id	sid	tid	token	lemma	pos
ejpam-5305	1	1	european	european	PROPN
ejpam-5305	1	2	journal	journal	PROPN
ejpam-5305	1	3	of	of	ADP
ejpam-5305	1	4	pure	pure	ADJ
ejpam-5305	1	5	and	and	CCONJ
ejpam-5305	1	6	applied	applied	ADJ
ejpam-5305	1	7	mathematics	mathematic	NOUN
ejpam-5305	1	8	2025	2025	NUM
ejpam-5305	1	9	,	,	PUNCT
ejpam-5305	1	10	vol	vol	NOUN
ejpam-5305	1	11	.	.	PROPN
ejpam-5305	1	12	18	18	NUM
ejpam-5305	1	13	,	,	PUNCT
ejpam-5305	1	14	issue	issue	NOUN
ejpam-5305	1	15	2	2	NUM
ejpam-5305	1	16	,	,	PUNCT
ejpam-5305	1	17	article	article	NOUN
ejpam-5305	1	18	number	number	NOUN
ejpam-5305	1	19	5305	5305	NUM
ejpam-5305	1	20	issn	issn	PROPN
ejpam-5305	1	21	1307	1307	NUM
ejpam-5305	1	22	-	-	SYM
ejpam-5305	1	23	5543	5543	NUM
ejpam-5305	1	24	–	–	PUNCT
ejpam-5305	1	25	ejpam.com	ejpam.com	X
ejpam-5305	1	26	published	publish	VERB
ejpam-5305	1	27	by	by	ADP
ejpam-5305	1	28	new	new	PROPN
ejpam-5305	1	29	york	york	PROPN
ejpam-5305	1	30	business	business	PROPN
ejpam-5305	1	31	global	global	PROPN
ejpam-5305	1	32	graceful	graceful	ADJ
ejpam-5305	1	33	labeling	labeling	NOUN
ejpam-5305	1	34	of	of	ADP
ejpam-5305	1	35	some	some	DET
ejpam-5305	1	36	spider	spider	NOUN
ejpam-5305	1	37	graphs	graph	NOUN
ejpam-5305	1	38	kittisak	kittisak	PROPN
ejpam-5305	1	39	saengsura1,∗	saengsura1,∗	PROPN
ejpam-5305	1	40	,	,	PUNCT
ejpam-5305	1	41	tiang	tiang	PROPN
ejpam-5305	1	42	poomsa	poomsa	PROPN
ejpam-5305	1	43	-	-	PUNCT
ejpam-5305	1	44	ard2	ard2	ADJ
ejpam-5305	1	45	1	1	NUM
ejpam-5305	1	46	algebras	algebra	NOUN
ejpam-5305	1	47	and	and	CCONJ
ejpam-5305	1	48	applications	application	NOUN
ejpam-5305	1	49	research	research	NOUN
ejpam-5305	1	50	unit	unit	NOUN
ejpam-5305	1	51	,	,	PUNCT
ejpam-5305	1	52	mahasarakham	mahasarakham	NOUN
ejpam-5305	1	53	universtity	universtity	NOUN
ejpam-5305	1	54	,	,	PUNCT
ejpam-5305	1	55	kantarawichai	kantarawichai	PROPN
ejpam-5305	1	56	,	,	PUNCT
ejpam-5305	1	57	mahasarakham	mahasarakham	PROPN
ejpam-5305	1	58	,	,	PUNCT
ejpam-5305	1	59	thailand	thailand	PROPN
ejpam-5305	1	60	2	2	NUM
ejpam-5305	1	61	department	department	NOUN
ejpam-5305	1	62	of	of	ADP
ejpam-5305	1	63	mathematics	mathematic	NOUN
ejpam-5305	1	64	,	,	PUNCT
ejpam-5305	1	65	faculty	faculty	NOUN
ejpam-5305	1	66	of	of	ADP
ejpam-5305	1	67	science	science	NOUN
ejpam-5305	1	68	,	,	PUNCT
ejpam-5305	1	69	khonkaen	khonkaen	PROPN
ejpam-5305	1	70	university	university	PROPN
ejpam-5305	1	71	,	,	PUNCT
ejpam-5305	1	72	khonkaen	khonkaen	PROPN
ejpam-5305	1	73	,	,	PUNCT
ejpam-5305	1	74	thailand	thailand	PROPN
ejpam-5305	1	75	abstract	abstract	PROPN
ejpam-5305	1	76	.	.	PUNCT
ejpam-5305	2	1	a	a	DET
ejpam-5305	2	2	graceful	graceful	ADJ
ejpam-5305	2	3	labeling	labeling	NOUN
ejpam-5305	2	4	of	of	ADP
ejpam-5305	2	5	a	a	DET
ejpam-5305	2	6	tree	tree	NOUN
ejpam-5305	2	7	t	t	NOUN
ejpam-5305	2	8	with	with	ADP
ejpam-5305	2	9	n	n	PRON
ejpam-5305	2	10	edges	edge	NOUN
ejpam-5305	2	11	is	be	AUX
ejpam-5305	2	12	a	a	DET
ejpam-5305	2	13	bijection	bijection	ADJ
ejpam-5305	2	14	f	f	NOUN
ejpam-5305	2	15	:	:	PUNCT
ejpam-5305	2	16	v	v	PROPN
ejpam-5305	2	17	(	(	PUNCT
ejpam-5305	2	18	t	t	NOUN
ejpam-5305	2	19	)	)	PUNCT
ejpam-5305	2	20	−→	−→	NOUN
ejpam-5305	2	21	{	{	PUNCT
ejpam-5305	2	22	0	0	NUM
ejpam-5305	2	23	,	,	PUNCT
ejpam-5305	2	24	1	1	NUM
ejpam-5305	2	25	,	,	PUNCT
ejpam-5305	2	26	2	2	NUM
ejpam-5305	2	27	,	,	PUNCT
ejpam-5305	2	28	.	.	PUNCT
ejpam-5305	2	29	.	.	PUNCT
ejpam-5305	2	30	.	.	PUNCT
ejpam-5305	3	1	n	n	CCONJ
ejpam-5305	3	2	}	}	PUNCT
ejpam-5305	3	3	such	such	ADJ
ejpam-5305	3	4	that	that	SCONJ
ejpam-5305	3	5	{	{	PUNCT
ejpam-5305	3	6	|f(u	|f(u	ADJ
ejpam-5305	3	7	)	)	PUNCT
ejpam-5305	3	8	−	−	PROPN
ejpam-5305	3	9	f(v)|	f(v)|	NOUN
ejpam-5305	3	10	:	:	PUNCT
ejpam-5305	3	11	uv	uv	PROPN
ejpam-5305	3	12	∈	∈	PROPN
ejpam-5305	3	13	e(t	e(t	PROPN
ejpam-5305	3	14	)	)	PUNCT
ejpam-5305	3	15	}	}	PUNCT
ejpam-5305	3	16	equal	equal	ADJ
ejpam-5305	3	17	to	to	ADP
ejpam-5305	3	18	{	{	PUNCT
ejpam-5305	3	19	1	1	NUM
ejpam-5305	3	20	,	,	PUNCT
ejpam-5305	3	21	2	2	NUM
ejpam-5305	3	22	,	,	PUNCT
ejpam-5305	3	23	3	3	NUM
ejpam-5305	3	24	,	,	PUNCT
ejpam-5305	3	25	.	.	PUNCT
ejpam-5305	3	26	.	.	PUNCT
ejpam-5305	3	27	.	.	PUNCT
ejpam-5305	3	28	,	,	PUNCT
ejpam-5305	3	29	n	n	CCONJ
ejpam-5305	3	30	}	}	PUNCT
ejpam-5305	3	31	.	.	PUNCT
ejpam-5305	4	1	a	a	DET
ejpam-5305	4	2	spider	spider	NOUN
ejpam-5305	4	3	graph	graph	NOUN
ejpam-5305	4	4	is	be	AUX
ejpam-5305	4	5	a	a	DET
ejpam-5305	4	6	tree	tree	NOUN
ejpam-5305	4	7	with	with	ADP
ejpam-5305	4	8	one	one	NUM
ejpam-5305	4	9	vertex	vertex	NOUN
ejpam-5305	4	10	of	of	ADP
ejpam-5305	4	11	degree	degree	NOUN
ejpam-5305	4	12	at	at	ADV
ejpam-5305	4	13	least	least	ADV
ejpam-5305	4	14	3	3	NUM
ejpam-5305	4	15	and	and	CCONJ
ejpam-5305	4	16	all	all	DET
ejpam-5305	4	17	others	other	NOUN
ejpam-5305	4	18	with	with	ADP
ejpam-5305	4	19	degree	degree	NOUN
ejpam-5305	4	20	at	at	ADP
ejpam-5305	4	21	most	most	ADV
ejpam-5305	4	22	2	2	NUM
ejpam-5305	4	23	.	.	X
ejpam-5305	5	1	we	we	PRON
ejpam-5305	5	2	show	show	VERB
ejpam-5305	5	3	that	that	SCONJ
ejpam-5305	5	4	some	some	DET
ejpam-5305	5	5	classes	class	NOUN
ejpam-5305	5	6	of	of	ADP
ejpam-5305	5	7	spider	spider	NOUN
ejpam-5305	5	8	graphs	graph	NOUN
ejpam-5305	5	9	admit	admit	VERB
ejpam-5305	5	10	graceful	graceful	ADJ
ejpam-5305	5	11	labeling	labeling	NOUN
ejpam-5305	5	12	.	.	PUNCT
ejpam-5305	6	1	2020	2020	NUM
ejpam-5305	6	2	mathematics	mathematic	NOUN
ejpam-5305	6	3	subject	subject	NOUN
ejpam-5305	6	4	classifications	classification	NOUN
ejpam-5305	6	5	:	:	PUNCT
ejpam-5305	6	6	05c05	05c05	NUM
ejpam-5305	6	7	,	,	PUNCT
ejpam-5305	6	8	05c15	05c15	NOUN
ejpam-5305	6	9	key	key	ADJ
ejpam-5305	6	10	words	word	NOUN
ejpam-5305	6	11	and	and	CCONJ
ejpam-5305	6	12	phrases	phrase	NOUN
ejpam-5305	6	13	:	:	PUNCT
ejpam-5305	6	14	spider	spider	NOUN
ejpam-5305	6	15	graph	graph	NOUN
ejpam-5305	6	16	,	,	PUNCT
ejpam-5305	6	17	graceful	graceful	ADJ
ejpam-5305	6	18	labeling	labeling	NOUN
ejpam-5305	6	19	,	,	PUNCT
ejpam-5305	6	20	inverse	inverse	ADJ
ejpam-5305	6	21	transformation	transformation	NOUN
ejpam-5305	6	22	1	1	NUM
ejpam-5305	6	23	.	.	PUNCT
ejpam-5305	7	1	introduction	introduction	NOUN
ejpam-5305	7	2	labeled	label	VERB
ejpam-5305	7	3	graphs	graph	NOUN
ejpam-5305	7	4	form	form	VERB
ejpam-5305	7	5	useful	useful	ADJ
ejpam-5305	7	6	models	model	NOUN
ejpam-5305	7	7	for	for	ADP
ejpam-5305	7	8	a	a	DET
ejpam-5305	7	9	wide	wide	ADJ
ejpam-5305	7	10	range	range	NOUN
ejpam-5305	7	11	of	of	ADP
ejpam-5305	7	12	disciplines	discipline	NOUN
ejpam-5305	7	13	and	and	CCONJ
ejpam-5305	7	14	applications	application	NOUN
ejpam-5305	7	15	such	such	ADJ
ejpam-5305	7	16	as	as	ADP
ejpam-5305	7	17	in	in	ADP
ejpam-5305	7	18	coding	code	VERB
ejpam-5305	7	19	theory	theory	NOUN
ejpam-5305	7	20	,	,	PUNCT
ejpam-5305	7	21	x	x	NOUN
ejpam-5305	7	22	-	-	NOUN
ejpam-5305	7	23	ray	ray	NOUN
ejpam-5305	7	24	crystallography	crystallography	NOUN
ejpam-5305	7	25	,	,	PUNCT
ejpam-5305	7	26	radar	radar	NOUN
ejpam-5305	7	27	,	,	PUNCT
ejpam-5305	7	28	astronomy	astronomy	NOUN
ejpam-5305	7	29	,	,	PUNCT
ejpam-5305	7	30	electronic	electronic	ADJ
ejpam-5305	7	31	circuit	circuit	NOUN
ejpam-5305	7	32	design	design	NOUN
ejpam-5305	7	33	and	and	CCONJ
ejpam-5305	7	34	communication	communication	NOUN
ejpam-5305	7	35	network	network	NOUN
ejpam-5305	7	36	addressing	address	VERB
ejpam-5305	7	37	[	[	X
ejpam-5305	7	38	1	1	NUM
ejpam-5305	7	39	]	]	PUNCT
ejpam-5305	7	40	.	.	PUNCT
ejpam-5305	8	1	a	a	DET
ejpam-5305	8	2	systematic	systematic	ADJ
ejpam-5305	8	3	presentation	presentation	NOUN
ejpam-5305	8	4	of	of	ADP
ejpam-5305	8	5	diverse	diverse	ADJ
ejpam-5305	8	6	applications	application	NOUN
ejpam-5305	8	7	of	of	ADP
ejpam-5305	8	8	graph	graph	NOUN
ejpam-5305	8	9	labeling	labeling	NOUN
ejpam-5305	8	10	is	be	AUX
ejpam-5305	8	11	presented	present	VERB
ejpam-5305	8	12	in	in	ADP
ejpam-5305	8	13	[	[	X
ejpam-5305	8	14	2	2	NUM
ejpam-5305	8	15	]	]	PUNCT
ejpam-5305	8	16	.	.	PUNCT
ejpam-5305	9	1	a	a	DET
ejpam-5305	9	2	graceful	graceful	ADJ
ejpam-5305	9	3	labeling	labeling	NOUN
ejpam-5305	9	4	of	of	ADP
ejpam-5305	9	5	a	a	DET
ejpam-5305	9	6	tree	tree	NOUN
ejpam-5305	9	7	is	be	AUX
ejpam-5305	9	8	a	a	DET
ejpam-5305	9	9	special	special	ADJ
ejpam-5305	9	10	kind	kind	NOUN
ejpam-5305	9	11	of	of	ADP
ejpam-5305	9	12	labeling	labeling	NOUN
ejpam-5305	9	13	graph	graph	NOUN
ejpam-5305	9	14	,	,	PUNCT
ejpam-5305	9	15	i.e.	i.e.	X
ejpam-5305	9	16	,	,	PUNCT
ejpam-5305	9	17	a	a	DET
ejpam-5305	9	18	graceful	graceful	ADJ
ejpam-5305	9	19	labeling	labeling	NOUN
ejpam-5305	9	20	f	f	PROPN
ejpam-5305	9	21	of	of	ADP
ejpam-5305	9	22	a	a	DET
ejpam-5305	9	23	tree	tree	NOUN
ejpam-5305	9	24	t	t	NOUN
ejpam-5305	9	25	is	be	AUX
ejpam-5305	9	26	a	a	DET
ejpam-5305	9	27	bijective	bijective	ADJ
ejpam-5305	9	28	function	function	NOUN
ejpam-5305	9	29	from	from	ADP
ejpam-5305	9	30	the	the	DET
ejpam-5305	9	31	set	set	NOUN
ejpam-5305	9	32	of	of	ADP
ejpam-5305	9	33	vertices	vertex	NOUN
ejpam-5305	9	34	v	v	X
ejpam-5305	9	35	(	(	PUNCT
ejpam-5305	9	36	t	t	PROPN
ejpam-5305	9	37	)	)	PUNCT
ejpam-5305	9	38	of	of	ADP
ejpam-5305	9	39	t	t	PROPN
ejpam-5305	9	40	to	to	ADP
ejpam-5305	9	41	the	the	DET
ejpam-5305	9	42	set	set	NOUN
ejpam-5305	9	43	{	{	PUNCT
ejpam-5305	9	44	0	0	NUM
ejpam-5305	9	45	,	,	PUNCT
ejpam-5305	9	46	1	1	NUM
ejpam-5305	9	47	,	,	PUNCT
ejpam-5305	9	48	2	2	NUM
ejpam-5305	9	49	,	,	PUNCT
ejpam-5305	9	50	.	.	PUNCT
ejpam-5305	9	51	.	.	PUNCT
ejpam-5305	10	1	.	.	PUNCT
ejpam-5305	11	1	,	,	PUNCT
ejpam-5305	11	2	|e(t	|e(t	PROPN
ejpam-5305	11	3	)	)	PUNCT
ejpam-5305	11	4	|	|	ADV
ejpam-5305	11	5	}	}	PUNCT
ejpam-5305	11	6	such	such	ADJ
ejpam-5305	11	7	that	that	SCONJ
ejpam-5305	11	8	when	when	SCONJ
ejpam-5305	11	9	each	each	DET
ejpam-5305	11	10	edge	edge	NOUN
ejpam-5305	11	11	xy	xy	PROPN
ejpam-5305	11	12	is	be	AUX
ejpam-5305	11	13	assigned	assign	VERB
ejpam-5305	11	14	the	the	DET
ejpam-5305	11	15	label	label	NOUN
ejpam-5305	11	16	|f(x	|f(x	PROPN
ejpam-5305	11	17	)	)	PUNCT
ejpam-5305	12	1	−	−	PROPN
ejpam-5305	13	1	f(y)|	f(y)|	PROPN
ejpam-5305	13	2	,	,	PUNCT
ejpam-5305	13	3	the	the	DET
ejpam-5305	13	4	resulting	result	VERB
ejpam-5305	13	5	edge	edge	NOUN
ejpam-5305	13	6	labels	label	NOUN
ejpam-5305	13	7	are	be	AUX
ejpam-5305	13	8	distinct	distinct	ADJ
ejpam-5305	13	9	.	.	PUNCT
ejpam-5305	14	1	there	there	PRON
ejpam-5305	14	2	are	be	VERB
ejpam-5305	14	3	other	other	ADJ
ejpam-5305	14	4	concepts	concept	NOUN
ejpam-5305	14	5	of	of	ADP
ejpam-5305	14	6	labeling	labeling	NOUN
ejpam-5305	14	7	graphs	graph	NOUN
ejpam-5305	14	8	that	that	PRON
ejpam-5305	14	9	are	be	AUX
ejpam-5305	14	10	equivalent	equivalent	ADJ
ejpam-5305	14	11	to	to	ADP
ejpam-5305	14	12	graceful	graceful	ADJ
ejpam-5305	14	13	labeling	labeling	NOUN
ejpam-5305	14	14	,	,	PUNCT
ejpam-5305	14	15	for	for	ADP
ejpam-5305	14	16	instance	instance	NOUN
ejpam-5305	14	17	,	,	PUNCT
ejpam-5305	14	18	edge	edge	VERB
ejpam-5305	14	19	antimagic	antimagic	ADJ
ejpam-5305	14	20	vertex	vertex	NOUN
ejpam-5305	14	21	labeling	labeling	NOUN
ejpam-5305	14	22	and	and	CCONJ
ejpam-5305	14	23	rainbow	rainbow	VERB
ejpam-5305	14	24	antimagic	antimagic	ADJ
ejpam-5305	14	25	labeling	labeling	NOUN
ejpam-5305	14	26	(	(	PUNCT
ejpam-5305	14	27	see	see	VERB
ejpam-5305	14	28	[	[	X
ejpam-5305	14	29	3	3	NUM
ejpam-5305	14	30	]	]	PUNCT
ejpam-5305	14	31	,	,	PUNCT
ejpam-5305	14	32	[	[	X
ejpam-5305	14	33	4	4	NUM
ejpam-5305	14	34	]	]	NUM
ejpam-5305	14	35	)	)	PUNCT
ejpam-5305	14	36	.	.	PUNCT
ejpam-5305	15	1	a	a	DET
ejpam-5305	15	2	tree	tree	NOUN
ejpam-5305	15	3	that	that	PRON
ejpam-5305	15	4	admits	admit	VERB
ejpam-5305	15	5	graceful	graceful	ADJ
ejpam-5305	15	6	labeling	labeling	NOUN
ejpam-5305	15	7	is	be	AUX
ejpam-5305	15	8	called	call	VERB
ejpam-5305	15	9	a	a	DET
ejpam-5305	15	10	graceful	graceful	ADJ
ejpam-5305	15	11	tree	tree	NOUN
ejpam-5305	15	12	.	.	PUNCT
ejpam-5305	16	1	in	in	ADP
ejpam-5305	16	2	1964	1964	NUM
ejpam-5305	16	3	,	,	PUNCT
ejpam-5305	16	4	rangel	rangel	NOUN
ejpam-5305	16	5	and	and	CCONJ
ejpam-5305	16	6	rosa	rosa	PROPN
ejpam-5305	16	7	(	(	PUNCT
ejpam-5305	16	8	see	see	VERB
ejpam-5305	16	9	[	[	X
ejpam-5305	16	10	5	5	NUM
ejpam-5305	16	11	]	]	PUNCT
ejpam-5305	16	12	,	,	PUNCT
ejpam-5305	17	1	[	[	X
ejpam-5305	17	2	6	6	NUM
ejpam-5305	17	3	]	]	PUNCT
ejpam-5305	17	4	)	)	PUNCT
ejpam-5305	17	5	gave	give	VERB
ejpam-5305	17	6	the	the	DET
ejpam-5305	17	7	famous	famous	ADJ
ejpam-5305	17	8	and	and	CCONJ
ejpam-5305	17	9	unsolved	unsolved	ADJ
ejpam-5305	17	10	graceful	graceful	ADJ
ejpam-5305	17	11	tree	tree	NOUN
ejpam-5305	17	12	conjecture	conjecture	NOUN
ejpam-5305	17	13	which	which	PRON
ejpam-5305	17	14	stated	state	VERB
ejpam-5305	17	15	that	that	SCONJ
ejpam-5305	17	16	all	all	DET
ejpam-5305	17	17	trees	tree	NOUN
ejpam-5305	17	18	are	be	AUX
ejpam-5305	17	19	graceful	graceful	ADJ
ejpam-5305	17	20	.	.	PUNCT
ejpam-5305	18	1	in	in	ADP
ejpam-5305	18	2	order	order	NOUN
ejpam-5305	18	3	to	to	PART
ejpam-5305	18	4	solve	solve	VERB
ejpam-5305	18	5	the	the	DET
ejpam-5305	18	6	graceful	graceful	ADJ
ejpam-5305	18	7	tree	tree	NOUN
ejpam-5305	18	8	conjecture	conjecture	NOUN
ejpam-5305	18	9	,	,	PUNCT
ejpam-5305	18	10	we	we	PRON
ejpam-5305	18	11	also	also	ADV
ejpam-5305	18	12	start	start	VERB
ejpam-5305	18	13	to	to	PART
ejpam-5305	18	14	consider	consider	VERB
ejpam-5305	18	15	a	a	DET
ejpam-5305	18	16	graceful	graceful	ADJ
ejpam-5305	18	17	labeling	labeling	NOUN
ejpam-5305	18	18	of	of	ADP
ejpam-5305	18	19	some	some	DET
ejpam-5305	18	20	graph	graph	NOUN
ejpam-5305	18	21	which	which	PRON
ejpam-5305	18	22	usually	usually	ADV
ejpam-5305	18	23	belongs	belong	VERB
ejpam-5305	18	24	to	to	ADP
ejpam-5305	18	25	a	a	DET
ejpam-5305	18	26	subgraph	subgraph	NOUN
ejpam-5305	18	27	of	of	ADP
ejpam-5305	18	28	a	a	DET
ejpam-5305	18	29	tree	tree	NOUN
ejpam-5305	18	30	such	such	ADJ
ejpam-5305	18	31	as	as	SCONJ
ejpam-5305	18	32	we	we	PRON
ejpam-5305	18	33	called	call	VERB
ejpam-5305	18	34	a	a	DET
ejpam-5305	18	35	spider	spider	NOUN
ejpam-5305	18	36	graph	graph	NOUN
ejpam-5305	18	37	.	.	PUNCT
ejpam-5305	19	1	the	the	DET
ejpam-5305	19	2	results	result	NOUN
ejpam-5305	19	3	of	of	ADP
ejpam-5305	19	4	graceful	graceful	ADJ
ejpam-5305	19	5	labeling	labeling	NOUN
ejpam-5305	19	6	on	on	ADP
ejpam-5305	19	7	spider	spider	NOUN
ejpam-5305	19	8	graphs	graph	NOUN
ejpam-5305	19	9	is	be	AUX
ejpam-5305	19	10	a	a	DET
ejpam-5305	19	11	step	step	NOUN
ejpam-5305	19	12	towards	towards	ADP
ejpam-5305	19	13	potentially	potentially	ADV
ejpam-5305	19	14	solving	solve	VERB
ejpam-5305	19	15	the	the	DET
ejpam-5305	19	16	graceful	graceful	ADJ
ejpam-5305	19	17	tree	tree	NOUN
ejpam-5305	19	18	conjecture	conjecture	NOUN
ejpam-5305	19	19	.	.	PUNCT
ejpam-5305	20	1	a	a	DET
ejpam-5305	20	2	spider	spider	NOUN
ejpam-5305	20	3	graph	graph	NOUN
ejpam-5305	20	4	is	be	AUX
ejpam-5305	20	5	a	a	DET
ejpam-5305	20	6	tree	tree	NOUN
ejpam-5305	20	7	with	with	ADP
ejpam-5305	20	8	one	one	NUM
ejpam-5305	20	9	vertex	vertex	NOUN
ejpam-5305	20	10	of	of	ADP
ejpam-5305	20	11	degree	degree	NOUN
ejpam-5305	20	12	at	at	ADV
ejpam-5305	20	13	least	least	ADV
ejpam-5305	20	14	3	3	NUM
ejpam-5305	20	15	and	and	CCONJ
ejpam-5305	20	16	all	all	DET
ejpam-5305	20	17	others	other	NOUN
ejpam-5305	20	18	with	with	ADP
ejpam-5305	20	19	degree	degree	NOUN
ejpam-5305	20	20	at	at	ADP
ejpam-5305	20	21	most	most	ADV
ejpam-5305	20	22	2	2	NUM
ejpam-5305	20	23	.	.	PUNCT
ejpam-5305	20	24	gillian	gillian	PROPN
ejpam-5305	21	1	[	[	X
ejpam-5305	21	2	1	1	NUM
ejpam-5305	21	3	]	]	PUNCT
ejpam-5305	21	4	has	have	AUX
ejpam-5305	21	5	noted	note	VERB
ejpam-5305	21	6	that	that	SCONJ
ejpam-5305	21	7	the	the	DET
ejpam-5305	21	8	special	special	ADJ
ejpam-5305	21	9	case	case	NOUN
ejpam-5305	21	10	of	of	ADP
ejpam-5305	21	11	the	the	DET
ejpam-5305	21	12	conjecture	conjecture	NOUN
ejpam-5305	21	13	regarding	regard	VERB
ejpam-5305	21	14	a	a	DET
ejpam-5305	21	15	spider	spider	NOUN
ejpam-5305	21	16	∗corresponding	∗corresponde	VERB
ejpam-5305	21	17	author	author	NOUN
ejpam-5305	21	18	.	.	PUNCT
ejpam-5305	22	1	doi	doi	NOUN
ejpam-5305	22	2	:	:	PUNCT
ejpam-5305	22	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5305	https://doi.org/10.29020/nybg.ejpam.v18i2.5305	VERB
ejpam-5305	22	4	email	email	NOUN
ejpam-5305	22	5	addresses	address	NOUN
ejpam-5305	22	6	:	:	PUNCT
ejpam-5305	23	1	kittisak.s@msu.ac.th	kittisak.s@msu.ac.th	PROPN
ejpam-5305	23	2	(	(	PUNCT
ejpam-5305	23	3	k.	k.	PROPN
ejpam-5305	23	4	saengsura	saengsura	PROPN
ejpam-5305	23	5	)	)	PUNCT
ejpam-5305	23	6	,	,	PUNCT
ejpam-5305	23	7	tiang.poom@gmail.com	tiang.poom@gmail.com	X
ejpam-5305	23	8	(	(	PUNCT
ejpam-5305	23	9	t.	t.	PROPN
ejpam-5305	23	10	poomsa	poomsa	PROPN
ejpam-5305	23	11	-	-	PUNCT
ejpam-5305	23	12	ard	ard	NOUN
ejpam-5305	23	13	)	)	PUNCT
ejpam-5305	23	14	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5305	23	15	1	1	NUM
ejpam-5305	23	16	copyright	copyright	NOUN
ejpam-5305	23	17	:	:	PUNCT
ejpam-5305	24	1	©	©	PROPN
ejpam-5305	24	2	2025	2025	NUM
ejpam-5305	24	3	the	the	DET
ejpam-5305	24	4	author(s	author(s	NOUN
ejpam-5305	24	5	)	)	PUNCT
ejpam-5305	24	6	.	.	PUNCT
ejpam-5305	25	1	(	(	PUNCT
ejpam-5305	25	2	cc	cc	NOUN
ejpam-5305	25	3	by	by	ADP
ejpam-5305	25	4	-	-	PUNCT
ejpam-5305	25	5	nc	nc	PROPN
ejpam-5305	25	6	4.0	4.0	NUM
ejpam-5305	25	7	)	)	PUNCT
ejpam-5305	25	8	k.	k.	PROPN
ejpam-5305	25	9	saengsura	saengsura	PROPN
ejpam-5305	25	10	,	,	PUNCT
ejpam-5305	25	11	t.	t.	PROPN
ejpam-5305	25	12	poomsa	poomsa	PROPN
ejpam-5305	25	13	-	-	PUNCT
ejpam-5305	25	14	ard	ard	PROPN
ejpam-5305	25	15	/	/	PUNCT
ejpam-5305	25	16	eur	eur	PROPN
ejpam-5305	25	17	.	.	PUNCT
ejpam-5305	26	1	j.	j.	PROPN
ejpam-5305	26	2	pure	pure	PROPN
ejpam-5305	26	3	appl	appl	PROPN
ejpam-5305	26	4	.	.	PROPN
ejpam-5305	26	5	math	math	PROPN
ejpam-5305	26	6	,	,	PUNCT
ejpam-5305	26	7	18	18	NUM
ejpam-5305	26	8	(	(	PUNCT
ejpam-5305	26	9	2	2	NUM
ejpam-5305	26	10	)	)	PUNCT
ejpam-5305	26	11	(	(	PUNCT
ejpam-5305	26	12	2025	2025	NUM
ejpam-5305	26	13	)	)	PUNCT
ejpam-5305	26	14	,	,	PUNCT
ejpam-5305	26	15	5305	5305	NUM
ejpam-5305	26	16	2	2	NUM
ejpam-5305	26	17	of	of	ADP
ejpam-5305	26	18	8	8	NUM
ejpam-5305	26	19	graph	graph	NOUN
ejpam-5305	26	20	is	be	AUX
ejpam-5305	26	21	still	still	ADV
ejpam-5305	26	22	open	open	ADJ
ejpam-5305	26	23	and	and	CCONJ
ejpam-5305	26	24	that	that	SCONJ
ejpam-5305	26	25	very	very	ADV
ejpam-5305	26	26	few	few	ADJ
ejpam-5305	26	27	classes	class	NOUN
ejpam-5305	26	28	of	of	ADP
ejpam-5305	26	29	spider	spider	NOUN
ejpam-5305	26	30	graphs	graph	NOUN
ejpam-5305	26	31	are	be	AUX
ejpam-5305	26	32	known	know	VERB
ejpam-5305	26	33	to	to	PART
ejpam-5305	26	34	be	be	AUX
ejpam-5305	26	35	graceful	graceful	ADJ
ejpam-5305	26	36	.	.	PUNCT
ejpam-5305	27	1	huang	huang	PROPN
ejpam-5305	27	2	et	et	PROPN
ejpam-5305	27	3	al	al	PROPN
ejpam-5305	27	4	.	.	PUNCT
ejpam-5305	28	1	[	[	X
ejpam-5305	28	2	7	7	X
ejpam-5305	28	3	]	]	PUNCT
ejpam-5305	28	4	proved	prove	VERB
ejpam-5305	28	5	that	that	SCONJ
ejpam-5305	28	6	all	all	DET
ejpam-5305	28	7	spider	spider	NOUN
ejpam-5305	28	8	graphs	graph	NOUN
ejpam-5305	28	9	with	with	ADP
ejpam-5305	28	10	three	three	NUM
ejpam-5305	28	11	or	or	CCONJ
ejpam-5305	28	12	four	four	NUM
ejpam-5305	28	13	legs	leg	NOUN
ejpam-5305	28	14	are	be	AUX
ejpam-5305	28	15	graceful	graceful	ADJ
ejpam-5305	28	16	.	.	PUNCT
ejpam-5305	29	1	bahls	bahls	PROPN
ejpam-5305	29	2	et	et	PROPN
ejpam-5305	29	3	al	al	PROPN
ejpam-5305	29	4	.	.	PUNCT
ejpam-5305	30	1	[	[	X
ejpam-5305	30	2	8	8	NUM
ejpam-5305	30	3	]	]	PUNCT
ejpam-5305	30	4	also	also	ADV
ejpam-5305	30	5	proved	prove	VERB
ejpam-5305	30	6	that	that	SCONJ
ejpam-5305	30	7	every	every	DET
ejpam-5305	30	8	spider	spider	NOUN
ejpam-5305	30	9	graph	graph	NOUN
ejpam-5305	30	10	in	in	ADP
ejpam-5305	30	11	which	which	PRON
ejpam-5305	30	12	the	the	DET
ejpam-5305	30	13	lengths	length	NOUN
ejpam-5305	30	14	of	of	ADP
ejpam-5305	30	15	any	any	DET
ejpam-5305	30	16	two	two	NUM
ejpam-5305	30	17	of	of	ADP
ejpam-5305	30	18	its	its	PRON
ejpam-5305	30	19	legs	leg	NOUN
ejpam-5305	30	20	differ	differ	VERB
ejpam-5305	30	21	by	by	ADP
ejpam-5305	30	22	at	at	ADP
ejpam-5305	30	23	most	most	ADJ
ejpam-5305	30	24	one	one	NUM
ejpam-5305	30	25	is	be	AUX
ejpam-5305	30	26	graceful	graceful	ADJ
ejpam-5305	30	27	.	.	PUNCT
ejpam-5305	31	1	jampachon	jampachon	PROPN
ejpam-5305	31	2	et	et	PROPN
ejpam-5305	31	3	al	al	PROPN
ejpam-5305	31	4	.	.	PUNCT
ejpam-5305	32	1	[	[	X
ejpam-5305	32	2	9	9	NUM
ejpam-5305	32	3	]	]	PUNCT
ejpam-5305	32	4	,	,	PUNCT
ejpam-5305	32	5	[	[	X
ejpam-5305	32	6	10	10	NUM
ejpam-5305	32	7	]	]	PUNCT
ejpam-5305	32	8	have	have	AUX
ejpam-5305	32	9	also	also	ADV
ejpam-5305	32	10	proven	prove	VERB
ejpam-5305	32	11	that	that	SCONJ
ejpam-5305	32	12	sn(k	sn(k	VERB
ejpam-5305	32	13	,	,	PUNCT
ejpam-5305	32	14	l	l	NOUN
ejpam-5305	32	15	,	,	PUNCT
ejpam-5305	32	16	m	m	NOUN
ejpam-5305	32	17	)	)	PUNCT
ejpam-5305	32	18	is	be	AUX
ejpam-5305	32	19	graceful	graceful	ADJ
ejpam-5305	32	20	,	,	PUNCT
ejpam-5305	32	21	where	where	SCONJ
ejpam-5305	32	22	sn(k	sn(k	VERB
ejpam-5305	32	23	,	,	PUNCT
ejpam-5305	32	24	l	l	NOUN
ejpam-5305	32	25	,	,	PUNCT
ejpam-5305	32	26	m	m	NOUN
ejpam-5305	32	27	)	)	PUNCT
ejpam-5305	32	28	is	be	AUX
ejpam-5305	32	29	defined	define	VERB
ejpam-5305	32	30	in	in	ADP
ejpam-5305	32	31	section	section	NOUN
ejpam-5305	32	32	2	2	NUM
ejpam-5305	32	33	.	.	PUNCT
ejpam-5305	32	34	recently	recently	ADV
ejpam-5305	32	35	,	,	PUNCT
