id	sid	tid	token	lemma	pos
ejpam-3890	1	1	european	european	PROPN
ejpam-3890	1	2	journal	journal	PROPN
ejpam-3890	1	3	of	of	ADP
ejpam-3890	1	4	pure	pure	ADJ
ejpam-3890	1	5	and	and	CCONJ
ejpam-3890	1	6	applied	apply	VERB
ejpam-3890	1	7	mathematics	mathematic	NOUN
ejpam-3890	1	8	vol	vol	NOUN
ejpam-3890	1	9	.	.	PUNCT
ejpam-3890	2	1	14	14	NUM
ejpam-3890	2	2	,	,	PUNCT
ejpam-3890	2	3	no	no	INTJ
ejpam-3890	2	4	.	.	NOUN
ejpam-3890	2	5	1	1	NUM
ejpam-3890	2	6	,	,	PUNCT
ejpam-3890	2	7	2021	2021	NUM
ejpam-3890	2	8	,	,	PUNCT
ejpam-3890	2	9	268	268	NUM
ejpam-3890	2	10	-	-	SYM
ejpam-3890	2	11	277	277	NUM
ejpam-3890	2	12	issn	issn	PROPN
ejpam-3890	2	13	1307	1307	NUM
ejpam-3890	2	14	-	-	SYM
ejpam-3890	2	15	5543	5543	NUM
ejpam-3890	2	16	–	–	PUNCT
ejpam-3890	2	17	ejpam.com	ejpam.com	X
ejpam-3890	2	18	published	publish	VERB
ejpam-3890	2	19	by	by	ADP
ejpam-3890	2	20	new	new	PROPN
ejpam-3890	2	21	york	york	PROPN
ejpam-3890	2	22	business	business	PROPN
ejpam-3890	2	23	global	global	ADJ
ejpam-3890	2	24	on	on	ADP
ejpam-3890	2	25	efficient	efficient	ADJ
ejpam-3890	2	26	zero	zero	NUM
ejpam-3890	2	27	ring	ring	NOUN
ejpam-3890	2	28	labeling	labeling	NOUN
ejpam-3890	2	29	and	and	CCONJ
ejpam-3890	2	30	restricted	restrict	VERB
ejpam-3890	2	31	zero	zero	NUM
ejpam-3890	2	32	ring	ring	NOUN
ejpam-3890	2	33	graphs	graph	NOUN
ejpam-3890	2	34	francis	francis	PROPN
ejpam-3890	2	35	joseph	joseph	PROPN
ejpam-3890	2	36	h.	h.	PROPN
ejpam-3890	2	37	campeña1,∗	campeña1,∗	PROPN
ejpam-3890	2	38	,	,	PUNCT
ejpam-3890	2	39	dhenmar	dhenmar	PROPN
ejpam-3890	2	40	e.	e.	PROPN
ejpam-3890	2	41	chua1,2	chua1,2	PROPN
ejpam-3890	2	42	,	,	PUNCT
ejpam-3890	2	43	floresto	floresto	PROPN
ejpam-3890	2	44	a.	a.	NOUN
ejpam-3890	2	45	franco	franco	PROPN
ejpam-3890	2	46	,	,	PUNCT
ejpam-3890	2	47	jr.1,3	jr.1,3	PROPN
ejpam-3890	2	48	,	,	PUNCT
ejpam-3890	2	49	jon	jon	PROPN
ejpam-3890	2	50	-	-	PUNCT
ejpam-3890	2	51	jon	jon	PROPN
ejpam-3890	2	52	a.	a.	NOUN
ejpam-3890	2	53	casica1	casica1	NOUN
ejpam-3890	2	54	1	1	NUM
ejpam-3890	2	55	mathematics	mathematic	NOUN
ejpam-3890	2	56	and	and	CCONJ
ejpam-3890	2	57	statistics	statistics	PROPN
ejpam-3890	2	58	department	department	PROPN
ejpam-3890	2	59	,	,	PUNCT
ejpam-3890	2	60	de	de	PROPN
ejpam-3890	2	61	la	la	X
ejpam-3890	2	62	salle	salle	PROPN
ejpam-3890	2	63	university	university	PROPN
ejpam-3890	2	64	,	,	PUNCT
ejpam-3890	2	65	manila	manila	PROPN
ejpam-3890	2	66	,	,	PUNCT
ejpam-3890	3	1	philippines	philippine	NOUN
ejpam-3890	3	2	2	2	NUM
ejpam-3890	3	3	mathematics	mathematic	NOUN
ejpam-3890	3	4	and	and	CCONJ
ejpam-3890	3	5	physics	physics	PROPN
ejpam-3890	3	6	department	department	PROPN
ejpam-3890	3	7	,	,	PUNCT
ejpam-3890	3	8	adamson	adamson	PROPN
ejpam-3890	3	9	university	university	PROPN
ejpam-3890	3	10	,	,	PUNCT
ejpam-3890	3	11	manila	manila	PROPN
ejpam-3890	3	12	,	,	PUNCT
ejpam-3890	3	13	philippines	philippine	NOUN
ejpam-3890	3	14	3	3	NUM
ejpam-3890	3	15	mathematics	mathematics	PROPN
ejpam-3890	3	16	department	department	NOUN
ejpam-3890	3	17	,	,	PUNCT
ejpam-3890	3	18	mariano	mariano	PROPN
ejpam-3890	3	19	marcos	marcos	PROPN
ejpam-3890	3	20	state	state	PROPN
ejpam-3890	3	21	university	university	PROPN
ejpam-3890	3	22	,	,	PUNCT
ejpam-3890	3	23	batac	batac	PROPN
ejpam-3890	3	24	ilocos	ilocos	PROPN
ejpam-3890	3	25	norte	norte	NOUN
ejpam-3890	3	26	,	,	PUNCT
ejpam-3890	3	27	philippines	philippine	NOUN
ejpam-3890	3	28	abstract	abstract	ADJ
ejpam-3890	3	29	.	.	PUNCT
ejpam-3890	4	1	in	in	ADP
ejpam-3890	4	2	[	[	X
ejpam-3890	4	3	3	3	NUM
ejpam-3890	4	4	]	]	PUNCT
ejpam-3890	4	5	,	,	PUNCT
ejpam-3890	4	6	acharya	acharya	PROPN
ejpam-3890	4	7	et	et	PROPN
ejpam-3890	4	8	al	al	PROPN
ejpam-3890	4	9	.	.	PROPN
ejpam-3890	4	10	introduced	introduce	VERB
ejpam-3890	4	11	the	the	DET
ejpam-3890	4	12	notion	notion	NOUN
ejpam-3890	4	13	of	of	ADP
ejpam-3890	4	14	a	a	DET
ejpam-3890	4	15	zero	zero	NUM
ejpam-3890	4	16	ring	ring	NOUN
ejpam-3890	4	17	labeling	labeling	NOUN
ejpam-3890	4	18	of	of	ADP
ejpam-3890	4	19	a	a	DET
ejpam-3890	4	20	connected	connected	ADJ
ejpam-3890	4	21	graph	graph	NOUN
ejpam-3890	4	22	g	g	NOUN
ejpam-3890	4	23	,	,	PUNCT
ejpam-3890	4	24	where	where	SCONJ
ejpam-3890	4	25	vertices	vertex	NOUN
ejpam-3890	4	26	are	be	AUX
ejpam-3890	4	27	labeled	label	VERB
ejpam-3890	4	28	by	by	ADP
ejpam-3890	4	29	the	the	DET
ejpam-3890	4	30	elements	element	NOUN
ejpam-3890	4	31	of	of	ADP
ejpam-3890	4	32	a	a	DET
ejpam-3890	4	33	zero	zero	NUM
ejpam-3890	4	34	ring	ring	NOUN
ejpam-3890	4	35	such	such	ADJ
ejpam-3890	4	36	that	that	SCONJ
ejpam-3890	4	37	the	the	DET
ejpam-3890	4	38	sum	sum	NOUN
ejpam-3890	4	39	of	of	ADP
ejpam-3890	4	40	the	the	DET
ejpam-3890	4	41	labels	label	NOUN
ejpam-3890	4	42	of	of	ADP
ejpam-3890	4	43	adjacent	adjacent	ADJ
ejpam-3890	4	44	vertices	vertex	NOUN
ejpam-3890	4	45	is	be	AUX
ejpam-3890	4	46	not	not	PART
ejpam-3890	4	47	the	the	DET
ejpam-3890	4	48	additive	additive	ADJ
ejpam-3890	4	49	identity	identity	NOUN
ejpam-3890	4	50	of	of	ADP
ejpam-3890	4	51	the	the	DET
ejpam-3890	4	52	ring	ring	NOUN
ejpam-3890	4	53	.	.	PUNCT
ejpam-3890	5	1	archarya	archarya	PROPN
ejpam-3890	5	2	and	and	CCONJ
ejpam-3890	5	3	pranjali	pranjali	VERB
ejpam-3890	5	4	[	[	X
ejpam-3890	5	5	1	1	X
ejpam-3890	5	6	]	]	PUNCT
ejpam-3890	5	7	also	also	ADV
ejpam-3890	5	8	constructed	construct	VERB
ejpam-3890	5	9	a	a	DET
ejpam-3890	5	10	graph	graph	NOUN
ejpam-3890	5	11	based	base	VERB
ejpam-3890	5	12	on	on	ADP
ejpam-3890	5	13	a	a	DET
ejpam-3890	5	14	finite	finite	ADJ
ejpam-3890	5	15	zero	zero	NUM
ejpam-3890	5	16	ring	ring	NOUN
ejpam-3890	5	17	called	call	VERB
ejpam-3890	5	18	the	the	DET
ejpam-3890	5	19	zero	zero	NUM
ejpam-3890	5	20	ring	ring	NOUN
ejpam-3890	5	21	graph	graph	NOUN
ejpam-3890	5	22	.	.	PUNCT
ejpam-3890	6	1	in	in	ADP
ejpam-3890	6	2	[	[	X
ejpam-3890	6	3	5	5	NUM
ejpam-3890	6	4	]	]	PUNCT
ejpam-3890	6	5	,	,	PUNCT
ejpam-3890	6	6	chua	chua	PROPN
ejpam-3890	6	7	et	et	PROPN
ejpam-3890	6	8	al	al	PROPN
ejpam-3890	6	9	.	.	PROPN
ejpam-3890	6	10	defined	define	VERB
ejpam-3890	6	11	a	a	DET
ejpam-3890	6	12	class	class	NOUN
ejpam-3890	6	13	of	of	ADP
ejpam-3890	6	14	zero	zero	NUM
ejpam-3890	6	15	ring	ring	NOUN
ejpam-3890	6	16	labeling	labeling	NOUN
ejpam-3890	6	17	called	call	VERB
ejpam-3890	6	18	efficient	efficient	ADJ
ejpam-3890	6	19	zero	zero	NUM
ejpam-3890	6	20	ring	ring	NOUN
ejpam-3890	6	21	labeling	labeling	NOUN
ejpam-3890	6	22	and	and	CCONJ
ejpam-3890	6	23	it	it	PRON
ejpam-3890	6	24	was	be	AUX
ejpam-3890	6	25	shown	show	VERB
ejpam-3890	6	26	that	that	SCONJ
ejpam-3890	6	27	a	a	DET
ejpam-3890	6	28	labeling	labeling	NOUN
ejpam-3890	6	29	scheme	scheme	NOUN
ejpam-3890	6	30	exists	exist	VERB
ejpam-3890	6	31	for	for	ADP
ejpam-3890	6	32	some	some	DET
ejpam-3890	6	33	families	family	NOUN
ejpam-3890	6	34	of	of	ADP
ejpam-3890	6	35	trees	tree	NOUN
ejpam-3890	6	36	.	.	PUNCT
ejpam-3890	7	1	in	in	ADP
ejpam-3890	7	2	this	this	DET
ejpam-3890	7	3	paper	paper	NOUN
ejpam-3890	7	4	,	,	PUNCT
ejpam-3890	7	5	we	we	PRON
ejpam-3890	7	6	provide	provide	VERB
ejpam-3890	7	7	an	an	DET
ejpam-3890	7	8	efficient	efficient	ADJ
ejpam-3890	7	9	zero	zero	NUM
ejpam-3890	7	10	ring	ring	NOUN
ejpam-3890	7	11	labeling	labeling	NOUN
ejpam-3890	7	12	for	for	ADP
ejpam-3890	7	13	some	some	DET
ejpam-3890	7	14	classes	class	NOUN
ejpam-3890	7	15	of	of	ADP
ejpam-3890	7	16	graphs	graph	NOUN
ejpam-3890	7	17	.	.	PUNCT
ejpam-3890	8	1	we	we	PRON
ejpam-3890	8	2	also	also	ADV
ejpam-3890	8	3	introduce	introduce	VERB
ejpam-3890	8	4	the	the	DET
ejpam-3890	8	5	notion	notion	NOUN
ejpam-3890	8	6	of	of	ADP
ejpam-3890	8	7	the	the	DET
ejpam-3890	8	8	restricted	restricted	ADJ
ejpam-3890	8	9	zero	zero	NUM
ejpam-3890	8	10	ring	ring	NOUN
ejpam-3890	8	11	graphs	graph	NOUN
ejpam-3890	8	12	and	and	CCONJ
ejpam-3890	8	13	use	use	VERB
ejpam-3890	8	14	them	they	PRON
ejpam-3890	8	15	to	to	PART
ejpam-3890	8	16	show	show	VERB
ejpam-3890	8	17	that	that	SCONJ
ejpam-3890	8	18	a	a	DET
ejpam-3890	8	19	zero	zero	NUM
ejpam-3890	8	20	ring	ring	NOUN
ejpam-3890	8	21	labeling	labeling	NOUN
ejpam-3890	8	22	exists	exist	VERB
ejpam-3890	8	23	for	for	ADP
ejpam-3890	8	24	some	some	DET
ejpam-3890	8	25	classes	class	NOUN
ejpam-3890	8	26	of	of	ADP
ejpam-3890	8	27	cactus	cactus	NOUN
ejpam-3890	8	28	graphs	graph	NOUN
ejpam-3890	8	29	.	.	PUNCT
ejpam-3890	9	1	2020	2020	NUM
ejpam-3890	9	2	mathematics	mathematic	NOUN
ejpam-3890	9	3	subject	subject	NOUN
ejpam-3890	9	4	classifications	classification	NOUN
ejpam-3890	9	5	:	:	PUNCT
ejpam-3890	9	6	05c2	05c2	NUM
ejpam-3890	9	7	,	,	PUNCT
ejpam-3890	9	8	05c78	05c78	NUM
ejpam-3890	9	9	key	key	ADJ
ejpam-3890	9	10	words	word	NOUN
ejpam-3890	9	11	and	and	CCONJ
ejpam-3890	9	12	phrases	phrase	NOUN
ejpam-3890	9	13	:	:	PUNCT
ejpam-3890	9	14	efficient	efficient	ADJ
ejpam-3890	9	15	zero	zero	NUM
ejpam-3890	9	16	ring	ring	NOUN
ejpam-3890	9	17	labeling	labeling	NOUN
ejpam-3890	9	18	,	,	PUNCT
ejpam-3890	9	19	zero	zero	NUM
ejpam-3890	9	20	ring	ring	NOUN
ejpam-3890	9	21	labeling	labeling	NOUN
ejpam-3890	9	22	,	,	PUNCT
ejpam-3890	9	23	zero	zero	NUM
ejpam-3890	9	24	ring	ring	NOUN
ejpam-3890	9	25	1	1	NUM
ejpam-3890	9	26	.	.	PUNCT
ejpam-3890	9	27	introduction	introduction	NOUN
ejpam-3890	9	28	one	one	NUM
ejpam-3890	9	29	of	of	ADP
ejpam-3890	9	30	the	the	DET
ejpam-3890	9	31	fields	field	NOUN
ejpam-3890	9	32	in	in	ADP
ejpam-3890	9	33	graph	graph	NOUN
ejpam-3890	9	34	theory	theory	NOUN
ejpam-3890	9	35	that	that	PRON
ejpam-3890	9	36	has	have	AUX
ejpam-3890	9	37	been	be	AUX
ejpam-3890	9	38	a	a	DET
ejpam-3890	9	39	study	study	NOUN
ejpam-3890	9	40	of	of	ADP
ejpam-3890	9	41	interest	interest	NOUN
ejpam-3890	9	42	of	of	ADP
ejpam-3890	9	43	many	many	ADJ
ejpam-3890	9	44	mathematical	mathematical	ADJ
ejpam-3890	9	45	researchers	researcher	NOUN
ejpam-3890	9	46	is	be	AUX
ejpam-3890	9	47	graph	graph	VERB
ejpam-3890	9	48	labeling	labeling	NOUN
ejpam-3890	9	49	.	.	PUNCT
ejpam-3890	10	1	most	most	ADJ
ejpam-3890	10	2	of	of	ADP
ejpam-3890	10	3	the	the	DET
ejpam-3890	10	4	methods	method	NOUN
ejpam-3890	10	5	or	or	CCONJ
ejpam-3890	10	6	schemes	scheme	NOUN
ejpam-3890	10	7	in	in	ADP
ejpam-3890	10	8	graph	graph	NOUN
ejpam-3890	10	9	labeling	labeling	NOUN
ejpam-3890	10	10	can	can	AUX
ejpam-3890	10	11	be	be	AUX
ejpam-3890	10	12	traced	trace	VERB
ejpam-3890	10	13	to	to	ADP
ejpam-3890	10	14	or	or	CCONJ
ejpam-3890	10	15	have	have	VERB
ejpam-3890	10	16	links	link	NOUN
ejpam-3890	10	17	to	to	ADP
ejpam-3890	10	18	a	a	DET
ejpam-3890	10	19	paper	paper	NOUN
ejpam-3890	10	20	by	by	ADP
ejpam-3890	10	21	rosa	rosa	PROPN
ejpam-3890	11	1	[	[	X
ejpam-3890	11	2	9	9	NUM
ejpam-3890	11	3	]	]	PUNCT
ejpam-3890	11	4	.	.	PUNCT
ejpam-3890	12	1	at	at	ADP
ejpam-3890	12	2	present	present	ADJ
ejpam-3890	12	3	,	,	PUNCT
ejpam-3890	12	4	over	over	ADP
ejpam-3890	12	5	200	200	NUM
ejpam-3890	12	6	graph	graph	NOUN
ejpam-3890	12	7	labeling	labeling	NOUN
ejpam-3890	12	8	techniques	technique	NOUN
ejpam-3890	12	9	have	have	AUX
ejpam-3890	12	10	been	be	AUX
ejpam-3890	12	11	studied	study	VERB
ejpam-3890	12	12	in	in	ADP
ejpam-3890	12	13	over	over	ADP
ejpam-3890	12	14	2500	2500	NUM
ejpam-3890	12	15	papers	paper	NOUN
ejpam-3890	12	16	.	.	PUNCT
ejpam-3890	13	1	for	for	ADP
ejpam-3890	13	2	an	an	DET
ejpam-3890	13	3	excellent	excellent	ADJ
ejpam-3890	13	4	survey	survey	NOUN
ejpam-3890	13	5	on	on	ADP
ejpam-3890	13	6	different	different	ADJ
ejpam-3890	13	7	labeling	labeling	NOUN
ejpam-3890	13	8	schemes	scheme	NOUN
ejpam-3890	13	9	on	on	ADP
ejpam-3890	13	10	graph	graph	NOUN
ejpam-3890	13	11	,	,	PUNCT
ejpam-3890	13	12	the	the	DET
ejpam-3890	13	13	readers	reader	NOUN
ejpam-3890	13	14	may	may	AUX
ejpam-3890	13	15	refer	refer	VERB
ejpam-3890	13	16	to	to	ADP
ejpam-3890	13	17	[	[	X
ejpam-3890	13	18	7	7	NUM
ejpam-3890	13	19	]	]	PUNCT
ejpam-3890	13	20	.	.	PUNCT
ejpam-3890	14	1	in	in	ADP
ejpam-3890	14	2	2014	2014	NUM
ejpam-3890	14	3	,	,	PUNCT
ejpam-3890	14	4	acharya	acharya	PROPN
ejpam-3890	14	5	et	et	PROPN
ejpam-3890	14	6	al	al	PROPN
ejpam-3890	14	7	.	.	PUNCT
ejpam-3890	15	1	[	[	X
ejpam-3890	15	2	3	3	NUM
ejpam-3890	15	3	]	]	PUNCT
ejpam-3890	15	4	introduced	introduce	VERB
ejpam-3890	15	5	zero	zero	NUM
ejpam-3890	15	6	ring	ring	NOUN
ejpam-3890	15	7	labeling	labeling	NOUN
ejpam-3890	15	8	.	.	PUNCT
ejpam-3890	16	1	in	in	ADP
ejpam-3890	16	2	this	this	DET
ejpam-3890	16	3	labeling	labeling	NOUN
ejpam-3890	16	4	,	,	PUNCT
ejpam-3890	16	5	each	each	DET
ejpam-3890	16	6	vertex	vertex	NOUN
ejpam-3890	16	7	is	be	AUX
ejpam-3890	16	8	assigned	assign	VERB
ejpam-3890	16	9	a	a	DET
ejpam-3890	16	10	unique	unique	ADJ
ejpam-3890	16	11	label	label	NOUN
ejpam-3890	16	12	from	from	ADP
ejpam-3890	16	13	a	a	DET
ejpam-3890	16	14	zero	zero	NUM
ejpam-3890	16	15	ring	ring	NOUN
ejpam-3890	16	16	such	such	ADJ
ejpam-3890	16	17	that	that	SCONJ
ejpam-3890	16	18	the	the	DET
ejpam-3890	16	19	sum	sum	NOUN
ejpam-3890	16	20	of	of	ADP
ejpam-3890	16	21	any	any	DET
ejpam-3890	16	22	two	two	NUM
ejpam-3890	16	23	adjacent	adjacent	ADJ
ejpam-3890	16	24	vertices	vertex	NOUN
ejpam-3890	16	25	is	be	AUX
ejpam-3890	16	26	not	not	PART
ejpam-3890	16	27	zero	zero	NUM
ejpam-3890	16	28	,	,	PUNCT
ejpam-3890	16	29	i.e.	i.e.	X
ejpam-3890	16	30	,	,	PUNCT
ejpam-3890	16	31	the	the	DET
ejpam-3890	16	32	additive	additive	ADJ
ejpam-3890	16	33	identity	identity	NOUN
ejpam-3890	16	34	of	of	ADP
ejpam-3890	16	35	the	the	DET
ejpam-3890	16	36	zero	zero	NUM
ejpam-3890	16	37	ring	ring	NOUN
ejpam-3890	16	38	.	.	PUNCT
ejpam-3890	17	1	it	it	PRON
ejpam-3890	17	2	was	be	AUX
ejpam-3890	17	3	proved	prove	VERB
ejpam-3890	17	4	that	that	SCONJ
ejpam-3890	17	5	every	every	DET
ejpam-3890	17	6	graph	graph	NOUN
ejpam-3890	17	7	admits	admit	VERB
ejpam-3890	17	8	a	a	DET
ejpam-3890	17	9	zero	zero	NUM
ejpam-3890	17	10	ring	ring	NOUN
ejpam-3890	17	11	labeling	labeling	NOUN
ejpam-3890	17	12	with	with	ADP
ejpam-3890	17	13	respect	respect	NOUN
ejpam-3890	17	14	to	to	ADP
ejpam-3890	17	15	some	some	DET
ejpam-3890	17	16	zero	zero	NUM
ejpam-3890	17	17	ring	ring	NOUN
ejpam-3890	17	18	.	.	PUNCT
ejpam-3890	18	1	the	the	DET
ejpam-3890	18	2	zero	zero	NUM
ejpam-3890	18	3	ring	ring	NOUN
ejpam-3890	18	4	index	index	NOUN
ejpam-3890	18	5	of	of	ADP
ejpam-3890	18	6	a	a	DET
ejpam-3890	18	7	graph	graph	NOUN
ejpam-3890	18	8	,	,	PUNCT
ejpam-3890	18	9	which	which	PRON
ejpam-3890	18	10	is	be	AUX
ejpam-3890	18	11	the	the	DET
ejpam-3890	18	12	smallest	small	ADJ
ejpam-3890	18	13	order	order	NOUN
ejpam-3890	18	14	of	of	ADP
ejpam-3890	18	15	a	a	DET
ejpam-3890	18	16	zero	zero	NUM
ejpam-3890	18	17	ring	ring	NOUN
ejpam-3890	18	18	in	in	ADP
ejpam-3890	18	19	which	which	PRON
ejpam-3890	18	20	the	the	DET
ejpam-3890	18	21	graph	graph	NOUN
ejpam-3890	18	22	admits	admit	VERB
ejpam-3890	18	23	a	a	DET
ejpam-3890	18	24	zero	zero	NUM
ejpam-3890	18	25	ring	ring	NOUN
ejpam-3890	18	26	labeling	labeling	NOUN
ejpam-3890	18	27	,	,	PUNCT
ejpam-3890	18	28	∗corresponding	∗corresponde	VERB
ejpam-3890	18	29	author	author	NOUN
ejpam-3890	18	30	.	.	PUNCT
ejpam-3890	19	1	doi	doi	NOUN
ejpam-3890	19	2	:	:	PUNCT
ejpam-3890	19	3	https://doi.org/10.29020/nybg.ejpam.v14i1.3890	https://doi.org/10.29020/nybg.ejpam.v14i1.3890	PRON
ejpam-3890	19	4	email	email	NOUN
ejpam-3890	19	5	addresses	address	NOUN
ejpam-3890	19	6	:	:	PUNCT
ejpam-3890	19	7	francis.campena@dlsu.edu.ph	francis.campena@dlsu.edu.ph	PROPN
ejpam-3890	19	8	(	(	PUNCT
ejpam-3890	19	9	f.j	f.j	PROPN
ejpam-3890	19	10	.	.	PROPN
ejpam-3890	19	11	campena	campena	NOUN
ejpam-3890	19	12	)	)	PUNCT
ejpam-3890	19	13	,	,	PUNCT
ejpam-3890	19	14	chuadhenmar@gmail.com	chuadhenmar@gmail.com	X
ejpam-3890	19	15	(	(	PUNCT
ejpam-3890	19	16	d.	d.	PROPN
ejpam-3890	19	17	chua	chua	PROPN
ejpam-3890	19	18	)	)	PUNCT
ejpam-3890	19	19	,	,	PUNCT
ejpam-3890	19	20	otserolf@yahoo.com	otserolf@yahoo.com	X
ejpam-3890	19	21	(	(	PUNCT
ejpam-3890	19	22	f.	f.	PROPN
ejpam-3890	19	23	franco	franco	PROPN
ejpam-3890	19	24	)	)	PUNCT
ejpam-3890	19	25	,	,	PUNCT
ejpam-3890	19	26	jon	jon	PROPN
ejpam-3890	19	27	-	-	PUNCT
ejpam-3890	19	28	jon	jon	PROPN
ejpam-3890	19	29	casica@dlsu.edu.ph	casica@dlsu.edu.ph	PROPN
ejpam-3890	19	30	(	(	PUNCT
ejpam-3890	19	31	j.	j.	PROPN
ejpam-3890	19	32	casica	casica	PROPN
ejpam-3890	19	33	)	)	PUNCT
ejpam-3890	19	34	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3890	20	1	268	268	NUM
ejpam-3890	20	2	c	c	X
ejpam-3890	20	3	©	©	PROPN
ejpam-3890	20	4	2021	2021	NUM
ejpam-3890	20	5	ejpam	ejpam	VERB
ejpam-3890	20	6	all	all	DET
ejpam-3890	20	7	rights	right	NOUN
ejpam-3890	20	8	reserved	reserve	VERB
ejpam-3890	20	9	.	.	PUNCT
ejpam-3890	21	1	f.j	f.j	NOUN
ejpam-3890	21	2	.	.	PROPN
ejpam-3890	22	1	campeña	campeña	PROPN
ejpam-3890	22	2	et	et	PROPN
ejpam-3890	22	3	al	al	PROPN
ejpam-3890	22	4	.	.	PUNCT
ejpam-3890	22	5	/	/	SYM
ejpam-3890	22	6	eur	eur	PROPN
ejpam-3890	22	7	.	.	PUNCT
ejpam-3890	23	1	j.	j.	PROPN
ejpam-3890	23	2	pure	pure	PROPN
ejpam-3890	23	3	appl	appl	PROPN
ejpam-3890	23	4	.	.	PROPN
ejpam-3890	23	5	math	math	PROPN
ejpam-3890	23	6	,	,	PUNCT
ejpam-3890	23	7	14	14	NUM
ejpam-3890	23	8	(	(	PUNCT
ejpam-3890	23	9	1	1	NUM
ejpam-3890	23	10	)	)	PUNCT
ejpam-3890	23	11	(	(	PUNCT
ejpam-3890	23	12	2021	2021	NUM
ejpam-3890	23	13	)	)	PUNCT
ejpam-3890	23	14	,	,	PUNCT
ejpam-3890	23	15	268	268	NUM
ejpam-3890	23	16	-	-	SYM
ejpam-3890	23	17	277	277	NUM
ejpam-3890	23	18	269	269	NUM
ejpam-3890	23	19	was	be	AUX
ejpam-3890	23	20	also	also	ADV
ejpam-3890	23	21	studied	study	VERB
ejpam-3890	23	22	for	for	ADP
ejpam-3890	23	23	some	some	DET
ejpam-3890	23	24	well	well	ADV
ejpam-3890	23	25	-	-	PUNCT
ejpam-3890	23	26	known	know	VERB
ejpam-3890	23	27	graphs	graph	NOUN
ejpam-3890	23	28	.	.	PUNCT
ejpam-3890	24	1	acharya	acharya	PROPN
ejpam-3890	24	2	et	et	PROPN
ejpam-3890	24	3	al	al	PROPN
ejpam-3890	24	4	.	.	PUNCT
ejpam-3890	25	1	[	[	X
ejpam-3890	25	2	2	2	NUM
ejpam-3890	25	3	]	]	PUNCT
ejpam-3890	25	4	determined	determine	VERB
ejpam-3890	25	5	a	a	DET
ejpam-3890	25	6	necessary	necessary	ADJ
ejpam-3890	25	7	and	and	CCONJ
ejpam-3890	25	8	sufficient	sufficient	ADJ
ejpam-3890	25	9	condition	condition	NOUN
ejpam-3890	25	10	for	for	ADP
ejpam-3890	25	11	a	a	DET
ejpam-3890	25	12	finite	finite	ADJ
ejpam-3890	25	13	graph	graph	NOUN
ejpam-3890	25	14	of	of	ADP
ejpam-3890	25	15	order	order	NOUN
ejpam-3890	25	16	n	n	PRON
ejpam-3890	25	17	to	to	PART
ejpam-3890	25	18	attain	attain	VERB
ejpam-3890	25	19	an	an	DET
ejpam-3890	25	20	optimal	optimal	ADJ
ejpam-3890	25	21	zero	zero	NUM
ejpam-3890	25	22	ring	ring	NOUN
ejpam-3890	25	23	index	index	NOUN
ejpam-3890	25	24	.	.	PUNCT
ejpam-3890	26	1	in	in	ADP
ejpam-3890	26	2	[	[	X
ejpam-3890	26	3	5	5	NUM
ejpam-3890	26	4	]	]	PUNCT
ejpam-3890	26	5	,	,	PUNCT
ejpam-3890	26	6	the	the	DET
ejpam-3890	26	7	notion	notion	NOUN
ejpam-3890	26	8	of	of	ADP
ejpam-3890	26	9	an	an	DET
ejpam-3890	26	10	efficient	efficient	ADJ
ejpam-3890	26	11	zero	zero	NUM
ejpam-3890	26	12	ring	ring	NOUN
ejpam-3890	26	13	labeling	labeling	NOUN
ejpam-3890	26	14	was	be	AUX
ejpam-3890	26	15	introduced	introduce	VERB
ejpam-3890	26	16	.	.	PUNCT
ejpam-3890	27	1	a	a	DET
ejpam-3890	27	2	zero	zero	NUM
ejpam-3890	27	3	ring	ring	NOUN
ejpam-3890	27	4	labeling	labeling	NOUN
ejpam-3890	27	5	of	of	ADP
ejpam-3890	27	6	a	a	DET
ejpam-3890	27	7	graph	graph	NOUN
ejpam-3890	27	8	is	be	AUX
ejpam-3890	27	9	called	call	VERB
ejpam-3890	27	10	efficient	efficient	ADJ
ejpam-3890	27	11	if	if	SCONJ
ejpam-3890	27	12	the	the	DET
ejpam-3890	27	13	cardinality	cardinality	NOUN
ejpam-3890	27	14	of	of	ADP
ejpam-3890	27	15	the	the	DET
ejpam-3890	27	16	set	set	NOUN
ejpam-3890	27	17	of	of	ADP
ejpam-3890	27	18	distinct	distinct	ADJ
ejpam-3890	27	19	sums	sum	NOUN
ejpam-3890	27	20	obtained	obtain	VERB
ejpam-3890	27	21	from	from	ADP
ejpam-3890	27	22	all	all	DET
ejpam-3890	27	23	adjacent	adjacent	ADJ
ejpam-3890	27	24	vertices	vertex	NOUN
ejpam-3890	27	25	is	be	AUX
ejpam-3890	27	26	equal	equal	ADJ
ejpam-3890	27	27	to	to	ADP
ejpam-3890	27	28	the	the	DET
ejpam-3890	27	29	maximum	maximum	ADJ
ejpam-3890	27	30	degree	degree	NOUN
ejpam-3890	27	31	of	of	ADP
ejpam-3890	27	32	the	the	DET
ejpam-3890	27	33	graph	graph	NOUN
ejpam-3890	27	34	.	.	PUNCT
ejpam-3890	28	1	it	it	PRON
ejpam-3890	28	2	was	be	AUX
ejpam-3890	28	3	shown	show	VERB
ejpam-3890	28	4	that	that	SCONJ
ejpam-3890	28	5	some	some	DET
ejpam-3890	28	6	trees	tree	NOUN
ejpam-3890	28	7	have	have	VERB
ejpam-3890	28	8	an	an	DET
ejpam-3890	28	9	efficient	efficient	ADJ
ejpam-3890	28	10	zero	zero	NUM
ejpam-3890	28	11	ring	ring	NOUN
ejpam-3890	28	12	labeling	labeling	NOUN
ejpam-3890	28	13	.	.	PUNCT
ejpam-3890	29	1	however	however	ADV
ejpam-3890	29	2	,	,	PUNCT
ejpam-3890	29	3	not	not	PART
ejpam-3890	29	4	all	all	DET
ejpam-3890	29	5	graphs	graph	NOUN
ejpam-3890	29	6	have	have	VERB
ejpam-3890	29	7	an	an	DET
ejpam-3890	29	8	efficient	efficient	ADJ
ejpam-3890	29	9	zero	zero	NUM
ejpam-3890	29	10	ring	ring	NOUN
ejpam-3890	29	11	labeling	labeling	NOUN
ejpam-3890	29	12	.	.	PUNCT
ejpam-3890	30	1	the	the	DET
ejpam-3890	30	2	paper	paper	NOUN
ejpam-3890	30	3	is	be	AUX
ejpam-3890	30	4	organized	organize	VERB
ejpam-3890	30	5	as	as	SCONJ
ejpam-3890	30	6	follows	follow	VERB
ejpam-3890	30	7	.	.	PUNCT
ejpam-3890	31	1	section	section	NOUN
ejpam-3890	31	2	2	2	NUM
ejpam-3890	31	3	provides	provide	VERB
ejpam-3890	31	4	some	some	DET
ejpam-3890	31	5	background	background	NOUN
ejpam-3890	31	6	on	on	ADP
ejpam-3890	31	7	the	the	DET
ejpam-3890	31	8	notion	notion	NOUN
ejpam-3890	31	9	of	of	ADP
ejpam-3890	31	10	zero	zero	NUM
ejpam-3890	31	11	ring	ring	NOUN
ejpam-3890	31	12	labeling	labeling	NOUN
ejpam-3890	31	13	and	and	CCONJ
ejpam-3890	31	14	efficient	efficient	ADJ
ejpam-3890	31	15	zero	zero	NUM
ejpam-3890	31	16	ring	ring	NOUN
ejpam-3890	31	17	labeling	labeling	NOUN
ejpam-3890	31	18	.	.	PUNCT
ejpam-3890	32	1	in	in	ADP
ejpam-3890	32	2	section	section	NOUN
ejpam-3890	32	3	3	3	NUM
ejpam-3890	32	4	,	,	PUNCT
ejpam-3890	32	5	we	we	PRON
ejpam-3890	32	6	show	show	VERB
ejpam-3890	32	7	that	that	SCONJ
ejpam-3890	32	8	the	the	DET
ejpam-3890	32	9	path	path	NOUN
ejpam-3890	32	10	graphs	graph	NOUN
ejpam-3890	32	11	and	and	CCONJ
ejpam-3890	32	12	the	the	DET
ejpam-3890	32	13	complete	complete	ADJ
ejpam-3890	32	14	graph	graph	NOUN
ejpam-3890	32	15	on	on	ADP
ejpam-3890	32	16	2k	2k	NUM
ejpam-3890	32	17	vertices	vertex	NOUN
ejpam-3890	32	18	admit	admit	VERB
ejpam-3890	32	19	an	an	DET
ejpam-3890	32	20	efficient	efficient	ADJ
ejpam-3890	32	21	zero	zero	NUM
ejpam-3890	32	22	ring	ring	NOUN
ejpam-3890	32	23	labeling	labeling	NOUN
ejpam-3890	32	24	.	.	PUNCT
ejpam-3890	33	1	moreover	moreover	ADV
ejpam-3890	33	2	,	,	PUNCT
ejpam-3890	33	3	an	an	DET
ejpam-3890	33	4	explicit	explicit	ADJ
ejpam-3890	33	5	labeling	labeling	NOUN
ejpam-3890	33	6	scheme	scheme	NOUN
ejpam-3890	33	7	for	for	ADP
ejpam-3890	33	8	path	path	NOUN
ejpam-3890	33	9	graphs	graph	NOUN
ejpam-3890	33	10	was	be	AUX
ejpam-3890	33	11	provided	provide	VERB
ejpam-3890	33	12	.	.	PUNCT
ejpam-3890	34	1	in	in	ADP
ejpam-3890	34	2	section	section	NOUN
ejpam-3890	34	3	4	4	NUM
ejpam-3890	34	4	,	,	PUNCT
ejpam-3890	34	5	we	we	PRON
ejpam-3890	34	6	construct	construct	VERB
ejpam-3890	34	7	restricted	restrict	VERB
ejpam-3890	34	8	zero	zero	NUM
ejpam-3890	34	9	ring	ring	NOUN
ejpam-3890	34	10	graphs	graph	NOUN
ejpam-3890	34	11	.	.	PUNCT
ejpam-3890	35	1	some	some	DET
ejpam-3890	35	2	edge	edge	NOUN
ejpam-3890	35	3	induced	induce	VERB
ejpam-3890	35	4	subgraphs	subgraph	NOUN
ejpam-3890	35	5	of	of	ADP
ejpam-3890	35	6	a	a	DET
ejpam-3890	35	7	zero	zero	NUM
ejpam-3890	35	8	ring	ring	NOUN
ejpam-3890	35	9	graph	graph	NOUN
ejpam-3890	35	10	were	be	AUX
ejpam-3890	35	11	identified	identify	VERB
ejpam-3890	35	12	and	and	CCONJ
ejpam-3890	35	13	shown	show	VERB
ejpam-3890	35	14	to	to	PART
ejpam-3890	35	15	admit	admit	VERB
ejpam-3890	35	16	an	an	DET
ejpam-3890	35	17	efficient	efficient	ADJ
ejpam-3890	35	18	zero	zero	NUM
ejpam-3890	35	19	ring	ring	NOUN
ejpam-3890	35	20	labeling	labeling	NOUN
ejpam-3890	35	21	.	.	PUNCT
ejpam-3890	36	1	2	2	X
ejpam-3890	36	2	.	.	X
ejpam-3890	36	3	preliminaries	preliminary	NOUN
ejpam-3890	36	4	for	for	ADP
ejpam-3890	36	5	completeness	completeness	NOUN
ejpam-3890	36	6	,	,	PUNCT
ejpam-3890	36	7	we	we	PRON
ejpam-3890	36	8	state	state	VERB
ejpam-3890	36	9	the	the	DET
ejpam-3890	36	10	following	follow	VERB
ejpam-3890	36	11	definitions	definition	NOUN
ejpam-3890	36	12	and	and	CCONJ
ejpam-3890	36	13	concepts	concept	NOUN
ejpam-3890	36	14	related	relate	VERB
ejpam-3890	36	15	to	to	ADP
ejpam-3890	36	16	zero	zero	NUM
ejpam-3890	36	17	ring	ring	NOUN
ejpam-3890	36	18	labeling	labeling	NOUN
ejpam-3890	36	19	and	and	CCONJ
ejpam-3890	36	20	some	some	DET
ejpam-3890	36	21	terminologies	terminology	NOUN
ejpam-3890	36	22	in	in	ADP
ejpam-3890	36	23	graph	graph	NOUN
ejpam-3890	36	24	theory	theory	NOUN
ejpam-3890	36	25	.	.	PUNCT
ejpam-3890	37	1	the	the	DET
ejpam-3890	37	2	graphs	graph	NOUN
ejpam-3890	37	3	considered	consider	VERB
ejpam-3890	37	4	in	in	ADP
ejpam-3890	37	5	the	the	DET
ejpam-3890	37	6	paper	paper	NOUN
ejpam-3890	37	7	are	be	AUX
ejpam-3890	37	8	finite	finite	ADJ
ejpam-3890	37	9	undirected	undirected	ADJ
ejpam-3890	37	10	simple	simple	ADJ
ejpam-3890	37	11	graph	graph	NOUN
ejpam-3890	37	12	.	.	PUNCT
ejpam-3890	38	1	a	a	DET
ejpam-3890	38	2	tadpole	tadpole	NOUN
ejpam-3890	38	3	graph	graph	NOUN
ejpam-3890	38	4	denoted	denote	VERB
ejpam-3890	38	5	by	by	ADP
ejpam-3890	38	6	tm	tm	PROPN
ejpam-3890	38	7	,	,	PUNCT
ejpam-3890	38	8	n	n	PROPN
ejpam-3890	38	9	is	be	AUX
ejpam-3890	38	10	a	a	DET
ejpam-3890	38	11	graph	graph	NOUN
ejpam-3890	38	12	obtained	obtain	VERB
ejpam-3890	38	13	from	from	ADP
ejpam-3890	38	14	a	a	DET
ejpam-3890	38	15	cycle	cycle	NOUN
ejpam-3890	38	16	cm	cm	NOUN
ejpam-3890	38	17	and	and	CCONJ
ejpam-3890	38	18	a	a	DET
ejpam-3890	38	19	path	path	NOUN
ejpam-3890	38	20	pn	pn	NOUN
ejpam-3890	38	21	by	by	ADP
ejpam-3890	38	22	joining	join	VERB
ejpam-3890	38	23	an	an	DET
ejpam-3890	38	24	end	end	NOUN
ejpam-3890	38	25	vertex	vertex	NOUN
ejpam-3890	38	26	of	of	ADP
ejpam-3890	38	27	the	the	DET
ejpam-3890	38	28	path	path	NOUN
ejpam-3890	38	29	to	to	ADP
ejpam-3890	38	30	a	a	DET
ejpam-3890	38	31	vertex	vertex	NOUN
ejpam-3890	38	32	in	in	ADP
ejpam-3890	38	33	the	the	DET
ejpam-3890	38	34	cycle	cycle	NOUN
ejpam-3890	38	35	with	with	ADP
ejpam-3890	38	36	an	an	DET
ejpam-3890	38	37	edge	edge	NOUN
ejpam-3890	38	38	.	.	PUNCT
ejpam-3890	39	1	a	a	DET
ejpam-3890	39	2	cactus	cactus	NOUN
ejpam-3890	39	3	graph	graph	NOUN
ejpam-3890	39	4	is	be	AUX
ejpam-3890	39	5	a	a	DET
ejpam-3890	39	6	connected	connected	ADJ
ejpam-3890	39	7	graph	graph	NOUN
ejpam-3890	39	8	in	in	ADP
ejpam-3890	39	9	which	which	PRON
ejpam-3890	39	10	any	any	DET
ejpam-3890	39	11	two	two	NUM
ejpam-3890	39	12	cycles	cycle	NOUN
ejpam-3890	39	13	have	have	VERB
ejpam-3890	39	14	at	at	ADP
ejpam-3890	39	15	most	most	ADV
ejpam-3890	39	16	one	one	NUM
ejpam-3890	39	17	vertex	vertex	NOUN
ejpam-3890	39	18	in	in	ADP
ejpam-3890	39	19	common	common	ADJ
ejpam-3890	39	20	.	.	PUNCT
ejpam-3890	40	1	equivalently	equivalently	ADV
ejpam-3890	40	2	,	,	PUNCT
ejpam-3890	40	3	any	any	DET
ejpam-3890	40	4	edge	edge	NOUN
ejpam-3890	40	5	of	of	ADP
ejpam-3890	40	6	a	a	DET
ejpam-3890	40	7	cactus	cactus	NOUN
ejpam-3890	40	8	graph	graph	NOUN
ejpam-3890	40	9	lies	lie	VERB
ejpam-3890	40	10	on	on	ADP
ejpam-3890	40	11	at	at	ADP
ejpam-3890	40	12	most	most	ADV
ejpam-3890	40	13	one	one	NUM
ejpam-3890	40	14	cycle	cycle	NOUN
ejpam-3890	40	15	.	.	PUNCT
ejpam-3890	41	1	a	a	DET
ejpam-3890	41	2	cactus	cactus	NOUN
ejpam-3890	41	3	graph	graph	NOUN
ejpam-3890	41	4	with	with	ADP
ejpam-3890	41	5	minimum	minimum	NOUN
ejpam-3890	41	6	degree	degree	NOUN
ejpam-3890	41	7	2	2	NUM
ejpam-3890	41	8	,	,	PUNCT
ejpam-3890	41	9	maximum	maximum	ADJ
ejpam-3890	41	10	degree	degree	NOUN
ejpam-3890	41	11	3	3	NUM
ejpam-3890	41	12	,	,	PUNCT
ejpam-3890	41	13	and	and	CCONJ
ejpam-3890	41	14	having	have	VERB
ejpam-3890	41	15	exactly	exactly	ADV
ejpam-3890	41	16	2	2	NUM
ejpam-3890	41	17	cycles	cycle	NOUN
ejpam-3890	41	18	is	be	AUX
ejpam-3890	41	19	a	a	DET
ejpam-3890	41	20	graph	graph	NOUN
ejpam-3890	41	21	that	that	PRON
ejpam-3890	41	22	can	can	AUX
ejpam-3890	41	23	be	be	AUX
ejpam-3890	41	24	constructed	construct	VERB
ejpam-3890	41	25	with	with	ADP
ejpam-3890	41	26	two	two	NUM
ejpam-3890	41	27	disjoint	disjoint	ADJ
ejpam-3890	41	28	cycles	cycle	NOUN
ejpam-3890	41	29	of	of	ADP
ejpam-3890	41	30	length	length	NOUN
ejpam-3890	41	31	n1	n1	NOUN
ejpam-3890	41	32	,	,	PUNCT
ejpam-3890	41	33	n2	n2	NOUN
ejpam-3890	41	34	and	and	CCONJ
ejpam-3890	41	35	a	a	DET
ejpam-3890	41	36	vertex	vertex	NOUN
ejpam-3890	41	37	from	from	ADP
ejpam-3890	41	38	one	one	NUM
ejpam-3890	41	39	cycle	cycle	NOUN
ejpam-3890	41	40	joined	join	VERB
ejpam-3890	41	41	to	to	ADP
ejpam-3890	41	42	a	a	DET
ejpam-3890	41	43	vertex	vertex	NOUN
ejpam-3890	41	44	of	of	ADP
ejpam-3890	41	45	the	the	DET
ejpam-3890	41	46	second	second	ADJ
ejpam-3890	41	47	cycle	cycle	NOUN
ejpam-3890	41	48	by	by	ADP
ejpam-3890	41	49	a	a	DET
ejpam-3890	41	50	path	path	NOUN
ejpam-3890	41	51	of	of	ADP
ejpam-3890	41	52	length	length	NOUN
ejpam-3890	41	53	m.	m.	NOUN
ejpam-3890	41	54	suppose	suppose	VERB
ejpam-3890	41	55	g	g	PROPN
ejpam-3890	41	56	and	and	CCONJ
ejpam-3890	41	57	h	h	NOUN
ejpam-3890	41	58	are	be	AUX
ejpam-3890	41	59	graphs	graph	NOUN
ejpam-3890	41	60	with	with	ADP
ejpam-3890	41	61	v	v	NOUN
ejpam-3890	41	62	(	(	PUNCT
ejpam-3890	41	63	g	g	NOUN
ejpam-3890	41	64	)	)	PUNCT
ejpam-3890	41	65	=	=	SYM
ejpam-3890	41	66	{	{	PUNCT
ejpam-3890	41	67	u0	u0	ADJ
ejpam-3890	41	68	,	,	PUNCT
ejpam-3890	41	69	u1	u1	NOUN
ejpam-3890	41	70	,	,	PUNCT
ejpam-3890	41	71	u2	u2	NOUN
ejpam-3890	41	72	,	,	PUNCT
ejpam-3890	41	73	...	...	PUNCT
ejpam-3890	41	74	,	,	PUNCT
ejpam-3890	41	75	um−1	um−1	NOUN
ejpam-3890	41	76	}	}	PUNCT
ejpam-3890	41	77	and	and	CCONJ
ejpam-3890	41	78	v	v	NOUN
ejpam-3890	41	79	(	(	PUNCT
ejpam-3890	41	80	h	h	NOUN
ejpam-3890	41	81	)	)	PUNCT
ejpam-3890	41	82	=	=	SYM
ejpam-3890	41	83	{	{	PUNCT
ejpam-3890	41	84	v0	v0	NOUN
ejpam-3890	41	85	,	,	PUNCT
ejpam-3890	41	86	v1	v1	NOUN
ejpam-3890	41	87	,	,	PUNCT
ejpam-3890	41	88	v2	v2	PROPN
ejpam-3890	41	89	,	,	PUNCT
ejpam-3890	41	90	...	...	PUNCT
