id	sid	tid	token	lemma	pos
ejpam-3780	1	1	european	european	PROPN
ejpam-3780	1	2	journal	journal	PROPN
ejpam-3780	1	3	of	of	ADP
ejpam-3780	1	4	pure	pure	ADJ
ejpam-3780	1	5	and	and	CCONJ
ejpam-3780	1	6	applied	apply	VERB
ejpam-3780	1	7	mathematics	mathematic	NOUN
ejpam-3780	1	8	vol	vol	NOUN
ejpam-3780	1	9	.	.	PROPN
ejpam-3780	2	1	13	13	NUM
ejpam-3780	2	2	,	,	PUNCT
ejpam-3780	2	3	no	no	INTJ
ejpam-3780	2	4	.	.	NOUN
ejpam-3780	2	5	3	3	NUM
ejpam-3780	2	6	,	,	PUNCT
ejpam-3780	2	7	2020	2020	NUM
ejpam-3780	2	8	,	,	PUNCT
ejpam-3780	2	9	674	674	NUM
ejpam-3780	2	10	-	-	SYM
ejpam-3780	2	11	696	696	NUM
ejpam-3780	2	12	issn	issn	PROPN
ejpam-3780	2	13	1307	1307	NUM
ejpam-3780	2	14	-	-	SYM
ejpam-3780	2	15	5543	5543	NUM
ejpam-3780	2	16	–	–	PUNCT
ejpam-3780	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3780	2	18	published	publish	VERB
ejpam-3780	2	19	by	by	ADP
ejpam-3780	2	20	new	new	PROPN
ejpam-3780	2	21	york	york	PROPN
ejpam-3780	2	22	business	business	PROPN
ejpam-3780	2	23	global	global	ADJ
ejpam-3780	2	24	efficient	efficient	ADJ
ejpam-3780	2	25	zero	zero	NUM
ejpam-3780	2	26	ring	ring	NOUN
ejpam-3780	2	27	labeling	labeling	NOUN
ejpam-3780	2	28	of	of	ADP
ejpam-3780	2	29	graphs	graph	NOUN
ejpam-3780	2	30	dhenmar	dhenmar	PROPN
ejpam-3780	2	31	e.	e.	PROPN
ejpam-3780	2	32	chua1,2,∗	chua1,2,∗	PROPN
ejpam-3780	2	33	,	,	PUNCT
ejpam-3780	2	34	francis	francis	PROPN
ejpam-3780	2	35	joseph	joseph	PROPN
ejpam-3780	2	36	h.	h.	PROPN
ejpam-3780	2	37	campeña1	campeña1	PROPN
ejpam-3780	2	38	,	,	PUNCT
ejpam-3780	2	39	floresto	floresto	PROPN
ejpam-3780	2	40	a.	a.	NOUN
ejpam-3780	2	41	franco	franco	PROPN
ejpam-3780	2	42	jr.1,3	jr.1,3	PROPN
ejpam-3780	2	43	1	1	NUM
ejpam-3780	2	44	mathematics	mathematic	NOUN
ejpam-3780	2	45	and	and	CCONJ
ejpam-3780	2	46	statistics	statistics	PROPN
ejpam-3780	2	47	department	department	PROPN
ejpam-3780	2	48	,	,	PUNCT
ejpam-3780	2	49	de	de	PROPN
ejpam-3780	2	50	la	la	X
ejpam-3780	2	51	salle	salle	PROPN
ejpam-3780	2	52	university	university	PROPN
ejpam-3780	2	53	,	,	PUNCT
ejpam-3780	2	54	manila	manila	PROPN
ejpam-3780	2	55	,	,	PUNCT
ejpam-3780	2	56	philippines	philippine	NOUN
ejpam-3780	2	57	2	2	NUM
ejpam-3780	2	58	mathematics	mathematic	NOUN
ejpam-3780	2	59	and	and	CCONJ
ejpam-3780	2	60	physics	physics	PROPN
ejpam-3780	2	61	department	department	PROPN
ejpam-3780	2	62	,	,	PUNCT
ejpam-3780	2	63	adamson	adamson	PROPN
ejpam-3780	2	64	university	university	PROPN
ejpam-3780	2	65	,	,	PUNCT
ejpam-3780	2	66	manila	manila	PROPN
ejpam-3780	2	67	,	,	PUNCT
ejpam-3780	2	68	philippines	philippine	NOUN
ejpam-3780	2	69	3	3	NUM
ejpam-3780	2	70	mathematics	mathematics	PROPN
ejpam-3780	2	71	department	department	NOUN
ejpam-3780	2	72	,	,	PUNCT
ejpam-3780	2	73	mariano	mariano	PROPN
ejpam-3780	2	74	marcos	marcos	PROPN
ejpam-3780	2	75	state	state	PROPN
ejpam-3780	2	76	university	university	PROPN
ejpam-3780	2	77	,	,	PUNCT
ejpam-3780	2	78	batac	batac	NOUN
ejpam-3780	2	79	,	,	PUNCT
ejpam-3780	2	80	ilocos	ilocos	PROPN
ejpam-3780	2	81	norte	norte	NOUN
ejpam-3780	2	82	,	,	PUNCT
ejpam-3780	2	83	philippines	philippine	NOUN
ejpam-3780	2	84	abstract	abstract	ADJ
ejpam-3780	2	85	.	.	PUNCT
ejpam-3780	3	1	a	a	DET
ejpam-3780	3	2	zero	zero	NUM
ejpam-3780	3	3	ring	ring	NOUN
ejpam-3780	3	4	is	be	AUX
ejpam-3780	3	5	a	a	DET
ejpam-3780	3	6	ring	ring	NOUN
ejpam-3780	3	7	in	in	ADP
ejpam-3780	3	8	which	which	PRON
ejpam-3780	3	9	the	the	DET
ejpam-3780	3	10	product	product	NOUN
ejpam-3780	3	11	of	of	ADP
ejpam-3780	3	12	any	any	DET
ejpam-3780	3	13	two	two	NUM
ejpam-3780	3	14	elements	element	NOUN
ejpam-3780	3	15	is	be	AUX
ejpam-3780	3	16	zero	zero	NUM
ejpam-3780	3	17	,	,	PUNCT
ejpam-3780	3	18	which	which	PRON
ejpam-3780	3	19	is	be	AUX
ejpam-3780	3	20	the	the	DET
ejpam-3780	3	21	additive	additive	ADJ
ejpam-3780	3	22	identity	identity	NOUN
ejpam-3780	3	23	.	.	PUNCT
ejpam-3780	4	1	a	a	DET
ejpam-3780	4	2	zero	zero	NUM
ejpam-3780	4	3	ring	ring	NOUN
ejpam-3780	4	4	labeling	labeling	NOUN
ejpam-3780	4	5	of	of	ADP
ejpam-3780	4	6	a	a	DET
ejpam-3780	4	7	graph	graph	NOUN
ejpam-3780	4	8	is	be	AUX
ejpam-3780	4	9	an	an	DET
ejpam-3780	4	10	assignment	assignment	NOUN
ejpam-3780	4	11	of	of	ADP
ejpam-3780	4	12	distinct	distinct	ADJ
ejpam-3780	4	13	elements	element	NOUN
ejpam-3780	4	14	of	of	ADP
ejpam-3780	4	15	a	a	DET
ejpam-3780	4	16	zero	zero	NUM
ejpam-3780	4	17	ring	ring	NOUN
ejpam-3780	4	18	to	to	ADP
ejpam-3780	4	19	the	the	DET
ejpam-3780	4	20	vertices	vertex	NOUN
ejpam-3780	4	21	of	of	ADP
ejpam-3780	4	22	the	the	DET
ejpam-3780	4	23	graph	graph	NOUN
ejpam-3780	4	24	such	such	ADJ
ejpam-3780	4	25	that	that	SCONJ
ejpam-3780	4	26	the	the	DET
ejpam-3780	4	27	sum	sum	NOUN
ejpam-3780	4	28	of	of	ADP
ejpam-3780	4	29	the	the	DET
ejpam-3780	4	30	labels	label	NOUN
ejpam-3780	4	31	of	of	ADP
ejpam-3780	4	32	any	any	DET
ejpam-3780	4	33	two	two	NUM
ejpam-3780	4	34	adjacent	adjacent	ADJ
ejpam-3780	4	35	vertices	vertex	NOUN
ejpam-3780	4	36	is	be	AUX
ejpam-3780	4	37	not	not	PART
ejpam-3780	4	38	the	the	DET
ejpam-3780	4	39	zero	zero	NUM
ejpam-3780	4	40	element	element	NOUN
ejpam-3780	4	41	in	in	ADP
ejpam-3780	4	42	the	the	DET
ejpam-3780	4	43	ring	ring	NOUN
ejpam-3780	4	44	.	.	PUNCT
ejpam-3780	5	1	given	give	VERB
ejpam-3780	5	2	a	a	DET
ejpam-3780	5	3	zero	zero	NUM
ejpam-3780	5	4	ring	ring	NOUN
ejpam-3780	5	5	labeling	labeling	NOUN
ejpam-3780	5	6	of	of	ADP
ejpam-3780	5	7	a	a	DET
ejpam-3780	5	8	graph	graph	NOUN
ejpam-3780	5	9	,	,	PUNCT
ejpam-3780	5	10	if	if	SCONJ
ejpam-3780	5	11	the	the	DET
ejpam-3780	5	12	cardinality	cardinality	NOUN
ejpam-3780	5	13	of	of	ADP
ejpam-3780	5	14	the	the	DET
ejpam-3780	5	15	set	set	NOUN
ejpam-3780	5	16	of	of	ADP
ejpam-3780	5	17	distinct	distinct	ADJ
ejpam-3780	5	18	sums	sum	NOUN
ejpam-3780	5	19	obtained	obtain	VERB
ejpam-3780	5	20	from	from	ADP
ejpam-3780	5	21	all	all	DET
ejpam-3780	5	22	adjacent	adjacent	ADJ
ejpam-3780	5	23	vertices	vertex	NOUN
ejpam-3780	5	24	is	be	AUX
ejpam-3780	5	25	equal	equal	ADJ
ejpam-3780	5	26	to	to	ADP
ejpam-3780	5	27	the	the	DET
ejpam-3780	5	28	maximum	maximum	ADJ
ejpam-3780	5	29	degree	degree	NOUN
ejpam-3780	5	30	of	of	ADP
ejpam-3780	5	31	the	the	DET
ejpam-3780	5	32	graph	graph	NOUN
ejpam-3780	5	33	,	,	PUNCT
ejpam-3780	5	34	then	then	ADV
ejpam-3780	5	35	the	the	DET
ejpam-3780	5	36	zero	zero	NUM
ejpam-3780	5	37	ring	ring	NOUN
ejpam-3780	5	38	labeling	labeling	NOUN
ejpam-3780	5	39	is	be	AUX
ejpam-3780	5	40	efficient	efficient	ADJ
ejpam-3780	5	41	.	.	PUNCT
ejpam-3780	6	1	in	in	ADP
ejpam-3780	6	2	this	this	DET
ejpam-3780	6	3	paper	paper	NOUN
ejpam-3780	6	4	,	,	PUNCT
ejpam-3780	6	5	we	we	PRON
ejpam-3780	6	6	showed	show	VERB
ejpam-3780	6	7	the	the	DET
ejpam-3780	6	8	existence	existence	NOUN
ejpam-3780	6	9	of	of	ADP
ejpam-3780	6	10	an	an	DET
ejpam-3780	6	11	efficient	efficient	ADJ
ejpam-3780	6	12	zero	zero	NUM
ejpam-3780	6	13	ring	ring	NOUN
ejpam-3780	6	14	labeling	labeling	NOUN
ejpam-3780	6	15	for	for	ADP
ejpam-3780	6	16	some	some	DET
ejpam-3780	6	17	classes	class	NOUN
ejpam-3780	6	18	of	of	ADP
ejpam-3780	6	19	trees	tree	NOUN
ejpam-3780	6	20	and	and	CCONJ
ejpam-3780	6	21	their	their	PRON
ejpam-3780	6	22	disjoint	disjoint	NOUN
ejpam-3780	6	23	union	union	NOUN
ejpam-3780	6	24	.	.	PUNCT
ejpam-3780	7	1	in	in	ADP
ejpam-3780	7	2	particular	particular	ADJ
ejpam-3780	7	3	,	,	PUNCT
ejpam-3780	7	4	we	we	PRON
ejpam-3780	7	5	showed	show	VERB
ejpam-3780	7	6	that	that	SCONJ
ejpam-3780	7	7	an	an	DET
ejpam-3780	7	8	efficient	efficient	ADJ
ejpam-3780	7	9	zero	zero	NUM
ejpam-3780	7	10	ring	ring	NOUN
ejpam-3780	7	11	labeling	labeling	NOUN
ejpam-3780	7	12	exists	exist	VERB
ejpam-3780	7	13	for	for	ADP
ejpam-3780	7	14	some	some	DET
ejpam-3780	7	15	families	family	NOUN
ejpam-3780	7	16	of	of	ADP
ejpam-3780	7	17	the	the	DET
ejpam-3780	7	18	following	follow	VERB
ejpam-3780	7	19	classes	class	NOUN
ejpam-3780	7	20	of	of	ADP
ejpam-3780	7	21	trees	tree	NOUN
ejpam-3780	7	22	:	:	PUNCT
ejpam-3780	7	23	path	path	NOUN
ejpam-3780	7	24	graphs	graph	NOUN
ejpam-3780	7	25	,	,	PUNCT
ejpam-3780	7	26	star	star	NOUN
ejpam-3780	7	27	graphs	graph	NOUN
ejpam-3780	7	28	,	,	PUNCT
ejpam-3780	7	29	bistars	bistar	NOUN
ejpam-3780	7	30	,	,	PUNCT
ejpam-3780	7	31	centipede	centipede	NOUN
ejpam-3780	7	32	graphs	graph	NOUN
ejpam-3780	7	33	,	,	PUNCT
ejpam-3780	7	34	caterpillars	caterpillar	NOUN
ejpam-3780	7	35	,	,	PUNCT
ejpam-3780	7	36	spiders	spider	NOUN
ejpam-3780	7	37	,	,	PUNCT
ejpam-3780	7	38	lobsters	lobster	NOUN
ejpam-3780	7	39	,	,	PUNCT
ejpam-3780	7	40	and	and	CCONJ
ejpam-3780	7	41	rooted	rooted	ADJ
ejpam-3780	7	42	trees	tree	NOUN
ejpam-3780	7	43	.	.	PUNCT
ejpam-3780	8	1	we	we	PRON
ejpam-3780	8	2	also	also	ADV
ejpam-3780	8	3	showed	show	VERB
ejpam-3780	8	4	results	result	NOUN
ejpam-3780	8	5	for	for	ADP
ejpam-3780	8	6	other	other	ADJ
ejpam-3780	8	7	common	common	ADJ
ejpam-3780	8	8	classes	class	NOUN
ejpam-3780	8	9	of	of	ADP
ejpam-3780	8	10	graphs	graph	NOUN
ejpam-3780	8	11	.	.	PUNCT
ejpam-3780	9	1	2020	2020	NUM
ejpam-3780	9	2	mathematics	mathematic	NOUN
ejpam-3780	9	3	subject	subject	NOUN
ejpam-3780	9	4	classifications	classification	NOUN
ejpam-3780	9	5	:	:	PUNCT
ejpam-3780	9	6	05c05	05c05	NUM
ejpam-3780	9	7	,	,	PUNCT
ejpam-3780	9	8	05c25	05c25	NUM
ejpam-3780	9	9	,	,	PUNCT
ejpam-3780	9	10	05c78	05c78	NUM
ejpam-3780	9	11	key	key	ADJ
ejpam-3780	9	12	words	word	NOUN
ejpam-3780	9	13	and	and	CCONJ
ejpam-3780	9	14	phrases	phrase	NOUN
ejpam-3780	9	15	:	:	PUNCT
ejpam-3780	9	16	efficient	efficient	ADJ
ejpam-3780	9	17	zero	zero	NUM
ejpam-3780	9	18	ring	ring	NOUN
ejpam-3780	9	19	labeling	labeling	NOUN
ejpam-3780	9	20	,	,	PUNCT
ejpam-3780	9	21	zero	zero	NUM
ejpam-3780	9	22	ring	ring	NOUN
ejpam-3780	9	23	labeling	labeling	NOUN
ejpam-3780	9	24	,	,	PUNCT
ejpam-3780	9	25	zero	zero	NUM
ejpam-3780	9	26	ring	ring	NOUN
ejpam-3780	9	27	1	1	NUM
ejpam-3780	9	28	.	.	PUNCT
ejpam-3780	10	1	introduction	introduction	NOUN
ejpam-3780	10	2	graph	graph	NOUN
ejpam-3780	10	3	labeling	labeling	NOUN
ejpam-3780	10	4	is	be	AUX
ejpam-3780	10	5	an	an	DET
ejpam-3780	10	6	assignment	assignment	NOUN
ejpam-3780	10	7	of	of	ADP
ejpam-3780	10	8	labels	label	NOUN
ejpam-3780	10	9	to	to	PART
ejpam-3780	10	10	vertices	vertex	NOUN
ejpam-3780	10	11	or	or	CCONJ
ejpam-3780	10	12	edges	edge	NOUN
ejpam-3780	10	13	of	of	ADP
ejpam-3780	10	14	a	a	DET
ejpam-3780	10	15	graph	graph	NOUN
ejpam-3780	10	16	.	.	PUNCT
ejpam-3780	11	1	interesting	interesting	ADJ
ejpam-3780	11	2	questions	question	NOUN
ejpam-3780	11	3	naturally	naturally	ADV
ejpam-3780	11	4	arise	arise	VERB
ejpam-3780	11	5	from	from	ADP
ejpam-3780	11	6	this	this	DET
ejpam-3780	11	7	topic	topic	NOUN
ejpam-3780	11	8	,	,	PUNCT
ejpam-3780	11	9	so	so	SCONJ
ejpam-3780	11	10	there	there	PRON
ejpam-3780	11	11	has	have	AUX
ejpam-3780	11	12	been	be	AUX
ejpam-3780	11	13	a	a	DET
ejpam-3780	11	14	vast	vast	ADJ
ejpam-3780	11	15	amount	amount	NOUN
ejpam-3780	11	16	of	of	ADP
ejpam-3780	11	17	literature	literature	NOUN
ejpam-3780	11	18	that	that	PRON
ejpam-3780	11	19	aimed	aim	VERB
ejpam-3780	11	20	to	to	PART
ejpam-3780	11	21	answer	answer	VERB
ejpam-3780	11	22	these	these	DET
ejpam-3780	11	23	questions	question	NOUN
ejpam-3780	11	24	while	while	SCONJ
ejpam-3780	11	25	consequently	consequently	ADV
ejpam-3780	11	26	introducing	introduce	VERB
ejpam-3780	11	27	new	new	ADJ
ejpam-3780	11	28	problems	problem	NOUN
ejpam-3780	11	29	.	.	PUNCT
ejpam-3780	12	1	in	in	ADP
ejpam-3780	12	2	2014	2014	NUM
ejpam-3780	12	3	,	,	PUNCT
ejpam-3780	12	4	acharya	acharya	PROPN
ejpam-3780	12	5	et	et	PROPN
ejpam-3780	12	6	al	al	PROPN
ejpam-3780	13	1	[	[	X
ejpam-3780	13	2	1	1	X
ejpam-3780	13	3	]	]	PUNCT
ejpam-3780	13	4	introduced	introduce	VERB
ejpam-3780	13	5	zero	zero	NUM
ejpam-3780	13	6	ring	ring	NOUN
ejpam-3780	13	7	labeling	labeling	NOUN
ejpam-3780	13	8	.	.	PUNCT
ejpam-3780	14	1	in	in	ADP
ejpam-3780	14	2	this	this	DET
ejpam-3780	14	3	labeling	labeling	NOUN
ejpam-3780	14	4	,	,	PUNCT
ejpam-3780	14	5	each	each	DET
ejpam-3780	14	6	vertex	vertex	NOUN
ejpam-3780	14	7	is	be	AUX
ejpam-3780	14	8	assigned	assign	VERB
ejpam-3780	14	9	a	a	DET
ejpam-3780	14	10	unique	unique	ADJ
ejpam-3780	14	11	label	label	NOUN
ejpam-3780	14	12	from	from	ADP
ejpam-3780	14	13	a	a	DET
ejpam-3780	14	14	zero	zero	NUM
ejpam-3780	14	15	ring	ring	NOUN
ejpam-3780	14	16	such	such	ADJ
ejpam-3780	14	17	that	that	SCONJ
ejpam-3780	14	18	the	the	DET
ejpam-3780	14	19	sum	sum	NOUN
ejpam-3780	14	20	of	of	ADP
ejpam-3780	14	21	any	any	DET
ejpam-3780	14	22	two	two	NUM
ejpam-3780	14	23	adjacent	adjacent	ADJ
ejpam-3780	14	24	vertices	vertex	NOUN
ejpam-3780	14	25	is	be	AUX
ejpam-3780	14	26	not	not	PART
ejpam-3780	14	27	zero	zero	NUM
ejpam-3780	14	28	,	,	PUNCT
ejpam-3780	14	29	i.e.	i.e.	X
ejpam-3780	14	30	,	,	PUNCT
ejpam-3780	14	31	the	the	DET
ejpam-3780	14	32	additive	additive	ADJ
ejpam-3780	14	33	identity	identity	NOUN
ejpam-3780	14	34	of	of	ADP
ejpam-3780	14	35	the	the	DET
ejpam-3780	14	36	zero	zero	NUM
ejpam-3780	14	37	ring	ring	NOUN
ejpam-3780	14	38	.	.	PUNCT
ejpam-3780	15	1	it	it	PRON
ejpam-3780	15	2	was	be	AUX
ejpam-3780	15	3	proved	prove	VERB
ejpam-3780	15	4	that	that	SCONJ
ejpam-3780	15	5	every	every	DET
ejpam-3780	15	6	graph	graph	NOUN
ejpam-3780	15	7	admits	admit	VERB
ejpam-3780	15	8	a	a	DET
ejpam-3780	15	9	zero	zero	NUM
ejpam-3780	15	10	ring	ring	NOUN
ejpam-3780	15	11	labeling	labeling	NOUN
ejpam-3780	15	12	with	with	ADP
ejpam-3780	15	13	respect	respect	NOUN
ejpam-3780	15	14	to	to	ADP
ejpam-3780	15	15	some	some	DET
ejpam-3780	15	16	zero	zero	NUM
ejpam-3780	15	17	ring	ring	NOUN
ejpam-3780	15	18	.	.	PUNCT
ejpam-3780	16	1	the	the	DET
ejpam-3780	16	2	zero	zero	NUM
ejpam-3780	16	3	ring	ring	NOUN
ejpam-3780	16	4	index	index	NOUN
ejpam-3780	16	5	of	of	ADP
ejpam-3780	16	6	a	a	DET
ejpam-3780	16	7	graph	graph	NOUN
ejpam-3780	16	8	,	,	PUNCT
ejpam-3780	16	9	which	which	PRON
ejpam-3780	16	10	is	be	AUX
ejpam-3780	16	11	the	the	DET
ejpam-3780	16	12	smallest	small	ADJ
ejpam-3780	16	13	order	order	NOUN
ejpam-3780	16	14	of	of	ADP
ejpam-3780	16	15	a	a	DET
ejpam-3780	16	16	zero	zero	NUM
ejpam-3780	16	17	ring	ring	NOUN
ejpam-3780	16	18	in	in	ADP
ejpam-3780	16	19	which	which	PRON
ejpam-3780	16	20	the	the	DET
ejpam-3780	16	21	graph	graph	NOUN
ejpam-3780	16	22	admits	admit	VERB
ejpam-3780	16	23	a	a	DET
ejpam-3780	16	24	zero	zero	NUM
ejpam-3780	16	25	ring	ring	NOUN
ejpam-3780	16	26	labeling	labeling	NOUN
ejpam-3780	16	27	,	,	PUNCT
ejpam-3780	16	28	was	be	AUX
ejpam-3780	16	29	also	also	ADV
ejpam-3780	16	30	studied	study	VERB
ejpam-3780	16	31	for	for	ADP
ejpam-3780	16	32	some	some	DET
ejpam-3780	16	33	well	well	ADV
ejpam-3780	16	34	-	-	PUNCT
ejpam-3780	16	35	known	know	VERB
ejpam-3780	16	36	graphs	graph	NOUN
ejpam-3780	16	37	.	.	PUNCT
ejpam-3780	17	1	pranjali	pranjali	PROPN
ejpam-3780	17	2	et	et	PROPN
ejpam-3780	17	3	al	al	PROPN
ejpam-3780	18	1	[	[	X
ejpam-3780	18	2	6	6	NUM
ejpam-3780	18	3	]	]	PUNCT
ejpam-3780	18	4	determined	determine	VERB
ejpam-3780	18	5	a	a	DET
ejpam-3780	18	6	necessary	necessary	ADJ
ejpam-3780	18	7	and	and	CCONJ
ejpam-3780	18	8	sufficient	sufficient	ADJ
ejpam-3780	18	9	condition	condition	NOUN
ejpam-3780	18	10	for	for	ADP
ejpam-3780	18	11	a	a	DET
ejpam-3780	18	12	finite	finite	ADJ
ejpam-3780	18	13	graph	graph	NOUN
ejpam-3780	18	14	of	of	ADP
ejpam-3780	18	15	order	order	NOUN
ejpam-3780	18	16	n	n	PRON
ejpam-3780	18	17	to	to	PART
ejpam-3780	18	18	attain	attain	VERB
ejpam-3780	18	19	an	an	DET
ejpam-3780	18	20	optimal	optimal	ADJ
ejpam-3780	18	21	zero	zero	NUM
ejpam-3780	18	22	ring	ring	NOUN
ejpam-3780	18	23	index	index	NOUN
ejpam-3780	18	24	of	of	ADP
ejpam-3780	18	25	n.	n.	NOUN
ejpam-3780	18	26	∗corresponding	∗corresponde	VERB
ejpam-3780	18	27	author	author	NOUN
ejpam-3780	18	28	.	.	PUNCT
ejpam-3780	19	1	doi	doi	NOUN
ejpam-3780	19	2	:	:	PUNCT
ejpam-3780	19	3	https://doi.org/10.29020/nybg.ejpam.v13i3.3780	https://doi.org/10.29020/nybg.ejpam.v13i3.3780	ADJ
ejpam-3780	19	4	email	email	NOUN
ejpam-3780	19	5	addresses	address	NOUN
ejpam-3780	19	6	:	:	PUNCT
ejpam-3780	19	7	chuadhenmar@gmail.com	chuadhenmar@gmail.com	X
ejpam-3780	19	8	(	(	PUNCT
ejpam-3780	19	9	d.	d.	PROPN
ejpam-3780	19	10	chua	chua	PROPN
ejpam-3780	19	11	)	)	PUNCT
ejpam-3780	19	12	,	,	PUNCT
ejpam-3780	19	13	francis.campena@dlsu.edu.ph	francis.campena@dlsu.edu.ph	PROPN
ejpam-3780	19	14	(	(	PUNCT
ejpam-3780	19	15	f.	f.	PROPN
ejpam-3780	19	16	campeña	campeña	PROPN
ejpam-3780	19	17	)	)	PUNCT
ejpam-3780	19	18	,	,	PUNCT
ejpam-3780	19	19	otserolf@yahoo.com	otserolf@yahoo.com	X
ejpam-3780	19	20	(	(	PUNCT
ejpam-3780	19	21	f.	f.	PROPN
ejpam-3780	19	22	franco	franco	PROPN
ejpam-3780	19	23	)	)	PUNCT
ejpam-3780	19	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-3780	20	1	674	674	NUM
ejpam-3780	20	2	c	c	X
ejpam-3780	20	3	©	©	NOUN
ejpam-3780	20	4	2020	2020	NUM
ejpam-3780	20	5	ejpam	ejpam	VERB
ejpam-3780	20	6	all	all	DET
ejpam-3780	20	7	rights	right	NOUN
ejpam-3780	20	8	reserved	reserve	VERB
ejpam-3780	20	9	.	.	PUNCT
ejpam-3780	21	1	d.	d.	PROPN
ejpam-3780	21	2	chua	chua	PROPN
ejpam-3780	21	3	,	,	PUNCT
ejpam-3780	21	4	f.	f.	PROPN
ejpam-3780	21	5	campeña	campeña	PROPN
ejpam-3780	21	6	,	,	PUNCT
ejpam-3780	21	7	f.	f.	PROPN
ejpam-3780	21	8	franco	franco	PROPN
ejpam-3780	21	9	/	/	SYM
ejpam-3780	21	10	eur	eur	PROPN
ejpam-3780	21	11	.	.	PUNCT
ejpam-3780	22	1	j.	j.	PROPN
ejpam-3780	22	2	pure	pure	PROPN
ejpam-3780	22	3	appl	appl	PROPN
ejpam-3780	22	4	.	.	PROPN
ejpam-3780	22	5	math	math	PROPN
ejpam-3780	22	6	,	,	PUNCT
ejpam-3780	22	7	13	13	NUM
ejpam-3780	22	8	(	(	PUNCT
ejpam-3780	22	9	3	3	NUM
ejpam-3780	22	10	)	)	PUNCT
ejpam-3780	22	11	(	(	PUNCT
ejpam-3780	22	12	2020	2020	NUM
ejpam-3780	22	13	)	)	PUNCT
ejpam-3780	22	14	,	,	PUNCT
ejpam-3780	22	15	674	674	NUM
ejpam-3780	22	16	-	-	SYM
ejpam-3780	22	17	696	696	NUM
ejpam-3780	22	18	675	675	NUM
ejpam-3780	22	19	dela	dela	PROPN
ejpam-3780	22	20	rosa	rosa	PROPN
ejpam-3780	22	21	-	-	PUNCT
ejpam-3780	22	22	reynera	reynera	NOUN
ejpam-3780	23	1	[	[	X
ejpam-3780	23	2	2	2	NUM
ejpam-3780	23	3	]	]	PUNCT
ejpam-3780	23	4	constructed	construct	VERB
ejpam-3780	23	5	optimal	optimal	ADJ
ejpam-3780	23	6	zero	zero	NUM
ejpam-3780	23	7	ring	ring	NOUN
ejpam-3780	23	8	labelings	labeling	NOUN
ejpam-3780	23	9	for	for	ADP
ejpam-3780	23	10	some	some	DET
ejpam-3780	23	11	classes	class	NOUN
ejpam-3780	23	12	of	of	ADP
ejpam-3780	23	13	graphs	graph	NOUN
ejpam-3780	23	14	.	.	PUNCT
ejpam-3780	24	1	included	include	VERB
ejpam-3780	24	2	are	be	AUX
ejpam-3780	24	3	labeling	label	VERB
ejpam-3780	24	4	schemes	scheme	NOUN
ejpam-3780	24	5	for	for	ADP
ejpam-3780	24	6	trees	tree	NOUN
ejpam-3780	24	7	and	and	CCONJ
ejpam-3780	24	8	cactus	cactus	NOUN
ejpam-3780	24	9	graphs	graph	NOUN
ejpam-3780	24	10	.	.	PUNCT
ejpam-3780	25	1	moreover	moreover	ADV
ejpam-3780	25	2	,	,	PUNCT
ejpam-3780	25	3	the	the	DET
ejpam-3780	25	4	zero	zero	NUM
ejpam-3780	25	5	ring	ring	NOUN
ejpam-3780	25	6	indices	index	NOUN
ejpam-3780	25	7	of	of	ADP
ejpam-3780	25	8	graphs	graph	NOUN
ejpam-3780	25	9	that	that	PRON
ejpam-3780	25	10	result	result	VERB
ejpam-3780	25	11	from	from	ADP
ejpam-3780	25	12	some	some	DET
ejpam-3780	25	13	graph	graph	NOUN
ejpam-3780	25	14	operations	operation	NOUN
ejpam-3780	25	15	were	be	AUX
ejpam-3780	25	16	also	also	ADV
ejpam-3780	25	17	determined	determine	VERB
ejpam-3780	25	18	.	.	PUNCT
ejpam-3780	26	1	in	in	ADP
ejpam-3780	26	2	this	this	DET
ejpam-3780	26	3	paper	paper	NOUN
ejpam-3780	26	4	,	,	PUNCT
ejpam-3780	26	5	we	we	PRON
ejpam-3780	26	6	extend	extend	VERB
ejpam-3780	26	7	the	the	DET
ejpam-3780	26	8	notion	notion	NOUN
ejpam-3780	26	9	of	of	ADP
ejpam-3780	26	10	zero	zero	NUM
ejpam-3780	26	11	ring	ring	NOUN
ejpam-3780	26	12	labeling	labeling	NOUN
ejpam-3780	26	13	further	far	ADV
ejpam-3780	26	14	.	.	PUNCT
ejpam-3780	27	1	can	can	AUX
ejpam-3780	27	2	a	a	DET
ejpam-3780	27	3	zero	zero	NUM
ejpam-3780	27	4	ring	ring	NOUN
ejpam-3780	27	5	labeling	labeling	NOUN
ejpam-3780	27	6	be	be	AUX
ejpam-3780	27	7	constructed	construct	VERB
ejpam-3780	27	8	for	for	ADP
ejpam-3780	27	9	a	a	DET
ejpam-3780	27	10	graph	graph	NOUN
ejpam-3780	27	11	such	such	ADJ
ejpam-3780	27	12	that	that	SCONJ
ejpam-3780	27	13	the	the	DET
ejpam-3780	27	14	number	number	NOUN
ejpam-3780	27	15	of	of	ADP
ejpam-3780	27	16	distinct	distinct	ADJ
ejpam-3780	27	17	sums	sum	NOUN
ejpam-3780	27	18	is	be	AUX
ejpam-3780	27	19	equal	equal	ADJ
ejpam-3780	27	20	to	to	ADP
ejpam-3780	27	21	some	some	DET
ejpam-3780	27	22	number	number	NOUN
ejpam-3780	27	23	?	?	PUNCT
ejpam-3780	28	1	in	in	ADP
ejpam-3780	28	2	particular	particular	ADJ
ejpam-3780	28	3	,	,	PUNCT
ejpam-3780	28	4	we	we	PRON
ejpam-3780	28	5	want	want	VERB
ejpam-3780	28	6	to	to	PART
ejpam-3780	28	7	determine	determine	VERB
ejpam-3780	28	8	a	a	DET
ejpam-3780	28	9	zero	zero	NUM
ejpam-3780	28	10	ring	ring	NOUN
ejpam-3780	28	11	labeling	labeling	NOUN
ejpam-3780	28	12	where	where	SCONJ
ejpam-3780	28	13	this	this	DET
ejpam-3780	28	14	number	number	NOUN
ejpam-3780	28	15	is	be	AUX
ejpam-3780	28	16	as	as	ADV
ejpam-3780	28	17	small	small	ADJ
ejpam-3780	28	18	as	as	ADP
ejpam-3780	28	19	possible	possible	ADJ
ejpam-3780	28	20	.	.	PUNCT
ejpam-3780	29	1	2	2	X
ejpam-3780	29	2	.	.	NUM
ejpam-3780	29	3	preliminaries	preliminary	NOUN
ejpam-3780	29	4	2.1	2.1	NUM
ejpam-3780	29	5	.	.	PUNCT
ejpam-3780	29	6	basic	basic	ADJ
ejpam-3780	29	7	definitions	definition	NOUN
ejpam-3780	29	8	we	we	PRON
ejpam-3780	29	9	now	now	ADV
ejpam-3780	29	10	present	present	VERB
ejpam-3780	29	11	some	some	DET
ejpam-3780	29	12	definitions	definition	NOUN
ejpam-3780	29	13	used	use	VERB
ejpam-3780	29	14	in	in	ADP
ejpam-3780	29	15	this	this	DET
ejpam-3780	29	16	paper	paper	NOUN
ejpam-3780	29	17	.	.	PUNCT
ejpam-3780	30	1	for	for	ADP
ejpam-3780	30	2	graph	graph	NOUN
ejpam-3780	30	3	theory	theory	NOUN
ejpam-3780	30	4	,	,	PUNCT
ejpam-3780	30	5	[	[	X
ejpam-3780	30	6	3	3	NUM
ejpam-3780	30	7	,	,	PUNCT
ejpam-3780	30	8	5	5	NUM
ejpam-3780	30	9	]	]	PUNCT
ejpam-3780	30	10	are	be	AUX
ejpam-3780	30	11	considered	consider	VERB
ejpam-3780	30	12	,	,	PUNCT
ejpam-3780	30	13	while	while	SCONJ
ejpam-3780	30	14	[	[	X
ejpam-3780	30	15	4	4	X
ejpam-3780	30	16	]	]	PUNCT
ejpam-3780	30	17	is	be	AUX
ejpam-3780	30	18	used	use	VERB
ejpam-3780	30	19	for	for	ADP
ejpam-3780	30	20	abstract	abstract	ADJ
ejpam-3780	30	21	algebra	algebra	NOUN
ejpam-3780	30	22	.	.	PUNCT
ejpam-3780	31	1	definition	definition	NOUN
ejpam-3780	31	2	1	1	NUM
ejpam-3780	31	3	.	.	PUNCT
ejpam-3780	32	1	a	a	DET
ejpam-3780	32	2	graph	graph	NOUN
ejpam-3780	32	3	g	g	NOUN
ejpam-3780	32	4	=	=	SYM
ejpam-3780	32	5	(	(	PUNCT
ejpam-3780	32	6	v	v	NOUN
ejpam-3780	32	7	,	,	PUNCT
ejpam-3780	32	8	e	e	NOUN
ejpam-3780	32	9	)	)	PUNCT
ejpam-3780	32	10	is	be	AUX
ejpam-3780	32	11	a	a	DET
ejpam-3780	32	12	pair	pair	NOUN
ejpam-3780	32	13	of	of	ADP
ejpam-3780	32	14	sets	set	NOUN
ejpam-3780	32	15	v	v	NOUN
ejpam-3780	32	16	and	and	CCONJ
ejpam-3780	32	17	e	e	NOUN
ejpam-3780	32	18	such	such	ADJ
ejpam-3780	32	19	that	that	SCONJ
ejpam-3780	32	20	e	e	NOUN
ejpam-3780	32	21	is	be	AUX
ejpam-3780	32	22	a	a	DET
ejpam-3780	32	23	set	set	NOUN
ejpam-3780	32	24	of	of	ADP
ejpam-3780	32	25	two	two	NUM
ejpam-3780	32	26	-	-	PUNCT
ejpam-3780	32	27	element	element	NOUN
ejpam-3780	32	28	subsets	subset	NOUN
ejpam-3780	32	29	of	of	ADP
ejpam-3780	32	30	v	v	NOUN
ejpam-3780	32	31	.	.	PUNCT
ejpam-3780	33	1	the	the	DET
ejpam-3780	33	2	elements	element	NOUN
ejpam-3780	33	3	of	of	ADP
ejpam-3780	33	4	v	v	NUM
ejpam-3780	33	5	are	be	AUX
ejpam-3780	33	6	the	the	DET
ejpam-3780	33	7	vertices	vertex	NOUN
ejpam-3780	33	8	of	of	ADP
ejpam-3780	33	9	g	g	NOUN
ejpam-3780	33	10	,	,	PUNCT
ejpam-3780	33	11	while	while	SCONJ
ejpam-3780	33	12	the	the	DET
ejpam-3780	33	13	elements	element	NOUN
ejpam-3780	33	14	of	of	ADP
ejpam-3780	33	15	e	e	NOUN
ejpam-3780	33	16	are	be	AUX
ejpam-3780	33	17	the	the	DET
ejpam-3780	33	18	edges	edge	NOUN
ejpam-3780	33	19	of	of	ADP
ejpam-3780	33	20	g.	g.	PROPN
ejpam-3780	33	21	the	the	DET
ejpam-3780	33	22	vertex	vertex	NOUN
ejpam-3780	33	23	set	set	NOUN
ejpam-3780	33	24	of	of	ADP
ejpam-3780	33	25	a	a	DET
ejpam-3780	33	26	graph	graph	NOUN
ejpam-3780	33	27	g	g	NOUN
ejpam-3780	33	28	is	be	AUX
ejpam-3780	33	29	denoted	denote	VERB
ejpam-3780	33	30	by	by	ADP
ejpam-3780	33	31	v	v	NOUN
ejpam-3780	33	32	(	(	PUNCT
ejpam-3780	33	33	g	g	NOUN
ejpam-3780	33	34	)	)	PUNCT
ejpam-3780	33	35	,	,	PUNCT
ejpam-3780	33	36	while	while	SCONJ
ejpam-3780	33	37	its	its	PRON
ejpam-3780	33	38	edge	edge	NOUN
ejpam-3780	33	39	set	set	NOUN
ejpam-3780	33	40	is	be	AUX
ejpam-3780	33	41	denoted	denote	VERB
ejpam-3780	33	42	by	by	ADP
ejpam-3780	33	43	e(g	e(g	PROPN
ejpam-3780	33	44	)	)	PUNCT
ejpam-3780	33	45	.	.	PUNCT
ejpam-3780	34	1	we	we	PRON
ejpam-3780	34	2	say	say	VERB
ejpam-3780	34	3	that	that	SCONJ
ejpam-3780	34	4	e	e	NOUN
ejpam-3780	34	5	is	be	AUX
ejpam-3780	34	6	an	an	DET
ejpam-3780	34	7	edge	edge	NOUN
ejpam-3780	34	8	at	at	ADP
ejpam-3780	34	9	v	v	NOUN
ejpam-3780	34	10	if	if	SCONJ
ejpam-3780	34	11	v	v	PROPN
ejpam-3780	34	12	∈	∈	PROPN
ejpam-3780	34	13	e.	e.	PROPN
ejpam-3780	35	1	an	an	DET
ejpam-3780	35	2	edge	edge	NOUN
ejpam-3780	35	3	{	{	PUNCT
ejpam-3780	35	4	u	u	NOUN
ejpam-3780	35	5	,	,	PUNCT
ejpam-3780	35	6	v	v	NOUN
ejpam-3780	35	7	}	}	PUNCT
ejpam-3780	35	8	can	can	AUX
ejpam-3780	35	9	be	be	AUX
ejpam-3780	35	10	written	write	VERB
ejpam-3780	35	11	as	as	ADP
ejpam-3780	35	12	uv	uv	NOUN
ejpam-3780	35	13	;	;	PUNCT
ejpam-3780	35	14	we	we	PRON
ejpam-3780	35	15	say	say	VERB
ejpam-3780	35	16	that	that	SCONJ
ejpam-3780	35	17	uv	uv	NOUN
ejpam-3780	35	18	joins	join	VERB
ejpam-3780	35	19	u	u	NOUN
ejpam-3780	35	20	and	and	CCONJ
ejpam-3780	35	21	v	v	NOUN
ejpam-3780	35	22	,	,	PUNCT
ejpam-3780	35	23	and	and	CCONJ
ejpam-3780	35	24	u	u	NOUN
ejpam-3780	35	25	and	and	CCONJ
ejpam-3780	35	26	v	v	NOUN
ejpam-3780	35	27	are	be	AUX
ejpam-3780	35	28	the	the	DET
ejpam-3780	35	29	ends	end	NOUN
ejpam-3780	35	30	of	of	ADP
ejpam-3780	35	31	uv	uv	NOUN
ejpam-3780	35	32	.	.	PUNCT
ejpam-3780	36	1	the	the	DET
ejpam-3780	36	2	number	number	NOUN
ejpam-3780	36	3	of	of	ADP
ejpam-3780	36	4	vertices	vertex	NOUN
ejpam-3780	36	5	of	of	ADP
ejpam-3780	36	6	a	a	DET
ejpam-3780	36	7	graph	graph	NOUN
ejpam-3780	36	8	g	g	NOUN
ejpam-3780	36	9	is	be	AUX
ejpam-3780	36	10	its	its	PRON
ejpam-3780	36	11	order	order	NOUN
ejpam-3780	36	12	,	,	PUNCT
ejpam-3780	36	13	denoted	denote	VERB
ejpam-3780	36	14	by	by	ADP
ejpam-3780	36	15	|g|	|g|	PROPN
ejpam-3780	36	16	.	.	PUNCT
ejpam-3780	37	1	the	the	DET
ejpam-3780	37	2	degree	degree	NOUN
ejpam-3780	37	3	of	of	ADP
ejpam-3780	37	4	a	a	DET
ejpam-3780	37	5	vertex	vertex	NOUN
ejpam-3780	37	6	v	v	NOUN
ejpam-3780	37	7	,	,	PUNCT
ejpam-3780	37	8	denoted	denote	VERB
ejpam-3780	37	9	by	by	ADP
ejpam-3780	37	10	d(v	d(v	PROPN
ejpam-3780	37	11	)	)	PUNCT
ejpam-3780	37	12	,	,	PUNCT
ejpam-3780	37	13	is	be	AUX
ejpam-3780	37	14	the	the	DET
ejpam-3780	37	15	number	number	NOUN
ejpam-3780	37	16	of	of	ADP
ejpam-3780	37	17	edges	edge	NOUN
ejpam-3780	37	18	at	at	ADP
ejpam-3780	37	19	v.	v.	ADP
ejpam-3780	37	20	two	two	NUM
ejpam-3780	37	21	vertices	vertex	NOUN
ejpam-3780	37	22	u	u	NOUN
ejpam-3780	37	23	and	and	CCONJ
ejpam-3780	37	24	v	v	NOUN
ejpam-3780	37	25	in	in	ADP
ejpam-3780	37	26	a	a	DET
ejpam-3780	37	27	graph	graph	NOUN
ejpam-3780	37	28	g	g	NOUN
ejpam-3780	37	29	are	be	AUX
ejpam-3780	37	30	adjacent	adjacent	ADJ
ejpam-3780	37	31	if	if	SCONJ
ejpam-3780	37	32	uv	uv	NOUN
ejpam-3780	37	33	is	be	AUX
ejpam-3780	37	34	an	an	DET
ejpam-3780	37	35	edge	edge	NOUN
ejpam-3780	37	36	in	in	ADP
ejpam-3780	37	37	g.	g.	PROPN
ejpam-3780	37	38	a	a	DET
ejpam-3780	37	39	graph	graph	NOUN
ejpam-3780	37	40	g	g	NOUN
ejpam-3780	37	41	of	of	ADP
ejpam-3780	37	42	order	order	NOUN
ejpam-3780	37	43	n	n	X
ejpam-3780	37	44	is	be	AUX
ejpam-3780	37	45	a	a	DET
ejpam-3780	37	46	complete	complete	ADJ
ejpam-3780	37	47	graph	graph	NOUN
ejpam-3780	37	48	,	,	PUNCT
ejpam-3780	37	49	denoted	denote	VERB
ejpam-3780	37	50	by	by	ADP
ejpam-3780	37	51	kn	kn	PROPN
ejpam-3780	37	52	,	,	PUNCT
ejpam-3780	37	53	if	if	SCONJ
ejpam-3780	37	54	for	for	ADP
ejpam-3780	37	55	any	any	DET
ejpam-3780	37	56	two	two	NUM
ejpam-3780	37	57	distinct	distinct	ADJ
ejpam-3780	37	58	vertices	vertex	NOUN
ejpam-3780	37	59	u	u	NOUN
ejpam-3780	37	60	and	and	CCONJ
ejpam-3780	37	61	v	v	NOUN
ejpam-3780	37	62	in	in	ADP
ejpam-3780	37	63	g	g	NOUN
ejpam-3780	37	64	,	,	PUNCT
ejpam-3780	37	65	uv	uv	PROPN
ejpam-3780	37	66	is	be	AUX
ejpam-3780	37	67	an	an	DET
ejpam-3780	37	68	edge	edge	NOUN
ejpam-3780	37	69	in	in	ADP
ejpam-3780	37	70	g.	g.	PROPN
ejpam-3780	37	71	definition	definition	NOUN
ejpam-3780	37	72	2	2	NUM
ejpam-3780	37	73	.	.	PUNCT
ejpam-3780	38	1	the	the	DET
ejpam-3780	38	2	maximum	maximum	ADJ
ejpam-3780	38	3	degree	degree	NOUN
ejpam-3780	38	4	of	of	ADP
ejpam-3780	38	5	a	a	DET
ejpam-3780	38	6	graph	graph	NOUN
ejpam-3780	38	7	g	g	NOUN
ejpam-3780	38	8	,	,	PUNCT
ejpam-3780	38	9	denoted	denote	VERB
ejpam-3780	38	10	by	by	ADP
ejpam-3780	38	11	∆(g	∆(g	PROPN
ejpam-3780	38	12	)	)	PUNCT
ejpam-3780	38	13	,	,	PUNCT
ejpam-3780	38	14	is	be	AUX
ejpam-3780	38	15	the	the	DET
ejpam-3780	38	16	highest	high	ADJ
ejpam-3780	38	17	degree	degree	NOUN
ejpam-3780	38	18	of	of	ADP
ejpam-3780	38	19	a	a	DET
ejpam-3780	38	20	vertex	vertex	NOUN
ejpam-3780	38	21	in	in	ADP
ejpam-3780	38	22	g.	g.	PROPN
ejpam-3780	38	23	definition	definition	NOUN
ejpam-3780	38	24	3	3	NUM
ejpam-3780	38	25	.	.	PUNCT
ejpam-3780	38	26	a	a	DET
ejpam-3780	38	27	path	path	NOUN
ejpam-3780	38	28	in	in	ADP
ejpam-3780	38	29	a	a	DET
ejpam-3780	38	30	graph	graph	NOUN
ejpam-3780	38	31	is	be	AUX
ejpam-3780	38	32	a	a	DET
ejpam-3780	38	33	subgraph	subgraph	NOUN
ejpam-3780	38	34	p	p	NOUN
ejpam-3780	38	35	=	=	PUNCT
ejpam-3780	39	1	[	[	X
ejpam-3780	39	2	v1	v1	NOUN
ejpam-3780	39	3	,	,	PUNCT
ejpam-3780	39	4	v2	v2	NOUN
ejpam-3780	39	5	,	,	PUNCT
ejpam-3780	39	6	.	.	PUNCT
ejpam-3780	39	7	.	.	PUNCT
ejpam-3780	40	1	.	.	PUNCT
ejpam-3780	41	1	,	,	PUNCT
ejpam-3780	41	2	vn	vn	X
ejpam-3780	41	3	]	]	X
ejpam-3780	41	4	such	such	ADJ
ejpam-3780	41	5	that	that	PRON
ejpam-3780	41	6	v	v	NOUN
ejpam-3780	41	7	(	(	PUNCT
ejpam-3780	41	8	p	p	NOUN
ejpam-3780	41	9	)	)	PUNCT
ejpam-3780	41	10	=	=	SYM
ejpam-3780	41	11	{	{	PUNCT
ejpam-3780	41	12	v1	v1	PROPN
ejpam-3780	41	13	,	,	PUNCT
ejpam-3780	41	14	v2	v2	PROPN
ejpam-3780	41	15	,	,	PUNCT
ejpam-3780	41	16	.	.	PUNCT
ejpam-3780	41	17	.	.	PUNCT
ejpam-3780	41	18	.	.	PUNCT
ejpam-3780	42	1	,	,	PUNCT
ejpam-3780	42	2	vn	vn	NOUN
ejpam-3780	42	3	}	}	PUNCT
ejpam-3780	42	4	and	and	CCONJ
ejpam-3780	42	5	e(p	e(p	PROPN
ejpam-3780	42	6	)	)	PUNCT
ejpam-3780	43	1	=	=	PRON
ejpam-3780	43	2	{	{	PUNCT
ejpam-3780	43	3	v1v2	v1v2	PROPN
ejpam-3780	43	4	,	,	PUNCT
ejpam-3780	43	5	v2v3	v2v3	PROPN
ejpam-3780	43	6	,	,	PUNCT
ejpam-3780	43	7	.	.	PUNCT
ejpam-3780	43	8	.	.	PUNCT
ejpam-3780	44	1	.	.	PUNCT
ejpam-3780	45	1	,	,	PUNCT
ejpam-3780	45	2	vn−1vn	vn−1vn	NUM
ejpam-3780	45	3	}	}	PUNCT
ejpam-3780	45	4	,	,	PUNCT
ejpam-3780	45	5	where	where	SCONJ
ejpam-3780	45	6	v1	v1	NOUN
ejpam-3780	45	7	,	,	PUNCT
ejpam-3780	45	8	v2	v2	NOUN
ejpam-3780	45	9	,	,	PUNCT
ejpam-3780	45	10	.	.	PUNCT
ejpam-3780	45	11	.	.	PUNCT
ejpam-3780	45	12	.	.	PUNCT
ejpam-3780	46	1	,	,	PUNCT
ejpam-3780	46	2	vn	vn	PROPN
ejpam-3780	46	3	are	be	AUX
ejpam-3780	46	4	distinct	distinct	ADJ
ejpam-3780	46	5	.	.	PUNCT
ejpam-3780	47	1	the	the	DET
ejpam-3780	47	2	number	number	NOUN
ejpam-3780	47	3	of	of	ADP
ejpam-3780	47	4	edges	edge	NOUN
ejpam-3780	47	5	in	in	ADP
ejpam-3780	47	6	a	a	DET
ejpam-3780	47	7	path	path	NOUN
ejpam-3780	47	8	is	be	AUX
ejpam-3780	47	9	its	its	PRON
ejpam-3780	47	10	length	length	NOUN
ejpam-3780	47	11	.	.	PUNCT
ejpam-3780	48	1	in	in	ADP
ejpam-3780	48	2	a	a	DET
ejpam-3780	48	3	path	path	NOUN
ejpam-3780	48	4	p	p	NOUN
ejpam-3780	48	5	=	=	PUNCT
ejpam-3780	49	1	[	[	X
ejpam-3780	49	2	v1	v1	NOUN
ejpam-3780	49	3	,	,	PUNCT
ejpam-3780	49	4	v2	v2	NOUN
ejpam-3780	49	5	,	,	PUNCT
ejpam-3780	49	6	.	.	PUNCT
ejpam-3780	49	7	.	.	PUNCT
ejpam-3780	50	1	.	.	PUNCT
ejpam-3780	51	1	,	,	PUNCT
ejpam-3780	51	2	vn	vn	X
ejpam-3780	51	3	]	]	X
ejpam-3780	51	4	,	,	PUNCT
ejpam-3780	51	5	the	the	DET
ejpam-3780	51	6	vertices	vertex	NOUN
ejpam-3780	51	7	v1	v1	VERB
ejpam-3780	51	8	and	and	CCONJ
ejpam-3780	51	9	vn	vn	PROPN
ejpam-3780	51	10	are	be	AUX
ejpam-3780	51	11	the	the	DET
ejpam-3780	51	12	endvertices	endvertice	NOUN
ejpam-3780	51	13	,	,	PUNCT
ejpam-3780	51	14	while	while	SCONJ
ejpam-3780	51	15	v2	v2	PROPN
ejpam-3780	51	16	,	,	PUNCT
ejpam-3780	51	17	v3	v3	PROPN
ejpam-3780	51	18	,	,	PUNCT
ejpam-3780	51	19	.	.	PUNCT
ejpam-3780	51	20	.	.	PUNCT
ejpam-3780	52	1	.	.	PUNCT
ejpam-3780	53	1	,	,	PUNCT
ejpam-3780	53	2	vn−1	vn−1	PROPN
ejpam-3780	53	3	are	be	AUX
ejpam-3780	53	4	the	the	DET
ejpam-3780	53	5	inner	inner	ADJ
ejpam-3780	53	6	vertices	vertex	NOUN
ejpam-3780	53	7	.	.	PUNCT
ejpam-3780	54	1	moreover	moreover	ADV
ejpam-3780	54	2	,	,	PUNCT
ejpam-3780	54	3	we	we	PRON
ejpam-3780	54	4	say	say	VERB
ejpam-3780	54	5	that	that	SCONJ
ejpam-3780	54	6	p	p	NOUN
ejpam-3780	54	7	is	be	AUX
ejpam-3780	54	8	a	a	DET
ejpam-3780	54	9	path	path	NOUN
ejpam-3780	54	10	between	between	ADP
ejpam-3780	54	11	v1	v1	PROPN
ejpam-3780	54	12	and	and	CCONJ
ejpam-3780	54	13	vn	vn	NOUN
ejpam-3780	54	14	.	.	PUNCT
ejpam-3780	55	1	the	the	DET
ejpam-3780	55	2	distance	distance	NOUN
ejpam-3780	55	3	between	between	ADP
ejpam-3780	55	4	two	two	NUM
ejpam-3780	55	5	vertices	vertex	NOUN
ejpam-3780	55	6	u	u	NOUN
ejpam-3780	55	7	and	and	CCONJ
ejpam-3780	55	8	v	v	NOUN
ejpam-3780	55	9	in	in	ADP
ejpam-3780	55	10	a	a	DET
ejpam-3780	55	11	graph	graph	NOUN
ejpam-3780	55	12	g	g	NOUN
ejpam-3780	55	13	is	be	AUX
ejpam-3780	55	14	the	the	DET
ejpam-3780	55	15	length	length	NOUN
ejpam-3780	55	16	of	of	ADP
ejpam-3780	55	17	the	the	DET
ejpam-3780	55	18	shortest	short	ADJ
ejpam-3780	55	19	path	path	NOUN
ejpam-3780	55	20	between	between	ADP
ejpam-3780	55	21	u	u	NOUN
ejpam-3780	55	22	and	and	CCONJ
ejpam-3780	55	23	v.	v.	ADP
ejpam-3780	55	24	definition	definition	NOUN
ejpam-3780	55	25	4	4	NUM
ejpam-3780	55	26	.	.	PUNCT
ejpam-3780	56	1	a	a	DET
ejpam-3780	56	2	graph	graph	NOUN
ejpam-3780	56	3	g	g	NOUN
ejpam-3780	56	4	is	be	AUX
ejpam-3780	56	5	connected	connect	VERB
ejpam-3780	56	6	if	if	SCONJ
ejpam-3780	56	7	for	for	ADP
ejpam-3780	56	8	any	any	DET
ejpam-3780	56	9	vertices	vertex	NOUN
ejpam-3780	56	10	u	u	NOUN
ejpam-3780	56	11	and	and	CCONJ
ejpam-3780	56	12	v	v	NOUN
ejpam-3780	56	13	in	in	ADP
ejpam-3780	56	14	g	g	NOUN
ejpam-3780	56	15	,	,	PUNCT
ejpam-3780	56	16	there	there	PRON
ejpam-3780	56	17	is	be	VERB
ejpam-3780	56	18	a	a	DET
ejpam-3780	56	19	path	path	NOUN
ejpam-3780	56	20	between	between	ADP
ejpam-3780	56	21	u	u	PROPN
ejpam-3780	56	22	and	and	CCONJ
ejpam-3780	56	23	v.	v.	ADP
ejpam-3780	56	24	a	a	DET
ejpam-3780	56	25	connected	connected	ADJ
ejpam-3780	56	26	subgraph	subgraph	NOUN
ejpam-3780	56	27	of	of	ADP
ejpam-3780	56	28	a	a	DET
ejpam-3780	56	29	graph	graph	NOUN
ejpam-3780	56	30	g	g	NOUN
ejpam-3780	56	31	is	be	AUX
ejpam-3780	56	32	a	a	DET
ejpam-3780	56	33	component	component	NOUN
ejpam-3780	56	34	of	of	ADP
ejpam-3780	56	35	g	g	PROPN
ejpam-3780	56	36	if	if	SCONJ
ejpam-3780	56	37	it	it	PRON
ejpam-3780	56	38	is	be	AUX
ejpam-3780	56	39	not	not	PART
ejpam-3780	56	40	a	a	DET
ejpam-3780	56	41	proper	proper	ADJ
ejpam-3780	56	42	subgraph	subgraph	NOUN
ejpam-3780	56	43	of	of	ADP
ejpam-3780	56	44	any	any	DET
ejpam-3780	56	45	other	other	ADJ
ejpam-3780	56	46	connected	connected	ADJ
ejpam-3780	56	47	subgraph	subgraph	NOUN
ejpam-3780	56	48	in	in	ADP
ejpam-3780	56	49	g.	g.	PROPN
ejpam-3780	56	50	d.	d.	PROPN
ejpam-3780	56	51	chua	chua	PROPN
ejpam-3780	56	52	,	,	PUNCT
ejpam-3780	56	53	f.	f.	PROPN
ejpam-3780	56	54	campeña	campeña	PROPN
ejpam-3780	56	55	,	,	PUNCT
ejpam-3780	56	56	f.	f.	PROPN
ejpam-3780	56	57	franco	franco	PROPN
ejpam-3780	56	58	/	/	SYM
ejpam-3780	56	59	eur	eur	PROPN
ejpam-3780	56	60	.	.	PUNCT
ejpam-3780	57	1	j.	j.	PROPN
ejpam-3780	57	2	pure	pure	PROPN
ejpam-3780	57	3	appl	appl	PROPN
ejpam-3780	57	4	.	.	PROPN
ejpam-3780	57	5	math	math	PROPN
ejpam-3780	57	6	,	,	PUNCT
ejpam-3780	57	7	13	13	NUM
ejpam-3780	57	8	(	(	PUNCT
ejpam-3780	57	9	3	3	NUM
ejpam-3780	57	10	)	)	PUNCT
ejpam-3780	57	11	(	(	PUNCT
ejpam-3780	57	12	2020	2020	NUM
ejpam-3780	57	13	)	)	PUNCT
ejpam-3780	57	14	,	,	PUNCT
ejpam-3780	57	15	674	674	NUM
ejpam-3780	57	16	-	-	SYM
ejpam-3780	57	17	696	696	NUM
ejpam-3780	57	18	676	676	NUM
ejpam-3780	57	19	2.2	2.2	NUM
ejpam-3780	57	20	.	.	PUNCT
ejpam-3780	58	1	trees	tree	NOUN
ejpam-3780	58	2	definition	definition	NOUN
ejpam-3780	58	3	5	5	NUM
ejpam-3780	58	4	.	.	PUNCT
ejpam-3780	59	1	a	a	DET
ejpam-3780	59	2	tree	tree	NOUN
ejpam-3780	59	3	is	be	AUX
ejpam-3780	59	4	a	a	DET
ejpam-3780	59	5	connected	connected	ADJ
ejpam-3780	59	6	graph	graph	NOUN
ejpam-3780	59	7	with	with	ADP
ejpam-3780	59	8	no	no	DET
ejpam-3780	59	9	cycle	cycle	NOUN
ejpam-3780	59	10	.	.	PUNCT
ejpam-3780	60	1	the	the	DET
ejpam-3780	60	2	vertices	vertex	NOUN
ejpam-3780	60	3	of	of	ADP
ejpam-3780	60	4	degree	degree	NOUN
ejpam-3780	60	5	one	one	NUM
ejpam-3780	60	6	in	in	ADP
ejpam-3780	60	7	a	a	DET
ejpam-3780	60	8	tree	tree	NOUN
ejpam-3780	60	9	are	be	AUX
ejpam-3780	60	10	its	its	PRON
ejpam-3780	60	11	leaves	leave	NOUN
ejpam-3780	60	12	.	.	PUNCT
ejpam-3780	61	1	definition	definition	NOUN
ejpam-3780	61	2	6	6	NUM
ejpam-3780	61	3	.	.	PUNCT
ejpam-3780	62	1	a	a	DET
ejpam-3780	62	2	path	path	NOUN
ejpam-3780	62	3	graph	graph	NOUN
ejpam-3780	62	4	pn	pn	PROPN
ejpam-3780	62	5	,	,	PUNCT
ejpam-3780	62	6	where	where	SCONJ
ejpam-3780	62	7	n	n	PRON
ejpam-3780	62	8	≥	≥	NOUN
ejpam-3780	62	9	2	2	NUM
ejpam-3780	62	10	,	,	PUNCT
ejpam-3780	62	11	is	be	AUX
ejpam-3780	62	12	a	a	DET
ejpam-3780	62	13	tree	tree	NOUN
ejpam-3780	62	14	with	with	ADP
ejpam-3780	62	15	two	two	NUM
ejpam-3780	62	16	leaves	leave	NOUN
ejpam-3780	62	17	and	and	CCONJ
ejpam-3780	62	18	n−2	n−2	PROPN
ejpam-3780	62	19	vertices	vertice	VERB
ejpam-3780	62	20	with	with	ADP
ejpam-3780	62	21	degree	degree	NOUN
ejpam-3780	62	22	two	two	NUM
ejpam-3780	62	23	.	.	PUNCT
ejpam-3780	63	1	definition	definition	NOUN
ejpam-3780	63	2	7	7	NUM
ejpam-3780	63	3	.	.	PUNCT
ejpam-3780	64	1	a	a	DET
ejpam-3780	64	2	star	star	NOUN
ejpam-3780	64	3	graph	graph	NOUN
ejpam-3780	64	4	sn	sn	PROPN
ejpam-3780	64	5	,	,	PUNCT
ejpam-3780	64	6	where	where	SCONJ
ejpam-3780	64	7	n	n	PRON
ejpam-3780	64	8	≥	≥	NOUN
ejpam-3780	64	9	1	1	NUM
ejpam-3780	64	10	,	,	PUNCT
ejpam-3780	64	11	is	be	AUX
ejpam-3780	64	12	a	a	DET
ejpam-3780	64	13	tree	tree	NOUN
ejpam-3780	64	14	with	with	ADP
ejpam-3780	64	15	one	one	NUM
ejpam-3780	64	16	vertex	vertex	NOUN
ejpam-3780	64	17	of	of	ADP
ejpam-3780	64	18	degree	degree	NOUN
ejpam-3780	64	19	n	n	CCONJ
ejpam-3780	64	20	−	−	PROPN
ejpam-3780	64	21	1	1	NUM
ejpam-3780	64	22	and	and	CCONJ
ejpam-3780	64	23	all	all	DET
ejpam-3780	64	24	others	other	NOUN
ejpam-3780	64	25	with	with	ADP
ejpam-3780	64	26	degree	degree	NOUN
ejpam-3780	64	27	one	one	NUM
ejpam-3780	64	28	.	.	PUNCT
ejpam-3780	65	1	the	the	DET
ejpam-3780	65	2	vertex	vertex	NOUN
ejpam-3780	65	3	with	with	ADP
ejpam-3780	65	4	the	the	DET
ejpam-3780	65	5	maximum	maximum	ADJ
ejpam-3780	65	6	degree	degree	NOUN
ejpam-3780	65	7	in	in	ADP
ejpam-3780	65	8	a	a	DET
ejpam-3780	65	9	star	star	NOUN
ejpam-3780	65	10	graph	graph	NOUN
ejpam-3780	65	11	is	be	AUX
ejpam-3780	65	12	its	its	PRON
ejpam-3780	65	13	center	center	NOUN
ejpam-3780	65	14	.	.	PUNCT
ejpam-3780	66	1	definition	definition	NOUN
ejpam-3780	66	2	8	8	NUM
ejpam-3780	66	3	.	.	PUNCT
ejpam-3780	67	1	a	a	DET
ejpam-3780	67	2	bistar	bistar	PROPN
ejpam-3780	67	3	bn	bn	NOUN
ejpam-3780	67	4	,	,	PUNCT
ejpam-3780	67	5	where	where	SCONJ
ejpam-3780	67	6	n	n	PRON
ejpam-3780	67	7	≥	≥	NOUN
ejpam-3780	67	8	1	1	NUM
ejpam-3780	67	9	,	,	PUNCT
ejpam-3780	67	10	is	be	AUX
ejpam-3780	67	11	a	a	DET
ejpam-3780	67	12	graph	graph	NOUN
ejpam-3780	67	13	formed	form	VERB
ejpam-3780	67	14	by	by	ADP
ejpam-3780	67	15	joining	join	VERB
ejpam-3780	67	16	the	the	DET
ejpam-3780	67	17	centers	center	NOUN
ejpam-3780	67	18	of	of	ADP
ejpam-3780	67	19	two	two	NUM
ejpam-3780	67	20	star	star	NOUN
ejpam-3780	67	21	graphs	graph	NOUN
ejpam-3780	67	22	sn	sn	PROPN
ejpam-3780	67	23	.	.	PUNCT
ejpam-3780	68	1	definition	definition	NOUN
ejpam-3780	68	2	9	9	NUM
ejpam-3780	68	3	.	.	PUNCT
ejpam-3780	69	1	an	an	DET
ejpam-3780	69	2	n	n	NOUN
ejpam-3780	69	3	-	-	PUNCT
ejpam-3780	69	4	centipede	centipede	NOUN
ejpam-3780	69	5	,	,	PUNCT
ejpam-3780	69	6	where	where	SCONJ
ejpam-3780	69	7	n	n	PRON
ejpam-3780	69	8	≥	≥	NOUN
ejpam-3780	69	9	1	1	NUM
ejpam-3780	69	10	,	,	PUNCT
ejpam-3780	69	11	is	be	AUX
ejpam-3780	69	12	a	a	DET
ejpam-3780	69	13	tree	tree	NOUN
ejpam-3780	69	14	with	with	ADP
ejpam-3780	69	15	vertex	vertex	NOUN
ejpam-3780	69	16	set	set	VERB
ejpam-3780	69	17	a	a	DET
ejpam-3780	69	18	∪	∪	ADJ
ejpam-3780	69	19	b	b	NOUN
ejpam-3780	69	20	,	,	PUNCT
ejpam-3780	69	21	where	where	SCONJ
ejpam-3780	69	22	a	a	DET
ejpam-3780	69	23	=	=	SYM
ejpam-3780	69	24	{	{	PUNCT
ejpam-3780	69	25	a1	a1	PROPN
ejpam-3780	69	26	,	,	PUNCT
ejpam-3780	69	27	a2	a2	PROPN
ejpam-3780	69	28	,	,	PUNCT
ejpam-3780	69	29	.	.	PUNCT
ejpam-3780	69	30	.	.	PUNCT
ejpam-3780	70	1	.	.	PUNCT
ejpam-3780	71	1	,	,	PUNCT
ejpam-3780	71	2	an	an	PRON
ejpam-3780	71	3	}	}	PUNCT
ejpam-3780	71	4	and	and	CCONJ
ejpam-3780	71	5	b	b	X
ejpam-3780	71	6	=	=	SYM
ejpam-3780	71	7	{	{	PUNCT
ejpam-3780	71	8	b1	b1	PROPN
ejpam-3780	71	9	,	,	PUNCT
ejpam-3780	71	10	b2	b2	NOUN
ejpam-3780	71	11	,	,	PUNCT
ejpam-3780	71	12	.	.	PUNCT
ejpam-3780	71	13	.	.	PUNCT
ejpam-3780	72	1	.	.	PUNCT
ejpam-3780	73	1	,	,	PUNCT
ejpam-3780	73	2	bn	bn	ADP
ejpam-3780	73	3	}	}	PUNCT
ejpam-3780	73	4	,	,	PUNCT
ejpam-3780	74	1	and	and	CCONJ
ejpam-3780	74	2	edge	edge	VERB
ejpam-3780	74	3	set	set	VERB
ejpam-3780	75	1	e	e	NOUN
ejpam-3780	75	2	=	=	NOUN
ejpam-3780	75	3	{	{	PUNCT
ejpam-3780	75	4	aibi	aibi	NOUN
ejpam-3780	75	5	:	:	PUNCT
ejpam-3780	75	6	1	1	NUM
ejpam-3780	75	7	≤	≤	NUM
ejpam-3780	75	8	i	i	PRON
ejpam-3780	75	9	≤	≤	NOUN
ejpam-3780	75	10	n	n	CCONJ
ejpam-3780	75	11	}	}	PUNCT
ejpam-3780	75	12	∪	∪	ADJ
ejpam-3780	75	13	{	{	PUNCT
ejpam-3780	75	14	aiai+1	aiai+1	NOUN
ejpam-3780	75	15	:	:	PUNCT
ejpam-3780	75	16	1	1	NUM
ejpam-3780	75	17	≤	≤	NUM
ejpam-3780	76	1	i	i	PRON
ejpam-3780	76	2	≤	≤	ADJ
ejpam-3780	76	3	n−	n−	NOUN
ejpam-3780	76	4	1	1	NUM
ejpam-3780	76	5	}	}	PUNCT
ejpam-3780	76	6	.	.	PUNCT
ejpam-3780	77	1	definition	definition	NOUN
ejpam-3780	77	2	10	10	NUM
ejpam-3780	77	3	.	.	PUNCT
ejpam-3780	78	1	a	a	DET
ejpam-3780	78	2	caterpillar	caterpillar	NOUN
ejpam-3780	78	3	is	be	AUX
ejpam-3780	78	4	a	a	DET
ejpam-3780	78	5	tree	tree	NOUN
ejpam-3780	78	6	in	in	ADP
ejpam-3780	78	7	which	which	PRON
ejpam-3780	78	8	all	all	DET
ejpam-3780	78	9	vertices	vertex	NOUN
ejpam-3780	78	10	are	be	AUX
ejpam-3780	78	11	within	within	ADP
ejpam-3780	78	12	distance	distance	NOUN
ejpam-3780	78	13	one	one	NUM
ejpam-3780	78	14	of	of	ADP
ejpam-3780	78	15	a	a	DET
ejpam-3780	78	16	central	central	ADJ
ejpam-3780	78	17	path	path	NOUN
ejpam-3780	78	18	.	.	PUNCT
ejpam-3780	79	1	we	we	PRON
ejpam-3780	79	2	define	define	VERB
ejpam-3780	79	3	a	a	DET
ejpam-3780	79	4	hanging	hang	VERB
ejpam-3780	79	5	leaf	leaf	NOUN
ejpam-3780	79	6	of	of	ADP
ejpam-3780	79	7	a	a	DET
ejpam-3780	79	8	vertex	vertex	NOUN
ejpam-3780	79	9	in	in	ADP
ejpam-3780	79	10	the	the	DET
ejpam-3780	79	11	central	central	ADJ
ejpam-3780	79	12	path	path	NOUN
ejpam-3780	79	13	as	as	ADP
ejpam-3780	79	14	a	a	DET
ejpam-3780	79	15	leaf	leaf	NOUN
ejpam-3780	79	16	that	that	PRON
ejpam-3780	79	17	is	be	AUX
ejpam-3780	79	18	adjacent	adjacent	ADJ
ejpam-3780	79	19	to	to	ADP
ejpam-3780	79	20	it	it	PRON
ejpam-3780	79	21	but	but	CCONJ
ejpam-3780	79	22	not	not	PART
ejpam-3780	79	23	a	a	DET
ejpam-3780	79	24	vertex	vertex	NOUN
ejpam-3780	79	25	in	in	ADP
ejpam-3780	79	26	the	the	DET
ejpam-3780	79	27	central	central	ADJ
ejpam-3780	79	28	path	path	NOUN
ejpam-3780	79	29	.	.	PUNCT
ejpam-3780	80	1	example	example	NOUN
ejpam-3780	81	1	1	1	NUM
ejpam-3780	81	2	.	.	PUNCT
ejpam-3780	81	3	the	the	DET
ejpam-3780	81	4	graph	graph	NOUN
ejpam-3780	81	5	in	in	ADP
ejpam-3780	81	6	fig	fig	NOUN
ejpam-3780	81	7	.	.	PUNCT
ejpam-3780	82	1	1	1	NUM
ejpam-3780	82	2	is	be	AUX
ejpam-3780	82	3	a	a	DET
ejpam-3780	82	4	caterpillar	caterpillar	NOUN
ejpam-3780	82	5	with	with	ADP
ejpam-3780	82	6	respect	respect	NOUN
ejpam-3780	82	7	to	to	ADP
ejpam-3780	82	8	any	any	PRON
ejpam-3780	82	9	of	of	ADP
ejpam-3780	82	10	the	the	DET
ejpam-3780	82	11	following	follow	VERB
ejpam-3780	82	12	central	central	ADJ
ejpam-3780	82	13	paths	path	NOUN
ejpam-3780	82	14	:	:	PUNCT
ejpam-3780	83	1	[	[	X
ejpam-3780	83	2	c	c	X
ejpam-3780	83	3	,	,	PUNCT
ejpam-3780	83	4	d	d	NOUN
ejpam-3780	83	5	,	,	PUNCT
ejpam-3780	83	6	e	e	NOUN
ejpam-3780	83	7	,	,	PUNCT
ejpam-3780	83	8	f	f	X
ejpam-3780	83	9	]	]	X
ejpam-3780	83	10	,	,	PUNCT
ejpam-3780	83	11	[	[	X
ejpam-3780	83	12	a	a	X
ejpam-3780	83	13	,	,	PUNCT
ejpam-3780	83	14	c	c	NOUN
ejpam-3780	83	15	,	,	PUNCT
ejpam-3780	83	16	d	d	NOUN
ejpam-3780	83	17	,	,	PUNCT
ejpam-3780	83	18	e	e	NOUN
ejpam-3780	83	19	,	,	PUNCT
ejpam-3780	83	20	f	f	X
ejpam-3780	83	21	]	]	X
ejpam-3780	83	22	,	,	PUNCT
ejpam-3780	83	23	[	[	X
ejpam-3780	83	24	g	g	X
ejpam-3780	83	25	,	,	PUNCT
ejpam-3780	83	26	c	c	X
ejpam-3780	83	27	,	,	PUNCT
ejpam-3780	83	28	d	d	NOUN
ejpam-3780	83	29	,	,	PUNCT
ejpam-3780	83	30	e	e	NOUN
ejpam-3780	83	31	,	,	PUNCT
ejpam-3780	83	32	f	f	X
ejpam-3780	83	33	]	]	PUNCT
ejpam-3780	83	34	,	,	PUNCT
ejpam-3780	84	1	[	[	X
ejpam-3780	84	2	c	c	X
ejpam-3780	84	3	,	,	PUNCT
ejpam-3780	84	4	d	d	NOUN
ejpam-3780	84	5	,	,	PUNCT
ejpam-3780	84	6	e	e	NOUN
ejpam-3780	84	7	,	,	PUNCT
ejpam-3780	84	8	f	f	PROPN
ejpam-3780	84	9	,	,	PUNCT
ejpam-3780	84	10	j	j	PROPN
ejpam-3780	84	11	]	]	X
ejpam-3780	84	12	,	,	PUNCT
ejpam-3780	84	13	[	[	X
ejpam-3780	84	14	a	a	X
ejpam-3780	84	15	,	,	PUNCT
ejpam-3780	84	16	c	c	NOUN
ejpam-3780	84	17	,	,	PUNCT
ejpam-3780	84	18	d	d	NOUN
ejpam-3780	84	19	,	,	PUNCT
ejpam-3780	84	20	e	e	NOUN
ejpam-3780	84	21	,	,	PUNCT
ejpam-3780	84	22	f	f	PROPN
ejpam-3780	84	23	,	,	PUNCT
ejpam-3780	84	24	j	j	PROPN
ejpam-3780	84	25	]	]	X
ejpam-3780	84	26	,	,	PUNCT
ejpam-3780	84	27	or	or	CCONJ
ejpam-3780	84	28	[	[	X
ejpam-3780	84	29	g	g	NOUN
ejpam-3780	84	30	,	,	PUNCT
ejpam-3780	84	31	c	c	X
ejpam-3780	84	32	,	,	PUNCT
ejpam-3780	84	33	d	d	NOUN
ejpam-3780	84	34	,	,	PUNCT
ejpam-3780	84	35	e	e	NOUN
ejpam-3780	84	36	,	,	PUNCT
ejpam-3780	84	37	f	f	PROPN
ejpam-3780	84	38	,	,	PUNCT
ejpam-3780	84	39	j	j	PROPN
ejpam-3780	84	40	]	]	X
ejpam-3780	84	41	.	.	PUNCT
ejpam-3780	85	1	the	the	DET
ejpam-3780	85	2	hanging	hang	VERB
ejpam-3780	85	3	leaves	leave	NOUN
ejpam-3780	85	4	of	of	ADP
ejpam-3780	85	5	d	d	NOUN
ejpam-3780	85	6	are	be	AUX
ejpam-3780	85	7	i	i	PRON
ejpam-3780	85	8	,	,	PUNCT
ejpam-3780	85	9	b	b	NOUN
ejpam-3780	85	10	,	,	PUNCT
ejpam-3780	85	11	and	and	CCONJ
ejpam-3780	86	1	h.	h.	PROPN
ejpam-3780	86	2	c	c	PROPN
ejpam-3780	87	1	d	d	PROPN
ejpam-3780	87	2	e	e	X
ejpam-3780	87	3	fc	fc	PROPN
ejpam-3780	88	1	a	a	PRON
ejpam-3780	88	2	c	c	NOUN
ejpam-3780	88	3	g	g	PROPN
ejpam-3780	88	4	d	d	PROPN
ejpam-3780	88	5	b	b	PROPN
ejpam-3780	88	6	d	d	NOUN
ejpam-3780	88	7	h	h	NOUN
ejpam-3780	89	1	d	d	NOUN
ejpam-3780	89	2	i	i	PRON
ejpam-3780	89	3	f	f	PROPN
ejpam-3780	89	4	j	j	PROPN
ejpam-3780	89	5	figure	figure	NOUN
ejpam-3780	89	6	1	1	NUM
ejpam-3780	89	7	:	:	PUNCT
ejpam-3780	89	8	a	a	DET
ejpam-3780	89	9	caterpillar	caterpillar	ADJ
ejpam-3780	89	10	definition	definition	NOUN
ejpam-3780	89	11	11	11	NUM
ejpam-3780	89	12	.	.	PUNCT
ejpam-3780	90	1	a	a	DET
ejpam-3780	90	2	spider	spider	NOUN
ejpam-3780	90	3	is	be	AUX
ejpam-3780	90	4	a	a	DET
