id	sid	tid	token	lemma	pos
ejpam-5499	1	1	european	european	PROPN
ejpam-5499	1	2	journal	journal	PROPN
ejpam-5499	1	3	of	of	ADP
ejpam-5499	1	4	pure	pure	ADJ
ejpam-5499	1	5	and	and	CCONJ
ejpam-5499	1	6	applied	applied	ADJ
ejpam-5499	1	7	mathematics	mathematic	NOUN
ejpam-5499	1	8	2025	2025	NUM
ejpam-5499	1	9	,	,	PUNCT
ejpam-5499	1	10	vol	vol	NOUN
ejpam-5499	1	11	.	.	PROPN
ejpam-5499	1	12	18	18	NUM
ejpam-5499	1	13	,	,	PUNCT
ejpam-5499	1	14	issue	issue	NOUN
ejpam-5499	1	15	2	2	NUM
ejpam-5499	1	16	,	,	PUNCT
ejpam-5499	1	17	article	article	NOUN
ejpam-5499	1	18	number	number	NOUN
ejpam-5499	1	19	5499	5499	NUM
ejpam-5499	1	20	issn	issn	PROPN
ejpam-5499	1	21	1307	1307	NUM
ejpam-5499	1	22	-	-	SYM
ejpam-5499	1	23	5543	5543	NUM
ejpam-5499	1	24	–	–	PUNCT
ejpam-5499	1	25	ejpam.com	ejpam.com	X
ejpam-5499	1	26	published	publish	VERB
ejpam-5499	1	27	by	by	ADP
ejpam-5499	1	28	new	new	PROPN
ejpam-5499	1	29	york	york	PROPN
ejpam-5499	1	30	business	business	PROPN
ejpam-5499	1	31	global	global	ADJ
ejpam-5499	1	32	1	1	NUM
ejpam-5499	1	33	middle	middle	ADJ
ejpam-5499	1	34	graph	graph	NOUN
ejpam-5499	1	35	of	of	ADP
ejpam-5499	1	36	the	the	DET
ejpam-5499	1	37	identity	identity	NOUN
ejpam-5499	1	38	graph	graph	NOUN
ejpam-5499	1	39	of	of	ADP
ejpam-5499	1	40	finite	finite	PROPN
ejpam-5499	1	41	cyclic2	cyclic2	PROPN
ejpam-5499	1	42	and	and	CCONJ
ejpam-5499	1	43	dihedral	dihedral	ADJ
ejpam-5499	1	44	groups3	groups3	NOUN
ejpam-5499	1	45	jiel	jiel	PROPN
ejpam-5499	1	46	mark	mark	PROPN
ejpam-5499	1	47	d.	d.	PROPN
ejpam-5499	1	48	jagmis1	jagmis1	PROPN
ejpam-5499	1	49	,	,	PUNCT
ejpam-5499	1	50	daryl	daryl	PROPN
ejpam-5499	1	51	m.	m.	PROPN
ejpam-5499	1	52	magpantay2,∗4	magpantay2,∗4	VERB
ejpam-5499	1	53	1	1	NUM
ejpam-5499	1	54	college	college	NOUN
ejpam-5499	1	55	of	of	ADP
ejpam-5499	1	56	arts	art	NOUN
ejpam-5499	1	57	and	and	CCONJ
ejpam-5499	1	58	sciences	science	NOUN
ejpam-5499	1	59	,	,	PUNCT
ejpam-5499	1	60	camarines	camarines	PROPN
ejpam-5499	1	61	sur	sur	PROPN
ejpam-5499	1	62	polytechnic	polytechnic	ADJ
ejpam-5499	1	63	colleges	college	NOUN
ejpam-5499	1	64	,	,	PUNCT
ejpam-5499	1	65	nabua	nabua	PROPN
ejpam-5499	1	66	,	,	PUNCT
ejpam-5499	1	67	camarines5	camarines5	PROPN
ejpam-5499	1	68	sur	sur	PROPN
ejpam-5499	1	69	,	,	PUNCT
ejpam-5499	1	70	philippines6	philippines6	PROPN
ejpam-5499	1	71	2	2	NUM
ejpam-5499	1	72	batangas	batangas	PROPN
ejpam-5499	1	73	state	state	NOUN
ejpam-5499	1	74	university	university	PROPN
ejpam-5499	1	75	the	the	DET
ejpam-5499	1	76	national	national	PROPN
ejpam-5499	1	77	engineering	engineering	PROPN
ejpam-5499	1	78	university	university	PROPN
ejpam-5499	1	79	,	,	PUNCT
ejpam-5499	1	80	pablo	pablo	PROPN
ejpam-5499	1	81	borbon	borbon	NOUN
ejpam-5499	1	82	campus,7	campus,7	VERB
ejpam-5499	1	83	batangas	batangas	PROPN
ejpam-5499	1	84	city	city	PROPN
ejpam-5499	1	85	,	,	PUNCT
ejpam-5499	1	86	batangas	batangas	PROPN
ejpam-5499	1	87	,	,	PUNCT
ejpam-5499	1	88	philippines8	philippines8	NOUN
ejpam-5499	1	89	9	9	NUM
ejpam-5499	1	90	abstract	abstract	NOUN
ejpam-5499	1	91	.	.	PUNCT
ejpam-5499	2	1	given	give	VERB
ejpam-5499	2	2	a	a	DET
ejpam-5499	2	3	group	group	NOUN
ejpam-5499	2	4	g	g	NOUN
ejpam-5499	2	5	with	with	ADP
ejpam-5499	2	6	e	e	PROPN
ejpam-5499	2	7	as	as	ADP
ejpam-5499	2	8	the	the	DET
ejpam-5499	2	9	identity	identity	NOUN
ejpam-5499	2	10	element	element	NOUN
ejpam-5499	2	11	,	,	PUNCT
ejpam-5499	2	12	the	the	DET
ejpam-5499	2	13	identity	identity	NOUN
ejpam-5499	2	14	graph	graph	NOUN
ejpam-5499	2	15	γg	γg	ADV
ejpam-5499	2	16	having	have	VERB
ejpam-5499	2	17	the	the	DET
ejpam-5499	2	18	vertex	vertex	NOUN
ejpam-5499	2	19	-	-	PUNCT
ejpam-5499	2	20	set	set	VERB
ejpam-5499	2	21	g	g	NOUN
ejpam-5499	2	22	and	and	CCONJ
ejpam-5499	2	23	the	the	DET
ejpam-5499	2	24	edge	edge	NOUN
ejpam-5499	2	25	-	-	PUNCT
ejpam-5499	2	26	set	set	VERB
ejpam-5499	2	27	e	e	NOUN
ejpam-5499	2	28	satisfies	satisfy	VERB
ejpam-5499	2	29	two	two	NUM
ejpam-5499	2	30	conditions	condition	NOUN
ejpam-5499	2	31	:	:	PUNCT
ejpam-5499	2	32	(	(	PUNCT
ejpam-5499	2	33	i	i	NOUN
ejpam-5499	2	34	)	)	PUNCT
ejpam-5499	2	35	for	for	ADP
ejpam-5499	2	36	every	every	DET
ejpam-5499	2	37	x	x	PROPN
ejpam-5499	2	38	,	,	PUNCT
ejpam-5499	2	39	y	y	PROPN
ejpam-5499	2	40	∈	∈	PROPN
ejpam-5499	2	41	g	g	NOUN
ejpam-5499	2	42	where	where	SCONJ
ejpam-5499	2	43	x	x	ADJ
ejpam-5499	2	44	̸=	̸=	PROPN
ejpam-5499	2	45	y	y	PROPN
ejpam-5499	2	46	,	,	PUNCT
ejpam-5499	2	47	x	x	X
ejpam-5499	2	48	and	and	CCONJ
ejpam-5499	2	49	y	y	PROPN
ejpam-5499	2	50	are	be	AUX
ejpam-5499	2	51	adjacent	adjacent	ADJ
ejpam-5499	2	52	in	in	ADP
ejpam-5499	2	53	γg	γg	ADV
ejpam-5499	2	54	if	if	SCONJ
ejpam-5499	2	55	and	and	CCONJ
ejpam-5499	2	56	only	only	ADV
ejpam-5499	2	57	if	if	SCONJ
ejpam-5499	2	58	xy	xy	PROPN
ejpam-5499	2	59	=	=	SYM
ejpam-5499	2	60	e	e	NOUN
ejpam-5499	2	61	;	;	PUNCT
ejpam-5499	2	62	(	(	PUNCT
ejpam-5499	2	63	ii	ii	NOUN
ejpam-5499	2	64	)	)	PUNCT
ejpam-5499	2	65	for	for	ADP
ejpam-5499	2	66	each	each	DET
ejpam-5499	2	67	x	x	SYM
ejpam-5499	2	68	∈	∈	PROPN
ejpam-5499	2	69	g	g	PROPN
ejpam-5499	2	70	,	,	PUNCT
ejpam-5499	2	71	x	x	PUNCT
ejpam-5499	2	72	and	and	CCONJ
ejpam-5499	2	73	e	e	NOUN
ejpam-5499	2	74	are	be	AUX
ejpam-5499	2	75	adjacent	adjacent	ADJ
ejpam-5499	2	76	in	in	ADP
ejpam-5499	2	77	γg	γg	PROPN
ejpam-5499	2	78	.	.	PUNCT
ejpam-5499	3	1	the	the	DET
ejpam-5499	3	2	middle	middle	ADJ
ejpam-5499	3	3	graph	graph	NOUN
ejpam-5499	3	4	of	of	ADP
ejpam-5499	3	5	g	g	PROPN
ejpam-5499	3	6	denoted	denote	VERB
ejpam-5499	3	7	by	by	ADP
ejpam-5499	3	8	m(g	m(g	PROPN
ejpam-5499	3	9	)	)	PUNCT
ejpam-5499	3	10	is	be	AUX
ejpam-5499	3	11	the	the	DET
ejpam-5499	3	12	graph	graph	NOUN
ejpam-5499	3	13	with	with	ADP
ejpam-5499	3	14	vertex	vertex	NOUN
ejpam-5499	3	15	set	set	VERB
ejpam-5499	3	16	v	v	NOUN
ejpam-5499	3	17	(	(	PUNCT
ejpam-5499	3	18	g	g	NOUN
ejpam-5499	3	19	)	)	PUNCT
ejpam-5499	3	20	∪	∪	ADP
ejpam-5499	3	21	e(g	e(g	PROPN
ejpam-5499	3	22	)	)	PUNCT
ejpam-5499	3	23	where	where	SCONJ
ejpam-5499	3	24	two	two	NUM
ejpam-5499	3	25	vertices	vertex	NOUN
ejpam-5499	3	26	will	will	AUX
ejpam-5499	3	27	be	be	AUX
ejpam-5499	3	28	adjacent	adjacent	ADJ
ejpam-5499	3	29	if	if	SCONJ
ejpam-5499	3	30	and	and	CCONJ
ejpam-5499	3	31	only	only	ADV
ejpam-5499	3	32	if	if	SCONJ
ejpam-5499	3	33	they	they	PRON
ejpam-5499	3	34	are	be	AUX
ejpam-5499	3	35	either	either	CCONJ
ejpam-5499	3	36	adjacent	adjacent	ADJ
ejpam-5499	3	37	edges	edge	NOUN
ejpam-5499	3	38	of	of	ADP
ejpam-5499	3	39	g	g	NOUN
ejpam-5499	3	40	or	or	CCONJ
ejpam-5499	3	41	one	one	NUM
ejpam-5499	3	42	is	be	AUX
ejpam-5499	3	43	a	a	DET
ejpam-5499	3	44	vertex	vertex	NOUN
ejpam-5499	3	45	and	and	CCONJ
ejpam-5499	3	46	the	the	DET
ejpam-5499	3	47	other	other	ADJ
ejpam-5499	3	48	is	be	AUX
ejpam-5499	3	49	an	an	DET
ejpam-5499	3	50	edge	edge	NOUN
ejpam-5499	3	51	incident	incident	NOUN
ejpam-5499	3	52	to	to	ADP
ejpam-5499	3	53	it	it	PRON
ejpam-5499	3	54	.	.	PUNCT
ejpam-5499	4	1	it	it	PRON
ejpam-5499	4	2	can	can	AUX
ejpam-5499	4	3	be	be	AUX
ejpam-5499	4	4	obtained	obtain	VERB
ejpam-5499	4	5	by	by	ADP
ejpam-5499	4	6	inserting	insert	VERB
ejpam-5499	4	7	a	a	DET
ejpam-5499	4	8	new	new	ADJ
ejpam-5499	4	9	vertex	vertex	NOUN
ejpam-5499	4	10	into	into	ADP
ejpam-5499	4	11	every	every	DET
ejpam-5499	4	12	edge	edge	NOUN
ejpam-5499	4	13	of	of	ADP
ejpam-5499	4	14	g	g	NOUN
ejpam-5499	4	15	and	and	CCONJ
ejpam-5499	4	16	connecting	connect	VERB
ejpam-5499	4	17	the	the	DET
ejpam-5499	4	18	new	new	ADJ
ejpam-5499	4	19	obtained	obtain	VERB
ejpam-5499	4	20	vertices	vertex	NOUN
ejpam-5499	4	21	if	if	SCONJ
ejpam-5499	4	22	they	they	PRON
ejpam-5499	4	23	are	be	AUX
ejpam-5499	4	24	adjacent	adjacent	ADJ
ejpam-5499	4	25	edges	edge	NOUN
ejpam-5499	4	26	in	in	ADP
ejpam-5499	4	27	g.	g.	PROPN
ejpam-5499	4	28	in	in	ADP
ejpam-5499	4	29	this	this	DET
ejpam-5499	4	30	paper	paper	NOUN
ejpam-5499	4	31	,	,	PUNCT
ejpam-5499	4	32	we	we	PRON
ejpam-5499	4	33	constructed	construct	VERB
ejpam-5499	4	34	the	the	DET
ejpam-5499	4	35	middle	middle	ADJ
ejpam-5499	4	36	graph	graph	NOUN
ejpam-5499	4	37	of	of	ADP
ejpam-5499	4	38	the	the	DET
ejpam-5499	4	39	identity	identity	NOUN
ejpam-5499	4	40	graph	graph	NOUN
ejpam-5499	4	41	particular	particular	ADJ
ejpam-5499	4	42	for	for	ADP
ejpam-5499	4	43	finite	finite	ADJ
ejpam-5499	4	44	cyclic	cyclic	ADJ
ejpam-5499	4	45	and	and	CCONJ
ejpam-5499	4	46	dihedral	dihedral	ADJ
ejpam-5499	4	47	groups	group	NOUN
ejpam-5499	4	48	.	.	PUNCT
ejpam-5499	5	1	some	some	DET
ejpam-5499	5	2	parameters	parameter	NOUN
ejpam-5499	5	3	of	of	ADP
ejpam-5499	5	4	a	a	DET
ejpam-5499	5	5	graph	graph	NOUN
ejpam-5499	5	6	such	such	ADJ
ejpam-5499	5	7	as	as	ADP
ejpam-5499	5	8	the	the	DET
ejpam-5499	5	9	size	size	NOUN
ejpam-5499	5	10	,	,	PUNCT
ejpam-5499	5	11	order	order	NOUN
ejpam-5499	5	12	,	,	PUNCT
ejpam-5499	5	13	graph	graph	NOUN
ejpam-5499	5	14	measurements	measurement	NOUN
ejpam-5499	5	15	,	,	PUNCT
ejpam-5499	5	16	independence	independence	NOUN
ejpam-5499	5	17	number	number	NOUN
ejpam-5499	5	18	,	,	PUNCT
ejpam-5499	5	19	domination	domination	NOUN
ejpam-5499	5	20	number	number	NOUN
ejpam-5499	5	21	,	,	PUNCT
ejpam-5499	5	22	vertex	vertex	NOUN
ejpam-5499	5	23	chromatic	chromatic	ADJ
ejpam-5499	5	24	number	number	NOUN
ejpam-5499	5	25	and	and	CCONJ
ejpam-5499	5	26	edge	edge	NOUN
ejpam-5499	5	27	chromatic	chromatic	ADJ
ejpam-5499	5	28	number	number	NOUN
ejpam-5499	5	29	were	be	AUX
ejpam-5499	5	30	also	also	ADV
ejpam-5499	5	31	investigated	investigate	VERB
ejpam-5499	5	32	.	.	PUNCT
ejpam-5499	6	1	2020	2020	NUM
ejpam-5499	6	2	mathematics	mathematic	NOUN
ejpam-5499	6	3	subject	subject	NOUN
ejpam-5499	6	4	classifications	classification	NOUN
ejpam-5499	6	5	:	:	PUNCT
ejpam-5499	6	6	0510	0510	NUM
ejpam-5499	6	7	key	key	ADJ
ejpam-5499	6	8	words	word	NOUN
ejpam-5499	6	9	and	and	CCONJ
ejpam-5499	6	10	phrases	phrase	NOUN
ejpam-5499	6	11	:	:	PUNCT
ejpam-5499	6	12	middle	middle	ADJ
ejpam-5499	6	13	graph	graph	NOUN
ejpam-5499	6	14	,	,	PUNCT
ejpam-5499	6	15	identity	identity	NOUN
ejpam-5499	6	16	graph	graph	NOUN
ejpam-5499	6	17	,	,	PUNCT
ejpam-5499	6	18	cyclic	cyclic	ADJ
ejpam-5499	6	19	groups	group	NOUN
ejpam-5499	6	20	,	,	PUNCT
ejpam-5499	6	21	dihedral	dihedral	ADJ
ejpam-5499	6	22	groups,11	groups,11	NOUN
ejpam-5499	6	23	graph	graph	VERB
ejpam-5499	6	24	properties12	properties12	NOUN
ejpam-5499	6	25	13	13	NUM
ejpam-5499	6	26	1	1	NUM
ejpam-5499	6	27	.	.	PUNCT
ejpam-5499	6	28	introduction14	introduction14	VERB
ejpam-5499	6	29	the	the	DET
ejpam-5499	6	30	linking	linking	NOUN
ejpam-5499	6	31	of	of	ADP
ejpam-5499	6	32	group	group	NOUN
ejpam-5499	6	33	theory	theory	NOUN
ejpam-5499	6	34	to	to	ADP
ejpam-5499	6	35	graph	graph	NOUN
ejpam-5499	6	36	theory	theory	NOUN
ejpam-5499	6	37	was	be	AUX
ejpam-5499	6	38	started	start	VERB
ejpam-5499	6	39	in	in	ADP
ejpam-5499	6	40	2009	2009	NUM
ejpam-5499	6	41	in	in	ADP
ejpam-5499	6	42	the	the	DET
ejpam-5499	6	43	book	book	NOUN
ejpam-5499	6	44	of	of	ADP
ejpam-5499	6	45	[	[	X
ejpam-5499	6	46	1	1	NUM
ejpam-5499	6	47	]	]	PUNCT
ejpam-5499	6	48	by15	by15	PROPN
ejpam-5499	6	49	creating	create	VERB
ejpam-5499	6	50	a	a	DET
ejpam-5499	6	51	new	new	ADJ
ejpam-5499	6	52	kind	kind	NOUN
ejpam-5499	6	53	of	of	ADP
ejpam-5499	6	54	graph	graph	NOUN
ejpam-5499	6	55	from	from	ADP
ejpam-5499	6	56	the	the	DET
ejpam-5499	6	57	concepts	concept	NOUN
ejpam-5499	6	58	of	of	ADP
ejpam-5499	6	59	group	group	NOUN
ejpam-5499	6	60	theory	theory	NOUN
ejpam-5499	6	61	.	.	PUNCT
ejpam-5499	7	1	they	they	PRON
ejpam-5499	7	2	represented	represent	VERB
ejpam-5499	7	3	the16	the16	PROPN
ejpam-5499	7	4	finite	finite	ADJ
ejpam-5499	7	5	groups	group	NOUN
ejpam-5499	7	6	in	in	ADP
ejpam-5499	7	7	terms	term	NOUN
ejpam-5499	7	8	of	of	ADP
ejpam-5499	7	9	graphs	graph	NOUN
ejpam-5499	7	10	which	which	PRON
ejpam-5499	7	11	they	they	PRON
ejpam-5499	7	12	called	call	VERB
ejpam-5499	7	13	identity	identity	NOUN
ejpam-5499	7	14	graphs	graph	NOUN
ejpam-5499	7	15	or	or	CCONJ
ejpam-5499	7	16	identity	identity	NOUN
ejpam-5499	7	17	graphs	graph	NOUN
ejpam-5499	7	18	since17	since17	VERB
ejpam-5499	7	19	the	the	DET
ejpam-5499	7	20	identity	identity	NOUN
ejpam-5499	7	21	element	element	NOUN
ejpam-5499	7	22	of	of	ADP
ejpam-5499	7	23	the	the	DET
ejpam-5499	7	24	group	group	NOUN
ejpam-5499	7	25	is	be	AUX
ejpam-5499	7	26	the	the	DET
ejpam-5499	7	27	main	main	ADJ
ejpam-5499	7	28	role	role	NOUN
ejpam-5499	7	29	in	in	ADP
ejpam-5499	7	30	order	order	NOUN
ejpam-5499	7	31	to	to	PART
ejpam-5499	7	32	create	create	VERB
ejpam-5499	7	33	a	a	DET
ejpam-5499	7	34	graph.18	graph.18	PROPN
ejpam-5499	7	35	19	19	NUM
ejpam-5499	7	36	a.d	a.d	PROPN
ejpam-5499	7	37	.	.	PROPN
ejpam-5499	7	38	godase	godase	PROPN
ejpam-5499	8	1	[	[	X
ejpam-5499	8	2	2	2	NUM
ejpam-5499	8	3	]	]	PUNCT
ejpam-5499	8	4	in	in	ADP
ejpam-5499	8	5	2015	2015	NUM
ejpam-5499	8	6	gave	give	VERB
ejpam-5499	8	7	some	some	DET
ejpam-5499	8	8	examples	example	NOUN
ejpam-5499	8	9	of	of	ADP
ejpam-5499	8	10	the	the	DET
ejpam-5499	8	11	identity	identity	NOUN
ejpam-5499	8	12	graphs	graph	NOUN
ejpam-5499	8	13	of	of	ADP
ejpam-5499	8	14	some	some	DET
ejpam-5499	8	15	finite20	finite20	NOUN
ejpam-5499	8	16	groups	group	NOUN
ejpam-5499	8	17	particular	particular	ADJ
ejpam-5499	8	18	in	in	ADP
ejpam-5499	8	19	finite	finite	ADJ
ejpam-5499	8	20	cyclic	cyclic	ADJ
ejpam-5499	8	21	and	and	CCONJ
ejpam-5499	8	22	dihedral	dihedral	ADJ
ejpam-5499	8	23	groups	group	NOUN
ejpam-5499	8	24	which	which	PRON
ejpam-5499	8	25	he	he	PRON
ejpam-5499	8	26	discovered	discover	VERB
ejpam-5499	8	27	that	that	SCONJ
ejpam-5499	8	28	the	the	DET
ejpam-5499	8	29	graph21	graph21	NOUN
ejpam-5499	8	30	formed	form	VERB
ejpam-5499	8	31	were	be	AUX
ejpam-5499	8	32	consists	consist	NOUN
ejpam-5499	8	33	of	of	ADP
ejpam-5499	8	34	lines	line	NOUN
ejpam-5499	8	35	and	and	CCONJ
ejpam-5499	8	36	triangles	triangle	NOUN
ejpam-5499	8	37	.	.	PUNCT
ejpam-5499	9	1	in	in	ADP
ejpam-5499	9	2	the	the	DET
ejpam-5499	9	3	papers	paper	NOUN
ejpam-5499	9	4	[	[	X
ejpam-5499	9	5	3	3	X
ejpam-5499	9	6	]	]	PUNCT
ejpam-5499	9	7	and	and	CCONJ
ejpam-5499	9	8	[	[	X
ejpam-5499	9	9	4	4	NUM
ejpam-5499	9	10	]	]	PUNCT
ejpam-5499	9	11	,	,	PUNCT
ejpam-5499	9	12	the	the	DET
ejpam-5499	9	13	authors	author	NOUN
ejpam-5499	9	14	further22	further22	ADV
ejpam-5499	9	15	∗corresponding	∗corresponde	VERB
ejpam-5499	9	16	author	author	NOUN
ejpam-5499	9	17	.	.	PUNCT
ejpam-5499	10	1	doi	doi	NOUN
ejpam-5499	10	2	:	:	PUNCT
ejpam-5499	10	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5499	https://doi.org/10.29020/nybg.ejpam.v18i2.5499	NUM
ejpam-5499	10	4	email	email	NOUN
ejpam-5499	10	5	addresses	address	NOUN
ejpam-5499	10	6	:	:	PUNCT
ejpam-5499	10	7	jagmis656@gmail.com	jagmis656@gmail.com	PROPN
ejpam-5499	10	8	(	(	PUNCT
ejpam-5499	10	9	j.	j.	PROPN
ejpam-5499	10	10	m.	m.	PROPN
ejpam-5499	10	11	jamis	jamis	PROPN
ejpam-5499	10	12	)	)	PUNCT
ejpam-5499	10	13	,	,	PUNCT
ejpam-5499	10	14	daryl.magpantay@g.batstate-u.edu.ph	daryl.magpantay@g.batstate-u.edu.ph	PROPN
ejpam-5499	10	15	(	(	PUNCT
ejpam-5499	10	16	d.	d.	PROPN
ejpam-5499	10	17	m.	m.	PROPN
ejpam-5499	10	18	magpantay	magpantay	PROPN
ejpam-5499	10	19	https://www.ejpam.com	https://www.ejpam.com	PROPN
ejpam-5499	10	20	1	1	NUM
ejpam-5499	11	1	copyright	copyright	NOUN
ejpam-5499	11	2	:	:	PUNCT
ejpam-5499	11	3	©	©	PROPN
ejpam-5499	11	4	2025	2025	NUM
ejpam-5499	11	5	the	the	DET
ejpam-5499	11	6	author(s	author(s	NOUN
ejpam-5499	11	7	)	)	PUNCT
ejpam-5499	11	8	.	.	PUNCT
ejpam-5499	12	1	(	(	PUNCT
ejpam-5499	12	2	cc	cc	NOUN
ejpam-5499	12	3	by	by	ADP
ejpam-5499	12	4	-	-	PUNCT
ejpam-5499	12	5	nc	nc	PROPN
ejpam-5499	12	6	4.0	4.0	NUM
ejpam-5499	12	7	)	)	PUNCT
ejpam-5499	12	8	j.	j.	PROPN
ejpam-5499	12	9	m.	m.	PROPN
ejpam-5499	12	10	jamis	jamis	PROPN
ejpam-5499	12	11	,	,	PUNCT
ejpam-5499	12	12	d.	d.	PROPN
ejpam-5499	12	13	m.	m.	PROPN
ejpam-5499	12	14	magpantay	magpantay	PROPN
ejpam-5499	12	15	/	/	SYM
ejpam-5499	12	16	eur	eur	PROPN
ejpam-5499	12	17	.	.	PUNCT
ejpam-5499	13	1	j.	j.	PROPN
ejpam-5499	13	2	pure	pure	PROPN
ejpam-5499	13	3	appl	appl	PROPN
ejpam-5499	13	4	.	.	PROPN
ejpam-5499	13	5	math	math	PROPN
ejpam-5499	13	6	,	,	PUNCT
ejpam-5499	13	7	18	18	NUM
ejpam-5499	13	8	(	(	PUNCT
ejpam-5499	13	9	2	2	NUM
ejpam-5499	13	10	)	)	PUNCT
ejpam-5499	13	11	(	(	PUNCT
ejpam-5499	13	12	2025	2025	NUM
ejpam-5499	13	13	)	)	PUNCT
ejpam-5499	13	14	,	,	PUNCT
ejpam-5499	13	15	5499	5499	NUM
ejpam-5499	13	16	2	2	NUM
ejpam-5499	13	17	of	of	ADP
ejpam-5499	13	18	18	18	NUM
ejpam-5499	13	19	studied	study	VERB
ejpam-5499	13	20	the	the	DET
ejpam-5499	13	21	said	say	VERB
ejpam-5499	13	22	graph	graph	NOUN
ejpam-5499	13	23	by	by	ADP
ejpam-5499	13	24	investigating	investigate	VERB
ejpam-5499	13	25	its	its	PRON
ejpam-5499	13	26	properties	property	NOUN
ejpam-5499	13	27	and	and	CCONJ
ejpam-5499	13	28	characteristics.23	characteristics.23	PROPN
ejpam-5499	13	29	24	24	NUM
ejpam-5499	13	30	another	another	DET
ejpam-5499	13	31	focus	focus	NOUN
ejpam-5499	13	32	of	of	ADP
ejpam-5499	13	33	studies	study	NOUN
ejpam-5499	13	34	in	in	ADP
ejpam-5499	13	35	graph	graph	NOUN
ejpam-5499	13	36	theory	theory	NOUN
ejpam-5499	13	37	is	be	AUX
ejpam-5499	13	38	the	the	DET
ejpam-5499	13	39	construction	construction	NOUN
ejpam-5499	13	40	new	new	ADJ
ejpam-5499	13	41	graphs	graph	NOUN
ejpam-5499	13	42	from	from	ADP
ejpam-5499	13	43	another25	another25	NOUN
ejpam-5499	13	44	graph	graph	NOUN
ejpam-5499	13	45	.	.	PUNCT
ejpam-5499	14	1	given	give	VERB
ejpam-5499	14	2	a	a	DET
ejpam-5499	14	3	graph	graph	NOUN
ejpam-5499	14	4	,	,	PUNCT
ejpam-5499	14	5	operation	operation	NOUN
ejpam-5499	14	6	will	will	AUX
ejpam-5499	14	7	be	be	AUX
ejpam-5499	14	8	defined	define	VERB
ejpam-5499	14	9	that	that	SCONJ
ejpam-5499	14	10	yields	yield	NOUN
ejpam-5499	14	11	to	to	ADP
ejpam-5499	14	12	another	another	DET
ejpam-5499	14	13	form	form	NOUN
ejpam-5499	14	14	of	of	ADP
ejpam-5499	14	15	graph	graph	NOUN
ejpam-5499	14	16	.	.	PUNCT
ejpam-5499	15	1	one26	one26	PROPN
ejpam-5499	15	2	particular	particular	ADJ
ejpam-5499	15	3	example	example	NOUN
ejpam-5499	15	4	is	be	AUX
ejpam-5499	15	5	the	the	DET
ejpam-5499	15	6	paper	paper	NOUN
ejpam-5499	15	7	of	of	ADP
ejpam-5499	15	8	akiyama	akiyama	PROPN
ejpam-5499	15	9	,	,	PUNCT
ejpam-5499	15	10	hamada	hamada	PROPN
ejpam-5499	15	11	,	,	PUNCT
ejpam-5499	15	12	and	and	CCONJ
ejpam-5499	15	13	yoshimura	yoshimura	VERB
ejpam-5499	16	1	[	[	X
ejpam-5499	16	2	5	5	NUM
ejpam-5499	16	3	]	]	PUNCT
ejpam-5499	16	4	which	which	PRON
ejpam-5499	16	5	introduced27	introduced27	VERB
ejpam-5499	16	6	the	the	DET
ejpam-5499	16	7	concept	concept	NOUN
ejpam-5499	16	8	of	of	ADP
ejpam-5499	16	9	the	the	DET
ejpam-5499	16	10	middle	middle	ADJ
ejpam-5499	16	11	graph	graph	NOUN
ejpam-5499	16	12	and	and	CCONJ
ejpam-5499	16	13	established	establish	VERB
ejpam-5499	16	14	some	some	DET
ejpam-5499	16	15	characterizations	characterization	NOUN
ejpam-5499	16	16	in	in	ADP
ejpam-5499	16	17	particular	particular	ADJ
ejpam-5499	16	18	for28	for28	NOUN
ejpam-5499	16	19	some	some	DET
ejpam-5499	16	20	common	common	ADJ
ejpam-5499	16	21	classes	class	NOUN
ejpam-5499	16	22	of	of	ADP
ejpam-5499	16	23	graphs	graph	NOUN
ejpam-5499	16	24	.	.	PUNCT
ejpam-5499	17	1	the	the	DET
ejpam-5499	17	2	middle	middle	ADJ
ejpam-5499	17	3	graph	graph	NOUN
ejpam-5499	17	4	of	of	ADP
ejpam-5499	17	5	a	a	DET
ejpam-5499	17	6	graph	graph	NOUN
ejpam-5499	17	7	is	be	AUX
ejpam-5499	17	8	obtained	obtain	VERB
ejpam-5499	17	9	by	by	ADP
ejpam-5499	17	10	inserting	insert	VERB
ejpam-5499	17	11	a29	a29	PROPN
ejpam-5499	17	12	new	new	ADJ
ejpam-5499	17	13	vertex	vertex	NOUN
ejpam-5499	17	14	into	into	ADP
ejpam-5499	17	15	every	every	DET
ejpam-5499	17	16	edge	edge	NOUN
ejpam-5499	17	17	of	of	ADP
ejpam-5499	17	18	the	the	DET
ejpam-5499	17	19	original	original	ADJ
ejpam-5499	17	20	graph	graph	NOUN
ejpam-5499	17	21	and	and	CCONJ
ejpam-5499	17	22	connecting	connect	VERB
ejpam-5499	17	23	the	the	DET
ejpam-5499	17	24	new	new	ADJ
ejpam-5499	17	25	obtained	obtain	VERB
ejpam-5499	17	26	vertices30	vertices30	NOUN
ejpam-5499	17	27	if	if	SCONJ
ejpam-5499	17	28	they	they	PRON
ejpam-5499	17	29	are	be	AUX
ejpam-5499	17	30	adjacent	adjacent	ADJ
ejpam-5499	17	31	edges	edge	NOUN
ejpam-5499	17	32	in	in	ADP
ejpam-5499	17	33	the	the	DET
ejpam-5499	17	34	original	original	ADJ
ejpam-5499	17	35	graph.31	graph.31	NOUN
ejpam-5499	17	36	32	32	NUM
ejpam-5499	17	37	the	the	DET
ejpam-5499	17	38	combination	combination	NOUN
ejpam-5499	17	39	of	of	ADP
ejpam-5499	17	40	two	two	NUM
ejpam-5499	17	41	concepts	concept	NOUN
ejpam-5499	17	42	motivates	motivate	VERB
ejpam-5499	17	43	the	the	DET
ejpam-5499	17	44	authors	author	NOUN
ejpam-5499	17	45	to	to	PART
ejpam-5499	17	46	explore	explore	VERB
ejpam-5499	17	47	the	the	DET
ejpam-5499	17	48	middle	middle	ADJ
ejpam-5499	17	49	graphs33	graphs33	NOUN
ejpam-5499	17	50	of	of	ADP
ejpam-5499	17	51	the	the	DET
ejpam-5499	17	52	identity	identity	NOUN
ejpam-5499	17	53	graph	graph	NOUN
ejpam-5499	17	54	of	of	ADP
ejpam-5499	17	55	the	the	DET
ejpam-5499	17	56	finite	finite	ADJ
ejpam-5499	17	57	cyclic	cyclic	ADJ
ejpam-5499	17	58	and	and	CCONJ
ejpam-5499	17	59	dihedral	dihedral	ADJ
ejpam-5499	17	60	groups	group	NOUN
ejpam-5499	17	61	.	.	PUNCT
ejpam-5499	18	1	this	this	PRON
ejpam-5499	18	2	is	be	AUX
ejpam-5499	18	3	in	in	ADP
ejpam-5499	18	4	parallel	parallel	ADJ
ejpam-5499	18	5	with34	with34	NOUN
ejpam-5499	18	6	the	the	DET
ejpam-5499	18	7	study	study	NOUN
ejpam-5499	18	8	of	of	ADP
ejpam-5499	18	9	murusegan	murusegan	PROPN
ejpam-5499	18	10	and	and	CCONJ
ejpam-5499	18	11	nair	nair	NOUN
ejpam-5499	19	1	[	[	X
ejpam-5499	19	2	6	6	NUM
ejpam-5499	19	3	]	]	PUNCT
ejpam-5499	19	4	which	which	PRON
ejpam-5499	19	5	discussed	discuss	VERB
ejpam-5499	19	6	the	the	DET
ejpam-5499	19	7	(	(	PUNCT
ejpam-5499	19	8	1	1	NUM
ejpam-5499	19	9	,	,	PUNCT
ejpam-5499	19	10	2)-domination	2)-domination	NUM
ejpam-5499	19	11	in	in	ADP
ejpam-5499	19	12	middle	middle	PROPN
ejpam-5499	19	13	and35	and35	PROPN
ejpam-5499	19	14	central	central	ADJ
ejpam-5499	19	15	graph	graph	NOUN
ejpam-5499	19	16	of	of	ADP
ejpam-5499	19	17	a	a	DET
ejpam-5499	19	18	star	star	NOUN
ejpam-5499	19	19	,	,	PUNCT
ejpam-5499	19	20	cycle	cycle	NOUN
ejpam-5499	19	21	and	and	CCONJ
ejpam-5499	19	22	path	path	NOUN
ejpam-5499	19	23	.	.	PUNCT
ejpam-5499	20	1	they	they	PRON
ejpam-5499	20	2	also	also	ADV
ejpam-5499	20	3	established	establish	VERB
ejpam-5499	20	4	some	some	DET
ejpam-5499	20	5	upperbounds	upperbound	NOUN
ejpam-5499	20	6	and36	and36	ADJ
ejpam-5499	20	7	lower	low	ADJ
ejpam-5499	20	8	bounds	bound	NOUN
ejpam-5499	20	9	.	.	PUNCT
ejpam-5499	21	1	later	later	ADV
ejpam-5499	21	2	on	on	ADP
ejpam-5499	21	3	2017	2017	NUM
ejpam-5499	21	4	,	,	PUNCT
ejpam-5499	21	5	they	they	PRON
ejpam-5499	21	6	investigated	investigate	VERB
ejpam-5499	21	7	the	the	DET
ejpam-5499	21	8	power	power	NOUN
ejpam-5499	21	9	domination	domination	NOUN
ejpam-5499	21	10	of	of	ADP
ejpam-5499	21	11	middle	middle	ADJ
ejpam-5499	21	12	graph	graph	NOUN
ejpam-5499	21	13	of37	of37	PROPN
ejpam-5499	21	14	path	path	NOUN
ejpam-5499	21	15	,	,	PUNCT
ejpam-5499	21	16	cycle	cycle	NOUN
ejpam-5499	21	17	and	and	CCONJ
ejpam-5499	21	18	star	star	NOUN
ejpam-5499	21	19	.	.	PUNCT
ejpam-5499	22	1	alib	alib	PROPN
ejpam-5499	22	2	et	et	PROPN
ejpam-5499	22	3	al	al	PROPN
ejpam-5499	22	4	.	.	PUNCT
ejpam-5499	23	1	[	[	X
ejpam-5499	23	2	7	7	X
ejpam-5499	23	3	]	]	PUNCT
ejpam-5499	23	4	presented	present	VERB
ejpam-5499	23	5	the	the	DET
ejpam-5499	23	6	construction	construction	NOUN
ejpam-5499	23	7	of	of	ADP
ejpam-5499	23	8	the	the	DET
ejpam-5499	23	9	central	central	ADJ
ejpam-5499	23	10	graph	graph	NOUN
ejpam-5499	23	11	of	of	ADP
ejpam-5499	23	12	the38	the38	ADJ
ejpam-5499	23	13	identity	identity	NOUN
ejpam-5499	23	14	graph	graph	NOUN
ejpam-5499	23	15	of	of	ADP
ejpam-5499	23	16	finite	finite	ADJ
ejpam-5499	23	17	cyclic	cyclic	PROPN
ejpam-5499	23	18	group	group	NOUN
ejpam-5499	23	19	and	and	CCONJ
ejpam-5499	23	20	investigated	investigate	VERB
ejpam-5499	23	21	some	some	PRON
ejpam-5499	23	22	of	of	ADP
ejpam-5499	23	23	its	its	PRON
ejpam-5499	23	24	graph	graph	NOUN
ejpam-5499	23	25	properties.39	properties.39	NOUN
ejpam-5499	23	26	40	40	NUM
ejpam-5499	23	27	this	this	DET
ejpam-5499	23	28	paper	paper	NOUN
ejpam-5499	23	29	presents	present	VERB
ejpam-5499	23	30	the	the	DET
ejpam-5499	23	31	construction	construction	NOUN
ejpam-5499	23	32	of	of	ADP
ejpam-5499	23	33	the	the	DET
ejpam-5499	23	34	middle	middle	ADJ
ejpam-5499	23	35	graphs	graph	NOUN
ejpam-5499	23	36	of	of	ADP
ejpam-5499	23	37	the	the	DET
ejpam-5499	23	38	identity	identity	NOUN
ejpam-5499	23	39	graph	graph	NOUN
ejpam-5499	23	40	of41	of41	PROPN
ejpam-5499	23	41	the	the	DET
ejpam-5499	23	42	finite	finite	ADJ
ejpam-5499	23	43	cyclic	cyclic	ADJ
ejpam-5499	23	44	and	and	CCONJ
ejpam-5499	23	45	dihedral	dihedral	ADJ
ejpam-5499	23	46	groups	group	NOUN
ejpam-5499	23	47	.	.	PUNCT
ejpam-5499	24	1	the	the	DET
ejpam-5499	24	2	properties	property	NOUN
ejpam-5499	24	3	particular	particular	ADJ
ejpam-5499	24	4	in	in	ADP
ejpam-5499	24	5	graph	graph	NOUN
ejpam-5499	24	6	measurements,42	measurements,42	ADJ
ejpam-5499	24	7	independence	independence	NOUN
ejpam-5499	24	8	number	number	NOUN
ejpam-5499	24	9	,	,	PUNCT
ejpam-5499	24	10	domination	domination	NOUN
ejpam-5499	24	11	number	number	NOUN
ejpam-5499	24	12	and	and	CCONJ
ejpam-5499	24	13	graph	graph	NOUN
ejpam-5499	24	14	coloring	coloring	NOUN
ejpam-5499	24	15	were	be	AUX
ejpam-5499	24	16	also	also	ADV
ejpam-5499	24	17	investigated.43	investigated.43	ADJ
ejpam-5499	24	18	2	2	NUM
ejpam-5499	24	19	.	.	PUNCT
ejpam-5499	24	20	preliminaries44	preliminaries44	NOUN
ejpam-5499	24	21	for	for	ADP
ejpam-5499	24	22	the	the	DET
ejpam-5499	24	23	purpose	purpose	NOUN
ejpam-5499	24	24	of	of	ADP
ejpam-5499	24	25	further	further	ADJ
ejpam-5499	24	26	understanding	understanding	NOUN
ejpam-5499	24	27	concepts	concept	NOUN
ejpam-5499	24	28	,	,	PUNCT
ejpam-5499	24	29	examples	example	NOUN
ejpam-5499	24	30	,	,	PUNCT
ejpam-5499	24	31	and	and	CCONJ
ejpam-5499	24	32	illustrations	illustration	NOUN
ejpam-5499	25	1	are45	are45	INTJ
ejpam-5499	25	2	given.46	given.46	PROPN
ejpam-5499	25	3	2.1	2.1	NUM
ejpam-5499	25	4	.	.	PUNCT
ejpam-5499	26	1	group	group	NOUN
ejpam-5499	26	2	theory47	theory47	NOUN
ejpam-5499	26	3	this	this	DET
ejpam-5499	26	4	section	section	NOUN
ejpam-5499	26	5	contains	contain	VERB
ejpam-5499	26	6	some	some	DET
ejpam-5499	26	7	basic	basic	ADJ
ejpam-5499	26	8	concepts	concept	NOUN
ejpam-5499	26	9	in	in	ADP
ejpam-5499	26	10	group	group	NOUN
ejpam-5499	26	11	theory	theory	NOUN
ejpam-5499	26	12	and	and	CCONJ
ejpam-5499	26	13	its	its	PRON
ejpam-5499	26	14	examples	example	NOUN
ejpam-5499	26	15	that	that	DET
ejpam-5499	26	16	will48	will48	NOUN
ejpam-5499	26	17	be	be	AUX
ejpam-5499	26	18	needed	need	VERB
ejpam-5499	26	19	in	in	ADP
ejpam-5499	26	20	the	the	DET
ejpam-5499	26	21	discussion	discussion	NOUN
ejpam-5499	26	22	of	of	ADP
ejpam-5499	26	23	the	the	DET
ejpam-5499	26	24	following	follow	VERB
ejpam-5499	26	25	chapters	chapter	NOUN
ejpam-5499	26	26	.	.	PUNCT
ejpam-5499	27	1	groups	group	NOUN
ejpam-5499	27	2	can	can	AUX
ejpam-5499	27	3	be	be	AUX
ejpam-5499	27	4	finite	finite	ADJ
ejpam-5499	27	5	or	or	CCONJ
ejpam-5499	27	6	infinite	infinite	VERB
ejpam-5499	27	7	.	.	PUNCT
ejpam-5499	28	1	in49	in49	ADJ
ejpam-5499	28	2	general	general	ADJ
ejpam-5499	28	3	,	,	PUNCT
ejpam-5499	28	4	groups	group	NOUN
ejpam-5499	28	5	can	can	AUX
ejpam-5499	28	6	be	be	AUX
ejpam-5499	28	7	classified	classify	VERB
ejpam-5499	28	8	into	into	ADP
ejpam-5499	28	9	two	two	NUM
ejpam-5499	28	10	categories	category	NOUN
ejpam-5499	28	11	,	,	PUNCT
ejpam-5499	28	12	these	these	PRON
ejpam-5499	28	13	are	be	AUX
ejpam-5499	28	14	the	the	DET
ejpam-5499	28	15	cyclic	cyclic	ADJ
ejpam-5499	28	16	and	and	CCONJ
ejpam-5499	28	17	noncyclic50	noncyclic50	NOUN
ejpam-5499	28	18	groups	group	NOUN
ejpam-5499	28	19	.	.	PUNCT
ejpam-5499	29	1	in	in	ADP
ejpam-5499	29	2	this	this	DET
ejpam-5499	29	3	paper	paper	NOUN
ejpam-5499	29	4	,	,	PUNCT
ejpam-5499	29	5	we	we	PRON
ejpam-5499	29	6	focus	focus	VERB
ejpam-5499	29	7	in	in	ADP
ejpam-5499	29	8	finite	finite	ADJ
ejpam-5499	29	9	cyclic	cyclic	ADJ
ejpam-5499	29	10	groups	group	NOUN
ejpam-5499	29	11	and	and	CCONJ
ejpam-5499	29	12	the	the	DET
ejpam-5499	29	13	dihedral	dihedral	ADJ
ejpam-5499	29	14	groups	group	NOUN
ejpam-5499	29	15	.	.	PUNCT
ejpam-5499	30	1	let	let	VERB
ejpam-5499	30	2	us51	us51	PROPN
ejpam-5499	30	3	start	start	VERB
ejpam-5499	30	4	by	by	ADP
ejpam-5499	30	5	defininng	defininng	PROPN
ejpam-5499	30	6	a	a	DET
ejpam-5499	30	7	binary	binary	ADJ
ejpam-5499	30	8	operation.52	operation.52	ADJ
ejpam-5499	30	9	definition	definition	NOUN
ejpam-5499	30	10	1	1	NUM
ejpam-5499	30	11	.	.	PUNCT
ejpam-5499	31	1	let	let	VERB
ejpam-5499	31	2	s	s	PRON
ejpam-5499	31	3	be	be	AUX
ejpam-5499	31	4	a	a	DET
ejpam-5499	31	5	set	set	NOUN
ejpam-5499	31	6	.	.	PUNCT
ejpam-5499	32	1	a	a	DET
ejpam-5499	32	2	binary	binary	PROPN
ejpam-5499	32	3	operation	operation	NOUN
ejpam-5499	32	4	∗	∗	NOUN
ejpam-5499	32	5	on	on	ADP
ejpam-5499	32	6	s	s	NOUN
ejpam-5499	32	7	is	be	AUX
ejpam-5499	32	8	a	a	DET
ejpam-5499	32	9	function	function	NOUN
ejpam-5499	32	10	that	that	PRON
ejpam-5499	32	11	assigns	assign	VERB
ejpam-5499	32	12	each53	each53	NOUN
ejpam-5499	32	13	ordered	order	VERB
ejpam-5499	32	14	pair	pair	NOUN
ejpam-5499	32	15	of	of	ADP
ejpam-5499	32	16	elements	element	NOUN
ejpam-5499	32	17	of	of	ADP
ejpam-5499	32	18	s	s	PRON
ejpam-5499	32	19	an	an	DET
ejpam-5499	32	20	element	element	NOUN
ejpam-5499	32	21	of	of	ADP
ejpam-5499	32	22	s.54	s.54	X
ejpam-5499	32	23	consider	consider	VERB
ejpam-5499	32	24	the	the	DET
ejpam-5499	32	25	set	set	NOUN
ejpam-5499	32	26	of	of	ADP
ejpam-5499	32	27	even	even	ADV
ejpam-5499	32	28	integers	integer	NOUN
ejpam-5499	32	29	s	s	PART
ejpam-5499	32	30	=	=	NOUN
ejpam-5499	32	31	2z	2z	NOUN
ejpam-5499	32	32	using	use	VERB
ejpam-5499	32	33	the	the	DET
ejpam-5499	32	34	addition	addition	NOUN
ejpam-5499	32	35	as	as	ADP
ejpam-5499	32	36	the	the	DET
ejpam-5499	32	37	operation	operation	NOUN
ejpam-5499	32	38	.	.	PUNCT
ejpam-5499	33	1	if	if	SCONJ
ejpam-5499	33	2	we55	we55	PROPN
ejpam-5499	33	3	take	take	VERB
ejpam-5499	33	4	a	a	DET
ejpam-5499	33	5	=	=	PUNCT
ejpam-5499	33	6	2k	2k	NUM
ejpam-5499	33	7	,	,	PUNCT
ejpam-5499	33	8	b	b	X
ejpam-5499	33	9	=	=	SYM
ejpam-5499	33	10	j	j	PROPN
ejpam-5499	33	11	∈	∈	PROPN
ejpam-5499	33	12	s	s	PROPN
ejpam-5499	33	13	,	,	PUNCT
ejpam-5499	33	14	for	for	ADP
ejpam-5499	33	15	some	some	DET
ejpam-5499	33	16	integers	integer	NOUN
ejpam-5499	33	17	k	k	PROPN
ejpam-5499	33	18	and	and	CCONJ
ejpam-5499	33	19	j	j	PROPN
ejpam-5499	33	20	,	,	PUNCT
ejpam-5499	33	21	a+	a+	PUNCT
ejpam-5499	33	22	b	b	X
ejpam-5499	33	23	=	=	SYM
ejpam-5499	33	24	2k+2j	2k+2j	NUM
ejpam-5499	33	25	=	=	SYM
ejpam-5499	33	26	2(k+	2(k+	NUM
ejpam-5499	33	27	j	j	NOUN
ejpam-5499	33	28	)	)	PUNCT
ejpam-5499	33	29	which	which	PRON
ejpam-5499	33	30	is	be	AUX
ejpam-5499	33	31	also56	also56	NOUN
ejpam-5499	33	32	an	an	DET
ejpam-5499	33	33	even	even	ADV
ejpam-5499	33	34	integer	integer	NOUN
ejpam-5499	33	35	.	.	PUNCT
ejpam-5499	34	1	thus	thus	ADV
ejpam-5499	34	2	,	,	PUNCT
ejpam-5499	34	3	addition	addition	NOUN
ejpam-5499	34	4	is	be	AUX
ejpam-5499	34	5	a	a	DET
ejpam-5499	34	6	binary	binary	ADJ
ejpam-5499	34	7	operation	operation	NOUN
ejpam-5499	34	8	in	in	ADP
ejpam-5499	34	9	s.57	s.57	PROPN
ejpam-5499	34	10	definition	definition	NOUN
ejpam-5499	34	11	2	2	NUM
ejpam-5499	34	12	.	.	PUNCT
ejpam-5499	35	1	a	a	DET
ejpam-5499	35	2	group	group	NOUN
ejpam-5499	35	3	is	be	AUX
ejpam-5499	35	4	a	a	DET
ejpam-5499	35	5	non	non	ADJ
ejpam-5499	35	6	-	-	ADJ
ejpam-5499	35	7	empty	empty	ADJ
ejpam-5499	35	8	set	set	VERB
ejpam-5499	35	9	g	g	NOUN
ejpam-5499	35	10	with	with	ADP
ejpam-5499	35	11	binary	binary	ADJ
ejpam-5499	35	12	operation	operation	NOUN
ejpam-5499	35	13	∗	∗	NOUN
ejpam-5499	35	14	such	such	ADJ
ejpam-5499	35	15	that58	that58	PROPN
ejpam-5499	35	16	i.	i.	PROPN
ejpam-5499	35	17	a	a	DET
ejpam-5499	35	18	∗	∗	X
ejpam-5499	35	19	(	(	PUNCT
ejpam-5499	35	20	b	b	NOUN
ejpam-5499	35	21	∗	∗	NOUN
ejpam-5499	35	22	c	c	NOUN
ejpam-5499	35	23	)	)	PUNCT
ejpam-5499	35	24	=	=	NOUN
ejpam-5499	35	25	(	(	PUNCT
ejpam-5499	35	26	a	a	DET
ejpam-5499	35	27	∗	∗	NOUN
ejpam-5499	35	28	b	b	NOUN
ejpam-5499	35	29	)	)	PUNCT
ejpam-5499	35	30	∗	∗	NOUN
ejpam-5499	35	31	c	c	NOUN
ejpam-5499	35	32	for	for	ADP
ejpam-5499	35	33	all	all	DET
ejpam-5499	35	34	a	a	DET
ejpam-5499	35	35	,	,	PUNCT
ejpam-5499	35	36	b	b	NOUN
ejpam-5499	35	37	,	,	PUNCT
ejpam-5499	35	38	c	c	PROPN
ejpam-5499	35	39	in	in	ADP
ejpam-5499	35	40	g	g	PROPN
ejpam-5499	35	41	(	(	PUNCT
ejpam-5499	35	42	associativity),59	associativity),59	NOUN
ejpam-5499	35	43	j.	j.	PROPN
ejpam-5499	35	44	m.	m.	PROPN
ejpam-5499	35	45	jamis	jamis	PROPN
ejpam-5499	35	46	,	,	PUNCT
ejpam-5499	35	47	d.	d.	PROPN
ejpam-5499	35	48	m.	m.	PROPN
ejpam-5499	35	49	magpantay	magpantay	PROPN
ejpam-5499	35	50	/	/	SYM
ejpam-5499	35	51	eur	eur	PROPN
ejpam-5499	35	52	.	.	PUNCT
ejpam-5499	36	1	j.	j.	PROPN
ejpam-5499	36	2	pure	pure	PROPN
ejpam-5499	36	3	appl	appl	PROPN
ejpam-5499	36	4	.	.	PROPN
ejpam-5499	36	5	math	math	PROPN
ejpam-5499	36	6	,	,	PUNCT
ejpam-5499	36	7	18	18	NUM
ejpam-5499	36	8	(	(	PUNCT
ejpam-5499	36	9	2	2	NUM
ejpam-5499	36	10	)	)	PUNCT
ejpam-5499	36	11	(	(	PUNCT
ejpam-5499	36	12	2025	2025	NUM
ejpam-5499	36	13	)	)	PUNCT
ejpam-5499	36	14	,	,	PUNCT
ejpam-5499	36	15	5499	5499	NUM
ejpam-5499	36	16	3	3	NUM
ejpam-5499	36	17	of	of	ADP
ejpam-5499	36	18	18	18	NUM
ejpam-5499	36	19	ii	ii	NOUN
ejpam-5499	36	20	.	.	PUNCT
ejpam-5499	37	1	there	there	PRON
ejpam-5499	37	2	is	be	VERB
ejpam-5499	37	3	an	an	DET
ejpam-5499	37	4	element	element	NOUN
ejpam-5499	37	5	e	e	NOUN
ejpam-5499	37	6	∈	∈	PROPN
ejpam-5499	37	7	g	g	PROPN
ejpam-5499	37	8	such	such	DET
ejpam-5499	37	9	that	that	SCONJ
ejpam-5499	37	10	a	a	DET
ejpam-5499	37	11	∗	∗	NOUN
ejpam-5499	37	12	e	e	NOUN
ejpam-5499	37	13	=	=	SYM
ejpam-5499	37	14	e	e	NOUN
ejpam-5499	37	15	∗	∗	VERB
ejpam-5499	37	16	a	a	DET
ejpam-5499	37	17	=	=	NOUN
ejpam-5499	37	18	a	a	PRON
ejpam-5499	37	19	for	for	ADP
ejpam-5499	37	20	all	all	DET
ejpam-5499	37	21	a	a	DET
ejpam-5499	37	22	∈	∈	PROPN
ejpam-5499	37	23	g	g	NOUN
ejpam-5499	37	24	(	(	PUNCT
ejpam-5499	37	25	existence	existence	PROPN
ejpam-5499	37	26	of60	of60	PROPN
ejpam-5499	37	27	identity),61	identity),61	VERB
ejpam-5499	37	28	iii	iii	PROPN
ejpam-5499	37	29	.	.	PUNCT
ejpam-5499	38	1	if	if	SCONJ
ejpam-5499	38	2	a	a	DET
ejpam-5499	38	3	∈	∈	PROPN
ejpam-5499	38	4	g	g	NOUN
ejpam-5499	38	5	,	,	PUNCT
ejpam-5499	38	6	then	then	ADV
ejpam-5499	38	7	there	there	PRON
ejpam-5499	38	8	is	be	VERB
ejpam-5499	38	9	an	an	DET
ejpam-5499	38	10	element	element	NOUN
ejpam-5499	38	11	a−1	a−1	PROPN
ejpam-5499	38	12	∈	∈	PROPN
ejpam-5499	38	13	g	g	ADP
ejpam-5499	38	14	such	such	ADJ
ejpam-5499	38	15	that	that	SCONJ
ejpam-5499	38	16	a∗a−1	a∗a−1	PROPN
ejpam-5499	38	17	=	=	SYM
ejpam-5499	38	18	a−1	a−1	PROPN
ejpam-5499	38	19	∗a	∗a	PROPN
ejpam-5499	38	20	=	=	SYM
ejpam-5499	38	21	e	e	X
ejpam-5499	38	22	(	(	PUNCT
ejpam-5499	38	23	existence62	existence62	NOUN
ejpam-5499	38	24	of	of	ADP
ejpam-5499	38	25	inverse).63	inverse).63	PROPN
ejpam-5499	38	26	example	example	NOUN
ejpam-5499	38	27	1	1	X
ejpam-5499	38	28	.	.	PUNCT
ejpam-5499	39	1	the	the	DET
ejpam-5499	39	2	set	set	NOUN
ejpam-5499	39	3	of	of	ADP
ejpam-5499	39	4	integers	integer	NOUN
ejpam-5499	39	5	is	be	AUX
ejpam-5499	39	6	a	a	DET
ejpam-5499	39	7	group	group	NOUN
ejpam-5499	39	8	under	under	ADP
ejpam-5499	39	9	the	the	DET
ejpam-5499	39	10	operation	operation	NOUN
ejpam-5499	39	11	of	of	ADP
ejpam-5499	39	12	ordinary	ordinary	ADJ
ejpam-5499	39	13	addition	addition	NOUN
ejpam-5499	39	14	.	.	PUNCT
ejpam-5499	40	1	note64	note64	NOUN
ejpam-5499	40	2	that	that	SCONJ
ejpam-5499	40	3	the	the	DET
ejpam-5499	40	4	set	set	NOUN
