	directly	ADV
ejpam-5770	35	16	receiving	receive	VERB
ejpam-5770	35	17	a	a	DET
ejpam-5770	35	18	service	service	NOUN
ejpam-5770	35	19	has	have	VERB
ejpam-5770	35	20	access	access	NOUN
ejpam-5770	35	21	to	to	ADP
ejpam-5770	35	22	all	all	DET
ejpam-5770	35	23	service	service	NOUN
ejpam-5770	35	24	types	type	NOUN
ejpam-5770	35	25	within	within	ADP
ejpam-5770	35	26	its	its	PRON
ejpam-5770	35	27	neighborhood	neighborhood	NOUN
ejpam-5770	35	28	.	.	PUNCT
ejpam-5770	36	1	in	in	ADP
ejpam-5770	36	2	the	the	DET
ejpam-5770	36	3	following	following	ADJ
ejpam-5770	36	4	years	year	NOUN
ejpam-5770	36	5	,	,	PUNCT
ejpam-5770	36	6	this	this	DET
ejpam-5770	36	7	concept	concept	NOUN
ejpam-5770	36	8	gained	gain	VERB
ejpam-5770	36	9	significant	significant	ADJ
ejpam-5770	36	10	attention	attention	NOUN
ejpam-5770	36	11	from	from	ADP
ejpam-5770	36	12	researchers	researcher	NOUN
ejpam-5770	36	13	,	,	PUNCT
ejpam-5770	36	14	leading	lead	VERB
ejpam-5770	36	15	to	to	ADP
ejpam-5770	36	16	numerous	numerous	ADJ
ejpam-5770	36	17	studies	study	NOUN
ejpam-5770	36	18	exploring	explore	VERB
ejpam-5770	36	19	its	its	PRON
ejpam-5770	36	20	properties	property	NOUN
ejpam-5770	36	21	and	and	CCONJ
ejpam-5770	36	22	applications	application	NOUN
ejpam-5770	36	23	,	,	PUNCT
ejpam-5770	36	24	as	as	SCONJ
ejpam-5770	36	25	discussed	discuss	VERB
ejpam-5770	36	26	in	in	ADP
ejpam-5770	36	27	[	[	X
ejpam-5770	36	28	4–11	4–11	NOUN
ejpam-5770	36	29	]	]	X
ejpam-5770	36	30	.	.	PUNCT
ejpam-5770	37	1	in	in	ADP
ejpam-5770	37	2	1997	1997	NUM
ejpam-5770	37	3	,	,	PUNCT
ejpam-5770	37	4	j.	j.	PROPN
ejpam-5770	37	5	e.	e.	PROPN
ejpam-5770	37	6	dunbar	dunbar	PROPN
ejpam-5770	37	7	et	et	PROPN
ejpam-5770	37	8	al	al	PROPN
ejpam-5770	37	9	.	.	PUNCT
ejpam-5770	38	1	[	[	X
ejpam-5770	38	2	12	12	NUM
ejpam-5770	38	3	]	]	PUNCT
ejpam-5770	38	4	introduced	introduce	VERB
ejpam-5770	38	5	the	the	DET
ejpam-5770	38	6	concept	concept	NOUN
ejpam-5770	38	7	of	of	ADP
ejpam-5770	38	8	a	a	DET
ejpam-5770	38	9	weakly	weakly	ADV
ejpam-5770	38	10	connected	connected	ADJ
ejpam-5770	38	11	dominating	dominating	NOUN
ejpam-5770	38	12	set	set	VERB
ejpam-5770	38	13	in	in	ADP
ejpam-5770	38	14	a	a	DET
ejpam-5770	38	15	connected	connected	ADJ
ejpam-5770	38	16	graph	graph	NOUN
ejpam-5770	38	17	and	and	CCONJ
ejpam-5770	38	18	examined	examine	VERB
ejpam-5770	38	19	the	the	DET
ejpam-5770	38	20	weakly	weakly	ADJ
ejpam-5770	38	21	connected	connected	ADJ
ejpam-5770	38	22	domination	domination	NOUN
ejpam-5770	38	23	number	number	NOUN
ejpam-5770	38	24	along	along	ADP
ejpam-5770	38	25	with	with	ADP
ejpam-5770	38	26	related	related	ADJ
ejpam-5770	38	27	parameters	parameter	NOUN
ejpam-5770	38	28	.	.	PUNCT
ejpam-5770	39	1	further	further	ADJ
ejpam-5770	39	2	insights	insight	NOUN
ejpam-5770	39	3	into	into	ADP
ejpam-5770	39	4	weakly	weakly	ADJ
ejpam-5770	39	5	connected	connected	ADJ
ejpam-5770	39	6	domination	domination	NOUN
ejpam-5770	39	7	can	can	AUX
ejpam-5770	39	8	be	be	AUX
ejpam-5770	39	9	found	find	VERB
ejpam-5770	39	10	in	in	ADP
ejpam-5770	39	11	[	[	X
ejpam-5770	39	12	13–19	13–19	NUM
ejpam-5770	39	13	]	]	PUNCT
ejpam-5770	39	14	.	.	PUNCT
ejpam-5770	40	1	rainbow	rainbow	PROPN
ejpam-5770	40	2	domination	domination	PROPN
ejpam-5770	40	3	extends	extend	VERB
ejpam-5770	40	4	to	to	PART
ejpam-5770	40	5	weakly	weakly	ADV
ejpam-5770	40	6	connected	connected	ADJ
ejpam-5770	40	7	domination	domination	NOUN
ejpam-5770	40	8	by	by	ADP
ejpam-5770	40	9	ensuring	ensure	VERB
ejpam-5770	40	10	resource	resource	NOUN
ejpam-5770	40	11	distribution	distribution	NOUN
ejpam-5770	40	12	while	while	SCONJ
ejpam-5770	40	13	maintaining	maintain	VERB
ejpam-5770	40	14	a	a	DET
ejpam-5770	40	15	weakly	weakly	ADV
ejpam-5770	40	16	connected	connected	ADJ
ejpam-5770	40	17	dominating	dominating	NOUN
ejpam-5770	40	18	set	set	NOUN
ejpam-5770	40	19	.	.	PUNCT
ejpam-5770	41	1	this	this	DET
ejpam-5770	41	2	integration	integration	NOUN
ejpam-5770	41	3	is	be	AUX
ejpam-5770	41	4	crucial	crucial	ADJ
ejpam-5770	41	5	for	for	ADP
ejpam-5770	41	6	networks	network	NOUN
ejpam-5770	41	7	that	that	PRON
ejpam-5770	41	8	require	require	VERB
ejpam-5770	41	9	both	both	CCONJ
ejpam-5770	41	10	connectivity	connectivity	NOUN
ejpam-5770	41	11	and	and	CCONJ
ejpam-5770	41	12	service	service	NOUN
ejpam-5770	41	13	diversity	diversity	NOUN
ejpam-5770	41	14	,	,	PUNCT
ejpam-5770	41	15	leading	lead	VERB
ejpam-5770	41	16	to	to	ADP
ejpam-5770	41	17	new	new	ADJ
ejpam-5770	41	18	parameters	parameter	NOUN
ejpam-5770	41	19	and	and	CCONJ
ejpam-5770	41	20	optimization	optimization	NOUN
ejpam-5770	41	21	techniques	technique	NOUN
ejpam-5770	41	22	in	in	ADP
ejpam-5770	41	23	graph	graph	NOUN
ejpam-5770	41	24	theory	theory	NOUN
ejpam-5770	41	25	.	.	PUNCT
ejpam-5770	42	1	in	in	ADP
ejpam-5770	42	2	this	this	DET
ejpam-5770	42	3	paper	paper	NOUN
ejpam-5770	42	4	,	,	PUNCT
ejpam-5770	42	5	we	we	PRON
ejpam-5770	42	6	initiate	initiate	VERB
ejpam-5770	42	7	the	the	DET
ejpam-5770	42	8	study	study	NOUN
ejpam-5770	42	9	of	of	ADP
ejpam-5770	42	10	the	the	DET
ejpam-5770	42	11	weakly	weakly	ADJ
ejpam-5770	42	12	connected	connected	ADJ
ejpam-5770	42	13	k	k	ADJ
ejpam-5770	42	14	-	-	PUNCT
ejpam-5770	42	15	rainbow	rainbow	NOUN
ejpam-5770	42	16	domination	domination	NOUN
ejpam-5770	42	17	parameter	parameter	NOUN
ejpam-5770	42	18	.	.	PUNCT
ejpam-5770	43	1	first	first	ADV
ejpam-5770	43	2	,	,	PUNCT
ejpam-5770	43	3	we	we	PRON
ejpam-5770	43	4	establish	establish	VERB
ejpam-5770	43	5	fundamental	fundamental	ADJ
ejpam-5770	43	6	properties	property	NOUN
ejpam-5770	43	7	and	and	CCONJ
ejpam-5770	43	8	bounds	bound	NOUN
ejpam-5770	43	9	for	for	ADP
ejpam-5770	43	10	weakly	weakly	ADV
ejpam-5770	43	11	connected	connected	ADJ
ejpam-5770	43	12	k	k	ADJ
ejpam-5770	43	13	-	-	PUNCT
ejpam-5770	43	14	rainbow	rainbow	NOUN
ejpam-5770	43	15	domination	domination	NOUN
ejpam-5770	43	16	.	.	PUNCT
ejpam-5770	44	1	then	then	ADV
ejpam-5770	44	2	,	,	PUNCT
ejpam-5770	44	3	we	we	PRON
ejpam-5770	44	4	determine	determine	VERB
ejpam-5770	44	5	the	the	DET
ejpam-5770	44	6	weakly	weakly	ADJ
ejpam-5770	44	7	connected	connected	ADJ
ejpam-5770	44	8	k	k	ADJ
ejpam-5770	44	9	-	-	PUNCT
ejpam-5770	44	10	rainbow	rainbow	NOUN
ejpam-5770	44	11	domination	domination	NOUN
ejpam-5770	44	12	number	number	NOUN
ejpam-5770	44	13	for	for	ADP
ejpam-5770	44	14	various	various	ADJ
ejpam-5770	44	15	classes	class	NOUN
ejpam-5770	44	16	of	of	ADP
ejpam-5770	44	17	graphs	graph	NOUN
ejpam-5770	44	18	.	.	PUNCT
ejpam-5770	45	1	furthermore	furthermore	ADV
ejpam-5770	45	2	,	,	PUNCT
ejpam-5770	45	3	we	we	PRON
ejpam-5770	45	4	characterize	characterize	VERB
ejpam-5770	45	5	the	the	DET
ejpam-5770	45	6	weakly	weakly	ADV
ejpam-5770	45	7	connected	connected	ADJ
ejpam-5770	45	8	krainbow	krainbow	NOUN
ejpam-5770	45	9	dominating	dominate	VERB
ejpam-5770	45	10	function	function	NOUN
ejpam-5770	45	11	under	under	ADP
ejpam-5770	45	12	the	the	DET
ejpam-5770	45	13	join	join	NOUN
ejpam-5770	45	14	of	of	ADP
ejpam-5770	45	15	graphs	graph	NOUN
ejpam-5770	45	16	and	and	CCONJ
ejpam-5770	45	17	determine	determine	VERB
ejpam-5770	45	18	the	the	DET
ejpam-5770	45	19	weakly	weakly	ADJ
ejpam-5770	45	20	connected	connected	ADJ
ejpam-5770	45	21	k	k	ADJ
ejpam-5770	45	22	-	-	PUNCT
ejpam-5770	45	23	rainbow	rainbow	NOUN
ejpam-5770	45	24	domination	domination	NOUN
ejpam-5770	45	25	number	number	NOUN
ejpam-5770	45	26	for	for	ADP
ejpam-5770	45	27	this	this	DET
ejpam-5770	45	28	binary	binary	ADJ
ejpam-5770	45	29	operation	operation	NOUN
ejpam-5770	45	30	.	.	PUNCT
ejpam-5770	46	1	2	2	X
ejpam-5770	46	2	.	.	X
ejpam-5770	46	3	terminology	terminology	NOUN
ejpam-5770	46	4	and	and	CCONJ
ejpam-5770	46	5	notation	notation	NOUN
ejpam-5770	46	6	for	for	ADP
ejpam-5770	46	7	general	general	ADJ
ejpam-5770	46	8	graph	graph	NOUN
ejpam-5770	46	9	theory	theory	NOUN
ejpam-5770	46	10	terminology	terminology	NOUN
ejpam-5770	46	11	,	,	PUNCT
ejpam-5770	46	12	we	we	PRON
ejpam-5770	46	13	adhere	adhere	VERB
ejpam-5770	46	14	to	to	ADP
ejpam-5770	46	15	the	the	DET
ejpam-5770	46	16	definitions	definition	NOUN
ejpam-5770	46	17	provided	provide	VERB
ejpam-5770	46	18	by	by	ADP
ejpam-5770	46	19	harary	harary	NOUN
ejpam-5770	46	20	in	in	ADP
ejpam-5770	46	21	[	[	X
ejpam-5770	46	22	20	20	NUM
ejpam-5770	46	23	]	]	PUNCT
ejpam-5770	46	24	.	.	PUNCT
ejpam-5770	47	1	let	let	VERB
ejpam-5770	47	2	g	g	PROPN
ejpam-5770	47	3	=	=	SYM
ejpam-5770	47	4	(	(	PUNCT
ejpam-5770	47	5	v	v	NOUN
ejpam-5770	47	6	,	,	PUNCT
ejpam-5770	47	7	e	e	NOUN
ejpam-5770	47	8	)	)	PUNCT
ejpam-5770	47	9	be	be	AUX
ejpam-5770	47	10	a	a	DET
ejpam-5770	47	11	simple	simple	ADJ
ejpam-5770	47	12	and	and	CCONJ
ejpam-5770	47	13	connected	connected	ADJ
ejpam-5770	47	14	graph	graph	NOUN
ejpam-5770	47	15	,	,	PUNCT
ejpam-5770	47	16	where	where	SCONJ
ejpam-5770	47	17	v	v	NOUN
ejpam-5770	47	18	=	=	SYM
ejpam-5770	47	19	v	v	NOUN
ejpam-5770	47	20	(	(	PUNCT
ejpam-5770	47	21	g	g	NOUN
ejpam-5770	47	22	)	)	PUNCT
ejpam-5770	47	23	represents	represent	VERB
ejpam-5770	47	24	the	the	DET
ejpam-5770	47	25	vertex	vertex	NOUN
ejpam-5770	47	26	set	set	NOUN
ejpam-5770	47	27	and	and	CCONJ
ejpam-5770	47	28	e	e	NOUN
ejpam-5770	47	29	=	=	PROPN
ejpam-5770	47	30	e(g	e(g	PROPN
ejpam-5770	47	31	)	)	PUNCT
ejpam-5770	47	32	denotes	denote	VERB
ejpam-5770	47	33	the	the	DET
ejpam-5770	47	34	edge	edge	NOUN
ejpam-5770	47	35	set	set	NOUN
ejpam-5770	47	36	of	of	ADP
ejpam-5770	47	37	g.	g.	PROPN
ejpam-5770	47	38	the	the	DET
ejpam-5770	47	39	number	number	NOUN
ejpam-5770	47	40	of	of	ADP
ejpam-5770	47	41	edges	edge	NOUN
ejpam-5770	47	42	incident	incident	NOUN
ejpam-5770	47	43	to	to	ADP
ejpam-5770	47	44	a	a	DET
ejpam-5770	47	45	vertex	vertex	NOUN
ejpam-5770	47	46	v	v	NOUN
ejpam-5770	47	47	is	be	AUX
ejpam-5770	47	48	called	call	VERB
ejpam-5770	47	49	its	its	PRON
ejpam-5770	47	50	degree	degree	NOUN
ejpam-5770	47	51	,	,	PUNCT
ejpam-5770	47	52	denoted	denote	VERB
ejpam-5770	47	53	as	as	ADP
ejpam-5770	47	54	deg(v	deg(v	NOUN
ejpam-5770	47	55	)	)	PUNCT
ejpam-5770	47	56	.	.	PUNCT
ejpam-5770	48	1	the	the	DET
ejpam-5770	48	2	maximum	maximum	ADJ
ejpam-5770	48	3	degree	degree	NOUN
ejpam-5770	48	4	of	of	ADP
ejpam-5770	48	5	g	g	NOUN
ejpam-5770	48	6	,	,	PUNCT
ejpam-5770	48	7	represented	represent	VERB
ejpam-5770	48	8	by	by	ADP
ejpam-5770	48	9	∆(g	∆(g	PROPN
ejpam-5770	48	10	)	)	PUNCT
ejpam-5770	48	11	,	,	PUNCT
ejpam-5770	48	12	is	be	AUX
ejpam-5770	48	13	given	give	VERB
ejpam-5770	48	14	by	by	ADP
ejpam-5770	48	15	∆(g	∆(g	NOUN
ejpam-5770	48	16	)	)	PUNCT
ejpam-5770	48	17	=	=	SYM
ejpam-5770	48	18	max{deg(v	max{deg(v	PROPN
ejpam-5770	48	19	)	)	PUNCT
ejpam-5770	48	20	:	:	PUNCT
ejpam-5770	48	21	v	v	X
ejpam-5770	48	22	∈	∈	PROPN
ejpam-5770	48	23	v	v	NOUN
ejpam-5770	48	24	(	(	PUNCT
ejpam-5770	48	25	g	g	NOUN
ejpam-5770	48	26	)	)	PUNCT
ejpam-5770	48	27	}	}	PUNCT
ejpam-5770	48	28	.	.	PUNCT
ejpam-5770	49	1	the	the	DET
ejpam-5770	49	2	open	open	ADJ
ejpam-5770	49	3	neighborhood	neighborhood	NOUN
ejpam-5770	49	4	of	of	ADP
ejpam-5770	49	5	a	a	DET
ejpam-5770	49	6	vertex	vertex	NOUN
ejpam-5770	49	7	u	u	NOUN
ejpam-5770	49	8	,	,	PUNCT
ejpam-5770	49	9	denoted	denote	VERB
ejpam-5770	49	10	by	by	ADP
ejpam-5770	49	11	ng(u	ng(u	NOUN
ejpam-5770	49	12	)	)	PUNCT
ejpam-5770	49	13	,	,	PUNCT
ejpam-5770	49	14	is	be	AUX
ejpam-5770	49	15	the	the	DET
ejpam-5770	49	16	set	set	NOUN
ejpam-5770	49	17	of	of	ADP
ejpam-5770	49	18	all	all	DET
ejpam-5770	49	19	vertices	vertex	NOUN
ejpam-5770	49	20	adjacent	adjacent	ADJ
ejpam-5770	49	21	to	to	ADP
ejpam-5770	49	22	u	u	PRON
ejpam-5770	49	23	,	,	PUNCT
ejpam-5770	49	24	i.e.	i.e.	X
ejpam-5770	49	25	,	,	PUNCT
ejpam-5770	49	26	ng(u	ng(u	NOUN
ejpam-5770	49	27	)	)	PUNCT
ejpam-5770	49	28	=	=	PRON
ejpam-5770	49	29	{	{	PUNCT
ejpam-5770	49	30	v	v	NUM
ejpam-5770	49	31	∈	∈	NOUN
ejpam-5770	49	32	v	v	NOUN
ejpam-5770	49	33	(	(	PUNCT
ejpam-5770	49	34	g	g	NOUN
ejpam-5770	49	35	)	)	PUNCT
ejpam-5770	49	36	:	:	PUNCT
ejpam-5770	49	37	uv	uv	PROPN
ejpam-5770	49	38	∈	∈	PROPN
ejpam-5770	49	39	e(g	e(g	PROPN
ejpam-5770	49	40	)	)	PUNCT
ejpam-5770	49	41	}	}	PUNCT
ejpam-5770	49	42	.	.	PUNCT
ejpam-5770	50	1	the	the	DET
ejpam-5770	50	2	closed	closed	ADJ
ejpam-5770	50	3	neighborhood	neighborhood	NOUN
ejpam-5770	50	4	of	of	ADP
ejpam-5770	50	5	u	u	NOUN
ejpam-5770	50	6	is	be	AUX
ejpam-5770	50	7	defined	define	VERB
ejpam-5770	50	8	as	as	ADP
ejpam-5770	50	9	ng[u	ng[u	PROPN
ejpam-5770	50	10	]	]	X
ejpam-5770	50	11	=	=	SYM
ejpam-5770	50	12	ng(u	ng(u	PROPN
ejpam-5770	50	13	)	)	PUNCT
ejpam-5770	50	14	∪	∪	NOUN
ejpam-5770	50	15	{	{	PUNCT
ejpam-5770	50	16	u	u	NOUN
ejpam-5770	50	17	}	}	PUNCT
ejpam-5770	50	18	.	.	PUNCT
ejpam-5770	51	1	similarly	similarly	ADV
ejpam-5770	51	2	,	,	PUNCT
ejpam-5770	51	3	for	for	ADP
ejpam-5770	51	4	a	a	DET
ejpam-5770	51	5	subset	subset	NOUN
ejpam-5770	51	6	s	s	VERB
ejpam-5770	51	7	⊆	⊆	NUM
ejpam-5770	51	8	v	v	NOUN
ejpam-5770	51	9	(	(	PUNCT
ejpam-5770	51	10	g	g	NOUN
ejpam-5770	51	11	)	)	PUNCT
ejpam-5770	51	12	,	,	PUNCT
ejpam-5770	51	13	the	the	DET
ejpam-5770	51	14	closed	closed	ADJ
ejpam-5770	51	15	neighborhood	neighborhood	NOUN
ejpam-5770	51	16	of	of	ADP
ejpam-5770	51	17	s	s	NOUN
ejpam-5770	51	18	is	be	AUX
ejpam-5770	51	19	given	give	VERB
ejpam-5770	51	20	by	by	ADP
ejpam-5770	51	21	ng[s	ng[	NOUN
ejpam-5770	51	22	]	]	PUNCT
ejpam-5770	51	23	=	=	PUNCT
ejpam-5770	51	24	⋃	⋃	VERB
ejpam-5770	51	25	v∈s	v∈s	ADJ
ejpam-5770	51	26	ng[v	ng[v	NOUN
ejpam-5770	51	27	]	]	PUNCT
ejpam-5770	51	28	.	.	PUNCT
ejpam-5770	52	1	the	the	DET
ejpam-5770	52	2	subgraph	subgraph	NOUN
ejpam-5770	52	3	weakly	weakly	ADV
ejpam-5770	52	4	induced	induce	VERB
ejpam-5770	52	5	by	by	ADP
ejpam-5770	52	6	a	a	DET
ejpam-5770	52	7	set	set	NOUN
ejpam-5770	52	8	s	s	PROPN
ejpam-5770	52	9	⊆	⊆	NUM
ejpam-5770	52	10	v	v	NOUN
ejpam-5770	52	11	(	(	PUNCT
ejpam-5770	52	12	g	g	NOUN
ejpam-5770	52	13	)	)	PUNCT
ejpam-5770	52	14	,	,	PUNCT
ejpam-5770	52	15	as	as	SCONJ
ejpam-5770	52	16	defined	define	VERB
ejpam-5770	52	17	by	by	ADP
ejpam-5770	52	18	e.	e.	PROPN
ejpam-5770	52	19	p.	p.	PROPN
ejpam-5770	52	20	sandueta	sandueta	PROPN
ejpam-5770	52	21	et	et	PROPN
ejpam-5770	52	22	al	al	PROPN
ejpam-5770	52	23	.	.	PUNCT
ejpam-5770	53	1	[	[	X
ejpam-5770	53	2	18	18	NUM
ejpam-5770	53	3	]	]	PUNCT
ejpam-5770	53	4	,	,	PUNCT
ejpam-5770	53	5	is	be	AUX
ejpam-5770	53	6	the	the	DET
ejpam-5770	53	7	graph	graph	NOUN
ejpam-5770	53	8	⟨s⟩w	⟨s⟩w	NOUN
ejpam-5770	53	9	=	=	PUNCT
ejpam-5770	53	10	(	(	PUNCT
ejpam-5770	53	11	ng[s	ng[s	PROPN
ejpam-5770	53	12	]	]	PUNCT
ejpam-5770	53	13	,	,	PUNCT
ejpam-5770	53	14	ew(s	ew(s	NOUN
ejpam-5770	53	15	)	)	PUNCT
ejpam-5770	53	16	)	)	PUNCT
ejpam-5770	53	17	,	,	PUNCT
ejpam-5770	53	18	where	where	SCONJ
ejpam-5770	53	19	the	the	DET
ejpam-5770	53	20	edge	edge	NOUN
ejpam-5770	53	21	set	set	NOUN
ejpam-5770	53	22	is	be	AUX
ejpam-5770	53	23	j.	j.	PROPN
ejpam-5770	53	24	j.	j.	PROPN
ejpam-5770	53	25	hamja	hamja	PROPN
ejpam-5770	53	26	et	et	PROPN
ejpam-5770	53	27	al	al	PROPN
ejpam-5770	53	28	.	.	PUNCT
ejpam-5770	53	29	/	/	SYM
ejpam-5770	53	30	eur	eur	PROPN
ejpam-5770	53	31	.	.	PUNCT
ejpam-5770	54	1	j.	j.	PROPN
ejpam-5770	54	2	pure	pure	PROPN
ejpam-5770	54	3	appl	appl	PROPN
ejpam-5770	54	4	.	.	PROPN
ejpam-5770	54	5	math	math	PROPN
ejpam-5770	54	6	,	,	PUNCT
ejpam-5770	54	7	18	18	NUM
ejpam-5770	54	8	(	(	PUNCT
ejpam-5770	54	9	2	2	NUM
ejpam-5770	54	10	)	)	PUNCT
ejpam-5770	54	11	(	(	PUNCT
ejpam-5770	54	12	2025	2025	NUM
ejpam-5770	54	13	)	)	PUNCT
ejpam-5770	54	14	,	,	PUNCT
ejpam-5770	54	15	5770	5770	NUM
ejpam-5770	54	16	3	3	NUM
ejpam-5770	54	17	of	of	ADP
ejpam-5770	54	18	11	11	NUM
ejpam-5770	54	19	given	give	VERB
ejpam-5770	54	20	by	by	ADP
ejpam-5770	54	21	ew(s	ew(s	NOUN
ejpam-5770	54	22	)	)	PUNCT
ejpam-5770	54	23	=	=	PRON
ejpam-5770	55	1	{	{	PUNCT
ejpam-5770	55	2	uv	uv	PROPN
ejpam-5770	55	3	∈	∈	PROPN
ejpam-5770	55	4	e(g	e(g	PROPN
ejpam-5770	55	5	)	)	PUNCT
ejpam-5770	55	6	:	:	PUNCT
ejpam-5770	56	1	u	u	PROPN
ejpam-5770	56	2	∈	∈	PROPN
ejpam-5770	56	3	s	s	X
ejpam-5770	56	4	or	or	CCONJ
ejpam-5770	56	5	v	v	ADP
ejpam-5770	56	6	∈	∈	NOUN
ejpam-5770	56	7	s	s	PART
ejpam-5770	56	8	}	}	PUNCT
ejpam-5770	56	9	.	.	PUNCT
ejpam-5770	57	1	a	a	DET
ejpam-5770	57	2	set	set	NOUN
ejpam-5770	57	3	s	s	NOUN
ejpam-5770	57	4	⊆	⊆	NUM
ejpam-5770	57	5	v	v	NOUN
ejpam-5770	57	6	(	(	PUNCT
ejpam-5770	57	7	g	g	NOUN
ejpam-5770	57	8	)	)	PUNCT
ejpam-5770	57	9	is	be	AUX
ejpam-5770	57	10	said	say	VERB
ejpam-5770	57	11	to	to	PART
ejpam-5770	57	12	be	be	AUX
ejpam-5770	57	13	a	a	DET
ejpam-5770	57	14	dominating	dominating	NOUN
ejpam-5770	57	15	set	set	NOUN
ejpam-5770	57	16	of	of	ADP
ejpam-5770	57	17	g	g	PROPN
ejpam-5770	57	18	if	if	SCONJ
ejpam-5770	57	19	ng[s	ng[	NOUN
ejpam-5770	57	20	]	]	PUNCT
ejpam-5770	57	21	=	=	SYM
ejpam-5770	57	22	v	v	NOUN
ejpam-5770	57	23	(	(	PUNCT
ejpam-5770	57	24	g	g	NOUN
ejpam-5770	57	25	)	)	PUNCT
ejpam-5770	57	26	.	.	PUNCT
ejpam-5770	58	1	a	a	DET
ejpam-5770	58	2	dominating	dominating	NOUN
ejpam-5770	58	3	set	set	NOUN
ejpam-5770	58	4	s	s	PART
ejpam-5770	58	5	is	be	AUX
ejpam-5770	58	6	called	call	VERB
ejpam-5770	58	7	a	a	DET
ejpam-5770	58	8	minimal	minimal	ADJ
ejpam-5770	58	9	dominating	dominating	NOUN
ejpam-5770	58	10	set	set	NOUN
ejpam-5770	58	11	if	if	SCONJ
ejpam-5770	58	12	no	no	DET
ejpam-5770	58	13	proper	proper	ADJ
ejpam-5770	58	14	subset	subset	NOUN
ejpam-5770	58	15	s′	s′	VERB
ejpam-5770	58	16	⊂	⊂	ADJ
ejpam-5770	58	17	s	s	X
ejpam-5770	58	18	is	be	AUX
ejpam-5770	58	19	itself	itself	PRON
ejpam-5770	58	20	a	a	DET
ejpam-5770	58	21	dominating	dominating	NOUN
ejpam-5770	58	22	set	set	NOUN
ejpam-5770	58	23	.	.	PUNCT
ejpam-5770	59	1	the	the	DET
ejpam-5770	59	2	domination	domination	NOUN
ejpam-5770	59	3	number	number	NOUN
ejpam-5770	59	4	,	,	PUNCT
ejpam-5770	59	5	denoted	denote	VERB
ejpam-5770	59	6	by	by	ADP
ejpam-5770	59	7	γ(g	γ(g	PROPN
ejpam-5770	59	8	)	)	PUNCT
ejpam-5770	59	9	is	be	AUX
ejpam-5770	59	10	the	the	DET
ejpam-5770	59	11	minimum	minimum	ADJ
ejpam-5770	59	12	cardinality	cardinality	NOUN
ejpam-5770	59	13	of	of	ADP
ejpam-5770	59	14	a	a	DET
ejpam-5770	59	15	dominating	dominating	NOUN
ejpam-5770	59	16	set	set	VERB
ejpam-5770	59	17	in	in	ADP
ejpam-5770	59	18	g.	g.	PROPN
ejpam-5770	59	19	a	a	DET
ejpam-5770	59	20	dominating	dominating	NOUN
ejpam-5770	59	21	set	set	NOUN
ejpam-5770	59	22	s	s	NOUN
ejpam-5770	59	23	with	with	ADP
ejpam-5770	59	24	|s|	|s|	PROPN
ejpam-5770	59	25	=	=	SYM
ejpam-5770	59	26	γ(g	γ(g	PROPN
ejpam-5770	59	27	)	)	PUNCT
ejpam-5770	59	28	is	be	AUX
ejpam-5770	59	29	referred	refer	VERB
ejpam-5770	59	30	to	to	ADP
ejpam-5770	59	31	as	as	ADP
ejpam-5770	59	32	a	a	DET
ejpam-5770	59	33	γ	γ	NOUN
ejpam-5770	59	34	-	-	PUNCT
ejpam-5770	59	35	set	set	NOUN
ejpam-5770	59	36	of	of	ADP
ejpam-5770	59	37	g.	g.	PROPN
ejpam-5770	59	38	a	a	DET
ejpam-5770	59	39	set	set	NOUN
ejpam-5770	59	40	s	s	PROPN
ejpam-5770	59	41	⊆	⊆	NUM
ejpam-5770	59	42	v	v	NOUN
ejpam-5770	59	43	(	(	PUNCT
ejpam-5770	59	44	g	g	NOUN
ejpam-5770	59	45	)	)	PUNCT
ejpam-5770	59	46	is	be	AUX
ejpam-5770	59	47	said	say	VERB
ejpam-5770	59	48	to	to	PART
ejpam-5770	59	49	be	be	AUX
ejpam-5770	59	50	a	a	DET
ejpam-5770	59	51	weakly	weakly	ADV
ejpam-5770	59	52	connected	connected	ADJ
ejpam-5770	59	53	dominating	dominating	NOUN
ejpam-5770	59	54	set	set	VERB
ejpam-5770	59	55	in	in	ADP
ejpam-5770	59	56	g	g	PROPN
ejpam-5770	59	57	if	if	SCONJ
ejpam-5770	59	58	s	s	NOUN
ejpam-5770	59	59	is	be	AUX
ejpam-5770	59	60	dominating	dominate	VERB
ejpam-5770	59	61	and	and	CCONJ
ejpam-5770	59	62	the	the	DET
ejpam-5770	59	63	subgraph	subgraph	PROPN
ejpam-5770	59	64	⟨s⟩w	⟨s⟩w	NOUN
ejpam-5770	59	65	weakly	weakly	ADV
ejpam-5770	59	66	induced	induce	VERB
ejpam-5770	59	67	by	by	ADP
ejpam-5770	59	68	s	s	PRON
ejpam-5770	59	69	is	be	AUX
ejpam-5770	59	70	connected	connect	VERB
ejpam-5770	59	71	.	.	PUNCT
ejpam-5770	60	1	the	the	DET
ejpam-5770	60	2	weakly	weakly	ADJ
ejpam-5770	60	3	connected	connected	ADJ
ejpam-5770	60	4	domination	domination	NOUN
ejpam-5770	60	5	number	number	NOUN
ejpam-5770	60	6	,	,	PUNCT
ejpam-5770	60	7	denoted	denote	VERB
ejpam-5770	60	8	by	by	ADP
ejpam-5770	60	9	γw(g	γw(g	NOUN
ejpam-5770	60	10	)	)	PUNCT
ejpam-5770	60	11	is	be	AUX
ejpam-5770	60	12	the	the	DET
ejpam-5770	60	13	minimum	minimum	ADJ
ejpam-5770	60	14	cardinality	cardinality	NOUN
ejpam-5770	60	15	among	among	ADP
ejpam-5770	60	16	all	all	DET
ejpam-5770	60	17	weakly	weakly	ADV
ejpam-5770	60	18	connected	connected	ADJ
ejpam-5770	60	19	dominating	dominating	NOUN
ejpam-5770	60	20	sets	set	NOUN
ejpam-5770	60	21	.	.	PUNCT
ejpam-5770	61	1	a	a	DET
ejpam-5770	61	2	weakly	weakly	ADV
ejpam-5770	61	3	connected	connected	ADJ
ejpam-5770	61	4	dominating	dominating	NOUN
ejpam-5770	61	5	set	set	NOUN
ejpam-5770	61	6	s	s	NOUN
ejpam-5770	61	7	with	with	ADP
ejpam-5770	61	8	|s|	|s|	NOUN
ejpam-5770	61	9	=	=	NOUN
ejpam-5770	61	10	γw(g	γw(g	X
ejpam-5770	61	11	)	)	PUNCT
ejpam-5770	61	12	is	be	AUX
ejpam-5770	61	13	referred	refer	VERB
ejpam-5770	61	14	to	to	ADP
ejpam-5770	61	15	as	as	ADP
ejpam-5770	61	16	a	a	DET
ejpam-5770	61	17	γw	γw	NOUN
ejpam-5770	61	18	-	-	PUNCT
ejpam-5770	61	19	set	set	NOUN
ejpam-5770	61	20	of	of	ADP
ejpam-5770	61	21	g	g	NOUN
ejpam-5770	61	22	,	,	PUNCT
ejpam-5770	61	23	as	as	SCONJ
ejpam-5770	61	24	defined	define	VERB
ejpam-5770	61	25	by	by	ADP
ejpam-5770	61	26	j.	j.	PROPN
ejpam-5770	61	27	e.	e.	PROPN
ejpam-5770	61	28	dunbar	dunbar	PROPN
ejpam-5770	61	29	et	et	PROPN
ejpam-5770	61	30	al	al	PROPN
ejpam-5770	61	31	.	.	PUNCT
ejpam-5770	62	1	in	in	ADP
ejpam-5770	62	2	[	[	X
ejpam-5770	62	3	12	12	NUM
ejpam-5770	62	4	]	]	PUNCT
ejpam-5770	62	5	.	.	PUNCT
ejpam-5770	63	1	a	a	DET
ejpam-5770	63	2	function	function	NOUN
ejpam-5770	63	3	f	f	NOUN
ejpam-5770	63	4	:	:	PUNCT
ejpam-5770	63	5	v	v	X
ejpam-5770	63	6	(	(	PUNCT
ejpam-5770	63	7	g	g	NOUN
ejpam-5770	63	8	)	)	PUNCT
ejpam-5770	63	9	→	→	SYM
ejpam-5770	64	1	p({1	p({1	PROPN
ejpam-5770	64	2	,	,	PUNCT
ejpam-5770	64	3	2	2	NUM
ejpam-5770	64	4	,	,	PUNCT
ejpam-5770	64	5	3	3	NUM
ejpam-5770	64	6	,	,	PUNCT
ejpam-5770	64	7	.	.	PUNCT
ejpam-5770	64	8	.	.	PUNCT
ejpam-5770	65	1	.	.	PUNCT
ejpam-5770	66	1	,	,	PUNCT
ejpam-5770	66	2	k	k	NOUN
ejpam-5770	66	3	}	}	PUNCT
ejpam-5770	66	4	)	)	PUNCT
ejpam-5770	66	5	assigns	assign	NOUN
ejpam-5770	66	6	to	to	ADP
ejpam-5770	66	7	each	each	DET
ejpam-5770	66	8	vertex	vertex	NOUN
ejpam-5770	66	9	of	of	ADP
ejpam-5770	66	10	a	a	DET
ejpam-5770	66	11	graph	graph	NOUN
ejpam-5770	66	12	g	g	ADP
ejpam-5770	66	13	a	a	DET
ejpam-5770	66	14	set	set	NOUN
ejpam-5770	66	15	of	of	ADP
ejpam-5770	66	16	colors	color	NOUN
ejpam-5770	66	17	chosen	choose	VERB
ejpam-5770	66	18	from	from	ADP
ejpam-5770	66	19	the	the	DET
ejpam-5770	66	20	set	set	NOUN
ejpam-5770	66	21	{	{	PUNCT
ejpam-5770	66	22	1	1	NUM
ejpam-5770	66	23	,	,	PUNCT
ejpam-5770	66	24	2	2	NUM
ejpam-5770	66	25	,	,	PUNCT
ejpam-5770	66	26	3	3	NUM
ejpam-5770	66	27	,	,	PUNCT
ejpam-5770	66	28	.	.	PUNCT
ejpam-5770	66	29	.	.	PUNCT
ejpam-5770	67	1	.	.	PUNCT
ejpam-5770	68	1	,	,	PUNCT
ejpam-5770	68	2	k	k	X
ejpam-5770	68	3	}	}	PUNCT
ejpam-5770	68	4	.	.	PUNCT
ejpam-5770	69	1	if	if	SCONJ
ejpam-5770	69	2	,	,	PUNCT
ejpam-5770	69	3	for	for	ADP
ejpam-5770	69	4	every	every	DET
ejpam-5770	69	5	vertex	vertex	NOUN
ejpam-5770	69	6	v	v	ADP
ejpam-5770	69	7	∈	∈	NOUN
ejpam-5770	69	8	v	v	NOUN
ejpam-5770	69	9	(	(	PUNCT
ejpam-5770	69	10	g	g	NOUN
ejpam-5770	69	11	)	)	PUNCT
ejpam-5770	69	12	such	such	ADJ
ejpam-5770	69	13	that	that	SCONJ
ejpam-5770	69	14	f(v	f(v	NOUN
ejpam-5770	69	15	)	)	PUNCT
ejpam-5770	69	16	=	=	SYM
ejpam-5770	69	17	∅	∅	NOUN
ejpam-5770	69	18	,	,	PUNCT
ejpam-5770	69	19	we	we	PRON
ejpam-5770	69	20	have	have	VERB
ejpam-5770	69	21	⋃	⋃	PROPN
ejpam-5770	69	22	u∈ng(v	u∈ng(v	PROPN
ejpam-5770	69	23	)	)	PUNCT
ejpam-5770	69	24	f(u	f(u	PROPN
ejpam-5770	69	25	)	)	PUNCT
ejpam-5770	69	26	=	=	PRON