ejpam-3890	41	91	,	,	PUNCT
ejpam-3890	41	92	vn−1	vn−1	ADJ
ejpam-3890	41	93	}	}	PUNCT
ejpam-3890	41	94	.	.	PUNCT
ejpam-3890	42	1	the	the	DET
ejpam-3890	42	2	cartesian	cartesian	ADJ
ejpam-3890	42	3	product	product	NOUN
ejpam-3890	42	4	g	g	PROPN
ejpam-3890	42	5	×h	×h	PROPN
ejpam-3890	42	6	of	of	ADP
ejpam-3890	42	7	graphs	graph	NOUN
ejpam-3890	42	8	g	g	NOUN
ejpam-3890	42	9	and	and	CCONJ
ejpam-3890	42	10	h	h	NOUN
ejpam-3890	42	11	is	be	AUX
ejpam-3890	42	12	the	the	DET
ejpam-3890	42	13	graph	graph	NOUN
ejpam-3890	42	14	with	with	ADP
ejpam-3890	42	15	vertex	vertex	NOUN
ejpam-3890	42	16	set	set	VERB
ejpam-3890	42	17	v	v	NOUN
ejpam-3890	42	18	(	(	PUNCT
ejpam-3890	42	19	g	g	NOUN
ejpam-3890	42	20	×h	×h	PROPN
ejpam-3890	42	21	)	)	PUNCT
ejpam-3890	42	22	=	=	SYM
ejpam-3890	42	23	v	v	X
ejpam-3890	42	24	(	(	PUNCT
ejpam-3890	42	25	g	g	NOUN
ejpam-3890	42	26	)	)	PUNCT
ejpam-3890	42	27	×	×	NOUN
ejpam-3890	42	28	v	v	NOUN
ejpam-3890	42	29	(	(	PUNCT
ejpam-3890	42	30	h	h	NOUN
ejpam-3890	42	31	)	)	PUNCT
ejpam-3890	42	32	and	and	CCONJ
ejpam-3890	42	33	e	e	X
ejpam-3890	42	34	=	=	SYM
ejpam-3890	42	35	(	(	PUNCT
ejpam-3890	42	36	ui	ui	PROPN
ejpam-3890	42	37	,	,	PUNCT
ejpam-3890	42	38	vj)(uk	vj)(uk	PROPN
ejpam-3890	42	39	,	,	PUNCT
ejpam-3890	42	40	vl	vl	NOUN
ejpam-3890	42	41	)	)	PUNCT
ejpam-3890	42	42	is	be	AUX
ejpam-3890	42	43	an	an	DET
ejpam-3890	42	44	edge	edge	NOUN
ejpam-3890	42	45	of	of	ADP
ejpam-3890	42	46	g	g	PROPN
ejpam-3890	42	47	×	×	PROPN
ejpam-3890	42	48	h	h	NOUN
ejpam-3890	42	49	if	if	SCONJ
ejpam-3890	43	1	and	and	CCONJ
ejpam-3890	43	2	only	only	ADV
ejpam-3890	43	3	if	if	SCONJ
ejpam-3890	43	4	either	either	CCONJ
ejpam-3890	43	5	i	i	PRON
ejpam-3890	43	6	=	=	SYM
ejpam-3890	43	7	k	k	PROPN
ejpam-3890	43	8	and	and	CCONJ
ejpam-3890	43	9	vjvl	vjvl	NOUN
ejpam-3890	43	10	∈	∈	PROPN
ejpam-3890	43	11	e(h	e(h	PROPN
ejpam-3890	43	12	)	)	PUNCT
ejpam-3890	43	13	;	;	PUNCT
ejpam-3890	43	14	or	or	CCONJ
ejpam-3890	43	15	j	j	PROPN
ejpam-3890	43	16	=	=	SYM
ejpam-3890	43	17	l	l	PROPN
ejpam-3890	43	18	and	and	CCONJ
ejpam-3890	43	19	uiuk	uiuk	PROPN
ejpam-3890	43	20	∈	∈	PROPN
ejpam-3890	43	21	e(g	e(g	PROPN
ejpam-3890	43	22	)	)	PUNCT
ejpam-3890	43	23	.	.	PUNCT
ejpam-3890	44	1	for	for	ADP
ejpam-3890	44	2	other	other	ADJ
ejpam-3890	44	3	notations	notation	NOUN
ejpam-3890	44	4	and	and	CCONJ
ejpam-3890	44	5	concepts	concept	NOUN
ejpam-3890	44	6	in	in	ADP
ejpam-3890	44	7	graphs	graph	NOUN
ejpam-3890	44	8	and	and	CCONJ
ejpam-3890	44	9	groups	group	NOUN
ejpam-3890	44	10	not	not	PART
ejpam-3890	44	11	explicitly	explicitly	ADV
ejpam-3890	44	12	stated	state	VERB
ejpam-3890	44	13	in	in	ADP
ejpam-3890	44	14	the	the	DET
ejpam-3890	44	15	paper	paper	NOUN
ejpam-3890	44	16	,	,	PUNCT
ejpam-3890	44	17	we	we	PRON
ejpam-3890	44	18	refer	refer	VERB
ejpam-3890	44	19	to	to	ADP
ejpam-3890	44	20	[	[	X
ejpam-3890	44	21	6	6	NUM
ejpam-3890	44	22	,	,	PUNCT
ejpam-3890	44	23	8	8	NUM
ejpam-3890	44	24	]	]	PUNCT
ejpam-3890	44	25	.	.	PUNCT
ejpam-3890	45	1	although	although	SCONJ
ejpam-3890	45	2	in	in	ADP
ejpam-3890	45	3	some	some	DET
ejpam-3890	45	4	literature	literature	NOUN
ejpam-3890	45	5	a	a	DET
ejpam-3890	45	6	zero	zero	NUM
ejpam-3890	45	7	ring	ring	NOUN
ejpam-3890	45	8	is	be	AUX
ejpam-3890	45	9	defined	define	VERB
ejpam-3890	45	10	as	as	ADP
ejpam-3890	45	11	the	the	DET
ejpam-3890	45	12	trivial	trivial	ADJ
ejpam-3890	45	13	ring	ring	NOUN
ejpam-3890	45	14	that	that	PRON
ejpam-3890	45	15	contains	contain	VERB
ejpam-3890	45	16	only	only	ADV
ejpam-3890	45	17	one	one	NUM
ejpam-3890	45	18	element	element	NOUN
ejpam-3890	45	19	,	,	PUNCT
ejpam-3890	45	20	in	in	ADP
ejpam-3890	45	21	this	this	DET
ejpam-3890	45	22	paper	paper	NOUN
ejpam-3890	45	23	we	we	PRON
ejpam-3890	45	24	follow	follow	VERB
ejpam-3890	45	25	the	the	DET
ejpam-3890	45	26	definition	definition	NOUN
ejpam-3890	45	27	used	use	VERB
ejpam-3890	45	28	by	by	ADP
ejpam-3890	45	29	acharya	acharya	PROPN
ejpam-3890	45	30	et	et	PROPN
ejpam-3890	45	31	al	al	PROPN
ejpam-3890	45	32	.	.	PUNCT
ejpam-3890	46	1	[	[	X
ejpam-3890	46	2	3	3	X
ejpam-3890	46	3	]	]	PUNCT
ejpam-3890	46	4	and	and	CCONJ
ejpam-3890	46	5	use	use	VERB
ejpam-3890	46	6	the	the	DET
ejpam-3890	46	7	term	term	NOUN
ejpam-3890	46	8	zero	zero	NUM
ejpam-3890	46	9	ring	ring	NOUN
ejpam-3890	46	10	for	for	ADP
ejpam-3890	46	11	a	a	DET
ejpam-3890	46	12	ring	ring	NOUN
ejpam-3890	46	13	with	with	ADP
ejpam-3890	46	14	additive	additive	ADJ
ejpam-3890	46	15	identity	identity	NOUN
ejpam-3890	46	16	0	0	NUM
ejpam-3890	46	17	such	such	ADJ
ejpam-3890	46	18	that	that	SCONJ
ejpam-3890	46	19	ab	ab	PROPN
ejpam-3890	46	20	=	=	NOUN
ejpam-3890	46	21	0	0	NUM
ejpam-3890	46	22	for	for	ADP
ejpam-3890	46	23	any	any	DET
ejpam-3890	46	24	a	a	NOUN
ejpam-3890	46	25	,	,	PUNCT
ejpam-3890	46	26	b	b	PROPN
ejpam-3890	46	27	∈	∈	PROPN
ejpam-3890	46	28	r.	r.	NOUN
ejpam-3890	46	29	this	this	DET
ejpam-3890	46	30	notion	notion	NOUN
ejpam-3890	46	31	of	of	ADP
ejpam-3890	46	32	a	a	DET
ejpam-3890	46	33	zero	zero	NUM
ejpam-3890	46	34	ring	ring	NOUN
ejpam-3890	46	35	was	be	AUX
ejpam-3890	46	36	previously	previously	ADV
ejpam-3890	46	37	defined	define	VERB
ejpam-3890	46	38	by	by	ADP
ejpam-3890	46	39	bourbaki	bourbaki	NOUN
ejpam-3890	46	40	in	in	ADP
ejpam-3890	46	41	[	[	X
ejpam-3890	46	42	4	4	NUM
ejpam-3890	46	43	]	]	PUNCT
ejpam-3890	46	44	and	and	CCONJ
ejpam-3890	46	45	called	call	VERB
ejpam-3890	46	46	it	it	PRON
ejpam-3890	46	47	a	a	DET
ejpam-3890	46	48	pseudo	pseudo	NOUN
ejpam-3890	46	49	ring	ring	NOUN
ejpam-3890	46	50	of	of	ADP
ejpam-3890	46	51	square	square	ADJ
ejpam-3890	46	52	zero	zero	NUM
ejpam-3890	46	53	.	.	PUNCT
ejpam-3890	47	1	note	note	VERB
ejpam-3890	47	2	that	that	SCONJ
ejpam-3890	47	3	a	a	DET
ejpam-3890	47	4	zero	zero	NUM
ejpam-3890	47	5	ring	ring	NOUN
ejpam-3890	47	6	r	r	NOUN
ejpam-3890	47	7	can	can	AUX
ejpam-3890	47	8	always	always	ADV
ejpam-3890	47	9	be	be	AUX
ejpam-3890	47	10	constructed	construct	VERB
ejpam-3890	47	11	from	from	ADP
ejpam-3890	47	12	an	an	DET
ejpam-3890	47	13	abelian	abelian	ADJ
ejpam-3890	47	14	group	group	NOUN
ejpam-3890	47	15	g	g	PROPN
ejpam-3890	47	16	,	,	PUNCT
ejpam-3890	47	17	by	by	ADP
ejpam-3890	47	18	defining	define	VERB
ejpam-3890	47	19	r	r	NOUN
ejpam-3890	47	20	as	as	ADP
ejpam-3890	47	21	the	the	DET
ejpam-3890	47	22	set	set	NOUN
ejpam-3890	47	23	of	of	ADP
ejpam-3890	47	24	the	the	DET
ejpam-3890	47	25	set	set	NOUN
ejpam-3890	47	26	of	of	ADP
ejpam-3890	47	27	all	all	DET
ejpam-3890	47	28	2×	2×	NUM
ejpam-3890	47	29	2	2	NUM
ejpam-3890	47	30	matrices	matrix	NOUN
ejpam-3890	47	31	of	of	ADP
ejpam-3890	47	32	the	the	DET
ejpam-3890	47	33	form	form	NOUN
ejpam-3890	47	34	r	r	NOUN
ejpam-3890	47	35	=	=	SYM
ejpam-3890	47	36	{	{	PUNCT
ejpam-3890	47	37	[	[	PUNCT
ejpam-3890	47	38	a	a	DET
ejpam-3890	47	39	−a	−a	NOUN
ejpam-3890	47	40	a	a	DET
ejpam-3890	47	41	−a	−a	NOUN
ejpam-3890	47	42	]	]	PUNCT
ejpam-3890	47	43	,	,	PUNCT
ejpam-3890	47	44	a	a	DET
ejpam-3890	47	45	∈	∈	PROPN
ejpam-3890	47	46	g	g	NOUN
ejpam-3890	47	47	}	}	PUNCT
ejpam-3890	47	48	.	.	PUNCT
ejpam-3890	48	1	we	we	PRON
ejpam-3890	48	2	use	use	VERB
ejpam-3890	48	3	the	the	DET
ejpam-3890	48	4	notation	notation	NOUN
ejpam-3890	48	5	aa	aa	PROPN
ejpam-3890	48	6	∈	∈	PROPN
ejpam-3890	48	7	r	r	NOUN
ejpam-3890	48	8	to	to	PART
ejpam-3890	48	9	identify	identify	VERB
ejpam-3890	48	10	the	the	DET
ejpam-3890	48	11	element	element	NOUN
ejpam-3890	48	12	of	of	ADP
ejpam-3890	48	13	the	the	DET
ejpam-3890	48	14	ring	ring	NOUN
ejpam-3890	48	15	associated	associate	VERB
ejpam-3890	48	16	with	with	ADP
ejpam-3890	48	17	the	the	DET
ejpam-3890	48	18	element	element	NOUN
ejpam-3890	48	19	a	a	DET
ejpam-3890	48	20	∈	∈	PROPN
ejpam-3890	48	21	g.	g.	NOUN
ejpam-3890	48	22	throughout	throughout	ADP
ejpam-3890	48	23	the	the	DET
ejpam-3890	48	24	paper	paper	NOUN
ejpam-3890	48	25	,	,	PUNCT
ejpam-3890	48	26	we	we	PRON
ejpam-3890	48	27	denote	denote	VERB
ejpam-3890	48	28	this	this	DET
ejpam-3890	48	29	zero	zero	NUM
ejpam-3890	48	30	ring	ring	NOUN
ejpam-3890	48	31	by	by	ADP
ejpam-3890	48	32	m0	m0	PROPN
ejpam-3890	48	33	2	2	NUM
ejpam-3890	48	34	(	(	PUNCT
ejpam-3890	48	35	g	g	NOUN
ejpam-3890	48	36	)	)	PUNCT
ejpam-3890	48	37	.	.	PUNCT
ejpam-3890	49	1	we	we	PRON
ejpam-3890	49	2	often	often	ADV
ejpam-3890	49	3	use	use	VERB
ejpam-3890	49	4	this	this	DET
ejpam-3890	49	5	notation	notation	NOUN
ejpam-3890	49	6	when	when	SCONJ
ejpam-3890	49	7	we	we	PRON
ejpam-3890	49	8	wish	wish	VERB
ejpam-3890	49	9	to	to	PART
ejpam-3890	49	10	be	be	AUX
ejpam-3890	49	11	specific	specific	ADJ
ejpam-3890	49	12	on	on	ADP
ejpam-3890	49	13	the	the	DET
ejpam-3890	49	14	ring	ring	NOUN
ejpam-3890	49	15	being	be	AUX
ejpam-3890	49	16	used	use	VERB
ejpam-3890	49	17	in	in	ADP
ejpam-3890	49	18	the	the	DET
ejpam-3890	49	19	labeling	labeling	NOUN
ejpam-3890	49	20	scheme	scheme	NOUN
ejpam-3890	49	21	.	.	PUNCT
ejpam-3890	50	1	f.j	f.j	PROPN
ejpam-3890	50	2	.	.	PROPN
ejpam-3890	51	1	campeña	campeña	PROPN
ejpam-3890	51	2	et	et	PROPN
ejpam-3890	51	3	al	al	PROPN
ejpam-3890	51	4	.	.	PUNCT
ejpam-3890	51	5	/	/	SYM
ejpam-3890	51	6	eur	eur	PROPN
ejpam-3890	51	7	.	.	PUNCT
ejpam-3890	52	1	j.	j.	PROPN
ejpam-3890	52	2	pure	pure	PROPN
ejpam-3890	52	3	appl	appl	PROPN
ejpam-3890	52	4	.	.	PROPN
ejpam-3890	52	5	math	math	PROPN
ejpam-3890	52	6	,	,	PUNCT
ejpam-3890	52	7	14	14	NUM
ejpam-3890	52	8	(	(	PUNCT
ejpam-3890	52	9	1	1	NUM
ejpam-3890	52	10	)	)	PUNCT
ejpam-3890	52	11	(	(	PUNCT
ejpam-3890	52	12	2021	2021	NUM
ejpam-3890	52	13	)	)	PUNCT
ejpam-3890	52	14	,	,	PUNCT
ejpam-3890	52	15	268	268	NUM
ejpam-3890	52	16	-	-	SYM
ejpam-3890	52	17	277	277	NUM
ejpam-3890	52	18	270	270	NUM
ejpam-3890	52	19	let	let	VERB
ejpam-3890	52	20	γ	γ	X
ejpam-3890	52	21	=	=	SYM
ejpam-3890	52	22	(	(	PUNCT
ejpam-3890	52	23	v	v	NOUN
ejpam-3890	52	24	,	,	PUNCT
ejpam-3890	52	25	e	e	NOUN
ejpam-3890	52	26	)	)	PUNCT
ejpam-3890	52	27	be	be	AUX
ejpam-3890	52	28	a	a	DET
ejpam-3890	52	29	graph	graph	NOUN
ejpam-3890	52	30	with	with	ADP
ejpam-3890	52	31	vertex	vertex	NOUN
ejpam-3890	52	32	set	set	VERB
ejpam-3890	52	33	v	v	NOUN
ejpam-3890	52	34	=	=	NOUN
ejpam-3890	52	35	:	:	PUNCT
ejpam-3890	52	36	v	v	NOUN
ejpam-3890	52	37	(	(	PUNCT
ejpam-3890	52	38	γ	γ	NOUN
ejpam-3890	52	39	)	)	PUNCT
ejpam-3890	52	40	and	and	CCONJ
ejpam-3890	52	41	edge	edge	NOUN
ejpam-3890	52	42	set	set	VERB
ejpam-3890	52	43	e	e	NOUN
ejpam-3890	52	44	=	=	NOUN
ejpam-3890	52	45	:	:	PUNCT
ejpam-3890	52	46	e(γ	e(γ	NOUN
ejpam-3890	52	47	)	)	PUNCT
ejpam-3890	52	48	,	,	PUNCT
ejpam-3890	52	49	and	and	CCONJ
ejpam-3890	52	50	let	let	VERB
ejpam-3890	52	51	r	r	PRON
ejpam-3890	52	52	be	be	AUX
ejpam-3890	52	53	a	a	DET
ejpam-3890	52	54	finite	finite	ADJ
ejpam-3890	52	55	zero	zero	NUM
ejpam-3890	52	56	ring	ring	NOUN
ejpam-3890	52	57	.	.	PUNCT
ejpam-3890	53	1	an	an	DET
ejpam-3890	53	2	injective	injective	ADJ
ejpam-3890	53	3	function	function	NOUN
ejpam-3890	53	4	f	f	NOUN
ejpam-3890	53	5	:	:	PUNCT
ejpam-3890	53	6	v	v	X
ejpam-3890	53	7	→	→	SYM
ejpam-3890	53	8	r	r	NOUN
ejpam-3890	53	9	is	be	AUX
ejpam-3890	53	10	called	call	VERB
ejpam-3890	53	11	a	a	DET
ejpam-3890	53	12	zero	zero	NUM
ejpam-3890	53	13	ring	ring	NOUN
ejpam-3890	53	14	labeling	labeling	NOUN
ejpam-3890	53	15	of	of	ADP
ejpam-3890	53	16	γ	γ	PROPN
ejpam-3890	53	17	if	if	SCONJ
ejpam-3890	53	18	f(u	f(u	PROPN
ejpam-3890	53	19	)	)	PUNCT
ejpam-3890	53	20	+	+	NUM
ejpam-3890	53	21	f(v	f(v	NOUN
ejpam-3890	53	22	)	)	PUNCT
ejpam-3890	53	23	6=	6=	ADP
ejpam-3890	53	24	0	0	NUM
ejpam-3890	53	25	for	for	ADP
ejpam-3890	53	26	every	every	DET
ejpam-3890	53	27	edge	edge	NOUN
ejpam-3890	53	28	uv	uv	NOUN
ejpam-3890	53	29	∈	∈	NOUN
ejpam-3890	53	30	e	e	NOUN
ejpam-3890	53	31	in	in	ADP
ejpam-3890	53	32	[	[	X
ejpam-3890	53	33	5	5	NUM
ejpam-3890	53	34	]	]	PUNCT
ejpam-3890	53	35	,	,	PUNCT
ejpam-3890	53	36	the	the	DET
ejpam-3890	53	37	authors	author	NOUN
ejpam-3890	53	38	defined	define	VERB
ejpam-3890	53	39	the	the	DET
ejpam-3890	53	40	notion	notion	NOUN
ejpam-3890	53	41	about	about	ADP
ejpam-3890	53	42	k	k	ADJ
ejpam-3890	53	43	-	-	ADJ
ejpam-3890	53	44	zero	zero	NUM
ejpam-3890	53	45	ring	ring	NOUN
ejpam-3890	53	46	labelings	labeling	NOUN
ejpam-3890	53	47	and	and	CCONJ
ejpam-3890	53	48	efficient	efficient	ADJ
ejpam-3890	53	49	zero	zero	NUM
ejpam-3890	53	50	ring	ring	NOUN
ejpam-3890	53	51	labeling	labeling	NOUN
ejpam-3890	53	52	of	of	ADP
ejpam-3890	53	53	a	a	DET
ejpam-3890	53	54	graph	graph	NOUN
ejpam-3890	53	55	.	.	PUNCT
ejpam-3890	54	1	a	a	DET
ejpam-3890	54	2	zero	zero	NUM
ejpam-3890	54	3	ring	ring	NOUN
ejpam-3890	54	4	labeling	labeling	NOUN
ejpam-3890	54	5	f	f	PROPN
ejpam-3890	54	6	of	of	ADP
ejpam-3890	54	7	a	a	DET
ejpam-3890	54	8	graph	graph	NOUN
ejpam-3890	54	9	γ	γ	X
ejpam-3890	54	10	=	=	SYM
ejpam-3890	54	11	(	(	PUNCT
ejpam-3890	54	12	v	v	NOUN
ejpam-3890	54	13	,	,	PUNCT
ejpam-3890	54	14	e	e	NOUN
ejpam-3890	54	15	)	)	PUNCT
ejpam-3890	54	16	is	be	AUX
ejpam-3890	54	17	called	call	VERB
ejpam-3890	54	18	a	a	DET
ejpam-3890	54	19	k	k	NOUN
ejpam-3890	54	20	-	-	ADJ
ejpam-3890	54	21	zero	zero	NUM
ejpam-3890	54	22	ring	ring	NOUN
ejpam-3890	54	23	labeling	labeling	NOUN
ejpam-3890	54	24	if	if	SCONJ
ejpam-3890	54	25	|k|	|k|	PROPN
ejpam-3890	54	26	=	=	SYM
ejpam-3890	54	27	|	|	ADV
ejpam-3890	54	28	{	{	PUNCT
ejpam-3890	54	29	f(u	f(u	PROPN
ejpam-3890	54	30	)	)	PUNCT
ejpam-3890	54	31	+	+	NUM
ejpam-3890	54	32	f(v	f(v	NOUN
ejpam-3890	54	33	)	)	PUNCT
ejpam-3890	54	34	:	:	PUNCT
ejpam-3890	54	35	uv	uv	NOUN
ejpam-3890	54	36	∈	∈	NOUN
ejpam-3890	54	37	e	e	NOUN
ejpam-3890	54	38	}	}	PUNCT
ejpam-3890	54	39	|	|	ADV
ejpam-3890	54	40	=	=	PUNCT
ejpam-3890	54	41	k.	k.	PROPN
ejpam-3890	55	1	if	if	SCONJ
ejpam-3890	55	2	|k|	|k|	PROPN
ejpam-3890	55	3	=	=	SYM
ejpam-3890	55	4	∆(γ	∆(γ	X
ejpam-3890	55	5	)	)	PUNCT
ejpam-3890	55	6	where	where	SCONJ
ejpam-3890	55	7	∆(γ	∆(γ	X
ejpam-3890	55	8	)	)	PUNCT
ejpam-3890	55	9	denotes	denote	VERB
ejpam-3890	55	10	the	the	DET
ejpam-3890	55	11	maximum	maximum	ADJ
ejpam-3890	55	12	degree	degree	NOUN
ejpam-3890	55	13	of	of	ADP
ejpam-3890	55	14	a	a	DET
ejpam-3890	55	15	vertex	vertex	NOUN
ejpam-3890	55	16	in	in	ADP
ejpam-3890	55	17	γ	γ	PROPN
ejpam-3890	55	18	,	,	PUNCT
ejpam-3890	55	19	then	then	ADV
ejpam-3890	55	20	the	the	DET
ejpam-3890	55	21	zero	zero	NUM
ejpam-3890	55	22	ring	ring	NOUN
ejpam-3890	55	23	labeling	labeling	NOUN
ejpam-3890	55	24	is	be	AUX
ejpam-3890	55	25	efficient	efficient	ADJ
ejpam-3890	55	26	.	.	PUNCT
ejpam-3890	56	1	it	it	PRON
ejpam-3890	56	2	was	be	AUX
ejpam-3890	56	3	shown	show	VERB
ejpam-3890	56	4	that	that	SCONJ
ejpam-3890	56	5	some	some	DET
ejpam-3890	56	6	families	family	NOUN
ejpam-3890	56	7	of	of	ADP
ejpam-3890	56	8	trees	tree	NOUN
ejpam-3890	56	9	have	have	VERB
ejpam-3890	56	10	an	an	DET
ejpam-3890	56	11	efficient	efficient	ADJ
ejpam-3890	56	12	zero	zero	NUM
ejpam-3890	56	13	ring	ring	NOUN
ejpam-3890	56	14	labeling	labeling	NOUN
ejpam-3890	56	15	.	.	PUNCT
ejpam-3890	57	1	although	although	SCONJ
ejpam-3890	57	2	it	it	PRON
ejpam-3890	57	3	was	be	AUX
ejpam-3890	57	4	shown	show	VERB
ejpam-3890	57	5	in	in	ADP
ejpam-3890	57	6	[	[	X
ejpam-3890	57	7	3	3	X
ejpam-3890	57	8	]	]	PUNCT
ejpam-3890	57	9	that	that	SCONJ
ejpam-3890	57	10	every	every	DET
ejpam-3890	57	11	finite	finite	ADJ
ejpam-3890	57	12	graph	graph	NOUN
ejpam-3890	57	13	admits	admit	VERB
ejpam-3890	57	14	a	a	DET
ejpam-3890	57	15	zero	zero	NUM
ejpam-3890	57	16	ring	ring	NOUN
ejpam-3890	57	17	labeling	labeling	NOUN
ejpam-3890	57	18	,	,	PUNCT
ejpam-3890	57	19	not	not	PART
ejpam-3890	57	20	all	all	DET
ejpam-3890	57	21	graphs	graph	NOUN
ejpam-3890	57	22	have	have	VERB
ejpam-3890	57	23	an	an	DET
ejpam-3890	57	24	efficient	efficient	ADJ
ejpam-3890	57	25	zero	zero	NUM
ejpam-3890	57	26	ring	ring	NOUN
ejpam-3890	57	27	labeling	labeling	NOUN
ejpam-3890	57	28	.	.	PUNCT
ejpam-3890	58	1	in	in	ADP
ejpam-3890	58	2	particular	particular	ADJ
ejpam-3890	58	3	it	it	PRON
ejpam-3890	58	4	was	be	AUX
ejpam-3890	58	5	shown	show	VERB
ejpam-3890	58	6	in	in	ADP
ejpam-3890	58	7	[	[	X
ejpam-3890	58	8	5	5	NUM
ejpam-3890	58	9	]	]	PUNCT
ejpam-3890	58	10	that	that	SCONJ
ejpam-3890	58	11	some	some	DET
ejpam-3890	58	12	cycles	cycle	NOUN
ejpam-3890	58	13	do	do	AUX
ejpam-3890	58	14	not	not	PART
ejpam-3890	58	15	have	have	VERB
ejpam-3890	58	16	an	an	DET
ejpam-3890	58	17	efficient	efficient	ADJ
ejpam-3890	58	18	zero	zero	NUM
ejpam-3890	58	19	ring	ring	NOUN
ejpam-3890	58	20	labeling	labeling	NOUN
ejpam-3890	58	21	.	.	PUNCT
ejpam-3890	59	1	theorem	theorem	NOUN
ejpam-3890	59	2	1	1	NUM
ejpam-3890	59	3	.	.	PUNCT
ejpam-3890	60	1	[	[	X
ejpam-3890	60	2	5	5	NUM
ejpam-3890	60	3	]	]	PUNCT
ejpam-3890	60	4	cycles	cycle	NOUN
ejpam-3890	60	5	of	of	ADP
ejpam-3890	60	6	odd	odd	ADJ
ejpam-3890	60	7	length	length	NOUN
ejpam-3890	60	8	do	do	AUX
ejpam-3890	60	9	not	not	PART
ejpam-3890	60	10	have	have	VERB
ejpam-3890	60	11	an	an	DET
ejpam-3890	60	12	efficient	efficient	ADJ
ejpam-3890	60	13	zero	zero	NUM
ejpam-3890	60	14	ring	ring	NOUN
ejpam-3890	60	15	labeling	labeling	NOUN
ejpam-3890	60	16	.	.	PUNCT
ejpam-3890	61	1	3	3	X
ejpam-3890	61	2	.	.	X
ejpam-3890	61	3	families	family	NOUN
ejpam-3890	61	4	of	of	ADP
ejpam-3890	61	5	graphs	graph	NOUN
ejpam-3890	61	6	with	with	ADP
ejpam-3890	61	7	an	an	DET
ejpam-3890	61	8	efficient	efficient	ADJ
ejpam-3890	61	9	zero	zero	NUM
ejpam-3890	61	10	ring	ring	NOUN
ejpam-3890	61	11	labeling	labeling	NOUN
ejpam-3890	61	12	in	in	ADP
ejpam-3890	61	13	this	this	DET
ejpam-3890	61	14	section	section	NOUN
ejpam-3890	61	15	,	,	PUNCT
ejpam-3890	61	16	we	we	PRON
ejpam-3890	61	17	show	show	VERB
ejpam-3890	61	18	that	that	SCONJ
ejpam-3890	61	19	several	several	ADJ
ejpam-3890	61	20	common	common	ADJ
ejpam-3890	61	21	families	family	NOUN
ejpam-3890	61	22	of	of	ADP
ejpam-3890	61	23	graphs	graph	NOUN
ejpam-3890	61	24	admit	admit	VERB
ejpam-3890	61	25	an	an	DET
ejpam-3890	61	26	efficient	efficient	ADJ
ejpam-3890	61	27	zero	zero	NUM
ejpam-3890	61	28	ring	ring	NOUN
ejpam-3890	61	29	labeling	labeling	NOUN
ejpam-3890	61	30	by	by	ADP
ejpam-3890	61	31	explicitly	explicitly	ADV
ejpam-3890	61	32	providing	provide	VERB
ejpam-3890	61	33	a	a	DET
ejpam-3890	61	34	labeling	labeling	NOUN
ejpam-3890	61	35	scheme	scheme	NOUN
ejpam-3890	61	36	.	.	PUNCT
ejpam-3890	62	1	in	in	ADP
ejpam-3890	62	2	[	[	X
ejpam-3890	62	3	5	5	NUM
ejpam-3890	62	4	]	]	PUNCT
ejpam-3890	62	5	,	,	PUNCT
ejpam-3890	62	6	it	it	PRON
ejpam-3890	62	7	was	be	AUX
ejpam-3890	62	8	shown	show	VERB
ejpam-3890	62	9	that	that	SCONJ
ejpam-3890	62	10	several	several	ADJ
ejpam-3890	62	11	families	family	NOUN
ejpam-3890	62	12	of	of	ADP
ejpam-3890	62	13	graphs	graph	NOUN
ejpam-3890	62	14	admit	admit	VERB
ejpam-3890	62	15	an	an	DET
ejpam-3890	62	16	efficient	efficient	ADJ
ejpam-3890	62	17	zero	zero	NUM
ejpam-3890	62	18	ring	ring	NOUN
ejpam-3890	62	19	labeling	labeling	NOUN
ejpam-3890	62	20	;	;	PUNCT
ejpam-3890	62	21	however	however	ADV
ejpam-3890	62	22	,	,	PUNCT
ejpam-3890	62	23	some	some	DET
ejpam-3890	62	24	labeling	labeling	NOUN
ejpam-3890	62	25	schemes	scheme	NOUN
ejpam-3890	62	26	were	be	AUX
ejpam-3890	62	27	not	not	PART
ejpam-3890	62	28	explicitly	explicitly	ADV
ejpam-3890	62	29	shown	show	VERB
ejpam-3890	62	30	,	,	PUNCT
ejpam-3890	62	31	particularly	particularly	ADV
ejpam-3890	62	32	for	for	ADP
ejpam-3890	62	33	paths	path	NOUN
ejpam-3890	62	34	and	and	CCONJ
ejpam-3890	62	35	complete	complete	ADJ
ejpam-3890	62	36	graphs	graph	NOUN
ejpam-3890	62	37	.	.	PUNCT
ejpam-3890	63	1	one	one	NUM
ejpam-3890	63	2	useful	useful	ADJ
ejpam-3890	63	3	result	result	NOUN
ejpam-3890	63	4	in	in	ADP
ejpam-3890	63	5	[	[	X
ejpam-3890	63	6	5	5	NUM
ejpam-3890	63	7	]	]	PUNCT
ejpam-3890	63	8	that	that	PRON
ejpam-3890	63	9	can	can	AUX
ejpam-3890	63	10	be	be	AUX
ejpam-3890	63	11	used	use	VERB
ejpam-3890	63	12	to	to	PART
ejpam-3890	63	13	show	show	VERB
ejpam-3890	63	14	that	that	SCONJ
ejpam-3890	63	15	a	a	DET
ejpam-3890	63	16	graph	graph	NOUN
ejpam-3890	63	17	γ	γ	PROPN
ejpam-3890	63	18	admits	admit	VERB
ejpam-3890	63	19	an	an	DET
ejpam-3890	63	20	efficient	efficient	ADJ
ejpam-3890	63	21	zero	zero	NUM
ejpam-3890	63	22	ring	ring	NOUN
ejpam-3890	63	23	labeling	labeling	NOUN
ejpam-3890	63	24	is	be	AUX
ejpam-3890	63	25	shown	show	VERB
ejpam-3890	63	26	in	in	ADP
ejpam-3890	63	27	the	the	DET
ejpam-3890	63	28	following	follow	VERB
ejpam-3890	63	29	theorem	theorem	PROPN
ejpam-3890	63	30	.	.	PUNCT
ejpam-3890	63	31	theorem	theorem	NOUN
ejpam-3890	63	32	2	2	NUM
ejpam-3890	63	33	.	.	PUNCT
ejpam-3890	64	1	[	[	X
ejpam-3890	64	2	5	5	NUM
ejpam-3890	64	3	]	]	PUNCT
ejpam-3890	64	4	.	.	PUNCT
ejpam-3890	65	1	if	if	SCONJ
ejpam-3890	65	2	γ	γ	X
ejpam-3890	65	3	is	be	AUX
ejpam-3890	65	4	a	a	DET
ejpam-3890	65	5	graph	graph	NOUN
ejpam-3890	65	6	that	that	PRON
ejpam-3890	65	7	admits	admit	VERB
ejpam-3890	65	8	an	an	DET
ejpam-3890	65	9	efficient	efficient	ADJ
ejpam-3890	65	10	zero	zero	NUM
ejpam-3890	65	11	ring	ring	NOUN
ejpam-3890	65	12	labeling	labeling	NOUN
ejpam-3890	65	13	,	,	PUNCT
ejpam-3890	65	14	then	then	ADV
ejpam-3890	65	15	any	any	DET
ejpam-3890	65	16	edge	edge	NOUN
ejpam-3890	65	17	induced	induce	VERB
ejpam-3890	65	18	subgraph	subgraph	NOUN
ejpam-3890	65	19	γ′	γ′	PROPN
ejpam-3890	65	20	of	of	ADP
ejpam-3890	65	21	γ	γ	PROPN
ejpam-3890	65	22	such	such	ADJ
ejpam-3890	65	23	that	that	DET
ejpam-3890	65	24	∆(γ′	∆(γ′	PROPN
ejpam-3890	65	25	)	)	PUNCT
ejpam-3890	65	26	=	=	SYM
ejpam-3890	65	27	∆(γ	∆(γ	X
ejpam-3890	65	28	)	)	PUNCT
ejpam-3890	65	29	has	have	VERB
ejpam-3890	65	30	an	an	DET
ejpam-3890	65	31	efficient	efficient	ADJ
ejpam-3890	65	32	zero	zero	NUM
ejpam-3890	65	33	ring	ring	NOUN
ejpam-3890	65	34	labeling	labeling	NOUN
ejpam-3890	65	35	.	.	PUNCT
ejpam-3890	66	1	in	in	ADP
ejpam-3890	66	2	[	[	X
ejpam-3890	66	3	5	5	NUM
ejpam-3890	66	4	]	]	PUNCT
ejpam-3890	66	5	,	,	PUNCT
ejpam-3890	66	6	it	it	PRON
ejpam-3890	66	7	was	be	AUX
ejpam-3890	66	8	shown	show	VERB
ejpam-3890	66	9	that	that	SCONJ
ejpam-3890	66	10	the	the	DET
ejpam-3890	66	11	path	path	NOUN
ejpam-3890	66	12	pn	pn	PROPN
ejpam-3890	66	13	admits	admit	VERB
ejpam-3890	66	14	an	an	DET
ejpam-3890	66	15	efficient	efficient	ADJ
ejpam-3890	66	16	zero	zero	NUM
ejpam-3890	66	17	ring	ring	NOUN
ejpam-3890	66	18	labeling	labeling	NOUN
ejpam-3890	66	19	,	,	PUNCT
ejpam-3890	66	20	which	which	PRON
ejpam-3890	66	21	is	be	AUX
ejpam-3890	66	22	just	just	ADV
ejpam-3890	66	23	a	a	DET
ejpam-3890	66	24	corollary	corollary	NOUN
ejpam-3890	66	25	to	to	ADP
ejpam-3890	66	26	the	the	DET
ejpam-3890	66	27	labeling	labeling	NOUN
ejpam-3890	66	28	scheme	scheme	NOUN
ejpam-3890	66	29	used	use	VERB
ejpam-3890	66	30	for	for	ADP
ejpam-3890	66	31	caterpillars	caterpillar	NOUN
ejpam-3890	66	32	.	.	PUNCT
ejpam-3890	67	1	we	we	PRON
ejpam-3890	67	2	now	now	ADV
ejpam-3890	67	3	give	give	VERB
ejpam-3890	67	4	a	a	DET
ejpam-3890	67	5	different	different	ADJ
ejpam-3890	67	6	and	and	CCONJ
ejpam-3890	67	7	explicit	explicit	ADJ
ejpam-3890	67	8	labeling	labeling	NOUN
ejpam-3890	67	9	scheme	scheme	NOUN
ejpam-3890	67	10	for	for	ADP
ejpam-3890	67	11	paths	path	NOUN
ejpam-3890	67	12	as	as	SCONJ
ejpam-3890	67	13	stated	state	VERB
ejpam-3890	67	14	in	in	ADP
ejpam-3890	67	15	the	the	DET
ejpam-3890	67	16	following	follow	VERB
ejpam-3890	67	17	theorem	theorem	PROPN
ejpam-3890	67	18	.	.	PUNCT
ejpam-3890	67	19	theorem	theorem	NOUN
ejpam-3890	67	20	3	3	NUM
ejpam-3890	67	21	.	.	PUNCT
ejpam-3890	68	1	the	the	DET
ejpam-3890	68	2	path	path	NOUN
ejpam-3890	68	3	pn	pn	PROPN
ejpam-3890	68	4	,	,	PUNCT
ejpam-3890	68	5	n	n	PRON
ejpam-3890	68	6	≥	≥	NOUN
ejpam-3890	68	7	1	1	NUM
ejpam-3890	68	8	is	be	AUX
ejpam-3890	68	9	an	an	DET
ejpam-3890	68	10	integer	integer	NOUN
ejpam-3890	68	11	,	,	PUNCT
ejpam-3890	68	12	admits	admit	VERB
ejpam-3890	68	13	an	an	DET
ejpam-3890	68	14	efficient	efficient	ADJ
ejpam-3890	68	15	zero	zero	NUM
ejpam-3890	68	16	ring	ring	NOUN
ejpam-3890	68	17	labeling	labeling	NOUN
ejpam-3890	68	18	.	.	PUNCT
ejpam-3890	69	1	proof	proof	NOUN
ejpam-3890	69	2	.	.	PUNCT
ejpam-3890	70	1	let	let	VERB
ejpam-3890	70	2	v0	v0	NOUN
ejpam-3890	70	3	,	,	PUNCT
ejpam-3890	70	4	v1	v1	NOUN
ejpam-3890	70	5	,	,	PUNCT
ejpam-3890	70	6	·	·	PUNCT
ejpam-3890	70	7	·	·	PUNCT
ejpam-3890	70	8	·	·	PUNCT
ejpam-3890	70	9	,	,	PUNCT
ejpam-3890	70	10	vn−1	vn−1	ADV
ejpam-3890	70	11	be	be	VERB
ejpam-3890	70	12	the	the	DET
ejpam-3890	70	13	vertices	vertex	NOUN
ejpam-3890	70	14	of	of	ADP
ejpam-3890	70	15	a	a	DET
ejpam-3890	70	16	path	path	NOUN
ejpam-3890	71	1	pn	pn	INTJ
ejpam-3890	71	2	whose	whose	DET
ejpam-3890	71	3	edges	edge	NOUN
ejpam-3890	71	4	are	be	AUX
ejpam-3890	71	5	of	of	ADP
ejpam-3890	71	6	the	the	DET
ejpam-3890	71	7	form	form	NOUN
ejpam-3890	71	8	(	(	PUNCT
ejpam-3890	71	9	vi	vi	NOUN
ejpam-3890	71	10	,	,	PUNCT
ejpam-3890	71	11	vi+1	vi+1	NOUN
ejpam-3890	71	12	)	)	PUNCT
ejpam-3890	71	13	where	where	SCONJ
ejpam-3890	71	14	i	i	PRON
ejpam-3890	71	15	=	=	NOUN
ejpam-3890	71	16	0	0	NUM
ejpam-3890	71	17	,	,	PUNCT
ejpam-3890	71	18	1	1	NUM
ejpam-3890	71	19	,	,	PUNCT
ejpam-3890	71	20	.	.	PUNCT
ejpam-3890	71	21	.	.	PUNCT
ejpam-3890	72	1	.	.	PUNCT
ejpam-3890	73	1	,	,	PUNCT
ejpam-3890	74	1	n	n	CCONJ
ejpam-3890	74	2	−	−	PROPN
ejpam-3890	74	3	2	2	X
ejpam-3890	74	4	.	.	X
ejpam-3890	75	1	we	we	PRON
ejpam-3890	75	2	consider	consider	VERB
ejpam-3890	75	3	the	the	DET
ejpam-3890	75	4	case	case	NOUN
ejpam-3890	75	5	when	when	SCONJ
ejpam-3890	75	6	n	n	PRON
ejpam-3890	75	7	is	be	AUX
ejpam-3890	75	8	even	even	ADV
ejpam-3890	75	9	or	or	CCONJ
ejpam-3890	75	10	odd	odd	ADJ
ejpam-3890	75	11	.	.	PUNCT
ejpam-3890	76	1	let	let	VERB
ejpam-3890	76	2	ai	ai	AUX
ejpam-3890	76	3	denote	denote	VERB
ejpam-3890	76	4	the	the	DET
ejpam-3890	76	5	matrix	matrix	NOUN
ejpam-3890	76	6	[	[	PUNCT
ejpam-3890	76	7	i	i	PRON
ejpam-3890	76	8	−i	−i	PROPN
ejpam-3890	76	9	i	i	PRON
ejpam-3890	76	10	−i	−i	ADV
ejpam-3890	76	11	]	]	PUNCT
ejpam-3890	77	1	∈m0	∈m0	ADJ
ejpam-3890	77	2	2	2	NUM
ejpam-3890	77	3	(	(	PUNCT
ejpam-3890	77	4	zn	zn	NUM
ejpam-3890	77	5	)	)	PUNCT
ejpam-3890	77	6	.	.	PUNCT
ejpam-3890	78	1	case	case	NOUN
ejpam-3890	78	2	1	1	X
ejpam-3890	78	3	.	.	PUNCT
ejpam-3890	79	1	let	let	VERB
ejpam-3890	79	2	n	n	PRON
ejpam-3890	79	3	>	>	X
ejpam-3890	79	4	1	1	NUM
ejpam-3890	79	5	be	be	AUX
ejpam-3890	79	6	even	even	ADV
ejpam-3890	79	7	.	.	PUNCT
ejpam-3890	80	1	define	define	VERB
ejpam-3890	80	2	a	a	DET
ejpam-3890	80	3	function	function	NOUN
ejpam-3890	80	4	f	f	NOUN
ejpam-3890	80	5	:	:	PUNCT
ejpam-3890	80	6	v	v	X
ejpam-3890	80	7	(	(	PUNCT
ejpam-3890	80	8	pn)→m0	pn)→m0	PROPN
ejpam-3890	80	9	2	2	NUM
ejpam-3890	80	10	(	(	PUNCT
ejpam-3890	80	11	zn	zn	NOUN
ejpam-3890	80	12	)	)	PUNCT
ejpam-3890	80	13	such	such	ADJ
ejpam-3890	80	14	that	that	SCONJ
ejpam-3890	80	15	f(vi	f(vi	NOUN
ejpam-3890	80	16	)	)	PUNCT
ejpam-3890	81	1	=	=	PRON
ejpam-3890	81	2	{	{	PUNCT
ejpam-3890	81	3	ai	ai	VERB
ejpam-3890	81	4	when	when	SCONJ
ejpam-3890	81	5	i	i	PRON
ejpam-3890	81	6	=	=	VERB
ejpam-3890	81	7	1	1	NUM
ejpam-3890	81	8	or	or	CCONJ
ejpam-3890	81	9	i	i	PRON
ejpam-3890	81	10	is	be	AUX
ejpam-3890	81	11	even	even	ADV
ejpam-3890	81	12	an+2−i	an+2−i	ADJ
ejpam-3890	81	13	if	if	SCONJ
ejpam-3890	81	14	i	i	PRON
ejpam-3890	81	15	is	be	AUX
ejpam-3890	81	16	odd	odd	ADJ
ejpam-3890	81	17	and	and	CCONJ
ejpam-3890	81	18	i	i	PRON
ejpam-3890	81	19	≥	≥	VERB
ejpam-3890	81	20	3	3	NUM
ejpam-3890	81	21	.	.	PUNCT
ejpam-3890	82	1	then	then	ADV
ejpam-3890	82	2	f	f	PROPN
ejpam-3890	82	3	is	be	AUX
ejpam-3890	82	4	an	an	DET
ejpam-3890	82	5	injective	injective	ADJ
ejpam-3890	82	6	function	function	NOUN
ejpam-3890	82	7	and	and	CCONJ
ejpam-3890	82	8	f(v0	f(v0	ADJ
ejpam-3890	82	9	)	)	PUNCT
ejpam-3890	83	1	+	+	SYM
ejpam-3890	83	2	f(v1	f(v1	ADJ
ejpam-3890	83	3	)	)	PUNCT
ejpam-3890	83	4	and	and	CCONJ
ejpam-3890	83	5	f(v1	f(v1	ADJ
ejpam-3890	83	6	)	)	PUNCT
ejpam-3890	83	7	+	+	SYM
ejpam-3890	83	8	f(v2	f(v2	NOUN
ejpam-3890	83	9	)	)	PUNCT
ejpam-3890	83	10	are	be	AUX
ejpam-3890	83	11	equal	equal	ADJ
ejpam-3890	83	12	to	to	ADP
ejpam-3890	83	13	a1	a1	NOUN
ejpam-3890	83	14	and	and	CCONJ
ejpam-3890	83	15	a3	a3	NOUN
ejpam-3890	83	16	,	,	PUNCT
ejpam-3890	83	17	respectively	respectively	ADV
ejpam-3890	83	18	.	.	PUNCT
ejpam-3890	84	1	furthermore	furthermore	ADV
ejpam-3890	84	2	,	,	PUNCT
ejpam-3890	84	3	we	we	PRON
ejpam-3890	84	4	have	have	VERB
ejpam-3890	84	5	f(vi	f(vi	NOUN
ejpam-3890	84	6	)	)	PUNCT
ejpam-3890	84	7	+	+	NUM
ejpam-3890	84	8	f(vi+1	f(vi+1	X
ejpam-3890	84	9	)	)	PUNCT
ejpam-3890	85	1	=	=	PRON
ejpam-3890	85	2	{	{	PUNCT
ejpam-3890	85	3	a1	a1	NOUN
ejpam-3890	85	4	when	when	SCONJ
ejpam-3890	85	5	i	i	PRON
ejpam-3890	85	6	is	be	AUX
ejpam-3890	85	7	even	even	ADV
ejpam-3890	85	8	,	,	PUNCT
ejpam-3890	85	9	i	i	PRON
ejpam-3890	85	10	≥	≥	NOUN
ejpam-3890	85	11	2	2	NUM
ejpam-3890	85	12	a3	a3	NOUN
ejpam-3890	85	13	if	if	SCONJ
ejpam-3890	85	14	i	i	PRON
ejpam-3890	85	15	is	be	AUX
ejpam-3890	85	16	odd	odd	ADJ
ejpam-3890	85	17	,	,	PUNCT
ejpam-3890	85	18	i	i	PRON
ejpam-3890	85	19	≥	≥	VERB
ejpam-3890	85	20	3	3	NUM
ejpam-3890	85	21	f.j	f.j	PROPN
ejpam-3890	85	22	.	.	PROPN
ejpam-3890	86	1	campeña	campeña	PROPN
ejpam-3890	86	2	et	et	PROPN
ejpam-3890	86	3	al	al	PROPN
ejpam-3890	86	4	.	.	PUNCT
ejpam-3890	86	5	/	/	SYM
ejpam-3890	86	6	eur	eur	PROPN
ejpam-3890	86	7	.	.	PUNCT
ejpam-3890	87	1	j.	j.	PROPN
ejpam-3890	87	2	pure	pure	PROPN
ejpam-3890	87	3	appl	appl	PROPN
ejpam-3890	87	4	.	.	PROPN
ejpam-3890	87	5	math	math	PROPN
ejpam-3890	87	6	,	,	PUNCT
ejpam-3890	87	7	14	14	NUM
ejpam-3890	87	8	(	(	PUNCT
ejpam-3890	87	9	1	1	NUM
ejpam-3890	87	10	)	)	PUNCT
ejpam-3890	87	11	(	(	PUNCT
ejpam-3890	87	12	2021	2021	NUM
ejpam-3890	87	13	)	)	PUNCT
ejpam-3890	87	14	,	,	PUNCT
ejpam-3890	87	15	268	268	NUM
ejpam-3890	87	16	-	-	SYM
ejpam-3890	87	17	277	277	NUM
ejpam-3890	87	18	271	271	NUM
ejpam-3890	87	19	which	which	PRON
ejpam-3890	87	20	shows	show	VERB
ejpam-3890	87	21	that	that	SCONJ
ejpam-3890	87	22	f(vi	f(vi	NOUN
ejpam-3890	87	23	)	)	PUNCT
ejpam-3890	87	24	+	+	NUM
ejpam-3890	87	25	f(vi+1	f(vi+1	PROPN
ejpam-3890	87	26	)	)	PUNCT
ejpam-3890	87	27	6=	6=	NUM
ejpam-3890	87	28	a0	a0	PROPN
ejpam-3890	87	29	for	for	ADP
ejpam-3890	87	30	all	all	DET
ejpam-3890	87	31	i	i	NOUN
ejpam-3890	87	32	=	=	NOUN
ejpam-3890	87	33	0	0	NUM
ejpam-3890	87	34	,	,	PUNCT