ejpam-5499	40	5	of	of	ADP
ejpam-5499	40	6	integers	integer	NOUN
ejpam-5499	40	7	under	under	ADP
ejpam-5499	40	8	the	the	DET
ejpam-5499	40	9	operation	operation	NOUN
ejpam-5499	40	10	of	of	ADP
ejpam-5499	40	11	addition	addition	NOUN
ejpam-5499	40	12	is	be	AUX
ejpam-5499	40	13	closed	closed	ADJ
ejpam-5499	40	14	,	,	PUNCT
ejpam-5499	40	15	associative	associative	ADJ
ejpam-5499	40	16	,	,	PUNCT
ejpam-5499	40	17	contains	contain	VERB
ejpam-5499	40	18	iden-65	iden-65	PROPN
ejpam-5499	40	19	tity	tity	NOUN
ejpam-5499	40	20	element	element	NOUN
ejpam-5499	40	21	0	0	NUM
ejpam-5499	40	22	,	,	PUNCT
ejpam-5499	40	23	and	and	CCONJ
ejpam-5499	40	24	for	for	ADP
ejpam-5499	40	25	any	any	DET
ejpam-5499	40	26	integer	integer	NOUN
ejpam-5499	40	27	a	a	PRON
ejpam-5499	40	28	it	it	PRON
ejpam-5499	40	29	has	have	VERB
ejpam-5499	40	30	an	an	DET
ejpam-5499	40	31	−a.66	−a.66	NUM
ejpam-5499	40	32	67	67	NUM
ejpam-5499	40	33	the	the	DET
ejpam-5499	40	34	set	set	NOUN
ejpam-5499	40	35	of	of	ADP
ejpam-5499	40	36	integers	integer	NOUN
ejpam-5499	40	37	under	under	ADP
ejpam-5499	40	38	ordinary	ordinary	ADJ
ejpam-5499	40	39	multiplication	multiplication	NOUN
ejpam-5499	40	40	is	be	AUX
ejpam-5499	40	41	not	not	PART
ejpam-5499	40	42	a	a	DET
ejpam-5499	40	43	group	group	NOUN
ejpam-5499	40	44	.	.	PUNCT
ejpam-5499	41	1	the	the	DET
ejpam-5499	41	2	third	third	ADJ
ejpam-5499	41	3	property68	property68	NOUN
ejpam-5499	41	4	fails	fail	VERB
ejpam-5499	41	5	since	since	SCONJ
ejpam-5499	41	6	there	there	PRON
ejpam-5499	41	7	is	be	VERB
ejpam-5499	41	8	no	no	DET
ejpam-5499	41	9	integer	integer	NOUN
ejpam-5499	41	10	b	b	NOUN
ejpam-5499	41	11	such	such	ADJ
ejpam-5499	41	12	that	that	DET
ejpam-5499	41	13	2b	2b	NUM
ejpam-5499	41	14	=	=	SYM
ejpam-5499	41	15	1	1	NUM
ejpam-5499	41	16	where	where	SCONJ
ejpam-5499	41	17	1	1	NUM
ejpam-5499	41	18	is	be	AUX
ejpam-5499	41	19	the	the	DET
ejpam-5499	41	20	identity	identity	NOUN
ejpam-5499	41	21	element.69	element.69	NOUN
ejpam-5499	41	22	70	70	NUM
ejpam-5499	41	23	now	now	ADV
ejpam-5499	41	24	we	we	PRON
ejpam-5499	41	25	will	will	AUX
ejpam-5499	41	26	define	define	VERB
ejpam-5499	41	27	a	a	DET
ejpam-5499	41	28	cyclic	cyclic	ADJ
ejpam-5499	41	29	subgroup,71	subgroup,71	NOUN
ejpam-5499	41	30	definition	definition	NOUN
ejpam-5499	41	31	3	3	X
ejpam-5499	41	32	.	.	PUNCT
ejpam-5499	42	1	if	if	SCONJ
ejpam-5499	42	2	g	g	PROPN
ejpam-5499	42	3	is	be	AUX
ejpam-5499	42	4	a	a	DET
ejpam-5499	42	5	group	group	NOUN
ejpam-5499	42	6	and	and	CCONJ
ejpam-5499	42	7	a	a	DET
ejpam-5499	42	8	∈	∈	PROPN
ejpam-5499	42	9	g	g	NOUN
ejpam-5499	42	10	,	,	PUNCT
ejpam-5499	42	11	then	then	ADV
ejpam-5499	42	12	the	the	DET
ejpam-5499	42	13	cyclic	cyclic	ADJ
ejpam-5499	42	14	subgroup	subgroup	NOUN
ejpam-5499	42	15	generated	generate	VERB
ejpam-5499	42	16	by	by	ADP
ejpam-5499	42	17	a	a	DET
ejpam-5499	42	18	is72	is72	PROPN
ejpam-5499	42	19	the	the	DET
ejpam-5499	42	20	set73	set73	PROPN
ejpam-5499	42	21	⟨a⟩	⟨a⟩	PROPN
ejpam-5499	42	22	=	=	PUNCT
ejpam-5499	42	23	{	{	PUNCT
ejpam-5499	42	24	an	an	NOUN
ejpam-5499	42	25	:	:	PUNCT
ejpam-5499	42	26	n	n	NOUN
ejpam-5499	42	27	∈	∈	PROPN
ejpam-5499	42	28	z	z	PROPN
ejpam-5499	42	29	}	}	PUNCT
ejpam-5499	42	30	if	if	SCONJ
ejpam-5499	42	31	,	,	PUNCT
ejpam-5499	42	32	in	in	ADP
ejpam-5499	42	33	g	g	NOUN
ejpam-5499	42	34	,	,	PUNCT
ejpam-5499	42	35	there	there	PRON
ejpam-5499	42	36	exists	exist	VERB
ejpam-5499	42	37	an	an	DET
ejpam-5499	42	38	element	element	NOUN
ejpam-5499	42	39	a	a	DET
ejpam-5499	42	40	such	such	ADJ
ejpam-5499	42	41	that	that	PRON
ejpam-5499	42	42	g	g	NOUN
ejpam-5499	42	43	=	=	SYM
ejpam-5499	42	44	⟨a⟩	⟨a⟩	PROPN
ejpam-5499	42	45	,	,	PUNCT
ejpam-5499	42	46	then	then	ADV
ejpam-5499	42	47	we	we	PRON
ejpam-5499	42	48	say	say	VERB
ejpam-5499	42	49	that	that	SCONJ
ejpam-5499	42	50	g	g	PROPN
ejpam-5499	42	51	is	be	AUX
ejpam-5499	42	52	a	a	DET
ejpam-5499	42	53	cyclic74	cyclic74	NOUN
ejpam-5499	42	54	group	group	NOUN
ejpam-5499	42	55	and	and	CCONJ
ejpam-5499	42	56	a	a	PRON
ejpam-5499	42	57	is	be	AUX
ejpam-5499	42	58	a	a	DET
ejpam-5499	42	59	generator	generator	NOUN
ejpam-5499	42	60	of	of	ADP
ejpam-5499	42	61	g.	g.	PROPN
ejpam-5499	42	62	we	we	PRON
ejpam-5499	42	63	may	may	AUX
ejpam-5499	42	64	also	also	ADV
ejpam-5499	42	65	say	say	VERB
ejpam-5499	42	66	that	that	SCONJ
ejpam-5499	42	67	g	g	PROPN
ejpam-5499	42	68	is	be	AUX
ejpam-5499	42	69	a	a	DET
ejpam-5499	42	70	group	group	NOUN
ejpam-5499	42	71	generated	generate	VERB
ejpam-5499	42	72	by	by	ADP
ejpam-5499	42	73	a.75	a.75	PROPN
ejpam-5499	42	74	if	if	SCONJ
ejpam-5499	42	75	no	no	DET
ejpam-5499	42	76	such	such	ADJ
ejpam-5499	42	77	element	element	NOUN
ejpam-5499	42	78	exists	exist	VERB
ejpam-5499	42	79	in	in	ADP
ejpam-5499	42	80	g	g	NOUN
ejpam-5499	42	81	,	,	PUNCT
ejpam-5499	42	82	then	then	ADV
ejpam-5499	42	83	g	g	PROPN
ejpam-5499	42	84	is	be	AUX
ejpam-5499	42	85	said	say	VERB
ejpam-5499	42	86	to	to	PART
ejpam-5499	42	87	be	be	AUX
ejpam-5499	42	88	a	a	DET
ejpam-5499	42	89	noncyclic	noncyclic	ADJ
ejpam-5499	42	90	group.76	group.76	PROPN
ejpam-5499	42	91	example	example	NOUN
ejpam-5499	43	1	2	2	X
ejpam-5499	43	2	.	.	PUNCT
ejpam-5499	43	3	let	let	VERB
ejpam-5499	43	4	g	g	NOUN
ejpam-5499	43	5	=	=	NOUN
ejpam-5499	43	6	z,+	z,+	NUM
ejpam-5499	43	7	.	.	PUNCT
ejpam-5499	44	1	then	then	ADV
ejpam-5499	44	2	g	g	PROPN
ejpam-5499	44	3	=	=	PUNCT
ejpam-5499	44	4	⟨1⟩	⟨1⟩	X
ejpam-5499	44	5	=	=	SYM
ejpam-5499	44	6	⟨−1⟩	⟨−1⟩	PROPN
ejpam-5499	44	7	,	,	PUNCT
ejpam-5499	44	8	so	so	SCONJ
ejpam-5499	44	9	g	g	PROPN
ejpam-5499	44	10	is	be	AUX
ejpam-5499	44	11	cyclic	cyclic	ADJ
ejpam-5499	44	12	.	.	PUNCT
ejpam-5499	45	1	on	on	ADP
ejpam-5499	45	2	the	the	DET
ejpam-5499	45	3	other	other	ADJ
ejpam-5499	45	4	hand,77	hand,77	NOUN
ejpam-5499	45	5	g	g	NOUN
ejpam-5499	45	6	=	=	PUNCT
ejpam-5499	45	7	q,+	q,+	NOUN
ejpam-5499	45	8	is	be	AUX
ejpam-5499	45	9	noncyclic	noncyclic	ADJ
ejpam-5499	45	10	,	,	PUNCT
ejpam-5499	45	11	since	since	SCONJ
ejpam-5499	45	12	there	there	PRON
ejpam-5499	45	13	is	be	VERB
ejpam-5499	45	14	no	no	DET
ejpam-5499	45	15	rational	rational	ADJ
ejpam-5499	45	16	number	number	NOUN
ejpam-5499	45	17	which	which	PRON
ejpam-5499	45	18	generates	generate	VERB
ejpam-5499	45	19	all	all	DET
ejpam-5499	45	20	possible78	possible78	ADJ
ejpam-5499	45	21	rational	rational	ADJ
ejpam-5499	45	22	numbers	number	NOUN
ejpam-5499	45	23	.	.	PUNCT
ejpam-5499	46	1	for	for	ADP
ejpam-5499	46	2	the	the	DET
ejpam-5499	46	3	same	same	ADJ
ejpam-5499	46	4	reason	reason	NOUN
ejpam-5499	46	5	,	,	PUNCT
ejpam-5499	46	6	g	g	NOUN
ejpam-5499	46	7	=	=	PUNCT
ejpam-5499	46	8	r,+	r,+	PRON
ejpam-5499	46	9	is	be	AUX
ejpam-5499	46	10	noncyclic79	noncyclic79	ADJ
ejpam-5499	46	11	another	another	DET
ejpam-5499	46	12	type	type	NOUN
ejpam-5499	46	13	of	of	ADP
ejpam-5499	46	14	finite	finite	ADJ
ejpam-5499	46	15	groups	group	NOUN
ejpam-5499	46	16	are	be	AUX
ejpam-5499	46	17	dihedral	dihedral	ADJ
ejpam-5499	46	18	groups	group	NOUN
ejpam-5499	46	19	which	which	PRON
ejpam-5499	46	20	belongs	belong	VERB
ejpam-5499	46	21	to	to	ADP
ejpam-5499	46	22	the	the	DET
ejpam-5499	46	23	classification80	classification80	NOUN
ejpam-5499	46	24	of	of	ADP
ejpam-5499	46	25	noncyclic.81	noncyclic.81	ADJ
ejpam-5499	46	26	definition	definition	NOUN
ejpam-5499	46	27	4	4	NUM
ejpam-5499	46	28	.	.	PUNCT
ejpam-5499	47	1	the	the	DET
ejpam-5499	47	2	group	group	NOUN
ejpam-5499	47	3	of	of	ADP
ejpam-5499	47	4	symmetries	symmetry	NOUN
ejpam-5499	47	5	of	of	ADP
ejpam-5499	47	6	an	an	DET
ejpam-5499	47	7	n	n	ADV
ejpam-5499	47	8	-	-	PUNCT
ejpam-5499	47	9	sided	sided	ADJ
ejpam-5499	47	10	regular	regular	ADJ
ejpam-5499	47	11	polygon	polygon	NOUN
ejpam-5499	47	12	for	for	ADP
ejpam-5499	47	13	n	n	PROPN
ejpam-5499	47	14	≥	≥	NUM
ejpam-5499	47	15	1	1	NUM
ejpam-5499	47	16	with82	with82	NOUN
ejpam-5499	47	17	rotations	rotation	NOUN
ejpam-5499	47	18	and	and	CCONJ
ejpam-5499	47	19	reflections	reflection	NOUN
ejpam-5499	47	20	is	be	AUX
ejpam-5499	47	21	termed	term	VERB
ejpam-5499	47	22	dihedral	dihedral	ADJ
ejpam-5499	47	23	group	group	NOUN
ejpam-5499	47	24	,	,	PUNCT
ejpam-5499	47	25	which	which	PRON
ejpam-5499	47	26	is	be	AUX
ejpam-5499	47	27	denoted	denote	VERB
ejpam-5499	47	28	as	as	ADP
ejpam-5499	47	29	dn	dn	PROPN
ejpam-5499	47	30	.	.	PUNCT
ejpam-5499	48	1	the	the	DET
ejpam-5499	48	2	order83	order83	NOUN
ejpam-5499	48	3	of	of	ADP
ejpam-5499	48	4	the	the	DET
ejpam-5499	48	5	dihedral	dihedral	ADJ
ejpam-5499	48	6	group	group	NOUN
ejpam-5499	48	7	is	be	AUX
ejpam-5499	48	8	2n.84	2n.84	NUM
ejpam-5499	48	9	for	for	ADP
ejpam-5499	48	10	n	n	X
ejpam-5499	48	11	≥	≥	NOUN
ejpam-5499	48	12	3	3	NUM
ejpam-5499	48	13	,	,	PUNCT
ejpam-5499	48	14	dn	dn	PROPN
ejpam-5499	48	15	is	be	AUX
ejpam-5499	48	16	the	the	DET
ejpam-5499	48	17	group	group	NOUN
ejpam-5499	48	18	of	of	ADP
ejpam-5499	48	19	symmetries	symmetry	NOUN
ejpam-5499	48	20	of	of	ADP
ejpam-5499	48	21	a	a	DET
ejpam-5499	48	22	regular	regular	ADJ
ejpam-5499	48	23	polygon	polygon	NOUN
ejpam-5499	48	24	with	with	ADP
ejpam-5499	48	25	n	n	NOUN
ejpam-5499	48	26	-	-	PUNCT
ejpam-5499	48	27	sides	side	NOUN
ejpam-5499	48	28	.	.	PUNCT
ejpam-5499	49	1	number85	number85	VERB
ejpam-5499	49	2	the	the	DET
ejpam-5499	49	3	vertices	vertex	NOUN
ejpam-5499	49	4	1	1	NUM
ejpam-5499	49	5	,	,	PUNCT
ejpam-5499	49	6	...	...	PUNCT
ejpam-5499	49	7	,	,	PUNCT
ejpam-5499	49	8	n	n	CCONJ
ejpam-5499	49	9	in	in	ADP
ejpam-5499	49	10	the	the	DET
ejpam-5499	49	11	counterclockwise	counterclockwise	NOUN
ejpam-5499	49	12	direction	direction	NOUN
ejpam-5499	49	13	.	.	PUNCT
ejpam-5499	50	1	let	let	VERB
ejpam-5499	50	2	r	r	NOUN
ejpam-5499	50	3	be	be	AUX
ejpam-5499	50	4	the	the	DET
ejpam-5499	50	5	rotation	rotation	NOUN
ejpam-5499	50	6	through	through	ADP
ejpam-5499	50	7	2π	2π	NOUN
ejpam-5499	50	8	/	/	SYM
ejpam-5499	50	9	n86	n86	NOUN
ejpam-5499	50	10	about	about	ADP
ejpam-5499	50	11	the	the	DET
ejpam-5499	50	12	centre	centre	NOUN
ejpam-5499	50	13	of	of	ADP
ejpam-5499	50	14	polygon	polygon	PROPN
ejpam-5499	50	15	(	(	PUNCT
ejpam-5499	51	1	so	so	ADV
ejpam-5499	51	2	i	i	PRON
ejpam-5499	51	3	7→	7→	NUM
ejpam-5499	52	1	i	i	PRON
ejpam-5499	52	2	+	+	CCONJ
ejpam-5499	52	3	1	1	NUM
ejpam-5499	52	4	mod	mod	NOUN
ejpam-5499	52	5	n	n	CCONJ
ejpam-5499	52	6	)	)	PUNCT
ejpam-5499	52	7	,	,	PUNCT
ejpam-5499	53	1	and	and	CCONJ
ejpam-5499	53	2	let	let	VERB
ejpam-5499	53	3	s	s	PRON
ejpam-5499	53	4	be	be	AUX
ejpam-5499	53	5	the	the	DET
ejpam-5499	53	6	reflection	reflection	NOUN
ejpam-5499	53	7	in	in	ADP
ejpam-5499	53	8	the87	the87	NOUN
ejpam-5499	53	9	line	line	NOUN
ejpam-5499	53	10	(=	(=	ADP
ejpam-5499	53	11	rotation	rotation	NOUN
ejpam-5499	53	12	about	about	ADP
ejpam-5499	53	13	the	the	DET
ejpam-5499	53	14	line	line	NOUN
ejpam-5499	53	15	)	)	PUNCT
ejpam-5499	53	16	through	through	ADP
ejpam-5499	53	17	the	the	DET
ejpam-5499	53	18	vertex	vertex	NOUN
ejpam-5499	53	19	1	1	NUM
ejpam-5499	53	20	and	and	CCONJ
ejpam-5499	53	21	the	the	DET
ejpam-5499	53	22	centre	centre	NOUN
ejpam-5499	53	23	of	of	ADP
ejpam-5499	53	24	the	the	DET
ejpam-5499	53	25	polygon	polygon	NOUN
ejpam-5499	53	26	(	(	PUNCT
ejpam-5499	54	1	so88	so88	PROPN
ejpam-5499	54	2	i	i	PRON
ejpam-5499	54	3	7→	7→	PROPN
ejpam-5499	54	4	n+	n+	NUM
ejpam-5499	54	5	2−	2−	NUM
ejpam-5499	54	6	i	i	PRON
ejpam-5499	54	7	mod	mod	PROPN
ejpam-5499	54	8	n	n	CCONJ
ejpam-5499	54	9	)	)	PUNCT
ejpam-5499	54	10	.	.	PUNCT
ejpam-5499	55	1	here	here	ADV
ejpam-5499	55	2	is	be	AUX
ejpam-5499	55	3	an	an	DET
ejpam-5499	55	4	illustration.89	illustration.89	PROPN
ejpam-5499	55	5	90	90	NUM
ejpam-5499	55	6	j.	j.	PROPN
ejpam-5499	55	7	m.	m.	PROPN
ejpam-5499	55	8	jamis	jamis	PROPN
ejpam-5499	55	9	,	,	PUNCT
ejpam-5499	55	10	d.	d.	PROPN
ejpam-5499	55	11	m.	m.	PROPN
ejpam-5499	55	12	magpantay	magpantay	PROPN
ejpam-5499	55	13	/	/	SYM
ejpam-5499	55	14	eur	eur	PROPN
ejpam-5499	55	15	.	.	PUNCT
ejpam-5499	56	1	j.	j.	PROPN
ejpam-5499	56	2	pure	pure	PROPN
ejpam-5499	56	3	appl	appl	PROPN
ejpam-5499	56	4	.	.	PROPN
ejpam-5499	56	5	math	math	PROPN
ejpam-5499	56	6	,	,	PUNCT
ejpam-5499	56	7	18	18	NUM
ejpam-5499	56	8	(	(	PUNCT
ejpam-5499	56	9	2	2	NUM
ejpam-5499	56	10	)	)	PUNCT
ejpam-5499	56	11	(	(	PUNCT
ejpam-5499	56	12	2025	2025	NUM
ejpam-5499	56	13	)	)	PUNCT
ejpam-5499	56	14	,	,	PUNCT
ejpam-5499	56	15	5499	5499	NUM
ejpam-5499	56	16	4	4	NUM
ejpam-5499	56	17	of	of	ADP
ejpam-5499	56	18	18	18	NUM
ejpam-5499	56	19	2.2	2.2	NUM
ejpam-5499	56	20	.	.	PUNCT
ejpam-5499	57	1	graph	graph	NOUN
ejpam-5499	57	2	theory91	theory91	NOUN
ejpam-5499	57	3	a	a	DET
ejpam-5499	57	4	graph	graph	NOUN
ejpam-5499	57	5	g	g	NOUN
ejpam-5499	57	6	is	be	AUX
ejpam-5499	57	7	an	an	DET
ejpam-5499	57	8	ordered	order	VERB
ejpam-5499	57	9	pair	pair	NOUN
ejpam-5499	57	10	g	g	NOUN
ejpam-5499	57	11	=	=	PUNCT
ejpam-5499	57	12	(	(	PUNCT
ejpam-5499	57	13	v	v	NOUN
ejpam-5499	57	14	(	(	PUNCT
ejpam-5499	57	15	g	g	NOUN
ejpam-5499	57	16	)	)	PUNCT
ejpam-5499	57	17	,	,	PUNCT
ejpam-5499	57	18	e(g	e(g	PROPN
ejpam-5499	57	19	)	)	PUNCT
ejpam-5499	57	20	)	)	PUNCT
ejpam-5499	57	21	where	where	SCONJ
ejpam-5499	57	22	v	v	X
ejpam-5499	57	23	(	(	PUNCT
ejpam-5499	57	24	g	g	NOUN
ejpam-5499	57	25	)	)	PUNCT
ejpam-5499	57	26	is	be	AUX
ejpam-5499	57	27	a	a	DET
ejpam-5499	57	28	nonempty	nonempty	ADV
ejpam-5499	57	29	set	set	VERB
ejpam-5499	57	30	of92	of92	PROPN
ejpam-5499	57	31	elements	element	NOUN
ejpam-5499	57	32	called	call	VERB
ejpam-5499	57	33	vertices	vertex	NOUN
ejpam-5499	57	34	,	,	PUNCT
ejpam-5499	57	35	and	and	CCONJ
ejpam-5499	57	36	e(g	e(g	NOUN
ejpam-5499	57	37	)	)	PUNCT
ejpam-5499	57	38	is	be	AUX
ejpam-5499	57	39	a	a	DET
ejpam-5499	57	40	set	set	NOUN
ejpam-5499	57	41	of	of	ADP
ejpam-5499	57	42	unordered	unordered	ADJ
ejpam-5499	57	43	pairs	pair	NOUN
ejpam-5499	57	44	of	of	ADP
ejpam-5499	57	45	vertices	vertex	NOUN
ejpam-5499	57	46	called	call	VERB
ejpam-5499	57	47	edges.93	edges.93	NUM
ejpam-5499	57	48	the	the	DET
ejpam-5499	57	49	number	number	NOUN
ejpam-5499	57	50	|v	|v	NOUN
ejpam-5499	57	51	(	(	PUNCT
ejpam-5499	57	52	g)|	g)|	PROPN
ejpam-5499	57	53	is	be	AUX
ejpam-5499	57	54	called	call	VERB
ejpam-5499	57	55	the	the	DET
ejpam-5499	57	56	order	order	NOUN
ejpam-5499	57	57	of	of	ADP
ejpam-5499	57	58	g	g	NOUN
ejpam-5499	57	59	and	and	CCONJ
ejpam-5499	57	60	the	the	DET
ejpam-5499	57	61	number	number	NOUN
ejpam-5499	57	62	|e(g)|	|e(g)|	PROPN
ejpam-5499	57	63	is	be	AUX
ejpam-5499	57	64	called	call	VERB
ejpam-5499	57	65	the	the	DET
ejpam-5499	57	66	size	size	NOUN
ejpam-5499	57	67	of94	of94	PROPN
ejpam-5499	57	68	g	g	ADP
ejpam-5499	57	69	.95	.95	NUM
ejpam-5499	57	70	connected	connect	VERB
ejpam-5499	57	71	graphs	graph	NOUN
ejpam-5499	57	72	are	be	AUX
ejpam-5499	57	73	the	the	DET
ejpam-5499	57	74	graphs	graph	NOUN
ejpam-5499	57	75	in	in	ADP
ejpam-5499	57	76	which	which	PRON
ejpam-5499	57	77	every	every	DET
ejpam-5499	57	78	two	two	NUM
ejpam-5499	57	79	vertices	vertex	NOUN
ejpam-5499	57	80	is	be	AUX
ejpam-5499	57	81	adjacent	adjacent	ADJ
ejpam-5499	57	82	to	to	ADP
ejpam-5499	57	83	each	each	DET
ejpam-5499	57	84	other.96	other.96	PROPN
ejpam-5499	58	1	if	if	SCONJ
ejpam-5499	58	2	[	[	X
ejpam-5499	58	3	u	u	NOUN
ejpam-5499	58	4	,	,	PUNCT
ejpam-5499	58	5	v	v	NOUN
ejpam-5499	58	6	]	]	PUNCT
ejpam-5499	58	7	is	be	AUX
ejpam-5499	58	8	an	an	DET
ejpam-5499	58	9	edge	edge	NOUN
ejpam-5499	58	10	of	of	ADP
ejpam-5499	58	11	g	g	NOUN
ejpam-5499	58	12	,	,	PUNCT
ejpam-5499	58	13	then	then	ADV
ejpam-5499	58	14	u	u	NOUN
ejpam-5499	58	15	and	and	CCONJ
ejpam-5499	58	16	v	v	NOUN
ejpam-5499	58	17	are	be	AUX
ejpam-5499	58	18	adjacent	adjacent	ADJ
ejpam-5499	58	19	vertices	vertex	NOUN
ejpam-5499	58	20	.	.	PUNCT
ejpam-5499	59	1	if	if	SCONJ
ejpam-5499	59	2	[	[	X
ejpam-5499	59	3	u	u	NOUN
ejpam-5499	59	4	,	,	PUNCT
ejpam-5499	59	5	v	v	NOUN
ejpam-5499	59	6	]	]	PUNCT
ejpam-5499	59	7	and	and	CCONJ
ejpam-5499	59	8	[	[	X
ejpam-5499	59	9	v	v	NOUN
ejpam-5499	59	10	,	,	PUNCT
ejpam-5499	59	11	w	w	NOUN
ejpam-5499	59	12	]	]	X
ejpam-5499	59	13	are97	are97	VERB
ejpam-5499	59	14	distinct	distinct	ADJ
ejpam-5499	59	15	edges	edge	NOUN
ejpam-5499	59	16	in	in	ADP
ejpam-5499	59	17	g	g	NOUN
ejpam-5499	59	18	,	,	PUNCT
ejpam-5499	59	19	then	then	ADV
ejpam-5499	59	20	[	[	X
ejpam-5499	59	21	u	u	NOUN
ejpam-5499	59	22	,	,	PUNCT
ejpam-5499	59	23	v	v	NOUN
ejpam-5499	59	24	]	]	PUNCT
ejpam-5499	59	25	and	and	CCONJ
ejpam-5499	59	26	[	[	X
ejpam-5499	59	27	v	v	NOUN
ejpam-5499	59	28	,	,	PUNCT
ejpam-5499	59	29	w	w	NOUN
ejpam-5499	59	30	]	]	PUNCT
ejpam-5499	59	31	are	be	AUX
ejpam-5499	59	32	adjacent	adjacent	ADJ
ejpam-5499	59	33	edges	edge	NOUN
ejpam-5499	59	34	.	.	PUNCT
ejpam-5499	60	1	the	the	DET
ejpam-5499	60	2	vertex	vertex	NOUN
ejpam-5499	60	3	u	u	NOUN
ejpam-5499	60	4	and	and	CCONJ
ejpam-5499	60	5	the	the	DET
ejpam-5499	60	6	edge98	edge98	NOUN
ejpam-5499	60	7	[	[	X
ejpam-5499	60	8	u	u	NOUN
ejpam-5499	60	9	,	,	PUNCT
ejpam-5499	60	10	v	v	NOUN
ejpam-5499	60	11	]	]	PUNCT
ejpam-5499	60	12	are	be	AUX
ejpam-5499	60	13	said	say	VERB
ejpam-5499	60	14	to	to	PART
ejpam-5499	60	15	be	be	AUX
ejpam-5499	60	16	incident	incident	NOUN
ejpam-5499	60	17	with	with	ADP
ejpam-5499	60	18	each	each	DET
ejpam-5499	60	19	other.99	other.99	PROPN
ejpam-5499	60	20	a	a	DET
ejpam-5499	60	21	graph	graph	NOUN
ejpam-5499	60	22	h	h	NOUN
ejpam-5499	60	23	is	be	AUX
ejpam-5499	60	24	called	call	VERB
ejpam-5499	60	25	a	a	DET
ejpam-5499	60	26	subgraph	subgraph	NOUN
ejpam-5499	60	27	of	of	ADP
ejpam-5499	60	28	a	a	DET
ejpam-5499	60	29	graph	graph	NOUN
ejpam-5499	60	30	g	g	NOUN
ejpam-5499	60	31	,	,	PUNCT
ejpam-5499	60	32	written	write	VERB
ejpam-5499	60	33	as	as	ADP
ejpam-5499	60	34	h	h	NOUN
ejpam-5499	60	35	⊆	⊆	NUM
ejpam-5499	60	36	g	g	NOUN
ejpam-5499	60	37	,	,	PUNCT
ejpam-5499	60	38	if	if	SCONJ
ejpam-5499	60	39	v	v	X
ejpam-5499	60	40	(	(	PUNCT
ejpam-5499	60	41	h	h	NOUN
ejpam-5499	60	42	)	)	PUNCT
ejpam-5499	60	43	⊆	⊆	NUM
ejpam-5499	60	44	v	v	NOUN
ejpam-5499	60	45	(	(	PUNCT
ejpam-5499	60	46	g)100	g)100	PROPN
ejpam-5499	60	47	and	and	CCONJ
ejpam-5499	60	48	e(h	e(h	PROPN
ejpam-5499	60	49	)	)	PUNCT
ejpam-5499	60	50	⊆	⊆	NUM
ejpam-5499	60	51	e(g	e(g	PROPN
ejpam-5499	60	52	)	)	PUNCT
ejpam-5499	60	53	.	.	PUNCT
ejpam-5499	61	1	the	the	DET
ejpam-5499	61	2	degree	degree	NOUN
ejpam-5499	61	3	of	of	ADP
ejpam-5499	61	4	a	a	DET
ejpam-5499	61	5	vertex	vertex	NOUN
ejpam-5499	61	6	v	v	NOUN
ejpam-5499	61	7	in	in	ADP
ejpam-5499	61	8	a	a	DET
ejpam-5499	61	9	graph	graph	NOUN
ejpam-5499	61	10	g	g	NOUN
ejpam-5499	61	11	denoted	denote	VERB
ejpam-5499	61	12	by	by	ADP
ejpam-5499	61	13	degg(v	degg(v	PROPN
ejpam-5499	61	14	)	)	PUNCT
ejpam-5499	61	15	or101	or101	NOUN
ejpam-5499	61	16	simply	simply	ADV
ejpam-5499	61	17	by	by	ADP
ejpam-5499	61	18	deg(v	deg(v	PROPN
ejpam-5499	61	19	)	)	PUNCT
ejpam-5499	61	20	is	be	AUX
ejpam-5499	61	21	the	the	DET
ejpam-5499	61	22	number	number	NOUN
ejpam-5499	61	23	of	of	ADP
ejpam-5499	61	24	vertices	vertex	NOUN
ejpam-5499	61	25	in	in	ADP
ejpam-5499	61	26	g	g	PROPN
ejpam-5499	61	27	that	that	PRON
ejpam-5499	61	28	are	be	AUX
ejpam-5499	61	29	adjacent	adjacent	ADJ
ejpam-5499	61	30	to	to	ADP
ejpam-5499	61	31	v.	v.	ADP
ejpam-5499	61	32	a	a	DET
ejpam-5499	61	33	vertex	vertex	NOUN
ejpam-5499	61	34	of	of	ADP
ejpam-5499	61	35	degree102	degree102	PROPN
ejpam-5499	61	36	0	0	NUM
ejpam-5499	61	37	is	be	AUX
ejpam-5499	61	38	referred	refer	VERB
ejpam-5499	61	39	to	to	ADP
ejpam-5499	61	40	as	as	ADP
ejpam-5499	61	41	an	an	DET
ejpam-5499	61	42	isolated	isolated	ADJ
ejpam-5499	61	43	vertex	vertex	NOUN
ejpam-5499	61	44	and	and	CCONJ
ejpam-5499	61	45	a	a	DET
ejpam-5499	61	46	vertex	vertex	NOUN
ejpam-5499	61	47	of	of	ADP
ejpam-5499	61	48	degree	degree	NOUN
ejpam-5499	61	49	1	1	NUM
ejpam-5499	61	50	is	be	AUX
ejpam-5499	61	51	an	an	DET
ejpam-5499	61	52	end	end	NOUN
ejpam-5499	61	53	-	-	PUNCT
ejpam-5499	61	54	vertex	vertex	NOUN
ejpam-5499	61	55	or	or	CCONJ
ejpam-5499	61	56	a103	a103	PUNCT
ejpam-5499	61	57	leaf	leaf	NOUN
ejpam-5499	61	58	.	.	PUNCT
ejpam-5499	62	1	an	an	DET
ejpam-5499	62	2	edge	edge	NOUN
ejpam-5499	62	3	incident	incident	NOUN
ejpam-5499	62	4	with	with	ADP
ejpam-5499	62	5	an	an	DET
ejpam-5499	62	6	end	end	NOUN
ejpam-5499	62	7	-	-	PUNCT
ejpam-5499	62	8	vertex	vertex	NOUN
ejpam-5499	62	9	is	be	AUX
ejpam-5499	62	10	called	call	VERB
ejpam-5499	62	11	a	a	DET
ejpam-5499	62	12	pendant	pendant	ADJ
ejpam-5499	62	13	edge.104	edge.104	NOUN
ejpam-5499	62	14	the	the	DET
ejpam-5499	62	15	distance	distance	NOUN
ejpam-5499	62	16	d(u	d(u	PROPN
ejpam-5499	62	17	,	,	PUNCT
ejpam-5499	62	18	v	v	NOUN
ejpam-5499	62	19	)	)	PUNCT
ejpam-5499	62	20	between	between	ADP
ejpam-5499	62	21	u	u	PROPN
ejpam-5499	62	22	,	,	PUNCT
ejpam-5499	62	23	v	v	PROPN
ejpam-5499	62	24	∈	∈	PROPN
ejpam-5499	62	25	v	v	NOUN
ejpam-5499	62	26	(	(	PUNCT
ejpam-5499	62	27	g	g	NOUN
ejpam-5499	62	28	)	)	PUNCT
ejpam-5499	62	29	is	be	AUX
ejpam-5499	62	30	the	the	DET
ejpam-5499	62	31	length	length	NOUN
ejpam-5499	62	32	of	of	ADP
ejpam-5499	62	33	a	a	DET
ejpam-5499	62	34	shortest	short	ADJ
ejpam-5499	62	35	u	u	NOUN
ejpam-5499	62	36	−	−	PROPN
ejpam-5499	62	37	v	v	ADP
ejpam-5499	62	38	path	path	NOUN
ejpam-5499	62	39	in105	in105	PROPN
ejpam-5499	62	40	the	the	DET
ejpam-5499	62	41	graph	graph	NOUN
ejpam-5499	62	42	g.	g.	VERB
ejpam-5499	62	43	the	the	DET
ejpam-5499	62	44	eccentricity	eccentricity	NOUN
ejpam-5499	62	45	of	of	ADP
ejpam-5499	62	46	a	a	DET
ejpam-5499	62	47	vertex	vertex	NOUN
ejpam-5499	62	48	u	u	NOUN
ejpam-5499	62	49	∈	∈	PROPN
ejpam-5499	62	50	v	v	ADP
ejpam-5499	62	51	(	(	PUNCT
ejpam-5499	62	52	g	g	NOUN
ejpam-5499	62	53	)	)	PUNCT
ejpam-5499	62	54	is	be	AUX
ejpam-5499	62	55	e(u	e(u	PROPN
ejpam-5499	62	56	)	)	PUNCT
ejpam-5499	63	1	=	=	SYM
ejpam-5499	63	2	max{d(u	max{d(u	PROPN
ejpam-5499	63	3	,	,	PUNCT
ejpam-5499	63	4	v)|u	v)|u	NOUN
ejpam-5499	63	5	∈	∈	PROPN
ejpam-5499	63	6	v	v	NOUN
ejpam-5499	63	7	(	(	PUNCT
ejpam-5499	63	8	g)}.106	g)}.106	X
ejpam-5499	63	9	the	the	DET
ejpam-5499	63	10	diameter	diameter	NOUN
ejpam-5499	63	11	of	of	ADP
ejpam-5499	63	12	a	a	DET
ejpam-5499	63	13	graph	graph	NOUN
ejpam-5499	63	14	g	g	PROPN
ejpam-5499	63	15	is	be	AUX
ejpam-5499	63	16	diam	diam	PROPN
ejpam-5499	63	17	=	=	PUNCT
ejpam-5499	63	18	max{e(u)|u	max{e(u)|u	PROPN
ejpam-5499	63	19	∈	∈	PROPN
ejpam-5499	63	20	v	v	NOUN
ejpam-5499	63	21	(	(	PUNCT
ejpam-5499	63	22	g	g	NOUN
ejpam-5499	63	23	)	)	PUNCT
ejpam-5499	63	24	}	}	PUNCT
ejpam-5499	63	25	.	.	PUNCT
ejpam-5499	64	1	the	the	DET
ejpam-5499	64	2	radius	radius	NOUN
ejpam-5499	64	3	of	of	ADP
ejpam-5499	64	4	a	a	DET
ejpam-5499	64	5	graph	graph	NOUN
ejpam-5499	64	6	g	g	ADP
ejpam-5499	64	7	is107	is107	PROPN
ejpam-5499	64	8	rad	rad	NOUN
ejpam-5499	64	9	=	=	SYM
ejpam-5499	64	10	min{e(u)|u	min{e(u)|u	PROPN
ejpam-5499	64	11	∈	∈	PROPN
ejpam-5499	64	12	v	v	NOUN
ejpam-5499	64	13	(	(	PUNCT
ejpam-5499	64	14	g	g	NOUN
ejpam-5499	64	15	)	)	PUNCT
ejpam-5499	64	16	}	}	PUNCT
ejpam-5499	64	17	.	.	PUNCT
ejpam-5499	65	1	if	if	SCONJ
ejpam-5499	65	2	e(u	e(u	PROPN
ejpam-5499	65	3	)	)	PUNCT
ejpam-5499	65	4	=	=	SYM
ejpam-5499	65	5	rad(g	rad(g	PROPN
ejpam-5499	65	6	)	)	PUNCT
ejpam-5499	65	7	,	,	PUNCT
ejpam-5499	65	8	the	the	DET
ejpam-5499	65	9	vertex	vertex	NOUN
ejpam-5499	65	10	u	u	NOUN
ejpam-5499	65	11	is	be	AUX
ejpam-5499	65	12	a	a	DET
ejpam-5499	65	13	central	central	ADJ
ejpam-5499	65	14	vertex	vertex	NOUN
ejpam-5499	65	15	.	.	PUNCT
ejpam-5499	66	1	the	the	DET
ejpam-5499	66	2	set108	set108	NOUN
ejpam-5499	66	3	of	of	ADP
ejpam-5499	66	4	all	all	DET
ejpam-5499	66	5	such	such	ADJ
ejpam-5499	66	6	vertices	vertex	NOUN
ejpam-5499	66	7	is	be	AUX
ejpam-5499	66	8	the	the	DET
ejpam-5499	66	9	center	center	NOUN
ejpam-5499	66	10	of	of	ADP
ejpam-5499	66	11	g.	g.	PROPN
ejpam-5499	66	12	the	the	DET
ejpam-5499	66	13	girth	girth	NOUN
ejpam-5499	66	14	of	of	ADP
ejpam-5499	66	15	a	a	DET
ejpam-5499	66	16	graph	graph	NOUN
ejpam-5499	66	17	g	g	NOUN
ejpam-5499	66	18	denoted	denote	VERB
ejpam-5499	66	19	by	by	ADP
ejpam-5499	66	20	gir(g	gir(g	PROPN
ejpam-5499	66	21	)	)	PUNCT
ejpam-5499	66	22	is	be	AUX
ejpam-5499	66	23	the109	the109	PROPN
ejpam-5499	66	24	length	length	NOUN
ejpam-5499	66	25	of	of	ADP
ejpam-5499	66	26	the	the	DET
ejpam-5499	66	27	shortest	short	ADJ
ejpam-5499	66	28	cycle	cycle	NOUN
ejpam-5499	66	29	(	(	PUNCT
ejpam-5499	66	30	if	if	SCONJ
ejpam-5499	66	31	any	any	PRON
ejpam-5499	66	32	)	)	PUNCT
ejpam-5499	66	33	in	in	ADP
ejpam-5499	66	34	g.110	g.110	ADJ
ejpam-5499	66	35	definition	definition	NOUN
ejpam-5499	66	36	5	5	NUM
ejpam-5499	66	37	.	.	PUNCT
ejpam-5499	67	1	given	give	VERB
ejpam-5499	67	2	a	a	DET
ejpam-5499	67	3	group	group	NOUN
ejpam-5499	67	4	g	g	NOUN
ejpam-5499	67	5	with	with	ADP
ejpam-5499	67	6	e	e	PROPN
ejpam-5499	67	7	as	as	ADP
ejpam-5499	67	8	the	the	DET
ejpam-5499	67	9	identity	identity	NOUN
ejpam-5499	67	10	element	element	NOUN
ejpam-5499	67	11	,	,	PUNCT
ejpam-5499	67	12	define	define	VERB
ejpam-5499	67	13	the	the	DET
ejpam-5499	67	14	identity	identity	NOUN
ejpam-5499	67	15	graph111	graph111	PROPN
ejpam-5499	67	16	γg	γg	PROPN
ejpam-5499	67	17	=	=	SYM
ejpam-5499	67	18	γ(g	γ(g	PROPN
ejpam-5499	67	19	,	,	PUNCT
ejpam-5499	67	20	e	e	NOUN
ejpam-5499	67	21	)	)	PUNCT
ejpam-5499	67	22	to	to	PART
ejpam-5499	67	23	have	have	VERB
ejpam-5499	67	24	the	the	DET
ejpam-5499	67	25	vertex	vertex	NOUN
ejpam-5499	67	26	-	-	PUNCT
ejpam-5499	67	27	set	set	VERB
ejpam-5499	67	28	g	g	NOUN
ejpam-5499	67	29	and	and	CCONJ
ejpam-5499	67	30	the	the	DET
ejpam-5499	67	31	edge	edge	NOUN
ejpam-5499	67	32	-	-	PUNCT
ejpam-5499	67	33	set	set	VERB
ejpam-5499	67	34	e	e	NOUN
ejpam-5499	67	35	satisfying	satisfy	VERB
ejpam-5499	67	36	two	two	NUM
ejpam-5499	67	37	conditions:112	conditions:112	X
ejpam-5499	67	38	(	(	PUNCT
ejpam-5499	67	39	i	i	NOUN
ejpam-5499	67	40	)	)	PUNCT
ejpam-5499	67	41	for	for	ADP
ejpam-5499	67	42	every	every	DET
ejpam-5499	67	43	x	x	PROPN
ejpam-5499	67	44	,	,	PUNCT
ejpam-5499	67	45	y	y	PROPN
ejpam-5499	67	46	∈	∈	PROPN
ejpam-5499	67	47	g	g	NOUN
ejpam-5499	67	48	where	where	SCONJ
ejpam-5499	67	49	x	x	ADJ
ejpam-5499	67	50	̸=	̸=	PROPN
ejpam-5499	67	51	y	y	PROPN
ejpam-5499	67	52	,	,	PUNCT
ejpam-5499	67	53	x	x	X
ejpam-5499	67	54	and	and	CCONJ
ejpam-5499	67	55	y	y	PROPN
ejpam-5499	67	56	are	be	AUX
ejpam-5499	67	57	adjacent	adjacent	ADJ
ejpam-5499	67	58	in	in	ADP
ejpam-5499	67	59	γg	γg	ADV
ejpam-5499	67	60	if	if	SCONJ
ejpam-5499	67	61	and	and	CCONJ
ejpam-5499	67	62	only	only	ADV
ejpam-5499	67	63	if	if	SCONJ
ejpam-5499	67	64	xy	xy	PROPN
ejpam-5499	67	65	=	=	SYM
ejpam-5499	67	66	e	e	PROPN
ejpam-5499	67	67	;	;	PUNCT
ejpam-5499	67	68	113	113	NUM
ejpam-5499	67	69	(	(	PUNCT
ejpam-5499	67	70	ii	ii	NOUN
ejpam-5499	67	71	)	)	PUNCT
ejpam-5499	67	72	for	for	ADP
ejpam-5499	67	73	each	each	DET
ejpam-5499	67	74	x	x	SYM
ejpam-5499	67	75	∈	∈	PROPN
ejpam-5499	67	76	g	g	PROPN
ejpam-5499	67	77	,	,	PUNCT
ejpam-5499	67	78	x	x	PUNCT
ejpam-5499	67	79	and	and	CCONJ
ejpam-5499	67	80	e	e	NOUN
ejpam-5499	67	81	are	be	AUX
ejpam-5499	67	82	adjacent	adjacent	ADJ
ejpam-5499	67	83	in	in	ADP
ejpam-5499	67	84	γg	γg	PROPN
ejpam-5499	67	85	.114	.114	NUM
ejpam-5499	67	86	two	two	NUM
ejpam-5499	67	87	important	important	ADJ
ejpam-5499	67	88	structures	structure	NOUN
ejpam-5499	67	89	in	in	ADP
ejpam-5499	67	90	identity	identity	NOUN
ejpam-5499	67	91	graph	graph	NOUN
ejpam-5499	67	92	are	be	AUX
ejpam-5499	67	93	lines	line	NOUN
ejpam-5499	67	94	and	and	CCONJ
ejpam-5499	67	95	triangles	triangle	NOUN
ejpam-5499	67	96	defines	define	NOUN
ejpam-5499	67	97	as	as	ADP
ejpam-5499	67	98	follows:115	follows:115	PROPN
ejpam-5499	67	99	definition	definition	NOUN
ejpam-5499	67	100	6	6	NUM
ejpam-5499	67	101	.	.	PUNCT
ejpam-5499	68	1	given	give	VERB
ejpam-5499	68	2	a	a	DET
ejpam-5499	68	3	group	group	NOUN
ejpam-5499	68	4	g	g	NOUN
ejpam-5499	68	5	,	,	PUNCT
ejpam-5499	68	6	a	a	DET
ejpam-5499	68	7	line	line	NOUN
ejpam-5499	68	8	in	in	ADP
ejpam-5499	68	9	the	the	DET
ejpam-5499	68	10	identity	identity	NOUN
ejpam-5499	68	11	graph	graph	NOUN
ejpam-5499	68	12	γg	γg	ADV
ejpam-5499	68	13	is	be	AUX
ejpam-5499	68	14	an	an	DET
ejpam-5499	68	15	edge	edge	NOUN
ejpam-5499	68	16	[	[	X
ejpam-5499	68	17	x	x	X
ejpam-5499	68	18	,	,	PUNCT
ejpam-5499	68	19	e	e	X
ejpam-5499	68	20	]	]	X
ejpam-5499	68	21	such	such	ADJ
ejpam-5499	68	22	that116	that116	PROPN
ejpam-5499	68	23	the	the	DET
ejpam-5499	68	24	degree	degree	NOUN
ejpam-5499	68	25	of	of	ADP
ejpam-5499	68	26	a	a	DET
ejpam-5499	68	27	vertex	vertex	NOUN
ejpam-5499	68	28	x	x	PRON
ejpam-5499	68	29	is	be	AUX
ejpam-5499	68	30	one	one	NUM
ejpam-5499	68	31	.	.	PUNCT
ejpam-5499	69	1	the	the	DET
ejpam-5499	69	2	number	number	NOUN
ejpam-5499	69	3	of	of	ADP
ejpam-5499	69	4	lines	line	NOUN
ejpam-5499	69	5	in	in	ADP
ejpam-5499	69	6	the	the	DET
ejpam-5499	69	7	identity	identity	NOUN
ejpam-5499	69	8	graph	graph	NOUN
ejpam-5499	69	9	γg	γg	ADV
ejpam-5499	69	10	is	be	AUX
ejpam-5499	69	11	denoted117	denoted117	PROPN
ejpam-5499	69	12	by	by	ADP
ejpam-5499	69	13	line(g).118	line(g).118	PROPN
ejpam-5499	69	14	definition	definition	NOUN
ejpam-5499	69	15	7	7	NUM
ejpam-5499	69	16	.	.	PUNCT
ejpam-5499	70	1	a	a	DET
ejpam-5499	70	2	triangle	triangle	NOUN
ejpam-5499	70	3	in	in	ADP
ejpam-5499	70	4	the	the	DET
ejpam-5499	70	5	identity	identity	NOUN
ejpam-5499	70	6	graph	graph	NOUN
ejpam-5499	70	7	γg	γg	ADV
ejpam-5499	70	8	is	be	AUX
ejpam-5499	70	9	a	a	DET
ejpam-5499	70	10	subgraph	subgraph	NOUN
ejpam-5499	70	11	which	which	PRON
ejpam-5499	70	12	is	be	AUX
ejpam-5499	70	13	isomorphic	isomorphic	ADJ
ejpam-5499	70	14	to119	to119	PROPN
ejpam-5499	70	15	the	the	DET
ejpam-5499	70	16	cycle	cycle	NOUN
ejpam-5499	70	17	of	of	ADP
ejpam-5499	70	18	length	length	NOUN
ejpam-5499	70	19	three	three	NUM
ejpam-5499	70	20	.	.	PUNCT
ejpam-5499	71	1	the	the	DET
ejpam-5499	71	2	number	number	NOUN
ejpam-5499	71	3	of	of	ADP
ejpam-5499	71	4	the	the	DET
ejpam-5499	71	5	triangles	triangle	NOUN
ejpam-5499	71	6	in	in	ADP
ejpam-5499	71	7	the	the	DET
ejpam-5499	71	8	identity	identity	NOUN
ejpam-5499	71	9	graph	graph	NOUN
ejpam-5499	71	10	is	be	AUX
ejpam-5499	71	11	denoted	denote	VERB
ejpam-5499	71	12	by120	by120	PROPN
ejpam-5499	71	13	tri(g).121	tri(g).121	VERB
ejpam-5499	71	14	to	to	PART
ejpam-5499	71	15	give	give	VERB
ejpam-5499	71	16	an	an	DET
ejpam-5499	71	17	example	example	NOUN
ejpam-5499	71	18	,	,	PUNCT
ejpam-5499	71	19	consider	consider	VERB
ejpam-5499	71	20	the	the	DET
ejpam-5499	71	21	identity	identity	NOUN
ejpam-5499	71	22	graphs	graph	NOUN
ejpam-5499	71	23	of	of	ADP
ejpam-5499	71	24	selected	select	VERB
ejpam-5499	71	25	cyclic	cyclic	ADJ
ejpam-5499	71	26	groups	group	NOUN
ejpam-5499	71	27	below.122	below.122	NOUN
ejpam-5499	71	28	123	123	NUM
ejpam-5499	71	29	to	to	PART
ejpam-5499	71	30	further	far	ADV
ejpam-5499	71	31	understand	understand	VERB
ejpam-5499	71	32	the	the	DET
ejpam-5499	71	33	properties	property	NOUN
ejpam-5499	71	34	of	of	ADP
ejpam-5499	71	35	groups	group	NOUN
ejpam-5499	71	36	and	and	CCONJ
ejpam-5499	71	37	graphs	graph	NOUN
ejpam-5499	71	38	,	,	PUNCT
ejpam-5499	71	39	here	here	ADV
ejpam-5499	71	40	are	be	AUX
ejpam-5499	71	41	some	some	DET
ejpam-5499	71	42	propositions.124	propositions.124	ADJ
ejpam-5499	71	43	propositions	proposition	NOUN
ejpam-5499	71	44	are	be	AUX
ejpam-5499	71	45	presented	present	VERB
ejpam-5499	71	46	without	without	ADP
ejpam-5499	71	47	proof	proof	NOUN
ejpam-5499	71	48	and	and	CCONJ
ejpam-5499	71	49	can	can	AUX
ejpam-5499	71	50	be	be	AUX
ejpam-5499	71	51	found	find	VERB
ejpam-5499	71	52	in	in	ADP
ejpam-5499	71	53	[	[	X
ejpam-5499	71	54	5	5	NUM
ejpam-5499	71	55	]	]	PUNCT
ejpam-5499	71	56	,	,	PUNCT
ejpam-5499	71	57	[	[	X
ejpam-5499	71	58	8	8	NUM
ejpam-5499	71	59	]	]	PUNCT
ejpam-5499	71	60	,	,	PUNCT
ejpam-5499	71	61	[	[	X
ejpam-5499	71	62	1	1	NUM
ejpam-5499	71	63	]	]	PUNCT
ejpam-5499	71	64	,	,	PUNCT
ejpam-5499	71	65	[	[	X
ejpam-5499	71	66	9	9	NUM
ejpam-5499	71	67	]	]	PUNCT
ejpam-5499	71	68	and	and	CCONJ
ejpam-5499	71	69	[	[	X
ejpam-5499	71	70	3].125	3].125	NUM
ejpam-5499	71	71	j.	j.	PROPN
ejpam-5499	71	72	m.	m.	PROPN
ejpam-5499	71	73	jamis	jamis	PROPN
ejpam-5499	71	74	,	,	PUNCT
ejpam-5499	71	75	d.	d.	PROPN
ejpam-5499	71	76	m.	m.	PROPN
ejpam-5499	71	77	magpantay	magpantay	PROPN
ejpam-5499	71	78	/	/	SYM
ejpam-5499	71	79	eur	eur	PROPN
ejpam-5499	71	80	.	.	PUNCT
ejpam-5499	72	1	j.	j.	PROPN
ejpam-5499	72	2	pure	pure	PROPN
ejpam-5499	72	3	appl	appl	PROPN
ejpam-5499	72	4	.	.	PROPN
ejpam-5499	72	5	math	math	PROPN
ejpam-5499	72	6	,	,	PUNCT
ejpam-5499	72	7	18	18	NUM
ejpam-5499	72	8	(	(	PUNCT
ejpam-5499	72	9	2	2	NUM
ejpam-5499	72	10	)	)	PUNCT
ejpam-5499	72	11	(	(	PUNCT
ejpam-5499	72	12	2025	2025	NUM
ejpam-5499	72	13	)	)	PUNCT
ejpam-5499	72	14	,	,	PUNCT
ejpam-5499	72	15	5499	5499	NUM
ejpam-5499	72	16	5	5	NUM
ejpam-5499	72	17	of	of	ADP
ejpam-5499	72	18	18	18	NUM
ejpam-5499	72	19	proposition	proposition	NOUN
ejpam-5499	72	20	1	1	NUM
ejpam-5499	72	21	.	.	PUNCT
ejpam-5499	73	1	for	for	ADP
ejpam-5499	73	2	a	a	DET
ejpam-5499	73	3	cyclic	cyclic	ADJ
ejpam-5499	73	4	groups	group	NOUN
ejpam-5499	73	5	cn	cn	VERB
ejpam-5499	73	6	of	of	ADP
ejpam-5499	73	7	order	order	NOUN
ejpam-5499	73	8	n	n	CCONJ
ejpam-5499	73	9	,	,	PUNCT
ejpam-5499	73	10	if	if	SCONJ
ejpam-5499	73	11	n	n	PRON
ejpam-5499	73	12	is	be	AUX
ejpam-5499	73	13	odd	odd	ADJ
ejpam-5499	73	14	,	,	PUNCT
ejpam-5499	73	15	then	then	ADV
ejpam-5499	73	16	line(cn	line(cn	NOUN
ejpam-5499	73	17	)	)	PUNCT
ejpam-5499	73	18	=	=	SYM
ejpam-5499	74	1	0	0	NUM
ejpam-5499	74	2	and126	and126	PROPN
ejpam-5499	74	3	tri(cn	tri(cn	NOUN
ejpam-5499	74	4	)	)	PUNCT
ejpam-5499	75	1	=	=	SYM
ejpam-5499	75	2	n−1	n−1	PROPN
ejpam-5499	75	3	2	2	NUM
ejpam-5499	75	4	.	.	PUNCT
ejpam-5499	76	1	if	if	SCONJ
ejpam-5499	76	2	n	n	PRON
ejpam-5499	76	3	is	be	AUX
ejpam-5499	76	4	even	even	ADV
ejpam-5499	76	5	,	,	PUNCT
ejpam-5499	76	6	then	then	ADV
ejpam-5499	76	7	line(cn	line(cn	VERB
ejpam-5499	76	8	)	)	PUNCT
ejpam-5499	76	9	=	=	SYM
ejpam-5499	76	10	1	1	NUM
ejpam-5499	76	11	and	and	CCONJ
ejpam-5499	76	12	tri(cn	tri(cn	NUM
ejpam-5499	76	13	)	)	PUNCT
ejpam-5499	77	1	=	=	SYM
ejpam-5499	77	2	n−2	n−2	PROPN
ejpam-5499	77	3	2	2	NUM
ejpam-5499	77	4	.127	.127	NUM
ejpam-5499	77	5	proposition	proposition	NOUN
ejpam-5499	77	6	2	2	NUM
ejpam-5499	77	7	.	.	PUNCT
ejpam-5499	78	1	if	if	SCONJ
ejpam-5499	78	2	cn	cn	PROPN
ejpam-5499	78	3	is	be	AUX
ejpam-5499	78	4	a	a	DET
ejpam-5499	78	5	cyclic	cyclic	ADJ
ejpam-5499	78	6	group	group	NOUN
ejpam-5499	78	7	of	of	ADP
ejpam-5499	78	8	order	order	NOUN
ejpam-5499	78	9	n	n	CCONJ
ejpam-5499	78	10	,	,	PUNCT
ejpam-5499	78	11	then	then	ADV
ejpam-5499	78	12	we	we	PRON
ejpam-5499	78	13	have128	have128	PROPN
ejpam-5499	78	14	|e(γcn)|	|e(γcn)|	NOUN
ejpam-5499	78	15	=	=	PUNCT
ejpam-5499	78	16	{	{	PUNCT
ejpam-5499	78	17	3(n−1	3(n−1	PROPN
ejpam-5499	78	18	)	)	PUNCT
ejpam-5499	78	19	2	2	NUM
ejpam-5499	78	20	,	,	PUNCT
ejpam-5499	78	21	if	if	SCONJ
ejpam-5499	78	22	n	n	PRON
ejpam-5499	78	23	is	be	AUX
ejpam-5499	78	24	odd	odd	ADJ
ejpam-5499	78	25	3(n−2	3(n−2	ADJ
ejpam-5499	78	26	)	)	PUNCT
ejpam-5499	78	27	2	2	NUM
ejpam-5499	79	1	+	+	NUM
ejpam-5499	79	2	1	1	NUM
ejpam-5499	79	3	,	,	PUNCT
ejpam-5499	79	4	if	if	SCONJ
ejpam-5499	79	5	n	n	PRON
ejpam-5499	79	6	is	be	AUX
ejpam-5499	79	7	even	even	ADV
ejpam-5499	79	8	.	.	PUNCT
ejpam-5499	80	1	129	129	NUM
ejpam-5499	80	2	proposition	proposition	NOUN
ejpam-5499	80	3	3	3	NUM
ejpam-5499	80	4	.	.	PUNCT
ejpam-5499	80	5	(	(	PUNCT
ejpam-5499	80	6	handshaking	handshake	VERB
ejpam-5499	80	7	lemma	lemma	PROPN
ejpam-5499	80	8	)	)	PUNCT
ejpam-5499	80	9	if	if	SCONJ
ejpam-5499	80	10	g	g	PROPN
ejpam-5499	80	11	is	be	AUX
ejpam-5499	80	12	a	a	DET
ejpam-5499	80	13	graph	graph	NOUN
ejpam-5499	80	14	of	of	ADP
ejpam-5499	80	15	size	size	NOUN
ejpam-5499	80	16	m	m	PROPN
ejpam-5499	80	17	,	,	PUNCT
ejpam-5499	80	18	then130	then130	PROPN
ejpam-5499	80	19	∑	∑	PUNCT
ejpam-5499	80	20	v∈v	v∈v	PROPN
ejpam-5499	80	21	(	(	PUNCT
ejpam-5499	80	22	g	g	NOUN
ejpam-5499	80	23	)	)	PUNCT
ejpam-5499	80	24	deg(v	deg(v	PROPN
ejpam-5499	80	25	)	)	PUNCT
ejpam-5499	80	26	=	=	PUNCT