ejpam-5305	32	36	panpa	panpa	PROPN
ejpam-5305	32	37	et	et	PROPN
ejpam-5305	32	38	al	al	PROPN
ejpam-5305	32	39	.	.	PUNCT
ejpam-5305	33	1	[	[	X
ejpam-5305	33	2	11	11	NUM
ejpam-5305	33	3	]	]	PUNCT
ejpam-5305	33	4	proved	prove	VERB
ejpam-5305	33	5	that	that	SCONJ
ejpam-5305	33	6	all	all	DET
ejpam-5305	33	7	spider	spider	NOUN
ejpam-5305	33	8	graphs	graph	NOUN
ejpam-5305	33	9	with	with	ADP
ejpam-5305	33	10	at	at	ADP
ejpam-5305	33	11	most	most	ADJ
ejpam-5305	33	12	four	four	NUM
ejpam-5305	33	13	legs	leg	NOUN
ejpam-5305	33	14	of	of	ADP
ejpam-5305	33	15	lengths	length	NOUN
ejpam-5305	33	16	greater	great	ADJ
ejpam-5305	33	17	than	than	ADP
ejpam-5305	33	18	one	one	NUM
ejpam-5305	33	19	are	be	AUX
ejpam-5305	33	20	graceful	graceful	ADJ
ejpam-5305	33	21	.	.	PUNCT
ejpam-5305	34	1	to	to	PART
ejpam-5305	34	2	prove	prove	VERB
ejpam-5305	34	3	our	our	PRON
ejpam-5305	34	4	results	result	NOUN
ejpam-5305	34	5	,	,	PUNCT
ejpam-5305	34	6	we	we	PRON
ejpam-5305	34	7	need	need	VERB
ejpam-5305	34	8	some	some	DET
ejpam-5305	34	9	terminology	terminology	NOUN
ejpam-5305	34	10	and	and	CCONJ
ejpam-5305	34	11	existence	existence	NOUN
ejpam-5305	34	12	results	result	NOUN
ejpam-5305	34	13	which	which	PRON
ejpam-5305	34	14	are	be	AUX
ejpam-5305	34	15	described	describe	VERB
ejpam-5305	34	16	below	below	ADV
ejpam-5305	34	17	.	.	PUNCT
ejpam-5305	35	1	let	let	VERB
ejpam-5305	35	2	t	t	NOUN
ejpam-5305	35	3	be	be	AUX
ejpam-5305	35	4	a	a	DET
ejpam-5305	35	5	tree	tree	NOUN
ejpam-5305	35	6	with	with	ADP
ejpam-5305	35	7	n	n	ADP
ejpam-5305	35	8	edges	edge	NOUN
ejpam-5305	35	9	.	.	PUNCT
ejpam-5305	36	1	a	a	DET
ejpam-5305	36	2	graceful	graceful	ADJ
ejpam-5305	36	3	labeling	labeling	NOUN
ejpam-5305	36	4	of	of	ADP
ejpam-5305	36	5	t	t	PROPN
ejpam-5305	36	6	is	be	AUX
ejpam-5305	36	7	a	a	DET
ejpam-5305	36	8	bijection	bijection	ADJ
ejpam-5305	36	9	f	f	NOUN
ejpam-5305	36	10	:	:	PUNCT
ejpam-5305	36	11	v	v	PROPN
ejpam-5305	36	12	(	(	PUNCT
ejpam-5305	36	13	t	t	PROPN
ejpam-5305	36	14	)	)	PUNCT
ejpam-5305	36	15	→	→	SYM
ejpam-5305	36	16	{	{	PUNCT
ejpam-5305	36	17	0	0	NUM
ejpam-5305	36	18	,	,	PUNCT
ejpam-5305	36	19	1	1	NUM
ejpam-5305	36	20	,	,	PUNCT
ejpam-5305	36	21	2	2	NUM
ejpam-5305	36	22	,	,	PUNCT
ejpam-5305	36	23	.	.	PUNCT
ejpam-5305	36	24	.	.	PUNCT
ejpam-5305	37	1	.	.	PUNCT
ejpam-5305	38	1	,	,	PUNCT
ejpam-5305	39	1	n	n	CCONJ
ejpam-5305	39	2	}	}	PUNCT
ejpam-5305	39	3	such	such	ADJ
ejpam-5305	39	4	that	that	SCONJ
ejpam-5305	39	5	when	when	SCONJ
ejpam-5305	39	6	each	each	DET
ejpam-5305	39	7	edge	edge	NOUN
ejpam-5305	39	8	xy	xy	PROPN
ejpam-5305	39	9	is	be	AUX
ejpam-5305	39	10	assigned	assign	VERB
ejpam-5305	39	11	the	the	DET
ejpam-5305	39	12	label	label	NOUN
ejpam-5305	39	13	|f(x	|f(x	PROPN
ejpam-5305	39	14	)	)	PUNCT
ejpam-5305	39	15	−	−	PROPN
ejpam-5305	40	1	f(y)|	f(y)|	PROPN
ejpam-5305	40	2	,	,	PUNCT
ejpam-5305	40	3	the	the	DET
ejpam-5305	40	4	edge	edge	NOUN
ejpam-5305	40	5	label	label	NOUN
ejpam-5305	40	6	set	set	NOUN
ejpam-5305	40	7	is	be	AUX
ejpam-5305	40	8	equal	equal	ADJ
ejpam-5305	40	9	to	to	ADP
ejpam-5305	40	10	{	{	PUNCT
ejpam-5305	40	11	1	1	NUM
ejpam-5305	40	12	,	,	PUNCT
ejpam-5305	40	13	2	2	NUM
ejpam-5305	40	14	,	,	PUNCT
ejpam-5305	40	15	3	3	NUM
ejpam-5305	40	16	,	,	PUNCT
ejpam-5305	40	17	.	.	PUNCT
ejpam-5305	40	18	.	.	PUNCT
ejpam-5305	41	1	.	.	PUNCT
ejpam-5305	41	2	,	,	PUNCT
ejpam-5305	41	3	n	n	CCONJ
ejpam-5305	41	4	}	}	PUNCT
ejpam-5305	41	5	.	.	PUNCT
ejpam-5305	42	1	a	a	DET
ejpam-5305	42	2	tree	tree	NOUN
ejpam-5305	42	3	with	with	ADP
ejpam-5305	42	4	a	a	DET
ejpam-5305	42	5	graceful	graceful	ADJ
ejpam-5305	42	6	labeling	labeling	NOUN
ejpam-5305	42	7	is	be	AUX
ejpam-5305	42	8	called	call	VERB
ejpam-5305	42	9	a	a	DET
ejpam-5305	42	10	graceful	graceful	ADJ
ejpam-5305	42	11	tree	tree	NOUN
ejpam-5305	42	12	.	.	PUNCT
ejpam-5305	43	1	in	in	ADP
ejpam-5305	43	2	[	[	X
ejpam-5305	43	3	12	12	NUM
ejpam-5305	43	4	]	]	X
ejpam-5305	43	5	hrnčiar	hrnčiar	NOUN
ejpam-5305	43	6	and	and	CCONJ
ejpam-5305	43	7	haviar	haviar	NOUN
ejpam-5305	43	8	proved	prove	VERB
ejpam-5305	43	9	lemma	lemma	PROPN
ejpam-5305	43	10	1	1	NUM
ejpam-5305	43	11	and	and	CCONJ
ejpam-5305	43	12	in	in	ADP
ejpam-5305	43	13	[	[	X
ejpam-5305	43	14	10	10	NUM
ejpam-5305	43	15	]	]	SYM
ejpam-5305	43	16	jampachon	jampachon	ADJ
ejpam-5305	43	17	and	and	CCONJ
ejpam-5305	43	18	poomsa	poomsa	PROPN
ejpam-5305	43	19	-	-	PUNCT
ejpam-5305	43	20	ard	ard	PROPN
ejpam-5305	43	21	proved	prove	VERB
ejpam-5305	43	22	lemma	lemma	PROPN
ejpam-5305	43	23	2	2	NUM
ejpam-5305	43	24	and	and	CCONJ
ejpam-5305	43	25	lemma	lemma	PROPN
ejpam-5305	43	26	3	3	X
ejpam-5305	43	27	.	.	PUNCT
ejpam-5305	44	1	lemma	lemma	PROPN
ejpam-5305	44	2	1	1	X
ejpam-5305	44	3	.	.	PUNCT
ejpam-5305	45	1	let	let	VERB
ejpam-5305	45	2	t	t	NOUN
ejpam-5305	45	3	be	be	AUX
ejpam-5305	45	4	a	a	DET
ejpam-5305	45	5	tree	tree	NOUN
ejpam-5305	45	6	with	with	ADP
ejpam-5305	45	7	n	n	ADP
ejpam-5305	45	8	edges	edge	NOUN
ejpam-5305	45	9	and	and	CCONJ
ejpam-5305	45	10	a	a	DET
ejpam-5305	45	11	graceful	graceful	ADJ
ejpam-5305	45	12	labeling	labeling	NOUN
ejpam-5305	45	13	f	f	NOUN
ejpam-5305	45	14	.	.	PUNCT
ejpam-5305	46	1	then	then	ADV
ejpam-5305	46	2	,	,	PUNCT
ejpam-5305	46	3	the	the	DET
ejpam-5305	46	4	function	function	NOUN
ejpam-5305	46	5	f∗	f∗	NOUN
ejpam-5305	46	6	:	:	PUNCT
ejpam-5305	46	7	v	v	X
ejpam-5305	46	8	(	(	PUNCT
ejpam-5305	46	9	t	t	PROPN
ejpam-5305	46	10	)	)	PUNCT
ejpam-5305	46	11	→	→	SYM
ejpam-5305	46	12	{	{	PUNCT
ejpam-5305	46	13	0	0	NUM
ejpam-5305	46	14	,	,	PUNCT
ejpam-5305	46	15	1	1	NUM
ejpam-5305	46	16	,	,	PUNCT
ejpam-5305	46	17	2	2	NUM
ejpam-5305	46	18	,	,	PUNCT
ejpam-5305	46	19	.	.	PUNCT
ejpam-5305	46	20	.	.	PUNCT
ejpam-5305	46	21	.	.	PUNCT
ejpam-5305	47	1	,	,	PUNCT
ejpam-5305	47	2	n	n	CCONJ
ejpam-5305	47	3	}	}	PUNCT
ejpam-5305	47	4	is	be	AUX
ejpam-5305	47	5	given	give	VERB
ejpam-5305	47	6	f∗(v	f∗(v	PROPN
ejpam-5305	47	7	)	)	PUNCT
ejpam-5305	48	1	=	=	NUM
ejpam-5305	48	2	n−	n−	PROPN
ejpam-5305	48	3	f(v	f(v	NOUN
ejpam-5305	48	4	)	)	PUNCT
ejpam-5305	48	5	is	be	AUX
ejpam-5305	48	6	also	also	ADV
ejpam-5305	48	7	a	a	DET
ejpam-5305	48	8	graceful	graceful	ADJ
ejpam-5305	48	9	labeling	labeling	NOUN
ejpam-5305	48	10	of	of	ADP
ejpam-5305	48	11	t.	t.	PROPN
ejpam-5305	48	12	lemma	lemma	PROPN
ejpam-5305	48	13	2	2	X
ejpam-5305	48	14	.	.	PUNCT
ejpam-5305	49	1	let	let	VERB
ejpam-5305	49	2	p2n	p2n	PROPN
ejpam-5305	49	3	be	be	AUX
ejpam-5305	49	4	a	a	DET
ejpam-5305	49	5	path	path	NOUN
ejpam-5305	49	6	graph	graph	NOUN
ejpam-5305	49	7	with	with	ADP
ejpam-5305	49	8	v	v	NOUN
ejpam-5305	49	9	(	(	PUNCT
ejpam-5305	49	10	p2n	p2n	PROPN
ejpam-5305	49	11	)	)	PUNCT
ejpam-5305	49	12	=	=	PRON
ejpam-5305	49	13	{	{	PUNCT
ejpam-5305	49	14	v1	v1	PROPN
ejpam-5305	49	15	,	,	PUNCT
ejpam-5305	49	16	v2	v2	PROPN
ejpam-5305	49	17	,	,	PUNCT
ejpam-5305	49	18	v3	v3	PROPN
ejpam-5305	49	19	,	,	PUNCT
ejpam-5305	49	20	.	.	PUNCT
ejpam-5305	49	21	.	.	PUNCT
ejpam-5305	50	1	.	.	PUNCT
ejpam-5305	51	1	,	,	PUNCT
ejpam-5305	51	2	v2n	v2n	NOUN
ejpam-5305	51	3	}	}	PUNCT
ejpam-5305	51	4	and	and	CCONJ
ejpam-5305	51	5	let	let	VERB
ejpam-5305	51	6	m	m	VERB
ejpam-5305	51	7	=	=	PRON
ejpam-5305	51	8	{	{	PUNCT
ejpam-5305	51	9	a+1	a+1	X
ejpam-5305	51	10	,	,	PUNCT
ejpam-5305	51	11	a+2	a+2	PRON
ejpam-5305	51	12	,	,	PUNCT
ejpam-5305	51	13	.	.	PUNCT
ejpam-5305	51	14	.	.	PUNCT
ejpam-5305	52	1	.	.	PUNCT
ejpam-5305	53	1	,	,	PUNCT
ejpam-5305	53	2	a+n	a+n	PROPN
ejpam-5305	53	3	,	,	PUNCT
ejpam-5305	53	4	m+a+1,m+a+2	m+a+1,m+a+2	PROPN
ejpam-5305	53	5	,	,	PUNCT
ejpam-5305	53	6	.	.	PUNCT
ejpam-5305	53	7	.	.	PUNCT
ejpam-5305	54	1	.	.	PUNCT
ejpam-5305	55	1	,	,	PUNCT
ejpam-5305	55	2	m+a+n	m+a+n	NOUN
ejpam-5305	55	3	}	}	PUNCT
ejpam-5305	55	4	,	,	PUNCT
ejpam-5305	55	5	m	m	VERB
ejpam-5305	55	6	≥	≥	NOUN
ejpam-5305	55	7	n	n	ADV
ejpam-5305	55	8	and	and	CCONJ
ejpam-5305	55	9	a	a	DET
ejpam-5305	55	10	≥	≥	NOUN
ejpam-5305	55	11	0	0	NUM
ejpam-5305	55	12	.	.	PUNCT
ejpam-5305	56	1	then	then	ADV
ejpam-5305	56	2	,	,	PUNCT
ejpam-5305	56	3	there	there	PRON
ejpam-5305	56	4	is	be	VERB
ejpam-5305	56	5	a	a	DET
ejpam-5305	56	6	bijective	bijective	ADJ
ejpam-5305	56	7	labeling	labeling	NOUN
ejpam-5305	56	8	f	f	NOUN
ejpam-5305	56	9	:	:	PUNCT
ejpam-5305	56	10	v	v	X
ejpam-5305	56	11	(	(	PUNCT
ejpam-5305	56	12	p2n	p2n	PROPN
ejpam-5305	56	13	)	)	PUNCT
ejpam-5305	56	14	→	→	SYM
ejpam-5305	56	15	m	m	VERB
ejpam-5305	56	16	such	such	ADJ
ejpam-5305	56	17	that	that	DET
ejpam-5305	56	18	f(v1	f(v1	NOUN
ejpam-5305	56	19	)	)	PUNCT
ejpam-5305	56	20	=	=	SYM
ejpam-5305	56	21	i	i	PRON
ejpam-5305	56	22	or	or	CCONJ
ejpam-5305	56	23	f(v2n	f(v2n	PROPN
ejpam-5305	56	24	)	)	PUNCT
ejpam-5305	56	25	=	=	SYM
ejpam-5305	57	1	i	i	PROPN
ejpam-5305	57	2	,	,	PUNCT
ejpam-5305	57	3	where	where	SCONJ
ejpam-5305	57	4	i	i	PRON
ejpam-5305	57	5	∈	∈	VERB
ejpam-5305	57	6	m	m	VERB
ejpam-5305	57	7	and	and	CCONJ
ejpam-5305	57	8	the	the	DET
ejpam-5305	57	9	edge	edge	NOUN
ejpam-5305	57	10	label	label	NOUN
ejpam-5305	57	11	set	set	VERB
ejpam-5305	57	12	is	be	AUX
ejpam-5305	57	13	{	{	PUNCT
ejpam-5305	57	14	m−	m−	PROPN
ejpam-5305	57	15	n+	n+	NUM
ejpam-5305	57	16	1,m−	1,m−	NUM
ejpam-5305	57	17	n+	n+	PUNCT
ejpam-5305	57	18	2	2	NUM
ejpam-5305	57	19	,	,	PUNCT
ejpam-5305	57	20	.	.	PUNCT
ejpam-5305	57	21	.	.	PUNCT
ejpam-5305	57	22	.	.	PUNCT
ejpam-5305	58	1	,	,	PUNCT
ejpam-5305	58	2	m+	m+	NUM
ejpam-5305	58	3	n−	n−	NOUN
ejpam-5305	58	4	1	1	NUM
ejpam-5305	58	5	}	}	PUNCT
ejpam-5305	58	6	.	.	PUNCT
ejpam-5305	59	1	lemma	lemma	PROPN
ejpam-5305	59	2	3	3	X
ejpam-5305	59	3	.	.	PUNCT
ejpam-5305	60	1	let	let	VERB
ejpam-5305	60	2	pn	pn	PART
ejpam-5305	60	3	be	be	AUX
ejpam-5305	60	4	a	a	DET
ejpam-5305	60	5	path	path	NOUN
ejpam-5305	60	6	graph	graph	NOUN
ejpam-5305	60	7	with	with	ADP
ejpam-5305	60	8	v	v	NOUN
ejpam-5305	60	9	(	(	PUNCT
ejpam-5305	60	10	pn	pn	NOUN
ejpam-5305	60	11	)	)	PUNCT
ejpam-5305	60	12	=	=	SYM
ejpam-5305	60	13	{	{	PUNCT
ejpam-5305	60	14	v1	v1	PROPN
ejpam-5305	60	15	,	,	PUNCT
ejpam-5305	60	16	v2	v2	PROPN
ejpam-5305	60	17	,	,	PUNCT
ejpam-5305	60	18	v3	v3	PROPN
ejpam-5305	60	19	,	,	PUNCT
ejpam-5305	60	20	.	.	PUNCT
ejpam-5305	60	21	.	.	PUNCT
ejpam-5305	61	1	.	.	PUNCT
ejpam-5305	62	1	,	,	PUNCT
ejpam-5305	62	2	vn	vn	X
ejpam-5305	62	3	}	}	PUNCT
ejpam-5305	62	4	and	and	CCONJ
ejpam-5305	62	5	let	let	VERB
ejpam-5305	62	6	m	m	VERB
ejpam-5305	62	7	=	=	PUNCT
ejpam-5305	62	8	{	{	PUNCT
ejpam-5305	62	9	m	m	PROPN
ejpam-5305	62	10	,	,	PUNCT
ejpam-5305	62	11	m+	m+	NOUN
ejpam-5305	62	12	1,m	1,m	NOUN
ejpam-5305	62	13	+	+	CCONJ
ejpam-5305	62	14	2	2	NUM
ejpam-5305	62	15	,	,	PUNCT
ejpam-5305	62	16	.	.	PUNCT
ejpam-5305	62	17	.	.	PUNCT
ejpam-5305	62	18	.	.	PUNCT
ejpam-5305	63	1	,	,	PUNCT
ejpam-5305	63	2	m	m	VERB
ejpam-5305	63	3	+	+	ADJ
ejpam-5305	63	4	n	n	CCONJ
ejpam-5305	63	5	−	−	PROPN
ejpam-5305	63	6	1	1	NUM
ejpam-5305	63	7	}	}	PUNCT
ejpam-5305	63	8	.	.	PUNCT
ejpam-5305	64	1	then	then	ADV
ejpam-5305	64	2	,	,	PUNCT
ejpam-5305	64	3	there	there	PRON
ejpam-5305	64	4	is	be	VERB
ejpam-5305	64	5	a	a	DET
ejpam-5305	64	6	bijective	bijective	ADJ
ejpam-5305	64	7	labeling	labeling	NOUN
ejpam-5305	64	8	f	f	NOUN
ejpam-5305	64	9	:	:	PUNCT
ejpam-5305	64	10	v	v	X
ejpam-5305	64	11	(	(	PUNCT
ejpam-5305	64	12	pn	pn	NOUN
ejpam-5305	64	13	)	)	PUNCT
ejpam-5305	64	14	→	→	SYM
ejpam-5305	64	15	m	m	VERB
ejpam-5305	64	16	such	such	ADJ
ejpam-5305	64	17	that	that	DET
ejpam-5305	64	18	f(v1	f(v1	NOUN
ejpam-5305	64	19	)	)	PUNCT
ejpam-5305	64	20	=	=	SYM
ejpam-5305	64	21	i	i	PROPN
ejpam-5305	64	22	or	or	CCONJ
ejpam-5305	64	23	f(vn	f(vn	PROPN
ejpam-5305	64	24	)	)	PUNCT
ejpam-5305	65	1	=	=	SYM
ejpam-5305	65	2	i	i	PROPN
ejpam-5305	65	3	,	,	PUNCT
ejpam-5305	65	4	where	where	SCONJ
ejpam-5305	65	5	i	i	PRON
ejpam-5305	65	6	∈	∈	VERB
ejpam-5305	65	7	m	m	VERB
ejpam-5305	65	8	and	and	CCONJ
ejpam-5305	65	9	the	the	DET
ejpam-5305	65	10	edge	edge	NOUN
ejpam-5305	65	11	label	label	NOUN
ejpam-5305	65	12	set	set	VERB
ejpam-5305	65	13	is	be	AUX
ejpam-5305	65	14	{	{	PUNCT
ejpam-5305	65	15	1	1	NUM
ejpam-5305	65	16	,	,	PUNCT
ejpam-5305	65	17	2	2	NUM
ejpam-5305	65	18	,	,	PUNCT
ejpam-5305	65	19	3	3	NUM
ejpam-5305	65	20	,	,	PUNCT
ejpam-5305	65	21	.	.	PUNCT
ejpam-5305	65	22	.	.	PUNCT
ejpam-5305	65	23	.	.	PUNCT
ejpam-5305	66	1	,	,	PUNCT
ejpam-5305	66	2	n−	n−	NOUN
ejpam-5305	66	3	1	1	NUM
ejpam-5305	66	4	}	}	PUNCT
ejpam-5305	66	5	.	.	PUNCT
ejpam-5305	67	1	let	let	VERB
ejpam-5305	67	2	t	t	NOUN
ejpam-5305	67	3	be	be	AUX
ejpam-5305	67	4	a	a	DET
ejpam-5305	67	5	tree	tree	NOUN
ejpam-5305	67	6	and	and	CCONJ
ejpam-5305	67	7	let	let	VERB
ejpam-5305	67	8	u	u	PRON
ejpam-5305	67	9	be	be	AUX
ejpam-5305	67	10	a	a	DET
ejpam-5305	67	11	leaf	leaf	NOUN
ejpam-5305	67	12	of	of	ADP
ejpam-5305	67	13	t	t	PROPN
ejpam-5305	67	14	.	.	PUNCT
ejpam-5305	68	1	let	let	VERB
ejpam-5305	68	2	t	t	PROPN
ejpam-5305	68	3	(	(	PUNCT
ejpam-5305	68	4	u	u	PROPN
ejpam-5305	68	5	,	,	PUNCT
ejpam-5305	68	6	u1	u1	NOUN
ejpam-5305	68	7	,	,	PUNCT
ejpam-5305	68	8	u2	u2	NOUN
ejpam-5305	68	9	,	,	PUNCT
ejpam-5305	68	10	.	.	PUNCT
ejpam-5305	68	11	.	.	PUNCT
ejpam-5305	69	1	.	.	PUNCT
ejpam-5305	70	1	,	,	PUNCT
ejpam-5305	70	2	un	un	AUX
ejpam-5305	70	3	)	)	PUNCT
ejpam-5305	70	4	be	be	VERB
ejpam-5305	70	5	the	the	DET
ejpam-5305	70	6	tree	tree	NOUN
ejpam-5305	70	7	obtained	obtain	VERB
ejpam-5305	70	8	from	from	ADP
ejpam-5305	70	9	t	t	NOUN
ejpam-5305	70	10	by	by	ADP
ejpam-5305	70	11	adding	add	VERB
ejpam-5305	70	12	the	the	DET
ejpam-5305	70	13	vertices	vertex	NOUN
ejpam-5305	70	14	u1	u1	NOUN
ejpam-5305	70	15	,	,	PUNCT
ejpam-5305	70	16	u2	u2	NOUN
ejpam-5305	70	17	,	,	PUNCT
ejpam-5305	70	18	u3	u3	NOUN
ejpam-5305	70	19	,	,	PUNCT
ejpam-5305	70	20	.	.	PUNCT
ejpam-5305	70	21	.	.	PUNCT
ejpam-5305	71	1	.	.	PUNCT
ejpam-5305	72	1	,	,	PUNCT
ejpam-5305	72	2	un	un	PROPN
ejpam-5305	72	3	and	and	CCONJ
ejpam-5305	72	4	the	the	DET
ejpam-5305	72	5	edges	edge	NOUN
ejpam-5305	72	6	uu1	uu1	PROPN
ejpam-5305	72	7	,	,	PUNCT
ejpam-5305	72	8	u1u2	u1u2	NOUN
ejpam-5305	72	9	,	,	PUNCT
ejpam-5305	72	10	.	.	PUNCT
ejpam-5305	72	11	.	.	PUNCT
ejpam-5305	72	12	.	.	PUNCT
ejpam-5305	73	1	,	,	PUNCT
ejpam-5305	73	2	un−1un	un−1un	PRON
ejpam-5305	73	3	.	.	PUNCT
ejpam-5305	74	1	in	in	ADP
ejpam-5305	74	2	[	[	X
ejpam-5305	74	3	13	13	NUM
ejpam-5305	74	4	]	]	SYM
ejpam-5305	74	5	sangsura	sangsura	NOUN
ejpam-5305	74	6	and	and	CCONJ
ejpam-5305	74	7	poomsa	poomsa	PROPN
ejpam-5305	74	8	-	-	PUNCT
ejpam-5305	74	9	ard	ard	PROPN
ejpam-5305	74	10	have	have	AUX
ejpam-5305	74	11	proved	prove	VERB
ejpam-5305	74	12	lemma	lemma	PROPN
ejpam-5305	74	13	4	4	X
ejpam-5305	74	14	.	.	PUNCT
ejpam-5305	75	1	lemma	lemma	PROPN
ejpam-5305	75	2	4	4	X
ejpam-5305	75	3	.	.	PUNCT
ejpam-5305	76	1	if	if	SCONJ
ejpam-5305	76	2	a	a	DET
ejpam-5305	76	3	tree	tree	NOUN
ejpam-5305	76	4	t	t	NOUN
ejpam-5305	76	5	has	have	VERB
ejpam-5305	76	6	a	a	DET
ejpam-5305	76	7	graceful	graceful	ADJ
ejpam-5305	76	8	labeling	labeling	NOUN
ejpam-5305	76	9	f	f	PRON
ejpam-5305	76	10	such	such	ADJ
ejpam-5305	76	11	that	that	DET
ejpam-5305	76	12	f(u	f(u	PROPN
ejpam-5305	76	13	)	)	PUNCT
ejpam-5305	77	1	=	=	SYM
ejpam-5305	77	2	0	0	NUM
ejpam-5305	77	3	,	,	PUNCT
ejpam-5305	77	4	where	where	SCONJ
ejpam-5305	77	5	u	u	NOUN
ejpam-5305	77	6	is	be	AUX
ejpam-5305	77	7	a	a	DET
ejpam-5305	77	8	leaf	leaf	NOUN
ejpam-5305	77	9	of	of	ADP
ejpam-5305	77	10	t	t	PROPN
ejpam-5305	77	11	,	,	PUNCT
ejpam-5305	77	12	then	then	ADV
ejpam-5305	77	13	t	t	PROPN
ejpam-5305	77	14	(	(	PUNCT
ejpam-5305	77	15	u	u	PROPN
ejpam-5305	77	16	,	,	PUNCT
ejpam-5305	77	17	u1	u1	NOUN
ejpam-5305	77	18	,	,	PUNCT
ejpam-5305	77	19	u2	u2	NOUN
ejpam-5305	77	20	,	,	PUNCT
ejpam-5305	77	21	.	.	PUNCT
ejpam-5305	77	22	.	.	PUNCT
ejpam-5305	77	23	.	.	PUNCT
ejpam-5305	78	1	,	,	PUNCT
ejpam-5305	78	2	un	un	PROPN
ejpam-5305	78	3	)	)	PUNCT
ejpam-5305	78	4	is	be	AUX
ejpam-5305	78	5	graceful	graceful	ADJ
ejpam-5305	78	6	for	for	ADP
ejpam-5305	78	7	n	n	X
ejpam-5305	78	8	≥	≥	NUM
ejpam-5305	78	9	1	1	NUM
ejpam-5305	78	10	.	.	PUNCT
ejpam-5305	78	11	a	a	DET
ejpam-5305	78	12	spider	spider	NOUN
ejpam-5305	78	13	graph	graph	NOUN
ejpam-5305	78	14	or	or	CCONJ
ejpam-5305	78	15	spider	spider	NOUN
ejpam-5305	78	16	is	be	AUX
ejpam-5305	78	17	a	a	DET
ejpam-5305	78	18	tree	tree	NOUN
ejpam-5305	78	19	with	with	ADP
ejpam-5305	78	20	at	at	ADP
ejpam-5305	78	21	most	most	ADV
ejpam-5305	78	22	one	one	NUM
ejpam-5305	78	23	vertex	vertex	NOUN
ejpam-5305	78	24	of	of	ADP
ejpam-5305	78	25	degree	degree	NOUN
ejpam-5305	78	26	greater	great	ADJ
ejpam-5305	78	27	than	than	ADP
ejpam-5305	78	28	2	2	NUM
ejpam-5305	78	29	and	and	CCONJ
ejpam-5305	78	30	this	this	DET
ejpam-5305	78	31	vertex	vertex	NOUN
ejpam-5305	78	32	is	be	AUX
ejpam-5305	78	33	called	call	VERB
ejpam-5305	78	34	the	the	DET
ejpam-5305	78	35	branch	branch	NOUN
ejpam-5305	78	36	vertex	vertex	NOUN
ejpam-5305	78	37	and	and	CCONJ
ejpam-5305	78	38	is	be	AUX
ejpam-5305	78	39	denoted	denote	VERB
ejpam-5305	78	40	by	by	ADP
ejpam-5305	78	41	v0	v0	PROPN
ejpam-5305	78	42	.	.	PUNCT
ejpam-5305	79	1	a	a	DET
ejpam-5305	79	2	leg	leg	NOUN
ejpam-5305	79	3	of	of	ADP
ejpam-5305	79	4	a	a	DET
ejpam-5305	79	5	spider	spider	NOUN
ejpam-5305	79	6	graph	graph	NOUN
ejpam-5305	79	7	is	be	AUX
ejpam-5305	79	8	a	a	DET
ejpam-5305	79	9	path	path	NOUN
ejpam-5305	79	10	from	from	ADP
ejpam-5305	79	11	the	the	DET
ejpam-5305	79	12	branch	branch	NOUN
ejpam-5305	79	13	vertex	vertex	NOUN
ejpam-5305	79	14	to	to	ADP
ejpam-5305	79	15	a	a	DET
ejpam-5305	79	16	leaf	leaf	NOUN
ejpam-5305	79	17	of	of	ADP
ejpam-5305	79	18	the	the	DET
ejpam-5305	79	19	tree	tree	NOUN
ejpam-5305	79	20	.	.	PUNCT
ejpam-5305	80	1	let	let	VERB
ejpam-5305	80	2	sn(m1,m2	sn(m1,m2	NOUN
ejpam-5305	80	3	,	,	PUNCT
ejpam-5305	80	4	.	.	PUNCT
ejpam-5305	80	5	.	.	PUNCT
ejpam-5305	81	1	.	.	PUNCT
ejpam-5305	82	1	,	,	PUNCT
ejpam-5305	82	2	mk	mk	PROPN
ejpam-5305	82	3	)	)	PUNCT
ejpam-5305	82	4	,	,	PUNCT
ejpam-5305	83	1	n	n	PRON
ejpam-5305	83	2	≥	≥	NOUN
ejpam-5305	83	3	k	k	NOUN
ejpam-5305	83	4	,	,	PUNCT
ejpam-5305	83	5	denote	denote	VERB
ejpam-5305	83	6	a	a	DET
ejpam-5305	83	7	spider	spider	NOUN
ejpam-5305	83	8	graph	graph	NOUN
ejpam-5305	83	9	of	of	ADP
ejpam-5305	83	10	n	n	NOUN
ejpam-5305	83	11	legs	leg	NOUN
ejpam-5305	83	12	such	such	ADJ
ejpam-5305	83	13	that	that	SCONJ
ejpam-5305	83	14	its	its	PRON
ejpam-5305	83	15	legs	leg	NOUN
ejpam-5305	83	16	have	have	VERB
ejpam-5305	83	17	length	length	NOUN
ejpam-5305	83	18	one	one	NUM
ejpam-5305	83	19	except	except	SCONJ
ejpam-5305	83	20	for	for	ADP
ejpam-5305	83	21	k	k	PROPN
ejpam-5305	83	22	legs	leg	NOUN
ejpam-5305	83	23	of	of	ADP
ejpam-5305	83	24	lengths	length	NOUN
ejpam-5305	83	25	m1,m2	m1,m2	PROPN
ejpam-5305	83	26	,	,	PUNCT
ejpam-5305	83	27	.	.	PUNCT
ejpam-5305	83	28	.	.	PUNCT
ejpam-5305	84	1	.	.	PUNCT
ejpam-5305	85	1	,	,	PUNCT
ejpam-5305	85	2	mk	mk	PROPN
ejpam-5305	85	3	,	,	PUNCT
ejpam-5305	85	4	where	where	SCONJ
ejpam-5305	85	5	mi	mi	PROPN
ejpam-5305	85	6	≥	≥	PROPN
ejpam-5305	85	7	2	2	NUM
ejpam-5305	85	8	for	for	ADP
ejpam-5305	85	9	all	all	DET
ejpam-5305	85	10	i	i	PRON
ejpam-5305	85	11	=	=	NOUN
ejpam-5305	85	12	1	1	NUM
ejpam-5305	85	13	,	,	PUNCT
ejpam-5305	85	14	2	2	NUM
ejpam-5305	85	15	,	,	PUNCT
ejpam-5305	85	16	.	.	PUNCT
ejpam-5305	85	17	.	.	PUNCT
ejpam-5305	85	18	.	.	PUNCT
ejpam-5305	86	1	,	,	PUNCT
ejpam-5305	86	2	k.	k.	PROPN
ejpam-5305	86	3	in	in	ADP
ejpam-5305	86	4	[	[	X
ejpam-5305	86	5	11	11	NUM
ejpam-5305	86	6	]	]	SYM
ejpam-5305	86	7	panpa	panpa	NOUN
ejpam-5305	86	8	and	and	CCONJ
ejpam-5305	86	9	poomsa	poomsa	PROPN
ejpam-5305	86	10	-	-	PUNCT
ejpam-5305	86	11	ard	ard	PROPN
ejpam-5305	86	12	proved	prove	VERB
ejpam-5305	86	13	lemma	lemma	PROPN
ejpam-5305	86	14	5	5	NUM
ejpam-5305	86	15	and	and	CCONJ
ejpam-5305	86	16	lemma	lemma	PROPN
ejpam-5305	86	17	6	6	NUM
ejpam-5305	86	18	.	.	PUNCT
ejpam-5305	87	1	lemma	lemma	PROPN
ejpam-5305	87	2	5	5	NUM
ejpam-5305	87	3	.	.	PUNCT
ejpam-5305	88	1	if	if	SCONJ
ejpam-5305	88	2	sk(m1,m2	sk(m1,m2	NUM
ejpam-5305	88	3	,	,	PUNCT
ejpam-5305	88	4	.	.	PUNCT
ejpam-5305	88	5	.	.	PUNCT
ejpam-5305	89	1	.	.	PUNCT
ejpam-5305	90	1	,	,	PUNCT
ejpam-5305	90	2	mk	mk	PROPN
ejpam-5305	90	3	)	)	PUNCT
ejpam-5305	90	4	has	have	VERB
ejpam-5305	90	5	a	a	DET
ejpam-5305	90	6	graceful	graceful	ADJ
ejpam-5305	90	7	labeling	labeling	NOUN
ejpam-5305	91	1	f	f	PRON
ejpam-5305	91	2	such	such	ADJ
ejpam-5305	91	3	that	that	DET
ejpam-5305	91	4	f(v0	f(v0	NOUN
ejpam-5305	91	5	)	)	PUNCT
ejpam-5305	92	1	=	=	SYM
ejpam-5305	92	2	0	0	NUM
ejpam-5305	92	3	,	,	PUNCT
ejpam-5305	92	4	then	then	ADV
ejpam-5305	92	5	there	there	PRON
ejpam-5305	92	6	is	be	VERB
ejpam-5305	92	7	a	a	DET
ejpam-5305	92	8	graceful	graceful	ADJ
ejpam-5305	92	9	labeling	labeling	NOUN
ejpam-5305	92	10	f	f	NOUN
ejpam-5305	92	11	′	′	NUM
ejpam-5305	92	12	of	of	ADP
ejpam-5305	92	13	sn(m1,m2	sn(m1,m2	NOUN
ejpam-5305	92	14	,	,	PUNCT
ejpam-5305	92	15	.	.	PUNCT
ejpam-5305	92	16	.	.	PUNCT
ejpam-5305	93	1	.	.	PUNCT
ejpam-5305	94	1	,	,	PUNCT
ejpam-5305	94	2	mk	mk	PROPN
ejpam-5305	94	3	)	)	PUNCT
ejpam-5305	95	1	such	such	ADJ
ejpam-5305	95	2	that	that	SCONJ
ejpam-5305	95	3	f	f	PROPN
ejpam-5305	95	4	′(v0	′(v0	PROPN
ejpam-5305	95	5	)	)	PUNCT
ejpam-5305	96	1	=	=	SYM
ejpam-5305	96	2	0	0	X
ejpam-5305	96	3	.	.	PUNCT
ejpam-5305	97	1	lemma	lemma	PROPN
ejpam-5305	97	2	6	6	NUM
ejpam-5305	97	3	.	.	PUNCT
ejpam-5305	98	1	if	if	SCONJ
ejpam-5305	98	2	sk(m1,m2	sk(m1,m2	NUM
ejpam-5305	98	3	,	,	PUNCT
ejpam-5305	98	4	.	.	PUNCT