ejpam-5770	70	1	{	{	PUNCT
ejpam-5770	70	2	1	1	NUM
ejpam-5770	70	3	,	,	PUNCT
ejpam-5770	70	4	2	2	NUM
ejpam-5770	70	5	,	,	PUNCT
ejpam-5770	70	6	3	3	NUM
ejpam-5770	70	7	,	,	PUNCT
ejpam-5770	70	8	.	.	PUNCT
ejpam-5770	70	9	.	.	PUNCT
ejpam-5770	71	1	.	.	PUNCT
ejpam-5770	72	1	,	,	PUNCT
ejpam-5770	72	2	k	k	X
ejpam-5770	72	3	}	}	PUNCT
ejpam-5770	72	4	,	,	PUNCT
ejpam-5770	72	5	then	then	ADV
ejpam-5770	72	6	f	f	PROPN
ejpam-5770	72	7	is	be	AUX
ejpam-5770	72	8	called	call	VERB
ejpam-5770	72	9	a	a	DET
ejpam-5770	72	10	k	k	ADJ
ejpam-5770	72	11	-	-	PUNCT
ejpam-5770	72	12	rainbow	rainbow	NOUN
ejpam-5770	72	13	dominating	dominating	NOUN
ejpam-5770	72	14	function	function	NOUN
ejpam-5770	72	15	(	(	PUNCT
ejpam-5770	72	16	krdf	krdf	PROPN
ejpam-5770	72	17	)	)	PUNCT
ejpam-5770	72	18	of	of	ADP
ejpam-5770	72	19	g.	g.	PROPN
ejpam-5770	72	20	the	the	DET
ejpam-5770	72	21	weight	weight	NOUN
ejpam-5770	72	22	ω(f	ω(f	PUNCT
ejpam-5770	72	23	)	)	PUNCT
ejpam-5770	72	24	of	of	ADP
ejpam-5770	72	25	f	f	PROPN
ejpam-5770	72	26	is	be	AUX
ejpam-5770	72	27	defined	define	VERB
ejpam-5770	72	28	as	as	ADP
ejpam-5770	72	29	ω(f	ω(f	ADJ
ejpam-5770	72	30	)	)	PUNCT
ejpam-5770	72	31	=	=	SYM
ejpam-5770	73	1	∑	∑	PUNCT
ejpam-5770	73	2	v∈v	v∈v	NOUN
ejpam-5770	73	3	(	(	PUNCT
ejpam-5770	73	4	g)|f(v)|	g)|f(v)|	PROPN
ejpam-5770	73	5	.	.	PUNCT
ejpam-5770	74	1	the	the	DET
ejpam-5770	74	2	k	k	ADJ
ejpam-5770	74	3	-	-	PUNCT
ejpam-5770	74	4	rainbow	rainbow	NOUN
ejpam-5770	74	5	domination	domination	NOUN
ejpam-5770	74	6	number	number	NOUN
ejpam-5770	74	7	of	of	ADP
ejpam-5770	74	8	g	g	NOUN
ejpam-5770	74	9	,	,	PUNCT
ejpam-5770	74	10	denoted	denote	VERB
ejpam-5770	74	11	by	by	ADP
ejpam-5770	74	12	γrk(g	γrk(g	PROPN
ejpam-5770	74	13	)	)	PUNCT
ejpam-5770	74	14	,	,	PUNCT
ejpam-5770	74	15	is	be	AUX
ejpam-5770	74	16	the	the	DET
ejpam-5770	74	17	minimum	minimum	ADJ
ejpam-5770	74	18	weight	weight	NOUN
ejpam-5770	74	19	of	of	ADP
ejpam-5770	74	20	a	a	DET
ejpam-5770	74	21	krdf	krdf	NOUN
ejpam-5770	74	22	.	.	PUNCT
ejpam-5770	75	1	a	a	DET
ejpam-5770	75	2	k	k	ADJ
ejpam-5770	75	3	-	-	PUNCT
ejpam-5770	75	4	rainbow	rainbow	NOUN
ejpam-5770	75	5	dominating	dominating	NOUN
ejpam-5770	75	6	function	function	NOUN
ejpam-5770	75	7	of	of	ADP
ejpam-5770	75	8	g	g	NOUN
ejpam-5770	75	9	with	with	ADP
ejpam-5770	75	10	weight	weight	NOUN
ejpam-5770	75	11	γrk(g	γrk(g	PROPN
ejpam-5770	75	12	)	)	PUNCT
ejpam-5770	75	13	,	,	PUNCT
ejpam-5770	75	14	i.e.	i.e.	X
ejpam-5770	75	15	,	,	PUNCT
ejpam-5770	75	16	ω(f	ω(f	NUM
ejpam-5770	75	17	)	)	PUNCT
ejpam-5770	75	18	=	=	SYM
ejpam-5770	75	19	γrk(g	γrk(g	PROPN
ejpam-5770	75	20	)	)	PUNCT
ejpam-5770	75	21	,	,	PUNCT
ejpam-5770	75	22	is	be	AUX
ejpam-5770	75	23	referred	refer	VERB
ejpam-5770	75	24	to	to	ADP
ejpam-5770	75	25	as	as	ADP
ejpam-5770	75	26	a	a	DET
ejpam-5770	75	27	γrk	γrk	NOUN
ejpam-5770	75	28	-	-	PUNCT
ejpam-5770	75	29	function	function	NOUN
ejpam-5770	75	30	of	of	ADP
ejpam-5770	75	31	g	g	NOUN
ejpam-5770	75	32	,	,	PUNCT
ejpam-5770	75	33	as	as	SCONJ
ejpam-5770	75	34	defined	define	VERB
ejpam-5770	75	35	by	by	ADP
ejpam-5770	75	36	b.	b.	PROPN
ejpam-5770	75	37	brešar	brešar	PROPN
ejpam-5770	75	38	et	et	PROPN
ejpam-5770	75	39	al	al	PROPN
ejpam-5770	75	40	.	.	PUNCT
ejpam-5770	76	1	in	in	ADP
ejpam-5770	76	2	[	[	X
ejpam-5770	76	3	3	3	NUM
ejpam-5770	76	4	]	]	PUNCT
ejpam-5770	76	5	.	.	PUNCT
ejpam-5770	77	1	a	a	DET
ejpam-5770	77	2	krdf	krdf	NOUN
ejpam-5770	77	3	f	f	X
ejpam-5770	77	4	:	:	PUNCT
ejpam-5770	77	5	v	v	X
ejpam-5770	77	6	(	(	PUNCT
ejpam-5770	77	7	g	g	NOUN
ejpam-5770	77	8	)	)	PUNCT
ejpam-5770	77	9	→	→	SYM
ejpam-5770	77	10	p({1	p({1	PROPN
ejpam-5770	77	11	,	,	PUNCT
ejpam-5770	77	12	2	2	NUM
ejpam-5770	77	13	,	,	PUNCT
ejpam-5770	77	14	.	.	PUNCT
ejpam-5770	77	15	.	.	PUNCT
ejpam-5770	77	16	.	.	PUNCT
ejpam-5770	78	1	,	,	PUNCT
ejpam-5770	78	2	k	k	X
ejpam-5770	78	3	}	}	PUNCT
ejpam-5770	78	4	)	)	PUNCT
ejpam-5770	78	5	is	be	AUX
ejpam-5770	78	6	said	say	VERB
ejpam-5770	78	7	to	to	PART
ejpam-5770	78	8	be	be	AUX
ejpam-5770	78	9	a	a	DET
ejpam-5770	78	10	weakly	weakly	ADV
ejpam-5770	78	11	connected	connected	ADJ
ejpam-5770	78	12	k	k	ADJ
ejpam-5770	78	13	-	-	PUNCT
ejpam-5770	78	14	rainbow	rainbow	NOUN
ejpam-5770	78	15	dominating	dominating	NOUN
ejpam-5770	78	16	function	function	NOUN
ejpam-5770	78	17	(	(	PUNCT
ejpam-5770	78	18	wckrdf	wckrdf	PROPN
ejpam-5770	78	19	)	)	PUNCT
ejpam-5770	78	20	if	if	SCONJ
ejpam-5770	78	21	the	the	DET
ejpam-5770	78	22	set	set	NOUN
ejpam-5770	78	23	s	s	AUX
ejpam-5770	78	24	=	=	PUNCT
ejpam-5770	78	25	{	{	PUNCT
ejpam-5770	78	26	v	v	NUM
ejpam-5770	78	27	∈	∈	NOUN
ejpam-5770	78	28	v	v	NOUN
ejpam-5770	78	29	(	(	PUNCT
ejpam-5770	78	30	g	g	NOUN
ejpam-5770	78	31	)	)	PUNCT
ejpam-5770	78	32	:	:	PUNCT
ejpam-5770	78	33	f(v	f(v	NOUN
ejpam-5770	78	34	)	)	PUNCT
ejpam-5770	78	35	̸=	̸=	NOUN
ejpam-5770	78	36	∅	∅	NOUN
ejpam-5770	78	37	}	}	PUNCT
ejpam-5770	78	38	is	be	AUX
ejpam-5770	78	39	weakly	weakly	ADV
ejpam-5770	78	40	connected	connected	ADJ
ejpam-5770	78	41	dominating	dominating	NOUN
ejpam-5770	78	42	.	.	PUNCT
ejpam-5770	79	1	the	the	DET
ejpam-5770	79	2	weight	weight	NOUN
ejpam-5770	79	3	ω(f	ω(f	PUNCT
ejpam-5770	79	4	)	)	PUNCT
ejpam-5770	79	5	of	of	ADP
ejpam-5770	79	6	f	f	PROPN
ejpam-5770	79	7	is	be	AUX
ejpam-5770	79	8	defined	define	VERB
ejpam-5770	79	9	as	as	ADP
ejpam-5770	79	10	ω(f	ω(f	ADJ
ejpam-5770	79	11	)	)	PUNCT
ejpam-5770	80	1	=	=	SYM
ejpam-5770	80	2	∑	∑	PUNCT
ejpam-5770	80	3	v∈v	v∈v	NOUN
ejpam-5770	80	4	(	(	PUNCT
ejpam-5770	80	5	g)|f(v)|	g)|f(v)|	PROPN
ejpam-5770	80	6	.	.	PUNCT
ejpam-5770	81	1	the	the	DET
ejpam-5770	81	2	weakly	weakly	ADV
ejpam-5770	81	3	connected	connected	ADJ
ejpam-5770	81	4	k	k	ADJ
ejpam-5770	81	5	-	-	PUNCT
ejpam-5770	81	6	rainbow	rainbow	NOUN
ejpam-5770	81	7	domination	domination	NOUN
ejpam-5770	81	8	number	number	NOUN
ejpam-5770	81	9	of	of	ADP
ejpam-5770	81	10	g	g	NOUN
ejpam-5770	81	11	,	,	PUNCT
ejpam-5770	81	12	denoted	denote	VERB
ejpam-5770	81	13	by	by	ADP
ejpam-5770	81	14	γwc	γwc	ADJ
ejpam-5770	81	15	rk	rk	PROPN
ejpam-5770	81	16	(	(	PUNCT
ejpam-5770	81	17	g	g	NOUN
ejpam-5770	81	18	)	)	PUNCT
ejpam-5770	81	19	,	,	PUNCT
ejpam-5770	81	20	is	be	AUX
ejpam-5770	81	21	the	the	DET
ejpam-5770	81	22	minimum	minimum	ADJ
ejpam-5770	81	23	weight	weight	NOUN
ejpam-5770	81	24	of	of	ADP
ejpam-5770	81	25	a	a	DET
ejpam-5770	81	26	wckrdf	wckrdf	NOUN
ejpam-5770	81	27	.	.	PUNCT
ejpam-5770	82	1	a	a	DET
ejpam-5770	82	2	weakly	weakly	ADV
ejpam-5770	82	3	connected	connected	ADJ
ejpam-5770	82	4	k	k	ADJ
ejpam-5770	82	5	-	-	PUNCT
ejpam-5770	82	6	rainbow	rainbow	NOUN
ejpam-5770	82	7	dominating	dominating	NOUN
ejpam-5770	82	8	function	function	NOUN
ejpam-5770	82	9	of	of	ADP
ejpam-5770	82	10	g	g	NOUN
ejpam-5770	82	11	with	with	ADP
ejpam-5770	82	12	weight	weight	NOUN
ejpam-5770	82	13	γwc	γwc	PROPN
ejpam-5770	82	14	rk	rk	PROPN
ejpam-5770	82	15	(	(	PUNCT
ejpam-5770	82	16	g	g	NOUN
ejpam-5770	82	17	)	)	PUNCT
ejpam-5770	82	18	,	,	PUNCT
ejpam-5770	82	19	i.e.	i.e.	X
ejpam-5770	82	20	,	,	PUNCT
ejpam-5770	82	21	ω(f	ω(f	NUM
ejpam-5770	82	22	)	)	PUNCT
ejpam-5770	82	23	=	=	SYM
ejpam-5770	82	24	γwc	γwc	ADJ
ejpam-5770	82	25	rk	rk	PROPN
ejpam-5770	82	26	(	(	PUNCT
ejpam-5770	82	27	g	g	NOUN
ejpam-5770	82	28	)	)	PUNCT
ejpam-5770	82	29	,	,	PUNCT
ejpam-5770	82	30	is	be	AUX
ejpam-5770	82	31	referred	refer	VERB
ejpam-5770	82	32	to	to	ADP
ejpam-5770	82	33	as	as	ADP
ejpam-5770	82	34	a	a	DET
ejpam-5770	82	35	γwc	γwc	ADJ
ejpam-5770	82	36	rk	rk	NOUN
ejpam-5770	82	37	-function	-function	NOUN
ejpam-5770	82	38	of	of	ADP
ejpam-5770	82	39	g.	g.	NOUN
ejpam-5770	82	40	clearly	clearly	ADV
ejpam-5770	82	41	,	,	PUNCT
ejpam-5770	82	42	when	when	SCONJ
ejpam-5770	82	43	k	k	PROPN
ejpam-5770	82	44	=	=	SYM
ejpam-5770	82	45	1	1	NUM
ejpam-5770	82	46	,	,	PUNCT
ejpam-5770	82	47	the	the	DET
ejpam-5770	82	48	weakly	weakly	ADV
ejpam-5770	82	49	connected	connected	ADJ
ejpam-5770	82	50	1	1	NUM
ejpam-5770	82	51	-	-	PUNCT
ejpam-5770	82	52	rainbow	rainbow	NOUN
ejpam-5770	82	53	domination	domination	NOUN
ejpam-5770	82	54	number	number	NOUN
ejpam-5770	82	55	γwc	γwc	PROPN
ejpam-5770	82	56	r1	r1	PROPN
ejpam-5770	82	57	(	(	PUNCT
ejpam-5770	82	58	g	g	NOUN
ejpam-5770	82	59	)	)	PUNCT
ejpam-5770	82	60	is	be	AUX
ejpam-5770	82	61	equivalent	equivalent	ADJ
ejpam-5770	82	62	to	to	ADP
ejpam-5770	82	63	the	the	DET
ejpam-5770	82	64	classical	classical	ADJ
ejpam-5770	82	65	weakly	weakly	ADJ
ejpam-5770	82	66	connected	connected	ADJ
ejpam-5770	82	67	domination	domination	NOUN
ejpam-5770	82	68	number	number	NOUN
ejpam-5770	82	69	γw(g	γw(g	PUNCT
ejpam-5770	82	70	)	)	PUNCT
ejpam-5770	82	71	of	of	ADP
ejpam-5770	82	72	g.	g.	PROPN
ejpam-5770	82	73	for	for	ADP
ejpam-5770	82	74	any	any	DET
ejpam-5770	82	75	graph	graph	NOUN
ejpam-5770	82	76	g	g	NOUN
ejpam-5770	82	77	and	and	CCONJ
ejpam-5770	82	78	a	a	DET
ejpam-5770	82	79	γwc	γwc	ADJ
ejpam-5770	82	80	rk	rk	NOUN
ejpam-5770	82	81	-function	-function	PROPN
ejpam-5770	82	82	f	f	NOUN
ejpam-5770	82	83	of	of	ADP
ejpam-5770	82	84	g	g	NOUN
ejpam-5770	82	85	,	,	PUNCT
ejpam-5770	82	86	set	set	VERB
ejpam-5770	82	87	v	v	PRON
ejpam-5770	82	88	f	f	X
ejpam-5770	83	1	i	i	PRON
ejpam-5770	83	2	=	=	PUNCT
ejpam-5770	83	3	{	{	PUNCT
ejpam-5770	83	4	x	x	PROPN
ejpam-5770	83	5	∈	∈	PROPN
ejpam-5770	83	6	v	v	NOUN
ejpam-5770	83	7	(	(	PUNCT
ejpam-5770	83	8	g	g	NOUN
ejpam-5770	83	9	)	)	PUNCT
ejpam-5770	83	10	:	:	PUNCT
ejpam-5770	84	1	|f(x)|	|f(x)|	NOUN
ejpam-5770	84	2	=	=	PUNCT
ejpam-5770	84	3	i	i	PROPN
ejpam-5770	84	4	}	}	PUNCT
ejpam-5770	84	5	for	for	ADP
ejpam-5770	84	6	i	i	PROPN
ejpam-5770	84	7	∈	∈	PROPN
ejpam-5770	84	8	{	{	PUNCT
ejpam-5770	84	9	1	1	NUM
ejpam-5770	84	10	,	,	PUNCT
ejpam-5770	84	11	2	2	NUM
ejpam-5770	84	12	,	,	PUNCT
ejpam-5770	84	13	.	.	PUNCT
ejpam-5770	84	14	.	.	PUNCT
ejpam-5770	84	15	.	.	PUNCT
ejpam-5770	85	1	,	,	PUNCT
ejpam-5770	85	2	k	k	X
ejpam-5770	85	3	}	}	PUNCT
ejpam-5770	85	4	.	.	PUNCT
ejpam-5770	86	1	3	3	X
ejpam-5770	86	2	.	.	X
ejpam-5770	86	3	preliminary	preliminary	ADJ
ejpam-5770	86	4	results	result	NOUN
ejpam-5770	86	5	we	we	PRON
ejpam-5770	86	6	begin	begin	VERB
ejpam-5770	86	7	this	this	DET
ejpam-5770	86	8	section	section	NOUN
ejpam-5770	86	9	by	by	ADP
ejpam-5770	86	10	presenting	present	VERB
ejpam-5770	86	11	some	some	DET
ejpam-5770	86	12	properties	property	NOUN
ejpam-5770	86	13	and	and	CCONJ
ejpam-5770	86	14	bounds	bound	NOUN
ejpam-5770	86	15	,	,	PUNCT
ejpam-5770	86	16	and	and	CCONJ
ejpam-5770	86	17	then	then	ADV
ejpam-5770	86	18	we	we	PRON
ejpam-5770	86	19	determine	determine	VERB
ejpam-5770	86	20	the	the	DET
ejpam-5770	86	21	weakly	weakly	ADJ
ejpam-5770	86	22	connected	connected	ADJ
ejpam-5770	86	23	k	k	ADJ
ejpam-5770	86	24	-	-	PUNCT
ejpam-5770	86	25	rainbow	rainbow	NOUN
ejpam-5770	86	26	domination	domination	NOUN
ejpam-5770	86	27	number	number	NOUN
ejpam-5770	86	28	of	of	ADP
ejpam-5770	86	29	g.	g.	PROPN
ejpam-5770	86	30	remark	remark	PROPN
ejpam-5770	86	31	1	1	NUM
ejpam-5770	86	32	.	.	PUNCT
ejpam-5770	87	1	every	every	DET
ejpam-5770	87	2	weakly	weakly	ADV
ejpam-5770	87	3	connected	connected	ADJ
ejpam-5770	87	4	k	k	ADJ
ejpam-5770	87	5	-	-	PUNCT
ejpam-5770	87	6	rainbow	rainbow	NOUN
ejpam-5770	87	7	dominating	dominating	NOUN
ejpam-5770	87	8	function	function	NOUN
ejpam-5770	87	9	f	f	PROPN
ejpam-5770	87	10	of	of	ADP
ejpam-5770	87	11	g	g	PROPN
ejpam-5770	87	12	is	be	AUX
ejpam-5770	87	13	also	also	ADV
ejpam-5770	87	14	a	a	DET
ejpam-5770	87	15	krainbow	krainbow	ADJ
ejpam-5770	87	16	dominating	dominating	NOUN
ejpam-5770	87	17	function	function	NOUN
ejpam-5770	87	18	of	of	ADP
ejpam-5770	87	19	g.	g.	PROPN
ejpam-5770	87	20	in	in	ADP
ejpam-5770	87	21	particular	particular	ADJ
ejpam-5770	87	22	,	,	PUNCT
ejpam-5770	87	23	γrk(g	γrk(g	PROPN
ejpam-5770	87	24	)	)	PUNCT
ejpam-5770	87	25	≤	≤	NOUN
ejpam-5770	87	26	γwc	γwc	ADJ
ejpam-5770	87	27	rk	rk	NOUN
ejpam-5770	87	28	(	(	PUNCT
ejpam-5770	87	29	g	g	NOUN
ejpam-5770	87	30	)	)	PUNCT
ejpam-5770	87	31	.	.	PUNCT
ejpam-5770	88	1	theorem	theorem	NOUN
ejpam-5770	88	2	1	1	X
ejpam-5770	88	3	.	.	PUNCT
ejpam-5770	89	1	let	let	VERB
ejpam-5770	89	2	g	g	PRON
ejpam-5770	89	3	be	be	AUX
ejpam-5770	89	4	a	a	DET
ejpam-5770	89	5	connected	connected	ADJ
ejpam-5770	89	6	graph	graph	NOUN
ejpam-5770	89	7	with	with	ADP
ejpam-5770	89	8	∆(g	∆(g	PROPN
ejpam-5770	89	9	)	)	PUNCT
ejpam-5770	89	10	≤	≤	NUM
ejpam-5770	90	1	k	k	NOUN
ejpam-5770	90	2	for	for	ADP
ejpam-5770	90	3	some	some	DET
ejpam-5770	90	4	positive	positive	ADJ
ejpam-5770	90	5	integer	integer	NOUN
ejpam-5770	90	6	k	k	PROPN
ejpam-5770	90	7	≥	≥	NUM
ejpam-5770	90	8	2	2	NUM
ejpam-5770	90	9	.	.	PUNCT
ejpam-5770	91	1	then	then	ADV
ejpam-5770	91	2	there	there	PRON
ejpam-5770	91	3	exists	exist	VERB
ejpam-5770	91	4	a	a	DET
ejpam-5770	91	5	γwc	γwc	ADJ
ejpam-5770	91	6	rk	rk	NOUN
ejpam-5770	91	7	-function	-function	PROPN
ejpam-5770	91	8	f	f	NOUN
ejpam-5770	91	9	of	of	ADP
ejpam-5770	91	10	g	g	PROPN
ejpam-5770	91	11	such	such	ADJ
ejpam-5770	91	12	that	that	SCONJ
ejpam-5770	92	1	|f(v)|	|f(v)|	ADP
ejpam-5770	92	2	<	<	X
ejpam-5770	92	3	k	k	PROPN
ejpam-5770	92	4	for	for	ADP
ejpam-5770	92	5	every	every	DET
ejpam-5770	92	6	vertex	vertex	NOUN
ejpam-5770	92	7	v	v	ADP
ejpam-5770	92	8	∈	∈	NOUN
ejpam-5770	92	9	v	v	NOUN
ejpam-5770	92	10	(	(	PUNCT
ejpam-5770	92	11	g	g	NOUN
ejpam-5770	92	12	)	)	PUNCT
ejpam-5770	92	13	.	.	PUNCT
ejpam-5770	93	1	j.	j.	PROPN
ejpam-5770	93	2	j.	j.	PROPN
ejpam-5770	93	3	hamja	hamja	PROPN
ejpam-5770	93	4	et	et	PROPN
ejpam-5770	93	5	al	al	PROPN
ejpam-5770	93	6	.	.	PUNCT
ejpam-5770	93	7	/	/	SYM
ejpam-5770	93	8	eur	eur	PROPN
ejpam-5770	93	9	.	.	PUNCT
ejpam-5770	94	1	j.	j.	PROPN
ejpam-5770	94	2	pure	pure	PROPN
ejpam-5770	94	3	appl	appl	PROPN
ejpam-5770	94	4	.	.	PROPN
ejpam-5770	94	5	math	math	PROPN
ejpam-5770	94	6	,	,	PUNCT
ejpam-5770	94	7	18	18	NUM
ejpam-5770	94	8	(	(	PUNCT
ejpam-5770	94	9	2	2	NUM
ejpam-5770	94	10	)	)	PUNCT
ejpam-5770	94	11	(	(	PUNCT
ejpam-5770	94	12	2025	2025	NUM
ejpam-5770	94	13	)	)	PUNCT
ejpam-5770	94	14	,	,	PUNCT
ejpam-5770	94	15	5770	5770	NUM
ejpam-5770	94	16	4	4	NUM
ejpam-5770	94	17	of	of	ADP
ejpam-5770	94	18	11	11	NUM
ejpam-5770	94	19	proof	proof	NOUN
ejpam-5770	94	20	.	.	PUNCT
ejpam-5770	95	1	let	let	VERB
ejpam-5770	95	2	g	g	PRON
ejpam-5770	95	3	be	be	AUX
ejpam-5770	95	4	a	a	DET
ejpam-5770	95	5	γwc	γwc	ADJ
ejpam-5770	95	6	rk	rk	NOUN
ejpam-5770	95	7	-function	-function	NOUN
ejpam-5770	95	8	of	of	ADP
ejpam-5770	95	9	g	g	NOUN
ejpam-5770	95	10	such	such	ADJ
ejpam-5770	95	11	that	that	SCONJ
ejpam-5770	95	12	|v	|v	PROPN
ejpam-5770	95	13	g	g	PROPN
ejpam-5770	95	14	k	k	PROPN
ejpam-5770	96	1	|	|	ADV
ejpam-5770	96	2	is	be	AUX
ejpam-5770	96	3	as	as	ADV
ejpam-5770	96	4	small	small	ADJ
ejpam-5770	96	5	as	as	ADP
ejpam-5770	96	6	possible	possible	ADJ
ejpam-5770	96	7	.	.	PUNCT
ejpam-5770	97	1	we	we	PRON
ejpam-5770	97	2	claim	claim	VERB
ejpam-5770	97	3	that	that	SCONJ
ejpam-5770	97	4	|f(v)|	|f(v)|	ADP
ejpam-5770	97	5	<	<	X
ejpam-5770	97	6	k	k	PROPN
ejpam-5770	97	7	for	for	ADP
ejpam-5770	97	8	every	every	DET
ejpam-5770	97	9	vertex	vertex	NOUN
ejpam-5770	97	10	v	v	ADP
ejpam-5770	97	11	∈	∈	NOUN
ejpam-5770	97	12	v	v	NOUN
ejpam-5770	97	13	(	(	PUNCT
ejpam-5770	97	14	g	g	NOUN
ejpam-5770	97	15	)	)	PUNCT
ejpam-5770	97	16	as	as	SCONJ
ejpam-5770	97	17	desired	desire	VERB
ejpam-5770	97	18	.	.	PUNCT
ejpam-5770	98	1	assume	assume	VERB
ejpam-5770	98	2	,	,	PUNCT
ejpam-5770	98	3	to	to	ADP
ejpam-5770	98	4	the	the	DET
ejpam-5770	98	5	contrary	contrary	NOUN
ejpam-5770	98	6	,	,	PUNCT
ejpam-5770	98	7	that	that	SCONJ
ejpam-5770	98	8	there	there	PRON
ejpam-5770	98	9	exists	exist	VERB
ejpam-5770	98	10	a	a	DET
ejpam-5770	98	11	vertex	vertex	NOUN
ejpam-5770	98	12	v	v	ADP
ejpam-5770	98	13	∈	∈	PROPN
ejpam-5770	98	14	v	v	NOUN
ejpam-5770	98	15	(	(	PUNCT
ejpam-5770	98	16	g	g	NOUN
ejpam-5770	98	17	)	)	PUNCT
ejpam-5770	98	18	such	such	ADJ
ejpam-5770	98	19	that	that	SCONJ
ejpam-5770	98	20	|g(v)|	|g(v)|	PROPN
ejpam-5770	98	21	=	=	SYM
ejpam-5770	98	22	k.	k.	PROPN
ejpam-5770	98	23	since	since	SCONJ
ejpam-5770	98	24	|g(v)|	|g(v)|	PROPN
ejpam-5770	98	25	=	=	SYM
ejpam-5770	98	26	k	k	PROPN
ejpam-5770	98	27	,	,	PUNCT
ejpam-5770	98	28	by	by	ADP
ejpam-5770	98	29	the	the	DET
ejpam-5770	98	30	definition	definition	NOUN
ejpam-5770	98	31	there	there	PRON
ejpam-5770	98	32	exists	exist	VERB
ejpam-5770	98	33	a	a	DET
ejpam-5770	98	34	vertex	vertex	NOUN
ejpam-5770	98	35	x1	x1	PROPN
ejpam-5770	98	36	∈	∈	PROPN
ejpam-5770	98	37	ng(v	ng(v	PUNCT
ejpam-5770	98	38	)	)	PUNCT
ejpam-5770	98	39	such	such	ADJ
ejpam-5770	98	40	that	that	DET
ejpam-5770	98	41	g(x1	g(x1	NOUN
ejpam-5770	98	42	)	)	PUNCT
ejpam-5770	98	43	=	=	PUNCT
ejpam-5770	98	44	∅.	∅.	NOUN
ejpam-5770	98	45	let	let	VERB
ejpam-5770	98	46	there	there	PRON
ejpam-5770	98	47	exist	exist	VERB
ejpam-5770	98	48	r	r	NOUN
ejpam-5770	98	49	vertices	vertex	NOUN
ejpam-5770	98	50	x1	x1	PROPN
ejpam-5770	98	51	,	,	PUNCT
ejpam-5770	98	52	x2	x2	PROPN
ejpam-5770	98	53	,	,	PUNCT
ejpam-5770	98	54	.	.	PUNCT
ejpam-5770	98	55	.	.	PUNCT
ejpam-5770	99	1	.	.	PUNCT
ejpam-5770	100	1	,	,	PUNCT
ejpam-5770	100	2	xr	xr	PROPN
ejpam-5770	100	3	∈	∈	PROPN
ejpam-5770	100	4	ng(v	ng(v	PUNCT
ejpam-5770	100	5	)	)	PUNCT
ejpam-5770	100	6	such	such	ADJ
ejpam-5770	100	7	that	that	SCONJ
ejpam-5770	100	8	g(xi	g(xi	PROPN
ejpam-5770	100	9	)	)	PUNCT
ejpam-5770	100	10	=	=	NOUN
ejpam-5770	100	11	∅	∅	NOUN
ejpam-5770	100	12	for	for	ADP
ejpam-5770	100	13	i	i	PRON
ejpam-5770	100	14	∈	∈	PROPN
ejpam-5770	100	15	{	{	PUNCT
ejpam-5770	100	16	1	1	NUM
ejpam-5770	100	17	,	,	PUNCT
ejpam-5770	100	18	2	2	NUM
ejpam-5770	100	19	,	,	PUNCT
ejpam-5770	100	20	.	.	PUNCT
ejpam-5770	100	21	.	.	PUNCT
ejpam-5770	100	22	.	.	PUNCT
ejpam-5770	101	1	,	,	PUNCT
ejpam-5770	101	2	r	r	X
ejpam-5770	101	3	}	}	PUNCT
ejpam-5770	101	4	,	,	PUNCT
ejpam-5770	101	5	where	where	SCONJ
ejpam-5770	101	6	r	r	NOUN
ejpam-5770	101	7	≤	≤	PUNCT
ejpam-5770	101	8	∆(g	∆(g	NOUN
ejpam-5770	101	9	)	)	PUNCT
ejpam-5770	101	10	≤	≤	PROPN
ejpam-5770	102	1	k.	k.	PROPN
ejpam-5770	103	1	now	now	ADV
ejpam-5770	103	2	,	,	PUNCT
ejpam-5770	103	3	define	define	VERB
ejpam-5770	103	4	a	a	DET
ejpam-5770	103	5	new	new	ADJ
ejpam-5770	103	6	function	function	NOUN
ejpam-5770	103	7	f	f	NOUN
ejpam-5770	103	8	:	:	PUNCT
ejpam-5770	103	9	v	v	X
ejpam-5770	103	10	(	(	PUNCT
ejpam-5770	103	11	g	g	NOUN
ejpam-5770	103	12	)	)	PUNCT
ejpam-5770	103	13	→	→	SYM
ejpam-5770	104	1	p({1	p({1	PROPN
ejpam-5770	104	2	,	,	PUNCT
ejpam-5770	104	3	2	2	NUM
ejpam-5770	104	4	,	,	PUNCT
ejpam-5770	104	5	.	.	PUNCT
ejpam-5770	104	6	.	.	PUNCT
ejpam-5770	105	1	.	.	PUNCT
ejpam-5770	106	1	,	,	PUNCT
ejpam-5770	106	2	k	k	X
ejpam-5770	106	3	}	}	PUNCT
ejpam-5770	106	4	)	)	PUNCT
ejpam-5770	106	5	by	by	ADP
ejpam-5770	106	6	f(xi	f(xi	NOUN
ejpam-5770	106	7	)	)	PUNCT
ejpam-5770	106	8	=	=	PRON
ejpam-5770	106	9	{	{	PUNCT
ejpam-5770	106	10	i	i	NOUN
ejpam-5770	106	11	}	}	PUNCT
ejpam-5770	106	12	for	for	ADP
ejpam-5770	106	13	i	i	PROPN
ejpam-5770	106	14	∈	∈	PROPN
ejpam-5770	106	15	{	{	PUNCT
ejpam-5770	106	16	1	1	NUM
ejpam-5770	106	17	,	,	PUNCT
ejpam-5770	106	18	2	2	NUM
ejpam-5770	106	19	,	,	PUNCT
ejpam-5770	106	20	.	.	PUNCT
ejpam-5770	106	21	.	.	PUNCT
ejpam-5770	107	1	.	.	PUNCT
ejpam-5770	108	1	,	,	PUNCT
ejpam-5770	108	2	r	r	NOUN
ejpam-5770	108	3	−	−	PROPN
ejpam-5770	108	4	1	1	NUM
ejpam-5770	108	5	}	}	PUNCT
ejpam-5770	108	6	,	,	PUNCT
ejpam-5770	108	7	f(xr	f(xr	PROPN
ejpam-5770	108	8	)	)	PUNCT
ejpam-5770	108	9	=	=	PRON
ejpam-5770	108	10	{	{	PUNCT
ejpam-5770	108	11	r	r	NOUN
ejpam-5770	108	12	,	,	PUNCT
ejpam-5770	108	13	r	r	NOUN
ejpam-5770	108	14	+	+	NOUN
ejpam-5770	108	15	1	1	NUM
ejpam-5770	108	16	,	,	PUNCT
ejpam-5770	108	17	.	.	PUNCT
ejpam-5770	108	18	.	.	PUNCT
ejpam-5770	108	19	.	.	PUNCT
ejpam-5770	109	1	,	,	PUNCT
ejpam-5770	109	2	k	k	X
ejpam-5770	109	3	}	}	PUNCT
ejpam-5770	109	4	,	,	PUNCT
ejpam-5770	109	5	f(v	f(v	PROPN
ejpam-5770	109	6	)	)	PUNCT
ejpam-5770	109	7	=	=	SYM
ejpam-5770	109	8	∅	∅	NOUN
ejpam-5770	109	9	,	,	PUNCT
ejpam-5770	109	10	and	and	CCONJ
ejpam-5770	109	11	f(u	f(u	PROPN
ejpam-5770	109	12	)	)	PUNCT
ejpam-5770	109	13	=	=	SYM
ejpam-5770	110	1	g(u	g(u	PROPN
ejpam-5770	110	2	)	)	PUNCT
ejpam-5770	110	3	for	for	ADP
ejpam-5770	110	4	all	all	DET
ejpam-5770	110	5	other	other	ADJ
ejpam-5770	110	6	vertices	vertex	NOUN
ejpam-5770	110	7	u	u	PROPN
ejpam-5770	110	8	∈	∈	PROPN
ejpam-5770	110	9	v	v	NOUN
ejpam-5770	110	10	(	(	PUNCT
ejpam-5770	110	11	g	g	NOUN
ejpam-5770	110	12	)	)	PUNCT
ejpam-5770	110	13	.	.	PUNCT
ejpam-5770	111	1	clearly	clearly	ADV
ejpam-5770	111	2	,	,	PUNCT
ejpam-5770	111	3	f	f	PROPN
ejpam-5770	111	4	is	be	AUX
ejpam-5770	111	5	a	a	DET
ejpam-5770	111	6	weakly	weakly	ADV
ejpam-5770	111	7	connected	connected	ADJ
ejpam-5770	111	8	k	k	ADJ
ejpam-5770	111	9	-	-	PUNCT
ejpam-5770	111	10	rainbow	rainbow	NOUN
ejpam-5770	111	11	dominating	dominating	NOUN
ejpam-5770	111	12	function	function	NOUN
ejpam-5770	111	13	of	of	ADP
ejpam-5770	111	14	g	g	NOUN
ejpam-5770	111	15	of	of	ADP
ejpam-5770	111	16	weight	weight	NOUN
ejpam-5770	111	17	ω(g	ω(g	NOUN
ejpam-5770	111	18	)	)	PUNCT
ejpam-5770	111	19	,	,	PUNCT
ejpam-5770	111	20	contradicting	contradict	VERB
ejpam-5770	111	21	the	the	DET
ejpam-5770	111	22	choice	choice	NOUN
ejpam-5770	111	23	of	of	ADP
ejpam-5770	111	24	g.	g.	PROPN
ejpam-5770	111	25	thus	thus	ADV
ejpam-5770	111	26	|f(v)|	|f(v)|	ADP
ejpam-5770	111	27	<	<	X
ejpam-5770	111	28	k	k	PROPN
ejpam-5770	111	29	for	for	ADP
ejpam-5770	111	30	every	every	DET
ejpam-5770	111	31	vertex	vertex	NOUN
ejpam-5770	111	32	v	v	ADP
ejpam-5770	111	33	∈	∈	NOUN
ejpam-5770	111	34	v	v	NOUN
ejpam-5770	111	35	(	(	PUNCT
ejpam-5770	111	36	g	g	NOUN
ejpam-5770	111	37	)	)	PUNCT
ejpam-5770	111	38	,	,	PUNCT
ejpam-5770	111	39	and	and	CCONJ
ejpam-5770	111	40	the	the	DET
ejpam-5770	111	41	proof	proof	NOUN
ejpam-5770	111	42	is	be	AUX
ejpam-5770	111	43	complete	complete	ADJ
ejpam-5770	111	44	.	.	PUNCT
ejpam-5770	112	1	theorem	theorem	NOUN
ejpam-5770	112	2	2	2	NUM
ejpam-5770	112	3	.	.	PUNCT
ejpam-5770	113	1	let	let	VERB
ejpam-5770	113	2	g	g	PRON
ejpam-5770	113	3	be	be	AUX
ejpam-5770	113	4	a	a	DET
ejpam-5770	113	5	connected	connected	ADJ
ejpam-5770	113	6	graph	graph	NOUN
ejpam-5770	113	7	.	.	PUNCT
ejpam-5770	114	1	then	then	ADV
ejpam-5770	114	2	γwc	γwc	PROPN
ejpam-5770	114	3	r2	r2	PROPN
ejpam-5770	114	4	(	(	PUNCT
ejpam-5770	114	5	g	g	NOUN
ejpam-5770	114	6	)	)	PUNCT
ejpam-5770	114	7	≥	≥	NOUN
ejpam-5770	114	8	γw(g	γw(g	PUNCT
ejpam-5770	114	9	)	)	PUNCT
ejpam-5770	114	10	.	.	PUNCT
ejpam-5770	115	1	proof	proof	NOUN
ejpam-5770	115	2	.	.	PUNCT
ejpam-5770	116	1	let	let	VERB
ejpam-5770	116	2	f	f	PRON
ejpam-5770	116	3	be	be	AUX
ejpam-5770	116	4	a	a	DET
ejpam-5770	116	5	γwc	γwc	ADJ
ejpam-5770	116	6	r2	r2	PROPN
ejpam-5770	116	7	-function	-function	NOUN
ejpam-5770	116	8	of	of	ADP
ejpam-5770	116	9	g.	g.	NOUN
ejpam-5770	116	10	by	by	ADP
ejpam-5770	116	11	definition	definition	NOUN
ejpam-5770	116	12	the	the	DET
ejpam-5770	116	13	set	set	NOUN
ejpam-5770	116	14	v	v	NOUN
ejpam-5770	116	15	(	(	PUNCT
ejpam-5770	116	16	g	g	NOUN
ejpam-5770	116	17	)	)	PUNCT
ejpam-5770	116	18	\	\	PROPN
ejpam-5770	116	19	v	v	ADP
ejpam-5770	116	20	f	f	PROPN
ejpam-5770	116	21	0	0	NUM
ejpam-5770	116	22	is	be	AUX
ejpam-5770	116	23	a	a	DET
ejpam-5770	116	24	weakly	weakly	ADV
ejpam-5770	116	25	connected	connected	ADJ
ejpam-5770	116	26	dominating	dominating	NOUN
ejpam-5770	116	27	set	set	NOUN
ejpam-5770	116	28	of	of	ADP
ejpam-5770	116	29	g	g	PROPN
ejpam-5770	116	30	and	and	CCONJ
ejpam-5770	116	31	thus	thus	ADV
ejpam-5770	116	32	γwc	γwc	ADJ
ejpam-5770	116	33	r2	r2	PROPN
ejpam-5770	116	34	(	(	PUNCT
ejpam-5770	116	35	g	g	NOUN
ejpam-5770	116	36	)	)	PUNCT
ejpam-5770	116	37	=	=	SYM
ejpam-5770	116	38	∑	∑	PUNCT
ejpam-5770	116	39	v∈v	v∈v	PROPN
ejpam-5770	116	40	(	(	PUNCT
ejpam-5770	116	41	g)\v	g)\v	NOUN
