id	sid	tid	token	lemma	pos
american_scientific_journal-1255	1	1	american	american	PROPN
american_scientific_journal-1255	1	2	scientific	scientific	ADJ
american_scientific_journal-1255	1	3	research	research	NOUN
american_scientific_journal-1255	1	4	journal	journal	NOUN
american_scientific_journal-1255	1	5	for	for	ADP
american_scientific_journal-1255	1	6	engineering	engineering	NOUN
american_scientific_journal-1255	1	7	,	,	PUNCT
american_scientific_journal-1255	1	8	technology	technology	NOUN
american_scientific_journal-1255	1	9	,	,	PUNCT
american_scientific_journal-1255	1	10	and	and	CCONJ
american_scientific_journal-1255	1	11	sciences	science	NOUN
american_scientific_journal-1255	1	12	(	(	PUNCT
american_scientific_journal-1255	1	13	asrjets	asrjet	NOUN
american_scientific_journal-1255	1	14	)	)	PUNCT
american_scientific_journal-1255	1	15	issn	issn	PROPN
american_scientific_journal-1255	1	16	(	(	PUNCT
american_scientific_journal-1255	1	17	print	print	NOUN
american_scientific_journal-1255	1	18	)	)	PUNCT
american_scientific_journal-1255	1	19	2313	2313	NUM
american_scientific_journal-1255	1	20	-	-	SYM
american_scientific_journal-1255	1	21	4410	4410	NUM
american_scientific_journal-1255	1	22	,	,	PUNCT
american_scientific_journal-1255	1	23	issn	issn	PROPN
american_scientific_journal-1255	1	24	(	(	PUNCT
american_scientific_journal-1255	1	25	online	online	ADJ
american_scientific_journal-1255	1	26	)	)	PUNCT
american_scientific_journal-1255	1	27	2313	2313	NUM
american_scientific_journal-1255	1	28	-	-	SYM
american_scientific_journal-1255	1	29	4402	4402	NUM
american_scientific_journal-1255	1	30	©	©	PROPN
american_scientific_journal-1255	1	31	global	global	ADJ
american_scientific_journal-1255	1	32	society	society	NOUN
american_scientific_journal-1255	1	33	of	of	ADP
american_scientific_journal-1255	1	34	scientific	scientific	ADJ
american_scientific_journal-1255	1	35	research	research	NOUN
american_scientific_journal-1255	1	36	and	and	CCONJ
american_scientific_journal-1255	1	37	researchers	researcher	NOUN
american_scientific_journal-1255	1	38	http://asrjetsjournal.org/	http://asrjetsjournal.org/	VERB
american_scientific_journal-1255	1	39	on	on	ADP
american_scientific_journal-1255	1	40	chromatic	chromatic	ADJ
american_scientific_journal-1255	1	41	number	number	NOUN
american_scientific_journal-1255	1	42	and	and	CCONJ
american_scientific_journal-1255	1	43	edge	edge	NOUN
american_scientific_journal-1255	1	44	-	-	PUNCT
american_scientific_journal-1255	1	45	chromatic	chromatic	ADJ
american_scientific_journal-1255	1	46	number	number	NOUN
american_scientific_journal-1255	1	47	of	of	ADP
american_scientific_journal-1255	1	48	the	the	DET
american_scientific_journal-1255	1	49	ottomar	ottomar	NOUN
american_scientific_journal-1255	1	50	graph	graph	NOUN
american_scientific_journal-1255	1	51	mark	mark	PROPN
american_scientific_journal-1255	1	52	caay	caay	VERB
american_scientific_journal-1255	1	53	a	a	DET
american_scientific_journal-1255	1	54	*	*	PROPN
american_scientific_journal-1255	1	55	,	,	PUNCT
american_scientific_journal-1255	1	56	esperanza	esperanza	PROPN
american_scientific_journal-1255	1	57	arugay	arugay	PROPN
american_scientific_journal-1255	1	58	b	b	PROPN
american_scientific_journal-1255	1	59	a	a	DET
american_scientific_journal-1255	1	60	convergys	convergys	NOUN
american_scientific_journal-1255	1	61	philippines	philippine	NOUN
american_scientific_journal-1255	1	62	,	,	PUNCT
american_scientific_journal-1255	1	63	apas	apa	NOUN
american_scientific_journal-1255	1	64	,	,	PUNCT
american_scientific_journal-1255	1	65	cebu	cebu	NOUN
american_scientific_journal-1255	1	66	city	city	NOUN
american_scientific_journal-1255	1	67	,	,	PUNCT
american_scientific_journal-1255	1	68	6000	6000	NUM
american_scientific_journal-1255	1	69	,	,	PUNCT
american_scientific_journal-1255	1	70	philippines	philippines	PROPN
american_scientific_journal-1255	1	71	b	b	PROPN
