id	sid	tid	token	lemma	pos
ejpam-5566	1	1	european	european	PROPN
ejpam-5566	1	2	journal	journal	PROPN
ejpam-5566	1	3	of	of	ADP
ejpam-5566	1	4	pure	pure	ADJ
ejpam-5566	1	5	and	and	CCONJ
ejpam-5566	1	6	applied	apply	VERB
ejpam-5566	1	7	mathematics	mathematic	NOUN
ejpam-5566	1	8	vol	vol	NOUN
ejpam-5566	1	9	.	.	PROPN
ejpam-5566	2	1	17	17	NUM
ejpam-5566	2	2	,	,	PUNCT
ejpam-5566	2	3	no	no	INTJ
ejpam-5566	2	4	.	.	NOUN
ejpam-5566	2	5	4	4	NUM
ejpam-5566	2	6	,	,	PUNCT
ejpam-5566	2	7	2024	2024	NUM
ejpam-5566	2	8	,	,	PUNCT
ejpam-5566	2	9	3815	3815	NUM
ejpam-5566	2	10	-	-	SYM
ejpam-5566	2	11	3825	3825	NUM
ejpam-5566	2	12	issn	issn	PROPN
ejpam-5566	2	13	1307	1307	NUM
ejpam-5566	2	14	-	-	SYM
ejpam-5566	2	15	5543	5543	NUM
ejpam-5566	2	16	–	–	PUNCT
ejpam-5566	3	1	ejpam.com	ejpam.com	X
ejpam-5566	3	2	published	publish	VERB
ejpam-5566	3	3	by	by	ADP
ejpam-5566	3	4	new	new	PROPN
ejpam-5566	3	5	york	york	PROPN
ejpam-5566	3	6	business	business	PROPN
ejpam-5566	3	7	global	global	PROPN
ejpam-5566	3	8	some	some	DET
ejpam-5566	3	9	generator	generator	NOUN
ejpam-5566	3	10	subgraphs	subgraphs	NOUN
ejpam-5566	3	11	of	of	ADP
ejpam-5566	3	12	the	the	DET
ejpam-5566	3	13	square	square	NOUN
ejpam-5566	3	14	of	of	ADP
ejpam-5566	3	15	a	a	DET
ejpam-5566	3	16	cycle	cycle	NOUN
ejpam-5566	3	17	realiza	realiza	PROPN
ejpam-5566	3	18	m.	m.	PROPN
ejpam-5566	3	19	mame	mame	PROPN
ejpam-5566	3	20	college	college	PROPN
ejpam-5566	3	21	of	of	ADP
ejpam-5566	3	22	teacher	teacher	NOUN
ejpam-5566	3	23	education	education	NOUN
ejpam-5566	3	24	,	,	PUNCT
ejpam-5566	3	25	batangas	batangas	PROPN
ejpam-5566	3	26	state	state	PROPN
ejpam-5566	3	27	university	university	PROPN
ejpam-5566	3	28	,	,	PUNCT
ejpam-5566	3	29	the	the	DET
ejpam-5566	3	30	national	national	PROPN
ejpam-5566	3	31	engineering	engineering	PROPN
ejpam-5566	3	32	university	university	PROPN
ejpam-5566	3	33	,	,	PUNCT
ejpam-5566	3	34	pablo	pablo	PROPN
ejpam-5566	3	35	borbon	borbon	PROPN
ejpam-5566	3	36	campus	campus	PROPN
ejpam-5566	3	37	,	,	PUNCT
ejpam-5566	3	38	batangas	batangas	PROPN
ejpam-5566	3	39	city	city	PROPN
ejpam-5566	3	40	,	,	PUNCT
ejpam-5566	3	41	philippines	philippine	NOUN
ejpam-5566	3	42	abstract	abstract	ADJ
ejpam-5566	3	43	.	.	PUNCT
ejpam-5566	4	1	graphs	graph	NOUN
ejpam-5566	4	2	considered	consider	VERB
ejpam-5566	4	3	in	in	ADP
ejpam-5566	4	4	this	this	DET
ejpam-5566	4	5	paper	paper	NOUN
ejpam-5566	4	6	are	be	AUX
ejpam-5566	4	7	finite	finite	ADJ
ejpam-5566	4	8	simple	simple	ADJ
ejpam-5566	4	9	graphs	graph	NOUN
ejpam-5566	4	10	,	,	PUNCT
ejpam-5566	4	11	which	which	PRON
ejpam-5566	4	12	have	have	VERB
ejpam-5566	4	13	no	no	DET
ejpam-5566	4	14	loops	loop	NOUN
ejpam-5566	4	15	and	and	CCONJ
ejpam-5566	4	16	multiple	multiple	ADJ
ejpam-5566	4	17	edges	edge	NOUN
ejpam-5566	4	18	.	.	PUNCT
ejpam-5566	5	1	let	let	VERB
ejpam-5566	5	2	g	g	PROPN
ejpam-5566	5	3	=	=	SYM
ejpam-5566	5	4	(	(	PUNCT
ejpam-5566	5	5	v	v	NOUN
ejpam-5566	5	6	(	(	PUNCT
ejpam-5566	5	7	g	g	NOUN
ejpam-5566	5	8	)	)	PUNCT
ejpam-5566	5	9	,	,	PUNCT
ejpam-5566	5	10	e(g	e(g	PROPN
ejpam-5566	5	11	)	)	PUNCT
ejpam-5566	5	12	)	)	PUNCT
ejpam-5566	6	1	be	be	AUX
ejpam-5566	6	2	a	a	DET
ejpam-5566	6	3	graph	graph	NOUN
ejpam-5566	6	4	with	with	ADP
ejpam-5566	6	5	e(g	e(g	NOUN
ejpam-5566	6	6	)	)	PUNCT
ejpam-5566	7	1	=	=	PRON
ejpam-5566	7	2	{	{	PUNCT
ejpam-5566	7	3	e1	e1	PROPN
ejpam-5566	7	4	,	,	PUNCT
ejpam-5566	7	5	e2	e2	PROPN
ejpam-5566	7	6	,	,	PUNCT
ejpam-5566	7	7	.	.	PUNCT
ejpam-5566	7	8	.	.	PUNCT
ejpam-5566	8	1	.	.	PUNCT
ejpam-5566	9	1	,	,	PUNCT
ejpam-5566	9	2	em	em	PRON
ejpam-5566	9	3	}	}	PUNCT
ejpam-5566	9	4	,	,	PUNCT
ejpam-5566	9	5	for	for	ADP
ejpam-5566	9	6	some	some	DET
ejpam-5566	9	7	positive	positive	ADJ
ejpam-5566	9	8	integer	integer	NOUN
ejpam-5566	9	9	m.	m.	NOUN
ejpam-5566	9	10	the	the	DET
ejpam-5566	9	11	edge	edge	NOUN
ejpam-5566	9	12	space	space	NOUN
ejpam-5566	9	13	of	of	ADP
ejpam-5566	9	14	g	g	NOUN
ejpam-5566	9	15	,	,	PUNCT
ejpam-5566	9	16	denoted	denote	VERB
ejpam-5566	9	17	by	by	ADP
ejpam-5566	9	18	e	e	PROPN
ejpam-5566	9	19	(	(	PUNCT
ejpam-5566	9	20	g	g	NOUN
ejpam-5566	9	21	)	)	PUNCT
ejpam-5566	9	22	,	,	PUNCT
ejpam-5566	9	23	is	be	AUX
ejpam-5566	9	24	a	a	DET
ejpam-5566	9	25	vector	vector	NOUN
ejpam-5566	9	26	space	space	NOUN
ejpam-5566	9	27	over	over	ADP
ejpam-5566	9	28	the	the	DET
ejpam-5566	9	29	field	field	NOUN
ejpam-5566	9	30	z2	z2	PROPN
ejpam-5566	9	31	.	.	PUNCT
ejpam-5566	10	1	the	the	DET
ejpam-5566	10	2	elements	element	NOUN
ejpam-5566	10	3	of	of	ADP
ejpam-5566	10	4	e	e	X
ejpam-5566	10	5	(	(	PUNCT
ejpam-5566	10	6	g	g	NOUN
ejpam-5566	10	7	)	)	PUNCT
ejpam-5566	10	8	are	be	AUX
ejpam-5566	10	9	all	all	DET
ejpam-5566	10	10	the	the	DET
ejpam-5566	10	11	subsets	subset	NOUN
ejpam-5566	10	12	of	of	ADP
ejpam-5566	10	13	e(g	e(g	PROPN
ejpam-5566	10	14	)	)	PUNCT
ejpam-5566	10	15	.	.	PUNCT
ejpam-5566	11	1	vector	vector	NOUN
ejpam-5566	11	2	addition	addition	NOUN
ejpam-5566	11	3	is	be	AUX
ejpam-5566	11	4	defined	define	VERB
ejpam-5566	11	5	as	as	ADP
ejpam-5566	11	6	x	x	PROPN
ejpam-5566	11	7	+	+	NOUN
ejpam-5566	11	8	y	y	NOUN
ejpam-5566	11	9	=	=	PUNCT
ejpam-5566	11	10	x	x	SYM
ejpam-5566	11	11	∆	∆	PROPN
ejpam-5566	11	12	y	y	PROPN
ejpam-5566	11	13	,	,	PUNCT
ejpam-5566	11	14	the	the	DET
ejpam-5566	11	15	symmetric	symmetric	ADJ
ejpam-5566	11	16	difference	difference	NOUN
ejpam-5566	11	17	of	of	ADP
ejpam-5566	11	18	sets	set	NOUN
ejpam-5566	11	19	x	x	PUNCT
ejpam-5566	11	20	and	and	CCONJ
ejpam-5566	11	21	y	y	PROPN
ejpam-5566	11	22	,	,	PUNCT
ejpam-5566	11	23	for	for	ADP
ejpam-5566	11	24	x	x	X
ejpam-5566	11	25	,	,	PUNCT
ejpam-5566	11	26	y	y	PROPN
ejpam-5566	11	27	∈	∈	PROPN
ejpam-5566	11	28	e	e	X
ejpam-5566	11	29	(	(	PUNCT
ejpam-5566	11	30	g	g	NOUN
ejpam-5566	11	31	)	)	PUNCT
ejpam-5566	11	32	.	.	PUNCT
ejpam-5566	12	1	scalar	scalar	ADJ
ejpam-5566	12	2	multiplication	multiplication	NOUN
ejpam-5566	12	3	is	be	AUX
ejpam-5566	12	4	defined	define	VERB
ejpam-5566	12	5	as	as	ADP
ejpam-5566	12	6	1	1	NUM
ejpam-5566	12	7	·	·	PUNCT
ejpam-5566	12	8	x	x	SYM
ejpam-5566	13	1	=	=	PUNCT
ejpam-5566	13	2	x	x	X
ejpam-5566	13	3	and	and	CCONJ
ejpam-5566	13	4	0	0	NUM
ejpam-5566	13	5	·	·	PUNCT
ejpam-5566	13	6	x	x	X
ejpam-5566	13	7	=	=	NOUN
ejpam-5566	13	8	∅	∅	NOUN
ejpam-5566	13	9	for	for	ADP
ejpam-5566	13	10	x	x	SYM
ejpam-5566	13	11	∈	∈	PROPN
ejpam-5566	13	12	e	e	X
ejpam-5566	13	13	(	(	PUNCT
ejpam-5566	13	14	g	g	NOUN
ejpam-5566	13	15	)	)	PUNCT
ejpam-5566	13	16	.	.	PUNCT
ejpam-5566	14	1	let	let	VERB
ejpam-5566	14	2	h	h	PRON
ejpam-5566	14	3	be	be	AUX
ejpam-5566	14	4	a	a	DET
ejpam-5566	14	5	subgraph	subgraph	NOUN
ejpam-5566	14	6	of	of	ADP
ejpam-5566	14	7	g.	g.	PROPN
ejpam-5566	14	8	the	the	DET
ejpam-5566	14	9	uniform	uniform	NOUN
ejpam-5566	14	10	set	set	NOUN
ejpam-5566	14	11	of	of	ADP
ejpam-5566	14	12	h	h	NOUN
ejpam-5566	14	13	with	with	ADP
ejpam-5566	14	14	respect	respect	NOUN
ejpam-5566	14	15	to	to	ADP
ejpam-5566	14	16	g	g	NOUN
ejpam-5566	14	17	,	,	PUNCT
ejpam-5566	14	18	denoted	denote	VERB
ejpam-5566	14	19	by	by	ADP
ejpam-5566	14	20	eh(g	eh(g	NOUN
ejpam-5566	14	21	)	)	PUNCT
ejpam-5566	14	22	,	,	PUNCT
ejpam-5566	14	23	is	be	AUX
ejpam-5566	14	24	the	the	DET
ejpam-5566	14	25	set	set	NOUN
ejpam-5566	14	26	of	of	ADP
ejpam-5566	14	27	all	all	DET
ejpam-5566	14	28	elements	element	NOUN
ejpam-5566	14	29	of	of	ADP
ejpam-5566	14	30	e	e	X
ejpam-5566	14	31	(	(	PUNCT
ejpam-5566	14	32	g	g	NOUN
ejpam-5566	14	33	)	)	PUNCT
ejpam-5566	14	34	that	that	PRON
ejpam-5566	14	35	induces	induce	VERB
ejpam-5566	14	36	a	a	DET
ejpam-5566	14	37	subgraph	subgraph	NOUN
ejpam-5566	14	38	isomorphic	isomorphic	ADJ
ejpam-5566	14	39	to	to	ADP
ejpam-5566	14	40	h.	h.	PROPN
ejpam-5566	14	41	the	the	DET
ejpam-5566	14	42	subspace	subspace	NOUN
ejpam-5566	14	43	of	of	ADP
ejpam-5566	14	44	e	e	PROPN
ejpam-5566	14	45	(	(	PUNCT
ejpam-5566	14	46	g	g	NOUN
ejpam-5566	14	47	)	)	PUNCT
ejpam-5566	14	48	generated	generate	VERB
ejpam-5566	14	49	by	by	ADP
ejpam-5566	14	50	eh(g	eh(g	NOUN
ejpam-5566	14	51	)	)	PUNCT
ejpam-5566	14	52	shall	shall	AUX
ejpam-5566	14	53	be	be	AUX
ejpam-5566	14	54	denoted	denote	VERB
ejpam-5566	14	55	by	by	ADP
ejpam-5566	14	56	eh(g	eh(g	NOUN
ejpam-5566	14	57	)	)	PUNCT
ejpam-5566	14	58	.	.	PUNCT
ejpam-5566	15	1	if	if	SCONJ
ejpam-5566	15	2	eh(g	eh(g	NOUN
ejpam-5566	15	3	)	)	PUNCT
ejpam-5566	15	4	is	be	AUX
ejpam-5566	15	5	a	a	DET
ejpam-5566	15	6	generating	generate	VERB
ejpam-5566	15	7	set	set	NOUN
ejpam-5566	15	8	,	,	PUNCT
ejpam-5566	15	9	that	that	PRON
ejpam-5566	15	10	is	be	AUX
ejpam-5566	15	11	eh(g	eh(g	NOUN
ejpam-5566	15	12	)	)	PUNCT
ejpam-5566	15	13	=	=	SYM
ejpam-5566	15	14	e	e	X
ejpam-5566	15	15	(	(	PUNCT
ejpam-5566	15	16	g	g	NOUN
ejpam-5566	15	17	)	)	PUNCT
ejpam-5566	15	18	,	,	PUNCT
ejpam-5566	15	19	then	then	ADV
ejpam-5566	15	20	h	h	PROPN
ejpam-5566	15	21	is	be	AUX
ejpam-5566	15	22	called	call	VERB
ejpam-5566	15	23	a	a	DET
ejpam-5566	15	24	generator	generator	NOUN
ejpam-5566	15	25	subgraph	subgraph	NOUN
ejpam-5566	15	26	of	of	ADP
ejpam-5566	15	27	g.	g.	PROPN
ejpam-5566	15	28	this	this	DET
ejpam-5566	15	29	paper	paper	NOUN
ejpam-5566	15	30	provides	provide	VERB
ejpam-5566	15	31	characterization	characterization	NOUN
ejpam-5566	15	32	for	for	ADP
ejpam-5566	15	33	the	the	DET
ejpam-5566	15	34	star	star	NOUN
ejpam-5566	15	35	graph	graph	NOUN
ejpam-5566	15	36	,	,	PUNCT
ejpam-5566	15	37	path	path	NOUN
ejpam-5566	15	38	graph	graph	NOUN
ejpam-5566	15	39	,	,	PUNCT
ejpam-5566	15	40	(	(	PUNCT
ejpam-5566	15	41	3	3	NUM
ejpam-5566	15	42	,	,	PUNCT
ejpam-5566	15	43	r)−	r)−	PROPN
ejpam-5566	15	44	tadpole	tadpole	NOUN
ejpam-5566	15	45	graph	graph	NOUN
ejpam-5566	15	46	,	,	PUNCT
ejpam-5566	15	47	and	and	CCONJ
ejpam-5566	15	48	kite	kite	NOUN
ejpam-5566	15	49	graph	graph	NOUN
ejpam-5566	15	50	ktr	ktr	PROPN
ejpam-5566	15	51	,	,	PUNCT
ejpam-5566	15	52	s	s	VERB
ejpam-5566	15	53	so	so	SCONJ
ejpam-5566	15	54	that	that	SCONJ
ejpam-5566	15	55	these	these	DET
ejpam-5566	15	56	classes	class	NOUN
ejpam-5566	15	57	of	of	ADP
ejpam-5566	15	58	graphs	graph	NOUN
ejpam-5566	15	59	are	be	AUX
ejpam-5566	15	60	generator	generator	NOUN
ejpam-5566	15	61	subgraphs	subgraph	NOUN
ejpam-5566	15	62	of	of	ADP
ejpam-5566	15	63	the	the	DET
ejpam-5566	15	64	square	square	NOUN
ejpam-5566	15	65	of	of	ADP
ejpam-5566	15	66	a	a	DET
ejpam-5566	15	67	cycle	cycle	NOUN
ejpam-5566	15	68	.	.	PUNCT
ejpam-5566	16	1	2020	2020	NUM
ejpam-5566	16	2	mathematics	mathematics	PROPN
ejpam-5566	16	3	subject	subject	NOUN
ejpam-5566	16	4	classifications	classification	NOUN
ejpam-5566	16	5	:	:	PUNCT
ejpam-5566	16	6	05c25	05c25	NUM
ejpam-5566	16	7	key	key	ADJ
ejpam-5566	16	8	words	word	NOUN
ejpam-5566	16	9	and	and	CCONJ
ejpam-5566	16	10	phrases	phrase	NOUN
ejpam-5566	16	11	:	:	PUNCT
ejpam-5566	16	12	edge	edge	NOUN
ejpam-5566	16	13	space	space	NOUN
ejpam-5566	16	14	of	of	ADP
ejpam-5566	16	15	a	a	DET
ejpam-5566	16	16	graph	graph	NOUN
ejpam-5566	16	17	,	,	PUNCT
ejpam-5566	16	18	generator	generator	NOUN
ejpam-5566	16	19	subgraph	subgraph	NOUN
ejpam-5566	16	20	,	,	PUNCT
ejpam-5566	16	21	square	square	NOUN
ejpam-5566	16	22	of	of	ADP
ejpam-5566	16	23	a	a	DET
ejpam-5566	16	24	cycle	cycle	NOUN
ejpam-5566	16	25	,	,	PUNCT
ejpam-5566	16	26	uniform	uniform	NOUN
ejpam-5566	16	27	set	set	VERB
ejpam-5566	16	28	1	1	NUM
ejpam-5566	16	29	.	.	PUNCT
ejpam-5566	17	1	introduction	introduction	NOUN
ejpam-5566	17	2	many	many	ADJ
ejpam-5566	17	3	interesting	interesting	ADJ
ejpam-5566	17	4	studies	study	NOUN
ejpam-5566	17	5	in	in	ADP
ejpam-5566	17	6	graph	graph	NOUN
ejpam-5566	17	7	theory	theory	NOUN
ejpam-5566	17	8	use	use	VERB
ejpam-5566	17	9	algebraic	algebraic	ADJ
ejpam-5566	17	10	structures	structure	NOUN
ejpam-5566	17	11	to	to	PART
ejpam-5566	17	12	define	define	VERB
ejpam-5566	17	13	new	new	ADJ
ejpam-5566	17	14	classes	class	NOUN
ejpam-5566	17	15	of	of	ADP
ejpam-5566	17	16	graphs	graph	NOUN
ejpam-5566	17	17	.	.	PUNCT
ejpam-5566	18	1	then	then	ADV
ejpam-5566	18	2	,	,	PUNCT
ejpam-5566	18	3	determine	determine	VERB
ejpam-5566	18	4	the	the	DET
ejpam-5566	18	5	characteristics	characteristic	NOUN
ejpam-5566	18	6	of	of	ADP
ejpam-5566	18	7	the	the	DET
ejpam-5566	18	8	new	new	ADJ
ejpam-5566	18	9	developed	develop	VERB
ejpam-5566	18	10	graphs	graph	NOUN
ejpam-5566	18	11	using	use	VERB
ejpam-5566	18	12	graphtheoretic	graphtheoretic	ADJ
ejpam-5566	18	13	properties	property	NOUN
ejpam-5566	18	14	.	.	PUNCT
ejpam-5566	19	1	for	for	ADP
ejpam-5566	19	2	example	example	NOUN
ejpam-5566	19	3	,	,	PUNCT
ejpam-5566	19	4	to	to	PART
ejpam-5566	19	5	mention	mention	VERB
ejpam-5566	19	6	some	some	PRON
ejpam-5566	19	7	,	,	PUNCT
ejpam-5566	19	8	the	the	DET
ejpam-5566	19	9	set	set	NOUN
ejpam-5566	19	10	of	of	ADP
ejpam-5566	19	11	k−	k−	PROPN
ejpam-5566	19	12	subset	subset	NOUN
ejpam-5566	19	13	of	of	ADP
ejpam-5566	19	14	an	an	DET
ejpam-5566	19	15	artibrary	artibrary	ADJ
ejpam-5566	19	16	set	set	NOUN
ejpam-5566	19	17	was	be	AUX
ejpam-5566	19	18	used	use	VERB
ejpam-5566	19	19	in	in	ADP
ejpam-5566	19	20	[	[	X
ejpam-5566	19	21	10	10	NUM
ejpam-5566	19	22	]	]	PUNCT
ejpam-5566	19	23	.	.	PUNCT
ejpam-5566	20	1	the	the	DET
ejpam-5566	20	2	notion	notion	NOUN
ejpam-5566	20	3	of	of	ADP
ejpam-5566	20	4	group	group	NOUN
ejpam-5566	20	5	was	be	AUX
ejpam-5566	20	6	used	use	VERB
ejpam-5566	20	7	in	in	ADP
ejpam-5566	20	8	[	[	X
ejpam-5566	20	9	1	1	NUM
ejpam-5566	20	10	]	]	PUNCT
ejpam-5566	20	11	.	.	PUNCT
ejpam-5566	21	1	in	in	ADP
ejpam-5566	21	2	[	[	X
ejpam-5566	21	3	2	2	NUM
ejpam-5566	21	4	]	]	PUNCT
ejpam-5566	21	5	,	,	PUNCT
ejpam-5566	21	6	the	the	DET
ejpam-5566	21	7	set	set	NOUN
ejpam-5566	21	8	of	of	ADP
ejpam-5566	21	9	all	all	DET
ejpam-5566	21	10	induced	induced	ADJ
ejpam-5566	21	11	subgraphs	subgraph	NOUN
ejpam-5566	21	12	were	be	AUX
ejpam-5566	21	13	utilized	utilize	VERB
ejpam-5566	21	14	to	to	PART
ejpam-5566	21	15	develop	develop	VERB
ejpam-5566	21	16	new	new	ADJ
ejpam-5566	21	17	classes	class	NOUN
ejpam-5566	21	18	of	of	ADP
ejpam-5566	21	19	graphs	graph	NOUN
ejpam-5566	21	20	.	.	PUNCT
ejpam-5566	22	1	there	there	PRON
ejpam-5566	22	2	are	be	VERB
ejpam-5566	22	3	several	several	ADJ
ejpam-5566	22	4	similar	similar	ADJ
ejpam-5566	22	5	studies	study	NOUN
ejpam-5566	22	6	that	that	PRON
ejpam-5566	22	7	can	can	AUX
ejpam-5566	22	8	be	be	AUX
ejpam-5566	22	9	found	find	VERB
ejpam-5566	22	10	in	in	ADP
ejpam-5566	22	11	the	the	DET
ejpam-5566	22	12	literature	literature	NOUN
ejpam-5566	22	13	,	,	PUNCT
ejpam-5566	22	14	although	although	SCONJ
ejpam-5566	22	15	some	some	PRON
ejpam-5566	22	16	uses	use	VERB
ejpam-5566	22	17	different	different	ADJ
ejpam-5566	22	18	algebraic	algebraic	ADJ
ejpam-5566	22	19	structures	structure	NOUN
ejpam-5566	22	20	.	.	PUNCT
ejpam-5566	23	1	the	the	DET
ejpam-5566	23	2	notion	notion	NOUN
ejpam-5566	23	3	of	of	ADP
ejpam-5566	23	4	the	the	DET
ejpam-5566	23	5	generator	generator	NOUN
ejpam-5566	23	6	subgraph	subgraph	NOUN
ejpam-5566	23	7	of	of	ADP
ejpam-5566	23	8	a	a	DET
ejpam-5566	23	9	graph	graph	NOUN
ejpam-5566	23	10	introduced	introduce	VERB
ejpam-5566	23	11	by	by	ADP
ejpam-5566	23	12	gervacio	gervacio	NOUN
ejpam-5566	23	13	in	in	ADP
ejpam-5566	23	14	2008	2008	NUM
ejpam-5566	23	15	links	link	NOUN
ejpam-5566	23	16	the	the	DET
ejpam-5566	23	17	graph	graph	NOUN
ejpam-5566	23	18	theory	theory	NOUN
ejpam-5566	23	19	with	with	ADP
ejpam-5566	23	20	algebra	algebra	NOUN
ejpam-5566	23	21	.	.	PUNCT
ejpam-5566	24	1	this	this	DET
ejpam-5566	24	2	notion	notion	NOUN
ejpam-5566	24	3	stems	stem	VERB
ejpam-5566	24	4	from	from	ADP
ejpam-5566	24	5	the	the	DET
ejpam-5566	24	6	theory	theory	NOUN
ejpam-5566	24	7	of	of	ADP
ejpam-5566	24	8	the	the	DET
ejpam-5566	24	9	edge	edge	NOUN
ejpam-5566	24	10	space	space	NOUN
ejpam-5566	24	11	of	of	ADP
ejpam-5566	24	12	a	a	DET
ejpam-5566	24	13	graph	graph	NOUN
ejpam-5566	24	14	.	.	PUNCT
ejpam-5566	25	1	in	in	ADP
ejpam-5566	25	2	this	this	DET
ejpam-5566	25	3	study	study	NOUN
ejpam-5566	25	4	,	,	PUNCT
ejpam-5566	25	5	graphs	graph	NOUN
ejpam-5566	25	6	considered	consider	VERB
ejpam-5566	25	7	are	be	AUX
ejpam-5566	25	8	finite	finite	ADJ
ejpam-5566	25	9	simple	simple	ADJ
ejpam-5566	25	10	undirected	undirected	ADJ
ejpam-5566	25	11	graphs	graph	NOUN
ejpam-5566	25	12	,	,	PUNCT
ejpam-5566	25	13	which	which	PRON
ejpam-5566	25	14	have	have	VERB
ejpam-5566	25	15	no	no	DET
ejpam-5566	25	16	loops	loop	NOUN
ejpam-5566	25	17	and	and	CCONJ
ejpam-5566	25	18	multiple	multiple	ADJ
ejpam-5566	25	19	edges	edge	NOUN
ejpam-5566	25	20	.	.	PUNCT
ejpam-5566	26	1	let	let	VERB
ejpam-5566	26	2	g	g	PRON
ejpam-5566	26	3	be	be	AUX
ejpam-5566	26	4	a	a	DET
ejpam-5566	26	5	graph	graph	NOUN
ejpam-5566	26	6	with	with	ADP
ejpam-5566	26	7	e(g	e(g	NOUN
ejpam-5566	26	8	)	)	PUNCT
ejpam-5566	27	1	=	=	PRON
ejpam-5566	27	2	{	{	PUNCT
ejpam-5566	27	3	e1	e1	PROPN
ejpam-5566	27	4	,	,	PUNCT
ejpam-5566	27	5	e2	e2	PROPN
ejpam-5566	27	6	,	,	PUNCT
ejpam-5566	27	7	.	.	PUNCT
ejpam-5566	27	8	.	.	PUNCT
ejpam-5566	28	1	.	.	PUNCT
ejpam-5566	29	1	,	,	PUNCT
ejpam-5566	29	2	em	em	PRON
ejpam-5566	29	3	}	}	PUNCT
ejpam-5566	29	4	,	,	PUNCT
ejpam-5566	29	5	for	for	ADP
ejpam-5566	29	6	some	some	DET
ejpam-5566	29	7	positive	positive	ADJ
ejpam-5566	29	8	integer	integer	NOUN
ejpam-5566	29	9	m.	m.	NOUN
ejpam-5566	29	10	the	the	DET
ejpam-5566	29	11	edge	edge	NOUN
ejpam-5566	29	12	space	space	NOUN
ejpam-5566	29	13	of	of	ADP
ejpam-5566	29	14	g	g	NOUN
ejpam-5566	29	15	,	,	PUNCT
ejpam-5566	29	16	denoted	denote	VERB
ejpam-5566	29	17	by	by	ADP
ejpam-5566	29	18	e	e	PROPN
ejpam-5566	29	19	(	(	PUNCT
ejpam-5566	29	20	g	g	NOUN
ejpam-5566	29	21	)	)	PUNCT
ejpam-5566	29	22	,	,	PUNCT
ejpam-5566	29	23	is	be	AUX
ejpam-5566	29	24	a	a	DET
ejpam-5566	29	25	vector	vector	NOUN
ejpam-5566	29	26	space	space	NOUN
ejpam-5566	29	27	over	over	ADP
ejpam-5566	29	28	the	the	DET
ejpam-5566	29	29	field	field	NOUN
ejpam-5566	29	30	z2	z2	NOUN
ejpam-5566	29	31	=	=	SYM
ejpam-5566	29	32	{	{	PUNCT
ejpam-5566	29	33	0	0	NUM
ejpam-5566	29	34	,	,	PUNCT
ejpam-5566	29	35	1	1	NUM
ejpam-5566	29	36	}	}	PUNCT
ejpam-5566	29	37	.	.	PUNCT
ejpam-5566	30	1	the	the	DET
ejpam-5566	30	2	elements	element	NOUN
ejpam-5566	30	3	doi	doi	NOUN
ejpam-5566	30	4	:	:	PUNCT
ejpam-5566	30	5	https://doi.org/10.29020/nybg.ejpam.v17i4.5566	https://doi.org/10.29020/nybg.ejpam.v17i4.5566	ADP
ejpam-5566	30	6	email	email	NOUN
ejpam-5566	30	7	address	address	NOUN
ejpam-5566	30	8	:	:	PUNCT
ejpam-5566	30	9	realiza.mame@g.batstate-u.edu.ph	realiza.mame@g.batstate-u.edu.ph	PROPN
ejpam-5566	30	10	(	(	PUNCT
ejpam-5566	30	11	r.	r.	PROPN
ejpam-5566	30	12	mame	mame	PROPN
ejpam-5566	30	13	)	)	PUNCT
ejpam-5566	30	14	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5566	30	15	3815	3815	NUM
ejpam-5566	30	16	copyright	copyright	NOUN
ejpam-5566	30	17	:	:	PUNCT
ejpam-5566	30	18	©	©	PROPN
ejpam-5566	30	19	2024	2024	NUM
ejpam-5566	30	20	the	the	DET
ejpam-5566	30	21	author(s	author(s	NOUN
ejpam-5566	30	22	)	)	PUNCT
ejpam-5566	30	23	.	.	PUNCT
ejpam-5566	31	1	(	(	PUNCT
ejpam-5566	31	2	cc	cc	NOUN
ejpam-5566	31	3	by	by	ADP
ejpam-5566	31	4	-	-	PUNCT
ejpam-5566	31	5	nc	nc	PROPN
ejpam-5566	31	6	4.0	4.0	NUM
ejpam-5566	31	7	)	)	PUNCT
ejpam-5566	31	8	r.	r.	PROPN
ejpam-5566	31	9	mame	mame	PROPN
ejpam-5566	31	10	/	/	SYM
ejpam-5566	31	11	eur	eur	PROPN
ejpam-5566	31	12	.	.	PUNCT
ejpam-5566	32	1	j.	j.	PROPN
ejpam-5566	32	2	pure	pure	PROPN
ejpam-5566	32	3	appl	appl	PROPN
ejpam-5566	32	4	.	.	PROPN
ejpam-5566	32	5	math	math	PROPN
ejpam-5566	32	6	,	,	PUNCT
ejpam-5566	32	7	17	17	NUM
ejpam-5566	32	8	(	(	PUNCT
ejpam-5566	32	9	4	4	NUM
ejpam-5566	32	10	)	)	PUNCT
ejpam-5566	32	11	(	(	PUNCT
ejpam-5566	32	12	2024	2024	NUM
ejpam-5566	32	13	)	)	PUNCT
ejpam-5566	32	14	,	,	PUNCT
ejpam-5566	32	15	3815	3815	NUM
ejpam-5566	32	16	-	-	SYM
ejpam-5566	32	17	3825	3825	NUM
ejpam-5566	32	18	3816	3816	NUM
ejpam-5566	32	19	of	of	ADP
ejpam-5566	32	20	e	e	PROPN
ejpam-5566	32	21	(	(	PUNCT
ejpam-5566	32	22	g	g	NOUN
ejpam-5566	32	23	)	)	PUNCT
ejpam-5566	32	24	are	be	AUX
ejpam-5566	32	25	all	all	DET
ejpam-5566	32	26	the	the	DET
ejpam-5566	32	27	subsets	subset	NOUN
ejpam-5566	32	28	of	of	ADP
ejpam-5566	32	29	e(g	e(g	PROPN
ejpam-5566	32	30	)	)	PUNCT
ejpam-5566	32	31	.	.	PUNCT
ejpam-5566	33	1	vector	vector	NOUN
ejpam-5566	33	2	addition	addition	NOUN
ejpam-5566	33	3	is	be	AUX
ejpam-5566	33	4	defined	define	VERB
ejpam-5566	33	5	as	as	ADP
ejpam-5566	33	6	x	x	PROPN
ejpam-5566	33	7	+	+	NUM
ejpam-5566	33	8	y	y	NOUN
ejpam-5566	33	9	=	=	PUNCT
ejpam-5566	33	10	x	x	SYM
ejpam-5566	33	11	∆	∆	PROPN
ejpam-5566	33	12	y	y	PROPN
ejpam-5566	33	13	,	,	PUNCT
ejpam-5566	33	14	the	the	DET
ejpam-5566	33	15	symmetric	symmetric	ADJ
ejpam-5566	33	16	difference	difference	NOUN
ejpam-5566	33	17	of	of	ADP
ejpam-5566	33	18	sets	set	NOUN
ejpam-5566	33	19	x	x	PUNCT
ejpam-5566	33	20	and	and	CCONJ
ejpam-5566	33	21	y	y	PROPN
ejpam-5566	33	22	,	,	PUNCT
ejpam-5566	33	23	for	for	ADP
ejpam-5566	33	24	x	x	X
ejpam-5566	33	25	,	,	PUNCT
ejpam-5566	33	26	y	y	PROPN
ejpam-5566	33	27	∈	∈	PROPN
ejpam-5566	33	28	e	e	X
ejpam-5566	33	29	(	(	PUNCT
ejpam-5566	33	30	g	g	NOUN
ejpam-5566	33	31	)	)	PUNCT
ejpam-5566	33	32	.	.	PUNCT
ejpam-5566	34	1	scalar	scalar	ADJ
ejpam-5566	34	2	multiplication	multiplication	NOUN
ejpam-5566	34	3	is	be	AUX
ejpam-5566	34	4	defined	define	VERB
ejpam-5566	34	5	as	as	ADP
ejpam-5566	34	6	1	1	NUM
ejpam-5566	34	7	·	·	SYM
ejpam-5566	34	8	x	x	X
ejpam-5566	35	1	=	=	PUNCT
ejpam-5566	35	2	x	x	X
ejpam-5566	35	3	and	and	CCONJ
ejpam-5566	35	4	0	0	NUM
ejpam-5566	35	5	·	·	PUNCT
ejpam-5566	35	6	x	x	X
ejpam-5566	35	7	=	=	NOUN
ejpam-5566	35	8	∅	∅	NOUN
ejpam-5566	35	9	for	for	ADP
ejpam-5566	35	10	x	x	SYM
ejpam-5566	35	11	∈	∈	PROPN
ejpam-5566	35	12	e	e	X
ejpam-5566	35	13	(	(	PUNCT
ejpam-5566	35	14	g	g	NOUN
ejpam-5566	35	15	)	)	PUNCT
ejpam-5566	35	16	.	.	PUNCT
ejpam-5566	36	1	the	the	DET
ejpam-5566	36	2	set	set	NOUN
ejpam-5566	36	3	s	s	PROPN
ejpam-5566	36	4	⊆	⊆	NUM
ejpam-5566	36	5	e	e	X
ejpam-5566	36	6	(	(	PUNCT
ejpam-5566	36	7	g	g	NOUN
ejpam-5566	36	8	)	)	PUNCT
ejpam-5566	36	9	is	be	AUX
ejpam-5566	36	10	called	call	VERB
ejpam-5566	36	11	a	a	DET
ejpam-5566	36	12	generating	generate	VERB
ejpam-5566	36	13	set	set	NOUN
ejpam-5566	36	14	if	if	SCONJ
ejpam-5566	36	15	every	every	DET
ejpam-5566	36	16	element	element	NOUN
ejpam-5566	36	17	of	of	ADP
ejpam-5566	36	18	e	e	PROPN
ejpam-5566	36	19	(	(	PUNCT
ejpam-5566	36	20	g	g	NOUN
ejpam-5566	36	21	)	)	PUNCT
ejpam-5566	36	22	is	be	AUX
ejpam-5566	36	23	a	a	DET
ejpam-5566	36	24	linear	linear	ADJ
ejpam-5566	36	25	combination	combination	NOUN
ejpam-5566	36	26	of	of	ADP
ejpam-5566	36	27	the	the	DET
ejpam-5566	36	28	elements	element	NOUN
ejpam-5566	36	29	of	of	ADP
ejpam-5566	36	30	s.	s.	PROPN
ejpam-5566	36	31	for	for	ADP
ejpam-5566	36	32	a	a	DET
ejpam-5566	36	33	non	non	ADJ
ejpam-5566	36	34	-	-	ADJ
ejpam-5566	36	35	empty	empty	ADJ
ejpam-5566	36	36	set	set	NOUN
ejpam-5566	36	37	x	x	SYM
ejpam-5566	36	38	⊆	⊆	NUM
ejpam-5566	36	39	e(g	e(g	NOUN
ejpam-5566	36	40	)	)	PUNCT
ejpam-5566	36	41	,	,	PUNCT
ejpam-5566	36	42	the	the	DET
ejpam-5566	36	43	smallest	small	ADJ
ejpam-5566	36	44	subgraph	subgraph	NOUN
ejpam-5566	36	45	of	of	ADP
ejpam-5566	36	46	g	g	NOUN
ejpam-5566	36	47	with	with	ADP
ejpam-5566	36	48	edge	edge	NOUN
ejpam-5566	36	49	set	set	NOUN
ejpam-5566	36	50	x	x	PUNCT
ejpam-5566	36	51	is	be	AUX
ejpam-5566	36	52	called	call	VERB
ejpam-5566	36	53	the	the	DET
ejpam-5566	36	54	edge	edge	NOUN
ejpam-5566	36	55	-	-	PUNCT
ejpam-5566	36	56	induced	induce	VERB
ejpam-5566	36	57	subgraph	subgraph	NOUN
ejpam-5566	36	58	of	of	ADP
ejpam-5566	36	59	g	g	NOUN
ejpam-5566	36	60	,	,	PUNCT
ejpam-5566	36	61	which	which	PRON
ejpam-5566	36	62	we	we	PRON
ejpam-5566	36	63	denote	denote	VERB
ejpam-5566	36	64	by	by	ADP
ejpam-5566	36	65	g[x	g[x	ADP
ejpam-5566	36	66	]	]	PUNCT
ejpam-5566	36	67	.	.	PUNCT
ejpam-5566	37	1	in	in	ADP
ejpam-5566	37	2	this	this	DET
ejpam-5566	37	3	paper	paper	NOUN
ejpam-5566	37	4	,	,	PUNCT
ejpam-5566	37	5	when	when	SCONJ
ejpam-5566	37	6	we	we	PRON
ejpam-5566	37	7	say	say	VERB
ejpam-5566	37	8	induced	induced	ADJ
ejpam-5566	37	9	subgraph	subgraph	NOUN
ejpam-5566	37	10	,	,	PUNCT
ejpam-5566	37	11	we	we	PRON
ejpam-5566	37	12	mean	mean	VERB
ejpam-5566	37	13	an	an	DET
ejpam-5566	37	14	edge	edge	NOUN
ejpam-5566	37	15	-	-	PUNCT
ejpam-5566	37	16	induced	induce	VERB
ejpam-5566	37	17	subgraph	subgraph	NOUN
ejpam-5566	37	18	of	of	ADP
ejpam-5566	37	19	a	a	DET
ejpam-5566	37	20	graph	graph	NOUN
ejpam-5566	37	21	.	.	PUNCT
ejpam-5566	38	1	let	let	VERB
ejpam-5566	38	2	h	h	PRON
ejpam-5566	38	3	be	be	AUX
ejpam-5566	38	4	a	a	DET
ejpam-5566	38	5	subgraph	subgraph	NOUN
ejpam-5566	38	6	of	of	ADP
ejpam-5566	38	7	g.the	g.the	DET
ejpam-5566	38	8	uniform	uniform	NOUN
ejpam-5566	38	9	set	set	NOUN
ejpam-5566	38	10	of	of	ADP
ejpam-5566	38	11	h	h	NOUN
ejpam-5566	38	12	with	with	ADP
ejpam-5566	38	13	respect	respect	NOUN
ejpam-5566	38	14	to	to	ADP
ejpam-5566	38	15	g	g	NOUN
ejpam-5566	38	16	,	,	PUNCT
ejpam-5566	38	17	denoted	denote	VERB
ejpam-5566	38	18	by	by	ADP
ejpam-5566	38	19	eh(g	eh(g	NOUN
ejpam-5566	38	20	)	)	PUNCT
ejpam-5566	38	21	,	,	PUNCT
ejpam-5566	38	22	is	be	AUX
ejpam-5566	38	23	the	the	DET
ejpam-5566	38	24	set	set	NOUN
ejpam-5566	38	25	of	of	ADP
ejpam-5566	38	26	all	all	DET
ejpam-5566	38	27	elements	element	NOUN
ejpam-5566	38	28	of	of	ADP
ejpam-5566	38	29	e	e	X
ejpam-5566	38	30	(	(	PUNCT
ejpam-5566	38	31	g	g	NOUN
ejpam-5566	38	32	)	)	PUNCT
ejpam-5566	38	33	that	that	PRON
ejpam-5566	38	34	induces	induce	VERB
ejpam-5566	38	35	a	a	DET
ejpam-5566	38	36	subgraph	subgraph	NOUN
ejpam-5566	38	37	isomorphic	isomorphic	ADJ
ejpam-5566	38	38	to	to	ADP
ejpam-5566	38	39	h.	h.	PROPN
ejpam-5566	38	40	the	the	DET
ejpam-5566	38	41	subspace	subspace	NOUN
ejpam-5566	38	42	of	of	ADP
ejpam-5566	38	43	e	e	PROPN
ejpam-5566	38	44	(	(	PUNCT
ejpam-5566	38	45	g	g	NOUN
ejpam-5566	38	46	)	)	PUNCT
ejpam-5566	38	47	generated	generate	VERB
ejpam-5566	38	48	by	by	ADP
ejpam-5566	38	49	eh(g	eh(g	NOUN
ejpam-5566	38	50	)	)	PUNCT
ejpam-5566	38	51	is	be	AUX
ejpam-5566	38	52	denoted	denote	VERB
ejpam-5566	38	53	by	by	ADP
ejpam-5566	38	54	eh(g	eh(g	NOUN
ejpam-5566	38	55	)	)	PUNCT
ejpam-5566	38	56	.	.	PUNCT
ejpam-5566	39	1	if	if	SCONJ
ejpam-5566	39	2	eh(g	eh(g	NOUN
ejpam-5566	39	3	)	)	PUNCT
ejpam-5566	39	4	is	be	AUX
ejpam-5566	39	5	a	a	DET
ejpam-5566	39	6	generating	generate	VERB
ejpam-5566	39	7	set	set	NOUN
ejpam-5566	39	8	,	,	PUNCT
ejpam-5566	39	9	that	that	PRON
ejpam-5566	39	10	is	be	AUX
ejpam-5566	39	11	eh(g	eh(g	NOUN
ejpam-5566	39	12	)	)	PUNCT
ejpam-5566	39	13	=	=	SYM
ejpam-5566	39	14	e	e	X
ejpam-5566	39	15	(	(	PUNCT
ejpam-5566	39	16	g	g	NOUN
ejpam-5566	39	17	)	)	PUNCT
ejpam-5566	39	18	,	,	PUNCT
ejpam-5566	39	19	then	then	ADV
ejpam-5566	39	20	h	h	PROPN
ejpam-5566	39	21	is	be	AUX
ejpam-5566	39	22	called	call	VERB
ejpam-5566	39	23	a	a	DET
ejpam-5566	39	24	generator	generator	NOUN
ejpam-5566	39	25	subgraph	subgraph	NOUN
ejpam-5566	39	26	of	of	ADP
ejpam-5566	39	27	g.	g.	PROPN
ejpam-5566	39	28	it	it	PRON
ejpam-5566	39	29	can	can	AUX
ejpam-5566	39	30	be	be	AUX
ejpam-5566	39	31	verified	verify	VERB
ejpam-5566	39	32	that	that	SCONJ
ejpam-5566	39	33	the	the	DET
ejpam-5566	39	34	set	set	NOUN
ejpam-5566	39	35	a	a	X
ejpam-5566	39	36	=	=	X
ejpam-5566	39	37	{	{	PUNCT
ejpam-5566	39	38	{	{	PUNCT
ejpam-5566	39	39	e1	e1	PROPN
ejpam-5566	39	40	}	}	PUNCT
ejpam-5566	39	41	,	,	PUNCT
ejpam-5566	39	42	{	{	PUNCT
ejpam-5566	39	43	e2	e2	PROPN
ejpam-5566	39	44	}	}	PUNCT
ejpam-5566	39	45	,	,	PUNCT
ejpam-5566	39	46	.	.	PUNCT
ejpam-5566	39	47	.	.	PUNCT
ejpam-5566	40	1	.	.	PUNCT
ejpam-5566	41	1	,	,	PUNCT
ejpam-5566	41	2	{	{	PUNCT
ejpam-5566	41	3	em	em	PRON
ejpam-5566	41	4	}	}	PUNCT
ejpam-5566	41	5	}	}	PUNCT
ejpam-5566	41	6	forms	form	VERB
ejpam-5566	41	7	a	a	DET
ejpam-5566	41	8	basis	basis	NOUN
ejpam-5566	41	9	of	of	ADP
ejpam-5566	41	10	e	e	PROPN
ejpam-5566	41	11	(	(	PUNCT
ejpam-5566	41	12	g	g	NOUN
ejpam-5566	41	13	)	)	PUNCT
ejpam-5566	41	14	.	.	PUNCT
ejpam-5566	42	1	hence	hence	ADV
ejpam-5566	42	2	,	,	PUNCT
ejpam-5566	42	3	dime	dime	NOUN
ejpam-5566	42	4	(	(	PUNCT
ejpam-5566	42	5	g	g	NOUN
ejpam-5566	42	6	)	)	PUNCT
ejpam-5566	42	7	=	=	SYM
ejpam-5566	42	8	m	m	PROPN
ejpam-5566	42	9	,	,	PUNCT
ejpam-5566	42	10	the	the	DET
ejpam-5566	42	11	size	size	NOUN
ejpam-5566	42	12	of	of	ADP
ejpam-5566	42	13	g.	g.	PROPN
ejpam-5566	42	14	the	the	DET
ejpam-5566	42	15	set	set	NOUN
ejpam-5566	42	16	a	a	PRON
ejpam-5566	42	17	is	be	AUX
ejpam-5566	42	18	called	call	VERB
ejpam-5566	42	19	the	the	DET
ejpam-5566	42	20	natural	natural	ADJ
ejpam-5566	42	21	basis	basis	NOUN
ejpam-5566	42	22	for	for	ADP
ejpam-5566	42	23	the	the	DET
