id	sid	tid	token	lemma	pos
ejpam-6425	1	1	european	european	PROPN
ejpam-6425	1	2	journal	journal	PROPN
ejpam-6425	1	3	of	of	ADP
ejpam-6425	1	4	pure	pure	ADJ
ejpam-6425	1	5	and	and	CCONJ
ejpam-6425	1	6	applied	applied	ADJ
ejpam-6425	1	7	mathematics	mathematic	NOUN
ejpam-6425	1	8	2025	2025	NUM
ejpam-6425	1	9	,	,	PUNCT
ejpam-6425	1	10	vol	vol	NOUN
ejpam-6425	1	11	.	.	PROPN
ejpam-6425	1	12	18	18	NUM
ejpam-6425	1	13	,	,	PUNCT
ejpam-6425	1	14	issue	issue	NOUN
ejpam-6425	1	15	4	4	NUM
ejpam-6425	1	16	,	,	PUNCT
ejpam-6425	1	17	article	article	NOUN
ejpam-6425	1	18	number	number	NOUN
ejpam-6425	1	19	6425	6425	NUM
ejpam-6425	1	20	issn	issn	PROPN
ejpam-6425	1	21	1307	1307	NUM
ejpam-6425	1	22	-	-	SYM
ejpam-6425	1	23	5543	5543	NUM
ejpam-6425	1	24	–	–	PUNCT
ejpam-6425	1	25	ejpam.com	ejpam.com	X
ejpam-6425	1	26	published	publish	VERB
ejpam-6425	1	27	by	by	ADP
ejpam-6425	1	28	new	new	PROPN
ejpam-6425	1	29	york	york	PROPN
ejpam-6425	1	30	business	business	PROPN
ejpam-6425	1	31	global	global	PROPN
ejpam-6425	1	32	on	on	ADP
ejpam-6425	1	33	the	the	DET
ejpam-6425	1	34	edge	edge	NOUN
ejpam-6425	1	35	-	-	PUNCT
ejpam-6425	1	36	gracefulness	gracefulness	NOUN
ejpam-6425	1	37	of	of	ADP
ejpam-6425	1	38	water	water	NOUN
ejpam-6425	1	39	wheel	wheel	NOUN
ejpam-6425	1	40	-	-	PUNCT
ejpam-6425	1	41	type	type	NOUN
ejpam-6425	1	42	graphs	graph	NOUN
ejpam-6425	1	43	and	and	CCONJ
ejpam-6425	1	44	their	their	PRON
ejpam-6425	1	45	splitting	splitting	NOUN
ejpam-6425	1	46	graphs	graph	NOUN
ejpam-6425	1	47	aaron	aaron	PROPN
ejpam-6425	1	48	d.	d.	PROPN
ejpam-6425	1	49	c.	c.	PROPN
ejpam-6425	1	50	angel1,∗	angel1,∗	PROPN
ejpam-6425	1	51	,	,	PUNCT
ejpam-6425	1	52	john	john	PROPN
ejpam-6425	1	53	rafael	rafael	PROPN
ejpam-6425	1	54	m.	m.	PROPN
ejpam-6425	1	55	antalan1	antalan1	PROPN
ejpam-6425	1	56	,	,	PUNCT
ejpam-6425	1	57	john	john	PROPN
ejpam-6425	1	58	loureynz	loureynz	PROPN
ejpam-6425	1	59	f.	f.	PROPN
ejpam-6425	1	60	gamurot2	gamurot2	PROPN
ejpam-6425	1	61	,	,	PUNCT
ejpam-6425	1	62	richard	richard	PROPN
ejpam-6425	1	63	p.	p.	PROPN
ejpam-6425	1	64	tagle1	tagle1	NOUN
ejpam-6425	2	1	1	1	NUM
ejpam-6425	2	2	faculty	faculty	NOUN
ejpam-6425	2	3	,	,	PUNCT
ejpam-6425	2	4	department	department	NOUN
ejpam-6425	2	5	of	of	ADP
ejpam-6425	2	6	mathematics	mathematics	PROPN
ejpam-6425	2	7	and	and	CCONJ
ejpam-6425	2	8	physics	physics	PROPN
ejpam-6425	2	9	,	,	PUNCT
ejpam-6425	2	10	college	college	NOUN
ejpam-6425	2	11	of	of	ADP
ejpam-6425	2	12	science	science	NOUN
ejpam-6425	2	13	,	,	PUNCT
ejpam-6425	2	14	central	central	ADJ
ejpam-6425	2	15	luzon	luzon	PROPN
ejpam-6425	2	16	state	state	PROPN
ejpam-6425	2	17	university	university	PROPN
ejpam-6425	2	18	(	(	PUNCT
ejpam-6425	2	19	3120	3120	NUM
ejpam-6425	2	20	)	)	PUNCT
ejpam-6425	2	21	,	,	PUNCT
ejpam-6425	2	22	science	science	NOUN
ejpam-6425	2	23	city	city	NOUN
ejpam-6425	2	24	of	of	ADP
ejpam-6425	2	25	muñoz	muñoz	PROPN
ejpam-6425	2	26	,	,	PUNCT
ejpam-6425	2	27	nueva	nueva	NOUN
ejpam-6425	2	28	ecija	ecija	PROPN
ejpam-6425	2	29	,	,	PUNCT
ejpam-6425	2	30	philippines	philippine	NOUN
ejpam-6425	2	31	2	2	NUM
ejpam-6425	2	32	alumni	alumnus	NOUN
ejpam-6425	2	33	,	,	PUNCT
ejpam-6425	2	34	department	department	NOUN
ejpam-6425	2	35	of	of	ADP
ejpam-6425	2	36	mathematics	mathematics	PROPN
ejpam-6425	2	37	and	and	CCONJ
ejpam-6425	2	38	physics	physics	PROPN
ejpam-6425	2	39	,	,	PUNCT
ejpam-6425	2	40	college	college	NOUN
ejpam-6425	2	41	of	of	ADP
ejpam-6425	2	42	science	science	NOUN
ejpam-6425	2	43	,	,	PUNCT
ejpam-6425	2	44	central	central	ADJ
ejpam-6425	2	45	luzon	luzon	PROPN
ejpam-6425	2	46	state	state	PROPN
ejpam-6425	2	47	university	university	PROPN
ejpam-6425	2	48	(	(	PUNCT
ejpam-6425	2	49	3120	3120	NUM
ejpam-6425	2	50	)	)	PUNCT
ejpam-6425	2	51	,	,	PUNCT
ejpam-6425	2	52	science	science	NOUN
ejpam-6425	2	53	city	city	NOUN
ejpam-6425	2	54	of	of	ADP
ejpam-6425	2	55	muñoz	muñoz	PROPN
ejpam-6425	2	56	,	,	PUNCT
ejpam-6425	2	57	nueva	nueva	NOUN
ejpam-6425	2	58	ecija	ecija	PROPN
ejpam-6425	2	59	,	,	PUNCT
ejpam-6425	2	60	philippines	philippine	NOUN
ejpam-6425	2	61	abstract	abstract	ADJ
ejpam-6425	2	62	.	.	PUNCT
ejpam-6425	3	1	a	a	DET
ejpam-6425	3	2	graph	graph	NOUN
ejpam-6425	3	3	g	g	NOUN
ejpam-6425	3	4	with	with	ADP
ejpam-6425	3	5	p	p	NOUN
ejpam-6425	3	6	vertices	vertex	NOUN
ejpam-6425	3	7	and	and	CCONJ
ejpam-6425	3	8	q	q	NOUN
ejpam-6425	3	9	edges	edge	NOUN
ejpam-6425	3	10	is	be	AUX
ejpam-6425	3	11	said	say	VERB
ejpam-6425	3	12	to	to	PART
ejpam-6425	3	13	be	be	AUX
ejpam-6425	3	14	edge	edge	NOUN
ejpam-6425	3	15	-	-	PUNCT
ejpam-6425	3	16	graceful	graceful	NOUN
ejpam-6425	3	17	if	if	SCONJ
ejpam-6425	3	18	its	its	PRON
ejpam-6425	3	19	edges	edge	NOUN
ejpam-6425	3	20	can	can	AUX
ejpam-6425	3	21	be	be	AUX
ejpam-6425	3	22	labeled	label	VERB
ejpam-6425	3	23	from	from	ADP
ejpam-6425	3	24	1	1	NUM
ejpam-6425	3	25	through	through	ADP
ejpam-6425	3	26	q	q	NOUN
ejpam-6425	3	27	,	,	PUNCT
ejpam-6425	3	28	in	in	ADP
ejpam-6425	3	29	such	such	DET
ejpam-6425	3	30	a	a	DET
ejpam-6425	3	31	way	way	NOUN
ejpam-6425	4	1	that	that	PRON
ejpam-6425	4	2	the	the	DET
ejpam-6425	4	3	labels	label	NOUN
ejpam-6425	4	4	induced	induce	VERB
ejpam-6425	4	5	on	on	ADP
ejpam-6425	4	6	the	the	DET
ejpam-6425	4	7	vertices	vertex	NOUN
ejpam-6425	4	8	by	by	ADP
ejpam-6425	4	9	adding	add	VERB
ejpam-6425	4	10	over	over	ADP
ejpam-6425	4	11	the	the	DET
ejpam-6425	4	12	labels	label	NOUN
ejpam-6425	4	13	of	of	ADP
ejpam-6425	4	14	incident	incident	NOUN
ejpam-6425	4	15	edges	edge	NOUN
ejpam-6425	4	16	modulo	modulo	ADV
ejpam-6425	4	17	p	p	NOUN
ejpam-6425	4	18	are	be	AUX
ejpam-6425	4	19	distinct	distinct	ADJ
ejpam-6425	5	1	.	.	PUNCT
ejpam-6425	6	1	lo	lo	PROPN
ejpam-6425	6	2	’s	’s	PART
ejpam-6425	6	3	theorem	theorem	NOUN
ejpam-6425	6	4	is	be	AUX
ejpam-6425	6	5	a	a	DET
ejpam-6425	6	6	known	know	VERB
ejpam-6425	6	7	result	result	NOUN
ejpam-6425	6	8	under	under	ADP
ejpam-6425	6	9	this	this	DET
ejpam-6425	6	10	topic	topic	NOUN
ejpam-6425	6	11	,	,	PUNCT
ejpam-6425	6	12	which	which	PRON
ejpam-6425	6	13	states	state	VERB
ejpam-6425	6	14	that	that	SCONJ
ejpam-6425	6	15	if	if	SCONJ
ejpam-6425	6	16	a	a	DET
ejpam-6425	6	17	graph	graph	NOUN
ejpam-6425	6	18	g	g	NOUN
ejpam-6425	6	19	with	with	ADP
ejpam-6425	6	20	p	p	NOUN
ejpam-6425	6	21	vertices	vertex	NOUN
ejpam-6425	6	22	and	and	CCONJ
ejpam-6425	6	23	q	q	NOUN
ejpam-6425	6	24	edges	edge	NOUN
ejpam-6425	6	25	is	be	AUX
ejpam-6425	6	26	edge	edge	NOUN
ejpam-6425	6	27	-	-	PUNCT
ejpam-6425	6	28	graceful	graceful	NOUN
ejpam-6425	6	29	,	,	PUNCT
ejpam-6425	6	30	then	then	ADV
ejpam-6425	6	31	p	p	NOUN
ejpam-6425	6	32	∣∣∣(q2+q−	∣∣∣(q2+q−	NOUN
ejpam-6425	6	33	p(p−1	p(p−1	NOUN
ejpam-6425	6	34	)	)	PUNCT
ejpam-6425	6	35	2	2	NUM
ejpam-6425	6	36	)	)	PUNCT
ejpam-6425	6	37	.	.	PUNCT
ejpam-6425	7	1	since	since	SCONJ
ejpam-6425	7	2	the	the	DET
ejpam-6425	7	3	introduction	introduction	NOUN
ejpam-6425	7	4	of	of	ADP
ejpam-6425	7	5	the	the	DET
ejpam-6425	7	6	concept	concept	NOUN
ejpam-6425	7	7	of	of	ADP
ejpam-6425	7	8	edge	edge	NOUN
ejpam-6425	7	9	-	-	PUNCT
ejpam-6425	7	10	graceful	graceful	NOUN
ejpam-6425	7	11	labeling	labeling	NOUN
ejpam-6425	7	12	,	,	PUNCT
ejpam-6425	7	13	several	several	ADJ
ejpam-6425	7	14	works	work	NOUN
ejpam-6425	7	15	on	on	ADP
ejpam-6425	7	16	the	the	DET
ejpam-6425	7	17	edge	edge	NOUN
ejpam-6425	7	18	-	-	PUNCT
ejpam-6425	7	19	graceful	graceful	NOUN
ejpam-6425	7	20	labeling	labeling	NOUN
ejpam-6425	7	21	of	of	ADP
ejpam-6425	7	22	specific	specific	ADJ
ejpam-6425	7	23	families	family	NOUN
ejpam-6425	7	24	of	of	ADP
ejpam-6425	7	25	graphs	graph	NOUN
ejpam-6425	7	26	have	have	AUX
ejpam-6425	7	27	been	be	AUX
ejpam-6425	7	28	conducted	conduct	VERB
ejpam-6425	7	29	.	.	PUNCT
ejpam-6425	8	1	we	we	PRON
ejpam-6425	8	2	recall	recall	VERB
ejpam-6425	8	3	the	the	DET
ejpam-6425	8	4	definition	definition	NOUN
ejpam-6425	8	5	of	of	ADP
ejpam-6425	8	6	water	water	NOUN
ejpam-6425	8	7	wheel	wheel	NOUN
ejpam-6425	8	8	graphs	graph	NOUN
ejpam-6425	8	9	(	(	PUNCT
ejpam-6425	8	10	wwn	wwn	NOUN
ejpam-6425	8	11	)	)	PUNCT
ejpam-6425	8	12	and	and	CCONJ
ejpam-6425	8	13	introduces	introduce	VERB
ejpam-6425	8	14	two	two	NUM
ejpam-6425	8	15	new	new	ADJ
ejpam-6425	8	16	related	relate	VERB
ejpam-6425	8	17	graphs	graph	NOUN
ejpam-6425	8	18	,	,	PUNCT
ejpam-6425	8	19	the	the	DET
ejpam-6425	8	20	right	right	ADJ
ejpam-6425	8	21	water	water	NOUN
ejpam-6425	8	22	wheel	wheel	NOUN
ejpam-6425	8	23	graphs	graph	NOUN
ejpam-6425	8	24	(	(	PUNCT
ejpam-6425	8	25	rwwn	rwwn	NOUN
ejpam-6425	8	26	)	)	PUNCT
ejpam-6425	8	27	and	and	CCONJ
ejpam-6425	8	28	left	leave	VERB
ejpam-6425	8	29	water	water	NOUN
ejpam-6425	8	30	wheel	wheel	NOUN
ejpam-6425	8	31	graphs	graph	NOUN
ejpam-6425	8	32	(	(	PUNCT
ejpam-6425	8	33	lwwn	lwwn	NOUN
ejpam-6425	8	34	)	)	PUNCT
ejpam-6425	8	35	.	.	PUNCT
ejpam-6425	9	1	next	next	ADV
ejpam-6425	9	2	,	,	PUNCT
ejpam-6425	9	3	using	use	VERB
ejpam-6425	9	4	lo	lo	PROPN
ejpam-6425	9	5	’s	’s	PART
ejpam-6425	9	6	theorem	theorem	NOUN
ejpam-6425	9	7	,	,	PUNCT
ejpam-6425	9	8	and	and	CCONJ
ejpam-6425	9	9	the	the	DET
ejpam-6425	9	10	concepts	concept	NOUN
ejpam-6425	9	11	of	of	ADP
ejpam-6425	9	12	divisibility	divisibility	NOUN
ejpam-6425	9	13	and	and	CCONJ
ejpam-6425	9	14	diophantine	diophantine	VERB
ejpam-6425	9	15	equations	equation	NOUN
ejpam-6425	9	16	,	,	PUNCT
ejpam-6425	9	17	we	we	PRON
ejpam-6425	9	18	proved	prove	VERB
ejpam-6425	9	19	the	the	DET
ejpam-6425	9	20	non	non	ADJ
ejpam-6425	9	21	-	-	ADJ
ejpam-6425	9	22	edge	edge	ADJ
ejpam-6425	9	23	-	-	PUNCT
ejpam-6425	9	24	gracefulness	gracefulness	NOUN
ejpam-6425	9	25	of	of	ADP
ejpam-6425	9	26	these	these	DET
ejpam-6425	9	27	graphs	graph	NOUN
ejpam-6425	9	28	.	.	PUNCT
ejpam-6425	10	1	then	then	ADV
ejpam-6425	10	2	,	,	PUNCT
ejpam-6425	10	3	using	use	VERB
ejpam-6425	10	4	the	the	DET
ejpam-6425	10	5	same	same	ADJ
ejpam-6425	10	6	concepts	concept	NOUN
ejpam-6425	10	7	,	,	PUNCT
ejpam-6425	10	8	we	we	PRON
ejpam-6425	10	9	determine	determine	VERB
ejpam-6425	10	10	all	all	DET
ejpam-6425	10	11	the	the	DET
ejpam-6425	10	12	edge	edge	NOUN
ejpam-6425	10	13	-	-	PUNCT
ejpam-6425	10	14	graceful	graceful	NOUN
ejpam-6425	10	15	splitting	splitting	NOUN
ejpam-6425	10	16	graphs	graph	NOUN
ejpam-6425	10	17	of	of	ADP
ejpam-6425	10	18	wwn	wwn	NOUN
ejpam-6425	10	19	,	,	PUNCT
ejpam-6425	10	20	rwwn	rwwn	NOUN
ejpam-6425	10	21	,	,	PUNCT
ejpam-6425	10	22	and	and	CCONJ
ejpam-6425	10	23	lwwn	lwwn	NOUN
ejpam-6425	10	24	,	,	PUNCT
ejpam-6425	10	25	denoted	denote	VERB
ejpam-6425	10	26	as	as	ADP
ejpam-6425	10	27	s(wwn	s(wwn	NOUN
ejpam-6425	10	28	)	)	PUNCT
ejpam-6425	10	29	,	,	PUNCT
ejpam-6425	10	30	s(rwwn	s(rwwn	NOUN
ejpam-6425	10	31	)	)	PUNCT
ejpam-6425	10	32	,	,	PUNCT
ejpam-6425	10	33	and	and	CCONJ
ejpam-6425	10	34	s(lwwn	s(lwwn	NOUN
ejpam-6425	10	35	)	)	PUNCT
ejpam-6425	10	36	,	,	PUNCT
ejpam-6425	10	37	respectively	respectively	ADV
ejpam-6425	10	38	.	.	PUNCT
ejpam-6425	11	1	finally	finally	ADV
ejpam-6425	11	2	,	,	PUNCT
ejpam-6425	11	3	python	python	PROPN
ejpam-6425	11	4	computer	computer	NOUN
ejpam-6425	11	5	programs	program	NOUN
ejpam-6425	11	6	were	be	AUX
ejpam-6425	11	7	created	create	VERB
ejpam-6425	11	8	and	and	CCONJ
ejpam-6425	11	9	used	use	VERB
ejpam-6425	11	10	to	to	PART
ejpam-6425	11	11	conclude	conclude	VERB
ejpam-6425	11	12	that	that	SCONJ
ejpam-6425	11	13	among	among	ADP
ejpam-6425	11	14	these	these	DET
ejpam-6425	11	15	graph	graph	NOUN
ejpam-6425	11	16	families	family	NOUN
ejpam-6425	11	17	,	,	PUNCT
ejpam-6425	11	18	only	only	ADV
ejpam-6425	11	19	s(ww3	s(ww3	PROPN
ejpam-6425	11	20	)	)	PUNCT
ejpam-6425	11	21	,	,	PUNCT
ejpam-6425	11	22	s(rww3	s(rww3	NOUN
ejpam-6425	11	23	)	)	PUNCT
ejpam-6425	11	24	,	,	PUNCT
ejpam-6425	11	25	and	and	CCONJ
ejpam-6425	11	26	s(lww3	s(lww3	NOUN
ejpam-6425	11	27	)	)	PUNCT
ejpam-6425	11	28	are	be	AUX
ejpam-6425	11	29	edge	edge	NOUN
ejpam-6425	11	30	-	-	PUNCT
ejpam-6425	11	31	graceful	graceful	ADJ
ejpam-6425	11	32	,	,	PUNCT
ejpam-6425	11	33	by	by	ADP
ejpam-6425	11	34	providing	provide	VERB
ejpam-6425	11	35	their	their	PRON
ejpam-6425	11	36	corresponding	corresponding	ADJ
ejpam-6425	11	37	edge	edge	NOUN
ejpam-6425	11	38	-	-	PUNCT
ejpam-6425	11	39	graceful	graceful	ADJ
ejpam-6425	11	40	labels	label	NOUN
ejpam-6425	11	41	.	.	PUNCT
ejpam-6425	12	1	2020	2020	NUM
ejpam-6425	12	2	mathematics	mathematic	NOUN
ejpam-6425	12	3	subject	subject	NOUN
ejpam-6425	12	4	classifications	classification	NOUN
ejpam-6425	12	5	:	:	PUNCT
ejpam-6425	12	6	05c76	05c76	NUM
ejpam-6425	12	7	,	,	PUNCT
ejpam-6425	12	8	05c78	05c78	NUM
ejpam-6425	12	9	,	,	PUNCT
ejpam-6425	12	10	11d09	11d09	NUM
ejpam-6425	12	11	,	,	PUNCT
ejpam-6425	12	12	11d45	11d45	NUM
ejpam-6425	12	13	,	,	PUNCT
ejpam-6425	12	14	68r10	68r10	NUM
ejpam-6425	12	15	key	key	ADJ
ejpam-6425	12	16	words	word	NOUN
ejpam-6425	12	17	and	and	CCONJ
ejpam-6425	12	18	phrases	phrase	NOUN
ejpam-6425	12	19	:	:	PUNCT
ejpam-6425	12	20	graph	graph	NOUN
ejpam-6425	12	21	labeling	labeling	NOUN
ejpam-6425	12	22	,	,	PUNCT
ejpam-6425	12	23	edge	edge	NOUN
ejpam-6425	12	24	-	-	PUNCT
ejpam-6425	12	25	graceful	graceful	NOUN
ejpam-6425	12	26	labeling	labeling	NOUN
ejpam-6425	12	27	,	,	PUNCT
ejpam-6425	12	28	diophantine	diophantine	VERB
ejpam-6425	12	29	equation	equation	NOUN
ejpam-6425	12	30	,	,	PUNCT
ejpam-6425	12	31	water	water	NOUN
ejpam-6425	12	32	wheel	wheel	NOUN
ejpam-6425	12	33	graph	graph	NOUN
ejpam-6425	12	34	1	1	NUM
ejpam-6425	12	35	.	.	PUNCT
ejpam-6425	12	36	introduction	introduction	NOUN
ejpam-6425	12	37	graph	graph	NOUN
ejpam-6425	12	38	theory	theory	NOUN
ejpam-6425	12	39	or	or	CCONJ
ejpam-6425	12	40	the	the	DET
ejpam-6425	12	41	study	study	NOUN
ejpam-6425	12	42	of	of	ADP
ejpam-6425	12	43	graphs	graph	NOUN
ejpam-6425	12	44	is	be	AUX
ejpam-6425	12	45	a	a	DET
ejpam-6425	12	46	vast	vast	ADJ
ejpam-6425	12	47	field	field	NOUN
ejpam-6425	12	48	in	in	ADP
ejpam-6425	12	49	mathematics	mathematic	NOUN
ejpam-6425	12	50	which	which	PRON
ejpam-6425	12	51	offers	offer	VERB
ejpam-6425	12	52	a	a	DET
ejpam-6425	12	53	plethora	plethora	NOUN
ejpam-6425	12	54	of	of	ADP
ejpam-6425	12	55	unique	unique	ADJ
ejpam-6425	12	56	and	and	CCONJ
ejpam-6425	12	57	interesting	interesting	ADJ
ejpam-6425	12	58	topics	topic	NOUN
ejpam-6425	12	59	.	.	PUNCT
ejpam-6425	13	1	one	one	NUM
ejpam-6425	13	2	of	of	ADP
ejpam-6425	13	3	these	these	DET
ejpam-6425	13	4	particular	particular	ADJ
ejpam-6425	13	5	topics	topic	NOUN
ejpam-6425	13	6	in	in	ADP
ejpam-6425	13	7	graph	graph	NOUN
ejpam-6425	13	8	theory	theory	NOUN
ejpam-6425	13	9	is	be	AUX
ejpam-6425	13	10	graph	graph	NOUN
ejpam-6425	13	11	labeling	labeling	NOUN
ejpam-6425	13	12	.	.	PUNCT
ejpam-6425	14	1	∗corresponding	∗corresponde	VERB
ejpam-6425	14	2	author	author	NOUN
ejpam-6425	14	3	.	.	PUNCT
ejpam-6425	15	1	doi	doi	NOUN
ejpam-6425	15	2	:	:	PUNCT
ejpam-6425	15	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6425	https://doi.org/10.29020/nybg.ejpam.v18i4.6425	PROPN
ejpam-6425	15	4	email	email	NOUN
ejpam-6425	15	5	addresses	address	NOUN
ejpam-6425	15	6	:	:	PUNCT
ejpam-6425	15	7	aaronangel1123@gmail.com	aaronangel1123@gmail.com	PROPN
ejpam-6425	15	8	,	,	PUNCT
ejpam-6425	15	9	aaron.angel@clsu2.edu.ph	aaron.angel@clsu2.edu.ph	PROPN
ejpam-6425	15	10	(	(	PUNCT
ejpam-6425	15	11	a.	a.	PROPN
ejpam-6425	15	12	d.	d.	PROPN
ejpam-6425	15	13	c.	c.	PROPN
ejpam-6425	15	14	angel	angel	PROPN
ejpam-6425	15	15	)	)	PUNCT
ejpam-6425	15	16	,	,	PUNCT
ejpam-6425	15	17	jrantalan@clsu.edu.ph	jrantalan@clsu.edu.ph	PROPN
ejpam-6425	15	18	(	(	PUNCT
ejpam-6425	15	19	j.	j.	PROPN
ejpam-6425	15	20	r.	r.	PROPN
ejpam-6425	15	21	m.	m.	PROPN
ejpam-6425	15	22	antalan	antalan	PROPN
ejpam-6425	15	23	)	)	PUNCT
ejpam-6425	15	24	,	,	PUNCT
ejpam-6425	15	25	gamurotloureynz@gmail.com	gamurotloureynz@gmail.com	X
ejpam-6425	15	26	(	(	PUNCT
ejpam-6425	15	27	j.	j.	PROPN
ejpam-6425	15	28	l.	l.	PROPN
ejpam-6425	15	29	f.	f.	PROPN
ejpam-6425	15	30	gamurot	gamurot	PROPN
ejpam-6425	15	31	)	)	PUNCT
ejpam-6425	15	32	,	,	PUNCT
ejpam-6425	15	33	richard_tagle@clsu.edu.ph	richard_tagle@clsu.edu.ph	PROPN
ejpam-6425	15	34	(	(	PUNCT
ejpam-6425	15	35	r.	r.	PROPN
ejpam-6425	15	36	p.	p.	PROPN
ejpam-6425	15	37	tagle	tagle	PROPN
ejpam-6425	15	38	)	)	PUNCT
ejpam-6425	15	39	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6425	16	1	1	1	NUM
ejpam-6425	16	2	copyright	copyright	NOUN
ejpam-6425	16	3	:	:	PUNCT
ejpam-6425	16	4	©	©	PROPN
ejpam-6425	16	5	2025	2025	NUM
ejpam-6425	16	6	the	the	DET
ejpam-6425	16	7	author(s	author(s	NOUN
ejpam-6425	16	8	)	)	PUNCT
ejpam-6425	16	9	.	.	PUNCT
ejpam-6425	17	1	(	(	PUNCT
ejpam-6425	17	2	cc	cc	NOUN
ejpam-6425	17	3	by	by	ADP
ejpam-6425	17	4	-	-	PUNCT
ejpam-6425	17	5	nc	nc	PROPN
ejpam-6425	17	6	4.0	4.0	NUM
ejpam-6425	17	7	)	)	PUNCT
ejpam-6425	17	8	a.	a.	PROPN
ejpam-6425	17	9	d.	d.	PROPN
ejpam-6425	17	10	c.	c.	PROPN
ejpam-6425	17	11	angel	angel	PROPN
ejpam-6425	17	12	et	et	PROPN
ejpam-6425	17	13	al	al	PROPN
ejpam-6425	17	14	.	.	PUNCT
ejpam-6425	17	15	/	/	SYM
ejpam-6425	17	16	eur	eur	PROPN
ejpam-6425	17	17	.	.	PUNCT
ejpam-6425	18	1	j.	j.	PROPN
ejpam-6425	18	2	pure	pure	PROPN
ejpam-6425	18	3	appl	appl	PROPN
ejpam-6425	18	4	.	.	PROPN
ejpam-6425	18	5	math	math	PROPN
ejpam-6425	18	6	,	,	PUNCT
ejpam-6425	18	7	18	18	NUM
ejpam-6425	18	8	(	(	PUNCT
ejpam-6425	18	9	4	4	NUM
ejpam-6425	18	10	)	)	PUNCT
ejpam-6425	18	11	(	(	PUNCT
ejpam-6425	18	12	2025	2025	NUM
ejpam-6425	18	13	)	)	PUNCT
ejpam-6425	18	14	,	,	PUNCT
ejpam-6425	18	15	6425	6425	NUM
ejpam-6425	18	16	2	2	NUM
ejpam-6425	18	17	of	of	ADP
ejpam-6425	18	18	33	33	NUM
ejpam-6425	18	19	in	in	ADP
ejpam-6425	18	20	the	the	DET
ejpam-6425	18	21	mid-1960s	mid-1960	NOUN
ejpam-6425	18	22	,	,	PUNCT
ejpam-6425	18	23	the	the	DET
ejpam-6425	18	24	concept	concept	NOUN
ejpam-6425	18	25	of	of	ADP
ejpam-6425	18	26	graph	graph	NOUN
ejpam-6425	18	27	labeling	labeling	NOUN
ejpam-6425	18	28	was	be	AUX
ejpam-6425	18	29	first	first	ADV
ejpam-6425	18	30	introduced	introduce	VERB
ejpam-6425	18	31	.	.	PUNCT
ejpam-6425	19	1	graph	graph	NOUN
ejpam-6425	19	2	labeling	labeling	NOUN
ejpam-6425	19	3	is	be	AUX
ejpam-6425	19	4	an	an	DET
ejpam-6425	19	5	assignment	assignment	NOUN
ejpam-6425	19	6	of	of	ADP
ejpam-6425	19	7	integers	integer	NOUN
ejpam-6425	19	8	to	to	ADP
ejpam-6425	19	9	the	the	DET
ejpam-6425	19	10	vertices	vertex	NOUN
ejpam-6425	19	11	or	or	CCONJ
ejpam-6425	19	12	edges	edge	NOUN
ejpam-6425	19	13	,	,	PUNCT
ejpam-6425	19	14	or	or	CCONJ
ejpam-6425	19	15	both	both	PRON
ejpam-6425	19	16	,	,	PUNCT
ejpam-6425	19	17	of	of	ADP
ejpam-6425	19	18	a	a	DET
ejpam-6425	19	19	graph	graph	NOUN
ejpam-6425	19	20	under	under	ADP
ejpam-6425	19	21	certain	certain	ADJ
ejpam-6425	19	22	conditions	condition	NOUN
ejpam-6425	19	23	[	[	X
ejpam-6425	19	24	1	1	NUM
ejpam-6425	19	25	]	]	PUNCT
ejpam-6425	19	26	.	.	PUNCT
ejpam-6425	20	1	in	in	ADP
ejpam-6425	20	2	the	the	DET
ejpam-6425	20	3	last	last	ADJ
ejpam-6425	20	4	60	60	NUM
ejpam-6425	20	5	years	year	NOUN
ejpam-6425	20	6	,	,	PUNCT
ejpam-6425	20	7	over	over	ADP
ejpam-6425	20	8	200	200	NUM
ejpam-6425	20	9	types	type	NOUN
ejpam-6425	20	10	or	or	CCONJ
ejpam-6425	20	11	variations	variation	NOUN
ejpam-6425	20	12	of	of	ADP
ejpam-6425	20	13	graph	graph	NOUN
ejpam-6425	20	14	labeling	labeling	NOUN
ejpam-6425	20	15	have	have	AUX
ejpam-6425	20	16	been	be	AUX
ejpam-6425	20	17	studied	study	VERB
ejpam-6425	20	18	and	and	CCONJ
ejpam-6425	20	19	about	about	ADV
ejpam-6425	20	20	2500	2500	NUM
ejpam-6425	20	21	articles	article	NOUN
ejpam-6425	20	22	have	have	AUX
ejpam-6425	20	23	been	be	AUX
ejpam-6425	20	24	published	publish	VERB
ejpam-6425	20	25	[	[	PUNCT
ejpam-6425	20	26	2	2	NUM
ejpam-6425	20	27	]	]	PUNCT
ejpam-6425	20	28	.	.	PUNCT
ejpam-6425	21	1	one	one	NUM
ejpam-6425	21	2	type	type	NOUN
ejpam-6425	21	3	of	of	ADP
ejpam-6425	21	4	graph	graph	NOUN
ejpam-6425	21	5	labeling	labeling	NOUN
ejpam-6425	21	6	is	be	AUX
ejpam-6425	21	7	graceful	graceful	ADJ
ejpam-6425	21	8	labeling	labeling	NOUN
ejpam-6425	21	9	.	.	PUNCT
ejpam-6425	22	1	a	a	DET
ejpam-6425	22	2	graceful	graceful	ADJ
ejpam-6425	22	3	labeling	labeling	NOUN
ejpam-6425	22	4	or	or	CCONJ
ejpam-6425	22	5	graceful	graceful	ADJ
ejpam-6425	22	6	numbering	numbering	NOUN
ejpam-6425	22	7	is	be	AUX
ejpam-6425	22	8	a	a	DET
ejpam-6425	22	9	special	special	ADJ
ejpam-6425	22	10	graph	graph	NOUN
ejpam-6425	22	11	labeling	labeling	NOUN
ejpam-6425	22	12	of	of	ADP
ejpam-6425	22	13	a	a	DET
ejpam-6425	22	14	graph	graph	NOUN
ejpam-6425	22	15	on	on	ADP
ejpam-6425	22	16	m	m	NOUN
ejpam-6425	22	17	edges	edge	NOUN
ejpam-6425	22	18	in	in	ADP
ejpam-6425	22	19	which	which	PRON
ejpam-6425	22	20	the	the	DET
ejpam-6425	22	21	vertices	vertex	NOUN
ejpam-6425	22	22	are	be	AUX
ejpam-6425	22	23	labeled	label	VERB
ejpam-6425	22	24	,	,	PUNCT
ejpam-6425	22	25	using	use	VERB
ejpam-6425	22	26	a	a	DET
ejpam-6425	22	27	subset	subset	NOUN
ejpam-6425	22	28	of	of	ADP
ejpam-6425	22	29	distinct	distinct	ADJ
ejpam-6425	22	30	nonnegative	nonnegative	ADJ
ejpam-6425	22	31	integers	integer	NOUN
ejpam-6425	22	32	from	from	ADP
ejpam-6425	22	33	0	0	NUM
ejpam-6425	22	34	to	to	ADP
ejpam-6425	22	35	m	m	PROPN
ejpam-6425	22	36	and	and	CCONJ
ejpam-6425	22	37	each	each	DET
ejpam-6425	22	38	edge	edge	NOUN
ejpam-6425	22	39	of	of	ADP
ejpam-6425	22	40	the	the	DET
ejpam-6425	22	41	graph	graph	NOUN
ejpam-6425	22	42	is	be	AUX
ejpam-6425	22	43	uniquely	uniquely	ADV
ejpam-6425	22	44	labeled	label	VERB
ejpam-6425	22	45	by	by	ADP
ejpam-6425	22	46	the	the	DET
ejpam-6425	22	47	absolute	absolute	ADJ
ejpam-6425	22	48	difference	difference	NOUN
ejpam-6425	22	49	between	between	ADP
ejpam-6425	22	50	the	the	DET
ejpam-6425	22	51	labels	label	NOUN
ejpam-6425	22	52	of	of	ADP
ejpam-6425	22	53	the	the	DET
ejpam-6425	22	54	vertices	vertex	NOUN
ejpam-6425	22	55	incident	incident	NOUN
ejpam-6425	22	56	to	to	ADP
ejpam-6425	22	57	it	it	PRON
ejpam-6425	22	58	.	.	PUNCT
ejpam-6425	23	1	if	if	SCONJ
ejpam-6425	23	2	the	the	DET
ejpam-6425	23	3	resulting	result	VERB
ejpam-6425	23	4	graph	graph	NOUN
ejpam-6425	23	5	edge	edge	NOUN
ejpam-6425	23	6	numbers	number	NOUN
ejpam-6425	23	7	run	run	VERB
ejpam-6425	23	8	from	from	ADP
ejpam-6425	23	9	1	1	NUM
ejpam-6425	23	10	to	to	ADP
ejpam-6425	23	11	m	m	PROPN
ejpam-6425	23	12	inclusive	inclusive	ADJ
ejpam-6425	23	13	,	,	PUNCT
ejpam-6425	23	14	it	it	PRON
ejpam-6425	23	15	is	be	AUX
ejpam-6425	23	16	a	a	DET
ejpam-6425	23	17	graceful	graceful	ADJ
ejpam-6425	23	18	labeling	labeling	NOUN
ejpam-6425	23	19	,	,	PUNCT
ejpam-6425	23	20	and	and	CCONJ
ejpam-6425	23	21	the	the	DET
ejpam-6425	23	22	graph	graph	NOUN
ejpam-6425	23	23	is	be	AUX
ejpam-6425	23	24	said	say	VERB
ejpam-6425	23	25	to	to	PART
ejpam-6425	23	26	be	be	AUX
ejpam-6425	23	27	a	a	DET
ejpam-6425	23	28	graceful	graceful	ADJ
ejpam-6425	23	29	graph	graph	NOUN
ejpam-6425	23	30	[	[	X
ejpam-6425	23	31	3	3	NUM
ejpam-6425	23	32	]	]	PUNCT
ejpam-6425	23	33	.	.	PUNCT
ejpam-6425	24	1	on	on	ADP
ejpam-6425	24	2	the	the	DET
ejpam-6425	24	3	other	other	ADJ
ejpam-6425	24	4	hand	hand	NOUN
ejpam-6425	24	5	,	,	PUNCT
ejpam-6425	24	6	the	the	DET
ejpam-6425	24	7	concepts	concept	NOUN
ejpam-6425	24	8	of	of	ADP
ejpam-6425	24	9	edge	edge	NOUN
ejpam-6425	24	10	-	-	PUNCT
ejpam-6425	24	11	gracefulness	gracefulness	NOUN
ejpam-6425	24	12	and	and	CCONJ
ejpam-6425	24	13	edge	edge	NOUN
ejpam-6425	24	14	-	-	PUNCT
ejpam-6425	24	15	graceful	graceful	NOUN
ejpam-6425	24	16	graphs	graph	NOUN
ejpam-6425	24	17	are	be	AUX
ejpam-6425	24	18	defined	define	VERB
ejpam-6425	24	19	and	and	CCONJ
ejpam-6425	24	20	introduced	introduce	VERB
ejpam-6425	24	21	by	by	ADP
ejpam-6425	24	22	lo	lo	PROPN
ejpam-6425	24	23	in	in	ADP
ejpam-6425	24	24	1985	1985	NUM
ejpam-6425	24	25	[	[	X
ejpam-6425	24	26	4	4	NUM
ejpam-6425	24	27	]	]	PUNCT
ejpam-6425	24	28	.	.	PUNCT
ejpam-6425	25	1	edge	edge	NOUN
ejpam-6425	25	2	-	-	PUNCT
ejpam-6425	25	3	graceful	graceful	NOUN
ejpam-6425	25	4	labeling	labeling	NOUN
ejpam-6425	25	5	is	be	AUX
ejpam-6425	25	6	considered	consider	VERB
ejpam-6425	25	7	as	as	ADP
ejpam-6425	25	8	a	a	DET
ejpam-6425	25	9	reversal	reversal	NOUN
ejpam-6425	25	10	of	of	ADP
ejpam-6425	25	11	graceful	graceful	ADJ
ejpam-6425	25	12	labeling	labeling	NOUN
ejpam-6425	25	13	because	because	SCONJ
ejpam-6425	25	14	it	it	PRON
ejpam-6425	25	15	labels	label	VERB
ejpam-6425	25	16	the	the	DET
ejpam-6425	25	17	edges	edge	NOUN
ejpam-6425	25	18	first	first	ADV
ejpam-6425	25	19	,	,	PUNCT
ejpam-6425	25	20	then	then	ADV
ejpam-6425	25	21	the	the	DET
ejpam-6425	25	22	labels	label	NOUN
ejpam-6425	25	23	of	of	ADP
ejpam-6425	25	24	the	the	DET
ejpam-6425	25	25	vertices	vertex	NOUN
ejpam-6425	25	26	would	would	AUX
ejpam-6425	25	27	depend	depend	VERB
ejpam-6425	25	28	on	on	ADP
ejpam-6425	25	29	the	the	DET
ejpam-6425	25	30	labels	label	NOUN
ejpam-6425	25	31	of	of	ADP
ejpam-6425	25	32	the	the	DET
ejpam-6425	25	33	edges	edge	NOUN
ejpam-6425	25	34	incident	incident	NOUN
ejpam-6425	25	35	to	to	ADP
ejpam-6425	25	36	them	they	PRON
ejpam-6425	25	37	.	.	PUNCT
ejpam-6425	26	1	that	that	PRON
ejpam-6425	26	2	is	be	AUX
ejpam-6425	26	3	,	,	PUNCT
ejpam-6425	26	4	a	a	DET
ejpam-6425	26	5	graph	graph	NOUN
ejpam-6425	26	6	g	g	NOUN
ejpam-6425	26	7	with	with	ADP
ejpam-6425	26	8	p	p	NOUN
ejpam-6425	26	9	vertices	vertex	NOUN
ejpam-6425	26	10	and	and	CCONJ
ejpam-6425	26	11	q	q	NOUN
ejpam-6425	26	12	edges	edge	NOUN
ejpam-6425	26	13	is	be	AUX
ejpam-6425	26	14	said	say	VERB
ejpam-6425	26	15	to	to	PART
ejpam-6425	26	16	be	be	AUX
ejpam-6425	26	17	edge	edge	NOUN
ejpam-6425	26	18	-	-	PUNCT
ejpam-6425	26	19	graceful	graceful	NOUN
ejpam-6425	26	20	if	if	SCONJ
ejpam-6425	26	21	the	the	DET
ejpam-6425	26	22	edges	edge	NOUN
ejpam-6425	26	23	can	can	AUX
ejpam-6425	26	24	be	be	AUX
ejpam-6425	26	25	labeled	label	VERB
ejpam-6425	26	26	from	from	ADP
ejpam-6425	26	27	1	1	NUM
ejpam-6425	26	28	through	through	ADP
ejpam-6425	26	29	q	q	NOUN
ejpam-6425	26	30	,	,	PUNCT
ejpam-6425	26	31	in	in	ADP
ejpam-6425	26	32	such	such	DET
ejpam-6425	26	33	a	a	DET
ejpam-6425	26	34	way	way	NOUN
ejpam-6425	27	1	that	that	PRON
ejpam-6425	27	2	the	the	DET
ejpam-6425	27	3	labels	label	NOUN
ejpam-6425	27	4	induced	induce	VERB
ejpam-6425	27	5	on	on	ADP
ejpam-6425	27	6	the	the	DET
ejpam-6425	27	7	vertices	vertex	NOUN
ejpam-6425	27	8	by	by	ADP
ejpam-6425	27	9	adding	add	VERB
ejpam-6425	27	10	over	over	ADP
ejpam-6425	27	11	the	the	DET
ejpam-6425	27	12	labels	label	NOUN
ejpam-6425	27	13	of	of	ADP
ejpam-6425	27	14	incident	incident	NOUN
ejpam-6425	27	15	edges	edge	NOUN
ejpam-6425	27	16	modulo	modulo	ADV
ejpam-6425	27	17	p	p	NOUN
ejpam-6425	27	18	are	be	AUX
ejpam-6425	27	19	distinct	distinct	ADJ
ejpam-6425	27	20	[	[	X
ejpam-6425	27	21	3	3	NUM
ejpam-6425	27	22	]	]	PUNCT
ejpam-6425	28	1	.	.	PUNCT
ejpam-6425	29	1	since	since	SCONJ
ejpam-6425	29	2	the	the	DET
ejpam-6425	29	3	introduction	introduction	NOUN
ejpam-6425	29	4	of	of	ADP
ejpam-6425	29	5	the	the	DET
ejpam-6425	29	6	concept	concept	NOUN
ejpam-6425	29	7	of	of	ADP
ejpam-6425	29	8	edge	edge	NOUN
ejpam-6425	29	9	-	-	PUNCT
ejpam-6425	29	10	graceful	graceful	NOUN
ejpam-6425	29	11	labeling	labeling	NOUN
ejpam-6425	29	12	in	in	ADP
ejpam-6425	29	13	1985	1985	NUM
ejpam-6425	29	14	,	,	PUNCT
ejpam-6425	29	15	several	several	ADJ
ejpam-6425	29	16	works	work	NOUN
ejpam-6425	29	17	on	on	ADP
ejpam-6425	29	18	the	the	DET
ejpam-6425	29	19	edge	edge	NOUN
ejpam-6425	29	20	-	-	PUNCT
ejpam-6425	29	21	graceful	graceful	NOUN
ejpam-6425	29	22	labeling	labeling	NOUN
ejpam-6425	29	23	of	of	ADP
ejpam-6425	29	24	specific	specific	ADJ
ejpam-6425	29	25	families	family	NOUN
ejpam-6425	29	26	of	of	ADP
ejpam-6425	29	27	graphs	graph	NOUN
ejpam-6425	29	28	have	have	AUX
ejpam-6425	29	29	been	be	AUX
ejpam-6425	29	30	conducted	conduct	VERB
ejpam-6425	29	31	.	.	PUNCT
ejpam-6425	30	1	for	for	ADP
ejpam-6425	30	2	instance	instance	NOUN
ejpam-6425	30	3	,	,	PUNCT
ejpam-6425	30	4	kuan	kuan	PROPN
ejpam-6425	30	5	et	et	PROPN
ejpam-6425	30	6	al	al	PROPN
ejpam-6425	30	7	.	.	PUNCT
ejpam-6425	31	1	[	[	X
ejpam-6425	31	2	5	5	NUM
ejpam-6425	31	3	]	]	PUNCT
ejpam-6425	31	4	studied	study	VERB
ejpam-6425	31	5	edge	edge	NOUN
ejpam-6425	31	6	-	-	PUNCT
ejpam-6425	31	7	graceful	graceful	ADJ
ejpam-6425	31	8	unicyclic	unicyclic	ADJ
ejpam-6425	31	9	graphs	graph	NOUN
ejpam-6425	31	10	.	.	PUNCT
ejpam-6425	32	1	lee	lee	PROPN
ejpam-6425	32	2	et	et	PROPN
ejpam-6425	32	3	al	al	PROPN
ejpam-6425	32	4	.	.	PUNCT
ejpam-6425	33	1	[	[	X
ejpam-6425	33	2	6	6	NUM
ejpam-6425	33	3	]	]	PUNCT
ejpam-6425	33	4	on	on	ADP
ejpam-6425	33	5	the	the	DET
ejpam-6425	33	6	other	other	ADJ
ejpam-6425	33	7	hand	hand	NOUN
ejpam-6425	33	8	,	,	PUNCT
ejpam-6425	33	9	studied	study	VERB
ejpam-6425	33	10	the	the	DET
ejpam-6425	33	11	edge	edge	NOUN
ejpam-6425	33	12	-	-	PUNCT
ejpam-6425	33	13	graceful	graceful	NOUN
ejpam-6425	33	14	labeling	labeling	NOUN
ejpam-6425	33	15	of	of	ADP
ejpam-6425	33	16	complete	complete	ADJ
ejpam-6425	33	17	graphs	graph	NOUN
ejpam-6425	33	18	.	.	PUNCT
ejpam-6425	34	1	the	the	DET
ejpam-6425	34	2	work	work	NOUN
ejpam-6425	34	3	of	of	ADP
ejpam-6425	34	4	small	small	ADJ
ejpam-6425	34	5	in	in	ADP
ejpam-6425	34	6	[	[	X
ejpam-6425	34	7	7	7	X
ejpam-6425	34	8	]	]	PUNCT
ejpam-6425	34	9	shows	show	VERB
ejpam-6425	34	10	that	that	SCONJ
ejpam-6425	34	11	regular	regular	ADJ
ejpam-6425	34	12	(	(	PUNCT
ejpam-6425	34	13	even	even	ADV
ejpam-6425	34	14	)	)	PUNCT
ejpam-6425	34	15	spider	spider	NOUN
ejpam-6425	34	16	graphs	graph	NOUN
ejpam-6425	34	17	are	be	AUX
ejpam-6425	34	18	edge	edge	NOUN
ejpam-6425	34	19	-	-	PUNCT
ejpam-6425	34	20	graceful	graceful	NOUN
ejpam-6425	34	21	.	.	PUNCT
ejpam-6425	35	1	in	in	ADP
ejpam-6425	35	2	1992	1992	NUM
ejpam-6425	35	3	,	,	PUNCT
ejpam-6425	35	4	cabannis	cabanni	NOUN
ejpam-6425	35	5	et	et	PROPN
ejpam-6425	35	6	al	al	PROPN
ejpam-6425	35	7	.	.	PUNCT
ejpam-6425	36	1	[	[	X
ejpam-6425	36	2	8	8	NUM
ejpam-6425	36	3	]	]	PUNCT
ejpam-6425	36	4	studied	study	VERB
ejpam-6425	36	5	the	the	DET
ejpam-6425	36	6	edge	edge	NOUN
ejpam-6425	36	7	-	-	PUNCT
ejpam-6425	36	8	gracefulness	gracefulness	NOUN
ejpam-6425	36	9	of	of	ADP
ejpam-6425	36	10	regular	regular	ADJ
ejpam-6425	36	11	graphs	graph	NOUN
ejpam-6425	36	12	and	and	CCONJ
ejpam-6425	36	13	trees	tree	NOUN
ejpam-6425	36	14	.	.	PUNCT
ejpam-6425	37	1	fast	fast	ADV
ejpam-6425	37	2	-	-	PUNCT
ejpam-6425	37	3	forward	forward	NOUN
ejpam-6425	37	4	to	to	ADP
ejpam-6425	37	5	2017	2017	NUM
ejpam-6425	37	6	,	,	PUNCT
ejpam-6425	37	7	venkatesan	venkatesan	ADJ
ejpam-6425	37	8	and	and	CCONJ
ejpam-6425	37	9	sekar	sekar	PROPN
ejpam-6425	38	1	[	[	X
ejpam-6425	38	2	3	3	X
ejpam-6425	38	3	]	]	PUNCT
ejpam-6425	38	4	attempted	attempt	VERB
ejpam-6425	38	5	to	to	PART
ejpam-6425	38	6	prove	prove	VERB
ejpam-6425	38	7	that	that	SCONJ
ejpam-6425	38	8	among	among	ADP
ejpam-6425	38	9	all	all	DET
ejpam-6425	38	10	wheel	wheel	NOUN
ejpam-6425	38	11	graphs	graph	NOUN
ejpam-6425	38	12	wn	wn	PROPN
ejpam-6425	38	13	,	,	PUNCT
ejpam-6425	38	14	only	only	ADV
ejpam-6425	38	15	w3	w3	PROPN
ejpam-6425	38	16	is	be	AUX
ejpam-6425	38	17	edge	edge	NOUN
ejpam-6425	38	18	-	-	PUNCT
ejpam-6425	38	19	graceful	graceful	NOUN
ejpam-6425	38	20	while	while	SCONJ
ejpam-6425	38	21	a	a	DET
ejpam-6425	38	22	complete	complete	ADJ
ejpam-6425	38	23	proof	proof	NOUN
ejpam-6425	38	24	of	of	ADP
ejpam-6425	38	25	their	their	PRON
ejpam-6425	38	26	result	result	NOUN
ejpam-6425	38	27	was	be	AUX
ejpam-6425	38	28	provided	provide	VERB
ejpam-6425	38	29	recently	recently	ADV
ejpam-6425	38	30	in	in	ADP
ejpam-6425	38	31	[	[	X
ejpam-6425	38	32	9	9	NUM
ejpam-6425	38	33	]	]	PUNCT
ejpam-6425	38	34	.	.	PUNCT
ejpam-6425	39	1	in	in	ADP
ejpam-6425	39	2	this	this	DET
ejpam-6425	39	3	paper	paper	NOUN
ejpam-6425	39	4	,	,	PUNCT
ejpam-6425	39	5	we	we	PRON
ejpam-6425	39	6	used	use	VERB
ejpam-6425	39	7	essential	essential	ADJ
ejpam-6425	39	8	graph	graph	NOUN
ejpam-6425	39	9	theory	theory	NOUN
ejpam-6425	39	10	concepts	concept	NOUN
ejpam-6425	39	11	to	to	PART
ejpam-6425	39	12	study	study	VERB
ejpam-6425	39	13	the	the	DET
ejpam-6425	39	14	basic	basic	ADJ
ejpam-6425	39	15	properties	property	NOUN
ejpam-6425	39	16	of	of	ADP
ejpam-6425	39	17	water	water	NOUN
ejpam-6425	39	18	wheel	wheel	NOUN
ejpam-6425	39	19	,	,	PUNCT
ejpam-6425	39	20	right	right	ADJ
ejpam-6425	39	21	water	water	NOUN
ejpam-6425	39	22	wheel	wheel	NOUN
ejpam-6425	39	23	,	,	PUNCT
ejpam-6425	39	24	and	and	CCONJ
ejpam-6425	39	25	left	leave	VERB
ejpam-6425	39	26	water	water	NOUN
ejpam-6425	39	27	wheel	wheel	NOUN
ejpam-6425	39	28	graphs	graph	NOUN
ejpam-6425	39	29	,	,	PUNCT
ejpam-6425	39	30	and	and	CCONJ
ejpam-6425	39	31	their	their	PRON
ejpam-6425	39	32	splitting	splitting	NOUN
ejpam-6425	39	33	graphs	graph	NOUN
ejpam-6425	39	34	.	.	PUNCT
ejpam-6425	40	1	in	in	ADP
ejpam-6425	40	2	particular	particular	ADJ
ejpam-6425	40	3	,	,	PUNCT
ejpam-6425	40	4	their	their	PRON
ejpam-6425	40	5	vertex	vertex	NOUN
ejpam-6425	40	6	and	and	CCONJ
ejpam-6425	40	7	edge	edge	NOUN
ejpam-6425	40	8	sets	set	NOUN
ejpam-6425	40	9	were	be	AUX
ejpam-6425	40	10	used	use	VERB
ejpam-6425	40	11	to	to	PART
ejpam-6425	40	12	apply	apply	VERB
ejpam-6425	40	13	lo	lo	PROPN
ejpam-6425	40	14	’s	’s	PART
ejpam-6425	40	15	theorem	theorem	NOUN
ejpam-6425	40	16	,	,	PUNCT
ejpam-6425	40	17	an	an	DET
ejpam-6425	40	18	initial	initial	ADJ
ejpam-6425	40	19	condition	condition	NOUN
ejpam-6425	40	20	that	that	PRON
ejpam-6425	40	21	must	must	AUX
ejpam-6425	40	22	be	be	AUX
ejpam-6425	40	23	satisfied	satisfied	ADJ
ejpam-6425	40	24	to	to	PART
ejpam-6425	40	25	conclude	conclude	VERB
ejpam-6425	40	26	that	that	SCONJ
ejpam-6425	40	27	a	a	DET
ejpam-6425	40	28	particular	particular	ADJ
ejpam-6425	40	29	graph	graph	NOUN
ejpam-6425	40	30	is	be	AUX
ejpam-6425	40	31	edge	edge	NOUN
ejpam-6425	40	32	-	-	PUNCT
ejpam-6425	40	33	graceful	graceful	NOUN
ejpam-6425	40	34	.	.	PUNCT
ejpam-6425	41	1	then	then	ADV
ejpam-6425	41	2	,	,	PUNCT
ejpam-6425	41	3	using	use	VERB
ejpam-6425	41	4	the	the	DET
ejpam-6425	41	5	definition	definition	NOUN
ejpam-6425	41	6	of	of	ADP
ejpam-6425	41	7	edge	edge	NOUN
ejpam-6425	41	8	-	-	PUNCT
ejpam-6425	41	9	graceful	graceful	NOUN
ejpam-6425	41	10	labeling	labeling	NOUN
ejpam-6425	41	11	,	,	PUNCT
ejpam-6425	41	12	and	and	CCONJ
ejpam-6425	41	13	some	some	DET
ejpam-6425	41	14	preliminary	preliminary	ADJ
ejpam-6425	41	15	concepts	concept	NOUN
ejpam-6425	41	16	in	in	ADP
ejpam-6425	41	17	number	number	NOUN
ejpam-6425	41	18	theory	theory	NOUN
ejpam-6425	41	19	,	,	PUNCT
ejpam-6425	41	20	which	which	PRON
ejpam-6425	41	21	include	include	VERB
ejpam-6425	41	22	the	the	DET
ejpam-6425	41	23	concept	concept	NOUN
ejpam-6425	41	24	of	of	ADP
ejpam-6425	41	25	the	the	DET
ejpam-6425	41	26	diophantine	diophantine	NOUN
ejpam-6425	41	27	equation	equation	NOUN
ejpam-6425	41	28	,	,	PUNCT
ejpam-6425	41	29	this	this	DET
ejpam-6425	41	30	paper	paper	NOUN
ejpam-6425	41	31	determined	determine	VERB
ejpam-6425	41	32	which	which	PRON
ejpam-6425	41	33	among	among	ADP
ejpam-6425	41	34	the	the	DET
ejpam-6425	41	35	said	say	VERB
ejpam-6425	41	36	graphs	graph	NOUN
ejpam-6425	41	37	satisfied	satisfied	ADJ
ejpam-6425	41	38	lo	lo	PROPN
ejpam-6425	41	39	’s	’s	PART
ejpam-6425	41	40	theorem	theorem	NOUN
ejpam-6425	41	41	.	.	PUNCT
ejpam-6425	42	1	if	if	SCONJ
ejpam-6425	42	2	a	a	DET
ejpam-6425	42	3	particular	particular	ADJ
ejpam-6425	42	4	graph	graph	NOUN
ejpam-6425	42	5	does	do	AUX
ejpam-6425	42	6	not	not	PART
ejpam-6425	42	7	satisfy	satisfy	VERB
ejpam-6425	42	8	lo	lo	PROPN
ejpam-6425	42	9	’s	’s	PART
ejpam-6425	42	10	theorem	theorem	NOUN
ejpam-6425	42	11	,	,	PUNCT
ejpam-6425	42	12	then	then	ADV
ejpam-6425	42	13	it	it	PRON
ejpam-6425	42	14	automatically	automatically	ADV
ejpam-6425	42	15	does	do	AUX
ejpam-6425	42	16	not	not	PART
ejpam-6425	42	17	qualify	qualify	VERB
ejpam-6425	42	18	to	to	PART
ejpam-6425	42	19	be	be	AUX
ejpam-6425	42	20	an	an	DET
ejpam-6425	42	21	edge	edge	NOUN
ejpam-6425	42	22	-	-	PUNCT
ejpam-6425	42	23	graceful	graceful	NOUN
ejpam-6425	42	24	graph	graph	NOUN
ejpam-6425	42	25	.	.	PUNCT
ejpam-6425	43	1	finally	finally	ADV
ejpam-6425	43	2	,	,	PUNCT
ejpam-6425	43	3	by	by	ADP
ejpam-6425	43	4	using	use	VERB
ejpam-6425	43	5	python	python	NOUN
ejpam-6425	43	6	computer	computer	NOUN
ejpam-6425	43	7	programs	program	NOUN
ejpam-6425	43	8	that	that	PRON
ejpam-6425	43	9	we	we	PRON
ejpam-6425	43	10	created	create	VERB
ejpam-6425	43	11	,	,	PUNCT
ejpam-6425	43	12	this	this	DET
ejpam-6425	43	13	paper	paper	NOUN
ejpam-6425	43	14	constructed	construct	VERB
ejpam-6425	43	15	actual	actual	ADJ
ejpam-6425	43	16	examples	example	NOUN
ejpam-6425	43	17	of	of	ADP
ejpam-6425	43	18	edge	edge	NOUN
ejpam-6425	43	19	-	-	PUNCT
ejpam-6425	43	20	graceful	graceful	NOUN
ejpam-6425	43	21	labeling	labeling	NOUN
ejpam-6425	43	22	of	of	ADP
ejpam-6425	43	23	the	the	DET
ejpam-6425	43	24	graphs	graph	NOUN
ejpam-6425	43	25	that	that	PRON
ejpam-6425	43	26	satisfy	satisfy	VERB
ejpam-6425	43	27	these	these	DET
ejpam-6425	43	28	initial	initial	ADJ
ejpam-6425	43	29	conditions	condition	NOUN
ejpam-6425	43	30	,	,	PUNCT
ejpam-6425	43	31	thereby	thereby	ADV
ejpam-6425	43	32	showing	show	VERB
ejpam-6425	43	33	that	that	SCONJ
ejpam-6425	43	34	the	the	DET
ejpam-6425	43	35	graphs	graph	NOUN
ejpam-6425	43	36	are	be	AUX
ejpam-6425	43	37	indeed	indeed	ADV
ejpam-6425	43	38	edge	edge	NOUN
ejpam-6425	43	39	-	-	PUNCT
ejpam-6425	43	40	graceful	graceful	NOUN
ejpam-6425	43	41	.	.	PUNCT
ejpam-6425	44	1	2	2	X
ejpam-6425	44	2	.	.	X
ejpam-6425	44	3	preliminaries	preliminary	NOUN
ejpam-6425	44	4	for	for	ADP
ejpam-6425	44	5	completeness	completeness	NOUN
ejpam-6425	44	6	,	,	PUNCT
ejpam-6425	44	7	we	we	PRON
ejpam-6425	44	8	provide	provide	VERB
ejpam-6425	44	9	all	all	DET
ejpam-6425	44	10	the	the	DET
ejpam-6425	44	11	preliminary	preliminary	ADJ
ejpam-6425	44	12	concepts	concept	NOUN
ejpam-6425	44	13	and	and	CCONJ
ejpam-6425	44	14	results	result	NOUN
ejpam-6425	44	15	necessary	necessary	ADJ
ejpam-6425	44	16	for	for	ADP
ejpam-6425	44	17	the	the	DET
ejpam-6425	44	18	clear	clear	ADJ
ejpam-6425	44	19	and	and	CCONJ
ejpam-6425	44	20	organized	organized	ADJ
ejpam-6425	44	21	presentation	presentation	NOUN
ejpam-6425	44	22	of	of	ADP
ejpam-6425	44	23	the	the	DET
ejpam-6425	44	24	results	result	NOUN
ejpam-6425	44	25	of	of	ADP
ejpam-6425	44	26	this	this	DET
ejpam-6425	44	27	study	study	NOUN
ejpam-6425	44	28	.	.	PUNCT
ejpam-6425	45	1	we	we	PRON
ejpam-6425	45	2	refer	refer	VERB
ejpam-6425	45	3	the	the	DET
ejpam-6425	45	4	readers	reader	NOUN
ejpam-6425	45	5	who	who	PRON
ejpam-6425	45	6	are	be	AUX
ejpam-6425	45	7	unfamiliar	unfamiliar	ADJ
ejpam-6425	45	8	with	with	ADP
ejpam-6425	45	9	graph	graph	NOUN
ejpam-6425	45	10	theory	theory	NOUN
ejpam-6425	45	11	concepts	concept	NOUN
ejpam-6425	45	12	that	that	PRON
ejpam-6425	45	13	were	be	AUX
ejpam-6425	45	14	briefly	briefly	ADV
ejpam-6425	45	15	stated	state	VERB
ejpam-6425	45	16	in	in	ADP
ejpam-6425	45	17	this	this	DET
ejpam-6425	45	18	paper	paper	NOUN
ejpam-6425	45	19	,	,	PUNCT
ejpam-6425	45	20	to	to	ADP
ejpam-6425	45	21	some	some	DET
ejpam-6425	45	22	standard	standard	ADJ
ejpam-6425	45	23	graph	graph	NOUN
ejpam-6425	45	24	theory	theory	NOUN
ejpam-6425	45	25	references	reference	NOUN
ejpam-6425	45	26	such	such	ADJ
ejpam-6425	45	27	as	as	ADP
ejpam-6425	45	28	gallian	gallian	ADJ
ejpam-6425	45	29	’s	’s	PART
ejpam-6425	45	30	“	"	PUNCT
ejpam-6425	45	31	a	a	DET
ejpam-6425	45	32	dynamic	dynamic	ADJ
ejpam-6425	45	33	survey	survey	NOUN
ejpam-6425	45	34	of	of	ADP
ejpam-6425	45	35	graph	graph	NOUN
ejpam-6425	45	36	a.	a.	PROPN
ejpam-6425	45	37	d.	d.	PROPN
ejpam-6425	45	38	c.	c.	PROPN
ejpam-6425	45	39	angel	angel	PROPN
ejpam-6425	45	40	et	et	PROPN
ejpam-6425	45	41	al	al	PROPN
ejpam-6425	45	42	.	.	PUNCT
ejpam-6425	45	43	/	/	SYM
ejpam-6425	45	44	eur	eur	PROPN
ejpam-6425	45	45	.	.	PUNCT