american_scientific_journal-1255	1	72	department	department	PROPN
american_scientific_journal-1255	1	73	of	of	ADP
american_scientific_journal-1255	1	74	mathematics	mathematics	PROPN
american_scientific_journal-1255	1	75	,	,	PUNCT
american_scientific_journal-1255	1	76	college	college	NOUN
american_scientific_journal-1255	1	77	of	of	ADP
american_scientific_journal-1255	1	78	science	science	NOUN
american_scientific_journal-1255	1	79	and	and	CCONJ
american_scientific_journal-1255	1	80	mathematics	mathematic	NOUN
american_scientific_journal-1255	1	81	,	,	PUNCT
american_scientific_journal-1255	1	82	mindanao	mindanao	PROPN
american_scientific_journal-1255	1	83	state	state	PROPN
american_scientific_journal-1255	1	84	university	university	PROPN
american_scientific_journal-1255	1	85	–	–	PUNCT
american_scientific_journal-1255	1	86	iligan	iligan	PROPN
american_scientific_journal-1255	1	87	institute	institute	PROPN
american_scientific_journal-1255	1	88	of	of	ADP
american_scientific_journal-1255	1	89	technology	technology	PROPN
american_scientific_journal-1255	1	90	,	,	PUNCT
american_scientific_journal-1255	1	91	tibanga	tibanga	PROPN
american_scientific_journal-1255	1	92	,	,	PUNCT
american_scientific_journal-1255	1	93	iligan	iligan	ADJ
american_scientific_journal-1255	1	94	city	city	NOUN
american_scientific_journal-1255	1	95	,	,	PUNCT
american_scientific_journal-1255	1	96	9200	9200	NUM
american_scientific_journal-1255	1	97	,	,	PUNCT
american_scientific_journal-1255	1	98	philippines	philippine	NOUN
american_scientific_journal-1255	1	99	aemail	aemail	NOUN
american_scientific_journal-1255	1	100	:	:	PUNCT
american_scientific_journal-1255	2	1	markycaay126@yahoo.com	markycaay126@yahoo.com	X
american_scientific_journal-1255	2	2	bemail	bemail	VERB
american_scientific_journal-1255	2	3	:	:	PUNCT
american_scientific_journal-1255	2	4	adamandeba@yahoo.com	adamandeba@yahoo.com	X
american_scientific_journal-1255	3	1	abstract	abstract	VERB
american_scientific_journal-1255	3	2	the	the	DET
american_scientific_journal-1255	3	3	path	path	NOUN
american_scientific_journal-1255	3	4	graph	graph	NOUN
american_scientific_journal-1255	3	5	pn	pn	PROPN
american_scientific_journal-1255	3	6	,	,	PUNCT
american_scientific_journal-1255	3	7	consists	consist	VERB
american_scientific_journal-1255	3	8	of	of	ADP
american_scientific_journal-1255	3	9	the	the	DET
american_scientific_journal-1255	3	10	vertex	vertex	NOUN
american_scientific_journal-1255	3	11	set	set	VERB
american_scientific_journal-1255	3	12	v	v	NOUN
american_scientific_journal-1255	3	13	=	=	SYM
american_scientific_journal-1255	3	14	{	{	PUNCT
american_scientific_journal-1255	3	15	1,2	1,2	NUM
american_scientific_journal-1255	3	16	,	,	PUNCT
american_scientific_journal-1255	3	17	…	…	PUNCT
american_scientific_journal-1255	3	18	,	,	PUNCT
american_scientific_journal-1255	3	19	n	n	CCONJ
american_scientific_journal-1255	3	20	}	}	PUNCT
american_scientific_journal-1255	3	21	and	and	CCONJ
american_scientific_journal-1255	3	22	the	the	DET
american_scientific_journal-1255	3	23	edge	edge	NOUN
american_scientific_journal-1255	3	24	set	set	NOUN
american_scientific_journal-1255	3	25	e	e	X
american_scientific_journal-1255	3	26	=	=	SYM
american_scientific_journal-1255	3	27	�	�	PROPN
american_scientific_journal-1255	3	28	{	{	PUNCT
american_scientific_journal-1255	3	29	1,2	1,2	NUM
american_scientific_journal-1255	3	30	}	}	PUNCT
american_scientific_journal-1255	3	31	,	,	PUNCT
american_scientific_journal-1255	3	32	{	{	PUNCT
american_scientific_journal-1255	3	33	2,3	2,3	NUM
american_scientific_journal-1255	3	34	}	}	PUNCT
american_scientific_journal-1255	3	35	,	,	PUNCT
american_scientific_journal-1255	3	36	.	.	PUNCT
american_scientific_journal-1255	3	37	.	.	PUNCT
american_scientific_journal-1255	4	1	.	.	PUNCT
american_scientific_journal-1255	5	1	,	,	PUNCT
american_scientific_journal-1255	5	2	{	{	PUNCT
american_scientific_journal-1255	5	3	n	n	CCONJ
american_scientific_journal-1255	5	4	−	−	PROPN