ejpam-5566	42	24	edge	edge	NOUN
ejpam-5566	42	25	space	space	NOUN
ejpam-5566	42	26	of	of	ADP
ejpam-5566	42	27	g	g	NOUN
ejpam-5566	42	28	,	,	PUNCT
ejpam-5566	42	29	as	as	SCONJ
ejpam-5566	42	30	adopted	adopt	VERB
ejpam-5566	42	31	from	from	ADP
ejpam-5566	42	32	[	[	X
ejpam-5566	42	33	7	7	NUM
ejpam-5566	42	34	]	]	PUNCT
ejpam-5566	42	35	.	.	PUNCT
ejpam-5566	43	1	clearly	clearly	ADV
ejpam-5566	43	2	,	,	PUNCT
ejpam-5566	43	3	eh(g	eh(g	NOUN
ejpam-5566	43	4	)	)	PUNCT
ejpam-5566	43	5	⊆	⊆	NUM
ejpam-5566	43	6	e	e	X
ejpam-5566	43	7	(	(	PUNCT
ejpam-5566	43	8	g	g	NOUN
ejpam-5566	43	9	)	)	PUNCT
ejpam-5566	43	10	.	.	PUNCT
ejpam-5566	44	1	to	to	PART
ejpam-5566	44	2	show	show	VERB
ejpam-5566	44	3	that	that	SCONJ
ejpam-5566	44	4	a	a	DET
ejpam-5566	44	5	subgraph	subgraph	NOUN
ejpam-5566	44	6	h	h	NOUN
ejpam-5566	44	7	is	be	AUX
ejpam-5566	44	8	a	a	DET
ejpam-5566	44	9	generator	generator	NOUN
ejpam-5566	44	10	subgraph	subgraph	NOUN
ejpam-5566	44	11	of	of	ADP
ejpam-5566	44	12	g	g	PROPN
ejpam-5566	44	13	,	,	PUNCT
ejpam-5566	44	14	it	it	PRON
ejpam-5566	44	15	is	be	AUX
ejpam-5566	44	16	sufficient	sufficient	ADJ
ejpam-5566	44	17	to	to	PART
ejpam-5566	44	18	show	show	VERB
ejpam-5566	44	19	that	that	SCONJ
ejpam-5566	44	20	e	e	PROPN
ejpam-5566	44	21	(	(	PUNCT
ejpam-5566	44	22	g	g	NOUN
ejpam-5566	44	23	)	)	PUNCT
ejpam-5566	44	24	⊆	⊆	NUM
ejpam-5566	44	25	eh(g	eh(g	NOUN
ejpam-5566	44	26	)	)	PUNCT
ejpam-5566	44	27	.	.	PUNCT
ejpam-5566	45	1	that	that	PRON
ejpam-5566	45	2	is	be	AUX
ejpam-5566	45	3	,	,	PUNCT
ejpam-5566	45	4	the	the	DET
ejpam-5566	45	5	basis	basis	NOUN
ejpam-5566	45	6	{	{	PUNCT
ejpam-5566	45	7	{	{	PUNCT
ejpam-5566	45	8	e1	e1	PROPN
ejpam-5566	45	9	}	}	PUNCT
ejpam-5566	45	10	,	,	PUNCT
ejpam-5566	45	11	{	{	PUNCT
ejpam-5566	45	12	e2	e2	PROPN
ejpam-5566	45	13	}	}	PUNCT
ejpam-5566	45	14	,	,	PUNCT
ejpam-5566	45	15	.	.	PUNCT
ejpam-5566	45	16	.	.	PUNCT
ejpam-5566	45	17	.	.	PUNCT
ejpam-5566	46	1	,	,	PUNCT
ejpam-5566	46	2	{	{	PUNCT
ejpam-5566	46	3	em	em	PRON
ejpam-5566	46	4	}	}	PUNCT
ejpam-5566	46	5	}	}	PUNCT
ejpam-5566	46	6	⊆	⊆	NUM
ejpam-5566	46	7	eh(g	eh(g	NOUN
ejpam-5566	46	8	)	)	PUNCT
ejpam-5566	46	9	.	.	PUNCT
ejpam-5566	47	1	equivalently	equivalently	ADV
ejpam-5566	47	2	,	,	PUNCT
ejpam-5566	47	3	we	we	PRON
ejpam-5566	47	4	have	have	VERB
ejpam-5566	47	5	the	the	DET
ejpam-5566	47	6	following	follow	VERB
ejpam-5566	47	7	useful	useful	ADJ
ejpam-5566	47	8	remark	remark	NOUN
ejpam-5566	47	9	.	.	PUNCT
ejpam-5566	48	1	remark	remark	NOUN
ejpam-5566	48	2	1	1	NUM
ejpam-5566	48	3	(	(	PUNCT
ejpam-5566	48	4	[	[	X
ejpam-5566	48	5	8	8	NUM
ejpam-5566	48	6	]	]	PUNCT
ejpam-5566	48	7	)	)	PUNCT
ejpam-5566	48	8	.	.	PUNCT
ejpam-5566	49	1	let	let	VERB
ejpam-5566	49	2	h	h	PRON
ejpam-5566	49	3	be	be	AUX
ejpam-5566	49	4	a	a	DET
ejpam-5566	49	5	subgraph	subgraph	NOUN
ejpam-5566	49	6	of	of	ADP
ejpam-5566	49	7	g.	g.	PROPN
ejpam-5566	50	1	then	then	ADV
ejpam-5566	50	2	h	h	PROPN
ejpam-5566	50	3	is	be	AUX
ejpam-5566	50	4	a	a	DET
ejpam-5566	50	5	generator	generator	NOUN
ejpam-5566	50	6	subgraph	subgraph	NOUN
ejpam-5566	50	7	of	of	ADP
ejpam-5566	50	8	g	g	PROPN
ejpam-5566	50	9	if	if	SCONJ
ejpam-5566	51	1	and	and	CCONJ
ejpam-5566	51	2	only	only	ADV
ejpam-5566	51	3	if	if	SCONJ
ejpam-5566	51	4	for	for	ADP
ejpam-5566	51	5	every	every	DET
ejpam-5566	51	6	e	e	PROPN
ejpam-5566	51	7	∈	∈	PROPN
ejpam-5566	51	8	e(g	e(g	PROPN
ejpam-5566	51	9	)	)	PUNCT
ejpam-5566	52	1	the	the	DET
ejpam-5566	52	2	singleton	singleton	NOUN
ejpam-5566	52	3	{	{	PUNCT
ejpam-5566	52	4	e	e	NOUN
ejpam-5566	52	5	}	}	PUNCT
ejpam-5566	52	6	∈	∈	PROPN
ejpam-5566	52	7	eh(g	eh(g	NOUN
ejpam-5566	52	8	)	)	PUNCT
ejpam-5566	52	9	.	.	PUNCT
ejpam-5566	53	1	readers	reader	NOUN
ejpam-5566	53	2	may	may	AUX
ejpam-5566	53	3	refer	refer	VERB
ejpam-5566	53	4	to	to	ADP
ejpam-5566	53	5	[	[	X
ejpam-5566	53	6	8	8	NUM
ejpam-5566	53	7	]	]	PUNCT
ejpam-5566	53	8	for	for	ADP
ejpam-5566	53	9	an	an	DET
ejpam-5566	53	10	illustration	illustration	NOUN
ejpam-5566	53	11	of	of	ADP
ejpam-5566	53	12	finding	find	VERB
ejpam-5566	53	13	the	the	DET
ejpam-5566	53	14	generator	generator	NOUN
ejpam-5566	53	15	subgraph	subgraph	NOUN
ejpam-5566	53	16	of	of	ADP
ejpam-5566	53	17	a	a	DET
ejpam-5566	53	18	graph	graph	NOUN
ejpam-5566	53	19	using	use	VERB
ejpam-5566	53	20	remark	remark	NOUN
ejpam-5566	53	21	1	1	NUM
ejpam-5566	53	22	.	.	PUNCT
ejpam-5566	54	1	in	in	ADP
ejpam-5566	54	2	[	[	X
ejpam-5566	54	3	8	8	NUM
ejpam-5566	54	4	]	]	PUNCT
ejpam-5566	54	5	,	,	PUNCT
ejpam-5566	54	6	the	the	DET
ejpam-5566	54	7	concept	concept	NOUN
ejpam-5566	54	8	of	of	ADP
ejpam-5566	54	9	even	even	ADV
ejpam-5566	54	10	edge	edge	NOUN
ejpam-5566	54	11	space	space	NOUN
ejpam-5566	54	12	e	e	X
ejpam-5566	54	13	∗(g	∗(g	PROPN
ejpam-5566	54	14	)	)	PUNCT
ejpam-5566	54	15	of	of	ADP
ejpam-5566	54	16	a	a	DET
ejpam-5566	54	17	graph	graph	NOUN
ejpam-5566	54	18	was	be	AUX
ejpam-5566	54	19	introduced	introduce	VERB
ejpam-5566	54	20	.	.	PUNCT
ejpam-5566	55	1	if	if	SCONJ
ejpam-5566	55	2	g	g	PROPN
ejpam-5566	55	3	is	be	AUX
ejpam-5566	55	4	a	a	DET
ejpam-5566	55	5	graph	graph	NOUN
ejpam-5566	55	6	with	with	ADP
ejpam-5566	55	7	size	size	NOUN
ejpam-5566	55	8	m	m	PROPN
ejpam-5566	55	9	,	,	PUNCT
ejpam-5566	55	10	it	it	PRON
ejpam-5566	55	11	was	be	AUX
ejpam-5566	55	12	shown	show	VERB
ejpam-5566	55	13	that	that	SCONJ
ejpam-5566	55	14	e	e	PROPN
ejpam-5566	55	15	∗(g	∗(g	PROPN
ejpam-5566	55	16	)	)	PUNCT
ejpam-5566	55	17	is	be	AUX
ejpam-5566	55	18	a	a	DET
ejpam-5566	55	19	maximal	maximal	ADJ
ejpam-5566	55	20	subspace	subspace	NOUN
ejpam-5566	55	21	of	of	ADP
ejpam-5566	55	22	the	the	DET
ejpam-5566	55	23	edge	edge	NOUN
ejpam-5566	55	24	space	space	NOUN
ejpam-5566	55	25	of	of	ADP
ejpam-5566	55	26	g	g	NOUN
ejpam-5566	55	27	with	with	ADP
ejpam-5566	55	28	dimension	dimension	NOUN
ejpam-5566	55	29	m−	m−	PROPN
ejpam-5566	55	30	1	1	NUM
ejpam-5566	55	31	.	.	PUNCT
ejpam-5566	56	1	the	the	DET
ejpam-5566	56	2	results	result	NOUN
ejpam-5566	56	3	on	on	ADP
ejpam-5566	56	4	the	the	DET
ejpam-5566	56	5	notion	notion	NOUN
ejpam-5566	56	6	of	of	ADP
ejpam-5566	56	7	even	even	ADV
ejpam-5566	56	8	edge	edge	NOUN
ejpam-5566	56	9	space	space	NOUN
ejpam-5566	56	10	are	be	AUX
ejpam-5566	56	11	useful	useful	ADJ
ejpam-5566	56	12	in	in	ADP
ejpam-5566	56	13	this	this	DET
ejpam-5566	56	14	study	study	NOUN
ejpam-5566	56	15	.	.	PUNCT
ejpam-5566	57	1	several	several	ADJ
ejpam-5566	57	2	studies	study	NOUN
ejpam-5566	57	3	on	on	ADP
ejpam-5566	57	4	this	this	DET
ejpam-5566	57	5	problem	problem	NOUN
ejpam-5566	57	6	focuses	focus	VERB
ejpam-5566	57	7	on	on	ADP
ejpam-5566	57	8	the	the	DET
ejpam-5566	57	9	determination	determination	NOUN
ejpam-5566	57	10	of	of	ADP
ejpam-5566	57	11	the	the	DET
ejpam-5566	57	12	generator	generator	NOUN
ejpam-5566	57	13	subgraphs	subgraphs	NOUN
ejpam-5566	57	14	of	of	ADP
ejpam-5566	57	15	some	some	DET
ejpam-5566	57	16	common	common	ADJ
ejpam-5566	57	17	classes	class	NOUN
ejpam-5566	57	18	of	of	ADP
ejpam-5566	57	19	graphs	graph	NOUN
ejpam-5566	57	20	,	,	PUNCT
ejpam-5566	57	21	see	see	VERB
ejpam-5566	57	22	[	[	X
ejpam-5566	57	23	11	11	NUM
ejpam-5566	57	24	]	]	PUNCT
ejpam-5566	57	25	,	,	PUNCT
ejpam-5566	57	26	[	[	X
ejpam-5566	57	27	8	8	NUM
ejpam-5566	57	28	]	]	PUNCT
ejpam-5566	57	29	,	,	PUNCT
ejpam-5566	57	30	[	[	X
ejpam-5566	57	31	6	6	NUM
ejpam-5566	57	32	]	]	PUNCT
ejpam-5566	57	33	,	,	PUNCT
ejpam-5566	57	34	[	[	X
ejpam-5566	57	35	5	5	NUM
ejpam-5566	57	36	]	]	PUNCT
ejpam-5566	57	37	,	,	PUNCT
ejpam-5566	57	38	[	[	X
ejpam-5566	57	39	7	7	NUM
ejpam-5566	57	40	]	]	PUNCT
ejpam-5566	57	41	.	.	PUNCT
ejpam-5566	58	1	it	it	PRON
ejpam-5566	58	2	can	can	AUX
ejpam-5566	58	3	be	be	AUX
ejpam-5566	58	4	noted	note	VERB
ejpam-5566	58	5	that	that	SCONJ
ejpam-5566	58	6	among	among	ADP
ejpam-5566	58	7	the	the	DET
ejpam-5566	58	8	classes	class	NOUN
ejpam-5566	58	9	of	of	ADP
ejpam-5566	58	10	graphs	graph	NOUN
ejpam-5566	58	11	being	be	AUX
ejpam-5566	58	12	studied	study	VERB
ejpam-5566	58	13	,	,	PUNCT
ejpam-5566	58	14	only	only	ADV
ejpam-5566	58	15	the	the	DET
ejpam-5566	58	16	generator	generator	NOUN
ejpam-5566	58	17	subgraphs	subgraphs	NOUN
ejpam-5566	58	18	of	of	ADP
ejpam-5566	58	19	the	the	DET
ejpam-5566	58	20	complete	complete	ADJ
ejpam-5566	58	21	graph	graph	NOUN
ejpam-5566	58	22	and	and	CCONJ
ejpam-5566	58	23	star	star	NOUN
ejpam-5566	58	24	graph	graph	NOUN
ejpam-5566	58	25	were	be	AUX
ejpam-5566	58	26	completely	completely	ADV
ejpam-5566	58	27	known	know	VERB
ejpam-5566	58	28	.	.	PUNCT
ejpam-5566	59	1	one	one	NUM
ejpam-5566	59	2	significant	significant	ADJ
ejpam-5566	59	3	result	result	NOUN
ejpam-5566	59	4	on	on	ADP
ejpam-5566	59	5	this	this	DET
ejpam-5566	59	6	problem	problem	NOUN
ejpam-5566	59	7	was	be	AUX
ejpam-5566	59	8	a	a	DET
ejpam-5566	59	9	necessary	necessary	ADJ
ejpam-5566	59	10	condition	condition	NOUN
ejpam-5566	59	11	that	that	SCONJ
ejpam-5566	59	12	the	the	DET
ejpam-5566	59	13	size	size	NOUN
ejpam-5566	59	14	of	of	ADP
ejpam-5566	59	15	a	a	DET
ejpam-5566	59	16	subgraph	subgraph	NOUN
ejpam-5566	59	17	h	h	NOUN
ejpam-5566	59	18	of	of	ADP
ejpam-5566	59	19	g	g	PROPN
ejpam-5566	59	20	must	must	AUX
ejpam-5566	59	21	be	be	AUX
ejpam-5566	59	22	odd	odd	ADJ
ejpam-5566	59	23	,	,	PUNCT
ejpam-5566	59	24	[	[	X
ejpam-5566	59	25	4	4	NUM
ejpam-5566	59	26	]	]	PUNCT
ejpam-5566	59	27	.	.	PUNCT
ejpam-5566	60	1	hence	hence	ADV
ejpam-5566	60	2	,	,	PUNCT
ejpam-5566	60	3	in	in	ADP
ejpam-5566	60	4	finding	find	VERB
ejpam-5566	60	5	the	the	DET
ejpam-5566	60	6	generator	generator	NOUN
ejpam-5566	60	7	subgraph	subgraph	NOUN
ejpam-5566	60	8	of	of	ADP
ejpam-5566	60	9	a	a	DET
ejpam-5566	60	10	graph	graph	NOUN
ejpam-5566	60	11	,	,	PUNCT
ejpam-5566	60	12	we	we	PRON
ejpam-5566	60	13	consider	consider	VERB
ejpam-5566	60	14	only	only	ADV
ejpam-5566	60	15	those	those	DET
ejpam-5566	60	16	subgraphs	subgraph	NOUN
ejpam-5566	60	17	with	with	ADP
ejpam-5566	60	18	odd	odd	ADJ
ejpam-5566	60	19	sizes	size	NOUN
ejpam-5566	60	20	.	.	PUNCT
ejpam-5566	61	1	equivalently	equivalently	ADV
ejpam-5566	61	2	,	,	PUNCT
ejpam-5566	61	3	we	we	PRON
ejpam-5566	61	4	have	have	VERB
ejpam-5566	61	5	the	the	DET
ejpam-5566	61	6	following	follow	VERB
ejpam-5566	61	7	theorem	theorem	VERB
ejpam-5566	61	8	.	.	PUNCT
ejpam-5566	61	9	theorem	theorem	NOUN
ejpam-5566	61	10	1	1	NUM
ejpam-5566	61	11	(	(	PUNCT
ejpam-5566	61	12	[	[	X
ejpam-5566	61	13	4	4	NUM
ejpam-5566	61	14	]	]	NUM
ejpam-5566	61	15	)	)	PUNCT
ejpam-5566	61	16	.	.	PUNCT
ejpam-5566	62	1	let	let	VERB
ejpam-5566	62	2	h	h	PRON
ejpam-5566	62	3	be	be	AUX
ejpam-5566	62	4	a	a	DET
ejpam-5566	62	5	subgraph	subgraph	NOUN
ejpam-5566	62	6	of	of	ADP
ejpam-5566	62	7	the	the	DET
ejpam-5566	62	8	graph	graph	NOUN
ejpam-5566	62	9	g.	g.	NOUN
ejpam-5566	62	10	if	if	SCONJ
ejpam-5566	62	11	h	h	NOUN
ejpam-5566	62	12	is	be	AUX
ejpam-5566	62	13	a	a	DET
ejpam-5566	62	14	generator	generator	NOUN
ejpam-5566	62	15	subgraph	subgraph	NOUN
ejpam-5566	62	16	of	of	ADP
ejpam-5566	62	17	g	g	PROPN
ejpam-5566	62	18	,	,	PUNCT
ejpam-5566	62	19	then	then	ADV
ejpam-5566	62	20	|e(h)|	|e(h)|	PROPN
ejpam-5566	62	21	is	be	AUX
ejpam-5566	62	22	odd	odd	ADJ
ejpam-5566	62	23	.	.	PUNCT
ejpam-5566	63	1	by	by	ADP
ejpam-5566	63	2	a	a	DET
ejpam-5566	63	3	graph	graph	NOUN
ejpam-5566	63	4	g	g	NOUN
ejpam-5566	63	5	,	,	PUNCT
ejpam-5566	63	6	we	we	PRON
ejpam-5566	63	7	mean	mean	VERB
ejpam-5566	63	8	an	an	DET
ejpam-5566	63	9	ordered	order	VERB
ejpam-5566	63	10	pair	pair	NOUN
ejpam-5566	63	11	(	(	PUNCT
ejpam-5566	63	12	v	v	NOUN
ejpam-5566	63	13	(	(	PUNCT
ejpam-5566	63	14	g	g	NOUN
ejpam-5566	63	15	)	)	PUNCT
ejpam-5566	63	16	,	,	PUNCT
ejpam-5566	63	17	e(g	e(g	PROPN
ejpam-5566	63	18	)	)	PUNCT
ejpam-5566	63	19	)	)	PUNCT
ejpam-5566	63	20	,	,	PUNCT
ejpam-5566	63	21	where	where	SCONJ
ejpam-5566	63	22	v	v	X
ejpam-5566	63	23	(	(	PUNCT
ejpam-5566	63	24	g	g	NOUN
ejpam-5566	63	25	)	)	PUNCT
ejpam-5566	63	26	is	be	AUX
ejpam-5566	63	27	a	a	DET
ejpam-5566	63	28	finite	finite	NOUN
ejpam-5566	63	29	nonempty	nonempty	NOUN
ejpam-5566	63	30	set	set	NOUN
ejpam-5566	63	31	of	of	ADP
ejpam-5566	63	32	elements	element	NOUN
ejpam-5566	63	33	called	call	VERB
ejpam-5566	63	34	vertices	vertex	NOUN
ejpam-5566	63	35	and	and	CCONJ
ejpam-5566	63	36	e(g	e(g	PROPN
ejpam-5566	63	37	)	)	PUNCT
ejpam-5566	63	38	is	be	AUX
ejpam-5566	63	39	a	a	DET
ejpam-5566	63	40	set	set	NOUN
ejpam-5566	63	41	of	of	ADP
ejpam-5566	63	42	2−	2−	NUM
ejpam-5566	63	43	subset	subset	NOUN
ejpam-5566	63	44	of	of	ADP
ejpam-5566	63	45	v	v	NOUN
ejpam-5566	63	46	(	(	PUNCT
ejpam-5566	63	47	g	g	NOUN
ejpam-5566	63	48	)	)	PUNCT
ejpam-5566	63	49	whose	whose	DET
ejpam-5566	63	50	elements	element	NOUN
ejpam-5566	63	51	are	be	AUX
ejpam-5566	63	52	called	call	VERB
ejpam-5566	63	53	edges	edge	NOUN
ejpam-5566	63	54	.	.	PUNCT
ejpam-5566	64	1	the	the	DET
ejpam-5566	64	2	sets	set	NOUN
ejpam-5566	64	3	v	v	ADP
ejpam-5566	64	4	(	(	PUNCT
ejpam-5566	64	5	g	g	NOUN
ejpam-5566	64	6	)	)	PUNCT
ejpam-5566	64	7	and	and	CCONJ
ejpam-5566	64	8	e(g	e(g	PROPN
ejpam-5566	64	9	)	)	PUNCT
ejpam-5566	64	10	are	be	AUX
ejpam-5566	64	11	called	call	VERB
ejpam-5566	64	12	the	the	DET
ejpam-5566	64	13	vertex	vertex	NOUN
ejpam-5566	64	14	set	set	NOUN
ejpam-5566	64	15	and	and	CCONJ
ejpam-5566	64	16	edge	edge	NOUN
ejpam-5566	64	17	set	set	NOUN
ejpam-5566	64	18	of	of	ADP
ejpam-5566	64	19	g	g	NOUN
ejpam-5566	64	20	,	,	PUNCT
ejpam-5566	64	21	respectively	respectively	ADV
ejpam-5566	64	22	.	.	PUNCT
ejpam-5566	65	1	the	the	DET
ejpam-5566	65	2	order	order	NOUN
ejpam-5566	65	3	of	of	ADP
ejpam-5566	65	4	g	g	PROPN
ejpam-5566	65	5	is	be	AUX
ejpam-5566	65	6	the	the	DET
ejpam-5566	65	7	cardinality	cardinality	NOUN
ejpam-5566	65	8	of	of	ADP
ejpam-5566	65	9	v	v	NOUN
ejpam-5566	65	10	(	(	PUNCT
ejpam-5566	65	11	g	g	NOUN
ejpam-5566	65	12	)	)	PUNCT
ejpam-5566	65	13	,	,	PUNCT
ejpam-5566	65	14	denoted	denote	VERB
ejpam-5566	65	15	by	by	ADP
ejpam-5566	65	16	|v	|v	PROPN
ejpam-5566	65	17	(	(	PUNCT
ejpam-5566	65	18	g)|	g)|	NOUN
ejpam-5566	65	19	,	,	PUNCT
ejpam-5566	65	20	and	and	CCONJ
ejpam-5566	65	21	the	the	DET
ejpam-5566	65	22	size	size	NOUN
ejpam-5566	65	23	of	of	ADP
ejpam-5566	65	24	g	g	PROPN
ejpam-5566	65	25	is	be	AUX
ejpam-5566	65	26	the	the	DET
ejpam-5566	65	27	cardinality	cardinality	NOUN
ejpam-5566	65	28	of	of	ADP
ejpam-5566	65	29	e(g	e(g	PROPN
ejpam-5566	65	30	)	)	PUNCT
ejpam-5566	65	31	,	,	PUNCT
ejpam-5566	65	32	denoted	denote	VERB
ejpam-5566	65	33	by	by	ADP
ejpam-5566	65	34	|e(g)|	|e(g)|	PROPN
ejpam-5566	65	35	.	.	PUNCT
ejpam-5566	66	1	if	if	SCONJ
ejpam-5566	66	2	[	[	X
ejpam-5566	66	3	x	x	X
ejpam-5566	66	4	,	,	PUNCT
ejpam-5566	66	5	y	y	PROPN
ejpam-5566	66	6	]	]	X
ejpam-5566	66	7	∈	∈	PROPN
ejpam-5566	66	8	e(g	e(g	PROPN
ejpam-5566	66	9	)	)	PUNCT
ejpam-5566	66	10	,	,	PUNCT
ejpam-5566	66	11	we	we	PRON
ejpam-5566	66	12	say	say	VERB
ejpam-5566	66	13	that	that	SCONJ
ejpam-5566	66	14	x	x	PRON
ejpam-5566	66	15	is	be	AUX
ejpam-5566	66	16	adjacent	adjacent	ADJ
ejpam-5566	66	17	to	to	ADP
ejpam-5566	66	18	y	y	PROPN
ejpam-5566	66	19	or	or	CCONJ
ejpam-5566	66	20	y	y	PROPN
ejpam-5566	66	21	is	be	AUX
ejpam-5566	66	22	adjacent	adjacent	ADJ
ejpam-5566	66	23	to	to	PART
ejpam-5566	66	24	x.	x.	VERB
ejpam-5566	66	25	for	for	ADP
ejpam-5566	66	26	the	the	DET
ejpam-5566	66	27	two	two	NUM
ejpam-5566	66	28	graphs	graph	NOUN
ejpam-5566	66	29	g	g	NOUN
ejpam-5566	66	30	and	and	CCONJ
ejpam-5566	66	31	h	h	NOUN
ejpam-5566	66	32	,	,	PUNCT
ejpam-5566	66	33	by	by	ADP
ejpam-5566	66	34	g	g	PROPN
ejpam-5566	66	35	≃	≃	PROPN
ejpam-5566	66	36	h	h	NOUN
ejpam-5566	66	37	,	,	PUNCT
ejpam-5566	66	38	we	we	PRON
ejpam-5566	66	39	mean	mean	VERB
ejpam-5566	66	40	g	g	PROPN
ejpam-5566	66	41	is	be	AUX
ejpam-5566	66	42	isomorphic	isomorphic	ADJ
ejpam-5566	66	43	to	to	ADP
ejpam-5566	66	44	h.	h.	PROPN
ejpam-5566	66	45	a	a	DET
ejpam-5566	66	46	vertex	vertex	NOUN
ejpam-5566	66	47	in	in	ADP
ejpam-5566	66	48	a	a	DET
ejpam-5566	66	49	graph	graph	NOUN
ejpam-5566	66	50	with	with	ADP
ejpam-5566	66	51	degree	degree	NOUN
ejpam-5566	66	52	1	1	NUM
ejpam-5566	66	53	is	be	AUX
ejpam-5566	66	54	called	call	VERB
ejpam-5566	66	55	a	a	DET
ejpam-5566	66	56	pendant	pendant	ADJ
ejpam-5566	66	57	vertex	vertex	NOUN
ejpam-5566	66	58	while	while	SCONJ
ejpam-5566	66	59	an	an	DET
ejpam-5566	66	60	edge	edge	NOUN
ejpam-5566	66	61	of	of	ADP
ejpam-5566	66	62	the	the	DET
ejpam-5566	66	63	graph	graph	NOUN
ejpam-5566	66	64	incident	incident	NOUN
ejpam-5566	66	65	to	to	ADP
ejpam-5566	66	66	a	a	DET
ejpam-5566	66	67	pendant	pendant	ADJ
ejpam-5566	66	68	vertex	vertex	NOUN
ejpam-5566	66	69	is	be	AUX
ejpam-5566	66	70	called	call	VERB
ejpam-5566	66	71	pendant	pendant	ADJ
ejpam-5566	66	72	edge	edge	NOUN
ejpam-5566	66	73	.	.	PUNCT
ejpam-5566	67	1	we	we	PRON
ejpam-5566	67	2	used	use	VERB
ejpam-5566	67	3	the	the	DET
ejpam-5566	67	4	usual	usual	ADJ
ejpam-5566	67	5	notations	notation	NOUN
ejpam-5566	67	6	for	for	ADP
ejpam-5566	67	7	some	some	DET
ejpam-5566	67	8	special	special	ADJ
ejpam-5566	67	9	classes	class	NOUN
ejpam-5566	67	10	of	of	ADP
ejpam-5566	67	11	graphs	graph	NOUN
ejpam-5566	67	12	,	,	PUNCT
ejpam-5566	67	13	kn	kn	PROPN
ejpam-5566	67	14	for	for	ADP
ejpam-5566	67	15	complete	complete	ADJ
ejpam-5566	67	16	graph	graph	NOUN
ejpam-5566	67	17	of	of	ADP
ejpam-5566	67	18	order	order	NOUN
ejpam-5566	67	19	n	n	CCONJ
ejpam-5566	67	20	,	,	PUNCT
ejpam-5566	67	21	pn	pn	PROPN
ejpam-5566	67	22	for	for	ADP
ejpam-5566	67	23	r.	r.	PROPN
ejpam-5566	67	24	mame	mame	PROPN
ejpam-5566	67	25	/	/	SYM
ejpam-5566	67	26	eur	eur	PROPN
ejpam-5566	67	27	.	.	PUNCT
ejpam-5566	68	1	j.	j.	PROPN
ejpam-5566	68	2	pure	pure	PROPN
ejpam-5566	68	3	appl	appl	PROPN
ejpam-5566	68	4	.	.	PROPN
ejpam-5566	68	5	math	math	PROPN
ejpam-5566	68	6	,	,	PUNCT
ejpam-5566	68	7	17	17	NUM
ejpam-5566	68	8	(	(	PUNCT
ejpam-5566	68	9	4	4	NUM
ejpam-5566	68	10	)	)	PUNCT
ejpam-5566	68	11	(	(	PUNCT
ejpam-5566	68	12	2024	2024	NUM
ejpam-5566	68	13	)	)	PUNCT
ejpam-5566	68	14	,	,	PUNCT
ejpam-5566	68	15	3815	3815	NUM
ejpam-5566	68	16	-	-	SYM
ejpam-5566	68	17	3825	3825	NUM
ejpam-5566	68	18	3817	3817	NUM
ejpam-5566	68	19	path	path	NOUN
ejpam-5566	68	20	of	of	ADP
ejpam-5566	68	21	order	order	NOUN
ejpam-5566	68	22	n	n	CCONJ
ejpam-5566	68	23	,	,	PUNCT
ejpam-5566	68	24	and	and	CCONJ
ejpam-5566	68	25	sn	sn	PROPN
ejpam-5566	68	26	for	for	ADP
ejpam-5566	68	27	star	star	NOUN
ejpam-5566	68	28	graph	graph	NOUN
ejpam-5566	68	29	of	of	ADP
ejpam-5566	68	30	order	order	NOUN
ejpam-5566	68	31	n+1	n+1	NOUN
ejpam-5566	68	32	.	.	PUNCT
ejpam-5566	69	1	some	some	DET
ejpam-5566	69	2	other	other	ADJ
ejpam-5566	69	3	classes	class	NOUN
ejpam-5566	69	4	of	of	ADP
ejpam-5566	69	5	graphs	graph	NOUN
ejpam-5566	69	6	,	,	PUNCT
ejpam-5566	69	7	which	which	PRON
ejpam-5566	69	8	were	be	AUX
ejpam-5566	69	9	identified	identify	VERB
ejpam-5566	69	10	to	to	PART
ejpam-5566	69	11	be	be	AUX
ejpam-5566	69	12	a	a	DET
ejpam-5566	69	13	generator	generator	NOUN
ejpam-5566	69	14	subgraphs	subgraphs	NOUN
ejpam-5566	69	15	of	of	ADP
ejpam-5566	69	16	the	the	DET
ejpam-5566	69	17	square	square	NOUN
ejpam-5566	69	18	of	of	ADP
ejpam-5566	69	19	a	a	DET
ejpam-5566	69	20	cycle	cycle	NOUN
ejpam-5566	69	21	,	,	PUNCT
ejpam-5566	69	22	are	be	AUX
ejpam-5566	69	23	defined	define	VERB
ejpam-5566	69	24	in	in	ADP
ejpam-5566	69	25	the	the	DET
ejpam-5566	69	26	appropriate	appropriate	ADJ
ejpam-5566	69	27	section	section	NOUN
ejpam-5566	69	28	of	of	ADP
ejpam-5566	69	29	this	this	DET
ejpam-5566	69	30	paper	paper	NOUN
ejpam-5566	69	31	.	.	PUNCT
ejpam-5566	70	1	for	for	ADP
ejpam-5566	70	2	other	other	ADJ
ejpam-5566	70	3	basic	basic	ADJ
ejpam-5566	70	4	concepts	concept	NOUN
ejpam-5566	70	5	in	in	ADP
ejpam-5566	70	6	graph	graph	NOUN
ejpam-5566	70	7	theory	theory	NOUN
ejpam-5566	70	8	,	,	PUNCT
ejpam-5566	70	9	readers	reader	NOUN
ejpam-5566	70	10	may	may	AUX
ejpam-5566	70	11	refer	refer	VERB
ejpam-5566	70	12	to	to	ADP
ejpam-5566	70	13	the	the	DET
ejpam-5566	70	14	book	book	NOUN
ejpam-5566	70	15	written	write	VERB
ejpam-5566	70	16	by	by	ADP
ejpam-5566	70	17	chartrand	chartrand	PROPN
ejpam-5566	70	18	&	&	CCONJ
ejpam-5566	70	19	zhang	zhang	PROPN
ejpam-5566	71	1	[	[	X
ejpam-5566	71	2	3	3	NUM
ejpam-5566	71	3	]	]	PUNCT
ejpam-5566	71	4	.	.	PUNCT
ejpam-5566	72	1	for	for	ADP
ejpam-5566	72	2	the	the	DET
ejpam-5566	72	3	algebra	algebra	NOUN
ejpam-5566	72	4	concepts	concept	NOUN
ejpam-5566	72	5	,	,	PUNCT
ejpam-5566	72	6	particularly	particularly	ADV
ejpam-5566	72	7	vector	vector	NOUN
ejpam-5566	72	8	spaces	space	NOUN
ejpam-5566	72	9	and	and	CCONJ
ejpam-5566	72	10	some	some	PRON
ejpam-5566	72	11	of	of	ADP
ejpam-5566	72	12	its	its	PRON
ejpam-5566	72	13	properties	property	NOUN
ejpam-5566	72	14	,	,	PUNCT
ejpam-5566	72	15	readers	reader	NOUN
ejpam-5566	72	16	may	may	AUX
ejpam-5566	72	17	refer	refer	VERB
ejpam-5566	72	18	to	to	ADP
ejpam-5566	72	19	the	the	DET
ejpam-5566	72	20	book	book	NOUN
ejpam-5566	72	21	written	write	VERB
ejpam-5566	72	22	by	by	ADP
ejpam-5566	72	23	e.d	e.d	PROPN
ejpam-5566	72	24	.	.	PROPN
ejpam-5566	72	25	nering	nere	VERB
ejpam-5566	73	1	[	[	X
ejpam-5566	73	2	9	9	NUM
ejpam-5566	73	3	]	]	PUNCT
ejpam-5566	73	4	.	.	PUNCT
ejpam-5566	74	1	let	let	VERB
ejpam-5566	74	2	x	x	PRON
ejpam-5566	74	3	,	,	PUNCT
ejpam-5566	74	4	y	y	PROPN
ejpam-5566	74	5	∈	∈	PROPN
ejpam-5566	74	6	v	v	NOUN
ejpam-5566	74	7	(	(	PUNCT
ejpam-5566	74	8	g	g	NOUN
ejpam-5566	74	9	)	)	PUNCT
ejpam-5566	74	10	.	.	PUNCT
ejpam-5566	75	1	the	the	DET
ejpam-5566	75	2	distance	distance	NOUN
ejpam-5566	75	3	between	between	ADP
ejpam-5566	75	4	x	x	PROPN
ejpam-5566	75	5	and	and	CCONJ
ejpam-5566	75	6	y	y	PROPN
ejpam-5566	75	7	,	,	PUNCT
ejpam-5566	75	8	denoted	denote	VERB
ejpam-5566	75	9	by	by	ADP
ejpam-5566	75	10	d(x	d(x	PROPN
ejpam-5566	75	11	,	,	PUNCT
ejpam-5566	75	12	y	y	PROPN
ejpam-5566	75	13	)	)	PUNCT
ejpam-5566	75	14	,	,	PUNCT
ejpam-5566	75	15	is	be	AUX
ejpam-5566	75	16	the	the	DET
ejpam-5566	75	17	length	length	NOUN
ejpam-5566	75	18	of	of	ADP
ejpam-5566	75	19	the	the	DET
ejpam-5566	75	20	shortest	short	ADJ
ejpam-5566	75	21	x	x	PUNCT
ejpam-5566	75	22	−	−	PROPN
ejpam-5566	75	23	y	y	PROPN
ejpam-5566	75	24	path	path	NOUN
ejpam-5566	75	25	.	.	PUNCT
ejpam-5566	76	1	let	let	VERB
ejpam-5566	76	2	cn	cn	PROPN
ejpam-5566	76	3	be	be	AUX
ejpam-5566	76	4	a	a	DET
ejpam-5566	76	5	cycle	cycle	NOUN
ejpam-5566	76	6	of	of	ADP
ejpam-5566	76	7	length	length	NOUN
ejpam-5566	76	8	n.	n.	PROPN
ejpam-5566	76	9	the	the	DET
ejpam-5566	76	10	square	square	NOUN
ejpam-5566	76	11	of	of	ADP
ejpam-5566	76	12	the	the	DET
ejpam-5566	76	13	cycle	cycle	NOUN
ejpam-5566	76	14	cn	cn	PROPN
ejpam-5566	76	15	,	,	PUNCT
ejpam-5566	76	16	denoted	denote	VERB
ejpam-5566	76	17	by	by	ADP
ejpam-5566	76	18	c2	c2	PROPN
ejpam-5566	76	19	n	n	CCONJ
ejpam-5566	76	20	,	,	PUNCT
ejpam-5566	76	21	is	be	AUX
ejpam-5566	76	22	the	the	DET
ejpam-5566	76	23	graph	graph	NOUN
ejpam-5566	76	24	obtained	obtain	VERB
ejpam-5566	76	25	from	from	ADP
ejpam-5566	76	26	cn	cn	PROPN
ejpam-5566	76	27	by	by	ADP
ejpam-5566	76	28	adding	add	VERB
ejpam-5566	76	29	the	the	DET
ejpam-5566	76	30	edge	edge	NOUN
ejpam-5566	76	31	[	[	X
ejpam-5566	76	32	x	x	X
ejpam-5566	76	33	,	,	PUNCT
ejpam-5566	76	34	y	y	PROPN
ejpam-5566	76	35	]	]	PUNCT
ejpam-5566	76	36	to	to	ADP
ejpam-5566	76	37	the	the	DET
ejpam-5566	76	38	cycle	cycle	NOUN
ejpam-5566	77	1	cn	cn	PROPN
ejpam-5566	77	2	if	if	SCONJ
ejpam-5566	77	3	and	and	CCONJ
ejpam-5566	77	4	only	only	ADV
ejpam-5566	77	5	if	if	SCONJ
ejpam-5566	77	6	d(x	d(x	PROPN
ejpam-5566	77	7	,	,	PUNCT
ejpam-5566	77	8	y	y	NOUN
ejpam-5566	77	9	)	)	PUNCT
ejpam-5566	77	10	=	=	SYM
ejpam-5566	77	11	2	2	X
ejpam-5566	77	12	.	.	PUNCT
ejpam-5566	77	13	examples	example	NOUN
ejpam-5566	77	14	of	of	ADP
ejpam-5566	77	15	square	square	NOUN
ejpam-5566	77	16	of	of	ADP
ejpam-5566	77	17	some	some	DET
ejpam-5566	77	18	cycle	cycle	NOUN
ejpam-5566	77	19	graphs	graph	NOUN
ejpam-5566	77	20	cn	cn	PROPN
ejpam-5566	77	21	are	be	AUX
ejpam-5566	77	22	given	give	VERB
ejpam-5566	77	23	in	in	ADP
ejpam-5566	77	24	figure	figure	NOUN
ejpam-5566	77	25	1	1	NUM
ejpam-5566	77	26	.	.	PUNCT
ejpam-5566	77	27	figure	figure	NOUN
ejpam-5566	77	28	1	1	NUM
ejpam-5566	77	29	:	:	PUNCT
ejpam-5566	77	30	illustrating	illustrate	VERB
ejpam-5566	77	31	the	the	DET
ejpam-5566	77	32	square	square	NOUN
ejpam-5566	77	33	of	of	ADP
ejpam-5566	77	34	different	different	ADJ
ejpam-5566	77	35	cycle	cycle	NOUN
ejpam-5566	77	36	graphs	graph	NOUN
ejpam-5566	77	37	this	this	DET
ejpam-5566	77	38	study	study	NOUN
ejpam-5566	77	39	focuses	focus	VERB
ejpam-5566	77	40	on	on	ADP
ejpam-5566	77	41	determining	determine	VERB
ejpam-5566	77	42	the	the	DET
ejpam-5566	77	43	generator	generator	NOUN
ejpam-5566	77	44	subgraphs	subgraphs	NOUN
ejpam-5566	77	45	of	of	ADP
ejpam-5566	77	46	the	the	DET
ejpam-5566	77	47	square	square	NOUN
ejpam-5566	77	48	of	of	ADP
ejpam-5566	77	49	the	the	DET
ejpam-5566	77	50	cycle	cycle	NOUN
ejpam-5566	77	51	.	.	PUNCT
ejpam-5566	78	1	at	at	ADP
ejpam-5566	78	2	first	first	ADV
ejpam-5566	78	3	,	,	PUNCT
ejpam-5566	78	4	we	we	PRON
ejpam-5566	78	5	provide	provide	VERB
ejpam-5566	78	6	the	the	DET
ejpam-5566	78	7	fixed	fix	VERB
ejpam-5566	78	8	labeling	labeling	NOUN
ejpam-5566	78	9	of	of	ADP
ejpam-5566	78	10	the	the	DET
ejpam-5566	78	11	square	square	NOUN
ejpam-5566	78	12	of	of	ADP
ejpam-5566	78	13	a	a	DET
ejpam-5566	78	14	cycle	cycle	NOUN
ejpam-5566	78	15	and	and	CCONJ
ejpam-5566	78	16	define	define	VERB
ejpam-5566	78	17	the	the	DET
ejpam-5566	78	18	edges	edge	NOUN
ejpam-5566	78	19	in	in	ADP
ejpam-5566	78	20	terms	term	NOUN
ejpam-5566	78	21	of	of	ADP
ejpam-5566	78	22	its	its	PRON
ejpam-5566	78	23	vertices	vertex	NOUN
ejpam-5566	78	24	.	.	PUNCT
ejpam-5566	79	1	then	then	ADV
ejpam-5566	79	2	,	,	PUNCT
ejpam-5566	79	3	use	use	VERB
ejpam-5566	79	4	some	some	DET
ejpam-5566	79	5	properties	property	NOUN
ejpam-5566	79	6	of	of	ADP
ejpam-5566	79	7	the	the	DET
ejpam-5566	79	8	square	square	NOUN
ejpam-5566	79	9	of	of	ADP
ejpam-5566	79	10	a	a	DET
ejpam-5566	79	11	cycle	cycle	NOUN
ejpam-5566	79	12	,	,	PUNCT
ejpam-5566	79	13	such	such	ADJ
ejpam-5566	79	14	as	as	ADP
ejpam-5566	79	15	rotational	rotational	ADJ
ejpam-5566	79	16	symmetry	symmetry	NOUN
ejpam-5566	79	17	to	to	PART
ejpam-5566	79	18	determine	determine	VERB
ejpam-5566	79	19	its	its	PRON
ejpam-5566	79	20	generator	generator	NOUN
ejpam-5566	79	21	subgraphs	subgraph	NOUN
ejpam-5566	79	22	.	.	PUNCT
ejpam-5566	80	1	finally	finally	ADV
ejpam-5566	80	2	,	,	PUNCT
ejpam-5566	80	3	some	some	DET
ejpam-5566	80	4	classes	class	NOUN
ejpam-5566	80	5	of	of	ADP
ejpam-5566	80	6	graphs	graph	NOUN
ejpam-5566	80	7	were	be	AUX
ejpam-5566	80	8	found	find	VERB
ejpam-5566	80	9	to	to	PART
ejpam-5566	80	10	be	be	AUX
ejpam-5566	80	11	generator	generator	NOUN
ejpam-5566	80	12	subgraphs	subgraph	NOUN
ejpam-5566	80	13	of	of	ADP
ejpam-5566	80	14	the	the	DET
ejpam-5566	80	15	square	square	NOUN
ejpam-5566	80	16	of	of	ADP
ejpam-5566	80	17	the	the	DET
ejpam-5566	80	18	cycle	cycle	NOUN
ejpam-5566	80	19	.	.	PUNCT
ejpam-5566	81	1	in	in	ADP
ejpam-5566	81	2	determining	determine	VERB
ejpam-5566	81	3	the	the	DET
ejpam-5566	81	4	dimension	dimension	NOUN
ejpam-5566	81	5	of	of	ADP
ejpam-5566	81	6	the	the	DET
ejpam-5566	81	7	edge	edge	NOUN
ejpam-5566	81	8	space	space	NOUN
ejpam-5566	81	9	of	of	ADP
ejpam-5566	81	10	the	the	DET
ejpam-5566	81	11	square	square	NOUN
ejpam-5566	81	12	of	of	ADP
ejpam-5566	81	13	a	a	DET
ejpam-5566	81	14	cycle	cycle	NOUN
ejpam-5566	81	15	graph	graph	NOUN
ejpam-5566	81	16	,	,	PUNCT
ejpam-5566	81	17	we	we	PRON
ejpam-5566	81	18	utilize	utilize	VERB
ejpam-5566	81	19	the	the	DET
ejpam-5566	81	20	following	follow	VERB
ejpam-5566	81	21	theorem	theorem	NOUN
ejpam-5566	81	22	,	,	PUNCT
ejpam-5566	81	23	a	a	DET
ejpam-5566	81	24	well	well	ADV
ejpam-5566	81	25	-	-	PUNCT
ejpam-5566	81	26	known	know	VERB
ejpam-5566	81	27	theorem	theorem	NOUN
ejpam-5566	81	28	in	in	ADP
ejpam-5566	81	29	graph	graph	NOUN
ejpam-5566	81	30	theory	theory	NOUN
ejpam-5566	81	31	.	.	PUNCT
ejpam-5566	82	1	theorem	theorem	NOUN
ejpam-5566	82	2	2	2	NUM
ejpam-5566	82	3	.	.	PUNCT
ejpam-5566	83	1	if	if	SCONJ
ejpam-5566	83	2	g	g	PROPN
ejpam-5566	83	3	is	be	AUX
ejpam-5566	83	4	a	a	DET
ejpam-5566	83	5	graph	graph	NOUN
ejpam-5566	83	6	of	of	ADP
ejpam-5566	83	7	size	size	NOUN
ejpam-5566	83	8	m	m	PROPN
ejpam-5566	83	9	,	,	PUNCT
ejpam-5566	83	10	then∑	then∑	NOUN
ejpam-5566	83	11	v∈v	v∈v	NOUN
ejpam-5566	83	12	(	(	PUNCT
ejpam-5566	83	13	g	g	NOUN
ejpam-5566	83	14	)	)	PUNCT
ejpam-5566	83	15	deg(v	deg(v	PROPN
ejpam-5566	83	16	)	)	PUNCT
ejpam-5566	83	17	=	=	SYM
ejpam-5566	83	18	2	2	NUM
ejpam-5566	83	19	m.	m.	NOUN
ejpam-5566	83	20	1.1	1.1	NUM
ejpam-5566	83	21	.	.	PUNCT