ejpam-5305	98	5	.	.	PUNCT
ejpam-5305	99	1	.	.	PUNCT
ejpam-5305	100	1	,	,	PUNCT
ejpam-5305	100	2	mk	mk	PROPN
ejpam-5305	100	3	)	)	PUNCT
ejpam-5305	100	4	has	have	VERB
ejpam-5305	100	5	a	a	DET
ejpam-5305	100	6	graceful	graceful	ADJ
ejpam-5305	100	7	labeling	labeling	NOUN
ejpam-5305	101	1	f	f	PRON
ejpam-5305	101	2	such	such	ADJ
ejpam-5305	101	3	that	that	DET
ejpam-5305	101	4	f(v0	f(v0	NOUN
ejpam-5305	101	5	)	)	PUNCT
ejpam-5305	102	1	=	=	SYM
ejpam-5305	102	2	0	0	NUM
ejpam-5305	102	3	,	,	PUNCT
ejpam-5305	102	4	then	then	ADV
ejpam-5305	102	5	sn(l	sn(l	ADJ
ejpam-5305	102	6	,	,	PUNCT
ejpam-5305	102	7	m1,m2	m1,m2	PROPN
ejpam-5305	102	8	,	,	PUNCT
ejpam-5305	102	9	.	.	PUNCT
ejpam-5305	102	10	.	.	PUNCT
ejpam-5305	102	11	.	.	PUNCT
ejpam-5305	103	1	,	,	PUNCT
ejpam-5305	103	2	mk	mk	PROPN
ejpam-5305	103	3	)	)	PUNCT
ejpam-5305	103	4	is	be	AUX
ejpam-5305	103	5	also	also	ADV
ejpam-5305	103	6	graceful	graceful	ADJ
ejpam-5305	103	7	.	.	PUNCT
ejpam-5305	104	1	k.	k.	PROPN
ejpam-5305	104	2	saengsura	saengsura	PROPN
ejpam-5305	104	3	,	,	PUNCT
ejpam-5305	104	4	t.	t.	PROPN
ejpam-5305	104	5	poomsa	poomsa	PROPN
ejpam-5305	104	6	-	-	PUNCT
ejpam-5305	104	7	ard	ard	PROPN
ejpam-5305	104	8	/	/	PUNCT
ejpam-5305	104	9	eur	eur	PROPN
ejpam-5305	104	10	.	.	PUNCT
ejpam-5305	105	1	j.	j.	PROPN
ejpam-5305	105	2	pure	pure	PROPN
ejpam-5305	105	3	appl	appl	PROPN
ejpam-5305	105	4	.	.	PROPN
ejpam-5305	105	5	math	math	PROPN
ejpam-5305	105	6	,	,	PUNCT
ejpam-5305	105	7	18	18	NUM
ejpam-5305	105	8	(	(	PUNCT
ejpam-5305	105	9	2	2	NUM
ejpam-5305	105	10	)	)	PUNCT
ejpam-5305	105	11	(	(	PUNCT
ejpam-5305	105	12	2025	2025	NUM
ejpam-5305	105	13	)	)	PUNCT
ejpam-5305	105	14	,	,	PUNCT
ejpam-5305	105	15	5305	5305	NUM
ejpam-5305	105	16	3	3	NUM
ejpam-5305	105	17	of	of	ADP
ejpam-5305	105	18	8	8	NUM
ejpam-5305	105	19	in	in	ADP
ejpam-5305	105	20	this	this	DET
ejpam-5305	105	21	paper	paper	NOUN
ejpam-5305	105	22	we	we	PRON
ejpam-5305	105	23	study	study	VERB
ejpam-5305	105	24	graceful	graceful	ADJ
ejpam-5305	105	25	labeling	labeling	NOUN
ejpam-5305	105	26	of	of	ADP
ejpam-5305	105	27	some	some	DET
ejpam-5305	105	28	class	class	NOUN
ejpam-5305	105	29	of	of	ADP
ejpam-5305	105	30	spider	spider	NOUN
ejpam-5305	105	31	graphs	graph	NOUN
ejpam-5305	105	32	whose	whose	DET
ejpam-5305	105	33	graceful	graceful	ADJ
ejpam-5305	105	34	labeling	labeling	NOUN
ejpam-5305	105	35	is	be	AUX
ejpam-5305	105	36	generated	generate	VERB
ejpam-5305	105	37	by	by	ADP
ejpam-5305	105	38	our	our	PRON
ejpam-5305	105	39	two	two	NUM
ejpam-5305	105	40	special	special	ADJ
ejpam-5305	105	41	labeling	labeling	NOUN
ejpam-5305	105	42	types	type	NOUN
ejpam-5305	105	43	.	.	PUNCT
ejpam-5305	106	1	now	now	ADV
ejpam-5305	106	2	,	,	PUNCT
ejpam-5305	106	3	we	we	PRON
ejpam-5305	106	4	consider	consider	VERB
ejpam-5305	106	5	the	the	DET
ejpam-5305	106	6	special	special	ADJ
ejpam-5305	106	7	labeling	labeling	NOUN
ejpam-5305	106	8	of	of	ADP
ejpam-5305	106	9	a	a	DET
ejpam-5305	106	10	graph	graph	NOUN
ejpam-5305	106	11	as	as	ADP
ejpam-5305	106	12	the	the	DET
ejpam-5305	106	13	following	following	NOUN
ejpam-5305	106	14	.	.	PUNCT
ejpam-5305	107	1	let	let	VERB
ejpam-5305	107	2	t	t	NOUN
ejpam-5305	107	3	be	be	AUX
ejpam-5305	107	4	a	a	DET
ejpam-5305	107	5	spider	spider	NOUN
ejpam-5305	107	6	graph	graph	NOUN
ejpam-5305	107	7	of	of	ADP
ejpam-5305	107	8	n	n	NOUN
ejpam-5305	107	9	legs	leg	NOUN
ejpam-5305	107	10	and	and	CCONJ
ejpam-5305	107	11	for	for	SCONJ
ejpam-5305	107	12	each	each	DET
ejpam-5305	107	13	i	i	PROPN
ejpam-5305	107	14	-	-	PUNCT
ejpam-5305	107	15	th	th	X
ejpam-5305	107	16	leg	leg	NOUN
ejpam-5305	107	17	of	of	ADP
ejpam-5305	107	18	t	t	PROPN
ejpam-5305	107	19	is	be	AUX
ejpam-5305	107	20	represented	represent	VERB
ejpam-5305	107	21	by	by	ADP
ejpam-5305	107	22	a	a	DET
ejpam-5305	107	23	path	path	NOUN
ejpam-5305	107	24	v	v	ADP
ejpam-5305	107	25	−	−	NOUN
ejpam-5305	107	26	vimi	vimi	VERB
ejpam-5305	107	27	where	where	SCONJ
ejpam-5305	107	28	an	an	DET
ejpam-5305	107	29	integer	integer	NOUN
ejpam-5305	107	30	mi	mi	PROPN
ejpam-5305	107	31	≥	≥	PROPN
ejpam-5305	107	32	2	2	NUM
ejpam-5305	107	33	for	for	ADP
ejpam-5305	107	34	all	all	DET
ejpam-5305	107	35	i	i	PRON
ejpam-5305	107	36	=	=	NOUN
ejpam-5305	107	37	1	1	NUM
ejpam-5305	107	38	,	,	PUNCT
ejpam-5305	107	39	2	2	NUM
ejpam-5305	107	40	,	,	PUNCT
ejpam-5305	107	41	.	.	PUNCT
ejpam-5305	107	42	.	.	PUNCT
ejpam-5305	108	1	.	.	PUNCT
ejpam-5305	109	1	,	,	PUNCT
ejpam-5305	109	2	n	n	CCONJ
ejpam-5305	109	3	as	as	SCONJ
ejpam-5305	109	4	shown	show	VERB
ejpam-5305	109	5	in	in	ADP
ejpam-5305	109	6	figure	figure	NOUN
ejpam-5305	109	7	1	1	NUM
ejpam-5305	109	8	.	.	PUNCT
ejpam-5305	110	1	it	it	PRON
ejpam-5305	110	2	can	can	AUX
ejpam-5305	110	3	be	be	AUX
ejpam-5305	110	4	seen	see	VERB
ejpam-5305	110	5	that	that	SCONJ
ejpam-5305	110	6	|v	|v	PROPN
ejpam-5305	110	7	(	(	PUNCT
ejpam-5305	110	8	t	t	NOUN
ejpam-5305	110	9	)	)	PUNCT
ejpam-5305	110	10	|	|	ADV
ejpam-5305	110	11	=	=	SYM
ejpam-5305	110	12	m1	m1	PROPN
ejpam-5305	110	13	+	+	NUM
ejpam-5305	110	14	m2	m2	PROPN
ejpam-5305	110	15	+	+	CCONJ
ejpam-5305	110	16	m3	m3	PROPN
ejpam-5305	110	17	+	+	CCONJ
ejpam-5305	110	18	.	.	PUNCT
ejpam-5305	110	19	.	.	PUNCT
ejpam-5305	110	20	.	.	PUNCT
ejpam-5305	111	1	+	+	CCONJ
ejpam-5305	111	2	mn	mn	PROPN
ejpam-5305	111	3	+	+	NOUN
ejpam-5305	111	4	1	1	X
ejpam-5305	111	5	.	.	X
ejpam-5305	111	6	we	we	PRON
ejpam-5305	111	7	now	now	ADV
ejpam-5305	111	8	introduce	introduce	VERB
ejpam-5305	111	9	a	a	DET
ejpam-5305	111	10	labeling	labeling	NOUN
ejpam-5305	111	11	f	f	PROPN
ejpam-5305	111	12	of	of	ADP
ejpam-5305	111	13	t	t	PROPN
ejpam-5305	111	14	,	,	PUNCT
ejpam-5305	111	15	which	which	PRON
ejpam-5305	111	16	will	will	AUX
ejpam-5305	111	17	be	be	AUX
ejpam-5305	111	18	used	use	VERB
ejpam-5305	111	19	to	to	PART
ejpam-5305	111	20	generate	generate	VERB
ejpam-5305	111	21	graceful	graceful	ADJ
ejpam-5305	111	22	labelings	labeling	NOUN
ejpam-5305	111	23	in	in	ADP
ejpam-5305	111	24	the	the	DET
ejpam-5305	111	25	following	following	ADJ
ejpam-5305	111	26	subsequent	subsequent	ADJ
ejpam-5305	111	27	part	part	NOUN
ejpam-5305	111	28	.	.	PUNCT
ejpam-5305	111	29	.	.	PUNCT
ejpam-5305	112	1	r	r	NOUN
ejpam-5305	112	2	rr	rr	PROPN
ejpam-5305	112	3	rr	rr	NOUN
ejpam-5305	113	1	r	r	NOUN
ejpam-5305	113	2	r	r	NOUN
ejpam-5305	113	3	...	...	PUNCT
ejpam-5305	114	1	r	r	NOUN
ejpam-5305	114	2	r	r	NOUN
ejpam-5305	114	3	r	r	NOUN
ejpam-5305	114	4	.	.	PUNCT
ejpam-5305	114	5	.	.	PUNCT
ejpam-5305	114	6	.	.	PUNCT
ejpam-5305	115	1	r	r	NOUN
ejpam-5305	115	2	rr	rr	PROPN
ejpam-5305	115	3	.	.	PUNCT
ejpam-5305	115	4	.	.	PUNCT
ejpam-5305	115	5	.	.	PUNCT
ejpam-5305	116	1	r	r	NOUN
ejpam-5305	116	2	r	r	NOUN
ejpam-5305	116	3	r	r	NOUN
ejpam-5305	116	4	.	.	PUNCT
ejpam-5305	116	5	.	.	PUNCT
ejpam-5305	116	6	.	.	PUNCT
ejpam-5305	117	1	r	r	NOUN
ejpam-5305	117	2	r	r	NOUN
ejpam-5305	117	3	.	.	PUNCT
ejpam-5305	117	4	.	.	PUNCT
ejpam-5305	117	5	.	.	PUNCT
ejpam-5305	117	6	.	.	PUNCT
ejpam-5305	117	7	.	.	PUNCT
ejpam-5305	117	8	.	.	PUNCT
ejpam-5305	117	9	.	.	PUNCT
ejpam-5305	117	10	.	.	PUNCT
ejpam-5305	117	11	.	.	PUNCT
ejpam-5305	117	12	.	.	PUNCT
ejpam-5305	117	13	.	.	PUNCT
ejpam-5305	117	14	.	.	PUNCT
ejpam-5305	117	15	.	.	PUNCT
ejpam-5305	117	16	.	.	PUNCT
ejpam-5305	117	17	.rr	.rr	PUNCT
ejpam-5305	118	1	rr	rr	NOUN
ejpam-5305	119	1	r	r	NOUN
ejpam-5305	119	2	rr	rr	NOUN
ejpam-5305	119	3	r	r	NOUN
ejpam-5305	119	4	rr	rr	PROPN
ejpam-5305	119	5	rr	rr	NOUN
ejpam-5305	119	6	r	r	PROPN
ejpam-5305	119	7	rr	rr	PROPN
ejpam-5305	119	8	�	�	PROPN
ejpam-5305	119	9	�	�	PROPN
ejpam-5305	119	10	�	�	PROPN
ejpam-5305	119	11	@	@	ADP
ejpam-5305	119	12	@	@	ADP
ejpam-5305	119	13	@	@	ADP
ejpam-5305	119	14	j	j	PROPN
ejpam-5305	119	15	j	j	PROPN
ejpam-5305	119	16	j	j	PROPN
ejpam-5305	119	17	jj	jj	PROPN
ejpam-5305	119	18	�	�	PROPN
ejpam-5305	119	19	�	�	PROPN
ejpam-5305	119	20	�	�	PROPN
ejpam-5305	119	21	�	�	PROPN
ejpam-5305	119	22	�	�	PROPN
ejpam-5305	119	23	�	�	PROPN
ejpam-5305	120	1	a	a	DET
ejpam-5305	120	2	a	a	PRON
ejpam-5305	120	3	a	a	DET
ejpam-5305	120	4	a	a	PRON
ejpam-5305	120	5	a	a	DET
ejpam-5305	120	6	a	a	DET
ejpam-5305	120	7	hhh	hhh	PROPN
ejpam-5305	120	8	�	�	PROPN
ejpam-5305	120	9	�	�	PROPN
ejpam-5305	120	10	�	�	PROPN
ejpam-5305	120	11	v	v	ADP
ejpam-5305	120	12	v11	v11	NOUN
ejpam-5305	120	13	v51	v51	NOUN
ejpam-5305	120	14	v31	v31	NOUN
ejpam-5305	120	15	v21	v21	NOUN
ejpam-5305	120	16	v12	v12	VERB
ejpam-5305	120	17	v52	v52	NOUN
ejpam-5305	120	18	v32	v32	VERB
ejpam-5305	120	19	v22	v22	NOUN
ejpam-5305	120	20	v1m1	v1m1	X
ejpam-5305	120	21	v5m5	v5m5	X
ejpam-5305	120	22	v3m3	v3m3	X
ejpam-5305	120	23	v2m2	v2m2	X
ejpam-5305	120	24	v1(m1−1	v1(m1−1	ADJ
ejpam-5305	120	25	)	)	PUNCT
ejpam-5305	120	26	v5(m5−1	v5(m5−1	PROPN
ejpam-5305	120	27	)	)	PUNCT
ejpam-5305	120	28	v3(m3−1	v3(m3−1	NOUN
ejpam-5305	120	29	)	)	PUNCT
ejpam-5305	120	30	v2(m2−1	v2(m2−1	NOUN
ejpam-5305	120	31	)	)	PUNCT
ejpam-5305	120	32	v41	v41	NOUN
ejpam-5305	120	33	v42	v42	PROPN
ejpam-5305	120	34	v4(m4−1	v4(m4−1	PROPN
ejpam-5305	120	35	)	)	PUNCT
ejpam-5305	120	36	v4m4	v4m4	DET
ejpam-5305	120	37	v(n−2)1	v(n−2)1	NOUN
ejpam-5305	120	38	v(n−2)2	v(n−2)2	NOUN
ejpam-5305	120	39	v(n−2)(mn−2−1	v(n−2)(mn−2−1	NOUN
ejpam-5305	120	40	)	)	PUNCT
ejpam-5305	120	41	v(n−2)mn−2	v(n−2)mn−2	ADV
ejpam-5305	120	42	v(n−1)1	v(n−1)1	PROPN
ejpam-5305	120	43	v(n−1)2	v(n−1)2	PROPN
ejpam-5305	120	44	v(n−1)(mn−1−1	v(n−1)(mn−1−1	PROPN
ejpam-5305	120	45	)	)	PUNCT
ejpam-5305	120	46	v(n−1)mn−1	v(n−1)mn−1	ADP
ejpam-5305	120	47	vn1	vn1	NOUN
ejpam-5305	120	48	vn2	vn2	NOUN
ejpam-5305	120	49	vn(mn−1	vn(mn−1	NUM
ejpam-5305	120	50	)	)	PUNCT
ejpam-5305	121	1	vnmn	vnmn	NOUN
ejpam-5305	121	2	figure	figure	NOUN
ejpam-5305	121	3	1	1	NUM
ejpam-5305	121	4	let	let	VERB
ejpam-5305	121	5	t	t	PROPN
ejpam-5305	121	6	′	′	NOUN
ejpam-5305	121	7	be	be	AUX
ejpam-5305	121	8	obtained	obtain	VERB
ejpam-5305	121	9	from	from	ADP
ejpam-5305	121	10	t	t	NOUN
ejpam-5305	121	11	by	by	ADP
ejpam-5305	121	12	deleting	delete	VERB
ejpam-5305	121	13	the	the	DET
ejpam-5305	121	14	edges	edge	NOUN
ejpam-5305	121	15	vv31	vv31	PROPN
ejpam-5305	121	16	,	,	PUNCT
ejpam-5305	121	17	vv41	vv41	PROPN
ejpam-5305	121	18	,	,	PUNCT
ejpam-5305	121	19	.	.	PUNCT
ejpam-5305	121	20	.	.	PUNCT
ejpam-5305	122	1	.	.	PUNCT
ejpam-5305	123	1	,	,	PUNCT
ejpam-5305	123	2	vv(n−1)1	vv(n−1)1	NOUN
ejpam-5305	123	3	,	,	PUNCT
ejpam-5305	123	4	vvn1	vvn1	NOUN
ejpam-5305	123	5	and	and	CCONJ
ejpam-5305	123	6	adding	add	VERB
ejpam-5305	123	7	the	the	DET
ejpam-5305	123	8	edges	edge	NOUN
ejpam-5305	123	9	v2m2v31	v2m2v31	NOUN
ejpam-5305	123	10	,	,	PUNCT
ejpam-5305	123	11	v3m3v41	v3m3v41	PROPN
ejpam-5305	123	12	,	,	PUNCT
ejpam-5305	123	13	.	.	PUNCT
ejpam-5305	123	14	.	.	PUNCT
ejpam-5305	124	1	.	.	PUNCT
ejpam-5305	125	1	,	,	PUNCT
ejpam-5305	125	2	v(n−2)m(n−2	v(n−2)m(n−2	PROPN
ejpam-5305	125	3	)	)	PUNCT
ejpam-5305	125	4	v(n−1)1	v(n−1)1	PROPN
ejpam-5305	125	5	,	,	PUNCT
ejpam-5305	125	6	v(n−1)m(n−1	v(n−1)m(n−1	PROPN
ejpam-5305	125	7	)	)	PUNCT
ejpam-5305	125	8	vn1	vn1	NOUN
ejpam-5305	125	9	.	.	PUNCT
ejpam-5305	126	1	then	then	ADV
ejpam-5305	126	2	we	we	PRON
ejpam-5305	126	3	get	get	VERB
ejpam-5305	126	4	t	t	PROPN
ejpam-5305	126	5	′	′	NUM
ejpam-5305	126	6	is	be	AUX
ejpam-5305	126	7	a	a	DET
ejpam-5305	126	8	path	path	NOUN
ejpam-5305	126	9	.	.	PUNCT
ejpam-5305	127	1	next	next	ADV
ejpam-5305	127	2	we	we	PRON
ejpam-5305	127	3	will	will	AUX
ejpam-5305	127	4	label	label	VERB
ejpam-5305	127	5	the	the	DET
ejpam-5305	127	6	vertices	vertex	NOUN
ejpam-5305	127	7	of	of	ADP
ejpam-5305	127	8	path	path	NOUN
ejpam-5305	127	9	t	t	PROPN
ejpam-5305	127	10	′	′	NUM
ejpam-5305	127	11	in	in	ADP
ejpam-5305	127	12	the	the	DET
ejpam-5305	127	13	following	following	ADJ
ejpam-5305	127	14	way	way	NOUN
ejpam-5305	127	15	.	.	PUNCT
ejpam-5305	128	1	case	case	NOUN
ejpam-5305	128	2	m1	m1	PROPN
ejpam-5305	128	3	is	be	AUX
ejpam-5305	128	4	odd	odd	ADJ
ejpam-5305	128	5	.	.	PUNCT
ejpam-5305	129	1	let	let	VERB
ejpam-5305	129	2	m1	m1	NOUN
ejpam-5305	129	3	=	=	NOUN
ejpam-5305	130	1	2p	2p	NOUN
ejpam-5305	130	2	+	+	CCONJ
ejpam-5305	130	3	1	1	X
ejpam-5305	130	4	.	.	PUNCT
ejpam-5305	130	5	then	then	ADV
ejpam-5305	130	6	we	we	PRON
ejpam-5305	130	7	label	label	VERB
ejpam-5305	130	8	the	the	DET
ejpam-5305	130	9	vertices	vertex	NOUN
ejpam-5305	130	10	of	of	ADP
ejpam-5305	130	11	t	t	NOUN
ejpam-5305	130	12	′	′	NUM
ejpam-5305	130	13	alternating	alternating	NOUN
ejpam-5305	130	14	between	between	ADP
ejpam-5305	130	15	the	the	DET
ejpam-5305	130	16	highest	high	ADJ
ejpam-5305	130	17	and	and	CCONJ
ejpam-5305	130	18	the	the	DET
ejpam-5305	130	19	lowest	low	ADJ
ejpam-5305	130	20	remaining	remain	VERB
ejpam-5305	130	21	unused	unused	ADJ
ejpam-5305	130	22	labels	label	NOUN
ejpam-5305	130	23	by	by	ADP
ejpam-5305	130	24	starting	start	VERB
ejpam-5305	130	25	with	with	ADP
ejpam-5305	130	26	f(v1m1	f(v1m1	NOUN
ejpam-5305	130	27	)	)	PUNCT
ejpam-5305	131	1	=	=	SYM
ejpam-5305	131	2	m1	m1	PROPN
ejpam-5305	132	1	+	+	PROPN
ejpam-5305	132	2	m2	m2	PROPN
ejpam-5305	132	3	+	+	X
ejpam-5305	132	4	.	.	PUNCT
ejpam-5305	132	5	.	.	PUNCT
ejpam-5305	133	1	.+mn	.+mn	PROPN
ejpam-5305	133	2	and	and	CCONJ
ejpam-5305	133	3	f(v1(m1−1	f(v1(m1−1	NUM
ejpam-5305	133	4	)	)	PUNCT
ejpam-5305	133	5	)	)	PUNCT
ejpam-5305	134	1	=	=	SYM
ejpam-5305	135	1	o.	o.	INTJ
ejpam-5305	135	2	then	then	ADV
ejpam-5305	135	3	we	we	PRON
ejpam-5305	135	4	get	get	VERB
ejpam-5305	135	5	f(v	f(v	NOUN
ejpam-5305	135	6	)	)	PUNCT
ejpam-5305	136	1	=	=	SYM
ejpam-5305	136	2	p.	p.	NOUN
ejpam-5305	136	3	case	case	NOUN
ejpam-5305	136	4	m1	m1	NOUN
ejpam-5305	136	5	is	be	AUX
ejpam-5305	136	6	even	even	ADV
ejpam-5305	136	7	.	.	PUNCT
ejpam-5305	137	1	let	let	VERB
ejpam-5305	137	2	m1	m1	PROPN
ejpam-5305	137	3	=	=	PUNCT
ejpam-5305	138	1	2p′.	2p′.	NUM
ejpam-5305	138	2	then	then	ADV
ejpam-5305	138	3	we	we	PRON
ejpam-5305	138	4	label	label	VERB
ejpam-5305	138	5	the	the	DET
ejpam-5305	138	6	vertices	vertex	NOUN
ejpam-5305	138	7	of	of	ADP
ejpam-5305	138	8	t	t	NOUN
ejpam-5305	138	9	′	′	NUM
ejpam-5305	138	10	alternating	alternating	NOUN
ejpam-5305	138	11	between	between	ADP
ejpam-5305	138	12	the	the	DET
ejpam-5305	138	13	lowest	low	ADJ
ejpam-5305	138	14	and	and	CCONJ
ejpam-5305	138	15	the	the	DET
ejpam-5305	138	16	highest	high	ADJ
ejpam-5305	138	17	remaining	remain	VERB
ejpam-5305	138	18	unused	unused	ADJ
ejpam-5305	138	19	labels	label	NOUN
ejpam-5305	138	20	by	by	ADP
ejpam-5305	138	21	starting	start	VERB
ejpam-5305	138	22	with	with	ADP
ejpam-5305	138	23	f(v1m1	f(v1m1	NOUN
ejpam-5305	138	24	)	)	PUNCT
ejpam-5305	138	25	=	=	SYM
ejpam-5305	138	26	0	0	NUM
ejpam-5305	138	27	and	and	CCONJ
ejpam-5305	138	28	f(v1(m1−1	f(v1(m1−1	NUM
ejpam-5305	138	29	)	)	PUNCT
ejpam-5305	138	30	)	)	PUNCT
ejpam-5305	139	1	=	=	SYM
ejpam-5305	139	2	m1	m1	PROPN
ejpam-5305	140	1	+	+	PROPN
ejpam-5305	140	2	m2	m2	PROPN
ejpam-5305	140	3	+	+	X
ejpam-5305	140	4	.	.	PUNCT
ejpam-5305	140	5	.	.	PUNCT
ejpam-5305	141	1	.+mn	.+mn	PROPN
ejpam-5305	141	2	.	.	PUNCT
ejpam-5305	142	1	then	then	ADV
ejpam-5305	142	2	we	we	PRON
ejpam-5305	142	3	get	get	VERB
ejpam-5305	142	4	f(v	f(v	NOUN
ejpam-5305	142	5	)	)	PUNCT
ejpam-5305	143	1	=	=	SYM
ejpam-5305	143	2	p′.	p′.	NOUN
ejpam-5305	143	3	notice	notice	VERB
ejpam-5305	143	4	that	that	SCONJ
ejpam-5305	143	5	the	the	DET
ejpam-5305	143	6	labeling	labeling	NOUN
ejpam-5305	143	7	in	in	ADP
ejpam-5305	143	8	two	two	NUM
ejpam-5305	143	9	cases	case	NOUN
ejpam-5305	143	10	of	of	ADP
ejpam-5305	143	11	t	t	NOUN
ejpam-5305	143	12	′	′	NUM
ejpam-5305	143	13	are	be	AUX
ejpam-5305	143	14	graceful	graceful	ADJ
ejpam-5305	143	15	.	.	PUNCT
ejpam-5305	144	1	we	we	PRON
ejpam-5305	144	2	will	will	AUX
ejpam-5305	144	3	use	use	VERB
ejpam-5305	144	4	the	the	DET
ejpam-5305	144	5	labeling	labeling	NOUN
ejpam-5305	144	6	in	in	ADP
ejpam-5305	144	7	two	two	NUM
ejpam-5305	144	8	cases	case	NOUN
ejpam-5305	144	9	to	to	PART
ejpam-5305	144	10	generate	generate	VERB
ejpam-5305	144	11	graceful	graceful	ADJ
ejpam-5305	144	12	labeling	labeling	NOUN
ejpam-5305	144	13	of	of	ADP
ejpam-5305	144	14	some	some	DET
ejpam-5305	144	15	spider	spider	NOUN
ejpam-5305	144	16	graphs	graph	NOUN
ejpam-5305	144	17	in	in	ADP
ejpam-5305	144	18	the	the	DET
ejpam-5305	144	19	later	later	ADJ
ejpam-5305	144	20	part	part	NOUN
ejpam-5305	144	21	.	.	PUNCT
ejpam-5305	145	1	for	for	ADP
ejpam-5305	145	2	convenience	convenience	NOUN
ejpam-5305	145	3	,	,	PUNCT
ejpam-5305	145	4	we	we	PRON
ejpam-5305	145	5	call	call	VERB
ejpam-5305	145	6	the	the	DET
ejpam-5305	145	7	labeling	labeling	NOUN
ejpam-5305	145	8	constructed	construct	VERB
ejpam-5305	145	9	in	in	ADP
ejpam-5305	145	10	the	the	DET
ejpam-5305	145	11	first	first	ADJ
ejpam-5305	145	12	case	case	NOUN
ejpam-5305	145	13	and	and	CCONJ
ejpam-5305	145	14	the	the	DET
ejpam-5305	145	15	second	second	ADJ
ejpam-5305	145	16	case	case	NOUN
ejpam-5305	145	17	as	as	ADP
ejpam-5305	145	18	type	type	NOUN
ejpam-5305	145	19	i	i	PRON
ejpam-5305	145	20	and	and	CCONJ
ejpam-5305	145	21	type	type	PROPN
ejpam-5305	145	22	ii	ii	PROPN
ejpam-5305	145	23	,	,	PUNCT
ejpam-5305	145	24	respectively	respectively	ADV
ejpam-5305	145	25	.	.	PUNCT
ejpam-5305	146	1	2	2	X
ejpam-5305	146	2	.	.	X
ejpam-5305	146	3	main	main	ADJ
ejpam-5305	146	4	results	result	NOUN
ejpam-5305	146	5	in	in	ADP
ejpam-5305	146	6	this	this	DET
ejpam-5305	146	7	section	section	NOUN
ejpam-5305	146	8	we	we	PRON
ejpam-5305	146	9	want	want	VERB
ejpam-5305	146	10	to	to	PART
ejpam-5305	146	11	show	show	VERB
ejpam-5305	146	12	that	that	SCONJ
ejpam-5305	146	13	some	some	DET
ejpam-5305	146	14	spider	spider	NOUN
ejpam-5305	146	15	graphs	graph	NOUN
ejpam-5305	146	16	are	be	AUX
ejpam-5305	146	17	graceful	graceful	ADJ
ejpam-5305	146	18	.	.	PUNCT
ejpam-5305	147	1	at	at	ADP
ejpam-5305	147	2	first	first	ADV
ejpam-5305	147	3	,	,	PUNCT
ejpam-5305	147	4	for	for	ADP
ejpam-5305	147	5	a	a	DET
ejpam-5305	147	6	spider	spider	NOUN
ejpam-5305	147	7	graph	graph	NOUN
ejpam-5305	147	8	that	that	PRON
ejpam-5305	147	9	has	have	VERB
ejpam-5305	147	10	n	n	DET
ejpam-5305	147	11	legs	leg	NOUN
ejpam-5305	147	12	and	and	CCONJ
ejpam-5305	147	13	there	there	PRON
ejpam-5305	147	14	exist	exist	VERB
ejpam-5305	147	15	n	n	CCONJ
ejpam-5305	147	16	−	−	NUM
ejpam-5305	147	17	2	2	NUM
ejpam-5305	147	18	legs	leg	NOUN
ejpam-5305	147	19	of	of	ADP
ejpam-5305	147	20	them	they	PRON
ejpam-5305	147	21	are	be	AUX
ejpam-5305	147	22	even	even	ADV
ejpam-5305	147	23	lengths	length	NOUN
ejpam-5305	147	24	as	as	ADP
ejpam-5305	147	25	the	the	DET
ejpam-5305	147	26	following	follow	VERB
ejpam-5305	147	27	theorem	theorem	NOUN
ejpam-5305	147	28	.	.	PUNCT
ejpam-5305	147	29	theorem	theorem	NOUN
ejpam-5305	147	30	1	1	NUM
ejpam-5305	147	31	.	.	PUNCT
ejpam-5305	148	1	let	let	VERB
ejpam-5305	148	2	t	t	PROPN
ejpam-5305	148	3	be	be	AUX
ejpam-5305	148	4	a	a	DET
ejpam-5305	148	5	spider	spider	NOUN
ejpam-5305	148	6	graph	graph	NOUN
ejpam-5305	148	7	of	of	ADP
ejpam-5305	148	8	n	n	NOUN
ejpam-5305	148	9	legs	leg	NOUN
ejpam-5305	148	10	with	with	ADP
ejpam-5305	148	11	lengths	length	NOUN
ejpam-5305	148	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	148	13	,	,	PUNCT
ejpam-5305	148	14	.	.	PUNCT
ejpam-5305	148	15	.	.	PUNCT
ejpam-5305	149	1	.	.	PUNCT
ejpam-5305	150	1	,	,	PUNCT
ejpam-5305	150	2	mn	mn	PROPN
ejpam-5305	150	3	.	.	PUNCT
ejpam-5305	151	1	if	if	SCONJ
ejpam-5305	151	2	mi+1	mi+1	NUM
ejpam-5305	151	3	=	=	SYM
ejpam-5305	151	4	2mi	2mi	NOUN
ejpam-5305	151	5	for	for	ADP
ejpam-5305	151	6	i	i	PRON
ejpam-5305	151	7	=	=	SYM
ejpam-5305	151	8	2	2	NUM
ejpam-5305	151	9	,	,	PUNCT
ejpam-5305	151	10	3	3	NUM
ejpam-5305	151	11	,	,	PUNCT
ejpam-5305	151	12	.	.	PUNCT
ejpam-5305	151	13	.	.	PUNCT
ejpam-5305	151	14	.	.	PUNCT
ejpam-5305	152	1	,	,	PUNCT
ejpam-5305	152	2	n−	n−	NOUN
ejpam-5305	152	3	1	1	NUM
ejpam-5305	152	4	,	,	PUNCT
ejpam-5305	152	5	then	then	ADV
ejpam-5305	152	6	t	t	PROPN
ejpam-5305	152	7	is	be	AUX
ejpam-5305	152	8	a	a	DET
ejpam-5305	152	9	graceful	graceful	ADJ
ejpam-5305	152	10	labeling	labeling	NOUN
ejpam-5305	152	11	graph	graph	NOUN
ejpam-5305	152	12	.	.	PUNCT
ejpam-5305	153	1	proof	proof	NOUN
ejpam-5305	153	2	.	.	PUNCT
ejpam-5305	154	1	let	let	VERB
ejpam-5305	154	2	t	t	NOUN
ejpam-5305	154	3	be	be	AUX
ejpam-5305	154	4	the	the	DET
ejpam-5305	154	5	spider	spider	NOUN
ejpam-5305	154	6	as	as	SCONJ
ejpam-5305	154	7	shown	show	VERB
ejpam-5305	154	8	in	in	ADP
ejpam-5305	154	9	figure	figure	NOUN
ejpam-5305	154	10	1	1	NUM
ejpam-5305	154	11	.	.	PUNCT
ejpam-5305	154	12	k.	k.	PROPN
ejpam-5305	154	13	saengsura	saengsura	PROPN
ejpam-5305	154	14	,	,	PUNCT
ejpam-5305	154	15	t.	t.	PROPN
ejpam-5305	154	16	poomsa	poomsa	PROPN
ejpam-5305	154	17	-	-	PUNCT
ejpam-5305	154	18	ard	ard	PROPN
ejpam-5305	154	19	/	/	PUNCT
ejpam-5305	154	20	eur	eur	PROPN
ejpam-5305	154	21	.	.	PUNCT
ejpam-5305	155	1	j.	j.	PROPN
ejpam-5305	155	2	pure	pure	PROPN
ejpam-5305	155	3	appl	appl	PROPN
ejpam-5305	155	4	.	.	PROPN
ejpam-5305	155	5	math	math	PROPN
ejpam-5305	155	6	,	,	PUNCT
ejpam-5305	155	7	18	18	NUM
ejpam-5305	155	8	(	(	PUNCT
ejpam-5305	155	9	2	2	NUM
ejpam-5305	155	10	)	)	PUNCT
ejpam-5305	155	11	(	(	PUNCT
ejpam-5305	155	12	2025	2025	NUM
ejpam-5305	155	13	)	)	PUNCT
ejpam-5305	155	14	,	,	PUNCT
ejpam-5305	155	15	5305	5305	NUM
ejpam-5305	155	16	4	4	NUM
ejpam-5305	155	17	of	of	ADP
ejpam-5305	155	18	8	8	NUM
ejpam-5305	155	19	case	case	NOUN
ejpam-5305	155	20	1	1	NUM