ejpam-3890	87	35	1	1	NUM
ejpam-3890	87	36	,	,	PUNCT
ejpam-3890	87	37	·	·	PUNCT
ejpam-3890	87	38	·	·	PUNCT
ejpam-3890	87	39	·	·	PUNCT
ejpam-3890	87	40	,	,	PUNCT
ejpam-3890	87	41	n−	n−	NOUN
ejpam-3890	87	42	2	2	NUM
ejpam-3890	87	43	.	.	PUNCT
ejpam-3890	88	1	since	since	SCONJ
ejpam-3890	88	2	k	k	PROPN
ejpam-3890	88	3	=	=	PUNCT
ejpam-3890	88	4	{	{	PUNCT
ejpam-3890	88	5	f(vi	f(vi	PROPN
ejpam-3890	88	6	)	)	PUNCT
ejpam-3890	88	7	+	+	NUM
ejpam-3890	88	8	f(vi+1|i	f(vi+1|i	NOUN
ejpam-3890	88	9	=	=	SYM
ejpam-3890	88	10	0	0	NUM
ejpam-3890	88	11	,	,	PUNCT
ejpam-3890	88	12	1	1	NUM
ejpam-3890	88	13	,	,	PUNCT
ejpam-3890	88	14	·	·	PUNCT
ejpam-3890	88	15	·	·	PUNCT
ejpam-3890	88	16	·	·	PUNCT
ejpam-3890	88	17	,	,	PUNCT
ejpam-3890	88	18	n−	n−	NOUN
ejpam-3890	88	19	2	2	NUM
ejpam-3890	88	20	}	}	PUNCT
ejpam-3890	88	21	=	=	SYM
ejpam-3890	88	22	{	{	PUNCT
ejpam-3890	88	23	a1	a1	NOUN
ejpam-3890	88	24	,	,	PUNCT
ejpam-3890	88	25	a3	a3	NOUN
ejpam-3890	88	26	}	}	PUNCT
ejpam-3890	88	27	,	,	PUNCT
ejpam-3890	88	28	f	f	PROPN
ejpam-3890	88	29	is	be	AUX
ejpam-3890	88	30	a	a	DET
ejpam-3890	88	31	2	2	NUM
ejpam-3890	88	32	-	-	SYM
ejpam-3890	88	33	zero	zero	NUM
ejpam-3890	88	34	ring	ring	NOUN
ejpam-3890	88	35	labeling	labeling	NOUN
ejpam-3890	88	36	of	of	ADP
ejpam-3890	88	37	pn	pn	PROPN
ejpam-3890	88	38	.	.	PROPN
ejpam-3890	88	39	case	case	NOUN
ejpam-3890	89	1	2	2	X
ejpam-3890	89	2	.	.	PUNCT
ejpam-3890	90	1	let	let	VERB
ejpam-3890	90	2	n	n	PRON
ejpam-3890	90	3	be	be	AUX
ejpam-3890	90	4	odd	odd	ADJ
ejpam-3890	90	5	.	.	PUNCT
ejpam-3890	91	1	define	define	VERB
ejpam-3890	91	2	a	a	DET
ejpam-3890	91	3	function	function	NOUN
ejpam-3890	91	4	f	f	NOUN
ejpam-3890	91	5	:	:	PUNCT
ejpam-3890	91	6	v	v	X
ejpam-3890	91	7	(	(	PUNCT
ejpam-3890	91	8	pn)→m0	pn)→m0	PROPN
ejpam-3890	91	9	2	2	NUM
ejpam-3890	91	10	(	(	PUNCT
ejpam-3890	91	11	zn	zn	NOUN
ejpam-3890	91	12	)	)	PUNCT
ejpam-3890	91	13	such	such	ADJ
ejpam-3890	91	14	that	that	SCONJ
ejpam-3890	91	15	when	when	SCONJ
ejpam-3890	91	16	n	n	PRON
ejpam-3890	91	17	is	be	AUX
ejpam-3890	91	18	of	of	ADP
ejpam-3890	91	19	the	the	DET
ejpam-3890	91	20	form	form	NOUN
ejpam-3890	91	21	4k	4k	NOUN
ejpam-3890	91	22	+	+	CCONJ
ejpam-3890	91	23	1	1	NUM
ejpam-3890	91	24	,	,	PUNCT
ejpam-3890	91	25	k	k	PROPN
ejpam-3890	91	26	∈	∈	PROPN
ejpam-3890	91	27	z+	z+	NUM
ejpam-3890	91	28	f(vi	f(vi	NOUN
ejpam-3890	91	29	)	)	PUNCT
ejpam-3890	91	30	=	=	PUNCT
ejpam-3890	91	31			NOUN
ejpam-3890	91	32	abn	abn	VERB
ejpam-3890	91	33	2	2	NUM
ejpam-3890	91	34	c−i+1	c−i+1	NOUN
ejpam-3890	91	35	for	for	ADP
ejpam-3890	91	36	i	i	PRON
ejpam-3890	91	37	=	=	SYM
ejpam-3890	91	38	0	0	NUM
ejpam-3890	91	39	,	,	PUNCT
ejpam-3890	91	40	2	2	NUM
ejpam-3890	91	41	,	,	PUNCT
ejpam-3890	91	42	·	·	PUNCT
ejpam-3890	91	43	·	·	PUNCT
ejpam-3890	91	44	·	·	PUNCT
ejpam-3890	91	45	,	,	PUNCT
ejpam-3890	91	46	bn2	bn2	NOUN
ejpam-3890	91	47	c	c	NOUN
ejpam-3890	91	48	−	−	PROPN
ejpam-3890	91	49	2	2	NUM
ejpam-3890	91	50	,	,	PUNCT
ejpam-3890	91	51	bn2	bn2	NOUN
ejpam-3890	91	52	c	c	NOUN
ejpam-3890	91	53	an+i−bn	an+i−bn	X
ejpam-3890	91	54	2	2	NUM
ejpam-3890	91	55	c+1	c+1	PROPN
ejpam-3890	91	56	for	for	ADP
ejpam-3890	91	57	i	i	PRON
ejpam-3890	91	58	=	=	NOUN
ejpam-3890	91	59	1	1	NUM
ejpam-3890	91	60	,	,	PUNCT
ejpam-3890	91	61	3	3	NUM
ejpam-3890	91	62	,	,	PUNCT
ejpam-3890	91	63	·	·	PUNCT
ejpam-3890	91	64	·	·	PUNCT
ejpam-3890	91	65	·	·	PUNCT
ejpam-3890	91	66	,	,	PUNCT
ejpam-3890	91	67	bn2	bn2	NOUN
ejpam-3890	91	68	c	c	NOUN
ejpam-3890	91	69	−	−	PROPN
ejpam-3890	91	70	5	5	NUM
ejpam-3890	91	71	,	,	PUNCT
ejpam-3890	91	72	bn2	bn2	NOUN
ejpam-3890	91	73	c	c	NOUN
ejpam-3890	91	74	−	−	PROPN
ejpam-3890	91	75	3	3	NUM
ejpam-3890	91	76	ai−bn	ai−bn	NOUN
ejpam-3890	91	77	2	2	NUM
ejpam-3890	91	78	c+1	c+1	NOUN
ejpam-3890	91	79	for	for	ADP
ejpam-3890	91	80	i	i	PRON
ejpam-3890	91	81	=	=	PUNCT
ejpam-3890	91	82	bn2	bn2	VERB
ejpam-3890	92	1	c	c	NOUN
ejpam-3890	92	2	−	−	PROPN
ejpam-3890	92	3	1	1	NUM
ejpam-3890	92	4	,	,	PUNCT
ejpam-3890	92	5	bn2	bn2	NOUN
ejpam-3890	92	6	c+	c+	VERB
ejpam-3890	92	7	1	1	NUM
ejpam-3890	92	8	,	,	PUNCT
ejpam-3890	92	9	bn2	bn2	NOUN
ejpam-3890	92	10	c+	c+	VERB
ejpam-3890	92	11	3	3	NUM
ejpam-3890	92	12	·	·	PUNCT
ejpam-3890	92	13	·	·	PUNCT
ejpam-3890	92	14	·	·	PUNCT
ejpam-3890	92	15	,	,	PUNCT
ejpam-3890	92	16	n−	n−	NOUN
ejpam-3890	92	17	2	2	NUM
ejpam-3890	92	18	an−i+bn	an−i+bn	ADP
ejpam-3890	92	19	2	2	NUM
ejpam-3890	92	20	c+1	c+1	PROPN
ejpam-3890	92	21	for	for	ADP
ejpam-3890	92	22	i	i	PRON
ejpam-3890	92	23	=	=	PUNCT
ejpam-3890	92	24	bn2	bn2	NOUN
ejpam-3890	92	25	c+	c+	VERB
ejpam-3890	92	26	2	2	NUM
ejpam-3890	92	27	,	,	PUNCT
ejpam-3890	92	28	bn2	bn2	NOUN
ejpam-3890	92	29	c+	c+	VERB
ejpam-3890	92	30	4	4	NUM
ejpam-3890	92	31	,	,	PUNCT
ejpam-3890	92	32	·	·	PUNCT
ejpam-3890	92	33	·	·	PUNCT
ejpam-3890	92	34	·	·	PUNCT
ejpam-3890	92	35	,	,	PUNCT
ejpam-3890	92	36	n−	n−	NOUN
ejpam-3890	92	37	1	1	NUM
ejpam-3890	92	38	and	and	CCONJ
ejpam-3890	92	39	when	when	SCONJ
ejpam-3890	92	40	n	n	X
ejpam-3890	92	41	is	be	AUX
ejpam-3890	92	42	of	of	ADP
ejpam-3890	92	43	the	the	DET
ejpam-3890	92	44	form	form	NOUN
ejpam-3890	92	45	4k	4k	NOUN
ejpam-3890	92	46	+	+	CCONJ
ejpam-3890	92	47	3	3	NUM
ejpam-3890	92	48	,	,	PUNCT
ejpam-3890	92	49	k	k	PROPN
ejpam-3890	92	50	∈	∈	PROPN
ejpam-3890	92	51	z+	z+	NUM
ejpam-3890	92	52	f(vi	f(vi	NOUN
ejpam-3890	92	53	)	)	PUNCT
ejpam-3890	92	54	=	=	PUNCT
ejpam-3890	92	55			NOUN
ejpam-3890	92	56	an+i−bn	an+i−bn	PUNCT
ejpam-3890	92	57	2	2	NUM
ejpam-3890	92	58	c+1	c+1	PROPN
ejpam-3890	92	59	for	for	ADP
ejpam-3890	92	60	i	i	PRON
ejpam-3890	92	61	=	=	SYM
ejpam-3890	92	62	0	0	NUM
ejpam-3890	92	63	,	,	PUNCT
ejpam-3890	92	64	2	2	NUM
ejpam-3890	92	65	,	,	PUNCT
ejpam-3890	92	66	·	·	PUNCT
ejpam-3890	92	67	·	·	PUNCT
ejpam-3890	92	68	·	·	PUNCT
ejpam-3890	92	69	,	,	PUNCT
ejpam-3890	92	70	bn2	bn2	NOUN
ejpam-3890	92	71	c	c	NOUN
ejpam-3890	92	72	−	−	PROPN
ejpam-3890	92	73	5	5	NUM
ejpam-3890	92	74	,	,	PUNCT
ejpam-3890	92	75	bn2	bn2	NOUN
ejpam-3890	92	76	c	c	NOUN
ejpam-3890	92	77	−	−	PROPN
ejpam-3890	92	78	3	3	NUM
ejpam-3890	92	79	abn	abn	NOUN
ejpam-3890	92	80	2	2	NUM
ejpam-3890	92	81	c−i+1	c−i+1	NOUN
ejpam-3890	92	82	for	for	ADP
ejpam-3890	92	83	i	i	PRON
ejpam-3890	92	84	=	=	SYM
ejpam-3890	92	85	1	1	NUM
ejpam-3890	92	86	,	,	PUNCT
ejpam-3890	92	87	3	3	NUM
ejpam-3890	92	88	·	·	PUNCT
ejpam-3890	92	89	·	·	PUNCT
ejpam-3890	92	90	·	·	PUNCT
ejpam-3890	92	91	,	,	PUNCT
ejpam-3890	92	92	bn2	bn2	VERB
ejpam-3890	92	93	c	c	X
ejpam-3890	92	94	,	,	PUNCT
ejpam-3890	92	95	b	b	PROPN
ejpam-3890	92	96	n	n	NUM
ejpam-3890	92	97	2	2	NUM
ejpam-3890	92	98	c	c	NOUN
ejpam-3890	92	99	−	−	PROPN
ejpam-3890	92	100	4	4	NUM
ejpam-3890	92	101	,	,	PUNCT
ejpam-3890	92	102	bn2	bn2	NOUN
ejpam-3890	92	103	c	c	NOUN
ejpam-3890	92	104	−	−	PROPN
ejpam-3890	92	105	2	2	NUM
ejpam-3890	92	106	ai−bn	ai−bn	NOUN
ejpam-3890	92	107	2	2	NUM
ejpam-3890	92	108	c+1	c+1	NOUN
ejpam-3890	92	109	for	for	ADP
ejpam-3890	92	110	i	i	PRON
ejpam-3890	92	111	=	=	PUNCT
ejpam-3890	92	112	bn2	bn2	VERB
ejpam-3890	93	1	c	c	NOUN
ejpam-3890	93	2	−	−	PROPN
ejpam-3890	93	3	1	1	NUM
ejpam-3890	93	4	,	,	PUNCT
ejpam-3890	93	5	bn2	bn2	NOUN
ejpam-3890	93	6	c+	c+	VERB
ejpam-3890	93	7	1	1	NUM
ejpam-3890	93	8	,	,	PUNCT
ejpam-3890	93	9	bn2	bn2	NOUN
ejpam-3890	93	10	c+	c+	VERB
ejpam-3890	93	11	3	3	NUM
ejpam-3890	93	12	·	·	PUNCT
ejpam-3890	93	13	·	·	PUNCT
ejpam-3890	93	14	·	·	PUNCT
ejpam-3890	93	15	,	,	PUNCT
ejpam-3890	93	16	n−	n−	NOUN
ejpam-3890	93	17	1	1	NUM
ejpam-3890	93	18	an−i+bn	an−i+bn	ADP
ejpam-3890	93	19	2	2	NUM
ejpam-3890	93	20	c+1	c+1	NOUN
ejpam-3890	93	21	for	for	ADP
ejpam-3890	93	22	i	i	PRON
ejpam-3890	93	23	=	=	PUNCT
ejpam-3890	93	24	bn2	bn2	NOUN
ejpam-3890	93	25	c+	c+	VERB
ejpam-3890	93	26	2	2	NUM
ejpam-3890	93	27	,	,	PUNCT
ejpam-3890	93	28	bn2	bn2	NOUN
ejpam-3890	93	29	c+	c+	VERB
ejpam-3890	93	30	4	4	NUM
ejpam-3890	93	31	,	,	PUNCT
ejpam-3890	93	32	·	·	PUNCT
ejpam-3890	93	33	·	·	PUNCT
ejpam-3890	93	34	·	·	PUNCT
ejpam-3890	93	35	,	,	PUNCT
ejpam-3890	93	36	n−	n−	NOUN
ejpam-3890	93	37	2	2	NUM
ejpam-3890	93	38	clearly	clearly	ADV
ejpam-3890	93	39	,	,	PUNCT
ejpam-3890	93	40	f	f	PROPN
ejpam-3890	93	41	is	be	AUX
ejpam-3890	93	42	injective	injective	ADJ
ejpam-3890	93	43	for	for	ADP
ejpam-3890	93	44	both	both	DET
ejpam-3890	93	45	subcases	subcase	NOUN
ejpam-3890	93	46	.	.	PUNCT
ejpam-3890	94	1	now	now	ADV
ejpam-3890	94	2	we	we	PRON
ejpam-3890	94	3	verify	verify	VERB
ejpam-3890	94	4	that	that	SCONJ
ejpam-3890	94	5	f	f	PROPN
ejpam-3890	94	6	is	be	AUX
ejpam-3890	94	7	an	an	DET
ejpam-3890	94	8	efficient	efficient	ADJ
ejpam-3890	94	9	zero	zero	NUM
ejpam-3890	94	10	ring	ring	NOUN
ejpam-3890	94	11	labeling	labeling	NOUN
ejpam-3890	94	12	.	.	PUNCT
ejpam-3890	95	1	we	we	PRON
ejpam-3890	95	2	show	show	VERB
ejpam-3890	95	3	that	that	SCONJ
ejpam-3890	95	4	the	the	DET
ejpam-3890	95	5	sums	sum	NOUN
ejpam-3890	95	6	f(vi	f(vi	NOUN
ejpam-3890	95	7	)	)	PUNCT
ejpam-3890	95	8	+	+	NUM
ejpam-3890	95	9	f(vi+1	f(vi+1	PROPN
ejpam-3890	95	10	)	)	PUNCT
ejpam-3890	95	11	6=	6=	NUM
ejpam-3890	95	12	a0	a0	NOUN
ejpam-3890	95	13	for	for	ADP
ejpam-3890	95	14	0	0	NUM
ejpam-3890	95	15	≤	≤	NUM
ejpam-3890	95	16	i	i	PRON
ejpam-3890	95	17	≤	≤	NOUN
ejpam-3890	95	18	n	n	CCONJ
ejpam-3890	95	19	−	−	PROPN
ejpam-3890	95	20	2	2	NUM
ejpam-3890	95	21	,	,	PUNCT
ejpam-3890	95	22	that	that	ADV
ejpam-3890	95	23	is	is	ADV
ejpam-3890	95	24	,	,	PUNCT
ejpam-3890	95	25	within	within	ADP
ejpam-3890	95	26	the	the	DET
ejpam-3890	95	27	intervals	interval	NOUN
ejpam-3890	95	28	0	0	NUM
ejpam-3890	95	29	≤	≤	NUM
ejpam-3890	96	1	i	i	PRON
ejpam-3890	96	2	<	<	X
ejpam-3890	96	3	bn2	bn2	VERB
ejpam-3890	97	1	c	c	NOUN
ejpam-3890	97	2	−	−	PROPN
ejpam-3890	97	3	2	2	NUM
ejpam-3890	97	4	,	,	PUNCT
ejpam-3890	97	5	bn2	bn2	NOUN
ejpam-3890	97	6	c	c	NOUN
ejpam-3890	97	7	−	−	PROPN
ejpam-3890	97	8	2	2	NUM
ejpam-3890	97	9	≤	≤	PUNCT
ejpam-3890	97	10	i	i	PRON
ejpam-3890	97	11	<	<	X
ejpam-3890	97	12	bn2	bn2	X
ejpam-3890	97	13	c+	c+	VERB
ejpam-3890	97	14	1	1	NUM
ejpam-3890	97	15	and	and	CCONJ
ejpam-3890	97	16	bn2	bn2	NOUN
ejpam-3890	97	17	c+	c+	VERB
ejpam-3890	97	18	1	1	NUM
ejpam-3890	97	19	≤	≤	NUM
ejpam-3890	98	1	i	i	PRON
ejpam-3890	98	2	≤	≤	ADJ
ejpam-3890	98	3	n−	n−	PROPN
ejpam-3890	98	4	2	2	NUM
ejpam-3890	98	5	.	.	PUNCT
ejpam-3890	99	1	for	for	ADP
ejpam-3890	99	2	the	the	DET
ejpam-3890	99	3	interval	interval	NOUN
ejpam-3890	99	4	0	0	NUM
ejpam-3890	99	5	≤	≤	PUNCT
ejpam-3890	100	1	i	i	PRON
ejpam-3890	100	2	<	<	X
ejpam-3890	100	3	bn2	bn2	VERB
ejpam-3890	100	4	c	c	NOUN
ejpam-3890	100	5	−	−	PROPN
ejpam-3890	100	6	2	2	NUM
ejpam-3890	100	7	,	,	PUNCT
ejpam-3890	100	8	the	the	DET
ejpam-3890	100	9	sums	sum	NOUN
ejpam-3890	100	10	f(vi	f(vi	NOUN
ejpam-3890	100	11	)	)	PUNCT
ejpam-3890	100	12	+	+	NUM
ejpam-3890	100	13	f(vi+1	f(vi+1	X
ejpam-3890	100	14	)	)	PUNCT
ejpam-3890	100	15	are	be	AUX
ejpam-3890	100	16	abn	abn	NOUN
ejpam-3890	100	17	2	2	NUM
ejpam-3890	100	18	c−i+1	c−i+1	NOUN
ejpam-3890	100	19	+	+	CCONJ
ejpam-3890	100	20	an+(i+1)−bn	an+(i+1)−bn	X
ejpam-3890	100	21	2	2	NUM
ejpam-3890	100	22	c+1	c+1	PROPN
ejpam-3890	100	23	=	=	NOUN
ejpam-3890	100	24	a3	a3	NOUN
ejpam-3890	100	25	when	when	SCONJ
ejpam-3890	100	26	i	i	PRON
ejpam-3890	100	27	is	be	AUX
ejpam-3890	100	28	odd	odd	ADJ
ejpam-3890	100	29	and	and	CCONJ
ejpam-3890	100	30	n	n	PRON
ejpam-3890	100	31	+	+	CCONJ
ejpam-3890	100	32	3	3	NUM
ejpam-3890	100	33	taken	take	VERB
ejpam-3890	100	34	modulo	modulo	NOUN
ejpam-3890	100	35	n	n	CCONJ
ejpam-3890	100	36	,	,	PUNCT
ejpam-3890	100	37	and	and	CCONJ
ejpam-3890	100	38	an+i−bn	an+i−bn	X
ejpam-3890	100	39	2	2	NUM
ejpam-3890	100	40	c+1	c+1	PROPN
ejpam-3890	100	41	+	+	NOUN
ejpam-3890	100	42	abn	abn	NOUN
ejpam-3890	100	43	2	2	NUM
ejpam-3890	100	44	c−(i+1)+1	c−(i+1)+1	NOUN
ejpam-3890	100	45	=	=	PUNCT
ejpam-3890	100	46	a1	a1	NOUN
ejpam-3890	100	47	when	when	SCONJ
ejpam-3890	100	48	i	i	PRON
ejpam-3890	100	49	is	be	AUX
ejpam-3890	100	50	even	even	ADV
ejpam-3890	100	51	and	and	CCONJ
ejpam-3890	100	52	n	n	PRON
ejpam-3890	101	1	+	+	PRON
ejpam-3890	101	2	1	1	NUM
ejpam-3890	101	3	is	be	AUX
ejpam-3890	101	4	taken	take	VERB
ejpam-3890	101	5	modulo	modulo	ADJ
ejpam-3890	101	6	n.	n.	NOUN
ejpam-3890	101	7	for	for	ADP
ejpam-3890	101	8	the	the	DET
ejpam-3890	101	9	interval	interval	NOUN
ejpam-3890	101	10	bn2	bn2	NOUN
ejpam-3890	102	1	c	c	NOUN
ejpam-3890	102	2	−	−	PROPN
ejpam-3890	102	3	2	2	NUM
ejpam-3890	102	4	≤	≤	PUNCT
ejpam-3890	102	5	i	i	PRON
ejpam-3890	102	6	<	<	X
ejpam-3890	102	7	bn2	bn2	X
ejpam-3890	102	8	c+	c+	VERB
ejpam-3890	102	9	1	1	NUM
ejpam-3890	102	10	,	,	PUNCT
ejpam-3890	102	11	we	we	PRON
ejpam-3890	102	12	have	have	VERB
ejpam-3890	102	13	the	the	DET
ejpam-3890	102	14	sums	sum	NOUN
ejpam-3890	102	15	abn	abn	NOUN
ejpam-3890	102	16	2	2	NUM
ejpam-3890	102	17	c−i+1	c−i+1	NOUN
ejpam-3890	102	18	+	+	CCONJ
ejpam-3890	102	19	ai+1−bn	ai+1−bn	SYM
ejpam-3890	102	20	2	2	NUM
ejpam-3890	102	21	c+1	c+1	PROPN
ejpam-3890	102	22	=	=	NOUN
ejpam-3890	102	23	a3	a3	NOUN
ejpam-3890	102	24	when	when	SCONJ
ejpam-3890	102	25	i	i	PRON
ejpam-3890	102	26	=	=	PUNCT
ejpam-3890	102	27	bn2	bn2	VERB
ejpam-3890	102	28	c	c	NOUN
ejpam-3890	102	29	−	−	PROPN
ejpam-3890	102	30	2	2	NUM
ejpam-3890	102	31	and	and	CCONJ
ejpam-3890	102	32	i	i	PRON
ejpam-3890	102	33	=	=	PUNCT
ejpam-3890	102	34	bn2	bn2	VERB
ejpam-3890	102	35	c	c	X
ejpam-3890	102	36	,	,	PUNCT
ejpam-3890	102	37	and	and	CCONJ
ejpam-3890	102	38	when	when	SCONJ
ejpam-3890	102	39	i	i	PRON
ejpam-3890	102	40	=	=	PUNCT
ejpam-3890	103	1	bn2	bn2	VERB
ejpam-3890	103	2	c	c	NOUN
ejpam-3890	103	3	−	−	PROPN
ejpam-3890	103	4	1	1	NUM
ejpam-3890	103	5	,	,	PUNCT
ejpam-3890	103	6	ai−bn	ai−bn	X
ejpam-3890	103	7	2	2	NUM
ejpam-3890	103	8	c+1	c+1	PROPN
ejpam-3890	103	9	+	+	NOUN
ejpam-3890	103	10	abn	abn	NOUN
ejpam-3890	103	11	2	2	NUM
ejpam-3890	103	12	c−(i+1)+1	c−(i+1)+1	NOUN
ejpam-3890	103	13	=	=	PUNCT
ejpam-3890	103	14	a1	a1	PROPN
ejpam-3890	103	15	.	.	PROPN
ejpam-3890	104	1	for	for	ADP
ejpam-3890	104	2	the	the	DET
ejpam-3890	104	3	interval	interval	NOUN
ejpam-3890	104	4	bn2	bn2	NOUN
ejpam-3890	104	5	c+	c+	VERB
ejpam-3890	104	6	1	1	NUM
ejpam-3890	104	7	≤	≤	NUM
ejpam-3890	105	1	i	i	PRON
ejpam-3890	105	2	≤	≤	ADJ
ejpam-3890	105	3	n−	n−	NOUN
ejpam-3890	105	4	2	2	NUM
ejpam-3890	105	5	,	,	PUNCT
ejpam-3890	105	6	we	we	PRON
ejpam-3890	105	7	have	have	VERB
ejpam-3890	105	8	the	the	DET
ejpam-3890	105	9	sums	sum	NOUN
ejpam-3890	105	10	ai−bn	ai−bn	NOUN
ejpam-3890	105	11	2	2	NUM
ejpam-3890	105	12	c+1	c+1	NUM
ejpam-3890	105	13	+	+	CCONJ
ejpam-3890	105	14	an−(i+1)+bn	an−(i+1)+bn	PROPN
ejpam-3890	105	15	2	2	NUM
ejpam-3890	105	16	c+1	c+1	PROPN
ejpam-3890	105	17	=	=	PUNCT
ejpam-3890	105	18	a1	a1	NOUN
ejpam-3890	105	19	when	when	SCONJ
ejpam-3890	105	20	i	i	PRON
ejpam-3890	105	21	is	be	AUX
ejpam-3890	105	22	odd	odd	ADJ
ejpam-3890	105	23	and	and	CCONJ
ejpam-3890	105	24	n	n	PRON
ejpam-3890	105	25	+	+	PRON
ejpam-3890	105	26	1	1	NUM
ejpam-3890	105	27	is	be	AUX
ejpam-3890	105	28	taken	take	VERB
ejpam-3890	105	29	modulo	modulo	NOUN
ejpam-3890	105	30	n	n	CCONJ
ejpam-3890	105	31	,	,	PUNCT
ejpam-3890	105	32	and	and	CCONJ
ejpam-3890	105	33	an−i+bn	an−i+bn	X
ejpam-3890	105	34	2	2	NUM
ejpam-3890	105	35	c+1	c+1	PROPN
ejpam-3890	105	36	+	+	CCONJ
ejpam-3890	105	37	ai+1−bn	ai+1−bn	NOUN
ejpam-3890	105	38	2	2	NUM
ejpam-3890	105	39	c+1	c+1	PROPN
ejpam-3890	105	40	=	=	NOUN
ejpam-3890	105	41	a3	a3	NOUN
ejpam-3890	105	42	when	when	SCONJ
ejpam-3890	105	43	i	i	PRON
ejpam-3890	105	44	is	be	AUX
ejpam-3890	105	45	even	even	ADV
ejpam-3890	105	46	and	and	CCONJ
ejpam-3890	105	47	n	n	PRON
ejpam-3890	105	48	+	+	CCONJ
ejpam-3890	105	49	3	3	NUM
ejpam-3890	105	50	is	be	AUX
ejpam-3890	105	51	taken	take	VERB
ejpam-3890	105	52	modulo	modulo	PROPN
ejpam-3890	105	53	n.	n.	PROPN
ejpam-3890	105	54	f.j	f.j	PROPN
ejpam-3890	105	55	.	.	PROPN
ejpam-3890	106	1	campeña	campeña	PROPN
ejpam-3890	106	2	et	et	PROPN
ejpam-3890	106	3	al	al	PROPN
ejpam-3890	106	4	.	.	PUNCT
ejpam-3890	106	5	/	/	SYM
ejpam-3890	106	6	eur	eur	PROPN
ejpam-3890	106	7	.	.	PUNCT
ejpam-3890	107	1	j.	j.	PROPN
ejpam-3890	107	2	pure	pure	PROPN
ejpam-3890	107	3	appl	appl	PROPN
ejpam-3890	107	4	.	.	PROPN
ejpam-3890	107	5	math	math	PROPN
ejpam-3890	107	6	,	,	PUNCT
ejpam-3890	107	7	14	14	NUM
ejpam-3890	107	8	(	(	PUNCT
ejpam-3890	107	9	1	1	NUM
ejpam-3890	107	10	)	)	PUNCT
ejpam-3890	107	11	(	(	PUNCT
ejpam-3890	107	12	2021	2021	NUM
ejpam-3890	107	13	)	)	PUNCT
ejpam-3890	107	14	,	,	PUNCT
ejpam-3890	107	15	268	268	NUM
ejpam-3890	107	16	-	-	SYM
ejpam-3890	107	17	277	277	NUM
ejpam-3890	107	18	272	272	NUM
ejpam-3890	107	19	..........	..........	PUNCT
ejpam-3890	107	20	.............................................	.............................................	PUNCT
ejpam-3890	107	21	.	.	PUNCT
ejpam-3890	108	1	..........	..........	PUNCT
ejpam-3890	108	2	.............................................	.............................................	PUNCT
ejpam-3890	108	3	.	.	PUNCT
ejpam-3890	109	1	..........	..........	PUNCT
ejpam-3890	109	2	.............................................	.............................................	PUNCT
ejpam-3890	109	3	.	.	PUNCT
ejpam-3890	110	1	..........	..........	PUNCT
ejpam-3890	110	2	.............................................	.............................................	PUNCT
ejpam-3890	110	3	.	.	PUNCT
ejpam-3890	111	1	..........	..........	PUNCT
ejpam-3890	111	2	.............................................	.............................................	PUNCT
ejpam-3890	111	3	.	.	PUNCT
ejpam-3890	112	1	..........	..........	PUNCT
ejpam-3890	112	2	.............................................	.............................................	PUNCT
ejpam-3890	112	3	.	.	PUNCT
ejpam-3890	113	1	..........	..........	PUNCT
ejpam-3890	113	2	.............................................	.............................................	PUNCT
ejpam-3890	113	3	.	.	PUNCT
ejpam-3890	114	1	..........	..........	PUNCT
ejpam-3890	114	2	.............................................	.............................................	PUNCT
ejpam-3890	114	3	.	.	PUNCT
ejpam-3890	115	1	..........	..........	PUNCT
ejpam-3890	115	2	.............................................	.............................................	PUNCT
ejpam-3890	115	3	.	.	PUNCT
ejpam-3890	116	1	..........	..........	PUNCT
ejpam-3890	116	2	.............................................	.............................................	PUNCT
ejpam-3890	116	3	.	.	PUNCT
ejpam-3890	117	1	.........	.........	PUNCT
ejpam-3890	117	2	........	........	PUNCT
ejpam-3890	117	3	........	........	PUNCT
ejpam-3890	117	4	........	........	PUNCT
ejpam-3890	118	1	........	........	PUNCT
ejpam-3890	118	2	.	.	PUNCT
ejpam-3890	119	1	.........	.........	PUNCT
ejpam-3890	119	2	........	........	PUNCT
ejpam-3890	119	3	........	........	PUNCT
ejpam-3890	119	4	........	........	PUNCT
ejpam-3890	120	1	........	........	PUNCT
ejpam-3890	120	2	.	.	PUNCT
ejpam-3890	121	1	.........	.........	PUNCT
ejpam-3890	121	2	........	........	PUNCT
ejpam-3890	121	3	........	........	PUNCT
ejpam-3890	121	4	........	........	PUNCT
ejpam-3890	122	1	........	........	PUNCT
ejpam-3890	122	2	.	.	PUNCT
ejpam-3890	123	1	.........	.........	PUNCT
ejpam-3890	123	2	........	........	PUNCT
ejpam-3890	123	3	........	........	PUNCT
ejpam-3890	123	4	........	........	PUNCT
ejpam-3890	124	1	........	........	PUNCT
ejpam-3890	124	2	.	.	PUNCT
ejpam-3890	125	1	.........	.........	PUNCT
ejpam-3890	125	2	........	........	PUNCT
ejpam-3890	125	3	........	........	PUNCT
ejpam-3890	125	4	........	........	PUNCT
ejpam-3890	126	1	........	........	PUNCT
ejpam-3890	126	2	.	.	PUNCT
ejpam-3890	127	1	..............	..............	PUNCT
ejpam-3890	127	2	.............	.............	PUNCT
ejpam-3890	127	3	.............	.............	PUNCT
ejpam-3890	127	4	.............	.............	PUNCT
ejpam-3890	127	5	.............	.............	PUNCT
ejpam-3890	128	1	..........	..........	PUNCT
ejpam-3890	128	2	..............	..............	PUNCT
ejpam-3890	128	3	.............	.............	PUNCT
ejpam-3890	128	4	.............	.............	PUNCT
ejpam-3890	128	5	.............	.............	PUNCT
ejpam-3890	128	6	.............	.............	PUNCT
ejpam-3890	129	1	..........	..........	PUNCT
ejpam-3890	129	2	..............	..............	PUNCT
ejpam-3890	129	3	.............	.............	PUNCT
ejpam-3890	129	4	.............	.............	PUNCT
ejpam-3890	129	5	.............	.............	PUNCT
ejpam-3890	129	6	.............	.............	PUNCT
ejpam-3890	130	1	..........	..........	PUNCT
ejpam-3890	130	2	..............	..............	PUNCT
ejpam-3890	130	3	.............	.............	PUNCT
ejpam-3890	130	4	.............	.............	PUNCT
ejpam-3890	130	5	.............	.............	PUNCT
ejpam-3890	130	6	.............	.............	PUNCT
ejpam-3890	131	1	..........	..........	PUNCT
ejpam-3890	131	2	a0	a0	PROPN
ejpam-3890	131	3	a2	a2	PROPN
ejpam-3890	131	4	a4	a4	PROPN
ejpam-3890	131	5	a6	a6	NOUN
ejpam-3890	131	6	a8	a8	PROPN
ejpam-3890	131	7	a1	a1	NOUN
ejpam-3890	131	8	a9	a9	PROPN
ejpam-3890	131	9	a7	a7	PROPN
ejpam-3890	131	10	a5	a5	PROPN
ejpam-3890	131	11	a3	a3	NOUN
ejpam-3890	131	12	figure	figure	NOUN
ejpam-3890	131	13	1	1	NUM
ejpam-3890	131	14	:	:	PUNCT
ejpam-3890	131	15	an	an	DET
ejpam-3890	131	16	efficient	efficient	ADJ
ejpam-3890	131	17	zero	zero	NUM
ejpam-3890	131	18	ring	ring	NOUN
ejpam-3890	131	19	labeling	labeling	NOUN
ejpam-3890	131	20	for	for	ADP
ejpam-3890	131	21	p10	p10	NOUN
ejpam-3890	131	22	using	use	VERB
ejpam-3890	131	23	the	the	DET
ejpam-3890	131	24	zero	zero	NUM
ejpam-3890	131	25	ring	ring	NOUN
ejpam-3890	131	26	r	r	NOUN
ejpam-3890	131	27	=	=	SYM
ejpam-3890	131	28	m0	m0	NOUN
ejpam-3890	131	29	2	2	NUM
ejpam-3890	131	30	(	(	PUNCT
ejpam-3890	131	31	z10	z10	NOUN
ejpam-3890	131	32	)	)	PUNCT
ejpam-3890	131	33	.	.	PUNCT
ejpam-3890	132	1	similar	similar	ADJ
ejpam-3890	132	2	computations	computation	NOUN
ejpam-3890	132	3	can	can	AUX
ejpam-3890	132	4	be	be	AUX
ejpam-3890	132	5	done	do	VERB
ejpam-3890	132	6	for	for	ADP
ejpam-3890	132	7	the	the	DET
ejpam-3890	132	8	case	case	NOUN
ejpam-3890	132	9	when	when	SCONJ
ejpam-3890	132	10	n	n	PRON
ejpam-3890	132	11	is	be	AUX
ejpam-3890	132	12	odd	odd	ADJ
ejpam-3890	132	13	and	and	CCONJ
ejpam-3890	132	14	of	of	ADP
ejpam-3890	132	15	the	the	DET
ejpam-3890	132	16	form	form	NOUN
ejpam-3890	132	17	4k	4k	NOUN
ejpam-3890	132	18	+	+	CCONJ
ejpam-3890	132	19	3	3	NUM
ejpam-3890	132	20	to	to	PART
ejpam-3890	132	21	yield	yield	VERB
ejpam-3890	132	22	the	the	DET
ejpam-3890	132	23	same	same	ADJ
ejpam-3890	132	24	result	result	NOUN
ejpam-3890	132	25	.	.	PUNCT
ejpam-3890	133	1	since	since	SCONJ
ejpam-3890	133	2	k	k	PROPN
ejpam-3890	133	3	=	=	PUNCT
ejpam-3890	133	4	{	{	PUNCT
ejpam-3890	133	5	f(vi	f(vi	PROPN
ejpam-3890	133	6	)	)	PUNCT
ejpam-3890	133	7	+	+	NUM
ejpam-3890	133	8	f(vi+1|i	f(vi+1|i	NOUN
ejpam-3890	133	9	=	=	SYM
ejpam-3890	133	10	0	0	NUM
ejpam-3890	133	11	,	,	PUNCT
ejpam-3890	133	12	1	1	NUM
ejpam-3890	133	13	,	,	PUNCT
ejpam-3890	133	14	·	·	PUNCT
ejpam-3890	133	15	·	·	PUNCT
ejpam-3890	133	16	·	·	PUNCT
ejpam-3890	133	17	,	,	PUNCT
ejpam-3890	133	18	n−	n−	NOUN
ejpam-3890	133	19	2	2	NUM
ejpam-3890	133	20	}	}	PUNCT
ejpam-3890	133	21	=	=	SYM
ejpam-3890	133	22	{	{	PUNCT
ejpam-3890	133	23	a1	a1	NOUN
ejpam-3890	133	24	,	,	PUNCT
ejpam-3890	133	25	a3	a3	NOUN
ejpam-3890	133	26	}	}	PUNCT
ejpam-3890	133	27	,	,	PUNCT
ejpam-3890	133	28	f	f	PROPN
ejpam-3890	133	29	is	be	AUX
ejpam-3890	133	30	a	a	DET
ejpam-3890	133	31	2	2	NUM
ejpam-3890	133	32	-	-	SYM
ejpam-3890	133	33	zero	zero	NUM
ejpam-3890	133	34	ring	ring	NOUN
ejpam-3890	133	35	labeling	labeling	NOUN
ejpam-3890	133	36	of	of	ADP
ejpam-3890	133	37	pn	pn	PROPN
ejpam-3890	133	38	.	.	PROPN
ejpam-3890	133	39	therefore	therefore	ADV
ejpam-3890	133	40	,	,	PUNCT
ejpam-3890	133	41	pn	pn	PROPN
ejpam-3890	133	42	admits	admit	VERB
ejpam-3890	133	43	an	an	DET
ejpam-3890	133	44	efficient	efficient	ADJ
ejpam-3890	133	45	zero	zero	NUM
ejpam-3890	133	46	ring	ring	NOUN
ejpam-3890	133	47	labeling	labeling	NOUN
ejpam-3890	133	48	.	.	PUNCT
ejpam-3890	134	1	the	the	DET
ejpam-3890	134	2	zero	zero	NUM
ejpam-3890	134	3	ring	ring	NOUN
ejpam-3890	134	4	labeling	labeling	NOUN
ejpam-3890	134	5	for	for	ADP
ejpam-3890	134	6	p10	p10	NOUN
ejpam-3890	134	7	shown	show	VERB
ejpam-3890	134	8	in	in	ADP
ejpam-3890	134	9	figure	figure	NOUN
ejpam-3890	134	10	1	1	NUM
ejpam-3890	134	11	using	use	VERB
ejpam-3890	134	12	the	the	DET
ejpam-3890	134	13	zero	zero	NUM
ejpam-3890	134	14	ring	ring	NOUN
ejpam-3890	134	15	m0	m0	NOUN
ejpam-3890	134	16	2	2	NUM
ejpam-3890	134	17	(	(	PUNCT
ejpam-3890	134	18	z10	z10	NOUN
ejpam-3890	134	19	)	)	PUNCT
ejpam-3890	134	20	is	be	AUX
ejpam-3890	134	21	an	an	DET
ejpam-3890	134	22	efficient	efficient	ADJ
ejpam-3890	134	23	zero	zero	NUM
ejpam-3890	134	24	ring	ring	NOUN
ejpam-3890	134	25	labeling	labeling	NOUN
ejpam-3890	134	26	where	where	SCONJ
ejpam-3890	134	27	k	k	NOUN
ejpam-3890	134	28	=	=	PRON
ejpam-3890	134	29	{	{	PUNCT
ejpam-3890	134	30	a1	a1	NOUN
ejpam-3890	134	31	,	,	PUNCT
ejpam-3890	134	32	a3	a3	NOUN
ejpam-3890	134	33	}	}	PUNCT
ejpam-3890	134	34	.	.	PUNCT
ejpam-3890	135	1	it	it	PRON
ejpam-3890	135	2	is	be	AUX
ejpam-3890	135	3	a	a	DET
ejpam-3890	135	4	well	well	ADV
ejpam-3890	135	5	-	-	PUNCT
ejpam-3890	135	6	known	know	VERB
ejpam-3890	135	7	result	result	NOUN
ejpam-3890	135	8	in	in	ADP
ejpam-3890	135	9	group	group	NOUN
ejpam-3890	135	10	theory	theory	NOUN
ejpam-3890	135	11	that	that	SCONJ
ejpam-3890	135	12	if	if	SCONJ
ejpam-3890	135	13	in	in	ADP
ejpam-3890	135	14	a	a	DET
ejpam-3890	135	15	finite	finite	ADJ
ejpam-3890	135	16	group	group	NOUN
ejpam-3890	135	17	g	g	PROPN
ejpam-3890	135	18	the	the	DET
ejpam-3890	135	19	order	order	NOUN
ejpam-3890	135	20	of	of	ADP
ejpam-3890	135	21	the	the	DET
ejpam-3890	135	22	non	non	ADJ
ejpam-3890	135	23	-	-	ADJ
ejpam-3890	135	24	identity	identity	ADJ
ejpam-3890	135	25	elements	element	NOUN
ejpam-3890	135	26	is	be	AUX
ejpam-3890	135	27	2	2	NUM
ejpam-3890	135	28	,	,	PUNCT
ejpam-3890	135	29	then	then	ADV
ejpam-3890	135	30	the	the	DET
ejpam-3890	135	31	order	order	NOUN
ejpam-3890	135	32	of	of	ADP
ejpam-3890	135	33	the	the	DET
ejpam-3890	135	34	group	group	NOUN
ejpam-3890	135	35	is	be	AUX
ejpam-3890	135	36	a	a	DET
ejpam-3890	135	37	power	power	NOUN
ejpam-3890	135	38	of	of	ADP
ejpam-3890	135	39	2	2	NUM
ejpam-3890	135	40	.	.	PUNCT
ejpam-3890	136	1	we	we	PRON
ejpam-3890	136	2	state	state	VERB
ejpam-3890	136	3	this	this	PRON
ejpam-3890	136	4	as	as	ADP
ejpam-3890	136	5	the	the	DET
ejpam-3890	136	6	following	follow	VERB
ejpam-3890	136	7	lemma	lemma	PROPN
ejpam-3890	136	8	.	.	PUNCT
ejpam-3890	137	1	lemma	lemma	PROPN
ejpam-3890	137	2	1	1	NUM
ejpam-3890	137	3	.	.	PUNCT
ejpam-3890	138	1	if	if	SCONJ
ejpam-3890	138	2	g	g	PROPN
ejpam-3890	138	3	is	be	AUX
ejpam-3890	138	4	a	a	DET
ejpam-3890	138	5	finite	finite	ADJ
ejpam-3890	138	6	group	group	NOUN
ejpam-3890	138	7	such	such	ADJ
ejpam-3890	138	8	that	that	SCONJ
ejpam-3890	138	9	every	every	DET
ejpam-3890	138	10	non	non	ADJ
ejpam-3890	138	11	-	-	ADJ
ejpam-3890	138	12	identity	identity	ADJ
ejpam-3890	138	13	element	element	NOUN
ejpam-3890	138	14	is	be	AUX
ejpam-3890	138	15	of	of	ADP
ejpam-3890	138	16	order	order	NOUN
ejpam-3890	138	17	2	2	NUM
ejpam-3890	138	18	,	,	PUNCT
ejpam-3890	138	19	then	then	ADV
ejpam-3890	138	20	|g|	|g|	PROPN
ejpam-3890	138	21	=	=	SYM
ejpam-3890	138	22	2k	2k	PROPN
ejpam-3890	138	23	for	for	ADP
ejpam-3890	138	24	some	some	DET
ejpam-3890	138	25	positive	positive	ADJ
ejpam-3890	138	26	integer	integer	NOUN
ejpam-3890	138	27	k.	k.	PROPN
ejpam-3890	138	28	now	now	ADV
ejpam-3890	138	29	consider	consider	VERB
ejpam-3890	138	30	a	a	DET
ejpam-3890	138	31	finite	finite	ADJ
ejpam-3890	138	32	group	group	NOUN
ejpam-3890	138	33	g	g	PROPN
ejpam-3890	138	34	with	with	ADP
ejpam-3890	138	35	identity	identity	NOUN
ejpam-3890	138	36	e	e	NOUN
ejpam-3890	138	37	such	such	ADJ
ejpam-3890	138	38	that	that	SCONJ
ejpam-3890	138	39	x2	x2	PROPN
ejpam-3890	138	40	=	=	PUNCT
ejpam-3890	138	41	e	e	PROPN
ejpam-3890	138	42	for	for	ADP
ejpam-3890	138	43	any	any	DET
ejpam-3890	138	44	x	x	SYM
ejpam-3890	138	45	∈	∈	PROPN
ejpam-3890	138	46	g.	g.	NOUN
ejpam-3890	138	47	note	note	VERB
ejpam-3890	138	48	that	that	SCONJ
ejpam-3890	138	49	by	by	ADP
ejpam-3890	138	50	lemma	lemma	PROPN
ejpam-3890	138	51	1	1	NUM
ejpam-3890	138	52	,	,	PUNCT
ejpam-3890	138	53	|g|	|g|	PROPN
ejpam-3890	138	54	=	=	SYM
ejpam-3890	138	55	2n	2n	NUM
ejpam-3890	138	56	for	for	ADP
ejpam-3890	138	57	n	n	PRON
ejpam-3890	138	58	≥	≥	NOUN
ejpam-3890	138	59	0	0	NUM
ejpam-3890	138	60	.	.	PUNCT
ejpam-3890	139	1	this	this	DET
ejpam-3890	139	2	group	group	NOUN
ejpam-3890	139	3	is	be	AUX
ejpam-3890	139	4	always	always	ADV
ejpam-3890	139	5	abelian	abelian	ADJ
ejpam-3890	139	6	.	.	PUNCT
ejpam-3890	140	1	thus	thus	ADV
ejpam-3890	140	2	,	,	PUNCT
ejpam-3890	140	3	a	a	DET
ejpam-3890	140	4	zero	zero	NUM
ejpam-3890	140	5	ring	ring	NOUN
ejpam-3890	140	6	r	r	NOUN
ejpam-3890	140	7	can	can	AUX
ejpam-3890	140	8	be	be	AUX
ejpam-3890	140	9	constructed	construct	VERB
ejpam-3890	140	10	from	from	ADP
ejpam-3890	140	11	g.	g.	PROPN
ejpam-3890	140	12	consider	consider	VERB
ejpam-3890	140	13	a	a	DET
ejpam-3890	140	14	zero	zero	NUM
ejpam-3890	140	15	ring	ring	NOUN
ejpam-3890	140	16	r	r	NOUN
ejpam-3890	140	17	=	=	SYM
ejpam-3890	140	18	m0	m0	NOUN
ejpam-3890	140	19	2	2	NUM
ejpam-3890	140	20	(	(	PUNCT
ejpam-3890	140	21	g	g	NOUN
ejpam-3890	140	22	)	)	PUNCT
ejpam-3890	140	23	where	where	SCONJ
ejpam-3890	140	24	g	g	PROPN
ejpam-3890	140	25	is	be	AUX
ejpam-3890	140	26	a	a	DET
ejpam-3890	140	27	finite	finite	ADJ
ejpam-3890	140	28	group	group	NOUN
ejpam-3890	140	29	whose	whose	DET
ejpam-3890	140	30	non	non	ADJ
ejpam-3890	140	31	-	-	ADJ
ejpam-3890	140	32	identity	identity	ADJ
ejpam-3890	140	33	elements	element	NOUN
ejpam-3890	140	34	have	have	VERB
ejpam-3890	140	35	order	order	NOUN
ejpam-3890	140	36	2	2	NUM
ejpam-3890	140	37	.	.	PUNCT
ejpam-3890	140	38	consider	consider	VERB
ejpam-3890	140	39	the	the	DET
ejpam-3890	140	40	complete	complete	ADJ