ejpam-5770	116	42	f	f	NOUN
ejpam-5770	116	43	0	0	PUNCT
ejpam-5770	117	1	|f(v)|	|f(v)|	ADP
ejpam-5770	117	2	≥	≥	NUM
ejpam-5770	117	3	∑	∑	PUNCT
ejpam-5770	117	4	v∈v	v∈v	PROPN
ejpam-5770	117	5	(	(	PUNCT
ejpam-5770	117	6	g)\v	g)\v	NOUN
ejpam-5770	117	7	f	f	PROPN
ejpam-5770	117	8	0	0	PROPN
ejpam-5770	117	9	1	1	NUM
ejpam-5770	117	10	=	=	SYM
ejpam-5770	117	11	|v	|v	X
ejpam-5770	117	12	(	(	PUNCT
ejpam-5770	117	13	g	g	NOUN
ejpam-5770	117	14	)	)	PUNCT
ejpam-5770	117	15	\	\	PROPN
ejpam-5770	118	1	v	v	ADP
ejpam-5770	118	2	f	f	PROPN
ejpam-5770	118	3	0	0	NUM
ejpam-5770	118	4	|	|	CCONJ
ejpam-5770	118	5	≥	≥	NOUN
ejpam-5770	118	6	γw(g	γw(g	PUNCT
ejpam-5770	118	7	)	)	PUNCT
ejpam-5770	118	8	,	,	PUNCT
ejpam-5770	118	9	as	as	SCONJ
ejpam-5770	118	10	desired	desire	VERB
ejpam-5770	118	11	.	.	PUNCT
ejpam-5770	119	1	the	the	DET
ejpam-5770	119	2	graph	graph	NOUN
ejpam-5770	119	3	g	g	PROPN
ejpam-5770	119	4	illustrated	illustrate	VERB
ejpam-5770	119	5	in	in	ADP
ejpam-5770	119	6	figure	figure	NOUN
ejpam-5770	119	7	1	1	NUM
ejpam-5770	119	8	demonstrates	demonstrate	VERB
ejpam-5770	119	9	that	that	SCONJ
ejpam-5770	119	10	the	the	DET
ejpam-5770	119	11	bound	bind	VERB
ejpam-5770	119	12	in	in	ADP
ejpam-5770	119	13	theorem	theorem	ADJ
ejpam-5770	119	14	2	2	NUM
ejpam-5770	119	15	is	be	AUX
ejpam-5770	119	16	sharp	sharp	ADJ
ejpam-5770	119	17	.	.	PUNCT
ejpam-5770	119	18	...	...	PUNCT
ejpam-5770	120	1	......	......	PUNCT
ejpam-5770	120	2	...	...	PUNCT
ejpam-5770	121	1	x1	x1	NUM
ejpam-5770	121	2	xn	xn	PUNCT
ejpam-5770	122	1	y1	y1	INTJ
ejpam-5770	122	2	yn	yn	PRON
ejpam-5770	122	3	g	g	NOUN
ejpam-5770	122	4	:	:	PUNCT
ejpam-5770	122	5	figure	figure	NOUN
ejpam-5770	122	6	1	1	NUM
ejpam-5770	122	7	:	:	PUNCT
ejpam-5770	122	8	a	a	DET
ejpam-5770	122	9	graph	graph	NOUN
ejpam-5770	122	10	g	g	NOUN
ejpam-5770	122	11	attaining	attain	VERB
ejpam-5770	122	12	the	the	DET
ejpam-5770	122	13	bound	bind	VERB
ejpam-5770	122	14	in	in	ADP
ejpam-5770	122	15	theorem	theorem	NOUN
ejpam-5770	122	16	2	2	NUM
ejpam-5770	122	17	the	the	DET
ejpam-5770	122	18	exact	exact	ADJ
ejpam-5770	122	19	values	value	NOUN
ejpam-5770	122	20	of	of	ADP
ejpam-5770	122	21	the	the	DET
ejpam-5770	122	22	k	k	ADJ
ejpam-5770	122	23	-	-	PUNCT
ejpam-5770	122	24	rainbow	rainbow	NOUN
ejpam-5770	122	25	domination	domination	NOUN
ejpam-5770	122	26	number	number	NOUN
ejpam-5770	122	27	for	for	ADP
ejpam-5770	122	28	k	k	PROPN
ejpam-5770	122	29	∈	∈	PROPN
ejpam-5770	122	30	{	{	PUNCT
ejpam-5770	122	31	2	2	NUM
ejpam-5770	122	32	,	,	PUNCT
ejpam-5770	122	33	3	3	NUM
ejpam-5770	122	34	}	}	PUNCT
ejpam-5770	122	35	of	of	ADP
ejpam-5770	122	36	paths	path	NOUN
ejpam-5770	122	37	and	and	CCONJ
ejpam-5770	122	38	cycles	cycle	NOUN
ejpam-5770	122	39	determined	determine	VERB
ejpam-5770	122	40	in	in	ADP
ejpam-5770	122	41	[	[	X
ejpam-5770	122	42	6	6	NUM
ejpam-5770	122	43	]	]	PUNCT
ejpam-5770	122	44	and	and	CCONJ
ejpam-5770	122	45	[	[	X
ejpam-5770	122	46	21	21	NUM
ejpam-5770	122	47	]	]	PUNCT
ejpam-5770	122	48	as	as	SCONJ
ejpam-5770	122	49	follows	follow	VERB
ejpam-5770	122	50	.	.	PUNCT
ejpam-5770	123	1	theorem	theorem	NOUN
ejpam-5770	123	2	3	3	NUM
ejpam-5770	123	3	.	.	PUNCT
ejpam-5770	123	4	(	(	PUNCT
ejpam-5770	123	5	i	i	NOUN
ejpam-5770	123	6	)	)	PUNCT
ejpam-5770	123	7	γr2(pn	γr2(pn	PROPN
ejpam-5770	123	8	)	)	PUNCT
ejpam-5770	123	9	=	=	PUNCT
ejpam-5770	124	1	⌊	⌊	VERB
ejpam-5770	124	2	n	n	ADV
ejpam-5770	124	3	2	2	NUM
ejpam-5770	124	4	⌋	⌋	NOUN
ejpam-5770	124	5	+	+	CCONJ
ejpam-5770	124	6	1	1	NUM
ejpam-5770	124	7	and	and	CCONJ
ejpam-5770	124	8	γr2(cn	γr2(cn	NOUN
ejpam-5770	124	9	)	)	PUNCT
ejpam-5770	124	10	=	=	PUNCT
ejpam-5770	125	1	⌊	⌊	VERB
ejpam-5770	125	2	n	n	ADV
ejpam-5770	125	3	2	2	NUM
ejpam-5770	125	4	⌋	⌋	NOUN
ejpam-5770	125	5	+	+	CCONJ
ejpam-5770	125	6	⌈	⌈	SYM
ejpam-5770	125	7	n	n	PRON
ejpam-5770	125	8	4	4	NUM
ejpam-5770	125	9	⌉	⌉	PART
ejpam-5770	125	10	−	−	ADP
ejpam-5770	125	11	⌊	⌊	PROPN
ejpam-5770	125	12	n	n	ADV
ejpam-5770	125	13	4	4	NUM
ejpam-5770	125	14	⌋	⌋	NOUN
ejpam-5770	125	15	.	.	PUNCT
ejpam-5770	126	1	(	(	PUNCT
ejpam-5770	126	2	ii	ii	NOUN
ejpam-5770	126	3	)	)	PUNCT
ejpam-5770	126	4	for	for	ADP
ejpam-5770	126	5	n	n	X
ejpam-5770	126	6	≥	≥	NUM
ejpam-5770	126	7	5	5	NUM
ejpam-5770	126	8	,	,	PUNCT
ejpam-5770	126	9	γr3(pn	γr3(pn	NOUN
ejpam-5770	126	10	)	)	PUNCT
ejpam-5770	126	11	=	=	PUNCT
ejpam-5770	126	12			PUNCT
ejpam-5770	126	13	⌈	⌈	NOUN
ejpam-5770	126	14	3n	3n	NUM
ejpam-5770	126	15	4	4	NUM
ejpam-5770	126	16	⌉	⌉	NOUN
ejpam-5770	126	17	+	+	CCONJ
ejpam-5770	126	18	1	1	NUM
ejpam-5770	126	19	if	if	SCONJ
ejpam-5770	126	20	n	n	PRON
ejpam-5770	126	21	≡	≡	PROPN
ejpam-5770	126	22	0	0	PUNCT
ejpam-5770	126	23	(	(	PUNCT
ejpam-5770	126	24	mod	mod	PROPN
ejpam-5770	126	25	4)⌈	4)⌈	PROPN
ejpam-5770	126	26	3n	3n	NUM
ejpam-5770	126	27	4	4	NUM
ejpam-5770	126	28	⌉	⌉	NOUN
ejpam-5770	126	29	if	if	SCONJ
ejpam-5770	126	30	n	n	PRON
ejpam-5770	126	31	≡	≡	PROPN
ejpam-5770	126	32	1	1	NUM
ejpam-5770	126	33	,	,	PUNCT
ejpam-5770	126	34	2	2	NUM
ejpam-5770	126	35	,	,	PUNCT
ejpam-5770	126	36	3	3	NUM
ejpam-5770	126	37	(	(	PUNCT
ejpam-5770	126	38	mod	mod	NOUN
ejpam-5770	126	39	4	4	NUM
ejpam-5770	126	40	)	)	PUNCT
ejpam-5770	126	41	.	.	PUNCT
ejpam-5770	127	1	(	(	PUNCT
ejpam-5770	127	2	iii	iii	NOUN
ejpam-5770	127	3	)	)	PUNCT
ejpam-5770	127	4	for	for	ADP
ejpam-5770	127	5	n	n	X
ejpam-5770	127	6	≥	≥	NUM
ejpam-5770	127	7	5	5	NUM
ejpam-5770	127	8	,	,	PUNCT
ejpam-5770	127	9	γr3(cn	γr3(cn	NUM
ejpam-5770	127	10	)	)	PUNCT
ejpam-5770	128	1	=	=	PUNCT
ejpam-5770	129	1	⌈	⌈	NOUN
ejpam-5770	129	2	3n	3n	NUM
ejpam-5770	129	3	4	4	NUM
ejpam-5770	129	4	⌉	⌉	X
ejpam-5770	129	5	.	.	PUNCT
ejpam-5770	130	1	j.	j.	PROPN
ejpam-5770	130	2	j.	j.	PROPN
ejpam-5770	130	3	hamja	hamja	PROPN
ejpam-5770	130	4	et	et	PROPN
ejpam-5770	130	5	al	al	PROPN
ejpam-5770	130	6	.	.	PUNCT
ejpam-5770	130	7	/	/	SYM
ejpam-5770	130	8	eur	eur	PROPN
ejpam-5770	130	9	.	.	PUNCT
ejpam-5770	131	1	j.	j.	PROPN
ejpam-5770	131	2	pure	pure	PROPN
ejpam-5770	131	3	appl	appl	PROPN
ejpam-5770	131	4	.	.	PROPN
ejpam-5770	131	5	math	math	PROPN
ejpam-5770	131	6	,	,	PUNCT
ejpam-5770	131	7	18	18	NUM
ejpam-5770	131	8	(	(	PUNCT
ejpam-5770	131	9	2	2	NUM
ejpam-5770	131	10	)	)	PUNCT
ejpam-5770	131	11	(	(	PUNCT
ejpam-5770	131	12	2025	2025	NUM
ejpam-5770	131	13	)	)	PUNCT
ejpam-5770	131	14	,	,	PUNCT
ejpam-5770	131	15	5770	5770	NUM
ejpam-5770	131	16	5	5	NUM
ejpam-5770	131	17	of	of	ADP
ejpam-5770	131	18	11	11	NUM
ejpam-5770	131	19	using	use	VERB
ejpam-5770	131	20	theorem	theorem	NOUN
ejpam-5770	131	21	3	3	NUM
ejpam-5770	131	22	and	and	CCONJ
ejpam-5770	131	23	remark	remark	NOUN
ejpam-5770	131	24	1	1	NUM
ejpam-5770	131	25	,	,	PUNCT
ejpam-5770	131	26	we	we	PRON
ejpam-5770	131	27	will	will	AUX
ejpam-5770	131	28	determine	determine	VERB
ejpam-5770	131	29	the	the	DET
ejpam-5770	131	30	weakly	weakly	ADJ
ejpam-5770	131	31	connected	connected	ADJ
ejpam-5770	131	32	k	k	ADJ
ejpam-5770	131	33	-	-	PUNCT
ejpam-5770	131	34	rainbow	rainbow	NOUN
ejpam-5770	131	35	domination	domination	NOUN
ejpam-5770	131	36	number	number	NOUN
ejpam-5770	131	37	for	for	ADP
ejpam-5770	131	38	k	k	PROPN
ejpam-5770	131	39	∈	∈	PROPN
ejpam-5770	131	40	{	{	PUNCT
ejpam-5770	131	41	2	2	NUM
ejpam-5770	131	42	,	,	PUNCT
ejpam-5770	131	43	3	3	NUM
ejpam-5770	131	44	}	}	PUNCT
ejpam-5770	131	45	for	for	ADP
ejpam-5770	131	46	paths	path	NOUN
ejpam-5770	131	47	pn	pn	NOUN
ejpam-5770	131	48	and	and	CCONJ
ejpam-5770	131	49	cycle	cycle	NOUN
ejpam-5770	131	50	cn	cn	PROPN
ejpam-5770	131	51	.	.	PROPN
ejpam-5770	131	52	proposition	proposition	NOUN
ejpam-5770	131	53	1	1	NUM
ejpam-5770	131	54	.	.	PUNCT
ejpam-5770	132	1	let	let	VERB
ejpam-5770	132	2	n	n	PRON
ejpam-5770	132	3	be	be	AUX
ejpam-5770	132	4	a	a	DET
ejpam-5770	132	5	positive	positive	ADJ
ejpam-5770	132	6	integer	integer	NOUN
ejpam-5770	132	7	.	.	PUNCT
ejpam-5770	133	1	then	then	ADV
ejpam-5770	133	2	(	(	PUNCT
ejpam-5770	133	3	i	i	NOUN
ejpam-5770	133	4	)	)	PUNCT
ejpam-5770	133	5	for	for	ADP
ejpam-5770	133	6	all	all	PRON
ejpam-5770	133	7	n	n	PRON
ejpam-5770	133	8	≥	≥	NUM
ejpam-5770	133	9	1	1	NUM
ejpam-5770	133	10	,	,	PUNCT
ejpam-5770	133	11	γwc	γwc	PROPN
ejpam-5770	133	12	r2	r2	PROPN
ejpam-5770	133	13	(	(	PUNCT
ejpam-5770	133	14	pn	pn	NOUN
ejpam-5770	133	15	)	)	PUNCT
ejpam-5770	133	16	=	=	PUNCT
ejpam-5770	133	17	⌊	⌊	VERB
ejpam-5770	133	18	n	n	ADV
ejpam-5770	133	19	2	2	NUM
ejpam-5770	133	20	⌋	⌋	NOUN
ejpam-5770	133	21	+	+	CCONJ
ejpam-5770	133	22	1	1	X
ejpam-5770	133	23	.	.	X
ejpam-5770	133	24	(	(	PUNCT
ejpam-5770	133	25	ii	ii	NOUN
ejpam-5770	133	26	)	)	PUNCT
ejpam-5770	133	27	for	for	ADP
ejpam-5770	133	28	all	all	DET
ejpam-5770	133	29	n	n	PRON
ejpam-5770	133	30	≥	≥	NUM
ejpam-5770	133	31	1	1	NUM
ejpam-5770	133	32	,	,	PUNCT
ejpam-5770	133	33	γwc	γwc	PROPN
ejpam-5770	133	34	r3	r3	PROPN
ejpam-5770	133	35	(	(	PUNCT
ejpam-5770	133	36	pn	pn	NOUN
ejpam-5770	133	37	)	)	PUNCT
ejpam-5770	133	38	=	=	PRON
ejpam-5770	133	39	{	{	PUNCT
ejpam-5770	133	40	⌈3n4	⌈3n4	NUM
ejpam-5770	133	41	⌉+	⌉+	NUM
ejpam-5770	133	42	1	1	NUM
ejpam-5770	133	43	if	if	SCONJ
ejpam-5770	133	44	n	n	PRON
ejpam-5770	133	45	≡	≡	PROPN
ejpam-5770	133	46	0	0	PUNCT
ejpam-5770	133	47	(	(	PUNCT
ejpam-5770	133	48	mod	mod	NOUN
ejpam-5770	133	49	4	4	NUM
ejpam-5770	133	50	)	)	PUNCT
ejpam-5770	133	51	⌈3n4	⌈3n4	NOUN
ejpam-5770	133	52	⌉	⌉	X
ejpam-5770	133	53	if	if	SCONJ
ejpam-5770	133	54	n	n	PRON
ejpam-5770	133	55	≡	≡	PROPN
ejpam-5770	133	56	1	1	NUM
ejpam-5770	133	57	,	,	PUNCT
ejpam-5770	133	58	2	2	NUM
ejpam-5770	133	59	,	,	PUNCT
ejpam-5770	133	60	3	3	NUM
ejpam-5770	133	61	(	(	PUNCT
ejpam-5770	133	62	mod	mod	NOUN
ejpam-5770	133	63	4	4	NUM
ejpam-5770	133	64	)	)	PUNCT
ejpam-5770	133	65	.	.	PUNCT
ejpam-5770	134	1	(	(	PUNCT
ejpam-5770	134	2	iii	iii	X
ejpam-5770	134	3	)	)	PUNCT
ejpam-5770	134	4	for	for	ADP
ejpam-5770	134	5	all	all	PRON
ejpam-5770	134	6	n	n	PRON
ejpam-5770	134	7	≥	≥	NOUN
ejpam-5770	134	8	3	3	NUM
ejpam-5770	134	9	,	,	PUNCT
ejpam-5770	134	10	γwc	γwc	PROPN
ejpam-5770	134	11	r2	r2	PROPN
ejpam-5770	134	12	(	(	PUNCT
ejpam-5770	134	13	cn	cn	PROPN
ejpam-5770	134	14	)	)	PUNCT
ejpam-5770	134	15	=	=	PUNCT
ejpam-5770	134	16	⌊	⌊	VERB
ejpam-5770	134	17	n	n	ADV
ejpam-5770	134	18	2	2	NUM
ejpam-5770	134	19	⌋	⌋	NOUN
ejpam-5770	134	20	+	+	CCONJ
ejpam-5770	134	21	⌈	⌈	SYM
ejpam-5770	134	22	n	n	PRON
ejpam-5770	134	23	4	4	NUM
ejpam-5770	134	24	⌉	⌉	PART
ejpam-5770	134	25	−	−	ADP
ejpam-5770	134	26	⌊	⌊	PROPN
ejpam-5770	134	27	n	n	ADV
ejpam-5770	134	28	4	4	NUM
ejpam-5770	134	29	⌋	⌋	NOUN
ejpam-5770	134	30	(	(	PUNCT
ejpam-5770	134	31	iv	iv	X
ejpam-5770	134	32	)	)	PUNCT
ejpam-5770	134	33	for	for	ADP
ejpam-5770	134	34	all	all	DET
ejpam-5770	134	35	n	n	PRON
ejpam-5770	134	36	≥	≥	NOUN
ejpam-5770	134	37	3	3	NUM
ejpam-5770	134	38	,	,	PUNCT
ejpam-5770	134	39	γwc	γwc	PROPN
ejpam-5770	134	40	r3	r3	PROPN
ejpam-5770	134	41	(	(	PUNCT
ejpam-5770	134	42	cn	cn	PROPN
ejpam-5770	134	43	)	)	PUNCT
ejpam-5770	134	44	=	=	PUNCT
ejpam-5770	135	1	⌈	⌈	NOUN
ejpam-5770	135	2	3n	3n	NUM
ejpam-5770	135	3	4	4	NUM
ejpam-5770	135	4	⌉	⌉	X
ejpam-5770	135	5	.	.	PUNCT
ejpam-5770	136	1	proof	proof	NOUN
ejpam-5770	136	2	.	.	PUNCT
ejpam-5770	137	1	let	let	VERB
ejpam-5770	137	2	pn	pn	VERB
ejpam-5770	137	3	=	=	PUNCT
ejpam-5770	138	1	[	[	X
ejpam-5770	138	2	v1	v1	NOUN
ejpam-5770	138	3	,	,	PUNCT
ejpam-5770	138	4	v2	v2	NOUN
ejpam-5770	138	5	,	,	PUNCT
ejpam-5770	138	6	.	.	PUNCT
ejpam-5770	138	7	.	.	PUNCT
ejpam-5770	139	1	.	.	PUNCT
ejpam-5770	140	1	,	,	PUNCT
ejpam-5770	140	2	vn−1	vn−1	PROPN
ejpam-5770	140	3	,	,	PUNCT
ejpam-5770	140	4	vn	vn	PART
ejpam-5770	140	5	]	]	X
ejpam-5770	140	6	be	be	AUX
ejpam-5770	140	7	a	a	DET
ejpam-5770	140	8	path	path	NOUN
ejpam-5770	140	9	of	of	ADP
ejpam-5770	140	10	order	order	NOUN
ejpam-5770	140	11	n	n	NOUN
ejpam-5770	140	12	and	and	CCONJ
ejpam-5770	140	13	cn	cn	ADJ
ejpam-5770	140	14	=	=	PUNCT
ejpam-5770	141	1	[	[	X
ejpam-5770	141	2	v1	v1	NOUN
ejpam-5770	141	3	,	,	PUNCT
ejpam-5770	141	4	v2	v2	NOUN
ejpam-5770	141	5	,	,	PUNCT
ejpam-5770	141	6	.	.	PUNCT
ejpam-5770	141	7	.	.	PUNCT
ejpam-5770	142	1	.	.	PUNCT
ejpam-5770	143	1	,	,	PUNCT
ejpam-5770	143	2	vn−1	vn−1	PROPN
ejpam-5770	143	3	,	,	PUNCT
ejpam-5770	143	4	vn	vn	NOUN
ejpam-5770	143	5	,	,	PUNCT
ejpam-5770	143	6	v1	v1	PROPN
ejpam-5770	143	7	]	]	PUNCT
ejpam-5770	143	8	be	be	VERB
ejpam-5770	143	9	a	a	DET
ejpam-5770	143	10	cycle	cycle	NOUN
ejpam-5770	143	11	of	of	ADP
ejpam-5770	143	12	order	order	NOUN
ejpam-5770	143	13	n.	n.	NOUN
ejpam-5770	143	14	we	we	PRON
ejpam-5770	143	15	first	first	ADV
ejpam-5770	143	16	establish	establish	VERB
ejpam-5770	143	17	the	the	DET
ejpam-5770	143	18	result	result	NOUN
ejpam-5770	143	19	(	(	PUNCT
ejpam-5770	143	20	i	i	NOUN
ejpam-5770	143	21	)	)	PUNCT
ejpam-5770	143	22	and	and	CCONJ
ejpam-5770	143	23	(	(	PUNCT
ejpam-5770	143	24	ii	ii	NOUN
ejpam-5770	143	25	)	)	PUNCT
ejpam-5770	143	26	for	for	ADP
ejpam-5770	143	27	paths	path	NOUN
ejpam-5770	143	28	.	.	PUNCT
ejpam-5770	144	1	for	for	ADP
ejpam-5770	144	2	k	k	PROPN
ejpam-5770	144	3	∈	∈	PROPN
ejpam-5770	144	4	{	{	PUNCT
ejpam-5770	144	5	2	2	NUM
ejpam-5770	144	6	,	,	PUNCT
ejpam-5770	144	7	3	3	NUM
ejpam-5770	144	8	}	}	PUNCT
ejpam-5770	144	9	,	,	PUNCT
ejpam-5770	144	10	define	define	VERB
ejpam-5770	144	11	the	the	DET
ejpam-5770	144	12	function	function	NOUN
ejpam-5770	144	13	fk	fk	INTJ
ejpam-5770	144	14	:	:	PUNCT
ejpam-5770	144	15	v	v	X
ejpam-5770	144	16	(	(	PUNCT
ejpam-5770	144	17	pn	pn	NOUN
ejpam-5770	144	18	)	)	PUNCT
ejpam-5770	144	19	→	→	SYM
ejpam-5770	144	20	p({1	p({1	PROPN
ejpam-5770	144	21	,	,	PUNCT
ejpam-5770	144	22	2	2	NUM
ejpam-5770	144	23	,	,	PUNCT
ejpam-5770	144	24	.	.	PUNCT
ejpam-5770	144	25	.	.	PUNCT
ejpam-5770	145	1	.	.	PUNCT
ejpam-5770	146	1	,	,	PUNCT
ejpam-5770	146	2	k	k	X
ejpam-5770	146	3	}	}	PUNCT
ejpam-5770	146	4	)	)	PUNCT
ejpam-5770	146	5	as	as	SCONJ
ejpam-5770	146	6	follows	follow	VERB
ejpam-5770	146	7	:	:	PUNCT
ejpam-5770	146	8	(	(	PUNCT
ejpam-5770	146	9	a	a	X
ejpam-5770	146	10	)	)	PUNCT
ejpam-5770	146	11	if	if	SCONJ
ejpam-5770	146	12	n	n	PRON
ejpam-5770	146	13	≡	≡	PROPN
ejpam-5770	146	14	1	1	NUM
ejpam-5770	146	15	(	(	PUNCT
ejpam-5770	146	16	mod	mod	NOUN
ejpam-5770	146	17	4	4	NUM
ejpam-5770	146	18	)	)	PUNCT
ejpam-5770	146	19	,	,	PUNCT
ejpam-5770	146	20	then	then	ADV
ejpam-5770	146	21	let	let	VERB
ejpam-5770	146	22	fk(v4i+1	fk(v4i+1	NOUN
ejpam-5770	146	23	)	)	PUNCT
ejpam-5770	146	24	=	=	PRON
ejpam-5770	147	1	{	{	PUNCT
ejpam-5770	147	2	1	1	NUM
ejpam-5770	147	3	}	}	PUNCT
ejpam-5770	147	4	for	for	ADP
ejpam-5770	147	5	0	0	NUM
ejpam-5770	147	6	≤	≤	NUM
ejpam-5770	147	7	i	i	PRON
ejpam-5770	147	8	≤	≤	ADJ
ejpam-5770	147	9	n−1	n−1	PROPN
ejpam-5770	147	10	4	4	NUM
ejpam-5770	147	11	,	,	PUNCT
ejpam-5770	147	12	fk(v4i+3	fk(v4i+3	NOUN
ejpam-5770	147	13	)	)	PUNCT
ejpam-5770	147	14	=	=	SYM
ejpam-5770	147	15	{	{	PUNCT
ejpam-5770	147	16	2	2	NUM
ejpam-5770	147	17	,	,	PUNCT
ejpam-5770	147	18	k	k	NOUN
ejpam-5770	147	19	}	}	PUNCT
ejpam-5770	147	20	for	for	ADP
ejpam-5770	147	21	0	0	NUM
ejpam-5770	147	22	≤	≤	NUM
ejpam-5770	148	1	i	i	PRON
ejpam-5770	148	2	≤	≤	NUM
ejpam-5770	148	3	n−5	n−5	PROPN
ejpam-5770	148	4	4	4	NUM
ejpam-5770	148	5	,	,	PUNCT
ejpam-5770	148	6	and	and	CCONJ
ejpam-5770	148	7	fk(x	fk(x	NOUN
ejpam-5770	148	8	)	)	PUNCT
ejpam-5770	148	9	=	=	NOUN
ejpam-5770	148	10	∅	∅	NOUN
ejpam-5770	148	11	otherwise	otherwise	ADV
ejpam-5770	148	12	.	.	PUNCT
ejpam-5770	149	1	(	(	PUNCT
ejpam-5770	149	2	b	b	X
ejpam-5770	149	3	)	)	PUNCT
ejpam-5770	149	4	if	if	SCONJ
ejpam-5770	149	5	n	n	PRON
ejpam-5770	149	6	≡	≡	PROPN
ejpam-5770	149	7	2	2	NUM
ejpam-5770	149	8	(	(	PUNCT
ejpam-5770	149	9	mod	mod	NOUN
ejpam-5770	149	10	4	4	NUM
ejpam-5770	149	11	)	)	PUNCT
ejpam-5770	149	12	,	,	PUNCT
ejpam-5770	149	13	then	then	ADV
ejpam-5770	149	14	let	let	VERB
ejpam-5770	149	15	fk(vn	fk(vn	PROPN
ejpam-5770	149	16	)	)	PUNCT
ejpam-5770	150	1	=	=	PUNCT
ejpam-5770	150	2	{	{	PUNCT
ejpam-5770	150	3	1	1	NUM
ejpam-5770	150	4	}	}	PUNCT
ejpam-5770	150	5	,	,	PUNCT
ejpam-5770	150	6	fk(v4i+1	fk(v4i+1	NOUN
ejpam-5770	150	7	)	)	PUNCT
ejpam-5770	150	8	=	=	PRON
ejpam-5770	150	9	{	{	PUNCT
ejpam-5770	150	10	1	1	NUM
ejpam-5770	150	11	}	}	PUNCT
ejpam-5770	150	12	for	for	ADP
ejpam-5770	150	13	0	0	NUM
ejpam-5770	150	14	≤	≤	NUM
ejpam-5770	151	1	i	i	PRON
ejpam-5770	151	2	≤	≤	ADJ
ejpam-5770	151	3	n−2	n−2	PROPN
ejpam-5770	151	4	4	4	NUM
ejpam-5770	151	5	,	,	PUNCT
ejpam-5770	151	6	fk(v4i+3	fk(v4i+3	NOUN
ejpam-5770	151	7	)	)	PUNCT
ejpam-5770	151	8	=	=	SYM
ejpam-5770	151	9	{	{	PUNCT
ejpam-5770	151	10	2	2	NUM
ejpam-5770	151	11	,	,	PUNCT
ejpam-5770	151	12	k	k	NOUN
ejpam-5770	151	13	}	}	PUNCT
ejpam-5770	151	14	for	for	ADP
ejpam-5770	151	15	0	0	NUM
ejpam-5770	151	16	≤	≤	NUM
ejpam-5770	151	17	i	i	PRON
ejpam-5770	151	18	≤	≤	PUNCT
ejpam-5770	151	19	n−6	n−6	PROPN
ejpam-5770	151	20	4	4	NUM
ejpam-5770	151	21	,	,	PUNCT
ejpam-5770	151	22	and	and	CCONJ
ejpam-5770	151	23	fk(x	fk(x	NOUN
ejpam-5770	151	24	)	)	PUNCT
ejpam-5770	152	1	=	=	NOUN
ejpam-5770	152	2	∅	∅	NOUN
ejpam-5770	152	3	otherwise	otherwise	ADV
ejpam-5770	152	4	.	.	PUNCT
ejpam-5770	153	1	(	(	PUNCT
ejpam-5770	153	2	c	c	X
ejpam-5770	153	3	)	)	PUNCT
ejpam-5770	153	4	if	if	SCONJ
ejpam-5770	153	5	n	n	NUM
ejpam-5770	153	6	≡	≡	PROPN
ejpam-5770	153	7	3	3	NUM
ejpam-5770	153	8	(	(	PUNCT
ejpam-5770	153	9	mod	mod	NOUN
ejpam-5770	153	10	4	4	NUM
ejpam-5770	153	11	)	)	PUNCT
ejpam-5770	153	12	,	,	PUNCT
ejpam-5770	153	13	then	then	ADV
ejpam-5770	153	14	let	let	VERB
ejpam-5770	153	15	fk(v4i+1	fk(v4i+1	NOUN
ejpam-5770	153	16	)	)	PUNCT
ejpam-5770	153	17	=	=	PRON
ejpam-5770	153	18	{	{	PUNCT
ejpam-5770	153	19	1	1	NUM
ejpam-5770	153	20	}	}	PUNCT
ejpam-5770	153	21	,	,	PUNCT
ejpam-5770	153	22	fk(v4i+3	fk(v4i+3	NOUN
ejpam-5770	153	23	)	)	PUNCT
ejpam-5770	154	1	=	=	SYM
ejpam-5770	154	2	{	{	PUNCT
ejpam-5770	154	3	2	2	NUM
ejpam-5770	154	4	,	,	PUNCT
ejpam-5770	154	5	k	k	NOUN
ejpam-5770	154	6	}	}	PUNCT
ejpam-5770	154	7	for	for	ADP
ejpam-5770	154	8	0	0	NUM
ejpam-5770	154	9	≤	≤	NOUN
ejpam-5770	154	10	i	i	PRON
ejpam-5770	154	11	≤	≤	ADJ
ejpam-5770	154	12	n−3	n−3	PROPN
ejpam-5770	154	13	4	4	NUM
ejpam-5770	154	14	and	and	CCONJ
ejpam-5770	154	15	fk(x	fk(x	NOUN
ejpam-5770	154	16	)	)	PUNCT
ejpam-5770	155	1	=	=	NOUN
ejpam-5770	155	2	∅	∅	NOUN
ejpam-5770	155	3	otherwise	otherwise	ADV
ejpam-5770	155	4	.	.	PUNCT
ejpam-5770	156	1	(	(	PUNCT
ejpam-5770	156	2	d	d	X
ejpam-5770	156	3	)	)	PUNCT
ejpam-5770	156	4	if	if	SCONJ
ejpam-5770	156	5	n	n	PRON
ejpam-5770	156	6	≡	≡	PROPN
ejpam-5770	156	7	0	0	PUNCT
ejpam-5770	156	8	(	(	PUNCT
ejpam-5770	156	9	mod	mod	PROPN
ejpam-5770	156	10	4	4	NUM
ejpam-5770	156	11	)	)	PUNCT
ejpam-5770	156	12	,	,	PUNCT
ejpam-5770	156	13	then	then	ADV
ejpam-5770	156	14	let	let	VERB
ejpam-5770	156	15	fk(vn	fk(vn	PROPN
ejpam-5770	156	16	)	)	PUNCT
ejpam-5770	157	1	=	=	PUNCT
ejpam-5770	157	2	{	{	PUNCT
ejpam-5770	157	3	1	1	NUM
ejpam-5770	157	4	}	}	PUNCT
ejpam-5770	157	5	,	,	PUNCT
ejpam-5770	157	6	fk(v4i+1	fk(v4i+1	NOUN
ejpam-5770	157	7	)	)	PUNCT
ejpam-5770	157	8	=	=	PRON
ejpam-5770	157	9	{	{	PUNCT
ejpam-5770	157	10	1	1	NUM
ejpam-5770	157	11	}	}	PUNCT
ejpam-5770	157	12	,	,	PUNCT
ejpam-5770	157	13	fk(v4i+3	fk(v4i+3	NOUN
ejpam-5770	157	14	)	)	PUNCT
ejpam-5770	157	15	=	=	SYM
ejpam-5770	157	16	{	{	PUNCT
ejpam-5770	157	17	2	2	NUM
ejpam-5770	157	18	,	,	PUNCT
ejpam-5770	157	19	k	k	NOUN
ejpam-5770	157	20	}	}	PUNCT
ejpam-5770	157	21	for	for	ADP
ejpam-5770	157	22	0	0	NUM
ejpam-5770	157	23	≤	≤	NUM
ejpam-5770	157	24	i	i	PRON
ejpam-5770	157	25	≤	≤	NUM
ejpam-5770	157	26	n−4	n−4	PROPN
ejpam-5770	157	27	4	4	NUM
ejpam-5770	157	28	and	and	CCONJ
ejpam-5770	157	29	fk(x	fk(x	NOUN
ejpam-5770	157	30	)	)	PUNCT
ejpam-5770	158	1	=	=	NOUN
ejpam-5770	158	2	∅	∅	NOUN
ejpam-5770	158	3	otherwise	otherwise	ADV
ejpam-5770	158	4	.	.	PUNCT
ejpam-5770	159	1	in	in	ADP
ejpam-5770	159	2	all	all	DET
ejpam-5770	159	3	cases	case	NOUN
ejpam-5770	159	4	,	,	PUNCT
ejpam-5770	159	5	fk	fk	INTJ
ejpam-5770	159	6	is	be	AUX
ejpam-5770	159	7	a	a	DET
ejpam-5770	159	8	wckrdf	wckrdf	NOUN
ejpam-5770	159	9	of	of	ADP
ejpam-5770	159	10	pn	pn	PROPN
ejpam-5770	159	11	of	of	ADP
ejpam-5770	159	12	weight	weight	NOUN
ejpam-5770	159	13	γrk(pn	γrk(pn	NOUN
ejpam-5770	159	14	)	)	PUNCT
ejpam-5770	159	15	and	and	CCONJ
ejpam-5770	159	16	thus	thus	ADV
ejpam-5770	159	17	γwc	γwc	ADJ
ejpam-5770	159	18	rk	rk	PROPN
ejpam-5770	159	19	(	(	PUNCT
ejpam-5770	159	20	pn	pn	PROPN
ejpam-5770	159	21	)	)	PUNCT
ejpam-5770	159	22	≤	≤	NOUN
ejpam-5770	159	23	γrk(pn	γrk(pn	NOUN
ejpam-5770	159	24	)	)	PUNCT
ejpam-5770	159	25	for	for	ADP
ejpam-5770	159	26	k	k	PROPN
ejpam-5770	159	27	∈	∈	PROPN
ejpam-5770	159	28	{	{	PUNCT
ejpam-5770	159	29	2	2	NUM
ejpam-5770	159	30	,	,	PUNCT
ejpam-5770	159	31	3	3	NUM
ejpam-5770	159	32	}	}	PUNCT
ejpam-5770	159	33	.	.	PUNCT
ejpam-5770	160	1	by	by	ADP
ejpam-5770	160	2	remark	remark	NOUN
ejpam-5770	160	3	1	1	NUM
ejpam-5770	160	4	,	,	PUNCT
ejpam-5770	160	5	we	we	PRON
ejpam-5770	160	6	obtain	obtain	VERB
ejpam-5770	160	7	γwc	γwc	ADJ
ejpam-5770	160	8	rk	rk	PROPN
ejpam-5770	160	9	(	(	PUNCT
ejpam-5770	160	10	pn	pn	NOUN
ejpam-5770	160	11	)	)	PUNCT
ejpam-5770	160	12	=	=	SYM
ejpam-5770	160	13	γrk(pn	γrk(pn	NOUN
ejpam-5770	160	14	)	)	PUNCT
ejpam-5770	160	15	for	for	ADP
ejpam-5770	160	16	k	k	PROPN
ejpam-5770	160	17	∈	∈	PROPN
ejpam-5770	160	18	{	{	PUNCT
ejpam-5770	160	19	2	2	NUM
ejpam-5770	160	20	,	,	PUNCT
ejpam-5770	160	21	3	3	NUM
ejpam-5770	160	22	}	}	PUNCT
ejpam-5770	160	23	,	,	PUNCT
ejpam-5770	160	24	and	and	CCONJ
ejpam-5770	160	25	theorem	theorem	VERB
ejpam-5770	160	26	3-(i	3-(i	NUM
ejpam-5770	160	27	,	,	PUNCT
ejpam-5770	160	28	ii	ii	NOUN
ejpam-5770	160	29	)	)	PUNCT
ejpam-5770	160	30	leads	lead	VERB
ejpam-5770	160	31	to	to	ADP
ejpam-5770	160	32	the	the	DET
ejpam-5770	160	33	desired	desire	VERB
ejpam-5770	160	34	values	value	NOUN
ejpam-5770	160	35	.	.	PUNCT
ejpam-5770	161	1	now	now	ADV
ejpam-5770	161	2	,	,	PUNCT
ejpam-5770	161	3	we	we	PRON
ejpam-5770	161	4	prove	prove	VERB
ejpam-5770	161	5	(	(	PUNCT
ejpam-5770	161	6	iii	iii	NOUN
ejpam-5770	161	7	)	)	PUNCT
ejpam-5770	161	8	and	and	CCONJ
ejpam-5770	161	9	(	(	PUNCT
ejpam-5770	161	10	iv	iv	X
ejpam-5770	161	11	)	)	PUNCT
ejpam-5770	161	12	for	for	ADP
ejpam-5770	161	13	cycles	cycle	NOUN
ejpam-5770	161	14	simultaneously	simultaneously	ADV
ejpam-5770	161	15	.	.	PUNCT
ejpam-5770	162	1	let	let	VERB
ejpam-5770	162	2	k	k	PROPN
ejpam-5770	162	3	∈	∈	PROPN
ejpam-5770	162	4	{	{	PUNCT
ejpam-5770	162	5	2	2	NUM
ejpam-5770	162	6	,	,	PUNCT
ejpam-5770	162	7	3	3	NUM
ejpam-5770	162	8	}	}	PUNCT
ejpam-5770	162	9	.	.	PUNCT
ejpam-5770	163	1	if	if	SCONJ
ejpam-5770	163	2	n	n	PRON
ejpam-5770	163	3	≡	≡	PROPN
ejpam-5770	163	4	0	0	PUNCT
ejpam-5770	163	5	(	(	PUNCT
ejpam-5770	163	6	mod	mod	PROPN
ejpam-5770	163	7	4	4	NUM
ejpam-5770	163	8	)	)	PUNCT
ejpam-5770	163	9	,	,	PUNCT
ejpam-5770	163	10	then	then	ADV
ejpam-5770	163	11	define	define	VERB
ejpam-5770	163	12	the	the	DET
ejpam-5770	163	13	function	function	NOUN
ejpam-5770	163	14	fk	fk	INTJ
ejpam-5770	163	15	:	:	PUNCT
ejpam-5770	163	16	v	v	X
ejpam-5770	163	17	(	(	PUNCT
ejpam-5770	163	18	cn	cn	PROPN
ejpam-5770	163	19	)	)	PUNCT
ejpam-5770	163	20	→	→	SYM
ejpam-5770	163	21	p({1	p({1	PROPN
ejpam-5770	163	22	,	,	PUNCT
ejpam-5770	163	23	2	2	NUM
ejpam-5770	163	24	,	,	PUNCT
ejpam-5770	163	25	.	.	PUNCT
ejpam-5770	163	26	.	.	PUNCT
ejpam-5770	164	1	.	.	PUNCT
ejpam-5770	165	1	,	,	PUNCT
ejpam-5770	165	2	k	k	X
ejpam-5770	165	3	}	}	PUNCT
ejpam-5770	165	4	)	)	PUNCT
ejpam-5770	165	5	by	by	ADP
ejpam-5770	165	6	fk(v4i+1	fk(v4i+1	NOUN
ejpam-5770	165	7	)	)	PUNCT
ejpam-5770	165	8	=	=	PRON
ejpam-5770	165	9	{	{	PUNCT
ejpam-5770	165	10	1	1	NUM
ejpam-5770	165	11	}	}	PUNCT
ejpam-5770	165	12	,	,	PUNCT
ejpam-5770	165	13	fk(v4i+3	fk(v4i+3	NOUN
ejpam-5770	165	14	)	)	PUNCT
ejpam-5770	165	15	=	=	SYM