ejpam-6425	46	1	j.	j.	PROPN
ejpam-6425	46	2	pure	pure	PROPN
ejpam-6425	46	3	appl	appl	PROPN
ejpam-6425	46	4	.	.	PROPN
ejpam-6425	46	5	math	math	PROPN
ejpam-6425	46	6	,	,	PUNCT
ejpam-6425	46	7	18	18	NUM
ejpam-6425	46	8	(	(	PUNCT
ejpam-6425	46	9	4	4	NUM
ejpam-6425	46	10	)	)	PUNCT
ejpam-6425	46	11	(	(	PUNCT
ejpam-6425	46	12	2025	2025	NUM
ejpam-6425	46	13	)	)	PUNCT
ejpam-6425	46	14	,	,	PUNCT
ejpam-6425	46	15	6425	6425	NUM
ejpam-6425	46	16	3	3	NUM
ejpam-6425	46	17	of	of	ADP
ejpam-6425	46	18	33	33	NUM
ejpam-6425	46	19	labeling	labeling	NOUN
ejpam-6425	46	20	”	"	PUNCT
ejpam-6425	47	1	[	[	X
ejpam-6425	47	2	1	1	NUM
ejpam-6425	47	3	]	]	PUNCT
ejpam-6425	47	4	and	and	CCONJ
ejpam-6425	47	5	gross	gross	PROPN
ejpam-6425	47	6	’s	’s	PART
ejpam-6425	47	7	“	"	PUNCT
ejpam-6425	47	8	graph	graph	NOUN
ejpam-6425	47	9	theory	theory	NOUN
ejpam-6425	47	10	and	and	CCONJ
ejpam-6425	47	11	its	its	PRON
ejpam-6425	47	12	applications”[10	applications”[10	X
ejpam-6425	47	13	]	]	PUNCT
ejpam-6425	47	14	.	.	PUNCT
ejpam-6425	48	1	2.1	2.1	NUM
ejpam-6425	48	2	.	.	PUNCT
ejpam-6425	49	1	essential	essential	ADJ
ejpam-6425	49	2	graph	graph	NOUN
ejpam-6425	49	3	theory	theory	NOUN
ejpam-6425	49	4	concepts	concept	NOUN
ejpam-6425	49	5	to	to	PART
ejpam-6425	49	6	begin	begin	VERB
ejpam-6425	49	7	this	this	DET
ejpam-6425	49	8	subsection	subsection	NOUN
ejpam-6425	49	9	,	,	PUNCT
ejpam-6425	49	10	we	we	PRON
ejpam-6425	49	11	briefly	briefly	ADV
ejpam-6425	49	12	recall	recall	VERB
ejpam-6425	49	13	some	some	DET
ejpam-6425	49	14	key	key	ADJ
ejpam-6425	49	15	concepts	concept	NOUN
ejpam-6425	49	16	in	in	ADP
ejpam-6425	49	17	graph	graph	NOUN
ejpam-6425	49	18	theory	theory	NOUN
ejpam-6425	49	19	that	that	PRON
ejpam-6425	49	20	are	be	AUX
ejpam-6425	49	21	essential	essential	ADJ
ejpam-6425	49	22	to	to	ADP
ejpam-6425	49	23	this	this	DET
ejpam-6425	49	24	paper	paper	NOUN
ejpam-6425	49	25	.	.	PUNCT
ejpam-6425	50	1	definition	definition	NOUN
ejpam-6425	50	2	1	1	NUM
ejpam-6425	50	3	.	.	PUNCT
ejpam-6425	51	1	a	a	DET
ejpam-6425	51	2	(	(	PUNCT
ejpam-6425	51	3	p	p	NOUN
ejpam-6425	51	4	,	,	PUNCT
ejpam-6425	51	5	q)-graph	q)-graph	NOUN
ejpam-6425	51	6	g(v	g(v	PROPN
ejpam-6425	51	7	,	,	PUNCT
ejpam-6425	51	8	e	e	NOUN
ejpam-6425	51	9	)	)	PUNCT
ejpam-6425	51	10	is	be	AUX
ejpam-6425	51	11	called	call	VERB
ejpam-6425	51	12	edge	edge	NOUN
ejpam-6425	51	13	-	-	PUNCT
ejpam-6425	51	14	graceful	graceful	NOUN
ejpam-6425	51	15	if	if	SCONJ
ejpam-6425	51	16	there	there	PRON
ejpam-6425	51	17	exist	exist	VERB
ejpam-6425	51	18	a	a	DET
ejpam-6425	51	19	bijection	bijection	NOUN
ejpam-6425	51	20	f	f	NOUN
ejpam-6425	51	21	:	:	PUNCT
ejpam-6425	51	22	e	e	X
ejpam-6425	51	23	→	→	PUNCT
ejpam-6425	51	24	{	{	PUNCT
ejpam-6425	51	25	1	1	NUM
ejpam-6425	51	26	,	,	PUNCT
ejpam-6425	51	27	2	2	NUM
ejpam-6425	51	28	,	,	PUNCT
ejpam-6425	51	29	....	....	PUNCT
ejpam-6425	51	30	,	,	PUNCT
ejpam-6425	51	31	q	q	X
ejpam-6425	51	32	}	}	PUNCT
ejpam-6425	51	33	,	,	PUNCT
ejpam-6425	51	34	such	such	ADJ
ejpam-6425	51	35	that	that	SCONJ
ejpam-6425	51	36	the	the	DET
ejpam-6425	51	37	induced	induced	ADJ
ejpam-6425	51	38	mapping	mapping	NOUN
ejpam-6425	51	39	f+	f+	NOUN
ejpam-6425	51	40	:	:	PUNCT
ejpam-6425	51	41	v	v	X
ejpam-6425	51	42	→	→	SYM
ejpam-6425	51	43	zp	zp	PROPN
ejpam-6425	51	44	,	,	PUNCT
ejpam-6425	51	45	defined	define	VERB
ejpam-6425	51	46	by	by	ADP
ejpam-6425	51	47	f+(u	f+(u	PUNCT
ejpam-6425	51	48	)	)	PUNCT
ejpam-6425	51	49	≡	≡	PROPN
ejpam-6425	51	50	(	(	PUNCT
ejpam-6425	51	51	∑	∑	ADV
ejpam-6425	51	52	v∈n(u	v∈n(u	PROPN
ejpam-6425	51	53	)	)	PUNCT
ejpam-6425	51	54	f(uv	f(uv	NOUN
ejpam-6425	51	55	)	)	PUNCT
ejpam-6425	51	56	)	)	PUNCT
ejpam-6425	51	57	(	(	PUNCT
ejpam-6425	51	58	mod	mod	PROPN
ejpam-6425	51	59	p	p	X
ejpam-6425	51	60	)	)	PUNCT
ejpam-6425	51	61	is	be	AUX
ejpam-6425	51	62	also	also	ADV
ejpam-6425	51	63	a	a	DET
ejpam-6425	51	64	bijection	bijection	NOUN
ejpam-6425	51	65	,	,	PUNCT
ejpam-6425	51	66	where	where	SCONJ
ejpam-6425	51	67	zp	zp	PROPN
ejpam-6425	51	68	is	be	AUX
ejpam-6425	51	69	the	the	DET
ejpam-6425	51	70	set	set	NOUN
ejpam-6425	51	71	of	of	ADP
ejpam-6425	51	72	integers	integer	NOUN
ejpam-6425	51	73	{	{	PUNCT
ejpam-6425	51	74	0	0	NUM
ejpam-6425	51	75	,	,	PUNCT
ejpam-6425	51	76	1	1	NUM
ejpam-6425	51	77	,	,	PUNCT
ejpam-6425	51	78	2	2	NUM
ejpam-6425	51	79	,	,	PUNCT
ejpam-6425	51	80	..	..	PUNCT
ejpam-6425	51	81	,	,	PUNCT
ejpam-6425	51	82	(	(	PUNCT
ejpam-6425	51	83	p	p	NOUN
ejpam-6425	51	84	−	−	PROPN
ejpam-6425	51	85	1	1	NUM
ejpam-6425	51	86	)	)	PUNCT
ejpam-6425	51	87	}	}	PUNCT
ejpam-6425	51	88	.	.	PUNCT
ejpam-6425	52	1	moreover	moreover	ADV
ejpam-6425	52	2	,	,	PUNCT
ejpam-6425	52	3	every	every	DET
ejpam-6425	52	4	f(uv	f(uv	NOUN
ejpam-6425	52	5	)	)	PUNCT
ejpam-6425	52	6	are	be	AUX
ejpam-6425	52	7	the	the	DET
ejpam-6425	52	8	labels	label	NOUN
ejpam-6425	52	9	of	of	ADP
ejpam-6425	52	10	the	the	DET
ejpam-6425	52	11	edges	edge	NOUN
ejpam-6425	52	12	with	with	ADP
ejpam-6425	52	13	endpoint	endpoint	NOUN
ejpam-6425	52	14	u	u	NOUN
ejpam-6425	52	15	in	in	ADP
ejpam-6425	52	16	g.	g.	PROPN
ejpam-6425	52	17	example	example	NOUN
ejpam-6425	53	1	1	1	X
ejpam-6425	53	2	.	.	PUNCT
ejpam-6425	53	3	an	an	DET
ejpam-6425	53	4	example	example	NOUN
ejpam-6425	53	5	of	of	ADP
ejpam-6425	53	6	an	an	DET
ejpam-6425	53	7	edge	edge	NOUN
ejpam-6425	53	8	-	-	PUNCT
ejpam-6425	53	9	graceful	graceful	NOUN
ejpam-6425	53	10	labeling	labeling	NOUN
ejpam-6425	53	11	of	of	ADP
ejpam-6425	53	12	a	a	DET
ejpam-6425	53	13	graph	graph	NOUN
ejpam-6425	53	14	is	be	AUX
ejpam-6425	53	15	shown	show	VERB
ejpam-6425	53	16	in	in	ADP
ejpam-6425	53	17	figure	figure	NOUN
ejpam-6425	53	18	1	1	NUM
ejpam-6425	53	19	.	.	PUNCT
ejpam-6425	54	1	first	first	ADV
ejpam-6425	54	2	,	,	PUNCT
ejpam-6425	54	3	the	the	DET
ejpam-6425	54	4	graph	graph	NOUN
ejpam-6425	54	5	involved	involve	VERB
ejpam-6425	54	6	,	,	PUNCT
ejpam-6425	54	7	has	have	VERB
ejpam-6425	54	8	seven	seven	NUM
ejpam-6425	54	9	edges	edge	NOUN
ejpam-6425	54	10	and	and	CCONJ
ejpam-6425	54	11	seven	seven	NUM
ejpam-6425	54	12	vertices	vertex	NOUN
ejpam-6425	54	13	.	.	PUNCT
ejpam-6425	55	1	by	by	ADP
ejpam-6425	55	2	following	follow	VERB
ejpam-6425	55	3	definition	definition	NOUN
ejpam-6425	55	4	1	1	NUM
ejpam-6425	55	5	,	,	PUNCT
ejpam-6425	55	6	the	the	DET
ejpam-6425	55	7	edges	edge	NOUN
ejpam-6425	55	8	of	of	ADP
ejpam-6425	55	9	the	the	DET
ejpam-6425	55	10	graph	graph	NOUN
ejpam-6425	55	11	were	be	AUX
ejpam-6425	55	12	labeled	label	VERB
ejpam-6425	55	13	using	use	VERB
ejpam-6425	55	14	distinct	distinct	ADJ
ejpam-6425	55	15	integers	integer	NOUN
ejpam-6425	55	16	from	from	ADP
ejpam-6425	55	17	1	1	NUM
ejpam-6425	55	18	to	to	ADP
ejpam-6425	55	19	7	7	NUM
ejpam-6425	55	20	,	,	PUNCT
ejpam-6425	55	21	and	and	CCONJ
ejpam-6425	55	22	the	the	DET
ejpam-6425	55	23	induced	induce	VERB
ejpam-6425	55	24	labels	label	NOUN
ejpam-6425	55	25	of	of	ADP
ejpam-6425	55	26	the	the	DET
ejpam-6425	55	27	vertices	vertex	NOUN
ejpam-6425	55	28	are	be	AUX
ejpam-6425	55	29	distinct	distinct	ADJ
ejpam-6425	55	30	integers	integer	NOUN
ejpam-6425	55	31	from	from	ADP
ejpam-6425	55	32	0	0	NUM
ejpam-6425	55	33	to	to	ADP
ejpam-6425	55	34	6	6	NUM
ejpam-6425	55	35	.	.	PUNCT
ejpam-6425	56	1	figure	figure	NOUN
ejpam-6425	56	2	1	1	NUM
ejpam-6425	56	3	:	:	PUNCT
ejpam-6425	56	4	an	an	DET
ejpam-6425	56	5	edge	edge	NOUN
ejpam-6425	56	6	-	-	PUNCT
ejpam-6425	56	7	graceful	graceful	NOUN
ejpam-6425	56	8	labeling	labeling	NOUN
ejpam-6425	56	9	of	of	ADP
ejpam-6425	56	10	cycle	cycle	NOUN
ejpam-6425	56	11	graph	graph	NOUN
ejpam-6425	56	12	c7	c7	PROPN
ejpam-6425	56	13	.	.	PUNCT
ejpam-6425	57	1	definition	definition	NOUN
ejpam-6425	57	2	2	2	NUM
ejpam-6425	57	3	(	(	PUNCT
ejpam-6425	57	4	[	[	X
ejpam-6425	57	5	11	11	NUM
ejpam-6425	57	6	]	]	NUM
ejpam-6425	57	7	)	)	PUNCT
ejpam-6425	57	8	.	.	PUNCT
ejpam-6425	58	1	let	let	VERB
ejpam-6425	58	2	n	n	PRON
ejpam-6425	58	3	≥	≥	X
ejpam-6425	58	4	3	3	NUM
ejpam-6425	58	5	be	be	AUX
ejpam-6425	58	6	an	an	DET
ejpam-6425	58	7	integer	integer	NOUN
ejpam-6425	58	8	.	.	PUNCT
ejpam-6425	59	1	a	a	DET
ejpam-6425	59	2	water	water	NOUN
ejpam-6425	59	3	wheel	wheel	NOUN
ejpam-6425	59	4	graph	graph	NOUN
ejpam-6425	59	5	denoted	denote	VERB
ejpam-6425	59	6	by	by	ADP
ejpam-6425	59	7	wwn	wwn	PROPN
ejpam-6425	59	8	,	,	PUNCT
ejpam-6425	59	9	is	be	AUX
ejpam-6425	59	10	a	a	DET
ejpam-6425	59	11	graph	graph	NOUN
ejpam-6425	59	12	with	with	ADP
ejpam-6425	59	13	•	•	NUM
ejpam-6425	59	14	v	v	NOUN
ejpam-6425	59	15	(	(	PUNCT
ejpam-6425	59	16	wwn	wwn	NOUN
ejpam-6425	59	17	)	)	PUNCT
ejpam-6425	59	18	=	=	PUNCT
ejpam-6425	59	19	{	{	PUNCT
ejpam-6425	59	20	u0	u0	ADJ
ejpam-6425	59	21	}	}	PUNCT
ejpam-6425	59	22	∪	∪	ADJ
ejpam-6425	59	23	{	{	PUNCT
ejpam-6425	59	24	ui	ui	PROPN
ejpam-6425	59	25	,	,	PUNCT
ejpam-6425	59	26	vi	vi	PROPN
ejpam-6425	59	27	,	,	PUNCT
ejpam-6425	59	28	wi	wi	PROPN
ejpam-6425	59	29	:	:	PUNCT
ejpam-6425	59	30	1	1	NUM
ejpam-6425	59	31	≤	≤	NUM
ejpam-6425	59	32	i	i	PRON
ejpam-6425	59	33	≤	≤	NOUN
ejpam-6425	59	34	n	n	CCONJ
ejpam-6425	59	35	}	}	PUNCT
ejpam-6425	59	36	;	;	PUNCT
ejpam-6425	59	37	and	and	CCONJ
ejpam-6425	59	38	•	•	NUM
ejpam-6425	59	39	e(wwn	e(wwn	NOUN
ejpam-6425	59	40	)	)	PUNCT
ejpam-6425	59	41	=	=	PRON
ejpam-6425	59	42	{	{	PUNCT
ejpam-6425	59	43	u0ui	u0ui	PROPN
ejpam-6425	59	44	,	,	PUNCT
ejpam-6425	59	45	u0vi	u0vi	NUM
ejpam-6425	59	46	,	,	PUNCT
ejpam-6425	59	47	uiwi	uiwi	NOUN
ejpam-6425	59	48	,	,	PUNCT
ejpam-6425	59	49	viwi	viwi	NOUN
ejpam-6425	59	50	:	:	PUNCT
ejpam-6425	59	51	1	1	NUM
ejpam-6425	59	52	≤	≤	NUM
ejpam-6425	59	53	i	i	PRON
ejpam-6425	59	54	≤	≤	NOUN
ejpam-6425	59	55	n	n	CCONJ
ejpam-6425	59	56	}	}	PUNCT
ejpam-6425	59	57	∪	∪	X
ejpam-6425	59	58	{	{	PUNCT
ejpam-6425	59	59	uivi+1	uivi+1	NOUN
ejpam-6425	59	60	:	:	SYM
ejpam-6425	59	61	1	1	NUM
ejpam-6425	59	62	≤	≤	NUM
ejpam-6425	60	1	i	i	PRON
ejpam-6425	60	2	≤	≤	ADJ
ejpam-6425	60	3	n−	n−	NOUN
ejpam-6425	60	4	1	1	NUM
ejpam-6425	60	5	}	}	PUNCT
ejpam-6425	60	6	∪	∪	ADJ
ejpam-6425	60	7	{	{	PUNCT
ejpam-6425	60	8	unv1	unv1	PROPN
ejpam-6425	60	9	}	}	PUNCT
ejpam-6425	60	10	.	.	PUNCT
ejpam-6425	61	1	thus	thus	ADV
ejpam-6425	61	2	,	,	PUNCT
ejpam-6425	61	3	|v	|v	PROPN
ejpam-6425	61	4	(	(	PUNCT
ejpam-6425	61	5	wwn)|	wwn)|	PROPN
ejpam-6425	61	6	=	=	SYM
ejpam-6425	61	7	3n+	3n+	NUM
ejpam-6425	61	8	1	1	NUM
ejpam-6425	61	9	and	and	CCONJ
ejpam-6425	61	10	|e(wwn)|	|e(wwn)|	NOUN
ejpam-6425	61	11	=	=	SYM
ejpam-6425	61	12	5n	5n	NUM
ejpam-6425	61	13	.	.	PUNCT
ejpam-6425	61	14	example	example	NOUN
ejpam-6425	62	1	2	2	NUM
ejpam-6425	62	2	.	.	PUNCT
ejpam-6425	63	1	an	an	DET
ejpam-6425	63	2	illustration	illustration	NOUN
ejpam-6425	63	3	of	of	ADP
ejpam-6425	63	4	water	water	NOUN
ejpam-6425	63	5	wheel	wheel	NOUN
ejpam-6425	63	6	graph	graph	NOUN
ejpam-6425	63	7	wwn	wwn	NOUN
ejpam-6425	63	8	is	be	AUX
ejpam-6425	63	9	shown	show	VERB
ejpam-6425	63	10	in	in	ADP
ejpam-6425	63	11	figure	figure	NOUN
ejpam-6425	63	12	2	2	NUM
ejpam-6425	63	13	.	.	PUNCT
ejpam-6425	63	14	a.	a.	PROPN
ejpam-6425	63	15	d.	d.	PROPN
ejpam-6425	63	16	c.	c.	PROPN
ejpam-6425	63	17	angel	angel	PROPN
ejpam-6425	63	18	et	et	PROPN
ejpam-6425	63	19	al	al	PROPN
ejpam-6425	63	20	.	.	PUNCT
ejpam-6425	63	21	/	/	SYM
ejpam-6425	63	22	eur	eur	PROPN
ejpam-6425	63	23	.	.	PUNCT
ejpam-6425	64	1	j.	j.	PROPN
ejpam-6425	64	2	pure	pure	PROPN
ejpam-6425	64	3	appl	appl	PROPN
ejpam-6425	64	4	.	.	PROPN
ejpam-6425	64	5	math	math	PROPN
ejpam-6425	64	6	,	,	PUNCT
ejpam-6425	64	7	18	18	NUM
ejpam-6425	64	8	(	(	PUNCT
ejpam-6425	64	9	4	4	NUM
ejpam-6425	64	10	)	)	PUNCT
ejpam-6425	64	11	(	(	PUNCT
ejpam-6425	64	12	2025	2025	NUM
ejpam-6425	64	13	)	)	PUNCT
ejpam-6425	64	14	,	,	PUNCT
ejpam-6425	64	15	6425	6425	NUM
ejpam-6425	64	16	4	4	NUM
ejpam-6425	64	17	of	of	ADP
ejpam-6425	64	18	33	33	NUM
ejpam-6425	64	19	figure	figure	NOUN
ejpam-6425	64	20	2	2	NUM
ejpam-6425	64	21	:	:	PUNCT
ejpam-6425	64	22	water	water	NOUN
ejpam-6425	64	23	wheel	wheel	NOUN
ejpam-6425	64	24	graph	graph	NOUN
ejpam-6425	64	25	wwn	wwn	PROPN
ejpam-6425	64	26	.	.	PUNCT
ejpam-6425	65	1	now	now	ADV
ejpam-6425	65	2	,	,	PUNCT
ejpam-6425	65	3	we	we	PRON
ejpam-6425	65	4	introduce	introduce	VERB
ejpam-6425	65	5	two	two	NUM
ejpam-6425	65	6	new	new	ADJ
ejpam-6425	65	7	water	water	NOUN
ejpam-6425	65	8	wheel	wheel	NOUN
ejpam-6425	65	9	-	-	PUNCT
ejpam-6425	65	10	related	relate	VERB
ejpam-6425	65	11	graphs	graph	NOUN
ejpam-6425	65	12	,	,	PUNCT
ejpam-6425	65	13	the	the	DET
ejpam-6425	65	14	right	right	ADJ
ejpam-6425	65	15	water	water	NOUN
ejpam-6425	65	16	wheel	wheel	NOUN
ejpam-6425	65	17	and	and	CCONJ
ejpam-6425	65	18	left	leave	VERB
ejpam-6425	65	19	water	water	NOUN
ejpam-6425	65	20	wheel	wheel	NOUN
ejpam-6425	65	21	graphs	graph	NOUN
ejpam-6425	65	22	.	.	PUNCT
ejpam-6425	66	1	definition	definition	NOUN
ejpam-6425	66	2	3	3	NUM
ejpam-6425	66	3	.	.	PUNCT
ejpam-6425	67	1	let	let	VERB
ejpam-6425	67	2	n	n	PRON
ejpam-6425	67	3	≥	≥	X
ejpam-6425	67	4	3	3	NUM
ejpam-6425	67	5	be	be	AUX
ejpam-6425	67	6	an	an	DET
ejpam-6425	67	7	integer	integer	NOUN
ejpam-6425	67	8	.	.	PUNCT
ejpam-6425	68	1	a	a	DET
ejpam-6425	68	2	right	right	ADJ
ejpam-6425	68	3	water	water	NOUN
ejpam-6425	68	4	wheel	wheel	NOUN
ejpam-6425	68	5	graph	graph	NOUN
ejpam-6425	68	6	denoted	denote	VERB
ejpam-6425	68	7	by	by	ADP
ejpam-6425	68	8	rwwn	rwwn	NOUN
ejpam-6425	68	9	,	,	PUNCT
ejpam-6425	68	10	is	be	AUX
ejpam-6425	68	11	obtained	obtain	VERB
ejpam-6425	68	12	by	by	ADP
ejpam-6425	68	13	adding	add	VERB
ejpam-6425	68	14	edges	edge	NOUN
ejpam-6425	68	15	wnv1	wnv1	NOUN
ejpam-6425	68	16	and	and	CCONJ
ejpam-6425	68	17	wivi+1	wivi+1	PROPN
ejpam-6425	68	18	,	,	PUNCT
ejpam-6425	68	19	for	for	ADP
ejpam-6425	68	20	1	1	NUM
ejpam-6425	68	21	≤	≤	NUM
ejpam-6425	69	1	i	i	PRON
ejpam-6425	69	2	≤	≤	ADJ
ejpam-6425	69	3	n−	n−	PROPN
ejpam-6425	69	4	1	1	NUM
ejpam-6425	69	5	,	,	PUNCT
ejpam-6425	69	6	to	to	ADP
ejpam-6425	69	7	the	the	DET
ejpam-6425	69	8	water	water	NOUN
ejpam-6425	69	9	wheel	wheel	NOUN
ejpam-6425	69	10	graph	graph	NOUN
ejpam-6425	69	11	wwn	wwn	PROPN
ejpam-6425	69	12	.	.	PUNCT
ejpam-6425	70	1	hence	hence	ADV
ejpam-6425	70	2	,	,	PUNCT
ejpam-6425	70	3	•	•	NOUN
ejpam-6425	70	4	v	v	NOUN
ejpam-6425	70	5	(	(	PUNCT
ejpam-6425	70	6	rwwn	rwwn	NOUN
ejpam-6425	70	7	)	)	PUNCT
ejpam-6425	70	8	=	=	SYM
ejpam-6425	70	9	v	v	X
ejpam-6425	70	10	(	(	PUNCT
ejpam-6425	70	11	wwn	wwn	NOUN
ejpam-6425	70	12	)	)	PUNCT
ejpam-6425	70	13	;	;	PUNCT
ejpam-6425	70	14	and	and	CCONJ
ejpam-6425	70	15	•	•	NUM
ejpam-6425	70	16	e(wwn	e(wwn	NOUN
ejpam-6425	70	17	)	)	PUNCT
ejpam-6425	70	18	=	=	SYM
ejpam-6425	71	1	e(wwn	e(wwn	PROPN
ejpam-6425	71	2	)	)	PUNCT
ejpam-6425	71	3	∪	∪	NOUN
ejpam-6425	71	4	{	{	PUNCT
ejpam-6425	71	5	wivi+1	wivi+1	NOUN
ejpam-6425	71	6	:	:	PUNCT
ejpam-6425	71	7	1	1	NUM
ejpam-6425	71	8	≤	≤	NUM
ejpam-6425	72	1	i	i	PRON
ejpam-6425	72	2	≤	≤	ADJ
ejpam-6425	72	3	n−	n−	NOUN
ejpam-6425	72	4	1	1	NUM
ejpam-6425	72	5	}	}	PUNCT
ejpam-6425	72	6	∪	∪	ADJ
ejpam-6425	72	7	{	{	PUNCT
ejpam-6425	72	8	wnv1	wnv1	NOUN
ejpam-6425	72	9	}	}	PUNCT
ejpam-6425	72	10	.	.	PUNCT
ejpam-6425	73	1	thus	thus	ADV
ejpam-6425	73	2	,	,	PUNCT
ejpam-6425	73	3	|v	|v	PROPN
ejpam-6425	73	4	(	(	PUNCT
ejpam-6425	73	5	rwwn)|	rwwn)|	NOUN
ejpam-6425	73	6	=	=	SYM
ejpam-6425	73	7	3n+	3n+	NUM
ejpam-6425	73	8	1	1	NUM
ejpam-6425	73	9	and	and	CCONJ
ejpam-6425	73	10	|e(rwwn)|	|e(rwwn)|	PROPN
ejpam-6425	73	11	=	=	SYM
ejpam-6425	73	12	6n	6n	PROPN
ejpam-6425	73	13	.	.	PUNCT
ejpam-6425	73	14	example	example	NOUN
ejpam-6425	73	15	3	3	NUM
ejpam-6425	73	16	.	.	PUNCT
ejpam-6425	73	17	an	an	DET
ejpam-6425	73	18	illustration	illustration	NOUN
ejpam-6425	73	19	of	of	ADP
ejpam-6425	73	20	right	right	ADJ
ejpam-6425	73	21	water	water	NOUN
ejpam-6425	73	22	wheel	wheel	NOUN
ejpam-6425	73	23	graph	graph	NOUN
ejpam-6425	73	24	rwwn	rwwn	NOUN
ejpam-6425	73	25	is	be	AUX
ejpam-6425	73	26	shown	show	VERB
ejpam-6425	73	27	in	in	ADP
ejpam-6425	73	28	figure	figure	NOUN
ejpam-6425	73	29	3	3	NUM
ejpam-6425	73	30	,	,	PUNCT
ejpam-6425	73	31	in	in	ADP
ejpam-6425	73	32	the	the	DET
ejpam-6425	73	33	next	next	ADJ
ejpam-6425	73	34	page	page	NOUN
ejpam-6425	73	35	.	.	PUNCT
ejpam-6425	74	1	a.	a.	PROPN
ejpam-6425	74	2	d.	d.	PROPN
ejpam-6425	74	3	c.	c.	PROPN
ejpam-6425	74	4	angel	angel	PROPN
ejpam-6425	74	5	et	et	PROPN
ejpam-6425	74	6	al	al	PROPN
ejpam-6425	74	7	.	.	PUNCT
ejpam-6425	74	8	/	/	SYM
ejpam-6425	74	9	eur	eur	PROPN
ejpam-6425	74	10	.	.	PUNCT
ejpam-6425	75	1	j.	j.	PROPN
ejpam-6425	75	2	pure	pure	PROPN
ejpam-6425	75	3	appl	appl	PROPN
ejpam-6425	75	4	.	.	PROPN
ejpam-6425	75	5	math	math	PROPN
ejpam-6425	75	6	,	,	PUNCT
ejpam-6425	75	7	18	18	NUM
ejpam-6425	75	8	(	(	PUNCT
ejpam-6425	75	9	4	4	NUM
ejpam-6425	75	10	)	)	PUNCT
ejpam-6425	75	11	(	(	PUNCT
ejpam-6425	75	12	2025	2025	NUM
ejpam-6425	75	13	)	)	PUNCT
ejpam-6425	75	14	,	,	PUNCT
ejpam-6425	75	15	6425	6425	NUM
ejpam-6425	75	16	5	5	NUM
ejpam-6425	75	17	of	of	ADP
ejpam-6425	75	18	33	33	NUM
ejpam-6425	75	19	figure	figure	NOUN
ejpam-6425	75	20	3	3	NUM
ejpam-6425	75	21	:	:	PUNCT
ejpam-6425	75	22	right	right	ADJ
ejpam-6425	75	23	water	water	NOUN
ejpam-6425	75	24	wheel	wheel	NOUN
ejpam-6425	75	25	graph	graph	NOUN
ejpam-6425	75	26	rwwn	rwwn	NOUN
ejpam-6425	75	27	.	.	PUNCT
ejpam-6425	76	1	definition	definition	NOUN
ejpam-6425	76	2	4	4	NUM
ejpam-6425	76	3	.	.	PUNCT
ejpam-6425	77	1	let	let	VERB
ejpam-6425	77	2	n	n	PRON
ejpam-6425	77	3	≥	≥	X
ejpam-6425	77	4	3	3	NUM
ejpam-6425	77	5	be	be	AUX
ejpam-6425	77	6	an	an	DET
ejpam-6425	77	7	integer	integer	NOUN
ejpam-6425	77	8	.	.	PUNCT
ejpam-6425	78	1	a	a	DET
ejpam-6425	78	2	left	left	ADJ
ejpam-6425	78	3	water	water	NOUN
ejpam-6425	78	4	wheel	wheel	NOUN
ejpam-6425	78	5	graph	graph	NOUN
ejpam-6425	78	6	denoted	denote	VERB
ejpam-6425	78	7	by	by	ADP
ejpam-6425	78	8	lwwn	lwwn	NOUN
ejpam-6425	78	9	,	,	PUNCT
ejpam-6425	78	10	is	be	AUX
ejpam-6425	78	11	obtained	obtain	VERB
ejpam-6425	78	12	by	by	ADP
ejpam-6425	78	13	adding	add	VERB
ejpam-6425	78	14	edges	edge	NOUN
ejpam-6425	78	15	w1un	w1un	X
ejpam-6425	78	16	and	and	CCONJ
ejpam-6425	78	17	wiui−1	wiui−1	PROPN
ejpam-6425	78	18	,	,	PUNCT
ejpam-6425	78	19	for	for	ADP
ejpam-6425	78	20	2	2	NUM
ejpam-6425	78	21	≤	≤	NOUN
ejpam-6425	78	22	i	i	PRON
ejpam-6425	78	23	≤	≤	NOUN
ejpam-6425	78	24	n	n	CCONJ
ejpam-6425	78	25	,	,	PUNCT
ejpam-6425	78	26	to	to	ADP
ejpam-6425	78	27	the	the	DET
ejpam-6425	78	28	water	water	NOUN
ejpam-6425	78	29	wheel	wheel	NOUN
ejpam-6425	78	30	graph	graph	NOUN
ejpam-6425	78	31	wwn	wwn	PROPN
ejpam-6425	78	32	.	.	PUNCT
ejpam-6425	79	1	hence	hence	ADV
ejpam-6425	79	2	,	,	PUNCT
ejpam-6425	79	3	•	•	NOUN
ejpam-6425	79	4	v	v	NOUN
ejpam-6425	79	5	(	(	PUNCT
ejpam-6425	79	6	rwwn	rwwn	NOUN
ejpam-6425	79	7	)	)	PUNCT
ejpam-6425	79	8	=	=	SYM
ejpam-6425	79	9	v	v	X
ejpam-6425	79	10	(	(	PUNCT
ejpam-6425	79	11	wwn	wwn	NOUN
ejpam-6425	79	12	)	)	PUNCT
ejpam-6425	79	13	;	;	PUNCT
ejpam-6425	79	14	and	and	CCONJ
ejpam-6425	79	15	•	•	NUM
ejpam-6425	79	16	e(wwn	e(wwn	NOUN
ejpam-6425	79	17	)	)	PUNCT
ejpam-6425	79	18	=	=	SYM
ejpam-6425	79	19	e(wwn	e(wwn	PROPN
ejpam-6425	79	20	)	)	PUNCT
ejpam-6425	79	21	∪	∪	NOUN
ejpam-6425	79	22	{	{	PUNCT
ejpam-6425	79	23	wiui−1	wiui−1	NOUN
ejpam-6425	79	24	:	:	PUNCT
ejpam-6425	79	25	2	2	NUM
ejpam-6425	79	26	≤	≤	NUM
ejpam-6425	79	27	i	i	PRON
ejpam-6425	79	28	≤	≤	NOUN
ejpam-6425	79	29	n	n	CCONJ
ejpam-6425	79	30	}	}	PUNCT
ejpam-6425	79	31	∪	∪	ADJ
ejpam-6425	79	32	{	{	PUNCT
ejpam-6425	79	33	w1un	w1un	X
ejpam-6425	79	34	}	}	PUNCT
ejpam-6425	79	35	.	.	PUNCT
ejpam-6425	80	1	thus	thus	ADV
ejpam-6425	80	2	,	,	PUNCT
ejpam-6425	80	3	|v	|v	PROPN
ejpam-6425	80	4	(	(	PUNCT
ejpam-6425	80	5	rwwn)|	rwwn)|	NOUN
ejpam-6425	80	6	=	=	SYM
ejpam-6425	80	7	3n+	3n+	NUM
ejpam-6425	80	8	1	1	NUM
ejpam-6425	80	9	and	and	CCONJ
ejpam-6425	80	10	|e(rwwn)|	|e(rwwn)|	PROPN
ejpam-6425	80	11	=	=	SYM
ejpam-6425	80	12	6n	6n	PROPN
ejpam-6425	80	13	.	.	PUNCT
ejpam-6425	80	14	example	example	NOUN
ejpam-6425	81	1	4	4	NUM
ejpam-6425	81	2	.	.	PUNCT
ejpam-6425	82	1	an	an	DET
ejpam-6425	82	2	illustration	illustration	NOUN
ejpam-6425	82	3	of	of	ADP
ejpam-6425	82	4	left	left	ADJ
ejpam-6425	82	5	water	water	NOUN
ejpam-6425	82	6	wheel	wheel	NOUN
ejpam-6425	82	7	graph	graph	NOUN
ejpam-6425	82	8	lwwn	lwwn	NOUN
ejpam-6425	82	9	is	be	AUX
ejpam-6425	82	10	shown	show	VERB
ejpam-6425	82	11	in	in	ADP
ejpam-6425	82	12	figure	figure	NOUN
ejpam-6425	82	13	4	4	NUM
ejpam-6425	82	14	,	,	PUNCT
ejpam-6425	82	15	in	in	ADP
ejpam-6425	82	16	the	the	DET
ejpam-6425	82	17	next	next	ADJ
ejpam-6425	82	18	page	page	NOUN
ejpam-6425	82	19	.	.	PUNCT
ejpam-6425	83	1	a.	a.	PROPN
ejpam-6425	83	2	d.	d.	PROPN
ejpam-6425	83	3	c.	c.	PROPN
ejpam-6425	83	4	angel	angel	PROPN
ejpam-6425	83	5	et	et	PROPN
ejpam-6425	83	6	al	al	PROPN
ejpam-6425	83	7	.	.	PUNCT
ejpam-6425	83	8	/	/	SYM
ejpam-6425	83	9	eur	eur	PROPN
ejpam-6425	83	10	.	.	PUNCT
ejpam-6425	84	1	j.	j.	PROPN
ejpam-6425	84	2	pure	pure	PROPN
ejpam-6425	84	3	appl	appl	PROPN
ejpam-6425	84	4	.	.	PROPN
ejpam-6425	84	5	math	math	PROPN
ejpam-6425	84	6	,	,	PUNCT
ejpam-6425	84	7	18	18	NUM
ejpam-6425	84	8	(	(	PUNCT
ejpam-6425	84	9	4	4	NUM
ejpam-6425	84	10	)	)	PUNCT
ejpam-6425	84	11	(	(	PUNCT
ejpam-6425	84	12	2025	2025	NUM
ejpam-6425	84	13	)	)	PUNCT
ejpam-6425	84	14	,	,	PUNCT
ejpam-6425	84	15	6425	6425	NUM
ejpam-6425	84	16	6	6	NUM
ejpam-6425	84	17	of	of	ADP
ejpam-6425	84	18	33	33	NUM
ejpam-6425	84	19	figure	figure	NOUN
ejpam-6425	84	20	4	4	NUM
ejpam-6425	84	21	:	:	PUNCT
ejpam-6425	84	22	left	leave	VERB
ejpam-6425	84	23	water	water	NOUN
ejpam-6425	84	24	wheel	wheel	NOUN
ejpam-6425	84	25	graph	graph	NOUN
ejpam-6425	84	26	lwwn	lwwn	NOUN
ejpam-6425	84	27	.	.	PUNCT
ejpam-6425	85	1	definition	definition	NOUN
ejpam-6425	85	2	5	5	NUM
ejpam-6425	85	3	(	(	PUNCT
ejpam-6425	85	4	[	[	X
ejpam-6425	85	5	12	12	NUM
ejpam-6425	85	6	]	]	PUNCT
ejpam-6425	85	7	)	)	PUNCT
ejpam-6425	85	8	.	.	PUNCT
ejpam-6425	86	1	suppose	suppose	VERB
ejpam-6425	86	2	we	we	PRON
ejpam-6425	86	3	have	have	VERB
ejpam-6425	86	4	a	a	DET
ejpam-6425	86	5	graph	graph	NOUN
ejpam-6425	86	6	g.	g.	NOUN
ejpam-6425	86	7	the	the	DET
ejpam-6425	86	8	splitting	splitting	NOUN
ejpam-6425	86	9	graph	graph	NOUN
ejpam-6425	86	10	of	of	ADP
ejpam-6425	86	11	g	g	PROPN
ejpam-6425	86	12	denoted	denote	VERB
ejpam-6425	86	13	by	by	ADP
ejpam-6425	86	14	s(g	s(g	PROPN
ejpam-6425	86	15	)	)	PUNCT
ejpam-6425	86	16	can	can	AUX
ejpam-6425	86	17	be	be	AUX
ejpam-6425	86	18	constructed	construct	VERB
ejpam-6425	86	19	by	by	ADP
ejpam-6425	86	20	introducing	introduce	VERB
ejpam-6425	86	21	a	a	DET
ejpam-6425	86	22	new	new	ADJ
ejpam-6425	86	23	vertex	vertex	NOUN
ejpam-6425	86	24	v′	v′	NOUN
ejpam-6425	86	25	for	for	ADP
ejpam-6425	86	26	each	each	DET
ejpam-6425	86	27	vertex	vertex	NOUN
ejpam-6425	86	28	v	v	ADP
ejpam-6425	86	29	∈	∈	PROPN
ejpam-6425	86	30	v	v	NOUN
ejpam-6425	86	31	(	(	PUNCT
ejpam-6425	86	32	g	g	NOUN
ejpam-6425	86	33	)	)	PUNCT
ejpam-6425	86	34	,	,	PUNCT
ejpam-6425	86	35	and	and	CCONJ
ejpam-6425	86	36	connecting	connect	VERB
ejpam-6425	86	37	v′	v′	NOUN
ejpam-6425	86	38	to	to	ADP
ejpam-6425	86	39	all	all	DET
ejpam-6425	86	40	the	the	DET
ejpam-6425	86	41	points	point	NOUN
ejpam-6425	86	42	in	in	ADP
ejpam-6425	86	43	g	g	PROPN
ejpam-6425	86	44	that	that	PRON
ejpam-6425	86	45	are	be	AUX
ejpam-6425	86	46	adjacent	adjacent	ADJ
ejpam-6425	86	47	to	to	ADP
ejpam-6425	86	48	v.	v.	ADP
ejpam-6425	86	49	example	example	NOUN
ejpam-6425	86	50	5	5	X
ejpam-6425	86	51	.	.	PUNCT
ejpam-6425	87	1	the	the	DET
ejpam-6425	87	2	path	path	NOUN
ejpam-6425	87	3	graph	graph	NOUN
ejpam-6425	87	4	p4	p4	ADJ
ejpam-6425	87	5	and	and	CCONJ
ejpam-6425	87	6	its	its	PRON
ejpam-6425	87	7	splitting	splitting	NOUN
ejpam-6425	87	8	graph	graph	NOUN
ejpam-6425	87	9	s(p4	s(p4	NOUN
ejpam-6425	87	10	)	)	PUNCT
ejpam-6425	87	11	is	be	AUX
ejpam-6425	87	12	shown	show	VERB
ejpam-6425	87	13	in	in	ADP
ejpam-6425	87	14	figure	figure	NOUN
ejpam-6425	87	15	5	5	NUM
ejpam-6425	87	16	figure	figure	NOUN
ejpam-6425	87	17	5	5	NUM
ejpam-6425	87	18	:	:	PUNCT
ejpam-6425	87	19	splitting	splitting	NOUN
ejpam-6425	87	20	graph	graph	NOUN
ejpam-6425	87	21	of	of	ADP
ejpam-6425	87	22	path	path	NOUN
ejpam-6425	87	23	graph	graph	NOUN
ejpam-6425	87	24	p4	p4	ADJ
ejpam-6425	87	25	a.	a.	PROPN
ejpam-6425	87	26	d.	d.	PROPN
ejpam-6425	87	27	c.	c.	PROPN
ejpam-6425	87	28	angel	angel	PROPN
ejpam-6425	87	29	et	et	PROPN
ejpam-6425	87	30	al	al	PROPN
ejpam-6425	87	31	.	.	PUNCT
ejpam-6425	87	32	/	/	SYM
ejpam-6425	87	33	eur	eur	PROPN
ejpam-6425	87	34	.	.	PUNCT
ejpam-6425	88	1	j.	j.	PROPN
ejpam-6425	88	2	pure	pure	PROPN
ejpam-6425	88	3	appl	appl	PROPN
ejpam-6425	88	4	.	.	PROPN
ejpam-6425	88	5	math	math	PROPN
ejpam-6425	88	6	,	,	PUNCT
ejpam-6425	88	7	18	18	NUM
ejpam-6425	88	8	(	(	PUNCT
ejpam-6425	88	9	4	4	NUM
ejpam-6425	88	10	)	)	PUNCT
ejpam-6425	88	11	(	(	PUNCT
ejpam-6425	88	12	2025	2025	NUM
ejpam-6425	88	13	)	)	PUNCT
ejpam-6425	88	14	,	,	PUNCT
ejpam-6425	88	15	6425	6425	NUM
ejpam-6425	88	16	7	7	NUM
ejpam-6425	88	17	of	of	ADP
ejpam-6425	88	18	33	33	NUM
ejpam-6425	88	19	remark	remark	NOUN
ejpam-6425	88	20	1	1	NUM
ejpam-6425	88	21	.	.	PUNCT
ejpam-6425	89	1	for	for	ADP
ejpam-6425	89	2	convenience	convenience	NOUN
ejpam-6425	89	3	with	with	ADP
ejpam-6425	89	4	the	the	DET
ejpam-6425	89	5	computer	computer	NOUN
ejpam-6425	89	6	programs	program	NOUN
ejpam-6425	89	7	used	use	VERB
ejpam-6425	89	8	in	in	ADP
ejpam-6425	89	9	the	the	DET
ejpam-6425	89	10	study	study	NOUN
ejpam-6425	89	11	,	,	PUNCT
ejpam-6425	89	12	we	we	PRON
ejpam-6425	89	13	represent	represent	VERB
ejpam-6425	89	14	the	the	DET
ejpam-6425	89	15	new	new	ADJ
ejpam-6425	89	16	vertices	vertex	NOUN
ejpam-6425	89	17	in	in	ADP
ejpam-6425	89	18	the	the	DET
ejpam-6425	89	19	splitting	splitting	NOUN
ejpam-6425	89	20	graph	graph	NOUN
ejpam-6425	89	21	with	with	ADP
ejpam-6425	89	22	capital	capital	NOUN
ejpam-6425	89	23	letters	letter	NOUN
ejpam-6425	89	24	.	.	PUNCT
ejpam-6425	90	1	for	for	ADP
ejpam-6425	90	2	example	example	NOUN
ejpam-6425	90	3	,	,	PUNCT
ejpam-6425	90	4	a	a	DET
ejpam-6425	90	5	new	new	ADJ
ejpam-6425	90	6	vertex	vertex	NOUN
ejpam-6425	90	7	w′	w′	NOUN
ejpam-6425	90	8	1	1	NUM
ejpam-6425	90	9	will	will	AUX
ejpam-6425	90	10	be	be	AUX
ejpam-6425	90	11	re	re	VERB
ejpam-6425	90	12	-	-	VERB
ejpam-6425	90	13	represented	represent	VERB
ejpam-6425	90	14	as	as	ADP
ejpam-6425	90	15	simply	simply	ADV
ejpam-6425	90	16	vertex	vertex	NOUN
ejpam-6425	90	17	w1	w1	NOUN
ejpam-6425	90	18	.	.	PUNCT
ejpam-6425	91	1	example	example	NOUN
ejpam-6425	91	2	6	6	NUM
ejpam-6425	91	3	.	.	PUNCT
ejpam-6425	91	4	to	to	PART
ejpam-6425	91	5	illustrate	illustrate	VERB
ejpam-6425	91	6	remark	remark	NOUN
ejpam-6425	91	7	1	1	NUM
ejpam-6425	91	8	,	,	PUNCT
ejpam-6425	91	9	the	the	DET
ejpam-6425	91	10	splitting	splitting	NOUN
ejpam-6425	91	11	graph	graph	NOUN
ejpam-6425	91	12	s(ww3	s(ww3	NOUN
ejpam-6425	91	13	)	)	PUNCT
ejpam-6425	91	14	is	be	AUX
ejpam-6425	91	15	shown	show	VERB
ejpam-6425	91	16	in	in	ADP
ejpam-6425	91	17	figure	figure	NOUN
ejpam-6425	91	18	6	6	NUM
ejpam-6425	91	19	.	.	PUNCT
ejpam-6425	92	1	the	the	DET
ejpam-6425	92	2	subgraph	subgraph	NOUN
ejpam-6425	92	3	colored	color	VERB
ejpam-6425	92	4	in	in	ADP
ejpam-6425	92	5	black	black	ADJ
ejpam-6425	92	6	is	be	AUX
ejpam-6425	92	7	the	the	DET
ejpam-6425	92	8	original	original	ADJ
ejpam-6425	92	9	water	water	NOUN
ejpam-6425	92	10	wheel	wheel	NOUN
ejpam-6425	92	11	graph	graph	NOUN
ejpam-6425	92	12	ww3	ww3	PROPN
ejpam-6425	92	13	,	,	PUNCT
ejpam-6425	92	14	while	while	SCONJ
ejpam-6425	92	15	the	the	DET
ejpam-6425	92	16	newly	newly	ADV
ejpam-6425	92	17	introduced	introduce	VERB
ejpam-6425	92	18	vertices	vertex	NOUN
ejpam-6425	92	19	and	and	CCONJ
ejpam-6425	92	20	edges	edge	NOUN
ejpam-6425	92	21	are	be	AUX
ejpam-6425	92	22	colored	color	VERB
ejpam-6425	92	23	in	in	ADP
ejpam-6425	92	24	blue	blue	ADJ
ejpam-6425	92	25	.	.	PUNCT
ejpam-6425	93	1	figure	figure	VERB
ejpam-6425	93	2	6	6	NUM
ejpam-6425	93	3	:	:	PUNCT
ejpam-6425	93	4	splitting	splitting	NOUN
ejpam-6425	93	5	graph	graph	NOUN
ejpam-6425	93	6	of	of	ADP
ejpam-6425	93	7	s(ww3	s(ww3	PROPN
ejpam-6425	93	8	)	)	PUNCT
ejpam-6425	93	9	we	we	PRON
ejpam-6425	93	10	now	now	ADV
ejpam-6425	93	11	introduce	introduce	VERB
ejpam-6425	93	12	a	a	DET
ejpam-6425	93	13	proposition	proposition	NOUN
ejpam-6425	93	14	,	,	PUNCT
ejpam-6425	93	15	used	use	VERB
ejpam-6425	93	16	in	in	ADP
ejpam-6425	93	17	this	this	DET
ejpam-6425	93	18	paper	paper	NOUN
ejpam-6425	93	19	,	,	PUNCT
ejpam-6425	93	20	regarding	regard	VERB
ejpam-6425	93	21	the	the	DET
ejpam-6425	93	22	order	order	NOUN
ejpam-6425	93	23	and	and	CCONJ
ejpam-6425	93	24	size	size	NOUN
ejpam-6425	93	25	of	of	ADP
ejpam-6425	93	26	splitting	splitting	NOUN
ejpam-6425	93	27	graphs	graph	NOUN
ejpam-6425	93	28	.	.	PUNCT
ejpam-6425	94	1	proposition	proposition	NOUN
ejpam-6425	94	2	1	1	NUM
ejpam-6425	94	3	(	(	PUNCT
ejpam-6425	94	4	[	[	X
ejpam-6425	94	5	12	12	NUM
ejpam-6425	94	6	]	]	PUNCT
ejpam-6425	94	7	)	)	PUNCT
ejpam-6425	94	8	.	.	PUNCT
ejpam-6425	95	1	suppose	suppose	VERB
ejpam-6425	95	2	that	that	SCONJ
ejpam-6425	95	3	g	g	PROPN
ejpam-6425	95	4	is	be	AUX
ejpam-6425	95	5	a	a	DET
ejpam-6425	95	6	(	(	PUNCT
ejpam-6425	95	7	p	p	NOUN
ejpam-6425	95	8	,	,	PUNCT
ejpam-6425	95	9	q)graph	q)graph	PROPN
ejpam-6425	95	10	,	,	PUNCT
ejpam-6425	95	11	then	then	ADV
ejpam-6425	95	12	its	its	PRON
ejpam-6425	95	13	splitting	splitting	NOUN
ejpam-6425	95	14	graph	graph	NOUN
ejpam-6425	95	15	s(g	s(g	PROPN
ejpam-6425	95	16	)	)	PUNCT
ejpam-6425	95	17	is	be	AUX
ejpam-6425	95	18	a	a	DET
ejpam-6425	95	19	(	(	PUNCT
ejpam-6425	95	20	2p	2p	NUM
ejpam-6425	95	21	,	,	PUNCT
ejpam-6425	95	22	3q)graph	3q)graph	NOUN
ejpam-6425	95	23	.	.	PUNCT
ejpam-6425	96	1	2.2	2.2	NUM
ejpam-6425	96	2	.	.	PUNCT
ejpam-6425	97	1	essential	essential	ADJ
ejpam-6425	97	2	number	number	NOUN
ejpam-6425	97	3	theory	theory	NOUN
ejpam-6425	97	4	concepts	concept	NOUN
ejpam-6425	97	5	in	in	ADP
ejpam-6425	97	6	this	this	DET
ejpam-6425	97	7	subsection	subsection	NOUN
ejpam-6425	97	8	,	,	PUNCT
ejpam-6425	97	9	we	we	PRON
ejpam-6425	97	10	recall	recall	VERB
ejpam-6425	97	11	the	the	DET
ejpam-6425	97	12	important	important	ADJ
ejpam-6425	97	13	concepts	concept	NOUN
ejpam-6425	97	14	and	and	CCONJ
ejpam-6425	97	15	results	result	NOUN
ejpam-6425	97	16	in	in	ADP
ejpam-6425	97	17	number	number	NOUN
ejpam-6425	97	18	theory	theory	NOUN
ejpam-6425	97	19	that	that	PRON
ejpam-6425	97	20	are	be	AUX
ejpam-6425	97	21	used	use	VERB
ejpam-6425	97	22	in	in	ADP
ejpam-6425	97	23	this	this	DET
ejpam-6425	97	24	paper	paper	NOUN
ejpam-6425	97	25	.	.	PUNCT
ejpam-6425	98	1	for	for	ADP
ejpam-6425	98	2	a	a	DET
ejpam-6425	98	3	more	more	ADV
ejpam-6425	98	4	detailed	detailed	ADJ
ejpam-6425	98	5	discussion	discussion	NOUN
ejpam-6425	98	6	of	of	ADP
ejpam-6425	98	7	basic	basic	ADJ
ejpam-6425	98	8	concepts	concept	NOUN
ejpam-6425	98	9	in	in	ADP
ejpam-6425	98	10	number	number	NOUN
ejpam-6425	98	11	theory	theory	NOUN
ejpam-6425	98	12	,	,	PUNCT
ejpam-6425	98	13	readers	reader	NOUN
ejpam-6425	98	14	may	may	AUX
ejpam-6425	98	15	refer	refer	VERB
ejpam-6425	98	16	to	to	ADP
ejpam-6425	98	17	burton	burton	PROPN
ejpam-6425	98	18	’s	’s	PART
ejpam-6425	98	19	“	"	PUNCT
ejpam-6425	98	20	elementary	elementary	ADJ
ejpam-6425	98	21	number	number	NOUN
ejpam-6425	98	22	theory”[13	theory”[13	NOUN
ejpam-6425	98	23	]	]	PUNCT
ejpam-6425	98	24	.	.	PUNCT
ejpam-6425	99	1	definition	definition	NOUN
ejpam-6425	99	2	6	6	NUM
ejpam-6425	99	3	.	.	PUNCT
ejpam-6425	100	1	let	let	VERB
ejpam-6425	100	2	a	a	PRON
ejpam-6425	100	3	and	and	CCONJ
ejpam-6425	100	4	b	b	NOUN
ejpam-6425	100	5	be	be	AUX
ejpam-6425	100	6	two	two	NUM
ejpam-6425	100	7	positive	positive	ADJ
ejpam-6425	100	8	integers	integer	NOUN
ejpam-6425	100	9	,	,	PUNCT
ejpam-6425	100	10	where	where	SCONJ
ejpam-6425	100	11	a	a	DET
ejpam-6425	100	12	≤	≤	PROPN
ejpam-6425	100	13	b.	b.	NOUN
ejpam-6425	101	1	we	we	PRON
ejpam-6425	101	2	say	say	VERB
ejpam-6425	101	3	that	that	SCONJ
ejpam-6425	101	4	a	a	DET
ejpam-6425	101	5	divides	divide	NOUN
ejpam-6425	101	6	b	b	NOUN
ejpam-6425	101	7	,	,	PUNCT
ejpam-6425	101	8	written	write	VERB
ejpam-6425	101	9	in	in	ADP
ejpam-6425	101	10	symbols	symbol	NOUN
ejpam-6425	101	11	by	by	ADP
ejpam-6425	101	12	a	a	DET
ejpam-6425	101	13	|	|	NOUN
ejpam-6425	101	14	b	b	NOUN
ejpam-6425	101	15	,	,	PUNCT
ejpam-6425	101	16	if	if	SCONJ
ejpam-6425	101	17	there	there	PRON
ejpam-6425	101	18	is	be	VERB
ejpam-6425	101	19	a	a	DET
ejpam-6425	101	20	positive	positive	ADJ
ejpam-6425	101	21	integer	integer	NOUN
ejpam-6425	101	22	c	c	NOUN
ejpam-6425	101	23	such	such	ADJ
ejpam-6425	101	24	that	that	DET
ejpam-6425	101	25	b	b	X
ejpam-6425	101	26	=	=	SYM
ejpam-6425	101	27	ac	ac	PROPN
ejpam-6425	101	28	.	.	PUNCT
ejpam-6425	102	1	otherwise	otherwise	ADV
ejpam-6425	102	2	,	,	PUNCT
ejpam-6425	102	3	we	we	PRON
ejpam-6425	102	4	say	say	VERB
ejpam-6425	102	5	that	that	SCONJ
ejpam-6425	102	6	a	a	PRON
ejpam-6425	102	7	does	do	AUX
ejpam-6425	102	8	not	not	PART
ejpam-6425	102	9	divide	divide	VERB
ejpam-6425	102	10	b	b	NOUN
ejpam-6425	102	11	,	,	PUNCT
ejpam-6425	102	12	denoted	denote	VERB
ejpam-6425	102	13	by	by	ADP
ejpam-6425	102	14	a	a	DET
ejpam-6425	102	15	∤	∤	PROPN
ejpam-6425	102	16	b.	b.	PROPN
ejpam-6425	103	1	now	now	ADV
ejpam-6425	103	2	,	,	PUNCT
ejpam-6425	103	3	if	if	SCONJ
ejpam-6425	103	4	a	a	DET
ejpam-6425	103	5	|	|	NOUN
ejpam-6425	103	6	b	b	NOUN
ejpam-6425	103	7	,	,	PUNCT
ejpam-6425	103	8	it	it	PRON
ejpam-6425	103	9	also	also	ADV
ejpam-6425	103	10	implies	imply	VERB
ejpam-6425	103	11	the	the	DET
ejpam-6425	103	12	following	following	NOUN
ejpam-6425	103	13	:	:	PUNCT
ejpam-6425	103	14	(	(	PUNCT
ejpam-6425	103	15	i	i	NOUN
ejpam-6425	103	16	)	)	PUNCT
ejpam-6425	103	17	b	b	PROPN
ejpam-6425	103	18	is	be	AUX
ejpam-6425	103	19	a	a	DET
ejpam-6425	103	20	multiple	multiple	NOUN
ejpam-6425	103	21	of	of	ADP
ejpam-6425	103	22	a	a	DET
ejpam-6425	103	23	,	,	PUNCT
ejpam-6425	103	24	(	(	PUNCT
ejpam-6425	103	25	ii	ii	NOUN
ejpam-6425	103	26	)	)	PUNCT
ejpam-6425	103	27	a	a	PRON
ejpam-6425	103	28	is	be	AUX
ejpam-6425	103	29	a	a	DET
ejpam-6425	103	30	divisor	divisor	NOUN
ejpam-6425	103	31	of	of	ADP
ejpam-6425	103	32	b	b	NOUN
ejpam-6425	103	33	,	,	PUNCT
ejpam-6425	103	34	and	and	CCONJ
ejpam-6425	103	35	(	(	PUNCT
ejpam-6425	103	36	iii	iii	X
ejpam-6425	103	37	)	)	PUNCT
ejpam-6425	103	38	a	a	PRON
ejpam-6425	103	39	is	be	AUX
ejpam-6425	103	40	a	a	DET
ejpam-6425	103	41	factor	factor	NOUN
ejpam-6425	103	42	of	of	ADP
ejpam-6425	103	43	b.	b.	PROPN
ejpam-6425	103	44	a.	a.	PROPN
ejpam-6425	103	45	d.	d.	PROPN
ejpam-6425	103	46	c.	c.	PROPN
ejpam-6425	103	47	angel	angel	PROPN
ejpam-6425	103	48	et	et	PROPN
ejpam-6425	103	49	al	al	PROPN
ejpam-6425	103	50	.	.	PUNCT
ejpam-6425	103	51	/	/	SYM
ejpam-6425	103	52	eur	eur	PROPN
ejpam-6425	103	53	.	.	PUNCT
ejpam-6425	104	1	j.	j.	PROPN
ejpam-6425	104	2	pure	pure	PROPN
ejpam-6425	104	3	appl	appl	PROPN
ejpam-6425	104	4	.	.	PROPN
ejpam-6425	104	5	math	math	PROPN
ejpam-6425	104	6	,	,	PUNCT
ejpam-6425	104	7	18	18	NUM
ejpam-6425	104	8	(	(	PUNCT
ejpam-6425	104	9	4	4	NUM
ejpam-6425	104	10	)	)	PUNCT
ejpam-6425	104	11	(	(	PUNCT
ejpam-6425	104	12	2025	2025	NUM
ejpam-6425	104	13	)	)	PUNCT
ejpam-6425	104	14	,	,	PUNCT
ejpam-6425	104	15	6425	6425	NUM
ejpam-6425	104	16	8	8	NUM
ejpam-6425	104	17	of	of	ADP
ejpam-6425	104	18	33	33	NUM
ejpam-6425	104	19	definition	definition	NOUN
ejpam-6425	104	20	7	7	NUM
ejpam-6425	104	21	.	.	PUNCT
ejpam-6425	105	1	let	let	VERB
ejpam-6425	105	2	p	p	PRON
ejpam-6425	105	3	be	be	AUX
ejpam-6425	105	4	a	a	DET
ejpam-6425	105	5	positive	positive	ADJ
ejpam-6425	105	6	integer	integer	NOUN
ejpam-6425	105	7	.	.	PUNCT
ejpam-6425	106	1	also	also	ADV
ejpam-6425	106	2	let	let	VERB
ejpam-6425	106	3	a	a	DET
ejpam-6425	106	4	,	,	PUNCT
ejpam-6425	106	5	b	b	X
ejpam-6425	106	6	∈	∈	PROPN
ejpam-6425	106	7	z.	z.	PROPN
ejpam-6425	107	1	we	we	PRON
ejpam-6425	107	2	can	can	AUX
ejpam-6425	107	3	say	say	VERB
ejpam-6425	107	4	that	that	SCONJ
ejpam-6425	107	5	a	a	PRON
ejpam-6425	107	6	is	be	AUX
ejpam-6425	107	7	congruent	congruent	ADJ
ejpam-6425	107	8	to	to	PART
ejpam-6425	107	9	b(modulo	b(modulo	VERB
ejpam-6425	107	10	p	p	X
ejpam-6425	107	11	)	)	PUNCT
ejpam-6425	107	12	if	if	SCONJ
ejpam-6425	107	13	p	p	PROPN
ejpam-6425	107	14	|	|	ADV
ejpam-6425	107	15	(	(	PUNCT
ejpam-6425	107	16	a−	a−	PROPN
ejpam-6425	107	17	b	b	NOUN
ejpam-6425	107	18	)	)	PUNCT
ejpam-6425	107	19	.	.	PUNCT
ejpam-6425	108	1	that	that	PRON
ejpam-6425	108	2	is	be	AUX
ejpam-6425	108	3	if	if	SCONJ
ejpam-6425	108	4	a	a	DET
ejpam-6425	108	5	=	=	NOUN
ejpam-6425	108	6	b+	b+	NUM
ejpam-6425	108	7	kp	kp	PROPN
ejpam-6425	108	8	for	for	ADP
ejpam-6425	108	9	some	some	DET
ejpam-6425	108	10	k	k	PROPN
ejpam-6425	108	11	∈	∈	PROPN
ejpam-6425	108	12	z.	z.	PROPN
ejpam-6425	108	13	definition	definition	NOUN
ejpam-6425	108	14	8	8	NUM
ejpam-6425	108	15	.	.	PUNCT
ejpam-6425	109	1	general	general	ADJ
ejpam-6425	109	2	quadratic	quadratic	ADJ
ejpam-6425	109	3	diophantine	diophantine	NOUN
ejpam-6425	109	4	equation	equation	NOUN
ejpam-6425	109	5	:	:	PUNCT
ejpam-6425	109	6	diophantine	diophantine	VERB
ejpam-6425	109	7	equations	equation	NOUN
ejpam-6425	109	8	are	be	AUX
ejpam-6425	109	9	equations	equation	NOUN
ejpam-6425	109	10	wherein	wherein	ADJ
ejpam-6425	109	11	only	only	ADV
ejpam-6425	109	12	integer	integer	NOUN
ejpam-6425	109	13	solutions	solution	NOUN
ejpam-6425	109	14	are	be	AUX
ejpam-6425	109	15	allowed	allow	VERB
ejpam-6425	109	16	.	.	PUNCT
ejpam-6425	110	1	a	a	DET
ejpam-6425	110	2	general	general	ADJ
ejpam-6425	110	3	quadratic	quadratic	ADJ
ejpam-6425	110	4	diophantine	diophantine	NOUN
ejpam-6425	110	5	equation	equation	NOUN
ejpam-6425	110	6	is	be	AUX
ejpam-6425	110	7	an	an	DET
ejpam-6425	110	8	equation	equation	NOUN
ejpam-6425	110	9	of	of	ADP
ejpam-6425	110	10	the	the	DET
ejpam-6425	110	11	form	form	NOUN
ejpam-6425	110	12	ax2	ax2	NOUN
ejpam-6425	110	13	+	+	CCONJ
ejpam-6425	110	14	bxy	bxy	NOUN
ejpam-6425	110	15	+	+	CCONJ
ejpam-6425	110	16	cy2	cy2	ADJ
ejpam-6425	110	17	+	+	CCONJ
ejpam-6425	110	18	dx+	dx+	ADJ
ejpam-6425	110	19	ey	ey	ADJ
ejpam-6425	110	20	+	+	CCONJ
ejpam-6425	110	21	f	f	X
ejpam-6425	110	22	=	=	SYM
ejpam-6425	110	23	0	0	PROPN
ejpam-6425	110	24	,	,	PUNCT
ejpam-6425	110	25	where	where	SCONJ
ejpam-6425	110	26	x	x	PUNCT
ejpam-6425	110	27	and	and	CCONJ
ejpam-6425	110	28	y	y	PROPN
ejpam-6425	110	29	are	be	AUX
ejpam-6425	110	30	both	both	DET
ejpam-6425	110	31	integers	integer	NOUN
ejpam-6425	110	32	.	.	PUNCT
ejpam-6425	111	1	2.3	2.3	NUM
ejpam-6425	111	2	.	.	PUNCT
ejpam-6425	112	1	essential	essential	ADJ
ejpam-6425	112	2	results	result	NOUN
ejpam-6425	112	3	used	use	VERB
ejpam-6425	112	4	in	in	ADP
ejpam-6425	112	5	this	this	DET
ejpam-6425	112	6	paper	paper	NOUN
ejpam-6425	112	7	this	this	DET
ejpam-6425	112	8	subsection	subsection	NOUN
ejpam-6425	112	9	discusses	discuss	VERB
ejpam-6425	112	10	the	the	DET
ejpam-6425	112	11	lo	lo	PROPN
ejpam-6425	112	12	’s	’s	PART
ejpam-6425	112	13	theorem	theorem	NOUN
ejpam-6425	112	14	,	,	PUNCT
ejpam-6425	112	15	which	which	PRON
ejpam-6425	112	16	is	be	AUX
ejpam-6425	112	17	the	the	DET
ejpam-6425	112	18	main	main	ADJ
ejpam-6425	112	19	theorem	theorem	NOUN
ejpam-6425	112	20	that	that	PRON
ejpam-6425	112	21	we	we	PRON
ejpam-6425	112	22	will	will	AUX
ejpam-6425	112	23	use	use	VERB
ejpam-6425	112	24	to	to	PART
ejpam-6425	112	25	obtain	obtain	VERB
ejpam-6425	112	26	the	the	DET
ejpam-6425	112	27	results	result	NOUN
ejpam-6425	112	28	of	of	ADP
ejpam-6425	112	29	this	this	DET
ejpam-6425	112	30	paper	paper	NOUN
ejpam-6425	112	31	.	.	PUNCT
ejpam-6425	113	1	furthermore	furthermore	ADV
ejpam-6425	113	2	,	,	PUNCT
ejpam-6425	113	3	some	some	DET
ejpam-6425	113	4	essential	essential	ADJ
ejpam-6425	113	5	transformations	transformation	NOUN