ejpam-5305	155	21	.	.	PUNCT
ejpam-5305	156	1	if	if	SCONJ
ejpam-5305	156	2	m1	m1	PROPN
ejpam-5305	156	3	is	be	AUX
ejpam-5305	156	4	odd	odd	ADJ
ejpam-5305	156	5	,	,	PUNCT
ejpam-5305	156	6	then	then	ADV
ejpam-5305	156	7	let	let	VERB
ejpam-5305	156	8	f	f	PRON
ejpam-5305	156	9	be	be	AUX
ejpam-5305	156	10	the	the	DET
ejpam-5305	156	11	labeling	labeling	NOUN
ejpam-5305	156	12	of	of	ADP
ejpam-5305	156	13	t	t	PROPN
ejpam-5305	156	14	of	of	ADP
ejpam-5305	156	15	type	type	NOUN
ejpam-5305	156	16	i.	i.	NOUN
ejpam-5305	156	17	consider	consider	VERB
ejpam-5305	156	18	|f(v31	|f(v31	NOUN
ejpam-5305	156	19	)	)	PUNCT
ejpam-5305	156	20	−	−	PROPN
ejpam-5305	156	21	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	156	22	.	.	PUNCT
ejpam-5305	157	1	since	since	SCONJ
ejpam-5305	157	2	m3	m3	PROPN
ejpam-5305	157	3	=	=	SYM
ejpam-5305	157	4	2m2	2m2	NUM
ejpam-5305	157	5	,	,	PUNCT
ejpam-5305	157	6	then	then	ADV
ejpam-5305	157	7	by	by	ADP
ejpam-5305	157	8	the	the	DET
ejpam-5305	157	9	way	way	NOUN
ejpam-5305	157	10	of	of	ADP
ejpam-5305	157	11	labeling	labeling	NOUN
ejpam-5305	157	12	of	of	ADP
ejpam-5305	157	13	t	t	PROPN
ejpam-5305	157	14	of	of	ADP
ejpam-5305	157	15	type	type	NOUN
ejpam-5305	157	16	i	i	PRON
ejpam-5305	157	17	there	there	PRON
ejpam-5305	157	18	exists	exist	VERB
ejpam-5305	157	19	v′	v′	PROPN
ejpam-5305	157	20	∈	∈	PROPN
ejpam-5305	157	21	{	{	PUNCT
ejpam-5305	157	22	v31	v31	NOUN
ejpam-5305	157	23	,	,	PUNCT
ejpam-5305	157	24	v32	v32	PROPN
ejpam-5305	157	25	,	,	PUNCT
ejpam-5305	157	26	.	.	PUNCT
ejpam-5305	157	27	.	.	PUNCT
ejpam-5305	158	1	.	.	PUNCT
ejpam-5305	159	1	,	,	PUNCT
ejpam-5305	159	2	v3m3	v3m3	X
ejpam-5305	159	3	}	}	PUNCT
ejpam-5305	159	4	such	such	ADJ
ejpam-5305	159	5	that	that	SCONJ
ejpam-5305	159	6	|f(v′	|f(v′	PROPN
ejpam-5305	159	7	)	)	PUNCT
ejpam-5305	159	8	−	−	PROPN
ejpam-5305	159	9	f(v)|	f(v)|	NOUN
ejpam-5305	159	10	=	=	NOUN
ejpam-5305	159	11	|f(v31	|f(v31	NOUN
ejpam-5305	159	12	)	)	PUNCT
ejpam-5305	159	13	−	−	PROPN
ejpam-5305	159	14	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	159	15	.	.	PUNCT
ejpam-5305	160	1	since	since	SCONJ
ejpam-5305	160	2	m3	m3	PROPN
ejpam-5305	160	3	is	be	AUX
ejpam-5305	160	4	even	even	ADV
ejpam-5305	160	5	,	,	PUNCT
ejpam-5305	160	6	then	then	ADV
ejpam-5305	160	7	by	by	ADP
ejpam-5305	160	8	lemma	lemma	PROPN
ejpam-5305	160	9	2	2	NUM
ejpam-5305	160	10	,	,	PUNCT
ejpam-5305	160	11	we	we	PRON
ejpam-5305	160	12	can	can	AUX
ejpam-5305	160	13	change	change	VERB
ejpam-5305	160	14	the	the	DET
ejpam-5305	160	15	labeling	labeling	NOUN
ejpam-5305	160	16	f	f	PROPN
ejpam-5305	160	17	at	at	ADP
ejpam-5305	160	18	v31	v31	PROPN
ejpam-5305	160	19	,	,	PUNCT
ejpam-5305	160	20	v32	v32	PROPN
ejpam-5305	160	21	,	,	PUNCT
ejpam-5305	160	22	.	.	PUNCT
ejpam-5305	160	23	.	.	PUNCT
ejpam-5305	161	1	.	.	PUNCT
ejpam-5305	162	1	,	,	PUNCT
ejpam-5305	162	2	v3m3	v3m3	X
ejpam-5305	162	3	to	to	PART
ejpam-5305	162	4	be	be	AUX
ejpam-5305	162	5	the	the	DET
ejpam-5305	162	6	labeling	labeling	NOUN
ejpam-5305	162	7	f	f	NOUN
ejpam-5305	162	8	′	′	NUM
ejpam-5305	162	9	such	such	ADJ
ejpam-5305	162	10	that	that	PRON
ejpam-5305	162	11	|f	|f	PROPN
ejpam-5305	162	12	′(v31	′(v31	PROPN
ejpam-5305	162	13	)	)	PUNCT
ejpam-5305	162	14	−	−	PROPN
ejpam-5305	162	15	f(v)|	f(v)|	NOUN
ejpam-5305	162	16	=	=	NOUN
ejpam-5305	162	17	|f(v31	|f(v31	NOUN
ejpam-5305	162	18	)	)	PUNCT
ejpam-5305	162	19	−	−	PROPN
ejpam-5305	162	20	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	162	21	,	,	PUNCT
ejpam-5305	162	22	the	the	DET
ejpam-5305	162	23	set	set	NOUN
ejpam-5305	162	24	of	of	ADP
ejpam-5305	162	25	vertices	vertex	NOUN
ejpam-5305	162	26	labeling	labeling	NOUN
ejpam-5305	162	27	of	of	ADP
ejpam-5305	162	28	f	f	PROPN
ejpam-5305	162	29	and	and	CCONJ
ejpam-5305	162	30	the	the	DET
ejpam-5305	162	31	set	set	NOUN
ejpam-5305	162	32	of	of	ADP
ejpam-5305	162	33	vertices	vertex	NOUN
ejpam-5305	162	34	labeling	labeling	NOUN
ejpam-5305	162	35	of	of	ADP
ejpam-5305	162	36	f	f	PROPN
ejpam-5305	162	37	′	′	NUM
ejpam-5305	162	38	are	be	AUX
ejpam-5305	162	39	the	the	DET
ejpam-5305	162	40	same	same	ADJ
ejpam-5305	162	41	and	and	CCONJ
ejpam-5305	162	42	the	the	DET
ejpam-5305	162	43	set	set	NOUN
ejpam-5305	162	44	of	of	ADP
ejpam-5305	162	45	edge	edge	NOUN
ejpam-5305	162	46	labeling	labeling	NOUN
ejpam-5305	162	47	of	of	ADP
ejpam-5305	162	48	f	f	PROPN
ejpam-5305	162	49	and	and	CCONJ
ejpam-5305	162	50	the	the	DET
ejpam-5305	162	51	set	set	NOUN
ejpam-5305	162	52	of	of	ADP
ejpam-5305	162	53	edge	edge	NOUN
ejpam-5305	162	54	labeling	labeling	NOUN
ejpam-5305	162	55	of	of	ADP
ejpam-5305	162	56	f	f	PROPN
ejpam-5305	162	57	′	′	NUM
ejpam-5305	162	58	are	be	AUX
ejpam-5305	162	59	the	the	DET
ejpam-5305	162	60	same	same	ADJ
ejpam-5305	162	61	.	.	PUNCT
ejpam-5305	163	1	further	far	ADV
ejpam-5305	163	2	,	,	PUNCT
ejpam-5305	163	3	for	for	SCONJ
ejpam-5305	163	4	any	any	DET
ejpam-5305	163	5	i	i	NOUN
ejpam-5305	163	6	,	,	PUNCT
ejpam-5305	163	7	4	4	NUM
ejpam-5305	163	8	≤	≤	NUM
ejpam-5305	163	9	i	i	PRON
ejpam-5305	163	10	≤	≤	NOUN
ejpam-5305	163	11	n	n	PRON
ejpam-5305	163	12	consider	consider	VERB
ejpam-5305	163	13	|f(vi1	|f(vi1	NOUN
ejpam-5305	163	14	)	)	PUNCT
ejpam-5305	163	15	−	−	PROPN
ejpam-5305	163	16	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	163	17	)	)	PUNCT
ejpam-5305	163	18	|	|	ADV
ejpam-5305	163	19	.	.	PUNCT
ejpam-5305	164	1	since	since	SCONJ
ejpam-5305	164	2	mi	mi	PROPN
ejpam-5305	164	3	=	=	PROPN
ejpam-5305	164	4	2mi−1	2mi−1	PROPN
ejpam-5305	164	5	,	,	PUNCT
ejpam-5305	164	6	then	then	ADV
ejpam-5305	164	7	by	by	ADP
ejpam-5305	164	8	the	the	DET
ejpam-5305	164	9	way	way	NOUN
ejpam-5305	164	10	of	of	ADP
ejpam-5305	164	11	labeling	labeling	NOUN
ejpam-5305	164	12	of	of	ADP
ejpam-5305	164	13	t	t	PROPN
ejpam-5305	164	14	of	of	ADP
ejpam-5305	164	15	type	type	NOUN
ejpam-5305	164	16	i	i	PRON
ejpam-5305	164	17	there	there	PRON
ejpam-5305	164	18	exists	exist	VERB
ejpam-5305	164	19	v	v	ADP
ejpam-5305	164	20	′′	′′	PROPN
ejpam-5305	164	21	∈	∈	PROPN
ejpam-5305	164	22	{	{	PUNCT
ejpam-5305	164	23	vi1	vi1	PROPN
ejpam-5305	164	24	,	,	PUNCT
ejpam-5305	164	25	vi2	vi2	NOUN
ejpam-5305	164	26	,	,	PUNCT
ejpam-5305	164	27	.	.	PUNCT
ejpam-5305	164	28	.	.	PUNCT
ejpam-5305	165	1	.	.	PUNCT
ejpam-5305	166	1	,	,	PUNCT
ejpam-5305	166	2	vimi	vimi	VERB
ejpam-5305	166	3	}	}	PUNCT
ejpam-5305	166	4	such	such	ADJ
ejpam-5305	166	5	that	that	SCONJ
ejpam-5305	166	6	|f(v′′	|f(v′′	PROPN
ejpam-5305	166	7	)	)	PUNCT
ejpam-5305	166	8	−	−	PROPN
ejpam-5305	166	9	f(v)|	f(v)|	NOUN
ejpam-5305	166	10	=	=	SYM
ejpam-5305	166	11	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	166	12	f(v(i−	f(v(i−	PROPN
ejpam-5305	167	1	1)mi−1)|	1)mi−1)|	NUM
ejpam-5305	167	2	.	.	PUNCT
ejpam-5305	168	1	since	since	SCONJ
ejpam-5305	168	2	mi	mi	PROPN
ejpam-5305	168	3	is	be	AUX
ejpam-5305	168	4	even	even	ADV
ejpam-5305	168	5	,	,	PUNCT
ejpam-5305	168	6	then	then	ADV
ejpam-5305	168	7	by	by	ADP
ejpam-5305	168	8	lemma	lemma	PROPN
ejpam-5305	168	9	2	2	NUM
ejpam-5305	168	10	,	,	PUNCT
ejpam-5305	168	11	we	we	PRON
ejpam-5305	168	12	can	can	AUX
ejpam-5305	168	13	change	change	VERB
ejpam-5305	168	14	labeling	labeling	NOUN
ejpam-5305	168	15	f	f	PROPN
ejpam-5305	168	16	at	at	ADP
ejpam-5305	168	17	vi1	vi1	PROPN
ejpam-5305	168	18	,	,	PUNCT
ejpam-5305	168	19	vi2	vi2	NOUN
ejpam-5305	168	20	,	,	PUNCT
ejpam-5305	168	21	.	.	PUNCT
ejpam-5305	168	22	.	.	PUNCT
ejpam-5305	169	1	.	.	PUNCT
ejpam-5305	170	1	,	,	PUNCT
ejpam-5305	170	2	vimi	vimi	VERB
ejpam-5305	170	3	to	to	PART
ejpam-5305	170	4	be	be	AUX
ejpam-5305	170	5	the	the	DET
ejpam-5305	170	6	labeling	labeling	NOUN
ejpam-5305	170	7	f	f	PROPN
ejpam-5305	170	8	′′	′′	PROPN
ejpam-5305	170	9	such	such	ADJ
ejpam-5305	170	10	that	that	SCONJ
ejpam-5305	170	11	|f	|f	PROPN
ejpam-5305	171	1	′′	′′	PROPN
ejpam-5305	171	2	(	(	PUNCT
ejpam-5305	171	3	vi1)−	vi1)−	ADJ
ejpam-5305	171	4	f(v)|	f(v)|	NOUN
ejpam-5305	171	5	=	=	SYM
ejpam-5305	171	6	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	171	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	171	8	)	)	PUNCT
ejpam-5305	171	9	|	|	ADV
ejpam-5305	171	10	the	the	DET
ejpam-5305	171	11	set	set	NOUN
ejpam-5305	171	12	of	of	ADP
ejpam-5305	171	13	vertices	vertex	NOUN
ejpam-5305	171	14	labeling	labeling	NOUN
ejpam-5305	171	15	of	of	ADP
ejpam-5305	171	16	f	f	PROPN
ejpam-5305	171	17	and	and	CCONJ
ejpam-5305	171	18	the	the	DET
ejpam-5305	171	19	set	set	NOUN
ejpam-5305	171	20	of	of	ADP
ejpam-5305	171	21	vertices	vertex	NOUN
ejpam-5305	171	22	labeling	labeling	NOUN
ejpam-5305	171	23	of	of	ADP
ejpam-5305	171	24	f	f	PROPN
ejpam-5305	171	25	′′	′′	PROPN
ejpam-5305	171	26	are	be	AUX
ejpam-5305	171	27	the	the	DET
ejpam-5305	171	28	same	same	ADJ
ejpam-5305	171	29	and	and	CCONJ
ejpam-5305	171	30	the	the	DET
ejpam-5305	171	31	set	set	NOUN
ejpam-5305	171	32	of	of	ADP
ejpam-5305	171	33	edge	edge	NOUN
ejpam-5305	171	34	labeling	labeling	NOUN
ejpam-5305	171	35	of	of	ADP
ejpam-5305	171	36	f	f	PROPN
ejpam-5305	171	37	and	and	CCONJ
ejpam-5305	171	38	the	the	DET
ejpam-5305	171	39	set	set	NOUN
ejpam-5305	171	40	of	of	ADP
ejpam-5305	171	41	edge	edge	NOUN
ejpam-5305	171	42	labeling	labeling	NOUN
ejpam-5305	171	43	of	of	ADP
ejpam-5305	171	44	f	f	PROPN
ejpam-5305	171	45	′′	′′	PROPN
ejpam-5305	171	46	are	be	AUX
ejpam-5305	171	47	the	the	DET
ejpam-5305	171	48	same	same	ADJ
ejpam-5305	171	49	.	.	PUNCT
ejpam-5305	172	1	hence	hence	ADV
ejpam-5305	172	2	t	t	PROPN
ejpam-5305	172	3	amit	amit	PROPN
ejpam-5305	172	4	graceful	graceful	ADJ
ejpam-5305	172	5	labeling	labeling	NOUN
ejpam-5305	172	6	.	.	PUNCT
ejpam-5305	173	1	case	case	NOUN
ejpam-5305	173	2	2	2	NUM
ejpam-5305	173	3	.	.	PUNCT
ejpam-5305	174	1	if	if	SCONJ
ejpam-5305	174	2	m1	m1	PROPN
ejpam-5305	174	3	is	be	AUX
ejpam-5305	174	4	even	even	ADV
ejpam-5305	174	5	,	,	PUNCT
ejpam-5305	174	6	then	then	ADV
ejpam-5305	174	7	let	let	VERB
ejpam-5305	174	8	f	f	PRON
ejpam-5305	174	9	be	be	AUX
ejpam-5305	174	10	the	the	DET
ejpam-5305	174	11	labeling	labeling	NOUN
ejpam-5305	174	12	of	of	ADP
ejpam-5305	174	13	t	t	PROPN
ejpam-5305	174	14	of	of	ADP
ejpam-5305	174	15	type	type	NOUN
ejpam-5305	174	16	ii	ii	PROPN
ejpam-5305	174	17	and	and	CCONJ
ejpam-5305	174	18	consider	consider	VERB
ejpam-5305	174	19	in	in	ADP
ejpam-5305	174	20	a	a	DET
ejpam-5305	174	21	same	same	ADJ
ejpam-5305	174	22	way	way	NOUN
ejpam-5305	174	23	as	as	ADP
ejpam-5305	174	24	the	the	DET
ejpam-5305	174	25	case	case	NOUN
ejpam-5305	174	26	1	1	NUM
ejpam-5305	174	27	,	,	PUNCT
ejpam-5305	174	28	we	we	PRON
ejpam-5305	174	29	get	get	VERB
ejpam-5305	174	30	t	t	PROPN
ejpam-5305	174	31	admit	admit	VERB
ejpam-5305	174	32	graceful	graceful	ADJ
ejpam-5305	174	33	labeling	labeling	NOUN
ejpam-5305	174	34	.	.	PUNCT
ejpam-5305	175	1	we	we	PRON
ejpam-5305	175	2	get	get	VERB
ejpam-5305	175	3	the	the	DET
ejpam-5305	175	4	second	second	ADJ
ejpam-5305	175	5	case	case	NOUN
ejpam-5305	175	6	as	as	ADP
ejpam-5305	175	7	the	the	DET
ejpam-5305	175	8	following	follow	VERB
ejpam-5305	175	9	theorem	theorem	NOUN
ejpam-5305	175	10	.	.	PUNCT
ejpam-5305	175	11	theorem	theorem	NOUN
ejpam-5305	175	12	2	2	NUM
ejpam-5305	175	13	.	.	PUNCT
ejpam-5305	176	1	let	let	VERB
ejpam-5305	176	2	t	t	PROPN
ejpam-5305	176	3	be	be	AUX
ejpam-5305	176	4	a	a	DET
ejpam-5305	176	5	spider	spider	NOUN
ejpam-5305	176	6	graph	graph	NOUN
ejpam-5305	176	7	of	of	ADP
ejpam-5305	176	8	n	n	NOUN
ejpam-5305	176	9	legs	leg	NOUN
ejpam-5305	176	10	with	with	ADP
ejpam-5305	176	11	lengths	length	NOUN
ejpam-5305	176	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	176	13	,	,	PUNCT
ejpam-5305	176	14	.	.	PUNCT
ejpam-5305	176	15	.	.	PUNCT
ejpam-5305	177	1	.	.	PUNCT
ejpam-5305	178	1	,	,	PUNCT
ejpam-5305	178	2	mn	mn	PROPN
ejpam-5305	178	3	.	.	PUNCT
ejpam-5305	179	1	if	if	SCONJ
ejpam-5305	179	2	mi+1	mi+1	NUM
ejpam-5305	179	3	=	=	PROPN
ejpam-5305	179	4	m2	m2	PROPN
ejpam-5305	179	5	+	+	PROPN
ejpam-5305	179	6	m3	m3	PROPN
ejpam-5305	179	7	+	+	X
ejpam-5305	179	8	.	.	PUNCT
ejpam-5305	179	9	.	.	PUNCT
ejpam-5305	180	1	.+mi	.+mi	PUNCT
ejpam-5305	181	1	+	+	CCONJ
ejpam-5305	181	2	1	1	NUM
ejpam-5305	181	3	,	,	PUNCT
ejpam-5305	181	4	i	i	PRON
ejpam-5305	181	5	=	=	NOUN
ejpam-5305	181	6	2	2	NUM
ejpam-5305	181	7	,	,	PUNCT
ejpam-5305	181	8	3	3	NUM
ejpam-5305	181	9	,	,	PUNCT
ejpam-5305	181	10	.	.	PUNCT
ejpam-5305	181	11	.	.	PUNCT
ejpam-5305	181	12	.	.	PUNCT
ejpam-5305	182	1	,	,	PUNCT
ejpam-5305	182	2	n−	n−	NOUN
ejpam-5305	182	3	1	1	NUM
ejpam-5305	182	4	,	,	PUNCT
ejpam-5305	182	5	then	then	ADV
ejpam-5305	182	6	t	t	PROPN
ejpam-5305	182	7	is	be	AUX
ejpam-5305	182	8	a	a	DET
ejpam-5305	182	9	graceful	graceful	ADJ
ejpam-5305	182	10	labeling	labeling	NOUN
ejpam-5305	182	11	graph	graph	NOUN
ejpam-5305	182	12	.	.	PUNCT
ejpam-5305	183	1	proof	proof	NOUN
ejpam-5305	183	2	.	.	PUNCT
ejpam-5305	184	1	let	let	VERB
ejpam-5305	184	2	t	t	NOUN
ejpam-5305	184	3	be	be	AUX
ejpam-5305	184	4	the	the	DET
ejpam-5305	184	5	spider	spider	NOUN
ejpam-5305	184	6	as	as	SCONJ
ejpam-5305	184	7	shown	show	VERB
ejpam-5305	184	8	in	in	ADP
ejpam-5305	184	9	figure	figure	NOUN
ejpam-5305	184	10	1	1	NUM
ejpam-5305	184	11	.	.	PUNCT
ejpam-5305	184	12	case	case	NOUN
ejpam-5305	184	13	1	1	X
ejpam-5305	184	14	.	.	PUNCT
ejpam-5305	185	1	if	if	SCONJ
ejpam-5305	185	2	m1	m1	PROPN
ejpam-5305	185	3	is	be	AUX
ejpam-5305	185	4	odd	odd	ADJ
ejpam-5305	185	5	,	,	PUNCT
ejpam-5305	185	6	then	then	ADV
ejpam-5305	185	7	let	let	VERB
ejpam-5305	185	8	f	f	PRON
ejpam-5305	185	9	be	be	AUX
ejpam-5305	185	10	the	the	DET
ejpam-5305	185	11	labeling	labeling	NOUN
ejpam-5305	185	12	of	of	ADP
ejpam-5305	185	13	t	t	PROPN
ejpam-5305	185	14	of	of	ADP
ejpam-5305	185	15	type	type	NOUN
ejpam-5305	185	16	i.	i.	NOUN
ejpam-5305	185	17	consider	consider	VERB
ejpam-5305	185	18	|f(v31	|f(v31	NOUN
ejpam-5305	185	19	)	)	PUNCT
ejpam-5305	185	20	−	−	PROPN
ejpam-5305	185	21	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	185	22	.	.	PUNCT
ejpam-5305	186	1	since	since	SCONJ
ejpam-5305	186	2	m3	m3	PROPN
ejpam-5305	186	3	=	=	PROPN
ejpam-5305	186	4	m2	m2	PROPN
ejpam-5305	186	5	+	+	PROPN
ejpam-5305	186	6	1	1	NUM
ejpam-5305	186	7	,	,	PUNCT
ejpam-5305	186	8	then	then	ADV
ejpam-5305	186	9	by	by	ADP
ejpam-5305	186	10	the	the	DET
ejpam-5305	186	11	way	way	NOUN
ejpam-5305	186	12	of	of	ADP
ejpam-5305	186	13	labeling	labeling	NOUN
ejpam-5305	186	14	of	of	ADP
ejpam-5305	186	15	t	t	PROPN
ejpam-5305	186	16	of	of	ADP
ejpam-5305	186	17	type	type	NOUN
ejpam-5305	186	18	i	i	PRON
ejpam-5305	186	19	we	we	PRON
ejpam-5305	186	20	have	have	VERB
ejpam-5305	186	21	|f(v3m3	|f(v3m3	NUM
ejpam-5305	186	22	)	)	PUNCT
ejpam-5305	187	1	−	−	PROPN
ejpam-5305	187	2	f(v)|	f(v)|	NOUN
ejpam-5305	187	3	=	=	NOUN
ejpam-5305	187	4	|f(v31	|f(v31	NOUN
ejpam-5305	187	5	)	)	PUNCT
ejpam-5305	187	6	−	−	PROPN
ejpam-5305	187	7	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	187	8	.	.	PUNCT
ejpam-5305	188	1	we	we	PRON
ejpam-5305	188	2	can	can	AUX
ejpam-5305	188	3	change	change	VERB
ejpam-5305	188	4	the	the	DET
ejpam-5305	188	5	labeling	labeling	NOUN
ejpam-5305	188	6	f	f	PROPN
ejpam-5305	188	7	at	at	ADP
ejpam-5305	188	8	v31	v31	PROPN
ejpam-5305	188	9	,	,	PUNCT
ejpam-5305	188	10	v32	v32	PROPN
ejpam-5305	188	11	,	,	PUNCT
ejpam-5305	188	12	.	.	PUNCT
ejpam-5305	188	13	.	.	PUNCT
ejpam-5305	189	1	.	.	PUNCT
ejpam-5305	190	1	,	,	PUNCT
ejpam-5305	190	2	v3m3	v3m3	X
ejpam-5305	190	3	to	to	PART
ejpam-5305	190	4	be	be	AUX
ejpam-5305	190	5	the	the	DET
ejpam-5305	190	6	labeling	labeling	NOUN
ejpam-5305	190	7	f	f	NOUN
ejpam-5305	190	8	′	′	NUM
ejpam-5305	190	9	such	such	ADJ
ejpam-5305	190	10	that	that	PRON
ejpam-5305	190	11	|f	|f	PROPN
ejpam-5305	190	12	′(v31)−	′(v31)−	NOUN
ejpam-5305	190	13	f(v)|	f(v)|	NOUN
ejpam-5305	190	14	=	=	SYM
ejpam-5305	190	15	|f(v31)−	|f(v31)−	SYM
ejpam-5305	190	16	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	190	17	by	by	ADP
ejpam-5305	190	18	reversing	reverse	VERB
ejpam-5305	190	19	the	the	DET
ejpam-5305	190	20	order	order	NOUN
ejpam-5305	190	21	of	of	ADP
ejpam-5305	190	22	theirs	theirs	PRON
ejpam-5305	190	23	labels	label	NOUN
ejpam-5305	190	24	.	.	PUNCT
ejpam-5305	191	1	further	far	ADV
ejpam-5305	191	2	,	,	PUNCT
ejpam-5305	191	3	for	for	ADP
ejpam-5305	191	4	any	any	DET
ejpam-5305	191	5	i	i	NOUN
ejpam-5305	191	6	,	,	PUNCT
ejpam-5305	191	7	4	4	NUM
ejpam-5305	191	8	≤	≤	NUM
ejpam-5305	191	9	i	i	PRON
ejpam-5305	191	10	≤	≤	NOUN
ejpam-5305	191	11	n	n	CCONJ
ejpam-5305	191	12	−	−	PROPN
ejpam-5305	191	13	1	1	NUM
ejpam-5305	191	14	consider	consider	VERB
ejpam-5305	191	15	|f(vi1	|f(vi1	NOUN
ejpam-5305	191	16	)	)	PUNCT
ejpam-5305	191	17	−	−	PROPN
ejpam-5305	191	18	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	191	19	)	)	PUNCT
ejpam-5305	192	1	|	|	ADV
ejpam-5305	192	2	.	.	PUNCT
ejpam-5305	193	1	since	since	SCONJ
ejpam-5305	193	2	mi	mi	PROPN
ejpam-5305	193	3	=	=	PROPN
ejpam-5305	193	4	m2	m2	PROPN
ejpam-5305	193	5	+	+	CCONJ
ejpam-5305	193	6	m3	m3	PROPN
ejpam-5305	193	7	+	+	CCONJ
ejpam-5305	193	8	.	.	PUNCT
ejpam-5305	193	9	.	.	PUNCT
ejpam-5305	193	10	.	.	PUNCT
ejpam-5305	194	1	+	+	CCONJ
ejpam-5305	194	2	mi−1	mi−1	PROPN
ejpam-5305	194	3	+	+	CCONJ
ejpam-5305	194	4	1	1	NUM
ejpam-5305	194	5	,	,	PUNCT
ejpam-5305	194	6	then	then	ADV
ejpam-5305	194	7	by	by	ADP
ejpam-5305	194	8	the	the	DET
ejpam-5305	194	9	way	way	NOUN
ejpam-5305	194	10	of	of	ADP
ejpam-5305	194	11	labeling	labeling	NOUN
ejpam-5305	194	12	of	of	ADP
ejpam-5305	194	13	t	t	PROPN
ejpam-5305	194	14	of	of	ADP
ejpam-5305	194	15	type	type	NOUN
ejpam-5305	194	16	i	i	PRON
ejpam-5305	194	17	we	we	PRON
ejpam-5305	194	18	have	have	AUX
ejpam-5305	194	19	|f(vimi)−f(v)|	|f(vimi)−f(v)|	VERB
ejpam-5305	194	20	=	=	PUNCT
ejpam-5305	194	21	|f(vi1)−f(v(i−	|f(vi1)−f(v(i−	PUNCT
ejpam-5305	194	22	1)mi−1)|	1)mi−1)|	NUM
ejpam-5305	194	23	.	.	PUNCT
ejpam-5305	195	1	we	we	PRON
ejpam-5305	195	2	can	can	AUX
ejpam-5305	195	3	change	change	VERB
ejpam-5305	195	4	labeling	labeling	NOUN
ejpam-5305	195	5	f	f	PROPN
ejpam-5305	195	6	at	at	ADP
ejpam-5305	195	7	vi1	vi1	PROPN
ejpam-5305	195	8	,	,	PUNCT
ejpam-5305	195	9	vi2	vi2	NOUN
ejpam-5305	195	10	,	,	PUNCT
ejpam-5305	195	11	.	.	PUNCT
ejpam-5305	195	12	.	.	PUNCT
ejpam-5305	196	1	.	.	PUNCT
ejpam-5305	197	1	,	,	PUNCT
ejpam-5305	197	2	vimi	vimi	VERB
ejpam-5305	197	3	to	to	PART
ejpam-5305	197	4	be	be	AUX
ejpam-5305	197	5	the	the	DET
ejpam-5305	197	6	labeling	labeling	NOUN
ejpam-5305	197	7	f	f	PROPN
ejpam-5305	197	8	′′	′′	PROPN
ejpam-5305	197	9	such	such	ADJ
ejpam-5305	197	10	that	that	SCONJ
ejpam-5305	197	11	|f	|f	PROPN
ejpam-5305	198	1	′′	′′	PROPN
ejpam-5305	198	2	(	(	PUNCT
ejpam-5305	198	3	vi1)−	vi1)−	ADJ
ejpam-5305	198	4	f(v)|	f(v)|	NOUN
ejpam-5305	198	5	=	=	SYM
ejpam-5305	198	6	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	198	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	198	8	)	)	PUNCT
ejpam-5305	198	9	|	|	ADV
ejpam-5305	198	10	by	by	ADP
ejpam-5305	198	11	reversing	reverse	VERB
ejpam-5305	198	12	the	the	DET
ejpam-5305	198	13	order	order	NOUN
ejpam-5305	198	14	of	of	ADP
ejpam-5305	198	15	theirs	theirs	PRON
ejpam-5305	198	16	labels	label	NOUN
ejpam-5305	198	17	.	.	PUNCT
ejpam-5305	199	1	hence	hence	ADV
ejpam-5305	199	2	t	t	PROPN
ejpam-5305	199	3	amit	amit	PROPN
ejpam-5305	199	4	graceful	graceful	ADJ
ejpam-5305	199	5	labeling	labeling	NOUN
ejpam-5305	199	6	.	.	PUNCT
ejpam-5305	200	1	case	case	NOUN
ejpam-5305	200	2	2	2	NUM
ejpam-5305	200	3	.	.	PUNCT
ejpam-5305	201	1	if	if	SCONJ
ejpam-5305	201	2	m1	m1	PROPN
ejpam-5305	201	3	is	be	AUX
ejpam-5305	201	4	even	even	ADV
ejpam-5305	201	5	,	,	PUNCT
ejpam-5305	201	6	then	then	ADV
ejpam-5305	201	7	let	let	VERB
ejpam-5305	201	8	f	f	PRON
ejpam-5305	201	9	be	be	AUX
ejpam-5305	201	10	the	the	DET
ejpam-5305	201	11	labeling	labeling	NOUN
ejpam-5305	201	12	of	of	ADP
ejpam-5305	201	13	t	t	PROPN
ejpam-5305	201	14	of	of	ADP
ejpam-5305	201	15	type	type	NOUN
ejpam-5305	201	16	ii	ii	PROPN
ejpam-5305	201	17	and	and	CCONJ
ejpam-5305	201	18	consider	consider	VERB
ejpam-5305	201	19	in	in	ADP
ejpam-5305	201	20	a	a	DET
ejpam-5305	201	21	similar	similar	ADJ
ejpam-5305	201	22	way	way	NOUN
ejpam-5305	201	23	as	as	ADP
ejpam-5305	201	24	the	the	DET
ejpam-5305	201	25	case	case	NOUN
ejpam-5305	201	26	1	1	NUM
ejpam-5305	201	27	,	,	PUNCT
ejpam-5305	201	28	we	we	PRON
ejpam-5305	201	29	get	get	VERB
ejpam-5305	201	30	t	t	PROPN
ejpam-5305	201	31	admit	admit	VERB
ejpam-5305	201	32	graceful	graceful	ADJ
ejpam-5305	201	33	labeling	labeling	NOUN
ejpam-5305	201	34	.	.	PUNCT
ejpam-5305	202	1	next	next	ADV
ejpam-5305	202	2	we	we	PRON
ejpam-5305	202	3	will	will	AUX
ejpam-5305	202	4	extend	extend	VERB
ejpam-5305	202	5	theorem	theorem	NOUN
ejpam-5305	202	6	1	1	NUM
ejpam-5305	202	7	to	to	PART
ejpam-5305	202	8	be	be	AUX
ejpam-5305	202	9	more	more	ADV
ejpam-5305	202	10	general	general	ADJ
ejpam-5305	202	11	.	.	PUNCT
ejpam-5305	203	1	theorem	theorem	NOUN
ejpam-5305	203	2	3	3	X
ejpam-5305	203	3	.	.	PUNCT
ejpam-5305	204	1	let	let	VERB
ejpam-5305	204	2	t	t	PROPN
ejpam-5305	204	3	be	be	AUX
ejpam-5305	204	4	a	a	DET
ejpam-5305	204	5	spider	spider	NOUN
ejpam-5305	204	6	graph	graph	NOUN
ejpam-5305	204	7	of	of	ADP
ejpam-5305	204	8	n	n	NOUN
ejpam-5305	204	9	legs	leg	NOUN
ejpam-5305	204	10	with	with	ADP
ejpam-5305	204	11	lengths	length	NOUN
ejpam-5305	204	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	204	13	,	,	PUNCT
ejpam-5305	204	14	.	.	PUNCT