ejpam-5566	84	1	some	some	DET
ejpam-5566	84	2	other	other	ADJ
ejpam-5566	84	3	known	know	VERB
ejpam-5566	84	4	results	result	NOUN
ejpam-5566	84	5	on	on	ADP
ejpam-5566	84	6	the	the	DET
ejpam-5566	84	7	generator	generator	NOUN
ejpam-5566	84	8	subgraph	subgraph	NOUN
ejpam-5566	84	9	of	of	ADP
ejpam-5566	84	10	a	a	DET
ejpam-5566	84	11	graph	graph	NOUN
ejpam-5566	84	12	this	this	DET
ejpam-5566	84	13	section	section	NOUN
ejpam-5566	84	14	provides	provide	VERB
ejpam-5566	84	15	some	some	DET
ejpam-5566	84	16	other	other	ADJ
ejpam-5566	84	17	results	result	NOUN
ejpam-5566	84	18	on	on	ADP
ejpam-5566	84	19	the	the	DET
ejpam-5566	84	20	generator	generator	NOUN
ejpam-5566	84	21	subgraph	subgraph	NOUN
ejpam-5566	84	22	of	of	ADP
ejpam-5566	84	23	a	a	DET
ejpam-5566	84	24	graph	graph	NOUN
ejpam-5566	84	25	.	.	PUNCT
ejpam-5566	85	1	these	these	DET
ejpam-5566	85	2	results	result	NOUN
ejpam-5566	85	3	are	be	AUX
ejpam-5566	85	4	useful	useful	ADJ
ejpam-5566	85	5	in	in	ADP
ejpam-5566	85	6	proving	prove	VERB
ejpam-5566	85	7	results	result	NOUN
ejpam-5566	85	8	of	of	ADP
ejpam-5566	85	9	this	this	DET
ejpam-5566	85	10	study	study	NOUN
ejpam-5566	85	11	.	.	PUNCT
ejpam-5566	86	1	the	the	DET
ejpam-5566	86	2	remaining	remain	VERB
ejpam-5566	86	3	theorems	theorem	NOUN
ejpam-5566	86	4	can	can	AUX
ejpam-5566	86	5	be	be	AUX
ejpam-5566	86	6	found	find	VERB
ejpam-5566	86	7	in	in	ADP
ejpam-5566	86	8	[	[	X
ejpam-5566	86	9	8	8	NUM
ejpam-5566	86	10	]	]	PUNCT
ejpam-5566	86	11	.	.	PUNCT
ejpam-5566	87	1	the	the	DET
ejpam-5566	87	2	first	first	ADJ
ejpam-5566	87	3	theorem	theorem	ADJ
ejpam-5566	87	4	states	state	NOUN
ejpam-5566	87	5	that	that	SCONJ
ejpam-5566	87	6	the	the	DET
ejpam-5566	87	7	path	path	NOUN
ejpam-5566	87	8	p2	p2	PROPN
ejpam-5566	87	9	is	be	AUX
ejpam-5566	87	10	a	a	DET
ejpam-5566	87	11	generator	generator	NOUN
ejpam-5566	87	12	subgraph	subgraph	NOUN
ejpam-5566	87	13	of	of	ADP
ejpam-5566	87	14	a	a	DET
ejpam-5566	87	15	nonempty	nonempty	ADJ
ejpam-5566	87	16	graph	graph	NOUN
ejpam-5566	87	17	g.	g.	NOUN
ejpam-5566	87	18	theorem	theorem	NOUN
ejpam-5566	87	19	3	3	X
ejpam-5566	87	20	.	.	PUNCT
ejpam-5566	88	1	let	let	VERB
ejpam-5566	88	2	g	g	PRON
ejpam-5566	88	3	be	be	AUX
ejpam-5566	88	4	a	a	DET
ejpam-5566	88	5	graph	graph	NOUN
ejpam-5566	88	6	with	with	ADP
ejpam-5566	88	7	|e(g)|	|e(g)|	PROPN
ejpam-5566	88	8	=	=	NOUN
ejpam-5566	88	9	m	m	PROPN
ejpam-5566	88	10	>	>	X
ejpam-5566	89	1	0	0	X
ejpam-5566	89	2	.	.	PUNCT
ejpam-5566	90	1	then	then	ADV
ejpam-5566	90	2	the	the	DET
ejpam-5566	90	3	path	path	NOUN
ejpam-5566	90	4	p2	p2	PROPN
ejpam-5566	90	5	is	be	AUX
ejpam-5566	90	6	a	a	DET
ejpam-5566	90	7	generator	generator	NOUN
ejpam-5566	90	8	subgraph	subgraph	NOUN
ejpam-5566	90	9	of	of	ADP
ejpam-5566	90	10	g.	g.	PROPN
ejpam-5566	90	11	in	in	ADP
ejpam-5566	90	12	[	[	X
ejpam-5566	90	13	8	8	NUM
ejpam-5566	90	14	]	]	PUNCT
ejpam-5566	90	15	,	,	PUNCT
ejpam-5566	90	16	it	it	PRON
ejpam-5566	90	17	was	be	AUX
ejpam-5566	90	18	found	find	VERB
ejpam-5566	90	19	that	that	SCONJ
ejpam-5566	90	20	the	the	DET
ejpam-5566	90	21	subspace	subspace	NOUN
ejpam-5566	90	22	generated	generate	VERB
ejpam-5566	90	23	by	by	ADP
ejpam-5566	90	24	set	set	NOUN
ejpam-5566	90	25	of	of	ADP
ejpam-5566	90	26	all	all	DET
ejpam-5566	90	27	elements	element	NOUN
ejpam-5566	90	28	of	of	ADP
ejpam-5566	90	29	e	e	X
ejpam-5566	90	30	(	(	PUNCT
ejpam-5566	90	31	g	g	NOUN
ejpam-5566	90	32	)	)	PUNCT
ejpam-5566	90	33	with	with	ADP
ejpam-5566	90	34	even	even	ADV
ejpam-5566	90	35	cardinality	cardinality	NOUN
ejpam-5566	90	36	has	have	AUX
ejpam-5566	90	37	dimension	dimension	NOUN
ejpam-5566	90	38	m−	m−	PROPN
ejpam-5566	90	39	1	1	NUM
ejpam-5566	90	40	,	,	PUNCT
ejpam-5566	90	41	where	where	SCONJ
ejpam-5566	90	42	m	m	NOUN
ejpam-5566	90	43	is	be	AUX
ejpam-5566	90	44	the	the	DET
ejpam-5566	90	45	size	size	NOUN
ejpam-5566	90	46	of	of	ADP
ejpam-5566	90	47	the	the	DET
ejpam-5566	90	48	graph	graph	NOUN
ejpam-5566	90	49	.	.	PUNCT
ejpam-5566	91	1	this	this	DET
ejpam-5566	91	2	vector	vector	NOUN
ejpam-5566	91	3	space	space	NOUN
ejpam-5566	91	4	r.	r.	PROPN
ejpam-5566	91	5	mame	mame	PROPN
ejpam-5566	91	6	/	/	SYM
ejpam-5566	91	7	eur	eur	PROPN
ejpam-5566	91	8	.	.	PUNCT
ejpam-5566	92	1	j.	j.	PROPN
ejpam-5566	92	2	pure	pure	PROPN
ejpam-5566	92	3	appl	appl	PROPN
ejpam-5566	92	4	.	.	PROPN
ejpam-5566	92	5	math	math	PROPN
ejpam-5566	92	6	,	,	PUNCT
ejpam-5566	92	7	17	17	NUM
ejpam-5566	92	8	(	(	PUNCT
ejpam-5566	92	9	4	4	NUM
ejpam-5566	92	10	)	)	PUNCT
ejpam-5566	92	11	(	(	PUNCT
ejpam-5566	92	12	2024	2024	NUM
ejpam-5566	92	13	)	)	PUNCT
ejpam-5566	92	14	,	,	PUNCT
ejpam-5566	92	15	3815	3815	NUM
ejpam-5566	92	16	-	-	SYM
ejpam-5566	92	17	3825	3825	NUM
ejpam-5566	92	18	3818	3818	NUM
ejpam-5566	92	19	is	be	AUX
ejpam-5566	92	20	called	call	VERB
ejpam-5566	92	21	even	even	ADV
ejpam-5566	92	22	edge	edge	NOUN
ejpam-5566	92	23	space	space	NOUN
ejpam-5566	92	24	of	of	ADP
ejpam-5566	92	25	graph	graph	NOUN
ejpam-5566	92	26	,	,	PUNCT
ejpam-5566	92	27	denoted	denote	VERB
ejpam-5566	92	28	by	by	ADP
ejpam-5566	92	29	e	e	PROPN
ejpam-5566	92	30	∗(g	∗(g	PROPN
ejpam-5566	92	31	)	)	PUNCT
ejpam-5566	92	32	.	.	PUNCT
ejpam-5566	93	1	equivalently	equivalently	ADV
ejpam-5566	93	2	,	,	PUNCT
ejpam-5566	93	3	the	the	DET
ejpam-5566	93	4	following	follow	VERB
ejpam-5566	93	5	theorem	theorem	NOUN
ejpam-5566	93	6	is	be	AUX
ejpam-5566	93	7	stated	state	VERB
ejpam-5566	93	8	below	below	ADV
ejpam-5566	93	9	.	.	PUNCT
ejpam-5566	94	1	theorem	theorem	VERB
ejpam-5566	94	2	4	4	NUM
ejpam-5566	94	3	(	(	PUNCT
ejpam-5566	94	4	[	[	NOUN
ejpam-5566	94	5	8	8	NUM
ejpam-5566	94	6	]	]	PUNCT
ejpam-5566	94	7	)	)	PUNCT
ejpam-5566	94	8	.	.	PUNCT
ejpam-5566	95	1	let	let	VERB
ejpam-5566	95	2	g	g	PRON
ejpam-5566	95	3	be	be	AUX
ejpam-5566	95	4	a	a	DET
ejpam-5566	95	5	graph	graph	NOUN
ejpam-5566	95	6	with	with	ADP
ejpam-5566	95	7	e(g	e(g	NOUN
ejpam-5566	95	8	)	)	PUNCT
ejpam-5566	96	1	=	=	PRON
ejpam-5566	96	2	{	{	PUNCT
ejpam-5566	96	3	e1	e1	PROPN
ejpam-5566	96	4	,	,	PUNCT
ejpam-5566	96	5	e2	e2	PROPN
ejpam-5566	96	6	,	,	PUNCT
ejpam-5566	96	7	.	.	PUNCT
ejpam-5566	96	8	.	.	PUNCT
ejpam-5566	97	1	.	.	PUNCT
ejpam-5566	98	1	,	,	PUNCT
ejpam-5566	98	2	em	em	PRON
ejpam-5566	98	3	}	}	PUNCT
ejpam-5566	98	4	.	.	PUNCT
ejpam-5566	99	1	then	then	ADV
ejpam-5566	99	2	e	e	X
ejpam-5566	99	3	∗(g	∗(g	PROPN
ejpam-5566	99	4	)	)	PUNCT
ejpam-5566	99	5	is	be	AUX
ejpam-5566	99	6	a	a	DET
ejpam-5566	99	7	subspace	subspace	NOUN
ejpam-5566	99	8	of	of	ADP
ejpam-5566	99	9	e	e	PROPN
ejpam-5566	99	10	(	(	PUNCT
ejpam-5566	99	11	g	g	NOUN
ejpam-5566	99	12	)	)	PUNCT
ejpam-5566	99	13	.	.	PUNCT
ejpam-5566	100	1	moreover	moreover	ADV
ejpam-5566	100	2	,	,	PUNCT
ejpam-5566	100	3	dime	dime	NOUN
ejpam-5566	100	4	∗(g	∗(g	PROPN
ejpam-5566	100	5	)	)	PUNCT
ejpam-5566	101	1	=	=	SYM
ejpam-5566	101	2	m−	m−	PROPN
ejpam-5566	101	3	1	1	NUM
ejpam-5566	101	4	.	.	PUNCT
ejpam-5566	102	1	also	also	ADV
ejpam-5566	102	2	,	,	PUNCT
ejpam-5566	102	3	they	they	PRON
ejpam-5566	102	4	found	find	VERB
ejpam-5566	102	5	a	a	DET
ejpam-5566	102	6	basis	basis	NOUN
ejpam-5566	102	7	for	for	ADP
ejpam-5566	102	8	e	e	PROPN
ejpam-5566	102	9	∗(g	∗(g	PROPN
ejpam-5566	102	10	)	)	PUNCT
ejpam-5566	102	11	,	,	PUNCT
ejpam-5566	102	12	which	which	PRON
ejpam-5566	102	13	is	be	AUX
ejpam-5566	102	14	stated	state	VERB
ejpam-5566	102	15	below	below	ADV
ejpam-5566	102	16	.	.	PUNCT
ejpam-5566	103	1	theorem	theorem	ADJ
ejpam-5566	103	2	5	5	NUM
ejpam-5566	103	3	(	(	PUNCT
ejpam-5566	103	4	[	[	NOUN
ejpam-5566	103	5	8	8	NUM
ejpam-5566	103	6	]	]	PUNCT
ejpam-5566	103	7	)	)	PUNCT
ejpam-5566	103	8	.	.	PUNCT
ejpam-5566	104	1	let	let	VERB
ejpam-5566	104	2	g	g	PRON
ejpam-5566	104	3	be	be	AUX
ejpam-5566	104	4	a	a	DET
ejpam-5566	104	5	graph	graph	NOUN
ejpam-5566	104	6	with	with	ADP
ejpam-5566	104	7	e(g	e(g	NOUN
ejpam-5566	104	8	)	)	PUNCT
ejpam-5566	105	1	=	=	PRON
ejpam-5566	105	2	{	{	PUNCT
ejpam-5566	105	3	e1	e1	PROPN
ejpam-5566	105	4	,	,	PUNCT
ejpam-5566	105	5	e2	e2	PROPN
ejpam-5566	105	6	,	,	PUNCT
ejpam-5566	105	7	.	.	PUNCT
ejpam-5566	105	8	.	.	PUNCT
ejpam-5566	106	1	.	.	PUNCT
ejpam-5566	107	1	,	,	PUNCT
ejpam-5566	107	2	em	em	PRON
ejpam-5566	107	3	}	}	PUNCT
ejpam-5566	107	4	and	and	CCONJ
ejpam-5566	107	5	define	define	VERB
ejpam-5566	107	6	b	b	NOUN
ejpam-5566	107	7	=	=	SYM
ejpam-5566	107	8	{	{	PUNCT
ejpam-5566	107	9	x1	x1	PROPN
ejpam-5566	107	10	,	,	PUNCT
ejpam-5566	107	11	x2	x2	PROPN
ejpam-5566	107	12	,	,	PUNCT
ejpam-5566	107	13	.	.	PUNCT
ejpam-5566	107	14	.	.	PUNCT
ejpam-5566	108	1	.	.	PUNCT
ejpam-5566	109	1	,	,	PUNCT
ejpam-5566	109	2	xm−1	xm−1	PROPN
ejpam-5566	109	3	}	}	PUNCT
ejpam-5566	109	4	,	,	PUNCT
ejpam-5566	109	5	where	where	SCONJ
ejpam-5566	109	6	x1	x1	ADV
ejpam-5566	109	7	=	=	SYM
ejpam-5566	109	8	{	{	PUNCT
ejpam-5566	109	9	e1	e1	PROPN
ejpam-5566	109	10	,	,	PUNCT
ejpam-5566	109	11	e2	e2	PROPN
ejpam-5566	109	12	}	}	PUNCT
ejpam-5566	109	13	,	,	PUNCT
ejpam-5566	109	14	x2	x2	PROPN
ejpam-5566	109	15	=	=	PRON
ejpam-5566	109	16	{	{	PUNCT
ejpam-5566	109	17	e1	e1	PROPN
ejpam-5566	109	18	,	,	PUNCT
ejpam-5566	109	19	e3	e3	NOUN
ejpam-5566	109	20	}	}	PUNCT
ejpam-5566	109	21	,	,	PUNCT
ejpam-5566	109	22	.	.	PUNCT
ejpam-5566	109	23	.	.	PUNCT
ejpam-5566	109	24	.	.	PUNCT
ejpam-5566	110	1	,	,	PUNCT
ejpam-5566	110	2	xm−1	xm−1	PROPN
ejpam-5566	110	3	=	=	PRON
ejpam-5566	110	4	{	{	PUNCT
ejpam-5566	110	5	e1	e1	PROPN
ejpam-5566	110	6	,	,	PUNCT
ejpam-5566	110	7	em	em	PRON
ejpam-5566	110	8	}	}	PUNCT
ejpam-5566	110	9	.	.	PUNCT
ejpam-5566	111	1	then	then	ADV
ejpam-5566	111	2	b	b	X
ejpam-5566	111	3	forms	form	VERB
ejpam-5566	111	4	a	a	DET
ejpam-5566	111	5	basis	basis	NOUN
ejpam-5566	111	6	for	for	ADP
ejpam-5566	111	7	e	e	NOUN
ejpam-5566	111	8	∗(g	∗(g	PROPN
ejpam-5566	111	9	)	)	PUNCT
ejpam-5566	111	10	.	.	PUNCT
ejpam-5566	112	1	finally	finally	ADV
ejpam-5566	112	2	,	,	PUNCT
ejpam-5566	112	3	in	in	ADP
ejpam-5566	112	4	[	[	X
ejpam-5566	112	5	8	8	NUM
ejpam-5566	112	6	]	]	PUNCT
ejpam-5566	112	7	,	,	PUNCT
ejpam-5566	112	8	they	they	PRON
ejpam-5566	112	9	determined	determine	VERB
ejpam-5566	112	10	some	some	DET
ejpam-5566	112	11	properties	property	NOUN
ejpam-5566	112	12	of	of	ADP
ejpam-5566	112	13	graphs	graph	NOUN
ejpam-5566	112	14	wherein	wherein	SCONJ
ejpam-5566	112	15	a	a	DET
ejpam-5566	112	16	star	star	NOUN
ejpam-5566	112	17	is	be	AUX
ejpam-5566	112	18	one	one	NUM
ejpam-5566	112	19	of	of	ADP
ejpam-5566	112	20	its	its	PRON
ejpam-5566	112	21	generator	generator	NOUN
ejpam-5566	112	22	subgraphs	subgraph	NOUN
ejpam-5566	112	23	.	.	PUNCT
ejpam-5566	113	1	theorem	theorem	NOUN
ejpam-5566	113	2	6	6	NUM
ejpam-5566	113	3	.	.	PUNCT
ejpam-5566	114	1	let	let	VERB
ejpam-5566	114	2	p	p	PRON
ejpam-5566	114	3	>	>	X
ejpam-5566	114	4	0	0	PUNCT
ejpam-5566	114	5	be	be	AUX
ejpam-5566	114	6	an	an	DET
ejpam-5566	114	7	odd	odd	ADJ
ejpam-5566	114	8	integer	integer	NOUN
ejpam-5566	114	9	.	.	PUNCT
ejpam-5566	115	1	if	if	SCONJ
ejpam-5566	115	2	g	g	PROPN
ejpam-5566	115	3	is	be	AUX
ejpam-5566	115	4	a	a	DET
ejpam-5566	115	5	graph	graph	NOUN
ejpam-5566	115	6	such	such	ADJ
ejpam-5566	115	7	that	that	PRON
ejpam-5566	115	8	for	for	ADP
ejpam-5566	115	9	every	every	DET
ejpam-5566	115	10	edge	edge	NOUN
ejpam-5566	115	11	[	[	X
ejpam-5566	115	12	a	a	X
ejpam-5566	115	13	,	,	PUNCT
ejpam-5566	115	14	b	b	NOUN
ejpam-5566	115	15	]	]	X
ejpam-5566	115	16	in	in	ADP
ejpam-5566	115	17	g	g	PROPN
ejpam-5566	115	18	either	either	CCONJ
ejpam-5566	115	19	deg(a	deg(a	PROPN
ejpam-5566	115	20	)	)	PUNCT
ejpam-5566	115	21	>	>	X
ejpam-5566	116	1	p	p	PROPN
ejpam-5566	116	2	or	or	CCONJ
ejpam-5566	116	3	deg(b	deg(b	NUM
ejpam-5566	116	4	)	)	PUNCT
ejpam-5566	116	5	>	>	X
ejpam-5566	117	1	p	p	X
ejpam-5566	117	2	,	,	PUNCT
ejpam-5566	117	3	then	then	ADV
ejpam-5566	117	4	star	star	NOUN
ejpam-5566	117	5	sp	sp	PROPN
ejpam-5566	117	6	is	be	AUX
ejpam-5566	117	7	a	a	DET
ejpam-5566	117	8	generator	generator	NOUN
ejpam-5566	117	9	subgraph	subgraph	NOUN
ejpam-5566	117	10	of	of	ADP
ejpam-5566	117	11	g.	g.	PROPN
ejpam-5566	117	12	below	below	ADV
ejpam-5566	117	13	is	be	AUX
ejpam-5566	117	14	an	an	DET
ejpam-5566	117	15	immediate	immediate	ADJ
ejpam-5566	117	16	consequence	consequence	NOUN
ejpam-5566	117	17	of	of	ADP
ejpam-5566	117	18	theorem	theorem	ADJ
ejpam-5566	117	19	6	6	NUM
ejpam-5566	117	20	.	.	PUNCT
ejpam-5566	117	21	corollary	corollary	ADJ
ejpam-5566	117	22	1	1	NUM
ejpam-5566	117	23	.	.	PUNCT
ejpam-5566	118	1	let	let	VERB
ejpam-5566	118	2	p	p	PRON
ejpam-5566	118	3	>	>	X
ejpam-5566	118	4	0	0	NUM
ejpam-5566	118	5	be	be	AUX
ejpam-5566	118	6	odd	odd	ADJ
ejpam-5566	118	7	.	.	PUNCT
ejpam-5566	119	1	if	if	SCONJ
ejpam-5566	119	2	g	g	PROPN
ejpam-5566	119	3	is	be	AUX
ejpam-5566	119	4	kregular	kregular	ADJ
ejpam-5566	119	5	and	and	CCONJ
ejpam-5566	119	6	k	k	X
ejpam-5566	119	7	>	>	X
ejpam-5566	119	8	p	p	X
ejpam-5566	119	9	then	then	ADV
ejpam-5566	119	10	star	star	NOUN
ejpam-5566	119	11	sp	sp	PROPN
ejpam-5566	119	12	is	be	AUX
ejpam-5566	119	13	a	a	DET
ejpam-5566	119	14	generator	generator	NOUN
ejpam-5566	119	15	subgraph	subgraph	NOUN
ejpam-5566	119	16	of	of	ADP
ejpam-5566	119	17	g.	g.	PROPN
ejpam-5566	119	18	the	the	DET
ejpam-5566	119	19	converse	converse	NOUN
ejpam-5566	119	20	of	of	ADP
ejpam-5566	119	21	theorem	theorem	NOUN
ejpam-5566	119	22	6	6	NUM
ejpam-5566	119	23	is	be	AUX
ejpam-5566	119	24	not	not	PART
ejpam-5566	119	25	true	true	ADJ
ejpam-5566	119	26	for	for	ADP
ejpam-5566	119	27	p	p	NOUN
ejpam-5566	119	28	=	=	SYM
ejpam-5566	119	29	1	1	NUM
ejpam-5566	119	30	since	since	SCONJ
ejpam-5566	119	31	star	star	NOUN
ejpam-5566	119	32	s1	s1	PROPN
ejpam-5566	119	33	≃	≃	NOUN
ejpam-5566	119	34	p2	p2	PROPN
ejpam-5566	119	35	is	be	AUX
ejpam-5566	119	36	a	a	DET
ejpam-5566	119	37	generator	generator	NOUN
ejpam-5566	119	38	subgraph	subgraph	NOUN
ejpam-5566	119	39	of	of	ADP
ejpam-5566	119	40	the	the	DET
ejpam-5566	119	41	graph	graph	NOUN
ejpam-5566	119	42	g	g	NOUN
ejpam-5566	119	43	=	=	PUNCT
ejpam-5566	119	44	kp2	kp2	PROPN
ejpam-5566	119	45	,	,	PUNCT
ejpam-5566	119	46	a	a	DET
ejpam-5566	119	47	graph	graph	NOUN
ejpam-5566	119	48	consisting	consist	VERB
ejpam-5566	119	49	of	of	ADP
ejpam-5566	119	50	k	k	PROPN
ejpam-5566	119	51	vertex	vertex	NOUN
ejpam-5566	119	52	-	-	PUNCT
ejpam-5566	119	53	disjoint	disjoint	NOUN
ejpam-5566	119	54	copies	copy	NOUN
ejpam-5566	119	55	of	of	ADP
ejpam-5566	119	56	p2	p2	NOUN
ejpam-5566	119	57	.	.	PUNCT
ejpam-5566	120	1	if	if	SCONJ
ejpam-5566	120	2	p	p	PRON
ejpam-5566	120	3	̸=	̸=	PROPN
ejpam-5566	120	4	1	1	NUM
ejpam-5566	120	5	,	,	PUNCT
ejpam-5566	120	6	we	we	PRON
ejpam-5566	120	7	have	have	VERB
ejpam-5566	120	8	the	the	DET
ejpam-5566	120	9	following	follow	VERB
ejpam-5566	120	10	result	result	NOUN
ejpam-5566	120	11	.	.	PUNCT
ejpam-5566	121	1	theorem	theorem	ADJ
ejpam-5566	121	2	7	7	NUM
ejpam-5566	121	3	.	.	PUNCT
ejpam-5566	122	1	let	let	VERB
ejpam-5566	122	2	p	p	PRON
ejpam-5566	122	3	>	>	X
ejpam-5566	122	4	1	1	NUM
ejpam-5566	122	5	be	be	AUX
ejpam-5566	122	6	odd	odd	ADJ
ejpam-5566	122	7	.	.	PUNCT
ejpam-5566	123	1	then	then	ADV
ejpam-5566	123	2	sp	sp	VERB
ejpam-5566	123	3	is	be	AUX
ejpam-5566	123	4	a	a	DET
ejpam-5566	123	5	generator	generator	NOUN
ejpam-5566	123	6	subgraph	subgraph	NOUN
ejpam-5566	123	7	of	of	ADP
ejpam-5566	123	8	g	g	PROPN
ejpam-5566	123	9	if	if	SCONJ
ejpam-5566	123	10	and	and	CCONJ
ejpam-5566	123	11	only	only	ADV
ejpam-5566	123	12	if	if	SCONJ
ejpam-5566	123	13	for	for	ADP
ejpam-5566	123	14	every	every	DET
ejpam-5566	123	15	edge	edge	NOUN
ejpam-5566	123	16	[	[	X
ejpam-5566	123	17	a	a	X
ejpam-5566	123	18	,	,	PUNCT
ejpam-5566	123	19	b	b	NOUN
ejpam-5566	123	20	]	]	X
ejpam-5566	123	21	in	in	ADP
ejpam-5566	123	22	g	g	PROPN
ejpam-5566	123	23	,	,	PUNCT
ejpam-5566	123	24	either	either	CCONJ
ejpam-5566	123	25	deg(a	deg(a	PROPN
ejpam-5566	123	26	)	)	PUNCT
ejpam-5566	123	27	>	>	X
ejpam-5566	124	1	p	p	PROPN
ejpam-5566	124	2	or	or	CCONJ
ejpam-5566	124	3	deg(b	deg(b	NUM
ejpam-5566	124	4	)	)	PUNCT
ejpam-5566	124	5	>	>	X
ejpam-5566	125	1	p.	p.	NOUN
ejpam-5566	125	2	2	2	NUM
ejpam-5566	125	3	.	.	PUNCT
ejpam-5566	125	4	results	result	VERB
ejpam-5566	125	5	the	the	DET
ejpam-5566	125	6	main	main	ADJ
ejpam-5566	125	7	results	result	NOUN
ejpam-5566	125	8	of	of	ADP
ejpam-5566	125	9	this	this	DET
ejpam-5566	125	10	study	study	NOUN
ejpam-5566	125	11	are	be	AUX
ejpam-5566	125	12	divided	divide	VERB
ejpam-5566	125	13	into	into	ADP
ejpam-5566	125	14	two	two	NUM
ejpam-5566	125	15	parts	part	NOUN
ejpam-5566	125	16	.	.	PUNCT
ejpam-5566	126	1	the	the	DET
ejpam-5566	126	2	first	first	ADJ
ejpam-5566	126	3	part	part	NOUN
ejpam-5566	126	4	investigated	investigate	VERB
ejpam-5566	126	5	the	the	DET
ejpam-5566	126	6	edge	edge	NOUN
ejpam-5566	126	7	space	space	NOUN
ejpam-5566	126	8	of	of	ADP
ejpam-5566	126	9	the	the	DET
ejpam-5566	126	10	square	square	NOUN
ejpam-5566	126	11	of	of	ADP
ejpam-5566	126	12	a	a	DET
ejpam-5566	126	13	cycle	cycle	NOUN
ejpam-5566	126	14	,	,	PUNCT
ejpam-5566	126	15	its	its	PRON
ejpam-5566	126	16	dimension	dimension	NOUN
ejpam-5566	126	17	and	and	CCONJ
ejpam-5566	126	18	discusses	discuss	VERB
ejpam-5566	126	19	some	some	DET
ejpam-5566	126	20	preliminary	preliminary	ADJ
ejpam-5566	126	21	results	result	NOUN
ejpam-5566	126	22	.	.	PUNCT
ejpam-5566	127	1	the	the	DET
ejpam-5566	127	2	second	second	ADJ
ejpam-5566	127	3	part	part	NOUN
ejpam-5566	127	4	provides	provide	VERB
ejpam-5566	127	5	some	some	DET
ejpam-5566	127	6	special	special	ADJ
ejpam-5566	127	7	classes	class	NOUN
ejpam-5566	127	8	of	of	ADP
ejpam-5566	127	9	graphs	graph	NOUN
ejpam-5566	127	10	which	which	PRON
ejpam-5566	127	11	are	be	AUX
ejpam-5566	127	12	generator	generator	NOUN
ejpam-5566	127	13	subgraphs	subgraph	NOUN
ejpam-5566	127	14	of	of	ADP
ejpam-5566	127	15	the	the	DET
ejpam-5566	127	16	square	square	NOUN
ejpam-5566	127	17	of	of	ADP
ejpam-5566	127	18	a	a	DET
ejpam-5566	127	19	cycle	cycle	NOUN
ejpam-5566	127	20	.	.	PUNCT
ejpam-5566	128	1	the	the	DET
ejpam-5566	128	2	preliminary	preliminary	ADJ
ejpam-5566	128	3	results	result	NOUN
ejpam-5566	128	4	in	in	ADP
ejpam-5566	128	5	the	the	DET
ejpam-5566	128	6	first	first	ADJ
ejpam-5566	128	7	part	part	NOUN
ejpam-5566	128	8	were	be	AUX
ejpam-5566	128	9	utilized	utilize	VERB
ejpam-5566	128	10	in	in	ADP
ejpam-5566	128	11	obtaining	obtain	VERB
ejpam-5566	128	12	the	the	DET
ejpam-5566	128	13	generator	generator	NOUN
ejpam-5566	128	14	subgraphs	subgraphs	NOUN
ejpam-5566	128	15	of	of	ADP
ejpam-5566	128	16	the	the	DET
ejpam-5566	128	17	square	square	NOUN
ejpam-5566	128	18	of	of	ADP
ejpam-5566	128	19	a	a	DET
ejpam-5566	128	20	cycle	cycle	NOUN
ejpam-5566	128	21	.	.	PUNCT
ejpam-5566	129	1	2.1	2.1	NUM
ejpam-5566	129	2	.	.	PUNCT
ejpam-5566	129	3	edge	edge	NOUN
ejpam-5566	129	4	space	space	NOUN
ejpam-5566	129	5	of	of	ADP
ejpam-5566	129	6	the	the	DET
ejpam-5566	129	7	square	square	NOUN
ejpam-5566	129	8	of	of	ADP
ejpam-5566	129	9	a	a	DET
ejpam-5566	129	10	cycle	cycle	NOUN
ejpam-5566	129	11	let	let	VERB
ejpam-5566	129	12	c2	c2	PROPN
ejpam-5566	129	13	n	n	PRON
ejpam-5566	129	14	denote	denote	VERB
ejpam-5566	129	15	the	the	DET
ejpam-5566	129	16	square	square	NOUN
ejpam-5566	129	17	of	of	ADP
ejpam-5566	129	18	a	a	DET
ejpam-5566	129	19	cycle	cycle	NOUN
ejpam-5566	129	20	of	of	ADP
ejpam-5566	129	21	order	order	NOUN
ejpam-5566	129	22	n.	n.	NOUN
ejpam-5566	129	23	let	let	VERB
ejpam-5566	129	24	v	v	X
ejpam-5566	129	25	(	(	PUNCT
ejpam-5566	129	26	c2	c2	PROPN
ejpam-5566	129	27	n	n	CCONJ
ejpam-5566	129	28	)	)	PUNCT
ejpam-5566	129	29	=	=	PRON
ejpam-5566	129	30	{	{	PUNCT
ejpam-5566	129	31	1	1	NUM
ejpam-5566	129	32	,	,	PUNCT
ejpam-5566	129	33	2	2	NUM
ejpam-5566	129	34	,	,	PUNCT
ejpam-5566	129	35	3	3	NUM
ejpam-5566	129	36	,	,	PUNCT
ejpam-5566	129	37	.	.	PUNCT
ejpam-5566	129	38	.	.	PUNCT
ejpam-5566	130	1	.	.	PUNCT
ejpam-5566	131	1	,	,	PUNCT
ejpam-5566	132	1	n	n	CCONJ
ejpam-5566	132	2	}	}	PUNCT
ejpam-5566	133	1	where	where	SCONJ
ejpam-5566	133	2	the	the	DET
ejpam-5566	133	3	sequence	sequence	NOUN
ejpam-5566	133	4	of	of	ADP
ejpam-5566	133	5	vertices	vertex	NOUN
ejpam-5566	133	6	[	[	X
ejpam-5566	133	7	1	1	NUM
ejpam-5566	133	8	,	,	PUNCT
ejpam-5566	133	9	2	2	NUM
ejpam-5566	133	10	,	,	PUNCT
ejpam-5566	133	11	3	3	NUM
ejpam-5566	133	12	,	,	PUNCT
ejpam-5566	134	1	.	.	PUNCT
ejpam-5566	134	2	.	.	PUNCT
ejpam-5566	135	1	.	.	PUNCT
ejpam-5566	136	1	,	,	PUNCT
ejpam-5566	136	2	n	n	CCONJ
ejpam-5566	136	3	]	]	PUNCT
ejpam-5566	136	4	forms	form	VERB
ejpam-5566	136	5	the	the	DET
ejpam-5566	136	6	cycle	cycle	NOUN
ejpam-5566	137	1	cn	cn	PROPN
ejpam-5566	137	2	.	.	PUNCT
ejpam-5566	138	1	we	we	PRON
ejpam-5566	138	2	shall	shall	AUX
ejpam-5566	138	3	assume	assume	VERB
ejpam-5566	138	4	that	that	SCONJ
ejpam-5566	138	5	the	the	DET
ejpam-5566	138	6	vertices	vertex	NOUN
ejpam-5566	138	7	1	1	NUM
ejpam-5566	138	8	,	,	PUNCT
ejpam-5566	138	9	2	2	NUM
ejpam-5566	138	10	,	,	PUNCT
ejpam-5566	138	11	3	3	NUM
ejpam-5566	138	12	,	,	PUNCT
ejpam-5566	138	13	.	.	PUNCT
ejpam-5566	138	14	.	.	PUNCT
ejpam-5566	139	1	.	.	PUNCT
ejpam-5566	140	1	,	,	PUNCT
ejpam-5566	140	2	n−1	n−1	PROPN
ejpam-5566	140	3	and	and	CCONJ
ejpam-5566	140	4	n	n	PROPN
ejpam-5566	140	5	are	be	AUX
ejpam-5566	140	6	arranged	arrange	VERB
ejpam-5566	140	7	in	in	ADP
ejpam-5566	140	8	increasing	increase	VERB
ejpam-5566	140	9	order	order	NOUN
ejpam-5566	140	10	in	in	ADP
ejpam-5566	140	11	a	a	DET
ejpam-5566	140	12	clockwise	clockwise	NOUN
ejpam-5566	140	13	direction	direction	NOUN
ejpam-5566	140	14	.	.	PUNCT
ejpam-5566	141	1	thus	thus	ADV
ejpam-5566	141	2	,	,	PUNCT
ejpam-5566	141	3	the	the	DET
ejpam-5566	141	4	edges	edge	NOUN
ejpam-5566	141	5	of	of	ADP
ejpam-5566	141	6	c2	c2	PROPN
ejpam-5566	141	7	n	n	NUM
ejpam-5566	141	8	are	be	AUX
ejpam-5566	141	9	of	of	ADP
ejpam-5566	141	10	the	the	DET
ejpam-5566	141	11	form	form	NOUN
ejpam-5566	142	1	[	[	X
ejpam-5566	142	2	i	i	X
ejpam-5566	142	3	,	,	PUNCT
ejpam-5566	142	4	i+1	i+1	ADV
ejpam-5566	142	5	]	]	PUNCT
ejpam-5566	142	6	and	and	CCONJ
ejpam-5566	143	1	[	[	X
ejpam-5566	143	2	i	i	X
ejpam-5566	143	3	,	,	PUNCT
ejpam-5566	143	4	i+2	i+2	PROPN
ejpam-5566	143	5	]	]	X
ejpam-5566	143	6	,	,	PUNCT
ejpam-5566	143	7	where	where	SCONJ
ejpam-5566	143	8	1	1	NUM
ejpam-5566	143	9	≤	≤	NUM
ejpam-5566	143	10	i	i	NOUN
ejpam-5566	143	11	≤	≤	ADJ
ejpam-5566	143	12	n	n	CCONJ
ejpam-5566	143	13	and	and	CCONJ
ejpam-5566	143	14	n+1	n+1	PROPN
ejpam-5566	143	15	=	=	SYM
ejpam-5566	143	16	1	1	NUM
ejpam-5566	143	17	&	&	CCONJ
ejpam-5566	143	18	n+2	n+2	NUM
ejpam-5566	143	19	=	=	SYM
ejpam-5566	143	20	2	2	X
ejpam-5566	143	21	.	.	X
ejpam-5566	143	22	define	define	VERB
ejpam-5566	143	23	ei	ei	NOUN
ejpam-5566	144	1	=	=	PUNCT
ejpam-5566	145	1	[	[	X
ejpam-5566	145	2	i	i	X
ejpam-5566	145	3	,	,	PUNCT
ejpam-5566	145	4	i+1	i+1	ADV
ejpam-5566	145	5	]	]	PUNCT
ejpam-5566	145	6	and	and	CCONJ
ejpam-5566	145	7	si	si	X
ejpam-5566	145	8	=	=	SYM
ejpam-5566	146	1	[	[	X
ejpam-5566	146	2	i	i	X
ejpam-5566	146	3	,	,	PUNCT
ejpam-5566	146	4	i+2	i+2	PROPN
ejpam-5566	146	5	]	]	PUNCT
ejpam-5566	146	6	.	.	PUNCT
ejpam-5566	147	1	although	although	SCONJ
ejpam-5566	147	2	[	[	X
ejpam-5566	147	3	i	i	X
ejpam-5566	147	4	,	,	PUNCT
ejpam-5566	147	5	i+1	i+1	ADV
ejpam-5566	147	6	]	]	X
ejpam-5566	147	7	=	=	PUNCT
ejpam-5566	148	1	[	[	X
ejpam-5566	148	2	i+1	i+1	X
ejpam-5566	148	3	,	,	PUNCT
ejpam-5566	148	4	i	i	PRON
ejpam-5566	148	5	]	]	PUNCT
ejpam-5566	148	6	and	and	CCONJ
ejpam-5566	148	7	[	[	X
ejpam-5566	148	8	i	i	X
ejpam-5566	148	9	,	,	PUNCT
ejpam-5566	148	10	i+2	i+2	X
ejpam-5566	148	11	]	]	PUNCT
ejpam-5566	148	12	=	=	PUNCT
ejpam-5566	149	1	[	[	X
ejpam-5566	149	2	i+2	i+2	X
ejpam-5566	149	3	,	,	PUNCT
ejpam-5566	149	4	i	i	PRON
ejpam-5566	149	5	]	]	X
ejpam-5566	149	6	,	,	PUNCT
ejpam-5566	149	7	for	for	ADP
ejpam-5566	149	8	isomorphism	isomorphism	NOUN
ejpam-5566	149	9	purposes	purpose	NOUN
ejpam-5566	149	10	,	,	PUNCT
ejpam-5566	149	11	we	we	PRON
ejpam-5566	149	12	shall	shall	AUX
ejpam-5566	149	13	observe	observe	VERB
ejpam-5566	149	14	the	the	DET
ejpam-5566	149	15	order	order	NOUN
ejpam-5566	149	16	of	of	ADP
ejpam-5566	149	17	the	the	DET
ejpam-5566	149	18	vertices	vertex	NOUN
ejpam-5566	149	19	in	in	ADP
ejpam-5566	149	20	the	the	DET
ejpam-5566	149	21	definition	definition	NOUN
ejpam-5566	149	22	of	of	ADP
ejpam-5566	149	23	ei	ei	NOUN
ejpam-5566	149	24	and	and	CCONJ
ejpam-5566	149	25	si	si	INTJ
ejpam-5566	149	26	.	.	PUNCT
ejpam-5566	149	27	thus	thus	ADV
ejpam-5566	149	28	,	,	PUNCT
ejpam-5566	149	29	by	by	ADP
ejpam-5566	149	30	the	the	DET
ejpam-5566	149	31	mapping	mapping	NOUN
ejpam-5566	149	32	of	of	ADP
ejpam-5566	149	33	vertices	vertex	NOUN
ejpam-5566	149	34	ei	ei	X
ejpam-5566	149	35	7→	7→	NUM
ejpam-5566	149	36	ej	ej	INTJ
ejpam-5566	149	37	,	,	PUNCT
ejpam-5566	149	38	we	we	PRON
ejpam-5566	149	39	mean	mean	VERB
ejpam-5566	149	40	the	the	DET
ejpam-5566	149	41	mapping	mapping	NOUN
ejpam-5566	149	42	of	of	ADP
ejpam-5566	149	43	vertices	vertex	NOUN
ejpam-5566	149	44	i	i	X
ejpam-5566	149	45	7→	7→	NUM
ejpam-5566	149	46	j	j	PROPN
ejpam-5566	149	47	r.	r.	PROPN
ejpam-5566	149	48	mame	mame	PROPN
ejpam-5566	149	49	/	/	SYM
ejpam-5566	149	50	eur	eur	PROPN
ejpam-5566	149	51	.	.	PUNCT
ejpam-5566	150	1	j.	j.	PROPN
ejpam-5566	150	2	pure	pure	PROPN
ejpam-5566	150	3	appl	appl	PROPN
ejpam-5566	150	4	.	.	PROPN
ejpam-5566	150	5	math	math	PROPN
ejpam-5566	150	6	,	,	PUNCT
ejpam-5566	150	7	17	17	NUM
ejpam-5566	150	8	(	(	PUNCT
ejpam-5566	150	9	4	4	NUM
ejpam-5566	150	10	)	)	PUNCT
ejpam-5566	150	11	(	(	PUNCT
ejpam-5566	150	12	2024	2024	NUM
ejpam-5566	150	13	)	)	PUNCT
ejpam-5566	150	14	,	,	PUNCT
ejpam-5566	150	15	3815	3815	NUM
ejpam-5566	150	16	-	-	SYM
ejpam-5566	150	17	3825	3825	NUM
ejpam-5566	150	18	3819	3819	NUM
ejpam-5566	150	19	....................................	....................................	PUNCT
ejpam-5566	150	20	....................................	....................................	PUNCT
ejpam-5566	151	1	....................................	....................................	PUNCT
ejpam-5566	151	2	....................................	....................................	PUNCT
ejpam-5566	152	1	....................................	....................................	PUNCT
ejpam-5566	152	2	....................................	....................................	PUNCT
ejpam-5566	153	1	....................................	....................................	PUNCT
ejpam-5566	153	2	....................................	....................................	PUNCT
ejpam-5566	154	1	....................................................................................................................................................................................................................................................	....................................................................................................................................................................................................................................................	PUNCT
ejpam-5566	154	2	..............	..............	PUNCT
ejpam-5566	154	3	.............	.............	PUNCT
ejpam-5566	154	4	.............	.............	PUNCT
ejpam-5566	154	5	.............	.............	PUNCT
ejpam-5566	154	6	.............	.............	PUNCT
ejpam-5566	154	7	.............	.............	PUNCT
ejpam-5566	154	8	.............	.............	PUNCT
ejpam-5566	154	9	.............	.............	PUNCT
ejpam-5566	154	10	.............	.............	PUNCT
ejpam-5566	155	1	....	....	PUNCT
ejpam-5566	155	2	..............	..............	PUNCT
ejpam-5566	155	3	.............	.............	PUNCT
ejpam-5566	155	4	.............	.............	PUNCT
ejpam-5566	155	5	.............	.............	PUNCT
ejpam-5566	155	6	.............	.............	PUNCT
ejpam-5566	155	7	.............	.............	PUNCT
ejpam-5566	155	8	.............	.............	PUNCT
ejpam-5566	155	9	.............	.............	PUNCT
ejpam-5566	155	10	.............	.............	PUNCT
ejpam-5566	155	11	..........	..........	PUNCT
ejpam-5566	155	12	................................	................................	PUNCT
ejpam-5566	155	13	................................	................................	PUNCT
ejpam-5566	155	14	................................	................................	PUNCT
ejpam-5566	155	15	....................	....................	PUNCT
ejpam-5566	155	16	.................................	.................................	PUNCT
ejpam-5566	155	17	................................	................................	PUNCT
ejpam-5566	155	18	................................	................................	PUNCT