american_scientific_journal-1255	5	5	1	1	NUM
american_scientific_journal-1255	5	6	,	,	PUNCT
american_scientific_journal-1255	5	7	n}	n}	PROPN
american_scientific_journal-1255	5	8	�	�	PROPN
american_scientific_journal-1255	5	9	.	.	PUNCT
american_scientific_journal-1255	6	1	the	the	DET
american_scientific_journal-1255	6	2	cycle	cycle	NOUN
american_scientific_journal-1255	6	3	graph	graph	NOUN
american_scientific_journal-1255	6	4	cn	cn	PROPN
american_scientific_journal-1255	6	5	,	,	PUNCT
american_scientific_journal-1255	6	6	is	be	AUX
american_scientific_journal-1255	6	7	the	the	DET
american_scientific_journal-1255	6	8	path	path	NOUN
american_scientific_journal-1255	6	9	graph	graph	NOUN
american_scientific_journal-1255	6	10	,	,	PUNCT
american_scientific_journal-1255	6	11	pn	pn	NOUN
american_scientific_journal-1255	6	12	with	with	ADP
american_scientific_journal-1255	6	13	an	an	DET
american_scientific_journal-1255	6	14	additional	additional	ADJ
american_scientific_journal-1255	6	15	edge	edge	NOUN
american_scientific_journal-1255	6	16	{	{	PUNCT
american_scientific_journal-1255	6	17	1	1	NUM
american_scientific_journal-1255	6	18	,	,	PUNCT
american_scientific_journal-1255	6	19	n	n	CCONJ
american_scientific_journal-1255	6	20	}	}	PUNCT
american_scientific_journal-1255	6	21	.	.	PUNCT
american_scientific_journal-1255	7	1	define	define	VERB
american_scientific_journal-1255	7	2	the	the	DET
american_scientific_journal-1255	7	3	ottomar	ottomar	NOUN
american_scientific_journal-1255	7	4	graph	graph	NOUN
american_scientific_journal-1255	7	5	,	,	PUNCT
american_scientific_journal-1255	7	6	denoted	denote	VERB
american_scientific_journal-1255	7	7	by	by	ADP
american_scientific_journal-1255	7	8	on	on	ADP
american_scientific_journal-1255	7	9	,	,	PUNCT
american_scientific_journal-1255	7	10	m	m	PROPN
american_scientific_journal-1255	7	11	,	,	PUNCT
american_scientific_journal-1255	7	12	to	to	PART
american_scientific_journal-1255	7	13	be	be	AUX
american_scientific_journal-1255	7	14	the	the	DET
american_scientific_journal-1255	7	15	graph	graph	NOUN
american_scientific_journal-1255	7	16	cn	cn	PROPN
american_scientific_journal-1255	7	17	,	,	PUNCT
american_scientific_journal-1255	7	18	n	n	X
american_scientific_journal-1255	7	19	ϵℤ+	ϵℤ+	PROPN
american_scientific_journal-1255	7	20	,	,	PUNCT
american_scientific_journal-1255	7	21	n	n	PRON
american_scientific_journal-1255	7	22	≥	≥	NOUN
american_scientific_journal-1255	7	23	3	3	NUM
american_scientific_journal-1255	7	24	,	,	PUNCT
american_scientific_journal-1255	7	25	with	with	ADP
american_scientific_journal-1255	7	26	a	a	DET
american_scientific_journal-1255	7	27	vertex	vertex	NOUN
american_scientific_journal-1255	7	28	connected	connect	VERB
american_scientific_journal-1255	7	29	by	by	ADP
american_scientific_journal-1255	7	30	a	a	DET
american_scientific_journal-1255	7	31	path	path	NOUN
american_scientific_journal-1255	7	32	p2	p2	NOUN
american_scientific_journal-1255	7	33	to	to	ADP
american_scientific_journal-1255	7	34	a	a	DET
american_scientific_journal-1255	7	35	vertex	vertex	NOUN
american_scientific_journal-1255	7	36	of	of	ADP
american_scientific_journal-1255	7	37	cm	cm	NOUN
american_scientific_journal-1255	7	38	,	,	PUNCT
american_scientific_journal-1255	7	39	m	m	VERB
american_scientific_journal-1255	7	40	ϵ	ϵ	PRON
american_scientific_journal-1255	7	41	ℤ+	ℤ+	ADV
american_scientific_journal-1255	7	42	,	,	PUNCT
american_scientific_journal-1255	7	43	n	n	PRON
american_scientific_journal-1255	7	44	≥	≥	NOUN
american_scientific_journal-1255	7	45	3	3	NUM
american_scientific_journal-1255	7	46	.	.	PUNCT
american_scientific_journal-1255	8	1	cn	cn	PROPN
american_scientific_journal-1255	8	2	is	be	AUX
american_scientific_journal-1255	8	3	called	call	VERB
american_scientific_journal-1255	8	4	the	the	DET
american_scientific_journal-1255	8	5	heart	heart	NOUN
american_scientific_journal-1255	8	6	while	while	SCONJ