ejpam-5305	204	15	.	.	PUNCT
ejpam-5305	205	1	.	.	PUNCT
ejpam-5305	206	1	,	,	PUNCT
ejpam-5305	206	2	mn	mn	PROPN
ejpam-5305	206	3	.	.	PUNCT
ejpam-5305	207	1	if	if	SCONJ
ejpam-5305	207	2	m3,m4	m3,m4	PROPN
ejpam-5305	207	3	,	,	PUNCT
ejpam-5305	207	4	.	.	PUNCT
ejpam-5305	207	5	.	.	PUNCT
ejpam-5305	208	1	.	.	PUNCT
ejpam-5305	209	1	,	,	PUNCT
ejpam-5305	209	2	mn	mn	PROPN
ejpam-5305	209	3	are	be	AUX
ejpam-5305	209	4	even	even	ADV
ejpam-5305	209	5	and	and	CCONJ
ejpam-5305	209	6	mi+1	mi+1	PRON
ejpam-5305	209	7	≥	≥	NOUN
ejpam-5305	209	8	2mi	2mi	NOUN
ejpam-5305	209	9	for	for	ADP
ejpam-5305	209	10	i	i	PRON
ejpam-5305	209	11	=	=	SYM
ejpam-5305	209	12	2	2	NUM
ejpam-5305	209	13	,	,	PUNCT
ejpam-5305	209	14	3	3	NUM
ejpam-5305	209	15	,	,	PUNCT
ejpam-5305	209	16	.	.	PUNCT
ejpam-5305	209	17	.	.	PUNCT
ejpam-5305	210	1	.	.	PUNCT
ejpam-5305	211	1	,	,	PUNCT
ejpam-5305	211	2	n	n	CCONJ
ejpam-5305	211	3	−	−	PROPN
ejpam-5305	211	4	1	1	NUM
ejpam-5305	211	5	,	,	PUNCT
ejpam-5305	211	6	then	then	ADV
ejpam-5305	211	7	t	t	PROPN
ejpam-5305	211	8	is	be	AUX
ejpam-5305	211	9	a	a	DET
ejpam-5305	211	10	graceful	graceful	ADJ
ejpam-5305	211	11	labeling	labeling	NOUN
ejpam-5305	211	12	graph	graph	NOUN
ejpam-5305	211	13	.	.	PUNCT
ejpam-5305	212	1	proof	proof	NOUN
ejpam-5305	212	2	.	.	PUNCT
ejpam-5305	213	1	let	let	VERB
ejpam-5305	213	2	t	t	NOUN
ejpam-5305	213	3	be	be	AUX
ejpam-5305	213	4	the	the	DET
ejpam-5305	213	5	spider	spider	NOUN
ejpam-5305	213	6	as	as	SCONJ
ejpam-5305	213	7	shown	show	VERB
ejpam-5305	213	8	in	in	ADP
ejpam-5305	213	9	figure	figure	NOUN
ejpam-5305	213	10	1	1	NUM
ejpam-5305	213	11	.	.	PUNCT
ejpam-5305	213	12	case	case	NOUN
ejpam-5305	213	13	1	1	X
ejpam-5305	213	14	.	.	PUNCT
ejpam-5305	214	1	if	if	SCONJ
ejpam-5305	214	2	m1	m1	PROPN
ejpam-5305	214	3	is	be	AUX
ejpam-5305	214	4	odd	odd	ADJ
ejpam-5305	214	5	,	,	PUNCT
ejpam-5305	214	6	then	then	ADV
ejpam-5305	214	7	let	let	VERB
ejpam-5305	214	8	f	f	PRON
ejpam-5305	214	9	be	be	AUX
ejpam-5305	214	10	the	the	DET
ejpam-5305	214	11	labeling	labeling	NOUN
ejpam-5305	214	12	of	of	ADP
ejpam-5305	214	13	t	t	PROPN
ejpam-5305	214	14	of	of	ADP
ejpam-5305	214	15	type	type	NOUN
ejpam-5305	214	16	i.	i.	NOUN
ejpam-5305	214	17	consider	consider	VERB
ejpam-5305	214	18	|f(v31	|f(v31	NOUN
ejpam-5305	214	19	)	)	PUNCT
ejpam-5305	214	20	−	−	PROPN
ejpam-5305	214	21	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	214	22	.	.	PUNCT
ejpam-5305	215	1	sincem3	sincem3	PROPN
ejpam-5305	215	2	≥	≥	NUM
ejpam-5305	215	3	2m2	2m2	NUM
ejpam-5305	215	4	,	,	PUNCT
ejpam-5305	215	5	then	then	ADV
ejpam-5305	215	6	by	by	ADP
ejpam-5305	215	7	the	the	DET
ejpam-5305	215	8	way	way	NOUN
ejpam-5305	215	9	of	of	ADP
ejpam-5305	215	10	labeling	labeling	NOUN
ejpam-5305	215	11	of	of	ADP
ejpam-5305	215	12	t	t	PROPN
ejpam-5305	215	13	of	of	ADP
ejpam-5305	215	14	type	type	NOUN
ejpam-5305	215	15	i	i	PRON
ejpam-5305	215	16	we	we	PRON
ejpam-5305	215	17	have	have	VERB
ejpam-5305	215	18	|f(v3(m2	|f(v3(m2	NUM
ejpam-5305	215	19	+	+	ADJ
ejpam-5305	215	20	1))−	1))−	NUM
ejpam-5305	215	21	f(v)|	f(v)|	NOUN
ejpam-5305	215	22	=	=	SYM
ejpam-5305	215	23	|f(v31)−f(v2m2)|	|f(v31)−f(v2m2)|	PROPN
ejpam-5305	215	24	.	.	PUNCT
ejpam-5305	216	1	since	since	SCONJ
ejpam-5305	216	2	m3	m3	PROPN
ejpam-5305	216	3	is	be	AUX
ejpam-5305	216	4	even	even	ADV
ejpam-5305	216	5	,	,	PUNCT
ejpam-5305	216	6	then	then	ADV
ejpam-5305	216	7	by	by	ADP
ejpam-5305	216	8	lemma	lemma	PROPN
ejpam-5305	216	9	2	2	NUM
ejpam-5305	216	10	,	,	PUNCT
ejpam-5305	216	11	we	we	PRON
ejpam-5305	216	12	can	can	AUX
ejpam-5305	216	13	change	change	VERB
ejpam-5305	216	14	the	the	DET
ejpam-5305	216	15	labeling	labeling	NOUN
ejpam-5305	216	16	f	f	PROPN
ejpam-5305	216	17	at	at	ADP
ejpam-5305	216	18	v31	v31	PROPN
ejpam-5305	216	19	,	,	PUNCT
ejpam-5305	216	20	v32	v32	PROPN
ejpam-5305	216	21	,	,	PUNCT
ejpam-5305	216	22	.	.	PUNCT
ejpam-5305	216	23	.	.	PUNCT
ejpam-5305	217	1	.	.	PUNCT
ejpam-5305	218	1	,	,	PUNCT
ejpam-5305	218	2	v3m3	v3m3	X
ejpam-5305	218	3	to	to	PART
ejpam-5305	218	4	be	be	AUX
ejpam-5305	218	5	the	the	DET
ejpam-5305	218	6	labeling	labeling	NOUN
ejpam-5305	218	7	f	f	NOUN
ejpam-5305	218	8	′	′	NUM
ejpam-5305	218	9	such	such	ADJ
ejpam-5305	218	10	that	that	PRON
ejpam-5305	218	11	|f	|f	PROPN
ejpam-5305	218	12	′(v31)−	′(v31)−	NOUN
ejpam-5305	218	13	f(v)|	f(v)|	NOUN
ejpam-5305	218	14	=	=	SYM
ejpam-5305	218	15	|f(v31)−	|f(v31)−	NUM
ejpam-5305	218	16	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	218	17	,	,	PUNCT
ejpam-5305	218	18	the	the	DET
ejpam-5305	218	19	set	set	NOUN
ejpam-5305	218	20	of	of	ADP
ejpam-5305	218	21	vertices	vertex	NOUN
ejpam-5305	218	22	labeling	labeling	NOUN
ejpam-5305	218	23	of	of	ADP
ejpam-5305	218	24	f	f	PROPN
ejpam-5305	218	25	and	and	CCONJ
ejpam-5305	218	26	the	the	DET
ejpam-5305	218	27	set	set	NOUN
ejpam-5305	218	28	of	of	ADP
ejpam-5305	218	29	vertices	vertex	NOUN
ejpam-5305	218	30	labeling	labeling	NOUN
ejpam-5305	218	31	of	of	ADP
ejpam-5305	218	32	f	f	PROPN
ejpam-5305	218	33	′	′	NUM
ejpam-5305	218	34	are	be	AUX
ejpam-5305	218	35	the	the	DET
ejpam-5305	218	36	same	same	ADJ
ejpam-5305	218	37	and	and	CCONJ
ejpam-5305	218	38	k.	k.	PROPN
ejpam-5305	218	39	saengsura	saengsura	PROPN
ejpam-5305	218	40	,	,	PUNCT
ejpam-5305	218	41	t.	t.	PROPN
ejpam-5305	218	42	poomsa	poomsa	PROPN
ejpam-5305	218	43	-	-	PUNCT
ejpam-5305	218	44	ard	ard	PROPN
ejpam-5305	218	45	/	/	PUNCT
ejpam-5305	218	46	eur	eur	PROPN
ejpam-5305	218	47	.	.	PUNCT
ejpam-5305	219	1	j.	j.	PROPN
ejpam-5305	219	2	pure	pure	PROPN
ejpam-5305	219	3	appl	appl	PROPN
ejpam-5305	219	4	.	.	PROPN
ejpam-5305	219	5	math	math	PROPN
ejpam-5305	219	6	,	,	PUNCT
ejpam-5305	219	7	18	18	NUM
ejpam-5305	219	8	(	(	PUNCT
ejpam-5305	219	9	2	2	NUM
ejpam-5305	219	10	)	)	PUNCT
ejpam-5305	219	11	(	(	PUNCT
ejpam-5305	219	12	2025	2025	NUM
ejpam-5305	219	13	)	)	PUNCT
ejpam-5305	219	14	,	,	PUNCT
ejpam-5305	219	15	5305	5305	NUM
ejpam-5305	219	16	5	5	NUM
ejpam-5305	219	17	of	of	ADP
ejpam-5305	219	18	8	8	NUM
ejpam-5305	219	19	the	the	DET
ejpam-5305	219	20	set	set	NOUN
ejpam-5305	219	21	of	of	ADP
ejpam-5305	219	22	edge	edge	NOUN
ejpam-5305	219	23	labeling	labeling	NOUN
ejpam-5305	219	24	of	of	ADP
ejpam-5305	219	25	f	f	PROPN
ejpam-5305	219	26	and	and	CCONJ
ejpam-5305	219	27	the	the	DET
ejpam-5305	219	28	set	set	NOUN
ejpam-5305	219	29	of	of	ADP
ejpam-5305	219	30	edge	edge	NOUN
ejpam-5305	219	31	labeling	labeling	NOUN
ejpam-5305	219	32	of	of	ADP
ejpam-5305	219	33	f	f	PROPN
ejpam-5305	219	34	′	′	NUM
ejpam-5305	219	35	are	be	AUX
ejpam-5305	219	36	the	the	DET
ejpam-5305	219	37	same	same	ADJ
ejpam-5305	219	38	.	.	PUNCT
ejpam-5305	220	1	further	far	ADV
ejpam-5305	220	2	,	,	PUNCT
ejpam-5305	220	3	for	for	SCONJ
ejpam-5305	220	4	any	any	DET
ejpam-5305	220	5	i	i	NOUN
ejpam-5305	220	6	,	,	PUNCT
ejpam-5305	220	7	4	4	NUM
ejpam-5305	220	8	≤	≤	NUM
ejpam-5305	220	9	i	i	X
ejpam-5305	220	10	≤	≤	NOUN
ejpam-5305	220	11	n	n	PRON
ejpam-5305	220	12	consider	consider	VERB
ejpam-5305	220	13	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	220	14	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	220	15	)	)	PUNCT
ejpam-5305	220	16	|	|	ADV
ejpam-5305	220	17	.	.	PUNCT
ejpam-5305	221	1	let	let	VERB
ejpam-5305	221	2	m′	m′	NOUN
ejpam-5305	221	3	=	=	SYM
ejpam-5305	221	4	m2	m2	PROPN
ejpam-5305	221	5	+	+	PROPN
ejpam-5305	221	6	m3	m3	PROPN
ejpam-5305	221	7	+	+	X
ejpam-5305	221	8	.	.	PUNCT
ejpam-5305	221	9	.	.	PUNCT
ejpam-5305	222	1	.+mi−1	.+mi−1	PROPN
ejpam-5305	222	2	.	.	PUNCT
ejpam-5305	223	1	since	since	SCONJ
ejpam-5305	223	2	mi	mi	PROPN
ejpam-5305	223	3	≥	≥	PROPN
ejpam-5305	223	4	2mi−1	2mi−1	PROPN
ejpam-5305	223	5	>	>	X
ejpam-5305	223	6	m′	m′	PROPN
ejpam-5305	223	7	,	,	PUNCT
ejpam-5305	223	8	then	then	ADV
ejpam-5305	223	9	by	by	ADP
ejpam-5305	223	10	the	the	DET
ejpam-5305	223	11	way	way	NOUN
ejpam-5305	223	12	of	of	ADP
ejpam-5305	223	13	labeling	labeling	NOUN
ejpam-5305	223	14	of	of	ADP
ejpam-5305	223	15	t	t	PROPN
ejpam-5305	223	16	of	of	ADP
ejpam-5305	223	17	type	type	NOUN
ejpam-5305	223	18	i	i	PRON
ejpam-5305	223	19	we	we	PRON
ejpam-5305	223	20	have	have	VERB
ejpam-5305	223	21	|f(vi(m′+1))−f(v)|	|f(vi(m′+1))−f(v)|	X
ejpam-5305	223	22	=	=	SYM
ejpam-5305	223	23	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	223	24	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	223	25	)	)	PUNCT
ejpam-5305	224	1	|	|	ADV
ejpam-5305	224	2	.	.	PUNCT
ejpam-5305	225	1	since	since	SCONJ
ejpam-5305	225	2	mi	mi	PROPN
ejpam-5305	225	3	is	be	AUX
ejpam-5305	225	4	even	even	ADV
ejpam-5305	225	5	,	,	PUNCT
ejpam-5305	225	6	then	then	ADV
ejpam-5305	225	7	by	by	ADP
ejpam-5305	225	8	lemma	lemma	PROPN
ejpam-5305	225	9	2	2	NUM
ejpam-5305	225	10	,	,	PUNCT
ejpam-5305	225	11	we	we	PRON
ejpam-5305	225	12	can	can	AUX
ejpam-5305	225	13	change	change	VERB
ejpam-5305	225	14	labeling	labeling	NOUN
ejpam-5305	225	15	f	f	PROPN
ejpam-5305	225	16	at	at	ADP
ejpam-5305	225	17	vi1	vi1	PROPN
ejpam-5305	225	18	,	,	PUNCT
ejpam-5305	225	19	vi2	vi2	NOUN
ejpam-5305	225	20	,	,	PUNCT
ejpam-5305	225	21	.	.	PUNCT
ejpam-5305	225	22	.	.	PUNCT
ejpam-5305	226	1	.	.	PUNCT
ejpam-5305	227	1	,	,	PUNCT
ejpam-5305	227	2	vimi	vimi	VERB
ejpam-5305	227	3	to	to	PART
ejpam-5305	227	4	be	be	AUX
ejpam-5305	227	5	the	the	DET
ejpam-5305	227	6	labeling	labeling	NOUN
ejpam-5305	227	7	f	f	PROPN
ejpam-5305	227	8	′′	′′	PROPN
ejpam-5305	227	9	such	such	ADJ
ejpam-5305	227	10	that	that	SCONJ
ejpam-5305	227	11	|f	|f	PROPN
ejpam-5305	228	1	′′	′′	PROPN
ejpam-5305	228	2	(	(	PUNCT
ejpam-5305	228	3	vi1)−	vi1)−	ADJ
ejpam-5305	228	4	f(v)|	f(v)|	NOUN
ejpam-5305	228	5	=	=	SYM
ejpam-5305	228	6	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	228	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	228	8	)	)	PUNCT
ejpam-5305	228	9	|	|	ADV
ejpam-5305	228	10	the	the	DET
ejpam-5305	228	11	set	set	NOUN
ejpam-5305	228	12	of	of	ADP
ejpam-5305	228	13	vertices	vertex	NOUN
ejpam-5305	228	14	labeling	labeling	NOUN
ejpam-5305	228	15	of	of	ADP
ejpam-5305	228	16	f	f	PROPN
ejpam-5305	228	17	and	and	CCONJ
ejpam-5305	228	18	the	the	DET
ejpam-5305	228	19	set	set	NOUN
ejpam-5305	228	20	of	of	ADP
ejpam-5305	228	21	vertices	vertex	NOUN
ejpam-5305	228	22	labeling	labeling	NOUN
ejpam-5305	228	23	of	of	ADP
ejpam-5305	228	24	f	f	PROPN
ejpam-5305	228	25	′′	′′	PROPN
ejpam-5305	228	26	are	be	AUX
ejpam-5305	228	27	the	the	DET
ejpam-5305	228	28	same	same	ADJ
ejpam-5305	228	29	and	and	CCONJ
ejpam-5305	228	30	the	the	DET
ejpam-5305	228	31	set	set	NOUN
ejpam-5305	228	32	of	of	ADP
ejpam-5305	228	33	edge	edge	NOUN
ejpam-5305	228	34	labeling	labeling	NOUN
ejpam-5305	228	35	of	of	ADP
ejpam-5305	228	36	f	f	PROPN
ejpam-5305	228	37	and	and	CCONJ
ejpam-5305	228	38	the	the	DET
ejpam-5305	228	39	set	set	NOUN
ejpam-5305	228	40	of	of	ADP
ejpam-5305	228	41	edge	edge	NOUN
ejpam-5305	228	42	labeling	labeling	NOUN
ejpam-5305	228	43	of	of	ADP
ejpam-5305	228	44	f	f	PROPN
ejpam-5305	228	45	′′	′′	PROPN
ejpam-5305	228	46	are	be	AUX
ejpam-5305	228	47	the	the	DET
ejpam-5305	228	48	same	same	ADJ
ejpam-5305	228	49	.	.	PUNCT
ejpam-5305	229	1	hence	hence	ADV
ejpam-5305	229	2	t	t	PROPN
ejpam-5305	229	3	amit	amit	PROPN
ejpam-5305	229	4	graceful	graceful	ADJ
ejpam-5305	229	5	labeling	labeling	NOUN
ejpam-5305	229	6	.	.	PUNCT
ejpam-5305	230	1	case	case	NOUN
ejpam-5305	230	2	2	2	NUM
ejpam-5305	230	3	.	.	PUNCT
ejpam-5305	231	1	if	if	SCONJ
ejpam-5305	231	2	m1	m1	PROPN
ejpam-5305	231	3	is	be	AUX
ejpam-5305	231	4	even	even	ADV
ejpam-5305	231	5	,	,	PUNCT
ejpam-5305	231	6	then	then	ADV
ejpam-5305	231	7	let	let	VERB
ejpam-5305	231	8	f	f	PRON
ejpam-5305	231	9	be	be	AUX
ejpam-5305	231	10	the	the	DET
ejpam-5305	231	11	labeling	labeling	NOUN
ejpam-5305	231	12	of	of	ADP
ejpam-5305	231	13	t	t	PROPN
ejpam-5305	231	14	of	of	ADP
ejpam-5305	231	15	type	type	NOUN
ejpam-5305	231	16	ii	ii	PROPN
ejpam-5305	231	17	and	and	CCONJ
ejpam-5305	231	18	consider	consider	VERB
ejpam-5305	231	19	similar	similar	ADJ
ejpam-5305	231	20	way	way	NOUN
ejpam-5305	231	21	as	as	SCONJ
ejpam-5305	231	22	the	the	DET
ejpam-5305	231	23	case	case	NOUN
ejpam-5305	231	24	1	1	NUM
ejpam-5305	231	25	we	we	PRON
ejpam-5305	231	26	get	get	VERB
ejpam-5305	231	27	t	t	PROPN
ejpam-5305	231	28	admit	admit	VERB
ejpam-5305	231	29	graceful	graceful	ADJ
ejpam-5305	231	30	labeling	labeling	NOUN
ejpam-5305	231	31	.	.	PUNCT
ejpam-5305	232	1	we	we	PRON
ejpam-5305	232	2	get	get	VERB
ejpam-5305	232	3	the	the	DET
ejpam-5305	232	4	fourth	fourth	ADJ
ejpam-5305	232	5	case	case	NOUN
ejpam-5305	232	6	as	as	ADP
ejpam-5305	232	7	the	the	DET
ejpam-5305	232	8	following	follow	VERB
ejpam-5305	232	9	theorem	theorem	NOUN
ejpam-5305	232	10	.	.	PUNCT
ejpam-5305	232	11	theorem	theorem	NOUN
ejpam-5305	232	12	4	4	NUM
ejpam-5305	232	13	.	.	PUNCT
ejpam-5305	233	1	let	let	VERB
ejpam-5305	233	2	t	t	PROPN
ejpam-5305	233	3	be	be	AUX
ejpam-5305	233	4	a	a	DET
ejpam-5305	233	5	spider	spider	NOUN
ejpam-5305	233	6	graph	graph	NOUN
ejpam-5305	233	7	of	of	ADP
ejpam-5305	233	8	n	n	NOUN
ejpam-5305	233	9	legs	leg	NOUN
ejpam-5305	233	10	with	with	ADP
ejpam-5305	233	11	lengths	length	NOUN
ejpam-5305	233	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	233	13	,	,	PUNCT
ejpam-5305	233	14	.	.	PUNCT
ejpam-5305	233	15	.	.	PUNCT
ejpam-5305	234	1	.	.	PUNCT
ejpam-5305	235	1	,	,	PUNCT
ejpam-5305	235	2	mn	mn	PROPN
ejpam-5305	235	3	.	.	PROPN
ejpam-5305	236	1	if	if	SCONJ
ejpam-5305	236	2	m2	m2	PROPN
ejpam-5305	236	3	=	=	PROPN
ejpam-5305	236	4	m3	m3	PROPN
ejpam-5305	236	5	=	=	SYM
ejpam-5305	236	6	mn	mn	PROPN
ejpam-5305	236	7	and	and	CCONJ
ejpam-5305	236	8	mi+1	mi+1	PRON
ejpam-5305	236	9	=	=	PROPN
ejpam-5305	236	10	m2	m2	PROPN
ejpam-5305	236	11	+	+	PROPN
ejpam-5305	236	12	m3	m3	PROPN
ejpam-5305	236	13	+	+	X
ejpam-5305	236	14	.	.	PUNCT
ejpam-5305	236	15	.	.	PUNCT
ejpam-5305	237	1	.+mi	.+mi	PROPN
ejpam-5305	237	2	,	,	PUNCT
ejpam-5305	237	3	i	i	PRON
ejpam-5305	237	4	=	=	NOUN
ejpam-5305	237	5	3	3	NUM
ejpam-5305	237	6	,	,	PUNCT
ejpam-5305	237	7	4	4	NUM
ejpam-5305	237	8	,	,	PUNCT
ejpam-5305	237	9	.	.	PUNCT
ejpam-5305	237	10	.	.	PUNCT
ejpam-5305	237	11	.	.	PUNCT
ejpam-5305	238	1	,	,	PUNCT
ejpam-5305	238	2	n−	n−	NOUN
ejpam-5305	238	3	2	2	NUM
ejpam-5305	238	4	,	,	PUNCT
ejpam-5305	238	5	then	then	ADV
ejpam-5305	238	6	t	t	PROPN
ejpam-5305	238	7	is	be	AUX
ejpam-5305	238	8	a	a	DET
ejpam-5305	238	9	graceful	graceful	ADJ
ejpam-5305	238	10	labeling	labeling	NOUN
ejpam-5305	238	11	graph	graph	NOUN
ejpam-5305	238	12	.	.	PUNCT
ejpam-5305	239	1	proof	proof	NOUN
ejpam-5305	239	2	.	.	PUNCT
ejpam-5305	240	1	let	let	VERB
ejpam-5305	240	2	t	t	NOUN
ejpam-5305	240	3	be	be	AUX
ejpam-5305	240	4	the	the	DET
ejpam-5305	240	5	spider	spider	NOUN
ejpam-5305	240	6	as	as	SCONJ
ejpam-5305	240	7	shown	show	VERB
ejpam-5305	240	8	in	in	ADP
ejpam-5305	240	9	the	the	DET
ejpam-5305	240	10	figure	figure	NOUN
ejpam-5305	240	11	1	1	NUM
ejpam-5305	240	12	.	.	PUNCT
ejpam-5305	240	13	case	case	NOUN
ejpam-5305	240	14	1	1	X
ejpam-5305	240	15	.	.	PUNCT
ejpam-5305	241	1	if	if	SCONJ
ejpam-5305	241	2	m1	m1	PROPN
ejpam-5305	241	3	is	be	AUX
ejpam-5305	241	4	odd	odd	ADJ
ejpam-5305	241	5	,	,	PUNCT
ejpam-5305	241	6	then	then	ADV
ejpam-5305	241	7	let	let	VERB
ejpam-5305	241	8	f	f	PRON
ejpam-5305	241	9	be	be	AUX
ejpam-5305	241	10	the	the	DET
ejpam-5305	241	11	labeling	labeling	NOUN
ejpam-5305	241	12	of	of	ADP
ejpam-5305	241	13	t	t	PROPN
ejpam-5305	241	14	of	of	ADP
ejpam-5305	241	15	type	type	NOUN
ejpam-5305	241	16	i.	i.	NOUN
ejpam-5305	241	17	consider	consider	VERB
ejpam-5305	241	18	|f(v31	|f(v31	NOUN
ejpam-5305	241	19	)	)	PUNCT
ejpam-5305	241	20	−	−	PROPN
ejpam-5305	241	21	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	241	22	.	.	PUNCT
ejpam-5305	242	1	since	since	SCONJ
ejpam-5305	242	2	m3	m3	PROPN
ejpam-5305	242	3	=	=	PROPN
ejpam-5305	242	4	m2	m2	PROPN
ejpam-5305	242	5	,	,	PUNCT
ejpam-5305	242	6	then	then	ADV
ejpam-5305	242	7	by	by	ADP
ejpam-5305	242	8	the	the	DET
ejpam-5305	242	9	way	way	NOUN
ejpam-5305	242	10	of	of	ADP
ejpam-5305	242	11	labeling	labeling	NOUN
ejpam-5305	242	12	of	of	ADP
ejpam-5305	242	13	t	t	PROPN
ejpam-5305	242	14	of	of	ADP
ejpam-5305	242	15	type	type	NOUN
ejpam-5305	242	16	i	i	PRON
ejpam-5305	242	17	we	we	PRON
ejpam-5305	242	18	have	have	VERB
ejpam-5305	242	19	|f(v41	|f(v41	NOUN
ejpam-5305	242	20	)	)	PUNCT
ejpam-5305	242	21	−	−	PROPN
ejpam-5305	242	22	f(v)|	f(v)|	PROPN
ejpam-5305	242	23	=	=	SYM
ejpam-5305	242	24	|f(v31)−f(v2m2)|	|f(v31)−f(v2m2)|	PROPN
ejpam-5305	242	25	.	.	PUNCT
ejpam-5305	243	1	further	far	ADV
ejpam-5305	243	2	,	,	PUNCT
ejpam-5305	243	3	for	for	SCONJ
ejpam-5305	243	4	any	any	DET
ejpam-5305	243	5	i	i	NOUN
ejpam-5305	243	6	,	,	PUNCT
ejpam-5305	243	7	4	4	NUM
ejpam-5305	243	8	≤	≤	NUM
ejpam-5305	243	9	i	i	PRON
ejpam-5305	243	10	≤	≤	PUNCT
ejpam-5305	243	11	n−1	n−1	PROPN
ejpam-5305	243	12	consider	consider	VERB
ejpam-5305	243	13	|f(vi1)−f(v(i−1)mi−1	|f(vi1)−f(v(i−1)mi−1	PROPN
ejpam-5305	243	14	)	)	PUNCT
ejpam-5305	243	15	|	|	ADV
ejpam-5305	243	16	.	.	PUNCT
ejpam-5305	244	1	since	since	SCONJ
ejpam-5305	244	2	mi	mi	PROPN
ejpam-5305	244	3	=	=	PROPN
ejpam-5305	244	4	m2	m2	PROPN
ejpam-5305	244	5	+	+	CCONJ
ejpam-5305	244	6	m3	m3	PROPN
ejpam-5305	244	7	+	+	CCONJ
ejpam-5305	244	8	.	.	PUNCT
ejpam-5305	244	9	.	.	PUNCT
ejpam-5305	244	10	.	.	PUNCT
ejpam-5305	245	1	+	+	CCONJ
ejpam-5305	245	2	mi−1	mi−1	PROPN
ejpam-5305	245	3	,	,	PUNCT
ejpam-5305	245	4	then	then	ADV
ejpam-5305	245	5	by	by	ADP
ejpam-5305	245	6	the	the	DET
ejpam-5305	245	7	way	way	NOUN
ejpam-5305	245	8	of	of	ADP
ejpam-5305	245	9	labeling	labeling	NOUN
ejpam-5305	245	10	of	of	ADP
ejpam-5305	245	11	t	t	PROPN
ejpam-5305	245	12	of	of	ADP
ejpam-5305	245	13	type	type	NOUN
ejpam-5305	246	1	i	i	PRON
ejpam-5305	246	2	we	we	PRON
ejpam-5305	246	3	have	have	AUX
ejpam-5305	246	4	|f(v(i+1)1	|f(v(i+1)1	VERB
ejpam-5305	246	5	)	)	PUNCT
ejpam-5305	246	6	−	−	PROPN
ejpam-5305	246	7	f(v)|	f(v)|	NOUN
ejpam-5305	246	8	=	=	SYM
ejpam-5305	246	9	|f(vi1	|f(vi1	NOUN
ejpam-5305	246	10	)	)	PUNCT
ejpam-5305	246	11	−	−	PROPN
ejpam-5305	246	12	f(v(i−	f(v(i−	NUM
ejpam-5305	247	1	1)mi−1	1)mi−1	PROPN
ejpam-5305	247	2	)	)	PUNCT
ejpam-5305	247	3	.	.	PUNCT
ejpam-5305	248	1	next	next	ADV
ejpam-5305	248	2	consider	consider	VERB
ejpam-5305	248	3	|f(vn1	|f(vn1	PROPN
ejpam-5305	248	4	)	)	PUNCT
ejpam-5305	248	5	−	−	PROPN
ejpam-5305	248	6	f(v(n−1)mn−1	f(v(n−1)mn−1	PROPN
ejpam-5305	248	7	)	)	PUNCT
ejpam-5305	249	1	|	|	ADV
ejpam-5305	249	2	.	.	PUNCT
ejpam-5305	249	3	sincem2	sincem2	X
ejpam-5305	250	1	=	=	SYM
ejpam-5305	250	2	m3	m3	PROPN
ejpam-5305	250	3	=	=	SYM
ejpam-5305	250	4	mn	mn	PROPN
ejpam-5305	250	5	,	,	PUNCT
ejpam-5305	250	6	then	then	ADV
ejpam-5305	250	7	by	by	ADP
ejpam-5305	250	8	the	the	DET
ejpam-5305	250	9	way	way	NOUN
ejpam-5305	250	10	of	of	ADP
ejpam-5305	250	11	labeling	labeling	NOUN
ejpam-5305	250	12	of	of	ADP
ejpam-5305	250	13	t	t	PROPN
ejpam-5305	250	14	of	of	ADP
ejpam-5305	250	15	type	type	NOUN
ejpam-5305	251	1	i	i	PRON
ejpam-5305	251	2	we	we	PRON
ejpam-5305	251	3	have	have	VERB
ejpam-5305	251	4	|f(v3m3)−f(v)|	|f(v3m3)−f(v)|	NOUN
ejpam-5305	251	5	=	=	SYM
ejpam-5305	251	6	|f(vn1)−	|f(vn1)−	ADJ
ejpam-5305	251	7	f(v(n−1)mn−1	f(v(n−1)mn−1	PROPN
ejpam-5305	251	8	)	)	PUNCT
ejpam-5305	252	1	|	|	ADV
ejpam-5305	252	2	.	.	PUNCT
ejpam-5305	253	1	we	we	PRON
ejpam-5305	253	2	can	can	AUX
ejpam-5305	253	3	change	change	VERB
ejpam-5305	253	4	labeling	labeling	NOUN
ejpam-5305	253	5	f	f	PROPN
ejpam-5305	253	6	at	at	ADP
ejpam-5305	253	7	v31	v31	PROPN
ejpam-5305	253	8	,	,	PUNCT
ejpam-5305	253	9	v32	v32	PROPN
ejpam-5305	253	10	,	,	PUNCT
ejpam-5305	253	11	.	.	PUNCT
ejpam-5305	253	12	.	.	PUNCT
ejpam-5305	254	1	.	.	PUNCT
ejpam-5305	255	1	,	,	PUNCT
ejpam-5305	255	2	v3m3	v3m3	X
ejpam-5305	255	3	to	to	PART
ejpam-5305	255	4	be	be	AUX
ejpam-5305	255	5	the	the	DET
ejpam-5305	255	6	labeling	labeling	NOUN
ejpam-5305	255	7	f	f	NOUN
ejpam-5305	255	8	′	′	NUM
ejpam-5305	255	9	such	such	ADJ
ejpam-5305	255	10	that	that	PRON
ejpam-5305	255	11	|f	|f	PROPN
ejpam-5305	255	12	′(v31	′(v31	PROPN
ejpam-5305	255	13	)	)	PUNCT
ejpam-5305	255	14	−	−	PROPN
ejpam-5305	255	15	f(v)|	f(v)|	NOUN
ejpam-5305	255	16	=	=	SYM
ejpam-5305	255	17	|f(vn1	|f(vn1	PROPN
ejpam-5305	255	18	)	)	PUNCT
ejpam-5305	255	19	−	−	PROPN
ejpam-5305	255	20	f(v(n−1)mn−1	f(v(n−1)mn−1	PROPN
ejpam-5305	255	21	)	)	PUNCT
ejpam-5305	255	22	|	|	ADV
ejpam-5305	255	23	by	by	ADP
ejpam-5305	255	24	reversing	reverse	VERB
ejpam-5305	255	25	the	the	DET
ejpam-5305	255	26	order	order	NOUN
ejpam-5305	255	27	of	of	ADP
ejpam-5305	255	28	theirs	theirs	PRON
ejpam-5305	255	29	labels	label	NOUN
ejpam-5305	255	30	.	.	PUNCT
ejpam-5305	256	1	hence	hence	ADV
ejpam-5305	256	2	t	t	PROPN
ejpam-5305	256	3	amit	amit	PROPN
ejpam-5305	256	4	graceful	graceful	ADJ
ejpam-5305	256	5	labeling	labeling	NOUN
ejpam-5305	256	6	.	.	PUNCT
ejpam-5305	257	1	case	case	NOUN
ejpam-5305	257	2	2	2	NUM
ejpam-5305	257	3	.	.	PUNCT
ejpam-5305	258	1	if	if	SCONJ
ejpam-5305	258	2	m1	m1	PROPN
ejpam-5305	258	3	is	be	AUX
ejpam-5305	258	4	even	even	ADV
ejpam-5305	258	5	,	,	PUNCT
ejpam-5305	258	6	then	then	ADV
ejpam-5305	258	7	let	let	VERB
ejpam-5305	258	8	f	f	PRON
ejpam-5305	258	9	be	be	AUX
ejpam-5305	258	10	the	the	DET
ejpam-5305	258	11	labeling	labeling	NOUN
ejpam-5305	258	12	of	of	ADP
ejpam-5305	258	13	t	t	PROPN
ejpam-5305	258	14	of	of	ADP
ejpam-5305	258	15	type	type	NOUN