ejpam-6425	113	6	of	of	ADP
ejpam-6425	113	7	quadratic	quadratic	ADJ
ejpam-6425	113	8	diophantine	diophantine	NOUN
ejpam-6425	113	9	equations	equation	NOUN
ejpam-6425	113	10	are	be	AUX
ejpam-6425	113	11	also	also	ADV
ejpam-6425	113	12	discussed	discuss	VERB
ejpam-6425	113	13	in	in	ADP
ejpam-6425	113	14	this	this	DET
ejpam-6425	113	15	section	section	NOUN
ejpam-6425	113	16	.	.	PUNCT
ejpam-6425	114	1	theorem	theorem	NOUN
ejpam-6425	114	2	1	1	NUM
ejpam-6425	114	3	.	.	PUNCT
ejpam-6425	115	1	lo	lo	PROPN
ejpam-6425	115	2	’s	’s	PART
ejpam-6425	115	3	theorem	theorem	NOUN
ejpam-6425	115	4	[	[	X
ejpam-6425	115	5	4	4	NUM
ejpam-6425	115	6	]	]	PUNCT
ejpam-6425	115	7	:	:	PUNCT
ejpam-6425	115	8	if	if	SCONJ
ejpam-6425	115	9	a	a	DET
ejpam-6425	115	10	graph	graph	NOUN
ejpam-6425	115	11	g	g	NOUN
ejpam-6425	115	12	of	of	ADP
ejpam-6425	115	13	p	p	NOUN
ejpam-6425	115	14	vertices	vertex	NOUN
ejpam-6425	115	15	and	and	CCONJ
ejpam-6425	115	16	q	q	NOUN
ejpam-6425	115	17	edges	edge	NOUN
ejpam-6425	115	18	is	be	AUX
ejpam-6425	115	19	edge	edge	NOUN
ejpam-6425	115	20	-	-	PUNCT
ejpam-6425	115	21	graceful	graceful	NOUN
ejpam-6425	115	22	,	,	PUNCT
ejpam-6425	115	23	then	then	ADV
ejpam-6425	115	24	p	p	X
ejpam-6425	115	25	∣∣∣(q2	∣∣∣(q2	PROPN
ejpam-6425	115	26	+	+	CCONJ
ejpam-6425	115	27	q	q	NOUN
ejpam-6425	115	28	−	−	NOUN
ejpam-6425	115	29	p(p−	p(p−	VERB
ejpam-6425	115	30	1	1	NUM
ejpam-6425	115	31	)	)	PUNCT
ejpam-6425	115	32	2	2	NUM
ejpam-6425	115	33	)	)	PUNCT
ejpam-6425	115	34	.	.	PUNCT
ejpam-6425	116	1	a	a	DET
ejpam-6425	116	2	complete	complete	ADJ
ejpam-6425	116	3	proof	proof	NOUN
ejpam-6425	116	4	of	of	ADP
ejpam-6425	116	5	theorem	theorem	ADJ
ejpam-6425	116	6	1(lo	1(lo	PROPN
ejpam-6425	116	7	’s	’s	PART
ejpam-6425	116	8	theorem	theorem	VERB
ejpam-6425	116	9	)	)	PUNCT
ejpam-6425	116	10	was	be	AUX
ejpam-6425	116	11	provided	provide	VERB
ejpam-6425	116	12	in	in	ADP
ejpam-6425	116	13	[	[	X
ejpam-6425	116	14	14	14	NUM
ejpam-6425	116	15	]	]	PUNCT
ejpam-6425	116	16	.	.	PUNCT
ejpam-6425	117	1	lemma	lemma	PROPN
ejpam-6425	117	2	1	1	NUM
ejpam-6425	117	3	.	.	PUNCT
ejpam-6425	118	1	[	[	X
ejpam-6425	118	2	15]the	15]the	DET
ejpam-6425	118	3	quadratic	quadratic	ADJ
ejpam-6425	118	4	diophantine	diophantine	NOUN
ejpam-6425	118	5	equation	equation	NOUN
ejpam-6425	118	6	ax2	ax2	NOUN
ejpam-6425	118	7	+	+	CCONJ
ejpam-6425	118	8	bxy	bxy	NOUN
ejpam-6425	118	9	+	+	CCONJ
ejpam-6425	118	10	cy2	cy2	ADJ
ejpam-6425	118	11	+	+	CCONJ
ejpam-6425	118	12	dx+	dx+	ADJ
ejpam-6425	118	13	ey	ey	ADJ
ejpam-6425	118	14	+	+	CCONJ
ejpam-6425	118	15	f	f	X
ejpam-6425	118	16	=	=	SYM
ejpam-6425	118	17	0	0	PROPN
ejpam-6425	118	18	,	,	PUNCT
ejpam-6425	118	19	(	(	PUNCT
ejpam-6425	118	20	1	1	X
ejpam-6425	118	21	)	)	PUNCT
ejpam-6425	118	22	where	where	SCONJ
ejpam-6425	118	23	a	a	DET
ejpam-6425	118	24	,	,	PUNCT
ejpam-6425	118	25	b	b	NOUN
ejpam-6425	118	26	,	,	PUNCT
ejpam-6425	118	27	c	c	NOUN
ejpam-6425	118	28	,	,	PUNCT
ejpam-6425	118	29	d	d	NOUN
ejpam-6425	118	30	,	,	PUNCT
ejpam-6425	118	31	e	e	NOUN
ejpam-6425	118	32	,	,	PUNCT
ejpam-6425	118	33	and	and	CCONJ
ejpam-6425	118	34	f	f	PROPN
ejpam-6425	118	35	are	be	AUX
ejpam-6425	118	36	integer	integer	NOUN
ejpam-6425	118	37	coefficients	coefficient	NOUN
ejpam-6425	118	38	,	,	PUNCT
ejpam-6425	118	39	and	and	CCONJ
ejpam-6425	118	40	x	x	AUX
ejpam-6425	118	41	and	and	CCONJ
ejpam-6425	118	42	y	y	PROPN
ejpam-6425	118	43	are	be	AUX
ejpam-6425	118	44	the	the	DET
ejpam-6425	118	45	unknown	unknown	ADJ
ejpam-6425	118	46	variables	variable	NOUN
ejpam-6425	118	47	can	can	AUX
ejpam-6425	118	48	be	be	AUX
ejpam-6425	118	49	reduced	reduce	VERB
ejpam-6425	118	50	to	to	ADP
ejpam-6425	118	51	x2	x2	PROPN
ejpam-6425	118	52	−dy	−dy	ADJ
ejpam-6425	118	53	2	2	NUM
ejpam-6425	118	54	=	=	SYM
ejpam-6425	118	55	n	n	X
ejpam-6425	118	56	,	,	PUNCT
ejpam-6425	118	57	(	(	PUNCT
ejpam-6425	118	58	2	2	X
ejpam-6425	118	59	)	)	PUNCT
ejpam-6425	118	60	where	where	SCONJ
ejpam-6425	118	61	x	x	X
ejpam-6425	118	62	=	=	PUNCT
ejpam-6425	118	63	dy	dy	NOUN
ejpam-6425	118	64	+	+	CCONJ
ejpam-6425	118	65	e	e	NOUN
ejpam-6425	118	66	,	,	PUNCT
ejpam-6425	118	67	y	y	PROPN
ejpam-6425	118	68	=	=	SYM
ejpam-6425	118	69	2ax+	2ax+	NUM
ejpam-6425	118	70	by	by	ADP
ejpam-6425	118	71	+	+	CCONJ
ejpam-6425	118	72	d	d	NOUN
ejpam-6425	118	73	,	,	PUNCT
ejpam-6425	118	74	and	and	CCONJ
ejpam-6425	118	75	n	n	PROPN
ejpam-6425	118	76	=	=	PROPN
ejpam-6425	118	77	e2	e2	PROPN
ejpam-6425	118	78	−df	−df	PROPN
ejpam-6425	118	79	.	.	PUNCT
ejpam-6425	119	1	note	note	VERB
ejpam-6425	119	2	that	that	SCONJ
ejpam-6425	119	3	in	in	ADP
ejpam-6425	119	4	the	the	DET
ejpam-6425	119	5	above	above	ADJ
ejpam-6425	119	6	transformation	transformation	NOUN
ejpam-6425	119	7	,	,	PUNCT
ejpam-6425	119	8	d	d	NOUN
ejpam-6425	119	9	=	=	PUNCT
ejpam-6425	119	10	b2	b2	PROPN
ejpam-6425	119	11	−	−	NOUN
ejpam-6425	119	12	4ac	4ac	NOUN
ejpam-6425	119	13	,	,	PUNCT
ejpam-6425	119	14	e	e	X
ejpam-6425	119	15	=	=	SYM
ejpam-6425	119	16	bd	bd	PROPN
ejpam-6425	119	17	−	−	PROPN
ejpam-6425	119	18	2ae	2ae	NOUN
ejpam-6425	119	19	,	,	PUNCT
ejpam-6425	119	20	and	and	CCONJ
ejpam-6425	119	21	f	f	PROPN
ejpam-6425	119	22	=	=	PROPN
ejpam-6425	119	23	d2	d2	PROPN
ejpam-6425	119	24	−	−	PROPN
ejpam-6425	119	25	4af	4af	PROPN
ejpam-6425	119	26	.	.	PUNCT
ejpam-6425	120	1	moreover	moreover	ADV
ejpam-6425	120	2	,	,	PUNCT
ejpam-6425	120	3	observe	observe	VERB
ejpam-6425	120	4	that	that	SCONJ
ejpam-6425	120	5	if	if	SCONJ
ejpam-6425	120	6	x	x	PRON
ejpam-6425	120	7	and	and	CCONJ
ejpam-6425	120	8	y	y	PROPN
ejpam-6425	120	9	is	be	AUX
ejpam-6425	120	10	a	a	DET
ejpam-6425	120	11	solution	solution	NOUN
ejpam-6425	120	12	to	to	ADP
ejpam-6425	120	13	equation	equation	NOUN
ejpam-6425	120	14	(	(	PUNCT
ejpam-6425	120	15	2	2	NUM
ejpam-6425	120	16	)	)	PUNCT
ejpam-6425	120	17	,	,	PUNCT
ejpam-6425	120	18	then	then	ADV
ejpam-6425	120	19	there	there	PRON
ejpam-6425	120	20	are	be	VERB
ejpam-6425	120	21	integers	integer	NOUN
ejpam-6425	120	22	x	x	PUNCT
ejpam-6425	120	23	and	and	CCONJ
ejpam-6425	120	24	y	y	PROPN
ejpam-6425	120	25	,	,	PUNCT
ejpam-6425	120	26	such	such	ADJ
ejpam-6425	120	27	that	that	SCONJ
ejpam-6425	120	28	x	x	X
ejpam-6425	120	29	=	=	PUNCT
ejpam-6425	120	30	y	y	PROPN
ejpam-6425	120	31	−	−	PROPN
ejpam-6425	120	32	by	by	ADP
ejpam-6425	120	33	−	−	PROPN
ejpam-6425	120	34	d	d	PROPN
ejpam-6425	120	35	2a	2a	NUM
ejpam-6425	120	36	and	and	CCONJ
ejpam-6425	120	37	y	y	NOUN
ejpam-6425	121	1	=	=	PUNCT
ejpam-6425	121	2	x	x	PUNCT
ejpam-6425	121	3	−	−	NOUN
ejpam-6425	121	4	e	e	X
ejpam-6425	121	5	d	d	PROPN
ejpam-6425	121	6	,	,	PUNCT
ejpam-6425	121	7	(	(	PUNCT
ejpam-6425	121	8	3	3	X
ejpam-6425	121	9	)	)	PUNCT
ejpam-6425	121	10	where	where	SCONJ
ejpam-6425	121	11	x	x	PUNCT
ejpam-6425	121	12	and	and	CCONJ
ejpam-6425	121	13	y	y	PROPN
ejpam-6425	121	14	in	in	ADP
ejpam-6425	121	15	equation	equation	NOUN
ejpam-6425	121	16	(	(	PUNCT
ejpam-6425	121	17	3	3	X
ejpam-6425	121	18	)	)	PUNCT
ejpam-6425	121	19	are	be	AUX
ejpam-6425	121	20	the	the	DET
ejpam-6425	121	21	integer	integer	NOUN
ejpam-6425	121	22	solutions	solution	NOUN
ejpam-6425	121	23	for	for	ADP
ejpam-6425	121	24	equation	equation	NOUN
ejpam-6425	121	25	(	(	PUNCT
ejpam-6425	121	26	1	1	NUM
ejpam-6425	121	27	)	)	PUNCT
ejpam-6425	121	28	.	.	PUNCT
ejpam-6425	122	1	remark	remark	NOUN
ejpam-6425	122	2	2	2	NUM
ejpam-6425	122	3	.	.	PUNCT
ejpam-6425	123	1	the	the	DET
ejpam-6425	123	2	diophantine	diophantine	NOUN
ejpam-6425	123	3	equations	equation	NOUN
ejpam-6425	123	4	involved	involve	VERB
ejpam-6425	123	5	in	in	ADP
ejpam-6425	123	6	this	this	DET
ejpam-6425	123	7	study	study	NOUN
ejpam-6425	123	8	have	have	VERB
ejpam-6425	123	9	c	c	NOUN
ejpam-6425	123	10	=	=	SYM
ejpam-6425	123	11	0	0	NUM
ejpam-6425	123	12	.	.	PUNCT
ejpam-6425	124	1	thus	thus	ADV
ejpam-6425	124	2	,	,	PUNCT
ejpam-6425	124	3	equation	equation	NOUN
ejpam-6425	124	4	(	(	PUNCT
ejpam-6425	124	5	2	2	X
ejpam-6425	124	6	)	)	PUNCT
ejpam-6425	124	7	will	will	AUX
ejpam-6425	124	8	just	just	ADV
ejpam-6425	124	9	be	be	AUX
ejpam-6425	124	10	written	write	VERB
ejpam-6425	124	11	as	as	ADP
ejpam-6425	124	12	x2	x2	PROPN
ejpam-6425	124	13	−	−	PROPN
ejpam-6425	124	14	(	(	PUNCT
ejpam-6425	124	15	by	by	ADP
ejpam-6425	124	16	)	)	PUNCT
ejpam-6425	124	17	2	2	NUM
ejpam-6425	124	18	=	=	SYM
ejpam-6425	124	19	n	n	X
ejpam-6425	124	20	(	(	PUNCT
ejpam-6425	124	21	4	4	X
ejpam-6425	124	22	)	)	PUNCT
ejpam-6425	124	23	observe	observe	VERB
ejpam-6425	124	24	that	that	SCONJ
ejpam-6425	124	25	we	we	PRON
ejpam-6425	124	26	can	can	AUX
ejpam-6425	124	27	factor	factor	VERB
ejpam-6425	124	28	the	the	DET
ejpam-6425	124	29	left	left	ADJ
ejpam-6425	124	30	side	side	NOUN
ejpam-6425	124	31	of	of	ADP
ejpam-6425	124	32	equation	equation	NOUN
ejpam-6425	124	33	(	(	PUNCT
ejpam-6425	124	34	4	4	NUM
ejpam-6425	124	35	)	)	PUNCT
ejpam-6425	124	36	as	as	ADP
ejpam-6425	124	37	(	(	PUNCT
ejpam-6425	124	38	x	x	X
ejpam-6425	124	39	+	+	CCONJ
ejpam-6425	124	40	by	by	ADP
ejpam-6425	124	41	)	)	PUNCT
ejpam-6425	124	42	(	(	PUNCT
ejpam-6425	124	43	x	x	X
ejpam-6425	124	44	−	−	NOUN
ejpam-6425	124	45	by	by	ADP
ejpam-6425	124	46	)	)	PUNCT
ejpam-6425	124	47	.	.	PUNCT
ejpam-6425	125	1	moreover	moreover	ADV
ejpam-6425	125	2	,	,	PUNCT
ejpam-6425	125	3	if	if	SCONJ
ejpam-6425	125	4	the	the	DET
ejpam-6425	125	5	pair	pair	NOUN
ejpam-6425	125	6	n1	n1	NOUN
ejpam-6425	125	7	and	and	CCONJ
ejpam-6425	125	8	n2	n2	NOUN
ejpam-6425	125	9	is	be	AUX
ejpam-6425	125	10	a	a	DET
ejpam-6425	125	11	factor	factor	NOUN
ejpam-6425	125	12	pair	pair	NOUN
ejpam-6425	125	13	of	of	ADP
ejpam-6425	125	14	n	n	PROPN
ejpam-6425	125	15	,	,	PUNCT
ejpam-6425	125	16	then	then	ADV
ejpam-6425	125	17	we	we	PRON
ejpam-6425	125	18	will	will	AUX
ejpam-6425	125	19	have	have	VERB
ejpam-6425	125	20	linear	linear	ADJ
ejpam-6425	125	21	system	system	NOUN
ejpam-6425	125	22	of	of	ADP
ejpam-6425	125	23	equation	equation	NOUN
ejpam-6425	125	24	,	,	PUNCT
ejpam-6425	125	25	(	(	PUNCT
ejpam-6425	125	26	x	x	X
ejpam-6425	125	27	+	+	CCONJ
ejpam-6425	125	28	by	by	ADP
ejpam-6425	125	29	)	)	PUNCT
ejpam-6425	125	30	=	=	SYM
ejpam-6425	125	31	n1	n1	PROPN
ejpam-6425	125	32	and	and	CCONJ
ejpam-6425	125	33	(	(	PUNCT
ejpam-6425	125	34	x	x	X
ejpam-6425	125	35	−	−	NOUN
ejpam-6425	125	36	by	by	ADP
ejpam-6425	125	37	)	)	PUNCT
ejpam-6425	125	38	=	=	SYM
ejpam-6425	125	39	n2	n2	PROPN
ejpam-6425	125	40	.	.	PUNCT
ejpam-6425	126	1	now	now	ADV
ejpam-6425	126	2	,	,	PUNCT
ejpam-6425	126	3	solving	solve	VERB
ejpam-6425	126	4	for	for	ADP
ejpam-6425	126	5	x	x	PUNCT
ejpam-6425	126	6	and	and	CCONJ
ejpam-6425	126	7	y	y	PROPN
ejpam-6425	126	8	,	,	PUNCT
ejpam-6425	126	9	we	we	PRON
ejpam-6425	126	10	will	will	AUX
ejpam-6425	126	11	have	have	VERB
ejpam-6425	126	12	,	,	PUNCT
ejpam-6425	126	13	x	x	PUNCT
ejpam-6425	126	14	=	=	SYM
ejpam-6425	126	15	n1	n1	PROPN
ejpam-6425	126	16	+	+	NOUN
ejpam-6425	126	17	n2	n2	ADJ
ejpam-6425	126	18	2	2	NUM
ejpam-6425	126	19	and	and	CCONJ
ejpam-6425	126	20	y	y	NOUN
ejpam-6425	126	21	=	=	SYM
ejpam-6425	126	22	n1	n1	PROPN
ejpam-6425	126	23	−n2	−n2	PROPN
ejpam-6425	126	24	2b	2b	NUM
ejpam-6425	126	25	a.	a.	PROPN
ejpam-6425	126	26	d.	d.	PROPN
ejpam-6425	126	27	c.	c.	PROPN
ejpam-6425	126	28	angel	angel	PROPN
ejpam-6425	126	29	et	et	PROPN
ejpam-6425	126	30	al	al	PROPN
ejpam-6425	126	31	.	.	PUNCT
ejpam-6425	126	32	/	/	SYM
ejpam-6425	126	33	eur	eur	PROPN
ejpam-6425	126	34	.	.	PUNCT
ejpam-6425	127	1	j.	j.	PROPN
ejpam-6425	127	2	pure	pure	PROPN
ejpam-6425	127	3	appl	appl	PROPN
ejpam-6425	127	4	.	.	PROPN
ejpam-6425	127	5	math	math	PROPN
ejpam-6425	127	6	,	,	PUNCT
ejpam-6425	127	7	18	18	NUM
ejpam-6425	127	8	(	(	PUNCT
ejpam-6425	127	9	4	4	NUM
ejpam-6425	127	10	)	)	PUNCT
ejpam-6425	127	11	(	(	PUNCT
ejpam-6425	127	12	2025	2025	NUM
ejpam-6425	127	13	)	)	PUNCT
ejpam-6425	127	14	,	,	PUNCT
ejpam-6425	127	15	6425	6425	NUM
ejpam-6425	127	16	9	9	NUM
ejpam-6425	127	17	of	of	ADP
ejpam-6425	127	18	33	33	NUM
ejpam-6425	127	19	having	having	AUX
ejpam-6425	127	20	discussed	discuss	VERB
ejpam-6425	127	21	all	all	DET
ejpam-6425	127	22	preliminary	preliminary	ADJ
ejpam-6425	127	23	information	information	NOUN
ejpam-6425	127	24	,	,	PUNCT
ejpam-6425	127	25	we	we	PRON
ejpam-6425	127	26	are	be	AUX
ejpam-6425	127	27	now	now	ADV
ejpam-6425	127	28	ready	ready	ADJ
ejpam-6425	127	29	to	to	PART
ejpam-6425	127	30	present	present	VERB
ejpam-6425	127	31	the	the	DET
ejpam-6425	127	32	results	result	NOUN
ejpam-6425	127	33	of	of	ADP
ejpam-6425	127	34	this	this	DET
ejpam-6425	127	35	study	study	NOUN
ejpam-6425	127	36	.	.	PUNCT
ejpam-6425	128	1	3	3	X
ejpam-6425	128	2	.	.	X
ejpam-6425	128	3	results	result	NOUN
ejpam-6425	128	4	we	we	PRON
ejpam-6425	128	5	begin	begin	VERB
ejpam-6425	128	6	the	the	DET
ejpam-6425	128	7	section	section	NOUN
ejpam-6425	128	8	by	by	ADP
ejpam-6425	128	9	providing	provide	VERB
ejpam-6425	128	10	results	result	NOUN
ejpam-6425	128	11	on	on	ADP
ejpam-6425	128	12	the	the	DET
ejpam-6425	128	13	non	non	ADJ
ejpam-6425	128	14	-	-	ADJ
ejpam-6425	128	15	edge	edge	ADJ
ejpam-6425	128	16	-	-	PUNCT
ejpam-6425	128	17	gracefulness	gracefulness	NOUN
ejpam-6425	128	18	of	of	ADP
ejpam-6425	128	19	water	water	NOUN
ejpam-6425	128	20	wheel	wheel	NOUN
ejpam-6425	128	21	,	,	PUNCT
ejpam-6425	128	22	right	right	ADJ
ejpam-6425	128	23	water	water	NOUN
ejpam-6425	128	24	wheel	wheel	NOUN
ejpam-6425	128	25	,	,	PUNCT
ejpam-6425	128	26	and	and	CCONJ
ejpam-6425	128	27	left	leave	VERB
ejpam-6425	128	28	water	water	NOUN
ejpam-6425	128	29	wheel	wheel	NOUN
ejpam-6425	128	30	graphs	graph	NOUN
ejpam-6425	128	31	.	.	PUNCT
ejpam-6425	129	1	3.1	3.1	NUM
ejpam-6425	129	2	.	.	PUNCT
ejpam-6425	129	3	non	non	ADJ
ejpam-6425	129	4	-	-	ADJ
ejpam-6425	129	5	edge	edge	ADJ
ejpam-6425	129	6	-	-	PUNCT
ejpam-6425	129	7	gracefulness	gracefulness	NOUN
ejpam-6425	129	8	of	of	ADP
ejpam-6425	129	9	water	water	NOUN
ejpam-6425	129	10	wheel	wheel	NOUN
ejpam-6425	129	11	graphs	graph	VERB
ejpam-6425	129	12	wwn	wwn	NOUN
ejpam-6425	129	13	,	,	PUNCT
ejpam-6425	129	14	right	right	ADJ
ejpam-6425	129	15	water	water	NOUN
ejpam-6425	129	16	wheel	wheel	NOUN
ejpam-6425	129	17	graphs	graph	NOUN
ejpam-6425	129	18	rwwn	rwwn	NOUN
ejpam-6425	129	19	,	,	PUNCT
ejpam-6425	129	20	and	and	CCONJ
ejpam-6425	129	21	left	leave	VERB
ejpam-6425	129	22	water	water	NOUN
ejpam-6425	129	23	wheel	wheel	NOUN
ejpam-6425	129	24	graphs	graph	NOUN
ejpam-6425	129	25	lwwn	lwwn	NOUN
ejpam-6425	129	26	.	.	PUNCT
ejpam-6425	130	1	theorem	theorem	NOUN
ejpam-6425	130	2	2	2	NUM
ejpam-6425	130	3	.	.	X
ejpam-6425	130	4	for	for	ADP
ejpam-6425	130	5	n	n	PRON
ejpam-6425	130	6	≥	≥	NUM
ejpam-6425	130	7	3	3	NUM
ejpam-6425	130	8	,	,	PUNCT
ejpam-6425	130	9	water	water	NOUN
ejpam-6425	130	10	wheel	wheel	NOUN
ejpam-6425	130	11	graph	graph	NOUN
ejpam-6425	130	12	wwn	wwn	NOUN
ejpam-6425	130	13	is	be	AUX
ejpam-6425	130	14	not	not	PART
ejpam-6425	130	15	edge	edge	NOUN
ejpam-6425	130	16	-	-	PUNCT
ejpam-6425	130	17	graceful	graceful	NOUN
ejpam-6425	130	18	.	.	PUNCT
ejpam-6425	131	1	proof	proof	NOUN
ejpam-6425	131	2	.	.	PUNCT
ejpam-6425	132	1	first	first	ADV
ejpam-6425	132	2	,	,	PUNCT
ejpam-6425	132	3	recall	recall	VERB
ejpam-6425	132	4	from	from	ADP
ejpam-6425	132	5	definition	definition	NOUN
ejpam-6425	132	6	2	2	NUM
ejpam-6425	132	7	that	that	PRON
ejpam-6425	132	8	water	water	NOUN
ejpam-6425	132	9	wheel	wheel	NOUN
ejpam-6425	132	10	graph	graph	NOUN
ejpam-6425	132	11	wwn	wwn	PROPN
ejpam-6425	132	12	has	have	VERB
ejpam-6425	132	13	3n+1	3n+1	PROPN
ejpam-6425	132	14	vertices	vertex	NOUN
ejpam-6425	132	15	and	and	CCONJ
ejpam-6425	132	16	5n	5n	NUM
ejpam-6425	132	17	edges	edge	NOUN
ejpam-6425	132	18	.	.	PUNCT
ejpam-6425	133	1	using	use	VERB
ejpam-6425	133	2	theorem	theorem	ADJ
ejpam-6425	133	3	1(lo	1(lo	PROPN
ejpam-6425	133	4	’s	’s	PART
ejpam-6425	133	5	theorem	theorem	NOUN
ejpam-6425	133	6	)	)	PUNCT
ejpam-6425	133	7	,	,	PUNCT
ejpam-6425	133	8	if	if	SCONJ
ejpam-6425	133	9	a	a	DET
ejpam-6425	133	10	graph	graph	NOUN
ejpam-6425	133	11	with	with	ADP
ejpam-6425	133	12	p	p	NOUN
ejpam-6425	133	13	vertices	vertex	NOUN
ejpam-6425	133	14	and	and	CCONJ
ejpam-6425	133	15	q	q	NOUN
ejpam-6425	133	16	edges	edge	NOUN
ejpam-6425	133	17	is	be	AUX
ejpam-6425	133	18	edge	edge	NOUN
ejpam-6425	133	19	-	-	PUNCT
ejpam-6425	133	20	graceful	graceful	NOUN
ejpam-6425	133	21	,	,	PUNCT
ejpam-6425	133	22	then	then	ADV
ejpam-6425	133	23	p	p	X
ejpam-6425	133	24	∣∣∣(q2	∣∣∣(q2	PROPN
ejpam-6425	133	25	+	+	CCONJ
ejpam-6425	133	26	q	q	NOUN
ejpam-6425	133	27	−	−	NOUN
ejpam-6425	133	28	p(p−	p(p−	VERB
ejpam-6425	133	29	1	1	NUM
ejpam-6425	133	30	)	)	PUNCT
ejpam-6425	133	31	2	2	NUM
ejpam-6425	133	32	)	)	PUNCT
ejpam-6425	133	33	.	.	PUNCT
ejpam-6425	134	1	(	(	PUNCT
ejpam-6425	134	2	5	5	NUM
ejpam-6425	134	3	)	)	PUNCT
ejpam-6425	134	4	thus	thus	ADV
ejpam-6425	134	5	,	,	PUNCT
ejpam-6425	134	6	we	we	PRON
ejpam-6425	134	7	substitute	substitute	VERB
ejpam-6425	134	8	p	p	NOUN
ejpam-6425	135	1	=	=	PROPN
ejpam-6425	135	2	3n+	3n+	NUM
ejpam-6425	135	3	1	1	NUM
ejpam-6425	135	4	and	and	CCONJ
ejpam-6425	135	5	q	q	NOUN
ejpam-6425	135	6	=	=	NOUN
ejpam-6425	135	7	5n	5n	NOUN
ejpam-6425	135	8	in	in	ADP
ejpam-6425	135	9	equation	equation	NOUN
ejpam-6425	135	10	(	(	PUNCT
ejpam-6425	135	11	5	5	NUM
ejpam-6425	135	12	)	)	PUNCT
ejpam-6425	135	13	.	.	PUNCT
ejpam-6425	136	1	that	that	PRON
ejpam-6425	136	2	is	be	AUX
ejpam-6425	136	3	,	,	PUNCT
ejpam-6425	136	4	(	(	PUNCT
ejpam-6425	136	5	3n+	3n+	NUM
ejpam-6425	136	6	1	1	NUM
ejpam-6425	136	7	)	)	PUNCT
ejpam-6425	137	1	∣∣∣((5n)2	∣∣∣((5n)2	PROPN
ejpam-6425	138	1	+	+	PUNCT
ejpam-6425	138	2	5n−	5n−	NUM
ejpam-6425	138	3	(	(	PUNCT
ejpam-6425	138	4	3n+	3n+	NUM
ejpam-6425	138	5	1)(3n	1)(3n	NUM
ejpam-6425	138	6	)	)	PUNCT
ejpam-6425	138	7	2	2	NUM
ejpam-6425	138	8	)	)	PUNCT
ejpam-6425	138	9	,	,	PUNCT
ejpam-6425	138	10	which	which	PRON
ejpam-6425	138	11	further	far	ADV
ejpam-6425	138	12	implies	imply	VERB
ejpam-6425	138	13	that	that	SCONJ
ejpam-6425	138	14	,	,	PUNCT
ejpam-6425	138	15	(	(	PUNCT
ejpam-6425	138	16	41n2	41n2	NUM
ejpam-6425	138	17	+	+	NOUN
ejpam-6425	138	18	7n	7n	NUM
ejpam-6425	138	19	6n+	6n+	NUM
ejpam-6425	138	20	2	2	NUM
ejpam-6425	138	21	)	)	PUNCT
ejpam-6425	138	22	∈	∈	PROPN
ejpam-6425	138	23	z.	z.	PROPN
ejpam-6425	139	1	this	this	PRON
ejpam-6425	139	2	means	mean	VERB
ejpam-6425	139	3	,	,	PUNCT
ejpam-6425	139	4	41n2	41n2	NUM
ejpam-6425	139	5	+	+	NOUN
ejpam-6425	139	6	7n	7n	NOUN
ejpam-6425	139	7	=	=	SYM
ejpam-6425	139	8	(	(	PUNCT
ejpam-6425	139	9	6n+	6n+	NUM
ejpam-6425	139	10	2)k	2)k	NOUN
ejpam-6425	139	11	=	=	SYM
ejpam-6425	139	12	6nk	6nk	NOUN
ejpam-6425	139	13	+	+	CCONJ
ejpam-6425	139	14	2k	2k	NUM
ejpam-6425	139	15	,	,	PUNCT
ejpam-6425	139	16	for	for	ADP
ejpam-6425	139	17	some	some	DET
ejpam-6425	139	18	k	k	PROPN
ejpam-6425	139	19	∈	∈	PROPN
ejpam-6425	139	20	z.	z.	PROPN
ejpam-6425	140	1	thus	thus	ADV
ejpam-6425	140	2	,	,	PUNCT
ejpam-6425	140	3	we	we	PRON
ejpam-6425	140	4	solve	solve	VERB
ejpam-6425	140	5	for	for	ADP
ejpam-6425	140	6	all	all	DET
ejpam-6425	140	7	the	the	DET
ejpam-6425	140	8	possible	possible	ADJ
ejpam-6425	140	9	integer	integer	NOUN
ejpam-6425	140	10	values	value	NOUN
ejpam-6425	140	11	of	of	ADP
ejpam-6425	140	12	n	n	PRON
ejpam-6425	140	13	and	and	CCONJ
ejpam-6425	140	14	k	k	X
ejpam-6425	140	15	in	in	ADP
ejpam-6425	140	16	the	the	DET
ejpam-6425	140	17	diophantine	diophantine	NOUN
ejpam-6425	140	18	equation	equation	NOUN
ejpam-6425	140	19	,	,	PUNCT
ejpam-6425	140	20	41n2	41n2	NUM
ejpam-6425	140	21	+	+	SYM
ejpam-6425	140	22	7n−	7n−	NUM
ejpam-6425	140	23	6nk	6nk	NOUN
ejpam-6425	140	24	−	−	NOUN
ejpam-6425	140	25	2k	2k	NOUN
ejpam-6425	140	26	=	=	SYM
ejpam-6425	140	27	0	0	NUM
ejpam-6425	140	28	,	,	PUNCT
ejpam-6425	140	29	(	(	PUNCT
ejpam-6425	140	30	6	6	NUM
ejpam-6425	140	31	)	)	PUNCT
ejpam-6425	140	32	where	where	SCONJ
ejpam-6425	140	33	n	n	X
ejpam-6425	140	34	,	,	PUNCT
ejpam-6425	140	35	k	k	PROPN
ejpam-6425	140	36	∈	∈	PROPN
ejpam-6425	140	37	z.	z.	PROPN
ejpam-6425	141	1	this	this	DET
ejpam-6425	141	2	diophantine	diophantine	NOUN
ejpam-6425	141	3	equation	equation	NOUN
ejpam-6425	141	4	is	be	AUX
ejpam-6425	141	5	of	of	ADP
ejpam-6425	141	6	the	the	DET
ejpam-6425	141	7	form	form	NOUN
ejpam-6425	141	8	,	,	PUNCT
ejpam-6425	141	9	an2	an2	PROPN
ejpam-6425	141	10	+	+	CCONJ
ejpam-6425	141	11	bnk	bnk	PROPN
ejpam-6425	141	12	+	+	CCONJ
ejpam-6425	141	13	ck2	ck2	NOUN
ejpam-6425	141	14	+	+	CCONJ
ejpam-6425	141	15	dn+	dn+	PROPN
ejpam-6425	142	1	ek	ek	PROPN
ejpam-6425	143	1	+	+	CCONJ
ejpam-6425	143	2	f	f	X
ejpam-6425	143	3	=	=	SYM
ejpam-6425	143	4	0	0	PROPN
ejpam-6425	143	5	,	,	PUNCT
ejpam-6425	143	6	where	where	SCONJ
ejpam-6425	143	7	a	a	DET
ejpam-6425	143	8	=	=	SYM
ejpam-6425	143	9	41	41	NUM
ejpam-6425	143	10	,	,	PUNCT
ejpam-6425	143	11	b	b	X
ejpam-6425	143	12	=	=	SYM
ejpam-6425	143	13	−6	−6	NOUN
ejpam-6425	143	14	,	,	PUNCT
ejpam-6425	143	15	c	c	NOUN
ejpam-6425	143	16	=	=	SYM
ejpam-6425	143	17	0	0	NUM
ejpam-6425	143	18	,	,	PUNCT
ejpam-6425	143	19	d	d	NOUN
ejpam-6425	143	20	=	=	SYM
ejpam-6425	143	21	7	7	NUM
ejpam-6425	143	22	,	,	PUNCT
ejpam-6425	143	23	e	e	NOUN
ejpam-6425	143	24	=	=	SYM
ejpam-6425	143	25	−2	−2	NOUN
ejpam-6425	143	26	,	,	PUNCT
ejpam-6425	143	27	and	and	CCONJ
ejpam-6425	143	28	f	f	PROPN
ejpam-6425	143	29	=	=	SYM
ejpam-6425	143	30	0	0	X
ejpam-6425	143	31	.	.	PUNCT
ejpam-6425	144	1	using	use	VERB
ejpam-6425	144	2	the	the	DET
ejpam-6425	144	3	transformation	transformation	NOUN
ejpam-6425	144	4	stated	state	VERB
ejpam-6425	144	5	in	in	ADP
ejpam-6425	144	6	lemma	lemma	PROPN
ejpam-6425	144	7	1	1	NUM
ejpam-6425	144	8	,	,	PUNCT
ejpam-6425	144	9	equation	equation	NOUN
ejpam-6425	144	10	(	(	PUNCT
ejpam-6425	144	11	6	6	NUM
ejpam-6425	144	12	)	)	PUNCT
ejpam-6425	144	13	can	can	AUX
ejpam-6425	144	14	be	be	AUX
ejpam-6425	144	15	reduced	reduce	VERB
ejpam-6425	144	16	to	to	ADP
ejpam-6425	144	17	x2	x2	PROPN
ejpam-6425	144	18	−	−	PROPN
ejpam-6425	144	19	36y2	36y2	NUM
ejpam-6425	144	20	=	=	SYM
ejpam-6425	144	21	13120	13120	NUM
ejpam-6425	144	22	.	.	PUNCT
ejpam-6425	145	1	(	(	PUNCT
ejpam-6425	145	2	7	7	NUM
ejpam-6425	145	3	)	)	PUNCT
ejpam-6425	145	4	where	where	SCONJ
ejpam-6425	145	5	y	y	PROPN
ejpam-6425	145	6	=	=	SYM
ejpam-6425	145	7	2an+	2an+	NUM
ejpam-6425	145	8	bk	bk	ADP
ejpam-6425	145	9	+	+	PROPN
ejpam-6425	145	10	d	d	X
ejpam-6425	145	11	=	=	SYM
ejpam-6425	145	12	82n−	82n−	NUM
ejpam-6425	145	13	6k	6k	NOUN
ejpam-6425	145	14	+	+	CCONJ
ejpam-6425	145	15	7	7	NUM
ejpam-6425	145	16	and	and	CCONJ
ejpam-6425	145	17	x	x	SYM
ejpam-6425	146	1	=	=	SYM
ejpam-6425	146	2	dk	dk	X
ejpam-6425	146	3	+	+	CCONJ
ejpam-6425	146	4	e	e	X
ejpam-6425	146	5	=	=	PROPN
ejpam-6425	146	6	36k	36k	X
ejpam-6425	146	7	+	+	CCONJ
ejpam-6425	146	8	122	122	NUM
ejpam-6425	146	9	.	.	PUNCT
ejpam-6425	146	10	by	by	ADP
ejpam-6425	146	11	factoring	factor	VERB
ejpam-6425	146	12	the	the	DET
ejpam-6425	146	13	left	left	ADJ
ejpam-6425	146	14	side	side	NOUN
ejpam-6425	146	15	of	of	ADP
ejpam-6425	146	16	the	the	DET
ejpam-6425	146	17	equation	equation	NOUN
ejpam-6425	146	18	(	(	PUNCT
ejpam-6425	146	19	7	7	NUM
ejpam-6425	146	20	)	)	PUNCT
ejpam-6425	146	21	,	,	PUNCT
ejpam-6425	146	22	we	we	PRON
ejpam-6425	146	23	will	will	AUX
ejpam-6425	146	24	have	have	VERB
ejpam-6425	146	25	,	,	PUNCT
ejpam-6425	146	26	(	(	PUNCT
ejpam-6425	146	27	x	x	SYM
ejpam-6425	146	28	+	+	X
ejpam-6425	146	29	(	(	PUNCT
ejpam-6425	146	30	−6)y	−6)y	NOUN
ejpam-6425	146	31	)	)	PUNCT
ejpam-6425	146	32	(	(	PUNCT
ejpam-6425	146	33	x	x	X
ejpam-6425	146	34	−	−	PROPN
ejpam-6425	146	35	(	(	PUNCT
ejpam-6425	146	36	−6)y	−6)y	NOUN
ejpam-6425	146	37	)	)	PUNCT
ejpam-6425	146	38	=	=	SYM
ejpam-6425	146	39	13120	13120	NUM
ejpam-6425	146	40	,	,	PUNCT
ejpam-6425	146	41	a.	a.	PROPN
ejpam-6425	146	42	d.	d.	PROPN
ejpam-6425	146	43	c.	c.	PROPN
ejpam-6425	146	44	angel	angel	PROPN
ejpam-6425	146	45	et	et	PROPN
ejpam-6425	146	46	al	al	PROPN
ejpam-6425	146	47	.	.	PUNCT
ejpam-6425	146	48	/	/	SYM
ejpam-6425	146	49	eur	eur	PROPN
ejpam-6425	146	50	.	.	PUNCT
ejpam-6425	147	1	j.	j.	PROPN
ejpam-6425	147	2	pure	pure	PROPN
ejpam-6425	147	3	appl	appl	PROPN
ejpam-6425	147	4	.	.	PROPN
ejpam-6425	147	5	math	math	PROPN
ejpam-6425	147	6	,	,	PUNCT
ejpam-6425	147	7	18	18	NUM
ejpam-6425	147	8	(	(	PUNCT
ejpam-6425	147	9	4	4	NUM
ejpam-6425	147	10	)	)	PUNCT
ejpam-6425	147	11	(	(	PUNCT
ejpam-6425	147	12	2025	2025	NUM
ejpam-6425	147	13	)	)	PUNCT
ejpam-6425	147	14	,	,	PUNCT
ejpam-6425	147	15	6425	6425	NUM
ejpam-6425	147	16	10	10	NUM
ejpam-6425	147	17	of	of	ADP
ejpam-6425	147	18	33	33	NUM
ejpam-6425	147	19	where	where	SCONJ
ejpam-6425	147	20	b	b	NOUN
ejpam-6425	147	21	=	=	SYM
ejpam-6425	147	22	−6	−6	NOUN
ejpam-6425	147	23	.	.	PUNCT
ejpam-6425	148	1	therefore	therefore	ADV
ejpam-6425	148	2	,	,	PUNCT
ejpam-6425	148	3	by	by	ADP
ejpam-6425	148	4	lemma	lemma	PROPN
ejpam-6425	148	5	1	1	NUM
ejpam-6425	148	6	and	and	CCONJ
ejpam-6425	148	7	remark	remark	NOUN
ejpam-6425	148	8	2	2	NUM
ejpam-6425	148	9	,	,	PUNCT
ejpam-6425	148	10	we	we	PRON
ejpam-6425	148	11	have	have	VERB
ejpam-6425	148	12	the	the	DET
ejpam-6425	148	13	following	following	NOUN
ejpam-6425	148	14	:	:	PUNCT
ejpam-6425	148	15	x	x	SYM
ejpam-6425	148	16	=	=	SYM
ejpam-6425	148	17	n1	n1	PROPN
ejpam-6425	148	18	+	+	NOUN
ejpam-6425	148	19	n2	n2	ADJ
ejpam-6425	148	20	2	2	NUM
ejpam-6425	148	21	,	,	PUNCT
ejpam-6425	148	22	y	y	PROPN
ejpam-6425	148	23	=	=	SYM
ejpam-6425	148	24	n1	n1	PROPN
ejpam-6425	149	1	−n2	−n2	PROPN
ejpam-6425	149	2	−12	−12	NOUN
ejpam-6425	149	3	,	,	PUNCT
ejpam-6425	149	4	k	k	X
ejpam-6425	150	1	=	=	PUNCT
ejpam-6425	150	2	x	x	SYM
ejpam-6425	150	3	−	−	PROPN
ejpam-6425	150	4	122	122	NUM
ejpam-6425	150	5	36	36	NUM
ejpam-6425	150	6	,	,	PUNCT
ejpam-6425	150	7	n	n	NOUN
ejpam-6425	150	8	=	=	SYM
ejpam-6425	150	9	y	y	PROPN
ejpam-6425	150	10	+	+	NUM
ejpam-6425	150	11	6k	6k	NOUN
ejpam-6425	150	12	−	−	PROPN
ejpam-6425	150	13	7	7	NUM
ejpam-6425	150	14	82	82	NUM
ejpam-6425	150	15	,	,	PUNCT
ejpam-6425	150	16	(	(	PUNCT
ejpam-6425	150	17	8)	8)	NUM
ejpam-6425	150	18	where	where	SCONJ
ejpam-6425	150	19	(	(	PUNCT
ejpam-6425	150	20	n1	n1	NOUN
ejpam-6425	150	21	,	,	PUNCT
ejpam-6425	150	22	n2	n2	ADJ
ejpam-6425	150	23	)	)	PUNCT
ejpam-6425	150	24	are	be	AUX
ejpam-6425	150	25	factor	factor	NOUN
ejpam-6425	150	26	pairs	pair	NOUN
ejpam-6425	150	27	of	of	ADP
ejpam-6425	150	28	n	n	NOUN
ejpam-6425	150	29	=	=	SYM
ejpam-6425	150	30	13120	13120	NUM
ejpam-6425	150	31	.	.	PUNCT
ejpam-6425	151	1	using	use	VERB
ejpam-6425	151	2	(	(	PUNCT
ejpam-6425	151	3	8)	8)	NUM
ejpam-6425	151	4	,	,	PUNCT
ejpam-6425	151	5	we	we	PRON
ejpam-6425	151	6	can	can	AUX
ejpam-6425	151	7	now	now	ADV
ejpam-6425	151	8	solve	solve	VERB
ejpam-6425	151	9	for	for	ADP
ejpam-6425	151	10	the	the	DET
ejpam-6425	151	11	integer	integer	NOUN
ejpam-6425	151	12	values	value	NOUN
ejpam-6425	151	13	of	of	ADP
ejpam-6425	151	14	n	n	PROPN
ejpam-6425	151	15	and	and	CCONJ
ejpam-6425	151	16	k	k	NOUN
ejpam-6425	151	17	,	,	PUNCT
ejpam-6425	151	18	by	by	ADP
ejpam-6425	151	19	considering	consider	VERB
ejpam-6425	151	20	all	all	DET
ejpam-6425	151	21	the	the	DET
ejpam-6425	151	22	factor	factor	NOUN
ejpam-6425	151	23	pairs	pair	NOUN
ejpam-6425	151	24	of	of	ADP
ejpam-6425	151	25	13120	13120	NUM
ejpam-6425	151	26	,	,	PUNCT
ejpam-6425	151	27	as	as	SCONJ
ejpam-6425	151	28	presented	present	VERB
ejpam-6425	151	29	in	in	ADP
ejpam-6425	151	30	table	table	NOUN
ejpam-6425	151	31	1	1	NUM
ejpam-6425	151	32	in	in	ADP
ejpam-6425	151	33	the	the	DET
ejpam-6425	151	34	appendices	appendix	NOUN
ejpam-6425	151	35	section	section	NOUN
ejpam-6425	151	36	.	.	PUNCT
ejpam-6425	152	1	table	table	NOUN
ejpam-6425	152	2	1	1	NUM
ejpam-6425	152	3	shows	show	VERB
ejpam-6425	152	4	that	that	SCONJ
ejpam-6425	152	5	there	there	PRON
ejpam-6425	152	6	are	be	VERB
ejpam-6425	152	7	4	4	NUM
ejpam-6425	152	8	possible	possible	ADJ
ejpam-6425	152	9	solutions	solution	NOUN
ejpam-6425	152	10	for	for	ADP
ejpam-6425	152	11	the	the	DET
ejpam-6425	152	12	diophantine	diophantine	NOUN
ejpam-6425	152	13	equation	equation	NOUN
ejpam-6425	152	14	41n2	41n2	NUM
ejpam-6425	152	15	+	+	NUM
ejpam-6425	152	16	7n−6nk−2k	7n−6nk−2k	NUM
ejpam-6425	152	17	=	=	SYM
ejpam-6425	152	18	0	0	X
ejpam-6425	152	19	.	.	PUNCT
ejpam-6425	153	1	these	these	PRON
ejpam-6425	153	2	are	be	AUX
ejpam-6425	153	3	:	:	PUNCT
ejpam-6425	153	4	(	(	PUNCT
ejpam-6425	153	5	n	n	CCONJ
ejpam-6425	153	6	,	,	PUNCT
ejpam-6425	153	7	k)=	k)=	X
ejpam-6425	153	8	(	(	PUNCT
ejpam-6425	153	9	0,0	0,0	NOUN
ejpam-6425	153	10	)	)	PUNCT
ejpam-6425	153	11	,	,	PUNCT
ejpam-6425	153	12	(	(	PUNCT
ejpam-6425	153	13	1,6	1,6	NUM
ejpam-6425	153	14	)	)	PUNCT
ejpam-6425	153	15	,	,	PUNCT
ejpam-6425	153	16	(	(	PUNCT
ejpam-6425	153	17	-7,-49	-7,-49	NOUN
ejpam-6425	153	18	)	)	PUNCT
ejpam-6425	153	19	,	,	PUNCT
ejpam-6425	153	20	and	and	CCONJ
ejpam-6425	153	21	(	(	PUNCT
ejpam-6425	153	22	-2	-2	INTJ
ejpam-6425	153	23	,	,	PUNCT
ejpam-6425	153	24	-15	-15	NUM
ejpam-6425	153	25	)	)	PUNCT
ejpam-6425	153	26	.	.	PUNCT
ejpam-6425	154	1	however	however	ADV
ejpam-6425	154	2	,	,	PUNCT
ejpam-6425	154	3	since	since	SCONJ
ejpam-6425	154	4	we	we	PRON
ejpam-6425	154	5	are	be	AUX
ejpam-6425	154	6	only	only	ADV
ejpam-6425	154	7	concern	concern	NOUN
ejpam-6425	154	8	with	with	ADP
ejpam-6425	154	9	possible	possible	ADJ
ejpam-6425	154	10	values	value	NOUN
ejpam-6425	154	11	of	of	ADP
ejpam-6425	154	12	n	n	PRON
ejpam-6425	154	13	for	for	ADP
ejpam-6425	154	14	water	water	NOUN
ejpam-6425	154	15	wheel	wheel	NOUN
ejpam-6425	154	16	graph	graph	NOUN
ejpam-6425	154	17	wwn	wwn	NOUN
ejpam-6425	154	18	,	,	PUNCT
ejpam-6425	154	19	then	then	ADV
ejpam-6425	154	20	by	by	ADP
ejpam-6425	154	21	definition	definition	NOUN
ejpam-6425	154	22	2	2	NUM
ejpam-6425	154	23	,	,	PUNCT
ejpam-6425	154	24	n	n	PRON
ejpam-6425	154	25	must	must	AUX
ejpam-6425	154	26	be	be	AUX
ejpam-6425	154	27	a	a	DET
ejpam-6425	154	28	positive	positive	ADJ
ejpam-6425	154	29	integer	integer	NOUN
ejpam-6425	154	30	and	and	CCONJ
ejpam-6425	154	31	n	n	PRON
ejpam-6425	154	32	≥	≥	NOUN
ejpam-6425	154	33	3	3	NUM
ejpam-6425	154	34	.	.	PUNCT
ejpam-6425	155	1	thus	thus	ADV
ejpam-6425	155	2	,	,	PUNCT
ejpam-6425	155	3	none	none	NOUN
ejpam-6425	155	4	of	of	ADP
ejpam-6425	155	5	our	our	PRON
ejpam-6425	155	6	4	4	NUM
ejpam-6425	155	7	solutions	solution	NOUN
ejpam-6425	155	8	satisfies	satisfie	NOUN
ejpam-6425	155	9	both	both	PRON
ejpam-6425	155	10	of	of	ADP
ejpam-6425	155	11	these	these	DET
ejpam-6425	155	12	conditions	condition	NOUN
ejpam-6425	155	13	.	.	PUNCT
ejpam-6425	156	1	hence	hence	ADV
ejpam-6425	156	2	,	,	PUNCT
ejpam-6425	156	3	we	we	PRON
ejpam-6425	156	4	have	have	AUX
ejpam-6425	156	5	shown	show	VERB
ejpam-6425	156	6	that	that	SCONJ
ejpam-6425	156	7	for	for	ADP
ejpam-6425	156	8	all	all	DET
ejpam-6425	156	9	n	n	PRON
ejpam-6425	156	10	≥	≥	NOUN
ejpam-6425	156	11	3	3	NUM
ejpam-6425	156	12	,	,	PUNCT
ejpam-6425	156	13	water	water	NOUN
ejpam-6425	156	14	wheel	wheel	NOUN
ejpam-6425	156	15	graph	graph	NOUN
ejpam-6425	156	16	wwn	wwn	PROPN
ejpam-6425	156	17	does	do	AUX
ejpam-6425	156	18	not	not	PART
ejpam-6425	156	19	satisfy	satisfy	VERB
ejpam-6425	156	20	the	the	DET
ejpam-6425	156	21	lo	lo	PROPN
ejpam-6425	156	22	’s	’s	PART
ejpam-6425	156	23	theorem	theorem	NOUN
ejpam-6425	156	24	.	.	PUNCT
ejpam-6425	157	1	therefore	therefore	ADV
ejpam-6425	157	2	,	,	PUNCT
ejpam-6425	157	3	we	we	PRON
ejpam-6425	157	4	can	can	AUX
ejpam-6425	157	5	conclude	conclude	VERB
ejpam-6425	157	6	that	that	PRON
ejpam-6425	157	7	for	for	ADP
ejpam-6425	157	8	all	all	DET
ejpam-6425	157	9	n	n	PRON
ejpam-6425	157	10	≥	≥	NOUN
ejpam-6425	157	11	3	3	NUM
ejpam-6425	157	12	,	,	PUNCT
ejpam-6425	157	13	water	water	NOUN
ejpam-6425	157	14	wheel	wheel	NOUN
ejpam-6425	157	15	graph	graph	NOUN
ejpam-6425	157	16	wwn	wwn	NOUN
ejpam-6425	157	17	is	be	AUX
ejpam-6425	157	18	not	not	PART
ejpam-6425	157	19	edge	edge	NOUN
ejpam-6425	157	20	-	-	PUNCT
ejpam-6425	157	21	graceful	graceful	NOUN
ejpam-6425	157	22	.	.	PUNCT
ejpam-6425	158	1	theorem	theorem	NOUN
ejpam-6425	158	2	3	3	NUM
ejpam-6425	158	3	.	.	X
ejpam-6425	158	4	for	for	ADP
ejpam-6425	158	5	n	n	X
ejpam-6425	158	6	≥	≥	NOUN
ejpam-6425	158	7	3	3	NUM
ejpam-6425	158	8	,	,	PUNCT
ejpam-6425	158	9	right	right	ADJ
ejpam-6425	158	10	water	water	NOUN
ejpam-6425	158	11	wheel	wheel	NOUN
ejpam-6425	158	12	graph	graph	NOUN
ejpam-6425	158	13	rwwn	rwwn	NOUN
ejpam-6425	158	14	is	be	AUX
ejpam-6425	158	15	not	not	PART
ejpam-6425	158	16	edge	edge	NOUN
ejpam-6425	158	17	-	-	PUNCT
ejpam-6425	158	18	graceful	graceful	NOUN
ejpam-6425	158	19	.	.	PUNCT
ejpam-6425	159	1	proof	proof	NOUN
ejpam-6425	159	2	.	.	PUNCT
ejpam-6425	160	1	recall	recall	VERB
ejpam-6425	160	2	from	from	ADP
ejpam-6425	160	3	definition	definition	NOUN
ejpam-6425	160	4	3	3	NUM
ejpam-6425	160	5	that	that	SCONJ
ejpam-6425	160	6	right	right	ADJ
ejpam-6425	160	7	water	water	NOUN
ejpam-6425	160	8	wheel	wheel	NOUN
ejpam-6425	160	9	graph	graph	NOUN
ejpam-6425	160	10	rwwn	rwwn	NOUN
ejpam-6425	160	11	has	have	VERB
ejpam-6425	160	12	3n	3n	NUM
ejpam-6425	160	13	+	+	SYM
ejpam-6425	160	14	1	1	NUM
ejpam-6425	160	15	vertices	vertex	NOUN
ejpam-6425	160	16	and	and	CCONJ
ejpam-6425	160	17	6n	6n	NOUN
ejpam-6425	160	18	edges	edge	NOUN
ejpam-6425	160	19	.	.	PUNCT
ejpam-6425	161	1	similar	similar	ADJ
ejpam-6425	161	2	to	to	ADP
ejpam-6425	161	3	the	the	DET
ejpam-6425	161	4	proof	proof	NOUN
ejpam-6425	161	5	of	of	ADP
ejpam-6425	161	6	theorem	theorem	NOUN
ejpam-6425	161	7	2	2	NUM
ejpam-6425	161	8	,	,	PUNCT
ejpam-6425	161	9	we	we	PRON
ejpam-6425	161	10	substitute	substitute	VERB
ejpam-6425	161	11	p	p	NOUN
ejpam-6425	162	1	=	=	NOUN
ejpam-6425	162	2	3n	3n	NOUN
ejpam-6425	163	1	+	+	SYM
ejpam-6425	163	2	1	1	NUM
ejpam-6425	163	3	and	and	CCONJ
ejpam-6425	163	4	q	q	NOUN
ejpam-6425	164	1	=	=	SYM
ejpam-6425	164	2	6n	6n	NOUN
ejpam-6425	164	3	in	in	ADP
ejpam-6425	164	4	theorem	theorem	ADJ
ejpam-6425	164	5	1(lo	1(lo	PROPN
ejpam-6425	164	6	’s	’s	PART
ejpam-6425	164	7	theorem	theorem	NOUN
ejpam-6425	164	8	)	)	PUNCT
ejpam-6425	164	9	.	.	PUNCT
ejpam-6425	165	1	that	that	PRON
ejpam-6425	165	2	is	be	AUX
ejpam-6425	165	3	,	,	PUNCT
ejpam-6425	165	4	(	(	PUNCT
ejpam-6425	165	5	3n+	3n+	NUM
ejpam-6425	165	6	1	1	NUM
ejpam-6425	165	7	)	)	PUNCT
ejpam-6425	165	8	∣∣∣((6n)2	∣∣∣((6n)2	PROPN
ejpam-6425	166	1	+	+	CCONJ
ejpam-6425	166	2	6n−	6n−	PROPN
ejpam-6425	166	3	(	(	PUNCT
ejpam-6425	166	4	3n+	3n+	NUM
ejpam-6425	166	5	1)(3n	1)(3n	NUM
ejpam-6425	166	6	)	)	PUNCT
ejpam-6425	166	7	2	2	NUM
ejpam-6425	166	8	)	)	PUNCT
ejpam-6425	166	9	,	,	PUNCT
ejpam-6425	166	10	which	which	PRON
ejpam-6425	166	11	further	far	ADV
ejpam-6425	166	12	implies	imply	VERB
ejpam-6425	166	13	that	that	SCONJ
ejpam-6425	166	14	,	,	PUNCT
ejpam-6425	166	15	(	(	PUNCT
ejpam-6425	166	16	63n2	63n2	NUM
ejpam-6425	166	17	+	+	NUM
ejpam-6425	166	18	9n	9n	NUM
ejpam-6425	166	19	6n+	6n+	NUM
ejpam-6425	166	20	2	2	NUM
ejpam-6425	166	21	)	)	PUNCT
ejpam-6425	166	22	∈	∈	PROPN
ejpam-6425	166	23	z.	z.	PROPN
ejpam-6425	166	24	this	this	PRON
ejpam-6425	166	25	means	mean	VERB
ejpam-6425	166	26	,	,	PUNCT
ejpam-6425	166	27	63n2	63n2	PROPN
ejpam-6425	166	28	+	+	NUM
ejpam-6425	166	29	9n	9n	NUM
ejpam-6425	166	30	=	=	SYM
ejpam-6425	166	31	(	(	PUNCT
ejpam-6425	166	32	6n+	6n+	NUM
ejpam-6425	166	33	2)k	2)k	NOUN
ejpam-6425	166	34	=	=	SYM
ejpam-6425	166	35	6nk	6nk	NOUN
ejpam-6425	166	36	+	+	CCONJ
ejpam-6425	166	37	2k	2k	NUM
ejpam-6425	166	38	,	,	PUNCT
ejpam-6425	166	39	for	for	ADP
ejpam-6425	166	40	some	some	DET
ejpam-6425	166	41	k	k	PROPN
ejpam-6425	166	42	∈	∈	PROPN
ejpam-6425	166	43	z.	z.	PROPN
ejpam-6425	167	1	thus	thus	ADV
ejpam-6425	167	2	,	,	PUNCT
ejpam-6425	167	3	we	we	PRON
ejpam-6425	167	4	solve	solve	VERB
ejpam-6425	167	5	for	for	ADP
ejpam-6425	167	6	all	all	DET
ejpam-6425	167	7	the	the	DET
ejpam-6425	167	8	possible	possible	ADJ
ejpam-6425	167	9	integer	integer	NOUN
ejpam-6425	167	10	values	value	NOUN
ejpam-6425	167	11	of	of	ADP
ejpam-6425	167	12	n	n	PRON
ejpam-6425	167	13	and	and	CCONJ
ejpam-6425	167	14	k	k	X
ejpam-6425	167	15	in	in	ADP
ejpam-6425	167	16	the	the	DET
ejpam-6425	167	17	diophantine	diophantine	NOUN
ejpam-6425	167	18	equation	equation	NOUN
ejpam-6425	167	19	,	,	PUNCT
ejpam-6425	167	20	63n2	63n2	PROPN
ejpam-6425	168	1	+	+	NUM
ejpam-6425	168	2	9n−	9n−	NUM
ejpam-6425	168	3	6nk	6nk	NOUN
ejpam-6425	168	4	−	−	NOUN
ejpam-6425	168	5	2k	2k	NOUN
ejpam-6425	168	6	=	=	SYM
ejpam-6425	168	7	0	0	NUM
ejpam-6425	168	8	,	,	PUNCT
ejpam-6425	168	9	(	(	PUNCT
ejpam-6425	168	10	9	9	X
ejpam-6425	168	11	)	)	PUNCT
ejpam-6425	168	12	where	where	SCONJ
ejpam-6425	168	13	n	n	X
ejpam-6425	168	14	,	,	PUNCT
ejpam-6425	168	15	k	k	PROPN
ejpam-6425	168	16	∈	∈	PROPN
ejpam-6425	168	17	z.	z.	PROPN
ejpam-6425	169	1	this	this	DET
ejpam-6425	169	2	diophantine	diophantine	NOUN
ejpam-6425	169	3	equation	equation	NOUN
ejpam-6425	169	4	is	be	AUX
ejpam-6425	169	5	of	of	ADP
ejpam-6425	169	6	the	the	DET
ejpam-6425	169	7	form	form	NOUN
ejpam-6425	169	8	,	,	PUNCT
ejpam-6425	169	9	an2	an2	PROPN
ejpam-6425	169	10	+	+	CCONJ
ejpam-6425	169	11	bnk	bnk	PROPN
ejpam-6425	169	12	+	+	CCONJ
ejpam-6425	169	13	ck2	ck2	NOUN
ejpam-6425	169	14	+	+	CCONJ
ejpam-6425	169	15	dn+	dn+	PROPN
ejpam-6425	170	1	ek	ek	PROPN
ejpam-6425	171	1	+	+	CCONJ
ejpam-6425	171	2	f	f	X
ejpam-6425	171	3	=	=	SYM
ejpam-6425	171	4	0	0	PROPN
ejpam-6425	171	5	,	,	PUNCT
ejpam-6425	171	6	where	where	SCONJ
ejpam-6425	171	7	a	a	DET
ejpam-6425	171	8	=	=	SYM
ejpam-6425	171	9	63	63	NUM
ejpam-6425	171	10	,	,	PUNCT
ejpam-6425	171	11	b	b	X
ejpam-6425	171	12	=	=	SYM
ejpam-6425	171	13	−6	−6	NOUN
ejpam-6425	171	14	,	,	PUNCT
ejpam-6425	171	15	c	c	NOUN
ejpam-6425	171	16	=	=	SYM
ejpam-6425	171	17	0	0	NUM
ejpam-6425	171	18	,	,	PUNCT
ejpam-6425	171	19	d	d	NOUN
ejpam-6425	171	20	=	=	SYM
ejpam-6425	171	21	9	9	NUM
ejpam-6425	171	22	,	,	PUNCT
ejpam-6425	171	23	e	e	NOUN
ejpam-6425	171	24	=	=	SYM
ejpam-6425	171	25	−2	−2	NOUN
ejpam-6425	171	26	,	,	PUNCT
ejpam-6425	171	27	and	and	CCONJ
ejpam-6425	171	28	f	f	PROPN
ejpam-6425	171	29	=	=	SYM
ejpam-6425	171	30	0	0	X
ejpam-6425	171	31	.	.	PUNCT
ejpam-6425	172	1	using	use	VERB
ejpam-6425	172	2	the	the	DET
ejpam-6425	172	3	transformation	transformation	NOUN
ejpam-6425	172	4	stated	state	VERB
ejpam-6425	172	5	in	in	ADP
ejpam-6425	172	6	lemma	lemma	PROPN
ejpam-6425	172	7	1	1	NUM
ejpam-6425	172	8	,	,	PUNCT
ejpam-6425	172	9	equation	equation	NOUN
ejpam-6425	172	10	(	(	PUNCT
ejpam-6425	172	11	9	9	NUM
ejpam-6425	172	12	)	)	PUNCT
ejpam-6425	172	13	can	can	AUX
ejpam-6425	172	14	be	be	AUX
ejpam-6425	172	15	reduced	reduce	VERB
ejpam-6425	172	16	to	to	ADP
ejpam-6425	172	17	x2	x2	PROPN
ejpam-6425	172	18	−	−	PROPN
ejpam-6425	172	19	36y2	36y2	NUM
ejpam-6425	172	20	=	=	SYM
ejpam-6425	172	21	36288	36288	NUM
ejpam-6425	172	22	.	.	PUNCT
ejpam-6425	173	1	(	(	PUNCT
ejpam-6425	173	2	10	10	NUM
ejpam-6425	173	3	)	)	PUNCT
ejpam-6425	173	4	a.	a.	PROPN
ejpam-6425	173	5	d.	d.	PROPN
ejpam-6425	173	6	c.	c.	PROPN
ejpam-6425	173	7	angel	angel	PROPN
ejpam-6425	173	8	et	et	PROPN
ejpam-6425	173	9	al	al	PROPN
ejpam-6425	173	10	.	.	PUNCT
ejpam-6425	173	11	/	/	SYM
ejpam-6425	173	12	eur	eur	PROPN
ejpam-6425	173	13	.	.	PUNCT
ejpam-6425	174	1	j.	j.	PROPN
ejpam-6425	174	2	pure	pure	PROPN
ejpam-6425	174	3	appl	appl	PROPN
ejpam-6425	174	4	.	.	PROPN
ejpam-6425	174	5	math	math	PROPN
ejpam-6425	174	6	,	,	PUNCT
ejpam-6425	174	7	18	18	NUM
ejpam-6425	174	8	(	(	PUNCT
ejpam-6425	174	9	4	4	NUM
ejpam-6425	174	10	)	)	PUNCT
ejpam-6425	174	11	(	(	PUNCT
ejpam-6425	174	12	2025	2025	NUM
ejpam-6425	174	13	)	)	PUNCT
ejpam-6425	174	14	,	,	PUNCT
ejpam-6425	174	15	6425	6425	NUM
ejpam-6425	174	16	11	11	NUM
ejpam-6425	174	17	of	of	ADP
ejpam-6425	174	18	33	33	NUM
ejpam-6425	174	19	where	where	SCONJ
ejpam-6425	174	20	y	y	NOUN
ejpam-6425	174	21	=	=	SYM
ejpam-6425	174	22	2an+	2an+	NOUN