american_scientific_journal-1255	8	7	cm	cm	PROPN
american_scientific_journal-1255	8	8	is	be	AUX
american_scientific_journal-1255	8	9	called	call	VERB
american_scientific_journal-1255	8	10	a	a	DET
american_scientific_journal-1255	8	11	foot	foot	NOUN
american_scientific_journal-1255	8	12	(	(	PUNCT
american_scientific_journal-1255	8	13	feet	foot	NOUN
american_scientific_journal-1255	8	14	for	for	ADP
american_scientific_journal-1255	8	15	plural	plural	ADJ
american_scientific_journal-1255	8	16	)	)	PUNCT
american_scientific_journal-1255	8	17	.	.	PUNCT
american_scientific_journal-1255	9	1	note	note	VERB
american_scientific_journal-1255	9	2	that	that	SCONJ
american_scientific_journal-1255	9	3	there	there	PRON
american_scientific_journal-1255	9	4	are	be	VERB
american_scientific_journal-1255	9	5	n	n	ADV
american_scientific_journal-1255	9	6	copies	copy	NOUN
american_scientific_journal-1255	9	7	of	of	ADP
american_scientific_journal-1255	9	8	cm	cm	NOUN
american_scientific_journal-1255	9	9	.	.	PUNCT
american_scientific_journal-1255	10	1	the	the	DET
american_scientific_journal-1255	10	2	chromatic	chromatic	ADJ
american_scientific_journal-1255	10	3	number	number	NOUN
american_scientific_journal-1255	10	4	of	of	ADP
american_scientific_journal-1255	10	5	a	a	DET
american_scientific_journal-1255	10	6	graph	graph	NOUN
american_scientific_journal-1255	10	7	g	g	NOUN
american_scientific_journal-1255	10	8	,	,	PUNCT
american_scientific_journal-1255	10	9	denoted	denote	VERB
american_scientific_journal-1255	10	10	by	by	ADP
american_scientific_journal-1255	10	11	χ(g	χ(g	PROPN
american_scientific_journal-1255	10	12	)	)	PUNCT
american_scientific_journal-1255	10	13	,	,	PUNCT
american_scientific_journal-1255	10	14	is	be	AUX
american_scientific_journal-1255	10	15	the	the	DET
american_scientific_journal-1255	10	16	minimum	minimum	ADJ
american_scientific_journal-1255	10	17	number	number	NOUN
american_scientific_journal-1255	10	18	of	of	ADP
american_scientific_journal-1255	10	19	colors	color	NOUN
american_scientific_journal-1255	10	20	the	the	DET
american_scientific_journal-1255	10	21	vertices	vertex	NOUN
american_scientific_journal-1255	10	22	of	of	ADP
american_scientific_journal-1255	10	23	g	g	NOUN
american_scientific_journal-1255	10	24	maybe	maybe	ADV
american_scientific_journal-1255	10	25	colored	color	VERB
american_scientific_journal-1255	10	26	such	such	ADJ
american_scientific_journal-1255	10	27	that	that	SCONJ
american_scientific_journal-1255	10	28	any	any	DET
american_scientific_journal-1255	10	29	two	two	NUM
american_scientific_journal-1255	10	30	adjacent	adjacent	ADJ
american_scientific_journal-1255	10	31	vertices	vertex	NOUN
american_scientific_journal-1255	10	32	have	have	VERB
american_scientific_journal-1255	10	33	different	different	ADJ
american_scientific_journal-1255	10	34	colors	color	NOUN
american_scientific_journal-1255	10	35	.	.	PUNCT
american_scientific_journal-1255	11	1	the	the	DET
american_scientific_journal-1255	11	2	edge	edge	NOUN
american_scientific_journal-1255	11	3	-	-	PUNCT
american_scientific_journal-1255	11	4	chromatic	chromatic	ADJ
american_scientific_journal-1255	11	5	number	number	NOUN
american_scientific_journal-1255	11	6	of	of	ADP
american_scientific_journal-1255	11	7	a	a	DET
american_scientific_journal-1255	11	8	graph	graph	NOUN
american_scientific_journal-1255	11	9	g	g	NOUN
american_scientific_journal-1255	11	10	,	,	PUNCT
american_scientific_journal-1255	11	11	denoted	denote	VERB
american_scientific_journal-1255	11	12	by	by	ADP
american_scientific_journal-1255	11	13	χe(g	χe(g	PROPN
american_scientific_journal-1255	11	14	)	)	PUNCT
american_scientific_journal-1255	11	15	,	,	PUNCT
american_scientific_journal-1255	11	16	is	be	AUX
american_scientific_journal-1255	11	17	the	the	DET
american_scientific_journal-1255	11	18	minimum	minimum	ADJ
american_scientific_journal-1255	11	19	number	number	NOUN
american_scientific_journal-1255	11	20	of	of	ADP