ejpam-5770	165	16	{	{	PUNCT
ejpam-5770	165	17	2	2	NUM
ejpam-5770	165	18	,	,	PUNCT
ejpam-5770	165	19	k	k	NOUN
ejpam-5770	165	20	}	}	PUNCT
ejpam-5770	165	21	for	for	ADP
ejpam-5770	165	22	0	0	NUM
ejpam-5770	165	23	≤	≤	NUM
ejpam-5770	166	1	i	i	PRON
ejpam-5770	166	2	≤	≤	NUM
ejpam-5770	166	3	n−4	n−4	PROPN
ejpam-5770	166	4	4	4	NUM
ejpam-5770	166	5	,	,	PUNCT
ejpam-5770	166	6	and	and	CCONJ
ejpam-5770	166	7	fk(x	fk(x	NOUN
ejpam-5770	166	8	)	)	PUNCT
ejpam-5770	166	9	=	=	NOUN
ejpam-5770	166	10	∅	∅	NOUN
ejpam-5770	166	11	otherwise	otherwise	ADV
ejpam-5770	166	12	.	.	PUNCT
ejpam-5770	167	1	if	if	SCONJ
ejpam-5770	167	2	n	n	X
ejpam-5770	167	3	̸≡	̸≡	VERB
ejpam-5770	167	4	0	0	PUNCT
ejpam-5770	167	5	(	(	PUNCT
ejpam-5770	167	6	mod	mod	PROPN
ejpam-5770	167	7	4	4	NUM
ejpam-5770	167	8	)	)	PUNCT
ejpam-5770	167	9	,	,	PUNCT
ejpam-5770	167	10	then	then	ADV
ejpam-5770	167	11	let	let	VERB
ejpam-5770	167	12	fk	fk	INTJ
ejpam-5770	167	13	be	be	AUX
ejpam-5770	167	14	the	the	DET
ejpam-5770	167	15	function	function	NOUN
ejpam-5770	167	16	defined	define	VERB
ejpam-5770	167	17	in	in	ADP
ejpam-5770	167	18	the	the	DET
ejpam-5770	167	19	above	above	ADJ
ejpam-5770	167	20	item	item	NOUN
ejpam-5770	167	21	(	(	PUNCT
ejpam-5770	167	22	i	i	NOUN
ejpam-5770	167	23	)	)	PUNCT
ejpam-5770	167	24	and	and	CCONJ
ejpam-5770	167	25	(	(	PUNCT
ejpam-5770	167	26	ii	ii	NOUN
ejpam-5770	167	27	)	)	PUNCT
ejpam-5770	167	28	depending	depend	VERB
ejpam-5770	167	29	on	on	ADP
ejpam-5770	167	30	n.	n.	NOUN
ejpam-5770	167	31	in	in	ADP
ejpam-5770	167	32	all	all	DET
ejpam-5770	167	33	cases	case	NOUN
ejpam-5770	167	34	,	,	PUNCT
ejpam-5770	167	35	fk	fk	INTJ
ejpam-5770	167	36	is	be	AUX
ejpam-5770	167	37	a	a	DET
ejpam-5770	167	38	wc2rdf	wc2rdf	NOUN
ejpam-5770	167	39	of	of	ADP
ejpam-5770	167	40	cn	cn	PROPN
ejpam-5770	167	41	of	of	ADP
ejpam-5770	167	42	weight	weight	NOUN
ejpam-5770	167	43	γrk(cn	γrk(cn	NOUN
ejpam-5770	167	44	)	)	PUNCT
ejpam-5770	167	45	,	,	PUNCT
ejpam-5770	167	46	and	and	CCONJ
ejpam-5770	167	47	thus	thus	ADV
ejpam-5770	167	48	γwc	γwc	ADJ
ejpam-5770	167	49	rk	rk	PROPN
ejpam-5770	167	50	(	(	PUNCT
ejpam-5770	167	51	cn	cn	PROPN
ejpam-5770	167	52	)	)	PUNCT
ejpam-5770	167	53	≤	≤	NUM
ejpam-5770	167	54	γrk(cn	γrk(cn	NOUN
ejpam-5770	167	55	)	)	PUNCT
ejpam-5770	167	56	for	for	ADP
ejpam-5770	167	57	k	k	PROPN
ejpam-5770	167	58	∈	∈	PROPN
ejpam-5770	167	59	{	{	PUNCT
ejpam-5770	167	60	2	2	NUM
ejpam-5770	167	61	,	,	PUNCT
ejpam-5770	167	62	3	3	NUM
ejpam-5770	167	63	}	}	PUNCT
ejpam-5770	167	64	.	.	PUNCT
ejpam-5770	168	1	now	now	ADV
ejpam-5770	168	2	,	,	PUNCT
ejpam-5770	168	3	remark	remark	NOUN
ejpam-5770	168	4	1	1	NUM
ejpam-5770	168	5	leads	lead	VERB
ejpam-5770	168	6	to	to	ADP
ejpam-5770	168	7	γwc	γwc	PROPN
ejpam-5770	168	8	rk	rk	PROPN
ejpam-5770	168	9	(	(	PUNCT
ejpam-5770	168	10	cn	cn	PROPN
ejpam-5770	168	11	)	)	PUNCT
ejpam-5770	168	12	=	=	SYM
ejpam-5770	168	13	γrk(cn	γrk(cn	NOUN
ejpam-5770	168	14	)	)	PUNCT
ejpam-5770	168	15	for	for	ADP
ejpam-5770	168	16	k	k	PROPN
ejpam-5770	168	17	∈	∈	PROPN
ejpam-5770	168	18	{	{	PUNCT
ejpam-5770	168	19	2	2	NUM
ejpam-5770	168	20	,	,	PUNCT
ejpam-5770	168	21	3	3	NUM
ejpam-5770	168	22	}	}	PUNCT
ejpam-5770	168	23	,	,	PUNCT
ejpam-5770	168	24	and	and	CCONJ
ejpam-5770	168	25	(	(	PUNCT
ejpam-5770	168	26	iii	iii	NOUN
ejpam-5770	168	27	)	)	PUNCT
ejpam-5770	168	28	and	and	CCONJ
ejpam-5770	168	29	(	(	PUNCT
ejpam-5770	168	30	iv	iv	X
ejpam-5770	168	31	)	)	PUNCT
ejpam-5770	168	32	follow	follow	VERB
ejpam-5770	168	33	from	from	ADP
ejpam-5770	168	34	theorem	theorem	ADJ
ejpam-5770	168	35	3-(i	3-(i	NUM
ejpam-5770	168	36	,	,	PUNCT
ejpam-5770	168	37	iii	iii	NOUN
ejpam-5770	168	38	)	)	PUNCT
ejpam-5770	168	39	.	.	PUNCT
ejpam-5770	169	1	j.	j.	PROPN
ejpam-5770	169	2	j.	j.	PROPN
ejpam-5770	169	3	hamja	hamja	PROPN
ejpam-5770	169	4	et	et	PROPN
ejpam-5770	169	5	al	al	PROPN
ejpam-5770	169	6	.	.	PUNCT
ejpam-5770	169	7	/	/	SYM
ejpam-5770	169	8	eur	eur	PROPN
ejpam-5770	169	9	.	.	PUNCT
ejpam-5770	170	1	j.	j.	PROPN
ejpam-5770	170	2	pure	pure	PROPN
ejpam-5770	170	3	appl	appl	PROPN
ejpam-5770	170	4	.	.	PROPN
ejpam-5770	170	5	math	math	PROPN
ejpam-5770	170	6	,	,	PUNCT
ejpam-5770	170	7	18	18	NUM
ejpam-5770	170	8	(	(	PUNCT
ejpam-5770	170	9	2	2	NUM
ejpam-5770	170	10	)	)	PUNCT
ejpam-5770	170	11	(	(	PUNCT
ejpam-5770	170	12	2025	2025	NUM
ejpam-5770	170	13	)	)	PUNCT
ejpam-5770	170	14	,	,	PUNCT
ejpam-5770	170	15	5770	5770	NUM
ejpam-5770	170	16	6	6	NUM
ejpam-5770	170	17	of	of	ADP
ejpam-5770	170	18	11	11	NUM
ejpam-5770	170	19	proposition	proposition	NOUN
ejpam-5770	170	20	2	2	NUM
ejpam-5770	170	21	.	.	PUNCT
ejpam-5770	171	1	let	let	VERB
ejpam-5770	171	2	k	k	PRON
ejpam-5770	171	3	be	be	AUX
ejpam-5770	171	4	a	a	DET
ejpam-5770	171	5	positive	positive	ADJ
ejpam-5770	171	6	integer	integer	NOUN
ejpam-5770	171	7	and	and	CCONJ
ejpam-5770	171	8	g	g	PROPN
ejpam-5770	171	9	be	be	AUX
ejpam-5770	171	10	a	a	DET
ejpam-5770	171	11	connected	connected	ADJ
ejpam-5770	171	12	graph	graph	NOUN
ejpam-5770	171	13	of	of	ADP
ejpam-5770	171	14	order	order	NOUN
ejpam-5770	171	15	n.	n.	NOUN
ejpam-5770	171	16	then	then	ADV
ejpam-5770	171	17	min{n	min{n	NOUN
ejpam-5770	171	18	,	,	PUNCT
ejpam-5770	171	19	k	k	NOUN
ejpam-5770	171	20	}	}	PUNCT
ejpam-5770	171	21	≤	≤	NUM
ejpam-5770	171	22	γwc	γwc	ADJ
ejpam-5770	171	23	rk	rk	NOUN
ejpam-5770	171	24	(	(	PUNCT
ejpam-5770	171	25	g	g	NOUN
ejpam-5770	171	26	)	)	PUNCT
ejpam-5770	171	27	≤	≤	NOUN
ejpam-5770	171	28	n.	n.	NOUN
ejpam-5770	171	29	in	in	ADP
ejpam-5770	171	30	particular	particular	ADJ
ejpam-5770	171	31	,	,	PUNCT
ejpam-5770	172	1	if	if	SCONJ
ejpam-5770	172	2	n	n	ADP
ejpam-5770	172	3	≤	≤	X
ejpam-5770	172	4	k	k	NOUN
ejpam-5770	172	5	,	,	PUNCT
ejpam-5770	172	6	then	then	ADV
ejpam-5770	172	7	γwc	γwc	ADJ
ejpam-5770	172	8	rk	rk	PROPN
ejpam-5770	172	9	(	(	PUNCT
ejpam-5770	172	10	g	g	NOUN
ejpam-5770	172	11	)	)	PUNCT
ejpam-5770	172	12	=	=	SYM
ejpam-5770	172	13	n.	n.	NOUN
ejpam-5770	172	14	proof	proof	NOUN
ejpam-5770	172	15	.	.	PUNCT
ejpam-5770	173	1	suppose	suppose	VERB
ejpam-5770	173	2	that	that	SCONJ
ejpam-5770	173	3	f	f	PROPN
ejpam-5770	173	4	is	be	AUX
ejpam-5770	173	5	a	a	DET
ejpam-5770	173	6	γwc	γwc	ADJ
ejpam-5770	173	7	rk	rk	NOUN
ejpam-5770	173	8	-function	-function	NOUN
ejpam-5770	173	9	of	of	ADP
ejpam-5770	173	10	g.	g.	PROPN
ejpam-5770	173	11	if	if	SCONJ
ejpam-5770	173	12	v	v	NUM
ejpam-5770	173	13	f	f	NOUN
ejpam-5770	173	14	0	0	NUM
ejpam-5770	174	1	=	=	NOUN
ejpam-5770	174	2	∅	∅	NOUN
ejpam-5770	174	3	,	,	PUNCT
ejpam-5770	174	4	then	then	ADV
ejpam-5770	174	5	we	we	PRON
ejpam-5770	174	6	have	have	VERB
ejpam-5770	174	7	γwc	γwc	ADJ
ejpam-5770	174	8	rk	rk	PROPN
ejpam-5770	174	9	(	(	PUNCT
ejpam-5770	174	10	g	g	NOUN
ejpam-5770	174	11	)	)	PUNCT
ejpam-5770	174	12	=	=	SYM
ejpam-5770	174	13	∑	∑	PUNCT
ejpam-5770	174	14	v∈v	v∈v	NOUN
ejpam-5770	174	15	(	(	PUNCT
ejpam-5770	174	16	g	g	NOUN
ejpam-5770	174	17	)	)	PUNCT
ejpam-5770	174	18	|f(v)|	|f(v)|	ADP
ejpam-5770	174	19	≥	≥	NUM
ejpam-5770	174	20	∑	∑	PUNCT
ejpam-5770	174	21	v∈v	v∈v	PROPN
ejpam-5770	174	22	(	(	PUNCT
ejpam-5770	174	23	g	g	NOUN
ejpam-5770	174	24	)	)	PUNCT
ejpam-5770	174	25	1	1	NUM
ejpam-5770	174	26	=	=	SYM
ejpam-5770	174	27	n.	n.	NOUN
ejpam-5770	174	28	assume	assume	VERB
ejpam-5770	175	1	that	that	SCONJ
ejpam-5770	175	2	v	v	X
ejpam-5770	175	3	f	f	PROPN
ejpam-5770	175	4	0	0	NUM
ejpam-5770	175	5	̸=	̸=	PROPN
ejpam-5770	175	6	∅	∅	NOUN
ejpam-5770	175	7	and	and	CCONJ
ejpam-5770	175	8	v	v	ADP
ejpam-5770	175	9	∈	∈	PROPN
ejpam-5770	175	10	v	v	ADP
ejpam-5770	175	11	f	f	PROPN
ejpam-5770	175	12	0	0	PROPN
ejpam-5770	175	13	.	.	PUNCT
ejpam-5770	176	1	then	then	ADV
ejpam-5770	176	2	we	we	PRON
ejpam-5770	176	3	have	have	VERB
ejpam-5770	176	4	⋃	⋃	PROPN
ejpam-5770	176	5	u∈ng(v	u∈ng(v	PROPN
ejpam-5770	176	6	)	)	PUNCT
ejpam-5770	176	7	f(u	f(u	PROPN
ejpam-5770	176	8	)	)	PUNCT
ejpam-5770	176	9	=	=	PRON
ejpam-5770	177	1	{	{	PUNCT
ejpam-5770	177	2	1	1	NUM
ejpam-5770	177	3	,	,	PUNCT
ejpam-5770	177	4	2	2	NUM
ejpam-5770	177	5	,	,	PUNCT
ejpam-5770	177	6	3	3	NUM
ejpam-5770	177	7	,	,	PUNCT
ejpam-5770	177	8	.	.	PUNCT
ejpam-5770	177	9	.	.	PUNCT
ejpam-5770	178	1	.	.	PUNCT
ejpam-5770	179	1	,	,	PUNCT
ejpam-5770	179	2	k	k	X
ejpam-5770	179	3	}	}	PUNCT
ejpam-5770	179	4	,	,	PUNCT
ejpam-5770	179	5	and	and	CCONJ
ejpam-5770	179	6	thus	thus	ADV
ejpam-5770	179	7	γwc	γwc	ADJ
ejpam-5770	179	8	rk	rk	PROPN
ejpam-5770	179	9	(	(	PUNCT
ejpam-5770	179	10	g	g	NOUN
ejpam-5770	179	11	)	)	PUNCT
ejpam-5770	179	12	=	=	SYM
ejpam-5770	179	13	∑	∑	PUNCT
ejpam-5770	179	14	u∈v	u∈v	NOUN
ejpam-5770	179	15	(	(	PUNCT
ejpam-5770	179	16	g	g	NOUN
ejpam-5770	179	17	)	)	PUNCT
ejpam-5770	179	18	|f(u)|	|f(u)|	PROPN
ejpam-5770	179	19	≥∑	≥∑	PROPN
ejpam-5770	179	20	u∈ng(v	u∈ng(v	PROPN
ejpam-5770	179	21	)	)	PUNCT
ejpam-5770	179	22	|f(u)|	|f(u)|	PROPN
ejpam-5770	179	23	≥	≥	PROPN
ejpam-5770	179	24	k.	k.	NOUN
ejpam-5770	179	25	combining	combine	VERB
ejpam-5770	179	26	the	the	DET
ejpam-5770	179	27	above	above	ADJ
ejpam-5770	179	28	inequalities	inequality	NOUN
ejpam-5770	179	29	,	,	PUNCT
ejpam-5770	179	30	we	we	PRON
ejpam-5770	179	31	get	get	VERB
ejpam-5770	179	32	min{n	min{n	NOUN
ejpam-5770	179	33	,	,	PUNCT
ejpam-5770	179	34	k	k	NOUN
ejpam-5770	179	35	}	}	PUNCT
ejpam-5770	179	36	≤	≤	NUM
ejpam-5770	179	37	γwc	γwc	ADJ
ejpam-5770	179	38	rk	rk	NOUN
ejpam-5770	179	39	(	(	PUNCT
ejpam-5770	179	40	g	g	NOUN
ejpam-5770	179	41	)	)	PUNCT
ejpam-5770	179	42	.	.	PUNCT
ejpam-5770	180	1	for	for	ADP
ejpam-5770	180	2	the	the	DET
ejpam-5770	180	3	upper	upper	ADJ
ejpam-5770	180	4	bound	bound	NOUN
ejpam-5770	180	5	,	,	PUNCT
ejpam-5770	180	6	consider	consider	VERB
ejpam-5770	180	7	the	the	DET
ejpam-5770	180	8	function	function	NOUN
ejpam-5770	180	9	g	g	NOUN
ejpam-5770	180	10	:	:	PUNCT
ejpam-5770	180	11	v	v	NOUN
ejpam-5770	180	12	(	(	PUNCT
ejpam-5770	180	13	g	g	NOUN
ejpam-5770	180	14	)	)	PUNCT
ejpam-5770	180	15	→	→	SYM
ejpam-5770	181	1	p({1	p({1	PROPN
ejpam-5770	181	2	,	,	PUNCT
ejpam-5770	181	3	2	2	NUM
ejpam-5770	181	4	,	,	PUNCT
ejpam-5770	181	5	.	.	PUNCT
ejpam-5770	181	6	.	.	PUNCT
ejpam-5770	182	1	.	.	PUNCT
ejpam-5770	183	1	,	,	PUNCT
ejpam-5770	183	2	k	k	X
ejpam-5770	183	3	}	}	PUNCT
ejpam-5770	183	4	)	)	PUNCT
ejpam-5770	183	5	defined	define	VERB
ejpam-5770	183	6	by	by	ADP
ejpam-5770	183	7	g(v	g(v	NOUN
ejpam-5770	183	8	)	)	PUNCT
ejpam-5770	183	9	=	=	SYM
ejpam-5770	184	1	{	{	PUNCT
ejpam-5770	184	2	1	1	NUM
ejpam-5770	184	3	}	}	PUNCT
ejpam-5770	184	4	for	for	ADP
ejpam-5770	184	5	all	all	PRON
ejpam-5770	184	6	v	v	ADP
ejpam-5770	184	7	∈	∈	NOUN
ejpam-5770	184	8	v	v	NOUN
ejpam-5770	184	9	(	(	PUNCT
ejpam-5770	184	10	g	g	NOUN
ejpam-5770	184	11	)	)	PUNCT
ejpam-5770	184	12	.	.	PUNCT
ejpam-5770	185	1	then	then	ADV
ejpam-5770	185	2	g	g	PROPN
ejpam-5770	185	3	is	be	AUX
ejpam-5770	185	4	a	a	DET
ejpam-5770	185	5	k	k	ADJ
ejpam-5770	185	6	-	-	PUNCT
ejpam-5770	185	7	rainbow	rainbow	NOUN
ejpam-5770	185	8	dominating	dominating	NOUN
ejpam-5770	185	9	function	function	NOUN
ejpam-5770	185	10	.	.	PUNCT
ejpam-5770	186	1	let	let	VERB
ejpam-5770	186	2	s	s	PRON
ejpam-5770	186	3	=	=	PUNCT
ejpam-5770	186	4	{	{	PUNCT
ejpam-5770	186	5	v	v	NUM
ejpam-5770	186	6	∈	∈	NOUN
ejpam-5770	186	7	v	v	NOUN
ejpam-5770	186	8	(	(	PUNCT
ejpam-5770	186	9	g	g	NOUN
ejpam-5770	186	10	)	)	PUNCT
ejpam-5770	186	11	:	:	PUNCT
ejpam-5770	187	1	g(v	g(v	X
ejpam-5770	187	2	)	)	PUNCT
ejpam-5770	187	3	̸=	̸=	PROPN
ejpam-5770	187	4	∅	∅	NOUN
ejpam-5770	187	5	}	}	PUNCT
ejpam-5770	187	6	.	.	PUNCT
ejpam-5770	188	1	observe	observe	VERB
ejpam-5770	188	2	that	that	SCONJ
ejpam-5770	188	3	s	s	VERB
ejpam-5770	188	4	=	=	SYM
ejpam-5770	188	5	v	v	X
ejpam-5770	188	6	(	(	PUNCT
ejpam-5770	188	7	g	g	NOUN
ejpam-5770	188	8	)	)	PUNCT
ejpam-5770	188	9	since	since	SCONJ
ejpam-5770	188	10	g(v	g(v	NOUN
ejpam-5770	188	11	)	)	PUNCT
ejpam-5770	188	12	=	=	SYM
ejpam-5770	188	13	{	{	PUNCT
ejpam-5770	188	14	1	1	NUM
ejpam-5770	188	15	}	}	PUNCT
ejpam-5770	188	16	for	for	ADP
ejpam-5770	188	17	all	all	PRON
ejpam-5770	188	18	v	v	ADP
ejpam-5770	188	19	∈	∈	NOUN
ejpam-5770	188	20	v	v	NOUN
ejpam-5770	188	21	(	(	PUNCT
ejpam-5770	188	22	g	g	NOUN
ejpam-5770	188	23	)	)	PUNCT
ejpam-5770	188	24	.	.	PUNCT
ejpam-5770	189	1	thus	thus	ADV
ejpam-5770	189	2	,	,	PUNCT
ejpam-5770	189	3	s	s	VERB
ejpam-5770	189	4	is	be	AUX
ejpam-5770	189	5	a	a	DET
ejpam-5770	189	6	weakly	weakly	ADV
ejpam-5770	189	7	connected	connected	ADJ
ejpam-5770	189	8	dominating	dominating	NOUN
ejpam-5770	189	9	set	set	NOUN
ejpam-5770	189	10	of	of	ADP
ejpam-5770	189	11	g.	g.	PROPN
ejpam-5770	189	12	it	it	PRON
ejpam-5770	189	13	would	would	AUX
ejpam-5770	189	14	imply	imply	VERB
ejpam-5770	189	15	that	that	SCONJ
ejpam-5770	189	16	g	g	PROPN
ejpam-5770	189	17	is	be	AUX
ejpam-5770	189	18	a	a	DET
ejpam-5770	189	19	weakly	weakly	ADV
ejpam-5770	189	20	connected	connected	ADJ
ejpam-5770	189	21	k	k	ADJ
ejpam-5770	189	22	-	-	PUNCT
ejpam-5770	189	23	rainbow	rainbow	NOUN
ejpam-5770	189	24	dominating	dominating	NOUN
ejpam-5770	189	25	function	function	NOUN
ejpam-5770	189	26	.	.	PUNCT
ejpam-5770	190	1	thus	thus	ADV
ejpam-5770	190	2	,	,	PUNCT
ejpam-5770	190	3	γwc	γwc	ADJ
ejpam-5770	190	4	rk	rk	PROPN
ejpam-5770	190	5	(	(	PUNCT
ejpam-5770	190	6	g	g	NOUN
ejpam-5770	190	7	)	)	PUNCT
ejpam-5770	190	8	≤	≤	NOUN
ejpam-5770	190	9	n.	n.	NOUN
ejpam-5770	190	10	clearly	clearly	ADV
ejpam-5770	190	11	,	,	PUNCT
ejpam-5770	190	12	if	if	SCONJ
ejpam-5770	190	13	n	n	PRON
ejpam-5770	190	14	≤	≤	X
ejpam-5770	190	15	k	k	NOUN
ejpam-5770	190	16	,	,	PUNCT
ejpam-5770	190	17	then	then	ADV
ejpam-5770	190	18	γwc	γwc	ADJ
ejpam-5770	190	19	rk	rk	PROPN
ejpam-5770	190	20	(	(	PUNCT
ejpam-5770	190	21	g	g	NOUN
ejpam-5770	190	22	)	)	PUNCT
ejpam-5770	190	23	=	=	SYM
ejpam-5770	191	1	n.	n.	PROPN
ejpam-5770	191	2	shao	shao	PROPN
ejpam-5770	191	3	et	et	PROPN
ejpam-5770	191	4	al	al	PROPN
ejpam-5770	191	5	.	.	PUNCT
ejpam-5770	192	1	in	in	ADP
ejpam-5770	192	2	[	[	X
ejpam-5770	192	3	21	21	NUM
ejpam-5770	192	4	]	]	PUNCT
ejpam-5770	192	5	proved	prove	VERB
ejpam-5770	192	6	the	the	DET
ejpam-5770	192	7	next	next	ADJ
ejpam-5770	192	8	result	result	NOUN
ejpam-5770	192	9	.	.	PUNCT
ejpam-5770	193	1	theorem	theorem	ADJ
ejpam-5770	193	2	4	4	NUM
ejpam-5770	193	3	.	.	X
ejpam-5770	194	1	for	for	ADP
ejpam-5770	194	2	positive	positive	ADJ
ejpam-5770	194	3	integers	integer	NOUN
ejpam-5770	194	4	n	n	PRON
ejpam-5770	194	5	and	and	CCONJ
ejpam-5770	194	6	k	k	PROPN
ejpam-5770	194	7	≥	≥	NUM
ejpam-5770	194	8	2	2	NUM
ejpam-5770	194	9	,	,	PUNCT
ejpam-5770	194	10	let	let	VERB
ejpam-5770	194	11	g	g	PRON
ejpam-5770	194	12	be	be	AUX
ejpam-5770	194	13	a	a	DET
ejpam-5770	194	14	connected	connected	ADJ
ejpam-5770	194	15	graph	graph	NOUN
ejpam-5770	194	16	of	of	ADP
ejpam-5770	194	17	order	order	NOUN
ejpam-5770	194	18	n	n	PRON
ejpam-5770	194	19	≥	≥	NOUN
ejpam-5770	194	20	k	k	X
ejpam-5770	194	21	with	with	ADP
ejpam-5770	194	22	k	k	PROPN
ejpam-5770	194	23	>	>	X
ejpam-5770	194	24	∆(g)2	∆(g)2	PROPN
ejpam-5770	194	25	.	.	PUNCT
ejpam-5770	195	1	then	then	ADV
ejpam-5770	195	2	γrk(g	γrk(g	NUM
ejpam-5770	195	3	)	)	PUNCT
ejpam-5770	195	4	=	=	VERB
ejpam-5770	196	1	n.	n.	NOUN
ejpam-5770	196	2	the	the	DET
ejpam-5770	196	3	next	next	ADJ
ejpam-5770	196	4	results	result	NOUN
ejpam-5770	196	5	are	be	AUX
ejpam-5770	196	6	direct	direct	ADJ
ejpam-5770	196	7	consequences	consequence	NOUN
ejpam-5770	196	8	of	of	ADP
ejpam-5770	196	9	theorem	theorem	ADJ
ejpam-5770	196	10	4	4	NUM
ejpam-5770	196	11	and	and	CCONJ
ejpam-5770	196	12	remark	remark	NOUN
ejpam-5770	196	13	1	1	NUM
ejpam-5770	196	14	.	.	PUNCT
ejpam-5770	196	15	corollary	corollary	ADJ
ejpam-5770	196	16	1	1	NUM
ejpam-5770	196	17	.	.	PUNCT
ejpam-5770	197	1	for	for	ADP
ejpam-5770	197	2	a	a	DET
ejpam-5770	197	3	positive	positive	ADJ
ejpam-5770	197	4	integer	integer	NOUN
ejpam-5770	197	5	n	n	PROPN
ejpam-5770	197	6	and	and	CCONJ
ejpam-5770	197	7	k	k	PROPN
ejpam-5770	197	8	≥	≥	NUM
ejpam-5770	197	9	2	2	NUM
ejpam-5770	197	10	,	,	PUNCT
ejpam-5770	197	11	let	let	VERB
ejpam-5770	197	12	g	g	PRON
ejpam-5770	197	13	be	be	AUX
ejpam-5770	197	14	a	a	DET
ejpam-5770	197	15	connected	connected	ADJ
ejpam-5770	197	16	graph	graph	NOUN
ejpam-5770	197	17	of	of	ADP
ejpam-5770	197	18	order	order	NOUN
ejpam-5770	197	19	n	n	PRON
ejpam-5770	197	20	≥	≥	NOUN
ejpam-5770	197	21	k	k	X
ejpam-5770	197	22	with	with	ADP
ejpam-5770	197	23	k	k	PROPN
ejpam-5770	197	24	>	>	X
ejpam-5770	197	25	∆(g)2	∆(g)2	PROPN
ejpam-5770	197	26	.	.	PUNCT
ejpam-5770	198	1	then	then	ADV
ejpam-5770	198	2	γwc	γwc	PROPN
ejpam-5770	198	3	rk	rk	PROPN
ejpam-5770	198	4	(	(	PUNCT
ejpam-5770	198	5	g	g	NOUN
ejpam-5770	198	6	)	)	PUNCT
ejpam-5770	198	7	=	=	VERB
ejpam-5770	198	8	n.	n.	NOUN
ejpam-5770	198	9	corollary	corollary	NOUN
ejpam-5770	198	10	2	2	NUM
ejpam-5770	198	11	.	.	PUNCT
ejpam-5770	199	1	for	for	ADP
ejpam-5770	199	2	positive	positive	ADJ
ejpam-5770	199	3	integers	integer	NOUN
ejpam-5770	199	4	n	n	PRON
ejpam-5770	199	5	and	and	CCONJ
ejpam-5770	199	6	k	k	PROPN
ejpam-5770	199	7	≥	≥	NUM
ejpam-5770	199	8	5	5	NUM
ejpam-5770	199	9	,	,	PUNCT
ejpam-5770	199	10	γwc	γwc	ADJ
ejpam-5770	199	11	rk	rk	PROPN
ejpam-5770	199	12	(	(	PUNCT
ejpam-5770	199	13	pn	pn	PROPN
ejpam-5770	199	14	)	)	PUNCT
ejpam-5770	199	15	=	=	SYM
ejpam-5770	199	16	γwc	γwc	PROPN
ejpam-5770	199	17	rk	rk	PROPN
ejpam-5770	199	18	(	(	PUNCT
ejpam-5770	199	19	cn	cn	PROPN
ejpam-5770	199	20	)	)	PUNCT
ejpam-5770	199	21	=	=	VERB
ejpam-5770	199	22	n.	n.	NOUN
ejpam-5770	199	23	the	the	DET
ejpam-5770	199	24	following	follow	VERB
ejpam-5770	199	25	result	result	NOUN
ejpam-5770	199	26	shows	show	VERB
ejpam-5770	199	27	that	that	SCONJ
ejpam-5770	199	28	the	the	DET
ejpam-5770	199	29	difference	difference	NOUN
ejpam-5770	199	30	γwc	γwc	PROPN
ejpam-5770	199	31	r2	r2	PROPN
ejpam-5770	199	32	(	(	PUNCT
ejpam-5770	199	33	g)−γr2(g	g)−γr2(g	PROPN
ejpam-5770	199	34	)	)	PUNCT
ejpam-5770	199	35	can	can	AUX
ejpam-5770	199	36	be	be	AUX
ejpam-5770	199	37	arbitrarily	arbitrarily	ADV
ejpam-5770	199	38	large	large	ADJ
ejpam-5770	199	39	.	.	PUNCT
ejpam-5770	200	1	a	a	DET
ejpam-5770	200	2	caterpillar	caterpillar	ADJ
ejpam-5770	200	3	c(n	c(n	NOUN
ejpam-5770	200	4	;	;	PUNCT
ejpam-5770	200	5	d1	d1	PROPN
ejpam-5770	200	6	,	,	PUNCT
ejpam-5770	200	7	.	.	PUNCT
ejpam-5770	200	8	.	.	PUNCT
ejpam-5770	200	9	.	.	PUNCT
ejpam-5770	201	1	,	,	PUNCT
ejpam-5770	201	2	dn	dn	PROPN
ejpam-5770	201	3	)	)	PUNCT
ejpam-5770	201	4	is	be	AUX
ejpam-5770	201	5	defined	define	VERB
ejpam-5770	201	6	as	as	ADP
ejpam-5770	201	7	a	a	DET
ejpam-5770	201	8	tree	tree	NOUN
ejpam-5770	201	9	in	in	ADP
ejpam-5770	201	10	which	which	PRON
ejpam-5770	201	11	removal	removal	NOUN
ejpam-5770	201	12	of	of	ADP
ejpam-5770	201	13	all	all	DET
ejpam-5770	201	14	its	its	PRON
ejpam-5770	201	15	leaves	leave	NOUN
ejpam-5770	201	16	yields	yield	VERB
ejpam-5770	201	17	a	a	DET
ejpam-5770	201	18	path	path	NOUN
ejpam-5770	201	19	pn	pn	NOUN
ejpam-5770	202	1	=	=	PUNCT
ejpam-5770	203	1	[	[	X
ejpam-5770	203	2	x1	x1	PROPN
ejpam-5770	203	3	,	,	PUNCT
ejpam-5770	203	4	x2	x2	PROPN
ejpam-5770	203	5	,	,	PUNCT
ejpam-5770	203	6	.	.	PUNCT
ejpam-5770	203	7	.	.	PUNCT
ejpam-5770	204	1	.	.	PUNCT
ejpam-5770	205	1	,	,	PUNCT
ejpam-5770	205	2	xn	xn	X
ejpam-5770	205	3	]	]	PUNCT
ejpam-5770	205	4	and	and	CCONJ
ejpam-5770	205	5	that	that	DET
ejpam-5770	205	6	di	di	NOUN
ejpam-5770	205	7	is	be	AUX
ejpam-5770	205	8	the	the	DET
ejpam-5770	205	9	number	number	NOUN
ejpam-5770	205	10	of	of	ADP
ejpam-5770	205	11	leaf	leaf	NOUN
ejpam-5770	205	12	neighbors	neighbor	NOUN
ejpam-5770	205	13	of	of	ADP
ejpam-5770	205	14	xi	xi	PROPN
ejpam-5770	205	15	for	for	ADP
ejpam-5770	205	16	each	each	DET
ejpam-5770	205	17	i.	i.	NOUN
ejpam-5770	205	18	the	the	DET
ejpam-5770	205	19	path	path	NOUN
ejpam-5770	205	20	pn	pn	PROPN
ejpam-5770	205	21	=	=	PUNCT
ejpam-5770	206	1	[	[	X
ejpam-5770	206	2	x1	x1	PROPN
ejpam-5770	206	3	,	,	PUNCT
ejpam-5770	206	4	x2	x2	PROPN
ejpam-5770	206	5	,	,	PUNCT
ejpam-5770	206	6	.	.	PUNCT
ejpam-5770	206	7	.	.	PUNCT
ejpam-5770	207	1	.	.	PUNCT
ejpam-5770	208	1	,	,	PUNCT
ejpam-5770	208	2	xn	xn	PROPN
ejpam-5770	208	3	]	]	X
ejpam-5770	208	4	is	be	AUX
ejpam-5770	208	5	called	call	VERB
ejpam-5770	208	6	the	the	DET
ejpam-5770	208	7	backbone	backbone	NOUN
ejpam-5770	208	8	of	of	ADP
ejpam-5770	208	9	the	the	DET
ejpam-5770	208	10	caterpillar	caterpillar	NOUN
ejpam-5770	208	11	.	.	PUNCT
ejpam-5770	209	1	theorem	theorem	NOUN
ejpam-5770	209	2	5	5	NUM
ejpam-5770	209	3	.	.	X
ejpam-5770	209	4	for	for	ADP
ejpam-5770	209	5	the	the	DET
ejpam-5770	209	6	integer	integer	PROPN
ejpam-5770	209	7	k	k	PROPN
ejpam-5770	209	8	≥	≥	NUM
ejpam-5770	209	9	2	2	NUM
ejpam-5770	209	10	and	and	CCONJ
ejpam-5770	209	11	every	every	DET
ejpam-5770	209	12	non	non	ADJ
ejpam-5770	209	13	-	-	ADJ
ejpam-5770	209	14	negative	negative	ADJ
ejpam-5770	209	15	integer	integer	NOUN
ejpam-5770	209	16	c	c	NOUN
ejpam-5770	209	17	,	,	PUNCT
ejpam-5770	209	18	there	there	PRON
ejpam-5770	209	19	exists	exist	VERB
ejpam-5770	209	20	a	a	DET
ejpam-5770	209	21	connected	connected	ADJ
ejpam-5770	209	22	graph	graph	NOUN
ejpam-5770	209	23	g	g	ADP
ejpam-5770	209	24	such	such	ADJ
ejpam-5770	209	25	that	that	DET
ejpam-5770	209	26	γwc	γwc	PROPN
ejpam-5770	209	27	r2	r2	PROPN
ejpam-5770	209	28	(	(	PUNCT
ejpam-5770	209	29	g)−	g)−	PROPN
ejpam-5770	209	30	γr2(g	γr2(g	PROPN
ejpam-5770	209	31	)	)	PUNCT
ejpam-5770	209	32	=	=	SYM
ejpam-5770	209	33	c.	c.	NOUN
ejpam-5770	209	34	proof	proof	NOUN
ejpam-5770	209	35	.	.	PUNCT
ejpam-5770	210	1	consider	consider	VERB
ejpam-5770	210	2	the	the	DET
ejpam-5770	210	3	caterpillar	caterpillar	NOUN
ejpam-5770	210	4	g	g	NOUN
ejpam-5770	210	5	=	=	SYM
ejpam-5770	210	6	c(3c	c(3c	NOUN
ejpam-5770	211	1	+	+	NOUN
ejpam-5770	211	2	1	1	NUM
ejpam-5770	211	3	;	;	PUNCT
ejpam-5770	211	4	2k	2k	NUM
ejpam-5770	211	5	,	,	PUNCT
ejpam-5770	211	6	0	0	NUM
ejpam-5770	211	7	,	,	PUNCT
ejpam-5770	211	8	0	0	NUM
ejpam-5770	211	9	,	,	PUNCT
ejpam-5770	211	10	2k	2k	NUM
ejpam-5770	211	11	,	,	PUNCT
ejpam-5770	211	12	.	.	PUNCT
ejpam-5770	211	13	.	.	PUNCT
ejpam-5770	212	1	.	.	PUNCT
ejpam-5770	213	1	,	,	PUNCT
ejpam-5770	213	2	0	0	NUM
ejpam-5770	213	3	,	,	PUNCT
ejpam-5770	213	4	0	0	NUM
ejpam-5770	213	5	,	,	PUNCT
ejpam-5770	213	6	2k	2k	NUM
ejpam-5770	213	7	)	)	PUNCT
ejpam-5770	213	8	with	with	ADP
ejpam-5770	213	9	backbone	backbone	NOUN
ejpam-5770	213	10	pn	pn	NOUN
ejpam-5770	213	11	=	=	PUNCT
ejpam-5770	214	1	[	[	X
ejpam-5770	214	2	x1	x1	PROPN
ejpam-5770	214	3	,	,	PUNCT
ejpam-5770	214	4	x2	x2	PROPN
ejpam-5770	214	5	,	,	PUNCT
ejpam-5770	214	6	.	.	PUNCT
ejpam-5770	214	7	.	.	PUNCT
ejpam-5770	214	8	.	.	PUNCT
ejpam-5770	215	1	x3c+1	x3c+1	PROPN
ejpam-5770	215	2	]	]	X
ejpam-5770	215	3	in	in	ADP
ejpam-5770	215	4	figure	figure	NOUN
ejpam-5770	215	5	2	2	NUM
ejpam-5770	215	6	.	.	PUNCT
ejpam-5770	216	1	it	it	PRON
ejpam-5770	216	2	is	be	AUX
ejpam-5770	216	3	easily	easily	ADV
ejpam-5770	216	4	seen	see	VERB
ejpam-5770	216	5	that	that	SCONJ
ejpam-5770	216	6	the	the	DET
ejpam-5770	216	7	function	function	NOUN
ejpam-5770	216	8	f	f	X
ejpam-5770	216	9	:	:	PUNCT
ejpam-5770	216	10	v	v	X
ejpam-5770	216	11	(	(	PUNCT
ejpam-5770	216	12	g	g	NOUN
ejpam-5770	216	13	)	)	PUNCT
ejpam-5770	216	14	→	→	SYM
ejpam-5770	216	15	p({1	p({1	PROPN
ejpam-5770	216	16	,	,	PUNCT
ejpam-5770	216	17	2	2	NUM
ejpam-5770	216	18	,	,	PUNCT
ejpam-5770	216	19	.	.	PUNCT
ejpam-5770	216	20	.	.	PUNCT
ejpam-5770	216	21	.	.	PUNCT
ejpam-5770	217	1	,	,	PUNCT
ejpam-5770	217	2	k	k	X
ejpam-5770	217	3	}	}	PUNCT
ejpam-5770	217	4	)	)	PUNCT
ejpam-5770	217	5	defined	define	VERB
ejpam-5770	217	6	by	by	ADP
ejpam-5770	217	7	f(x3i+1	f(x3i+1	NOUN
ejpam-5770	217	8	)	)	PUNCT
ejpam-5770	217	9	=	=	SYM
ejpam-5770	218	1	{	{	PUNCT
ejpam-5770	218	2	1	1	NUM
ejpam-5770	218	3	,	,	PUNCT
ejpam-5770	218	4	2	2	NUM
ejpam-5770	218	5	,	,	PUNCT