ejpam-5566	155	19	.........................	.........................	PUNCT
ejpam-5566	155	20	..........................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................	PUNCT
ejpam-5566	155	21	..............	..............	PUNCT
ejpam-5566	155	22	.............	.............	PUNCT
ejpam-5566	155	23	.............	.............	PUNCT
ejpam-5566	155	24	.............	.............	PUNCT
ejpam-5566	155	25	.............	.............	PUNCT
ejpam-5566	155	26	.............	.............	PUNCT
ejpam-5566	155	27	.............	.............	PUNCT
ejpam-5566	155	28	.............	.............	PUNCT
ejpam-5566	155	29	.............	.............	PUNCT
ejpam-5566	155	30	.............	.............	PUNCT
ejpam-5566	155	31	.............	.............	PUNCT
ejpam-5566	155	32	.............	.............	PUNCT
ejpam-5566	155	33	.............	.............	PUNCT
ejpam-5566	155	34	.............	.............	PUNCT
ejpam-5566	155	35	.............	.............	PUNCT
ejpam-5566	155	36	.............	.............	PUNCT
ejpam-5566	155	37	.............	.............	PUNCT
ejpam-5566	155	38	............	............	PUNCT
ejpam-5566	155	39	...................	...................	PUNCT
ejpam-5566	155	40	..................	..................	PUNCT
ejpam-5566	156	1	..................	..................	PUNCT
ejpam-5566	156	2	..................	..................	PUNCT
ejpam-5566	157	1	..................	..................	PUNCT
ejpam-5566	157	2	..................	..................	PUNCT
ejpam-5566	158	1	..................	..................	PUNCT
ejpam-5566	158	2	..................	..................	PUNCT
ejpam-5566	159	1	..................	..................	PUNCT
ejpam-5566	159	2	..................	..................	PUNCT
ejpam-5566	159	3	..................	..................	PUNCT
ejpam-5566	159	4	..................	..................	PUNCT
ejpam-5566	160	1	.................	.................	PUNCT
ejpam-5566	160	2	...................	...................	PUNCT
ejpam-5566	161	1	..................	..................	PUNCT
ejpam-5566	161	2	..................	..................	PUNCT
ejpam-5566	162	1	..................	..................	PUNCT
ejpam-5566	162	2	..................	..................	PUNCT
ejpam-5566	163	1	..................	..................	PUNCT
ejpam-5566	163	2	..................	..................	PUNCT
ejpam-5566	164	1	..................	..................	PUNCT
ejpam-5566	164	2	..................	..................	PUNCT
ejpam-5566	165	1	..................	..................	PUNCT
ejpam-5566	165	2	..................	..................	PUNCT
ejpam-5566	166	1	..................	..................	PUNCT
ejpam-5566	166	2	.................	.................	PUNCT
ejpam-5566	166	3	..........................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................	PUNCT
ejpam-5566	166	4	....................................................................................................................................................................................................................................................................................................................................................................	....................................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-5566	166	5	........	........	PUNCT
ejpam-5566	167	1	........	........	PUNCT
ejpam-5566	167	2	........	........	PUNCT
ejpam-5566	168	1	e1	e1	PROPN
ejpam-5566	168	2	e2	e2	PROPN
ejpam-5566	168	3	e3	e3	NOUN
ejpam-5566	168	4	e4	e4	PROPN
ejpam-5566	168	5	en−1	en−1	PROPN
ejpam-5566	168	6	en	en	PROPN
ejpam-5566	168	7	e5	e5	PROPN
ejpam-5566	168	8	sn	sn	PROPN
ejpam-5566	168	9	1	1	NUM
ejpam-5566	168	10	2	2	NUM
ejpam-5566	168	11	3	3	NUM
ejpam-5566	168	12	4	4	NUM
ejpam-5566	168	13	5	5	NUM
ejpam-5566	168	14	n−	n−	NOUN
ejpam-5566	168	15	1	1	NUM
ejpam-5566	168	16	n	n	DET
ejpam-5566	168	17	6	6	NUM
ejpam-5566	168	18	sn−1	sn−1	PROPN
ejpam-5566	168	19	s1	s1	PROPN
ejpam-5566	168	20	s2	s2	PROPN
ejpam-5566	168	21	s3	s3	PROPN
ejpam-5566	168	22	s4	s4	PROPN
ejpam-5566	168	23	figure	figure	NOUN
ejpam-5566	168	24	2	2	NUM
ejpam-5566	168	25	:	:	PUNCT
ejpam-5566	168	26	the	the	DET
ejpam-5566	168	27	labeling	labeling	NOUN
ejpam-5566	168	28	of	of	ADP
ejpam-5566	168	29	c2	c2	PROPN
ejpam-5566	168	30	n	n	PROPN
ejpam-5566	168	31	and	and	CCONJ
ejpam-5566	168	32	i+	i+	NUM
ejpam-5566	168	33	1	1	NUM
ejpam-5566	168	34	7→	7→	NUM
ejpam-5566	168	35	j	j	NOUN
ejpam-5566	168	36	+	+	NOUN
ejpam-5566	168	37	1	1	X
ejpam-5566	168	38	.	.	PUNCT
ejpam-5566	168	39	here	here	ADV
ejpam-5566	168	40	,	,	PUNCT
ejpam-5566	168	41	subscripts	subscript	NOUN
ejpam-5566	168	42	are	be	AUX
ejpam-5566	168	43	taken	take	VERB
ejpam-5566	168	44	modulo	modulo	ADJ
ejpam-5566	168	45	n.	n.	NOUN
ejpam-5566	168	46	unless	unless	SCONJ
ejpam-5566	168	47	otherwise	otherwise	ADV
ejpam-5566	168	48	stated	state	VERB
ejpam-5566	168	49	,	,	PUNCT
ejpam-5566	168	50	we	we	PRON
ejpam-5566	168	51	shall	shall	AUX
ejpam-5566	168	52	use	use	VERB
ejpam-5566	168	53	this	this	DET
ejpam-5566	168	54	labeling	labeling	NOUN
ejpam-5566	168	55	throughout	throughout	ADP
ejpam-5566	168	56	the	the	DET
ejpam-5566	168	57	discussion	discussion	NOUN
ejpam-5566	168	58	of	of	ADP
ejpam-5566	168	59	this	this	DET
ejpam-5566	168	60	paper	paper	NOUN
ejpam-5566	168	61	and	and	CCONJ
ejpam-5566	168	62	we	we	PRON
ejpam-5566	168	63	shall	shall	AUX
ejpam-5566	168	64	call	call	VERB
ejpam-5566	168	65	this	this	PRON
ejpam-5566	168	66	the	the	DET
ejpam-5566	168	67	labeling	labeling	NOUN
ejpam-5566	168	68	of	of	ADP
ejpam-5566	168	69	c2	c2	PROPN
ejpam-5566	168	70	n.	n.	PROPN
ejpam-5566	168	71	figure	figure	NOUN
ejpam-5566	168	72	2	2	NUM
ejpam-5566	168	73	represents	represent	VERB
ejpam-5566	168	74	the	the	DET
ejpam-5566	168	75	labeling	labeling	NOUN
ejpam-5566	168	76	of	of	ADP
ejpam-5566	168	77	c2	c2	PROPN
ejpam-5566	168	78	n.	n.	PROPN
ejpam-5566	168	79	remark	remark	PROPN
ejpam-5566	168	80	2	2	NUM
ejpam-5566	168	81	.	.	PUNCT
ejpam-5566	169	1	the	the	DET
ejpam-5566	169	2	following	follow	VERB
ejpam-5566	169	3	statements	statement	NOUN
ejpam-5566	169	4	hold	hold	VERB
ejpam-5566	169	5	.	.	PUNCT
ejpam-5566	170	1	i.	i.	PROPN
ejpam-5566	170	2	c2	c2	PROPN
ejpam-5566	170	3	3	3	NUM
ejpam-5566	170	4	≃	≃	NOUN
ejpam-5566	170	5	k3	k3	PROPN
ejpam-5566	170	6	,	,	PUNCT
ejpam-5566	170	7	c	c	NOUN
ejpam-5566	170	8	2	2	NUM
ejpam-5566	170	9	4	4	NUM
ejpam-5566	170	10	≃	≃	ADJ
ejpam-5566	170	11	k4	k4	NOUN
ejpam-5566	170	12	,	,	PUNCT
ejpam-5566	170	13	and	and	CCONJ
ejpam-5566	170	14	c2	c2	PROPN
ejpam-5566	170	15	5	5	NUM
ejpam-5566	170	16	≃	≃	PROPN
ejpam-5566	170	17	k5	k5	PROPN
ejpam-5566	170	18	.	.	PUNCT
ejpam-5566	170	19	ii	ii	PROPN
ejpam-5566	170	20	.	.	PUNCT
ejpam-5566	171	1	let	let	VERB
ejpam-5566	171	2	n	n	PRON
ejpam-5566	171	3	be	be	AUX
ejpam-5566	171	4	a	a	DET
ejpam-5566	171	5	positive	positive	ADJ
ejpam-5566	171	6	integer	integer	NOUN
ejpam-5566	171	7	.	.	PUNCT
ejpam-5566	172	1	if	if	SCONJ
ejpam-5566	172	2	n	n	NUM
ejpam-5566	172	3	≥	≥	NOUN
ejpam-5566	172	4	5	5	NUM
ejpam-5566	172	5	then	then	ADV
ejpam-5566	172	6	c2	c2	PROPN
ejpam-5566	172	7	n	n	PART
ejpam-5566	172	8	is	be	AUX
ejpam-5566	172	9	a	a	DET
ejpam-5566	172	10	4−	4−	NUM
ejpam-5566	172	11	regular	regular	ADJ
ejpam-5566	172	12	graph	graph	NOUN
ejpam-5566	172	13	.	.	PUNCT
ejpam-5566	173	1	in	in	ADP
ejpam-5566	173	2	this	this	DET
ejpam-5566	173	3	study	study	NOUN
ejpam-5566	173	4	we	we	PRON
ejpam-5566	173	5	consider	consider	VERB
ejpam-5566	173	6	the	the	DET
ejpam-5566	173	7	square	square	NOUN
ejpam-5566	173	8	of	of	ADP
ejpam-5566	173	9	cn	cn	PROPN
ejpam-5566	173	10	where	where	SCONJ
ejpam-5566	173	11	n	n	CCONJ
ejpam-5566	173	12	>	>	X
ejpam-5566	173	13	5	5	NUM
ejpam-5566	173	14	since	since	SCONJ
ejpam-5566	173	15	c2	c2	PROPN
ejpam-5566	173	16	n	n	PART
ejpam-5566	173	17	is	be	AUX
ejpam-5566	173	18	a	a	DET
ejpam-5566	173	19	complete	complete	ADJ
ejpam-5566	173	20	graph	graph	NOUN
ejpam-5566	173	21	of	of	ADP
ejpam-5566	173	22	order	order	NOUN
ejpam-5566	173	23	n	n	NOUN
ejpam-5566	173	24	if	if	SCONJ
ejpam-5566	173	25	n	n	NOUN
ejpam-5566	173	26	=	=	SYM
ejpam-5566	173	27	3	3	NUM
ejpam-5566	173	28	,	,	PUNCT
ejpam-5566	173	29	4,&5	4,&5	NUM
ejpam-5566	173	30	and	and	CCONJ
ejpam-5566	173	31	the	the	DET
ejpam-5566	173	32	generator	generator	NOUN
ejpam-5566	173	33	subgraphs	subgraph	NOUN
ejpam-5566	173	34	of	of	ADP
ejpam-5566	173	35	complete	complete	ADJ
ejpam-5566	173	36	graphs	graph	NOUN
ejpam-5566	173	37	were	be	AUX
ejpam-5566	173	38	completely	completely	ADV
ejpam-5566	173	39	known	know	VERB
ejpam-5566	173	40	by	by	ADP
ejpam-5566	173	41	gervacio	gervacio	NOUN
ejpam-5566	173	42	[	[	X
ejpam-5566	173	43	4	4	NUM
ejpam-5566	173	44	]	]	PUNCT
ejpam-5566	173	45	.	.	PUNCT
ejpam-5566	174	1	next	next	ADV
ejpam-5566	174	2	,	,	PUNCT
ejpam-5566	174	3	we	we	PRON
ejpam-5566	174	4	determine	determine	VERB
ejpam-5566	174	5	the	the	DET
ejpam-5566	174	6	dimension	dimension	NOUN
ejpam-5566	174	7	of	of	ADP
ejpam-5566	174	8	the	the	DET
ejpam-5566	174	9	edge	edge	NOUN
ejpam-5566	174	10	space	space	NOUN
ejpam-5566	174	11	of	of	ADP
ejpam-5566	174	12	c2	c2	PROPN
ejpam-5566	174	13	n.	n.	PROPN
ejpam-5566	174	14	theorem	theorem	VERB
ejpam-5566	174	15	8	8	NUM
ejpam-5566	174	16	.	.	PUNCT
ejpam-5566	175	1	let	let	VERB
ejpam-5566	175	2	n	n	PRON
ejpam-5566	175	3	≥	≥	NUM
ejpam-5566	175	4	5	5	NUM
ejpam-5566	175	5	be	be	AUX
ejpam-5566	175	6	an	an	DET
ejpam-5566	175	7	integer	integer	NOUN
ejpam-5566	175	8	.	.	PUNCT
ejpam-5566	176	1	then	then	ADV
ejpam-5566	176	2	dime	dime	VERB
ejpam-5566	176	3	(	(	PUNCT
ejpam-5566	176	4	c2	c2	PROPN
ejpam-5566	176	5	n	n	CCONJ
ejpam-5566	176	6	)	)	PUNCT
ejpam-5566	176	7	=	=	SYM
ejpam-5566	176	8	2n	2n	NUM
ejpam-5566	176	9	.	.	PUNCT
ejpam-5566	177	1	proof	proof	NOUN
ejpam-5566	177	2	.	.	PUNCT
ejpam-5566	178	1	to	to	PART
ejpam-5566	178	2	show	show	VERB
ejpam-5566	178	3	that	that	DET
ejpam-5566	178	4	dime	dime	NOUN
ejpam-5566	178	5	(	(	PUNCT
ejpam-5566	178	6	c2	c2	PROPN
ejpam-5566	178	7	n	n	CCONJ
ejpam-5566	178	8	)	)	PUNCT
ejpam-5566	178	9	=	=	SYM
ejpam-5566	178	10	2n	2n	NUM
ejpam-5566	178	11	,	,	PUNCT
ejpam-5566	178	12	it	it	PRON
ejpam-5566	178	13	is	be	AUX
ejpam-5566	178	14	enough	enough	ADJ
ejpam-5566	178	15	to	to	PART
ejpam-5566	178	16	show	show	VERB
ejpam-5566	178	17	that	that	SCONJ
ejpam-5566	178	18	the	the	DET
ejpam-5566	178	19	size	size	NOUN
ejpam-5566	178	20	of	of	ADP
ejpam-5566	178	21	c2	c2	PROPN
ejpam-5566	178	22	n	n	PART
ejpam-5566	178	23	is	be	AUX
ejpam-5566	178	24	2n	2n	NUM
ejpam-5566	178	25	.	.	PUNCT
ejpam-5566	179	1	let	let	VERB
ejpam-5566	179	2	v	v	X
ejpam-5566	179	3	(	(	PUNCT
ejpam-5566	179	4	c2	c2	PROPN
ejpam-5566	179	5	n	n	CCONJ
ejpam-5566	179	6	)	)	PUNCT
ejpam-5566	179	7	=	=	NOUN
ejpam-5566	179	8	{	{	PUNCT
ejpam-5566	179	9	v1	v1	PROPN
ejpam-5566	179	10	,	,	PUNCT
ejpam-5566	179	11	v2	v2	PROPN
ejpam-5566	179	12	,	,	PUNCT
ejpam-5566	179	13	v3	v3	PROPN
ejpam-5566	179	14	,	,	PUNCT
ejpam-5566	179	15	.	.	PUNCT
ejpam-5566	179	16	.	.	PUNCT
ejpam-5566	180	1	.	.	PUNCT
ejpam-5566	181	1	,	,	PUNCT
ejpam-5566	181	2	vn	vn	PROPN
ejpam-5566	181	3	}	}	PUNCT
ejpam-5566	181	4	.	.	PUNCT
ejpam-5566	182	1	by	by	ADP
ejpam-5566	182	2	theorem	theorem	NOUN
ejpam-5566	182	3	2	2	NUM
ejpam-5566	182	4	,	,	PUNCT
ejpam-5566	182	5	2|e(c2	2|e(c2	NUM
ejpam-5566	182	6	n)|	n)|	NOUN
ejpam-5566	183	1	=	=	SYM
ejpam-5566	183	2	∑n	∑n	PROPN
ejpam-5566	183	3	i=1	i=1	PROPN
ejpam-5566	183	4	deg(vi	deg(vi	NOUN
ejpam-5566	183	5	)	)	PUNCT
ejpam-5566	183	6	.	.	PUNCT
ejpam-5566	184	1	by	by	ADP
ejpam-5566	184	2	remark	remark	NOUN
ejpam-5566	184	3	2	2	NUM
ejpam-5566	184	4	,	,	PUNCT
ejpam-5566	184	5	c2	c2	PROPN
ejpam-5566	184	6	n	n	PART
ejpam-5566	184	7	is	be	AUX
ejpam-5566	184	8	a	a	DET
ejpam-5566	184	9	4−	4−	NUM
ejpam-5566	184	10	regular	regular	ADJ
ejpam-5566	184	11	graph	graph	NOUN
ejpam-5566	184	12	.	.	PUNCT
ejpam-5566	185	1	hence	hence	ADV
ejpam-5566	185	2	,	,	PUNCT
ejpam-5566	185	3	2|e(c2	2|e(c2	NUM
ejpam-5566	185	4	n)|	n)|	NOUN
ejpam-5566	185	5	=	=	SYM
ejpam-5566	186	1	∑n	∑n	PROPN
ejpam-5566	186	2	i=1	i=1	PROPN
ejpam-5566	186	3	4	4	NUM
ejpam-5566	186	4	=	=	NOUN
ejpam-5566	186	5	4n	4n	NOUN
ejpam-5566	186	6	.	.	PUNCT
ejpam-5566	187	1	this	this	PRON
ejpam-5566	187	2	implies	imply	VERB
ejpam-5566	187	3	that	that	SCONJ
ejpam-5566	187	4	|e(c2	|e(c2	VERB
ejpam-5566	187	5	n)|	n)|	NOUN
ejpam-5566	187	6	=	=	SYM
ejpam-5566	187	7	2n	2n	NUM
ejpam-5566	187	8	.	.	PUNCT
ejpam-5566	188	1	the	the	DET
ejpam-5566	188	2	following	follow	VERB
ejpam-5566	188	3	remark	remark	NOUN
ejpam-5566	188	4	and	and	CCONJ
ejpam-5566	188	5	lemma	lemma	PROPN
ejpam-5566	188	6	are	be	AUX
ejpam-5566	188	7	simple	simple	ADJ
ejpam-5566	188	8	observations	observation	NOUN
ejpam-5566	188	9	.	.	PUNCT
ejpam-5566	189	1	remark	remark	PROPN
ejpam-5566	189	2	3	3	NUM
ejpam-5566	189	3	.	.	PUNCT
ejpam-5566	190	1	let	let	VERB
ejpam-5566	190	2	h	h	PRON
ejpam-5566	190	3	be	be	AUX
ejpam-5566	190	4	a	a	DET
ejpam-5566	190	5	subgraph	subgraph	NOUN
ejpam-5566	190	6	of	of	ADP
ejpam-5566	190	7	c2	c2	PROPN
ejpam-5566	190	8	n.	n.	PROPN
ejpam-5566	191	1	if	if	SCONJ
ejpam-5566	191	2	{	{	PUNCT
ejpam-5566	191	3	ei	ei	NOUN
ejpam-5566	191	4	,	,	PUNCT
ejpam-5566	191	5	si	si	NOUN
ejpam-5566	191	6	}	}	PUNCT
ejpam-5566	191	7	∈	∈	PROPN
ejpam-5566	191	8	e	e	X
ejpam-5566	191	9	(	(	PUNCT
ejpam-5566	191	10	c2	c2	PROPN
ejpam-5566	191	11	n	n	CCONJ
ejpam-5566	191	12	)	)	PUNCT
ejpam-5566	191	13	for	for	ADP
ejpam-5566	191	14	some	some	DET
ejpam-5566	191	15	integer	integer	NOUN
ejpam-5566	191	16	i	i	PRON
ejpam-5566	191	17	,	,	PUNCT
ejpam-5566	191	18	1	1	NUM
ejpam-5566	191	19	≤	≤	NUM
ejpam-5566	191	20	i	i	PRON
ejpam-5566	191	21	≤	≤	PROPN
ejpam-5566	191	22	n	n	CCONJ
ejpam-5566	191	23	,	,	PUNCT
ejpam-5566	191	24	then	then	ADV
ejpam-5566	191	25	by	by	ADP
ejpam-5566	191	26	rotational	rotational	ADJ
ejpam-5566	191	27	symmetry	symmetry	NOUN
ejpam-5566	191	28	on	on	ADP
ejpam-5566	191	29	c2	c2	PROPN
ejpam-5566	191	30	n	n	CCONJ
ejpam-5566	191	31	,	,	PUNCT
ejpam-5566	191	32	{	{	PUNCT
ejpam-5566	191	33	ei	ei	NOUN
ejpam-5566	191	34	,	,	PUNCT
ejpam-5566	191	35	si	si	NOUN
ejpam-5566	191	36	}	}	PUNCT
ejpam-5566	191	37	∈	∈	PROPN
ejpam-5566	191	38	e	e	X
ejpam-5566	191	39	(	(	PUNCT
ejpam-5566	191	40	c2	c2	PROPN
ejpam-5566	191	41	n	n	CCONJ
ejpam-5566	191	42	)	)	PUNCT
ejpam-5566	191	43	for	for	ADP
ejpam-5566	191	44	all	all	DET
ejpam-5566	191	45	i.	i.	NOUN
ejpam-5566	191	46	similarly	similarly	ADV
ejpam-5566	191	47	,	,	PUNCT
ejpam-5566	191	48	if	if	SCONJ
ejpam-5566	191	49	{	{	PUNCT
ejpam-5566	191	50	ei	ei	NOUN
ejpam-5566	191	51	,	,	PUNCT
ejpam-5566	191	52	si−1	si−1	PROPN
ejpam-5566	191	53	}	}	PUNCT
ejpam-5566	191	54	∈	∈	PROPN
ejpam-5566	191	55	e	e	X
ejpam-5566	191	56	(	(	PUNCT
ejpam-5566	191	57	c2	c2	PROPN
ejpam-5566	191	58	n	n	CCONJ
ejpam-5566	191	59	)	)	PUNCT
ejpam-5566	191	60	for	for	ADP
ejpam-5566	191	61	some	some	DET
ejpam-5566	191	62	integer	integer	NOUN
ejpam-5566	191	63	i	i	PRON
ejpam-5566	191	64	,	,	PUNCT
ejpam-5566	191	65	1	1	NUM
ejpam-5566	191	66	≤	≤	NUM
ejpam-5566	191	67	i	i	PRON
ejpam-5566	191	68	≤	≤	PROPN
ejpam-5566	191	69	n	n	CCONJ
ejpam-5566	191	70	,	,	PUNCT
ejpam-5566	191	71	then	then	ADV
ejpam-5566	191	72	{	{	PUNCT
ejpam-5566	191	73	ei	ei	PROPN
ejpam-5566	191	74	,	,	PUNCT
ejpam-5566	191	75	si−1	si−1	PROPN
ejpam-5566	191	76	}	}	PUNCT
ejpam-5566	191	77	∈	∈	PROPN
ejpam-5566	191	78	e	e	X
ejpam-5566	191	79	(	(	PUNCT
ejpam-5566	191	80	c2	c2	PROPN
ejpam-5566	191	81	n	n	CCONJ
ejpam-5566	191	82	)	)	PUNCT
ejpam-5566	191	83	for	for	ADP
ejpam-5566	191	84	all	all	DET
ejpam-5566	191	85	i.	i.	PROPN
ejpam-5566	191	86	lemma	lemma	PROPN
ejpam-5566	191	87	1	1	X
ejpam-5566	191	88	.	.	PUNCT
ejpam-5566	191	89	let	let	VERB
ejpam-5566	191	90	h	h	PRON
ejpam-5566	191	91	be	be	AUX
ejpam-5566	191	92	a	a	DET
ejpam-5566	191	93	subgraph	subgraph	NOUN
ejpam-5566	191	94	of	of	ADP
ejpam-5566	191	95	c2	c2	PROPN
ejpam-5566	191	96	n.	n.	PROPN
ejpam-5566	191	97	then	then	ADV
ejpam-5566	191	98	{	{	PUNCT
ejpam-5566	191	99	ei	ei	PROPN
ejpam-5566	191	100	,	,	PUNCT
ejpam-5566	191	101	si	si	NOUN
ejpam-5566	191	102	}	}	PUNCT
ejpam-5566	191	103	∈	∈	PROPN
ejpam-5566	191	104	eh(c2	eh(c2	NOUN
ejpam-5566	191	105	n	n	CCONJ
ejpam-5566	191	106	)	)	PUNCT
ejpam-5566	191	107	if	if	SCONJ
ejpam-5566	191	108	and	and	CCONJ
ejpam-5566	191	109	only	only	ADV
ejpam-5566	191	110	if	if	SCONJ
ejpam-5566	191	111	{	{	PUNCT
ejpam-5566	191	112	ei	ei	NOUN
ejpam-5566	191	113	,	,	PUNCT
ejpam-5566	191	114	si−1	si−1	PROPN
ejpam-5566	191	115	}	}	PUNCT
ejpam-5566	191	116	∈	∈	PROPN
ejpam-5566	191	117	eh(c2	eh(c2	NOUN
ejpam-5566	191	118	n	n	CCONJ
ejpam-5566	191	119	)	)	PUNCT
ejpam-5566	191	120	where	where	SCONJ
ejpam-5566	191	121	1	1	NUM
ejpam-5566	191	122	≤	≤	NUM
ejpam-5566	191	123	i	i	PRON
ejpam-5566	191	124	≤	≤	ADJ
ejpam-5566	191	125	n.	n.	NOUN
ejpam-5566	191	126	proof	proof	NOUN
ejpam-5566	191	127	.	.	PUNCT
ejpam-5566	192	1	let	let	VERB
ejpam-5566	192	2	us	we	PRON
ejpam-5566	192	3	consider	consider	VERB
ejpam-5566	192	4	the	the	DET
ejpam-5566	192	5	labeling	labeling	NOUN
ejpam-5566	192	6	of	of	ADP
ejpam-5566	192	7	c2	c2	PROPN
ejpam-5566	192	8	n.	n.	PROPN
ejpam-5566	192	9	assume	assume	VERB
ejpam-5566	192	10	that	that	SCONJ
ejpam-5566	192	11	{	{	PUNCT
ejpam-5566	192	12	ei	ei	NOUN
ejpam-5566	192	13	,	,	PUNCT
ejpam-5566	192	14	si	si	NOUN
ejpam-5566	192	15	}	}	PUNCT
ejpam-5566	192	16	∈	∈	PROPN
ejpam-5566	192	17	eh(c2	eh(c2	NOUN
ejpam-5566	192	18	n	n	CCONJ
ejpam-5566	192	19	)	)	PUNCT
ejpam-5566	192	20	.	.	PUNCT
ejpam-5566	193	1	then	then	ADV
ejpam-5566	193	2	{	{	PUNCT
ejpam-5566	193	3	ei	ei	PROPN
ejpam-5566	193	4	,	,	PUNCT
ejpam-5566	193	5	si	si	NOUN
ejpam-5566	193	6	}	}	PUNCT
ejpam-5566	193	7	=	=	X
ejpam-5566	193	8	h1∆h2∆	h1∆h2∆	X
ejpam-5566	193	9	·	·	PUNCT
ejpam-5566	193	10	·	·	PUNCT
ejpam-5566	193	11	·	·	PUNCT
ejpam-5566	193	12	∆hk	∆hk	NOUN
ejpam-5566	193	13	,	,	PUNCT
ejpam-5566	193	14	where	where	SCONJ
ejpam-5566	193	15	hj	hj	PROPN
ejpam-5566	193	16	∈	∈	PROPN
ejpam-5566	193	17	eh(c2	eh(c2	X
ejpam-5566	193	18	n	n	CCONJ
ejpam-5566	193	19	)	)	PUNCT
ejpam-5566	193	20	,	,	PUNCT
ejpam-5566	193	21	j	j	PROPN
ejpam-5566	193	22	and	and	CCONJ
ejpam-5566	193	23	k	k	PROPN
ejpam-5566	193	24	are	be	AUX
ejpam-5566	193	25	positive	positive	ADJ
ejpam-5566	193	26	integers	integer	NOUN
ejpam-5566	193	27	,	,	PUNCT
ejpam-5566	193	28	where	where	SCONJ
ejpam-5566	193	29	r.	r.	PROPN
ejpam-5566	193	30	mame	mame	PROPN
ejpam-5566	193	31	/	/	SYM
ejpam-5566	193	32	eur	eur	PROPN
ejpam-5566	193	33	.	.	PUNCT
ejpam-5566	194	1	j.	j.	PROPN
ejpam-5566	194	2	pure	pure	PROPN
ejpam-5566	194	3	appl	appl	PROPN
ejpam-5566	194	4	.	.	PROPN
ejpam-5566	194	5	math	math	PROPN
ejpam-5566	194	6	,	,	PUNCT
ejpam-5566	194	7	17	17	NUM
ejpam-5566	194	8	(	(	PUNCT
ejpam-5566	194	9	4	4	NUM
ejpam-5566	194	10	)	)	PUNCT
ejpam-5566	194	11	(	(	PUNCT
ejpam-5566	194	12	2024	2024	NUM
ejpam-5566	194	13	)	)	PUNCT
ejpam-5566	194	14	,	,	PUNCT
ejpam-5566	194	15	3815	3815	NUM
ejpam-5566	194	16	-	-	SYM
ejpam-5566	194	17	3825	3825	NUM
ejpam-5566	194	18	3820	3820	NUM
ejpam-5566	194	19	1	1	NUM
ejpam-5566	194	20	≤	≤	NUM
ejpam-5566	194	21	j	j	PROPN
ejpam-5566	194	22	≤	≤	PROPN
ejpam-5566	194	23	k.	k.	PROPN
ejpam-5566	194	24	define	define	VERB
ejpam-5566	194	25	the	the	DET
ejpam-5566	194	26	mapping	mapping	NOUN
ejpam-5566	194	27	ϕ	ϕ	NOUN
ejpam-5566	194	28	:	:	PUNCT
ejpam-5566	194	29	hj	hj	VERB
ejpam-5566	194	30	−→	−→	NOUN
ejpam-5566	194	31	h	h	NOUN
ejpam-5566	195	1	′	′	NUM
ejpam-5566	195	2	j	j	NOUN
ejpam-5566	195	3	by	by	ADP
ejpam-5566	195	4	ϕ(ei	ϕ(ei	PROPN
ejpam-5566	195	5	)	)	PUNCT
ejpam-5566	195	6	=	=	SYM
ejpam-5566	195	7	en−(i−1	en−(i−1	PROPN
ejpam-5566	195	8	)	)	PUNCT
ejpam-5566	195	9	and	and	CCONJ
ejpam-5566	195	10	ϕ(si	ϕ(si	NUM
ejpam-5566	195	11	)	)	PUNCT
ejpam-5566	196	1	=	=	PUNCT
ejpam-5566	196	2	sn−i	sn−i	ADV
ejpam-5566	196	3	.	.	PUNCT
ejpam-5566	197	1	here	here	ADV
ejpam-5566	197	2	,	,	PUNCT
ejpam-5566	197	3	subscripts	subscript	NOUN
ejpam-5566	197	4	are	be	AUX
ejpam-5566	197	5	taken	take	VERB
ejpam-5566	197	6	modulo	modulo	ADJ
ejpam-5566	197	7	n.	n.	NOUN
ejpam-5566	197	8	it	it	PRON
ejpam-5566	197	9	can	can	AUX
ejpam-5566	197	10	be	be	AUX
ejpam-5566	197	11	verified	verify	VERB
ejpam-5566	197	12	that	that	SCONJ
ejpam-5566	197	13	ϕ	ϕ	NOUN
ejpam-5566	197	14	is	be	AUX
ejpam-5566	197	15	an	an	DET
ejpam-5566	197	16	isomorphism	isomorphism	NOUN
ejpam-5566	197	17	.	.	PUNCT
ejpam-5566	198	1	thus	thus	ADV
ejpam-5566	198	2	,	,	PUNCT
ejpam-5566	198	3	h	h	NOUN
ejpam-5566	198	4	′	′	NUM
ejpam-5566	199	1	j	j	PROPN
ejpam-5566	199	2	∈	∈	PROPN
ejpam-5566	199	3	eh(c2	eh(c2	X
ejpam-5566	199	4	n	n	CCONJ
ejpam-5566	199	5	)	)	PUNCT
ejpam-5566	199	6	for	for	ADP
ejpam-5566	199	7	all	all	DET
ejpam-5566	199	8	j.	j.	PROPN
ejpam-5566	199	9	hence	hence	PROPN
ejpam-5566	199	10	,	,	PUNCT
ejpam-5566	199	11	{	{	PUNCT
ejpam-5566	199	12	ϕ(ei	ϕ(ei	NOUN
ejpam-5566	199	13	)	)	PUNCT
ejpam-5566	199	14	,	,	PUNCT
ejpam-5566	199	15	ϕ(si	ϕ(si	PROPN
ejpam-5566	199	16	)	)	PUNCT
ejpam-5566	199	17	}	}	PUNCT
ejpam-5566	199	18	=	=	SYM
ejpam-5566	199	19	{	{	PUNCT
ejpam-5566	199	20	en−(i−1	en−(i−1	PROPN
ejpam-5566	199	21	)	)	PUNCT
ejpam-5566	199	22	,	,	PUNCT
ejpam-5566	199	23	sn−i	sn−i	ADV
ejpam-5566	199	24	}	}	PUNCT
ejpam-5566	199	25	=	=	SYM
ejpam-5566	199	26	{	{	PUNCT
ejpam-5566	199	27	e(n−i+1	e(n−i+1	NOUN
ejpam-5566	199	28	)	)	PUNCT
ejpam-5566	199	29	,	,	PUNCT
ejpam-5566	199	30	sn−i	sn−i	ADV
ejpam-5566	199	31	}	}	PUNCT
ejpam-5566	199	32	=	=	PUNCT
ejpam-5566	199	33	h	h	NOUN
ejpam-5566	200	1	′	′	NOUN
ejpam-5566	200	2	1∆h	1∆h	NUM
ejpam-5566	201	1	′	′	NUM
ejpam-5566	201	2	2∆	2∆	NUM
ejpam-5566	201	3	·	·	PUNCT
ejpam-5566	201	4	·	·	PUNCT
ejpam-5566	201	5	·	·	PUNCT
ejpam-5566	202	1	∆h	∆h	NUM
ejpam-5566	202	2	′	′	NUM
ejpam-5566	202	3	k	k	PROPN
ejpam-5566	202	4	∈	∈	PROPN
ejpam-5566	202	5	eh(c2	eh(c2	NOUN
ejpam-5566	202	6	n	n	CCONJ
ejpam-5566	202	7	)	)	PUNCT
ejpam-5566	202	8	.	.	PUNCT
ejpam-5566	203	1	by	by	ADP
ejpam-5566	203	2	remark	remark	NOUN
ejpam-5566	203	3	3	3	NUM
ejpam-5566	203	4	,	,	PUNCT
ejpam-5566	203	5	{	{	PUNCT
ejpam-5566	203	6	ei	ei	PROPN
ejpam-5566	203	7	,	,	PUNCT
ejpam-5566	203	8	si−1	si−1	PROPN
ejpam-5566	203	9	}	}	PUNCT
ejpam-5566	203	10	∈	∈	PROPN
ejpam-5566	203	11	eh(c2	eh(c2	NOUN
ejpam-5566	203	12	n	n	CCONJ
ejpam-5566	203	13	)	)	PUNCT
ejpam-5566	203	14	.	.	PUNCT
ejpam-5566	204	1	for	for	ADP
ejpam-5566	204	2	the	the	DET
ejpam-5566	204	3	converse	converse	NOUN
ejpam-5566	204	4	,	,	PUNCT
ejpam-5566	204	5	the	the	DET
ejpam-5566	204	6	proof	proof	NOUN
ejpam-5566	204	7	is	be	AUX
ejpam-5566	204	8	similar	similar	ADJ
ejpam-5566	204	9	.	.	PUNCT
ejpam-5566	205	1	the	the	DET
ejpam-5566	205	2	next	next	ADJ
ejpam-5566	205	3	result	result	NOUN
ejpam-5566	205	4	is	be	AUX
ejpam-5566	205	5	an	an	DET
ejpam-5566	205	6	extension	extension	NOUN
ejpam-5566	205	7	of	of	ADP
ejpam-5566	205	8	the	the	DET
ejpam-5566	205	9	above	above	ADJ
ejpam-5566	205	10	lemma	lemma	PROPN
ejpam-5566	205	11	.	.	PUNCT
ejpam-5566	206	1	lemma	lemma	PROPN
ejpam-5566	206	2	2	2	X
ejpam-5566	206	3	.	.	PUNCT
ejpam-5566	207	1	let	let	VERB
ejpam-5566	207	2	h	h	PRON
ejpam-5566	207	3	be	be	AUX
ejpam-5566	207	4	a	a	DET
ejpam-5566	207	5	subgraph	subgraph	NOUN
ejpam-5566	207	6	of	of	ADP
ejpam-5566	207	7	c2	c2	PROPN
ejpam-5566	207	8	n.	n.	PROPN
ejpam-5566	208	1	if	if	SCONJ
ejpam-5566	208	2	{	{	PUNCT
ejpam-5566	208	3	ei	ei	NOUN
ejpam-5566	208	4	,	,	PUNCT
ejpam-5566	208	5	si	si	NOUN
ejpam-5566	208	6	}	}	PUNCT
ejpam-5566	208	7	∈	∈	PROPN
ejpam-5566	208	8	eh(c2	eh(c2	NOUN
ejpam-5566	208	9	n	n	CCONJ
ejpam-5566	208	10	)	)	PUNCT
ejpam-5566	208	11	for	for	ADP
ejpam-5566	208	12	some	some	DET
ejpam-5566	208	13	integer	integer	NOUN
ejpam-5566	208	14	i	i	PRON
ejpam-5566	208	15	,	,	PUNCT
ejpam-5566	208	16	where	where	SCONJ
ejpam-5566	208	17	1	1	NUM
ejpam-5566	208	18	≤	≤	NUM
ejpam-5566	208	19	i	i	PRON
ejpam-5566	208	20	≤	≤	PROPN
ejpam-5566	208	21	n	n	CCONJ
ejpam-5566	208	22	,	,	PUNCT
ejpam-5566	208	23	then	then	ADV
ejpam-5566	208	24	e	e	NOUN
ejpam-5566	208	25	∗(c2	∗(c2	NOUN
ejpam-5566	208	26	n	n	CCONJ
ejpam-5566	208	27	)	)	PUNCT
ejpam-5566	208	28	⊆	⊆	PROPN
ejpam-5566	208	29	eh(c2	eh(c2	NUM
ejpam-5566	208	30	n	n	CCONJ
ejpam-5566	208	31	)	)	PUNCT
ejpam-5566	208	32	.	.	PUNCT
ejpam-5566	209	1	proof	proof	NOUN
ejpam-5566	209	2	.	.	PUNCT
ejpam-5566	210	1	consider	consider	VERB
ejpam-5566	210	2	the	the	DET
ejpam-5566	210	3	labeling	labeling	NOUN
ejpam-5566	210	4	of	of	ADP
ejpam-5566	210	5	c2	c2	PROPN
ejpam-5566	210	6	n	n	CCONJ
ejpam-5566	210	7	,	,	PUNCT
ejpam-5566	210	8	let	let	VERB
ejpam-5566	210	9	b	b	NOUN
ejpam-5566	210	10	=	=	PRON
ejpam-5566	210	11	{	{	PUNCT
ejpam-5566	210	12	{	{	PUNCT
ejpam-5566	210	13	e1	e1	PROPN
ejpam-5566	210	14	,	,	PUNCT
ejpam-5566	210	15	e2	e2	PROPN
ejpam-5566	210	16	}	}	PUNCT
ejpam-5566	210	17	,	,	PUNCT
ejpam-5566	210	18	{	{	PUNCT
ejpam-5566	210	19	e1	e1	NOUN
ejpam-5566	210	20	,	,	PUNCT
ejpam-5566	210	21	e3	e3	NOUN
ejpam-5566	210	22	}	}	PUNCT
ejpam-5566	210	23	,	,	PUNCT
ejpam-5566	210	24	.	.	PUNCT
ejpam-5566	210	25	.	.	PUNCT
ejpam-5566	211	1	.	.	PUNCT
ejpam-5566	212	1	,	,	PUNCT
ejpam-5566	212	2	{	{	PUNCT
ejpam-5566	212	3	e1	e1	NOUN
ejpam-5566	212	4	,	,	PUNCT
ejpam-5566	212	5	en	en	ADP
ejpam-5566	212	6	}	}	PUNCT
ejpam-5566	212	7	,	,	PUNCT
ejpam-5566	212	8	{	{	PUNCT
ejpam-5566	212	9	e1	e1	NOUN
ejpam-5566	212	10	,	,	PUNCT
ejpam-5566	212	11	s1	s1	NOUN
ejpam-5566	212	12	}	}	PUNCT
ejpam-5566	212	13	,	,	PUNCT
ejpam-5566	212	14	{	{	PUNCT
ejpam-5566	212	15	e1	e1	NOUN
ejpam-5566	212	16	,	,	PUNCT
ejpam-5566	212	17	s2	s2	NOUN
ejpam-5566	212	18	}	}	PUNCT
ejpam-5566	212	19	,	,	PUNCT
ejpam-5566	212	20	.	.	PUNCT
ejpam-5566	212	21	.	.	PUNCT
ejpam-5566	213	1	.	.	PUNCT
ejpam-5566	214	1	,	,	PUNCT
ejpam-5566	214	2	{	{	PUNCT
ejpam-5566	214	3	e1	e1	NOUN
ejpam-5566	214	4	,	,	PUNCT
ejpam-5566	214	5	sn	sn	PROPN
ejpam-5566	214	6	}	}	PUNCT
ejpam-5566	214	7	}	}	PUNCT
ejpam-5566	214	8	.	.	PUNCT
ejpam-5566	215	1	by	by	ADP
ejpam-5566	215	2	theorem	theorem	NOUN
ejpam-5566	215	3	5	5	NUM
ejpam-5566	215	4	,	,	PUNCT
ejpam-5566	215	5	b	b	NOUN
ejpam-5566	215	6	forms	form	VERB
ejpam-5566	215	7	a	a	DET
ejpam-5566	215	8	basis	basis	NOUN
ejpam-5566	215	9	for	for	ADP
ejpam-5566	215	10	e	e	NOUN
ejpam-5566	215	11	∗(c2	∗(c2	NOUN
ejpam-5566	215	12	n	n	CCONJ
ejpam-5566	215	13	)	)	PUNCT
ejpam-5566	215	14	so	so	SCONJ
ejpam-5566	215	15	it	it	PRON
ejpam-5566	215	16	is	be	AUX
ejpam-5566	215	17	enough	enough	ADJ
ejpam-5566	215	18	to	to	PART
ejpam-5566	215	19	show	show	VERB
ejpam-5566	215	20	that	that	SCONJ
ejpam-5566	215	21	b	b	PROPN
ejpam-5566	215	22	⊆	⊆	NUM
ejpam-5566	215	23	eh(c2	eh(c2	X
ejpam-5566	215	24	n	n	CCONJ
ejpam-5566	215	25	)	)	PUNCT
ejpam-5566	215	26	.	.	PUNCT
ejpam-5566	216	1	let	let	VERB
ejpam-5566	216	2	x	x	SYM
ejpam-5566	216	3	∈	∈	PROPN
ejpam-5566	216	4	b.	b.	PROPN
ejpam-5566	217	1	then	then	ADV
ejpam-5566	217	2	x	x	X
ejpam-5566	217	3	=	=	PRON
ejpam-5566	217	4	{	{	PUNCT
ejpam-5566	217	5	e1	e1	PROPN
ejpam-5566	217	6	,	,	PUNCT
ejpam-5566	217	7	ei	ei	NOUN
ejpam-5566	217	8	}	}	PUNCT
ejpam-5566	217	9	or	or	CCONJ
ejpam-5566	217	10	x	x	X
ejpam-5566	217	11	=	=	SYM
ejpam-5566	217	12	{	{	PUNCT
ejpam-5566	217	13	e1	e1	PROPN
ejpam-5566	217	14	,	,	PUNCT
ejpam-5566	217	15	si	si	INTJ
ejpam-5566	217	16	}	}	PUNCT
ejpam-5566	217	17	for	for	ADP
ejpam-5566	217	18	some	some	DET
ejpam-5566	217	19	i	i	PRON
ejpam-5566	217	20	,	,	PUNCT
ejpam-5566	217	21	where	where	SCONJ
ejpam-5566	217	22	1	1	NUM
ejpam-5566	217	23	≤	≤	NUM
ejpam-5566	217	24	i	i	PRON
ejpam-5566	217	25	≤	≤	PROPN
ejpam-5566	217	26	n.	n.	NOUN
ejpam-5566	217	27	first	first	ADV
ejpam-5566	217	28	,	,	PUNCT
ejpam-5566	217	29	we	we	PRON
ejpam-5566	217	30	show	show	VERB
ejpam-5566	217	31	that	that	SCONJ
ejpam-5566	217	32	{	{	PUNCT
ejpam-5566	217	33	e1	e1	NOUN
ejpam-5566	217	34	,	,	PUNCT
ejpam-5566	217	35	ei	ei	NOUN
ejpam-5566	217	36	}	}	PUNCT
ejpam-5566	217	37	∈	∈	PROPN
ejpam-5566	217	38	eh(c2	eh(c2	NOUN
ejpam-5566	217	39	n	n	CCONJ
ejpam-5566	217	40	)	)	PUNCT
ejpam-5566	217	41	.	.	PUNCT
ejpam-5566	218	1	since	since	SCONJ
ejpam-5566	218	2	{	{	PUNCT
ejpam-5566	218	3	ei	ei	NOUN
ejpam-5566	218	4	,	,	PUNCT
ejpam-5566	218	5	si	si	NOUN
ejpam-5566	218	6	}	}	PUNCT
ejpam-5566	218	7	∈	∈	PROPN
ejpam-5566	218	8	eh(c2	eh(c2	NOUN
ejpam-5566	218	9	n	n	CCONJ
ejpam-5566	218	10	)	)	PUNCT
ejpam-5566	218	11	for	for	ADP
ejpam-5566	218	12	some	some	DET
ejpam-5566	218	13	i	i	PROPN
ejpam-5566	218	14	,	,	PUNCT