ejpam-5305	258	16	ii	ii	PROPN
ejpam-5305	258	17	and	and	CCONJ
ejpam-5305	258	18	consider	consider	VERB
ejpam-5305	258	19	in	in	ADP
ejpam-5305	258	20	a	a	DET
ejpam-5305	258	21	similar	similar	ADJ
ejpam-5305	258	22	way	way	NOUN
ejpam-5305	258	23	as	as	ADP
ejpam-5305	258	24	the	the	DET
ejpam-5305	258	25	case	case	NOUN
ejpam-5305	258	26	1	1	NUM
ejpam-5305	258	27	,	,	PUNCT
ejpam-5305	258	28	we	we	PRON
ejpam-5305	258	29	get	get	VERB
ejpam-5305	258	30	t	t	PROPN
ejpam-5305	258	31	admit	admit	VERB
ejpam-5305	258	32	graceful	graceful	ADJ
ejpam-5305	258	33	labeling	labeling	NOUN
ejpam-5305	258	34	.	.	PUNCT
ejpam-5305	259	1	next	next	ADV
ejpam-5305	259	2	we	we	PRON
ejpam-5305	259	3	will	will	AUX
ejpam-5305	259	4	extend	extend	VERB
ejpam-5305	259	5	theorem	theorem	NOUN
ejpam-5305	259	6	2	2	NUM
ejpam-5305	259	7	to	to	PART
ejpam-5305	259	8	be	be	AUX
ejpam-5305	259	9	more	more	ADV
ejpam-5305	259	10	general	general	ADJ
ejpam-5305	259	11	.	.	PUNCT
ejpam-5305	260	1	theorem	theorem	NOUN
ejpam-5305	260	2	5	5	NUM
ejpam-5305	260	3	.	.	PUNCT
ejpam-5305	261	1	let	let	VERB
ejpam-5305	261	2	t	t	PROPN
ejpam-5305	261	3	be	be	AUX
ejpam-5305	261	4	a	a	DET
ejpam-5305	261	5	spider	spider	NOUN
ejpam-5305	261	6	graph	graph	NOUN
ejpam-5305	261	7	of	of	ADP
ejpam-5305	261	8	n	n	NOUN
ejpam-5305	261	9	legs	leg	NOUN
ejpam-5305	261	10	with	with	ADP
ejpam-5305	261	11	lengths	length	NOUN
ejpam-5305	261	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	261	13	,	,	PUNCT
ejpam-5305	261	14	.	.	PUNCT
ejpam-5305	261	15	.	.	PUNCT
ejpam-5305	262	1	.	.	PUNCT
ejpam-5305	263	1	,	,	PUNCT
ejpam-5305	263	2	mn	mn	PROPN
ejpam-5305	263	3	.	.	PUNCT
ejpam-5305	264	1	if	if	SCONJ
ejpam-5305	264	2	m3,m4	m3,m4	PROPN
ejpam-5305	264	3	,	,	PUNCT
ejpam-5305	264	4	.	.	PUNCT
ejpam-5305	264	5	.	.	PUNCT
ejpam-5305	265	1	.	.	PUNCT
ejpam-5305	266	1	,	,	PUNCT
ejpam-5305	266	2	mn	mn	PROPN
ejpam-5305	266	3	are	be	AUX
ejpam-5305	266	4	even	even	ADV
ejpam-5305	266	5	and	and	CCONJ
ejpam-5305	266	6	mi+1	mi+1	DET
ejpam-5305	266	7	≥	≥	NOUN
ejpam-5305	266	8	m2	m2	PROPN
ejpam-5305	266	9	+	+	CCONJ
ejpam-5305	266	10	m3	m3	PROPN
ejpam-5305	266	11	+	+	CCONJ
ejpam-5305	266	12	.	.	PUNCT
ejpam-5305	266	13	.	.	PUNCT
ejpam-5305	266	14	.	.	PUNCT
ejpam-5305	267	1	+	+	CCONJ
ejpam-5305	267	2	mi	mi	PROPN
ejpam-5305	267	3	+	+	CCONJ
ejpam-5305	267	4	1	1	NUM
ejpam-5305	267	5	for	for	ADP
ejpam-5305	267	6	i	i	PRON
ejpam-5305	267	7	=	=	SYM
ejpam-5305	267	8	2	2	NUM
ejpam-5305	267	9	,	,	PUNCT
ejpam-5305	267	10	3	3	NUM
ejpam-5305	267	11	,	,	PUNCT
ejpam-5305	267	12	.	.	PUNCT
ejpam-5305	267	13	.	.	PUNCT
ejpam-5305	268	1	.	.	PUNCT
ejpam-5305	269	1	,	,	PUNCT
ejpam-5305	269	2	n	n	CCONJ
ejpam-5305	269	3	−	−	PROPN
ejpam-5305	269	4	1	1	NUM
ejpam-5305	269	5	,	,	PUNCT
ejpam-5305	269	6	then	then	ADV
ejpam-5305	269	7	t	t	PROPN
ejpam-5305	269	8	is	be	AUX
ejpam-5305	269	9	a	a	DET
ejpam-5305	269	10	graceful	graceful	ADJ
ejpam-5305	269	11	labeling	labeling	NOUN
ejpam-5305	269	12	graph	graph	NOUN
ejpam-5305	269	13	.	.	PUNCT
ejpam-5305	270	1	proof	proof	NOUN
ejpam-5305	270	2	.	.	PUNCT
ejpam-5305	271	1	let	let	VERB
ejpam-5305	271	2	t	t	NOUN
ejpam-5305	271	3	be	be	AUX
ejpam-5305	271	4	the	the	DET
ejpam-5305	271	5	spider	spider	NOUN
ejpam-5305	271	6	as	as	SCONJ
ejpam-5305	271	7	shown	show	VERB
ejpam-5305	271	8	in	in	ADP
ejpam-5305	271	9	figure	figure	NOUN
ejpam-5305	271	10	1	1	NUM
ejpam-5305	271	11	.	.	PUNCT
ejpam-5305	271	12	case	case	NOUN
ejpam-5305	271	13	1	1	X
ejpam-5305	271	14	.	.	PUNCT
ejpam-5305	272	1	if	if	SCONJ
ejpam-5305	272	2	m1	m1	PROPN
ejpam-5305	272	3	is	be	AUX
ejpam-5305	272	4	odd	odd	ADJ
ejpam-5305	272	5	,	,	PUNCT
ejpam-5305	272	6	then	then	ADV
ejpam-5305	272	7	let	let	VERB
ejpam-5305	272	8	f	f	PRON
ejpam-5305	272	9	be	be	AUX
ejpam-5305	272	10	the	the	DET
ejpam-5305	272	11	labeling	labeling	NOUN
ejpam-5305	272	12	of	of	ADP
ejpam-5305	272	13	t	t	PROPN
ejpam-5305	272	14	of	of	ADP
ejpam-5305	272	15	type	type	NOUN
ejpam-5305	272	16	i.	i.	NOUN
ejpam-5305	272	17	consider	consider	VERB
ejpam-5305	272	18	|f(v31	|f(v31	NOUN
ejpam-5305	272	19	)	)	PUNCT
ejpam-5305	272	20	−	−	PROPN
ejpam-5305	272	21	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	272	22	.	.	PUNCT
ejpam-5305	273	1	since	since	SCONJ
ejpam-5305	273	2	m3	m3	PROPN
ejpam-5305	273	3	≥	≥	PROPN
ejpam-5305	273	4	m2	m2	PROPN
ejpam-5305	273	5	+	+	PROPN
ejpam-5305	273	6	1	1	NUM
ejpam-5305	273	7	,	,	PUNCT
ejpam-5305	273	8	then	then	ADV
ejpam-5305	273	9	by	by	ADP
ejpam-5305	273	10	the	the	DET
ejpam-5305	273	11	way	way	NOUN
ejpam-5305	273	12	of	of	ADP
ejpam-5305	273	13	labeling	labeling	NOUN
ejpam-5305	273	14	of	of	ADP
ejpam-5305	273	15	t	t	PROPN
ejpam-5305	273	16	of	of	ADP
ejpam-5305	273	17	type	type	NOUN
ejpam-5305	273	18	i	i	PRON
ejpam-5305	273	19	we	we	PRON
ejpam-5305	273	20	have	have	VERB
ejpam-5305	273	21	|f(v3(m2	|f(v3(m2	NUM
ejpam-5305	273	22	+	+	ADJ
ejpam-5305	273	23	1))−	1))−	NUM
ejpam-5305	273	24	f(v)|	f(v)|	NOUN
ejpam-5305	273	25	=	=	SYM
ejpam-5305	273	26	|f(v31)−	|f(v31)−	NUM
ejpam-5305	273	27	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	273	28	.	.	PUNCT
ejpam-5305	274	1	if	if	SCONJ
ejpam-5305	274	2	m3	m3	PROPN
ejpam-5305	274	3	=	=	PROPN
ejpam-5305	274	4	m2	m2	PROPN
ejpam-5305	274	5	+	+	PROPN
ejpam-5305	274	6	1	1	NUM
ejpam-5305	274	7	,	,	PUNCT
ejpam-5305	274	8	then	then	ADV
ejpam-5305	274	9	we	we	PRON
ejpam-5305	274	10	can	can	AUX
ejpam-5305	274	11	change	change	VERB
ejpam-5305	274	12	labeling	labeling	NOUN
ejpam-5305	274	13	f	f	PROPN
ejpam-5305	274	14	at	at	ADP
ejpam-5305	274	15	v31	v31	PROPN
ejpam-5305	274	16	,	,	PUNCT
ejpam-5305	274	17	v32	v32	PROPN
ejpam-5305	274	18	,	,	PUNCT
ejpam-5305	274	19	.	.	PUNCT
ejpam-5305	274	20	.	.	PUNCT
ejpam-5305	275	1	.	.	PUNCT
ejpam-5305	276	1	,	,	PUNCT
ejpam-5305	276	2	v3m3	v3m3	X
ejpam-5305	276	3	to	to	PART
ejpam-5305	276	4	be	be	AUX
ejpam-5305	276	5	the	the	DET
ejpam-5305	276	6	labeling	labeling	NOUN
ejpam-5305	276	7	f	f	NOUN
ejpam-5305	276	8	′	′	NUM
ejpam-5305	276	9	such	such	ADJ
ejpam-5305	276	10	that	that	PRON
ejpam-5305	276	11	|f	|f	PROPN
ejpam-5305	276	12	′(v31	′(v31	PROPN
ejpam-5305	276	13	)	)	PUNCT
ejpam-5305	276	14	−	−	PROPN
ejpam-5305	276	15	f(v)|	f(v)|	NOUN
ejpam-5305	276	16	=	=	NOUN
ejpam-5305	276	17	|f(v31	|f(v31	NOUN
ejpam-5305	276	18	)	)	PUNCT
ejpam-5305	276	19	−	−	NOUN
ejpam-5305	276	20	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	276	21	by	by	ADP
ejpam-5305	276	22	reversing	reverse	VERB
ejpam-5305	276	23	the	the	DET
ejpam-5305	276	24	order	order	NOUN
ejpam-5305	276	25	of	of	ADP
ejpam-5305	276	26	theirs	theirs	PRON
ejpam-5305	276	27	labels	label	NOUN
ejpam-5305	276	28	.	.	PUNCT
ejpam-5305	277	1	suppose	suppose	VERB
ejpam-5305	277	2	that	that	SCONJ
ejpam-5305	277	3	m3	m3	PROPN
ejpam-5305	277	4	>	>	X
ejpam-5305	277	5	m2	m2	PROPN
ejpam-5305	278	1	+	+	PROPN
ejpam-5305	278	2	1	1	X
ejpam-5305	278	3	.	.	PUNCT
ejpam-5305	278	4	since	since	SCONJ
ejpam-5305	278	5	m3	m3	PROPN
ejpam-5305	278	6	is	be	AUX
ejpam-5305	278	7	even	even	ADV
ejpam-5305	278	8	,	,	PUNCT
ejpam-5305	278	9	then	then	ADV
ejpam-5305	278	10	by	by	ADP
ejpam-5305	278	11	lemma	lemma	PROPN
ejpam-5305	278	12	2	2	NUM
ejpam-5305	278	13	,	,	PUNCT
ejpam-5305	278	14	we	we	PRON
ejpam-5305	278	15	can	can	AUX
ejpam-5305	278	16	change	change	VERB
ejpam-5305	278	17	the	the	DET
ejpam-5305	278	18	labeling	labeling	NOUN
ejpam-5305	278	19	f	f	PROPN
ejpam-5305	278	20	at	at	ADP
ejpam-5305	278	21	v31	v31	PROPN
ejpam-5305	278	22	,	,	PUNCT
ejpam-5305	278	23	v32	v32	PROPN
ejpam-5305	278	24	,	,	PUNCT
ejpam-5305	278	25	.	.	PUNCT
ejpam-5305	278	26	.	.	PUNCT
ejpam-5305	279	1	.	.	PUNCT
ejpam-5305	280	1	,	,	PUNCT
ejpam-5305	280	2	v3m3	v3m3	X
ejpam-5305	280	3	to	to	PART
ejpam-5305	280	4	be	be	AUX
ejpam-5305	280	5	the	the	DET
ejpam-5305	280	6	labeling	labeling	NOUN
ejpam-5305	280	7	f	f	PROPN
ejpam-5305	280	8	′′	′′	PROPN
ejpam-5305	280	9	such	such	ADJ
ejpam-5305	280	10	that	that	SCONJ
ejpam-5305	280	11	|f	|f	PROPN
ejpam-5305	281	1	′′	′′	PROPN
ejpam-5305	281	2	(	(	PUNCT
ejpam-5305	281	3	v31	v31	PROPN
ejpam-5305	281	4	)	)	PUNCT
ejpam-5305	281	5	−	−	PROPN
ejpam-5305	281	6	f(v)|	f(v)|	NOUN
ejpam-5305	281	7	=	=	NOUN
ejpam-5305	281	8	|f(v31	|f(v31	NOUN
ejpam-5305	281	9	)	)	PUNCT
ejpam-5305	281	10	−	−	PROPN
ejpam-5305	281	11	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	281	12	,	,	PUNCT
ejpam-5305	281	13	the	the	DET
ejpam-5305	281	14	set	set	NOUN
ejpam-5305	281	15	of	of	ADP
ejpam-5305	281	16	vertices	vertex	NOUN
ejpam-5305	281	17	labeling	labeling	NOUN
ejpam-5305	281	18	of	of	ADP
ejpam-5305	281	19	f	f	PROPN
ejpam-5305	281	20	and	and	CCONJ
ejpam-5305	281	21	the	the	DET
ejpam-5305	281	22	set	set	NOUN
ejpam-5305	281	23	of	of	ADP
ejpam-5305	281	24	vertices	vertex	NOUN
ejpam-5305	281	25	labeling	labeling	NOUN
ejpam-5305	281	26	of	of	ADP
ejpam-5305	281	27	f	f	PROPN
ejpam-5305	281	28	′′	′′	PROPN
ejpam-5305	281	29	are	be	AUX
ejpam-5305	281	30	the	the	DET
ejpam-5305	281	31	same	same	ADJ
ejpam-5305	281	32	and	and	CCONJ
ejpam-5305	281	33	the	the	DET
ejpam-5305	281	34	set	set	NOUN
ejpam-5305	281	35	of	of	ADP
ejpam-5305	281	36	edge	edge	NOUN
ejpam-5305	281	37	labeling	labeling	NOUN
ejpam-5305	281	38	of	of	ADP
ejpam-5305	281	39	f	f	PROPN
ejpam-5305	281	40	and	and	CCONJ
ejpam-5305	281	41	k.	k.	PROPN
ejpam-5305	281	42	saengsura	saengsura	PROPN
ejpam-5305	281	43	,	,	PUNCT
ejpam-5305	281	44	t.	t.	PROPN
ejpam-5305	281	45	poomsa	poomsa	PROPN
ejpam-5305	281	46	-	-	PUNCT
ejpam-5305	281	47	ard	ard	PROPN
ejpam-5305	281	48	/	/	PUNCT
ejpam-5305	281	49	eur	eur	PROPN
ejpam-5305	281	50	.	.	PUNCT
ejpam-5305	282	1	j.	j.	PROPN
ejpam-5305	282	2	pure	pure	PROPN
ejpam-5305	282	3	appl	appl	PROPN
ejpam-5305	282	4	.	.	PROPN
ejpam-5305	282	5	math	math	PROPN
ejpam-5305	282	6	,	,	PUNCT
ejpam-5305	282	7	18	18	NUM
ejpam-5305	282	8	(	(	PUNCT
ejpam-5305	282	9	2	2	NUM
ejpam-5305	282	10	)	)	PUNCT
ejpam-5305	282	11	(	(	PUNCT
ejpam-5305	282	12	2025	2025	NUM
ejpam-5305	282	13	)	)	PUNCT
ejpam-5305	282	14	,	,	PUNCT
ejpam-5305	282	15	5305	5305	NUM
ejpam-5305	282	16	6	6	NUM
ejpam-5305	282	17	of	of	ADP
ejpam-5305	282	18	8	8	NUM
ejpam-5305	282	19	the	the	DET
ejpam-5305	282	20	set	set	NOUN
ejpam-5305	282	21	of	of	ADP
ejpam-5305	282	22	edge	edge	NOUN
ejpam-5305	282	23	labeling	labeling	NOUN
ejpam-5305	282	24	of	of	ADP
ejpam-5305	282	25	f	f	PROPN
ejpam-5305	282	26	′′	′′	PROPN
ejpam-5305	282	27	are	be	AUX
ejpam-5305	282	28	the	the	DET
ejpam-5305	282	29	same	same	ADJ
ejpam-5305	282	30	.	.	PUNCT
ejpam-5305	283	1	further	far	ADV
ejpam-5305	283	2	,	,	PUNCT
ejpam-5305	283	3	for	for	SCONJ
ejpam-5305	283	4	any	any	DET
ejpam-5305	283	5	i	i	NOUN
ejpam-5305	283	6	,	,	PUNCT
ejpam-5305	283	7	4	4	NUM
ejpam-5305	283	8	≤	≤	NUM
ejpam-5305	283	9	i	i	PRON
ejpam-5305	283	10	≤	≤	NOUN
ejpam-5305	283	11	n	n	PRON
ejpam-5305	283	12	consider	consider	VERB
ejpam-5305	283	13	|f(vi1	|f(vi1	NOUN
ejpam-5305	283	14	)	)	PUNCT
ejpam-5305	283	15	−	−	PROPN
ejpam-5305	283	16	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	283	17	)	)	PUNCT
ejpam-5305	283	18	|	|	ADV
ejpam-5305	283	19	.	.	PUNCT
ejpam-5305	284	1	let	let	VERB
ejpam-5305	284	2	m′	m′	NOUN
ejpam-5305	284	3	=	=	SYM
ejpam-5305	284	4	m2	m2	PROPN
ejpam-5305	284	5	+	+	CCONJ
ejpam-5305	284	6	m3	m3	PROPN
ejpam-5305	284	7	+	+	CCONJ
ejpam-5305	284	8	.	.	PUNCT
ejpam-5305	284	9	.	.	PUNCT
ejpam-5305	284	10	.	.	PUNCT
ejpam-5305	285	1	+	+	CCONJ
ejpam-5305	286	1	mi−1	mi−1	PROPN
ejpam-5305	286	2	.	.	PROPN
ejpam-5305	287	1	since	since	SCONJ
ejpam-5305	287	2	mi	mi	PROPN
ejpam-5305	287	3	≥	≥	PROPN
ejpam-5305	287	4	m′	m′	NOUN
ejpam-5305	287	5	+	+	CCONJ
ejpam-5305	287	6	1	1	NUM
ejpam-5305	287	7	,	,	PUNCT
ejpam-5305	287	8	then	then	ADV
ejpam-5305	287	9	by	by	ADP
ejpam-5305	287	10	the	the	DET
ejpam-5305	287	11	way	way	NOUN
ejpam-5305	287	12	of	of	ADP
ejpam-5305	287	13	labeling	labeling	NOUN
ejpam-5305	287	14	of	of	ADP
ejpam-5305	287	15	t	t	PROPN
ejpam-5305	287	16	of	of	ADP
ejpam-5305	287	17	type	type	NOUN
ejpam-5305	287	18	i	i	PRON
ejpam-5305	287	19	we	we	PRON
ejpam-5305	287	20	have	have	VERB
ejpam-5305	287	21	|f(vm′+1	|f(vm′+1	NOUN
ejpam-5305	287	22	)	)	PUNCT
ejpam-5305	288	1	−	−	PROPN
ejpam-5305	288	2	f(v)|	f(v)|	NOUN
ejpam-5305	288	3	=	=	SYM
ejpam-5305	288	4	|f(vi1	|f(vi1	NOUN
ejpam-5305	288	5	)	)	PUNCT
ejpam-5305	288	6	−	−	PROPN
ejpam-5305	288	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	288	8	)	)	PUNCT
ejpam-5305	288	9	|	|	ADV
ejpam-5305	288	10	.	.	PUNCT
ejpam-5305	289	1	if	if	SCONJ
ejpam-5305	289	2	mi	mi	PROPN
ejpam-5305	289	3	=	=	NOUN
ejpam-5305	289	4	m′	m′	PROPN
ejpam-5305	289	5	+	+	CCONJ
ejpam-5305	289	6	1	1	NUM
ejpam-5305	289	7	,	,	PUNCT
ejpam-5305	289	8	then	then	ADV
ejpam-5305	289	9	we	we	PRON
ejpam-5305	289	10	can	can	AUX
ejpam-5305	289	11	change	change	VERB
ejpam-5305	289	12	labeling	labeling	NOUN
ejpam-5305	289	13	f	f	PROPN
ejpam-5305	289	14	at	at	ADP
ejpam-5305	289	15	vi1	vi1	PROPN
ejpam-5305	289	16	,	,	PUNCT
ejpam-5305	289	17	vi2	vi2	NOUN
ejpam-5305	289	18	,	,	PUNCT
ejpam-5305	289	19	.	.	PUNCT
ejpam-5305	289	20	.	.	PUNCT
ejpam-5305	290	1	.	.	PUNCT
ejpam-5305	291	1	,	,	PUNCT
ejpam-5305	291	2	vimi	vimi	VERB
ejpam-5305	291	3	to	to	PART
ejpam-5305	291	4	be	be	AUX
ejpam-5305	291	5	the	the	DET
ejpam-5305	291	6	labeling	labeling	NOUN
ejpam-5305	291	7	f	f	X
ejpam-5305	291	8	′′′	′′′	PROPN
ejpam-5305	291	9	such	such	ADJ
ejpam-5305	291	10	that	that	PRON
ejpam-5305	291	11	|f	|f	PROPN
ejpam-5305	292	1	′′′	′′′	PROPN
ejpam-5305	292	2	(	(	PUNCT
ejpam-5305	292	3	vi1)−	vi1)−	ADJ
ejpam-5305	292	4	f(v)|	f(v)|	NOUN
ejpam-5305	292	5	=	=	SYM
ejpam-5305	292	6	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	292	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	292	8	)	)	PUNCT
ejpam-5305	293	1	|	|	ADV
ejpam-5305	293	2	by	by	ADP
ejpam-5305	293	3	reversing	reverse	VERB
ejpam-5305	293	4	the	the	DET
ejpam-5305	293	5	order	order	NOUN
ejpam-5305	293	6	of	of	ADP
ejpam-5305	293	7	theirs	theirs	PRON
ejpam-5305	293	8	labels	label	NOUN
ejpam-5305	293	9	.	.	PUNCT
ejpam-5305	294	1	suppose	suppose	VERB
ejpam-5305	294	2	thatmi	thatmi	PROPN
ejpam-5305	294	3	>	>	X
ejpam-5305	294	4	m′+1	m′+1	PROPN
ejpam-5305	294	5	.	.	PUNCT
ejpam-5305	295	1	sincemi	sincemi	PROPN
ejpam-5305	295	2	is	be	AUX
ejpam-5305	295	3	even	even	ADV
ejpam-5305	295	4	,	,	PUNCT
ejpam-5305	295	5	then	then	ADV
ejpam-5305	295	6	by	by	ADP
ejpam-5305	295	7	lemma	lemma	PROPN
ejpam-5305	295	8	2	2	NUM
ejpam-5305	295	9	,	,	PUNCT
ejpam-5305	295	10	we	we	PRON
ejpam-5305	295	11	can	can	AUX
ejpam-5305	295	12	change	change	VERB
ejpam-5305	295	13	labeling	labeling	NOUN
ejpam-5305	295	14	f	f	PROPN
ejpam-5305	295	15	at	at	ADP
ejpam-5305	295	16	vi1	vi1	PROPN
ejpam-5305	295	17	,	,	PUNCT
ejpam-5305	295	18	vi2	vi2	NOUN
ejpam-5305	295	19	,	,	PUNCT
ejpam-5305	295	20	.	.	PUNCT
ejpam-5305	295	21	.	.	PUNCT
ejpam-5305	296	1	.	.	PUNCT
ejpam-5305	297	1	,	,	PUNCT
ejpam-5305	297	2	vimi	vimi	VERB
ejpam-5305	297	3	to	to	PART
ejpam-5305	297	4	be	be	AUX
ejpam-5305	297	5	the	the	DET
ejpam-5305	297	6	labeling	labeling	NOUN
ejpam-5305	297	7	f	f	NOUN
ejpam-5305	297	8	iv	iv	NUM
ejpam-5305	297	9	such	such	ADJ
ejpam-5305	297	10	that	that	SCONJ
ejpam-5305	297	11	|f	|f	PROPN
ejpam-5305	297	12	iv(vi1)−	iv(vi1)−	PROPN
ejpam-5305	297	13	f(v)|	f(v)|	NOUN
ejpam-5305	297	14	=	=	SYM
ejpam-5305	297	15	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	297	16	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	297	17	)	)	PUNCT
ejpam-5305	297	18	|	|	ADV
ejpam-5305	297	19	the	the	DET
ejpam-5305	297	20	set	set	NOUN
ejpam-5305	297	21	of	of	ADP
ejpam-5305	297	22	vertices	vertex	NOUN
ejpam-5305	297	23	labeling	labeling	NOUN
ejpam-5305	297	24	of	of	ADP
ejpam-5305	297	25	f	f	PROPN
ejpam-5305	297	26	and	and	CCONJ
ejpam-5305	297	27	the	the	DET
ejpam-5305	297	28	set	set	NOUN
ejpam-5305	297	29	of	of	ADP
ejpam-5305	297	30	vertices	vertex	NOUN
ejpam-5305	297	31	labeling	labeling	NOUN
ejpam-5305	297	32	of	of	ADP
ejpam-5305	297	33	f	f	PROPN
ejpam-5305	297	34	iv	iv	NOUN
ejpam-5305	297	35	are	be	AUX
ejpam-5305	297	36	the	the	DET
ejpam-5305	297	37	same	same	ADJ
ejpam-5305	297	38	and	and	CCONJ
ejpam-5305	297	39	the	the	DET
ejpam-5305	297	40	set	set	NOUN
ejpam-5305	297	41	of	of	ADP
ejpam-5305	297	42	edge	edge	NOUN
ejpam-5305	297	43	labeling	labeling	NOUN
ejpam-5305	297	44	of	of	ADP
ejpam-5305	297	45	f	f	PROPN
ejpam-5305	297	46	and	and	CCONJ
ejpam-5305	297	47	the	the	DET
ejpam-5305	297	48	set	set	NOUN
ejpam-5305	297	49	of	of	ADP
ejpam-5305	297	50	edge	edge	NOUN
ejpam-5305	297	51	labeling	labeling	NOUN
ejpam-5305	297	52	of	of	ADP
ejpam-5305	297	53	f	f	PROPN
ejpam-5305	297	54	iv	iv	NOUN
ejpam-5305	297	55	are	be	AUX
ejpam-5305	297	56	the	the	DET
ejpam-5305	297	57	same	same	ADJ
ejpam-5305	297	58	.	.	PUNCT
ejpam-5305	298	1	hence	hence	ADV
ejpam-5305	298	2	t	t	PROPN
ejpam-5305	298	3	amit	amit	PROPN
ejpam-5305	298	4	graceful	graceful	ADJ
ejpam-5305	298	5	labeling	labeling	NOUN
ejpam-5305	298	6	.	.	PUNCT
ejpam-5305	299	1	case	case	NOUN
ejpam-5305	299	2	2	2	NUM
ejpam-5305	299	3	.	.	PUNCT
ejpam-5305	300	1	if	if	SCONJ
ejpam-5305	300	2	m1	m1	PROPN
ejpam-5305	300	3	is	be	AUX
ejpam-5305	300	4	even	even	ADV
ejpam-5305	300	5	,	,	PUNCT
ejpam-5305	300	6	then	then	ADV
ejpam-5305	300	7	let	let	VERB
ejpam-5305	300	8	f	f	PRON
ejpam-5305	300	9	be	be	AUX
ejpam-5305	300	10	the	the	DET
ejpam-5305	300	11	labeling	labeling	NOUN
ejpam-5305	300	12	of	of	ADP
ejpam-5305	300	13	t	t	PROPN
ejpam-5305	300	14	of	of	ADP
ejpam-5305	300	15	type	type	NOUN
ejpam-5305	300	16	ii	ii	PROPN
ejpam-5305	300	17	and	and	CCONJ
ejpam-5305	300	18	consider	consider	VERB
ejpam-5305	300	19	in	in	ADP
ejpam-5305	300	20	a	a	DET
ejpam-5305	300	21	similar	similar	ADJ
ejpam-5305	300	22	way	way	NOUN
ejpam-5305	300	23	as	as	ADP
ejpam-5305	300	24	the	the	DET
ejpam-5305	300	25	case	case	NOUN
ejpam-5305	300	26	1	1	NUM
ejpam-5305	300	27	,	,	PUNCT
ejpam-5305	300	28	we	we	PRON
ejpam-5305	300	29	get	get	VERB
ejpam-5305	300	30	t	t	PROPN
ejpam-5305	300	31	admit	admit	VERB
ejpam-5305	300	32	graceful	graceful	ADJ
ejpam-5305	300	33	labeling	labeling	NOUN
ejpam-5305	300	34	.	.	PUNCT
ejpam-5305	301	1	next	next	ADV
ejpam-5305	301	2	we	we	PRON
ejpam-5305	301	3	will	will	AUX
ejpam-5305	301	4	combine	combine	VERB
ejpam-5305	301	5	the	the	DET
ejpam-5305	301	6	results	result	NOUN
ejpam-5305	301	7	of	of	ADP
ejpam-5305	301	8	theorem	theorem	ADJ
ejpam-5305	301	9	2	2	NUM
ejpam-5305	301	10	and	and	CCONJ
ejpam-5305	301	11	theorem	theorem	VERB
ejpam-5305	301	12	5	5	NUM
ejpam-5305	301	13	as	as	ADP
ejpam-5305	301	14	the	the	DET
ejpam-5305	301	15	following	follow	VERB
ejpam-5305	301	16	theorem	theorem	NOUN
ejpam-5305	301	17	.	.	PUNCT
ejpam-5305	301	18	theorem	theorem	NOUN
ejpam-5305	301	19	6	6	NUM
ejpam-5305	301	20	.	.	PUNCT
ejpam-5305	302	1	let	let	VERB
ejpam-5305	302	2	t	t	PROPN
ejpam-5305	302	3	be	be	AUX
ejpam-5305	302	4	a	a	DET
ejpam-5305	302	5	spider	spider	NOUN
ejpam-5305	302	6	graph	graph	NOUN
ejpam-5305	302	7	of	of	ADP
ejpam-5305	302	8	n	n	NOUN
ejpam-5305	302	9	legs	leg	NOUN
ejpam-5305	302	10	with	with	ADP
ejpam-5305	302	11	lengths	length	NOUN
ejpam-5305	302	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	302	13	,	,	PUNCT
ejpam-5305	302	14	.	.	PUNCT
ejpam-5305	302	15	.	.	PUNCT
ejpam-5305	303	1	.	.	PUNCT
ejpam-5305	304	1	,	,	PUNCT
ejpam-5305	304	2	mn	mn	PROPN
ejpam-5305	304	3	.	.	PUNCT
ejpam-5305	305	1	if	if	SCONJ
ejpam-5305	305	2	t	t	PROPN
ejpam-5305	305	3	satisfies	satisfy	VERB
ejpam-5305	305	4	the	the	DET
ejpam-5305	305	5	following	follow	VERB
ejpam-5305	305	6	conditions	condition	NOUN
ejpam-5305	305	7	:	:	PUNCT
ejpam-5305	305	8	(	(	PUNCT
ejpam-5305	305	9	i	i	NOUN
ejpam-5305	305	10	)	)	PUNCT
ejpam-5305	305	11	if	if	SCONJ
ejpam-5305	305	12	mi	mi	PROPN
ejpam-5305	305	13	is	be	AUX
ejpam-5305	305	14	odd	odd	ADJ
ejpam-5305	305	15	,	,	PUNCT
ejpam-5305	305	16	then	then	ADV
ejpam-5305	305	17	mi	mi	PROPN
ejpam-5305	305	18	=	=	PROPN
ejpam-5305	305	19	m2	m2	PROPN
ejpam-5305	305	20	+	+	PROPN
ejpam-5305	305	21	m3	m3	PROPN
ejpam-5305	305	22	+	+	X
ejpam-5305	305	23	.	.	PUNCT
ejpam-5305	305	24	.	.	PUNCT
ejpam-5305	306	1	.+mi−1	.+mi−1	PROPN
ejpam-5305	307	1	+	+	CCONJ
ejpam-5305	307	2	1	1	NUM
ejpam-5305	307	3	,	,	PUNCT
ejpam-5305	307	4	(	(	PUNCT
ejpam-5305	307	5	ii	ii	NOUN
ejpam-5305	307	6	)	)	PUNCT
ejpam-5305	307	7	if	if	SCONJ
ejpam-5305	307	8	mi	mi	PROPN
ejpam-5305	307	9	is	be	AUX
ejpam-5305	307	10	even	even	ADV
ejpam-5305	307	11	,	,	PUNCT
ejpam-5305	307	12	then	then	ADV
ejpam-5305	307	13	mi	mi	PROPN
ejpam-5305	307	14	≥	≥	PROPN
ejpam-5305	307	15	m2	m2	PROPN
ejpam-5305	307	16	+	+	PROPN
ejpam-5305	307	17	m3	m3	PROPN
ejpam-5305	307	18	+	+	X
ejpam-5305	307	19	.	.	PUNCT
ejpam-5305	307	20	.	.	PUNCT
ejpam-5305	308	1	.+mi−1	.+mi−1	PROPN
ejpam-5305	309	1	+	+	CCONJ
ejpam-5305	309	2	1	1	NUM
ejpam-5305	309	3	,	,	PUNCT
ejpam-5305	309	4	for	for	ADP
ejpam-5305	309	5	any	any	DET
ejpam-5305	309	6	i	i	NOUN
ejpam-5305	309	7	=	=	NOUN
ejpam-5305	309	8	3	3	NUM
ejpam-5305	309	9	,	,	PUNCT
ejpam-5305	309	10	4	4	NUM
ejpam-5305	309	11	,	,	PUNCT
ejpam-5305	309	12	.	.	PUNCT
ejpam-5305	309	13	.	.	PUNCT
ejpam-5305	310	1	.	.	PUNCT
ejpam-5305	311	1	,	,	PUNCT
ejpam-5305	311	2	n	n	CCONJ
ejpam-5305	311	3	,	,	PUNCT
ejpam-5305	311	4	then	then	ADV
ejpam-5305	311	5	t	t	PROPN
ejpam-5305	311	6	is	be	AUX
ejpam-5305	311	7	a	a	DET
ejpam-5305	311	8	graceful	graceful	ADJ