ejpam-3780	90	5	tree	tree	NOUN
ejpam-3780	90	6	with	with	ADP
ejpam-3780	90	7	a	a	DET
ejpam-3780	90	8	unique	unique	ADJ
ejpam-3780	90	9	vertex	vertex	NOUN
ejpam-3780	90	10	of	of	ADP
ejpam-3780	90	11	degree	degree	NOUN
ejpam-3780	90	12	at	at	ADV
ejpam-3780	90	13	least	least	ADV
ejpam-3780	90	14	three	three	NUM
ejpam-3780	90	15	and	and	CCONJ
ejpam-3780	90	16	all	all	DET
ejpam-3780	90	17	others	other	NOUN
ejpam-3780	90	18	with	with	ADP
ejpam-3780	90	19	degree	degree	NOUN
ejpam-3780	90	20	at	at	ADP
ejpam-3780	90	21	most	most	ADV
ejpam-3780	90	22	two	two	NUM
ejpam-3780	90	23	.	.	PUNCT
ejpam-3780	91	1	we	we	PRON
ejpam-3780	91	2	define	define	VERB
ejpam-3780	91	3	the	the	DET
ejpam-3780	91	4	head	head	NOUN
ejpam-3780	91	5	of	of	ADP
ejpam-3780	91	6	a	a	DET
ejpam-3780	91	7	spider	spider	NOUN
ejpam-3780	91	8	as	as	ADP
ejpam-3780	91	9	the	the	DET
ejpam-3780	91	10	vertex	vertex	NOUN
ejpam-3780	91	11	with	with	ADP
ejpam-3780	91	12	the	the	DET
ejpam-3780	91	13	maximum	maximum	ADJ
ejpam-3780	91	14	degree	degree	NOUN
ejpam-3780	91	15	and	and	CCONJ
ejpam-3780	91	16	a	a	DET
ejpam-3780	91	17	leg	leg	NOUN
ejpam-3780	91	18	as	as	ADP
ejpam-3780	91	19	a	a	DET
ejpam-3780	91	20	component	component	NOUN
ejpam-3780	91	21	of	of	ADP
ejpam-3780	91	22	the	the	DET
ejpam-3780	91	23	graph	graph	NOUN
ejpam-3780	91	24	that	that	PRON
ejpam-3780	91	25	is	be	AUX
ejpam-3780	91	26	obtained	obtain	VERB
ejpam-3780	91	27	by	by	ADP
ejpam-3780	91	28	removing	remove	VERB
ejpam-3780	91	29	the	the	DET
ejpam-3780	91	30	head	head	NOUN
ejpam-3780	91	31	;	;	PUNCT
ejpam-3780	91	32	that	that	ADV
ejpam-3780	91	33	is	is	ADV
ejpam-3780	91	34	,	,	PUNCT
ejpam-3780	91	35	a	a	DET
ejpam-3780	91	36	leg	leg	NOUN
ejpam-3780	91	37	is	be	AUX
ejpam-3780	91	38	a	a	DET
ejpam-3780	91	39	path	path	NOUN
ejpam-3780	91	40	.	.	PUNCT
ejpam-3780	92	1	definition	definition	NOUN
ejpam-3780	92	2	12	12	NUM
ejpam-3780	92	3	.	.	PUNCT
ejpam-3780	93	1	a	a	DET
ejpam-3780	93	2	lobster	lobster	NOUN
ejpam-3780	93	3	is	be	AUX
ejpam-3780	93	4	a	a	DET
ejpam-3780	93	5	tree	tree	NOUN
ejpam-3780	93	6	in	in	ADP
ejpam-3780	93	7	which	which	PRON
ejpam-3780	93	8	all	all	DET
ejpam-3780	93	9	vertices	vertex	NOUN
ejpam-3780	93	10	are	be	AUX
ejpam-3780	93	11	within	within	ADP
ejpam-3780	93	12	distance	distance	NOUN
ejpam-3780	93	13	two	two	NUM
ejpam-3780	93	14	of	of	ADP
ejpam-3780	93	15	a	a	DET
ejpam-3780	93	16	central	central	ADJ
ejpam-3780	93	17	path	path	NOUN
ejpam-3780	93	18	.	.	PUNCT
ejpam-3780	94	1	d.	d.	PROPN
ejpam-3780	94	2	chua	chua	PROPN
ejpam-3780	94	3	,	,	PUNCT
ejpam-3780	94	4	f.	f.	PROPN
ejpam-3780	94	5	campeña	campeña	PROPN
ejpam-3780	94	6	,	,	PUNCT
ejpam-3780	94	7	f.	f.	PROPN
ejpam-3780	94	8	franco	franco	PROPN
ejpam-3780	94	9	/	/	SYM
ejpam-3780	94	10	eur	eur	PROPN
ejpam-3780	94	11	.	.	PUNCT
ejpam-3780	95	1	j.	j.	PROPN
ejpam-3780	95	2	pure	pure	PROPN
ejpam-3780	95	3	appl	appl	PROPN
ejpam-3780	95	4	.	.	PROPN
ejpam-3780	95	5	math	math	PROPN
ejpam-3780	95	6	,	,	PUNCT
ejpam-3780	95	7	13	13	NUM
ejpam-3780	95	8	(	(	PUNCT
ejpam-3780	95	9	3	3	NUM
ejpam-3780	95	10	)	)	PUNCT
ejpam-3780	95	11	(	(	PUNCT
ejpam-3780	95	12	2020	2020	NUM
ejpam-3780	95	13	)	)	PUNCT
ejpam-3780	95	14	,	,	PUNCT
ejpam-3780	95	15	674	674	NUM
ejpam-3780	95	16	-	-	SYM
ejpam-3780	95	17	696	696	NUM
ejpam-3780	95	18	677	677	NUM
ejpam-3780	95	19	we	we	PRON
ejpam-3780	95	20	define	define	VERB
ejpam-3780	95	21	an	an	DET
ejpam-3780	95	22	elbow	elbow	NOUN
ejpam-3780	95	23	of	of	ADP
ejpam-3780	95	24	a	a	DET
ejpam-3780	95	25	vertex	vertex	NOUN
ejpam-3780	95	26	in	in	ADP
ejpam-3780	95	27	the	the	DET
ejpam-3780	95	28	central	central	ADJ
ejpam-3780	95	29	path	path	NOUN
ejpam-3780	95	30	of	of	ADP
ejpam-3780	95	31	a	a	DET
ejpam-3780	95	32	lobster	lobster	NOUN
ejpam-3780	95	33	as	as	ADP
ejpam-3780	95	34	a	a	DET
ejpam-3780	95	35	vertex	vertex	NOUN
ejpam-3780	95	36	that	that	PRON
ejpam-3780	95	37	is	be	AUX
ejpam-3780	95	38	adjacent	adjacent	ADJ
ejpam-3780	95	39	to	to	ADP
ejpam-3780	95	40	it	it	PRON
ejpam-3780	95	41	and	and	CCONJ
ejpam-3780	95	42	whose	whose	DET
ejpam-3780	95	43	distance	distance	NOUN
ejpam-3780	95	44	from	from	ADP
ejpam-3780	95	45	the	the	DET
ejpam-3780	95	46	central	central	ADJ
ejpam-3780	95	47	path	path	NOUN
ejpam-3780	95	48	is	be	AUX
ejpam-3780	95	49	one	one	NUM
ejpam-3780	95	50	.	.	PUNCT
ejpam-3780	96	1	moreover	moreover	ADV
ejpam-3780	96	2	,	,	PUNCT
ejpam-3780	96	3	we	we	PRON
ejpam-3780	96	4	define	define	VERB
ejpam-3780	96	5	a	a	DET
ejpam-3780	96	6	hanging	hang	VERB
ejpam-3780	96	7	leaf	leaf	NOUN
ejpam-3780	96	8	of	of	ADP
ejpam-3780	96	9	an	an	DET
ejpam-3780	96	10	elbow	elbow	NOUN
ejpam-3780	96	11	as	as	ADP
ejpam-3780	96	12	a	a	DET
ejpam-3780	96	13	vertex	vertex	NOUN
ejpam-3780	96	14	that	that	PRON
ejpam-3780	96	15	is	be	AUX
ejpam-3780	96	16	adjacent	adjacent	ADJ
ejpam-3780	96	17	to	to	ADP
ejpam-3780	96	18	it	it	PRON
ejpam-3780	96	19	and	and	CCONJ
ejpam-3780	96	20	whose	whose	DET
ejpam-3780	96	21	distance	distance	NOUN
ejpam-3780	96	22	from	from	ADP
ejpam-3780	96	23	the	the	DET
ejpam-3780	96	24	central	central	ADJ
ejpam-3780	96	25	path	path	NOUN
ejpam-3780	96	26	is	be	AUX
ejpam-3780	96	27	two	two	NUM
ejpam-3780	96	28	.	.	PUNCT
ejpam-3780	97	1	definition	definition	NOUN
ejpam-3780	97	2	13	13	NUM
ejpam-3780	97	3	.	.	PUNCT
ejpam-3780	98	1	a	a	DET
ejpam-3780	98	2	rooted	rooted	ADJ
ejpam-3780	98	3	tree	tree	NOUN
ejpam-3780	98	4	is	be	AUX
ejpam-3780	98	5	a	a	DET
ejpam-3780	98	6	tree	tree	NOUN
ejpam-3780	98	7	in	in	ADP
ejpam-3780	98	8	which	which	PRON
ejpam-3780	98	9	a	a	DET
ejpam-3780	98	10	particular	particular	ADJ
ejpam-3780	98	11	vertex	vertex	NOUN
ejpam-3780	98	12	is	be	AUX
ejpam-3780	98	13	designated	designate	VERB
ejpam-3780	98	14	as	as	ADP
ejpam-3780	98	15	the	the	DET
ejpam-3780	98	16	root	root	NOUN
ejpam-3780	98	17	.	.	PUNCT
ejpam-3780	99	1	the	the	DET
ejpam-3780	99	2	level	level	NOUN
ejpam-3780	99	3	of	of	ADP
ejpam-3780	99	4	a	a	DET
ejpam-3780	99	5	vertex	vertex	NOUN
ejpam-3780	99	6	in	in	ADP
ejpam-3780	99	7	a	a	DET
ejpam-3780	99	8	rooted	rooted	ADJ
ejpam-3780	99	9	tree	tree	NOUN
ejpam-3780	99	10	is	be	AUX
ejpam-3780	99	11	its	its	PRON
ejpam-3780	99	12	distance	distance	NOUN
ejpam-3780	99	13	from	from	ADP
ejpam-3780	99	14	the	the	DET
ejpam-3780	99	15	root	root	NOUN
ejpam-3780	99	16	,	,	PUNCT
ejpam-3780	99	17	and	and	CCONJ
ejpam-3780	99	18	the	the	DET
ejpam-3780	99	19	height	height	NOUN
ejpam-3780	99	20	of	of	ADP
ejpam-3780	99	21	a	a	DET
ejpam-3780	99	22	rooted	rooted	ADJ
ejpam-3780	99	23	tree	tree	NOUN
ejpam-3780	99	24	is	be	AUX
ejpam-3780	99	25	the	the	DET
ejpam-3780	99	26	maximum	maximum	ADJ
ejpam-3780	99	27	level	level	NOUN
ejpam-3780	99	28	of	of	ADP
ejpam-3780	99	29	a	a	DET
ejpam-3780	99	30	vertex	vertex	NOUN
ejpam-3780	99	31	.	.	PUNCT
ejpam-3780	100	1	a	a	DET
ejpam-3780	100	2	child	child	NOUN
ejpam-3780	100	3	(	(	PUNCT
ejpam-3780	100	4	plural	plural	ADJ
ejpam-3780	100	5	:	:	PUNCT
ejpam-3780	100	6	children	child	NOUN
ejpam-3780	100	7	)	)	PUNCT
ejpam-3780	100	8	of	of	ADP
ejpam-3780	100	9	a	a	DET
ejpam-3780	100	10	vertex	vertex	NOUN
ejpam-3780	100	11	of	of	ADP
ejpam-3780	100	12	level	level	NOUN
ejpam-3780	100	13	n	n	NOUN
ejpam-3780	100	14	in	in	ADP
ejpam-3780	100	15	a	a	DET
ejpam-3780	100	16	rooted	rooted	ADJ
ejpam-3780	100	17	tree	tree	NOUN
ejpam-3780	100	18	is	be	AUX
ejpam-3780	100	19	an	an	DET
ejpam-3780	100	20	adjacent	adjacent	ADJ
ejpam-3780	100	21	vertex	vertex	NOUN
ejpam-3780	100	22	whose	whose	DET
ejpam-3780	100	23	level	level	NOUN
ejpam-3780	100	24	is	be	AUX
ejpam-3780	100	25	n	n	NOUN
ejpam-3780	100	26	+	+	NUM
ejpam-3780	100	27	1	1	NUM
ejpam-3780	100	28	.	.	X
ejpam-3780	100	29	2.3	2.3	NUM
ejpam-3780	100	30	.	.	PUNCT
ejpam-3780	101	1	common	common	ADJ
ejpam-3780	101	2	classes	class	NOUN
ejpam-3780	101	3	of	of	ADP
ejpam-3780	101	4	graphs	graph	NOUN
ejpam-3780	101	5	we	we	PRON
ejpam-3780	101	6	now	now	ADV
ejpam-3780	101	7	define	define	VERB
ejpam-3780	101	8	other	other	ADJ
ejpam-3780	101	9	common	common	ADJ
ejpam-3780	101	10	classes	class	NOUN
ejpam-3780	101	11	of	of	ADP
ejpam-3780	101	12	graphs	graph	NOUN
ejpam-3780	101	13	that	that	PRON
ejpam-3780	101	14	are	be	AUX
ejpam-3780	101	15	considered	consider	VERB
ejpam-3780	101	16	in	in	ADP
ejpam-3780	101	17	this	this	DET
ejpam-3780	101	18	paper	paper	NOUN
ejpam-3780	101	19	.	.	PUNCT
ejpam-3780	102	1	definition	definition	NOUN
ejpam-3780	102	2	14	14	NUM
ejpam-3780	102	3	.	.	PUNCT
ejpam-3780	103	1	a	a	DET
ejpam-3780	103	2	cycle	cycle	NOUN
ejpam-3780	103	3	graph	graph	NOUN
ejpam-3780	103	4	cn	cn	NOUN
ejpam-3780	104	1	=	=	PUNCT
ejpam-3780	105	1	[	[	X
ejpam-3780	105	2	v1	v1	NOUN
ejpam-3780	105	3	,	,	PUNCT
ejpam-3780	105	4	v2	v2	NOUN
ejpam-3780	105	5	,	,	PUNCT
ejpam-3780	105	6	.	.	PUNCT
ejpam-3780	105	7	.	.	PUNCT
ejpam-3780	106	1	.	.	PUNCT
ejpam-3780	107	1	,	,	PUNCT
ejpam-3780	107	2	vn	vn	X
ejpam-3780	107	3	,	,	PUNCT
ejpam-3780	107	4	v1	v1	PROPN
ejpam-3780	107	5	]	]	PUNCT
ejpam-3780	107	6	is	be	AUX
ejpam-3780	107	7	a	a	DET
ejpam-3780	107	8	graph	graph	NOUN
ejpam-3780	107	9	with	with	ADP
ejpam-3780	107	10	vertex	vertex	NOUN
ejpam-3780	107	11	set	set	VERB
ejpam-3780	107	12	v	v	NOUN
ejpam-3780	107	13	=	=	SYM
ejpam-3780	107	14	{	{	PUNCT
ejpam-3780	107	15	v1	v1	PROPN
ejpam-3780	107	16	,	,	PUNCT
ejpam-3780	107	17	v2	v2	PROPN
ejpam-3780	107	18	,	,	PUNCT
ejpam-3780	107	19	.	.	PUNCT
ejpam-3780	107	20	.	.	PUNCT
ejpam-3780	108	1	.	.	PUNCT
ejpam-3780	109	1	,	,	PUNCT
ejpam-3780	109	2	vn	vn	INTJ
ejpam-3780	109	3	}	}	PUNCT
ejpam-3780	109	4	and	and	CCONJ
ejpam-3780	109	5	edge	edge	VERB
ejpam-3780	109	6	set	set	VERB
ejpam-3780	109	7	e	e	NOUN
ejpam-3780	109	8	=	=	PRON
ejpam-3780	109	9	{	{	PUNCT
ejpam-3780	109	10	v1v2	v1v2	PROPN
ejpam-3780	109	11	,	,	PUNCT
ejpam-3780	109	12	v2v3	v2v3	PROPN
ejpam-3780	109	13	,	,	PUNCT
ejpam-3780	109	14	.	.	PUNCT
ejpam-3780	109	15	.	.	PUNCT
ejpam-3780	109	16	.	.	PUNCT
ejpam-3780	110	1	,	,	PUNCT
ejpam-3780	110	2	vn−1vn	vn−1vn	NUM
ejpam-3780	110	3	}	}	PUNCT
ejpam-3780	110	4	∪	∪	X
ejpam-3780	110	5	{	{	PUNCT
ejpam-3780	110	6	v1vn	v1vn	NOUN
ejpam-3780	110	7	}	}	PUNCT
ejpam-3780	110	8	,	,	PUNCT
ejpam-3780	110	9	where	where	SCONJ
ejpam-3780	110	10	n	n	PRON
ejpam-3780	110	11	≥	≥	NOUN
ejpam-3780	110	12	3	3	NUM
ejpam-3780	110	13	.	.	PUNCT
ejpam-3780	110	14	definition	definition	NOUN
ejpam-3780	110	15	15	15	NUM
ejpam-3780	110	16	.	.	PUNCT
ejpam-3780	111	1	a	a	DET
ejpam-3780	111	2	complete	complete	ADJ
ejpam-3780	111	3	bipartite	bipartite	NOUN
ejpam-3780	111	4	graph	graph	NOUN
ejpam-3780	111	5	km	km	PROPN
ejpam-3780	111	6	,	,	PUNCT
ejpam-3780	111	7	n	n	PRON
ejpam-3780	111	8	is	be	AUX
ejpam-3780	111	9	a	a	DET
ejpam-3780	111	10	graph	graph	NOUN
ejpam-3780	111	11	with	with	ADP
ejpam-3780	111	12	vertex	vertex	NOUN
ejpam-3780	111	13	set	set	VERB
ejpam-3780	111	14	v	v	NOUN
ejpam-3780	111	15	=	=	SYM
ejpam-3780	111	16	{	{	PUNCT
ejpam-3780	111	17	x1	x1	PROPN
ejpam-3780	111	18	,	,	PUNCT
ejpam-3780	111	19	x2	x2	PROPN
ejpam-3780	111	20	,	,	PUNCT
ejpam-3780	111	21	.	.	PUNCT
ejpam-3780	111	22	.	.	PUNCT
ejpam-3780	112	1	.	.	PUNCT
ejpam-3780	113	1	,	,	PUNCT
ejpam-3780	113	2	xm	xm	X
ejpam-3780	113	3	}	}	PUNCT
ejpam-3780	113	4	∪	∪	ADJ
ejpam-3780	113	5	{	{	PUNCT
ejpam-3780	113	6	y1	y1	NOUN
ejpam-3780	113	7	,	,	PUNCT
ejpam-3780	113	8	y2	y2	PROPN
ejpam-3780	113	9	,	,	PUNCT
ejpam-3780	113	10	.	.	PUNCT
ejpam-3780	113	11	.	.	PUNCT
ejpam-3780	114	1	.	.	PUNCT
ejpam-3780	115	1	,	,	PUNCT
ejpam-3780	115	2	yn	yn	PROPN
ejpam-3780	115	3	}	}	PUNCT
ejpam-3780	115	4	,	,	PUNCT
ejpam-3780	115	5	where	where	SCONJ
ejpam-3780	115	6	m	m	PROPN
ejpam-3780	115	7	≥	≥	VERB
ejpam-3780	115	8	1	1	NUM
ejpam-3780	115	9	and	and	CCONJ
ejpam-3780	115	10	n	n	PRON
ejpam-3780	115	11	≥	≥	NOUN
ejpam-3780	115	12	1	1	NUM
ejpam-3780	115	13	,	,	PUNCT
ejpam-3780	115	14	and	and	CCONJ
ejpam-3780	115	15	edge	edge	VERB
ejpam-3780	115	16	set	set	VERB
ejpam-3780	115	17	e	e	NOUN
ejpam-3780	115	18	=	=	PUNCT
ejpam-3780	115	19	{	{	PUNCT
ejpam-3780	115	20	xiyj	xiyj	NOUN
ejpam-3780	115	21	:	:	PUNCT
ejpam-3780	115	22	1	1	NUM
ejpam-3780	115	23	≤	≤	NUM
ejpam-3780	115	24	i	i	X
ejpam-3780	116	1	≤	≤	NOUN
ejpam-3780	116	2	m	m	VERB
ejpam-3780	116	3	and	and	CCONJ
ejpam-3780	116	4	1	1	NUM
ejpam-3780	116	5	≤	≤	NUM
ejpam-3780	116	6	j	j	PROPN
ejpam-3780	116	7	≤	≤	PROPN
ejpam-3780	116	8	n	n	CCONJ
ejpam-3780	116	9	}	}	PUNCT
ejpam-3780	116	10	.	.	PUNCT
ejpam-3780	117	1	definition	definition	NOUN
ejpam-3780	117	2	16	16	NUM
ejpam-3780	117	3	.	.	PUNCT
ejpam-3780	118	1	a	a	DET
ejpam-3780	118	2	fan	fan	NOUN
ejpam-3780	118	3	graph	graph	NOUN
ejpam-3780	118	4	fn	fn	NOUN
ejpam-3780	118	5	is	be	AUX
ejpam-3780	118	6	a	a	DET
ejpam-3780	118	7	graph	graph	NOUN
ejpam-3780	118	8	that	that	PRON
ejpam-3780	118	9	is	be	AUX
ejpam-3780	118	10	obtained	obtain	VERB
ejpam-3780	118	11	by	by	ADP
ejpam-3780	118	12	joining	join	VERB
ejpam-3780	118	13	all	all	DET
ejpam-3780	118	14	vertices	vertex	NOUN
ejpam-3780	118	15	of	of	ADP
ejpam-3780	118	16	pn	pn	NOUN
ejpam-3780	118	17	to	to	ADP
ejpam-3780	118	18	another	another	DET
ejpam-3780	118	19	vertex	vertex	NOUN
ejpam-3780	118	20	.	.	PUNCT
ejpam-3780	119	1	definition	definition	NOUN
ejpam-3780	119	2	17	17	NUM
ejpam-3780	119	3	.	.	PUNCT
ejpam-3780	120	1	a	a	DET
ejpam-3780	120	2	wheel	wheel	NOUN
ejpam-3780	120	3	graph	graph	NOUN
ejpam-3780	120	4	wn	wn	PROPN
ejpam-3780	120	5	is	be	AUX
ejpam-3780	120	6	a	a	DET
ejpam-3780	120	7	graph	graph	NOUN
ejpam-3780	120	8	that	that	PRON
ejpam-3780	120	9	is	be	AUX
ejpam-3780	120	10	obtained	obtain	VERB
ejpam-3780	120	11	by	by	ADP
ejpam-3780	120	12	joining	join	VERB
ejpam-3780	120	13	all	all	DET
ejpam-3780	120	14	vertices	vertex	NOUN
ejpam-3780	120	15	of	of	ADP
ejpam-3780	120	16	cn	cn	PROPN
ejpam-3780	120	17	to	to	ADP
ejpam-3780	120	18	another	another	DET
ejpam-3780	120	19	vertex	vertex	NOUN
ejpam-3780	120	20	.	.	PUNCT
ejpam-3780	121	1	definition	definition	NOUN
ejpam-3780	121	2	18	18	NUM
ejpam-3780	121	3	.	.	PUNCT
ejpam-3780	122	1	a	a	DET
ejpam-3780	122	2	friendship	friendship	NOUN
ejpam-3780	122	3	graph	graph	NOUN
ejpam-3780	122	4	tn	tn	NOUN
ejpam-3780	122	5	is	be	AUX
ejpam-3780	122	6	a	a	DET
ejpam-3780	122	7	graph	graph	NOUN
ejpam-3780	122	8	that	that	PRON
ejpam-3780	122	9	consists	consist	VERB
ejpam-3780	122	10	of	of	ADP
ejpam-3780	122	11	n	n	NOUN
ejpam-3780	122	12	copies	copy	NOUN
ejpam-3780	122	13	of	of	ADP
ejpam-3780	122	14	c3	c3	PROPN
ejpam-3780	122	15	having	have	VERB
ejpam-3780	122	16	exactly	exactly	ADV
ejpam-3780	122	17	one	one	NUM
ejpam-3780	122	18	common	common	ADJ
ejpam-3780	122	19	vertex	vertex	NOUN
ejpam-3780	122	20	.	.	PUNCT
ejpam-3780	123	1	2.4	2.4	NUM
ejpam-3780	123	2	.	.	PUNCT
ejpam-3780	124	1	efficient	efficient	ADJ
ejpam-3780	124	2	zero	zero	NUM
ejpam-3780	124	3	ring	ring	NOUN
ejpam-3780	124	4	labeling	labeling	NOUN
ejpam-3780	124	5	we	we	PRON
ejpam-3780	124	6	now	now	ADV
ejpam-3780	124	7	define	define	VERB
ejpam-3780	124	8	a	a	DET
ejpam-3780	124	9	vertex	vertex	NOUN
ejpam-3780	124	10	labeling	labeling	NOUN
ejpam-3780	124	11	of	of	ADP
ejpam-3780	124	12	a	a	DET
ejpam-3780	124	13	graph	graph	NOUN
ejpam-3780	124	14	which	which	PRON
ejpam-3780	124	15	was	be	AUX
ejpam-3780	124	16	introduced	introduce	VERB
ejpam-3780	124	17	by	by	ADP
ejpam-3780	124	18	acharya	acharya	PROPN
ejpam-3780	124	19	et	et	PROPN
ejpam-3780	124	20	al	al	PROPN
ejpam-3780	125	1	[	[	X
ejpam-3780	125	2	1	1	NUM
ejpam-3780	125	3	,	,	PUNCT
ejpam-3780	125	4	6	6	NUM
ejpam-3780	125	5	]	]	PUNCT
ejpam-3780	125	6	called	call	VERB
ejpam-3780	125	7	zero	zero	NUM
ejpam-3780	125	8	ring	ring	NOUN
ejpam-3780	125	9	labeling	labeling	NOUN
ejpam-3780	125	10	.	.	PUNCT
ejpam-3780	126	1	moreover	moreover	ADV
ejpam-3780	126	2	,	,	PUNCT
ejpam-3780	126	3	this	this	DET
ejpam-3780	126	4	paper	paper	NOUN
ejpam-3780	126	5	introduces	introduce	VERB
ejpam-3780	126	6	a	a	DET
ejpam-3780	126	7	variation	variation	NOUN
ejpam-3780	126	8	of	of	ADP
ejpam-3780	126	9	this	this	DET
ejpam-3780	126	10	vertex	vertex	NOUN
ejpam-3780	126	11	labeling	labeling	NOUN
ejpam-3780	126	12	called	call	VERB
ejpam-3780	126	13	k	k	PROPN
ejpam-3780	126	14	-	-	ADJ
ejpam-3780	126	15	zero	zero	NUM
ejpam-3780	126	16	ring	ring	NOUN
ejpam-3780	126	17	labeling	labeling	NOUN
ejpam-3780	126	18	and	and	CCONJ
ejpam-3780	126	19	efficient	efficient	ADJ
ejpam-3780	126	20	zero	zero	NUM
ejpam-3780	126	21	ring	ring	NOUN
ejpam-3780	126	22	labeling	labeling	NOUN
ejpam-3780	126	23	of	of	ADP
ejpam-3780	126	24	graphs	graph	NOUN
ejpam-3780	126	25	.	.	PUNCT
ejpam-3780	127	1	in	in	ADP
ejpam-3780	127	2	this	this	DET
ejpam-3780	127	3	study	study	NOUN
ejpam-3780	127	4	,	,	PUNCT
ejpam-3780	127	5	the	the	DET
ejpam-3780	127	6	zero	zero	NUM
ejpam-3780	127	7	ring	ring	NOUN
ejpam-3780	127	8	that	that	PRON
ejpam-3780	127	9	will	will	AUX
ejpam-3780	127	10	be	be	AUX
ejpam-3780	127	11	used	use	VERB
ejpam-3780	127	12	in	in	ADP
ejpam-3780	127	13	the	the	DET
ejpam-3780	127	14	vertex	vertex	NOUN
ejpam-3780	127	15	labelings	labeling	NOUN
ejpam-3780	127	16	is	be	AUX
ejpam-3780	127	17	the	the	DET
ejpam-3780	127	18	zero	zero	NUM
ejpam-3780	127	19	ring	ring	NOUN
ejpam-3780	127	20	m0	m0	NOUN
ejpam-3780	127	21	2	2	NUM
ejpam-3780	127	22	(	(	PUNCT
ejpam-3780	127	23	zn	zn	NUM
ejpam-3780	127	24	)	)	PUNCT
ejpam-3780	127	25	.	.	PUNCT
ejpam-3780	128	1	definition	definition	NOUN
ejpam-3780	128	2	19	19	NUM
ejpam-3780	128	3	.	.	PUNCT
ejpam-3780	129	1	let	let	VERB
ejpam-3780	129	2	r	r	PRON
ejpam-3780	129	3	be	be	AUX
ejpam-3780	129	4	a	a	DET
ejpam-3780	129	5	ring	ring	NOUN
ejpam-3780	129	6	with	with	ADP
ejpam-3780	129	7	additive	additive	ADJ
ejpam-3780	129	8	identity	identity	NOUN
ejpam-3780	129	9	0	0	NUM
ejpam-3780	129	10	.	.	PUNCT
ejpam-3780	130	1	if	if	SCONJ
ejpam-3780	130	2	ab	ab	PROPN
ejpam-3780	130	3	=	=	NOUN
ejpam-3780	130	4	0	0	NUM
ejpam-3780	130	5	for	for	ADP
ejpam-3780	130	6	any	any	DET
ejpam-3780	130	7	a	a	NOUN
ejpam-3780	130	8	,	,	PUNCT
ejpam-3780	130	9	b	b	X
ejpam-3780	130	10	∈	∈	PROPN
ejpam-3780	130	11	r	r	NOUN
ejpam-3780	130	12	,	,	PUNCT
ejpam-3780	130	13	then	then	ADV
ejpam-3780	130	14	r	r	NOUN
ejpam-3780	130	15	is	be	AUX
ejpam-3780	130	16	a	a	DET
ejpam-3780	130	17	zero	zero	NUM
ejpam-3780	130	18	ring	ring	NOUN
ejpam-3780	130	19	.	.	PUNCT
ejpam-3780	131	1	let	let	VERB
ejpam-3780	131	2	r	r	PRON
ejpam-3780	131	3	be	be	AUX
ejpam-3780	131	4	a	a	DET
ejpam-3780	131	5	ring	ring	NOUN
ejpam-3780	131	6	with	with	ADP
ejpam-3780	131	7	additive	additive	ADJ
ejpam-3780	131	8	identity	identity	NOUN
ejpam-3780	131	9	0	0	NUM
ejpam-3780	131	10	.	.	PUNCT
ejpam-3780	132	1	we	we	PRON
ejpam-3780	132	2	denote	denote	VERB
ejpam-3780	132	3	by	by	ADP
ejpam-3780	132	4	m0	m0	PROPN
ejpam-3780	132	5	2	2	NUM
ejpam-3780	132	6	(	(	PUNCT
ejpam-3780	132	7	r	r	NOUN
ejpam-3780	132	8	)	)	PUNCT
ejpam-3780	132	9	the	the	DET
ejpam-3780	132	10	set	set	NOUN
ejpam-3780	132	11	of	of	ADP
ejpam-3780	132	12	all	all	DET
ejpam-3780	132	13	2	2	NUM
ejpam-3780	132	14	×	×	NOUN
ejpam-3780	132	15	2	2	NUM
ejpam-3780	132	16	matrices	matrix	NOUN
ejpam-3780	132	17	of	of	ADP
ejpam-3780	132	18	the	the	DET
ejpam-3780	132	19	form	form	NOUN
ejpam-3780	132	20	[	[	PUNCT
ejpam-3780	132	21	a	a	DET
ejpam-3780	132	22	−a	−a	NOUN
ejpam-3780	132	23	a	a	DET
ejpam-3780	132	24	−a	−a	NOUN
ejpam-3780	132	25	]	]	PUNCT
ejpam-3780	132	26	,	,	PUNCT
ejpam-3780	132	27	a	a	DET
ejpam-3780	132	28	∈	∈	PROPN
ejpam-3780	132	29	r.	r.	PROPN
ejpam-3780	132	30	d.	d.	PROPN
ejpam-3780	132	31	chua	chua	PROPN
ejpam-3780	132	32	,	,	PUNCT
ejpam-3780	132	33	f.	f.	PROPN
ejpam-3780	132	34	campeña	campeña	PROPN
ejpam-3780	132	35	,	,	PUNCT
ejpam-3780	132	36	f.	f.	PROPN
ejpam-3780	132	37	franco	franco	PROPN
ejpam-3780	132	38	/	/	SYM
ejpam-3780	132	39	eur	eur	PROPN
ejpam-3780	132	40	.	.	PUNCT
ejpam-3780	133	1	j.	j.	PROPN
ejpam-3780	133	2	pure	pure	PROPN
ejpam-3780	133	3	appl	appl	PROPN
ejpam-3780	133	4	.	.	PROPN
ejpam-3780	133	5	math	math	PROPN
ejpam-3780	133	6	,	,	PUNCT
ejpam-3780	133	7	13	13	NUM
ejpam-3780	133	8	(	(	PUNCT
ejpam-3780	133	9	3	3	NUM
ejpam-3780	133	10	)	)	PUNCT
ejpam-3780	133	11	(	(	PUNCT
ejpam-3780	133	12	2020	2020	NUM
ejpam-3780	133	13	)	)	PUNCT
ejpam-3780	133	14	,	,	PUNCT
ejpam-3780	133	15	674	674	NUM
ejpam-3780	133	16	-	-	SYM
ejpam-3780	133	17	696	696	NUM
ejpam-3780	133	18	678	678	NUM
ejpam-3780	133	19	it	it	PRON
ejpam-3780	133	20	can	can	AUX
ejpam-3780	133	21	be	be	AUX
ejpam-3780	133	22	verified	verify	VERB
ejpam-3780	133	23	that	that	SCONJ
ejpam-3780	133	24	m0	m0	NOUN
ejpam-3780	133	25	2	2	NUM
ejpam-3780	133	26	(	(	PUNCT
ejpam-3780	133	27	r	r	NOUN
ejpam-3780	133	28	)	)	PUNCT
ejpam-3780	133	29	is	be	AUX
ejpam-3780	133	30	a	a	DET
ejpam-3780	133	31	ring	ring	NOUN
ejpam-3780	133	32	under	under	ADP
ejpam-3780	133	33	matrix	matrix	NOUN
ejpam-3780	133	34	addition	addition	NOUN
ejpam-3780	133	35	and	and	CCONJ
ejpam-3780	133	36	matrix	matrix	NOUN
ejpam-3780	133	37	multiplication	multiplication	NOUN
ejpam-3780	133	38	with	with	ADP
ejpam-3780	133	39	additive	additive	ADJ
ejpam-3780	133	40	identity	identity	NOUN
ejpam-3780	133	41	[	[	PUNCT
ejpam-3780	133	42	0	0	NUM
ejpam-3780	133	43	0	0	NUM
ejpam-3780	133	44	0	0	NUM
ejpam-3780	133	45	0	0	NUM
ejpam-3780	133	46	]	]	PUNCT
ejpam-3780	133	47	.	.	PUNCT
ejpam-3780	134	1	since	since	SCONJ
ejpam-3780	134	2	for	for	ADP
ejpam-3780	134	3	any	any	DET
ejpam-3780	134	4	a	a	NOUN
ejpam-3780	134	5	,	,	PUNCT
ejpam-3780	134	6	b	b	X
ejpam-3780	134	7	∈	∈	PROPN
ejpam-3780	134	8	r	r	NOUN
ejpam-3780	134	9	,	,	PUNCT
ejpam-3780	134	10	[	[	PUNCT
ejpam-3780	134	11	a	a	DET
ejpam-3780	134	12	−a	−a	NOUN
ejpam-3780	134	13	a	a	DET
ejpam-3780	134	14	−a	−a	NOUN
ejpam-3780	134	15	]	]	PUNCT
ejpam-3780	134	16	[	[	PUNCT
ejpam-3780	134	17	b	b	X
ejpam-3780	134	18	−b	−b	ADP
ejpam-3780	134	19	b	b	X
ejpam-3780	134	20	−b	−b	ADJ
ejpam-3780	134	21	]	]	PUNCT
ejpam-3780	135	1	=	=	PUNCT
ejpam-3780	135	2	[	[	PUNCT
ejpam-3780	135	3	ab−	ab−	NUM
ejpam-3780	135	4	ab	ab	PROPN
ejpam-3780	135	5	−ab	−ab	PROPN
ejpam-3780	136	1	+	+	CCONJ
ejpam-3780	136	2	ab	ab	PROPN
ejpam-3780	136	3	ab−	ab−	NUM
ejpam-3780	136	4	ab	ab	PROPN
ejpam-3780	136	5	−ab	−ab	NOUN
ejpam-3780	136	6	+	+	CCONJ
ejpam-3780	136	7	ab	ab	X
ejpam-3780	136	8	]	]	PUNCT
ejpam-3780	137	1	=	=	PUNCT
ejpam-3780	137	2	[	[	PUNCT
ejpam-3780	137	3	0	0	NUM
ejpam-3780	137	4	0	0	NUM
ejpam-3780	137	5	0	0	NUM
ejpam-3780	137	6	0	0	NUM
ejpam-3780	137	7	]	]	PUNCT
ejpam-3780	137	8	,	,	PUNCT
ejpam-3780	137	9	(	(	PUNCT
ejpam-3780	137	10	1	1	X
ejpam-3780	137	11	)	)	PUNCT
ejpam-3780	137	12	m0	m0	NOUN
ejpam-3780	137	13	2	2	NUM
ejpam-3780	137	14	(	(	PUNCT
ejpam-3780	137	15	r	r	NOUN
ejpam-3780	137	16	)	)	PUNCT
ejpam-3780	137	17	is	be	AUX
ejpam-3780	137	18	a	a	DET
ejpam-3780	137	19	zero	zero	NUM
ejpam-3780	137	20	ring	ring	NOUN
ejpam-3780	137	21	.	.	PUNCT
ejpam-3780	138	1	since	since	SCONJ
ejpam-3780	138	2	zn	zn	PROPN
ejpam-3780	138	3	is	be	AUX
ejpam-3780	138	4	a	a	DET
ejpam-3780	138	5	ring	ring	NOUN
ejpam-3780	138	6	,	,	PUNCT
ejpam-3780	138	7	it	it	PRON
ejpam-3780	138	8	follows	follow	VERB
ejpam-3780	138	9	that	that	SCONJ
ejpam-3780	138	10	m0	m0	PROPN
ejpam-3780	138	11	2	2	NUM
ejpam-3780	138	12	(	(	PUNCT
ejpam-3780	138	13	zn	zn	NOUN
ejpam-3780	138	14	)	)	PUNCT
ejpam-3780	138	15	is	be	AUX
ejpam-3780	138	16	a	a	DET
ejpam-3780	138	17	zero	zero	NUM
ejpam-3780	138	18	ring	ring	NOUN
ejpam-3780	138	19	.	.	PUNCT
ejpam-3780	139	1	we	we	PRON
ejpam-3780	139	2	use	use	VERB
ejpam-3780	139	3	ai	ai	VERB
ejpam-3780	139	4	to	to	PART
ejpam-3780	139	5	denote	denote	VERB
ejpam-3780	139	6	the	the	DET
ejpam-3780	139	7	matrix	matrix	NOUN
ejpam-3780	139	8	[	[	PUNCT
ejpam-3780	139	9	i	i	PRON
ejpam-3780	139	10	−i	−i	PROPN
ejpam-3780	140	1	i	i	PRON
ejpam-3780	140	2	−i	−i	ADV
ejpam-3780	140	3	]	]	PUNCT
ejpam-3780	140	4	,	,	PUNCT
ejpam-3780	140	5	i	i	PRON
ejpam-3780	140	6	∈	∈	PROPN
ejpam-3780	141	1	zn	zn	X
ejpam-3780	141	2	.	.	PUNCT
ejpam-3780	142	1	definition	definition	NOUN
ejpam-3780	142	2	20	20	NUM
ejpam-3780	142	3	.	.	PUNCT
ejpam-3780	143	1	let	let	VERB
ejpam-3780	143	2	g	g	PRON
ejpam-3780	143	3	be	be	AUX
ejpam-3780	143	4	a	a	DET
ejpam-3780	143	5	graph	graph	NOUN
ejpam-3780	143	6	,	,	PUNCT
ejpam-3780	143	7	and	and	CCONJ
ejpam-3780	143	8	let	let	VERB
ejpam-3780	143	9	r0	r0	NOUN
ejpam-3780	143	10	be	be	AUX
ejpam-3780	143	11	a	a	DET
ejpam-3780	143	12	finite	finite	ADJ
ejpam-3780	143	13	zero	zero	NUM
ejpam-3780	143	14	ring	ring	NOUN
ejpam-3780	143	15	.	.	PUNCT
ejpam-3780	144	1	an	an	DET
ejpam-3780	144	2	injective	injective	ADJ
ejpam-3780	144	3	function	function	NOUN
ejpam-3780	144	4	f	f	NOUN
ejpam-3780	144	5	:	:	PUNCT
ejpam-3780	144	6	v	v	NOUN
ejpam-3780	144	7	(	(	PUNCT
ejpam-3780	144	8	g)→	g)→	NOUN
ejpam-3780	144	9	r0	r0	NOUN
ejpam-3780	144	10	is	be	AUX
ejpam-3780	144	11	a	a	DET
ejpam-3780	144	12	zero	zero	NUM
ejpam-3780	144	13	ring	ring	NOUN
ejpam-3780	144	14	labeling	labeling	NOUN
ejpam-3780	144	15	of	of	ADP
ejpam-3780	144	16	g	g	PROPN
ejpam-3780	144	17	if	if	SCONJ
ejpam-3780	144	18	f(u	f(u	PROPN
ejpam-3780	144	19	)	)	PUNCT
ejpam-3780	144	20	+	+	NUM
ejpam-3780	144	21	f(v	f(v	NOUN
ejpam-3780	144	22	)	)	PUNCT
ejpam-3780	144	23	6=	6=	ADP
ejpam-3780	144	24	0	0	NUM
ejpam-3780	144	25	for	for	ADP
ejpam-3780	144	26	every	every	DET
ejpam-3780	144	27	uv	uv	PROPN
ejpam-3780	144	28	∈	∈	PROPN
ejpam-3780	144	29	e(g	e(g	PROPN
ejpam-3780	144	30	)	)	PUNCT
ejpam-3780	144	31	.	.	PUNCT
ejpam-3780	145	1	given	give	VERB
ejpam-3780	145	2	a	a	DET
ejpam-3780	145	3	zero	zero	NUM
ejpam-3780	145	4	ring	ring	NOUN
ejpam-3780	145	5	labeling	labeling	NOUN
ejpam-3780	145	6	,	,	PUNCT
ejpam-3780	145	7	if	if	SCONJ
ejpam-3780	145	8	the	the	DET
ejpam-3780	145	9	order	order	NOUN
ejpam-3780	145	10	of	of	ADP
ejpam-3780	145	11	the	the	DET
ejpam-3780	145	12	zero	zero	NUM
ejpam-3780	145	13	ring	ring	NOUN
ejpam-3780	145	14	is	be	AUX
ejpam-3780	145	15	equal	equal	ADJ
ejpam-3780	145	16	to	to	ADP
ejpam-3780	145	17	the	the	DET
ejpam-3780	145	18	order	order	NOUN
ejpam-3780	145	19	of	of	ADP
ejpam-3780	145	20	the	the	DET
ejpam-3780	145	21	graph	graph	NOUN
ejpam-3780	145	22	,	,	PUNCT
ejpam-3780	145	23	we	we	PRON
ejpam-3780	145	24	say	say	VERB
ejpam-3780	145	25	that	that	SCONJ
ejpam-3780	145	26	the	the	DET
ejpam-3780	145	27	labeling	labeling	NOUN
ejpam-3780	145	28	is	be	AUX
ejpam-3780	145	29	optimal	optimal	ADJ
ejpam-3780	145	30	.	.	PUNCT
ejpam-3780	146	1	example	example	NOUN
ejpam-3780	146	2	2	2	NUM
ejpam-3780	146	3	.	.	X
ejpam-3780	146	4	figure	figure	NOUN
ejpam-3780	146	5	2	2	NUM
ejpam-3780	146	6	shows	show	VERB
ejpam-3780	146	7	a	a	DET
ejpam-3780	146	8	zero	zero	NUM
ejpam-3780	146	9	ring	ring	NOUN
ejpam-3780	146	10	labeling	labeling	NOUN
ejpam-3780	146	11	of	of	ADP
ejpam-3780	146	12	c5	c5	PROPN
ejpam-3780	146	13	using	use	VERB
ejpam-3780	146	14	m0	m0	PROPN
ejpam-3780	146	15	2	2	NUM
ejpam-3780	146	16	(	(	PUNCT
ejpam-3780	146	17	z5	z5	PROPN
ejpam-3780	146	18	)	)	PUNCT
ejpam-3780	146	19	.	.	PUNCT
ejpam-3780	147	1	since	since	SCONJ
ejpam-3780	147	2	|c5|	|c5|	NOUN
ejpam-3780	147	3	=	=	SYM
ejpam-3780	147	4	|m0	|m0	NOUN
ejpam-3780	147	5	2	2	NUM
ejpam-3780	147	6	(	(	PUNCT
ejpam-3780	147	7	z5)|	z5)|	PROPN
ejpam-3780	147	8	=	=	SYM
ejpam-3780	147	9	5	5	NUM
ejpam-3780	147	10	,	,	PUNCT
ejpam-3780	147	11	this	this	DET
ejpam-3780	147	12	labeling	labeling	NOUN
ejpam-3780	147	13	is	be	AUX
ejpam-3780	147	14	optimal	optimal	ADJ
ejpam-3780	147	15	.	.	PUNCT
ejpam-3780	148	1	a3	a3	PROPN
ejpam-3780	148	2	a0	a0	PROPN
ejpam-3780	148	3	a1	a1	PROPN
ejpam-3780	148	4	a2	a2	PROPN
ejpam-3780	148	5	a4	a4	NOUN
ejpam-3780	148	6	figure	figure	NOUN
ejpam-3780	148	7	2	2	NUM
ejpam-3780	148	8	:	:	PUNCT
ejpam-3780	148	9	zero	zero	NUM
ejpam-3780	148	10	ring	ring	NOUN
ejpam-3780	148	11	labeling	labeling	NOUN
ejpam-3780	148	12	of	of	ADP
ejpam-3780	148	13	c5	c5	PROPN
ejpam-3780	148	14	using	use	VERB
ejpam-3780	148	15	m0	m0	PROPN
ejpam-3780	148	16	2	2	NUM
ejpam-3780	148	17	(	(	PUNCT
ejpam-3780	148	18	z5	z5	X
ejpam-3780	148	19	)	)	PUNCT
ejpam-3780	148	20	given	give	VERB
ejpam-3780	148	21	a	a	DET
ejpam-3780	148	22	graph	graph	NOUN
ejpam-3780	148	23	g	g	NOUN
ejpam-3780	148	24	with	with	ADP
ejpam-3780	148	25	zero	zero	NUM
ejpam-3780	148	26	ring	ring	NOUN
ejpam-3780	148	27	labeling	labeling	NOUN
ejpam-3780	148	28	f	f	NOUN
ejpam-3780	148	29	,	,	PUNCT
ejpam-3780	148	30	we	we	PRON
ejpam-3780	148	31	consider	consider	VERB
ejpam-3780	148	32	the	the	DET
ejpam-3780	148	33	set	set	NOUN
ejpam-3780	148	34	k	k	PROPN
ejpam-3780	149	1	=	=	PRON
ejpam-3780	149	2	{	{	PUNCT
ejpam-3780	149	3	f(u	f(u	PROPN
ejpam-3780	149	4	)	)	PUNCT
ejpam-3780	149	5	+	+	NUM
ejpam-3780	149	6	f(v	f(v	NOUN
ejpam-3780	149	7	)	)	PUNCT
ejpam-3780	149	8	:	:	PUNCT
ejpam-3780	149	9	uv	uv	PROPN
ejpam-3780	149	10	∈	∈	PROPN
ejpam-3780	149	11	e(g	e(g	PROPN
ejpam-3780	149	12	)	)	PUNCT
ejpam-3780	149	13	}	}	PUNCT
ejpam-3780	149	14	.	.	PUNCT
ejpam-3780	150	1	since	since	SCONJ
ejpam-3780	150	2	it	it	PRON
ejpam-3780	150	3	was	be	AUX
ejpam-3780	150	4	shown	show	VERB
ejpam-3780	150	5	in	in	ADP
ejpam-3780	150	6	[	[	X
ejpam-3780	150	7	1	1	X
ejpam-3780	150	8	]	]	PUNCT
ejpam-3780	150	9	that	that	SCONJ
ejpam-3780	150	10	every	every	DET
ejpam-3780	150	11	graph	graph	NOUN
ejpam-3780	150	12	admits	admit	VERB
ejpam-3780	150	13	a	a	DET
ejpam-3780	150	14	zero	zero	NUM
ejpam-3780	150	15	ring	ring	NOUN
ejpam-3780	150	16	labeling	labeling	NOUN
ejpam-3780	150	17	,	,	PUNCT
ejpam-3780	150	18	we	we	PRON
ejpam-3780	150	19	are	be	AUX
ejpam-3780	150	20	interested	interested	ADJ
ejpam-3780	150	21	in	in	ADP
ejpam-3780	150	22	finding	find	VERB
ejpam-3780	150	23	zero	zero	NUM
ejpam-3780	150	24	ring	ring	NOUN
ejpam-3780	150	25	labelings	labeling	NOUN
ejpam-3780	150	26	for	for	ADP
ejpam-3780	150	27	g	g	PROPN
ejpam-3780	150	28	such	such	ADJ
ejpam-3780	150	29	that	that	SCONJ
ejpam-3780	150	30	|k|	|k|	NOUN
ejpam-3780	150	31	is	be	AUX
ejpam-3780	150	32	as	as	ADV
ejpam-3780	150	33	small	small	ADJ
ejpam-3780	150	34	as	as	ADP
ejpam-3780	150	35	possible	possible	ADJ
ejpam-3780	150	36	.	.	PUNCT
ejpam-3780	151	1	definition	definition	NOUN
ejpam-3780	151	2	21	21	NUM
ejpam-3780	151	3	.	.	PUNCT
ejpam-3780	152	1	let	let	VERB
ejpam-3780	152	2	g	g	PRON
ejpam-3780	152	3	be	be	AUX
ejpam-3780	152	4	a	a	DET
ejpam-3780	152	5	graph	graph	NOUN
ejpam-3780	152	6	with	with	ADP
ejpam-3780	152	7	zero	zero	NUM
ejpam-3780	152	8	ring	ring	NOUN
ejpam-3780	152	9	labeling	labeling	NOUN
ejpam-3780	152	10	f	f	NOUN
ejpam-3780	152	11	,	,	PUNCT
ejpam-3780	152	12	and	and	CCONJ
ejpam-3780	152	13	let	let	VERB
ejpam-3780	152	14	k	k	PROPN
ejpam-3780	152	15	=	=	PRON
ejpam-3780	152	16	{	{	PUNCT
ejpam-3780	152	17	f(u)+f(v	f(u)+f(v	PROPN
ejpam-3780	152	18	)	)	PUNCT
ejpam-3780	152	19	:	:	PUNCT
ejpam-3780	152	20	uv	uv	PROPN
ejpam-3780	152	21	∈	∈	PROPN
ejpam-3780	152	22	e(g	e(g	PROPN
ejpam-3780	152	23	)	)	PUNCT
ejpam-3780	152	24	}	}	PUNCT
ejpam-3780	152	25	.	.	PUNCT
ejpam-3780	153	1	a	a	DET
ejpam-3780	153	2	zero	zero	NUM
ejpam-3780	153	3	ring	ring	NOUN
ejpam-3780	153	4	labeling	labeling	NOUN
ejpam-3780	153	5	f	f	PROPN
ejpam-3780	153	6	of	of	ADP
ejpam-3780	153	7	g	g	PROPN
ejpam-3780	153	8	is	be	AUX
ejpam-3780	153	9	a	a	DET
ejpam-3780	153	10	k	k	ADJ
ejpam-3780	153	11	-	-	ADJ
ejpam-3780	153	12	zero	zero	NUM
ejpam-3780	153	13	ring	ring	NOUN
ejpam-3780	153	14	labeling	labeling	NOUN
ejpam-3780	153	15	if	if	SCONJ
ejpam-3780	153	16	|k|	|k|	PROPN
ejpam-3780	153	17	=	=	SYM
ejpam-3780	153	18	k.	k.	PROPN
ejpam-3780	154	1	if	if	SCONJ
ejpam-3780	154	2	|k|	|k|	PROPN
ejpam-3780	154	3	=	=	SYM
ejpam-3780	154	4	∆(g	∆(g	PROPN
ejpam-3780	154	5	)	)	PUNCT
ejpam-3780	154	6	,	,	PUNCT
ejpam-3780	154	7	then	then	ADV
ejpam-3780	154	8	the	the	DET
ejpam-3780	154	9	zero	zero	NUM
ejpam-3780	154	10	ring	ring	NOUN
ejpam-3780	154	11	labeling	labeling	NOUN
ejpam-3780	154	12	is	be	AUX
ejpam-3780	154	13	efficient	efficient	ADJ
ejpam-3780	154	14	.	.	PUNCT
ejpam-3780	155	1	example	example	NOUN
ejpam-3780	156	1	3	3	NUM
ejpam-3780	156	2	.	.	X
ejpam-3780	156	3	figure	figure	NOUN
ejpam-3780	156	4	3	3	NUM
ejpam-3780	156	5	shows	show	VERB
ejpam-3780	156	6	a	a	DET
ejpam-3780	156	7	4	4	NUM
ejpam-3780	156	8	-	-	SYM
ejpam-3780	156	9	zero	zero	NUM
ejpam-3780	156	10	ring	ring	NOUN
ejpam-3780	156	11	labeling	labeling	NOUN
ejpam-3780	156	12	of	of	ADP
ejpam-3780	156	13	the	the	DET
ejpam-3780	156	14	diamond	diamond	NOUN
ejpam-3780	156	15	graph	graph	NOUN
ejpam-3780	156	16	using	use	VERB
ejpam-3780	156	17	m0	m0	PROPN
ejpam-3780	156	18	2	2	NUM
ejpam-3780	156	19	(	(	PUNCT
ejpam-3780	156	20	z8	z8	NOUN
ejpam-3780	156	21	)	)	PUNCT
ejpam-3780	156	22	.	.	PUNCT
ejpam-3780	157	1	in	in	ADP
ejpam-3780	157	2	this	this	DET
ejpam-3780	157	3	labeling	labeling	NOUN
ejpam-3780	157	4	,	,	PUNCT
ejpam-3780	157	5	the	the	DET
ejpam-3780	157	6	set	set	NOUN
ejpam-3780	157	7	of	of	ADP
ejpam-3780	157	8	sums	sum	NOUN
ejpam-3780	157	9	is	be	AUX
ejpam-3780	157	10	k	k	NOUN
ejpam-3780	157	11	=	=	PUNCT
ejpam-3780	157	12	{	{	PUNCT
ejpam-3780	157	13	a1	a1	PROPN
ejpam-3780	157	14	,	,	PUNCT
ejpam-3780	157	15	a2	a2	PROPN
ejpam-3780	157	16	,	,	PUNCT
ejpam-3780	157	17	a5	a5	PROPN
ejpam-3780	157	18	,	,	PUNCT
ejpam-3780	157	19	a7	a7	PROPN
ejpam-3780	157	20	}	}	PUNCT
ejpam-3780	157	21	and	and	CCONJ
ejpam-3780	157	22	thus	thus	ADV
ejpam-3780	157	23	|k|	|k|	NOUN
ejpam-3780	157	24	=	=	SYM
ejpam-3780	157	25	4	4	X
ejpam-3780	157	26	.	.	X
ejpam-3780	157	27	figure	figure	VERB
ejpam-3780	157	28	4	4	NUM
ejpam-3780	157	29	shows	show	VERB
ejpam-3780	157	30	an	an	DET
ejpam-3780	157	31	efficient	efficient	ADJ
ejpam-3780	157	32	zero	zero	NUM
ejpam-3780	157	33	ring	ring	NOUN
ejpam-3780	157	34	labeling	labeling	NOUN
ejpam-3780	157	35	of	of	ADP
ejpam-3780	157	36	the	the	DET
ejpam-3780	157	37	butterfly	butterfly	NOUN
ejpam-3780	157	38	graph	graph	NOUN
ejpam-3780	157	39	g	g	NOUN
ejpam-3780	157	40	using	use	VERB
ejpam-3780	157	41	m0	m0	PROPN
ejpam-3780	157	42	2	2	NUM
ejpam-3780	157	43	(	(	PUNCT
ejpam-3780	157	44	z10	z10	NOUN
ejpam-3780	157	45	)	)	PUNCT
ejpam-3780	157	46	.	.	PUNCT
ejpam-3780	158	1	in	in	ADP
ejpam-3780	158	2	this	this	DET
ejpam-3780	158	3	labeling	labeling	NOUN
ejpam-3780	158	4	,	,	PUNCT
ejpam-3780	158	5	the	the	DET
ejpam-3780	158	6	set	set	NOUN
ejpam-3780	158	7	of	of	ADP
ejpam-3780	158	8	sums	sum	NOUN
ejpam-3780	158	9	is	be	AUX
ejpam-3780	158	10	k	k	NOUN
ejpam-3780	158	11	=	=	PUNCT
ejpam-3780	158	12	{	{	PUNCT
ejpam-3780	158	13	a2	a2	PROPN
ejpam-3780	158	14	,	,	PUNCT
ejpam-3780	158	15	a3	a3	NOUN
ejpam-3780	158	16	,	,	PUNCT
ejpam-3780	158	17	a7	a7	PROPN
ejpam-3780	158	18	,	,	PUNCT
ejpam-3780	158	19	a9	a9	PROPN
ejpam-3780	158	20	}	}	PUNCT
ejpam-3780	158	21	and	and	CCONJ
ejpam-3780	158	22	thus	thus	ADV
ejpam-3780	158	23	|k|	|k|	PROPN
ejpam-3780	158	24	=	=	SYM
ejpam-3780	158	25	∆(g	∆(g	PROPN
ejpam-3780	158	26	)	)	PUNCT
ejpam-3780	158	27	=	=	SYM
ejpam-3780	159	1	4	4	X
ejpam-3780	159	2	.	.	X
ejpam-3780	159	3	d.	d.	PROPN
ejpam-3780	159	4	chua	chua	PROPN
ejpam-3780	159	5	,	,	PUNCT
ejpam-3780	159	6	f.	f.	PROPN
ejpam-3780	159	7	campeña	campeña	PROPN
ejpam-3780	159	8	,	,	PUNCT
ejpam-3780	159	9	f.	f.	PROPN
ejpam-3780	159	10	franco	franco	PROPN
ejpam-3780	159	11	/	/	SYM
ejpam-3780	159	12	eur	eur	PROPN
ejpam-3780	159	13	.	.	PUNCT
ejpam-3780	160	1	j.	j.	PROPN
ejpam-3780	160	2	pure	pure	PROPN
ejpam-3780	160	3	appl	appl	PROPN
ejpam-3780	160	4	.	.	PROPN
ejpam-3780	160	5	math	math	PROPN
ejpam-3780	160	6	,	,	PUNCT
ejpam-3780	160	7	13	13	NUM
ejpam-3780	160	8	(	(	PUNCT
ejpam-3780	160	9	3	3	NUM
ejpam-3780	160	10	)	)	PUNCT
ejpam-3780	160	11	(	(	PUNCT
ejpam-3780	160	12	2020	2020	NUM
ejpam-3780	160	13	)	)	PUNCT
ejpam-3780	160	14	,	,	PUNCT
ejpam-3780	160	15	674	674	NUM
ejpam-3780	160	16	-	-	SYM
ejpam-3780	160	17	696	696	NUM
ejpam-3780	160	18	679	679	NUM
ejpam-3780	160	19	a0	a0	PROPN
ejpam-3780	160	20	a2	a2	PROPN
ejpam-3780	160	21	a3	a3	PROPN
ejpam-3780	160	22	a7	a7	PROPN
ejpam-3780	160	23	a2	a2	PROPN
ejpam-3780	160	24	a7	a7	PROPN
ejpam-3780	160	25	figure	figure	NOUN
ejpam-3780	160	26	3	3	NUM
ejpam-3780	160	27	:	:	SYM
ejpam-3780	160	28	4	4	NUM
ejpam-3780	160	29	-	-	SYM
ejpam-3780	160	30	zero	zero	NUM
ejpam-3780	160	31	ring	ring	NOUN
ejpam-3780	160	32	labeling	labeling	NOUN
ejpam-3780	160	33	of	of	ADP
ejpam-3780	160	34	the	the	DET
ejpam-3780	160	35	diamond	diamond	NOUN
ejpam-3780	160	36	graph	graph	NOUN
ejpam-3780	160	37	using	use	VERB
ejpam-3780	160	38	m0	m0	PROPN
ejpam-3780	160	39	2	2	NUM
ejpam-3780	160	40	(	(	PUNCT
ejpam-3780	160	41	z8	z8	NOUN
ejpam-3780	160	42	)	)	PUNCT
ejpam-3780	160	43	a0	a0	PROPN
ejpam-3780	160	44	a2	a2	PROPN
ejpam-3780	160	45	a7	a7	PROPN
ejpam-3780	160	46	a0	a0	PROPN
ejpam-3780	160	47	a3	a3	PROPN
ejpam-3780	160	48	a9	a9	PROPN
ejpam-3780	160	49	figure	figure	NOUN
ejpam-3780	160	50	4	4	NUM
ejpam-3780	160	51	:	:	PUNCT
ejpam-3780	160	52	efficient	efficient	ADJ
ejpam-3780	160	53	zero	zero	NUM
ejpam-3780	160	54	ring	ring	NOUN
ejpam-3780	160	55	labeling	labeling	NOUN
ejpam-3780	160	56	of	of	ADP
ejpam-3780	160	57	the	the	DET
ejpam-3780	160	58	butterfly	butterfly	NOUN
ejpam-3780	160	59	graph	graph	NOUN
ejpam-3780	160	60	using	use	VERB
ejpam-3780	160	61	m0	m0	PROPN
ejpam-3780	160	62	2	2	NUM
ejpam-3780	160	63	(	(	PUNCT
ejpam-3780	160	64	z10	z10	NOUN
ejpam-3780	160	65	)	)	PUNCT
ejpam-3780	160	66	3	3	NUM
ejpam-3780	160	67	.	.	PUNCT
ejpam-3780	160	68	results	result	NOUN
ejpam-3780	160	69	theorem	theorem	VERB
ejpam-3780	160	70	1	1	X
ejpam-3780	160	71	.	.	PUNCT
ejpam-3780	161	1	let	let	VERB
ejpam-3780	161	2	g	g	PRON
ejpam-3780	161	3	be	be	AUX
ejpam-3780	161	4	a	a	DET
ejpam-3780	161	5	graph	graph	NOUN
ejpam-3780	161	6	,	,	PUNCT
ejpam-3780	161	7	and	and	CCONJ
ejpam-3780	161	8	let	let	VERB
ejpam-3780	161	9	f	f	PRON
ejpam-3780	161	10	:	:	PUNCT
ejpam-3780	161	11	g	g	PROPN
ejpam-3780	161	12	→	→	SYM
ejpam-3780	161	13	r0	r0	NOUN
ejpam-3780	161	14	be	be	AUX
ejpam-3780	161	15	a	a	DET
ejpam-3780	161	16	zero	zero	NUM
ejpam-3780	161	17	ring	ring	NOUN
ejpam-3780	161	18	labeling	labeling	NOUN
ejpam-3780	161	19	.	.	PUNCT
ejpam-3780	162	1	if	if	SCONJ
ejpam-3780	162	2	k	k	PROPN
ejpam-3780	162	3	=	=	PRON
ejpam-3780	162	4	{	{	PUNCT
ejpam-3780	162	5	f(u	f(u	PROPN
ejpam-3780	162	6	)	)	PUNCT
ejpam-3780	162	7	+	+	NUM
ejpam-3780	162	8	f(v	f(v	NOUN
ejpam-3780	162	9	)	)	PUNCT
ejpam-3780	162	10	:	:	PUNCT
ejpam-3780	162	11	uv	uv	PROPN
ejpam-3780	162	12	∈	∈	PROPN
ejpam-3780	162	13	e(g	e(g	PROPN
ejpam-3780	162	14	)	)	PUNCT
ejpam-3780	162	15	}	}	PUNCT
ejpam-3780	162	16	,	,	PUNCT
ejpam-3780	162	17	then	then	ADV
ejpam-3780	162	18	∆(g	∆(g	PROPN
ejpam-3780	162	19	)	)	PUNCT
ejpam-3780	162	20	≤	≤	PUNCT
ejpam-3780	162	21	|k|	|k|	NOUN
ejpam-3780	162	22	≤	≤	NUM
ejpam-3780	162	23	|r0|	|r0|	PROPN
ejpam-3780	162	24	−	−	PROPN
ejpam-3780	163	1	1	1	X
ejpam-3780	163	2	.	.	PUNCT
ejpam-3780	164	1	proof	proof	NOUN
ejpam-3780	164	2	.	.	PUNCT
ejpam-3780	165	1	since	since	SCONJ
ejpam-3780	165	2	f	f	PROPN
ejpam-3780	165	3	is	be	AUX
ejpam-3780	165	4	a	a	DET
ejpam-3780	165	5	zero	zero	NUM
ejpam-3780	165	6	ring	ring	NOUN
ejpam-3780	165	7	labeling	labeling	NOUN
ejpam-3780	165	8	,	,	PUNCT
ejpam-3780	165	9	0	0	NUM
ejpam-3780	165	10	/∈	/∈	PUNCT
ejpam-3780	166	1	k.	k.	PROPN
ejpam-3780	167	1	furthermore	furthermore	ADV
ejpam-3780	167	2	,	,	PUNCT
ejpam-3780	167	3	r0	r0	NOUN
ejpam-3780	167	4	contains	contain	VERB
ejpam-3780	167	5	every	every	DET
ejpam-3780	167	6	sum	sum	NOUN
ejpam-3780	167	7	f(u	f(u	PROPN
ejpam-3780	167	8	)	)	PUNCT
ejpam-3780	167	9	+	+	NUM
ejpam-3780	167	10	f(v	f(v	NOUN
ejpam-3780	167	11	)	)	PUNCT
ejpam-3780	167	12	where	where	SCONJ
ejpam-3780	167	13	uv	uv	PROPN
ejpam-3780	167	14	∈	∈	PROPN
ejpam-3780	167	15	e(g	e(g	PROPN
ejpam-3780	167	16	)	)	PUNCT
ejpam-3780	167	17	,	,	PUNCT
ejpam-3780	167	18	hence	hence	ADV
ejpam-3780	167	19	k	k	PROPN
ejpam-3780	167	20	⊂	⊂	PROPN
ejpam-3780	167	21	r0	r0	PROPN
ejpam-3780	167	22	and	and	CCONJ
ejpam-3780	167	23	|k|	|k|	NOUN
ejpam-3780	167	24	≤	≤	NOUN
ejpam-3780	167	25	|r0|	|r0|	NOUN
ejpam-3780	167	26	−	−	PROPN
ejpam-3780	168	1	1	1	X
ejpam-3780	168	2	.	.	PUNCT
ejpam-3780	169	1	let	let	VERB
ejpam-3780	169	2	v	v	NUM
ejpam-3780	169	3	∈	∈	PROPN
ejpam-3780	169	4	v	v	NOUN
ejpam-3780	169	5	(	(	PUNCT
ejpam-3780	169	6	g	g	NOUN
ejpam-3780	169	7	)	)	PUNCT
ejpam-3780	169	8	and	and	CCONJ
ejpam-3780	169	9	let	let	VERB
ejpam-3780	169	10	n(v	n(v	NOUN
ejpam-3780	169	11	)	)	PUNCT
ejpam-3780	169	12	=	=	PRON
ejpam-3780	169	13	{	{	PUNCT
ejpam-3780	169	14	v1	v1	PROPN
ejpam-3780	169	15	,	,	PUNCT
ejpam-3780	169	16	v2	v2	PROPN
ejpam-3780	169	17	,	,	PUNCT
ejpam-3780	169	18	.	.	PUNCT
ejpam-3780	169	19	.	.	PUNCT
ejpam-3780	170	1	.	.	PUNCT
ejpam-3780	171	1	,	,	PUNCT
ejpam-3780	171	2	vm	vm	AUX
ejpam-3780	171	3	}	}	PUNCT
ejpam-3780	171	4	be	be	AUX
ejpam-3780	171	5	the	the	DET
ejpam-3780	171	6	set	set	NOUN
ejpam-3780	171	7	of	of	ADP
ejpam-3780	171	8	vertices	vertex	NOUN
ejpam-3780	171	9	that	that	PRON
ejpam-3780	171	10	are	be	AUX
ejpam-3780	171	11	adjacent	adjacent	ADJ
ejpam-3780	171	12	to	to	ADP
ejpam-3780	171	13	v.	v.	PROPN
ejpam-3780	171	14	since	since	SCONJ
ejpam-3780	171	15	f	f	PROPN
ejpam-3780	171	16	is	be	AUX
ejpam-3780	171	17	injective	injective	ADJ
ejpam-3780	171	18	,	,	PUNCT
ejpam-3780	171	19	it	it	PRON
ejpam-3780	171	20	follows	follow	VERB
ejpam-3780	171	21	that	that	SCONJ
ejpam-3780	171	22	the	the	DET
ejpam-3780	171	23	number	number	NOUN
ejpam-3780	171	24	of	of	ADP
ejpam-3780	171	25	distinct	distinct	ADJ
ejpam-3780	171	26	sums	sum	NOUN
ejpam-3780	171	27	f(v	f(v	NOUN
ejpam-3780	171	28	)	)	PUNCT
ejpam-3780	172	1	+	+	CCONJ
ejpam-3780	172	2	f(vi	f(vi	NOUN
ejpam-3780	172	3	)	)	PUNCT
ejpam-3780	172	4	for	for	ADP
ejpam-3780	172	5	i	i	PROPN
ejpam-3780	172	6	=	=	SYM
ejpam-3780	172	7	1	1	NUM
ejpam-3780	172	8	,	,	PUNCT
ejpam-3780	172	9	2	2	NUM
ejpam-3780	172	10	,	,	PUNCT
ejpam-3780	172	11	3	3	NUM
ejpam-3780	172	12	,	,	PUNCT
ejpam-3780	172	13	.	.	PUNCT
ejpam-3780	172	14	.	.	PUNCT
ejpam-3780	173	1	.	.	PUNCT
ejpam-3780	174	1	,	,	PUNCT
ejpam-3780	174	2	m	m	AUX
ejpam-3780	174	3	generated	generate	VERB
ejpam-3780	174	4	from	from	ADP
ejpam-3780	174	5	edges	edge	NOUN
ejpam-3780	174	6	that	that	PRON
ejpam-3780	174	7	end	end	VERB
ejpam-3780	174	8	at	at	ADP
ejpam-3780	174	9	v	v	NOUN
ejpam-3780	174	10	is	be	AUX
ejpam-3780	174	11	equal	equal	ADJ
ejpam-3780	174	12	to	to	ADP
ejpam-3780	174	13	|n(v)|	|n(v)|	PROPN
ejpam-3780	174	14	=	=	SYM
ejpam-3780	174	15	m	m	PROPN
ejpam-3780	174	16	,	,	PUNCT
ejpam-3780	174	17	which	which	PRON
ejpam-3780	174	18	is	be	AUX
ejpam-3780	174	19	the	the	DET
ejpam-3780	174	20	degree	degree	NOUN
ejpam-3780	174	21	of	of	ADP
ejpam-3780	174	22	v.	v.	CCONJ
ejpam-3780	174	23	clearly	clearly	ADV
ejpam-3780	174	24	,	,	PUNCT
ejpam-3780	174	25	the	the	DET
ejpam-3780	174	26	maximum	maximum	ADJ
ejpam-3780	174	27	degree	degree	NOUN
ejpam-3780	174	28	of	of	ADP
ejpam-3780	174	29	g	g	PROPN
ejpam-3780	174	30	is	be	AUX
ejpam-3780	174	31	the	the	DET
ejpam-3780	174	32	minimum	minimum	ADJ
ejpam-3780	174	33	number	number	NOUN
ejpam-3780	174	34	of	of	ADP
ejpam-3780	174	35	elements	element	NOUN
ejpam-3780	174	36	of	of	ADP
ejpam-3780	174	37	k.	k.	PROPN
ejpam-3780	174	38	theorem	theorem	PROPN
ejpam-3780	174	39	2	2	NUM
ejpam-3780	174	40	.	.	PUNCT
ejpam-3780	175	1	if	if	SCONJ
ejpam-3780	175	2	g	g	PROPN
ejpam-3780	175	3	is	be	AUX
ejpam-3780	175	4	a	a	DET
ejpam-3780	175	5	graph	graph	NOUN
ejpam-3780	175	6	with	with	ADP
ejpam-3780	175	7	an	an	DET
ejpam-3780	175	8	efficient	efficient	ADJ
ejpam-3780	175	9	zero	zero	NUM
ejpam-3780	175	10	ring	ring	NOUN
ejpam-3780	175	11	labeling	labeling	NOUN
ejpam-3780	175	12	,	,	PUNCT
ejpam-3780	175	13	then	then	ADV
ejpam-3780	175	14	any	any	DET
ejpam-3780	175	15	edge	edge	NOUN
ejpam-3780	175	16	-	-	PUNCT
ejpam-3780	175	17	induced	induce	VERB
ejpam-3780	175	18	subgraph	subgraph	NOUN
ejpam-3780	175	19	h	h	NOUN
ejpam-3780	175	20	of	of	ADP
ejpam-3780	175	21	g	g	PROPN
ejpam-3780	175	22	such	such	ADJ
ejpam-3780	175	23	that	that	DET
ejpam-3780	175	24	∆(g	∆(g	NOUN
ejpam-3780	175	25	)	)	PUNCT
ejpam-3780	175	26	=	=	SYM
ejpam-3780	175	27	∆(h	∆(h	NOUN
ejpam-3780	175	28	)	)	PUNCT
ejpam-3780	175	29	has	have	VERB
ejpam-3780	175	30	an	an	DET
ejpam-3780	175	31	efficient	efficient	ADJ
ejpam-3780	175	32	zero	zero	NUM
ejpam-3780	175	33	ring	ring	NOUN
ejpam-3780	175	34	labeling	labeling	NOUN
ejpam-3780	175	35	.	.	PUNCT
ejpam-3780	176	1	proof	proof	NOUN
ejpam-3780	176	2	.	.	PUNCT
ejpam-3780	177	1	let	let	VERB
ejpam-3780	177	2	g	g	PRON
ejpam-3780	177	3	be	be	AUX
ejpam-3780	177	4	a	a	DET
ejpam-3780	177	5	graph	graph	NOUN
ejpam-3780	177	6	with	with	ADP
ejpam-3780	177	7	efficient	efficient	ADJ
ejpam-3780	177	8	zero	zero	NUM
ejpam-3780	177	9	ring	ring	NOUN
ejpam-3780	177	10	labeling	labeling	NOUN
ejpam-3780	178	1	f	f	NOUN
ejpam-3780	178	2	:	:	PUNCT
ejpam-3780	178	3	v	v	X
ejpam-3780	178	4	(	(	PUNCT
ejpam-3780	178	5	g	g	NOUN
ejpam-3780	178	6	)	)	PUNCT
ejpam-3780	178	7	→	→	SYM
ejpam-3780	178	8	r0	r0	NOUN
ejpam-3780	178	9	,	,	PUNCT
ejpam-3780	178	10	and	and	CCONJ
ejpam-3780	178	11	let	let	VERB
ejpam-3780	178	12	h	h	PRON
ejpam-3780	178	13	be	be	AUX
ejpam-3780	178	14	an	an	DET
ejpam-3780	178	15	edge	edge	NOUN
ejpam-3780	178	16	-	-	PUNCT
ejpam-3780	178	17	induced	induce	VERB
ejpam-3780	178	18	subgraph	subgraph	NOUN
ejpam-3780	178	19	of	of	ADP
ejpam-3780	178	20	g	g	PROPN
ejpam-3780	178	21	such	such	ADJ
ejpam-3780	178	22	that	that	DET
ejpam-3780	178	23	∆(g	∆(g	NOUN
ejpam-3780	178	24	)	)	PUNCT
ejpam-3780	178	25	=	=	SYM
ejpam-3780	178	26	∆(h	∆(h	NOUN
ejpam-3780	178	27	)	)	PUNCT
ejpam-3780	178	28	.	.	PUNCT
ejpam-3780	179	1	consider	consider	VERB
ejpam-3780	179	2	the	the	DET
ejpam-3780	179	3	restriction	restriction	NOUN
ejpam-3780	179	4	function	function	VERB
ejpam-3780	179	5	f	f	PROPN
ejpam-3780	179	6	|h	|h	X
ejpam-3780	179	7	of	of	ADP
ejpam-3780	179	8	f	f	PROPN
ejpam-3780	179	9	to	to	ADP
ejpam-3780	179	10	the	the	DET
ejpam-3780	179	11	vertex	vertex	NOUN
ejpam-3780	179	12	set	set	NOUN
ejpam-3780	179	13	of	of	ADP
ejpam-3780	179	14	h	h	NOUN
ejpam-3780	179	15	;	;	PUNCT
ejpam-3780	179	16	that	that	PRON
ejpam-3780	179	17	is	is	ADV
ejpam-3780	179	18	,	,	PUNCT
ejpam-3780	179	19	f	f	PROPN
ejpam-3780	179	20	|h(v	|h(v	PROPN
ejpam-3780	179	21	)	)	PUNCT
ejpam-3780	179	22	=	=	SYM
ejpam-3780	179	23	f(v	f(v	NOUN
ejpam-3780	179	24	)	)	PUNCT
ejpam-3780	179	25	for	for	ADP
ejpam-3780	179	26	all	all	DET
ejpam-3780	179	27	v	v	ADP
ejpam-3780	179	28	∈	∈	NOUN
ejpam-3780	179	29	v	v	NOUN
ejpam-3780	179	30	(	(	PUNCT
ejpam-3780	179	31	h	h	NOUN
ejpam-3780	179	32	)	)	PUNCT
ejpam-3780	179	33	.	.	PUNCT
ejpam-3780	180	1	since	since	SCONJ
ejpam-3780	180	2	f	f	PROPN
ejpam-3780	180	3	is	be	AUX
ejpam-3780	180	4	injective	injective	ADJ
ejpam-3780	180	5	,	,	PUNCT
ejpam-3780	180	6	f	f	PROPN
ejpam-3780	180	7	|h	|h	PROPN
ejpam-3780	180	8	is	be	AUX
ejpam-3780	180	9	injective	injective	ADJ
ejpam-3780	180	10	.	.	PUNCT
ejpam-3780	181	1	also	also	ADV
ejpam-3780	181	2	,	,	PUNCT
ejpam-3780	181	3	0	0	NUM
ejpam-3780	181	4	/∈	/∈	PUNCT
ejpam-3780	182	1	k	k	X
ejpam-3780	182	2	=	=	PRON
ejpam-3780	182	3	{	{	PUNCT
ejpam-3780	182	4	f(u	f(u	PROPN
ejpam-3780	182	5	)	)	PUNCT
ejpam-3780	182	6	+	+	NUM
ejpam-3780	182	7	f(v	f(v	NOUN
ejpam-3780	182	8	)	)	PUNCT
ejpam-3780	182	9	:	:	PUNCT
ejpam-3780	182	10	uv	uv	PROPN
ejpam-3780	182	11	∈	∈	PROPN
ejpam-3780	182	12	e(g	e(g	PROPN
ejpam-3780	182	13	)	)	PUNCT
ejpam-3780	182	14	}	}	PUNCT
ejpam-3780	182	15	implies	imply	VERB
ejpam-3780	182	16	that	that	SCONJ
ejpam-3780	182	17	0	0	NUM
ejpam-3780	183	1	/∈	/∈	PUNCT
ejpam-3780	184	1	kh	kh	PROPN
ejpam-3780	184	2	=	=	PRON
ejpam-3780	185	1	{	{	PUNCT
ejpam-3780	185	2	f	f	PROPN
ejpam-3780	185	3	|h(u	|h(u	PROPN
ejpam-3780	185	4	)	)	PUNCT
ejpam-3780	186	1	+	+	CCONJ