ejpam-5770	218	6	.	.	PUNCT
ejpam-5770	218	7	.	.	PUNCT
ejpam-5770	219	1	.	.	PUNCT
ejpam-5770	220	1	,	,	PUNCT
ejpam-5770	220	2	k	k	X
ejpam-5770	220	3	}	}	PUNCT
ejpam-5770	220	4	for	for	ADP
ejpam-5770	220	5	0	0	NUM
ejpam-5770	220	6	≤	≤	NUM
ejpam-5770	220	7	i	i	PRON
ejpam-5770	220	8	≤	≤	NOUN
ejpam-5770	220	9	c	c	PROPN
ejpam-5770	220	10	and	and	CCONJ
ejpam-5770	220	11	f(x	f(x	PROPN
ejpam-5770	220	12	)	)	PUNCT
ejpam-5770	221	1	=	=	NOUN
ejpam-5770	221	2	∅	∅	NOUN
ejpam-5770	221	3	for	for	ADP
ejpam-5770	221	4	other	other	ADJ
ejpam-5770	221	5	vertices	vertex	NOUN
ejpam-5770	221	6	,	,	PUNCT
ejpam-5770	221	7	is	be	AUX
ejpam-5770	221	8	the	the	DET
ejpam-5770	221	9	unique	unique	ADJ
ejpam-5770	221	10	γrk	γrk	NOUN
ejpam-5770	221	11	-	-	PUNCT
ejpam-5770	221	12	function	function	NOUN
ejpam-5770	221	13	of	of	ADP
ejpam-5770	221	14	g	g	NOUN
ejpam-5770	221	15	of	of	ADP
ejpam-5770	221	16	weight	weight	NOUN
ejpam-5770	221	17	k(c	k(c	PROPN
ejpam-5770	222	1	+	+	CCONJ
ejpam-5770	222	2	1	1	NUM
ejpam-5770	222	3	)	)	PUNCT
ejpam-5770	222	4	,	,	PUNCT
ejpam-5770	222	5	and	and	CCONJ
ejpam-5770	222	6	the	the	DET
ejpam-5770	222	7	function	function	NOUN
ejpam-5770	222	8	g	g	NOUN
ejpam-5770	222	9	:	:	PUNCT
ejpam-5770	222	10	v	v	NOUN
ejpam-5770	222	11	(	(	PUNCT
ejpam-5770	222	12	g	g	NOUN
ejpam-5770	222	13	)	)	PUNCT
ejpam-5770	222	14	→	→	SYM
ejpam-5770	222	15	p({1	p({1	PROPN
ejpam-5770	222	16	,	,	PUNCT
ejpam-5770	222	17	2	2	NUM
ejpam-5770	222	18	,	,	PUNCT
ejpam-5770	222	19	.	.	PUNCT
ejpam-5770	222	20	.	.	PUNCT
ejpam-5770	223	1	.	.	PUNCT
ejpam-5770	224	1	,	,	PUNCT
ejpam-5770	224	2	k	k	X
ejpam-5770	224	3	}	}	PUNCT
ejpam-5770	224	4	)	)	PUNCT
ejpam-5770	224	5	defined	define	VERB
ejpam-5770	224	6	by	by	ADP
ejpam-5770	224	7	g(x3i+1	g(x3i+1	NOUN
ejpam-5770	224	8	)	)	PUNCT
ejpam-5770	225	1	=	=	SYM
ejpam-5770	225	2	{	{	PUNCT
ejpam-5770	225	3	1	1	NUM
ejpam-5770	225	4	,	,	PUNCT
ejpam-5770	225	5	2	2	NUM
ejpam-5770	225	6	,	,	PUNCT
ejpam-5770	225	7	.	.	PUNCT
ejpam-5770	225	8	.	.	PUNCT
ejpam-5770	225	9	.	.	PUNCT
ejpam-5770	226	1	,	,	PUNCT
ejpam-5770	226	2	k	k	X
ejpam-5770	226	3	}	}	PUNCT
ejpam-5770	226	4	for	for	ADP
ejpam-5770	226	5	0	0	NUM
ejpam-5770	226	6	≤	≤	NUM
ejpam-5770	226	7	i	i	PRON
ejpam-5770	226	8	≤	≤	NOUN
ejpam-5770	226	9	c	c	X
ejpam-5770	226	10	,	,	PUNCT
ejpam-5770	226	11	g(x3i	g(x3i	PUNCT
ejpam-5770	226	12	)	)	PUNCT
ejpam-5770	226	13	=	=	SYM
ejpam-5770	226	14	{	{	PUNCT
ejpam-5770	226	15	1	1	NUM
ejpam-5770	226	16	}	}	PUNCT
ejpam-5770	226	17	for	for	ADP
ejpam-5770	226	18	1	1	NUM
ejpam-5770	226	19	≤	≤	NUM
ejpam-5770	226	20	i	i	PRON
ejpam-5770	227	1	≤	≤	NOUN
ejpam-5770	227	2	c	c	X
ejpam-5770	227	3	,	,	PUNCT
ejpam-5770	227	4	and	and	CCONJ
ejpam-5770	227	5	g(x	g(x	NOUN
ejpam-5770	227	6	)	)	PUNCT
ejpam-5770	228	1	=	=	NOUN
ejpam-5770	228	2	∅	∅	NOUN
ejpam-5770	228	3	for	for	ADP
ejpam-5770	228	4	other	other	ADJ
ejpam-5770	228	5	vertices	vertex	NOUN
ejpam-5770	228	6	,	,	PUNCT
ejpam-5770	228	7	is	be	AUX
ejpam-5770	228	8	a	a	DET
ejpam-5770	228	9	γwc	γwc	ADJ
ejpam-5770	228	10	rk	rk	NOUN
ejpam-5770	228	11	-function	-function	NOUN
ejpam-5770	228	12	of	of	ADP
ejpam-5770	228	13	g	g	NOUN
ejpam-5770	228	14	of	of	ADP
ejpam-5770	228	15	weight	weight	NOUN
ejpam-5770	228	16	k(c+1)+	k(c+1)+	PROPN
ejpam-5770	228	17	c.	c.	PROPN
ejpam-5770	228	18	thus	thus	ADV
ejpam-5770	228	19	,	,	PUNCT
ejpam-5770	228	20	γwc	γwc	PROPN
ejpam-5770	228	21	r2	r2	PROPN
ejpam-5770	228	22	(	(	PUNCT
ejpam-5770	228	23	g)−	g)−	PROPN
ejpam-5770	228	24	γr2(g	γr2(g	PROPN
ejpam-5770	228	25	)	)	PUNCT
ejpam-5770	228	26	=	=	SYM
ejpam-5770	228	27	c	c	NOUN
ejpam-5770	228	28	and	and	CCONJ
ejpam-5770	228	29	the	the	DET
ejpam-5770	228	30	proof	proof	NOUN
ejpam-5770	228	31	is	be	AUX
ejpam-5770	228	32	complete	complete	ADJ
ejpam-5770	228	33	.	.	PUNCT
ejpam-5770	229	1	j.	j.	PROPN
ejpam-5770	229	2	j.	j.	PROPN
ejpam-5770	229	3	hamja	hamja	PROPN
ejpam-5770	229	4	et	et	PROPN
ejpam-5770	229	5	al	al	PROPN
ejpam-5770	229	6	.	.	PUNCT
ejpam-5770	229	7	/	/	SYM
ejpam-5770	229	8	eur	eur	PROPN
ejpam-5770	229	9	.	.	PUNCT
ejpam-5770	230	1	j.	j.	PROPN
ejpam-5770	230	2	pure	pure	PROPN
ejpam-5770	230	3	appl	appl	PROPN
ejpam-5770	230	4	.	.	PROPN
ejpam-5770	230	5	math	math	PROPN
ejpam-5770	230	6	,	,	PUNCT
ejpam-5770	230	7	18	18	NUM
ejpam-5770	230	8	(	(	PUNCT
ejpam-5770	230	9	2	2	NUM
ejpam-5770	230	10	)	)	PUNCT
ejpam-5770	230	11	(	(	PUNCT
ejpam-5770	230	12	2025	2025	NUM
ejpam-5770	230	13	)	)	PUNCT
ejpam-5770	230	14	,	,	PUNCT
ejpam-5770	230	15	5770	5770	NUM
ejpam-5770	230	16	7	7	NUM
ejpam-5770	230	17	of	of	ADP
ejpam-5770	230	18	11	11	NUM
ejpam-5770	230	19	x1	x1	NOUN
ejpam-5770	231	1	x2	x2	NOUN
ejpam-5770	231	2	x3	x3	PROPN
ejpam-5770	232	1	x4	x4	PROPN
ejpam-5770	232	2	x3c−1	x3c−1	PROPN
ejpam-5770	232	3	x3c	x3c	PROPN
ejpam-5770	232	4	x3c+1	x3c+1	PROPN
ejpam-5770	232	5	.	.	PUNCT
ejpam-5770	232	6	.	.	PUNCT
ejpam-5770	232	7	.	.	PUNCT
ejpam-5770	232	8	.	.	PUNCT
ejpam-5770	232	9	.	.	PUNCT
ejpam-5770	232	10	.	.	PUNCT
ejpam-5770	232	11	.	.	PUNCT
ejpam-5770	232	12	.	.	PUNCT
ejpam-5770	232	13	.	.	PUNCT
ejpam-5770	232	14	.	.	PUNCT
ejpam-5770	232	15	.	.	PUNCT
ejpam-5770	233	1	.	.	PUNCT
ejpam-5770	234	1	figure	figure	VERB
ejpam-5770	234	2	2	2	NUM
ejpam-5770	234	3	:	:	PUNCT
ejpam-5770	234	4	a	a	DET
ejpam-5770	234	5	caterpillar	caterpillar	ADJ
ejpam-5770	234	6	g	g	NOUN
ejpam-5770	234	7	=	=	SYM
ejpam-5770	234	8	c(3c+	c(3c+	NOUN
ejpam-5770	234	9	1	1	NUM
ejpam-5770	234	10	;	;	PUNCT
ejpam-5770	234	11	2k	2k	NUM
ejpam-5770	234	12	,	,	PUNCT
ejpam-5770	234	13	0	0	NUM
ejpam-5770	234	14	,	,	PUNCT
ejpam-5770	234	15	0	0	NUM
ejpam-5770	234	16	,	,	PUNCT
ejpam-5770	234	17	2k	2k	NUM
ejpam-5770	234	18	,	,	PUNCT
ejpam-5770	234	19	.	.	PUNCT
ejpam-5770	234	20	.	.	PUNCT
ejpam-5770	235	1	.	.	PUNCT
ejpam-5770	236	1	,	,	PUNCT
ejpam-5770	236	2	0	0	NUM
ejpam-5770	236	3	,	,	PUNCT
ejpam-5770	236	4	0	0	NUM
ejpam-5770	236	5	,	,	PUNCT
ejpam-5770	236	6	2k	2k	NUM
ejpam-5770	236	7	)	)	PUNCT
ejpam-5770	236	8	4	4	NUM
ejpam-5770	236	9	.	.	PUNCT
ejpam-5770	237	1	graphs	graph	NOUN
ejpam-5770	237	2	with	with	ADP
ejpam-5770	237	3	γwc	γwc	ADJ
ejpam-5770	237	4	rk	rk	PROPN
ejpam-5770	237	5	(	(	PUNCT
ejpam-5770	237	6	g	g	NOUN
ejpam-5770	237	7	)	)	PUNCT
ejpam-5770	237	8	=	=	SYM
ejpam-5770	238	1	k	k	X
ejpam-5770	238	2	in	in	ADP
ejpam-5770	238	3	this	this	DET
ejpam-5770	238	4	section	section	NOUN
ejpam-5770	238	5	,	,	PUNCT
ejpam-5770	238	6	we	we	PRON
ejpam-5770	238	7	characterize	characterize	VERB
ejpam-5770	238	8	all	all	DET
ejpam-5770	238	9	graphs	graph	NOUN
ejpam-5770	238	10	g	g	PROPN
ejpam-5770	238	11	with	with	ADP
ejpam-5770	238	12	γwc	γwc	ADJ
ejpam-5770	238	13	rk	rk	PROPN
ejpam-5770	238	14	(	(	PUNCT
ejpam-5770	238	15	g	g	NOUN
ejpam-5770	238	16	)	)	PUNCT
ejpam-5770	238	17	=	=	SYM
ejpam-5770	238	18	k.	k.	PROPN
ejpam-5770	238	19	theorem	theorem	VERB
ejpam-5770	238	20	6	6	NUM
ejpam-5770	238	21	.	.	PUNCT
ejpam-5770	239	1	let	let	VERB
ejpam-5770	239	2	k	k	PROPN
ejpam-5770	239	3	≥	≥	NUM
ejpam-5770	239	4	1	1	NUM
ejpam-5770	239	5	be	be	AUX
ejpam-5770	239	6	an	an	DET
ejpam-5770	239	7	integer	integer	NOUN
ejpam-5770	239	8	,	,	PUNCT
ejpam-5770	239	9	and	and	CCONJ
ejpam-5770	239	10	let	let	VERB
ejpam-5770	239	11	g	g	PRON
ejpam-5770	239	12	be	be	AUX
ejpam-5770	239	13	a	a	DET
ejpam-5770	239	14	connected	connected	ADJ
ejpam-5770	239	15	graph	graph	NOUN
ejpam-5770	239	16	of	of	ADP
ejpam-5770	239	17	order	order	NOUN
ejpam-5770	239	18	n	n	PRON
ejpam-5770	239	19	≥	≥	NOUN
ejpam-5770	239	20	k.	k.	PROPN
ejpam-5770	240	1	then	then	ADV
ejpam-5770	240	2	γwc	γwc	PROPN
ejpam-5770	240	3	rk	rk	PROPN
ejpam-5770	240	4	(	(	PUNCT
ejpam-5770	240	5	g	g	NOUN
ejpam-5770	240	6	)	)	PUNCT
ejpam-5770	240	7	=	=	SYM
ejpam-5770	241	1	k	k	NOUN
ejpam-5770	242	1	if	if	SCONJ
ejpam-5770	242	2	and	and	CCONJ
ejpam-5770	242	3	only	only	ADV
ejpam-5770	242	4	if	if	SCONJ
ejpam-5770	242	5	n	n	PROPN
ejpam-5770	242	6	=	=	SYM
ejpam-5770	242	7	k	k	PROPN
ejpam-5770	242	8	or	or	CCONJ
ejpam-5770	242	9	n	n	PROPN
ejpam-5770	242	10	>	>	X
ejpam-5770	242	11	k	k	NOUN
ejpam-5770	242	12	and	and	CCONJ
ejpam-5770	242	13	there	there	PRON
ejpam-5770	242	14	exists	exist	VERB
ejpam-5770	242	15	a	a	DET
ejpam-5770	242	16	set	set	NOUN
ejpam-5770	242	17	x	x	X
ejpam-5770	242	18	=	=	SYM
ejpam-5770	242	19	{	{	PUNCT
ejpam-5770	242	20	x1	x1	PROPN
ejpam-5770	242	21	,	,	PUNCT
ejpam-5770	242	22	x2	x2	PROPN
ejpam-5770	242	23	,	,	PUNCT
ejpam-5770	242	24	.	.	PUNCT
ejpam-5770	242	25	.	.	PUNCT
ejpam-5770	243	1	.	.	PUNCT
ejpam-5770	244	1	,	,	PUNCT
ejpam-5770	244	2	xm	xm	PROPN
ejpam-5770	244	3	}	}	PUNCT
ejpam-5770	244	4	of	of	ADP
ejpam-5770	244	5	vertices	vertex	NOUN
ejpam-5770	244	6	with	with	ADP
ejpam-5770	244	7	1	1	NUM
ejpam-5770	244	8	≤	≤	NUM
ejpam-5770	244	9	|x|	|x|	PROPN
ejpam-5770	244	10	≤	≤	NOUN
ejpam-5770	245	1	k	k	PRON
ejpam-5770	245	2	such	such	ADJ
ejpam-5770	245	3	that	that	PRON
ejpam-5770	245	4	(	(	PUNCT
ejpam-5770	245	5	v	v	NOUN
ejpam-5770	245	6	(	(	PUNCT
ejpam-5770	245	7	g	g	NOUN
ejpam-5770	245	8	)	)	PUNCT
ejpam-5770	245	9	\x	\x	NOUN
ejpam-5770	245	10	)	)	PUNCT
ejpam-5770	245	11	⊆	⊆	NUM
ejpam-5770	245	12	⋂m	⋂m	NOUN
ejpam-5770	245	13	i=1ng(xi	i=1ng(xi	NOUN
ejpam-5770	245	14	)	)	PUNCT
ejpam-5770	245	15	.	.	PUNCT
ejpam-5770	246	1	proof	proof	NOUN
ejpam-5770	246	2	.	.	PUNCT
ejpam-5770	247	1	let	let	VERB
ejpam-5770	247	2	n	n	NOUN
ejpam-5770	247	3	=	=	SYM
ejpam-5770	247	4	k	k	PROPN
ejpam-5770	247	5	or	or	CCONJ
ejpam-5770	247	6	n	n	PROPN
ejpam-5770	247	7	>	>	X
ejpam-5770	247	8	k	k	NOUN
ejpam-5770	247	9	and	and	CCONJ
ejpam-5770	247	10	there	there	PRON
ejpam-5770	247	11	exists	exist	VERB
ejpam-5770	247	12	a	a	DET
ejpam-5770	247	13	set	set	NOUN
ejpam-5770	247	14	x	x	X
ejpam-5770	247	15	=	=	SYM
ejpam-5770	247	16	{	{	PUNCT
ejpam-5770	247	17	x1	x1	PROPN
ejpam-5770	247	18	,	,	PUNCT
ejpam-5770	247	19	x2	x2	PROPN
ejpam-5770	247	20	,	,	PUNCT
ejpam-5770	247	21	.	.	PUNCT
ejpam-5770	247	22	.	.	PUNCT
ejpam-5770	248	1	.	.	PUNCT
ejpam-5770	249	1	,	,	PUNCT
ejpam-5770	249	2	xm	xm	PROPN
ejpam-5770	249	3	}	}	PUNCT
ejpam-5770	249	4	of	of	ADP
ejpam-5770	249	5	vertices	vertex	NOUN
ejpam-5770	249	6	with	with	ADP
ejpam-5770	249	7	1	1	NUM
ejpam-5770	249	8	≤	≤	NUM
ejpam-5770	249	9	|x|	|x|	PROPN
ejpam-5770	249	10	≤	≤	NOUN
ejpam-5770	250	1	k	k	PRON
ejpam-5770	250	2	such	such	ADJ
ejpam-5770	250	3	that	that	PRON
ejpam-5770	250	4	(	(	PUNCT
ejpam-5770	250	5	v	v	NOUN
ejpam-5770	250	6	(	(	PUNCT
ejpam-5770	250	7	g	g	NOUN
ejpam-5770	250	8	)	)	PUNCT
ejpam-5770	250	9	\x	\x	NOUN
ejpam-5770	250	10	)	)	PUNCT
ejpam-5770	250	11	⊆	⊆	NUM
ejpam-5770	250	12	⋂m	⋂m	NOUN
ejpam-5770	250	13	i=1ng(xi	i=1ng(xi	NOUN
ejpam-5770	250	14	)	)	PUNCT
ejpam-5770	250	15	.	.	PUNCT
ejpam-5770	251	1	by	by	ADP
ejpam-5770	251	2	proposition	proposition	NOUN
ejpam-5770	251	3	2	2	NUM
ejpam-5770	251	4	,	,	PUNCT
ejpam-5770	251	5	we	we	PRON
ejpam-5770	251	6	have	have	VERB
ejpam-5770	251	7	γwc	γwc	ADJ
ejpam-5770	251	8	rk	rk	PROPN
ejpam-5770	251	9	(	(	PUNCT
ejpam-5770	251	10	g	g	NOUN
ejpam-5770	251	11	)	)	PUNCT
ejpam-5770	251	12	≥	≥	NOUN
ejpam-5770	251	13	k.	k.	INTJ
ejpam-5770	252	1	if	if	SCONJ
ejpam-5770	252	2	n	n	PROPN
ejpam-5770	252	3	=	=	SYM
ejpam-5770	252	4	k	k	NOUN
ejpam-5770	252	5	,	,	PUNCT
ejpam-5770	252	6	then	then	ADV
ejpam-5770	252	7	obviously	obviously	ADV
ejpam-5770	252	8	γwc	γwc	ADJ
ejpam-5770	252	9	rk	rk	PROPN
ejpam-5770	252	10	(	(	PUNCT
ejpam-5770	252	11	g	g	NOUN
ejpam-5770	252	12	)	)	PUNCT
ejpam-5770	252	13	=	=	SYM
ejpam-5770	253	1	k.	k.	PROPN
ejpam-5770	253	2	assume	assume	VERB
ejpam-5770	253	3	that	that	SCONJ
ejpam-5770	253	4	n	n	PROPN
ejpam-5770	253	5	>	>	X
ejpam-5770	253	6	k.	k.	PROPN
ejpam-5770	254	1	then	then	ADV
ejpam-5770	254	2	the	the	DET
ejpam-5770	254	3	function	function	NOUN
ejpam-5770	254	4	f	f	X
ejpam-5770	254	5	:	:	PUNCT
ejpam-5770	254	6	v	v	X
ejpam-5770	254	7	(	(	PUNCT
ejpam-5770	254	8	g	g	NOUN
ejpam-5770	254	9	)	)	PUNCT
ejpam-5770	254	10	→	→	SYM
ejpam-5770	254	11	p({1	p({1	PROPN
ejpam-5770	254	12	,	,	PUNCT
ejpam-5770	254	13	2	2	NUM
ejpam-5770	254	14	,	,	PUNCT
ejpam-5770	254	15	.	.	PUNCT
ejpam-5770	254	16	.	.	PUNCT
ejpam-5770	254	17	.	.	PUNCT
ejpam-5770	255	1	,	,	PUNCT
ejpam-5770	255	2	k	k	X
ejpam-5770	255	3	}	}	PUNCT
ejpam-5770	255	4	)	)	PUNCT
ejpam-5770	255	5	defined	define	VERB
ejpam-5770	255	6	by	by	ADP
ejpam-5770	255	7	f(xi	f(xi	NOUN
ejpam-5770	255	8	)	)	PUNCT
ejpam-5770	256	1	=	=	PRON
ejpam-5770	256	2	{	{	PUNCT
ejpam-5770	256	3	i	i	NOUN
ejpam-5770	256	4	}	}	PUNCT
ejpam-5770	256	5	for	for	ADP
ejpam-5770	256	6	1	1	NUM
ejpam-5770	256	7	≤	≤	NUM
ejpam-5770	256	8	i	i	PRON
ejpam-5770	256	9	≤	≤	NOUN
ejpam-5770	256	10	m	m	VERB
ejpam-5770	256	11	−	−	PROPN
ejpam-5770	256	12	1	1	NUM
ejpam-5770	256	13	,	,	PUNCT
ejpam-5770	256	14	f(xm	f(xm	PROPN
ejpam-5770	256	15	)	)	PUNCT
ejpam-5770	256	16	=	=	SYM
ejpam-5770	256	17	{	{	PUNCT
ejpam-5770	256	18	t	t	PROPN
ejpam-5770	256	19	,	,	PUNCT
ejpam-5770	256	20	t	t	PROPN
ejpam-5770	256	21	+	+	NOUN
ejpam-5770	256	22	1	1	NUM
ejpam-5770	256	23	,	,	PUNCT
ejpam-5770	256	24	.	.	PUNCT
ejpam-5770	256	25	.	.	PUNCT
ejpam-5770	256	26	.	.	PUNCT
ejpam-5770	257	1	,	,	PUNCT
ejpam-5770	257	2	k	k	X
ejpam-5770	257	3	}	}	PUNCT
ejpam-5770	257	4	and	and	CCONJ
ejpam-5770	257	5	f(x	f(x	PROPN
ejpam-5770	257	6	)	)	PUNCT
ejpam-5770	258	1	=	=	NOUN
ejpam-5770	258	2	∅	∅	NOUN
ejpam-5770	258	3	otherwise	otherwise	ADV
ejpam-5770	258	4	,	,	PUNCT
ejpam-5770	258	5	is	be	AUX
ejpam-5770	258	6	clearly	clearly	ADV
ejpam-5770	258	7	a	a	DET
ejpam-5770	258	8	wckrdf	wckrdf	NOUN
ejpam-5770	258	9	of	of	ADP
ejpam-5770	258	10	g	g	NOUN
ejpam-5770	258	11	of	of	ADP
ejpam-5770	258	12	weight	weight	NOUN
ejpam-5770	258	13	k	k	PROPN
ejpam-5770	258	14	,	,	PUNCT
ejpam-5770	258	15	and	and	CCONJ
ejpam-5770	258	16	so	so	ADV
ejpam-5770	258	17	γwc	γwc	ADJ
ejpam-5770	258	18	rk	rk	PROPN
ejpam-5770	258	19	(	(	PUNCT
ejpam-5770	258	20	g	g	NOUN
ejpam-5770	258	21	)	)	PUNCT
ejpam-5770	258	22	≤	≤	NOUN
ejpam-5770	258	23	k.	k.	PROPN
ejpam-5770	259	1	thus	thus	ADV
ejpam-5770	259	2	,	,	PUNCT
ejpam-5770	259	3	γwc	γwc	ADJ
ejpam-5770	259	4	rk	rk	PROPN
ejpam-5770	259	5	(	(	PUNCT
ejpam-5770	259	6	g	g	NOUN
ejpam-5770	259	7	)	)	PUNCT
ejpam-5770	259	8	=	=	VERB
ejpam-5770	259	9	k.	k.	PROPN
ejpam-5770	259	10	conversely	conversely	ADV
ejpam-5770	259	11	,	,	PUNCT
ejpam-5770	259	12	assume	assume	VERB
ejpam-5770	259	13	that	that	SCONJ
ejpam-5770	259	14	γwc	γwc	ADJ
ejpam-5770	259	15	rk	rk	PROPN
ejpam-5770	259	16	(	(	PUNCT
ejpam-5770	259	17	g	g	NOUN
ejpam-5770	259	18	)	)	PUNCT
ejpam-5770	259	19	=	=	VERB
ejpam-5770	260	1	k.	k.	PROPN
ejpam-5770	260	2	let	let	VERB
ejpam-5770	260	3	f	f	PRON
ejpam-5770	260	4	be	be	AUX
ejpam-5770	260	5	a	a	DET
ejpam-5770	260	6	γwc	γwc	ADJ
ejpam-5770	260	7	rk	rk	NOUN
ejpam-5770	260	8	-function	-function	NOUN
ejpam-5770	260	9	of	of	ADP
ejpam-5770	260	10	g.	g.	PROPN
ejpam-5770	261	1	if	if	SCONJ
ejpam-5770	261	2	v	v	NUM
ejpam-5770	261	3	f	f	NOUN
ejpam-5770	261	4	0	0	NUM
ejpam-5770	261	5	=	=	NOUN
ejpam-5770	261	6	∅	∅	NOUN
ejpam-5770	261	7	,	,	PUNCT
ejpam-5770	261	8	then	then	ADV
ejpam-5770	261	9	we	we	PRON
ejpam-5770	261	10	have	have	VERB
ejpam-5770	261	11	that	that	PRON
ejpam-5770	261	12	n	n	PROPN
ejpam-5770	261	13	=	=	SYM
ejpam-5770	261	14	k.	k.	PROPN
ejpam-5770	261	15	assume	assume	VERB
ejpam-5770	261	16	that	that	SCONJ
ejpam-5770	261	17	v	v	X
ejpam-5770	261	18	f	f	PROPN
ejpam-5770	261	19	0	0	NUM
ejpam-5770	261	20	̸=	̸=	PROPN
ejpam-5770	261	21	∅	∅	NOUN
ejpam-5770	261	22	and	and	CCONJ
ejpam-5770	261	23	let	let	VERB
ejpam-5770	261	24	u	u	PRON
ejpam-5770	261	25	∈	∈	PROPN
ejpam-5770	261	26	v	v	ADP
ejpam-5770	261	27	f	f	PROPN
ejpam-5770	261	28	0	0	NUM
ejpam-5770	261	29	.	.	PUNCT
ejpam-5770	262	1	by	by	ADP
ejpam-5770	262	2	definition	definition	NOUN
ejpam-5770	262	3	,	,	PUNCT
ejpam-5770	262	4	∪m	∪m	PUNCT
ejpam-5770	262	5	i=1f(xi	i=1f(xi	PROPN
ejpam-5770	262	6	)	)	PUNCT
ejpam-5770	262	7	=	=	PUNCT
ejpam-5770	262	8	{	{	PUNCT
ejpam-5770	262	9	1	1	NUM
ejpam-5770	262	10	,	,	PUNCT
ejpam-5770	262	11	2	2	NUM
ejpam-5770	262	12	,	,	PUNCT
ejpam-5770	262	13	.	.	PUNCT
ejpam-5770	262	14	.	.	PUNCT
ejpam-5770	262	15	.	.	PUNCT
ejpam-5770	263	1	,	,	PUNCT
ejpam-5770	263	2	k	k	X
ejpam-5770	263	3	}	}	PUNCT
ejpam-5770	263	4	.	.	PUNCT
ejpam-5770	264	1	now	now	ADV
ejpam-5770	264	2	let	let	VERB
ejpam-5770	264	3	x1	x1	NUM
ejpam-5770	264	4	,	,	PUNCT
ejpam-5770	264	5	x2	x2	PROPN
ejpam-5770	264	6	,	,	PUNCT
ejpam-5770	264	7	.	.	PUNCT
ejpam-5770	264	8	.	.	PUNCT
ejpam-5770	265	1	.	.	PUNCT
ejpam-5770	266	1	,	,	PUNCT
ejpam-5770	266	2	xm	xm	PROPN
ejpam-5770	266	3	be	be	AUX
ejpam-5770	266	4	all	all	DET
ejpam-5770	266	5	vertices	vertex	NOUN
ejpam-5770	266	6	in	in	ADP
ejpam-5770	266	7	ng(u	ng(u	NOUN
ejpam-5770	266	8	)	)	PUNCT
ejpam-5770	266	9	such	such	ADJ
ejpam-5770	266	10	that	that	DET
ejpam-5770	266	11	f(xi	f(xi	X
ejpam-5770	266	12	)	)	PUNCT
ejpam-5770	266	13	̸=	̸=	PROPN
ejpam-5770	266	14	∅	∅	NOUN
ejpam-5770	266	15	for	for	ADP
ejpam-5770	266	16	1	1	NUM
ejpam-5770	266	17	≤	≤	NUM
ejpam-5770	266	18	i	i	PRON
ejpam-5770	266	19	≤	≤	NUM
ejpam-5770	266	20	m.	m.	NOUN
ejpam-5770	266	21	it	it	PRON
ejpam-5770	266	22	follows	follow	VERB
ejpam-5770	266	23	from	from	ADP
ejpam-5770	266	24	the	the	DET
ejpam-5770	266	25	condition	condition	NOUN
ejpam-5770	266	26	γwc	γwc	PROPN
ejpam-5770	266	27	rk	rk	PROPN
ejpam-5770	266	28	(	(	PUNCT
ejpam-5770	266	29	g	g	NOUN
ejpam-5770	266	30	)	)	PUNCT
ejpam-5770	266	31	=	=	PUNCT
ejpam-5770	267	1	k	k	PROPN
ejpam-5770	268	1	that	that	SCONJ
ejpam-5770	268	2	∪m	∪m	VERB
ejpam-5770	268	3	i=1|f(xi)|	i=1|f(xi)|	NOUN
ejpam-5770	268	4	=	=	SYM
ejpam-5770	268	5	k	k	NOUN
ejpam-5770	268	6	,	,	PUNCT
ejpam-5770	268	7	1	1	NUM
ejpam-5770	268	8	≤	≤	NUM
ejpam-5770	268	9	i	i	X
ejpam-5770	268	10	≤	≤	NOUN
ejpam-5770	268	11	m	m	ADP
ejpam-5770	268	12	,	,	PUNCT
ejpam-5770	268	13	and	and	CCONJ
ejpam-5770	268	14	(	(	PUNCT
ejpam-5770	268	15	v	v	NOUN
ejpam-5770	268	16	(	(	PUNCT
ejpam-5770	268	17	g	g	NOUN
ejpam-5770	268	18	)	)	PUNCT
ejpam-5770	268	19	\x	\x	NOUN
ejpam-5770	268	20	)	)	PUNCT
ejpam-5770	269	1	⊆	⊆	NUM
ejpam-5770	269	2	⋂m	⋂m	NOUN
ejpam-5770	269	3	i=1ng(xi	i=1ng(xi	NOUN
ejpam-5770	269	4	)	)	PUNCT
ejpam-5770	269	5	.	.	PUNCT
ejpam-5770	270	1	this	this	PRON
ejpam-5770	270	2	completes	complete	VERB
ejpam-5770	270	3	the	the	DET
ejpam-5770	270	4	proof	proof	NOUN
ejpam-5770	270	5	.	.	PUNCT
ejpam-5770	271	1	as	as	ADP
ejpam-5770	271	2	a	a	DET
ejpam-5770	271	3	consequence	consequence	NOUN
ejpam-5770	271	4	of	of	ADP
ejpam-5770	271	5	proposition	proposition	NOUN
ejpam-5770	271	6	2	2	NUM
ejpam-5770	271	7	and	and	CCONJ
ejpam-5770	271	8	theorem	theorem	VERB
ejpam-5770	271	9	6	6	NUM
ejpam-5770	271	10	,	,	PUNCT
ejpam-5770	271	11	we	we	PRON
ejpam-5770	271	12	have	have	VERB
ejpam-5770	271	13	the	the	DET
ejpam-5770	271	14	following	following	NOUN
ejpam-5770	271	15	.	.	PUNCT
ejpam-5770	272	1	corollary	corollary	ADJ
ejpam-5770	272	2	3	3	X
ejpam-5770	272	3	.	.	PUNCT
ejpam-5770	273	1	let	let	VERB
ejpam-5770	273	2	n	n	PRON
ejpam-5770	274	1	and	and	CCONJ
ejpam-5770	274	2	k	k	PROPN
ejpam-5770	274	3	be	be	AUX
ejpam-5770	274	4	positive	positive	ADJ
ejpam-5770	274	5	integers	integer	NOUN
ejpam-5770	274	6	.	.	PUNCT
ejpam-5770	275	1	then	then	ADV
ejpam-5770	275	2	γwc	γwc	PROPN
ejpam-5770	275	3	rk	rk	PROPN
ejpam-5770	275	4	(	(	PUNCT
ejpam-5770	275	5	kn	kn	PROPN
ejpam-5770	275	6	)	)	PUNCT
ejpam-5770	275	7	=	=	PRON
ejpam-5770	275	8	{	{	PUNCT
ejpam-5770	275	9	k	k	X
ejpam-5770	275	10	if	if	SCONJ
ejpam-5770	275	11	n	n	PROPN
ejpam-5770	275	12	≥	≥	X
ejpam-5770	275	13	k	k	NOUN
ejpam-5770	275	14	,	,	PUNCT
ejpam-5770	275	15	n	n	CCONJ
ejpam-5770	275	16	if	if	SCONJ
ejpam-5770	275	17	n	n	PROPN
ejpam-5770	275	18	<	<	X
ejpam-5770	275	19	k.	k.	PROPN
ejpam-5770	275	20	proposition	proposition	PROPN
ejpam-5770	275	21	3	3	X
ejpam-5770	275	22	.	.	PUNCT
ejpam-5770	276	1	let	let	VERB
ejpam-5770	276	2	m	m	PRON
ejpam-5770	276	3	,	,	PUNCT
ejpam-5770	276	4	n	n	CCONJ
ejpam-5770	276	5	,	,	PUNCT
ejpam-5770	276	6	and	and	CCONJ
ejpam-5770	276	7	k	k	PROPN
ejpam-5770	276	8	be	be	AUX
ejpam-5770	276	9	positive	positive	ADJ
ejpam-5770	276	10	integers	integer	NOUN
ejpam-5770	276	11	with	with	ADP
ejpam-5770	276	12	k	k	PROPN
ejpam-5770	276	13	≥	≥	NUM
ejpam-5770	276	14	1	1	NUM
ejpam-5770	276	15	and	and	CCONJ
ejpam-5770	276	16	m	m	PROPN
ejpam-5770	276	17	≤	≤	NOUN
ejpam-5770	276	18	n.	n.	NOUN
ejpam-5770	276	19	then	then	ADV
ejpam-5770	276	20	γwc	γwc	PROPN
ejpam-5770	276	21	rk	rk	PROPN
ejpam-5770	276	22	(	(	PUNCT
ejpam-5770	276	23	km	km	PROPN
ejpam-5770	276	24	,	,	PUNCT
ejpam-5770	276	25	n	n	CCONJ
ejpam-5770	276	26	)	)	PUNCT
ejpam-5770	277	1	=	=	SYM
ejpam-5770	278	1			PRON
ejpam-5770	278	2	m+	m+	NOUN
ejpam-5770	278	3	n	n	NOUN
ejpam-5770	278	4	if	if	SCONJ
ejpam-5770	278	5	m+	m+	NUM
ejpam-5770	278	6	n	n	NOUN
ejpam-5770	278	7	≤	≤	NOUN
ejpam-5770	278	8	k	k	PROPN
ejpam-5770	278	9	,	,	PUNCT
ejpam-5770	278	10	2k	2k	NOUN
ejpam-5770	278	11	if	if	SCONJ
ejpam-5770	278	12	m	m	PROPN
ejpam-5770	278	13	≥	≥	NOUN
ejpam-5770	278	14	2k	2k	NUM
ejpam-5770	278	15	,	,	PUNCT
ejpam-5770	278	16	max{m	max{m	PROPN
ejpam-5770	278	17	,	,	PUNCT
ejpam-5770	278	18	k	k	NOUN
ejpam-5770	278	19	}	}	PUNCT
ejpam-5770	278	20	if	if	SCONJ
ejpam-5770	278	21	m+	m+	NUM
ejpam-5770	278	22	n	n	CCONJ
ejpam-5770	278	23	>	>	X
ejpam-5770	278	24	k	k	PROPN
ejpam-5770	278	25	and	and	CCONJ
ejpam-5770	278	26	m	m	VERB
ejpam-5770	278	27	<	<	X
ejpam-5770	278	28	2k	2k	NUM
ejpam-5770	278	29	.	.	PUNCT
ejpam-5770	279	1	proof	proof	NOUN
ejpam-5770	279	2	.	.	PUNCT
ejpam-5770	280	1	suppose	suppose	VERB
ejpam-5770	280	2	km	km	PROPN
ejpam-5770	280	3	,	,	PUNCT
ejpam-5770	280	4	n	n	PRON
ejpam-5770	280	5	is	be	AUX
ejpam-5770	280	6	a	a	DET
ejpam-5770	280	7	complete	complete	ADJ
ejpam-5770	280	8	bipartite	bipartite	NOUN
ejpam-5770	280	9	graph	graph	NOUN
ejpam-5770	280	10	with	with	ADP
ejpam-5770	280	11	m	m	PROPN
ejpam-5770	280	12	,	,	PUNCT
ejpam-5770	280	13	n	n	PRON
ejpam-5770	280	14	vertices	vertex	NOUN
ejpam-5770	280	15	.	.	PUNCT
ejpam-5770	281	1	let	let	VERB
ejpam-5770	281	2	x	x	PRON
ejpam-5770	281	3	and	and	CCONJ
ejpam-5770	281	4	y	y	PROPN
ejpam-5770	281	5	be	be	AUX
ejpam-5770	281	6	the	the	DET
ejpam-5770	281	7	two	two	NUM
ejpam-5770	281	8	partite	partite	ADJ
ejpam-5770	281	9	sets	set	NOUN
ejpam-5770	281	10	of	of	ADP
ejpam-5770	281	11	km	km	PROPN
ejpam-5770	281	12	,	,	PUNCT
ejpam-5770	281	13	n	n	CCONJ
ejpam-5770	281	14	,	,	PUNCT
ejpam-5770	281	15	where	where	SCONJ
ejpam-5770	281	16	x	x	X
ejpam-5770	281	17	=	=	PRON
ejpam-5770	281	18	{	{	PUNCT
ejpam-5770	281	19	x1	x1	PROPN
ejpam-5770	281	20	,	,	PUNCT
ejpam-5770	281	21	x2	x2	PROPN
ejpam-5770	281	22	,	,	PUNCT
ejpam-5770	281	23	.	.	PUNCT
ejpam-5770	281	24	.	.	PUNCT
ejpam-5770	282	1	.	.	PUNCT
ejpam-5770	283	1	,	,	PUNCT
ejpam-5770	283	2	xm	xm	PROPN
ejpam-5770	283	3	}	}	PUNCT
ejpam-5770	283	4	and	and	CCONJ
ejpam-5770	283	5	y	y	PROPN
ejpam-5770	283	6	=	=	SYM
ejpam-5770	283	7	{	{	PUNCT
ejpam-5770	283	8	y1	y1	PROPN
ejpam-5770	283	9	,	,	PUNCT
ejpam-5770	283	10	y2	y2	PROPN
ejpam-5770	283	11	,	,	PUNCT
ejpam-5770	283	12	.	.	PUNCT
ejpam-5770	283	13	.	.	PUNCT
ejpam-5770	284	1	.	.	PUNCT
ejpam-5770	285	1	,	,	PUNCT
ejpam-5770	285	2	yn	yn	PROPN
ejpam-5770	285	3	}	}	PUNCT
ejpam-5770	285	4	.	.	PUNCT
ejpam-5770	286	1	if	if	SCONJ
ejpam-5770	286	2	m	m	VERB
ejpam-5770	286	3	+	+	ADP
ejpam-5770	286	4	n	n	CCONJ
ejpam-5770	286	5	≤	≤	ADV
ejpam-5770	286	6	k	k	NOUN
ejpam-5770	286	7	,	,	PUNCT
ejpam-5770	286	8	then	then	ADV
ejpam-5770	286	9	we	we	PRON
ejpam-5770	286	10	have	have	VERB
ejpam-5770	286	11	γwc	γwc	ADJ
ejpam-5770	286	12	rk	rk	PROPN
ejpam-5770	286	13	(	(	PUNCT
ejpam-5770	286	14	km	km	PROPN
ejpam-5770	286	15	,	,	PUNCT
ejpam-5770	286	16	n	n	CCONJ
ejpam-5770	286	17	)	)	PUNCT
ejpam-5770	286	18	=	=	SYM