ejpam-5566	218	15	by	by	ADP
ejpam-5566	218	16	remark	remark	NOUN
ejpam-5566	218	17	3	3	NUM
ejpam-5566	218	18	,	,	PUNCT
ejpam-5566	218	19	{	{	PUNCT
ejpam-5566	218	20	ei	ei	NOUN
ejpam-5566	218	21	,	,	PUNCT
ejpam-5566	218	22	si	si	NOUN
ejpam-5566	218	23	}	}	PUNCT
ejpam-5566	218	24	∈	∈	PROPN
ejpam-5566	218	25	eh(c2	eh(c2	NOUN
ejpam-5566	218	26	n	n	CCONJ
ejpam-5566	218	27	)	)	PUNCT
ejpam-5566	218	28	for	for	ADP
ejpam-5566	218	29	all	all	DET
ejpam-5566	218	30	i.	i.	NOUN
ejpam-5566	218	31	by	by	ADP
ejpam-5566	218	32	lemma	lemma	PROPN
ejpam-5566	218	33	1	1	NUM
ejpam-5566	218	34	,	,	PUNCT
ejpam-5566	218	35	{	{	PUNCT
ejpam-5566	218	36	ei+1	ei+1	PROPN
ejpam-5566	218	37	,	,	PUNCT
ejpam-5566	218	38	si	si	ADJ
ejpam-5566	218	39	}	}	PUNCT
ejpam-5566	218	40	∈	∈	PROPN
ejpam-5566	218	41	eh(c2	eh(c2	NOUN
ejpam-5566	218	42	n	n	CCONJ
ejpam-5566	218	43	)	)	PUNCT
ejpam-5566	218	44	for	for	ADP
ejpam-5566	218	45	all	all	DET
ejpam-5566	218	46	i.	i.	NOUN
ejpam-5566	218	47	now	now	ADV
ejpam-5566	218	48	,	,	PUNCT
ejpam-5566	218	49	{	{	PUNCT
ejpam-5566	218	50	ei	ei	X
ejpam-5566	218	51	,	,	PUNCT
ejpam-5566	218	52	ei+1	ei+1	NOUN
ejpam-5566	218	53	}	}	PUNCT
ejpam-5566	218	54	=	=	SYM
ejpam-5566	218	55	{	{	PUNCT
ejpam-5566	218	56	ei	ei	NOUN
ejpam-5566	218	57	,	,	PUNCT
ejpam-5566	218	58	si}∆{ei+1	si}∆{ei+1	ADJ
ejpam-5566	218	59	,	,	PUNCT
ejpam-5566	218	60	si	si	ADJ
ejpam-5566	218	61	}	}	PUNCT
ejpam-5566	218	62	∈	∈	PROPN
ejpam-5566	218	63	eh(c2	eh(c2	NOUN
ejpam-5566	218	64	n	n	CCONJ
ejpam-5566	218	65	)	)	PUNCT
ejpam-5566	218	66	for	for	ADP
ejpam-5566	218	67	all	all	DET
ejpam-5566	218	68	i.	i.	NOUN
ejpam-5566	218	69	in	in	ADP
ejpam-5566	218	70	particular	particular	ADJ
ejpam-5566	218	71	,	,	PUNCT
ejpam-5566	218	72	{	{	PUNCT
ejpam-5566	218	73	e1	e1	NOUN
ejpam-5566	218	74	,	,	PUNCT
ejpam-5566	218	75	e2	e2	PROPN
ejpam-5566	218	76	}	}	PUNCT
ejpam-5566	218	77	∈	∈	PROPN
ejpam-5566	218	78	eh(c2	eh(c2	NOUN
ejpam-5566	218	79	n	n	CCONJ
ejpam-5566	218	80	)	)	PUNCT
ejpam-5566	218	81	.	.	PUNCT
ejpam-5566	219	1	thus	thus	ADV
ejpam-5566	219	2	,	,	PUNCT
ejpam-5566	219	3	for	for	ADP
ejpam-5566	219	4	3	3	NUM
ejpam-5566	219	5	≤	≤	NUM
ejpam-5566	219	6	i	i	PRON
ejpam-5566	219	7	≤	≤	NOUN
ejpam-5566	219	8	n	n	CCONJ
ejpam-5566	219	9	,	,	PUNCT
ejpam-5566	219	10	x	x	PUNCT
ejpam-5566	219	11	=	=	PRON
ejpam-5566	219	12	{	{	PUNCT
ejpam-5566	219	13	e1	e1	PROPN
ejpam-5566	219	14	,	,	PUNCT
ejpam-5566	219	15	ei	ei	NOUN
ejpam-5566	219	16	}	}	PUNCT
ejpam-5566	219	17	=	=	SYM
ejpam-5566	219	18	{	{	PUNCT
ejpam-5566	219	19	e1	e1	PROPN
ejpam-5566	219	20	,	,	PUNCT
ejpam-5566	219	21	e2}∆{e2	e2}∆{e2	PROPN
ejpam-5566	219	22	,	,	PUNCT
ejpam-5566	219	23	e3}∆	e3}∆	PROPN
ejpam-5566	219	24	·	·	PUNCT
ejpam-5566	219	25	·	·	PUNCT
ejpam-5566	219	26	·	·	PUNCT
ejpam-5566	219	27	∆{ei−1	∆{ei−1	PROPN
ejpam-5566	219	28	,	,	PUNCT
ejpam-5566	219	29	ei	ei	NOUN
ejpam-5566	219	30	}	}	PUNCT
ejpam-5566	219	31	∈	∈	PROPN
ejpam-5566	219	32	eh(c2	eh(c2	NOUN
ejpam-5566	219	33	n	n	CCONJ
ejpam-5566	219	34	)	)	PUNCT
ejpam-5566	219	35	.	.	PUNCT
ejpam-5566	220	1	next	next	ADV
ejpam-5566	220	2	,	,	PUNCT
ejpam-5566	220	3	we	we	PRON
ejpam-5566	220	4	show	show	VERB
ejpam-5566	220	5	that	that	SCONJ
ejpam-5566	220	6	{	{	PUNCT
ejpam-5566	220	7	e1	e1	NOUN
ejpam-5566	220	8	,	,	PUNCT
ejpam-5566	220	9	si	si	ADJ
ejpam-5566	220	10	}	}	PUNCT
ejpam-5566	220	11	∈	∈	PROPN
ejpam-5566	220	12	eh(c2	eh(c2	NOUN
ejpam-5566	220	13	n	n	CCONJ
ejpam-5566	220	14	)	)	PUNCT
ejpam-5566	220	15	.	.	PUNCT
ejpam-5566	221	1	clearly	clearly	ADV
ejpam-5566	221	2	,	,	PUNCT
ejpam-5566	221	3	{	{	PUNCT
ejpam-5566	221	4	e1	e1	NOUN
ejpam-5566	221	5	,	,	PUNCT
ejpam-5566	221	6	s1	s1	NOUN
ejpam-5566	221	7	}	}	PUNCT
ejpam-5566	221	8	∈	∈	PROPN
ejpam-5566	221	9	eh(c2	eh(c2	NOUN
ejpam-5566	221	10	n	n	CCONJ
ejpam-5566	221	11	)	)	PUNCT
ejpam-5566	221	12	.	.	PUNCT
ejpam-5566	222	1	now	now	ADV
ejpam-5566	222	2	,	,	PUNCT
ejpam-5566	222	3	for	for	ADP
ejpam-5566	222	4	2	2	NUM
ejpam-5566	222	5	≤	≤	NOUN
ejpam-5566	222	6	i	i	PRON
ejpam-5566	222	7	≤	≤	PROPN
ejpam-5566	222	8	n	n	CCONJ
ejpam-5566	222	9	,	,	PUNCT
ejpam-5566	222	10	we	we	PRON
ejpam-5566	222	11	have	have	VERB
ejpam-5566	222	12	{	{	PUNCT
ejpam-5566	222	13	e1	e1	NOUN
ejpam-5566	222	14	,	,	PUNCT
ejpam-5566	222	15	si	si	NOUN
ejpam-5566	222	16	}	}	PUNCT
ejpam-5566	222	17	=	=	SYM
ejpam-5566	222	18	{	{	PUNCT
ejpam-5566	222	19	e1	e1	PROPN
ejpam-5566	222	20	,	,	PUNCT
ejpam-5566	222	21	ei}∆{ei	ei}∆{ei	PROPN
ejpam-5566	222	22	,	,	PUNCT
ejpam-5566	222	23	si	si	NOUN
ejpam-5566	222	24	}	}	PUNCT
ejpam-5566	222	25	∈	∈	PROPN
ejpam-5566	222	26	eh(c2	eh(c2	NOUN
ejpam-5566	222	27	n	n	CCONJ
ejpam-5566	222	28	)	)	PUNCT
ejpam-5566	222	29	.	.	PUNCT
ejpam-5566	223	1	thus	thus	ADV
ejpam-5566	223	2	,	,	PUNCT
ejpam-5566	223	3	x	x	PROPN
ejpam-5566	223	4	∈	∈	PROPN
ejpam-5566	223	5	eh(c2	eh(c2	NOUN
ejpam-5566	223	6	n	n	CCONJ
ejpam-5566	223	7	)	)	PUNCT
ejpam-5566	223	8	.	.	PUNCT
ejpam-5566	224	1	therefore	therefore	ADV
ejpam-5566	224	2	,	,	PUNCT
ejpam-5566	224	3	b	b	PROPN
ejpam-5566	224	4	⊆	⊆	NUM
ejpam-5566	224	5	eh(c2	eh(c2	NUM
ejpam-5566	224	6	n	n	CCONJ
ejpam-5566	224	7	)	)	PUNCT
ejpam-5566	224	8	.	.	PUNCT
ejpam-5566	225	1	now	now	ADV
ejpam-5566	225	2	,	,	PUNCT
ejpam-5566	225	3	we	we	PRON
ejpam-5566	225	4	give	give	VERB
ejpam-5566	225	5	necessary	necessary	ADJ
ejpam-5566	225	6	and	and	CCONJ
ejpam-5566	225	7	sufficient	sufficient	ADJ
ejpam-5566	225	8	conditions	condition	NOUN
ejpam-5566	225	9	for	for	SCONJ
ejpam-5566	225	10	a	a	DET
ejpam-5566	225	11	subgraph	subgraph	NOUN
ejpam-5566	225	12	h	h	NOUN
ejpam-5566	225	13	to	to	PART
ejpam-5566	225	14	be	be	AUX
ejpam-5566	225	15	a	a	DET
ejpam-5566	225	16	generator	generator	NOUN
ejpam-5566	225	17	subgraph	subgraph	NOUN
ejpam-5566	225	18	of	of	ADP
ejpam-5566	225	19	c2	c2	PROPN
ejpam-5566	225	20	n.	n.	PROPN
ejpam-5566	225	21	lemma	lemma	PROPN
ejpam-5566	226	1	3	3	X
ejpam-5566	226	2	.	.	PUNCT
ejpam-5566	226	3	let	let	VERB
ejpam-5566	226	4	h	h	PRON
ejpam-5566	226	5	be	be	AUX
ejpam-5566	226	6	a	a	DET
ejpam-5566	226	7	subgraph	subgraph	NOUN
ejpam-5566	226	8	of	of	ADP
ejpam-5566	226	9	c2	c2	PROPN
ejpam-5566	226	10	n	n	PROPN
ejpam-5566	226	11	where	where	SCONJ
ejpam-5566	226	12	|e(h)|	|e(h)|	PROPN
ejpam-5566	226	13	is	be	AUX
ejpam-5566	226	14	odd	odd	ADJ
ejpam-5566	226	15	.	.	PUNCT
ejpam-5566	227	1	then	then	ADV
ejpam-5566	227	2	h	h	PROPN
ejpam-5566	227	3	is	be	AUX
ejpam-5566	227	4	a	a	DET
ejpam-5566	227	5	generator	generator	NOUN
ejpam-5566	227	6	subgraph	subgraph	NOUN
ejpam-5566	227	7	of	of	ADP
ejpam-5566	227	8	c2	c2	PROPN
ejpam-5566	227	9	n	n	PROPN
ejpam-5566	227	10	if	if	SCONJ
ejpam-5566	228	1	and	and	CCONJ
ejpam-5566	228	2	only	only	ADV
ejpam-5566	228	3	if	if	SCONJ
ejpam-5566	228	4	{	{	PUNCT
ejpam-5566	228	5	ei	ei	NOUN
ejpam-5566	228	6	,	,	PUNCT
ejpam-5566	228	7	si	si	NOUN
ejpam-5566	228	8	}	}	PUNCT
ejpam-5566	228	9	∈	∈	PROPN
ejpam-5566	228	10	eh(c2	eh(c2	NOUN
ejpam-5566	228	11	n	n	CCONJ
ejpam-5566	228	12	)	)	PUNCT
ejpam-5566	228	13	for	for	ADP
ejpam-5566	228	14	some	some	DET
ejpam-5566	228	15	integer	integer	NOUN
ejpam-5566	228	16	i	i	PRON
ejpam-5566	228	17	,	,	PUNCT
ejpam-5566	228	18	where	where	SCONJ
ejpam-5566	228	19	1	1	NUM
ejpam-5566	228	20	≤	≤	NUM
ejpam-5566	228	21	i	i	PRON
ejpam-5566	228	22	≤	≤	ADJ
ejpam-5566	228	23	n.	n.	NOUN
ejpam-5566	228	24	proof	proof	NOUN
ejpam-5566	228	25	.	.	PUNCT
ejpam-5566	228	26	let	let	VERB
ejpam-5566	228	27	us	we	PRON
ejpam-5566	228	28	consider	consider	VERB
ejpam-5566	228	29	the	the	DET
ejpam-5566	228	30	labeling	labeling	NOUN
ejpam-5566	228	31	of	of	ADP
ejpam-5566	228	32	c2	c2	PROPN
ejpam-5566	228	33	n.	n.	PROPN
ejpam-5566	228	34	assume	assume	VERB
ejpam-5566	228	35	that	that	SCONJ
ejpam-5566	228	36	h	h	NOUN
ejpam-5566	228	37	is	be	AUX
ejpam-5566	228	38	a	a	DET
ejpam-5566	228	39	generator	generator	NOUN
ejpam-5566	228	40	subgraph	subgraph	NOUN
ejpam-5566	228	41	of	of	ADP
ejpam-5566	228	42	c2	c2	PROPN
ejpam-5566	228	43	n.	n.	PROPN
ejpam-5566	228	44	then	then	ADV
ejpam-5566	228	45	{	{	PUNCT
ejpam-5566	228	46	ei	ei	PROPN
ejpam-5566	228	47	,	,	PUNCT
ejpam-5566	228	48	si	si	NOUN
ejpam-5566	228	49	}	}	PUNCT
ejpam-5566	228	50	∈	∈	PROPN
ejpam-5566	228	51	eh(c2	eh(c2	NOUN
ejpam-5566	228	52	n	n	CCONJ
ejpam-5566	228	53	)	)	PUNCT
ejpam-5566	228	54	for	for	ADP
ejpam-5566	228	55	all	all	DET
ejpam-5566	228	56	i.	i.	NOUN
ejpam-5566	228	57	conversely	conversely	ADV
ejpam-5566	228	58	,	,	PUNCT
ejpam-5566	228	59	let	let	VERB
ejpam-5566	228	60	h	h	PRON
ejpam-5566	228	61	be	be	AUX
ejpam-5566	228	62	a	a	DET
ejpam-5566	228	63	subgraph	subgraph	NOUN
ejpam-5566	228	64	of	of	ADP
ejpam-5566	228	65	c2	c2	PROPN
ejpam-5566	228	66	n.	n.	PROPN
ejpam-5566	228	67	we	we	PRON
ejpam-5566	228	68	show	show	VERB
ejpam-5566	228	69	{	{	PUNCT
ejpam-5566	228	70	ei	ei	NOUN
ejpam-5566	228	71	}	}	PUNCT
ejpam-5566	228	72	,	,	PUNCT
ejpam-5566	228	73	{	{	PUNCT
ejpam-5566	228	74	si	si	ADJ
ejpam-5566	228	75	}	}	PUNCT
ejpam-5566	228	76	∈	∈	PROPN
ejpam-5566	228	77	eh(c2	eh(c2	NOUN
ejpam-5566	228	78	n	n	CCONJ
ejpam-5566	228	79	)	)	PUNCT
ejpam-5566	228	80	for	for	ADP
ejpam-5566	228	81	all	all	DET
ejpam-5566	228	82	i	i	PRON
ejpam-5566	228	83	,	,	PUNCT
ejpam-5566	228	84	where	where	SCONJ
ejpam-5566	228	85	1	1	NUM
ejpam-5566	228	86	≤	≤	NUM
ejpam-5566	228	87	i	i	PRON
ejpam-5566	228	88	≤	≤	PROPN
ejpam-5566	228	89	n.	n.	NOUN
ejpam-5566	228	90	since	since	SCONJ
ejpam-5566	228	91	{	{	PUNCT
ejpam-5566	228	92	ei	ei	NOUN
ejpam-5566	228	93	,	,	PUNCT
ejpam-5566	228	94	si	si	NOUN
ejpam-5566	228	95	}	}	PUNCT
ejpam-5566	228	96	∈	∈	PROPN
ejpam-5566	228	97	eh(c2	eh(c2	NOUN
ejpam-5566	228	98	n	n	CCONJ
ejpam-5566	228	99	)	)	PUNCT
ejpam-5566	228	100	for	for	ADP
ejpam-5566	228	101	some	some	DET
ejpam-5566	228	102	integer	integer	NOUN
ejpam-5566	228	103	i	i	PRON
ejpam-5566	228	104	,	,	PUNCT
ejpam-5566	228	105	there	there	PRON
ejpam-5566	228	106	exists	exist	VERB
ejpam-5566	228	107	a	a	DET
ejpam-5566	228	108	∈	∈	PROPN
ejpam-5566	228	109	eh(c2	eh(c2	NOUN
ejpam-5566	228	110	n	n	CCONJ
ejpam-5566	228	111	)	)	PUNCT
ejpam-5566	228	112	,	,	PUNCT
ejpam-5566	228	113	such	such	ADJ
ejpam-5566	228	114	that	that	SCONJ
ejpam-5566	228	115	ei	ei	NOUN
ejpam-5566	228	116	∈	∈	PROPN
ejpam-5566	228	117	a.	a.	NOUN
ejpam-5566	228	118	observe	observe	VERB
ejpam-5566	228	119	that	that	SCONJ
ejpam-5566	228	120	|a|	|a|	PROPN
ejpam-5566	228	121	is	be	AUX
ejpam-5566	228	122	odd	odd	ADJ
ejpam-5566	228	123	since	since	SCONJ
ejpam-5566	228	124	|e(h)|	|e(h)|	PROPN
ejpam-5566	228	125	is	be	AUX
ejpam-5566	228	126	odd	odd	ADJ
ejpam-5566	228	127	.	.	PUNCT
ejpam-5566	229	1	define	define	VERB
ejpam-5566	229	2	b	b	NOUN
ejpam-5566	229	3	=	=	SYM
ejpam-5566	229	4	a\{ei	a\{ei	NOUN
ejpam-5566	229	5	}	}	PUNCT
ejpam-5566	229	6	.	.	PUNCT
ejpam-5566	230	1	then	then	ADV
ejpam-5566	230	2	|b|	|b|	PROPN
ejpam-5566	230	3	is	be	AUX
ejpam-5566	230	4	even	even	ADV
ejpam-5566	230	5	so	so	ADV
ejpam-5566	230	6	b	b	NOUN
ejpam-5566	230	7	∈	∈	PROPN
ejpam-5566	230	8	e	e	NOUN
ejpam-5566	230	9	∗(c2	∗(c2	NOUN
ejpam-5566	230	10	n	n	CCONJ
ejpam-5566	230	11	)	)	PUNCT
ejpam-5566	230	12	.	.	PUNCT
ejpam-5566	231	1	by	by	ADP
ejpam-5566	231	2	lemma	lemma	PROPN
ejpam-5566	231	3	2	2	NUM
ejpam-5566	231	4	,	,	PUNCT
ejpam-5566	231	5	b	b	PROPN
ejpam-5566	231	6	∈	∈	PROPN
ejpam-5566	231	7	eh(c2	eh(c2	NOUN
ejpam-5566	231	8	n	n	CCONJ
ejpam-5566	231	9	)	)	PUNCT
ejpam-5566	231	10	.	.	PUNCT
ejpam-5566	232	1	thus	thus	ADV
ejpam-5566	232	2	,	,	PUNCT
ejpam-5566	232	3	{	{	PUNCT
ejpam-5566	232	4	ei	ei	NOUN
ejpam-5566	232	5	}	}	PUNCT
ejpam-5566	232	6	=	=	PUNCT
ejpam-5566	232	7	a∆b	a∆b	NOUN
ejpam-5566	232	8	∈	∈	PROPN
ejpam-5566	232	9	eh(c2	eh(c2	NOUN
ejpam-5566	232	10	n	n	CCONJ
ejpam-5566	232	11	)	)	PUNCT
ejpam-5566	232	12	.	.	PUNCT
ejpam-5566	233	1	by	by	ADP
ejpam-5566	233	2	rotational	rotational	ADJ
ejpam-5566	233	3	symmetry	symmetry	NOUN
ejpam-5566	233	4	on	on	ADP
ejpam-5566	233	5	c2	c2	PROPN
ejpam-5566	233	6	n	n	CCONJ
ejpam-5566	233	7	,	,	PUNCT
ejpam-5566	233	8	{	{	PUNCT
ejpam-5566	233	9	ei	ei	NOUN
ejpam-5566	233	10	}	}	PUNCT
ejpam-5566	233	11	∈	∈	PROPN
ejpam-5566	233	12	eh(c2	eh(c2	NOUN
ejpam-5566	233	13	n	n	CCONJ
ejpam-5566	233	14	)	)	PUNCT
ejpam-5566	233	15	for	for	ADP
ejpam-5566	233	16	all	all	DET
ejpam-5566	233	17	i.	i.	NOUN
ejpam-5566	233	18	in	in	ADP
ejpam-5566	233	19	similar	similar	ADJ
ejpam-5566	233	20	argument	argument	NOUN
ejpam-5566	233	21	,	,	PUNCT
ejpam-5566	233	22	we	we	PRON
ejpam-5566	233	23	can	can	AUX
ejpam-5566	233	24	show	show	VERB
ejpam-5566	233	25	that	that	SCONJ
ejpam-5566	233	26	{	{	PUNCT
ejpam-5566	233	27	si	si	ADJ
ejpam-5566	233	28	}	}	PUNCT
ejpam-5566	233	29	∈	∈	PROPN
ejpam-5566	233	30	eh(c2	eh(c2	NOUN
ejpam-5566	233	31	n	n	CCONJ
ejpam-5566	233	32	)	)	PUNCT
ejpam-5566	233	33	for	for	ADP
ejpam-5566	233	34	all	all	DET
ejpam-5566	233	35	i.	i.	NOUN
ejpam-5566	233	36	by	by	ADP
ejpam-5566	233	37	remark	remark	NOUN
ejpam-5566	233	38	1	1	NUM
ejpam-5566	233	39	,	,	PUNCT
ejpam-5566	233	40	h	h	NOUN
ejpam-5566	233	41	is	be	AUX
ejpam-5566	233	42	a	a	DET
ejpam-5566	233	43	generator	generator	NOUN
ejpam-5566	233	44	subgraph	subgraph	NOUN
ejpam-5566	233	45	of	of	ADP
ejpam-5566	233	46	c2	c2	PROPN
ejpam-5566	233	47	n.	n.	PROPN
ejpam-5566	233	48	2.2	2.2	NUM
ejpam-5566	233	49	.	.	PUNCT
ejpam-5566	234	1	generator	generator	NOUN
ejpam-5566	234	2	subgraphs	subgraph	NOUN
ejpam-5566	234	3	of	of	ADP
ejpam-5566	234	4	the	the	DET
ejpam-5566	234	5	square	square	NOUN
ejpam-5566	234	6	of	of	ADP
ejpam-5566	234	7	a	a	DET
ejpam-5566	234	8	cycle	cycle	NOUN
ejpam-5566	234	9	first	first	ADV
ejpam-5566	234	10	,	,	PUNCT
ejpam-5566	234	11	we	we	PRON
ejpam-5566	234	12	determine	determine	VERB
ejpam-5566	234	13	the	the	DET
ejpam-5566	234	14	necessary	necessary	ADJ
ejpam-5566	234	15	and	and	CCONJ
ejpam-5566	234	16	sufficient	sufficient	ADJ
ejpam-5566	234	17	conditions	condition	NOUN
ejpam-5566	234	18	for	for	ADP
ejpam-5566	234	19	star	star	NOUN
ejpam-5566	234	20	graph	graph	NOUN
ejpam-5566	234	21	sq	sq	INTJ
ejpam-5566	234	22	to	to	PART
ejpam-5566	234	23	be	be	AUX
ejpam-5566	234	24	a	a	DET
ejpam-5566	234	25	generator	generator	NOUN
ejpam-5566	234	26	subgraph	subgraph	NOUN
ejpam-5566	234	27	of	of	ADP
ejpam-5566	234	28	c2	c2	PROPN
ejpam-5566	234	29	n.	n.	PROPN
ejpam-5566	234	30	theorem	theorem	VERB
ejpam-5566	234	31	9	9	NUM
ejpam-5566	234	32	.	.	PUNCT
ejpam-5566	235	1	let	let	VERB
ejpam-5566	235	2	q	q	NOUN
ejpam-5566	235	3	and	and	CCONJ
ejpam-5566	235	4	n	n	CCONJ
ejpam-5566	235	5	be	be	AUX
ejpam-5566	235	6	positive	positive	ADJ
ejpam-5566	235	7	integers	integer	NOUN
ejpam-5566	235	8	.	.	PUNCT
ejpam-5566	236	1	then	then	ADV
ejpam-5566	236	2	the	the	DET
ejpam-5566	236	3	star	star	NOUN
ejpam-5566	236	4	sq	sq	PROPN
ejpam-5566	236	5	is	be	AUX
ejpam-5566	236	6	a	a	DET
ejpam-5566	236	7	generator	generator	NOUN
ejpam-5566	236	8	subgraph	subgraph	NOUN
ejpam-5566	236	9	of	of	ADP
ejpam-5566	236	10	c2	c2	PROPN
ejpam-5566	236	11	n	n	PROPN
ejpam-5566	236	12	if	if	SCONJ
ejpam-5566	237	1	and	and	CCONJ
ejpam-5566	237	2	only	only	ADV
ejpam-5566	237	3	if	if	SCONJ
ejpam-5566	237	4	q	q	NOUN
ejpam-5566	237	5	=	=	SYM
ejpam-5566	237	6	1	1	NUM
ejpam-5566	237	7	or	or	CCONJ
ejpam-5566	237	8	q	q	ADJ
ejpam-5566	237	9	=	=	SYM
ejpam-5566	237	10	3	3	X
ejpam-5566	237	11	.	.	PUNCT
ejpam-5566	237	12	proof	proof	NOUN
ejpam-5566	237	13	.	.	PUNCT
ejpam-5566	238	1	assume	assume	VERB
ejpam-5566	238	2	that	that	SCONJ
ejpam-5566	238	3	sq	sq	PROPN
ejpam-5566	238	4	is	be	AUX
ejpam-5566	238	5	a	a	DET
ejpam-5566	238	6	generator	generator	NOUN
ejpam-5566	238	7	subgraph	subgraph	NOUN
ejpam-5566	238	8	of	of	ADP
ejpam-5566	238	9	c2	c2	PROPN
ejpam-5566	238	10	n.	n.	PROPN
ejpam-5566	238	11	we	we	PRON
ejpam-5566	238	12	show	show	VERB
ejpam-5566	238	13	that	that	SCONJ
ejpam-5566	238	14	q	q	NOUN
ejpam-5566	239	1	=	=	SYM
ejpam-5566	239	2	1	1	NUM
ejpam-5566	239	3	or	or	CCONJ
ejpam-5566	239	4	q	q	ADJ
ejpam-5566	239	5	=	=	NOUN
ejpam-5566	239	6	3	3	X
ejpam-5566	239	7	.	.	PUNCT
ejpam-5566	239	8	suppose	suppose	VERB
ejpam-5566	239	9	on	on	ADP
ejpam-5566	239	10	the	the	DET
ejpam-5566	239	11	contrary	contrary	NOUN
ejpam-5566	239	12	,	,	PUNCT
ejpam-5566	239	13	q	q	PROPN
ejpam-5566	239	14	̸=	̸=	PROPN
ejpam-5566	239	15	1	1	NUM
ejpam-5566	239	16	and	and	CCONJ
ejpam-5566	239	17	q	q	PROPN
ejpam-5566	239	18	̸=	̸=	PROPN
ejpam-5566	239	19	3	3	NUM
ejpam-5566	239	20	.	.	PUNCT
ejpam-5566	239	21	note	note	VERB
ejpam-5566	239	22	that	that	SCONJ
ejpam-5566	239	23	c2	c2	PROPN
ejpam-5566	239	24	n	n	PART
ejpam-5566	239	25	is	be	AUX
ejpam-5566	239	26	4	4	NUM
ejpam-5566	239	27	-	-	PUNCT
ejpam-5566	239	28	regular	regular	ADJ
ejpam-5566	239	29	.	.	PUNCT
ejpam-5566	240	1	then	then	ADV
ejpam-5566	240	2	by	by	ADP
ejpam-5566	240	3	theorem	theorem	PROPN
ejpam-5566	240	4	1	1	NUM
ejpam-5566	240	5	r.	r.	PROPN
ejpam-5566	240	6	mame	mame	PROPN
ejpam-5566	240	7	/	/	SYM
ejpam-5566	240	8	eur	eur	PROPN
ejpam-5566	240	9	.	.	PUNCT
ejpam-5566	241	1	j.	j.	PROPN
ejpam-5566	241	2	pure	pure	PROPN
ejpam-5566	241	3	appl	appl	PROPN
ejpam-5566	241	4	.	.	PROPN
ejpam-5566	241	5	math	math	PROPN
ejpam-5566	241	6	,	,	PUNCT
ejpam-5566	241	7	17	17	NUM
ejpam-5566	241	8	(	(	PUNCT
ejpam-5566	241	9	4	4	NUM
ejpam-5566	241	10	)	)	PUNCT
ejpam-5566	241	11	(	(	PUNCT
ejpam-5566	241	12	2024	2024	NUM
ejpam-5566	241	13	)	)	PUNCT
ejpam-5566	241	14	,	,	PUNCT
ejpam-5566	241	15	3815	3815	NUM
ejpam-5566	241	16	-	-	SYM
ejpam-5566	241	17	3825	3825	NUM
ejpam-5566	241	18	3821	3821	NUM
ejpam-5566	241	19	,	,	PUNCT
ejpam-5566	241	20	q	q	X
ejpam-5566	241	21	<	<	X
ejpam-5566	241	22	4	4	X
ejpam-5566	241	23	.	.	PUNCT
ejpam-5566	242	1	this	this	PRON
ejpam-5566	242	2	implies	imply	VERB
ejpam-5566	242	3	that	that	PRON
ejpam-5566	242	4	q	q	X
ejpam-5566	242	5	=	=	ADJ
ejpam-5566	242	6	2	2	NUM
ejpam-5566	242	7	.	.	PUNCT
ejpam-5566	242	8	thus	thus	ADV
ejpam-5566	242	9	|e(sq)|	|e(sq)|	NUM
ejpam-5566	242	10	=	=	SYM
ejpam-5566	242	11	2	2	NUM
ejpam-5566	242	12	,	,	PUNCT
ejpam-5566	242	13	which	which	PRON
ejpam-5566	242	14	is	be	AUX
ejpam-5566	242	15	even	even	ADV
ejpam-5566	242	16	.	.	PUNCT
ejpam-5566	243	1	this	this	PRON
ejpam-5566	243	2	is	be	AUX
ejpam-5566	243	3	a	a	DET
ejpam-5566	243	4	contradiction	contradiction	NOUN
ejpam-5566	243	5	in	in	ADP
ejpam-5566	243	6	view	view	NOUN
ejpam-5566	243	7	of	of	ADP
ejpam-5566	243	8	theorem	theorem	NOUN
ejpam-5566	243	9	1	1	NUM
ejpam-5566	243	10	.	.	PUNCT
ejpam-5566	244	1	therefore	therefore	ADV
ejpam-5566	244	2	,	,	PUNCT
ejpam-5566	244	3	q	q	NOUN
ejpam-5566	244	4	=	=	SYM
ejpam-5566	244	5	1	1	NUM
ejpam-5566	244	6	or	or	CCONJ
ejpam-5566	244	7	q	q	ADJ
ejpam-5566	244	8	=	=	NOUN
ejpam-5566	244	9	3	3	X
ejpam-5566	244	10	.	.	PUNCT
ejpam-5566	245	1	conversely	conversely	ADV
ejpam-5566	245	2	,	,	PUNCT
ejpam-5566	245	3	suppose	suppose	VERB
ejpam-5566	245	4	q	q	X
ejpam-5566	245	5	=	=	SYM
ejpam-5566	245	6	1	1	NUM
ejpam-5566	245	7	or	or	CCONJ
ejpam-5566	245	8	q	q	ADJ
ejpam-5566	245	9	=	=	NOUN
ejpam-5566	245	10	3	3	X
ejpam-5566	245	11	.	.	X
ejpam-5566	246	1	we	we	PRON
ejpam-5566	246	2	show	show	VERB
ejpam-5566	246	3	that	that	SCONJ
ejpam-5566	246	4	sq	sq	PROPN
ejpam-5566	246	5	is	be	AUX
ejpam-5566	246	6	a	a	DET
ejpam-5566	246	7	generator	generator	NOUN
ejpam-5566	246	8	subgraph	subgraph	NOUN
ejpam-5566	246	9	of	of	ADP
ejpam-5566	246	10	c2	c2	PROPN
ejpam-5566	246	11	n.	n.	PROPN
ejpam-5566	246	12	case	case	NOUN
ejpam-5566	246	13	1	1	NUM
ejpam-5566	246	14	,	,	PUNCT
ejpam-5566	246	15	q	q	NOUN
ejpam-5566	246	16	=	=	NOUN
ejpam-5566	246	17	1	1	X
ejpam-5566	246	18	.	.	PUNCT
ejpam-5566	246	19	then	then	ADV
ejpam-5566	246	20	sq	sq	PROPN
ejpam-5566	246	21	is	be	AUX
ejpam-5566	246	22	isomorphic	isomorphic	ADJ
ejpam-5566	246	23	to	to	ADP
ejpam-5566	246	24	p2	p2	PROPN
ejpam-5566	246	25	.	.	PUNCT
ejpam-5566	247	1	by	by	ADP
ejpam-5566	247	2	theorem	theorem	NOUN
ejpam-5566	247	3	3	3	NUM
ejpam-5566	247	4	,	,	PUNCT
ejpam-5566	247	5	sq	sq	X
ejpam-5566	247	6	is	be	AUX
ejpam-5566	247	7	a	a	DET
ejpam-5566	247	8	generator	generator	NOUN
ejpam-5566	247	9	subgraph	subgraph	NOUN
ejpam-5566	247	10	of	of	ADP
ejpam-5566	247	11	c2	c2	PROPN
ejpam-5566	247	12	n.	n.	PROPN
ejpam-5566	247	13	for	for	ADP
ejpam-5566	247	14	case	case	NOUN
ejpam-5566	247	15	2	2	NUM
ejpam-5566	247	16	,	,	PUNCT
ejpam-5566	247	17	q	q	NOUN
ejpam-5566	247	18	=	=	NOUN
ejpam-5566	247	19	3	3	NUM
ejpam-5566	247	20	,	,	PUNCT
ejpam-5566	247	21	since	since	SCONJ
ejpam-5566	247	22	c2	c2	PROPN
ejpam-5566	247	23	n	n	PART
ejpam-5566	247	24	is	be	AUX
ejpam-5566	247	25	a	a	DET
ejpam-5566	247	26	4	4	NUM
ejpam-5566	247	27	-	-	PUNCT
ejpam-5566	247	28	regular	regular	ADJ
ejpam-5566	247	29	graph	graph	NOUN
ejpam-5566	247	30	and	and	CCONJ
ejpam-5566	247	31	q	q	NOUN
ejpam-5566	247	32	<	<	X
ejpam-5566	247	33	4	4	NUM
ejpam-5566	247	34	then	then	ADV
ejpam-5566	247	35	the	the	DET
ejpam-5566	247	36	star	star	NOUN
ejpam-5566	247	37	sq	sq	PROPN
ejpam-5566	247	38	is	be	AUX
ejpam-5566	247	39	a	a	DET
ejpam-5566	247	40	generator	generator	NOUN
ejpam-5566	247	41	subgraph	subgraph	NOUN
ejpam-5566	247	42	of	of	ADP
ejpam-5566	247	43	c2	c2	PROPN
ejpam-5566	247	44	n	n	PROPN
ejpam-5566	247	45	in	in	ADP
ejpam-5566	247	46	view	view	NOUN
ejpam-5566	247	47	of	of	ADP
ejpam-5566	247	48	corollary	corollary	ADJ
ejpam-5566	247	49	1	1	NUM
ejpam-5566	247	50	.	.	PUNCT
ejpam-5566	248	1	the	the	DET
ejpam-5566	248	2	theorem	theorem	NOUN
ejpam-5566	248	3	below	below	ADP
ejpam-5566	248	4	determines	determine	VERB
ejpam-5566	248	5	the	the	DET
ejpam-5566	248	6	necessary	necessary	ADJ
ejpam-5566	248	7	and	and	CCONJ
ejpam-5566	248	8	sufficient	sufficient	ADJ
ejpam-5566	248	9	conditions	condition	NOUN
ejpam-5566	248	10	for	for	ADP
ejpam-5566	248	11	the	the	DET
ejpam-5566	248	12	path	path	NOUN
ejpam-5566	248	13	pk	pk	NOUN
ejpam-5566	248	14	to	to	PART
ejpam-5566	248	15	be	be	AUX
ejpam-5566	248	16	a	a	DET
ejpam-5566	248	17	generator	generator	NOUN
ejpam-5566	248	18	subgraph	subgraph	NOUN
ejpam-5566	248	19	of	of	ADP
ejpam-5566	248	20	c2	c2	PROPN
ejpam-5566	248	21	n.	n.	PROPN
ejpam-5566	248	22	theorem	theorem	VERB
ejpam-5566	248	23	10	10	NUM
ejpam-5566	248	24	.	.	PUNCT
ejpam-5566	249	1	let	let	VERB
ejpam-5566	249	2	k	k	NOUN
ejpam-5566	249	3	and	and	CCONJ
ejpam-5566	249	4	n	n	PROPN
ejpam-5566	249	5	are	be	AUX
ejpam-5566	249	6	positive	positive	ADJ
ejpam-5566	249	7	integers	integer	NOUN
ejpam-5566	249	8	.	.	PUNCT
ejpam-5566	250	1	then	then	ADV
ejpam-5566	250	2	the	the	DET
ejpam-5566	250	3	path	path	NOUN
ejpam-5566	250	4	pk	pk	NOUN
ejpam-5566	250	5	is	be	AUX
ejpam-5566	250	6	a	a	DET
ejpam-5566	250	7	generator	generator	NOUN
ejpam-5566	250	8	subgraph	subgraph	NOUN
ejpam-5566	250	9	of	of	ADP
ejpam-5566	250	10	c2	c2	PROPN
ejpam-5566	250	11	n	n	PROPN
ejpam-5566	250	12	if	if	SCONJ
ejpam-5566	251	1	and	and	CCONJ
ejpam-5566	251	2	only	only	ADV
ejpam-5566	251	3	if	if	SCONJ
ejpam-5566	251	4	k	k	PROPN
ejpam-5566	251	5	is	be	AUX
ejpam-5566	251	6	even	even	ADV
ejpam-5566	251	7	and	and	CCONJ
ejpam-5566	252	1	2	2	NUM
ejpam-5566	252	2	≤	≤	NUM
ejpam-5566	252	3	k	k	X
ejpam-5566	252	4	≤	≤	PROPN
ejpam-5566	252	5	n.	n.	NOUN
ejpam-5566	252	6	proof	proof	NOUN
ejpam-5566	252	7	.	.	PUNCT
ejpam-5566	253	1	assume	assume	VERB
ejpam-5566	253	2	that	that	SCONJ
ejpam-5566	253	3	pk	pk	NOUN
ejpam-5566	253	4	is	be	AUX
ejpam-5566	253	5	a	a	DET
ejpam-5566	253	6	generator	generator	NOUN
ejpam-5566	253	7	subgraph	subgraph	NOUN
ejpam-5566	253	8	of	of	ADP
ejpam-5566	253	9	c2	c2	PROPN
ejpam-5566	253	10	n.	n.	PROPN
ejpam-5566	253	11	then	then	ADV
ejpam-5566	253	12	by	by	ADP
ejpam-5566	253	13	theorem	theorem	NOUN
ejpam-5566	253	14	1	1	NUM
ejpam-5566	253	15	,	,	PUNCT
ejpam-5566	253	16	|e(pk)|	|e(pk)|	PRON
ejpam-5566	253	17	must	must	AUX
ejpam-5566	253	18	be	be	AUX
ejpam-5566	253	19	odd	odd	ADJ
ejpam-5566	253	20	.	.	PUNCT
ejpam-5566	254	1	this	this	PRON
ejpam-5566	254	2	implies	imply	VERB
ejpam-5566	254	3	that	that	SCONJ
ejpam-5566	254	4	k	k	PROPN
ejpam-5566	254	5	is	be	AUX
ejpam-5566	254	6	even	even	ADV
ejpam-5566	254	7	.	.	PUNCT
ejpam-5566	255	1	next	next	ADV
ejpam-5566	255	2	,	,	PUNCT
ejpam-5566	255	3	we	we	PRON
ejpam-5566	255	4	claim	claim	VERB
ejpam-5566	255	5	that	that	SCONJ
ejpam-5566	255	6	2	2	NUM
ejpam-5566	255	7	≤	≤	NUM
ejpam-5566	255	8	k	k	PROPN
ejpam-5566	255	9	≤	≤	PROPN
ejpam-5566	255	10	n.	n.	NOUN
ejpam-5566	255	11	suppose	suppose	VERB
ejpam-5566	255	12	not	not	PART
ejpam-5566	255	13	,	,	PUNCT
ejpam-5566	255	14	then	then	ADV
ejpam-5566	255	15	either	either	CCONJ
ejpam-5566	255	16	k	k	PROPN
ejpam-5566	255	17	=	=	SYM
ejpam-5566	255	18	1	1	NUM
ejpam-5566	255	19	or	or	CCONJ
ejpam-5566	255	20	k	k	ADJ
ejpam-5566	255	21	>	>	X
ejpam-5566	255	22	n.	n.	PROPN
ejpam-5566	255	23	if	if	SCONJ
ejpam-5566	255	24	k	k	PROPN
ejpam-5566	255	25	=	=	SYM
ejpam-5566	255	26	1	1	NUM
ejpam-5566	255	27	,	,	PUNCT
ejpam-5566	255	28	then	then	ADV
ejpam-5566	255	29	pk	pk	NOUN
ejpam-5566	255	30	has	have	VERB
ejpam-5566	255	31	no	no	DET
ejpam-5566	255	32	edge	edge	NOUN
ejpam-5566	255	33	so	so	ADV
ejpam-5566	255	34	epk	epk	PROPN
ejpam-5566	255	35	(	(	PUNCT
ejpam-5566	255	36	c2	c2	PROPN
ejpam-5566	255	37	n	n	CCONJ
ejpam-5566	255	38	)	)	PUNCT
ejpam-5566	255	39	=	=	PUNCT
ejpam-5566	255	40	∅.	∅.	NOUN
ejpam-5566	255	41	if	if	SCONJ
ejpam-5566	255	42	k	k	PROPN
ejpam-5566	255	43	>	>	X
ejpam-5566	255	44	n	n	CCONJ
ejpam-5566	255	45	,	,	PUNCT
ejpam-5566	255	46	then	then	ADV
ejpam-5566	255	47	pk	pk	NOUN
ejpam-5566	255	48	is	be	AUX
ejpam-5566	255	49	not	not	PART
ejpam-5566	255	50	a	a	DET
ejpam-5566	255	51	subgraph	subgraph	NOUN
ejpam-5566	255	52	of	of	ADP
ejpam-5566	255	53	c2	c2	PROPN
ejpam-5566	255	54	n	n	CCONJ
ejpam-5566	255	55	,	,	PUNCT
ejpam-5566	255	56	so	so	ADV
ejpam-5566	255	57	epk	epk	PROPN
ejpam-5566	255	58	(	(	PUNCT
ejpam-5566	255	59	c2	c2	PROPN
ejpam-5566	255	60	n	n	CCONJ
ejpam-5566	255	61	)	)	PUNCT
ejpam-5566	255	62	=	=	NOUN
ejpam-5566	255	63	∅	∅	NOUN
ejpam-5566	255	64	also	also	ADV
ejpam-5566	255	65	.	.	PUNCT
ejpam-5566	256	1	in	in	ADP
ejpam-5566	256	2	either	either	DET
ejpam-5566	256	3	case	case	NOUN
ejpam-5566	256	4	epk	epk	X
ejpam-5566	256	5	(	(	PUNCT
ejpam-5566	256	6	c2	c2	PROPN
ejpam-5566	256	7	n	n	CCONJ
ejpam-5566	256	8	)	)	PUNCT
ejpam-5566	256	9	=	=	NOUN
ejpam-5566	256	10	∅	∅	NOUN
ejpam-5566	256	11	=	=	NOUN
ejpam-5566	256	12	̸	̸	NUM
ejpam-5566	256	13	e	e	NOUN
ejpam-5566	256	14	(	(	PUNCT
ejpam-5566	256	15	c2	c2	PROPN
ejpam-5566	256	16	n	n	CCONJ
ejpam-5566	256	17	)	)	PUNCT
ejpam-5566	256	18	.	.	PUNCT
ejpam-5566	257	1	this	this	PRON
ejpam-5566	257	2	is	be	AUX
ejpam-5566	257	3	a	a	DET
ejpam-5566	257	4	contradiction	contradiction	NOUN
ejpam-5566	257	5	to	to	ADP
ejpam-5566	257	6	the	the	DET
ejpam-5566	257	7	assumption	assumption	NOUN
ejpam-5566	257	8	that	that	SCONJ
ejpam-5566	257	9	pk	pk	NOUN
ejpam-5566	257	10	is	be	AUX
ejpam-5566	257	11	a	a	DET
ejpam-5566	257	12	generator	generator	NOUN
ejpam-5566	257	13	subgraph	subgraph	NOUN
ejpam-5566	257	14	of	of	ADP
ejpam-5566	257	15	c2	c2	PROPN
ejpam-5566	257	16	n.	n.	PROPN
ejpam-5566	257	17	conversely	conversely	ADV
ejpam-5566	257	18	,	,	PUNCT
ejpam-5566	257	19	since	since	SCONJ
ejpam-5566	257	20	k	k	PROPN
ejpam-5566	257	21	is	be	AUX
ejpam-5566	257	22	even	even	ADV
ejpam-5566	257	23	,	,	PUNCT
ejpam-5566	257	24	|e(pk)|	|e(pk)|	PRON
ejpam-5566	257	25	is	be	AUX
ejpam-5566	257	26	odd	odd	ADJ
ejpam-5566	257	27	.	.	PUNCT
ejpam-5566	258	1	if	if	SCONJ
ejpam-5566	258	2	k	k	PROPN
ejpam-5566	258	3	=	=	SYM
ejpam-5566	258	4	2	2	NUM
ejpam-5566	258	5	,	,	PUNCT
ejpam-5566	258	6	then	then	ADV
ejpam-5566	258	7	pk	pk	NOUN
ejpam-5566	258	8	≃	≃	ADJ
ejpam-5566	258	9	p2	p2	NOUN
ejpam-5566	258	10	.	.	PUNCT
ejpam-5566	259	1	by	by	ADP
ejpam-5566	259	2	theorem	theorem	NOUN
ejpam-5566	259	3	3	3	NUM
ejpam-5566	259	4	,	,	PUNCT
ejpam-5566	259	5	pk	pk	NOUN
ejpam-5566	259	6	is	be	AUX
ejpam-5566	259	7	a	a	DET
ejpam-5566	259	8	generator	generator	NOUN
ejpam-5566	259	9	subgraph	subgraph	NOUN
ejpam-5566	259	10	of	of	ADP
ejpam-5566	259	11	c2	c2	PROPN
ejpam-5566	259	12	n.let	n.let	VERB
ejpam-5566	259	13	us	we	PRON
ejpam-5566	259	14	assume	assume	VERB
ejpam-5566	259	15	that	that	SCONJ
ejpam-5566	259	16	2	2	NUM
ejpam-5566	259	17	<	<	X
ejpam-5566	259	18	k	k	PROPN
ejpam-5566	259	19	≤	≤	PROPN
ejpam-5566	259	20	n.	n.	NOUN
ejpam-5566	259	21	consider	consider	VERB
ejpam-5566	259	22	the	the	DET
ejpam-5566	259	23	labeling	labeling	NOUN
ejpam-5566	259	24	of	of	ADP
ejpam-5566	259	25	c2	c2	PROPN
ejpam-5566	259	26	n.	n.	PROPN
ejpam-5566	259	27	let	let	VERB
ejpam-5566	259	28	a	a	DET
ejpam-5566	259	29	=	=	SYM
ejpam-5566	259	30	{	{	PUNCT
ejpam-5566	259	31	e1	e1	PROPN