american_scientific_journal-1255	11	21	colors	color	NOUN
american_scientific_journal-1255	11	22	the	the	DET
american_scientific_journal-1255	11	23	edges	edge	NOUN
american_scientific_journal-1255	11	24	of	of	ADP
american_scientific_journal-1255	11	25	g	g	NOUN
american_scientific_journal-1255	11	26	maybe	maybe	ADV
american_scientific_journal-1255	11	27	colored	color	VERB
american_scientific_journal-1255	11	28	such	such	ADJ
american_scientific_journal-1255	11	29	that	that	SCONJ
american_scientific_journal-1255	11	30	any	any	DET
american_scientific_journal-1255	11	31	two	two	NUM
american_scientific_journal-1255	11	32	incident	incident	NOUN
american_scientific_journal-1255	11	33	edges	edge	NOUN
american_scientific_journal-1255	11	34	have	have	VERB
american_scientific_journal-1255	11	35	different	different	ADJ
american_scientific_journal-1255	11	36	colors	color	NOUN
american_scientific_journal-1255	11	37	.	.	PUNCT
american_scientific_journal-1255	12	1	the	the	DET
american_scientific_journal-1255	12	2	chromatic	chromatic	ADJ
american_scientific_journal-1255	12	3	number	number	NOUN
american_scientific_journal-1255	12	4	and	and	CCONJ
american_scientific_journal-1255	12	5	the	the	DET
american_scientific_journal-1255	12	6	edge	edge	NOUN
american_scientific_journal-1255	12	7	-	-	PUNCT
american_scientific_journal-1255	12	8	chromatic	chromatic	ADJ
american_scientific_journal-1255	12	9	number	number	NOUN
american_scientific_journal-1255	12	10	of	of	ADP
american_scientific_journal-1255	12	11	the	the	DET
american_scientific_journal-1255	12	12	ottomar	ottomar	NOUN
american_scientific_journal-1255	12	13	graph	graph	NOUN
american_scientific_journal-1255	12	14	are	be	AUX
american_scientific_journal-1255	12	15	determined	determine	VERB
american_scientific_journal-1255	12	16	.	.	PUNCT
american_scientific_journal-1255	13	1	when	when	SCONJ
american_scientific_journal-1255	13	2	will	will	AUX
american_scientific_journal-1255	13	3	the	the	DET
american_scientific_journal-1255	13	4	two	two	NUM
american_scientific_journal-1255	13	5	invariants	invariant	NOUN
american_scientific_journal-1255	13	6	be	be	AUX
american_scientific_journal-1255	13	7	equal	equal	ADJ
american_scientific_journal-1255	13	8	or	or	CCONJ
american_scientific_journal-1255	13	9	when	when	SCONJ
american_scientific_journal-1255	13	10	will	will	AUX
american_scientific_journal-1255	13	11	they	they	PRON
american_scientific_journal-1255	13	12	be	be	AUX
american_scientific_journal-1255	13	13	unequal	unequal	ADJ
american_scientific_journal-1255	13	14	?	?	PUNCT
american_scientific_journal-1255	14	1	when	when	SCONJ
american_scientific_journal-1255	14	2	the	the	DET
american_scientific_journal-1255	14	3	connecting	connect	VERB
american_scientific_journal-1255	14	4	path	path	NOUN
american_scientific_journal-1255	14	5	pk	pk	NOUN
american_scientific_journal-1255	14	6	has	have	VERB
american_scientific_journal-1255	14	7	order	order	NOUN
american_scientific_journal-1255	14	8	greater	great	ADJ
american_scientific_journal-1255	14	9	than	than	ADP
american_scientific_journal-1255	14	10	2	2	NUM
american_scientific_journal-1255	14	11	,	,	PUNCT
american_scientific_journal-1255	14	12	what	what	PRON
american_scientific_journal-1255	14	13	happens	happen	VERB
american_scientific_journal-1255	14	14	to	to	ADP
american_scientific_journal-1255	14	15	the	the	DET
american_scientific_journal-1255	14	16	value	value	NOUN
american_scientific_journal-1255	14	17	of	of	ADP
american_scientific_journal-1255	14	18	χ(g	χ(g	PROPN
american_scientific_journal-1255	14	19	)	)	PUNCT
american_scientific_journal-1255	14	20	and	and	CCONJ
american_scientific_journal-1255	14	21	χe(g	χe(g	NUM
american_scientific_journal-1255	14	22	)	)	PUNCT
american_scientific_journal-1255	14	23	?	?	PUNCT
american_scientific_journal-1255	15	1	also	also	ADV
american_scientific_journal-1255	15	2	in	in	ADP
american_scientific_journal-1255	15	3	the	the	DET