ejpam-5499	81	1	2m.131	2m.131	NUM
ejpam-5499	81	2	proposition	proposition	NOUN
ejpam-5499	81	3	4	4	NUM
ejpam-5499	81	4	.	.	PUNCT
ejpam-5499	82	1	a	a	DET
ejpam-5499	82	2	nontrivial	nontrivial	ADJ
ejpam-5499	82	3	connected	connect	VERB
ejpam-5499	82	4	graph	graph	NOUN
ejpam-5499	82	5	g	g	PROPN
ejpam-5499	82	6	is	be	AUX
ejpam-5499	82	7	eulerian	eulerian	ADJ
ejpam-5499	82	8	if	if	SCONJ
ejpam-5499	83	1	and	and	CCONJ
ejpam-5499	83	2	only	only	ADV
ejpam-5499	83	3	if	if	SCONJ
ejpam-5499	83	4	every	every	DET
ejpam-5499	83	5	vertex132	vertex132	PROPN
ejpam-5499	83	6	of	of	ADP
ejpam-5499	83	7	g	g	PROPN
ejpam-5499	83	8	has	have	AUX
ejpam-5499	83	9	even	even	ADV
ejpam-5499	83	10	degree.133	degree.133	VERB
ejpam-5499	83	11	at	at	ADP
ejpam-5499	83	12	this	this	DET
ejpam-5499	83	13	point	point	NOUN
ejpam-5499	83	14	,	,	PUNCT
ejpam-5499	83	15	we	we	PRON
ejpam-5499	83	16	will	will	AUX
ejpam-5499	83	17	formally	formally	ADV
ejpam-5499	83	18	define	define	VERB
ejpam-5499	83	19	the	the	DET
ejpam-5499	83	20	main	main	ADJ
ejpam-5499	83	21	focus	focus	NOUN
ejpam-5499	83	22	of	of	ADP
ejpam-5499	83	23	the	the	DET
ejpam-5499	83	24	study	study	NOUN
ejpam-5499	83	25	,	,	PUNCT
ejpam-5499	83	26	the	the	DET
ejpam-5499	83	27	graph	graph	NOUN
ejpam-5499	83	28	operation134	operation134	PROPN
ejpam-5499	83	29	middle	middle	ADJ
ejpam-5499	83	30	graph	graph	NOUN
ejpam-5499	83	31	of	of	ADP
ejpam-5499	83	32	a	a	DET
ejpam-5499	83	33	graph.135	graph.135	NOUN
ejpam-5499	83	34	definition	definition	NOUN
ejpam-5499	83	35	8	8	NUM
ejpam-5499	83	36	.	.	PUNCT
ejpam-5499	84	1	let	let	VERB
ejpam-5499	84	2	g	g	PROPN
ejpam-5499	84	3	=	=	SYM
ejpam-5499	84	4	(	(	PUNCT
ejpam-5499	84	5	v	v	NOUN
ejpam-5499	84	6	(	(	PUNCT
ejpam-5499	84	7	g	g	NOUN
ejpam-5499	84	8	)	)	PUNCT
ejpam-5499	84	9	,	,	PUNCT
ejpam-5499	84	10	e(g	e(g	PROPN
ejpam-5499	84	11	)	)	PUNCT
ejpam-5499	84	12	)	)	PUNCT
ejpam-5499	85	1	be	be	AUX
ejpam-5499	85	2	a	a	DET
ejpam-5499	85	3	simple	simple	ADJ
ejpam-5499	85	4	graph	graph	NOUN
ejpam-5499	85	5	.	.	PUNCT
ejpam-5499	86	1	the	the	DET
ejpam-5499	86	2	middle	middle	ADJ
ejpam-5499	86	3	graph	graph	NOUN
ejpam-5499	86	4	of	of	ADP
ejpam-5499	86	5	g	g	PROPN
ejpam-5499	86	6	denoted136	denoted136	PROPN
ejpam-5499	86	7	by	by	ADP
ejpam-5499	86	8	m(g	m(g	PROPN
ejpam-5499	86	9	)	)	PUNCT
ejpam-5499	86	10	is	be	AUX
ejpam-5499	86	11	the	the	DET
ejpam-5499	86	12	graph	graph	NOUN
ejpam-5499	86	13	whose	whose	DET
ejpam-5499	86	14	vertex	vertex	NOUN
ejpam-5499	86	15	set	set	NOUN
ejpam-5499	86	16	is	be	AUX
ejpam-5499	86	17	v	v	NOUN
ejpam-5499	86	18	(	(	PUNCT
ejpam-5499	86	19	g	g	NOUN
ejpam-5499	86	20	)	)	PUNCT
ejpam-5499	86	21	∪	∪	ADP
ejpam-5499	86	22	e(g	e(g	PROPN
ejpam-5499	86	23	)	)	PUNCT
ejpam-5499	86	24	where	where	SCONJ
ejpam-5499	86	25	two	two	NUM
ejpam-5499	86	26	vertices	vertex	NOUN
ejpam-5499	86	27	are	be	AUX
ejpam-5499	86	28	adjacent137	adjacent137	PROPN
ejpam-5499	86	29	if	if	SCONJ
ejpam-5499	86	30	(	(	PUNCT
ejpam-5499	86	31	1	1	X
ejpam-5499	86	32	)	)	PUNCT
ejpam-5499	86	33	they	they	PRON
ejpam-5499	86	34	are	be	AUX
ejpam-5499	86	35	either	either	CCONJ
ejpam-5499	86	36	adjacent	adjacent	ADJ
ejpam-5499	86	37	edges	edge	NOUN
ejpam-5499	86	38	of	of	ADP
ejpam-5499	86	39	g	g	NOUN
ejpam-5499	86	40	or	or	CCONJ
ejpam-5499	86	41	(	(	PUNCT
ejpam-5499	86	42	2	2	X
ejpam-5499	86	43	)	)	PUNCT
ejpam-5499	86	44	one	one	NOUN
ejpam-5499	86	45	is	be	AUX
ejpam-5499	86	46	a	a	DET
ejpam-5499	86	47	vertex	vertex	NOUN
ejpam-5499	86	48	and	and	CCONJ
ejpam-5499	86	49	the	the	DET
ejpam-5499	86	50	other	other	ADJ
ejpam-5499	86	51	is	be	AUX
ejpam-5499	86	52	an	an	DET
ejpam-5499	86	53	edge138	edge138	PROPN
ejpam-5499	86	54	incident	incident	NOUN
ejpam-5499	86	55	to	to	ADP
ejpam-5499	86	56	it.139	it.139	PROPN
ejpam-5499	86	57	for	for	ADP
ejpam-5499	86	58	example,140	example,140	PROPN
ejpam-5499	86	59	example	example	NOUN
ejpam-5499	86	60	3	3	X
ejpam-5499	86	61	.	.	X
ejpam-5499	86	62	consider	consider	VERB
ejpam-5499	86	63	the	the	DET
ejpam-5499	86	64	graphs	graph	NOUN
ejpam-5499	86	65	c4	c4	NOUN
ejpam-5499	86	66	.	.	PUNCT
ejpam-5499	87	1	the	the	DET
ejpam-5499	87	2	middle	middle	ADJ
ejpam-5499	87	3	graph	graph	NOUN
ejpam-5499	87	4	of	of	ADP
ejpam-5499	87	5	c4	c4	NOUN
ejpam-5499	87	6	,	,	PUNCT
ejpam-5499	87	7	m(c4	m(c4	NOUN
ejpam-5499	87	8	)	)	PUNCT
ejpam-5499	87	9	,	,	PUNCT
ejpam-5499	87	10	is	be	AUX
ejpam-5499	87	11	given	give	VERB
ejpam-5499	87	12	by141	by141	NOUN
ejpam-5499	87	13	.142	.142	NUM
ejpam-5499	87	14	the	the	DET
ejpam-5499	87	15	middle	middle	ADJ
ejpam-5499	87	16	graph	graph	NOUN
ejpam-5499	87	17	m(c4	m(c4	NOUN
ejpam-5499	87	18	)	)	PUNCT
ejpam-5499	87	19	of	of	ADP
ejpam-5499	87	20	c4	c4	NOUN
ejpam-5499	87	21	is	be	AUX
ejpam-5499	87	22	a	a	DET
ejpam-5499	87	23	graph	graph	NOUN
ejpam-5499	87	24	with	with	ADP
ejpam-5499	87	25	v	v	NOUN
ejpam-5499	87	26	(	(	PUNCT
ejpam-5499	87	27	m(c4	m(c4	NOUN
ejpam-5499	87	28	)	)	PUNCT
ejpam-5499	87	29	)	)	PUNCT
ejpam-5499	88	1	=	=	PRON
ejpam-5499	88	2	{	{	PUNCT
ejpam-5499	88	3	a	a	PRON
ejpam-5499	88	4	,	,	PUNCT
ejpam-5499	88	5	b	b	NOUN
ejpam-5499	88	6	,	,	PUNCT
ejpam-5499	88	7	c	c	NOUN
ejpam-5499	88	8	,	,	PUNCT
ejpam-5499	88	9	d	d	NOUN
ejpam-5499	88	10	,	,	PUNCT
ejpam-5499	88	11	1	1	NUM
ejpam-5499	88	12	,	,	PUNCT
ejpam-5499	88	13	2	2	NUM
ejpam-5499	88	14	,	,	PUNCT
ejpam-5499	88	15	3	3	NUM
ejpam-5499	88	16	,	,	PUNCT
ejpam-5499	88	17	4	4	NUM
ejpam-5499	88	18	}	}	PUNCT
ejpam-5499	88	19	and143	and143	PROPN
ejpam-5499	88	20	two	two	NUM
ejpam-5499	88	21	vertices	vertex	NOUN
ejpam-5499	88	22	is	be	AUX
ejpam-5499	88	23	adjacent	adjacent	ADJ
ejpam-5499	88	24	if	if	SCONJ
ejpam-5499	88	25	and	and	CCONJ
ejpam-5499	88	26	only	only	ADV
ejpam-5499	88	27	if	if	SCONJ
ejpam-5499	88	28	they	they	PRON
ejpam-5499	88	29	are	be	AUX
ejpam-5499	88	30	adjacent	adjacent	ADJ
ejpam-5499	88	31	edges	edge	NOUN
ejpam-5499	88	32	of	of	ADP
ejpam-5499	88	33	c4	c4	NOUN
ejpam-5499	88	34	or	or	CCONJ
ejpam-5499	88	35	one	one	NUM
ejpam-5499	88	36	is	be	AUX
ejpam-5499	88	37	a	a	DET
ejpam-5499	88	38	vertex	vertex	NOUN
ejpam-5499	88	39	and144	and144	PROPN
ejpam-5499	88	40	the	the	DET
ejpam-5499	88	41	other	other	ADJ
ejpam-5499	88	42	is	be	AUX
ejpam-5499	88	43	an	an	DET
ejpam-5499	88	44	edge	edge	NOUN
ejpam-5499	88	45	incident	incident	NOUN
ejpam-5499	88	46	to	to	ADP
ejpam-5499	88	47	it	it	PRON
ejpam-5499	88	48	.	.	PUNCT
ejpam-5499	89	1	for	for	ADP
ejpam-5499	89	2	instance	instance	NOUN
ejpam-5499	89	3	the	the	DET
ejpam-5499	89	4	vertices	vertex	NOUN
ejpam-5499	89	5	4	4	NUM
ejpam-5499	89	6	and	and	CCONJ
ejpam-5499	89	7	1	1	NUM
ejpam-5499	89	8	in	in	ADP
ejpam-5499	89	9	m(c4	m(c4	NOUN
ejpam-5499	89	10	)	)	PUNCT
ejpam-5499	89	11	are	be	AUX
ejpam-5499	89	12	adjacent145	adjacent145	PROPN
ejpam-5499	89	13	since	since	SCONJ
ejpam-5499	89	14	they	they	PRON
ejpam-5499	89	15	are	be	AUX
ejpam-5499	89	16	adjacent	adjacent	ADJ
ejpam-5499	89	17	edges	edge	NOUN
ejpam-5499	89	18	in	in	ADP
ejpam-5499	89	19	c4	c4	NOUN
ejpam-5499	89	20	.	.	PUNCT
ejpam-5499	90	1	also	also	ADV
ejpam-5499	90	2	the	the	DET
ejpam-5499	90	3	vertices	vertex	NOUN
ejpam-5499	90	4	3	3	NUM
ejpam-5499	90	5	and	and	CCONJ
ejpam-5499	90	6	d	d	NOUN
ejpam-5499	90	7	in	in	ADP
ejpam-5499	90	8	m(c4	m(c4	NOUN
ejpam-5499	90	9	)	)	PUNCT
ejpam-5499	90	10	are	be	AUX
ejpam-5499	90	11	adjacent	adjacent	ADJ
ejpam-5499	90	12	since146	since146	PROPN
ejpam-5499	90	13	3	3	NUM
ejpam-5499	90	14	is	be	AUX
ejpam-5499	90	15	an	an	DET
ejpam-5499	90	16	edge	edge	NOUN
ejpam-5499	90	17	incident	incident	NOUN
ejpam-5499	90	18	to	to	ADP
ejpam-5499	90	19	a	a	DET
ejpam-5499	90	20	vertex	vertex	NOUN
ejpam-5499	90	21	d	d	NOUN
ejpam-5499	90	22	in	in	ADP
ejpam-5499	90	23	c4.147	c4.147	PROPN
ejpam-5499	90	24	3	3	NUM
ejpam-5499	90	25	.	.	PUNCT
ejpam-5499	91	1	the	the	DET
ejpam-5499	91	2	middle	middle	ADJ
ejpam-5499	91	3	graph	graph	NOUN
ejpam-5499	91	4	of	of	ADP
ejpam-5499	91	5	γcn148	γcn148	PROPN
ejpam-5499	91	6	below	below	ADV
ejpam-5499	91	7	is	be	AUX
ejpam-5499	91	8	the	the	DET
ejpam-5499	91	9	structure	structure	NOUN
ejpam-5499	91	10	of	of	ADP
ejpam-5499	91	11	the	the	DET
ejpam-5499	91	12	middle	middle	ADJ
ejpam-5499	91	13	graph	graph	NOUN
ejpam-5499	91	14	of	of	ADP
ejpam-5499	91	15	the	the	DET
ejpam-5499	91	16	identity	identity	NOUN
ejpam-5499	91	17	graph	graph	NOUN
ejpam-5499	91	18	of	of	ADP
ejpam-5499	91	19	the	the	DET
ejpam-5499	91	20	finite	finite	NOUN
ejpam-5499	91	21	cyclic149	cyclic149	PROPN
ejpam-5499	91	22	groups	group	NOUN
ejpam-5499	91	23	.	.	PUNCT
ejpam-5499	92	1	for	for	ADP
ejpam-5499	92	2	easy	easy	ADJ
ejpam-5499	92	3	reference	reference	NOUN
ejpam-5499	92	4	,	,	PUNCT
ejpam-5499	92	5	we	we	PRON
ejpam-5499	92	6	refer	refer	VERB
ejpam-5499	92	7	to	to	ADP
ejpam-5499	92	8	the	the	DET
ejpam-5499	92	9	middle	middle	ADJ
ejpam-5499	92	10	graph	graph	NOUN
ejpam-5499	92	11	of	of	ADP
ejpam-5499	92	12	the	the	DET
ejpam-5499	92	13	identity	identity	NOUN
ejpam-5499	92	14	graph	graph	NOUN
ejpam-5499	92	15	as	as	SCONJ
ejpam-5499	92	16	mig.150	mig.150	NOUN
ejpam-5499	92	17	this	this	PRON
ejpam-5499	92	18	will	will	AUX
ejpam-5499	92	19	be	be	AUX
ejpam-5499	92	20	used	use	VERB
ejpam-5499	92	21	for	for	ADP
ejpam-5499	92	22	the	the	DET
ejpam-5499	92	23	rest	rest	NOUN
ejpam-5499	92	24	of	of	ADP
ejpam-5499	92	25	this	this	DET
ejpam-5499	92	26	paper.151	paper.151	ADJ
ejpam-5499	92	27	definition	definition	NOUN
ejpam-5499	92	28	9	9	NUM
ejpam-5499	92	29	.	.	PUNCT
ejpam-5499	93	1	let	let	VERB
ejpam-5499	93	2	cn	cn	PROPN
ejpam-5499	93	3	be	be	AUX
ejpam-5499	93	4	a	a	DET
ejpam-5499	93	5	finite	finite	ADJ
ejpam-5499	93	6	group	group	NOUN
ejpam-5499	93	7	of	of	ADP
ejpam-5499	93	8	order	order	NOUN
ejpam-5499	93	9	n	n	NOUN
ejpam-5499	93	10	and	and	CCONJ
ejpam-5499	93	11	γcn	γcn	PROPN
ejpam-5499	93	12	be	be	VERB
ejpam-5499	93	13	the	the	DET
ejpam-5499	93	14	identity	identity	NOUN
ejpam-5499	93	15	graph	graph	NOUN
ejpam-5499	93	16	of	of	ADP
ejpam-5499	93	17	cn	cn	PROPN
ejpam-5499	93	18	.	.	PUNCT
ejpam-5499	93	19	the152	the152	NOUN
ejpam-5499	93	20	middle	middle	ADJ
ejpam-5499	93	21	graph	graph	NOUN
ejpam-5499	93	22	of	of	ADP
ejpam-5499	93	23	γcn	γcn	NOUN
ejpam-5499	93	24	denoted	denote	VERB
ejpam-5499	93	25	by	by	ADP
ejpam-5499	93	26	m(γcn	m(γcn	PROPN
ejpam-5499	93	27	)	)	PUNCT
ejpam-5499	93	28	is	be	AUX
ejpam-5499	93	29	the	the	DET
ejpam-5499	93	30	graph	graph	NOUN
ejpam-5499	93	31	whose	whose	DET
ejpam-5499	93	32	vertex	vertex	NOUN
ejpam-5499	93	33	set	set	NOUN
ejpam-5499	93	34	is	be	AUX
ejpam-5499	93	35	v	v	NOUN
ejpam-5499	93	36	(	(	PUNCT
ejpam-5499	93	37	γcn)∪e(γcn)153	γcn)∪e(γcn)153	VERB
ejpam-5499	93	38	where	where	SCONJ
ejpam-5499	93	39	two	two	NUM
ejpam-5499	93	40	vertices	vertex	NOUN
ejpam-5499	93	41	are	be	AUX
ejpam-5499	93	42	adjacent	adjacent	ADJ
ejpam-5499	93	43	if	if	SCONJ
ejpam-5499	93	44	(	(	PUNCT
ejpam-5499	93	45	1	1	X
ejpam-5499	93	46	)	)	PUNCT
ejpam-5499	93	47	they	they	PRON
ejpam-5499	93	48	are	be	AUX
ejpam-5499	93	49	either	either	CCONJ
ejpam-5499	93	50	adjacent	adjacent	ADJ
ejpam-5499	93	51	edges	edge	NOUN
ejpam-5499	93	52	of	of	ADP
ejpam-5499	93	53	γcn	γcn	PROPN
ejpam-5499	93	54	or	or	CCONJ
ejpam-5499	93	55	(	(	PUNCT
ejpam-5499	93	56	2	2	X
ejpam-5499	93	57	)	)	PUNCT
ejpam-5499	93	58	one	one	NOUN
ejpam-5499	93	59	is154	is154	VERB
ejpam-5499	93	60	a	a	DET
ejpam-5499	93	61	vertex	vertex	NOUN
ejpam-5499	93	62	and	and	CCONJ
ejpam-5499	93	63	the	the	DET
ejpam-5499	93	64	other	other	ADJ
ejpam-5499	93	65	is	be	AUX
ejpam-5499	93	66	an	an	DET
ejpam-5499	93	67	edge	edge	NOUN
ejpam-5499	93	68	incident	incident	NOUN
ejpam-5499	93	69	to	to	ADP
ejpam-5499	93	70	it.155	it.155	PROPN
ejpam-5499	93	71	j.	j.	PROPN
ejpam-5499	93	72	m.	m.	PROPN
ejpam-5499	93	73	jamis	jamis	PROPN
ejpam-5499	93	74	,	,	PUNCT
ejpam-5499	93	75	d.	d.	PROPN
ejpam-5499	93	76	m.	m.	PROPN
ejpam-5499	93	77	magpantay	magpantay	PROPN
ejpam-5499	93	78	/	/	SYM
ejpam-5499	93	79	eur	eur	PROPN
ejpam-5499	93	80	.	.	PUNCT
ejpam-5499	94	1	j.	j.	PROPN
ejpam-5499	94	2	pure	pure	PROPN
ejpam-5499	94	3	appl	appl	PROPN
ejpam-5499	94	4	.	.	PROPN
ejpam-5499	94	5	math	math	PROPN
ejpam-5499	94	6	,	,	PUNCT
ejpam-5499	94	7	18	18	NUM
ejpam-5499	94	8	(	(	PUNCT
ejpam-5499	94	9	2	2	NUM
ejpam-5499	94	10	)	)	PUNCT
ejpam-5499	94	11	(	(	PUNCT
ejpam-5499	94	12	2025	2025	NUM
ejpam-5499	94	13	)	)	PUNCT
ejpam-5499	94	14	,	,	PUNCT
ejpam-5499	94	15	5499	5499	NUM
ejpam-5499	94	16	6	6	NUM
ejpam-5499	94	17	of	of	ADP
ejpam-5499	94	18	18	18	NUM
ejpam-5499	94	19	in	in	ADP
ejpam-5499	94	20	this	this	DET
ejpam-5499	94	21	study	study	NOUN
ejpam-5499	94	22	,	,	PUNCT
ejpam-5499	94	23	the	the	DET
ejpam-5499	94	24	vertex	vertex	NOUN
ejpam-5499	94	25	-	-	PUNCT
ejpam-5499	94	26	set	set	VERB
ejpam-5499	94	27	and	and	CCONJ
ejpam-5499	94	28	edge	edge	NOUN
ejpam-5499	94	29	-	-	PUNCT
ejpam-5499	94	30	set	set	VERB
ejpam-5499	94	31	notations	notation	NOUN
ejpam-5499	94	32	of	of	ADP
ejpam-5499	94	33	m(γcn	m(γcn	PROPN
ejpam-5499	94	34	)	)	PUNCT
ejpam-5499	94	35	are	be	AUX
ejpam-5499	94	36	fixed	fix	VERB
ejpam-5499	94	37	.	.	PUNCT
ejpam-5499	95	1	here	here	ADV
ejpam-5499	95	2	are	be	AUX
ejpam-5499	95	3	the156	the156	PROPN
ejpam-5499	95	4	following	follow	VERB
ejpam-5499	95	5	steps	step	NOUN
ejpam-5499	95	6	on	on	ADP
ejpam-5499	95	7	how	how	SCONJ
ejpam-5499	95	8	to	to	PART
ejpam-5499	95	9	construct	construct	VERB
ejpam-5499	95	10	the	the	DET
ejpam-5499	95	11	middle	middle	ADJ
ejpam-5499	95	12	graph	graph	NOUN
ejpam-5499	95	13	of	of	ADP
ejpam-5499	95	14	the	the	DET
ejpam-5499	95	15	identity	identity	NOUN
ejpam-5499	95	16	graph	graph	NOUN
ejpam-5499	95	17	of	of	ADP
ejpam-5499	95	18	finite	finite	PROPN
ejpam-5499	95	19	cyclic157	cyclic157	PROPN
ejpam-5499	95	20	groups.158	groups.158	PROPN
ejpam-5499	95	21	step	step	VERB
ejpam-5499	95	22	1	1	NUM
ejpam-5499	95	23	.	.	PUNCT
ejpam-5499	96	1	draw	draw	VERB
ejpam-5499	96	2	the	the	DET
ejpam-5499	96	3	identity	identity	NOUN
ejpam-5499	96	4	graph	graph	NOUN
ejpam-5499	96	5	of	of	ADP
ejpam-5499	96	6	finite	finite	ADJ
ejpam-5499	96	7	cyclic	cyclic	PROPN
ejpam-5499	96	8	group	group	NOUN
ejpam-5499	96	9	γcn	γcn	PROPN
ejpam-5499	96	10	for	for	ADP
ejpam-5499	96	11	n	n	PRON
ejpam-5499	96	12	≥	≥	NUM
ejpam-5499	96	13	2	2	NUM
ejpam-5499	96	14	.	.	PUNCT
ejpam-5499	96	15	set	set	VERB
ejpam-5499	96	16	the	the	DET
ejpam-5499	96	17	vertices159	vertices159	PROPN
ejpam-5499	96	18	v	v	NOUN
ejpam-5499	96	19	(	(	PUNCT
ejpam-5499	96	20	γcn	γcn	PROPN
ejpam-5499	96	21	)	)	PUNCT
ejpam-5499	97	1	=	=	PRON
ejpam-5499	97	2	{	{	PUNCT
ejpam-5499	97	3	e}∪{xi|1	e}∪{xi|1	NOUN
ejpam-5499	97	4	≤	≤	VERB
ejpam-5499	97	5	i	i	PRON
ejpam-5499	97	6	≤	≤	NOUN
ejpam-5499	97	7	n−1	n−1	PROPN
ejpam-5499	97	8	}	}	PUNCT
ejpam-5499	97	9	where	where	SCONJ
ejpam-5499	97	10	e	e	NOUN
ejpam-5499	97	11	is	be	AUX
ejpam-5499	97	12	the	the	DET
ejpam-5499	97	13	identity	identity	NOUN
ejpam-5499	97	14	element	element	NOUN
ejpam-5499	97	15	of	of	ADP
ejpam-5499	97	16	the	the	DET
ejpam-5499	97	17	cyclic	cyclic	ADJ
ejpam-5499	97	18	group160	group160	PROPN
ejpam-5499	97	19	cn	cn	PROPN
ejpam-5499	97	20	.	.	PROPN
ejpam-5499	98	1	set	set	VERB
ejpam-5499	98	2	the	the	DET
ejpam-5499	98	3	edges	edge	NOUN
ejpam-5499	98	4	e(γcn	e(γcn	PROPN
ejpam-5499	98	5	)	)	PUNCT
ejpam-5499	98	6	=	=	PRON
ejpam-5499	99	1	{	{	PUNCT
ejpam-5499	99	2	zi|zi	zi|zi	NOUN
ejpam-5499	99	3	=	=	PUNCT
ejpam-5499	100	1	[	[	X
ejpam-5499	100	2	xi	xi	ADP
ejpam-5499	100	3	,	,	PUNCT
ejpam-5499	100	4	e]for	e]for	ADP
ejpam-5499	100	5	all1	all1	NOUN
ejpam-5499	100	6	≤	≤	PROPN
ejpam-5499	101	1	i	i	PRON
ejpam-5499	101	2	≤	≤	ADJ
ejpam-5499	101	3	n−	n−	NOUN
ejpam-5499	101	4	1}161	1}161	ADP
ejpam-5499	101	5	⋃	⋃	PUNCT
ejpam-5499	101	6	{	{	PUNCT
ejpam-5499	101	7	yp|yp	yp|yp	X
ejpam-5499	101	8	=	=	PUNCT
ejpam-5499	102	1	[	[	X
ejpam-5499	102	2	xi	xi	X
ejpam-5499	102	3	,	,	PUNCT
ejpam-5499	102	4	xi+1]andp	xi+1]andp	X
ejpam-5499	102	5	=	=	PUNCT
ejpam-5499	102	6	i+1	i+1	NUM
ejpam-5499	102	7	2	2	NUM
ejpam-5499	102	8	}	}	PUNCT
ejpam-5499	102	9	{	{	PUNCT
ejpam-5499	102	10	for	for	ADP
ejpam-5499	102	11	all	all	DET
ejpam-5499	102	12	odd	odd	ADJ
ejpam-5499	102	13	1	1	NUM
ejpam-5499	102	14	≤	≤	NUM
ejpam-5499	102	15	i	i	PRON
ejpam-5499	102	16	≤	≤	ADJ
ejpam-5499	102	17	n−	n−	NOUN
ejpam-5499	102	18	2	2	NUM
ejpam-5499	102	19	}	}	PUNCT
ejpam-5499	102	20	,	,	PUNCT
ejpam-5499	102	21	if	if	SCONJ
ejpam-5499	102	22	n	n	PRON
ejpam-5499	102	23	is	be	AUX
ejpam-5499	102	24	odd	odd	ADJ
ejpam-5499	102	25	for	for	ADP
ejpam-5499	102	26	all	all	DET
ejpam-5499	102	27	odd	odd	ADJ
ejpam-5499	102	28	1	1	NUM
ejpam-5499	102	29	≤	≤	NUM
ejpam-5499	102	30	i	i	PRON
ejpam-5499	102	31	≤	≤	ADJ
ejpam-5499	102	32	n−	n−	NOUN
ejpam-5499	102	33	3	3	NUM
ejpam-5499	102	34	}	}	PUNCT
ejpam-5499	102	35	,	,	PUNCT
ejpam-5499	102	36	if	if	SCONJ
ejpam-5499	102	37	n	n	PRON
ejpam-5499	102	38	is	be	AUX
ejpam-5499	102	39	even	even	ADV
ejpam-5499	102	40	.	.	PUNCT
ejpam-5499	103	1	162	162	NUM
ejpam-5499	103	2	step	step	NOUN
ejpam-5499	103	3	2	2	NUM
ejpam-5499	103	4	.	.	PUNCT
ejpam-5499	103	5	subdivide	subdivide	VERB
ejpam-5499	103	6	the	the	DET
ejpam-5499	103	7	edges	edge	NOUN
ejpam-5499	103	8	of	of	ADP
ejpam-5499	103	9	the	the	DET
ejpam-5499	103	10	original	original	ADJ
ejpam-5499	103	11	graph	graph	NOUN
ejpam-5499	103	12	.	.	PUNCT
ejpam-5499	104	1	the	the	DET
ejpam-5499	104	2	additional	additional	ADJ
ejpam-5499	104	3	new	new	ADJ
ejpam-5499	104	4	vertices	vertex	NOUN
ejpam-5499	104	5	will	will	AUX
ejpam-5499	104	6	be	be	AUX
ejpam-5499	104	7	ob-163	ob-163	VERB
ejpam-5499	104	8	tained	taine	VERB
ejpam-5499	104	9	by	by	ADP
ejpam-5499	104	10	subdividing	subdivide	VERB
ejpam-5499	104	11	the	the	DET
ejpam-5499	104	12	edges	edge	NOUN
ejpam-5499	104	13	and	and	CCONJ
ejpam-5499	104	14	the	the	DET
ejpam-5499	104	15	second	second	ADJ
ejpam-5499	104	16	condition	condition	NOUN
ejpam-5499	104	17	of	of	ADP
ejpam-5499	104	18	the	the	DET
ejpam-5499	104	19	middle	middle	ADJ
ejpam-5499	104	20	graph	graph	NOUN
ejpam-5499	104	21	will164	will164	PROPN
ejpam-5499	104	22	automatically	automatically	ADV
ejpam-5499	104	23	be	be	AUX
ejpam-5499	104	24	satisfied.165	satisfied.165	ADJ
ejpam-5499	104	25	step	step	NOUN
ejpam-5499	104	26	3	3	NUM
ejpam-5499	104	27	.	.	PUNCT
ejpam-5499	104	28	connect	connect	VERB
ejpam-5499	104	29	the	the	DET
ejpam-5499	104	30	new	new	ADJ
ejpam-5499	104	31	obtained	obtain	VERB
ejpam-5499	104	32	vertices	vertex	NOUN
ejpam-5499	104	33	to	to	ADP
ejpam-5499	104	34	each	each	DET
ejpam-5499	104	35	other	other	ADJ
ejpam-5499	104	36	if	if	SCONJ
ejpam-5499	104	37	they	they	PRON
ejpam-5499	104	38	are	be	AUX
ejpam-5499	104	39	adjacent	adjacent	ADJ
ejpam-5499	104	40	edges	edge	NOUN
ejpam-5499	104	41	in	in	ADP
ejpam-5499	104	42	the166	the166	PROPN
ejpam-5499	104	43	original	original	ADJ
ejpam-5499	104	44	graph.167	graph.167	NOUN
ejpam-5499	104	45	consider	consider	VERB
ejpam-5499	104	46	the	the	DET
ejpam-5499	104	47	following	follow	VERB
ejpam-5499	104	48	example,168	example,168	PROPN
ejpam-5499	104	49	example	example	NOUN
ejpam-5499	104	50	4	4	NUM
ejpam-5499	104	51	.	.	PUNCT
ejpam-5499	105	1	let	let	VERB
ejpam-5499	105	2	γc3	γc3	PROPN
ejpam-5499	105	3	be	be	AUX
ejpam-5499	105	4	the	the	DET
ejpam-5499	105	5	identity	identity	NOUN
ejpam-5499	105	6	graph	graph	NOUN
ejpam-5499	105	7	of	of	ADP
ejpam-5499	105	8	c3	c3	PROPN
ejpam-5499	105	9	with	with	ADP
ejpam-5499	105	10	vertices	vertex	NOUN
ejpam-5499	105	11	v	v	PROPN
ejpam-5499	105	12	(	(	PUNCT
ejpam-5499	105	13	c3	c3	NOUN
ejpam-5499	105	14	)	)	PUNCT
ejpam-5499	105	15	=	=	SYM
ejpam-5499	105	16	{	{	PUNCT
ejpam-5499	105	17	e	e	NOUN
ejpam-5499	105	18	,	,	PUNCT
ejpam-5499	105	19	x1	x1	PROPN
ejpam-5499	105	20	,	,	PUNCT
ejpam-5499	105	21	x2	x2	PROPN
ejpam-5499	105	22	}	}	PUNCT
ejpam-5499	105	23	and169	and169	PROPN
ejpam-5499	105	24	e(c3	e(c3	NOUN
ejpam-5499	105	25	)	)	PUNCT
ejpam-5499	106	1	=	=	SYM
ejpam-5499	106	2	{	{	PUNCT
ejpam-5499	106	3	z1	z1	PROPN
ejpam-5499	106	4	,	,	PUNCT
ejpam-5499	106	5	z2	z2	PROPN
ejpam-5499	106	6	,	,	PUNCT
ejpam-5499	106	7	y1	y1	NOUN
ejpam-5499	106	8	}	}	PUNCT
ejpam-5499	106	9	shown	show	VERB
ejpam-5499	106	10	in	in	ADP
ejpam-5499	106	11	the	the	DET
ejpam-5499	106	12	figure	figure	NOUN
ejpam-5499	106	13	below	below	ADV
ejpam-5499	106	14	and	and	CCONJ
ejpam-5499	106	15	its	its	PRON
ejpam-5499	106	16	corresponding	corresponding	ADJ
ejpam-5499	106	17	mig.170	mig.170	NOUN
ejpam-5499	106	18	.171	.171	NUM
ejpam-5499	106	19	3.1	3.1	NUM
ejpam-5499	106	20	.	.	PUNCT
ejpam-5499	107	1	the	the	DET
ejpam-5499	107	2	middle	middle	ADJ
ejpam-5499	107	3	graph	graph	NOUN
ejpam-5499	107	4	of	of	ADP
ejpam-5499	107	5	γcn	γcn	PROPN
ejpam-5499	107	6	where	where	SCONJ
ejpam-5499	107	7	n	n	X
ejpam-5499	107	8	is	be	AUX
ejpam-5499	107	9	odd172	odd172	PROPN
ejpam-5499	107	10	for	for	ADP
ejpam-5499	107	11	the	the	DET
ejpam-5499	107	12	general	general	ADJ
ejpam-5499	107	13	structure	structure	NOUN
ejpam-5499	107	14	of	of	ADP
ejpam-5499	107	15	the	the	DET
ejpam-5499	107	16	middle	middle	ADJ
ejpam-5499	107	17	graph	graph	NOUN
ejpam-5499	107	18	of	of	ADP
ejpam-5499	107	19	γcn	γcn	PROPN
ejpam-5499	107	20	where	where	SCONJ
ejpam-5499	107	21	n	n	PRON
ejpam-5499	107	22	is	be	AUX
ejpam-5499	107	23	odd	odd	ADJ
ejpam-5499	107	24	,	,	PUNCT
ejpam-5499	107	25	set	set	VERB
ejpam-5499	107	26	first	first	ADJ
ejpam-5499	107	27	the173	the173	PROPN
ejpam-5499	107	28	vertices	vertex	NOUN
ejpam-5499	107	29	of	of	ADP
ejpam-5499	107	30	the	the	DET
ejpam-5499	107	31	identity	identity	NOUN
ejpam-5499	107	32	graph	graph	NOUN
ejpam-5499	107	33	of	of	ADP
ejpam-5499	107	34	γcn	γcn	PROPN
ejpam-5499	107	35	as174	as174	PROPN
ejpam-5499	107	36	v	v	PROPN
ejpam-5499	107	37	(	(	PUNCT
ejpam-5499	107	38	γcn	γcn	PROPN
ejpam-5499	107	39	)	)	PUNCT
ejpam-5499	107	40	=	=	PRON
ejpam-5499	108	1	{	{	PUNCT
ejpam-5499	108	2	e	e	NOUN
ejpam-5499	108	3	}	}	PUNCT
ejpam-5499	108	4	∪	∪	ADJ
ejpam-5499	108	5	{	{	PUNCT
ejpam-5499	108	6	xi|1	xi|1	NOUN
ejpam-5499	108	7	≤	≤	NOUN
ejpam-5499	109	1	i	i	PRON
ejpam-5499	109	2	≤	≤	ADJ
ejpam-5499	109	3	n−	n−	NOUN
ejpam-5499	109	4	1	1	NUM
ejpam-5499	109	5	}	}	PUNCT
ejpam-5499	109	6	and175	and175	PROPN
ejpam-5499	109	7	e(γcn	e(γcn	PROPN
ejpam-5499	109	8	)	)	PUNCT
ejpam-5499	110	1	=	=	PRON
ejpam-5499	111	1	{	{	PUNCT
ejpam-5499	111	2	zi|1	zi|1	NOUN
ejpam-5499	111	3	≤	≤	NOUN
ejpam-5499	111	4	i	i	PRON
ejpam-5499	111	5	≤	≤	ADJ
ejpam-5499	111	6	n−	n−	NOUN
ejpam-5499	111	7	1	1	NUM
ejpam-5499	111	8	}	}	PUNCT
ejpam-5499	111	9	∪	∪	X
ejpam-5499	111	10	{	{	PUNCT
ejpam-5499	111	11	yp|p	yp|p	PROPN
ejpam-5499	111	12	=	=	PUNCT
ejpam-5499	112	1	i+	i+	PUNCT
ejpam-5499	112	2	1	1	NUM
ejpam-5499	112	3	2	2	NUM
ejpam-5499	112	4	for	for	ADP
ejpam-5499	112	5	all	all	DET
ejpam-5499	112	6	odd	odd	ADJ
ejpam-5499	112	7	1	1	NUM
ejpam-5499	112	8	≤	≤	NUM
ejpam-5499	112	9	i	i	PRON
ejpam-5499	112	10	≤	≤	ADJ
ejpam-5499	112	11	n−	n−	NOUN
ejpam-5499	112	12	2	2	NUM
ejpam-5499	112	13	}	}	PUNCT
ejpam-5499	112	14	here	here	ADV
ejpam-5499	112	15	is	be	AUX
ejpam-5499	112	16	the	the	DET
ejpam-5499	112	17	pictorial	pictorial	ADJ
ejpam-5499	112	18	representation,176	representation,176	PROPN
ejpam-5499	112	19	j.	j.	PROPN
ejpam-5499	112	20	m.	m.	PROPN
ejpam-5499	112	21	jamis	jamis	PROPN
ejpam-5499	112	22	,	,	PUNCT
ejpam-5499	112	23	d.	d.	PROPN
ejpam-5499	112	24	m.	m.	PROPN
ejpam-5499	112	25	magpantay	magpantay	PROPN
ejpam-5499	112	26	/	/	SYM
ejpam-5499	112	27	eur	eur	PROPN
ejpam-5499	112	28	.	.	PUNCT
ejpam-5499	113	1	j.	j.	PROPN
ejpam-5499	113	2	pure	pure	PROPN
ejpam-5499	113	3	appl	appl	PROPN
ejpam-5499	113	4	.	.	PROPN
ejpam-5499	113	5	math	math	PROPN
ejpam-5499	113	6	,	,	PUNCT
ejpam-5499	113	7	18	18	NUM
ejpam-5499	113	8	(	(	PUNCT
ejpam-5499	113	9	2	2	NUM
ejpam-5499	113	10	)	)	PUNCT
ejpam-5499	113	11	(	(	PUNCT
ejpam-5499	113	12	2025	2025	NUM
ejpam-5499	113	13	)	)	PUNCT
ejpam-5499	113	14	,	,	PUNCT
ejpam-5499	113	15	5499	5499	NUM
ejpam-5499	113	16	7	7	NUM
ejpam-5499	113	17	of	of	ADP
ejpam-5499	113	18	18	18	NUM
ejpam-5499	113	19	.177	.177	NUM
ejpam-5499	113	20	to	to	PART
ejpam-5499	113	21	generalize	generalize	VERB
ejpam-5499	113	22	the	the	DET
ejpam-5499	113	23	vertex	vertex	NOUN
ejpam-5499	113	24	set	set	NOUN
ejpam-5499	113	25	and	and	CCONJ
ejpam-5499	113	26	edge	edge	NOUN
ejpam-5499	113	27	set	set	NOUN
ejpam-5499	113	28	of	of	ADP
ejpam-5499	113	29	m(γcn	m(γcn	PROPN
ejpam-5499	113	30	)	)	PUNCT
ejpam-5499	113	31	for	for	ADP
ejpam-5499	113	32	n	n	X
ejpam-5499	113	33	is	be	AUX
ejpam-5499	113	34	odd	odd	ADJ
ejpam-5499	113	35	,	,	PUNCT
ejpam-5499	113	36	we	we	PRON
ejpam-5499	113	37	now	now	ADV
ejpam-5499	113	38	have;178	have;178	ADV
ejpam-5499	113	39	v	v	NOUN
ejpam-5499	113	40	(	(	PUNCT
ejpam-5499	113	41	m(γcn	m(γcn	NOUN
ejpam-5499	113	42	)	)	PUNCT
ejpam-5499	113	43	)	)	PUNCT
ejpam-5499	114	1	=	=	PRON
ejpam-5499	114	2	{	{	PUNCT
ejpam-5499	114	3	e	e	NOUN
ejpam-5499	114	4	}	}	PUNCT
ejpam-5499	114	5	⋃	⋃	ADV
ejpam-5499	114	6	{	{	PUNCT
ejpam-5499	114	7	xi|1	xi|1	NOUN
ejpam-5499	114	8	≤	≤	NOUN
ejpam-5499	115	1	i	i	PRON
ejpam-5499	115	2	≤	≤	ADJ
ejpam-5499	115	3	n−	n−	NOUN
ejpam-5499	115	4	1	1	NUM
ejpam-5499	115	5	}	}	PUNCT
ejpam-5499	115	6	⋃	⋃	NOUN
ejpam-5499	115	7	{	{	PUNCT
ejpam-5499	115	8	zi|1	zi|1	NOUN
ejpam-5499	115	9	≤	≤	NOUN
ejpam-5499	115	10	i	i	PRON
ejpam-5499	115	11	≤	≤	ADJ
ejpam-5499	115	12	n−	n−	NOUN
ejpam-5499	115	13	1}⋃	1}⋃	NUM
ejpam-5499	115	14	{	{	PUNCT
ejpam-5499	115	15	yp|p	yp|p	PROPN
ejpam-5499	116	1	=	=	PUNCT
ejpam-5499	117	1	i+1	i+1	NUM
ejpam-5499	117	2	2	2	NUM
ejpam-5499	117	3	for	for	ADP
ejpam-5499	117	4	all	all	DET
ejpam-5499	117	5	odd	odd	ADJ
ejpam-5499	117	6	1	1	NUM
ejpam-5499	117	7	≤	≤	NUM
ejpam-5499	117	8	i	i	PRON
ejpam-5499	117	9	≤	≤	ADJ
ejpam-5499	117	10	n−	n−	NOUN
ejpam-5499	117	11	2	2	NUM
ejpam-5499	117	12	}	}	PUNCT
ejpam-5499	117	13	and179	and179	VERB
ejpam-5499	117	14	e(m(γcn	e(m(γcn	NUM
ejpam-5499	117	15	)	)	PUNCT
ejpam-5499	117	16	)	)	PUNCT
ejpam-5499	118	1	=	=	PRON
ejpam-5499	118	2	{	{	PUNCT
ejpam-5499	119	1	[	[	X
ejpam-5499	119	2	e	e	NOUN
ejpam-5499	119	3	,	,	PUNCT
ejpam-5499	119	4	zi]|1	zi]|1	NOUN
ejpam-5499	119	5	≤	≤	NUM
ejpam-5499	120	1	i	i	PRON
ejpam-5499	120	2	≤	≤	ADJ
ejpam-5499	120	3	n−	n−	NOUN
ejpam-5499	120	4	1}⋃	1}⋃	NUM
ejpam-5499	120	5	{	{	PUNCT
ejpam-5499	120	6	[	[	X
ejpam-5499	120	7	zi	zi	X
ejpam-5499	120	8	,	,	PUNCT
ejpam-5499	120	9	zj	zj	PROPN
ejpam-5499	120	10	]	]	X
ejpam-5499	120	11	|1	|1	NUM
ejpam-5499	120	12	≤	≤	NUM
ejpam-5499	121	1	i	i	PRON
ejpam-5499	121	2	≤	≤	ADJ
ejpam-5499	121	3	n−	n−	PROPN
ejpam-5499	121	4	1	1	NUM
ejpam-5499	121	5	,	,	PUNCT
ejpam-5499	121	6	1	1	NUM
ejpam-5499	121	7	≤	≤	NUM
ejpam-5499	121	8	j	j	PROPN
ejpam-5499	121	9	≤	≤	PROPN
ejpam-5499	121	10	n−	n−	PROPN
ejpam-5499	121	11	1	1	NUM
ejpam-5499	121	12	,	,	PUNCT
ejpam-5499	121	13	i	i	PRON
ejpam-5499	121	14	̸=	̸=	PROPN
ejpam-5499	121	15	j}⋃	j}⋃	PROPN
ejpam-5499	121	16	{	{	PUNCT
ejpam-5499	121	17	[	[	X
ejpam-5499	121	18	xi	xi	X
ejpam-5499	121	19	,	,	PUNCT
ejpam-5499	121	20	zi]|1	zi]|1	NOUN
ejpam-5499	121	21	≤	≤	NUM
ejpam-5499	121	22	i	i	PRON
ejpam-5499	121	23	≤	≤	ADJ
ejpam-5499	121	24	n−	n−	PROPN
ejpam-5499	121	25	1	1	NUM
ejpam-5499	121	26	,	,	PUNCT
ejpam-5499	121	27	1	1	NUM
ejpam-5499	121	28	≤	≤	NUM
ejpam-5499	121	29	j	j	PROPN
ejpam-5499	121	30	≤	≤	PROPN
ejpam-5499	121	31	n−	n−	PROPN
ejpam-5499	121	32	1}⋃	1}⋃	NUM
ejpam-5499	121	33	{	{	PUNCT
ejpam-5499	121	34	[	[	X
ejpam-5499	121	35	yp	yp	X
ejpam-5499	121	36	,	,	PUNCT
ejpam-5499	121	37	zi]|p	zi]|p	PROPN
ejpam-5499	121	38	=	=	PUNCT
ejpam-5499	121	39	i+1	i+1	SYM
ejpam-5499	121	40	2	2	NUM
ejpam-5499	121	41	for	for	ADP
ejpam-5499	121	42	all	all	DET
ejpam-5499	121	43	odd1	odd1	NOUN
ejpam-5499	121	44	≤	≤	ADV
ejpam-5499	121	45	i	i	PRON
ejpam-5499	121	46	≤	≤	ADJ
ejpam-5499	121	47	n−	n−	NOUN
ejpam-5499	121	48	2	2	NUM
ejpam-5499	121	49	}	}	PUNCT
ejpam-5499	121	50	⋃	⋃	NOUN
ejpam-5499	121	51	{	{	PUNCT
ejpam-5499	121	52	[	[	X
ejpam-5499	121	53	yp	yp	PROPN
ejpam-5499	121	54	,	,	PUNCT
ejpam-5499	121	55	zi+1]|p	zi+1]|p	X
ejpam-5499	121	56	=	=	SYM
ejpam-5499	121	57	i+1	i+1	SYM
ejpam-5499	121	58	2	2	NUM
ejpam-5499	121	59	for	for	ADP
ejpam-5499	121	60	all	all	DET
ejpam-5499	121	61	odd1	odd1	NOUN
ejpam-5499	121	62	≤	≤	ADV
ejpam-5499	122	1	i	i	PRON
ejpam-5499	122	2	≤	≤	ADJ
ejpam-5499	122	3	n−	n−	NOUN
ejpam-5499	122	4	2	2	NUM
ejpam-5499	122	5	}	}	PUNCT
ejpam-5499	122	6	⋃	⋃	NOUN
ejpam-5499	122	7	{	{	PUNCT
ejpam-5499	122	8	[	[	X
ejpam-5499	122	9	yp	yp	PROPN
ejpam-5499	122	10	,	,	PUNCT
ejpam-5499	122	11	xi]|p	xi]|p	X
ejpam-5499	122	12	=	=	PUNCT
ejpam-5499	123	1	i+1	i+1	SYM
ejpam-5499	123	2	2	2	NUM
ejpam-5499	123	3	for	for	ADP
ejpam-5499	123	4	all	all	DET
ejpam-5499	123	5	odd1	odd1	NOUN
ejpam-5499	123	6	≤	≤	ADV
ejpam-5499	124	1	i	i	PRON
ejpam-5499	124	2	≤	≤	ADJ
ejpam-5499	124	3	n−	n−	NOUN
ejpam-5499	124	4	2	2	NUM
ejpam-5499	124	5	}	}	PUNCT
ejpam-5499	124	6	⋃	⋃	NOUN
ejpam-5499	124	7	{	{	PUNCT
ejpam-5499	124	8	[	[	X
ejpam-5499	124	9	yp	yp	X
ejpam-5499	124	10	,	,	PUNCT
ejpam-5499	124	11	xi+1]|p	xi+1]|p	PROPN
ejpam-5499	124	12	=	=	PUNCT
ejpam-5499	124	13	i+1	i+1	SYM
ejpam-5499	124	14	2	2	NUM
ejpam-5499	124	15	for	for	ADP
ejpam-5499	124	16	all	all	DET
ejpam-5499	124	17	odd1	odd1	NOUN
ejpam-5499	124	18	≤	≤	ADV
ejpam-5499	125	1	i	i	PRON
ejpam-5499	125	2	≤	≤	ADJ
ejpam-5499	125	3	n−	n−	NOUN
ejpam-5499	125	4	2	2	NUM
ejpam-5499	125	5	}	}	PUNCT
ejpam-5499	125	6	here	here	ADV
ejpam-5499	125	7	is	be	AUX
ejpam-5499	125	8	the	the	DET
ejpam-5499	125	9	pictorial	pictorial	ADJ
ejpam-5499	125	10	representation,180	representation,180	PROPN
ejpam-5499	125	11	.181	.181	NUM
ejpam-5499	126	1	the	the	DET
ejpam-5499	126	2	degree	degree	NOUN
ejpam-5499	126	3	of	of	ADP
ejpam-5499	126	4	the	the	DET
ejpam-5499	126	5	vertices	vertex	NOUN
ejpam-5499	126	6	of	of	ADP
ejpam-5499	126	7	m(γcn	m(γcn	PROPN
ejpam-5499	126	8	)	)	PUNCT
ejpam-5499	126	9	where	where	SCONJ
ejpam-5499	126	10	n	n	PRON
ejpam-5499	126	11	is	be	AUX
ejpam-5499	126	12	odd	odd	ADJ
ejpam-5499	126	13	is	be	AUX
ejpam-5499	126	14	summarized	summarize	VERB
ejpam-5499	126	15	below:182	below:182	PROPN
ejpam-5499	126	16	j.	j.	PROPN
ejpam-5499	126	17	m.	m.	PROPN
ejpam-5499	126	18	jamis	jamis	PROPN
ejpam-5499	126	19	,	,	PUNCT
ejpam-5499	126	20	d.	d.	PROPN
ejpam-5499	126	21	m.	m.	PROPN
ejpam-5499	126	22	magpantay	magpantay	PROPN
ejpam-5499	126	23	/	/	SYM
ejpam-5499	126	24	eur	eur	PROPN
ejpam-5499	126	25	.	.	PUNCT
ejpam-5499	127	1	j.	j.	PROPN
ejpam-5499	127	2	pure	pure	PROPN
ejpam-5499	127	3	appl	appl	PROPN
ejpam-5499	127	4	.	.	PROPN
ejpam-5499	127	5	math	math	PROPN
ejpam-5499	127	6	,	,	PUNCT
ejpam-5499	127	7	18	18	NUM
ejpam-5499	127	8	(	(	PUNCT
ejpam-5499	127	9	2	2	NUM
ejpam-5499	127	10	)	)	PUNCT
ejpam-5499	127	11	(	(	PUNCT
ejpam-5499	127	12	2025	2025	NUM
ejpam-5499	127	13	)	)	PUNCT
ejpam-5499	127	14	,	,	PUNCT
ejpam-5499	127	15	5499	5499	NUM
ejpam-5499	127	16	8	8	NUM
ejpam-5499	127	17	of	of	ADP
ejpam-5499	127	18	18	18	NUM
ejpam-5499	127	19	a.	a.	NOUN
ejpam-5499	127	20	deg(e	deg(e	NOUN
ejpam-5499	127	21	)	)	PUNCT
ejpam-5499	128	1	=	=	SYM
ejpam-5499	128	2	n−	n−	PROPN
ejpam-5499	128	3	1183	1183	NUM
ejpam-5499	128	4	b.	b.	PROPN
ejpam-5499	128	5	deg(xi	deg(xi	PROPN
ejpam-5499	128	6	)	)	PUNCT
ejpam-5499	129	1	=	=	SYM
ejpam-5499	129	2	2	2	NUM
ejpam-5499	129	3	,	,	PUNCT
ejpam-5499	129	4	1	1	NUM
ejpam-5499	129	5	≤	≤	NUM
ejpam-5499	129	6	i	i	PRON
ejpam-5499	129	7	≤	≤	ADJ
ejpam-5499	129	8	n−	n−	PROPN
ejpam-5499	129	9	1184	1184	NUM
ejpam-5499	129	10	c.	c.	PROPN
ejpam-5499	129	11	deg(yp	deg(yp	NOUN
ejpam-5499	129	12	)	)	PUNCT
ejpam-5499	129	13	=	=	SYM
ejpam-5499	129	14	4	4	NUM
ejpam-5499	129	15	,	,	PUNCT
ejpam-5499	129	16	1	1	NUM
ejpam-5499	129	17	≤	≤	NOUN
ejpam-5499	129	18	p	p	NOUN
ejpam-5499	129	19	≤	≤	NUM
ejpam-5499	129	20	i+1	i+1	NUM
ejpam-5499	129	21	2	2	NUM
ejpam-5499	129	22	for	for	ADP
ejpam-5499	129	23	all	all	DET
ejpam-5499	129	24	odd	odd	ADJ
ejpam-5499	129	25	1	1	NUM
ejpam-5499	129	26	≤	≤	NUM
ejpam-5499	129	27	i	i	PRON
ejpam-5499	129	28	≤	≤	ADJ
ejpam-5499	129	29	n−	n−	NOUN
ejpam-5499	129	30	2185	2185	NUM
ejpam-5499	129	31	d.	d.	PROPN
ejpam-5499	129	32	deg(zi	deg(zi	X
ejpam-5499	129	33	)	)	PUNCT
ejpam-5499	129	34	=	=	SYM
ejpam-5499	129	35	n+	n+	PUNCT
ejpam-5499	129	36	1	1	NUM
ejpam-5499	129	37	,	,	PUNCT
ejpam-5499	129	38	1	1	NUM
ejpam-5499	129	39	≤	≤	NUM
ejpam-5499	129	40	i	i	PRON
ejpam-5499	129	41	≤	≤	ADJ
ejpam-5499	129	42	n−	n−	NOUN
ejpam-5499	129	43	1.186	1.186	NUM
ejpam-5499	129	44	for	for	ADP
ejpam-5499	129	45	the	the	DET
ejpam-5499	129	46	summation	summation	NOUN
ejpam-5499	129	47	of	of	ADP
ejpam-5499	129	48	all	all	PRON
ejpam-5499	129	49	of	of	ADP
ejpam-5499	129	50	the	the	DET
ejpam-5499	129	51	degrees	degree	NOUN
ejpam-5499	129	52	of	of	ADP
ejpam-5499	129	53	the	the	DET
ejpam-5499	129	54	vertices	vertex	NOUN
ejpam-5499	129	55	of	of	ADP
ejpam-5499	129	56	m(γcn	m(γcn	PROPN
ejpam-5499	129	57	)	)	PUNCT
ejpam-5499	129	58	for	for	ADP
ejpam-5499	129	59	n	n	X
ejpam-5499	129	60	is	be	AUX
ejpam-5499	129	61	odd	odd	ADJ
ejpam-5499	129	62	,	,	PUNCT
ejpam-5499	129	63	we187	we187	PROPN
ejpam-5499	129	64	have188	have188	PROPN
ejpam-5499	129	65	∑	∑	PUNCT
ejpam-5499	129	66	v∈v	v∈v	PROPN
ejpam-5499	129	67	(	(	PUNCT
ejpam-5499	129	68	m(γcn	m(γcn	PROPN
ejpam-5499	129	69	)	)	PUNCT
ejpam-5499	129	70	)	)	PUNCT
ejpam-5499	130	1	deg(v	deg(v	PROPN
ejpam-5499	130	2	)	)	PUNCT
ejpam-5499	130	3	=	=	PUNCT
ejpam-5499	130	4	(	(	PUNCT
ejpam-5499	130	5	n−	n−	NOUN
ejpam-5499	130	6	1	1	NUM
ejpam-5499	130	7	)	)	PUNCT
ejpam-5499	131	1	+	+	CCONJ
ejpam-5499	132	1	2(n−	2(n−	NUM
ejpam-5499	132	2	1	1	NUM
ejpam-5499	132	3	)	)	PUNCT
ejpam-5499	132	4	+	+	CCONJ
ejpam-5499	132	5	4(n−1	4(n−1	NOUN
ejpam-5499	132	6	2	2	NUM
ejpam-5499	132	7	)	)	PUNCT
ejpam-5499	132	8	+	+	CCONJ
ejpam-5499	132	9	(	(	PUNCT
ejpam-5499	132	10	n+	n+	NUM
ejpam-5499	132	11	1)(n−	1)(n−	NUM
ejpam-5499	132	12	1	1	NUM
ejpam-5499	132	13	)	)	PUNCT
ejpam-5499	132	14	=	=	VERB
ejpam-5499	132	15	n−	n−	NOUN
ejpam-5499	132	16	1	1	NUM
ejpam-5499	132	17	+	+	CCONJ
ejpam-5499	132	18	2n−	2n−	NUM
ejpam-5499	132	19	2	2	NUM
ejpam-5499	132	20	+	+	NOUN
ejpam-5499	132	21	2n−	2n−	PROPN
ejpam-5499	132	22	2	2	NUM
ejpam-5499	132	23	+	+	NOUN
ejpam-5499	132	24	n2	n2	ADJ
ejpam-5499	132	25	−	−	PROPN
ejpam-5499	132	26	1	1	NUM
ejpam-5499	132	27	=	=	SYM
ejpam-5499	132	28	n2	n2	NOUN
ejpam-5499	132	29	+	+	X
ejpam-5499	132	30	5n−	5n−	NUM
ejpam-5499	132	31	6	6	NUM
ejpam-5499	132	32	.	.	NOUN
ejpam-5499	132	33	3.2	3.2	NUM
ejpam-5499	132	34	.	.	PUNCT
ejpam-5499	133	1	the	the	DET
ejpam-5499	133	2	middle	middle	ADJ
ejpam-5499	133	3	graph	graph	NOUN
ejpam-5499	133	4	of	of	ADP
ejpam-5499	133	5	γcn	γcn	PROPN
ejpam-5499	133	6	,	,	PUNCT
ejpam-5499	133	7	where	where	SCONJ
ejpam-5499	133	8	n	n	X
ejpam-5499	133	9	is	be	AUX
ejpam-5499	133	10	even189	even189	PROPN
ejpam-5499	133	11	for	for	ADP
ejpam-5499	133	12	the	the	DET
ejpam-5499	133	13	general	general	ADJ
ejpam-5499	133	14	structure	structure	NOUN
ejpam-5499	133	15	of	of	ADP
ejpam-5499	133	16	the	the	DET
ejpam-5499	133	17	middle	middle	ADJ
ejpam-5499	133	18	graph	graph	NOUN
ejpam-5499	133	19	of	of	ADP
ejpam-5499	133	20	γcn	γcn	PROPN
ejpam-5499	133	21	where	where	SCONJ
ejpam-5499	133	22	n	n	PRON
ejpam-5499	133	23	is	be	AUX
ejpam-5499	133	24	even	even	ADV
ejpam-5499	133	25	,	,	PUNCT
ejpam-5499	133	26	set	set	VERB
ejpam-5499	133	27	first	first	ADJ
ejpam-5499	133	28	the190	the190	NOUN
ejpam-5499	133	29	vertices	vertex	NOUN
ejpam-5499	133	30	of	of	ADP
ejpam-5499	133	31	the	the	DET
ejpam-5499	133	32	identity	identity	NOUN
ejpam-5499	133	33	graph	graph	NOUN
ejpam-5499	133	34	of	of	ADP
ejpam-5499	133	35	γcn	γcn	PROPN
ejpam-5499	133	36	as191	as191	PROPN
ejpam-5499	133	37	v	v	PROPN
ejpam-5499	133	38	(	(	PUNCT
ejpam-5499	133	39	γcn	γcn	PROPN
ejpam-5499	133	40	)	)	PUNCT
ejpam-5499	133	41	=	=	PRON
ejpam-5499	133	42	{	{	PUNCT
ejpam-5499	133	43	e	e	NOUN
ejpam-5499	133	44	}	}	PUNCT
ejpam-5499	133	45	∪	∪	ADJ