ejpam-3890	140	41	graph	graph	NOUN
ejpam-3890	140	42	γ	γ	X
ejpam-3890	140	43	=	=	SYM
ejpam-3890	140	44	(	(	PUNCT
ejpam-3890	140	45	v	v	NOUN
ejpam-3890	140	46	,	,	PUNCT
ejpam-3890	140	47	e	e	NOUN
ejpam-3890	140	48	)	)	PUNCT
ejpam-3890	140	49	with	with	ADP
ejpam-3890	140	50	2n	2n	NUM
ejpam-3890	140	51	vertices	vertex	NOUN
ejpam-3890	140	52	and	and	CCONJ
ejpam-3890	140	53	any	any	DET
ejpam-3890	140	54	zero	zero	NUM
ejpam-3890	140	55	ring	ring	NOUN
ejpam-3890	140	56	labeling	labeling	NOUN
ejpam-3890	141	1	f	f	NOUN
ejpam-3890	141	2	:	:	PUNCT
ejpam-3890	141	3	v	v	X
ejpam-3890	141	4	→	→	SYM
ejpam-3890	141	5	r	r	NOUN
ejpam-3890	141	6	on	on	ADP
ejpam-3890	141	7	γ	γ	PROPN
ejpam-3890	141	8	.	.	PROPN
ejpam-3890	142	1	since	since	SCONJ
ejpam-3890	142	2	f	f	PROPN
ejpam-3890	142	3	is	be	AUX
ejpam-3890	142	4	injective	injective	ADJ
ejpam-3890	142	5	,	,	PUNCT
ejpam-3890	142	6	the	the	DET
ejpam-3890	142	7	sum	sum	NOUN
ejpam-3890	142	8	of	of	ADP
ejpam-3890	142	9	any	any	DET
ejpam-3890	142	10	two	two	NUM
ejpam-3890	142	11	distinct	distinct	ADJ
ejpam-3890	142	12	elements	element	NOUN
ejpam-3890	142	13	of	of	ADP
ejpam-3890	142	14	r	r	NOUN
ejpam-3890	142	15	is	be	AUX
ejpam-3890	142	16	not	not	PART
ejpam-3890	142	17	the	the	DET
ejpam-3890	142	18	identity	identity	NOUN
ejpam-3890	142	19	and	and	CCONJ
ejpam-3890	142	20	{	{	PUNCT
ejpam-3890	142	21	f(u	f(u	PROPN
ejpam-3890	142	22	)	)	PUNCT
ejpam-3890	142	23	+	+	NUM
ejpam-3890	142	24	f(v	f(v	NOUN
ejpam-3890	142	25	)	)	PUNCT
ejpam-3890	142	26	:	:	PUNCT
ejpam-3890	142	27	uv	uv	NOUN
ejpam-3890	142	28	∈	∈	NOUN
ejpam-3890	142	29	e	e	NOUN
ejpam-3890	142	30	}	}	PUNCT
ejpam-3890	142	31	=	=	SYM
ejpam-3890	142	32	r	r	NOUN
ejpam-3890	142	33	−	−	PROPN
ejpam-3890	142	34	{	{	PUNCT
ejpam-3890	142	35	a0	a0	NOUN
ejpam-3890	142	36	}	}	PUNCT
ejpam-3890	142	37	,	,	PUNCT
ejpam-3890	142	38	hence	hence	ADV
ejpam-3890	142	39	f	f	PROPN
ejpam-3890	142	40	is	be	AUX
ejpam-3890	142	41	an	an	DET
ejpam-3890	142	42	efficient	efficient	ADJ
ejpam-3890	142	43	zero	zero	NUM
ejpam-3890	142	44	ring	ring	NOUN
ejpam-3890	142	45	labeling	labeling	NOUN
ejpam-3890	142	46	of	of	ADP
ejpam-3890	142	47	the	the	DET
ejpam-3890	142	48	complete	complete	ADJ
ejpam-3890	142	49	graph	graph	NOUN
ejpam-3890	142	50	.	.	PUNCT
ejpam-3890	143	1	from	from	ADP
ejpam-3890	143	2	this	this	DET
ejpam-3890	143	3	observation	observation	NOUN
ejpam-3890	143	4	,	,	PUNCT
ejpam-3890	143	5	the	the	DET
ejpam-3890	143	6	next	next	ADJ
ejpam-3890	143	7	theorem	theorem	NOUN
ejpam-3890	143	8	follows	follow	VERB
ejpam-3890	143	9	.	.	PUNCT
ejpam-3890	144	1	theorem	theorem	ADJ
ejpam-3890	144	2	4	4	NUM
ejpam-3890	144	3	.	.	PUNCT
ejpam-3890	145	1	the	the	DET
ejpam-3890	145	2	complete	complete	ADJ
ejpam-3890	145	3	graph	graph	NOUN
ejpam-3890	145	4	on	on	ADP
ejpam-3890	145	5	2n	2n	NUM
ejpam-3890	145	6	vertices	vertex	NOUN
ejpam-3890	145	7	,	,	PUNCT
ejpam-3890	145	8	n	n	PRON
ejpam-3890	145	9	≥	≥	NOUN
ejpam-3890	145	10	0	0	NUM
ejpam-3890	145	11	,	,	PUNCT
ejpam-3890	145	12	denoted	denote	VERB
ejpam-3890	145	13	by	by	ADP
ejpam-3890	145	14	k2n	k2n	PROPN
ejpam-3890	145	15	admits	admit	VERB
ejpam-3890	145	16	an	an	DET
ejpam-3890	145	17	efficient	efficient	ADJ
ejpam-3890	145	18	zero	zero	NUM
ejpam-3890	145	19	ring	ring	NOUN
ejpam-3890	145	20	labeling	labeling	NOUN
ejpam-3890	145	21	.	.	PUNCT
ejpam-3890	146	1	4	4	X
ejpam-3890	146	2	.	.	NUM
ejpam-3890	146	3	restricted	restrict	VERB
ejpam-3890	146	4	zero	zero	NUM
ejpam-3890	146	5	ring	ring	NOUN
ejpam-3890	146	6	graphs	graph	NOUN
ejpam-3890	146	7	in	in	ADP
ejpam-3890	146	8	[	[	X
ejpam-3890	146	9	1	1	NUM
ejpam-3890	146	10	]	]	PUNCT
ejpam-3890	146	11	,	,	PUNCT
ejpam-3890	146	12	acharya	acharya	PROPN
ejpam-3890	146	13	and	and	CCONJ
ejpam-3890	146	14	pranjali	pranjali	PROPN
ejpam-3890	146	15	defined	define	VERB
ejpam-3890	146	16	the	the	DET
ejpam-3890	146	17	notion	notion	NOUN
ejpam-3890	146	18	of	of	ADP
ejpam-3890	146	19	a	a	DET
ejpam-3890	146	20	zero	zero	NUM
ejpam-3890	146	21	ring	ring	NOUN
ejpam-3890	146	22	graph	graph	NOUN
ejpam-3890	146	23	as	as	SCONJ
ejpam-3890	146	24	follows	follow	VERB
ejpam-3890	146	25	.	.	PUNCT
ejpam-3890	147	1	let	let	VERB
ejpam-3890	147	2	r	r	PRON
ejpam-3890	147	3	be	be	AUX
ejpam-3890	147	4	a	a	DET
ejpam-3890	147	5	finite	finite	ADJ
ejpam-3890	147	6	zero	zero	NUM
ejpam-3890	147	7	ring	ring	NOUN
ejpam-3890	147	8	.	.	PUNCT
ejpam-3890	148	1	the	the	DET
ejpam-3890	148	2	zero	zero	NUM
ejpam-3890	148	3	ring	ring	NOUN
ejpam-3890	148	4	graph	graph	NOUN
ejpam-3890	148	5	γ(r	γ(r	PROPN
ejpam-3890	148	6	)	)	PUNCT
ejpam-3890	148	7	is	be	AUX
ejpam-3890	148	8	a	a	DET
ejpam-3890	148	9	simple	simple	ADJ
ejpam-3890	148	10	undirected	undirected	ADJ
ejpam-3890	148	11	graph	graph	NOUN
ejpam-3890	148	12	whose	whose	DET
ejpam-3890	148	13	vertices	vertex	NOUN
ejpam-3890	148	14	are	be	AUX
ejpam-3890	148	15	the	the	DET
ejpam-3890	148	16	elements	element	NOUN
ejpam-3890	148	17	of	of	ADP
ejpam-3890	148	18	r	r	NOUN
ejpam-3890	148	19	and	and	CCONJ
ejpam-3890	148	20	two	two	NUM
ejpam-3890	148	21	distinct	distinct	ADJ
ejpam-3890	148	22	vertices	vertex	NOUN
ejpam-3890	148	23	x	x	X
ejpam-3890	148	24	and	and	CCONJ
ejpam-3890	148	25	y	y	PROPN
ejpam-3890	148	26	are	be	AUX
ejpam-3890	148	27	adjacent	adjacent	ADJ
ejpam-3890	148	28	if	if	SCONJ
ejpam-3890	148	29	and	and	CCONJ
ejpam-3890	148	30	only	only	ADV
ejpam-3890	148	31	if	if	SCONJ
ejpam-3890	148	32	x+	x+	PROPN
ejpam-3890	148	33	y	y	PROPN
ejpam-3890	148	34	6=	6=	PROPN
ejpam-3890	148	35	0	0	NUM
ejpam-3890	148	36	where	where	SCONJ
ejpam-3890	148	37	0	0	NUM
ejpam-3890	148	38	is	be	AUX
ejpam-3890	148	39	the	the	DET
ejpam-3890	148	40	additive	additive	ADJ
ejpam-3890	148	41	identity	identity	NOUN
ejpam-3890	148	42	of	of	ADP
ejpam-3890	148	43	r.	r.	PROPN
ejpam-3890	148	44	in	in	ADP
ejpam-3890	148	45	this	this	DET
ejpam-3890	148	46	study	study	NOUN
ejpam-3890	148	47	,	,	PUNCT
ejpam-3890	148	48	we	we	PRON
ejpam-3890	148	49	define	define	VERB
ejpam-3890	148	50	a	a	DET
ejpam-3890	148	51	variation	variation	NOUN
ejpam-3890	148	52	of	of	ADP
ejpam-3890	148	53	this	this	DET
ejpam-3890	148	54	graph	graph	NOUN
ejpam-3890	148	55	called	call	VERB
ejpam-3890	148	56	restricted	restricted	ADJ
ejpam-3890	148	57	zero	zero	NUM
ejpam-3890	148	58	ring	ring	NOUN
ejpam-3890	148	59	graph	graph	NOUN
ejpam-3890	148	60	.	.	PUNCT
ejpam-3890	149	1	let	let	VERB
ejpam-3890	149	2	s	s	PRON
ejpam-3890	149	3	⊆	⊆	NUM
ejpam-3890	149	4	r	r	NOUN
ejpam-3890	149	5	−	−	NOUN
ejpam-3890	149	6	{	{	PUNCT
ejpam-3890	149	7	0	0	NUM
ejpam-3890	149	8	}	}	PUNCT
ejpam-3890	149	9	for	for	ADP
ejpam-3890	149	10	any	any	DET
ejpam-3890	149	11	zero	zero	NUM
ejpam-3890	149	12	ring	ring	NOUN
ejpam-3890	149	13	r.	r.	NOUN
ejpam-3890	149	14	we	we	PRON
ejpam-3890	149	15	define	define	VERB
ejpam-3890	149	16	the	the	DET
ejpam-3890	149	17	restricted	restricted	ADJ
ejpam-3890	149	18	zero	zero	NUM
ejpam-3890	149	19	ring	ring	NOUN
ejpam-3890	149	20	graph	graph	NOUN
ejpam-3890	149	21	of	of	ADP
ejpam-3890	149	22	r	r	NOUN
ejpam-3890	149	23	in	in	ADP
ejpam-3890	149	24	s	s	PRON
ejpam-3890	149	25	denoted	denote	VERB
ejpam-3890	149	26	by	by	ADP
ejpam-3890	149	27	γs(r	γs(r	NOUN
ejpam-3890	149	28	)	)	PUNCT
ejpam-3890	149	29	=	=	SYM
ejpam-3890	149	30	(	(	PUNCT
ejpam-3890	149	31	v	v	NOUN
ejpam-3890	149	32	,	,	PUNCT
ejpam-3890	149	33	e	e	NOUN
ejpam-3890	149	34	)	)	PUNCT
ejpam-3890	149	35	to	to	PART
ejpam-3890	149	36	be	be	AUX
ejpam-3890	149	37	the	the	DET
ejpam-3890	149	38	graph	graph	NOUN
ejpam-3890	149	39	with	with	ADP
ejpam-3890	149	40	v	v	NOUN
ejpam-3890	149	41	=	=	SYM
ejpam-3890	149	42	r	r	NOUN
ejpam-3890	149	43	and	and	CCONJ
ejpam-3890	149	44	edge	edge	NOUN
ejpam-3890	149	45	set	set	NOUN
ejpam-3890	149	46	given	give	VERB
ejpam-3890	149	47	by	by	ADP
ejpam-3890	149	48	e	e	NOUN
ejpam-3890	149	49	=	=	SYM
ejpam-3890	149	50	{	{	PUNCT
ejpam-3890	149	51	(	(	PUNCT
ejpam-3890	149	52	u	u	NOUN
ejpam-3890	149	53	,	,	PUNCT
ejpam-3890	149	54	v	v	NOUN
ejpam-3890	149	55	)	)	PUNCT
ejpam-3890	149	56	:	:	PUNCT
ejpam-3890	150	1	u	u	NOUN
ejpam-3890	150	2	+	+	NOUN
ejpam-3890	150	3	v	v	NUM
ejpam-3890	150	4	∈	∈	NOUN
ejpam-3890	150	5	s	s	PART
ejpam-3890	150	6	}	}	PUNCT
ejpam-3890	150	7	.	.	PUNCT
ejpam-3890	151	1	in	in	ADP
ejpam-3890	151	2	view	view	NOUN
ejpam-3890	151	3	of	of	ADP
ejpam-3890	151	4	the	the	DET
ejpam-3890	151	5	above	above	ADJ
ejpam-3890	151	6	definition	definition	NOUN
ejpam-3890	151	7	,	,	PUNCT
ejpam-3890	151	8	if	if	SCONJ
ejpam-3890	151	9	s	s	VERB
ejpam-3890	151	10	=	=	VERB
ejpam-3890	151	11	r−	r−	PROPN
ejpam-3890	151	12	{	{	PUNCT
ejpam-3890	151	13	0	0	NUM
ejpam-3890	151	14	}	}	PUNCT
ejpam-3890	151	15	,	,	PUNCT
ejpam-3890	151	16	then	then	ADV
ejpam-3890	151	17	the	the	DET
ejpam-3890	151	18	graph	graph	NOUN
ejpam-3890	151	19	γs(r	γs(r	PUNCT
ejpam-3890	151	20	)	)	PUNCT
ejpam-3890	151	21	is	be	AUX
ejpam-3890	151	22	isomorphic	isomorphic	ADJ
ejpam-3890	151	23	to	to	ADP
ejpam-3890	151	24	the	the	DET
ejpam-3890	151	25	zero	zero	NUM
ejpam-3890	151	26	ring	ring	NOUN
ejpam-3890	151	27	graph	graph	NOUN
ejpam-3890	151	28	γ(r	γ(r	PROPN
ejpam-3890	151	29	)	)	PUNCT
ejpam-3890	151	30	.	.	PUNCT
ejpam-3890	152	1	consider	consider	VERB
ejpam-3890	152	2	the	the	DET
ejpam-3890	152	3	zero	zero	NUM
ejpam-3890	152	4	ring	ring	NOUN
ejpam-3890	152	5	r	r	NOUN
ejpam-3890	152	6	=	=	SYM
ejpam-3890	152	7	m0	m0	NOUN
ejpam-3890	152	8	2	2	NUM
ejpam-3890	152	9	(	(	PUNCT
ejpam-3890	152	10	z4	z4	PROPN
ejpam-3890	152	11	)	)	PUNCT
ejpam-3890	152	12	,	,	PUNCT
ejpam-3890	152	13	the	the	DET
ejpam-3890	152	14	following	follow	VERB
ejpam-3890	152	15	are	be	AUX
ejpam-3890	152	16	examples	example	NOUN
ejpam-3890	152	17	of	of	ADP
ejpam-3890	152	18	restricted	restricted	ADJ
ejpam-3890	152	19	zero	zero	NUM
ejpam-3890	152	20	ring	ring	NOUN
ejpam-3890	152	21	graphs	graph	NOUN
ejpam-3890	152	22	on	on	ADP
ejpam-3890	152	23	r	r	NOUN
ejpam-3890	152	24	where	where	SCONJ
ejpam-3890	152	25	|s|	|s|	PROPN
ejpam-3890	152	26	=	=	SYM
ejpam-3890	152	27	2	2	X
ejpam-3890	152	28	.	.	X
ejpam-3890	152	29	note	note	VERB
ejpam-3890	152	30	that	that	SCONJ
ejpam-3890	152	31	if	if	SCONJ
ejpam-3890	152	32	s1	s1	PROPN
ejpam-3890	152	33	,	,	PUNCT
ejpam-3890	152	34	s2	s2	NOUN
ejpam-3890	152	35	⊂	⊂	X
ejpam-3890	152	36	r−{0	r−{0	ADV
ejpam-3890	152	37	}	}	PUNCT
ejpam-3890	152	38	,	,	PUNCT
ejpam-3890	152	39	the	the	DET
ejpam-3890	152	40	graphs	graph	NOUN
ejpam-3890	152	41	γs1(r),γs2(r	γs1(r),γs2(r	PROPN
ejpam-3890	152	42	)	)	PUNCT
ejpam-3890	152	43	are	be	AUX
ejpam-3890	152	44	not	not	PART
ejpam-3890	152	45	necessarily	necessarily	ADV
ejpam-3890	152	46	isomorphic	isomorphic	ADJ
ejpam-3890	152	47	even	even	ADV
ejpam-3890	152	48	if	if	SCONJ
ejpam-3890	152	49	|s1|	|s1|	NOUN
ejpam-3890	152	50	=	=	SYM
ejpam-3890	152	51	|s2|	|s2|	NOUN
ejpam-3890	152	52	.	.	PUNCT
ejpam-3890	153	1	f.j	f.j	PROPN
ejpam-3890	153	2	.	.	PROPN
ejpam-3890	154	1	campeña	campeña	PROPN
ejpam-3890	154	2	et	et	PROPN
ejpam-3890	154	3	al	al	PROPN
ejpam-3890	154	4	.	.	PUNCT
ejpam-3890	154	5	/	/	SYM
ejpam-3890	154	6	eur	eur	PROPN
ejpam-3890	154	7	.	.	PUNCT
ejpam-3890	155	1	j.	j.	PROPN
ejpam-3890	155	2	pure	pure	PROPN
ejpam-3890	155	3	appl	appl	PROPN
ejpam-3890	155	4	.	.	PROPN
ejpam-3890	155	5	math	math	PROPN
ejpam-3890	155	6	,	,	PUNCT
ejpam-3890	155	7	14	14	NUM
ejpam-3890	155	8	(	(	PUNCT
ejpam-3890	155	9	1	1	NUM
ejpam-3890	155	10	)	)	PUNCT
ejpam-3890	155	11	(	(	PUNCT
ejpam-3890	155	12	2021	2021	NUM
ejpam-3890	155	13	)	)	PUNCT
ejpam-3890	155	14	,	,	PUNCT
ejpam-3890	155	15	268	268	NUM
ejpam-3890	155	16	-	-	SYM
ejpam-3890	155	17	277	277	NUM
ejpam-3890	155	18	273	273	NUM
ejpam-3890	155	19	..........	..........	PUNCT
ejpam-3890	155	20	.............................................	.............................................	PUNCT
ejpam-3890	155	21	.	.	PUNCT
ejpam-3890	156	1	..........	..........	PUNCT
ejpam-3890	156	2	.............................................	.............................................	PUNCT
ejpam-3890	156	3	.	.	PUNCT
ejpam-3890	157	1	..........	..........	PUNCT
ejpam-3890	157	2	.............................................	.............................................	PUNCT
ejpam-3890	157	3	.	.	PUNCT
ejpam-3890	158	1	..........	..........	PUNCT
ejpam-3890	158	2	.............................................	.............................................	PUNCT
ejpam-3890	158	3	.	.	PUNCT
ejpam-3890	159	1	.........	.........	PUNCT
ejpam-3890	159	2	........	........	PUNCT
ejpam-3890	159	3	........	........	PUNCT
ejpam-3890	159	4	........	........	PUNCT
ejpam-3890	159	5	........	........	PUNCT
ejpam-3890	159	6	........	........	PUNCT
ejpam-3890	160	1	........	........	PUNCT
ejpam-3890	160	2	........	........	PUNCT
ejpam-3890	161	1	.....	.....	PUNCT
ejpam-3890	161	2	.........	.........	PUNCT
ejpam-3890	161	3	........	........	PUNCT
ejpam-3890	161	4	........	........	PUNCT
ejpam-3890	161	5	........	........	PUNCT
ejpam-3890	161	6	........	........	PUNCT
ejpam-3890	161	7	........	........	PUNCT
ejpam-3890	161	8	........	........	PUNCT
ejpam-3890	161	9	........	........	PUNCT
ejpam-3890	161	10	...............................................................................................................	...............................................................................................................	PUNCT
ejpam-3890	162	1	..........	..........	PUNCT
ejpam-3890	162	2	.............................................	.............................................	PUNCT
ejpam-3890	162	3	.	.	PUNCT
ejpam-3890	163	1	..........	..........	PUNCT
ejpam-3890	163	2	.............................................	.............................................	PUNCT
ejpam-3890	163	3	.	.	PUNCT
ejpam-3890	164	1	..........	..........	PUNCT
ejpam-3890	164	2	.............................................	.............................................	PUNCT
ejpam-3890	164	3	.	.	PUNCT
ejpam-3890	165	1	..........	..........	PUNCT
ejpam-3890	165	2	.............................................	.............................................	PUNCT
ejpam-3890	165	3	.	.	PUNCT
ejpam-3890	166	1	.........	.........	PUNCT
ejpam-3890	166	2	........	........	PUNCT
ejpam-3890	166	3	........	........	PUNCT
ejpam-3890	166	4	........	........	PUNCT
ejpam-3890	166	5	........	........	PUNCT
ejpam-3890	166	6	........	........	PUNCT
ejpam-3890	167	1	........	........	PUNCT
ejpam-3890	167	2	........	........	PUNCT
ejpam-3890	168	1	.....	.....	PUNCT
ejpam-3890	168	2	.........	.........	PUNCT
ejpam-3890	168	3	........	........	PUNCT
ejpam-3890	168	4	........	........	PUNCT
ejpam-3890	168	5	........	........	PUNCT
ejpam-3890	168	6	........	........	PUNCT
ejpam-3890	168	7	........	........	PUNCT
ejpam-3890	168	8	........	........	PUNCT
ejpam-3890	168	9	........	........	PUNCT
ejpam-3890	169	1	.....	.....	PUNCT
ejpam-3890	169	2	......................................................................	......................................................................	PUNCT
ejpam-3890	169	3	......................................................................	......................................................................	PUNCT
ejpam-3890	170	1	..........	..........	PUNCT
ejpam-3890	170	2	.............................................	.............................................	PUNCT
ejpam-3890	170	3	.	.	PUNCT
ejpam-3890	171	1	..........	..........	PUNCT
ejpam-3890	171	2	.............................................	.............................................	PUNCT
ejpam-3890	171	3	.	.	PUNCT
ejpam-3890	172	1	..........	..........	PUNCT
ejpam-3890	172	2	.............................................	.............................................	PUNCT
ejpam-3890	172	3	.	.	PUNCT
ejpam-3890	173	1	..........	..........	PUNCT
ejpam-3890	173	2	.............................................	.............................................	PUNCT
ejpam-3890	174	1	.......................................................................................................................................................................................................................................................	.......................................................................................................................................................................................................................................................	PROPN
ejpam-3890	175	1	a0	a0	PROPN
ejpam-3890	175	2	a1	a1	PROPN
ejpam-3890	175	3	a2	a2	PROPN
ejpam-3890	175	4	a3	a3	PROPN
ejpam-3890	175	5	a0	a0	PROPN
ejpam-3890	175	6	a1	a1	PROPN
ejpam-3890	175	7	a2	a2	PROPN
ejpam-3890	175	8	a3	a3	PROPN
ejpam-3890	175	9	a0	a0	PROPN
ejpam-3890	175	10	a1	a1	PROPN
ejpam-3890	175	11	a2	a2	PROPN
ejpam-3890	175	12	a3	a3	PROPN
ejpam-3890	175	13	γ{a1,a2}(r	γ{a1,a2}(r	PROPN
ejpam-3890	175	14	)	)	PUNCT
ejpam-3890	175	15	γ{a1,a3}(r	γ{a1,a3}(r	PROPN
ejpam-3890	175	16	)	)	PUNCT
ejpam-3890	175	17	γ{a2,a3}(r	γ{a2,a3}(r	NOUN
ejpam-3890	175	18	)	)	PUNCT
ejpam-3890	175	19	figure	figure	NOUN
ejpam-3890	175	20	2	2	NUM
ejpam-3890	175	21	:	:	PUNCT
ejpam-3890	175	22	some	some	PRON
ejpam-3890	175	23	restricted	restrict	VERB
ejpam-3890	175	24	zero	zero	NUM
ejpam-3890	175	25	ring	ring	NOUN
ejpam-3890	175	26	graphs	graph	NOUN
ejpam-3890	175	27	in	in	ADP
ejpam-3890	175	28	m0	m0	PROPN
ejpam-3890	175	29	2	2	NUM
ejpam-3890	175	30	(	(	PUNCT
ejpam-3890	175	31	z4	z4	PROPN
ejpam-3890	175	32	)	)	PUNCT
ejpam-3890	175	33	.	.	PUNCT
ejpam-3890	176	1	the	the	DET
ejpam-3890	176	2	following	follow	VERB
ejpam-3890	176	3	lemma	lemma	PROPN
ejpam-3890	176	4	follows	follow	VERB
ejpam-3890	176	5	directly	directly	ADV
ejpam-3890	176	6	from	from	ADP
ejpam-3890	176	7	the	the	DET
ejpam-3890	176	8	definition	definition	NOUN
ejpam-3890	176	9	of	of	ADP
ejpam-3890	176	10	the	the	DET
ejpam-3890	176	11	restricted	restricted	ADJ
ejpam-3890	176	12	zero	zero	NUM
ejpam-3890	176	13	ring	ring	NOUN
ejpam-3890	176	14	graph	graph	NOUN
ejpam-3890	176	15	and	and	CCONJ
ejpam-3890	176	16	the	the	DET
ejpam-3890	176	17	fact	fact	NOUN
ejpam-3890	176	18	that	that	SCONJ
ejpam-3890	176	19	r	r	NOUN
ejpam-3890	176	20	together	together	ADV
ejpam-3890	176	21	with	with	ADP
ejpam-3890	176	22	the	the	DET
ejpam-3890	176	23	addition	addition	NOUN
ejpam-3890	176	24	operation	operation	NOUN
ejpam-3890	176	25	on	on	ADP
ejpam-3890	176	26	the	the	DET
ejpam-3890	176	27	ring	ring	NOUN
ejpam-3890	176	28	is	be	AUX
ejpam-3890	176	29	an	an	DET
ejpam-3890	176	30	abelian	abelian	ADJ
ejpam-3890	176	31	group	group	NOUN
ejpam-3890	176	32	.	.	PUNCT
ejpam-3890	177	1	lemma	lemma	PROPN
ejpam-3890	177	2	2	2	X
ejpam-3890	177	3	.	.	PUNCT
ejpam-3890	177	4	consider	consider	VERB
ejpam-3890	177	5	a	a	DET
ejpam-3890	177	6	zero	zero	NUM
ejpam-3890	177	7	ring	ring	NOUN
ejpam-3890	177	8	r	r	NOUN
ejpam-3890	177	9	with	with	ADP
ejpam-3890	177	10	additive	additive	ADJ
ejpam-3890	177	11	identity	identity	NOUN
ejpam-3890	177	12	0	0	NUM
ejpam-3890	177	13	and	and	CCONJ
ejpam-3890	177	14	a	a	DET
ejpam-3890	177	15	non	non	ADJ
ejpam-3890	177	16	-	-	ADJ
ejpam-3890	177	17	empty	empty	ADJ
ejpam-3890	177	18	s	s	PART
ejpam-3890	177	19	⊆	⊆	NUM
ejpam-3890	177	20	r−{0	r−{0	NOUN
ejpam-3890	177	21	}	}	PUNCT
ejpam-3890	177	22	such	such	ADJ
ejpam-3890	177	23	that	that	SCONJ
ejpam-3890	177	24	|r|	|r|	NOUN
ejpam-3890	177	25	=	=	SYM
ejpam-3890	177	26	n	n	PROPN
ejpam-3890	177	27	and	and	CCONJ
ejpam-3890	177	28	|s|	|s|	PROPN
ejpam-3890	177	29	=	=	SYM
ejpam-3890	177	30	m	m	NOUN
ejpam-3890	177	31	≥	≥	NOUN
ejpam-3890	177	32	1	1	NUM
ejpam-3890	177	33	.	.	PUNCT
ejpam-3890	178	1	then	then	ADV
ejpam-3890	178	2	the	the	DET
ejpam-3890	178	3	graph	graph	NOUN
ejpam-3890	178	4	γs(r	γs(r	PUNCT
ejpam-3890	178	5	)	)	PUNCT
ejpam-3890	178	6	=	=	SYM
ejpam-3890	179	1	(	(	PUNCT
ejpam-3890	179	2	v	v	NOUN
ejpam-3890	179	3	,	,	PUNCT
ejpam-3890	179	4	e	e	NOUN
ejpam-3890	179	5	)	)	PUNCT
ejpam-3890	179	6	satisfies	satisfy	VERB
ejpam-3890	179	7	the	the	DET
ejpam-3890	179	8	following	following	NOUN
ejpam-3890	179	9	:	:	PUNCT
ejpam-3890	179	10	(	(	PUNCT
ejpam-3890	179	11	i	i	NOUN
ejpam-3890	179	12	)	)	PUNCT
ejpam-3890	179	13	∆(γs(r	∆(γs(r	ADJ
ejpam-3890	179	14	)	)	PUNCT
ejpam-3890	179	15	)	)	PUNCT
ejpam-3890	180	1	=	=	SYM
ejpam-3890	180	2	m	m	PROPN
ejpam-3890	180	3	(	(	PUNCT
ejpam-3890	180	4	ii	ii	NOUN
ejpam-3890	180	5	)	)	PUNCT
ejpam-3890	180	6	γs(r	γs(r	PUNCT
ejpam-3890	180	7	)	)	PUNCT
ejpam-3890	180	8	is	be	AUX
ejpam-3890	180	9	a	a	DET
ejpam-3890	180	10	disconnected	disconnected	ADJ
ejpam-3890	180	11	graph	graph	NOUN
ejpam-3890	180	12	for	for	ADP
ejpam-3890	180	13	n	n	NOUN
ejpam-3890	180	14	>	>	SYM
ejpam-3890	180	15	2	2	NUM
ejpam-3890	180	16	and	and	CCONJ
ejpam-3890	180	17	m	m	NOUN
ejpam-3890	180	18	=	=	ADJ
ejpam-3890	180	19	1	1	X
ejpam-3890	180	20	.	.	PUNCT
ejpam-3890	181	1	proof	proof	NOUN
ejpam-3890	181	2	.	.	PUNCT
ejpam-3890	182	1	for	for	ADP
ejpam-3890	182	2	(	(	PUNCT
ejpam-3890	182	3	i	i	NOUN
ejpam-3890	182	4	)	)	PUNCT
ejpam-3890	182	5	:	:	PUNCT
ejpam-3890	182	6	note	note	VERB
ejpam-3890	182	7	that	that	SCONJ
ejpam-3890	182	8	for	for	ADP
ejpam-3890	182	9	any	any	DET
ejpam-3890	182	10	s	s	PART
ejpam-3890	182	11	=	=	PUNCT
ejpam-3890	182	12	{	{	PUNCT
ejpam-3890	182	13	s1	s1	NOUN
ejpam-3890	182	14	,	,	PUNCT
ejpam-3890	182	15	.	.	PUNCT
ejpam-3890	182	16	.	.	PUNCT
ejpam-3890	182	17	.	.	PUNCT
ejpam-3890	183	1	,	,	PUNCT
ejpam-3890	183	2	sm	sm	INTJ
ejpam-3890	183	3	}	}	PUNCT
ejpam-3890	183	4	⊆	⊆	NUM
ejpam-3890	183	5	r	r	NOUN
ejpam-3890	183	6	−	−	NOUN
ejpam-3890	183	7	{	{	PUNCT
ejpam-3890	183	8	0	0	NUM
ejpam-3890	183	9	}	}	PUNCT
ejpam-3890	183	10	,	,	PUNCT
ejpam-3890	183	11	the	the	DET
ejpam-3890	183	12	edge	edge	NOUN
ejpam-3890	183	13	(	(	PUNCT
ejpam-3890	183	14	0	0	NUM
ejpam-3890	183	15	,	,	PUNCT
ejpam-3890	183	16	si	si	NOUN
ejpam-3890	183	17	)	)	PUNCT
ejpam-3890	183	18	∈	∈	PROPN
ejpam-3890	183	19	e(γs(r	e(γs(r	PROPN
ejpam-3890	183	20	)	)	PUNCT
ejpam-3890	183	21	)	)	PUNCT
ejpam-3890	183	22	by	by	ADP
ejpam-3890	183	23	the	the	DET
ejpam-3890	183	24	definition	definition	NOUN
ejpam-3890	183	25	of	of	ADP
ejpam-3890	183	26	γs(r	γs(r	PROPN
ejpam-3890	183	27	)	)	PUNCT
ejpam-3890	183	28	.	.	PUNCT
ejpam-3890	184	1	suppose	suppose	VERB
ejpam-3890	184	2	there	there	PRON
ejpam-3890	184	3	is	be	VERB
ejpam-3890	184	4	a	a	DET
ejpam-3890	184	5	vertex	vertex	NOUN
ejpam-3890	184	6	v	v	NOUN
ejpam-3890	184	7	in	in	ADP
ejpam-3890	184	8	the	the	DET
ejpam-3890	184	9	graph	graph	NOUN
ejpam-3890	184	10	with	with	ADP
ejpam-3890	184	11	neighbor	neighbor	NOUN
ejpam-3890	184	12	set	set	VERB
ejpam-3890	184	13	n(v	n(v	PROPN
ejpam-3890	184	14	)	)	PUNCT
ejpam-3890	185	1	=	=	PRON
ejpam-3890	185	2	{	{	PUNCT
ejpam-3890	185	3	u1	u1	NOUN
ejpam-3890	185	4	,	,	PUNCT
ejpam-3890	185	5	.	.	PUNCT
ejpam-3890	185	6	.	.	PUNCT
ejpam-3890	185	7	.	.	PUNCT
ejpam-3890	186	1	,	,	PUNCT
ejpam-3890	186	2	uk	uk	PROPN
ejpam-3890	186	3	}	}	PUNCT
ejpam-3890	186	4	where	where	SCONJ
ejpam-3890	186	5	k	k	PROPN
ejpam-3890	186	6	>	>	X
ejpam-3890	186	7	m.	m.	NOUN
ejpam-3890	186	8	since	since	SCONJ
ejpam-3890	186	9	{	{	PUNCT
ejpam-3890	186	10	v	v	NOUN
ejpam-3890	186	11	+	+	NOUN
ejpam-3890	186	12	ui	ui	NOUN
ejpam-3890	186	13	:	:	PUNCT
ejpam-3890	187	1	i	i	NOUN
ejpam-3890	187	2	=	=	NOUN
ejpam-3890	187	3	1	1	NUM
ejpam-3890	187	4	,	,	PUNCT
ejpam-3890	187	5	2	2	NUM
ejpam-3890	187	6	.	.	PUNCT
ejpam-3890	187	7	.	.	PUNCT
ejpam-3890	187	8	.	.	PUNCT
ejpam-3890	188	1	,	,	PUNCT
ejpam-3890	188	2	k	k	X
ejpam-3890	188	3	}	}	PUNCT
ejpam-3890	188	4	⊂	⊂	PROPN
ejpam-3890	188	5	s	s	PART
ejpam-3890	188	6	,	,	PUNCT
ejpam-3890	188	7	there	there	PRON
ejpam-3890	188	8	is	be	VERB
ejpam-3890	188	9	at	at	ADV
ejpam-3890	188	10	least	least	ADJ
ejpam-3890	188	11	one	one	NUM
ejpam-3890	188	12	pair	pair	NOUN
ejpam-3890	188	13	of	of	ADP
ejpam-3890	188	14	distinct	distinct	ADJ
ejpam-3890	188	15	vertices	vertex	NOUN
ejpam-3890	188	16	u1	u1	NOUN
ejpam-3890	188	17	,	,	PUNCT
ejpam-3890	188	18	u2	u2	NOUN
ejpam-3890	188	19	where	where	SCONJ
ejpam-3890	188	20	v	v	NOUN
ejpam-3890	188	21	+	+	NOUN
ejpam-3890	188	22	u1	u1	NOUN
ejpam-3890	188	23	=	=	SYM
ejpam-3890	188	24	si	si	PROPN
ejpam-3890	188	25	=	=	SYM
ejpam-3890	188	26	v	v	PROPN
ejpam-3890	188	27	+	+	CCONJ
ejpam-3890	188	28	u2	u2	NOUN
ejpam-3890	188	29	for	for	ADP
ejpam-3890	188	30	some	some	DET
ejpam-3890	188	31	si	si	PROPN
ejpam-3890	188	32	∈	∈	PROPN
ejpam-3890	188	33	s.	s.	PROPN
ejpam-3890	188	34	then	then	ADV
ejpam-3890	188	35	ui	ui	PROPN
ejpam-3890	188	36	=	=	PROPN
ejpam-3890	188	37	u2	u2	PROPN
ejpam-3890	188	38	,	,	PUNCT
ejpam-3890	188	39	which	which	PRON
ejpam-3890	188	40	is	be	AUX
ejpam-3890	188	41	a	a	DET
ejpam-3890	188	42	contradiction	contradiction	NOUN
ejpam-3890	188	43	.	.	PUNCT
ejpam-3890	189	1	hence	hence	ADV
ejpam-3890	189	2	,	,	PUNCT
ejpam-3890	189	3	the	the	DET
ejpam-3890	189	4	maximum	maximum	ADJ
ejpam-3890	189	5	degree	degree	NOUN
ejpam-3890	189	6	of	of	ADP
ejpam-3890	189	7	a	a	DET
ejpam-3890	189	8	vertex	vertex	NOUN
ejpam-3890	189	9	is	be	AUX
ejpam-3890	189	10	m.	m.	NOUN
ejpam-3890	189	11	statement	statement	NOUN
ejpam-3890	189	12	(	(	PUNCT
ejpam-3890	189	13	ii	ii	NOUN
ejpam-3890	189	14	)	)	PUNCT
ejpam-3890	189	15	follows	follow	VERB
ejpam-3890	189	16	directly	directly	ADV
ejpam-3890	189	17	from	from	ADP
ejpam-3890	189	18	(	(	PUNCT
ejpam-3890	189	19	i	i	NOUN
ejpam-3890	189	20	)	)	PUNCT
ejpam-3890	189	21	for	for	ADP
ejpam-3890	189	22	any	any	DET
ejpam-3890	189	23	zero	zero	NUM
ejpam-3890	189	24	ring	ring	NOUN
ejpam-3890	189	25	r	r	NOUN
ejpam-3890	189	26	with	with	ADP
ejpam-3890	189	27	|r|	|r|	PROPN
ejpam-3890	189	28	>	>	X
ejpam-3890	189	29	2	2	NUM
ejpam-3890	189	30	.	.	PUNCT
ejpam-3890	190	1	the	the	DET
ejpam-3890	190	2	following	follow	VERB
ejpam-3890	190	3	are	be	AUX
ejpam-3890	190	4	some	some	DET
ejpam-3890	190	5	examples	example	NOUN
ejpam-3890	190	6	of	of	ADP
ejpam-3890	190	7	restricted	restrict	VERB
ejpam-3890	190	8	zero	zero	NUM
ejpam-3890	190	9	ring	ring	NOUN
ejpam-3890	190	10	graphs	graph	NOUN
ejpam-3890	190	11	.	.	PUNCT
ejpam-3890	191	1	..........	..........	PUNCT
ejpam-3890	191	2	.............................................	.............................................	PUNCT
ejpam-3890	191	3	.	.	PUNCT
ejpam-3890	192	1	..........	..........	PUNCT
ejpam-3890	192	2	.............................................	.............................................	PUNCT
ejpam-3890	192	3	.	.	PUNCT
ejpam-3890	193	1	..........	..........	PUNCT
ejpam-3890	193	2	.............................................	.............................................	PUNCT
ejpam-3890	193	3	.	.	PUNCT
ejpam-3890	194	1	..........	..........	PUNCT
ejpam-3890	194	2	.............................................	.............................................	PUNCT
ejpam-3890	194	3	.	.	PUNCT
ejpam-3890	195	1	..........	..........	PUNCT
ejpam-3890	195	2	.............................................	.............................................	PUNCT
ejpam-3890	195	3	.	.	PUNCT
ejpam-3890	196	1	..........	..........	PUNCT
ejpam-3890	196	2	.............................................	.............................................	PUNCT
ejpam-3890	196	3	.	.	PUNCT
ejpam-3890	197	1	..........	..........	PUNCT
ejpam-3890	197	2	.............................................	.............................................	PUNCT
ejpam-3890	197	3	.	.	PUNCT
ejpam-3890	198	1	..........	..........	PUNCT
ejpam-3890	198	2	.............................................	.............................................	PUNCT
ejpam-3890	198	3	.	.	PUNCT
ejpam-3890	199	1	..........	..........	PUNCT
ejpam-3890	199	2	.............................................	.............................................	PUNCT
ejpam-3890	199	3	.	.	PUNCT
ejpam-3890	200	1	..........	..........	PUNCT
ejpam-3890	200	2	.............................................	.............................................	PUNCT
ejpam-3890	200	3	.	.	PUNCT
ejpam-3890	201	1	.........	.........	PUNCT
ejpam-3890	201	2	........	........	PUNCT
ejpam-3890	201	3	........	........	PUNCT
ejpam-3890	201	4	........	........	PUNCT
ejpam-3890	202	1	........	........	PUNCT
ejpam-3890	202	2	.	.	PUNCT
ejpam-3890	203	1	.........	.........	PUNCT
ejpam-3890	203	2	........	........	PUNCT
ejpam-3890	203	3	........	........	PUNCT
ejpam-3890	203	4	........	........	PUNCT
ejpam-3890	203	5	........	........	PUNCT
ejpam-3890	203	6	.	.	PUNCT
ejpam-3890	203	7	........................................................	........................................................	PUNCT
ejpam-3890	204	1	........................................................	........................................................	PUNCT
ejpam-3890	204	2	........................................................	........................................................	PUNCT
ejpam-3890	205	1	.........	.........	PUNCT
ejpam-3890	205	2	........	........	PUNCT
ejpam-3890	205	3	........	........	PUNCT
ejpam-3890	205	4	........	........	PUNCT
ejpam-3890	206	1	........	........	PUNCT
ejpam-3890	206	2	.	.	PUNCT
ejpam-3890	207	1	.........	.........	PUNCT
ejpam-3890	207	2	........	........	PUNCT
ejpam-3890	207	3	........	........	PUNCT
ejpam-3890	207	4	........	........	PUNCT
ejpam-3890	207	5	........	........	PUNCT
ejpam-3890	207	6	.	.	PUNCT
ejpam-3890	208	1	............................................................................	............................................................................	PUNCT
ejpam-3890	209	1	a0	a0	PROPN
ejpam-3890	209	2	a1	a1	PROPN
ejpam-3890	209	3	a2	a2	PROPN
ejpam-3890	209	4	a3	a3	PROPN
ejpam-3890	209	5	a4	a4	PROPN
ejpam-3890	209	6	γ{a1}(m	γ{a1}(m	PROPN
ejpam-3890	209	7	0	0	NUM
ejpam-3890	209	8	2	2	NUM
ejpam-3890	209	9	(	(	PUNCT
ejpam-3890	209	10	z5	z5	NOUN
ejpam-3890	209	11	)	)	PUNCT
ejpam-3890	209	12	)	)	PUNCT
ejpam-3890	209	13	a1	a1	NOUN
ejpam-3890	209	14	a2	a2	PROPN
ejpam-3890	209	15	a3	a3	NOUN
ejpam-3890	209	16	a0	a0	PROPN
ejpam-3890	209	17	a4	a4	PROPN
ejpam-3890	209	18	γ{a2,a3,a4}(m	γ{a2,a3,a4}(m	PROPN
ejpam-3890	209	19	0	0	NUM