ejpam-6425	174	23	bk	bk	ADP
ejpam-6425	174	24	+	+	PROPN
ejpam-6425	174	25	d	d	X
ejpam-6425	174	26	=	=	SYM
ejpam-6425	174	27	126n−	126n−	NUM
ejpam-6425	174	28	6k	6k	NOUN
ejpam-6425	174	29	+	+	CCONJ
ejpam-6425	174	30	9	9	NUM
ejpam-6425	174	31	and	and	CCONJ
ejpam-6425	174	32	x	x	SYM
ejpam-6425	175	1	=	=	SYM
ejpam-6425	175	2	dk	dk	X
ejpam-6425	175	3	+	+	CCONJ
ejpam-6425	175	4	e	e	X
ejpam-6425	175	5	=	=	PROPN
ejpam-6425	175	6	36k	36k	X
ejpam-6425	175	7	+	+	SYM
ejpam-6425	175	8	198	198	NUM
ejpam-6425	175	9	.	.	PUNCT
ejpam-6425	175	10	by	by	ADP
ejpam-6425	175	11	factoring	factor	VERB
ejpam-6425	175	12	the	the	DET
ejpam-6425	175	13	left	left	ADJ
ejpam-6425	175	14	side	side	NOUN
ejpam-6425	175	15	of	of	ADP
ejpam-6425	175	16	the	the	DET
ejpam-6425	175	17	equation	equation	NOUN
ejpam-6425	175	18	(	(	PUNCT
ejpam-6425	175	19	10	10	NUM
ejpam-6425	175	20	)	)	PUNCT
ejpam-6425	175	21	,	,	PUNCT
ejpam-6425	175	22	we	we	PRON
ejpam-6425	175	23	will	will	AUX
ejpam-6425	175	24	have	have	VERB
ejpam-6425	175	25	,	,	PUNCT
ejpam-6425	175	26	(	(	PUNCT
ejpam-6425	175	27	x	x	SYM
ejpam-6425	175	28	+	+	X
ejpam-6425	175	29	(	(	PUNCT
ejpam-6425	175	30	−6)y	−6)y	NOUN
ejpam-6425	175	31	)	)	PUNCT
ejpam-6425	175	32	(	(	PUNCT
ejpam-6425	175	33	x	x	X
ejpam-6425	175	34	−	−	PROPN
ejpam-6425	175	35	(	(	PUNCT
ejpam-6425	175	36	−6)y	−6)y	NOUN
ejpam-6425	175	37	)	)	PUNCT
ejpam-6425	175	38	=	=	SYM
ejpam-6425	175	39	36288	36288	NUM
ejpam-6425	175	40	,	,	PUNCT
ejpam-6425	175	41	where	where	SCONJ
ejpam-6425	175	42	b	b	X
ejpam-6425	175	43	=	=	SYM
ejpam-6425	175	44	−6	−6	NOUN
ejpam-6425	175	45	.	.	PUNCT
ejpam-6425	176	1	therefore	therefore	ADV
ejpam-6425	176	2	,	,	PUNCT
ejpam-6425	176	3	by	by	ADP
ejpam-6425	176	4	lemma	lemma	PROPN
ejpam-6425	176	5	1	1	NUM
ejpam-6425	176	6	and	and	CCONJ
ejpam-6425	176	7	remark	remark	NOUN
ejpam-6425	176	8	2	2	NUM
ejpam-6425	176	9	,	,	PUNCT
ejpam-6425	176	10	we	we	PRON
ejpam-6425	176	11	have	have	VERB
ejpam-6425	176	12	the	the	DET
ejpam-6425	176	13	following	following	NOUN
ejpam-6425	176	14	:	:	PUNCT
ejpam-6425	176	15	x	x	SYM
ejpam-6425	176	16	=	=	SYM
ejpam-6425	176	17	n1	n1	PROPN
ejpam-6425	176	18	+	+	NOUN
ejpam-6425	176	19	n2	n2	ADJ
ejpam-6425	176	20	2	2	NUM
ejpam-6425	176	21	,	,	PUNCT
ejpam-6425	176	22	y	y	PROPN
ejpam-6425	176	23	=	=	SYM
ejpam-6425	176	24	n1	n1	PROPN
ejpam-6425	177	1	−n2	−n2	PROPN
ejpam-6425	177	2	−12	−12	NOUN
ejpam-6425	177	3	,	,	PUNCT
ejpam-6425	177	4	k	k	X
ejpam-6425	178	1	=	=	PUNCT
ejpam-6425	178	2	x	x	SYM
ejpam-6425	178	3	−	−	PROPN
ejpam-6425	178	4	198	198	NUM
ejpam-6425	178	5	36	36	NUM
ejpam-6425	178	6	,	,	PUNCT
ejpam-6425	178	7	n	n	NOUN
ejpam-6425	178	8	=	=	SYM
ejpam-6425	178	9	y	y	PROPN
ejpam-6425	178	10	+	+	NUM
ejpam-6425	178	11	6k	6k	NOUN
ejpam-6425	178	12	−	−	NOUN
ejpam-6425	178	13	9	9	NUM
ejpam-6425	178	14	126	126	NUM
ejpam-6425	178	15	,	,	PUNCT
ejpam-6425	178	16	(	(	PUNCT
ejpam-6425	178	17	11	11	NUM
ejpam-6425	178	18	)	)	PUNCT
ejpam-6425	178	19	where	where	SCONJ
ejpam-6425	178	20	(	(	PUNCT
ejpam-6425	178	21	n1	n1	NOUN
ejpam-6425	178	22	,	,	PUNCT
ejpam-6425	178	23	n2	n2	ADJ
ejpam-6425	178	24	)	)	PUNCT
ejpam-6425	178	25	are	be	AUX
ejpam-6425	178	26	factor	factor	NOUN
ejpam-6425	178	27	pairs	pair	NOUN
ejpam-6425	178	28	of	of	ADP
ejpam-6425	178	29	n	n	NOUN
ejpam-6425	178	30	=	=	SYM
ejpam-6425	178	31	36288	36288	NUM
ejpam-6425	178	32	.	.	PUNCT
ejpam-6425	179	1	using	use	VERB
ejpam-6425	179	2	(	(	PUNCT
ejpam-6425	179	3	11	11	NUM
ejpam-6425	179	4	)	)	PUNCT
ejpam-6425	179	5	,	,	PUNCT
ejpam-6425	179	6	we	we	PRON
ejpam-6425	179	7	can	can	AUX
ejpam-6425	179	8	now	now	ADV
ejpam-6425	179	9	solve	solve	VERB
ejpam-6425	179	10	for	for	ADP
ejpam-6425	179	11	the	the	DET
ejpam-6425	179	12	integer	integer	NOUN
ejpam-6425	179	13	values	value	NOUN
ejpam-6425	179	14	of	of	ADP
ejpam-6425	179	15	n	n	PROPN
ejpam-6425	179	16	and	and	CCONJ
ejpam-6425	179	17	k	k	NOUN
ejpam-6425	179	18	,	,	PUNCT
ejpam-6425	179	19	by	by	ADP
ejpam-6425	179	20	considering	consider	VERB
ejpam-6425	179	21	all	all	DET
ejpam-6425	179	22	the	the	DET
ejpam-6425	179	23	factor	factor	NOUN
ejpam-6425	179	24	pairs	pair	NOUN
ejpam-6425	179	25	of	of	ADP
ejpam-6425	179	26	36288	36288	NUM
ejpam-6425	179	27	,	,	PUNCT
ejpam-6425	179	28	as	as	SCONJ
ejpam-6425	179	29	presented	present	VERB
ejpam-6425	179	30	in	in	ADP
ejpam-6425	179	31	table	table	NOUN
ejpam-6425	179	32	2	2	NUM
ejpam-6425	179	33	in	in	ADP
ejpam-6425	179	34	the	the	DET
ejpam-6425	179	35	appendices	appendix	NOUN
ejpam-6425	179	36	section	section	NOUN
ejpam-6425	179	37	.	.	PUNCT
ejpam-6425	180	1	table	table	NOUN
ejpam-6425	180	2	2	2	NUM
ejpam-6425	180	3	shows	show	VERB
ejpam-6425	180	4	that	that	SCONJ
ejpam-6425	180	5	there	there	PRON
ejpam-6425	180	6	are	be	VERB
ejpam-6425	180	7	two	two	NUM
ejpam-6425	180	8	possible	possible	ADJ
ejpam-6425	180	9	solutions	solution	NOUN
ejpam-6425	180	10	for	for	ADP
ejpam-6425	180	11	the	the	DET
ejpam-6425	180	12	diophantine	diophantine	NOUN
ejpam-6425	180	13	equation	equation	NOUN
ejpam-6425	180	14	63n2	63n2	NUM
ejpam-6425	181	1	+	+	NUM
ejpam-6425	181	2	9n	9n	NUM
ejpam-6425	182	1	−	−	NOUN
ejpam-6425	182	2	6nk	6nk	NOUN
ejpam-6425	182	3	−	−	NOUN
ejpam-6425	182	4	2k	2k	NOUN
ejpam-6425	182	5	=	=	SYM
ejpam-6425	182	6	0	0	X
ejpam-6425	182	7	.	.	PUNCT
ejpam-6425	183	1	these	these	PRON
ejpam-6425	183	2	are	be	AUX
ejpam-6425	183	3	:	:	PUNCT
ejpam-6425	183	4	(	(	PUNCT
ejpam-6425	183	5	n	n	CCONJ
ejpam-6425	183	6	,	,	PUNCT
ejpam-6425	183	7	k)=	k)=	X
ejpam-6425	183	8	(	(	PUNCT
ejpam-6425	183	9	0,0	0,0	NOUN
ejpam-6425	183	10	)	)	PUNCT
ejpam-6425	183	11	,	,	PUNCT
ejpam-6425	183	12	and	and	CCONJ
ejpam-6425	183	13	(	(	PUNCT
ejpam-6425	183	14	1	1	NUM
ejpam-6425	183	15	,	,	PUNCT
ejpam-6425	183	16	9	9	NUM
ejpam-6425	183	17	)	)	PUNCT
ejpam-6425	183	18	.	.	PUNCT
ejpam-6425	184	1	however	however	ADV
ejpam-6425	184	2	,	,	PUNCT
ejpam-6425	184	3	since	since	SCONJ
ejpam-6425	184	4	we	we	PRON
ejpam-6425	184	5	are	be	AUX
ejpam-6425	184	6	only	only	ADV
ejpam-6425	184	7	concern	concern	NOUN
ejpam-6425	184	8	with	with	ADP
ejpam-6425	184	9	possible	possible	ADJ
ejpam-6425	184	10	values	value	NOUN
ejpam-6425	184	11	of	of	ADP
ejpam-6425	184	12	n	n	NOUN
ejpam-6425	184	13	for	for	ADP
ejpam-6425	184	14	right	right	ADJ
ejpam-6425	184	15	water	water	NOUN
ejpam-6425	184	16	wheel	wheel	NOUN
ejpam-6425	184	17	graph	graph	NOUN
ejpam-6425	184	18	rwwn	rwwn	NOUN
ejpam-6425	184	19	,	,	PUNCT
ejpam-6425	184	20	then	then	ADV
ejpam-6425	184	21	by	by	ADP
ejpam-6425	184	22	definition	definition	NOUN
ejpam-6425	184	23	3	3	NUM
ejpam-6425	184	24	,	,	PUNCT
ejpam-6425	184	25	n	n	PRON
ejpam-6425	184	26	must	must	AUX
ejpam-6425	184	27	be	be	AUX
ejpam-6425	184	28	a	a	DET
ejpam-6425	184	29	positive	positive	ADJ
ejpam-6425	184	30	integer	integer	NOUN
ejpam-6425	184	31	and	and	CCONJ
ejpam-6425	184	32	n	n	PRON
ejpam-6425	184	33	≥	≥	NOUN
ejpam-6425	184	34	3	3	NUM
ejpam-6425	184	35	.	.	PUNCT
ejpam-6425	185	1	thus	thus	ADV
ejpam-6425	185	2	,	,	PUNCT
ejpam-6425	185	3	none	none	NOUN
ejpam-6425	185	4	of	of	ADP
ejpam-6425	185	5	our	our	PRON
ejpam-6425	185	6	two	two	NUM
ejpam-6425	185	7	solutions	solution	NOUN
ejpam-6425	185	8	satisfies	satisfie	NOUN
ejpam-6425	185	9	both	both	PRON
ejpam-6425	185	10	of	of	ADP
ejpam-6425	185	11	these	these	DET
ejpam-6425	185	12	conditions	condition	NOUN
ejpam-6425	185	13	.	.	PUNCT
ejpam-6425	186	1	hence	hence	ADV
ejpam-6425	186	2	,	,	PUNCT
ejpam-6425	186	3	we	we	PRON
ejpam-6425	186	4	have	have	AUX
ejpam-6425	186	5	shown	show	VERB
ejpam-6425	186	6	that	that	SCONJ
ejpam-6425	186	7	for	for	ADP
ejpam-6425	186	8	all	all	DET
ejpam-6425	186	9	n	n	PRON
ejpam-6425	186	10	≥	≥	NOUN
ejpam-6425	186	11	3	3	NUM
ejpam-6425	186	12	,	,	PUNCT
ejpam-6425	186	13	right	right	ADJ
ejpam-6425	186	14	water	water	NOUN
ejpam-6425	186	15	wheel	wheel	NOUN
ejpam-6425	186	16	graph	graph	NOUN
ejpam-6425	186	17	rwwn	rwwn	NOUN
ejpam-6425	186	18	does	do	AUX
ejpam-6425	186	19	not	not	PART
ejpam-6425	186	20	satisfy	satisfy	VERB
ejpam-6425	186	21	the	the	DET
ejpam-6425	186	22	lo	lo	PROPN
ejpam-6425	186	23	’s	’s	PART
ejpam-6425	186	24	theorem	theorem	NOUN
ejpam-6425	186	25	.	.	PUNCT
ejpam-6425	187	1	therefore	therefore	ADV
ejpam-6425	187	2	,	,	PUNCT
ejpam-6425	187	3	we	we	PRON
ejpam-6425	187	4	conclude	conclude	VERB
ejpam-6425	187	5	that	that	SCONJ
ejpam-6425	187	6	for	for	ADP
ejpam-6425	187	7	all	all	DET
ejpam-6425	187	8	n	n	PRON
ejpam-6425	187	9	≥	≥	NOUN
ejpam-6425	187	10	3	3	NUM
ejpam-6425	187	11	,	,	PUNCT
ejpam-6425	187	12	right	right	ADJ
ejpam-6425	187	13	water	water	NOUN
ejpam-6425	187	14	wheel	wheel	NOUN
ejpam-6425	187	15	graph	graph	NOUN
ejpam-6425	187	16	rwwn	rwwn	NOUN
ejpam-6425	187	17	is	be	AUX
ejpam-6425	187	18	not	not	PART
ejpam-6425	187	19	edge	edge	NOUN
ejpam-6425	187	20	-	-	PUNCT
ejpam-6425	187	21	graceful	graceful	NOUN
ejpam-6425	187	22	.	.	PUNCT
ejpam-6425	188	1	theorem	theorem	ADJ
ejpam-6425	188	2	4	4	NUM
ejpam-6425	188	3	.	.	PUNCT
ejpam-6425	188	4	for	for	ADP
ejpam-6425	188	5	n	n	PRON
ejpam-6425	188	6	≥	≥	NUM
ejpam-6425	188	7	3	3	NUM
ejpam-6425	188	8	,	,	PUNCT
ejpam-6425	188	9	left	leave	VERB
ejpam-6425	188	10	water	water	NOUN
ejpam-6425	188	11	wheel	wheel	NOUN
ejpam-6425	188	12	graph	graph	NOUN
ejpam-6425	188	13	lwwn	lwwn	NOUN
ejpam-6425	188	14	is	be	AUX
ejpam-6425	188	15	not	not	PART
ejpam-6425	188	16	edge	edge	NOUN
ejpam-6425	188	17	-	-	PUNCT
ejpam-6425	188	18	graceful	graceful	NOUN
ejpam-6425	188	19	.	.	PUNCT
ejpam-6425	189	1	proof	proof	NOUN
ejpam-6425	189	2	.	.	PUNCT
ejpam-6425	190	1	recall	recall	VERB
ejpam-6425	190	2	from	from	ADP
ejpam-6425	190	3	definition	definition	NOUN
ejpam-6425	190	4	4	4	NUM
ejpam-6425	190	5	that	that	PRON
ejpam-6425	190	6	left	leave	VERB
ejpam-6425	190	7	water	water	NOUN
ejpam-6425	190	8	wheel	wheel	NOUN
ejpam-6425	190	9	graph	graph	NOUN
ejpam-6425	190	10	lwwn	lwwn	NOUN
ejpam-6425	190	11	,	,	PUNCT
ejpam-6425	190	12	also	also	ADV
ejpam-6425	190	13	has	have	VERB
ejpam-6425	190	14	3n+	3n+	NUM
ejpam-6425	190	15	1	1	NUM
ejpam-6425	190	16	vertices	vertex	NOUN
ejpam-6425	190	17	and	and	CCONJ
ejpam-6425	190	18	6n	6n	NUM
ejpam-6425	190	19	edges	edge	NOUN
ejpam-6425	190	20	,	,	PUNCT
ejpam-6425	190	21	the	the	DET
ejpam-6425	190	22	same	same	ADJ
ejpam-6425	190	23	with	with	ADP
ejpam-6425	190	24	right	right	ADJ
ejpam-6425	190	25	water	water	NOUN
ejpam-6425	190	26	wheel	wheel	NOUN
ejpam-6425	190	27	graph	graph	NOUN
ejpam-6425	190	28	rwwn	rwwn	NOUN
ejpam-6425	190	29	.	.	PUNCT
ejpam-6425	191	1	since	since	SCONJ
ejpam-6425	191	2	lwwn	lwwn	NOUN
ejpam-6425	191	3	has	have	VERB
ejpam-6425	191	4	the	the	DET
ejpam-6425	191	5	same	same	ADJ
ejpam-6425	191	6	size	size	NOUN
ejpam-6425	191	7	and	and	CCONJ
ejpam-6425	191	8	order	order	NOUN
ejpam-6425	191	9	as	as	ADP
ejpam-6425	191	10	rwwn	rwwn	NOUN
ejpam-6425	191	11	,	,	PUNCT
ejpam-6425	191	12	then	then	ADV
ejpam-6425	191	13	the	the	DET
ejpam-6425	191	14	proof	proof	NOUN
ejpam-6425	191	15	of	of	ADP
ejpam-6425	191	16	theorem	theorem	NOUN
ejpam-6425	191	17	4	4	NUM
ejpam-6425	191	18	will	will	AUX
ejpam-6425	191	19	be	be	AUX
ejpam-6425	191	20	the	the	DET
ejpam-6425	191	21	same	same	ADJ
ejpam-6425	191	22	as	as	ADP
ejpam-6425	191	23	the	the	DET
ejpam-6425	191	24	proof	proof	NOUN
ejpam-6425	191	25	of	of	ADP
ejpam-6425	191	26	theorem	theorem	NOUN
ejpam-6425	191	27	3	3	X
ejpam-6425	191	28	.	.	PUNCT
ejpam-6425	192	1	therefore	therefore	ADV
ejpam-6425	192	2	,	,	PUNCT
ejpam-6425	192	3	we	we	PRON
ejpam-6425	192	4	already	already	ADV
ejpam-6425	192	5	have	have	VERB
ejpam-6425	192	6	enough	enough	ADJ
ejpam-6425	192	7	evidence	evidence	NOUN
ejpam-6425	192	8	to	to	PART
ejpam-6425	192	9	conclude	conclude	VERB
ejpam-6425	192	10	that	that	PRON
ejpam-6425	192	11	for	for	ADP
ejpam-6425	192	12	all	all	DET
ejpam-6425	192	13	n	n	PRON
ejpam-6425	192	14	≥	≥	NOUN
ejpam-6425	192	15	3	3	NUM
ejpam-6425	192	16	,	,	PUNCT
ejpam-6425	192	17	no	no	DET
ejpam-6425	192	18	left	left	ADJ
ejpam-6425	192	19	water	water	NOUN
ejpam-6425	192	20	wheel	wheel	NOUN
ejpam-6425	192	21	graph	graph	NOUN
ejpam-6425	192	22	lwwn	lwwn	NOUN
ejpam-6425	192	23	is	be	AUX
ejpam-6425	192	24	edge	edge	NOUN
ejpam-6425	192	25	-	-	PUNCT
ejpam-6425	192	26	graceful	graceful	NOUN
ejpam-6425	192	27	.	.	PUNCT
ejpam-6425	193	1	we	we	PRON
ejpam-6425	193	2	now	now	ADV
ejpam-6425	193	3	present	present	VERB
ejpam-6425	193	4	our	our	PRON
ejpam-6425	193	5	results	result	NOUN
ejpam-6425	193	6	regarding	regard	VERB
ejpam-6425	193	7	the	the	DET
ejpam-6425	193	8	edge	edge	NOUN
ejpam-6425	193	9	-	-	PUNCT
ejpam-6425	193	10	gracefulness	gracefulness	NOUN
ejpam-6425	193	11	of	of	ADP
ejpam-6425	193	12	the	the	DET
ejpam-6425	193	13	splitting	splitting	NOUN
ejpam-6425	193	14	graphs	graph	NOUN
ejpam-6425	193	15	of	of	ADP
ejpam-6425	193	16	water	water	NOUN
ejpam-6425	193	17	wheel	wheel	NOUN
ejpam-6425	193	18	,	,	PUNCT
ejpam-6425	193	19	right	right	ADJ
ejpam-6425	193	20	water	water	NOUN
ejpam-6425	193	21	wheel	wheel	NOUN
ejpam-6425	193	22	,	,	PUNCT
ejpam-6425	193	23	and	and	CCONJ
ejpam-6425	193	24	left	leave	VERB
ejpam-6425	193	25	water	water	NOUN
ejpam-6425	193	26	wheel	wheel	NOUN
ejpam-6425	193	27	graphs	graph	NOUN
ejpam-6425	193	28	.	.	PUNCT
ejpam-6425	194	1	3.2	3.2	NUM
ejpam-6425	194	2	.	.	PUNCT
ejpam-6425	195	1	on	on	ADP
ejpam-6425	195	2	edge	edge	NOUN
ejpam-6425	195	3	-	-	PUNCT
ejpam-6425	195	4	gracefulness	gracefulness	NOUN
ejpam-6425	195	5	of	of	ADP
ejpam-6425	195	6	the	the	DET
ejpam-6425	195	7	splitting	splitting	NOUN
ejpam-6425	195	8	graphs	graph	NOUN
ejpam-6425	195	9	of	of	ADP
ejpam-6425	195	10	water	water	NOUN
ejpam-6425	195	11	wheel	wheel	NOUN
ejpam-6425	195	12	graphs	graph	NOUN
ejpam-6425	195	13	s(wwn	s(wwn	NOUN
ejpam-6425	195	14	)	)	PUNCT
ejpam-6425	195	15	,	,	PUNCT
ejpam-6425	195	16	right	right	ADJ
ejpam-6425	195	17	water	water	NOUN
ejpam-6425	195	18	wheel	wheel	NOUN
ejpam-6425	195	19	graphs	graph	NOUN
ejpam-6425	195	20	s(rwwn	s(rwwn	NOUN
ejpam-6425	195	21	)	)	PUNCT
ejpam-6425	195	22	,	,	PUNCT
ejpam-6425	195	23	and	and	CCONJ
ejpam-6425	195	24	left	leave	VERB
ejpam-6425	195	25	water	water	NOUN
ejpam-6425	195	26	wheel	wheel	NOUN
ejpam-6425	195	27	graphs	graph	NOUN
ejpam-6425	195	28	s(lwwn	s(lwwn	NOUN
ejpam-6425	195	29	)	)	PUNCT
ejpam-6425	195	30	.	.	PUNCT
ejpam-6425	196	1	theorem	theorem	NOUN
ejpam-6425	196	2	5	5	NUM
ejpam-6425	196	3	.	.	PUNCT
ejpam-6425	197	1	among	among	ADP
ejpam-6425	197	2	all	all	DET
ejpam-6425	197	3	splitting	splitting	NOUN
ejpam-6425	197	4	graphs	graph	NOUN
ejpam-6425	197	5	of	of	ADP
ejpam-6425	197	6	water	water	NOUN
ejpam-6425	197	7	wheel	wheel	NOUN
ejpam-6425	197	8	graphs	graph	NOUN
ejpam-6425	197	9	,	,	PUNCT
ejpam-6425	197	10	only	only	ADV
ejpam-6425	197	11	s(ww3	s(ww3	PROPN
ejpam-6425	197	12	)	)	PUNCT
ejpam-6425	197	13	is	be	AUX
ejpam-6425	197	14	edgegraceful	edgegraceful	ADJ
ejpam-6425	197	15	.	.	PUNCT
ejpam-6425	198	1	proof	proof	NOUN
ejpam-6425	198	2	.	.	PUNCT
ejpam-6425	199	1	first	first	ADV
ejpam-6425	199	2	,	,	PUNCT
ejpam-6425	199	3	using	use	VERB
ejpam-6425	199	4	definiton	definiton	PROPN
ejpam-6425	199	5	2	2	NUM
ejpam-6425	199	6	and	and	CCONJ
ejpam-6425	199	7	proposition	proposition	NOUN
ejpam-6425	199	8	1	1	NUM
ejpam-6425	199	9	,	,	PUNCT
ejpam-6425	199	10	the	the	DET
ejpam-6425	199	11	splitting	splitting	NOUN
ejpam-6425	199	12	graph	graph	NOUN
ejpam-6425	199	13	s(wwn	s(wwn	NOUN
ejpam-6425	199	14	)	)	PUNCT
ejpam-6425	199	15	has	have	VERB
ejpam-6425	199	16	6n+	6n+	NUM
ejpam-6425	199	17	2	2	NUM
ejpam-6425	199	18	vertices	vertex	NOUN
ejpam-6425	199	19	and	and	CCONJ
ejpam-6425	199	20	15n	15n	NOUN
ejpam-6425	199	21	edges	edge	NOUN
ejpam-6425	199	22	.	.	PUNCT
ejpam-6425	200	1	hence	hence	ADV
ejpam-6425	200	2	,	,	PUNCT
ejpam-6425	200	3	we	we	PRON
ejpam-6425	200	4	substitute	substitute	VERB
ejpam-6425	200	5	p	p	NOUN
ejpam-6425	200	6	=	=	PUNCT
ejpam-6425	200	7	6n+	6n+	NUM
ejpam-6425	200	8	2	2	NUM
ejpam-6425	200	9	and	and	CCONJ
ejpam-6425	200	10	q	q	NOUN
ejpam-6425	200	11	=	=	NOUN
ejpam-6425	200	12	15n	15n	PROPN
ejpam-6425	200	13	in	in	ADP
ejpam-6425	200	14	theorem	theorem	PROPN
ejpam-6425	200	15	1(lo	1(lo	PROPN
ejpam-6425	200	16	’s	’s	PART
ejpam-6425	200	17	theorem	theorem	NOUN
ejpam-6425	200	18	)	)	PUNCT
ejpam-6425	200	19	.	.	PUNCT
ejpam-6425	201	1	that	that	PRON
ejpam-6425	201	2	is	be	AUX
ejpam-6425	201	3	,	,	PUNCT
ejpam-6425	201	4	(	(	PUNCT
ejpam-6425	201	5	6n+	6n+	NUM
ejpam-6425	201	6	2	2	NUM
ejpam-6425	201	7	)	)	PUNCT
ejpam-6425	201	8	∣∣∣((15n)2	∣∣∣((15n)2	NOUN
ejpam-6425	202	1	+	+	CCONJ
ejpam-6425	202	2	15n−	15n−	NUM
ejpam-6425	202	3	(	(	PUNCT
ejpam-6425	202	4	6n+	6n+	NUM
ejpam-6425	202	5	2)(6n+	2)(6n+	NUM
ejpam-6425	202	6	1	1	NUM
ejpam-6425	202	7	)	)	PUNCT
ejpam-6425	202	8	2	2	NUM
ejpam-6425	202	9	)	)	PUNCT
ejpam-6425	202	10	,	,	PUNCT
ejpam-6425	202	11	a.	a.	PROPN
ejpam-6425	202	12	d.	d.	PROPN
ejpam-6425	202	13	c.	c.	PROPN
ejpam-6425	202	14	angel	angel	PROPN
ejpam-6425	202	15	et	et	PROPN
ejpam-6425	202	16	al	al	PROPN
ejpam-6425	202	17	.	.	PUNCT
ejpam-6425	202	18	/	/	SYM
ejpam-6425	202	19	eur	eur	PROPN
ejpam-6425	202	20	.	.	PUNCT
ejpam-6425	203	1	j.	j.	PROPN
ejpam-6425	203	2	pure	pure	PROPN
ejpam-6425	203	3	appl	appl	PROPN
ejpam-6425	203	4	.	.	PROPN
ejpam-6425	203	5	math	math	PROPN
ejpam-6425	203	6	,	,	PUNCT
ejpam-6425	203	7	18	18	NUM
ejpam-6425	203	8	(	(	PUNCT
ejpam-6425	203	9	4	4	NUM
ejpam-6425	203	10	)	)	PUNCT
ejpam-6425	203	11	(	(	PUNCT
ejpam-6425	203	12	2025	2025	NUM
ejpam-6425	203	13	)	)	PUNCT
ejpam-6425	203	14	,	,	PUNCT
ejpam-6425	203	15	6425	6425	NUM
ejpam-6425	203	16	12	12	NUM
ejpam-6425	203	17	of	of	ADP
ejpam-6425	203	18	33	33	NUM
ejpam-6425	203	19	which	which	PRON
ejpam-6425	203	20	further	far	ADV
ejpam-6425	203	21	implies	imply	VERB
ejpam-6425	203	22	that	that	SCONJ
ejpam-6425	203	23	,	,	PUNCT
ejpam-6425	203	24	(	(	PUNCT
ejpam-6425	203	25	207n2	207n2	NUM
ejpam-6425	204	1	+	+	CCONJ
ejpam-6425	204	2	6n−	6n−	NUM
ejpam-6425	204	3	1	1	NUM
ejpam-6425	204	4	6n+	6n+	NUM
ejpam-6425	204	5	2	2	NUM
ejpam-6425	204	6	)	)	PUNCT
ejpam-6425	204	7	∈	∈	PROPN
ejpam-6425	204	8	z.	z.	PROPN
ejpam-6425	205	1	this	this	PRON
ejpam-6425	205	2	means	mean	VERB
ejpam-6425	205	3	,	,	PUNCT
ejpam-6425	205	4	207n2	207n2	NUM
ejpam-6425	205	5	+	+	CCONJ
ejpam-6425	206	1	6n−	6n−	NUM
ejpam-6425	206	2	1	1	NUM
ejpam-6425	206	3	=	=	SYM
ejpam-6425	206	4	(	(	PUNCT
ejpam-6425	206	5	6n+	6n+	NUM
ejpam-6425	206	6	2)k	2)k	NOUN
ejpam-6425	206	7	=	=	SYM
ejpam-6425	206	8	6nk	6nk	NOUN
ejpam-6425	206	9	+	+	CCONJ
ejpam-6425	206	10	2k	2k	NUM
ejpam-6425	206	11	,	,	PUNCT
ejpam-6425	206	12	for	for	ADP
ejpam-6425	206	13	some	some	DET
ejpam-6425	206	14	k	k	PROPN
ejpam-6425	206	15	∈	∈	PROPN
ejpam-6425	206	16	z.	z.	PROPN
ejpam-6425	207	1	thus	thus	ADV
ejpam-6425	207	2	,	,	PUNCT
ejpam-6425	207	3	we	we	PRON
ejpam-6425	207	4	solve	solve	VERB
ejpam-6425	207	5	for	for	ADP
ejpam-6425	207	6	all	all	DET
ejpam-6425	207	7	the	the	DET
ejpam-6425	207	8	possible	possible	ADJ
ejpam-6425	207	9	integer	integer	NOUN
ejpam-6425	207	10	values	value	NOUN
ejpam-6425	207	11	of	of	ADP
ejpam-6425	207	12	n	n	PRON
ejpam-6425	207	13	and	and	CCONJ
ejpam-6425	207	14	k	k	X
ejpam-6425	207	15	in	in	ADP
ejpam-6425	207	16	the	the	DET
ejpam-6425	207	17	diophantine	diophantine	NOUN
ejpam-6425	207	18	equation	equation	NOUN
ejpam-6425	207	19	,	,	PUNCT
ejpam-6425	207	20	207n2	207n2	NUM
ejpam-6425	207	21	+	+	CCONJ
ejpam-6425	208	1	6n−	6n−	PROPN
ejpam-6425	208	2	1−	1−	NUM
ejpam-6425	208	3	6nk	6nk	NOUN
ejpam-6425	208	4	−	−	NOUN
ejpam-6425	208	5	2k	2k	NOUN
ejpam-6425	208	6	=	=	SYM
ejpam-6425	208	7	0	0	NUM
ejpam-6425	208	8	,	,	PUNCT
ejpam-6425	208	9	(	(	PUNCT
ejpam-6425	208	10	12	12	NUM
ejpam-6425	208	11	)	)	PUNCT
ejpam-6425	208	12	where	where	SCONJ
ejpam-6425	208	13	n	n	X
ejpam-6425	208	14	,	,	PUNCT
ejpam-6425	208	15	k	k	PROPN
ejpam-6425	208	16	∈	∈	PROPN
ejpam-6425	208	17	z.	z.	PROPN
ejpam-6425	209	1	this	this	DET
ejpam-6425	209	2	diophantine	diophantine	NOUN
ejpam-6425	209	3	equation	equation	NOUN
ejpam-6425	209	4	is	be	AUX
ejpam-6425	209	5	of	of	ADP
ejpam-6425	209	6	the	the	DET
ejpam-6425	209	7	form	form	NOUN
ejpam-6425	209	8	,	,	PUNCT
ejpam-6425	209	9	an2	an2	PROPN
ejpam-6425	209	10	+	+	CCONJ
ejpam-6425	209	11	bnk	bnk	PROPN
ejpam-6425	209	12	+	+	CCONJ
ejpam-6425	209	13	ck2	ck2	NOUN
ejpam-6425	209	14	+	+	CCONJ
ejpam-6425	209	15	dn+	dn+	PROPN
ejpam-6425	210	1	ek	ek	PROPN
ejpam-6425	211	1	+	+	CCONJ
ejpam-6425	211	2	f	f	X
ejpam-6425	211	3	=	=	SYM
ejpam-6425	211	4	0	0	PROPN
ejpam-6425	211	5	,	,	PUNCT
ejpam-6425	211	6	where	where	SCONJ
ejpam-6425	211	7	a	a	DET
ejpam-6425	211	8	=	=	SYM
ejpam-6425	211	9	207	207	NUM
ejpam-6425	211	10	,	,	PUNCT
ejpam-6425	211	11	b	b	X
ejpam-6425	211	12	=	=	SYM
ejpam-6425	211	13	−6	−6	NOUN
ejpam-6425	211	14	,	,	PUNCT
ejpam-6425	211	15	c	c	NOUN
ejpam-6425	211	16	=	=	SYM
ejpam-6425	211	17	0	0	NUM
ejpam-6425	211	18	,	,	PUNCT
ejpam-6425	211	19	d	d	NOUN
ejpam-6425	211	20	=	=	SYM
ejpam-6425	211	21	6	6	NUM
ejpam-6425	211	22	,	,	PUNCT
ejpam-6425	211	23	e	e	NOUN
ejpam-6425	211	24	=	=	SYM
ejpam-6425	211	25	−2	−2	NOUN
ejpam-6425	211	26	,	,	PUNCT
ejpam-6425	211	27	and	and	CCONJ
ejpam-6425	211	28	f	f	PROPN
ejpam-6425	211	29	=	=	SYM
ejpam-6425	211	30	−1	−1	NOUN
ejpam-6425	211	31	.	.	PUNCT
ejpam-6425	212	1	using	use	VERB
ejpam-6425	212	2	the	the	DET
ejpam-6425	212	3	transformation	transformation	NOUN
ejpam-6425	212	4	stated	state	VERB
ejpam-6425	212	5	in	in	ADP
ejpam-6425	212	6	lemma	lemma	PROPN
ejpam-6425	212	7	1	1	NUM
ejpam-6425	212	8	,	,	PUNCT
ejpam-6425	212	9	equation	equation	NOUN
ejpam-6425	212	10	(	(	PUNCT
ejpam-6425	212	11	12	12	NUM
ejpam-6425	212	12	)	)	PUNCT
ejpam-6425	212	13	can	can	AUX
ejpam-6425	212	14	be	be	AUX
ejpam-6425	212	15	reduced	reduce	VERB
ejpam-6425	212	16	to	to	ADP
ejpam-6425	212	17	x2	x2	PROPN
ejpam-6425	212	18	−	−	PROPN
ejpam-6425	212	19	36y2	36y2	NUM
ejpam-6425	212	20	=	=	SYM
ejpam-6425	212	21	596160	596160	NUM
ejpam-6425	212	22	.	.	PUNCT
ejpam-6425	213	1	(	(	PUNCT
ejpam-6425	213	2	13	13	NUM
ejpam-6425	213	3	)	)	PUNCT
ejpam-6425	213	4	where	where	SCONJ
ejpam-6425	213	5	y	y	PROPN
ejpam-6425	213	6	=	=	SYM
ejpam-6425	213	7	2an+	2an+	NUM
ejpam-6425	213	8	bk	bk	ADP
ejpam-6425	213	9	+	+	PROPN
ejpam-6425	213	10	d	d	X
ejpam-6425	213	11	=	=	SYM
ejpam-6425	213	12	414n−	414n−	NUM
ejpam-6425	213	13	6k	6k	NOUN
ejpam-6425	213	14	+	+	CCONJ
ejpam-6425	213	15	6	6	NUM
ejpam-6425	213	16	and	and	CCONJ
ejpam-6425	213	17	x	x	SYM
ejpam-6425	214	1	=	=	SYM
ejpam-6425	214	2	dk	dk	X
ejpam-6425	214	3	+	+	CCONJ
ejpam-6425	214	4	e	e	X
ejpam-6425	214	5	=	=	PROPN
ejpam-6425	214	6	36k	36k	NOUN
ejpam-6425	214	7	+	+	CCONJ
ejpam-6425	214	8	792	792	NUM
ejpam-6425	214	9	.	.	PUNCT
ejpam-6425	214	10	by	by	ADP
ejpam-6425	214	11	factoring	factor	VERB
ejpam-6425	214	12	the	the	DET
ejpam-6425	214	13	left	left	ADJ
ejpam-6425	214	14	side	side	NOUN
ejpam-6425	214	15	of	of	ADP
ejpam-6425	214	16	the	the	DET
ejpam-6425	214	17	equation	equation	NOUN
ejpam-6425	214	18	(	(	PUNCT
ejpam-6425	214	19	13	13	NUM
ejpam-6425	214	20	)	)	PUNCT
ejpam-6425	214	21	,	,	PUNCT
ejpam-6425	214	22	we	we	PRON
ejpam-6425	214	23	will	will	AUX
ejpam-6425	214	24	have	have	VERB
ejpam-6425	214	25	,	,	PUNCT
ejpam-6425	214	26	(	(	PUNCT
ejpam-6425	214	27	x	x	SYM
ejpam-6425	214	28	+	+	X
ejpam-6425	214	29	(	(	PUNCT
ejpam-6425	214	30	−6)y	−6)y	NOUN
ejpam-6425	214	31	)	)	PUNCT
ejpam-6425	214	32	(	(	PUNCT
ejpam-6425	214	33	x	x	X
ejpam-6425	214	34	−	−	PROPN
ejpam-6425	214	35	(	(	PUNCT
ejpam-6425	214	36	−6)y	−6)y	NOUN
ejpam-6425	214	37	)	)	PUNCT
ejpam-6425	214	38	=	=	SYM
ejpam-6425	214	39	596160	596160	NUM
ejpam-6425	214	40	,	,	PUNCT
ejpam-6425	214	41	where	where	SCONJ
ejpam-6425	214	42	b	b	X
ejpam-6425	214	43	=	=	SYM
ejpam-6425	214	44	−6	−6	NOUN
ejpam-6425	214	45	.	.	PUNCT
ejpam-6425	215	1	therefore	therefore	ADV
ejpam-6425	215	2	,	,	PUNCT
ejpam-6425	215	3	by	by	ADP
ejpam-6425	215	4	lemma	lemma	PROPN
ejpam-6425	215	5	1	1	NUM
ejpam-6425	215	6	and	and	CCONJ
ejpam-6425	215	7	remark	remark	NOUN
ejpam-6425	215	8	2	2	NUM
ejpam-6425	215	9	,	,	PUNCT
ejpam-6425	215	10	we	we	PRON
ejpam-6425	215	11	have	have	VERB
ejpam-6425	215	12	the	the	DET
ejpam-6425	215	13	following	following	NOUN
ejpam-6425	215	14	:	:	PUNCT
ejpam-6425	215	15	x	x	SYM
ejpam-6425	215	16	=	=	SYM
ejpam-6425	215	17	n1	n1	PROPN
ejpam-6425	215	18	+	+	NOUN
ejpam-6425	215	19	n2	n2	ADJ
ejpam-6425	215	20	2	2	NUM
ejpam-6425	215	21	,	,	PUNCT
ejpam-6425	215	22	y	y	PROPN
ejpam-6425	215	23	=	=	SYM
ejpam-6425	215	24	n1	n1	PROPN
ejpam-6425	216	1	−n2	−n2	PROPN
ejpam-6425	216	2	−12	−12	NOUN
ejpam-6425	216	3	,	,	PUNCT
ejpam-6425	216	4	k	k	X
ejpam-6425	217	1	=	=	PUNCT
ejpam-6425	217	2	x	x	SYM
ejpam-6425	217	3	−	−	NOUN
ejpam-6425	217	4	792	792	NUM
ejpam-6425	217	5	36	36	NUM
ejpam-6425	217	6	,	,	PUNCT
ejpam-6425	217	7	n	n	NOUN
ejpam-6425	217	8	=	=	SYM
ejpam-6425	217	9	y	y	PROPN
ejpam-6425	217	10	+	+	NUM
ejpam-6425	217	11	6k	6k	NOUN
ejpam-6425	217	12	−	−	PROPN
ejpam-6425	217	13	6	6	NUM
ejpam-6425	217	14	414	414	NUM
ejpam-6425	217	15	,	,	PUNCT
ejpam-6425	217	16	(	(	PUNCT
ejpam-6425	217	17	14	14	NUM
ejpam-6425	217	18	)	)	PUNCT
ejpam-6425	217	19	where	where	SCONJ
ejpam-6425	217	20	(	(	PUNCT
ejpam-6425	217	21	n1	n1	NOUN
ejpam-6425	217	22	,	,	PUNCT
ejpam-6425	217	23	n2	n2	ADJ
ejpam-6425	217	24	)	)	PUNCT
ejpam-6425	217	25	are	be	AUX
ejpam-6425	217	26	the	the	DET
ejpam-6425	217	27	factor	factor	NOUN
ejpam-6425	217	28	pairs	pair	NOUN
ejpam-6425	217	29	of	of	ADP
ejpam-6425	217	30	n	n	NOUN
ejpam-6425	217	31	=	=	SYM
ejpam-6425	217	32	596160	596160	NUM
ejpam-6425	217	33	.	.	PUNCT
ejpam-6425	218	1	using	use	VERB
ejpam-6425	218	2	(	(	PUNCT
ejpam-6425	218	3	14	14	NUM
ejpam-6425	218	4	)	)	PUNCT
ejpam-6425	218	5	,	,	PUNCT
ejpam-6425	218	6	we	we	PRON
ejpam-6425	218	7	can	can	AUX
ejpam-6425	218	8	now	now	ADV
ejpam-6425	218	9	solve	solve	VERB
ejpam-6425	218	10	for	for	ADP
ejpam-6425	218	11	the	the	DET
ejpam-6425	218	12	integer	integer	NOUN
ejpam-6425	218	13	values	value	NOUN
ejpam-6425	218	14	of	of	ADP
ejpam-6425	218	15	n	n	PROPN
ejpam-6425	218	16	and	and	CCONJ
ejpam-6425	218	17	k	k	NOUN
ejpam-6425	218	18	,	,	PUNCT
ejpam-6425	218	19	by	by	ADP
ejpam-6425	218	20	considering	consider	VERB
ejpam-6425	218	21	all	all	DET
ejpam-6425	218	22	the	the	DET
ejpam-6425	218	23	factor	factor	NOUN
ejpam-6425	218	24	pairs	pair	NOUN
ejpam-6425	218	25	of	of	ADP
ejpam-6425	218	26	596160	596160	NUM
ejpam-6425	218	27	,	,	PUNCT
ejpam-6425	218	28	as	as	SCONJ
ejpam-6425	218	29	presented	present	VERB
ejpam-6425	218	30	in	in	ADP
ejpam-6425	218	31	table	table	NOUN
ejpam-6425	218	32	3	3	NUM
ejpam-6425	218	33	in	in	ADP
ejpam-6425	218	34	the	the	DET
ejpam-6425	218	35	appendices	appendix	NOUN
ejpam-6425	218	36	section	section	NOUN
ejpam-6425	218	37	.	.	PUNCT
ejpam-6425	219	1	table	table	NOUN
ejpam-6425	219	2	3	3	NUM
ejpam-6425	219	3	shows	show	VERB
ejpam-6425	219	4	that	that	SCONJ
ejpam-6425	219	5	there	there	PRON
ejpam-6425	219	6	are	be	VERB
ejpam-6425	219	7	2	2	NUM
ejpam-6425	219	8	possible	possible	ADJ
ejpam-6425	219	9	solutions	solution	NOUN
ejpam-6425	219	10	for	for	ADP
ejpam-6425	219	11	the	the	DET
ejpam-6425	219	12	diophantine	diophantine	NOUN
ejpam-6425	219	13	equation	equation	NOUN
ejpam-6425	219	14	207n2	207n2	NUM
ejpam-6425	219	15	+	+	CCONJ
ejpam-6425	220	1	6n−	6n−	PROPN
ejpam-6425	220	2	1−	1−	NUM
ejpam-6425	220	3	6nk−	6nk−	NUM
ejpam-6425	220	4	2k	2k	NUM
ejpam-6425	220	5	=	=	SYM
ejpam-6425	220	6	0	0	X
ejpam-6425	220	7	.	.	PUNCT
ejpam-6425	221	1	these	these	PRON
ejpam-6425	221	2	are	be	AUX
ejpam-6425	221	3	:	:	PUNCT
ejpam-6425	221	4	(	(	PUNCT
ejpam-6425	221	5	n	n	CCONJ
ejpam-6425	221	6	,	,	PUNCT
ejpam-6425	221	7	k)=	k)=	X
ejpam-6425	221	8	(	(	PUNCT
ejpam-6425	221	9	3,94	3,94	NUM
ejpam-6425	221	10	)	)	PUNCT
ejpam-6425	221	11	,	,	PUNCT
ejpam-6425	221	12	and	and	CCONJ
ejpam-6425	221	13	(	(	PUNCT
ejpam-6425	221	14	-1,-50	-1,-50	ADJ
ejpam-6425	221	15	)	)	PUNCT
ejpam-6425	221	16	.	.	PUNCT
ejpam-6425	222	1	however	however	ADV
ejpam-6425	222	2	,	,	PUNCT
ejpam-6425	222	3	since	since	SCONJ
ejpam-6425	222	4	we	we	PRON
ejpam-6425	222	5	are	be	AUX
ejpam-6425	222	6	only	only	ADV
ejpam-6425	222	7	concern	concern	NOUN
ejpam-6425	222	8	with	with	ADP
ejpam-6425	222	9	possible	possible	ADJ
ejpam-6425	222	10	values	value	NOUN
ejpam-6425	222	11	of	of	ADP
ejpam-6425	222	12	n	n	PROPN
ejpam-6425	222	13	for	for	ADP
ejpam-6425	222	14	s(wwn	s(wwn	NOUN
ejpam-6425	222	15	)	)	PUNCT
ejpam-6425	222	16	,	,	PUNCT
ejpam-6425	222	17	then	then	ADV
ejpam-6425	222	18	only	only	ADV
ejpam-6425	222	19	(	(	PUNCT
ejpam-6425	222	20	3,94	3,94	NUM
ejpam-6425	222	21	)	)	PUNCT
ejpam-6425	222	22	,	,	PUNCT
ejpam-6425	222	23	where	where	SCONJ
ejpam-6425	222	24	n	n	X
ejpam-6425	222	25	=	=	SYM
ejpam-6425	222	26	3	3	NUM
ejpam-6425	222	27	,	,	PUNCT
ejpam-6425	222	28	is	be	AUX
ejpam-6425	222	29	a	a	DET
ejpam-6425	222	30	valid	valid	ADJ
ejpam-6425	222	31	solution	solution	NOUN
ejpam-6425	222	32	under	under	ADP
ejpam-6425	222	33	this	this	DET
ejpam-6425	222	34	conditions	condition	NOUN
ejpam-6425	222	35	.	.	PUNCT
ejpam-6425	223	1	therefore	therefore	ADV
ejpam-6425	223	2	,	,	PUNCT
ejpam-6425	223	3	we	we	PRON
ejpam-6425	223	4	have	have	AUX
ejpam-6425	223	5	shown	show	VERB
ejpam-6425	223	6	that	that	SCONJ
ejpam-6425	223	7	s(ww3	s(ww3	NOUN
ejpam-6425	223	8	)	)	PUNCT
ejpam-6425	223	9	is	be	AUX
ejpam-6425	223	10	the	the	DET
ejpam-6425	223	11	only	only	ADJ
ejpam-6425	223	12	splitting	splitting	NOUN
ejpam-6425	223	13	graph	graph	NOUN
ejpam-6425	223	14	of	of	ADP
ejpam-6425	223	15	water	water	NOUN
ejpam-6425	223	16	wheel	wheel	NOUN
ejpam-6425	223	17	graph	graph	NOUN
ejpam-6425	223	18	that	that	PRON
ejpam-6425	223	19	satisfies	satisfy	VERB
ejpam-6425	223	20	the	the	DET
ejpam-6425	223	21	lo	lo	PROPN
ejpam-6425	223	22	’s	’s	PART
ejpam-6425	223	23	theorem	theorem	ADJ
ejpam-6425	223	24	.	.	PROPN
ejpam-6425	223	25	to	to	PART
ejpam-6425	223	26	complete	complete	VERB
ejpam-6425	223	27	this	this	DET
ejpam-6425	223	28	proof	proof	NOUN
ejpam-6425	223	29	,	,	PUNCT
ejpam-6425	223	30	an	an	DET
ejpam-6425	223	31	actual	actual	ADJ
ejpam-6425	223	32	example	example	NOUN
ejpam-6425	223	33	of	of	ADP
ejpam-6425	223	34	an	an	DET
ejpam-6425	223	35	edge	edge	NOUN
ejpam-6425	223	36	-	-	PUNCT
ejpam-6425	223	37	graceful	graceful	NOUN
ejpam-6425	223	38	labeling	labeling	NOUN
ejpam-6425	223	39	of	of	ADP
ejpam-6425	223	40	s(ww3	s(ww3	PROPN
ejpam-6425	223	41	)	)	PUNCT
ejpam-6425	223	42	must	must	AUX
ejpam-6425	223	43	be	be	AUX
ejpam-6425	223	44	provided	provide	VERB
ejpam-6425	223	45	.	.	PUNCT
ejpam-6425	224	1	s(ww3	s(ww3	PROPN
ejpam-6425	224	2	)	)	PUNCT
ejpam-6425	224	3	is	be	AUX
ejpam-6425	224	4	a	a	DET
ejpam-6425	224	5	very	very	ADV
ejpam-6425	224	6	large	large	ADJ
ejpam-6425	224	7	graph	graph	NOUN
ejpam-6425	224	8	with	with	ADP
ejpam-6425	224	9	20	20	NUM
ejpam-6425	224	10	vertices	vertex	NOUN
ejpam-6425	224	11	and	and	CCONJ
ejpam-6425	224	12	45	45	NUM
ejpam-6425	224	13	edges	edge	NOUN
ejpam-6425	224	14	.	.	PUNCT
ejpam-6425	225	1	by	by	ADP
ejpam-6425	225	2	permutation	permutation	NOUN
ejpam-6425	225	3	,	,	PUNCT
ejpam-6425	225	4	there	there	PRON
ejpam-6425	225	5	is	be	VERB
ejpam-6425	225	6	a	a	DET
ejpam-6425	225	7	total	total	NOUN
ejpam-6425	225	8	of	of	ADP
ejpam-6425	225	9	45	45	NUM
ejpam-6425	225	10	!	!	PUNCT
ejpam-6425	226	1	possible	possible	ADJ
ejpam-6425	226	2	labeling	labeling	NOUN
ejpam-6425	226	3	for	for	ADP
ejpam-6425	226	4	its	its	PRON
ejpam-6425	226	5	edges	edge	NOUN
ejpam-6425	226	6	.	.	PUNCT
ejpam-6425	227	1	thus	thus	ADV
ejpam-6425	227	2	,	,	PUNCT
ejpam-6425	227	3	the	the	DET
ejpam-6425	227	4	researchers	researcher	NOUN
ejpam-6425	227	5	created	create	VERB
ejpam-6425	227	6	and	and	CCONJ
ejpam-6425	227	7	used	use	VERB
ejpam-6425	227	8	a	a	DET
ejpam-6425	227	9	python	python	NOUN
ejpam-6425	227	10	computer	computer	NOUN
ejpam-6425	227	11	program	program	NOUN
ejpam-6425	227	12	to	to	PART
ejpam-6425	227	13	generate	generate	VERB
ejpam-6425	227	14	an	an	DET
ejpam-6425	227	15	edge	edge	NOUN
ejpam-6425	227	16	-	-	PUNCT
ejpam-6425	227	17	graceful	graceful	NOUN
ejpam-6425	227	18	labeling	labeling	NOUN
ejpam-6425	227	19	for	for	ADP
ejpam-6425	227	20	the	the	DET
ejpam-6425	227	21	said	say	VERB
ejpam-6425	227	22	graph	graph	NOUN
ejpam-6425	227	23	.	.	PUNCT
ejpam-6425	228	1	one	one	NUM
ejpam-6425	228	2	of	of	ADP
ejpam-6425	228	3	the	the	DET
ejpam-6425	228	4	results	result	NOUN
ejpam-6425	228	5	from	from	ADP
ejpam-6425	228	6	the	the	DET
ejpam-6425	228	7	computer	computer	NOUN
ejpam-6425	228	8	program	program	NOUN
ejpam-6425	228	9	is	be	AUX
ejpam-6425	228	10	the	the	DET
ejpam-6425	228	11	labeling	labeling	NOUN
ejpam-6425	228	12	shown	show	VERB
ejpam-6425	228	13	in	in	ADP
ejpam-6425	228	14	figure	figure	NOUN
ejpam-6425	228	15	7	7	NUM
ejpam-6425	228	16	.	.	PUNCT
ejpam-6425	228	17	a.	a.	PROPN
ejpam-6425	228	18	d.	d.	PROPN
ejpam-6425	228	19	c.	c.	PROPN
ejpam-6425	228	20	angel	angel	PROPN
ejpam-6425	228	21	et	et	PROPN
ejpam-6425	228	22	al	al	PROPN
ejpam-6425	228	23	.	.	PUNCT
ejpam-6425	228	24	/	/	SYM
ejpam-6425	228	25	eur	eur	PROPN
ejpam-6425	228	26	.	.	PUNCT
ejpam-6425	229	1	j.	j.	PROPN
ejpam-6425	229	2	pure	pure	PROPN
ejpam-6425	229	3	appl	appl	PROPN
ejpam-6425	229	4	.	.	PROPN
ejpam-6425	229	5	math	math	PROPN
ejpam-6425	229	6	,	,	PUNCT
ejpam-6425	229	7	18	18	NUM
ejpam-6425	229	8	(	(	PUNCT
ejpam-6425	229	9	4	4	NUM
ejpam-6425	229	10	)	)	PUNCT
ejpam-6425	229	11	(	(	PUNCT
ejpam-6425	229	12	2025	2025	NUM
ejpam-6425	229	13	)	)	PUNCT
ejpam-6425	229	14	,	,	PUNCT
ejpam-6425	229	15	6425	6425	NUM
ejpam-6425	229	16	13	13	NUM
ejpam-6425	229	17	of	of	ADP
ejpam-6425	229	18	33	33	NUM
ejpam-6425	229	19	figure	figure	NOUN
ejpam-6425	229	20	7	7	NUM
ejpam-6425	229	21	:	:	PUNCT
ejpam-6425	229	22	an	an	DET
ejpam-6425	229	23	edge	edge	NOUN
ejpam-6425	229	24	-	-	PUNCT
ejpam-6425	229	25	graceful	graceful	NOUN
ejpam-6425	229	26	labeling	labeling	NOUN
ejpam-6425	229	27	of	of	ADP
ejpam-6425	229	28	s(ww3	s(ww3	PROPN
ejpam-6425	229	29	)	)	PUNCT
ejpam-6425	229	30	as	as	SCONJ
ejpam-6425	229	31	shown	show	VERB
ejpam-6425	229	32	in	in	ADP
ejpam-6425	229	33	figure	figure	NOUN
ejpam-6425	229	34	7	7	NUM
ejpam-6425	229	35	,	,	PUNCT
ejpam-6425	229	36	the	the	DET
ejpam-6425	229	37	45	45	NUM
ejpam-6425	229	38	edges	edge	NOUN
ejpam-6425	229	39	of	of	ADP
ejpam-6425	229	40	s(ww3	s(ww3	PROPN
ejpam-6425	229	41	)	)	PUNCT
ejpam-6425	229	42	are	be	AUX
ejpam-6425	229	43	labeled	label	VERB
ejpam-6425	229	44	using	use	VERB
ejpam-6425	229	45	distinct	distinct	ADJ
ejpam-6425	229	46	integers	integer	NOUN
ejpam-6425	229	47	from	from	ADP
ejpam-6425	229	48	1	1	NUM
ejpam-6425	229	49	to	to	ADP
ejpam-6425	229	50	45	45	NUM
ejpam-6425	229	51	.	.	PUNCT
ejpam-6425	230	1	then	then	ADV
ejpam-6425	230	2	,	,	PUNCT
ejpam-6425	230	3	by	by	ADP
ejpam-6425	230	4	definition	definition	NOUN
ejpam-6425	230	5	1	1	NUM
ejpam-6425	230	6	,	,	PUNCT
ejpam-6425	230	7	the	the	DET
ejpam-6425	230	8	20	20	NUM
ejpam-6425	230	9	vertices	vertex	NOUN
ejpam-6425	230	10	are	be	AUX
ejpam-6425	230	11	successfully	successfully	ADV
ejpam-6425	230	12	labeled	label	VERB
ejpam-6425	230	13	using	use	VERB
ejpam-6425	230	14	distinct	distinct	ADJ
ejpam-6425	230	15	integers	integer	NOUN
ejpam-6425	230	16	from	from	ADP
ejpam-6425	230	17	0	0	NUM
ejpam-6425	230	18	to	to	ADP
ejpam-6425	230	19	19	19	NUM
ejpam-6425	230	20	.	.	PUNCT
ejpam-6425	231	1	therefore	therefore	ADV
ejpam-6425	231	2	,	,	PUNCT
ejpam-6425	231	3	the	the	DET
ejpam-6425	231	4	labeled	label	VERB
ejpam-6425	231	5	graph	graph	NOUN
ejpam-6425	231	6	in	in	ADP
ejpam-6425	231	7	figure	figure	NOUN
ejpam-6425	231	8	7	7	NUM
ejpam-6425	231	9	is	be	AUX
ejpam-6425	231	10	indeed	indeed	ADV
ejpam-6425	231	11	an	an	DET
ejpam-6425	231	12	edge	edge	NOUN
ejpam-6425	231	13	-	-	PUNCT
ejpam-6425	231	14	graceful	graceful	NOUN
ejpam-6425	231	15	labeling	labeling	NOUN
ejpam-6425	231	16	of	of	ADP
ejpam-6425	231	17	s(ww3	s(ww3	PROPN
ejpam-6425	231	18	)	)	PUNCT
ejpam-6425	231	19	.	.	PUNCT
ejpam-6425	232	1	since	since	SCONJ
ejpam-6425	232	2	all	all	DET
ejpam-6425	232	3	the	the	DET
ejpam-6425	232	4	needed	need	VERB
ejpam-6425	232	5	conditions	condition	NOUN
ejpam-6425	232	6	are	be	AUX
ejpam-6425	232	7	satisfied	satisfied	ADJ
ejpam-6425	232	8	,	,	PUNCT
ejpam-6425	232	9	then	then	ADV
ejpam-6425	232	10	we	we	PRON
ejpam-6425	232	11	can	can	AUX
ejpam-6425	232	12	now	now	ADV
ejpam-6425	232	13	conclude	conclude	VERB
ejpam-6425	232	14	that	that	SCONJ
ejpam-6425	232	15	s(ww3	s(ww3	PROPN
ejpam-6425	232	16	)	)	PUNCT
ejpam-6425	232	17	is	be	AUX
ejpam-6425	232	18	the	the	DET
ejpam-6425	232	19	only	only	ADJ
ejpam-6425	232	20	edge	edge	NOUN
ejpam-6425	232	21	-	-	PUNCT
ejpam-6425	232	22	graceful	graceful	NOUN
ejpam-6425	232	23	splitting	splitting	NOUN
ejpam-6425	232	24	graph	graph	NOUN
ejpam-6425	232	25	of	of	ADP
ejpam-6425	232	26	water	water	NOUN
ejpam-6425	232	27	wheel	wheel	NOUN
ejpam-6425	232	28	graphs	graph	NOUN
ejpam-6425	232	29	.	.	PUNCT
ejpam-6425	233	1	theorem	theorem	VERB
ejpam-6425	233	2	6	6	NUM
ejpam-6425	233	3	.	.	PUNCT
ejpam-6425	234	1	among	among	ADP
ejpam-6425	234	2	all	all	DET
ejpam-6425	234	3	splitting	splitting	NOUN
ejpam-6425	234	4	graphs	graph	NOUN
ejpam-6425	234	5	of	of	ADP
ejpam-6425	234	6	right	right	ADJ
ejpam-6425	234	7	water	water	NOUN
ejpam-6425	234	8	wheel	wheel	NOUN
ejpam-6425	234	9	graphs	graph	NOUN
ejpam-6425	234	10	,	,	PUNCT
ejpam-6425	234	11	only	only	ADV
ejpam-6425	234	12	s(rww3	s(rww3	X
ejpam-6425	234	13	)	)	PUNCT
ejpam-6425	234	14	is	be	AUX
ejpam-6425	234	15	edge	edge	NOUN
ejpam-6425	234	16	-	-	PUNCT
ejpam-6425	234	17	graceful	graceful	NOUN
ejpam-6425	234	18	.	.	PUNCT
ejpam-6425	235	1	proof	proof	NOUN
ejpam-6425	235	2	.	.	PUNCT
ejpam-6425	236	1	first	first	ADV
ejpam-6425	236	2	,	,	PUNCT
ejpam-6425	236	3	using	use	VERB
ejpam-6425	236	4	definiton	definiton	PROPN
ejpam-6425	236	5	3	3	NUM
ejpam-6425	236	6	and	and	CCONJ
ejpam-6425	236	7	proposition	proposition	NOUN
ejpam-6425	236	8	1	1	NUM
ejpam-6425	236	9	,	,	PUNCT
ejpam-6425	236	10	the	the	DET
ejpam-6425	236	11	splitting	splitting	NOUN
ejpam-6425	236	12	graph	graph	NOUN
ejpam-6425	236	13	s(rwwn	s(rwwn	NOUN
ejpam-6425	236	14	)	)	PUNCT
ejpam-6425	236	15	has	have	VERB
ejpam-6425	236	16	6n+	6n+	NUM
ejpam-6425	236	17	2	2	NUM
ejpam-6425	236	18	vertices	vertex	NOUN
ejpam-6425	236	19	and	and	CCONJ
ejpam-6425	236	20	18n	18n	NOUN
ejpam-6425	236	21	edges	edge	NOUN
ejpam-6425	236	22	.	.	PUNCT
ejpam-6425	237	1	hence	hence	ADV
ejpam-6425	237	2	,	,	PUNCT
ejpam-6425	237	3	we	we	PRON
ejpam-6425	237	4	substitute	substitute	VERB
ejpam-6425	237	5	p	p	NOUN
ejpam-6425	237	6	=	=	PUNCT
ejpam-6425	237	7	6n+	6n+	NUM
ejpam-6425	237	8	2	2	NUM
ejpam-6425	237	9	and	and	CCONJ
ejpam-6425	237	10	q	q	NOUN
ejpam-6425	237	11	=	=	NOUN
ejpam-6425	237	12	18n	18n	NOUN
ejpam-6425	237	13	in	in	ADP
ejpam-6425	237	14	theorem	theorem	ADJ
ejpam-6425	237	15	1(lo	1(lo	PROPN
ejpam-6425	237	16	’s	’s	PART
ejpam-6425	237	17	theorem	theorem	NOUN
ejpam-6425	237	18	)	)	PUNCT
ejpam-6425	237	19	.	.	PUNCT
ejpam-6425	238	1	that	that	PRON
ejpam-6425	238	2	is	be	AUX
ejpam-6425	238	3	,	,	PUNCT
ejpam-6425	238	4	(	(	PUNCT
ejpam-6425	238	5	6n+	6n+	NUM
ejpam-6425	238	6	2	2	NUM
ejpam-6425	238	7	)	)	PUNCT
ejpam-6425	238	8	∣∣∣((18n)2	∣∣∣((18n)2	NOUN
ejpam-6425	239	1	+	+	CCONJ
ejpam-6425	239	2	18n−	18n−	NUM
ejpam-6425	239	3	(	(	PUNCT
ejpam-6425	239	4	6n+	6n+	NUM
ejpam-6425	239	5	2)(6n+	2)(6n+	NUM
ejpam-6425	239	6	1	1	NUM
ejpam-6425	239	7	)	)	PUNCT
ejpam-6425	239	8	2	2	NUM
ejpam-6425	239	9	)	)	PUNCT
ejpam-6425	239	10	,	,	PUNCT
ejpam-6425	239	11	a.	a.	PROPN
ejpam-6425	239	12	d.	d.	PROPN
ejpam-6425	239	13	c.	c.	PROPN
ejpam-6425	239	14	angel	angel	PROPN
ejpam-6425	239	15	et	et	PROPN
ejpam-6425	239	16	al	al	PROPN