ejpam-3780	186	2	f	f	PROPN
ejpam-3780	186	3	|h(v	|h(v	PROPN
ejpam-3780	186	4	)	)	PUNCT
ejpam-3780	186	5	:	:	PUNCT
ejpam-3780	186	6	uv	uv	NOUN
ejpam-3780	186	7	∈	∈	PROPN
ejpam-3780	186	8	e(h	e(h	PROPN
ejpam-3780	186	9	)	)	PUNCT
ejpam-3780	186	10	}	}	PUNCT
ejpam-3780	186	11	.	.	PUNCT
ejpam-3780	187	1	it	it	PRON
ejpam-3780	187	2	remains	remain	VERB
ejpam-3780	187	3	to	to	PART
ejpam-3780	187	4	show	show	VERB
ejpam-3780	187	5	that	that	SCONJ
ejpam-3780	187	6	|kh	|kh	NUM
ejpam-3780	187	7	|	|	NOUN
ejpam-3780	187	8	=	=	SYM
ejpam-3780	187	9	∆(h	∆(h	NOUN
ejpam-3780	187	10	)	)	PUNCT
ejpam-3780	187	11	.	.	PUNCT
ejpam-3780	188	1	since	since	SCONJ
ejpam-3780	188	2	∆(g	∆(g	NOUN
ejpam-3780	188	3	)	)	PUNCT
ejpam-3780	188	4	=	=	SYM
ejpam-3780	188	5	∆(h	∆(h	NOUN
ejpam-3780	188	6	)	)	PUNCT
ejpam-3780	188	7	,	,	PUNCT
ejpam-3780	188	8	there	there	PRON
ejpam-3780	188	9	exists	exist	VERB
ejpam-3780	188	10	at	at	ADV
ejpam-3780	188	11	least	least	ADV
ejpam-3780	188	12	one	one	NUM
ejpam-3780	188	13	vertex	vertex	NOUN
ejpam-3780	188	14	v	v	NOUN
ejpam-3780	188	15	in	in	ADP
ejpam-3780	188	16	v	v	NOUN
ejpam-3780	188	17	(	(	PUNCT
ejpam-3780	188	18	g	g	NOUN
ejpam-3780	188	19	)	)	PUNCT
ejpam-3780	188	20	where	where	SCONJ
ejpam-3780	188	21	d(v	d(v	ADJ
ejpam-3780	188	22	)	)	PUNCT
ejpam-3780	188	23	=	=	SYM
ejpam-3780	188	24	∆(g	∆(g	PROPN
ejpam-3780	188	25	)	)	PUNCT
ejpam-3780	188	26	such	such	ADJ
ejpam-3780	188	27	that	that	DET
ejpam-3780	188	28	v	v	NOUN
ejpam-3780	188	29	is	be	AUX
ejpam-3780	188	30	also	also	ADV
ejpam-3780	188	31	in	in	ADP
ejpam-3780	188	32	h	h	NOUN
ejpam-3780	188	33	where	where	SCONJ
ejpam-3780	188	34	d(v	d(v	ADJ
ejpam-3780	188	35	)	)	PUNCT
ejpam-3780	188	36	=	=	SYM
ejpam-3780	188	37	∆(g	∆(g	PROPN
ejpam-3780	188	38	)	)	PUNCT
ejpam-3780	188	39	.	.	PUNCT
ejpam-3780	189	1	then	then	ADV
ejpam-3780	189	2	|kh	|kh	ADJ
ejpam-3780	189	3	|	|	NOUN
ejpam-3780	189	4	=	=	SYM
ejpam-3780	189	5	∆(g	∆(g	NOUN
ejpam-3780	189	6	)	)	PUNCT
ejpam-3780	189	7	=	=	SYM
ejpam-3780	189	8	∆(h	∆(h	NOUN
ejpam-3780	189	9	)	)	PUNCT
ejpam-3780	189	10	.	.	PUNCT
ejpam-3780	190	1	theorem	theorem	NOUN
ejpam-3780	190	2	3	3	X
ejpam-3780	190	3	.	.	PUNCT
ejpam-3780	191	1	let	let	VERB
ejpam-3780	191	2	g	g	PRON
ejpam-3780	191	3	be	be	AUX
ejpam-3780	191	4	a	a	DET
ejpam-3780	191	5	graph	graph	NOUN
ejpam-3780	191	6	with	with	ADP
ejpam-3780	192	1	n	n	ADP
ejpam-3780	192	2	vertices	vertex	NOUN
ejpam-3780	192	3	,	,	PUNCT
ejpam-3780	192	4	and	and	CCONJ
ejpam-3780	192	5	let	let	VERB
ejpam-3780	192	6	f	f	NOUN
ejpam-3780	192	7	:	:	PUNCT
ejpam-3780	192	8	v	v	X
ejpam-3780	192	9	(	(	PUNCT
ejpam-3780	192	10	g	g	NOUN
ejpam-3780	192	11	)	)	PUNCT
ejpam-3780	192	12	→	→	PUNCT
ejpam-3780	192	13	r0	r0	NOUN
ejpam-3780	192	14	be	be	AUX
ejpam-3780	192	15	a	a	DET
ejpam-3780	192	16	zero	zero	NUM
ejpam-3780	192	17	ring	ring	NOUN
ejpam-3780	192	18	labeling	labeling	NOUN
ejpam-3780	192	19	.	.	PUNCT
ejpam-3780	193	1	if	if	SCONJ
ejpam-3780	193	2	f	f	PROPN
ejpam-3780	193	3	is	be	AUX
ejpam-3780	193	4	optimal	optimal	ADJ
ejpam-3780	193	5	and	and	CCONJ
ejpam-3780	193	6	∆(g	∆(g	NOUN
ejpam-3780	193	7	)	)	PUNCT
ejpam-3780	194	1	=	=	PUNCT
ejpam-3780	194	2	n−	n−	NOUN
ejpam-3780	194	3	1	1	NUM
ejpam-3780	194	4	,	,	PUNCT
ejpam-3780	194	5	then	then	ADV
ejpam-3780	194	6	f	f	PROPN
ejpam-3780	194	7	is	be	AUX
ejpam-3780	194	8	efficient	efficient	ADJ
ejpam-3780	194	9	.	.	PUNCT
ejpam-3780	195	1	proof	proof	NOUN
ejpam-3780	195	2	.	.	PUNCT
ejpam-3780	196	1	let	let	VERB
ejpam-3780	196	2	k	k	NOUN
ejpam-3780	196	3	=	=	PRON
ejpam-3780	196	4	{	{	PUNCT
ejpam-3780	196	5	f(u	f(u	PROPN
ejpam-3780	196	6	)	)	PUNCT
ejpam-3780	196	7	+	+	NUM
ejpam-3780	196	8	f(v	f(v	NOUN
ejpam-3780	196	9	)	)	PUNCT
ejpam-3780	196	10	:	:	PUNCT
ejpam-3780	196	11	uv	uv	PROPN
ejpam-3780	196	12	∈	∈	PROPN
ejpam-3780	196	13	e(g	e(g	PROPN
ejpam-3780	196	14	)	)	PUNCT
ejpam-3780	196	15	}	}	PUNCT
ejpam-3780	196	16	.	.	PUNCT
ejpam-3780	197	1	to	to	PART
ejpam-3780	197	2	show	show	VERB
ejpam-3780	197	3	that	that	SCONJ
ejpam-3780	197	4	f	f	PROPN
ejpam-3780	197	5	is	be	AUX
ejpam-3780	197	6	efficient	efficient	ADJ
ejpam-3780	197	7	,	,	PUNCT
ejpam-3780	197	8	we	we	PRON
ejpam-3780	197	9	need	need	VERB
ejpam-3780	197	10	to	to	PART
ejpam-3780	197	11	show	show	VERB
ejpam-3780	197	12	that	that	SCONJ
ejpam-3780	197	13	|k|	|k|	NOUN
ejpam-3780	197	14	=	=	SYM
ejpam-3780	197	15	n−	n−	PROPN
ejpam-3780	197	16	1	1	NUM
ejpam-3780	197	17	.	.	PUNCT
ejpam-3780	198	1	since	since	SCONJ
ejpam-3780	198	2	f	f	PROPN
ejpam-3780	198	3	is	be	AUX
ejpam-3780	198	4	optimal	optimal	ADJ
ejpam-3780	198	5	,	,	PUNCT
ejpam-3780	198	6	|r0|	|r0|	X
ejpam-3780	198	7	=	=	SYM
ejpam-3780	198	8	n.	n.	NOUN
ejpam-3780	198	9	by	by	ADP
ejpam-3780	198	10	theorem	theorem	NOUN
ejpam-3780	198	11	1	1	NUM
ejpam-3780	198	12	,	,	PUNCT
ejpam-3780	198	13	∆(g	∆(g	NOUN
ejpam-3780	198	14	)	)	PUNCT
ejpam-3780	198	15	≤	≤	PUNCT
ejpam-3780	198	16	|k|	|k|	NOUN
ejpam-3780	198	17	≤	≤	NUM
ejpam-3780	198	18	|r0|	|r0|	NOUN
ejpam-3780	198	19	−	−	PROPN
ejpam-3780	199	1	1	1	NUM
ejpam-3780	199	2	.	.	PUNCT
ejpam-3780	200	1	but	but	CCONJ
ejpam-3780	200	2	∆(g	∆(g	NOUN
ejpam-3780	200	3	)	)	PUNCT
ejpam-3780	200	4	=	=	SYM
ejpam-3780	200	5	|r0|	|r0|	NOUN
ejpam-3780	200	6	−	−	NOUN
ejpam-3780	201	1	1	1	NUM
ejpam-3780	201	2	=	=	SYM
ejpam-3780	201	3	n−	n−	NOUN
ejpam-3780	201	4	1	1	NUM
ejpam-3780	201	5	.	.	PUNCT
ejpam-3780	202	1	therefore	therefore	ADV
ejpam-3780	202	2	,	,	PUNCT
ejpam-3780	202	3	|k|	|k|	NOUN
ejpam-3780	202	4	=	=	SYM
ejpam-3780	202	5	n−	n−	NOUN
ejpam-3780	202	6	1	1	X
ejpam-3780	202	7	.	.	PUNCT
ejpam-3780	203	1	we	we	PRON
ejpam-3780	203	2	now	now	ADV
ejpam-3780	203	3	look	look	VERB
ejpam-3780	203	4	at	at	ADP
ejpam-3780	203	5	the	the	DET
ejpam-3780	203	6	efficient	efficient	ADJ
ejpam-3780	203	7	zero	zero	NUM
ejpam-3780	203	8	ring	ring	NOUN
ejpam-3780	203	9	labeling	labeling	NOUN
ejpam-3780	203	10	of	of	ADP
ejpam-3780	203	11	some	some	DET
ejpam-3780	203	12	classes	class	NOUN
ejpam-3780	203	13	of	of	ADP
ejpam-3780	203	14	trees	tree	NOUN
ejpam-3780	203	15	and	and	CCONJ
ejpam-3780	203	16	other	other	ADJ
ejpam-3780	203	17	common	common	ADJ
ejpam-3780	203	18	classes	class	NOUN
ejpam-3780	203	19	of	of	ADP
ejpam-3780	203	20	graphs	graph	NOUN
ejpam-3780	203	21	.	.	PUNCT
ejpam-3780	204	1	theorem	theorem	ADJ
ejpam-3780	204	2	4	4	NUM
ejpam-3780	204	3	.	.	PUNCT
ejpam-3780	205	1	let	let	VERB
ejpam-3780	205	2	g	g	PRON
ejpam-3780	205	3	be	be	AUX
ejpam-3780	205	4	a	a	DET
ejpam-3780	205	5	caterpillar	caterpillar	NOUN
ejpam-3780	205	6	in	in	ADP
ejpam-3780	205	7	which	which	PRON
ejpam-3780	205	8	each	each	DET
ejpam-3780	205	9	vertex	vertex	NOUN
ejpam-3780	205	10	in	in	ADP
ejpam-3780	205	11	the	the	DET
ejpam-3780	205	12	central	central	ADJ
ejpam-3780	205	13	path	path	NOUN
ejpam-3780	205	14	has	have	VERB
ejpam-3780	205	15	an	an	DET
ejpam-3780	205	16	equal	equal	ADJ
ejpam-3780	205	17	number	number	NOUN
ejpam-3780	205	18	of	of	ADP
ejpam-3780	205	19	hanging	hang	VERB
ejpam-3780	205	20	leaves	leave	NOUN
ejpam-3780	205	21	.	.	PUNCT
ejpam-3780	206	1	then	then	ADV
ejpam-3780	206	2	g	g	PROPN
ejpam-3780	206	3	has	have	VERB
ejpam-3780	206	4	an	an	DET
ejpam-3780	206	5	efficient	efficient	ADJ
ejpam-3780	206	6	zero	zero	NUM
ejpam-3780	206	7	ring	ring	NOUN
ejpam-3780	206	8	labeling	labeling	NOUN
ejpam-3780	206	9	.	.	PUNCT
ejpam-3780	207	1	d.	d.	PROPN
ejpam-3780	207	2	chua	chua	PROPN
ejpam-3780	207	3	,	,	PUNCT
ejpam-3780	207	4	f.	f.	PROPN
ejpam-3780	207	5	campeña	campeña	PROPN
ejpam-3780	207	6	,	,	PUNCT
ejpam-3780	207	7	f.	f.	PROPN
ejpam-3780	207	8	franco	franco	PROPN
ejpam-3780	207	9	/	/	SYM
ejpam-3780	207	10	eur	eur	PROPN
ejpam-3780	207	11	.	.	PUNCT
ejpam-3780	208	1	j.	j.	PROPN
ejpam-3780	208	2	pure	pure	PROPN
ejpam-3780	208	3	appl	appl	PROPN
ejpam-3780	208	4	.	.	PROPN
ejpam-3780	208	5	math	math	PROPN
ejpam-3780	208	6	,	,	PUNCT
ejpam-3780	208	7	13	13	NUM
ejpam-3780	208	8	(	(	PUNCT
ejpam-3780	208	9	3	3	NUM
ejpam-3780	208	10	)	)	PUNCT
ejpam-3780	208	11	(	(	PUNCT
ejpam-3780	208	12	2020	2020	NUM
ejpam-3780	208	13	)	)	PUNCT
ejpam-3780	208	14	,	,	PUNCT
ejpam-3780	208	15	674	674	NUM
ejpam-3780	208	16	-	-	SYM
ejpam-3780	208	17	696	696	NUM
ejpam-3780	208	18	680	680	NUM
ejpam-3780	208	19	proof	proof	NOUN
ejpam-3780	208	20	.	.	PUNCT
ejpam-3780	209	1	let	let	VERB
ejpam-3780	209	2	[	[	X
ejpam-3780	209	3	w1	w1	NOUN
ejpam-3780	209	4	,	,	PUNCT
ejpam-3780	209	5	w2	w2	NOUN
ejpam-3780	209	6	,	,	PUNCT
ejpam-3780	209	7	.	.	PUNCT
ejpam-3780	209	8	.	.	PUNCT
ejpam-3780	210	1	.	.	PUNCT
ejpam-3780	211	1	,	,	PUNCT
ejpam-3780	211	2	wn	wn	PROPN
ejpam-3780	211	3	]	]	X
ejpam-3780	211	4	denote	denote	VERB
ejpam-3780	211	5	the	the	DET
ejpam-3780	211	6	central	central	ADJ
ejpam-3780	211	7	path	path	NOUN
ejpam-3780	211	8	of	of	ADP
ejpam-3780	211	9	a	a	DET
ejpam-3780	211	10	caterpillar	caterpillar	ADJ
ejpam-3780	211	11	g	g	NOUN
ejpam-3780	211	12	and	and	CCONJ
ejpam-3780	211	13	suppose	suppose	VERB
ejpam-3780	211	14	that	that	SCONJ
ejpam-3780	211	15	each	each	DET
ejpam-3780	211	16	wi	wi	PROPN
ejpam-3780	211	17	,	,	PUNCT
ejpam-3780	211	18	1	1	NUM
ejpam-3780	211	19	≤	≤	NUM
ejpam-3780	211	20	i	i	PRON
ejpam-3780	211	21	≤	≤	NOUN
ejpam-3780	211	22	n	n	CCONJ
ejpam-3780	211	23	,	,	PUNCT
ejpam-3780	211	24	has	have	VERB
ejpam-3780	211	25	r	r	NOUN
ejpam-3780	211	26	hanging	hang	VERB
ejpam-3780	211	27	leaves	leave	NOUN
ejpam-3780	211	28	.	.	PUNCT
ejpam-3780	212	1	denote	denote	VERB
ejpam-3780	212	2	the	the	DET
ejpam-3780	212	3	hanging	hang	VERB
ejpam-3780	212	4	leaves	leave	NOUN
ejpam-3780	212	5	of	of	ADP
ejpam-3780	212	6	wi	wi	PROPN
ejpam-3780	212	7	by	by	ADP
ejpam-3780	212	8	wj	wj	PROPN
ejpam-3780	213	1	i	i	PRON
ejpam-3780	213	2	,	,	PUNCT
ejpam-3780	213	3	where	where	SCONJ
ejpam-3780	213	4	j	j	PROPN
ejpam-3780	213	5	=	=	SYM
ejpam-3780	213	6	1	1	NUM
ejpam-3780	213	7	,	,	PUNCT
ejpam-3780	213	8	2	2	NUM
ejpam-3780	213	9	,	,	PUNCT
ejpam-3780	213	10	.	.	PUNCT
ejpam-3780	213	11	.	.	PUNCT
ejpam-3780	213	12	.	.	PUNCT
ejpam-3780	214	1	,	,	PUNCT
ejpam-3780	214	2	r.	r.	PROPN
ejpam-3780	214	3	case	case	NOUN
ejpam-3780	214	4	1	1	NUM
ejpam-3780	214	5	:	:	PUNCT
ejpam-3780	214	6	suppose	suppose	VERB
ejpam-3780	214	7	n	n	PROPN
ejpam-3780	214	8	=	=	SYM
ejpam-3780	214	9	1	1	X
ejpam-3780	214	10	.	.	PUNCT
ejpam-3780	215	1	in	in	ADP
ejpam-3780	215	2	this	this	DET
ejpam-3780	215	3	case	case	NOUN
ejpam-3780	215	4	,	,	PUNCT
ejpam-3780	215	5	∆(g	∆(g	NOUN
ejpam-3780	215	6	)	)	PUNCT
ejpam-3780	215	7	=	=	NOUN
ejpam-3780	215	8	d(w1	d(w1	X
ejpam-3780	215	9	)	)	PUNCT
ejpam-3780	215	10	=	=	VERB
ejpam-3780	215	11	r.	r.	NOUN
ejpam-3780	215	12	define	define	VERB
ejpam-3780	215	13	a	a	DET
ejpam-3780	215	14	function	function	NOUN
ejpam-3780	215	15	f	f	NOUN
ejpam-3780	215	16	:	:	PUNCT
ejpam-3780	215	17	v	v	NOUN
ejpam-3780	215	18	(	(	PUNCT
ejpam-3780	215	19	g)→	g)→	NOUN
ejpam-3780	215	20	m0	m0	NOUN
ejpam-3780	215	21	2	2	NUM
ejpam-3780	215	22	(	(	PUNCT
ejpam-3780	215	23	zr+1	zr+1	NUM
ejpam-3780	215	24	)	)	PUNCT
ejpam-3780	215	25	such	such	ADJ
ejpam-3780	215	26	that	that	DET
ejpam-3780	215	27	f(w1	f(w1	NOUN
ejpam-3780	215	28	)	)	PUNCT
ejpam-3780	215	29	=	=	SYM
ejpam-3780	215	30	a0	a0	PROPN
ejpam-3780	215	31	and	and	CCONJ
ejpam-3780	215	32	f(wj	f(wj	PROPN
ejpam-3780	215	33	1	1	NUM
ejpam-3780	215	34	)	)	PUNCT
ejpam-3780	215	35	=	=	SYM
ejpam-3780	215	36	aj	aj	PROPN
ejpam-3780	215	37	.	.	PUNCT
ejpam-3780	216	1	clearly	clearly	ADV
ejpam-3780	216	2	,	,	PUNCT
ejpam-3780	216	3	f	f	PROPN
ejpam-3780	216	4	is	be	AUX
ejpam-3780	216	5	injective	injective	ADJ
ejpam-3780	216	6	.	.	PUNCT
ejpam-3780	217	1	let	let	VERB
ejpam-3780	217	2	k	k	NOUN
ejpam-3780	217	3	=	=	PRON
ejpam-3780	217	4	{	{	PUNCT
ejpam-3780	217	5	f(u	f(u	PROPN
ejpam-3780	217	6	)	)	PUNCT
ejpam-3780	217	7	+	+	NUM
ejpam-3780	217	8	f(v	f(v	NOUN
ejpam-3780	217	9	)	)	PUNCT
ejpam-3780	217	10	:	:	PUNCT
ejpam-3780	217	11	uv	uv	PROPN
ejpam-3780	217	12	∈	∈	PROPN
ejpam-3780	217	13	e(g	e(g	PROPN
ejpam-3780	217	14	)	)	PUNCT
ejpam-3780	217	15	}	}	PUNCT
ejpam-3780	217	16	.	.	PUNCT
ejpam-3780	218	1	to	to	PART
ejpam-3780	218	2	show	show	VERB
ejpam-3780	218	3	that	that	SCONJ
ejpam-3780	218	4	f	f	PROPN
ejpam-3780	218	5	is	be	AUX
ejpam-3780	218	6	an	an	DET
ejpam-3780	218	7	efficient	efficient	ADJ
ejpam-3780	218	8	zero	zero	NUM
ejpam-3780	218	9	ring	ring	NOUN
ejpam-3780	218	10	labeling	labeling	NOUN
ejpam-3780	218	11	of	of	ADP
ejpam-3780	218	12	g	g	NOUN
ejpam-3780	218	13	,	,	PUNCT
ejpam-3780	218	14	we	we	PRON
ejpam-3780	218	15	need	need	VERB
ejpam-3780	218	16	to	to	PART
ejpam-3780	218	17	show	show	VERB
ejpam-3780	218	18	that	that	SCONJ
ejpam-3780	218	19	|k|	|k|	NOUN
ejpam-3780	218	20	=	=	SYM
ejpam-3780	218	21	r	r	NOUN
ejpam-3780	218	22	and	and	CCONJ
ejpam-3780	218	23	a0	a0	PROPN
ejpam-3780	218	24	/∈	/∈	PROPN
ejpam-3780	218	25	k.	k.	PROPN
ejpam-3780	218	26	for	for	ADP
ejpam-3780	218	27	the	the	DET
ejpam-3780	218	28	set	set	NOUN
ejpam-3780	218	29	of	of	ADP
ejpam-3780	218	30	sums	sum	NOUN
ejpam-3780	218	31	,	,	PUNCT
ejpam-3780	218	32	we	we	PRON
ejpam-3780	218	33	obtain	obtain	VERB
ejpam-3780	218	34	f(w1	f(w1	ADV
ejpam-3780	218	35	)	)	PUNCT
ejpam-3780	219	1	+	+	CCONJ
ejpam-3780	219	2	f(wj	f(wj	NUM
ejpam-3780	219	3	1	1	NUM
ejpam-3780	219	4	)	)	PUNCT
ejpam-3780	219	5	=	=	SYM
ejpam-3780	219	6	a0	a0	PROPN
ejpam-3780	219	7	+	+	CCONJ
ejpam-3780	219	8	aj	aj	PROPN
ejpam-3780	219	9	=	=	PROPN
ejpam-3780	219	10	aj	aj	PROPN
ejpam-3780	219	11	6=	6=	PROPN
ejpam-3780	219	12	a0	a0	PROPN
ejpam-3780	219	13	(	(	PUNCT
ejpam-3780	219	14	2	2	NUM
ejpam-3780	219	15	)	)	PUNCT
ejpam-3780	219	16	for	for	ADP
ejpam-3780	219	17	j	j	PROPN
ejpam-3780	219	18	=	=	SYM
ejpam-3780	219	19	1	1	NUM
ejpam-3780	219	20	,	,	PUNCT
ejpam-3780	219	21	2	2	NUM
ejpam-3780	219	22	,	,	PUNCT
ejpam-3780	219	23	.	.	PUNCT
ejpam-3780	219	24	.	.	PUNCT
ejpam-3780	219	25	.	.	PUNCT
ejpam-3780	220	1	,	,	PUNCT
ejpam-3780	220	2	r.	r.	PROPN
ejpam-3780	220	3	then	then	ADV
ejpam-3780	220	4	a0	a0	PROPN
ejpam-3780	220	5	/∈	/∈	PROPN
ejpam-3780	221	1	k.	k.	PROPN
ejpam-3780	222	1	moreover	moreover	ADV
ejpam-3780	222	2	,	,	PUNCT
ejpam-3780	222	3	k	k	PROPN
ejpam-3780	222	4	=	=	PRON
ejpam-3780	222	5	{	{	PUNCT
ejpam-3780	222	6	a1	a1	PROPN
ejpam-3780	222	7	,	,	PUNCT
ejpam-3780	222	8	a2	a2	PROPN
ejpam-3780	222	9	,	,	PUNCT
ejpam-3780	222	10	.	.	PUNCT
ejpam-3780	222	11	.	.	PUNCT
ejpam-3780	223	1	.	.	PUNCT
ejpam-3780	224	1	,	,	PUNCT
ejpam-3780	224	2	ar	ar	PROPN
ejpam-3780	224	3	}	}	PUNCT
ejpam-3780	224	4	(	(	PUNCT
ejpam-3780	224	5	3	3	NUM
ejpam-3780	224	6	)	)	PUNCT
ejpam-3780	224	7	and	and	CCONJ
ejpam-3780	224	8	thus	thus	ADV
ejpam-3780	224	9	|k|	|k|	PROPN
ejpam-3780	224	10	=	=	SYM
ejpam-3780	224	11	r.	r.	PROPN
ejpam-3780	224	12	therefore	therefore	ADV
ejpam-3780	224	13	,	,	PUNCT
ejpam-3780	224	14	f	f	PROPN
ejpam-3780	224	15	is	be	AUX
ejpam-3780	224	16	an	an	DET
ejpam-3780	224	17	efficient	efficient	ADJ
ejpam-3780	224	18	zero	zero	NUM
ejpam-3780	224	19	ring	ring	NOUN
ejpam-3780	224	20	labeling	labeling	NOUN
ejpam-3780	224	21	of	of	ADP
ejpam-3780	224	22	g.	g.	PROPN
ejpam-3780	224	23	case	case	NOUN
ejpam-3780	224	24	2	2	NUM
ejpam-3780	224	25	:	:	PUNCT
ejpam-3780	224	26	suppose	suppose	VERB
ejpam-3780	224	27	n	n	PROPN
ejpam-3780	224	28	=	=	SYM
ejpam-3780	224	29	2	2	X
ejpam-3780	224	30	.	.	PUNCT
ejpam-3780	225	1	in	in	ADP
ejpam-3780	225	2	this	this	DET
ejpam-3780	225	3	case	case	NOUN
ejpam-3780	225	4	,	,	PUNCT
ejpam-3780	225	5	∆(g	∆(g	NOUN
ejpam-3780	225	6	)	)	PUNCT
ejpam-3780	225	7	=	=	NOUN
ejpam-3780	225	8	d(w1	d(w1	X
ejpam-3780	225	9	)	)	PUNCT
ejpam-3780	225	10	=	=	SYM
ejpam-3780	225	11	d(w2	d(w2	NOUN
ejpam-3780	225	12	)	)	PUNCT
ejpam-3780	225	13	=	=	SYM
ejpam-3780	226	1	r+1	r+1	PROPN
ejpam-3780	226	2	.	.	PUNCT
ejpam-3780	226	3	define	define	VERB
ejpam-3780	226	4	a	a	DET
ejpam-3780	226	5	function	function	NOUN
ejpam-3780	226	6	f	f	NOUN
ejpam-3780	226	7	:	:	PUNCT
ejpam-3780	226	8	v	v	X
ejpam-3780	226	9	(	(	PUNCT
ejpam-3780	226	10	g)→m0	g)→m0	NOUN
ejpam-3780	226	11	2	2	NUM
ejpam-3780	226	12	(	(	PUNCT
ejpam-3780	226	13	z2r+2	z2r+2	NOUN
ejpam-3780	226	14	)	)	PUNCT
ejpam-3780	226	15	such	such	ADJ
ejpam-3780	226	16	that	that	SCONJ
ejpam-3780	226	17	f(wi	f(wi	NOUN
ejpam-3780	226	18	)	)	PUNCT
ejpam-3780	226	19	=	=	SYM
ejpam-3780	227	1	ai(r+1	ai(r+1	VERB
ejpam-3780	227	2	)	)	PUNCT
ejpam-3780	227	3	and	and	CCONJ
ejpam-3780	227	4	f(wj	f(wj	PROPN
ejpam-3780	227	5	i	i	NOUN
ejpam-3780	227	6	)	)	PUNCT
ejpam-3780	227	7	=	=	SYM
ejpam-3780	227	8	a(i−1)(r+1)+j	a(i−1)(r+1)+j	NOUN
ejpam-3780	227	9	.	.	PUNCT
ejpam-3780	228	1	clearly	clearly	ADV
ejpam-3780	228	2	,	,	PUNCT
ejpam-3780	228	3	f	f	PROPN
ejpam-3780	228	4	is	be	AUX
ejpam-3780	228	5	injective	injective	ADJ
ejpam-3780	228	6	.	.	PUNCT
ejpam-3780	229	1	let	let	VERB
ejpam-3780	229	2	k	k	NOUN
ejpam-3780	229	3	=	=	PRON
ejpam-3780	229	4	{	{	PUNCT
ejpam-3780	229	5	f(u	f(u	PROPN
ejpam-3780	229	6	)	)	PUNCT
ejpam-3780	229	7	+	+	NUM
ejpam-3780	229	8	f(v	f(v	NOUN
ejpam-3780	229	9	)	)	PUNCT
ejpam-3780	229	10	:	:	PUNCT
ejpam-3780	229	11	uv	uv	PROPN
ejpam-3780	229	12	∈	∈	PROPN
ejpam-3780	229	13	e(g	e(g	PROPN
ejpam-3780	229	14	)	)	PUNCT
ejpam-3780	229	15	}	}	PUNCT
ejpam-3780	229	16	.	.	PUNCT
ejpam-3780	230	1	to	to	PART
ejpam-3780	230	2	show	show	VERB
ejpam-3780	230	3	that	that	SCONJ
ejpam-3780	230	4	f	f	PROPN
ejpam-3780	230	5	is	be	AUX
ejpam-3780	230	6	an	an	DET
ejpam-3780	230	7	efficient	efficient	ADJ
ejpam-3780	230	8	zero	zero	NUM
ejpam-3780	230	9	ring	ring	NOUN
ejpam-3780	230	10	labeling	labeling	NOUN
ejpam-3780	230	11	of	of	ADP
ejpam-3780	230	12	g	g	NOUN
ejpam-3780	230	13	,	,	PUNCT
ejpam-3780	230	14	we	we	PRON
ejpam-3780	230	15	need	need	VERB
ejpam-3780	230	16	to	to	PART
ejpam-3780	230	17	show	show	VERB
ejpam-3780	230	18	that	that	SCONJ
ejpam-3780	230	19	|k|	|k|	NOUN
ejpam-3780	230	20	=	=	SYM
ejpam-3780	231	1	r	r	NOUN
ejpam-3780	231	2	+	+	NUM
ejpam-3780	231	3	1	1	NUM
ejpam-3780	231	4	and	and	CCONJ
ejpam-3780	231	5	a0	a0	PROPN
ejpam-3780	231	6	/∈	/∈	PROPN
ejpam-3780	231	7	k.	k.	PROPN
ejpam-3780	232	1	for	for	ADP
ejpam-3780	232	2	the	the	DET
ejpam-3780	232	3	set	set	NOUN
ejpam-3780	232	4	of	of	ADP
ejpam-3780	232	5	sums	sum	NOUN
ejpam-3780	232	6	,	,	PUNCT
ejpam-3780	232	7	we	we	PRON
ejpam-3780	232	8	obtain	obtain	VERB
ejpam-3780	232	9	f(wi	f(wi	NOUN
ejpam-3780	232	10	)	)	PUNCT
ejpam-3780	233	1	+	+	NUM
ejpam-3780	233	2	f(wj	f(wj	PROPN
ejpam-3780	233	3	i	i	NOUN
ejpam-3780	233	4	)	)	PUNCT
ejpam-3780	233	5	=	=	PUNCT
ejpam-3780	234	1	ai(r+1	ai(r+1	VERB
ejpam-3780	234	2	)	)	PUNCT
ejpam-3780	234	3	+	+	NUM
ejpam-3780	234	4	a(i−1)(r+1)+j	a(i−1)(r+1)+j	NOUN
ejpam-3780	234	5	=	=	SYM
ejpam-3780	234	6	a2ir+2i−r−1+j	a2ir+2i−r−1+j	NOUN
ejpam-3780	234	7	=	=	SYM
ejpam-3780	234	8	aj−r−1	aj−r−1	PROPN
ejpam-3780	234	9	(	(	PUNCT
ejpam-3780	234	10	4	4	NUM
ejpam-3780	234	11	)	)	PUNCT
ejpam-3780	234	12	for	for	ADP
ejpam-3780	234	13	j	j	PROPN
ejpam-3780	234	14	=	=	SYM
ejpam-3780	234	15	1	1	NUM
ejpam-3780	234	16	,	,	PUNCT
ejpam-3780	234	17	2	2	NUM
ejpam-3780	234	18	,	,	PUNCT
ejpam-3780	234	19	.	.	PUNCT
ejpam-3780	234	20	.	.	PUNCT
ejpam-3780	235	1	.	.	PUNCT
ejpam-3780	236	1	,	,	PUNCT
ejpam-3780	236	2	r	r	NOUN
ejpam-3780	236	3	,	,	PUNCT
ejpam-3780	236	4	and	and	CCONJ
ejpam-3780	236	5	f(w1	f(w1	X
ejpam-3780	236	6	)	)	PUNCT
ejpam-3780	237	1	+	+	NUM
ejpam-3780	237	2	f(w2	f(w2	X
ejpam-3780	237	3	)	)	PUNCT
ejpam-3780	237	4	=	=	SYM
ejpam-3780	238	1	ar+1	ar+1	INTJ
ejpam-3780	238	2	+	+	CCONJ
ejpam-3780	238	3	a2r+2	a2r+2	PROPN
ejpam-3780	238	4	=	=	SYM
ejpam-3780	238	5	a3r+3	a3r+3	PROPN
ejpam-3780	238	6	=	=	SYM
ejpam-3780	238	7	ar+1	ar+1	PROPN
ejpam-3780	238	8	6=	6=	NUM
ejpam-3780	238	9	a0	a0	PROPN
ejpam-3780	238	10	(	(	PUNCT
ejpam-3780	238	11	5	5	NUM
ejpam-3780	238	12	)	)	PUNCT
ejpam-3780	238	13	it	it	PRON
ejpam-3780	238	14	remains	remain	VERB
ejpam-3780	238	15	to	to	PART
ejpam-3780	238	16	show	show	VERB
ejpam-3780	238	17	that	that	SCONJ
ejpam-3780	238	18	aj−r−1	aj−r−1	PROPN
ejpam-3780	238	19	6=	6=	NUM
ejpam-3780	238	20	a0	a0	PROPN
ejpam-3780	238	21	for	for	ADP
ejpam-3780	238	22	j	j	PROPN
ejpam-3780	238	23	=	=	SYM
ejpam-3780	238	24	1	1	NUM
ejpam-3780	238	25	,	,	PUNCT
ejpam-3780	238	26	2	2	NUM
ejpam-3780	238	27	,	,	PUNCT
ejpam-3780	238	28	.	.	PUNCT
ejpam-3780	238	29	.	.	PUNCT
ejpam-3780	238	30	.	.	PUNCT
ejpam-3780	239	1	,	,	PUNCT
ejpam-3780	239	2	r.	r.	NOUN
ejpam-3780	239	3	by	by	ADP
ejpam-3780	239	4	substitution	substitution	NOUN
ejpam-3780	239	5	,	,	PUNCT
ejpam-3780	239	6	we	we	PRON
ejpam-3780	239	7	obtain	obtain	VERB
ejpam-3780	239	8	the	the	DET
ejpam-3780	239	9	sums	sum	NOUN
ejpam-3780	239	10	a−r	a−r	PROPN
ejpam-3780	239	11	,	,	PUNCT
ejpam-3780	239	12	a1−r	a1−r	PROPN
ejpam-3780	239	13	,	,	PUNCT
ejpam-3780	239	14	.	.	PUNCT
ejpam-3780	239	15	.	.	PUNCT
ejpam-3780	240	1	.	.	PUNCT
ejpam-3780	241	1	,	,	PUNCT
ejpam-3780	241	2	a−1	a−1	PROPN
ejpam-3780	241	3	,	,	PUNCT
ejpam-3780	241	4	which	which	PRON
ejpam-3780	241	5	are	be	AUX
ejpam-3780	241	6	equal	equal	ADJ
ejpam-3780	241	7	to	to	ADP
ejpam-3780	241	8	ar+2	ar+2	NUM
ejpam-3780	241	9	,	,	PUNCT
ejpam-3780	241	10	ar+3	ar+3	PROPN
ejpam-3780	241	11	,	,	PUNCT
ejpam-3780	241	12	.	.	PUNCT
ejpam-3780	241	13	.	.	PUNCT
ejpam-3780	241	14	.	.	PUNCT
ejpam-3780	242	1	,	,	PUNCT
ejpam-3780	242	2	a2r+1	a2r+1	NOUN
ejpam-3780	242	3	,	,	PUNCT
ejpam-3780	242	4	respectively	respectively	ADV
ejpam-3780	242	5	.	.	PUNCT
ejpam-3780	243	1	for	for	ADP
ejpam-3780	243	2	r	r	NOUN
ejpam-3780	243	3	+	+	CCONJ
ejpam-3780	243	4	2	2	NUM
ejpam-3780	243	5	≤	≤	NUM
ejpam-3780	243	6	m	m	VERB
ejpam-3780	243	7	≤	≤	NOUN
ejpam-3780	243	8	2r	2r	NUM
ejpam-3780	244	1	+	+	CCONJ
ejpam-3780	244	2	1	1	NUM
ejpam-3780	244	3	,	,	PUNCT
ejpam-3780	244	4	we	we	PRON
ejpam-3780	244	5	have	have	VERB
ejpam-3780	244	6	0	0	NUM
ejpam-3780	244	7	<	<	X
ejpam-3780	244	8	m	m	X
ejpam-3780	244	9	<	<	X
ejpam-3780	244	10	2r	2r	NUM
ejpam-3780	245	1	+	+	CCONJ
ejpam-3780	245	2	2	2	NUM
ejpam-3780	245	3	,	,	PUNCT
ejpam-3780	245	4	hence	hence	ADV
ejpam-3780	245	5	m	m	VERB
ejpam-3780	245	6	6≡	6≡	NUM
ejpam-3780	245	7	0	0	NUM
ejpam-3780	246	1	(	(	PUNCT
ejpam-3780	246	2	mod	mod	PROPN
ejpam-3780	246	3	2r	2r	NUM
ejpam-3780	246	4	+	+	CCONJ
ejpam-3780	246	5	2	2	NUM
ejpam-3780	246	6	)	)	PUNCT
ejpam-3780	246	7	.	.	PUNCT
ejpam-3780	247	1	thus	thus	ADV
ejpam-3780	247	2	,	,	PUNCT
ejpam-3780	247	3	am	be	AUX
ejpam-3780	247	4	6=	6=	NUM
ejpam-3780	247	5	a0	a0	NOUN
ejpam-3780	247	6	for	for	ADP
ejpam-3780	247	7	m	m	PROPN
ejpam-3780	247	8	=	=	NOUN
ejpam-3780	247	9	r	r	NOUN
ejpam-3780	247	10	+	+	NUM
ejpam-3780	247	11	2	2	NUM
ejpam-3780	247	12	,	,	PUNCT
ejpam-3780	247	13	r	r	NOUN
ejpam-3780	247	14	+	+	NOUN
ejpam-3780	247	15	3	3	NUM
ejpam-3780	247	16	,	,	PUNCT
ejpam-3780	247	17	.	.	PUNCT
ejpam-3780	247	18	.	.	PUNCT
ejpam-3780	248	1	.	.	PUNCT
ejpam-3780	249	1	,	,	PUNCT
ejpam-3780	249	2	2r	2r	NUM
ejpam-3780	249	3	+	+	CCONJ
ejpam-3780	250	1	1	1	X
ejpam-3780	250	2	.	.	X
ejpam-3780	250	3	then	then	ADV
ejpam-3780	250	4	a0	a0	PROPN
ejpam-3780	250	5	/∈	/∈	PROPN
ejpam-3780	250	6	k.	k.	PROPN
ejpam-3780	251	1	moreover	moreover	ADV
ejpam-3780	251	2	,	,	PUNCT
ejpam-3780	251	3	k	k	PROPN
ejpam-3780	251	4	=	=	PRON
ejpam-3780	251	5	{	{	PUNCT
ejpam-3780	251	6	ar+1	ar+1	NOUN
ejpam-3780	251	7	,	,	PUNCT
ejpam-3780	251	8	ar+2	ar+2	NOUN
ejpam-3780	251	9	,	,	PUNCT
ejpam-3780	251	10	ar+3	ar+3	PROPN
ejpam-3780	251	11	,	,	PUNCT
ejpam-3780	251	12	.	.	PUNCT
ejpam-3780	251	13	.	.	PUNCT
ejpam-3780	251	14	.	.	PUNCT
ejpam-3780	252	1	,	,	PUNCT
ejpam-3780	252	2	a2r+1	a2r+1	ADJ
ejpam-3780	252	3	}	}	PUNCT
ejpam-3780	252	4	(	(	PUNCT
ejpam-3780	252	5	6	6	NUM
ejpam-3780	252	6	)	)	PUNCT
ejpam-3780	252	7	and	and	CCONJ
ejpam-3780	252	8	thus	thus	ADV
ejpam-3780	252	9	|k|	|k|	NOUN
ejpam-3780	253	1	=	=	SYM
ejpam-3780	253	2	r	r	NOUN
ejpam-3780	253	3	+	+	NOUN
ejpam-3780	253	4	1	1	NUM
ejpam-3780	253	5	.	.	PUNCT
ejpam-3780	254	1	therefore	therefore	ADV
ejpam-3780	254	2	,	,	PUNCT
ejpam-3780	254	3	f	f	PROPN
ejpam-3780	254	4	is	be	AUX
ejpam-3780	254	5	an	an	DET
ejpam-3780	254	6	efficient	efficient	ADJ
ejpam-3780	254	7	zero	zero	NUM
ejpam-3780	254	8	ring	ring	NOUN
ejpam-3780	254	9	labeling	labeling	NOUN
ejpam-3780	254	10	of	of	ADP
ejpam-3780	254	11	g.	g.	PROPN
ejpam-3780	254	12	case	case	NOUN
ejpam-3780	254	13	3	3	X
ejpam-3780	254	14	:	:	PUNCT
ejpam-3780	254	15	suppose	suppose	VERB
ejpam-3780	254	16	n	n	PRON
ejpam-3780	254	17	≥	≥	NUM
ejpam-3780	254	18	3	3	NUM
ejpam-3780	254	19	.	.	PUNCT
ejpam-3780	255	1	in	in	ADP
ejpam-3780	255	2	this	this	DET
ejpam-3780	255	3	case	case	NOUN
ejpam-3780	255	4	,	,	PUNCT
ejpam-3780	255	5	∆(g	∆(g	NOUN
ejpam-3780	255	6	)	)	PUNCT
ejpam-3780	255	7	=	=	SYM
ejpam-3780	255	8	d(wi	d(wi	PROPN
ejpam-3780	255	9	)	)	PUNCT
ejpam-3780	255	10	,	,	PUNCT
ejpam-3780	255	11	where	where	SCONJ
ejpam-3780	255	12	i	i	PRON
ejpam-3780	255	13	6=	6=	NOUN
ejpam-3780	255	14	1	1	NUM
ejpam-3780	255	15	and	and	CCONJ
ejpam-3780	255	16	i	i	PRON
ejpam-3780	255	17	6=	6=	PROPN
ejpam-3780	256	1	n.	n.	PROPN
ejpam-3780	256	2	then	then	ADV
ejpam-3780	256	3	∆(g	∆(g	NOUN
ejpam-3780	256	4	)	)	PUNCT
ejpam-3780	257	1	=	=	SYM
ejpam-3780	257	2	r	r	NOUN
ejpam-3780	257	3	+	+	NOUN
ejpam-3780	257	4	2	2	NUM
ejpam-3780	257	5	.	.	PUNCT
ejpam-3780	257	6	define	define	VERB
ejpam-3780	257	7	a	a	DET
ejpam-3780	257	8	function	function	NOUN
ejpam-3780	257	9	f	f	NOUN
ejpam-3780	257	10	:	:	PUNCT
ejpam-3780	257	11	v	v	X
ejpam-3780	257	12	(	(	PUNCT
ejpam-3780	257	13	g)→m0	g)→m0	NOUN
ejpam-3780	257	14	2	2	NUM
ejpam-3780	257	15	(	(	PUNCT
ejpam-3780	257	16	znr+n	znr+n	PROPN
ejpam-3780	257	17	)	)	PUNCT
ejpam-3780	257	18	such	such	ADJ
ejpam-3780	257	19	that	that	SCONJ
ejpam-3780	257	20	f(wi	f(wi	NOUN
ejpam-3780	257	21	)	)	PUNCT
ejpam-3780	257	22	=	=	SYM
ejpam-3780	257	23	a(r+1	a(r+1	PROPN
ejpam-3780	257	24	)	)	PUNCT
ejpam-3780	257	25	(	(	PUNCT
ejpam-3780	257	26	2n−i−1	2n−i−1	PROPN
ejpam-3780	257	27	2	2	NUM
ejpam-3780	257	28	)	)	PUNCT
ejpam-3780	257	29	if	if	SCONJ
ejpam-3780	257	30	i	i	PRON
ejpam-3780	257	31	is	be	AUX
ejpam-3780	257	32	odd	odd	ADJ
ejpam-3780	257	33	a(r+1	a(r+1	PUNCT
ejpam-3780	257	34	)	)	PUNCT
ejpam-3780	257	35	(	(	PUNCT
ejpam-3780	257	36	i−2	i−2	NOUN
ejpam-3780	257	37	2	2	NUM
ejpam-3780	257	38	)	)	PUNCT
ejpam-3780	257	39	if	if	SCONJ
ejpam-3780	257	40	i	i	PRON
ejpam-3780	257	41	is	be	AUX
ejpam-3780	257	42	even	even	ADV
ejpam-3780	257	43	(	(	PUNCT
ejpam-3780	257	44	7	7	NUM
ejpam-3780	257	45	)	)	PUNCT
ejpam-3780	257	46	and	and	CCONJ
ejpam-3780	257	47	f(wj	f(wj	PROPN
ejpam-3780	257	48	i	i	NOUN
ejpam-3780	257	49	)	)	PUNCT
ejpam-3780	258	1	=	=	SYM
ejpam-3780	258	2	a(r+1	a(r+1	PROPN
ejpam-3780	258	3	)	)	PUNCT
ejpam-3780	258	4	(	(	PUNCT
ejpam-3780	258	5	i−3	i−3	PROPN
ejpam-3780	258	6	2	2	NUM
ejpam-3780	258	7	)	)	PUNCT
ejpam-3780	259	1	+	+	NOUN
ejpam-3780	259	2	j	j	NOUN
ejpam-3780	259	3	if	if	SCONJ
ejpam-3780	259	4	i	i	PRON
ejpam-3780	259	5	is	be	AUX
ejpam-3780	259	6	odd	odd	ADJ
ejpam-3780	259	7	a(r+1	a(r+1	PUNCT
ejpam-3780	259	8	)	)	PUNCT
ejpam-3780	259	9	(	(	PUNCT
ejpam-3780	259	10	2n−i−2	2n−i−2	NUM
ejpam-3780	259	11	2	2	NUM
ejpam-3780	259	12	)	)	PUNCT
ejpam-3780	260	1	+	+	NOUN
ejpam-3780	260	2	j	j	NOUN
ejpam-3780	260	3	if	if	SCONJ
ejpam-3780	260	4	i	i	PRON
ejpam-3780	260	5	is	be	AUX
ejpam-3780	260	6	even	even	ADV
ejpam-3780	260	7	.	.	PUNCT
ejpam-3780	261	1	(	(	PUNCT
ejpam-3780	261	2	8)	8)	NUM
ejpam-3780	261	3	d.	d.	PROPN
ejpam-3780	261	4	chua	chua	PROPN
ejpam-3780	261	5	,	,	PUNCT
ejpam-3780	261	6	f.	f.	PROPN
ejpam-3780	261	7	campeña	campeña	PROPN
ejpam-3780	261	8	,	,	PUNCT
ejpam-3780	261	9	f.	f.	PROPN
ejpam-3780	261	10	franco	franco	PROPN
ejpam-3780	261	11	/	/	SYM
ejpam-3780	261	12	eur	eur	PROPN
ejpam-3780	261	13	.	.	PUNCT
ejpam-3780	262	1	j.	j.	PROPN
ejpam-3780	262	2	pure	pure	PROPN
ejpam-3780	262	3	appl	appl	PROPN
ejpam-3780	262	4	.	.	PROPN
ejpam-3780	262	5	math	math	PROPN
ejpam-3780	262	6	,	,	PUNCT
ejpam-3780	262	7	13	13	NUM
ejpam-3780	262	8	(	(	PUNCT
ejpam-3780	262	9	3	3	NUM
ejpam-3780	262	10	)	)	PUNCT
ejpam-3780	262	11	(	(	PUNCT
ejpam-3780	262	12	2020	2020	NUM
ejpam-3780	262	13	)	)	PUNCT
ejpam-3780	262	14	,	,	PUNCT
ejpam-3780	262	15	674	674	NUM
ejpam-3780	262	16	-	-	SYM
ejpam-3780	262	17	696	696	NUM
ejpam-3780	262	18	681	681	NUM
ejpam-3780	262	19	clearly	clearly	ADV
ejpam-3780	262	20	,	,	PUNCT
ejpam-3780	262	21	f	f	PROPN
ejpam-3780	262	22	is	be	AUX
ejpam-3780	262	23	injective	injective	ADJ
ejpam-3780	262	24	.	.	PUNCT
ejpam-3780	263	1	let	let	VERB
ejpam-3780	263	2	k	k	NOUN
ejpam-3780	263	3	=	=	PRON
ejpam-3780	263	4	{	{	PUNCT
ejpam-3780	263	5	f(u	f(u	PROPN
ejpam-3780	263	6	)	)	PUNCT
ejpam-3780	263	7	+	+	NUM
ejpam-3780	263	8	f(v	f(v	NOUN
ejpam-3780	263	9	)	)	PUNCT
ejpam-3780	263	10	:	:	PUNCT
ejpam-3780	263	11	uv	uv	PROPN
ejpam-3780	263	12	∈	∈	PROPN
ejpam-3780	263	13	e(g	e(g	PROPN
ejpam-3780	263	14	)	)	PUNCT
ejpam-3780	263	15	}	}	PUNCT
ejpam-3780	263	16	.	.	PUNCT
ejpam-3780	264	1	to	to	PART
ejpam-3780	264	2	show	show	VERB
ejpam-3780	264	3	that	that	SCONJ
ejpam-3780	264	4	f	f	PROPN
ejpam-3780	264	5	is	be	AUX
ejpam-3780	264	6	an	an	DET
ejpam-3780	264	7	efficient	efficient	ADJ
ejpam-3780	264	8	zero	zero	NUM
ejpam-3780	264	9	ring	ring	NOUN
ejpam-3780	264	10	labeling	labeling	NOUN
ejpam-3780	264	11	of	of	ADP
ejpam-3780	264	12	g	g	NOUN
ejpam-3780	264	13	,	,	PUNCT
ejpam-3780	264	14	we	we	PRON
ejpam-3780	264	15	need	need	VERB
ejpam-3780	264	16	to	to	PART
ejpam-3780	264	17	show	show	VERB
ejpam-3780	264	18	that	that	SCONJ
ejpam-3780	264	19	|k|	|k|	NOUN
ejpam-3780	264	20	=	=	SYM
ejpam-3780	265	1	r	r	NOUN
ejpam-3780	265	2	+	+	NUM
ejpam-3780	265	3	2	2	NUM
ejpam-3780	265	4	and	and	CCONJ
ejpam-3780	265	5	a0	a0	PROPN
ejpam-3780	265	6	/∈	/∈	PROPN
ejpam-3780	265	7	k.	k.	PROPN
ejpam-3780	266	1	for	for	ADP
ejpam-3780	266	2	adjacent	adjacent	ADJ
ejpam-3780	266	3	vertices	vertex	NOUN
ejpam-3780	266	4	in	in	ADP
ejpam-3780	266	5	the	the	DET
ejpam-3780	266	6	central	central	ADJ
ejpam-3780	266	7	path	path	NOUN
ejpam-3780	266	8	,	,	PUNCT
ejpam-3780	266	9	we	we	PRON
ejpam-3780	266	10	obtain	obtain	VERB
ejpam-3780	266	11	the	the	DET
ejpam-3780	266	12	sums	sum	NOUN
ejpam-3780	266	13	f(wi	f(wi	NOUN
ejpam-3780	266	14	)	)	PUNCT
ejpam-3780	266	15	+	+	NUM
ejpam-3780	266	16	f(wi+1	f(wi+1	X
ejpam-3780	266	17	)	)	PUNCT
ejpam-3780	266	18	=	=	SYM
ejpam-3780	267	1	a(r+1	a(r+1	PROPN
ejpam-3780	267	2	)	)	PUNCT
ejpam-3780	268	1	(	(	PUNCT
ejpam-3780	268	2	2n−i−1	2n−i−1	PROPN
ejpam-3780	268	3	2	2	NUM
ejpam-3780	268	4	)	)	PUNCT
ejpam-3780	268	5	+	+	CCONJ
ejpam-3780	268	6	a	a	DET
ejpam-3780	268	7	(	(	PUNCT
ejpam-3780	268	8	r+1	r+1	NOUN
ejpam-3780	268	9	)	)	PUNCT
ejpam-3780	268	10	(	(	PUNCT
ejpam-3780	268	11	(	(	PUNCT
ejpam-3780	268	12	i+1)−2	i+1)−2	INTJ
ejpam-3780	268	13	2	2	X
ejpam-3780	268	14	)	)	PUNCT
ejpam-3780	268	15	=	=	SYM
ejpam-3780	268	16	a(r+1)(n−1	a(r+1)(n−1	NOUN
ejpam-3780	268	17	)	)	PUNCT
ejpam-3780	268	18	(	(	PUNCT
ejpam-3780	268	19	9	9	X
ejpam-3780	268	20	)	)	PUNCT
ejpam-3780	268	21	if	if	SCONJ
ejpam-3780	268	22	i	i	PRON
ejpam-3780	268	23	is	be	AUX
ejpam-3780	268	24	odd	odd	ADJ
ejpam-3780	268	25	,	,	PUNCT
ejpam-3780	268	26	and	and	CCONJ
ejpam-3780	268	27	f(wi	f(wi	NOUN
ejpam-3780	268	28	)	)	PUNCT
ejpam-3780	268	29	+	+	NUM
ejpam-3780	268	30	f(wi+1	f(wi+1	X
ejpam-3780	268	31	)	)	PUNCT
ejpam-3780	268	32	=	=	SYM
ejpam-3780	268	33	a(r+1	a(r+1	PROPN
ejpam-3780	268	34	)	)	PUNCT
ejpam-3780	268	35	(	(	PUNCT
ejpam-3780	268	36	i−2	i−2	NOUN
ejpam-3780	268	37	2	2	NUM
ejpam-3780	268	38	)	)	PUNCT
ejpam-3780	268	39	+	+	CCONJ
ejpam-3780	268	40	a	a	DET
ejpam-3780	268	41	(	(	PUNCT
ejpam-3780	268	42	r+1	r+1	NOUN
ejpam-3780	268	43	)	)	PUNCT
ejpam-3780	268	44	(	(	PUNCT
ejpam-3780	268	45	2n−(i+1)−1	2n−(i+1)−1	NOUN
ejpam-3780	268	46	2	2	NUM
ejpam-3780	268	47	)	)	PUNCT
ejpam-3780	268	48	=	=	PUNCT
ejpam-3780	268	49	a(r+1)(n−2	a(r+1)(n−2	PROPN
ejpam-3780	268	50	)	)	PUNCT
ejpam-3780	268	51	(	(	PUNCT
ejpam-3780	268	52	10	10	NUM
ejpam-3780	268	53	)	)	PUNCT
ejpam-3780	268	54	if	if	SCONJ
ejpam-3780	268	55	i	i	PRON
ejpam-3780	268	56	is	be	AUX
ejpam-3780	268	57	even	even	ADV
ejpam-3780	268	58	.	.	PUNCT
ejpam-3780	269	1	note	note	VERB
ejpam-3780	269	2	that	that	SCONJ
ejpam-3780	269	3	a(r+1)(n−1	a(r+1)(n−1	NOUN
ejpam-3780	269	4	)	)	PUNCT
ejpam-3780	269	5	6=	6=	NUM
ejpam-3780	269	6	a0	a0	PROPN
ejpam-3780	269	7	;	;	PUNCT
ejpam-3780	269	8	otherwise	otherwise	ADV
ejpam-3780	269	9	,	,	PUNCT
ejpam-3780	269	10	either	either	CCONJ
ejpam-3780	269	11	r	r	PROPN
ejpam-3780	269	12	≡	≡	PROPN
ejpam-3780	269	13	−1	−1	NOUN
ejpam-3780	269	14	(	(	PUNCT
ejpam-3780	269	15	mod	mod	PROPN
ejpam-3780	269	16	nr	nr	PROPN
ejpam-3780	269	17	+	+	PROPN
ejpam-3780	269	18	n	n	CCONJ
ejpam-3780	269	19	)	)	PUNCT
ejpam-3780	269	20	,	,	PUNCT
ejpam-3780	269	21	which	which	PRON
ejpam-3780	269	22	is	be	AUX
ejpam-3780	269	23	a	a	DET
ejpam-3780	269	24	contradiction	contradiction	NOUN
ejpam-3780	269	25	by	by	ADP
ejpam-3780	269	26	definition	definition	NOUN
ejpam-3780	269	27	of	of	ADP
ejpam-3780	269	28	r	r	NOUN
ejpam-3780	269	29	,	,	PUNCT
ejpam-3780	269	30	or	or	CCONJ
ejpam-3780	269	31	n	n	PRON
ejpam-3780	269	32	≡	≡	PROPN
ejpam-3780	269	33	1	1	NUM
ejpam-3780	269	34	(	(	PUNCT
ejpam-3780	269	35	mod	mod	PROPN
ejpam-3780	269	36	nr+n	nr+n	PROPN
ejpam-3780	269	37	)	)	PUNCT
ejpam-3780	269	38	,	,	PUNCT
ejpam-3780	269	39	which	which	PRON
ejpam-3780	269	40	is	be	AUX
ejpam-3780	269	41	also	also	ADV
ejpam-3780	269	42	a	a	DET
ejpam-3780	269	43	contradiction	contradiction	NOUN
ejpam-3780	269	44	by	by	ADP
ejpam-3780	269	45	definition	definition	NOUN
ejpam-3780	269	46	of	of	ADP
ejpam-3780	269	47	n	n	PROPN
ejpam-3780	269	48	in	in	ADP
ejpam-3780	269	49	this	this	DET
ejpam-3780	269	50	case	case	NOUN
ejpam-3780	269	51	.	.	PUNCT
ejpam-3780	270	1	similarly	similarly	ADV
ejpam-3780	270	2	,	,	PUNCT
ejpam-3780	270	3	a(r+1)(n−2	a(r+1)(n−2	PROPN
ejpam-3780	270	4	)	)	PUNCT
ejpam-3780	270	5	6=	6=	NUM
ejpam-3780	270	6	a0	a0	PROPN
ejpam-3780	270	7	.	.	PUNCT
ejpam-3780	271	1	for	for	ADP
ejpam-3780	271	2	pairs	pair	NOUN
ejpam-3780	271	3	of	of	ADP
ejpam-3780	271	4	adjacent	adjacent	ADJ
ejpam-3780	271	5	hanging	hang	VERB
ejpam-3780	271	6	leaf	leaf	NOUN
ejpam-3780	271	7	and	and	CCONJ
ejpam-3780	271	8	vertex	vertex	NOUN
ejpam-3780	271	9	in	in	ADP
ejpam-3780	271	10	the	the	DET
ejpam-3780	271	11	central	central	ADJ
ejpam-3780	271	12	path	path	NOUN
ejpam-3780	271	13	,	,	PUNCT
ejpam-3780	271	14	we	we	PRON
ejpam-3780	271	15	obtain	obtain	VERB
ejpam-3780	271	16	the	the	DET
ejpam-3780	271	17	sums	sum	NOUN
ejpam-3780	271	18	f(wi	f(wi	NOUN
ejpam-3780	271	19	)	)	PUNCT
ejpam-3780	272	1	+	+	NUM
ejpam-3780	273	1	f(wj	f(wj	PROPN
ejpam-3780	273	2	i	i	NOUN
ejpam-3780	273	3	)	)	PUNCT
ejpam-3780	274	1	=	=	SYM
ejpam-3780	274	2	a(r+1	a(r+1	PROPN
ejpam-3780	274	3	)	)	PUNCT
ejpam-3780	274	4	(	(	PUNCT
ejpam-3780	274	5	2n−i−1	2n−i−1	PROPN
ejpam-3780	274	6	2	2	NUM
ejpam-3780	274	7	)	)	PUNCT
ejpam-3780	274	8	+	+	CCONJ
ejpam-3780	274	9	a(r+1	a(r+1	X
ejpam-3780	274	10	)	)	PUNCT
ejpam-3780	274	11	(	(	PUNCT
ejpam-3780	274	12	i−3	i−3	PROPN
ejpam-3780	274	13	2	2	NUM
ejpam-3780	274	14	)	)	PUNCT
ejpam-3780	274	15	+	+	NUM
ejpam-3780	274	16	j	j	NOUN
ejpam-3780	274	17	=	=	SYM
ejpam-3780	274	18	a(r+1)(n−2)+j	a(r+1)(n−2)+j	PROPN
ejpam-3780	274	19	(	(	PUNCT
ejpam-3780	274	20	11	11	NUM
ejpam-3780	274	21	)	)	PUNCT
ejpam-3780	274	22	for	for	ADP
ejpam-3780	274	23	j	j	PROPN
ejpam-3780	274	24	=	=	SYM
ejpam-3780	274	25	1	1	NUM
ejpam-3780	274	26	,	,	PUNCT
ejpam-3780	274	27	2	2	NUM
ejpam-3780	274	28	,	,	PUNCT
ejpam-3780	274	29	.	.	PUNCT
ejpam-3780	274	30	.	.	PUNCT
ejpam-3780	274	31	.	.	PUNCT
ejpam-3780	275	1	,	,	PUNCT
ejpam-3780	275	2	r	r	NOUN
ejpam-3780	275	3	if	if	SCONJ
ejpam-3780	275	4	i	i	PRON
ejpam-3780	275	5	is	be	AUX
ejpam-3780	275	6	odd	odd	ADJ
ejpam-3780	275	7	,	,	PUNCT
ejpam-3780	275	8	and	and	CCONJ
ejpam-3780	275	9	f(wi	f(wi	NOUN
ejpam-3780	275	10	)	)	PUNCT
ejpam-3780	275	11	+	+	NUM
ejpam-3780	275	12	f(wj	f(wj	PROPN
ejpam-3780	275	13	i	i	NOUN
ejpam-3780	275	14	)	)	PUNCT
ejpam-3780	275	15	=	=	SYM
ejpam-3780	276	1	a(r+1	a(r+1	PROPN
ejpam-3780	276	2	)	)	PUNCT
ejpam-3780	276	3	(	(	PUNCT
ejpam-3780	276	4	i−2	i−2	NOUN
ejpam-3780	276	5	2	2	NUM
ejpam-3780	276	6	)	)	PUNCT
ejpam-3780	277	1	+	+	CCONJ
ejpam-3780	277	2	a(r+1	a(r+1	X
ejpam-3780	277	3	)	)	PUNCT
ejpam-3780	277	4	(	(	PUNCT
ejpam-3780	277	5	2n−i−2	2n−i−2	NUM
ejpam-3780	277	6	2	2	NUM
ejpam-3780	277	7	)	)	PUNCT
ejpam-3780	278	1	+	+	NUM
ejpam-3780	278	2	j	j	NOUN
ejpam-3780	278	3	=	=	SYM
ejpam-3780	278	4	a(r+1)(n−2)+j	a(r+1)(n−2)+j	PROPN
ejpam-3780	278	5	(	(	PUNCT
ejpam-3780	278	6	12	12	NUM
ejpam-3780	278	7	)	)	PUNCT
ejpam-3780	278	8	for	for	ADP
ejpam-3780	278	9	j	j	PROPN
ejpam-3780	278	10	=	=	SYM
ejpam-3780	278	11	1	1	NUM
ejpam-3780	278	12	,	,	PUNCT
ejpam-3780	278	13	2	2	NUM
ejpam-3780	278	14	,	,	PUNCT
ejpam-3780	278	15	.	.	PUNCT
ejpam-3780	278	16	.	.	PUNCT
ejpam-3780	278	17	.	.	PUNCT
ejpam-3780	279	1	,	,	PUNCT
ejpam-3780	279	2	r	r	NOUN
ejpam-3780	279	3	if	if	SCONJ
ejpam-3780	279	4	i	i	PRON
ejpam-3780	279	5	is	be	AUX
ejpam-3780	279	6	even	even	ADV
ejpam-3780	279	7	.	.	PUNCT
ejpam-3780	280	1	we	we	PRON
ejpam-3780	280	2	show	show	VERB
ejpam-3780	280	3	that	that	SCONJ
ejpam-3780	280	4	a(r+1)(n−2)+j	a(r+1)(n−2)+j	PROPN
ejpam-3780	280	5	6=	6=	PROPN
ejpam-3780	280	6	a0	a0	PROPN
ejpam-3780	280	7	.	.	PROPN
ejpam-3780	281	1	assume	assume	VERB
ejpam-3780	281	2	that	that	SCONJ
ejpam-3780	281	3	a(r+1)(n−2)+j	a(r+1)(n−2)+j	PROPN
ejpam-3780	281	4	=	=	SYM
ejpam-3780	281	5	a0	a0	PROPN
ejpam-3780	281	6	for	for	ADP
ejpam-3780	281	7	some	some	DET
ejpam-3780	281	8	j.	j.	PROPN
ejpam-3780	281	9	but	but	CCONJ
ejpam-3780	281	10	this	this	PRON
ejpam-3780	281	11	implies	imply	VERB
ejpam-3780	281	12	that	that	SCONJ
ejpam-3780	281	13	j	j	PROPN
ejpam-3780	281	14	≡	≡	PROPN
ejpam-3780	281	15	2r	2r	NUM
ejpam-3780	282	1	+	+	CCONJ
ejpam-3780	282	2	2	2	NUM
ejpam-3780	282	3	(	(	PUNCT
ejpam-3780	282	4	mod	mod	PROPN
ejpam-3780	282	5	nr	nr	PROPN
ejpam-3780	282	6	+	+	PROPN
ejpam-3780	282	7	n	n	CCONJ
ejpam-3780	282	8	)	)	PUNCT
ejpam-3780	282	9	.	.	PUNCT
ejpam-3780	283	1	this	this	PRON
ejpam-3780	283	2	is	be	AUX
ejpam-3780	283	3	a	a	DET
ejpam-3780	283	4	contradiction	contradiction	NOUN
ejpam-3780	283	5	since	since	SCONJ
ejpam-3780	283	6	n	n	NUM
ejpam-3780	283	7	is	be	AUX
ejpam-3780	283	8	at	at	ADV
ejpam-3780	283	9	least	least	ADJ
ejpam-3780	283	10	3	3	NUM
ejpam-3780	283	11	in	in	ADP
ejpam-3780	283	12	this	this	DET
ejpam-3780	283	13	case	case	NOUN
ejpam-3780	283	14	,	,	PUNCT
ejpam-3780	283	15	and	and	CCONJ
ejpam-3780	283	16	j	j	PROPN
ejpam-3780	283	17	can	can	AUX
ejpam-3780	283	18	not	not	PART
ejpam-3780	283	19	be	be	AUX
ejpam-3780	283	20	greater	great	ADJ
ejpam-3780	283	21	than	than	ADP
ejpam-3780	283	22	r	r	NOUN
ejpam-3780	283	23	by	by	ADP
ejpam-3780	283	24	its	its	PRON
ejpam-3780	283	25	definition	definition	NOUN
ejpam-3780	283	26	.	.	PUNCT
ejpam-3780	284	1	then	then	ADV
ejpam-3780	284	2	a0	a0	PROPN
ejpam-3780	284	3	/∈	/∈	PROPN
ejpam-3780	285	1	k.	k.	PROPN
ejpam-3780	286	1	moreover	moreover	ADV
ejpam-3780	286	2	,	,	PUNCT
ejpam-3780	286	3	k	k	PROPN
ejpam-3780	286	4	=	=	X
ejpam-3780	286	5	{	{	PUNCT
ejpam-3780	286	6	a(r+1)(n−2	a(r+1)(n−2	NOUN
ejpam-3780	286	7	)	)	PUNCT
ejpam-3780	286	8	,	,	PUNCT
ejpam-3780	286	9	a(r+1)(n−2)+1	a(r+1)(n−2)+1	NOUN
ejpam-3780	286	10	,	,	PUNCT
ejpam-3780	286	11	.	.	PUNCT
ejpam-3780	286	12	.	.	PUNCT
ejpam-3780	287	1	.	.	PUNCT
ejpam-3780	288	1	,	,	PUNCT
ejpam-3780	288	2	a(r+1)(n−2)+r	a(r+1)(n−2)+r	PROPN
ejpam-3780	288	3	,	,	PUNCT
ejpam-3780	288	4	a(r+1)(n−1	a(r+1)(n−1	PROPN
ejpam-3780	288	5	)	)	PUNCT
ejpam-3780	288	6	}	}	PUNCT
ejpam-3780	288	7	(	(	PUNCT
ejpam-3780	288	8	13	13	NUM
ejpam-3780	288	9	)	)	PUNCT
ejpam-3780	288	10	and	and	CCONJ
ejpam-3780	288	11	thus	thus	ADV
ejpam-3780	288	12	|k|	|k|	NOUN
ejpam-3780	288	13	=	=	SYM
ejpam-3780	289	1	r	r	NOUN
ejpam-3780	289	2	+	+	PROPN
ejpam-3780	289	3	2	2	NUM
ejpam-3780	289	4	.	.	X
ejpam-3780	290	1	therefore	therefore	ADV
ejpam-3780	290	2	,	,	PUNCT
ejpam-3780	290	3	f	f	PROPN
ejpam-3780	290	4	is	be	AUX
ejpam-3780	290	5	an	an	DET
ejpam-3780	290	6	efficient	efficient	ADJ
ejpam-3780	290	7	zero	zero	NUM
ejpam-3780	290	8	ring	ring	NOUN
ejpam-3780	290	9	labeling	labeling	NOUN
ejpam-3780	290	10	of	of	ADP
ejpam-3780	290	11	g.	g.	PROPN
ejpam-3780	290	12	example	example	PROPN
ejpam-3780	290	13	4	4	NUM
ejpam-3780	290	14	.	.	PUNCT
ejpam-3780	290	15	figure	figure	NOUN
ejpam-3780	290	16	5	5	NUM
ejpam-3780	290	17	shows	show	VERB
ejpam-3780	290	18	an	an	DET
ejpam-3780	290	19	efficient	efficient	ADJ
ejpam-3780	290	20	zero	zero	NUM
ejpam-3780	290	21	ring	ring	NOUN
ejpam-3780	290	22	labeling	labeling	NOUN
ejpam-3780	290	23	of	of	ADP
ejpam-3780	290	24	a	a	DET
ejpam-3780	290	25	caterpillar	caterpillar	ADJ
ejpam-3780	290	26	g	g	NOUN
ejpam-3780	290	27	with	with	ADP
ejpam-3780	290	28	a	a	DET
ejpam-3780	290	29	central	central	ADJ
ejpam-3780	290	30	path	path	NOUN
ejpam-3780	290	31	with	with	ADP
ejpam-3780	290	32	four	four	NUM
ejpam-3780	290	33	vertices	vertex	NOUN
ejpam-3780	290	34	,	,	PUNCT
ejpam-3780	290	35	where	where	SCONJ
ejpam-3780	290	36	each	each	DET
ejpam-3780	290	37	vertex	vertex	NOUN
ejpam-3780	290	38	in	in	ADP
ejpam-3780	290	39	the	the	DET
ejpam-3780	290	40	central	central	ADJ
ejpam-3780	290	41	path	path	NOUN
ejpam-3780	290	42	has	have	VERB
ejpam-3780	290	43	three	three	NUM
ejpam-3780	290	44	hanging	hang	VERB
ejpam-3780	290	45	leaves	leave	NOUN
ejpam-3780	290	46	,	,	PUNCT
ejpam-3780	290	47	using	use	VERB
ejpam-3780	290	48	m0	m0	PROPN
ejpam-3780	290	49	2	2	NUM
ejpam-3780	290	50	(	(	PUNCT
ejpam-3780	290	51	z16	z16	NOUN
ejpam-3780	290	52	)	)	PUNCT
ejpam-3780	290	53	.	.	PUNCT
ejpam-3780	291	1	in	in	ADP
ejpam-3780	291	2	this	this	DET
ejpam-3780	291	3	labeling	labeling	NOUN
ejpam-3780	291	4	,	,	PUNCT
ejpam-3780	291	5	the	the	DET
ejpam-3780	291	6	set	set	NOUN
ejpam-3780	291	7	of	of	ADP
ejpam-3780	291	8	sums	sum	NOUN
ejpam-3780	291	9	is	be	AUX
ejpam-3780	291	10	k	k	NOUN
ejpam-3780	291	11	=	=	PUNCT
ejpam-3780	291	12	{	{	PUNCT
ejpam-3780	291	13	a8	a8	PROPN
ejpam-3780	291	14	,	,	PUNCT
ejpam-3780	291	15	a9	a9	PROPN
ejpam-3780	291	16	,	,	PUNCT
ejpam-3780	291	17	a10	a10	PROPN
ejpam-3780	291	18	,	,	PUNCT
ejpam-3780	291	19	a11	a11	PROPN
ejpam-3780	291	20	,	,	PUNCT
ejpam-3780	291	21	a12	a12	NOUN
ejpam-3780	291	22	}	}	PUNCT
ejpam-3780	291	23	and	and	CCONJ
ejpam-3780	291	24	thus	thus	ADV
ejpam-3780	291	25	|k|	|k|	PROPN
ejpam-3780	291	26	=	=	SYM
ejpam-3780	291	27	∆(g	∆(g	PROPN
ejpam-3780	291	28	)	)	PUNCT
ejpam-3780	291	29	=	=	SYM
ejpam-3780	292	1	5	5	X
ejpam-3780	292	2	.	.	PUNCT
ejpam-3780	292	3	corollary	corollary	ADJ
ejpam-3780	292	4	1	1	NUM
ejpam-3780	292	5	.	.	PUNCT
ejpam-3780	293	1	a	a	DET
ejpam-3780	293	2	path	path	NOUN
ejpam-3780	293	3	graph	graph	NOUN
ejpam-3780	293	4	has	have	VERB
ejpam-3780	293	5	an	an	DET
ejpam-3780	293	6	efficient	efficient	ADJ
ejpam-3780	293	7	zero	zero	NUM
ejpam-3780	293	8	ring	ring	NOUN
ejpam-3780	293	9	labeling	labeling	NOUN
ejpam-3780	293	10	.	.	PUNCT
ejpam-3780	294	1	proof	proof	NOUN
ejpam-3780	294	2	.	.	PUNCT
ejpam-3780	295	1	let	let	VERB
ejpam-3780	295	2	g	g	NOUN
ejpam-3780	295	3	be	be	AUX
ejpam-3780	295	4	the	the	DET
ejpam-3780	295	5	path	path	NOUN
ejpam-3780	295	6	graph	graph	NOUN
ejpam-3780	295	7	pn	pn	AUX
ejpam-3780	295	8	.	.	PROPN
ejpam-3780	295	9	consider	consider	VERB
ejpam-3780	295	10	g	g	NOUN
ejpam-3780	295	11	as	as	ADP
ejpam-3780	295	12	its	its	PRON
ejpam-3780	295	13	own	own	ADJ
ejpam-3780	295	14	central	central	ADJ
ejpam-3780	295	15	path	path	NOUN
ejpam-3780	295	16	.	.	PUNCT
ejpam-3780	296	1	then	then	ADV
ejpam-3780	296	2	g	g	PROPN
ejpam-3780	296	3	is	be	AUX
ejpam-3780	296	4	a	a	DET
ejpam-3780	296	5	caterpillar	caterpillar	NOUN
ejpam-3780	296	6	with	with	ADP
ejpam-3780	296	7	a	a	DET
ejpam-3780	296	8	central	central	ADJ
ejpam-3780	296	9	path	path	NOUN
ejpam-3780	296	10	with	with	ADP
ejpam-3780	296	11	n	n	ADP
ejpam-3780	296	12	vertices	vertex	NOUN
ejpam-3780	296	13	,	,	PUNCT
ejpam-3780	296	14	where	where	SCONJ
ejpam-3780	296	15	each	each	DET
ejpam-3780	296	16	vertex	vertex	NOUN
ejpam-3780	296	17	in	in	ADP
ejpam-3780	296	18	the	the	DET
ejpam-3780	296	19	central	central	ADJ
ejpam-3780	296	20	path	path	NOUN
ejpam-3780	296	21	has	have	VERB
ejpam-3780	296	22	no	no	DET
ejpam-3780	296	23	hanging	hang	VERB
ejpam-3780	296	24	leaf	leaf	NOUN
ejpam-3780	296	25	.	.	PUNCT
ejpam-3780	297	1	by	by	ADP
ejpam-3780	297	2	theorem	theorem	NOUN
ejpam-3780	297	3	4	4	NUM
ejpam-3780	297	4	,	,	PUNCT
ejpam-3780	297	5	g	g	PROPN
ejpam-3780	297	6	has	have	VERB
ejpam-3780	297	7	an	an	DET
ejpam-3780	297	8	efficient	efficient	ADJ
ejpam-3780	297	9	zero	zero	NUM
ejpam-3780	297	10	ring	ring	NOUN
ejpam-3780	297	11	labeling	labeling	NOUN
ejpam-3780	297	12	.	.	PUNCT
ejpam-3780	298	1	example	example	NOUN
ejpam-3780	298	2	5	5	NUM
ejpam-3780	298	3	.	.	PUNCT
ejpam-3780	298	4	figure	figure	VERB
ejpam-3780	298	5	6	6	NUM
ejpam-3780	298	6	shows	show	VERB
ejpam-3780	298	7	an	an	DET
ejpam-3780	298	8	efficient	efficient	ADJ
ejpam-3780	298	9	zero	zero	NUM
ejpam-3780	298	10	ring	ring	NOUN
ejpam-3780	298	11	labeling	labeling	NOUN
ejpam-3780	298	12	of	of	ADP
ejpam-3780	298	13	p10	p10	NOUN
ejpam-3780	298	14	using	use	VERB
ejpam-3780	298	15	m0	m0	PROPN
ejpam-3780	298	16	2	2	NUM
ejpam-3780	298	17	(	(	PUNCT
ejpam-3780	298	18	z10	z10	NOUN
ejpam-3780	298	19	)	)	PUNCT
ejpam-3780	298	20	.	.	PUNCT
ejpam-3780	299	1	in	in	ADP
ejpam-3780	299	2	this	this	DET
ejpam-3780	299	3	labeling	labeling	NOUN
ejpam-3780	299	4	,	,	PUNCT
ejpam-3780	299	5	the	the	DET
ejpam-3780	299	6	set	set	NOUN
ejpam-3780	299	7	of	of	ADP
ejpam-3780	299	8	sums	sum	NOUN
ejpam-3780	299	9	is	be	AUX
ejpam-3780	299	10	k	k	NOUN
ejpam-3780	299	11	=	=	PUNCT
ejpam-3780	299	12	{	{	PUNCT
ejpam-3780	299	13	a8	a8	PROPN
ejpam-3780	299	14	,	,	PUNCT
ejpam-3780	299	15	a9	a9	NOUN
ejpam-3780	299	16	}	}	PUNCT
ejpam-3780	299	17	and	and	CCONJ
ejpam-3780	299	18	thus	thus	ADV
ejpam-3780	299	19	|k|	|k|	PROPN
ejpam-3780	299	20	=	=	SYM
ejpam-3780	299	21	∆(p10	∆(p10	NOUN
ejpam-3780	299	22	)	)	PUNCT
ejpam-3780	299	23	=	=	SYM
ejpam-3780	299	24	2	2	X
ejpam-3780	299	25	.	.	X
ejpam-3780	299	26	d.	d.	PROPN
ejpam-3780	299	27	chua	chua	PROPN
ejpam-3780	299	28	,	,	PUNCT
ejpam-3780	299	29	f.	f.	PROPN
ejpam-3780	299	30	campeña	campeña	PROPN
ejpam-3780	299	31	,	,	PUNCT
ejpam-3780	299	32	f.	f.	PROPN
ejpam-3780	299	33	franco	franco	PROPN
ejpam-3780	299	34	/	/	SYM
ejpam-3780	299	35	eur	eur	PROPN
ejpam-3780	299	36	.	.	PUNCT
ejpam-3780	300	1	j.	j.	PROPN
ejpam-3780	300	2	pure	pure	PROPN
ejpam-3780	300	3	appl	appl	PROPN
ejpam-3780	300	4	.	.	PROPN
ejpam-3780	300	5	math	math	PROPN
ejpam-3780	300	6	,	,	PUNCT
ejpam-3780	300	7	13	13	NUM
ejpam-3780	300	8	(	(	PUNCT
ejpam-3780	300	9	3	3	NUM
ejpam-3780	300	10	)	)	PUNCT
ejpam-3780	300	11	(	(	PUNCT
ejpam-3780	300	12	2020	2020	NUM
ejpam-3780	300	13	)	)	PUNCT
ejpam-3780	300	14	,	,	PUNCT
ejpam-3780	300	15	674	674	NUM
ejpam-3780	300	16	-	-	SYM
ejpam-3780	300	17	696	696	NUM
ejpam-3780	300	18	682	682	NUM
ejpam-3780	300	19	a12	a12	NOUN
ejpam-3780	300	20	a0	a0	PROPN
ejpam-3780	300	21	a8	a8	PROPN
ejpam-3780	300	22	a4a12	a4a12	PROPN
ejpam-3780	300	23	a13	a13	PROPN
ejpam-3780	300	24	a12	a12	PROPN
ejpam-3780	300	25	a14	a14	PROPN
ejpam-3780	300	26	a12	a12	PROPN
ejpam-3780	300	27	a15	a15	PROPN
ejpam-3780	300	28	a0	a0	PROPN
ejpam-3780	300	29	a11	a11	PROPN
ejpam-3780	300	30	a0	a0	PROPN
ejpam-3780	300	31	a9	a9	PROPN
ejpam-3780	300	32	a0	a0	PROPN
ejpam-3780	300	33	a10	a10	PROPN
ejpam-3780	300	34	a8	a8	PROPN
ejpam-3780	300	35	a1	a1	PROPN
ejpam-3780	300	36	a8	a8	PROPN
ejpam-3780	300	37	a2	a2	PROPN