ejpam-3890	209	20	2	2	NUM
ejpam-3890	209	21	(	(	PUNCT
ejpam-3890	209	22	z5	z5	NOUN
ejpam-3890	209	23	)	)	PUNCT
ejpam-3890	209	24	)	)	PUNCT
ejpam-3890	209	25	figure	figure	NOUN
ejpam-3890	209	26	3	3	NUM
ejpam-3890	209	27	:	:	PUNCT
ejpam-3890	209	28	some	some	PRON
ejpam-3890	209	29	restricted	restrict	VERB
ejpam-3890	209	30	zero	zero	NUM
ejpam-3890	209	31	ring	ring	NOUN
ejpam-3890	209	32	graphs	graph	NOUN
ejpam-3890	209	33	in	in	ADP
ejpam-3890	209	34	m0	m0	PROPN
ejpam-3890	209	35	2	2	NUM
ejpam-3890	209	36	(	(	PUNCT
ejpam-3890	209	37	zn	zn	NUM
ejpam-3890	209	38	)	)	PUNCT
ejpam-3890	209	39	.	.	PUNCT
ejpam-3890	210	1	now	now	ADV
ejpam-3890	210	2	we	we	PRON
ejpam-3890	210	3	consider	consider	VERB
ejpam-3890	210	4	the	the	DET
ejpam-3890	210	5	restricted	restricted	ADJ
ejpam-3890	210	6	zero	zero	NUM
ejpam-3890	210	7	ring	ring	NOUN
ejpam-3890	210	8	graph	graph	NOUN
ejpam-3890	210	9	γs(r	γs(r	PUNCT
ejpam-3890	210	10	)	)	PUNCT
ejpam-3890	210	11	where	where	SCONJ
ejpam-3890	210	12	r	r	NOUN
ejpam-3890	210	13	=	=	SYM
ejpam-3890	210	14	m0	m0	NOUN
ejpam-3890	210	15	2	2	NUM
ejpam-3890	210	16	(	(	PUNCT
ejpam-3890	210	17	zn	zn	NUM
ejpam-3890	210	18	)	)	PUNCT
ejpam-3890	210	19	.	.	PUNCT
ejpam-3890	211	1	the	the	DET
ejpam-3890	211	2	following	follow	VERB
ejpam-3890	211	3	lemma	lemma	PROPN
ejpam-3890	211	4	can	can	AUX
ejpam-3890	211	5	be	be	AUX
ejpam-3890	211	6	easily	easily	ADV
ejpam-3890	211	7	verified	verify	VERB
ejpam-3890	211	8	from	from	ADP
ejpam-3890	211	9	the	the	DET
ejpam-3890	211	10	definition	definition	NOUN
ejpam-3890	211	11	of	of	ADP
ejpam-3890	211	12	γs(r	γs(r	PUNCT
ejpam-3890	211	13	)	)	PUNCT
ejpam-3890	211	14	and	and	CCONJ
ejpam-3890	211	15	the	the	DET
ejpam-3890	211	16	fact	fact	NOUN
ejpam-3890	211	17	that	that	SCONJ
ejpam-3890	211	18	r	r	NOUN
ejpam-3890	211	19	together	together	ADV
ejpam-3890	211	20	with	with	ADP
ejpam-3890	211	21	the	the	DET
ejpam-3890	211	22	additive	additive	ADJ
ejpam-3890	211	23	operation	operation	NOUN
ejpam-3890	211	24	in	in	ADP
ejpam-3890	211	25	the	the	DET
ejpam-3890	211	26	ring	ring	NOUN
ejpam-3890	211	27	is	be	AUX
ejpam-3890	211	28	an	an	DET
ejpam-3890	211	29	abelian	abelian	ADJ
ejpam-3890	211	30	group	group	NOUN
ejpam-3890	211	31	.	.	PUNCT
ejpam-3890	212	1	lemma	lemma	PROPN
ejpam-3890	213	1	3	3	X
ejpam-3890	213	2	.	.	X
ejpam-3890	213	3	if	if	SCONJ
ejpam-3890	213	4	|s|	|s|	PROPN
ejpam-3890	213	5	=	=	SYM
ejpam-3890	213	6	1	1	NUM
ejpam-3890	213	7	,	,	PUNCT
ejpam-3890	213	8	then	then	ADV
ejpam-3890	213	9	γs(m0	γs(m0	INTJ
ejpam-3890	213	10	2	2	NUM
ejpam-3890	213	11	(	(	PUNCT
ejpam-3890	213	12	zn	zn	NOUN
ejpam-3890	213	13	)	)	PUNCT
ejpam-3890	213	14	)	)	PUNCT
ejpam-3890	213	15	is	be	AUX
ejpam-3890	213	16	a	a	DET
ejpam-3890	213	17	disconnected	disconnected	ADJ
ejpam-3890	213	18	graph	graph	NOUN
ejpam-3890	213	19	.	.	PUNCT
ejpam-3890	214	1	in	in	ADP
ejpam-3890	214	2	particular	particular	ADJ
ejpam-3890	214	3	,	,	PUNCT
ejpam-3890	214	4	the	the	DET
ejpam-3890	214	5	graph	graph	NOUN
ejpam-3890	214	6	γs(m0	γs(m0	INTJ
ejpam-3890	214	7	2	2	NUM
ejpam-3890	214	8	(	(	PUNCT
ejpam-3890	214	9	zn	zn	NOUN
ejpam-3890	214	10	)	)	PUNCT
ejpam-3890	214	11	)	)	PUNCT
ejpam-3890	214	12	consists	consist	VERB
ejpam-3890	214	13	of	of	ADP
ejpam-3890	214	14	at	at	ADP
ejpam-3890	214	15	most	most	ADJ
ejpam-3890	214	16	bn2	bn2	NOUN
ejpam-3890	214	17	c	c	X
ejpam-3890	214	18	disjoint	disjoint	NOUN
ejpam-3890	214	19	copies	copy	NOUN
ejpam-3890	214	20	of	of	ADP
ejpam-3890	214	21	k2	k2	NOUN
ejpam-3890	214	22	.	.	PUNCT
ejpam-3890	215	1	proof	proof	NOUN
ejpam-3890	215	2	.	.	PUNCT
ejpam-3890	216	1	by	by	ADP
ejpam-3890	216	2	lemma	lemma	PROPN
ejpam-3890	216	3	2	2	NUM
ejpam-3890	216	4	,	,	PUNCT
ejpam-3890	216	5	the	the	DET
ejpam-3890	216	6	maximum	maximum	ADJ
ejpam-3890	216	7	degree	degree	NOUN
ejpam-3890	216	8	of	of	ADP
ejpam-3890	216	9	a	a	DET
ejpam-3890	216	10	vertex	vertex	NOUN
ejpam-3890	216	11	in	in	ADP
ejpam-3890	216	12	γs(m0	γs(m0	PROPN
ejpam-3890	216	13	2	2	NUM
ejpam-3890	216	14	(	(	PUNCT
ejpam-3890	216	15	zn	zn	NOUN
ejpam-3890	216	16	)	)	PUNCT
ejpam-3890	216	17	)	)	PUNCT
ejpam-3890	216	18	is	be	AUX
ejpam-3890	216	19	1	1	NUM
ejpam-3890	216	20	.	.	PUNCT
ejpam-3890	217	1	thus	thus	ADV
ejpam-3890	217	2	,	,	PUNCT
ejpam-3890	217	3	the	the	DET
ejpam-3890	217	4	graph	graph	NOUN
ejpam-3890	217	5	should	should	AUX
ejpam-3890	217	6	consist	consist	VERB
ejpam-3890	217	7	only	only	ADV
ejpam-3890	217	8	of	of	ADP
ejpam-3890	217	9	disjoint	disjoint	ADJ
ejpam-3890	217	10	edges	edge	NOUN
ejpam-3890	217	11	.	.	PUNCT
ejpam-3890	218	1	for	for	ADP
ejpam-3890	218	2	an	an	DET
ejpam-3890	218	3	integer	integer	NOUN
ejpam-3890	218	4	n	n	PRON
ejpam-3890	218	5	≥	≥	NOUN
ejpam-3890	218	6	2	2	NUM
ejpam-3890	218	7	,	,	PUNCT
ejpam-3890	218	8	the	the	DET
ejpam-3890	218	9	maximum	maximum	ADJ
ejpam-3890	218	10	number	number	NOUN
ejpam-3890	218	11	of	of	ADP
ejpam-3890	218	12	disjoint	disjoint	NOUN
ejpam-3890	218	13	edges	edge	NOUN
ejpam-3890	218	14	is	be	AUX
ejpam-3890	218	15	bn2	bn2	VERB
ejpam-3890	218	16	c.	c.	PROPN
ejpam-3890	218	17	f.j	f.j	PROPN
ejpam-3890	218	18	.	.	PROPN
ejpam-3890	219	1	campeña	campeña	PROPN
ejpam-3890	219	2	et	et	PROPN
ejpam-3890	219	3	al	al	PROPN
ejpam-3890	219	4	.	.	PUNCT
ejpam-3890	219	5	/	/	SYM
ejpam-3890	219	6	eur	eur	PROPN
ejpam-3890	219	7	.	.	PUNCT
ejpam-3890	220	1	j.	j.	PROPN
ejpam-3890	220	2	pure	pure	PROPN
ejpam-3890	220	3	appl	appl	PROPN
ejpam-3890	220	4	.	.	PROPN
ejpam-3890	220	5	math	math	PROPN
ejpam-3890	220	6	,	,	PUNCT
ejpam-3890	220	7	14	14	NUM
ejpam-3890	220	8	(	(	PUNCT
ejpam-3890	220	9	1	1	NUM
ejpam-3890	220	10	)	)	PUNCT
ejpam-3890	220	11	(	(	PUNCT
ejpam-3890	220	12	2021	2021	NUM
ejpam-3890	220	13	)	)	PUNCT
ejpam-3890	220	14	,	,	PUNCT
ejpam-3890	220	15	268	268	NUM
ejpam-3890	220	16	-	-	SYM
ejpam-3890	220	17	277	277	NUM
ejpam-3890	220	18	274	274	NUM
ejpam-3890	220	19	theorem	theorem	NOUN
ejpam-3890	220	20	5	5	NUM
ejpam-3890	220	21	.	.	PUNCT
ejpam-3890	220	22	consider	consider	VERB
ejpam-3890	220	23	the	the	DET
ejpam-3890	220	24	zero	zero	NUM
ejpam-3890	220	25	ring	ring	NOUN
ejpam-3890	220	26	r	r	NOUN
ejpam-3890	220	27	=	=	SYM
ejpam-3890	220	28	m0	m0	NOUN
ejpam-3890	220	29	2	2	NUM
ejpam-3890	220	30	(	(	PUNCT
ejpam-3890	220	31	zn	zn	NOUN
ejpam-3890	220	32	)	)	PUNCT
ejpam-3890	220	33	and	and	CCONJ
ejpam-3890	220	34	s	s	X
ejpam-3890	220	35	⊂	⊂	X
ejpam-3890	220	36	r	r	NOUN
ejpam-3890	220	37	−	−	PROPN
ejpam-3890	220	38	{	{	PUNCT
ejpam-3890	220	39	0	0	NUM
ejpam-3890	220	40	}	}	PUNCT
ejpam-3890	220	41	,	,	PUNCT
ejpam-3890	220	42	where	where	SCONJ
ejpam-3890	220	43	|s|	|s|	VERB
ejpam-3890	220	44	=	=	VERB
ejpam-3890	220	45	m	m	VERB
ejpam-3890	220	46	>	>	X
ejpam-3890	220	47	1	1	X
ejpam-3890	220	48	.	.	PUNCT
ejpam-3890	221	1	let	let	VERB
ejpam-3890	221	2	ne	ne	VERB
ejpam-3890	221	3	=	=	PUNCT
ejpam-3890	221	4	|{ai	|{ai	PROPN
ejpam-3890	221	5	∈	∈	PROPN
ejpam-3890	221	6	s	s	PART
ejpam-3890	221	7	:	:	PUNCT
ejpam-3890	221	8	i	i	PRON
ejpam-3890	222	1	even	even	ADV
ejpam-3890	222	2	}	}	PUNCT
ejpam-3890	222	3	|	|	ADV
ejpam-3890	222	4	and	and	CCONJ
ejpam-3890	222	5	no	no	DET
ejpam-3890	222	6	=	=	NOUN
ejpam-3890	222	7	|{ai	|{ai	NOUN
ejpam-3890	222	8	∈	∈	PROPN
ejpam-3890	222	9	s	s	PART
ejpam-3890	222	10	:	:	PUNCT
ejpam-3890	222	11	i	i	PRON
ejpam-3890	222	12	odd	odd	ADJ
ejpam-3890	222	13	}	}	PUNCT
ejpam-3890	222	14	|	|	ADV
ejpam-3890	222	15	.	.	PUNCT
ejpam-3890	223	1	then	then	ADV
ejpam-3890	223	2	(	(	PUNCT
ejpam-3890	223	3	i	i	NOUN
ejpam-3890	223	4	)	)	PUNCT
ejpam-3890	223	5	if	if	SCONJ
ejpam-3890	223	6	n	n	PRON
ejpam-3890	223	7	is	be	AUX
ejpam-3890	223	8	odd	odd	ADJ
ejpam-3890	223	9	,	,	PUNCT
ejpam-3890	223	10	then	then	ADV
ejpam-3890	223	11	the	the	DET
ejpam-3890	223	12	number	number	NOUN
ejpam-3890	223	13	of	of	ADP
ejpam-3890	223	14	edges	edge	NOUN
ejpam-3890	223	15	in	in	ADP
ejpam-3890	223	16	γs(r	γs(r	NOUN
ejpam-3890	223	17	)	)	PUNCT
ejpam-3890	223	18	is	be	AUX
ejpam-3890	223	19	m(n−1	m(n−1	ADJ
ejpam-3890	223	20	)	)	PUNCT
ejpam-3890	223	21	2	2	NUM
ejpam-3890	223	22	.	.	PUNCT
ejpam-3890	224	1	(	(	PUNCT
ejpam-3890	224	2	ii	ii	NOUN
ejpam-3890	224	3	)	)	PUNCT
ejpam-3890	224	4	if	if	SCONJ
ejpam-3890	224	5	n	n	PRON
ejpam-3890	224	6	is	be	AUX
ejpam-3890	224	7	even	even	ADV
ejpam-3890	224	8	,	,	PUNCT
ejpam-3890	224	9	then	then	ADV
ejpam-3890	224	10	the	the	DET
ejpam-3890	224	11	number	number	NOUN
ejpam-3890	224	12	of	of	ADP
ejpam-3890	224	13	edges	edge	NOUN
ejpam-3890	224	14	in	in	ADP
ejpam-3890	224	15	γs(r	γs(r	NOUN
ejpam-3890	224	16	)	)	PUNCT
ejpam-3890	224	17	is	be	AUX
ejpam-3890	224	18	ne(n−	ne(n−	VERB
ejpam-3890	224	19	2	2	NUM
ejpam-3890	224	20	)	)	PUNCT
ejpam-3890	224	21	2	2	NUM
ejpam-3890	224	22	+	+	CCONJ
ejpam-3890	224	23	no(n	no(n	X
ejpam-3890	224	24	)	)	PUNCT
ejpam-3890	224	25	2	2	NUM
ejpam-3890	224	26	.	.	PUNCT
ejpam-3890	225	1	proof	proof	NOUN
ejpam-3890	225	2	.	.	PUNCT
ejpam-3890	226	1	consider	consider	VERB
ejpam-3890	226	2	the	the	DET
ejpam-3890	226	3	additive	additive	ADJ
ejpam-3890	226	4	group	group	NOUN
ejpam-3890	226	5	table	table	NOUN
ejpam-3890	226	6	of	of	ADP
ejpam-3890	226	7	zn	zn	PROPN
ejpam-3890	226	8	.	.	PUNCT
ejpam-3890	227	1	if	if	SCONJ
ejpam-3890	227	2	n	n	NOUN
ejpam-3890	227	3	is	be	AUX
ejpam-3890	227	4	odd	odd	ADJ
ejpam-3890	227	5	,	,	PUNCT
ejpam-3890	227	6	then	then	ADV
ejpam-3890	227	7	every	every	DET
ejpam-3890	227	8	element	element	NOUN
ejpam-3890	227	9	of	of	ADP
ejpam-3890	227	10	the	the	DET
ejpam-3890	227	11	group	group	NOUN
ejpam-3890	227	12	occurs	occur	VERB
ejpam-3890	227	13	exactly	exactly	ADV
ejpam-3890	227	14	(	(	PUNCT
ejpam-3890	227	15	n−1	n−1	PROPN
ejpam-3890	227	16	)	)	PUNCT
ejpam-3890	227	17	2	2	NUM
ejpam-3890	227	18	times	time	NOUN
ejpam-3890	227	19	above	above	ADP
ejpam-3890	227	20	the	the	DET
ejpam-3890	227	21	main	main	ADJ
ejpam-3890	227	22	diagonal	diagonal	ADJ
ejpam-3890	227	23	entries	entry	NOUN
ejpam-3890	227	24	of	of	ADP
ejpam-3890	227	25	the	the	DET
ejpam-3890	227	26	table	table	NOUN
ejpam-3890	227	27	.	.	PUNCT
ejpam-3890	228	1	thus	thus	ADV
ejpam-3890	228	2	,	,	PUNCT
ejpam-3890	228	3	if	if	SCONJ
ejpam-3890	228	4	|s|	|s|	PROPN
ejpam-3890	228	5	=	=	VERB
ejpam-3890	228	6	m	m	VERB
ejpam-3890	228	7	then	then	ADV
ejpam-3890	228	8	there	there	PRON
ejpam-3890	228	9	are	be	VERB
ejpam-3890	228	10	m(n−1	m(n−1	ADJ
ejpam-3890	228	11	)	)	PUNCT
ejpam-3890	228	12	2	2	NUM
ejpam-3890	228	13	distinct	distinct	ADJ
ejpam-3890	228	14	pairs	pair	NOUN
ejpam-3890	228	15	of	of	ADP
ejpam-3890	228	16	indices	index	NOUN
ejpam-3890	228	17	i	i	PRON
ejpam-3890	228	18	,	,	PUNCT
ejpam-3890	228	19	j	j	PROPN
ejpam-3890	228	20	such	such	ADJ
ejpam-3890	228	21	that	that	PRON
ejpam-3890	228	22	ai	ai	VERB
ejpam-3890	228	23	+	+	PROPN
ejpam-3890	228	24	aj	aj	PROPN
ejpam-3890	228	25	∈	∈	PROPN
ejpam-3890	228	26	s.	s.	PROPN
ejpam-3890	229	1	now	now	ADV
ejpam-3890	229	2	,	,	PUNCT
ejpam-3890	229	3	if	if	SCONJ
ejpam-3890	229	4	n	n	PRON
ejpam-3890	229	5	is	be	AUX
ejpam-3890	229	6	even	even	ADV
ejpam-3890	229	7	,	,	PUNCT
ejpam-3890	229	8	the	the	DET
ejpam-3890	229	9	number	number	NOUN
ejpam-3890	229	10	of	of	ADP
ejpam-3890	229	11	occurrences	occurrence	NOUN
ejpam-3890	229	12	of	of	ADP
ejpam-3890	229	13	an	an	DET
ejpam-3890	229	14	odd	odd	ADJ
ejpam-3890	229	15	index	index	NOUN
ejpam-3890	229	16	above	above	ADP
ejpam-3890	229	17	the	the	DET
ejpam-3890	229	18	main	main	ADJ
ejpam-3890	229	19	diagonal	diagonal	ADJ
ejpam-3890	229	20	entries	entry	NOUN
ejpam-3890	229	21	of	of	ADP
ejpam-3890	229	22	the	the	DET
ejpam-3890	229	23	table	table	NOUN
ejpam-3890	229	24	is	be	AUX
ejpam-3890	229	25	n	n	PRON
ejpam-3890	229	26	2	2	NUM
ejpam-3890	229	27	while	while	SCONJ
ejpam-3890	229	28	there	there	PRON
ejpam-3890	229	29	are	be	VERB
ejpam-3890	229	30	(	(	PUNCT
ejpam-3890	229	31	n−2	n−2	PROPN
ejpam-3890	229	32	)	)	PUNCT
ejpam-3890	229	33	2	2	NUM
ejpam-3890	229	34	occurrences	occurrence	NOUN
ejpam-3890	229	35	of	of	ADP
ejpam-3890	229	36	an	an	DET
ejpam-3890	229	37	odd	odd	ADJ
ejpam-3890	229	38	index	index	NOUN
ejpam-3890	229	39	above	above	ADP
ejpam-3890	229	40	the	the	DET
ejpam-3890	229	41	main	main	ADJ
ejpam-3890	229	42	diagonal	diagonal	ADJ
ejpam-3890	229	43	entries	entry	NOUN
ejpam-3890	229	44	.	.	PUNCT
ejpam-3890	230	1	hence	hence	ADV
ejpam-3890	230	2	,	,	PUNCT
ejpam-3890	230	3	if	if	SCONJ
ejpam-3890	230	4	there	there	PRON
ejpam-3890	230	5	are	be	VERB
ejpam-3890	230	6	ne	ne	NOUN
ejpam-3890	230	7	,	,	PUNCT
ejpam-3890	230	8	no	no	ADV
ejpam-3890	230	9	even	even	ADV
ejpam-3890	230	10	and	and	CCONJ
ejpam-3890	230	11	odd	odd	ADJ
ejpam-3890	230	12	indices	index	NOUN
ejpam-3890	230	13	respectively	respectively	ADV
ejpam-3890	230	14	in	in	ADP
ejpam-3890	230	15	s	s	PROPN
ejpam-3890	230	16	,	,	PUNCT
ejpam-3890	230	17	there	there	PRON
ejpam-3890	230	18	is	be	VERB
ejpam-3890	230	19	a	a	DET
ejpam-3890	230	20	total	total	NOUN
ejpam-3890	230	21	of	of	ADP
ejpam-3890	230	22	ne(n−	ne(n−	X
ejpam-3890	230	23	2	2	NUM
ejpam-3890	230	24	)	)	PUNCT
ejpam-3890	230	25	2	2	NUM
ejpam-3890	230	26	+	+	CCONJ
ejpam-3890	230	27	no(n	no(n	X
ejpam-3890	230	28	)	)	PUNCT
ejpam-3890	230	29	2	2	NUM
ejpam-3890	230	30	distinct	distinct	ADJ
ejpam-3890	230	31	pairs	pair	NOUN
ejpam-3890	230	32	of	of	ADP
ejpam-3890	230	33	indices	index	NOUN
ejpam-3890	230	34	i	i	PRON
ejpam-3890	230	35	,	,	PUNCT
ejpam-3890	230	36	j	j	PROPN
ejpam-3890	230	37	whose	whose	DET
ejpam-3890	230	38	sum	sum	NOUN
ejpam-3890	230	39	is	be	AUX
ejpam-3890	230	40	an	an	DET
ejpam-3890	230	41	index	index	NOUN
ejpam-3890	230	42	in	in	ADP
ejpam-3890	230	43	s.	s.	PROPN
ejpam-3890	230	44	.	.	PUNCT
ejpam-3890	231	1	in	in	ADP
ejpam-3890	231	2	theorem	theorem	NOUN
ejpam-3890	231	3	3	3	NUM
ejpam-3890	231	4	,	,	PUNCT
ejpam-3890	231	5	it	it	PRON
ejpam-3890	231	6	was	be	AUX
ejpam-3890	231	7	shown	show	VERB
ejpam-3890	231	8	that	that	SCONJ
ejpam-3890	231	9	pn	pn	PROPN
ejpam-3890	231	10	admits	admit	VERB
ejpam-3890	231	11	an	an	DET
ejpam-3890	231	12	efficient	efficient	ADJ
ejpam-3890	231	13	zero	zero	NUM
ejpam-3890	231	14	ring	ring	NOUN
ejpam-3890	231	15	labeling	labeling	NOUN
ejpam-3890	231	16	by	by	ADP
ejpam-3890	231	17	providing	provide	VERB
ejpam-3890	231	18	an	an	DET
ejpam-3890	231	19	explicit	explicit	ADJ
ejpam-3890	231	20	labeling	labeling	NOUN
ejpam-3890	231	21	of	of	ADP
ejpam-3890	231	22	its	its	PRON
ejpam-3890	231	23	vertices	vertex	NOUN
ejpam-3890	231	24	.	.	PUNCT
ejpam-3890	232	1	moreover	moreover	ADV
ejpam-3890	232	2	,	,	PUNCT
ejpam-3890	232	3	given	give	VERB
ejpam-3890	232	4	the	the	DET
ejpam-3890	232	5	zero	zero	NUM
ejpam-3890	232	6	ring	ring	NOUN
ejpam-3890	232	7	m0	m0	NOUN
ejpam-3890	232	8	2	2	NUM
ejpam-3890	232	9	(	(	PUNCT
ejpam-3890	232	10	zn	zn	NUM
ejpam-3890	232	11	)	)	PUNCT
ejpam-3890	232	12	,	,	PUNCT
ejpam-3890	232	13	the	the	DET
ejpam-3890	232	14	restricted	restricted	ADJ
ejpam-3890	232	15	zero	zero	NUM
ejpam-3890	232	16	ring	ring	NOUN
ejpam-3890	232	17	graph	graph	NOUN
ejpam-3890	232	18	γs(m0	γs(m0	X
ejpam-3890	232	19	2	2	NUM
ejpam-3890	232	20	(	(	PUNCT
ejpam-3890	232	21	zn	zn	NOUN
ejpam-3890	232	22	)	)	PUNCT
ejpam-3890	232	23	)	)	PUNCT
ejpam-3890	232	24	where	where	SCONJ
ejpam-3890	232	25	s	s	VERB
ejpam-3890	232	26	=	=	PUNCT
ejpam-3890	232	27	{	{	PUNCT
ejpam-3890	232	28	a1	a1	PROPN
ejpam-3890	232	29	,	,	PUNCT
ejpam-3890	232	30	a2	a2	PROPN
ejpam-3890	232	31	}	}	PUNCT
ejpam-3890	232	32	is	be	AUX
ejpam-3890	232	33	isomorphic	isomorphic	ADJ
ejpam-3890	232	34	to	to	PART
ejpam-3890	232	35	pn	pn	PROPN
ejpam-3890	232	36	as	as	SCONJ
ejpam-3890	232	37	shown	show	VERB
ejpam-3890	232	38	in	in	ADP
ejpam-3890	232	39	figure	figure	NOUN
ejpam-3890	232	40	4	4	NUM
ejpam-3890	232	41	.	.	PUNCT
ejpam-3890	232	42	..........	..........	PUNCT
ejpam-3890	232	43	.............................................	.............................................	PUNCT
ejpam-3890	232	44	.	.	PUNCT
ejpam-3890	233	1	..........	..........	PUNCT
ejpam-3890	233	2	.............................................	.............................................	PUNCT
ejpam-3890	233	3	.	.	PUNCT
ejpam-3890	234	1	..........	..........	PUNCT
ejpam-3890	234	2	.............................................	.............................................	PUNCT
ejpam-3890	234	3	.	.	PUNCT
ejpam-3890	235	1	..........	..........	PUNCT
ejpam-3890	235	2	.............................................	.............................................	PUNCT
ejpam-3890	235	3	.	.	PUNCT
ejpam-3890	236	1	..........	..........	PUNCT
ejpam-3890	236	2	.............................................	.............................................	PUNCT
ejpam-3890	236	3	.	.	PUNCT
ejpam-3890	237	1	..........	..........	PUNCT
ejpam-3890	237	2	.............................................	.............................................	PUNCT
ejpam-3890	237	3	.	.	PUNCT
ejpam-3890	238	1	..........	..........	PUNCT
ejpam-3890	238	2	.............................................	.............................................	PUNCT
ejpam-3890	238	3	.	.	PUNCT
ejpam-3890	239	1	..........	..........	PUNCT
ejpam-3890	239	2	.............................................	.............................................	PUNCT
ejpam-3890	239	3	.	.	PUNCT
ejpam-3890	240	1	..........	..........	PUNCT
ejpam-3890	240	2	.............................................	.............................................	PUNCT
ejpam-3890	240	3	.	.	PUNCT
ejpam-3890	241	1	..........	..........	PUNCT
ejpam-3890	241	2	.............................................	.............................................	PUNCT
ejpam-3890	241	3	.	.	PUNCT
ejpam-3890	242	1	.........	.........	PUNCT
ejpam-3890	242	2	........	........	PUNCT
ejpam-3890	242	3	........	........	PUNCT
ejpam-3890	242	4	........	........	PUNCT
ejpam-3890	243	1	........	........	PUNCT
ejpam-3890	243	2	.	.	PUNCT
ejpam-3890	244	1	.........	.........	PUNCT
ejpam-3890	244	2	........	........	PUNCT
ejpam-3890	244	3	........	........	PUNCT
ejpam-3890	244	4	........	........	PUNCT
ejpam-3890	245	1	........	........	PUNCT
ejpam-3890	245	2	.	.	PUNCT
ejpam-3890	246	1	.........	.........	PUNCT
ejpam-3890	246	2	........	........	PUNCT
ejpam-3890	246	3	........	........	PUNCT
ejpam-3890	246	4	........	........	PUNCT
ejpam-3890	247	1	........	........	PUNCT
ejpam-3890	247	2	.	.	PUNCT
ejpam-3890	248	1	.........	.........	PUNCT
ejpam-3890	248	2	........	........	PUNCT
ejpam-3890	248	3	........	........	PUNCT
ejpam-3890	248	4	........	........	PUNCT
ejpam-3890	249	1	........	........	PUNCT
ejpam-3890	249	2	.	.	PUNCT
ejpam-3890	250	1	.........	.........	PUNCT
ejpam-3890	250	2	........	........	PUNCT
ejpam-3890	250	3	........	........	PUNCT
ejpam-3890	250	4	........	........	PUNCT
ejpam-3890	250	5	........	........	PUNCT
ejpam-3890	250	6	.............................................................................	.............................................................................	PUNCT
ejpam-3890	250	7	............................................................................	............................................................................	PUNCT
ejpam-3890	251	1	............................................................................	............................................................................	PUNCT
ejpam-3890	252	1	a0	a0	PROPN
ejpam-3890	252	2	a1	a1	PROPN
ejpam-3890	252	3	a2	a2	PROPN
ejpam-3890	252	4	a3	a3	NOUN
ejpam-3890	252	5	an−2	an−2	PROPN
ejpam-3890	252	6	2	2	NUM
ejpam-3890	252	7	an	an	DET
ejpam-3890	252	8	2	2	NUM
ejpam-3890	252	9	an+2	an+2	NUM
ejpam-3890	252	10	2	2	NUM
ejpam-3890	252	11	an+4	an+4	NUM
ejpam-3890	252	12	2	2	NUM
ejpam-3890	252	13	·	·	PUNCT
ejpam-3890	252	14	·	·	PUNCT
ejpam-3890	252	15	·	·	PUNCT
ejpam-3890	252	16	·	·	PUNCT
ejpam-3890	252	17	·	·	PUNCT
ejpam-3890	252	18	·	·	PUNCT
ejpam-3890	253	1	an−1an−2	an−1an−2	CCONJ
ejpam-3890	253	2	figure	figure	NOUN
ejpam-3890	253	3	4	4	NUM
ejpam-3890	253	4	:	:	PUNCT
ejpam-3890	253	5	the	the	DET
ejpam-3890	253	6	graphs	graph	NOUN
ejpam-3890	253	7	γ{a1,a2}(m	γ{a1,a2}(m	ADJ
ejpam-3890	253	8	0	0	NUM
ejpam-3890	253	9	2	2	NUM
ejpam-3890	253	10	(	(	PUNCT
ejpam-3890	253	11	zn	zn	NOUN
ejpam-3890	253	12	)	)	PUNCT
ejpam-3890	253	13	)	)	PUNCT
ejpam-3890	254	1	lemma	lemma	PROPN
ejpam-3890	254	2	4	4	X
ejpam-3890	254	3	.	.	PUNCT
ejpam-3890	255	1	let	let	VERB
ejpam-3890	255	2	r	r	NOUN
ejpam-3890	255	3	=	=	SYM
ejpam-3890	255	4	m0	m0	NOUN
ejpam-3890	255	5	2	2	NUM
ejpam-3890	255	6	(	(	PUNCT
ejpam-3890	255	7	zn	zn	NOUN
ejpam-3890	255	8	)	)	PUNCT
ejpam-3890	255	9	and	and	CCONJ
ejpam-3890	255	10	si	si	X
ejpam-3890	256	1	=	=	NOUN
ejpam-3890	256	2	{	{	PUNCT
ejpam-3890	256	3	ai	ai	NOUN
ejpam-3890	256	4	,	,	PUNCT
ejpam-3890	256	5	ai+1	ai+1	NOUN
ejpam-3890	256	6	}	}	PUNCT
ejpam-3890	256	7	where	where	SCONJ
ejpam-3890	256	8	0	0	NUM
ejpam-3890	256	9	6∈	6∈	NOUN
ejpam-3890	256	10	si	si	INTJ
ejpam-3890	256	11	.	.	PUNCT
ejpam-3890	257	1	then	then	ADV
ejpam-3890	257	2	γsi(r	γsi(r	PROPN
ejpam-3890	257	3	)	)	PUNCT
ejpam-3890	257	4	∼=	∼=	PROPN
ejpam-3890	257	5	γsj	γsj	NOUN
ejpam-3890	257	6	(	(	PUNCT
ejpam-3890	257	7	r	r	NOUN
ejpam-3890	257	8	)	)	PUNCT
ejpam-3890	257	9	for	for	ADP
ejpam-3890	257	10	i	i	PRON
ejpam-3890	257	11	,	,	PUNCT
ejpam-3890	257	12	j	j	PROPN
ejpam-3890	257	13	∈	∈	PROPN
ejpam-3890	257	14	{	{	PUNCT
ejpam-3890	257	15	1	1	NUM
ejpam-3890	257	16	,	,	PUNCT
ejpam-3890	257	17	2	2	NUM
ejpam-3890	257	18	,	,	PUNCT
ejpam-3890	257	19	.	.	PUNCT
ejpam-3890	257	20	.	.	PUNCT
ejpam-3890	257	21	.	.	PUNCT
ejpam-3890	258	1	,	,	PUNCT
ejpam-3890	258	2	n−	n−	NOUN
ejpam-3890	258	3	2	2	NUM
ejpam-3890	258	4	}	}	PUNCT
ejpam-3890	258	5	.	.	PUNCT
ejpam-3890	259	1	proof	proof	NOUN
ejpam-3890	259	2	.	.	PUNCT
ejpam-3890	260	1	since	since	SCONJ
ejpam-3890	260	2	γ{1,2}(r	γ{1,2}(r	NUM
ejpam-3890	260	3	)	)	PUNCT
ejpam-3890	260	4	∼=	∼=	PROPN
ejpam-3890	260	5	pn	pn	NOUN
ejpam-3890	260	6	,	,	PUNCT
ejpam-3890	260	7	we	we	PRON
ejpam-3890	260	8	need	need	VERB
ejpam-3890	260	9	to	to	PART
ejpam-3890	260	10	show	show	VERB
ejpam-3890	260	11	that	that	SCONJ
ejpam-3890	260	12	each	each	PRON
ejpam-3890	260	13	of	of	ADP
ejpam-3890	260	14	γsi(r	γsi(r	PROPN
ejpam-3890	260	15	)	)	PUNCT
ejpam-3890	260	16	∼=	∼=	PROPN
ejpam-3890	260	17	pn	pn	NOUN
ejpam-3890	260	18	.	.	PUNCT
ejpam-3890	260	19	let	let	VERB
ejpam-3890	260	20	n	n	PRON
ejpam-3890	260	21	be	be	AUX
ejpam-3890	260	22	even	even	ADV
ejpam-3890	260	23	.	.	PUNCT
ejpam-3890	261	1	for	for	ADP
ejpam-3890	261	2	the	the	DET
ejpam-3890	261	3	set	set	NOUN
ejpam-3890	261	4	si	si	NOUN
ejpam-3890	261	5	,	,	PUNCT
ejpam-3890	261	6	consider	consider	VERB
ejpam-3890	261	7	the	the	DET
ejpam-3890	261	8	arrangement	arrangement	NOUN
ejpam-3890	261	9	of	of	ADP
ejpam-3890	261	10	the	the	DET
ejpam-3890	261	11	elements	element	NOUN
ejpam-3890	261	12	of	of	ADP
ejpam-3890	261	13	r	r	NOUN
ejpam-3890	261	14	in	in	ADP
ejpam-3890	261	15	a	a	DET
ejpam-3890	261	16	matrix	matrix	NOUN
ejpam-3890	261	17	of	of	ADP
ejpam-3890	261	18	size	size	NOUN
ejpam-3890	261	19	2	2	NUM
ejpam-3890	261	20	×	×	NOUN
ejpam-3890	261	21	n	n	PRON
ejpam-3890	261	22	2	2	NUM
ejpam-3890	261	23	in	in	ADP
ejpam-3890	261	24	the	the	DET
ejpam-3890	261	25	order	order	NOUN
ejpam-3890	261	26	a0	a0	NOUN
ejpam-3890	261	27	,	,	PUNCT
ejpam-3890	261	28	a1	a1	PROPN
ejpam-3890	261	29	,	,	PUNCT
ejpam-3890	261	30	.	.	PUNCT
ejpam-3890	261	31	.	.	PUNCT
ejpam-3890	262	1	.	.	PUNCT
ejpam-3890	263	1	,	,	PUNCT
ejpam-3890	263	2	an−1	an−1	ADJ
ejpam-3890	263	3	in	in	ADP
ejpam-3890	263	4	a	a	DET
ejpam-3890	263	5	counterclockwise	counterclockwise	NOUN
ejpam-3890	263	6	manner	manner	NOUN
ejpam-3890	263	7	such	such	ADJ
ejpam-3890	263	8	that	that	SCONJ
ejpam-3890	263	9	the	the	DET
ejpam-3890	263	10	entry	entry	NOUN
ejpam-3890	263	11	of	of	ADP
ejpam-3890	263	12	the	the	DET
ejpam-3890	263	13	first	first	ADJ
ejpam-3890	263	14	row	row	NOUN
ejpam-3890	263	15	column	column	NOUN
ejpam-3890	263	16	b	b	NOUN
ejpam-3890	263	17	i2c	i2c	NOUN
ejpam-3890	263	18	+	+	CCONJ
ejpam-3890	263	19	1	1	NUM
ejpam-3890	263	20	is	be	AUX
ejpam-3890	263	21	a0	a0	PROPN
ejpam-3890	263	22	.	.	PUNCT
ejpam-3890	264	1	in	in	ADP
ejpam-3890	264	2	this	this	DET
ejpam-3890	264	3	arrangement	arrangement	NOUN
ejpam-3890	264	4	,	,	PUNCT
ejpam-3890	264	5	the	the	DET
ejpam-3890	264	6	sum	sum	NOUN
ejpam-3890	264	7	of	of	ADP
ejpam-3890	264	8	the	the	DET
ejpam-3890	264	9	entries	entry	NOUN
ejpam-3890	264	10	for	for	ADP
ejpam-3890	264	11	each	each	DET
ejpam-3890	264	12	column	column	NOUN
ejpam-3890	264	13	taken	take	VERB
ejpam-3890	264	14	under	under	ADP
ejpam-3890	264	15	addition	addition	NOUN
ejpam-3890	264	16	modulo	modulo	NOUN
ejpam-3890	264	17	n	n	PRON
ejpam-3890	264	18	is	be	AUX
ejpam-3890	264	19	ai	ai	VERB
ejpam-3890	264	20	if	if	SCONJ
ejpam-3890	264	21	i	i	PRON
ejpam-3890	264	22	is	be	AUX
ejpam-3890	264	23	odd	odd	ADJ
ejpam-3890	264	24	and	and	CCONJ
ejpam-3890	264	25	ai+1	ai+1	INTJ
ejpam-3890	264	26	if	if	SCONJ
ejpam-3890	264	27	i	i	PRON
ejpam-3890	264	28	is	be	AUX
ejpam-3890	264	29	even	even	ADV
ejpam-3890	264	30	.	.	PUNCT
ejpam-3890	265	1	the	the	DET
ejpam-3890	265	2	sum	sum	NOUN
ejpam-3890	265	3	of	of	ADP
ejpam-3890	265	4	the	the	DET
ejpam-3890	265	5	(	(	PUNCT
ejpam-3890	265	6	1	1	NUM
ejpam-3890	265	7	,	,	PUNCT
ejpam-3890	265	8	j)-entry	j)-entry	NOUN
ejpam-3890	265	9	and	and	CCONJ
ejpam-3890	265	10	(	(	PUNCT
ejpam-3890	265	11	2	2	NUM
ejpam-3890	265	12	,	,	PUNCT
ejpam-3890	265	13	j	j	PROPN
ejpam-3890	265	14	+	+	CCONJ
ejpam-3890	265	15	1	1	X
ejpam-3890	265	16	)	)	PUNCT
ejpam-3890	265	17	entry	entry	NOUN
ejpam-3890	265	18	is	be	AUX
ejpam-3890	265	19	ai+1	ai+1	NUM
ejpam-3890	265	20	for	for	ADP
ejpam-3890	265	21	j	j	PROPN
ejpam-3890	265	22	=	=	SYM
ejpam-3890	265	23	1	1	NUM
ejpam-3890	265	24	,	,	PUNCT
ejpam-3890	265	25	2	2	NUM
ejpam-3890	265	26	,	,	PUNCT
ejpam-3890	265	27	.	.	PUNCT
ejpam-3890	265	28	.	.	PUNCT
ejpam-3890	266	1	.	.	PUNCT
ejpam-3890	267	1	,	,	PUNCT
ejpam-3890	267	2	n2	n2	ADJ
ejpam-3890	267	3	−	−	PROPN
ejpam-3890	267	4	1	1	NUM
ejpam-3890	267	5	if	if	SCONJ
ejpam-3890	267	6	i	i	PRON
ejpam-3890	267	7	is	be	AUX
ejpam-3890	267	8	odd	odd	ADJ
ejpam-3890	267	9	,	,	PUNCT
ejpam-3890	267	10	and	and	CCONJ
ejpam-3890	267	11	the	the	DET
ejpam-3890	267	12	sum	sum	NOUN
ejpam-3890	267	13	of	of	ADP
ejpam-3890	267	14	the	the	DET
ejpam-3890	267	15	(	(	PUNCT
ejpam-3890	267	16	2	2	NUM
ejpam-3890	267	17	,	,	PUNCT
ejpam-3890	267	18	j)-entry	j)-entry	NOUN
ejpam-3890	267	19	and	and	CCONJ
ejpam-3890	267	20	(	(	PUNCT
ejpam-3890	267	21	1	1	NUM
ejpam-3890	267	22	,	,	PUNCT
ejpam-3890	267	23	j	j	PROPN
ejpam-3890	267	24	+	+	CCONJ
ejpam-3890	267	25	1	1	X
ejpam-3890	267	26	)	)	PUNCT
ejpam-3890	267	27	entry	entry	NOUN
ejpam-3890	267	28	is	be	AUX
ejpam-3890	267	29	ai	ai	VERB
ejpam-3890	267	30	for	for	ADP
ejpam-3890	267	31	j	j	PROPN
ejpam-3890	267	32	=	=	SYM
ejpam-3890	267	33	1	1	NUM
ejpam-3890	267	34	,	,	PUNCT
ejpam-3890	267	35	2	2	NUM
ejpam-3890	267	36	,	,	PUNCT
ejpam-3890	267	37	.	.	PUNCT
ejpam-3890	267	38	.	.	PUNCT
ejpam-3890	267	39	.	.	PUNCT
ejpam-3890	268	1	,	,	PUNCT
ejpam-3890	268	2	n2	n2	ADJ
ejpam-3890	268	3	−	−	PROPN
ejpam-3890	268	4	1	1	NUM
ejpam-3890	268	5	if	if	SCONJ
ejpam-3890	268	6	i	i	PRON
ejpam-3890	268	7	is	be	AUX
ejpam-3890	268	8	even	even	ADV
ejpam-3890	268	9	.	.	PUNCT
ejpam-3890	269	1	with	with	ADP
ejpam-3890	269	2	this	this	DET
ejpam-3890	269	3	arrangement	arrangement	NOUN
ejpam-3890	269	4	of	of	ADP
ejpam-3890	269	5	the	the	DET
ejpam-3890	269	6	elements	element	NOUN
ejpam-3890	269	7	of	of	ADP
ejpam-3890	269	8	r	r	NOUN
ejpam-3890	269	9	,	,	PUNCT
ejpam-3890	269	10	it	it	PRON
ejpam-3890	269	11	is	be	AUX
ejpam-3890	269	12	easy	easy	ADJ
ejpam-3890	269	13	to	to	PART
ejpam-3890	269	14	see	see	VERB
ejpam-3890	269	15	that	that	SCONJ
ejpam-3890	269	16	the	the	DET
ejpam-3890	269	17	graph	graph	NOUN
ejpam-3890	269	18	γsi(r	γsi(r	PROPN
ejpam-3890	269	19	)	)	PUNCT
ejpam-3890	269	20	is	be	AUX
ejpam-3890	269	21	a	a	DET
ejpam-3890	269	22	path	path	NOUN
ejpam-3890	269	23	on	on	ADP
ejpam-3890	269	24	n	n	DET
ejpam-3890	269	25	vertices	vertex	NOUN
ejpam-3890	269	26	.	.	PUNCT
ejpam-3890	270	1	let	let	VERB
ejpam-3890	270	2	n	n	PRON
ejpam-3890	270	3	be	be	AUX
ejpam-3890	270	4	an	an	DET
ejpam-3890	270	5	odd	odd	ADJ
ejpam-3890	270	6	integer	integer	NOUN
ejpam-3890	270	7	,	,	PUNCT
ejpam-3890	270	8	we	we	PRON
ejpam-3890	270	9	now	now	ADV
ejpam-3890	270	10	consider	consider	VERB
ejpam-3890	270	11	the	the	DET
ejpam-3890	270	12	arrangement	arrangement	NOUN
ejpam-3890	270	13	of	of	ADP
ejpam-3890	270	14	the	the	DET
ejpam-3890	270	15	elements	element	NOUN
ejpam-3890	270	16	of	of	ADP
ejpam-3890	270	17	r	r	NOUN
ejpam-3890	270	18	in	in	ADP
ejpam-3890	270	19	array	array	NOUN
ejpam-3890	270	20	with	with	ADP
ejpam-3890	270	21	2	2	NUM
ejpam-3890	270	22	rows	row	NOUN
ejpam-3890	270	23	such	such	ADJ
ejpam-3890	270	24	that	that	SCONJ
ejpam-3890	270	25	the	the	DET
ejpam-3890	270	26	first	first	ADJ
ejpam-3890	270	27	row	row	NOUN
ejpam-3890	270	28	contains	contain	VERB
ejpam-3890	270	29	bn2	bn2	NOUN
ejpam-3890	270	30	c	c	NOUN
ejpam-3890	270	31	columns	column	NOUN
ejpam-3890	270	32	when	when	SCONJ
ejpam-3890	270	33	i	i	PRON
ejpam-3890	270	34	is	be	AUX
ejpam-3890	270	35	odd	odd	ADJ
ejpam-3890	270	36	and	and	CCONJ
ejpam-3890	270	37	dn2	dn2	NOUN
ejpam-3890	270	38	e	e	NOUN
ejpam-3890	270	39	columns	column	NOUN
ejpam-3890	270	40	when	when	SCONJ
ejpam-3890	270	41	i	i	PRON
ejpam-3890	270	42	is	be	AUX
ejpam-3890	270	43	even	even	ADV
ejpam-3890	270	44	.	.	PUNCT
ejpam-3890	271	1	the	the	DET
ejpam-3890	271	2	second	second	ADJ
ejpam-3890	271	3	row	row	NOUN
ejpam-3890	271	4	always	always	ADV
ejpam-3890	271	5	contains	contain	VERB
ejpam-3890	271	6	dn2	dn2	NOUN
ejpam-3890	271	7	e	e	NOUN
ejpam-3890	271	8	columns	column	NOUN
ejpam-3890	271	9	.	.	PUNCT
ejpam-3890	272	1	the	the	DET
ejpam-3890	272	2	elements	element	NOUN
ejpam-3890	272	3	of	of	ADP
ejpam-3890	272	4	r	r	NOUN
ejpam-3890	272	5	are	be	AUX
ejpam-3890	272	6	arranged	arrange	VERB
ejpam-3890	272	7	in	in	ADP
ejpam-3890	272	8	the	the	DET
ejpam-3890	272	9	following	follow	VERB
ejpam-3890	272	10	order	order	NOUN