ejpam-5770	287	1	m	m	VERB
ejpam-5770	287	2	+	+	NOUN
ejpam-5770	287	3	n	n	CCONJ
ejpam-5770	287	4	by	by	ADP
ejpam-5770	287	5	proposition	proposition	NOUN
ejpam-5770	287	6	2	2	NUM
ejpam-5770	287	7	.	.	PUNCT
ejpam-5770	288	1	hence	hence	ADV
ejpam-5770	288	2	we	we	PRON
ejpam-5770	288	3	assume	assume	VERB
ejpam-5770	288	4	that	that	SCONJ
ejpam-5770	288	5	m+	m+	PRON
ejpam-5770	288	6	n	n	CCONJ
ejpam-5770	288	7	>	>	PUNCT
ejpam-5770	288	8	k.	k.	PROPN
ejpam-5770	288	9	consider	consider	VERB
ejpam-5770	288	10	the	the	DET
ejpam-5770	288	11	following	follow	VERB
ejpam-5770	288	12	cases	case	NOUN
ejpam-5770	288	13	:	:	PUNCT
ejpam-5770	288	14	j.	j.	PROPN
ejpam-5770	288	15	j.	j.	PROPN
ejpam-5770	288	16	hamja	hamja	PROPN
ejpam-5770	289	1	et	et	PROPN
ejpam-5770	289	2	al	al	PROPN
ejpam-5770	289	3	.	.	PUNCT
ejpam-5770	289	4	/	/	SYM
ejpam-5770	289	5	eur	eur	PROPN
ejpam-5770	289	6	.	.	PUNCT
ejpam-5770	290	1	j.	j.	PROPN
ejpam-5770	290	2	pure	pure	PROPN
ejpam-5770	290	3	appl	appl	PROPN
ejpam-5770	290	4	.	.	PROPN
ejpam-5770	290	5	math	math	PROPN
ejpam-5770	290	6	,	,	PUNCT
ejpam-5770	290	7	18	18	NUM
ejpam-5770	290	8	(	(	PUNCT
ejpam-5770	290	9	2	2	NUM
ejpam-5770	290	10	)	)	PUNCT
ejpam-5770	290	11	(	(	PUNCT
ejpam-5770	290	12	2025	2025	NUM
ejpam-5770	290	13	)	)	PUNCT
ejpam-5770	290	14	,	,	PUNCT
ejpam-5770	290	15	5770	5770	NUM
ejpam-5770	290	16	8	8	NUM
ejpam-5770	290	17	of	of	ADP
ejpam-5770	290	18	11	11	NUM
ejpam-5770	290	19	case	case	NOUN
ejpam-5770	290	20	1	1	NUM
ejpam-5770	290	21	.	.	PUNCT
ejpam-5770	291	1	m	m	PROPN
ejpam-5770	291	2	≥	≥	NOUN
ejpam-5770	291	3	2k	2k	NUM
ejpam-5770	291	4	.	.	PUNCT
ejpam-5770	292	1	let	let	VERB
ejpam-5770	292	2	f	f	NOUN
ejpam-5770	292	3	:	:	PUNCT
ejpam-5770	292	4	v	v	X
ejpam-5770	292	5	(	(	PUNCT
ejpam-5770	292	6	km	km	PROPN
ejpam-5770	292	7	,	,	PUNCT
ejpam-5770	292	8	n	n	CCONJ
ejpam-5770	292	9	)	)	PUNCT
ejpam-5770	292	10	→	→	SYM
ejpam-5770	293	1	p	p	X
ejpam-5770	293	2	(	(	PUNCT
ejpam-5770	293	3	{	{	PUNCT
ejpam-5770	293	4	1	1	NUM
ejpam-5770	293	5	,	,	PUNCT
ejpam-5770	293	6	2	2	NUM
ejpam-5770	293	7	,	,	PUNCT
ejpam-5770	293	8	.	.	PUNCT
ejpam-5770	293	9	.	.	PUNCT
ejpam-5770	293	10	.	.	PUNCT
ejpam-5770	294	1	,	,	PUNCT
ejpam-5770	294	2	k	k	X
ejpam-5770	294	3	}	}	PUNCT
ejpam-5770	294	4	)	)	PUNCT
ejpam-5770	294	5	be	be	AUX
ejpam-5770	294	6	a	a	DET
ejpam-5770	294	7	function	function	NOUN
ejpam-5770	294	8	defined	define	VERB
ejpam-5770	294	9	by	by	ADP
ejpam-5770	294	10	f(x1	f(x1	ADJ
ejpam-5770	294	11	)	)	PUNCT
ejpam-5770	294	12	=	=	SYM
ejpam-5770	294	13	f(y1	f(y1	NOUN
ejpam-5770	294	14	)	)	PUNCT
ejpam-5770	294	15	=	=	SYM
ejpam-5770	294	16	{	{	PUNCT
ejpam-5770	294	17	1	1	NUM
ejpam-5770	294	18	,	,	PUNCT
ejpam-5770	294	19	2	2	NUM
ejpam-5770	294	20	,	,	PUNCT
ejpam-5770	294	21	.	.	PUNCT
ejpam-5770	294	22	.	.	PUNCT
ejpam-5770	295	1	.	.	PUNCT
ejpam-5770	296	1	,	,	PUNCT
ejpam-5770	296	2	k	k	X
ejpam-5770	296	3	}	}	PUNCT
ejpam-5770	296	4	,	,	PUNCT
ejpam-5770	296	5	and	and	CCONJ
ejpam-5770	296	6	f(v	f(v	NOUN
ejpam-5770	296	7	)	)	PUNCT
ejpam-5770	296	8	=	=	PUNCT
ejpam-5770	296	9	∅	∅	NOUN
ejpam-5770	296	10	for	for	ADP
ejpam-5770	296	11	every	every	DET
ejpam-5770	296	12	vertex	vertex	NOUN
ejpam-5770	296	13	v	v	ADP
ejpam-5770	296	14	∈	∈	PROPN
ejpam-5770	296	15	v	v	NOUN
ejpam-5770	296	16	(	(	PUNCT
ejpam-5770	296	17	km	km	PROPN
ejpam-5770	296	18	,	,	PUNCT
ejpam-5770	296	19	n)\{x1	n)\{x1	NOUN
ejpam-5770	296	20	,	,	PUNCT
ejpam-5770	296	21	y1	y1	NOUN
ejpam-5770	296	22	}	}	PUNCT
ejpam-5770	296	23	.	.	PUNCT
ejpam-5770	297	1	then	then	ADV
ejpam-5770	297	2	⋃	⋃	PROPN
ejpam-5770	297	3	u∈ng(v	u∈ng(v	PROPN
ejpam-5770	297	4	)	)	PUNCT
ejpam-5770	297	5	f(u	f(u	PROPN
ejpam-5770	297	6	)	)	PUNCT
ejpam-5770	297	7	=	=	PRON
ejpam-5770	298	1	{	{	PUNCT
ejpam-5770	298	2	1	1	NUM
ejpam-5770	298	3	,	,	PUNCT
ejpam-5770	298	4	2	2	NUM
ejpam-5770	298	5	,	,	PUNCT
ejpam-5770	298	6	.	.	PUNCT
ejpam-5770	298	7	.	.	PUNCT
ejpam-5770	299	1	.	.	PUNCT
ejpam-5770	300	1	,	,	PUNCT
ejpam-5770	300	2	k	k	X
ejpam-5770	300	3	}	}	PUNCT
ejpam-5770	300	4	for	for	ADP
ejpam-5770	300	5	every	every	DET
ejpam-5770	300	6	v	v	NUM
ejpam-5770	300	7	∈	∈	PROPN
ejpam-5770	300	8	v	v	NOUN
ejpam-5770	300	9	(	(	PUNCT
ejpam-5770	300	10	km	km	PROPN
ejpam-5770	300	11	,	,	PUNCT
ejpam-5770	300	12	n	n	CCONJ
ejpam-5770	300	13	)	)	PUNCT
ejpam-5770	300	14	\	\	NOUN
ejpam-5770	300	15	{	{	PUNCT
ejpam-5770	300	16	x1	x1	PROPN
ejpam-5770	300	17	,	,	PUNCT
ejpam-5770	300	18	y1	y1	NOUN
ejpam-5770	300	19	}	}	PUNCT
ejpam-5770	300	20	.	.	PUNCT
ejpam-5770	301	1	observe	observe	VERB
ejpam-5770	301	2	that	that	SCONJ
ejpam-5770	301	3	the	the	DET
ejpam-5770	301	4	subgraph	subgraph	NOUN
ejpam-5770	301	5	⟨{x1	⟨{x1	PROPN
ejpam-5770	301	6	,	,	PUNCT
ejpam-5770	301	7	y1}⟩w	y1}⟩w	VERB
ejpam-5770	301	8	induced	induce	VERB
ejpam-5770	301	9	by	by	ADP
ejpam-5770	301	10	{	{	PUNCT
ejpam-5770	301	11	x1	x1	PROPN
ejpam-5770	301	12	,	,	PUNCT
ejpam-5770	301	13	y1	y1	NOUN
ejpam-5770	301	14	}	}	PUNCT
ejpam-5770	301	15	is	be	AUX
ejpam-5770	301	16	connected	connect	VERB
ejpam-5770	301	17	and	and	CCONJ
ejpam-5770	301	18	nkm	nkm	NOUN
ejpam-5770	301	19	,	,	PUNCT
ejpam-5770	301	20	n	n	PRON
ejpam-5770	301	21	[	[	X
ejpam-5770	301	22	{	{	PUNCT
ejpam-5770	301	23	x1	x1	PROPN
ejpam-5770	301	24	,	,	PUNCT
ejpam-5770	301	25	y1	y1	PROPN
ejpam-5770	301	26	}	}	PUNCT
ejpam-5770	301	27	]	]	PUNCT
ejpam-5770	301	28	=	=	SYM
ejpam-5770	301	29	v	v	X
ejpam-5770	301	30	(	(	PUNCT
ejpam-5770	301	31	km	km	PROPN
ejpam-5770	301	32	,	,	PUNCT
ejpam-5770	301	33	n	n	CCONJ
ejpam-5770	301	34	)	)	PUNCT
ejpam-5770	301	35	.	.	PUNCT
ejpam-5770	302	1	thus	thus	ADV
ejpam-5770	302	2	,	,	PUNCT
ejpam-5770	302	3	by	by	ADP
ejpam-5770	302	4	definition	definition	NOUN
ejpam-5770	302	5	,	,	PUNCT
ejpam-5770	302	6	f	f	PROPN
ejpam-5770	302	7	is	be	AUX
ejpam-5770	302	8	a	a	DET
ejpam-5770	302	9	weakly	weakly	ADV
ejpam-5770	302	10	connected	connected	ADJ
ejpam-5770	302	11	k	k	ADJ
ejpam-5770	302	12	-	-	PUNCT
ejpam-5770	302	13	rainbow	rainbow	NOUN
ejpam-5770	302	14	dominating	dominating	NOUN
ejpam-5770	302	15	function	function	NOUN
ejpam-5770	302	16	of	of	ADP
ejpam-5770	302	17	km	km	PROPN
ejpam-5770	302	18	,	,	PUNCT
ejpam-5770	302	19	n	n	PRON
ejpam-5770	302	20	of	of	ADP
ejpam-5770	302	21	weight	weight	NOUN
ejpam-5770	302	22	ω(f	ω(f	PUNCT
ejpam-5770	302	23	)	)	PUNCT
ejpam-5770	302	24	=	=	SYM
ejpam-5770	303	1	∑	∑	PUNCT
ejpam-5770	303	2	v∈v	v∈v	PROPN
ejpam-5770	303	3	(	(	PUNCT
ejpam-5770	303	4	km	km	NOUN
ejpam-5770	303	5	,	,	PUNCT
ejpam-5770	303	6	n	n	CCONJ
ejpam-5770	303	7	)	)	PUNCT
ejpam-5770	303	8	|f(v)|	|f(v)|	PROPN
ejpam-5770	303	9	=	=	ADJ
ejpam-5770	303	10	2k	2k	NUM
ejpam-5770	303	11	.	.	PUNCT
ejpam-5770	304	1	this	this	PRON
ejpam-5770	304	2	implies	imply	VERB
ejpam-5770	304	3	that	that	SCONJ
ejpam-5770	304	4	γwc	γwc	ADJ
ejpam-5770	304	5	rk	rk	PROPN
ejpam-5770	304	6	(	(	PUNCT
ejpam-5770	304	7	km	km	PROPN
ejpam-5770	304	8	,	,	PUNCT
ejpam-5770	304	9	n	n	CCONJ
ejpam-5770	304	10	)	)	PUNCT
ejpam-5770	304	11	≤	≤	NOUN
ejpam-5770	304	12	2k	2k	NUM
ejpam-5770	304	13	.	.	PUNCT
ejpam-5770	305	1	now	now	ADV
ejpam-5770	305	2	,	,	PUNCT
ejpam-5770	305	3	let	let	VERB
ejpam-5770	305	4	f∗	f∗	NOUN
ejpam-5770	305	5	be	be	AUX
ejpam-5770	305	6	any	any	DET
ejpam-5770	305	7	γwc	γwc	ADJ
ejpam-5770	305	8	rk	rk	NOUN
ejpam-5770	305	9	-function	-function	NOUN
ejpam-5770	305	10	of	of	ADP
ejpam-5770	305	11	km	km	NOUN
ejpam-5770	305	12	,	,	PUNCT
ejpam-5770	305	13	n.	n.	NOUN
ejpam-5770	305	14	if	if	SCONJ
ejpam-5770	305	15	for	for	ADP
ejpam-5770	305	16	every	every	DET
ejpam-5770	305	17	vertex	vertex	NOUN
ejpam-5770	305	18	x	x	SYM
ejpam-5770	305	19	∈	∈	NOUN
ejpam-5770	305	20	x	x	SYM
ejpam-5770	305	21	,	,	PUNCT
ejpam-5770	305	22	f∗(x	f∗(x	NOUN
ejpam-5770	305	23	)	)	PUNCT
ejpam-5770	305	24	̸=	̸=	PROPN
ejpam-5770	305	25	∅	∅	NOUN
ejpam-5770	305	26	or	or	CCONJ
ejpam-5770	305	27	for	for	ADP
ejpam-5770	305	28	every	every	DET
ejpam-5770	305	29	vertex	vertex	NOUN
ejpam-5770	305	30	y	y	PROPN
ejpam-5770	305	31	∈	∈	PROPN
ejpam-5770	305	32	y	y	PROPN
ejpam-5770	305	33	,	,	PUNCT
ejpam-5770	305	34	f∗(y	f∗(y	PROPN
ejpam-5770	305	35	)	)	PUNCT
ejpam-5770	305	36	̸=	̸=	NOUN
ejpam-5770	305	37	∅	∅	NOUN
ejpam-5770	305	38	,	,	PUNCT
ejpam-5770	305	39	then	then	ADV
ejpam-5770	305	40	clearly	clearly	ADV
ejpam-5770	305	41	ω(f∗	ω(f∗	NUM
ejpam-5770	305	42	)	)	PUNCT
ejpam-5770	305	43	≥	≥	PROPN
ejpam-5770	305	44	m	m	PROPN
ejpam-5770	305	45	≥	≥	NOUN
ejpam-5770	305	46	2k	2k	NUM
ejpam-5770	305	47	.	.	PUNCT
ejpam-5770	306	1	hence	hence	ADV
ejpam-5770	306	2	,	,	PUNCT
ejpam-5770	306	3	assume	assume	VERB
ejpam-5770	306	4	that	that	SCONJ
ejpam-5770	306	5	there	there	PRON
ejpam-5770	306	6	are	be	VERB
ejpam-5770	306	7	two	two	NUM
ejpam-5770	306	8	vertices	vertex	NOUN
ejpam-5770	306	9	x	x	X
ejpam-5770	306	10	∈	∈	NOUN
ejpam-5770	306	11	x	x	X
ejpam-5770	306	12	and	and	CCONJ
ejpam-5770	306	13	y	y	PROPN
ejpam-5770	306	14	∈	∈	PROPN
ejpam-5770	306	15	y	y	PROPN
ejpam-5770	306	16	such	such	ADJ
ejpam-5770	306	17	that	that	SCONJ
ejpam-5770	306	18	f∗(x	f∗(x	NOUN
ejpam-5770	306	19	)	)	PUNCT
ejpam-5770	306	20	=	=	SYM
ejpam-5770	306	21	∅	∅	NOUN
ejpam-5770	306	22	and	and	CCONJ
ejpam-5770	306	23	f∗(y	f∗(y	PROPN
ejpam-5770	306	24	)	)	PUNCT
ejpam-5770	306	25	=	=	PUNCT
ejpam-5770	306	26	∅.	∅.	NOUN
ejpam-5770	306	27	by	by	ADP
ejpam-5770	306	28	definition	definition	NOUN
ejpam-5770	306	29	,	,	PUNCT
ejpam-5770	306	30	we	we	PRON
ejpam-5770	306	31	have	have	VERB
ejpam-5770	306	32	ω(f∗	ω(f∗	NUM
ejpam-5770	306	33	)	)	PUNCT
ejpam-5770	306	34	=	=	SYM
ejpam-5770	306	35	∑	∑	PUNCT
ejpam-5770	306	36	v∈v	v∈v	PROPN
ejpam-5770	306	37	(	(	PUNCT
ejpam-5770	306	38	km	km	NOUN
ejpam-5770	306	39	,	,	PUNCT
ejpam-5770	306	40	n	n	CCONJ
ejpam-5770	306	41	)	)	PUNCT
ejpam-5770	306	42	|f∗(v)|	|f∗(v)|	PROPN
ejpam-5770	306	43	=	=	SYM
ejpam-5770	306	44	∑	∑	PUNCT
ejpam-5770	306	45	v∈x	v∈x	PROPN
ejpam-5770	306	46	|f∗(v)|+	|f∗(v)|+	NUM
ejpam-5770	306	47	∑	∑	PROPN
ejpam-5770	306	48	v∈y	v∈y	NOUN
ejpam-5770	306	49	|f∗(v)|	|f∗(v)|	PROPN
ejpam-5770	306	50	=	=	SYM
ejpam-5770	306	51	∑	∑	PUNCT
ejpam-5770	306	52	v∈nkm	v∈nkm	PROPN
ejpam-5770	306	53	,	,	PUNCT
ejpam-5770	306	54	n	n	CCONJ
ejpam-5770	306	55	(	(	PUNCT
ejpam-5770	306	56	y	y	NOUN
ejpam-5770	306	57	)	)	PUNCT
ejpam-5770	306	58	|f∗(v)|+	|f∗(v)|+	PUNCT
ejpam-5770	307	1	∑	∑	PROPN
ejpam-5770	307	2	v∈nkm	v∈nkm	PROPN
ejpam-5770	307	3	,	,	PUNCT
ejpam-5770	307	4	n	n	CCONJ
ejpam-5770	307	5	(	(	PUNCT
ejpam-5770	307	6	x	x	X
ejpam-5770	307	7	)	)	PUNCT
ejpam-5770	307	8	|f∗(v)|	|f∗(v)|	PROPN
ejpam-5770	307	9	≥	≥	NUM
ejpam-5770	307	10	2k	2k	NUM
ejpam-5770	307	11	.	.	PUNCT
ejpam-5770	308	1	therefore	therefore	ADV
ejpam-5770	308	2	,	,	PUNCT
ejpam-5770	308	3	γwc	γwc	ADJ
ejpam-5770	308	4	rk	rk	PROPN
ejpam-5770	308	5	(	(	PUNCT
ejpam-5770	308	6	km	km	PROPN
ejpam-5770	308	7	,	,	PUNCT
ejpam-5770	308	8	n	n	CCONJ
ejpam-5770	308	9	)	)	PUNCT
ejpam-5770	308	10	≥	≥	NOUN
ejpam-5770	308	11	2k	2k	NUM
ejpam-5770	308	12	,	,	PUNCT
ejpam-5770	308	13	and	and	CCONJ
ejpam-5770	308	14	as	as	ADP
ejpam-5770	308	15	a	a	DET
ejpam-5770	308	16	result	result	NOUN
ejpam-5770	308	17	,	,	PUNCT
ejpam-5770	308	18	we	we	PRON
ejpam-5770	308	19	conclude	conclude	VERB
ejpam-5770	308	20	γwc	γwc	ADJ
ejpam-5770	308	21	rk	rk	PROPN
ejpam-5770	308	22	(	(	PUNCT
ejpam-5770	308	23	km	km	PROPN
ejpam-5770	308	24	,	,	PUNCT
ejpam-5770	308	25	n	n	CCONJ
ejpam-5770	308	26	)	)	PUNCT
ejpam-5770	308	27	=	=	SYM
ejpam-5770	308	28	2k	2k	NUM
ejpam-5770	308	29	.	.	PUNCT
ejpam-5770	309	1	case	case	NOUN
ejpam-5770	309	2	2	2	NUM
ejpam-5770	309	3	.	.	PUNCT
ejpam-5770	309	4	m+	m+	NUM
ejpam-5770	309	5	n	n	CCONJ
ejpam-5770	309	6	>	>	X
ejpam-5770	309	7	k	k	PROPN
ejpam-5770	309	8	and	and	CCONJ
ejpam-5770	309	9	m	m	VERB
ejpam-5770	309	10	<	<	X
ejpam-5770	309	11	2k	2k	NUM
ejpam-5770	309	12	.	.	PUNCT
ejpam-5770	310	1	if	if	SCONJ
ejpam-5770	310	2	m	m	VERB
ejpam-5770	310	3	≤	≤	ADJ
ejpam-5770	311	1	k	k	X
ejpam-5770	311	2	,	,	PUNCT
ejpam-5770	311	3	then	then	ADV
ejpam-5770	311	4	by	by	ADP
ejpam-5770	311	5	theorem	theorem	NOUN
ejpam-5770	311	6	6	6	NUM
ejpam-5770	311	7	,	,	PUNCT
ejpam-5770	311	8	we	we	PRON
ejpam-5770	311	9	have	have	VERB
ejpam-5770	311	10	γwc	γwc	ADJ
ejpam-5770	311	11	rk	rk	PROPN
ejpam-5770	311	12	(	(	PUNCT
ejpam-5770	311	13	km	km	PROPN
ejpam-5770	311	14	,	,	PUNCT
ejpam-5770	311	15	n	n	CCONJ
ejpam-5770	311	16	)	)	PUNCT
ejpam-5770	311	17	=	=	SYM
ejpam-5770	312	1	k	k	NOUN
ejpam-5770	312	2	=	=	SYM
ejpam-5770	312	3	max{m	max{m	PROPN
ejpam-5770	312	4	,	,	PUNCT
ejpam-5770	312	5	k	k	NOUN
ejpam-5770	312	6	}	}	PUNCT
ejpam-5770	312	7	.	.	PUNCT
ejpam-5770	313	1	assume	assume	VERB
ejpam-5770	313	2	that	that	SCONJ
ejpam-5770	313	3	k	k	PROPN
ejpam-5770	313	4	<	<	X
ejpam-5770	313	5	m	m	X
ejpam-5770	313	6	<	<	X
ejpam-5770	313	7	2k	2k	NUM
ejpam-5770	313	8	.	.	PUNCT
ejpam-5770	314	1	let	let	VERB
ejpam-5770	314	2	f	f	NOUN
ejpam-5770	314	3	:	:	PUNCT
ejpam-5770	314	4	v	v	X
ejpam-5770	314	5	(	(	PUNCT
ejpam-5770	314	6	km	km	PROPN
ejpam-5770	314	7	,	,	PUNCT
ejpam-5770	314	8	n	n	CCONJ
ejpam-5770	314	9	)	)	PUNCT
ejpam-5770	314	10	→	→	SYM
ejpam-5770	314	11	p({1	p({1	PROPN
ejpam-5770	314	12	,	,	PUNCT
ejpam-5770	314	13	2	2	NUM
ejpam-5770	314	14	,	,	PUNCT
ejpam-5770	314	15	.	.	PUNCT
ejpam-5770	314	16	.	.	PUNCT
ejpam-5770	315	1	.	.	PUNCT
ejpam-5770	316	1	,	,	PUNCT
ejpam-5770	316	2	k	k	X
ejpam-5770	316	3	}	}	PUNCT
ejpam-5770	316	4	)	)	PUNCT
ejpam-5770	316	5	be	be	AUX
ejpam-5770	316	6	a	a	DET
ejpam-5770	316	7	function	function	NOUN
ejpam-5770	316	8	defined	define	VERB
ejpam-5770	316	9	by	by	ADP
ejpam-5770	316	10	f(xi	f(xi	NOUN
ejpam-5770	316	11	)	)	PUNCT
ejpam-5770	316	12	=	=	PRON
ejpam-5770	316	13	{	{	PUNCT
ejpam-5770	316	14	i	i	NOUN
ejpam-5770	316	15	}	}	PUNCT
ejpam-5770	316	16	for	for	ADP
ejpam-5770	316	17	1	1	NUM
ejpam-5770	316	18	≤	≤	NUM
ejpam-5770	317	1	i	i	NOUN
ejpam-5770	317	2	≤	≤	NOUN
ejpam-5770	317	3	k	k	X
ejpam-5770	317	4	and	and	CCONJ
ejpam-5770	317	5	f(xi	f(xi	PROPN
ejpam-5770	317	6	)	)	PUNCT
ejpam-5770	317	7	=	=	PRON
ejpam-5770	317	8	{	{	PUNCT
ejpam-5770	317	9	1	1	NUM
ejpam-5770	317	10	}	}	PUNCT
ejpam-5770	317	11	for	for	ADP
ejpam-5770	317	12	each	each	DET
ejpam-5770	317	13	i	i	PRON
ejpam-5770	317	14	∈	∈	PROPN
ejpam-5770	317	15	{	{	PUNCT
ejpam-5770	317	16	k	k	NOUN
ejpam-5770	317	17	+	+	PROPN
ejpam-5770	317	18	1	1	NUM
ejpam-5770	317	19	,	,	PUNCT
ejpam-5770	317	20	.	.	PUNCT
ejpam-5770	317	21	.	.	PUNCT
ejpam-5770	317	22	.	.	PUNCT
ejpam-5770	318	1	,	,	PUNCT
ejpam-5770	318	2	m	m	VERB
ejpam-5770	318	3	}	}	PUNCT
ejpam-5770	318	4	and	and	CCONJ
ejpam-5770	318	5	f(y	f(y	NOUN
ejpam-5770	318	6	)	)	PUNCT
ejpam-5770	318	7	=	=	NOUN
ejpam-5770	318	8	∅	∅	NOUN
ejpam-5770	318	9	for	for	ADP
ejpam-5770	318	10	each	each	DET
ejpam-5770	318	11	y	y	PROPN
ejpam-5770	318	12	∈	∈	PROPN
ejpam-5770	318	13	y	y	PROPN
ejpam-5770	318	14	.	.	PUNCT
ejpam-5770	319	1	as	as	ADP
ejpam-5770	319	2	case	case	NOUN
ejpam-5770	319	3	1	1	NUM
ejpam-5770	319	4	,	,	PUNCT
ejpam-5770	319	5	we	we	PRON
ejpam-5770	319	6	observe	observe	VERB
ejpam-5770	319	7	that	that	SCONJ
ejpam-5770	319	8	f	f	PROPN
ejpam-5770	319	9	is	be	AUX
ejpam-5770	319	10	a	a	DET
ejpam-5770	319	11	wckrdf	wckrdf	NOUN
ejpam-5770	319	12	of	of	ADP
ejpam-5770	319	13	km	km	PROPN
ejpam-5770	319	14	,	,	PUNCT
ejpam-5770	319	15	n	n	CCONJ
ejpam-5770	319	16	,	,	PUNCT
ejpam-5770	319	17	and	and	CCONJ
ejpam-5770	319	18	thus	thus	ADV
ejpam-5770	319	19	γwc	γwc	ADJ
ejpam-5770	319	20	rk	rk	PROPN
ejpam-5770	319	21	(	(	PUNCT
ejpam-5770	319	22	km	km	PROPN
ejpam-5770	319	23	,	,	PUNCT
ejpam-5770	319	24	n	n	CCONJ
ejpam-5770	319	25	)	)	PUNCT
ejpam-5770	319	26	≤	≤	NOUN
ejpam-5770	319	27	m.	m.	NOUN
ejpam-5770	319	28	using	use	VERB
ejpam-5770	319	29	a	a	DET
ejpam-5770	319	30	similar	similar	ADJ
ejpam-5770	319	31	argument	argument	NOUN
ejpam-5770	319	32	as	as	ADP
ejpam-5770	319	33	in	in	ADP
ejpam-5770	319	34	case	case	NOUN
ejpam-5770	319	35	1	1	NUM
ejpam-5770	319	36	,	,	PUNCT
ejpam-5770	319	37	we	we	PRON
ejpam-5770	319	38	can	can	AUX
ejpam-5770	319	39	see	see	VERB
ejpam-5770	319	40	that	that	DET
ejpam-5770	319	41	γwc	γwc	ADJ
ejpam-5770	319	42	rk	rk	PROPN
ejpam-5770	319	43	(	(	PUNCT
ejpam-5770	319	44	km	km	PROPN
ejpam-5770	319	45	,	,	PUNCT
ejpam-5770	319	46	n	n	CCONJ
ejpam-5770	319	47	)	)	PUNCT
ejpam-5770	319	48	=	=	SYM
ejpam-5770	319	49	m.	m.	NOUN
ejpam-5770	319	50	thus	thus	ADV
ejpam-5770	319	51	,	,	PUNCT
ejpam-5770	319	52	γwc	γwc	ADJ
ejpam-5770	319	53	rk	rk	PROPN
ejpam-5770	319	54	(	(	PUNCT
ejpam-5770	319	55	km	km	PROPN
ejpam-5770	319	56	,	,	PUNCT
ejpam-5770	319	57	n	n	CCONJ
ejpam-5770	319	58	)	)	PUNCT
ejpam-5770	319	59	=	=	SYM
ejpam-5770	319	60	max{m	max{m	NOUN
ejpam-5770	319	61	,	,	PUNCT
ejpam-5770	319	62	k	k	NOUN
ejpam-5770	319	63	}	}	PUNCT
ejpam-5770	319	64	.	.	PUNCT
ejpam-5770	320	1	this	this	PRON
ejpam-5770	320	2	completes	complete	VERB
ejpam-5770	320	3	the	the	DET
ejpam-5770	320	4	proof	proof	NOUN
ejpam-5770	320	5	.	.	PUNCT
ejpam-5770	321	1	5	5	X
ejpam-5770	321	2	.	.	X
ejpam-5770	321	3	join	join	VERB
ejpam-5770	321	4	of	of	ADP
ejpam-5770	321	5	graphs	graph	NOUN
ejpam-5770	321	6	harary	harary	NOUN
ejpam-5770	321	7	[	[	X
ejpam-5770	321	8	20	20	NUM
ejpam-5770	321	9	]	]	PUNCT
ejpam-5770	321	10	defined	define	VERB
ejpam-5770	321	11	the	the	DET
ejpam-5770	321	12	join	join	NOUN
ejpam-5770	321	13	of	of	ADP
ejpam-5770	321	14	two	two	NUM
ejpam-5770	321	15	graphs	graph	NOUN
ejpam-5770	321	16	g	g	NOUN
ejpam-5770	321	17	and	and	CCONJ
ejpam-5770	321	18	h	h	NOUN
ejpam-5770	321	19	,	,	PUNCT
ejpam-5770	321	20	denoted	denote	VERB
ejpam-5770	321	21	by	by	ADP
ejpam-5770	321	22	g+h	g+h	PROPN
ejpam-5770	321	23	,	,	PUNCT
ejpam-5770	321	24	as	as	ADP
ejpam-5770	321	25	the	the	DET
ejpam-5770	321	26	graph	graph	NOUN
ejpam-5770	321	27	with	with	ADP
ejpam-5770	321	28	v	v	NOUN
ejpam-5770	321	29	(	(	PUNCT
ejpam-5770	321	30	g+h	g+h	NOUN
ejpam-5770	321	31	)	)	PUNCT
ejpam-5770	321	32	=	=	SYM
ejpam-5770	321	33	v	v	X
ejpam-5770	321	34	(	(	PUNCT
ejpam-5770	321	35	g)∪v	g)∪v	NOUN
ejpam-5770	321	36	(	(	PUNCT
ejpam-5770	321	37	h	h	NOUN
ejpam-5770	321	38	)	)	PUNCT
ejpam-5770	321	39	and	and	CCONJ
ejpam-5770	321	40	e(g+h	e(g+h	NUM
ejpam-5770	321	41	)	)	PUNCT
ejpam-5770	322	1	=	=	PUNCT
ejpam-5770	322	2	e(g)∪e(h)∪{uv	e(g)∪e(h)∪{uv	X
ejpam-5770	322	3	:	:	PUNCT
ejpam-5770	322	4	u	u	PROPN
ejpam-5770	322	5	∈	∈	PROPN
ejpam-5770	322	6	v	v	NOUN
ejpam-5770	322	7	(	(	PUNCT
ejpam-5770	322	8	g	g	NOUN
ejpam-5770	322	9	)	)	PUNCT
ejpam-5770	322	10	,	,	PUNCT
ejpam-5770	322	11	v	v	X
ejpam-5770	322	12	∈	∈	PROPN
ejpam-5770	322	13	v	v	NOUN
ejpam-5770	322	14	(	(	PUNCT
ejpam-5770	322	15	h	h	NOUN
ejpam-5770	322	16	)	)	PUNCT
ejpam-5770	322	17	}	}	PUNCT
ejpam-5770	322	18	.	.	PUNCT
ejpam-5770	323	1	theorem	theorem	VERB
ejpam-5770	323	2	7	7	NUM
ejpam-5770	323	3	.	.	PUNCT
ejpam-5770	324	1	let	let	VERB
ejpam-5770	324	2	m	m	PRON
ejpam-5770	324	3	,	,	PUNCT
ejpam-5770	324	4	n	n	CCONJ
ejpam-5770	324	5	,	,	PUNCT
ejpam-5770	324	6	and	and	CCONJ
ejpam-5770	324	7	k	k	PROPN
ejpam-5770	324	8	be	be	AUX
ejpam-5770	324	9	positive	positive	ADJ
ejpam-5770	324	10	integers	integer	NOUN
ejpam-5770	324	11	such	such	ADJ
ejpam-5770	324	12	that	that	DET
ejpam-5770	324	13	min{m	min{m	NOUN
ejpam-5770	324	14	,	,	PUNCT
ejpam-5770	324	15	n	n	CCONJ
ejpam-5770	324	16	}	}	PUNCT
ejpam-5770	324	17	≥	≥	X
ejpam-5770	324	18	k	k	NOUN
ejpam-5770	324	19	,	,	PUNCT
ejpam-5770	324	20	and	and	CCONJ
ejpam-5770	324	21	let	let	VERB
ejpam-5770	324	22	g	g	NOUN
ejpam-5770	324	23	and	and	CCONJ
ejpam-5770	324	24	h	h	NOUN
ejpam-5770	324	25	be	be	AUX
ejpam-5770	324	26	graphs	graph	NOUN
ejpam-5770	324	27	of	of	ADP
ejpam-5770	324	28	order	order	NOUN
ejpam-5770	324	29	m	m	VERB
ejpam-5770	324	30	and	and	CCONJ
ejpam-5770	324	31	n	n	CCONJ
ejpam-5770	324	32	,	,	PUNCT
ejpam-5770	324	33	respectively	respectively	ADV
ejpam-5770	324	34	.	.	PUNCT
ejpam-5770	325	1	then	then	ADV
ejpam-5770	325	2	γwc	γwc	PROPN
ejpam-5770	325	3	rk	rk	PROPN
ejpam-5770	325	4	(	(	PUNCT
ejpam-5770	325	5	g	g	PROPN
ejpam-5770	325	6	+	+	NOUN
ejpam-5770	325	7	h	h	NOUN
ejpam-5770	325	8	)	)	PUNCT
ejpam-5770	325	9	=	=	SYM
ejpam-5770	326	1	k	k	NOUN
ejpam-5770	326	2	if	if	SCONJ
ejpam-5770	326	3	and	and	CCONJ
ejpam-5770	326	4	only	only	ADV
ejpam-5770	326	5	if	if	SCONJ
ejpam-5770	326	6	γwc	γwc	ADJ
ejpam-5770	326	7	rk	rk	PROPN
ejpam-5770	326	8	(	(	PUNCT
ejpam-5770	326	9	g	g	NOUN
ejpam-5770	326	10	)	)	PUNCT
ejpam-5770	326	11	=	=	SYM
ejpam-5770	326	12	k	k	PROPN
ejpam-5770	326	13	or	or	CCONJ
ejpam-5770	326	14	γwc	γwc	ADJ
ejpam-5770	326	15	rk	rk	PROPN
ejpam-5770	326	16	(	(	PUNCT
ejpam-5770	326	17	h	h	NOUN
ejpam-5770	326	18	)	)	PUNCT
ejpam-5770	326	19	=	=	SYM
ejpam-5770	326	20	k.	k.	PROPN
ejpam-5770	327	1	j.	j.	PROPN
ejpam-5770	327	2	j.	j.	PROPN
ejpam-5770	327	3	hamja	hamja	PROPN
ejpam-5770	327	4	et	et	PROPN
ejpam-5770	327	5	al	al	PROPN
ejpam-5770	327	6	.	.	PUNCT
ejpam-5770	327	7	/	/	SYM
ejpam-5770	327	8	eur	eur	PROPN
ejpam-5770	327	9	.	.	PUNCT
ejpam-5770	328	1	j.	j.	PROPN
ejpam-5770	328	2	pure	pure	PROPN
ejpam-5770	328	3	appl	appl	PROPN
ejpam-5770	328	4	.	.	PROPN
ejpam-5770	328	5	math	math	PROPN
ejpam-5770	328	6	,	,	PUNCT
ejpam-5770	328	7	18	18	NUM
ejpam-5770	328	8	(	(	PUNCT
ejpam-5770	328	9	2	2	NUM
ejpam-5770	328	10	)	)	PUNCT
ejpam-5770	328	11	(	(	PUNCT
ejpam-5770	328	12	2025	2025	NUM
ejpam-5770	328	13	)	)	PUNCT
ejpam-5770	328	14	,	,	PUNCT
ejpam-5770	328	15	5770	5770	NUM
ejpam-5770	328	16	9	9	NUM
ejpam-5770	328	17	of	of	ADP
ejpam-5770	328	18	11	11	NUM
ejpam-5770	328	19	proof	proof	NOUN
ejpam-5770	328	20	.	.	PUNCT
ejpam-5770	329	1	if	if	SCONJ
ejpam-5770	329	2	γwc	γwc	ADJ
ejpam-5770	329	3	rk	rk	PROPN
ejpam-5770	329	4	(	(	PUNCT
ejpam-5770	329	5	g	g	NOUN
ejpam-5770	329	6	)	)	PUNCT
ejpam-5770	329	7	=	=	SYM
ejpam-5770	329	8	k	k	X
ejpam-5770	329	9	(	(	PUNCT
ejpam-5770	329	10	the	the	DET
ejpam-5770	329	11	case	case	NOUN
ejpam-5770	329	12	γwc	γwc	ADJ
ejpam-5770	329	13	rk	rk	PROPN
ejpam-5770	329	14	(	(	PUNCT
ejpam-5770	329	15	h	h	NOUN
ejpam-5770	329	16	)	)	PUNCT
ejpam-5770	329	17	=	=	SYM
ejpam-5770	330	1	k	k	PROPN
ejpam-5770	330	2	is	be	AUX
ejpam-5770	330	3	similar	similar	ADJ
ejpam-5770	330	4	)	)	PUNCT
ejpam-5770	330	5	,	,	PUNCT
ejpam-5770	330	6	then	then	ADV
ejpam-5770	330	7	by	by	ADP
ejpam-5770	330	8	theorem	theorem	NOUN
ejpam-5770	330	9	6	6	NUM
ejpam-5770	330	10	,	,	PUNCT
ejpam-5770	330	11	there	there	PRON
ejpam-5770	330	12	exists	exist	VERB
ejpam-5770	330	13	a	a	DET
ejpam-5770	330	14	set	set	NOUN
ejpam-5770	330	15	x	x	SYM
ejpam-5770	330	16	⊆	⊆	NUM
ejpam-5770	330	17	v	v	ADP
ejpam-5770	330	18	(	(	PUNCT
ejpam-5770	330	19	g	g	NOUN
ejpam-5770	330	20	)	)	PUNCT
ejpam-5770	330	21	such	such	ADJ
ejpam-5770	330	22	that	that	SCONJ
ejpam-5770	330	23	|x|	|x|	PROPN
ejpam-5770	330	24	≤	≤	PROPN
ejpam-5770	330	25	k	k	PROPN
ejpam-5770	330	26	and	and	CCONJ
ejpam-5770	330	27	each	each	DET
ejpam-5770	330	28	vertex	vertex	NOUN
ejpam-5770	330	29	in	in	ADP
ejpam-5770	330	30	v	v	NOUN
ejpam-5770	330	31	(	(	PUNCT
ejpam-5770	330	32	g	g	NOUN
ejpam-5770	330	33	)	)	PUNCT
ejpam-5770	330	34	\x	\x	NOUN
ejpam-5770	330	35	is	be	AUX
ejpam-5770	330	36	adjacent	adjacent	ADJ
ejpam-5770	330	37	to	to	ADP
ejpam-5770	330	38	all	all	DET
ejpam-5770	330	39	vertices	vertex	NOUN
ejpam-5770	330	40	in	in	ADP
ejpam-5770	330	41	x.	x.	NOUN
ejpam-5770	330	42	by	by	ADP
ejpam-5770	330	43	definition	definition	NOUN
ejpam-5770	330	44	of	of	ADP
ejpam-5770	330	45	the	the	DET
ejpam-5770	330	46	join	join	NOUN
ejpam-5770	330	47	of	of	ADP
ejpam-5770	330	48	graphs	graph	NOUN
ejpam-5770	330	49	,	,	PUNCT
ejpam-5770	330	50	each	each	DET
ejpam-5770	330	51	vertex	vertex	NOUN
ejpam-5770	330	52	in	in	ADP
ejpam-5770	330	53	v	v	NUM
ejpam-5770	330	54	(	(	PUNCT
ejpam-5770	330	55	h	h	NOUN
ejpam-5770	330	56	)	)	PUNCT