american_scientific_journal-1255	15	4	paper	paper	NOUN
american_scientific_journal-1255	15	5	,	,	PUNCT
american_scientific_journal-1255	15	6	the	the	DET
american_scientific_journal-1255	15	7	other	other	ADJ
american_scientific_journal-1255	15	8	coloring	coloring	NOUN
american_scientific_journal-1255	15	9	invariants	invariant	NOUN
american_scientific_journal-1255	15	10	are	be	AUX
american_scientific_journal-1255	15	11	compared	compare	VERB
american_scientific_journal-1255	15	12	and	and	CCONJ
american_scientific_journal-1255	15	13	investigated	investigate	VERB
american_scientific_journal-1255	15	14	with	with	ADP
american_scientific_journal-1255	15	15	chromatic	chromatic	ADJ
american_scientific_journal-1255	15	16	number	number	NOUN
american_scientific_journal-1255	15	17	and	and	CCONJ
american_scientific_journal-1255	15	18	edge	edge	NOUN
american_scientific_journal-1255	15	19	-	-	PUNCT
american_scientific_journal-1255	15	20	chromatic	chromatic	ADJ
american_scientific_journal-1255	15	21	number	number	NOUN
american_scientific_journal-1255	15	22	.	.	PUNCT
american_scientific_journal-1255	16	1	keywords	keyword	NOUN
american_scientific_journal-1255	16	2	:	:	PUNCT
american_scientific_journal-1255	16	3	path	path	NOUN
american_scientific_journal-1255	16	4	;	;	PUNCT
american_scientific_journal-1255	16	5	cycle	cycle	NOUN
american_scientific_journal-1255	16	6	;	;	PUNCT
american_scientific_journal-1255	16	7	chromatic	chromatic	ADJ
american_scientific_journal-1255	16	8	number	number	NOUN
american_scientific_journal-1255	16	9	;	;	PUNCT
american_scientific_journal-1255	16	10	edge	edge	NOUN
american_scientific_journal-1255	16	11	-	-	PUNCT
american_scientific_journal-1255	16	12	chromatic	chromatic	ADJ
american_scientific_journal-1255	16	13	number	number	NOUN
american_scientific_journal-1255	16	14	;	;	PUNCT
american_scientific_journal-1255	16	15	ottomar	ottomar	NOUN
american_scientific_journal-1255	16	16	graph	graph	NOUN
american_scientific_journal-1255	16	17	;	;	PUNCT
american_scientific_journal-1255	16	18	generalized	generalized	ADJ
american_scientific_journal-1255	16	19	ottomar	ottomar	NOUN
american_scientific_journal-1255	16	20	graph	graph	NOUN
american_scientific_journal-1255	16	21	.	.	PUNCT
american_scientific_journal-1255	17	1	1	1	X
american_scientific_journal-1255	17	2	.	.	X
american_scientific_journal-1255	17	3	introduction	introduction	NOUN
american_scientific_journal-1255	17	4	a	a	DET
american_scientific_journal-1255	17	5	pair	pair	NOUN
american_scientific_journal-1255	17	6	𝐺	𝐺	NOUN
american_scientific_journal-1255	17	7	=	=	NOUN
american_scientific_journal-1255	17	8	(	(	PUNCT
american_scientific_journal-1255	17	9	𝑉,𝐸	𝑉,𝐸	ADJ
american_scientific_journal-1255	17	10	)	)	PUNCT
american_scientific_journal-1255	17	11	with	with	ADP
american_scientific_journal-1255	17	12	𝐸	𝐸	PROPN
american_scientific_journal-1255	17	13	⊆	⊆	NUM
american_scientific_journal-1255	17	14	𝐸(𝑉	𝐸(𝑉	PROPN
american_scientific_journal-1255	17	15	)	)	PUNCT
american_scientific_journal-1255	17	16	is	be	AUX
american_scientific_journal-1255	17	17	called	call	VERB
american_scientific_journal-1255	17	18	a	a	DET
american_scientific_journal-1255	17	19	graph	graph	NOUN
american_scientific_journal-1255	17	20	(	(	PUNCT
american_scientific_journal-1255	17	21	on	on	ADP
american_scientific_journal-1255	17	22	𝑉	𝑉	PROPN
american_scientific_journal-1255	17	23	)	)	PUNCT
american_scientific_journal-1255	17	24	.	.	PUNCT
american_scientific_journal-1255	18	1	the	the	DET
american_scientific_journal-1255	18	2	elements	element	NOUN
american_scientific_journal-1255	18	3	of	of	ADP
american_scientific_journal-1255	18	4	𝑉	𝑉	PROPN
american_scientific_journal-1255	18	5	are	be	AUX
american_scientific_journal-1255	18	6	the	the	DET
american_scientific_journal-1255	18	7	vertices	vertex	NOUN
american_scientific_journal-1255	18	8	of	of	ADP
american_scientific_journal-1255	18	9	𝐺	𝐺	PROPN
american_scientific_journal-1255	18	10	,	,	PUNCT