ejpam-5566	259	32	,	,	PUNCT
ejpam-5566	259	33	e2	e2	PROPN
ejpam-5566	259	34	,	,	PUNCT
ejpam-5566	259	35	e3	e3	NOUN
ejpam-5566	259	36	,	,	PUNCT
ejpam-5566	259	37	.	.	PUNCT
ejpam-5566	259	38	.	.	PUNCT
ejpam-5566	260	1	.	.	PUNCT
ejpam-5566	261	1	,	,	PUNCT
ejpam-5566	261	2	ek	ek	X
ejpam-5566	261	3	}	}	PUNCT
ejpam-5566	261	4	and	and	CCONJ
ejpam-5566	261	5	define	define	VERB
ejpam-5566	261	6	b	b	NOUN
ejpam-5566	261	7	=	=	PUNCT
ejpam-5566	261	8	a\{e2}∪{s1	a\{e2}∪{s1	NOUN
ejpam-5566	261	9	}	}	PUNCT
ejpam-5566	261	10	.	.	PUNCT
ejpam-5566	262	1	it	it	PRON
ejpam-5566	262	2	can	can	AUX
ejpam-5566	262	3	be	be	AUX
ejpam-5566	262	4	verified	verify	VERB
ejpam-5566	262	5	that	that	SCONJ
ejpam-5566	262	6	a	a	DET
ejpam-5566	262	7	,	,	PUNCT
ejpam-5566	262	8	b	b	PROPN
ejpam-5566	262	9	∈	∈	PROPN
ejpam-5566	262	10	epk	epk	X
ejpam-5566	262	11	(	(	PUNCT
ejpam-5566	262	12	c2	c2	PROPN
ejpam-5566	262	13	n	n	CCONJ
ejpam-5566	262	14	)	)	PUNCT
ejpam-5566	262	15	,	,	PUNCT
ejpam-5566	262	16	as	as	SCONJ
ejpam-5566	262	17	shown	show	VERB
ejpam-5566	262	18	in	in	ADP
ejpam-5566	262	19	figure	figure	NOUN
ejpam-5566	262	20	3	3	NUM
ejpam-5566	262	21	.	.	PUNCT
ejpam-5566	262	22	....................................	....................................	PUNCT
ejpam-5566	262	23	....................................	....................................	PUNCT
ejpam-5566	262	24	....................................	....................................	PUNCT
ejpam-5566	263	1	....................................	....................................	PUNCT
ejpam-5566	263	2	....................................	....................................	PUNCT
ejpam-5566	264	1	....................................	....................................	PUNCT
ejpam-5566	264	2	....................................	....................................	PUNCT
ejpam-5566	265	1	............................................................................................................................................................	............................................................................................................................................................	PUNCT
ejpam-5566	265	2	..............................................................................	..............................................................................	PUNCT
ejpam-5566	265	3	..............................................................................	..............................................................................	PUNCT
ejpam-5566	266	1	..............	..............	PUNCT
ejpam-5566	266	2	.............	.............	PUNCT
ejpam-5566	266	3	.............	.............	PUNCT
ejpam-5566	266	4	.............	.............	PUNCT
ejpam-5566	266	5	.............	.............	PUNCT
ejpam-5566	266	6	............	............	PUNCT
ejpam-5566	267	1	e1	e1	NOUN
ejpam-5566	267	2	e2	e2	PROPN
ejpam-5566	267	3	e3	e3	NOUN
ejpam-5566	267	4	ek−1	ek−1	NOUN
ejpam-5566	267	5	ek	ek	PROPN
ejpam-5566	267	6	c2	c2	PROPN
ejpam-5566	267	7	n[a	n[a	PROPN
ejpam-5566	267	8	]	]	PUNCT
ejpam-5566	267	9	....................................	....................................	PUNCT
ejpam-5566	267	10	....................................	....................................	PUNCT
ejpam-5566	267	11	....................................	....................................	PUNCT
ejpam-5566	267	12	....................................	....................................	PUNCT
ejpam-5566	267	13	....................................	....................................	PUNCT
ejpam-5566	267	14	....................................	....................................	PUNCT
ejpam-5566	267	15	....................................	....................................	PUNCT
ejpam-5566	267	16	..............................................................................	..............................................................................	PUNCT
ejpam-5566	267	17	..............................................................................	..............................................................................	PUNCT
ejpam-5566	267	18	..............................................................................	..............................................................................	PUNCT
ejpam-5566	267	19	..............	..............	PUNCT
ejpam-5566	267	20	.............	.............	PUNCT
ejpam-5566	267	21	.............	.............	PUNCT
ejpam-5566	267	22	.............	.............	PUNCT
ejpam-5566	267	23	.............	.............	PUNCT
ejpam-5566	267	24	............	............	PUNCT
ejpam-5566	267	25	........................................................................................................................................................	........................................................................................................................................................	PUNCT
ejpam-5566	267	26	........	........	PUNCT
ejpam-5566	267	27	........	........	PUNCT
ejpam-5566	268	1	........	........	PUNCT
ejpam-5566	269	1	e1	e1	VERB
ejpam-5566	269	2	e3	e3	NOUN
ejpam-5566	269	3	ek−1	ek−1	NOUN
ejpam-5566	269	4	ek	ek	PROPN
ejpam-5566	269	5	s1	s1	PROPN
ejpam-5566	269	6	c2	c2	PROPN
ejpam-5566	269	7	n[b	n[b	PROPN
ejpam-5566	269	8	]	]	PUNCT
ejpam-5566	269	9	........	........	PUNCT
ejpam-5566	269	10	........	........	PUNCT
ejpam-5566	270	1	........	........	PUNCT
ejpam-5566	270	2	figure	figure	VERB
ejpam-5566	270	3	3	3	NUM
ejpam-5566	270	4	:	:	PUNCT
ejpam-5566	270	5	illustrating	illustrate	VERB
ejpam-5566	270	6	the	the	DET
ejpam-5566	270	7	subgraphs	subgraph	NOUN
ejpam-5566	270	8	c2	c2	PROPN
ejpam-5566	270	9	n[a	n[a	PROPN
ejpam-5566	270	10	]	]	PUNCT
ejpam-5566	270	11	and	and	CCONJ
ejpam-5566	270	12	c2	c2	PROPN
ejpam-5566	270	13	n[b	n[b	PROPN
ejpam-5566	270	14	]	]	PUNCT
ejpam-5566	270	15	now	now	ADV
ejpam-5566	270	16	,	,	PUNCT
ejpam-5566	270	17	a∆b	a∆b	NOUN
ejpam-5566	270	18	=	=	SYM
ejpam-5566	270	19	{	{	PUNCT
ejpam-5566	270	20	e2	e2	PROPN
ejpam-5566	270	21	,	,	PUNCT
ejpam-5566	270	22	s1	s1	NOUN
ejpam-5566	270	23	}	}	PUNCT
ejpam-5566	270	24	∈	∈	PROPN
ejpam-5566	270	25	epk	epk	X
ejpam-5566	270	26	(	(	PUNCT
ejpam-5566	270	27	c2	c2	PROPN
ejpam-5566	270	28	n	n	CCONJ
ejpam-5566	270	29	)	)	PUNCT
ejpam-5566	270	30	.	.	PUNCT
ejpam-5566	271	1	by	by	ADP
ejpam-5566	271	2	lemma	lemma	PROPN
ejpam-5566	271	3	2	2	NUM
ejpam-5566	271	4	,	,	PUNCT
ejpam-5566	271	5	{	{	PUNCT
ejpam-5566	271	6	e1	e1	NOUN
ejpam-5566	271	7	,	,	PUNCT
ejpam-5566	271	8	s1	s1	NOUN
ejpam-5566	271	9	}	}	PUNCT
ejpam-5566	271	10	∈	∈	PROPN
ejpam-5566	271	11	epk	epk	X
ejpam-5566	271	12	(	(	PUNCT
ejpam-5566	271	13	c2	c2	PROPN
ejpam-5566	271	14	n	n	CCONJ
ejpam-5566	271	15	)	)	PUNCT
ejpam-5566	271	16	.	.	PUNCT
ejpam-5566	272	1	by	by	ADP
ejpam-5566	272	2	lemma	lemma	PROPN
ejpam-5566	272	3	3.4.3	3.4.3	PROPN
ejpam-5566	272	4	,	,	PUNCT
ejpam-5566	272	5	pk	pk	NOUN
ejpam-5566	272	6	is	be	AUX
ejpam-5566	272	7	a	a	DET
ejpam-5566	272	8	generator	generator	NOUN
ejpam-5566	272	9	subgraph	subgraph	NOUN
ejpam-5566	272	10	of	of	ADP
ejpam-5566	272	11	c2	c2	PROPN
ejpam-5566	272	12	n.	n.	PROPN
ejpam-5566	272	13	we	we	PRON
ejpam-5566	272	14	consider	consider	VERB
ejpam-5566	272	15	another	another	DET
ejpam-5566	272	16	class	class	NOUN
ejpam-5566	272	17	of	of	ADP
ejpam-5566	272	18	subgraphs	subgraph	NOUN
ejpam-5566	272	19	of	of	ADP
ejpam-5566	272	20	c2	c2	PROPN
ejpam-5566	272	21	n	n	CCONJ
ejpam-5566	272	22	,	,	PUNCT
ejpam-5566	272	23	the	the	DET
ejpam-5566	272	24	tadpole	tadpole	NOUN
ejpam-5566	272	25	graph	graph	NOUN
ejpam-5566	272	26	.	.	PUNCT
ejpam-5566	273	1	the	the	DET
ejpam-5566	273	2	(	(	PUNCT
ejpam-5566	273	3	k	k	NOUN
ejpam-5566	273	4	,	,	PUNCT
ejpam-5566	273	5	r)tadpole	r)tadpole	NOUN
ejpam-5566	273	6	graph	graph	NOUN
ejpam-5566	273	7	,	,	PUNCT
ejpam-5566	273	8	denoted	denote	VERB
ejpam-5566	273	9	by	by	ADP
ejpam-5566	273	10	tk	tk	PROPN
ejpam-5566	273	11	,	,	PUNCT
ejpam-5566	273	12	r	r	PROPN
ejpam-5566	273	13	,	,	PUNCT
ejpam-5566	273	14	is	be	AUX
ejpam-5566	273	15	the	the	DET
ejpam-5566	273	16	graph	graph	NOUN
ejpam-5566	273	17	obtained	obtain	VERB
ejpam-5566	273	18	by	by	ADP
ejpam-5566	273	19	joining	join	VERB
ejpam-5566	273	20	a	a	DET
ejpam-5566	273	21	cycle	cycle	NOUN
ejpam-5566	273	22	graph	graph	NOUN
ejpam-5566	273	23	ck	ck	INTJ
ejpam-5566	273	24	to	to	ADP
ejpam-5566	273	25	a	a	DET
ejpam-5566	273	26	path	path	NOUN
ejpam-5566	273	27	graph	graph	NOUN
ejpam-5566	273	28	pr	pr	NOUN
ejpam-5566	273	29	with	with	ADP
ejpam-5566	273	30	an	an	DET
ejpam-5566	273	31	edge	edge	NOUN
ejpam-5566	273	32	[	[	X
ejpam-5566	273	33	a	a	X
ejpam-5566	273	34	,	,	PUNCT
ejpam-5566	273	35	b	b	NOUN
ejpam-5566	273	36	]	]	X
ejpam-5566	273	37	where	where	SCONJ
ejpam-5566	273	38	a	a	DET
ejpam-5566	273	39	∈	∈	PROPN
ejpam-5566	273	40	v	v	NOUN
ejpam-5566	273	41	(	(	PUNCT
ejpam-5566	273	42	ck	ck	NOUN
ejpam-5566	273	43	)	)	PUNCT
ejpam-5566	273	44	and	and	CCONJ
ejpam-5566	273	45	b	b	X
ejpam-5566	273	46	∈	∈	PROPN
ejpam-5566	273	47	v	v	NOUN
ejpam-5566	273	48	(	(	PUNCT
ejpam-5566	273	49	pr	pr	NOUN
ejpam-5566	273	50	)	)	PUNCT
ejpam-5566	273	51	,	,	PUNCT
ejpam-5566	273	52	deg(b	deg(b	NUM
ejpam-5566	273	53	)	)	PUNCT
ejpam-5566	273	54	in	in	ADP
ejpam-5566	273	55	pr	pr	NOUN
ejpam-5566	273	56	is	be	AUX
ejpam-5566	273	57	either	either	CCONJ
ejpam-5566	273	58	0	0	NUM
ejpam-5566	273	59	or	or	CCONJ
ejpam-5566	273	60	1	1	NUM
ejpam-5566	273	61	.	.	X
ejpam-5566	274	1	for	for	ADP
ejpam-5566	274	2	instance	instance	NOUN
ejpam-5566	274	3	the	the	DET
ejpam-5566	274	4	graphs	graph	NOUN
ejpam-5566	274	5	t6,2	t6,2	PROPN
ejpam-5566	274	6	and	and	CCONJ
ejpam-5566	274	7	t8,1	t8,1	PROPN
ejpam-5566	274	8	are	be	AUX
ejpam-5566	274	9	shown	show	VERB
ejpam-5566	274	10	in	in	ADP
ejpam-5566	274	11	figure	figure	NOUN
ejpam-5566	274	12	4	4	NUM
ejpam-5566	274	13	.	.	PUNCT
ejpam-5566	275	1	first	first	ADV
ejpam-5566	275	2	,	,	PUNCT
ejpam-5566	275	3	we	we	PRON
ejpam-5566	275	4	investigated	investigate	VERB
ejpam-5566	275	5	the	the	DET
ejpam-5566	275	6	tadpole	tadpole	NOUN
ejpam-5566	275	7	graph	graph	NOUN
ejpam-5566	275	8	t3,2	t3,2	PROPN
ejpam-5566	275	9	.	.	PUNCT
ejpam-5566	276	1	the	the	DET
ejpam-5566	276	2	result	result	NOUN
ejpam-5566	276	3	is	be	AUX
ejpam-5566	276	4	stated	state	VERB
ejpam-5566	276	5	below	below	ADV
ejpam-5566	276	6	.	.	PUNCT
ejpam-5566	277	1	theorem	theorem	VERB
ejpam-5566	277	2	11	11	NUM
ejpam-5566	277	3	.	.	PUNCT
ejpam-5566	278	1	the	the	DET
ejpam-5566	278	2	tadpole	tadpole	NOUN
ejpam-5566	278	3	graph	graph	NOUN
ejpam-5566	278	4	t3,2	t3,2	PROPN
ejpam-5566	278	5	is	be	AUX
ejpam-5566	278	6	a	a	DET
ejpam-5566	278	7	generator	generator	NOUN
ejpam-5566	278	8	subgraph	subgraph	NOUN
ejpam-5566	278	9	of	of	ADP
ejpam-5566	278	10	c2	c2	PROPN
ejpam-5566	278	11	n.	n.	PROPN
ejpam-5566	278	12	proof	proof	PROPN
ejpam-5566	278	13	.	.	PUNCT
ejpam-5566	279	1	consider	consider	VERB
ejpam-5566	279	2	the	the	DET
ejpam-5566	279	3	labeling	labeling	NOUN
ejpam-5566	279	4	of	of	ADP
ejpam-5566	279	5	c2	c2	PROPN
ejpam-5566	279	6	n.	n.	PROPN
ejpam-5566	279	7	let	let	VERB
ejpam-5566	279	8	a	a	DET
ejpam-5566	279	9	=	=	SYM
ejpam-5566	279	10	{	{	PUNCT
ejpam-5566	279	11	e1	e1	PROPN
ejpam-5566	279	12	,	,	PUNCT
ejpam-5566	279	13	e2	e2	PROPN
ejpam-5566	279	14	,	,	PUNCT
ejpam-5566	279	15	e3	e3	NOUN
ejpam-5566	279	16	,	,	PUNCT
ejpam-5566	279	17	e4	e4	PROPN
ejpam-5566	279	18	,	,	PUNCT
ejpam-5566	279	19	s1	s1	NOUN
ejpam-5566	279	20	}	}	PUNCT
ejpam-5566	279	21	and	and	CCONJ
ejpam-5566	279	22	define	define	VERB
ejpam-5566	279	23	b	b	NOUN
ejpam-5566	279	24	=	=	NOUN
ejpam-5566	279	25	a\{e3}∪	a\{e3}∪	PROPN
ejpam-5566	279	26	{	{	PUNCT
ejpam-5566	279	27	s3	s3	PROPN
ejpam-5566	279	28	}	}	PUNCT
ejpam-5566	279	29	.	.	PUNCT
ejpam-5566	280	1	as	as	SCONJ
ejpam-5566	280	2	shown	show	VERB
ejpam-5566	280	3	in	in	ADP
ejpam-5566	280	4	figure	figure	NOUN
ejpam-5566	280	5	5	5	NUM
ejpam-5566	280	6	,	,	PUNCT
ejpam-5566	280	7	it	it	PRON
ejpam-5566	280	8	can	can	AUX
ejpam-5566	280	9	be	be	AUX
ejpam-5566	280	10	observed	observe	VERB
ejpam-5566	280	11	that	that	SCONJ
ejpam-5566	280	12	c2	c2	PROPN
ejpam-5566	280	13	n[a	n[a	PROPN
ejpam-5566	280	14	]	]	PUNCT
ejpam-5566	280	15	≃	≃	VERB
ejpam-5566	280	16	t3,2	t3,2	PROPN
ejpam-5566	280	17	and	and	CCONJ
ejpam-5566	280	18	c2	c2	PROPN
ejpam-5566	280	19	n[b	n[b	PROPN
ejpam-5566	280	20	]	]	PUNCT
ejpam-5566	280	21	≃	≃	PROPN
ejpam-5566	280	22	t3,2	t3,2	PROPN
ejpam-5566	280	23	.	.	PUNCT
ejpam-5566	281	1	thus	thus	ADV
ejpam-5566	281	2	r.	r.	PROPN
ejpam-5566	281	3	mame	mame	PROPN
ejpam-5566	281	4	/	/	SYM
ejpam-5566	281	5	eur	eur	PROPN
ejpam-5566	281	6	.	.	PUNCT
ejpam-5566	282	1	j.	j.	PROPN
ejpam-5566	282	2	pure	pure	PROPN
ejpam-5566	282	3	appl	appl	PROPN
ejpam-5566	282	4	.	.	PROPN
ejpam-5566	282	5	math	math	PROPN
ejpam-5566	282	6	,	,	PUNCT
ejpam-5566	282	7	17	17	NUM
ejpam-5566	282	8	(	(	PUNCT
ejpam-5566	282	9	4	4	NUM
ejpam-5566	282	10	)	)	PUNCT
ejpam-5566	282	11	(	(	PUNCT
ejpam-5566	282	12	2024	2024	NUM
ejpam-5566	282	13	)	)	PUNCT
ejpam-5566	282	14	,	,	PUNCT
ejpam-5566	282	15	3815	3815	NUM
ejpam-5566	282	16	-	-	SYM
ejpam-5566	282	17	3825	3825	NUM
ejpam-5566	282	18	3822	3822	NUM
ejpam-5566	282	19	..............................	..............................	PUNCT
ejpam-5566	283	1	......	......	PUNCT
ejpam-5566	283	2	..............................	..............................	PUNCT
ejpam-5566	284	1	......	......	PUNCT
ejpam-5566	284	2	.........	.........	PUNCT
ejpam-5566	284	3	........	........	PUNCT
ejpam-5566	284	4	........	........	PUNCT
ejpam-5566	284	5	........	........	PUNCT
ejpam-5566	284	6	........	........	PUNCT
ejpam-5566	285	1	......	......	PUNCT
ejpam-5566	285	2	.................	.................	PUNCT
ejpam-5566	286	1	................	................	PUNCT
ejpam-5566	286	2	..............	..............	PUNCT
ejpam-5566	286	3	..............................	..............................	PUNCT
ejpam-5566	286	4	......	......	PUNCT
ejpam-5566	286	5	...............................................	...............................................	PUNCT
ejpam-5566	286	6	..............................	..............................	PUNCT
ejpam-5566	287	1	......	......	PUNCT
ejpam-5566	287	2	...............................................	...............................................	PUNCT
ejpam-5566	287	3	..............................	..............................	PUNCT
ejpam-5566	288	1	......	......	PUNCT
ejpam-5566	288	2	.................	.................	PUNCT
ejpam-5566	289	1	................	................	PUNCT
ejpam-5566	289	2	..............	..............	PUNCT
ejpam-5566	289	3	..............................	..............................	PUNCT
ejpam-5566	290	1	......	......	PUNCT
ejpam-5566	290	2	.........	.........	PUNCT
ejpam-5566	290	3	........	........	PUNCT
ejpam-5566	290	4	........	........	PUNCT
ejpam-5566	290	5	........	........	PUNCT
ejpam-5566	290	6	........	........	PUNCT
ejpam-5566	291	1	......	......	PUNCT
ejpam-5566	291	2	.........	.........	PUNCT
ejpam-5566	291	3	........	........	PUNCT
ejpam-5566	291	4	........	........	PUNCT
ejpam-5566	291	5	........	........	PUNCT
ejpam-5566	291	6	........	........	PUNCT
ejpam-5566	292	1	......	......	PUNCT
ejpam-5566	292	2	..............................	..............................	PUNCT
ejpam-5566	293	1	......	......	PUNCT
ejpam-5566	293	2	.........	.........	PUNCT
ejpam-5566	293	3	........	........	PUNCT
ejpam-5566	293	4	........	........	PUNCT
ejpam-5566	293	5	........	........	PUNCT
ejpam-5566	293	6	........	........	PUNCT
ejpam-5566	294	1	......	......	PUNCT
ejpam-5566	294	2	..............................	..............................	PUNCT
ejpam-5566	295	1	......	......	PUNCT
ejpam-5566	295	2	..............................	..............................	PUNCT
ejpam-5566	296	1	......	......	PUNCT
ejpam-5566	296	2	t6,2	t6,2	INTJ
ejpam-5566	296	3	....................................	....................................	PUNCT
ejpam-5566	296	4	....................................	....................................	PUNCT
ejpam-5566	296	5	....................................	....................................	PUNCT
ejpam-5566	297	1	....................................	....................................	PUNCT
ejpam-5566	297	2	....................................	....................................	PUNCT
ejpam-5566	298	1	....................................	....................................	PUNCT
ejpam-5566	298	2	....................................	....................................	PUNCT
ejpam-5566	299	1	....................................	....................................	PUNCT
ejpam-5566	299	2	....................................	....................................	PUNCT
ejpam-5566	300	1	..................................	..................................	PUNCT
ejpam-5566	300	2	..................................	..................................	PUNCT
ejpam-5566	300	3	..................................	..................................	PUNCT
ejpam-5566	300	4	..................................	..................................	PUNCT
ejpam-5566	301	1	..........	..........	PUNCT
ejpam-5566	301	2	.........	.........	PUNCT
ejpam-5566	302	1	.........	.........	PUNCT
ejpam-5566	302	2	......	......	PUNCT
ejpam-5566	303	1	..........	..........	PUNCT
ejpam-5566	303	2	.........	.........	PUNCT
ejpam-5566	304	1	.........	.........	PUNCT
ejpam-5566	304	2	......	......	PUNCT
ejpam-5566	304	3	......................	......................	PUNCT
ejpam-5566	304	4	............	............	PUNCT
ejpam-5566	304	5	......................	......................	PUNCT
ejpam-5566	304	6	............	............	PUNCT
ejpam-5566	304	7	.........	.........	PUNCT
ejpam-5566	304	8	........	........	PUNCT
ejpam-5566	304	9	........	........	PUNCT
ejpam-5566	304	10	........	........	PUNCT
ejpam-5566	304	11	........	........	PUNCT
ejpam-5566	305	1	......	......	PUNCT
ejpam-5566	306	1	t8,1	t8,1	ADJ
ejpam-5566	306	2	figure	figure	NOUN
ejpam-5566	306	3	4	4	NUM
ejpam-5566	306	4	:	:	PUNCT
ejpam-5566	306	5	illustrating	illustrate	VERB
ejpam-5566	306	6	the	the	DET
ejpam-5566	306	7	graphs	graph	NOUN
ejpam-5566	306	8	t6,2	t6,2	PROPN
ejpam-5566	306	9	and	and	CCONJ
ejpam-5566	306	10	t8,1	t8,1	PROPN
ejpam-5566	306	11	....................................	....................................	PUNCT
ejpam-5566	306	12	....................................	....................................	PUNCT
ejpam-5566	306	13	....................................	....................................	PUNCT
ejpam-5566	306	14	....................................	....................................	PUNCT
ejpam-5566	306	15	....................................	....................................	PUNCT
ejpam-5566	307	1	............................................................................................................................................................	............................................................................................................................................................	PUNCT
ejpam-5566	307	2	..............	..............	PUNCT
ejpam-5566	307	3	.............	.............	PUNCT
ejpam-5566	307	4	.............	.............	PUNCT
ejpam-5566	307	5	.............	.............	PUNCT
ejpam-5566	307	6	.............	.............	PUNCT
ejpam-5566	307	7	............	............	PUNCT
ejpam-5566	307	8	.................................	.................................	PUNCT
ejpam-5566	307	9	................................	................................	PUNCT
ejpam-5566	307	10	.............	.............	PUNCT
ejpam-5566	307	11	........................................................................................................................................................	........................................................................................................................................................	PUNCT
ejpam-5566	308	1	e1	e1	PROPN
ejpam-5566	308	2	e2	e2	PROPN
ejpam-5566	308	3	e3	e3	NOUN
ejpam-5566	308	4	e4	e4	PROPN
ejpam-5566	308	5	s1	s1	PROPN
ejpam-5566	308	6	c2	c2	PROPN
ejpam-5566	308	7	n[a	n[a	PROPN
ejpam-5566	308	8	]	]	PUNCT
ejpam-5566	308	9	....................................	....................................	PUNCT
ejpam-5566	308	10	....................................	....................................	PUNCT
ejpam-5566	309	1	....................................	....................................	PUNCT
ejpam-5566	309	2	....................................	....................................	PUNCT
ejpam-5566	310	1	....................................	....................................	PUNCT
ejpam-5566	310	2	............................................................................................................................................................	............................................................................................................................................................	PUNCT
ejpam-5566	310	3	.................................	.................................	PUNCT
ejpam-5566	310	4	................................	................................	PUNCT
ejpam-5566	310	5	.............	.............	PUNCT
ejpam-5566	310	6	........................................................................................................................................................	........................................................................................................................................................	PUNCT
ejpam-5566	310	7	..................	..................	PUNCT
ejpam-5566	311	1	..................	..................	PUNCT
ejpam-5566	311	2	.................	.................	PUNCT
ejpam-5566	312	1	..................	..................	PUNCT
ejpam-5566	312	2	.................	.................	PUNCT
ejpam-5566	313	1	..................	..................	PUNCT
ejpam-5566	313	2	..................	..................	PUNCT
ejpam-5566	314	1	..................	..................	PUNCT
ejpam-5566	315	1	..........	..........	PUNCT
ejpam-5566	316	1	e1	e1	PROPN
ejpam-5566	316	2	e2	e2	PROPN
ejpam-5566	316	3	e4	e4	PROPN
ejpam-5566	316	4	c2	c2	PROPN
ejpam-5566	316	5	n[b	n[b	PROPN
ejpam-5566	316	6	]	]	PUNCT
ejpam-5566	316	7	s1	s1	PROPN
ejpam-5566	316	8	s3	s3	PROPN
ejpam-5566	316	9	figure	figure	NOUN
ejpam-5566	316	10	5	5	NUM
ejpam-5566	316	11	:	:	PUNCT
ejpam-5566	316	12	illustrating	illustrate	VERB
ejpam-5566	316	13	the	the	DET
ejpam-5566	316	14	subgraphs	subgraph	NOUN
ejpam-5566	316	15	c2	c2	PROPN
ejpam-5566	316	16	n[a	n[a	PROPN
ejpam-5566	316	17	]	]	PUNCT
ejpam-5566	316	18	and	and	CCONJ
ejpam-5566	316	19	c2	c2	PROPN
ejpam-5566	316	20	n[b	n[b	PROPN
ejpam-5566	316	21	]	]	X
ejpam-5566	316	22	a	a	PRON
ejpam-5566	316	23	,	,	PUNCT
ejpam-5566	316	24	b	b	X
ejpam-5566	316	25	∈	∈	PROPN
ejpam-5566	316	26	et3,2(c	et3,2(c	PROPN
ejpam-5566	316	27	2	2	NUM
ejpam-5566	316	28	n	n	CCONJ
ejpam-5566	316	29	)	)	PUNCT
ejpam-5566	316	30	.	.	PUNCT
ejpam-5566	317	1	now	now	ADV
ejpam-5566	317	2	,	,	PUNCT
ejpam-5566	317	3	a∆b	a∆b	NOUN
ejpam-5566	317	4	=	=	PUNCT
ejpam-5566	317	5	{	{	PUNCT
ejpam-5566	317	6	e3	e3	NOUN
ejpam-5566	317	7	,	,	PUNCT
ejpam-5566	317	8	s3	s3	PROPN
ejpam-5566	317	9	}	}	PUNCT
ejpam-5566	317	10	∈	∈	PROPN
ejpam-5566	317	11	et3,2(c	et3,2(c	PROPN
ejpam-5566	317	12	2	2	NUM
ejpam-5566	317	13	n	n	NUM
ejpam-5566	317	14	)	)	PUNCT
ejpam-5566	317	15	.	.	PUNCT
ejpam-5566	318	1	by	by	ADP
ejpam-5566	318	2	lemma	lemma	PROPN
ejpam-5566	318	3	3	3	NUM
ejpam-5566	318	4	,	,	PUNCT
ejpam-5566	318	5	t3,2	t3,2	PROPN
ejpam-5566	318	6	is	be	AUX
ejpam-5566	318	7	a	a	DET
ejpam-5566	318	8	generator	generator	NOUN
ejpam-5566	318	9	subgraph	subgraph	NOUN
ejpam-5566	318	10	of	of	ADP
ejpam-5566	318	11	c2	c2	PROPN
ejpam-5566	318	12	n.	n.	PROPN
ejpam-5566	318	13	the	the	DET
ejpam-5566	318	14	next	next	ADJ
ejpam-5566	318	15	result	result	NOUN
ejpam-5566	318	16	gives	give	VERB
ejpam-5566	318	17	the	the	DET
ejpam-5566	318	18	characterization	characterization	NOUN
ejpam-5566	318	19	for	for	ADP
ejpam-5566	318	20	the	the	DET
ejpam-5566	318	21	tadpole	tadpole	NOUN
ejpam-5566	318	22	t3,r	t3,r	PROPN
ejpam-5566	318	23	so	so	SCONJ
ejpam-5566	318	24	that	that	SCONJ
ejpam-5566	318	25	it	it	PRON
ejpam-5566	318	26	is	be	AUX
ejpam-5566	318	27	a	a	DET
ejpam-5566	318	28	generator	generator	NOUN
ejpam-5566	318	29	subgraph	subgraph	NOUN
ejpam-5566	318	30	of	of	ADP
ejpam-5566	318	31	c2	c2	PROPN
ejpam-5566	318	32	n.	n.	PROPN
ejpam-5566	318	33	theorem	theorem	VERB
ejpam-5566	318	34	12	12	NUM
ejpam-5566	318	35	.	.	PUNCT
ejpam-5566	319	1	let	let	VERB
ejpam-5566	319	2	n	n	NOUN
ejpam-5566	319	3	and	and	CCONJ
ejpam-5566	319	4	r	r	NOUN
ejpam-5566	319	5	be	be	AUX
ejpam-5566	319	6	positive	positive	ADJ
ejpam-5566	319	7	integers	integer	NOUN
ejpam-5566	319	8	.	.	PUNCT
ejpam-5566	320	1	then	then	ADV
ejpam-5566	320	2	the	the	DET
ejpam-5566	320	3	tadpole	tadpole	NOUN
ejpam-5566	320	4	graph	graph	NOUN
ejpam-5566	320	5	t3,r	t3,r	PROPN
ejpam-5566	320	6	is	be	AUX
ejpam-5566	320	7	a	a	DET
ejpam-5566	320	8	generator	generator	NOUN
ejpam-5566	320	9	subgraph	subgraph	NOUN
ejpam-5566	320	10	of	of	ADP
ejpam-5566	320	11	c2	c2	PROPN
ejpam-5566	320	12	n	n	PROPN
ejpam-5566	320	13	if	if	SCONJ
ejpam-5566	321	1	and	and	CCONJ
ejpam-5566	321	2	only	only	ADV
ejpam-5566	321	3	if	if	SCONJ
ejpam-5566	321	4	r	r	NOUN
ejpam-5566	321	5	is	be	AUX
ejpam-5566	321	6	even	even	ADV
ejpam-5566	321	7	and	and	CCONJ
ejpam-5566	322	1	2	2	NUM
ejpam-5566	322	2	≤	≤	NOUN
ejpam-5566	322	3	r	r	NOUN
ejpam-5566	322	4	≤	≤	NUM
ejpam-5566	322	5	n−	n−	NOUN
ejpam-5566	322	6	3	3	NUM
ejpam-5566	322	7	.	.	PUNCT
ejpam-5566	323	1	proof	proof	NOUN
ejpam-5566	323	2	.	.	PUNCT
ejpam-5566	324	1	assume	assume	VERB
ejpam-5566	324	2	that	that	SCONJ
ejpam-5566	324	3	the	the	DET
ejpam-5566	324	4	tadpole	tadpole	NOUN
ejpam-5566	324	5	graph	graph	NOUN
ejpam-5566	324	6	t3,r	t3,r	PROPN
ejpam-5566	324	7	is	be	AUX
ejpam-5566	324	8	a	a	DET
ejpam-5566	324	9	generator	generator	NOUN
ejpam-5566	324	10	subgraph	subgraph	NOUN
ejpam-5566	324	11	of	of	ADP
ejpam-5566	324	12	c2	c2	PROPN
ejpam-5566	324	13	n.	n.	PROPN
ejpam-5566	324	14	then	then	ADV
ejpam-5566	324	15	the	the	DET
ejpam-5566	324	16	size	size	NOUN
ejpam-5566	324	17	of	of	ADP
ejpam-5566	324	18	t3,r	t3,r	PROPN
ejpam-5566	324	19	must	must	AUX
ejpam-5566	324	20	be	be	AUX
ejpam-5566	324	21	odd	odd	ADJ
ejpam-5566	324	22	in	in	ADP
ejpam-5566	324	23	view	view	NOUN
ejpam-5566	324	24	of	of	ADP
ejpam-5566	324	25	theorem	theorem	NOUN
ejpam-5566	324	26	1	1	NUM
ejpam-5566	324	27	.	.	PUNCT
ejpam-5566	325	1	it	it	PRON
ejpam-5566	325	2	follows	follow	VERB
ejpam-5566	325	3	that	that	SCONJ
ejpam-5566	325	4	r	r	NOUN
ejpam-5566	325	5	is	be	AUX
ejpam-5566	325	6	even	even	ADV
ejpam-5566	325	7	.	.	PUNCT
ejpam-5566	326	1	we	we	PRON
ejpam-5566	326	2	claim	claim	VERB
ejpam-5566	326	3	that	that	SCONJ
ejpam-5566	326	4	2	2	NUM
ejpam-5566	326	5	≤	≤	NOUN
ejpam-5566	326	6	r	r	NOUN
ejpam-5566	326	7	≤	≤	NOUN
ejpam-5566	326	8	n	n	CCONJ
ejpam-5566	326	9	−	−	PROPN
ejpam-5566	326	10	3	3	X
ejpam-5566	326	11	.	.	PUNCT
ejpam-5566	326	12	suppose	suppose	VERB
ejpam-5566	326	13	,	,	PUNCT
ejpam-5566	326	14	on	on	ADP
ejpam-5566	326	15	the	the	DET
ejpam-5566	326	16	contrary	contrary	NOUN
ejpam-5566	326	17	,	,	PUNCT
ejpam-5566	326	18	r	r	NOUN
ejpam-5566	326	19	=	=	SYM
ejpam-5566	326	20	1	1	NUM
ejpam-5566	326	21	or	or	CCONJ
ejpam-5566	326	22	r	r	NOUN
ejpam-5566	326	23	>	>	X
ejpam-5566	326	24	n	n	CCONJ
ejpam-5566	327	1	−	−	NOUN
ejpam-5566	328	1	3	3	X
ejpam-5566	328	2	.	.	PUNCT
ejpam-5566	329	1	if	if	SCONJ
ejpam-5566	329	2	r	r	NOUN
ejpam-5566	329	3	=	=	NOUN
ejpam-5566	329	4	1	1	NUM
ejpam-5566	329	5	then	then	ADV
ejpam-5566	329	6	t3,r	t3,r	PROPN
ejpam-5566	329	7	consists	consist	VERB
ejpam-5566	329	8	of	of	ADP
ejpam-5566	329	9	four	four	NUM
ejpam-5566	329	10	edges	edge	NOUN
ejpam-5566	329	11	,	,	PUNCT
ejpam-5566	329	12	which	which	PRON
ejpam-5566	329	13	are	be	AUX
ejpam-5566	329	14	even	even	ADV
ejpam-5566	329	15	.	.	PUNCT
ejpam-5566	330	1	this	this	PRON
ejpam-5566	330	2	is	be	AUX
ejpam-5566	330	3	a	a	DET
ejpam-5566	330	4	contradiction	contradiction	NOUN
ejpam-5566	330	5	by	by	ADP
ejpam-5566	330	6	theorem	theorem	NOUN
ejpam-5566	330	7	1	1	NUM