ejpam-3890	272	11	a0	a0	PROPN
ejpam-3890	272	12	,	,	PUNCT
ejpam-3890	272	13	a1	a1	PROPN
ejpam-3890	272	14	,	,	PUNCT
ejpam-3890	272	15	.	.	PUNCT
ejpam-3890	272	16	.	.	PUNCT
ejpam-3890	273	1	.	.	PUNCT
ejpam-3890	274	1	,	,	PUNCT
ejpam-3890	274	2	an−1	an−1	ADJ
ejpam-3890	274	3	in	in	ADP
ejpam-3890	274	4	a	a	DET
ejpam-3890	274	5	clockwise	clockwise	NOUN
ejpam-3890	274	6	manner	manner	NOUN
ejpam-3890	274	7	such	such	ADJ
ejpam-3890	274	8	that	that	SCONJ
ejpam-3890	274	9	the	the	DET
ejpam-3890	274	10	entry	entry	NOUN
ejpam-3890	274	11	of	of	ADP
ejpam-3890	274	12	the	the	DET
ejpam-3890	274	13	first	first	ADJ
ejpam-3890	274	14	row	row	NOUN
ejpam-3890	274	15	column	column	NOUN
ejpam-3890	274	16	b	b	PROPN
ejpam-3890	275	1	i2c+	i2c+	VERB
ejpam-3890	275	2	1	1	NUM
ejpam-3890	275	3	is	be	AUX
ejpam-3890	275	4	a0	a0	PROPN
ejpam-3890	275	5	.	.	PUNCT
ejpam-3890	276	1	in	in	ADP
ejpam-3890	276	2	a	a	DET
ejpam-3890	276	3	similar	similar	ADJ
ejpam-3890	276	4	manner	manner	NOUN
ejpam-3890	276	5	,	,	PUNCT
ejpam-3890	276	6	the	the	DET
ejpam-3890	276	7	sum	sum	NOUN
ejpam-3890	276	8	of	of	ADP
ejpam-3890	276	9	the	the	DET
ejpam-3890	276	10	entries	entry	NOUN
ejpam-3890	276	11	for	for	ADP
ejpam-3890	276	12	f.j	f.j	PROPN
ejpam-3890	276	13	.	.	PROPN
ejpam-3890	276	14	campeña	campeña	PROPN
ejpam-3890	276	15	et	et	PROPN
ejpam-3890	276	16	al	al	PROPN
ejpam-3890	276	17	.	.	PUNCT
ejpam-3890	276	18	/	/	SYM
ejpam-3890	276	19	eur	eur	PROPN
ejpam-3890	276	20	.	.	PUNCT
ejpam-3890	277	1	j.	j.	PROPN
ejpam-3890	277	2	pure	pure	PROPN
ejpam-3890	277	3	appl	appl	PROPN
ejpam-3890	277	4	.	.	PROPN
ejpam-3890	277	5	math	math	PROPN
ejpam-3890	277	6	,	,	PUNCT
ejpam-3890	277	7	14	14	NUM
ejpam-3890	277	8	(	(	PUNCT
ejpam-3890	277	9	1	1	NUM
ejpam-3890	277	10	)	)	PUNCT
ejpam-3890	277	11	(	(	PUNCT
ejpam-3890	277	12	2021	2021	NUM
ejpam-3890	277	13	)	)	PUNCT
ejpam-3890	277	14	,	,	PUNCT
ejpam-3890	277	15	268	268	NUM
ejpam-3890	277	16	-	-	SYM
ejpam-3890	277	17	277	277	NUM
ejpam-3890	277	18	275	275	NUM
ejpam-3890	277	19	each	each	DET
ejpam-3890	277	20	column	column	NOUN
ejpam-3890	277	21	taken	take	VERB
ejpam-3890	277	22	under	under	ADP
ejpam-3890	277	23	addition	addition	NOUN
ejpam-3890	277	24	modulo	modulo	NOUN
ejpam-3890	277	25	n	n	PRON
ejpam-3890	277	26	is	be	AUX
ejpam-3890	277	27	ai	ai	VERB
ejpam-3890	277	28	if	if	SCONJ
ejpam-3890	277	29	i	i	PRON
ejpam-3890	277	30	is	be	AUX
ejpam-3890	277	31	odd	odd	ADJ
ejpam-3890	277	32	and	and	CCONJ
ejpam-3890	277	33	ai+1	ai+1	INTJ
ejpam-3890	277	34	if	if	SCONJ
ejpam-3890	277	35	i	i	PRON
ejpam-3890	277	36	is	be	AUX
ejpam-3890	277	37	even	even	ADV
ejpam-3890	277	38	.	.	PUNCT
ejpam-3890	278	1	using	use	VERB
ejpam-3890	278	2	the	the	DET
ejpam-3890	278	3	same	same	ADJ
ejpam-3890	278	4	arguments	argument	NOUN
ejpam-3890	278	5	in	in	ADP
ejpam-3890	278	6	the	the	DET
ejpam-3890	278	7	case	case	NOUN
ejpam-3890	278	8	for	for	ADP
ejpam-3890	278	9	an	an	DET
ejpam-3890	278	10	even	even	ADJ
ejpam-3890	278	11	n	n	CCONJ
ejpam-3890	278	12	,	,	PUNCT
ejpam-3890	278	13	the	the	DET
ejpam-3890	278	14	sum	sum	NOUN
ejpam-3890	278	15	of	of	ADP
ejpam-3890	278	16	the	the	DET
ejpam-3890	278	17	(	(	PUNCT
ejpam-3890	278	18	1	1	NUM
ejpam-3890	278	19	,	,	PUNCT
ejpam-3890	278	20	j)-entry	j)-entry	NOUN
ejpam-3890	278	21	and	and	CCONJ
ejpam-3890	278	22	(	(	PUNCT
ejpam-3890	278	23	2	2	NUM
ejpam-3890	278	24	,	,	PUNCT
ejpam-3890	278	25	j	j	PROPN
ejpam-3890	278	26	+	+	NOUN
ejpam-3890	278	27	1)-entry	1)-entry	NUM
ejpam-3890	278	28	is	be	AUX
ejpam-3890	278	29	ai+1	ai+1	NUM
ejpam-3890	278	30	for	for	ADP
ejpam-3890	278	31	j	j	PROPN
ejpam-3890	278	32	=	=	SYM
ejpam-3890	278	33	1	1	NUM
ejpam-3890	278	34	,	,	PUNCT
ejpam-3890	278	35	2	2	NUM
ejpam-3890	278	36	,	,	PUNCT
ejpam-3890	278	37	.	.	PUNCT
ejpam-3890	278	38	.	.	PUNCT
ejpam-3890	279	1	.	.	PUNCT
ejpam-3890	280	1	,	,	PUNCT
ejpam-3890	280	2	n2	n2	PROPN
ejpam-3890	280	3	−1	−1	NOUN
ejpam-3890	280	4	if	if	SCONJ
ejpam-3890	280	5	i	i	PRON
ejpam-3890	280	6	is	be	AUX
ejpam-3890	280	7	odd	odd	ADJ
ejpam-3890	280	8	,	,	PUNCT
ejpam-3890	280	9	and	and	CCONJ
ejpam-3890	280	10	the	the	DET
ejpam-3890	280	11	sum	sum	NOUN
ejpam-3890	280	12	of	of	ADP
ejpam-3890	280	13	the	the	DET
ejpam-3890	280	14	(	(	PUNCT
ejpam-3890	280	15	2	2	NUM
ejpam-3890	280	16	,	,	PUNCT
ejpam-3890	280	17	j)-entry	j)-entry	NOUN
ejpam-3890	280	18	and	and	CCONJ
ejpam-3890	280	19	(	(	PUNCT
ejpam-3890	280	20	1	1	NUM
ejpam-3890	280	21	,	,	PUNCT
ejpam-3890	280	22	j+1)-entry	j+1)-entry	NOUN
ejpam-3890	280	23	is	be	AUX
ejpam-3890	280	24	ai	ai	VERB
ejpam-3890	280	25	for	for	ADP
ejpam-3890	280	26	j	j	PROPN
ejpam-3890	280	27	=	=	SYM
ejpam-3890	280	28	1	1	NUM
ejpam-3890	280	29	,	,	PUNCT
ejpam-3890	280	30	2	2	NUM
ejpam-3890	280	31	,	,	PUNCT
ejpam-3890	280	32	.	.	PUNCT
ejpam-3890	280	33	.	.	PUNCT
ejpam-3890	281	1	.	.	PUNCT
ejpam-3890	282	1	,	,	PUNCT
ejpam-3890	282	2	n2	n2	ADJ
ejpam-3890	282	3	−	−	PROPN
ejpam-3890	282	4	1	1	NUM
ejpam-3890	282	5	if	if	SCONJ
ejpam-3890	282	6	i	i	PRON
ejpam-3890	282	7	is	be	AUX
ejpam-3890	282	8	even	even	ADV
ejpam-3890	282	9	.	.	PUNCT
ejpam-3890	283	1	..........	..........	PUNCT
ejpam-3890	283	2	.............................................	.............................................	PUNCT
ejpam-3890	283	3	.	.	PUNCT
ejpam-3890	284	1	..........	..........	PUNCT
ejpam-3890	284	2	.............................................	.............................................	PUNCT
ejpam-3890	284	3	.	.	PUNCT
ejpam-3890	285	1	..........	..........	PUNCT
ejpam-3890	285	2	.............................................	.............................................	PUNCT
ejpam-3890	285	3	.	.	PUNCT
ejpam-3890	286	1	..........	..........	PUNCT
ejpam-3890	286	2	.............................................	.............................................	PUNCT
ejpam-3890	286	3	.	.	PUNCT
ejpam-3890	287	1	..........	..........	PUNCT
ejpam-3890	287	2	.............................................	.............................................	PUNCT
ejpam-3890	287	3	.	.	PUNCT
ejpam-3890	288	1	..........	..........	PUNCT
ejpam-3890	288	2	.............................................	.............................................	PUNCT
ejpam-3890	288	3	.	.	PUNCT
ejpam-3890	289	1	..........	..........	PUNCT
ejpam-3890	289	2	.............................................	.............................................	PUNCT
ejpam-3890	289	3	.	.	PUNCT
ejpam-3890	290	1	..........	..........	PUNCT
ejpam-3890	290	2	.............................................	.............................................	PUNCT
ejpam-3890	290	3	.	.	PUNCT
ejpam-3890	291	1	.........	.........	PUNCT
ejpam-3890	291	2	........	........	PUNCT
ejpam-3890	291	3	........	........	PUNCT
ejpam-3890	291	4	........	........	PUNCT
ejpam-3890	292	1	........	........	PUNCT
ejpam-3890	292	2	.	.	PUNCT
ejpam-3890	293	1	.........	.........	PUNCT
ejpam-3890	293	2	........	........	PUNCT
ejpam-3890	293	3	........	........	PUNCT
ejpam-3890	293	4	........	........	PUNCT
ejpam-3890	294	1	........	........	PUNCT
ejpam-3890	294	2	.	.	PUNCT
ejpam-3890	295	1	.........	.........	PUNCT
ejpam-3890	295	2	........	........	PUNCT
ejpam-3890	295	3	........	........	PUNCT
ejpam-3890	295	4	........	........	PUNCT
ejpam-3890	296	1	........	........	PUNCT
ejpam-3890	296	2	.	.	PUNCT
ejpam-3890	297	1	.........	.........	PUNCT
ejpam-3890	297	2	........	........	PUNCT
ejpam-3890	297	3	........	........	PUNCT
ejpam-3890	297	4	........	........	PUNCT
ejpam-3890	298	1	........	........	PUNCT
ejpam-3890	298	2	..................................................................	..................................................................	PUNCT
ejpam-3890	299	1	.................................................................	.................................................................	PUNCT
ejpam-3890	299	2	.................................................................	.................................................................	PUNCT
ejpam-3890	300	1	a0	a0	PROPN
ejpam-3890	300	2	a1	a1	PROPN
ejpam-3890	300	3	a2	a2	PROPN
ejpam-3890	300	4	a3	a3	PROPN
ejpam-3890	300	5	a4	a4	PROPN
ejpam-3890	300	6	a5a6a7	a5a6a7	PROPN
ejpam-3890	300	7	γs1(r	γs1(r	PROPN
ejpam-3890	300	8	)	)	PUNCT
ejpam-3890	300	9	..........	..........	PUNCT
ejpam-3890	300	10	.............................................	.............................................	PUNCT
ejpam-3890	300	11	.	.	PUNCT
ejpam-3890	301	1	..........	..........	PUNCT
ejpam-3890	301	2	.............................................	.............................................	PUNCT
ejpam-3890	301	3	.	.	PUNCT
ejpam-3890	302	1	..........	..........	PUNCT
ejpam-3890	302	2	.............................................	.............................................	PUNCT
ejpam-3890	302	3	.	.	PUNCT
ejpam-3890	303	1	..........	..........	PUNCT
ejpam-3890	303	2	.............................................	.............................................	PUNCT
ejpam-3890	303	3	.	.	PUNCT
ejpam-3890	304	1	..........	..........	PUNCT
ejpam-3890	304	2	.............................................	.............................................	PUNCT
ejpam-3890	304	3	.	.	PUNCT
ejpam-3890	305	1	..........	..........	PUNCT
ejpam-3890	305	2	.............................................	.............................................	PUNCT
ejpam-3890	305	3	.	.	PUNCT
ejpam-3890	306	1	..........	..........	PUNCT
ejpam-3890	306	2	.............................................	.............................................	PUNCT
ejpam-3890	306	3	.	.	PUNCT
ejpam-3890	307	1	..........	..........	PUNCT
ejpam-3890	307	2	.............................................	.............................................	PUNCT
ejpam-3890	307	3	.	.	PUNCT
ejpam-3890	308	1	.........	.........	PUNCT
ejpam-3890	308	2	........	........	PUNCT
ejpam-3890	308	3	........	........	PUNCT
ejpam-3890	308	4	........	........	PUNCT
ejpam-3890	309	1	........	........	PUNCT
ejpam-3890	309	2	.	.	PUNCT
ejpam-3890	310	1	.........	.........	PUNCT
ejpam-3890	310	2	........	........	PUNCT
ejpam-3890	310	3	........	........	PUNCT
ejpam-3890	310	4	........	........	PUNCT
ejpam-3890	311	1	........	........	PUNCT
ejpam-3890	311	2	.	.	PUNCT
ejpam-3890	312	1	.........	.........	PUNCT
ejpam-3890	312	2	........	........	PUNCT
ejpam-3890	312	3	........	........	PUNCT
ejpam-3890	312	4	........	........	PUNCT
ejpam-3890	313	1	........	........	PUNCT
ejpam-3890	313	2	.	.	PUNCT
ejpam-3890	314	1	.........	.........	PUNCT
ejpam-3890	314	2	........	........	PUNCT
ejpam-3890	314	3	........	........	PUNCT
ejpam-3890	314	4	........	........	PUNCT
ejpam-3890	315	1	........	........	PUNCT
ejpam-3890	315	2	.	.	PUNCT
ejpam-3890	315	3	............	............	PUNCT
ejpam-3890	316	1	...........	...........	PUNCT
ejpam-3890	316	2	...........	...........	PUNCT
ejpam-3890	316	3	...........	...........	PUNCT
ejpam-3890	316	4	...........	...........	PUNCT
ejpam-3890	316	5	.........	.........	PUNCT
ejpam-3890	316	6	............	............	PUNCT
ejpam-3890	316	7	...........	...........	PUNCT
ejpam-3890	316	8	...........	...........	PUNCT
ejpam-3890	316	9	...........	...........	PUNCT
ejpam-3890	316	10	...........	...........	PUNCT
ejpam-3890	316	11	.........	.........	PUNCT
ejpam-3890	316	12	............	............	PUNCT
ejpam-3890	316	13	...........	...........	PUNCT
ejpam-3890	316	14	...........	...........	PUNCT
ejpam-3890	316	15	...........	...........	PUNCT
ejpam-3890	316	16	...........	...........	PUNCT
ejpam-3890	316	17	.........	.........	PUNCT
ejpam-3890	316	18	a0a1	a0a1	X
ejpam-3890	316	19	a2	a2	PROPN
ejpam-3890	316	20	a3	a3	PROPN
ejpam-3890	316	21	a4	a4	PROPN
ejpam-3890	316	22	a5	a5	PROPN
ejpam-3890	316	23	a6a7	a6a7	PUNCT
ejpam-3890	316	24	γs2(r	γs2(r	PROPN
ejpam-3890	316	25	)	)	PUNCT
ejpam-3890	316	26	..........	..........	PUNCT
ejpam-3890	316	27	.............................................	.............................................	PUNCT
ejpam-3890	316	28	.	.	PUNCT
ejpam-3890	317	1	..........	..........	PUNCT
ejpam-3890	317	2	.............................................	.............................................	PUNCT
ejpam-3890	317	3	.	.	PUNCT
ejpam-3890	318	1	..........	..........	PUNCT
ejpam-3890	318	2	.............................................	.............................................	PUNCT
ejpam-3890	318	3	.	.	PUNCT
ejpam-3890	319	1	..........	..........	PUNCT
ejpam-3890	319	2	.............................................	.............................................	PUNCT
ejpam-3890	319	3	.	.	PUNCT
ejpam-3890	320	1	..........	..........	PUNCT
ejpam-3890	320	2	.............................................	.............................................	PUNCT
ejpam-3890	320	3	.	.	PUNCT
ejpam-3890	321	1	..........	..........	PUNCT
ejpam-3890	321	2	.............................................	.............................................	PUNCT
ejpam-3890	321	3	.	.	PUNCT
ejpam-3890	322	1	..........	..........	PUNCT
ejpam-3890	322	2	.............................................	.............................................	PUNCT
ejpam-3890	322	3	.	.	PUNCT
ejpam-3890	323	1	..........	..........	PUNCT
ejpam-3890	323	2	.............................................	.............................................	PUNCT
ejpam-3890	323	3	.	.	PUNCT
ejpam-3890	324	1	.........	.........	PUNCT
ejpam-3890	324	2	........	........	PUNCT
ejpam-3890	324	3	........	........	PUNCT
ejpam-3890	324	4	........	........	PUNCT
ejpam-3890	325	1	........	........	PUNCT
ejpam-3890	325	2	.	.	PUNCT
ejpam-3890	326	1	.........	.........	PUNCT
ejpam-3890	326	2	........	........	PUNCT
ejpam-3890	326	3	........	........	PUNCT
ejpam-3890	326	4	........	........	PUNCT
ejpam-3890	327	1	........	........	PUNCT
ejpam-3890	327	2	.	.	PUNCT
ejpam-3890	328	1	.........	.........	PUNCT
ejpam-3890	328	2	........	........	PUNCT
ejpam-3890	328	3	........	........	PUNCT
ejpam-3890	328	4	........	........	PUNCT
ejpam-3890	329	1	........	........	PUNCT
ejpam-3890	329	2	.	.	PUNCT
ejpam-3890	330	1	.........	.........	PUNCT
ejpam-3890	330	2	........	........	PUNCT
ejpam-3890	330	3	........	........	PUNCT
ejpam-3890	330	4	........	........	PUNCT
ejpam-3890	331	1	........	........	PUNCT
ejpam-3890	331	2	..................................................................	..................................................................	PUNCT
ejpam-3890	332	1	.................................................................	.................................................................	PUNCT
ejpam-3890	332	2	.................................................................	.................................................................	PUNCT
ejpam-3890	333	1	a0a1	a0a1	X
ejpam-3890	333	2	a2	a2	PROPN
ejpam-3890	333	3	a3	a3	PROPN
ejpam-3890	333	4	a4	a4	PROPN
ejpam-3890	333	5	a5	a5	PROPN
ejpam-3890	333	6	a6a7	a6a7	X
ejpam-3890	333	7	γs3(r	γs3(r	NOUN
ejpam-3890	333	8	)	)	PUNCT
ejpam-3890	333	9	..........	..........	PUNCT
ejpam-3890	333	10	.............................................	.............................................	PUNCT
ejpam-3890	333	11	.	.	PUNCT
ejpam-3890	334	1	..........	..........	PUNCT
ejpam-3890	334	2	.............................................	.............................................	PUNCT
ejpam-3890	334	3	.	.	PUNCT
ejpam-3890	335	1	..........	..........	PUNCT
ejpam-3890	335	2	.............................................	.............................................	PUNCT
ejpam-3890	335	3	.	.	PUNCT
ejpam-3890	336	1	..........	..........	PUNCT
ejpam-3890	336	2	.............................................	.............................................	PUNCT
ejpam-3890	336	3	.	.	PUNCT
ejpam-3890	337	1	..........	..........	PUNCT
ejpam-3890	337	2	.............................................	.............................................	PUNCT
ejpam-3890	337	3	.	.	PUNCT
ejpam-3890	338	1	..........	..........	PUNCT
ejpam-3890	338	2	.............................................	.............................................	PUNCT
ejpam-3890	338	3	.	.	PUNCT
ejpam-3890	339	1	..........	..........	PUNCT
ejpam-3890	339	2	.............................................	.............................................	PUNCT
ejpam-3890	339	3	.	.	PUNCT
ejpam-3890	340	1	..........	..........	PUNCT
ejpam-3890	340	2	.............................................	.............................................	PUNCT
ejpam-3890	340	3	.	.	PUNCT
ejpam-3890	341	1	.........	.........	PUNCT
ejpam-3890	341	2	........	........	PUNCT
ejpam-3890	341	3	........	........	PUNCT
ejpam-3890	341	4	........	........	PUNCT
ejpam-3890	342	1	........	........	PUNCT
ejpam-3890	342	2	.	.	PUNCT
ejpam-3890	343	1	.........	.........	PUNCT
ejpam-3890	343	2	........	........	PUNCT
ejpam-3890	343	3	........	........	PUNCT
ejpam-3890	343	4	........	........	PUNCT
ejpam-3890	344	1	........	........	PUNCT
ejpam-3890	344	2	.	.	PUNCT
ejpam-3890	345	1	.........	.........	PUNCT
ejpam-3890	345	2	........	........	PUNCT
ejpam-3890	345	3	........	........	PUNCT
ejpam-3890	345	4	........	........	PUNCT
ejpam-3890	346	1	........	........	PUNCT
ejpam-3890	346	2	.	.	PUNCT
ejpam-3890	347	1	.........	.........	PUNCT
ejpam-3890	347	2	........	........	PUNCT
ejpam-3890	347	3	........	........	PUNCT
ejpam-3890	347	4	........	........	PUNCT
ejpam-3890	348	1	........	........	PUNCT
ejpam-3890	348	2	.	.	PUNCT
ejpam-3890	348	3	............	............	PUNCT
ejpam-3890	349	1	...........	...........	PUNCT
ejpam-3890	349	2	...........	...........	PUNCT
ejpam-3890	349	3	...........	...........	PUNCT
ejpam-3890	349	4	...........	...........	PUNCT
ejpam-3890	349	5	.........	.........	PUNCT
ejpam-3890	349	6	............	............	PUNCT
ejpam-3890	349	7	...........	...........	PUNCT
ejpam-3890	349	8	...........	...........	PUNCT
ejpam-3890	349	9	...........	...........	PUNCT
ejpam-3890	349	10	...........	...........	PUNCT
ejpam-3890	349	11	.........	.........	PUNCT
ejpam-3890	349	12	............	............	PUNCT
ejpam-3890	349	13	...........	...........	PUNCT
ejpam-3890	349	14	...........	...........	PUNCT
ejpam-3890	349	15	...........	...........	PUNCT
ejpam-3890	349	16	...........	...........	PUNCT
ejpam-3890	349	17	.........	.........	PUNCT
ejpam-3890	350	1	a2	a2	PROPN
ejpam-3890	350	2	a3	a3	PROPN
ejpam-3890	350	3	a4	a4	PROPN
ejpam-3890	350	4	a5	a5	PROPN
ejpam-3890	350	5	a6	a6	PROPN
ejpam-3890	350	6	a7a0a1	a7a0a1	PROPN
ejpam-3890	350	7	γs4(r	γs4(r	PROPN
ejpam-3890	350	8	)	)	PUNCT
ejpam-3890	350	9	..........	..........	PUNCT
ejpam-3890	351	1	.............................................	.............................................	PUNCT
ejpam-3890	351	2	.	.	PUNCT
ejpam-3890	352	1	..........	..........	PUNCT
ejpam-3890	352	2	.............................................	.............................................	PUNCT
ejpam-3890	352	3	.	.	PUNCT
ejpam-3890	353	1	..........	..........	PUNCT
ejpam-3890	353	2	.............................................	.............................................	PUNCT
ejpam-3890	353	3	.	.	PUNCT
ejpam-3890	354	1	..........	..........	PUNCT
ejpam-3890	354	2	.............................................	.............................................	PUNCT
ejpam-3890	354	3	.	.	PUNCT
ejpam-3890	355	1	..........	..........	PUNCT
ejpam-3890	355	2	.............................................	.............................................	PUNCT
ejpam-3890	355	3	.	.	PUNCT
ejpam-3890	356	1	..........	..........	PUNCT
ejpam-3890	356	2	.............................................	.............................................	PUNCT
ejpam-3890	356	3	.	.	PUNCT
ejpam-3890	357	1	..........	..........	PUNCT
ejpam-3890	357	2	.............................................	.............................................	PUNCT
ejpam-3890	357	3	.	.	PUNCT
ejpam-3890	358	1	..........	..........	PUNCT
ejpam-3890	358	2	.............................................	.............................................	PUNCT
ejpam-3890	358	3	.	.	PUNCT
ejpam-3890	359	1	.........	.........	PUNCT
ejpam-3890	359	2	........	........	PUNCT
ejpam-3890	359	3	........	........	PUNCT
ejpam-3890	359	4	........	........	PUNCT
ejpam-3890	360	1	........	........	PUNCT
ejpam-3890	360	2	.	.	PUNCT
ejpam-3890	361	1	.........	.........	PUNCT
ejpam-3890	361	2	........	........	PUNCT
ejpam-3890	361	3	........	........	PUNCT
ejpam-3890	361	4	........	........	PUNCT
ejpam-3890	362	1	........	........	PUNCT
ejpam-3890	362	2	.	.	PUNCT
ejpam-3890	363	1	.........	.........	PUNCT
ejpam-3890	363	2	........	........	PUNCT
ejpam-3890	363	3	........	........	PUNCT
ejpam-3890	363	4	........	........	PUNCT
ejpam-3890	364	1	........	........	PUNCT
ejpam-3890	364	2	.	.	PUNCT
ejpam-3890	365	1	.........	.........	PUNCT
ejpam-3890	365	2	........	........	PUNCT
ejpam-3890	365	3	........	........	PUNCT
ejpam-3890	365	4	........	........	PUNCT
ejpam-3890	366	1	........	........	PUNCT
ejpam-3890	366	2	..................................................................	..................................................................	PUNCT
ejpam-3890	367	1	.................................................................	.................................................................	PUNCT
ejpam-3890	367	2	.................................................................	.................................................................	PUNCT
ejpam-3890	368	1	a2	a2	PROPN
ejpam-3890	368	2	a3	a3	PROPN
ejpam-3890	368	3	a4	a4	PROPN
ejpam-3890	368	4	a5	a5	PROPN
ejpam-3890	368	5	a6	a6	PROPN
ejpam-3890	368	6	a7a0a1	a7a0a1	PROPN
ejpam-3890	368	7	γs5(r	γs5(r	PROPN
ejpam-3890	368	8	)	)	PUNCT
ejpam-3890	368	9	..........	..........	PUNCT
ejpam-3890	368	10	.............................................	.............................................	PUNCT
ejpam-3890	368	11	.	.	PUNCT
ejpam-3890	369	1	..........	..........	PUNCT
ejpam-3890	369	2	.............................................	.............................................	PUNCT
ejpam-3890	369	3	.	.	PUNCT
ejpam-3890	370	1	..........	..........	PUNCT
ejpam-3890	370	2	.............................................	.............................................	PUNCT
ejpam-3890	370	3	.	.	PUNCT
ejpam-3890	371	1	..........	..........	PUNCT
ejpam-3890	371	2	.............................................	.............................................	PUNCT
ejpam-3890	371	3	.	.	PUNCT
ejpam-3890	372	1	..........	..........	PUNCT
ejpam-3890	372	2	.............................................	.............................................	PUNCT
ejpam-3890	372	3	.	.	PUNCT
ejpam-3890	373	1	..........	..........	PUNCT
ejpam-3890	373	2	.............................................	.............................................	PUNCT
ejpam-3890	373	3	.	.	PUNCT
ejpam-3890	374	1	..........	..........	PUNCT
ejpam-3890	374	2	.............................................	.............................................	PUNCT
ejpam-3890	374	3	.	.	PUNCT
ejpam-3890	375	1	..........	..........	PUNCT
ejpam-3890	375	2	.............................................	.............................................	PUNCT
ejpam-3890	375	3	.	.	PUNCT
ejpam-3890	376	1	.........	.........	PUNCT
ejpam-3890	376	2	........	........	PUNCT
ejpam-3890	376	3	........	........	PUNCT
ejpam-3890	376	4	........	........	PUNCT
ejpam-3890	377	1	........	........	PUNCT
ejpam-3890	377	2	.	.	PUNCT
ejpam-3890	378	1	.........	.........	PUNCT
ejpam-3890	378	2	........	........	PUNCT
ejpam-3890	378	3	........	........	PUNCT
ejpam-3890	378	4	........	........	PUNCT
ejpam-3890	379	1	........	........	PUNCT
ejpam-3890	379	2	.	.	PUNCT
ejpam-3890	380	1	.........	.........	PUNCT
ejpam-3890	380	2	........	........	PUNCT
ejpam-3890	380	3	........	........	PUNCT
ejpam-3890	380	4	........	........	PUNCT
ejpam-3890	381	1	........	........	PUNCT
ejpam-3890	381	2	.	.	PUNCT
ejpam-3890	382	1	.........	.........	PUNCT
ejpam-3890	382	2	........	........	PUNCT
ejpam-3890	382	3	........	........	PUNCT
ejpam-3890	382	4	........	........	PUNCT
ejpam-3890	383	1	........	........	PUNCT
ejpam-3890	383	2	.	.	PUNCT
ejpam-3890	383	3	............	............	PUNCT
ejpam-3890	384	1	...........	...........	PUNCT
ejpam-3890	384	2	...........	...........	PUNCT
ejpam-3890	384	3	...........	...........	PUNCT
ejpam-3890	384	4	...........	...........	PUNCT
ejpam-3890	384	5	.........	.........	PUNCT
ejpam-3890	384	6	............	............	PUNCT
ejpam-3890	384	7	...........	...........	PUNCT
ejpam-3890	384	8	...........	...........	PUNCT
ejpam-3890	384	9	...........	...........	PUNCT
ejpam-3890	384	10	...........	...........	PUNCT
ejpam-3890	384	11	.........	.........	PUNCT
ejpam-3890	384	12	............	............	PUNCT
ejpam-3890	384	13	...........	...........	PUNCT
ejpam-3890	384	14	...........	...........	PUNCT
ejpam-3890	384	15	...........	...........	PUNCT
ejpam-3890	384	16	...........	...........	PUNCT
ejpam-3890	384	17	.........	.........	PUNCT
ejpam-3890	385	1	a3	a3	PROPN
ejpam-3890	385	2	a4	a4	PROPN
ejpam-3890	385	3	a5	a5	PROPN
ejpam-3890	385	4	a6	a6	PROPN
ejpam-3890	385	5	a7	a7	PROPN
ejpam-3890	385	6	a0a1a2	a0a1a2	PROPN
ejpam-3890	385	7	γs6(r	γs6(r	PROPN
ejpam-3890	385	8	)	)	PUNCT
ejpam-3890	385	9	figure	figure	NOUN
ejpam-3890	385	10	5	5	NUM
ejpam-3890	385	11	:	:	PUNCT
ejpam-3890	385	12	the	the	DET
ejpam-3890	385	13	graphs	graph	NOUN
ejpam-3890	385	14	γsi(r	γsi(r	X
ejpam-3890	385	15	)	)	PUNCT
ejpam-3890	385	16	for	for	ADP
ejpam-3890	385	17	i	i	PROPN
ejpam-3890	385	18	=	=	NOUN
ejpam-3890	385	19	1	1	NUM
ejpam-3890	385	20	,	,	PUNCT
ejpam-3890	385	21	.	.	PUNCT
ejpam-3890	385	22	.	.	PUNCT
ejpam-3890	385	23	.	.	PUNCT
ejpam-3890	386	1	,	,	PUNCT
ejpam-3890	386	2	6	6	NUM
ejpam-3890	386	3	respectively	respectively	ADV
ejpam-3890	386	4	where	where	SCONJ
ejpam-3890	386	5	si	si	PROPN
ejpam-3890	386	6	=	=	PRON
ejpam-3890	386	7	{	{	PUNCT
ejpam-3890	386	8	i	i	NOUN
ejpam-3890	386	9	,	,	PUNCT
ejpam-3890	386	10	i	i	PRON
ejpam-3890	386	11	+	+	NOUN
ejpam-3890	386	12	1	1	NUM
ejpam-3890	386	13	}	}	PUNCT
ejpam-3890	386	14	,	,	PUNCT
ejpam-3890	386	15	and	and	CCONJ
ejpam-3890	386	16	r	r	NOUN
ejpam-3890	386	17	=	=	SYM
ejpam-3890	386	18	m0	m0	NOUN
ejpam-3890	386	19	2	2	NUM
ejpam-3890	386	20	(	(	PUNCT
ejpam-3890	386	21	z8	z8	NOUN
ejpam-3890	386	22	)	)	PUNCT
ejpam-3890	386	23	.	.	PUNCT
ejpam-3890	387	1	the	the	DET
ejpam-3890	387	2	graph	graph	NOUN
ejpam-3890	387	3	γs(m0	γs(m0	INTJ
ejpam-3890	387	4	2	2	NUM
ejpam-3890	387	5	(	(	PUNCT
ejpam-3890	387	6	zn	zn	NOUN
ejpam-3890	387	7	)	)	PUNCT
ejpam-3890	387	8	)	)	PUNCT
ejpam-3890	387	9	where	where	SCONJ
ejpam-3890	387	10	s	s	VERB
ejpam-3890	387	11	=	=	PUNCT
ejpam-3890	387	12	{	{	PUNCT
ejpam-3890	387	13	a1	a1	PROPN
ejpam-3890	387	14	,	,	PUNCT
ejpam-3890	387	15	a2	a2	PROPN
ejpam-3890	387	16	,	,	PUNCT
ejpam-3890	387	17	a3	a3	NOUN
ejpam-3890	387	18	}	}	PUNCT
ejpam-3890	387	19	is	be	AUX
ejpam-3890	387	20	shown	show	VERB
ejpam-3890	387	21	in	in	ADP
ejpam-3890	387	22	figure	figure	NOUN
ejpam-3890	387	23	6	6	NUM
ejpam-3890	387	24	when	when	SCONJ
ejpam-3890	387	25	n	n	X
ejpam-3890	387	26	is	be	AUX
ejpam-3890	387	27	even	even	ADV
ejpam-3890	387	28	.	.	PUNCT
ejpam-3890	388	1	..........	..........	PUNCT
ejpam-3890	388	2	.............................................	.............................................	PUNCT
ejpam-3890	388	3	.	.	PUNCT
ejpam-3890	389	1	..........	..........	PUNCT
ejpam-3890	389	2	.............................................	.............................................	PUNCT
ejpam-3890	389	3	.	.	PUNCT
ejpam-3890	390	1	..........	..........	PUNCT
ejpam-3890	390	2	.............................................	.............................................	PUNCT
ejpam-3890	390	3	.	.	PUNCT
ejpam-3890	391	1	..........	..........	PUNCT
ejpam-3890	391	2	.............................................	.............................................	PUNCT
ejpam-3890	391	3	.	.	PUNCT
ejpam-3890	392	1	..........	..........	PUNCT
ejpam-3890	392	2	.............................................	.............................................	PUNCT
ejpam-3890	392	3	.	.	PUNCT
ejpam-3890	393	1	..........	..........	PUNCT
ejpam-3890	393	2	.............................................	.............................................	PUNCT
ejpam-3890	393	3	.	.	PUNCT
ejpam-3890	394	1	..........	..........	PUNCT
ejpam-3890	394	2	.............................................	.............................................	PUNCT
ejpam-3890	394	3	.	.	PUNCT
ejpam-3890	395	1	..........	..........	PUNCT
ejpam-3890	395	2	.............................................	.............................................	PUNCT
ejpam-3890	395	3	.	.	PUNCT
ejpam-3890	396	1	..........	..........	PUNCT
ejpam-3890	396	2	.............................................	.............................................	PUNCT
ejpam-3890	396	3	.	.	PUNCT
ejpam-3890	397	1	..........	..........	PUNCT
ejpam-3890	397	2	.............................................	.............................................	PUNCT
ejpam-3890	397	3	.	.	PUNCT
ejpam-3890	398	1	..........	..........	PUNCT
ejpam-3890	398	2	.............................................	.............................................	PUNCT
ejpam-3890	398	3	.	.	PUNCT
ejpam-3890	399	1	..........	..........	PUNCT
ejpam-3890	399	2	.............................................	.............................................	PUNCT
ejpam-3890	399	3	..............................................	..............................................	PUNCT
ejpam-3890	400	1	............	............	PUNCT
ejpam-3890	400	2	...........	...........	PUNCT
ejpam-3890	400	3	...........	...........	PUNCT
ejpam-3890	400	4	...........	...........	PUNCT
ejpam-3890	400	5	.........	.........	PUNCT
ejpam-3890	400	6	........	........	PUNCT
ejpam-3890	400	7	........	........	PUNCT
ejpam-3890	400	8	........	........	PUNCT