ejpam-6425	239	17	.	.	PUNCT
ejpam-6425	239	18	/	/	SYM
ejpam-6425	239	19	eur	eur	PROPN
ejpam-6425	239	20	.	.	PUNCT
ejpam-6425	240	1	j.	j.	PROPN
ejpam-6425	240	2	pure	pure	PROPN
ejpam-6425	240	3	appl	appl	PROPN
ejpam-6425	240	4	.	.	PROPN
ejpam-6425	240	5	math	math	PROPN
ejpam-6425	240	6	,	,	PUNCT
ejpam-6425	240	7	18	18	NUM
ejpam-6425	240	8	(	(	PUNCT
ejpam-6425	240	9	4	4	NUM
ejpam-6425	240	10	)	)	PUNCT
ejpam-6425	240	11	(	(	PUNCT
ejpam-6425	240	12	2025	2025	NUM
ejpam-6425	240	13	)	)	PUNCT
ejpam-6425	240	14	,	,	PUNCT
ejpam-6425	240	15	6425	6425	NUM
ejpam-6425	240	16	14	14	NUM
ejpam-6425	240	17	of	of	ADP
ejpam-6425	240	18	33	33	NUM
ejpam-6425	240	19	which	which	PRON
ejpam-6425	240	20	further	far	ADV
ejpam-6425	240	21	implies	imply	VERB
ejpam-6425	240	22	that	that	SCONJ
ejpam-6425	240	23	,	,	PUNCT
ejpam-6425	240	24	(	(	PUNCT
ejpam-6425	240	25	306n2	306n2	NUM
ejpam-6425	240	26	+	+	NOUN
ejpam-6425	240	27	9n−	9n−	NUM
ejpam-6425	240	28	1	1	NUM
ejpam-6425	240	29	6n+	6n+	NUM
ejpam-6425	240	30	2	2	NUM
ejpam-6425	240	31	)	)	PUNCT
ejpam-6425	240	32	∈	∈	PROPN
ejpam-6425	240	33	z.	z.	PROPN
ejpam-6425	240	34	this	this	PRON
ejpam-6425	240	35	means	mean	VERB
ejpam-6425	240	36	,	,	PUNCT
ejpam-6425	240	37	306n2	306n2	NUM
ejpam-6425	240	38	+	+	NOUN
ejpam-6425	240	39	9n−	9n−	NUM
ejpam-6425	240	40	1	1	NUM
ejpam-6425	240	41	=	=	SYM
ejpam-6425	240	42	(	(	PUNCT
ejpam-6425	240	43	6n+	6n+	NUM
ejpam-6425	240	44	2)k	2)k	NOUN
ejpam-6425	240	45	=	=	SYM
ejpam-6425	240	46	6nk	6nk	NOUN
ejpam-6425	240	47	+	+	CCONJ
ejpam-6425	240	48	2k	2k	NUM
ejpam-6425	240	49	,	,	PUNCT
ejpam-6425	240	50	for	for	ADP
ejpam-6425	240	51	some	some	DET
ejpam-6425	240	52	k	k	PROPN
ejpam-6425	240	53	∈	∈	PROPN
ejpam-6425	240	54	z.	z.	PROPN
ejpam-6425	241	1	thus	thus	ADV
ejpam-6425	241	2	,	,	PUNCT
ejpam-6425	241	3	we	we	PRON
ejpam-6425	241	4	solve	solve	VERB
ejpam-6425	241	5	for	for	ADP
ejpam-6425	241	6	all	all	DET
ejpam-6425	241	7	the	the	DET
ejpam-6425	241	8	possible	possible	ADJ
ejpam-6425	241	9	integer	integer	NOUN
ejpam-6425	241	10	values	value	NOUN
ejpam-6425	241	11	of	of	ADP
ejpam-6425	241	12	n	n	PRON
ejpam-6425	241	13	and	and	CCONJ
ejpam-6425	241	14	k	k	X
ejpam-6425	241	15	in	in	ADP
ejpam-6425	241	16	the	the	DET
ejpam-6425	241	17	diophantine	diophantine	NOUN
ejpam-6425	241	18	equation	equation	NOUN
ejpam-6425	241	19	,	,	PUNCT
ejpam-6425	241	20	306n2	306n2	NUM
ejpam-6425	241	21	+	+	SYM
ejpam-6425	241	22	9n−	9n−	NUM
ejpam-6425	241	23	1−	1−	NUM
ejpam-6425	241	24	6nk	6nk	NOUN
ejpam-6425	241	25	−	−	NOUN
ejpam-6425	241	26	2k	2k	NOUN
ejpam-6425	241	27	=	=	SYM
ejpam-6425	241	28	0	0	NUM
ejpam-6425	241	29	,	,	PUNCT
ejpam-6425	241	30	(	(	PUNCT
ejpam-6425	241	31	15	15	NUM
ejpam-6425	241	32	)	)	PUNCT
ejpam-6425	241	33	where	where	SCONJ
ejpam-6425	241	34	n	n	X
ejpam-6425	241	35	,	,	PUNCT
ejpam-6425	241	36	k	k	PROPN
ejpam-6425	241	37	∈	∈	PROPN
ejpam-6425	241	38	z.	z.	PROPN
ejpam-6425	242	1	this	this	DET
ejpam-6425	242	2	diophantine	diophantine	NOUN
ejpam-6425	242	3	equation	equation	NOUN
ejpam-6425	242	4	is	be	AUX
ejpam-6425	242	5	of	of	ADP
ejpam-6425	242	6	the	the	DET
ejpam-6425	242	7	form	form	NOUN
ejpam-6425	242	8	,	,	PUNCT
ejpam-6425	242	9	an2	an2	PROPN
ejpam-6425	242	10	+	+	CCONJ
ejpam-6425	242	11	bnk	bnk	PROPN
ejpam-6425	242	12	+	+	CCONJ
ejpam-6425	242	13	ck2	ck2	NOUN
ejpam-6425	242	14	+	+	CCONJ
ejpam-6425	242	15	dn+	dn+	PROPN
ejpam-6425	243	1	ek	ek	PROPN
ejpam-6425	244	1	+	+	CCONJ
ejpam-6425	244	2	f	f	X
ejpam-6425	244	3	=	=	SYM
ejpam-6425	244	4	0	0	PROPN
ejpam-6425	244	5	,	,	PUNCT
ejpam-6425	244	6	where	where	SCONJ
ejpam-6425	244	7	a	a	DET
ejpam-6425	244	8	=	=	SYM
ejpam-6425	244	9	306	306	NUM
ejpam-6425	244	10	,	,	PUNCT
ejpam-6425	244	11	b	b	X
ejpam-6425	244	12	=	=	SYM
ejpam-6425	244	13	−6	−6	NOUN
ejpam-6425	244	14	,	,	PUNCT
ejpam-6425	244	15	c	c	NOUN
ejpam-6425	244	16	=	=	SYM
ejpam-6425	244	17	0	0	NUM
ejpam-6425	244	18	,	,	PUNCT
ejpam-6425	244	19	d	d	NOUN
ejpam-6425	244	20	=	=	SYM
ejpam-6425	244	21	9	9	NUM
ejpam-6425	244	22	,	,	PUNCT
ejpam-6425	244	23	e	e	NOUN
ejpam-6425	244	24	=	=	SYM
ejpam-6425	244	25	−2	−2	NOUN
ejpam-6425	244	26	,	,	PUNCT
ejpam-6425	244	27	and	and	CCONJ
ejpam-6425	244	28	f	f	PROPN
ejpam-6425	244	29	=	=	SYM
ejpam-6425	244	30	−1	−1	NOUN
ejpam-6425	244	31	.	.	PUNCT
ejpam-6425	245	1	using	use	VERB
ejpam-6425	245	2	the	the	DET
ejpam-6425	245	3	transformation	transformation	NOUN
ejpam-6425	245	4	stated	state	VERB
ejpam-6425	245	5	in	in	ADP
ejpam-6425	245	6	lemma	lemma	PROPN
ejpam-6425	245	7	1	1	NUM
ejpam-6425	245	8	,	,	PUNCT
ejpam-6425	245	9	equation	equation	NOUN
ejpam-6425	245	10	(	(	PUNCT
ejpam-6425	245	11	15	15	NUM
ejpam-6425	245	12	)	)	PUNCT
ejpam-6425	245	13	can	can	AUX
ejpam-6425	245	14	be	be	AUX
ejpam-6425	245	15	reduced	reduce	VERB
ejpam-6425	245	16	to	to	ADP
ejpam-6425	245	17	x2	x2	PROPN
ejpam-6425	245	18	−	−	PROPN
ejpam-6425	245	19	36y2	36y2	NUM
ejpam-6425	245	20	=	=	SYM
ejpam-6425	245	21	1321920	1321920	NUM
ejpam-6425	245	22	.	.	PUNCT
ejpam-6425	246	1	(	(	PUNCT
ejpam-6425	246	2	16	16	NUM
ejpam-6425	246	3	)	)	PUNCT
ejpam-6425	246	4	where	where	SCONJ
ejpam-6425	246	5	y	y	PROPN
ejpam-6425	246	6	=	=	SYM
ejpam-6425	246	7	2an+	2an+	NUM
ejpam-6425	246	8	bk	bk	ADP
ejpam-6425	246	9	+	+	PROPN
ejpam-6425	246	10	d	d	X
ejpam-6425	246	11	=	=	SYM
ejpam-6425	246	12	612n−	612n−	NUM
ejpam-6425	246	13	6k	6k	NOUN
ejpam-6425	246	14	+	+	CCONJ
ejpam-6425	246	15	9	9	NUM
ejpam-6425	246	16	and	and	CCONJ
ejpam-6425	246	17	x	x	SYM
ejpam-6425	246	18	=	=	SYM
ejpam-6425	246	19	dk	dk	X
ejpam-6425	246	20	+	+	CCONJ
ejpam-6425	246	21	e	e	X
ejpam-6425	246	22	=	=	PROPN
ejpam-6425	246	23	36k	36k	NUM
ejpam-6425	246	24	+	+	CCONJ
ejpam-6425	246	25	1170	1170	NUM
ejpam-6425	246	26	.	.	PUNCT
ejpam-6425	247	1	by	by	ADP
ejpam-6425	247	2	factoring	factor	VERB
ejpam-6425	247	3	the	the	DET
ejpam-6425	247	4	left	left	ADJ
ejpam-6425	247	5	side	side	NOUN
ejpam-6425	247	6	of	of	ADP
ejpam-6425	247	7	the	the	DET
ejpam-6425	247	8	equation	equation	NOUN
ejpam-6425	247	9	(	(	PUNCT
ejpam-6425	247	10	16	16	NUM
ejpam-6425	247	11	)	)	PUNCT
ejpam-6425	247	12	,	,	PUNCT
ejpam-6425	247	13	we	we	PRON
ejpam-6425	247	14	will	will	AUX
ejpam-6425	247	15	have	have	VERB
ejpam-6425	247	16	,	,	PUNCT
ejpam-6425	247	17	(	(	PUNCT
ejpam-6425	247	18	x	x	SYM
ejpam-6425	247	19	+	+	X
ejpam-6425	247	20	(	(	PUNCT
ejpam-6425	247	21	−6)y	−6)y	NOUN
ejpam-6425	247	22	)	)	PUNCT
ejpam-6425	247	23	(	(	PUNCT
ejpam-6425	247	24	x	x	X
ejpam-6425	247	25	−	−	PROPN
ejpam-6425	247	26	(	(	PUNCT
ejpam-6425	247	27	−6)y	−6)y	NOUN
ejpam-6425	247	28	)	)	PUNCT
ejpam-6425	248	1	=	=	SYM
ejpam-6425	248	2	1321920	1321920	NUM
ejpam-6425	248	3	,	,	PUNCT
ejpam-6425	248	4	where	where	SCONJ
ejpam-6425	248	5	b	b	X
ejpam-6425	248	6	=	=	SYM
ejpam-6425	248	7	−6	−6	NOUN
ejpam-6425	248	8	.	.	PUNCT
ejpam-6425	249	1	therefore	therefore	ADV
ejpam-6425	249	2	,	,	PUNCT
ejpam-6425	249	3	by	by	ADP
ejpam-6425	249	4	lemma	lemma	PROPN
ejpam-6425	249	5	1	1	NUM
ejpam-6425	249	6	and	and	CCONJ
ejpam-6425	249	7	remark	remark	NOUN
ejpam-6425	249	8	2	2	NUM
ejpam-6425	249	9	,	,	PUNCT
ejpam-6425	249	10	we	we	PRON
ejpam-6425	249	11	have	have	VERB
ejpam-6425	249	12	the	the	DET
ejpam-6425	249	13	following	following	NOUN
ejpam-6425	249	14	:	:	PUNCT
ejpam-6425	249	15	x	x	SYM
ejpam-6425	249	16	=	=	SYM
ejpam-6425	249	17	n1	n1	PROPN
ejpam-6425	249	18	+	+	NOUN
ejpam-6425	249	19	n2	n2	ADJ
ejpam-6425	249	20	2	2	NUM
ejpam-6425	249	21	,	,	PUNCT
ejpam-6425	249	22	y	y	PROPN
ejpam-6425	249	23	=	=	SYM
ejpam-6425	249	24	n1	n1	PROPN
ejpam-6425	250	1	−n2	−n2	PROPN
ejpam-6425	250	2	−12	−12	NOUN
ejpam-6425	250	3	,	,	PUNCT
ejpam-6425	250	4	k	k	X
ejpam-6425	251	1	=	=	PUNCT
ejpam-6425	251	2	x	x	SYM
ejpam-6425	251	3	−	−	PROPN
ejpam-6425	251	4	1170	1170	NUM
ejpam-6425	251	5	36	36	NUM
ejpam-6425	251	6	,	,	PUNCT
ejpam-6425	251	7	n	n	NOUN
ejpam-6425	251	8	=	=	SYM
ejpam-6425	251	9	y	y	PROPN
ejpam-6425	251	10	+	+	NUM
ejpam-6425	251	11	6k	6k	NOUN
ejpam-6425	251	12	−	−	NOUN
ejpam-6425	251	13	9	9	NUM
ejpam-6425	251	14	612	612	NUM
ejpam-6425	251	15	,	,	PUNCT
ejpam-6425	251	16	(	(	PUNCT
ejpam-6425	251	17	17	17	NUM
ejpam-6425	251	18	)	)	PUNCT
ejpam-6425	251	19	where	where	SCONJ
ejpam-6425	251	20	(	(	PUNCT
ejpam-6425	251	21	n1	n1	NOUN
ejpam-6425	251	22	,	,	PUNCT
ejpam-6425	251	23	n2	n2	ADJ
ejpam-6425	251	24	)	)	PUNCT
ejpam-6425	251	25	are	be	AUX
ejpam-6425	251	26	the	the	DET
ejpam-6425	251	27	factor	factor	NOUN
ejpam-6425	251	28	pairs	pair	NOUN
ejpam-6425	251	29	of	of	ADP
ejpam-6425	251	30	n	n	NOUN
ejpam-6425	251	31	=	=	SYM
ejpam-6425	251	32	1321920	1321920	NUM
ejpam-6425	251	33	.	.	PUNCT
ejpam-6425	252	1	using	use	VERB
ejpam-6425	252	2	(	(	PUNCT
ejpam-6425	252	3	17	17	NUM
ejpam-6425	252	4	)	)	PUNCT
ejpam-6425	252	5	,	,	PUNCT
ejpam-6425	252	6	we	we	PRON
ejpam-6425	252	7	can	can	AUX
ejpam-6425	252	8	now	now	ADV
ejpam-6425	252	9	solve	solve	VERB
ejpam-6425	252	10	for	for	ADP
ejpam-6425	252	11	the	the	DET
ejpam-6425	252	12	integer	integer	NOUN
ejpam-6425	252	13	values	value	NOUN
ejpam-6425	252	14	of	of	ADP
ejpam-6425	252	15	n	n	PROPN
ejpam-6425	252	16	and	and	CCONJ
ejpam-6425	252	17	k	k	NOUN
ejpam-6425	252	18	,	,	PUNCT
ejpam-6425	252	19	by	by	ADP
ejpam-6425	252	20	considering	consider	VERB
ejpam-6425	252	21	all	all	DET
ejpam-6425	252	22	the	the	DET
ejpam-6425	252	23	factor	factor	NOUN
ejpam-6425	252	24	pairs	pair	NOUN
ejpam-6425	252	25	of	of	ADP
ejpam-6425	252	26	1321920	1321920	NUM
ejpam-6425	252	27	,	,	PUNCT
ejpam-6425	252	28	as	as	SCONJ
ejpam-6425	252	29	presented	present	VERB
ejpam-6425	252	30	in	in	ADP
ejpam-6425	252	31	table	table	NOUN
ejpam-6425	252	32	4	4	NUM
ejpam-6425	252	33	in	in	ADP
ejpam-6425	252	34	the	the	DET
ejpam-6425	252	35	appendices	appendix	NOUN
ejpam-6425	252	36	section	section	NOUN
ejpam-6425	252	37	.	.	PUNCT
ejpam-6425	253	1	table	table	NOUN
ejpam-6425	253	2	4	4	NUM
ejpam-6425	253	3	shows	show	VERB
ejpam-6425	253	4	that	that	SCONJ
ejpam-6425	253	5	there	there	PRON
ejpam-6425	253	6	are	be	VERB
ejpam-6425	253	7	2	2	NUM
ejpam-6425	253	8	possible	possible	ADJ
ejpam-6425	253	9	solutions	solution	NOUN
ejpam-6425	253	10	for	for	ADP
ejpam-6425	253	11	the	the	DET
ejpam-6425	253	12	diophantine	diophantine	NOUN
ejpam-6425	253	13	equation	equation	NOUN
ejpam-6425	253	14	306n2	306n2	NUM
ejpam-6425	253	15	+	+	CCONJ
ejpam-6425	253	16	9n−1−6nk−2k	9n−1−6nk−2k	NUM
ejpam-6425	253	17	=	=	SYM
ejpam-6425	253	18	0	0	X
ejpam-6425	253	19	.	.	PUNCT
ejpam-6425	254	1	these	these	PRON
ejpam-6425	254	2	are	be	AUX
ejpam-6425	254	3	:	:	PUNCT
ejpam-6425	254	4	(	(	PUNCT
ejpam-6425	254	5	n	n	CCONJ
ejpam-6425	254	6	,	,	PUNCT
ejpam-6425	254	7	k)=	k)=	X
ejpam-6425	254	8	(	(	PUNCT
ejpam-6425	254	9	3,139	3,139	NUM
ejpam-6425	254	10	)	)	PUNCT
ejpam-6425	254	11	,	,	PUNCT
ejpam-6425	254	12	and	and	CCONJ
ejpam-6425	254	13	(	(	PUNCT
ejpam-6425	254	14	-1,-74	-1,-74	NOUN
ejpam-6425	254	15	)	)	PUNCT
ejpam-6425	254	16	.	.	PUNCT
ejpam-6425	255	1	however	however	ADV
ejpam-6425	255	2	,	,	PUNCT
ejpam-6425	255	3	since	since	SCONJ
ejpam-6425	255	4	we	we	PRON
ejpam-6425	255	5	are	be	AUX
ejpam-6425	255	6	only	only	ADV
ejpam-6425	255	7	concern	concern	NOUN
ejpam-6425	255	8	with	with	ADP
ejpam-6425	255	9	possible	possible	ADJ
ejpam-6425	255	10	values	value	NOUN
ejpam-6425	255	11	of	of	ADP
ejpam-6425	255	12	n	n	NOUN
ejpam-6425	255	13	for	for	ADP
ejpam-6425	255	14	s(rwwn	s(rwwn	NOUN
ejpam-6425	255	15	)	)	PUNCT
ejpam-6425	255	16	,	,	PUNCT
ejpam-6425	255	17	then	then	ADV
ejpam-6425	255	18	only	only	ADV
ejpam-6425	255	19	(	(	PUNCT
ejpam-6425	255	20	3,139	3,139	NUM
ejpam-6425	255	21	)	)	PUNCT
ejpam-6425	255	22	,	,	PUNCT
ejpam-6425	255	23	where	where	SCONJ
ejpam-6425	255	24	n	n	X
ejpam-6425	255	25	=	=	SYM
ejpam-6425	255	26	3	3	NUM
ejpam-6425	255	27	,	,	PUNCT
ejpam-6425	255	28	is	be	AUX
ejpam-6425	255	29	a	a	DET
ejpam-6425	255	30	valid	valid	ADJ
ejpam-6425	255	31	solution	solution	NOUN
ejpam-6425	255	32	under	under	ADP
ejpam-6425	255	33	this	this	DET
ejpam-6425	255	34	conditions	condition	NOUN
ejpam-6425	255	35	.	.	PUNCT
ejpam-6425	256	1	therefore	therefore	ADV
ejpam-6425	256	2	,	,	PUNCT
ejpam-6425	256	3	we	we	PRON
ejpam-6425	256	4	have	have	AUX
ejpam-6425	256	5	shown	show	VERB
ejpam-6425	256	6	that	that	SCONJ
ejpam-6425	256	7	s(rww3	s(rww3	NOUN
ejpam-6425	256	8	)	)	PUNCT
ejpam-6425	256	9	is	be	AUX
ejpam-6425	256	10	the	the	DET
ejpam-6425	256	11	only	only	ADJ
ejpam-6425	256	12	splitting	splitting	NOUN
ejpam-6425	256	13	graph	graph	NOUN
ejpam-6425	256	14	of	of	ADP
ejpam-6425	256	15	right	right	ADJ
ejpam-6425	256	16	water	water	NOUN
ejpam-6425	256	17	wheel	wheel	NOUN
ejpam-6425	256	18	graph	graph	NOUN
ejpam-6425	256	19	that	that	PRON
ejpam-6425	256	20	satisfies	satisfy	VERB
ejpam-6425	256	21	the	the	DET
ejpam-6425	256	22	lo	lo	PROPN
ejpam-6425	256	23	’s	’s	PART
ejpam-6425	256	24	theorem	theorem	ADJ
ejpam-6425	256	25	.	.	PROPN
ejpam-6425	256	26	similar	similar	ADJ
ejpam-6425	256	27	to	to	ADP
ejpam-6425	256	28	theorem	theorem	NOUN
ejpam-6425	256	29	5	5	NUM
ejpam-6425	256	30	,	,	PUNCT
ejpam-6425	256	31	s(rww3	s(rww3	X
ejpam-6425	256	32	)	)	PUNCT
ejpam-6425	256	33	is	be	AUX
ejpam-6425	256	34	a	a	DET
ejpam-6425	256	35	very	very	ADV
ejpam-6425	256	36	large	large	ADJ
ejpam-6425	256	37	graph	graph	NOUN
ejpam-6425	256	38	with	with	ADP
ejpam-6425	256	39	20	20	NUM
ejpam-6425	256	40	vertices	vertex	NOUN
ejpam-6425	256	41	and	and	CCONJ
ejpam-6425	256	42	54	54	NUM
ejpam-6425	256	43	edges	edge	NOUN
ejpam-6425	256	44	.	.	PUNCT
ejpam-6425	257	1	by	by	ADP
ejpam-6425	257	2	permutation	permutation	NOUN
ejpam-6425	257	3	,	,	PUNCT
ejpam-6425	257	4	there	there	PRON
ejpam-6425	257	5	is	be	VERB
ejpam-6425	257	6	a	a	DET
ejpam-6425	257	7	total	total	NOUN
ejpam-6425	257	8	of	of	ADP
ejpam-6425	257	9	54	54	NUM
ejpam-6425	257	10	!	!	PUNCT
ejpam-6425	258	1	possible	possible	ADJ
ejpam-6425	258	2	labeling	labeling	NOUN
ejpam-6425	258	3	for	for	ADP
ejpam-6425	258	4	its	its	PRON
ejpam-6425	258	5	edges	edge	NOUN
ejpam-6425	258	6	.	.	PUNCT
ejpam-6425	259	1	thus	thus	ADV
ejpam-6425	259	2	,	,	PUNCT
ejpam-6425	259	3	the	the	DET
ejpam-6425	259	4	researchers	researcher	NOUN
ejpam-6425	259	5	created	create	VERB
ejpam-6425	259	6	and	and	CCONJ
ejpam-6425	259	7	used	use	VERB
ejpam-6425	259	8	a	a	DET
ejpam-6425	259	9	python	python	NOUN
ejpam-6425	259	10	computer	computer	NOUN
ejpam-6425	259	11	program	program	NOUN
ejpam-6425	259	12	to	to	PART
ejpam-6425	259	13	generate	generate	VERB
ejpam-6425	259	14	an	an	DET
ejpam-6425	259	15	edge	edge	NOUN
ejpam-6425	259	16	-	-	PUNCT
ejpam-6425	259	17	graceful	graceful	NOUN
ejpam-6425	259	18	labeling	labeling	NOUN
ejpam-6425	259	19	for	for	ADP
ejpam-6425	259	20	the	the	DET
ejpam-6425	259	21	said	say	VERB
ejpam-6425	259	22	graph	graph	NOUN
ejpam-6425	259	23	.	.	PUNCT
ejpam-6425	260	1	one	one	NUM
ejpam-6425	260	2	of	of	ADP
ejpam-6425	260	3	the	the	DET
ejpam-6425	260	4	results	result	NOUN
ejpam-6425	260	5	from	from	ADP
ejpam-6425	260	6	the	the	DET
ejpam-6425	260	7	computer	computer	NOUN
ejpam-6425	260	8	program	program	NOUN
ejpam-6425	260	9	is	be	AUX
ejpam-6425	260	10	the	the	DET
ejpam-6425	260	11	labeling	labeling	NOUN
ejpam-6425	260	12	shown	show	VERB
ejpam-6425	260	13	in	in	ADP
ejpam-6425	260	14	figure	figure	NOUN
ejpam-6425	260	15	8	8	NUM
ejpam-6425	260	16	.	.	PUNCT
ejpam-6425	260	17	a.	a.	PROPN
ejpam-6425	260	18	d.	d.	PROPN
ejpam-6425	260	19	c.	c.	PROPN
ejpam-6425	260	20	angel	angel	PROPN
ejpam-6425	260	21	et	et	PROPN
ejpam-6425	260	22	al	al	PROPN
ejpam-6425	260	23	.	.	PUNCT
ejpam-6425	260	24	/	/	SYM
ejpam-6425	260	25	eur	eur	PROPN
ejpam-6425	260	26	.	.	PUNCT
ejpam-6425	261	1	j.	j.	PROPN
ejpam-6425	261	2	pure	pure	PROPN
ejpam-6425	261	3	appl	appl	PROPN
ejpam-6425	261	4	.	.	PROPN
ejpam-6425	261	5	math	math	PROPN
ejpam-6425	261	6	,	,	PUNCT
ejpam-6425	261	7	18	18	NUM
ejpam-6425	261	8	(	(	PUNCT
ejpam-6425	261	9	4	4	NUM
ejpam-6425	261	10	)	)	PUNCT
ejpam-6425	261	11	(	(	PUNCT
ejpam-6425	261	12	2025	2025	NUM
ejpam-6425	261	13	)	)	PUNCT
ejpam-6425	261	14	,	,	PUNCT
ejpam-6425	261	15	6425	6425	NUM
ejpam-6425	261	16	15	15	NUM
ejpam-6425	261	17	of	of	ADP
ejpam-6425	261	18	33	33	NUM
ejpam-6425	261	19	figure	figure	NOUN
ejpam-6425	261	20	8	8	NUM
ejpam-6425	261	21	:	:	PUNCT
ejpam-6425	261	22	an	an	DET
ejpam-6425	261	23	edge	edge	NOUN
ejpam-6425	261	24	-	-	PUNCT
ejpam-6425	261	25	graceful	graceful	NOUN
ejpam-6425	261	26	labeling	labeling	NOUN
ejpam-6425	261	27	of	of	ADP
ejpam-6425	261	28	s(rww3	s(rww3	NOUN
ejpam-6425	261	29	)	)	PUNCT
ejpam-6425	261	30	as	as	SCONJ
ejpam-6425	261	31	shown	show	VERB
ejpam-6425	261	32	in	in	ADP
ejpam-6425	261	33	figure	figure	NOUN
ejpam-6425	261	34	8	8	NUM
ejpam-6425	261	35	,	,	PUNCT
ejpam-6425	261	36	the	the	DET
ejpam-6425	261	37	54	54	NUM
ejpam-6425	261	38	edges	edge	NOUN
ejpam-6425	261	39	of	of	ADP
ejpam-6425	261	40	s(rww3	s(rww3	NOUN
ejpam-6425	261	41	)	)	PUNCT
ejpam-6425	261	42	are	be	AUX
ejpam-6425	261	43	labeled	label	VERB
ejpam-6425	261	44	using	use	VERB
ejpam-6425	261	45	distinct	distinct	ADJ
ejpam-6425	261	46	integers	integer	NOUN
ejpam-6425	261	47	from	from	ADP
ejpam-6425	261	48	1	1	NUM
ejpam-6425	261	49	to	to	ADP
ejpam-6425	261	50	54	54	NUM
ejpam-6425	261	51	.	.	PUNCT
ejpam-6425	262	1	then	then	ADV
ejpam-6425	262	2	,	,	PUNCT
ejpam-6425	262	3	by	by	ADP
ejpam-6425	262	4	definition	definition	NOUN
ejpam-6425	262	5	1	1	NUM
ejpam-6425	262	6	,	,	PUNCT
ejpam-6425	262	7	the	the	DET
ejpam-6425	262	8	20	20	NUM
ejpam-6425	262	9	vertices	vertex	NOUN
ejpam-6425	262	10	are	be	AUX
ejpam-6425	262	11	successfully	successfully	ADV
ejpam-6425	262	12	labeled	label	VERB
ejpam-6425	262	13	using	use	VERB
ejpam-6425	262	14	distinct	distinct	ADJ
ejpam-6425	262	15	integers	integer	NOUN
ejpam-6425	262	16	from	from	ADP
ejpam-6425	262	17	0	0	NUM
ejpam-6425	262	18	to	to	ADP
ejpam-6425	262	19	19	19	NUM
ejpam-6425	262	20	.	.	PUNCT
ejpam-6425	263	1	therefore	therefore	ADV
ejpam-6425	263	2	,	,	PUNCT
ejpam-6425	263	3	the	the	DET
ejpam-6425	263	4	labeled	label	VERB
ejpam-6425	263	5	graph	graph	NOUN
ejpam-6425	263	6	in	in	ADP
ejpam-6425	263	7	figure	figure	NOUN
ejpam-6425	263	8	8	8	NUM
ejpam-6425	263	9	is	be	AUX
ejpam-6425	263	10	indeed	indeed	ADV
ejpam-6425	263	11	an	an	DET
ejpam-6425	263	12	edge	edge	NOUN
ejpam-6425	263	13	-	-	PUNCT
ejpam-6425	263	14	graceful	graceful	NOUN
ejpam-6425	263	15	labeling	labeling	NOUN
ejpam-6425	263	16	of	of	ADP
ejpam-6425	263	17	s(rww3	s(rww3	NOUN
ejpam-6425	263	18	)	)	PUNCT
ejpam-6425	263	19	.	.	PUNCT
ejpam-6425	264	1	since	since	SCONJ
ejpam-6425	264	2	all	all	DET
ejpam-6425	264	3	the	the	DET
ejpam-6425	264	4	needed	need	VERB
ejpam-6425	264	5	conditions	condition	NOUN
ejpam-6425	264	6	are	be	AUX
ejpam-6425	264	7	satisfied	satisfied	ADJ
ejpam-6425	264	8	,	,	PUNCT
ejpam-6425	264	9	then	then	ADV
ejpam-6425	264	10	we	we	PRON
ejpam-6425	264	11	can	can	AUX
ejpam-6425	264	12	now	now	ADV
ejpam-6425	264	13	conclude	conclude	VERB
ejpam-6425	264	14	that	that	PRON
ejpam-6425	264	15	s(rww3	s(rww3	ADP
ejpam-6425	264	16	)	)	PUNCT
ejpam-6425	264	17	is	be	AUX
ejpam-6425	264	18	the	the	DET
ejpam-6425	264	19	only	only	ADJ
ejpam-6425	264	20	edge	edge	NOUN
ejpam-6425	264	21	-	-	PUNCT
ejpam-6425	264	22	graceful	graceful	NOUN
ejpam-6425	264	23	splitting	splitting	NOUN
ejpam-6425	264	24	graph	graph	NOUN
ejpam-6425	264	25	of	of	ADP
ejpam-6425	264	26	right	right	ADJ
ejpam-6425	264	27	water	water	NOUN
ejpam-6425	264	28	wheel	wheel	NOUN
ejpam-6425	264	29	graphs	graph	NOUN
ejpam-6425	264	30	.	.	PUNCT
ejpam-6425	265	1	theorem	theorem	VERB
ejpam-6425	265	2	7	7	NUM
ejpam-6425	265	3	.	.	PUNCT
ejpam-6425	266	1	among	among	ADP
ejpam-6425	266	2	all	all	DET
ejpam-6425	266	3	splitting	splitting	NOUN
ejpam-6425	266	4	graphs	graph	NOUN
ejpam-6425	266	5	of	of	ADP
ejpam-6425	266	6	left	left	ADJ
ejpam-6425	266	7	water	water	NOUN
ejpam-6425	266	8	wheel	wheel	NOUN
ejpam-6425	266	9	graphs	graph	NOUN
ejpam-6425	266	10	,	,	PUNCT
ejpam-6425	266	11	only	only	ADV
ejpam-6425	266	12	s(lww3	s(lww3	VERB
ejpam-6425	266	13	)	)	PUNCT
ejpam-6425	266	14	is	be	AUX
ejpam-6425	266	15	edge	edge	NOUN
ejpam-6425	266	16	-	-	PUNCT
ejpam-6425	266	17	graceful	graceful	NOUN
ejpam-6425	266	18	.	.	PUNCT
ejpam-6425	267	1	proof	proof	NOUN
ejpam-6425	267	2	.	.	PUNCT
ejpam-6425	268	1	first	first	ADV
ejpam-6425	268	2	,	,	PUNCT
ejpam-6425	268	3	using	use	VERB
ejpam-6425	268	4	definiton	definiton	PROPN
ejpam-6425	268	5	4	4	NUM
ejpam-6425	268	6	and	and	CCONJ
ejpam-6425	268	7	proposition	proposition	NOUN
ejpam-6425	268	8	1	1	NUM
ejpam-6425	268	9	,	,	PUNCT
ejpam-6425	268	10	the	the	DET
ejpam-6425	268	11	splitting	splitting	NOUN
ejpam-6425	268	12	graph	graph	NOUN
ejpam-6425	268	13	s(lwwn	s(lwwn	NOUN
ejpam-6425	268	14	)	)	PUNCT
ejpam-6425	268	15	also	also	ADV
ejpam-6425	268	16	has	have	VERB
ejpam-6425	268	17	6n+	6n+	NUM
ejpam-6425	268	18	2	2	NUM
ejpam-6425	268	19	vertices	vertex	NOUN
ejpam-6425	268	20	and	and	CCONJ
ejpam-6425	268	21	18n	18n	NOUN
ejpam-6425	268	22	edges	edge	NOUN
ejpam-6425	268	23	,	,	PUNCT
ejpam-6425	268	24	the	the	DET
ejpam-6425	268	25	same	same	ADJ
ejpam-6425	268	26	as	as	ADP
ejpam-6425	268	27	s(rwwn	s(rwwn	NOUN
ejpam-6425	268	28	)	)	PUNCT
ejpam-6425	268	29	.	.	PUNCT
ejpam-6425	269	1	since	since	SCONJ
ejpam-6425	269	2	the	the	DET
ejpam-6425	269	3	order	order	NOUN
ejpam-6425	269	4	and	and	CCONJ
ejpam-6425	269	5	size	size	NOUN
ejpam-6425	269	6	of	of	ADP
ejpam-6425	269	7	s(rwwn	s(rwwn	NOUN
ejpam-6425	269	8	)	)	PUNCT
ejpam-6425	269	9	and	and	CCONJ
ejpam-6425	269	10	s(lwwn	s(lwwn	NOUN
ejpam-6425	269	11	)	)	PUNCT
ejpam-6425	269	12	are	be	AUX
ejpam-6425	269	13	the	the	DET
ejpam-6425	269	14	same	same	ADJ
ejpam-6425	269	15	,	,	PUNCT
ejpam-6425	269	16	then	then	ADV
ejpam-6425	269	17	the	the	DET
ejpam-6425	269	18	proof	proof	NOUN
ejpam-6425	269	19	of	of	ADP
ejpam-6425	269	20	theorem	theorem	ADJ
ejpam-6425	269	21	7	7	NUM
ejpam-6425	269	22	will	will	AUX
ejpam-6425	269	23	just	just	ADV
ejpam-6425	269	24	be	be	AUX
ejpam-6425	269	25	similar	similar	ADJ
ejpam-6425	269	26	to	to	ADP
ejpam-6425	269	27	the	the	DET
ejpam-6425	269	28	proof	proof	NOUN
ejpam-6425	269	29	of	of	ADP
ejpam-6425	269	30	theorem	theorem	ADJ
ejpam-6425	269	31	6	6	NUM
ejpam-6425	269	32	.	.	PUNCT
ejpam-6425	269	33	a.	a.	PROPN
ejpam-6425	269	34	d.	d.	PROPN
ejpam-6425	269	35	c.	c.	PROPN
ejpam-6425	269	36	angel	angel	PROPN
ejpam-6425	269	37	et	et	PROPN
ejpam-6425	269	38	al	al	PROPN
ejpam-6425	269	39	.	.	PUNCT
ejpam-6425	269	40	/	/	SYM
ejpam-6425	269	41	eur	eur	PROPN
ejpam-6425	269	42	.	.	PUNCT
ejpam-6425	270	1	j.	j.	PROPN
ejpam-6425	270	2	pure	pure	PROPN
ejpam-6425	270	3	appl	appl	PROPN
ejpam-6425	270	4	.	.	PROPN
ejpam-6425	270	5	math	math	PROPN
ejpam-6425	270	6	,	,	PUNCT
ejpam-6425	270	7	18	18	NUM
ejpam-6425	270	8	(	(	PUNCT
ejpam-6425	270	9	4	4	NUM
ejpam-6425	270	10	)	)	PUNCT
ejpam-6425	270	11	(	(	PUNCT
ejpam-6425	270	12	2025	2025	NUM
ejpam-6425	270	13	)	)	PUNCT
ejpam-6425	270	14	,	,	PUNCT
ejpam-6425	270	15	6425	6425	NUM
ejpam-6425	270	16	16	16	NUM
ejpam-6425	270	17	of	of	ADP
ejpam-6425	270	18	33	33	NUM
ejpam-6425	270	19	with	with	ADP
ejpam-6425	270	20	the	the	DET
ejpam-6425	270	21	proof	proof	NOUN
ejpam-6425	270	22	of	of	ADP
ejpam-6425	270	23	theorem	theorem	NOUN
ejpam-6425	270	24	6	6	NUM
ejpam-6425	270	25	,	,	PUNCT
ejpam-6425	270	26	we	we	PRON
ejpam-6425	270	27	have	have	AUX
ejpam-6425	270	28	also	also	ADV
ejpam-6425	270	29	shown	show	VERB
ejpam-6425	270	30	that	that	SCONJ
ejpam-6425	270	31	s(lww3	s(lww3	VERB
ejpam-6425	270	32	)	)	PUNCT
ejpam-6425	270	33	is	be	AUX
ejpam-6425	270	34	the	the	DET
ejpam-6425	270	35	only	only	ADJ
ejpam-6425	270	36	splitting	splitting	NOUN
ejpam-6425	270	37	graph	graph	NOUN
ejpam-6425	270	38	of	of	ADP
ejpam-6425	270	39	left	left	ADJ
ejpam-6425	270	40	water	water	NOUN
ejpam-6425	270	41	wheel	wheel	NOUN
ejpam-6425	270	42	graph	graph	NOUN
ejpam-6425	270	43	that	that	PRON
ejpam-6425	270	44	satisfies	satisfy	VERB
ejpam-6425	270	45	the	the	DET
ejpam-6425	270	46	lo	lo	PROPN
ejpam-6425	270	47	’s	’s	PART
ejpam-6425	270	48	theorem	theorem	ADJ
ejpam-6425	270	49	.	.	PROPN
ejpam-6425	271	1	similar	similar	ADJ
ejpam-6425	271	2	to	to	ADP
ejpam-6425	271	3	the	the	DET
ejpam-6425	271	4	preceding	precede	VERB
ejpam-6425	271	5	theorems	theorem	NOUN
ejpam-6425	271	6	,	,	PUNCT
ejpam-6425	271	7	an	an	DET
ejpam-6425	271	8	actual	actual	ADJ
ejpam-6425	271	9	edge	edge	NOUN
ejpam-6425	271	10	-	-	PUNCT
ejpam-6425	271	11	graceful	graceful	NOUN
ejpam-6425	271	12	labeling	labeling	NOUN
ejpam-6425	271	13	of	of	ADP
ejpam-6425	271	14	s(lww3	s(lww3	NOUN
ejpam-6425	271	15	)	)	PUNCT
ejpam-6425	271	16	must	must	AUX
ejpam-6425	271	17	be	be	AUX
ejpam-6425	271	18	provided	provide	VERB
ejpam-6425	271	19	,	,	PUNCT
ejpam-6425	271	20	in	in	ADP
ejpam-6425	271	21	order	order	NOUN
ejpam-6425	271	22	to	to	PART
ejpam-6425	271	23	complete	complete	VERB
ejpam-6425	271	24	this	this	DET
ejpam-6425	271	25	proof	proof	NOUN
ejpam-6425	271	26	.	.	PUNCT
ejpam-6425	272	1	the	the	DET
ejpam-6425	272	2	same	same	ADJ
ejpam-6425	272	3	with	with	ADP
ejpam-6425	272	4	s(rww3	s(rww3	NOUN
ejpam-6425	272	5	)	)	PUNCT
ejpam-6425	272	6	,	,	PUNCT
ejpam-6425	272	7	splitting	splitting	NOUN
ejpam-6425	272	8	graph	graph	NOUN
ejpam-6425	272	9	s(lww3	s(lww3	VERB
ejpam-6425	272	10	)	)	PUNCT
ejpam-6425	272	11	also	also	ADV
ejpam-6425	272	12	has	have	VERB
ejpam-6425	272	13	20	20	NUM
ejpam-6425	272	14	vertices	vertex	NOUN
ejpam-6425	272	15	and	and	CCONJ
ejpam-6425	272	16	54	54	NUM
ejpam-6425	272	17	edges	edge	NOUN
ejpam-6425	272	18	.	.	PUNCT
ejpam-6425	273	1	thus	thus	ADV
ejpam-6425	273	2	,	,	PUNCT
ejpam-6425	273	3	a	a	DET
ejpam-6425	273	4	python	python	NOUN
ejpam-6425	273	5	computer	computer	NOUN
ejpam-6425	273	6	program	program	NOUN
ejpam-6425	273	7	was	be	AUX
ejpam-6425	273	8	created	create	VERB
ejpam-6425	273	9	and	and	CCONJ
ejpam-6425	273	10	used	use	VERB
ejpam-6425	273	11	by	by	ADP
ejpam-6425	273	12	the	the	DET
ejpam-6425	273	13	researchers	researcher	NOUN
ejpam-6425	273	14	to	to	PART
ejpam-6425	273	15	generate	generate	VERB
ejpam-6425	273	16	an	an	DET
ejpam-6425	273	17	edge	edge	NOUN
ejpam-6425	273	18	-	-	PUNCT
ejpam-6425	273	19	graceful	graceful	NOUN
ejpam-6425	273	20	labeling	labeling	NOUN
ejpam-6425	273	21	of	of	ADP
ejpam-6425	273	22	the	the	DET
ejpam-6425	273	23	said	say	VERB
ejpam-6425	273	24	graph	graph	NOUN
ejpam-6425	273	25	.	.	PUNCT
ejpam-6425	274	1	one	one	NUM
ejpam-6425	274	2	of	of	ADP
ejpam-6425	274	3	the	the	DET
ejpam-6425	274	4	results	result	NOUN
ejpam-6425	274	5	from	from	ADP
ejpam-6425	274	6	the	the	DET
ejpam-6425	274	7	computer	computer	NOUN
ejpam-6425	274	8	program	program	NOUN
ejpam-6425	274	9	is	be	AUX
ejpam-6425	274	10	the	the	DET
ejpam-6425	274	11	labeling	labeling	NOUN
ejpam-6425	274	12	shown	show	VERB
ejpam-6425	274	13	in	in	ADP
ejpam-6425	274	14	figure	figure	NOUN
ejpam-6425	274	15	9	9	NUM
ejpam-6425	274	16	.	.	PUNCT
ejpam-6425	274	17	figure	figure	VERB
ejpam-6425	274	18	9	9	NUM
ejpam-6425	274	19	:	:	PUNCT
ejpam-6425	274	20	an	an	DET
ejpam-6425	274	21	edge	edge	NOUN
ejpam-6425	274	22	-	-	PUNCT
ejpam-6425	274	23	graceful	graceful	NOUN
ejpam-6425	274	24	labeling	labeling	NOUN
ejpam-6425	274	25	of	of	ADP
ejpam-6425	274	26	s(lww3	s(lww3	NOUN
ejpam-6425	274	27	)	)	PUNCT
ejpam-6425	274	28	as	as	SCONJ
ejpam-6425	274	29	shown	show	VERB
ejpam-6425	274	30	in	in	ADP
ejpam-6425	274	31	figure	figure	NOUN
ejpam-6425	274	32	9	9	NUM
ejpam-6425	274	33	,	,	PUNCT
ejpam-6425	274	34	the	the	DET
ejpam-6425	274	35	54	54	NUM
ejpam-6425	274	36	edges	edge	NOUN
ejpam-6425	274	37	of	of	ADP
ejpam-6425	274	38	s(lww3	s(lww3	NOUN
ejpam-6425	274	39	)	)	PUNCT
ejpam-6425	274	40	are	be	AUX
ejpam-6425	274	41	labeled	label	VERB
ejpam-6425	274	42	using	use	VERB
ejpam-6425	274	43	distinct	distinct	ADJ
ejpam-6425	274	44	integers	integer	NOUN
ejpam-6425	274	45	from	from	ADP
ejpam-6425	274	46	1	1	NUM
ejpam-6425	274	47	to	to	ADP
ejpam-6425	274	48	54	54	NUM
ejpam-6425	274	49	.	.	PUNCT
ejpam-6425	275	1	then	then	ADV
ejpam-6425	275	2	,	,	PUNCT
ejpam-6425	275	3	by	by	ADP
ejpam-6425	275	4	definition	definition	NOUN
ejpam-6425	275	5	1	1	NUM
ejpam-6425	275	6	,	,	PUNCT
ejpam-6425	275	7	the	the	DET
ejpam-6425	275	8	20	20	NUM
ejpam-6425	275	9	vertices	vertex	NOUN
ejpam-6425	275	10	are	be	AUX
ejpam-6425	275	11	successfully	successfully	ADV
ejpam-6425	275	12	labeled	label	VERB
ejpam-6425	275	13	using	use	VERB
ejpam-6425	275	14	distinct	distinct	ADJ
ejpam-6425	275	15	integers	integer	NOUN
ejpam-6425	275	16	from	from	ADP
ejpam-6425	275	17	0	0	NUM
ejpam-6425	275	18	to	to	ADP
ejpam-6425	275	19	19	19	NUM
ejpam-6425	275	20	.	.	PUNCT
ejpam-6425	276	1	therefore	therefore	ADV
ejpam-6425	276	2	,	,	PUNCT
ejpam-6425	276	3	the	the	DET
ejpam-6425	276	4	labeled	label	VERB
ejpam-6425	276	5	graph	graph	NOUN
ejpam-6425	276	6	in	in	ADP
ejpam-6425	276	7	figure	figure	NOUN
ejpam-6425	276	8	9	9	NUM
ejpam-6425	276	9	is	be	AUX
ejpam-6425	276	10	indeed	indeed	ADV
ejpam-6425	276	11	an	an	DET
ejpam-6425	276	12	edge	edge	NOUN
ejpam-6425	276	13	-	-	PUNCT
ejpam-6425	276	14	graceful	graceful	NOUN
ejpam-6425	276	15	labeling	labeling	NOUN
ejpam-6425	276	16	of	of	ADP
ejpam-6425	276	17	s(lww3	s(lww3	NOUN
ejpam-6425	276	18	)	)	PUNCT
ejpam-6425	276	19	.	.	PUNCT
ejpam-6425	277	1	since	since	SCONJ
ejpam-6425	277	2	all	all	DET
ejpam-6425	277	3	the	the	DET
ejpam-6425	277	4	needed	need	VERB
ejpam-6425	277	5	conditions	condition	NOUN
ejpam-6425	277	6	are	be	AUX
ejpam-6425	277	7	satisfied	satisfied	ADJ
ejpam-6425	277	8	,	,	PUNCT
ejpam-6425	277	9	then	then	ADV
ejpam-6425	277	10	we	we	PRON
ejpam-6425	277	11	can	can	AUX
ejpam-6425	277	12	now	now	ADV
ejpam-6425	277	13	conclude	conclude	VERB
ejpam-6425	277	14	that	that	SCONJ
ejpam-6425	277	15	s(lww3	s(lww3	VERB
ejpam-6425	277	16	)	)	PUNCT
ejpam-6425	277	17	is	be	AUX
ejpam-6425	277	18	the	the	DET
ejpam-6425	277	19	only	only	ADJ
ejpam-6425	277	20	edge	edge	NOUN
ejpam-6425	277	21	-	-	PUNCT
ejpam-6425	277	22	graceful	graceful	NOUN
ejpam-6425	277	23	splitting	splitting	NOUN
ejpam-6425	277	24	graph	graph	NOUN
ejpam-6425	277	25	of	of	ADP
ejpam-6425	277	26	left	left	ADJ
ejpam-6425	277	27	water	water	NOUN
ejpam-6425	277	28	wheel	wheel	NOUN
ejpam-6425	277	29	graphs	graph	NOUN
ejpam-6425	277	30	.	.	PUNCT
ejpam-6425	278	1	a.	a.	PROPN
ejpam-6425	278	2	d.	d.	PROPN
ejpam-6425	278	3	c.	c.	PROPN
ejpam-6425	278	4	angel	angel	PROPN
ejpam-6425	278	5	et	et	PROPN
ejpam-6425	278	6	al	al	PROPN
ejpam-6425	278	7	.	.	PUNCT
ejpam-6425	278	8	/	/	SYM
ejpam-6425	278	9	eur	eur	PROPN
ejpam-6425	278	10	.	.	PUNCT
ejpam-6425	279	1	j.	j.	PROPN
ejpam-6425	279	2	pure	pure	PROPN
ejpam-6425	279	3	appl	appl	PROPN
ejpam-6425	279	4	.	.	PROPN
ejpam-6425	279	5	math	math	PROPN
ejpam-6425	279	6	,	,	PUNCT
ejpam-6425	279	7	18	18	NUM
ejpam-6425	279	8	(	(	PUNCT
ejpam-6425	279	9	4	4	NUM
ejpam-6425	279	10	)	)	PUNCT
ejpam-6425	279	11	(	(	PUNCT
ejpam-6425	279	12	2025	2025	NUM
ejpam-6425	279	13	)	)	PUNCT
ejpam-6425	279	14	,	,	PUNCT
ejpam-6425	279	15	6425	6425	NUM
ejpam-6425	279	16	17	17	NUM
ejpam-6425	279	17	of	of	ADP
ejpam-6425	279	18	33	33	NUM
ejpam-6425	279	19	4	4	NUM
ejpam-6425	279	20	.	.	PUNCT
ejpam-6425	279	21	conclusion	conclusion	NOUN
ejpam-6425	279	22	in	in	ADP
ejpam-6425	279	23	this	this	DET
ejpam-6425	279	24	paper	paper	NOUN
ejpam-6425	279	25	,	,	PUNCT
ejpam-6425	279	26	two	two	NUM
ejpam-6425	279	27	new	new	ADJ
ejpam-6425	279	28	graphs	graph	NOUN
ejpam-6425	279	29	are	be	AUX
ejpam-6425	279	30	introduced	introduce	VERB
ejpam-6425	279	31	,	,	PUNCT
ejpam-6425	279	32	namely	namely	ADV
ejpam-6425	279	33	the	the	DET
ejpam-6425	279	34	right	right	ADJ
ejpam-6425	279	35	water	water	NOUN
ejpam-6425	279	36	wheel	wheel	NOUN
ejpam-6425	279	37	graph	graph	NOUN
ejpam-6425	279	38	rwwn	rwwn	NOUN
ejpam-6425	279	39	and	and	CCONJ
ejpam-6425	279	40	the	the	DET
ejpam-6425	279	41	left	left	ADJ
ejpam-6425	279	42	water	water	NOUN
ejpam-6425	279	43	wheel	wheel	NOUN
ejpam-6425	279	44	graph	graph	NOUN
ejpam-6425	279	45	lwwn	lwwn	NOUN
ejpam-6425	279	46	.	.	PUNCT
ejpam-6425	280	1	using	use	VERB
ejpam-6425	280	2	lo	lo	PROPN
ejpam-6425	280	3	’s	’s	PART
ejpam-6425	280	4	theorem	theorem	NOUN
ejpam-6425	280	5	,	,	PUNCT
ejpam-6425	280	6	and	and	CCONJ
ejpam-6425	280	7	the	the	DET
ejpam-6425	280	8	concepts	concept	NOUN
ejpam-6425	280	9	of	of	ADP
ejpam-6425	280	10	divisibility	divisibility	NOUN
ejpam-6425	280	11	and	and	CCONJ
ejpam-6425	280	12	diophantine	diophantine	VERB
ejpam-6425	280	13	equations	equation	NOUN
ejpam-6425	280	14	,	,	PUNCT
ejpam-6425	280	15	we	we	PRON
ejpam-6425	280	16	have	have	AUX
ejpam-6425	280	17	proven	prove	VERB
ejpam-6425	280	18	the	the	DET
ejpam-6425	280	19	non	non	ADJ
ejpam-6425	280	20	-	-	ADJ
ejpam-6425	280	21	edge	edge	ADJ
ejpam-6425	280	22	gracefulness	gracefulness	NOUN
ejpam-6425	280	23	of	of	ADP
ejpam-6425	280	24	water	water	NOUN
ejpam-6425	280	25	wheel	wheel	NOUN
ejpam-6425	280	26	graphs	graph	VERB
ejpam-6425	280	27	wwn	wwn	NOUN
ejpam-6425	280	28	,	,	PUNCT
ejpam-6425	280	29	right	right	ADJ
ejpam-6425	280	30	water	water	NOUN
ejpam-6425	280	31	wheel	wheel	NOUN
ejpam-6425	280	32	graphs	graph	NOUN
ejpam-6425	280	33	rwwn	rwwn	NOUN
ejpam-6425	280	34	,	,	PUNCT
ejpam-6425	280	35	and	and	CCONJ
ejpam-6425	280	36	left	leave	VERB
ejpam-6425	280	37	water	water	NOUN
ejpam-6425	280	38	wheel	wheel	NOUN
ejpam-6425	280	39	graphs	graph	NOUN
ejpam-6425	280	40	lwwn	lwwn	NOUN
ejpam-6425	280	41	.	.	PUNCT
ejpam-6425	281	1	using	use	VERB
ejpam-6425	281	2	similar	similar	ADJ
ejpam-6425	281	3	concepts	concept	NOUN
ejpam-6425	281	4	and	and	CCONJ
ejpam-6425	281	5	custom	custom	ADV
ejpam-6425	281	6	-	-	PUNCT
ejpam-6425	281	7	developed	develop	VERB
ejpam-6425	281	8	python	python	NOUN
ejpam-6425	281	9	computer	computer	NOUN
ejpam-6425	281	10	programs	program	NOUN
ejpam-6425	281	11	,	,	PUNCT
ejpam-6425	281	12	we	we	PRON
ejpam-6425	281	13	have	have	AUX
ejpam-6425	281	14	also	also	ADV
ejpam-6425	281	15	successfully	successfully	ADV
ejpam-6425	281	16	investigated	investigate	VERB
ejpam-6425	281	17	the	the	DET
ejpam-6425	281	18	edge	edge	NOUN
ejpam-6425	281	19	-	-	PUNCT
ejpam-6425	281	20	gracefulness	gracefulness	NOUN
ejpam-6425	281	21	of	of	ADP
ejpam-6425	281	22	the	the	DET
ejpam-6425	281	23	splitting	splitting	NOUN
ejpam-6425	281	24	graphs	graph	NOUN
ejpam-6425	281	25	of	of	ADP
ejpam-6425	281	26	water	water	NOUN
ejpam-6425	281	27	wheel	wheel	NOUN
ejpam-6425	281	28	,	,	PUNCT
ejpam-6425	281	29	right	right	ADJ
ejpam-6425	281	30	water	water	NOUN
ejpam-6425	281	31	wheel	wheel	NOUN
ejpam-6425	281	32	,	,	PUNCT
ejpam-6425	281	33	and	and	CCONJ
ejpam-6425	281	34	left	leave	VERB
ejpam-6425	281	35	water	water	NOUN
ejpam-6425	281	36	wheel	wheel	NOUN
ejpam-6425	281	37	graphs	graph	NOUN
ejpam-6425	281	38	.	.	PUNCT
ejpam-6425	282	1	in	in	ADP
ejpam-6425	282	2	theorem	theorem	NOUN
ejpam-6425	282	3	5	5	NUM
ejpam-6425	282	4	,	,	PUNCT
ejpam-6425	282	5	we	we	PRON
ejpam-6425	282	6	proved	prove	VERB
ejpam-6425	282	7	that	that	SCONJ
ejpam-6425	282	8	s(ww3	s(ww3	NOUN
ejpam-6425	282	9	)	)	PUNCT
ejpam-6425	282	10	is	be	AUX
ejpam-6425	282	11	the	the	DET
ejpam-6425	282	12	only	only	ADJ
ejpam-6425	282	13	edge	edge	NOUN
ejpam-6425	282	14	-	-	PUNCT
ejpam-6425	282	15	graceful	graceful	NOUN
ejpam-6425	282	16	splitting	splitting	NOUN
ejpam-6425	282	17	graph	graph	NOUN
ejpam-6425	282	18	of	of	ADP
ejpam-6425	282	19	water	water	NOUN
ejpam-6425	282	20	wheel	wheel	NOUN
ejpam-6425	282	21	graphs	graph	NOUN
ejpam-6425	282	22	.	.	PUNCT
ejpam-6425	283	1	in	in	ADP
ejpam-6425	283	2	theorem	theorem	NOUN
ejpam-6425	283	3	6	6	NUM
ejpam-6425	283	4	,	,	PUNCT
ejpam-6425	283	5	we	we	PRON
ejpam-6425	283	6	showed	show	VERB
ejpam-6425	283	7	that	that	SCONJ
ejpam-6425	283	8	s(rww3	s(rww3	NOUN
ejpam-6425	283	9	)	)	PUNCT
ejpam-6425	283	10	is	be	AUX
ejpam-6425	283	11	the	the	DET
ejpam-6425	283	12	only	only	ADJ
ejpam-6425	283	13	edge	edge	NOUN
ejpam-6425	283	14	-	-	PUNCT
ejpam-6425	283	15	graceful	graceful	NOUN
ejpam-6425	283	16	splitting	splitting	NOUN
ejpam-6425	283	17	graph	graph	NOUN
ejpam-6425	283	18	of	of	ADP
ejpam-6425	283	19	right	right	ADJ
ejpam-6425	283	20	water	water	NOUN
ejpam-6425	283	21	wheel	wheel	NOUN
ejpam-6425	283	22	graphs	graph	NOUN
ejpam-6425	283	23	.	.	PUNCT
ejpam-6425	284	1	finally	finally	ADV
ejpam-6425	284	2	,	,	PUNCT
ejpam-6425	284	3	we	we	PRON
ejpam-6425	284	4	proved	prove	VERB
ejpam-6425	284	5	in	in	ADP
ejpam-6425	284	6	theorem	theorem	NOUN
ejpam-6425	284	7	7	7	NUM
ejpam-6425	284	8	that	that	SCONJ
ejpam-6425	284	9	s(lww3	s(lww3	VERB
ejpam-6425	284	10	)	)	PUNCT
ejpam-6425	284	11	is	be	AUX
ejpam-6425	284	12	the	the	DET
ejpam-6425	284	13	only	only	ADJ
ejpam-6425	284	14	edge	edge	NOUN
ejpam-6425	284	15	-	-	PUNCT
ejpam-6425	284	16	graceful	graceful	NOUN
ejpam-6425	284	17	splitting	splitting	NOUN
ejpam-6425	284	18	graph	graph	NOUN
ejpam-6425	284	19	of	of	ADP
ejpam-6425	284	20	left	left	ADJ
ejpam-6425	284	21	water	water	NOUN
ejpam-6425	284	22	wheel	wheel	NOUN
ejpam-6425	284	23	graphs	graph	NOUN
ejpam-6425	284	24	.	.	PUNCT
ejpam-6425	285	1	acknowledgements	acknowledgement	VERB
ejpam-6425	285	2	the	the	DET
ejpam-6425	285	3	completion	completion	NOUN
ejpam-6425	285	4	of	of	ADP
ejpam-6425	285	5	this	this	DET
ejpam-6425	285	6	paper	paper	NOUN
ejpam-6425	285	7	would	would	AUX
ejpam-6425	285	8	not	not	PART
ejpam-6425	285	9	be	be	AUX
ejpam-6425	285	10	possible	possible	ADJ
ejpam-6425	285	11	without	without	ADP
ejpam-6425	285	12	the	the	DET
ejpam-6425	285	13	help	help	NOUN
ejpam-6425	285	14	and	and	CCONJ
ejpam-6425	285	15	support	support	NOUN
ejpam-6425	285	16	of	of	ADP
ejpam-6425	285	17	the	the	DET
ejpam-6425	285	18	following	follow	VERB
ejpam-6425	285	19	people	people	NOUN
ejpam-6425	285	20	and	and	CCONJ
ejpam-6425	285	21	organizations	organization	NOUN
ejpam-6425	285	22	:	:	PUNCT
ejpam-6425	285	23	central	central	PROPN
ejpam-6425	285	24	luzon	luzon	PROPN
ejpam-6425	285	25	state	state	PROPN
ejpam-6425	285	26	university	university	PROPN
ejpam-6425	285	27	(	(	PUNCT
ejpam-6425	285	28	clsu	clsu	NOUN
ejpam-6425	285	29	)	)	PUNCT
ejpam-6425	285	30	,	,	PUNCT
ejpam-6425	285	31	clsu	clsu	NOUN
ejpam-6425	285	32	,	,	PUNCT
ejpam-6425	285	33	college	college	NOUN
ejpam-6425	285	34	of	of	ADP
ejpam-6425	285	35	science	science	NOUN
ejpam-6425	285	36	,	,	PUNCT
ejpam-6425	285	37	department	department	NOUN
ejpam-6425	285	38	of	of	ADP
ejpam-6425	285	39	mathematics	mathematics	PROPN
ejpam-6425	285	40	and	and	CCONJ
ejpam-6425	285	41	physics	physics	PROPN
ejpam-6425	285	42	,	,	PUNCT
ejpam-6425	285	43	department	department	NOUN
ejpam-6425	285	44	of	of	ADP
ejpam-6425	285	45	science	science	NOUN
ejpam-6425	285	46	and	and	CCONJ
ejpam-6425	285	47	technology	technology	NOUN
ejpam-6425	285	48	-	-	PUNCT
ejpam-6425	285	49	science	science	NOUN
ejpam-6425	285	50	education	education	PROPN
ejpam-6425	285	51	institute	institute	NOUN
ejpam-6425	285	52	(	(	PUNCT
ejpam-6425	285	53	dost	dost	NOUN
ejpam-6425	285	54	-	-	PUNCT
ejpam-6425	285	55	sei	sei	ADJ
ejpam-6425	285	56	)	)	PUNCT
ejpam-6425	285	57	,	,	PUNCT
ejpam-6425	285	58	so	so	ADV
ejpam-6425	285	59	our	our	PRON
ejpam-6425	285	60	heartfelt	heartfelt	ADJ
ejpam-6425	285	61	gratitude	gratitude	NOUN
ejpam-6425	285	62	to	to	ADP
ejpam-6425	285	63	them	they	PRON
ejpam-6425	285	64	.	.	PUNCT
ejpam-6425	286	1	we	we	PRON
ejpam-6425	286	2	also	also	ADV
ejpam-6425	286	3	express	express	VERB
ejpam-6425	286	4	our	our	PRON
ejpam-6425	286	5	gratitude	gratitude	NOUN
ejpam-6425	286	6	to	to	ADP
ejpam-6425	286	7	various	various	ADJ
ejpam-6425	286	8	reviewers	reviewer	NOUN
ejpam-6425	286	9	for	for	ADP
ejpam-6425	286	10	their	their	PRON
ejpam-6425	286	11	comments	comment	NOUN
ejpam-6425	286	12	and	and	CCONJ
ejpam-6425	286	13	suggestions	suggestion	NOUN