ejpam-5566	330	8	.	.	PUNCT
ejpam-5566	331	1	if	if	SCONJ
ejpam-5566	331	2	r	r	NOUN
ejpam-5566	331	3	>	>	X
ejpam-5566	331	4	n	n	CCONJ
ejpam-5566	331	5	−	−	PROPN
ejpam-5566	331	6	3	3	NUM
ejpam-5566	331	7	,	,	PUNCT
ejpam-5566	331	8	then	then	ADV
ejpam-5566	331	9	the	the	DET
ejpam-5566	331	10	order	order	NOUN
ejpam-5566	331	11	of	of	ADP
ejpam-5566	331	12	t3,r	t3,r	PROPN
ejpam-5566	331	13	is	be	AUX
ejpam-5566	331	14	greater	great	ADJ
ejpam-5566	331	15	than	than	ADP
ejpam-5566	331	16	n.	n.	NOUN
ejpam-5566	331	17	meaning	meaning	NOUN
ejpam-5566	331	18	,	,	PUNCT
ejpam-5566	331	19	t3,r	t3,r	PROPN
ejpam-5566	331	20	is	be	AUX
ejpam-5566	331	21	not	not	PART
ejpam-5566	331	22	a	a	DET
ejpam-5566	331	23	subgraph	subgraph	NOUN
ejpam-5566	331	24	of	of	ADP
ejpam-5566	331	25	c2	c2	PROPN
ejpam-5566	331	26	n.	n.	PROPN
ejpam-5566	331	27	again	again	ADV
ejpam-5566	331	28	,	,	PUNCT
ejpam-5566	331	29	a	a	DET
ejpam-5566	331	30	contradiction	contradiction	NOUN
ejpam-5566	331	31	to	to	ADP
ejpam-5566	331	32	the	the	DET
ejpam-5566	331	33	assumption	assumption	NOUN
ejpam-5566	331	34	that	that	SCONJ
ejpam-5566	331	35	t3,r	t3,r	PROPN
ejpam-5566	331	36	is	be	AUX
ejpam-5566	331	37	a	a	DET
ejpam-5566	331	38	generator	generator	NOUN
ejpam-5566	331	39	subgraph	subgraph	NOUN
ejpam-5566	331	40	of	of	ADP
ejpam-5566	331	41	c2	c2	PROPN
ejpam-5566	331	42	n.	n.	PROPN
ejpam-5566	331	43	conversely	conversely	ADV
ejpam-5566	331	44	,	,	PUNCT
ejpam-5566	331	45	assume	assume	VERB
ejpam-5566	331	46	that	that	SCONJ
ejpam-5566	331	47	r	r	NOUN
ejpam-5566	331	48	is	be	AUX
ejpam-5566	331	49	even	even	ADV
ejpam-5566	331	50	and	and	CCONJ
ejpam-5566	332	1	2	2	NUM
ejpam-5566	332	2	≤	≤	NOUN
ejpam-5566	332	3	r	r	NOUN
ejpam-5566	332	4	≤	≤	PUNCT
ejpam-5566	332	5	n−3	n−3	PROPN
ejpam-5566	332	6	.	.	PUNCT
ejpam-5566	333	1	we	we	PRON
ejpam-5566	333	2	show	show	VERB
ejpam-5566	333	3	that	that	SCONJ
ejpam-5566	333	4	t3,r	t3,r	PROPN
ejpam-5566	333	5	is	be	AUX
ejpam-5566	333	6	a	a	DET
ejpam-5566	333	7	generator	generator	NOUN
ejpam-5566	333	8	subgraph	subgraph	NOUN
ejpam-5566	333	9	of	of	ADP
ejpam-5566	333	10	c2	c2	PROPN
ejpam-5566	333	11	n.	n.	PROPN
ejpam-5566	333	12	since	since	SCONJ
ejpam-5566	333	13	r	r	NOUN
ejpam-5566	333	14	is	be	AUX
ejpam-5566	333	15	even	even	ADV
ejpam-5566	333	16	,	,	PUNCT
ejpam-5566	333	17	then	then	ADV
ejpam-5566	333	18	the	the	DET
ejpam-5566	333	19	size	size	NOUN
ejpam-5566	333	20	of	of	ADP
ejpam-5566	333	21	the	the	DET
ejpam-5566	333	22	t3,r	t3,r	PROPN
ejpam-5566	333	23	is	be	AUX
ejpam-5566	333	24	odd	odd	ADJ
ejpam-5566	333	25	.	.	PUNCT
ejpam-5566	334	1	if	if	SCONJ
ejpam-5566	334	2	r	r	NOUN
ejpam-5566	334	3	=	=	SYM
ejpam-5566	334	4	2	2	NUM
ejpam-5566	334	5	,	,	PUNCT
ejpam-5566	334	6	then	then	ADV
ejpam-5566	334	7	t3,r	t3,r	PROPN
ejpam-5566	334	8	is	be	AUX
ejpam-5566	334	9	a	a	DET
ejpam-5566	334	10	generator	generator	NOUN
ejpam-5566	334	11	subgraph	subgraph	NOUN
ejpam-5566	334	12	of	of	ADP
ejpam-5566	334	13	c2	c2	PROPN
ejpam-5566	334	14	n	n	X
ejpam-5566	334	15	by	by	ADP
ejpam-5566	334	16	theorem	theorem	NOUN
ejpam-5566	334	17	11	11	NUM
ejpam-5566	334	18	.	.	PUNCT
ejpam-5566	335	1	let	let	VERB
ejpam-5566	335	2	us	we	PRON
ejpam-5566	335	3	assume	assume	VERB
ejpam-5566	335	4	that	that	SCONJ
ejpam-5566	335	5	2	2	NUM
ejpam-5566	335	6	<	<	X
ejpam-5566	335	7	r	r	NOUN
ejpam-5566	335	8	≤	≤	NUM
ejpam-5566	335	9	n−	n−	NOUN
ejpam-5566	335	10	3	3	NUM
ejpam-5566	335	11	.	.	PUNCT
ejpam-5566	335	12	consider	consider	VERB
ejpam-5566	335	13	the	the	DET
ejpam-5566	335	14	labeling	labeling	NOUN
ejpam-5566	335	15	of	of	ADP
ejpam-5566	335	16	c2	c2	PROPN
ejpam-5566	335	17	n.	n.	PROPN
ejpam-5566	335	18	let	let	VERB
ejpam-5566	335	19	a	a	DET
ejpam-5566	335	20	=	=	PUNCT
ejpam-5566	335	21	{	{	PUNCT
ejpam-5566	335	22	en−1	en−1	PROPN
ejpam-5566	335	23	,	,	PUNCT
ejpam-5566	335	24	sn−1	sn−1	PROPN
ejpam-5566	335	25	,	,	PUNCT
ejpam-5566	335	26	sn	sn	PROPN
ejpam-5566	335	27	,	,	PUNCT
ejpam-5566	335	28	e1	e1	PROPN
ejpam-5566	335	29	,	,	PUNCT
ejpam-5566	335	30	e2	e2	PROPN
ejpam-5566	335	31	,	,	PUNCT
ejpam-5566	335	32	.	.	PUNCT
ejpam-5566	335	33	.	.	PUNCT
ejpam-5566	336	1	.	.	PUNCT
ejpam-5566	337	1	,	,	PUNCT
ejpam-5566	337	2	er}and	er}and	PUNCT
ejpam-5566	337	3	define	define	VERB
ejpam-5566	337	4	b	b	NOUN
ejpam-5566	337	5	=	=	PUNCT
ejpam-5566	337	6	a\{e1	a\{e1	NOUN
ejpam-5566	337	7	}	}	PUNCT
ejpam-5566	337	8	∪	∪	X
ejpam-5566	337	9	{	{	PUNCT
ejpam-5566	337	10	sn	sn	NOUN
ejpam-5566	337	11	}	}	PUNCT
ejpam-5566	337	12	.	.	PUNCT
ejpam-5566	338	1	it	it	PRON
ejpam-5566	338	2	can	can	AUX
ejpam-5566	338	3	be	be	AUX
ejpam-5566	338	4	verified	verify	VERB
ejpam-5566	338	5	that	that	SCONJ
ejpam-5566	338	6	a	a	DET
ejpam-5566	338	7	,	,	PUNCT
ejpam-5566	338	8	b	b	X
ejpam-5566	338	9	∈	∈	NOUN
ejpam-5566	338	10	et3,r(c	et3,r(c	ADP
ejpam-5566	338	11	2	2	NUM
ejpam-5566	338	12	n	n	CCONJ
ejpam-5566	338	13	)	)	PUNCT
ejpam-5566	338	14	,	,	PUNCT
ejpam-5566	338	15	as	as	SCONJ
ejpam-5566	338	16	shown	show	VERB
ejpam-5566	338	17	in	in	ADP
ejpam-5566	338	18	figure	figure	NOUN
ejpam-5566	338	19	6	6	NUM
ejpam-5566	338	20	.	.	PUNCT
ejpam-5566	338	21	now	now	ADV
ejpam-5566	338	22	,	,	PUNCT
ejpam-5566	338	23	a∆b	a∆b	NOUN
ejpam-5566	338	24	=	=	SYM
ejpam-5566	338	25	{	{	PUNCT
ejpam-5566	338	26	e1	e1	PROPN
ejpam-5566	338	27	,	,	PUNCT
ejpam-5566	338	28	sn	sn	NOUN
ejpam-5566	338	29	}	}	PUNCT
ejpam-5566	338	30	∈	∈	PROPN
ejpam-5566	338	31	et3,r(c	et3,r(c	ADP
ejpam-5566	338	32	2	2	NUM
ejpam-5566	338	33	n	n	CCONJ
ejpam-5566	338	34	)	)	PUNCT
ejpam-5566	338	35	.	.	PUNCT
ejpam-5566	339	1	by	by	ADP
ejpam-5566	339	2	lemma	lemma	PROPN
ejpam-5566	339	3	1	1	NUM
ejpam-5566	339	4	,	,	PUNCT
ejpam-5566	339	5	{	{	PUNCT
ejpam-5566	339	6	e1	e1	NOUN
ejpam-5566	339	7	,	,	PUNCT
ejpam-5566	339	8	sn	sn	NOUN
ejpam-5566	339	9	}	}	PUNCT
ejpam-5566	339	10	∈	∈	PROPN
ejpam-5566	339	11	e	e	X
ejpam-5566	339	12	(	(	PUNCT
ejpam-5566	339	13	c2	c2	PROPN
ejpam-5566	339	14	n	n	CCONJ
ejpam-5566	339	15	)	)	PUNCT
ejpam-5566	339	16	.	.	PUNCT
ejpam-5566	340	1	by	by	ADP
ejpam-5566	340	2	lemma	lemma	PROPN
ejpam-5566	340	3	3	3	NUM
ejpam-5566	340	4	,	,	PUNCT
ejpam-5566	340	5	t3,r	t3,r	PROPN
ejpam-5566	340	6	is	be	AUX
ejpam-5566	340	7	a	a	DET
ejpam-5566	340	8	generator	generator	NOUN
ejpam-5566	340	9	subgraph	subgraph	NOUN
ejpam-5566	340	10	of	of	ADP
ejpam-5566	340	11	c2	c2	PROPN
ejpam-5566	340	12	n.	n.	PROPN
ejpam-5566	340	13	by	by	ADP
ejpam-5566	340	14	a	a	DET
ejpam-5566	340	15	kite	kite	NOUN
ejpam-5566	340	16	graph	graph	NOUN
ejpam-5566	340	17	,	,	PUNCT
ejpam-5566	340	18	denoted	denote	VERB
ejpam-5566	340	19	by	by	ADP
ejpam-5566	340	20	ktr	ktr	PROPN
ejpam-5566	340	21	,	,	PUNCT
ejpam-5566	340	22	s	s	PART
ejpam-5566	340	23	,	,	PUNCT
ejpam-5566	340	24	we	we	PRON
ejpam-5566	340	25	mean	mean	VERB
ejpam-5566	340	26	a	a	DET
ejpam-5566	340	27	graph	graph	NOUN
ejpam-5566	340	28	formed	form	VERB
ejpam-5566	340	29	by	by	ADP
ejpam-5566	340	30	joining	join	VERB
ejpam-5566	340	31	a	a	DET
ejpam-5566	340	32	path	path	NOUN
ejpam-5566	340	33	graph	graph	NOUN
ejpam-5566	340	34	pr	pr	NOUN
ejpam-5566	340	35	,	,	PUNCT
ejpam-5566	340	36	a	a	DET
ejpam-5566	340	37	path	path	NOUN
ejpam-5566	340	38	graph	graph	NOUN
ejpam-5566	340	39	ps	ps	NOUN
ejpam-5566	340	40	and	and	CCONJ
ejpam-5566	340	41	a	a	DET
ejpam-5566	340	42	cycle	cycle	NOUN
ejpam-5566	340	43	c3	c3	NOUN
ejpam-5566	340	44	with	with	ADP
ejpam-5566	340	45	two	two	NUM
ejpam-5566	340	46	edges	edge	NOUN
ejpam-5566	340	47	.	.	PUNCT
ejpam-5566	341	1	one	one	NUM
ejpam-5566	341	2	edge	edge	NOUN
ejpam-5566	341	3	joins	join	VERB
ejpam-5566	341	4	one	one	NUM
ejpam-5566	341	5	vertex	vertex	NOUN
ejpam-5566	341	6	of	of	ADP
ejpam-5566	341	7	c3	c3	PROPN
ejpam-5566	341	8	to	to	ADP
ejpam-5566	341	9	a	a	DET
ejpam-5566	341	10	r.	r.	PROPN
ejpam-5566	341	11	mame	mame	PROPN
ejpam-5566	341	12	/	/	SYM
ejpam-5566	341	13	eur	eur	PROPN
ejpam-5566	341	14	.	.	PUNCT
ejpam-5566	342	1	j.	j.	PROPN
ejpam-5566	342	2	pure	pure	PROPN
ejpam-5566	342	3	appl	appl	PROPN
ejpam-5566	342	4	.	.	PROPN
ejpam-5566	342	5	math	math	PROPN
ejpam-5566	342	6	,	,	PUNCT
ejpam-5566	342	7	17	17	NUM
ejpam-5566	342	8	(	(	PUNCT
ejpam-5566	342	9	4	4	NUM
ejpam-5566	342	10	)	)	PUNCT
ejpam-5566	342	11	(	(	PUNCT
ejpam-5566	342	12	2024	2024	NUM
ejpam-5566	342	13	)	)	PUNCT
ejpam-5566	342	14	,	,	PUNCT
ejpam-5566	342	15	3815	3815	NUM
ejpam-5566	342	16	-	-	SYM
ejpam-5566	342	17	3825	3825	NUM
ejpam-5566	342	18	3823	3823	NUM
ejpam-5566	342	19	....................................	....................................	PUNCT
ejpam-5566	342	20	....................................	....................................	PUNCT
ejpam-5566	343	1	....................................	....................................	PUNCT
ejpam-5566	343	2	....................................	....................................	PUNCT
ejpam-5566	344	1	....................................	....................................	PUNCT
ejpam-5566	344	2	....................................	....................................	PUNCT
ejpam-5566	345	1	....................................	....................................	PUNCT
ejpam-5566	345	2	....................................	....................................	PUNCT
ejpam-5566	346	1	..................................................................................................................................................................................................................................................................	..................................................................................................................................................................................................................................................................	PUNCT
ejpam-5566	346	2	.............	.............	PUNCT
ejpam-5566	346	3	.............	.............	PUNCT
ejpam-5566	346	4	.............	.............	PUNCT
ejpam-5566	346	5	.............	.............	PUNCT
ejpam-5566	346	6	.............	.............	PUNCT
ejpam-5566	346	7	.............	.............	PUNCT
ejpam-5566	346	8	.............	.............	PUNCT
ejpam-5566	346	9	.............	.............	PUNCT
ejpam-5566	346	10	..........	..........	PUNCT
ejpam-5566	346	11	................................	................................	PUNCT
ejpam-5566	346	12	................................	................................	PUNCT
ejpam-5566	346	13	................................	................................	PUNCT
ejpam-5566	346	14	....................	....................	PUNCT
ejpam-5566	346	15	.................................	.................................	PUNCT
ejpam-5566	346	16	................................	................................	PUNCT
ejpam-5566	346	17	................................	................................	PUNCT
ejpam-5566	346	18	.........................	.........................	PUNCT
ejpam-5566	346	19	...................	...................	PUNCT
ejpam-5566	346	20	..................	..................	PUNCT
ejpam-5566	347	1	..................	..................	PUNCT
ejpam-5566	347	2	..................	..................	PUNCT
ejpam-5566	348	1	..................	..................	PUNCT
ejpam-5566	348	2	..................	..................	PUNCT
ejpam-5566	349	1	..................	..................	PUNCT
ejpam-5566	349	2	..................	..................	PUNCT
ejpam-5566	350	1	..................	..................	PUNCT
ejpam-5566	350	2	..................	..................	PUNCT
ejpam-5566	351	1	..................	..................	PUNCT
ejpam-5566	351	2	..................	..................	PUNCT
ejpam-5566	352	1	.................	.................	PUNCT
ejpam-5566	353	1	..........................................................................................................................	..........................................................................................................................	PUNCT
ejpam-5566	354	1	e1	e1	PROPN
ejpam-5566	354	2	e2	e2	PROPN
ejpam-5566	354	3	er−1	er−1	PROPN
ejpam-5566	354	4	en−1	en−1	PROPN
ejpam-5566	354	5	en	en	PROPN
ejpam-5566	354	6	c2	c2	PROPN
ejpam-5566	354	7	n[a	n[a	PROPN
ejpam-5566	354	8	]	]	X
ejpam-5566	354	9	er	er	INTJ
ejpam-5566	354	10	sn−1	sn−1	PROPN
ejpam-5566	354	11	....................................	....................................	PUNCT
ejpam-5566	354	12	....................................	....................................	PUNCT
ejpam-5566	354	13	....................................	....................................	PUNCT
ejpam-5566	354	14	....................................	....................................	PUNCT
ejpam-5566	354	15	....................................	....................................	PUNCT
ejpam-5566	354	16	....................................	....................................	PUNCT
ejpam-5566	354	17	....................................	....................................	PUNCT
ejpam-5566	354	18	....................................	....................................	PUNCT
ejpam-5566	354	19	........................................................................................................................................	........................................................................................................................................	PUNCT
ejpam-5566	354	20	.............	.............	PUNCT
ejpam-5566	354	21	.............	.............	PUNCT
ejpam-5566	354	22	.............	.............	PUNCT
ejpam-5566	354	23	.............	.............	PUNCT
ejpam-5566	354	24	.............	.............	PUNCT
ejpam-5566	354	25	.............	.............	PUNCT
ejpam-5566	354	26	.............	.............	PUNCT
ejpam-5566	354	27	.............	.............	PUNCT
ejpam-5566	354	28	..........	..........	PUNCT
ejpam-5566	354	29	................................	................................	PUNCT
ejpam-5566	354	30	................................	................................	PUNCT
ejpam-5566	354	31	................................	................................	PUNCT
ejpam-5566	354	32	....................	....................	PUNCT
ejpam-5566	354	33	.................................	.................................	PUNCT
ejpam-5566	354	34	................................	................................	PUNCT
ejpam-5566	354	35	................................	................................	PUNCT
ejpam-5566	354	36	.........................	.........................	PUNCT
ejpam-5566	354	37	...................	...................	PUNCT
ejpam-5566	354	38	..................	..................	PUNCT
ejpam-5566	354	39	..................	..................	PUNCT
ejpam-5566	354	40	..................	..................	PUNCT
ejpam-5566	354	41	..................	..................	PUNCT
ejpam-5566	354	42	..................	..................	PUNCT
ejpam-5566	354	43	..................	..................	PUNCT
ejpam-5566	354	44	..................	..................	PUNCT
ejpam-5566	354	45	..................	..................	PUNCT
ejpam-5566	354	46	..................	..................	PUNCT
ejpam-5566	354	47	..................	..................	PUNCT
ejpam-5566	354	48	..................	..................	PUNCT
ejpam-5566	354	49	.................	.................	PUNCT
ejpam-5566	355	1	..........................................................................................................................	..........................................................................................................................	PUNCT
ejpam-5566	355	2	..........................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................	PUNCT
ejpam-5566	356	1	e2	e2	PROPN
ejpam-5566	356	2	er−1	er−1	PROPN
ejpam-5566	356	3	en−1	en−1	PROPN
ejpam-5566	356	4	en	en	ADP
ejpam-5566	356	5	sn	sn	PROPN
ejpam-5566	356	6	sn−1	sn−1	PROPN
ejpam-5566	356	7	c2	c2	PROPN
ejpam-5566	356	8	n[b	n[b	PROPN
ejpam-5566	356	9	]	]	X
ejpam-5566	356	10	er	er	INTJ
ejpam-5566	356	11	sn−1	sn−1	PROPN
ejpam-5566	356	12	........	........	PUNCT
ejpam-5566	356	13	........	........	PUNCT
ejpam-5566	356	14	........	........	PUNCT
ejpam-5566	356	15	........	........	PUNCT
ejpam-5566	356	16	........	........	PUNCT
ejpam-5566	357	1	........	........	PUNCT
ejpam-5566	357	2	figure	figure	VERB
ejpam-5566	357	3	6	6	NUM
ejpam-5566	357	4	:	:	PUNCT
ejpam-5566	357	5	illustrating	illustrate	VERB
ejpam-5566	357	6	the	the	DET
ejpam-5566	357	7	subgraphs	subgraph	NOUN
ejpam-5566	357	8	c2	c2	PROPN
ejpam-5566	357	9	n[a	n[a	PROPN
ejpam-5566	357	10	]	]	PUNCT
ejpam-5566	357	11	and	and	CCONJ
ejpam-5566	357	12	c2	c2	PROPN
ejpam-5566	357	13	n[b	n[b	PROPN
ejpam-5566	357	14	]	]	SYM
ejpam-5566	357	15	vertex	vertex	NOUN
ejpam-5566	357	16	of	of	ADP
ejpam-5566	357	17	pr	pr	NOUN
ejpam-5566	357	18	whose	whose	DET
ejpam-5566	357	19	degree	degree	NOUN
ejpam-5566	357	20	in	in	ADP
ejpam-5566	357	21	pr	pr	NOUN
ejpam-5566	357	22	is	be	AUX
ejpam-5566	357	23	either	either	CCONJ
ejpam-5566	357	24	0	0	NUM
ejpam-5566	357	25	or	or	CCONJ
ejpam-5566	357	26	1	1	NUM
ejpam-5566	357	27	.	.	PUNCT
ejpam-5566	358	1	the	the	DET
ejpam-5566	358	2	second	second	ADJ
ejpam-5566	358	3	edge	edge	NOUN
ejpam-5566	358	4	joins	join	VERB
ejpam-5566	358	5	another	another	DET
ejpam-5566	358	6	vertex	vertex	NOUN
ejpam-5566	358	7	of	of	ADP
ejpam-5566	358	8	c3	c3	PROPN
ejpam-5566	358	9	to	to	ADP
ejpam-5566	358	10	a	a	DET
ejpam-5566	358	11	vertex	vertex	NOUN
ejpam-5566	358	12	of	of	ADP
ejpam-5566	358	13	ps	ps	NOUN
ejpam-5566	358	14	whose	whose	DET
ejpam-5566	358	15	degree	degree	NOUN
ejpam-5566	358	16	in	in	ADP
ejpam-5566	358	17	ps	ps	PROPN
ejpam-5566	358	18	is	be	AUX
ejpam-5566	358	19	either	either	CCONJ
ejpam-5566	358	20	0	0	NUM
ejpam-5566	358	21	or	or	CCONJ
ejpam-5566	358	22	1	1	NUM
ejpam-5566	358	23	.	.	PUNCT
ejpam-5566	359	1	some	some	DET
ejpam-5566	359	2	examples	example	NOUN
ejpam-5566	359	3	of	of	ADP
ejpam-5566	359	4	kites	kite	NOUN
ejpam-5566	359	5	are	be	AUX
ejpam-5566	359	6	shown	show	VERB
ejpam-5566	359	7	in	in	ADP
ejpam-5566	359	8	figure	figure	NOUN
ejpam-5566	359	9	7	7	NUM
ejpam-5566	359	10	.	.	PUNCT
ejpam-5566	359	11	............................................................................................................................................	............................................................................................................................................	PUNCT
ejpam-5566	359	12	.............................................................................................	.............................................................................................	PUNCT
ejpam-5566	360	1	....................................	....................................	PUNCT
ejpam-5566	360	2	...........	...........	PUNCT
ejpam-5566	361	1	..........	..........	PUNCT
ejpam-5566	361	2	..........	..........	PUNCT
ejpam-5566	362	1	..........	..........	PUNCT
ejpam-5566	362	2	..........	..........	PUNCT
ejpam-5566	363	1	..........	..........	PUNCT
ejpam-5566	363	2	..........	..........	PUNCT
ejpam-5566	364	1	..........	..........	PUNCT
ejpam-5566	364	2	..........	..........	PUNCT
ejpam-5566	365	1	..	..	PUNCT
ejpam-5566	365	2	..	..	PUNCT
ejpam-5566	365	3	..................................	..................................	PUNCT
ejpam-5566	366	1	....................................	....................................	PUNCT
ejpam-5566	366	2	.........	.........	PUNCT
ejpam-5566	366	3	........	........	PUNCT
ejpam-5566	366	4	........	........	PUNCT
ejpam-5566	366	5	........	........	PUNCT
ejpam-5566	366	6	........	........	PUNCT
ejpam-5566	366	7	.......	.......	PUNCT
ejpam-5566	367	1	....................................	....................................	PUNCT
ejpam-5566	367	2	....................................	....................................	PUNCT
ejpam-5566	368	1	................................................	................................................	PUNCT
ejpam-5566	368	2	....................................	....................................	PUNCT
ejpam-5566	369	1	....................................	....................................	PUNCT
ejpam-5566	369	2	kt1,1	kt1,1	PROPN
ejpam-5566	369	3	............................................................................................................................................	............................................................................................................................................	PUNCT
ejpam-5566	369	4	.............................................................................................	.............................................................................................	PUNCT
ejpam-5566	370	1	....................................	....................................	PUNCT
ejpam-5566	370	2	...........	...........	PUNCT
ejpam-5566	371	1	..........	..........	PUNCT
ejpam-5566	371	2	..........	..........	PUNCT
ejpam-5566	372	1	..........	..........	PUNCT
ejpam-5566	372	2	..........	..........	PUNCT
ejpam-5566	373	1	..........	..........	PUNCT
ejpam-5566	373	2	..........	..........	PUNCT
ejpam-5566	374	1	..........	..........	PUNCT
ejpam-5566	374	2	..........	..........	PUNCT
ejpam-5566	375	1	..	..	PUNCT
ejpam-5566	375	2	..	..	PUNCT
ejpam-5566	375	3	..................................	..................................	PUNCT
ejpam-5566	376	1	....................................	....................................	PUNCT
ejpam-5566	376	2	.........	.........	PUNCT
ejpam-5566	376	3	........	........	PUNCT
ejpam-5566	376	4	........	........	PUNCT
ejpam-5566	376	5	........	........	PUNCT
ejpam-5566	376	6	........	........	PUNCT
ejpam-5566	376	7	.......	.......	PUNCT
ejpam-5566	377	1	....................................	....................................	PUNCT
ejpam-5566	377	2	....................................	....................................	PUNCT
ejpam-5566	378	1	................................................	................................................	PUNCT
ejpam-5566	378	2	....................................	....................................	PUNCT
ejpam-5566	379	1	................................................	................................................	PUNCT
ejpam-5566	379	2	....................................	....................................	PUNCT
ejpam-5566	380	1	................................................	................................................	PUNCT
ejpam-5566	380	2	....................................	....................................	PUNCT
ejpam-5566	381	1	....................................	....................................	PUNCT
ejpam-5566	381	2	kt1,3	kt1,3	NOUN
ejpam-5566	381	3	............................................................................................................................................	............................................................................................................................................	PUNCT
ejpam-5566	381	4	.............................................................................................	.............................................................................................	PUNCT
ejpam-5566	382	1	....................................	....................................	PUNCT
ejpam-5566	382	2	...........	...........	PUNCT
ejpam-5566	383	1	..........	..........	PUNCT
ejpam-5566	383	2	..........	..........	PUNCT
ejpam-5566	384	1	..........	..........	PUNCT
ejpam-5566	384	2	..........	..........	PUNCT
ejpam-5566	385	1	..........	..........	PUNCT
ejpam-5566	385	2	..........	..........	PUNCT
ejpam-5566	386	1	..........	..........	PUNCT
ejpam-5566	386	2	..........	..........	PUNCT
ejpam-5566	387	1	..	..	PUNCT
ejpam-5566	387	2	..	..	PUNCT
ejpam-5566	387	3	..................................	..................................	PUNCT
ejpam-5566	388	1	....................................	....................................	PUNCT
ejpam-5566	388	2	.........	.........	PUNCT
ejpam-5566	388	3	........	........	PUNCT
ejpam-5566	388	4	........	........	PUNCT
ejpam-5566	388	5	........	........	PUNCT
ejpam-5566	388	6	........	........	PUNCT
ejpam-5566	388	7	.......	.......	PUNCT
ejpam-5566	388	8	....................................	....................................	PUNCT
ejpam-5566	388	9	.........	.........	PUNCT
ejpam-5566	388	10	........	........	PUNCT
ejpam-5566	388	11	........	........	PUNCT
ejpam-5566	388	12	........	........	PUNCT
ejpam-5566	388	13	........	........	PUNCT
ejpam-5566	388	14	.......	.......	PUNCT
ejpam-5566	389	1	....................................	....................................	PUNCT
ejpam-5566	389	2	....................................	....................................	PUNCT
ejpam-5566	390	1	................................................	................................................	PUNCT
ejpam-5566	390	2	....................................	....................................	PUNCT
ejpam-5566	391	1	................................................	................................................	PUNCT
ejpam-5566	391	2	....................................	....................................	PUNCT
ejpam-5566	392	1	....................................	....................................	PUNCT
ejpam-5566	393	1	kt2,2	kt2,2	NOUN
ejpam-5566	393	2	figure	figure	NOUN
ejpam-5566	393	3	7	7	NUM
ejpam-5566	393	4	:	:	PUNCT
ejpam-5566	393	5	illustrating	illustrate	VERB
ejpam-5566	393	6	the	the	DET
ejpam-5566	393	7	graphs	graph	NOUN
ejpam-5566	393	8	kt2,2	kt2,2	NOUN
ejpam-5566	393	9	,	,	PUNCT
ejpam-5566	393	10	kt1,3	kt1,3	NOUN
ejpam-5566	393	11	and	and	CCONJ
ejpam-5566	393	12	kt1,1	kt1,1	VERB
ejpam-5566	393	13	the	the	DET
ejpam-5566	393	14	following	follow	VERB
ejpam-5566	393	15	remark	remark	NOUN
ejpam-5566	393	16	can	can	AUX
ejpam-5566	393	17	be	be	AUX
ejpam-5566	393	18	easily	easily	ADV
ejpam-5566	393	19	observed	observe	VERB
ejpam-5566	393	20	.	.	PUNCT
ejpam-5566	394	1	remark	remark	NOUN
ejpam-5566	394	2	4	4	NUM
ejpam-5566	394	3	.	.	PUNCT
ejpam-5566	395	1	the	the	DET
ejpam-5566	395	2	size	size	NOUN
ejpam-5566	395	3	and	and	CCONJ
ejpam-5566	395	4	order	order	NOUN
ejpam-5566	395	5	of	of	ADP
ejpam-5566	395	6	the	the	DET
ejpam-5566	395	7	graph	graph	NOUN
ejpam-5566	395	8	ktr	ktr	PROPN
ejpam-5566	395	9	,	,	PUNCT
ejpam-5566	395	10	s	s	PART
ejpam-5566	395	11	is	be	AUX
ejpam-5566	395	12	r	r	NOUN
ejpam-5566	395	13	+	+	SYM
ejpam-5566	395	14	s+	s+	NUM
ejpam-5566	395	15	3	3	NUM
ejpam-5566	395	16	.	.	PUNCT
ejpam-5566	396	1	thus	thus	ADV
ejpam-5566	396	2	,	,	PUNCT
ejpam-5566	396	3	ktr	ktr	PROPN
ejpam-5566	396	4	,	,	PUNCT
ejpam-5566	396	5	s	s	PART
ejpam-5566	396	6	is	be	AUX
ejpam-5566	396	7	a	a	DET
ejpam-5566	396	8	subgraph	subgraph	NOUN
ejpam-5566	396	9	of	of	ADP
ejpam-5566	396	10	c2	c2	PROPN
ejpam-5566	396	11	n	n	PROPN
ejpam-5566	396	12	if	if	SCONJ
ejpam-5566	397	1	and	and	CCONJ
ejpam-5566	397	2	only	only	ADV
ejpam-5566	397	3	if	if	SCONJ
ejpam-5566	397	4	r	r	NOUN
ejpam-5566	397	5	+	+	SYM
ejpam-5566	397	6	s	s	PART
ejpam-5566	397	7	≤	≤	NUM
ejpam-5566	397	8	n−	n−	NOUN
ejpam-5566	397	9	3	3	NUM
ejpam-5566	397	10	.	.	PUNCT
ejpam-5566	398	1	first	first	ADV
ejpam-5566	398	2	we	we	PRON
ejpam-5566	398	3	show	show	VERB
ejpam-5566	398	4	that	that	SCONJ
ejpam-5566	398	5	kt1,1	kt1,1	NOUN
ejpam-5566	398	6	is	be	AUX
ejpam-5566	398	7	a	a	DET
ejpam-5566	398	8	generator	generator	NOUN
ejpam-5566	398	9	subgraph	subgraph	NOUN
ejpam-5566	398	10	of	of	ADP
ejpam-5566	398	11	c2	c2	PROPN
ejpam-5566	398	12	n	n	PRON
ejpam-5566	398	13	theorem	theorem	VERB
ejpam-5566	398	14	13	13	NUM
ejpam-5566	398	15	.	.	PUNCT
ejpam-5566	399	1	the	the	DET
ejpam-5566	399	2	kite	kite	NOUN
ejpam-5566	399	3	graph	graph	NOUN
ejpam-5566	399	4	kt1,1	kt1,1	PROPN
ejpam-5566	399	5	is	be	AUX
ejpam-5566	399	6	a	a	DET
ejpam-5566	399	7	generator	generator	NOUN
ejpam-5566	399	8	subgraph	subgraph	NOUN
ejpam-5566	399	9	of	of	ADP
ejpam-5566	399	10	c2	c2	PROPN
ejpam-5566	399	11	n.	n.	PROPN
ejpam-5566	399	12	proof	proof	PROPN
ejpam-5566	399	13	.	.	PUNCT
ejpam-5566	400	1	consider	consider	VERB
ejpam-5566	400	2	the	the	DET
ejpam-5566	400	3	labeling	labeling	NOUN
ejpam-5566	400	4	of	of	ADP
ejpam-5566	400	5	c2	c2	PROPN
ejpam-5566	400	6	n.	n.	PROPN
ejpam-5566	400	7	let	let	VERB