ejpam-5499	133	46	{	{	PUNCT
ejpam-5499	133	47	xi|1	xi|1	NOUN
ejpam-5499	133	48	≤	≤	NOUN
ejpam-5499	133	49	i	i	PRON
ejpam-5499	133	50	≤	≤	ADJ
ejpam-5499	133	51	n−	n−	NOUN
ejpam-5499	133	52	1	1	NUM
ejpam-5499	133	53	}	}	PUNCT
ejpam-5499	133	54	and192	and192	PROPN
ejpam-5499	133	55	e(γcn	e(γcn	PROPN
ejpam-5499	133	56	)	)	PUNCT
ejpam-5499	133	57	=	=	PRON
ejpam-5499	134	1	{	{	PUNCT
ejpam-5499	134	2	zi|1	zi|1	NOUN
ejpam-5499	134	3	≤	≤	NOUN
ejpam-5499	134	4	i	i	PRON
ejpam-5499	134	5	≤	≤	ADJ
ejpam-5499	134	6	n−	n−	NOUN
ejpam-5499	134	7	1	1	NUM
ejpam-5499	134	8	}	}	PUNCT
ejpam-5499	134	9	∪	∪	X
ejpam-5499	134	10	{	{	PUNCT
ejpam-5499	134	11	yp|p	yp|p	PROPN
ejpam-5499	134	12	=	=	PUNCT
ejpam-5499	135	1	i+1	i+1	NUM
ejpam-5499	135	2	2	2	NUM
ejpam-5499	135	3	for	for	ADP
ejpam-5499	135	4	all	all	DET
ejpam-5499	135	5	odd	odd	ADJ
ejpam-5499	135	6	1	1	NUM
ejpam-5499	135	7	≤	≤	NUM
ejpam-5499	135	8	i	i	PRON
ejpam-5499	135	9	≤	≤	ADJ
ejpam-5499	135	10	n−	n−	NOUN
ejpam-5499	135	11	3	3	NUM
ejpam-5499	135	12	}	}	PUNCT
ejpam-5499	135	13	here	here	ADV
ejpam-5499	135	14	is	be	AUX
ejpam-5499	135	15	the	the	DET
ejpam-5499	135	16	pictorial	pictorial	ADJ
ejpam-5499	135	17	representation,193	representation,193	NOUN
ejpam-5499	135	18	.194	.194	NUM
ejpam-5499	135	19	to	to	PART
ejpam-5499	135	20	generalize	generalize	VERB
ejpam-5499	135	21	the	the	DET
ejpam-5499	135	22	vertex	vertex	NOUN
ejpam-5499	135	23	set	set	NOUN
ejpam-5499	135	24	of	of	ADP
ejpam-5499	135	25	m(γcn	m(γcn	PROPN
ejpam-5499	135	26	)	)	PUNCT
ejpam-5499	135	27	where	where	SCONJ
ejpam-5499	135	28	n	n	PRON
ejpam-5499	135	29	is	be	AUX
ejpam-5499	135	30	odd	odd	ADJ
ejpam-5499	135	31	,	,	PUNCT
ejpam-5499	135	32	we	we	PRON
ejpam-5499	135	33	have;195	have;195	PROPN
ejpam-5499	135	34	v	v	X
ejpam-5499	135	35	(	(	PUNCT
ejpam-5499	135	36	m(γcn	m(γcn	NOUN
ejpam-5499	135	37	)	)	PUNCT
ejpam-5499	135	38	)	)	PUNCT
ejpam-5499	136	1	=	=	PRON
ejpam-5499	136	2	{	{	PUNCT
ejpam-5499	136	3	e	e	NOUN
ejpam-5499	136	4	}	}	PUNCT
ejpam-5499	136	5	⋃	⋃	ADV
ejpam-5499	136	6	{	{	PUNCT
ejpam-5499	136	7	xi|1	xi|1	NOUN
ejpam-5499	136	8	≤	≤	NOUN
ejpam-5499	137	1	i	i	PRON
ejpam-5499	137	2	≤	≤	ADJ
ejpam-5499	137	3	n−	n−	NOUN
ejpam-5499	137	4	1	1	NUM
ejpam-5499	137	5	}	}	PUNCT
ejpam-5499	137	6	⋃	⋃	NOUN
ejpam-5499	137	7	{	{	PUNCT
ejpam-5499	137	8	zi|1	zi|1	NOUN
ejpam-5499	137	9	≤	≤	NOUN
ejpam-5499	137	10	i	i	PRON
ejpam-5499	137	11	≤	≤	ADJ
ejpam-5499	137	12	n−	n−	NOUN
ejpam-5499	137	13	1}⋃	1}⋃	NUM
ejpam-5499	137	14	{	{	PUNCT
ejpam-5499	137	15	yp|p	yp|p	PROPN
ejpam-5499	138	1	=	=	PUNCT
ejpam-5499	139	1	i+1	i+1	NUM
ejpam-5499	139	2	2	2	NUM
ejpam-5499	139	3	for	for	ADP
ejpam-5499	139	4	all	all	DET
ejpam-5499	139	5	odd	odd	ADJ
ejpam-5499	139	6	1	1	NUM
ejpam-5499	139	7	≤	≤	NUM
ejpam-5499	139	8	i	i	PRON
ejpam-5499	139	9	≤	≤	ADJ
ejpam-5499	139	10	n−	n−	NOUN
ejpam-5499	139	11	3	3	NUM
ejpam-5499	139	12	}	}	PUNCT
ejpam-5499	139	13	and196	and196	NOUN
ejpam-5499	139	14	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	139	15	)	)	PUNCT
ejpam-5499	139	16	)	)	PUNCT
ejpam-5499	140	1	=	=	PRON
ejpam-5499	140	2	{	{	PUNCT
ejpam-5499	141	1	[	[	X
ejpam-5499	141	2	e	e	NOUN
ejpam-5499	141	3	,	,	PUNCT
ejpam-5499	141	4	zi]|1	zi]|1	NOUN
ejpam-5499	141	5	≤	≤	NUM
ejpam-5499	142	1	i	i	PRON
ejpam-5499	142	2	≤	≤	ADJ
ejpam-5499	142	3	n−	n−	NOUN
ejpam-5499	142	4	1	1	NUM
ejpam-5499	142	5	}	}	PUNCT
ejpam-5499	142	6	j.	j.	PROPN
ejpam-5499	142	7	m.	m.	PROPN
ejpam-5499	142	8	jamis	jamis	PROPN
ejpam-5499	142	9	,	,	PUNCT
ejpam-5499	142	10	d.	d.	PROPN
ejpam-5499	142	11	m.	m.	PROPN
ejpam-5499	142	12	magpantay	magpantay	PROPN
ejpam-5499	142	13	/	/	SYM
ejpam-5499	142	14	eur	eur	PROPN
ejpam-5499	142	15	.	.	PUNCT
ejpam-5499	143	1	j.	j.	PROPN
ejpam-5499	143	2	pure	pure	PROPN
ejpam-5499	143	3	appl	appl	PROPN
ejpam-5499	143	4	.	.	PROPN
ejpam-5499	143	5	math	math	PROPN
ejpam-5499	143	6	,	,	PUNCT
ejpam-5499	143	7	18	18	NUM
ejpam-5499	143	8	(	(	PUNCT
ejpam-5499	143	9	2	2	NUM
ejpam-5499	143	10	)	)	PUNCT
ejpam-5499	143	11	(	(	PUNCT
ejpam-5499	143	12	2025	2025	NUM
ejpam-5499	143	13	)	)	PUNCT
ejpam-5499	143	14	,	,	PUNCT
ejpam-5499	143	15	5499	5499	NUM
ejpam-5499	143	16	9	9	NUM
ejpam-5499	143	17	of	of	ADP
ejpam-5499	143	18	18⋃	18⋃	NUM
ejpam-5499	143	19	{	{	PUNCT
ejpam-5499	143	20	[	[	X
ejpam-5499	143	21	zi	zi	PROPN
ejpam-5499	143	22	,	,	PUNCT
ejpam-5499	143	23	zj	zj	PROPN
ejpam-5499	143	24	]	]	X
ejpam-5499	143	25	|1	|1	NUM
ejpam-5499	143	26	≤	≤	NUM
ejpam-5499	144	1	i	i	PRON
ejpam-5499	144	2	≤	≤	ADJ
ejpam-5499	144	3	n−	n−	PROPN
ejpam-5499	144	4	1	1	NUM
ejpam-5499	144	5	,	,	PUNCT
ejpam-5499	144	6	1	1	NUM
ejpam-5499	144	7	≤	≤	NUM
ejpam-5499	144	8	j	j	PROPN
ejpam-5499	144	9	≤	≤	PROPN
ejpam-5499	144	10	n−	n−	PROPN
ejpam-5499	144	11	1	1	NUM
ejpam-5499	144	12	,	,	PUNCT
ejpam-5499	144	13	i	i	PRON
ejpam-5499	144	14	̸=	̸=	PROPN
ejpam-5499	144	15	j}⋃	j}⋃	PROPN
ejpam-5499	144	16	{	{	PUNCT
ejpam-5499	144	17	[	[	X
ejpam-5499	144	18	xi	xi	X
ejpam-5499	144	19	,	,	PUNCT
ejpam-5499	144	20	zi]|1	zi]|1	NOUN
ejpam-5499	144	21	≤	≤	NUM
ejpam-5499	144	22	i	i	PRON
ejpam-5499	144	23	≤	≤	ADJ
ejpam-5499	144	24	n−	n−	PROPN
ejpam-5499	144	25	1	1	NUM
ejpam-5499	144	26	,	,	PUNCT
ejpam-5499	144	27	1	1	NUM
ejpam-5499	144	28	≤	≤	NUM
ejpam-5499	144	29	j	j	PROPN
ejpam-5499	144	30	≤	≤	PROPN
ejpam-5499	144	31	n−	n−	PROPN
ejpam-5499	144	32	1}⋃	1}⋃	NUM
ejpam-5499	144	33	{	{	PUNCT
ejpam-5499	144	34	[	[	X
ejpam-5499	144	35	yp	yp	X
ejpam-5499	144	36	,	,	PUNCT
ejpam-5499	144	37	zi]|p	zi]|p	PROPN
ejpam-5499	144	38	=	=	PUNCT
ejpam-5499	144	39	i+1	i+1	SYM
ejpam-5499	144	40	2	2	NUM
ejpam-5499	144	41	for	for	ADP
ejpam-5499	144	42	all	all	DET
ejpam-5499	144	43	odd1	odd1	NOUN
ejpam-5499	144	44	≤	≤	ADV
ejpam-5499	144	45	i	i	PRON
ejpam-5499	144	46	≤	≤	ADJ
ejpam-5499	144	47	n−	n−	NOUN
ejpam-5499	144	48	3	3	NUM
ejpam-5499	144	49	}	}	PUNCT
ejpam-5499	144	50	⋃	⋃	NOUN
ejpam-5499	144	51	{	{	PUNCT
ejpam-5499	144	52	[	[	X
ejpam-5499	144	53	yp	yp	PROPN
ejpam-5499	144	54	,	,	PUNCT
ejpam-5499	144	55	zi+1]|p	zi+1]|p	X
ejpam-5499	144	56	=	=	SYM
ejpam-5499	144	57	i+1	i+1	SYM
ejpam-5499	144	58	2	2	NUM
ejpam-5499	144	59	for	for	ADP
ejpam-5499	144	60	all	all	DET
ejpam-5499	144	61	odd1	odd1	NOUN
ejpam-5499	144	62	≤	≤	ADV
ejpam-5499	145	1	i	i	PRON
ejpam-5499	145	2	≤	≤	ADJ
ejpam-5499	145	3	n−	n−	NOUN
ejpam-5499	145	4	3	3	NUM
ejpam-5499	145	5	}	}	PUNCT
ejpam-5499	145	6	⋃	⋃	NOUN
ejpam-5499	145	7	{	{	PUNCT
ejpam-5499	145	8	[	[	X
ejpam-5499	145	9	yp	yp	PROPN
ejpam-5499	145	10	,	,	PUNCT
ejpam-5499	145	11	xi]|p	xi]|p	X
ejpam-5499	145	12	=	=	PUNCT
ejpam-5499	146	1	i+1	i+1	SYM
ejpam-5499	146	2	2	2	NUM
ejpam-5499	146	3	for	for	ADP
ejpam-5499	146	4	all	all	DET
ejpam-5499	146	5	odd1	odd1	NOUN
ejpam-5499	146	6	≤	≤	ADV
ejpam-5499	147	1	i	i	PRON
ejpam-5499	147	2	≤	≤	ADJ
ejpam-5499	147	3	n−	n−	NOUN
ejpam-5499	147	4	3	3	NUM
ejpam-5499	147	5	}	}	PUNCT
ejpam-5499	147	6	⋃	⋃	NOUN
ejpam-5499	147	7	{	{	PUNCT
ejpam-5499	147	8	[	[	X
ejpam-5499	147	9	yp	yp	X
ejpam-5499	147	10	,	,	PUNCT
ejpam-5499	147	11	xi+1]|p	xi+1]|p	PROPN
ejpam-5499	147	12	=	=	PUNCT
ejpam-5499	147	13	i+1	i+1	SYM
ejpam-5499	147	14	2	2	NUM
ejpam-5499	147	15	for	for	ADP
ejpam-5499	147	16	all	all	DET
ejpam-5499	147	17	odd1	odd1	NOUN
ejpam-5499	147	18	≤	≤	ADV
ejpam-5499	148	1	i	i	PRON
ejpam-5499	148	2	≤	≤	ADJ
ejpam-5499	148	3	n−	n−	NOUN
ejpam-5499	148	4	3	3	NUM
ejpam-5499	148	5	}	}	PUNCT
ejpam-5499	148	6	here	here	ADV
ejpam-5499	148	7	is	be	AUX
ejpam-5499	148	8	the	the	DET
ejpam-5499	148	9	pictorial	pictorial	ADJ
ejpam-5499	148	10	representation,197	representation,197	PROPN
ejpam-5499	148	11	.198	.198	NUM
ejpam-5499	148	12	the	the	DET
ejpam-5499	148	13	degree	degree	NOUN
ejpam-5499	148	14	of	of	ADP
ejpam-5499	148	15	the	the	DET
ejpam-5499	148	16	vertices	vertex	NOUN
ejpam-5499	148	17	of	of	ADP
ejpam-5499	148	18	m(γcn	m(γcn	PROPN
ejpam-5499	148	19	)	)	PUNCT
ejpam-5499	148	20	where	where	SCONJ
ejpam-5499	148	21	n	n	PRON
ejpam-5499	148	22	is	be	AUX
ejpam-5499	148	23	even	even	ADV
ejpam-5499	148	24	is	be	AUX
ejpam-5499	148	25	summarized	summarize	VERB
ejpam-5499	148	26	below:199	below:199	NOUN
ejpam-5499	148	27	a.	a.	NOUN
ejpam-5499	148	28	deg(e	deg(e	NOUN
ejpam-5499	148	29	)	)	PUNCT
ejpam-5499	149	1	=	=	SYM
ejpam-5499	149	2	n−	n−	PROPN
ejpam-5499	149	3	1200	1200	NUM
ejpam-5499	149	4	b.	b.	PROPN
ejpam-5499	149	5	deg(xi	deg(xi	PROPN
ejpam-5499	149	6	)	)	PUNCT
ejpam-5499	149	7	=	=	SYM
ejpam-5499	149	8	2	2	NUM
ejpam-5499	149	9	,	,	PUNCT
ejpam-5499	149	10	1	1	NUM
ejpam-5499	149	11	≤	≤	NUM
ejpam-5499	149	12	i	i	PRON
ejpam-5499	149	13	≤	≤	ADJ
ejpam-5499	149	14	n−	n−	PROPN
ejpam-5499	149	15	2201	2201	NUM
ejpam-5499	149	16	c.	c.	NOUN
ejpam-5499	149	17	deg(xn−1	deg(xn−1	PROPN
ejpam-5499	149	18	)	)	PUNCT
ejpam-5499	149	19	=	=	SYM
ejpam-5499	149	20	1202	1202	NUM
ejpam-5499	149	21	d.	d.	PROPN
ejpam-5499	149	22	deg(yp	deg(yp	PROPN
ejpam-5499	149	23	)	)	PUNCT
ejpam-5499	149	24	=	=	SYM
ejpam-5499	149	25	4	4	NUM
ejpam-5499	149	26	,	,	PUNCT
ejpam-5499	149	27	1	1	NUM
ejpam-5499	149	28	≤	≤	NOUN
ejpam-5499	149	29	p	p	ADJ
ejpam-5499	149	30	≤	≤	ADJ
ejpam-5499	149	31	n−2	n−2	PROPN
ejpam-5499	149	32	2203	2203	NUM
ejpam-5499	149	33	e.	e.	PROPN
ejpam-5499	149	34	deg(zi	deg(zi	PROPN
ejpam-5499	149	35	)	)	PUNCT
ejpam-5499	149	36	=	=	SYM
ejpam-5499	149	37	n+	n+	PUNCT
ejpam-5499	149	38	1	1	NUM
ejpam-5499	149	39	,	,	PUNCT
ejpam-5499	149	40	1	1	NUM
ejpam-5499	149	41	≤	≤	NUM
ejpam-5499	149	42	i	i	PRON
ejpam-5499	149	43	≤	≤	ADJ
ejpam-5499	149	44	n−	n−	PROPN
ejpam-5499	149	45	2204	2204	NUM
ejpam-5499	149	46	f.	f.	PROPN
ejpam-5499	149	47	deg(zn−1	deg(zn−1	PROPN
ejpam-5499	149	48	)	)	PUNCT
ejpam-5499	149	49	=	=	SYM
ejpam-5499	149	50	n205	n205	NUM
ejpam-5499	149	51	to	to	PART
ejpam-5499	149	52	sum	sum	VERB
ejpam-5499	149	53	up	up	ADP
ejpam-5499	149	54	all	all	DET
ejpam-5499	149	55	the	the	DET
ejpam-5499	149	56	degrees	degree	NOUN
ejpam-5499	149	57	of	of	ADP
ejpam-5499	149	58	the	the	DET
ejpam-5499	149	59	vertices	vertex	NOUN
ejpam-5499	149	60	of	of	ADP
ejpam-5499	149	61	m(γcn	m(γcn	PROPN
ejpam-5499	149	62	)	)	PUNCT
ejpam-5499	149	63	where	where	SCONJ
ejpam-5499	149	64	n	n	X
ejpam-5499	149	65	is	be	AUX
ejpam-5499	149	66	even	even	ADV
ejpam-5499	149	67	we	we	PRON
ejpam-5499	149	68	have:206	have:206	INTJ
ejpam-5499	149	69	∑	∑	PUNCT
ejpam-5499	149	70	v∈v	v∈v	PROPN
ejpam-5499	149	71	(	(	PUNCT
ejpam-5499	149	72	m(γcn	m(γcn	PROPN
ejpam-5499	149	73	)	)	PUNCT
ejpam-5499	149	74	)	)	PUNCT
ejpam-5499	150	1	deg(v	deg(v	PROPN
ejpam-5499	150	2	)	)	PUNCT
ejpam-5499	150	3	=	=	PUNCT
ejpam-5499	150	4	(	(	PUNCT
ejpam-5499	150	5	n−	n−	NOUN
ejpam-5499	150	6	1	1	NUM
ejpam-5499	150	7	)	)	PUNCT
ejpam-5499	151	1	+	+	CCONJ
ejpam-5499	151	2	1	1	NUM
ejpam-5499	151	3	+	+	CCONJ
ejpam-5499	151	4	n+	n+	PUNCT
ejpam-5499	152	1	2(n−	2(n−	NUM
ejpam-5499	152	2	2	2	NUM
ejpam-5499	152	3	)	)	PUNCT
ejpam-5499	152	4	+	+	NUM
ejpam-5499	152	5	4(n−2	4(n−2	NOUN
ejpam-5499	152	6	2	2	NUM
ejpam-5499	152	7	)	)	PUNCT
ejpam-5499	152	8	+	+	CCONJ
ejpam-5499	152	9	(	(	PUNCT
ejpam-5499	152	10	n+	n+	NUM
ejpam-5499	152	11	1)(n−	1)(n−	NUM
ejpam-5499	152	12	2	2	NUM
ejpam-5499	152	13	)	)	PUNCT
ejpam-5499	152	14	=	=	SYM
ejpam-5499	152	15	2n+	2n+	NUM
ejpam-5499	153	1	2n−	2n−	NUM
ejpam-5499	153	2	4	4	NUM
ejpam-5499	153	3	+	+	NOUN
ejpam-5499	153	4	2n−	2n−	PROPN
ejpam-5499	153	5	4	4	NUM
ejpam-5499	153	6	+	+	NOUN
ejpam-5499	153	7	n2	n2	ADJ
ejpam-5499	153	8	−	−	PROPN
ejpam-5499	153	9	n−	n−	NOUN
ejpam-5499	153	10	2	2	NUM
ejpam-5499	153	11	=	=	NOUN
ejpam-5499	153	12	n2	n2	NOUN
ejpam-5499	153	13	+	+	X
ejpam-5499	153	14	5n−	5n−	NUM
ejpam-5499	153	15	10	10	NUM
ejpam-5499	153	16	j.	j.	PROPN
ejpam-5499	153	17	m.	m.	PROPN
ejpam-5499	153	18	jamis	jamis	PROPN
ejpam-5499	153	19	,	,	PUNCT
ejpam-5499	153	20	d.	d.	PROPN
ejpam-5499	153	21	m.	m.	PROPN
ejpam-5499	153	22	magpantay	magpantay	PROPN
ejpam-5499	153	23	/	/	SYM
ejpam-5499	153	24	eur	eur	PROPN
ejpam-5499	153	25	.	.	PUNCT
ejpam-5499	154	1	j.	j.	PROPN
ejpam-5499	154	2	pure	pure	PROPN
ejpam-5499	154	3	appl	appl	PROPN
ejpam-5499	154	4	.	.	PROPN
ejpam-5499	154	5	math	math	PROPN
ejpam-5499	154	6	,	,	PUNCT
ejpam-5499	154	7	18	18	NUM
ejpam-5499	154	8	(	(	PUNCT
ejpam-5499	154	9	2	2	NUM
ejpam-5499	154	10	)	)	PUNCT
ejpam-5499	154	11	(	(	PUNCT
ejpam-5499	154	12	2025	2025	NUM
ejpam-5499	154	13	)	)	PUNCT
ejpam-5499	154	14	,	,	PUNCT
ejpam-5499	154	15	5499	5499	NUM
ejpam-5499	154	16	10	10	NUM
ejpam-5499	154	17	of	of	ADP
ejpam-5499	154	18	18	18	NUM
ejpam-5499	154	19	theorem	theorem	NOUN
ejpam-5499	154	20	1	1	NUM
ejpam-5499	154	21	.	.	PUNCT
ejpam-5499	155	1	let	let	VERB
ejpam-5499	155	2	cn	cn	PROPN
ejpam-5499	155	3	be	be	AUX
ejpam-5499	155	4	a	a	DET
ejpam-5499	155	5	cyclic	cyclic	ADJ
ejpam-5499	155	6	group	group	NOUN
ejpam-5499	155	7	of	of	ADP
ejpam-5499	155	8	order	order	NOUN
ejpam-5499	155	9	n	n	NOUN
ejpam-5499	155	10	and	and	CCONJ
ejpam-5499	155	11	m(γcn	m(γcn	PROPN
ejpam-5499	155	12	)	)	PUNCT
ejpam-5499	155	13	be	be	VERB
ejpam-5499	155	14	the	the	DET
ejpam-5499	155	15	mig	mig	NOUN
ejpam-5499	155	16	of	of	ADP
ejpam-5499	155	17	cn	cn	PROPN
ejpam-5499	155	18	for207	for207	PROPN
ejpam-5499	155	19	n	n	PROPN
ejpam-5499	155	20	≥	≥	NOUN
ejpam-5499	155	21	2	2	NUM
ejpam-5499	155	22	.	.	PUNCT
ejpam-5499	156	1	the	the	DET
ejpam-5499	156	2	order	order	NOUN
ejpam-5499	156	3	of	of	ADP
ejpam-5499	156	4	m(γcn	m(γcn	PROPN
ejpam-5499	156	5	)	)	PUNCT
ejpam-5499	156	6	is208	is208	ADJ
ejpam-5499	156	7	|v	|v	PROPN
ejpam-5499	156	8	(	(	PUNCT
ejpam-5499	156	9	m(γcn))|	m(γcn))|	PROPN
ejpam-5499	156	10	=	=	SYM
ejpam-5499	156	11	{	{	PUNCT
ejpam-5499	156	12	5n−3	5n−3	NUM
ejpam-5499	156	13	2	2	NUM
ejpam-5499	156	14	,	,	PUNCT
ejpam-5499	156	15	if	if	SCONJ
ejpam-5499	156	16	n	n	PRON
ejpam-5499	156	17	is	be	AUX
ejpam-5499	156	18	odd	odd	ADJ
ejpam-5499	156	19	5n−4	5n−4	NUM
ejpam-5499	156	20	2	2	NUM
ejpam-5499	156	21	,	,	PUNCT
ejpam-5499	156	22	if	if	SCONJ
ejpam-5499	156	23	n	n	PRON
ejpam-5499	156	24	is	be	AUX
ejpam-5499	156	25	even	even	ADV
ejpam-5499	156	26	.	.	PUNCT
ejpam-5499	157	1	209	209	NUM
ejpam-5499	157	2	proof.210	proof.210	PROPN
ejpam-5499	157	3	i.	i.	NOUN
ejpam-5499	157	4	for	for	ADP
ejpam-5499	157	5	n	n	PROPN
ejpam-5499	157	6	is	be	AUX
ejpam-5499	157	7	odd,211	odd,211	PROPN
ejpam-5499	157	8	|v	|v	PROPN
ejpam-5499	157	9	(	(	PUNCT
ejpam-5499	157	10	m(γcn)|	m(γcn)|	NOUN
ejpam-5499	157	11	=	=	SYM
ejpam-5499	157	12	|v	|v	PROPN
ejpam-5499	157	13	(	(	PUNCT
ejpam-5499	157	14	γcn)|+	γcn)|+	NOUN
ejpam-5499	157	15	|e(γcn)|	|e(γcn)|	X
ejpam-5499	157	16	=	=	SYM
ejpam-5499	157	17	n+	n+	PROPN
ejpam-5499	157	18	3(n−	3(n−	NUM
ejpam-5499	157	19	1	1	NUM
ejpam-5499	157	20	)	)	PUNCT
ejpam-5499	157	21	2	2	NUM
ejpam-5499	157	22	=	=	SYM
ejpam-5499	157	23	2n+	2n+	NUM
ejpam-5499	158	1	3n−	3n−	NUM
ejpam-5499	158	2	3	3	NUM
ejpam-5499	158	3	2	2	NUM
ejpam-5499	158	4	=	=	SYM
ejpam-5499	158	5	5n−	5n−	NUM
ejpam-5499	158	6	3	3	NUM
ejpam-5499	158	7	2	2	NUM
ejpam-5499	158	8	ii	ii	NOUN
ejpam-5499	158	9	.	.	PUNCT
ejpam-5499	159	1	for	for	ADP
ejpam-5499	159	2	n	n	PRON
ejpam-5499	159	3	is	be	AUX
ejpam-5499	159	4	even,212	even,212	PROPN
ejpam-5499	159	5	|v	|v	X
ejpam-5499	159	6	(	(	PUNCT
ejpam-5499	159	7	m(γcn)|	m(γcn)|	NOUN
ejpam-5499	159	8	=	=	SYM
ejpam-5499	159	9	|v	|v	PROPN
ejpam-5499	159	10	(	(	PUNCT
ejpam-5499	159	11	γcn)|+	γcn)|+	NOUN
ejpam-5499	159	12	|e(γcn)|	|e(γcn)|	X
ejpam-5499	159	13	=	=	SYM
ejpam-5499	159	14	n+	n+	NOUN
ejpam-5499	159	15	[	[	PUNCT
ejpam-5499	159	16	3(n−	3(n−	NUM
ejpam-5499	159	17	2	2	NUM
ejpam-5499	159	18	)	)	PUNCT
ejpam-5499	159	19	2	2	NUM
ejpam-5499	160	1	+	+	CCONJ
ejpam-5499	160	2	1	1	NUM
ejpam-5499	160	3	]	]	X
ejpam-5499	160	4	=	=	SYM
ejpam-5499	160	5	(	(	PUNCT
ejpam-5499	160	6	2n+	2n+	NUM
ejpam-5499	160	7	3n−	3n−	NUM
ejpam-5499	160	8	6	6	NUM
ejpam-5499	160	9	)	)	PUNCT
ejpam-5499	160	10	+	+	CCONJ
ejpam-5499	160	11	2	2	NUM
ejpam-5499	160	12	2	2	NUM
ejpam-5499	160	13	=	=	SYM
ejpam-5499	160	14	5n−	5n−	NUM
ejpam-5499	160	15	4	4	NUM
ejpam-5499	160	16	2	2	NUM
ejpam-5499	160	17	now	now	ADV
ejpam-5499	160	18	,	,	PUNCT
ejpam-5499	160	19	for	for	ADP
ejpam-5499	160	20	the	the	DET
ejpam-5499	160	21	size	size	NOUN
ejpam-5499	160	22	of	of	ADP
ejpam-5499	160	23	the	the	DET
ejpam-5499	160	24	middle	middle	ADJ
ejpam-5499	160	25	graph	graph	NOUN
ejpam-5499	160	26	of	of	ADP
ejpam-5499	160	27	identity	identity	NOUN
ejpam-5499	160	28	graph	graph	NOUN
ejpam-5499	160	29	of	of	ADP
ejpam-5499	160	30	a	a	DET
ejpam-5499	160	31	cyclic	cyclic	ADJ
ejpam-5499	160	32	group	group	NOUN
ejpam-5499	160	33	,	,	PUNCT
ejpam-5499	160	34	refer	refer	VERB
ejpam-5499	160	35	to	to	ADP
ejpam-5499	160	36	the213	the213	PROPN
ejpam-5499	160	37	theorem	theorem	ADJ
ejpam-5499	160	38	below.214	below.214	ADJ
ejpam-5499	160	39	theorem	theorem	NOUN
ejpam-5499	160	40	2	2	NUM
ejpam-5499	160	41	.	.	PUNCT
ejpam-5499	160	42	the	the	DET
ejpam-5499	160	43	mig	mig	NOUN
ejpam-5499	160	44	of	of	ADP
ejpam-5499	160	45	a	a	DET
ejpam-5499	160	46	cyclic	cyclic	ADJ
ejpam-5499	160	47	group	group	NOUN
ejpam-5499	160	48	cn	cn	NOUN
ejpam-5499	160	49	of	of	ADP
ejpam-5499	160	50	order	order	NOUN
ejpam-5499	160	51	n	n	PRON
ejpam-5499	160	52	has	have	AUX
ejpam-5499	160	53	size215	size215	VERB
ejpam-5499	160	54	|e(m(γcn))|	|e(m(γcn))|	NOUN
ejpam-5499	160	55	=	=	PUNCT
ejpam-5499	160	56	{	{	PUNCT
ejpam-5499	160	57	n2	n2	ADJ
ejpam-5499	160	58	+	+	NOUN
ejpam-5499	160	59	5n−6	5n−6	NOUN
ejpam-5499	160	60	2	2	NUM
ejpam-5499	160	61	,	,	PUNCT
ejpam-5499	160	62	if	if	SCONJ
ejpam-5499	160	63	n	n	PRON
ejpam-5499	160	64	is	be	AUX
ejpam-5499	160	65	odd	odd	ADJ
ejpam-5499	160	66	n2	n2	ADJ
ejpam-5499	160	67	+	+	PROPN
ejpam-5499	160	68	5n−10	5n−10	NOUN
ejpam-5499	160	69	2	2	NUM
ejpam-5499	160	70	,	,	PUNCT
ejpam-5499	160	71	if	if	SCONJ
ejpam-5499	160	72	n	n	PRON
ejpam-5499	160	73	is	be	AUX
ejpam-5499	160	74	even	even	ADV
ejpam-5499	160	75	.	.	PUNCT
ejpam-5499	161	1	216	216	NUM
ejpam-5499	161	2	proof	proof	NOUN
ejpam-5499	161	3	.	.	PUNCT
ejpam-5499	162	1	to	to	PART
ejpam-5499	162	2	prove	prove	VERB
ejpam-5499	162	3	this	this	PRON
ejpam-5499	162	4	,	,	PUNCT
ejpam-5499	162	5	we	we	PRON
ejpam-5499	162	6	need	need	VERB
ejpam-5499	162	7	to	to	PART
ejpam-5499	162	8	consider	consider	VERB
ejpam-5499	162	9	two	two	NUM
ejpam-5499	162	10	cases.217	cases.217	ADJ
ejpam-5499	162	11	i.	i.	NOUN
ejpam-5499	162	12	first	first	ADV
ejpam-5499	162	13	we	we	PRON
ejpam-5499	162	14	will	will	AUX
ejpam-5499	162	15	consider	consider	VERB
ejpam-5499	162	16	if	if	SCONJ
ejpam-5499	162	17	n	n	PRON
ejpam-5499	162	18	is	be	AUX
ejpam-5499	162	19	odd	odd	ADJ
ejpam-5499	162	20	.	.	PUNCT
ejpam-5499	163	1	from	from	ADP
ejpam-5499	163	2	the	the	DET
ejpam-5499	163	3	summation	summation	NOUN
ejpam-5499	163	4	of	of	ADP
ejpam-5499	163	5	all	all	PRON
ejpam-5499	163	6	of	of	ADP
ejpam-5499	163	7	the	the	DET
ejpam-5499	163	8	degrees	degree	NOUN
ejpam-5499	163	9	of218	of218	PROPN
ejpam-5499	163	10	the	the	DET
ejpam-5499	163	11	vertices	vertex	NOUN
ejpam-5499	163	12	where	where	SCONJ
ejpam-5499	163	13	n	n	PRON
ejpam-5499	163	14	is	be	AUX
ejpam-5499	163	15	odd	odd	ADJ
ejpam-5499	163	16	,	,	PUNCT
ejpam-5499	163	17	∑	∑	ADV
ejpam-5499	163	18	v∈v	v∈v	NOUN
ejpam-5499	163	19	(	(	PUNCT
ejpam-5499	163	20	m(γcn	m(γcn	PROPN
ejpam-5499	163	21	)	)	PUNCT
ejpam-5499	163	22	)	)	PUNCT
ejpam-5499	164	1	deg(v	deg(v	X
ejpam-5499	164	2	)	)	PUNCT
ejpam-5499	164	3	=	=	SYM
ejpam-5499	164	4	n2	n2	NOUN
ejpam-5499	164	5	+	+	NUM
ejpam-5499	164	6	5n	5n	NUM
ejpam-5499	164	7	−	−	NOUN
ejpam-5499	164	8	6	6	NUM
ejpam-5499	164	9	.	.	PUNCT
ejpam-5499	165	1	by	by	ADP
ejpam-5499	165	2	theorem219	theorem219	PROPN
ejpam-5499	165	3	3	3	NUM
ejpam-5499	165	4	,	,	PUNCT
ejpam-5499	165	5	for	for	ADP
ejpam-5499	165	6	a	a	DET
ejpam-5499	165	7	graph	graph	NOUN
ejpam-5499	165	8	of	of	ADP
ejpam-5499	165	9	size	size	NOUN
ejpam-5499	165	10	m	m	PROPN
ejpam-5499	165	11	,	,	PUNCT
ejpam-5499	165	12	∑	∑	PUNCT
ejpam-5499	165	13	v∈v	v∈v	NOUN
ejpam-5499	165	14	(	(	PUNCT
ejpam-5499	165	15	m(γcn	m(γcn	PROPN
ejpam-5499	165	16	)	)	PUNCT
ejpam-5499	165	17	)	)	PUNCT
ejpam-5499	166	1	deg(v	deg(v	X
ejpam-5499	166	2	)	)	PUNCT
ejpam-5499	166	3	=	=	SYM
ejpam-5499	166	4	2	2	NUM
ejpam-5499	166	5	m.	m.	NOUN
ejpam-5499	166	6	by	by	ADP
ejpam-5499	166	7	substitution	substitution	NOUN
ejpam-5499	166	8	,	,	PUNCT
ejpam-5499	166	9	we	we	PRON
ejpam-5499	166	10	have220	have220	PROPN
ejpam-5499	166	11	n2	n2	PROPN
ejpam-5499	166	12	+	+	X
ejpam-5499	166	13	5n−	5n−	NUM
ejpam-5499	166	14	6	6	NUM
ejpam-5499	166	15	=	=	SYM
ejpam-5499	166	16	2	2	NUM
ejpam-5499	166	17	m.	m.	NOUN
ejpam-5499	166	18	thus	thus	ADV
ejpam-5499	166	19	m	m	NOUN
ejpam-5499	166	20	=	=	SYM
ejpam-5499	166	21	n2	n2	NOUN
ejpam-5499	166	22	+	+	NOUN
ejpam-5499	166	23	5n−6	5n−6	NUM
ejpam-5499	166	24	2	2	NUM
ejpam-5499	166	25	.221	.221	NUM
ejpam-5499	166	26	ii	ii	NOUN
ejpam-5499	166	27	.	.	PUNCT
ejpam-5499	167	1	for	for	ADP
ejpam-5499	167	2	n	n	NUM
ejpam-5499	167	3	is	be	AUX
ejpam-5499	167	4	even	even	ADV
ejpam-5499	167	5	,	,	PUNCT
ejpam-5499	167	6	using	use	VERB
ejpam-5499	167	7	the	the	DET
ejpam-5499	167	8	summary	summary	NOUN
ejpam-5499	167	9	of	of	ADP
ejpam-5499	167	10	the	the	DET
ejpam-5499	167	11	degree	degree	NOUN
ejpam-5499	167	12	of	of	ADP
ejpam-5499	167	13	the	the	DET
ejpam-5499	167	14	vertices	vertex	NOUN
ejpam-5499	167	15	where	where	SCONJ
ejpam-5499	167	16	n	n	PRON
ejpam-5499	167	17	is	be	AUX
ejpam-5499	167	18	even,222	even,222	PROPN
ejpam-5499	167	19	∑	∑	PUNCT
ejpam-5499	167	20	v∈v	v∈v	PROPN
ejpam-5499	167	21	(	(	PUNCT
ejpam-5499	167	22	m(γcn	m(γcn	PROPN
ejpam-5499	167	23	)	)	PUNCT
ejpam-5499	167	24	)	)	PUNCT
ejpam-5499	168	1	deg(v	deg(v	X
ejpam-5499	168	2	)	)	PUNCT
ejpam-5499	168	3	=	=	SYM
ejpam-5499	168	4	n2	n2	PROPN
ejpam-5499	168	5	+	+	NOUN
ejpam-5499	168	6	5n−10	5n−10	NUM
ejpam-5499	168	7	.	.	PUNCT
ejpam-5499	168	8	also	also	ADV
ejpam-5499	168	9	by	by	ADP
ejpam-5499	168	10	theorem	theorem	NOUN
ejpam-5499	168	11	3	3	NUM
ejpam-5499	168	12	,	,	PUNCT
ejpam-5499	168	13	the	the	DET
ejpam-5499	168	14	sum	sum	NOUN
ejpam-5499	168	15	of	of	ADP
ejpam-5499	168	16	all	all	PRON
ejpam-5499	168	17	of	of	ADP
ejpam-5499	168	18	its	its	PRON
ejpam-5499	168	19	vertices223	vertices223	PROPN
ejpam-5499	168	20	is	be	AUX
ejpam-5499	168	21	∑	∑	PUNCT
ejpam-5499	168	22	v∈v	v∈v	PROPN
ejpam-5499	168	23	(	(	PUNCT
ejpam-5499	168	24	m(γcn	m(γcn	PROPN
ejpam-5499	168	25	)	)	PUNCT
ejpam-5499	168	26	)	)	PUNCT
ejpam-5499	169	1	deg(v	deg(v	X
ejpam-5499	169	2	)	)	PUNCT
ejpam-5499	169	3	=	=	SYM
ejpam-5499	170	1	2	2	NUM
ejpam-5499	170	2	m	m	NOUN
ejpam-5499	170	3	where	where	SCONJ
ejpam-5499	170	4	m	m	NOUN
ejpam-5499	170	5	is	be	AUX
ejpam-5499	170	6	the	the	DET
ejpam-5499	170	7	size	size	NOUN
ejpam-5499	170	8	of	of	ADP
ejpam-5499	170	9	the	the	DET
ejpam-5499	170	10	graph	graph	NOUN
ejpam-5499	170	11	.	.	PUNCT
ejpam-5499	171	1	by	by	ADP
ejpam-5499	171	2	substitution	substitution	NOUN
ejpam-5499	171	3	,	,	PUNCT
ejpam-5499	171	4	we224	we224	PROPN
ejpam-5499	171	5	have	have	AUX
ejpam-5499	171	6	n2	n2	NOUN
ejpam-5499	171	7	+	+	X
ejpam-5499	172	1	5n−	5n−	NUM
ejpam-5499	172	2	10	10	NUM
ejpam-5499	172	3	=	=	SYM
ejpam-5499	172	4	2	2	NUM
ejpam-5499	172	5	m.	m.	NOUN
ejpam-5499	172	6	thus	thus	ADV
ejpam-5499	172	7	m	m	NOUN
ejpam-5499	172	8	=	=	SYM
ejpam-5499	172	9	n2	n2	NOUN
ejpam-5499	172	10	+	+	PROPN
ejpam-5499	172	11	5n−10	5n−10	NUM
ejpam-5499	172	12	2	2	NUM
ejpam-5499	172	13	.225	.225	NUM
ejpam-5499	172	14	j.	j.	PROPN
ejpam-5499	172	15	m.	m.	PROPN
ejpam-5499	172	16	jamis	jamis	PROPN
ejpam-5499	172	17	,	,	PUNCT
ejpam-5499	172	18	d.	d.	PROPN
ejpam-5499	172	19	m.	m.	PROPN
ejpam-5499	172	20	magpantay	magpantay	PROPN
ejpam-5499	172	21	/	/	SYM
ejpam-5499	172	22	eur	eur	PROPN
ejpam-5499	172	23	.	.	PUNCT
ejpam-5499	173	1	j.	j.	PROPN
ejpam-5499	173	2	pure	pure	PROPN
ejpam-5499	173	3	appl	appl	PROPN
ejpam-5499	173	4	.	.	PROPN
ejpam-5499	173	5	math	math	PROPN
ejpam-5499	173	6	,	,	PUNCT
ejpam-5499	173	7	18	18	NUM
ejpam-5499	173	8	(	(	PUNCT
ejpam-5499	173	9	2	2	NUM
ejpam-5499	173	10	)	)	PUNCT
ejpam-5499	173	11	(	(	PUNCT
ejpam-5499	173	12	2025	2025	NUM
ejpam-5499	173	13	)	)	PUNCT
ejpam-5499	173	14	,	,	PUNCT
ejpam-5499	173	15	5499	5499	NUM
ejpam-5499	173	16	11	11	NUM
ejpam-5499	173	17	of	of	ADP
ejpam-5499	173	18	18	18	NUM
ejpam-5499	173	19	4	4	NUM
ejpam-5499	173	20	.	.	PUNCT
ejpam-5499	173	21	properties	property	NOUN
ejpam-5499	173	22	of	of	ADP
ejpam-5499	173	23	m(γcn	m(γcn	PROPN
ejpam-5499	173	24	)	)	PUNCT
ejpam-5499	173	25	on	on	ADP
ejpam-5499	173	26	some	some	DET
ejpam-5499	173	27	parameters226	parameters226	PROPN
ejpam-5499	173	28	in	in	ADP
ejpam-5499	173	29	this	this	DET
ejpam-5499	173	30	section	section	NOUN
ejpam-5499	173	31	,	,	PUNCT
ejpam-5499	173	32	we	we	PRON
ejpam-5499	173	33	will	will	AUX
ejpam-5499	173	34	explore	explore	VERB
ejpam-5499	173	35	the	the	DET
ejpam-5499	173	36	graphical	graphical	ADJ
ejpam-5499	173	37	properties	property	NOUN
ejpam-5499	173	38	of	of	ADP
ejpam-5499	173	39	migs	mig	NOUN
ejpam-5499	173	40	to	to	PART
ejpam-5499	173	41	further	further	VERB
ejpam-5499	173	42	understand227	understand227	PROPN
ejpam-5499	173	43	its	its	PRON
ejpam-5499	173	44	structure.228	structure.228	PROPN
ejpam-5499	173	45	4.1	4.1	NUM
ejpam-5499	173	46	.	.	PUNCT
ejpam-5499	174	1	distance	distance	NOUN
ejpam-5499	174	2	between	between	ADP
ejpam-5499	174	3	two	two	NUM
ejpam-5499	174	4	vertcices229	vertcices229	PROPN
ejpam-5499	174	5	theorem	theorem	NOUN
ejpam-5499	174	6	3	3	X
ejpam-5499	174	7	.	.	PUNCT
ejpam-5499	175	1	let	let	AUX
ejpam-5499	175	2	m(γcn	m(γcn	PRON
ejpam-5499	175	3	)	)	PUNCT
ejpam-5499	175	4	be	be	AUX
ejpam-5499	175	5	the	the	DET
ejpam-5499	175	6	middle	middle	ADJ
ejpam-5499	175	7	graph	graph	NOUN
ejpam-5499	175	8	of	of	ADP
ejpam-5499	175	9	γcn	γcn	PROPN
ejpam-5499	175	10	for	for	ADP
ejpam-5499	175	11	n	n	X
ejpam-5499	175	12	≥	≥	NUM
ejpam-5499	175	13	4	4	NUM
ejpam-5499	175	14	.	.	PUNCT
ejpam-5499	176	1	the	the	DET
ejpam-5499	176	2	distance230	distance230	PROPN
ejpam-5499	176	3	i.	i.	PROPN
ejpam-5499	176	4	d(xi	d(xi	PROPN
ejpam-5499	176	5	,	,	PUNCT
ejpam-5499	176	6	a	a	PRON
ejpam-5499	176	7	)	)	PUNCT
ejpam-5499	176	8	≤	≤	NOUN
ejpam-5499	176	9	3	3	NUM
ejpam-5499	176	10	for	for	ADP
ejpam-5499	176	11	all	all	DET
ejpam-5499	176	12	a	a	PRON
ejpam-5499	176	13	∈	∈	PROPN
ejpam-5499	176	14	v	v	NOUN
ejpam-5499	176	15	(	(	PUNCT
ejpam-5499	176	16	m(γcn	m(γcn	NOUN
ejpam-5499	176	17	)	)	PUNCT
ejpam-5499	176	18	)	)	PUNCT
ejpam-5499	176	19	\	\	PROPN
ejpam-5499	177	1	xi	xi	X
ejpam-5499	177	2	where	where	SCONJ
ejpam-5499	177	3	1	1	NUM
ejpam-5499	177	4	≤	≤	NUM
ejpam-5499	177	5	i	i	NOUN
ejpam-5499	177	6	≤	≤	NUM
ejpam-5499	177	7	n−	n−	PROPN
ejpam-5499	177	8	1,231	1,231	NUM
ejpam-5499	177	9	ii	ii	NOUN
ejpam-5499	177	10	.	.	PUNCT
ejpam-5499	178	1	d(yp	d(yp	NOUN
ejpam-5499	178	2	,	,	PUNCT
ejpam-5499	178	3	a	a	PRON
ejpam-5499	178	4	)	)	PUNCT
ejpam-5499	178	5	≤	≤	NOUN
ejpam-5499	178	6	3	3	NUM
ejpam-5499	178	7	for	for	ADP
ejpam-5499	178	8	all	all	DET
ejpam-5499	178	9	a	a	PRON
ejpam-5499	178	10	∈	∈	PROPN
ejpam-5499	178	11	v	v	NOUN
ejpam-5499	178	12	(	(	PUNCT
ejpam-5499	178	13	m(γcn	m(γcn	NOUN
ejpam-5499	178	14	)	)	PUNCT
ejpam-5499	178	15	)	)	PUNCT
ejpam-5499	178	16	\	\	PROPN
ejpam-5499	179	1	yp	yp	PROPN
ejpam-5499	179	2	for	for	ADP
ejpam-5499	179	3	all	all	DET
ejpam-5499	179	4	1	1	NUM
ejpam-5499	179	5	≤	≤	NOUN
ejpam-5499	179	6	p	p	NOUN
ejpam-5499	179	7	≤	≤	NUM
ejpam-5499	179	8	n−1	n−1	PROPN
ejpam-5499	179	9	2	2	NUM
ejpam-5499	179	10	if	if	SCONJ
ejpam-5499	179	11	n	n	NOUN
ejpam-5499	179	12	is	be	AUX
ejpam-5499	179	13	odd	odd	ADJ
ejpam-5499	179	14	and232	and232	PROPN
ejpam-5499	179	15	1	1	NUM
ejpam-5499	179	16	≤	≤	NOUN
ejpam-5499	179	17	p	p	NOUN
ejpam-5499	179	18	≤	≤	NOUN
ejpam-5499	179	19	n−2	n−2	PROPN
ejpam-5499	179	20	2	2	NUM
ejpam-5499	179	21	if	if	SCONJ
ejpam-5499	179	22	n	n	NOUN
ejpam-5499	179	23	is	be	AUX
ejpam-5499	179	24	even,233	even,233	PROPN
ejpam-5499	179	25	iii	iii	NOUN
ejpam-5499	179	26	.	.	PUNCT
ejpam-5499	179	27	d(zi	d(zi	NOUN
ejpam-5499	179	28	,	,	PUNCT
ejpam-5499	179	29	a	a	PRON
ejpam-5499	179	30	)	)	PUNCT
ejpam-5499	179	31	≤	≤	NUM
ejpam-5499	179	32	2	2	NUM
ejpam-5499	179	33	for	for	ADP
ejpam-5499	179	34	a	a	DET
ejpam-5499	179	35	∈	∈	PROPN
ejpam-5499	179	36	v	v	NOUN
ejpam-5499	179	37	(	(	PUNCT
ejpam-5499	179	38	m(γcn	m(γcn	NOUN
ejpam-5499	179	39	)	)	PUNCT
ejpam-5499	179	40	)	)	PUNCT
ejpam-5499	179	41	\	\	PROPN
ejpam-5499	179	42	zi	zi	NOUN
ejpam-5499	180	1	where	where	SCONJ
ejpam-5499	180	2	1	1	NUM
ejpam-5499	180	3	≤	≤	NUM
ejpam-5499	180	4	i	i	PRON
ejpam-5499	180	5	≤	≤	NUM
ejpam-5499	180	6	n−	n−	NOUN
ejpam-5499	180	7	1,234	1,234	NUM
ejpam-5499	180	8	iv	iv	NOUN
ejpam-5499	180	9	.	.	PUNCT
ejpam-5499	181	1	d(e	d(e	PROPN
ejpam-5499	181	2	,	,	PUNCT
ejpam-5499	181	3	a	a	PRON
ejpam-5499	181	4	)	)	PUNCT
ejpam-5499	181	5	≤	≤	NOUN
ejpam-5499	182	1	2for	2for	ADP
ejpam-5499	182	2	a	a	DET
ejpam-5499	182	3	∈	∈	PROPN
ejpam-5499	182	4	v	v	NOUN
ejpam-5499	182	5	(	(	PUNCT
ejpam-5499	182	6	m(γcn	m(γcn	NOUN
ejpam-5499	182	7	)	)	PUNCT
ejpam-5499	182	8	)	)	PUNCT
ejpam-5499	182	9	\	\	PROPN
ejpam-5499	182	10	e.235	e.235	NUM
ejpam-5499	182	11	proof	proof	NOUN
ejpam-5499	182	12	.	.	PUNCT
ejpam-5499	183	1	we	we	PRON
ejpam-5499	183	2	divided	divide	VERB
ejpam-5499	183	3	it	it	PRON
ejpam-5499	183	4	into	into	ADP
ejpam-5499	183	5	four	four	NUM
ejpam-5499	183	6	cases:236	cases:236	X
ejpam-5499	183	7	i.	i.	NOUN
ejpam-5499	183	8	let	let	VERB
ejpam-5499	183	9	a	a	DET
ejpam-5499	183	10	∈	∈	PROPN
ejpam-5499	183	11	v	v	NOUN
ejpam-5499	183	12	(	(	PUNCT
ejpam-5499	183	13	m(γcn	m(γcn	NOUN
ejpam-5499	183	14	)	)	PUNCT
ejpam-5499	183	15	)	)	PUNCT
ejpam-5499	183	16	\	\	PROPN
ejpam-5499	184	1	xi	xi	X
ejpam-5499	184	2	.	.	PUNCT
ejpam-5499	185	1	if	if	SCONJ
ejpam-5499	185	2	a	a	DET
ejpam-5499	185	3	=	=	SYM
ejpam-5499	185	4	xi+1	xi+1	NOUN
ejpam-5499	185	5	,	,	PUNCT
ejpam-5499	185	6	then	then	ADV
ejpam-5499	185	7	the	the	DET
ejpam-5499	185	8	distance	distance	NOUN
ejpam-5499	185	9	d(xi	d(xi	PROPN
ejpam-5499	185	10	,	,	PUNCT
ejpam-5499	185	11	xi+1	xi+1	NUM
ejpam-5499	185	12	)	)	PUNCT
ejpam-5499	185	13	=	=	SYM
ejpam-5499	185	14	2	2	NUM
ejpam-5499	185	15	since	since	SCONJ
ejpam-5499	185	16	both237	both237	PROPN
ejpam-5499	186	1	[	[	X
ejpam-5499	186	2	xi	xi	X
ejpam-5499	186	3	,	,	PUNCT
ejpam-5499	186	4	yi+1	yi+1	PROPN
ejpam-5499	186	5	]	]	PUNCT
ejpam-5499	186	6	and	and	CCONJ
ejpam-5499	186	7	[	[	X
ejpam-5499	186	8	xi+1	xi+1	X
ejpam-5499	186	9	,	,	PUNCT
ejpam-5499	186	10	yi+1	yi+1	X
ejpam-5499	186	11	]	]	X
ejpam-5499	186	12	∈	∈	PROPN
ejpam-5499	186	13	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	186	14	)	)	PUNCT
ejpam-5499	186	15	)	)	PUNCT
ejpam-5499	187	1	and	and	CCONJ
ejpam-5499	188	1	[	[	X
ejpam-5499	188	2	xi	xi	X
ejpam-5499	188	3	,	,	PUNCT
ejpam-5499	188	4	xi+1	xi+1	PROPN
ejpam-5499	188	5	]	]	PUNCT
ejpam-5499	188	6	/∈	/∈	PUNCT
ejpam-5499	188	7	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	188	8	)	)	PUNCT
ejpam-5499	188	9	)	)	PUNCT
ejpam-5499	188	10	.	.	PUNCT
ejpam-5499	189	1	if	if	SCONJ
ejpam-5499	189	2	a	a	DET
ejpam-5499	189	3	=	=	PUNCT
ejpam-5499	189	4	xj	xj	PROPN
ejpam-5499	189	5	such238	such238	PROPN
ejpam-5499	189	6	that	that	SCONJ
ejpam-5499	189	7	j	j	PROPN
ejpam-5499	189	8	̸=	̸=	PROPN
ejpam-5499	189	9	i	i	PRON
ejpam-5499	189	10	or	or	CCONJ
ejpam-5499	189	11	i+	i+	NUM
ejpam-5499	189	12	1	1	NUM
ejpam-5499	189	13	,	,	PUNCT
ejpam-5499	189	14	then	then	ADV
ejpam-5499	189	15	[	[	X
ejpam-5499	189	16	xj	xj	X
ejpam-5499	189	17	,	,	PUNCT
ejpam-5499	189	18	zj	zj	X
ejpam-5499	189	19	]	]	PUNCT
ejpam-5499	189	20	∈	∈	PROPN
ejpam-5499	189	21	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	189	22	)	)	PUNCT
ejpam-5499	189	23	)	)	PUNCT
ejpam-5499	189	24	.	.	PUNCT
ejpam-5499	190	1	also	also	ADV
ejpam-5499	190	2	note	note	VERB
ejpam-5499	190	3	that	that	SCONJ
ejpam-5499	190	4	[	[	X
ejpam-5499	190	5	xi	xi	X
ejpam-5499	190	6	,	,	PUNCT
ejpam-5499	190	7	zi	zi	NOUN
ejpam-5499	190	8	]	]	X
ejpam-5499	190	9	∈	∈	PROPN
ejpam-5499	190	10	e(m(γcn))239	e(m(γcn))239	NOUN
ejpam-5499	190	11	and	and	CCONJ
ejpam-5499	190	12	[	[	X
ejpam-5499	190	13	zi	zi	PROPN
ejpam-5499	190	14	,	,	PUNCT
ejpam-5499	190	15	zj	zj	X
ejpam-5499	190	16	]	]	PUNCT
ejpam-5499	190	17	∈	∈	PROPN
ejpam-5499	190	18	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	190	19	)	)	PUNCT
ejpam-5499	190	20	)	)	PUNCT
ejpam-5499	190	21	.	.	PUNCT
ejpam-5499	191	1	now	now	ADV
ejpam-5499	191	2	since	since	SCONJ
ejpam-5499	191	3	neither	neither	CCONJ
ejpam-5499	191	4	[	[	X
ejpam-5499	191	5	xi	xi	PROPN
ejpam-5499	191	6	,	,	PUNCT
ejpam-5499	191	7	zj	zj	X
ejpam-5499	191	8	]	]	PUNCT
ejpam-5499	191	9	nor	nor	CCONJ
ejpam-5499	191	10	[	[	X
ejpam-5499	191	11	xj	xj	PROPN
ejpam-5499	191	12	,	,	PUNCT
ejpam-5499	191	13	zi	zi	PROPN
ejpam-5499	191	14	]	]	PUNCT
ejpam-5499	191	15	∈	∈	PROPN
ejpam-5499	191	16	e(m(γcn)),240	e(m(γcn)),240	NOUN
ejpam-5499	191	17	then	then	ADV
ejpam-5499	191	18	the	the	DET
ejpam-5499	191	19	shortest	short	ADJ
ejpam-5499	191	20	path	path	NOUN
ejpam-5499	191	21	from	from	ADP
ejpam-5499	191	22	xi	xi	PROPN
ejpam-5499	191	23	to	to	ADP
ejpam-5499	191	24	xj	xj	PROPN
ejpam-5499	191	25	is	be	AUX
ejpam-5499	191	26	the	the	DET
ejpam-5499	191	27	path	path	NOUN
ejpam-5499	191	28	xi	xi	PROPN
ejpam-5499	191	29	,	,	PUNCT
ejpam-5499	191	30	zi	zi	PROPN
ejpam-5499	191	31	,	,	PUNCT
ejpam-5499	191	32	zj	zj	PROPN
ejpam-5499	191	33	,	,	PUNCT
ejpam-5499	191	34	xj	xj	PROPN
ejpam-5499	191	35	of	of	ADP
ejpam-5499	191	36	length	length	NOUN
ejpam-5499	191	37	3	3	NUM
ejpam-5499	191	38	.	.	PUNCT
ejpam-5499	192	1	thus	thus	ADV
ejpam-5499	192	2	the241	the241	VERB
ejpam-5499	192	3	distance	distance	NOUN
ejpam-5499	192	4	d(xi	d(xi	PROPN
ejpam-5499	192	5	,	,	PUNCT
ejpam-5499	192	6	xj	xj	PROPN
ejpam-5499	192	7	)	)	PUNCT
ejpam-5499	192	8	=	=	SYM
ejpam-5499	193	1	3	3	X
ejpam-5499	193	2	.	.	NOUN
ejpam-5499	193	3	similar	similar	ADJ
ejpam-5499	193	4	argument	argument	NOUN
ejpam-5499	194	1	if	if	SCONJ
ejpam-5499	194	2	a	a	DET
ejpam-5499	194	3	=	=	SYM
ejpam-5499	194	4	yp	yp	PROPN
ejpam-5499	194	5	.	.	PUNCT
ejpam-5499	195	1	hence	hence	ADV
ejpam-5499	195	2	the	the	DET
ejpam-5499	195	3	distance	distance	NOUN
ejpam-5499	195	4	d(xi	d(xi	PROPN
ejpam-5499	195	5	,	,	PUNCT
ejpam-5499	195	6	yp	yp	X
ejpam-5499	195	7	)	)	PUNCT
ejpam-5499	195	8	=	=	NUM
ejpam-5499	195	9	3.242	3.242	NUM
ejpam-5499	195	10	if	if	SCONJ
ejpam-5499	195	11	a	a	DET
ejpam-5499	195	12	=	=	SYM
ejpam-5499	195	13	zi	zi	NOUN
ejpam-5499	195	14	,	,	PUNCT
ejpam-5499	195	15	then	then	ADV
ejpam-5499	195	16	d(xi	d(xi	PROPN
ejpam-5499	195	17	,	,	PUNCT
ejpam-5499	195	18	zi	zi	NOUN
ejpam-5499	195	19	)	)	PUNCT
ejpam-5499	195	20	=	=	PUNCT
ejpam-5499	195	21	1	1	NUM
ejpam-5499	195	22	since	since	SCONJ
ejpam-5499	195	23	[	[	X
ejpam-5499	195	24	xi	xi	PROPN
ejpam-5499	195	25	,	,	PUNCT
ejpam-5499	195	26	zi	zi	NOUN
ejpam-5499	195	27	]	]	X
ejpam-5499	195	28	]	]	X
ejpam-5499	195	29	∈	∈	PROPN
ejpam-5499	195	30	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	195	31	)	)	PUNCT
ejpam-5499	195	32	)	)	PUNCT
ejpam-5499	195	33	.	.	PUNCT
ejpam-5499	196	1	if	if	SCONJ
ejpam-5499	196	2	a	a	DET
ejpam-5499	196	3	=	=	X
ejpam-5499	196	4	zj	zj	INTJ
ejpam-5499	196	5	such	such	ADJ
ejpam-5499	196	6	that	that	SCONJ
ejpam-5499	196	7	i	i	PRON
ejpam-5499	196	8	̸=	̸=	PROPN
ejpam-5499	197	1	j,243	j,243	ADV
ejpam-5499	197	2	then	then	ADV
ejpam-5499	198	1	[	[	X
ejpam-5499	198	2	zi	zi	NOUN
ejpam-5499	198	3	,	,	PUNCT
ejpam-5499	198	4	zj	zj	X
ejpam-5499	198	5	]	]	PUNCT
ejpam-5499	198	6	∈	∈	PROPN
ejpam-5499	198	7	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	198	8	)	)	PUNCT
ejpam-5499	198	9	)	)	PUNCT
ejpam-5499	198	10	.	.	PUNCT
ejpam-5499	199	1	also	also	ADV
ejpam-5499	199	2	since	since	SCONJ
ejpam-5499	199	3	[	[	X
ejpam-5499	199	4	xi	xi	X
ejpam-5499	199	5	,	,	PUNCT
ejpam-5499	199	6	zi	zi	NOUN
ejpam-5499	199	7	]	]	X
ejpam-5499	199	8	∈	∈	PROPN