ejpam-3890	400	9	........	........	PUNCT
ejpam-3890	400	10	........	........	PUNCT
ejpam-3890	400	11	........	........	PUNCT
ejpam-3890	400	12	........	........	PUNCT
ejpam-3890	401	1	.....	.....	PUNCT
ejpam-3890	401	2	.........	.........	PUNCT
ejpam-3890	401	3	........	........	PUNCT
ejpam-3890	401	4	........	........	PUNCT
ejpam-3890	401	5	........	........	PUNCT
ejpam-3890	401	6	........	........	PUNCT
ejpam-3890	401	7	........	........	PUNCT
ejpam-3890	401	8	........	........	PUNCT
ejpam-3890	401	9	........	........	PUNCT
ejpam-3890	402	1	.....	.....	PUNCT
ejpam-3890	402	2	.........	.........	PUNCT
ejpam-3890	402	3	........	........	PUNCT
ejpam-3890	402	4	........	........	PUNCT
ejpam-3890	402	5	........	........	PUNCT
ejpam-3890	402	6	........	........	PUNCT
ejpam-3890	402	7	........	........	PUNCT
ejpam-3890	402	8	........	........	PUNCT
ejpam-3890	402	9	........	........	PUNCT
ejpam-3890	403	1	.....	.....	PUNCT
ejpam-3890	403	2	............	............	PUNCT
ejpam-3890	403	3	...........	...........	PUNCT
ejpam-3890	403	4	...........	...........	PUNCT
ejpam-3890	403	5	...........	...........	PUNCT
ejpam-3890	403	6	...........	...........	PUNCT
ejpam-3890	403	7	...........	...........	PUNCT
ejpam-3890	403	8	...........	...........	PUNCT
ejpam-3890	403	9	...........	...........	PUNCT
ejpam-3890	403	10	...........	...........	PUNCT
ejpam-3890	403	11	................................................................................................................	................................................................................................................	PUNCT
ejpam-3890	403	12	............	............	PUNCT
ejpam-3890	403	13	...........	...........	PUNCT
ejpam-3890	403	14	...........	...........	PUNCT
ejpam-3890	403	15	...........	...........	PUNCT
ejpam-3890	403	16	...........	...........	PUNCT
ejpam-3890	403	17	...........	...........	PUNCT
ejpam-3890	403	18	...........	...........	PUNCT
ejpam-3890	403	19	...........	...........	PUNCT
ejpam-3890	403	20	...........	...........	PUNCT
ejpam-3890	403	21	................................................................................................................	................................................................................................................	PUNCT
ejpam-3890	403	22	............	............	PUNCT
ejpam-3890	403	23	...........	...........	PUNCT
ejpam-3890	403	24	...........	...........	PUNCT
ejpam-3890	403	25	...........	...........	PUNCT
ejpam-3890	403	26	...........	...........	PUNCT
ejpam-3890	403	27	...........	...........	PUNCT
ejpam-3890	403	28	...........	...........	PUNCT
ejpam-3890	403	29	...........	...........	PUNCT
ejpam-3890	403	30	...........	...........	PUNCT
ejpam-3890	403	31	......	......	PUNCT
ejpam-3890	403	32	.........	.........	PUNCT
ejpam-3890	403	33	........	........	PUNCT
ejpam-3890	403	34	........	........	PUNCT
ejpam-3890	403	35	........	........	PUNCT
ejpam-3890	403	36	........	........	PUNCT
ejpam-3890	403	37	........	........	PUNCT
ejpam-3890	403	38	........	........	PUNCT
ejpam-3890	403	39	........	........	PUNCT
ejpam-3890	404	1	.....	.....	PUNCT
ejpam-3890	404	2	..........................................................................................................	..........................................................................................................	PUNCT
ejpam-3890	404	3	.......	.......	PUNCT
ejpam-3890	404	4	........	........	PUNCT
ejpam-3890	404	5	........	........	PUNCT
ejpam-3890	404	6	........	........	PUNCT
ejpam-3890	404	7	........	........	PUNCT
ejpam-3890	404	8	........	........	PUNCT
ejpam-3890	404	9	........	........	PUNCT
ejpam-3890	404	10	........	........	PUNCT
ejpam-3890	404	11	.......	.......	PUNCT
ejpam-3890	404	12	............	............	PUNCT
ejpam-3890	404	13	...........	...........	PUNCT
ejpam-3890	404	14	...........	...........	PUNCT
ejpam-3890	404	15	...........	...........	PUNCT
ejpam-3890	404	16	.............................................	.............................................	PUNCT
ejpam-3890	404	17	a1	a1	NOUN
ejpam-3890	404	18	an+2	an+2	PROPN
ejpam-3890	404	19	2	2	NUM
ejpam-3890	404	20	a0	a0	PROPN
ejpam-3890	404	21	a2	a2	PROPN
ejpam-3890	404	22	a3	a3	PROPN
ejpam-3890	404	23	a4	a4	PROPN
ejpam-3890	404	24	an−1	an−1	PROPN
ejpam-3890	404	25	an−2	an−2	PROPN
ejpam-3890	404	26	an	an	DET
ejpam-3890	404	27	2	2	NUM
ejpam-3890	404	28	an+4	an+4	NUM
ejpam-3890	404	29	2	2	NUM
ejpam-3890	404	30	·	·	PUNCT
ejpam-3890	404	31	·	·	PUNCT
ejpam-3890	404	32	·	·	PUNCT
ejpam-3890	404	33	·	·	PUNCT
ejpam-3890	404	34	·	·	PUNCT
ejpam-3890	405	1	·	·	PUNCT
ejpam-3890	405	2	figure	figure	VERB
ejpam-3890	405	3	6	6	NUM
ejpam-3890	405	4	:	:	PUNCT
ejpam-3890	405	5	the	the	DET
ejpam-3890	405	6	graph	graph	NOUN
ejpam-3890	405	7	γs(r	γs(r	PUNCT
ejpam-3890	405	8	)	)	PUNCT
ejpam-3890	405	9	where	where	SCONJ
ejpam-3890	405	10	s	s	VERB
ejpam-3890	405	11	=	=	PUNCT
ejpam-3890	405	12	{	{	PUNCT
ejpam-3890	405	13	a1	a1	PROPN
ejpam-3890	405	14	,	,	PUNCT
ejpam-3890	405	15	a2	a2	PROPN
ejpam-3890	405	16	,	,	PUNCT
ejpam-3890	405	17	a3	a3	NOUN
ejpam-3890	405	18	}	}	PUNCT
ejpam-3890	405	19	,	,	PUNCT
ejpam-3890	405	20	and	and	CCONJ
ejpam-3890	405	21	r	r	NOUN
ejpam-3890	405	22	=	=	SYM
ejpam-3890	405	23	m0	m0	NOUN
ejpam-3890	405	24	2	2	NUM
ejpam-3890	405	25	(	(	PUNCT
ejpam-3890	405	26	zn	zn	NUM
ejpam-3890	405	27	)	)	PUNCT
ejpam-3890	405	28	.	.	PUNCT
ejpam-3890	406	1	the	the	DET
ejpam-3890	406	2	graph	graph	NOUN
ejpam-3890	406	3	γs(m0	γs(m0	INTJ
ejpam-3890	406	4	2	2	NUM
ejpam-3890	406	5	(	(	PUNCT
ejpam-3890	406	6	zn	zn	NOUN
ejpam-3890	406	7	)	)	PUNCT
ejpam-3890	406	8	)	)	PUNCT
ejpam-3890	406	9	where	where	SCONJ
ejpam-3890	406	10	s	s	VERB
ejpam-3890	406	11	=	=	PUNCT
ejpam-3890	406	12	{	{	PUNCT
ejpam-3890	406	13	a1	a1	PROPN
ejpam-3890	406	14	,	,	PUNCT
ejpam-3890	406	15	a2	a2	PROPN
ejpam-3890	406	16	,	,	PUNCT
ejpam-3890	406	17	a3	a3	NOUN
ejpam-3890	406	18	}	}	PUNCT
ejpam-3890	406	19	is	be	AUX
ejpam-3890	406	20	shown	show	VERB
ejpam-3890	406	21	in	in	ADP
ejpam-3890	406	22	figure	figure	NOUN
ejpam-3890	406	23	7	7	NUM
ejpam-3890	406	24	when	when	SCONJ
ejpam-3890	406	25	n	n	X
ejpam-3890	406	26	is	be	AUX
ejpam-3890	406	27	odd	odd	ADJ
ejpam-3890	406	28	.	.	PUNCT
ejpam-3890	406	29	..........	..........	PUNCT
ejpam-3890	406	30	.............................................	.............................................	PUNCT
ejpam-3890	406	31	.	.	PUNCT
ejpam-3890	407	1	..........	..........	PUNCT
ejpam-3890	407	2	.............................................	.............................................	PUNCT
ejpam-3890	407	3	.	.	PUNCT
ejpam-3890	408	1	..........	..........	PUNCT
ejpam-3890	408	2	.............................................	.............................................	PUNCT
ejpam-3890	408	3	.	.	PUNCT
ejpam-3890	409	1	..........	..........	PUNCT
ejpam-3890	409	2	.............................................	.............................................	PUNCT
ejpam-3890	409	3	.	.	PUNCT
ejpam-3890	410	1	..........	..........	PUNCT
ejpam-3890	410	2	.............................................	.............................................	PUNCT
ejpam-3890	410	3	.	.	PUNCT
ejpam-3890	411	1	..........	..........	PUNCT
ejpam-3890	411	2	.............................................	.............................................	PUNCT
ejpam-3890	411	3	.	.	PUNCT
ejpam-3890	412	1	..........	..........	PUNCT
ejpam-3890	412	2	.............................................	.............................................	PUNCT
ejpam-3890	412	3	.	.	PUNCT
ejpam-3890	413	1	..........	..........	PUNCT
ejpam-3890	413	2	.............................................	.............................................	PUNCT
ejpam-3890	413	3	.	.	PUNCT
ejpam-3890	414	1	..........	..........	PUNCT
ejpam-3890	414	2	.............................................	.............................................	PUNCT
ejpam-3890	414	3	.	.	PUNCT
ejpam-3890	415	1	..........	..........	PUNCT
ejpam-3890	415	2	.............................................	.............................................	PUNCT
ejpam-3890	415	3	.	.	PUNCT
ejpam-3890	416	1	..........	..........	PUNCT
ejpam-3890	416	2	.............................................	.............................................	PUNCT
ejpam-3890	416	3	.	.	PUNCT
ejpam-3890	417	1	.............................................	.............................................	PUNCT
ejpam-3890	417	2	............	............	PUNCT
ejpam-3890	417	3	...........	...........	PUNCT
ejpam-3890	417	4	...........	...........	PUNCT
ejpam-3890	417	5	...........	...........	PUNCT
ejpam-3890	417	6	.........	.........	PUNCT
ejpam-3890	417	7	........	........	PUNCT
ejpam-3890	417	8	........	........	PUNCT
ejpam-3890	417	9	........	........	PUNCT
ejpam-3890	417	10	........	........	PUNCT
ejpam-3890	417	11	........	........	PUNCT
ejpam-3890	417	12	........	........	PUNCT
ejpam-3890	417	13	........	........	PUNCT
ejpam-3890	418	1	.....	.....	PUNCT
ejpam-3890	418	2	.........	.........	PUNCT
ejpam-3890	418	3	........	........	PUNCT
ejpam-3890	418	4	........	........	PUNCT
ejpam-3890	418	5	........	........	PUNCT
ejpam-3890	418	6	........	........	PUNCT
ejpam-3890	418	7	........	........	PUNCT
ejpam-3890	418	8	........	........	PUNCT
ejpam-3890	418	9	........	........	PUNCT
ejpam-3890	419	1	.....	.....	PUNCT
ejpam-3890	419	2	.........	.........	PUNCT
ejpam-3890	419	3	........	........	PUNCT
ejpam-3890	419	4	........	........	PUNCT
ejpam-3890	419	5	........	........	PUNCT
ejpam-3890	419	6	........	........	PUNCT
ejpam-3890	419	7	........	........	PUNCT
ejpam-3890	419	8	........	........	PUNCT
ejpam-3890	419	9	........	........	PUNCT
ejpam-3890	420	1	.....	.....	PUNCT
ejpam-3890	420	2	............	............	PUNCT
ejpam-3890	420	3	...........	...........	PUNCT
ejpam-3890	420	4	...........	...........	PUNCT
ejpam-3890	420	5	...........	...........	PUNCT
ejpam-3890	420	6	...........	...........	PUNCT
ejpam-3890	420	7	...........	...........	PUNCT
ejpam-3890	420	8	...........	...........	PUNCT
ejpam-3890	420	9	...........	...........	PUNCT
ejpam-3890	420	10	...........	...........	PUNCT
ejpam-3890	420	11	................................................................................................................	................................................................................................................	PUNCT
ejpam-3890	420	12	............	............	PUNCT
ejpam-3890	420	13	...........	...........	PUNCT
ejpam-3890	420	14	...........	...........	PUNCT
ejpam-3890	420	15	...........	...........	PUNCT
ejpam-3890	420	16	...........	...........	PUNCT
ejpam-3890	420	17	...........	...........	PUNCT
ejpam-3890	420	18	...........	...........	PUNCT
ejpam-3890	420	19	...........	...........	PUNCT
ejpam-3890	420	20	...........	...........	PUNCT
ejpam-3890	420	21	................................................................................................................	................................................................................................................	PUNCT
ejpam-3890	420	22	............	............	PUNCT
ejpam-3890	420	23	...........	...........	PUNCT
ejpam-3890	420	24	...........	...........	PUNCT
ejpam-3890	420	25	...........	...........	PUNCT
ejpam-3890	420	26	...........	...........	PUNCT
ejpam-3890	420	27	...........	...........	PUNCT
ejpam-3890	420	28	...........	...........	PUNCT
ejpam-3890	420	29	...........	...........	PUNCT
ejpam-3890	420	30	...........	...........	PUNCT
ejpam-3890	420	31	......	......	PUNCT
ejpam-3890	420	32	.........	.........	PUNCT
ejpam-3890	420	33	........	........	PUNCT
ejpam-3890	420	34	........	........	PUNCT
ejpam-3890	420	35	........	........	PUNCT
ejpam-3890	420	36	........	........	PUNCT
ejpam-3890	420	37	........	........	PUNCT
ejpam-3890	420	38	........	........	PUNCT
ejpam-3890	420	39	........	........	PUNCT
ejpam-3890	421	1	.....	.....	PUNCT
ejpam-3890	421	2	..........................................................................................................	..........................................................................................................	PUNCT
ejpam-3890	421	3	.......	.......	PUNCT
ejpam-3890	421	4	........	........	PUNCT
ejpam-3890	421	5	........	........	PUNCT
ejpam-3890	421	6	........	........	PUNCT
ejpam-3890	421	7	........	........	PUNCT
ejpam-3890	421	8	........	........	PUNCT
ejpam-3890	421	9	........	........	PUNCT
ejpam-3890	421	10	........	........	PUNCT
ejpam-3890	421	11	.......	.......	PUNCT
ejpam-3890	422	1	a1	a1	PROPN
ejpam-3890	422	2	a0	a0	PROPN
ejpam-3890	422	3	a2	a2	PROPN
ejpam-3890	422	4	a3	a3	PROPN
ejpam-3890	422	5	a4	a4	PROPN
ejpam-3890	422	6	an−1	an−1	ADJ
ejpam-3890	422	7	an−2	an−2	PROPN
ejpam-3890	422	8	an−1	an−1	ADJ
ejpam-3890	422	9	2	2	NUM
ejpam-3890	422	10	an+5	an+5	SYM
ejpam-3890	422	11	2	2	NUM
ejpam-3890	422	12	an+1	an+1	NOUN
ejpam-3890	422	13	2	2	NUM
ejpam-3890	422	14	an+3	an+3	NOUN
ejpam-3890	422	15	2	2	NUM
ejpam-3890	422	16	·	·	PUNCT
ejpam-3890	422	17	·	·	PUNCT
ejpam-3890	422	18	·	·	PUNCT
ejpam-3890	422	19	·	·	PUNCT
ejpam-3890	422	20	·	·	PUNCT
ejpam-3890	422	21	·	·	PUNCT
ejpam-3890	422	22	figure	figure	VERB
ejpam-3890	422	23	7	7	NUM
ejpam-3890	422	24	:	:	PUNCT
ejpam-3890	422	25	the	the	DET
ejpam-3890	422	26	graph	graph	NOUN
ejpam-3890	422	27	γs(r	γs(r	PUNCT
ejpam-3890	422	28	)	)	PUNCT
ejpam-3890	422	29	where	where	SCONJ
ejpam-3890	422	30	s	s	VERB
ejpam-3890	422	31	=	=	PUNCT
ejpam-3890	422	32	{	{	PUNCT
ejpam-3890	422	33	a1	a1	PROPN
ejpam-3890	422	34	,	,	PUNCT
ejpam-3890	422	35	a2	a2	PROPN
ejpam-3890	422	36	,	,	PUNCT
ejpam-3890	422	37	a3	a3	NOUN
ejpam-3890	422	38	}	}	PUNCT
ejpam-3890	422	39	,	,	PUNCT
ejpam-3890	422	40	and	and	CCONJ
ejpam-3890	422	41	r	r	NOUN
ejpam-3890	422	42	=	=	SYM
ejpam-3890	422	43	m0	m0	NOUN
ejpam-3890	422	44	2	2	NUM
ejpam-3890	422	45	(	(	PUNCT
ejpam-3890	422	46	zn	zn	NUM
ejpam-3890	422	47	)	)	PUNCT
ejpam-3890	422	48	.	.	PUNCT
ejpam-3890	423	1	note	note	VERB
ejpam-3890	423	2	that	that	SCONJ
ejpam-3890	423	3	theorem	theorem	NOUN
ejpam-3890	423	4	2	2	NUM
ejpam-3890	423	5	can	can	AUX
ejpam-3890	423	6	be	be	AUX
ejpam-3890	423	7	used	use	VERB
ejpam-3890	423	8	to	to	PART
ejpam-3890	423	9	determine	determine	VERB
ejpam-3890	423	10	if	if	SCONJ
ejpam-3890	423	11	a	a	DET
ejpam-3890	423	12	graph	graph	NOUN
ejpam-3890	423	13	admits	admit	VERB
ejpam-3890	423	14	an	an	DET
ejpam-3890	423	15	efficient	efficient	ADJ
ejpam-3890	423	16	zero	zero	NUM
ejpam-3890	423	17	ring	ring	NOUN
ejpam-3890	423	18	labeling	labeling	NOUN
ejpam-3890	423	19	.	.	PUNCT
ejpam-3890	424	1	if	if	SCONJ
ejpam-3890	424	2	a	a	DET
ejpam-3890	424	3	graph	graph	NOUN
ejpam-3890	424	4	γ	γ	PROPN
ejpam-3890	424	5	can	can	AUX
ejpam-3890	424	6	be	be	AUX
ejpam-3890	424	7	shown	show	VERB
ejpam-3890	424	8	to	to	PART
ejpam-3890	424	9	be	be	AUX
ejpam-3890	424	10	an	an	DET
ejpam-3890	424	11	edge	edge	NOUN
ejpam-3890	424	12	induced	induce	VERB
ejpam-3890	424	13	subgraph	subgraph	NOUN
ejpam-3890	424	14	of	of	ADP
ejpam-3890	424	15	γs(r	γs(r	PROPN
ejpam-3890	424	16	)	)	PUNCT
ejpam-3890	424	17	for	for	ADP
ejpam-3890	424	18	some	some	DET
ejpam-3890	424	19	subset	subset	NOUN
ejpam-3890	424	20	s	s	PROPN
ejpam-3890	424	21	of	of	ADP
ejpam-3890	424	22	r−{0	r−{0	VERB
ejpam-3890	424	23	}	}	PUNCT
ejpam-3890	424	24	,	,	PUNCT
ejpam-3890	424	25	such	such	ADJ
ejpam-3890	424	26	that	that	SCONJ
ejpam-3890	424	27	the	the	DET
ejpam-3890	424	28	maximum	maximum	ADJ
ejpam-3890	424	29	degree	degree	NOUN
ejpam-3890	424	30	of	of	ADP
ejpam-3890	424	31	a	a	DET
ejpam-3890	424	32	vertex	vertex	NOUN
ejpam-3890	424	33	in	in	ADP
ejpam-3890	424	34	γ	γ	X
ejpam-3890	424	35	=	=	SYM
ejpam-3890	424	36	|s|	|s|	PROPN
ejpam-3890	424	37	,	,	PUNCT
ejpam-3890	424	38	then	then	ADV
ejpam-3890	424	39	γ	γ	PROPN
ejpam-3890	424	40	admits	admit	VERB
ejpam-3890	424	41	an	an	DET
ejpam-3890	424	42	efficient	efficient	ADJ
ejpam-3890	424	43	zero	zero	NUM
ejpam-3890	424	44	ring	ring	NOUN
ejpam-3890	424	45	labeling	labeling	NOUN
ejpam-3890	424	46	.	.	PUNCT
ejpam-3890	425	1	in	in	ADP
ejpam-3890	425	2	particular	particular	ADJ
ejpam-3890	425	3	,	,	PUNCT
ejpam-3890	425	4	the	the	DET
ejpam-3890	425	5	following	follow	VERB
ejpam-3890	425	6	graphs	graph	NOUN
ejpam-3890	425	7	can	can	AUX
ejpam-3890	425	8	be	be	AUX
ejpam-3890	425	9	embedded	embed	VERB
ejpam-3890	425	10	in	in	ADP
ejpam-3890	425	11	the	the	DET
ejpam-3890	425	12	restricted	restricted	ADJ
ejpam-3890	425	13	zero	zero	NUM
ejpam-3890	425	14	ring	ring	NOUN
ejpam-3890	425	15	graph	graph	NOUN
ejpam-3890	425	16	γs(r	γs(r	PUNCT
ejpam-3890	425	17	)	)	PUNCT
ejpam-3890	425	18	where	where	SCONJ
ejpam-3890	425	19	s	s	VERB
ejpam-3890	425	20	=	=	PUNCT
ejpam-3890	425	21	{	{	PUNCT
ejpam-3890	425	22	a1	a1	PROPN
ejpam-3890	425	23	,	,	PUNCT
ejpam-3890	425	24	a2	a2	PROPN
ejpam-3890	425	25	,	,	PUNCT
ejpam-3890	425	26	a3	a3	NOUN
ejpam-3890	425	27	}	}	PUNCT
ejpam-3890	425	28	and	and	CCONJ
ejpam-3890	425	29	r	r	NOUN
ejpam-3890	425	30	=	=	SYM
ejpam-3890	425	31	m0	m0	NOUN
ejpam-3890	425	32	2	2	NUM
ejpam-3890	425	33	(	(	PUNCT
ejpam-3890	425	34	zm	zm	PROPN
ejpam-3890	425	35	)	)	PUNCT
ejpam-3890	425	36	:	:	PUNCT
ejpam-3890	425	37	tadpole	tadpole	NOUN
ejpam-3890	425	38	graphs	graph	NOUN
ejpam-3890	425	39	,	,	PUNCT
ejpam-3890	425	40	cactus	cactus	NOUN
ejpam-3890	425	41	graphs	graph	NOUN
ejpam-3890	425	42	with	with	ADP
ejpam-3890	425	43	exactly	exactly	ADV
ejpam-3890	425	44	2	2	NUM
ejpam-3890	425	45	cycles	cycle	NOUN
ejpam-3890	425	46	and	and	CCONJ
ejpam-3890	425	47	maximum	maximum	ADJ
ejpam-3890	425	48	degree	degree	NOUN
ejpam-3890	425	49	3	3	NUM
ejpam-3890	425	50	,	,	PUNCT
ejpam-3890	425	51	minimum	minimum	ADJ
ejpam-3890	425	52	degree	degree	NOUN
ejpam-3890	425	53	2	2	NUM
ejpam-3890	425	54	and	and	CCONJ
ejpam-3890	425	55	the	the	DET
ejpam-3890	425	56	cartesian	cartesian	ADJ
ejpam-3890	425	57	product	product	NOUN
ejpam-3890	425	58	of	of	ADP
ejpam-3890	425	59	p2×pn	p2×pn	PROPN
ejpam-3890	425	60	for	for	ADP
ejpam-3890	425	61	n	n	X
ejpam-3890	425	62	≥	≥	NOUN
ejpam-3890	425	63	3	3	NUM
ejpam-3890	425	64	.	.	PUNCT
ejpam-3890	426	1	thus	thus	ADV
ejpam-3890	426	2	,	,	PUNCT
ejpam-3890	426	3	these	these	DET
ejpam-3890	426	4	graphs	graph	NOUN
ejpam-3890	426	5	admit	admit	VERB
ejpam-3890	426	6	an	an	DET
ejpam-3890	426	7	efficient	efficient	ADJ
ejpam-3890	426	8	zero	zero	NUM
ejpam-3890	426	9	ring	ring	NOUN
ejpam-3890	426	10	labeling	labeling	NOUN
ejpam-3890	426	11	for	for	ADP
ejpam-3890	426	12	some	some	DET
ejpam-3890	426	13	appropriate	appropriate	ADJ
ejpam-3890	426	14	zero	zero	NUM
ejpam-3890	426	15	ring	ring	NOUN
ejpam-3890	426	16	r.	r.	PROPN
ejpam-3890	426	17	f.j	f.j	PROPN
ejpam-3890	426	18	.	.	PUNCT
ejpam-3890	427	1	campeña	campeña	PROPN
ejpam-3890	427	2	et	et	PROPN
ejpam-3890	427	3	al	al	PROPN
ejpam-3890	427	4	.	.	PUNCT
ejpam-3890	427	5	/	/	SYM
ejpam-3890	427	6	eur	eur	PROPN
ejpam-3890	427	7	.	.	PUNCT
ejpam-3890	428	1	j.	j.	PROPN
ejpam-3890	428	2	pure	pure	PROPN
ejpam-3890	428	3	appl	appl	PROPN
ejpam-3890	428	4	.	.	PROPN
ejpam-3890	428	5	math	math	PROPN
ejpam-3890	428	6	,	,	PUNCT
ejpam-3890	428	7	14	14	NUM
ejpam-3890	428	8	(	(	PUNCT
ejpam-3890	428	9	1	1	NUM
ejpam-3890	428	10	)	)	PUNCT
ejpam-3890	428	11	(	(	PUNCT
ejpam-3890	428	12	2021	2021	NUM
ejpam-3890	428	13	)	)	PUNCT
ejpam-3890	428	14	,	,	PUNCT
ejpam-3890	428	15	268	268	NUM
ejpam-3890	428	16	-	-	SYM
ejpam-3890	428	17	277	277	NUM
ejpam-3890	428	18	276	276	NUM
ejpam-3890	428	19	theorem	theorem	NOUN
ejpam-3890	428	20	6	6	NUM
ejpam-3890	428	21	.	.	PUNCT
ejpam-3890	429	1	the	the	DET
ejpam-3890	429	2	following	follow	VERB
ejpam-3890	429	3	graphs	graph	NOUN
ejpam-3890	429	4	admit	admit	VERB
ejpam-3890	429	5	an	an	DET
ejpam-3890	429	6	efficient	efficient	ADJ
ejpam-3890	429	7	zero	zero	NUM
ejpam-3890	429	8	ring	ring	NOUN
ejpam-3890	429	9	labeling	labeling	NOUN
ejpam-3890	429	10	:	:	PUNCT
ejpam-3890	429	11	(	(	PUNCT
ejpam-3890	429	12	i	i	NOUN
ejpam-3890	429	13	)	)	PUNCT
ejpam-3890	429	14	tadpole	tadpole	PROPN
ejpam-3890	429	15	graphs	graph	VERB
ejpam-3890	429	16	tn	tn	PROPN
ejpam-3890	429	17	,	,	PUNCT
ejpam-3890	429	18	m	m	PROPN
ejpam-3890	429	19	(	(	PUNCT
ejpam-3890	429	20	ii	ii	NOUN
ejpam-3890	429	21	)	)	PUNCT
ejpam-3890	429	22	cactus	cactus	NOUN
ejpam-3890	429	23	graph	graph	NOUN
ejpam-3890	429	24	with	with	ADP
ejpam-3890	429	25	exactly	exactly	ADV
ejpam-3890	429	26	two	two	NUM
ejpam-3890	429	27	cycles	cycle	NOUN
ejpam-3890	429	28	having	have	VERB
ejpam-3890	429	29	a	a	DET
ejpam-3890	429	30	minimum	minimum	ADJ
ejpam-3890	429	31	and	and	CCONJ
ejpam-3890	429	32	maximum	maximum	ADJ
ejpam-3890	429	33	degree	degree	NOUN
ejpam-3890	429	34	of	of	ADP
ejpam-3890	429	35	a	a	DET
ejpam-3890	429	36	vertex	vertex	NOUN
ejpam-3890	429	37	2	2	NUM
ejpam-3890	429	38	and	and	CCONJ
ejpam-3890	429	39	3	3	NUM
ejpam-3890	429	40	,	,	PUNCT
ejpam-3890	429	41	respectively	respectively	ADV
ejpam-3890	429	42	.	.	PUNCT
ejpam-3890	430	1	(	(	PUNCT
ejpam-3890	430	2	iii	iii	X
ejpam-3890	430	3	)	)	PUNCT
ejpam-3890	430	4	p2	p2	PROPN
ejpam-3890	430	5	×	×	PROPN
ejpam-3890	430	6	pn	pn	NOUN
ejpam-3890	430	7	for	for	ADP
ejpam-3890	430	8	any	any	DET
ejpam-3890	430	9	n	n	PRON
ejpam-3890	430	10	≥	≥	NOUN
ejpam-3890	430	11	3	3	NUM
ejpam-3890	430	12	proof	proof	NOUN
ejpam-3890	430	13	.	.	PUNCT
ejpam-3890	431	1	the	the	DET
ejpam-3890	431	2	proof	proof	NOUN
ejpam-3890	431	3	follows	follow	VERB
ejpam-3890	431	4	from	from	ADP
ejpam-3890	431	5	the	the	DET
ejpam-3890	431	6	fact	fact	NOUN
ejpam-3890	431	7	that	that	SCONJ
ejpam-3890	431	8	these	these	DET
ejpam-3890	431	9	graphs	graph	NOUN
ejpam-3890	431	10	are	be	AUX
ejpam-3890	431	11	edge	edge	NOUN
ejpam-3890	431	12	induced	induced	ADJ
ejpam-3890	431	13	subgraphs	subgraph	NOUN
ejpam-3890	431	14	of	of	ADP
ejpam-3890	431	15	the	the	DET
ejpam-3890	431	16	graph	graph	NOUN
ejpam-3890	431	17	γs(m0	γs(m0	INTJ
ejpam-3890	431	18	2	2	NUM
ejpam-3890	431	19	(	(	PUNCT
ejpam-3890	431	20	zn	zn	NOUN
ejpam-3890	431	21	)	)	PUNCT
ejpam-3890	431	22	)	)	PUNCT
ejpam-3890	431	23	for	for	ADP
ejpam-3890	431	24	some	some	DET
ejpam-3890	431	25	positive	positive	ADJ
ejpam-3890	431	26	integer	integer	NOUN
ejpam-3890	431	27	n	n	NOUN
ejpam-3890	431	28	and	and	CCONJ
ejpam-3890	431	29	s	s	PART
ejpam-3890	431	30	=	=	NOUN
ejpam-3890	431	31	{	{	PUNCT
ejpam-3890	431	32	a1	a1	PROPN
ejpam-3890	431	33	,	,	PUNCT
ejpam-3890	431	34	a2	a2	PROPN
ejpam-3890	431	35	,	,	PUNCT
ejpam-3890	431	36	a3	a3	NOUN
ejpam-3890	431	37	}	}	PUNCT
ejpam-3890	431	38	.	.	PUNCT
ejpam-3890	432	1	thus	thus	ADV
ejpam-3890	432	2	,	,	PUNCT
ejpam-3890	432	3	by	by	ADP
ejpam-3890	432	4	theorem	theorem	NOUN
ejpam-3890	432	5	2	2	NUM
ejpam-3890	432	6	,	,	PUNCT
ejpam-3890	432	7	these	these	DET
ejpam-3890	432	8	graphs	graph	NOUN
ejpam-3890	432	9	admit	admit	VERB
ejpam-3890	432	10	an	an	DET
ejpam-3890	432	11	efficient	efficient	ADJ
ejpam-3890	432	12	zero	zero	NUM
ejpam-3890	432	13	ring	ring	NOUN
ejpam-3890	432	14	labeling	labeling	NOUN
ejpam-3890	432	15	.	.	PUNCT
ejpam-3890	433	1	in	in	ADP
ejpam-3890	433	2	particular	particular	ADJ
ejpam-3890	433	3	,	,	PUNCT
ejpam-3890	433	4	the	the	DET
ejpam-3890	433	5	graph	graph	NOUN
ejpam-3890	433	6	tn	tn	PROPN
ejpam-3890	433	7	,	,	PUNCT
ejpam-3890	433	8	m	m	VERB
ejpam-3890	433	9	is	be	AUX
ejpam-3890	433	10	an	an	DET
ejpam-3890	433	11	edge	edge	NOUN
ejpam-3890	433	12	induced	induce	VERB
ejpam-3890	433	13	subgraph	subgraph	NOUN
ejpam-3890	433	14	of	of	ADP
ejpam-3890	433	15	γs(m0	γs(m0	PROPN
ejpam-3890	433	16	2	2	PROPN
ejpam-3890	433	17	(	(	PUNCT
ejpam-3890	433	18	zk	zk	PROPN
ejpam-3890	433	19	)	)	PUNCT
ejpam-3890	433	20	)	)	PUNCT
ejpam-3890	433	21	where	where	SCONJ
ejpam-3890	433	22	k	k	PROPN
ejpam-3890	433	23	≥	≥	X
ejpam-3890	433	24	n+m	n+m	NUM
ejpam-3890	433	25	and	and	CCONJ
ejpam-3890	433	26	s	s	VERB
ejpam-3890	433	27	=	=	SYM
ejpam-3890	433	28	{	{	PUNCT
ejpam-3890	433	29	a1	a1	PROPN
ejpam-3890	433	30	,	,	PUNCT
ejpam-3890	433	31	a2	a2	PROPN
ejpam-3890	433	32	,	,	PUNCT
ejpam-3890	433	33	a3	a3	NOUN
ejpam-3890	433	34	}	}	PUNCT
ejpam-3890	433	35	.	.	PUNCT
ejpam-3890	434	1	suppose	suppose	VERB
ejpam-3890	434	2	γ	γ	NOUN
ejpam-3890	434	3	is	be	AUX
ejpam-3890	434	4	a	a	DET
ejpam-3890	434	5	cactus	cactus	NOUN
ejpam-3890	434	6	graph	graph	NOUN
ejpam-3890	434	7	with	with	ADP
ejpam-3890	434	8	exactly	exactly	ADV
ejpam-3890	434	9	two	two	NUM
ejpam-3890	434	10	cycles	cycle	NOUN
ejpam-3890	434	11	,	,	PUNCT
ejpam-3890	434	12	and	and	CCONJ
ejpam-3890	434	13	has	have	VERB
ejpam-3890	434	14	minimum	minimum	ADJ
ejpam-3890	434	15	and	and	CCONJ
ejpam-3890	434	16	maximum	maximum	ADJ
ejpam-3890	434	17	degree	degree	NOUN
ejpam-3890	434	18	of	of	ADP
ejpam-3890	434	19	a	a	DET
ejpam-3890	434	20	vertex	vertex	NOUN
ejpam-3890	434	21	equal	equal	ADJ
ejpam-3890	434	22	to	to	ADP
ejpam-3890	434	23	2	2	NUM
ejpam-3890	434	24	and	and	CCONJ
ejpam-3890	434	25	3	3	NUM
ejpam-3890	434	26	,	,	PUNCT
ejpam-3890	434	27	respectively	respectively	ADV
ejpam-3890	434	28	.	.	PUNCT
ejpam-3890	435	1	if	if	SCONJ
ejpam-3890	435	2	the	the	DET
ejpam-3890	435	3	cycles	cycle	NOUN
ejpam-3890	435	4	in	in	ADP
ejpam-3890	435	5	γ	γ	NOUN
ejpam-3890	435	6	have	have	VERB
ejpam-3890	435	7	lengths	length	NOUN
ejpam-3890	435	8	n1	n1	ADJ
ejpam-3890	435	9	and	and	CCONJ
ejpam-3890	435	10	n2	n2	ADJ
ejpam-3890	435	11	and	and	CCONJ
ejpam-3890	435	12	are	be	AUX
ejpam-3890	435	13	connected	connect	VERB
ejpam-3890	435	14	by	by	ADP
ejpam-3890	435	15	a	a	DET
ejpam-3890	435	16	path	path	NOUN
ejpam-3890	435	17	of	of	ADP
ejpam-3890	435	18	length	length	NOUN
ejpam-3890	435	19	m	m	PROPN
ejpam-3890	435	20	,	,	PUNCT
ejpam-3890	435	21	then	then	ADV
ejpam-3890	435	22	the	the	DET
ejpam-3890	435	23	graph	graph	NOUN
ejpam-3890	435	24	γ	γ	PROPN
ejpam-3890	435	25	is	be	AUX
ejpam-3890	435	26	an	an	DET
ejpam-3890	435	27	edge	edge	NOUN
ejpam-3890	435	28	induced	induce	VERB
ejpam-3890	435	29	subgraph	subgraph	NOUN
ejpam-3890	435	30	of	of	ADP
ejpam-3890	435	31	γs(m0	γs(m0	PROPN
ejpam-3890	435	32	2	2	PROPN
ejpam-3890	435	33	(	(	PUNCT
ejpam-3890	435	34	zk	zk	PROPN
ejpam-3890	435	35	)	)	PUNCT
ejpam-3890	435	36	)	)	PUNCT
ejpam-3890	435	37	where	where	SCONJ
ejpam-3890	435	38	s	s	VERB
ejpam-3890	435	39	=	=	PUNCT
ejpam-3890	435	40	{	{	PUNCT
ejpam-3890	435	41	a1	a1	PROPN
ejpam-3890	435	42	,	,	PUNCT
ejpam-3890	435	43	a2	a2	PROPN
ejpam-3890	435	44	,	,	PUNCT
ejpam-3890	435	45	a3	a3	NOUN
ejpam-3890	435	46	}	}	PUNCT
ejpam-3890	435	47	and	and	CCONJ
ejpam-3890	435	48	for	for	ADP
ejpam-3890	435	49	k	k	PROPN
ejpam-3890	435	50	≥	≥	PROPN
ejpam-3890	435	51	n1	n1	PROPN
ejpam-3890	435	52	+	+	CCONJ
ejpam-3890	435	53	n2	n2	PROPN
ejpam-3890	435	54	+	+	CCONJ
ejpam-3890	435	55	m−	m−	PROPN
ejpam-3890	435	56	1	1	NUM
ejpam-3890	435	57	.	.	PUNCT
ejpam-3890	436	1	the	the	DET
ejpam-3890	436	2	graph	graph	NOUN
ejpam-3890	436	3	p2	p2	PROPN
ejpam-3890	436	4	×	×	NOUN
ejpam-3890	436	5	pn	pn	NOUN
ejpam-3890	436	6	can	can	AUX
ejpam-3890	436	7	be	be	AUX
ejpam-3890	436	8	embedded	embed	VERB
ejpam-3890	436	9	in	in	ADP
ejpam-3890	436	10	the	the	DET
ejpam-3890	436	11	restricted	restricted	ADJ
ejpam-3890	436	12	zero	zero	NUM
ejpam-3890	436	13	ring	ring	NOUN
ejpam-3890	436	14	graph	graph	NOUN
ejpam-3890	436	15	γa(m0	γa(m0	ADP
ejpam-3890	436	16	2	2	NUM
ejpam-3890	436	17	(	(	PUNCT
ejpam-3890	436	18	zk	zk	PROPN
ejpam-3890	436	19	)	)	PUNCT
ejpam-3890	436	20	)	)	PUNCT
ejpam-3890	436	21	where	where	SCONJ
ejpam-3890	436	22	s	s	VERB
ejpam-3890	436	23	=	=	PUNCT
ejpam-3890	436	24	{	{	PUNCT
ejpam-3890	436	25	a1	a1	PROPN
ejpam-3890	436	26	,	,	PUNCT
ejpam-3890	436	27	a2	a2	PROPN
ejpam-3890	436	28	,	,	PUNCT
ejpam-3890	436	29	a3	a3	NOUN
ejpam-3890	436	30	}	}	PUNCT
ejpam-3890	436	31	and	and	CCONJ
ejpam-3890	436	32	for	for	ADP
ejpam-3890	436	33	k	k	PROPN
ejpam-3890	436	34	=	=	SYM
ejpam-3890	436	35	2n	2n	NUM
ejpam-3890	437	1	+	+	CCONJ
ejpam-3890	437	2	2	2	X
ejpam-3890	437	3	.	.	PUNCT
ejpam-3890	437	4	...................................................................................	...................................................................................	PUNCT
ejpam-3890	437	5	...................................................................................	...................................................................................	PUNCT
ejpam-3890	437	6	...................................................................................	...................................................................................	PUNCT
ejpam-3890	437	7	...................................................................................	...................................................................................	PUNCT
ejpam-3890	437	8	....................................	....................................	PUNCT
ejpam-3890	437	9	....................................	....................................	PUNCT
ejpam-3890	437	10	...................................................................................	...................................................................................	PUNCT
ejpam-3890	437	11	...............................................	...............................................	PUNCT
ejpam-3890	437	12	....................................	....................................	PUNCT
ejpam-3890	437	13	...........	...........	PUNCT
ejpam-3890	437	14	..........	..........	PUNCT
ejpam-3890	437	15	..........	..........	PUNCT
ejpam-3890	438	1	..........	..........	PUNCT
ejpam-3890	438	2	......	......	PUNCT
ejpam-3890	439	1	....................................	....................................	PUNCT
ejpam-3890	439	2	.........	.........	PUNCT
ejpam-3890	439	3	........	........	PUNCT
ejpam-3890	439	4	........	........	PUNCT
ejpam-3890	439	5	........	........	PUNCT
ejpam-3890	439	6	........	........	PUNCT
ejpam-3890	440	1	......	......	PUNCT
ejpam-3890	440	2	....................................	....................................	PUNCT