ejpam-6425	286	14	that	that	PRON
ejpam-6425	286	15	helped	help	VERB
ejpam-6425	286	16	to	to	PART
ejpam-6425	286	17	improve	improve	VERB
ejpam-6425	286	18	the	the	DET
ejpam-6425	286	19	overall	overall	ADJ
ejpam-6425	286	20	content	content	NOUN
ejpam-6425	286	21	of	of	ADP
ejpam-6425	286	22	this	this	DET
ejpam-6425	286	23	paper	paper	NOUN
ejpam-6425	286	24	.	.	PUNCT
ejpam-6425	287	1	references	reference	NOUN
ejpam-6425	287	2	[	[	X
ejpam-6425	287	3	1	1	X
ejpam-6425	287	4	]	]	X
ejpam-6425	287	5	joseph	joseph	PROPN
ejpam-6425	287	6	a.	a.	PROPN
ejpam-6425	287	7	gallian	gallian	PROPN
ejpam-6425	287	8	.	.	PUNCT
ejpam-6425	288	1	a	a	DET
ejpam-6425	288	2	dynamic	dynamic	ADJ
ejpam-6425	288	3	survey	survey	NOUN
ejpam-6425	288	4	of	of	ADP
ejpam-6425	288	5	graph	graph	NOUN
ejpam-6425	288	6	labeling	labeling	NOUN
ejpam-6425	288	7	.	.	PUNCT
ejpam-6425	289	1	electronic	electronic	ADJ
ejpam-6425	289	2	journal	journal	NOUN
ejpam-6425	289	3	of	of	ADP
ejpam-6425	289	4	combinatorics	combinatoric	NOUN
ejpam-6425	289	5	,	,	PUNCT
ejpam-6425	289	6	6(25):4–623	6(25):4–623	NUM
ejpam-6425	289	7	,	,	PUNCT
ejpam-6425	289	8	2022	2022	NUM
ejpam-6425	289	9	.	.	PUNCT
ejpam-6425	290	1	[	[	X
ejpam-6425	290	2	2	2	X
ejpam-6425	290	3	]	]	PUNCT
ejpam-6425	290	4	s.	s.	PROPN
ejpam-6425	290	5	ramachandran	ramachandran	PROPN
ejpam-6425	290	6	and	and	CCONJ
ejpam-6425	290	7	t.	t.	PROPN
ejpam-6425	290	8	gnanaseelan	gnanaseelan	PROPN
ejpam-6425	290	9	.	.	PUNCT
ejpam-6425	291	1	prime	prime	ADJ
ejpam-6425	291	2	labeling	labeling	NOUN
ejpam-6425	291	3	on	on	ADP
ejpam-6425	291	4	some	some	DET
ejpam-6425	291	5	cycle	cycle	NOUN
ejpam-6425	291	6	related	relate	VERB
ejpam-6425	291	7	graphs	graph	NOUN
ejpam-6425	291	8	.	.	PUNCT
ejpam-6425	292	1	advances	advance	NOUN
ejpam-6425	292	2	and	and	CCONJ
ejpam-6425	292	3	applications	application	NOUN
ejpam-6425	292	4	in	in	ADP
ejpam-6425	292	5	mathematical	mathematical	ADJ
ejpam-6425	292	6	sciences	science	NOUN
ejpam-6425	292	7	,	,	PUNCT
ejpam-6425	292	8	21(12):6711–6719	21(12):6711–6719	NUM
ejpam-6425	292	9	,	,	PUNCT
ejpam-6425	292	10	2022	2022	NUM
ejpam-6425	292	11	.	.	PUNCT
ejpam-6425	293	1	[	[	X
ejpam-6425	293	2	3	3	X
ejpam-6425	293	3	]	]	X
ejpam-6425	293	4	s.	s.	PROPN
ejpam-6425	293	5	venkatesan	venkatesan	PROPN
ejpam-6425	293	6	and	and	CCONJ
ejpam-6425	293	7	p.	p.	PROPN
ejpam-6425	293	8	sekar	sekar	PROPN
ejpam-6425	293	9	.	.	PUNCT
ejpam-6425	294	1	not	not	PART
ejpam-6425	294	2	all	all	DET
ejpam-6425	294	3	wheel	wheel	NOUN
ejpam-6425	294	4	graphs	graph	NOUN
ejpam-6425	294	5	were	be	AUX
ejpam-6425	294	6	edge	edge	NOUN
ejpam-6425	294	7	-	-	PUNCT
ejpam-6425	294	8	graceful	graceful	NOUN
ejpam-6425	294	9	.	.	PUNCT
ejpam-6425	295	1	journal	journal	PROPN
ejpam-6425	295	2	homepage	homepage	NOUN
ejpam-6425	295	3	:	:	PUNCT
ejpam-6425	295	4	http://www.ijesm.co.in	http://www.ijesm.co.in	PROPN
ejpam-6425	295	5	,	,	PUNCT
ejpam-6425	295	6	6:5	6:5	NUM
ejpam-6425	295	7	,	,	PUNCT
ejpam-6425	295	8	2017	2017	NUM
ejpam-6425	295	9	.	.	PUNCT
ejpam-6425	296	1	[	[	X
ejpam-6425	296	2	4	4	NUM
ejpam-6425	296	3	]	]	PUNCT
ejpam-6425	296	4	s.-p	s.-p	PROPN
ejpam-6425	296	5	.	.	PUNCT
ejpam-6425	297	1	lo	lo	PROPN
ejpam-6425	297	2	.	.	PUNCT
ejpam-6425	298	1	on	on	ADP
ejpam-6425	298	2	edge	edge	NOUN
ejpam-6425	298	3	-	-	PUNCT
ejpam-6425	298	4	graceful	graceful	NOUN
ejpam-6425	298	5	labelings	labeling	NOUN
ejpam-6425	298	6	of	of	ADP
ejpam-6425	298	7	graphs	graph	NOUN
ejpam-6425	298	8	.	.	PUNCT
ejpam-6425	299	1	congressus	congressus	PROPN
ejpam-6425	299	2	numerantium	numerantium	PROPN
ejpam-6425	299	3	,	,	PUNCT
ejpam-6425	299	4	50:231–241	50:231–241	PROPN
ejpam-6425	299	5	,	,	PUNCT
ejpam-6425	299	6	1985	1985	NUM
ejpam-6425	299	7	.	.	PUNCT
ejpam-6425	300	1	[	[	X
ejpam-6425	300	2	5	5	NUM
ejpam-6425	300	3	]	]	PUNCT
ejpam-6425	300	4	q.	q.	PROPN
ejpam-6425	300	5	kuan	kuan	PROPN
ejpam-6425	300	6	,	,	PUNCT
ejpam-6425	300	7	s.	s.	PROPN
ejpam-6425	300	8	lee	lee	PROPN
ejpam-6425	300	9	,	,	PUNCT
ejpam-6425	300	10	j.	j.	PROPN
ejpam-6425	300	11	mitchem	mitchem	PROPN
ejpam-6425	300	12	,	,	PUNCT
ejpam-6425	300	13	and	and	CCONJ
ejpam-6425	300	14	a.	a.	PROPN
ejpam-6425	300	15	wang	wang	PROPN
ejpam-6425	300	16	.	.	PUNCT
ejpam-6425	301	1	on	on	ADP
ejpam-6425	301	2	edge	edge	NOUN
ejpam-6425	301	3	-	-	PUNCT
ejpam-6425	301	4	graceful	graceful	ADJ
ejpam-6425	301	5	unicyclic	unicyclic	ADJ
ejpam-6425	301	6	graphs	graph	NOUN
ejpam-6425	301	7	.	.	PUNCT
ejpam-6425	302	1	congressus	congressus	PROPN
ejpam-6425	302	2	numerantium	numerantium	PROPN
ejpam-6425	302	3	,	,	PUNCT
ejpam-6425	302	4	61:65–74	61:65–74	PROPN
ejpam-6425	302	5	,	,	PUNCT
ejpam-6425	302	6	1988	1988	NUM
ejpam-6425	302	7	.	.	PUNCT
ejpam-6425	303	1	[	[	X
ejpam-6425	303	2	6	6	NUM
ejpam-6425	303	3	]	]	PUNCT
ejpam-6425	303	4	l.	l.	PROPN
ejpam-6425	303	5	lee	lee	PROPN
ejpam-6425	303	6	,	,	PUNCT
ejpam-6425	303	7	s.	s.	PROPN
ejpam-6425	303	8	lee	lee	PROPN
ejpam-6425	303	9	,	,	PUNCT
ejpam-6425	303	10	and	and	CCONJ
ejpam-6425	303	11	g.	g.	PROPN
ejpam-6425	303	12	muty	muty	PROPN
ejpam-6425	303	13	.	.	PUNCT
ejpam-6425	304	1	on	on	ADP
ejpam-6425	304	2	edge	edge	NOUN
ejpam-6425	304	3	-	-	PUNCT
ejpam-6425	304	4	graceful	graceful	NOUN
ejpam-6425	304	5	labelings	labeling	NOUN
ejpam-6425	304	6	of	of	ADP
ejpam-6425	304	7	complete	complete	ADJ
ejpam-6425	304	8	graphs	graph	NOUN
ejpam-6425	304	9	:	:	PUNCT
ejpam-6425	304	10	solutions	solution	NOUN
ejpam-6425	304	11	of	of	ADP
ejpam-6425	304	12	lo	lo	PROPN
ejpam-6425	304	13	’s	’s	PART
ejpam-6425	304	14	conjecture	conjecture	NOUN
ejpam-6425	304	15	.	.	PUNCT
ejpam-6425	305	1	congressus	congressus	PROPN
ejpam-6425	305	2	numerantium	numerantium	PROPN
ejpam-6425	305	3	,	,	PUNCT
ejpam-6425	305	4	62:225–233	62:225–233	PROPN
ejpam-6425	305	5	,	,	PUNCT
ejpam-6425	305	6	1988	1988	NUM
ejpam-6425	305	7	.	.	PUNCT
ejpam-6425	306	1	[	[	X
ejpam-6425	306	2	7	7	X
ejpam-6425	306	3	]	]	X
ejpam-6425	306	4	d.	d.	PROPN
ejpam-6425	306	5	small	small	PROPN
ejpam-6425	306	6	.	.	PUNCT
ejpam-6425	307	1	regular	regular	ADJ
ejpam-6425	307	2	(	(	PUNCT
ejpam-6425	307	3	even	even	ADV
ejpam-6425	307	4	)	)	PUNCT
ejpam-6425	307	5	spider	spider	NOUN
ejpam-6425	307	6	graphs	graph	NOUN
ejpam-6425	307	7	are	be	AUX
ejpam-6425	307	8	edge	edge	NOUN
ejpam-6425	307	9	-	-	PUNCT
ejpam-6425	307	10	graceful	graceful	NOUN
ejpam-6425	307	11	.	.	PUNCT
ejpam-6425	308	1	congressus	congressus	PROPN
ejpam-6425	308	2	numerantium	numerantium	PROPN
ejpam-6425	308	3	,	,	PUNCT
ejpam-6425	308	4	74:247–254	74:247–254	PROPN
ejpam-6425	308	5	,	,	PUNCT
ejpam-6425	308	6	1990	1990	NUM
ejpam-6425	308	7	.	.	PUNCT
ejpam-6425	309	1	[	[	X
ejpam-6425	309	2	8	8	NUM
ejpam-6425	309	3	]	]	PUNCT
ejpam-6425	309	4	s.	s.	PROPN
ejpam-6425	309	5	cabaniss	cabaniss	PROPN
ejpam-6425	309	6	,	,	PUNCT
ejpam-6425	309	7	r.	r.	PROPN
ejpam-6425	309	8	low	low	PROPN
ejpam-6425	309	9	,	,	PUNCT
ejpam-6425	309	10	and	and	CCONJ
ejpam-6425	309	11	j.	j.	PROPN
ejpam-6425	309	12	mitchem	mitchem	PROPN
ejpam-6425	309	13	.	.	PUNCT
ejpam-6425	310	1	on	on	ADP
ejpam-6425	310	2	edge	edge	NOUN
ejpam-6425	310	3	-	-	PUNCT
ejpam-6425	310	4	graceful	graceful	NOUN
ejpam-6425	310	5	regular	regular	ADJ
ejpam-6425	310	6	graphs	graph	NOUN
ejpam-6425	310	7	and	and	CCONJ
ejpam-6425	310	8	trees	tree	NOUN
ejpam-6425	310	9	.	.	PUNCT
ejpam-6425	311	1	ars	ars	PROPN
ejpam-6425	311	2	combinatoria	combinatoria	PROPN
ejpam-6425	311	3	,	,	PUNCT
ejpam-6425	311	4	34:129–142	34:129–142	PROPN
ejpam-6425	311	5	,	,	PUNCT
ejpam-6425	311	6	1992	1992	NUM
ejpam-6425	311	7	.	.	PUNCT
ejpam-6425	312	1	[	[	X
ejpam-6425	312	2	9	9	NUM
ejpam-6425	312	3	]	]	PUNCT
ejpam-6425	312	4	a.	a.	NOUN
ejpam-6425	312	5	angel	angel	NOUN
ejpam-6425	312	6	,	,	PUNCT
ejpam-6425	312	7	j.	j.	PROPN
ejpam-6425	312	8	l.	l.	PROPN
ejpam-6425	312	9	gamurot	gamurot	PROPN
ejpam-6425	312	10	,	,	PUNCT
ejpam-6425	312	11	j.	j.	PROPN
ejpam-6425	312	12	r.	r.	PROPN
ejpam-6425	312	13	antalan	antalan	PROPN
ejpam-6425	312	14	,	,	PUNCT
ejpam-6425	312	15	and	and	CCONJ
ejpam-6425	312	16	r.	r.	PROPN
ejpam-6425	312	17	tagle	tagle	NOUN
ejpam-6425	312	18	.	.	PUNCT
ejpam-6425	313	1	on	on	ADP
ejpam-6425	313	2	edge	edge	NOUN
ejpam-6425	313	3	-	-	PUNCT
ejpam-6425	313	4	gracefulness	gracefulness	NOUN
ejpam-6425	313	5	of	of	ADP
ejpam-6425	313	6	wheel	wheel	NOUN
ejpam-6425	313	7	a.	a.	PROPN
ejpam-6425	313	8	d.	d.	PROPN
ejpam-6425	313	9	c.	c.	PROPN
ejpam-6425	313	10	angel	angel	PROPN
ejpam-6425	313	11	et	et	PROPN
ejpam-6425	313	12	al	al	PROPN
ejpam-6425	313	13	.	.	PUNCT
ejpam-6425	313	14	/	/	SYM
ejpam-6425	313	15	eur	eur	PROPN
ejpam-6425	313	16	.	.	PUNCT
ejpam-6425	314	1	j.	j.	PROPN
ejpam-6425	314	2	pure	pure	PROPN
ejpam-6425	314	3	appl	appl	PROPN
ejpam-6425	314	4	.	.	PROPN
ejpam-6425	314	5	math	math	PROPN
ejpam-6425	314	6	,	,	PUNCT
ejpam-6425	314	7	18	18	NUM
ejpam-6425	314	8	(	(	PUNCT
ejpam-6425	314	9	4	4	NUM
ejpam-6425	314	10	)	)	PUNCT
ejpam-6425	314	11	(	(	PUNCT
ejpam-6425	314	12	2025	2025	NUM
ejpam-6425	314	13	)	)	PUNCT
ejpam-6425	314	14	,	,	PUNCT
ejpam-6425	314	15	6425	6425	NUM
ejpam-6425	314	16	18	18	NUM
ejpam-6425	314	17	of	of	ADP
ejpam-6425	314	18	33	33	NUM
ejpam-6425	314	19	graphs	graph	NOUN
ejpam-6425	314	20	.	.	PUNCT
ejpam-6425	315	1	communications	communication	NOUN
ejpam-6425	315	2	in	in	ADP
ejpam-6425	315	3	combinatorics	combinatoric	NOUN
ejpam-6425	315	4	,	,	PUNCT
ejpam-6425	315	5	cryptography	cryptography	NOUN
ejpam-6425	315	6	&	&	CCONJ
ejpam-6425	315	7	computer	computer	PROPN
ejpam-6425	315	8	science	science	NOUN
ejpam-6425	315	9	,	,	PUNCT
ejpam-6425	315	10	pages	page	NOUN
ejpam-6425	315	11	213–219	213–219	NUM
ejpam-6425	315	12	,	,	PUNCT
ejpam-6425	315	13	2024	2024	NUM
ejpam-6425	315	14	.	.	PUNCT
ejpam-6425	316	1	[	[	X
ejpam-6425	316	2	10	10	NUM
ejpam-6425	316	3	]	]	X
ejpam-6425	316	4	jonathan	jonathan	PROPN
ejpam-6425	316	5	l.	l.	PROPN
ejpam-6425	316	6	gross	gross	PROPN
ejpam-6425	316	7	,	,	PUNCT
ejpam-6425	316	8	jay	jay	PROPN
ejpam-6425	316	9	yellen	yellen	VERB
ejpam-6425	316	10	,	,	PUNCT
ejpam-6425	316	11	and	and	CCONJ
ejpam-6425	316	12	mark	mark	PROPN
ejpam-6425	316	13	anderson	anderson	PROPN
ejpam-6425	316	14	.	.	PUNCT
ejpam-6425	317	1	graph	graph	NOUN
ejpam-6425	317	2	theory	theory	NOUN
ejpam-6425	317	3	and	and	CCONJ
ejpam-6425	317	4	its	its	PRON
ejpam-6425	317	5	applications	application	NOUN
ejpam-6425	317	6	.	.	PUNCT
ejpam-6425	318	1	chapman	chapman	NOUN
ejpam-6425	318	2	and	and	CCONJ
ejpam-6425	318	3	hall	hall	PROPN
ejpam-6425	318	4	/	/	SYM
ejpam-6425	318	5	crc	crc	NOUN
ejpam-6425	318	6	,	,	PUNCT
ejpam-6425	318	7	2018	2018	NUM
ejpam-6425	318	8	.	.	PUNCT
ejpam-6425	319	1	[	[	X
ejpam-6425	319	2	11	11	NUM
ejpam-6425	319	3	]	]	PUNCT
ejpam-6425	319	4	mohammed	mohammed	PROPN
ejpam-6425	319	5	aljohani	aljohani	PROPN
ejpam-6425	319	6	and	and	CCONJ
ejpam-6425	319	7	salama	salama	PROPN
ejpam-6425	319	8	nagy	nagy	PROPN
ejpam-6425	319	9	daoud	daoud	PROPN
ejpam-6425	319	10	.	.	PUNCT
ejpam-6425	320	1	the	the	DET
ejpam-6425	320	2	edge	edge	NOUN
ejpam-6425	320	3	odd	odd	ADJ
ejpam-6425	320	4	graceful	graceful	ADJ
ejpam-6425	320	5	labeling	labeling	NOUN
ejpam-6425	320	6	of	of	ADP
ejpam-6425	320	7	water	water	NOUN
ejpam-6425	320	8	wheel	wheel	NOUN
ejpam-6425	320	9	graphs	graph	NOUN
ejpam-6425	320	10	.	.	PUNCT
ejpam-6425	321	1	axioms	axiom	NOUN
ejpam-6425	321	2	,	,	PUNCT
ejpam-6425	321	3	14(1):5	14(1):5	NUM
ejpam-6425	321	4	,	,	PUNCT
ejpam-6425	321	5	2024	2024	NUM
ejpam-6425	321	6	.	.	PUNCT
ejpam-6425	322	1	[	[	X
ejpam-6425	322	2	12	12	NUM
ejpam-6425	322	3	]	]	X
ejpam-6425	322	4	e.	e.	PROPN
ejpam-6425	322	5	sampathkumar	sampathkumar	PROPN
ejpam-6425	322	6	and	and	CCONJ
ejpam-6425	322	7	h.	h.	PROPN
ejpam-6425	322	8	b.	b.	PROPN
ejpam-6425	322	9	walikar	walikar	PROPN
ejpam-6425	322	10	.	.	PUNCT
ejpam-6425	323	1	on	on	ADP
ejpam-6425	323	2	the	the	DET
ejpam-6425	323	3	splitting	splitting	NOUN
ejpam-6425	323	4	graph	graph	NOUN
ejpam-6425	323	5	of	of	ADP
ejpam-6425	323	6	a	a	DET
ejpam-6425	323	7	graph	graph	NOUN
ejpam-6425	323	8	.	.	PUNCT
ejpam-6425	323	9	karnatak	karnatak	PROPN
ejpam-6425	323	10	university	university	PROPN
ejpam-6425	323	11	science	science	NOUN
ejpam-6425	323	12	,	,	PUNCT
ejpam-6425	323	13	25(13):13–16	25(13):13–16	NOUN
ejpam-6425	323	14	,	,	PUNCT
ejpam-6425	323	15	1980	1980	NUM
ejpam-6425	323	16	.	.	PUNCT
ejpam-6425	324	1	[	[	X
ejpam-6425	324	2	13	13	NUM
ejpam-6425	324	3	]	]	X
ejpam-6425	324	4	david	david	PROPN
ejpam-6425	324	5	burton	burton	PROPN
ejpam-6425	324	6	.	.	PUNCT
ejpam-6425	325	1	ebook	ebook	PROPN
ejpam-6425	325	2	:	:	PUNCT
ejpam-6425	325	3	elementary	elementary	ADJ
ejpam-6425	325	4	number	number	NOUN
ejpam-6425	325	5	theory	theory	NOUN
ejpam-6425	325	6	.	.	PUNCT
ejpam-6425	326	1	mcgraw	mcgraw	PROPN
ejpam-6425	326	2	hill	hill	PROPN
ejpam-6425	326	3	,	,	PUNCT
ejpam-6425	326	4	2010	2010	NUM
ejpam-6425	326	5	.	.	PUNCT
ejpam-6425	327	1	[	[	X
ejpam-6425	327	2	14	14	NUM
ejpam-6425	327	3	]	]	X
ejpam-6425	327	4	aaron	aaron	PROPN
ejpam-6425	327	5	dc	dc	PROPN
ejpam-6425	327	6	angel	angel	PROPN
ejpam-6425	327	7	,	,	PUNCT
ejpam-6425	327	8	john	john	PROPN
ejpam-6425	327	9	rafael	rafael	PROPN
ejpam-6425	327	10	m.	m.	PROPN
ejpam-6425	327	11	antalan	antalan	PROPN
ejpam-6425	327	12	,	,	PUNCT
ejpam-6425	327	13	john	john	PROPN
ejpam-6425	327	14	loureynz	loureynz	PROPN
ejpam-6425	327	15	f.	f.	PROPN
ejpam-6425	327	16	gamurot	gamurot	PROPN
ejpam-6425	327	17	,	,	PUNCT
ejpam-6425	327	18	and	and	CCONJ
ejpam-6425	327	19	richard	richard	PROPN
ejpam-6425	327	20	p.	p.	PROPN
ejpam-6425	327	21	tagle	tagle	NOUN
ejpam-6425	327	22	.	.	PUNCT
ejpam-6425	328	1	edge	edge	NOUN
ejpam-6425	328	2	-	-	PUNCT
ejpam-6425	328	3	graceful	graceful	NOUN
ejpam-6425	328	4	usual	usual	ADJ
ejpam-6425	328	5	fan	fan	NOUN
ejpam-6425	328	6	graphs	graph	NOUN
ejpam-6425	328	7	,	,	PUNCT
ejpam-6425	328	8	2024	2024	NUM
ejpam-6425	328	9	.	.	PUNCT
ejpam-6425	329	1	arxiv	arxiv	PROPN
ejpam-6425	329	2	preprint	preprint	PROPN
ejpam-6425	329	3	arxiv:2412.08338	arxiv:2412.08338	PROPN
ejpam-6425	329	4	.	.	PUNCT
ejpam-6425	330	1	[	[	X
ejpam-6425	330	2	15	15	NUM
ejpam-6425	330	3	]	]	X
ejpam-6425	330	4	bal	bal	PROPN
ejpam-6425	330	5	bahadur	bahadur	PROPN
ejpam-6425	330	6	tamang	tamang	PROPN
ejpam-6425	330	7	.	.	PUNCT
ejpam-6425	331	1	on	on	ADP
ejpam-6425	331	2	the	the	DET
ejpam-6425	331	3	study	study	NOUN
ejpam-6425	331	4	of	of	ADP
ejpam-6425	331	5	quadratic	quadratic	ADJ
ejpam-6425	331	6	diophantine	diophantine	NOUN
ejpam-6425	331	7	equations	equation	NOUN
ejpam-6425	331	8	.	.	PUNCT
ejpam-6425	332	1	phd	phd	NOUN
ejpam-6425	332	2	thesis	thesis	PROPN
ejpam-6425	332	3	,	,	PUNCT
ejpam-6425	332	4	tribhuvan	tribhuvan	PROPN
ejpam-6425	332	5	university	university	PROPN
ejpam-6425	332	6	kathmandu	kathmandu	NOUN
ejpam-6425	332	7	,	,	PUNCT
ejpam-6425	332	8	2021	2021	NUM
ejpam-6425	332	9	.	.	PUNCT
ejpam-6425	333	1	a.	a.	PROPN
ejpam-6425	333	2	d.	d.	PROPN
ejpam-6425	333	3	c.	c.	PROPN
ejpam-6425	333	4	angel	angel	PROPN
ejpam-6425	333	5	et	et	PROPN
ejpam-6425	333	6	al	al	PROPN
ejpam-6425	333	7	.	.	PUNCT
ejpam-6425	333	8	/	/	SYM
ejpam-6425	333	9	eur	eur	PROPN
ejpam-6425	333	10	.	.	PUNCT
ejpam-6425	334	1	j.	j.	PROPN
ejpam-6425	334	2	pure	pure	PROPN
ejpam-6425	334	3	appl	appl	PROPN
ejpam-6425	334	4	.	.	PROPN
ejpam-6425	334	5	math	math	PROPN
ejpam-6425	334	6	,	,	PUNCT
ejpam-6425	334	7	18	18	NUM
ejpam-6425	334	8	(	(	PUNCT
ejpam-6425	334	9	4	4	NUM
ejpam-6425	334	10	)	)	PUNCT
ejpam-6425	334	11	(	(	PUNCT
ejpam-6425	334	12	2025	2025	NUM
ejpam-6425	334	13	)	)	PUNCT
ejpam-6425	334	14	,	,	PUNCT
ejpam-6425	334	15	6425	6425	NUM
ejpam-6425	334	16	19	19	NUM
ejpam-6425	334	17	of	of	ADP
ejpam-6425	334	18	33	33	NUM
ejpam-6425	334	19	appendices	appendix	NOUN
ejpam-6425	334	20	this	this	DET
ejpam-6425	334	21	section	section	NOUN
ejpam-6425	334	22	shows	show	VERB
ejpam-6425	334	23	all	all	DET
ejpam-6425	334	24	the	the	DET
ejpam-6425	334	25	numerical	numerical	ADJ
ejpam-6425	334	26	tables	table	NOUN
ejpam-6425	334	27	and	and	CCONJ
ejpam-6425	334	28	computer	computer	NOUN
ejpam-6425	334	29	programs	program	NOUN
ejpam-6425	334	30	that	that	PRON
ejpam-6425	334	31	are	be	AUX
ejpam-6425	334	32	used	use	VERB
ejpam-6425	334	33	for	for	ADP
ejpam-6425	334	34	this	this	DET
ejpam-6425	334	35	study	study	NOUN
ejpam-6425	334	36	.	.	PUNCT
ejpam-6425	335	1	we	we	PRON
ejpam-6425	335	2	begin	begin	VERB
ejpam-6425	335	3	by	by	ADP
ejpam-6425	335	4	presenting	present	VERB
ejpam-6425	335	5	the	the	DET
ejpam-6425	335	6	numerical	numerical	ADJ
ejpam-6425	335	7	tables	table	NOUN
ejpam-6425	335	8	used	use	VERB
ejpam-6425	335	9	in	in	ADP
ejpam-6425	335	10	this	this	DET
ejpam-6425	335	11	study	study	NOUN
ejpam-6425	335	12	.	.	PUNCT
ejpam-6425	336	1	these	these	DET
ejpam-6425	336	2	tables	table	NOUN
ejpam-6425	336	3	show	show	VERB
ejpam-6425	336	4	the	the	DET
ejpam-6425	336	5	complete	complete	ADJ
ejpam-6425	336	6	set	set	NOUN
ejpam-6425	336	7	of	of	ADP
ejpam-6425	336	8	solutions	solution	NOUN
ejpam-6425	336	9	to	to	ADP
ejpam-6425	336	10	the	the	DET
ejpam-6425	336	11	diophantine	diophantine	NOUN
ejpam-6425	336	12	equations	equation	NOUN
ejpam-6425	336	13	that	that	PRON
ejpam-6425	336	14	are	be	AUX
ejpam-6425	336	15	involved	involve	VERB
ejpam-6425	336	16	in	in	ADP
ejpam-6425	336	17	the	the	DET
ejpam-6425	336	18	study	study	NOUN
ejpam-6425	336	19	.	.	PUNCT
ejpam-6425	337	1	each	each	DET
ejpam-6425	337	2	table	table	NOUN
ejpam-6425	337	3	corresponds	correspond	VERB
ejpam-6425	337	4	to	to	ADP
ejpam-6425	337	5	a	a	DET
ejpam-6425	337	6	particular	particular	ADJ
ejpam-6425	337	7	diophantine	diophantine	NOUN
ejpam-6425	337	8	equation	equation	NOUN
ejpam-6425	337	9	associated	associate	VERB
ejpam-6425	337	10	with	with	ADP
ejpam-6425	337	11	the	the	DET
ejpam-6425	337	12	main	main	ADJ
ejpam-6425	337	13	theorems	theorem	NOUN
ejpam-6425	337	14	,	,	PUNCT
ejpam-6425	337	15	providing	provide	VERB
ejpam-6425	337	16	detailed	detailed	ADJ
ejpam-6425	337	17	numerical	numerical	ADJ
ejpam-6425	337	18	evidence	evidence	NOUN
ejpam-6425	337	19	that	that	PRON
ejpam-6425	337	20	supports	support	VERB
ejpam-6425	337	21	the	the	DET
ejpam-6425	337	22	theoretical	theoretical	ADJ
ejpam-6425	337	23	results	result	NOUN
ejpam-6425	337	24	established	establish	VERB
ejpam-6425	337	25	in	in	ADP
ejpam-6425	337	26	the	the	DET
ejpam-6425	337	27	earlier	early	ADJ
ejpam-6425	337	28	sections	section	NOUN
ejpam-6425	337	29	.	.	PUNCT
ejpam-6425	338	1	presented	present	VERB
ejpam-6425	338	2	below	below	ADV
ejpam-6425	338	3	,	,	PUNCT
ejpam-6425	338	4	is	be	AUX
ejpam-6425	338	5	the	the	DET
ejpam-6425	338	6	complete	complete	ADJ
ejpam-6425	338	7	table	table	NOUN
ejpam-6425	338	8	of	of	ADP
ejpam-6425	338	9	solutions	solution	NOUN
ejpam-6425	338	10	for	for	ADP
ejpam-6425	338	11	the	the	DET
ejpam-6425	338	12	diophantine	diophantine	NOUN
ejpam-6425	338	13	equation	equation	NOUN
ejpam-6425	338	14	41n2	41n2	NUM
ejpam-6425	339	1	+	+	CCONJ
ejpam-6425	340	1	7n−	7n−	NUM
ejpam-6425	340	2	6nk	6nk	NOUN
ejpam-6425	340	3	−	−	NOUN
ejpam-6425	340	4	2k	2k	NOUN
ejpam-6425	340	5	=	=	SYM
ejpam-6425	340	6	0	0	NUM
ejpam-6425	340	7	,	,	PUNCT
ejpam-6425	340	8	relevant	relevant	ADJ
ejpam-6425	340	9	to	to	PART
ejpam-6425	340	10	theorem	theorem	VERB
ejpam-6425	340	11	2	2	NUM
ejpam-6425	340	12	.	.	NOUN
ejpam-6425	340	13	table	table	NOUN
ejpam-6425	340	14	1	1	NUM
ejpam-6425	340	15	:	:	PUNCT
ejpam-6425	340	16	solutions	solution	NOUN
ejpam-6425	340	17	for	for	ADP
ejpam-6425	340	18	41n2	41n2	NUM
ejpam-6425	341	1	+	+	PROPN
ejpam-6425	342	1	7n−	7n−	NUM
ejpam-6425	342	2	6nk	6nk	NOUN
ejpam-6425	342	3	−	−	NOUN
ejpam-6425	342	4	2k	2k	NOUN
ejpam-6425	342	5	=	=	SYM
ejpam-6425	342	6	0	0	NUM
ejpam-6425	342	7	n1	n1	PROPN
ejpam-6425	342	8	n2	n2	NOUN
ejpam-6425	342	9	x	x	PUNCT
ejpam-6425	342	10	y	y	PROPN
ejpam-6425	342	11	n	n	CCONJ
ejpam-6425	342	12	k	k	PROPN
ejpam-6425	342	13	n1	n1	PROPN
ejpam-6425	342	14	n2	n2	NOUN
ejpam-6425	342	15	x	x	PUNCT
ejpam-6425	342	16	y	y	PROPN
ejpam-6425	342	17	n	n	CCONJ
ejpam-6425	342	18	k	k	PROPN
ejpam-6425	342	19	1	1	NUM
ejpam-6425	342	20	13120	13120	NUM
ejpam-6425	342	21	6560.5	6560.5	NUM
ejpam-6425	342	22	1093.25	1093.25	NUM
ejpam-6425	342	23	26.33	26.33	NUM
ejpam-6425	342	24	178.85	178.85	NUM
ejpam-6425	342	25	-1	-1	ADP
ejpam-6425	342	26	-13120	-13120	PROPN
ejpam-6425	342	27	-6560.5	-6560.5	PUNCT
ejpam-6425	343	1	-1093.25	-1093.25	NOUN
ejpam-6425	343	2	-27	-27	SYM
ejpam-6425	343	3	-185.63	-185.63	PROPN
ejpam-6425	343	4	2	2	NUM
ejpam-6425	343	5	6560	6560	NUM
ejpam-6425	343	6	3281	3281	NUM
ejpam-6425	343	7	546.5	546.5	NUM
ejpam-6425	343	8	13	13	NUM
ejpam-6425	343	9	87.75	87.75	NUM
ejpam-6425	343	10	-2	-2	NOUN
ejpam-6425	344	1	-6560	-6560	PROPN
ejpam-6425	344	2	-3281	-3281	PUNCT
ejpam-6425	345	1	-546.5	-546.5	INTJ
ejpam-6425	345	2	-13.67	-13.67	PROPN
ejpam-6425	345	3	-94.53	-94.53	PROPN
ejpam-6425	345	4	4	4	NUM
ejpam-6425	345	5	3280	3280	NUM
ejpam-6425	345	6	1642	1642	NUM
ejpam-6425	345	7	273	273	NUM
ejpam-6425	345	8	6.33	6.33	NUM
ejpam-6425	345	9	42.22	42.22	NUM
ejpam-6425	345	10	-4	-4	PUNCT
ejpam-6425	345	11	-3280	-3280	PROPN
ejpam-6425	345	12	-1642	-1642	PROPN
ejpam-6425	345	13	-273	-273	PROPN
ejpam-6425	346	1	-7	-7	NOUN
ejpam-6425	346	2	-49	-49	PROPN
ejpam-6425	346	3	5	5	NUM
ejpam-6425	346	4	2624	2624	NUM
ejpam-6425	346	5	1314.5	1314.5	NUM
ejpam-6425	346	6	218.25	218.25	NUM
ejpam-6425	346	7	5	5	NUM
ejpam-6425	346	8	33.13	33.13	NUM
ejpam-6425	346	9	-5	-5	PUNCT
ejpam-6425	346	10	-2624	-2624	PUNCT
ejpam-6425	346	11	-1314.5	-1314.5	PROPN
ejpam-6425	346	12	-218.25	-218.25	PROPN
ejpam-6425	346	13	-5.67	-5.67	NUM
ejpam-6425	347	1	-39.90	-39.90	NUM
ejpam-6425	347	2	8	8	NUM
ejpam-6425	347	3	1640	1640	NUM
ejpam-6425	347	4	824	824	NUM
ejpam-6425	347	5	136	136	NUM
ejpam-6425	347	6	3	3	NUM
ejpam-6425	347	7	19.5	19.5	NUM
ejpam-6425	347	8	-8	-8	NUM
ejpam-6425	347	9	-1640	-1640	PROPN
ejpam-6425	347	10	-824	-824	PROPN
ejpam-6425	347	11	-136	-136	PROPN
ejpam-6425	347	12	-3.67	-3.67	NOUN
ejpam-6425	347	13	-26.28	-26.28	NUM
ejpam-6425	347	14	10	10	NUM
ejpam-6425	347	15	1312	1312	NUM
ejpam-6425	347	16	661	661	NUM
ejpam-6425	347	17	108.5	108.5	NUM
ejpam-6425	347	18	2.33	2.33	NUM
ejpam-6425	347	19	14.97	14.97	NUM
ejpam-6425	347	20	-10	-10	PUNCT
ejpam-6425	347	21	-1312	-1312	PROPN
ejpam-6425	347	22	-661	-661	PROPN
ejpam-6425	347	23	-108.5	-108.5	INTJ
ejpam-6425	347	24	-3	-3	PUNCT
ejpam-6425	347	25	-21.75	-21.75	PROPN
ejpam-6425	347	26	16	16	NUM
ejpam-6425	347	27	820	820	NUM
ejpam-6425	347	28	418	418	NUM
ejpam-6425	347	29	67	67	NUM
ejpam-6425	347	30	1.33	1.33	NUM
ejpam-6425	347	31	8.22	8.22	NUM
ejpam-6425	347	32	-16	-16	PUNCT
ejpam-6425	348	1	-820	-820	NUM
ejpam-6425	348	2	-418	-418	PROPN
ejpam-6425	348	3	-67	-67	PUNCT
ejpam-6425	348	4	-2	-2	NOUN
ejpam-6425	348	5	-15	-15	NUM
ejpam-6425	348	6	20	20	NUM
ejpam-6425	348	7	656	656	NUM
ejpam-6425	348	8	338	338	NUM
ejpam-6425	348	9	53	53	NUM
ejpam-6425	348	10	1	1	NUM
ejpam-6425	348	11	6	6	NUM
ejpam-6425	348	12	-20	-20	SYM
ejpam-6425	348	13	-656	-656	NUM
ejpam-6425	349	1	-338	-338	PROPN
ejpam-6425	349	2	-53	-53	PROPN
ejpam-6425	349	3	-1.67	-1.67	PRON
ejpam-6425	349	4	-12.78	-12.78	NOUN
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ejpam-6425	349	8	31.5	31.5	NUM
ejpam-6425	349	9	0.5	0.5	NUM
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ejpam-6425	349	11	-32	-32	PUNCT
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ejpam-6425	349	16	-9.53	-9.53	NUM
ejpam-6425	349	17	40	40	NUM
ejpam-6425	349	18	328	328	NUM
ejpam-6425	349	19	184	184	NUM
ejpam-6425	349	20	24	24	NUM
ejpam-6425	349	21	0.33	0.33	NUM
ejpam-6425	349	22	1.72	1.72	NUM
ejpam-6425	349	23	-40	-40	PUNCT
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ejpam-6425	350	4	-1	-1	INTJ
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ejpam-6425	351	4	180.5	180.5	NUM
ejpam-6425	351	5	23.25	23.25	NUM
ejpam-6425	351	6	0.32	0.32	NUM
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ejpam-6425	351	8	-41	-41	SYM
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ejpam-6425	351	16	134.5	134.5	NUM
ejpam-6425	351	17	11.75	11.75	NUM
ejpam-6425	351	18	0.08	0.08	NUM
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ejpam-6425	351	20	-64	-64	PUNCT
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ejpam-6425	352	5	80	80	NUM
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ejpam-6425	352	7	122	122	NUM
ejpam-6425	352	8	7	7	NUM
ejpam-6425	352	9	0	0	NUM
ejpam-6425	352	10	0	0	NUM
ejpam-6425	352	11	-80	-80	PUNCT
ejpam-6425	353	1	-164	-164	PROPN
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ejpam-6425	353	3	-7	-7	INTJ
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ejpam-6425	353	8	121	121	NUM
ejpam-6425	353	9	6.5	6.5	NUM
ejpam-6425	353	10	-0.01	-0.01	NUM
ejpam-6425	353	11	-0.03	-0.03	NUM
ejpam-6425	353	12	-82	-82	PUNCT
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ejpam-6425	355	5	1	1	NUM
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ejpam-6425	355	18	3281	3281	NUM
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ejpam-6425	355	21	87.75	87.75	NUM
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ejpam-6425	357	5	4	4	NUM
ejpam-6425	357	6	1642	1642	NUM
ejpam-6425	357	7	-273	-273	NOUN
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ejpam-6425	358	4	-49	-49	NUM
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ejpam-6425	360	12	-20	-20	PUNCT
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ejpam-6425	360	30	40	40	NUM
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ejpam-6425	362	12	-41	-41	PROPN
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ejpam-6425	362	17	205	205	NUM
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ejpam-6425	362	24	-64	-64	PUNCT
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ejpam-6425	363	5	164	164	NUM
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ejpam-6425	363	29	a.	a.	PROPN
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ejpam-6425	363	36	/	/	SYM
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ejpam-6425	364	1	j.	j.	PROPN
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ejpam-6425	364	3	appl	appl	PROPN
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ejpam-6425	364	5	math	math	PROPN
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ejpam-6425	364	8	(	(	PUNCT
ejpam-6425	364	9	4	4	NUM
ejpam-6425	364	10	)	)	PUNCT
ejpam-6425	364	11	(	(	PUNCT
ejpam-6425	364	12	2025	2025	NUM
ejpam-6425	364	13	)	)	PUNCT
ejpam-6425	364	14	,	,	PUNCT
ejpam-6425	364	15	6425	6425	NUM
ejpam-6425	364	16	20	20	NUM
ejpam-6425	364	17	of	of	ADP
ejpam-6425	364	18	33	33	NUM
ejpam-6425	364	19	presented	present	VERB
ejpam-6425	364	20	below	below	ADV
ejpam-6425	364	21	,	,	PUNCT
ejpam-6425	364	22	is	be	AUX
ejpam-6425	364	23	the	the	DET
ejpam-6425	364	24	complete	complete	ADJ
ejpam-6425	364	25	table	table	NOUN
ejpam-6425	364	26	of	of	ADP
ejpam-6425	364	27	solutions	solution	NOUN
ejpam-6425	364	28	for	for	ADP
ejpam-6425	364	29	the	the	DET
ejpam-6425	364	30	diophantine	diophantine	NOUN
ejpam-6425	364	31	equation	equation	NOUN
ejpam-6425	364	32	63n2	63n2	NUM
ejpam-6425	365	1	+	+	NUM
ejpam-6425	365	2	9n−	9n−	NUM
ejpam-6425	365	3	6nk	6nk	NOUN
ejpam-6425	365	4	−	−	NOUN
ejpam-6425	365	5	2k	2k	NOUN
ejpam-6425	365	6	=	=	SYM
ejpam-6425	365	7	0	0	NUM
ejpam-6425	365	8	,	,	PUNCT
ejpam-6425	365	9	relevant	relevant	ADJ
ejpam-6425	365	10	to	to	PART
ejpam-6425	365	11	theorem	theorem	VERB
ejpam-6425	365	12	3	3	NUM
ejpam-6425	365	13	.	.	NOUN
ejpam-6425	365	14	table	table	NOUN
ejpam-6425	365	15	2	2	NUM
ejpam-6425	365	16	:	:	PUNCT
ejpam-6425	365	17	solutions	solution	NOUN
ejpam-6425	365	18	for	for	ADP
ejpam-6425	365	19	63n2	63n2	PROPN
ejpam-6425	365	20	+	+	PROPN
ejpam-6425	365	21	9n−	9n−	NUM
ejpam-6425	365	22	6nk	6nk	NOUN
ejpam-6425	365	23	−	−	NOUN
ejpam-6425	365	24	2k	2k	NOUN
ejpam-6425	365	25	=	=	SYM
ejpam-6425	365	26	0	0	NUM
ejpam-6425	365	27	n1	n1	PROPN
ejpam-6425	365	28	n2	n2	NOUN
ejpam-6425	365	29	x	x	PUNCT
ejpam-6425	365	30	y	y	PROPN
ejpam-6425	365	31	n	n	CCONJ
ejpam-6425	365	32	k	k	PROPN
ejpam-6425	365	33	n1	n1	PROPN
ejpam-6425	365	34	n2	n2	NOUN
ejpam-6425	365	35	x	x	PUNCT
ejpam-6425	365	36	y	y	PROPN
ejpam-6425	365	37	n	n	CCONJ
ejpam-6425	365	38	k	k	PROPN
ejpam-6425	365	39	1	1	NUM
ejpam-6425	365	40	36288	36288	NUM
ejpam-6425	365	41	18144.5	18144.5	NUM
ejpam-6425	365	42	3023.92	3023.92	NUM
ejpam-6425	365	43	47.67	47.67	NUM
ejpam-6425	365	44	498.51	498.51	NUM
ejpam-6425	365	45	-1	-1	ADP
ejpam-6425	365	46	-36288	-36288	PROPN
ejpam-6425	365	47	-18144.5	-18144.5	PROPN
ejpam-6425	365	48	-3023.92	-3023.92	PROPN
ejpam-6425	366	1	-48.33	-48.33	PROPN
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ejpam-6425	366	4	18144	18144	NUM
ejpam-6425	366	5	9073	9073	NUM
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ejpam-6425	366	7	23.67	23.67	NUM
ejpam-6425	366	8	246.53	246.53	NUM
ejpam-6425	366	9	-2	-2	NOUN
ejpam-6425	367	1	-18144	-18144	PROPN
ejpam-6425	367	2	-9073	-9073	INTJ
ejpam-6425	368	1	-1511.83	-1511.83	PROPN
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ejpam-6425	368	4	3	3	NUM
ejpam-6425	368	5	12096	12096	NUM
ejpam-6425	368	6	6049.5	6049.5	NUM
ejpam-6425	368	7	1007.75	1007.75	NUM
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ejpam-6425	368	10	-3	-3	PUNCT
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ejpam-6425	368	12	-6049.5	-6049.5	PUNCT
ejpam-6425	369	1	-1007.75	-1007.75	PROPN
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ejpam-6425	369	4	4	4	NUM
ejpam-6425	369	5	9072	9072	NUM
ejpam-6425	369	6	4538	4538	NUM
ejpam-6425	369	7	755.67	755.67	NUM
ejpam-6425	369	8	11.67	11.67	NUM
ejpam-6425	369	9	120.56	120.56	NUM
ejpam-6425	369	10	-4	-4	SYM
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ejpam-6425	369	12	-4538	-4538	PROPN
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ejpam-6425	369	21	78.58	78.58	NUM
ejpam-6425	369	22	-6	-6	NOUN
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ejpam-6425	370	1	-3027	-3027	PROPN
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ejpam-6425	425	25	-288	-288	PROPN
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ejpam-6425	425	33	198	198	NUM
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ejpam-6425	426	1	-252	-252	PROPN
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ejpam-6425	427	5	-11	-11	NUM
ejpam-6425	427	6	224	224	NUM
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ejpam-6425	427	12	-224	-224	PROPN
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ejpam-6425	427	20	192	192	NUM
ejpam-6425	427	21	-4	-4	SYM
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ejpam-6425	427	23	-0.17	-0.17	PROPN
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ejpam-6425	427	25	-168	-168	PROPN
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ejpam-6425	427	34	-0.08	-0.08	PRON
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ejpam-6425	427	46	the	the	DET
ejpam-6425	427	47	complete	complete	ADJ
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ejpam-6425	427	54	equation	equation	NOUN
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ejpam-6425	427	56	+	+	CCONJ
ejpam-6425	428	1	6n−	6n−	PROPN
ejpam-6425	428	2	1−	1−	NUM
ejpam-6425	428	3	6nk	6nk	NOUN
ejpam-6425	428	4	−	−	NOUN
ejpam-6425	428	5	2k	2k	NOUN
ejpam-6425	428	6	=	=	SYM
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ejpam-6425	428	10	to	to	PART
ejpam-6425	428	11	theorem	theorem	VERB
ejpam-6425	428	12	5	5	NUM
ejpam-6425	428	13	.	.	PUNCT
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ejpam-6425	428	15	3	3	NUM
ejpam-6425	428	16	:	:	PUNCT
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ejpam-6425	428	20	+	+	PROPN
ejpam-6425	428	21	6n−	6n−	PROPN
ejpam-6425	428	22	1−	1−	NUM
ejpam-6425	428	23	6nk	6nk	NOUN
ejpam-6425	429	1	−	−	NOUN
ejpam-6425	429	2	2k	2k	NOUN
ejpam-6425	429	3	=	=	SYM
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ejpam-6425	429	19	298080.5	298080.5	NUM
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ejpam-6425	430	10	-2	-2	NOUN
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ejpam-6425	433	8	59.67	59.67	NUM
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ejpam-6425	440	5	74520	74520	NUM
ejpam-6425	440	6	37264	37264	NUM
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ejpam-6425	440	8	29.67	29.67	NUM
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ejpam-6425	440	10	-8	-8	NUM
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ejpam-6425	442	8	-9	-9	PROPN
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ejpam-6425	444	4	4967.17	4967.17	NUM
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ejpam-6425	446	7	4139	4139	NUM
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ejpam-6425	446	9	668.17	668.17	NUM
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ejpam-6425	447	4	39744	39744	NUM
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ejpam-6425	447	16	37260	37260	NUM
ejpam-6425	447	17	18638	18638	NUM
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ejpam-6425	447	21	-16	-16	SYM
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ejpam-6425	449	4	33120	33120	NUM
ejpam-6425	449	5	16569	16569	NUM
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ejpam-6425	450	8	392.28	392.28	NUM
ejpam-6425	450	9	-20	-20	NUM
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ejpam-6425	453	5	12971.5	12971.5	NUM
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ejpam-6425	453	9	-23	-23	PUNCT
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ejpam-6425	453	16	24840	24840	NUM
ejpam-6425	453	17	12432	12432	NUM
ejpam-6425	453	18	2068	2068	NUM
ejpam-6425	453	19	9.67	9.67	NUM
ejpam-6425	453	20	323.33	323.33	NUM
ejpam-6425	453	21	-24	-24	NOUN
ejpam-6425	454	1	-24840	-24840	NOUN
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ejpam-6425	454	3	-2068	-2068	PROPN
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ejpam-6425	455	4	22080	22080	NUM
ejpam-6425	455	5	11053.5	11053.5	NUM
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ejpam-6425	455	9	-27	-27	PUNCT
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ejpam-6425	456	4	30	30	NUM
ejpam-6425	456	5	19872	19872	NUM
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ejpam-6425	456	9	254.42	254.42	NUM
ejpam-6425	456	10	-30	-30	PUNCT
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ejpam-6425	456	16	a.	a.	PROPN
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ejpam-6425	457	8	(	(	PUNCT
ejpam-6425	457	9	4	4	NUM
ejpam-6425	457	10	)	)	PUNCT
ejpam-6425	457	11	(	(	PUNCT
ejpam-6425	457	12	2025	2025	NUM
ejpam-6425	457	13	)	)	PUNCT
ejpam-6425	457	14	,	,	PUNCT
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ejpam-6425	457	16	22	22	NUM
ejpam-6425	457	17	of	of	ADP
ejpam-6425	457	18	33	33	NUM
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ejpam-6425	457	21	9331	9331	NUM
ejpam-6425	458	1	1549.83	1549.83	NUM
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ejpam-6425	511	13	)	)	PUNCT
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ejpam-6425	517	10	-74520	-74520	PROPN
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ejpam-6425	541	8	(	(	PUNCT
ejpam-6425	541	9	4	4	NUM
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ejpam-6425	541	11	(	(	PUNCT
ejpam-6425	541	12	2025	2025	NUM
ejpam-6425	541	13	)	)	PUNCT
ejpam-6425	541	14	,	,	PUNCT
ejpam-6425	541	15	6425	6425	NUM
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ejpam-6425	541	17	of	of	ADP
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ejpam-6425	541	20	345	345	NUM
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ejpam-6425	543	1	-1620	-1620	INTJ
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ejpam-6425	547	1	-432	-432	NOUN
ejpam-6425	547	2	-906	-906	SYM
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ejpam-6425	548	1	-878	-878	PROPN
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ejpam-6425	548	12	-480	-480	PROPN
ejpam-6425	548	13	-861	-861	PROPN
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ejpam-6425	549	1	-540	-540	PROPN
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ejpam-6425	554	10	)	)	PUNCT
ejpam-6425	554	11	(	(	PUNCT
ejpam-6425	554	12	2025	2025	NUM
ejpam-6425	554	13	)	)	PUNCT
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ejpam-6425	555	2	1−	1−	NUM
ejpam-6425	555	3	6nk	6nk	NOUN
ejpam-6425	555	4	−	−	NOUN
ejpam-6425	555	5	2k	2k	NOUN
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ejpam-6425	556	1	:	:	PUNCT
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ejpam-6425	556	5	+	+	PROPN
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ejpam-6425	556	7	1−	1−	NUM
ejpam-6425	556	8	6nk	6nk	NOUN
ejpam-6425	556	9	−	−	NOUN
ejpam-6425	556	10	2k	2k	NOUN
ejpam-6425	556	11	=	=	SYM
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ejpam-6425	631	10	-648	-648	PROPN
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ejpam-6425	665	13	)	)	PUNCT
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ejpam-6425	667	10	-24493.5	-24493.5	X
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ejpam-6425	704	13	)	)	PUNCT
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ejpam-6425	704	17	of	of	ADP
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ejpam-6425	715	6	1296	1296	NUM
ejpam-6425	715	7	1020	1020	NUM
ejpam-6425	715	8	1158	1158	NUM
ejpam-6425	715	9	-23	-23	SYM
ejpam-6425	715	10	-0.06	-0.06	NUM
ejpam-6425	715	11	-0.33	-0.33	PROPN
ejpam-6425	715	12	-1296	-1296	PROPN
ejpam-6425	715	13	-1020	-1020	PROPN
ejpam-6425	715	14	-1158	-1158	PROPN
ejpam-6425	715	15	23	23	NUM
ejpam-6425	715	16	-0.61	-0.61	NOUN
ejpam-6425	715	17	-64.67	-64.67	PROPN
ejpam-6425	715	18	1224	1224	NUM
ejpam-6425	715	19	1080	1080	NUM
ejpam-6425	715	20	1152	1152	NUM
ejpam-6425	715	21	-12	-12	SYM
ejpam-6425	715	22	-0.04	-0.04	NUM
ejpam-6425	715	23	-0.5	-0.5	X
ejpam-6425	715	24	-1224	-1224	PROPN
ejpam-6425	716	1	-1080	-1080	PROPN
ejpam-6425	716	2	-1152	-1152	PROPN
ejpam-6425	716	3	12	12	NUM
ejpam-6425	716	4	-0.63	-0.63	NUM
ejpam-6425	716	5	-64.5	-64.5	NUM
ejpam-6425	716	6	1215	1215	NUM
ejpam-6425	716	7	1088	1088	NUM
ejpam-6425	716	8	1151.5	1151.5	NUM
ejpam-6425	716	9	-10.6	-10.6	NOUN
ejpam-6425	716	10	-0.04	-0.04	NUM
ejpam-6425	716	11	-0.51	-0.51	NOUN
ejpam-6425	716	12	-1215	-1215	NUM
ejpam-6425	717	1	-1088	-1088	PROPN
ejpam-6425	717	2	-1151.5	-1151.5	NUM
ejpam-6425	717	3	10.6	10.6	NUM
ejpam-6425	717	4	-0.63	-0.63	NOUN
ejpam-6425	717	5	-64.49	-64.49	PROPN
ejpam-6425	717	6	next	next	ADV
ejpam-6425	717	7	,	,	PUNCT
ejpam-6425	717	8	we	we	PRON
ejpam-6425	717	9	present	present	VERB
ejpam-6425	717	10	the	the	DET
ejpam-6425	717	11	computer	computer	NOUN
ejpam-6425	717	12	programs	program	NOUN
ejpam-6425	717	13	that	that	PRON
ejpam-6425	717	14	are	be	AUX
ejpam-6425	717	15	used	use	VERB
ejpam-6425	717	16	in	in	ADP
ejpam-6425	717	17	the	the	DET
ejpam-6425	717	18	study	study	NOUN
ejpam-6425	717	19	.	.	PUNCT
ejpam-6425	718	1	these	these	PRON
ejpam-6425	718	2	are	be	AUX
ejpam-6425	718	3	python	python	NOUN
ejpam-6425	718	4	computer	computer	NOUN
ejpam-6425	718	5	programs	program	NOUN
ejpam-6425	718	6	that	that	PRON
ejpam-6425	718	7	were	be	AUX
ejpam-6425	718	8	used	use	VERB
ejpam-6425	718	9	to	to	PART
ejpam-6425	718	10	generate	generate	VERB
ejpam-6425	718	11	actual	actual	ADJ
ejpam-6425	718	12	edge	edge	NOUN
ejpam-6425	718	13	-	-	PUNCT
ejpam-6425	718	14	graceful	graceful	NOUN
ejpam-6425	718	15	labeling	labeling	NOUN
ejpam-6425	718	16	of	of	ADP
ejpam-6425	718	17	needed	need	VERB
ejpam-6425	718	18	particular	particular	ADJ
ejpam-6425	718	19	graphs	graph	NOUN
ejpam-6425	718	20	,	,	PUNCT
ejpam-6425	718	21	in	in	ADP
ejpam-6425	718	22	order	order	NOUN