ejpam-5499	199	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	199	10	)	)	PUNCT
ejpam-5499	199	11	)	)	PUNCT
ejpam-5499	199	12	,	,	PUNCT
ejpam-5499	199	13	then	then	ADV
ejpam-5499	199	14	we	we	PRON
ejpam-5499	199	15	have	have	VERB
ejpam-5499	199	16	a	a	DET
ejpam-5499	199	17	path244	path244	PROPN
ejpam-5499	199	18	xi	xi	PROPN
ejpam-5499	199	19	,	,	PUNCT
ejpam-5499	199	20	zi	zi	PROPN
ejpam-5499	199	21	,	,	PUNCT
ejpam-5499	199	22	zj	zj	PROPN
ejpam-5499	199	23	from	from	ADP
ejpam-5499	199	24	xi	xi	PROPN
ejpam-5499	199	25	to	to	ADP
ejpam-5499	199	26	zj	zj	PROPN
ejpam-5499	199	27	and	and	CCONJ
ejpam-5499	199	28	this	this	PRON
ejpam-5499	199	29	is	be	AUX
ejpam-5499	199	30	the	the	DET
ejpam-5499	199	31	shortest	short	ADJ
ejpam-5499	199	32	since	since	SCONJ
ejpam-5499	199	33	[	[	X
ejpam-5499	199	34	xi	xi	PROPN
ejpam-5499	199	35	,	,	PUNCT
ejpam-5499	199	36	zj	zj	X
ejpam-5499	199	37	]	]	PUNCT
ejpam-5499	199	38	/∈	/∈	PUNCT
ejpam-5499	199	39	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	199	40	)	)	PUNCT
ejpam-5499	199	41	)	)	PUNCT
ejpam-5499	199	42	.	.	PUNCT
ejpam-5499	200	1	thus	thus	ADV
ejpam-5499	200	2	the245	the245	NUM
ejpam-5499	200	3	distance	distance	NOUN
ejpam-5499	200	4	d(xi	d(xi	PROPN
ejpam-5499	200	5	,	,	PUNCT
ejpam-5499	200	6	zj	zj	X
ejpam-5499	200	7	)	)	PUNCT
ejpam-5499	200	8	=	=	SYM
ejpam-5499	200	9	2	2	X
ejpam-5499	200	10	.	.	PUNCT
ejpam-5499	200	11	lastly	lastly	ADV
ejpam-5499	200	12	if	if	SCONJ
ejpam-5499	200	13	a	a	DET
ejpam-5499	200	14	=	=	SYM
ejpam-5499	200	15	e	e	NOUN
ejpam-5499	200	16	,	,	PUNCT
ejpam-5499	200	17	the	the	DET
ejpam-5499	200	18	argument	argument	NOUN
ejpam-5499	200	19	is	be	AUX
ejpam-5499	200	20	similar	similar	ADJ
ejpam-5499	200	21	to	to	ADP
ejpam-5499	200	22	a	a	DET
ejpam-5499	200	23	=	=	X
ejpam-5499	200	24	zj	zj	PROPN
ejpam-5499	200	25	.	.	PUNCT
ejpam-5499	201	1	hence	hence	ADV
ejpam-5499	201	2	the246	the246	PROPN
ejpam-5499	201	3	distance	distance	PROPN
ejpam-5499	201	4	d(xi	d(xi	PROPN
ejpam-5499	201	5	,	,	PUNCT
ejpam-5499	201	6	a	a	PRON
ejpam-5499	201	7	)	)	PUNCT
ejpam-5499	201	8	≤	≤	NOUN
ejpam-5499	201	9	3	3	NUM
ejpam-5499	201	10	for	for	ADP
ejpam-5499	201	11	all	all	DET
ejpam-5499	201	12	a	a	PRON
ejpam-5499	201	13	∈	∈	PROPN
ejpam-5499	201	14	v	v	NOUN
ejpam-5499	201	15	(	(	PUNCT
ejpam-5499	201	16	m(γcn	m(γcn	NOUN
ejpam-5499	201	17	)	)	PUNCT
ejpam-5499	201	18	)	)	PUNCT
ejpam-5499	201	19	\	\	PROPN
ejpam-5499	201	20	xi.247	xi.247	PROPN
ejpam-5499	201	21	ii	ii	PROPN
ejpam-5499	201	22	.	.	PUNCT
ejpam-5499	202	1	the	the	DET
ejpam-5499	202	2	proof	proof	NOUN
ejpam-5499	202	3	for	for	ADP
ejpam-5499	202	4	the	the	DET
ejpam-5499	202	5	distance	distance	NOUN
ejpam-5499	202	6	d(yp	d(yp	NOUN
ejpam-5499	202	7	,	,	PUNCT
ejpam-5499	202	8	a	a	PRON
ejpam-5499	202	9	)	)	PUNCT
ejpam-5499	202	10	≤	≤	NOUN
ejpam-5499	202	11	3	3	NUM
ejpam-5499	202	12	is	be	AUX
ejpam-5499	202	13	analogous	analogous	ADJ
ejpam-5499	202	14	to	to	PART
ejpam-5499	202	15	case	case	VERB
ejpam-5499	202	16	i.248	i.248	ADJ
ejpam-5499	202	17	iii	iii	X
ejpam-5499	202	18	.	.	PUNCT
ejpam-5499	203	1	let	let	VERB
ejpam-5499	203	2	a	a	DET
ejpam-5499	203	3	∈	∈	PROPN
ejpam-5499	203	4	v	v	NOUN
ejpam-5499	203	5	(	(	PUNCT
ejpam-5499	203	6	m(γcn	m(γcn	NOUN
ejpam-5499	203	7	)	)	PUNCT
ejpam-5499	203	8	)	)	PUNCT
ejpam-5499	204	1	\	\	PROPN
ejpam-5499	204	2	zi	zi	PROPN
ejpam-5499	204	3	.	.	PUNCT
ejpam-5499	205	1	if	if	SCONJ
ejpam-5499	205	2	a	a	DET
ejpam-5499	205	3	=	=	X
ejpam-5499	205	4	zj	zj	INTJ
ejpam-5499	205	5	such	such	ADJ
ejpam-5499	205	6	that	that	SCONJ
ejpam-5499	205	7	i	i	PRON
ejpam-5499	205	8	̸=	̸=	PROPN
ejpam-5499	205	9	j	j	PROPN
ejpam-5499	205	10	,	,	PUNCT
ejpam-5499	205	11	then	then	ADV
ejpam-5499	205	12	d(zi	d(zi	NOUN
ejpam-5499	205	13	,	,	PUNCT
ejpam-5499	205	14	zj	zj	NOUN
ejpam-5499	205	15	)	)	PUNCT
ejpam-5499	205	16	=	=	SYM
ejpam-5499	206	1	1	1	NUM
ejpam-5499	206	2	since249	since249	PROPN
ejpam-5499	206	3	[	[	X
ejpam-5499	206	4	zi	zi	PROPN
ejpam-5499	206	5	,	,	PUNCT
ejpam-5499	206	6	zj	zj	X
ejpam-5499	206	7	]	]	PUNCT
ejpam-5499	206	8	∈	∈	PROPN
ejpam-5499	206	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	206	10	)	)	PUNCT
ejpam-5499	206	11	)	)	PUNCT
ejpam-5499	206	12	.	.	PUNCT
ejpam-5499	207	1	similar	similar	ADJ
ejpam-5499	207	2	argument	argument	NOUN
ejpam-5499	207	3	if	if	SCONJ
ejpam-5499	207	4	a	a	PRON
ejpam-5499	207	5	=	=	X
ejpam-5499	207	6	e.	e.	PROPN
ejpam-5499	207	7	if	if	SCONJ
ejpam-5499	207	8	a	a	PRON
ejpam-5499	207	9	=	=	SYM
ejpam-5499	207	10	xi	xi	PROPN
ejpam-5499	207	11	,	,	PUNCT
ejpam-5499	207	12	then	then	ADV
ejpam-5499	207	13	clearly	clearly	ADV
ejpam-5499	207	14	d(xi	d(xi	PROPN
ejpam-5499	207	15	,	,	PUNCT
ejpam-5499	207	16	zi	zi	NOUN
ejpam-5499	207	17	)	)	PUNCT
ejpam-5499	207	18	=	=	NOUN
ejpam-5499	208	1	250	250	NUM
ejpam-5499	208	2	1	1	NUM
ejpam-5499	208	3	.	.	PUNCT
ejpam-5499	209	1	now	now	ADV
ejpam-5499	209	2	suppose	suppose	VERB
ejpam-5499	209	3	a	a	DET
ejpam-5499	209	4	=	=	X
ejpam-5499	209	5	xj	xj	PROPN
ejpam-5499	209	6	where	where	SCONJ
ejpam-5499	209	7	i	i	PRON
ejpam-5499	209	8	̸=	̸=	PROPN
ejpam-5499	209	9	j	j	PROPN
ejpam-5499	209	10	,	,	PUNCT
ejpam-5499	209	11	note	note	VERB
ejpam-5499	209	12	that	that	SCONJ
ejpam-5499	209	13	[	[	X
ejpam-5499	209	14	xj	xj	X
ejpam-5499	209	15	,	,	PUNCT
ejpam-5499	209	16	zj	zj	X
ejpam-5499	209	17	]	]	PUNCT
ejpam-5499	209	18	∈	∈	PROPN
ejpam-5499	209	19	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	209	20	)	)	PUNCT
ejpam-5499	209	21	)	)	PUNCT
ejpam-5499	209	22	and	and	CCONJ
ejpam-5499	209	23	also251	also251	PROPN
ejpam-5499	210	1	[	[	X
ejpam-5499	210	2	zi	zi	PROPN
ejpam-5499	210	3	,	,	PUNCT
ejpam-5499	210	4	zj	zj	X
ejpam-5499	210	5	]	]	PUNCT
ejpam-5499	210	6	∈	∈	PROPN
ejpam-5499	210	7	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	210	8	)	)	PUNCT
ejpam-5499	210	9	)	)	PUNCT
ejpam-5499	210	10	,	,	PUNCT
ejpam-5499	210	11	it	it	PRON
ejpam-5499	210	12	follows	follow	VERB
ejpam-5499	210	13	that	that	SCONJ
ejpam-5499	210	14	zi	zi	NOUN
ejpam-5499	210	15	,	,	PUNCT
ejpam-5499	210	16	zj	zj	PROPN
ejpam-5499	210	17	,	,	PUNCT
ejpam-5499	210	18	xj	xj	PROPN
ejpam-5499	210	19	is	be	AUX
ejpam-5499	210	20	a	a	DET
ejpam-5499	210	21	shortest	short	ADJ
ejpam-5499	210	22	path	path	NOUN
ejpam-5499	210	23	from	from	ADP
ejpam-5499	210	24	zi	zi	PROPN
ejpam-5499	210	25	to	to	ADP
ejpam-5499	210	26	xj	xj	PROPN
ejpam-5499	210	27	since252	since252	PROPN
ejpam-5499	211	1	[	[	X
ejpam-5499	211	2	zi	zi	PROPN
ejpam-5499	211	3	,	,	PUNCT
ejpam-5499	211	4	xj	xj	PROPN
ejpam-5499	211	5	]	]	PUNCT
ejpam-5499	211	6	/∈	/∈	PUNCT
ejpam-5499	211	7	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	211	8	)	)	PUNCT
ejpam-5499	211	9	)	)	PUNCT
ejpam-5499	211	10	.	.	PUNCT
ejpam-5499	212	1	thus	thus	ADV
ejpam-5499	212	2	the	the	DET
ejpam-5499	212	3	distance	distance	NOUN
ejpam-5499	212	4	d(zi	d(zi	NOUN
ejpam-5499	212	5	,	,	PUNCT
ejpam-5499	212	6	xj	xj	PROPN
ejpam-5499	212	7	)	)	PUNCT
ejpam-5499	212	8	=	=	SYM
ejpam-5499	213	1	2	2	X
ejpam-5499	213	2	.	.	PUNCT
ejpam-5499	214	1	lastly	lastly	ADV
ejpam-5499	214	2	,	,	PUNCT
ejpam-5499	214	3	if	if	SCONJ
ejpam-5499	214	4	a	a	PRON
ejpam-5499	214	5	=	=	X
ejpam-5499	214	6	yp	yp	X
ejpam-5499	214	7	where253	where253	PROPN
ejpam-5499	214	8	p	p	NOUN
ejpam-5499	215	1	=	=	PUNCT
ejpam-5499	215	2	i+1	i+1	NUM
ejpam-5499	215	3	2	2	NUM
ejpam-5499	215	4	,	,	PUNCT
ejpam-5499	215	5	then	then	ADV
ejpam-5499	215	6	clearly	clearly	ADV
ejpam-5499	215	7	d(zi	d(zi	VERB
ejpam-5499	215	8	,	,	PUNCT
ejpam-5499	215	9	y	y	PROPN
ejpam-5499	215	10	i+1	i+1	NUM
ejpam-5499	215	11	2	2	X
ejpam-5499	215	12	)	)	PUNCT
ejpam-5499	215	13	=	=	SYM
ejpam-5499	216	1	1	1	X
ejpam-5499	216	2	.	.	X
ejpam-5499	216	3	for	for	ADP
ejpam-5499	216	4	p	p	NOUN
ejpam-5499	216	5	=	=	SYM
ejpam-5499	216	6	j+1	j+1	ADJ
ejpam-5499	216	7	2	2	NUM
ejpam-5499	216	8	where	where	SCONJ
ejpam-5499	216	9	i	i	PRON
ejpam-5499	216	10	̸=	̸=	PROPN
ejpam-5499	216	11	j	j	PROPN
ejpam-5499	216	12	,	,	PUNCT
ejpam-5499	216	13	it	it	PRON
ejpam-5499	216	14	follows	follow	VERB
ejpam-5499	216	15	that254	that254	PROPN
ejpam-5499	217	1	[	[	X
ejpam-5499	217	2	y	y	X
ejpam-5499	217	3	j+1	j+1	ADJ
ejpam-5499	217	4	2	2	NUM
ejpam-5499	217	5	,	,	PUNCT
ejpam-5499	217	6	zj	zj	X
ejpam-5499	217	7	]	]	PUNCT
ejpam-5499	217	8	∈	∈	PROPN
ejpam-5499	217	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	217	10	)	)	PUNCT
ejpam-5499	217	11	)	)	PUNCT
ejpam-5499	218	1	and	and	CCONJ
ejpam-5499	218	2	we	we	PRON
ejpam-5499	218	3	know	know	VERB
ejpam-5499	218	4	that	that	SCONJ
ejpam-5499	218	5	[	[	X
ejpam-5499	218	6	zi	zi	NOUN
ejpam-5499	218	7	,	,	PUNCT
ejpam-5499	218	8	zj	zj	X
ejpam-5499	218	9	]	]	PUNCT
ejpam-5499	218	10	∈	∈	PROPN
ejpam-5499	218	11	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	218	12	)	)	PUNCT
ejpam-5499	218	13	)	)	PUNCT
ejpam-5499	218	14	,	,	PUNCT
ejpam-5499	218	15	thus	thus	ADV
ejpam-5499	218	16	a	a	DET
ejpam-5499	218	17	shortest255	shortest255	PROPN
ejpam-5499	218	18	path	path	NOUN
ejpam-5499	218	19	from	from	ADP
ejpam-5499	218	20	zi	zi	NOUN
ejpam-5499	218	21	to	to	ADP
ejpam-5499	218	22	y	y	PROPN
ejpam-5499	218	23	i+1	i+1	NUM
ejpam-5499	218	24	2	2	NUM
ejpam-5499	218	25	is	be	AUX
ejpam-5499	218	26	zi	zi	NOUN
ejpam-5499	218	27	,	,	PUNCT
ejpam-5499	218	28	zj	zj	PROPN
ejpam-5499	218	29	,	,	PUNCT
ejpam-5499	218	30	y	y	PROPN
ejpam-5499	218	31	i+1	i+1	NUM
ejpam-5499	218	32	2	2	NUM
ejpam-5499	218	33	since	since	SCONJ
ejpam-5499	218	34	[	[	X
ejpam-5499	218	35	y	y	PROPN
ejpam-5499	218	36	i+1	i+1	NUM
ejpam-5499	218	37	2	2	NUM
ejpam-5499	218	38	,	,	PUNCT
ejpam-5499	218	39	zi	zi	NOUN
ejpam-5499	218	40	]	]	PUNCT
ejpam-5499	218	41	/∈	/∈	PUNCT
ejpam-5499	218	42	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	218	43	)	)	PUNCT
ejpam-5499	218	44	)	)	PUNCT
ejpam-5499	218	45	.	.	PUNCT
ejpam-5499	219	1	hence	hence	ADV
ejpam-5499	219	2	the	the	DET
ejpam-5499	219	3	distance256	distance256	PROPN
ejpam-5499	219	4	d(zi	d(zi	NOUN
ejpam-5499	219	5	,	,	PUNCT
ejpam-5499	219	6	yp	yp	PROPN
ejpam-5499	219	7	)	)	PUNCT
ejpam-5499	219	8	≤	≤	NUM
ejpam-5499	219	9	2	2	NUM
ejpam-5499	219	10	.	.	PUNCT
ejpam-5499	220	1	therefore	therefore	ADV
ejpam-5499	220	2	the	the	DET
ejpam-5499	220	3	distance	distance	NOUN
ejpam-5499	220	4	d(zi	d(zi	NOUN
ejpam-5499	220	5	,	,	PUNCT
ejpam-5499	220	6	a	a	PRON
ejpam-5499	220	7	)	)	PUNCT
ejpam-5499	220	8	≤	≤	NOUN
ejpam-5499	220	9	2	2	NUM
ejpam-5499	220	10	for	for	ADP
ejpam-5499	220	11	all	all	DET
ejpam-5499	220	12	a	a	PRON
ejpam-5499	220	13	∈	∈	PROPN
ejpam-5499	220	14	v	v	NOUN
ejpam-5499	220	15	(	(	PUNCT
ejpam-5499	220	16	m(γcn	m(γcn	NOUN
ejpam-5499	220	17	)	)	PUNCT
ejpam-5499	220	18	)	)	PUNCT
ejpam-5499	220	19	\	\	PROPN
ejpam-5499	221	1	zi.257	zi.257	NOUN
ejpam-5499	221	2	iv	iv	NUM
ejpam-5499	221	3	.	.	PUNCT
ejpam-5499	222	1	let	let	AUX
ejpam-5499	222	2	m(γcn	m(γcn	PRON
ejpam-5499	222	3	)	)	PUNCT
ejpam-5499	222	4	be	be	AUX
ejpam-5499	222	5	the	the	DET
ejpam-5499	222	6	middle	middle	ADJ
ejpam-5499	222	7	graph	graph	NOUN
ejpam-5499	222	8	of	of	ADP
ejpam-5499	222	9	γcn	γcn	NOUN
ejpam-5499	222	10	and	and	CCONJ
ejpam-5499	222	11	let	let	VERB
ejpam-5499	222	12	a	a	DET
ejpam-5499	222	13	∈	∈	PROPN
ejpam-5499	222	14	v	v	NOUN
ejpam-5499	222	15	(	(	PUNCT
ejpam-5499	222	16	m(γcn	m(γcn	NOUN
ejpam-5499	222	17	)	)	PUNCT
ejpam-5499	222	18	)	)	PUNCT
ejpam-5499	222	19	\	\	PROPN
ejpam-5499	223	1	e.	e.	PROPN
ejpam-5499	224	1	if	if	SCONJ
ejpam-5499	224	2	a	a	DET
ejpam-5499	224	3	=	=	NOUN
ejpam-5499	224	4	zi,258	zi,258	NOUN
ejpam-5499	224	5	then	then	ADV
ejpam-5499	224	6	d(e	d(e	PROPN
ejpam-5499	224	7	,	,	PUNCT
ejpam-5499	224	8	zi	zi	NOUN
ejpam-5499	224	9	)	)	PUNCT
ejpam-5499	224	10	=	=	PUNCT
ejpam-5499	224	11	1	1	NUM
ejpam-5499	224	12	since	since	SCONJ
ejpam-5499	224	13	[	[	X
ejpam-5499	224	14	e	e	X
ejpam-5499	224	15	,	,	PUNCT
ejpam-5499	224	16	zi	zi	NOUN
ejpam-5499	224	17	]	]	X
ejpam-5499	224	18	∈	∈	PROPN
ejpam-5499	224	19	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	224	20	)	)	PUNCT
ejpam-5499	224	21	)	)	PUNCT
ejpam-5499	224	22	.	.	PUNCT
ejpam-5499	225	1	now	now	ADV
ejpam-5499	225	2	if	if	SCONJ
ejpam-5499	225	3	a	a	DET
ejpam-5499	225	4	=	=	X
ejpam-5499	225	5	xi	xi	PROPN
ejpam-5499	225	6	for	for	ADP
ejpam-5499	225	7	1	1	NUM
ejpam-5499	225	8	≤	≤	NUM
ejpam-5499	225	9	i	i	PRON
ejpam-5499	225	10	≤	≤	NOUN
ejpam-5499	225	11	n	n	CCONJ
ejpam-5499	225	12	−	−	PROPN
ejpam-5499	225	13	1	1	NUM
ejpam-5499	225	14	,	,	PUNCT
ejpam-5499	225	15	then259	then259	PROPN
ejpam-5499	225	16	j.	j.	PROPN
ejpam-5499	225	17	m.	m.	PROPN
ejpam-5499	225	18	jamis	jamis	PROPN
ejpam-5499	225	19	,	,	PUNCT
ejpam-5499	225	20	d.	d.	PROPN
ejpam-5499	225	21	m.	m.	PROPN
ejpam-5499	225	22	magpantay	magpantay	PROPN
ejpam-5499	225	23	/	/	SYM
ejpam-5499	225	24	eur	eur	PROPN
ejpam-5499	225	25	.	.	PUNCT
ejpam-5499	226	1	j.	j.	PROPN
ejpam-5499	226	2	pure	pure	PROPN
ejpam-5499	226	3	appl	appl	PROPN
ejpam-5499	226	4	.	.	PROPN
ejpam-5499	226	5	math	math	PROPN
ejpam-5499	226	6	,	,	PUNCT
ejpam-5499	226	7	18	18	NUM
ejpam-5499	226	8	(	(	PUNCT
ejpam-5499	226	9	2	2	NUM
ejpam-5499	226	10	)	)	PUNCT
ejpam-5499	226	11	(	(	PUNCT
ejpam-5499	226	12	2025	2025	NUM
ejpam-5499	226	13	)	)	PUNCT
ejpam-5499	226	14	,	,	PUNCT
ejpam-5499	226	15	5499	5499	NUM
ejpam-5499	226	16	12	12	NUM
ejpam-5499	226	17	of	of	ADP
ejpam-5499	226	18	18	18	NUM
ejpam-5499	226	19	[	[	X
ejpam-5499	226	20	xi	xi	X
ejpam-5499	226	21	,	,	PUNCT
ejpam-5499	226	22	zi	zi	NOUN
ejpam-5499	226	23	]	]	X
ejpam-5499	226	24	∈	∈	PROPN
ejpam-5499	226	25	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	226	26	)	)	PUNCT
ejpam-5499	226	27	)	)	PUNCT
ejpam-5499	227	1	also	also	ADV
ejpam-5499	227	2	since	since	SCONJ
ejpam-5499	227	3	[	[	X
ejpam-5499	227	4	e	e	X
ejpam-5499	227	5	,	,	PUNCT
ejpam-5499	227	6	zi	zi	NOUN
ejpam-5499	227	7	]	]	X
ejpam-5499	227	8	∈	∈	PROPN
ejpam-5499	227	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	227	10	)	)	PUNCT
ejpam-5499	227	11	)	)	PUNCT
ejpam-5499	227	12	then	then	ADV
ejpam-5499	227	13	we	we	PRON
ejpam-5499	227	14	have	have	VERB
ejpam-5499	227	15	a	a	DET
ejpam-5499	227	16	path	path	NOUN
ejpam-5499	227	17	e	e	NOUN
ejpam-5499	227	18	,	,	PUNCT
ejpam-5499	227	19	zi	zi	NOUN
ejpam-5499	227	20	,	,	PUNCT
ejpam-5499	227	21	xi	xi	X
ejpam-5499	227	22	of260	of260	PROPN
ejpam-5499	227	23	length	length	NOUN
ejpam-5499	227	24	2	2	NUM
ejpam-5499	227	25	from	from	ADP
ejpam-5499	227	26	e	e	PRON
ejpam-5499	227	27	to	to	ADP
ejpam-5499	227	28	xi	xi	PROPN
ejpam-5499	227	29	.	.	PUNCT
ejpam-5499	228	1	and	and	CCONJ
ejpam-5499	228	2	also	also	ADV
ejpam-5499	228	3	since	since	SCONJ
ejpam-5499	228	4	[	[	X
ejpam-5499	228	5	e	e	NOUN
ejpam-5499	228	6	,	,	PUNCT
ejpam-5499	228	7	xi	xi	X
ejpam-5499	228	8	]	]	PUNCT
ejpam-5499	228	9	/∈	/∈	PUNCT
ejpam-5499	228	10	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	228	11	)	)	PUNCT
ejpam-5499	228	12	)	)	PUNCT
ejpam-5499	228	13	,	,	PUNCT
ejpam-5499	228	14	thus	thus	ADV
ejpam-5499	228	15	this	this	PRON
ejpam-5499	228	16	is	be	AUX
ejpam-5499	228	17	the	the	DET
ejpam-5499	228	18	shortest261	shortest261	ADJ
ejpam-5499	228	19	path	path	NOUN
ejpam-5499	228	20	from	from	ADP
ejpam-5499	228	21	e	e	NOUN
ejpam-5499	228	22	to	to	ADP
ejpam-5499	228	23	xi	xi	PROPN
ejpam-5499	228	24	.	.	PUNCT
ejpam-5499	229	1	hence	hence	ADV
ejpam-5499	229	2	d(e	d(e	PROPN
ejpam-5499	229	3	,	,	PUNCT
ejpam-5499	229	4	xi	xi	X
ejpam-5499	229	5	)	)	PUNCT
ejpam-5499	229	6	=	=	SYM
ejpam-5499	230	1	2	2	X
ejpam-5499	230	2	.	.	PUNCT
ejpam-5499	230	3	finally	finally	ADV
ejpam-5499	230	4	if	if	SCONJ
ejpam-5499	230	5	a	a	DET
ejpam-5499	230	6	=	=	X
ejpam-5499	230	7	yp	yp	PROPN
ejpam-5499	230	8	for	for	ADP
ejpam-5499	230	9	p	p	NOUN
ejpam-5499	230	10	=	=	PROPN
ejpam-5499	230	11	i+1	i+1	SYM
ejpam-5499	230	12	2	2	NUM
ejpam-5499	230	13	for	for	ADP
ejpam-5499	230	14	all	all	DET
ejpam-5499	230	15	odd262	odd262	PROPN
ejpam-5499	230	16	1	1	NUM
ejpam-5499	230	17	≤	≤	NUM
ejpam-5499	230	18	i	i	PRON
ejpam-5499	230	19	≤	≤	PROPN
ejpam-5499	230	20	n−2	n−2	PROPN
ejpam-5499	230	21	,	,	PUNCT
ejpam-5499	230	22	then	then	ADV
ejpam-5499	230	23	[	[	X
ejpam-5499	230	24	yp	yp	PROPN
ejpam-5499	230	25	,	,	PUNCT
ejpam-5499	230	26	zi	zi	NOUN
ejpam-5499	230	27	]	]	X
ejpam-5499	230	28	∈	∈	PROPN
ejpam-5499	230	29	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	230	30	)	)	PUNCT
ejpam-5499	230	31	)	)	PUNCT
ejpam-5499	230	32	.	.	PUNCT
ejpam-5499	231	1	the	the	DET
ejpam-5499	231	2	other	other	ADJ
ejpam-5499	231	3	arguments	argument	NOUN
ejpam-5499	231	4	are	be	AUX
ejpam-5499	231	5	similar	similar	ADJ
ejpam-5499	231	6	for	for	ADP
ejpam-5499	231	7	a	a	DET
ejpam-5499	231	8	=	=	X
ejpam-5499	231	9	xi.263	xi.263	PUNCT
ejpam-5499	231	10	thus	thus	ADV
ejpam-5499	231	11	d(e	d(e	PROPN
ejpam-5499	231	12	,	,	PUNCT
ejpam-5499	231	13	yp	yp	X
ejpam-5499	231	14	)	)	PUNCT
ejpam-5499	231	15	=	=	SYM
ejpam-5499	232	1	2	2	X
ejpam-5499	232	2	.	.	X
ejpam-5499	232	3	therefore	therefore	ADV
ejpam-5499	232	4	the	the	DET
ejpam-5499	232	5	distance	distance	NOUN
ejpam-5499	232	6	d(e	d(e	PROPN
ejpam-5499	232	7	,	,	PUNCT
ejpam-5499	232	8	a	a	PRON
ejpam-5499	232	9	)	)	PUNCT
ejpam-5499	232	10	≤	≤	NOUN
ejpam-5499	232	11	2	2	NUM
ejpam-5499	232	12	for	for	ADP
ejpam-5499	232	13	all	all	DET
ejpam-5499	232	14	a	a	PRON
ejpam-5499	232	15	∈	∈	PROPN
ejpam-5499	232	16	v	v	NOUN
ejpam-5499	232	17	(	(	PUNCT
ejpam-5499	232	18	m(γcn	m(γcn	NOUN
ejpam-5499	232	19	)	)	PUNCT
ejpam-5499	232	20	)	)	PUNCT
ejpam-5499	232	21	\	\	NOUN
ejpam-5499	232	22	e.264	e.264	VERB
ejpam-5499	232	23	4.2	4.2	NUM
ejpam-5499	232	24	.	.	PUNCT
ejpam-5499	233	1	eccentricity	eccentricity	NOUN
ejpam-5499	233	2	of	of	ADP
ejpam-5499	233	3	the	the	DET
ejpam-5499	233	4	vertcices265	vertcices265	NUM
ejpam-5499	233	5	theorem	theorem	ADJ
ejpam-5499	233	6	4	4	NUM
ejpam-5499	233	7	.	.	PUNCT
ejpam-5499	234	1	let	let	AUX
ejpam-5499	234	2	m(γcn	m(γcn	PRON
ejpam-5499	234	3	)	)	PUNCT
ejpam-5499	234	4	be	be	AUX
ejpam-5499	234	5	the	the	DET
ejpam-5499	234	6	middle	middle	ADJ
ejpam-5499	234	7	graph	graph	NOUN
ejpam-5499	234	8	of	of	ADP
ejpam-5499	234	9	γcn	γcn	PROPN
ejpam-5499	234	10	.	.	PUNCT
ejpam-5499	235	1	the	the	DET
ejpam-5499	235	2	eccentricity	eccentricity	NOUN
ejpam-5499	235	3	of	of	ADP
ejpam-5499	235	4	the	the	DET
ejpam-5499	235	5	vertices266	vertices266	PROPN
ejpam-5499	235	6	i.	i.	NOUN
ejpam-5499	235	7	e(xi	e(xi	PROPN
ejpam-5499	235	8	)	)	PUNCT
ejpam-5499	235	9	=	=	PRON
ejpam-5499	235	10	{	{	PUNCT
ejpam-5499	235	11	2	2	NUM
ejpam-5499	235	12	,	,	PUNCT
ejpam-5499	235	13	if	if	SCONJ
ejpam-5499	235	14	n=2	n=2	ADV
ejpam-5499	235	15	or	or	CCONJ
ejpam-5499	235	16	3	3	NUM
ejpam-5499	235	17	3	3	NUM
ejpam-5499	235	18	,	,	PUNCT
ejpam-5499	235	19	if	if	SCONJ
ejpam-5499	235	20	n	n	PRON
ejpam-5499	235	21	≥	≥	VERB
ejpam-5499	235	22	4	4	NUM
ejpam-5499	235	23	267	267	NUM
ejpam-5499	235	24	for	for	ADP
ejpam-5499	235	25	all	all	DET
ejpam-5499	235	26	1	1	NUM
ejpam-5499	235	27	≤	≤	NUM
ejpam-5499	235	28	i	i	PRON
ejpam-5499	235	29	≤	≤	NOUN
ejpam-5499	235	30	n−	n−	PROPN
ejpam-5499	235	31	1,268	1,268	NUM
ejpam-5499	235	32	ii	ii	NOUN
ejpam-5499	235	33	.	.	PUNCT
ejpam-5499	236	1	e(yp	e(yp	PUNCT
ejpam-5499	236	2	)	)	PUNCT
ejpam-5499	237	1	=	=	PRON
ejpam-5499	237	2	{	{	PUNCT
ejpam-5499	237	3	2	2	NUM
ejpam-5499	237	4	,	,	PUNCT
ejpam-5499	237	5	if	if	SCONJ
ejpam-5499	237	6	n=3	n=3	PUNCT
ejpam-5499	237	7	3	3	NUM
ejpam-5499	237	8	,	,	PUNCT
ejpam-5499	237	9	for	for	ADP
ejpam-5499	237	10	n	n	NUM
ejpam-5499	237	11	≥	≥	NUM
ejpam-5499	237	12	4	4	NUM
ejpam-5499	237	13	269	269	NUM
ejpam-5499	237	14	for	for	ADP
ejpam-5499	237	15	all	all	DET
ejpam-5499	237	16	1	1	NUM
ejpam-5499	237	17	≤	≤	NOUN
ejpam-5499	237	18	p	p	NOUN
ejpam-5499	237	19	≤	≤	NOUN
ejpam-5499	237	20	n−2	n−2	PROPN
ejpam-5499	237	21	2	2	NUM
ejpam-5499	237	22	,	,	PUNCT
ejpam-5499	237	23	270	270	NUM
ejpam-5499	237	24	iii	iii	NOUN
ejpam-5499	237	25	.	.	PUNCT
ejpam-5499	237	26	e(zi	e(zi	PROPN
ejpam-5499	237	27	)	)	PUNCT
ejpam-5499	237	28	=	=	PRON
ejpam-5499	237	29	{	{	PUNCT
ejpam-5499	237	30	1	1	NUM
ejpam-5499	237	31	,	,	PUNCT
ejpam-5499	237	32	if	if	SCONJ
ejpam-5499	237	33	n=2	n=2	ADV
ejpam-5499	237	34	or	or	CCONJ
ejpam-5499	237	35	3	3	NUM
ejpam-5499	237	36	2	2	NUM
ejpam-5499	237	37	,	,	PUNCT
ejpam-5499	237	38	if	if	SCONJ
ejpam-5499	237	39	n	n	PRON
ejpam-5499	237	40	≥	≥	VERB
ejpam-5499	237	41	4	4	NUM
ejpam-5499	237	42	271	271	NUM
ejpam-5499	237	43	for	for	ADP
ejpam-5499	237	44	all	all	DET
ejpam-5499	237	45	1	1	NUM
ejpam-5499	237	46	≤	≤	NUM
ejpam-5499	237	47	i	i	PRON
ejpam-5499	237	48	≤	≤	ADJ
ejpam-5499	237	49	n−	n−	NOUN
ejpam-5499	237	50	1,272	1,272	NUM
ejpam-5499	237	51	iv	iv	NOUN
ejpam-5499	237	52	.	.	PUNCT
ejpam-5499	237	53	e(e	e(e	NOUN
ejpam-5499	237	54	)	)	PUNCT
ejpam-5499	237	55	=	=	SYM
ejpam-5499	237	56	2273	2273	NUM
ejpam-5499	237	57	proof	proof	NOUN
ejpam-5499	237	58	.	.	PUNCT
ejpam-5499	238	1	let	let	AUX
ejpam-5499	238	2	m(γcn	m(γcn	PRON
ejpam-5499	238	3	)	)	PUNCT
ejpam-5499	238	4	be	be	AUX
ejpam-5499	238	5	the	the	DET
ejpam-5499	238	6	middle	middle	ADJ
ejpam-5499	238	7	graph	graph	NOUN
ejpam-5499	238	8	of	of	ADP
ejpam-5499	238	9	γcn	γcn	PROPN
ejpam-5499	238	10	.274	.274	NUM
ejpam-5499	238	11	i.	i.	NOUN
ejpam-5499	238	12	for	for	ADP
ejpam-5499	238	13	n	n	NOUN
ejpam-5499	238	14	=	=	SYM
ejpam-5499	238	15	2	2	NUM
ejpam-5499	238	16	,	,	PUNCT
ejpam-5499	238	17	it	it	PRON
ejpam-5499	238	18	is	be	AUX
ejpam-5499	238	19	very	very	ADV
ejpam-5499	238	20	obvious	obvious	ADJ
ejpam-5499	238	21	since	since	SCONJ
ejpam-5499	238	22	the	the	DET
ejpam-5499	238	23	middle	middle	ADJ
ejpam-5499	238	24	graph	graph	NOUN
ejpam-5499	238	25	m(γc2	m(γc2	NOUN
ejpam-5499	238	26	)	)	PUNCT
ejpam-5499	238	27	is	be	AUX
ejpam-5499	238	28	isomorphic	isomorphic	ADJ
ejpam-5499	238	29	to	to	ADP
ejpam-5499	238	30	a	a	DET
ejpam-5499	238	31	path275	path275	PROPN
ejpam-5499	238	32	p3	p3	PROPN
ejpam-5499	238	33	with	with	ADP
ejpam-5499	238	34	the	the	DET
ejpam-5499	238	35	vertex	vertex	NOUN
ejpam-5499	238	36	set	set	VERB
ejpam-5499	238	37	v	v	NOUN
ejpam-5499	238	38	(	(	PUNCT
ejpam-5499	238	39	m(γc2	m(γc2	PROPN
ejpam-5499	238	40	)	)	PUNCT
ejpam-5499	238	41	=	=	SYM
ejpam-5499	238	42	{	{	PUNCT
ejpam-5499	238	43	e	e	NOUN
ejpam-5499	238	44	,	,	PUNCT
ejpam-5499	238	45	x1	x1	PROPN
ejpam-5499	238	46	,	,	PUNCT
ejpam-5499	238	47	z1	z1	NOUN
ejpam-5499	238	48	}	}	PUNCT
ejpam-5499	238	49	and	and	CCONJ
ejpam-5499	238	50	edges276	edges276	PROPN
ejpam-5499	238	51	e(m(γc2	e(m(γc2	PROPN
ejpam-5499	238	52	)	)	PUNCT
ejpam-5499	238	53	=	=	PRON
ejpam-5499	239	1	{	{	PUNCT
ejpam-5499	239	2	[	[	X
ejpam-5499	239	3	e	e	NOUN
ejpam-5499	239	4	,	,	PUNCT
ejpam-5499	239	5	z1	z1	NOUN
ejpam-5499	239	6	]	]	X
ejpam-5499	239	7	,	,	PUNCT
ejpam-5499	239	8	[	[	X
ejpam-5499	239	9	z1	z1	VERB
ejpam-5499	239	10	,	,	PUNCT
ejpam-5499	239	11	x1]}.277	x1]}.277	PROPN
ejpam-5499	239	12	ii	ii	PROPN
ejpam-5499	239	13	.	.	PUNCT
ejpam-5499	240	1	for	for	ADP
ejpam-5499	240	2	n	n	NOUN
ejpam-5499	240	3	=	=	SYM
ejpam-5499	240	4	3	3	NUM
ejpam-5499	240	5	,	,	PUNCT
ejpam-5499	240	6	note	note	VERB
ejpam-5499	240	7	that	that	SCONJ
ejpam-5499	240	8	γc3	γc3	PROPN
ejpam-5499	240	9	is	be	AUX
ejpam-5499	240	10	isomorphic	isomorphic	ADJ
ejpam-5499	240	11	to	to	ADP
ejpam-5499	240	12	a	a	DET
ejpam-5499	240	13	cycle	cycle	NOUN
ejpam-5499	240	14	c3	c3	NOUN
ejpam-5499	240	15	with	with	ADP
ejpam-5499	240	16	the	the	DET
ejpam-5499	240	17	vertex	vertex	NOUN
ejpam-5499	240	18	set	set	VERB
ejpam-5499	240	19	v	v	NOUN
ejpam-5499	240	20	(	(	PUNCT
ejpam-5499	240	21	γc3	γc3	PROPN
ejpam-5499	240	22	)	)	PUNCT
ejpam-5499	241	1	=	=	NOUN
ejpam-5499	241	2	278	278	NUM
ejpam-5499	241	3	{	{	PUNCT
ejpam-5499	241	4	e	e	NOUN
ejpam-5499	241	5	,	,	PUNCT
ejpam-5499	241	6	x1	x1	PROPN
ejpam-5499	241	7	,	,	PUNCT
ejpam-5499	241	8	x2	x2	ADJ
ejpam-5499	241	9	}	}	PUNCT
ejpam-5499	241	10	and	and	CCONJ
ejpam-5499	241	11	edge	edge	VERB
ejpam-5499	241	12	set	set	VERB
ejpam-5499	241	13	e(γc3	e(γc3	PROPN
ejpam-5499	241	14	)	)	PUNCT
ejpam-5499	241	15	=	=	PRON
ejpam-5499	241	16	{	{	PUNCT
ejpam-5499	241	17	z1	z1	NOUN
ejpam-5499	241	18	=	=	PUNCT
ejpam-5499	242	1	[	[	X
ejpam-5499	242	2	e	e	NOUN
ejpam-5499	242	3	,	,	PUNCT
ejpam-5499	242	4	x1	x1	PROPN
ejpam-5499	242	5	]	]	X
ejpam-5499	242	6	,	,	PUNCT
ejpam-5499	242	7	z2	z2	PROPN
ejpam-5499	242	8	=	=	PUNCT
ejpam-5499	243	1	[	[	X
ejpam-5499	243	2	e	e	NOUN
ejpam-5499	243	3	,	,	PUNCT
ejpam-5499	243	4	x2	x2	PROPN
ejpam-5499	243	5	]	]	X
ejpam-5499	243	6	,	,	PUNCT
ejpam-5499	243	7	y1	y1	INTJ
ejpam-5499	243	8	=	=	PUNCT
ejpam-5499	244	1	[	[	X
ejpam-5499	244	2	x1x2	x1x2	X
ejpam-5499	244	3	]	]	X
ejpam-5499	244	4	}	}	PUNCT
ejpam-5499	244	5	.	.	PUNCT
ejpam-5499	245	1	now	now	ADV
ejpam-5499	245	2	for279	for279	PROPN
ejpam-5499	245	3	m(γc3	m(γc3	PROPN
ejpam-5499	245	4	)	)	PUNCT
ejpam-5499	245	5	,	,	PUNCT
ejpam-5499	245	6	the	the	DET
ejpam-5499	245	7	vertex	vertex	NOUN
ejpam-5499	245	8	set	set	VERB
ejpam-5499	245	9	v	v	NOUN
ejpam-5499	245	10	(	(	PUNCT
ejpam-5499	245	11	m(γc3	m(γc3	PROPN
ejpam-5499	245	12	)	)	PUNCT
ejpam-5499	245	13	)	)	PUNCT
ejpam-5499	246	1	=	=	PRON
ejpam-5499	246	2	{	{	PUNCT
ejpam-5499	246	3	e	e	NOUN
ejpam-5499	246	4	,	,	PUNCT
ejpam-5499	246	5	x1	x1	PROPN
ejpam-5499	246	6	,	,	PUNCT
ejpam-5499	246	7	x2	x2	PROPN
ejpam-5499	246	8	,	,	PUNCT
ejpam-5499	246	9	z1	z1	PROPN
ejpam-5499	246	10	,	,	PUNCT
ejpam-5499	246	11	z2	z2	PROPN
ejpam-5499	246	12	,	,	PUNCT
ejpam-5499	246	13	y1	y1	NOUN
ejpam-5499	246	14	}	}	PUNCT
ejpam-5499	246	15	where	where	SCONJ
ejpam-5499	246	16	the	the	DET
ejpam-5499	246	17	vertices	vertex	NOUN
ejpam-5499	246	18	x1	x1	PROPN
ejpam-5499	246	19	,	,	PUNCT
ejpam-5499	246	20	x2280	x2280	PROPN
ejpam-5499	246	21	and	and	CCONJ
ejpam-5499	246	22	e	e	NOUN
ejpam-5499	246	23	are	be	AUX
ejpam-5499	246	24	the	the	DET
ejpam-5499	246	25	corner	corner	NOUN
ejpam-5499	246	26	vertices	vertex	NOUN
ejpam-5499	246	27	and	and	CCONJ
ejpam-5499	246	28	the	the	DET
ejpam-5499	246	29	vertices	vertex	NOUN
ejpam-5499	246	30	z1	z1	VERB
ejpam-5499	246	31	,	,	PUNCT
ejpam-5499	246	32	z2	z2	NOUN
ejpam-5499	246	33	and	and	CCONJ
ejpam-5499	246	34	y1	y1	NOUN
ejpam-5499	246	35	are	be	AUX
ejpam-5499	246	36	the	the	DET
ejpam-5499	246	37	inner	inner	ADJ
ejpam-5499	246	38	vertices.281	vertices.281	ADJ
ejpam-5499	246	39	note	note	NOUN
ejpam-5499	247	1	that	that	SCONJ
ejpam-5499	247	2	[	[	X
ejpam-5499	247	3	x1	x1	ADJ
ejpam-5499	247	4	,	,	PUNCT
ejpam-5499	247	5	z1	z1	PROPN
ejpam-5499	247	6	]	]	PUNCT
ejpam-5499	247	7	and	and	CCONJ
ejpam-5499	247	8	[	[	X
ejpam-5499	247	9	z1	z1	ADJ
ejpam-5499	247	10	,	,	PUNCT
ejpam-5499	247	11	z2	z2	PROPN
ejpam-5499	247	12	]	]	X
ejpam-5499	247	13	∈	∈	PROPN
ejpam-5499	247	14	e(γc3	e(γc3	PROPN
ejpam-5499	247	15	)	)	PUNCT
ejpam-5499	247	16	but	but	CCONJ
ejpam-5499	247	17	[	[	X
ejpam-5499	247	18	x1	x1	X
ejpam-5499	247	19	,	,	PUNCT
ejpam-5499	247	20	x2	x2	PROPN
ejpam-5499	247	21	]	]	PUNCT
ejpam-5499	247	22	/∈	/∈	PUNCT
ejpam-5499	247	23	e(γc3	e(γc3	PROPN
ejpam-5499	247	24	)	)	PUNCT
ejpam-5499	247	25	thus	thus	ADV
ejpam-5499	247	26	the	the	DET
ejpam-5499	247	27	distance	distance	NOUN
ejpam-5499	247	28	from282	from282	PROPN
ejpam-5499	247	29	x1	x1	PROPN
ejpam-5499	247	30	to	to	ADP
ejpam-5499	247	31	z2	z2	PROPN
ejpam-5499	247	32	is	be	AUX
ejpam-5499	247	33	2	2	NUM
ejpam-5499	247	34	same	same	ADJ
ejpam-5499	247	35	with	with	ADP
ejpam-5499	247	36	x2	x2	PROPN
ejpam-5499	247	37	to	to	ADP
ejpam-5499	247	38	x1	x1	PROPN
ejpam-5499	247	39	and	and	CCONJ
ejpam-5499	247	40	e	e	NOUN
ejpam-5499	247	41	to	to	ADP
ejpam-5499	247	42	y1	y1	PROPN
ejpam-5499	247	43	.	.	PUNCT
ejpam-5499	248	1	also	also	ADV
ejpam-5499	248	2	the	the	DET
ejpam-5499	248	3	distance	distance	NOUN
ejpam-5499	248	4	of	of	ADP
ejpam-5499	248	5	each	each	DET
ejpam-5499	248	6	corner283	corner283	PROPN
ejpam-5499	248	7	vertex	vertex	NOUN
ejpam-5499	248	8	to	to	ADP
ejpam-5499	248	9	each	each	DET
ejpam-5499	248	10	other	other	ADJ
ejpam-5499	248	11	is	be	AUX
ejpam-5499	248	12	2	2	NUM
ejpam-5499	248	13	.	.	PUNCT
ejpam-5499	249	1	now	now	ADV
ejpam-5499	249	2	for	for	ADP
ejpam-5499	249	3	the	the	DET
ejpam-5499	249	4	distance	distance	NOUN
ejpam-5499	249	5	of	of	ADP
ejpam-5499	249	6	all	all	DET
ejpam-5499	249	7	inner	inner	ADJ
ejpam-5499	249	8	vertices	vertex	NOUN
ejpam-5499	249	9	to	to	ADP
ejpam-5499	249	10	each	each	DET
ejpam-5499	249	11	other	other	ADJ
ejpam-5499	249	12	is284	is284	PROPN
ejpam-5499	249	13	1	1	NUM
ejpam-5499	249	14	since	since	SCONJ
ejpam-5499	249	15	they	they	PRON
ejpam-5499	249	16	are	be	AUX
ejpam-5499	249	17	adjacent	adjacent	ADJ
ejpam-5499	249	18	edges	edge	NOUN
ejpam-5499	249	19	in	in	ADP
ejpam-5499	249	20	γc3	γc3	PROPN
ejpam-5499	249	21	.	.	PUNCT
ejpam-5499	250	1	thus	thus	ADV
ejpam-5499	250	2	the	the	DET
ejpam-5499	250	3	maximum	maximum	ADJ
ejpam-5499	250	4	distance	distance	NOUN
ejpam-5499	250	5	or	or	CCONJ
ejpam-5499	250	6	eccentricity285	eccentricity285	PROPN
ejpam-5499	250	7	e(x1	e(x1	NOUN
ejpam-5499	250	8	)	)	PUNCT
ejpam-5499	250	9	=	=	SYM
ejpam-5499	250	10	e(x2	e(x2	NOUN
ejpam-5499	250	11	)	)	PUNCT
ejpam-5499	250	12	=	=	SYM
ejpam-5499	250	13	e(e	e(e	NOUN
ejpam-5499	250	14	)	)	PUNCT
ejpam-5499	250	15	=	=	SYM
ejpam-5499	250	16	e(z1	e(z1	ADJ
ejpam-5499	250	17	)	)	PUNCT
ejpam-5499	250	18	=	=	SYM
ejpam-5499	250	19	e(z2	e(z2	NOUN
ejpam-5499	250	20	)	)	PUNCT
ejpam-5499	250	21	=	=	SYM
ejpam-5499	250	22	e(y1	e(y1	X
ejpam-5499	250	23	)	)	PUNCT
ejpam-5499	250	24	=	=	NOUN
ejpam-5499	250	25	2.286	2.286	NUM
ejpam-5499	250	26	iii	iii	NOUN
ejpam-5499	250	27	.	.	PUNCT
ejpam-5499	251	1	for	for	ADP
ejpam-5499	251	2	n	n	X
ejpam-5499	251	3	≥	≥	NUM
ejpam-5499	251	4	4	4	NUM
ejpam-5499	251	5	,	,	PUNCT
ejpam-5499	251	6	the	the	DET
ejpam-5499	251	7	proof	proof	NOUN
ejpam-5499	251	8	follows	follow	VERB
ejpam-5499	251	9	from	from	ADP
ejpam-5499	251	10	theorem	theorem	ADJ
ejpam-5499	251	11	3.287	3.287	NUM
ejpam-5499	251	12	j.	j.	PROPN
ejpam-5499	251	13	m.	m.	PROPN
ejpam-5499	251	14	jamis	jamis	PROPN
ejpam-5499	251	15	,	,	PUNCT
ejpam-5499	251	16	d.	d.	PROPN
ejpam-5499	251	17	m.	m.	PROPN
ejpam-5499	251	18	magpantay	magpantay	PROPN
ejpam-5499	251	19	/	/	SYM
ejpam-5499	251	20	eur	eur	PROPN
ejpam-5499	251	21	.	.	PUNCT
ejpam-5499	252	1	j.	j.	PROPN
ejpam-5499	252	2	pure	pure	PROPN
ejpam-5499	252	3	appl	appl	PROPN
ejpam-5499	252	4	.	.	PROPN
ejpam-5499	252	5	math	math	PROPN
ejpam-5499	252	6	,	,	PUNCT
ejpam-5499	252	7	18	18	NUM
ejpam-5499	252	8	(	(	PUNCT
ejpam-5499	252	9	2	2	NUM
ejpam-5499	252	10	)	)	PUNCT
ejpam-5499	252	11	(	(	PUNCT
ejpam-5499	252	12	2025	2025	NUM
ejpam-5499	252	13	)	)	PUNCT
ejpam-5499	252	14	,	,	PUNCT
ejpam-5499	252	15	5499	5499	NUM
ejpam-5499	252	16	13	13	NUM
ejpam-5499	252	17	of	of	ADP
ejpam-5499	252	18	18	18	NUM
ejpam-5499	252	19	4.3	4.3	NUM
ejpam-5499	252	20	.	.	PUNCT
ejpam-5499	253	1	radius288	radius288	PROPN
ejpam-5499	253	2	theorem	theorem	VERB
ejpam-5499	253	3	5	5	NUM
ejpam-5499	253	4	.	.	PUNCT
ejpam-5499	254	1	if	if	SCONJ
ejpam-5499	254	2	m(γcn	m(γcn	PROPN
ejpam-5499	254	3	)	)	PUNCT
ejpam-5499	254	4	be	be	VERB
ejpam-5499	254	5	the	the	DET
ejpam-5499	254	6	middle	middle	ADJ
ejpam-5499	254	7	graph	graph	NOUN
ejpam-5499	254	8	of	of	ADP
ejpam-5499	254	9	γcn	γcn	PROPN
ejpam-5499	254	10	,	,	PUNCT
ejpam-5499	254	11	then289	then289	PROPN
ejpam-5499	254	12	rad(m(γcn	rad(m(γcn	NUM
ejpam-5499	254	13	)	)	PUNCT
ejpam-5499	254	14	)	)	PUNCT
ejpam-5499	255	1	=	=	PRON
ejpam-5499	255	2	{	{	PUNCT
ejpam-5499	255	3	1	1	NUM
ejpam-5499	255	4	,	,	PUNCT
ejpam-5499	255	5	if	if	SCONJ
ejpam-5499	255	6	n=2	n=2	ADV
ejpam-5499	255	7	2	2	NUM
ejpam-5499	255	8	,	,	PUNCT
ejpam-5499	255	9	for	for	ADP
ejpam-5499	255	10	n	n	NUM
ejpam-5499	255	11	≥	≥	NOUN
ejpam-5499	255	12	3	3	NUM
ejpam-5499	255	13	and	and	CCONJ
ejpam-5499	255	14	diam(m(γcn	diam(m(γcn	PROPN
ejpam-5499	255	15	)	)	PUNCT
ejpam-5499	255	16	)	)	PUNCT
ejpam-5499	256	1	=	=	PRON
ejpam-5499	256	2	{	{	PUNCT
ejpam-5499	256	3	2	2	NUM
ejpam-5499	256	4	,	,	PUNCT
ejpam-5499	256	5	if	if	SCONJ
ejpam-5499	256	6	n=2	n=2	ADV
ejpam-5499	256	7	or	or	CCONJ
ejpam-5499	256	8	3	3	NUM
ejpam-5499	256	9	3	3	NUM
ejpam-5499	256	10	,	,	PUNCT
ejpam-5499	256	11	for	for	ADP
ejpam-5499	256	12	n	n	PRON
ejpam-5499	256	13	≥	≥	NUM
ejpam-5499	256	14	4	4	NUM
ejpam-5499	256	15	.	.	NOUN
ejpam-5499	256	16	290	290	NUM
ejpam-5499	256	17	proof	proof	NOUN
ejpam-5499	256	18	.	.	PUNCT
ejpam-5499	257	1	let	let	AUX
ejpam-5499	257	2	m(γcn	m(γcn	PRON
ejpam-5499	257	3	)	)	PUNCT
ejpam-5499	257	4	be	be	AUX
ejpam-5499	257	5	the	the	DET
ejpam-5499	257	6	middle	middle	ADJ
ejpam-5499	257	7	graph	graph	NOUN
ejpam-5499	257	8	of	of	ADP
ejpam-5499	257	9	γcn	γcn	PROPN
ejpam-5499	257	10	.	.	PUNCT
ejpam-5499	258	1	we	we	PRON
ejpam-5499	258	2	divided	divide	VERB
ejpam-5499	258	3	it	it	PRON
ejpam-5499	258	4	into	into	ADP
ejpam-5499	258	5	three	three	NUM
ejpam-5499	258	6	cases:291	cases:291	PROPN
ejpam-5499	258	7	i.	i.	NOUN
ejpam-5499	258	8	for	for	ADP
ejpam-5499	258	9	n	n	NOUN
ejpam-5499	258	10	=	=	SYM
ejpam-5499	258	11	2	2	NUM
ejpam-5499	258	12	,	,	PUNCT
ejpam-5499	258	13	sincem(γc2	sincem(γc2	NOUN
ejpam-5499	258	14	)	)	PUNCT
ejpam-5499	258	15	is	be	AUX
ejpam-5499	258	16	isomorphic	isomorphic	ADJ
ejpam-5499	258	17	to	to	ADP
ejpam-5499	258	18	a	a	DET
ejpam-5499	258	19	path	path	NOUN
ejpam-5499	258	20	of	of	ADP
ejpam-5499	258	21	length	length	NOUN
ejpam-5499	258	22	3	3	NUM
ejpam-5499	258	23	,	,	PUNCT
ejpam-5499	258	24	clearly	clearly	ADV
ejpam-5499	258	25	rad(m(γc2	rad(m(γc2	NOUN
ejpam-5499	258	26	)	)	PUNCT
ejpam-5499	258	27	)	)	PUNCT
ejpam-5499	259	1	=	=	SYM
ejpam-5499	259	2	1292	1292	NUM
ejpam-5499	259	3	and	and	CCONJ
ejpam-5499	259	4	the	the	DET
ejpam-5499	259	5	diameter	diameter	NOUN
ejpam-5499	259	6	diam(m(γc2	diam(m(γc2	NOUN
ejpam-5499	259	7	)	)	PUNCT
ejpam-5499	259	8	)	)	PUNCT
ejpam-5499	260	1	=	=	PUNCT
ejpam-5499	260	2	2.293	2.293	NUM
ejpam-5499	260	3	ii	ii	NOUN
ejpam-5499	260	4	.	.	PUNCT
ejpam-5499	261	1	for	for	ADP
ejpam-5499	261	2	n	n	NOUN
ejpam-5499	261	3	=	=	SYM
ejpam-5499	261	4	3	3	NUM
ejpam-5499	261	5	,	,	PUNCT
ejpam-5499	261	6	by	by	ADP
ejpam-5499	261	7	theorem	theorem	NOUN
ejpam-5499	261	8	4	4	NUM
ejpam-5499	261	9	,	,	PUNCT
ejpam-5499	261	10	the	the	DET
ejpam-5499	261	11	eccentricity	eccentricity	NOUN
ejpam-5499	261	12	of	of	ADP
ejpam-5499	261	13	all	all	DET
ejpam-5499	261	14	the	the	DET
ejpam-5499	261	15	vertices	vertex	NOUN
ejpam-5499	261	16	is	be	AUX
ejpam-5499	261	17	equal	equal	ADJ
ejpam-5499	261	18	to	to	ADP
ejpam-5499	261	19	2	2	NUM
ejpam-5499	261	20	,	,	PUNCT
ejpam-5499	261	21	thus	thus	ADV
ejpam-5499	261	22	the294	the294	NOUN
ejpam-5499	261	23	radius	radius	NOUN
ejpam-5499	261	24	and	and	CCONJ
ejpam-5499	261	25	diameter	diameter	NOUN
ejpam-5499	261	26	rad(m(γc3	rad(m(γc3	PROPN
ejpam-5499	261	27	)	)	PUNCT
ejpam-5499	261	28	)	)	PUNCT
ejpam-5499	262	1	=	=	SYM
ejpam-5499	262	2	2	2	NUM
ejpam-5499	262	3	and	and	CCONJ
ejpam-5499	262	4	diam(m(γc3	diam(m(γc3	NOUN
ejpam-5499	262	5	)	)	PUNCT
ejpam-5499	262	6	)	)	PUNCT
ejpam-5499	263	1	=	=	SYM
ejpam-5499	263	2	2	2	NUM
ejpam-5499	263	3	respectively.295	respectively.295	NOUN
ejpam-5499	263	4	iii	iii	NOUN
ejpam-5499	263	5	.	.	PUNCT
ejpam-5499	264	1	for	for	ADP
ejpam-5499	264	2	n	n	X
ejpam-5499	264	3	≥	≥	NOUN
ejpam-5499	264	4	4	4	NUM
ejpam-5499	264	5	,	,	PUNCT
ejpam-5499	264	6	from	from	ADP
ejpam-5499	264	7	theorem	theorem	ADJ
ejpam-5499	264	8	4	4	NUM
ejpam-5499	264	9	,	,	PUNCT
ejpam-5499	264	10	e(zi	e(zi	NUM
ejpam-5499	264	11	)	)	PUNCT
ejpam-5499	264	12	=	=	SYM
ejpam-5499	264	13	2	2	NUM
ejpam-5499	264	14	for	for	ADP
ejpam-5499	264	15	all	all	DET
ejpam-5499	264	16	1	1	NUM
ejpam-5499	264	17	≤	≤	NUM
ejpam-5499	264	18	i	i	PRON
ejpam-5499	264	19	≤	≤	ADJ
ejpam-5499	264	20	n−	n−	NOUN
ejpam-5499	264	21	1	1	NUM
ejpam-5499	264	22	and	and	CCONJ
ejpam-5499	264	23	e(e	e(e	NOUN
ejpam-5499	264	24	)	)	PUNCT
ejpam-5499	264	25	=	=	SYM
ejpam-5499	264	26	2	2	NUM
ejpam-5499	264	27	and	and	CCONJ
ejpam-5499	264	28	this	this	DET
ejpam-5499	264	29	is296	is296	NOUN