american_scientific_journal-1255	18	11	and	and	CCONJ
american_scientific_journal-1255	18	12	those	those	PRON
american_scientific_journal-1255	18	13	of	of	ADP
american_scientific_journal-1255	18	14	𝐸	𝐸	PRON
american_scientific_journal-1255	18	15	the	the	DET
american_scientific_journal-1255	18	16	edges	edge	NOUN
american_scientific_journal-1255	18	17	of	of	ADP
american_scientific_journal-1255	18	18	𝐺.	𝐺.	NOUN
american_scientific_journal-1255	18	19	-----------------------------------------------------------------------	-----------------------------------------------------------------------	PUNCT
american_scientific_journal-1255	19	1	*	*	PUNCT
american_scientific_journal-1255	19	2	corresponding	correspond	VERB
american_scientific_journal-1255	19	3	author	author	NOUN
american_scientific_journal-1255	19	4	98	98	NUM
american_scientific_journal-1255	19	5	http://asrjetsjournal.org/	http://asrjetsjournal.org/	PROPN
american_scientific_journal-1255	19	6	american	american	ADJ
american_scientific_journal-1255	19	7	scientific	scientific	ADJ
american_scientific_journal-1255	19	8	research	research	NOUN
american_scientific_journal-1255	19	9	journal	journal	NOUN
american_scientific_journal-1255	19	10	for	for	ADP
american_scientific_journal-1255	19	11	engineering	engineering	NOUN
american_scientific_journal-1255	19	12	,	,	PUNCT
american_scientific_journal-1255	19	13	technology	technology	NOUN
american_scientific_journal-1255	19	14	,	,	PUNCT
american_scientific_journal-1255	19	15	and	and	CCONJ
american_scientific_journal-1255	19	16	sciences	science	NOUN
american_scientific_journal-1255	19	17	(	(	PUNCT
american_scientific_journal-1255	19	18	asrjets	asrjet	NOUN
american_scientific_journal-1255	19	19	)	)	PUNCT
american_scientific_journal-1255	19	20	(	(	PUNCT
american_scientific_journal-1255	19	21	2016	2016	NUM
american_scientific_journal-1255	19	22	)	)	PUNCT
american_scientific_journal-1255	19	23	volume	volume	NOUN
american_scientific_journal-1255	19	24	16	16	NUM
american_scientific_journal-1255	19	25	,	,	PUNCT
american_scientific_journal-1255	19	26	no	no	DET
american_scientific_journal-1255	19	27	1	1	NUM
american_scientific_journal-1255	19	28	,	,	PUNCT
american_scientific_journal-1255	19	29	pp	pp	ADV
american_scientific_journal-1255	19	30	98	98	NUM
american_scientific_journal-1255	19	31	-	-	SYM
american_scientific_journal-1255	19	32	107	107	NUM
american_scientific_journal-1255	19	33	the	the	DET
american_scientific_journal-1255	19	34	vertex	vertex	NOUN
american_scientific_journal-1255	19	35	set	set	NOUN
american_scientific_journal-1255	19	36	of	of	ADP
american_scientific_journal-1255	19	37	a	a	DET
american_scientific_journal-1255	19	38	graph	graph	NOUN
american_scientific_journal-1255	19	39	𝐺	𝐺	NOUN
american_scientific_journal-1255	19	40	is	be	AUX
american_scientific_journal-1255	19	41	denoted	denote	VERB
american_scientific_journal-1255	19	42	by	by	ADP
american_scientific_journal-1255	19	43	𝑉𝐺	𝑉𝐺	PROPN
american_scientific_journal-1255	19	44	and	and	CCONJ
american_scientific_journal-1255	19	45	its	its	PRON
american_scientific_journal-1255	19	46	edge	edge	NOUN
american_scientific_journal-1255	19	47	set	set	VERB
american_scientific_journal-1255	19	48	by	by	ADP
american_scientific_journal-1255	19	49	𝐸𝐺	𝐸𝐺	PROPN
american_scientific_journal-1255	19	50	.	.	PUNCT
american_scientific_journal-1255	20	1	therefore	therefore	ADV
american_scientific_journal-1255	20	2	𝐺	𝐺	PROPN
american_scientific_journal-1255	20	3	=	=	SYM
american_scientific_journal-1255	20	4	(	(	PUNCT
american_scientific_journal-1255	20	5	𝑉𝐺	𝑉𝐺	PROPN
american_scientific_journal-1255	20	6	,	,	PUNCT
american_scientific_journal-1255	20	7	𝐸𝐺	𝐸𝐺	PROPN
american_scientific_journal-1255	20	8	)	)	PUNCT
american_scientific_journal-1255	20	9	.	.	PUNCT
american_scientific_journal-1255	21	1	the	the	DET
american_scientific_journal-1255	21	2	path	path	NOUN
american_scientific_journal-1255	21	3	graph	graph	NOUN