ejpam-5566	400	8	a	a	PRON
ejpam-5566	400	9	=	=	PUNCT
ejpam-5566	400	10	{	{	PUNCT
ejpam-5566	400	11	en−1	en−1	PROPN
ejpam-5566	400	12	,	,	PUNCT
ejpam-5566	400	13	sn−1	sn−1	PROPN
ejpam-5566	400	14	,	,	PUNCT
ejpam-5566	400	15	en	en	X
ejpam-5566	400	16	,	,	PUNCT
ejpam-5566	400	17	en−2	en−2	PROPN
ejpam-5566	400	18	,	,	PUNCT
ejpam-5566	400	19	e1	e1	PROPN
ejpam-5566	400	20	}	}	PUNCT
ejpam-5566	400	21	.	.	PUNCT
ejpam-5566	401	1	define	define	VERB
ejpam-5566	401	2	b	b	NOUN
ejpam-5566	401	3	=	=	PUNCT
ejpam-5566	401	4	a\{e1	a\{e1	NOUN
ejpam-5566	401	5	}	}	PUNCT
ejpam-5566	401	6	∪	∪	ADJ
ejpam-5566	401	7	{	{	PUNCT
ejpam-5566	401	8	s1	s1	NOUN
ejpam-5566	401	9	}	}	PUNCT
ejpam-5566	401	10	.	.	PUNCT
ejpam-5566	402	1	it	it	PRON
ejpam-5566	402	2	can	can	AUX
ejpam-5566	402	3	be	be	AUX
ejpam-5566	402	4	verified	verify	VERB
ejpam-5566	402	5	that	that	SCONJ
ejpam-5566	402	6	a	a	DET
ejpam-5566	402	7	,	,	PUNCT
ejpam-5566	402	8	b	b	PROPN
ejpam-5566	402	9	∈	∈	NOUN
ejpam-5566	402	10	ekt1,1(c	ekt1,1(c	ADJ
ejpam-5566	402	11	2	2	NUM
ejpam-5566	402	12	n	n	NUM
ejpam-5566	402	13	)	)	PUNCT
ejpam-5566	402	14	.	.	PUNCT
ejpam-5566	403	1	thus	thus	ADV
ejpam-5566	403	2	,	,	PUNCT
ejpam-5566	403	3	a	a	DET
ejpam-5566	403	4	+	+	NOUN
ejpam-5566	403	5	b	b	NOUN
ejpam-5566	403	6	=	=	SYM
ejpam-5566	403	7	{	{	PUNCT
ejpam-5566	403	8	e1	e1	PROPN
ejpam-5566	403	9	,	,	PUNCT
ejpam-5566	403	10	s1	s1	NOUN
ejpam-5566	403	11	}	}	PUNCT
ejpam-5566	403	12	∈	∈	PROPN
ejpam-5566	403	13	ekt1,1(c	ekt1,1(c	ADJ
ejpam-5566	403	14	2	2	NUM
ejpam-5566	403	15	n	n	NUM
ejpam-5566	403	16	)	)	PUNCT
ejpam-5566	403	17	.	.	PUNCT
ejpam-5566	404	1	clearly	clearly	ADV
ejpam-5566	404	2	,	,	PUNCT
ejpam-5566	404	3	the	the	DET
ejpam-5566	404	4	size	size	NOUN
ejpam-5566	404	5	of	of	ADP
ejpam-5566	404	6	kt1,1	kt1,1	NOUN
ejpam-5566	404	7	is	be	AUX
ejpam-5566	404	8	5	5	NUM
ejpam-5566	404	9	,	,	PUNCT
ejpam-5566	404	10	which	which	PRON
ejpam-5566	404	11	is	be	AUX
ejpam-5566	404	12	odd	odd	ADJ
ejpam-5566	404	13	.	.	PUNCT
ejpam-5566	405	1	by	by	ADP
ejpam-5566	405	2	lemma	lemma	PROPN
ejpam-5566	405	3	3	3	NUM
ejpam-5566	405	4	,	,	PUNCT
ejpam-5566	405	5	kt1,1	kt1,1	PROPN
ejpam-5566	405	6	is	be	AUX
ejpam-5566	405	7	a	a	DET
ejpam-5566	405	8	generator	generator	NOUN
ejpam-5566	405	9	subgraph	subgraph	NOUN
ejpam-5566	405	10	of	of	ADP
ejpam-5566	405	11	c2	c2	PROPN
ejpam-5566	405	12	n.	n.	PROPN
ejpam-5566	405	13	the	the	DET
ejpam-5566	405	14	next	next	ADJ
ejpam-5566	405	15	result	result	NOUN
ejpam-5566	405	16	gives	give	VERB
ejpam-5566	405	17	the	the	DET
ejpam-5566	405	18	criteria	criterion	NOUN
ejpam-5566	405	19	for	for	ADP
ejpam-5566	405	20	the	the	DET
ejpam-5566	405	21	subgraph	subgraph	PROPN
ejpam-5566	405	22	ktr	ktr	PROPN
ejpam-5566	405	23	,	,	PUNCT
ejpam-5566	405	24	s	s	VERB
ejpam-5566	405	25	to	to	PART
ejpam-5566	405	26	be	be	AUX
ejpam-5566	405	27	a	a	DET
ejpam-5566	405	28	generator	generator	NOUN
ejpam-5566	405	29	subgraph	subgraph	NOUN
ejpam-5566	405	30	of	of	ADP
ejpam-5566	405	31	c2	c2	PROPN
ejpam-5566	405	32	n.	n.	PROPN
ejpam-5566	405	33	references	reference	NOUN
ejpam-5566	405	34	3824	3824	NUM
ejpam-5566	405	35	theorem	theorem	VERB
ejpam-5566	405	36	14	14	NUM
ejpam-5566	405	37	.	.	PUNCT
ejpam-5566	406	1	let	let	VERB
ejpam-5566	406	2	r	r	NOUN
ejpam-5566	406	3	and	and	CCONJ
ejpam-5566	406	4	s	s	AUX
ejpam-5566	406	5	be	be	AUX
ejpam-5566	406	6	positive	positive	ADJ
ejpam-5566	406	7	integers	integer	NOUN
ejpam-5566	406	8	.	.	PUNCT
ejpam-5566	407	1	then	then	ADV
ejpam-5566	407	2	the	the	DET
ejpam-5566	407	3	kite	kite	NOUN
ejpam-5566	407	4	graph	graph	NOUN
ejpam-5566	407	5	ktr	ktr	PROPN
ejpam-5566	407	6	,	,	PUNCT
ejpam-5566	407	7	s	s	PART
ejpam-5566	407	8	is	be	AUX
ejpam-5566	407	9	a	a	DET
ejpam-5566	407	10	generator	generator	NOUN
ejpam-5566	407	11	subgraph	subgraph	NOUN
ejpam-5566	407	12	of	of	ADP
ejpam-5566	407	13	c2	c2	PROPN
ejpam-5566	407	14	n	n	PROPN
ejpam-5566	407	15	if	if	SCONJ
ejpam-5566	408	1	and	and	CCONJ
ejpam-5566	408	2	only	only	ADV
ejpam-5566	408	3	if	if	SCONJ
ejpam-5566	408	4	r	r	NOUN
ejpam-5566	408	5	+	+	SYM
ejpam-5566	408	6	s	s	NOUN
ejpam-5566	408	7	is	be	AUX
ejpam-5566	408	8	even	even	ADV
ejpam-5566	408	9	and	and	CCONJ
ejpam-5566	408	10	r	r	NOUN
ejpam-5566	408	11	+	+	SYM
ejpam-5566	408	12	s	s	PART
ejpam-5566	408	13	≤	≤	NUM
ejpam-5566	408	14	n−	n−	NOUN
ejpam-5566	408	15	3	3	NUM
ejpam-5566	408	16	.	.	PUNCT
ejpam-5566	409	1	proof	proof	NOUN
ejpam-5566	409	2	.	.	PUNCT
ejpam-5566	410	1	assume	assume	VERB
ejpam-5566	410	2	that	that	SCONJ
ejpam-5566	410	3	the	the	DET
ejpam-5566	410	4	kite	kite	NOUN
ejpam-5566	410	5	ktr	ktr	PROPN
ejpam-5566	410	6	,	,	PUNCT
ejpam-5566	410	7	s	s	PART
ejpam-5566	410	8	is	be	AUX
ejpam-5566	410	9	a	a	DET
ejpam-5566	410	10	generator	generator	NOUN
ejpam-5566	410	11	subgraph	subgraph	NOUN
ejpam-5566	410	12	of	of	ADP
ejpam-5566	410	13	c2	c2	PROPN
ejpam-5566	410	14	n.	n.	PROPN
ejpam-5566	410	15	by	by	ADP
ejpam-5566	410	16	theorem	theorem	NOUN
ejpam-5566	410	17	1	1	NUM
ejpam-5566	410	18	,	,	PUNCT
ejpam-5566	410	19	the	the	DET
ejpam-5566	410	20	size	size	NOUN
ejpam-5566	410	21	of	of	ADP
ejpam-5566	410	22	ktr	ktr	PROPN
ejpam-5566	410	23	,	,	PUNCT
ejpam-5566	410	24	s	s	PART
ejpam-5566	410	25	is	be	AUX
ejpam-5566	410	26	odd	odd	ADJ
ejpam-5566	410	27	.	.	PUNCT
ejpam-5566	411	1	it	it	PRON
ejpam-5566	411	2	follows	follow	VERB
ejpam-5566	411	3	that	that	SCONJ
ejpam-5566	411	4	that	that	PRON
ejpam-5566	411	5	r+	r+	PRON
ejpam-5566	411	6	s	s	NOUN
ejpam-5566	411	7	is	be	AUX
ejpam-5566	411	8	even	even	ADV
ejpam-5566	411	9	in	in	ADP
ejpam-5566	411	10	view	view	NOUN
ejpam-5566	411	11	of	of	ADP
ejpam-5566	411	12	remark	remark	NOUN
ejpam-5566	411	13	4	4	NUM
ejpam-5566	411	14	.	.	PUNCT
ejpam-5566	412	1	we	we	PRON
ejpam-5566	412	2	claim	claim	VERB
ejpam-5566	412	3	that	that	SCONJ
ejpam-5566	412	4	r+s	r+	VERB
ejpam-5566	412	5	≤	≤	ADJ
ejpam-5566	412	6	n−3	n−3	PROPN
ejpam-5566	412	7	.	.	PUNCT
ejpam-5566	412	8	suppose	suppose	VERB
ejpam-5566	412	9	r+s	r+	NOUN
ejpam-5566	412	10	>	>	X
ejpam-5566	413	1	n−3	n−3	PROPN
ejpam-5566	413	2	,	,	PUNCT
ejpam-5566	413	3	then	then	ADV
ejpam-5566	413	4	by	by	ADP
ejpam-5566	413	5	remark	remark	NOUN
ejpam-5566	413	6	4	4	NUM
ejpam-5566	413	7	,	,	PUNCT
ejpam-5566	413	8	ktr	ktr	PROPN
ejpam-5566	413	9	,	,	PUNCT
ejpam-5566	413	10	s	s	PART
ejpam-5566	413	11	is	be	AUX
ejpam-5566	413	12	not	not	PART
ejpam-5566	413	13	a	a	DET
ejpam-5566	413	14	subgraph	subgraph	NOUN
ejpam-5566	413	15	of	of	ADP
ejpam-5566	413	16	c2	c2	PROPN
ejpam-5566	413	17	n.	n.	PROPN
ejpam-5566	413	18	this	this	PRON
ejpam-5566	413	19	is	be	AUX
ejpam-5566	413	20	a	a	DET
ejpam-5566	413	21	contradiction	contradiction	NOUN
ejpam-5566	413	22	.	.	PUNCT
ejpam-5566	414	1	conversely	conversely	ADV
ejpam-5566	414	2	,	,	PUNCT
ejpam-5566	414	3	since	since	SCONJ
ejpam-5566	414	4	r+s	r+s	NUM
ejpam-5566	414	5	is	be	AUX
ejpam-5566	414	6	even	even	ADV
ejpam-5566	414	7	and	and	CCONJ
ejpam-5566	414	8	r+s	r+	VERB
ejpam-5566	414	9	≤	≤	ADJ
ejpam-5566	414	10	n−3	n−3	PROPN
ejpam-5566	414	11	,	,	PUNCT
ejpam-5566	414	12	the	the	DET
ejpam-5566	414	13	size	size	NOUN
ejpam-5566	414	14	of	of	ADP
ejpam-5566	414	15	ktr	ktr	PROPN
ejpam-5566	414	16	,	,	PUNCT
ejpam-5566	414	17	s	s	PART
ejpam-5566	414	18	is	be	AUX
ejpam-5566	414	19	odd	odd	ADJ
ejpam-5566	414	20	and	and	CCONJ
ejpam-5566	414	21	ktr	ktr	PROPN
ejpam-5566	414	22	,	,	PUNCT
ejpam-5566	414	23	s	s	PART
ejpam-5566	414	24	is	be	AUX
ejpam-5566	414	25	a	a	DET
ejpam-5566	414	26	subgraph	subgraph	NOUN
ejpam-5566	414	27	of	of	ADP
ejpam-5566	414	28	c2	c2	PROPN
ejpam-5566	414	29	n.	n.	PROPN
ejpam-5566	414	30	so	so	ADV
ejpam-5566	414	31	the	the	DET
ejpam-5566	414	32	uniform	uniform	PROPN
ejpam-5566	414	33	set	set	PROPN
ejpam-5566	414	34	ektr	ektr	PROPN
ejpam-5566	414	35	,	,	PUNCT
ejpam-5566	414	36	s(c	s(c	PROPN
ejpam-5566	414	37	2	2	NUM
ejpam-5566	414	38	n	n	CCONJ
ejpam-5566	414	39	)	)	PUNCT
ejpam-5566	414	40	is	be	AUX
ejpam-5566	414	41	not	not	PART
ejpam-5566	414	42	empty	empty	ADJ
ejpam-5566	414	43	.	.	PUNCT
ejpam-5566	415	1	if	if	SCONJ
ejpam-5566	415	2	r	r	NOUN
ejpam-5566	415	3	=	=	SYM
ejpam-5566	415	4	s	s	NOUN
ejpam-5566	415	5	=	=	SYM
ejpam-5566	415	6	1	1	NUM
ejpam-5566	415	7	then	then	ADV
ejpam-5566	415	8	ktr	ktr	PROPN
ejpam-5566	415	9	,	,	PUNCT
ejpam-5566	415	10	s	s	PART
ejpam-5566	415	11	is	be	AUX
ejpam-5566	415	12	a	a	DET
ejpam-5566	415	13	generator	generator	NOUN
ejpam-5566	415	14	subgraph	subgraph	NOUN
ejpam-5566	415	15	of	of	ADP
ejpam-5566	415	16	c2	c2	PROPN
ejpam-5566	415	17	n	n	PROPN
ejpam-5566	415	18	in	in	ADP
ejpam-5566	415	19	view	view	NOUN
ejpam-5566	415	20	of	of	ADP
ejpam-5566	415	21	theorem	theorem	NOUN
ejpam-5566	415	22	13	13	NUM
ejpam-5566	415	23	.	.	PUNCT
ejpam-5566	416	1	let	let	VERB
ejpam-5566	416	2	us	we	PRON
ejpam-5566	416	3	assume	assume	VERB
ejpam-5566	416	4	that	that	SCONJ
ejpam-5566	416	5	r	r	NOUN
ejpam-5566	416	6	,	,	PUNCT
ejpam-5566	416	7	s	s	PART
ejpam-5566	416	8	>	>	X
ejpam-5566	416	9	1	1	X
ejpam-5566	416	10	.	.	PUNCT
ejpam-5566	416	11	consider	consider	VERB
ejpam-5566	416	12	the	the	DET
ejpam-5566	416	13	labeling	labeling	NOUN
ejpam-5566	416	14	of	of	ADP
ejpam-5566	416	15	c2	c2	PROPN
ejpam-5566	416	16	n	n	CCONJ
ejpam-5566	416	17	,	,	PUNCT
ejpam-5566	416	18	let	let	VERB
ejpam-5566	416	19	a	a	DET
ejpam-5566	416	20	=	=	PUNCT
ejpam-5566	416	21	{	{	PUNCT
ejpam-5566	416	22	en	en	X
ejpam-5566	416	23	,	,	PUNCT
ejpam-5566	416	24	e1	e1	PROPN
ejpam-5566	416	25	,	,	PUNCT
ejpam-5566	416	26	sn	sn	PROPN
ejpam-5566	416	27	,	,	PUNCT
ejpam-5566	416	28	e2	e2	PROPN
ejpam-5566	416	29	,	,	PUNCT
ejpam-5566	416	30	e3	e3	NOUN
ejpam-5566	416	31	,	,	PUNCT
ejpam-5566	416	32	.	.	PUNCT
ejpam-5566	416	33	.	.	PUNCT
ejpam-5566	417	1	.	.	PUNCT
ejpam-5566	418	1	,	,	PUNCT
ejpam-5566	418	2	er+1	er+1	PROPN
ejpam-5566	418	3	,	,	PUNCT
ejpam-5566	418	4	en−1	en−1	PROPN
ejpam-5566	418	5	,	,	PUNCT
ejpam-5566	418	6	en−2	en−2	PROPN
ejpam-5566	418	7	,	,	PUNCT
ejpam-5566	418	8	.	.	PUNCT
ejpam-5566	418	9	.	.	PUNCT
ejpam-5566	418	10	.	.	PUNCT
ejpam-5566	419	1	,	,	PUNCT
ejpam-5566	419	2	en−s	en−s	ADJ
ejpam-5566	419	3	}	}	PUNCT
ejpam-5566	419	4	.	.	PUNCT
ejpam-5566	420	1	define	define	VERB
ejpam-5566	420	2	b	b	NOUN
ejpam-5566	420	3	=	=	SYM
ejpam-5566	420	4	a\{e2	a\{e2	X
ejpam-5566	420	5	}	}	PUNCT
ejpam-5566	420	6	∪	∪	NOUN
ejpam-5566	420	7	{	{	PUNCT
ejpam-5566	420	8	s1	s1	NOUN
ejpam-5566	420	9	}	}	PUNCT
ejpam-5566	420	10	.	.	PUNCT
ejpam-5566	421	1	it	it	PRON
ejpam-5566	421	2	can	can	AUX
ejpam-5566	421	3	be	be	AUX
ejpam-5566	421	4	verified	verify	VERB
ejpam-5566	421	5	that	that	SCONJ
ejpam-5566	421	6	a	a	DET
ejpam-5566	421	7	,	,	PUNCT
ejpam-5566	421	8	b	b	PROPN
ejpam-5566	421	9	∈	∈	PROPN
ejpam-5566	421	10	ektr	ektr	NOUN
ejpam-5566	421	11	,	,	PUNCT
ejpam-5566	421	12	s(c	s(c	PROPN
ejpam-5566	421	13	2	2	NUM
ejpam-5566	421	14	n	n	CCONJ
ejpam-5566	421	15	)	)	PUNCT
ejpam-5566	421	16	.	.	PUNCT
ejpam-5566	422	1	thus	thus	ADV
ejpam-5566	422	2	,	,	PUNCT
ejpam-5566	422	3	a	a	DET
ejpam-5566	422	4	+	+	X
ejpam-5566	422	5	b	b	NOUN
ejpam-5566	422	6	=	=	SYM
ejpam-5566	422	7	{	{	PUNCT
ejpam-5566	422	8	e2	e2	PROPN
ejpam-5566	422	9	,	,	PUNCT
ejpam-5566	422	10	s1	s1	NOUN
ejpam-5566	422	11	}	}	PUNCT
ejpam-5566	422	12	∈	∈	PROPN
ejpam-5566	422	13	ektr	ektr	NOUN
ejpam-5566	422	14	,	,	PUNCT
ejpam-5566	422	15	s(c	s(c	PROPN
ejpam-5566	422	16	2	2	NUM
ejpam-5566	422	17	n	n	CCONJ
ejpam-5566	422	18	)	)	PUNCT
ejpam-5566	422	19	.	.	PUNCT
ejpam-5566	423	1	by	by	ADP
ejpam-5566	423	2	lemma	lemma	PROPN
ejpam-5566	423	3	1	1	NUM
ejpam-5566	423	4	,	,	PUNCT
ejpam-5566	423	5	{	{	PUNCT
ejpam-5566	423	6	e1	e1	NOUN
ejpam-5566	423	7	,	,	PUNCT
ejpam-5566	423	8	s1	s1	NOUN
ejpam-5566	423	9	}	}	PUNCT
ejpam-5566	423	10	∈	∈	PROPN
ejpam-5566	423	11	ektr	ektr	NOUN
ejpam-5566	423	12	,	,	PUNCT
ejpam-5566	423	13	s(c	s(c	PROPN
ejpam-5566	423	14	2	2	NUM
ejpam-5566	423	15	n	n	CCONJ
ejpam-5566	423	16	)	)	PUNCT
ejpam-5566	423	17	.	.	PUNCT
ejpam-5566	424	1	by	by	ADP
ejpam-5566	424	2	lemma	lemma	PROPN
ejpam-5566	424	3	3,ktr	3,ktr	PROPN
ejpam-5566	424	4	,	,	PUNCT
ejpam-5566	424	5	s	s	PART
ejpam-5566	424	6	is	be	AUX
ejpam-5566	424	7	a	a	DET
ejpam-5566	424	8	generator	generator	NOUN
ejpam-5566	424	9	subgraph	subgraph	NOUN
ejpam-5566	424	10	of	of	ADP
ejpam-5566	424	11	c2	c2	PROPN
ejpam-5566	424	12	n.	n.	PROPN
ejpam-5566	424	13	3	3	X
ejpam-5566	424	14	.	.	X
ejpam-5566	425	1	summary	summary	NOUN
ejpam-5566	425	2	and	and	CCONJ
ejpam-5566	425	3	conclusions	conclusion	NOUN
ejpam-5566	425	4	some	some	DET
ejpam-5566	425	5	classes	class	NOUN
ejpam-5566	425	6	of	of	ADP
ejpam-5566	425	7	generator	generator	NOUN
ejpam-5566	425	8	subgraphs	subgraph	NOUN
ejpam-5566	425	9	of	of	ADP
ejpam-5566	425	10	c2	c2	PROPN
ejpam-5566	425	11	n	n	PRON
ejpam-5566	425	12	were	be	AUX
ejpam-5566	425	13	found	find	VERB
ejpam-5566	425	14	,	,	PUNCT
ejpam-5566	425	15	such	such	ADJ
ejpam-5566	425	16	as	as	ADP
ejpam-5566	425	17	the	the	DET
ejpam-5566	425	18	star	star	NOUN
ejpam-5566	425	19	graphs	graph	NOUN
ejpam-5566	425	20	,	,	PUNCT
ejpam-5566	425	21	path	path	NOUN
ejpam-5566	425	22	graphs,(3	graphs,(3	NOUN
ejpam-5566	425	23	,	,	PUNCT
ejpam-5566	425	24	r)−	r)−	PROPN
ejpam-5566	425	25	tadpole	tadpole	NOUN
ejpam-5566	425	26	graphs	graph	NOUN
ejpam-5566	425	27	and	and	CCONJ
ejpam-5566	425	28	kite	kite	NOUN
ejpam-5566	425	29	graphs	graph	NOUN
ejpam-5566	425	30	.	.	PUNCT
ejpam-5566	426	1	characterization	characterization	NOUN
ejpam-5566	426	2	for	for	ADP
ejpam-5566	426	3	the	the	DET
ejpam-5566	426	4	generator	generator	NOUN
ejpam-5566	426	5	subgraphs	subgraphs	NOUN
ejpam-5566	426	6	of	of	ADP
ejpam-5566	426	7	the	the	DET
ejpam-5566	426	8	square	square	NOUN
ejpam-5566	426	9	of	of	ADP
ejpam-5566	426	10	a	a	DET
ejpam-5566	426	11	cycle	cycle	NOUN
ejpam-5566	426	12	is	be	AUX
ejpam-5566	426	13	still	still	ADV
ejpam-5566	426	14	open	open	ADJ
ejpam-5566	426	15	.	.	PUNCT
ejpam-5566	427	1	acknowledgements	acknowledgement	NOUN
ejpam-5566	427	2	part	part	NOUN
ejpam-5566	427	3	of	of	ADP
ejpam-5566	427	4	this	this	DET
ejpam-5566	427	5	work	work	NOUN
ejpam-5566	427	6	was	be	AUX
ejpam-5566	427	7	done	do	VERB
ejpam-5566	427	8	while	while	SCONJ
ejpam-5566	427	9	the	the	DET
ejpam-5566	427	10	author	author	NOUN
ejpam-5566	427	11	was	be	AUX
ejpam-5566	427	12	taking	take	VERB
ejpam-5566	427	13	ph.d	ph.d	PROPN
ejpam-5566	427	14	.	.	PUNCT
ejpam-5566	428	1	fellowship	fellowship	NOUN
ejpam-5566	428	2	at	at	ADP
ejpam-5566	428	3	the	the	DET
ejpam-5566	428	4	batangas	batangas	PROPN
ejpam-5566	428	5	state	state	PROPN
ejpam-5566	428	6	university	university	PROPN
ejpam-5566	428	7	through	through	ADP
ejpam-5566	428	8	the	the	DET
ejpam-5566	428	9	faculty	faculty	NOUN
ejpam-5566	428	10	scholarship	scholarship	NOUN
ejpam-5566	428	11	grant	grant	NOUN
ejpam-5566	428	12	of	of	ADP
ejpam-5566	428	13	the	the	DET
ejpam-5566	428	14	batangas	batangas	PROPN
ejpam-5566	428	15	state	state	PROPN
ejpam-5566	428	16	university	university	PROPN
ejpam-5566	428	17	under	under	ADP
ejpam-5566	428	18	the	the	DET
ejpam-5566	428	19	research	research	NOUN
ejpam-5566	428	20	supervision	supervision	NOUN
ejpam-5566	428	21	of	of	ADP
ejpam-5566	428	22	dr	dr	PROPN
ejpam-5566	428	23	.	.	PROPN
ejpam-5566	428	24	neil	neil	PROPN
ejpam-5566	428	25	m.	m.	PROPN
ejpam-5566	428	26	mame	mame	PROPN
ejpam-5566	428	27	.	.	PUNCT
ejpam-5566	429	1	also	also	ADV
ejpam-5566	429	2	,	,	PUNCT
ejpam-5566	429	3	the	the	DET
ejpam-5566	429	4	author	author	NOUN
ejpam-5566	429	5	wishes	wish	VERB
ejpam-5566	429	6	to	to	PART
ejpam-5566	429	7	thank	thank	VERB
ejpam-5566	429	8	the	the	DET
ejpam-5566	429	9	anonymous	anonymous	ADJ
ejpam-5566	429	10	reviewers	reviewer	NOUN
ejpam-5566	429	11	for	for	ADP
ejpam-5566	429	12	their	their	PRON
ejpam-5566	429	13	comments	comment	NOUN
ejpam-5566	429	14	and	and	CCONJ
ejpam-5566	429	15	suggestions	suggestion	NOUN
ejpam-5566	429	16	,	,	PUNCT
ejpam-5566	429	17	which	which	PRON
ejpam-5566	429	18	have	have	AUX
ejpam-5566	429	19	significantly	significantly	ADV
ejpam-5566	429	20	improved	improve	VERB
ejpam-5566	429	21	this	this	DET
ejpam-5566	429	22	manuscript	manuscript	NOUN
ejpam-5566	429	23	.	.	PUNCT
ejpam-5566	430	1	references	reference	NOUN
ejpam-5566	430	2	[	[	X
ejpam-5566	430	3	1	1	NUM
ejpam-5566	430	4	]	]	PUNCT
ejpam-5566	430	5	c.	c.	PROPN
ejpam-5566	430	6	alib	alib	PROPN
ejpam-5566	430	7	and	and	CCONJ
ejpam-5566	430	8	d.	d.	PROPN
ejpam-5566	430	9	magpantay	magpantay	PROPN
ejpam-5566	430	10	.	.	PUNCT
ejpam-5566	431	1	some	some	DET
ejpam-5566	431	2	parameters	parameter	NOUN
ejpam-5566	431	3	of	of	ADP
ejpam-5566	431	4	the	the	DET
ejpam-5566	431	5	central	central	ADJ
ejpam-5566	431	6	graphs	graph	NOUN
ejpam-5566	431	7	of	of	ADP
ejpam-5566	431	8	the	the	DET
ejpam-5566	431	9	identity	identity	NOUN
ejpam-5566	431	10	graphs	graph	NOUN
ejpam-5566	431	11	of	of	ADP
ejpam-5566	431	12	finite	finite	ADJ
ejpam-5566	431	13	cyclic	cyclic	ADJ
ejpam-5566	431	14	groups	group	NOUN
ejpam-5566	431	15	.	.	PUNCT
ejpam-5566	432	1	european	european	ADJ
ejpam-5566	432	2	journal	journal	PROPN
ejpam-5566	432	3	of	of	ADP
ejpam-5566	432	4	pure	pure	ADJ
ejpam-5566	432	5	and	and	CCONJ
ejpam-5566	432	6	applied	applied	ADJ
ejpam-5566	432	7	mathematics	mathematic	NOUN
ejpam-5566	432	8	,	,	PUNCT
ejpam-5566	432	9	15(3):1098–1112	15(3):1098–1112	NUM
ejpam-5566	432	10	,	,	PUNCT
ejpam-5566	432	11	2022	2022	NUM
ejpam-5566	432	12	.	.	PUNCT
ejpam-5566	433	1	[	[	X
ejpam-5566	433	2	2	2	X
ejpam-5566	433	3	]	]	PUNCT
ejpam-5566	433	4	j.	j.	PROPN
ejpam-5566	433	5	bonifacio	bonifacio	PROPN
ejpam-5566	433	6	,	,	PUNCT
ejpam-5566	433	7	c.	c.	PROPN
ejpam-5566	433	8	adaya	adaya	PROPN
ejpam-5566	433	9	,	,	PUNCT
ejpam-5566	433	10	and	and	CCONJ
ejpam-5566	433	11	d.	d.	PROPN
ejpam-5566	433	12	magpantay	magpantay	PROPN
ejpam-5566	433	13	.	.	PUNCT
ejpam-5566	434	1	on	on	ADP
ejpam-5566	434	2	the	the	DET
ejpam-5566	434	3	j−	j−	PROPN
ejpam-5566	434	4	edge	edge	NOUN
ejpam-5566	434	5	intersection	intersection	NOUN
ejpam-5566	434	6	graph	graph	NOUN
ejpam-5566	434	7	of	of	ADP
ejpam-5566	434	8	cycle	cycle	NOUN
ejpam-5566	434	9	graph	graph	NOUN
ejpam-5566	434	10	.	.	PUNCT
ejpam-5566	435	1	european	european	PROPN
ejpam-5566	435	2	journal	journal	PROPN
ejpam-5566	435	3	of	of	ADP
ejpam-5566	435	4	pure	pure	ADJ
ejpam-5566	435	5	and	and	CCONJ
ejpam-5566	435	6	applied	applied	ADJ
ejpam-5566	435	7	mathematics	mathematic	NOUN
ejpam-5566	435	8	,	,	PUNCT
ejpam-5566	435	9	16(4):2476–2498	16(4):2476–2498	NUM
ejpam-5566	435	10	,	,	PUNCT
ejpam-5566	435	11	2023	2023	NUM
ejpam-5566	435	12	.	.	PUNCT
ejpam-5566	436	1	[	[	X
ejpam-5566	436	2	3	3	X
ejpam-5566	436	3	]	]	X
ejpam-5566	436	4	g.	g.	PROPN
ejpam-5566	436	5	chartrand	chartrand	PROPN
ejpam-5566	436	6	and	and	CCONJ
ejpam-5566	436	7	p.	p.	PROPN
ejpam-5566	436	8	zhang	zhang	PROPN
ejpam-5566	436	9	.	.	PUNCT
ejpam-5566	437	1	introduction	introduction	NOUN
ejpam-5566	437	2	to	to	AUX
ejpam-5566	437	3	graph	graph	NOUN
ejpam-5566	437	4	theory	theory	NOUN
ejpam-5566	437	5	.	.	PUNCT
ejpam-5566	438	1	mc	mc	PROPN
ejpam-5566	438	2	graw	graw	PROPN
ejpam-5566	438	3	hill	hill	PROPN
ejpam-5566	438	4	,	,	PUNCT
ejpam-5566	438	5	singapore	singapore	PROPN
ejpam-5566	438	6	,	,	PUNCT
ejpam-5566	438	7	2005	2005	NUM
ejpam-5566	438	8	.	.	PUNCT
ejpam-5566	439	1	[	[	X
ejpam-5566	439	2	4	4	X
ejpam-5566	439	3	]	]	X
ejpam-5566	439	4	s.	s.	PROPN
ejpam-5566	439	5	gervacio	gervacio	PROPN
ejpam-5566	439	6	.	.	PUNCT
ejpam-5566	440	1	generator	generator	NOUN
ejpam-5566	440	2	graphs	graph	NOUN
ejpam-5566	440	3	.	.	PUNCT
ejpam-5566	441	1	in	in	ADP
ejpam-5566	441	2	4th	4th	ADJ
ejpam-5566	441	3	international	international	ADJ
ejpam-5566	441	4	conference	conference	NOUN
ejpam-5566	441	5	on	on	ADP
ejpam-5566	441	6	combinatorial	combinatorial	ADJ
ejpam-5566	441	7	mathematics	mathematic	NOUN
ejpam-5566	441	8	and	and	CCONJ
ejpam-5566	441	9	combinatorial	combinatorial	ADJ
ejpam-5566	441	10	computing	computing	NOUN
ejpam-5566	441	11	.	.	PUNCT
ejpam-5566	441	12	,	,	PUNCT
ejpam-5566	441	13	u.a	u.a	PROPN
ejpam-5566	441	14	.	.	PROPN
ejpam-5566	441	15	,	,	PUNCT
ejpam-5566	441	16	new	new	PROPN
ejpam-5566	441	17	zealand	zealand	PROPN
ejpam-5566	441	18	,	,	PUNCT
ejpam-5566	441	19	2008	2008	NUM
ejpam-5566	441	20	.	.	PUNCT
ejpam-5566	442	1	references	reference	NOUN
ejpam-5566	442	2	3825	3825	NUM
ejpam-5566	443	1	[	[	X
ejpam-5566	443	2	5	5	NUM
ejpam-5566	443	3	]	]	PUNCT
ejpam-5566	443	4	s.	s.	PROPN
ejpam-5566	443	5	gervacio	gervacio	PROPN
ejpam-5566	443	6	.	.	PUNCT
ejpam-5566	444	1	determination	determination	NOUN
ejpam-5566	444	2	of	of	ADP
ejpam-5566	444	3	uniform	uniform	ADJ
ejpam-5566	444	4	generating	generating	NOUN
ejpam-5566	444	5	sets	set	NOUN
ejpam-5566	444	6	of	of	ADP
ejpam-5566	444	7	the	the	DET
ejpam-5566	444	8	edge	edge	NOUN
ejpam-5566	444	9	space	space	NOUN
ejpam-5566	444	10	of	of	ADP
ejpam-5566	444	11	some	some	DET
ejpam-5566	444	12	graphs	graph	NOUN
ejpam-5566	444	13	.	.	PUNCT
ejpam-5566	445	1	research	research	NOUN
ejpam-5566	445	2	project	project	NOUN
ejpam-5566	445	3	under	under	ADP
ejpam-5566	445	4	urco	urco	NOUN
ejpam-5566	445	5	,	,	PUNCT
ejpam-5566	445	6	de	de	X
ejpam-5566	445	7	la	la	X
ejpam-5566	445	8	salle	salle	PROPN
ejpam-5566	445	9	university	university	PROPN
ejpam-5566	445	10	-	-	PUNCT
ejpam-5566	445	11	manila	manila	PROPN
ejpam-5566	445	12	,	,	PUNCT
ejpam-5566	445	13	2009	2009	NUM
ejpam-5566	445	14	.	.	PUNCT
ejpam-5566	446	1	[	[	X
ejpam-5566	446	2	6	6	NUM
ejpam-5566	446	3	]	]	PUNCT
ejpam-5566	446	4	s.	s.	PROPN
ejpam-5566	446	5	gervacio	gervacio	PROPN
ejpam-5566	446	6	and	and	CCONJ
ejpam-5566	446	7	n.	n.	PROPN
ejpam-5566	446	8	mame	mame	PROPN
ejpam-5566	446	9	.	.	PUNCT
ejpam-5566	447	1	universal	universal	ADJ
ejpam-5566	447	2	and	and	CCONJ
ejpam-5566	447	3	primitive	primitive	ADJ
ejpam-5566	447	4	graphs	graph	NOUN
ejpam-5566	447	5	.	.	PUNCT
ejpam-5566	448	1	in	in	ADP
ejpam-5566	448	2	proc	proc	PROPN
ejpam-5566	448	3	.	.	PROPN
ejpam-5566	448	4	,	,	PUNCT
ejpam-5566	448	5	osaka	osaka	PROPN
ejpam-5566	448	6	university	university	PROPN
ejpam-5566	448	7	de	de	X
ejpam-5566	448	8	la	la	PROPN
ejpam-5566	448	9	salle	salle	PROPN
ejpam-5566	448	10	university	university	PROPN
ejpam-5566	448	11	academic	academic	ADJ
ejpam-5566	448	12	research	research	NOUN
ejpam-5566	448	13	workshop	workshop	NOUN
ejpam-5566	448	14	.	.	PUNCT
ejpam-5566	448	15	,	,	PUNCT
ejpam-5566	448	16	pages	page	NOUN
ejpam-5566	448	17	51–54	51–54	NUM
ejpam-5566	448	18	,	,	PUNCT
ejpam-5566	448	19	manila	manila	PROPN
ejpam-5566	448	20	,	,	PUNCT
ejpam-5566	448	21	philippines	philippine	NOUN
ejpam-5566	448	22	,	,	PUNCT
ejpam-5566	448	23	2007	2007	NUM
ejpam-5566	448	24	.	.	PUNCT
ejpam-5566	449	1	osaka	osaka	PROPN
ejpam-5566	449	2	universityde	universityde	PROPN
ejpam-5566	449	3	la	la	PROPN
ejpam-5566	449	4	salle	salle	PROPN
ejpam-5566	449	5	university	university	PROPN
ejpam-5566	449	6	-	-	PUNCT
ejpam-5566	449	7	manila	manila	PROPN
ejpam-5566	449	8	.	.	PUNCT
ejpam-5566	450	1	[	[	X
ejpam-5566	450	2	7	7	X
ejpam-5566	450	3	]	]	X
ejpam-5566	450	4	s.	s.	PROPN
ejpam-5566	450	5	gervacio	gervacio	PROPN
ejpam-5566	450	6	,	,	PUNCT
ejpam-5566	450	7	m.	m.	NOUN
ejpam-5566	450	8	valdez	valdez	PROPN
ejpam-5566	450	9	,	,	PUNCT
ejpam-5566	450	10	and	and	CCONJ
ejpam-5566	450	11	d.	d.	PROPN
ejpam-5566	450	12	bengo	bengo	PROPN
ejpam-5566	450	13	.	.	PUNCT
ejpam-5566	451	1	generator	generator	NOUN
ejpam-5566	451	2	subgraphs	subgraph	NOUN
ejpam-5566	451	3	of	of	ADP
ejpam-5566	451	4	fans	fan	NOUN
ejpam-5566	451	5	and	and	CCONJ
ejpam-5566	451	6	wheels	wheel	NOUN
ejpam-5566	451	7	.	.	PUNCT
ejpam-5566	452	1	in	in	ADP
ejpam-5566	452	2	proc	proc	PROPN
ejpam-5566	452	3	.	.	PROPN
ejpam-5566	452	4	,	,	PUNCT
ejpam-5566	452	5	osaka	osaka	PROPN
ejpam-5566	452	6	university	university	PROPN
ejpam-5566	452	7	de	de	X
ejpam-5566	452	8	la	la	PROPN
ejpam-5566	452	9	salle	salle	PROPN
ejpam-5566	452	10	university	university	PROPN
ejpam-5566	452	11	academic	academic	ADJ
ejpam-5566	452	12	research	research	NOUN
ejpam-5566	452	13	workshop	workshop	NOUN
ejpam-5566	452	14	.	.	PUNCT
ejpam-5566	452	15	,	,	PUNCT
ejpam-5566	452	16	manila	manila	PROPN
ejpam-5566	452	17	,	,	PUNCT
ejpam-5566	452	18	philippines	philippine	NOUN
ejpam-5566	452	19	,	,	PUNCT
ejpam-5566	452	20	2008	2008	NUM
ejpam-5566	452	21	.	.	PUNCT
ejpam-5566	453	1	osaka	osaka	PROPN
ejpam-5566	453	2	universityde	universityde	PROPN
ejpam-5566	453	3	la	la	PROPN
ejpam-5566	453	4	salle	salle	PROPN
ejpam-5566	453	5	university	university	PROPN
ejpam-5566	453	6	-	-	PUNCT
ejpam-5566	453	7	manila	manila	PROPN
ejpam-5566	453	8	.	.	PUNCT
ejpam-5566	454	1	[	[	X
ejpam-5566	454	2	8	8	NUM
ejpam-5566	454	3	]	]	X
ejpam-5566	454	4	n.	n.	NOUN
ejpam-5566	454	5	mame	mame	PROPN
ejpam-5566	454	6	and	and	CCONJ
ejpam-5566	454	7	s.	s.	PROPN
ejpam-5566	454	8	gervacio	gervacio	PROPN
ejpam-5566	454	9	.	.	PUNCT
ejpam-5566	455	1	a	a	DET
ejpam-5566	455	2	note	note	NOUN
ejpam-5566	455	3	on	on	ADP
ejpam-5566	455	4	generator	generator	NOUN
ejpam-5566	455	5	subgraph	subgraph	NOUN
ejpam-5566	455	6	of	of	ADP
ejpam-5566	455	7	a	a	DET
ejpam-5566	455	8	graph	graph	NOUN
ejpam-5566	455	9	.	.	PUNCT
ejpam-5566	456	1	electronic	electronic	ADJ
ejpam-5566	456	2	journal	journal	NOUN
ejpam-5566	456	3	of	of	ADP
ejpam-5566	456	4	graph	graph	NOUN
ejpam-5566	456	5	theory	theory	NOUN
ejpam-5566	456	6	and	and	CCONJ
ejpam-5566	456	7	applications	application	NOUN
ejpam-5566	456	8	,	,	PUNCT
ejpam-5566	456	9	8(3):17–27	8(3):17–27	NUM
ejpam-5566	456	10	,	,	PUNCT
ejpam-5566	456	11	2020	2020	NUM
ejpam-5566	456	12	.	.	PUNCT
ejpam-5566	457	1	[	[	X
ejpam-5566	457	2	9	9	NUM
ejpam-5566	457	3	]	]	X
ejpam-5566	457	4	e.	e.	PROPN
ejpam-5566	457	5	nering	nering	PROPN
ejpam-5566	457	6	.	.	PUNCT
ejpam-5566	458	1	linear	linear	PROPN
ejpam-5566	458	2	algebra	algebra	NOUN
ejpam-5566	458	3	and	and	CCONJ
ejpam-5566	458	4	matrix	matrix	NOUN
ejpam-5566	458	5	theory	theory	NOUN
ejpam-5566	458	6	.	.	PUNCT
ejpam-5566	459	1	john	john	PROPN
ejpam-5566	459	2	wiley	wiley	PROPN
ejpam-5566	459	3	and	and	CCONJ
ejpam-5566	459	4	sons	son	NOUN
ejpam-5566	459	5	,	,	PUNCT
ejpam-5566	459	6	usa	usa	PROPN
ejpam-5566	459	7	,	,	PUNCT
ejpam-5566	459	8	1970	1970	NUM
ejpam-5566	459	9	.	.	PUNCT
ejpam-5566	460	1	[	[	X
ejpam-5566	460	2	10	10	NUM
ejpam-5566	460	3	]	]	PUNCT
ejpam-5566	460	4	m.	m.	NOUN
ejpam-5566	460	5	pelagio	pelagio	PROPN
ejpam-5566	460	6	,	,	PUNCT
ejpam-5566	460	7	k.	k.	PROPN
ejpam-5566	460	8	mendoza	mendoza	PROPN
ejpam-5566	460	9	,	,	PUNCT
ejpam-5566	460	10	and	and	CCONJ
ejpam-5566	460	11	n.	n.	PROPN
ejpam-5566	460	12	mame	mame	PROPN
ejpam-5566	460	13	.	.	PUNCT
ejpam-5566	461	1	on	on	ADP
ejpam-5566	461	2	the	the	DET
ejpam-5566	461	3	k−	k−	PROPN
ejpam-5566	461	4	restricted	restrict	VERB
ejpam-5566	461	5	intersection	intersection	NOUN
ejpam-5566	461	6	graph	graph	NOUN
ejpam-5566	461	7	.	.	PUNCT
ejpam-5566	462	1	european	european	PROPN
ejpam-5566	462	2	journal	journal	PROPN
ejpam-5566	462	3	of	of	ADP
ejpam-5566	462	4	pure	pure	ADJ
ejpam-5566	462	5	and	and	CCONJ
ejpam-5566	462	6	applied	applied	ADJ
ejpam-5566	462	7	mathematics	mathematic	NOUN
ejpam-5566	462	8	,	,	PUNCT
ejpam-5566	462	9	17(3):1779–1803	17(3):1779–1803	NUM
ejpam-5566	462	10	,	,	PUNCT
ejpam-5566	462	11	2024	2024	NUM
ejpam-5566	462	12	.	.	PUNCT
ejpam-5566	463	1	[	[	X
ejpam-5566	463	2	11	11	NUM
ejpam-5566	463	3	]	]	X
ejpam-5566	463	4	l.	l.	PROPN
ejpam-5566	463	5	ruivivar	ruivivar	PROPN
ejpam-5566	463	6	.	.	PUNCT
ejpam-5566	464	1	some	some	DET
ejpam-5566	464	2	generator	generator	NOUN
ejpam-5566	464	3	subgraphs	subgraphs	NOUN
ejpam-5566	464	4	of	of	ADP
ejpam-5566	464	5	the	the	DET
ejpam-5566	464	6	complete	complete	ADJ
ejpam-5566	464	7	bipartite	bipartite	NOUN
ejpam-5566	464	8	graph	graph	NOUN
ejpam-5566	464	9	.	.	PUNCT
ejpam-5566	465	1	in	in	ADP
ejpam-5566	465	2	proc	proc	PROPN
ejpam-5566	465	3	.	.	PUNCT
ejpam-5566	465	4	,	,	PUNCT
ejpam-5566	465	5	5th	5th	ADJ
ejpam-5566	465	6	asian	asian	ADJ
ejpam-5566	465	7	mathematical	mathematical	ADJ
ejpam-5566	465	8	conference	conference	NOUN
ejpam-5566	465	9	.	.	PUNCT
ejpam-5566	465	10	,	,	PUNCT
ejpam-5566	465	11	pages	page	NOUN
ejpam-5566	465	12	516–520	516–520	NUM
ejpam-5566	465	13	,	,	PUNCT
ejpam-5566	465	14	malaysia	malaysia	PROPN
ejpam-5566	465	15	,	,	PUNCT
ejpam-5566	465	16	2009	2009	NUM
ejpam-5566	465	17	.	.	PUNCT
ejpam-5566	466	1	south	south	PROPN
ejpam-5566	466	2	east	east	PROPN
ejpam-5566	466	3	asian	asian	PROPN
ejpam-5566	466	4	mathematical	mathematical	ADJ
ejpam-5566	466	5	society	society	NOUN
ejpam-5566	466	6	.	.	PUNCT