ejpam-5499	264	30	the	the	DET
ejpam-5499	264	31	minimum	minimum	NOUN
ejpam-5499	264	32	eccentricity	eccentricity	NOUN
ejpam-5499	264	33	,	,	PUNCT
ejpam-5499	264	34	thus	thus	ADV
ejpam-5499	264	35	the	the	DET
ejpam-5499	264	36	radius	radius	NOUN
ejpam-5499	264	37	rad(m(γcn	rad(m(γcn	VERB
ejpam-5499	264	38	)	)	PUNCT
ejpam-5499	264	39	)	)	PUNCT
ejpam-5499	265	1	=	=	SYM
ejpam-5499	265	2	2	2	X
ejpam-5499	265	3	.	.	PUNCT
ejpam-5499	265	4	also	also	ADV
ejpam-5499	265	5	using	use	VERB
ejpam-5499	265	6	the	the	DET
ejpam-5499	265	7	same297	same297	PROPN
ejpam-5499	265	8	reference	reference	NOUN
ejpam-5499	265	9	,	,	PUNCT
ejpam-5499	265	10	the	the	DET
ejpam-5499	265	11	vertices	vertex	NOUN
ejpam-5499	265	12	with	with	ADP
ejpam-5499	265	13	the	the	DET
ejpam-5499	265	14	maximum	maximum	ADJ
ejpam-5499	265	15	eccentricities	eccentricity	NOUN
ejpam-5499	265	16	are	be	AUX
ejpam-5499	265	17	the	the	DET
ejpam-5499	265	18	vertices	vertex	NOUN
ejpam-5499	265	19	xi	xi	X
ejpam-5499	265	20	and	and	CCONJ
ejpam-5499	265	21	yi298	yi298	PROPN
ejpam-5499	265	22	with	with	ADP
ejpam-5499	265	23	e(xi	e(xi	NUM
ejpam-5499	265	24	)	)	PUNCT
ejpam-5499	265	25	=	=	SYM
ejpam-5499	265	26	3	3	NUM
ejpam-5499	265	27	for	for	ADP
ejpam-5499	265	28	all	all	DET
ejpam-5499	265	29	1	1	NUM
ejpam-5499	265	30	≤	≤	NUM
ejpam-5499	266	1	i	i	PRON
ejpam-5499	266	2	≤	≤	ADJ
ejpam-5499	266	3	n−	n−	NOUN
ejpam-5499	266	4	1	1	NUM
ejpam-5499	266	5	and	and	CCONJ
ejpam-5499	266	6	e(yp	e(yp	NUM
ejpam-5499	266	7	)	)	PUNCT
ejpam-5499	266	8	=	=	SYM
ejpam-5499	266	9	3	3	NUM
ejpam-5499	266	10	for	for	ADP
ejpam-5499	266	11	all	all	DET
ejpam-5499	266	12	1	1	NUM
ejpam-5499	266	13	≤	≤	NOUN
ejpam-5499	266	14	p	p	NOUN
ejpam-5499	266	15	≤	≤	NUM
ejpam-5499	266	16	n−1	n−1	PROPN
ejpam-5499	266	17	2	2	NUM
ejpam-5499	266	18	if	if	SCONJ
ejpam-5499	266	19	n	n	NOUN
ejpam-5499	266	20	is	be	AUX
ejpam-5499	266	21	odd	odd	ADJ
ejpam-5499	266	22	and299	and299	PROPN
ejpam-5499	266	23	1	1	NUM
ejpam-5499	266	24	≤	≤	NOUN
ejpam-5499	266	25	p	p	NOUN
ejpam-5499	266	26	≤	≤	NOUN
ejpam-5499	266	27	n−2	n−2	PROPN
ejpam-5499	266	28	2	2	NUM
ejpam-5499	266	29	if	if	SCONJ
ejpam-5499	266	30	n	n	PRON
ejpam-5499	266	31	is	be	AUX
ejpam-5499	266	32	even	even	ADV
ejpam-5499	266	33	.	.	PUNCT
ejpam-5499	267	1	hence	hence	ADV
ejpam-5499	267	2	the	the	DET
ejpam-5499	267	3	diameter	diameter	NOUN
ejpam-5499	267	4	diam(m(γcn	diam(m(γcn	PROPN
ejpam-5499	267	5	)	)	PUNCT
ejpam-5499	267	6	)	)	PUNCT
ejpam-5499	268	1	=	=	PUNCT
ejpam-5499	268	2	3.300	3.300	NUM
ejpam-5499	268	3	4.4	4.4	NUM
ejpam-5499	268	4	.	.	PUNCT
ejpam-5499	269	1	central	central	ADJ
ejpam-5499	269	2	vertices301	vertices301	PROPN
ejpam-5499	269	3	theorem	theorem	VERB
ejpam-5499	269	4	6	6	NUM
ejpam-5499	269	5	.	.	PUNCT
ejpam-5499	269	6	for	for	ADP
ejpam-5499	269	7	the	the	DET
ejpam-5499	269	8	middle	middle	ADJ
ejpam-5499	269	9	graph	graph	NOUN
ejpam-5499	269	10	m(γcn	m(γcn	PROPN
ejpam-5499	269	11	)	)	PUNCT
ejpam-5499	269	12	for	for	ADP
ejpam-5499	269	13	n	n	X
ejpam-5499	269	14	≥	≥	NUM
ejpam-5499	269	15	4	4	NUM
ejpam-5499	269	16	,	,	PUNCT
ejpam-5499	269	17	the	the	DET
ejpam-5499	269	18	central	central	ADJ
ejpam-5499	269	19	vertices	vertex	NOUN
ejpam-5499	269	20	are	be	AUX
ejpam-5499	269	21	the	the	DET
ejpam-5499	269	22	vertices302	vertices302	PROPN
ejpam-5499	269	23	zi	zi	PROPN
ejpam-5499	269	24	∈	∈	PROPN
ejpam-5499	269	25	v	v	PROPN
ejpam-5499	269	26	(	(	PUNCT
ejpam-5499	269	27	m(γcn	m(γcn	NOUN
ejpam-5499	269	28	)	)	PUNCT
ejpam-5499	269	29	)	)	PUNCT
ejpam-5499	269	30	and	and	CCONJ
ejpam-5499	269	31	e.303	e.303	PRON
ejpam-5499	269	32	proof	proof	NOUN
ejpam-5499	269	33	.	.	PUNCT
ejpam-5499	270	1	from	from	ADP
ejpam-5499	270	2	theorem	theorem	ADJ
ejpam-5499	270	3	4	4	NUM
ejpam-5499	270	4	,	,	PUNCT
ejpam-5499	270	5	the	the	DET
ejpam-5499	270	6	eccentricity	eccentricity	NOUN
ejpam-5499	270	7	e(zi	e(zi	NOUN
ejpam-5499	270	8	)	)	PUNCT
ejpam-5499	270	9	=	=	SYM
ejpam-5499	270	10	2	2	NUM
ejpam-5499	270	11	for	for	ADP
ejpam-5499	270	12	all	all	DET
ejpam-5499	270	13	i	i	PRON
ejpam-5499	270	14	,	,	PUNCT
ejpam-5499	270	15	1	1	NUM
ejpam-5499	270	16	≤	≤	NUM
ejpam-5499	270	17	i	i	X
ejpam-5499	270	18	≤	≤	ADV
ejpam-5499	270	19	n−1	n−1	PROPN
ejpam-5499	270	20	and	and	CCONJ
ejpam-5499	270	21	e(e	e(e	NOUN
ejpam-5499	270	22	)	)	PUNCT
ejpam-5499	270	23	=	=	PUNCT
ejpam-5499	271	1	2.304	2.304	NUM
ejpam-5499	271	2	now	now	ADV
ejpam-5499	271	3	by	by	ADP
ejpam-5499	271	4	theorem	theorem	NOUN
ejpam-5499	271	5	5	5	NUM
ejpam-5499	271	6	,	,	PUNCT
ejpam-5499	271	7	the	the	DET
ejpam-5499	271	8	radius	radius	NOUN
ejpam-5499	271	9	rad(m(γcn	rad(m(γcn	VERB
ejpam-5499	271	10	)	)	PUNCT
ejpam-5499	271	11	)	)	PUNCT
ejpam-5499	272	1	=	=	SYM
ejpam-5499	272	2	e(zi	e(zi	PROPN
ejpam-5499	272	3	)	)	PUNCT
ejpam-5499	272	4	=	=	SYM
ejpam-5499	272	5	e(e	e(e	NOUN
ejpam-5499	272	6	)	)	PUNCT
ejpam-5499	272	7	=	=	SYM
ejpam-5499	272	8	2	2	NUM
ejpam-5499	272	9	for	for	ADP
ejpam-5499	272	10	all	all	DET
ejpam-5499	272	11	1	1	NUM
ejpam-5499	272	12	≤	≤	NUM
ejpam-5499	273	1	i	i	PRON
ejpam-5499	273	2	≤	≤	ADJ
ejpam-5499	273	3	n−	n−	NOUN
ejpam-5499	273	4	1	1	NUM
ejpam-5499	273	5	.	.	PUNCT
ejpam-5499	274	1	the305	the305	ADJ
ejpam-5499	274	2	ramaining	ramaining	NOUN
ejpam-5499	274	3	vertices	vertice	VERB
ejpam-5499	274	4	xi′s	xi′s	PROPN
ejpam-5499	274	5	and	and	CCONJ
ejpam-5499	274	6	yp′s	yp′s	PROPN
ejpam-5499	274	7	has	have	VERB
ejpam-5499	274	8	the	the	DET
ejpam-5499	274	9	eccentricity	eccentricity	NOUN
ejpam-5499	274	10	of	of	ADP
ejpam-5499	274	11	3	3	NUM
ejpam-5499	274	12	.	.	PUNCT
ejpam-5499	275	1	thus	thus	ADV
ejpam-5499	275	2	the	the	DET
ejpam-5499	275	3	central	central	ADJ
ejpam-5499	275	4	vertices	vertex	NOUN
ejpam-5499	275	5	are	be	AUX
ejpam-5499	275	6	all306	all306	ADJ
ejpam-5499	275	7	the	the	DET
ejpam-5499	275	8	zi	zi	PROPN
ejpam-5499	275	9	∈	∈	PROPN
ejpam-5499	275	10	v	v	PROPN
ejpam-5499	275	11	(	(	PUNCT
ejpam-5499	275	12	m(γcn	m(γcn	NOUN
ejpam-5499	275	13	)	)	PUNCT
ejpam-5499	275	14	)	)	PUNCT
ejpam-5499	275	15	and	and	CCONJ
ejpam-5499	275	16	e.307	e.307	NOUN
ejpam-5499	275	17	4.5	4.5	NUM
ejpam-5499	275	18	.	.	PUNCT
ejpam-5499	276	1	center308	center308	PROPN
ejpam-5499	276	2	theorem	theorem	VERB
ejpam-5499	276	3	7	7	NUM
ejpam-5499	276	4	.	.	PUNCT
ejpam-5499	277	1	the	the	DET
ejpam-5499	277	2	set	set	NOUN
ejpam-5499	277	3	of	of	ADP
ejpam-5499	277	4	vertices	vertex	NOUN
ejpam-5499	277	5	cen(m(γcn	cen(m(γcn	NUM
ejpam-5499	277	6	)	)	PUNCT
ejpam-5499	277	7	)	)	PUNCT
ejpam-5499	278	1	=	=	PRON
ejpam-5499	278	2	{	{	PUNCT
ejpam-5499	278	3	{	{	PUNCT
ejpam-5499	278	4	zi|1	zi|1	NOUN
ejpam-5499	278	5	≤	≤	NOUN
ejpam-5499	278	6	i	i	PRON
ejpam-5499	278	7	≤	≤	ADJ
ejpam-5499	278	8	n−	n−	NOUN
ejpam-5499	278	9	1	1	NUM
ejpam-5499	278	10	}	}	PUNCT
ejpam-5499	278	11	⋃	⋃	NOUN
ejpam-5499	278	12	{	{	PUNCT
ejpam-5499	278	13	e	e	NOUN
ejpam-5499	278	14	}	}	PUNCT
ejpam-5499	278	15	}	}	PUNCT
ejpam-5499	278	16	is	be	AUX
ejpam-5499	278	17	the	the	DET
ejpam-5499	278	18	center309	center309	PROPN
ejpam-5499	278	19	of	of	ADP
ejpam-5499	278	20	m(γcn	m(γcn	PROPN
ejpam-5499	278	21	)	)	PUNCT
ejpam-5499	278	22	for	for	ADP
ejpam-5499	278	23	n	n	NUM
ejpam-5499	278	24	≥	≥	NOUN
ejpam-5499	278	25	4.310	4.310	NUM
ejpam-5499	278	26	proof	proof	NOUN
ejpam-5499	278	27	.	.	PUNCT
ejpam-5499	279	1	the	the	DET
ejpam-5499	279	2	proof	proof	NOUN
ejpam-5499	279	3	of	of	ADP
ejpam-5499	279	4	this	this	DET
ejpam-5499	279	5	theorem	theorem	NOUN
ejpam-5499	279	6	follows	follow	VERB
ejpam-5499	279	7	from	from	ADP
ejpam-5499	279	8	theorem	theorem	ADJ
ejpam-5499	279	9	6.311	6.311	NUM
ejpam-5499	279	10	4.6	4.6	NUM
ejpam-5499	279	11	.	.	PUNCT
ejpam-5499	280	1	complete	complete	ADJ
ejpam-5499	280	2	subgraph312	subgraph312	PROPN
ejpam-5499	280	3	theorem	theorem	NOUN
ejpam-5499	280	4	8	8	NUM
ejpam-5499	280	5	.	.	PUNCT
ejpam-5499	281	1	let	let	VERB
ejpam-5499	281	2	h	h	PRON
ejpam-5499	281	3	be	be	AUX
ejpam-5499	281	4	a	a	DET
ejpam-5499	281	5	subgraph	subgraph	NOUN
ejpam-5499	281	6	induced	induce	VERB
ejpam-5499	281	7	by	by	ADP
ejpam-5499	281	8	cen(m(γcn	cen(m(γcn	PROPN
ejpam-5499	281	9	)	)	PUNCT
ejpam-5499	281	10	)	)	PUNCT
ejpam-5499	281	11	for	for	ADP
ejpam-5499	281	12	n	n	X
ejpam-5499	281	13	≥	≥	NUM
ejpam-5499	281	14	4	4	NUM
ejpam-5499	281	15	,	,	PUNCT
ejpam-5499	281	16	then	then	ADV
ejpam-5499	281	17	h	h	PROPN
ejpam-5499	281	18	is	be	AUX
ejpam-5499	281	19	a313	a313	PROPN
ejpam-5499	281	20	complete	complete	ADJ
ejpam-5499	281	21	subgraph	subgraph	NOUN
ejpam-5499	281	22	of	of	ADP
ejpam-5499	281	23	order	order	NOUN
ejpam-5499	281	24	n	n	PROPN
ejpam-5499	281	25	.314	.314	NUM
ejpam-5499	281	26	j.	j.	PROPN
ejpam-5499	281	27	m.	m.	PROPN
ejpam-5499	281	28	jamis	jamis	PROPN
ejpam-5499	281	29	,	,	PUNCT
ejpam-5499	281	30	d.	d.	PROPN
ejpam-5499	281	31	m.	m.	PROPN
ejpam-5499	281	32	magpantay	magpantay	PROPN
ejpam-5499	281	33	/	/	SYM
ejpam-5499	281	34	eur	eur	PROPN
ejpam-5499	281	35	.	.	PUNCT
ejpam-5499	282	1	j.	j.	PROPN
ejpam-5499	282	2	pure	pure	PROPN
ejpam-5499	282	3	appl	appl	PROPN
ejpam-5499	282	4	.	.	PROPN
ejpam-5499	282	5	math	math	PROPN
ejpam-5499	282	6	,	,	PUNCT
ejpam-5499	282	7	18	18	NUM
ejpam-5499	282	8	(	(	PUNCT
ejpam-5499	282	9	2	2	NUM
ejpam-5499	282	10	)	)	PUNCT
ejpam-5499	282	11	(	(	PUNCT
ejpam-5499	282	12	2025	2025	NUM
ejpam-5499	282	13	)	)	PUNCT
ejpam-5499	282	14	,	,	PUNCT
ejpam-5499	282	15	5499	5499	NUM
ejpam-5499	282	16	14	14	NUM
ejpam-5499	282	17	of	of	ADP
ejpam-5499	282	18	18	18	NUM
ejpam-5499	282	19	proof.315	proof.315	PROPN
ejpam-5499	282	20	letm(γcn	letm(γcn	NOUN
ejpam-5499	282	21	)	)	PUNCT
ejpam-5499	282	22	be	be	AUX
ejpam-5499	282	23	the	the	DET
ejpam-5499	282	24	middle	middle	ADJ
ejpam-5499	282	25	graph	graph	NOUN
ejpam-5499	282	26	of	of	ADP
ejpam-5499	282	27	γcn	γcn	PROPN
ejpam-5499	282	28	for	for	ADP
ejpam-5499	282	29	n	n	X
ejpam-5499	282	30	≥	≥	NUM
ejpam-5499	282	31	4	4	NUM
ejpam-5499	282	32	.	.	PUNCT
ejpam-5499	283	1	supposeh	supposeh	PROPN
ejpam-5499	283	2	is	be	AUX
ejpam-5499	283	3	a	a	DET
ejpam-5499	283	4	subgraph	subgraph	NOUN
ejpam-5499	283	5	induced	induce	VERB
ejpam-5499	283	6	by316	by316	PROPN
ejpam-5499	283	7	cen(m(γcn	cen(m(γcn	NUM
ejpam-5499	283	8	)	)	PUNCT
ejpam-5499	283	9	)	)	PUNCT
ejpam-5499	283	10	,	,	PUNCT
ejpam-5499	283	11	then	then	ADV
ejpam-5499	283	12	v	v	X
ejpam-5499	283	13	(	(	PUNCT
ejpam-5499	283	14	h	h	NOUN
ejpam-5499	283	15	)	)	PUNCT
ejpam-5499	283	16	=	=	PRON
ejpam-5499	283	17	{	{	PUNCT
ejpam-5499	283	18	{	{	PUNCT
ejpam-5499	283	19	zi|1	zi|1	NOUN
ejpam-5499	283	20	≤	≤	NOUN
ejpam-5499	283	21	i	i	PRON
ejpam-5499	283	22	≤	≤	NOUN
ejpam-5499	283	23	n−1	n−1	PROPN
ejpam-5499	283	24	}	}	PUNCT
ejpam-5499	283	25	⋃	⋃	NOUN
ejpam-5499	283	26	{	{	PUNCT
ejpam-5499	283	27	e	e	NOUN
ejpam-5499	283	28	}	}	PUNCT
ejpam-5499	283	29	}	}	PUNCT
ejpam-5499	283	30	=	=	SYM
ejpam-5499	283	31	z∪{e	z∪{e	NOUN
ejpam-5499	283	32	}	}	PUNCT
ejpam-5499	283	33	and	and	CCONJ
ejpam-5499	283	34	clearly	clearly	ADV
ejpam-5499	283	35	|v	|v	VERB
ejpam-5499	283	36	(	(	PUNCT
ejpam-5499	283	37	h)|	h)|	NOUN
ejpam-5499	283	38	=	=	PUNCT
ejpam-5499	283	39	n.317	n.317	ADV
ejpam-5499	283	40	note	note	VERB
ejpam-5499	283	41	that	that	SCONJ
ejpam-5499	283	42	for	for	ADP
ejpam-5499	283	43	every	every	DET
ejpam-5499	283	44	vertex	vertex	NOUN
ejpam-5499	283	45	zi	zi	NOUN
ejpam-5499	283	46	and	and	CCONJ
ejpam-5499	283	47	zj	zj	PROPN
ejpam-5499	283	48	element	element	NOUN
ejpam-5499	283	49	of	of	ADP
ejpam-5499	283	50	z	z	NOUN
ejpam-5499	283	51	where	where	SCONJ
ejpam-5499	283	52	i	i	PRON
ejpam-5499	283	53	̸=	̸=	PROPN
ejpam-5499	283	54	j	j	PROPN
ejpam-5499	283	55	,	,	PUNCT
ejpam-5499	283	56	[	[	X
ejpam-5499	283	57	zi	zi	NOUN
ejpam-5499	283	58	,	,	PUNCT
ejpam-5499	283	59	zj	zj	X
ejpam-5499	283	60	]	]	PUNCT
ejpam-5499	283	61	∈	∈	PROPN
ejpam-5499	283	62	e(m(γcn))318	e(m(γcn))318	NOUN
ejpam-5499	283	63	and	and	CCONJ
ejpam-5499	283	64	also	also	ADV
ejpam-5499	283	65	[	[	X
ejpam-5499	283	66	zi	zi	NOUN
ejpam-5499	283	67	,	,	PUNCT
ejpam-5499	283	68	e	e	X
ejpam-5499	283	69	]	]	X
ejpam-5499	283	70	∈	∈	PROPN
ejpam-5499	283	71	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	283	72	)	)	PUNCT
ejpam-5499	283	73	)	)	PUNCT
ejpam-5499	283	74	thus	thus	ADV
ejpam-5499	283	75	degh(zi	degh(zi	VERB
ejpam-5499	283	76	)	)	PUNCT
ejpam-5499	283	77	=	=	SYM
ejpam-5499	283	78	degh(e	degh(e	NOUN
ejpam-5499	283	79	)	)	PUNCT
ejpam-5499	283	80	=	=	SYM
ejpam-5499	283	81	n	n	CCONJ
ejpam-5499	283	82	−	−	NOUN
ejpam-5499	283	83	1	1	NUM
ejpam-5499	283	84	.	.	PUNCT
ejpam-5499	284	1	consequently	consequently	ADV
ejpam-5499	284	2	,	,	PUNCT
ejpam-5499	284	3	the	the	DET
ejpam-5499	284	4	size319	size319	PROPN
ejpam-5499	284	5	|e(h)|	|e(h)|	PROPN
ejpam-5499	284	6	=	=	PUNCT
ejpam-5499	284	7	|e(kn)|	|e(kn)|	NOUN
ejpam-5499	284	8	=	=	SYM
ejpam-5499	284	9	n(n−1	n(n−1	NUM
ejpam-5499	284	10	)	)	PUNCT
ejpam-5499	284	11	2	2	NUM
ejpam-5499	284	12	.	.	PUNCT
ejpam-5499	285	1	hence	hence	ADV
ejpam-5499	285	2	h	h	PROPN
ejpam-5499	285	3	is	be	AUX
ejpam-5499	285	4	a	a	DET
ejpam-5499	285	5	complete	complete	ADJ
ejpam-5499	285	6	graph	graph	NOUN
ejpam-5499	285	7	of	of	ADP
ejpam-5499	285	8	order	order	NOUN
ejpam-5499	285	9	n.320	n.320	NOUN
ejpam-5499	285	10	4.7	4.7	NUM
ejpam-5499	285	11	.	.	PUNCT
ejpam-5499	286	1	circumference321	circumference321	PROPN
ejpam-5499	286	2	theorem	theorem	VERB
ejpam-5499	286	3	9	9	NUM
ejpam-5499	286	4	.	.	PUNCT
ejpam-5499	287	1	if	if	SCONJ
ejpam-5499	287	2	m(γcn	m(γcn	PROPN
ejpam-5499	287	3	)	)	PUNCT
ejpam-5499	287	4	be	be	VERB
ejpam-5499	287	5	the	the	DET
ejpam-5499	287	6	middle	middle	ADJ
ejpam-5499	287	7	graph	graph	NOUN
ejpam-5499	287	8	of	of	ADP
ejpam-5499	287	9	γcn	γcn	PROPN
ejpam-5499	287	10	,	,	PUNCT
ejpam-5499	287	11	then	then	ADV
ejpam-5499	287	12	the	the	DET
ejpam-5499	287	13	circumference322	circumference322	PROPN
ejpam-5499	287	14	c(m(γcn	c(m(γcn	PROPN
ejpam-5499	287	15	)	)	PUNCT
ejpam-5499	287	16	)	)	PUNCT
ejpam-5499	288	1	=	=	PRON
ejpam-5499	288	2	{	{	PUNCT
ejpam-5499	288	3	5n−3	5n−3	NUM
ejpam-5499	288	4	2	2	NUM
ejpam-5499	288	5	,	,	PUNCT
ejpam-5499	288	6	if	if	SCONJ
ejpam-5499	288	7	n	n	PRON
ejpam-5499	288	8	is	be	AUX
ejpam-5499	288	9	odd	odd	ADJ
ejpam-5499	288	10	,	,	PUNCT
ejpam-5499	288	11	5n−6	5n−6	NUM
ejpam-5499	288	12	2	2	NUM
ejpam-5499	288	13	,	,	PUNCT
ejpam-5499	288	14	if	if	SCONJ
ejpam-5499	288	15	n	n	PRON
ejpam-5499	288	16	is	be	AUX
ejpam-5499	288	17	even	even	ADV
ejpam-5499	288	18	.	.	PUNCT
ejpam-5499	289	1	323	323	NUM
ejpam-5499	289	2	proof	proof	NOUN
ejpam-5499	289	3	.	.	PUNCT
ejpam-5499	290	1	let	let	AUX
ejpam-5499	290	2	m(γcn	m(γcn	PRON
ejpam-5499	290	3	)	)	PUNCT
ejpam-5499	290	4	be	be	AUX
ejpam-5499	290	5	the	the	DET
ejpam-5499	290	6	middle	middle	ADJ
ejpam-5499	290	7	graph	graph	NOUN
ejpam-5499	290	8	of	of	ADP
ejpam-5499	290	9	γcn	γcn	PROPN
ejpam-5499	290	10	.	.	PUNCT
ejpam-5499	291	1	we	we	PRON
ejpam-5499	291	2	divided	divide	VERB
ejpam-5499	291	3	it	it	PRON
ejpam-5499	291	4	into	into	ADP
ejpam-5499	291	5	two	two	NUM
ejpam-5499	291	6	cases.324	cases.324	PROPN
ejpam-5499	291	7	i.	i.	NOUN
ejpam-5499	291	8	for	for	ADP
ejpam-5499	291	9	n	n	PROPN
ejpam-5499	291	10	is	be	AUX
ejpam-5499	291	11	odd	odd	ADJ
ejpam-5499	291	12	,	,	PUNCT
ejpam-5499	291	13	we	we	PRON
ejpam-5499	291	14	will	will	AUX
ejpam-5499	291	15	find	find	VERB
ejpam-5499	291	16	a	a	DET
ejpam-5499	291	17	cycle	cycle	NOUN
ejpam-5499	291	18	c	c	NOUN
ejpam-5499	291	19	⊆	⊆	NUM
ejpam-5499	291	20	m(γcn	m(γcn	PROPN
ejpam-5499	291	21	)	)	PUNCT
ejpam-5499	291	22	with	with	ADP
ejpam-5499	291	23	the	the	DET
ejpam-5499	291	24	largest	large	ADJ
ejpam-5499	291	25	size	size	NOUN
ejpam-5499	291	26	.	.	PUNCT
ejpam-5499	292	1	from	from	ADP
ejpam-5499	292	2	the325	the325	PROPN
ejpam-5499	292	3	structure	structure	NOUN
ejpam-5499	292	4	of	of	ADP
ejpam-5499	292	5	m(γcn	m(γcn	PROPN
ejpam-5499	292	6	)	)	PUNCT
ejpam-5499	292	7	where	where	SCONJ
ejpam-5499	292	8	n	n	PRON
ejpam-5499	292	9	is	be	AUX
ejpam-5499	292	10	odd	odd	ADJ
ejpam-5499	292	11	,	,	PUNCT
ejpam-5499	292	12	since	since	SCONJ
ejpam-5499	292	13	[	[	X
ejpam-5499	292	14	e	e	X
ejpam-5499	292	15	,	,	PUNCT
ejpam-5499	292	16	zi	zi	NOUN
ejpam-5499	292	17	]	]	X
ejpam-5499	292	18	∈	∈	PROPN
ejpam-5499	292	19	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	292	20	)	)	PUNCT
ejpam-5499	292	21	)	)	PUNCT
ejpam-5499	292	22	for	for	ADP
ejpam-5499	292	23	1	1	NUM
ejpam-5499	292	24	≤	≤	NUM
ejpam-5499	292	25	i	i	PRON
ejpam-5499	292	26	≤	≤	NOUN
ejpam-5499	293	1	n	n	CCONJ
ejpam-5499	293	2	−	−	PROPN
ejpam-5499	293	3	1,326	1,326	NUM
ejpam-5499	293	4	take	take	NOUN
ejpam-5499	293	5	e	e	NOUN
ejpam-5499	293	6	as	as	ADP
ejpam-5499	293	7	our	our	PRON
ejpam-5499	293	8	initial	initial	ADJ
ejpam-5499	293	9	vertex	vertex	NOUN
ejpam-5499	293	10	,	,	PUNCT
ejpam-5499	293	11	followed	follow	VERB
ejpam-5499	293	12	by	by	ADP
ejpam-5499	293	13	z1	z1	PROPN
ejpam-5499	293	14	.	.	PUNCT
ejpam-5499	294	1	note	note	VERB
ejpam-5499	294	2	that	that	SCONJ
ejpam-5499	294	3	[	[	X
ejpam-5499	294	4	zi	zi	NOUN
ejpam-5499	294	5	,	,	PUNCT
ejpam-5499	294	6	xi	xi	X
ejpam-5499	294	7	]	]	X
ejpam-5499	294	8	∈	∈	PROPN
ejpam-5499	294	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	294	10	)	)	PUNCT
ejpam-5499	294	11	)	)	PUNCT
ejpam-5499	294	12	,	,	PUNCT
ejpam-5499	294	13	then327	then327	PROPN
ejpam-5499	294	14	we	we	PRON
ejpam-5499	294	15	have	have	VERB
ejpam-5499	294	16	a	a	DET
ejpam-5499	294	17	path	path	NOUN
ejpam-5499	294	18	e	e	NOUN
ejpam-5499	294	19	,	,	PUNCT
ejpam-5499	294	20	z1	z1	PROPN
ejpam-5499	294	21	,	,	PUNCT
ejpam-5499	294	22	x1	x1	PROPN
ejpam-5499	294	23	.	.	PUNCT
ejpam-5499	295	1	now	now	ADV
ejpam-5499	295	2	also	also	ADV
ejpam-5499	295	3	[	[	X
ejpam-5499	295	4	xi	xi	X
ejpam-5499	295	5	,	,	PUNCT
ejpam-5499	295	6	y	y	PROPN
ejpam-5499	295	7	i+1	i+1	NUM
ejpam-5499	295	8	2	2	NUM
ejpam-5499	295	9	]	]	PUNCT
ejpam-5499	295	10	and	and	CCONJ
ejpam-5499	295	11	[	[	X
ejpam-5499	295	12	xi+1	xi+1	NOUN
ejpam-5499	295	13	,	,	PUNCT
ejpam-5499	295	14	y	y	PROPN
ejpam-5499	295	15	i+1	i+1	NUM
ejpam-5499	295	16	2	2	NUM
ejpam-5499	295	17	]	]	PUNCT
ejpam-5499	295	18	∈	∈	PROPN
ejpam-5499	295	19	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	295	20	)	)	PUNCT
ejpam-5499	295	21	)	)	PUNCT
ejpam-5499	295	22	,	,	PUNCT
ejpam-5499	295	23	then328	then328	NUM
ejpam-5499	295	24	we	we	PRON
ejpam-5499	295	25	can	can	AUX
ejpam-5499	295	26	extend	extend	VERB
ejpam-5499	295	27	our	our	PRON
ejpam-5499	295	28	path	path	NOUN
ejpam-5499	295	29	to	to	ADP
ejpam-5499	295	30	e	e	NOUN
ejpam-5499	295	31	,	,	PUNCT
ejpam-5499	295	32	z1	z1	PROPN
ejpam-5499	295	33	,	,	PUNCT
ejpam-5499	295	34	x1	x1	PROPN
ejpam-5499	295	35	,	,	PUNCT
ejpam-5499	295	36	y1	y1	PROPN
ejpam-5499	295	37	,	,	PUNCT
ejpam-5499	295	38	x2	x2	PROPN
ejpam-5499	295	39	,	,	PUNCT
ejpam-5499	295	40	z2	z2	PROPN
ejpam-5499	295	41	,	,	PUNCT
ejpam-5499	295	42	.	.	PUNCT
ejpam-5499	296	1	also	also	ADV
ejpam-5499	296	2	since	since	SCONJ
ejpam-5499	296	3	[	[	X
ejpam-5499	296	4	zi	zi	PROPN
ejpam-5499	296	5	,	,	PUNCT
ejpam-5499	296	6	zi	zi	NOUN
ejpam-5499	296	7	]	]	X
ejpam-5499	296	8	∈	∈	PROPN
ejpam-5499	296	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	296	10	)	)	PUNCT
ejpam-5499	296	11	)	)	PUNCT
ejpam-5499	296	12	,	,	PUNCT
ejpam-5499	296	13	we329	we329	PROPN
ejpam-5499	296	14	can	can	AUX
ejpam-5499	296	15	have	have	VERB
ejpam-5499	296	16	e	e	NOUN
ejpam-5499	296	17	,	,	PUNCT
ejpam-5499	296	18	z1	z1	PROPN
ejpam-5499	296	19	,	,	PUNCT
ejpam-5499	296	20	x1	x1	PROPN
ejpam-5499	296	21	,	,	PUNCT
ejpam-5499	296	22	y1	y1	PROPN
ejpam-5499	296	23	,	,	PUNCT
ejpam-5499	296	24	x2	x2	PROPN
ejpam-5499	296	25	,	,	PUNCT
ejpam-5499	296	26	z2	z2	PROPN
ejpam-5499	296	27	,	,	PUNCT
ejpam-5499	296	28	z3	z3	PROPN
ejpam-5499	296	29	then	then	ADV
ejpam-5499	296	30	repeat	repeat	VERB
ejpam-5499	296	31	the	the	DET
ejpam-5499	296	32	process	process	NOUN
ejpam-5499	296	33	.	.	PUNCT
ejpam-5499	297	1	by	by	ADP
ejpam-5499	297	2	continuing	continue	VERB
ejpam-5499	297	3	,	,	PUNCT
ejpam-5499	297	4	we	we	PRON
ejpam-5499	297	5	now	now	ADV
ejpam-5499	297	6	have330	have330	PROPN
ejpam-5499	297	7	e	e	PROPN
ejpam-5499	297	8	,	,	PUNCT
ejpam-5499	297	9	z1	z1	PROPN
ejpam-5499	297	10	,	,	PUNCT
ejpam-5499	297	11	x1	x1	PROPN
ejpam-5499	297	12	,	,	PUNCT
ejpam-5499	297	13	y1	y1	PROPN
ejpam-5499	297	14	,	,	PUNCT
ejpam-5499	297	15	x2	x2	PROPN
ejpam-5499	297	16	,	,	PUNCT
ejpam-5499	297	17	z2	z2	PROPN
ejpam-5499	297	18	,	,	PUNCT
ejpam-5499	297	19	z3	z3	PROPN
ejpam-5499	297	20	,	,	PUNCT
ejpam-5499	297	21	x3	x3	ADJ
ejpam-5499	297	22	,	,	PUNCT
ejpam-5499	297	23	y2	y2	PROPN
ejpam-5499	297	24	,	,	PUNCT
ejpam-5499	297	25	x4,331	x4,331	PROPN
ejpam-5499	297	26	z4	z4	PROPN
ejpam-5499	297	27	,	,	PUNCT
ejpam-5499	297	28	...	...	PUNCT
ejpam-5499	297	29	zi	zi	NOUN
ejpam-5499	297	30	,	,	PUNCT
ejpam-5499	297	31	xi	xi	PROPN
ejpam-5499	297	32	,	,	PUNCT
ejpam-5499	297	33	y	y	PROPN
ejpam-5499	297	34	i+1	i+1	NUM
ejpam-5499	297	35	2	2	NUM
ejpam-5499	297	36	,	,	PUNCT
ejpam-5499	297	37	xi+1	xi+1	NOUN
ejpam-5499	297	38	,	,	PUNCT
ejpam-5499	297	39	zi+1	zi+1	NUM
ejpam-5499	297	40	,	,	PUNCT
ejpam-5499	297	41	...	...	PUNCT
ejpam-5499	297	42	zn−2	zn−2	PROPN
ejpam-5499	297	43	,	,	PUNCT
ejpam-5499	297	44	xn−2	xn−2	PROPN
ejpam-5499	297	45	,	,	PUNCT
ejpam-5499	297	46	yn−1	yn−1	ADV
ejpam-5499	297	47	2	2	NUM
ejpam-5499	297	48	,	,	PUNCT
ejpam-5499	297	49	xn−1,332	xn−1,332	PROPN
ejpam-5499	297	50	zn−1	zn−1	PROPN
ejpam-5499	297	51	.	.	PUNCT
ejpam-5499	298	1	now	now	ADV
ejpam-5499	298	2	we	we	PRON
ejpam-5499	298	3	can	can	AUX
ejpam-5499	298	4	connect	connect	VERB
ejpam-5499	298	5	zn−1	zn−1	PROPN
ejpam-5499	298	6	to	to	ADP
ejpam-5499	298	7	e	e	VERB
ejpam-5499	298	8	since	since	SCONJ
ejpam-5499	298	9	[	[	X
ejpam-5499	298	10	e	e	X
ejpam-5499	298	11	,	,	PUNCT
ejpam-5499	298	12	zi	zi	NOUN
ejpam-5499	298	13	]	]	X
ejpam-5499	298	14	∈	∈	PROPN
ejpam-5499	298	15	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	298	16	)	)	PUNCT
ejpam-5499	298	17	)	)	PUNCT
ejpam-5499	298	18	for	for	ADP
ejpam-5499	298	19	1	1	NUM
ejpam-5499	298	20	≤	≤	NUM
ejpam-5499	298	21	i	i	PRON
ejpam-5499	298	22	≤	≤	NOUN
ejpam-5499	298	23	n	n	CCONJ
ejpam-5499	298	24	−	−	PROPN
ejpam-5499	298	25	1.333	1.333	NUM
ejpam-5499	298	26	clearly	clearly	ADV
ejpam-5499	298	27	,	,	PUNCT
ejpam-5499	298	28	c	c	NOUN
ejpam-5499	298	29	:	:	PUNCT
ejpam-5499	298	30	e	e	X
ejpam-5499	298	31	,	,	PUNCT
ejpam-5499	298	32	z1	z1	PROPN
ejpam-5499	298	33	,	,	PUNCT
ejpam-5499	298	34	x1	x1	PROPN
ejpam-5499	298	35	,	,	PUNCT
ejpam-5499	298	36	y1	y1	PROPN
ejpam-5499	298	37	,	,	PUNCT
ejpam-5499	298	38	x2	x2	PROPN
ejpam-5499	298	39	,	,	PUNCT
ejpam-5499	298	40	z2	z2	PROPN
ejpam-5499	298	41	,	,	PUNCT
ejpam-5499	298	42	z3	z3	PROPN
ejpam-5499	298	43	,	,	PUNCT
ejpam-5499	298	44	x3	x3	ADJ
ejpam-5499	298	45	,	,	PUNCT
ejpam-5499	298	46	y2	y2	PROPN
ejpam-5499	298	47	,	,	PUNCT
ejpam-5499	298	48	x4	x4	PROPN
ejpam-5499	298	49	,	,	PUNCT
ejpam-5499	298	50	z4	z4	PROPN
ejpam-5499	298	51	,	,	PUNCT
ejpam-5499	298	52	...	...	PUNCT
ejpam-5499	298	53	zi	zi	NOUN
ejpam-5499	298	54	,	,	PUNCT
ejpam-5499	298	55	xi	xi	PROPN
ejpam-5499	298	56	,	,	PUNCT
ejpam-5499	298	57	y	y	PROPN
ejpam-5499	298	58	i+1	i+1	NUM
ejpam-5499	298	59	2	2	NUM
ejpam-5499	298	60	,	,	PUNCT
ejpam-5499	298	61	xi+1	xi+1	NOUN
ejpam-5499	298	62	,	,	PUNCT
ejpam-5499	298	63	zi+1	zi+1	NUM
ejpam-5499	298	64	,	,	PUNCT
ejpam-5499	298	65	...	...	PUNCT
ejpam-5499	299	1	zn−2,334	zn−2,334	PROPN
ejpam-5499	299	2	xn−2	xn−2	PROPN
ejpam-5499	299	3	,	,	PUNCT
ejpam-5499	299	4	yn−1	yn−1	ADJ
ejpam-5499	299	5	2	2	NUM
ejpam-5499	299	6	,	,	PUNCT
ejpam-5499	299	7	xn−1	xn−1	PROPN
ejpam-5499	299	8	,	,	PUNCT
ejpam-5499	299	9	zn−1	zn−1	PROPN
ejpam-5499	299	10	,	,	PUNCT
ejpam-5499	299	11	e	e	X
ejpam-5499	299	12	is	be	AUX
ejpam-5499	299	13	a	a	DET
ejpam-5499	299	14	cycle	cycle	NOUN
ejpam-5499	299	15	since	since	SCONJ
ejpam-5499	299	16	no	no	DET
ejpam-5499	299	17	vertex	vertex	NOUN
ejpam-5499	299	18	is	be	AUX
ejpam-5499	299	19	repeated	repeat	VERB
ejpam-5499	299	20	except	except	SCONJ
ejpam-5499	299	21	for	for	ADP
ejpam-5499	299	22	the	the	DET
ejpam-5499	299	23	first	first	ADJ
ejpam-5499	299	24	and335	and335	NOUN
ejpam-5499	299	25	the	the	DET
ejpam-5499	299	26	last	last	ADJ
ejpam-5499	299	27	.	.	PUNCT
ejpam-5499	300	1	to	to	PART
ejpam-5499	300	2	compute	compute	VERB
ejpam-5499	300	3	the	the	DET
ejpam-5499	300	4	length	length	NOUN
ejpam-5499	300	5	,	,	PUNCT
ejpam-5499	300	6	we	we	PRON
ejpam-5499	300	7	have	have	VERB
ejpam-5499	300	8	to	to	PART
ejpam-5499	300	9	compute	compute	VERB
ejpam-5499	300	10	its	its	PRON
ejpam-5499	300	11	order	order	NOUN
ejpam-5499	300	12	since	since	SCONJ
ejpam-5499	300	13	the	the	DET
ejpam-5499	300	14	length	length	NOUN
ejpam-5499	300	15	of336	of336	PROPN
ejpam-5499	300	16	a	a	DET
ejpam-5499	300	17	cycle	cycle	NOUN
ejpam-5499	300	18	is	be	AUX
ejpam-5499	300	19	equal	equal	ADJ
ejpam-5499	300	20	to	to	ADP
ejpam-5499	300	21	its	its	PRON
ejpam-5499	300	22	order	order	NOUN
ejpam-5499	300	23	.	.	PUNCT
ejpam-5499	301	1	so	so	ADV
ejpam-5499	301	2	|e|	|e|	ADJ
ejpam-5499	301	3	=	=	SYM
ejpam-5499	301	4	1	1	NUM
ejpam-5499	301	5	,	,	PUNCT
ejpam-5499	301	6	|zi′s|	|zi′s|	X
ejpam-5499	301	7	=	=	SYM
ejpam-5499	301	8	n	n	CCONJ
ejpam-5499	301	9	−	−	PROPN
ejpam-5499	301	10	1	1	NUM
ejpam-5499	301	11	,	,	PUNCT
ejpam-5499	301	12	|xi′s|	|xi′s|	NOUN
ejpam-5499	301	13	=	=	SYM
ejpam-5499	301	14	n	n	NUM
ejpam-5499	301	15	−	−	PROPN
ejpam-5499	301	16	1	1	NUM
ejpam-5499	301	17	,	,	PUNCT
ejpam-5499	301	18	|yi′s|	|yi′s|	ADJ
ejpam-5499	301	19	=	=	SYM
ejpam-5499	301	20	n−2	n−2	PROPN
ejpam-5499	301	21	2	2	NUM
ejpam-5499	301	22	.337	.337	NUM
ejpam-5499	301	23	thus	thus	ADV
ejpam-5499	301	24	|v	|v	PROPN
ejpam-5499	301	25	(	(	PUNCT
ejpam-5499	301	26	c)|	c)|	NOUN
ejpam-5499	301	27	=	=	NOUN
ejpam-5499	301	28	1	1	NUM
ejpam-5499	301	29	+	+	CCONJ
ejpam-5499	301	30	(	(	PUNCT
ejpam-5499	301	31	n	n	CCONJ
ejpam-5499	301	32	−	−	PROPN
ejpam-5499	301	33	1	1	NUM
ejpam-5499	301	34	)	)	PUNCT
ejpam-5499	301	35	+	+	CCONJ
ejpam-5499	301	36	(	(	PUNCT
ejpam-5499	301	37	n	n	CCONJ
ejpam-5499	301	38	−	−	PROPN
ejpam-5499	301	39	1	1	NUM
ejpam-5499	301	40	)	)	PUNCT
ejpam-5499	301	41	+	+	CCONJ
ejpam-5499	301	42	(	(	PUNCT
ejpam-5499	301	43	n−1	n−1	PROPN
ejpam-5499	301	44	2	2	NUM
ejpam-5499	301	45	)	)	PUNCT
ejpam-5499	301	46	=	=	PUNCT
ejpam-5499	302	1	5n−3	5n−3	NUM
ejpam-5499	302	2	2	2	NUM
ejpam-5499	302	3	.	.	PUNCT
ejpam-5499	302	4	note	note	VERB
ejpam-5499	302	5	that	that	SCONJ
ejpam-5499	302	6	c	c	NOUN
ejpam-5499	302	7	⊆	⊆	NUM
ejpam-5499	302	8	m(γcn),338	m(γcn),338	NOUN
ejpam-5499	302	9	thus	thus	ADV
ejpam-5499	302	10	|v	|v	PROPN
ejpam-5499	302	11	(	(	PUNCT
ejpam-5499	302	12	c)|	c)|	PROPN
ejpam-5499	302	13	≤	≤	PROPN
ejpam-5499	302	14	|v	|v	PROPN
ejpam-5499	302	15	(	(	PUNCT
ejpam-5499	302	16	m(γcn))|	m(γcn))|	PROPN
ejpam-5499	302	17	.	.	PUNCT
ejpam-5499	303	1	and	and	CCONJ
ejpam-5499	303	2	from	from	ADP
ejpam-5499	303	3	theorem	theorem	NOUN
ejpam-5499	303	4	1	1	NUM
ejpam-5499	303	5	the	the	DET
ejpam-5499	303	6	order	order	NOUN
ejpam-5499	303	7	|v	|v	X
ejpam-5499	303	8	(	(	PUNCT
ejpam-5499	303	9	m(γcn))|	m(γcn))|	PROPN
ejpam-5499	303	10	=	=	SYM
ejpam-5499	303	11	5n−3	5n−3	NUM
ejpam-5499	303	12	2339	2339	NUM
ejpam-5499	303	13	if	if	SCONJ
ejpam-5499	303	14	n	n	PRON
ejpam-5499	303	15	is	be	AUX
ejpam-5499	303	16	odd	odd	ADJ
ejpam-5499	303	17	.	.	PUNCT
ejpam-5499	304	1	it	it	PRON
ejpam-5499	304	2	implies	imply	VERB
ejpam-5499	304	3	that	that	SCONJ
ejpam-5499	304	4	|v	|v	PROPN
ejpam-5499	304	5	(	(	PUNCT
ejpam-5499	304	6	c)|	c)|	PROPN
ejpam-5499	304	7	=	=	SYM
ejpam-5499	304	8	|v	|v	PROPN
ejpam-5499	304	9	(	(	PUNCT
ejpam-5499	304	10	m(γcn))|	m(γcn))|	PROPN
ejpam-5499	304	11	=	=	SYM
ejpam-5499	304	12	5n−3	5n−3	NUM
ejpam-5499	304	13	2	2	NUM
ejpam-5499	304	14	,	,	PUNCT
ejpam-5499	304	15	hence	hence	ADV
ejpam-5499	304	16	the	the	DET
ejpam-5499	304	17	length	length	NOUN
ejpam-5499	304	18	of	of	ADP
ejpam-5499	304	19	the340	the340	PROPN
ejpam-5499	304	20	maximum	maximum	ADJ
ejpam-5499	304	21	cycle	cycle	NOUN
ejpam-5499	304	22	in	in	ADP
ejpam-5499	304	23	m(γcn	m(γcn	PROPN
ejpam-5499	304	24	)	)	PUNCT
ejpam-5499	304	25	=	=	PUNCT
ejpam-5499	305	1	5n−3	5n−3	NUM
ejpam-5499	305	2	2	2	NUM
ejpam-5499	305	3	.341	.341	NUM
ejpam-5499	305	4	2	2	NUM
ejpam-5499	305	5	.	.	PUNCT
ejpam-5499	306	1	for	for	ADP
ejpam-5499	306	2	n	n	NUM
ejpam-5499	306	3	is	be	AUX
ejpam-5499	306	4	even	even	ADV
ejpam-5499	306	5	,	,	PUNCT
ejpam-5499	306	6	we	we	PRON
ejpam-5499	306	7	will	will	AUX
ejpam-5499	306	8	find	find	VERB
ejpam-5499	306	9	a	a	DET
ejpam-5499	306	10	cycle	cycle	NOUN
ejpam-5499	306	11	c	c	NOUN
ejpam-5499	306	12	⊆	⊆	NUM
ejpam-5499	306	13	m(γcn	m(γcn	NOUN
ejpam-5499	306	14	)	)	PUNCT
ejpam-5499	306	15	of	of	ADP
ejpam-5499	306	16	maximum	maximum	ADJ
ejpam-5499	306	17	length	length	NOUN
ejpam-5499	306	18	.	.	PUNCT
ejpam-5499	307	1	from	from	ADP
ejpam-5499	307	2	the342	the342	PROPN
ejpam-5499	307	3	structure	structure	NOUN
ejpam-5499	307	4	ofm(γcn	ofm(γcn	NOUN
ejpam-5499	307	5	)	)	PUNCT
ejpam-5499	307	6	where	where	SCONJ
ejpam-5499	307	7	n	n	PRON
ejpam-5499	307	8	is	be	AUX
ejpam-5499	307	9	even	even	ADV
ejpam-5499	307	10	,	,	PUNCT
ejpam-5499	307	11	since	since	SCONJ
ejpam-5499	307	12	[	[	X
ejpam-5499	307	13	e	e	X
ejpam-5499	307	14	,	,	PUNCT
ejpam-5499	307	15	zi	zi	NOUN
ejpam-5499	307	16	]	]	X
ejpam-5499	307	17	∈	∈	PROPN
ejpam-5499	307	18	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	307	19	)	)	PUNCT
ejpam-5499	307	20	)	)	PUNCT
ejpam-5499	307	21	for	for	ADP
ejpam-5499	307	22	1	1	NUM
ejpam-5499	307	23	≤	≤	NUM
ejpam-5499	308	1	i	i	PRON
ejpam-5499	308	2	≤	≤	PROPN
ejpam-5499	308	3	n−1	n−1	PROPN
ejpam-5499	308	4	,	,	PUNCT
ejpam-5499	308	5	then343	then343	PROPN
ejpam-5499	308	6	choose	choose	VERB
ejpam-5499	308	7	e	e	PROPN
ejpam-5499	308	8	as	as	ADP
ejpam-5499	308	9	our	our	PRON
ejpam-5499	308	10	initial	initial	ADJ
ejpam-5499	308	11	vertex	vertex	NOUN
ejpam-5499	308	12	followed	follow	VERB
ejpam-5499	308	13	by	by	ADP
ejpam-5499	308	14	zn−1	zn−1	PROPN
ejpam-5499	308	15	,	,	PUNCT
ejpam-5499	308	16	then	then	ADV
ejpam-5499	308	17	z1	z1	PROPN
ejpam-5499	308	18	since	since	SCONJ
ejpam-5499	308	19	[	[	X
ejpam-5499	308	20	zi	zi	PROPN
ejpam-5499	308	21	,	,	PUNCT
ejpam-5499	308	22	zj	zj	X
ejpam-5499	308	23	]	]	PUNCT
ejpam-5499	308	24	∈	∈	PROPN
ejpam-5499	308	25	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	308	26	)	)	PUNCT
ejpam-5499	308	27	)	)	PUNCT
ejpam-5499	309	1	so344	so344	NOUN
ejpam-5499	309	2	that	that	PRON
ejpam-5499	309	3	we	we	PRON
ejpam-5499	309	4	have	have	VERB
ejpam-5499	309	5	a	a	DET
ejpam-5499	309	6	path	path	NOUN
ejpam-5499	309	7	e	e	NOUN
ejpam-5499	309	8	,	,	PUNCT
ejpam-5499	309	9	zn−1	zn−1	PROPN
ejpam-5499	309	10	,	,	PUNCT
ejpam-5499	309	11	z1	z1	PROPN
ejpam-5499	309	12	.	.	PUNCT
ejpam-5499	310	1	also	also	ADV
ejpam-5499	310	2	since	since	SCONJ
ejpam-5499	310	3	[	[	X
ejpam-5499	310	4	zi	zi	NOUN
ejpam-5499	310	5	,	,	PUNCT
ejpam-5499	310	6	xi	xi	X
ejpam-5499	310	7	]	]	X
ejpam-5499	310	8	∈	∈	PROPN
ejpam-5499	310	9	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	310	10	)	)	PUNCT
ejpam-5499	310	11	)	)	PUNCT
ejpam-5499	310	12	,	,	PUNCT
ejpam-5499	310	13	then	then	ADV
ejpam-5499	310	14	we	we	PRON
ejpam-5499	310	15	can	can	AUX
ejpam-5499	310	16	extend345	extend345	PROPN
ejpam-5499	310	17	it	it	PRON
ejpam-5499	310	18	to	to	ADP
ejpam-5499	310	19	e	e	PROPN
ejpam-5499	310	20	,	,	PUNCT
ejpam-5499	310	21	zn−1	zn−1	PROPN
ejpam-5499	310	22	,	,	PUNCT
ejpam-5499	310	23	z1	z1	NOUN
ejpam-5499	310	24	,	,	PUNCT
ejpam-5499	310	25	x1	x1	PROPN
ejpam-5499	310	26	,	,	PUNCT
ejpam-5499	310	27	.	.	PUNCT
ejpam-5499	311	1	similar	similar	ADJ
ejpam-5499	311	2	to	to	ADP
ejpam-5499	311	3	case	case	NOUN
ejpam-5499	311	4	1	1	NUM
ejpam-5499	311	5	,	,	PUNCT
ejpam-5499	311	6	[	[	X
ejpam-5499	311	7	xi	xi	X
ejpam-5499	311	8	,	,	PUNCT
ejpam-5499	311	9	y	y	PROPN
ejpam-5499	311	10	i+1	i+1	NUM
ejpam-5499	311	11	2	2	NUM
ejpam-5499	311	12	]	]	PUNCT
ejpam-5499	311	13	and	and	CCONJ
ejpam-5499	311	14	[	[	X
ejpam-5499	311	15	xi+1	xi+1	NOUN
ejpam-5499	311	16	,	,	PUNCT
ejpam-5499	311	17	y	y	PROPN
ejpam-5499	311	18	i+1	i+1	NUM
ejpam-5499	311	19	2	2	NUM
ejpam-5499	311	20	]	]	PUNCT
ejpam-5499	311	21	∈	∈	PROPN
ejpam-5499	311	22	e(m(γcn	e(m(γcn	PROPN
ejpam-5499	311	23	)	)	PUNCT
ejpam-5499	311	24	)	)	PUNCT
ejpam-5499	312	1	for346	for346	PROPN
ejpam-5499	312	2	all	all	DET
ejpam-5499	312	3	odd	odd	ADJ
ejpam-5499	312	4	1	1	NUM
ejpam-5499	312	5	≤	≤	NUM
ejpam-5499	312	6	i	i	PRON
ejpam-5499	312	7	≤	≤	NOUN
ejpam-5499	312	8	n	n	CCONJ
ejpam-5499	312	9	−	−	PROPN
ejpam-5499	312	10	3	3	NUM
ejpam-5499	312	11	,	,	PUNCT
ejpam-5499	312	12	thus	thus	ADV
ejpam-5499	312	13	we	we	PRON
ejpam-5499	312	14	have	have	VERB
ejpam-5499	312	15	e	e	NOUN
ejpam-5499	312	16	,	,	PUNCT
ejpam-5499	312	17	zn−1	zn−1	PROPN
ejpam-5499	312	18	,	,	PUNCT
ejpam-5499	312	19	z1	z1	PROPN
ejpam-5499	312	20	,	,	PUNCT
ejpam-5499	312	21	x1	x1	PROPN
ejpam-5499	312	22	,	,	PUNCT
ejpam-5499	313	1	y	y	PROPN
ejpam-5499	313	2	−	−	PROPN
ejpam-5499	313	3	1	1	NUM
ejpam-5499	313	4	,	,	PUNCT
ejpam-5499	313	5	x2	x2	PROPN
ejpam-5499	313	6	.	.	PUNCT
ejpam-5499	314	1	now	now	ADV
ejpam-5499	314	2	choose	choose	VERB
ejpam-5499	314	3	z2	z2	PROPN
ejpam-5499	314	4	as	as	ADP
ejpam-5499	314	5	our347	our347	PROPN
ejpam-5499	314	6	next	next	ADJ
ejpam-5499	314	7	vertex	vertex	NOUN
ejpam-5499	314	8	so	so	SCONJ
ejpam-5499	314	9	that	that	SCONJ
ejpam-5499	314	10	we	we	PRON
ejpam-5499	314	11	have	have	VERB
ejpam-5499	314	12	e	e	NOUN
ejpam-5499	314	13	,	,	PUNCT
ejpam-5499	314	14	zn−1	zn−1	PROPN
ejpam-5499	314	15	,	,	PUNCT
ejpam-5499	314	16	z1	z1	PROPN
ejpam-5499	314	17	,	,	PUNCT
ejpam-5499	314	18	x1	x1	PROPN
ejpam-5499	314	19	,	,	PUNCT
ejpam-5499	314	20	y−1	y−1	PROPN
ejpam-5499	314	21	,	,	PUNCT
ejpam-5499	314	22	x2	x2	PROPN
ejpam-5499	314	23	,	,	PUNCT
ejpam-5499	314	24	z2	z2	PROPN
ejpam-5499	314	25	.	.	PUNCT
ejpam-5499	315	1	by	by	ADP
ejpam-5499	315	2	continuing	continue	VERB
ejpam-5499	315	3	the	the	DET
ejpam-5499	315	4	process	process	NOUN
ejpam-5499	315	5	,	,	PUNCT
ejpam-5499	315	6	we348	we348	PROPN
ejpam-5499	315	7	now	now	ADV
ejpam-5499	315	8	have	have	VERB
ejpam-5499	315	9	a	a	DET
ejpam-5499	315	10	cycle349	cycle349	PROPN
ejpam-5499	315	11	c	c	PROPN
ejpam-5499	315	12	:	:	PUNCT
ejpam-5499	315	13	e	e	X
ejpam-5499	315	14	,	,	PUNCT
ejpam-5499	315	15	zn−1	zn−1	PROPN
ejpam-5499	315	16	,	,	PUNCT
ejpam-5499	315	17	z1	z1	PROPN
ejpam-5499	315	18	,	,	PUNCT
ejpam-5499	315	19	x1	x1	PROPN
ejpam-5499	315	20	,	,	PUNCT
ejpam-5499	315	21	y	y	PROPN
ejpam-5499	315	22	−	−	PROPN
ejpam-5499	315	23	1	1	NUM
ejpam-5499	315	24	,	,	PUNCT
ejpam-5499	315	25	x2	x2	PROPN
ejpam-5499	315	26	,	,	PUNCT
ejpam-5499	315	27	z2	z2	PROPN
ejpam-5499	315	28	,	,	PUNCT
ejpam-5499	315	29	z3	z3	PROPN
ejpam-5499	315	30	,	,	PUNCT
ejpam-5499	315	31	x3	x3	ADJ
ejpam-5499	315	32	,	,	PUNCT
ejpam-5499	315	33	y2	y2	PROPN
ejpam-5499	315	34	,	,	PUNCT
ejpam-5499	315	35	x4	x4	PROPN
ejpam-5499	315	36	,	,	PUNCT
ejpam-5499	315	37	z4	z4	PROPN
ejpam-5499	315	38	,	,	PUNCT
ejpam-5499	315	39	...	...	PUNCT
ejpam-5499	315	40	zi	zi	NOUN
ejpam-5499	315	41	,	,	PUNCT
ejpam-5499	315	42	xi	xi	PROPN
ejpam-5499	315	43	,	,	PUNCT
ejpam-5499	315	44	y	y	PROPN
ejpam-5499	315	45	i+1	i+1	NUM
ejpam-5499	315	46	2	2	NUM
ejpam-5499	315	47	,	,	PUNCT
ejpam-5499	315	48	xi+1	xi+1	NOUN
ejpam-5499	315	49	,	,	PUNCT
ejpam-5499	315	50	zi+1	zi+1	NUM
ejpam-5499	315	51	,	,	PUNCT
ejpam-5499	315	52	...	...	PUNCT
ejpam-5499	315	53	zn−3	zn−3	PROPN
ejpam-5499	315	54	,	,	PUNCT
ejpam-5499	315	55	xn−3,350	xn−3,350	PROPN
ejpam-5499	315	56	yn−2	yn−2	PROPN
ejpam-5499	315	57	2	2	NUM
ejpam-5499	315	58	,	,	PUNCT
ejpam-5499	315	59	xn−2	xn−2	PROPN
ejpam-5499	315	60	,	,	PUNCT
ejpam-5499	315	61	zn−2	zn−2	PROPN
ejpam-5499	315	62	,	,	PUNCT
ejpam-5499	315	63	e.	e.	PROPN
ejpam-5499	315	64	to	to	PART
ejpam-5499	315	65	compute	compute	VERB