ejpam-3780	300	38	a8	a8	PROPN
ejpam-3780	300	39	a3	a3	PROPN
ejpam-3780	300	40	a4	a4	NUM
ejpam-3780	300	41	a7	a7	PROPN
ejpam-3780	300	42	a4	a4	PROPN
ejpam-3780	300	43	a5	a5	NOUN
ejpam-3780	300	44	a4	a4	NOUN
ejpam-3780	300	45	a6	a6	NOUN
ejpam-3780	300	46	figure	figure	NOUN
ejpam-3780	300	47	5	5	NUM
ejpam-3780	300	48	:	:	PUNCT
ejpam-3780	300	49	efficient	efficient	ADJ
ejpam-3780	300	50	zero	zero	NUM
ejpam-3780	300	51	ring	ring	NOUN
ejpam-3780	300	52	labeling	labeling	NOUN
ejpam-3780	300	53	of	of	ADP
ejpam-3780	300	54	a	a	DET
ejpam-3780	300	55	caterpillar	caterpillar	NOUN
ejpam-3780	300	56	using	use	VERB
ejpam-3780	300	57	m0	m0	PROPN
ejpam-3780	300	58	2	2	NUM
ejpam-3780	300	59	(	(	PUNCT
ejpam-3780	300	60	z16	z16	NOUN
ejpam-3780	300	61	)	)	PUNCT
ejpam-3780	300	62	a9	a9	PROPN
ejpam-3780	300	63	a0	a0	PROPN
ejpam-3780	300	64	a8	a8	PROPN
ejpam-3780	300	65	a1	a1	PROPN
ejpam-3780	300	66	a7	a7	PROPN
ejpam-3780	300	67	a2	a2	PROPN
ejpam-3780	300	68	a6	a6	PROPN
ejpam-3780	300	69	a3	a3	PROPN
ejpam-3780	300	70	a5	a5	PROPN
ejpam-3780	300	71	a4	a4	NOUN
ejpam-3780	300	72	figure	figure	NOUN
ejpam-3780	300	73	6	6	NUM
ejpam-3780	300	74	:	:	PUNCT
ejpam-3780	300	75	efficient	efficient	ADJ
ejpam-3780	300	76	zero	zero	NUM
ejpam-3780	300	77	ring	ring	NOUN
ejpam-3780	300	78	labeling	labeling	NOUN
ejpam-3780	300	79	of	of	ADP
ejpam-3780	300	80	p10	p10	NOUN
ejpam-3780	300	81	using	use	VERB
ejpam-3780	300	82	m0	m0	PROPN
ejpam-3780	300	83	2	2	NUM
ejpam-3780	300	84	(	(	PUNCT
ejpam-3780	300	85	z10	z10	NOUN
ejpam-3780	300	86	)	)	PUNCT
ejpam-3780	300	87	corollary	corollary	NOUN
ejpam-3780	300	88	2	2	NUM
ejpam-3780	300	89	.	.	PUNCT
ejpam-3780	301	1	a	a	DET
ejpam-3780	301	2	star	star	NOUN
ejpam-3780	301	3	graph	graph	NOUN
ejpam-3780	301	4	has	have	VERB
ejpam-3780	301	5	an	an	DET
ejpam-3780	301	6	efficient	efficient	ADJ
ejpam-3780	301	7	zero	zero	NUM
ejpam-3780	301	8	ring	ring	NOUN
ejpam-3780	301	9	labeling	labeling	NOUN
ejpam-3780	301	10	.	.	PUNCT
ejpam-3780	302	1	proof	proof	NOUN
ejpam-3780	302	2	.	.	PUNCT
ejpam-3780	303	1	let	let	VERB
ejpam-3780	303	2	g	g	NOUN
ejpam-3780	303	3	be	be	AUX
ejpam-3780	303	4	the	the	DET
ejpam-3780	303	5	star	star	NOUN
ejpam-3780	303	6	graph	graph	NOUN
ejpam-3780	303	7	sn	sn	PROPN
ejpam-3780	303	8	.	.	PUNCT
ejpam-3780	303	9	consider	consider	VERB
ejpam-3780	303	10	the	the	DET
ejpam-3780	303	11	center	center	NOUN
ejpam-3780	303	12	of	of	ADP
ejpam-3780	303	13	g	g	PROPN
ejpam-3780	303	14	as	as	ADP
ejpam-3780	303	15	its	its	PRON
ejpam-3780	303	16	central	central	ADJ
ejpam-3780	303	17	path	path	NOUN
ejpam-3780	303	18	.	.	PUNCT
ejpam-3780	304	1	then	then	ADV
ejpam-3780	304	2	g	g	PROPN
ejpam-3780	304	3	is	be	AUX
ejpam-3780	304	4	a	a	DET
ejpam-3780	304	5	caterpillar	caterpillar	NOUN
ejpam-3780	304	6	with	with	ADP
ejpam-3780	304	7	a	a	DET
ejpam-3780	304	8	central	central	ADJ
ejpam-3780	304	9	path	path	NOUN
ejpam-3780	304	10	with	with	ADP
ejpam-3780	304	11	one	one	NUM
ejpam-3780	304	12	vertex	vertex	NOUN
ejpam-3780	304	13	,	,	PUNCT
ejpam-3780	304	14	and	and	CCONJ
ejpam-3780	304	15	this	this	DET
ejpam-3780	304	16	vertex	vertex	NOUN
ejpam-3780	304	17	has	have	VERB
ejpam-3780	304	18	n−	n−	PROPN
ejpam-3780	304	19	1	1	NUM
ejpam-3780	304	20	hanging	hang	VERB
ejpam-3780	304	21	leaves	leave	NOUN
ejpam-3780	304	22	.	.	PUNCT
ejpam-3780	305	1	vacuously	vacuously	ADV
ejpam-3780	305	2	,	,	PUNCT
ejpam-3780	305	3	each	each	DET
ejpam-3780	305	4	vertex	vertex	NOUN
ejpam-3780	305	5	in	in	ADP
ejpam-3780	305	6	the	the	DET
ejpam-3780	305	7	central	central	ADJ
ejpam-3780	305	8	path	path	NOUN
ejpam-3780	305	9	of	of	ADP
ejpam-3780	305	10	g	g	PROPN
ejpam-3780	305	11	has	have	VERB
ejpam-3780	305	12	an	an	DET
ejpam-3780	305	13	equal	equal	ADJ
ejpam-3780	305	14	number	number	NOUN
ejpam-3780	305	15	of	of	ADP
ejpam-3780	305	16	hanging	hang	VERB
ejpam-3780	305	17	leaves	leave	NOUN
ejpam-3780	305	18	.	.	PUNCT
ejpam-3780	306	1	by	by	ADP
ejpam-3780	306	2	theorem	theorem	NOUN
ejpam-3780	306	3	4	4	NUM
ejpam-3780	306	4	,	,	PUNCT
ejpam-3780	306	5	g	g	PROPN
ejpam-3780	306	6	has	have	VERB
ejpam-3780	306	7	an	an	DET
ejpam-3780	306	8	efficient	efficient	ADJ
ejpam-3780	306	9	zero	zero	NUM
ejpam-3780	306	10	ring	ring	NOUN
ejpam-3780	306	11	labeling	labeling	NOUN
ejpam-3780	306	12	.	.	PUNCT
ejpam-3780	307	1	example	example	NOUN
ejpam-3780	308	1	6	6	NUM
ejpam-3780	308	2	.	.	PUNCT
ejpam-3780	308	3	figure	figure	VERB
ejpam-3780	308	4	7	7	NUM
ejpam-3780	308	5	shows	show	VERB
ejpam-3780	308	6	an	an	DET
ejpam-3780	308	7	efficient	efficient	ADJ
ejpam-3780	308	8	zero	zero	NUM
ejpam-3780	308	9	ring	ring	NOUN
ejpam-3780	308	10	labeling	labeling	NOUN
ejpam-3780	308	11	of	of	ADP
ejpam-3780	308	12	s9	s9	NOUN
ejpam-3780	308	13	using	use	VERB
ejpam-3780	308	14	m0	m0	PROPN
ejpam-3780	308	15	2	2	NUM
ejpam-3780	308	16	(	(	PUNCT
ejpam-3780	308	17	z9	z9	PROPN
ejpam-3780	308	18	)	)	PUNCT
ejpam-3780	308	19	.	.	PUNCT
ejpam-3780	309	1	in	in	ADP
ejpam-3780	309	2	this	this	DET
ejpam-3780	309	3	labeling	labeling	NOUN
ejpam-3780	309	4	,	,	PUNCT
ejpam-3780	309	5	the	the	DET
ejpam-3780	309	6	set	set	NOUN
ejpam-3780	309	7	of	of	ADP
ejpam-3780	309	8	sums	sum	NOUN
ejpam-3780	309	9	is	be	AUX
ejpam-3780	309	10	k	k	NOUN
ejpam-3780	309	11	=	=	PUNCT
ejpam-3780	309	12	{	{	PUNCT
ejpam-3780	309	13	a1	a1	PROPN
ejpam-3780	309	14	,	,	PUNCT
ejpam-3780	309	15	a2	a2	PROPN
ejpam-3780	309	16	,	,	PUNCT
ejpam-3780	309	17	a3	a3	NOUN
ejpam-3780	309	18	,	,	PUNCT
ejpam-3780	309	19	a4	a4	PROPN
ejpam-3780	309	20	,	,	PUNCT
ejpam-3780	309	21	a5	a5	NOUN
ejpam-3780	309	22	,	,	PUNCT
ejpam-3780	309	23	a6	a6	NOUN
ejpam-3780	309	24	,	,	PUNCT
ejpam-3780	309	25	a7	a7	PROPN
ejpam-3780	309	26	,	,	PUNCT
ejpam-3780	309	27	a8	a8	PROPN
ejpam-3780	309	28	}	}	PUNCT
ejpam-3780	309	29	and	and	CCONJ
ejpam-3780	309	30	thus	thus	ADV
ejpam-3780	309	31	|k|	|k|	NOUN
ejpam-3780	309	32	=	=	SYM
ejpam-3780	309	33	∆(s9	∆(s9	NUM
ejpam-3780	309	34	)	)	PUNCT
ejpam-3780	309	35	=	=	SYM
ejpam-3780	309	36	8	8	X
ejpam-3780	309	37	.	.	X
ejpam-3780	310	1	a0	a0	PROPN
ejpam-3780	310	2	a1	a1	PROPN
ejpam-3780	310	3	a0	a0	PROPN
ejpam-3780	310	4	a2	a2	PROPN
ejpam-3780	310	5	a0	a0	PROPN
ejpam-3780	310	6	a3	a3	PROPN
ejpam-3780	310	7	a0	a0	PROPN
ejpam-3780	310	8	a4	a4	PROPN
ejpam-3780	310	9	a0	a0	PROPN
ejpam-3780	310	10	a8	a8	PROPN
ejpam-3780	310	11	a0	a0	PROPN
ejpam-3780	310	12	a7	a7	PROPN
ejpam-3780	310	13	a0	a0	PROPN
ejpam-3780	310	14	a6	a6	PROPN
ejpam-3780	310	15	a0	a0	PROPN
ejpam-3780	310	16	a5	a5	PROPN
ejpam-3780	310	17	figure	figure	NOUN
ejpam-3780	310	18	7	7	NUM
ejpam-3780	310	19	:	:	PUNCT
ejpam-3780	310	20	efficient	efficient	ADJ
ejpam-3780	310	21	zero	zero	NUM
ejpam-3780	310	22	ring	ring	NOUN
ejpam-3780	310	23	labeling	labeling	NOUN
ejpam-3780	310	24	of	of	ADP
ejpam-3780	310	25	s9	s9	NOUN
ejpam-3780	310	26	using	use	VERB
ejpam-3780	310	27	m0	m0	PROPN
ejpam-3780	310	28	2	2	NUM
ejpam-3780	310	29	(	(	PUNCT
ejpam-3780	310	30	z9	z9	PROPN
ejpam-3780	310	31	)	)	PUNCT
ejpam-3780	310	32	corollary	corollary	ADJ
ejpam-3780	310	33	3	3	NUM
ejpam-3780	310	34	.	.	PUNCT
ejpam-3780	311	1	a	a	DET
ejpam-3780	311	2	bistar	bistar	NOUN
ejpam-3780	311	3	has	have	VERB
ejpam-3780	311	4	an	an	DET
ejpam-3780	311	5	efficient	efficient	ADJ
ejpam-3780	311	6	zero	zero	NUM
ejpam-3780	311	7	ring	ring	NOUN
ejpam-3780	311	8	labeling	labeling	NOUN
ejpam-3780	311	9	.	.	PUNCT
ejpam-3780	312	1	proof	proof	NOUN
ejpam-3780	312	2	.	.	PUNCT
ejpam-3780	313	1	let	let	VERB
ejpam-3780	313	2	g	g	NOUN
ejpam-3780	313	3	be	be	AUX
ejpam-3780	313	4	the	the	DET
ejpam-3780	313	5	bistar	bistar	PROPN
ejpam-3780	313	6	bn	bn	PROPN
ejpam-3780	313	7	.	.	PUNCT
ejpam-3780	314	1	consider	consider	VERB
ejpam-3780	314	2	the	the	DET
ejpam-3780	314	3	path	path	NOUN
ejpam-3780	314	4	joining	join	VERB
ejpam-3780	314	5	the	the	DET
ejpam-3780	314	6	centers	center	NOUN
ejpam-3780	314	7	of	of	ADP
ejpam-3780	314	8	the	the	DET
ejpam-3780	314	9	two	two	NUM
ejpam-3780	314	10	star	star	NOUN
ejpam-3780	314	11	graphs	graph	NOUN
ejpam-3780	314	12	sn	sn	PROPN
ejpam-3780	314	13	in	in	ADP
ejpam-3780	314	14	g	g	PROPN
ejpam-3780	314	15	as	as	ADP
ejpam-3780	314	16	its	its	PRON
ejpam-3780	314	17	central	central	ADJ
ejpam-3780	314	18	path	path	NOUN
ejpam-3780	314	19	.	.	PUNCT
ejpam-3780	315	1	then	then	ADV
ejpam-3780	315	2	g	g	PROPN
ejpam-3780	315	3	is	be	AUX
ejpam-3780	315	4	a	a	DET
ejpam-3780	315	5	caterpillar	caterpillar	NOUN
ejpam-3780	315	6	with	with	ADP
ejpam-3780	315	7	a	a	DET
ejpam-3780	315	8	central	central	ADJ
ejpam-3780	315	9	path	path	NOUN
ejpam-3780	315	10	with	with	ADP
ejpam-3780	315	11	two	two	NUM
ejpam-3780	315	12	vertices	vertex	NOUN
ejpam-3780	315	13	,	,	PUNCT
ejpam-3780	315	14	where	where	SCONJ
ejpam-3780	315	15	each	each	DET
ejpam-3780	315	16	vertex	vertex	NOUN
ejpam-3780	315	17	in	in	ADP
ejpam-3780	315	18	the	the	DET
ejpam-3780	315	19	central	central	ADJ
ejpam-3780	315	20	path	path	NOUN
ejpam-3780	315	21	has	have	VERB
ejpam-3780	315	22	n	n	CCONJ
ejpam-3780	315	23	−	−	PROPN
ejpam-3780	315	24	1	1	NUM
ejpam-3780	315	25	hanging	hanging	NOUN
ejpam-3780	315	26	leaves	leave	NOUN
ejpam-3780	315	27	.	.	PUNCT
ejpam-3780	316	1	by	by	ADP
ejpam-3780	316	2	theorem	theorem	NOUN
ejpam-3780	316	3	4	4	NUM
ejpam-3780	316	4	,	,	PUNCT
ejpam-3780	316	5	g	g	PROPN
ejpam-3780	316	6	has	have	VERB
ejpam-3780	316	7	an	an	DET
ejpam-3780	316	8	efficient	efficient	ADJ
ejpam-3780	316	9	zero	zero	NUM
ejpam-3780	316	10	ring	ring	NOUN
ejpam-3780	316	11	labeling	labeling	NOUN
ejpam-3780	316	12	.	.	PUNCT
ejpam-3780	317	1	example	example	NOUN
ejpam-3780	318	1	7	7	NUM
ejpam-3780	318	2	.	.	PUNCT
ejpam-3780	318	3	figure	figure	NOUN
ejpam-3780	318	4	8	8	NUM
ejpam-3780	318	5	shows	show	VERB
ejpam-3780	318	6	an	an	DET
ejpam-3780	318	7	efficient	efficient	ADJ
ejpam-3780	318	8	zero	zero	NUM
ejpam-3780	318	9	ring	ring	NOUN
ejpam-3780	318	10	labeling	labeling	NOUN
ejpam-3780	318	11	of	of	ADP
ejpam-3780	318	12	b9	b9	NOUN
ejpam-3780	318	13	using	use	VERB
ejpam-3780	318	14	m0	m0	PROPN
ejpam-3780	318	15	2	2	NUM
ejpam-3780	318	16	(	(	PUNCT
ejpam-3780	318	17	z18	z18	NOUN
ejpam-3780	318	18	)	)	PUNCT
ejpam-3780	318	19	.	.	PUNCT
ejpam-3780	319	1	in	in	ADP
ejpam-3780	319	2	this	this	DET
ejpam-3780	319	3	labeling	labeling	NOUN
ejpam-3780	319	4	,	,	PUNCT
ejpam-3780	319	5	the	the	DET
ejpam-3780	319	6	set	set	NOUN
ejpam-3780	319	7	of	of	ADP
ejpam-3780	319	8	sums	sum	NOUN
ejpam-3780	319	9	is	be	AUX
ejpam-3780	319	10	k	k	NOUN
ejpam-3780	319	11	=	=	PUNCT
ejpam-3780	319	12	{	{	PUNCT
ejpam-3780	319	13	a9	a9	PROPN
ejpam-3780	319	14	,	,	PUNCT
ejpam-3780	319	15	a10	a10	PROPN
ejpam-3780	319	16	,	,	PUNCT
ejpam-3780	319	17	a11	a11	PROPN
ejpam-3780	319	18	,	,	PUNCT
ejpam-3780	319	19	a12	a12	PROPN
ejpam-3780	319	20	,	,	PUNCT
ejpam-3780	319	21	a13	a13	PROPN
ejpam-3780	319	22	,	,	PUNCT
ejpam-3780	319	23	a14	a14	PROPN
ejpam-3780	319	24	,	,	PUNCT
ejpam-3780	319	25	a15	a15	NOUN
ejpam-3780	319	26	,	,	PUNCT
ejpam-3780	319	27	a16	a16	PROPN
ejpam-3780	319	28	,	,	PUNCT
ejpam-3780	319	29	a17	a17	NOUN
ejpam-3780	319	30	}	}	PUNCT
ejpam-3780	319	31	and	and	CCONJ
ejpam-3780	319	32	thus	thus	ADV
ejpam-3780	319	33	|k|	|k|	NOUN
ejpam-3780	319	34	=	=	SYM
ejpam-3780	319	35	∆(b9	∆(b9	X
ejpam-3780	319	36	)	)	PUNCT
ejpam-3780	319	37	=	=	SYM
ejpam-3780	319	38	9	9	X
ejpam-3780	319	39	.	.	X
ejpam-3780	319	40	d.	d.	PROPN
ejpam-3780	319	41	chua	chua	PROPN
ejpam-3780	319	42	,	,	PUNCT
ejpam-3780	319	43	f.	f.	PROPN
ejpam-3780	319	44	campeña	campeña	PROPN
ejpam-3780	319	45	,	,	PUNCT
ejpam-3780	319	46	f.	f.	PROPN
ejpam-3780	319	47	franco	franco	PROPN
ejpam-3780	319	48	/	/	SYM
ejpam-3780	319	49	eur	eur	PROPN
ejpam-3780	319	50	.	.	PUNCT
ejpam-3780	320	1	j.	j.	PROPN
ejpam-3780	320	2	pure	pure	PROPN
ejpam-3780	320	3	appl	appl	PROPN
ejpam-3780	320	4	.	.	PROPN
ejpam-3780	320	5	math	math	PROPN
ejpam-3780	320	6	,	,	PUNCT
ejpam-3780	320	7	13	13	NUM
ejpam-3780	320	8	(	(	PUNCT
ejpam-3780	320	9	3	3	NUM
ejpam-3780	320	10	)	)	PUNCT
ejpam-3780	320	11	(	(	PUNCT
ejpam-3780	320	12	2020	2020	NUM
ejpam-3780	320	13	)	)	PUNCT
ejpam-3780	320	14	,	,	PUNCT
ejpam-3780	320	15	674	674	NUM
ejpam-3780	320	16	-	-	SYM
ejpam-3780	320	17	696	696	NUM
ejpam-3780	320	18	683	683	NUM
ejpam-3780	320	19	a9	a9	NOUN
ejpam-3780	320	20	a1	a1	NOUN
ejpam-3780	320	21	a9	a9	PROPN
ejpam-3780	320	22	a2	a2	PROPN
ejpam-3780	320	23	a9	a9	PROPN
ejpam-3780	320	24	a3	a3	NOUN
ejpam-3780	320	25	a9	a9	PROPN
ejpam-3780	320	26	a4	a4	NOUN
ejpam-3780	320	27	a9	a9	NOUN
ejpam-3780	320	28	a8	a8	PROPN
ejpam-3780	320	29	a9	a9	PROPN
ejpam-3780	320	30	a7	a7	PROPN
ejpam-3780	320	31	a9	a9	PROPN
ejpam-3780	320	32	a6	a6	PROPN
ejpam-3780	320	33	a9	a9	PROPN
ejpam-3780	320	34	a5	a5	PROPN
ejpam-3780	320	35	a9	a9	PROPN
ejpam-3780	320	36	a0a0	a0a0	PUNCT
ejpam-3780	320	37	a10	a10	PROPN
ejpam-3780	320	38	a0	a0	PROPN
ejpam-3780	320	39	a11	a11	PROPN
ejpam-3780	320	40	a0	a0	PROPN
ejpam-3780	320	41	a12	a12	PROPN
ejpam-3780	320	42	a0	a0	PROPN
ejpam-3780	320	43	a13	a13	PROPN
ejpam-3780	320	44	a0	a0	PROPN
ejpam-3780	320	45	a17	a17	PROPN
ejpam-3780	320	46	a0	a0	PROPN
ejpam-3780	320	47	a16	a16	PROPN
ejpam-3780	320	48	a0	a0	PROPN
ejpam-3780	320	49	a15	a15	PROPN
ejpam-3780	320	50	a0	a0	PROPN
ejpam-3780	320	51	a14	a14	PROPN
ejpam-3780	320	52	figure	figure	NOUN
ejpam-3780	320	53	8	8	NUM
ejpam-3780	320	54	:	:	PUNCT
ejpam-3780	320	55	efficient	efficient	ADJ
ejpam-3780	320	56	zero	zero	NUM
ejpam-3780	320	57	ring	ring	NOUN
ejpam-3780	320	58	labeling	labeling	NOUN
ejpam-3780	320	59	of	of	ADP
ejpam-3780	320	60	b9	b9	NOUN
ejpam-3780	320	61	using	use	VERB
ejpam-3780	320	62	m0	m0	PROPN
ejpam-3780	320	63	2	2	NUM
ejpam-3780	320	64	(	(	PUNCT
ejpam-3780	320	65	z18	z18	NOUN
ejpam-3780	320	66	)	)	PUNCT
ejpam-3780	320	67	theorem	theorem	NOUN
ejpam-3780	320	68	5	5	NUM
ejpam-3780	320	69	.	.	PUNCT
ejpam-3780	321	1	a	a	DET
ejpam-3780	321	2	centipede	centipede	NOUN
ejpam-3780	321	3	graph	graph	NOUN
ejpam-3780	321	4	has	have	VERB
ejpam-3780	321	5	an	an	DET
ejpam-3780	321	6	efficient	efficient	ADJ
ejpam-3780	321	7	zero	zero	NUM
ejpam-3780	321	8	ring	ring	NOUN
ejpam-3780	321	9	labeling	labeling	NOUN
ejpam-3780	321	10	.	.	PUNCT
ejpam-3780	322	1	proof	proof	NOUN
ejpam-3780	322	2	.	.	PUNCT
ejpam-3780	323	1	let	let	VERB
ejpam-3780	323	2	g	g	PRON
ejpam-3780	323	3	be	be	AUX
ejpam-3780	323	4	an	an	DET
ejpam-3780	323	5	n	n	NOUN
ejpam-3780	323	6	-	-	PUNCT
ejpam-3780	323	7	centipede	centipede	NOUN
ejpam-3780	323	8	.	.	PUNCT
ejpam-3780	324	1	by	by	ADP
ejpam-3780	324	2	definition	definition	NOUN
ejpam-3780	324	3	of	of	ADP
ejpam-3780	324	4	a	a	DET
ejpam-3780	324	5	centipede	centipede	NOUN
ejpam-3780	324	6	,	,	PUNCT
ejpam-3780	324	7	we	we	PRON
ejpam-3780	324	8	can	can	AUX
ejpam-3780	324	9	find	find	VERB
ejpam-3780	324	10	a	a	DET
ejpam-3780	324	11	central	central	ADJ
ejpam-3780	324	12	path	path	NOUN
ejpam-3780	324	13	in	in	ADP
ejpam-3780	324	14	g	g	PROPN
ejpam-3780	324	15	such	such	ADJ
ejpam-3780	324	16	that	that	SCONJ
ejpam-3780	324	17	each	each	DET
ejpam-3780	324	18	vertex	vertex	NOUN
ejpam-3780	324	19	in	in	ADP
ejpam-3780	324	20	the	the	DET
ejpam-3780	324	21	central	central	ADJ
ejpam-3780	324	22	path	path	NOUN
ejpam-3780	324	23	has	have	VERB
ejpam-3780	324	24	1	1	NUM
ejpam-3780	324	25	hanging	hang	VERB
ejpam-3780	324	26	leaf	leaf	NOUN
ejpam-3780	324	27	.	.	PUNCT
ejpam-3780	325	1	by	by	ADP
ejpam-3780	325	2	theorem	theorem	NOUN
ejpam-3780	325	3	4	4	NUM
ejpam-3780	325	4	,	,	PUNCT
ejpam-3780	325	5	g	g	PROPN
ejpam-3780	325	6	has	have	VERB
ejpam-3780	325	7	an	an	DET
ejpam-3780	325	8	efficient	efficient	ADJ
ejpam-3780	325	9	zero	zero	NUM
ejpam-3780	325	10	ring	ring	NOUN
ejpam-3780	325	11	labeling	labeling	NOUN
ejpam-3780	325	12	.	.	PUNCT
ejpam-3780	326	1	example	example	NOUN
ejpam-3780	327	1	8	8	NUM
ejpam-3780	327	2	.	.	PUNCT
ejpam-3780	327	3	figure	figure	NOUN
ejpam-3780	327	4	9	9	NUM
ejpam-3780	327	5	shows	show	VERB
ejpam-3780	327	6	an	an	DET
ejpam-3780	327	7	efficient	efficient	ADJ
ejpam-3780	327	8	zero	zero	NUM
ejpam-3780	327	9	ring	ring	NOUN
ejpam-3780	327	10	labeling	labeling	NOUN
ejpam-3780	327	11	of	of	ADP
ejpam-3780	327	12	a	a	DET
ejpam-3780	327	13	6	6	NUM
ejpam-3780	327	14	-	-	PUNCT
ejpam-3780	327	15	centipede	centipede	NOUN
ejpam-3780	327	16	using	use	VERB
ejpam-3780	327	17	m0	m0	NOUN
ejpam-3780	327	18	2	2	NUM
ejpam-3780	327	19	(	(	PUNCT
ejpam-3780	327	20	z12	z12	PROPN
ejpam-3780	327	21	)	)	PUNCT
ejpam-3780	327	22	.	.	PUNCT
ejpam-3780	328	1	in	in	ADP
ejpam-3780	328	2	this	this	DET
ejpam-3780	328	3	labeling	labeling	NOUN
ejpam-3780	328	4	,	,	PUNCT
ejpam-3780	328	5	the	the	DET
ejpam-3780	328	6	set	set	NOUN
ejpam-3780	328	7	of	of	ADP
ejpam-3780	328	8	sums	sum	NOUN
ejpam-3780	328	9	is	be	AUX
ejpam-3780	328	10	k	k	NOUN
ejpam-3780	328	11	=	=	PUNCT
ejpam-3780	328	12	{	{	PUNCT
ejpam-3780	328	13	a8	a8	PROPN
ejpam-3780	328	14	,	,	PUNCT
ejpam-3780	328	15	a9	a9	NOUN
ejpam-3780	328	16	,	,	PUNCT
ejpam-3780	328	17	a10	a10	NOUN
ejpam-3780	328	18	}	}	PUNCT
ejpam-3780	328	19	and	and	CCONJ
ejpam-3780	328	20	thus	thus	ADV
ejpam-3780	328	21	|k|	|k|	NOUN
ejpam-3780	328	22	=	=	SYM
ejpam-3780	328	23	3	3	NUM
ejpam-3780	328	24	,	,	PUNCT
ejpam-3780	328	25	which	which	PRON
ejpam-3780	328	26	is	be	AUX
ejpam-3780	328	27	the	the	DET
ejpam-3780	328	28	maximum	maximum	ADJ
ejpam-3780	328	29	degree	degree	NOUN
ejpam-3780	328	30	of	of	ADP
ejpam-3780	328	31	the	the	DET
ejpam-3780	328	32	graph	graph	NOUN
ejpam-3780	328	33	.	.	PUNCT
ejpam-3780	329	1	a10	a10	PROPN
ejpam-3780	329	2	a0	a0	PROPN
ejpam-3780	329	3	a8	a8	PROPN
ejpam-3780	329	4	a2	a2	PROPN
ejpam-3780	329	5	a6	a6	PROPN
ejpam-3780	329	6	a4a10	a4a10	PROPN
ejpam-3780	329	7	a11	a11	PROPN
ejpam-3780	329	8	a0	a0	PROPN
ejpam-3780	329	9	a9	a9	PROPN
ejpam-3780	329	10	a8	a8	PROPN
ejpam-3780	329	11	a1	a1	NOUN
ejpam-3780	329	12	a2	a2	PROPN
ejpam-3780	329	13	a7	a7	PROPN
ejpam-3780	329	14	a6	a6	PROPN
ejpam-3780	329	15	a3	a3	PROPN
ejpam-3780	329	16	a4	a4	PROPN
ejpam-3780	329	17	a5	a5	NOUN
ejpam-3780	329	18	figure	figure	NOUN
ejpam-3780	329	19	9	9	NUM
ejpam-3780	329	20	:	:	PUNCT
ejpam-3780	329	21	efficient	efficient	ADJ
ejpam-3780	329	22	zero	zero	NUM
ejpam-3780	329	23	ring	ring	NOUN
ejpam-3780	329	24	labeling	labeling	NOUN
ejpam-3780	329	25	of	of	ADP
ejpam-3780	329	26	a	a	DET
ejpam-3780	329	27	6	6	NUM
ejpam-3780	329	28	-	-	PUNCT
ejpam-3780	329	29	centipede	centipede	NOUN
ejpam-3780	329	30	using	use	VERB
ejpam-3780	329	31	m0	m0	NOUN
ejpam-3780	329	32	2	2	NUM
ejpam-3780	329	33	(	(	PUNCT
ejpam-3780	329	34	z12	z12	PROPN
ejpam-3780	329	35	)	)	PUNCT
ejpam-3780	329	36	lemma	lemma	PROPN
ejpam-3780	329	37	1	1	NUM
ejpam-3780	329	38	.	.	PUNCT
ejpam-3780	330	1	let	let	VERB
ejpam-3780	330	2	g	g	PRON
ejpam-3780	330	3	be	be	AUX
ejpam-3780	330	4	a	a	DET
ejpam-3780	330	5	caterpillar	caterpillar	NOUN
ejpam-3780	330	6	with	with	ADP
ejpam-3780	330	7	at	at	ADV
ejpam-3780	330	8	least	least	ADJ
ejpam-3780	330	9	3	3	NUM
ejpam-3780	330	10	vertices	vertex	NOUN
ejpam-3780	330	11	.	.	PUNCT
ejpam-3780	331	1	then	then	ADV
ejpam-3780	331	2	g	g	PROPN
ejpam-3780	331	3	is	be	AUX
ejpam-3780	331	4	a	a	DET
ejpam-3780	331	5	caterpillar	caterpillar	NOUN
ejpam-3780	331	6	with	with	ADP
ejpam-3780	331	7	respect	respect	NOUN
ejpam-3780	331	8	to	to	ADP
ejpam-3780	331	9	some	some	DET
ejpam-3780	331	10	central	central	ADJ
ejpam-3780	331	11	path	path	NOUN
ejpam-3780	331	12	[	[	X
ejpam-3780	331	13	w1	w1	NOUN
ejpam-3780	331	14	,	,	PUNCT
ejpam-3780	331	15	w2	w2	NOUN
ejpam-3780	331	16	,	,	PUNCT
ejpam-3780	331	17	.	.	PUNCT
ejpam-3780	331	18	.	.	PUNCT
ejpam-3780	332	1	.	.	PUNCT
ejpam-3780	333	1	,	,	PUNCT
ejpam-3780	333	2	wn	wn	PROPN
ejpam-3780	333	3	]	]	X
ejpam-3780	333	4	,	,	PUNCT
ejpam-3780	333	5	where	where	SCONJ
ejpam-3780	333	6	n	n	PRON
ejpam-3780	333	7	≥	≥	X
ejpam-3780	333	8	3	3	NUM
ejpam-3780	333	9	and	and	CCONJ
ejpam-3780	333	10	w1	w1	PROPN
ejpam-3780	333	11	and	and	CCONJ
ejpam-3780	333	12	wn	wn	PROPN
ejpam-3780	333	13	have	have	VERB
ejpam-3780	333	14	no	no	DET
ejpam-3780	333	15	hanging	hang	VERB
ejpam-3780	333	16	leaf	leaf	NOUN
ejpam-3780	333	17	.	.	PUNCT
ejpam-3780	334	1	proof	proof	NOUN
ejpam-3780	334	2	.	.	PUNCT
ejpam-3780	335	1	let	let	VERB
ejpam-3780	335	2	g	g	PRON
ejpam-3780	335	3	be	be	AUX
ejpam-3780	335	4	a	a	DET
ejpam-3780	335	5	caterpillar	caterpillar	NOUN
ejpam-3780	335	6	with	with	ADP
ejpam-3780	335	7	at	at	ADV
ejpam-3780	335	8	least	least	ADJ
ejpam-3780	335	9	3	3	NUM
ejpam-3780	335	10	vertices	vertex	NOUN
ejpam-3780	335	11	,	,	PUNCT
ejpam-3780	335	12	and	and	CCONJ
ejpam-3780	335	13	let	let	VERB
ejpam-3780	335	14	[	[	X
ejpam-3780	335	15	v1	v1	VERB
ejpam-3780	335	16	,	,	PUNCT
ejpam-3780	335	17	v2	v2	NOUN
ejpam-3780	335	18	,	,	PUNCT
ejpam-3780	335	19	.	.	PUNCT
ejpam-3780	335	20	.	.	PUNCT
ejpam-3780	336	1	.	.	PUNCT
ejpam-3780	337	1	,	,	PUNCT
ejpam-3780	337	2	vm	vm	PROPN
ejpam-3780	337	3	]	]	X
ejpam-3780	337	4	denote	denote	VERB
ejpam-3780	337	5	its	its	PRON
ejpam-3780	337	6	central	central	ADJ
ejpam-3780	337	7	path	path	NOUN
ejpam-3780	337	8	.	.	PUNCT
ejpam-3780	338	1	suppose	suppose	VERB
ejpam-3780	338	2	each	each	DET
ejpam-3780	338	3	vertex	vertex	NOUN
ejpam-3780	338	4	vi	vi	PROPN
ejpam-3780	338	5	has	have	VERB
ejpam-3780	338	6	ri	ri	PROPN
ejpam-3780	338	7	hanging	hang	VERB
ejpam-3780	338	8	leaves	leave	NOUN
ejpam-3780	338	9	,	,	PUNCT
ejpam-3780	338	10	and	and	CCONJ
ejpam-3780	338	11	let	let	VERB
ejpam-3780	338	12	vji	vji	VERB
ejpam-3780	338	13	,	,	PUNCT
ejpam-3780	338	14	where	where	SCONJ
ejpam-3780	338	15	j	j	PROPN
ejpam-3780	338	16	=	=	SYM
ejpam-3780	338	17	1	1	NUM
ejpam-3780	338	18	,	,	PUNCT
ejpam-3780	338	19	2	2	NUM
ejpam-3780	338	20	,	,	PUNCT
ejpam-3780	338	21	.	.	PUNCT
ejpam-3780	338	22	.	.	PUNCT
ejpam-3780	339	1	.	.	PUNCT
ejpam-3780	340	1	,	,	PUNCT
ejpam-3780	340	2	ri	ri	PROPN
ejpam-3780	340	3	,	,	PUNCT
ejpam-3780	340	4	denote	denote	VERB
ejpam-3780	340	5	the	the	DET
ejpam-3780	340	6	hanging	hang	VERB
ejpam-3780	340	7	leaves	leave	NOUN
ejpam-3780	340	8	of	of	ADP
ejpam-3780	340	9	vi	vi	PROPN
ejpam-3780	340	10	.	.	PUNCT
ejpam-3780	340	11	consider	consider	VERB
ejpam-3780	340	12	v1	v1	NOUN
ejpam-3780	340	13	,	,	PUNCT
ejpam-3780	340	14	v2	v2	NOUN
ejpam-3780	340	15	,	,	PUNCT
ejpam-3780	340	16	.	.	PUNCT
ejpam-3780	340	17	.	.	PUNCT
ejpam-3780	341	1	.	.	PUNCT
ejpam-3780	342	1	,	,	PUNCT
ejpam-3780	342	2	vm	vm	PROPN
ejpam-3780	342	3	as	as	ADP
ejpam-3780	342	4	vertices	vertex	NOUN
ejpam-3780	342	5	in	in	ADP
ejpam-3780	342	6	a	a	DET
ejpam-3780	342	7	central	central	ADJ
ejpam-3780	342	8	path	path	NOUN
ejpam-3780	342	9	p	p	NOUN
ejpam-3780	342	10	,	,	PUNCT
ejpam-3780	342	11	along	along	ADP
ejpam-3780	342	12	with	with	ADP
ejpam-3780	342	13	v11	v11	NOUN
ejpam-3780	342	14	if	if	SCONJ
ejpam-3780	342	15	r1	r1	PROPN
ejpam-3780	342	16	≥	≥	NUM
ejpam-3780	342	17	1	1	NUM
ejpam-3780	342	18	,	,	PUNCT
ejpam-3780	342	19	and	and	CCONJ
ejpam-3780	342	20	vrmm	vrmm	VERB
ejpam-3780	342	21	if	if	SCONJ
ejpam-3780	342	22	rm	rm	PROPN
ejpam-3780	342	23	≥	≥	PROPN
ejpam-3780	342	24	1	1	NUM
ejpam-3780	342	25	.	.	PUNCT
ejpam-3780	343	1	thus	thus	ADV
ejpam-3780	343	2	,	,	PUNCT
ejpam-3780	343	3	p	p	PROPN
ejpam-3780	343	4	is	be	AUX
ejpam-3780	343	5	one	one	NUM
ejpam-3780	343	6	of	of	ADP
ejpam-3780	343	7	the	the	DET
ejpam-3780	343	8	following	following	NOUN
ejpam-3780	343	9	:	:	PUNCT
ejpam-3780	344	1	[	[	X
ejpam-3780	344	2	v1	v1	NOUN
ejpam-3780	344	3	,	,	PUNCT
ejpam-3780	344	4	v2	v2	NOUN
ejpam-3780	344	5	,	,	PUNCT
ejpam-3780	344	6	.	.	PUNCT
ejpam-3780	344	7	.	.	PUNCT
ejpam-3780	344	8	.	.	PUNCT
ejpam-3780	344	9	,	,	PUNCT
ejpam-3780	344	10	vm	vm	PROPN
ejpam-3780	344	11	]	]	X
ejpam-3780	344	12	,	,	PUNCT
ejpam-3780	344	13	[	[	X
ejpam-3780	344	14	v11	v11	NOUN
ejpam-3780	344	15	,	,	PUNCT
ejpam-3780	344	16	v1	v1	NOUN
ejpam-3780	344	17	,	,	PUNCT
ejpam-3780	344	18	v2	v2	NOUN
ejpam-3780	344	19	,	,	PUNCT
ejpam-3780	344	20	.	.	PUNCT
ejpam-3780	344	21	.	.	PUNCT
ejpam-3780	344	22	.	.	PUNCT
ejpam-3780	344	23	,	,	PUNCT
ejpam-3780	344	24	vm	vm	PROPN
ejpam-3780	344	25	]	]	X
ejpam-3780	344	26	,	,	PUNCT
ejpam-3780	344	27	[	[	X
ejpam-3780	344	28	v1	v1	NOUN
ejpam-3780	344	29	,	,	PUNCT
ejpam-3780	344	30	v2	v2	NOUN
ejpam-3780	344	31	,	,	PUNCT
ejpam-3780	344	32	.	.	PUNCT
ejpam-3780	344	33	.	.	PUNCT
ejpam-3780	344	34	.	.	PUNCT
ejpam-3780	345	1	,	,	PUNCT
ejpam-3780	345	2	vm	vm	NOUN
ejpam-3780	345	3	,	,	PUNCT
ejpam-3780	345	4	vrmm	vrmm	VERB
ejpam-3780	345	5	]	]	PUNCT
ejpam-3780	345	6	,	,	PUNCT
ejpam-3780	345	7	[	[	X
ejpam-3780	345	8	v11	v11	NOUN
ejpam-3780	345	9	,	,	PUNCT
ejpam-3780	345	10	v1	v1	NOUN
ejpam-3780	345	11	,	,	PUNCT
ejpam-3780	345	12	v2	v2	NOUN
ejpam-3780	345	13	,	,	PUNCT
ejpam-3780	345	14	.	.	PUNCT
ejpam-3780	345	15	.	.	PUNCT
ejpam-3780	346	1	.	.	PUNCT
ejpam-3780	347	1	,	,	PUNCT
ejpam-3780	347	2	vm	vm	NOUN
ejpam-3780	347	3	,	,	PUNCT
ejpam-3780	347	4	vrmm	vrmm	VERB
ejpam-3780	347	5	]	]	PUNCT
ejpam-3780	347	6	.	.	PUNCT
ejpam-3780	348	1	in	in	ADP
ejpam-3780	348	2	any	any	DET
ejpam-3780	348	3	case	case	NOUN
ejpam-3780	348	4	,	,	PUNCT
ejpam-3780	348	5	the	the	DET
ejpam-3780	348	6	number	number	NOUN
ejpam-3780	348	7	of	of	ADP
ejpam-3780	348	8	vertices	vertex	NOUN
ejpam-3780	348	9	in	in	ADP
ejpam-3780	348	10	p	p	PROPN
ejpam-3780	348	11	is	be	AUX
ejpam-3780	348	12	at	at	ADV
ejpam-3780	348	13	least	least	ADJ
ejpam-3780	348	14	three	three	NUM
ejpam-3780	348	15	,	,	PUNCT
ejpam-3780	348	16	and	and	CCONJ
ejpam-3780	348	17	its	its	PRON
ejpam-3780	348	18	endvertices	endvertice	NOUN
ejpam-3780	348	19	have	have	VERB
ejpam-3780	348	20	no	no	DET
ejpam-3780	348	21	hanging	hang	VERB
ejpam-3780	348	22	leaf	leaf	NOUN
ejpam-3780	348	23	.	.	PUNCT
ejpam-3780	349	1	clearly	clearly	ADV
ejpam-3780	349	2	,	,	PUNCT
ejpam-3780	349	3	each	each	DET
ejpam-3780	349	4	vertex	vertex	NOUN
ejpam-3780	349	5	in	in	ADP
ejpam-3780	349	6	g	g	PROPN
ejpam-3780	349	7	is	be	AUX
ejpam-3780	349	8	within	within	ADP
ejpam-3780	349	9	distance	distance	NOUN
ejpam-3780	349	10	one	one	NUM
ejpam-3780	349	11	from	from	ADP
ejpam-3780	349	12	p	p	NOUN
ejpam-3780	349	13	and	and	CCONJ
ejpam-3780	349	14	thus	thus	ADV
ejpam-3780	349	15	g	g	PROPN
ejpam-3780	349	16	is	be	AUX
ejpam-3780	349	17	a	a	DET
ejpam-3780	349	18	caterpillar	caterpillar	NOUN
ejpam-3780	349	19	with	with	ADP
ejpam-3780	349	20	respect	respect	NOUN
ejpam-3780	349	21	to	to	ADP
ejpam-3780	349	22	central	central	ADJ
ejpam-3780	349	23	path	path	NOUN
ejpam-3780	349	24	p	p	PROPN
ejpam-3780	349	25	.	.	PUNCT
ejpam-3780	350	1	theorem	theorem	ADJ
ejpam-3780	350	2	6	6	NUM
ejpam-3780	350	3	.	.	PUNCT
ejpam-3780	351	1	a	a	DET
ejpam-3780	351	2	caterpillar	caterpillar	NOUN
ejpam-3780	351	3	has	have	VERB
ejpam-3780	351	4	an	an	DET
ejpam-3780	351	5	efficient	efficient	ADJ
ejpam-3780	351	6	zero	zero	NUM
ejpam-3780	351	7	ring	ring	NOUN
ejpam-3780	351	8	labeling	labeling	NOUN
ejpam-3780	351	9	.	.	PUNCT
ejpam-3780	352	1	d.	d.	PROPN
ejpam-3780	352	2	chua	chua	PROPN
ejpam-3780	352	3	,	,	PUNCT
ejpam-3780	352	4	f.	f.	PROPN
ejpam-3780	352	5	campeña	campeña	PROPN
ejpam-3780	352	6	,	,	PUNCT
ejpam-3780	352	7	f.	f.	PROPN
ejpam-3780	352	8	franco	franco	PROPN
ejpam-3780	352	9	/	/	SYM
ejpam-3780	352	10	eur	eur	PROPN
ejpam-3780	352	11	.	.	PUNCT
ejpam-3780	353	1	j.	j.	PROPN
ejpam-3780	353	2	pure	pure	PROPN
ejpam-3780	353	3	appl	appl	PROPN
ejpam-3780	353	4	.	.	PROPN
ejpam-3780	353	5	math	math	PROPN
ejpam-3780	353	6	,	,	PUNCT
ejpam-3780	353	7	13	13	NUM
ejpam-3780	353	8	(	(	PUNCT
ejpam-3780	353	9	3	3	NUM
ejpam-3780	353	10	)	)	PUNCT
ejpam-3780	353	11	(	(	PUNCT
ejpam-3780	353	12	2020	2020	NUM
ejpam-3780	353	13	)	)	PUNCT
ejpam-3780	353	14	,	,	PUNCT
ejpam-3780	353	15	674	674	NUM
ejpam-3780	353	16	-	-	SYM
ejpam-3780	353	17	696	696	NUM
ejpam-3780	353	18	684	684	NUM
ejpam-3780	353	19	proof	proof	NOUN
ejpam-3780	353	20	.	.	PUNCT
ejpam-3780	354	1	let	let	VERB
ejpam-3780	354	2	g	g	PRON
ejpam-3780	354	3	be	be	AUX
ejpam-3780	354	4	a	a	DET
ejpam-3780	354	5	caterpillar	caterpillar	NOUN
ejpam-3780	354	6	with	with	ADP
ejpam-3780	354	7	at	at	ADV
ejpam-3780	354	8	least	least	ADV
ejpam-3780	354	9	three	three	NUM
ejpam-3780	354	10	vertices	vertex	NOUN
ejpam-3780	354	11	.	.	PUNCT
ejpam-3780	355	1	by	by	ADP
ejpam-3780	355	2	lemma	lemma	PROPN
ejpam-3780	355	3	1	1	NUM
ejpam-3780	355	4	,	,	PUNCT
ejpam-3780	355	5	g	g	PROPN
ejpam-3780	355	6	is	be	AUX
ejpam-3780	355	7	a	a	DET
ejpam-3780	355	8	caterpillar	caterpillar	NOUN
ejpam-3780	355	9	with	with	ADP
ejpam-3780	355	10	respect	respect	NOUN
ejpam-3780	355	11	to	to	ADP
ejpam-3780	355	12	some	some	DET
ejpam-3780	355	13	central	central	ADJ
ejpam-3780	355	14	path	path	NOUN
ejpam-3780	355	15	[	[	X
ejpam-3780	355	16	w1	w1	NOUN
ejpam-3780	355	17	,	,	PUNCT
ejpam-3780	355	18	w2	w2	NOUN
ejpam-3780	355	19	,	,	PUNCT
ejpam-3780	355	20	.	.	PUNCT
ejpam-3780	355	21	.	.	PUNCT
ejpam-3780	356	1	.	.	PUNCT
ejpam-3780	357	1	,	,	PUNCT
ejpam-3780	357	2	wn	wn	PROPN
ejpam-3780	357	3	]	]	X
ejpam-3780	357	4	,	,	PUNCT
ejpam-3780	357	5	where	where	SCONJ
ejpam-3780	357	6	n	n	PRON
ejpam-3780	357	7	≥	≥	X
ejpam-3780	357	8	3	3	NUM
ejpam-3780	357	9	and	and	CCONJ
ejpam-3780	357	10	w1	w1	PROPN
ejpam-3780	357	11	and	and	CCONJ
ejpam-3780	357	12	wn	wn	PROPN
ejpam-3780	357	13	have	have	VERB
ejpam-3780	357	14	no	no	DET
ejpam-3780	357	15	hanging	hang	VERB
ejpam-3780	357	16	leaf	leaf	NOUN
ejpam-3780	357	17	.	.	PUNCT
ejpam-3780	358	1	for	for	ADP
ejpam-3780	358	2	i	i	PRON
ejpam-3780	358	3	=	=	SYM
ejpam-3780	358	4	2	2	NUM
ejpam-3780	358	5	,	,	PUNCT
ejpam-3780	358	6	3	3	NUM
ejpam-3780	358	7	,	,	PUNCT
ejpam-3780	358	8	.	.	PUNCT
ejpam-3780	358	9	.	.	PUNCT
ejpam-3780	359	1	.	.	PUNCT
ejpam-3780	360	1	,	,	PUNCT
ejpam-3780	360	2	n−	n−	NOUN
ejpam-3780	360	3	1	1	NUM
ejpam-3780	360	4	,	,	PUNCT
ejpam-3780	360	5	suppose	suppose	VERB
ejpam-3780	360	6	wi	wi	PROPN
ejpam-3780	360	7	has	have	VERB
ejpam-3780	360	8	ri	ri	PROPN
ejpam-3780	360	9	hanging	hang	VERB
ejpam-3780	360	10	leaves	leave	NOUN
ejpam-3780	360	11	,	,	PUNCT
ejpam-3780	360	12	and	and	CCONJ
ejpam-3780	360	13	let	let	VERB
ejpam-3780	360	14	wj	wj	PROPN
ejpam-3780	360	15	i	i	PRON
ejpam-3780	360	16	,	,	PUNCT
ejpam-3780	360	17	where	where	SCONJ
ejpam-3780	360	18	j	j	PROPN
ejpam-3780	360	19	=	=	SYM
ejpam-3780	360	20	1	1	NUM
ejpam-3780	360	21	,	,	PUNCT
ejpam-3780	360	22	2	2	NUM
ejpam-3780	360	23	,	,	PUNCT
ejpam-3780	360	24	.	.	PUNCT
ejpam-3780	360	25	.	.	PUNCT
ejpam-3780	361	1	.	.	PUNCT
ejpam-3780	362	1	,	,	PUNCT
ejpam-3780	362	2	ri	ri	PROPN
ejpam-3780	362	3	,	,	PUNCT
ejpam-3780	362	4	denote	denote	VERB
ejpam-3780	362	5	the	the	DET
ejpam-3780	362	6	hanging	hang	VERB
ejpam-3780	362	7	leaves	leave	NOUN
ejpam-3780	362	8	of	of	ADP
ejpam-3780	362	9	wi	wi	PROPN
ejpam-3780	362	10	.	.	PUNCT
ejpam-3780	363	1	suppose	suppose	VERB
ejpam-3780	363	2	rk	rk	PRON
ejpam-3780	363	3	is	be	AUX
ejpam-3780	363	4	the	the	DET
ejpam-3780	363	5	maximum	maximum	ADJ
ejpam-3780	363	6	number	number	NOUN
ejpam-3780	363	7	of	of	ADP
ejpam-3780	363	8	hanging	hang	VERB
ejpam-3780	363	9	leaves	leave	NOUN
ejpam-3780	363	10	of	of	ADP
ejpam-3780	363	11	a	a	DET
ejpam-3780	363	12	vertex	vertex	NOUN
ejpam-3780	363	13	in	in	ADP
ejpam-3780	363	14	the	the	DET
ejpam-3780	363	15	central	central	ADJ
ejpam-3780	363	16	path	path	NOUN
ejpam-3780	363	17	;	;	PUNCT
ejpam-3780	363	18	that	that	PRON
ejpam-3780	363	19	is	is	ADV
ejpam-3780	363	20	,	,	PUNCT
ejpam-3780	363	21	rk	rk	PROPN
ejpam-3780	363	22	≥	≥	X
ejpam-3780	363	23	ri	ri	NOUN
ejpam-3780	363	24	for	for	ADP
ejpam-3780	363	25	i	i	PROPN
ejpam-3780	363	26	=	=	SYM
ejpam-3780	363	27	2	2	NUM
ejpam-3780	363	28	,	,	PUNCT
ejpam-3780	363	29	3	3	NUM
ejpam-3780	363	30	,	,	PUNCT
ejpam-3780	363	31	.	.	PUNCT
ejpam-3780	363	32	.	.	PUNCT
ejpam-3780	364	1	.	.	PUNCT
ejpam-3780	365	1	,	,	PUNCT
ejpam-3780	365	2	n−	n−	NOUN
ejpam-3780	365	3	1	1	NUM
ejpam-3780	365	4	.	.	PUNCT
ejpam-3780	366	1	then	then	ADV
ejpam-3780	366	2	∆(g	∆(g	PROPN
ejpam-3780	366	3	)	)	PUNCT
ejpam-3780	367	1	=	=	VERB
ejpam-3780	367	2	rk	rk	NOUN
ejpam-3780	367	3	+	+	NOUN
ejpam-3780	367	4	2	2	X
ejpam-3780	367	5	.	.	X
ejpam-3780	367	6	consider	consider	VERB
ejpam-3780	367	7	a	a	DET
ejpam-3780	367	8	caterpillar	caterpillar	ADJ
ejpam-3780	367	9	h	h	NOUN
ejpam-3780	367	10	with	with	ADP
ejpam-3780	367	11	a	a	DET
ejpam-3780	367	12	central	central	ADJ
ejpam-3780	367	13	path	path	NOUN
ejpam-3780	367	14	with	with	ADP
ejpam-3780	367	15	n	n	ADP
ejpam-3780	367	16	vertices	vertex	NOUN
ejpam-3780	367	17	,	,	PUNCT
ejpam-3780	367	18	and	and	CCONJ
ejpam-3780	367	19	where	where	SCONJ
ejpam-3780	367	20	each	each	DET
ejpam-3780	367	21	vertex	vertex	NOUN
ejpam-3780	367	22	in	in	ADP
ejpam-3780	367	23	the	the	DET
ejpam-3780	367	24	central	central	ADJ
ejpam-3780	367	25	path	path	NOUN
ejpam-3780	367	26	has	have	AUX
ejpam-3780	367	27	rk	rk	NOUN
ejpam-3780	367	28	hanging	hang	VERB
ejpam-3780	367	29	leaves	leave	NOUN
ejpam-3780	367	30	.	.	PUNCT
ejpam-3780	368	1	let	let	VERB
ejpam-3780	368	2	[	[	X
ejpam-3780	368	3	v1	v1	VERB
ejpam-3780	368	4	,	,	PUNCT
ejpam-3780	368	5	v2	v2	NOUN
ejpam-3780	368	6	,	,	PUNCT
ejpam-3780	368	7	.	.	PUNCT
ejpam-3780	368	8	.	.	PUNCT
ejpam-3780	369	1	.	.	PUNCT
ejpam-3780	370	1	,	,	PUNCT
ejpam-3780	370	2	vn	vn	PROPN
ejpam-3780	370	3	]	]	X
ejpam-3780	370	4	denote	denote	VERB
ejpam-3780	370	5	its	its	PRON
ejpam-3780	370	6	central	central	ADJ
ejpam-3780	370	7	path	path	NOUN
ejpam-3780	370	8	,	,	PUNCT
ejpam-3780	370	9	and	and	CCONJ
ejpam-3780	370	10	let	let	VERB
ejpam-3780	370	11	vji	vji	VERB
ejpam-3780	370	12	,	,	PUNCT
ejpam-3780	370	13	where	where	SCONJ
ejpam-3780	370	14	j	j	PROPN
ejpam-3780	370	15	=	=	SYM
ejpam-3780	370	16	1	1	NUM
ejpam-3780	370	17	,	,	PUNCT
ejpam-3780	370	18	2	2	NUM
ejpam-3780	370	19	,	,	PUNCT
ejpam-3780	370	20	.	.	PUNCT
ejpam-3780	370	21	.	.	PUNCT
ejpam-3780	371	1	.	.	PUNCT
ejpam-3780	372	1	,	,	PUNCT
ejpam-3780	372	2	rk	rk	NOUN
ejpam-3780	372	3	,	,	PUNCT
ejpam-3780	372	4	denote	denote	VERB
ejpam-3780	372	5	the	the	DET
ejpam-3780	372	6	hanging	hang	VERB
ejpam-3780	372	7	leaves	leave	NOUN
ejpam-3780	372	8	of	of	ADP
ejpam-3780	372	9	vi	vi	PROPN
ejpam-3780	372	10	.	.	PUNCT
ejpam-3780	372	11	by	by	ADP
ejpam-3780	372	12	theorem	theorem	NOUN
ejpam-3780	372	13	4	4	NUM
ejpam-3780	372	14	,	,	PUNCT
ejpam-3780	372	15	h	h	NOUN
ejpam-3780	372	16	has	have	VERB
ejpam-3780	372	17	an	an	DET
ejpam-3780	372	18	efficient	efficient	ADJ
ejpam-3780	372	19	zero	zero	NUM
ejpam-3780	372	20	ring	ring	NOUN
ejpam-3780	372	21	labeling	labeling	NOUN
ejpam-3780	372	22	h.	h.	NOUN
ejpam-3780	372	23	define	define	VERB
ejpam-3780	372	24	a	a	DET
ejpam-3780	372	25	function	function	NOUN
ejpam-3780	373	1	f	f	NOUN
ejpam-3780	373	2	:	:	PUNCT
ejpam-3780	373	3	v	v	X
ejpam-3780	373	4	(	(	PUNCT
ejpam-3780	373	5	g	g	NOUN
ejpam-3780	373	6	)	)	PUNCT
ejpam-3780	373	7	→	→	SYM
ejpam-3780	373	8	m0	m0	PROPN
ejpam-3780	373	9	2	2	NUM
ejpam-3780	373	10	(	(	PUNCT
ejpam-3780	373	11	znrk+n−rk	znrk+n−rk	PROPN
ejpam-3780	373	12	)	)	PUNCT
ejpam-3780	373	13	such	such	ADJ
ejpam-3780	373	14	that	that	SCONJ
ejpam-3780	373	15	f(wi	f(wi	NOUN
ejpam-3780	373	16	)	)	PUNCT
ejpam-3780	373	17	=	=	SYM
ejpam-3780	373	18	h(vi	h(vi	NOUN
ejpam-3780	373	19	)	)	PUNCT
ejpam-3780	373	20	and	and	CCONJ
ejpam-3780	373	21	f(wj	f(wj	PROPN
ejpam-3780	373	22	i	i	NOUN
ejpam-3780	373	23	)	)	PUNCT
ejpam-3780	373	24	=	=	PUNCT
ejpam-3780	374	1	h(wj	h(wj	ADV
ejpam-3780	374	2	i	i	INTJ
ejpam-3780	374	3	)	)	PUNCT
ejpam-3780	374	4	.	.	PUNCT
ejpam-3780	375	1	since	since	SCONJ
ejpam-3780	375	2	h	h	NOUN
ejpam-3780	375	3	is	be	AUX
ejpam-3780	375	4	injective	injective	ADJ
ejpam-3780	375	5	,	,	PUNCT
ejpam-3780	375	6	it	it	PRON
ejpam-3780	375	7	follows	follow	VERB
ejpam-3780	375	8	that	that	SCONJ
ejpam-3780	375	9	f	f	PROPN
ejpam-3780	375	10	is	be	AUX
ejpam-3780	375	11	also	also	ADV
ejpam-3780	375	12	injective	injective	ADJ
ejpam-3780	375	13	.	.	PUNCT
ejpam-3780	376	1	g	g	NOUN
ejpam-3780	376	2	is	be	AUX
ejpam-3780	376	3	an	an	DET
ejpam-3780	376	4	edge	edge	NOUN
ejpam-3780	376	5	-	-	PUNCT
ejpam-3780	376	6	induced	induce	VERB
ejpam-3780	376	7	subgraph	subgraph	NOUN
ejpam-3780	376	8	of	of	ADP
ejpam-3780	376	9	h	h	NOUN
ejpam-3780	376	10	,	,	PUNCT
ejpam-3780	376	11	so	so	ADV
ejpam-3780	376	12	a0	a0	PROPN
ejpam-3780	376	13	/∈	/∈	PUNCT
ejpam-3780	377	1	kh	kh	PROPN
ejpam-3780	377	2	=	=	X
ejpam-3780	377	3	{	{	PUNCT
ejpam-3780	377	4	h(u	h(u	PROPN
ejpam-3780	377	5	)	)	PUNCT
ejpam-3780	378	1	+	+	CCONJ
ejpam-3780	378	2	h(v	h(v	NOUN
ejpam-3780	378	3	)	)	PUNCT
ejpam-3780	378	4	:	:	PUNCT
ejpam-3780	378	5	uv	uv	NOUN
ejpam-3780	378	6	∈	∈	PROPN
ejpam-3780	378	7	e(h	e(h	PROPN
ejpam-3780	378	8	)	)	PUNCT
ejpam-3780	378	9	}	}	PUNCT
ejpam-3780	378	10	implies	imply	VERB
ejpam-3780	378	11	that	that	SCONJ
ejpam-3780	378	12	a0	a0	PROPN
ejpam-3780	378	13	/∈	/∈	PUNCT
ejpam-3780	379	1	k	k	X
ejpam-3780	379	2	=	=	PRON
ejpam-3780	379	3	{	{	PUNCT
ejpam-3780	379	4	f(u	f(u	PROPN
ejpam-3780	379	5	)	)	PUNCT
ejpam-3780	379	6	+	+	NUM
ejpam-3780	379	7	f(v	f(v	NOUN
ejpam-3780	379	8	)	)	PUNCT
ejpam-3780	379	9	:	:	PUNCT
ejpam-3780	379	10	uv	uv	PROPN
ejpam-3780	379	11	∈	∈	PROPN
ejpam-3780	379	12	e(g	e(g	PROPN
ejpam-3780	379	13	)	)	PUNCT
ejpam-3780	379	14	}	}	PUNCT
ejpam-3780	379	15	.	.	PUNCT
ejpam-3780	380	1	to	to	PART
ejpam-3780	380	2	show	show	VERB
ejpam-3780	380	3	that	that	SCONJ
ejpam-3780	380	4	f	f	PROPN
ejpam-3780	380	5	is	be	AUX
ejpam-3780	380	6	an	an	DET
ejpam-3780	380	7	efficient	efficient	ADJ
ejpam-3780	380	8	zero	zero	NUM
ejpam-3780	380	9	ring	ring	NOUN
ejpam-3780	380	10	labeling	labeling	NOUN
ejpam-3780	380	11	of	of	ADP
ejpam-3780	380	12	g	g	NOUN
ejpam-3780	380	13	,	,	PUNCT
ejpam-3780	380	14	it	it	PRON
ejpam-3780	380	15	remains	remain	VERB
ejpam-3780	380	16	to	to	PART
ejpam-3780	380	17	show	show	VERB
ejpam-3780	380	18	that	that	SCONJ
ejpam-3780	380	19	|k|	|k|	NOUN
ejpam-3780	380	20	=	=	SYM
ejpam-3780	380	21	∆(g	∆(g	PROPN
ejpam-3780	380	22	)	)	PUNCT
ejpam-3780	380	23	.	.	PUNCT
ejpam-3780	381	1	using	use	VERB
ejpam-3780	381	2	the	the	DET
ejpam-3780	381	3	labeling	labeling	NOUN
ejpam-3780	381	4	in	in	ADP
ejpam-3780	381	5	the	the	DET
ejpam-3780	381	6	proof	proof	NOUN
ejpam-3780	381	7	of	of	ADP
ejpam-3780	381	8	theorem	theorem	ADJ
ejpam-3780	381	9	4	4	NUM
ejpam-3780	381	10	,	,	PUNCT
ejpam-3780	381	11	we	we	PRON
ejpam-3780	381	12	obtain	obtain	VERB
ejpam-3780	381	13	k	k	X
ejpam-3780	381	14	=	=	PRON
ejpam-3780	381	15	{	{	PUNCT
ejpam-3780	381	16	a(rk+1)(n−2	a(rk+1)(n−2	PROPN
ejpam-3780	381	17	)	)	PUNCT
ejpam-3780	381	18	,	,	PUNCT
ejpam-3780	381	19	a(rk+1)(n−2)+1	a(rk+1)(n−2)+1	NOUN
ejpam-3780	381	20	,	,	PUNCT
ejpam-3780	381	21	.	.	PUNCT
ejpam-3780	381	22	.	.	PUNCT
ejpam-3780	382	1	.	.	PUNCT
ejpam-3780	383	1	,	,	PUNCT
ejpam-3780	383	2	a(rk+1)(n−2)+rk	a(rk+1)(n−2)+rk	INTJ
ejpam-3780	383	3	,	,	PUNCT
ejpam-3780	383	4	a(rk+1)(n−1	a(rk+1)(n−1	PROPN
ejpam-3780	383	5	)	)	PUNCT
ejpam-3780	383	6	}	}	PUNCT
ejpam-3780	383	7	.	.	PUNCT
ejpam-3780	384	1	(	(	PUNCT
ejpam-3780	384	2	14	14	NUM
ejpam-3780	384	3	)	)	PUNCT
ejpam-3780	384	4	thus	thus	ADV
ejpam-3780	384	5	,	,	PUNCT
ejpam-3780	384	6	|k|	|k|	NOUN
ejpam-3780	384	7	=	=	SYM
ejpam-3780	384	8	rk	rk	NOUN
ejpam-3780	384	9	+	+	CCONJ
ejpam-3780	384	10	2	2	NUM
ejpam-3780	384	11	=	=	SYM
ejpam-3780	384	12	∆(g	∆(g	NOUN
ejpam-3780	384	13	)	)	PUNCT
ejpam-3780	384	14	.	.	PUNCT
ejpam-3780	385	1	example	example	NOUN
ejpam-3780	385	2	9	9	NUM
ejpam-3780	385	3	.	.	X
ejpam-3780	385	4	figure	figure	NOUN
ejpam-3780	385	5	10	10	NUM
ejpam-3780	385	6	shows	show	VERB
ejpam-3780	385	7	an	an	DET
ejpam-3780	385	8	efficient	efficient	ADJ
ejpam-3780	385	9	zero	zero	NUM
ejpam-3780	385	10	ring	ring	NOUN
ejpam-3780	385	11	labeling	labeling	NOUN
ejpam-3780	385	12	of	of	ADP
ejpam-3780	385	13	a	a	DET
ejpam-3780	385	14	caterpillar	caterpillar	ADJ
ejpam-3780	385	15	g	g	NOUN
ejpam-3780	385	16	with	with	ADP
ejpam-3780	385	17	maximum	maximum	ADJ
ejpam-3780	385	18	degree	degree	NOUN
ejpam-3780	385	19	∆(g	∆(g	NOUN
ejpam-3780	385	20	)	)	PUNCT
ejpam-3780	385	21	=	=	NOUN
ejpam-3780	386	1	6	6	NUM
ejpam-3780	386	2	using	use	VERB
ejpam-3780	386	3	m0	m0	NOUN
ejpam-3780	386	4	2	2	NUM
ejpam-3780	386	5	(	(	PUNCT
ejpam-3780	386	6	z31	z31	PROPN
ejpam-3780	386	7	)	)	PUNCT
ejpam-3780	386	8	.	.	PUNCT
ejpam-3780	387	1	in	in	ADP
ejpam-3780	387	2	this	this	DET
ejpam-3780	387	3	labeling	labeling	NOUN
ejpam-3780	387	4	,	,	PUNCT
ejpam-3780	387	5	the	the	DET
ejpam-3780	387	6	set	set	NOUN
ejpam-3780	387	7	of	of	ADP
ejpam-3780	387	8	sums	sum	NOUN
ejpam-3780	387	9	is	be	AUX
ejpam-3780	387	10	k	k	NOUN
ejpam-3780	387	11	=	=	PUNCT
ejpam-3780	387	12	{	{	PUNCT
ejpam-3780	387	13	a15	a15	PROPN
ejpam-3780	387	14	,	,	PUNCT
ejpam-3780	387	15	a16	a16	PROPN
ejpam-3780	387	16	,	,	PUNCT
ejpam-3780	387	17	a17	a17	PROPN
ejpam-3780	387	18	,	,	PUNCT
ejpam-3780	387	19	a18	a18	PROPN
ejpam-3780	387	20	,	,	PUNCT
ejpam-3780	387	21	a19	a19	PROPN
ejpam-3780	387	22	,	,	PUNCT
ejpam-3780	387	23	a20	a20	PROPN
ejpam-3780	387	24	}	}	PUNCT
ejpam-3780	387	25	and	and	CCONJ
ejpam-3780	387	26	thus	thus	ADV
ejpam-3780	387	27	|k|	|k|	PROPN
ejpam-3780	387	28	=	=	SYM
ejpam-3780	387	29	∆(g	∆(g	PROPN
ejpam-3780	387	30	)	)	PUNCT
ejpam-3780	387	31	=	=	SYM
ejpam-3780	388	1	6	6	X
ejpam-3780	388	2	.	.	X
ejpam-3780	388	3	a0	a0	PROPN
ejpam-3780	388	4	a25	a25	PROPN
ejpam-3780	388	5	a5	a5	PROPN
ejpam-3780	388	6	a20	a20	PROPN
ejpam-3780	388	7	a10a0	a10a0	PROPN
ejpam-3780	388	8	a30	a30	PROPN
ejpam-3780	388	9	a0	a0	PROPN
ejpam-3780	388	10	a26	a26	PROPN
ejpam-3780	388	11	a0	a0	PROPN
ejpam-3780	388	12	a27	a27	PROPN
ejpam-3780	388	13	a25	a25	PROPN
ejpam-3780	388	14	a1	a1	PROPN
ejpam-3780	388	15	a5	a5	PROPN
ejpam-3780	388	16	a21	a21	PROPN
ejpam-3780	388	17	a5	a5	PROPN
ejpam-3780	388	18	a22	a22	PROPN
ejpam-3780	388	19	a20	a20	PROPN
ejpam-3780	388	20	a6	a6	PROPN
ejpam-3780	388	21	a20	a20	PROPN
ejpam-3780	388	22	a7	a7	PROPN
ejpam-3780	388	23	a20	a20	PROPN
ejpam-3780	388	24	a8	a8	PROPN
ejpam-3780	388	25	a20	a20	PROPN
ejpam-3780	388	26	a9	a9	PROPN
ejpam-3780	388	27	a10	a10	X
ejpam-3780	388	28	a16	a16	PROPN
ejpam-3780	388	29	a10	a10	PROPN
ejpam-3780	388	30	a15	a15	PROPN
ejpam-3780	388	31	figure	figure	NOUN
ejpam-3780	388	32	10	10	NUM
ejpam-3780	388	33	:	:	PUNCT
ejpam-3780	388	34	efficient	efficient	ADJ
ejpam-3780	388	35	zero	zero	NUM
ejpam-3780	388	36	ring	ring	NOUN
ejpam-3780	388	37	labeling	labeling	NOUN
ejpam-3780	388	38	of	of	ADP
ejpam-3780	388	39	a	a	DET
ejpam-3780	388	40	caterpillar	caterpillar	NOUN
ejpam-3780	388	41	using	use	VERB
ejpam-3780	388	42	m0	m0	PROPN
ejpam-3780	388	43	2	2	NUM
ejpam-3780	388	44	(	(	PUNCT
ejpam-3780	388	45	z31	z31	NOUN
ejpam-3780	388	46	)	)	PUNCT
ejpam-3780	388	47	theorem	theorem	VERB
ejpam-3780	388	48	7	7	NUM
ejpam-3780	388	49	.	.	PUNCT
ejpam-3780	389	1	a	a	DET
ejpam-3780	389	2	spider	spider	NOUN
ejpam-3780	389	3	has	have	VERB
ejpam-3780	389	4	an	an	DET
ejpam-3780	389	5	efficient	efficient	ADJ
ejpam-3780	389	6	zero	zero	NUM
ejpam-3780	389	7	ring	ring	NOUN
ejpam-3780	389	8	labeling	labeling	NOUN
ejpam-3780	389	9	.	.	PUNCT
ejpam-3780	390	1	proof	proof	NOUN
ejpam-3780	390	2	.	.	PUNCT
ejpam-3780	391	1	let	let	VERB
ejpam-3780	391	2	g	g	PRON
ejpam-3780	391	3	be	be	AUX
ejpam-3780	391	4	a	a	DET
ejpam-3780	391	5	spider	spider	NOUN
ejpam-3780	391	6	with	with	ADP
ejpam-3780	391	7	n	n	ADP
ejpam-3780	391	8	legs	leg	NOUN
ejpam-3780	391	9	,	,	PUNCT
ejpam-3780	391	10	where	where	SCONJ
ejpam-3780	391	11	n	n	PRON
ejpam-3780	391	12	≥	≥	NOUN
ejpam-3780	391	13	3	3	NUM
ejpam-3780	391	14	.	.	PUNCT
ejpam-3780	392	1	then	then	ADV
ejpam-3780	392	2	∆(g	∆(g	PROPN
ejpam-3780	392	3	)	)	PUNCT
ejpam-3780	393	1	=	=	VERB
ejpam-3780	393	2	n.	n.	NOUN
ejpam-3780	393	3	let	let	VERB
ejpam-3780	393	4	the	the	DET
ejpam-3780	393	5	head	head	NOUN
ejpam-3780	393	6	of	of	ADP
ejpam-3780	393	7	g	g	PROPN
ejpam-3780	393	8	be	be	AUX
ejpam-3780	393	9	denoted	denote	VERB
ejpam-3780	393	10	by	by	ADP
ejpam-3780	393	11	w	w	PROPN
ejpam-3780	393	12	,	,	PUNCT
ejpam-3780	393	13	and	and	CCONJ
ejpam-3780	393	14	let	let	VERB
ejpam-3780	393	15	w1	w1	NOUN
ejpam-3780	393	16	1	1	NUM
ejpam-3780	393	17	,	,	PUNCT
ejpam-3780	393	18	w	w	PROPN
ejpam-3780	393	19	1	1	NUM
ejpam-3780	393	20	2	2	NUM
ejpam-3780	393	21	,	,	PUNCT
ejpam-3780	393	22	.	.	PUNCT
ejpam-3780	393	23	.	.	PUNCT
ejpam-3780	394	1	.	.	PUNCT
ejpam-3780	395	1	,	,	PUNCT
ejpam-3780	395	2	w	w	PROPN
ejpam-3780	395	3	1	1	NUM
ejpam-3780	395	4	n	n	PRON
ejpam-3780	395	5	denote	denote	VERB
ejpam-3780	395	6	the	the	DET
ejpam-3780	395	7	vertices	vertex	NOUN
ejpam-3780	395	8	that	that	PRON
ejpam-3780	395	9	are	be	AUX
ejpam-3780	395	10	adjacent	adjacent	ADJ
ejpam-3780	395	11	to	to	ADP
ejpam-3780	395	12	w.	w.	PROPN
ejpam-3780	395	13	suppose	suppose	VERB
ejpam-3780	395	14	each	each	DET
ejpam-3780	395	15	leg	leg	NOUN
ejpam-3780	395	16	containing	contain	VERB
ejpam-3780	395	17	w1	w1	NOUN
ejpam-3780	395	18	i	i	PRON
ejpam-3780	395	19	has	have	VERB
ejpam-3780	395	20	ri	ri	NOUN
ejpam-3780	395	21	vertices	vertex	NOUN
ejpam-3780	395	22	,	,	PUNCT
ejpam-3780	395	23	and	and	CCONJ
ejpam-3780	395	24	let	let	VERB
ejpam-3780	395	25	[	[	PUNCT
ejpam-3780	395	26	w1	w1	NOUN
ejpam-3780	396	1	i	i	PRON
ejpam-3780	396	2	,	,	PUNCT
ejpam-3780	396	3	w	w	PROPN
ejpam-3780	396	4	2	2	NUM
ejpam-3780	396	5	i	i	NOUN
ejpam-3780	396	6	,	,	PUNCT
ejpam-3780	396	7	.	.	PUNCT
ejpam-3780	396	8	.	.	PUNCT
ejpam-3780	396	9	.	.	PUNCT
ejpam-3780	397	1	,	,	PUNCT
ejpam-3780	398	1	w	w	PROPN
ejpam-3780	398	2	ri	ri	PROPN
ejpam-3780	399	1	i	i	PRON
ejpam-3780	399	2	]	]	PUNCT
ejpam-3780	399	3	denote	denote	VERB
ejpam-3780	399	4	the	the	DET
ejpam-3780	399	5	leg	leg	NOUN
ejpam-3780	399	6	containing	contain	VERB
ejpam-3780	399	7	w1	w1	NOUN
ejpam-3780	399	8	i	i	PRON
ejpam-3780	399	9	.	.	PUNCT
ejpam-3780	400	1	d.	d.	PROPN
ejpam-3780	400	2	chua	chua	PROPN
ejpam-3780	400	3	,	,	PUNCT
ejpam-3780	400	4	f.	f.	PROPN
ejpam-3780	400	5	campeña	campeña	PROPN
ejpam-3780	400	6	,	,	PUNCT
ejpam-3780	400	7	f.	f.	PROPN
ejpam-3780	400	8	franco	franco	PROPN
ejpam-3780	400	9	/	/	SYM
ejpam-3780	400	10	eur	eur	PROPN
ejpam-3780	400	11	.	.	PUNCT
ejpam-3780	401	1	j.	j.	PROPN
ejpam-3780	401	2	pure	pure	PROPN
ejpam-3780	401	3	appl	appl	PROPN
ejpam-3780	401	4	.	.	PROPN
ejpam-3780	401	5	math	math	PROPN
ejpam-3780	401	6	,	,	PUNCT
ejpam-3780	401	7	13	13	NUM
ejpam-3780	401	8	(	(	PUNCT
ejpam-3780	401	9	3	3	NUM
ejpam-3780	401	10	)	)	PUNCT
ejpam-3780	401	11	(	(	PUNCT
ejpam-3780	401	12	2020	2020	NUM
ejpam-3780	401	13	)	)	PUNCT
ejpam-3780	401	14	,	,	PUNCT
ejpam-3780	401	15	674	674	NUM
ejpam-3780	401	16	-	-	SYM
ejpam-3780	401	17	696	696	NUM
ejpam-3780	401	18	685	685	NUM
ejpam-3780	401	19	suppose	suppose	VERB
ejpam-3780	401	20	rk	rk	PROPN
ejpam-3780	401	21	=	=	PROPN
ejpam-3780	401	22	max{r1	max{r1	NOUN
ejpam-3780	401	23	,	,	PUNCT
ejpam-3780	401	24	r2	r2	NOUN
ejpam-3780	401	25	,	,	PUNCT
ejpam-3780	401	26	.	.	PUNCT
ejpam-3780	401	27	.	.	PUNCT
ejpam-3780	401	28	.	.	PUNCT
ejpam-3780	402	1	,	,	PUNCT
ejpam-3780	402	2	rn	rn	PROPN
ejpam-3780	402	3	}	}	PUNCT
ejpam-3780	402	4	.	.	PUNCT
ejpam-3780	403	1	define	define	VERB
ejpam-3780	403	2	a	a	DET
ejpam-3780	403	3	function	function	NOUN
ejpam-3780	403	4	f	f	NOUN
ejpam-3780	403	5	:	:	PUNCT
ejpam-3780	403	6	v	v	X
ejpam-3780	403	7	(	(	PUNCT
ejpam-3780	403	8	g	g	NOUN
ejpam-3780	403	9	)	)	PUNCT
ejpam-3780	403	10	→	→	SYM
ejpam-3780	403	11	m0	m0	PROPN
ejpam-3780	403	12	2	2	NUM
ejpam-3780	403	13	(	(	PUNCT
ejpam-3780	403	14	z(n−1)(2rk+1	z(n−1)(2rk+1	NOUN
ejpam-3780	403	15	)	)	PUNCT
ejpam-3780	403	16	)	)	PUNCT
ejpam-3780	403	17	such	such	ADJ
ejpam-3780	403	18	that	that	SCONJ
ejpam-3780	403	19	f(w	f(w	NOUN
ejpam-3780	403	20	)	)	PUNCT
ejpam-3780	403	21	=	=	SYM
ejpam-3780	403	22	a0	a0	PROPN
ejpam-3780	403	23	,	,	PUNCT
ejpam-3780	403	24	f(wj	f(wj	PROPN