ejpam-6425	718	23	to	to	PART
ejpam-6425	718	24	complete	complete	VERB
ejpam-6425	718	25	the	the	DET
ejpam-6425	718	26	proof	proof	NOUN
ejpam-6425	718	27	of	of	ADP
ejpam-6425	718	28	the	the	DET
ejpam-6425	718	29	main	main	ADJ
ejpam-6425	718	30	results	result	NOUN
ejpam-6425	718	31	of	of	ADP
ejpam-6425	718	32	this	this	DET
ejpam-6425	718	33	study	study	NOUN
ejpam-6425	718	34	.	.	PUNCT
ejpam-6425	719	1	provided	provide	VERB
ejpam-6425	719	2	below	below	ADV
ejpam-6425	719	3	,	,	PUNCT
ejpam-6425	719	4	is	be	AUX
ejpam-6425	719	5	the	the	DET
ejpam-6425	719	6	python	python	NOUN
ejpam-6425	719	7	program	program	NOUN
ejpam-6425	719	8	used	use	VERB
ejpam-6425	719	9	to	to	PART
ejpam-6425	719	10	generate	generate	VERB
ejpam-6425	719	11	an	an	DET
ejpam-6425	719	12	actual	actual	ADJ
ejpam-6425	719	13	edge	edge	NOUN
ejpam-6425	719	14	-	-	PUNCT
ejpam-6425	719	15	graceful	graceful	NOUN
ejpam-6425	719	16	labeling	labeling	NOUN
ejpam-6425	719	17	of	of	ADP
ejpam-6425	719	18	s(ww3	s(ww3	PROPN
ejpam-6425	719	19	)	)	PUNCT
ejpam-6425	719	20	.	.	PUNCT
ejpam-6425	720	1	1	1	NUM
ejpam-6425	720	2	#	#	SYM
ejpam-6425	720	3	python	python	NOUN
ejpam-6425	720	4	program	program	NOUN
ejpam-6425	720	5	for	for	ADP
ejpam-6425	720	6	s(ww_3	s(ww_3	ADJ
ejpam-6425	720	7	)	)	PUNCT
ejpam-6425	720	8	2	2	NUM
ejpam-6425	720	9	from	from	ADP
ejpam-6425	720	10	itertools	itertools	PROPN
ejpam-6425	720	11	import	import	NOUN
ejpam-6425	720	12	permutations	permutation	NOUN
ejpam-6425	720	13	3	3	NUM
ejpam-6425	720	14	4	4	NUM
ejpam-6425	720	15	numbers	number	NOUN
ejpam-6425	720	16	=	=	PUNCT
ejpam-6425	721	1	[	[	X
ejpam-6425	721	2	1	1	NUM
ejpam-6425	721	3	,	,	PUNCT
ejpam-6425	721	4	2	2	NUM
ejpam-6425	721	5	,	,	PUNCT
ejpam-6425	721	6	3	3	NUM
ejpam-6425	721	7	,	,	PUNCT
ejpam-6425	721	8	4	4	NUM
ejpam-6425	721	9	,	,	PUNCT
ejpam-6425	721	10	5	5	NUM
ejpam-6425	721	11	,	,	PUNCT
ejpam-6425	721	12	6	6	NUM
ejpam-6425	721	13	,	,	PUNCT
ejpam-6425	721	14	7	7	NUM
ejpam-6425	721	15	,	,	PUNCT
ejpam-6425	721	16	8	8	NUM
ejpam-6425	721	17	,	,	PUNCT
ejpam-6425	721	18	9	9	NUM
ejpam-6425	721	19	,	,	PUNCT
ejpam-6425	721	20	10	10	NUM
ejpam-6425	721	21	,	,	PUNCT
ejpam-6425	721	22	11	11	NUM
ejpam-6425	721	23	,	,	PUNCT
ejpam-6425	721	24	12	12	NUM
ejpam-6425	721	25	,	,	PUNCT
ejpam-6425	721	26	13	13	NUM
ejpam-6425	721	27	,	,	PUNCT
ejpam-6425	721	28	14	14	NUM
ejpam-6425	721	29	,	,	PUNCT
ejpam-6425	721	30	15	15	NUM
ejpam-6425	721	31	,	,	PUNCT
ejpam-6425	721	32	16	16	NUM
ejpam-6425	721	33	,	,	PUNCT
ejpam-6425	721	34	17	17	NUM
ejpam-6425	721	35	,	,	PUNCT
ejpam-6425	721	36	18	18	NUM
ejpam-6425	721	37	,	,	PUNCT
ejpam-6425	721	38	19	19	NUM
ejpam-6425	721	39	,	,	PUNCT
ejpam-6425	721	40	20	20	NUM
ejpam-6425	721	41	,	,	PUNCT
ejpam-6425	721	42	21	21	NUM
ejpam-6425	721	43	,	,	PUNCT
ejpam-6425	721	44	22	22	NUM
ejpam-6425	721	45	,	,	PUNCT
ejpam-6425	721	46	23	23	NUM
ejpam-6425	721	47	,	,	PUNCT
ejpam-6425	721	48	24	24	NUM
ejpam-6425	721	49	,	,	PUNCT
ejpam-6425	721	50	25	25	NUM
ejpam-6425	721	51	,	,	PUNCT
ejpam-6425	721	52	26	26	NUM
ejpam-6425	721	53	,	,	PUNCT
ejpam-6425	721	54	27	27	NUM
ejpam-6425	721	55	,	,	PUNCT
ejpam-6425	721	56	28	28	NUM
ejpam-6425	721	57	,	,	PUNCT
ejpam-6425	721	58	29	29	NUM
ejpam-6425	721	59	,	,	PUNCT
ejpam-6425	721	60	30	30	NUM
ejpam-6425	721	61	,	,	PUNCT
ejpam-6425	721	62	31	31	NUM
ejpam-6425	721	63	,	,	PUNCT
ejpam-6425	721	64	32	32	NUM
ejpam-6425	721	65	,	,	PUNCT
ejpam-6425	721	66	33	33	NUM
ejpam-6425	721	67	,	,	PUNCT
ejpam-6425	721	68	34	34	NUM
ejpam-6425	721	69	,	,	PUNCT
ejpam-6425	721	70	35	35	NUM
ejpam-6425	721	71	,	,	PUNCT
ejpam-6425	721	72	36	36	NUM
ejpam-6425	721	73	,	,	PUNCT
ejpam-6425	721	74	37	37	NUM
ejpam-6425	721	75	,	,	PUNCT
ejpam-6425	721	76	38	38	NUM
ejpam-6425	721	77	,	,	PUNCT
ejpam-6425	721	78	39	39	NUM
ejpam-6425	721	79	,	,	PUNCT
ejpam-6425	721	80	40	40	NUM
ejpam-6425	721	81	,	,	PUNCT
ejpam-6425	721	82	41	41	NUM
ejpam-6425	721	83	,	,	PUNCT
ejpam-6425	721	84	42	42	NUM
ejpam-6425	721	85	,	,	PUNCT
ejpam-6425	721	86	43	43	NUM
ejpam-6425	721	87	,	,	PUNCT
ejpam-6425	721	88	44	44	NUM
ejpam-6425	721	89	,	,	PUNCT
ejpam-6425	721	90	45	45	NUM
ejpam-6425	721	91	]	]	SYM
ejpam-6425	721	92	5	5	NUM
ejpam-6425	721	93	6	6	NUM
ejpam-6425	721	94	#	#	NOUN
ejpam-6425	721	95	function	function	NOUN
ejpam-6425	721	96	to	to	PART
ejpam-6425	721	97	compute	compute	VERB
ejpam-6425	721	98	u0	u0	ADJ
ejpam-6425	721	99	to	to	PART
ejpam-6425	721	100	v1	v1	VERB
ejpam-6425	721	101	7	7	NUM
ejpam-6425	721	102	def	def	ADJ
ejpam-6425	721	103	compute_v_values(w1u1	compute_v_values(w1u1	PROPN
ejpam-6425	721	104	,	,	PUNCT
ejpam-6425	721	105	u1v2	u1v2	NOUN
ejpam-6425	721	106	,	,	PUNCT
ejpam-6425	721	107	v2w2	v2w2	X
ejpam-6425	721	108	,	,	PUNCT
ejpam-6425	721	109	w2u2	w2u2	NOUN
ejpam-6425	721	110	,	,	PUNCT
ejpam-6425	721	111	u2v3	u2v3	ADV
ejpam-6425	721	112	,	,	PUNCT
ejpam-6425	721	113	v3w3	v3w3	NOUN
ejpam-6425	721	114	,	,	PUNCT
ejpam-6425	721	115	w3u3	w3u3	ADJ
ejpam-6425	721	116	,	,	PUNCT
ejpam-6425	721	117	u3v1	u3v1	PROPN
ejpam-6425	721	118	,	,	PUNCT
ejpam-6425	721	119	v1w1	v1w1	PROPN
ejpam-6425	721	120	,	,	PUNCT
ejpam-6425	721	121	u0u1	u0u1	NOUN
ejpam-6425	721	122	,	,	PUNCT
ejpam-6425	721	123	u0v2	u0v2	NOUN
ejpam-6425	721	124	,	,	PUNCT
ejpam-6425	721	125	u0u2	u0u2	PRON
ejpam-6425	721	126	,	,	PUNCT
ejpam-6425	721	127	u0v3	u0v3	X
ejpam-6425	721	128	,	,	PUNCT
ejpam-6425	721	129	u0u3	u0u3	PROPN
ejpam-6425	721	130	,	,	PUNCT
ejpam-6425	721	131	u0v1	u0v1	X
ejpam-6425	721	132	,	,	PUNCT
ejpam-6425	721	133	w1v1	w1v1	X
ejpam-6425	721	134	,	,	PUNCT
ejpam-6425	721	135	w1u1	w1u1	NOUN
ejpam-6425	721	136	,	,	PUNCT
ejpam-6425	721	137	u1w1	u1w1	PROPN
ejpam-6425	721	138	,	,	PUNCT
ejpam-6425	721	139	u1u0	u1u0	ADP
ejpam-6425	721	140	,	,	PUNCT
ejpam-6425	721	141	u1v2	u1v2	NOUN
ejpam-6425	721	142	,	,	PUNCT
ejpam-6425	721	143	v2u1	v2u1	X
ejpam-6425	721	144	,	,	PUNCT
ejpam-6425	721	145	v2u0	v2u0	NOUN
ejpam-6425	721	146	,	,	PUNCT
ejpam-6425	721	147	v2w2	v2w2	NOUN
ejpam-6425	721	148	,	,	PUNCT
ejpam-6425	721	149	w2v2	w2v2	ADJ
ejpam-6425	721	150	,	,	PUNCT
ejpam-6425	721	151	w2u2	w2u2	NOUN
ejpam-6425	721	152	,	,	PUNCT
ejpam-6425	721	153	u2w2	u2w2	X
ejpam-6425	721	154	,	,	PUNCT
ejpam-6425	721	155	u2u0	u2u0	PROPN
ejpam-6425	721	156	,	,	PUNCT
ejpam-6425	721	157	u2v3	u2v3	NOUN
ejpam-6425	721	158	,	,	PUNCT
ejpam-6425	721	159	v3u2	v3u2	SYM
ejpam-6425	721	160	,	,	PUNCT
ejpam-6425	721	161	v3u0	v3u0	X
ejpam-6425	721	162	,	,	PUNCT
ejpam-6425	721	163	v3w3	v3w3	NOUN
ejpam-6425	721	164	,	,	PUNCT
ejpam-6425	721	165	w3v3	w3v3	NOUN
ejpam-6425	721	166	,	,	PUNCT
ejpam-6425	721	167	w3u3	w3u3	PROPN
ejpam-6425	721	168	,	,	PUNCT
ejpam-6425	721	169	u3w3	u3w3	NOUN
ejpam-6425	721	170	,	,	PUNCT
ejpam-6425	721	171	u3u0	u3u0	NOUN
ejpam-6425	721	172	,	,	PUNCT
ejpam-6425	721	173	u3v1	u3v1	PROPN
ejpam-6425	721	174	,	,	PUNCT
ejpam-6425	721	175	v1u3	v1u3	X
ejpam-6425	721	176	,	,	PUNCT
ejpam-6425	721	177	v1u0	v1u0	PROPN
ejpam-6425	721	178	,	,	PUNCT
ejpam-6425	721	179	v1w1	v1w1	NOUN
ejpam-6425	721	180	,	,	PUNCT
ejpam-6425	721	181	u0v1	u0v1	ADJ
ejpam-6425	721	182	,	,	PUNCT
ejpam-6425	721	183	u0u3	u0u3	PROPN
ejpam-6425	721	184	,	,	PUNCT
ejpam-6425	721	185	u0v3	u0v3	X
ejpam-6425	721	186	,	,	PUNCT
ejpam-6425	721	187	u0u2	u0u2	SYM
ejpam-6425	721	188	,	,	PUNCT
ejpam-6425	721	189	u0v2	u0v2	NOUN
ejpam-6425	721	190	,	,	PUNCT
ejpam-6425	721	191	u0u1	u0u1	X
ejpam-6425	721	192	):	):	PUNCT
ejpam-6425	721	193	8	8	NUM
ejpam-6425	721	194	u0	u0	NOUN
ejpam-6425	721	195	=	=	PUNCT
ejpam-6425	721	196	(	(	PUNCT
ejpam-6425	721	197	u0v1	u0v1	X
ejpam-6425	722	1	+	+	CCONJ
ejpam-6425	722	2	u0u1	u0u1	PROPN
ejpam-6425	722	3	+	+	X
ejpam-6425	722	4	u0v2	u0v2	X
ejpam-6425	722	5	+	+	NUM
ejpam-6425	722	6	u0u2	u0u2	PUNCT
ejpam-6425	722	7	+	+	NUM
ejpam-6425	722	8	u0v3	u0v3	ADJ
ejpam-6425	722	9	+	+	NUM
ejpam-6425	722	10	u0u3	u0u3	PROPN
ejpam-6425	722	11	+	+	CCONJ
ejpam-6425	722	12	v1u0	v1u0	PROPN
ejpam-6425	722	13	+	+	CCONJ
ejpam-6425	722	14	u1u0	u1u0	PROPN
ejpam-6425	722	15	+	+	ADJ
ejpam-6425	722	16	v2u0	v2u0	PROPN
ejpam-6425	722	17	+	+	NUM
ejpam-6425	722	18	u2u0	u2u0	X
ejpam-6425	722	19	+	+	CCONJ
ejpam-6425	722	20	v3u0	v3u0	X
ejpam-6425	722	21	+	+	CCONJ
ejpam-6425	722	22	u3u0	u3u0	X
ejpam-6425	722	23	)	)	PUNCT
ejpam-6425	722	24	%	%	NOUN
ejpam-6425	722	25	20	20	NUM
ejpam-6425	722	26	9	9	NUM
ejpam-6425	722	27	w1	w1	NOUN
ejpam-6425	722	28	=	=	SYM
ejpam-6425	722	29	(	(	PUNCT
ejpam-6425	722	30	v1w1	v1w1	X
ejpam-6425	722	31	+	+	X
ejpam-6425	722	32	w1u1	w1u1	X
ejpam-6425	722	33	+	+	NUM
ejpam-6425	722	34	v1w1	v1w1	PROPN
ejpam-6425	722	35	+	+	SYM
ejpam-6425	722	36	u1w1	u1w1	ADJ
ejpam-6425	722	37	)	)	PUNCT
ejpam-6425	722	38	%	%	NOUN
ejpam-6425	722	39	20	20	NUM
ejpam-6425	722	40	10	10	NUM
ejpam-6425	722	41	u1	u1	NOUN
ejpam-6425	722	42	=	=	SYM
ejpam-6425	722	43	(	(	PUNCT
ejpam-6425	722	44	w1u1	w1u1	X
ejpam-6425	722	45	+	+	SYM
ejpam-6425	722	46	u1v2	u1v2	PUNCT
ejpam-6425	722	47	+	+	CCONJ
ejpam-6425	722	48	u0u1	u0u1	PUNCT
ejpam-6425	723	1	+	+	PUNCT
ejpam-6425	723	2	w1u1	w1u1	X
ejpam-6425	723	3	+	+	CCONJ
ejpam-6425	723	4	v2u1	v2u1	X
ejpam-6425	723	5	+	+	CCONJ
ejpam-6425	723	6	u0u1	u0u1	NOUN
ejpam-6425	723	7	)	)	PUNCT
ejpam-6425	723	8	%	%	NOUN
ejpam-6425	723	9	20	20	NUM
ejpam-6425	723	10	11	11	NUM
ejpam-6425	723	11	v2	v2	NOUN
ejpam-6425	723	12	=	=	SYM
ejpam-6425	723	13	(	(	PUNCT
ejpam-6425	723	14	u1v2	u1v2	X
ejpam-6425	724	1	+	+	X
ejpam-6425	724	2	v2w2	v2w2	X
ejpam-6425	724	3	+	+	X
ejpam-6425	724	4	u0v2	u0v2	X
ejpam-6425	724	5	+	+	SYM
ejpam-6425	724	6	u1v2	u1v2	PUNCT
ejpam-6425	724	7	+	+	CCONJ
ejpam-6425	724	8	w2v2	w2v2	X
ejpam-6425	724	9	+	+	CCONJ
ejpam-6425	724	10	u0v2	u0v2	NOUN
ejpam-6425	724	11	)	)	PUNCT
ejpam-6425	724	12	%	%	NOUN
ejpam-6425	724	13	20	20	NUM
ejpam-6425	724	14	12	12	NUM
ejpam-6425	724	15	w2	w2	NOUN
ejpam-6425	724	16	=	=	SYM
ejpam-6425	724	17	(	(	PUNCT
ejpam-6425	724	18	v2w2	v2w2	X
ejpam-6425	724	19	+	+	X
ejpam-6425	724	20	w2u2	w2u2	X
ejpam-6425	724	21	+	+	X
ejpam-6425	724	22	v2w2	v2w2	X
ejpam-6425	724	23	+	+	X
ejpam-6425	724	24	u2w2	u2w2	X
ejpam-6425	724	25	)	)	PUNCT
ejpam-6425	724	26	%	%	NOUN
ejpam-6425	724	27	20	20	NUM
ejpam-6425	724	28	13	13	NUM
ejpam-6425	724	29	u2	u2	NOUN
ejpam-6425	724	30	=	=	PUNCT
ejpam-6425	724	31	(	(	PUNCT
ejpam-6425	724	32	w2u2	w2u2	X
ejpam-6425	725	1	+	+	CCONJ
ejpam-6425	725	2	u2v3	u2v3	X
ejpam-6425	725	3	+	+	X
ejpam-6425	725	4	u0u2	u0u2	VERB
ejpam-6425	725	5	+	+	PUNCT
ejpam-6425	725	6	w2u2	w2u2	X
ejpam-6425	725	7	+	+	NOUN
ejpam-6425	725	8	v3u2	v3u2	VERB
ejpam-6425	725	9	+	+	CCONJ
ejpam-6425	725	10	u0u2	u0u2	X
ejpam-6425	725	11	)	)	PUNCT
ejpam-6425	725	12	%	%	NOUN
ejpam-6425	725	13	20	20	NUM
ejpam-6425	725	14	14	14	NUM
ejpam-6425	725	15	v3	v3	PROPN
ejpam-6425	725	16	=	=	SYM
ejpam-6425	725	17	(	(	PUNCT
ejpam-6425	725	18	u2v3	u2v3	X
ejpam-6425	725	19	+	+	NOUN
ejpam-6425	725	20	v3w3	v3w3	X
ejpam-6425	726	1	+	+	CCONJ
ejpam-6425	726	2	u0v3	u0v3	X
ejpam-6425	726	3	+	+	CCONJ
ejpam-6425	726	4	u2v3	u2v3	X
ejpam-6425	726	5	+	+	CCONJ
ejpam-6425	726	6	w3v3	w3v3	X
ejpam-6425	726	7	+	+	CCONJ
ejpam-6425	726	8	u0v3	u0v3	ADJ
ejpam-6425	726	9	)	)	PUNCT
ejpam-6425	726	10	%	%	NOUN
ejpam-6425	726	11	20	20	NUM
ejpam-6425	726	12	15	15	NUM
ejpam-6425	726	13	w3	w3	NOUN
ejpam-6425	726	14	=	=	SYM
ejpam-6425	726	15	(	(	PUNCT
ejpam-6425	726	16	v3w3	v3w3	X
ejpam-6425	726	17	+	+	CCONJ
ejpam-6425	726	18	w3u3	w3u3	X
ejpam-6425	726	19	+	+	PUNCT
ejpam-6425	726	20	v3w3	v3w3	X
ejpam-6425	726	21	+	+	SYM
ejpam-6425	726	22	u3w3	u3w3	NOUN
ejpam-6425	726	23	)	)	PUNCT
ejpam-6425	726	24	%	%	NOUN
ejpam-6425	726	25	20	20	NUM
ejpam-6425	726	26	16	16	NUM
ejpam-6425	726	27	u3	u3	NOUN
ejpam-6425	726	28	=	=	SYM
ejpam-6425	726	29	(	(	PUNCT
ejpam-6425	726	30	w3u3	w3u3	X
ejpam-6425	726	31	+	+	SYM
ejpam-6425	726	32	u3v1	u3v1	PROPN
ejpam-6425	726	33	+	+	X
ejpam-6425	726	34	u0u3	u0u3	PROPN
ejpam-6425	726	35	+	+	CCONJ
ejpam-6425	726	36	w3u3	w3u3	X
ejpam-6425	726	37	+	+	CCONJ
ejpam-6425	726	38	v1u3	v1u3	X
ejpam-6425	726	39	+	+	X
ejpam-6425	726	40	u0u3	u0u3	ADJ
ejpam-6425	726	41	)	)	PUNCT
ejpam-6425	726	42	%	%	NOUN
ejpam-6425	726	43	20	20	NUM
ejpam-6425	726	44	a.	a.	PROPN
ejpam-6425	726	45	d.	d.	PROPN
ejpam-6425	726	46	c.	c.	PROPN
ejpam-6425	726	47	angel	angel	PROPN
ejpam-6425	726	48	et	et	PROPN
ejpam-6425	726	49	al	al	PROPN
ejpam-6425	726	50	.	.	PUNCT
ejpam-6425	726	51	/	/	SYM
ejpam-6425	726	52	eur	eur	PROPN
ejpam-6425	726	53	.	.	PUNCT
ejpam-6425	727	1	j.	j.	PROPN
ejpam-6425	727	2	pure	pure	PROPN
ejpam-6425	727	3	appl	appl	PROPN
ejpam-6425	727	4	.	.	PROPN
ejpam-6425	727	5	math	math	PROPN
ejpam-6425	727	6	,	,	PUNCT
ejpam-6425	727	7	18	18	NUM
ejpam-6425	727	8	(	(	PUNCT
ejpam-6425	727	9	4	4	NUM
ejpam-6425	727	10	)	)	PUNCT
ejpam-6425	727	11	(	(	PUNCT
ejpam-6425	727	12	2025	2025	NUM
ejpam-6425	727	13	)	)	PUNCT
ejpam-6425	727	14	,	,	PUNCT
ejpam-6425	727	15	6425	6425	NUM
ejpam-6425	727	16	29	29	NUM
ejpam-6425	727	17	of	of	ADP
ejpam-6425	727	18	33	33	NUM
ejpam-6425	727	19	17	17	NUM
ejpam-6425	727	20	v1	v1	NOUN
ejpam-6425	727	21	=	=	SYM
ejpam-6425	727	22	(	(	PUNCT
ejpam-6425	727	23	u3v1	u3v1	PROPN
ejpam-6425	727	24	+	+	CCONJ
ejpam-6425	727	25	v1w1	v1w1	PROPN
ejpam-6425	728	1	+	+	CCONJ
ejpam-6425	728	2	u0v1	u0v1	ADJ
ejpam-6425	728	3	+	+	CCONJ
ejpam-6425	728	4	u3v1	u3v1	PROPN
ejpam-6425	728	5	+	+	CCONJ
ejpam-6425	728	6	w1v1	w1v1	X
ejpam-6425	728	7	+	+	NUM
ejpam-6425	728	8	u0v1	u0v1	NOUN
ejpam-6425	728	9	)	)	PUNCT
ejpam-6425	728	10	%	%	NOUN
ejpam-6425	728	11	20	20	NUM
ejpam-6425	728	12	18	18	NUM
ejpam-6425	728	13	u0	u0	NOUN
ejpam-6425	728	14	=	=	PUNCT
ejpam-6425	728	15	(	(	PUNCT
ejpam-6425	728	16	u0v1	u0v1	X
ejpam-6425	729	1	+	+	CCONJ
ejpam-6425	729	2	u0u3	u0u3	PROPN
ejpam-6425	730	1	+	+	X
ejpam-6425	730	2	u0v3	u0v3	X
ejpam-6425	730	3	+	+	NOUN
ejpam-6425	730	4	u0u2	u0u2	VERB
ejpam-6425	730	5	+	+	NUM
ejpam-6425	730	6	u0v2	u0v2	ADJ
ejpam-6425	730	7	+	+	NOUN
ejpam-6425	730	8	u0u1	u0u1	NOUN
ejpam-6425	730	9	)	)	PUNCT
ejpam-6425	730	10	%	%	NOUN
ejpam-6425	730	11	20	20	NUM
ejpam-6425	730	12	19	19	NUM
ejpam-6425	730	13	w1	w1	NOUN
ejpam-6425	730	14	=	=	SYM
ejpam-6425	730	15	(	(	PUNCT
ejpam-6425	730	16	w1v1	w1v1	X
ejpam-6425	730	17	+	+	NUM
ejpam-6425	730	18	w1u1	w1u1	NOUN
ejpam-6425	730	19	)	)	PUNCT
ejpam-6425	730	20	%	%	NOUN
ejpam-6425	730	21	20	20	NUM
ejpam-6425	730	22	20	20	NUM
ejpam-6425	730	23	u1	u1	NOUN
ejpam-6425	730	24	=	=	SYM
ejpam-6425	730	25	(	(	PUNCT
ejpam-6425	730	26	u1w1	u1w1	X
ejpam-6425	730	27	+	+	CCONJ
ejpam-6425	730	28	u1u0	u1u0	PROPN
ejpam-6425	730	29	+	+	NUM
ejpam-6425	730	30	u1v2	u1v2	NOUN
ejpam-6425	730	31	)	)	PUNCT
ejpam-6425	730	32	%	%	NOUN
ejpam-6425	730	33	20	20	NUM
ejpam-6425	730	34	21	21	NUM
ejpam-6425	730	35	v2	v2	NOUN
ejpam-6425	730	36	=	=	SYM
ejpam-6425	730	37	(	(	PUNCT
ejpam-6425	730	38	v2u1	v2u1	X
ejpam-6425	730	39	+	+	ADP
ejpam-6425	730	40	v2u0	v2u0	X
ejpam-6425	730	41	+	+	NUM
ejpam-6425	730	42	v2w2	v2w2	NOUN
ejpam-6425	730	43	)	)	PUNCT
ejpam-6425	730	44	%	%	NOUN
ejpam-6425	730	45	20	20	NUM
ejpam-6425	730	46	22	22	NUM
ejpam-6425	730	47	w2	w2	NOUN
ejpam-6425	730	48	=	=	SYM
ejpam-6425	730	49	(	(	PUNCT
ejpam-6425	730	50	w2v2	w2v2	X
ejpam-6425	730	51	+	+	X
ejpam-6425	730	52	w2u2	w2u2	NOUN
ejpam-6425	730	53	)	)	PUNCT
ejpam-6425	730	54	%	%	NOUN
ejpam-6425	730	55	20	20	NUM
ejpam-6425	730	56	23	23	NUM
ejpam-6425	730	57	u2	u2	NOUN
ejpam-6425	730	58	=	=	PUNCT
ejpam-6425	730	59	(	(	PUNCT
ejpam-6425	730	60	u2w2	u2w2	X
ejpam-6425	730	61	+	+	NUM
ejpam-6425	730	62	u2u0	u2u0	X
ejpam-6425	730	63	+	+	X
ejpam-6425	730	64	u2v3	u2v3	NOUN
ejpam-6425	730	65	)	)	PUNCT
ejpam-6425	730	66	%	%	NOUN
ejpam-6425	730	67	20	20	NUM
ejpam-6425	730	68	24	24	NUM
ejpam-6425	730	69	v3	v3	PROPN
ejpam-6425	730	70	=	=	SYM
ejpam-6425	730	71	(	(	PUNCT
ejpam-6425	730	72	v3u2	v3u2	X
ejpam-6425	730	73	+	+	CCONJ
ejpam-6425	730	74	v3u0	v3u0	X
ejpam-6425	730	75	+	+	CCONJ
ejpam-6425	730	76	v3w3	v3w3	NOUN
ejpam-6425	730	77	)	)	PUNCT
ejpam-6425	730	78	%	%	NOUN
ejpam-6425	730	79	20	20	NUM
ejpam-6425	730	80	25	25	NUM
ejpam-6425	730	81	w3	w3	NOUN
ejpam-6425	730	82	=	=	SYM
ejpam-6425	730	83	(	(	PUNCT
ejpam-6425	730	84	w3v3	w3v3	X
ejpam-6425	730	85	+	+	CCONJ
ejpam-6425	730	86	w3u3	w3u3	NOUN
ejpam-6425	730	87	)	)	PUNCT
ejpam-6425	730	88	%	%	NOUN
ejpam-6425	730	89	20	20	NUM
ejpam-6425	730	90	26	26	NUM
ejpam-6425	730	91	u3	u3	NOUN
ejpam-6425	730	92	=	=	SYM
ejpam-6425	730	93	(	(	PUNCT
ejpam-6425	730	94	u3w3	u3w3	VERB
ejpam-6425	730	95	+	+	CCONJ
ejpam-6425	730	96	u3u0	u3u0	SYM
ejpam-6425	730	97	+	+	NUM
ejpam-6425	730	98	u3v1	u3v1	X
ejpam-6425	730	99	)	)	PUNCT
ejpam-6425	730	100	%	%	NOUN
ejpam-6425	730	101	20	20	NUM
ejpam-6425	730	102	27	27	NUM
ejpam-6425	730	103	v1	v1	NOUN
ejpam-6425	730	104	=	=	SYM
ejpam-6425	730	105	(	(	PUNCT
ejpam-6425	730	106	v1u3	v1u3	X
ejpam-6425	730	107	+	+	X
ejpam-6425	730	108	v1u0	v1u0	PROPN
ejpam-6425	730	109	+	+	CCONJ
ejpam-6425	730	110	v1w1	v1w1	NOUN
ejpam-6425	730	111	)	)	PUNCT
ejpam-6425	730	112	%	%	NOUN
ejpam-6425	730	113	20	20	NUM
ejpam-6425	730	114	28	28	NUM
ejpam-6425	730	115	return	return	NOUN
ejpam-6425	730	116	u0	u0	ADJ
ejpam-6425	730	117	,	,	PUNCT
ejpam-6425	730	118	w1	w1	NOUN
ejpam-6425	730	119	,	,	PUNCT
ejpam-6425	730	120	u1	u1	NOUN
ejpam-6425	730	121	,	,	PUNCT
ejpam-6425	730	122	v2	v2	NOUN
ejpam-6425	730	123	,	,	PUNCT
ejpam-6425	730	124	w2	w2	NOUN
ejpam-6425	730	125	,	,	PUNCT
ejpam-6425	730	126	u2	u2	PROPN
ejpam-6425	730	127	,	,	PUNCT
ejpam-6425	730	128	v3	v3	PROPN
ejpam-6425	730	129	,	,	PUNCT
ejpam-6425	730	130	w3	w3	PROPN
ejpam-6425	730	131	,	,	PUNCT
ejpam-6425	730	132	u3	u3	NOUN
ejpam-6425	730	133	,	,	PUNCT
ejpam-6425	730	134	v1	v1	NOUN
ejpam-6425	730	135	,	,	PUNCT
ejpam-6425	730	136	u0	u0	ADJ
ejpam-6425	730	137	,	,	PUNCT
ejpam-6425	730	138	w1	w1	NOUN
ejpam-6425	730	139	,	,	PUNCT
ejpam-6425	730	140	u1	u1	NOUN
ejpam-6425	730	141	,	,	PUNCT
ejpam-6425	730	142	v2	v2	NOUN
ejpam-6425	730	143	,	,	PUNCT
ejpam-6425	730	144	w2	w2	NOUN
ejpam-6425	730	145	,	,	PUNCT
ejpam-6425	730	146	u2	u2	PROPN
ejpam-6425	730	147	,	,	PUNCT
ejpam-6425	730	148	v3	v3	PROPN
ejpam-6425	730	149	,	,	PUNCT
ejpam-6425	730	150	w3	w3	PROPN
ejpam-6425	730	151	,	,	PUNCT
ejpam-6425	730	152	u3	u3	PROPN
ejpam-6425	730	153	,	,	PUNCT
ejpam-6425	730	154	v1	v1	VERB
ejpam-6425	730	155	29	29	NUM
ejpam-6425	730	156	30	30	NUM
ejpam-6425	730	157	#	#	NOUN
ejpam-6425	730	158	backtracking	backtrack	VERB
ejpam-6425	730	159	function	function	NOUN
ejpam-6425	730	160	to	to	PART
ejpam-6425	730	161	find	find	VERB
ejpam-6425	730	162	valid	valid	ADJ
ejpam-6425	730	163	permutations	permutation	NOUN
ejpam-6425	730	164	31	31	NUM
ejpam-6425	730	165	def	def	ADJ
ejpam-6425	730	166	backtrack(perm	backtrack(perm	NOUN
ejpam-6425	730	167	,	,	PUNCT
ejpam-6425	730	168	used	use	VERB
ejpam-6425	730	169	):	):	PUNCT
ejpam-6425	730	170	32	32	NUM
ejpam-6425	730	171	#	#	NOUN
ejpam-6425	730	172	if	if	SCONJ
ejpam-6425	730	173	we	we	PRON
ejpam-6425	730	174	have	have	VERB
ejpam-6425	730	175	a	a	DET
ejpam-6425	730	176	complete	complete	ADJ
ejpam-6425	730	177	permutation	permutation	NOUN
ejpam-6425	730	178	,	,	PUNCT
ejpam-6425	730	179	verify	verify	VERB
ejpam-6425	730	180	u0	u0	ADJ
ejpam-6425	730	181	to	to	PART
ejpam-6425	730	182	v1	v1	VERB
ejpam-6425	730	183	33	33	NUM
ejpam-6425	730	184	if	if	SCONJ
ejpam-6425	730	185	len(perm	len(perm	NOUN
ejpam-6425	730	186	)	)	PUNCT
ejpam-6425	730	187	=	=	NOUN
ejpam-6425	731	1	=	=	SYM
ejpam-6425	731	2	45	45	NUM
ejpam-6425	731	3	:	:	SYM
ejpam-6425	731	4	34	34	NUM
ejpam-6425	731	5	v_values	v_value	NOUN
ejpam-6425	731	6	=	=	SYM
ejpam-6425	731	7	compute_v_values(*perm	compute_v_values(*perm	NOUN
ejpam-6425	731	8	)	)	PUNCT
ejpam-6425	731	9	35	35	NUM
ejpam-6425	731	10	#	#	NOUN
ejpam-6425	731	11	verify	verify	NOUN
ejpam-6425	731	12	if	if	SCONJ
ejpam-6425	731	13	u0	u0	ADJ
ejpam-6425	731	14	to	to	PART
ejpam-6425	731	15	v1	v1	VERB
ejpam-6425	731	16	are	be	AUX
ejpam-6425	731	17	distinct	distinct	ADJ
ejpam-6425	731	18	36	36	NUM
ejpam-6425	731	19	if	if	SCONJ
ejpam-6425	731	20	len(set(v_values	len(set(v_value	NOUN
ejpam-6425	731	21	)	)	PUNCT
ejpam-6425	731	22	)	)	PUNCT
ejpam-6425	732	1	=	=	SYM
ejpam-6425	732	2	=	=	SYM
ejpam-6425	732	3	20	20	NUM
ejpam-6425	732	4	:	:	SYM
ejpam-6425	732	5	37	37	NUM
ejpam-6425	732	6	print(f"permutation	print(f"permutation	NOUN
ejpam-6425	732	7	:	:	PUNCT
ejpam-6425	732	8	{	{	PUNCT
ejpam-6425	732	9	perm	perm	NOUN
ejpam-6425	732	10	}	}	PUNCT
ejpam-6425	732	11	,	,	PUNCT
ejpam-6425	732	12	u0	u0	ADJ
ejpam-6425	732	13	to	to	PART
ejpam-6425	732	14	v1	v1	VERB
ejpam-6425	732	15	:	:	PUNCT
ejpam-6425	732	16	{	{	PUNCT
ejpam-6425	732	17	v_values	v_value	NOUN
ejpam-6425	732	18	}	}	PUNCT
ejpam-6425	732	19	"	"	PUNCT
ejpam-6425	732	20	)	)	PUNCT
ejpam-6425	732	21	38	38	NUM
ejpam-6425	732	22	return	return	VERB
ejpam-6425	732	23	39	39	NUM
ejpam-6425	732	24	40	40	NUM
ejpam-6425	732	25	#	#	NOUN
ejpam-6425	732	26	try	try	VERB
ejpam-6425	732	27	to	to	PART
ejpam-6425	732	28	add	add	VERB
ejpam-6425	732	29	each	each	DET
ejpam-6425	732	30	unused	unused	ADJ
ejpam-6425	732	31	number	number	NOUN
ejpam-6425	732	32	to	to	ADP
ejpam-6425	732	33	the	the	DET
ejpam-6425	732	34	permutation	permutation	NOUN
ejpam-6425	732	35	41	41	NUM
ejpam-6425	732	36	for	for	ADP
ejpam-6425	732	37	num	num	ADJ
ejpam-6425	732	38	in	in	ADP
ejpam-6425	732	39	numbers	number	NOUN
ejpam-6425	732	40	:	:	PUNCT
ejpam-6425	732	41	42	42	NUM
ejpam-6425	732	42	if	if	SCONJ
ejpam-6425	732	43	not	not	PART
ejpam-6425	732	44	used[num	used[num	ADV
ejpam-6425	733	1	1	1	NUM
ejpam-6425	733	2	]	]	NOUN
ejpam-6425	733	3	:	:	PUNCT
ejpam-6425	733	4	#	#	NOUN
ejpam-6425	733	5	check	check	NOUN
ejpam-6425	733	6	if	if	SCONJ
ejpam-6425	733	7	number	number	NOUN
ejpam-6425	733	8	is	be	AUX
ejpam-6425	733	9	not	not	PART
ejpam-6425	733	10	used	use	VERB
ejpam-6425	733	11	43	43	NUM
ejpam-6425	733	12	#	#	NOUN
ejpam-6425	733	13	mark	mark	NOUN
ejpam-6425	733	14	the	the	DET
ejpam-6425	733	15	number	number	NOUN
ejpam-6425	733	16	as	as	SCONJ
ejpam-6425	733	17	used	use	VERB
ejpam-6425	733	18	44	44	NUM
ejpam-6425	733	19	used[num	used[num	ADJ
ejpam-6425	733	20	1	1	NUM
ejpam-6425	734	1	]	]	PUNCT
ejpam-6425	734	2	=	=	PUNCT
ejpam-6425	734	3	true	true	ADJ
ejpam-6425	734	4	45	45	NUM
ejpam-6425	734	5	perm.append(num	perm.append(num	ADJ
ejpam-6425	734	6	)	)	PUNCT
ejpam-6425	734	7	46	46	NUM
ejpam-6425	735	1	47	47	NUM
ejpam-6425	735	2	#	#	NOUN
ejpam-6425	735	3	recursively	recursively	ADV
ejpam-6425	735	4	try	try	VERB
ejpam-6425	735	5	the	the	DET
ejpam-6425	735	6	next	next	ADJ
ejpam-6425	735	7	number	number	NOUN
ejpam-6425	735	8	48	48	NUM
ejpam-6425	735	9	backtrack(perm	backtrack(perm	NOUN
ejpam-6425	735	10	,	,	PUNCT
ejpam-6425	735	11	used	use	VERB
ejpam-6425	735	12	)	)	PUNCT
ejpam-6425	735	13	49	49	NUM
ejpam-6425	735	14	50	50	NUM
ejpam-6425	735	15	#	#	NOUN
ejpam-6425	735	16	undo	undo	VERB
ejpam-6425	735	17	the	the	DET
ejpam-6425	735	18	choice	choice	NOUN
ejpam-6425	735	19	(	(	PUNCT
ejpam-6425	735	20	backtrack	backtrack	NOUN
ejpam-6425	735	21	)	)	PUNCT
ejpam-6425	735	22	51	51	NUM
ejpam-6425	735	23	perm.pop	perm.pop	NOUN
ejpam-6425	735	24	(	(	PUNCT
ejpam-6425	735	25	)	)	PUNCT
ejpam-6425	735	26	52	52	NUM
ejpam-6425	735	27	used[num	used[num	ADJ
ejpam-6425	735	28	1	1	NUM
ejpam-6425	735	29	]	]	PUNCT
ejpam-6425	735	30	=	=	SYM
ejpam-6425	735	31	false	false	ADJ
ejpam-6425	735	32	53	53	NUM
ejpam-6425	735	33	54	54	NUM
ejpam-6425	735	34	#	#	NOUN
ejpam-6425	735	35	start	start	VERB
ejpam-6425	735	36	the	the	DET
ejpam-6425	735	37	backtracking	backtrack	VERB
ejpam-6425	735	38	process	process	NOUN
ejpam-6425	735	39	55	55	NUM
ejpam-6425	735	40	def	def	ADJ
ejpam-6425	735	41	find_valid_permutations	find_valid_permutation	NOUN
ejpam-6425	735	42	(	(	PUNCT
ejpam-6425	735	43	):	):	PUNCT
ejpam-6425	735	44	56	56	NUM
ejpam-6425	735	45	used	use	VERB
ejpam-6425	735	46	=	=	X
ejpam-6425	736	1	[	[	X
ejpam-6425	736	2	false	false	X
ejpam-6425	736	3	]	]	X
ejpam-6425	736	4	*	*	SYM
ejpam-6425	736	5	45	45	NUM
ejpam-6425	736	6	#	#	NOUN
ejpam-6425	736	7	keeps	keep	VERB
ejpam-6425	736	8	track	track	NOUN
ejpam-6425	736	9	of	of	ADP
ejpam-6425	736	10	used	used	ADJ
ejpam-6425	736	11	numbers	number	NOUN
ejpam-6425	736	12	57	57	NUM
ejpam-6425	736	13	backtrack	backtrack	NOUN
ejpam-6425	736	14	(	(	PUNCT
ejpam-6425	736	15	[	[	X
ejpam-6425	736	16	]	]	X
ejpam-6425	736	17	,	,	PUNCT
ejpam-6425	736	18	used	use	VERB
ejpam-6425	736	19	)	)	PUNCT
ejpam-6425	736	20	58	58	NUM
ejpam-6425	736	21	59	59	NUM
ejpam-6425	736	22	#	#	NOUN
ejpam-6425	736	23	start	start	NOUN
ejpam-6425	736	24	finding	find	VERB
ejpam-6425	736	25	the	the	DET
ejpam-6425	736	26	valid	valid	ADJ
ejpam-6425	736	27	permutations	permutation	NOUN
ejpam-6425	736	28	60	60	NUM
ejpam-6425	736	29	find_valid_permutations	find_valid_permutation	NOUN
ejpam-6425	736	30	(	(	PUNCT
ejpam-6425	736	31	)	)	PUNCT
ejpam-6425	736	32	provided	provide	VERB
ejpam-6425	736	33	below	below	ADV
ejpam-6425	736	34	,	,	PUNCT
ejpam-6425	736	35	is	be	AUX
ejpam-6425	736	36	the	the	DET
ejpam-6425	736	37	python	python	NOUN
ejpam-6425	736	38	program	program	NOUN
ejpam-6425	736	39	used	use	VERB
ejpam-6425	736	40	to	to	PART
ejpam-6425	736	41	generate	generate	VERB
ejpam-6425	736	42	an	an	DET
ejpam-6425	736	43	actual	actual	ADJ
ejpam-6425	736	44	edge	edge	NOUN
ejpam-6425	736	45	-	-	PUNCT
ejpam-6425	736	46	graceful	graceful	NOUN
ejpam-6425	736	47	labeling	labeling	NOUN
ejpam-6425	736	48	of	of	ADP
ejpam-6425	736	49	s(rww3	s(rww3	NOUN
ejpam-6425	736	50	)	)	PUNCT
ejpam-6425	736	51	.	.	PUNCT
ejpam-6425	737	1	a.	a.	PROPN
ejpam-6425	737	2	d.	d.	PROPN
ejpam-6425	737	3	c.	c.	PROPN
ejpam-6425	737	4	angel	angel	PROPN
ejpam-6425	737	5	et	et	PROPN
ejpam-6425	737	6	al	al	PROPN
ejpam-6425	737	7	.	.	PUNCT
ejpam-6425	737	8	/	/	SYM
ejpam-6425	737	9	eur	eur	PROPN
ejpam-6425	737	10	.	.	PUNCT
ejpam-6425	738	1	j.	j.	PROPN
ejpam-6425	738	2	pure	pure	PROPN
ejpam-6425	738	3	appl	appl	PROPN
ejpam-6425	738	4	.	.	PROPN
ejpam-6425	738	5	math	math	PROPN
ejpam-6425	738	6	,	,	PUNCT
ejpam-6425	738	7	18	18	NUM
ejpam-6425	738	8	(	(	PUNCT
ejpam-6425	738	9	4	4	NUM
ejpam-6425	738	10	)	)	PUNCT
ejpam-6425	738	11	(	(	PUNCT
ejpam-6425	738	12	2025	2025	NUM
ejpam-6425	738	13	)	)	PUNCT
ejpam-6425	738	14	,	,	PUNCT
ejpam-6425	738	15	6425	6425	NUM
ejpam-6425	738	16	30	30	NUM
ejpam-6425	738	17	of	of	ADP
ejpam-6425	738	18	33	33	NUM
ejpam-6425	738	19	1	1	NUM
ejpam-6425	738	20	#	#	NUM
ejpam-6425	738	21	python	python	NOUN
ejpam-6425	738	22	program	program	NOUN
ejpam-6425	738	23	for	for	ADP
ejpam-6425	738	24	s(rww_3	s(rww_3	ADJ
ejpam-6425	738	25	)	)	PUNCT
ejpam-6425	738	26	2	2	NUM
ejpam-6425	738	27	from	from	ADP
ejpam-6425	738	28	itertools	itertools	PROPN
ejpam-6425	738	29	import	import	NOUN
ejpam-6425	738	30	permutations	permutation	NOUN
ejpam-6425	738	31	3	3	NUM
ejpam-6425	738	32	4	4	NUM
ejpam-6425	738	33	numbers	number	NOUN
ejpam-6425	738	34	=	=	PUNCT
ejpam-6425	739	1	[	[	X
ejpam-6425	739	2	1	1	NUM
ejpam-6425	739	3	,	,	PUNCT
ejpam-6425	739	4	2	2	NUM
ejpam-6425	739	5	,	,	PUNCT
ejpam-6425	739	6	3	3	NUM
ejpam-6425	739	7	,	,	PUNCT
ejpam-6425	739	8	4	4	NUM
ejpam-6425	739	9	,	,	PUNCT
ejpam-6425	739	10	5	5	NUM
ejpam-6425	739	11	,	,	PUNCT
ejpam-6425	739	12	6	6	NUM
ejpam-6425	739	13	,	,	PUNCT
ejpam-6425	739	14	7	7	NUM
ejpam-6425	739	15	,	,	PUNCT
ejpam-6425	739	16	8	8	NUM
ejpam-6425	739	17	,	,	PUNCT
ejpam-6425	739	18	9	9	NUM
ejpam-6425	739	19	,	,	PUNCT
ejpam-6425	739	20	10	10	NUM
ejpam-6425	739	21	,	,	PUNCT
ejpam-6425	739	22	11	11	NUM
ejpam-6425	739	23	,	,	PUNCT
ejpam-6425	739	24	12	12	NUM
ejpam-6425	739	25	,	,	PUNCT
ejpam-6425	739	26	13	13	NUM
ejpam-6425	739	27	,	,	PUNCT
ejpam-6425	739	28	14	14	NUM
ejpam-6425	739	29	,	,	PUNCT
ejpam-6425	739	30	15	15	NUM
ejpam-6425	739	31	,	,	PUNCT
ejpam-6425	739	32	16	16	NUM
ejpam-6425	739	33	,	,	PUNCT
ejpam-6425	739	34	17	17	NUM
ejpam-6425	739	35	,	,	PUNCT
ejpam-6425	739	36	18	18	NUM
ejpam-6425	739	37	,	,	PUNCT
ejpam-6425	739	38	19	19	NUM
ejpam-6425	739	39	,	,	PUNCT
ejpam-6425	739	40	20	20	NUM
ejpam-6425	739	41	,	,	PUNCT
ejpam-6425	739	42	21	21	NUM
ejpam-6425	739	43	,	,	PUNCT
ejpam-6425	739	44	22	22	NUM
ejpam-6425	739	45	,	,	PUNCT
ejpam-6425	739	46	23	23	NUM
ejpam-6425	739	47	,	,	PUNCT
ejpam-6425	739	48	24	24	NUM
ejpam-6425	739	49	,	,	PUNCT
ejpam-6425	739	50	25	25	NUM
ejpam-6425	739	51	,	,	PUNCT
ejpam-6425	739	52	26	26	NUM
ejpam-6425	739	53	,	,	PUNCT
ejpam-6425	739	54	27	27	NUM
ejpam-6425	739	55	,	,	PUNCT
ejpam-6425	739	56	28	28	NUM
ejpam-6425	739	57	,	,	PUNCT
ejpam-6425	739	58	29	29	NUM
ejpam-6425	739	59	,	,	PUNCT
ejpam-6425	739	60	30	30	NUM
ejpam-6425	739	61	,	,	PUNCT
ejpam-6425	739	62	31	31	NUM
ejpam-6425	739	63	,	,	PUNCT
ejpam-6425	739	64	32	32	NUM
ejpam-6425	739	65	,	,	PUNCT
ejpam-6425	739	66	33	33	NUM
ejpam-6425	739	67	,	,	PUNCT
ejpam-6425	739	68	34	34	NUM
ejpam-6425	739	69	,	,	PUNCT
ejpam-6425	739	70	35	35	NUM
ejpam-6425	739	71	,	,	PUNCT
ejpam-6425	739	72	36	36	NUM
ejpam-6425	739	73	,	,	PUNCT
ejpam-6425	739	74	37	37	NUM
ejpam-6425	739	75	,	,	PUNCT
ejpam-6425	739	76	38	38	NUM
ejpam-6425	739	77	,	,	PUNCT
ejpam-6425	739	78	39	39	NUM
ejpam-6425	739	79	,	,	PUNCT
ejpam-6425	739	80	40	40	NUM
ejpam-6425	739	81	,	,	PUNCT
ejpam-6425	739	82	41	41	NUM
ejpam-6425	739	83	,	,	PUNCT
ejpam-6425	739	84	42	42	NUM
ejpam-6425	739	85	,	,	PUNCT
ejpam-6425	739	86	43	43	NUM
ejpam-6425	739	87	,	,	PUNCT
ejpam-6425	739	88	44	44	NUM
ejpam-6425	739	89	,	,	PUNCT
ejpam-6425	739	90	45	45	NUM
ejpam-6425	739	91	,	,	PUNCT
ejpam-6425	739	92	46	46	NUM
ejpam-6425	739	93	,	,	PUNCT
ejpam-6425	739	94	47	47	NUM
ejpam-6425	739	95	,	,	PUNCT
ejpam-6425	739	96	48	48	NUM
ejpam-6425	739	97	,	,	PUNCT
ejpam-6425	739	98	49	49	NUM
ejpam-6425	739	99	,	,	PUNCT
ejpam-6425	739	100	50	50	NUM
ejpam-6425	739	101	,	,	PUNCT
ejpam-6425	739	102	51	51	NUM
ejpam-6425	739	103	,	,	PUNCT
ejpam-6425	739	104	52	52	NUM
ejpam-6425	739	105	,	,	PUNCT
ejpam-6425	739	106	53	53	NUM
ejpam-6425	739	107	,	,	PUNCT
ejpam-6425	739	108	54	54	NUM
ejpam-6425	739	109	]	]	SYM
ejpam-6425	739	110	5	5	NUM
ejpam-6425	739	111	6	6	NUM
ejpam-6425	739	112	#	#	NOUN
ejpam-6425	739	113	function	function	NOUN
ejpam-6425	739	114	to	to	PART
ejpam-6425	739	115	compute	compute	VERB
ejpam-6425	739	116	u0	u0	ADJ
ejpam-6425	739	117	to	to	PART
ejpam-6425	739	118	v1	v1	VERB
ejpam-6425	739	119	7	7	NUM
ejpam-6425	739	120	def	def	ADJ
ejpam-6425	739	121	compute_v_values(w1u1	compute_v_values(w1u1	PROPN
ejpam-6425	739	122	,	,	PUNCT
ejpam-6425	739	123	u1v2	u1v2	NOUN
ejpam-6425	739	124	,	,	PUNCT
ejpam-6425	739	125	v2w2	v2w2	X
ejpam-6425	739	126	,	,	PUNCT
ejpam-6425	739	127	w2u2	w2u2	NOUN
ejpam-6425	739	128	,	,	PUNCT
ejpam-6425	739	129	u2v3	u2v3	ADV
ejpam-6425	739	130	,	,	PUNCT
ejpam-6425	739	131	v3w3	v3w3	NOUN
ejpam-6425	739	132	,	,	PUNCT
ejpam-6425	739	133	w3u3	w3u3	ADJ
ejpam-6425	739	134	,	,	PUNCT
ejpam-6425	739	135	u3v1	u3v1	PROPN
ejpam-6425	739	136	,	,	PUNCT
ejpam-6425	739	137	v1w1	v1w1	PROPN
ejpam-6425	739	138	,	,	PUNCT
ejpam-6425	739	139	u0u1	u0u1	NOUN
ejpam-6425	739	140	,	,	PUNCT
ejpam-6425	739	141	u0v2	u0v2	NOUN
ejpam-6425	739	142	,	,	PUNCT
ejpam-6425	739	143	u0u2	u0u2	PRON
ejpam-6425	739	144	,	,	PUNCT
ejpam-6425	739	145	u0v3	u0v3	X
ejpam-6425	739	146	,	,	PUNCT
ejpam-6425	739	147	u0u3	u0u3	PROPN
ejpam-6425	739	148	,	,	PUNCT
ejpam-6425	739	149	u0v1	u0v1	X
ejpam-6425	739	150	,	,	PUNCT
ejpam-6425	739	151	w1v1	w1v1	X
ejpam-6425	739	152	,	,	PUNCT
ejpam-6425	739	153	w1v2	w1v2	X
ejpam-6425	739	154	,	,	PUNCT
ejpam-6425	739	155	u1w1	u1w1	PROPN
ejpam-6425	739	156	,	,	PUNCT
ejpam-6425	739	157	u1v2	u1v2	ADJ
ejpam-6425	739	158	,	,	PUNCT
ejpam-6425	739	159	v2w1	v2w1	X
ejpam-6425	739	160	,	,	PUNCT
ejpam-6425	739	161	v2u1	v2u1	X
ejpam-6425	739	162	,	,	PUNCT
ejpam-6425	739	163	v2w2	v2w2	X
ejpam-6425	739	164	,	,	PUNCT
ejpam-6425	739	165	w2v2	w2v2	PROPN
ejpam-6425	739	166	,	,	PUNCT
ejpam-6425	739	167	w2v3	w2v3	NOUN
ejpam-6425	739	168	,	,	PUNCT
ejpam-6425	739	169	u2w2	u2w2	PROPN
ejpam-6425	739	170	,	,	PUNCT
ejpam-6425	739	171	u2v3	u2v3	NOUN
ejpam-6425	739	172	,	,	PUNCT
ejpam-6425	739	173	v3w2	v3w2	PROPN
ejpam-6425	739	174	,	,	PUNCT
ejpam-6425	739	175	v3u2	v3u2	PRON
ejpam-6425	739	176	,	,	PUNCT
ejpam-6425	739	177	v3w3	v3w3	X
ejpam-6425	739	178	,	,	PUNCT
ejpam-6425	739	179	w3v3	w3v3	NOUN
ejpam-6425	739	180	,	,	PUNCT
ejpam-6425	739	181	w3v1	w3v1	ADV
ejpam-6425	739	182	,	,	PUNCT
ejpam-6425	739	183	u3w3	u3w3	NOUN
ejpam-6425	739	184	,	,	PUNCT
ejpam-6425	739	185	u3v1	u3v1	ADJ
ejpam-6425	739	186	,	,	PUNCT
ejpam-6425	739	187	v1w3	v1w3	X
ejpam-6425	739	188	,	,	PUNCT
ejpam-6425	739	189	v1u3	v1u3	X
ejpam-6425	739	190	,	,	PUNCT
ejpam-6425	739	191	v1w1	v1w1	NOUN
ejpam-6425	739	192	,	,	PUNCT
ejpam-6425	739	193	u0v1	u0v1	ADJ
ejpam-6425	739	194	,	,	PUNCT
ejpam-6425	739	195	u0u3	u0u3	PROPN
ejpam-6425	739	196	,	,	PUNCT
ejpam-6425	739	197	u0v3	u0v3	X
ejpam-6425	739	198	,	,	PUNCT
ejpam-6425	739	199	u0u2	u0u2	SYM
ejpam-6425	739	200	,	,	PUNCT
ejpam-6425	739	201	u0v2	u0v2	NOUN
ejpam-6425	739	202	,	,	PUNCT
ejpam-6425	739	203	u0u1	u0u1	ADV
ejpam-6425	739	204	,	,	PUNCT
ejpam-6425	739	205	w3v1	w3v1	PROPN
ejpam-6425	739	206	,	,	PUNCT
ejpam-6425	739	207	w2v3	w2v3	NOUN
ejpam-6425	739	208	,	,	PUNCT
ejpam-6425	739	209	w1v2	w1v2	X
ejpam-6425	739	210	,	,	PUNCT
ejpam-6425	739	211	v1u0	v1u0	PROPN
ejpam-6425	739	212	,	,	PUNCT
ejpam-6425	739	213	u3u0	u3u0	NOUN
ejpam-6425	739	214	,	,	PUNCT
ejpam-6425	739	215	w3u3	w3u3	PROPN
ejpam-6425	739	216	,	,	PUNCT
ejpam-6425	739	217	v3u0	v3u0	NOUN
ejpam-6425	739	218	,	,	PUNCT
ejpam-6425	739	219	u2u0	u2u0	NOUN
ejpam-6425	739	220	,	,	PUNCT
ejpam-6425	739	221	w2u2	w2u2	NOUN
ejpam-6425	739	222	,	,	PUNCT
ejpam-6425	739	223	v2u0	v2u0	NOUN
ejpam-6425	739	224	,	,	PUNCT
ejpam-6425	739	225	u1u0	u1u0	NOUN
ejpam-6425	739	226	,	,	PUNCT
ejpam-6425	739	227	w1u1	w1u1	NOUN
ejpam-6425	739	228	):	):	PUNCT
ejpam-6425	739	229	8	8	NUM
ejpam-6425	739	230	u0	u0	NOUN
ejpam-6425	739	231	=	=	PUNCT
ejpam-6425	739	232	(	(	PUNCT
ejpam-6425	739	233	u0v1	u0v1	X
ejpam-6425	740	1	+	+	CCONJ
ejpam-6425	740	2	u0u1	u0u1	PROPN
ejpam-6425	740	3	+	+	X
ejpam-6425	740	4	u0v2	u0v2	X
ejpam-6425	740	5	+	+	NUM
ejpam-6425	740	6	u0u2	u0u2	PUNCT
ejpam-6425	740	7	+	+	NUM
ejpam-6425	740	8	u0v3	u0v3	ADJ
ejpam-6425	740	9	+	+	NUM
ejpam-6425	740	10	u0u3	u0u3	PROPN
ejpam-6425	740	11	+	+	CCONJ
ejpam-6425	740	12	v1u0	v1u0	PROPN
ejpam-6425	740	13	+	+	CCONJ
ejpam-6425	740	14	u1u0	u1u0	PROPN
ejpam-6425	740	15	+	+	ADJ
ejpam-6425	740	16	v2u0	v2u0	PROPN
ejpam-6425	740	17	+	+	NUM
ejpam-6425	740	18	u2u0	u2u0	X
ejpam-6425	740	19	+	+	CCONJ
ejpam-6425	740	20	v3u0	v3u0	X
ejpam-6425	740	21	+	+	CCONJ
ejpam-6425	740	22	u3u0	u3u0	X
ejpam-6425	740	23	)	)	PUNCT
ejpam-6425	740	24	%	%	NOUN
ejpam-6425	740	25	20	20	NUM
ejpam-6425	740	26	9	9	NUM
ejpam-6425	740	27	w1	w1	NOUN
ejpam-6425	740	28	=	=	SYM
ejpam-6425	740	29	(	(	PUNCT
ejpam-6425	740	30	v1w1	v1w1	X
ejpam-6425	740	31	+	+	X
ejpam-6425	740	32	w1u1	w1u1	X
ejpam-6425	741	1	+	+	SYM
ejpam-6425	741	2	w1v2	w1v2	X
ejpam-6425	741	3	+	+	CCONJ
ejpam-6425	741	4	v1w1	v1w1	PROPN
ejpam-6425	742	1	+	+	CCONJ
ejpam-6425	742	2	u1w1	u1w1	ADJ
ejpam-6425	742	3	+	+	NUM
ejpam-6425	742	4	v2w1	v2w1	X
ejpam-6425	742	5	)	)	PUNCT
ejpam-6425	742	6	%	%	NOUN
ejpam-6425	742	7	20	20	NUM
ejpam-6425	742	8	10	10	NUM
ejpam-6425	742	9	u1	u1	NOUN
ejpam-6425	742	10	=	=	SYM
ejpam-6425	742	11	(	(	PUNCT
ejpam-6425	742	12	w1u1	w1u1	X
ejpam-6425	743	1	+	+	CCONJ
ejpam-6425	743	2	u0u1	u0u1	PROPN
ejpam-6425	743	3	+	+	NUM
ejpam-6425	743	4	u1v2	u1v2	X
ejpam-6425	743	5	+	+	X
ejpam-6425	743	6	w1u1	w1u1	X
ejpam-6425	743	7	+	+	CCONJ
ejpam-6425	743	8	u0u1	u0u1	PROPN
ejpam-6425	743	9	+	+	NUM
ejpam-6425	743	10	v2u1	v2u1	X
ejpam-6425	743	11	)	)	PUNCT
ejpam-6425	743	12	%	%	NOUN
ejpam-6425	743	13	20	20	NUM
ejpam-6425	743	14	11	11	NUM
ejpam-6425	743	15	v2	v2	NOUN
ejpam-6425	743	16	=	=	SYM
ejpam-6425	743	17	(	(	PUNCT
ejpam-6425	743	18	w1v2	w1v2	X
ejpam-6425	743	19	+	+	X
ejpam-6425	743	20	u0v2	u0v2	X
ejpam-6425	743	21	+	+	NUM
ejpam-6425	743	22	u1v2	u1v2	PUNCT
ejpam-6425	744	1	+	+	CCONJ
ejpam-6425	744	2	v2w2	v2w2	X
ejpam-6425	744	3	+	+	X
ejpam-6425	744	4	w1v2	w1v2	X
ejpam-6425	744	5	+	+	CCONJ
ejpam-6425	744	6	u0v2	u0v2	X
ejpam-6425	744	7	+	+	SYM
ejpam-6425	744	8	u1v2	u1v2	X
ejpam-6425	744	9	+	+	CCONJ
ejpam-6425	744	10	w2v2	w2v2	NOUN
ejpam-6425	744	11	)	)	PUNCT
ejpam-6425	744	12	%	%	NOUN
ejpam-6425	744	13	20	20	NUM
ejpam-6425	744	14	12	12	NUM
ejpam-6425	744	15	w2	w2	NOUN
ejpam-6425	744	16	=	=	SYM
ejpam-6425	744	17	(	(	PUNCT
ejpam-6425	744	18	v2w2	v2w2	X
ejpam-6425	744	19	+	+	PUNCT
ejpam-6425	744	20	w2u2	w2u2	X
ejpam-6425	745	1	+	+	CCONJ
ejpam-6425	745	2	w2v3	w2v3	X
ejpam-6425	745	3	+	+	CCONJ
ejpam-6425	745	4	v2w2	v2w2	X
ejpam-6425	745	5	+	+	X
ejpam-6425	745	6	u2w2	u2w2	ADJ
ejpam-6425	745	7	+	+	X
ejpam-6425	745	8	v3w2	v3w2	NOUN
ejpam-6425	745	9	)	)	PUNCT
ejpam-6425	745	10	%	%	NOUN
ejpam-6425	745	11	20	20	NUM
ejpam-6425	745	12	13	13	NUM
ejpam-6425	745	13	u2	u2	NOUN
ejpam-6425	745	14	=	=	PUNCT
ejpam-6425	745	15	(	(	PUNCT
ejpam-6425	745	16	w2u2	w2u2	X
ejpam-6425	745	17	+	+	X
ejpam-6425	745	18	u0u2	u0u2	VERB
ejpam-6425	746	1	+	+	SYM
ejpam-6425	746	2	u2v3	u2v3	X
ejpam-6425	746	3	+	+	X
ejpam-6425	746	4	w2u2	w2u2	X
ejpam-6425	746	5	+	+	NUM
ejpam-6425	746	6	u0u2	u0u2	SYM
ejpam-6425	746	7	+	+	NUM
ejpam-6425	746	8	v3u2	v3u2	X
ejpam-6425	746	9	)	)	PUNCT
ejpam-6425	746	10	%	%	NOUN
ejpam-6425	746	11	20	20	NUM
ejpam-6425	746	12	14	14	NUM
ejpam-6425	746	13	v3	v3	PROPN
ejpam-6425	746	14	=	=	PUNCT
ejpam-6425	746	15	(	(	PUNCT
ejpam-6425	746	16	w2v3	w2v3	PROPN
ejpam-6425	747	1	+	+	CCONJ
ejpam-6425	747	2	u2v3	u2v3	PROPN
ejpam-6425	747	3	+	+	X
ejpam-6425	747	4	u0v3	u0v3	X
ejpam-6425	747	5	+	+	NOUN
ejpam-6425	747	6	v3w3	v3w3	X
ejpam-6425	747	7	+	+	NOUN
ejpam-6425	747	8	w2v3	w2v3	NOUN
ejpam-6425	747	9	+	+	CCONJ
ejpam-6425	747	10	u2v3	u2v3	PROPN
ejpam-6425	747	11	+	+	X
ejpam-6425	747	12	u0v3	u0v3	X
ejpam-6425	747	13	+	+	X
ejpam-6425	747	14	w3v3	w3v3	X
ejpam-6425	747	15	)	)	PUNCT
ejpam-6425	747	16	%	%	NOUN
ejpam-6425	747	17	20	20	NUM
ejpam-6425	747	18	15	15	NUM
ejpam-6425	747	19	w3	w3	NOUN
ejpam-6425	747	20	=	=	SYM
ejpam-6425	747	21	(	(	PUNCT
ejpam-6425	747	22	v3w3	v3w3	X
ejpam-6425	747	23	+	+	CCONJ
ejpam-6425	747	24	w3u3	w3u3	X
ejpam-6425	747	25	+	+	CCONJ
ejpam-6425	747	26	w3v1	w3v1	NOUN
ejpam-6425	747	27	+	+	X
ejpam-6425	747	28	v3w3	v3w3	X
ejpam-6425	747	29	+	+	CCONJ
ejpam-6425	747	30	u3w3	u3w3	VERB
ejpam-6425	747	31	+	+	CCONJ
ejpam-6425	747	32	v1w3	v1w3	X
ejpam-6425	747	33	)	)	PUNCT
ejpam-6425	747	34	%	%	NOUN
ejpam-6425	747	35	20	20	NUM
ejpam-6425	747	36	16	16	NUM
ejpam-6425	747	37	u3	u3	NOUN
ejpam-6425	747	38	=	=	SYM
ejpam-6425	747	39	(	(	PUNCT
ejpam-6425	747	40	w3u3	w3u3	X
ejpam-6425	747	41	+	+	SYM
ejpam-6425	747	42	u0u3	u0u3	PROPN
ejpam-6425	747	43	+	+	X
ejpam-6425	747	44	u3v1	u3v1	X
ejpam-6425	747	45	+	+	CCONJ
ejpam-6425	747	46	w3u3	w3u3	X
ejpam-6425	747	47	+	+	SYM
ejpam-6425	747	48	u0u3	u0u3	PROPN
ejpam-6425	747	49	+	+	SYM
ejpam-6425	747	50	v1u3	v1u3	X
ejpam-6425	747	51	)	)	PUNCT
ejpam-6425	747	52	%	%	NOUN
ejpam-6425	747	53	20	20	NUM
ejpam-6425	747	54	17	17	NUM
ejpam-6425	747	55	v1	v1	NOUN
ejpam-6425	747	56	=	=	SYM
ejpam-6425	747	57	(	(	PUNCT
ejpam-6425	747	58	u3v1	u3v1	PROPN
ejpam-6425	748	1	+	+	CCONJ
ejpam-6425	748	2	u0v1	u0v1	ADJ
ejpam-6425	749	1	+	+	CCONJ
ejpam-6425	749	2	v1w1	v1w1	PROPN
ejpam-6425	749	3	+	+	NOUN
ejpam-6425	749	4	w3v1	w3v1	ADJ
ejpam-6425	749	5	+	+	X
ejpam-6425	749	6	u3v1	u3v1	PROPN
ejpam-6425	750	1	+	+	CCONJ
ejpam-6425	750	2	u0v1	u0v1	X
ejpam-6425	751	1	+	+	CCONJ
ejpam-6425	751	2	w1v1	w1v1	X
ejpam-6425	752	1	+	+	NUM
ejpam-6425	752	2	w3v1	w3v1	NOUN
ejpam-6425	752	3	)	)	PUNCT
ejpam-6425	752	4	%	%	NOUN
ejpam-6425	752	5	20	20	NUM
ejpam-6425	752	6	18	18	NUM
ejpam-6425	752	7	u0	u0	NOUN
ejpam-6425	752	8	=	=	PUNCT
ejpam-6425	752	9	(	(	PUNCT