ejpam-5499	315	66	the	the	DET
ejpam-5499	315	67	order	order	NOUN
ejpam-5499	315	68	of	of	ADP
ejpam-5499	315	69	c	c	NOUN
ejpam-5499	315	70	,	,	PUNCT
ejpam-5499	315	71	we	we	PRON
ejpam-5499	315	72	have351	have351	VERB
ejpam-5499	315	73	j.	j.	PROPN
ejpam-5499	315	74	m.	m.	PROPN
ejpam-5499	315	75	jamis	jamis	PROPN
ejpam-5499	315	76	,	,	PUNCT
ejpam-5499	315	77	d.	d.	PROPN
ejpam-5499	315	78	m.	m.	PROPN
ejpam-5499	315	79	magpantay	magpantay	PROPN
ejpam-5499	315	80	/	/	SYM
ejpam-5499	315	81	eur	eur	PROPN
ejpam-5499	315	82	.	.	PUNCT
ejpam-5499	316	1	j.	j.	PROPN
ejpam-5499	316	2	pure	pure	PROPN
ejpam-5499	316	3	appl	appl	PROPN
ejpam-5499	316	4	.	.	PROPN
ejpam-5499	316	5	math	math	PROPN
ejpam-5499	316	6	,	,	PUNCT
ejpam-5499	316	7	18	18	NUM
ejpam-5499	316	8	(	(	PUNCT
ejpam-5499	316	9	2	2	NUM
ejpam-5499	316	10	)	)	PUNCT
ejpam-5499	316	11	(	(	PUNCT
ejpam-5499	316	12	2025	2025	NUM
ejpam-5499	316	13	)	)	PUNCT
ejpam-5499	316	14	,	,	PUNCT
ejpam-5499	316	15	5499	5499	NUM
ejpam-5499	316	16	15	15	NUM
ejpam-5499	316	17	of	of	ADP
ejpam-5499	316	18	18	18	NUM
ejpam-5499	316	19	v	v	NOUN
ejpam-5499	316	20	(	(	PUNCT
ejpam-5499	316	21	c	c	NOUN
ejpam-5499	316	22	)	)	PUNCT
ejpam-5499	316	23	=	=	SYM
ejpam-5499	316	24	{	{	PUNCT
ejpam-5499	316	25	e	e	NOUN
ejpam-5499	316	26	}	}	PUNCT
ejpam-5499	316	27	⋃	⋃	NOUN
ejpam-5499	316	28	{	{	PUNCT
ejpam-5499	316	29	zi|1	zi|1	NOUN
ejpam-5499	316	30	≤	≤	NOUN
ejpam-5499	316	31	i	i	PRON
ejpam-5499	316	32	≤	≤	NOUN
ejpam-5499	316	33	n	n	CCONJ
ejpam-5499	316	34	−	−	PROPN
ejpam-5499	316	35	1	1	NUM
ejpam-5499	316	36	}	}	PUNCT
ejpam-5499	316	37	⋃	⋃	NOUN
ejpam-5499	316	38	{	{	PUNCT
ejpam-5499	316	39	xi|1	xi|1	NOUN
ejpam-5499	316	40	≤	≤	NOUN
ejpam-5499	317	1	i	i	PRON
ejpam-5499	317	2	≤	≤	NOUN
ejpam-5499	317	3	n	n	CCONJ
ejpam-5499	317	4	−	−	PROPN
ejpam-5499	317	5	2	2	NUM
ejpam-5499	317	6	}	}	PUNCT
ejpam-5499	317	7	⋃	⋃	NOUN
ejpam-5499	317	8	{	{	PUNCT
ejpam-5499	317	9	yp|1	yp|1	PROPN
ejpam-5499	317	10	≤	≤	PROPN
ejpam-5499	317	11	p	p	NOUN
ejpam-5499	317	12	≤	≤	NOUN
ejpam-5499	317	13	n−2	n−2	PROPN
ejpam-5499	317	14	2	2	NUM
ejpam-5499	317	15	}	}	PUNCT
ejpam-5499	317	16	.	.	PUNCT
ejpam-5499	318	1	implies352	implies352	PROPN
ejpam-5499	318	2	that353	that353	PROPN
ejpam-5499	318	3	|v	|v	PROPN
ejpam-5499	318	4	(	(	PUNCT
ejpam-5499	318	5	c)|	c)|	NOUN
ejpam-5499	318	6	=	=	NOUN
ejpam-5499	318	7	1	1	NUM
ejpam-5499	318	8	+	+	CCONJ
ejpam-5499	318	9	(	(	PUNCT
ejpam-5499	318	10	n−	n−	NOUN
ejpam-5499	318	11	1	1	NUM
ejpam-5499	318	12	)	)	PUNCT
ejpam-5499	318	13	+	+	CCONJ
ejpam-5499	318	14	(	(	PUNCT
ejpam-5499	318	15	n−	n−	NOUN
ejpam-5499	318	16	2	2	NUM
ejpam-5499	318	17	)	)	PUNCT
ejpam-5499	318	18	+	+	CCONJ
ejpam-5499	318	19	(	(	PUNCT
ejpam-5499	318	20	n−	n−	NOUN
ejpam-5499	318	21	2	2	NUM
ejpam-5499	318	22	2	2	NUM
ejpam-5499	318	23	)	)	PUNCT
ejpam-5499	318	24	=	=	PUNCT
ejpam-5499	319	1	2n−	2n−	NUM
ejpam-5499	319	2	2	2	NUM
ejpam-5499	319	3	+	+	NUM
ejpam-5499	319	4	n−	n−	NOUN
ejpam-5499	319	5	2	2	NUM
ejpam-5499	319	6	2	2	NUM
ejpam-5499	319	7	=	=	SYM
ejpam-5499	319	8	4n−	4n−	NUM
ejpam-5499	319	9	4	4	NUM
ejpam-5499	319	10	+	+	NUM
ejpam-5499	319	11	n−	n−	NOUN
ejpam-5499	319	12	2	2	NUM
ejpam-5499	319	13	2	2	NUM
ejpam-5499	319	14	=	=	SYM
ejpam-5499	319	15	5n−	5n−	NUM
ejpam-5499	319	16	6	6	NUM
ejpam-5499	319	17	2	2	NUM
ejpam-5499	319	18	.	.	PUNCT
ejpam-5499	320	1	by	by	ADP
ejpam-5499	320	2	theorem	theorem	NOUN
ejpam-5499	320	3	13	13	NUM
ejpam-5499	320	4	,	,	PUNCT
ejpam-5499	320	5	there	there	PRON
ejpam-5499	320	6	exist	exist	VERB
ejpam-5499	320	7	exactly	exactly	ADV
ejpam-5499	320	8	one	one	NUM
ejpam-5499	320	9	vertex	vertex	NOUN
ejpam-5499	320	10	v	v	ADP
ejpam-5499	320	11	∈	∈	NOUN
ejpam-5499	320	12	v	v	NOUN
ejpam-5499	320	13	(	(	PUNCT
ejpam-5499	320	14	m(γcn	m(γcn	NOUN
ejpam-5499	320	15	)	)	PUNCT
ejpam-5499	320	16	)	)	PUNCT
ejpam-5499	320	17	of	of	ADP
ejpam-5499	320	18	degree	degree	NOUN
ejpam-5499	320	19	1	1	NUM
ejpam-5499	320	20	.	.	PUNCT
ejpam-5499	321	1	thus354	thus354	X
ejpam-5499	321	2	clearly	clearly	ADV
ejpam-5499	321	3	v	v	NOUN
ejpam-5499	321	4	/∈	/∈	NOUN
ejpam-5499	322	1	v	v	NOUN
ejpam-5499	322	2	(	(	PUNCT
ejpam-5499	322	3	c	c	NOUN
ejpam-5499	322	4	)	)	PUNCT
ejpam-5499	322	5	.	.	PUNCT
ejpam-5499	323	1	it	it	PRON
ejpam-5499	323	2	follows	follow	VERB
ejpam-5499	323	3	that355	that355	PROPN
ejpam-5499	323	4	|v	|v	PROPN
ejpam-5499	323	5	(	(	PUNCT
ejpam-5499	323	6	c)|	c)|	PROPN
ejpam-5499	323	7	≤	≤	PROPN
ejpam-5499	323	8	|v	|v	PROPN
ejpam-5499	323	9	(	(	PUNCT
ejpam-5499	323	10	m(γcn))|	m(γcn))|	PROPN
ejpam-5499	323	11	−	−	PROPN
ejpam-5499	323	12	1	1	NUM
ejpam-5499	323	13	≤	≤	NOUN
ejpam-5499	323	14	5n−	5n−	NUM
ejpam-5499	323	15	4	4	NUM
ejpam-5499	323	16	2	2	NUM
ejpam-5499	323	17	−	−	NOUN
ejpam-5499	323	18	1	1	NUM
ejpam-5499	323	19	≤	≤	NOUN
ejpam-5499	323	20	5n−	5n−	NUM
ejpam-5499	323	21	4−	4−	NUM
ejpam-5499	323	22	2	2	NUM
ejpam-5499	323	23	2	2	NUM
ejpam-5499	323	24	≤	≤	NOUN
ejpam-5499	323	25	5n−	5n−	NUM
ejpam-5499	323	26	6	6	NUM
ejpam-5499	323	27	2	2	NUM
ejpam-5499	323	28	.	.	PUNCT
ejpam-5499	324	1	thus	thus	ADV
ejpam-5499	324	2	|v	|v	X
ejpam-5499	324	3	(	(	PUNCT
ejpam-5499	324	4	ck)|	ck)|	X
ejpam-5499	324	5	=	=	SYM
ejpam-5499	324	6	|v	|v	PROPN
ejpam-5499	324	7	(	(	PUNCT
ejpam-5499	324	8	m(γcn))|	m(γcn))|	PROPN
ejpam-5499	324	9	−	−	PROPN
ejpam-5499	324	10	1	1	NUM
ejpam-5499	324	11	=	=	SYM
ejpam-5499	324	12	5n−6	5n−6	NUM
ejpam-5499	324	13	2	2	NUM
ejpam-5499	324	14	.	.	PUNCT
ejpam-5499	325	1	hence	hence	ADV
ejpam-5499	325	2	5n−6	5n−6	NUM
ejpam-5499	325	3	2	2	NUM
ejpam-5499	325	4	is	be	AUX
ejpam-5499	325	5	the	the	DET
ejpam-5499	325	6	maximum	maximum	ADJ
ejpam-5499	325	7	length	length	NOUN
ejpam-5499	325	8	of	of	ADP
ejpam-5499	325	9	a356	a356	PROPN
ejpam-5499	325	10	cycle	cycle	NOUN
ejpam-5499	325	11	contained	contain	VERB
ejpam-5499	325	12	in	in	ADP
ejpam-5499	325	13	(	(	PUNCT
ejpam-5499	325	14	m(γcn	m(γcn	PROPN
ejpam-5499	325	15	)	)	PUNCT
ejpam-5499	325	16	.	.	PUNCT
ejpam-5499	326	1	therefore	therefore	ADV
ejpam-5499	326	2	the	the	DET
ejpam-5499	326	3	circumference	circumference	NOUN
ejpam-5499	326	4	c(m(γcn	c(m(γcn	PROPN
ejpam-5499	326	5	)	)	PUNCT
ejpam-5499	326	6	)	)	PUNCT
ejpam-5499	327	1	=	=	PUNCT
ejpam-5499	327	2	5n−6	5n−6	NUM
ejpam-5499	327	3	2	2	NUM
ejpam-5499	327	4	,	,	PUNCT
ejpam-5499	327	5	if	if	SCONJ
ejpam-5499	327	6	n	n	NOUN
ejpam-5499	327	7	is357	is357	PROPN
ejpam-5499	327	8	even.358	even.358	NOUN
ejpam-5499	327	9	4.8	4.8	NUM
ejpam-5499	327	10	.	.	PUNCT
ejpam-5499	328	1	girth359	girth359	PROPN
ejpam-5499	328	2	theorem	theorem	VERB
ejpam-5499	328	3	10	10	NUM
ejpam-5499	328	4	.	.	PUNCT
ejpam-5499	329	1	if	if	SCONJ
ejpam-5499	329	2	m(γcn	m(γcn	PROPN
ejpam-5499	329	3	)	)	PUNCT
ejpam-5499	329	4	be	be	VERB
ejpam-5499	329	5	a	a	DET
ejpam-5499	329	6	middle	middle	ADJ
ejpam-5499	329	7	graph	graph	NOUN
ejpam-5499	329	8	of	of	ADP
ejpam-5499	329	9	a	a	DET
ejpam-5499	329	10	γcn	γcn	NOUN
ejpam-5499	329	11	,	,	PUNCT
ejpam-5499	329	12	then	then	ADV
ejpam-5499	329	13	the	the	DET
ejpam-5499	329	14	gir(m(γcn	gir(m(γcn	NOUN
ejpam-5499	329	15	)	)	PUNCT
ejpam-5499	329	16	)	)	PUNCT
ejpam-5499	330	1	=	=	SYM
ejpam-5499	330	2	3	3	NUM
ejpam-5499	330	3	for360	for360	PROPN
ejpam-5499	330	4	n	n	PRON
ejpam-5499	330	5	≥	≥	NOUN
ejpam-5499	330	6	3.361	3.361	NUM
ejpam-5499	330	7	proof	proof	NOUN
ejpam-5499	330	8	.	.	PUNCT
ejpam-5499	331	1	let	let	VERB
ejpam-5499	331	2	e	e	PRON
ejpam-5499	331	3	be	be	AUX
ejpam-5499	331	4	the	the	DET
ejpam-5499	331	5	vertex	vertex	NOUN
ejpam-5499	331	6	representing	represent	VERB
ejpam-5499	331	7	the	the	DET
ejpam-5499	331	8	identity	identity	NOUN
ejpam-5499	331	9	element	element	NOUN
ejpam-5499	331	10	in	in	ADP
ejpam-5499	331	11	γcn	γcn	PROPN
ejpam-5499	331	12	.	.	PUNCT
ejpam-5499	332	1	now	now	ADV
ejpam-5499	332	2	for	for	ADP
ejpam-5499	332	3	γcn	γcn	PROPN
ejpam-5499	332	4	,	,	PUNCT
ejpam-5499	332	5	there362	there362	PROPN
ejpam-5499	332	6	are	be	AUX
ejpam-5499	332	7	n−	n−	NOUN
ejpam-5499	332	8	1	1	NUM
ejpam-5499	332	9	edges	edge	NOUN
ejpam-5499	332	10	incident	incident	NOUN
ejpam-5499	332	11	to	to	PART
ejpam-5499	332	12	e.	e.	PROPN
ejpam-5499	332	13	pick	pick	VERB
ejpam-5499	332	14	any	any	DET
ejpam-5499	332	15	edges	edge	NOUN
ejpam-5499	332	16	namely	namely	ADV
ejpam-5499	332	17	z1	z1	ADJ
ejpam-5499	332	18	,	,	PUNCT
ejpam-5499	332	19	z2	z2	PROPN
ejpam-5499	332	20	.	.	PUNCT
ejpam-5499	333	1	by	by	ADP
ejpam-5499	333	2	the	the	DET
ejpam-5499	333	3	definition	definition	NOUN
ejpam-5499	333	4	8	8	NUM
ejpam-5499	333	5	of	of	ADP
ejpam-5499	333	6	mig,363	mig,363	NOUN
ejpam-5499	333	7	z1	z1	NOUN
ejpam-5499	333	8	and	and	CCONJ
ejpam-5499	333	9	z2	z2	NOUN
ejpam-5499	333	10	are	be	AUX
ejpam-5499	333	11	vertices	vertex	NOUN
ejpam-5499	333	12	in	in	ADP
ejpam-5499	333	13	m(γcn	m(γcn	PROPN
ejpam-5499	333	14	)	)	PUNCT
ejpam-5499	333	15	.	.	PUNCT
ejpam-5499	334	1	also	also	ADV
ejpam-5499	334	2	by	by	ADP
ejpam-5499	334	3	the	the	DET
ejpam-5499	334	4	condition	condition	NOUN
ejpam-5499	334	5	(	(	PUNCT
ejpam-5499	334	6	1	1	NUM
ejpam-5499	334	7	)	)	PUNCT
ejpam-5499	334	8	,	,	PUNCT
ejpam-5499	334	9	[	[	X
ejpam-5499	334	10	z1	z1	ADJ
ejpam-5499	334	11	,	,	PUNCT
ejpam-5499	334	12	z2	z2	PROPN
ejpam-5499	334	13	]	]	PUNCT
ejpam-5499	334	14	is	be	AUX
ejpam-5499	334	15	an	an	DET
ejpam-5499	334	16	edge	edge	NOUN
ejpam-5499	334	17	in	in	ADP
ejpam-5499	334	18	m(γcn).364	m(γcn).364	NOUN
ejpam-5499	334	19	now	now	ADV
ejpam-5499	334	20	by	by	ADP
ejpam-5499	334	21	the	the	DET
ejpam-5499	334	22	condition	condition	NOUN
ejpam-5499	334	23	(	(	PUNCT
ejpam-5499	334	24	2	2	NUM
ejpam-5499	334	25	)	)	PUNCT
ejpam-5499	334	26	,	,	PUNCT
ejpam-5499	335	1	[	[	X
ejpam-5499	335	2	z1	z1	X
ejpam-5499	335	3	,	,	PUNCT
ejpam-5499	335	4	e	e	NOUN
ejpam-5499	335	5	]	]	PUNCT
ejpam-5499	335	6	and	and	CCONJ
ejpam-5499	335	7	[	[	X
ejpam-5499	335	8	z2	z2	PROPN
ejpam-5499	335	9	,	,	PUNCT
ejpam-5499	335	10	e	e	X
ejpam-5499	335	11	]	]	X
ejpam-5499	335	12	are	be	AUX
ejpam-5499	335	13	edges	edge	NOUN
ejpam-5499	335	14	in	in	ADP
ejpam-5499	335	15	m(γcn	m(γcn	PROPN
ejpam-5499	335	16	)	)	PUNCT
ejpam-5499	335	17	.	.	PUNCT
ejpam-5499	336	1	thus	thus	ADV
ejpam-5499	336	2	z1	z1	VERB
ejpam-5499	336	3	,	,	PUNCT
ejpam-5499	336	4	z2	z2	PROPN
ejpam-5499	336	5	,	,	PUNCT
ejpam-5499	336	6	e	e	NOUN
ejpam-5499	336	7	,	,	PUNCT
ejpam-5499	336	8	z1	z1	PROPN
ejpam-5499	336	9	is	be	AUX
ejpam-5499	336	10	a365	a365	PROPN
ejpam-5499	336	11	cycle	cycle	NOUN
ejpam-5499	336	12	of	of	ADP
ejpam-5499	336	13	length	length	NOUN
ejpam-5499	336	14	3	3	NUM
ejpam-5499	336	15	in	in	ADP
ejpam-5499	336	16	m(γcn	m(γcn	PROPN
ejpam-5499	336	17	)	)	PUNCT
ejpam-5499	336	18	.	.	PUNCT
ejpam-5499	337	1	therefore	therefore	ADV
ejpam-5499	337	2	the	the	DET
ejpam-5499	337	3	girth	girth	NOUN
ejpam-5499	337	4	gir(m(γcn	gir(m(γcn	PROPN
ejpam-5499	337	5	)	)	PUNCT
ejpam-5499	337	6	)	)	PUNCT
ejpam-5499	338	1	=	=	SYM
ejpam-5499	338	2	3	3	NUM
ejpam-5499	338	3	for	for	ADP
ejpam-5499	338	4	n	n	NUM
ejpam-5499	338	5	≥	≥	NOUN
ejpam-5499	338	6	3.366	3.366	NUM
ejpam-5499	338	7	4.9	4.9	NUM
ejpam-5499	338	8	.	.	PUNCT
ejpam-5499	339	1	clique	clique	NOUN
ejpam-5499	339	2	number367	number367	PROPN
ejpam-5499	339	3	theorem	theorem	VERB
ejpam-5499	339	4	11	11	NUM
ejpam-5499	339	5	.	.	PUNCT
ejpam-5499	340	1	if	if	SCONJ
ejpam-5499	340	2	m(γcn	m(γcn	PROPN
ejpam-5499	340	3	)	)	PUNCT
ejpam-5499	340	4	be	be	VERB
ejpam-5499	340	5	a	a	DET
ejpam-5499	340	6	middle	middle	ADJ
ejpam-5499	340	7	graph	graph	NOUN
ejpam-5499	340	8	of	of	ADP
ejpam-5499	340	9	a	a	DET
ejpam-5499	340	10	γcn	γcn	NOUN
ejpam-5499	340	11	,	,	PUNCT
ejpam-5499	340	12	then	then	ADV
ejpam-5499	340	13	the	the	DET
ejpam-5499	340	14	clique	clique	ADJ
ejpam-5499	340	15	number	number	NOUN
ejpam-5499	340	16	ω[m(γcn	ω[m(γcn	PROPN
ejpam-5499	340	17	)	)	PUNCT
ejpam-5499	340	18	]	]	PUNCT
ejpam-5499	341	1	=	=	PUNCT
ejpam-5499	341	2	368	368	NUM
ejpam-5499	341	3	n	n	NOUN
ejpam-5499	341	4	for	for	ADP
ejpam-5499	341	5	n	n	NUM
ejpam-5499	341	6	≥	≥	NOUN
ejpam-5499	341	7	3.369	3.369	NUM
ejpam-5499	341	8	proof	proof	NOUN
ejpam-5499	341	9	.	.	PUNCT
ejpam-5499	342	1	let	let	AUX
ejpam-5499	342	2	m(γcn	m(γcn	PRON
ejpam-5499	342	3	)	)	PUNCT
ejpam-5499	342	4	be	be	AUX
ejpam-5499	342	5	the	the	DET
ejpam-5499	342	6	middle	middle	ADJ
ejpam-5499	342	7	graph	graph	NOUN
ejpam-5499	342	8	of	of	ADP
ejpam-5499	342	9	γcn	γcn	PROPN
ejpam-5499	342	10	.	.	PUNCT
ejpam-5499	343	1	we	we	PRON
ejpam-5499	343	2	will	will	AUX
ejpam-5499	343	3	show	show	VERB
ejpam-5499	343	4	that	that	SCONJ
ejpam-5499	343	5	the	the	DET
ejpam-5499	343	6	clique	clique	NOUN
ejpam-5499	343	7	num-370	num-370	INTJ
ejpam-5499	343	8	ber	ber	PROPN
ejpam-5499	343	9	ω[m(γcn	ω[m(γcn	PROPN
ejpam-5499	343	10	)	)	PUNCT
ejpam-5499	343	11	]	]	PUNCT
ejpam-5499	344	1	=	=	PUNCT
ejpam-5499	344	2	n.	n.	NOUN
ejpam-5499	344	3	note	note	VERB
ejpam-5499	344	4	that	that	SCONJ
ejpam-5499	344	5	from	from	ADP
ejpam-5499	344	6	theorem	theorem	ADJ
ejpam-5499	344	7	8	8	NUM
ejpam-5499	344	8	,	,	PUNCT
ejpam-5499	344	9	the	the	DET
ejpam-5499	344	10	subgraph	subgraph	NOUN
ejpam-5499	344	11	induced	induce	VERB
ejpam-5499	344	12	by	by	ADP
ejpam-5499	344	13	the	the	DET
ejpam-5499	344	14	center371	center371	PROPN
ejpam-5499	344	15	cen(m(γcn	cen(m(γcn	NUM
ejpam-5499	344	16	)	)	PUNCT
ejpam-5499	344	17	)	)	PUNCT
ejpam-5499	345	1	=	=	PRON
ejpam-5499	345	2	{	{	PUNCT
ejpam-5499	345	3	zi|1	zi|1	NOUN
ejpam-5499	345	4	≤	≤	NOUN
ejpam-5499	345	5	i	i	PRON
ejpam-5499	345	6	≤	≤	NOUN
ejpam-5499	345	7	n	n	CCONJ
ejpam-5499	345	8	−	−	PROPN
ejpam-5499	345	9	1	1	NUM
ejpam-5499	345	10	}	}	PUNCT
ejpam-5499	345	11	⋃	⋃	NOUN
ejpam-5499	345	12	{	{	PUNCT
ejpam-5499	345	13	e	e	NOUN
ejpam-5499	345	14	}	}	PUNCT
ejpam-5499	345	15	is	be	AUX
ejpam-5499	345	16	a	a	DET
ejpam-5499	345	17	complete	complete	ADJ
ejpam-5499	345	18	graph	graph	NOUN
ejpam-5499	345	19	of	of	ADP
ejpam-5499	345	20	order	order	NOUN
ejpam-5499	345	21	n.	n.	PROPN
ejpam-5499	345	22	suppose	suppose	VERB
ejpam-5499	345	23	there372	there372	PROPN
ejpam-5499	345	24	is	be	AUX
ejpam-5499	345	25	another	another	DET
ejpam-5499	345	26	complete	complete	ADJ
ejpam-5499	345	27	subgraph	subgraph	NOUN
ejpam-5499	345	28	km	km	NOUN
ejpam-5499	345	29	of	of	ADP
ejpam-5499	345	30	order	order	NOUN
ejpam-5499	345	31	m	m	VERB
ejpam-5499	345	32	where	where	SCONJ
ejpam-5499	345	33	m	m	VERB
ejpam-5499	345	34	≥	≥	NOUN
ejpam-5499	345	35	n.	n.	NOUN
ejpam-5499	345	36	but	but	CCONJ
ejpam-5499	345	37	from	from	ADP
ejpam-5499	345	38	the	the	DET
ejpam-5499	345	39	summary	summary	NOUN
ejpam-5499	345	40	of373	of373	VERB
ejpam-5499	345	41	the	the	DET
ejpam-5499	345	42	degree	degree	NOUN
ejpam-5499	345	43	of	of	ADP
ejpam-5499	345	44	the	the	DET
ejpam-5499	345	45	vertices	vertex	NOUN
ejpam-5499	345	46	,	,	PUNCT
ejpam-5499	345	47	the	the	DET
ejpam-5499	345	48	degree	degree	NOUN
ejpam-5499	345	49	deg(xi	deg(xi	NOUN
ejpam-5499	345	50	)	)	PUNCT
ejpam-5499	345	51	=	=	SYM
ejpam-5499	345	52	2	2	NUM
ejpam-5499	345	53	for	for	ADP
ejpam-5499	345	54	1	1	NUM
ejpam-5499	345	55	≤	≤	NUM
ejpam-5499	345	56	i	i	PRON
ejpam-5499	345	57	≤	≤	NOUN
ejpam-5499	345	58	n	n	CCONJ
ejpam-5499	345	59	−	−	PROPN
ejpam-5499	345	60	1	1	NUM
ejpam-5499	345	61	and	and	CCONJ
ejpam-5499	345	62	deg(yp	deg(yp	NOUN
ejpam-5499	345	63	)	)	PUNCT
ejpam-5499	345	64	=	=	SYM
ejpam-5499	345	65	4	4	NUM
ejpam-5499	345	66	for374	for374	NOUN
ejpam-5499	345	67	j.	j.	PROPN
ejpam-5499	345	68	m.	m.	PROPN
ejpam-5499	345	69	jamis	jamis	PROPN
ejpam-5499	345	70	,	,	PUNCT
ejpam-5499	345	71	d.	d.	PROPN
ejpam-5499	345	72	m.	m.	PROPN
ejpam-5499	345	73	magpantay	magpantay	PROPN
ejpam-5499	345	74	/	/	SYM
ejpam-5499	345	75	eur	eur	PROPN
ejpam-5499	345	76	.	.	PUNCT
ejpam-5499	346	1	j.	j.	PROPN
ejpam-5499	346	2	pure	pure	PROPN
ejpam-5499	346	3	appl	appl	PROPN
ejpam-5499	346	4	.	.	PROPN
ejpam-5499	346	5	math	math	PROPN
ejpam-5499	346	6	,	,	PUNCT
ejpam-5499	346	7	18	18	NUM
ejpam-5499	346	8	(	(	PUNCT
ejpam-5499	346	9	2	2	NUM
ejpam-5499	346	10	)	)	PUNCT
ejpam-5499	346	11	(	(	PUNCT
ejpam-5499	346	12	2025	2025	NUM
ejpam-5499	346	13	)	)	PUNCT
ejpam-5499	346	14	,	,	PUNCT
ejpam-5499	346	15	5499	5499	NUM
ejpam-5499	346	16	16	16	NUM
ejpam-5499	346	17	of	of	ADP
ejpam-5499	346	18	18	18	NUM
ejpam-5499	346	19	1	1	NUM
ejpam-5499	346	20	≤	≤	NOUN
ejpam-5499	346	21	p	p	NOUN
ejpam-5499	346	22	≤	≤	NUM
ejpam-5499	346	23	n−1	n−1	PROPN
ejpam-5499	346	24	2	2	NUM
ejpam-5499	346	25	if	if	SCONJ
ejpam-5499	346	26	n	n	NOUN
ejpam-5499	346	27	is	be	AUX
ejpam-5499	346	28	odd	odd	ADJ
ejpam-5499	346	29	and	and	CCONJ
ejpam-5499	346	30	1	1	NUM
ejpam-5499	346	31	≤	≤	NOUN
ejpam-5499	346	32	p	p	ADJ
ejpam-5499	346	33	≤	≤	NOUN
ejpam-5499	346	34	n−2	n−2	PROPN
ejpam-5499	346	35	2	2	NUM
ejpam-5499	346	36	if	if	SCONJ
ejpam-5499	346	37	n	n	PRON
ejpam-5499	346	38	is	be	AUX
ejpam-5499	346	39	even	even	ADV
ejpam-5499	346	40	.	.	PUNCT
ejpam-5499	347	1	thus	thus	ADV
ejpam-5499	347	2	clearly	clearly	ADV
ejpam-5499	347	3	xi′s	xi′s	PROPN
ejpam-5499	347	4	and	and	CCONJ
ejpam-5499	347	5	yp′s	yp′	NOUN
ejpam-5499	347	6	are	be	AUX
ejpam-5499	347	7	not375	not375	PROPN
ejpam-5499	347	8	elements	element	NOUN
ejpam-5499	347	9	of	of	ADP
ejpam-5499	347	10	v	v	NOUN
ejpam-5499	347	11	(	(	PUNCT
ejpam-5499	347	12	km	km	PROPN
ejpam-5499	347	13	)	)	PUNCT
ejpam-5499	347	14	.	.	PUNCT
ejpam-5499	348	1	now	now	ADV
ejpam-5499	348	2	we	we	PRON
ejpam-5499	348	3	are	be	AUX
ejpam-5499	348	4	left	leave	VERB
ejpam-5499	348	5	with	with	ADP
ejpam-5499	348	6	the	the	DET
ejpam-5499	348	7	vertices	vertex	NOUN
ejpam-5499	348	8	zi	zi	NOUN
ejpam-5499	348	9	for	for	ADP
ejpam-5499	348	10	1	1	NUM
ejpam-5499	348	11	≤	≤	NUM
ejpam-5499	348	12	i	i	PRON
ejpam-5499	348	13	≤	≤	NOUN
ejpam-5499	348	14	n	n	CCONJ
ejpam-5499	348	15	−	−	PROPN
ejpam-5499	348	16	1	1	NUM
ejpam-5499	349	1	and	and	CCONJ
ejpam-5499	349	2	e	e	PROPN
ejpam-5499	349	3	since376	since376	PROPN
ejpam-5499	349	4	v	v	PROPN
ejpam-5499	349	5	(	(	PUNCT
ejpam-5499	349	6	m(γcn	m(γcn	NOUN
ejpam-5499	349	7	)	)	PUNCT
ejpam-5499	349	8	)	)	PUNCT
ejpam-5499	349	9	\	\	NOUN
ejpam-5499	350	1	{	{	PUNCT
ejpam-5499	350	2	xi	xi	X
ejpam-5499	350	3	∪	∪	PROPN
ejpam-5499	350	4	yp	yp	PROPN
ejpam-5499	350	5	}	}	PUNCT
ejpam-5499	350	6	=	=	SYM
ejpam-5499	350	7	{	{	PUNCT
ejpam-5499	350	8	zi	zi	NOUN
ejpam-5499	350	9	}	}	PUNCT
ejpam-5499	350	10	∪	∪	X
ejpam-5499	350	11	{	{	PUNCT
ejpam-5499	350	12	e	e	NOUN
ejpam-5499	350	13	}	}	PUNCT
ejpam-5499	350	14	for	for	ADP
ejpam-5499	350	15	all	all	DET
ejpam-5499	350	16	1	1	NUM
ejpam-5499	350	17	≤	≤	NUM
ejpam-5499	350	18	i	i	PRON
ejpam-5499	350	19	≤	≤	NOUN
ejpam-5499	350	20	n	n	CCONJ
ejpam-5499	350	21	−	−	PROPN
ejpam-5499	350	22	1	1	NUM
ejpam-5499	350	23	and	and	CCONJ
ejpam-5499	350	24	1	1	NUM
ejpam-5499	350	25	≤	≤	NOUN
ejpam-5499	350	26	p	p	NOUN
ejpam-5499	350	27	≤	≤	NUM
ejpam-5499	350	28	n−1	n−1	PROPN
ejpam-5499	350	29	2	2	NUM
ejpam-5499	350	30	if	if	SCONJ
ejpam-5499	350	31	n	n	PRON
ejpam-5499	350	32	is	be	AUX
ejpam-5499	350	33	odd377	odd377	ADJ
ejpam-5499	350	34	and	and	CCONJ
ejpam-5499	350	35	1	1	NUM
ejpam-5499	350	36	≤	≤	NOUN
ejpam-5499	350	37	p	p	ADJ
ejpam-5499	350	38	≤	≤	NOUN
ejpam-5499	350	39	n−2	n−2	PROPN
ejpam-5499	350	40	2	2	NUM
ejpam-5499	350	41	if	if	SCONJ
ejpam-5499	350	42	n	n	PRON
ejpam-5499	350	43	is	be	AUX
ejpam-5499	350	44	even	even	ADV
ejpam-5499	350	45	which	which	PRON
ejpam-5499	350	46	is	be	AUX
ejpam-5499	350	47	clearly	clearly	ADV
ejpam-5499	350	48	the	the	DET
ejpam-5499	350	49	center	center	NOUN
ejpam-5499	350	50	cen(m(γcn	cen(m(γcn	NUM
ejpam-5499	350	51	)	)	PUNCT
ejpam-5499	350	52	)	)	PUNCT
ejpam-5499	350	53	.	.	PUNCT
ejpam-5499	351	1	hence	hence	ADV
ejpam-5499	351	2	it	it	PRON
ejpam-5499	351	3	is	be	AUX
ejpam-5499	351	4	not378	not378	PROPN
ejpam-5499	351	5	possible	possible	ADJ
ejpam-5499	351	6	to	to	PART
ejpam-5499	351	7	have	have	VERB
ejpam-5499	351	8	a	a	DET
ejpam-5499	351	9	complete	complete	ADJ
ejpam-5499	351	10	subgraph	subgraph	NOUN
ejpam-5499	351	11	km	km	NOUN
ejpam-5499	351	12	of	of	ADP
ejpam-5499	351	13	order	order	NOUN
ejpam-5499	351	14	m	m	VERB
ejpam-5499	351	15	where	where	SCONJ
ejpam-5499	351	16	m	m	VERB
ejpam-5499	351	17	≥	≥	PROPN
ejpam-5499	351	18	n.	n.	NOUN
ejpam-5499	351	19	therefore	therefore	ADV
ejpam-5499	351	20	the	the	DET
ejpam-5499	351	21	clique379	clique379	PROPN
ejpam-5499	351	22	number	number	NOUN
ejpam-5499	351	23	ω[m(γcn	ω[m(γcn	NOUN
ejpam-5499	351	24	)	)	PUNCT
ejpam-5499	351	25	]	]	PUNCT
ejpam-5499	352	1	=	=	SYM
ejpam-5499	352	2	n.380	n.380	VERB
ejpam-5499	352	3	4.10	4.10	NUM
ejpam-5499	352	4	.	.	PUNCT
ejpam-5499	353	1	independence	independence	NOUN
ejpam-5499	353	2	number381	number381	PROPN
ejpam-5499	353	3	theorem	theorem	NOUN
ejpam-5499	353	4	12	12	NUM
ejpam-5499	353	5	.	.	PUNCT
ejpam-5499	354	1	the	the	DET
ejpam-5499	354	2	indepedendence	indepedendence	NOUN
ejpam-5499	354	3	number	number	NOUN
ejpam-5499	354	4	α(m(γcn	α(m(γcn	NUM
ejpam-5499	354	5	)	)	PUNCT
ejpam-5499	354	6	)	)	PUNCT
ejpam-5499	355	1	=	=	SYM
ejpam-5499	356	1	n.382	n.382	NOUN
ejpam-5499	356	2	proof	proof	NOUN
ejpam-5499	356	3	.	.	PUNCT
ejpam-5499	357	1	to	to	PART
ejpam-5499	357	2	start	start	VERB
ejpam-5499	357	3	with	with	ADP
ejpam-5499	357	4	,	,	PUNCT
ejpam-5499	357	5	we	we	PRON
ejpam-5499	357	6	know	know	VERB
ejpam-5499	357	7	that	that	SCONJ
ejpam-5499	357	8	from	from	ADP
ejpam-5499	357	9	theorem	theorem	ADJ
ejpam-5499	357	10	11	11	NUM
ejpam-5499	357	11	,	,	PUNCT
ejpam-5499	357	12	the	the	DET
ejpam-5499	357	13	largest	large	ADJ
ejpam-5499	357	14	complete	complete	ADJ
ejpam-5499	357	15	graph	graph	NOUN
ejpam-5499	357	16	k383	k383	PROPN
ejpam-5499	357	17	contained	contain	VERB
ejpam-5499	357	18	in	in	ADP
ejpam-5499	357	19	(	(	PUNCT
ejpam-5499	357	20	m(γcn	m(γcn	X
ejpam-5499	357	21	)	)	PUNCT
ejpam-5499	357	22	has	have	VERB
ejpam-5499	357	23	order	order	NOUN
ejpam-5499	357	24	n	n	NOUN
ejpam-5499	357	25	so	so	SCONJ
ejpam-5499	357	26	that	that	SCONJ
ejpam-5499	357	27	the	the	DET
ejpam-5499	357	28	clique	clique	NOUN
ejpam-5499	357	29	number	number	NOUN
ejpam-5499	357	30	ω(m(γcn	ω(m(γcn	NUM
ejpam-5499	357	31	)	)	PUNCT
ejpam-5499	357	32	)	)	PUNCT
ejpam-5499	358	1	=	=	PUNCT
ejpam-5499	358	2	n.	n.	NOUN
ejpam-5499	358	3	note	note	NOUN
ejpam-5499	358	4	that384	that384	VERB
ejpam-5499	358	5	the	the	DET
ejpam-5499	358	6	vertices	vertex	NOUN
ejpam-5499	358	7	v	v	X
ejpam-5499	358	8	(	(	PUNCT
ejpam-5499	358	9	k	k	NOUN
ejpam-5499	358	10	)	)	PUNCT
ejpam-5499	358	11	=	=	SYM
ejpam-5499	359	1	{	{	PUNCT
ejpam-5499	359	2	zi|1	zi|1	NOUN
ejpam-5499	359	3	≤	≤	NOUN
ejpam-5499	359	4	i	i	PRON
ejpam-5499	359	5	≤	≤	NOUN
ejpam-5499	359	6	n	n	CCONJ
ejpam-5499	359	7	−	−	PROPN
ejpam-5499	359	8	1	1	NUM
ejpam-5499	359	9	}	}	PUNCT
ejpam-5499	359	10	⋃	⋃	NOUN
ejpam-5499	359	11	{	{	PUNCT
ejpam-5499	359	12	e	e	NOUN
ejpam-5499	359	13	}	}	PUNCT
ejpam-5499	359	14	.	.	PUNCT
ejpam-5499	360	1	let	let	VERB
ejpam-5499	360	2	s	s	PRON
ejpam-5499	360	3	be	be	AUX
ejpam-5499	360	4	an	an	DET
ejpam-5499	360	5	independent	independent	ADJ
ejpam-5499	360	6	set	set	NOUN
ejpam-5499	360	7	that	that	PRON
ejpam-5499	360	8	has	have	VERB
ejpam-5499	360	9	a385	a385	PROPN
ejpam-5499	360	10	maximum	maximum	ADJ
ejpam-5499	360	11	number	number	NOUN
ejpam-5499	360	12	of	of	ADP
ejpam-5499	360	13	elements	element	NOUN
ejpam-5499	360	14	.	.	PUNCT
ejpam-5499	361	1	it	it	PRON
ejpam-5499	361	2	follows	follow	VERB
ejpam-5499	361	3	that	that	SCONJ
ejpam-5499	361	4	exactly	exactly	ADV
ejpam-5499	361	5	one	one	NUM
ejpam-5499	361	6	vertex	vertex	NOUN
ejpam-5499	361	7	v	v	NOUN
ejpam-5499	361	8	(	(	PUNCT
ejpam-5499	361	9	k	k	NOUN
ejpam-5499	361	10	)	)	PUNCT
ejpam-5499	361	11	must	must	AUX
ejpam-5499	361	12	be	be	AUX
ejpam-5499	361	13	in	in	ADP
ejpam-5499	361	14	s.386	s.386	NOUN
ejpam-5499	361	15	thus	thus	ADV
ejpam-5499	361	16	it	it	PRON
ejpam-5499	361	17	is	be	AUX
ejpam-5499	361	18	either	either	CCONJ
ejpam-5499	361	19	e	e	X
ejpam-5499	361	20	only	only	ADV
ejpam-5499	361	21	or	or	CCONJ
ejpam-5499	361	22	one	one	NUM
ejpam-5499	361	23	of	of	ADP
ejpam-5499	361	24	the	the	DET
ejpam-5499	361	25	zi′s	zi′s	PROPN
ejpam-5499	361	26	only	only	ADV
ejpam-5499	361	27	.	.	PUNCT
ejpam-5499	362	1	for	for	ADP
ejpam-5499	362	2	n	n	PRON
ejpam-5499	362	3	is	be	AUX
ejpam-5499	362	4	odd	odd	ADJ
ejpam-5499	362	5	,	,	PUNCT
ejpam-5499	362	6	we	we	PRON
ejpam-5499	362	7	have	have	VERB
ejpam-5499	362	8	two	two	NUM
ejpam-5499	362	9	cases.387	cases.387	NOUN
ejpam-5499	362	10	1	1	NUM
ejpam-5499	362	11	.	.	PUNCT
ejpam-5499	363	1	let	let	VERB
ejpam-5499	363	2	zj	zj	INTJ
ejpam-5499	363	3	be	be	AUX
ejpam-5499	363	4	a	a	DET
ejpam-5499	363	5	fixed	fix	VERB
ejpam-5499	363	6	element	element	NOUN
ejpam-5499	363	7	in	in	ADP
ejpam-5499	363	8	s	s	PROPN
ejpam-5499	363	9	,	,	PUNCT
ejpam-5499	363	10	then	then	ADV
ejpam-5499	363	11	it	it	PRON
ejpam-5499	363	12	follows	follow	VERB
ejpam-5499	363	13	that	that	SCONJ
ejpam-5499	363	14	xj	xj	PROPN
ejpam-5499	363	15	and	and	CCONJ
ejpam-5499	363	16	yq	yq	PROPN
ejpam-5499	363	17	are	be	AUX
ejpam-5499	363	18	not	not	PART
ejpam-5499	363	19	in	in	ADP
ejpam-5499	363	20	s	s	PRON
ejpam-5499	363	21	for	for	ADP
ejpam-5499	363	22	any	any	DET
ejpam-5499	363	23	fixed388	fixed388	PROPN
ejpam-5499	363	24	xj	xj	PROPN
ejpam-5499	363	25	and	and	CCONJ
ejpam-5499	363	26	yq	yq	PROPN
ejpam-5499	363	27	such	such	ADJ
ejpam-5499	363	28	that	that	SCONJ
ejpam-5499	363	29	yq	yq	PROPN
ejpam-5499	363	30	=	=	PUNCT
ejpam-5499	364	1	[	[	X
ejpam-5499	364	2	xj	xj	PROPN
ejpam-5499	364	3	,	,	PUNCT
ejpam-5499	364	4	xj	xj	PROPN
ejpam-5499	364	5	+	+	PROPN
ejpam-5499	364	6	1	1	X
ejpam-5499	364	7	]	]	X
ejpam-5499	364	8	∈	∈	PROPN
ejpam-5499	364	9	v	v	NOUN
ejpam-5499	364	10	(	(	PUNCT
ejpam-5499	364	11	γcn	γcn	NOUN
ejpam-5499	364	12	)	)	PUNCT
ejpam-5499	364	13	but	but	CCONJ
ejpam-5499	364	14	xj+1	xj+1	NUM
ejpam-5499	364	15	∈	∈	PROPN
ejpam-5499	364	16	s	s	X
ejpam-5499	364	17	since	since	SCONJ
ejpam-5499	364	18	zj	zj	PROPN
ejpam-5499	364	19	is	be	AUX
ejpam-5499	364	20	not	not	PART
ejpam-5499	364	21	adjacent389	adjacent389	PROPN
ejpam-5499	364	22	to	to	ADP
ejpam-5499	364	23	xj+1	xj+1	NUM
ejpam-5499	364	24	.	.	PUNCT
ejpam-5499	365	1	now	now	ADV
ejpam-5499	365	2	we	we	PRON
ejpam-5499	365	3	are	be	AUX
ejpam-5499	365	4	left	leave	VERB
ejpam-5499	365	5	with	with	ADP
ejpam-5499	365	6	the	the	DET
ejpam-5499	365	7	set	set	NOUN
ejpam-5499	365	8	of	of	ADP
ejpam-5499	365	9	vertices	vertex	NOUN
ejpam-5499	365	10	x	x	X
ejpam-5499	365	11	=	=	SYM
ejpam-5499	365	12	{	{	PUNCT
ejpam-5499	365	13	xi|1	xi|1	NOUN
ejpam-5499	365	14	≤	≤	NOUN
ejpam-5499	366	1	i	i	PRON
ejpam-5499	366	2	≤	≤	ADJ
ejpam-5499	366	3	n−	n−	NOUN
ejpam-5499	366	4	1	1	NUM
ejpam-5499	366	5	}	}	PUNCT
ejpam-5499	366	6	\	\	NOUN
ejpam-5499	366	7	{	{	PUNCT
ejpam-5499	366	8	xj	xj	PROPN
ejpam-5499	366	9	,	,	PUNCT
ejpam-5499	366	10	xj+1}390	xj+1}390	X
ejpam-5499	366	11	for	for	ADP
ejpam-5499	366	12	all	all	PRON
ejpam-5499	366	13	odd	odd	ADJ
ejpam-5499	366	14	1	1	NUM
ejpam-5499	366	15	≤	≤	NUM
ejpam-5499	366	16	j	j	PROPN
ejpam-5499	366	17	≤	≤	PROPN
ejpam-5499	366	18	n−2	n−2	PROPN
ejpam-5499	366	19	and	and	CCONJ
ejpam-5499	366	20	the	the	DET
ejpam-5499	366	21	set	set	NOUN
ejpam-5499	366	22	y	y	PROPN
ejpam-5499	366	23	=	=	SYM
ejpam-5499	366	24	yp|1	yp|1	PROPN
ejpam-5499	366	25	≤	≤	NOUN
ejpam-5499	366	26	p	p	NOUN
ejpam-5499	366	27	≤	≤	NUM
ejpam-5499	366	28	n−1	n−1	PROPN
ejpam-5499	366	29	2	2	NUM
ejpam-5499	366	30	\yq	\yq	NOUN
ejpam-5499	366	31	.	.	PUNCT
ejpam-5499	367	1	but	but	CCONJ
ejpam-5499	367	2	note	note	VERB
ejpam-5499	367	3	that	that	SCONJ
ejpam-5499	367	4	for	for	ADP
ejpam-5499	367	5	every391	every391	PROPN
ejpam-5499	367	6	yp	yp	PROPN
ejpam-5499	367	7	,	,	PUNCT
ejpam-5499	367	8	there	there	PRON
ejpam-5499	367	9	are	be	VERB
ejpam-5499	367	10	exactly	exactly	ADV
ejpam-5499	367	11	2	2	NUM
ejpam-5499	367	12	xi′s	xi′s	PRON
ejpam-5499	367	13	adjacent	adjacent	ADJ
ejpam-5499	367	14	to	to	ADP
ejpam-5499	367	15	it	it	PRON
ejpam-5499	367	16	.	.	PUNCT
ejpam-5499	368	1	thus	thus	ADV
ejpam-5499	368	2	for	for	SCONJ
ejpam-5499	368	3	every	every	DET
ejpam-5499	368	4	yp	yp	PROPN
ejpam-5499	368	5	∈	∈	PROPN
ejpam-5499	368	6	s	s	VERB
ejpam-5499	368	7	there	there	PRON
ejpam-5499	368	8	are	be	VERB
ejpam-5499	368	9	exactly	exactly	ADV
ejpam-5499	368	10	2392	2392	NUM
ejpam-5499	368	11	xi	xi	X
ejpam-5499	368	12	/∈	/∈	PUNCT
ejpam-5499	369	1	s	s	VERB
ejpam-5499	370	1	so	so	ADV
ejpam-5499	370	2	that	that	SCONJ
ejpam-5499	370	3	s	s	VERB
ejpam-5499	370	4	=	=	SYM
ejpam-5499	370	5	{	{	PUNCT
ejpam-5499	370	6	zj	zj	PROPN
ejpam-5499	370	7	,	,	PUNCT
ejpam-5499	370	8	xi+1}∪y	xi+1}∪y	PROPN
ejpam-5499	370	9	.	.	PUNCT
ejpam-5499	371	1	hence	hence	ADV
ejpam-5499	371	2	|s|	|s|	PROPN
ejpam-5499	371	3	=	=	SYM
ejpam-5499	371	4	|{zj	|{zj	PROPN
ejpam-5499	371	5	,	,	PUNCT
ejpam-5499	371	6	xi+1}|∪|y	xi+1}|∪|y	PROPN
ejpam-5499	372	1	|	|	NOUN
ejpam-5499	372	2	=	=	NOUN
ejpam-5499	372	3	2	2	NUM
ejpam-5499	372	4	+	+	NUM
ejpam-5499	372	5	n−1	n−1	PROPN
ejpam-5499	372	6	2	2	NUM
ejpam-5499	372	7	−1	−1	NOUN
ejpam-5499	372	8	=	=	SYM
ejpam-5499	372	9	n+1	n+1	PROPN
ejpam-5499	372	10	2	2	NUM
ejpam-5499	372	11	.393	.393	NUM
ejpam-5499	372	12	now	now	ADV
ejpam-5499	372	13	if	if	SCONJ
ejpam-5499	372	14	we	we	PRON
ejpam-5499	372	15	choose	choose	VERB
ejpam-5499	372	16	xi	xi	ADP
ejpam-5499	372	17	∈	∈	PROPN
ejpam-5499	372	18	s	s	PART
ejpam-5499	372	19	,	,	PUNCT
ejpam-5499	372	20	it	it	PRON
ejpam-5499	372	21	follows	follow	VERB
ejpam-5499	372	22	that	that	SCONJ
ejpam-5499	372	23	yp	yp	PROPN
ejpam-5499	372	24	is	be	AUX
ejpam-5499	372	25	not	not	PART
ejpam-5499	372	26	in	in	ADP
ejpam-5499	372	27	s	s	PRON
ejpam-5499	372	28	for	for	ADP
ejpam-5499	372	29	yp	yp	PROPN
ejpam-5499	372	30	=	=	PUNCT
ejpam-5499	373	1	[	[	X
ejpam-5499	373	2	xi	xi	X
ejpam-5499	373	3	,	,	PUNCT
ejpam-5499	373	4	xi+1	xi+1	PROPN
ejpam-5499	373	5	.	.	PUNCT
ejpam-5499	374	1	but	but	CCONJ
ejpam-5499	374	2	since394	since394	PROPN
ejpam-5499	374	3	xi	xi	PROPN
ejpam-5499	374	4	is	be	AUX
ejpam-5499	374	5	not	not	PART
ejpam-5499	374	6	adjacent	adjacent	ADJ
ejpam-5499	374	7	to	to	ADP
ejpam-5499	374	8	xi+1	xi+1	PROPN
ejpam-5499	374	9	,	,	PUNCT
ejpam-5499	374	10	then	then	ADV
ejpam-5499	374	11	xi+1	xi+1	PROPN
ejpam-5499	374	12	must	must	AUX
ejpam-5499	374	13	be	be	AUX
ejpam-5499	374	14	in	in	ADP
ejpam-5499	374	15	s	s	PRON
ejpam-5499	374	16	so	so	SCONJ
ejpam-5499	374	17	that	that	SCONJ
ejpam-5499	374	18	s	s	VERB
ejpam-5499	374	19	=	=	SYM
ejpam-5499	374	20	{	{	PUNCT
ejpam-5499	374	21	zj	zj	PROPN
ejpam-5499	374	22	,	,	PUNCT
ejpam-5499	374	23	xj+1	xj+1	NUM
ejpam-5499	374	24	}	}	PUNCT
ejpam-5499	374	25	∪x	∪x	NUM
ejpam-5499	374	26	.	.	PUNCT
ejpam-5499	375	1	thus395	thus395	PROPN
ejpam-5499	375	2	|s|	|s|	PROPN
ejpam-5499	375	3	=	=	SYM
ejpam-5499	375	4	|{zj	|{zj	PROPN
ejpam-5499	375	5	,	,	PUNCT
ejpam-5499	375	6	xj+1}|	xj+1}|	X
ejpam-5499	375	7	∪	∪	ADP
ejpam-5499	375	8	|x|	|x|	PROPN
ejpam-5499	375	9	=	=	SYM
ejpam-5499	375	10	2	2	NUM
ejpam-5499	375	11	+	+	NUM
ejpam-5499	375	12	n−	n−	NOUN
ejpam-5499	375	13	3	3	NUM
ejpam-5499	375	14	=	=	SYM
ejpam-5499	375	15	n−	n−	NOUN
ejpam-5499	375	16	1.396	1.396	NUM
ejpam-5499	375	17	2	2	NUM
ejpam-5499	375	18	.	.	PUNCT
ejpam-5499	375	19	suppose	suppose	VERB
ejpam-5499	375	20	e	e	X
ejpam-5499	375	21	∈	∈	PROPN
ejpam-5499	375	22	s	s	VERB
ejpam-5499	375	23	then	then	ADV
ejpam-5499	375	24	clearly	clearly	ADV
ejpam-5499	375	25	each	each	PRON
ejpam-5499	375	26	of	of	ADP
ejpam-5499	375	27	the	the	DET
ejpam-5499	375	28	zi	zi	PROPN
ejpam-5499	375	29	for	for	ADP
ejpam-5499	375	30	all	all	DET
ejpam-5499	375	31	1	1	NUM
ejpam-5499	375	32	≤	≤	NUM
ejpam-5499	375	33	i	i	PRON
ejpam-5499	375	34	≤	≤	NOUN
ejpam-5499	375	35	n	n	CCONJ
ejpam-5499	375	36	−	−	PROPN
ejpam-5499	375	37	1	1	NUM
ejpam-5499	375	38	is	be	AUX
ejpam-5499	375	39	not	not	PART
ejpam-5499	375	40	in	in	ADP
ejpam-5499	375	41	s.397	s.397	NUM
ejpam-5499	375	42	now	now	ADV
ejpam-5499	375	43	we	we	PRON
ejpam-5499	375	44	are	be	AUX
ejpam-5499	375	45	left	leave	VERB
ejpam-5499	375	46	with	with	ADP
ejpam-5499	375	47	the	the	DET
ejpam-5499	375	48	set	set	NOUN
ejpam-5499	375	49	of	of	ADP
ejpam-5499	375	50	vertices	vertex	NOUN
ejpam-5499	375	51	x	x	X
ejpam-5499	375	52	=	=	SYM
ejpam-5499	375	53	{	{	PUNCT
ejpam-5499	375	54	xi|1	xi|1	NOUN
ejpam-5499	375	55	≤	≤	NOUN
ejpam-5499	376	1	i	i	PRON
ejpam-5499	376	2	≤	≤	NOUN
ejpam-5499	376	3	n	n	CCONJ
ejpam-5499	376	4	−	−	PROPN
ejpam-5499	376	5	1	1	NUM
ejpam-5499	376	6	}	}	PUNCT
ejpam-5499	376	7	and	and	CCONJ
ejpam-5499	376	8	the	the	DET
ejpam-5499	376	9	set398	set398	PROPN
ejpam-5499	376	10	y	y	PROPN
ejpam-5499	376	11	=	=	SYM
ejpam-5499	376	12	yp|1	yp|1	PROPN
ejpam-5499	376	13	≤	≤	NOUN
ejpam-5499	376	14	p	p	NOUN
ejpam-5499	376	15	≤	≤	NUM
ejpam-5499	376	16	n−1	n−1	PROPN
ejpam-5499	376	17	2	2	NUM
ejpam-5499	376	18	.	.	PUNCT
ejpam-5499	376	19	similar	similar	ADJ
ejpam-5499	376	20	to	to	ADP
ejpam-5499	376	21	case	case	NOUN
ejpam-5499	376	22	1	1	NUM
ejpam-5499	376	23	,	,	PUNCT
ejpam-5499	376	24	for	for	ADP
ejpam-5499	376	25	every	every	DET
ejpam-5499	376	26	yp	yp	NOUN
ejpam-5499	376	27	,	,	PUNCT
ejpam-5499	376	28	there	there	PRON
ejpam-5499	376	29	are	be	VERB
ejpam-5499	376	30	exactly	exactly	ADV
ejpam-5499	376	31	2	2	NUM
ejpam-5499	376	32	xi′s399	xi′s399	NOUN
ejpam-5499	376	33	adjacent	adjacent	ADJ
ejpam-5499	376	34	to	to	ADP
ejpam-5499	376	35	it	it	PRON
ejpam-5499	376	36	.	.	PUNCT
ejpam-5499	377	1	thus	thus	ADV
ejpam-5499	377	2	for	for	SCONJ
ejpam-5499	377	3	every	every	DET
ejpam-5499	377	4	yp	yp	PROPN
ejpam-5499	377	5	∈	∈	PROPN
ejpam-5499	377	6	s	s	VERB
ejpam-5499	377	7	there	there	PRON
ejpam-5499	377	8	are	be	VERB
ejpam-5499	377	9	exactly	exactly	ADV
ejpam-5499	377	10	2	2	NUM
ejpam-5499	377	11	xi	xi	NUM
ejpam-5499	377	12	/∈	/∈	PUNCT
ejpam-5499	378	1	s	s	VERB
ejpam-5499	379	1	so	so	ADV
ejpam-5499	379	2	that	that	SCONJ
ejpam-5499	379	3	s	s	VERB
ejpam-5499	379	4	=	=	X
ejpam-5499	379	5	{	{	PUNCT
ejpam-5499	379	6	e}∪y	e}∪y	NOUN
ejpam-5499	379	7	.400	.400	NUM
ejpam-5499	379	8	hence	hence	ADV
ejpam-5499	379	9	|s|	|s|	PROPN
ejpam-5499	379	10	=	=	SYM
ejpam-5499	379	11	|{e}|	|{e}|	PROPN
ejpam-5499	379	12	∪	∪	ADJ
ejpam-5499	379	13	|y	|y	NOUN
ejpam-5499	379	14	|	|	NOUN
ejpam-5499	379	15	=	=	SYM
ejpam-5499	379	16	1	1	NUM
ejpam-5499	379	17	+	+	CCONJ
ejpam-5499	379	18	n−1	n−1	PROPN
ejpam-5499	379	19	2	2	NUM
ejpam-5499	379	20	=	=	SYM
ejpam-5499	379	21	n+1	n+1	PROPN
ejpam-5499	379	22	2	2	NUM
ejpam-5499	379	23	.	.	PUNCT
ejpam-5499	380	1	now	now	ADV
ejpam-5499	380	2	if	if	SCONJ
ejpam-5499	380	3	we	we	PRON
ejpam-5499	380	4	choose	choose	VERB
ejpam-5499	380	5	xi	xi	ADP
ejpam-5499	380	6	∈	∈	PROPN
ejpam-5499	380	7	s	s	PART
ejpam-5499	380	8	,	,	PUNCT
ejpam-5499	380	9	it	it	PRON
ejpam-5499	380	10	follows	follow	VERB
ejpam-5499	380	11	that401	that401	PROPN
ejpam-5499	380	12	yp	yp	PROPN
ejpam-5499	380	13	is	be	AUX
ejpam-5499	380	14	not	not	PART
ejpam-5499	380	15	in	in	ADP
ejpam-5499	380	16	s	s	PRON
ejpam-5499	380	17	for	for	ADP
ejpam-5499	380	18	yp	yp	PROPN
ejpam-5499	380	19	=	=	PUNCT
ejpam-5499	381	1	[	[	X
ejpam-5499	381	2	xi	xi	X
ejpam-5499	381	3	,	,	PUNCT
ejpam-5499	381	4	xi+1	xi+1	PROPN
ejpam-5499	381	5	.	.	PUNCT
ejpam-5499	382	1	but	but	CCONJ
ejpam-5499	382	2	since	since	SCONJ
ejpam-5499	382	3	xi	xi	PROPN
ejpam-5499	382	4	is	be	AUX
ejpam-5499	382	5	not	not	PART
ejpam-5499	382	6	adjacent	adjacent	ADJ
ejpam-5499	382	7	to	to	ADP
ejpam-5499	382	8	xi+1	xi+1	PROPN
ejpam-5499	382	9	,	,	PUNCT
ejpam-5499	382	10	then	then	ADV
ejpam-5499	382	11	xi+1	xi+1	PROPN
ejpam-5499	382	12	must402	must402	PROPN
ejpam-5499	382	13	be	be	AUX
ejpam-5499	382	14	in	in	ADP
ejpam-5499	382	15	s	s	PRON
ejpam-5499	382	16	so	so	SCONJ
ejpam-5499	382	17	that	that	SCONJ
ejpam-5499	382	18	s	s	VERB
ejpam-5499	382	19	=	=	X
ejpam-5499	382	20	{	{	PUNCT
ejpam-5499	382	21	e	e	NOUN
ejpam-5499	382	22	}	}	PUNCT
ejpam-5499	382	23	∪x	∪x	NOUN
ejpam-5499	382	24	.	.	PUNCT
ejpam-5499	383	1	thus	thus	ADV