ejpam-3780	403	25	i	i	NOUN
ejpam-3780	403	26	)	)	PUNCT
ejpam-3780	404	1	=	=	PUNCT
ejpam-3780	405	1	a	a	NUM
ejpam-3780	405	2	(	(	PUNCT
ejpam-3780	405	3	n−1	n−1	PROPN
ejpam-3780	405	4	)	)	PUNCT
ejpam-3780	405	5	(	(	PUNCT
ejpam-3780	405	6	2rk−j+1	2rk−j+1	NUM
ejpam-3780	405	7	2	2	NUM
ejpam-3780	405	8	)	)	PUNCT
ejpam-3780	406	1	+	+	ADP
ejpam-3780	406	2	i−1	i−1	PROPN
ejpam-3780	406	3	if	if	SCONJ
ejpam-3780	406	4	j	j	PROPN
ejpam-3780	406	5	is	be	AUX
ejpam-3780	406	6	odd	odd	ADJ
ejpam-3780	406	7	a(n−1	a(n−1	ADV
ejpam-3780	406	8	)	)	PUNCT
ejpam-3780	406	9	(	(	PUNCT
ejpam-3780	406	10	j	j	PROPN
ejpam-3780	406	11	2)−i+1	2)−i+1	INTJ
ejpam-3780	406	12	if	if	SCONJ
ejpam-3780	406	13	j	j	PROPN
ejpam-3780	406	14	is	be	AUX
ejpam-3780	406	15	even	even	ADV
ejpam-3780	406	16	(	(	PUNCT
ejpam-3780	406	17	15	15	NUM
ejpam-3780	406	18	)	)	PUNCT
ejpam-3780	406	19	for	for	ADP
ejpam-3780	406	20	i	i	PROPN
ejpam-3780	406	21	=	=	SYM
ejpam-3780	406	22	1	1	NUM
ejpam-3780	406	23	,	,	PUNCT
ejpam-3780	406	24	2	2	NUM
ejpam-3780	406	25	,	,	PUNCT
ejpam-3780	406	26	.	.	PUNCT
ejpam-3780	406	27	.	.	PUNCT
ejpam-3780	406	28	.	.	PUNCT
ejpam-3780	407	1	,	,	PUNCT
ejpam-3780	407	2	n−	n−	NOUN
ejpam-3780	407	3	1	1	NUM
ejpam-3780	407	4	,	,	PUNCT
ejpam-3780	407	5	and	and	CCONJ
ejpam-3780	407	6	f(wj	f(wj	PROPN
ejpam-3780	407	7	n	n	CCONJ
ejpam-3780	407	8	)	)	PUNCT
ejpam-3780	407	9	=	=	PUNCT
ejpam-3780	408	1			PROPN
ejpam-3780	408	2	a	a	DET
ejpam-3780	408	3	(	(	PUNCT
ejpam-3780	408	4	n−1	n−1	PROPN
ejpam-3780	408	5	)	)	PUNCT
ejpam-3780	408	6	(	(	PUNCT
ejpam-3780	408	7	2rk+j+1	2rk+j+1	NUM
ejpam-3780	408	8	2	2	X
ejpam-3780	408	9	)	)	PUNCT
ejpam-3780	408	10	if	if	SCONJ
ejpam-3780	408	11	j	j	PROPN
ejpam-3780	408	12	is	be	AUX
ejpam-3780	408	13	odd	odd	ADJ
ejpam-3780	408	14	a	a	DET
ejpam-3780	408	15	(	(	PUNCT
ejpam-3780	408	16	n−1	n−1	PROPN
ejpam-3780	408	17	)	)	PUNCT
ejpam-3780	408	18	(	(	PUNCT
ejpam-3780	408	19	4rk−j+2	4rk−j+2	NUM
ejpam-3780	408	20	2	2	X
ejpam-3780	408	21	)	)	PUNCT
ejpam-3780	408	22	if	if	SCONJ
ejpam-3780	408	23	j	j	PROPN
ejpam-3780	408	24	is	be	AUX
ejpam-3780	408	25	even	even	ADV
ejpam-3780	408	26	.	.	PUNCT
ejpam-3780	409	1	(	(	PUNCT
ejpam-3780	409	2	16	16	NUM
ejpam-3780	409	3	)	)	PUNCT
ejpam-3780	409	4	clearly	clearly	ADV
ejpam-3780	409	5	,	,	PUNCT
ejpam-3780	409	6	f	f	PROPN
ejpam-3780	409	7	is	be	AUX
ejpam-3780	409	8	injective	injective	ADJ
ejpam-3780	409	9	.	.	PUNCT
ejpam-3780	410	1	let	let	VERB
ejpam-3780	410	2	k	k	NOUN
ejpam-3780	410	3	=	=	PRON
ejpam-3780	410	4	{	{	PUNCT
ejpam-3780	410	5	f(u	f(u	PROPN
ejpam-3780	410	6	)	)	PUNCT
ejpam-3780	410	7	+	+	NUM
ejpam-3780	410	8	f(v	f(v	NOUN
ejpam-3780	410	9	)	)	PUNCT
ejpam-3780	410	10	:	:	PUNCT
ejpam-3780	410	11	uv	uv	PROPN
ejpam-3780	410	12	∈	∈	PROPN
ejpam-3780	410	13	e(g	e(g	PROPN
ejpam-3780	410	14	)	)	PUNCT
ejpam-3780	410	15	}	}	PUNCT
ejpam-3780	410	16	.	.	PUNCT
ejpam-3780	411	1	to	to	PART
ejpam-3780	411	2	show	show	VERB
ejpam-3780	411	3	that	that	SCONJ
ejpam-3780	411	4	f	f	PROPN
ejpam-3780	411	5	is	be	AUX
ejpam-3780	411	6	an	an	DET
ejpam-3780	411	7	efficient	efficient	ADJ
ejpam-3780	411	8	zero	zero	NUM
ejpam-3780	411	9	ring	ring	NOUN
ejpam-3780	411	10	labeling	labeling	NOUN
ejpam-3780	411	11	of	of	ADP
ejpam-3780	411	12	g	g	NOUN
ejpam-3780	411	13	,	,	PUNCT
ejpam-3780	411	14	we	we	PRON
ejpam-3780	411	15	need	need	VERB
ejpam-3780	411	16	to	to	PART
ejpam-3780	411	17	show	show	VERB
ejpam-3780	411	18	that	that	SCONJ
ejpam-3780	411	19	|k|	|k|	PROPN
ejpam-3780	411	20	=	=	SYM
ejpam-3780	411	21	n	n	PROPN
ejpam-3780	411	22	and	and	CCONJ
ejpam-3780	411	23	a0	a0	PROPN
ejpam-3780	411	24	/∈	/∈	PROPN
ejpam-3780	411	25	k.	k.	PROPN
ejpam-3780	411	26	for	for	ADP
ejpam-3780	411	27	pairs	pair	NOUN
ejpam-3780	411	28	of	of	ADP
ejpam-3780	411	29	adjacent	adjacent	ADJ
ejpam-3780	411	30	head	head	NOUN
ejpam-3780	411	31	and	and	CCONJ
ejpam-3780	411	32	vertex	vertex	NOUN
ejpam-3780	411	33	in	in	ADP
ejpam-3780	411	34	a	a	DET
ejpam-3780	411	35	leg	leg	NOUN
ejpam-3780	411	36	,	,	PUNCT
ejpam-3780	411	37	we	we	PRON
ejpam-3780	411	38	obtain	obtain	VERB
ejpam-3780	411	39	the	the	DET
ejpam-3780	411	40	sums	sum	NOUN
ejpam-3780	411	41	f(w	f(w	PROPN
ejpam-3780	411	42	)	)	PUNCT
ejpam-3780	412	1	+	+	SYM
ejpam-3780	412	2	f(w1	f(w1	VERB
ejpam-3780	412	3	i	i	PRON
ejpam-3780	412	4	)	)	PUNCT
ejpam-3780	413	1	=	=	SYM
ejpam-3780	413	2	a0	a0	PROPN
ejpam-3780	413	3	+	+	CCONJ
ejpam-3780	413	4	a	a	DET
ejpam-3780	413	5	(	(	PUNCT
ejpam-3780	413	6	n−1	n−1	PROPN
ejpam-3780	413	7	)	)	PUNCT
ejpam-3780	413	8	(	(	PUNCT
ejpam-3780	413	9	2rk−1	2rk−1	NUM
ejpam-3780	413	10	+	+	NOUN
ejpam-3780	413	11	1	1	NUM
ejpam-3780	413	12	2	2	NUM
ejpam-3780	413	13	)	)	PUNCT
ejpam-3780	414	1	+	+	ADP
ejpam-3780	414	2	i−1	i−1	PROPN
ejpam-3780	414	3	=	=	SYM
ejpam-3780	414	4	a(n−1)(rk)+i−1	a(n−1)(rk)+i−1	PROPN
ejpam-3780	414	5	(	(	PUNCT
ejpam-3780	414	6	17	17	NUM
ejpam-3780	414	7	)	)	PUNCT
ejpam-3780	414	8	for	for	ADP
ejpam-3780	414	9	i	i	PROPN
ejpam-3780	414	10	=	=	SYM
ejpam-3780	414	11	1	1	NUM
ejpam-3780	414	12	,	,	PUNCT
ejpam-3780	414	13	2	2	NUM
ejpam-3780	414	14	,	,	PUNCT
ejpam-3780	414	15	.	.	PUNCT
ejpam-3780	414	16	.	.	PUNCT
ejpam-3780	414	17	.	.	PUNCT
ejpam-3780	415	1	,	,	PUNCT
ejpam-3780	415	2	n−	n−	NOUN
ejpam-3780	415	3	1	1	NUM
ejpam-3780	415	4	,	,	PUNCT
ejpam-3780	415	5	and	and	CCONJ
ejpam-3780	415	6	f(w	f(w	PROPN
ejpam-3780	415	7	)	)	PUNCT
ejpam-3780	415	8	+	+	NUM
ejpam-3780	415	9	f(w1	f(w1	NOUN
ejpam-3780	415	10	n	n	CCONJ
ejpam-3780	415	11	)	)	PUNCT
ejpam-3780	415	12	=	=	SYM
ejpam-3780	415	13	a0	a0	PROPN
ejpam-3780	415	14	+	+	CCONJ
ejpam-3780	415	15	a(n−1)(rk+1	a(n−1)(rk+1	VERB
ejpam-3780	415	16	)	)	PUNCT
ejpam-3780	415	17	=	=	PRON
ejpam-3780	415	18	a(n−1)(rk+1	a(n−1)(rk+1	VERB
ejpam-3780	415	19	)	)	PUNCT
ejpam-3780	415	20	=	=	SYM
ejpam-3780	416	1	a(n−1)(rk)+n−1	a(n−1)(rk)+n−1	PROPN
ejpam-3780	416	2	.	.	PUNCT
ejpam-3780	416	3	(	(	PUNCT
ejpam-3780	416	4	18	18	NUM
ejpam-3780	416	5	)	)	PUNCT
ejpam-3780	416	6	thus	thus	ADV
ejpam-3780	416	7	,	,	PUNCT
ejpam-3780	416	8	we	we	PRON
ejpam-3780	416	9	have	have	VERB
ejpam-3780	416	10	the	the	DET
ejpam-3780	416	11	sums	sum	NOUN
ejpam-3780	416	12	a(n−1)(rk	a(n−1)(rk	PROPN
ejpam-3780	416	13	)	)	PUNCT
ejpam-3780	416	14	,	,	PUNCT
ejpam-3780	416	15	a(n−1)(rk)+1	a(n−1)(rk)+1	NOUN
ejpam-3780	416	16	,	,	PUNCT
ejpam-3780	416	17	a(n−1)(rk)+2	a(n−1)(rk)+2	X
ejpam-3780	416	18	,	,	PUNCT
ejpam-3780	416	19	.	.	PUNCT
ejpam-3780	416	20	.	.	PUNCT
ejpam-3780	417	1	.	.	PUNCT
ejpam-3780	418	1	,	,	PUNCT
ejpam-3780	418	2	a(n−1)(rk)+n−2	a(n−1)(rk)+n−2	PROPN
ejpam-3780	418	3	,	,	PUNCT
ejpam-3780	418	4	a(n−1)(rk)+n−1	a(n−1)(rk)+n−1	PROPN
ejpam-3780	418	5	.	.	PROPN
ejpam-3780	418	6	for	for	ADP
ejpam-3780	418	7	adjacent	adjacent	ADJ
ejpam-3780	418	8	vertices	vertex	NOUN
ejpam-3780	418	9	in	in	ADP
ejpam-3780	418	10	a	a	DET
ejpam-3780	418	11	leg	leg	NOUN
ejpam-3780	418	12	,	,	PUNCT
ejpam-3780	418	13	we	we	PRON
ejpam-3780	418	14	obtain	obtain	VERB
ejpam-3780	418	15	the	the	DET
ejpam-3780	418	16	sums	sum	NOUN
ejpam-3780	418	17	f(wj	f(wj	PROPN
ejpam-3780	418	18	i	i	NOUN
ejpam-3780	418	19	)	)	PUNCT
ejpam-3780	419	1	+	+	CCONJ
ejpam-3780	419	2	f(wj+1	f(wj+1	PROPN
ejpam-3780	419	3	i	i	NOUN
ejpam-3780	419	4	)	)	PUNCT
ejpam-3780	419	5	=	=	PUNCT
ejpam-3780	420	1	a	a	DET
ejpam-3780	420	2	(	(	PUNCT
ejpam-3780	420	3	n−1	n−1	PROPN
ejpam-3780	420	4	)	)	PUNCT
ejpam-3780	420	5	(	(	PUNCT
ejpam-3780	420	6	2rk−j+1	2rk−j+1	NUM
ejpam-3780	420	7	2	2	NUM
ejpam-3780	420	8	)	)	PUNCT
ejpam-3780	421	1	+	+	ADP
ejpam-3780	421	2	i−1	i−1	PROPN
ejpam-3780	421	3	+	+	CCONJ
ejpam-3780	421	4	a(n−1	a(n−1	PROPN
ejpam-3780	421	5	)	)	PUNCT
ejpam-3780	421	6	(	(	PUNCT
ejpam-3780	421	7	j+1	j+1	ADV
ejpam-3780	421	8	2	2	X
ejpam-3780	421	9	)	)	PUNCT
ejpam-3780	421	10	−i+1	−i+1	NOUN
ejpam-3780	421	11	=	=	PRON
ejpam-3780	421	12	a(n−1)(rk+1	a(n−1)(rk+1	VERB
ejpam-3780	421	13	)	)	PUNCT
ejpam-3780	421	14	=	=	SYM
ejpam-3780	421	15	a(n−1)(rk)+n−1	a(n−1)(rk)+n−1	X
ejpam-3780	421	16	(	(	PUNCT
ejpam-3780	421	17	19	19	NUM
ejpam-3780	421	18	)	)	PUNCT
ejpam-3780	421	19	if	if	SCONJ
ejpam-3780	421	20	j	j	PROPN
ejpam-3780	421	21	is	be	AUX
ejpam-3780	421	22	odd	odd	ADJ
ejpam-3780	421	23	,	,	PUNCT
ejpam-3780	421	24	and	and	CCONJ
ejpam-3780	421	25	f(wj	f(wj	PROPN
ejpam-3780	421	26	i	i	PROPN
ejpam-3780	421	27	)	)	PUNCT
ejpam-3780	422	1	+	+	CCONJ
ejpam-3780	422	2	f(wj+1	f(wj+1	PROPN
ejpam-3780	422	3	i	i	NOUN
ejpam-3780	422	4	)	)	PUNCT
ejpam-3780	422	5	=	=	PUNCT
ejpam-3780	422	6	a(n−1	a(n−1	PROPN
ejpam-3780	422	7	)	)	PUNCT
ejpam-3780	422	8	(	(	PUNCT
ejpam-3780	422	9	j	j	PROPN
ejpam-3780	422	10	2)−i+1	2)−i+1	PROPN
ejpam-3780	422	11	+	+	CCONJ
ejpam-3780	422	12	a	a	DET
ejpam-3780	422	13	(	(	PUNCT
ejpam-3780	422	14	n−1	n−1	PROPN
ejpam-3780	422	15	)	)	PUNCT
ejpam-3780	422	16	(	(	PUNCT
ejpam-3780	422	17	2rk−(j+1)+1	2rk−(j+1)+1	NUM
ejpam-3780	422	18	2	2	NUM
ejpam-3780	422	19	)	)	PUNCT
ejpam-3780	423	1	+	+	ADP
ejpam-3780	423	2	i−1	i−1	PROPN
ejpam-3780	423	3	=	=	SYM
ejpam-3780	423	4	a(n−1)(rk	a(n−1)(rk	PROPN
ejpam-3780	423	5	)	)	PUNCT
ejpam-3780	423	6	(	(	PUNCT
ejpam-3780	423	7	20	20	NUM
ejpam-3780	423	8	)	)	PUNCT
ejpam-3780	423	9	if	if	SCONJ
ejpam-3780	423	10	j	j	PROPN
ejpam-3780	423	11	is	be	AUX
ejpam-3780	423	12	even	even	ADV
ejpam-3780	423	13	,	,	PUNCT
ejpam-3780	423	14	for	for	ADP
ejpam-3780	423	15	i	i	PROPN
ejpam-3780	423	16	=	=	SYM
ejpam-3780	423	17	1	1	NUM
ejpam-3780	423	18	,	,	PUNCT
ejpam-3780	423	19	2	2	NUM
ejpam-3780	423	20	,	,	PUNCT
ejpam-3780	423	21	.	.	PUNCT
ejpam-3780	423	22	.	.	PUNCT
ejpam-3780	423	23	.	.	PUNCT
ejpam-3780	424	1	,	,	PUNCT
ejpam-3780	424	2	n−	n−	NOUN
ejpam-3780	424	3	1	1	NUM
ejpam-3780	424	4	.	.	PUNCT
ejpam-3780	425	1	moreover	moreover	ADV
ejpam-3780	425	2	,	,	PUNCT
ejpam-3780	425	3	we	we	PRON
ejpam-3780	425	4	have	have	VERB
ejpam-3780	425	5	f(wj	f(wj	NOUN
ejpam-3780	425	6	n	n	CCONJ
ejpam-3780	425	7	)	)	PUNCT
ejpam-3780	425	8	+	+	CCONJ
ejpam-3780	425	9	f(wj+1	f(wj+1	PROPN
ejpam-3780	425	10	n	n	NOUN
ejpam-3780	425	11	)	)	PUNCT
ejpam-3780	425	12	=	=	SYM
ejpam-3780	426	1	a	a	DET
ejpam-3780	426	2	(	(	PUNCT
ejpam-3780	426	3	n−1	n−1	PROPN
ejpam-3780	426	4	)	)	PUNCT
ejpam-3780	426	5	(	(	PUNCT
ejpam-3780	426	6	2rk+j+1	2rk+j+1	NUM
ejpam-3780	426	7	2	2	NUM
ejpam-3780	426	8	)	)	PUNCT
ejpam-3780	426	9	+	+	CCONJ
ejpam-3780	426	10	a	a	DET
ejpam-3780	426	11	(	(	PUNCT
ejpam-3780	426	12	n−1	n−1	PROPN
ejpam-3780	426	13	)	)	PUNCT
ejpam-3780	426	14	(	(	PUNCT
ejpam-3780	426	15	4rk−(j+1)+2	4rk−(j+1)+2	NUM
ejpam-3780	426	16	2	2	NUM
ejpam-3780	426	17	)	)	PUNCT
ejpam-3780	426	18	=	=	SYM
ejpam-3780	426	19	a(n−1)(3rk+1	a(n−1)(3rk+1	PROPN
ejpam-3780	426	20	)	)	PUNCT
ejpam-3780	426	21	=	=	SYM
ejpam-3780	427	1	a(n−1)(rk	a(n−1)(rk	NOUN
ejpam-3780	427	2	)	)	PUNCT
ejpam-3780	427	3	(	(	PUNCT
ejpam-3780	427	4	21	21	NUM
ejpam-3780	427	5	)	)	PUNCT
ejpam-3780	427	6	if	if	SCONJ
ejpam-3780	427	7	j	j	PROPN
ejpam-3780	427	8	is	be	AUX
ejpam-3780	427	9	odd	odd	ADJ
ejpam-3780	427	10	,	,	PUNCT
ejpam-3780	427	11	and	and	CCONJ
ejpam-3780	427	12	f(wj	f(wj	PROPN
ejpam-3780	427	13	n	n	CCONJ
ejpam-3780	427	14	)	)	PUNCT
ejpam-3780	427	15	+	+	CCONJ
ejpam-3780	427	16	f(wj+1	f(wj+1	PROPN
ejpam-3780	427	17	n	n	NOUN
ejpam-3780	427	18	)	)	PUNCT
ejpam-3780	427	19	=	=	SYM
ejpam-3780	428	1	a	a	DET
ejpam-3780	428	2	(	(	PUNCT
ejpam-3780	428	3	n−1	n−1	PROPN
ejpam-3780	428	4	)	)	PUNCT
ejpam-3780	428	5	(	(	PUNCT
ejpam-3780	428	6	4rk−j+2	4rk−j+2	NUM
ejpam-3780	428	7	2	2	NUM
ejpam-3780	428	8	)	)	PUNCT
ejpam-3780	429	1	+	+	CCONJ
ejpam-3780	429	2	a	a	DET
ejpam-3780	429	3	(	(	PUNCT
ejpam-3780	429	4	n−1	n−1	PROPN
ejpam-3780	429	5	)	)	PUNCT
ejpam-3780	429	6	(	(	PUNCT
ejpam-3780	429	7	2rk+(j+1)+1	2rk+(j+1)+1	NUM
ejpam-3780	429	8	2	2	NUM
ejpam-3780	429	9	)	)	PUNCT
ejpam-3780	429	10	=	=	SYM
ejpam-3780	429	11	a(n−1)(3rk+2	a(n−1)(3rk+2	NUM
ejpam-3780	429	12	)	)	PUNCT
ejpam-3780	429	13	=	=	PRON
ejpam-3780	429	14	a(n−1)(rk+1	a(n−1)(rk+1	VERB
ejpam-3780	429	15	)	)	PUNCT
ejpam-3780	429	16	=	=	SYM
ejpam-3780	429	17	a(n−1)(rk)+n−1	a(n−1)(rk)+n−1	X
ejpam-3780	429	18	(	(	PUNCT
ejpam-3780	429	19	22	22	NUM
ejpam-3780	429	20	)	)	PUNCT
ejpam-3780	429	21	d.	d.	PROPN
ejpam-3780	429	22	chua	chua	PROPN
ejpam-3780	429	23	,	,	PUNCT
ejpam-3780	429	24	f.	f.	PROPN
ejpam-3780	429	25	campeña	campeña	PROPN
ejpam-3780	429	26	,	,	PUNCT
ejpam-3780	429	27	f.	f.	PROPN
ejpam-3780	429	28	franco	franco	PROPN
ejpam-3780	429	29	/	/	SYM
ejpam-3780	429	30	eur	eur	PROPN
ejpam-3780	429	31	.	.	PUNCT
ejpam-3780	430	1	j.	j.	PROPN
ejpam-3780	430	2	pure	pure	PROPN
ejpam-3780	430	3	appl	appl	PROPN
ejpam-3780	430	4	.	.	PROPN
ejpam-3780	430	5	math	math	PROPN
ejpam-3780	430	6	,	,	PUNCT
ejpam-3780	430	7	13	13	NUM
ejpam-3780	430	8	(	(	PUNCT
ejpam-3780	430	9	3	3	NUM
ejpam-3780	430	10	)	)	PUNCT
ejpam-3780	430	11	(	(	PUNCT
ejpam-3780	430	12	2020	2020	NUM
ejpam-3780	430	13	)	)	PUNCT
ejpam-3780	430	14	,	,	PUNCT
ejpam-3780	430	15	674	674	NUM
ejpam-3780	430	16	-	-	SYM
ejpam-3780	430	17	696	696	NUM
ejpam-3780	430	18	686	686	NUM
ejpam-3780	430	19	if	if	SCONJ
ejpam-3780	430	20	j	j	PROPN
ejpam-3780	430	21	is	be	AUX
ejpam-3780	430	22	even	even	ADV
ejpam-3780	430	23	.	.	PUNCT
ejpam-3780	431	1	then	then	ADV
ejpam-3780	431	2	k	k	PROPN
ejpam-3780	431	3	=	=	PRON
ejpam-3780	431	4	{	{	PUNCT
ejpam-3780	431	5	a(n−1)(rk	a(n−1)(rk	NOUN
ejpam-3780	431	6	)	)	PUNCT
ejpam-3780	431	7	,	,	PUNCT
ejpam-3780	431	8	a(n−1)(rk)+1	a(n−1)(rk)+1	ADV
ejpam-3780	431	9	,	,	PUNCT
ejpam-3780	431	10	.	.	PUNCT
ejpam-3780	431	11	.	.	PUNCT
ejpam-3780	431	12	.	.	PUNCT
ejpam-3780	432	1	,	,	PUNCT
ejpam-3780	432	2	a(n−1)(rk)+n−2	a(n−1)(rk)+n−2	PROPN
ejpam-3780	432	3	,	,	PUNCT
ejpam-3780	432	4	a(n−1)(rk)+n−1	a(n−1)(rk)+n−1	PROPN
ejpam-3780	432	5	}	}	PUNCT
ejpam-3780	432	6	,	,	PUNCT
ejpam-3780	432	7	(	(	PUNCT
ejpam-3780	432	8	23	23	NUM
ejpam-3780	432	9	)	)	PUNCT
ejpam-3780	432	10	hence	hence	ADV
ejpam-3780	432	11	|k|	|k|	NOUN
ejpam-3780	432	12	=	=	SYM
ejpam-3780	432	13	n.	n.	NOUN
ejpam-3780	432	14	for	for	ADP
ejpam-3780	432	15	m	m	PROPN
ejpam-3780	432	16	=	=	SYM
ejpam-3780	432	17	0	0	NUM
ejpam-3780	432	18	,	,	PUNCT
ejpam-3780	432	19	1	1	NUM
ejpam-3780	432	20	,	,	PUNCT
ejpam-3780	432	21	2	2	NUM
ejpam-3780	432	22	,	,	PUNCT
ejpam-3780	432	23	.	.	PUNCT
ejpam-3780	432	24	.	.	PUNCT
ejpam-3780	433	1	.	.	PUNCT
ejpam-3780	434	1	,	,	PUNCT
ejpam-3780	434	2	n	n	CCONJ
ejpam-3780	434	3	−	−	PROPN
ejpam-3780	434	4	1	1	NUM
ejpam-3780	434	5	,	,	PUNCT
ejpam-3780	434	6	it	it	PRON
ejpam-3780	434	7	is	be	AUX
ejpam-3780	434	8	clear	clear	ADJ
ejpam-3780	434	9	that	that	SCONJ
ejpam-3780	434	10	(	(	PUNCT
ejpam-3780	434	11	n	n	NUM
ejpam-3780	434	12	−	−	PROPN
ejpam-3780	434	13	1)(rk	1)(rk	NUM
ejpam-3780	434	14	)	)	PUNCT
ejpam-3780	435	1	+	+	CCONJ
ejpam-3780	435	2	m	m	VERB
ejpam-3780	435	3	>	>	X
ejpam-3780	435	4	0	0	PUNCT
ejpam-3780	436	1	and	and	CCONJ
ejpam-3780	436	2	(	(	PUNCT
ejpam-3780	436	3	n	n	CCONJ
ejpam-3780	436	4	−	−	PROPN
ejpam-3780	436	5	1)(rk	1)(rk	NUM
ejpam-3780	436	6	)	)	PUNCT
ejpam-3780	437	1	+	+	CCONJ
ejpam-3780	437	2	m	m	VERB
ejpam-3780	437	3	<	<	X
ejpam-3780	437	4	(	(	PUNCT
ejpam-3780	437	5	n	n	CCONJ
ejpam-3780	437	6	−	−	PROPN
ejpam-3780	437	7	1)(rk	1)(rk	NUM
ejpam-3780	437	8	)	)	PUNCT
ejpam-3780	437	9	+	+	CCONJ
ejpam-3780	437	10	(	(	PUNCT
ejpam-3780	437	11	n	n	ADV
ejpam-3780	437	12	−	−	PROPN
ejpam-3780	437	13	1)(rk	1)(rk	NUM
ejpam-3780	438	1	+	+	CCONJ
ejpam-3780	438	2	1	1	NUM
ejpam-3780	438	3	)	)	PUNCT
ejpam-3780	438	4	=	=	SYM
ejpam-3780	438	5	(	(	PUNCT
ejpam-3780	438	6	n	n	CCONJ
ejpam-3780	438	7	−	−	PROPN
ejpam-3780	438	8	1)(2rk	1)(2rk	NUM
ejpam-3780	438	9	+	+	NOUN
ejpam-3780	438	10	1	1	NUM
ejpam-3780	438	11	)	)	PUNCT
ejpam-3780	438	12	.	.	PUNCT
ejpam-3780	439	1	then	then	ADV
ejpam-3780	439	2	a0	a0	PROPN
ejpam-3780	439	3	/∈	/∈	PROPN
ejpam-3780	439	4	k.	k.	PROPN
ejpam-3780	440	1	therefore	therefore	ADV
ejpam-3780	440	2	,	,	PUNCT
ejpam-3780	440	3	f	f	PROPN
ejpam-3780	440	4	is	be	AUX
ejpam-3780	440	5	an	an	DET
ejpam-3780	440	6	efficient	efficient	ADJ
ejpam-3780	440	7	zero	zero	NUM
ejpam-3780	440	8	ring	ring	NOUN
ejpam-3780	440	9	labeling	labeling	NOUN
ejpam-3780	440	10	of	of	ADP
ejpam-3780	440	11	g.	g.	PROPN
ejpam-3780	440	12	example	example	PROPN
ejpam-3780	440	13	10	10	NUM
ejpam-3780	440	14	.	.	PUNCT
ejpam-3780	441	1	figure	figure	VERB
ejpam-3780	441	2	11	11	NUM
ejpam-3780	441	3	shows	show	VERB
ejpam-3780	441	4	an	an	DET
ejpam-3780	441	5	efficient	efficient	ADJ
ejpam-3780	441	6	zero	zero	NUM
ejpam-3780	441	7	ring	ring	NOUN
ejpam-3780	441	8	labeling	labeling	NOUN
ejpam-3780	441	9	of	of	ADP
ejpam-3780	441	10	a	a	DET
ejpam-3780	441	11	spider	spider	NOUN
ejpam-3780	441	12	with	with	ADP
ejpam-3780	441	13	4	4	NUM
ejpam-3780	441	14	legs	leg	NOUN
ejpam-3780	441	15	using	use	VERB
ejpam-3780	441	16	m0	m0	PROPN
ejpam-3780	441	17	2	2	NUM
ejpam-3780	441	18	(	(	PUNCT
ejpam-3780	441	19	z33	z33	NOUN
ejpam-3780	441	20	)	)	PUNCT
ejpam-3780	441	21	.	.	PUNCT
ejpam-3780	442	1	in	in	ADP
ejpam-3780	442	2	this	this	DET
ejpam-3780	442	3	labeling	labeling	NOUN
ejpam-3780	442	4	,	,	PUNCT
ejpam-3780	442	5	the	the	DET
ejpam-3780	442	6	set	set	NOUN
ejpam-3780	442	7	of	of	ADP
ejpam-3780	442	8	sums	sum	NOUN
ejpam-3780	442	9	is	be	AUX
ejpam-3780	442	10	k	k	NOUN
ejpam-3780	442	11	=	=	PUNCT
ejpam-3780	442	12	{	{	PUNCT
ejpam-3780	442	13	a15	a15	PROPN
ejpam-3780	442	14	,	,	PUNCT
ejpam-3780	442	15	a16	a16	PROPN
ejpam-3780	442	16	,	,	PUNCT
ejpam-3780	442	17	a17	a17	PROPN
ejpam-3780	442	18	,	,	PUNCT
ejpam-3780	442	19	a18	a18	PROPN
ejpam-3780	442	20	}	}	PUNCT
ejpam-3780	442	21	and	and	CCONJ
ejpam-3780	442	22	thus	thus	ADV
ejpam-3780	442	23	|k|	|k|	NOUN
ejpam-3780	442	24	=	=	SYM
ejpam-3780	442	25	4	4	X
ejpam-3780	442	26	.	.	X
ejpam-3780	442	27	a0	a0	PROPN
ejpam-3780	442	28	a15	a15	PROPN
ejpam-3780	442	29	a3	a3	PROPN
ejpam-3780	442	30	a12	a12	PROPN
ejpam-3780	442	31	a6	a6	PROPN
ejpam-3780	442	32	a9	a9	PROPN
ejpam-3780	442	33	a0	a0	PROPN
ejpam-3780	442	34	a16	a16	PROPN
ejpam-3780	442	35	a2	a2	PROPN
ejpam-3780	442	36	a13	a13	PROPN
ejpam-3780	442	37	a5	a5	PROPN
ejpam-3780	442	38	a0	a0	PROPN
ejpam-3780	442	39	a17	a17	PROPN
ejpam-3780	442	40	a1	a1	PROPN
ejpam-3780	442	41	a14	a14	PROPN
ejpam-3780	442	42	a0	a0	PROPN
ejpam-3780	442	43	a18	a18	PROPN
ejpam-3780	442	44	a30	a30	PROPN
ejpam-3780	442	45	a21	a21	PROPN
ejpam-3780	442	46	a27	a27	PROPN
ejpam-3780	442	47	a24	a24	PROPN
ejpam-3780	442	48	figure	figure	NOUN
ejpam-3780	442	49	11	11	NUM
ejpam-3780	442	50	:	:	PUNCT
ejpam-3780	442	51	efficient	efficient	ADJ
ejpam-3780	442	52	zero	zero	NUM
ejpam-3780	442	53	ring	ring	NOUN
ejpam-3780	442	54	labeling	labeling	NOUN
ejpam-3780	442	55	of	of	ADP
ejpam-3780	442	56	a	a	DET
ejpam-3780	442	57	spider	spider	NOUN
ejpam-3780	442	58	using	use	VERB
ejpam-3780	442	59	m0	m0	PROPN
ejpam-3780	442	60	2	2	NUM
ejpam-3780	442	61	(	(	PUNCT
ejpam-3780	442	62	z33	z33	NOUN
ejpam-3780	442	63	)	)	PUNCT
ejpam-3780	442	64	lemma	lemma	PROPN
ejpam-3780	443	1	2	2	NUM
ejpam-3780	443	2	.	.	PUNCT
ejpam-3780	443	3	let	let	VERB
ejpam-3780	443	4	g	g	PRON
ejpam-3780	443	5	be	be	AUX
ejpam-3780	443	6	a	a	DET
ejpam-3780	443	7	lobster	lobster	NOUN
ejpam-3780	443	8	with	with	ADP
ejpam-3780	443	9	at	at	ADV
ejpam-3780	443	10	least	least	ADV
ejpam-3780	443	11	three	three	NUM
ejpam-3780	443	12	vertices	vertex	NOUN
ejpam-3780	443	13	.	.	PUNCT
ejpam-3780	444	1	then	then	ADV
ejpam-3780	444	2	g	g	PROPN
ejpam-3780	444	3	is	be	AUX
ejpam-3780	444	4	a	a	DET
ejpam-3780	444	5	lobster	lobster	NOUN
ejpam-3780	444	6	with	with	ADP
ejpam-3780	444	7	respect	respect	NOUN
ejpam-3780	444	8	to	to	ADP
ejpam-3780	444	9	some	some	DET
ejpam-3780	444	10	central	central	ADJ
ejpam-3780	444	11	path	path	NOUN
ejpam-3780	445	1	[	[	X
ejpam-3780	445	2	v1	v1	NOUN
ejpam-3780	445	3	,	,	PUNCT
ejpam-3780	445	4	v2	v2	NOUN
ejpam-3780	445	5	,	,	PUNCT
ejpam-3780	445	6	.	.	PUNCT
ejpam-3780	445	7	.	.	PUNCT
ejpam-3780	445	8	.	.	PUNCT
ejpam-3780	446	1	,	,	PUNCT
ejpam-3780	446	2	vm	vm	PROPN
ejpam-3780	446	3	]	]	X
ejpam-3780	446	4	,	,	PUNCT
ejpam-3780	446	5	where	where	SCONJ
ejpam-3780	446	6	m	m	PROPN
ejpam-3780	446	7	≥	≥	VERB
ejpam-3780	446	8	3	3	NUM
ejpam-3780	446	9	and	and	CCONJ
ejpam-3780	446	10	v1	v1	VERB
ejpam-3780	446	11	and	and	CCONJ
ejpam-3780	446	12	vm	vm	PROPN
ejpam-3780	446	13	have	have	VERB
ejpam-3780	446	14	no	no	DET
ejpam-3780	446	15	elbow	elbow	NOUN
ejpam-3780	446	16	.	.	PUNCT
ejpam-3780	447	1	proof	proof	NOUN
ejpam-3780	447	2	.	.	PUNCT
ejpam-3780	448	1	let	let	VERB
ejpam-3780	448	2	g	g	PRON
ejpam-3780	448	3	be	be	AUX
ejpam-3780	448	4	a	a	DET
ejpam-3780	448	5	lobster	lobster	NOUN
ejpam-3780	448	6	with	with	ADP
ejpam-3780	448	7	at	at	ADV
ejpam-3780	448	8	least	least	ADV
ejpam-3780	448	9	three	three	NUM
ejpam-3780	448	10	vertices	vertex	NOUN
ejpam-3780	448	11	,	,	PUNCT
ejpam-3780	448	12	and	and	CCONJ
ejpam-3780	448	13	let	let	VERB
ejpam-3780	448	14	[	[	X
ejpam-3780	448	15	w1	w1	NOUN
ejpam-3780	448	16	,	,	PUNCT
ejpam-3780	448	17	w2	w2	NOUN
ejpam-3780	448	18	,	,	PUNCT
ejpam-3780	448	19	.	.	PUNCT
ejpam-3780	448	20	.	.	PUNCT
ejpam-3780	449	1	.	.	PUNCT
ejpam-3780	450	1	,	,	PUNCT
ejpam-3780	450	2	wn	wn	PROPN
ejpam-3780	450	3	]	]	X
ejpam-3780	450	4	denote	denote	VERB
ejpam-3780	450	5	its	its	PRON
ejpam-3780	450	6	central	central	ADJ
ejpam-3780	450	7	path	path	NOUN
ejpam-3780	450	8	.	.	PUNCT
ejpam-3780	451	1	suppose	suppose	VERB
ejpam-3780	451	2	each	each	DET
ejpam-3780	451	3	vertex	vertex	NOUN
ejpam-3780	451	4	wi	wi	PROPN
ejpam-3780	451	5	has	have	VERB
ejpam-3780	451	6	ri	ri	PROPN
ejpam-3780	451	7	elbows	elbow	NOUN
ejpam-3780	451	8	,	,	PUNCT
ejpam-3780	451	9	and	and	CCONJ
ejpam-3780	451	10	let	let	VERB
ejpam-3780	451	11	wi	wi	PROPN
ejpam-3780	451	12	,	,	PUNCT
ejpam-3780	451	13	j	j	PROPN
ejpam-3780	451	14	,	,	PUNCT
ejpam-3780	451	15	where	where	SCONJ
ejpam-3780	451	16	j	j	PROPN
ejpam-3780	451	17	=	=	SYM
ejpam-3780	451	18	1	1	NUM
ejpam-3780	451	19	,	,	PUNCT
ejpam-3780	451	20	2	2	NUM
ejpam-3780	451	21	,	,	PUNCT
ejpam-3780	451	22	.	.	PUNCT
ejpam-3780	451	23	.	.	PUNCT
ejpam-3780	452	1	.	.	PUNCT
ejpam-3780	453	1	,	,	PUNCT
ejpam-3780	453	2	ri	ri	PROPN
ejpam-3780	453	3	,	,	PUNCT
ejpam-3780	453	4	denote	denote	VERB
ejpam-3780	453	5	the	the	DET
ejpam-3780	453	6	elbows	elbow	NOUN
ejpam-3780	453	7	of	of	ADP
ejpam-3780	453	8	wi	wi	PROPN
ejpam-3780	453	9	.	.	PUNCT
ejpam-3780	454	1	suppose	suppose	VERB
ejpam-3780	454	2	each	each	DET
ejpam-3780	454	3	elbow	elbow	NOUN
ejpam-3780	454	4	wi	wi	PROPN
ejpam-3780	454	5	,	,	PUNCT
ejpam-3780	454	6	j	j	PROPN
ejpam-3780	454	7	has	have	VERB
ejpam-3780	454	8	si	si	PROPN
ejpam-3780	454	9	,	,	PUNCT
ejpam-3780	454	10	j	j	PROPN
ejpam-3780	454	11	hanging	hanging	NOUN
ejpam-3780	454	12	leaves	leave	VERB
ejpam-3780	454	13	,	,	PUNCT
ejpam-3780	454	14	and	and	CCONJ
ejpam-3780	454	15	let	let	VERB
ejpam-3780	454	16	wi	wi	PROPN
ejpam-3780	454	17	,	,	PUNCT
ejpam-3780	454	18	j	j	PROPN
ejpam-3780	454	19	,	,	PUNCT
ejpam-3780	454	20	k	k	PROPN
ejpam-3780	454	21	,	,	PUNCT
ejpam-3780	454	22	where	where	SCONJ
ejpam-3780	454	23	k	k	PROPN
ejpam-3780	454	24	=	=	SYM
ejpam-3780	454	25	1	1	NUM
ejpam-3780	454	26	,	,	PUNCT
ejpam-3780	454	27	2	2	NUM
ejpam-3780	454	28	,	,	PUNCT
ejpam-3780	454	29	.	.	PUNCT
ejpam-3780	454	30	.	.	PUNCT
ejpam-3780	455	1	.	.	PUNCT
ejpam-3780	456	1	,	,	PUNCT
ejpam-3780	456	2	si	si	X
ejpam-3780	456	3	,	,	PUNCT
ejpam-3780	456	4	j	j	PROPN
ejpam-3780	456	5	,	,	PUNCT
ejpam-3780	456	6	denote	denote	VERB
ejpam-3780	456	7	the	the	DET
ejpam-3780	456	8	hanging	hang	VERB
ejpam-3780	456	9	leaves	leave	NOUN
ejpam-3780	456	10	of	of	ADP
ejpam-3780	456	11	wi	wi	PROPN
ejpam-3780	456	12	,	,	PUNCT
ejpam-3780	456	13	j	j	PROPN
ejpam-3780	456	14	.	.	PUNCT
ejpam-3780	457	1	consider	consider	VERB
ejpam-3780	457	2	w1	w1	NOUN
ejpam-3780	457	3	,	,	PUNCT
ejpam-3780	457	4	w2	w2	NOUN
ejpam-3780	457	5	,	,	PUNCT
ejpam-3780	457	6	.	.	PUNCT
ejpam-3780	457	7	.	.	PUNCT
ejpam-3780	458	1	.	.	PUNCT
ejpam-3780	459	1	,	,	PUNCT
ejpam-3780	459	2	wn	wn	INTJ
ejpam-3780	459	3	as	as	ADP
ejpam-3780	459	4	vertices	vertex	NOUN
ejpam-3780	459	5	in	in	ADP
ejpam-3780	459	6	a	a	DET
ejpam-3780	459	7	central	central	ADJ
ejpam-3780	459	8	path	path	NOUN
ejpam-3780	459	9	p	p	NOUN
ejpam-3780	459	10	,	,	PUNCT
ejpam-3780	459	11	along	along	ADP
ejpam-3780	459	12	with	with	ADP
ejpam-3780	459	13	w1,1	w1,1	PROPN
ejpam-3780	459	14	if	if	SCONJ
ejpam-3780	459	15	r1	r1	PROPN
ejpam-3780	459	16	≥	≥	NUM
ejpam-3780	459	17	1	1	NUM
ejpam-3780	459	18	,	,	PUNCT
ejpam-3780	459	19	w1,1,1	w1,1,1	AUX
ejpam-3780	459	20	if	if	SCONJ
ejpam-3780	459	21	s1,1	s1,1	PROPN
ejpam-3780	459	22	≥	≥	NOUN
ejpam-3780	459	23	1	1	NUM
ejpam-3780	459	24	,	,	PUNCT
ejpam-3780	459	25	wn,1	wn,1	PROPN
ejpam-3780	459	26	if	if	SCONJ
ejpam-3780	459	27	rn	rn	PROPN
ejpam-3780	459	28	≥	≥	PROPN
ejpam-3780	459	29	1	1	NUM
ejpam-3780	459	30	,	,	PUNCT
ejpam-3780	459	31	and	and	CCONJ
ejpam-3780	459	32	wn,1,1	wn,1,1	VERB
ejpam-3780	459	33	if	if	SCONJ
ejpam-3780	459	34	sn,1	sn,1	PROPN
ejpam-3780	459	35	≥	≥	NOUN
ejpam-3780	459	36	1	1	NUM
ejpam-3780	459	37	.	.	PUNCT
ejpam-3780	460	1	thus	thus	ADV
ejpam-3780	460	2	,	,	PUNCT
ejpam-3780	460	3	p	p	PROPN
ejpam-3780	460	4	is	be	AUX
ejpam-3780	460	5	one	one	NUM
ejpam-3780	460	6	of	of	ADP
ejpam-3780	460	7	the	the	DET
ejpam-3780	460	8	following	following	NOUN
ejpam-3780	460	9	:	:	PUNCT
ejpam-3780	461	1	•	•	NUM
ejpam-3780	462	1	[	[	X
ejpam-3780	462	2	w1	w1	NOUN
ejpam-3780	462	3	,	,	PUNCT
ejpam-3780	462	4	w2	w2	NOUN
ejpam-3780	462	5	,	,	PUNCT
ejpam-3780	462	6	.	.	PUNCT
ejpam-3780	462	7	.	.	PUNCT
ejpam-3780	462	8	.	.	PUNCT
ejpam-3780	463	1	,	,	PUNCT
ejpam-3780	463	2	wn	wn	X
ejpam-3780	463	3	]	]	X
ejpam-3780	463	4	•	•	NOUN
ejpam-3780	463	5	[	[	X
ejpam-3780	463	6	w1,1	w1,1	NOUN
ejpam-3780	463	7	,	,	PUNCT
ejpam-3780	463	8	w1	w1	NOUN
ejpam-3780	463	9	,	,	PUNCT
ejpam-3780	463	10	w2	w2	NOUN
ejpam-3780	463	11	,	,	PUNCT
ejpam-3780	463	12	.	.	PUNCT
ejpam-3780	463	13	.	.	PUNCT
ejpam-3780	463	14	.	.	PUNCT
ejpam-3780	464	1	,	,	PUNCT
ejpam-3780	464	2	wn	wn	X
ejpam-3780	464	3	]	]	X
ejpam-3780	464	4	•	•	ADP
ejpam-3780	465	1	[	[	X
ejpam-3780	465	2	w1,1,1	w1,1,1	NOUN
ejpam-3780	465	3	,	,	PUNCT
ejpam-3780	465	4	w1,1	w1,1	NOUN
ejpam-3780	465	5	,	,	PUNCT
ejpam-3780	465	6	w1	w1	NOUN
ejpam-3780	465	7	,	,	PUNCT
ejpam-3780	465	8	w2	w2	NOUN
ejpam-3780	465	9	,	,	PUNCT
ejpam-3780	465	10	.	.	PUNCT
ejpam-3780	465	11	.	.	PUNCT
ejpam-3780	466	1	.	.	PUNCT
ejpam-3780	467	1	,	,	PUNCT
ejpam-3780	467	2	wn	wn	PROPN
ejpam-3780	467	3	]	]	X
ejpam-3780	467	4	•	•	NUM
ejpam-3780	468	1	[	[	X
ejpam-3780	468	2	w1	w1	NOUN
ejpam-3780	468	3	,	,	PUNCT
ejpam-3780	468	4	w2	w2	NOUN
ejpam-3780	468	5	,	,	PUNCT
ejpam-3780	468	6	.	.	PUNCT
ejpam-3780	468	7	.	.	PUNCT
ejpam-3780	468	8	.	.	PUNCT
ejpam-3780	469	1	,	,	PUNCT
ejpam-3780	469	2	wn	wn	PROPN
ejpam-3780	469	3	,	,	PUNCT
ejpam-3780	469	4	wn,1	wn,1	PROPN
ejpam-3780	469	5	]	]	PUNCT
ejpam-3780	469	6	•	•	PUNCT
ejpam-3780	470	1	[	[	X
ejpam-3780	470	2	w1,1	w1,1	NOUN
ejpam-3780	470	3	,	,	PUNCT
ejpam-3780	470	4	w1	w1	NOUN
ejpam-3780	470	5	,	,	PUNCT
ejpam-3780	470	6	w2	w2	NOUN
ejpam-3780	470	7	,	,	PUNCT
ejpam-3780	470	8	.	.	PUNCT
ejpam-3780	470	9	.	.	PUNCT
ejpam-3780	471	1	.	.	PUNCT
ejpam-3780	472	1	,	,	PUNCT
ejpam-3780	472	2	wn	wn	PROPN
ejpam-3780	472	3	,	,	PUNCT
ejpam-3780	472	4	wn,1	wn,1	PROPN
ejpam-3780	472	5	]	]	PUNCT
ejpam-3780	472	6	•	•	ADP
ejpam-3780	473	1	[	[	X
ejpam-3780	473	2	w1,1,1	w1,1,1	NOUN
ejpam-3780	473	3	,	,	PUNCT
ejpam-3780	473	4	w1,1	w1,1	NOUN
ejpam-3780	473	5	,	,	PUNCT
ejpam-3780	473	6	w1	w1	NOUN
ejpam-3780	473	7	,	,	PUNCT
ejpam-3780	473	8	w2	w2	NOUN
ejpam-3780	473	9	,	,	PUNCT
ejpam-3780	473	10	.	.	PUNCT
ejpam-3780	473	11	.	.	PUNCT
ejpam-3780	474	1	.	.	PUNCT
ejpam-3780	475	1	,	,	PUNCT
ejpam-3780	475	2	wn	wn	PROPN
ejpam-3780	475	3	,	,	PUNCT
ejpam-3780	475	4	wn,1	wn,1	PROPN
ejpam-3780	475	5	]	]	PUNCT
ejpam-3780	475	6	•	•	NUM
ejpam-3780	476	1	[	[	X
ejpam-3780	476	2	w1	w1	NOUN
ejpam-3780	476	3	,	,	PUNCT
ejpam-3780	476	4	w2	w2	NOUN
ejpam-3780	476	5	,	,	PUNCT
ejpam-3780	476	6	.	.	PUNCT
ejpam-3780	476	7	.	.	PUNCT
ejpam-3780	476	8	.	.	PUNCT
ejpam-3780	477	1	,	,	PUNCT
ejpam-3780	477	2	wn	wn	PROPN
ejpam-3780	477	3	,	,	PUNCT
ejpam-3780	477	4	wn,1	wn,1	PROPN
ejpam-3780	477	5	,	,	PUNCT
ejpam-3780	477	6	wn,1,1	wn,1,1	NOUN
ejpam-3780	477	7	,	,	PUNCT
ejpam-3780	477	8	]	]	PUNCT
ejpam-3780	477	9	d.	d.	PROPN
ejpam-3780	477	10	chua	chua	PROPN
ejpam-3780	477	11	,	,	PUNCT
ejpam-3780	477	12	f.	f.	PROPN
ejpam-3780	477	13	campeña	campeña	PROPN
ejpam-3780	477	14	,	,	PUNCT
ejpam-3780	477	15	f.	f.	PROPN
ejpam-3780	477	16	franco	franco	PROPN
ejpam-3780	477	17	/	/	SYM
ejpam-3780	477	18	eur	eur	PROPN
ejpam-3780	477	19	.	.	PUNCT
ejpam-3780	478	1	j.	j.	PROPN
ejpam-3780	478	2	pure	pure	PROPN
ejpam-3780	478	3	appl	appl	PROPN
ejpam-3780	478	4	.	.	PROPN
ejpam-3780	478	5	math	math	PROPN
ejpam-3780	478	6	,	,	PUNCT
ejpam-3780	478	7	13	13	NUM
ejpam-3780	478	8	(	(	PUNCT
ejpam-3780	478	9	3	3	NUM
ejpam-3780	478	10	)	)	PUNCT
ejpam-3780	478	11	(	(	PUNCT
ejpam-3780	478	12	2020	2020	NUM
ejpam-3780	478	13	)	)	PUNCT
ejpam-3780	478	14	,	,	PUNCT
ejpam-3780	478	15	674	674	NUM
ejpam-3780	478	16	-	-	SYM
ejpam-3780	478	17	696	696	NUM
ejpam-3780	478	18	687	687	NUM
ejpam-3780	478	19	•	•	NOUN
ejpam-3780	478	20	[	[	X
ejpam-3780	478	21	w1,1	w1,1	NOUN
ejpam-3780	478	22	,	,	PUNCT
ejpam-3780	478	23	w1	w1	NOUN
ejpam-3780	478	24	,	,	PUNCT
ejpam-3780	478	25	w2	w2	NOUN
ejpam-3780	478	26	,	,	PUNCT
ejpam-3780	478	27	.	.	PUNCT
ejpam-3780	478	28	.	.	PUNCT
ejpam-3780	479	1	.	.	PUNCT
ejpam-3780	480	1	,	,	PUNCT
ejpam-3780	480	2	wn	wn	PROPN
ejpam-3780	480	3	,	,	PUNCT
ejpam-3780	480	4	wn,1	wn,1	PROPN
ejpam-3780	480	5	,	,	PUNCT
ejpam-3780	480	6	wn,1,1	wn,1,1	NOUN
ejpam-3780	480	7	]	]	X
ejpam-3780	480	8	•	•	ADP
ejpam-3780	481	1	[	[	X
ejpam-3780	481	2	w1,1,1	w1,1,1	NOUN
ejpam-3780	481	3	,	,	PUNCT
ejpam-3780	481	4	w1,1	w1,1	NOUN
ejpam-3780	481	5	,	,	PUNCT
ejpam-3780	481	6	w1	w1	NOUN
ejpam-3780	481	7	,	,	PUNCT
ejpam-3780	481	8	w2	w2	NOUN
ejpam-3780	481	9	,	,	PUNCT
ejpam-3780	481	10	.	.	PUNCT
ejpam-3780	481	11	.	.	PUNCT
ejpam-3780	482	1	.	.	PUNCT
ejpam-3780	483	1	,	,	PUNCT
ejpam-3780	483	2	wn	wn	PROPN
ejpam-3780	483	3	,	,	PUNCT
ejpam-3780	483	4	wn,1	wn,1	PROPN
ejpam-3780	483	5	,	,	PUNCT
ejpam-3780	483	6	wn,1,1	wn,1,1	NOUN
ejpam-3780	483	7	]	]	PUNCT
ejpam-3780	483	8	in	in	ADP
ejpam-3780	483	9	any	any	DET
ejpam-3780	483	10	case	case	NOUN
ejpam-3780	483	11	,	,	PUNCT
ejpam-3780	483	12	the	the	DET
ejpam-3780	483	13	number	number	NOUN
ejpam-3780	483	14	of	of	ADP
ejpam-3780	483	15	vertices	vertex	NOUN
ejpam-3780	483	16	in	in	ADP
ejpam-3780	483	17	p	p	PROPN
ejpam-3780	483	18	is	be	AUX
ejpam-3780	483	19	at	at	ADV
ejpam-3780	483	20	least	least	ADJ
ejpam-3780	483	21	three	three	NUM
ejpam-3780	483	22	,	,	PUNCT
ejpam-3780	483	23	and	and	CCONJ
ejpam-3780	483	24	its	its	PRON
ejpam-3780	483	25	endvertices	endvertice	NOUN
ejpam-3780	483	26	have	have	VERB
ejpam-3780	483	27	no	no	DET
ejpam-3780	483	28	elbow	elbow	NOUN
ejpam-3780	483	29	.	.	PUNCT
ejpam-3780	484	1	clearly	clearly	ADV
ejpam-3780	484	2	,	,	PUNCT
ejpam-3780	484	3	each	each	DET
ejpam-3780	484	4	vertex	vertex	NOUN
ejpam-3780	484	5	in	in	ADP
ejpam-3780	484	6	g	g	PROPN
ejpam-3780	484	7	is	be	AUX
ejpam-3780	484	8	within	within	ADP
ejpam-3780	484	9	distance	distance	NOUN
ejpam-3780	484	10	two	two	NUM
ejpam-3780	484	11	from	from	ADP
ejpam-3780	484	12	p	p	NOUN
ejpam-3780	484	13	and	and	CCONJ
ejpam-3780	484	14	thus	thus	ADV
ejpam-3780	484	15	g	g	PROPN
ejpam-3780	484	16	is	be	AUX
ejpam-3780	484	17	a	a	DET
ejpam-3780	484	18	lobster	lobster	NOUN
ejpam-3780	484	19	with	with	ADP
ejpam-3780	484	20	respect	respect	NOUN
ejpam-3780	484	21	to	to	ADP
ejpam-3780	484	22	central	central	ADJ
ejpam-3780	484	23	path	path	NOUN
ejpam-3780	484	24	p	p	PROPN
ejpam-3780	484	25	.	.	PUNCT
ejpam-3780	485	1	theorem	theorem	ADJ
ejpam-3780	485	2	8	8	NUM
ejpam-3780	485	3	.	.	PUNCT
ejpam-3780	486	1	let	let	VERB
ejpam-3780	486	2	g	g	PRON
ejpam-3780	486	3	be	be	AUX
ejpam-3780	486	4	a	a	DET
ejpam-3780	486	5	lobster	lobster	NOUN
ejpam-3780	486	6	in	in	ADP
ejpam-3780	486	7	which	which	PRON
ejpam-3780	486	8	each	each	DET
ejpam-3780	486	9	elbow	elbow	NOUN
ejpam-3780	486	10	has	have	VERB
ejpam-3780	486	11	at	at	ADP
ejpam-3780	486	12	most	most	ADV
ejpam-3780	486	13	one	one	NUM
ejpam-3780	486	14	hanging	hang	VERB
ejpam-3780	486	15	leaf	leaf	NOUN
ejpam-3780	486	16	.	.	PUNCT
ejpam-3780	487	1	then	then	ADV
ejpam-3780	487	2	g	g	PROPN
ejpam-3780	487	3	has	have	VERB
ejpam-3780	487	4	an	an	DET
ejpam-3780	487	5	efficient	efficient	ADJ
ejpam-3780	487	6	zero	zero	NUM
ejpam-3780	487	7	ring	ring	NOUN
ejpam-3780	487	8	labeling	labeling	NOUN
ejpam-3780	487	9	.	.	PUNCT
ejpam-3780	488	1	proof	proof	NOUN
ejpam-3780	488	2	.	.	PUNCT
ejpam-3780	489	1	let	let	VERB
ejpam-3780	489	2	g	g	PRON
ejpam-3780	489	3	be	be	AUX
ejpam-3780	489	4	a	a	DET
ejpam-3780	489	5	lobster	lobster	NOUN
ejpam-3780	489	6	with	with	ADP
ejpam-3780	489	7	at	at	ADV
ejpam-3780	489	8	least	least	ADV
ejpam-3780	489	9	three	three	NUM
ejpam-3780	489	10	vertices	vertex	NOUN
ejpam-3780	489	11	in	in	ADP
ejpam-3780	489	12	which	which	PRON
ejpam-3780	489	13	each	each	DET
ejpam-3780	489	14	elbow	elbow	NOUN
ejpam-3780	489	15	has	have	VERB
ejpam-3780	489	16	at	at	ADP
ejpam-3780	489	17	most	most	ADV
ejpam-3780	489	18	one	one	NUM
ejpam-3780	489	19	hanging	hang	VERB
ejpam-3780	489	20	leaf	leaf	NOUN
ejpam-3780	489	21	.	.	PUNCT
ejpam-3780	490	1	by	by	ADP
ejpam-3780	490	2	lemma	lemma	PROPN
ejpam-3780	490	3	2	2	NUM
ejpam-3780	490	4	,	,	PUNCT
ejpam-3780	490	5	g	g	PROPN
ejpam-3780	490	6	is	be	AUX
ejpam-3780	490	7	a	a	DET
ejpam-3780	490	8	lobster	lobster	NOUN
ejpam-3780	490	9	with	with	ADP
ejpam-3780	490	10	respect	respect	NOUN
ejpam-3780	490	11	to	to	ADP
ejpam-3780	490	12	some	some	DET
ejpam-3780	490	13	central	central	ADJ
ejpam-3780	490	14	path	path	NOUN
ejpam-3780	491	1	[	[	X
ejpam-3780	491	2	w1	w1	NOUN
ejpam-3780	491	3	,	,	PUNCT
ejpam-3780	491	4	w2	w2	NOUN
ejpam-3780	491	5	,	,	PUNCT
ejpam-3780	491	6	.	.	PUNCT
ejpam-3780	491	7	.	.	PUNCT
ejpam-3780	492	1	.	.	PUNCT
ejpam-3780	493	1	,	,	PUNCT
ejpam-3780	493	2	wn	wn	PROPN
ejpam-3780	493	3	]	]	X
ejpam-3780	493	4	,	,	PUNCT
ejpam-3780	493	5	where	where	SCONJ
ejpam-3780	493	6	n	n	PRON
ejpam-3780	493	7	≥	≥	X
ejpam-3780	493	8	3	3	NUM
ejpam-3780	493	9	and	and	CCONJ
ejpam-3780	493	10	w1	w1	PROPN
ejpam-3780	493	11	and	and	CCONJ
ejpam-3780	493	12	wn	wn	PROPN
ejpam-3780	493	13	have	have	VERB
ejpam-3780	493	14	no	no	DET
ejpam-3780	493	15	elbow	elbow	NOUN
ejpam-3780	493	16	.	.	PUNCT
ejpam-3780	494	1	clearly	clearly	ADV
ejpam-3780	494	2	,	,	PUNCT
ejpam-3780	494	3	each	each	DET
ejpam-3780	494	4	elbow	elbow	NOUN
ejpam-3780	494	5	has	have	VERB
ejpam-3780	494	6	at	at	ADP
ejpam-3780	494	7	most	most	ADV
ejpam-3780	494	8	one	one	NUM
ejpam-3780	494	9	hanging	hang	VERB
ejpam-3780	494	10	leaf	leaf	NOUN
ejpam-3780	494	11	with	with	ADP
ejpam-3780	494	12	respect	respect	NOUN
ejpam-3780	494	13	to	to	ADP
ejpam-3780	494	14	this	this	DET
ejpam-3780	494	15	central	central	ADJ
ejpam-3780	494	16	path	path	NOUN
ejpam-3780	494	17	.	.	PUNCT
ejpam-3780	495	1	suppose	suppose	VERB
ejpam-3780	495	2	each	each	DET
ejpam-3780	495	3	vertex	vertex	NOUN
ejpam-3780	495	4	wi	wi	PROPN
ejpam-3780	495	5	has	have	VERB
ejpam-3780	495	6	ri	ri	PROPN
ejpam-3780	495	7	elbows	elbow	NOUN
ejpam-3780	495	8	,	,	PUNCT
ejpam-3780	495	9	and	and	CCONJ
ejpam-3780	495	10	let	let	VERB
ejpam-3780	495	11	wi	wi	PROPN
ejpam-3780	495	12	,	,	PUNCT
ejpam-3780	495	13	j	j	PROPN
ejpam-3780	495	14	,	,	PUNCT
ejpam-3780	495	15	where	where	SCONJ
ejpam-3780	495	16	j	j	PROPN
ejpam-3780	495	17	=	=	SYM
ejpam-3780	495	18	1	1	NUM
ejpam-3780	495	19	,	,	PUNCT
ejpam-3780	495	20	2	2	NUM
ejpam-3780	495	21	,	,	PUNCT
ejpam-3780	495	22	.	.	PUNCT
ejpam-3780	495	23	.	.	PUNCT
ejpam-3780	496	1	.	.	PUNCT
ejpam-3780	497	1	,	,	PUNCT
ejpam-3780	497	2	ri	ri	PROPN
ejpam-3780	497	3	,	,	PUNCT
ejpam-3780	497	4	denote	denote	VERB
ejpam-3780	497	5	the	the	DET
ejpam-3780	497	6	elbows	elbow	NOUN
ejpam-3780	497	7	of	of	ADP
ejpam-3780	497	8	wi	wi	PROPN
ejpam-3780	497	9	.	.	PUNCT
ejpam-3780	498	1	if	if	SCONJ
ejpam-3780	498	2	it	it	PRON
ejpam-3780	498	3	exists	exist	VERB
ejpam-3780	498	4	,	,	PUNCT
ejpam-3780	498	5	let	let	VERB
ejpam-3780	498	6	wi	wi	PROPN
ejpam-3780	498	7	,	,	PUNCT
ejpam-3780	498	8	j,1	j,1	NOUN
ejpam-3780	498	9	denote	denote	VERB
ejpam-3780	498	10	the	the	DET
ejpam-3780	498	11	hanging	hang	VERB
ejpam-3780	498	12	leaf	leaf	NOUN
ejpam-3780	498	13	of	of	ADP
ejpam-3780	498	14	wi	wi	PROPN
ejpam-3780	498	15	,	,	PUNCT
ejpam-3780	498	16	j	j	PROPN
ejpam-3780	498	17	.	.	PUNCT
ejpam-3780	499	1	suppose	suppose	VERB
ejpam-3780	499	2	rk	rk	PRON
ejpam-3780	499	3	is	be	AUX
ejpam-3780	499	4	the	the	DET
ejpam-3780	499	5	maximum	maximum	ADJ
ejpam-3780	499	6	number	number	NOUN
ejpam-3780	499	7	of	of	ADP
ejpam-3780	499	8	elbows	elbow	NOUN
ejpam-3780	499	9	of	of	ADP
ejpam-3780	499	10	a	a	DET
ejpam-3780	499	11	vertex	vertex	NOUN
ejpam-3780	499	12	in	in	ADP
ejpam-3780	499	13	the	the	DET
ejpam-3780	499	14	central	central	ADJ
ejpam-3780	499	15	path	path	NOUN
ejpam-3780	499	16	;	;	PUNCT
ejpam-3780	499	17	that	that	PRON
ejpam-3780	499	18	is	is	ADV
ejpam-3780	499	19	,	,	PUNCT
ejpam-3780	499	20	rk	rk	PROPN
ejpam-3780	499	21	≥	≥	X
ejpam-3780	499	22	ri	ri	NOUN
ejpam-3780	499	23	for	for	ADP
ejpam-3780	499	24	i	i	PRON
ejpam-3780	499	25	=	=	NOUN
ejpam-3780	499	26	1	1	NUM
ejpam-3780	499	27	,	,	PUNCT
ejpam-3780	499	28	2	2	NUM
ejpam-3780	499	29	,	,	PUNCT
ejpam-3780	499	30	.	.	PUNCT
ejpam-3780	499	31	.	.	PUNCT
ejpam-3780	500	1	.	.	PUNCT
ejpam-3780	501	1	,	,	PUNCT
ejpam-3780	501	2	n.	n.	NOUN
ejpam-3780	501	3	then	then	ADV
ejpam-3780	501	4	∆(g	∆(g	NOUN
ejpam-3780	501	5	)	)	PUNCT
ejpam-3780	502	1	=	=	VERB
ejpam-3780	502	2	rk	rk	NOUN
ejpam-3780	502	3	+	+	NOUN
ejpam-3780	502	4	2	2	X
ejpam-3780	502	5	.	.	PUNCT
ejpam-3780	502	6	define	define	VERB
ejpam-3780	502	7	a	a	DET
ejpam-3780	502	8	function	function	NOUN
ejpam-3780	502	9	f	f	NOUN
ejpam-3780	502	10	:	:	PUNCT
ejpam-3780	502	11	v	v	X
ejpam-3780	502	12	(	(	PUNCT
ejpam-3780	502	13	g	g	NOUN
ejpam-3780	502	14	)	)	PUNCT
ejpam-3780	502	15	→	→	SYM
ejpam-3780	502	16	m0	m0	NOUN
ejpam-3780	502	17	2	2	NUM
ejpam-3780	502	18	(	(	PUNCT
ejpam-3780	502	19	z2nrk+n−2rk	z2nrk+n−2rk	NOUN
ejpam-3780	502	20	)	)	PUNCT
ejpam-3780	502	21	such	such	ADJ
ejpam-3780	502	22	that	that	SCONJ
ejpam-3780	502	23	f(wi	f(wi	NOUN
ejpam-3780	502	24	)	)	PUNCT
ejpam-3780	502	25	=	=	SYM
ejpam-3780	502	26	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	502	27	)	)	PUNCT
ejpam-3780	502	28	(	(	PUNCT
ejpam-3780	502	29	2n−i−1	2n−i−1	PROPN
ejpam-3780	502	30	2	2	NUM
ejpam-3780	502	31	)	)	PUNCT
ejpam-3780	502	32	if	if	SCONJ
ejpam-3780	502	33	i	i	PRON
ejpam-3780	502	34	is	be	AUX
ejpam-3780	502	35	odd	odd	ADJ
ejpam-3780	502	36	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	502	37	)	)	PUNCT
ejpam-3780	502	38	(	(	PUNCT
ejpam-3780	502	39	i−2	i−2	NOUN
ejpam-3780	502	40	2	2	NUM
ejpam-3780	502	41	)	)	PUNCT
ejpam-3780	502	42	if	if	SCONJ
ejpam-3780	502	43	i	i	PRON
ejpam-3780	502	44	is	be	AUX
ejpam-3780	502	45	even	even	ADV
ejpam-3780	502	46	,	,	PUNCT
ejpam-3780	502	47	(	(	PUNCT
ejpam-3780	502	48	24	24	NUM
ejpam-3780	502	49	)	)	PUNCT
ejpam-3780	502	50	f(wi	f(wi	PROPN
ejpam-3780	502	51	,	,	PUNCT
ejpam-3780	502	52	j	j	NOUN
ejpam-3780	502	53	)	)	PUNCT
ejpam-3780	502	54	=	=	SYM
ejpam-3780	502	55	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	502	56	)	)	PUNCT
ejpam-3780	502	57	(	(	PUNCT
ejpam-3780	502	58	i−3	i−3	PROPN
ejpam-3780	502	59	2	2	NUM
ejpam-3780	502	60	)	)	PUNCT
ejpam-3780	503	1	+	+	NOUN
ejpam-3780	503	2	j	j	NOUN
ejpam-3780	503	3	if	if	SCONJ
ejpam-3780	503	4	i	i	PRON
ejpam-3780	503	5	is	be	AUX
ejpam-3780	503	6	odd	odd	ADJ
ejpam-3780	503	7	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	503	8	)	)	PUNCT
ejpam-3780	503	9	(	(	PUNCT
ejpam-3780	503	10	2n−i−2	2n−i−2	NUM
ejpam-3780	503	11	2	2	NUM
ejpam-3780	503	12	)	)	PUNCT
ejpam-3780	504	1	+	+	NOUN
ejpam-3780	504	2	j	j	NOUN
ejpam-3780	504	3	if	if	SCONJ
ejpam-3780	504	4	i	i	PRON
ejpam-3780	504	5	is	be	AUX
ejpam-3780	504	6	even	even	ADV
ejpam-3780	504	7	.	.	PUNCT
ejpam-3780	505	1	,	,	PUNCT
ejpam-3780	505	2	(	(	PUNCT
ejpam-3780	505	3	25	25	NUM
ejpam-3780	505	4	)	)	PUNCT
ejpam-3780	505	5	and	and	CCONJ
ejpam-3780	505	6	f(wi	f(wi	ADJ
ejpam-3780	505	7	,	,	PUNCT
ejpam-3780	505	8	j,1	j,1	X
ejpam-3780	505	9	)	)	PUNCT
ejpam-3780	505	10	=	=	SYM
ejpam-3780	505	11	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	505	12	)	)	PUNCT
ejpam-3780	505	13	(	(	PUNCT
ejpam-3780	505	14	2n−i+1	2n−i+1	NUM
ejpam-3780	505	15	2	2	NUM
ejpam-3780	505	16	)	)	PUNCT
ejpam-3780	505	17	−j	−j	NOUN
ejpam-3780	505	18	if	if	SCONJ
ejpam-3780	505	19	i	i	PRON
ejpam-3780	505	20	is	be	AUX
ejpam-3780	505	21	odd	odd	ADJ
ejpam-3780	505	22	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	505	23	)	)	PUNCT
ejpam-3780	505	24	(	(	PUNCT
ejpam-3780	505	25	i	i	PRON
ejpam-3780	505	26	2)−j	2)−j	NUM
ejpam-3780	505	27	if	if	SCONJ
ejpam-3780	505	28	i	i	PRON
ejpam-3780	505	29	is	be	AUX
ejpam-3780	505	30	even	even	ADV
ejpam-3780	505	31	.	.	PUNCT
ejpam-3780	506	1	(	(	PUNCT
ejpam-3780	506	2	26	26	NUM
ejpam-3780	506	3	)	)	PUNCT
ejpam-3780	506	4	clearly	clearly	ADV
ejpam-3780	506	5	,	,	PUNCT
ejpam-3780	506	6	f	f	PROPN
ejpam-3780	506	7	is	be	AUX
ejpam-3780	506	8	injective	injective	ADJ
ejpam-3780	506	9	.	.	PUNCT
ejpam-3780	507	1	let	let	VERB
ejpam-3780	507	2	k	k	NOUN
ejpam-3780	507	3	=	=	PRON
ejpam-3780	507	4	{	{	PUNCT
ejpam-3780	507	5	f(u	f(u	PROPN
ejpam-3780	507	6	)	)	PUNCT
ejpam-3780	507	7	+	+	NUM
ejpam-3780	507	8	f(v	f(v	NOUN
ejpam-3780	507	9	)	)	PUNCT
ejpam-3780	507	10	:	:	PUNCT
ejpam-3780	507	11	uv	uv	PROPN
ejpam-3780	507	12	∈	∈	PROPN
ejpam-3780	507	13	e(g	e(g	PROPN
ejpam-3780	507	14	)	)	PUNCT
ejpam-3780	507	15	}	}	PUNCT
ejpam-3780	507	16	.	.	PUNCT
ejpam-3780	508	1	to	to	PART
ejpam-3780	508	2	show	show	VERB
ejpam-3780	508	3	that	that	SCONJ
ejpam-3780	508	4	f	f	PROPN
ejpam-3780	508	5	is	be	AUX
ejpam-3780	508	6	an	an	DET
ejpam-3780	508	7	efficient	efficient	ADJ
ejpam-3780	508	8	zero	zero	NUM
ejpam-3780	508	9	ring	ring	NOUN
ejpam-3780	508	10	labeling	labeling	NOUN
ejpam-3780	508	11	of	of	ADP
ejpam-3780	508	12	g	g	NOUN
ejpam-3780	508	13	,	,	PUNCT
ejpam-3780	508	14	we	we	PRON
ejpam-3780	508	15	need	need	VERB
ejpam-3780	508	16	to	to	PART
ejpam-3780	508	17	show	show	VERB
ejpam-3780	508	18	that	that	SCONJ
ejpam-3780	508	19	|k|	|k|	NOUN
ejpam-3780	508	20	=	=	PRON
ejpam-3780	508	21	rk	rk	NOUN
ejpam-3780	508	22	+	+	CCONJ
ejpam-3780	508	23	2	2	NUM
ejpam-3780	508	24	and	and	CCONJ
ejpam-3780	508	25	a0	a0	PROPN
ejpam-3780	508	26	/∈	/∈	PROPN
ejpam-3780	508	27	k.	k.	PROPN
ejpam-3780	508	28	for	for	ADP
ejpam-3780	508	29	adjacent	adjacent	ADJ
ejpam-3780	508	30	vertices	vertex	NOUN
ejpam-3780	508	31	in	in	ADP
ejpam-3780	508	32	the	the	DET
ejpam-3780	508	33	central	central	ADJ
ejpam-3780	508	34	path	path	NOUN
ejpam-3780	508	35	,	,	PUNCT
ejpam-3780	508	36	we	we	PRON
ejpam-3780	508	37	obtain	obtain	VERB
ejpam-3780	508	38	the	the	DET
ejpam-3780	508	39	sums	sum	NOUN
ejpam-3780	508	40	f(wi	f(wi	NOUN
ejpam-3780	508	41	)	)	PUNCT
ejpam-3780	508	42	+	+	NUM
ejpam-3780	508	43	f(wi+1	f(wi+1	X
ejpam-3780	508	44	)	)	PUNCT
ejpam-3780	508	45	=	=	SYM
ejpam-3780	509	1	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	509	2	)	)	PUNCT
ejpam-3780	509	3	(	(	PUNCT
ejpam-3780	509	4	2n−i−1	2n−i−1	PROPN
ejpam-3780	509	5	2	2	NUM
ejpam-3780	509	6	)	)	PUNCT
ejpam-3780	509	7	+	+	CCONJ
ejpam-3780	509	8	a	a	DET
ejpam-3780	509	9	(	(	PUNCT
ejpam-3780	509	10	2rk+1	2rk+1	NUM
ejpam-3780	509	11	)	)	PUNCT
ejpam-3780	509	12	(	(	PUNCT
ejpam-3780	509	13	(	(	PUNCT
ejpam-3780	509	14	i+1)−2	i+1)−2	INTJ
ejpam-3780	509	15	2	2	X
ejpam-3780	509	16	)	)	PUNCT
ejpam-3780	509	17	=	=	SYM
ejpam-3780	509	18	a(2rk+1)(n−1	a(2rk+1)(n−1	PROPN
ejpam-3780	509	19	)	)	PUNCT
ejpam-3780	509	20	(	(	PUNCT
ejpam-3780	509	21	27	27	NUM
ejpam-3780	509	22	)	)	PUNCT
ejpam-3780	509	23	if	if	SCONJ
ejpam-3780	509	24	i	i	PRON
ejpam-3780	509	25	is	be	AUX
ejpam-3780	509	26	odd	odd	ADJ
ejpam-3780	509	27	,	,	PUNCT
ejpam-3780	509	28	and	and	CCONJ
ejpam-3780	509	29	f(wi	f(wi	NOUN
ejpam-3780	509	30	)	)	PUNCT
ejpam-3780	509	31	+	+	NUM
ejpam-3780	509	32	f(wi+1	f(wi+1	X
ejpam-3780	509	33	)	)	PUNCT
ejpam-3780	509	34	=	=	SYM
ejpam-3780	510	1	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	510	2	)	)	PUNCT
ejpam-3780	510	3	(	(	PUNCT
ejpam-3780	510	4	2n−i−1	2n−i−1	PROPN
ejpam-3780	510	5	2	2	NUM
ejpam-3780	510	6	)	)	PUNCT
ejpam-3780	510	7	+	+	CCONJ
ejpam-3780	510	8	a	a	DET
ejpam-3780	510	9	(	(	PUNCT
ejpam-3780	510	10	2rk+1	2rk+1	NUM
ejpam-3780	510	11	)	)	PUNCT
ejpam-3780	510	12	(	(	PUNCT
ejpam-3780	510	13	2n−(i+1)−1	2n−(i+1)−1	NOUN
ejpam-3780	510	14	2	2	NUM
ejpam-3780	510	15	)	)	PUNCT
ejpam-3780	510	16	=	=	SYM
ejpam-3780	511	1	a(2rk+1)(n−2	a(2rk+1)(n−2	PROPN
ejpam-3780	511	2	)	)	PUNCT
ejpam-3780	511	3	(	(	PUNCT
ejpam-3780	511	4	28	28	NUM
ejpam-3780	511	5	)	)	PUNCT
ejpam-3780	511	6	if	if	SCONJ
ejpam-3780	511	7	i	i	PRON
ejpam-3780	511	8	is	be	AUX
ejpam-3780	511	9	even	even	ADV
ejpam-3780	511	10	.	.	PUNCT
ejpam-3780	512	1	thus	thus	ADV
ejpam-3780	512	2	,	,	PUNCT
ejpam-3780	512	3	there	there	PRON
ejpam-3780	512	4	are	be	VERB
ejpam-3780	512	5	only	only	ADV
ejpam-3780	512	6	two	two	NUM
ejpam-3780	512	7	distinct	distinct	ADJ
ejpam-3780	512	8	sums	sum	NOUN
ejpam-3780	512	9	obtained	obtain	VERB
ejpam-3780	512	10	for	for	ADP
ejpam-3780	512	11	adjacent	adjacent	ADJ
ejpam-3780	512	12	vertices	vertex	NOUN
ejpam-3780	512	13	in	in	ADP
ejpam-3780	512	14	the	the	DET
ejpam-3780	512	15	central	central	ADJ
ejpam-3780	512	16	path	path	NOUN
ejpam-3780	512	17	.	.	PUNCT
ejpam-3780	513	1	for	for	ADP
ejpam-3780	513	2	pairs	pair	NOUN
ejpam-3780	513	3	of	of	ADP
ejpam-3780	513	4	adjacent	adjacent	ADJ
ejpam-3780	513	5	elbow	elbow	NOUN
ejpam-3780	513	6	and	and	CCONJ
ejpam-3780	513	7	vertex	vertex	NOUN
ejpam-3780	513	8	in	in	ADP
ejpam-3780	513	9	the	the	DET
ejpam-3780	513	10	central	central	ADJ
ejpam-3780	513	11	path	path	NOUN
ejpam-3780	513	12	,	,	PUNCT
ejpam-3780	513	13	we	we	PRON
ejpam-3780	513	14	obtain	obtain	VERB
ejpam-3780	513	15	the	the	DET
ejpam-3780	513	16	sums	sum	NOUN
ejpam-3780	513	17	f(wi	f(wi	NOUN
ejpam-3780	513	18	)	)	PUNCT
ejpam-3780	514	1	+	+	CCONJ
ejpam-3780	515	1	f(wi	f(wi	ADJ
ejpam-3780	515	2	,	,	PUNCT
ejpam-3780	515	3	j	j	NOUN
ejpam-3780	515	4	)	)	PUNCT
ejpam-3780	515	5	=	=	PUNCT
ejpam-3780	515	6	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	515	7	)	)	PUNCT
ejpam-3780	515	8	(	(	PUNCT
ejpam-3780	515	9	i−2	i−2	NOUN
ejpam-3780	515	10	2	2	NUM
ejpam-3780	515	11	)	)	PUNCT
ejpam-3780	515	12	+	+	CCONJ
ejpam-3780	515	13	a(2rk+1	a(2rk+1	NOUN
ejpam-3780	515	14	)	)	PUNCT
ejpam-3780	515	15	(	(	PUNCT
ejpam-3780	515	16	i−3	i−3	PROPN
ejpam-3780	515	17	2	2	NUM
ejpam-3780	515	18	)	)	PUNCT
ejpam-3780	516	1	+	+	NUM
ejpam-3780	516	2	j	j	NOUN
ejpam-3780	516	3	=	=	SYM
ejpam-3780	516	4	a(2rk+1)(n−2)+j	a(2rk+1)(n−2)+j	PROPN
ejpam-3780	516	5	(	(	PUNCT
ejpam-3780	516	6	29	29	NUM
ejpam-3780	516	7	)	)	PUNCT
ejpam-3780	516	8	d.	d.	PROPN
ejpam-3780	516	9	chua	chua	PROPN
ejpam-3780	516	10	,	,	PUNCT
ejpam-3780	516	11	f.	f.	PROPN
ejpam-3780	516	12	campeña	campeña	PROPN
ejpam-3780	516	13	,	,	PUNCT
ejpam-3780	516	14	f.	f.	PROPN
ejpam-3780	516	15	franco	franco	PROPN
ejpam-3780	516	16	/	/	SYM
ejpam-3780	516	17	eur	eur	PROPN
ejpam-3780	516	18	.	.	PUNCT
ejpam-3780	517	1	j.	j.	PROPN
ejpam-3780	517	2	pure	pure	PROPN
ejpam-3780	517	3	appl	appl	PROPN
ejpam-3780	517	4	.	.	PROPN