ejpam-5305	311	9	labeling	labeling	NOUN
ejpam-5305	311	10	graph	graph	NOUN
ejpam-5305	311	11	.	.	PUNCT
ejpam-5305	312	1	proof	proof	NOUN
ejpam-5305	312	2	.	.	PUNCT
ejpam-5305	313	1	let	let	VERB
ejpam-5305	313	2	t	t	NOUN
ejpam-5305	313	3	be	be	AUX
ejpam-5305	313	4	the	the	DET
ejpam-5305	313	5	spider	spider	NOUN
ejpam-5305	313	6	as	as	SCONJ
ejpam-5305	313	7	shown	show	VERB
ejpam-5305	313	8	in	in	ADP
ejpam-5305	313	9	figure	figure	NOUN
ejpam-5305	313	10	1	1	NUM
ejpam-5305	313	11	.	.	PUNCT
ejpam-5305	313	12	case	case	NOUN
ejpam-5305	313	13	1	1	X
ejpam-5305	313	14	.	.	PUNCT
ejpam-5305	314	1	if	if	SCONJ
ejpam-5305	314	2	m1	m1	PROPN
ejpam-5305	314	3	is	be	AUX
ejpam-5305	314	4	odd	odd	ADJ
ejpam-5305	314	5	,	,	PUNCT
ejpam-5305	314	6	then	then	ADV
ejpam-5305	314	7	let	let	VERB
ejpam-5305	314	8	f	f	PRON
ejpam-5305	314	9	be	be	AUX
ejpam-5305	314	10	the	the	DET
ejpam-5305	314	11	labeling	labeling	NOUN
ejpam-5305	314	12	of	of	ADP
ejpam-5305	314	13	t	t	PROPN
ejpam-5305	314	14	of	of	ADP
ejpam-5305	314	15	type	type	NOUN
ejpam-5305	314	16	i.	i.	NOUN
ejpam-5305	314	17	for	for	ADP
ejpam-5305	314	18	any	any	DET
ejpam-5305	314	19	mi	mi	PROPN
ejpam-5305	314	20	,	,	PUNCT
ejpam-5305	314	21	i	i	NOUN
ejpam-5305	314	22	=	=	NOUN
ejpam-5305	314	23	3	3	NUM
ejpam-5305	314	24	,	,	PUNCT
ejpam-5305	314	25	4	4	NUM
ejpam-5305	314	26	,	,	PUNCT
ejpam-5305	314	27	.	.	PUNCT
ejpam-5305	314	28	.	.	PUNCT
ejpam-5305	315	1	.	.	PUNCT
ejpam-5305	316	1	,	,	PUNCT
ejpam-5305	316	2	n	n	PRON
ejpam-5305	316	3	consider	consider	VERB
ejpam-5305	316	4	|f(vi1	|f(vi1	NOUN
ejpam-5305	316	5	)	)	PUNCT
ejpam-5305	316	6	−	−	PROPN
ejpam-5305	316	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	316	8	)	)	PUNCT
ejpam-5305	316	9	|	|	ADV
ejpam-5305	316	10	.	.	PUNCT
ejpam-5305	317	1	let	let	VERB
ejpam-5305	317	2	m′	m′	NOUN
ejpam-5305	317	3	=	=	SYM
ejpam-5305	317	4	m2	m2	PROPN
ejpam-5305	317	5	+	+	CCONJ
ejpam-5305	317	6	m3	m3	PROPN
ejpam-5305	317	7	+	+	CCONJ
ejpam-5305	317	8	.	.	PUNCT
ejpam-5305	317	9	.	.	PUNCT
ejpam-5305	317	10	.	.	PUNCT
ejpam-5305	318	1	+	+	CCONJ
ejpam-5305	319	1	mi−1	mi−1	PROPN
ejpam-5305	319	2	.	.	PROPN
ejpam-5305	320	1	since	since	SCONJ
ejpam-5305	320	2	mi	mi	PROPN
ejpam-5305	320	3	≥	≥	PROPN
ejpam-5305	320	4	m′+1	m′+1	PROPN
ejpam-5305	320	5	,	,	PUNCT
ejpam-5305	320	6	then	then	ADV
ejpam-5305	320	7	by	by	ADP
ejpam-5305	320	8	the	the	DET
ejpam-5305	320	9	way	way	NOUN
ejpam-5305	320	10	of	of	ADP
ejpam-5305	320	11	labeling	labeling	NOUN
ejpam-5305	320	12	of	of	ADP
ejpam-5305	320	13	t	t	PROPN
ejpam-5305	320	14	of	of	ADP
ejpam-5305	320	15	type	type	NOUN
ejpam-5305	320	16	i	i	PRON
ejpam-5305	320	17	we	we	PRON
ejpam-5305	320	18	have	have	VERB
ejpam-5305	320	19	|f(vm′+1)−f(v)|	|f(vm′+1)−f(v)|	NOUN
ejpam-5305	320	20	=	=	SYM
ejpam-5305	320	21	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	320	22	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	320	23	)	)	PUNCT
ejpam-5305	321	1	|	|	ADV
ejpam-5305	321	2	.	.	PUNCT
ejpam-5305	322	1	if	if	SCONJ
ejpam-5305	322	2	mi	mi	PROPN
ejpam-5305	322	3	is	be	AUX
ejpam-5305	322	4	odd	odd	ADJ
ejpam-5305	322	5	,	,	PUNCT
ejpam-5305	322	6	then	then	ADV
ejpam-5305	322	7	by	by	ADP
ejpam-5305	322	8	(	(	PUNCT
ejpam-5305	322	9	i	i	NOUN
ejpam-5305	322	10	)	)	PUNCT
ejpam-5305	322	11	we	we	PRON
ejpam-5305	322	12	have	have	VERB
ejpam-5305	322	13	mi	mi	PROPN
ejpam-5305	322	14	=	=	PROPN
ejpam-5305	322	15	m′+1	m′+1	PROPN
ejpam-5305	322	16	.	.	PUNCT
ejpam-5305	323	1	we	we	PRON
ejpam-5305	323	2	can	can	AUX
ejpam-5305	323	3	change	change	VERB
ejpam-5305	323	4	labeling	labeling	NOUN
ejpam-5305	323	5	f	f	PROPN
ejpam-5305	323	6	at	at	ADP
ejpam-5305	323	7	vi1	vi1	PROPN
ejpam-5305	323	8	,	,	PUNCT
ejpam-5305	323	9	vi2	vi2	NOUN
ejpam-5305	323	10	,	,	PUNCT
ejpam-5305	323	11	.	.	PUNCT
ejpam-5305	323	12	.	.	PUNCT
ejpam-5305	324	1	.	.	PUNCT
ejpam-5305	325	1	,	,	PUNCT
ejpam-5305	325	2	vimi	vimi	VERB
ejpam-5305	325	3	to	to	PART
ejpam-5305	325	4	be	be	AUX
ejpam-5305	325	5	the	the	DET
ejpam-5305	325	6	labeling	labeling	NOUN
ejpam-5305	326	1	f	f	NOUN
ejpam-5305	326	2	′	′	NUM
ejpam-5305	327	1	such	such	ADJ
ejpam-5305	327	2	that	that	SCONJ
ejpam-5305	327	3	|f	|f	PROPN
ejpam-5305	327	4	′(vi1	′(vi1	PROPN
ejpam-5305	327	5	)	)	PUNCT
ejpam-5305	327	6	−	−	PROPN
ejpam-5305	327	7	f(v)|	f(v)|	NOUN
ejpam-5305	327	8	=	=	SYM
ejpam-5305	327	9	|f(vi1	|f(vi1	NOUN
ejpam-5305	327	10	)	)	PUNCT
ejpam-5305	327	11	−	−	PROPN
ejpam-5305	327	12	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	327	13	)	)	PUNCT
ejpam-5305	327	14	|	|	ADV
ejpam-5305	327	15	by	by	ADP
ejpam-5305	327	16	reversing	reverse	VERB
ejpam-5305	327	17	the	the	DET
ejpam-5305	327	18	order	order	NOUN
ejpam-5305	327	19	of	of	ADP
ejpam-5305	327	20	theirs	theirs	PRON
ejpam-5305	327	21	labels	label	NOUN
ejpam-5305	327	22	.	.	PUNCT
ejpam-5305	328	1	if	if	SCONJ
ejpam-5305	328	2	mi	mi	PROPN
ejpam-5305	328	3	is	be	AUX
ejpam-5305	328	4	even	even	ADV
ejpam-5305	328	5	,	,	PUNCT
ejpam-5305	328	6	then	then	ADV
ejpam-5305	328	7	by	by	ADP
ejpam-5305	328	8	(	(	PUNCT
ejpam-5305	328	9	ii	ii	NOUN
ejpam-5305	328	10	)	)	PUNCT
ejpam-5305	328	11	we	we	PRON
ejpam-5305	328	12	have	have	VERB
ejpam-5305	328	13	mi	mi	PROPN
ejpam-5305	328	14	≥	≥	PROPN
ejpam-5305	328	15	m′	m′	NOUN
ejpam-5305	328	16	+	+	CCONJ
ejpam-5305	329	1	1	1	X
ejpam-5305	329	2	.	.	X
ejpam-5305	329	3	if	if	SCONJ
ejpam-5305	329	4	mi	mi	PROPN
ejpam-5305	329	5	=	=	NOUN
ejpam-5305	329	6	m′	m′	PROPN
ejpam-5305	330	1	+	+	CCONJ
ejpam-5305	330	2	1	1	NUM
ejpam-5305	330	3	,	,	PUNCT
ejpam-5305	330	4	then	then	ADV
ejpam-5305	330	5	we	we	PRON
ejpam-5305	330	6	can	can	AUX
ejpam-5305	330	7	change	change	VERB
ejpam-5305	330	8	labeling	labeling	NOUN
ejpam-5305	330	9	f	f	PROPN
ejpam-5305	330	10	at	at	ADP
ejpam-5305	330	11	vi1	vi1	PROPN
ejpam-5305	330	12	,	,	PUNCT
ejpam-5305	330	13	vi2	vi2	NOUN
ejpam-5305	330	14	,	,	PUNCT
ejpam-5305	330	15	.	.	PUNCT
ejpam-5305	330	16	.	.	PUNCT
ejpam-5305	331	1	.	.	PUNCT
ejpam-5305	332	1	,	,	PUNCT
ejpam-5305	332	2	vimi	vimi	VERB
ejpam-5305	332	3	to	to	PART
ejpam-5305	332	4	be	be	AUX
ejpam-5305	332	5	the	the	DET
ejpam-5305	332	6	labeling	labeling	NOUN
ejpam-5305	332	7	f	f	PROPN
ejpam-5305	332	8	′′	′′	PROPN
ejpam-5305	332	9	such	such	ADJ
ejpam-5305	332	10	that	that	SCONJ
ejpam-5305	332	11	|f	|f	PROPN
ejpam-5305	333	1	′′	′′	PROPN
ejpam-5305	333	2	(	(	PUNCT
ejpam-5305	333	3	vi1)−	vi1)−	ADJ
ejpam-5305	333	4	f(v)|	f(v)|	NOUN
ejpam-5305	333	5	=	=	SYM
ejpam-5305	333	6	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	333	7	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	333	8	)	)	PUNCT
ejpam-5305	333	9	|	|	ADV
ejpam-5305	333	10	by	by	ADP
ejpam-5305	333	11	reversing	reverse	VERB
ejpam-5305	333	12	the	the	DET
ejpam-5305	333	13	order	order	NOUN
ejpam-5305	333	14	of	of	ADP
ejpam-5305	333	15	theirs	theirs	PRON
ejpam-5305	333	16	labels	label	NOUN
ejpam-5305	333	17	.	.	PUNCT
ejpam-5305	334	1	suppose	suppose	VERB
ejpam-5305	334	2	thatmi	thatmi	PROPN
ejpam-5305	334	3	>	>	X
ejpam-5305	334	4	m′+1	m′+1	PROPN
ejpam-5305	334	5	.	.	PUNCT
ejpam-5305	335	1	sincemi	sincemi	PROPN
ejpam-5305	335	2	is	be	AUX
ejpam-5305	335	3	even	even	ADV
ejpam-5305	335	4	,	,	PUNCT
ejpam-5305	335	5	then	then	ADV
ejpam-5305	335	6	by	by	ADP
ejpam-5305	335	7	lemma	lemma	PROPN
ejpam-5305	335	8	2	2	NUM
ejpam-5305	335	9	,	,	PUNCT
ejpam-5305	335	10	we	we	PRON
ejpam-5305	335	11	can	can	AUX
ejpam-5305	335	12	change	change	VERB
ejpam-5305	335	13	labeling	labeling	NOUN
ejpam-5305	335	14	f	f	PROPN
ejpam-5305	335	15	at	at	ADP
ejpam-5305	335	16	vi1	vi1	PROPN
ejpam-5305	335	17	,	,	PUNCT
ejpam-5305	335	18	vi2	vi2	NOUN
ejpam-5305	335	19	,	,	PUNCT
ejpam-5305	335	20	.	.	PUNCT
ejpam-5305	335	21	.	.	PUNCT
ejpam-5305	336	1	.	.	PUNCT
ejpam-5305	337	1	,	,	PUNCT
ejpam-5305	337	2	vimi	vimi	VERB
ejpam-5305	337	3	to	to	PART
ejpam-5305	337	4	be	be	AUX
ejpam-5305	337	5	the	the	DET
ejpam-5305	337	6	labeling	labeling	NOUN
ejpam-5305	337	7	f	f	X
ejpam-5305	337	8	′′′	′′′	PROPN
ejpam-5305	337	9	such	such	ADJ
ejpam-5305	337	10	that	that	PRON
ejpam-5305	337	11	|f	|f	PROPN
ejpam-5305	337	12	′′′	′′′	PROPN
ejpam-5305	337	13	(	(	PUNCT
ejpam-5305	337	14	vi1)−	vi1)−	ADJ
ejpam-5305	337	15	f(v)|	f(v)|	NOUN
ejpam-5305	337	16	=	=	SYM
ejpam-5305	337	17	|f(vi1)−	|f(vi1)−	NOUN
ejpam-5305	337	18	f(v(i−1)mi−1	f(v(i−1)mi−1	PROPN
ejpam-5305	337	19	)	)	PUNCT
ejpam-5305	337	20	|	|	ADV
ejpam-5305	337	21	the	the	DET
ejpam-5305	337	22	set	set	NOUN
ejpam-5305	337	23	of	of	ADP
ejpam-5305	337	24	vertices	vertex	NOUN
ejpam-5305	337	25	labeling	labeling	NOUN
ejpam-5305	337	26	of	of	ADP
ejpam-5305	337	27	f	f	PROPN
ejpam-5305	337	28	and	and	CCONJ
ejpam-5305	337	29	the	the	DET
ejpam-5305	337	30	set	set	NOUN
ejpam-5305	337	31	of	of	ADP
ejpam-5305	337	32	vertices	vertex	NOUN
ejpam-5305	337	33	labeling	labeling	NOUN
ejpam-5305	337	34	of	of	ADP
ejpam-5305	337	35	f	f	PROPN
ejpam-5305	337	36	′′′	′′′	PROPN
ejpam-5305	337	37	are	be	AUX
ejpam-5305	337	38	the	the	DET
ejpam-5305	337	39	same	same	ADJ
ejpam-5305	337	40	and	and	CCONJ
ejpam-5305	337	41	the	the	DET
ejpam-5305	337	42	set	set	NOUN
ejpam-5305	337	43	of	of	ADP
ejpam-5305	337	44	edge	edge	NOUN
ejpam-5305	337	45	labeling	labeling	NOUN
ejpam-5305	337	46	of	of	ADP
ejpam-5305	337	47	f	f	PROPN
ejpam-5305	337	48	and	and	CCONJ
ejpam-5305	337	49	the	the	DET
ejpam-5305	337	50	set	set	NOUN
ejpam-5305	337	51	of	of	ADP
ejpam-5305	337	52	edge	edge	NOUN
ejpam-5305	337	53	labeling	labeling	NOUN
ejpam-5305	337	54	of	of	ADP
ejpam-5305	337	55	f	f	PROPN
ejpam-5305	337	56	′′′	′′′	PROPN
ejpam-5305	337	57	are	be	AUX
ejpam-5305	337	58	the	the	DET
ejpam-5305	337	59	same	same	ADJ
ejpam-5305	337	60	.	.	PUNCT
ejpam-5305	338	1	hence	hence	ADV
ejpam-5305	338	2	t	t	PROPN
ejpam-5305	338	3	amit	amit	PROPN
ejpam-5305	338	4	graceful	graceful	ADJ
ejpam-5305	338	5	labeling	labeling	NOUN
ejpam-5305	338	6	.	.	PUNCT
ejpam-5305	339	1	case	case	NOUN
ejpam-5305	339	2	2	2	NUM
ejpam-5305	339	3	.	.	PUNCT
ejpam-5305	340	1	if	if	SCONJ
ejpam-5305	340	2	m1	m1	PROPN
ejpam-5305	340	3	is	be	AUX
ejpam-5305	340	4	even	even	ADV
ejpam-5305	340	5	,	,	PUNCT
ejpam-5305	340	6	then	then	ADV
ejpam-5305	340	7	let	let	VERB
ejpam-5305	340	8	f	f	PRON
ejpam-5305	340	9	be	be	AUX
ejpam-5305	340	10	the	the	DET
ejpam-5305	340	11	labeling	labeling	NOUN
ejpam-5305	340	12	of	of	ADP
ejpam-5305	340	13	t	t	PROPN
ejpam-5305	340	14	of	of	ADP
ejpam-5305	340	15	type	type	NOUN
ejpam-5305	340	16	ii	ii	PROPN
ejpam-5305	340	17	and	and	CCONJ
ejpam-5305	340	18	consider	consider	VERB
ejpam-5305	340	19	in	in	ADP
ejpam-5305	340	20	a	a	DET
ejpam-5305	340	21	similar	similar	ADJ
ejpam-5305	340	22	way	way	NOUN
ejpam-5305	340	23	as	as	ADP
ejpam-5305	340	24	the	the	DET
ejpam-5305	340	25	case	case	NOUN
ejpam-5305	340	26	1	1	NUM
ejpam-5305	340	27	,	,	PUNCT
ejpam-5305	340	28	we	we	PRON
ejpam-5305	340	29	get	get	VERB
ejpam-5305	340	30	t	t	PROPN
ejpam-5305	340	31	admit	admit	VERB
ejpam-5305	340	32	graceful	graceful	ADJ
ejpam-5305	340	33	labeling	labeling	NOUN
ejpam-5305	340	34	.	.	PUNCT
ejpam-5305	341	1	next	next	ADV
ejpam-5305	341	2	we	we	PRON
ejpam-5305	341	3	will	will	AUX
ejpam-5305	341	4	consider	consider	VERB
ejpam-5305	341	5	a	a	DET
ejpam-5305	341	6	spider	spider	NOUN
ejpam-5305	341	7	graph	graph	NOUN
ejpam-5305	341	8	in	in	ADP
ejpam-5305	341	9	which	which	PRON
ejpam-5305	341	10	all	all	PRON
ejpam-5305	341	11	of	of	ADP
ejpam-5305	341	12	its	its	PRON
ejpam-5305	341	13	legs	leg	NOUN
ejpam-5305	341	14	except	except	SCONJ
ejpam-5305	341	15	one	one	NUM
ejpam-5305	341	16	are	be	AUX
ejpam-5305	341	17	equal	equal	ADJ
ejpam-5305	341	18	.	.	PUNCT
ejpam-5305	342	1	theorem	theorem	ADJ
ejpam-5305	342	2	7	7	NUM
ejpam-5305	342	3	.	.	PUNCT
ejpam-5305	343	1	let	let	VERB
ejpam-5305	343	2	t	t	PROPN
ejpam-5305	343	3	be	be	AUX
ejpam-5305	343	4	a	a	DET
ejpam-5305	343	5	spider	spider	NOUN
ejpam-5305	343	6	graph	graph	NOUN
ejpam-5305	343	7	of	of	ADP
ejpam-5305	343	8	n	n	NOUN
ejpam-5305	343	9	legs	leg	NOUN
ejpam-5305	343	10	with	with	ADP
ejpam-5305	343	11	lengths	length	NOUN
ejpam-5305	343	12	m1,m2,m3	m1,m2,m3	ADJ
ejpam-5305	343	13	,	,	PUNCT
ejpam-5305	343	14	.	.	PUNCT
ejpam-5305	343	15	.	.	PUNCT
ejpam-5305	344	1	.	.	PUNCT
ejpam-5305	345	1	,	,	PUNCT
ejpam-5305	345	2	mn	mn	PROPN
ejpam-5305	345	3	.	.	PROPN
ejpam-5305	346	1	if	if	SCONJ
ejpam-5305	346	2	m2	m2	PROPN
ejpam-5305	346	3	=	=	PROPN
ejpam-5305	346	4	m3	m3	PROPN
ejpam-5305	346	5	=	=	PUNCT
ejpam-5305	346	6	.	.	PUNCT
ejpam-5305	346	7	.	.	PUNCT
ejpam-5305	346	8	.	.	PUNCT
ejpam-5305	347	1	=	=	SYM
ejpam-5305	347	2	mn	mn	PROPN
ejpam-5305	347	3	,	,	PUNCT
ejpam-5305	347	4	then	then	ADV
ejpam-5305	347	5	t	t	PROPN
ejpam-5305	347	6	is	be	AUX
ejpam-5305	347	7	a	a	DET
ejpam-5305	347	8	graceful	graceful	ADJ
ejpam-5305	347	9	labeling	labeling	NOUN
ejpam-5305	347	10	graph	graph	NOUN
ejpam-5305	347	11	.	.	PUNCT
ejpam-5305	348	1	proof	proof	NOUN
ejpam-5305	348	2	.	.	PUNCT
ejpam-5305	349	1	let	let	VERB
ejpam-5305	349	2	t	t	NOUN
ejpam-5305	349	3	be	be	AUX
ejpam-5305	349	4	the	the	DET
ejpam-5305	349	5	spider	spider	NOUN
ejpam-5305	349	6	as	as	SCONJ
ejpam-5305	349	7	shown	show	VERB
ejpam-5305	349	8	in	in	ADP
ejpam-5305	349	9	figure	figure	NOUN
ejpam-5305	349	10	1	1	NUM
ejpam-5305	349	11	.	.	PUNCT
ejpam-5305	349	12	k.	k.	PROPN
ejpam-5305	349	13	saengsura	saengsura	PROPN
ejpam-5305	349	14	,	,	PUNCT
ejpam-5305	349	15	t.	t.	PROPN
ejpam-5305	349	16	poomsa	poomsa	PROPN
ejpam-5305	349	17	-	-	PUNCT
ejpam-5305	349	18	ard	ard	PROPN
ejpam-5305	349	19	/	/	PUNCT
ejpam-5305	349	20	eur	eur	PROPN
ejpam-5305	349	21	.	.	PUNCT
ejpam-5305	350	1	j.	j.	PROPN
ejpam-5305	350	2	pure	pure	PROPN
ejpam-5305	350	3	appl	appl	PROPN
ejpam-5305	350	4	.	.	PROPN
ejpam-5305	350	5	math	math	PROPN
ejpam-5305	350	6	,	,	PUNCT
ejpam-5305	350	7	18	18	NUM
ejpam-5305	350	8	(	(	PUNCT
ejpam-5305	350	9	2	2	NUM
ejpam-5305	350	10	)	)	PUNCT
ejpam-5305	350	11	(	(	PUNCT
ejpam-5305	350	12	2025	2025	NUM
ejpam-5305	350	13	)	)	PUNCT
ejpam-5305	350	14	,	,	PUNCT
ejpam-5305	350	15	5305	5305	NUM
ejpam-5305	350	16	7	7	NUM
ejpam-5305	350	17	of	of	ADP
ejpam-5305	350	18	8	8	NUM
ejpam-5305	350	19	case	case	NOUN
ejpam-5305	350	20	1	1	NUM
ejpam-5305	350	21	.	.	PUNCT
ejpam-5305	351	1	if	if	SCONJ
ejpam-5305	351	2	m1	m1	PROPN
ejpam-5305	351	3	is	be	AUX
ejpam-5305	351	4	odd	odd	ADJ
ejpam-5305	351	5	,	,	PUNCT
ejpam-5305	351	6	then	then	ADV
ejpam-5305	351	7	let	let	VERB
ejpam-5305	351	8	f	f	PRON
ejpam-5305	351	9	be	be	AUX
ejpam-5305	351	10	the	the	DET
ejpam-5305	351	11	labeling	labeling	NOUN
ejpam-5305	351	12	of	of	ADP
ejpam-5305	351	13	t	t	PROPN
ejpam-5305	351	14	of	of	ADP
ejpam-5305	351	15	type	type	NOUN
ejpam-5305	351	16	i.	i.	NOUN
ejpam-5305	351	17	since	since	SCONJ
ejpam-5305	351	18	m2	m2	PROPN
ejpam-5305	351	19	=	=	PROPN
ejpam-5305	351	20	m3	m3	PROPN
ejpam-5305	351	21	=	=	PUNCT
ejpam-5305	351	22	.	.	PUNCT
ejpam-5305	351	23	.	.	PUNCT
ejpam-5305	351	24	.	.	PUNCT
ejpam-5305	352	1	=	=	SYM
ejpam-5305	352	2	mn	mn	PROPN
ejpam-5305	352	3	,	,	PUNCT
ejpam-5305	352	4	then	then	ADV
ejpam-5305	352	5	by	by	ADP
ejpam-5305	352	6	the	the	DET
ejpam-5305	352	7	way	way	NOUN
ejpam-5305	352	8	of	of	ADP
ejpam-5305	352	9	labeling	labeling	NOUN
ejpam-5305	352	10	of	of	ADP
ejpam-5305	352	11	t	t	PROPN
ejpam-5305	352	12	of	of	ADP
ejpam-5305	352	13	type	type	NOUN
ejpam-5305	352	14	i	i	PRON
ejpam-5305	352	15	for	for	ADP
ejpam-5305	352	16	n	n	PROPN
ejpam-5305	352	17	is	be	AUX
ejpam-5305	352	18	odd	odd	ADJ
ejpam-5305	352	19	we	we	PRON
ejpam-5305	352	20	have	have	VERB
ejpam-5305	352	21	|f(v41)−	|f(v41)−	ADJ
ejpam-5305	352	22	f(v)|	f(v)|	NOUN
ejpam-5305	352	23	=	=	SYM
ejpam-5305	352	24	|f(v31)−	|f(v31)−	NUM
ejpam-5305	352	25	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	352	26	,	,	PUNCT
ejpam-5305	352	27	|f(v61)−	|f(v61)−	X
ejpam-5305	352	28	f(v)|	f(v)|	NOUN
ejpam-5305	352	29	=	=	SYM
ejpam-5305	352	30	|f(v41)−	|f(v41)−	PROPN
ejpam-5305	352	31	f(v3m3)|	f(v3m3)|	PROPN
ejpam-5305	352	32	,	,	PUNCT
ejpam-5305	352	33	.	.	PUNCT
ejpam-5305	352	34	.	.	PUNCT
ejpam-5305	353	1	.	.	PUNCT
ejpam-5305	353	2	,	,	PUNCT
ejpam-5305	353	3	|f(v(n−1)1)−	|f(v(n−1)1)−	NOUN
ejpam-5305	353	4	f(v)|	f(v)|	NOUN
ejpam-5305	353	5	=	=	SYM
ejpam-5305	353	6	|f(v((n−1)/2	|f(v((n−1)/2	NUM
ejpam-5305	353	7	+	+	PROPN
ejpam-5305	353	8	1)1)−	1)1)−	NUM
ejpam-5305	353	9	f(v((n−1)/2)m(n−1)/2	f(v((n−1)/2)m(n−1)/2	PUNCT
ejpam-5305	353	10	)	)	PUNCT
ejpam-5305	354	1	|	|	ADV
ejpam-5305	354	2	and	and	CCONJ
ejpam-5305	354	3	|f(vnmn)−	|f(vnmn)−	X
ejpam-5305	354	4	f(v)|	f(v)|	NOUN
ejpam-5305	354	5	=	=	SYM
ejpam-5305	354	6	|f(v((n−1)/2	|f(v((n−1)/2	NUM
ejpam-5305	354	7	+	+	ADJ
ejpam-5305	354	8	2)1)−	2)1)−	VERB
ejpam-5305	354	9	f(v((n−1)/2	f(v((n−1)/2	NOUN
ejpam-5305	354	10	+	+	NOUN
ejpam-5305	354	11	1)m(n−1)/2	1)m(n−1)/2	NUM
ejpam-5305	354	12	+	+	ADJ
ejpam-5305	354	13	1	1	NUM
ejpam-5305	354	14	)	)	PUNCT
ejpam-5305	354	15	|	|	ADV
ejpam-5305	354	16	,	,	PUNCT
ejpam-5305	354	17	|f(v(n−2)mn−2	|f(v(n−2)mn−2	NOUN
ejpam-5305	354	18	)	)	PUNCT
ejpam-5305	354	19	−f(v)|	−f(v)|	NOUN
ejpam-5305	354	20	=	=	SYM
ejpam-5305	354	21	|f(v((n−1)/2	|f(v((n−1)/2	NUM
ejpam-5305	354	22	+	+	NOUN
ejpam-5305	354	23	3)1)−f(v((n−1)/2	3)1)−f(v((n−1)/2	NUM
ejpam-5305	354	24	+	+	ADJ
ejpam-5305	354	25	2)m(n−1)/2	2)m(n−1)/2	NUM
ejpam-5305	354	26	+	+	ADJ
ejpam-5305	354	27	2	2	NUM
ejpam-5305	354	28	)	)	PUNCT
ejpam-5305	354	29	|	|	ADV
ejpam-5305	354	30	,	,	PUNCT
ejpam-5305	354	31	.	.	PUNCT
ejpam-5305	354	32	.	.	PUNCT
ejpam-5305	355	1	.	.	PUNCT
ejpam-5305	356	1	,	,	PUNCT
ejpam-5305	356	2	|f(v3m3)−f(v)|	|f(v3m3)−f(v)|	NOUN
ejpam-5305	356	3	=	=	SYM
ejpam-5305	356	4	|f(vn1)−f(v(n−1)mn−1	|f(vn1)−f(v(n−1)mn−1	ADJ
ejpam-5305	356	5	)	)	PUNCT
ejpam-5305	356	6	|	|	ADV
ejpam-5305	356	7	.	.	PUNCT
ejpam-5305	357	1	for	for	ADP
ejpam-5305	357	2	each	each	DET
ejpam-5305	357	3	i	i	PRON
ejpam-5305	357	4	,	,	PUNCT
ejpam-5305	357	5	i	i	PRON
ejpam-5305	357	6	=	=	NOUN
ejpam-5305	357	7	3	3	NUM
ejpam-5305	357	8	,	,	PUNCT
ejpam-5305	357	9	5	5	NUM
ejpam-5305	357	10	,	,	PUNCT
ejpam-5305	357	11	.	.	PUNCT
ejpam-5305	357	12	.	.	PUNCT
ejpam-5305	357	13	.	.	PUNCT
ejpam-5305	358	1	,	,	PUNCT
ejpam-5305	358	2	n	n	CCONJ
ejpam-5305	358	3	,	,	PUNCT
ejpam-5305	358	4	if	if	SCONJ
ejpam-5305	358	5	we	we	PRON
ejpam-5305	358	6	change	change	VERB
ejpam-5305	358	7	the	the	DET
ejpam-5305	358	8	labeling	labeling	NOUN
ejpam-5305	358	9	f	f	PROPN
ejpam-5305	358	10	at	at	ADP
ejpam-5305	358	11	vi1	vi1	PROPN
ejpam-5305	358	12	,	,	PUNCT
ejpam-5305	358	13	vi2	vi2	NOUN
ejpam-5305	358	14	,	,	PUNCT
ejpam-5305	358	15	.	.	PUNCT
ejpam-5305	359	1	.	.	PUNCT
ejpam-5305	360	1	.	.	PUNCT
ejpam-5305	361	1	,	,	PUNCT
ejpam-5305	361	2	vimi	vimi	VERB
ejpam-5305	361	3	by	by	ADP
ejpam-5305	361	4	reversing	reverse	VERB
ejpam-5305	361	5	the	the	DET
ejpam-5305	361	6	order	order	NOUN
ejpam-5305	361	7	of	of	ADP
ejpam-5305	361	8	theirs	theirs	PRON
ejpam-5305	361	9	labels	label	NOUN
ejpam-5305	361	10	,	,	PUNCT
ejpam-5305	361	11	then	then	ADV
ejpam-5305	361	12	we	we	PRON
ejpam-5305	361	13	get	get	VERB
ejpam-5305	361	14	t	t	PROPN
ejpam-5305	361	15	is	be	AUX
ejpam-5305	361	16	graceful	graceful	ADJ
ejpam-5305	361	17	.	.	PUNCT
ejpam-5305	362	1	for	for	ADP
ejpam-5305	362	2	n	n	NUM
ejpam-5305	362	3	is	be	AUX
ejpam-5305	362	4	even	even	ADV
ejpam-5305	362	5	we	we	PRON
ejpam-5305	362	6	have	have	VERB
ejpam-5305	362	7	|f(v41)−	|f(v41)−	ADJ
ejpam-5305	362	8	f(v)|	f(v)|	NOUN
ejpam-5305	362	9	=	=	SYM
ejpam-5305	362	10	|f(v31)−	|f(v31)−	NUM
ejpam-5305	362	11	f(v2m2)|	f(v2m2)|	NOUN
ejpam-5305	362	12	,	,	PUNCT
ejpam-5305	362	13	|f(v61)−	|f(v61)−	X
ejpam-5305	362	14	f(v)|	f(v)|	NOUN
ejpam-5305	362	15	=	=	SYM
ejpam-5305	362	16	|f(v41)−	|f(v41)−	PROPN
ejpam-5305	362	17	f(v3m3)|	f(v3m3)|	PROPN
ejpam-5305	362	18	,	,	PUNCT
ejpam-5305	362	19	.	.	PUNCT
ejpam-5305	362	20	.	.	PUNCT
ejpam-5305	363	1	.	.	PUNCT
ejpam-5305	364	1	,	,	PUNCT
ejpam-5305	364	2	|f(vn1)−	|f(vn1)−	VERB
ejpam-5305	364	3	f(v)|	f(v)|	NOUN
ejpam-5305	364	4	=	=	SYM
ejpam-5305	364	5	|f(v(n/2	|f(v(n/2	NOUN
ejpam-5305	364	6	+	+	NOUN
ejpam-5305	364	7	1)1)−	1)1)−	PROPN
ejpam-5305	364	8	f(v(n/2)mn/2	f(v(n/2)mn/2	NOUN
ejpam-5305	364	9	)	)	PUNCT
ejpam-5305	365	1	|	|	ADV
ejpam-5305	365	2	and	and	CCONJ
ejpam-5305	365	3	|f(v(n−1)mn−1	|f(v(n−1)mn−1	ADJ
ejpam-5305	365	4	)	)	PUNCT
ejpam-5305	365	5	−	−	PROPN
ejpam-5305	365	6	f(v)|	f(v)|	NOUN
ejpam-5305	365	7	=	=	NOUN
ejpam-5305	365	8	|f(v(n/2	|f(v(n/2	NOUN
ejpam-5305	365	9	+	+	NOUN
ejpam-5305	365	10	2)1)−f(v(n/2	2)1)−f(v(n/2	NOUN
ejpam-5305	365	11	+	+	NOUN
ejpam-5305	365	12	1)mn/2	1)mn/2	NUM
ejpam-5305	365	13	+	+	ADJ
ejpam-5305	365	14	1	1	NUM
ejpam-5305	365	15	)	)	PUNCT
ejpam-5305	365	16	|	|	ADV
ejpam-5305	365	17	,	,	PUNCT
ejpam-5305	365	18	|f(v(n−3)mn−3	|f(v(n−3)mn−3	PROPN
ejpam-5305	365	19	)	)	PUNCT
ejpam-5305	365	20	−f(v)|	−f(v)|	NOUN
ejpam-5305	365	21	=	=	NOUN
ejpam-5305	365	22	|f(v(n/2	|f(v(n/2	NOUN
ejpam-5305	365	23	+	+	NOUN
ejpam-5305	365	24	3)1)−f(v(n/2	3)1)−f(v(n/2	NOUN
ejpam-5305	365	25	+	+	NOUN
ejpam-5305	365	26	2)mn/2	2)mn/2	NOUN
ejpam-5305	365	27	+	+	ADJ
ejpam-5305	365	28	2	2	NUM
ejpam-5305	365	29	)	)	PUNCT
ejpam-5305	365	30	|	|	ADV
ejpam-5305	365	31	,	,	PUNCT
ejpam-5305	365	32	.	.	PUNCT
ejpam-5305	365	33	.	.	PUNCT
ejpam-5305	366	1	.	.	PUNCT
ejpam-5305	367	1	,	,	PUNCT
ejpam-5305	367	2	|f(v3m3)−f(v)|	|f(v3m3)−f(v)|	NOUN
ejpam-5305	367	3	=	=	SYM