ejpam-5770	330	57	is	be	AUX
ejpam-5770	330	58	adjacent	adjacent	ADJ
ejpam-5770	330	59	to	to	ADP
ejpam-5770	330	60	all	all	DET
ejpam-5770	330	61	vertices	vertex	NOUN
ejpam-5770	330	62	in	in	ADP
ejpam-5770	330	63	x	x	NOUN
ejpam-5770	330	64	,	,	PUNCT
ejpam-5770	330	65	and	and	CCONJ
ejpam-5770	330	66	again	again	ADV
ejpam-5770	330	67	theorem	theorem	VERB
ejpam-5770	330	68	6	6	NUM
ejpam-5770	330	69	leads	lead	NOUN
ejpam-5770	330	70	to	to	ADP
ejpam-5770	330	71	γwc	γwc	PROPN
ejpam-5770	330	72	rk	rk	PROPN
ejpam-5770	330	73	(	(	PUNCT
ejpam-5770	330	74	g+h	g+h	PROPN
ejpam-5770	330	75	)	)	PUNCT
ejpam-5770	331	1	=	=	VERB
ejpam-5770	331	2	k.	k.	PROPN
ejpam-5770	332	1	conversely	conversely	ADV
ejpam-5770	332	2	,	,	PUNCT
ejpam-5770	332	3	assume	assume	VERB
ejpam-5770	332	4	that	that	SCONJ
ejpam-5770	332	5	γwc	γwc	ADJ
ejpam-5770	332	6	rk	rk	PROPN
ejpam-5770	332	7	(	(	PUNCT
ejpam-5770	332	8	g+h	g+h	PROPN
ejpam-5770	332	9	)	)	PUNCT
ejpam-5770	332	10	=	=	PUNCT
ejpam-5770	333	1	k.	k.	PROPN
ejpam-5770	333	2	using	use	VERB
ejpam-5770	333	3	theorem	theorem	NOUN
ejpam-5770	333	4	6	6	NUM
ejpam-5770	333	5	,	,	PUNCT
ejpam-5770	333	6	it	it	PRON
ejpam-5770	333	7	follows	follow	VERB
ejpam-5770	333	8	that	that	SCONJ
ejpam-5770	333	9	there	there	PRON
ejpam-5770	333	10	is	be	VERB
ejpam-5770	333	11	a	a	DET
ejpam-5770	333	12	set	set	NOUN
ejpam-5770	333	13	x	x	PUNCT
ejpam-5770	333	14	of	of	ADP
ejpam-5770	333	15	vertices	vertex	NOUN
ejpam-5770	333	16	v	v	NOUN
ejpam-5770	333	17	(	(	PUNCT
ejpam-5770	333	18	g+h	g+h	NOUN
ejpam-5770	333	19	)	)	PUNCT
ejpam-5770	333	20	such	such	ADJ
ejpam-5770	333	21	that	that	SCONJ
ejpam-5770	333	22	|x|	|x|	PROPN
ejpam-5770	333	23	≤	≤	PROPN
ejpam-5770	333	24	k	k	PROPN
ejpam-5770	333	25	and	and	CCONJ
ejpam-5770	333	26	each	each	DET
ejpam-5770	333	27	vertex	vertex	NOUN
ejpam-5770	333	28	in	in	ADP
ejpam-5770	333	29	v	v	NOUN
ejpam-5770	333	30	(	(	PUNCT
ejpam-5770	333	31	g+h)\x	g+h)\x	PROPN
ejpam-5770	333	32	is	be	AUX
ejpam-5770	333	33	adjacent	adjacent	ADJ
ejpam-5770	333	34	to	to	ADP
ejpam-5770	333	35	all	all	DET
ejpam-5770	333	36	vertices	vertex	NOUN
ejpam-5770	333	37	in	in	ADP
ejpam-5770	333	38	x.	x.	NOUN
ejpam-5770	333	39	without	without	ADP
ejpam-5770	333	40	loss	loss	NOUN
ejpam-5770	333	41	of	of	ADP
ejpam-5770	333	42	generality	generality	NOUN
ejpam-5770	333	43	,	,	PUNCT
ejpam-5770	333	44	we	we	PRON
ejpam-5770	333	45	may	may	AUX
ejpam-5770	333	46	assume	assume	VERB
ejpam-5770	333	47	that	that	SCONJ
ejpam-5770	333	48	x	x	PRON
ejpam-5770	333	49	∩v	∩v	NOUN
ejpam-5770	333	50	(	(	PUNCT
ejpam-5770	333	51	g	g	NOUN
ejpam-5770	333	52	)	)	PUNCT
ejpam-5770	333	53	̸=	̸=	PROPN
ejpam-5770	333	54	∅.	∅.	ADV
ejpam-5770	333	55	since	since	SCONJ
ejpam-5770	333	56	|x	|x	NOUN
ejpam-5770	333	57	∩	∩	PROPN
ejpam-5770	333	58	v	v	X
ejpam-5770	333	59	(	(	PUNCT
ejpam-5770	333	60	g)|	g)|	NOUN
ejpam-5770	333	61	≤	≤	ADJ
ejpam-5770	333	62	k	k	PROPN
ejpam-5770	333	63	and	and	CCONJ
ejpam-5770	333	64	each	each	DET
ejpam-5770	333	65	vertex	vertex	NOUN
ejpam-5770	333	66	in	in	ADP
ejpam-5770	333	67	v	v	NOUN
ejpam-5770	333	68	(	(	PUNCT
ejpam-5770	333	69	g	g	NOUN
ejpam-5770	333	70	)	)	PUNCT
ejpam-5770	333	71	\x	\x	NOUN
ejpam-5770	333	72	is	be	AUX
ejpam-5770	333	73	adjacent	adjacent	ADJ
ejpam-5770	333	74	to	to	ADP
ejpam-5770	333	75	all	all	DET
ejpam-5770	333	76	vertices	vertex	NOUN
ejpam-5770	333	77	in	in	ADP
ejpam-5770	333	78	x	x	NOUN
ejpam-5770	333	79	,	,	PUNCT
ejpam-5770	333	80	we	we	PRON
ejpam-5770	333	81	deduce	deduce	VERB
ejpam-5770	333	82	from	from	ADP
ejpam-5770	333	83	theorem	theorem	NOUN
ejpam-5770	333	84	6	6	NUM
ejpam-5770	333	85	that	that	SCONJ
ejpam-5770	333	86	γwc	γwc	ADJ
ejpam-5770	333	87	rk	rk	PROPN
ejpam-5770	333	88	(	(	PUNCT
ejpam-5770	333	89	g	g	NOUN
ejpam-5770	333	90	)	)	PUNCT
ejpam-5770	333	91	=	=	VERB
ejpam-5770	334	1	k.	k.	PROPN
ejpam-5770	335	1	this	this	PRON
ejpam-5770	335	2	completes	complete	VERB
ejpam-5770	335	3	the	the	DET
ejpam-5770	335	4	proof	proof	NOUN
ejpam-5770	335	5	.	.	PUNCT
ejpam-5770	336	1	theorem	theorem	ADJ
ejpam-5770	336	2	8	8	NUM
ejpam-5770	336	3	.	.	PUNCT
ejpam-5770	337	1	let	let	VERB
ejpam-5770	337	2	m	m	PRON
ejpam-5770	337	3	,	,	PUNCT
ejpam-5770	337	4	n	n	CCONJ
ejpam-5770	337	5	,	,	PUNCT
ejpam-5770	337	6	and	and	CCONJ
ejpam-5770	337	7	k	k	PROPN
ejpam-5770	337	8	be	be	AUX
ejpam-5770	337	9	positive	positive	ADJ
ejpam-5770	337	10	integers	integer	NOUN
ejpam-5770	337	11	such	such	ADJ
ejpam-5770	337	12	that	that	DET
ejpam-5770	337	13	min{m	min{m	NOUN
ejpam-5770	337	14	,	,	PUNCT
ejpam-5770	337	15	n	n	CCONJ
ejpam-5770	337	16	}	}	PUNCT
ejpam-5770	337	17	≥	≥	X
ejpam-5770	337	18	k	k	NOUN
ejpam-5770	337	19	,	,	PUNCT
ejpam-5770	337	20	and	and	CCONJ
ejpam-5770	337	21	let	let	VERB
ejpam-5770	337	22	g	g	NOUN
ejpam-5770	337	23	and	and	CCONJ
ejpam-5770	337	24	h	h	NOUN
ejpam-5770	337	25	be	be	AUX
ejpam-5770	337	26	connected	connect	VERB
ejpam-5770	337	27	graphs	graph	NOUN
ejpam-5770	337	28	of	of	ADP
ejpam-5770	337	29	order	order	NOUN
ejpam-5770	337	30	m	m	VERB
ejpam-5770	337	31	and	and	CCONJ
ejpam-5770	337	32	n	n	CCONJ
ejpam-5770	337	33	,	,	PUNCT
ejpam-5770	337	34	respectively	respectively	ADV
ejpam-5770	337	35	.	.	PUNCT
ejpam-5770	338	1	then	then	ADV
ejpam-5770	338	2	γwc	γwc	PROPN
ejpam-5770	338	3	rk	rk	PROPN
ejpam-5770	338	4	(	(	PUNCT
ejpam-5770	338	5	g+h	g+h	PROPN
ejpam-5770	338	6	)	)	PUNCT
ejpam-5770	338	7	≤	≤	NOUN
ejpam-5770	339	1	min{2k	min{2k	NOUN
ejpam-5770	339	2	,	,	PUNCT
ejpam-5770	339	3	min{γwc	min{γwc	ADJ
ejpam-5770	339	4	rk	rk	NOUN
ejpam-5770	339	5	(	(	PUNCT
ejpam-5770	339	6	g	g	NOUN
ejpam-5770	339	7	)	)	PUNCT
ejpam-5770	339	8	,	,	PUNCT
ejpam-5770	339	9	γwc	γwc	PROPN
ejpam-5770	339	10	rk	rk	PROPN
ejpam-5770	339	11	(	(	PUNCT
ejpam-5770	339	12	h	h	NOUN
ejpam-5770	339	13	)	)	PUNCT
ejpam-5770	339	14	}	}	PUNCT
ejpam-5770	339	15	}	}	PUNCT
ejpam-5770	339	16	.	.	PUNCT
ejpam-5770	340	1	proof	proof	NOUN
ejpam-5770	340	2	.	.	PUNCT
ejpam-5770	341	1	let	let	VERB
ejpam-5770	341	2	first	first	ADJ
ejpam-5770	341	3	x	x	NOUN
ejpam-5770	341	4	,	,	PUNCT
ejpam-5770	341	5	y	y	PROPN
ejpam-5770	341	6	be	be	VERB
ejpam-5770	341	7	two	two	NUM
ejpam-5770	341	8	vertices	vertex	NOUN
ejpam-5770	341	9	of	of	ADP
ejpam-5770	341	10	g	g	PROPN
ejpam-5770	342	1	+	+	CCONJ
ejpam-5770	342	2	h	h	NOUN
ejpam-5770	342	3	such	such	ADJ
ejpam-5770	342	4	that	that	SCONJ
ejpam-5770	342	5	x	x	SYM
ejpam-5770	342	6	∈	∈	NOUN
ejpam-5770	342	7	v	v	X
ejpam-5770	342	8	(	(	PUNCT
ejpam-5770	342	9	g	g	NOUN
ejpam-5770	342	10	)	)	PUNCT
ejpam-5770	342	11	and	and	CCONJ
ejpam-5770	342	12	y	y	PROPN
ejpam-5770	342	13	∈	∈	PROPN
ejpam-5770	342	14	v	v	ADP
ejpam-5770	342	15	(	(	PUNCT
ejpam-5770	342	16	h	h	NOUN
ejpam-5770	342	17	)	)	PUNCT
ejpam-5770	342	18	,	,	PUNCT
ejpam-5770	342	19	and	and	CCONJ
ejpam-5770	342	20	define	define	VERB
ejpam-5770	342	21	the	the	DET
ejpam-5770	342	22	function	function	NOUN
ejpam-5770	343	1	f	f	NOUN
ejpam-5770	343	2	:	:	PUNCT
ejpam-5770	343	3	v	v	X
ejpam-5770	343	4	(	(	PUNCT
ejpam-5770	343	5	g	g	PROPN
ejpam-5770	343	6	+	+	NOUN
ejpam-5770	343	7	h	h	NOUN
ejpam-5770	343	8	)	)	PUNCT
ejpam-5770	343	9	→	→	SYM
ejpam-5770	344	1	p({1	p({1	PROPN
ejpam-5770	344	2	,	,	PUNCT
ejpam-5770	344	3	2	2	NUM
ejpam-5770	344	4	,	,	PUNCT
ejpam-5770	344	5	.	.	PUNCT
ejpam-5770	344	6	.	.	PUNCT
ejpam-5770	345	1	.	.	PUNCT
ejpam-5770	346	1	,	,	PUNCT
ejpam-5770	346	2	k	k	X
ejpam-5770	346	3	}	}	PUNCT
ejpam-5770	346	4	)	)	PUNCT
ejpam-5770	346	5	by	by	ADP
ejpam-5770	346	6	f(x	f(x	PROPN
ejpam-5770	346	7	)	)	PUNCT
ejpam-5770	346	8	=	=	SYM
ejpam-5770	346	9	f(y	f(y	NOUN
ejpam-5770	346	10	)	)	PUNCT
ejpam-5770	346	11	=	=	PRON
ejpam-5770	346	12	{	{	PUNCT
ejpam-5770	346	13	1	1	NUM
ejpam-5770	346	14	,	,	PUNCT
ejpam-5770	346	15	2	2	NUM
ejpam-5770	346	16	,	,	PUNCT
ejpam-5770	346	17	.	.	PUNCT
ejpam-5770	346	18	.	.	PUNCT
ejpam-5770	347	1	.	.	PUNCT
ejpam-5770	348	1	,	,	PUNCT
ejpam-5770	348	2	k	k	X
ejpam-5770	348	3	}	}	PUNCT
ejpam-5770	348	4	and	and	CCONJ
ejpam-5770	348	5	f(z	f(z	PROPN
ejpam-5770	348	6	)	)	PUNCT
ejpam-5770	348	7	=	=	NOUN
ejpam-5770	348	8	∅	∅	NOUN
ejpam-5770	348	9	or	or	CCONJ
ejpam-5770	348	10	all	all	DET
ejpam-5770	348	11	vertices	vertex	NOUN
ejpam-5770	348	12	z	z	PROPN
ejpam-5770	348	13	∈	∈	PROPN
ejpam-5770	348	14	(	(	PUNCT
ejpam-5770	348	15	v	v	NOUN
ejpam-5770	348	16	(	(	PUNCT
ejpam-5770	348	17	g	g	NOUN
ejpam-5770	348	18	)	)	PUNCT
ejpam-5770	348	19	∪	∪	NOUN
ejpam-5770	348	20	v	v	NOUN
ejpam-5770	348	21	(	(	PUNCT
ejpam-5770	348	22	h	h	NOUN
ejpam-5770	348	23	)	)	PUNCT
ejpam-5770	348	24	)	)	PUNCT
ejpam-5770	348	25	\	\	NOUN
ejpam-5770	349	1	{	{	PUNCT
ejpam-5770	349	2	x	x	NOUN
ejpam-5770	349	3	,	,	PUNCT
ejpam-5770	349	4	y	y	PROPN
ejpam-5770	349	5	}	}	PUNCT
ejpam-5770	349	6	.	.	PUNCT
ejpam-5770	350	1	clearly	clearly	ADV
ejpam-5770	350	2	,	,	PUNCT
ejpam-5770	350	3	f	f	PROPN
ejpam-5770	350	4	is	be	AUX
ejpam-5770	350	5	a	a	DET
ejpam-5770	350	6	wckrdf	wckrdf	NOUN
ejpam-5770	350	7	of	of	ADP
ejpam-5770	350	8	g+h	g+h	PROPN
ejpam-5770	350	9	and	and	CCONJ
ejpam-5770	350	10	thus	thus	ADV
ejpam-5770	350	11	γwc	γwc	ADJ
ejpam-5770	350	12	rk	rk	PROPN
ejpam-5770	350	13	(	(	PUNCT
ejpam-5770	350	14	g+h	g+h	PROPN
ejpam-5770	350	15	)	)	PUNCT
ejpam-5770	350	16	≤	≤	NOUN
ejpam-5770	350	17	2k	2k	NUM
ejpam-5770	350	18	.	.	PUNCT
ejpam-5770	351	1	now	now	ADV
ejpam-5770	351	2	assume	assume	VERB
ejpam-5770	351	3	,	,	PUNCT
ejpam-5770	351	4	without	without	ADP
ejpam-5770	351	5	loss	loss	NOUN
ejpam-5770	351	6	of	of	ADP
ejpam-5770	351	7	generality	generality	NOUN
ejpam-5770	351	8	,	,	PUNCT
ejpam-5770	351	9	that	that	SCONJ
ejpam-5770	351	10	γwc	γwc	ADJ
ejpam-5770	351	11	rk	rk	PROPN
ejpam-5770	351	12	(	(	PUNCT
ejpam-5770	351	13	g	g	NOUN
ejpam-5770	351	14	)	)	PUNCT
ejpam-5770	351	15	=	=	SYM
ejpam-5770	351	16	min{γwc	min{γwc	ADJ
ejpam-5770	351	17	rk	rk	NOUN
ejpam-5770	351	18	(	(	PUNCT
ejpam-5770	351	19	g	g	NOUN
ejpam-5770	351	20	)	)	PUNCT
ejpam-5770	351	21	,	,	PUNCT
ejpam-5770	351	22	γwc	γwc	PROPN
ejpam-5770	351	23	rk	rk	PROPN
ejpam-5770	351	24	(	(	PUNCT
ejpam-5770	351	25	h	h	NOUN
ejpam-5770	351	26	)	)	PUNCT
ejpam-5770	351	27	}	}	PUNCT
ejpam-5770	351	28	and	and	CCONJ
ejpam-5770	351	29	let	let	VERB
ejpam-5770	351	30	f	f	PRON
ejpam-5770	351	31	be	be	AUX
ejpam-5770	351	32	a	a	DET
ejpam-5770	351	33	γwc	γwc	ADJ
ejpam-5770	351	34	rk	rk	NOUN
ejpam-5770	351	35	-function	-function	NOUN
ejpam-5770	351	36	of	of	ADP
ejpam-5770	351	37	g.	g.	PROPN
ejpam-5770	351	38	since	since	SCONJ
ejpam-5770	351	39	m	m	PROPN
ejpam-5770	351	40	≥	≥	PROPN
ejpam-5770	351	41	k	k	NOUN
ejpam-5770	351	42	,	,	PUNCT
ejpam-5770	351	43	we	we	PRON
ejpam-5770	351	44	may	may	AUX
ejpam-5770	351	45	assume	assume	VERB
ejpam-5770	351	46	that	that	SCONJ
ejpam-5770	351	47	∪x∈v	∪x∈v	PROPN
ejpam-5770	351	48	(	(	PUNCT
ejpam-5770	351	49	g)f(x	g)f(x	PROPN
ejpam-5770	351	50	)	)	PUNCT
ejpam-5770	351	51	=	=	PUNCT
ejpam-5770	351	52	{	{	PUNCT
ejpam-5770	351	53	1	1	NUM
ejpam-5770	351	54	,	,	PUNCT
ejpam-5770	351	55	2	2	NUM
ejpam-5770	351	56	,	,	PUNCT
ejpam-5770	351	57	.	.	PUNCT
ejpam-5770	351	58	.	.	PUNCT
ejpam-5770	351	59	.	.	PUNCT
ejpam-5770	352	1	,	,	PUNCT
ejpam-5770	352	2	k	k	X
ejpam-5770	352	3	}	}	PUNCT
ejpam-5770	352	4	.	.	PUNCT
ejpam-5770	353	1	since	since	SCONJ
ejpam-5770	353	2	each	each	DET
ejpam-5770	353	3	vertex	vertex	NOUN
ejpam-5770	353	4	in	in	ADP
ejpam-5770	353	5	v	v	NUM
ejpam-5770	353	6	(	(	PUNCT
ejpam-5770	353	7	h	h	NOUN
ejpam-5770	353	8	)	)	PUNCT
ejpam-5770	353	9	is	be	AUX
ejpam-5770	353	10	adjacent	adjacent	ADJ
ejpam-5770	353	11	to	to	ADP
ejpam-5770	353	12	all	all	DET
ejpam-5770	353	13	vertices	vertex	NOUN
ejpam-5770	353	14	of	of	ADP
ejpam-5770	353	15	v	v	NOUN
ejpam-5770	353	16	(	(	PUNCT
ejpam-5770	353	17	g	g	NOUN
ejpam-5770	353	18	)	)	PUNCT
ejpam-5770	353	19	in	in	ADP
ejpam-5770	353	20	g+h	g+h	PROPN
ejpam-5770	353	21	,	,	PUNCT
ejpam-5770	353	22	f	f	PROPN
ejpam-5770	353	23	is	be	AUX
ejpam-5770	353	24	a	a	DET
ejpam-5770	353	25	wckrdf	wckrdf	NOUN
ejpam-5770	353	26	of	of	ADP
ejpam-5770	353	27	g+h	g+h	PROPN
ejpam-5770	353	28	and	and	CCONJ
ejpam-5770	353	29	thus	thus	ADV
ejpam-5770	353	30	γwc	γwc	ADJ
ejpam-5770	353	31	rk	rk	PROPN
ejpam-5770	353	32	(	(	PUNCT
ejpam-5770	353	33	g+h	g+h	PROPN
ejpam-5770	353	34	)	)	PUNCT
ejpam-5770	353	35	≤	≤	NUM
ejpam-5770	353	36	min{γwc	min{γwc	ADJ
ejpam-5770	353	37	rk	rk	NOUN
ejpam-5770	353	38	(	(	PUNCT
ejpam-5770	353	39	g	g	NOUN
ejpam-5770	353	40	)	)	PUNCT
ejpam-5770	353	41	,	,	PUNCT
ejpam-5770	353	42	γwc	γwc	PROPN
ejpam-5770	353	43	rk	rk	PROPN
ejpam-5770	353	44	(	(	PUNCT
ejpam-5770	353	45	h	h	NOUN
ejpam-5770	353	46	)	)	PUNCT
ejpam-5770	353	47	}	}	PUNCT
ejpam-5770	353	48	.	.	PUNCT
ejpam-5770	354	1	this	this	PRON
ejpam-5770	354	2	proves	prove	VERB
ejpam-5770	354	3	the	the	DET
ejpam-5770	354	4	upper	upper	ADJ
ejpam-5770	354	5	bound	bind	VERB
ejpam-5770	354	6	.	.	PUNCT
ejpam-5770	355	1	open	open	ADJ
ejpam-5770	355	2	questions	question	NOUN
ejpam-5770	355	3	and	and	CCONJ
ejpam-5770	355	4	problems	problem	NOUN
ejpam-5770	355	5	:	:	PUNCT
ejpam-5770	355	6	we	we	PRON
ejpam-5770	355	7	conclude	conclude	VERB
ejpam-5770	355	8	this	this	DET
ejpam-5770	355	9	paper	paper	NOUN
ejpam-5770	355	10	with	with	ADP
ejpam-5770	355	11	some	some	DET
ejpam-5770	355	12	open	open	ADJ
ejpam-5770	355	13	problem	problem	NOUN
ejpam-5770	355	14	and	and	CCONJ
ejpam-5770	355	15	perspective	perspective	NOUN
ejpam-5770	355	16	related	relate	VERB
ejpam-5770	355	17	to	to	ADP
ejpam-5770	355	18	our	our	PRON
ejpam-5770	355	19	work	work	NOUN
ejpam-5770	355	20	.	.	PUNCT
ejpam-5770	356	1	problem	problem	NOUN
ejpam-5770	356	2	1	1	NUM
ejpam-5770	356	3	.	.	PUNCT
ejpam-5770	357	1	for	for	ADP
ejpam-5770	357	2	positive	positive	ADJ
ejpam-5770	357	3	integer	integer	NOUN
ejpam-5770	357	4	k	k	PROPN
ejpam-5770	357	5	≥	≥	NUM
ejpam-5770	357	6	3	3	NUM
ejpam-5770	357	7	,	,	PUNCT
ejpam-5770	357	8	characterize	characterize	VERB
ejpam-5770	357	9	the	the	DET
ejpam-5770	357	10	graphs	graph	NOUN
ejpam-5770	357	11	g	g	ADP
ejpam-5770	357	12	of	of	ADP
ejpam-5770	357	13	order	order	NOUN
ejpam-5770	357	14	n	n	PRON
ejpam-5770	357	15	such	such	ADJ
ejpam-5770	357	16	that	that	SCONJ
ejpam-5770	357	17	γwc	γwc	ADJ
ejpam-5770	357	18	rk	rk	PROPN
ejpam-5770	357	19	(	(	PUNCT
ejpam-5770	357	20	g	g	NOUN
ejpam-5770	357	21	)	)	PUNCT
ejpam-5770	357	22	=	=	VERB
ejpam-5770	357	23	n.	n.	NOUN
ejpam-5770	357	24	problem	problem	NOUN
ejpam-5770	357	25	2	2	NUM
ejpam-5770	357	26	.	.	X
ejpam-5770	358	1	for	for	ADP
ejpam-5770	358	2	positive	positive	ADJ
ejpam-5770	358	3	integer	integer	NOUN
ejpam-5770	358	4	k	k	PROPN
ejpam-5770	358	5	≥	≥	NUM
ejpam-5770	358	6	2	2	NUM
ejpam-5770	358	7	,	,	PUNCT
ejpam-5770	358	8	characterize	characterize	VERB
ejpam-5770	358	9	the	the	DET
ejpam-5770	358	10	graphs	graph	NOUN
ejpam-5770	358	11	g	g	ADP
ejpam-5770	358	12	of	of	ADP
ejpam-5770	358	13	order	order	NOUN
ejpam-5770	358	14	n	n	PRON
ejpam-5770	358	15	such	such	ADJ
ejpam-5770	358	16	that	that	SCONJ
ejpam-5770	358	17	γwc	γwc	ADJ
ejpam-5770	358	18	rk	rk	PROPN
ejpam-5770	358	19	(	(	PUNCT
ejpam-5770	358	20	g	g	NOUN
ejpam-5770	358	21	)	)	PUNCT
ejpam-5770	358	22	=	=	PUNCT
ejpam-5770	359	1	k	k	PROPN
ejpam-5770	359	2	+	+	PUNCT
ejpam-5770	359	3	ℓ	ℓ	NOUN
ejpam-5770	359	4	for	for	ADP
ejpam-5770	359	5	some	some	DET
ejpam-5770	359	6	positive	positive	ADJ
ejpam-5770	359	7	integer	integer	NOUN
ejpam-5770	359	8	ℓ.	ℓ.	NOUN
ejpam-5770	359	9	problem	problem	NOUN
ejpam-5770	359	10	3	3	NUM
ejpam-5770	359	11	.	.	PUNCT
ejpam-5770	359	12	determine	determine	VERB
ejpam-5770	359	13	nordhaus	nordhaus	NOUN
ejpam-5770	359	14	-	-	PUNCT
ejpam-5770	359	15	gaddum	gaddum	NOUN
ejpam-5770	359	16	type	type	NOUN
ejpam-5770	359	17	results	result	NOUN
ejpam-5770	359	18	for	for	ADP
ejpam-5770	359	19	γwc	γwc	ADJ
ejpam-5770	359	20	rk	rk	PROPN
ejpam-5770	359	21	(	(	PUNCT
ejpam-5770	359	22	g	g	NOUN
ejpam-5770	359	23	)	)	PUNCT
ejpam-5770	359	24	.	.	PUNCT
ejpam-5770	360	1	problem	problem	NOUN
ejpam-5770	360	2	4	4	NUM
ejpam-5770	360	3	.	.	PUNCT
ejpam-5770	360	4	design	design	VERB
ejpam-5770	360	5	an	an	DET
ejpam-5770	360	6	algorithm	algorithm	NOUN
ejpam-5770	360	7	for	for	ADP
ejpam-5770	360	8	computing	compute	VERB
ejpam-5770	360	9	the	the	DET
ejpam-5770	360	10	value	value	NOUN
ejpam-5770	360	11	of	of	ADP
ejpam-5770	360	12	γwc	γwc	ADJ
ejpam-5770	360	13	rk	rk	PROPN
ejpam-5770	360	14	(	(	PUNCT
ejpam-5770	360	15	t	t	PROPN
ejpam-5770	360	16	)	)	PUNCT
ejpam-5770	360	17	for	for	ADP
ejpam-5770	360	18	any	any	DET
ejpam-5770	360	19	tree	tree	NOUN
ejpam-5770	360	20	t	t	PROPN
ejpam-5770	360	21	and	and	CCONJ
ejpam-5770	360	22	k	k	PROPN
ejpam-5770	360	23	≥	≥	NUM
ejpam-5770	360	24	2	2	NUM
ejpam-5770	360	25	.	.	NOUN
ejpam-5770	360	26	6	6	NUM
ejpam-5770	360	27	.	.	X
ejpam-5770	360	28	conclusion	conclusion	NOUN
ejpam-5770	360	29	in	in	ADP
ejpam-5770	360	30	this	this	DET
ejpam-5770	360	31	paper	paper	NOUN
ejpam-5770	360	32	,	,	PUNCT
ejpam-5770	360	33	we	we	PRON
ejpam-5770	360	34	have	have	AUX
ejpam-5770	360	35	introduced	introduce	VERB
ejpam-5770	360	36	and	and	CCONJ
ejpam-5770	360	37	investigated	investigate	VERB
ejpam-5770	360	38	the	the	DET
ejpam-5770	360	39	weakly	weakly	ADV
ejpam-5770	360	40	connected	connected	ADJ
ejpam-5770	360	41	k	k	ADJ
ejpam-5770	360	42	-	-	PUNCT
ejpam-5770	360	43	rainbow	rainbow	NOUN
ejpam-5770	360	44	domination	domination	NOUN
ejpam-5770	360	45	parameter	parameter	NOUN
ejpam-5770	360	46	in	in	ADP
ejpam-5770	360	47	graphs	graph	NOUN
ejpam-5770	360	48	.	.	PUNCT
ejpam-5770	361	1	we	we	PRON
ejpam-5770	361	2	established	establish	VERB
ejpam-5770	361	3	fundamental	fundamental	ADJ
ejpam-5770	361	4	properties	property	NOUN
ejpam-5770	361	5	and	and	CCONJ
ejpam-5770	361	6	derived	derive	VERB
ejpam-5770	361	7	bounds	bound	NOUN
ejpam-5770	361	8	for	for	ADP
ejpam-5770	361	9	the	the	DET
ejpam-5770	361	10	weakly	weakly	ADJ
ejpam-5770	361	11	connected	connected	ADJ
ejpam-5770	361	12	k	k	ADJ
ejpam-5770	361	13	-	-	PUNCT
ejpam-5770	361	14	rainbow	rainbow	NOUN
ejpam-5770	361	15	domination	domination	NOUN
ejpam-5770	361	16	number	number	NOUN
ejpam-5770	361	17	γwc	γwc	PROPN
ejpam-5770	361	18	rk	rk	PROPN
ejpam-5770	361	19	(	(	PUNCT
ejpam-5770	361	20	g	g	NOUN
ejpam-5770	361	21	)	)	PUNCT
ejpam-5770	361	22	.	.	PUNCT
ejpam-5770	362	1	moreover	moreover	ADV
ejpam-5770	362	2	,	,	PUNCT
ejpam-5770	362	3	we	we	PRON
ejpam-5770	362	4	determined	determine	VERB
ejpam-5770	362	5	the	the	DET
ejpam-5770	362	6	exact	exact	ADJ
ejpam-5770	362	7	values	value	NOUN
ejpam-5770	362	8	of	of	ADP
ejpam-5770	362	9	γwc	γwc	ADJ
ejpam-5770	362	10	rk	rk	PROPN
ejpam-5770	362	11	(	(	PUNCT
ejpam-5770	362	12	g	g	NOUN
ejpam-5770	362	13	)	)	PUNCT
ejpam-5770	362	14	for	for	ADP
ejpam-5770	362	15	various	various	ADJ
ejpam-5770	362	16	graph	graph	NOUN
ejpam-5770	362	17	classes	class	NOUN
ejpam-5770	362	18	,	,	PUNCT
ejpam-5770	362	19	providing	provide	VERB
ejpam-5770	362	20	insights	insight	NOUN
ejpam-5770	362	21	into	into	ADP
ejpam-5770	362	22	their	their	PRON
ejpam-5770	362	23	structural	structural	ADJ
ejpam-5770	362	24	dependencies	dependency	NOUN
ejpam-5770	362	25	.	.	PUNCT
ejpam-5770	363	1	additionally	additionally	ADV
ejpam-5770	363	2	,	,	PUNCT
ejpam-5770	363	3	we	we	PRON
ejpam-5770	363	4	examined	examine	VERB
ejpam-5770	363	5	the	the	DET
ejpam-5770	363	6	weakly	weakly	ADV
ejpam-5770	363	7	connected	connected	ADJ
ejpam-5770	363	8	k	k	ADJ
ejpam-5770	363	9	-	-	PUNCT
ejpam-5770	363	10	rainbow	rainbow	NOUN
ejpam-5770	363	11	domination	domination	NOUN
ejpam-5770	363	12	behavior	behavior	NOUN
ejpam-5770	363	13	under	under	ADP
ejpam-5770	363	14	the	the	DET
ejpam-5770	363	15	join	join	NOUN
ejpam-5770	363	16	operation	operation	NOUN
ejpam-5770	363	17	of	of	ADP
ejpam-5770	363	18	graphs	graph	NOUN
ejpam-5770	363	19	.	.	PUNCT
ejpam-5770	364	1	j.	j.	PROPN
ejpam-5770	364	2	j.	j.	PROPN
ejpam-5770	364	3	hamja	hamja	PROPN
ejpam-5770	364	4	et	et	PROPN
ejpam-5770	364	5	al	al	PROPN
ejpam-5770	364	6	.	.	PUNCT
ejpam-5770	364	7	/	/	SYM
ejpam-5770	364	8	eur	eur	PROPN
ejpam-5770	364	9	.	.	PUNCT
ejpam-5770	365	1	j.	j.	PROPN
ejpam-5770	365	2	pure	pure	PROPN
ejpam-5770	365	3	appl	appl	PROPN
ejpam-5770	365	4	.	.	PROPN
ejpam-5770	365	5	math	math	PROPN
ejpam-5770	365	6	,	,	PUNCT
ejpam-5770	365	7	18	18	NUM
ejpam-5770	365	8	(	(	PUNCT
ejpam-5770	365	9	2	2	NUM
ejpam-5770	365	10	)	)	PUNCT
ejpam-5770	365	11	(	(	PUNCT
ejpam-5770	365	12	2025	2025	NUM
ejpam-5770	365	13	)	)	PUNCT
ejpam-5770	365	14	,	,	PUNCT
ejpam-5770	365	15	5770	5770	NUM
ejpam-5770	365	16	10	10	NUM
ejpam-5770	365	17	of	of	ADP
ejpam-5770	365	18	11	11	NUM
ejpam-5770	365	19	acknowledgements	acknowledgement	NOUN
ejpam-5770	365	20	the	the	DET
ejpam-5770	365	21	authors	author	NOUN
ejpam-5770	365	22	sincerely	sincerely	ADV
ejpam-5770	365	23	thank	thank	VERB
ejpam-5770	365	24	the	the	DET
ejpam-5770	365	25	reviewers	reviewer	NOUN
ejpam-5770	365	26	for	for	ADP
ejpam-5770	365	27	their	their	PRON
ejpam-5770	365	28	valuable	valuable	ADJ
ejpam-5770	365	29	comments	comment	NOUN
ejpam-5770	365	30	and	and	CCONJ
ejpam-5770	365	31	suggestions	suggestion	NOUN
ejpam-5770	365	32	,	,	PUNCT
ejpam-5770	365	33	which	which	PRON
ejpam-5770	365	34	helped	help	VERB
ejpam-5770	365	35	improve	improve	VERB
ejpam-5770	365	36	the	the	DET
ejpam-5770	365	37	quality	quality	NOUN
ejpam-5770	365	38	of	of	ADP
ejpam-5770	365	39	this	this	DET
ejpam-5770	365	40	paper	paper	NOUN
ejpam-5770	365	41	.	.	PUNCT
ejpam-5770	366	1	they	they	PRON
ejpam-5770	366	2	also	also	ADV
ejpam-5770	366	3	acknowledge	acknowledge	VERB
ejpam-5770	366	4	the	the	DET
ejpam-5770	366	5	financial	financial	ADJ
ejpam-5770	366	6	support	support	NOUN
ejpam-5770	366	7	from	from	ADP
ejpam-5770	366	8	the	the	DET
ejpam-5770	366	9	department	department	NOUN
ejpam-5770	366	10	of	of	ADP
ejpam-5770	366	11	science	science	NOUN
ejpam-5770	366	12	and	and	CCONJ
ejpam-5770	366	13	technology	technology	NOUN
ejpam-5770	366	14	—	—	PUNCT
ejpam-5770	366	15	accelerated	accelerate	VERB
ejpam-5770	366	16	science	science	NOUN
ejpam-5770	366	17	and	and	CCONJ
ejpam-5770	366	18	technology	technology	NOUN
ejpam-5770	366	19	human	human	ADJ
ejpam-5770	366	20	resource	resource	NOUN
ejpam-5770	366	21	development	development	NOUN
ejpam-5770	366	22	program	program	NOUN
ejpam-5770	366	23	(	(	PUNCT
ejpam-5770	366	24	dost	dost	NOUN
ejpam-5770	366	25	-	-	PUNCT
ejpam-5770	366	26	asthrdp	asthrdp	NOUN
ejpam-5770	366	27	)	)	PUNCT
ejpam-5770	366	28	,	,	PUNCT
ejpam-5770	366	29	mindanao	mindanao	PROPN
ejpam-5770	366	30	state	state	PROPN
ejpam-5770	366	31	university	university	PROPN
ejpam-5770	366	32	—	—	PUNCT
ejpam-5770	366	33	tawi	tawi	NOUN
ejpam-5770	366	34	-	-	PUNCT
ejpam-5770	366	35	tawi	tawi	PROPN
ejpam-5770	366	36	college	college	PROPN
ejpam-5770	366	37	of	of	ADP
ejpam-5770	366	38	technology	technology	NOUN
ejpam-5770	366	39	and	and	CCONJ
ejpam-5770	366	40	oceanography	oceanography	NOUN
ejpam-5770	366	41	(	(	PUNCT
ejpam-5770	366	42	msu	msu	PROPN
ejpam-5770	366	43	-	-	PUNCT
ejpam-5770	366	44	tcto	tcto	NOUN
ejpam-5770	366	45	)	)	PUNCT
ejpam-5770	366	46	,	,	PUNCT
ejpam-5770	366	47	and	and	CCONJ
ejpam-5770	366	48	mindanao	mindanao	PROPN
ejpam-5770	366	49	state	state	PROPN
ejpam-5770	366	50	university	university	PROPN
ejpam-5770	366	51	—	—	PUNCT
ejpam-5770	366	52	iligan	iligan	PROPN
ejpam-5770	366	53	institute	institute	PROPN
ejpam-5770	366	54	of	of	ADP
ejpam-5770	366	55	technology	technology	PROPN
ejpam-5770	366	56	(	(	PUNCT
ejpam-5770	366	57	msu	msu	PROPN
ejpam-5770	366	58	-	-	PUNCT
ejpam-5770	366	59	iit	iit	NOUN
ejpam-5770	366	60	)	)	PUNCT
ejpam-5770	366	61	,	,	PUNCT
ejpam-5770	366	62	which	which	PRON
ejpam-5770	366	63	made	make	VERB
ejpam-5770	366	64	this	this	DET
ejpam-5770	366	65	publication	publication	NOUN