ejpam-3780	517	5	math	math	PROPN
ejpam-3780	517	6	,	,	PUNCT
ejpam-3780	517	7	13	13	NUM
ejpam-3780	517	8	(	(	PUNCT
ejpam-3780	517	9	3	3	NUM
ejpam-3780	517	10	)	)	PUNCT
ejpam-3780	517	11	(	(	PUNCT
ejpam-3780	517	12	2020	2020	NUM
ejpam-3780	517	13	)	)	PUNCT
ejpam-3780	517	14	,	,	PUNCT
ejpam-3780	517	15	674	674	NUM
ejpam-3780	517	16	-	-	SYM
ejpam-3780	517	17	696	696	NUM
ejpam-3780	517	18	688	688	NUM
ejpam-3780	517	19	for	for	ADP
ejpam-3780	517	20	j	j	PROPN
ejpam-3780	517	21	=	=	SYM
ejpam-3780	517	22	1	1	NUM
ejpam-3780	517	23	,	,	PUNCT
ejpam-3780	517	24	2	2	NUM
ejpam-3780	517	25	,	,	PUNCT
ejpam-3780	517	26	.	.	PUNCT
ejpam-3780	517	27	.	.	PUNCT
ejpam-3780	518	1	.	.	PUNCT
ejpam-3780	519	1	,	,	PUNCT
ejpam-3780	519	2	ri	ri	INTJ
ejpam-3780	519	3	if	if	SCONJ
ejpam-3780	519	4	i	i	PRON
ejpam-3780	519	5	is	be	AUX
ejpam-3780	519	6	odd	odd	ADJ
ejpam-3780	519	7	,	,	PUNCT
ejpam-3780	519	8	and	and	CCONJ
ejpam-3780	519	9	f(wi	f(wi	NOUN
ejpam-3780	519	10	)	)	PUNCT
ejpam-3780	519	11	+	+	CCONJ
ejpam-3780	519	12	f(wi	f(wi	ADJ
ejpam-3780	519	13	,	,	PUNCT
ejpam-3780	519	14	j	j	NOUN
ejpam-3780	519	15	)	)	PUNCT
ejpam-3780	519	16	=	=	PUNCT
ejpam-3780	519	17	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	519	18	)	)	PUNCT
ejpam-3780	519	19	(	(	PUNCT
ejpam-3780	519	20	i−2	i−2	NOUN
ejpam-3780	519	21	2	2	NUM
ejpam-3780	519	22	)	)	PUNCT
ejpam-3780	519	23	+	+	CCONJ
ejpam-3780	519	24	a(2rk+1	a(2rk+1	NOUN
ejpam-3780	519	25	)	)	PUNCT
ejpam-3780	519	26	(	(	PUNCT
ejpam-3780	519	27	i−3	i−3	PROPN
ejpam-3780	519	28	2	2	NUM
ejpam-3780	519	29	)	)	PUNCT
ejpam-3780	520	1	+	+	NUM
ejpam-3780	520	2	j	j	NOUN
ejpam-3780	520	3	=	=	SYM
ejpam-3780	520	4	a(2rk+1)(n−2)+j	a(2rk+1)(n−2)+j	PROPN
ejpam-3780	520	5	(	(	PUNCT
ejpam-3780	520	6	30	30	NUM
ejpam-3780	520	7	)	)	PUNCT
ejpam-3780	520	8	for	for	ADP
ejpam-3780	520	9	j	j	PROPN
ejpam-3780	520	10	=	=	SYM
ejpam-3780	520	11	1	1	NUM
ejpam-3780	520	12	,	,	PUNCT
ejpam-3780	520	13	2	2	NUM
ejpam-3780	520	14	,	,	PUNCT
ejpam-3780	520	15	.	.	PUNCT
ejpam-3780	520	16	.	.	PUNCT
ejpam-3780	520	17	.	.	PUNCT
ejpam-3780	521	1	,	,	PUNCT
ejpam-3780	521	2	ri	ri	INTJ
ejpam-3780	521	3	if	if	SCONJ
ejpam-3780	521	4	i	i	PRON
ejpam-3780	521	5	is	be	AUX
ejpam-3780	521	6	even	even	ADV
ejpam-3780	521	7	.	.	PUNCT
ejpam-3780	522	1	since	since	SCONJ
ejpam-3780	522	2	rk	rk	NOUN
ejpam-3780	522	3	is	be	AUX
ejpam-3780	522	4	the	the	DET
ejpam-3780	522	5	maximum	maximum	ADJ
ejpam-3780	522	6	number	number	NOUN
ejpam-3780	522	7	of	of	ADP
ejpam-3780	522	8	elbows	elbow	NOUN
ejpam-3780	522	9	of	of	ADP
ejpam-3780	522	10	a	a	DET
ejpam-3780	522	11	vertex	vertex	NOUN
ejpam-3780	522	12	in	in	ADP
ejpam-3780	522	13	the	the	DET
ejpam-3780	522	14	central	central	ADJ
ejpam-3780	522	15	path	path	NOUN
ejpam-3780	522	16	,	,	PUNCT
ejpam-3780	522	17	the	the	DET
ejpam-3780	522	18	obtained	obtain	VERB
ejpam-3780	522	19	sums	sum	NOUN
ejpam-3780	522	20	for	for	ADP
ejpam-3780	522	21	pairs	pair	NOUN
ejpam-3780	522	22	of	of	ADP
ejpam-3780	522	23	adjacent	adjacent	ADJ
ejpam-3780	522	24	elbow	elbow	NOUN
ejpam-3780	522	25	and	and	CCONJ
ejpam-3780	522	26	vertex	vertex	NOUN
ejpam-3780	522	27	in	in	ADP
ejpam-3780	522	28	the	the	DET
ejpam-3780	522	29	central	central	ADJ
ejpam-3780	522	30	path	path	NOUN
ejpam-3780	522	31	are	be	AUX
ejpam-3780	522	32	a(2rk+1)(n−2)+1	a(2rk+1)(n−2)+1	NOUN
ejpam-3780	522	33	,	,	PUNCT
ejpam-3780	522	34	a(2rk+1)(n−2)+2	a(2rk+1)(n−2)+2	NOUN
ejpam-3780	522	35	,	,	PUNCT
ejpam-3780	522	36	.	.	PUNCT
ejpam-3780	522	37	.	.	PUNCT
ejpam-3780	523	1	.	.	PUNCT
ejpam-3780	524	1	,	,	PUNCT
ejpam-3780	524	2	a(2rk+1)(n−2)+rk	a(2rk+1)(n−2)+rk	INTJ
ejpam-3780	524	3	.	.	PUNCT
ejpam-3780	525	1	for	for	ADP
ejpam-3780	525	2	pairs	pair	NOUN
ejpam-3780	525	3	of	of	ADP
ejpam-3780	525	4	adjacent	adjacent	ADJ
ejpam-3780	525	5	hanging	hang	VERB
ejpam-3780	525	6	leaf	leaf	NOUN
ejpam-3780	525	7	and	and	CCONJ
ejpam-3780	525	8	elbow	elbow	VERB
ejpam-3780	525	9	,	,	PUNCT
ejpam-3780	525	10	we	we	PRON
ejpam-3780	525	11	obtain	obtain	VERB
ejpam-3780	525	12	the	the	DET
ejpam-3780	525	13	sums	sum	NOUN
ejpam-3780	525	14	f(wi	f(wi	PROPN
ejpam-3780	525	15	,	,	PUNCT
ejpam-3780	525	16	j	j	NOUN
ejpam-3780	525	17	)	)	PUNCT
ejpam-3780	525	18	+	+	CCONJ
ejpam-3780	525	19	f(wi	f(wi	ADJ
ejpam-3780	525	20	,	,	PUNCT
ejpam-3780	525	21	j,1	j,1	NOUN
ejpam-3780	525	22	)	)	PUNCT
ejpam-3780	525	23	=	=	SYM
ejpam-3780	526	1	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	526	2	)	)	PUNCT
ejpam-3780	526	3	(	(	PUNCT
ejpam-3780	526	4	i−3	i−3	PROPN
ejpam-3780	526	5	2	2	NUM
ejpam-3780	526	6	)	)	PUNCT
ejpam-3780	527	1	+	+	NOUN
ejpam-3780	527	2	j	j	PROPN
ejpam-3780	527	3	+	+	CCONJ
ejpam-3780	527	4	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	527	5	)	)	PUNCT
ejpam-3780	527	6	(	(	PUNCT
ejpam-3780	527	7	2n−i+1	2n−i+1	NUM
ejpam-3780	527	8	2	2	NUM
ejpam-3780	527	9	)	)	PUNCT
ejpam-3780	527	10	−j	−j	NOUN
ejpam-3780	527	11	=	=	SYM
ejpam-3780	527	12	a(2rk+1)(n−1	a(2rk+1)(n−1	X
ejpam-3780	527	13	)	)	PUNCT
ejpam-3780	527	14	(	(	PUNCT
ejpam-3780	527	15	31	31	NUM
ejpam-3780	527	16	)	)	PUNCT
ejpam-3780	527	17	if	if	SCONJ
ejpam-3780	527	18	i	i	PRON
ejpam-3780	527	19	is	be	AUX
ejpam-3780	527	20	odd	odd	ADJ
ejpam-3780	527	21	,	,	PUNCT
ejpam-3780	527	22	and	and	CCONJ
ejpam-3780	527	23	f(wi	f(wi	ADJ
ejpam-3780	527	24	,	,	PUNCT
ejpam-3780	527	25	j	j	NOUN
ejpam-3780	527	26	)	)	PUNCT
ejpam-3780	527	27	+	+	CCONJ
ejpam-3780	527	28	f(wi	f(wi	ADJ
ejpam-3780	527	29	,	,	PUNCT
ejpam-3780	527	30	j,1	j,1	NOUN
ejpam-3780	527	31	)	)	PUNCT
ejpam-3780	527	32	=	=	SYM
ejpam-3780	527	33	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	527	34	)	)	PUNCT
ejpam-3780	527	35	(	(	PUNCT
ejpam-3780	527	36	2n−i−2	2n−i−2	NUM
ejpam-3780	527	37	2	2	NUM
ejpam-3780	527	38	)	)	PUNCT
ejpam-3780	528	1	+	+	NOUN
ejpam-3780	528	2	j	j	NOUN
ejpam-3780	528	3	+	+	CCONJ
ejpam-3780	528	4	a(2rk+1	a(2rk+1	PROPN
ejpam-3780	528	5	)	)	PUNCT
ejpam-3780	528	6	(	(	PUNCT
ejpam-3780	528	7	i	i	PRON
ejpam-3780	528	8	2)−j	2)−j	NUM
ejpam-3780	528	9	=	=	SYM
ejpam-3780	528	10	a(2rk+1)(n−1	a(2rk+1)(n−1	PROPN
ejpam-3780	528	11	)	)	PUNCT
ejpam-3780	528	12	(	(	PUNCT
ejpam-3780	528	13	32	32	NUM
ejpam-3780	528	14	)	)	PUNCT
ejpam-3780	528	15	if	if	SCONJ
ejpam-3780	528	16	i	i	PRON
ejpam-3780	528	17	is	be	AUX
ejpam-3780	528	18	even	even	ADV
ejpam-3780	528	19	.	.	PUNCT
ejpam-3780	529	1	thus	thus	ADV
ejpam-3780	529	2	,	,	PUNCT
ejpam-3780	529	3	the	the	DET
ejpam-3780	529	4	sum	sum	NOUN
ejpam-3780	529	5	of	of	ADP
ejpam-3780	529	6	any	any	DET
ejpam-3780	529	7	pair	pair	NOUN
ejpam-3780	529	8	of	of	ADP
ejpam-3780	529	9	adjacent	adjacent	ADJ
ejpam-3780	529	10	elbow	elbow	NOUN
ejpam-3780	529	11	and	and	CCONJ
ejpam-3780	529	12	hanging	hang	VERB
ejpam-3780	529	13	leaf	leaf	NOUN
ejpam-3780	529	14	is	be	AUX
ejpam-3780	529	15	equal	equal	ADJ
ejpam-3780	529	16	.	.	PUNCT
ejpam-3780	530	1	then	then	ADV
ejpam-3780	530	2	k	k	PROPN
ejpam-3780	530	3	=	=	PUNCT
ejpam-3780	530	4	{	{	PUNCT
ejpam-3780	530	5	a(2rk+1)(n−2	a(2rk+1)(n−2	PROPN
ejpam-3780	530	6	)	)	PUNCT
ejpam-3780	530	7	,	,	PUNCT
ejpam-3780	530	8	a(2rk+1)(n−2)+1	a(2rk+1)(n−2)+1	NOUN
ejpam-3780	530	9	,	,	PUNCT
ejpam-3780	530	10	.	.	PUNCT
ejpam-3780	530	11	.	.	PUNCT
ejpam-3780	530	12	.	.	PUNCT
ejpam-3780	531	1	,	,	PUNCT
ejpam-3780	531	2	a(2rk+1)(n−2)+rk	a(2rk+1)(n−2)+rk	INTJ
ejpam-3780	531	3	,	,	PUNCT
ejpam-3780	531	4	a(2rk+1)(n−1	a(2rk+1)(n−1	PROPN
ejpam-3780	531	5	)	)	PUNCT
ejpam-3780	531	6	}	}	PUNCT
ejpam-3780	531	7	,	,	PUNCT
ejpam-3780	531	8	(	(	PUNCT
ejpam-3780	531	9	33	33	NUM
ejpam-3780	531	10	)	)	PUNCT
ejpam-3780	531	11	hence	hence	ADV
ejpam-3780	531	12	|k|	|k|	NOUN
ejpam-3780	531	13	=	=	PRON
ejpam-3780	531	14	rk	rk	NOUN
ejpam-3780	531	15	+	+	NOUN
ejpam-3780	531	16	2	2	X
ejpam-3780	531	17	.	.	PUNCT
ejpam-3780	531	18	to	to	PART
ejpam-3780	531	19	show	show	VERB
ejpam-3780	531	20	that	that	DET
ejpam-3780	531	21	a0	a0	PROPN
ejpam-3780	531	22	/∈	/∈	PUNCT
ejpam-3780	532	1	k	k	PROPN
ejpam-3780	532	2	,	,	PUNCT
ejpam-3780	532	3	it	it	PRON
ejpam-3780	532	4	is	be	AUX
ejpam-3780	532	5	sufficient	sufficient	ADJ
ejpam-3780	532	6	to	to	PART
ejpam-3780	532	7	show	show	VERB
ejpam-3780	532	8	that	that	SCONJ
ejpam-3780	532	9	(	(	PUNCT
ejpam-3780	532	10	2rk	2rk	ADJ
ejpam-3780	532	11	+	+	CCONJ
ejpam-3780	532	12	1)(n−	1)(n−	NUM
ejpam-3780	532	13	2	2	NUM
ejpam-3780	532	14	)	)	PUNCT
ejpam-3780	532	15	,	,	PUNCT
ejpam-3780	532	16	(	(	PUNCT
ejpam-3780	532	17	2rk	2rk	NOUN
ejpam-3780	532	18	+	+	CCONJ
ejpam-3780	532	19	1)(n−	1)(n−	NUM
ejpam-3780	532	20	2	2	NUM
ejpam-3780	532	21	)	)	PUNCT
ejpam-3780	532	22	+	+	NUM
ejpam-3780	533	1	1	1	NUM
ejpam-3780	533	2	,	,	PUNCT
ejpam-3780	533	3	(	(	PUNCT
ejpam-3780	533	4	2rk+1)(n−2)+2	2rk+1)(n−2)+2	NUM
ejpam-3780	533	5	,	,	PUNCT
ejpam-3780	533	6	.	.	PUNCT
ejpam-3780	533	7	.	.	PUNCT
ejpam-3780	533	8	.	.	PUNCT
ejpam-3780	534	1	,	,	PUNCT
ejpam-3780	534	2	(	(	PUNCT
ejpam-3780	534	3	2rk+1)(n−2)+rk	2rk+1)(n−2)+rk	NUM
ejpam-3780	534	4	,	,	PUNCT
ejpam-3780	534	5	(	(	PUNCT
ejpam-3780	534	6	2rk+1)(n−1	2rk+1)(n−1	NUM
ejpam-3780	534	7	)	)	PUNCT
ejpam-3780	534	8	are	be	AUX
ejpam-3780	534	9	greater	great	ADJ
ejpam-3780	534	10	than	than	ADP
ejpam-3780	534	11	0	0	NUM
ejpam-3780	534	12	but	but	CCONJ
ejpam-3780	534	13	less	less	ADJ
ejpam-3780	534	14	than	than	ADP
ejpam-3780	534	15	2nrk	2nrk	NUM
ejpam-3780	534	16	+	+	NOUN
ejpam-3780	534	17	n−2rk	n−2rk	ADJ
ejpam-3780	534	18	.	.	PUNCT
ejpam-3780	535	1	the	the	DET
ejpam-3780	535	2	central	central	ADJ
ejpam-3780	535	3	path	path	NOUN
ejpam-3780	535	4	has	have	VERB
ejpam-3780	535	5	at	at	ADV
ejpam-3780	535	6	least	least	ADJ
ejpam-3780	535	7	3	3	NUM
ejpam-3780	535	8	vertices	vertex	NOUN
ejpam-3780	535	9	,	,	PUNCT
ejpam-3780	535	10	so	so	SCONJ
ejpam-3780	535	11	n	n	NUM
ejpam-3780	535	12	≥	≥	NOUN
ejpam-3780	535	13	3	3	NUM
ejpam-3780	535	14	,	,	PUNCT
ejpam-3780	535	15	and	and	CCONJ
ejpam-3780	535	16	rk	rk	NOUN
ejpam-3780	535	17	is	be	AUX
ejpam-3780	535	18	at	at	ADP
ejpam-3780	535	19	least	least	ADJ
ejpam-3780	535	20	zero	zero	NUM
ejpam-3780	535	21	by	by	ADP
ejpam-3780	535	22	its	its	PRON
ejpam-3780	535	23	definition	definition	NOUN
ejpam-3780	535	24	.	.	PUNCT
ejpam-3780	536	1	then	then	ADV
ejpam-3780	536	2	(	(	PUNCT
ejpam-3780	536	3	2rk+1)(n−2)+m	2rk+1)(n−2)+m	NUM
ejpam-3780	536	4	for	for	ADP
ejpam-3780	536	5	m	m	PROPN
ejpam-3780	536	6	=	=	SYM
ejpam-3780	536	7	0	0	NUM
ejpam-3780	536	8	,	,	PUNCT
ejpam-3780	536	9	1	1	NUM
ejpam-3780	536	10	,	,	PUNCT
ejpam-3780	536	11	.	.	PUNCT
ejpam-3780	536	12	.	.	PUNCT
ejpam-3780	537	1	.	.	PUNCT
ejpam-3780	538	1	,	,	PUNCT
ejpam-3780	538	2	rk	rk	PRON
ejpam-3780	538	3	and	and	CCONJ
ejpam-3780	538	4	(	(	PUNCT
ejpam-3780	538	5	2rk+1)(n−1	2rk+1)(n−1	NUM
ejpam-3780	538	6	)	)	PUNCT
ejpam-3780	538	7	are	be	AUX
ejpam-3780	538	8	greater	great	ADJ
ejpam-3780	538	9	than	than	ADP
ejpam-3780	538	10	zero	zero	NUM
ejpam-3780	538	11	.	.	PUNCT
ejpam-3780	539	1	moreover	moreover	ADV
ejpam-3780	539	2	,	,	PUNCT
ejpam-3780	539	3	(	(	PUNCT
ejpam-3780	539	4	2rk	2rk	ADJ
ejpam-3780	539	5	+	+	CCONJ
ejpam-3780	539	6	1)(n−2	1)(n−2	NUM
ejpam-3780	539	7	)	)	PUNCT
ejpam-3780	540	1	+	+	VERB
ejpam-3780	540	2	m	m	PRON
ejpam-3780	540	3	<	<	X
ejpam-3780	540	4	(	(	PUNCT
ejpam-3780	540	5	2rk	2rk	ADJ
ejpam-3780	540	6	+	+	CCONJ
ejpam-3780	540	7	1)(n−2	1)(n−2	NUM
ejpam-3780	540	8	)	)	PUNCT
ejpam-3780	541	1	+	+	CCONJ
ejpam-3780	541	2	(	(	PUNCT
ejpam-3780	541	3	2rk	2rk	ADJ
ejpam-3780	541	4	+	+	CCONJ
ejpam-3780	541	5	1	1	X
ejpam-3780	541	6	)	)	PUNCT
ejpam-3780	541	7	=	=	NOUN
ejpam-3780	541	8	(	(	PUNCT
ejpam-3780	541	9	2rk	2rk	ADJ
ejpam-3780	541	10	+	+	CCONJ
ejpam-3780	541	11	1)(n−1	1)(n−1	NUM
ejpam-3780	541	12	)	)	PUNCT
ejpam-3780	541	13	for	for	ADP
ejpam-3780	541	14	m	m	PROPN
ejpam-3780	541	15	=	=	SYM
ejpam-3780	541	16	0	0	NUM
ejpam-3780	541	17	,	,	PUNCT
ejpam-3780	541	18	1	1	NUM
ejpam-3780	541	19	,	,	PUNCT
ejpam-3780	541	20	.	.	PUNCT
ejpam-3780	541	21	.	.	PUNCT
ejpam-3780	541	22	.	.	PUNCT
ejpam-3780	542	1	,	,	PUNCT
ejpam-3780	542	2	rk	rk	VERB
ejpam-3780	542	3	.	.	PUNCT
ejpam-3780	543	1	since	since	SCONJ
ejpam-3780	543	2	(	(	PUNCT
ejpam-3780	543	3	2rk	2rk	ADJ
ejpam-3780	543	4	+	+	CCONJ
ejpam-3780	543	5	1)(n−	1)(n−	NUM
ejpam-3780	543	6	1	1	NUM
ejpam-3780	543	7	)	)	PUNCT
ejpam-3780	543	8	=	=	PRON
ejpam-3780	543	9	(	(	PUNCT
ejpam-3780	543	10	2rk	2rk	ADJ
ejpam-3780	543	11	+	+	CCONJ
ejpam-3780	543	12	1)(n)−	1)(n)−	NUM
ejpam-3780	543	13	2rk−	2rk−	NUM
ejpam-3780	543	14	1	1	NUM
ejpam-3780	543	15	<	<	X
ejpam-3780	543	16	(	(	PUNCT
ejpam-3780	543	17	2rk	2rk	ADJ
ejpam-3780	543	18	+	+	CCONJ
ejpam-3780	543	19	1)(n)−	1)(n)−	NUM
ejpam-3780	543	20	2rk	2rk	NOUN
ejpam-3780	544	1	=	=	PUNCT
ejpam-3780	544	2	2nrk	2nrk	NUM
ejpam-3780	544	3	+	+	NUM
ejpam-3780	544	4	n−	n−	NOUN
ejpam-3780	544	5	2rk	2rk	ADJ
ejpam-3780	545	1	,	,	PUNCT
ejpam-3780	545	2	it	it	PRON
ejpam-3780	545	3	follows	follow	VERB
ejpam-3780	545	4	that	that	DET
ejpam-3780	545	5	a0	a0	PROPN
ejpam-3780	545	6	/∈	/∈	PROPN
ejpam-3780	545	7	k.	k.	PROPN
ejpam-3780	545	8	example	example	NOUN
ejpam-3780	546	1	11	11	NUM
ejpam-3780	546	2	.	.	PUNCT
ejpam-3780	547	1	figure	figure	VERB
ejpam-3780	547	2	12	12	NUM
ejpam-3780	547	3	shows	show	VERB
ejpam-3780	547	4	an	an	DET
ejpam-3780	547	5	efficient	efficient	ADJ
ejpam-3780	547	6	zero	zero	NUM
ejpam-3780	547	7	ring	ring	NOUN
ejpam-3780	547	8	labeling	labeling	NOUN
ejpam-3780	547	9	of	of	ADP
ejpam-3780	547	10	a	a	DET
ejpam-3780	547	11	lobster	lobster	NOUN
ejpam-3780	547	12	g	g	PROPN
ejpam-3780	547	13	in	in	ADP
ejpam-3780	547	14	which	which	PRON
ejpam-3780	547	15	each	each	DET
ejpam-3780	547	16	elbow	elbow	NOUN
ejpam-3780	547	17	has	have	AUX
ejpam-3780	547	18	at	at	ADP
ejpam-3780	547	19	most	most	ADV
ejpam-3780	547	20	one	one	NUM
ejpam-3780	547	21	hanging	hang	VERB
ejpam-3780	547	22	leaf	leaf	NOUN
ejpam-3780	547	23	.	.	PUNCT
ejpam-3780	548	1	in	in	ADP
ejpam-3780	548	2	this	this	DET
ejpam-3780	548	3	labeling	labeling	NOUN
ejpam-3780	548	4	,	,	PUNCT
ejpam-3780	548	5	the	the	DET
ejpam-3780	548	6	set	set	NOUN
ejpam-3780	548	7	of	of	ADP
ejpam-3780	548	8	sums	sum	NOUN
ejpam-3780	548	9	is	be	AUX
ejpam-3780	548	10	k	k	NOUN
ejpam-3780	548	11	=	=	PUNCT
ejpam-3780	548	12	{	{	PUNCT
ejpam-3780	548	13	a54	a54	PROPN
ejpam-3780	548	14	,	,	PUNCT
ejpam-3780	548	15	a55	a55	PROPN
ejpam-3780	548	16	,	,	PUNCT
ejpam-3780	548	17	a56	a56	PROPN
ejpam-3780	548	18	,	,	PUNCT
ejpam-3780	548	19	a57	a57	PROPN
ejpam-3780	548	20	,	,	PUNCT
ejpam-3780	548	21	a58	a58	PROPN
ejpam-3780	548	22	,	,	PUNCT
ejpam-3780	548	23	a63	a63	NOUN
ejpam-3780	548	24	}	}	PUNCT
ejpam-3780	548	25	and	and	CCONJ
ejpam-3780	548	26	thus	thus	ADV
ejpam-3780	548	27	|k|	|k|	NOUN
ejpam-3780	548	28	=	=	SYM
ejpam-3780	548	29	6	6	NUM
ejpam-3780	548	30	=	=	SYM
ejpam-3780	548	31	∆(g	∆(g	NOUN
ejpam-3780	548	32	)	)	PUNCT
ejpam-3780	548	33	.	.	PUNCT
ejpam-3780	549	1	theorem	theorem	NOUN
ejpam-3780	549	2	9	9	NUM
ejpam-3780	549	3	.	.	PUNCT
ejpam-3780	550	1	let	let	VERB
ejpam-3780	550	2	g	g	PRON
ejpam-3780	550	3	be	be	AUX
ejpam-3780	550	4	a	a	DET
ejpam-3780	550	5	rooted	rooted	ADJ
ejpam-3780	550	6	tree	tree	NOUN
ejpam-3780	550	7	whose	whose	DET
ejpam-3780	550	8	root	root	NOUN
ejpam-3780	550	9	has	have	VERB
ejpam-3780	550	10	the	the	DET
ejpam-3780	550	11	maximum	maximum	ADJ
ejpam-3780	550	12	degree	degree	NOUN
ejpam-3780	550	13	.	.	PUNCT
ejpam-3780	551	1	if	if	SCONJ
ejpam-3780	551	2	the	the	DET
ejpam-3780	551	3	height	height	NOUN
ejpam-3780	551	4	of	of	ADP
ejpam-3780	551	5	g	g	PROPN
ejpam-3780	551	6	is	be	AUX
ejpam-3780	551	7	at	at	ADP
ejpam-3780	551	8	most	most	ADV
ejpam-3780	551	9	two	two	NUM
ejpam-3780	551	10	,	,	PUNCT
ejpam-3780	551	11	then	then	ADV
ejpam-3780	551	12	g	g	PROPN
ejpam-3780	551	13	has	have	VERB
ejpam-3780	551	14	an	an	DET
ejpam-3780	551	15	efficient	efficient	ADJ
ejpam-3780	551	16	zero	zero	NUM
ejpam-3780	551	17	ring	ring	NOUN
ejpam-3780	551	18	labeling	labeling	NOUN
ejpam-3780	551	19	.	.	PUNCT
ejpam-3780	552	1	proof	proof	NOUN
ejpam-3780	552	2	.	.	PUNCT
ejpam-3780	553	1	let	let	VERB
ejpam-3780	553	2	g	g	PRON
ejpam-3780	553	3	be	be	AUX
ejpam-3780	553	4	a	a	DET
ejpam-3780	553	5	rooted	rooted	ADJ
ejpam-3780	553	6	tree	tree	NOUN
ejpam-3780	553	7	of	of	ADP
ejpam-3780	553	8	height	height	NOUN
ejpam-3780	553	9	two	two	NUM
ejpam-3780	553	10	and	and	CCONJ
ejpam-3780	553	11	with	with	ADP
ejpam-3780	553	12	root	root	NOUN
ejpam-3780	553	13	w	w	ADP
ejpam-3780	553	14	such	such	ADJ
ejpam-3780	553	15	that	that	DET
ejpam-3780	553	16	d(w	d(w	PROPN
ejpam-3780	553	17	)	)	PUNCT
ejpam-3780	553	18	=	=	SYM
ejpam-3780	553	19	n	n	PROPN
ejpam-3780	553	20	=	=	SYM
ejpam-3780	553	21	∆(g	∆(g	NOUN
ejpam-3780	553	22	)	)	PUNCT
ejpam-3780	553	23	=	=	PUNCT
ejpam-3780	553	24	n.	n.	NOUN
ejpam-3780	553	25	if	if	SCONJ
ejpam-3780	553	26	n	n	PRON
ejpam-3780	553	27	is	be	AUX
ejpam-3780	553	28	one	one	NUM
ejpam-3780	553	29	or	or	CCONJ
ejpam-3780	553	30	two	two	NUM
ejpam-3780	553	31	,	,	PUNCT
ejpam-3780	553	32	then	then	ADV
ejpam-3780	553	33	g	g	PROPN
ejpam-3780	553	34	is	be	AUX
ejpam-3780	553	35	a	a	DET
ejpam-3780	553	36	caterpillar	caterpillar	NOUN
ejpam-3780	553	37	.	.	PUNCT
ejpam-3780	554	1	thus	thus	ADV
ejpam-3780	554	2	,	,	PUNCT
ejpam-3780	554	3	by	by	ADP
ejpam-3780	554	4	theorem	theorem	NOUN
ejpam-3780	554	5	6	6	NUM
ejpam-3780	554	6	,	,	PUNCT
ejpam-3780	554	7	g	g	PROPN
ejpam-3780	554	8	has	have	VERB
ejpam-3780	554	9	an	an	DET
ejpam-3780	554	10	efficient	efficient	ADJ
ejpam-3780	554	11	zero	zero	NUM
ejpam-3780	554	12	ring	ring	NOUN
ejpam-3780	554	13	labeling	labeling	NOUN
ejpam-3780	554	14	.	.	PUNCT
ejpam-3780	555	1	suppose	suppose	VERB
ejpam-3780	555	2	n	n	PRON
ejpam-3780	555	3	is	be	AUX
ejpam-3780	555	4	at	at	ADV
ejpam-3780	555	5	least	least	ADJ
ejpam-3780	555	6	three	three	NUM
ejpam-3780	555	7	,	,	PUNCT
ejpam-3780	555	8	and	and	CCONJ
ejpam-3780	555	9	there	there	PRON
ejpam-3780	555	10	exists	exist	VERB
ejpam-3780	555	11	at	at	ADP
ejpam-3780	555	12	least	least	ADV
ejpam-3780	555	13	one	one	NUM
ejpam-3780	555	14	vertex	vertex	NOUN
ejpam-3780	555	15	of	of	ADP
ejpam-3780	555	16	level	level	NOUN
ejpam-3780	555	17	two	two	NUM
ejpam-3780	555	18	.	.	PUNCT
ejpam-3780	556	1	denote	denote	VERB
ejpam-3780	556	2	the	the	DET
ejpam-3780	556	3	children	child	NOUN
ejpam-3780	556	4	of	of	ADP
ejpam-3780	556	5	w	w	NOUN
ejpam-3780	556	6	by	by	ADP
ejpam-3780	556	7	w1	w1	NOUN
ejpam-3780	556	8	,	,	PUNCT
ejpam-3780	556	9	w2	w2	NOUN
ejpam-3780	556	10	,	,	PUNCT
ejpam-3780	556	11	.	.	PUNCT
ejpam-3780	556	12	.	.	PUNCT
ejpam-3780	557	1	.	.	PUNCT
ejpam-3780	558	1	,	,	PUNCT
ejpam-3780	558	2	wn	wn	PROPN
ejpam-3780	558	3	.	.	PUNCT
ejpam-3780	558	4	since	since	SCONJ
ejpam-3780	558	5	w	w	PROPN
ejpam-3780	558	6	has	have	VERB
ejpam-3780	558	7	n	n	NUM
ejpam-3780	558	8	children	child	NOUN
ejpam-3780	558	9	and	and	CCONJ
ejpam-3780	558	10	∆(g	∆(g	NOUN
ejpam-3780	558	11	)	)	PUNCT
ejpam-3780	558	12	=	=	SYM
ejpam-3780	559	1	n	n	CCONJ
ejpam-3780	559	2	,	,	PUNCT
ejpam-3780	559	3	every	every	DET
ejpam-3780	559	4	child	child	NOUN
ejpam-3780	559	5	of	of	ADP
ejpam-3780	559	6	w	w	PROPN
ejpam-3780	559	7	has	have	VERB
ejpam-3780	559	8	at	at	ADP
ejpam-3780	559	9	most	most	ADJ
ejpam-3780	559	10	n−	n−	NOUN
ejpam-3780	559	11	1	1	NUM
ejpam-3780	559	12	children	child	NOUN
ejpam-3780	559	13	.	.	PUNCT
ejpam-3780	560	1	suppose	suppose	VERB
ejpam-3780	560	2	wi	wi	PROPN
ejpam-3780	560	3	has	have	VERB
ejpam-3780	560	4	ri	ri	PROPN
ejpam-3780	560	5	children	child	NOUN
ejpam-3780	560	6	,	,	PUNCT
ejpam-3780	560	7	and	and	CCONJ
ejpam-3780	560	8	let	let	VERB
ejpam-3780	560	9	wi	wi	PROPN
ejpam-3780	560	10	,	,	PUNCT
ejpam-3780	560	11	j	j	PROPN
ejpam-3780	560	12	,	,	PUNCT
ejpam-3780	560	13	where	where	SCONJ
ejpam-3780	560	14	j	j	PROPN
ejpam-3780	560	15	=	=	SYM
ejpam-3780	560	16	1	1	NUM
ejpam-3780	560	17	,	,	PUNCT
ejpam-3780	560	18	2	2	NUM
ejpam-3780	560	19	,	,	PUNCT
ejpam-3780	560	20	.	.	PUNCT
ejpam-3780	560	21	.	.	PUNCT
ejpam-3780	561	1	.	.	PUNCT
ejpam-3780	562	1	,	,	PUNCT
ejpam-3780	562	2	ri	ri	PROPN
ejpam-3780	562	3	denote	denote	VERB
ejpam-3780	562	4	the	the	DET
ejpam-3780	562	5	children	child	NOUN
ejpam-3780	562	6	of	of	ADP
ejpam-3780	562	7	wi	wi	PROPN
ejpam-3780	562	8	.	.	PUNCT
ejpam-3780	563	1	define	define	VERB
ejpam-3780	563	2	a	a	DET
ejpam-3780	563	3	function	function	NOUN
ejpam-3780	563	4	f	f	NOUN
ejpam-3780	563	5	:	:	PUNCT
ejpam-3780	563	6	v	v	X
ejpam-3780	563	7	(	(	PUNCT
ejpam-3780	563	8	g	g	NOUN
ejpam-3780	563	9	)	)	PUNCT
ejpam-3780	563	10	→	→	SYM
ejpam-3780	563	11	m0	m0	PROPN
ejpam-3780	563	12	2	2	NUM
ejpam-3780	563	13	(	(	PUNCT
ejpam-3780	563	14	z2n+1−1	z2n+1−1	NOUN
ejpam-3780	563	15	)	)	PUNCT
ejpam-3780	563	16	such	such	ADJ
ejpam-3780	563	17	that	that	SCONJ
ejpam-3780	563	18	f(w	f(w	NOUN
ejpam-3780	563	19	)	)	PUNCT
ejpam-3780	563	20	=	=	SYM
ejpam-3780	563	21	a0	a0	PROPN
ejpam-3780	563	22	,	,	PUNCT
ejpam-3780	563	23	f(wi	f(wi	NOUN
ejpam-3780	563	24	)	)	PUNCT
ejpam-3780	563	25	=	=	SYM
ejpam-3780	563	26	a2i+1−3	a2i+1−3	ADJ
ejpam-3780	563	27	,	,	PUNCT
ejpam-3780	563	28	and	and	CCONJ
ejpam-3780	563	29	f(wi	f(wi	NOUN
ejpam-3780	563	30	,	,	PUNCT
ejpam-3780	563	31	j	j	NOUN
ejpam-3780	563	32	)	)	PUNCT
ejpam-3780	563	33	=	=	PUNCT
ejpam-3780	564	1			PRON
ejpam-3780	564	2	a2i+j+1−2i+1	a2i+j+1−2i+1	VERB
ejpam-3780	564	3	if	if	SCONJ
ejpam-3780	564	4	i	i	PRON
ejpam-3780	564	5	+	+	X
ejpam-3780	564	6	j	j	X
ejpam-3780	564	7	<	<	X
ejpam-3780	564	8	n	n	PROPN
ejpam-3780	564	9	+	+	CCONJ
ejpam-3780	564	10	1	1	NUM
ejpam-3780	564	11	a2n+1−2i+1	a2n+1−2i+1	NOUN
ejpam-3780	564	12	+	+	NOUN
ejpam-3780	564	13	3	3	NUM
ejpam-3780	564	14	if	if	SCONJ
ejpam-3780	564	15	i	i	PRON
ejpam-3780	564	16	+	+	X
ejpam-3780	564	17	j	j	NOUN
ejpam-3780	564	18	=	=	SYM
ejpam-3780	565	1	n	n	PROPN
ejpam-3780	565	2	+	+	CCONJ
ejpam-3780	565	3	1	1	NUM
ejpam-3780	565	4	a2n+1−2i+1	a2n+1−2i+1	NOUN
ejpam-3780	565	5	+	+	NOUN
ejpam-3780	565	6	2i+j−n+1−1	2i+j−n+1−1	NOUN
ejpam-3780	565	7	if	if	SCONJ
ejpam-3780	565	8	i	i	PRON
ejpam-3780	565	9	+	+	NUM
ejpam-3780	565	10	j	j	PROPN
ejpam-3780	565	11	>	>	X
ejpam-3780	565	12	n	n	PROPN
ejpam-3780	566	1	+	+	ADV
ejpam-3780	566	2	1	1	NUM
ejpam-3780	566	3	.	.	PUNCT
ejpam-3780	567	1	(	(	PUNCT
ejpam-3780	567	2	34	34	NUM
ejpam-3780	567	3	)	)	PUNCT
ejpam-3780	567	4	d.	d.	PROPN
ejpam-3780	567	5	chua	chua	PROPN
ejpam-3780	567	6	,	,	PUNCT
ejpam-3780	567	7	f.	f.	PROPN
ejpam-3780	567	8	campeña	campeña	PROPN
ejpam-3780	567	9	,	,	PUNCT
ejpam-3780	567	10	f.	f.	PROPN
ejpam-3780	567	11	franco	franco	PROPN
ejpam-3780	567	12	/	/	SYM
ejpam-3780	567	13	eur	eur	PROPN
ejpam-3780	567	14	.	.	PUNCT
ejpam-3780	568	1	j.	j.	PROPN
ejpam-3780	568	2	pure	pure	PROPN
ejpam-3780	568	3	appl	appl	PROPN
ejpam-3780	568	4	.	.	PROPN
ejpam-3780	568	5	math	math	PROPN
ejpam-3780	568	6	,	,	PUNCT
ejpam-3780	568	7	13	13	NUM
ejpam-3780	568	8	(	(	PUNCT
ejpam-3780	568	9	3	3	NUM
ejpam-3780	568	10	)	)	PUNCT
ejpam-3780	568	11	(	(	PUNCT
ejpam-3780	568	12	2020	2020	NUM
ejpam-3780	568	13	)	)	PUNCT
ejpam-3780	568	14	,	,	PUNCT
ejpam-3780	568	15	674	674	NUM
ejpam-3780	568	16	-	-	SYM
ejpam-3780	568	17	696	696	NUM
ejpam-3780	568	18	689	689	NUM
ejpam-3780	568	19	a0	a0	NOUN
ejpam-3780	568	20	a54	a54	NOUN
ejpam-3780	568	21	a9	a9	PROPN
ejpam-3780	568	22	a45	a45	PROPN
ejpam-3780	568	23	a18a0	a18a0	PROPN
ejpam-3780	568	24	a63	a63	NOUN
ejpam-3780	568	25	a0	a0	PROPN
ejpam-3780	568	26	a55	a55	PROPN
ejpam-3780	568	27	a18	a18	PROPN
ejpam-3780	568	28	a36	a36	PROPN
ejpam-3780	568	29	a27	a27	PROPN
ejpam-3780	568	30	a54	a54	PROPN
ejpam-3780	568	31	a1	a1	NOUN
ejpam-3780	568	32	a62	a62	PROPN
ejpam-3780	568	33	a54	a54	PROPN
ejpam-3780	568	34	a2	a2	PROPN
ejpam-3780	568	35	a61	a61	PROPN
ejpam-3780	568	36	a54	a54	PROPN
ejpam-3780	568	37	a3	a3	NOUN
ejpam-3780	568	38	a9	a9	PROPN
ejpam-3780	568	39	a46	a46	VERB
ejpam-3780	568	40	a9	a9	PROPN
ejpam-3780	568	41	a47	a47	NOUN
ejpam-3780	568	42	a9	a9	PROPN
ejpam-3780	568	43	a48	a48	ADP
ejpam-3780	568	44	a45	a45	PROPN
ejpam-3780	568	45	a10	a10	PROPN
ejpam-3780	568	46	a53	a53	PROPN
ejpam-3780	568	47	a45	a45	PROPN
ejpam-3780	568	48	a11	a11	PROPN
ejpam-3780	568	49	a52	a52	PROPN
ejpam-3780	568	50	a45	a45	PROPN
ejpam-3780	568	51	a12	a12	PROPN
ejpam-3780	568	52	a51	a51	PROPN
ejpam-3780	568	53	a45	a45	PROPN
ejpam-3780	568	54	a13	a13	PROPN
ejpam-3780	568	55	a50	a50	NOUN
ejpam-3780	568	56	a18	a18	PROPN
ejpam-3780	568	57	a37	a37	PROPN
ejpam-3780	568	58	a26	a26	PROPN
ejpam-3780	568	59	a18	a18	PROPN
ejpam-3780	568	60	a38	a38	PROPN
ejpam-3780	568	61	a25	a25	PROPN
ejpam-3780	568	62	figure	figure	NOUN
ejpam-3780	568	63	12	12	NUM
ejpam-3780	568	64	:	:	PUNCT
ejpam-3780	568	65	efficient	efficient	ADJ
ejpam-3780	568	66	zero	zero	NUM
ejpam-3780	568	67	ring	ring	NOUN
ejpam-3780	568	68	labeling	labeling	NOUN
ejpam-3780	568	69	of	of	ADP
ejpam-3780	568	70	a	a	DET
ejpam-3780	568	71	lobster	lobster	NOUN
ejpam-3780	568	72	using	use	VERB
ejpam-3780	568	73	m0	m0	PROPN
ejpam-3780	568	74	2	2	NUM
ejpam-3780	568	75	(	(	PUNCT
ejpam-3780	568	76	z64	z64	NOUN
ejpam-3780	568	77	)	)	PUNCT
ejpam-3780	568	78	clearly	clearly	ADV
ejpam-3780	568	79	,	,	PUNCT
ejpam-3780	568	80	f	f	PROPN
ejpam-3780	568	81	is	be	AUX
ejpam-3780	568	82	injective	injective	ADJ
ejpam-3780	568	83	.	.	PUNCT
ejpam-3780	569	1	let	let	VERB
ejpam-3780	569	2	k	k	NOUN
ejpam-3780	569	3	=	=	PRON
ejpam-3780	569	4	{	{	PUNCT
ejpam-3780	569	5	f(u	f(u	PROPN
ejpam-3780	569	6	)	)	PUNCT
ejpam-3780	569	7	+	+	NUM
ejpam-3780	569	8	f(v	f(v	NOUN
ejpam-3780	569	9	)	)	PUNCT
ejpam-3780	569	10	:	:	PUNCT
ejpam-3780	569	11	uv	uv	PROPN
ejpam-3780	569	12	∈	∈	PROPN
ejpam-3780	569	13	e(g	e(g	PROPN
ejpam-3780	569	14	)	)	PUNCT
ejpam-3780	569	15	}	}	PUNCT
ejpam-3780	569	16	.	.	PUNCT
ejpam-3780	570	1	to	to	PART
ejpam-3780	570	2	show	show	VERB
ejpam-3780	570	3	that	that	SCONJ
ejpam-3780	570	4	f	f	PROPN
ejpam-3780	570	5	is	be	AUX
ejpam-3780	570	6	an	an	DET
ejpam-3780	570	7	efficient	efficient	ADJ
ejpam-3780	570	8	zero	zero	NUM
ejpam-3780	570	9	ring	ring	NOUN
ejpam-3780	570	10	labeling	labeling	NOUN
ejpam-3780	570	11	of	of	ADP
ejpam-3780	570	12	g	g	NOUN
ejpam-3780	570	13	,	,	PUNCT
ejpam-3780	570	14	we	we	PRON
ejpam-3780	570	15	need	need	VERB
ejpam-3780	570	16	to	to	PART
ejpam-3780	570	17	show	show	VERB
ejpam-3780	570	18	that	that	SCONJ
ejpam-3780	570	19	|k|	|k|	NOUN
ejpam-3780	570	20	=	=	SYM
ejpam-3780	570	21	∆(g	∆(g	PROPN
ejpam-3780	570	22	)	)	PUNCT
ejpam-3780	570	23	=	=	SYM
ejpam-3780	570	24	n	n	PROPN
ejpam-3780	570	25	and	and	CCONJ
ejpam-3780	570	26	a0	a0	PROPN
ejpam-3780	570	27	/∈	/∈	PROPN
ejpam-3780	571	1	k.	k.	PROPN
ejpam-3780	572	1	for	for	ADP
ejpam-3780	572	2	pairs	pair	NOUN
ejpam-3780	572	3	of	of	ADP
ejpam-3780	572	4	adjacent	adjacent	ADJ
ejpam-3780	572	5	root	root	NOUN
ejpam-3780	572	6	and	and	CCONJ
ejpam-3780	572	7	vertex	vertex	NOUN
ejpam-3780	572	8	of	of	ADP
ejpam-3780	572	9	level	level	NOUN
ejpam-3780	572	10	one	one	NUM
ejpam-3780	572	11	,	,	PUNCT
ejpam-3780	572	12	we	we	PRON
ejpam-3780	572	13	obtain	obtain	VERB
ejpam-3780	572	14	f(w	f(w	NUM
ejpam-3780	572	15	)	)	PUNCT
ejpam-3780	573	1	+	+	CCONJ
ejpam-3780	573	2	f(wi	f(wi	NOUN
ejpam-3780	573	3	)	)	PUNCT
ejpam-3780	573	4	=	=	SYM
ejpam-3780	573	5	a0	a0	PROPN
ejpam-3780	573	6	+	+	CCONJ
ejpam-3780	573	7	a2i+1−3	a2i+1−3	ADJ
ejpam-3780	573	8	=	=	SYM
ejpam-3780	573	9	a2i+1−3	a2i+1−3	ADJ
ejpam-3780	573	10	(	(	PUNCT
ejpam-3780	573	11	35	35	NUM
ejpam-3780	573	12	)	)	PUNCT
ejpam-3780	573	13	for	for	ADP
ejpam-3780	573	14	i	i	PROPN
ejpam-3780	573	15	=	=	SYM
ejpam-3780	573	16	1	1	NUM
ejpam-3780	573	17	,	,	PUNCT
ejpam-3780	573	18	2	2	NUM
ejpam-3780	573	19	,	,	PUNCT
ejpam-3780	573	20	.	.	PUNCT
ejpam-3780	573	21	.	.	PUNCT
ejpam-3780	574	1	.	.	PUNCT
ejpam-3780	575	1	,	,	PUNCT
ejpam-3780	575	2	n.	n.	PROPN
ejpam-3780	575	3	then	then	ADV
ejpam-3780	575	4	we	we	PRON
ejpam-3780	575	5	have	have	VERB
ejpam-3780	575	6	the	the	DET
ejpam-3780	575	7	sums	sum	NOUN
ejpam-3780	575	8	a1	a1	NOUN
ejpam-3780	575	9	,	,	PUNCT
ejpam-3780	575	10	a5	a5	NOUN
ejpam-3780	575	11	,	,	PUNCT
ejpam-3780	575	12	.	.	PUNCT
ejpam-3780	575	13	.	.	PUNCT
ejpam-3780	576	1	.	.	PUNCT
ejpam-3780	577	1	,	,	PUNCT
ejpam-3780	577	2	a2n+1−3	a2n+1−3	PROPN
ejpam-3780	577	3	.	.	PUNCT
ejpam-3780	578	1	for	for	ADP
ejpam-3780	578	2	pairs	pair	NOUN
ejpam-3780	578	3	of	of	ADP
ejpam-3780	578	4	adjacent	adjacent	ADJ
ejpam-3780	578	5	vertex	vertex	NOUN
ejpam-3780	578	6	of	of	ADP
ejpam-3780	578	7	level	level	NOUN
ejpam-3780	578	8	one	one	NUM
ejpam-3780	578	9	and	and	CCONJ
ejpam-3780	578	10	vertex	vertex	NOUN
ejpam-3780	578	11	of	of	ADP
ejpam-3780	578	12	level	level	NOUN
ejpam-3780	578	13	two	two	NUM
ejpam-3780	578	14	,	,	PUNCT
ejpam-3780	578	15	we	we	PRON
ejpam-3780	578	16	have	have	VERB
ejpam-3780	578	17	three	three	NUM
ejpam-3780	578	18	cases	case	NOUN
ejpam-3780	578	19	.	.	PUNCT
ejpam-3780	579	1	case	case	NOUN
ejpam-3780	579	2	1	1	NUM
ejpam-3780	579	3	:	:	PUNCT
ejpam-3780	579	4	suppose	suppose	VERB
ejpam-3780	579	5	i	i	PRON
ejpam-3780	579	6	+	+	NOUN
ejpam-3780	579	7	j	j	X
ejpam-3780	579	8	<	<	X
ejpam-3780	579	9	n	n	PROPN
ejpam-3780	579	10	+	+	NOUN
ejpam-3780	579	11	1	1	NUM
ejpam-3780	579	12	.	.	PUNCT
ejpam-3780	580	1	then	then	ADV
ejpam-3780	580	2	we	we	PRON
ejpam-3780	580	3	obtain	obtain	VERB
ejpam-3780	580	4	f(wi	f(wi	NOUN
ejpam-3780	580	5	)	)	PUNCT
ejpam-3780	581	1	+	+	NUM
ejpam-3780	582	1	f(wj	f(wj	PROPN
ejpam-3780	582	2	i	i	NOUN
ejpam-3780	582	3	)	)	PUNCT
ejpam-3780	582	4	=	=	PUNCT
ejpam-3780	583	1	a2i+1−3	a2i+1−3	ADJ
ejpam-3780	583	2	+	+	NUM
ejpam-3780	583	3	a2i+j+1−2i+1	a2i+j+1−2i+1	NOUN
ejpam-3780	583	4	=	=	SYM
ejpam-3780	583	5	a2i+j+1−3	a2i+j+1−3	PROPN
ejpam-3780	583	6	.	.	PUNCT
ejpam-3780	584	1	(	(	PUNCT
ejpam-3780	584	2	36	36	NUM
ejpam-3780	584	3	)	)	PUNCT
ejpam-3780	584	4	since	since	SCONJ
ejpam-3780	584	5	each	each	PRON
ejpam-3780	584	6	of	of	ADP
ejpam-3780	584	7	i	i	PRON
ejpam-3780	584	8	and	and	CCONJ
ejpam-3780	584	9	j	j	PROPN
ejpam-3780	584	10	are	be	AUX
ejpam-3780	584	11	at	at	ADV
ejpam-3780	584	12	least	least	ADJ
ejpam-3780	584	13	one	one	NUM
ejpam-3780	584	14	,	,	PUNCT
ejpam-3780	584	15	we	we	PRON
ejpam-3780	584	16	have	have	VERB
ejpam-3780	584	17	the	the	DET
ejpam-3780	584	18	sums	sum	NOUN
ejpam-3780	584	19	a2m+1−3	a2m+1−3	ADJ
ejpam-3780	584	20	for	for	ADP
ejpam-3780	584	21	m	m	PROPN
ejpam-3780	584	22	=	=	SYM
ejpam-3780	584	23	2	2	NUM
ejpam-3780	584	24	,	,	PUNCT
ejpam-3780	584	25	3	3	NUM
ejpam-3780	584	26	,	,	PUNCT
ejpam-3780	584	27	.	.	PUNCT
ejpam-3780	584	28	.	.	PUNCT
ejpam-3780	585	1	.	.	PUNCT
ejpam-3780	586	1	,	,	PUNCT
ejpam-3780	586	2	n.	n.	NOUN
ejpam-3780	586	3	these	these	PRON
ejpam-3780	586	4	are	be	AUX
ejpam-3780	586	5	a5	a5	NOUN
ejpam-3780	586	6	,	,	PUNCT
ejpam-3780	586	7	a13	a13	NOUN
ejpam-3780	586	8	,	,	PUNCT
ejpam-3780	586	9	.	.	PUNCT
ejpam-3780	586	10	.	.	PUNCT
ejpam-3780	587	1	.	.	PUNCT
ejpam-3780	588	1	,	,	PUNCT
ejpam-3780	588	2	a2n+1−3	a2n+1−3	PROPN
ejpam-3780	588	3	.	.	PUNCT
ejpam-3780	589	1	note	note	VERB
ejpam-3780	589	2	that	that	SCONJ
ejpam-3780	589	3	these	these	DET
ejpam-3780	589	4	sums	sum	NOUN
ejpam-3780	589	5	are	be	AUX
ejpam-3780	589	6	same	same	ADJ
ejpam-3780	589	7	with	with	ADP
ejpam-3780	589	8	those	those	PRON
ejpam-3780	589	9	from	from	ADP
ejpam-3780	589	10	pairs	pair	NOUN
ejpam-3780	589	11	of	of	ADP
ejpam-3780	589	12	adjacent	adjacent	ADJ
ejpam-3780	589	13	root	root	NOUN
ejpam-3780	589	14	and	and	CCONJ
ejpam-3780	589	15	vertex	vertex	NOUN
ejpam-3780	589	16	of	of	ADP
ejpam-3780	589	17	level	level	NOUN
ejpam-3780	589	18	one	one	NUM
ejpam-3780	589	19	,	,	PUNCT
ejpam-3780	589	20	only	only	ADV
ejpam-3780	589	21	that	that	PRON
ejpam-3780	589	22	a1	a1	NOUN
ejpam-3780	589	23	is	be	AUX
ejpam-3780	589	24	not	not	PART
ejpam-3780	589	25	included	include	VERB
ejpam-3780	589	26	.	.	PUNCT
ejpam-3780	590	1	case	case	NOUN
ejpam-3780	590	2	2	2	NUM
ejpam-3780	590	3	:	:	PUNCT
ejpam-3780	590	4	suppose	suppose	VERB
ejpam-3780	590	5	i	i	PRON
ejpam-3780	590	6	+	+	NUM
ejpam-3780	590	7	j	j	NOUN
ejpam-3780	590	8	=	=	SYM
ejpam-3780	590	9	n	n	PROPN
ejpam-3780	590	10	+	+	NOUN
ejpam-3780	590	11	1	1	X
ejpam-3780	590	12	.	.	PUNCT
ejpam-3780	591	1	then	then	ADV
ejpam-3780	591	2	we	we	PRON
ejpam-3780	591	3	obtain	obtain	VERB
ejpam-3780	591	4	f(wi	f(wi	NOUN
ejpam-3780	591	5	)	)	PUNCT
ejpam-3780	592	1	+	+	NUM
ejpam-3780	593	1	f(wj	f(wj	PROPN
ejpam-3780	593	2	i	i	NOUN
ejpam-3780	593	3	)	)	PUNCT
ejpam-3780	593	4	=	=	PUNCT
ejpam-3780	594	1	a2i+1−3	a2i+1−3	ADJ
ejpam-3780	595	1	+	+	SYM
ejpam-3780	595	2	a2n+1−2i+1	a2n+1−2i+1	ADJ
ejpam-3780	595	3	+	+	NOUN
ejpam-3780	595	4	3	3	NUM
ejpam-3780	595	5	=	=	SYM
ejpam-3780	595	6	a2n+1	a2n+1	PROPN
ejpam-3780	595	7	=	=	SYM
ejpam-3780	595	8	a1	a1	NOUN
ejpam-3780	595	9	.	.	PUNCT
ejpam-3780	596	1	(	(	PUNCT
ejpam-3780	596	2	37	37	NUM
ejpam-3780	596	3	)	)	PUNCT
ejpam-3780	596	4	thus	thus	ADV
ejpam-3780	596	5	,	,	PUNCT
ejpam-3780	596	6	in	in	ADP
ejpam-3780	596	7	this	this	DET
ejpam-3780	596	8	case	case	NOUN
ejpam-3780	596	9	,	,	PUNCT
ejpam-3780	596	10	there	there	PRON
ejpam-3780	596	11	is	be	VERB
ejpam-3780	596	12	only	only	ADV
ejpam-3780	596	13	one	one	NUM
ejpam-3780	596	14	possible	possible	ADJ
ejpam-3780	596	15	sum	sum	NOUN
ejpam-3780	596	16	,	,	PUNCT
ejpam-3780	596	17	which	which	PRON
ejpam-3780	596	18	is	be	AUX
ejpam-3780	596	19	a1	a1	PROPN
ejpam-3780	596	20	.	.	PUNCT
ejpam-3780	596	21	case	case	NOUN
ejpam-3780	596	22	3	3	X
ejpam-3780	596	23	:	:	PUNCT
ejpam-3780	596	24	suppose	suppose	VERB
ejpam-3780	596	25	i	i	PRON
ejpam-3780	596	26	+	+	NUM
ejpam-3780	596	27	j	j	PROPN
ejpam-3780	596	28	>	>	X
ejpam-3780	596	29	n	n	PROPN
ejpam-3780	597	1	+	+	NOUN
ejpam-3780	597	2	1	1	X
ejpam-3780	597	3	.	.	PUNCT
ejpam-3780	597	4	then	then	ADV
ejpam-3780	597	5	we	we	PRON
ejpam-3780	597	6	obtain	obtain	VERB
ejpam-3780	597	7	f(wi	f(wi	NOUN
ejpam-3780	597	8	)	)	PUNCT
ejpam-3780	598	1	+	+	NUM
ejpam-3780	599	1	f(wj	f(wj	PROPN
ejpam-3780	599	2	i	i	NOUN
ejpam-3780	599	3	)	)	PUNCT
ejpam-3780	599	4	=	=	PUNCT
ejpam-3780	600	1	a2i+1−3	a2i+1−3	ADJ
ejpam-3780	601	1	+	+	SYM
ejpam-3780	601	2	a2n+1−2i+1	a2n+1−2i+1	VERB
ejpam-3780	601	3	+	+	NOUN
ejpam-3780	601	4	2i+j−n+1−1	2i+j−n+1−1	NOUN
ejpam-3780	601	5	=	=	SYM
ejpam-3780	601	6	a2n+1	a2n+1	VERB
ejpam-3780	601	7	+	+	PROPN
ejpam-3780	601	8	2i+j−n+1−4	2i+j−n+1−4	NUM
ejpam-3780	601	9	=	=	SYM
ejpam-3780	601	10	a2i+j−n+1−3	a2i+j−n+1−3	ADJ
ejpam-3780	601	11	.	.	PUNCT
ejpam-3780	602	1	(	(	PUNCT
ejpam-3780	602	2	38	38	NUM
ejpam-3780	602	3	)	)	PUNCT
ejpam-3780	602	4	since	since	SCONJ
ejpam-3780	602	5	i	i	PRON
ejpam-3780	602	6	+	+	NUM
ejpam-3780	602	7	j	j	PROPN
ejpam-3780	602	8	>	>	X
ejpam-3780	602	9	n	n	PROPN
ejpam-3780	602	10	+	+	NOUN
ejpam-3780	602	11	1	1	NUM
ejpam-3780	602	12	,	,	PUNCT
ejpam-3780	602	13	it	it	PRON
ejpam-3780	602	14	follows	follow	VERB
ejpam-3780	602	15	that	that	SCONJ
ejpam-3780	602	16	i	i	PRON
ejpam-3780	602	17	+	+	NUM
ejpam-3780	602	18	j	j	PROPN
ejpam-3780	602	19	is	be	AUX
ejpam-3780	602	20	at	at	ADP
ejpam-3780	602	21	least	least	ADJ
ejpam-3780	602	22	n+2	n+2	NUM
ejpam-3780	602	23	.	.	PUNCT
ejpam-3780	603	1	also	also	ADV
ejpam-3780	603	2	,	,	PUNCT
ejpam-3780	603	3	i	i	PRON
ejpam-3780	603	4	is	be	AUX
ejpam-3780	603	5	at	at	ADP
ejpam-3780	603	6	most	most	ADJ
ejpam-3780	603	7	n	n	ADJ
ejpam-3780	603	8	and	and	CCONJ
ejpam-3780	603	9	j	j	PROPN
ejpam-3780	603	10	is	be	AUX
ejpam-3780	603	11	at	at	ADP
ejpam-3780	603	12	most	most	ADJ
ejpam-3780	603	13	n−	n−	NOUN
ejpam-3780	603	14	1	1	NUM
ejpam-3780	603	15	,	,	PUNCT
ejpam-3780	603	16	so	so	ADV
ejpam-3780	603	17	i+	i+	ADV
ejpam-3780	603	18	j	j	PROPN
ejpam-3780	603	19	is	be	AUX
ejpam-3780	603	20	at	at	ADP
ejpam-3780	603	21	most	most	ADJ
ejpam-3780	604	1	2n−	2n−	NUM
ejpam-3780	604	2	1	1	NUM
ejpam-3780	604	3	.	.	PUNCT
ejpam-3780	605	1	hence	hence	ADV
ejpam-3780	605	2	,	,	PUNCT
ejpam-3780	605	3	2i+j−n+1−	2i+j−n+1−	PROPN
ejpam-3780	605	4	3	3	NUM
ejpam-3780	605	5	≥	≥	NOUN
ejpam-3780	605	6	2(n+2)−n+1−	2(n+2)−n+1−	NUM
ejpam-3780	605	7	3	3	NUM
ejpam-3780	605	8	=	=	SYM
ejpam-3780	605	9	23−	23−	NUM
ejpam-3780	605	10	3	3	NUM
ejpam-3780	605	11	=	=	SYM
ejpam-3780	605	12	5	5	NUM
ejpam-3780	605	13	and	and	CCONJ
ejpam-3780	605	14	2i+j−n+1−	2i+j−n+1−	NUM
ejpam-3780	605	15	3	3	NUM
ejpam-3780	605	16	≤	≤	NOUN
ejpam-3780	605	17	2(2n−1)−n+1−	2(2n−1)−n+1−	NUM
ejpam-3780	605	18	3	3	NUM
ejpam-3780	605	19	=	=	SYM
ejpam-3780	605	20	2n−	2n−	PROPN
ejpam-3780	605	21	3	3	NUM
ejpam-3780	605	22	.	.	PUNCT
ejpam-3780	606	1	then	then	ADV
ejpam-3780	606	2	we	we	PRON
ejpam-3780	606	3	have	have	VERB
ejpam-3780	606	4	the	the	DET
ejpam-3780	606	5	sums	sum	NOUN
ejpam-3780	606	6	a5	a5	PROPN
ejpam-3780	606	7	,	,	PUNCT
ejpam-3780	606	8	a13	a13	NOUN
ejpam-3780	606	9	,	,	PUNCT
ejpam-3780	606	10	.	.	PUNCT
ejpam-3780	606	11	.	.	PUNCT
ejpam-3780	607	1	.	.	PUNCT
ejpam-3780	608	1	,	,	PUNCT
ejpam-3780	608	2	a2n−3	a2n−3	PROPN
ejpam-3780	608	3	.	.	PUNCT
ejpam-3780	608	4	d.	d.	PROPN
ejpam-3780	608	5	chua	chua	PROPN
ejpam-3780	608	6	,	,	PUNCT
ejpam-3780	608	7	f.	f.	PROPN
ejpam-3780	608	8	campeña	campeña	PROPN
ejpam-3780	608	9	,	,	PUNCT
ejpam-3780	608	10	f.	f.	PROPN
ejpam-3780	608	11	franco	franco	PROPN
ejpam-3780	608	12	/	/	SYM
ejpam-3780	608	13	eur	eur	PROPN
ejpam-3780	608	14	.	.	PUNCT
ejpam-3780	609	1	j.	j.	PROPN
ejpam-3780	609	2	pure	pure	PROPN
ejpam-3780	609	3	appl	appl	PROPN
ejpam-3780	609	4	.	.	PROPN
ejpam-3780	609	5	math	math	PROPN
ejpam-3780	609	6	,	,	PUNCT
ejpam-3780	609	7	13	13	NUM
ejpam-3780	609	8	(	(	PUNCT
ejpam-3780	609	9	3	3	NUM
ejpam-3780	609	10	)	)	PUNCT
ejpam-3780	609	11	(	(	PUNCT
ejpam-3780	609	12	2020	2020	NUM
ejpam-3780	609	13	)	)	PUNCT
ejpam-3780	609	14	,	,	PUNCT
ejpam-3780	609	15	674	674	NUM
ejpam-3780	609	16	-	-	SYM
ejpam-3780	609	17	696	696	NUM
ejpam-3780	609	18	690	690	NUM
ejpam-3780	609	19	it	it	PRON
ejpam-3780	609	20	can	can	AUX
ejpam-3780	609	21	be	be	AUX
ejpam-3780	609	22	observed	observe	VERB
ejpam-3780	609	23	that	that	SCONJ
ejpam-3780	609	24	these	these	DET
ejpam-3780	609	25	sums	sum	NOUN
ejpam-3780	609	26	are	be	AUX
ejpam-3780	609	27	same	same	ADJ
ejpam-3780	609	28	with	with	ADP
ejpam-3780	609	29	those	those	PRON
ejpam-3780	609	30	from	from	ADP
ejpam-3780	609	31	pairs	pair	NOUN
ejpam-3780	609	32	of	of	ADP
ejpam-3780	609	33	adjacent	adjacent	ADJ
ejpam-3780	609	34	root	root	NOUN
ejpam-3780	609	35	and	and	CCONJ
ejpam-3780	609	36	vertex	vertex	NOUN
ejpam-3780	609	37	of	of	ADP
ejpam-3780	609	38	level	level	NOUN
ejpam-3780	609	39	one	one	NUM
ejpam-3780	609	40	,	,	PUNCT
ejpam-3780	609	41	only	only	ADV
ejpam-3780	609	42	that	that	DET
ejpam-3780	609	43	a1	a1	NOUN
ejpam-3780	609	44	and	and	CCONJ
ejpam-3780	609	45	a2n+1−3	a2n+1−3	ADJ
ejpam-3780	609	46	are	be	AUX
ejpam-3780	609	47	not	not	PART
ejpam-3780	609	48	included	include	VERB
ejpam-3780	609	49	.	.	PUNCT
ejpam-3780	610	1	therefore	therefore	ADV
ejpam-3780	610	2	,	,	PUNCT
ejpam-3780	610	3	k	k	PROPN
ejpam-3780	610	4	=	=	PRON
ejpam-3780	610	5	{	{	PUNCT
ejpam-3780	610	6	a1	a1	PROPN
ejpam-3780	610	7	,	,	PUNCT
ejpam-3780	610	8	a5	a5	NOUN
ejpam-3780	610	9	,	,	PUNCT
ejpam-3780	610	10	.	.	PUNCT
ejpam-3780	610	11	.	.	PUNCT
ejpam-3780	610	12	.	.	PUNCT
ejpam-3780	611	1	,	,	PUNCT
ejpam-3780	611	2	a2n+1−3	a2n+1−3	ADJ
ejpam-3780	611	3	}	}	PUNCT
ejpam-3780	611	4	,	,	PUNCT
ejpam-3780	611	5	(	(	PUNCT
ejpam-3780	611	6	39	39	NUM
ejpam-3780	611	7	)	)	PUNCT
ejpam-3780	611	8	hence	hence	ADV
ejpam-3780	611	9	|k|	|k|	NOUN
ejpam-3780	611	10	=	=	SYM
ejpam-3780	611	11	n.	n.	NOUN
ejpam-3780	611	12	for	for	ADP
ejpam-3780	611	13	m	m	PROPN
ejpam-3780	611	14	=	=	SYM
ejpam-3780	611	15	1	1	NUM
ejpam-3780	611	16	,	,	PUNCT
ejpam-3780	611	17	5	5	NUM
ejpam-3780	611	18	,	,	PUNCT
ejpam-3780	611	19	.	.	PUNCT
ejpam-3780	611	20	.	.	PUNCT
ejpam-3780	612	1	.	.	PUNCT
ejpam-3780	613	1	,	,	PUNCT
ejpam-3780	613	2	2n+1	2n+1	PROPN
ejpam-3780	613	3	−	−	NOUN
ejpam-3780	613	4	3	3	NUM
ejpam-3780	613	5	,	,	PUNCT
ejpam-3780	613	6	it	it	PRON
ejpam-3780	613	7	is	be	AUX
ejpam-3780	613	8	also	also	ADV
ejpam-3780	613	9	clear	clear	ADJ
ejpam-3780	613	10	that	that	SCONJ
ejpam-3780	613	11	m	m	VERB
ejpam-3780	613	12	>	>	X
ejpam-3780	613	13	0	0	PUNCT
ejpam-3780	614	1	and	and	CCONJ
ejpam-3780	614	2	m	m	PRON
ejpam-3780	614	3	<	<	X
ejpam-3780	614	4	2n+1	2n+1	PROPN
ejpam-3780	614	5	−	−	NOUN
ejpam-3780	614	6	1	1	NUM
ejpam-3780	614	7	.	.	PUNCT
ejpam-3780	615	1	thus	thus	ADV
ejpam-3780	615	2	,	,	PUNCT
ejpam-3780	615	3	a0	a0	PROPN
ejpam-3780	615	4	/∈	/∈	PROPN
ejpam-3780	615	5	k.	k.	PROPN
ejpam-3780	615	6	example	example	NOUN
ejpam-3780	616	1	12	12	NUM
ejpam-3780	616	2	.	.	PUNCT
ejpam-3780	617	1	figure	figure	NOUN
ejpam-3780	617	2	13	13	NUM
ejpam-3780	617	3	shows	show	VERB
ejpam-3780	617	4	an	an	DET
ejpam-3780	617	5	efficient	efficient	ADJ
ejpam-3780	617	6	zero	zero	NUM
ejpam-3780	617	7	ring	ring	NOUN
ejpam-3780	617	8	labeling	labeling	NOUN
ejpam-3780	617	9	of	of	ADP
ejpam-3780	617	10	a	a	DET
ejpam-3780	617	11	rooted	rooted	ADJ
ejpam-3780	617	12	tree	tree	NOUN
ejpam-3780	617	13	g	g	NOUN
ejpam-3780	617	14	with	with	ADP
ejpam-3780	617	15	a	a	DET
ejpam-3780	617	16	height	height	NOUN
ejpam-3780	617	17	of	of	ADP
ejpam-3780	617	18	two	two	NUM
ejpam-3780	617	19	.	.	PUNCT
ejpam-3780	618	1	in	in	ADP
ejpam-3780	618	2	this	this	DET
ejpam-3780	618	3	labeling	labeling	NOUN
ejpam-3780	618	4	,	,	PUNCT
ejpam-3780	618	5	the	the	DET
ejpam-3780	618	6	set	set	NOUN
ejpam-3780	618	7	of	of	ADP
ejpam-3780	618	8	sums	sum	NOUN
ejpam-3780	618	9	is	be	AUX
ejpam-3780	618	10	k	k	NOUN
ejpam-3780	618	11	=	=	PUNCT
ejpam-3780	618	12	{	{	PUNCT
ejpam-3780	618	13	a1	a1	PROPN
ejpam-3780	618	14	,	,	PUNCT
ejpam-3780	618	15	a5	a5	NOUN
ejpam-3780	618	16	,	,	PUNCT
ejpam-3780	618	17	a13	a13	PROPN
ejpam-3780	618	18	,	,	PUNCT
ejpam-3780	618	19	a29	a29	PROPN
ejpam-3780	618	20	}	}	PUNCT
ejpam-3780	618	21	and	and	CCONJ
ejpam-3780	618	22	thus	thus	ADV
ejpam-3780	618	23	|k|	|k|	NOUN
ejpam-3780	618	24	=	=	SYM
ejpam-3780	618	25	4	4	NUM
ejpam-3780	618	26	=	=	SYM
ejpam-3780	618	27	∆(g	∆(g	NOUN
ejpam-3780	618	28	)	)	PUNCT
ejpam-3780	618	29	.	.	PUNCT
ejpam-3780	619	1	a0	a0	PROPN
ejpam-3780	619	2	a1	a1	PROPN
ejpam-3780	619	3	a0	a0	PROPN
ejpam-3780	619	4	a5	a5	PROPN
ejpam-3780	619	5	a0	a0	PROPN
ejpam-3780	619	6	a13	a13	PROPN
ejpam-3780	619	7	a0	a0	PROPN
ejpam-3780	619	8	a29	a29	PROPN
ejpam-3780	619	9	a1	a1	PROPN
ejpam-3780	619	10	a4	a4	NOUN
ejpam-3780	619	11	a1	a1	NOUN
ejpam-3780	619	12	a12	a12	NOUN
ejpam-3780	619	13	a1	a1	PROPN
ejpam-3780	619	14	a28	a28	PROPN
ejpam-3780	619	15	a5	a5	PROPN
ejpam-3780	619	16	a8	a8	PROPN
ejpam-3780	619	17	a5	a5	PROPN
ejpam-3780	619	18	a24	a24	PROPN
ejpam-3780	619	19	a5	a5	PROPN
ejpam-3780	619	20	a27	a27	PROPN
ejpam-3780	619	21	a13	a13	PROPN
ejpam-3780	619	22	a16	a16	PROPN
ejpam-3780	619	23	a13	a13	PROPN
ejpam-3780	619	24	a19	a19	PROPN
ejpam-3780	619	25	a13	a13	PROPN
ejpam-3780	619	26	a23	a23	PROPN
ejpam-3780	619	27	a29	a29	PROPN
ejpam-3780	619	28	a3	a3	PROPN
ejpam-3780	619	29	a29	a29	PROPN
ejpam-3780	619	30	a7	a7	PROPN
ejpam-3780	619	31	a29	a29	PROPN
ejpam-3780	619	32	a15	a15	PROPN
ejpam-3780	619	33	figure	figure	NOUN
ejpam-3780	619	34	13	13	NUM
ejpam-3780	619	35	:	:	PUNCT
ejpam-3780	619	36	efficient	efficient	ADJ
ejpam-3780	619	37	zero	zero	NUM
ejpam-3780	619	38	ring	ring	NOUN
ejpam-3780	619	39	labeling	labeling	NOUN
ejpam-3780	619	40	of	of	ADP
ejpam-3780	619	41	a	a	DET
ejpam-3780	619	42	rooted	rooted	ADJ
ejpam-3780	619	43	tree	tree	NOUN
ejpam-3780	619	44	with	with	ADP
ejpam-3780	619	45	a	a	DET
ejpam-3780	619	46	height	height	NOUN
ejpam-3780	619	47	of	of	ADP
ejpam-3780	619	48	two	two	NUM
ejpam-3780	619	49	using	use	VERB
ejpam-3780	619	50	m0	m0	PROPN
ejpam-3780	619	51	2	2	NUM
ejpam-3780	619	52	(	(	PUNCT
ejpam-3780	619	53	z31	z31	NOUN
ejpam-3780	619	54	)	)	PUNCT
ejpam-3780	619	55	theorem	theorem	VERB
ejpam-3780	619	56	10	10	NUM
ejpam-3780	619	57	.	.	PUNCT
ejpam-3780	620	1	let	let	VERB
ejpam-3780	620	2	g	g	PRON
ejpam-3780	620	3	be	be	AUX
ejpam-3780	620	4	a	a	DET
ejpam-3780	620	5	disjoint	disjoint	NOUN
ejpam-3780	620	6	union	union	NOUN
ejpam-3780	620	7	of	of	ADP
ejpam-3780	620	8	a	a	DET
ejpam-3780	620	9	finite	finite	ADJ
ejpam-3780	620	10	number	number	NOUN
ejpam-3780	620	11	of	of	ADP
ejpam-3780	620	12	caterpillars	caterpillar	NOUN
ejpam-3780	620	13	.	.	PUNCT
ejpam-3780	621	1	then	then	ADV
ejpam-3780	621	2	g	g	PROPN
ejpam-3780	621	3	has	have	VERB
ejpam-3780	621	4	an	an	DET
ejpam-3780	621	5	efficient	efficient	ADJ
ejpam-3780	621	6	zero	zero	NUM
ejpam-3780	621	7	ring	ring	NOUN
ejpam-3780	621	8	labeling	labeling	NOUN
ejpam-3780	621	9	.	.	PUNCT
ejpam-3780	622	1	proof	proof	NOUN
ejpam-3780	622	2	.	.	PUNCT
ejpam-3780	623	1	let	let	VERB
ejpam-3780	623	2	g	g	PRON
ejpam-3780	623	3	be	be	AUX
ejpam-3780	623	4	a	a	DET
ejpam-3780	623	5	disjoint	disjoint	NOUN
ejpam-3780	623	6	union	union	NOUN
ejpam-3780	623	7	of	of	ADP
ejpam-3780	623	8	p	p	PROPN
ejpam-3780	623	9	caterpillars	caterpillar	NOUN
ejpam-3780	623	10	.	.	PUNCT
ejpam-3780	624	1	we	we	PRON
ejpam-3780	624	2	show	show	VERB
ejpam-3780	624	3	that	that	SCONJ
ejpam-3780	624	4	g	g	PROPN
ejpam-3780	624	5	has	have	VERB
ejpam-3780	624	6	an	an	DET
ejpam-3780	624	7	efficient	efficient	ADJ
ejpam-3780	624	8	zero	zero	NUM
ejpam-3780	624	9	ring	ring	NOUN
ejpam-3780	624	10	labeling	labeling	NOUN
ejpam-3780	624	11	.	.	PUNCT
ejpam-3780	625	1	case	case	NOUN
ejpam-3780	625	2	1	1	NUM
ejpam-3780	625	3	:	:	PUNCT
ejpam-3780	625	4	the	the	DET
ejpam-3780	625	5	order	order	NOUN
ejpam-3780	625	6	of	of	ADP
ejpam-3780	625	7	each	each	DET
ejpam-3780	625	8	caterpillar	caterpillar	NOUN
ejpam-3780	625	9	is	be	AUX
ejpam-3780	625	10	one	one	NUM
ejpam-3780	625	11	or	or	CCONJ
ejpam-3780	625	12	two	two	NUM
ejpam-3780	625	13	;	;	PUNCT
ejpam-3780	625	14	that	that	PRON
ejpam-3780	625	15	is	is	ADV
ejpam-3780	625	16	,	,	PUNCT
ejpam-3780	625	17	each	each	DET
ejpam-3780	625	18	caterpillar	caterpillar	NOUN
ejpam-3780	625	19	is	be	AUX
ejpam-3780	625	20	either	either	CCONJ
ejpam-3780	625	21	the	the	DET
ejpam-3780	625	22	trivial	trivial	ADJ
ejpam-3780	625	23	graph	graph	NOUN
ejpam-3780	625	24	or	or	CCONJ
ejpam-3780	625	25	p2	p2	NOUN
ejpam-3780	625	26	.	.	PUNCT
ejpam-3780	626	1	if	if	SCONJ
ejpam-3780	626	2	all	all	DET
ejpam-3780	626	3	p	p	NOUN
ejpam-3780	626	4	caterpillars	caterpillar	NOUN
ejpam-3780	626	5	are	be	AUX
ejpam-3780	626	6	trivial	trivial	ADJ
ejpam-3780	626	7	,	,	PUNCT
ejpam-3780	626	8	then	then	ADV
ejpam-3780	626	9	∆(g	∆(g	NOUN
ejpam-3780	626	10	)	)	PUNCT
ejpam-3780	626	11	=	=	SYM
ejpam-3780	627	1	0	0	X
ejpam-3780	627	2	.	.	PUNCT
ejpam-3780	628	1	we	we	PRON
ejpam-3780	628	2	label	label	VERB
ejpam-3780	628	3	the	the	DET
ejpam-3780	628	4	vertices	vertex	NOUN
ejpam-3780	628	5	by	by	ADP
ejpam-3780	628	6	a0	a0	PROPN
ejpam-3780	628	7	,	,	PUNCT
ejpam-3780	628	8	a1	a1	PROPN
ejpam-3780	628	9	,	,	PUNCT
ejpam-3780	628	10	a2	a2	PROPN
ejpam-3780	628	11	,	,	PUNCT
ejpam-3780	628	12	.	.	PUNCT
ejpam-3780	628	13	.	.	PUNCT
ejpam-3780	629	1	.	.	PUNCT
ejpam-3780	630	1	,	,	PUNCT
ejpam-3780	630	2	ap−1	ap−1	INTJ
ejpam-3780	630	3	.	.	PUNCT
ejpam-3780	631	1	clearly	clearly	ADV
ejpam-3780	631	2	,	,	PUNCT
ejpam-3780	631	3	the	the	DET
ejpam-3780	631	4	labeling	labeling	NOUN
ejpam-3780	631	5	is	be	AUX
ejpam-3780	631	6	injective	injective	ADJ
ejpam-3780	631	7	.	.	PUNCT
ejpam-3780	632	1	moreover	moreover	ADV
ejpam-3780	632	2	,	,	PUNCT
ejpam-3780	632	3	the	the	DET
ejpam-3780	632	4	set	set	NOUN
ejpam-3780	632	5	of	of	ADP
ejpam-3780	632	6	sums	sum	NOUN
ejpam-3780	632	7	is	be	AUX
ejpam-3780	632	8	empty	empty	ADJ
ejpam-3780	632	9	and	and	CCONJ
ejpam-3780	632	10	thus	thus	ADV
ejpam-3780	632	11	a0	a0	PROPN
ejpam-3780	632	12	/∈	/∈	PROPN
ejpam-3780	633	1	k.	k.	PROPN
ejpam-3780	634	1	we	we	PRON
ejpam-3780	634	2	are	be	AUX