ejpam-5499	383	2	|s|	|s|	PROPN
ejpam-5499	383	3	=	=	SYM
ejpam-5499	383	4	|{e}|	|{e}|	PROPN
ejpam-5499	383	5	∪	∪	ADP
ejpam-5499	383	6	|x|	|x|	PROPN
ejpam-5499	383	7	=	=	SYM
ejpam-5499	383	8	1	1	NUM
ejpam-5499	383	9	+	+	NUM
ejpam-5499	383	10	n−	n−	NOUN
ejpam-5499	383	11	1	1	NUM
ejpam-5499	383	12	=	=	NOUN
ejpam-5499	383	13	n.	n.	NOUN
ejpam-5499	383	14	therefore	therefore	ADV
ejpam-5499	383	15	the403	the403	PROPN
ejpam-5499	383	16	cardinality	cardinality	NOUN
ejpam-5499	383	17	of	of	ADP
ejpam-5499	383	18	the	the	DET
ejpam-5499	383	19	maximum	maximum	ADJ
ejpam-5499	383	20	independent	independent	ADJ
ejpam-5499	383	21	set	set	NOUN
ejpam-5499	383	22	in	in	ADP
ejpam-5499	383	23	(	(	PUNCT
ejpam-5499	383	24	m(γcn	m(γcn	PROPN
ejpam-5499	383	25	)	)	PUNCT
ejpam-5499	383	26	is	be	AUX
ejpam-5499	383	27	n	n	PRON
ejpam-5499	383	28	for	for	ADP
ejpam-5499	383	29	n	n	X
ejpam-5499	383	30	is	be	AUX
ejpam-5499	383	31	odd.404	odd.404	NOUN
ejpam-5499	383	32	the	the	DET
ejpam-5499	383	33	proof	proof	NOUN
ejpam-5499	383	34	for	for	ADP
ejpam-5499	383	35	n	n	NUM
ejpam-5499	383	36	is	be	AUX
ejpam-5499	383	37	even	even	ADV
ejpam-5499	383	38	is	be	AUX
ejpam-5499	383	39	analogous	analogous	ADJ
ejpam-5499	383	40	for	for	ADP
ejpam-5499	383	41	n	n	NUM
ejpam-5499	383	42	is	be	AUX
ejpam-5499	383	43	odd.405	odd.405	PROPN
ejpam-5499	383	44	4.11	4.11	NUM
ejpam-5499	383	45	.	.	PUNCT
ejpam-5499	384	1	other	other	ADJ
ejpam-5499	384	2	properties406	properties406	PROPN
ejpam-5499	384	3	theorem	theorem	NOUN
ejpam-5499	384	4	13	13	NUM
ejpam-5499	384	5	.	.	PUNCT
ejpam-5499	385	1	if	if	SCONJ
ejpam-5499	385	2	n	n	NOUN
ejpam-5499	385	3	is	be	AUX
ejpam-5499	385	4	even	even	ADV
ejpam-5499	385	5	,	,	PUNCT
ejpam-5499	385	6	then	then	ADV
ejpam-5499	385	7	the	the	DET
ejpam-5499	385	8	middle	middle	ADJ
ejpam-5499	385	9	graph	graph	NOUN
ejpam-5499	385	10	m(γcn	m(γcn	PROPN
ejpam-5499	385	11	)	)	PUNCT
ejpam-5499	385	12	contains	contain	VERB
ejpam-5499	385	13	exactly	exactly	ADV
ejpam-5499	385	14	one	one	NUM
ejpam-5499	385	15	vertex	vertex	NOUN
ejpam-5499	385	16	of407	of407	PRON
ejpam-5499	385	17	degree	degree	NOUN
ejpam-5499	385	18	1.408	1.408	NUM
ejpam-5499	385	19	j.	j.	PROPN
ejpam-5499	385	20	m.	m.	PROPN
ejpam-5499	385	21	jamis	jamis	PROPN
ejpam-5499	385	22	,	,	PUNCT
ejpam-5499	385	23	d.	d.	PROPN
ejpam-5499	385	24	m.	m.	PROPN
ejpam-5499	385	25	magpantay	magpantay	PROPN
ejpam-5499	385	26	/	/	SYM
ejpam-5499	385	27	eur	eur	PROPN
ejpam-5499	385	28	.	.	PUNCT
ejpam-5499	386	1	j.	j.	PROPN
ejpam-5499	386	2	pure	pure	PROPN
ejpam-5499	386	3	appl	appl	PROPN
ejpam-5499	386	4	.	.	PROPN
ejpam-5499	386	5	math	math	PROPN
ejpam-5499	386	6	,	,	PUNCT
ejpam-5499	386	7	18	18	NUM
ejpam-5499	386	8	(	(	PUNCT
ejpam-5499	386	9	2	2	NUM
ejpam-5499	386	10	)	)	PUNCT
ejpam-5499	386	11	(	(	PUNCT
ejpam-5499	386	12	2025	2025	NUM
ejpam-5499	386	13	)	)	PUNCT
ejpam-5499	386	14	,	,	PUNCT
ejpam-5499	386	15	5499	5499	NUM
ejpam-5499	386	16	17	17	NUM
ejpam-5499	386	17	of	of	ADP
ejpam-5499	386	18	18	18	NUM
ejpam-5499	386	19	proof	proof	NOUN
ejpam-5499	386	20	.	.	PUNCT
ejpam-5499	387	1	by	by	ADP
ejpam-5499	387	2	theorem	theorem	NOUN
ejpam-5499	387	3	1	1	NUM
ejpam-5499	387	4	,	,	PUNCT
ejpam-5499	387	5	the	the	DET
ejpam-5499	387	6	line	line	NOUN
ejpam-5499	387	7	in	in	ADP
ejpam-5499	387	8	the	the	DET
ejpam-5499	387	9	identity	identity	NOUN
ejpam-5499	387	10	graph	graph	NOUN
ejpam-5499	387	11	γcn	γcn	NOUN
ejpam-5499	387	12	is	be	AUX
ejpam-5499	387	13	1	1	NUM
ejpam-5499	387	14	if	if	SCONJ
ejpam-5499	387	15	n	n	PRON
ejpam-5499	387	16	is	be	AUX
ejpam-5499	387	17	even	even	ADV
ejpam-5499	387	18	.	.	PUNCT
ejpam-5499	388	1	now	now	ADV
ejpam-5499	388	2	let	let	VERB
ejpam-5499	388	3	x409	x409	PUNCT
ejpam-5499	388	4	be	be	AUX
ejpam-5499	388	5	the	the	DET
ejpam-5499	388	6	vertex	vertex	NOUN
ejpam-5499	388	7	of	of	ADP
ejpam-5499	388	8	degree	degree	NOUN
ejpam-5499	388	9	1	1	NUM
ejpam-5499	388	10	in	in	ADP
ejpam-5499	388	11	γcn	γcn	PROPN
ejpam-5499	388	12	where	where	SCONJ
ejpam-5499	388	13	n	n	PRON
ejpam-5499	388	14	is	be	AUX
ejpam-5499	388	15	even	even	ADV
ejpam-5499	388	16	and	and	CCONJ
ejpam-5499	388	17	let	let	VERB
ejpam-5499	388	18	a	a	PRON
ejpam-5499	388	19	be	be	AUX
ejpam-5499	388	20	the	the	DET
ejpam-5499	388	21	edge	edge	NOUN
ejpam-5499	388	22	connecting	connect	VERB
ejpam-5499	388	23	x	x	PUNCT
ejpam-5499	388	24	to410	to410	ADJ
ejpam-5499	388	25	another	another	DET
ejpam-5499	388	26	vertex	vertex	NOUN
ejpam-5499	388	27	say	say	VERB
ejpam-5499	388	28	y.	y.	NOUN
ejpam-5499	388	29	now	now	ADV
ejpam-5499	388	30	by	by	ADP
ejpam-5499	388	31	the	the	DET
ejpam-5499	388	32	definition	definition	NOUN
ejpam-5499	388	33	of	of	ADP
ejpam-5499	388	34	the	the	DET
ejpam-5499	388	35	middle	middle	ADJ
ejpam-5499	388	36	graph	graph	NOUN
ejpam-5499	388	37	,	,	PUNCT
ejpam-5499	388	38	a	a	PRON
ejpam-5499	388	39	will	will	AUX
ejpam-5499	388	40	become	become	VERB
ejpam-5499	388	41	a	a	DET
ejpam-5499	388	42	vertex411	vertex411	PROPN
ejpam-5499	388	43	in	in	ADP
ejpam-5499	388	44	m(γcn	m(γcn	PROPN
ejpam-5499	388	45	)	)	PUNCT
ejpam-5499	388	46	.	.	PUNCT
ejpam-5499	389	1	also	also	ADV
ejpam-5499	389	2	by	by	ADP
ejpam-5499	389	3	condition	condition	NOUN
ejpam-5499	389	4	(	(	PUNCT
ejpam-5499	389	5	2	2	NUM
ejpam-5499	389	6	)	)	PUNCT
ejpam-5499	389	7	from	from	ADP
ejpam-5499	389	8	the	the	DET
ejpam-5499	389	9	definition	definition	NOUN
ejpam-5499	389	10	of	of	ADP
ejpam-5499	389	11	mig	mig	NOUN
ejpam-5499	389	12	in	in	ADP
ejpam-5499	389	13	definition	definition	NOUN
ejpam-5499	389	14	8	8	NUM
ejpam-5499	389	15	,	,	PUNCT
ejpam-5499	389	16	[	[	X
ejpam-5499	389	17	x	x	X
ejpam-5499	389	18	,	,	PUNCT
ejpam-5499	389	19	a	a	PRON
ejpam-5499	389	20	]	]	PUNCT
ejpam-5499	389	21	is	be	AUX
ejpam-5499	389	22	an412	an412	ADJ
ejpam-5499	389	23	edge	edge	NOUN
ejpam-5499	389	24	.	.	PUNCT
ejpam-5499	390	1	suppose	suppose	VERB
ejpam-5499	390	2	there	there	PRON
ejpam-5499	390	3	exists	exist	VERB
ejpam-5499	390	4	another	another	DET
ejpam-5499	390	5	vertex	vertex	NOUN
ejpam-5499	390	6	w	w	ADP
ejpam-5499	390	7	such	such	ADJ
ejpam-5499	390	8	that	that	SCONJ
ejpam-5499	390	9	[	[	X
ejpam-5499	390	10	x	x	X
ejpam-5499	390	11	,	,	PUNCT
ejpam-5499	390	12	w	w	PROPN
ejpam-5499	390	13	]	]	X
ejpam-5499	390	14	is	be	AUX
ejpam-5499	390	15	an	an	DET
ejpam-5499	390	16	edge	edge	NOUN
ejpam-5499	390	17	in	in	ADP
ejpam-5499	390	18	m(γcn	m(γcn	PROPN
ejpam-5499	390	19	)	)	PUNCT
ejpam-5499	390	20	.	.	PUNCT
ejpam-5499	391	1	by413	by413	PROPN
ejpam-5499	392	1	(	(	PUNCT
ejpam-5499	392	2	2	2	X
ejpam-5499	392	3	)	)	PUNCT
ejpam-5499	392	4	in	in	ADP
ejpam-5499	392	5	the	the	DET
ejpam-5499	392	6	definition	definition	NOUN
ejpam-5499	392	7	of	of	ADP
ejpam-5499	392	8	the	the	DET
ejpam-5499	392	9	mig	mig	NOUN
ejpam-5499	392	10	,	,	PUNCT
ejpam-5499	392	11	it	it	PRON
ejpam-5499	392	12	follows	follow	VERB
ejpam-5499	392	13	that	that	SCONJ
ejpam-5499	392	14	w	w	NOUN
ejpam-5499	392	15	is	be	AUX
ejpam-5499	392	16	an	an	DET
ejpam-5499	392	17	edge	edge	NOUN
ejpam-5499	392	18	in	in	ADP
ejpam-5499	392	19	γcn	γcn	NOUN
ejpam-5499	392	20	that	that	PRON
ejpam-5499	392	21	is	be	AUX
ejpam-5499	392	22	incident	incident	NOUN
ejpam-5499	392	23	to	to	ADP
ejpam-5499	392	24	x414	x414	NUM
ejpam-5499	392	25	which	which	PRON
ejpam-5499	392	26	contradicts	contradict	VERB
ejpam-5499	392	27	the	the	DET
ejpam-5499	392	28	fact	fact	NOUN
ejpam-5499	392	29	that	that	SCONJ
ejpam-5499	392	30	x	x	PRON
ejpam-5499	392	31	has	have	AUX
ejpam-5499	392	32	degree	degree	NOUN
ejpam-5499	392	33	1	1	NUM
ejpam-5499	392	34	in	in	ADP
ejpam-5499	392	35	γcn	γcn	PROPN
ejpam-5499	392	36	.	.	PUNCT
ejpam-5499	393	1	thus	thus	ADV
ejpam-5499	393	2	m(γcn	m(γcn	X
ejpam-5499	393	3	)	)	PUNCT
ejpam-5499	393	4	contains	contain	VERB
ejpam-5499	393	5	a	a	DET
ejpam-5499	393	6	vertex	vertex	NOUN
ejpam-5499	393	7	of415	of415	NOUN
ejpam-5499	393	8	degree	degree	NOUN
ejpam-5499	393	9	1	1	NUM
ejpam-5499	393	10	.	.	PUNCT
ejpam-5499	393	11	suppose	suppose	VERB
ejpam-5499	393	12	there	there	PRON
ejpam-5499	393	13	exists	exist	VERB
ejpam-5499	393	14	another	another	DET
ejpam-5499	393	15	vertex	vertex	NOUN
ejpam-5499	393	16	u	u	NOUN
ejpam-5499	393	17	where	where	SCONJ
ejpam-5499	393	18	u	u	NOUN
ejpam-5499	393	19	̸=	̸=	PROPN
ejpam-5499	393	20	x	x	PUNCT
ejpam-5499	393	21	in	in	ADP
ejpam-5499	393	22	m(γcn	m(γcn	PROPN
ejpam-5499	393	23	)	)	PUNCT
ejpam-5499	393	24	of	of	ADP
ejpam-5499	393	25	degree	degree	NOUN
ejpam-5499	393	26	1	1	NUM
ejpam-5499	393	27	,	,	PUNCT
ejpam-5499	393	28	then416	then416	NUM
ejpam-5499	393	29	u	u	NOUN
ejpam-5499	393	30	can	can	AUX
ejpam-5499	393	31	not	not	PART
ejpam-5499	393	32	be	be	AUX
ejpam-5499	393	33	in	in	ADP
ejpam-5499	393	34	e(γcn	e(γcn	PROPN
ejpam-5499	393	35	)	)	PUNCT
ejpam-5499	393	36	since	since	SCONJ
ejpam-5499	393	37	there	there	PRON
ejpam-5499	393	38	are	be	VERB
ejpam-5499	393	39	two	two	NUM
ejpam-5499	393	40	edges	edge	NOUN
ejpam-5499	393	41	namely	namely	ADV
ejpam-5499	393	42	q	q	NOUN
ejpam-5499	393	43	and	and	CCONJ
ejpam-5499	393	44	r	r	NOUN
ejpam-5499	393	45	incident	incident	NOUN
ejpam-5499	393	46	to	to	ADP
ejpam-5499	393	47	u	u	NOUN
ejpam-5499	393	48	and	and	CCONJ
ejpam-5499	393	49	by	by	ADP
ejpam-5499	393	50	(	(	PUNCT
ejpam-5499	393	51	2)417	2)417	NUM
ejpam-5499	393	52	of	of	ADP
ejpam-5499	393	53	definition	definition	NOUN
ejpam-5499	393	54	8	8	NUM
ejpam-5499	393	55	,	,	PUNCT
ejpam-5499	393	56	u	u	PRON
ejpam-5499	393	57	will	will	AUX
ejpam-5499	393	58	become	become	VERB
ejpam-5499	393	59	a	a	DET
ejpam-5499	393	60	vertex	vertex	NOUN
ejpam-5499	393	61	adjacent	adjacent	ADJ
ejpam-5499	393	62	to	to	ADP
ejpam-5499	393	63	q	q	PROPN
ejpam-5499	393	64	and	and	CCONJ
ejpam-5499	393	65	r.	r.	PROPN
ejpam-5499	393	66	thus	thus	ADV
ejpam-5499	393	67	it	it	PRON
ejpam-5499	393	68	must	must	AUX
ejpam-5499	393	69	be	be	AUX
ejpam-5499	393	70	u	u	PROPN
ejpam-5499	393	71	∈	∈	PROPN
ejpam-5499	393	72	v	v	NOUN
ejpam-5499	393	73	(	(	PUNCT
ejpam-5499	393	74	γcn).418	γcn).418	NOUN
ejpam-5499	393	75	note	note	VERB
ejpam-5499	393	76	that	that	SCONJ
ejpam-5499	393	77	γcn	γcn	PROPN
ejpam-5499	393	78	contains	contain	VERB
ejpam-5499	393	79	only	only	ADV
ejpam-5499	393	80	one	one	NUM
ejpam-5499	393	81	line	line	NOUN
ejpam-5499	393	82	,	,	PUNCT
ejpam-5499	393	83	it	it	PRON
ejpam-5499	393	84	follows	follow	VERB
ejpam-5499	393	85	that	that	SCONJ
ejpam-5499	393	86	u	u	PRON
ejpam-5499	393	87	lies	lie	VERB
ejpam-5499	393	88	in	in	ADP
ejpam-5499	393	89	some	some	DET
ejpam-5499	393	90	triangles	triangle	NOUN
ejpam-5499	393	91	of	of	ADP
ejpam-5499	393	92	γcn	γcn	PROPN
ejpam-5499	393	93	which419	which419	PROPN
ejpam-5499	393	94	implies	imply	VERB
ejpam-5499	393	95	that	that	SCONJ
ejpam-5499	393	96	atleast	atleast	ADJ
ejpam-5499	393	97	two	two	NUM
ejpam-5499	393	98	edges	edge	NOUN
ejpam-5499	393	99	say	say	VERB
ejpam-5499	393	100	s	s	PRON
ejpam-5499	393	101	and	and	CCONJ
ejpam-5499	393	102	t	t	PROPN
ejpam-5499	393	103	are	be	AUX
ejpam-5499	393	104	incident	incident	NOUN
ejpam-5499	393	105	to	to	ADP
ejpam-5499	393	106	u	u	PRON
ejpam-5499	393	107	that	that	PRON
ejpam-5499	393	108	will	will	AUX
ejpam-5499	393	109	eventually	eventually	ADV
ejpam-5499	393	110	become420	become420	NOUN
ejpam-5499	393	111	vertices	vertex	NOUN
ejpam-5499	393	112	in	in	ADP
ejpam-5499	393	113	m(γcn	m(γcn	PROPN
ejpam-5499	393	114	)	)	PUNCT
ejpam-5499	393	115	.	.	PUNCT
ejpam-5499	394	1	by	by	ADP
ejpam-5499	394	2	the	the	DET
ejpam-5499	394	3	(	(	PUNCT
ejpam-5499	394	4	2	2	NUM
ejpam-5499	394	5	)	)	PUNCT
ejpam-5499	394	6	of	of	ADP
ejpam-5499	394	7	definition	definition	NOUN
ejpam-5499	394	8	8	8	NUM
ejpam-5499	394	9	,	,	PUNCT
ejpam-5499	394	10	s	s	PRON
ejpam-5499	394	11	and	and	CCONJ
ejpam-5499	394	12	t	t	PROPN
ejpam-5499	394	13	will	will	AUX
ejpam-5499	394	14	be	be	AUX
ejpam-5499	394	15	vertices	vertex	NOUN
ejpam-5499	394	16	adjacent	adjacent	ADJ
ejpam-5499	394	17	to	to	ADP
ejpam-5499	394	18	u	u	PRON
ejpam-5499	394	19	which421	which421	PROPN
ejpam-5499	394	20	contradicts	contradict	VERB
ejpam-5499	394	21	that	that	SCONJ
ejpam-5499	394	22	u	u	PROPN
ejpam-5499	394	23	has	have	VERB
ejpam-5499	394	24	of	of	ADP
ejpam-5499	394	25	degree	degree	NOUN
ejpam-5499	394	26	1	1	NUM
ejpam-5499	394	27	.	.	PUNCT
ejpam-5499	395	1	hence	hence	ADV
ejpam-5499	395	2	there	there	PRON
ejpam-5499	395	3	is	be	VERB
ejpam-5499	395	4	only	only	ADV
ejpam-5499	395	5	one	one	NUM
ejpam-5499	395	6	vertex	vertex	NOUN
ejpam-5499	395	7	in	in	ADP
ejpam-5499	395	8	m(γcn	m(γcn	PROPN
ejpam-5499	395	9	)	)	PUNCT
ejpam-5499	395	10	of	of	ADP
ejpam-5499	395	11	degree	degree	NOUN
ejpam-5499	395	12	1.422	1.422	NUM
ejpam-5499	395	13	theorem	theorem	NOUN
ejpam-5499	395	14	14	14	NUM
ejpam-5499	395	15	.	.	PUNCT
ejpam-5499	396	1	every	every	DET
ejpam-5499	396	2	vertex	vertex	NOUN
ejpam-5499	396	3	of	of	ADP
ejpam-5499	396	4	m(γcn	m(γcn	PROPN
ejpam-5499	396	5	)	)	PUNCT
ejpam-5499	396	6	has	have	VERB
ejpam-5499	396	7	an	an	DET
ejpam-5499	396	8	even	even	ADJ
ejpam-5499	396	9	degree	degree	NOUN
ejpam-5499	396	10	if	if	SCONJ
ejpam-5499	396	11	n	n	PRON
ejpam-5499	396	12	is	be	AUX
ejpam-5499	396	13	odd.423	odd.423	NOUN
ejpam-5499	396	14	proof	proof	NOUN
ejpam-5499	396	15	.	.	PUNCT
ejpam-5499	397	1	the	the	DET
ejpam-5499	397	2	summary	summary	NOUN
ejpam-5499	397	3	of	of	ADP
ejpam-5499	397	4	the	the	DET
ejpam-5499	397	5	degree	degree	NOUN
ejpam-5499	397	6	of	of	ADP
ejpam-5499	397	7	the	the	DET
ejpam-5499	397	8	vertices	vertex	NOUN
ejpam-5499	397	9	ofm(γcn	ofm(γcn	NOUN
ejpam-5499	397	10	)	)	PUNCT
ejpam-5499	397	11	whwre	whwre	NOUN
ejpam-5499	397	12	n	n	PART
ejpam-5499	397	13	is	be	AUX
ejpam-5499	397	14	odd	odd	ADJ
ejpam-5499	397	15	is	be	AUX
ejpam-5499	397	16	sufficient424	sufficient424	PROPN
ejpam-5499	397	17	enough	enough	ADJ
ejpam-5499	397	18	to	to	PART
ejpam-5499	397	19	prove	prove	VERB
ejpam-5499	397	20	this	this	DET
ejpam-5499	397	21	theorem.425	theorem.425	NOUN
ejpam-5499	397	22	theorem	theorem	NOUN
ejpam-5499	397	23	15	15	NUM
ejpam-5499	397	24	.	.	PUNCT
ejpam-5499	398	1	the	the	DET
ejpam-5499	398	2	middle	middle	ADJ
ejpam-5499	398	3	graph	graph	NOUN
ejpam-5499	398	4	m(γcn	m(γcn	PROPN
ejpam-5499	398	5	)	)	PUNCT
ejpam-5499	398	6	is	be	AUX
ejpam-5499	398	7	eulerian	eulerian	ADJ
ejpam-5499	398	8	if	if	SCONJ
ejpam-5499	398	9	n	n	NOUN
ejpam-5499	398	10	is	be	AUX
ejpam-5499	398	11	odd.426	odd.426	NOUN
ejpam-5499	398	12	proof.427	proof.427	NOUN
ejpam-5499	398	13	let	let	VERB
ejpam-5499	398	14	m(γcn	m(γcn	PRON
ejpam-5499	398	15	)	)	PUNCT
ejpam-5499	398	16	be	be	AUX
ejpam-5499	398	17	the	the	DET
ejpam-5499	398	18	middle	middle	ADJ
ejpam-5499	398	19	graph	graph	NOUN
ejpam-5499	398	20	of	of	ADP
ejpam-5499	398	21	γcn	γcn	PROPN
ejpam-5499	398	22	.	.	PUNCT
ejpam-5499	399	1	suppose	suppose	VERB
ejpam-5499	399	2	n	n	PRON
ejpam-5499	399	3	is	be	AUX
ejpam-5499	399	4	odd	odd	ADJ
ejpam-5499	399	5	,	,	PUNCT
ejpam-5499	399	6	by	by	ADP
ejpam-5499	399	7	theorem	theorem	NOUN
ejpam-5499	399	8	14	14	NUM
ejpam-5499	399	9	,	,	PUNCT
ejpam-5499	399	10	every428	every428	DET
ejpam-5499	399	11	vertex	vertex	NOUN
ejpam-5499	399	12	of	of	ADP
ejpam-5499	399	13	m(γcn	m(γcn	PROPN
ejpam-5499	399	14	)	)	PUNCT
ejpam-5499	399	15	is	be	AUX
ejpam-5499	399	16	of	of	ADP
ejpam-5499	399	17	even	even	ADJ
ejpam-5499	399	18	degree	degree	NOUN
ejpam-5499	399	19	.	.	PUNCT
ejpam-5499	400	1	thus	thus	ADV
ejpam-5499	400	2	by	by	ADP
ejpam-5499	400	3	theorem	theorem	ADJ
ejpam-5499	400	4	4	4	NUM
ejpam-5499	400	5	,	,	PUNCT
ejpam-5499	400	6	m(γcn	m(γcn	PROPN
ejpam-5499	400	7	)	)	PUNCT
ejpam-5499	400	8	is	be	AUX
ejpam-5499	400	9	eulerian.429	eulerian.429	PROPN
ejpam-5499	400	10	theorem	theorem	ADJ
ejpam-5499	400	11	16	16	NUM
ejpam-5499	400	12	.	.	PUNCT
ejpam-5499	401	1	if	if	SCONJ
ejpam-5499	401	2	n	n	NOUN
ejpam-5499	401	3	is	be	AUX
ejpam-5499	401	4	odd	odd	ADJ
ejpam-5499	401	5	,	,	PUNCT
ejpam-5499	401	6	then	then	ADV
ejpam-5499	401	7	the	the	DET
ejpam-5499	401	8	middle	middle	ADJ
ejpam-5499	401	9	graph	graph	NOUN
ejpam-5499	401	10	m(γcn	m(γcn	PROPN
ejpam-5499	401	11	)	)	PUNCT
ejpam-5499	401	12	is	be	AUX
ejpam-5499	401	13	hamiltonian.430	hamiltonian.430	ADJ
ejpam-5499	401	14	proof	proof	NOUN
ejpam-5499	401	15	.	.	PUNCT
ejpam-5499	402	1	from	from	ADP
ejpam-5499	402	2	theorem	theorem	ADJ
ejpam-5499	402	3	9	9	NUM
ejpam-5499	402	4	,	,	PUNCT
ejpam-5499	402	5	if	if	SCONJ
ejpam-5499	402	6	n	n	PRON
ejpam-5499	402	7	is	be	AUX
ejpam-5499	402	8	odd	odd	ADJ
ejpam-5499	402	9	,	,	PUNCT
ejpam-5499	402	10	the	the	DET
ejpam-5499	402	11	order	order	NOUN
ejpam-5499	402	12	of	of	ADP
ejpam-5499	402	13	the	the	DET
ejpam-5499	402	14	largest	large	ADJ
ejpam-5499	402	15	cycle	cycle	NOUN
ejpam-5499	402	16	|v	|v	NOUN
ejpam-5499	402	17	(	(	PUNCT
ejpam-5499	402	18	c)|	c)|	PROPN
ejpam-5499	402	19	contained	contain	VERB
ejpam-5499	402	20	in431	in431	PROPN
ejpam-5499	402	21	m(γcn	m(γcn	PROPN
ejpam-5499	402	22	)	)	PUNCT
ejpam-5499	402	23	is	be	AUX
ejpam-5499	402	24	5n−3	5n−3	NUM
ejpam-5499	402	25	2	2	NUM
ejpam-5499	402	26	=	=	SYM
ejpam-5499	402	27	|m(γcn)|	|m(γcn)|	VERB
ejpam-5499	402	28	.	.	PUNCT
ejpam-5499	403	1	thus	thus	ADV
ejpam-5499	403	2	c	c	PROPN
ejpam-5499	403	3	is	be	AUX
ejpam-5499	403	4	hamiltonian	hamiltonian	ADJ
ejpam-5499	403	5	cycle	cycle	NOUN
ejpam-5499	403	6	.	.	PUNCT
ejpam-5499	404	1	consequently	consequently	ADV
ejpam-5499	404	2	,	,	PUNCT
ejpam-5499	404	3	m(γcn	m(γcn	PROPN
ejpam-5499	404	4	)	)	PUNCT
ejpam-5499	404	5	where432	where432	PROPN
ejpam-5499	404	6	n	n	PROPN
ejpam-5499	404	7	is	be	AUX
ejpam-5499	404	8	odd	odd	ADJ
ejpam-5499	404	9	is	be	AUX
ejpam-5499	404	10	a	a	DET
ejpam-5499	404	11	hamiltonian	hamiltonian	ADJ
ejpam-5499	404	12	graph.433	graph.433	NOUN
ejpam-5499	404	13	5	5	NUM
ejpam-5499	404	14	.	.	PUNCT
ejpam-5499	404	15	conclusion	conclusion	NOUN
ejpam-5499	404	16	and	and	CCONJ
ejpam-5499	404	17	recommendations434	recommendations434	PROPN
ejpam-5499	404	18	this	this	DET
ejpam-5499	404	19	paper	paper	NOUN
ejpam-5499	404	20	focuses	focus	VERB
ejpam-5499	404	21	on	on	ADP
ejpam-5499	404	22	the	the	DET
ejpam-5499	404	23	middle	middle	ADJ
ejpam-5499	404	24	graph	graph	NOUN
ejpam-5499	404	25	of	of	ADP
ejpam-5499	404	26	the	the	DET
ejpam-5499	404	27	identity	identity	NOUN
ejpam-5499	404	28	graph	graph	NOUN
ejpam-5499	404	29	of	of	ADP
ejpam-5499	404	30	finite	finite	ADJ
ejpam-5499	404	31	cyclic	cyclic	ADJ
ejpam-5499	404	32	and435	and435	PROPN
ejpam-5499	404	33	dihedral	dihedral	ADJ
ejpam-5499	404	34	groups	group	NOUN
ejpam-5499	404	35	denoted	denote	VERB
ejpam-5499	404	36	by	by	ADP
ejpam-5499	404	37	m(γcn	m(γcn	PROPN
ejpam-5499	404	38	)	)	PUNCT
ejpam-5499	404	39	and	and	CCONJ
ejpam-5499	404	40	m(γdn	m(γdn	PROPN
ejpam-5499	404	41	)	)	PUNCT
ejpam-5499	404	42	respectively	respectively	ADV
ejpam-5499	404	43	.	.	PUNCT
ejpam-5499	405	1	these	these	DET
ejpam-5499	405	2	graphs	graph	NOUN
ejpam-5499	405	3	are	be	AUX
ejpam-5499	405	4	simple,436	simple,436	PROPN
ejpam-5499	405	5	finite	finite	NOUN
ejpam-5499	405	6	,	,	PUNCT
ejpam-5499	405	7	connected	connected	ADJ
ejpam-5499	405	8	and	and	CCONJ
ejpam-5499	405	9	undirected	undirected	ADJ
ejpam-5499	405	10	graphs	graph	NOUN
ejpam-5499	405	11	.	.	PUNCT
ejpam-5499	406	1	the	the	DET
ejpam-5499	406	2	concept	concept	NOUN
ejpam-5499	406	3	of	of	ADP
ejpam-5499	406	4	the	the	DET
ejpam-5499	406	5	identity	identity	NOUN
ejpam-5499	406	6	graph	graph	NOUN
ejpam-5499	406	7	and	and	CCONJ
ejpam-5499	406	8	middle437	middle437	NOUN
ejpam-5499	406	9	graph	graph	NOUN
ejpam-5499	406	10	is	be	AUX
ejpam-5499	406	11	introduced	introduce	VERB
ejpam-5499	406	12	in	in	ADP
ejpam-5499	406	13	this	this	DET
ejpam-5499	406	14	paper.438	paper.438	NOUN
ejpam-5499	406	15	using	use	VERB
ejpam-5499	406	16	these	these	DET
ejpam-5499	406	17	concepts	concept	NOUN
ejpam-5499	406	18	,	,	PUNCT
ejpam-5499	406	19	the	the	DET
ejpam-5499	406	20	middle	middle	ADJ
ejpam-5499	406	21	graphs	graph	NOUN
ejpam-5499	406	22	m(γcn	m(γcn	PROPN
ejpam-5499	406	23	)	)	PUNCT
ejpam-5499	406	24	and	and	CCONJ
ejpam-5499	406	25	m(γdn	m(γdn	PROPN
ejpam-5499	406	26	)	)	PUNCT
ejpam-5499	406	27	were	be	AUX
ejpam-5499	406	28	constructed	construct	VERB
ejpam-5499	406	29	and439	and439	ADJ
ejpam-5499	406	30	the	the	DET
ejpam-5499	406	31	labeling	labeling	NOUN
ejpam-5499	406	32	for	for	ADP
ejpam-5499	406	33	the	the	DET
ejpam-5499	406	34	vertices	vertex	NOUN
ejpam-5499	406	35	and	and	CCONJ
ejpam-5499	406	36	edges	edge	NOUN
ejpam-5499	406	37	were	be	AUX
ejpam-5499	406	38	discussed	discuss	VERB
ejpam-5499	406	39	.	.	PUNCT
ejpam-5499	407	1	in	in	ADP
ejpam-5499	407	2	addition	addition	NOUN
ejpam-5499	407	3	,	,	PUNCT
ejpam-5499	407	4	some	some	DET
ejpam-5499	407	5	parameters	parameter	NOUN
ejpam-5499	407	6	such440	such440	PROPN
ejpam-5499	407	7	as	as	ADP
ejpam-5499	407	8	the	the	DET
ejpam-5499	407	9	size	size	NOUN
ejpam-5499	407	10	and	and	CCONJ
ejpam-5499	407	11	order	order	NOUN
ejpam-5499	407	12	were	be	AUX
ejpam-5499	407	13	easily	easily	ADV
ejpam-5499	407	14	shown	show	VERB
ejpam-5499	407	15	using	use	VERB
ejpam-5499	407	16	the	the	DET
ejpam-5499	407	17	construction	construction	NOUN
ejpam-5499	407	18	and	and	CCONJ
ejpam-5499	407	19	other	other	ADJ
ejpam-5499	407	20	existing	exist	VERB
ejpam-5499	407	21	theorems441	theorems441	PROPN
ejpam-5499	407	22	and	and	CCONJ
ejpam-5499	407	23	propositions	proposition	NOUN
ejpam-5499	407	24	.	.	PUNCT
ejpam-5499	408	1	it	it	PRON
ejpam-5499	408	2	is	be	AUX
ejpam-5499	408	3	also	also	ADV
ejpam-5499	408	4	found	find	VERB
ejpam-5499	408	5	that	that	SCONJ
ejpam-5499	408	6	the	the	DET
ejpam-5499	408	7	middle	middle	ADJ
ejpam-5499	408	8	graph	graph	NOUN
ejpam-5499	408	9	m(γcn	m(γcn	PROPN
ejpam-5499	408	10	)	)	PUNCT
ejpam-5499	408	11	is	be	AUX
ejpam-5499	408	12	both	both	PRON
ejpam-5499	408	13	eulerian	eulerian	ADJ
ejpam-5499	408	14	and442	and442	NOUN
ejpam-5499	408	15	hamiltonian	hamiltonian	NOUN
ejpam-5499	408	16	if	if	SCONJ
ejpam-5499	408	17	n	n	NOUN
ejpam-5499	408	18	is	be	AUX
ejpam-5499	408	19	odd	odd	ADJ
ejpam-5499	408	20	.	.	PUNCT
ejpam-5499	409	1	other	other	ADJ
ejpam-5499	409	2	properties	property	NOUN
ejpam-5499	409	3	on	on	ADP
ejpam-5499	409	4	some	some	DET
ejpam-5499	409	5	parameters	parameter	NOUN
ejpam-5499	409	6	is	be	AUX
ejpam-5499	409	7	summarized	summarize	VERB
ejpam-5499	409	8	in	in	ADP
ejpam-5499	409	9	table	table	NOUN
ejpam-5499	409	10	?	?	PUNCT
ejpam-5499	409	11	?	?	PUNCT
ejpam-5499	409	12	.443	.443	NUM
ejpam-5499	409	13	here	here	ADV
ejpam-5499	409	14	is	be	AUX
ejpam-5499	409	15	the	the	DET
ejpam-5499	409	16	table	table	NOUN
ejpam-5499	409	17	for	for	ADP
ejpam-5499	409	18	the	the	DET
ejpam-5499	409	19	properties	property	NOUN
ejpam-5499	409	20	of	of	ADP
ejpam-5499	409	21	m(γcn	m(γcn	PROPN
ejpam-5499	409	22	)	)	PUNCT
ejpam-5499	409	23	and	and	CCONJ
ejpam-5499	409	24	m(γdn	m(γdn	PROPN
ejpam-5499	409	25	)	)	PUNCT
ejpam-5499	409	26	on	on	ADP
ejpam-5499	409	27	some	some	DET
ejpam-5499	409	28	parameters	parameter	NOUN
ejpam-5499	409	29	of	of	ADP
ejpam-5499	409	30	a444	a444	PROPN
ejpam-5499	409	31	graph.445	graph.445	NOUN
ejpam-5499	409	32	j.	j.	PROPN
ejpam-5499	409	33	m.	m.	PROPN
ejpam-5499	409	34	jamis	jamis	PROPN
ejpam-5499	409	35	,	,	PUNCT
ejpam-5499	409	36	d.	d.	PROPN
ejpam-5499	409	37	m.	m.	PROPN
ejpam-5499	409	38	magpantay	magpantay	PROPN
ejpam-5499	409	39	/	/	SYM
ejpam-5499	409	40	eur	eur	PROPN
ejpam-5499	409	41	.	.	PUNCT
ejpam-5499	410	1	j.	j.	PROPN
ejpam-5499	410	2	pure	pure	PROPN
ejpam-5499	410	3	appl	appl	PROPN
ejpam-5499	410	4	.	.	PROPN
ejpam-5499	410	5	math	math	PROPN
ejpam-5499	410	6	,	,	PUNCT
ejpam-5499	410	7	18	18	NUM
ejpam-5499	410	8	(	(	PUNCT
ejpam-5499	410	9	2	2	NUM
ejpam-5499	410	10	)	)	PUNCT
ejpam-5499	410	11	(	(	PUNCT
ejpam-5499	410	12	2025	2025	NUM
ejpam-5499	410	13	)	)	PUNCT
ejpam-5499	410	14	,	,	PUNCT
ejpam-5499	410	15	5499	5499	NUM
ejpam-5499	410	16	18	18	NUM
ejpam-5499	410	17	of	of	ADP
ejpam-5499	410	18	18	18	NUM
ejpam-5499	410	19	parameters	parameter	NOUN
ejpam-5499	410	20	values	value	NOUN
ejpam-5499	410	21	gir(m(γcn	gir(m(γcn	NOUN
ejpam-5499	410	22	)	)	PUNCT
ejpam-5499	410	23	)	)	PUNCT
ejpam-5499	410	24	3	3	NUM
ejpam-5499	410	25	ω[m(γcn	ω[m(γcn	NOUN
ejpam-5499	410	26	)	)	PUNCT
ejpam-5499	410	27	]	]	PUNCT
ejpam-5499	411	1	n	n	PRON
ejpam-5499	411	2	α(m(γcn	α(m(γcn	NUM
ejpam-5499	411	3	)	)	PUNCT
ejpam-5499	411	4	)	)	PUNCT
ejpam-5499	412	1	n	n	PRON
ejpam-5499	412	2	γ(m(γcn	γ(m(γcn	NOUN
ejpam-5499	412	3	)	)	PUNCT
ejpam-5499	412	4	)	)	PUNCT
ejpam-5499	413	1	n+1	n+1	ADV
ejpam-5499	413	2	2	2	NUM
ejpam-5499	413	3	if	if	SCONJ
ejpam-5499	413	4	n	n	NOUN
ejpam-5499	413	5	is	be	AUX
ejpam-5499	413	6	odd	odd	ADJ
ejpam-5499	413	7	and	and	CCONJ
ejpam-5499	413	8	n	n	DET
ejpam-5499	413	9	2	2	NUM
ejpam-5499	413	10	if	if	SCONJ
ejpam-5499	413	11	n	n	PRON
ejpam-5499	413	12	is	be	AUX
ejpam-5499	413	13	even	even	ADV
ejpam-5499	413	14	χ(m(γcn	χ(m(γcn	NOUN
ejpam-5499	413	15	)	)	PUNCT
ejpam-5499	413	16	)	)	PUNCT
ejpam-5499	413	17	n	n	DET
ejpam-5499	413	18	χ′(m(γcn	χ′(m(γcn	NOUN
ejpam-5499	413	19	)	)	PUNCT
ejpam-5499	413	20	)	)	PUNCT
ejpam-5499	414	1	n+	n+	PUNCT
ejpam-5499	414	2	2	2	NUM
ejpam-5499	414	3	if	if	SCONJ
ejpam-5499	414	4	n	n	NOUN
ejpam-5499	414	5	is	be	AUX
ejpam-5499	414	6	odd	odd	ADJ
ejpam-5499	414	7	and	and	CCONJ
ejpam-5499	414	8	n+	n+	NUM
ejpam-5499	414	9	1	1	NUM
ejpam-5499	414	10	if	if	SCONJ
ejpam-5499	414	11	n	n	PRON
ejpam-5499	414	12	is	be	AUX
ejpam-5499	414	13	even	even	ADV
ejpam-5499	414	14	gir(m(γdn	gir(m(γdn	NOUN
ejpam-5499	414	15	)	)	PUNCT
ejpam-5499	414	16	)	)	PUNCT
ejpam-5499	414	17	3	3	NUM
ejpam-5499	414	18	ω[m(γdn	ω[m(γdn	NOUN
ejpam-5499	414	19	)	)	PUNCT
ejpam-5499	414	20	]	]	PUNCT
ejpam-5499	415	1	2n	2n	NUM
ejpam-5499	415	2	α(m(γdn	α(m(γdn	NUM
ejpam-5499	415	3	)	)	PUNCT
ejpam-5499	415	4	)	)	PUNCT
ejpam-5499	415	5	2n	2n	NUM
ejpam-5499	415	6	γ(m(γdn	γ(m(γdn	NUM
ejpam-5499	415	7	)	)	PUNCT
ejpam-5499	415	8	)	)	PUNCT
ejpam-5499	415	9	3n−1	3n−1	NUM
ejpam-5499	415	10	2	2	NUM
ejpam-5499	415	11	if	if	SCONJ
ejpam-5499	415	12	n	n	NOUN
ejpam-5499	415	13	is	be	AUX
ejpam-5499	415	14	odd	odd	ADJ
ejpam-5499	415	15	and	and	CCONJ
ejpam-5499	415	16	3n	3n	NUM
ejpam-5499	415	17	2	2	NUM
ejpam-5499	415	18	if	if	SCONJ
ejpam-5499	415	19	n	n	NOUN
ejpam-5499	415	20	is	be	AUX
ejpam-5499	415	21	even	even	ADV
ejpam-5499	415	22	.	.	PUNCT
ejpam-5499	416	1	χ(m(γdn	χ(m(γdn	PROPN
ejpam-5499	416	2	)	)	PUNCT
ejpam-5499	416	3	)	)	PUNCT
ejpam-5499	416	4	2n	2n	NUM
ejpam-5499	416	5	χ′(m(γdn	χ′(m(γdn	NOUN
ejpam-5499	416	6	)	)	PUNCT
ejpam-5499	416	7	)	)	PUNCT
ejpam-5499	417	1	2n+	2n+	NUM
ejpam-5499	417	2	1	1	NUM
ejpam-5499	417	3	446	446	NUM
ejpam-5499	417	4	the	the	DET
ejpam-5499	417	5	problem	problem	NOUN
ejpam-5499	417	6	on	on	ADP
ejpam-5499	417	7	the	the	DET
ejpam-5499	417	8	identity	identity	NOUN
ejpam-5499	417	9	graphs	graph	NOUN
ejpam-5499	417	10	is	be	AUX
ejpam-5499	417	11	still	still	ADV
ejpam-5499	417	12	open	open	ADJ
ejpam-5499	417	13	.	.	PUNCT
ejpam-5499	418	1	for	for	ADP
ejpam-5499	418	2	instance	instance	NOUN
ejpam-5499	418	3	,	,	PUNCT
ejpam-5499	418	4	the	the	DET
ejpam-5499	418	5	middle	middle	ADJ
ejpam-5499	418	6	graph	graph	NOUN
ejpam-5499	418	7	of447	of447	PROPN
ejpam-5499	418	8	the	the	DET
ejpam-5499	418	9	identity	identity	NOUN
ejpam-5499	418	10	graph	graph	NOUN
ejpam-5499	418	11	of	of	ADP
ejpam-5499	418	12	symmetric	symmetric	ADJ
ejpam-5499	418	13	groups	group	NOUN
ejpam-5499	418	14	is	be	AUX
ejpam-5499	418	15	also	also	ADV
ejpam-5499	418	16	interesting	interesting	ADJ
ejpam-5499	418	17	to	to	ADP
ejpam-5499	418	18	investigate.448	investigate.448	PROPN
ejpam-5499	418	19	references449	references449	PROPN
ejpam-5499	419	1	[	[	X
ejpam-5499	419	2	1	1	X
ejpam-5499	419	3	]	]	PUNCT
ejpam-5499	419	4	w.	w.	PROPN
ejpam-5499	419	5	b.	b.	PROPN
ejpam-5499	420	1	v.	v.	ADP
ejpam-5499	420	2	kandasamy	kandasamy	PROPN
ejpam-5499	420	3	and	and	CCONJ
ejpam-5499	420	4	f.	f.	PROPN
ejpam-5499	420	5	smarandache	smarandache	PROPN
ejpam-5499	420	6	.	.	PUNCT
ejpam-5499	421	1	groups	group	NOUN
ejpam-5499	421	2	as	as	ADP
ejpam-5499	421	3	graphs	graph	NOUN
ejpam-5499	421	4	.	.	PUNCT
ejpam-5499	422	1	editura	editura	NOUN
ejpam-5499	422	2	cuart,450	cuart,450	PROPN
ejpam-5499	422	3	slobozia	slobozia	PROPN
ejpam-5499	422	4	,	,	PUNCT
ejpam-5499	422	5	2009.451	2009.451	NUM
ejpam-5499	423	1	[	[	X
ejpam-5499	423	2	2	2	NUM
ejpam-5499	423	3	]	]	PUNCT
ejpam-5499	423	4	a.	a.	NOUN
ejpam-5499	423	5	d.	d.	PROPN
ejpam-5499	423	6	godase	godase	PROPN
ejpam-5499	423	7	.	.	PUNCT
ejpam-5499	424	1	unit	unit	NOUN
ejpam-5499	424	2	graph	graph	NOUN
ejpam-5499	424	3	of	of	ADP
ejpam-5499	424	4	some	some	DET
ejpam-5499	424	5	finite	finite	ADJ
ejpam-5499	424	6	group	group	NOUN
ejpam-5499	424	7	.	.	PUNCT
ejpam-5499	425	1	international	international	ADJ
ejpam-5499	425	2	journal	journal	PROPN
ejpam-5499	425	3	of	of	ADP
ejpam-5499	425	4	universal452	universal452	PROPN
ejpam-5499	425	5	science	science	NOUN
ejpam-5499	425	6	and	and	CCONJ
ejpam-5499	425	7	technology	technology	NOUN
ejpam-5499	425	8	,	,	PUNCT
ejpam-5499	425	9	1(1):12–18	1(1):12–18	NUM
ejpam-5499	425	10	,	,	PUNCT
ejpam-5499	425	11	2015.453	2015.453	NUM
ejpam-5499	425	12	[	[	SYM
ejpam-5499	425	13	3	3	NUM
ejpam-5499	425	14	]	]	X
ejpam-5499	425	15	n.	n.	PROPN
ejpam-5499	425	16	f.	f.	PROPN
ejpam-5499	425	17	yalcin	yalcin	PROPN
ejpam-5499	425	18	and	and	CCONJ
ejpam-5499	425	19	y.	y.	PROPN
ejpam-5499	425	20	kirgil	kirgil	PROPN
ejpam-5499	425	21	.	.	PUNCT
ejpam-5499	426	1	identity	identity	NOUN
ejpam-5499	426	2	graph	graph	NOUN
ejpam-5499	426	3	of	of	ADP
ejpam-5499	426	4	finite	finite	ADJ
ejpam-5499	426	5	cyclic	cyclic	ADJ
ejpam-5499	426	6	groups	group	NOUN
ejpam-5499	426	7	.	.	PUNCT
ejpam-5499	427	1	journal	journal	PROPN
ejpam-5499	427	2	of	of	ADP
ejpam-5499	427	3	balikesir454	balikesir454	PROPN
ejpam-5499	427	4	university	university	PROPN
ejpam-5499	427	5	institute	institute	PROPN
ejpam-5499	427	6	of	of	ADP
ejpam-5499	427	7	science	science	NOUN
ejpam-5499	427	8	and	and	CCONJ
ejpam-5499	427	9	technology	technology	NOUN
ejpam-5499	427	10	,	,	PUNCT
ejpam-5499	427	11	21(2):614–624	21(2):614–624	NOUN
ejpam-5499	427	12	,	,	PUNCT
ejpam-5499	427	13	2019.455	2019.455	PROPN
ejpam-5499	427	14	[	[	X
ejpam-5499	427	15	4	4	X
ejpam-5499	427	16	]	]	PUNCT
ejpam-5499	427	17	j.	j.	PROPN
ejpam-5499	427	18	u.	u.	PROPN
ejpam-5499	427	19	jeeshma	jeeshma	PROPN
ejpam-5499	427	20	.	.	PUNCT
ejpam-5499	428	1	coloring	color	VERB
ejpam-5499	428	2	for	for	ADP
ejpam-5499	428	3	the	the	DET
ejpam-5499	428	4	identity	identity	NOUN
ejpam-5499	428	5	graphs	graph	NOUN
ejpam-5499	428	6	of	of	ADP
ejpam-5499	428	7	groups	group	NOUN
ejpam-5499	428	8	.	.	PUNCT
ejpam-5499	429	1	international	international	ADJ
ejpam-5499	429	2	research456	research456	PROPN
ejpam-5499	429	3	journal	journal	NOUN
ejpam-5499	429	4	of	of	ADP
ejpam-5499	429	5	engineering	engineering	NOUN
ejpam-5499	429	6	and	and	CCONJ
ejpam-5499	429	7	technology	technology	NOUN
ejpam-5499	429	8	,	,	PUNCT
ejpam-5499	429	9	7(7):3459–3462	7(7):3459–3462	NUM
ejpam-5499	429	10	,	,	PUNCT
ejpam-5499	429	11	2020.457	2020.457	NUM
ejpam-5499	430	1	[	[	X
ejpam-5499	430	2	5	5	X
ejpam-5499	430	3	]	]	PUNCT
ejpam-5499	430	4	j.	j.	PROPN
ejpam-5499	430	5	akiyama	akiyama	PROPN
ejpam-5499	430	6	,	,	PUNCT
ejpam-5499	430	7	t.	t.	PROPN
ejpam-5499	430	8	hamada	hamada	PROPN
ejpam-5499	430	9	,	,	PUNCT
ejpam-5499	430	10	and	and	CCONJ
ejpam-5499	430	11	i.	i.	PROPN
ejpam-5499	430	12	yoshimora	yoshimora	PROPN
ejpam-5499	430	13	.	.	PUNCT
ejpam-5499	431	1	on	on	ADP
ejpam-5499	431	2	characterization	characterization	NOUN
ejpam-5499	431	3	of	of	ADP
ejpam-5499	431	4	the	the	DET
ejpam-5499	431	5	middle	middle	ADJ
ejpam-5499	431	6	graphs.458	graphs.458	PROPN
ejpam-5499	431	7	https://www.researchgate.net/publication/269002499	https://www.researchgate.net/publication/269002499	NOUN
ejpam-5499	431	8	,	,	PUNCT
ejpam-5499	431	9	1975.459	1975.459	PROPN
ejpam-5499	432	1	[	[	X
ejpam-5499	432	2	6	6	NUM
ejpam-5499	432	3	]	]	X
ejpam-5499	432	4	n.	n.	PROPN
ejpam-5499	432	5	murugesan	murugesan	PROPN
ejpam-5499	432	6	and	and	CCONJ
ejpam-5499	432	7	d.	d.	PROPN
ejpam-5499	432	8	s.	s.	PROPN
ejpam-5499	432	9	nair	nair	PROPN
ejpam-5499	432	10	.	.	PUNCT
ejpam-5499	433	1	power	power	NOUN
ejpam-5499	433	2	domination	domination	NOUN
ejpam-5499	433	3	of	of	ADP
ejpam-5499	433	4	middle	middle	ADJ
ejpam-5499	433	5	graph	graph	NOUN
ejpam-5499	433	6	of	of	ADP
ejpam-5499	433	7	central	central	ADJ
ejpam-5499	433	8	graph460	graph460	PROPN
ejpam-5499	433	9	of	of	ADP
ejpam-5499	433	10	path	path	NOUN
ejpam-5499	433	11	,	,	PUNCT
ejpam-5499	433	12	cycle	cycle	NOUN
ejpam-5499	433	13	,	,	PUNCT
ejpam-5499	433	14	and	and	CCONJ
ejpam-5499	433	15	star	star	NOUN
ejpam-5499	433	16	.	.	PUNCT
ejpam-5499	434	1	international	international	ADJ
ejpam-5499	434	2	journal	journal	PROPN
ejpam-5499	434	3	of	of	ADP
ejpam-5499	434	4	pure	pure	ADJ
ejpam-5499	434	5	and	and	CCONJ
ejpam-5499	434	6	applied	apply	VERB
ejpam-5499	434	7	mathematics,461	mathematics,461	ADJ
ejpam-5499	434	8	115(6):121–126	115(6):121–126	NOUN
ejpam-5499	434	9	,	,	PUNCT
ejpam-5499	434	10	2017.462	2017.462	PROPN
ejpam-5499	435	1	[	[	X
ejpam-5499	435	2	7	7	X
ejpam-5499	435	3	]	]	X
ejpam-5499	435	4	c.	c.	PROPN
ejpam-5499	435	5	alib	alib	PROPN
ejpam-5499	435	6	and	and	CCONJ
ejpam-5499	435	7	d.	d.	PROPN
ejpam-5499	435	8	magpantay	magpantay	PROPN
ejpam-5499	435	9	.	.	PUNCT
ejpam-5499	436	1	on	on	ADP
ejpam-5499	436	2	some	some	DET
ejpam-5499	436	3	parameters	parameter	NOUN
ejpam-5499	436	4	of	of	ADP
ejpam-5499	436	5	the	the	DET
ejpam-5499	436	6	central	central	ADJ
ejpam-5499	436	7	graphs	graph	NOUN
ejpam-5499	436	8	of	of	ADP
ejpam-5499	436	9	the	the	DET
ejpam-5499	436	10	identity463	identity463	PROPN
ejpam-5499	436	11	graphs	graph	NOUN
ejpam-5499	436	12	of	of	ADP
ejpam-5499	436	13	finite	finite	ADJ
ejpam-5499	436	14	cyclic	cyclic	ADJ
ejpam-5499	436	15	groups	group	NOUN
ejpam-5499	436	16	.	.	PUNCT
ejpam-5499	437	1	european	european	ADJ
ejpam-5499	437	2	journal	journal	PROPN
ejpam-5499	437	3	of	of	ADP
ejpam-5499	437	4	pure	pure	ADJ
ejpam-5499	437	5	and	and	CCONJ
ejpam-5499	437	6	applied	apply	VERB
ejpam-5499	437	7	mathematics,464	mathematics,464	PROPN
ejpam-5499	437	8	15(4):1888–1904	15(4):1888–1904	NUM
ejpam-5499	437	9	,	,	PUNCT
ejpam-5499	437	10	2022.465	2022.465	NUM
ejpam-5499	437	11	[	[	X
ejpam-5499	437	12	8	8	NUM
ejpam-5499	437	13	]	]	X
ejpam-5499	437	14	g.	g.	PROPN
ejpam-5499	437	15	chartrand	chartrand	PROPN
ejpam-5499	437	16	and	and	CCONJ
ejpam-5499	437	17	p.	p.	PROPN
ejpam-5499	437	18	zhang	zhang	PROPN
ejpam-5499	437	19	.	.	PUNCT
ejpam-5499	438	1	a	a	DET
ejpam-5499	438	2	first	first	ADJ
ejpam-5499	438	3	course	course	NOUN
ejpam-5499	438	4	in	in	ADP
ejpam-5499	438	5	graph	graph	NOUN
ejpam-5499	438	6	theory	theory	NOUN
ejpam-5499	438	7	.	.	PUNCT
ejpam-5499	439	1	dover	dover	PROPN
ejpam-5499	439	2	publications,466	publications,466	PROPN
ejpam-5499	439	3	mineola	mineola	PROPN
ejpam-5499	439	4	,	,	PUNCT
ejpam-5499	439	5	ny	ny	PROPN
ejpam-5499	439	6	,	,	PUNCT
ejpam-5499	439	7	2012.467	2012.467	NUM
ejpam-5499	439	8	[	[	X
ejpam-5499	439	9	9	9	NUM
ejpam-5499	439	10	]	]	X
ejpam-5499	439	11	w.	w.	PROPN
ejpam-5499	439	12	somnuek	somnuek	PROPN
ejpam-5499	439	13	.	.	PUNCT
ejpam-5499	440	1	counting	count	VERB
ejpam-5499	440	2	lines	line	NOUN
ejpam-5499	440	3	and	and	CCONJ
ejpam-5499	440	4	triangle	triangle	VERB
ejpam-5499	440	5	in	in	ADP
ejpam-5499	440	6	a	a	DET
ejpam-5499	440	7	unit	unit	NOUN
ejpam-5499	440	8	graph	graph	NOUN
ejpam-5499	440	9	.	.	PUNCT
ejpam-5499	441	1	current	current	ADJ
ejpam-5499	441	2	applied	apply	VERB
ejpam-5499	441	3	science468	science468	PROPN
ejpam-5499	441	4	and	and	CCONJ
ejpam-5499	441	5	technology	technology	NOUN
ejpam-5499	441	6	,	,	PUNCT
ejpam-5499	441	7	18(2):148–155	18(2):148–155	NUM
ejpam-5499	441	8	,	,	PUNCT
ejpam-5499	441	9	2018.469	2018.469	NUM