ejpam-5770	366	66	possible	possible	ADJ
ejpam-5770	366	67	.	.	PUNCT
ejpam-5770	367	1	references	reference	NOUN
ejpam-5770	367	2	[	[	X
ejpam-5770	367	3	1	1	X
ejpam-5770	367	4	]	]	PUNCT
ejpam-5770	367	5	t.	t.	PROPN
ejpam-5770	367	6	w.	w.	PROPN
ejpam-5770	367	7	haynes	haynes	PROPN
ejpam-5770	367	8	,	,	PUNCT
ejpam-5770	367	9	s.	s.	PROPN
ejpam-5770	367	10	t.	t.	PROPN
ejpam-5770	367	11	hedetniemi	hedetniemi	PROPN
ejpam-5770	367	12	,	,	PUNCT
ejpam-5770	367	13	and	and	CCONJ
ejpam-5770	367	14	m.	m.	PROPN
ejpam-5770	367	15	a.	a.	PROPN
ejpam-5770	367	16	henning	henning	PROPN
ejpam-5770	367	17	.	.	PUNCT
ejpam-5770	368	1	topics	topic	NOUN
ejpam-5770	368	2	in	in	ADP
ejpam-5770	368	3	domination	domination	NOUN
ejpam-5770	368	4	in	in	ADP
ejpam-5770	368	5	graphs	graph	NOUN
ejpam-5770	368	6	,	,	PUNCT
ejpam-5770	368	7	volume	volume	NOUN
ejpam-5770	368	8	64	64	NUM
ejpam-5770	368	9	of	of	ADP
ejpam-5770	368	10	developments	development	NOUN
ejpam-5770	368	11	in	in	ADP
ejpam-5770	368	12	mathematics	mathematic	NOUN
ejpam-5770	368	13	.	.	PUNCT
ejpam-5770	369	1	springer	springer	NOUN
ejpam-5770	369	2	,	,	PUNCT
ejpam-5770	369	3	2020	2020	NUM
ejpam-5770	369	4	.	.	PUNCT
ejpam-5770	370	1	[	[	X
ejpam-5770	370	2	2	2	X
ejpam-5770	370	3	]	]	PUNCT
ejpam-5770	370	4	s.	s.	PROPN
ejpam-5770	370	5	m.	m.	PROPN
ejpam-5770	370	6	hedetniemi	hedetniemi	PROPN
ejpam-5770	370	7	,	,	PUNCT
ejpam-5770	370	8	s.	s.	PROPN
ejpam-5770	370	9	t.	t.	PROPN
ejpam-5770	370	10	hedetniemi	hedetniemi	PROPN
ejpam-5770	370	11	,	,	PUNCT
ejpam-5770	370	12	and	and	CCONJ
ejpam-5770	370	13	t.	t.	PROPN
ejpam-5770	370	14	v.	v.	PROPN
ejpam-5770	370	15	wimer	wimer	PROPN
ejpam-5770	370	16	.	.	PUNCT
ejpam-5770	371	1	linear	linear	ADJ
ejpam-5770	371	2	time	time	NOUN
ejpam-5770	371	3	resource	resource	NOUN
ejpam-5770	371	4	allocation	allocation	NOUN
ejpam-5770	371	5	for	for	ADP
ejpam-5770	371	6	trees	tree	NOUN
ejpam-5770	371	7	.	.	PUNCT
ejpam-5770	372	1	technical	technical	ADJ
ejpam-5770	372	2	report	report	PROPN
ejpam-5770	372	3	uri-014	uri-014	PROPN
ejpam-5770	372	4	,	,	PUNCT
ejpam-5770	372	5	dept	dept	PROPN
ejpam-5770	372	6	.	.	PROPN
ejpam-5770	373	1	mathematical	mathematical	PROPN
ejpam-5770	373	2	sciences	sciences	PROPN
ejpam-5770	373	3	,	,	PUNCT
ejpam-5770	373	4	clemson	clemson	NOUN
ejpam-5770	373	5	univ	univ	PROPN
ejpam-5770	373	6	.	.	PROPN
ejpam-5770	373	7	,	,	PUNCT
ejpam-5770	373	8	1987	1987	NUM
ejpam-5770	373	9	.	.	PUNCT
ejpam-5770	373	10	presented	present	VERB
ejpam-5770	373	11	at	at	ADP
ejpam-5770	373	12	southeastern	southeastern	ADJ
ejpam-5770	373	13	conf	conf	NOUN
ejpam-5770	373	14	.	.	PUNCT
ejpam-5770	374	1	on	on	ADP
ejpam-5770	374	2	combinatorics	combinatoric	NOUN
ejpam-5770	374	3	,	,	PUNCT
ejpam-5770	374	4	graph	graph	NOUN
ejpam-5770	374	5	theory	theory	NOUN
ejpam-5770	374	6	and	and	CCONJ
ejpam-5770	374	7	computing	computing	NOUN
ejpam-5770	374	8	,	,	PUNCT
ejpam-5770	374	9	boca	boca	PROPN
ejpam-5770	374	10	raton	raton	PROPN
ejpam-5770	374	11	,	,	PUNCT
ejpam-5770	374	12	fl	fl	PROPN
ejpam-5770	374	13	,	,	PUNCT
ejpam-5770	374	14	1987	1987	NUM
ejpam-5770	374	15	.	.	PUNCT
ejpam-5770	375	1	[	[	X
ejpam-5770	375	2	3	3	X
ejpam-5770	375	3	]	]	X
ejpam-5770	375	4	b.	b.	PROPN
ejpam-5770	375	5	brešar	brešar	PROPN
ejpam-5770	375	6	,	,	PUNCT
ejpam-5770	375	7	m.	m.	NOUN
ejpam-5770	375	8	a.	a.	PROPN
ejpam-5770	375	9	henning	henning	PROPN
ejpam-5770	375	10	,	,	PUNCT
ejpam-5770	375	11	and	and	CCONJ
ejpam-5770	375	12	d.	d.	PROPN
ejpam-5770	375	13	f.	f.	PROPN
ejpam-5770	375	14	rall	rall	PROPN
ejpam-5770	375	15	.	.	PUNCT
ejpam-5770	376	1	rainbow	rainbow	PROPN
ejpam-5770	376	2	domination	domination	NOUN
ejpam-5770	376	3	in	in	ADP
ejpam-5770	376	4	graphs	graph	NOUN
ejpam-5770	376	5	.	.	PUNCT
ejpam-5770	377	1	taiwanese	taiwanese	ADJ
ejpam-5770	377	2	journal	journal	NOUN
ejpam-5770	377	3	of	of	ADP
ejpam-5770	377	4	mathematics	mathematic	NOUN
ejpam-5770	377	5	,	,	PUNCT
ejpam-5770	377	6	12:213–225	12:213–225	NUM
ejpam-5770	377	7	,	,	PUNCT
ejpam-5770	377	8	2008	2008	NUM
ejpam-5770	377	9	.	.	PUNCT
ejpam-5770	378	1	[	[	X
ejpam-5770	378	2	4	4	X
ejpam-5770	378	3	]	]	X
ejpam-5770	378	4	h.	h.	NOUN
ejpam-5770	378	5	abdollahzadeh	abdollahzadeh	PROPN
ejpam-5770	378	6	ahangar	ahangar	NOUN
ejpam-5770	378	7	,	,	PUNCT
ejpam-5770	378	8	j.	j.	PROPN
ejpam-5770	378	9	amjadi	amjadi	PROPN
ejpam-5770	378	10	,	,	PUNCT
ejpam-5770	378	11	n.	n.	PROPN
ejpam-5770	378	12	jafari	jafari	PROPN
ejpam-5770	378	13	rad	rad	PROPN
ejpam-5770	378	14	,	,	PUNCT
ejpam-5770	378	15	and	and	CCONJ
ejpam-5770	378	16	v.	v.	ADP
ejpam-5770	378	17	d.	d.	PROPN
ejpam-5770	378	18	samodivkin	samodivkin	PROPN
ejpam-5770	378	19	.	.	PUNCT
ejpam-5770	379	1	total	total	ADJ
ejpam-5770	379	2	k	k	ADJ
ejpam-5770	379	3	-	-	PUNCT
ejpam-5770	379	4	rainbow	rainbow	NOUN
ejpam-5770	379	5	domination	domination	NOUN
ejpam-5770	379	6	numbers	number	NOUN
ejpam-5770	379	7	in	in	ADP
ejpam-5770	379	8	graphs	graph	NOUN
ejpam-5770	379	9	.	.	PUNCT
ejpam-5770	380	1	communications	communication	NOUN
ejpam-5770	380	2	in	in	ADP
ejpam-5770	380	3	combinatorics	combinatoric	NOUN
ejpam-5770	380	4	and	and	CCONJ
ejpam-5770	380	5	optimization	optimization	NOUN
ejpam-5770	380	6	,	,	PUNCT
ejpam-5770	380	7	3:37–50	3:37–50	NUM
ejpam-5770	380	8	,	,	PUNCT
ejpam-5770	380	9	2018	2018	NUM
ejpam-5770	380	10	.	.	PUNCT
ejpam-5770	381	1	[	[	X
ejpam-5770	381	2	5	5	X
ejpam-5770	381	3	]	]	PUNCT
ejpam-5770	381	4	j.	j.	PROPN
ejpam-5770	381	5	amjadi	amjadi	PROPN
ejpam-5770	381	6	,	,	PUNCT
ejpam-5770	381	7	n.	n.	PROPN
ejpam-5770	381	8	dehgardi	dehgardi	PROPN
ejpam-5770	381	9	,	,	PUNCT
ejpam-5770	381	10	m.	m.	NOUN
ejpam-5770	381	11	furuya	furuya	PROPN
ejpam-5770	381	12	,	,	PUNCT
ejpam-5770	381	13	and	and	CCONJ
ejpam-5770	381	14	s.	s.	PROPN
ejpam-5770	381	15	m.	m.	PROPN
ejpam-5770	381	16	sheikholeslami	sheikholeslami	PROPN
ejpam-5770	381	17	.	.	PUNCT
ejpam-5770	382	1	a	a	DET
ejpam-5770	382	2	sufficient	sufficient	ADJ
ejpam-5770	382	3	condition	condition	NOUN
ejpam-5770	382	4	for	for	ADP
ejpam-5770	382	5	large	large	ADJ
ejpam-5770	382	6	rainbow	rainbow	NOUN
ejpam-5770	382	7	domination	domination	NOUN
ejpam-5770	382	8	number	number	NOUN
ejpam-5770	382	9	.	.	PUNCT
ejpam-5770	383	1	international	international	ADJ
ejpam-5770	383	2	journal	journal	PROPN
ejpam-5770	383	3	of	of	ADP
ejpam-5770	383	4	computer	computer	NOUN
ejpam-5770	383	5	mathematics	mathematic	NOUN
ejpam-5770	383	6	:	:	PUNCT
ejpam-5770	383	7	computer	computer	NOUN
ejpam-5770	383	8	systems	system	NOUN
ejpam-5770	383	9	theory	theory	NOUN
ejpam-5770	383	10	,	,	PUNCT
ejpam-5770	383	11	2:1–17	2:1–17	NUM
ejpam-5770	383	12	,	,	PUNCT
ejpam-5770	383	13	2017	2017	NUM
ejpam-5770	383	14	.	.	PUNCT
ejpam-5770	384	1	[	[	X
ejpam-5770	384	2	6	6	NUM
ejpam-5770	384	3	]	]	PUNCT
ejpam-5770	384	4	b.	b.	PROPN
ejpam-5770	384	5	brešar	brešar	PROPN
ejpam-5770	384	6	and	and	CCONJ
ejpam-5770	384	7	t.	t.	PROPN
ejpam-5770	384	8	k.	k.	PROPN
ejpam-5770	384	9	šumenjak	šumenjak	PROPN
ejpam-5770	384	10	.	.	PUNCT
ejpam-5770	385	1	note	note	NOUN
ejpam-5770	385	2	on	on	ADP
ejpam-5770	385	3	the	the	DET
ejpam-5770	385	4	2	2	NUM
ejpam-5770	385	5	-	-	PUNCT
ejpam-5770	385	6	rainbow	rainbow	NOUN
ejpam-5770	385	7	domination	domination	NOUN
ejpam-5770	385	8	in	in	ADP
ejpam-5770	385	9	graphs	graph	NOUN
ejpam-5770	385	10	.	.	PUNCT
ejpam-5770	386	1	discrete	discrete	ADJ
ejpam-5770	386	2	applied	apply	VERB
ejpam-5770	386	3	mathematics	mathematic	NOUN
ejpam-5770	386	4	,	,	PUNCT
ejpam-5770	386	5	155:2394–2400	155:2394–2400	NUM
ejpam-5770	386	6	,	,	PUNCT
ejpam-5770	386	7	2007	2007	NUM
ejpam-5770	386	8	.	.	PUNCT
ejpam-5770	387	1	[	[	X
ejpam-5770	387	2	7	7	X
ejpam-5770	387	3	]	]	X
ejpam-5770	387	4	s.	s.	PROPN
ejpam-5770	387	5	fujita	fujita	PROPN
ejpam-5770	387	6	,	,	PUNCT
ejpam-5770	387	7	m.	m.	NOUN
ejpam-5770	387	8	furuya	furuya	PROPN
ejpam-5770	387	9	,	,	PUNCT
ejpam-5770	387	10	and	and	CCONJ
ejpam-5770	387	11	c.	c.	PROPN
ejpam-5770	387	12	magnant	magnant	PROPN
ejpam-5770	387	13	.	.	PUNCT
ejpam-5770	388	1	general	general	ADJ
ejpam-5770	388	2	bounds	bound	VERB
ejpam-5770	388	3	on	on	ADP
ejpam-5770	388	4	rainbow	rainbow	NOUN
ejpam-5770	388	5	domination	domination	NOUN
ejpam-5770	388	6	numbers	number	NOUN
ejpam-5770	388	7	.	.	PUNCT
ejpam-5770	389	1	graphs	graph	NOUN
ejpam-5770	389	2	and	and	CCONJ
ejpam-5770	389	3	combinatorics	combinatoric	NOUN
ejpam-5770	389	4	,	,	PUNCT
ejpam-5770	389	5	31:601–613	31:601–613	NUM
ejpam-5770	389	6	,	,	PUNCT
ejpam-5770	389	7	2015	2015	NUM
ejpam-5770	389	8	.	.	PUNCT
ejpam-5770	390	1	[	[	X
ejpam-5770	390	2	8	8	NUM
ejpam-5770	390	3	]	]	X
ejpam-5770	390	4	b.	b.	PROPN
ejpam-5770	390	5	kuzman	kuzman	PROPN
ejpam-5770	390	6	.	.	PUNCT
ejpam-5770	391	1	on	on	ADP
ejpam-5770	391	2	k	k	ADJ
ejpam-5770	391	3	-	-	PUNCT
ejpam-5770	391	4	rainbow	rainbow	NOUN
ejpam-5770	391	5	domination	domination	NOUN
ejpam-5770	391	6	in	in	ADP
ejpam-5770	391	7	regular	regular	ADJ
ejpam-5770	391	8	graphs	graph	NOUN
ejpam-5770	391	9	.	.	PUNCT
ejpam-5770	392	1	discrete	discrete	ADJ
ejpam-5770	392	2	applied	apply	VERB
ejpam-5770	392	3	mathematics	mathematic	NOUN
ejpam-5770	392	4	,	,	PUNCT
ejpam-5770	392	5	184:454–464	184:454–464	NUM
ejpam-5770	392	6	,	,	PUNCT
ejpam-5770	392	7	2020	2020	NUM
ejpam-5770	392	8	.	.	PUNCT
ejpam-5770	393	1	[	[	X
ejpam-5770	393	2	9	9	NUM
ejpam-5770	393	3	]	]	X
ejpam-5770	393	4	d.	d.	PROPN
ejpam-5770	393	5	meierling	meierling	PROPN
ejpam-5770	393	6	,	,	PUNCT
ejpam-5770	393	7	s.	s.	PROPN
ejpam-5770	393	8	m.	m.	PROPN
ejpam-5770	393	9	sheikholeslami	sheikholeslami	PROPN
ejpam-5770	393	10	,	,	PUNCT
ejpam-5770	393	11	and	and	CCONJ
ejpam-5770	393	12	l.	l.	PROPN
ejpam-5770	393	13	volkmann	volkmann	PROPN
ejpam-5770	393	14	.	.	PUNCT
ejpam-5770	394	1	nordhaus	nordhaus	PROPN
ejpam-5770	394	2	-	-	PUNCT
ejpam-5770	394	3	gaddum	gaddum	PROPN
ejpam-5770	394	4	bounds	bound	VERB
ejpam-5770	394	5	on	on	ADP
ejpam-5770	394	6	the	the	DET
ejpam-5770	394	7	k	k	ADJ
ejpam-5770	394	8	-	-	PUNCT
ejpam-5770	394	9	rainbow	rainbow	NOUN
ejpam-5770	394	10	domatic	domatic	ADJ
ejpam-5770	394	11	number	number	NOUN
ejpam-5770	394	12	of	of	ADP
ejpam-5770	394	13	a	a	DET
ejpam-5770	394	14	graph	graph	NOUN
ejpam-5770	394	15	.	.	PUNCT
ejpam-5770	395	1	applied	apply	VERB
ejpam-5770	395	2	mathematics	mathematics	NOUN
ejpam-5770	395	3	letters	letter	NOUN
ejpam-5770	395	4	,	,	PUNCT
ejpam-5770	395	5	24:1758–1761	24:1758–1761	NUM
ejpam-5770	395	6	,	,	PUNCT
ejpam-5770	395	7	2011	2011	NUM
ejpam-5770	395	8	.	.	PUNCT
ejpam-5770	396	1	[	[	X
ejpam-5770	396	2	10	10	NUM
ejpam-5770	396	3	]	]	PUNCT
ejpam-5770	396	4	a.	a.	NOUN
ejpam-5770	396	5	mahmoodi	mahmoodi	PROPN
ejpam-5770	396	6	and	and	CCONJ
ejpam-5770	396	7	l.	l.	PROPN
ejpam-5770	396	8	volkmann	volkmann	PROPN
ejpam-5770	396	9	.	.	PUNCT
ejpam-5770	397	1	outer	outer	ADJ
ejpam-5770	397	2	-	-	PUNCT
ejpam-5770	397	3	independent	independent	ADJ
ejpam-5770	397	4	total	total	ADJ
ejpam-5770	397	5	2	2	NUM
ejpam-5770	397	6	-	-	PUNCT
ejpam-5770	397	7	rainbow	rainbow	NOUN
ejpam-5770	397	8	dominating	dominating	NOUN
ejpam-5770	397	9	functions	function	NOUN
ejpam-5770	397	10	in	in	ADP
ejpam-5770	397	11	graphs	graph	NOUN
ejpam-5770	397	12	.	.	PUNCT
ejpam-5770	398	1	communications	communication	NOUN
ejpam-5770	398	2	in	in	ADP
ejpam-5770	398	3	combinatorics	combinatoric	NOUN
ejpam-5770	398	4	and	and	CCONJ
ejpam-5770	398	5	optimization	optimization	NOUN
ejpam-5770	398	6	,	,	PUNCT
ejpam-5770	398	7	8:431–444	8:431–444	NUM
ejpam-5770	398	8	,	,	PUNCT
ejpam-5770	398	9	2023	2023	NUM
ejpam-5770	398	10	.	.	PUNCT
ejpam-5770	399	1	[	[	X
ejpam-5770	399	2	11	11	NUM
ejpam-5770	399	3	]	]	PUNCT
ejpam-5770	399	4	r.	r.	PROPN
ejpam-5770	399	5	y.	y.	PROPN
ejpam-5770	399	6	salkhori	salkhori	PROPN
ejpam-5770	399	7	,	,	PUNCT
ejpam-5770	399	8	e.	e.	PROPN
ejpam-5770	399	9	vatandoost	vatandoost	PROPN
ejpam-5770	399	10	,	,	PUNCT
ejpam-5770	399	11	and	and	CCONJ
ejpam-5770	399	12	a.	a.	NOUN
ejpam-5770	399	13	behtoei	behtoei	PROPN
ejpam-5770	399	14	.	.	PUNCT
ejpam-5770	400	1	2	2	NUM
ejpam-5770	400	2	-	-	PUNCT
ejpam-5770	400	3	rainbow	rainbow	NOUN
ejpam-5770	400	4	domination	domination	NOUN
ejpam-5770	400	5	number	number	NOUN
ejpam-5770	400	6	of	of	ADP
ejpam-5770	400	7	the	the	DET
ejpam-5770	400	8	subdivision	subdivision	NOUN
ejpam-5770	400	9	of	of	ADP
ejpam-5770	400	10	graphs	graph	NOUN
ejpam-5770	400	11	.	.	PUNCT
ejpam-5770	401	1	communications	communication	NOUN
ejpam-5770	401	2	in	in	ADP
ejpam-5770	401	3	combinatorics	combinatoric	NOUN
ejpam-5770	401	4	and	and	CCONJ
ejpam-5770	401	5	optimization	optimization	NOUN
ejpam-5770	401	6	.	.	PUNCT
ejpam-5770	402	1	in	in	ADP
ejpam-5770	402	2	press	press	NOUN
ejpam-5770	402	3	.	.	PUNCT
ejpam-5770	403	1	[	[	X
ejpam-5770	403	2	12	12	NUM
ejpam-5770	403	3	]	]	PUNCT
ejpam-5770	403	4	j.	j.	PROPN
ejpam-5770	403	5	e.	e.	PROPN
ejpam-5770	403	6	dunbar	dunbar	PROPN
ejpam-5770	403	7	,	,	PUNCT
ejpam-5770	403	8	j.	j.	PROPN
ejpam-5770	403	9	w.	w.	PROPN
ejpam-5770	403	10	grossman	grossman	PROPN
ejpam-5770	403	11	,	,	PUNCT
ejpam-5770	403	12	j.	j.	PROPN
ejpam-5770	403	13	h.	h.	PROPN
ejpam-5770	403	14	hattingh	hattingh	PROPN
ejpam-5770	403	15	,	,	PUNCT
ejpam-5770	403	16	s.	s.	PROPN
ejpam-5770	403	17	t.	t.	PROPN
ejpam-5770	403	18	hedetniemi	hedetniemi	PROPN
ejpam-5770	403	19	,	,	PUNCT
ejpam-5770	403	20	and	and	CCONJ
ejpam-5770	403	21	a.	a.	NOUN
ejpam-5770	403	22	a.	a.	PROPN
ejpam-5770	403	23	mcrae	mcrae	PROPN
ejpam-5770	403	24	.	.	PUNCT
ejpam-5770	404	1	on	on	ADP
ejpam-5770	404	2	weakly	weakly	ADJ
ejpam-5770	404	3	connected	connected	ADJ
ejpam-5770	404	4	domination	domination	NOUN
ejpam-5770	404	5	in	in	ADP
ejpam-5770	404	6	graphs	graph	NOUN
ejpam-5770	404	7	.	.	PUNCT
ejpam-5770	405	1	discrete	discrete	ADJ
ejpam-5770	405	2	mathematics	mathematic	NOUN
ejpam-5770	405	3	,	,	PUNCT
ejpam-5770	405	4	167:261–269	167:261–269	NUM
ejpam-5770	405	5	,	,	PUNCT
ejpam-5770	405	6	1997	1997	NUM
ejpam-5770	405	7	.	.	PUNCT
ejpam-5770	406	1	j.	j.	PROPN
ejpam-5770	406	2	j.	j.	PROPN
ejpam-5770	406	3	hamja	hamja	PROPN
ejpam-5770	406	4	et	et	PROPN
ejpam-5770	406	5	al	al	PROPN
ejpam-5770	406	6	.	.	PUNCT
ejpam-5770	406	7	/	/	SYM
ejpam-5770	406	8	eur	eur	PROPN
ejpam-5770	406	9	.	.	PUNCT
ejpam-5770	407	1	j.	j.	PROPN
ejpam-5770	407	2	pure	pure	PROPN
ejpam-5770	407	3	appl	appl	PROPN
ejpam-5770	407	4	.	.	PROPN
ejpam-5770	407	5	math	math	PROPN
ejpam-5770	407	6	,	,	PUNCT
ejpam-5770	407	7	18	18	NUM
ejpam-5770	407	8	(	(	PUNCT
ejpam-5770	407	9	2	2	NUM
ejpam-5770	407	10	)	)	PUNCT
ejpam-5770	407	11	(	(	PUNCT
ejpam-5770	407	12	2025	2025	NUM
ejpam-5770	407	13	)	)	PUNCT
ejpam-5770	407	14	,	,	PUNCT
ejpam-5770	407	15	5770	5770	NUM
ejpam-5770	407	16	11	11	NUM
ejpam-5770	407	17	of	of	ADP
ejpam-5770	407	18	11	11	NUM
ejpam-5770	407	19	[	[	SYM
ejpam-5770	407	20	13	13	NUM
ejpam-5770	407	21	]	]	PUNCT
ejpam-5770	407	22	g.	g.	PROPN
ejpam-5770	407	23	s.	s.	PROPN
ejpam-5770	407	24	domke	domke	PROPN
ejpam-5770	407	25	,	,	PUNCT
ejpam-5770	407	26	j.	j.	PROPN
ejpam-5770	407	27	h.	h.	PROPN
ejpam-5770	407	28	hattingh	hattingh	PROPN
ejpam-5770	407	29	,	,	PUNCT
ejpam-5770	407	30	and	and	CCONJ
ejpam-5770	407	31	l.	l.	PROPN
ejpam-5770	407	32	r.	r.	PROPN
ejpam-5770	407	33	markus	markus	PROPN
ejpam-5770	407	34	.	.	PUNCT
ejpam-5770	408	1	on	on	ADP
ejpam-5770	408	2	weakly	weakly	ADJ
ejpam-5770	408	3	connected	connected	ADJ
ejpam-5770	408	4	domination	domination	NOUN
ejpam-5770	408	5	in	in	ADP
ejpam-5770	408	6	graphs	graphs	PROPN
ejpam-5770	408	7	ii	ii	PROPN
ejpam-5770	408	8	.	.	PROPN
ejpam-5770	408	9	discrete	discrete	ADJ
ejpam-5770	408	10	mathematics	mathematic	NOUN
ejpam-5770	408	11	,	,	PUNCT
ejpam-5770	408	12	305:112–122	305:112–122	NUM
ejpam-5770	408	13	,	,	PUNCT
ejpam-5770	408	14	2005	2005	NUM
ejpam-5770	408	15	.	.	PUNCT
ejpam-5770	409	1	[	[	X
ejpam-5770	409	2	14	14	NUM
ejpam-5770	409	3	]	]	PUNCT
ejpam-5770	409	4	j.	j.	PROPN
ejpam-5770	409	5	j.	j.	PROPN
ejpam-5770	409	6	hamja	hamja	PROPN
ejpam-5770	409	7	,	,	PUNCT
ejpam-5770	409	8	i.	i.	PROPN
ejpam-5770	409	9	s.	s.	PROPN
ejpam-5770	409	10	aniversario	aniversario	PROPN
ejpam-5770	409	11	,	,	PUNCT
ejpam-5770	409	12	and	and	CCONJ
ejpam-5770	409	13	h.	h.	PROPN
ejpam-5770	409	14	m.	m.	PROPN
ejpam-5770	409	15	rara	rara	PROPN
ejpam-5770	409	16	.	.	PUNCT
ejpam-5770	410	1	weakly	weakly	ADV
ejpam-5770	410	2	connected	connected	ADJ
ejpam-5770	410	3	closed	close	VERB
ejpam-5770	410	4	geodetic	geodetic	ADJ
ejpam-5770	410	5	domination	domination	NOUN
ejpam-5770	410	6	in	in	ADP
ejpam-5770	410	7	graphs	graph	NOUN
ejpam-5770	410	8	under	under	ADP
ejpam-5770	410	9	some	some	DET
ejpam-5770	410	10	binary	binary	ADJ
ejpam-5770	410	11	operations	operation	NOUN
ejpam-5770	410	12	.	.	PUNCT
ejpam-5770	411	1	european	european	ADJ
ejpam-5770	411	2	journal	journal	PROPN
ejpam-5770	411	3	of	of	ADP
ejpam-5770	411	4	pure	pure	ADJ
ejpam-5770	411	5	and	and	CCONJ
ejpam-5770	411	6	applied	applied	ADJ
ejpam-5770	411	7	mathematics	mathematic	NOUN
ejpam-5770	411	8	,	,	PUNCT
ejpam-5770	411	9	15(2):736–752	15(2):736–752	PROPN
ejpam-5770	411	10	,	,	PUNCT
ejpam-5770	411	11	2023	2023	NUM
ejpam-5770	411	12	.	.	PUNCT
ejpam-5770	412	1	[	[	X
ejpam-5770	412	2	15	15	NUM
ejpam-5770	412	3	]	]	X
ejpam-5770	412	4	j.	j.	PROPN
ejpam-5770	412	5	j.	j.	PROPN
ejpam-5770	412	6	hamja	hamja	PROPN
ejpam-5770	412	7	,	,	PUNCT
ejpam-5770	412	8	i.	i.	PROPN
ejpam-5770	412	9	s.	s.	PROPN
ejpam-5770	412	10	aniversario	aniversario	PROPN
ejpam-5770	412	11	,	,	PUNCT
ejpam-5770	412	12	and	and	CCONJ
ejpam-5770	412	13	c.	c.	PROPN
ejpam-5770	412	14	i.	i.	PROPN
ejpam-5770	412	15	merca	merca	PROPN
ejpam-5770	412	16	.	.	PUNCT
ejpam-5770	413	1	weakly	weakly	ADV
ejpam-5770	413	2	connected	connect	VERB
ejpam-5770	413	3	hop	hop	NOUN
ejpam-5770	413	4	domination	domination	NOUN
ejpam-5770	413	5	in	in	ADP
ejpam-5770	413	6	graphs	graph	NOUN
ejpam-5770	413	7	resulting	result	VERB
ejpam-5770	413	8	from	from	ADP
ejpam-5770	413	9	some	some	DET
ejpam-5770	413	10	binary	binary	ADJ
ejpam-5770	413	11	operations	operation	NOUN
ejpam-5770	413	12	.	.	PUNCT
ejpam-5770	414	1	european	european	ADJ
ejpam-5770	414	2	journal	journal	PROPN
ejpam-5770	414	3	of	of	ADP
ejpam-5770	414	4	pure	pure	ADJ
ejpam-5770	414	5	and	and	CCONJ
ejpam-5770	414	6	applied	applied	ADJ
ejpam-5770	414	7	mathematics	mathematic	NOUN
ejpam-5770	414	8	,	,	PUNCT
ejpam-5770	414	9	16(1):454–464	16(1):454–464	PROPN
ejpam-5770	414	10	,	,	PUNCT
ejpam-5770	414	11	2023	2023	NUM
ejpam-5770	414	12	.	.	PUNCT
ejpam-5770	415	1	[	[	X
ejpam-5770	415	2	16	16	NUM
ejpam-5770	415	3	]	]	PUNCT
ejpam-5770	415	4	m.	m.	NOUN
ejpam-5770	415	5	lemańska	lemańska	NOUN
ejpam-5770	415	6	and	and	CCONJ
ejpam-5770	415	7	a.	a.	NOUN
ejpam-5770	415	8	patyk	patyk	NOUN
ejpam-5770	415	9	.	.	PUNCT
ejpam-5770	416	1	weakly	weakly	ADJ
ejpam-5770	416	2	connected	connected	ADJ
ejpam-5770	416	3	domination	domination	NOUN
ejpam-5770	416	4	critical	critical	ADJ
ejpam-5770	416	5	graphs	graph	NOUN
ejpam-5770	416	6	.	.	PUNCT
ejpam-5770	417	1	opuscula	opuscula	PROPN
ejpam-5770	417	2	mathematica	mathematica	PROPN
ejpam-5770	417	3	,	,	PUNCT
ejpam-5770	417	4	28:325–330	28:325–330	PROPN
ejpam-5770	417	5	,	,	PUNCT
ejpam-5770	417	6	2008	2008	NUM
ejpam-5770	417	7	.	.	PUNCT
ejpam-5770	418	1	[	[	X
ejpam-5770	418	2	17	17	NUM
ejpam-5770	418	3	]	]	X
ejpam-5770	418	4	j.	j.	PROPN
ejpam-5770	418	5	raczek	raczek	PROPN
ejpam-5770	418	6	and	and	CCONJ
ejpam-5770	418	7	j.	j.	PROPN
ejpam-5770	418	8	cyman	cyman	PROPN
ejpam-5770	418	9	.	.	PUNCT
ejpam-5770	419	1	weakly	weakly	ADJ
ejpam-5770	419	2	connected	connect	VERB
ejpam-5770	419	3	roma	roma	PROPN
ejpam-5770	419	4	domination	domination	NOUN
ejpam-5770	419	5	in	in	ADP
ejpam-5770	419	6	graphs	graph	NOUN
ejpam-5770	419	7	.	.	PUNCT
ejpam-5770	420	1	discrete	discrete	ADJ
ejpam-5770	420	2	applied	apply	VERB
ejpam-5770	420	3	mathematics	mathematic	NOUN
ejpam-5770	420	4	,	,	PUNCT
ejpam-5770	420	5	267:151–159	267:151–159	NUM
ejpam-5770	420	6	,	,	PUNCT
ejpam-5770	420	7	2019	2019	NUM
ejpam-5770	420	8	.	.	PUNCT
ejpam-5770	421	1	[	[	X
ejpam-5770	421	2	18	18	NUM
ejpam-5770	421	3	]	]	PUNCT
ejpam-5770	421	4	e.	e.	PROPN
ejpam-5770	421	5	p.	p.	PROPN
ejpam-5770	421	6	sandueta	sandueta	PROPN
ejpam-5770	421	7	and	and	CCONJ
ejpam-5770	421	8	s.	s.	PROPN
ejpam-5770	421	9	r.	r.	PROPN
ejpam-5770	421	10	canoy	canoy	PROPN
ejpam-5770	421	11	jr	jr	PROPN
ejpam-5770	421	12	.	.	PROPN
ejpam-5770	421	13	weakly	weakly	ADJ
ejpam-5770	421	14	connected	connected	ADJ
ejpam-5770	421	15	domination	domination	NOUN
ejpam-5770	421	16	in	in	ADP
ejpam-5770	421	17	graphs	graph	NOUN
ejpam-5770	421	18	resulting	result	VERB
ejpam-5770	421	19	from	from	ADP
ejpam-5770	421	20	some	some	DET
ejpam-5770	421	21	graph	graph	NOUN
ejpam-5770	421	22	operations	operation	NOUN
ejpam-5770	421	23	.	.	PUNCT
ejpam-5770	422	1	international	international	ADJ
ejpam-5770	422	2	mathematical	mathematical	PROPN
ejpam-5770	422	3	forum	forum	PROPN
ejpam-5770	422	4	,	,	PUNCT
ejpam-5770	422	5	6:1031–1035	6:1031–1035	NUM
ejpam-5770	422	6	,	,	PUNCT
ejpam-5770	422	7	2011	2011	NUM
ejpam-5770	422	8	.	.	PUNCT
ejpam-5770	423	1	[	[	X
ejpam-5770	423	2	19	19	NUM
ejpam-5770	423	3	]	]	X
ejpam-5770	423	4	v.	v.	ADP
ejpam-5770	423	5	swaminathan	swaminathan	ADV
ejpam-5770	423	6	.	.	PUNCT
ejpam-5770	424	1	weakly	weakly	ADJ
ejpam-5770	424	2	connected	connected	ADJ
ejpam-5770	424	3	domination	domination	NOUN
ejpam-5770	424	4	in	in	ADP
ejpam-5770	424	5	graphs	graph	NOUN
ejpam-5770	424	6	.	.	PUNCT
ejpam-5770	425	1	electronic	electronic	ADJ
ejpam-5770	425	2	notes	note	NOUN
ejpam-5770	425	3	in	in	ADP
ejpam-5770	425	4	discrete	discrete	ADJ
ejpam-5770	425	5	mathematics	mathematic	NOUN
ejpam-5770	425	6	,	,	PUNCT
ejpam-5770	425	7	33:67–73	33:67–73	NUM
ejpam-5770	425	8	,	,	PUNCT
ejpam-5770	425	9	2009	2009	NUM
ejpam-5770	425	10	.	.	PUNCT
ejpam-5770	426	1	[	[	X
ejpam-5770	426	2	20	20	NUM
ejpam-5770	426	3	]	]	PUNCT
ejpam-5770	426	4	f.	f.	PROPN
ejpam-5770	426	5	harary	harary	PROPN
ejpam-5770	426	6	.	.	PUNCT
ejpam-5770	427	1	graph	graph	NOUN
ejpam-5770	427	2	theory	theory	NOUN
ejpam-5770	427	3	.	.	PUNCT
ejpam-5770	428	1	addison	addison	PROPN
ejpam-5770	428	2	-	-	PUNCT
ejpam-5770	428	3	wesley	wesley	PROPN
ejpam-5770	428	4	publishing	publishing	PROPN
ejpam-5770	428	5	company	company	PROPN
ejpam-5770	428	6	,	,	PUNCT
ejpam-5770	428	7	inc	inc	PROPN
ejpam-5770	428	8	.	.	PROPN
ejpam-5770	428	9	,	,	PUNCT
ejpam-5770	428	10	massachusetts	massachusetts	PROPN
ejpam-5770	428	11	,	,	PUNCT
ejpam-5770	428	12	1969	1969	NUM
ejpam-5770	428	13	.	.	PUNCT
ejpam-5770	429	1	[	[	X
ejpam-5770	429	2	21	21	NUM
ejpam-5770	429	3	]	]	PUNCT
ejpam-5770	429	4	z.	z.	PROPN
ejpam-5770	429	5	shao	shao	PROPN
ejpam-5770	429	6	,	,	PUNCT
ejpam-5770	429	7	m.	m.	PROPN
ejpam-5770	429	8	liang	liang	PROPN
ejpam-5770	429	9	,	,	PUNCT
ejpam-5770	429	10	c.	c.	PROPN
ejpam-5770	429	11	yin	yin	PROPN
ejpam-5770	429	12	,	,	PUNCT
ejpam-5770	429	13	x.	x.	PROPN
ejpam-5770	429	14	xu	xu	PROPN
ejpam-5770	429	15	,	,	PUNCT
ejpam-5770	429	16	p.	p.	NOUN
ejpam-5770	429	17	pavlič	pavlič	PROPN
ejpam-5770	429	18	,	,	PUNCT
ejpam-5770	429	19	and	and	CCONJ
ejpam-5770	429	20	j.	j.	PROPN
ejpam-5770	429	21	žerovnik	žerovnik	VERB
ejpam-5770	429	22	.	.	PUNCT
ejpam-5770	430	1	on	on	ADP
ejpam-5770	430	2	rainbow	rainbow	PROPN
ejpam-5770	430	3	domination	domination	NOUN
ejpam-5770	430	4	numbers	number	NOUN
ejpam-5770	430	5	of	of	ADP
ejpam-5770	430	6	graphs	graph	NOUN
ejpam-5770	430	7	.	.	PUNCT
ejpam-5770	431	1	information	information	NOUN
ejpam-5770	431	2	sciences	sciences	PROPN
ejpam-5770	431	3	,	,	PUNCT
ejpam-5770	431	4	254:225–234	254:225–234	NUM
ejpam-5770	431	5	,	,	PUNCT
ejpam-5770	431	6	2014	2014	NUM
ejpam-5770	431	7	.	.	PUNCT