ejpam-3780	634	3	done	do	VERB
ejpam-3780	634	4	.	.	PUNCT
ejpam-3780	635	1	suppose	suppose	VERB
ejpam-3780	635	2	at	at	ADP
ejpam-3780	635	3	least	least	ADV
ejpam-3780	635	4	one	one	NUM
ejpam-3780	635	5	caterpillar	caterpillar	NOUN
ejpam-3780	635	6	is	be	AUX
ejpam-3780	635	7	not	not	PART
ejpam-3780	635	8	trivial	trivial	ADJ
ejpam-3780	635	9	.	.	PUNCT
ejpam-3780	636	1	then	then	ADV
ejpam-3780	636	2	∆(g	∆(g	PROPN
ejpam-3780	636	3	)	)	PUNCT
ejpam-3780	636	4	=	=	SYM
ejpam-3780	637	1	1	1	X
ejpam-3780	637	2	.	.	PUNCT
ejpam-3780	637	3	let	let	VERB
ejpam-3780	637	4	g1	g1	PROPN
ejpam-3780	637	5	,	,	PUNCT
ejpam-3780	637	6	g2	g2	PROPN
ejpam-3780	637	7	,	,	PUNCT
ejpam-3780	637	8	.	.	PUNCT
ejpam-3780	637	9	.	.	PUNCT
ejpam-3780	638	1	.	.	PUNCT
ejpam-3780	639	1	,	,	PUNCT
ejpam-3780	639	2	gm	gm	PROPN
ejpam-3780	639	3	denote	denote	VERB
ejpam-3780	639	4	the	the	DET
ejpam-3780	639	5	caterpillars	caterpillar	NOUN
ejpam-3780	639	6	with	with	ADP
ejpam-3780	639	7	two	two	NUM
ejpam-3780	639	8	vertices	vertex	NOUN
ejpam-3780	639	9	,	,	PUNCT
ejpam-3780	639	10	and	and	CCONJ
ejpam-3780	639	11	let	let	VERB
ejpam-3780	639	12	gm+1	gm+1	NUM
ejpam-3780	639	13	,	,	PUNCT
ejpam-3780	639	14	gm+2	gm+2	NOUN
ejpam-3780	639	15	,	,	PUNCT
ejpam-3780	639	16	.	.	PUNCT
ejpam-3780	639	17	.	.	PUNCT
ejpam-3780	640	1	.	.	PUNCT
ejpam-3780	641	1	,	,	PUNCT
ejpam-3780	641	2	gp	gp	PROPN
ejpam-3780	641	3	denote	denote	VERB
ejpam-3780	641	4	the	the	DET
ejpam-3780	641	5	trivial	trivial	ADJ
ejpam-3780	641	6	caterpillars	caterpillar	NOUN
ejpam-3780	641	7	,	,	PUNCT
ejpam-3780	641	8	where	where	SCONJ
ejpam-3780	641	9	1	1	NUM
ejpam-3780	641	10	≤	≤	NUM
ejpam-3780	641	11	m	m	VERB
ejpam-3780	641	12	≤	≤	NOUN
ejpam-3780	642	1	p.	p.	NOUN
ejpam-3780	642	2	let	let	VERB
ejpam-3780	642	3	v	v	X
ejpam-3780	642	4	(	(	PUNCT
ejpam-3780	642	5	gi	gi	INTJ
ejpam-3780	642	6	)	)	PUNCT
ejpam-3780	642	7	=	=	NOUN
ejpam-3780	642	8	{	{	PUNCT
ejpam-3780	642	9	wi−1	wi−1	PROPN
ejpam-3780	642	10	,	,	PUNCT
ejpam-3780	642	11	w2m−i+1	w2m−i+1	ADV
ejpam-3780	642	12	}	}	PUNCT
ejpam-3780	642	13	for	for	ADP
ejpam-3780	642	14	i	i	PROPN
ejpam-3780	642	15	=	=	NOUN
ejpam-3780	642	16	1	1	NUM
ejpam-3780	642	17	,	,	PUNCT
ejpam-3780	642	18	2	2	NUM
ejpam-3780	642	19	,	,	PUNCT
ejpam-3780	642	20	.	.	PUNCT
ejpam-3780	642	21	.	.	PUNCT
ejpam-3780	643	1	.	.	PUNCT
ejpam-3780	644	1	,	,	PUNCT
ejpam-3780	644	2	m	m	PROPN
ejpam-3780	644	3	,	,	PUNCT
ejpam-3780	645	1	and	and	CCONJ
ejpam-3780	645	2	let	let	VERB
ejpam-3780	645	3	v	v	X
ejpam-3780	645	4	(	(	PUNCT
ejpam-3780	645	5	gi	gi	INTJ
ejpam-3780	645	6	)	)	PUNCT
ejpam-3780	645	7	=	=	PRON
ejpam-3780	645	8	{	{	PUNCT
ejpam-3780	645	9	wi+m	wi+m	PROPN
ejpam-3780	645	10	}	}	PUNCT
ejpam-3780	645	11	for	for	ADP
ejpam-3780	645	12	i	i	PRON
ejpam-3780	645	13	=	=	NOUN
ejpam-3780	645	14	m	m	VERB
ejpam-3780	645	15	+	+	NOUN
ejpam-3780	645	16	1,m	1,m	ADJ
ejpam-3780	646	1	+	+	NUM
ejpam-3780	646	2	2	2	NUM
ejpam-3780	646	3	,	,	PUNCT
ejpam-3780	646	4	.	.	PUNCT
ejpam-3780	646	5	.	.	PUNCT
ejpam-3780	647	1	.	.	PUNCT
ejpam-3780	648	1	,	,	PUNCT
ejpam-3780	648	2	p.	p.	NOUN
ejpam-3780	648	3	define	define	VERB
ejpam-3780	648	4	a	a	DET
ejpam-3780	648	5	function	function	NOUN
ejpam-3780	648	6	f	f	NOUN
ejpam-3780	648	7	:	:	PUNCT
ejpam-3780	648	8	v	v	X
ejpam-3780	648	9	(	(	PUNCT
ejpam-3780	648	10	g	g	NOUN
ejpam-3780	648	11	)	)	PUNCT
ejpam-3780	648	12	→	→	SYM
ejpam-3780	648	13	m0	m0	PROPN
ejpam-3780	648	14	2	2	NUM
ejpam-3780	648	15	(	(	PUNCT
ejpam-3780	648	16	zm+p+1	zm+p+1	NUM
ejpam-3780	648	17	)	)	PUNCT
ejpam-3780	648	18	such	such	ADJ
ejpam-3780	648	19	that	that	SCONJ
ejpam-3780	648	20	f(wi	f(wi	NOUN
ejpam-3780	648	21	)	)	PUNCT
ejpam-3780	648	22	=	=	VERB
ejpam-3780	648	23	ai	ai	VERB
ejpam-3780	648	24	.	.	PUNCT
ejpam-3780	649	1	let	let	VERB
ejpam-3780	649	2	k	k	NOUN
ejpam-3780	649	3	=	=	PRON
ejpam-3780	649	4	{	{	PUNCT
ejpam-3780	649	5	f(u	f(u	PROPN
ejpam-3780	649	6	)	)	PUNCT
ejpam-3780	649	7	+	+	NUM
ejpam-3780	649	8	f(v	f(v	NOUN
ejpam-3780	649	9	)	)	PUNCT
ejpam-3780	649	10	:	:	PUNCT
ejpam-3780	649	11	uv	uv	PROPN
ejpam-3780	649	12	∈	∈	PROPN
ejpam-3780	649	13	e(g	e(g	PROPN
ejpam-3780	649	14	)	)	PUNCT
ejpam-3780	649	15	}	}	PUNCT
ejpam-3780	649	16	.	.	PUNCT
ejpam-3780	650	1	clearly	clearly	ADV
ejpam-3780	650	2	,	,	PUNCT
ejpam-3780	650	3	f	f	PROPN
ejpam-3780	650	4	is	be	AUX
ejpam-3780	650	5	injective	injective	ADJ
ejpam-3780	650	6	.	.	PUNCT
ejpam-3780	651	1	to	to	PART
ejpam-3780	651	2	show	show	VERB
ejpam-3780	651	3	that	that	SCONJ
ejpam-3780	651	4	f	f	PROPN
ejpam-3780	651	5	is	be	AUX
ejpam-3780	651	6	an	an	DET
ejpam-3780	651	7	efficient	efficient	ADJ
ejpam-3780	651	8	zero	zero	NUM
ejpam-3780	651	9	ring	ring	NOUN
ejpam-3780	651	10	labeling	labeling	NOUN
ejpam-3780	651	11	of	of	ADP
ejpam-3780	651	12	g	g	NOUN
ejpam-3780	651	13	,	,	PUNCT
ejpam-3780	651	14	we	we	PRON
ejpam-3780	651	15	need	need	VERB
ejpam-3780	651	16	to	to	PART
ejpam-3780	651	17	show	show	VERB
ejpam-3780	651	18	that	that	SCONJ
ejpam-3780	651	19	|k|	|k|	NOUN
ejpam-3780	651	20	=	=	SYM
ejpam-3780	651	21	∆(g	∆(g	PROPN
ejpam-3780	651	22	)	)	PUNCT
ejpam-3780	651	23	=	=	SYM
ejpam-3780	651	24	1	1	NUM
ejpam-3780	651	25	and	and	CCONJ
ejpam-3780	651	26	a0	a0	PROPN
ejpam-3780	651	27	/∈	/∈	PROPN
ejpam-3780	652	1	k.	k.	PROPN
ejpam-3780	653	1	we	we	PRON
ejpam-3780	653	2	obtain	obtain	VERB
ejpam-3780	653	3	the	the	DET
ejpam-3780	653	4	sums	sum	NOUN
ejpam-3780	653	5	f(wi−1	f(wi−1	NOUN
ejpam-3780	653	6	)	)	PUNCT
ejpam-3780	653	7	+	+	NUM
ejpam-3780	653	8	f(w2m−i+1	f(w2m−i+1	NOUN
ejpam-3780	653	9	)	)	PUNCT
ejpam-3780	653	10	=	=	PUNCT
ejpam-3780	654	1	ai−1	ai−1	PROPN
ejpam-3780	654	2	+	+	NUM
ejpam-3780	654	3	a2m−i+1	a2m−i+1	X
ejpam-3780	654	4	=	=	SYM
ejpam-3780	654	5	a2	a2	PROPN
ejpam-3780	654	6	m	m	PROPN
ejpam-3780	654	7	(	(	PUNCT
ejpam-3780	654	8	40	40	NUM
ejpam-3780	654	9	)	)	PUNCT
ejpam-3780	654	10	d.	d.	PROPN
ejpam-3780	654	11	chua	chua	PROPN
ejpam-3780	654	12	,	,	PUNCT
ejpam-3780	654	13	f.	f.	PROPN
ejpam-3780	654	14	campeña	campeña	PROPN
ejpam-3780	654	15	,	,	PUNCT
ejpam-3780	654	16	f.	f.	PROPN
ejpam-3780	654	17	franco	franco	PROPN
ejpam-3780	654	18	/	/	SYM
ejpam-3780	654	19	eur	eur	PROPN
ejpam-3780	654	20	.	.	PUNCT
ejpam-3780	655	1	j.	j.	PROPN
ejpam-3780	655	2	pure	pure	PROPN
ejpam-3780	655	3	appl	appl	PROPN
ejpam-3780	655	4	.	.	PROPN
ejpam-3780	655	5	math	math	PROPN
ejpam-3780	655	6	,	,	PUNCT
ejpam-3780	655	7	13	13	NUM
ejpam-3780	655	8	(	(	PUNCT
ejpam-3780	655	9	3	3	NUM
ejpam-3780	655	10	)	)	PUNCT
ejpam-3780	655	11	(	(	PUNCT
ejpam-3780	655	12	2020	2020	NUM
ejpam-3780	655	13	)	)	PUNCT
ejpam-3780	655	14	,	,	PUNCT
ejpam-3780	655	15	674	674	NUM
ejpam-3780	655	16	-	-	SYM
ejpam-3780	655	17	696	696	NUM
ejpam-3780	655	18	691	691	NUM
ejpam-3780	655	19	for	for	ADP
ejpam-3780	655	20	i	i	PRON
ejpam-3780	655	21	=	=	NOUN
ejpam-3780	655	22	1	1	NUM
ejpam-3780	655	23	,	,	PUNCT
ejpam-3780	655	24	2	2	NUM
ejpam-3780	655	25	,	,	PUNCT
ejpam-3780	655	26	.	.	PUNCT
ejpam-3780	655	27	.	.	PUNCT
ejpam-3780	655	28	.	.	PUNCT
ejpam-3780	656	1	,	,	PUNCT
ejpam-3780	656	2	m.	m.	NOUN
ejpam-3780	656	3	then	then	ADV
ejpam-3780	656	4	k	k	PROPN
ejpam-3780	656	5	=	=	PUNCT
ejpam-3780	656	6	{	{	PUNCT
ejpam-3780	656	7	a2	a2	PROPN
ejpam-3780	656	8	m	m	PROPN
ejpam-3780	656	9	}	}	PUNCT
ejpam-3780	656	10	and	and	CCONJ
ejpam-3780	656	11	thus	thus	ADV
ejpam-3780	656	12	|k|	|k|	NOUN
ejpam-3780	656	13	=	=	SYM
ejpam-3780	656	14	1	1	X
ejpam-3780	656	15	.	.	PUNCT
ejpam-3780	657	1	also	also	ADV
ejpam-3780	657	2	,	,	PUNCT
ejpam-3780	657	3	a2	a2	PROPN
ejpam-3780	657	4	m	m	PROPN
ejpam-3780	657	5	6=	6=	NOUN
ejpam-3780	657	6	a0	a0	PROPN
ejpam-3780	657	7	since	since	SCONJ
ejpam-3780	657	8	m	m	PROPN
ejpam-3780	657	9	>	>	X
ejpam-3780	657	10	0	0	PUNCT
ejpam-3780	657	11	and	and	CCONJ
ejpam-3780	657	12	m	m	VERB
ejpam-3780	657	13	<	<	X
ejpam-3780	657	14	m	m	VERB
ejpam-3780	657	15	+	+	X
ejpam-3780	657	16	p	p	X
ejpam-3780	658	1	+	+	CCONJ
ejpam-3780	658	2	1	1	NUM
ejpam-3780	658	3	≤	≤	NUM
ejpam-3780	658	4	m	m	VERB
ejpam-3780	658	5	+	+	NOUN
ejpam-3780	658	6	m	m	VERB
ejpam-3780	658	7	+	+	ADJ
ejpam-3780	658	8	1	1	NUM
ejpam-3780	658	9	=	=	SYM
ejpam-3780	658	10	2	2	NUM
ejpam-3780	658	11	m	m	NOUN
ejpam-3780	658	12	+	+	NOUN
ejpam-3780	658	13	1	1	NUM
ejpam-3780	658	14	.	.	X
ejpam-3780	658	15	case	case	NOUN
ejpam-3780	658	16	2	2	NUM
ejpam-3780	658	17	:	:	PUNCT
ejpam-3780	658	18	the	the	DET
ejpam-3780	658	19	order	order	NOUN
ejpam-3780	658	20	of	of	ADP
ejpam-3780	658	21	at	at	ADV
ejpam-3780	658	22	least	least	ADV
ejpam-3780	658	23	one	one	NUM
ejpam-3780	658	24	caterpillar	caterpillar	NOUN
ejpam-3780	658	25	is	be	AUX
ejpam-3780	658	26	at	at	ADV
ejpam-3780	658	27	least	least	ADJ
ejpam-3780	658	28	three	three	NUM
ejpam-3780	658	29	.	.	PUNCT
ejpam-3780	659	1	applying	apply	VERB
ejpam-3780	659	2	lemma	lemma	PROPN
ejpam-3780	659	3	1	1	NUM
ejpam-3780	659	4	,	,	PUNCT
ejpam-3780	659	5	we	we	PRON
ejpam-3780	659	6	denote	denote	VERB
ejpam-3780	659	7	the	the	DET
ejpam-3780	659	8	vertices	vertex	NOUN
ejpam-3780	659	9	of	of	ADP
ejpam-3780	659	10	each	each	DET
ejpam-3780	659	11	caterpillar	caterpillar	NOUN
ejpam-3780	659	12	such	such	ADJ
ejpam-3780	659	13	that	that	SCONJ
ejpam-3780	659	14	the	the	DET
ejpam-3780	659	15	endvertices	endvertice	NOUN
ejpam-3780	659	16	of	of	ADP
ejpam-3780	659	17	the	the	DET
ejpam-3780	659	18	central	central	ADJ
ejpam-3780	659	19	path	path	NOUN
ejpam-3780	659	20	have	have	VERB
ejpam-3780	659	21	no	no	DET
ejpam-3780	659	22	hanging	hang	VERB
ejpam-3780	659	23	leaf	leaf	NOUN
ejpam-3780	659	24	.	.	PUNCT
ejpam-3780	660	1	let	let	VERB
ejpam-3780	660	2	g1	g1	PROPN
ejpam-3780	660	3	,	,	PUNCT
ejpam-3780	660	4	g2	g2	PROPN
ejpam-3780	660	5	,	,	PUNCT
ejpam-3780	660	6	.	.	PUNCT
ejpam-3780	660	7	.	.	PUNCT
ejpam-3780	661	1	.	.	PUNCT
ejpam-3780	662	1	,	,	PUNCT
ejpam-3780	662	2	gp	gp	PROPN
ejpam-3780	662	3	denote	denote	VERB
ejpam-3780	662	4	the	the	DET
ejpam-3780	662	5	caterpillars	caterpillar	NOUN
ejpam-3780	662	6	.	.	PUNCT
ejpam-3780	663	1	suppose	suppose	VERB
ejpam-3780	663	2	the	the	DET
ejpam-3780	663	3	length	length	NOUN
ejpam-3780	663	4	of	of	ADP
ejpam-3780	663	5	the	the	DET
ejpam-3780	663	6	central	central	ADJ
ejpam-3780	663	7	path	path	NOUN
ejpam-3780	663	8	of	of	ADP
ejpam-3780	663	9	gi	gi	PROPN
ejpam-3780	663	10	has	have	VERB
ejpam-3780	663	11	si	si	NOUN
ejpam-3780	663	12	vertices	vertex	NOUN
ejpam-3780	663	13	.	.	PUNCT
ejpam-3780	664	1	denote	denote	VERB
ejpam-3780	664	2	the	the	DET
ejpam-3780	664	3	central	central	ADJ
ejpam-3780	664	4	path	path	NOUN
ejpam-3780	664	5	of	of	ADP
ejpam-3780	664	6	g1	g1	NOUN
ejpam-3780	664	7	by	by	ADP
ejpam-3780	664	8	[	[	X
ejpam-3780	664	9	w1	w1	NOUN
ejpam-3780	664	10	,	,	PUNCT
ejpam-3780	664	11	w2	w2	NOUN
ejpam-3780	664	12	,	,	PUNCT
ejpam-3780	664	13	.	.	PUNCT
ejpam-3780	664	14	.	.	PUNCT
ejpam-3780	665	1	.	.	PUNCT
ejpam-3780	666	1	,	,	PUNCT
ejpam-3780	666	2	ws1	ws1	X
ejpam-3780	666	3	]	]	X
ejpam-3780	666	4	,	,	PUNCT
ejpam-3780	666	5	and	and	CCONJ
ejpam-3780	666	6	the	the	DET
ejpam-3780	666	7	central	central	ADJ
ejpam-3780	666	8	path	path	NOUN
ejpam-3780	666	9	of	of	ADP
ejpam-3780	666	10	gi	gi	NOUN
ejpam-3780	666	11	by	by	ADP
ejpam-3780	666	12	[	[	X
ejpam-3780	666	13	wq+1	wq+1	PROPN
ejpam-3780	666	14	,	,	PUNCT
ejpam-3780	666	15	wq+2	wq+2	ADJ
ejpam-3780	666	16	,	,	PUNCT
ejpam-3780	666	17	.	.	PUNCT
ejpam-3780	666	18	.	.	PUNCT
ejpam-3780	667	1	.	.	PUNCT
ejpam-3780	668	1	,	,	PUNCT
ejpam-3780	668	2	wq+si	wq+si	X
ejpam-3780	668	3	]	]	PUNCT
ejpam-3780	668	4	for	for	ADP
ejpam-3780	668	5	i	i	PROPN
ejpam-3780	668	6	=	=	SYM
ejpam-3780	668	7	2	2	NUM
ejpam-3780	668	8	,	,	PUNCT
ejpam-3780	668	9	3	3	NUM
ejpam-3780	668	10	,	,	PUNCT
ejpam-3780	668	11	.	.	PUNCT
ejpam-3780	668	12	.	.	PUNCT
ejpam-3780	669	1	.	.	PUNCT
ejpam-3780	670	1	,	,	PUNCT
ejpam-3780	671	1	p	p	X
ejpam-3780	671	2	,	,	PUNCT
ejpam-3780	671	3	where	where	SCONJ
ejpam-3780	671	4	q	q	NOUN
ejpam-3780	671	5	=	=	SYM
ejpam-3780	671	6	s1	s1	PROPN
ejpam-3780	671	7	+	+	CCONJ
ejpam-3780	671	8	s2	s2	NOUN
ejpam-3780	671	9	+	+	CCONJ
ejpam-3780	671	10	·	·	PUNCT
ejpam-3780	671	11	·	·	PUNCT
ejpam-3780	671	12	·	·	PUNCT
ejpam-3780	671	13	+	+	NUM
ejpam-3780	671	14	si−1	si−1	PROPN
ejpam-3780	671	15	.	.	PUNCT
ejpam-3780	671	16	suppose	suppose	VERB
ejpam-3780	671	17	the	the	DET
ejpam-3780	671	18	number	number	NOUN
ejpam-3780	671	19	of	of	ADP
ejpam-3780	671	20	hanging	hang	VERB
ejpam-3780	671	21	leaves	leave	NOUN
ejpam-3780	671	22	of	of	ADP
ejpam-3780	671	23	wi	wi	PROPN
ejpam-3780	671	24	is	be	AUX
ejpam-3780	671	25	ri	ri	PROPN
ejpam-3780	671	26	,	,	PUNCT
ejpam-3780	671	27	and	and	CCONJ
ejpam-3780	671	28	denote	denote	VERB
ejpam-3780	671	29	the	the	DET
ejpam-3780	671	30	hanging	hang	VERB
ejpam-3780	671	31	leaves	leave	NOUN
ejpam-3780	671	32	of	of	ADP
ejpam-3780	671	33	wi	wi	PROPN
ejpam-3780	671	34	by	by	ADP
ejpam-3780	671	35	wj	wj	PROPN
ejpam-3780	672	1	i	i	PRON
ejpam-3780	672	2	,	,	PUNCT
ejpam-3780	672	3	where	where	SCONJ
ejpam-3780	672	4	j	j	PROPN
ejpam-3780	672	5	=	=	SYM
ejpam-3780	672	6	1	1	NUM
ejpam-3780	672	7	,	,	PUNCT
ejpam-3780	672	8	2	2	NUM
ejpam-3780	672	9	,	,	PUNCT
ejpam-3780	672	10	.	.	PUNCT
ejpam-3780	672	11	.	.	PUNCT
ejpam-3780	672	12	.	.	PUNCT
ejpam-3780	673	1	,	,	PUNCT
ejpam-3780	673	2	ri	ri	AUX
ejpam-3780	673	3	.	.	PROPN
ejpam-3780	673	4	suppose	suppose	VERB
ejpam-3780	673	5	the	the	DET
ejpam-3780	673	6	maximum	maximum	ADJ
ejpam-3780	673	7	number	number	NOUN
ejpam-3780	673	8	of	of	ADP
ejpam-3780	673	9	hanging	hang	VERB
ejpam-3780	673	10	leaves	leave	NOUN
ejpam-3780	673	11	of	of	ADP
ejpam-3780	673	12	a	a	DET
ejpam-3780	673	13	vertex	vertex	NOUN
ejpam-3780	673	14	in	in	ADP
ejpam-3780	673	15	g	g	PROPN
ejpam-3780	673	16	is	be	AUX
ejpam-3780	673	17	rk	rk	NOUN
ejpam-3780	673	18	.	.	PUNCT
ejpam-3780	674	1	since	since	SCONJ
ejpam-3780	674	2	the	the	DET
ejpam-3780	674	3	endvertices	endvertice	NOUN
ejpam-3780	674	4	in	in	ADP
ejpam-3780	674	5	the	the	DET
ejpam-3780	674	6	central	central	ADJ
ejpam-3780	674	7	path	path	NOUN
ejpam-3780	674	8	of	of	ADP
ejpam-3780	674	9	each	each	DET
ejpam-3780	674	10	caterpillar	caterpillar	NOUN
ejpam-3780	674	11	in	in	ADP
ejpam-3780	674	12	g	g	PROPN
ejpam-3780	674	13	have	have	VERB
ejpam-3780	674	14	no	no	DET
ejpam-3780	674	15	hanging	hang	VERB
ejpam-3780	674	16	leaf	leaf	NOUN
ejpam-3780	674	17	and	and	CCONJ
ejpam-3780	674	18	there	there	PRON
ejpam-3780	674	19	is	be	VERB
ejpam-3780	674	20	at	at	ADV
ejpam-3780	674	21	least	least	ADJ
ejpam-3780	674	22	one	one	NUM
ejpam-3780	674	23	caterpillar	caterpillar	NOUN
ejpam-3780	674	24	with	with	ADP
ejpam-3780	674	25	at	at	ADV
ejpam-3780	674	26	least	least	ADV
ejpam-3780	674	27	three	three	NUM
ejpam-3780	674	28	vertices	vertex	NOUN
ejpam-3780	674	29	,	,	PUNCT
ejpam-3780	674	30	it	it	PRON
ejpam-3780	674	31	follows	follow	VERB
ejpam-3780	674	32	that	that	SCONJ
ejpam-3780	674	33	∆(g	∆(g	NOUN
ejpam-3780	674	34	)	)	PUNCT
ejpam-3780	675	1	=	=	SYM
ejpam-3780	675	2	rk	rk	NOUN
ejpam-3780	675	3	+	+	NOUN
ejpam-3780	675	4	2	2	X
ejpam-3780	675	5	.	.	X
ejpam-3780	675	6	consider	consider	VERB
ejpam-3780	675	7	a	a	DET
ejpam-3780	675	8	caterpillar	caterpillar	ADJ
ejpam-3780	675	9	h	h	NOUN
ejpam-3780	675	10	such	such	ADJ
ejpam-3780	675	11	that	that	PRON
ejpam-3780	675	12	v	v	NOUN
ejpam-3780	675	13	(	(	PUNCT
ejpam-3780	675	14	h	h	NOUN
ejpam-3780	675	15	)	)	PUNCT
ejpam-3780	675	16	=	=	NOUN
ejpam-3780	675	17	v	v	X
ejpam-3780	675	18	(	(	PUNCT
ejpam-3780	675	19	g	g	NOUN
ejpam-3780	675	20	)	)	PUNCT
ejpam-3780	675	21	and	and	CCONJ
ejpam-3780	675	22	e(h	e(h	NOUN
ejpam-3780	675	23	)	)	PUNCT
ejpam-3780	675	24	=	=	PUNCT
ejpam-3780	675	25	e(g)∪{{wqi	e(g)∪{{wqi	NOUN
ejpam-3780	675	26	,	,	PUNCT
ejpam-3780	675	27	wqi	wqi	VERB
ejpam-3780	675	28	+1	+1	PROPN
ejpam-3780	675	29	}	}	PUNCT
ejpam-3780	675	30	:	:	PUNCT
ejpam-3780	676	1	i	i	NOUN
ejpam-3780	676	2	=	=	NOUN
ejpam-3780	676	3	1	1	NUM
ejpam-3780	676	4	,	,	PUNCT
ejpam-3780	676	5	2	2	NUM
ejpam-3780	676	6	,	,	PUNCT
ejpam-3780	676	7	.	.	PUNCT
ejpam-3780	676	8	.	.	PUNCT
ejpam-3780	676	9	.	.	PUNCT
ejpam-3780	677	1	p−	p−	NOUN
ejpam-3780	677	2	1	1	NUM
ejpam-3780	677	3	}	}	PUNCT
ejpam-3780	677	4	,	,	PUNCT
ejpam-3780	677	5	where	where	SCONJ
ejpam-3780	677	6	qi	qi	NOUN
ejpam-3780	677	7	=	=	SYM
ejpam-3780	677	8	s1	s1	PROPN
ejpam-3780	677	9	+	+	CCONJ
ejpam-3780	677	10	s2	s2	PROPN
ejpam-3780	677	11	·	·	PUNCT
ejpam-3780	677	12	·	·	PUNCT
ejpam-3780	677	13	·	·	PUNCT
ejpam-3780	677	14	+	+	NUM
ejpam-3780	677	15	si	si	AUX
ejpam-3780	677	16	.	.	PUNCT
ejpam-3780	677	17	by	by	ADP
ejpam-3780	677	18	theorem	theorem	NOUN
ejpam-3780	677	19	6	6	NUM
ejpam-3780	677	20	,	,	PUNCT
ejpam-3780	677	21	h	h	NOUN
ejpam-3780	677	22	has	have	VERB
ejpam-3780	677	23	an	an	DET
ejpam-3780	677	24	efficient	efficient	ADJ
ejpam-3780	677	25	zero	zero	NUM
ejpam-3780	677	26	ring	ring	NOUN
ejpam-3780	677	27	labeling	labeling	NOUN
ejpam-3780	677	28	h.	h.	NOUN
ejpam-3780	677	29	define	define	VERB
ejpam-3780	677	30	a	a	DET
ejpam-3780	677	31	function	function	NOUN
ejpam-3780	678	1	f	f	NOUN
ejpam-3780	678	2	:	:	PUNCT
ejpam-3780	678	3	v	v	X
ejpam-3780	678	4	(	(	PUNCT
ejpam-3780	678	5	g	g	NOUN
ejpam-3780	678	6	)	)	PUNCT
ejpam-3780	678	7	→	→	SYM
ejpam-3780	678	8	m0	m0	PROPN
ejpam-3780	678	9	2	2	NUM
ejpam-3780	678	10	(	(	PUNCT
ejpam-3780	678	11	znrk+n−rk	znrk+n−rk	PROPN
ejpam-3780	678	12	)	)	PUNCT
ejpam-3780	678	13	such	such	ADJ
ejpam-3780	678	14	that	that	DET
ejpam-3780	678	15	f(a	f(a	NOUN
ejpam-3780	678	16	)	)	PUNCT
ejpam-3780	678	17	=	=	SYM
ejpam-3780	678	18	h(a	h(a	PROPN
ejpam-3780	678	19	)	)	PUNCT
ejpam-3780	678	20	for	for	ADP
ejpam-3780	678	21	all	all	DET
ejpam-3780	678	22	a	a	DET
ejpam-3780	678	23	∈	∈	PROPN
ejpam-3780	678	24	v	v	NOUN
ejpam-3780	678	25	(	(	PUNCT
ejpam-3780	678	26	g	g	NOUN
ejpam-3780	678	27	)	)	PUNCT
ejpam-3780	678	28	.	.	PUNCT
ejpam-3780	679	1	since	since	SCONJ
ejpam-3780	679	2	h	h	NOUN
ejpam-3780	679	3	is	be	AUX
ejpam-3780	679	4	injective	injective	ADJ
ejpam-3780	679	5	,	,	PUNCT
ejpam-3780	679	6	it	it	PRON
ejpam-3780	679	7	follows	follow	VERB
ejpam-3780	679	8	that	that	SCONJ
ejpam-3780	679	9	f	f	PROPN
ejpam-3780	679	10	is	be	AUX
ejpam-3780	679	11	also	also	ADV
ejpam-3780	679	12	injective	injective	ADJ
ejpam-3780	679	13	.	.	PUNCT
ejpam-3780	680	1	g	g	NOUN
ejpam-3780	680	2	is	be	AUX
ejpam-3780	680	3	an	an	DET
ejpam-3780	680	4	edge	edge	NOUN
ejpam-3780	680	5	-	-	PUNCT
ejpam-3780	680	6	induced	induce	VERB
ejpam-3780	680	7	subgraph	subgraph	NOUN
ejpam-3780	680	8	of	of	ADP
ejpam-3780	680	9	h	h	NOUN
ejpam-3780	680	10	,	,	PUNCT
ejpam-3780	680	11	so	so	ADV
ejpam-3780	680	12	a0	a0	PROPN
ejpam-3780	680	13	/∈	/∈	PUNCT
ejpam-3780	681	1	kh	kh	PROPN
ejpam-3780	681	2	=	=	X
ejpam-3780	681	3	{	{	PUNCT
ejpam-3780	681	4	h(u	h(u	PROPN
ejpam-3780	681	5	)	)	PUNCT
ejpam-3780	682	1	+	+	CCONJ
ejpam-3780	682	2	h(v	h(v	NOUN
ejpam-3780	682	3	)	)	PUNCT
ejpam-3780	682	4	:	:	PUNCT
ejpam-3780	682	5	uv	uv	NOUN
ejpam-3780	682	6	∈	∈	PROPN
ejpam-3780	682	7	e(h	e(h	PROPN
ejpam-3780	682	8	)	)	PUNCT
ejpam-3780	682	9	}	}	PUNCT
ejpam-3780	682	10	implies	imply	VERB
ejpam-3780	682	11	that	that	SCONJ
ejpam-3780	682	12	a0	a0	PROPN
ejpam-3780	682	13	/∈	/∈	PUNCT
ejpam-3780	683	1	k	k	X
ejpam-3780	683	2	=	=	PRON
ejpam-3780	683	3	{	{	PUNCT
ejpam-3780	683	4	f(u	f(u	PROPN
ejpam-3780	683	5	)	)	PUNCT
ejpam-3780	683	6	+	+	NUM
ejpam-3780	683	7	f(v	f(v	NOUN
ejpam-3780	683	8	)	)	PUNCT
ejpam-3780	683	9	:	:	PUNCT
ejpam-3780	683	10	uv	uv	PROPN
ejpam-3780	683	11	∈	∈	PROPN
ejpam-3780	683	12	e(g	e(g	PROPN
ejpam-3780	683	13	)	)	PUNCT
ejpam-3780	683	14	}	}	PUNCT
ejpam-3780	683	15	.	.	PUNCT
ejpam-3780	684	1	to	to	PART
ejpam-3780	684	2	show	show	VERB
ejpam-3780	684	3	that	that	SCONJ
ejpam-3780	684	4	f	f	PROPN
ejpam-3780	684	5	is	be	AUX
ejpam-3780	684	6	an	an	DET
ejpam-3780	684	7	efficient	efficient	ADJ
ejpam-3780	684	8	zero	zero	NUM
ejpam-3780	684	9	ring	ring	NOUN
ejpam-3780	684	10	labeling	labeling	NOUN
ejpam-3780	684	11	of	of	ADP
ejpam-3780	684	12	g	g	NOUN
ejpam-3780	684	13	,	,	PUNCT
ejpam-3780	684	14	it	it	PRON
ejpam-3780	684	15	remains	remain	VERB
ejpam-3780	684	16	to	to	PART
ejpam-3780	684	17	show	show	VERB
ejpam-3780	684	18	that	that	SCONJ
ejpam-3780	684	19	|k|	|k|	NOUN
ejpam-3780	684	20	=	=	SYM
ejpam-3780	684	21	∆(g	∆(g	PROPN
ejpam-3780	684	22	)	)	PUNCT
ejpam-3780	685	1	=	=	SYM
ejpam-3780	685	2	rk	rk	NOUN
ejpam-3780	685	3	+	+	NOUN
ejpam-3780	685	4	2	2	X
ejpam-3780	685	5	.	.	PUNCT
ejpam-3780	685	6	using	use	VERB
ejpam-3780	685	7	the	the	DET
ejpam-3780	685	8	labeling	labeling	NOUN
ejpam-3780	685	9	in	in	ADP
ejpam-3780	685	10	the	the	DET
ejpam-3780	685	11	proof	proof	NOUN
ejpam-3780	685	12	of	of	ADP
ejpam-3780	685	13	theorem	theorem	ADJ
ejpam-3780	685	14	4	4	NUM
ejpam-3780	685	15	,	,	PUNCT
ejpam-3780	685	16	we	we	PRON
ejpam-3780	685	17	obtain	obtain	VERB
ejpam-3780	685	18	k	k	X
ejpam-3780	685	19	=	=	PRON
ejpam-3780	685	20	{	{	PUNCT
ejpam-3780	685	21	a(rk+1)(q−2	a(rk+1)(q−2	NOUN
ejpam-3780	685	22	)	)	PUNCT
ejpam-3780	685	23	,	,	PUNCT
ejpam-3780	685	24	a(rk+1)(q−2)+1	a(rk+1)(q−2)+1	NOUN
ejpam-3780	685	25	,	,	PUNCT
ejpam-3780	685	26	.	.	PUNCT
ejpam-3780	685	27	.	.	PUNCT
ejpam-3780	686	1	.	.	PUNCT
ejpam-3780	687	1	,	,	PUNCT
ejpam-3780	687	2	a(rk+1)(q−2)+rk	a(rk+1)(q−2)+rk	PROPN
ejpam-3780	687	3	,	,	PUNCT
ejpam-3780	687	4	a(rk+1)(q−1	a(rk+1)(q−1	PROPN
ejpam-3780	687	5	)	)	PUNCT
ejpam-3780	687	6	}	}	PUNCT
ejpam-3780	687	7	,	,	PUNCT
ejpam-3780	687	8	(	(	PUNCT
ejpam-3780	687	9	41	41	NUM
ejpam-3780	687	10	)	)	PUNCT
ejpam-3780	688	1	where	where	SCONJ
ejpam-3780	688	2	q	q	NOUN
ejpam-3780	688	3	=	=	SYM
ejpam-3780	688	4	s1	s1	PROPN
ejpam-3780	688	5	+	+	CCONJ
ejpam-3780	688	6	s2	s2	NOUN
ejpam-3780	688	7	+	+	CCONJ
ejpam-3780	688	8	·	·	PUNCT
ejpam-3780	688	9	·	·	PUNCT
ejpam-3780	688	10	·	·	PUNCT
ejpam-3780	688	11	+	+	NUM
ejpam-3780	688	12	sp	sp	NOUN
ejpam-3780	688	13	.	.	PUNCT
ejpam-3780	688	14	thus	thus	ADV
ejpam-3780	688	15	,	,	PUNCT
ejpam-3780	688	16	|k|	|k|	NOUN
ejpam-3780	688	17	=	=	SYM
ejpam-3780	688	18	rk	rk	NOUN
ejpam-3780	688	19	+	+	CCONJ
ejpam-3780	688	20	2	2	NUM
ejpam-3780	688	21	=	=	SYM
ejpam-3780	688	22	∆(g	∆(g	NOUN
ejpam-3780	688	23	)	)	PUNCT
ejpam-3780	688	24	.	.	PUNCT
ejpam-3780	689	1	example	example	NOUN
ejpam-3780	690	1	13	13	NUM
ejpam-3780	690	2	.	.	PUNCT
ejpam-3780	691	1	figure	figure	NOUN
ejpam-3780	691	2	14	14	NUM
ejpam-3780	691	3	shows	show	VERB
ejpam-3780	691	4	an	an	DET
ejpam-3780	691	5	efficient	efficient	ADJ
ejpam-3780	691	6	zero	zero	NUM
ejpam-3780	691	7	ring	ring	NOUN
ejpam-3780	691	8	labeling	labeling	NOUN
ejpam-3780	691	9	of	of	ADP
ejpam-3780	691	10	a	a	DET
ejpam-3780	691	11	disjoint	disjoint	NOUN
ejpam-3780	691	12	union	union	NOUN
ejpam-3780	691	13	of	of	ADP
ejpam-3780	691	14	three	three	NUM
ejpam-3780	691	15	caterpillars	caterpillar	NOUN
ejpam-3780	691	16	using	use	VERB
ejpam-3780	691	17	m0	m0	PROPN
ejpam-3780	691	18	2	2	NUM
ejpam-3780	691	19	(	(	PUNCT
ejpam-3780	691	20	z66	z66	PROPN
ejpam-3780	691	21	)	)	PUNCT
ejpam-3780	691	22	.	.	PUNCT
ejpam-3780	692	1	in	in	ADP
ejpam-3780	692	2	this	this	DET
ejpam-3780	692	3	labeling	labeling	NOUN
ejpam-3780	692	4	,	,	PUNCT
ejpam-3780	692	5	the	the	DET
ejpam-3780	692	6	set	set	NOUN
ejpam-3780	692	7	of	of	ADP
ejpam-3780	692	8	sums	sum	NOUN
ejpam-3780	692	9	is	be	AUX
ejpam-3780	692	10	k	k	NOUN
ejpam-3780	692	11	=	=	PUNCT
ejpam-3780	692	12	{	{	PUNCT
ejpam-3780	692	13	a60	a60	PROPN
ejpam-3780	692	14	,	,	PUNCT
ejpam-3780	692	15	a61	a61	PROPN
ejpam-3780	692	16	,	,	PUNCT
ejpam-3780	692	17	a62	a62	PROPN
ejpam-3780	692	18	,	,	PUNCT
ejpam-3780	692	19	a63	a63	NOUN
ejpam-3780	692	20	,	,	PUNCT
ejpam-3780	692	21	a64	a64	NOUN
ejpam-3780	692	22	,	,	PUNCT
ejpam-3780	692	23	a65	a65	NOUN
ejpam-3780	692	24	}	}	PUNCT
ejpam-3780	692	25	and	and	CCONJ
ejpam-3780	692	26	thus	thus	ADV
ejpam-3780	692	27	|k|	|k|	NOUN
ejpam-3780	692	28	=	=	SYM
ejpam-3780	692	29	6	6	NUM
ejpam-3780	692	30	,	,	PUNCT
ejpam-3780	692	31	which	which	PRON
ejpam-3780	692	32	is	be	AUX
ejpam-3780	692	33	the	the	DET
ejpam-3780	692	34	maximum	maximum	ADJ
ejpam-3780	692	35	degree	degree	NOUN
ejpam-3780	692	36	of	of	ADP
ejpam-3780	692	37	the	the	DET
ejpam-3780	692	38	graph	graph	NOUN
ejpam-3780	692	39	.	.	PUNCT
ejpam-3780	693	1	theorem	theorem	NOUN
ejpam-3780	693	2	11	11	NUM
ejpam-3780	693	3	.	.	PUNCT
ejpam-3780	694	1	let	let	VERB
ejpam-3780	694	2	g	g	PRON
ejpam-3780	694	3	be	be	AUX
ejpam-3780	694	4	a	a	DET
ejpam-3780	694	5	disjoint	disjoint	NOUN
ejpam-3780	694	6	union	union	NOUN
ejpam-3780	694	7	of	of	ADP
ejpam-3780	694	8	a	a	DET
ejpam-3780	694	9	finite	finite	ADJ
ejpam-3780	694	10	number	number	NOUN
ejpam-3780	694	11	of	of	ADP
ejpam-3780	694	12	lobsters	lobster	NOUN
ejpam-3780	694	13	in	in	ADP
ejpam-3780	694	14	which	which	PRON
ejpam-3780	694	15	each	each	DET
ejpam-3780	694	16	elbow	elbow	NOUN
ejpam-3780	694	17	has	have	VERB
ejpam-3780	694	18	at	at	ADP
ejpam-3780	694	19	most	most	ADV
ejpam-3780	694	20	one	one	NUM
ejpam-3780	694	21	hanging	hang	VERB
ejpam-3780	694	22	leaf	leaf	NOUN
ejpam-3780	694	23	.	.	PUNCT
ejpam-3780	695	1	then	then	ADV
ejpam-3780	695	2	g	g	PROPN
ejpam-3780	695	3	has	have	VERB
ejpam-3780	695	4	an	an	DET
ejpam-3780	695	5	efficient	efficient	ADJ
ejpam-3780	695	6	zero	zero	NUM
ejpam-3780	695	7	ring	ring	NOUN
ejpam-3780	695	8	labeling	labeling	NOUN
ejpam-3780	695	9	.	.	PUNCT
ejpam-3780	696	1	proof	proof	NOUN
ejpam-3780	696	2	.	.	PUNCT
ejpam-3780	697	1	let	let	VERB
ejpam-3780	697	2	g	g	PRON
ejpam-3780	697	3	be	be	AUX
ejpam-3780	697	4	a	a	DET
ejpam-3780	697	5	disjoint	disjoint	NOUN
ejpam-3780	697	6	union	union	NOUN
ejpam-3780	697	7	of	of	ADP
ejpam-3780	697	8	p	p	NOUN
ejpam-3780	697	9	lobsters	lobster	NOUN
ejpam-3780	697	10	.	.	PUNCT
ejpam-3780	698	1	we	we	PRON
ejpam-3780	698	2	show	show	VERB
ejpam-3780	698	3	that	that	SCONJ
ejpam-3780	698	4	g	g	PROPN
ejpam-3780	698	5	has	have	VERB
ejpam-3780	698	6	an	an	DET
ejpam-3780	698	7	efficient	efficient	ADJ
ejpam-3780	698	8	zero	zero	NUM
ejpam-3780	698	9	ring	ring	NOUN
ejpam-3780	698	10	labeling	labeling	NOUN
ejpam-3780	698	11	.	.	PUNCT
ejpam-3780	699	1	if	if	SCONJ
ejpam-3780	699	2	the	the	DET
ejpam-3780	699	3	order	order	NOUN
ejpam-3780	699	4	of	of	ADP
ejpam-3780	699	5	each	each	DET
ejpam-3780	699	6	lobster	lobster	NOUN
ejpam-3780	699	7	is	be	AUX
ejpam-3780	699	8	one	one	NUM
ejpam-3780	699	9	or	or	CCONJ
ejpam-3780	699	10	two	two	NUM
ejpam-3780	699	11	,	,	PUNCT
ejpam-3780	699	12	then	then	ADV
ejpam-3780	699	13	by	by	ADP
ejpam-3780	699	14	theorem	theorem	NOUN
ejpam-3780	699	15	10	10	NUM
ejpam-3780	699	16	,	,	PUNCT
ejpam-3780	699	17	g	g	PROPN
ejpam-3780	699	18	has	have	VERB
ejpam-3780	699	19	an	an	DET
ejpam-3780	699	20	efficient	efficient	ADJ
ejpam-3780	699	21	zero	zero	NUM
ejpam-3780	699	22	ring	ring	NOUN
ejpam-3780	699	23	labeling	labeling	NOUN
ejpam-3780	699	24	.	.	PUNCT
ejpam-3780	700	1	suppose	suppose	VERB
ejpam-3780	700	2	the	the	DET
ejpam-3780	700	3	order	order	NOUN
ejpam-3780	700	4	of	of	ADP
ejpam-3780	700	5	at	at	ADV
ejpam-3780	700	6	least	least	ADV
ejpam-3780	700	7	one	one	NUM
ejpam-3780	700	8	lobster	lobster	NOUN
ejpam-3780	700	9	is	be	AUX
ejpam-3780	700	10	at	at	ADV
ejpam-3780	700	11	least	least	ADJ
ejpam-3780	700	12	three	three	NUM
ejpam-3780	700	13	.	.	PUNCT
ejpam-3780	701	1	applying	apply	VERB
ejpam-3780	701	2	lemma	lemma	PROPN
ejpam-3780	701	3	2	2	NUM
ejpam-3780	701	4	,	,	PUNCT
ejpam-3780	701	5	we	we	PRON
ejpam-3780	701	6	denote	denote	VERB
ejpam-3780	701	7	the	the	DET
ejpam-3780	701	8	vertices	vertex	NOUN
ejpam-3780	701	9	of	of	ADP
ejpam-3780	701	10	each	each	DET
ejpam-3780	701	11	lobster	lobster	NOUN
ejpam-3780	701	12	such	such	ADJ
ejpam-3780	701	13	that	that	SCONJ
ejpam-3780	701	14	the	the	DET
ejpam-3780	701	15	endvertices	endvertice	NOUN
ejpam-3780	701	16	of	of	ADP
ejpam-3780	701	17	the	the	DET
ejpam-3780	701	18	central	central	ADJ
ejpam-3780	701	19	path	path	NOUN
ejpam-3780	701	20	have	have	VERB
ejpam-3780	701	21	no	no	DET
ejpam-3780	701	22	hanging	hang	VERB
ejpam-3780	701	23	leaf	leaf	NOUN
ejpam-3780	701	24	.	.	PUNCT
ejpam-3780	702	1	let	let	VERB
ejpam-3780	702	2	g1	g1	PROPN
ejpam-3780	702	3	,	,	PUNCT
ejpam-3780	702	4	g2	g2	PROPN
ejpam-3780	702	5	,	,	PUNCT
ejpam-3780	702	6	.	.	PUNCT
ejpam-3780	702	7	.	.	PUNCT
ejpam-3780	703	1	.	.	PUNCT
ejpam-3780	704	1	,	,	PUNCT
ejpam-3780	704	2	gp	gp	PROPN
ejpam-3780	704	3	denote	denote	VERB
ejpam-3780	704	4	the	the	DET
ejpam-3780	704	5	disjoint	disjoint	ADJ
ejpam-3780	704	6	lobster	lobster	NOUN
ejpam-3780	704	7	graphs	graph	NOUN
ejpam-3780	704	8	.	.	PUNCT
ejpam-3780	705	1	suppose	suppose	VERB
ejpam-3780	705	2	the	the	DET
ejpam-3780	705	3	length	length	NOUN
ejpam-3780	705	4	of	of	ADP
ejpam-3780	705	5	the	the	DET
ejpam-3780	705	6	central	central	ADJ
ejpam-3780	705	7	path	path	NOUN
ejpam-3780	705	8	of	of	ADP
ejpam-3780	705	9	gi	gi	PROPN
ejpam-3780	705	10	has	have	VERB
ejpam-3780	705	11	si	si	NOUN
ejpam-3780	705	12	vertices	vertex	NOUN
ejpam-3780	705	13	.	.	PUNCT
ejpam-3780	706	1	denote	denote	VERB
ejpam-3780	706	2	the	the	DET
ejpam-3780	706	3	central	central	ADJ
ejpam-3780	706	4	path	path	NOUN
ejpam-3780	706	5	of	of	ADP
ejpam-3780	706	6	g1	g1	NOUN
ejpam-3780	706	7	by	by	ADP
ejpam-3780	706	8	[	[	X
ejpam-3780	706	9	w1	w1	NOUN
ejpam-3780	706	10	,	,	PUNCT
ejpam-3780	706	11	w2	w2	NOUN
ejpam-3780	706	12	,	,	PUNCT
ejpam-3780	706	13	.	.	PUNCT
ejpam-3780	706	14	.	.	PUNCT
ejpam-3780	707	1	.	.	PUNCT
ejpam-3780	708	1	,	,	PUNCT
ejpam-3780	708	2	ws1	ws1	X
ejpam-3780	708	3	]	]	X
ejpam-3780	708	4	,	,	PUNCT
ejpam-3780	708	5	and	and	CCONJ
ejpam-3780	708	6	the	the	DET
ejpam-3780	708	7	central	central	ADJ
ejpam-3780	708	8	path	path	NOUN
ejpam-3780	708	9	of	of	ADP
ejpam-3780	708	10	gi	gi	NOUN
ejpam-3780	708	11	by	by	ADP
ejpam-3780	708	12	[	[	X
ejpam-3780	708	13	wq+1	wq+1	PROPN
ejpam-3780	708	14	,	,	PUNCT
ejpam-3780	708	15	wq+2	wq+2	ADJ
ejpam-3780	708	16	,	,	PUNCT
ejpam-3780	708	17	.	.	PUNCT
ejpam-3780	708	18	.	.	PUNCT
ejpam-3780	709	1	.	.	PUNCT
ejpam-3780	710	1	,	,	PUNCT
ejpam-3780	710	2	wq+si	wq+si	X
ejpam-3780	710	3	]	]	PUNCT
ejpam-3780	710	4	for	for	ADP
ejpam-3780	710	5	i	i	PROPN
ejpam-3780	710	6	=	=	SYM
ejpam-3780	710	7	2	2	NUM
ejpam-3780	710	8	,	,	PUNCT
ejpam-3780	710	9	3	3	NUM
ejpam-3780	710	10	,	,	PUNCT
ejpam-3780	710	11	.	.	PUNCT
ejpam-3780	710	12	.	.	PUNCT
ejpam-3780	711	1	.	.	PUNCT
ejpam-3780	712	1	,	,	PUNCT
ejpam-3780	713	1	p	p	X
ejpam-3780	713	2	,	,	PUNCT
ejpam-3780	713	3	where	where	SCONJ
ejpam-3780	713	4	q	q	NOUN
ejpam-3780	713	5	=	=	SYM
ejpam-3780	713	6	s1+s2	s1+s2	PROPN
ejpam-3780	713	7	+	+	PROPN
ejpam-3780	713	8	·	·	PUNCT
ejpam-3780	713	9	·	·	PUNCT
ejpam-3780	713	10	·	·	PUNCT
ejpam-3780	713	11	+	+	NUM
ejpam-3780	713	12	si−1	si−1	PROPN
ejpam-3780	713	13	.	.	PUNCT
ejpam-3780	714	1	d.	d.	PROPN
ejpam-3780	714	2	chua	chua	PROPN
ejpam-3780	714	3	,	,	PUNCT
ejpam-3780	714	4	f.	f.	PROPN
ejpam-3780	714	5	campeña	campeña	PROPN
ejpam-3780	714	6	,	,	PUNCT
ejpam-3780	714	7	f.	f.	PROPN
ejpam-3780	714	8	franco	franco	PROPN
ejpam-3780	714	9	/	/	SYM
ejpam-3780	714	10	eur	eur	PROPN
ejpam-3780	714	11	.	.	PUNCT
ejpam-3780	715	1	j.	j.	PROPN
ejpam-3780	715	2	pure	pure	PROPN
ejpam-3780	715	3	appl	appl	PROPN
ejpam-3780	715	4	.	.	PROPN
ejpam-3780	715	5	math	math	PROPN
ejpam-3780	715	6	,	,	PUNCT
ejpam-3780	715	7	13	13	NUM
ejpam-3780	715	8	(	(	PUNCT
ejpam-3780	715	9	3	3	NUM
ejpam-3780	715	10	)	)	PUNCT
ejpam-3780	715	11	(	(	PUNCT
ejpam-3780	715	12	2020	2020	NUM
ejpam-3780	715	13	)	)	PUNCT
ejpam-3780	715	14	,	,	PUNCT
ejpam-3780	715	15	674	674	NUM
ejpam-3780	715	16	-	-	SYM
ejpam-3780	715	17	696	696	NUM
ejpam-3780	715	18	692	692	NUM
ejpam-3780	715	19	a0	a0	NOUN
ejpam-3780	715	20	a60	a60	PROPN
ejpam-3780	715	21	a5a0	a5a0	ADP
ejpam-3780	715	22	a65	a65	PROPN
ejpam-3780	715	23	a0	a0	PROPN
ejpam-3780	715	24	a61	a61	PROPN
ejpam-3780	715	25	a0	a0	PROPN
ejpam-3780	715	26	a62	a62	PROPN
ejpam-3780	715	27	a60	a60	PROPN
ejpam-3780	715	28	a1	a1	PROPN
ejpam-3780	715	29	a60	a60	PROPN
ejpam-3780	715	30	a2	a2	PROPN
ejpam-3780	715	31	a5	a5	PROPN
ejpam-3780	715	32	a56	a56	PROPN
ejpam-3780	715	33	a5	a5	PROPN
ejpam-3780	715	34	a57	a57	PROPN
ejpam-3780	715	35	a5	a5	PROPN
ejpam-3780	715	36	a58	a58	PROPN
ejpam-3780	715	37	a5	a5	PROPN
ejpam-3780	715	38	a55	a55	PROPN
ejpam-3780	715	39	a50	a50	NOUN
ejpam-3780	715	40	a10	a10	PROPN
ejpam-3780	715	41	a50	a50	PROPN
ejpam-3780	715	42	a11	a11	PROPN
ejpam-3780	715	43	a50	a50	PROPN
ejpam-3780	715	44	a12	a12	PROPN
ejpam-3780	715	45	a50	a50	NOUN
ejpam-3780	715	46	a15	a15	PROPN
ejpam-3780	715	47	a20	a20	PROPN
ejpam-3780	715	48	a40	a40	PROPN
ejpam-3780	715	49	a25	a25	PROPN
ejpam-3780	715	50	a35a20	a35a20	PROPN
ejpam-3780	715	51	a45	a45	PROPN
ejpam-3780	715	52	a20	a20	PROPN
ejpam-3780	715	53	a41	a41	PROPN
ejpam-3780	715	54	a20	a20	PROPN
ejpam-3780	715	55	a42	a42	PROPN
ejpam-3780	715	56	a20	a20	PROPN
ejpam-3780	715	57	a43	a43	PROPN
ejpam-3780	715	58	a20	a20	PROPN
ejpam-3780	715	59	a44	a44	PROPN
ejpam-3780	715	60	a40	a40	PROPN
ejpam-3780	715	61	a21	a21	PROPN
ejpam-3780	715	62	a40	a40	PROPN
ejpam-3780	715	63	a22	a22	PROPN
ejpam-3780	715	64	a25	a25	PROPN
ejpam-3780	715	65	a36	a36	PROPN
ejpam-3780	715	66	a25	a25	PROPN
ejpam-3780	715	67	a37	a37	PROPN
ejpam-3780	715	68	a35	a35	PROPN
ejpam-3780	715	69	a26	a26	PROPN
ejpam-3780	715	70	a35	a35	PROPN
ejpam-3780	715	71	a27	a27	PROPN
ejpam-3780	715	72	a35	a35	PROPN
ejpam-3780	715	73	a30	a30	NOUN
ejpam-3780	715	74	figure	figure	NOUN
ejpam-3780	715	75	14	14	NUM
ejpam-3780	715	76	:	:	PUNCT
ejpam-3780	715	77	efficient	efficient	ADJ
ejpam-3780	715	78	zero	zero	NUM
ejpam-3780	715	79	ring	ring	NOUN
ejpam-3780	715	80	labeling	labeling	NOUN
ejpam-3780	715	81	of	of	ADP
ejpam-3780	715	82	a	a	DET
ejpam-3780	715	83	disjoint	disjoint	NOUN
ejpam-3780	715	84	union	union	NOUN
ejpam-3780	715	85	of	of	ADP
ejpam-3780	715	86	three	three	NUM
ejpam-3780	715	87	caterpillars	caterpillar	NOUN
ejpam-3780	715	88	using	use	VERB
ejpam-3780	715	89	m0	m0	PROPN
ejpam-3780	715	90	2	2	NUM
ejpam-3780	715	91	(	(	PUNCT
ejpam-3780	715	92	z66	z66	PROPN
ejpam-3780	715	93	)	)	PUNCT
ejpam-3780	715	94	suppose	suppose	VERB
ejpam-3780	715	95	the	the	DET
ejpam-3780	715	96	number	number	NOUN
ejpam-3780	715	97	of	of	ADP
ejpam-3780	715	98	elbows	elbow	NOUN
ejpam-3780	715	99	of	of	ADP
ejpam-3780	715	100	wi	wi	PROPN
ejpam-3780	715	101	is	be	AUX
ejpam-3780	715	102	ri	ri	PROPN
ejpam-3780	715	103	,	,	PUNCT
ejpam-3780	715	104	and	and	CCONJ
ejpam-3780	715	105	denote	denote	VERB
ejpam-3780	715	106	the	the	DET
ejpam-3780	715	107	elbows	elbow	NOUN
ejpam-3780	715	108	of	of	ADP
ejpam-3780	715	109	wi	wi	PROPN
ejpam-3780	715	110	by	by	ADP
ejpam-3780	715	111	wi	wi	PROPN
ejpam-3780	715	112	,	,	PUNCT
ejpam-3780	715	113	j	j	PROPN
ejpam-3780	715	114	,	,	PUNCT
ejpam-3780	715	115	where	where	SCONJ
ejpam-3780	715	116	j	j	PROPN
ejpam-3780	715	117	=	=	SYM
ejpam-3780	715	118	1	1	NUM
ejpam-3780	715	119	,	,	PUNCT
ejpam-3780	715	120	2	2	NUM
ejpam-3780	715	121	,	,	PUNCT
ejpam-3780	715	122	.	.	PUNCT
ejpam-3780	715	123	.	.	PUNCT
ejpam-3780	716	1	.	.	PUNCT
ejpam-3780	717	1	,	,	PUNCT
ejpam-3780	717	2	ri	ri	PROPN
ejpam-3780	717	3	.	.	PUNCT
ejpam-3780	718	1	if	if	SCONJ
ejpam-3780	718	2	it	it	PRON
ejpam-3780	718	3	exists	exist	VERB
ejpam-3780	718	4	,	,	PUNCT
ejpam-3780	718	5	denote	denote	VERB
ejpam-3780	718	6	the	the	DET
ejpam-3780	718	7	hanging	hang	VERB
ejpam-3780	718	8	leaf	leaf	NOUN
ejpam-3780	718	9	of	of	ADP
ejpam-3780	718	10	wi	wi	PROPN
ejpam-3780	718	11	,	,	PUNCT
ejpam-3780	718	12	j	j	PROPN
ejpam-3780	718	13	by	by	ADP
ejpam-3780	718	14	wi	wi	PROPN
ejpam-3780	718	15	,	,	PUNCT
ejpam-3780	718	16	j,1	j,1	PROPN
ejpam-3780	718	17	.	.	PUNCT
ejpam-3780	719	1	suppose	suppose	VERB
ejpam-3780	719	2	the	the	DET
ejpam-3780	719	3	maximum	maximum	ADJ
ejpam-3780	719	4	number	number	NOUN
ejpam-3780	719	5	of	of	ADP
ejpam-3780	719	6	elbows	elbow	NOUN
ejpam-3780	719	7	of	of	ADP
ejpam-3780	719	8	a	a	DET
ejpam-3780	719	9	vertex	vertex	NOUN
ejpam-3780	719	10	in	in	ADP
ejpam-3780	719	11	g	g	PROPN
ejpam-3780	719	12	is	be	AUX
ejpam-3780	719	13	rk	rk	NOUN
ejpam-3780	719	14	.	.	PUNCT
ejpam-3780	720	1	since	since	SCONJ
ejpam-3780	720	2	the	the	DET
ejpam-3780	720	3	endvertices	endvertice	NOUN
ejpam-3780	720	4	in	in	ADP
ejpam-3780	720	5	the	the	DET
ejpam-3780	720	6	central	central	ADJ
ejpam-3780	720	7	path	path	NOUN
ejpam-3780	720	8	of	of	ADP
ejpam-3780	720	9	each	each	DET
ejpam-3780	720	10	lobster	lobster	NOUN
ejpam-3780	720	11	in	in	ADP
ejpam-3780	720	12	g	g	PROPN
ejpam-3780	720	13	have	have	VERB
ejpam-3780	720	14	no	no	DET
ejpam-3780	720	15	elbow	elbow	NOUN
ejpam-3780	720	16	and	and	CCONJ
ejpam-3780	720	17	there	there	PRON
ejpam-3780	720	18	is	be	VERB
ejpam-3780	720	19	at	at	ADV
ejpam-3780	720	20	least	least	ADJ
ejpam-3780	720	21	one	one	NUM
ejpam-3780	720	22	lobster	lobster	NOUN
ejpam-3780	720	23	with	with	ADP
ejpam-3780	720	24	at	at	ADV
ejpam-3780	720	25	least	least	ADV
ejpam-3780	720	26	three	three	NUM
ejpam-3780	720	27	vertices	vertex	NOUN
ejpam-3780	720	28	,	,	PUNCT
ejpam-3780	720	29	it	it	PRON
ejpam-3780	720	30	follows	follow	VERB
ejpam-3780	720	31	that	that	SCONJ
ejpam-3780	720	32	∆(g	∆(g	NOUN
ejpam-3780	720	33	)	)	PUNCT
ejpam-3780	721	1	=	=	SYM
ejpam-3780	721	2	rk	rk	NOUN
ejpam-3780	721	3	+	+	NOUN
ejpam-3780	721	4	2	2	X
ejpam-3780	721	5	.	.	X
ejpam-3780	721	6	consider	consider	VERB
ejpam-3780	721	7	a	a	DET
ejpam-3780	721	8	lobster	lobster	NOUN
ejpam-3780	721	9	h	h	PROPN
ejpam-3780	722	1	such	such	ADJ
ejpam-3780	722	2	that	that	PRON
ejpam-3780	722	3	v	v	NOUN
ejpam-3780	722	4	(	(	PUNCT
ejpam-3780	722	5	h	h	NOUN
ejpam-3780	722	6	)	)	PUNCT
ejpam-3780	722	7	=	=	NOUN
ejpam-3780	722	8	v	v	X
ejpam-3780	722	9	(	(	PUNCT
ejpam-3780	722	10	g	g	NOUN
ejpam-3780	722	11	)	)	PUNCT
ejpam-3780	722	12	and	and	CCONJ
ejpam-3780	722	13	e(h	e(h	PROPN
ejpam-3780	722	14	)	)	PUNCT
ejpam-3780	722	15	=	=	SYM
ejpam-3780	722	16	e(g	e(g	PROPN
ejpam-3780	722	17	)	)	PUNCT
ejpam-3780	722	18	∪	∪	X
ejpam-3780	722	19	{	{	PUNCT
ejpam-3780	722	20	{	{	PUNCT
ejpam-3780	722	21	wqi	wqi	NOUN
ejpam-3780	722	22	,	,	PUNCT
ejpam-3780	722	23	wqi	wqi	VERB
ejpam-3780	722	24	+	+	CCONJ
ejpam-3780	722	25	1	1	NUM
ejpam-3780	722	26	}	}	PUNCT
ejpam-3780	722	27	:	:	PUNCT
ejpam-3780	723	1	i	i	PRON
ejpam-3780	723	2	=	=	NOUN
ejpam-3780	723	3	1	1	NUM
ejpam-3780	723	4	,	,	PUNCT
ejpam-3780	723	5	2	2	NUM
ejpam-3780	723	6	,	,	PUNCT
ejpam-3780	723	7	.	.	PUNCT
ejpam-3780	723	8	.	.	PUNCT
ejpam-3780	723	9	.	.	PUNCT
ejpam-3780	724	1	p−	p−	NOUN
ejpam-3780	724	2	1	1	NUM
ejpam-3780	724	3	}	}	PUNCT
ejpam-3780	724	4	,	,	PUNCT
ejpam-3780	724	5	where	where	SCONJ
ejpam-3780	724	6	qi	qi	NOUN
ejpam-3780	724	7	=	=	SYM
ejpam-3780	724	8	s1	s1	PROPN
ejpam-3780	724	9	+	+	CCONJ
ejpam-3780	724	10	s2	s2	PROPN
ejpam-3780	724	11	·	·	PUNCT
ejpam-3780	724	12	·	·	PUNCT
ejpam-3780	724	13	·	·	PUNCT
ejpam-3780	724	14	+	+	NUM
ejpam-3780	724	15	si	si	AUX
ejpam-3780	724	16	.	.	PUNCT
ejpam-3780	724	17	by	by	ADP
ejpam-3780	724	18	theorem	theorem	NOUN
ejpam-3780	724	19	8	8	NUM
ejpam-3780	724	20	,	,	PUNCT
ejpam-3780	724	21	h	h	NOUN
ejpam-3780	724	22	has	have	VERB
ejpam-3780	724	23	an	an	DET
ejpam-3780	724	24	efficient	efficient	ADJ
ejpam-3780	724	25	zero	zero	NUM
ejpam-3780	724	26	ring	ring	NOUN
ejpam-3780	724	27	labeling	labeling	NOUN
ejpam-3780	724	28	h.	h.	NOUN
ejpam-3780	724	29	define	define	VERB
ejpam-3780	724	30	a	a	DET
ejpam-3780	724	31	function	function	NOUN
ejpam-3780	725	1	f	f	NOUN
ejpam-3780	725	2	:	:	PUNCT
ejpam-3780	725	3	v	v	X
ejpam-3780	725	4	(	(	PUNCT
ejpam-3780	725	5	g	g	NOUN
ejpam-3780	725	6	)	)	PUNCT
ejpam-3780	725	7	→	→	SYM
ejpam-3780	725	8	m0	m0	NOUN
ejpam-3780	725	9	2	2	NUM
ejpam-3780	725	10	(	(	PUNCT
ejpam-3780	725	11	z2nrk+n−2rk	z2nrk+n−2rk	NOUN
ejpam-3780	725	12	)	)	PUNCT
ejpam-3780	725	13	such	such	ADJ
ejpam-3780	725	14	that	that	DET
ejpam-3780	725	15	f(a	f(a	NOUN
ejpam-3780	725	16	)	)	PUNCT
ejpam-3780	725	17	=	=	SYM
ejpam-3780	725	18	h(a	h(a	PROPN
ejpam-3780	725	19	)	)	PUNCT
ejpam-3780	725	20	for	for	ADP
ejpam-3780	725	21	all	all	DET
ejpam-3780	725	22	a	a	DET
ejpam-3780	725	23	∈	∈	PROPN
ejpam-3780	725	24	v	v	NOUN
ejpam-3780	725	25	(	(	PUNCT
ejpam-3780	725	26	g	g	NOUN
ejpam-3780	725	27	)	)	PUNCT
ejpam-3780	725	28	.	.	PUNCT
ejpam-3780	726	1	since	since	SCONJ
ejpam-3780	726	2	h	h	NOUN
ejpam-3780	726	3	is	be	AUX
ejpam-3780	726	4	injective	injective	ADJ
ejpam-3780	726	5	,	,	PUNCT
ejpam-3780	726	6	it	it	PRON
ejpam-3780	726	7	follows	follow	VERB
ejpam-3780	726	8	that	that	SCONJ
ejpam-3780	726	9	f	f	PROPN
ejpam-3780	726	10	is	be	AUX
ejpam-3780	726	11	also	also	ADV
ejpam-3780	726	12	injective	injective	ADJ
ejpam-3780	726	13	.	.	PUNCT
ejpam-3780	727	1	g	g	NOUN
ejpam-3780	727	2	is	be	AUX
ejpam-3780	727	3	an	an	DET
ejpam-3780	727	4	edge	edge	NOUN
ejpam-3780	727	5	-	-	PUNCT
ejpam-3780	727	6	induced	induce	VERB
ejpam-3780	727	7	subgraph	subgraph	NOUN
ejpam-3780	727	8	of	of	ADP
ejpam-3780	727	9	h	h	NOUN
ejpam-3780	727	10	,	,	PUNCT
ejpam-3780	727	11	so	so	ADV
ejpam-3780	727	12	a0	a0	PROPN
ejpam-3780	727	13	/∈	/∈	PUNCT
ejpam-3780	728	1	kh	kh	PROPN
ejpam-3780	728	2	=	=	X
ejpam-3780	728	3	{	{	PUNCT
ejpam-3780	728	4	h(u	h(u	PROPN
ejpam-3780	728	5	)	)	PUNCT
ejpam-3780	729	1	+	+	CCONJ
ejpam-3780	729	2	h(v	h(v	NOUN
ejpam-3780	729	3	)	)	PUNCT
ejpam-3780	729	4	:	:	PUNCT
ejpam-3780	729	5	uv	uv	NOUN
ejpam-3780	729	6	∈	∈	PROPN
ejpam-3780	729	7	e(h	e(h	PROPN
ejpam-3780	729	8	)	)	PUNCT
ejpam-3780	729	9	}	}	PUNCT
ejpam-3780	729	10	implies	imply	VERB
ejpam-3780	729	11	that	that	SCONJ
ejpam-3780	729	12	a0	a0	PROPN
ejpam-3780	729	13	/∈	/∈	PUNCT
ejpam-3780	730	1	k	k	X
ejpam-3780	730	2	=	=	PRON
ejpam-3780	730	3	{	{	PUNCT
ejpam-3780	730	4	f(u	f(u	PROPN
ejpam-3780	730	5	)	)	PUNCT
ejpam-3780	730	6	+	+	NUM
ejpam-3780	730	7	f(v	f(v	NOUN
ejpam-3780	730	8	)	)	PUNCT
ejpam-3780	730	9	:	:	PUNCT
ejpam-3780	730	10	uv	uv	PROPN
ejpam-3780	730	11	∈	∈	PROPN
ejpam-3780	730	12	e(g	e(g	PROPN
ejpam-3780	730	13	)	)	PUNCT
ejpam-3780	730	14	}	}	PUNCT
ejpam-3780	730	15	.	.	PUNCT
ejpam-3780	731	1	to	to	PART
ejpam-3780	731	2	show	show	VERB
ejpam-3780	731	3	that	that	SCONJ
ejpam-3780	731	4	f	f	PROPN
ejpam-3780	731	5	is	be	AUX
ejpam-3780	731	6	an	an	DET
ejpam-3780	731	7	efficient	efficient	ADJ
ejpam-3780	731	8	zero	zero	NUM
ejpam-3780	731	9	ring	ring	NOUN
ejpam-3780	731	10	labeling	labeling	NOUN
ejpam-3780	731	11	of	of	ADP
ejpam-3780	731	12	g	g	NOUN
ejpam-3780	731	13	,	,	PUNCT
ejpam-3780	731	14	it	it	PRON
ejpam-3780	731	15	remains	remain	VERB
ejpam-3780	731	16	to	to	PART
ejpam-3780	731	17	show	show	VERB
ejpam-3780	731	18	that	that	SCONJ
ejpam-3780	731	19	|k|	|k|	NOUN
ejpam-3780	731	20	=	=	SYM
ejpam-3780	731	21	∆(g	∆(g	PROPN
ejpam-3780	731	22	)	)	PUNCT
ejpam-3780	732	1	=	=	SYM
ejpam-3780	732	2	rk	rk	NOUN
ejpam-3780	732	3	+	+	NOUN
ejpam-3780	732	4	2	2	X
ejpam-3780	732	5	.	.	PUNCT
ejpam-3780	732	6	using	use	VERB
ejpam-3780	732	7	the	the	DET
ejpam-3780	732	8	labeling	labeling	NOUN
ejpam-3780	732	9	in	in	ADP
ejpam-3780	732	10	the	the	DET
ejpam-3780	732	11	proof	proof	NOUN
ejpam-3780	732	12	of	of	ADP
ejpam-3780	732	13	theorem	theorem	NOUN
ejpam-3780	732	14	8	8	NUM
ejpam-3780	732	15	,	,	PUNCT
ejpam-3780	732	16	we	we	PRON
ejpam-3780	732	17	obtain	obtain	VERB
ejpam-3780	732	18	k	k	X
ejpam-3780	732	19	=	=	SYM
ejpam-3780	732	20	{	{	PUNCT
ejpam-3780	732	21	a(2rk+1)(q−2	a(2rk+1)(q−2	PROPN
ejpam-3780	732	22	)	)	PUNCT
ejpam-3780	732	23	,	,	PUNCT
ejpam-3780	732	24	a(2rk+1)(q−2)+1	a(2rk+1)(q−2)+1	PROPN
ejpam-3780	732	25	,	,	PUNCT
ejpam-3780	732	26	.	.	PUNCT
ejpam-3780	732	27	.	.	PUNCT
ejpam-3780	733	1	.	.	PUNCT
ejpam-3780	734	1	,	,	PUNCT
ejpam-3780	734	2	a(2rk+1)(q−2)+rk	a(2rk+1)(q−2)+rk	INTJ
ejpam-3780	734	3	,	,	PUNCT
ejpam-3780	734	4	a(2rk+1)(q−1	a(2rk+1)(q−1	PROPN
ejpam-3780	734	5	)	)	PUNCT
ejpam-3780	734	6	}	}	PUNCT
ejpam-3780	734	7	,	,	PUNCT
ejpam-3780	734	8	(	(	PUNCT
ejpam-3780	734	9	42	42	NUM
ejpam-3780	734	10	)	)	PUNCT
ejpam-3780	734	11	where	where	SCONJ
ejpam-3780	734	12	q	q	NOUN
ejpam-3780	734	13	=	=	SYM
ejpam-3780	734	14	s1	s1	PROPN
ejpam-3780	734	15	+	+	CCONJ
ejpam-3780	734	16	s2	s2	NOUN
ejpam-3780	734	17	+	+	CCONJ
ejpam-3780	734	18	·	·	PUNCT
ejpam-3780	734	19	·	·	PUNCT
ejpam-3780	734	20	·	·	PUNCT
ejpam-3780	734	21	+	+	NUM
ejpam-3780	734	22	sp	sp	NOUN
ejpam-3780	734	23	.	.	PUNCT
ejpam-3780	734	24	thus	thus	ADV
ejpam-3780	734	25	,	,	PUNCT
ejpam-3780	734	26	|k|	|k|	NOUN
ejpam-3780	734	27	=	=	SYM
ejpam-3780	734	28	rk	rk	NOUN
ejpam-3780	734	29	+	+	CCONJ
ejpam-3780	734	30	2	2	NUM
ejpam-3780	734	31	=	=	SYM
ejpam-3780	734	32	∆(g	∆(g	NOUN
ejpam-3780	734	33	)	)	PUNCT
ejpam-3780	734	34	.	.	PUNCT
ejpam-3780	735	1	example	example	NOUN
ejpam-3780	736	1	14	14	NUM
ejpam-3780	736	2	.	.	PUNCT
ejpam-3780	737	1	figure	figure	VERB
ejpam-3780	737	2	15	15	NUM
ejpam-3780	737	3	shows	show	VERB
ejpam-3780	737	4	an	an	DET
ejpam-3780	737	5	efficient	efficient	ADJ
ejpam-3780	737	6	zero	zero	NUM
ejpam-3780	737	7	ring	ring	NOUN
ejpam-3780	737	8	labeling	labeling	NOUN
ejpam-3780	737	9	of	of	ADP
ejpam-3780	737	10	a	a	DET
ejpam-3780	737	11	disjoint	disjoint	NOUN
ejpam-3780	737	12	union	union	NOUN
ejpam-3780	737	13	of	of	ADP
ejpam-3780	737	14	two	two	NUM
ejpam-3780	737	15	lobsters	lobster	NOUN
ejpam-3780	737	16	using	use	VERB
ejpam-3780	737	17	m0	m0	PROPN
ejpam-3780	737	18	2	2	NUM
ejpam-3780	737	19	(	(	PUNCT
ejpam-3780	737	20	z57	z57	NOUN
ejpam-3780	737	21	)	)	PUNCT
ejpam-3780	737	22	.	.	PUNCT
ejpam-3780	738	1	in	in	ADP
ejpam-3780	738	2	this	this	DET
ejpam-3780	738	3	labeling	labeling	NOUN
ejpam-3780	738	4	,	,	PUNCT
ejpam-3780	738	5	the	the	DET
ejpam-3780	738	6	set	set	NOUN
ejpam-3780	738	7	of	of	ADP
ejpam-3780	738	8	sums	sum	NOUN
ejpam-3780	738	9	is	be	AUX
ejpam-3780	738	10	k	k	NOUN
ejpam-3780	738	11	=	=	X
ejpam-3780	738	12	{	{	PUNCT
ejpam-3780	738	13	a49	a49	PROPN
ejpam-3780	738	14	,	,	PUNCT
ejpam-3780	738	15	a50	a50	NOUN
ejpam-3780	738	16	,	,	PUNCT
ejpam-3780	738	17	a51	a51	PROPN
ejpam-3780	738	18	,	,	PUNCT
ejpam-3780	738	19	a52	a52	PROPN
ejpam-3780	738	20	,	,	PUNCT
ejpam-3780	738	21	a56	a56	PROPN
ejpam-3780	738	22	}	}	PUNCT
ejpam-3780	738	23	and	and	CCONJ
ejpam-3780	738	24	thus	thus	ADV
ejpam-3780	738	25	|k|	|k|	NOUN
ejpam-3780	738	26	=	=	SYM
ejpam-3780	738	27	5	5	NUM
ejpam-3780	738	28	,	,	PUNCT
ejpam-3780	738	29	which	which	PRON
ejpam-3780	738	30	is	be	AUX
ejpam-3780	738	31	the	the	DET
ejpam-3780	738	32	maximum	maximum	ADJ
ejpam-3780	738	33	degree	degree	NOUN
ejpam-3780	738	34	of	of	ADP
ejpam-3780	738	35	the	the	DET
ejpam-3780	738	36	graph	graph	NOUN
ejpam-3780	738	37	.	.	PUNCT
ejpam-3780	739	1	theorem	theorem	NOUN
ejpam-3780	739	2	12	12	NUM
ejpam-3780	739	3	.	.	PUNCT
ejpam-3780	740	1	a	a	DET
ejpam-3780	740	2	cycle	cycle	NOUN
ejpam-3780	740	3	graph	graph	NOUN
ejpam-3780	740	4	with	with	ADP
ejpam-3780	740	5	n	n	DET
ejpam-3780	740	6	vertices	vertex	NOUN
ejpam-3780	740	7	has	have	VERB
ejpam-3780	740	8	an	an	DET
ejpam-3780	740	9	efficient	efficient	ADJ
ejpam-3780	740	10	zero	zero	NUM
ejpam-3780	740	11	ring	ring	NOUN
ejpam-3780	740	12	labeling	labeling	NOUN
ejpam-3780	740	13	if	if	SCONJ
ejpam-3780	740	14	n	n	PRON
ejpam-3780	740	15	is	be	AUX
ejpam-3780	740	16	even	even	ADV
ejpam-3780	740	17	.	.	PUNCT
ejpam-3780	741	1	d.	d.	PROPN
ejpam-3780	741	2	chua	chua	PROPN
ejpam-3780	741	3	,	,	PUNCT
ejpam-3780	741	4	f.	f.	PROPN
ejpam-3780	741	5	campeña	campeña	PROPN
ejpam-3780	741	6	,	,	PUNCT
ejpam-3780	741	7	f.	f.	PROPN
ejpam-3780	741	8	franco	franco	PROPN
ejpam-3780	741	9	/	/	SYM
ejpam-3780	741	10	eur	eur	PROPN
ejpam-3780	741	11	.	.	PUNCT
ejpam-3780	742	1	j.	j.	PROPN
ejpam-3780	742	2	pure	pure	PROPN
ejpam-3780	742	3	appl	appl	PROPN
ejpam-3780	742	4	.	.	PROPN
ejpam-3780	742	5	math	math	PROPN