ejpam-5305	367	4	|f(vn1)−f(v(n−1)mn−1	|f(vn1)−f(v(n−1)mn−1	ADJ
ejpam-5305	367	5	)	)	PUNCT
ejpam-5305	367	6	|	|	ADV
ejpam-5305	367	7	.	.	PUNCT
ejpam-5305	368	1	for	for	ADP
ejpam-5305	368	2	each	each	DET
ejpam-5305	368	3	i	i	PRON
ejpam-5305	368	4	,	,	PUNCT
ejpam-5305	368	5	i	i	PRON
ejpam-5305	368	6	=	=	NOUN
ejpam-5305	368	7	3	3	NUM
ejpam-5305	368	8	,	,	PUNCT
ejpam-5305	368	9	5	5	NUM
ejpam-5305	368	10	,	,	PUNCT
ejpam-5305	368	11	.	.	PUNCT
ejpam-5305	368	12	.	.	PUNCT
ejpam-5305	368	13	.	.	PUNCT
ejpam-5305	369	1	,	,	PUNCT
ejpam-5305	369	2	n−1	n−1	PROPN
ejpam-5305	369	3	,	,	PUNCT
ejpam-5305	369	4	if	if	SCONJ
ejpam-5305	369	5	we	we	PRON
ejpam-5305	369	6	change	change	VERB
ejpam-5305	369	7	the	the	DET
ejpam-5305	369	8	labeling	labeling	NOUN
ejpam-5305	369	9	f	f	PROPN
ejpam-5305	369	10	at	at	ADP
ejpam-5305	369	11	vi1	vi1	PROPN
ejpam-5305	369	12	,	,	PUNCT
ejpam-5305	369	13	vi2	vi2	NOUN
ejpam-5305	369	14	,	,	PUNCT
ejpam-5305	369	15	.	.	PUNCT
ejpam-5305	369	16	.	.	PUNCT
ejpam-5305	370	1	.	.	PUNCT
ejpam-5305	371	1	,	,	PUNCT
ejpam-5305	371	2	vimi	vimi	VERB
ejpam-5305	371	3	by	by	ADP
ejpam-5305	371	4	reversing	reverse	VERB
ejpam-5305	371	5	the	the	DET
ejpam-5305	371	6	order	order	NOUN
ejpam-5305	371	7	of	of	ADP
ejpam-5305	371	8	theirs	theirs	PRON
ejpam-5305	371	9	labels	label	NOUN
ejpam-5305	371	10	,	,	PUNCT
ejpam-5305	371	11	then	then	ADV
ejpam-5305	371	12	we	we	PRON
ejpam-5305	371	13	get	get	VERB
ejpam-5305	371	14	t	t	PROPN
ejpam-5305	371	15	is	be	AUX
ejpam-5305	371	16	graceful	graceful	ADJ
ejpam-5305	371	17	.	.	PUNCT
ejpam-5305	372	1	case	case	NOUN
ejpam-5305	372	2	2	2	X
ejpam-5305	372	3	.	.	PUNCT
ejpam-5305	373	1	if	if	SCONJ
ejpam-5305	373	2	m1	m1	PROPN
ejpam-5305	373	3	is	be	AUX
ejpam-5305	373	4	even	even	ADV
ejpam-5305	373	5	,	,	PUNCT
ejpam-5305	373	6	then	then	ADV
ejpam-5305	373	7	let	let	VERB
ejpam-5305	373	8	f	f	PRON
ejpam-5305	373	9	be	be	AUX
ejpam-5305	373	10	the	the	DET
ejpam-5305	373	11	labeling	labeling	NOUN
ejpam-5305	373	12	of	of	ADP
ejpam-5305	373	13	t	t	PROPN
ejpam-5305	373	14	of	of	ADP
ejpam-5305	373	15	type	type	NOUN
ejpam-5305	373	16	ii	ii	PROPN
ejpam-5305	373	17	and	and	CCONJ
ejpam-5305	373	18	consider	consider	VERB
ejpam-5305	373	19	similar	similar	ADJ
ejpam-5305	373	20	as	as	ADP
ejpam-5305	373	21	the	the	DET
ejpam-5305	373	22	case	case	NOUN
ejpam-5305	373	23	1	1	NUM
ejpam-5305	373	24	we	we	PRON
ejpam-5305	373	25	get	get	VERB
ejpam-5305	373	26	t	t	PROPN
ejpam-5305	373	27	admit	admit	VERB
ejpam-5305	373	28	graceful	graceful	ADJ
ejpam-5305	373	29	labeling	labeling	NOUN
ejpam-5305	373	30	.	.	PUNCT
ejpam-5305	374	1	3	3	X
ejpam-5305	374	2	.	.	X
ejpam-5305	374	3	conclusion	conclusion	NOUN
ejpam-5305	374	4	in	in	ADP
ejpam-5305	374	5	this	this	DET
ejpam-5305	374	6	paper	paper	NOUN
ejpam-5305	374	7	,	,	PUNCT
ejpam-5305	374	8	we	we	PRON
ejpam-5305	374	9	obtain	obtain	VERB
ejpam-5305	374	10	that	that	SCONJ
ejpam-5305	374	11	certain	certain	ADJ
ejpam-5305	374	12	spider	spider	NOUN
ejpam-5305	374	13	graphs	graph	NOUN
ejpam-5305	374	14	,	,	PUNCT
ejpam-5305	374	15	whose	whose	DET
ejpam-5305	374	16	labeling	labeling	NOUN
ejpam-5305	374	17	is	be	AUX
ejpam-5305	374	18	constructed	construct	VERB
ejpam-5305	374	19	in	in	ADP
ejpam-5305	374	20	the	the	DET
ejpam-5305	374	21	forms	form	NOUN
ejpam-5305	374	22	of	of	ADP
ejpam-5305	374	23	type	type	NOUN
ejpam-5305	374	24	i	i	PRON
ejpam-5305	374	25	or	or	CCONJ
ejpam-5305	374	26	type	type	PROPN
ejpam-5305	374	27	ii	ii	PROPN
ejpam-5305	374	28	,	,	PUNCT
ejpam-5305	374	29	admit	admit	VERB
ejpam-5305	374	30	graceful	graceful	ADJ
ejpam-5305	374	31	labeling	labeling	NOUN
ejpam-5305	374	32	whenever	whenever	SCONJ
ejpam-5305	374	33	the	the	DET
ejpam-5305	374	34	lengths	length	NOUN
ejpam-5305	374	35	of	of	ADP
ejpam-5305	374	36	their	their	PRON
ejpam-5305	374	37	legs	leg	NOUN
ejpam-5305	374	38	satisfy	satisfy	VERB
ejpam-5305	374	39	the	the	DET
ejpam-5305	374	40	conditions	condition	NOUN
ejpam-5305	374	41	presented	present	VERB
ejpam-5305	374	42	in	in	ADP
ejpam-5305	374	43	theorems	theorem	NOUN
ejpam-5305	374	44	1	1	NUM
ejpam-5305	374	45	through	through	ADP
ejpam-5305	374	46	7	7	NUM
ejpam-5305	374	47	.	.	PUNCT
ejpam-5305	375	1	acknowledgements	acknowledgement	NOUN
ejpam-5305	375	2	this	this	DET
ejpam-5305	375	3	research	research	NOUN
ejpam-5305	375	4	was	be	AUX
ejpam-5305	375	5	financially	financially	ADV
ejpam-5305	375	6	supported	support	VERB
ejpam-5305	375	7	by	by	ADP
ejpam-5305	375	8	mahasarakham	mahasarakham	PROPN
ejpam-5305	375	9	university	university	PROPN
ejpam-5305	375	10	.	.	PUNCT
ejpam-5305	376	1	the	the	DET
ejpam-5305	376	2	authors	author	NOUN
ejpam-5305	376	3	wish	wish	VERB
ejpam-5305	376	4	to	to	PART
ejpam-5305	376	5	extend	extend	VERB
ejpam-5305	376	6	appreciation	appreciation	NOUN
ejpam-5305	376	7	to	to	ADP
ejpam-5305	376	8	the	the	DET
ejpam-5305	376	9	faculty	faculty	NOUN
ejpam-5305	376	10	of	of	ADP
ejpam-5305	376	11	science	science	NOUN
ejpam-5305	376	12	at	at	ADP
ejpam-5305	376	13	mahasarakham	mahasarakham	PROPN
ejpam-5305	376	14	university	university	PROPN
ejpam-5305	376	15	for	for	ADP
ejpam-5305	376	16	providing	provide	VERB
ejpam-5305	376	17	research	research	NOUN
ejpam-5305	376	18	facilities	facility	NOUN
ejpam-5305	376	19	.	.	PUNCT
ejpam-5305	377	1	the	the	DET
ejpam-5305	377	2	authors	author	NOUN
ejpam-5305	377	3	are	be	AUX
ejpam-5305	377	4	immensely	immensely	ADV
ejpam-5305	377	5	grateful	grateful	ADJ
ejpam-5305	377	6	to	to	ADP
ejpam-5305	377	7	the	the	DET
ejpam-5305	377	8	reviewer	reviewer	NOUN
ejpam-5305	377	9	for	for	ADP
ejpam-5305	377	10	their	their	PRON
ejpam-5305	377	11	valuable	valuable	ADJ
ejpam-5305	377	12	suggestions	suggestion	NOUN
ejpam-5305	377	13	,	,	PUNCT
ejpam-5305	377	14	which	which	PRON
ejpam-5305	377	15	have	have	AUX
ejpam-5305	377	16	greatly	greatly	ADV
ejpam-5305	377	17	helped	help	VERB
ejpam-5305	377	18	improve	improve	VERB
ejpam-5305	377	19	this	this	DET
ejpam-5305	377	20	work	work	NOUN
ejpam-5305	377	21	.	.	PUNCT
ejpam-5305	378	1	references	reference	NOUN
ejpam-5305	378	2	[	[	X
ejpam-5305	378	3	1	1	X
ejpam-5305	378	4	]	]	PUNCT
ejpam-5305	378	5	j.	j.	PROPN
ejpam-5305	378	6	a.	a.	PROPN
ejpam-5305	378	7	gillian	gillian	PROPN
ejpam-5305	378	8	.	.	PUNCT
ejpam-5305	379	1	a	a	DET
ejpam-5305	379	2	dynamic	dynamic	ADJ
ejpam-5305	379	3	survey	survey	NOUN
ejpam-5305	379	4	of	of	ADP
ejpam-5305	379	5	graph	graph	NOUN
ejpam-5305	379	6	labeling	labeling	NOUN
ejpam-5305	379	7	.	.	PUNCT
ejpam-5305	380	1	the	the	DET
ejpam-5305	380	2	electronic	electronic	ADJ
ejpam-5305	380	3	journal	journal	NOUN
ejpam-5305	380	4	of	of	ADP
ejpam-5305	380	5	combinatorics	combinatoric	NOUN
ejpam-5305	380	6	,	,	PUNCT
ejpam-5305	380	7	16	16	NUM
ejpam-5305	380	8	:	:	PUNCT
ejpam-5305	380	9	ds6	ds6	NOUN
ejpam-5305	380	10	,	,	PUNCT
ejpam-5305	380	11	2015	2015	NUM
ejpam-5305	380	12	.	.	PUNCT
ejpam-5305	381	1	[	[	X
ejpam-5305	381	2	2	2	X
ejpam-5305	381	3	]	]	X
ejpam-5305	381	4	g.	g.	PROPN
ejpam-5305	381	5	s.	s.	PROPN
ejpam-5305	381	6	bloom	bloom	PROPN
ejpam-5305	381	7	and	and	CCONJ
ejpam-5305	381	8	s.	s.	PROPN
ejpam-5305	381	9	w.	w.	PROPN
ejpam-5305	381	10	golomb	golomb	PROPN
ejpam-5305	381	11	.	.	PUNCT
ejpam-5305	382	1	applications	application	NOUN
ejpam-5305	382	2	of	of	ADP
ejpam-5305	382	3	numbered	number	VERB
ejpam-5305	382	4	undirected	undirected	ADJ
ejpam-5305	382	5	graphs	graph	NOUN
ejpam-5305	382	6	.	.	PUNCT
ejpam-5305	383	1	proceedings	proceeding	NOUN
ejpam-5305	383	2	of	of	ADP
ejpam-5305	383	3	the	the	DET
ejpam-5305	383	4	ieee	ieee	NOUN
ejpam-5305	383	5	,	,	PUNCT
ejpam-5305	383	6	65(4):562–570	65(4):562–570	NOUN
ejpam-5305	383	7	,	,	PUNCT
ejpam-5305	383	8	1977	1977	NUM
ejpam-5305	383	9	.	.	PUNCT
ejpam-5305	384	1	[	[	X
ejpam-5305	384	2	3	3	X
ejpam-5305	384	3	]	]	X
ejpam-5305	384	4	b.	b.	PROPN
ejpam-5305	384	5	j.	j.	PROPN
ejpam-5305	384	6	septory	septory	PROPN
ejpam-5305	384	7	,	,	PUNCT
ejpam-5305	384	8	l.	l.	PROPN
ejpam-5305	384	9	susilowati	susilowati	PROPN
ejpam-5305	384	10	,	,	PUNCT
ejpam-5305	384	11	dafik	dafik	NOUN
ejpam-5305	384	12	,	,	PUNCT
ejpam-5305	384	13	v.	v.	ADP
ejpam-5305	384	14	lokesha	lokesha	PROPN
ejpam-5305	384	15	,	,	PUNCT
ejpam-5305	384	16	and	and	CCONJ
ejpam-5305	384	17	v.	v.	ADP
ejpam-5305	384	18	nagamani	nagamani	NOUN
ejpam-5305	384	19	.	.	PUNCT
ejpam-5305	385	1	on	on	ADP
ejpam-5305	385	2	the	the	DET
ejpam-5305	385	3	study	study	NOUN
ejpam-5305	385	4	of	of	ADP
ejpam-5305	385	5	rainbow	rainbow	NOUN
ejpam-5305	385	6	antimagic	antimagic	ADJ
ejpam-5305	385	7	connection	connection	NOUN
ejpam-5305	385	8	number	number	NOUN
ejpam-5305	385	9	of	of	ADP
ejpam-5305	385	10	corona	corona	NOUN
ejpam-5305	385	11	product	product	NOUN
ejpam-5305	385	12	of	of	ADP
ejpam-5305	385	13	graphs	graph	NOUN
ejpam-5305	385	14	.	.	PUNCT
ejpam-5305	386	1	european	european	ADJ
ejpam-5305	386	2	journal	journal	PROPN
ejpam-5305	386	3	of	of	ADP
ejpam-5305	386	4	pure	pure	ADJ
ejpam-5305	386	5	and	and	CCONJ
ejpam-5305	386	6	applied	applied	ADJ
ejpam-5305	386	7	mathematics	mathematic	NOUN
ejpam-5305	386	8	,	,	PUNCT
ejpam-5305	386	9	16(1):271–285	16(1):271–285	NUM
ejpam-5305	386	10	,	,	PUNCT
ejpam-5305	386	11	2023	2023	NUM
ejpam-5305	386	12	.	.	PUNCT
ejpam-5305	387	1	[	[	X
ejpam-5305	387	2	4	4	X
ejpam-5305	387	3	]	]	PUNCT
ejpam-5305	387	4	b.	b.	PROPN
ejpam-5305	387	5	j.	j.	PROPN
ejpam-5305	387	6	septory	septory	PROPN
ejpam-5305	387	7	,	,	PUNCT
ejpam-5305	387	8	l.	l.	PROPN
ejpam-5305	387	9	susilowati	susilowati	PROPN
ejpam-5305	387	10	,	,	PUNCT
ejpam-5305	387	11	dafik	dafik	NOUN
ejpam-5305	387	12	,	,	PUNCT
ejpam-5305	387	13	and	and	CCONJ
ejpam-5305	387	14	v.	v.	ADP
ejpam-5305	387	15	lokesha	lokesha	PROPN
ejpam-5305	387	16	.	.	PUNCT
ejpam-5305	388	1	on	on	ADP
ejpam-5305	388	2	the	the	DET
ejpam-5305	388	3	study	study	NOUN
ejpam-5305	388	4	of	of	ADP
ejpam-5305	388	5	rainbow	rainbow	NOUN
ejpam-5305	388	6	antimagic	antimagic	ADJ
ejpam-5305	388	7	connection	connection	NOUN
ejpam-5305	388	8	number	number	NOUN
ejpam-5305	388	9	of	of	ADP
ejpam-5305	388	10	comb	comb	NOUN
ejpam-5305	388	11	product	product	NOUN
ejpam-5305	388	12	of	of	ADP
ejpam-5305	388	13	friendship	friendship	NOUN
ejpam-5305	388	14	graph	graph	NOUN
ejpam-5305	388	15	and	and	CCONJ
ejpam-5305	388	16	tree	tree	NOUN
ejpam-5305	388	17	.	.	PUNCT
ejpam-5305	389	1	symmetry	symmetry	NOUN
ejpam-5305	389	2	,	,	PUNCT
ejpam-5305	389	3	15(1):42	15(1):42	NUM
ejpam-5305	389	4	,	,	PUNCT
ejpam-5305	389	5	2023	2023	NUM
ejpam-5305	389	6	.	.	PUNCT
ejpam-5305	390	1	k.	k.	PROPN
ejpam-5305	390	2	saengsura	saengsura	PROPN
ejpam-5305	390	3	,	,	PUNCT
ejpam-5305	390	4	t.	t.	PROPN
ejpam-5305	390	5	poomsa	poomsa	PROPN
ejpam-5305	390	6	-	-	PUNCT
ejpam-5305	390	7	ard	ard	PROPN
ejpam-5305	390	8	/	/	PUNCT
ejpam-5305	390	9	eur	eur	PROPN
ejpam-5305	390	10	.	.	PUNCT
ejpam-5305	391	1	j.	j.	PROPN
ejpam-5305	391	2	pure	pure	PROPN
ejpam-5305	391	3	appl	appl	PROPN
ejpam-5305	391	4	.	.	PROPN
ejpam-5305	391	5	math	math	PROPN
ejpam-5305	391	6	,	,	PUNCT
ejpam-5305	391	7	18	18	NUM
ejpam-5305	391	8	(	(	PUNCT
ejpam-5305	391	9	2	2	NUM
ejpam-5305	391	10	)	)	PUNCT
ejpam-5305	391	11	(	(	PUNCT
ejpam-5305	391	12	2025	2025	NUM
ejpam-5305	391	13	)	)	PUNCT
ejpam-5305	391	14	,	,	PUNCT
ejpam-5305	391	15	5305	5305	NUM
ejpam-5305	391	16	8	8	NUM
ejpam-5305	391	17	of	of	ADP
ejpam-5305	391	18	8	8	NUM
ejpam-5305	391	19	[	[	SYM
ejpam-5305	391	20	5	5	NUM
ejpam-5305	391	21	]	]	PUNCT
ejpam-5305	391	22	g.	g.	NOUN
ejpam-5305	391	23	ringel	ringel	NOUN
ejpam-5305	391	24	.	.	PUNCT
ejpam-5305	392	1	theory	theory	NOUN
ejpam-5305	392	2	of	of	ADP
ejpam-5305	392	3	graphs	graph	NOUN
ejpam-5305	392	4	and	and	CCONJ
ejpam-5305	392	5	its	its	PRON
ejpam-5305	392	6	applications	application	NOUN
ejpam-5305	392	7	.	.	PUNCT
ejpam-5305	393	1	in	in	ADP
ejpam-5305	393	2	proceedings	proceeding	NOUN
ejpam-5305	393	3	of	of	ADP
ejpam-5305	393	4	the	the	DET
ejpam-5305	393	5	symposium	symposium	NOUN
ejpam-5305	393	6	smolenice	smolenice	NOUN
ejpam-5305	393	7	,	,	PUNCT
ejpam-5305	393	8	page	page	NOUN
ejpam-5305	393	9	162	162	NUM
ejpam-5305	393	10	,	,	PUNCT
ejpam-5305	393	11	prague	prague	NOUN
ejpam-5305	393	12	,	,	PUNCT
ejpam-5305	393	13	1964	1964	NUM
ejpam-5305	393	14	.	.	PUNCT
ejpam-5305	394	1	czechoslovak	czechoslovak	PROPN
ejpam-5305	394	2	academy	academy	PROPN
ejpam-5305	394	3	of	of	ADP
ejpam-5305	394	4	sciences	sciences	PROPN
ejpam-5305	394	5	.	.	PUNCT
ejpam-5305	395	1	[	[	X
ejpam-5305	395	2	6	6	NUM
ejpam-5305	395	3	]	]	PUNCT
ejpam-5305	395	4	a.	a.	PROPN
ejpam-5305	395	5	rosa	rosa	PROPN
ejpam-5305	395	6	.	.	PUNCT
ejpam-5305	396	1	on	on	ADP
ejpam-5305	396	2	certain	certain	ADJ
ejpam-5305	396	3	valuations	valuation	NOUN
ejpam-5305	396	4	of	of	ADP
ejpam-5305	396	5	the	the	DET
ejpam-5305	396	6	vertices	vertex	NOUN
ejpam-5305	396	7	of	of	ADP
ejpam-5305	396	8	a	a	DET
ejpam-5305	396	9	graph	graph	NOUN
ejpam-5305	396	10	.	.	PUNCT
ejpam-5305	397	1	in	in	ADP
ejpam-5305	397	2	theory	theory	NOUN
ejpam-5305	397	3	of	of	ADP
ejpam-5305	397	4	graphs	graph	NOUN
ejpam-5305	397	5	,	,	PUNCT
ejpam-5305	397	6	international	international	ADJ
ejpam-5305	397	7	symposium	symposium	NOUN
ejpam-5305	397	8	,	,	PUNCT
ejpam-5305	397	9	rome	rome	PROPN
ejpam-5305	397	10	,	,	PUNCT
ejpam-5305	397	11	july	july	PROPN
ejpam-5305	397	12	1966	1966	NUM
ejpam-5305	397	13	,	,	PUNCT
ejpam-5305	397	14	pages	page	NOUN
ejpam-5305	397	15	349–355	349–355	NUM
ejpam-5305	397	16	,	,	PUNCT
ejpam-5305	397	17	new	new	PROPN
ejpam-5305	397	18	york	york	PROPN
ejpam-5305	397	19	,	,	PUNCT
ejpam-5305	397	20	1966	1966	NUM
ejpam-5305	397	21	.	.	PUNCT
ejpam-5305	398	1	gordon	gordon	PROPN
ejpam-5305	398	2	and	and	CCONJ
ejpam-5305	398	3	breach	breach	NOUN
ejpam-5305	398	4	.	.	PUNCT
ejpam-5305	399	1	[	[	X
ejpam-5305	399	2	7	7	X
ejpam-5305	399	3	]	]	X
ejpam-5305	399	4	c.	c.	PROPN
ejpam-5305	399	5	huang	huang	PROPN
ejpam-5305	399	6	,	,	PUNCT
ejpam-5305	399	7	a.	a.	PROPN
ejpam-5305	399	8	kotzig	kotzig	PROPN
ejpam-5305	399	9	,	,	PUNCT
ejpam-5305	399	10	and	and	CCONJ
ejpam-5305	399	11	a.	a.	PROPN
ejpam-5305	399	12	rosa	rosa	PROPN
ejpam-5305	399	13	.	.	PUNCT
ejpam-5305	400	1	further	further	ADJ
ejpam-5305	400	2	results	result	NOUN
ejpam-5305	400	3	on	on	ADP
ejpam-5305	400	4	tree	tree	NOUN
ejpam-5305	400	5	labellings	labelling	NOUN
ejpam-5305	400	6	.	.	PUNCT
ejpam-5305	401	1	utilitas	utilitas	PROPN
ejpam-5305	401	2	mathematica	mathematica	PROPN
ejpam-5305	401	3	,	,	PUNCT
ejpam-5305	401	4	21:31–48	21:31–48	NUM
ejpam-5305	401	5	,	,	PUNCT
ejpam-5305	401	6	1982	1982	NUM
ejpam-5305	401	7	.	.	PUNCT
ejpam-5305	402	1	[	[	X
ejpam-5305	402	2	8	8	NUM
ejpam-5305	402	3	]	]	PUNCT
ejpam-5305	402	4	p.	p.	NOUN
ejpam-5305	402	5	bahls	bahls	PROPN
ejpam-5305	402	6	,	,	PUNCT
ejpam-5305	402	7	s.	s.	PROPN
ejpam-5305	402	8	lake	lake	PROPN
ejpam-5305	402	9	,	,	PUNCT
ejpam-5305	402	10	and	and	CCONJ
ejpam-5305	402	11	a.	a.	PROPN
ejpam-5305	402	12	wertheim	wertheim	PROPN
ejpam-5305	402	13	.	.	PUNCT
ejpam-5305	403	1	gracefulness	gracefulness	NOUN
ejpam-5305	403	2	of	of	ADP
ejpam-5305	403	3	families	family	NOUN
ejpam-5305	403	4	of	of	ADP
ejpam-5305	403	5	spiders	spider	NOUN
ejpam-5305	403	6	.	.	PUNCT
ejpam-5305	404	1	involve	involve	NOUN
ejpam-5305	404	2	,	,	PUNCT
ejpam-5305	404	3	3(3):241–247	3(3):241–247	NUM
ejpam-5305	404	4	,	,	PUNCT
ejpam-5305	404	5	2010	2010	NUM
ejpam-5305	404	6	.	.	PUNCT
ejpam-5305	405	1	[	[	X
ejpam-5305	405	2	9	9	NUM
ejpam-5305	405	3	]	]	PUNCT
ejpam-5305	405	4	p.	p.	NOUN
ejpam-5305	405	5	jampachon	jampachon	PROPN
ejpam-5305	405	6	,	,	PUNCT
ejpam-5305	405	7	k.	k.	PROPN
ejpam-5305	405	8	nakprasit	nakprasit	PROPN
ejpam-5305	405	9	,	,	PUNCT
ejpam-5305	405	10	and	and	CCONJ
ejpam-5305	405	11	t.	t.	PROPN
ejpam-5305	405	12	poomsa	poomsa	PROPN
ejpam-5305	405	13	-	-	PUNCT
ejpam-5305	405	14	ard	ard	PROPN
ejpam-5305	405	15	.	.	PUNCT
ejpam-5305	405	16	graceful	graceful	ADJ
ejpam-5305	405	17	labeling	labeling	NOUN
ejpam-5305	405	18	of	of	ADP
ejpam-5305	405	19	some	some	DET
ejpam-5305	405	20	classes	class	NOUN
ejpam-5305	405	21	of	of	ADP
ejpam-5305	405	22	spider	spider	NOUN
ejpam-5305	405	23	graphs	graph	NOUN
ejpam-5305	405	24	with	with	ADP
ejpam-5305	405	25	three	three	NUM
ejpam-5305	405	26	legs	leg	NOUN
ejpam-5305	405	27	greater	great	ADJ
ejpam-5305	405	28	than	than	ADP
ejpam-5305	405	29	one	one	NUM
ejpam-5305	405	30	.	.	PUNCT
ejpam-5305	406	1	thai	thai	PROPN
ejpam-5305	406	2	journal	journal	PROPN
ejpam-5305	406	3	of	of	ADP
ejpam-5305	406	4	mathematics	mathematic	NOUN
ejpam-5305	406	5	,	,	PUNCT
ejpam-5305	406	6	12(3):621–630	12(3):621–630	PROPN
ejpam-5305	406	7	,	,	PUNCT
ejpam-5305	406	8	2014	2014	NUM
ejpam-5305	406	9	.	.	PUNCT
ejpam-5305	407	1	[	[	X
ejpam-5305	407	2	10	10	NUM
ejpam-5305	407	3	]	]	PUNCT
ejpam-5305	407	4	p.	p.	NOUN
ejpam-5305	407	5	jampachon	jampachon	PROPN
ejpam-5305	407	6	and	and	CCONJ
ejpam-5305	407	7	t.	t.	PROPN
ejpam-5305	407	8	poomsa	poomsa	PROPN
ejpam-5305	407	9	-	-	PUNCT
ejpam-5305	407	10	ard	ard	PROPN
ejpam-5305	407	11	.	.	PUNCT
ejpam-5305	408	1	graceful	graceful	ADJ
ejpam-5305	408	2	labeling	labeling	NOUN
ejpam-5305	408	3	of	of	ADP
ejpam-5305	408	4	spider	spider	NOUN
ejpam-5305	408	5	graphs	graph	NOUN
ejpam-5305	408	6	with	with	ADP
ejpam-5305	408	7	three	three	NUM
ejpam-5305	408	8	legs	leg	NOUN
ejpam-5305	408	9	of	of	ADP
ejpam-5305	408	10	lengths	length	NOUN
ejpam-5305	408	11	greater	great	ADJ
ejpam-5305	408	12	than	than	ADP
ejpam-5305	408	13	one	one	NUM
ejpam-5305	408	14	.	.	PUNCT
ejpam-5305	409	1	far	far	PROPN
ejpam-5305	409	2	east	east	PROPN
ejpam-5305	409	3	journal	journal	PROPN
ejpam-5305	409	4	of	of	ADP
ejpam-5305	409	5	mathematical	mathematical	ADJ
ejpam-5305	409	6	sciences	science	NOUN
ejpam-5305	409	7	,	,	PUNCT
ejpam-5305	409	8	100(1):51–64	100(1):51–64	NUM
ejpam-5305	409	9	,	,	PUNCT
ejpam-5305	409	10	2016	2016	NUM
ejpam-5305	409	11	.	.	PUNCT
ejpam-5305	410	1	[	[	X
ejpam-5305	410	2	11	11	NUM
ejpam-5305	410	3	]	]	PUNCT
ejpam-5305	410	4	a.	a.	NOUN
ejpam-5305	410	5	panpa	panpa	NOUN
ejpam-5305	410	6	and	and	CCONJ
ejpam-5305	410	7	t.	t.	PROPN
ejpam-5305	410	8	poomsa	poomsa	PROPN
ejpam-5305	410	9	-	-	PUNCT
ejpam-5305	410	10	ard	ard	PROPN
ejpam-5305	410	11	.	.	PUNCT
ejpam-5305	411	1	on	on	ADP
ejpam-5305	411	2	graceful	graceful	ADJ
ejpam-5305	411	3	spider	spider	NOUN
ejpam-5305	411	4	graphs	graph	NOUN
ejpam-5305	411	5	with	with	ADP
ejpam-5305	411	6	at	at	ADP
ejpam-5305	411	7	most	most	ADJ
ejpam-5305	411	8	four	four	NUM
ejpam-5305	411	9	legs	leg	NOUN
ejpam-5305	411	10	of	of	ADP
ejpam-5305	411	11	length	length	NOUN
ejpam-5305	411	12	greater	great	ADJ
ejpam-5305	411	13	than	than	ADP
ejpam-5305	411	14	one	one	NUM
ejpam-5305	411	15	.	.	PUNCT
ejpam-5305	412	1	journal	journal	NOUN
ejpam-5305	412	2	of	of	ADP
ejpam-5305	412	3	applied	apply	VERB
ejpam-5305	412	4	mathematics	mathematic	NOUN
ejpam-5305	412	5	,	,	PUNCT
ejpam-5305	412	6	2016:5327026	2016:5327026	NUM
ejpam-5305	412	7	,	,	PUNCT
ejpam-5305	412	8	2016	2016	NUM
ejpam-5305	412	9	.	.	PUNCT
ejpam-5305	413	1	[	[	X
ejpam-5305	413	2	12	12	NUM
ejpam-5305	413	3	]	]	PUNCT
ejpam-5305	413	4	p.	p.	NOUN
ejpam-5305	413	5	hrnčiar	hrnčiar	NOUN
ejpam-5305	413	6	and	and	CCONJ
ejpam-5305	413	7	a.	a.	NOUN
ejpam-5305	413	8	haviar	haviar	PROPN
ejpam-5305	413	9	.	.	PUNCT
ejpam-5305	414	1	all	all	DET
ejpam-5305	414	2	trees	tree	NOUN
ejpam-5305	414	3	of	of	ADP
ejpam-5305	414	4	diameter	diameter	NOUN
ejpam-5305	414	5	five	five	NUM
ejpam-5305	414	6	are	be	AUX
ejpam-5305	414	7	graceful	graceful	ADJ
ejpam-5305	414	8	.	.	PUNCT
ejpam-5305	415	1	discrete	discrete	ADJ
ejpam-5305	415	2	mathematics	mathematic	NOUN
ejpam-5305	415	3	,	,	PUNCT
ejpam-5305	415	4	233(1	233(1	NUM
ejpam-5305	415	5	-	-	SYM
ejpam-5305	415	6	3):133–150	3):133–150	NUM
ejpam-5305	415	7	,	,	PUNCT
ejpam-5305	415	8	2001	2001	NUM
ejpam-5305	415	9	.	.	PUNCT
ejpam-5305	416	1	[	[	X
ejpam-5305	416	2	13	13	NUM
ejpam-5305	416	3	]	]	PUNCT
ejpam-5305	416	4	k.	k.	NOUN
ejpam-5305	416	5	saengsura	saengsura	PROPN
ejpam-5305	416	6	and	and	CCONJ
ejpam-5305	416	7	t.	t.	PROPN
ejpam-5305	416	8	poomsa	poomsa	PROPN
ejpam-5305	416	9	-	-	PUNCT
ejpam-5305	416	10	ard	ard	PROPN
ejpam-5305	416	11	.	.	PUNCT
ejpam-5305	416	12	graceful	graceful	ADJ
ejpam-5305	416	13	labeling	labeling	NOUN
ejpam-5305	416	14	of	of	ADP
ejpam-5305	416	15	the	the	DET
ejpam-5305	416	16	classes	class	NOUN
ejpam-5305	416	17	of	of	ADP
ejpam-5305	416	18	spider	spider	NOUN
ejpam-5305	416	19	graphs	graph	NOUN
ejpam-5305	416	20	with	with	ADP
ejpam-5305	416	21	four	four	NUM
ejpam-5305	416	22	legs	leg	NOUN
ejpam-5305	416	23	of	of	ADP
ejpam-5305	416	24	lengths	length	NOUN
ejpam-5305	416	25	greater	great	ADJ
ejpam-5305	416	26	than	than	ADP
ejpam-5305	416	27	one	one	NUM
ejpam-5305	416	28	sn(k	sn(k	NOUN
ejpam-5305	416	29	,	,	PUNCT
ejpam-5305	416	30	l	l	NOUN
ejpam-5305	416	31	,	,	PUNCT
ejpam-5305	416	32	m	m	PROPN
ejpam-5305	416	33	,	,	PUNCT
ejpam-5305	416	34	2	2	NUM
ejpam-5305	416	35	)	)	PUNCT
ejpam-5305	416	36	.	.	PUNCT
ejpam-5305	417	1	far	far	PROPN
ejpam-5305	417	2	east	east	PROPN
ejpam-5305	417	3	journal	journal	PROPN
ejpam-5305	417	4	of	of	ADP
ejpam-5305	417	5	mathematical	mathematical	ADJ
ejpam-5305	417	6	sciences	science	NOUN
ejpam-5305	417	7	,	,	PUNCT
ejpam-5305	417	8	100(2):255–269	100(2):255–269	NUM
ejpam-5305	417	9	,	,	PUNCT
ejpam-5305	417	10	2016	2016	NUM
ejpam-5305	417	11	.	.	PUNCT