ejpam-6425	752	10	u0v1	u0v1	X
ejpam-6425	752	11	+	+	CCONJ
ejpam-6425	752	12	u0u1	u0u1	PROPN
ejpam-6425	752	13	+	+	X
ejpam-6425	752	14	u0v2	u0v2	X
ejpam-6425	752	15	+	+	NUM
ejpam-6425	752	16	u0u2	u0u2	VERB
ejpam-6425	752	17	+	+	NUM
ejpam-6425	752	18	u0v3	u0v3	X
ejpam-6425	752	19	+	+	X
ejpam-6425	752	20	u0u3	u0u3	ADJ
ejpam-6425	752	21	)	)	PUNCT
ejpam-6425	752	22	%	%	NOUN
ejpam-6425	752	23	20	20	NUM
ejpam-6425	752	24	19	19	NUM
ejpam-6425	752	25	w1	w1	NOUN
ejpam-6425	752	26	=	=	SYM
ejpam-6425	752	27	(	(	PUNCT
ejpam-6425	752	28	w1v1	w1v1	X
ejpam-6425	752	29	+	+	X
ejpam-6425	752	30	w1u1	w1u1	X
ejpam-6425	752	31	+	+	SYM
ejpam-6425	752	32	w1v2	w1v2	X
ejpam-6425	752	33	)	)	PUNCT
ejpam-6425	752	34	%	%	NOUN
ejpam-6425	752	35	20	20	NUM
ejpam-6425	752	36	20	20	NUM
ejpam-6425	752	37	u1	u1	NOUN
ejpam-6425	752	38	=	=	SYM
ejpam-6425	752	39	(	(	PUNCT
ejpam-6425	752	40	u1w1	u1w1	X
ejpam-6425	752	41	+	+	CCONJ
ejpam-6425	752	42	u1u0	u1u0	PROPN
ejpam-6425	752	43	+	+	NUM
ejpam-6425	752	44	u1v2	u1v2	NOUN
ejpam-6425	752	45	)	)	PUNCT
ejpam-6425	752	46	%	%	NOUN
ejpam-6425	752	47	20	20	NUM
ejpam-6425	752	48	21	21	NUM
ejpam-6425	752	49	v2	v2	NOUN
ejpam-6425	752	50	=	=	SYM
ejpam-6425	752	51	(	(	PUNCT
ejpam-6425	752	52	v2w1	v2w1	X
ejpam-6425	752	53	+	+	CCONJ
ejpam-6425	752	54	v2u1	v2u1	X
ejpam-6425	752	55	+	+	CCONJ
ejpam-6425	752	56	v2u0	v2u0	X
ejpam-6425	752	57	+	+	NUM
ejpam-6425	752	58	v2w2	v2w2	NOUN
ejpam-6425	752	59	)	)	PUNCT
ejpam-6425	752	60	%	%	NOUN
ejpam-6425	752	61	20	20	NUM
ejpam-6425	752	62	22	22	NUM
ejpam-6425	752	63	w2	w2	NOUN
ejpam-6425	752	64	=	=	SYM
ejpam-6425	752	65	(	(	PUNCT
ejpam-6425	752	66	w2v2	w2v2	X
ejpam-6425	752	67	+	+	X
ejpam-6425	752	68	w2u2	w2u2	X
ejpam-6425	752	69	+	+	CCONJ
ejpam-6425	752	70	w2v3	w2v3	NOUN
ejpam-6425	752	71	)	)	PUNCT
ejpam-6425	752	72	%	%	NOUN
ejpam-6425	752	73	20	20	NUM
ejpam-6425	752	74	23	23	NUM
ejpam-6425	752	75	u2	u2	NOUN
ejpam-6425	752	76	=	=	PUNCT
ejpam-6425	752	77	(	(	PUNCT
ejpam-6425	752	78	u2w2	u2w2	X
ejpam-6425	752	79	+	+	NUM
ejpam-6425	752	80	u2u0	u2u0	X
ejpam-6425	752	81	+	+	X
ejpam-6425	752	82	u2v3	u2v3	NOUN
ejpam-6425	752	83	)	)	PUNCT
ejpam-6425	752	84	%	%	NOUN
ejpam-6425	752	85	20	20	NUM
ejpam-6425	752	86	24	24	NUM
ejpam-6425	752	87	v3	v3	PROPN
ejpam-6425	752	88	=	=	SYM
ejpam-6425	752	89	(	(	PUNCT
ejpam-6425	752	90	v3w2	v3w2	PUNCT
ejpam-6425	752	91	+	+	NUM
ejpam-6425	752	92	v3u2	v3u2	NUM
ejpam-6425	753	1	+	+	CCONJ
ejpam-6425	753	2	v3u0	v3u0	X
ejpam-6425	753	3	+	+	CCONJ
ejpam-6425	753	4	v3w3	v3w3	NOUN
ejpam-6425	753	5	)	)	PUNCT
ejpam-6425	753	6	%	%	NOUN
ejpam-6425	753	7	20	20	NUM
ejpam-6425	753	8	25	25	NUM
ejpam-6425	753	9	w3	w3	NOUN
ejpam-6425	753	10	=	=	SYM
ejpam-6425	753	11	(	(	PUNCT
ejpam-6425	753	12	w3v3	w3v3	X
ejpam-6425	753	13	+	+	CCONJ
ejpam-6425	753	14	w3u3	w3u3	X
ejpam-6425	753	15	+	+	X
ejpam-6425	753	16	w3v1	w3v1	NOUN
ejpam-6425	753	17	)	)	PUNCT
ejpam-6425	753	18	%	%	NOUN
ejpam-6425	753	19	20	20	NUM
ejpam-6425	753	20	26	26	NUM
ejpam-6425	753	21	u3	u3	NOUN
ejpam-6425	753	22	=	=	SYM
ejpam-6425	753	23	(	(	PUNCT
ejpam-6425	753	24	u3w3	u3w3	VERB
ejpam-6425	753	25	+	+	CCONJ
ejpam-6425	753	26	u3u0	u3u0	SYM
ejpam-6425	753	27	+	+	NUM
ejpam-6425	753	28	u3v1	u3v1	X
ejpam-6425	753	29	)	)	PUNCT
ejpam-6425	753	30	%	%	NOUN
ejpam-6425	753	31	20	20	NUM
ejpam-6425	753	32	27	27	NUM
ejpam-6425	753	33	v1	v1	NOUN
ejpam-6425	753	34	=	=	SYM
ejpam-6425	753	35	(	(	PUNCT
ejpam-6425	753	36	v1w3	v1w3	X
ejpam-6425	753	37	+	+	X
ejpam-6425	753	38	v1u3	v1u3	X
ejpam-6425	753	39	+	+	X
ejpam-6425	753	40	v1u0	v1u0	PROPN
ejpam-6425	753	41	+	+	CCONJ
ejpam-6425	753	42	v1w1	v1w1	NOUN
ejpam-6425	753	43	)	)	PUNCT
ejpam-6425	753	44	%	%	NOUN
ejpam-6425	753	45	20	20	NUM
ejpam-6425	753	46	28	28	NUM
ejpam-6425	753	47	return	return	NOUN
ejpam-6425	753	48	u0	u0	ADJ
ejpam-6425	753	49	,	,	PUNCT
ejpam-6425	753	50	w1	w1	NOUN
ejpam-6425	753	51	,	,	PUNCT
ejpam-6425	753	52	u1	u1	NOUN
ejpam-6425	753	53	,	,	PUNCT
ejpam-6425	753	54	v2	v2	NOUN
ejpam-6425	753	55	,	,	PUNCT
ejpam-6425	753	56	w2	w2	NOUN
ejpam-6425	753	57	,	,	PUNCT
ejpam-6425	753	58	u2	u2	PROPN
ejpam-6425	753	59	,	,	PUNCT
ejpam-6425	753	60	v3	v3	PROPN
ejpam-6425	753	61	,	,	PUNCT
ejpam-6425	753	62	w3	w3	PROPN
ejpam-6425	753	63	,	,	PUNCT
ejpam-6425	753	64	u3	u3	NOUN
ejpam-6425	753	65	,	,	PUNCT
ejpam-6425	753	66	v1	v1	NOUN
ejpam-6425	753	67	,	,	PUNCT
ejpam-6425	753	68	u0	u0	ADJ
ejpam-6425	753	69	,	,	PUNCT
ejpam-6425	753	70	w1	w1	NOUN
ejpam-6425	753	71	,	,	PUNCT
ejpam-6425	753	72	u1	u1	NOUN
ejpam-6425	753	73	,	,	PUNCT
ejpam-6425	753	74	v2	v2	NOUN
ejpam-6425	753	75	,	,	PUNCT
ejpam-6425	753	76	w2	w2	NOUN
ejpam-6425	753	77	,	,	PUNCT
ejpam-6425	753	78	u2	u2	PROPN
ejpam-6425	753	79	,	,	PUNCT
ejpam-6425	753	80	v3	v3	PROPN
ejpam-6425	753	81	,	,	PUNCT
ejpam-6425	753	82	w3	w3	PROPN
ejpam-6425	753	83	,	,	PUNCT
ejpam-6425	753	84	u3	u3	PROPN
ejpam-6425	753	85	,	,	PUNCT
ejpam-6425	753	86	v1	v1	VERB
ejpam-6425	753	87	29	29	NUM
ejpam-6425	753	88	30	30	NUM
ejpam-6425	753	89	#	#	NOUN
ejpam-6425	753	90	backtracking	backtrack	VERB
ejpam-6425	753	91	function	function	NOUN
ejpam-6425	753	92	to	to	PART
ejpam-6425	753	93	find	find	VERB
ejpam-6425	753	94	valid	valid	ADJ
ejpam-6425	753	95	permutations	permutation	NOUN
ejpam-6425	753	96	31	31	NUM
ejpam-6425	753	97	def	def	ADJ
ejpam-6425	753	98	backtrack(perm	backtrack(perm	NOUN
ejpam-6425	753	99	,	,	PUNCT
ejpam-6425	753	100	used	use	VERB
ejpam-6425	753	101	):	):	PUNCT
ejpam-6425	753	102	32	32	NUM
ejpam-6425	753	103	#	#	NOUN
ejpam-6425	753	104	if	if	SCONJ
ejpam-6425	753	105	we	we	PRON
ejpam-6425	753	106	have	have	VERB
ejpam-6425	753	107	a	a	DET
ejpam-6425	753	108	complete	complete	ADJ
ejpam-6425	753	109	permutation	permutation	NOUN
ejpam-6425	753	110	,	,	PUNCT
ejpam-6425	753	111	verify	verify	VERB
ejpam-6425	753	112	u0	u0	ADJ
ejpam-6425	753	113	to	to	PART
ejpam-6425	753	114	v1	v1	VERB
ejpam-6425	753	115	33	33	NUM
ejpam-6425	753	116	if	if	SCONJ
ejpam-6425	753	117	len(perm	len(perm	NOUN
ejpam-6425	753	118	)	)	PUNCT
ejpam-6425	753	119	=	=	NOUN
ejpam-6425	753	120	=	=	SYM
ejpam-6425	753	121	54	54	NUM
ejpam-6425	753	122	:	:	SYM
ejpam-6425	753	123	34	34	NUM
ejpam-6425	753	124	v_values	v_value	NOUN
ejpam-6425	753	125	=	=	SYM
ejpam-6425	753	126	compute_v_values(*perm	compute_v_values(*perm	NOUN
ejpam-6425	753	127	)	)	PUNCT
ejpam-6425	753	128	35	35	NUM
ejpam-6425	753	129	#	#	NOUN
ejpam-6425	753	130	verify	verify	NOUN
ejpam-6425	753	131	if	if	SCONJ
ejpam-6425	753	132	u0	u0	ADJ
ejpam-6425	753	133	to	to	PART
ejpam-6425	753	134	v1	v1	VERB
ejpam-6425	753	135	are	be	AUX
ejpam-6425	753	136	distinct	distinct	ADJ
ejpam-6425	753	137	36	36	NUM
ejpam-6425	753	138	if	if	SCONJ
ejpam-6425	753	139	len(set(v_values	len(set(v_value	NOUN
ejpam-6425	753	140	)	)	PUNCT
ejpam-6425	753	141	)	)	PUNCT
ejpam-6425	754	1	=	=	SYM
ejpam-6425	754	2	=	=	SYM
ejpam-6425	754	3	20	20	NUM
ejpam-6425	754	4	:	:	SYM
ejpam-6425	754	5	37	37	NUM
ejpam-6425	754	6	print(f"permutation	print(f"permutation	NOUN
ejpam-6425	754	7	:	:	PUNCT
ejpam-6425	754	8	{	{	PUNCT
ejpam-6425	754	9	perm	perm	NOUN
ejpam-6425	754	10	}	}	PUNCT
ejpam-6425	754	11	,	,	PUNCT
ejpam-6425	754	12	u0	u0	ADJ
ejpam-6425	754	13	to	to	PART
ejpam-6425	754	14	v1	v1	VERB
ejpam-6425	754	15	:	:	PUNCT
ejpam-6425	754	16	{	{	PUNCT
ejpam-6425	754	17	v_values	v_value	NOUN
ejpam-6425	754	18	}	}	PUNCT
ejpam-6425	754	19	"	"	PUNCT
ejpam-6425	754	20	)	)	PUNCT
ejpam-6425	754	21	a.	a.	PROPN
ejpam-6425	754	22	d.	d.	PROPN
ejpam-6425	754	23	c.	c.	PROPN
ejpam-6425	754	24	angel	angel	PROPN
ejpam-6425	754	25	et	et	PROPN
ejpam-6425	754	26	al	al	PROPN
ejpam-6425	754	27	.	.	PUNCT
ejpam-6425	754	28	/	/	SYM
ejpam-6425	754	29	eur	eur	PROPN
ejpam-6425	754	30	.	.	PUNCT
ejpam-6425	755	1	j.	j.	PROPN
ejpam-6425	755	2	pure	pure	PROPN
ejpam-6425	755	3	appl	appl	PROPN
ejpam-6425	755	4	.	.	PROPN
ejpam-6425	755	5	math	math	PROPN
ejpam-6425	755	6	,	,	PUNCT
ejpam-6425	755	7	18	18	NUM
ejpam-6425	755	8	(	(	PUNCT
ejpam-6425	755	9	4	4	NUM
ejpam-6425	755	10	)	)	PUNCT
ejpam-6425	755	11	(	(	PUNCT
ejpam-6425	755	12	2025	2025	NUM
ejpam-6425	755	13	)	)	PUNCT
ejpam-6425	755	14	,	,	PUNCT
ejpam-6425	755	15	6425	6425	NUM
ejpam-6425	755	16	31	31	NUM
ejpam-6425	755	17	of	of	ADP
ejpam-6425	755	18	33	33	NUM
ejpam-6425	755	19	38	38	NUM
ejpam-6425	755	20	return	return	NOUN
ejpam-6425	755	21	39	39	NUM
ejpam-6425	755	22	40	40	NUM
ejpam-6425	755	23	#	#	NOUN
ejpam-6425	755	24	try	try	VERB
ejpam-6425	755	25	to	to	PART
ejpam-6425	755	26	add	add	VERB
ejpam-6425	755	27	each	each	DET
ejpam-6425	755	28	unused	unused	ADJ
ejpam-6425	755	29	number	number	NOUN
ejpam-6425	755	30	to	to	ADP
ejpam-6425	755	31	the	the	DET
ejpam-6425	755	32	permutation	permutation	NOUN
ejpam-6425	755	33	41	41	NUM
ejpam-6425	755	34	for	for	ADP
ejpam-6425	755	35	num	num	ADJ
ejpam-6425	755	36	in	in	ADP
ejpam-6425	755	37	numbers	number	NOUN
ejpam-6425	755	38	:	:	PUNCT
ejpam-6425	755	39	42	42	NUM
ejpam-6425	755	40	if	if	SCONJ
ejpam-6425	755	41	not	not	PART
ejpam-6425	755	42	used[num	used[num	ADV
ejpam-6425	756	1	1	1	NUM
ejpam-6425	756	2	]	]	NOUN
ejpam-6425	756	3	:	:	PUNCT
ejpam-6425	756	4	#	#	NOUN
ejpam-6425	756	5	check	check	NOUN
ejpam-6425	756	6	if	if	SCONJ
ejpam-6425	756	7	number	number	NOUN
ejpam-6425	756	8	is	be	AUX
ejpam-6425	756	9	not	not	PART
ejpam-6425	756	10	used	use	VERB
ejpam-6425	756	11	43	43	NUM
ejpam-6425	756	12	#	#	NOUN
ejpam-6425	756	13	mark	mark	NOUN
ejpam-6425	756	14	the	the	DET
ejpam-6425	756	15	number	number	NOUN
ejpam-6425	756	16	as	as	SCONJ
ejpam-6425	756	17	used	use	VERB
ejpam-6425	756	18	44	44	NUM
ejpam-6425	756	19	used[num	used[num	ADJ
ejpam-6425	756	20	1	1	NUM
ejpam-6425	757	1	]	]	PUNCT
ejpam-6425	757	2	=	=	PUNCT
ejpam-6425	757	3	true	true	ADJ
ejpam-6425	757	4	45	45	NUM
ejpam-6425	757	5	perm.append(num	perm.append(num	ADJ
ejpam-6425	757	6	)	)	PUNCT
ejpam-6425	757	7	46	46	NUM
ejpam-6425	758	1	47	47	NUM
ejpam-6425	758	2	#	#	NOUN
ejpam-6425	758	3	recursively	recursively	ADV
ejpam-6425	758	4	try	try	VERB
ejpam-6425	758	5	the	the	DET
ejpam-6425	758	6	next	next	ADJ
ejpam-6425	758	7	number	number	NOUN
ejpam-6425	758	8	48	48	NUM
ejpam-6425	758	9	backtrack(perm	backtrack(perm	NOUN
ejpam-6425	758	10	,	,	PUNCT
ejpam-6425	758	11	used	use	VERB
ejpam-6425	758	12	)	)	PUNCT
ejpam-6425	758	13	49	49	NUM
ejpam-6425	758	14	50	50	NUM
ejpam-6425	758	15	#	#	NOUN
ejpam-6425	758	16	undo	undo	VERB
ejpam-6425	758	17	the	the	DET
ejpam-6425	758	18	choice	choice	NOUN
ejpam-6425	758	19	(	(	PUNCT
ejpam-6425	758	20	backtrack	backtrack	NOUN
ejpam-6425	758	21	)	)	PUNCT
ejpam-6425	758	22	51	51	NUM
ejpam-6425	758	23	perm.pop	perm.pop	NOUN
ejpam-6425	758	24	(	(	PUNCT
ejpam-6425	758	25	)	)	PUNCT
ejpam-6425	758	26	52	52	NUM
ejpam-6425	758	27	used[num	used[num	ADJ
ejpam-6425	758	28	1	1	NUM
ejpam-6425	758	29	]	]	PUNCT
ejpam-6425	758	30	=	=	SYM
ejpam-6425	758	31	false	false	ADJ
ejpam-6425	758	32	53	53	NUM
ejpam-6425	758	33	54	54	NUM
ejpam-6425	758	34	#	#	NOUN
ejpam-6425	758	35	start	start	VERB
ejpam-6425	758	36	the	the	DET
ejpam-6425	758	37	backtracking	backtrack	VERB
ejpam-6425	758	38	process	process	NOUN
ejpam-6425	758	39	55	55	NUM
ejpam-6425	758	40	def	def	ADJ
ejpam-6425	758	41	find_valid_permutations	find_valid_permutation	NOUN
ejpam-6425	758	42	(	(	PUNCT
ejpam-6425	758	43	):	):	PUNCT
ejpam-6425	758	44	56	56	NUM
ejpam-6425	758	45	used	use	VERB
ejpam-6425	758	46	=	=	X
ejpam-6425	759	1	[	[	X
ejpam-6425	759	2	false	false	X
ejpam-6425	759	3	]	]	X
ejpam-6425	759	4	*	*	SYM
ejpam-6425	759	5	54	54	NUM
ejpam-6425	759	6	#	#	NOUN
ejpam-6425	759	7	keeps	keep	VERB
ejpam-6425	759	8	track	track	NOUN
ejpam-6425	759	9	of	of	ADP
ejpam-6425	759	10	used	used	ADJ
ejpam-6425	759	11	numbers	number	NOUN
ejpam-6425	759	12	57	57	NUM
ejpam-6425	759	13	backtrack	backtrack	NOUN
ejpam-6425	759	14	(	(	PUNCT
ejpam-6425	759	15	[	[	X
ejpam-6425	759	16	]	]	X
ejpam-6425	759	17	,	,	PUNCT
ejpam-6425	759	18	used	use	VERB
ejpam-6425	759	19	)	)	PUNCT
ejpam-6425	759	20	58	58	NUM
ejpam-6425	759	21	59	59	NUM
ejpam-6425	759	22	#	#	NOUN
ejpam-6425	759	23	start	start	NOUN
ejpam-6425	759	24	finding	find	VERB
ejpam-6425	759	25	the	the	DET
ejpam-6425	759	26	valid	valid	ADJ
ejpam-6425	759	27	permutations	permutation	NOUN
ejpam-6425	759	28	60	60	NUM
ejpam-6425	759	29	find_valid_permutations	find_valid_permutation	NOUN
ejpam-6425	759	30	(	(	PUNCT
ejpam-6425	759	31	)	)	PUNCT
ejpam-6425	759	32	provided	provide	VERB
ejpam-6425	759	33	below	below	ADV
ejpam-6425	759	34	,	,	PUNCT
ejpam-6425	759	35	is	be	AUX
ejpam-6425	759	36	the	the	DET
ejpam-6425	759	37	python	python	NOUN
ejpam-6425	759	38	program	program	NOUN
ejpam-6425	759	39	used	use	VERB
ejpam-6425	759	40	to	to	PART
ejpam-6425	759	41	generate	generate	VERB
ejpam-6425	759	42	an	an	DET
ejpam-6425	759	43	actual	actual	ADJ
ejpam-6425	759	44	edge	edge	NOUN
ejpam-6425	759	45	-	-	PUNCT
ejpam-6425	759	46	graceful	graceful	NOUN
ejpam-6425	759	47	labeling	labeling	NOUN
ejpam-6425	759	48	of	of	ADP
ejpam-6425	759	49	s(lww3	s(lww3	NOUN
ejpam-6425	759	50	)	)	PUNCT
ejpam-6425	759	51	.	.	PUNCT
ejpam-6425	760	1	1	1	NUM
ejpam-6425	760	2	#	#	SYM
ejpam-6425	760	3	python	python	NOUN
ejpam-6425	760	4	program	program	NOUN
ejpam-6425	760	5	for	for	ADP
ejpam-6425	760	6	s(lww_3	s(lww_3	ADJ
ejpam-6425	760	7	)	)	PUNCT
ejpam-6425	760	8	2	2	NUM
ejpam-6425	760	9	from	from	ADP
ejpam-6425	760	10	itertools	itertools	PROPN
ejpam-6425	760	11	import	import	NOUN
ejpam-6425	760	12	permutations	permutation	NOUN
ejpam-6425	760	13	3	3	NUM
ejpam-6425	760	14	4	4	NUM
ejpam-6425	760	15	numbers	number	NOUN
ejpam-6425	760	16	=	=	PUNCT
ejpam-6425	761	1	[	[	X
ejpam-6425	761	2	1	1	NUM
ejpam-6425	761	3	,	,	PUNCT
ejpam-6425	761	4	2	2	NUM
ejpam-6425	761	5	,	,	PUNCT
ejpam-6425	761	6	3	3	NUM
ejpam-6425	761	7	,	,	PUNCT
ejpam-6425	761	8	4	4	NUM
ejpam-6425	761	9	,	,	PUNCT
ejpam-6425	761	10	5	5	NUM
ejpam-6425	761	11	,	,	PUNCT
ejpam-6425	761	12	6	6	NUM
ejpam-6425	761	13	,	,	PUNCT
ejpam-6425	761	14	7	7	NUM
ejpam-6425	761	15	,	,	PUNCT
ejpam-6425	761	16	8	8	NUM
ejpam-6425	761	17	,	,	PUNCT
ejpam-6425	761	18	9	9	NUM
ejpam-6425	761	19	,	,	PUNCT
ejpam-6425	761	20	10	10	NUM
ejpam-6425	761	21	,	,	PUNCT
ejpam-6425	761	22	11	11	NUM
ejpam-6425	761	23	,	,	PUNCT
ejpam-6425	761	24	12	12	NUM
ejpam-6425	761	25	,	,	PUNCT
ejpam-6425	761	26	13	13	NUM
ejpam-6425	761	27	,	,	PUNCT
ejpam-6425	761	28	14	14	NUM
ejpam-6425	761	29	,	,	PUNCT
ejpam-6425	761	30	15	15	NUM
ejpam-6425	761	31	,	,	PUNCT
ejpam-6425	761	32	16	16	NUM
ejpam-6425	761	33	,	,	PUNCT
ejpam-6425	761	34	17	17	NUM
ejpam-6425	761	35	,	,	PUNCT
ejpam-6425	761	36	18	18	NUM
ejpam-6425	761	37	,	,	PUNCT
ejpam-6425	761	38	19	19	NUM
ejpam-6425	761	39	,	,	PUNCT
ejpam-6425	761	40	20	20	NUM
ejpam-6425	761	41	,	,	PUNCT
ejpam-6425	761	42	21	21	NUM
ejpam-6425	761	43	,	,	PUNCT
ejpam-6425	761	44	22	22	NUM
ejpam-6425	761	45	,	,	PUNCT
ejpam-6425	761	46	23	23	NUM
ejpam-6425	761	47	,	,	PUNCT
ejpam-6425	761	48	24	24	NUM
ejpam-6425	761	49	,	,	PUNCT
ejpam-6425	761	50	25	25	NUM
ejpam-6425	761	51	,	,	PUNCT
ejpam-6425	761	52	26	26	NUM
ejpam-6425	761	53	,	,	PUNCT
ejpam-6425	761	54	27	27	NUM
ejpam-6425	761	55	,	,	PUNCT
ejpam-6425	761	56	28	28	NUM
ejpam-6425	761	57	,	,	PUNCT
ejpam-6425	761	58	29	29	NUM
ejpam-6425	761	59	,	,	PUNCT
ejpam-6425	761	60	30	30	NUM
ejpam-6425	761	61	,	,	PUNCT
ejpam-6425	761	62	31	31	NUM
ejpam-6425	761	63	,	,	PUNCT
ejpam-6425	761	64	32	32	NUM
ejpam-6425	761	65	,	,	PUNCT
ejpam-6425	761	66	33	33	NUM
ejpam-6425	761	67	,	,	PUNCT
ejpam-6425	761	68	34	34	NUM
ejpam-6425	761	69	,	,	PUNCT
ejpam-6425	761	70	35	35	NUM
ejpam-6425	761	71	,	,	PUNCT
ejpam-6425	761	72	36	36	NUM
ejpam-6425	761	73	,	,	PUNCT
ejpam-6425	761	74	37	37	NUM
ejpam-6425	761	75	,	,	PUNCT
ejpam-6425	761	76	38	38	NUM
ejpam-6425	761	77	,	,	PUNCT
ejpam-6425	761	78	39	39	NUM
ejpam-6425	761	79	,	,	PUNCT
ejpam-6425	761	80	40	40	NUM
ejpam-6425	761	81	,	,	PUNCT
ejpam-6425	761	82	41	41	NUM
ejpam-6425	761	83	,	,	PUNCT
ejpam-6425	761	84	42	42	NUM
ejpam-6425	761	85	,	,	PUNCT
ejpam-6425	761	86	43	43	NUM
ejpam-6425	761	87	,	,	PUNCT
ejpam-6425	761	88	44	44	NUM
ejpam-6425	761	89	,	,	PUNCT
ejpam-6425	761	90	45	45	NUM
ejpam-6425	761	91	,	,	PUNCT
ejpam-6425	761	92	46	46	NUM
ejpam-6425	761	93	,	,	PUNCT
ejpam-6425	761	94	47	47	NUM
ejpam-6425	761	95	,	,	PUNCT
ejpam-6425	761	96	48	48	NUM
ejpam-6425	761	97	,	,	PUNCT
ejpam-6425	761	98	49	49	NUM
ejpam-6425	761	99	,	,	PUNCT
ejpam-6425	761	100	50	50	NUM
ejpam-6425	761	101	,	,	PUNCT
ejpam-6425	761	102	51	51	NUM
ejpam-6425	761	103	,	,	PUNCT
ejpam-6425	761	104	52	52	NUM
ejpam-6425	761	105	,	,	PUNCT
ejpam-6425	761	106	53	53	NUM
ejpam-6425	761	107	,	,	PUNCT
ejpam-6425	761	108	54	54	NUM
ejpam-6425	761	109	]	]	SYM
ejpam-6425	761	110	5	5	NUM
ejpam-6425	761	111	6	6	NUM
ejpam-6425	761	112	#	#	NOUN
ejpam-6425	761	113	function	function	NOUN
ejpam-6425	761	114	to	to	PART
ejpam-6425	761	115	compute	compute	VERB
ejpam-6425	761	116	u0	u0	ADJ
ejpam-6425	761	117	to	to	PART
ejpam-6425	761	118	v1	v1	VERB
ejpam-6425	761	119	7	7	NUM
ejpam-6425	761	120	def	def	ADJ
ejpam-6425	761	121	compute_v_values(v1w1	compute_v_values(v1w1	NOUN
ejpam-6425	761	122	,	,	PUNCT
ejpam-6425	761	123	u3v1	u3v1	ADJ
ejpam-6425	761	124	,	,	PUNCT
ejpam-6425	761	125	w3u3	w3u3	PROPN
ejpam-6425	761	126	,	,	PUNCT
ejpam-6425	761	127	v3w3	v3w3	NOUN
ejpam-6425	761	128	,	,	PUNCT
ejpam-6425	761	129	u2v3	u2v3	NOUN
ejpam-6425	761	130	,	,	PUNCT
ejpam-6425	761	131	w2u2	w2u2	NOUN
ejpam-6425	761	132	,	,	PUNCT
ejpam-6425	761	133	v2w2	v2w2	X
ejpam-6425	761	134	,	,	PUNCT
ejpam-6425	761	135	u1v2	u1v2	NOUN
ejpam-6425	761	136	,	,	PUNCT
ejpam-6425	761	137	w1u1	w1u1	NOUN
ejpam-6425	761	138	,	,	PUNCT
ejpam-6425	761	139	u0v1	u0v1	PROPN
ejpam-6425	761	140	,	,	PUNCT
ejpam-6425	761	141	u0u3	u0u3	PROPN
ejpam-6425	761	142	,	,	PUNCT
ejpam-6425	761	143	u0v3	u0v3	X
ejpam-6425	761	144	,	,	PUNCT
ejpam-6425	761	145	u0u2	u0u2	SYM
ejpam-6425	761	146	,	,	PUNCT
ejpam-6425	761	147	u0v2	u0v2	NOUN
ejpam-6425	761	148	,	,	PUNCT
ejpam-6425	761	149	u0u1	u0u1	NOUN
ejpam-6425	761	150	,	,	PUNCT
ejpam-6425	761	151	w1u1	w1u1	NOUN
ejpam-6425	761	152	,	,	PUNCT
ejpam-6425	761	153	w1u3	w1u3	NOUN
ejpam-6425	761	154	,	,	PUNCT
ejpam-6425	761	155	v1w1	v1w1	PROPN
ejpam-6425	761	156	,	,	PUNCT
ejpam-6425	761	157	v1u3	v1u3	X
ejpam-6425	761	158	,	,	PUNCT
ejpam-6425	761	159	u3w1	u3w1	NOUN
ejpam-6425	761	160	,	,	PUNCT
ejpam-6425	761	161	u3v1	u3v1	ADJ
ejpam-6425	761	162	,	,	PUNCT
ejpam-6425	761	163	u3w3	u3w3	NOUN
ejpam-6425	761	164	,	,	PUNCT
ejpam-6425	761	165	w3u3	w3u3	PROPN
ejpam-6425	761	166	,	,	PUNCT
ejpam-6425	761	167	w3u2	w3u2	PRON
ejpam-6425	761	168	,	,	PUNCT
ejpam-6425	761	169	v3w3	v3w3	NOUN
ejpam-6425	761	170	,	,	PUNCT
ejpam-6425	761	171	v3u2	v3u2	X
ejpam-6425	761	172	,	,	PUNCT
ejpam-6425	761	173	u2w3	u2w3	PROPN
ejpam-6425	761	174	,	,	PUNCT
ejpam-6425	761	175	u2v3	u2v3	NOUN
ejpam-6425	761	176	,	,	PUNCT
ejpam-6425	761	177	u2w2	u2w2	ADJ
ejpam-6425	761	178	,	,	PUNCT
ejpam-6425	761	179	w2u2	w2u2	NOUN
ejpam-6425	761	180	,	,	PUNCT
ejpam-6425	761	181	w2u1	w2u1	PROPN
ejpam-6425	761	182	,	,	PUNCT
ejpam-6425	761	183	v2w2	v2w2	X
ejpam-6425	761	184	,	,	PUNCT
ejpam-6425	761	185	v2u1	v2u1	X
ejpam-6425	761	186	,	,	PUNCT
ejpam-6425	761	187	u1w2	u1w2	NOUN
ejpam-6425	761	188	,	,	PUNCT
ejpam-6425	761	189	u1v2	u1v2	NOUN
ejpam-6425	761	190	,	,	PUNCT
ejpam-6425	761	191	u1w1	u1w1	PROPN
ejpam-6425	761	192	,	,	PUNCT
ejpam-6425	761	193	u0u1	u0u1	NOUN
ejpam-6425	761	194	,	,	PUNCT
ejpam-6425	761	195	u0v2	u0v2	NOUN
ejpam-6425	761	196	,	,	PUNCT
ejpam-6425	761	197	u0u2	u0u2	PRON
ejpam-6425	761	198	,	,	PUNCT
ejpam-6425	761	199	u0v3	u0v3	X
ejpam-6425	761	200	,	,	PUNCT
ejpam-6425	761	201	u0u3	u0u3	PROPN
ejpam-6425	761	202	,	,	PUNCT
ejpam-6425	761	203	u0v1	u0v1	X
ejpam-6425	761	204	,	,	PUNCT
ejpam-6425	761	205	w2u1	w2u1	PROPN
ejpam-6425	761	206	,	,	PUNCT
ejpam-6425	761	207	w3u2	w3u2	PRON
ejpam-6425	761	208	,	,	PUNCT
ejpam-6425	761	209	w1u3	w1u3	NOUN
ejpam-6425	761	210	,	,	PUNCT
ejpam-6425	761	211	u1u0	u1u0	PROPN
ejpam-6425	761	212	,	,	PUNCT
ejpam-6425	761	213	v2u0	v2u0	NOUN
ejpam-6425	761	214	,	,	PUNCT
ejpam-6425	761	215	w2v2	w2v2	PROPN
ejpam-6425	761	216	,	,	PUNCT
ejpam-6425	761	217	u2u0	u2u0	PROPN
ejpam-6425	761	218	,	,	PUNCT
ejpam-6425	761	219	v3u0	v3u0	X
ejpam-6425	761	220	,	,	PUNCT
ejpam-6425	761	221	w3v3	w3v3	NOUN
ejpam-6425	761	222	,	,	PUNCT
ejpam-6425	761	223	u3u0	u3u0	NOUN
ejpam-6425	761	224	,	,	PUNCT
ejpam-6425	761	225	v1u0	v1u0	PROPN
ejpam-6425	761	226	,	,	PUNCT
ejpam-6425	761	227	w1v1	w1v1	X
ejpam-6425	761	228	):	):	PUNCT
ejpam-6425	761	229	8	8	NUM
ejpam-6425	761	230	u0	u0	NOUN
ejpam-6425	761	231	=	=	PUNCT
ejpam-6425	761	232	(	(	PUNCT
ejpam-6425	761	233	u0v1	u0v1	X
ejpam-6425	762	1	+	+	CCONJ
ejpam-6425	762	2	u0u1	u0u1	PROPN
ejpam-6425	762	3	+	+	X
ejpam-6425	762	4	u0v2	u0v2	X
ejpam-6425	762	5	+	+	NUM
ejpam-6425	762	6	u0u2	u0u2	PUNCT
ejpam-6425	762	7	+	+	NUM
ejpam-6425	762	8	u0v3	u0v3	ADJ
ejpam-6425	762	9	+	+	NUM
ejpam-6425	762	10	u0u3	u0u3	PROPN
ejpam-6425	762	11	+	+	CCONJ
ejpam-6425	762	12	v1u0	v1u0	PROPN
ejpam-6425	762	13	+	+	CCONJ
ejpam-6425	762	14	u1u0	u1u0	PROPN
ejpam-6425	762	15	+	+	ADJ
ejpam-6425	762	16	v2u0	v2u0	PROPN
ejpam-6425	762	17	+	+	NUM
ejpam-6425	762	18	u2u0	u2u0	X
ejpam-6425	762	19	+	+	CCONJ
ejpam-6425	762	20	v3u0	v3u0	X
ejpam-6425	762	21	+	+	CCONJ
ejpam-6425	762	22	u3u0	u3u0	X
ejpam-6425	762	23	)	)	PUNCT
ejpam-6425	762	24	%	%	NOUN
ejpam-6425	762	25	20	20	NUM
ejpam-6425	762	26	9	9	NUM
ejpam-6425	762	27	w1	w1	NOUN
ejpam-6425	762	28	=	=	SYM
ejpam-6425	762	29	(	(	PUNCT
ejpam-6425	762	30	v1w1	v1w1	X
ejpam-6425	762	31	+	+	X
ejpam-6425	762	32	w1u1	w1u1	X
ejpam-6425	762	33	+	+	NUM
ejpam-6425	762	34	w1u3	w1u3	NOUN
ejpam-6425	762	35	+	+	CCONJ
ejpam-6425	762	36	v1w1	v1w1	PROPN
ejpam-6425	763	1	+	+	CCONJ
ejpam-6425	763	2	u1w1	u1w1	ADJ
ejpam-6425	763	3	+	+	NUM
ejpam-6425	763	4	u3w1	u3w1	NOUN
ejpam-6425	763	5	)	)	PUNCT
ejpam-6425	763	6	%	%	NOUN
ejpam-6425	763	7	20	20	NUM
ejpam-6425	763	8	10	10	NUM
ejpam-6425	763	9	u1	u1	NOUN
ejpam-6425	763	10	=	=	SYM
ejpam-6425	763	11	(	(	PUNCT
ejpam-6425	763	12	w1u1	w1u1	X
ejpam-6425	764	1	+	+	SYM
ejpam-6425	764	2	u1v2	u1v2	PUNCT
ejpam-6425	764	3	+	+	CCONJ
ejpam-6425	764	4	u0u1	u0u1	PUNCT
ejpam-6425	765	1	+	+	NUM
ejpam-6425	765	2	w2u1	w2u1	PUNCT
ejpam-6425	765	3	+	+	PUNCT
ejpam-6425	765	4	w1u1	w1u1	X
ejpam-6425	765	5	+	+	CCONJ
ejpam-6425	765	6	v2u1	v2u1	X
ejpam-6425	766	1	+	+	CCONJ
ejpam-6425	766	2	u0u1	u0u1	PUNCT
ejpam-6425	766	3	+	+	NUM
ejpam-6425	766	4	w2u1	w2u1	X
ejpam-6425	766	5	)	)	PUNCT
ejpam-6425	766	6	%	%	NOUN
ejpam-6425	766	7	20	20	NUM
ejpam-6425	766	8	11	11	NUM
ejpam-6425	766	9	v2	v2	NOUN
ejpam-6425	766	10	=	=	SYM
ejpam-6425	766	11	(	(	PUNCT
ejpam-6425	766	12	u1v2	u1v2	X
ejpam-6425	766	13	+	+	X
ejpam-6425	766	14	v2w2	v2w2	X
ejpam-6425	766	15	+	+	X
ejpam-6425	766	16	u0v2	u0v2	X
ejpam-6425	766	17	+	+	SYM
ejpam-6425	766	18	u1v2	u1v2	PUNCT
ejpam-6425	766	19	+	+	CCONJ
ejpam-6425	766	20	w2v2	w2v2	X
ejpam-6425	766	21	+	+	CCONJ
ejpam-6425	766	22	u0v2	u0v2	NOUN
ejpam-6425	766	23	)	)	PUNCT
ejpam-6425	766	24	%	%	NOUN
ejpam-6425	766	25	20	20	NUM
ejpam-6425	766	26	12	12	NUM
ejpam-6425	766	27	w2	w2	NOUN
ejpam-6425	766	28	=	=	SYM
ejpam-6425	766	29	(	(	PUNCT
ejpam-6425	766	30	v2w2	v2w2	X
ejpam-6425	766	31	+	+	PUNCT
ejpam-6425	766	32	w2u2	w2u2	X
ejpam-6425	767	1	+	+	CCONJ
ejpam-6425	767	2	w2u1	w2u1	PUNCT
ejpam-6425	767	3	+	+	CCONJ
ejpam-6425	767	4	v2w2	v2w2	X
ejpam-6425	767	5	+	+	CCONJ
ejpam-6425	767	6	u2w2	u2w2	ADJ
ejpam-6425	767	7	+	+	NUM
ejpam-6425	767	8	u1w2	u1w2	NOUN
ejpam-6425	767	9	)	)	PUNCT
ejpam-6425	767	10	%	%	NOUN
ejpam-6425	767	11	20	20	NUM
ejpam-6425	767	12	a.	a.	PROPN
ejpam-6425	767	13	d.	d.	PROPN
ejpam-6425	767	14	c.	c.	PROPN
ejpam-6425	767	15	angel	angel	PROPN
ejpam-6425	767	16	et	et	PROPN
ejpam-6425	767	17	al	al	PROPN
ejpam-6425	767	18	.	.	PUNCT
ejpam-6425	767	19	/	/	SYM
ejpam-6425	767	20	eur	eur	PROPN
ejpam-6425	767	21	.	.	PUNCT
ejpam-6425	768	1	j.	j.	PROPN
ejpam-6425	768	2	pure	pure	PROPN
ejpam-6425	768	3	appl	appl	PROPN
ejpam-6425	768	4	.	.	PROPN
ejpam-6425	768	5	math	math	PROPN
ejpam-6425	768	6	,	,	PUNCT
ejpam-6425	768	7	18	18	NUM
ejpam-6425	768	8	(	(	PUNCT
ejpam-6425	768	9	4	4	NUM
ejpam-6425	768	10	)	)	PUNCT
ejpam-6425	768	11	(	(	PUNCT
ejpam-6425	768	12	2025	2025	NUM
ejpam-6425	768	13	)	)	PUNCT
ejpam-6425	768	14	,	,	PUNCT
ejpam-6425	768	15	6425	6425	NUM
ejpam-6425	768	16	32	32	NUM
ejpam-6425	768	17	of	of	ADP
ejpam-6425	768	18	33	33	NUM
ejpam-6425	768	19	13	13	NUM
ejpam-6425	768	20	u2	u2	NOUN
ejpam-6425	768	21	=	=	PUNCT
ejpam-6425	768	22	(	(	PUNCT
ejpam-6425	768	23	w2u2	w2u2	X
ejpam-6425	768	24	+	+	CCONJ
ejpam-6425	768	25	u2v3	u2v3	X
ejpam-6425	768	26	+	+	X
ejpam-6425	768	27	u0u2	u0u2	PUNCT
ejpam-6425	768	28	+	+	NUM
ejpam-6425	768	29	w3u2	w3u2	PRON
ejpam-6425	768	30	+	+	PUNCT
ejpam-6425	768	31	w2u2	w2u2	X
ejpam-6425	768	32	+	+	CCONJ
ejpam-6425	768	33	v3u2	v3u2	VERB
ejpam-6425	769	1	+	+	CCONJ
ejpam-6425	769	2	u0u2	u0u2	PUNCT
ejpam-6425	769	3	+	+	NUM
ejpam-6425	769	4	w3u2	w3u2	X
ejpam-6425	769	5	)	)	PUNCT
ejpam-6425	769	6	%	%	NOUN
ejpam-6425	769	7	20	20	NUM
ejpam-6425	769	8	14	14	NUM
ejpam-6425	769	9	v3	v3	PROPN
ejpam-6425	769	10	=	=	SYM
ejpam-6425	769	11	(	(	PUNCT
ejpam-6425	769	12	u2v3	u2v3	X
ejpam-6425	769	13	+	+	NOUN
ejpam-6425	769	14	v3w3	v3w3	X
ejpam-6425	770	1	+	+	CCONJ
ejpam-6425	770	2	u0v3	u0v3	X
ejpam-6425	770	3	+	+	CCONJ
ejpam-6425	770	4	u2v3	u2v3	X
ejpam-6425	770	5	+	+	CCONJ
ejpam-6425	770	6	w3v3	w3v3	X
ejpam-6425	770	7	+	+	CCONJ
ejpam-6425	770	8	u0v3	u0v3	ADJ
ejpam-6425	770	9	)	)	PUNCT
ejpam-6425	770	10	%	%	NOUN
ejpam-6425	770	11	20	20	NUM
ejpam-6425	770	12	15	15	NUM
ejpam-6425	770	13	w3	w3	NOUN
ejpam-6425	770	14	=	=	SYM
ejpam-6425	770	15	(	(	PUNCT
ejpam-6425	770	16	v3w3	v3w3	X
ejpam-6425	770	17	+	+	CCONJ
ejpam-6425	770	18	w3u3	w3u3	X
ejpam-6425	770	19	+	+	CCONJ
ejpam-6425	770	20	w3u2	w3u2	PRON
ejpam-6425	771	1	+	+	PUNCT
ejpam-6425	771	2	v3w3	v3w3	X
ejpam-6425	771	3	+	+	CCONJ
ejpam-6425	771	4	u3w3	u3w3	VERB
ejpam-6425	771	5	+	+	CCONJ
ejpam-6425	771	6	u2w3	u2w3	NOUN
ejpam-6425	771	7	)	)	PUNCT
ejpam-6425	771	8	%	%	NOUN
ejpam-6425	771	9	20	20	NUM
ejpam-6425	771	10	16	16	NUM
ejpam-6425	771	11	u3	u3	NOUN
ejpam-6425	771	12	=	=	SYM
ejpam-6425	771	13	(	(	PUNCT
ejpam-6425	771	14	w3u3	w3u3	X
ejpam-6425	772	1	+	+	SYM
ejpam-6425	772	2	u3v1	u3v1	PROPN
ejpam-6425	772	3	+	+	X
ejpam-6425	772	4	u0u3	u0u3	PROPN
ejpam-6425	772	5	+	+	X
ejpam-6425	772	6	w1u3	w1u3	NOUN
ejpam-6425	772	7	+	+	CCONJ
ejpam-6425	772	8	w3u3	w3u3	X
ejpam-6425	772	9	+	+	CCONJ
ejpam-6425	772	10	v1u3	v1u3	X
ejpam-6425	772	11	+	+	X
ejpam-6425	772	12	u0u3	u0u3	PROPN
ejpam-6425	772	13	+	+	NUM
ejpam-6425	772	14	w1u3	w1u3	NOUN
ejpam-6425	772	15	)	)	PUNCT
ejpam-6425	772	16	%	%	NOUN
ejpam-6425	772	17	20	20	NUM
ejpam-6425	772	18	17	17	NUM
ejpam-6425	772	19	v1	v1	NOUN
ejpam-6425	772	20	=	=	SYM
ejpam-6425	772	21	(	(	PUNCT
ejpam-6425	772	22	u3v1	u3v1	PROPN
ejpam-6425	772	23	+	+	CCONJ
ejpam-6425	772	24	v1w1	v1w1	PROPN
ejpam-6425	773	1	+	+	CCONJ
ejpam-6425	773	2	u0v1	u0v1	ADJ
ejpam-6425	773	3	+	+	CCONJ
ejpam-6425	773	4	u3v1	u3v1	PROPN
ejpam-6425	773	5	+	+	CCONJ
ejpam-6425	773	6	w1v1	w1v1	X
ejpam-6425	773	7	+	+	NUM
ejpam-6425	773	8	u0v1	u0v1	NOUN
ejpam-6425	773	9	)	)	PUNCT
ejpam-6425	773	10	%	%	NOUN
ejpam-6425	773	11	20	20	NUM
ejpam-6425	773	12	18	18	NUM
ejpam-6425	773	13	u0	u0	NOUN
ejpam-6425	773	14	=	=	PUNCT
ejpam-6425	773	15	(	(	PUNCT
ejpam-6425	773	16	u0v1	u0v1	X
ejpam-6425	774	1	+	+	CCONJ
ejpam-6425	774	2	u0u3	u0u3	PROPN
ejpam-6425	775	1	+	+	X
ejpam-6425	775	2	u0v3	u0v3	X
ejpam-6425	775	3	+	+	NOUN
ejpam-6425	775	4	u0u2	u0u2	VERB
ejpam-6425	775	5	+	+	NUM
ejpam-6425	775	6	u0v2	u0v2	ADJ
ejpam-6425	775	7	+	+	NUM
ejpam-6425	775	8	u0u1	u0u1	NOUN
ejpam-6425	775	9	)	)	PUNCT
ejpam-6425	775	10	%	%	NOUN
ejpam-6425	775	11	20	20	NUM
ejpam-6425	775	12	19	19	NUM
ejpam-6425	775	13	w1	w1	NOUN
ejpam-6425	775	14	=	=	SYM
ejpam-6425	775	15	(	(	PUNCT
ejpam-6425	775	16	w1u1	w1u1	X
ejpam-6425	775	17	+	+	CCONJ
ejpam-6425	775	18	w1v1	w1v1	X
ejpam-6425	775	19	+	+	NUM
ejpam-6425	775	20	w1u3	w1u3	NOUN
ejpam-6425	775	21	)	)	PUNCT
ejpam-6425	775	22	%	%	NOUN
ejpam-6425	775	23	20	20	NUM
ejpam-6425	775	24	20	20	NUM
ejpam-6425	775	25	u1	u1	NOUN
ejpam-6425	775	26	=	=	SYM
ejpam-6425	775	27	(	(	PUNCT
ejpam-6425	775	28	u1w1	u1w1	X
ejpam-6425	775	29	+	+	PUNCT
ejpam-6425	775	30	u1u0	u1u0	PROPN
ejpam-6425	775	31	+	+	NUM
ejpam-6425	775	32	u1v2	u1v2	ADJ
ejpam-6425	775	33	+	+	NUM
ejpam-6425	775	34	u1w2	u1w2	NOUN
ejpam-6425	775	35	)	)	PUNCT
ejpam-6425	775	36	%	%	NOUN
ejpam-6425	775	37	20	20	NUM
ejpam-6425	775	38	21	21	NUM
ejpam-6425	775	39	v2	v2	NOUN
ejpam-6425	775	40	=	=	SYM
ejpam-6425	775	41	(	(	PUNCT
ejpam-6425	775	42	v2u1	v2u1	X
ejpam-6425	775	43	+	+	ADP
ejpam-6425	775	44	v2u0	v2u0	X
ejpam-6425	775	45	+	+	NUM
ejpam-6425	775	46	v2w2	v2w2	NOUN
ejpam-6425	775	47	)	)	PUNCT
ejpam-6425	775	48	%	%	NOUN
ejpam-6425	775	49	20	20	NUM
ejpam-6425	775	50	22	22	NUM
ejpam-6425	775	51	w2	w2	NOUN
ejpam-6425	775	52	=	=	PUNCT
ejpam-6425	775	53	(	(	PUNCT
ejpam-6425	775	54	w2u1	w2u1	X
ejpam-6425	775	55	+	+	PUNCT
ejpam-6425	775	56	w2v2	w2v2	X
ejpam-6425	775	57	+	+	CCONJ
ejpam-6425	775	58	w2u2	w2u2	NOUN
ejpam-6425	775	59	)	)	PUNCT
ejpam-6425	775	60	%	%	NOUN
ejpam-6425	775	61	20	20	NUM
ejpam-6425	775	62	23	23	NUM
ejpam-6425	775	63	u2	u2	NOUN
ejpam-6425	775	64	=	=	PUNCT
ejpam-6425	775	65	(	(	PUNCT
ejpam-6425	775	66	u2w2	u2w2	X
ejpam-6425	775	67	+	+	NUM
ejpam-6425	775	68	u2u0	u2u0	X
ejpam-6425	775	69	+	+	X
ejpam-6425	775	70	u2v3	u2v3	PROPN
ejpam-6425	775	71	+	+	ADJ
ejpam-6425	775	72	u2w3	u2w3	NOUN
ejpam-6425	775	73	)	)	PUNCT
ejpam-6425	775	74	%	%	NOUN
ejpam-6425	775	75	20	20	NUM
ejpam-6425	775	76	24	24	NUM
ejpam-6425	775	77	v3	v3	PROPN
ejpam-6425	775	78	=	=	SYM
ejpam-6425	775	79	(	(	PUNCT
ejpam-6425	775	80	v3u2	v3u2	X
ejpam-6425	775	81	+	+	CCONJ
ejpam-6425	775	82	v3u0	v3u0	X
ejpam-6425	775	83	+	+	CCONJ
ejpam-6425	775	84	v3w3	v3w3	NOUN
ejpam-6425	775	85	)	)	PUNCT
ejpam-6425	775	86	%	%	NOUN
ejpam-6425	775	87	20	20	NUM
ejpam-6425	775	88	25	25	NUM
ejpam-6425	775	89	w3	w3	NOUN
ejpam-6425	775	90	=	=	SYM
ejpam-6425	775	91	(	(	PUNCT
ejpam-6425	775	92	w3u2	w3u2	X
ejpam-6425	775	93	+	+	X
ejpam-6425	775	94	w3v3	w3v3	X
ejpam-6425	775	95	+	+	CCONJ
ejpam-6425	775	96	w3u3	w3u3	NOUN
ejpam-6425	775	97	)	)	PUNCT
ejpam-6425	775	98	%	%	NOUN
ejpam-6425	775	99	20	20	NUM
ejpam-6425	775	100	26	26	NUM
ejpam-6425	775	101	u3	u3	NOUN
ejpam-6425	775	102	=	=	SYM
ejpam-6425	775	103	(	(	PUNCT
ejpam-6425	775	104	u3w3	u3w3	VERB
ejpam-6425	775	105	+	+	CCONJ
ejpam-6425	775	106	u3u0	u3u0	ADJ
ejpam-6425	775	107	+	+	X
ejpam-6425	775	108	u3v1	u3v1	X
ejpam-6425	775	109	+	+	ADJ
ejpam-6425	775	110	u3w1	u3w1	NOUN
ejpam-6425	775	111	)	)	PUNCT
ejpam-6425	775	112	%	%	NOUN
ejpam-6425	775	113	20	20	NUM
ejpam-6425	775	114	27	27	NUM
ejpam-6425	775	115	v1	v1	NOUN
ejpam-6425	775	116	=	=	SYM
ejpam-6425	775	117	(	(	PUNCT
ejpam-6425	775	118	v1u3	v1u3	X
ejpam-6425	775	119	+	+	X
ejpam-6425	775	120	v1u0	v1u0	PROPN
ejpam-6425	776	1	+	+	CCONJ
ejpam-6425	776	2	v1w1	v1w1	NOUN
ejpam-6425	776	3	)	)	PUNCT
ejpam-6425	776	4	%	%	NOUN
ejpam-6425	776	5	20	20	NUM
ejpam-6425	776	6	28	28	NUM
ejpam-6425	776	7	return	return	NOUN
ejpam-6425	776	8	u0	u0	ADJ
ejpam-6425	776	9	,	,	PUNCT
ejpam-6425	776	10	w1	w1	NOUN
ejpam-6425	776	11	,	,	PUNCT
ejpam-6425	776	12	u1	u1	NOUN
ejpam-6425	776	13	,	,	PUNCT
ejpam-6425	776	14	v2	v2	NOUN
ejpam-6425	776	15	,	,	PUNCT
ejpam-6425	776	16	w2	w2	NOUN
ejpam-6425	776	17	,	,	PUNCT
ejpam-6425	776	18	u2	u2	PROPN
ejpam-6425	776	19	,	,	PUNCT
ejpam-6425	776	20	v3	v3	PROPN
ejpam-6425	776	21	,	,	PUNCT
ejpam-6425	776	22	w3	w3	PROPN
ejpam-6425	776	23	,	,	PUNCT
ejpam-6425	776	24	u3	u3	NOUN
ejpam-6425	776	25	,	,	PUNCT
ejpam-6425	776	26	v1	v1	NOUN
ejpam-6425	776	27	,	,	PUNCT
ejpam-6425	776	28	u0	u0	ADJ
ejpam-6425	776	29	,	,	PUNCT
ejpam-6425	776	30	w1	w1	NOUN
ejpam-6425	776	31	,	,	PUNCT
ejpam-6425	776	32	u1	u1	NOUN
ejpam-6425	776	33	,	,	PUNCT
ejpam-6425	776	34	v2	v2	NOUN
ejpam-6425	776	35	,	,	PUNCT
ejpam-6425	776	36	w2	w2	NOUN
ejpam-6425	776	37	,	,	PUNCT
ejpam-6425	776	38	u2	u2	PROPN
ejpam-6425	776	39	,	,	PUNCT
ejpam-6425	776	40	v3	v3	PROPN
ejpam-6425	776	41	,	,	PUNCT
ejpam-6425	776	42	w3	w3	PROPN
ejpam-6425	776	43	,	,	PUNCT
ejpam-6425	776	44	u3	u3	PROPN
ejpam-6425	776	45	,	,	PUNCT
ejpam-6425	776	46	v1	v1	VERB
ejpam-6425	776	47	29	29	NUM
ejpam-6425	776	48	30	30	NUM
ejpam-6425	776	49	#	#	NOUN
ejpam-6425	776	50	backtracking	backtrack	VERB
ejpam-6425	776	51	function	function	NOUN
ejpam-6425	776	52	to	to	PART
ejpam-6425	776	53	find	find	VERB
ejpam-6425	776	54	valid	valid	ADJ
ejpam-6425	776	55	permutations	permutation	NOUN
ejpam-6425	776	56	31	31	NUM
ejpam-6425	776	57	def	def	ADJ
ejpam-6425	776	58	backtrack(perm	backtrack(perm	NOUN
ejpam-6425	776	59	,	,	PUNCT
ejpam-6425	776	60	used	use	VERB
ejpam-6425	776	61	):	):	PUNCT
ejpam-6425	776	62	32	32	NUM
ejpam-6425	776	63	#	#	NOUN
ejpam-6425	776	64	if	if	SCONJ
ejpam-6425	776	65	we	we	PRON
ejpam-6425	776	66	have	have	VERB
ejpam-6425	776	67	a	a	DET
ejpam-6425	776	68	complete	complete	ADJ
ejpam-6425	776	69	permutation	permutation	NOUN
ejpam-6425	776	70	,	,	PUNCT
ejpam-6425	776	71	verify	verify	VERB
ejpam-6425	776	72	u0	u0	ADJ
ejpam-6425	776	73	to	to	PART
ejpam-6425	776	74	v1	v1	VERB
ejpam-6425	776	75	33	33	NUM
ejpam-6425	776	76	if	if	SCONJ
ejpam-6425	776	77	len(perm	len(perm	NOUN
ejpam-6425	776	78	)	)	PUNCT
ejpam-6425	776	79	=	=	NOUN
ejpam-6425	776	80	=	=	SYM
ejpam-6425	776	81	54	54	NUM
ejpam-6425	776	82	:	:	SYM
ejpam-6425	776	83	34	34	NUM
ejpam-6425	776	84	v_values	v_value	NOUN
ejpam-6425	776	85	=	=	SYM
ejpam-6425	776	86	compute_v_values(*perm	compute_v_values(*perm	NOUN
ejpam-6425	776	87	)	)	PUNCT
ejpam-6425	776	88	35	35	NUM
ejpam-6425	776	89	#	#	NOUN
ejpam-6425	776	90	verify	verify	NOUN
ejpam-6425	776	91	if	if	SCONJ
ejpam-6425	776	92	u0	u0	ADJ
ejpam-6425	776	93	to	to	PART
ejpam-6425	776	94	v1	v1	VERB
ejpam-6425	776	95	are	be	AUX
ejpam-6425	776	96	distinct	distinct	ADJ
ejpam-6425	776	97	36	36	NUM
ejpam-6425	776	98	if	if	SCONJ
ejpam-6425	776	99	len(set(v_values	len(set(v_value	NOUN
ejpam-6425	776	100	)	)	PUNCT
ejpam-6425	776	101	)	)	PUNCT
ejpam-6425	777	1	=	=	SYM
ejpam-6425	777	2	=	=	SYM
ejpam-6425	777	3	20	20	NUM
ejpam-6425	777	4	:	:	SYM
ejpam-6425	777	5	37	37	NUM
ejpam-6425	777	6	print(f"permutation	print(f"permutation	NOUN
ejpam-6425	777	7	:	:	PUNCT
ejpam-6425	777	8	{	{	PUNCT
ejpam-6425	777	9	perm	perm	NOUN
ejpam-6425	777	10	}	}	PUNCT
ejpam-6425	777	11	,	,	PUNCT
ejpam-6425	777	12	u0	u0	ADJ
ejpam-6425	777	13	to	to	PART
ejpam-6425	777	14	v1	v1	VERB
ejpam-6425	777	15	:	:	PUNCT
ejpam-6425	777	16	{	{	PUNCT
ejpam-6425	777	17	v_values	v_value	NOUN
ejpam-6425	777	18	}	}	PUNCT
ejpam-6425	777	19	"	"	PUNCT
ejpam-6425	777	20	)	)	PUNCT
ejpam-6425	777	21	38	38	NUM
ejpam-6425	777	22	return	return	VERB
ejpam-6425	777	23	39	39	NUM
ejpam-6425	777	24	40	40	NUM
ejpam-6425	777	25	#	#	NOUN
ejpam-6425	777	26	try	try	VERB
ejpam-6425	777	27	to	to	PART
ejpam-6425	777	28	add	add	VERB
ejpam-6425	777	29	each	each	DET
ejpam-6425	777	30	unused	unused	ADJ
ejpam-6425	777	31	number	number	NOUN
ejpam-6425	777	32	to	to	ADP
ejpam-6425	777	33	the	the	DET
ejpam-6425	777	34	permutation	permutation	NOUN
ejpam-6425	777	35	41	41	NUM
ejpam-6425	777	36	for	for	ADP
ejpam-6425	777	37	num	num	ADJ
ejpam-6425	777	38	in	in	ADP
ejpam-6425	777	39	numbers	number	NOUN
ejpam-6425	777	40	:	:	PUNCT
ejpam-6425	777	41	42	42	NUM
ejpam-6425	777	42	if	if	SCONJ
ejpam-6425	777	43	not	not	PART
ejpam-6425	777	44	used[num	used[num	ADV
ejpam-6425	778	1	1	1	NUM
ejpam-6425	778	2	]	]	NOUN
ejpam-6425	778	3	:	:	PUNCT
ejpam-6425	778	4	#	#	NOUN
ejpam-6425	778	5	check	check	NOUN
ejpam-6425	778	6	if	if	SCONJ
ejpam-6425	778	7	number	number	NOUN
ejpam-6425	778	8	is	be	AUX
ejpam-6425	778	9	not	not	PART
ejpam-6425	778	10	used	use	VERB
ejpam-6425	778	11	43	43	NUM
ejpam-6425	778	12	#	#	NOUN
ejpam-6425	778	13	mark	mark	NOUN
ejpam-6425	778	14	the	the	DET
ejpam-6425	778	15	number	number	NOUN
ejpam-6425	778	16	as	as	SCONJ
ejpam-6425	778	17	used	use	VERB
ejpam-6425	778	18	44	44	NUM
ejpam-6425	778	19	used[num	used[num	ADJ
ejpam-6425	778	20	1	1	NUM
ejpam-6425	779	1	]	]	PUNCT
ejpam-6425	779	2	=	=	PUNCT
ejpam-6425	779	3	true	true	ADJ
ejpam-6425	779	4	45	45	NUM
ejpam-6425	779	5	perm.append(num	perm.append(num	ADJ
ejpam-6425	779	6	)	)	PUNCT
ejpam-6425	779	7	46	46	NUM
ejpam-6425	780	1	47	47	NUM
ejpam-6425	780	2	#	#	NOUN
ejpam-6425	780	3	recursively	recursively	ADV
ejpam-6425	780	4	try	try	VERB
ejpam-6425	780	5	the	the	DET
ejpam-6425	780	6	next	next	ADJ
ejpam-6425	780	7	number	number	NOUN
ejpam-6425	780	8	48	48	NUM
ejpam-6425	780	9	backtrack(perm	backtrack(perm	NOUN
ejpam-6425	780	10	,	,	PUNCT
ejpam-6425	780	11	used	use	VERB
ejpam-6425	780	12	)	)	PUNCT
ejpam-6425	780	13	49	49	NUM
ejpam-6425	780	14	50	50	NUM
ejpam-6425	780	15	#	#	NOUN
ejpam-6425	780	16	undo	undo	VERB
ejpam-6425	780	17	the	the	DET
ejpam-6425	780	18	choice	choice	NOUN
ejpam-6425	780	19	(	(	PUNCT
ejpam-6425	780	20	backtrack	backtrack	NOUN
ejpam-6425	780	21	)	)	PUNCT
ejpam-6425	780	22	51	51	NUM
ejpam-6425	780	23	perm.pop	perm.pop	NOUN
ejpam-6425	780	24	(	(	PUNCT
ejpam-6425	780	25	)	)	PUNCT
ejpam-6425	780	26	52	52	NUM
ejpam-6425	780	27	used[num	used[num	ADJ
ejpam-6425	780	28	1	1	NUM
ejpam-6425	780	29	]	]	PUNCT
ejpam-6425	780	30	=	=	SYM
ejpam-6425	780	31	false	false	ADJ
ejpam-6425	780	32	53	53	NUM
ejpam-6425	780	33	54	54	NUM
ejpam-6425	780	34	#	#	NOUN
ejpam-6425	780	35	start	start	VERB
ejpam-6425	780	36	the	the	DET
ejpam-6425	780	37	backtracking	backtrack	VERB
ejpam-6425	780	38	process	process	NOUN
ejpam-6425	780	39	55	55	NUM
ejpam-6425	780	40	def	def	ADJ
ejpam-6425	780	41	find_valid_permutations	find_valid_permutation	NOUN
ejpam-6425	780	42	(	(	PUNCT
ejpam-6425	780	43	):	):	PUNCT
ejpam-6425	780	44	56	56	NUM
ejpam-6425	780	45	used	use	VERB
ejpam-6425	780	46	=	=	X
ejpam-6425	781	1	[	[	X
ejpam-6425	781	2	false	false	X
ejpam-6425	781	3	]	]	X
ejpam-6425	781	4	*	*	SYM
ejpam-6425	781	5	54	54	NUM
ejpam-6425	781	6	#	#	NOUN
ejpam-6425	781	7	keeps	keep	VERB
ejpam-6425	781	8	track	track	NOUN
ejpam-6425	781	9	of	of	ADP
ejpam-6425	781	10	used	used	ADJ
ejpam-6425	781	11	numbers	number	NOUN
ejpam-6425	781	12	57	57	NUM
ejpam-6425	781	13	backtrack	backtrack	NOUN
ejpam-6425	781	14	(	(	PUNCT
ejpam-6425	781	15	[	[	X
ejpam-6425	781	16	]	]	X
ejpam-6425	781	17	,	,	PUNCT
ejpam-6425	781	18	used	use	VERB
ejpam-6425	781	19	)	)	PUNCT
ejpam-6425	781	20	58	58	NUM
ejpam-6425	781	21	59	59	NUM
ejpam-6425	781	22	#	#	NOUN
ejpam-6425	781	23	start	start	NOUN
ejpam-6425	781	24	finding	find	VERB
ejpam-6425	781	25	the	the	DET
ejpam-6425	781	26	valid	valid	ADJ
ejpam-6425	781	27	permutations	permutation	NOUN
ejpam-6425	781	28	a.	a.	PROPN
ejpam-6425	781	29	d.	d.	PROPN
ejpam-6425	781	30	c.	c.	PROPN
ejpam-6425	781	31	angel	angel	PROPN
ejpam-6425	781	32	et	et	PROPN
ejpam-6425	781	33	al	al	PROPN
ejpam-6425	781	34	.	.	PUNCT
ejpam-6425	781	35	/	/	SYM
ejpam-6425	781	36	eur	eur	PROPN
ejpam-6425	781	37	.	.	PUNCT
ejpam-6425	782	1	j.	j.	PROPN
ejpam-6425	782	2	pure	pure	PROPN
ejpam-6425	782	3	appl	appl	PROPN
ejpam-6425	782	4	.	.	PROPN
ejpam-6425	782	5	math	math	PROPN
ejpam-6425	782	6	,	,	PUNCT
ejpam-6425	782	7	18	18	NUM
ejpam-6425	782	8	(	(	PUNCT
ejpam-6425	782	9	4	4	NUM
ejpam-6425	782	10	)	)	PUNCT
ejpam-6425	782	11	(	(	PUNCT
ejpam-6425	782	12	2025	2025	NUM
ejpam-6425	782	13	)	)	PUNCT
ejpam-6425	782	14	,	,	PUNCT
ejpam-6425	782	15	6425	6425	NUM
ejpam-6425	782	16	33	33	NUM
ejpam-6425	782	17	of	of	ADP
ejpam-6425	782	18	33	33	NUM
ejpam-6425	782	19	60	60	NUM
ejpam-6425	782	20	find_valid_permutations	find_valid_permutation	NOUN
ejpam-6425	782	21	(	(	PUNCT
ejpam-6425	782	22	)	)	PUNCT