ejpam-3780	742	6	,	,	PUNCT
ejpam-3780	742	7	13	13	NUM
ejpam-3780	742	8	(	(	PUNCT
ejpam-3780	742	9	3	3	NUM
ejpam-3780	742	10	)	)	PUNCT
ejpam-3780	742	11	(	(	PUNCT
ejpam-3780	742	12	2020	2020	NUM
ejpam-3780	742	13	)	)	PUNCT
ejpam-3780	742	14	,	,	PUNCT
ejpam-3780	742	15	674	674	NUM
ejpam-3780	742	16	-	-	SYM
ejpam-3780	742	17	696	696	NUM
ejpam-3780	742	18	693	693	NUM
ejpam-3780	742	19	a49	a49	PROPN
ejpam-3780	742	20	a7	a7	PROPN
ejpam-3780	742	21	a42a49	a42a49	PROPN
ejpam-3780	742	22	a0	a0	PROPN
ejpam-3780	742	23	a49	a49	PROPN
ejpam-3780	742	24	a1	a1	PROPN
ejpam-3780	742	25	a49	a49	PROPN
ejpam-3780	742	26	a2	a2	PROPN
ejpam-3780	742	27	a7	a7	PROPN
ejpam-3780	742	28	a43	a43	PROPN
ejpam-3780	742	29	a7	a7	PROPN
ejpam-3780	742	30	a44	a44	PROPN
ejpam-3780	742	31	a42	a42	PROPN
ejpam-3780	742	32	a8	a8	PROPN
ejpam-3780	742	33	a42	a42	PROPN
ejpam-3780	742	34	a9	a9	PROPN
ejpam-3780	742	35	a42	a42	PROPN
ejpam-3780	742	36	a50	a50	NOUN
ejpam-3780	742	37	a42	a42	PROPN
ejpam-3780	742	38	a14	a14	PROPN
ejpam-3780	742	39	a21	a21	PROPN
ejpam-3780	742	40	a35	a35	PROPN
ejpam-3780	742	41	a21	a21	PROPN
ejpam-3780	742	42	a29	a29	PROPN
ejpam-3780	742	43	a21	a21	PROPN
ejpam-3780	742	44	a30	a30	PROPN
ejpam-3780	742	45	a21	a21	PROPN
ejpam-3780	742	46	a28	a28	PROPN
ejpam-3780	742	47	a0	a0	PROPN
ejpam-3780	742	48	a56	a56	PROPN
ejpam-3780	742	49	a1	a1	PROPN
ejpam-3780	742	50	a55	a55	NOUN
ejpam-3780	742	51	a2	a2	PROPN
ejpam-3780	742	52	a54	a54	NOUN
ejpam-3780	742	53	a8	a8	PROPN
ejpam-3780	742	54	a48	a48	PROPN
ejpam-3780	742	55	a9	a9	PROPN
ejpam-3780	742	56	a47	a47	NOUN
ejpam-3780	742	57	a10	a10	NOUN
ejpam-3780	742	58	a46	a46	PROPN
ejpam-3780	742	59	a44	a44	PROPN
ejpam-3780	742	60	a12	a12	PROPN
ejpam-3780	742	61	a29	a29	PROPN
ejpam-3780	742	62	a27	a27	PROPN
ejpam-3780	742	63	a30	a30	PROPN
ejpam-3780	742	64	a26	a26	PROPN
ejpam-3780	742	65	figure	figure	NOUN
ejpam-3780	742	66	15	15	NUM
ejpam-3780	742	67	:	:	PUNCT
ejpam-3780	742	68	efficient	efficient	ADJ
ejpam-3780	742	69	zero	zero	NUM
ejpam-3780	742	70	ring	ring	NOUN
ejpam-3780	742	71	labeling	labeling	NOUN
ejpam-3780	742	72	of	of	ADP
ejpam-3780	742	73	a	a	DET
ejpam-3780	742	74	disjoint	disjoint	NOUN
ejpam-3780	742	75	union	union	NOUN
ejpam-3780	742	76	of	of	ADP
ejpam-3780	742	77	two	two	NUM
ejpam-3780	742	78	lobsters	lobster	NOUN
ejpam-3780	742	79	using	use	VERB
ejpam-3780	742	80	m0	m0	PROPN
ejpam-3780	742	81	2	2	NUM
ejpam-3780	742	82	(	(	PUNCT
ejpam-3780	742	83	z57	z57	ADJ
ejpam-3780	742	84	)	)	PUNCT
ejpam-3780	742	85	proof	proof	NOUN
ejpam-3780	742	86	.	.	PUNCT
ejpam-3780	743	1	let	let	VERB
ejpam-3780	743	2	cn	cn	PROPN
ejpam-3780	743	3	=	=	PUNCT
ejpam-3780	744	1	[	[	X
ejpam-3780	744	2	w1	w1	NOUN
ejpam-3780	744	3	,	,	PUNCT
ejpam-3780	744	4	w2	w2	NOUN
ejpam-3780	744	5	,	,	PUNCT
ejpam-3780	744	6	.	.	PUNCT
ejpam-3780	744	7	.	.	PUNCT
ejpam-3780	744	8	.	.	PUNCT
ejpam-3780	745	1	,	,	PUNCT
ejpam-3780	745	2	wn	wn	PROPN
ejpam-3780	745	3	,	,	PUNCT
ejpam-3780	745	4	w1	w1	PROPN
ejpam-3780	745	5	]	]	PUNCT
ejpam-3780	745	6	,	,	PUNCT
ejpam-3780	745	7	where	where	SCONJ
ejpam-3780	745	8	n	n	PRON
ejpam-3780	745	9	is	be	AUX
ejpam-3780	745	10	even	even	ADV
ejpam-3780	745	11	.	.	PUNCT
ejpam-3780	746	1	clearly	clearly	ADV
ejpam-3780	746	2	,	,	PUNCT
ejpam-3780	746	3	∆(cn	∆(cn	NOUN
ejpam-3780	746	4	)	)	PUNCT
ejpam-3780	746	5	=	=	SYM
ejpam-3780	747	1	2	2	X
ejpam-3780	747	2	.	.	PUNCT
ejpam-3780	747	3	define	define	VERB
ejpam-3780	747	4	a	a	DET
ejpam-3780	747	5	function	function	NOUN
ejpam-3780	747	6	f	f	NOUN
ejpam-3780	747	7	:	:	PUNCT
ejpam-3780	747	8	v	v	X
ejpam-3780	747	9	(	(	PUNCT
ejpam-3780	747	10	cn)→m0	cn)→m0	X
ejpam-3780	747	11	2	2	NUM
ejpam-3780	747	12	(	(	PUNCT
ejpam-3780	747	13	zn	zn	NOUN
ejpam-3780	747	14	)	)	PUNCT
ejpam-3780	747	15	such	such	ADJ
ejpam-3780	747	16	that	that	SCONJ
ejpam-3780	747	17	f(wi	f(wi	NOUN
ejpam-3780	747	18	)	)	PUNCT
ejpam-3780	747	19	=	=	NOUN
ejpam-3780	747	20	{	{	PUNCT
ejpam-3780	747	21	ai−1	ai−1	PROPN
ejpam-3780	747	22	if	if	SCONJ
ejpam-3780	747	23	i	i	PRON
ejpam-3780	747	24	is	be	AUX
ejpam-3780	747	25	odd	odd	ADJ
ejpam-3780	747	26	an−i+3	an−i+3	NOUN
ejpam-3780	747	27	if	if	SCONJ
ejpam-3780	747	28	i	i	PRON
ejpam-3780	747	29	is	be	AUX
ejpam-3780	747	30	even	even	ADV
ejpam-3780	747	31	.	.	PUNCT
ejpam-3780	748	1	(	(	PUNCT
ejpam-3780	748	2	43	43	NUM
ejpam-3780	748	3	)	)	PUNCT
ejpam-3780	748	4	clearly	clearly	ADV
ejpam-3780	748	5	,	,	PUNCT
ejpam-3780	748	6	f	f	PROPN
ejpam-3780	748	7	is	be	AUX
ejpam-3780	748	8	injective	injective	ADJ
ejpam-3780	748	9	.	.	PUNCT
ejpam-3780	749	1	let	let	VERB
ejpam-3780	749	2	k	k	NOUN
ejpam-3780	749	3	=	=	PRON
ejpam-3780	749	4	{	{	PUNCT
ejpam-3780	749	5	f(u	f(u	PROPN
ejpam-3780	749	6	)	)	PUNCT
ejpam-3780	749	7	+	+	NUM
ejpam-3780	749	8	f(v	f(v	NOUN
ejpam-3780	749	9	)	)	PUNCT
ejpam-3780	749	10	:	:	PUNCT
ejpam-3780	749	11	uv	uv	NOUN
ejpam-3780	749	12	∈	∈	PROPN
ejpam-3780	749	13	e(cn	e(cn	NOUN
ejpam-3780	749	14	)	)	PUNCT
ejpam-3780	749	15	}	}	PUNCT
ejpam-3780	749	16	.	.	PUNCT
ejpam-3780	750	1	we	we	PRON
ejpam-3780	750	2	obtain	obtain	VERB
ejpam-3780	750	3	f(wi	f(wi	NOUN
ejpam-3780	750	4	)	)	PUNCT
ejpam-3780	750	5	+	+	NUM
ejpam-3780	750	6	f(wi+1	f(wi+1	X
ejpam-3780	750	7	)	)	PUNCT
ejpam-3780	750	8	=	=	SYM
ejpam-3780	751	1	ai−1	ai−1	PROPN
ejpam-3780	751	2	+	+	NUM
ejpam-3780	751	3	an−(i+1)+3	an−(i+1)+3	PROPN
ejpam-3780	751	4	=	=	SYM
ejpam-3780	751	5	an+1	an+1	NOUN
ejpam-3780	751	6	=	=	SYM
ejpam-3780	751	7	a1	a1	PROPN
ejpam-3780	751	8	6=	6=	PROPN
ejpam-3780	751	9	a0	a0	PROPN
ejpam-3780	751	10	(	(	PUNCT
ejpam-3780	751	11	44	44	NUM
ejpam-3780	751	12	)	)	PUNCT
ejpam-3780	751	13	if	if	SCONJ
ejpam-3780	751	14	i	i	PRON
ejpam-3780	751	15	is	be	AUX
ejpam-3780	751	16	odd	odd	ADJ
ejpam-3780	751	17	,	,	PUNCT
ejpam-3780	751	18	and	and	CCONJ
ejpam-3780	751	19	f(wi	f(wi	NOUN
ejpam-3780	751	20	)	)	PUNCT
ejpam-3780	751	21	+	+	NUM
ejpam-3780	751	22	f(wi+1	f(wi+1	X
ejpam-3780	751	23	)	)	PUNCT
ejpam-3780	751	24	=	=	SYM
ejpam-3780	751	25	an−i+3	an−i+3	NOUN
ejpam-3780	751	26	+	+	CCONJ
ejpam-3780	751	27	a(i+1)−1	a(i+1)−1	NOUN
ejpam-3780	751	28	=	=	SYM
ejpam-3780	751	29	an+3	an+3	NOUN
ejpam-3780	751	30	=	=	NOUN
ejpam-3780	751	31	a3	a3	PROPN
ejpam-3780	751	32	6=	6=	NUM
ejpam-3780	751	33	a0	a0	PROPN
ejpam-3780	751	34	(	(	PUNCT
ejpam-3780	751	35	45	45	NUM
ejpam-3780	751	36	)	)	PUNCT
ejpam-3780	751	37	if	if	SCONJ
ejpam-3780	751	38	i	i	PRON
ejpam-3780	751	39	is	be	AUX
ejpam-3780	751	40	even	even	ADV
ejpam-3780	751	41	.	.	PUNCT
ejpam-3780	752	1	thus	thus	ADV
ejpam-3780	752	2	,	,	PUNCT
ejpam-3780	752	3	k	k	PROPN
ejpam-3780	752	4	=	=	PRON
ejpam-3780	752	5	{	{	PUNCT
ejpam-3780	752	6	a1	a1	NOUN
ejpam-3780	752	7	,	,	PUNCT
ejpam-3780	752	8	a3	a3	NOUN
ejpam-3780	752	9	}	}	PUNCT
ejpam-3780	752	10	,	,	PUNCT
ejpam-3780	752	11	hence	hence	ADV
ejpam-3780	752	12	|k|	|k|	NOUN
ejpam-3780	752	13	=	=	SYM
ejpam-3780	752	14	2	2	NUM
ejpam-3780	752	15	=	=	SYM
ejpam-3780	752	16	∆(cn	∆(cn	NOUN
ejpam-3780	752	17	)	)	PUNCT
ejpam-3780	752	18	.	.	PUNCT
ejpam-3780	753	1	therefore	therefore	ADV
ejpam-3780	753	2	,	,	PUNCT
ejpam-3780	753	3	f	f	PROPN
ejpam-3780	753	4	is	be	AUX
ejpam-3780	753	5	an	an	DET
ejpam-3780	753	6	efficient	efficient	ADJ
ejpam-3780	753	7	zero	zero	NUM
ejpam-3780	753	8	ring	ring	NOUN
ejpam-3780	753	9	labeling	labeling	NOUN
ejpam-3780	753	10	of	of	ADP
ejpam-3780	753	11	cn	cn	PROPN
ejpam-3780	753	12	.	.	PROPN
ejpam-3780	753	13	example	example	NOUN
ejpam-3780	754	1	15	15	NUM
ejpam-3780	754	2	.	.	PUNCT
ejpam-3780	755	1	figure	figure	NOUN
ejpam-3780	755	2	16	16	NUM
ejpam-3780	755	3	shows	show	VERB
ejpam-3780	755	4	an	an	DET
ejpam-3780	755	5	efficient	efficient	ADJ
ejpam-3780	755	6	zero	zero	NUM
ejpam-3780	755	7	ring	ring	NOUN
ejpam-3780	755	8	labeling	labeling	NOUN
ejpam-3780	755	9	of	of	ADP
ejpam-3780	755	10	c8	c8	PROPN
ejpam-3780	755	11	using	use	VERB
ejpam-3780	755	12	m0	m0	PROPN
ejpam-3780	755	13	2	2	NUM
ejpam-3780	755	14	(	(	PUNCT
ejpam-3780	755	15	z8	z8	NOUN
ejpam-3780	755	16	)	)	PUNCT
ejpam-3780	755	17	.	.	PUNCT
ejpam-3780	756	1	in	in	ADP
ejpam-3780	756	2	this	this	DET
ejpam-3780	756	3	labeling	labeling	NOUN
ejpam-3780	756	4	,	,	PUNCT
ejpam-3780	756	5	the	the	DET
ejpam-3780	756	6	set	set	NOUN
ejpam-3780	756	7	of	of	ADP
ejpam-3780	756	8	sums	sum	NOUN
ejpam-3780	756	9	is	be	AUX
ejpam-3780	756	10	k	k	NOUN
ejpam-3780	756	11	=	=	PUNCT
ejpam-3780	756	12	{	{	PUNCT
ejpam-3780	756	13	a1	a1	NOUN
ejpam-3780	756	14	,	,	PUNCT
ejpam-3780	756	15	a3	a3	NOUN
ejpam-3780	756	16	}	}	PUNCT
ejpam-3780	756	17	and	and	CCONJ
ejpam-3780	756	18	thus	thus	ADV
ejpam-3780	756	19	|k|	|k|	NOUN
ejpam-3780	756	20	=	=	SYM
ejpam-3780	756	21	2	2	NUM
ejpam-3780	756	22	=	=	SYM
ejpam-3780	756	23	∆(c8	∆(c8	NUM
ejpam-3780	756	24	)	)	PUNCT
ejpam-3780	756	25	.	.	PUNCT
ejpam-3780	757	1	a1	a1	NOUN
ejpam-3780	757	2	a0a3	a0a3	PUNCT
ejpam-3780	757	3	a6	a6	PROPN
ejpam-3780	757	4	a5	a5	PROPN
ejpam-3780	757	5	a4	a4	PROPN
ejpam-3780	757	6	a7	a7	PROPN
ejpam-3780	757	7	a2	a2	PROPN
ejpam-3780	757	8	figure	figure	NOUN
ejpam-3780	757	9	16	16	NUM
ejpam-3780	757	10	:	:	PUNCT
ejpam-3780	757	11	efficient	efficient	ADJ
ejpam-3780	757	12	zero	zero	NUM
ejpam-3780	757	13	ring	ring	NOUN
ejpam-3780	757	14	labeling	labeling	NOUN
ejpam-3780	757	15	of	of	ADP
ejpam-3780	757	16	c8	c8	PROPN
ejpam-3780	757	17	using	use	VERB
ejpam-3780	757	18	m0	m0	PROPN
ejpam-3780	757	19	2	2	NUM
ejpam-3780	757	20	(	(	PUNCT
ejpam-3780	757	21	z8	z8	NOUN
ejpam-3780	757	22	)	)	PUNCT
ejpam-3780	757	23	d.	d.	PROPN
ejpam-3780	757	24	chua	chua	PROPN
ejpam-3780	757	25	,	,	PUNCT
ejpam-3780	757	26	f.	f.	PROPN
ejpam-3780	757	27	campeña	campeña	PROPN
ejpam-3780	757	28	,	,	PUNCT
ejpam-3780	757	29	f.	f.	PROPN
ejpam-3780	757	30	franco	franco	PROPN
ejpam-3780	757	31	/	/	SYM
ejpam-3780	757	32	eur	eur	PROPN
ejpam-3780	757	33	.	.	PUNCT
ejpam-3780	758	1	j.	j.	PROPN
ejpam-3780	758	2	pure	pure	PROPN
ejpam-3780	758	3	appl	appl	PROPN
ejpam-3780	758	4	.	.	PROPN
ejpam-3780	758	5	math	math	PROPN
ejpam-3780	758	6	,	,	PUNCT
ejpam-3780	758	7	13	13	NUM
ejpam-3780	758	8	(	(	PUNCT
ejpam-3780	758	9	3	3	NUM
ejpam-3780	758	10	)	)	PUNCT
ejpam-3780	758	11	(	(	PUNCT
ejpam-3780	758	12	2020	2020	NUM
ejpam-3780	758	13	)	)	PUNCT
ejpam-3780	758	14	,	,	PUNCT
ejpam-3780	758	15	674	674	NUM
ejpam-3780	758	16	-	-	SYM
ejpam-3780	758	17	696	696	NUM
ejpam-3780	758	18	694	694	NUM
ejpam-3780	758	19	theorem	theorem	VERB
ejpam-3780	758	20	13	13	NUM
ejpam-3780	758	21	.	.	PUNCT
ejpam-3780	759	1	a	a	DET
ejpam-3780	759	2	cycle	cycle	NOUN
ejpam-3780	759	3	graph	graph	NOUN
ejpam-3780	759	4	with	with	ADP
ejpam-3780	759	5	n	n	DET
ejpam-3780	759	6	vertices	vertex	NOUN
ejpam-3780	759	7	has	have	VERB
ejpam-3780	759	8	no	no	DET
ejpam-3780	759	9	efficient	efficient	ADJ
ejpam-3780	759	10	zero	zero	NUM
ejpam-3780	759	11	ring	ring	NOUN
ejpam-3780	759	12	labeling	labeling	NOUN
ejpam-3780	759	13	if	if	SCONJ
ejpam-3780	759	14	n	n	PRON
ejpam-3780	759	15	is	be	AUX
ejpam-3780	759	16	odd	odd	ADJ
ejpam-3780	759	17	.	.	PUNCT
ejpam-3780	760	1	proof	proof	NOUN
ejpam-3780	760	2	.	.	PUNCT
ejpam-3780	761	1	let	let	VERB
ejpam-3780	761	2	cn	cn	PROPN
ejpam-3780	761	3	=	=	PUNCT
ejpam-3780	762	1	[	[	X
ejpam-3780	762	2	w1	w1	NOUN
ejpam-3780	762	3	,	,	PUNCT
ejpam-3780	762	4	w2	w2	NOUN
ejpam-3780	762	5	,	,	PUNCT
ejpam-3780	762	6	.	.	PUNCT
ejpam-3780	762	7	.	.	PUNCT
ejpam-3780	762	8	.	.	PUNCT
ejpam-3780	763	1	,	,	PUNCT
ejpam-3780	763	2	wn	wn	PROPN
ejpam-3780	763	3	,	,	PUNCT
ejpam-3780	763	4	w1	w1	PROPN
ejpam-3780	763	5	]	]	PUNCT
ejpam-3780	763	6	,	,	PUNCT
ejpam-3780	763	7	where	where	SCONJ
ejpam-3780	763	8	n	n	PRON
ejpam-3780	763	9	is	be	AUX
ejpam-3780	763	10	odd	odd	ADJ
ejpam-3780	763	11	.	.	PUNCT
ejpam-3780	764	1	clearly	clearly	ADV
ejpam-3780	764	2	,	,	PUNCT
ejpam-3780	764	3	∆(cn	∆(cn	NOUN
ejpam-3780	764	4	)	)	PUNCT
ejpam-3780	764	5	=	=	SYM
ejpam-3780	764	6	2	2	X
ejpam-3780	764	7	.	.	X
ejpam-3780	764	8	assume	assume	VERB
ejpam-3780	764	9	that	that	SCONJ
ejpam-3780	764	10	cn	cn	PROPN
ejpam-3780	764	11	has	have	VERB
ejpam-3780	764	12	an	an	DET
ejpam-3780	764	13	efficient	efficient	ADJ
ejpam-3780	764	14	zero	zero	NUM
ejpam-3780	764	15	ring	ring	NOUN
ejpam-3780	764	16	labeling	labeling	NOUN
ejpam-3780	764	17	f	f	NOUN
ejpam-3780	764	18	:	:	PUNCT
ejpam-3780	764	19	v	v	X
ejpam-3780	764	20	(	(	PUNCT
ejpam-3780	764	21	cn)→	cn)→	NOUN
ejpam-3780	764	22	r0	r0	NOUN
ejpam-3780	764	23	.	.	PUNCT
ejpam-3780	765	1	since	since	SCONJ
ejpam-3780	765	2	w1	w1	NOUN
ejpam-3780	765	3	and	and	CCONJ
ejpam-3780	765	4	w2	w2	NOUN
ejpam-3780	765	5	are	be	AUX
ejpam-3780	765	6	adjacent	adjacent	ADJ
ejpam-3780	765	7	,	,	PUNCT
ejpam-3780	765	8	f(w1	f(w1	ADV
ejpam-3780	765	9	)	)	PUNCT
ejpam-3780	765	10	+	+	NUM
ejpam-3780	765	11	f(w2	f(w2	X
ejpam-3780	765	12	)	)	PUNCT
ejpam-3780	765	13	∈	∈	PROPN
ejpam-3780	766	1	k	k	NOUN
ejpam-3780	767	1	=	=	PRON
ejpam-3780	768	1	{	{	PUNCT
ejpam-3780	768	2	f(u	f(u	PROPN
ejpam-3780	768	3	)	)	PUNCT
ejpam-3780	769	1	+	+	NUM
ejpam-3780	769	2	f(v	f(v	NOUN
ejpam-3780	769	3	)	)	PUNCT
ejpam-3780	769	4	:	:	PUNCT
ejpam-3780	769	5	uv	uv	NOUN
ejpam-3780	769	6	∈	∈	PROPN
ejpam-3780	769	7	e(cn	e(cn	NOUN
ejpam-3780	769	8	)	)	PUNCT
ejpam-3780	769	9	}	}	PUNCT
ejpam-3780	769	10	.	.	PUNCT
ejpam-3780	770	1	it	it	PRON
ejpam-3780	770	2	follows	follow	VERB
ejpam-3780	770	3	that	that	DET
ejpam-3780	770	4	f(w2	f(w2	NOUN
ejpam-3780	770	5	)	)	PUNCT
ejpam-3780	771	1	+	+	CCONJ
ejpam-3780	771	2	f(w3	f(w3	NUM
ejpam-3780	771	3	)	)	PUNCT
ejpam-3780	771	4	6=	6=	SYM
ejpam-3780	771	5	f(w1	f(w1	NOUN
ejpam-3780	771	6	)	)	PUNCT
ejpam-3780	771	7	+	+	NUM
ejpam-3780	771	8	f(w2	f(w2	NOUN
ejpam-3780	771	9	)	)	PUNCT
ejpam-3780	771	10	;	;	PUNCT
ejpam-3780	771	11	otherwise	otherwise	ADV
ejpam-3780	771	12	,	,	PUNCT
ejpam-3780	771	13	f(w1	f(w1	ADV
ejpam-3780	771	14	)	)	PUNCT
ejpam-3780	771	15	=	=	SYM
ejpam-3780	772	1	f(w3	f(w3	PROPN
ejpam-3780	772	2	)	)	PUNCT
ejpam-3780	772	3	,	,	PUNCT
ejpam-3780	772	4	which	which	PRON
ejpam-3780	772	5	is	be	AUX
ejpam-3780	772	6	not	not	PART
ejpam-3780	772	7	possible	possible	ADJ
ejpam-3780	772	8	since	since	SCONJ
ejpam-3780	772	9	f	f	PROPN
ejpam-3780	772	10	is	be	AUX
ejpam-3780	772	11	injective	injective	ADJ
ejpam-3780	772	12	.	.	PUNCT
ejpam-3780	773	1	k	k	PROPN
ejpam-3780	773	2	has	have	VERB
ejpam-3780	773	3	only	only	ADV
ejpam-3780	773	4	two	two	NUM
ejpam-3780	773	5	elements	element	NOUN
ejpam-3780	773	6	,	,	PUNCT
ejpam-3780	773	7	so	so	SCONJ
ejpam-3780	773	8	k	k	PROPN
ejpam-3780	773	9	=	=	PRON
ejpam-3780	773	10	{	{	PUNCT
ejpam-3780	773	11	f(w1	f(w1	NOUN
ejpam-3780	773	12	)	)	PUNCT
ejpam-3780	773	13	+	+	NUM
ejpam-3780	773	14	f(w2	f(w2	NOUN
ejpam-3780	773	15	)	)	PUNCT
ejpam-3780	773	16	,	,	PUNCT
ejpam-3780	773	17	f(w2	f(w2	NUM
ejpam-3780	773	18	)	)	PUNCT
ejpam-3780	773	19	+	+	CCONJ
ejpam-3780	773	20	f(w3	f(w3	NUM
ejpam-3780	773	21	)	)	PUNCT
ejpam-3780	773	22	}	}	PUNCT
ejpam-3780	773	23	.	.	PUNCT
ejpam-3780	774	1	similarly	similarly	ADV
ejpam-3780	774	2	,	,	PUNCT
ejpam-3780	774	3	f(w3	f(w3	PROPN
ejpam-3780	774	4	)	)	PUNCT
ejpam-3780	774	5	+	+	CCONJ
ejpam-3780	774	6	f(w4	f(w4	NOUN
ejpam-3780	774	7	)	)	PUNCT
ejpam-3780	774	8	6=	6=	NOUN
ejpam-3780	774	9	f(w2	f(w2	NOUN
ejpam-3780	774	10	)	)	PUNCT
ejpam-3780	775	1	+	+	CCONJ
ejpam-3780	775	2	f(w3	f(w3	NUM
ejpam-3780	775	3	)	)	PUNCT
ejpam-3780	775	4	.	.	PUNCT
ejpam-3780	776	1	then	then	ADV
ejpam-3780	776	2	f(w3	f(w3	NUM
ejpam-3780	776	3	)	)	PUNCT
ejpam-3780	777	1	+	+	CCONJ
ejpam-3780	777	2	f(w4	f(w4	NOUN
ejpam-3780	777	3	)	)	PUNCT
ejpam-3780	777	4	=	=	SYM
ejpam-3780	777	5	f(w1	f(w1	NOUN
ejpam-3780	777	6	)	)	PUNCT
ejpam-3780	777	7	+	+	NUM
ejpam-3780	777	8	f(w2	f(w2	NOUN
ejpam-3780	777	9	)	)	PUNCT
ejpam-3780	777	10	.	.	PUNCT
ejpam-3780	778	1	continuing	continue	VERB
ejpam-3780	778	2	in	in	ADP
ejpam-3780	778	3	this	this	DET
ejpam-3780	778	4	manner	manner	NOUN
ejpam-3780	778	5	,	,	PUNCT
ejpam-3780	778	6	f(wn	f(wn	NOUN
ejpam-3780	778	7	)	)	PUNCT
ejpam-3780	778	8	+	+	NUM
ejpam-3780	778	9	f(w1	f(w1	NOUN
ejpam-3780	778	10	)	)	PUNCT
ejpam-3780	778	11	would	would	AUX
ejpam-3780	778	12	be	be	AUX
ejpam-3780	778	13	equal	equal	ADJ
ejpam-3780	778	14	to	to	ADP
ejpam-3780	778	15	f(w1	f(w1	NOUN
ejpam-3780	778	16	)	)	PUNCT
ejpam-3780	778	17	+	+	NUM
ejpam-3780	778	18	f(w2	f(w2	NOUN
ejpam-3780	778	19	)	)	PUNCT
ejpam-3780	778	20	since	since	SCONJ
ejpam-3780	778	21	n	n	NOUN
ejpam-3780	778	22	odd	odd	ADJ
ejpam-3780	778	23	.	.	PUNCT
ejpam-3780	779	1	but	but	CCONJ
ejpam-3780	779	2	this	this	PRON
ejpam-3780	779	3	implies	imply	VERB
ejpam-3780	779	4	that	that	PRON
ejpam-3780	779	5	f(w2	f(w2	NOUN
ejpam-3780	779	6	)	)	PUNCT
ejpam-3780	779	7	=	=	SYM
ejpam-3780	779	8	f(wn	f(wn	NOUN
ejpam-3780	779	9	)	)	PUNCT
ejpam-3780	779	10	.	.	PUNCT
ejpam-3780	780	1	this	this	PRON
ejpam-3780	780	2	is	be	AUX
ejpam-3780	780	3	a	a	DET
ejpam-3780	780	4	contradiction	contradiction	NOUN
ejpam-3780	780	5	since	since	SCONJ
ejpam-3780	780	6	f	f	PROPN
ejpam-3780	780	7	is	be	AUX
ejpam-3780	780	8	injective	injective	ADJ
ejpam-3780	780	9	.	.	PUNCT
ejpam-3780	781	1	theorem	theorem	ADJ
ejpam-3780	781	2	14	14	NUM
ejpam-3780	781	3	.	.	PUNCT
ejpam-3780	782	1	a	a	DET
ejpam-3780	782	2	complete	complete	ADJ
ejpam-3780	782	3	bipartite	bipartite	NOUN
ejpam-3780	782	4	graph	graph	NOUN
ejpam-3780	782	5	has	have	VERB
ejpam-3780	782	6	an	an	DET
ejpam-3780	782	7	efficient	efficient	ADJ
ejpam-3780	782	8	zero	zero	NUM
ejpam-3780	782	9	ring	ring	NOUN
ejpam-3780	782	10	labeling	labeling	NOUN
ejpam-3780	782	11	.	.	PUNCT
ejpam-3780	783	1	proof	proof	NOUN
ejpam-3780	783	2	.	.	PUNCT
ejpam-3780	784	1	let	let	VERB
ejpam-3780	784	2	g	g	PROPN
ejpam-3780	784	3	=	=	PROPN
ejpam-3780	784	4	km	km	PROPN
ejpam-3780	784	5	,	,	PUNCT
ejpam-3780	784	6	n	n	PRON
ejpam-3780	784	7	be	be	VERB
ejpam-3780	784	8	a	a	DET
ejpam-3780	784	9	complete	complete	ADJ
ejpam-3780	784	10	bipartite	bipartite	NOUN
ejpam-3780	784	11	graph	graph	NOUN
ejpam-3780	784	12	with	with	ADP
ejpam-3780	784	13	vertex	vertex	NOUN
ejpam-3780	784	14	set	set	NOUN
ejpam-3780	784	15	{	{	PUNCT
ejpam-3780	784	16	x1	x1	PROPN
ejpam-3780	784	17	,	,	PUNCT
ejpam-3780	784	18	x2	x2	PROPN
ejpam-3780	784	19	,	,	PUNCT
ejpam-3780	784	20	.	.	PUNCT
ejpam-3780	784	21	.	.	PUNCT
ejpam-3780	785	1	.	.	PUNCT
ejpam-3780	786	1	,	,	PUNCT
ejpam-3780	786	2	xm	xm	X
ejpam-3780	786	3	}	}	PUNCT
ejpam-3780	786	4	∪	∪	ADJ
ejpam-3780	786	5	{	{	PUNCT
ejpam-3780	786	6	y1	y1	NOUN
ejpam-3780	786	7	,	,	PUNCT
ejpam-3780	786	8	y2	y2	PROPN
ejpam-3780	786	9	,	,	PUNCT
ejpam-3780	786	10	.	.	PUNCT
ejpam-3780	786	11	.	.	PUNCT
ejpam-3780	787	1	.	.	PUNCT
ejpam-3780	788	1	,	,	PUNCT
ejpam-3780	788	2	yn	yn	PRON
ejpam-3780	788	3	}	}	PUNCT
ejpam-3780	788	4	and	and	CCONJ
ejpam-3780	788	5	edge	edge	NOUN
ejpam-3780	788	6	set	set	NOUN
ejpam-3780	788	7	{	{	PUNCT
ejpam-3780	788	8	xiyj	xiyj	NOUN
ejpam-3780	788	9	:	:	PUNCT
ejpam-3780	788	10	1	1	NUM
ejpam-3780	788	11	≤	≤	NUM
ejpam-3780	788	12	i	i	X
ejpam-3780	789	1	≤	≤	NOUN
ejpam-3780	789	2	m	m	VERB
ejpam-3780	789	3	and	and	CCONJ
ejpam-3780	789	4	1	1	NUM
ejpam-3780	789	5	≤	≤	NUM
ejpam-3780	789	6	j	j	PROPN
ejpam-3780	789	7	≤	≤	PROPN
ejpam-3780	789	8	n	n	CCONJ
ejpam-3780	789	9	}	}	PUNCT
ejpam-3780	789	10	.	.	PUNCT
ejpam-3780	790	1	without	without	ADP
ejpam-3780	790	2	loss	loss	NOUN
ejpam-3780	790	3	of	of	ADP
ejpam-3780	790	4	generality	generality	NOUN
ejpam-3780	790	5	,	,	PUNCT
ejpam-3780	790	6	suppose	suppose	VERB
ejpam-3780	790	7	m	m	PRON
ejpam-3780	790	8	≥	≥	PROPN
ejpam-3780	790	9	n.	n.	NOUN
ejpam-3780	790	10	then	then	ADV
ejpam-3780	790	11	∆(g	∆(g	NOUN
ejpam-3780	790	12	)	)	PUNCT
ejpam-3780	791	1	=	=	SYM
ejpam-3780	791	2	m.	m.	NOUN
ejpam-3780	791	3	define	define	VERB
ejpam-3780	791	4	a	a	DET
ejpam-3780	791	5	function	function	NOUN
ejpam-3780	791	6	f	f	NOUN
ejpam-3780	791	7	:	:	PUNCT
ejpam-3780	791	8	v	v	X
ejpam-3780	791	9	(	(	PUNCT
ejpam-3780	791	10	g	g	NOUN
ejpam-3780	791	11	)	)	PUNCT
ejpam-3780	791	12	→	→	SYM
ejpam-3780	791	13	m0	m0	PROPN
ejpam-3780	791	14	2	2	NUM
ejpam-3780	791	15	(	(	PUNCT
ejpam-3780	791	16	z2	z2	NOUN
ejpam-3780	791	17	m	m	NOUN
ejpam-3780	791	18	)	)	PUNCT
ejpam-3780	791	19	such	such	ADJ
ejpam-3780	791	20	that	that	DET
ejpam-3780	791	21	f(xi	f(xi	PROPN
ejpam-3780	791	22	)	)	PUNCT
ejpam-3780	791	23	=	=	SYM
ejpam-3780	791	24	a2i−2	a2i−2	PROPN
ejpam-3780	791	25	and	and	CCONJ
ejpam-3780	791	26	f(yj	f(yj	NOUN
ejpam-3780	791	27	)	)	PUNCT
ejpam-3780	791	28	=	=	SYM
ejpam-3780	791	29	a2j−1	a2j−1	PROPN
ejpam-3780	791	30	.	.	PUNCT
ejpam-3780	792	1	clearly	clearly	ADV
ejpam-3780	792	2	,	,	PUNCT
ejpam-3780	792	3	f	f	PROPN
ejpam-3780	792	4	is	be	AUX
ejpam-3780	792	5	injective	injective	ADJ
ejpam-3780	792	6	.	.	PUNCT
ejpam-3780	793	1	let	let	VERB
ejpam-3780	793	2	k	k	NOUN
ejpam-3780	793	3	=	=	PRON
ejpam-3780	793	4	{	{	PUNCT
ejpam-3780	793	5	f(u	f(u	PROPN
ejpam-3780	793	6	)	)	PUNCT
ejpam-3780	793	7	+	+	NUM
ejpam-3780	793	8	f(v	f(v	NOUN
ejpam-3780	793	9	)	)	PUNCT
ejpam-3780	793	10	:	:	PUNCT
ejpam-3780	793	11	uv	uv	PROPN
ejpam-3780	793	12	∈	∈	PROPN
ejpam-3780	793	13	e(g	e(g	PROPN
ejpam-3780	793	14	)	)	PUNCT
ejpam-3780	793	15	}	}	PUNCT
ejpam-3780	793	16	.	.	PUNCT
ejpam-3780	794	1	we	we	PRON
ejpam-3780	794	2	obtain	obtain	VERB
ejpam-3780	794	3	f(xi	f(xi	NOUN
ejpam-3780	794	4	)	)	PUNCT
ejpam-3780	795	1	+	+	NUM
ejpam-3780	795	2	f(yj	f(yj	NOUN
ejpam-3780	795	3	)	)	PUNCT
ejpam-3780	795	4	=	=	SYM
ejpam-3780	795	5	a2i−2	a2i−2	PROPN
ejpam-3780	795	6	+	+	CCONJ
ejpam-3780	795	7	a2j−1	a2j−1	PROPN
ejpam-3780	795	8	=	=	SYM
ejpam-3780	795	9	a2i+2j−3	a2i+2j−3	PROPN
ejpam-3780	795	10	(	(	PUNCT
ejpam-3780	795	11	46	46	NUM
ejpam-3780	795	12	)	)	PUNCT
ejpam-3780	795	13	for	for	ADP
ejpam-3780	795	14	i	i	PROPN
ejpam-3780	795	15	=	=	SYM
ejpam-3780	795	16	1	1	NUM
ejpam-3780	795	17	,	,	PUNCT
ejpam-3780	795	18	2	2	NUM
ejpam-3780	795	19	,	,	PUNCT
ejpam-3780	795	20	.	.	PUNCT
ejpam-3780	795	21	.	.	PUNCT
ejpam-3780	795	22	.	.	PUNCT
ejpam-3780	796	1	,	,	PUNCT
ejpam-3780	796	2	m	m	PROPN
ejpam-3780	796	3	and	and	CCONJ
ejpam-3780	796	4	j	j	PROPN
ejpam-3780	796	5	=	=	SYM
ejpam-3780	796	6	1	1	NUM
ejpam-3780	796	7	,	,	PUNCT
ejpam-3780	796	8	2	2	NUM
ejpam-3780	796	9	,	,	PUNCT
ejpam-3780	796	10	.	.	PUNCT
ejpam-3780	796	11	.	.	PUNCT
ejpam-3780	797	1	.	.	PUNCT
ejpam-3780	798	1	,	,	PUNCT
ejpam-3780	798	2	n.	n.	NOUN
ejpam-3780	798	3	then	then	ADV
ejpam-3780	798	4	k	k	PROPN
ejpam-3780	798	5	=	=	PUNCT
ejpam-3780	798	6	{	{	PUNCT
ejpam-3780	798	7	a1	a1	PROPN
ejpam-3780	798	8	,	,	PUNCT
ejpam-3780	798	9	a3	a3	NOUN
ejpam-3780	798	10	,	,	PUNCT
ejpam-3780	798	11	.	.	PUNCT
ejpam-3780	798	12	.	.	PUNCT
ejpam-3780	799	1	.	.	PUNCT
ejpam-3780	800	1	,	,	PUNCT
ejpam-3780	800	2	a2m−1	a2m−1	PROPN
ejpam-3780	800	3	}	}	PUNCT
ejpam-3780	800	4	,	,	PUNCT
ejpam-3780	800	5	hence	hence	ADV
ejpam-3780	800	6	|k|	|k|	NOUN
ejpam-3780	800	7	=	=	SYM
ejpam-3780	800	8	m.	m.	NOUN
ejpam-3780	800	9	moreover	moreover	ADV
ejpam-3780	800	10	,	,	PUNCT
ejpam-3780	800	11	since	since	SCONJ
ejpam-3780	800	12	m	m	PROPN
ejpam-3780	800	13	is	be	AUX
ejpam-3780	800	14	at	at	ADP
ejpam-3780	800	15	least	least	ADJ
ejpam-3780	800	16	one	one	NUM
ejpam-3780	800	17	,	,	PUNCT
ejpam-3780	800	18	a0	a0	PROPN
ejpam-3780	800	19	/∈	/∈	PROPN
ejpam-3780	801	1	k.	k.	PROPN
ejpam-3780	802	1	thus	thus	ADV
ejpam-3780	802	2	,	,	PUNCT
ejpam-3780	802	3	f	f	PROPN
ejpam-3780	802	4	is	be	AUX
ejpam-3780	802	5	an	an	DET
ejpam-3780	802	6	efficient	efficient	ADJ
ejpam-3780	802	7	zero	zero	NUM
ejpam-3780	802	8	ring	ring	NOUN
ejpam-3780	802	9	labeling	labeling	NOUN
ejpam-3780	802	10	of	of	ADP
ejpam-3780	802	11	g.	g.	PROPN
ejpam-3780	802	12	example	example	PROPN
ejpam-3780	803	1	16	16	NUM
ejpam-3780	803	2	.	.	PUNCT
ejpam-3780	804	1	figure	figure	VERB
ejpam-3780	804	2	17	17	NUM
ejpam-3780	804	3	shows	show	VERB
ejpam-3780	804	4	an	an	DET
ejpam-3780	804	5	efficient	efficient	ADJ
ejpam-3780	804	6	zero	zero	NUM
ejpam-3780	804	7	ring	ring	NOUN
ejpam-3780	804	8	labeling	labeling	NOUN
ejpam-3780	804	9	of	of	ADP
ejpam-3780	804	10	k5,4	k5,4	PROPN
ejpam-3780	804	11	using	use	VERB
ejpam-3780	804	12	m0	m0	PROPN
ejpam-3780	804	13	2	2	NUM
ejpam-3780	804	14	(	(	PUNCT
ejpam-3780	804	15	z10	z10	NOUN
ejpam-3780	804	16	)	)	PUNCT
ejpam-3780	804	17	.	.	PUNCT
ejpam-3780	805	1	in	in	ADP
ejpam-3780	805	2	this	this	DET
ejpam-3780	805	3	labeling	labeling	NOUN
ejpam-3780	805	4	,	,	PUNCT
ejpam-3780	805	5	the	the	DET
ejpam-3780	805	6	set	set	NOUN
ejpam-3780	805	7	of	of	ADP
ejpam-3780	805	8	sums	sum	NOUN
ejpam-3780	805	9	is	be	AUX
ejpam-3780	805	10	k	k	NOUN
ejpam-3780	805	11	=	=	PUNCT
ejpam-3780	805	12	{	{	PUNCT
ejpam-3780	805	13	a1	a1	PROPN
ejpam-3780	805	14	,	,	PUNCT
ejpam-3780	805	15	a3	a3	NOUN
ejpam-3780	805	16	,	,	PUNCT
ejpam-3780	805	17	a5	a5	PROPN
ejpam-3780	805	18	,	,	PUNCT
ejpam-3780	805	19	a7	a7	PROPN
ejpam-3780	805	20	,	,	PUNCT
ejpam-3780	805	21	a9	a9	PROPN
ejpam-3780	805	22	}	}	PUNCT
ejpam-3780	805	23	and	and	CCONJ
ejpam-3780	805	24	thus	thus	ADV
ejpam-3780	805	25	|k|	|k|	NOUN
ejpam-3780	805	26	=	=	SYM
ejpam-3780	805	27	5	5	NUM
ejpam-3780	805	28	=	=	SYM
ejpam-3780	805	29	∆(k5,4	∆(k5,4	PROPN
ejpam-3780	805	30	)	)	PUNCT
ejpam-3780	805	31	.	.	PUNCT
ejpam-3780	806	1	a0	a0	PROPN
ejpam-3780	806	2	a1	a1	PROPN
ejpam-3780	806	3	a0	a0	PROPN
ejpam-3780	806	4	a3	a3	PROPN
ejpam-3780	806	5	a0	a0	PROPN
ejpam-3780	806	6	a5	a5	PROPN
ejpam-3780	806	7	a0	a0	PROPN
ejpam-3780	806	8	a7	a7	PROPN
ejpam-3780	806	9	a2	a2	PROPN
ejpam-3780	806	10	a1	a1	PROPN
ejpam-3780	806	11	a2	a2	PROPN
ejpam-3780	806	12	a3	a3	PROPN
ejpam-3780	806	13	a2	a2	PROPN
ejpam-3780	806	14	a5	a5	PROPN
ejpam-3780	806	15	a2	a2	PROPN
ejpam-3780	806	16	a7	a7	PROPN
ejpam-3780	806	17	a4	a4	NOUN
ejpam-3780	806	18	a1	a1	NOUN
ejpam-3780	806	19	a4	a4	NOUN
ejpam-3780	806	20	a3	a3	NOUN
ejpam-3780	806	21	a4	a4	NOUN
ejpam-3780	806	22	a5	a5	NOUN
ejpam-3780	806	23	a4	a4	NOUN
ejpam-3780	806	24	a7	a7	PROPN
ejpam-3780	806	25	a6	a6	NOUN
ejpam-3780	806	26	a1	a1	NOUN
ejpam-3780	806	27	a6	a6	PROPN
ejpam-3780	806	28	a3	a3	PROPN
ejpam-3780	806	29	a6	a6	PROPN
ejpam-3780	806	30	a5	a5	PROPN
ejpam-3780	806	31	a6	a6	PROPN
ejpam-3780	806	32	a7	a7	PROPN
ejpam-3780	806	33	a8	a8	PROPN
ejpam-3780	806	34	a1	a1	NOUN
ejpam-3780	806	35	a8	a8	PROPN
ejpam-3780	806	36	a3	a3	NOUN
ejpam-3780	806	37	a8	a8	PROPN
ejpam-3780	806	38	a5	a5	PROPN
ejpam-3780	806	39	a8	a8	PROPN
ejpam-3780	806	40	a7	a7	PROPN
ejpam-3780	806	41	figure	figure	NOUN
ejpam-3780	806	42	17	17	NUM
ejpam-3780	806	43	:	:	PUNCT
ejpam-3780	806	44	efficient	efficient	ADJ
ejpam-3780	806	45	zero	zero	NUM
ejpam-3780	806	46	ring	ring	NOUN
ejpam-3780	806	47	labeling	labeling	NOUN
ejpam-3780	806	48	of	of	ADP
ejpam-3780	806	49	k5,4	k5,4	PROPN
ejpam-3780	806	50	using	use	VERB
ejpam-3780	806	51	m0	m0	PROPN
ejpam-3780	806	52	2	2	NUM
ejpam-3780	806	53	(	(	PUNCT
ejpam-3780	806	54	z10	z10	NOUN
ejpam-3780	806	55	)	)	PUNCT
ejpam-3780	806	56	theorem	theorem	NOUN
ejpam-3780	806	57	15	15	NUM
ejpam-3780	806	58	.	.	PUNCT
ejpam-3780	807	1	a	a	DET
ejpam-3780	807	2	fan	fan	NOUN
ejpam-3780	807	3	graph	graph	NOUN
ejpam-3780	807	4	fn	fn	NOUN
ejpam-3780	807	5	has	have	VERB
ejpam-3780	807	6	an	an	DET
ejpam-3780	807	7	efficient	efficient	ADJ
ejpam-3780	807	8	zero	zero	NUM
ejpam-3780	807	9	ring	ring	NOUN
ejpam-3780	807	10	labeling	labeling	NOUN
ejpam-3780	807	11	for	for	ADP
ejpam-3780	807	12	n	n	X
ejpam-3780	807	13	≥	≥	NOUN
ejpam-3780	807	14	3	3	NUM
ejpam-3780	807	15	.	.	PUNCT
ejpam-3780	808	1	proof	proof	NOUN
ejpam-3780	808	2	.	.	PUNCT
ejpam-3780	809	1	it	it	PRON
ejpam-3780	809	2	was	be	AUX
ejpam-3780	809	3	shown	show	VERB
ejpam-3780	809	4	in	in	ADP
ejpam-3780	809	5	[	[	X
ejpam-3780	809	6	2	2	X
ejpam-3780	809	7	]	]	PUNCT
ejpam-3780	809	8	that	that	PRON
ejpam-3780	809	9	fn	fn	PROPN
ejpam-3780	809	10	has	have	VERB
ejpam-3780	809	11	an	an	DET
ejpam-3780	809	12	optimal	optimal	ADJ
ejpam-3780	809	13	zero	zero	NUM
ejpam-3780	809	14	ring	ring	NOUN
ejpam-3780	809	15	labeling	labeling	NOUN
ejpam-3780	809	16	for	for	ADP
ejpam-3780	809	17	n	n	PRON
ejpam-3780	809	18	≥	≥	NOUN
ejpam-3780	809	19	3	3	NUM
ejpam-3780	809	20	.	.	PUNCT
ejpam-3780	810	1	by	by	ADP
ejpam-3780	810	2	definition	definition	NOUN
ejpam-3780	810	3	of	of	ADP
ejpam-3780	810	4	fn	fn	NOUN
ejpam-3780	810	5	,	,	PUNCT
ejpam-3780	810	6	it	it	PRON
ejpam-3780	810	7	has	have	VERB
ejpam-3780	810	8	a	a	DET
ejpam-3780	810	9	vertex	vertex	NOUN
ejpam-3780	810	10	with	with	ADP
ejpam-3780	810	11	degree	degree	NOUN
ejpam-3780	810	12	|fn|	|fn|	INTJ
ejpam-3780	811	1	−	−	NOUN
ejpam-3780	811	2	1	1	NUM
ejpam-3780	811	3	.	.	PUNCT
ejpam-3780	812	1	thus	thus	ADV
ejpam-3780	812	2	,	,	PUNCT
ejpam-3780	812	3	by	by	ADP
ejpam-3780	812	4	theorem	theorem	NOUN
ejpam-3780	812	5	3	3	NUM
ejpam-3780	812	6	,	,	PUNCT
ejpam-3780	812	7	fn	fn	PROPN
ejpam-3780	812	8	has	have	VERB
ejpam-3780	812	9	an	an	DET
ejpam-3780	812	10	efficient	efficient	ADJ
ejpam-3780	812	11	zero	zero	NUM
ejpam-3780	812	12	ring	ring	NOUN
ejpam-3780	812	13	labeling	labeling	NOUN
ejpam-3780	812	14	for	for	ADP
ejpam-3780	812	15	n	n	PRON
ejpam-3780	812	16	≥	≥	NUM
ejpam-3780	812	17	3	3	NUM
ejpam-3780	812	18	.	.	PUNCT
ejpam-3780	812	19	theorem	theorem	VERB
ejpam-3780	812	20	16	16	NUM
ejpam-3780	812	21	.	.	PUNCT
ejpam-3780	813	1	a	a	DET
ejpam-3780	813	2	wheel	wheel	NOUN
ejpam-3780	813	3	graph	graph	NOUN
ejpam-3780	813	4	wn	wn	PROPN
ejpam-3780	813	5	has	have	VERB
ejpam-3780	813	6	an	an	DET
ejpam-3780	813	7	efficient	efficient	ADJ
ejpam-3780	813	8	zero	zero	NUM
ejpam-3780	813	9	ring	ring	NOUN
ejpam-3780	813	10	labeling	labeling	NOUN
ejpam-3780	813	11	for	for	ADP
ejpam-3780	813	12	n	n	PRON
ejpam-3780	813	13	≥	≥	NOUN
ejpam-3780	813	14	3	3	NUM
ejpam-3780	813	15	.	.	PUNCT
ejpam-3780	813	16	d.	d.	PROPN
ejpam-3780	813	17	chua	chua	PROPN
ejpam-3780	813	18	,	,	PUNCT
ejpam-3780	813	19	f.	f.	PROPN
ejpam-3780	813	20	campeña	campeña	PROPN
ejpam-3780	813	21	,	,	PUNCT
ejpam-3780	813	22	f.	f.	PROPN
ejpam-3780	813	23	franco	franco	PROPN
ejpam-3780	813	24	/	/	SYM
ejpam-3780	813	25	eur	eur	PROPN
ejpam-3780	813	26	.	.	PUNCT
ejpam-3780	814	1	j.	j.	PROPN
ejpam-3780	814	2	pure	pure	PROPN
ejpam-3780	814	3	appl	appl	PROPN
ejpam-3780	814	4	.	.	PROPN
ejpam-3780	814	5	math	math	PROPN
ejpam-3780	814	6	,	,	PUNCT
ejpam-3780	814	7	13	13	NUM
ejpam-3780	814	8	(	(	PUNCT
ejpam-3780	814	9	3	3	NUM
ejpam-3780	814	10	)	)	PUNCT
ejpam-3780	814	11	(	(	PUNCT
ejpam-3780	814	12	2020	2020	NUM
ejpam-3780	814	13	)	)	PUNCT
ejpam-3780	814	14	,	,	PUNCT
ejpam-3780	814	15	674	674	NUM
ejpam-3780	814	16	-	-	SYM
ejpam-3780	814	17	696	696	NUM
ejpam-3780	814	18	695	695	NUM
ejpam-3780	814	19	proof	proof	NOUN
ejpam-3780	814	20	.	.	PUNCT
ejpam-3780	815	1	it	it	PRON
ejpam-3780	815	2	was	be	AUX
ejpam-3780	815	3	shown	show	VERB
ejpam-3780	815	4	in	in	ADP
ejpam-3780	815	5	[	[	X
ejpam-3780	815	6	2	2	X
ejpam-3780	815	7	]	]	PUNCT
ejpam-3780	815	8	that	that	SCONJ
ejpam-3780	815	9	wn	wn	PROPN
ejpam-3780	815	10	has	have	VERB
ejpam-3780	815	11	an	an	DET
ejpam-3780	815	12	optimal	optimal	ADJ
ejpam-3780	815	13	zero	zero	NUM
ejpam-3780	815	14	ring	ring	NOUN
ejpam-3780	815	15	labeling	labeling	NOUN
ejpam-3780	815	16	for	for	ADP
ejpam-3780	815	17	n	n	PRON
ejpam-3780	815	18	≥	≥	NOUN
ejpam-3780	815	19	3	3	NUM
ejpam-3780	815	20	.	.	PUNCT
ejpam-3780	816	1	by	by	ADP
ejpam-3780	816	2	definition	definition	NOUN
ejpam-3780	816	3	of	of	ADP
ejpam-3780	816	4	wn	wn	PROPN
ejpam-3780	816	5	,	,	PUNCT
ejpam-3780	816	6	it	it	PRON
ejpam-3780	816	7	has	have	VERB
ejpam-3780	816	8	a	a	DET
ejpam-3780	816	9	vertex	vertex	NOUN
ejpam-3780	816	10	with	with	ADP
ejpam-3780	816	11	degree	degree	NOUN
ejpam-3780	816	12	|wn|	|wn|	PROPN
ejpam-3780	816	13	−	−	PROPN
ejpam-3780	817	1	1	1	NUM
ejpam-3780	817	2	.	.	PUNCT
ejpam-3780	817	3	thus	thus	ADV
ejpam-3780	817	4	,	,	PUNCT
ejpam-3780	817	5	by	by	ADP
ejpam-3780	817	6	theorem	theorem	NOUN
ejpam-3780	817	7	3	3	NUM
ejpam-3780	817	8	,	,	PUNCT
ejpam-3780	817	9	wn	wn	PROPN
ejpam-3780	817	10	has	have	VERB
ejpam-3780	817	11	an	an	DET
ejpam-3780	817	12	efficient	efficient	ADJ
ejpam-3780	817	13	zero	zero	NUM
ejpam-3780	817	14	ring	ring	NOUN
ejpam-3780	817	15	labeling	labeling	NOUN
ejpam-3780	817	16	for	for	ADP
ejpam-3780	817	17	n	n	PRON
ejpam-3780	817	18	≥	≥	NUM
ejpam-3780	817	19	3	3	NUM
ejpam-3780	817	20	.	.	PUNCT
ejpam-3780	817	21	theorem	theorem	VERB
ejpam-3780	817	22	17	17	NUM
ejpam-3780	817	23	.	.	PUNCT
ejpam-3780	818	1	a	a	DET
ejpam-3780	818	2	friendship	friendship	NOUN
ejpam-3780	818	3	graph	graph	NOUN
ejpam-3780	818	4	tn	tn	NOUN
ejpam-3780	818	5	has	have	VERB
ejpam-3780	818	6	an	an	DET
ejpam-3780	818	7	efficient	efficient	ADJ
ejpam-3780	818	8	zero	zero	NUM
ejpam-3780	818	9	ring	ring	NOUN
ejpam-3780	818	10	labeling	labeling	NOUN
ejpam-3780	818	11	for	for	ADP
ejpam-3780	818	12	n	n	PRON
ejpam-3780	818	13	≥	≥	NUM
ejpam-3780	818	14	2	2	NUM
ejpam-3780	818	15	.	.	PUNCT
ejpam-3780	819	1	proof	proof	NOUN
ejpam-3780	819	2	.	.	PUNCT
ejpam-3780	820	1	it	it	PRON
ejpam-3780	820	2	was	be	AUX
ejpam-3780	820	3	shown	show	VERB
ejpam-3780	820	4	in	in	ADP
ejpam-3780	820	5	[	[	X
ejpam-3780	820	6	2	2	X
ejpam-3780	820	7	]	]	PUNCT
ejpam-3780	820	8	that	that	SCONJ
ejpam-3780	820	9	tn	tn	PROPN
ejpam-3780	820	10	has	have	VERB
ejpam-3780	820	11	an	an	DET
ejpam-3780	820	12	optimal	optimal	ADJ
ejpam-3780	820	13	zero	zero	NUM
ejpam-3780	820	14	ring	ring	NOUN
ejpam-3780	820	15	labeling	labeling	NOUN
ejpam-3780	820	16	for	for	ADP
ejpam-3780	820	17	n	n	PRON
ejpam-3780	820	18	≥	≥	NUM
ejpam-3780	820	19	2	2	NUM
ejpam-3780	820	20	.	.	PUNCT
ejpam-3780	820	21	by	by	ADP
ejpam-3780	820	22	definition	definition	NOUN
ejpam-3780	820	23	of	of	ADP
ejpam-3780	820	24	tn	tn	PROPN
ejpam-3780	820	25	,	,	PUNCT
ejpam-3780	820	26	it	it	PRON
ejpam-3780	820	27	has	have	VERB
ejpam-3780	820	28	a	a	DET
ejpam-3780	820	29	vertex	vertex	NOUN
ejpam-3780	820	30	with	with	ADP
ejpam-3780	820	31	degree	degree	NOUN
ejpam-3780	820	32	|tn|	|tn|	NOUN
ejpam-3780	820	33	−	−	NOUN
ejpam-3780	820	34	1	1	NUM
ejpam-3780	820	35	.	.	PUNCT
ejpam-3780	821	1	thus	thus	ADV
ejpam-3780	821	2	,	,	PUNCT
ejpam-3780	821	3	by	by	ADP
ejpam-3780	821	4	theorem	theorem	NOUN
ejpam-3780	821	5	3	3	NUM
ejpam-3780	821	6	,	,	PUNCT
ejpam-3780	821	7	tn	tn	PROPN
ejpam-3780	821	8	has	have	VERB
ejpam-3780	821	9	an	an	DET
ejpam-3780	821	10	efficient	efficient	ADJ
ejpam-3780	821	11	zero	zero	NUM
ejpam-3780	821	12	ring	ring	NOUN
ejpam-3780	821	13	labeling	labeling	NOUN
ejpam-3780	821	14	for	for	ADP
ejpam-3780	821	15	n	n	PRON
ejpam-3780	821	16	≥	≥	NUM
ejpam-3780	821	17	2	2	NUM
ejpam-3780	821	18	.	.	PUNCT
ejpam-3780	821	19	example	example	NOUN
ejpam-3780	821	20	17	17	NUM
ejpam-3780	821	21	.	.	PUNCT
ejpam-3780	822	1	figure	figure	NOUN
ejpam-3780	822	2	18	18	NUM
ejpam-3780	822	3	shows	show	VERB
ejpam-3780	822	4	an	an	DET
ejpam-3780	822	5	efficient	efficient	ADJ
ejpam-3780	822	6	zero	zero	NUM
ejpam-3780	822	7	ring	ring	NOUN
ejpam-3780	822	8	labeling	labeling	NOUN
ejpam-3780	822	9	of	of	ADP
ejpam-3780	822	10	f5	f5	NOUN
ejpam-3780	822	11	using	use	VERB
ejpam-3780	822	12	m0	m0	PROPN
ejpam-3780	822	13	2	2	NUM
ejpam-3780	822	14	(	(	PUNCT
ejpam-3780	822	15	z6	z6	PROPN
ejpam-3780	822	16	)	)	PUNCT
ejpam-3780	822	17	.	.	PUNCT
ejpam-3780	823	1	figure	figure	VERB
ejpam-3780	823	2	19	19	NUM
ejpam-3780	823	3	shows	show	VERB
ejpam-3780	823	4	an	an	DET
ejpam-3780	823	5	efficient	efficient	ADJ
ejpam-3780	823	6	zero	zero	NUM
ejpam-3780	823	7	ring	ring	NOUN
ejpam-3780	823	8	labeling	labeling	NOUN
ejpam-3780	823	9	of	of	ADP
ejpam-3780	823	10	w8	w8	NOUN
ejpam-3780	823	11	using	use	VERB
ejpam-3780	823	12	m0	m0	PROPN
ejpam-3780	823	13	2	2	NUM
ejpam-3780	823	14	(	(	PUNCT
ejpam-3780	823	15	z8	z8	NOUN
ejpam-3780	823	16	)	)	PUNCT
ejpam-3780	823	17	.	.	PUNCT
ejpam-3780	824	1	figure	figure	NOUN
ejpam-3780	824	2	20	20	NUM
ejpam-3780	824	3	shows	show	VERB
ejpam-3780	824	4	an	an	DET
ejpam-3780	824	5	efficient	efficient	ADJ
ejpam-3780	824	6	zero	zero	NUM
ejpam-3780	824	7	ring	ring	NOUN
ejpam-3780	824	8	labeling	labeling	NOUN
ejpam-3780	824	9	of	of	ADP
ejpam-3780	824	10	t4	t4	PROPN
ejpam-3780	824	11	using	use	VERB
ejpam-3780	824	12	m0	m0	PROPN
ejpam-3780	824	13	2	2	NUM
ejpam-3780	824	14	(	(	PUNCT
ejpam-3780	824	15	z9	z9	PROPN
ejpam-3780	824	16	)	)	PUNCT
ejpam-3780	824	17	.	.	PUNCT
ejpam-3780	825	1	in	in	ADP
ejpam-3780	825	2	these	these	DET
ejpam-3780	825	3	labelings	labeling	NOUN
ejpam-3780	825	4	,	,	PUNCT
ejpam-3780	825	5	the	the	DET
ejpam-3780	825	6	set	set	NOUN
ejpam-3780	825	7	of	of	ADP
ejpam-3780	825	8	sums	sum	NOUN
ejpam-3780	825	9	is	be	AUX
ejpam-3780	825	10	the	the	DET
ejpam-3780	825	11	difference	difference	NOUN
ejpam-3780	825	12	of	of	ADP
ejpam-3780	825	13	the	the	DET
ejpam-3780	825	14	zero	zero	NUM
ejpam-3780	825	15	ring	ring	NOUN
ejpam-3780	825	16	used	use	VERB
ejpam-3780	825	17	and	and	CCONJ
ejpam-3780	825	18	{	{	PUNCT
ejpam-3780	825	19	a0	a0	NOUN
ejpam-3780	825	20	}	}	PUNCT
ejpam-3780	825	21	.	.	PUNCT
ejpam-3780	826	1	a1	a1	NOUN
ejpam-3780	826	2	a2	a2	PROPN
ejpam-3780	826	3	a3	a3	PROPN
ejpam-3780	826	4	a4	a4	PROPN
ejpam-3780	826	5	a5	a5	PROPN
ejpam-3780	826	6	a0	a0	PROPN
ejpam-3780	826	7	a1	a1	PROPN
ejpam-3780	826	8	a0	a0	PROPN
ejpam-3780	826	9	a2	a2	PROPN
ejpam-3780	826	10	a0	a0	PROPN
ejpam-3780	826	11	a3	a3	PROPN
ejpam-3780	826	12	a0	a0	PROPN
ejpam-3780	826	13	a4	a4	PROPN
ejpam-3780	826	14	a0	a0	PROPN
ejpam-3780	826	15	a5	a5	PROPN
ejpam-3780	826	16	figure	figure	NOUN
ejpam-3780	826	17	18	18	NUM
ejpam-3780	826	18	:	:	PUNCT
ejpam-3780	826	19	efficient	efficient	ADJ
ejpam-3780	826	20	zero	zero	NUM
ejpam-3780	826	21	ring	ring	NOUN
ejpam-3780	826	22	labeling	labeling	NOUN
ejpam-3780	826	23	of	of	ADP
ejpam-3780	826	24	f5	f5	NOUN
ejpam-3780	826	25	using	use	VERB
ejpam-3780	826	26	m0	m0	PROPN
ejpam-3780	826	27	2	2	NUM
ejpam-3780	826	28	(	(	PUNCT
ejpam-3780	826	29	z6	z6	PROPN
ejpam-3780	826	30	)	)	PUNCT
ejpam-3780	826	31	a2	a2	PROPN
ejpam-3780	826	32	a1a5	a1a5	PROPN
ejpam-3780	826	33	a8	a8	PROPN
ejpam-3780	826	34	a7	a7	PROPN
ejpam-3780	826	35	a6	a6	PROPN
ejpam-3780	826	36	a4	a4	PROPN
ejpam-3780	826	37	a3	a3	NOUN
ejpam-3780	826	38	a0	a0	PROPN
ejpam-3780	826	39	a1	a1	PROPN
ejpam-3780	826	40	a0	a0	PROPN
ejpam-3780	826	41	a5	a5	PROPN
ejpam-3780	826	42	a0	a0	PROPN
ejpam-3780	826	43	a8	a8	PROPN
ejpam-3780	826	44	a0	a0	PROPN
ejpam-3780	826	45	a7	a7	PROPN
ejpam-3780	826	46	a0	a0	PROPN
ejpam-3780	826	47	a6	a6	PROPN
ejpam-3780	826	48	a0	a0	PROPN
ejpam-3780	826	49	a4	a4	PROPN
ejpam-3780	826	50	a0	a0	PROPN
ejpam-3780	826	51	a3	a3	PROPN
ejpam-3780	826	52	a0	a0	PROPN
ejpam-3780	826	53	a2	a2	PROPN
ejpam-3780	826	54	figure	figure	NOUN
ejpam-3780	826	55	19	19	NUM
ejpam-3780	826	56	:	:	PUNCT
ejpam-3780	826	57	efficient	efficient	ADJ
ejpam-3780	826	58	zero	zero	NUM
ejpam-3780	826	59	ring	ring	NOUN
ejpam-3780	826	60	labeling	labeling	NOUN
ejpam-3780	826	61	of	of	ADP
ejpam-3780	826	62	w8	w8	NOUN
ejpam-3780	826	63	using	use	VERB
ejpam-3780	826	64	m0	m0	PROPN
ejpam-3780	826	65	2	2	NUM
ejpam-3780	826	66	(	(	PUNCT
ejpam-3780	826	67	z9	z9	PROPN
ejpam-3780	826	68	)	)	PUNCT
ejpam-3780	826	69	a2	a2	PROPN
ejpam-3780	826	70	a1a8	a1a8	PROPN
ejpam-3780	826	71	a7	a7	PROPN
ejpam-3780	826	72	a6	a6	PROPN
ejpam-3780	826	73	a5	a5	PROPN
ejpam-3780	826	74	a4	a4	PROPN
ejpam-3780	826	75	a3	a3	NOUN
ejpam-3780	826	76	a0	a0	PROPN
ejpam-3780	826	77	a1	a1	PROPN
ejpam-3780	826	78	a0	a0	PROPN
ejpam-3780	826	79	a8	a8	PROPN
ejpam-3780	826	80	a0	a0	PROPN
ejpam-3780	826	81	a7	a7	PROPN
ejpam-3780	826	82	a0	a0	PROPN
ejpam-3780	826	83	a6	a6	PROPN
ejpam-3780	826	84	a0	a0	PROPN
ejpam-3780	826	85	a5	a5	PROPN
ejpam-3780	826	86	a0	a0	PROPN
ejpam-3780	826	87	a4	a4	PROPN
ejpam-3780	826	88	a0	a0	PROPN
ejpam-3780	826	89	a3	a3	PROPN
ejpam-3780	826	90	a0	a0	PROPN
ejpam-3780	826	91	a2	a2	PROPN
ejpam-3780	826	92	figure	figure	NOUN
ejpam-3780	826	93	20	20	NUM
ejpam-3780	826	94	:	:	PUNCT
ejpam-3780	826	95	efficient	efficient	ADJ
ejpam-3780	826	96	zero	zero	NUM
ejpam-3780	826	97	ring	ring	NOUN
ejpam-3780	826	98	labeling	labeling	NOUN
ejpam-3780	826	99	of	of	ADP
ejpam-3780	826	100	t4	t4	PROPN
ejpam-3780	826	101	using	use	VERB
ejpam-3780	826	102	m0	m0	PROPN
ejpam-3780	826	103	2	2	NUM
ejpam-3780	826	104	(	(	PUNCT
ejpam-3780	826	105	z9	z9	PROPN
ejpam-3780	826	106	)	)	PUNCT
ejpam-3780	826	107	acknowledgements	acknowledgement	NOUN
ejpam-3780	827	1	the	the	DET
ejpam-3780	827	2	authors	author	NOUN
ejpam-3780	827	3	are	be	AUX
ejpam-3780	827	4	grateful	grateful	ADJ
ejpam-3780	827	5	to	to	ADP
ejpam-3780	827	6	the	the	DET
ejpam-3780	827	7	blind	blind	ADJ
ejpam-3780	827	8	peer	peer	NOUN
ejpam-3780	827	9	reviewers	reviewer	NOUN
ejpam-3780	827	10	who	who	PRON
ejpam-3780	827	11	helped	help	VERB
ejpam-3780	827	12	improve	improve	VERB
ejpam-3780	827	13	the	the	DET
ejpam-3780	827	14	paper	paper	NOUN
ejpam-3780	827	15	,	,	PUNCT
ejpam-3780	827	16	as	as	ADV
ejpam-3780	827	17	well	well	ADV
ejpam-3780	827	18	as	as	ADP
ejpam-3780	827	19	dr	dr	PROPN
ejpam-3780	827	20	.	.	PROPN
ejpam-3780	827	21	severino	severino	PROPN
ejpam-3780	827	22	v.	v.	ADP
ejpam-3780	827	23	gervacio	gervacio	PROPN
ejpam-3780	827	24	,	,	PUNCT
ejpam-3780	827	25	dr	dr	PROPN
ejpam-3780	827	26	.	.	PROPN
ejpam-3780	827	27	leonor	leonor	PROPN
ejpam-3780	827	28	a.	a.	PROPN
ejpam-3780	827	29	ruivivar	ruivivar	PROPN
ejpam-3780	827	30	,	,	PUNCT
ejpam-3780	827	31	and	and	CCONJ
ejpam-3780	827	32	dr	dr	PROPN
ejpam-3780	827	33	.	.	PROPN
ejpam-3780	827	34	neil	neil	PROPN
ejpam-3780	827	35	m.	m.	PROPN
ejpam-3780	827	36	mame	mame	PROPN
ejpam-3780	827	37	who	who	PRON
ejpam-3780	827	38	contributed	contribute	VERB
ejpam-3780	827	39	invaluable	invaluable	ADJ
ejpam-3780	827	40	comments	comment	NOUN
ejpam-3780	827	41	and	and	CCONJ
ejpam-3780	827	42	suggestions	suggestion	NOUN
ejpam-3780	827	43	over	over	ADP
ejpam-3780	827	44	the	the	DET
ejpam-3780	827	45	course	course	NOUN
ejpam-3780	827	46	of	of	ADP
ejpam-3780	827	47	this	this	DET
ejpam-3780	827	48	study	study	NOUN
ejpam-3780	827	49	.	.	PUNCT
ejpam-3780	828	1	references	reference	NOUN
ejpam-3780	828	2	696	696	NUM
ejpam-3780	828	3	this	this	DET
ejpam-3780	828	4	work	work	NOUN
ejpam-3780	828	5	was	be	AUX
ejpam-3780	828	6	funded	fund	VERB
ejpam-3780	828	7	by	by	ADP
ejpam-3780	828	8	the	the	DET
ejpam-3780	828	9	department	department	PROPN
ejpam-3780	828	10	of	of	ADP
ejpam-3780	828	11	science	science	NOUN
ejpam-3780	828	12	and	and	CCONJ
ejpam-3780	828	13	technology	technology	NOUN
ejpam-3780	828	14	-	-	PUNCT
ejpam-3780	828	15	science	science	NOUN
ejpam-3780	828	16	education	education	PROPN
ejpam-3780	828	17	institute	institute	NOUN
ejpam-3780	828	18	(	(	PUNCT
ejpam-3780	828	19	dost	dost	NOUN
ejpam-3780	828	20	-	-	PUNCT
ejpam-3780	828	21	sei	sei	NOUN
ejpam-3780	828	22	)	)	PUNCT
ejpam-3780	828	23	and	and	CCONJ
ejpam-3780	828	24	the	the	DET
ejpam-3780	828	25	commission	commission	NOUN
ejpam-3780	828	26	on	on	ADP
ejpam-3780	828	27	higher	high	ADJ
ejpam-3780	828	28	education	education	NOUN
ejpam-3780	828	29	(	(	PUNCT
ejpam-3780	828	30	ched	che	VERB
ejpam-3780	828	31	)	)	PUNCT
ejpam-3780	828	32	of	of	ADP
ejpam-3780	828	33	the	the	DET
ejpam-3780	828	34	philippine	philippine	ADJ
ejpam-3780	828	35	government	government	NOUN
ejpam-3780	828	36	.	.	PUNCT
ejpam-3780	829	1	references	reference	NOUN
ejpam-3780	829	2	[	[	X
ejpam-3780	829	3	1	1	NUM
ejpam-3780	829	4	]	]	PUNCT
ejpam-3780	829	5	mukti	mukti	PROPN
ejpam-3780	829	6	acharya	acharya	PROPN
ejpam-3780	829	7	,	,	PUNCT
ejpam-3780	829	8	pranjali	pranjali	PROPN
ejpam-3780	829	9	,	,	PUNCT
ejpam-3780	829	10	and	and	CCONJ
ejpam-3780	829	11	purnima	purnima	PROPN
ejpam-3780	829	12	gupta	gupta	PROPN
ejpam-3780	829	13	.	.	PUNCT
ejpam-3780	830	1	zero	zero	NUM
ejpam-3780	830	2	ring	ring	NOUN
ejpam-3780	830	3	labeling	labeling	NOUN
ejpam-3780	830	4	of	of	ADP
ejpam-3780	830	5	graphs	graph	NOUN
ejpam-3780	830	6	.	.	PUNCT
ejpam-3780	831	1	electronic	electronic	ADJ
ejpam-3780	831	2	notes	note	NOUN
ejpam-3780	831	3	in	in	ADP
ejpam-3780	831	4	discrete	discrete	ADJ
ejpam-3780	831	5	mathematics	mathematic	NOUN
ejpam-3780	831	6	,	,	PUNCT
ejpam-3780	831	7	48:65–72	48:65–72	PROPN
ejpam-3780	831	8	,	,	PUNCT
ejpam-3780	831	9	july	july	PROPN
ejpam-3780	831	10	2015	2015	NUM
ejpam-3780	831	11	.	.	PUNCT
ejpam-3780	832	1	[	[	X
ejpam-3780	832	2	2	2	X
ejpam-3780	832	3	]	]	PUNCT
ejpam-3780	832	4	michelle	michelle	PROPN
ejpam-3780	832	5	dela	dela	PROPN
ejpam-3780	832	6	rosa	rosa	PROPN
ejpam-3780	832	7	-	-	PUNCT
ejpam-3780	832	8	reynera	reynera	NOUN
ejpam-3780	832	9	.	.	PUNCT
ejpam-3780	833	1	on	on	ADP
ejpam-3780	833	2	graphs	graph	NOUN
ejpam-3780	833	3	of	of	ADP
ejpam-3780	833	4	minimum	minimum	ADJ
ejpam-3780	833	5	zero	zero	NUM
ejpam-3780	833	6	ring	ring	NOUN
ejpam-3780	833	7	index	index	NOUN
ejpam-3780	833	8	.	.	PUNCT
ejpam-3780	834	1	phd	phd	NOUN
ejpam-3780	834	2	thesis	thesis	PROPN
ejpam-3780	834	3	,	,	PUNCT
ejpam-3780	834	4	august	august	PROPN
ejpam-3780	834	5	2018	2018	NUM
ejpam-3780	834	6	.	.	PUNCT
ejpam-3780	835	1	[	[	X
ejpam-3780	835	2	3	3	X
ejpam-3780	835	3	]	]	PUNCT
ejpam-3780	835	4	reinhard	reinhard	NOUN
ejpam-3780	835	5	diestel	diestel	NOUN
ejpam-3780	835	6	.	.	PUNCT
ejpam-3780	836	1	graph	graph	NOUN
ejpam-3780	836	2	theory	theory	NOUN
ejpam-3780	836	3	.	.	PUNCT
ejpam-3780	837	1	springer	springer	NOUN
ejpam-3780	837	2	,	,	PUNCT
ejpam-3780	837	3	4th	4th	ADJ
ejpam-3780	837	4	edition	edition	NOUN
ejpam-3780	837	5	,	,	PUNCT
ejpam-3780	837	6	2010	2010	NUM
ejpam-3780	837	7	.	.	PUNCT
ejpam-3780	838	1	[	[	X
ejpam-3780	838	2	4	4	X
ejpam-3780	838	3	]	]	X
ejpam-3780	838	4	joseph	joseph	PROPN
ejpam-3780	838	5	a.	a.	PROPN
ejpam-3780	838	6	gallian	gallian	PROPN
ejpam-3780	838	7	.	.	PUNCT
ejpam-3780	839	1	contemporary	contemporary	ADJ
ejpam-3780	839	2	abstract	abstract	ADJ
ejpam-3780	839	3	algebra	algebra	PROPN
ejpam-3780	839	4	.	.	PUNCT
ejpam-3780	840	1	brooks	brooks	PROPN
ejpam-3780	840	2	/	/	SYM
ejpam-3780	840	3	cole	cole	PROPN
ejpam-3780	840	4	,	,	PUNCT
ejpam-3780	840	5	8th	8th	ADJ
ejpam-3780	840	6	edition	edition	NOUN
ejpam-3780	840	7	,	,	PUNCT
ejpam-3780	840	8	2013	2013	NUM
ejpam-3780	840	9	.	.	PUNCT
ejpam-3780	841	1	[	[	X
ejpam-3780	841	2	5	5	NUM
ejpam-3780	841	3	]	]	X
ejpam-3780	841	4	frank	frank	PROPN
ejpam-3780	841	5	harary	harary	PROPN
ejpam-3780	841	6	.	.	PUNCT
ejpam-3780	842	1	graph	graph	NOUN
ejpam-3780	842	2	theory	theory	NOUN
ejpam-3780	842	3	.	.	PUNCT
ejpam-3780	843	1	addison	addison	PROPN
ejpam-3780	843	2	-	-	PUNCT
ejpam-3780	843	3	wesley	wesley	PROPN
ejpam-3780	843	4	,	,	PUNCT
ejpam-3780	843	5	1969	1969	NUM
ejpam-3780	843	6	.	.	PUNCT
ejpam-3780	844	1	[	[	X
ejpam-3780	844	2	6	6	NUM
ejpam-3780	844	3	]	]	PUNCT
ejpam-3780	844	4	pranjali	pranjali	PROPN
ejpam-3780	844	5	,	,	PUNCT
ejpam-3780	844	6	mukti	mukti	PROPN
ejpam-3780	844	7	acharya	acharya	PROPN
ejpam-3780	844	8	,	,	PUNCT
ejpam-3780	844	9	and	and	CCONJ
ejpam-3780	844	10	purnima	purnima	PROPN
ejpam-3780	844	11	gupta	gupta	PROPN
ejpam-3780	844	12	.	.	PUNCT
ejpam-3780	845	1	further	further	ADJ
ejpam-3780	845	2	results	result	NOUN
ejpam-3780	845	3	on	on	ADP
ejpam-3780	845	4	zero	zero	NUM
ejpam-3780	845	5	ring	ring	NOUN
ejpam-3780	845	6	labeling	labeling	NOUN
ejpam-3780	845	7	of	of	ADP
ejpam-3780	845	8	graphs	graph	NOUN
ejpam-3780	845	9	.	.	PUNCT
ejpam-3780	846	1	bulletin	bulletin	NOUN
ejpam-3780	846	2	of	of	ADP
ejpam-3780	846	3	the	the	DET
ejpam-3780	846	4	international	international	ADJ
ejpam-3780	846	5	mathematical	mathematical	ADJ
ejpam-3780	846	6	virtual	virtual	PROPN
ejpam-3780	846	7	institute	institute	PROPN
ejpam-3780	846	8	,	,	PUNCT
ejpam-3780	846	9	5:205–210	5:205–210	PROPN
ejpam-3780	846	10	,	,	PUNCT
ejpam-3780	846	11	2014	2014	NUM
ejpam-3780	846	12	.	.	PUNCT