ejpam-3890	440	3	...............................................	...............................................	PUNCT
ejpam-3890	441	1	....................................	....................................	PUNCT
ejpam-3890	441	2	...................	...................	PUNCT
ejpam-3890	442	1	..................	..................	PUNCT
ejpam-3890	443	1	..........	..........	PUNCT
ejpam-3890	443	2	..	..	PUNCT
ejpam-3890	443	3	.................................................................................	.................................................................................	PUNCT
ejpam-3890	443	4	...................................................................................	...................................................................................	PUNCT
ejpam-3890	444	1	a0	a0	PROPN
ejpam-3890	444	2	a1	a1	PROPN
ejpam-3890	444	3	a2	a2	PROPN
ejpam-3890	444	4	a11	a11	PROPN
ejpam-3890	444	5	a4	a4	PROPN
ejpam-3890	444	6	a9	a9	PROPN
ejpam-3890	444	7	a6	a6	NOUN
ejpam-3890	444	8	a7	a7	PROPN
ejpam-3890	444	9	a8	a8	PROPN
ejpam-3890	444	10	a5	a5	PROPN
ejpam-3890	444	11	a10	a10	PROPN
ejpam-3890	444	12	a3	a3	NOUN
ejpam-3890	444	13	...............	...............	PUNCT
ejpam-3890	444	14	........................................	........................................	PUNCT
ejpam-3890	444	15	....................................	....................................	PUNCT
ejpam-3890	444	16	....................................	....................................	PUNCT
ejpam-3890	444	17	....................................	....................................	PUNCT
ejpam-3890	444	18	....................................	....................................	PUNCT
ejpam-3890	444	19	....................................	....................................	PUNCT
ejpam-3890	444	20	....................................	....................................	PUNCT
ejpam-3890	444	21	....................................	....................................	PUNCT
ejpam-3890	444	22	....................................	....................................	PUNCT
ejpam-3890	444	23	....................................	....................................	PUNCT
ejpam-3890	444	24	....................................	....................................	PUNCT
ejpam-3890	444	25	.........	.........	PUNCT
ejpam-3890	444	26	........	........	PUNCT
ejpam-3890	444	27	........	........	PUNCT
ejpam-3890	444	28	........	........	PUNCT
ejpam-3890	444	29	........	........	PUNCT
ejpam-3890	444	30	........	........	PUNCT
ejpam-3890	444	31	........	........	PUNCT
ejpam-3890	444	32	........	........	PUNCT
ejpam-3890	444	33	........	........	PUNCT
ejpam-3890	444	34	...	...	PUNCT
ejpam-3890	444	35	.........	.........	PUNCT
ejpam-3890	444	36	........	........	PUNCT
ejpam-3890	444	37	........	........	PUNCT
ejpam-3890	444	38	........	........	PUNCT
ejpam-3890	444	39	........	........	PUNCT
ejpam-3890	444	40	........	........	PUNCT
ejpam-3890	444	41	........	........	PUNCT
ejpam-3890	444	42	........	........	PUNCT
ejpam-3890	444	43	........	........	PUNCT
ejpam-3890	444	44	...	...	PUNCT
ejpam-3890	444	45	.........	.........	PUNCT
ejpam-3890	444	46	........	........	PUNCT
ejpam-3890	444	47	........	........	PUNCT
ejpam-3890	444	48	........	........	PUNCT
ejpam-3890	444	49	........	........	PUNCT
ejpam-3890	444	50	........	........	PUNCT
ejpam-3890	444	51	........	........	PUNCT
ejpam-3890	444	52	........	........	PUNCT
ejpam-3890	444	53	........	........	PUNCT
ejpam-3890	444	54	...	...	PUNCT
ejpam-3890	444	55	.........	.........	PUNCT
ejpam-3890	444	56	........	........	PUNCT
ejpam-3890	444	57	........	........	PUNCT
ejpam-3890	444	58	........	........	PUNCT
ejpam-3890	444	59	........	........	PUNCT
ejpam-3890	444	60	........	........	PUNCT
ejpam-3890	444	61	........	........	PUNCT
ejpam-3890	444	62	........	........	PUNCT
ejpam-3890	444	63	........	........	PUNCT
ejpam-3890	444	64	...	...	PUNCT
ejpam-3890	444	65	.........	.........	PUNCT
ejpam-3890	444	66	........	........	PUNCT
ejpam-3890	444	67	........	........	PUNCT
ejpam-3890	444	68	........	........	PUNCT
ejpam-3890	444	69	........	........	PUNCT
ejpam-3890	444	70	........	........	PUNCT
ejpam-3890	444	71	........	........	PUNCT
ejpam-3890	444	72	........	........	PUNCT
ejpam-3890	444	73	........	........	PUNCT
ejpam-3890	444	74	...	...	PUNCT
ejpam-3890	444	75	............	............	PUNCT
ejpam-3890	444	76	...........	...........	PUNCT
ejpam-3890	444	77	...........	...........	PUNCT
ejpam-3890	444	78	...........	...........	PUNCT
ejpam-3890	444	79	...........	...........	PUNCT
ejpam-3890	444	80	...........	...........	PUNCT
ejpam-3890	444	81	...........	...........	PUNCT
ejpam-3890	444	82	...........	...........	PUNCT
ejpam-3890	444	83	...........	...........	PUNCT
ejpam-3890	444	84	...........	...........	PUNCT
ejpam-3890	444	85	............	............	PUNCT
ejpam-3890	444	86	...........	...........	PUNCT
ejpam-3890	444	87	...........	...........	PUNCT
ejpam-3890	444	88	...........	...........	PUNCT
ejpam-3890	444	89	...........	...........	PUNCT
ejpam-3890	444	90	...........	...........	PUNCT
ejpam-3890	444	91	...........	...........	PUNCT
ejpam-3890	444	92	...........	...........	PUNCT
ejpam-3890	444	93	...........	...........	PUNCT
ejpam-3890	444	94	...........	...........	PUNCT
ejpam-3890	444	95	............	............	PUNCT
ejpam-3890	444	96	...........	...........	PUNCT
ejpam-3890	444	97	...........	...........	PUNCT
ejpam-3890	444	98	...........	...........	PUNCT
ejpam-3890	444	99	...........	...........	PUNCT
ejpam-3890	444	100	...........	...........	PUNCT
ejpam-3890	444	101	...........	...........	PUNCT
ejpam-3890	444	102	...........	...........	PUNCT
ejpam-3890	444	103	...........	...........	PUNCT
ejpam-3890	444	104	...........	...........	PUNCT
ejpam-3890	444	105	............	............	PUNCT
ejpam-3890	444	106	...........	...........	PUNCT
ejpam-3890	444	107	...........	...........	PUNCT
ejpam-3890	444	108	...........	...........	PUNCT
ejpam-3890	444	109	...........	...........	PUNCT
ejpam-3890	444	110	...........	...........	PUNCT
ejpam-3890	444	111	...........	...........	PUNCT
ejpam-3890	444	112	...........	...........	PUNCT
ejpam-3890	444	113	...........	...........	PUNCT
ejpam-3890	444	114	...........	...........	PUNCT
ejpam-3890	444	115	...............................................................................................................	...............................................................................................................	PUNCT
ejpam-3890	444	116	...............................................................................................................	...............................................................................................................	PUNCT
ejpam-3890	444	117	...............................................................................................................	...............................................................................................................	PUNCT
ejpam-3890	444	118	...............................................................................................................	...............................................................................................................	PROPN
ejpam-3890	445	1	a0	a0	PROPN
ejpam-3890	445	2	a11	a11	PROPN
ejpam-3890	445	3	a10	a10	PROPN
ejpam-3890	445	4	a9	a9	PROPN
ejpam-3890	445	5	a8	a8	PROPN
ejpam-3890	445	6	a2	a2	PROPN
ejpam-3890	445	7	a3	a3	PROPN
ejpam-3890	445	8	a4	a4	PROPN
ejpam-3890	445	9	a5	a5	PROPN
ejpam-3890	445	10	a6	a6	NOUN
ejpam-3890	445	11	....................................	....................................	PUNCT
ejpam-3890	445	12	..................................................................................................	..................................................................................................	PUNCT
ejpam-3890	446	1	..............................................................	..............................................................	PUNCT
ejpam-3890	446	2	....................................	....................................	PUNCT
ejpam-3890	447	1	..........	..........	PUNCT
ejpam-3890	447	2	.........	.........	PUNCT
ejpam-3890	448	1	.........	.........	PUNCT
ejpam-3890	448	2	.........	.........	PUNCT
ejpam-3890	449	1	.........	.........	PUNCT
ejpam-3890	449	2	.........	.........	PUNCT
ejpam-3890	449	3	.......	.......	PUNCT
ejpam-3890	449	4	....................................	....................................	PUNCT
ejpam-3890	449	5	...............	...............	PUNCT
ejpam-3890	450	1	..............	..............	PUNCT
ejpam-3890	450	2	..............	..............	PUNCT
ejpam-3890	451	1	..............	..............	PUNCT
ejpam-3890	451	2	.....	.....	PUNCT
ejpam-3890	451	3	...	...	PUNCT
ejpam-3890	451	4	...............................................................................................	...............................................................................................	PUNCT
ejpam-3890	452	1	..................................................................................................	..................................................................................................	PUNCT
ejpam-3890	452	2	....................................	....................................	PUNCT
ejpam-3890	453	1	..................................................................................................	..................................................................................................	PUNCT
ejpam-3890	453	2	..............................................................	..............................................................	PUNCT
ejpam-3890	454	1	....................................	....................................	PUNCT
ejpam-3890	454	2	..........	..........	PUNCT
ejpam-3890	455	1	.........	.........	PUNCT
ejpam-3890	455	2	.........	.........	PUNCT
ejpam-3890	456	1	.........	.........	PUNCT
ejpam-3890	456	2	.........	.........	PUNCT
ejpam-3890	457	1	.........	.........	PUNCT
ejpam-3890	457	2	.......	.......	PUNCT
ejpam-3890	457	3	....................................	....................................	PUNCT
ejpam-3890	457	4	...............	...............	PUNCT
ejpam-3890	458	1	..............	..............	PUNCT
ejpam-3890	458	2	..............	..............	PUNCT
ejpam-3890	459	1	..............	..............	PUNCT
ejpam-3890	459	2	.....	.....	PUNCT
ejpam-3890	459	3	...	...	PUNCT
ejpam-3890	459	4	...............................................................................................	...............................................................................................	PUNCT
ejpam-3890	459	5	..................................................................................................................................................................................................................	..................................................................................................................................................................................................................	PUNCT
ejpam-3890	460	1	..................................................................................................	..................................................................................................	PUNCT
ejpam-3890	460	2	..............................................................	..............................................................	PUNCT
ejpam-3890	461	1	a1a10	a1a10	PROPN
ejpam-3890	461	2	a0	a0	PROPN
ejpam-3890	461	3	a11	a11	PROPN
ejpam-3890	461	4	a9	a9	PROPN
ejpam-3890	461	5	a2	a2	PROPN
ejpam-3890	461	6	a8	a8	PROPN
ejpam-3890	461	7	a3	a3	PROPN
ejpam-3890	461	8	a5	a5	PROPN
ejpam-3890	461	9	a6	a6	PROPN
ejpam-3890	461	10	a7	a7	PROPN
ejpam-3890	461	11	a4	a4	PROPN
ejpam-3890	461	12	figure	figure	NOUN
ejpam-3890	461	13	8	8	NUM
ejpam-3890	461	14	:	:	PUNCT
ejpam-3890	461	15	efficient	efficient	ADJ
ejpam-3890	461	16	zero	zero	NUM
ejpam-3890	461	17	ring	ring	NOUN
ejpam-3890	461	18	labelings	labeling	NOUN
ejpam-3890	461	19	of	of	ADP
ejpam-3890	461	20	tadpole	tadpole	PROPN
ejpam-3890	461	21	graph	graph	NOUN
ejpam-3890	461	22	t7,5	t7,5	PROPN
ejpam-3890	461	23	,	,	PUNCT
ejpam-3890	461	24	cartesian	cartesian	ADJ
ejpam-3890	461	25	product	product	NOUN
ejpam-3890	461	26	p2	p2	VERB
ejpam-3890	461	27	×	×	NOUN
ejpam-3890	461	28	p5	p5	ADJ
ejpam-3890	461	29	and	and	CCONJ
ejpam-3890	461	30	cactus	cactus	NOUN
ejpam-3890	461	31	graph	graph	NOUN
ejpam-3890	461	32	by	by	ADP
ejpam-3890	461	33	r	r	NOUN
ejpam-3890	461	34	=	=	SYM
ejpam-3890	461	35	m0	m0	NOUN
ejpam-3890	461	36	2	2	NUM
ejpam-3890	461	37	(	(	PUNCT
ejpam-3890	461	38	z12	z12	PROPN
ejpam-3890	461	39	)	)	PUNCT
ejpam-3890	461	40	figure	figure	NOUN
ejpam-3890	461	41	8	8	NUM
ejpam-3890	461	42	shows	show	VERB
ejpam-3890	461	43	an	an	DET
ejpam-3890	461	44	efficient	efficient	ADJ
ejpam-3890	461	45	zero	zero	NUM
ejpam-3890	461	46	ring	ring	NOUN
ejpam-3890	461	47	labeling	labeling	NOUN
ejpam-3890	461	48	of	of	ADP
ejpam-3890	461	49	t7,5	t7,5	PROPN
ejpam-3890	461	50	,	,	PUNCT
ejpam-3890	461	51	p2	p2	VERB
ejpam-3890	461	52	×	×	NOUN
ejpam-3890	461	53	p5	p5	ADJ
ejpam-3890	461	54	and	and	CCONJ
ejpam-3890	461	55	a	a	DET
ejpam-3890	461	56	cactus	cactus	NOUN
ejpam-3890	461	57	graph	graph	NOUN
ejpam-3890	461	58	with	with	ADP
ejpam-3890	461	59	exactly	exactly	ADV
ejpam-3890	461	60	two	two	NUM
ejpam-3890	461	61	cycles	cycle	NOUN
ejpam-3890	461	62	as	as	ADP
ejpam-3890	461	63	edge	edge	NOUN
ejpam-3890	461	64	induced	induce	VERB
ejpam-3890	461	65	subgraphs	subgraph	NOUN
ejpam-3890	461	66	of	of	ADP
ejpam-3890	461	67	γs(r	γs(r	NUM
ejpam-3890	461	68	)	)	PUNCT
ejpam-3890	461	69	,	,	PUNCT
ejpam-3890	461	70	where	where	SCONJ
ejpam-3890	461	71	s	s	VERB
ejpam-3890	461	72	=	=	PUNCT
ejpam-3890	461	73	{	{	PUNCT
ejpam-3890	461	74	a1	a1	PROPN
ejpam-3890	461	75	,	,	PUNCT
ejpam-3890	461	76	a2	a2	PROPN
ejpam-3890	461	77	,	,	PUNCT
ejpam-3890	461	78	a3	a3	NOUN
ejpam-3890	461	79	}	}	PUNCT
ejpam-3890	461	80	and	and	CCONJ
ejpam-3890	461	81	r	r	NOUN
ejpam-3890	461	82	=	=	SYM
ejpam-3890	461	83	m0	m0	NOUN
ejpam-3890	461	84	2	2	NUM
ejpam-3890	461	85	(	(	PUNCT
ejpam-3890	461	86	z12	z12	PROPN
ejpam-3890	461	87	)	)	PUNCT
ejpam-3890	461	88	.	.	PUNCT
ejpam-3890	462	1	so	so	ADV
ejpam-3890	462	2	far	far	ADV
ejpam-3890	462	3	,	,	PUNCT
ejpam-3890	462	4	the	the	DET
ejpam-3890	462	5	only	only	ADJ
ejpam-3890	462	6	known	know	VERB
ejpam-3890	462	7	families	family	NOUN
ejpam-3890	462	8	of	of	ADP
ejpam-3890	462	9	graphs	graph	NOUN
ejpam-3890	462	10	that	that	PRON
ejpam-3890	462	11	do	do	AUX
ejpam-3890	462	12	not	not	PART
ejpam-3890	462	13	admit	admit	VERB
ejpam-3890	462	14	an	an	DET
ejpam-3890	462	15	efficient	efficient	ADJ
ejpam-3890	462	16	zero	zero	NUM
ejpam-3890	462	17	ring	ring	NOUN
ejpam-3890	462	18	labeling	labeling	NOUN
ejpam-3890	462	19	are	be	AUX
ejpam-3890	462	20	the	the	DET
ejpam-3890	462	21	odd	odd	ADJ
ejpam-3890	462	22	cycles	cycle	NOUN
ejpam-3890	462	23	.	.	PUNCT
ejpam-3890	463	1	in	in	ADP
ejpam-3890	463	2	[	[	X
ejpam-3890	463	3	5	5	NUM
ejpam-3890	463	4	]	]	PUNCT
ejpam-3890	463	5	,	,	PUNCT
ejpam-3890	463	6	several	several	ADJ
ejpam-3890	463	7	families	family	NOUN
ejpam-3890	463	8	of	of	ADP
ejpam-3890	463	9	trees	tree	NOUN
ejpam-3890	463	10	have	have	AUX
ejpam-3890	463	11	been	be	AUX
ejpam-3890	463	12	shown	show	VERB
ejpam-3890	463	13	to	to	PART
ejpam-3890	463	14	admit	admit	VERB
ejpam-3890	463	15	an	an	DET
ejpam-3890	463	16	efficient	efficient	ADJ
ejpam-3890	463	17	zero	zero	NUM
ejpam-3890	463	18	ring	ring	NOUN
ejpam-3890	463	19	labeling	labeling	NOUN
ejpam-3890	463	20	.	.	PUNCT
ejpam-3890	464	1	the	the	DET
ejpam-3890	464	2	next	next	ADJ
ejpam-3890	464	3	logical	logical	ADJ
ejpam-3890	464	4	question	question	NOUN
ejpam-3890	464	5	on	on	ADP
ejpam-3890	464	6	this	this	PRON
ejpam-3890	464	7	is	be	AUX
ejpam-3890	464	8	to	to	PART
ejpam-3890	464	9	ask	ask	VERB
ejpam-3890	464	10	whether	whether	SCONJ
ejpam-3890	464	11	all	all	DET
ejpam-3890	464	12	trees	tree	NOUN
ejpam-3890	464	13	admit	admit	VERB
ejpam-3890	464	14	an	an	DET
ejpam-3890	464	15	efficient	efficient	ADJ
ejpam-3890	464	16	zero	zero	NUM
ejpam-3890	464	17	ring	ring	NOUN
ejpam-3890	464	18	labeling	labeling	NOUN
ejpam-3890	464	19	,	,	PUNCT
ejpam-3890	464	20	and	and	CCONJ
ejpam-3890	464	21	in	in	ADP
ejpam-3890	464	22	[	[	X
ejpam-3890	464	23	5	5	X
ejpam-3890	464	24	]	]	PUNCT
ejpam-3890	464	25	it	it	PRON
ejpam-3890	464	26	was	be	AUX
ejpam-3890	464	27	conjectured	conjecture	VERB
ejpam-3890	464	28	that	that	SCONJ
ejpam-3890	464	29	all	all	DET
ejpam-3890	464	30	trees	tree	NOUN
ejpam-3890	464	31	admit	admit	VERB
ejpam-3890	464	32	an	an	DET
ejpam-3890	464	33	efficient	efficient	ADJ
ejpam-3890	464	34	zero	zero	NUM
ejpam-3890	464	35	ring	ring	NOUN
ejpam-3890	464	36	labeling	labeling	NOUN
ejpam-3890	464	37	.	.	PUNCT
ejpam-3890	465	1	if	if	SCONJ
ejpam-3890	465	2	the	the	DET
ejpam-3890	465	3	conjecture	conjecture	NOUN
ejpam-3890	465	4	that	that	SCONJ
ejpam-3890	465	5	all	all	DET
ejpam-3890	465	6	trees	tree	NOUN
ejpam-3890	465	7	admit	admit	VERB
ejpam-3890	465	8	an	an	DET
ejpam-3890	465	9	efficient	efficient	ADJ
ejpam-3890	465	10	zero	zero	NUM
ejpam-3890	465	11	ring	ring	NOUN
ejpam-3890	465	12	labeling	labeling	NOUN
ejpam-3890	465	13	is	be	AUX
ejpam-3890	465	14	proved	prove	VERB
ejpam-3890	465	15	to	to	PART
ejpam-3890	465	16	be	be	AUX
ejpam-3890	465	17	true	true	ADJ
ejpam-3890	465	18	,	,	PUNCT
ejpam-3890	465	19	then	then	ADV
ejpam-3890	465	20	the	the	DET
ejpam-3890	465	21	cycle	cycle	NOUN
ejpam-3890	465	22	graph	graph	NOUN
ejpam-3890	465	23	on	on	ADP
ejpam-3890	465	24	an	an	DET
ejpam-3890	465	25	odd	odd	ADJ
ejpam-3890	465	26	number	number	NOUN
ejpam-3890	465	27	of	of	ADP
ejpam-3890	465	28	vertices	vertex	NOUN
ejpam-3890	465	29	is	be	AUX
ejpam-3890	465	30	the	the	DET
ejpam-3890	465	31	smallest	small	ADJ
ejpam-3890	465	32	graph	graph	NOUN
ejpam-3890	465	33	,	,	PUNCT
ejpam-3890	465	34	in	in	ADP
ejpam-3890	465	35	terms	term	NOUN
ejpam-3890	465	36	of	of	ADP
ejpam-3890	465	37	the	the	DET
ejpam-3890	465	38	number	number	NOUN
ejpam-3890	465	39	of	of	ADP
ejpam-3890	465	40	edges	edge	NOUN
ejpam-3890	465	41	,	,	PUNCT
ejpam-3890	465	42	that	that	PRON
ejpam-3890	465	43	does	do	AUX
ejpam-3890	465	44	not	not	PART
ejpam-3890	465	45	admit	admit	VERB
ejpam-3890	465	46	an	an	DET
ejpam-3890	465	47	efficient	efficient	ADJ
ejpam-3890	465	48	zero	zero	NUM
ejpam-3890	465	49	ring	ring	NOUN
ejpam-3890	465	50	labeling	labeling	NOUN
ejpam-3890	465	51	.	.	PUNCT
ejpam-3890	466	1	references	reference	NOUN
ejpam-3890	466	2	277	277	NUM
ejpam-3890	466	3	problem	problem	NOUN
ejpam-3890	466	4	:	:	PUNCT
ejpam-3890	466	5	let	let	VERB
ejpam-3890	466	6	s	s	NOUN
ejpam-3890	466	7	,	,	PUNCT
ejpam-3890	466	8	t	t	PROPN
ejpam-3890	466	9	⊆	⊆	NUM
ejpam-3890	466	10	r	r	NOUN
ejpam-3890	466	11	−	−	PROPN
ejpam-3890	466	12	{	{	PUNCT
ejpam-3890	466	13	0	0	NUM
ejpam-3890	466	14	}	}	PUNCT
ejpam-3890	466	15	such	such	ADJ
ejpam-3890	466	16	that	that	SCONJ
ejpam-3890	466	17	|s|	|s|	PROPN
ejpam-3890	466	18	=	=	SYM
ejpam-3890	466	19	|t	|t	NOUN
ejpam-3890	466	20	|	|	NOUN
ejpam-3890	466	21	.	.	PUNCT
ejpam-3890	467	1	determine	determine	VERB
ejpam-3890	467	2	necessary	necessary	ADJ
ejpam-3890	467	3	and	and	CCONJ
ejpam-3890	467	4	sufficient	sufficient	ADJ
ejpam-3890	467	5	conditions	condition	NOUN
ejpam-3890	467	6	such	such	ADJ
ejpam-3890	467	7	that	that	PRON
ejpam-3890	467	8	γs(r	γs(r	PUNCT
ejpam-3890	467	9	)	)	PUNCT
ejpam-3890	467	10	∼=	∼=	PROPN
ejpam-3890	467	11	γt	γt	NOUN
ejpam-3890	467	12	(	(	PUNCT
ejpam-3890	467	13	r	r	NOUN
ejpam-3890	467	14	)	)	PUNCT
ejpam-3890	467	15	.	.	PUNCT
ejpam-3890	468	1	acknowledgements	acknowledgement	NOUN
ejpam-3890	468	2	the	the	DET
ejpam-3890	468	3	authors	author	NOUN
ejpam-3890	468	4	would	would	AUX
ejpam-3890	468	5	like	like	VERB
ejpam-3890	468	6	to	to	PART
ejpam-3890	468	7	thank	thank	VERB
ejpam-3890	468	8	the	the	DET
ejpam-3890	468	9	referees	referee	NOUN
ejpam-3890	468	10	for	for	ADP
ejpam-3890	468	11	the	the	DET
ejpam-3890	468	12	invaluable	invaluable	ADJ
ejpam-3890	468	13	comments	comment	NOUN
ejpam-3890	468	14	and	and	CCONJ
ejpam-3890	468	15	suggestions	suggestion	NOUN
ejpam-3890	468	16	that	that	PRON
ejpam-3890	468	17	led	lead	VERB
ejpam-3890	468	18	to	to	ADP
ejpam-3890	468	19	this	this	DET
ejpam-3890	468	20	improved	improve	VERB
ejpam-3890	468	21	version	version	NOUN
ejpam-3890	468	22	of	of	ADP
ejpam-3890	468	23	the	the	DET
ejpam-3890	468	24	paper	paper	NOUN
ejpam-3890	468	25	.	.	PUNCT
ejpam-3890	469	1	the	the	DET
ejpam-3890	469	2	study	study	NOUN
ejpam-3890	469	3	is	be	AUX
ejpam-3890	469	4	partially	partially	ADV
ejpam-3890	469	5	funded	fund	VERB
ejpam-3890	469	6	by	by	ADP
ejpam-3890	469	7	the	the	DET
ejpam-3890	469	8	philippine	philippine	ADJ
ejpam-3890	469	9	government	government	NOUN
ejpam-3890	469	10	,	,	PUNCT
ejpam-3890	469	11	particularly	particularly	ADV
ejpam-3890	469	12	the	the	DET
ejpam-3890	469	13	commission	commission	NOUN
ejpam-3890	469	14	on	on	ADP
ejpam-3890	469	15	higher	high	ADJ
ejpam-3890	469	16	education	education	NOUN
ejpam-3890	469	17	and	and	CCONJ
ejpam-3890	469	18	department	department	NOUN
ejpam-3890	469	19	of	of	ADP
ejpam-3890	469	20	science	science	NOUN
ejpam-3890	469	21	and	and	CCONJ
ejpam-3890	469	22	technology	technology	NOUN
ejpam-3890	469	23	-	-	PUNCT
ejpam-3890	469	24	science	science	NOUN
ejpam-3890	469	25	education	education	NOUN
ejpam-3890	469	26	institute	institute	PROPN
ejpam-3890	469	27	.	.	PUNCT
ejpam-3890	470	1	references	reference	NOUN
ejpam-3890	470	2	[	[	X
ejpam-3890	470	3	1	1	NUM
ejpam-3890	470	4	]	]	PUNCT
ejpam-3890	470	5	mukti	mukti	PROPN
ejpam-3890	470	6	acharya	acharya	PROPN
ejpam-3890	470	7	and	and	CCONJ
ejpam-3890	470	8	pranjali	pranjali	PROPN
ejpam-3890	470	9	.	.	PUNCT
ejpam-3890	471	1	graphs	graph	NOUN
ejpam-3890	471	2	associated	associate	VERB
ejpam-3890	471	3	with	with	ADP
ejpam-3890	471	4	finite	finite	NOUN
ejpam-3890	471	5	zero	zero	NUM
ejpam-3890	471	6	ring	ring	NOUN
ejpam-3890	471	7	.	.	PUNCT
ejpam-3890	472	1	general	general	ADJ
ejpam-3890	472	2	mathematics	mathematics	PROPN
ejpam-3890	472	3	notes	note	NOUN
ejpam-3890	472	4	,	,	PUNCT
ejpam-3890	472	5	24(2):53–69	24(2):53–69	NUM
ejpam-3890	472	6	,	,	PUNCT
ejpam-3890	472	7	2014	2014	NUM
ejpam-3890	472	8	.	.	PUNCT
ejpam-3890	473	1	[	[	X
ejpam-3890	473	2	2	2	X
ejpam-3890	473	3	]	]	PUNCT
ejpam-3890	473	4	mukti	mukti	PROPN
ejpam-3890	473	5	acharya	acharya	PROPN
ejpam-3890	473	6	,	,	PUNCT
ejpam-3890	473	7	pranjali	pranjali	PROPN
ejpam-3890	473	8	,	,	PUNCT
ejpam-3890	473	9	and	and	CCONJ
ejpam-3890	473	10	purnima	purnima	PROPN
ejpam-3890	473	11	gupta	gupta	PROPN
ejpam-3890	473	12	.	.	PUNCT
ejpam-3890	474	1	further	further	ADJ
ejpam-3890	474	2	results	result	NOUN
ejpam-3890	474	3	on	on	ADP
ejpam-3890	474	4	zero	zero	NUM
ejpam-3890	474	5	ring	ring	NOUN
ejpam-3890	474	6	labeling	labeling	NOUN
ejpam-3890	474	7	of	of	ADP
ejpam-3890	474	8	graph	graph	NOUN
ejpam-3890	474	9	.	.	PUNCT
ejpam-3890	475	1	bulletin	bulletin	NOUN
ejpam-3890	475	2	of	of	ADP
ejpam-3890	475	3	the	the	DET
ejpam-3890	475	4	international	international	ADJ
ejpam-3890	475	5	mathematical	mathematical	ADJ
ejpam-3890	475	6	virtual	virtual	PROPN
ejpam-3890	475	7	institute	institute	NOUN
ejpam-3890	475	8	,	,	PUNCT
ejpam-3890	475	9	15:205–210	15:205–210	PROPN
ejpam-3890	475	10	,	,	PUNCT
ejpam-3890	475	11	2014	2014	NUM
ejpam-3890	475	12	.	.	PUNCT
ejpam-3890	476	1	[	[	X
ejpam-3890	476	2	3	3	X
ejpam-3890	476	3	]	]	X
ejpam-3890	476	4	mukti	mukti	PROPN
ejpam-3890	476	5	acharya	acharya	PROPN
ejpam-3890	476	6	,	,	PUNCT
ejpam-3890	476	7	pranjali	pranjali	PROPN
ejpam-3890	476	8	,	,	PUNCT
ejpam-3890	476	9	and	and	CCONJ
ejpam-3890	476	10	purnima	purnima	PROPN
ejpam-3890	476	11	gupta	gupta	PROPN
ejpam-3890	476	12	.	.	PUNCT
ejpam-3890	477	1	zero	zero	NUM
ejpam-3890	477	2	ring	ring	NOUN
ejpam-3890	477	3	labeling	labeling	NOUN
ejpam-3890	477	4	of	of	ADP
ejpam-3890	477	5	graphs	graph	NOUN
ejpam-3890	477	6	.	.	PUNCT
ejpam-3890	478	1	electronic	electronic	ADJ
ejpam-3890	478	2	notes	note	NOUN
ejpam-3890	478	3	in	in	ADP
ejpam-3890	478	4	discrete	discrete	ADJ
ejpam-3890	478	5	mathematics	mathematic	NOUN
ejpam-3890	478	6	,	,	PUNCT
ejpam-3890	478	7	48:65–72	48:65–72	PROPN
ejpam-3890	478	8	,	,	PUNCT
ejpam-3890	478	9	2015	2015	NUM
ejpam-3890	478	10	.	.	PUNCT
ejpam-3890	479	1	[	[	X
ejpam-3890	479	2	4	4	X
ejpam-3890	479	3	]	]	X
ejpam-3890	479	4	nicolas	nicolas	PROPN
ejpam-3890	479	5	bourbaki	bourbaki	PROPN
ejpam-3890	479	6	.	.	PUNCT
ejpam-3890	480	1	elements	element	NOUN
ejpam-3890	480	2	of	of	ADP
ejpam-3890	480	3	mathematics	mathematics	NOUN
ejpam-3890	480	4	algebra	algebra	PROPN
ejpam-3890	480	5	1	1	NUM
ejpam-3890	480	6	.	.	PUNCT
ejpam-3890	480	7	springer	springer	PROPN
ejpam-3890	480	8	,	,	PUNCT
ejpam-3890	480	9	germany	germany	PROPN
ejpam-3890	480	10	,	,	PUNCT
ejpam-3890	480	11	1998	1998	NUM
ejpam-3890	480	12	.	.	PUNCT
ejpam-3890	481	1	[	[	X
ejpam-3890	481	2	5	5	NUM
ejpam-3890	481	3	]	]	X
ejpam-3890	481	4	dhenmar	dhenmar	PROPN
ejpam-3890	481	5	e.	e.	PROPN
ejpam-3890	481	6	chua	chua	PROPN
ejpam-3890	481	7	,	,	PUNCT
ejpam-3890	481	8	francis	francis	PROPN
ejpam-3890	481	9	joseph	joseph	PROPN
ejpam-3890	481	10	h.	h.	PROPN
ejpam-3890	481	11	campeña	campeña	PROPN
ejpam-3890	481	12	,	,	PUNCT
ejpam-3890	481	13	and	and	CCONJ
ejpam-3890	481	14	jr	jr	PROPN
ejpam-3890	481	15	.	.	PROPN
ejpam-3890	481	16	floresto	floresto	PROPN
ejpam-3890	481	17	a.	a.	PROPN
ejpam-3890	481	18	franco	franco	PROPN
ejpam-3890	481	19	.	.	PUNCT
ejpam-3890	482	1	efficient	efficient	ADJ
ejpam-3890	482	2	zero	zero	NUM
ejpam-3890	482	3	ring	ring	NOUN
ejpam-3890	482	4	labeling	labeling	NOUN
ejpam-3890	482	5	of	of	ADP
ejpam-3890	482	6	graphs	graph	NOUN
ejpam-3890	482	7	.	.	PUNCT
ejpam-3890	483	1	european	european	ADJ
ejpam-3890	483	2	journal	journal	PROPN
ejpam-3890	483	3	of	of	ADP
ejpam-3890	483	4	pure	pure	ADJ
ejpam-3890	483	5	and	and	CCONJ
ejpam-3890	483	6	applied	applied	ADJ
ejpam-3890	483	7	mathematics	mathematic	NOUN
ejpam-3890	483	8	,	,	PUNCT
ejpam-3890	483	9	13(3):674–696	13(3):674–696	NUM
ejpam-3890	483	10	,	,	PUNCT
ejpam-3890	483	11	2020	2020	NUM
ejpam-3890	483	12	.	.	PUNCT
ejpam-3890	484	1	[	[	X
ejpam-3890	484	2	6	6	NUM
ejpam-3890	484	3	]	]	PUNCT
ejpam-3890	484	4	reinhard	reinhard	NOUN
ejpam-3890	484	5	diestel	diestel	NOUN
ejpam-3890	484	6	.	.	PUNCT
ejpam-3890	485	1	graph	graph	NOUN
ejpam-3890	485	2	theory	theory	NOUN
ejpam-3890	485	3	.	.	PUNCT
ejpam-3890	486	1	springer	springer	NOUN
ejpam-3890	486	2	-	-	PUNCT
ejpam-3890	486	3	verlag	verlag	PROPN
ejpam-3890	486	4	berlin	berlin	PROPN
ejpam-3890	486	5	heidelberg	heidelberg	PROPN
ejpam-3890	486	6	,	,	PUNCT
ejpam-3890	486	7	germany	germany	PROPN
ejpam-3890	486	8	,	,	PUNCT
ejpam-3890	486	9	2017	2017	NUM
ejpam-3890	486	10	.	.	PUNCT
ejpam-3890	487	1	[	[	X
ejpam-3890	487	2	7	7	X
ejpam-3890	487	3	]	]	X
ejpam-3890	487	4	joseph	joseph	PROPN
ejpam-3890	487	5	a.	a.	PROPN
ejpam-3890	487	6	gallian	gallian	PROPN
ejpam-3890	487	7	.	.	PUNCT
ejpam-3890	488	1	a	a	DET
ejpam-3890	488	2	dynamic	dynamic	ADJ
ejpam-3890	488	3	survey	survey	NOUN
ejpam-3890	488	4	of	of	ADP
ejpam-3890	488	5	graph	graph	NOUN
ejpam-3890	488	6	labelling	labelling	NOUN
ejpam-3890	488	7	.	.	PUNCT
ejpam-3890	489	1	electronic	electronic	ADJ
ejpam-3890	489	2	journal	journal	NOUN
ejpam-3890	489	3	of	of	ADP
ejpam-3890	489	4	combinatoric	combinatoric	NOUN
ejpam-3890	489	5	,	,	PUNCT
ejpam-3890	489	6	pages	page	NOUN
ejpam-3890	489	7	1–535	1–535	NUM
ejpam-3890	489	8	,	,	PUNCT
ejpam-3890	489	9	2019	2019	NUM
ejpam-3890	489	10	.	.	PUNCT
ejpam-3890	490	1	[	[	X
ejpam-3890	490	2	8	8	NUM
ejpam-3890	490	3	]	]	X
ejpam-3890	490	4	frank	frank	PROPN
ejpam-3890	490	5	harary	harary	PROPN
ejpam-3890	490	6	.	.	PUNCT
ejpam-3890	491	1	graph	graph	NOUN
ejpam-3890	491	2	theory	theory	NOUN
ejpam-3890	491	3	.	.	PUNCT
ejpam-3890	492	1	addison	addison	PROPN
ejpam-3890	492	2	-	-	PUNCT
ejpam-3890	492	3	wesley	wesley	PROPN
ejpam-3890	492	4	,	,	PUNCT
ejpam-3890	492	5	1969	1969	NUM
ejpam-3890	492	6	.	.	PUNCT
ejpam-3890	493	1	[	[	X
ejpam-3890	493	2	9	9	NUM
ejpam-3890	493	3	]	]	X
ejpam-3890	493	4	alexander	alexander	PROPN
ejpam-3890	493	5	rosa	rosa	PROPN
ejpam-3890	493	6	.	.	PROPN
ejpam-3890	494	1	on	on	ADP
ejpam-3890	494	2	certain	certain	ADJ
ejpam-3890	494	3	valuations	valuation	NOUN
ejpam-3890	494	4	of	of	ADP
ejpam-3890	494	5	the	the	DET
ejpam-3890	494	6	vertices	vertex	NOUN
ejpam-3890	494	7	of	of	ADP
ejpam-3890	494	8	a	a	DET
ejpam-3890	494	9	graph	graph	NOUN
ejpam-3890	494	10	.	.	PUNCT
ejpam-3890	495	1	in	in	ADP
ejpam-3890	495	2	b.n	b.n	PROPN
ejpam-3890	495	3	.	.	PROPN
ejpam-3890	495	4	petrox	petrox	PROPN
ejpam-3890	495	5	and	and	CCONJ
ejpam-3890	495	6	f.	f.	PROPN
ejpam-3890	495	7	csaki	csaki	PROPN
ejpam-3890	495	8	,	,	PUNCT
ejpam-3890	495	9	editors	editor	NOUN
ejpam-3890	495	10	,	,	PUNCT
ejpam-3890	495	11	theory	theory	NOUN
ejpam-3890	495	12	of	of	ADP
ejpam-3890	495	13	graphs	graph	NOUN
ejpam-3890	495	14	international	international	ADJ
ejpam-3890	495	15	symposium	symposium	NOUN
ejpam-3890	495	16	.	.	PUNCT
ejpam-3890	495	17	,	,	PUNCT
ejpam-3890	495	18	pages	page	NOUN
ejpam-3890	495	19	349–335	349–335	NUM
ejpam-3890	495	20	,	,	PUNCT
ejpam-3890	495	21	rome	rome	PROPN
ejpam-3890	495	22	,	,	PUNCT
ejpam-3890	495	23	1966	1966	NUM
ejpam-3890	495	24	.	.	PUNCT
ejpam-3890	496	1	gordon	gordon	PROPN
ejpam-3890	496	2	and	and	CCONJ
ejpam-3890	496	3	breach	breach	PROPN
ejpam-3890	496	4	,	,	PUNCT
ejpam-3890	496	5	n.y	n.y	PROPN
ejpam-3890	496	6	.	.	PROPN
ejpam-3890	496	7	and	and	CCONJ
ejpam-3890	496	8	dunod	dunod	PROPN
ejpam-3890	496	9	and	and	CCONJ
ejpam-3890	496	10	paris	paris	PROPN
ejpam-3890	496	11	.	.	PUNCT