american_scientific_journal-1255	21	4	𝑃𝑛	𝑃𝑛	PROPN
american_scientific_journal-1255	21	5	,	,	PUNCT
american_scientific_journal-1255	21	6	consists	consist	VERB
american_scientific_journal-1255	21	7	of	of	ADP
american_scientific_journal-1255	21	8	the	the	DET
american_scientific_journal-1255	21	9	vertex	vertex	NOUN
american_scientific_journal-1255	21	10	set	set	VERB
american_scientific_journal-1255	21	11	𝑉	𝑉	PROPN
american_scientific_journal-1255	21	12	=	=	PUNCT
american_scientific_journal-1255	21	13	{	{	PUNCT
american_scientific_journal-1255	21	14	1	1	NUM
american_scientific_journal-1255	21	15	,	,	PUNCT
american_scientific_journal-1255	21	16	2	2	NUM
american_scientific_journal-1255	21	17	,	,	PUNCT
american_scientific_journal-1255	21	18	.	.	PUNCT
american_scientific_journal-1255	21	19	.	.	PUNCT
american_scientific_journal-1255	21	20	.	.	PUNCT
american_scientific_journal-1255	22	1	,	,	PUNCT
american_scientific_journal-1255	22	2	𝑛	𝑛	X
american_scientific_journal-1255	22	3	}	}	PUNCT
american_scientific_journal-1255	22	4	and	and	CCONJ
american_scientific_journal-1255	22	5	the	the	DET
american_scientific_journal-1255	22	6	edge	edge	NOUN
american_scientific_journal-1255	22	7	set	set	VERB
american_scientific_journal-1255	22	8	𝐸	𝐸	NOUN
american_scientific_journal-1255	22	9	=	=	SYM
american_scientific_journal-1255	22	10	{	{	PUNCT
american_scientific_journal-1255	22	11	{	{	PUNCT
american_scientific_journal-1255	22	12	1,2	1,2	NUM
american_scientific_journal-1255	22	13	}	}	PUNCT
american_scientific_journal-1255	22	14	,	,	PUNCT
american_scientific_journal-1255	22	15	{	{	PUNCT
american_scientific_journal-1255	22	16	2,3	2,3	NUM
american_scientific_journal-1255	22	17	}	}	PUNCT
american_scientific_journal-1255	22	18	,	,	PUNCT
american_scientific_journal-1255	22	19	.	.	PUNCT
american_scientific_journal-1255	22	20	.	.	PUNCT
american_scientific_journal-1255	22	21	.	.	PUNCT
american_scientific_journal-1255	23	1	,	,	PUNCT
american_scientific_journal-1255	23	2	{	{	PUNCT
american_scientific_journal-1255	23	3	𝑛	𝑛	PRON
american_scientific_journal-1255	23	4	−	−	PROPN
american_scientific_journal-1255	23	5	1,𝑛	1,𝑛	NUM
american_scientific_journal-1255	23	6	}	}	PUNCT
american_scientific_journal-1255	23	7	}	}	PUNCT
american_scientific_journal-1255	23	8	.	.	PUNCT
american_scientific_journal-1255	24	1	the	the	DET
american_scientific_journal-1255	24	2	cycle	cycle	NOUN
american_scientific_journal-1255	24	3	graph	graph	NOUN
american_scientific_journal-1255	24	4	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	24	5	,	,	PUNCT
american_scientific_journal-1255	24	6	is	be	AUX
american_scientific_journal-1255	24	7	the	the	DET
american_scientific_journal-1255	24	8	path	path	NOUN
american_scientific_journal-1255	24	9	graph	graph	NOUN
american_scientific_journal-1255	24	10	,	,	PUNCT
american_scientific_journal-1255	24	11	𝑃𝑛	𝑃𝑛	NOUN
american_scientific_journal-1255	24	12	with	with	ADP
american_scientific_journal-1255	24	13	an	an	DET
american_scientific_journal-1255	24	14	additional	additional	ADJ
american_scientific_journal-1255	24	15	edge	edge	NOUN
american_scientific_journal-1255	24	16	{	{	PUNCT
american_scientific_journal-1255	24	17	1,𝑛	1,𝑛	NUM
american_scientific_journal-1255	24	18	}	}	PUNCT
american_scientific_journal-1255	24	19	.	.	PUNCT
american_scientific_journal-1255	25	1	the	the	DET
american_scientific_journal-1255	25	2	chromatic	chromatic	ADJ
american_scientific_journal-1255	25	3	number	number	NOUN
american_scientific_journal-1255	25	4	of	of	ADP
american_scientific_journal-1255	25	5	a	a	DET
american_scientific_journal-1255	25	6	graph	graph	NOUN
american_scientific_journal-1255	25	7	𝐺	𝐺	NOUN
american_scientific_journal-1255	25	8	,	,	PUNCT
american_scientific_journal-1255	25	9	denoted	denote	VERB
american_scientific_journal-1255	25	10	by	by	ADP
american_scientific_journal-1255	25	11	𝜒(𝐺	𝜒(𝐺	NOUN
american_scientific_journal-1255	25	12	)	)	PUNCT
american_scientific_journal-1255	25	13	,	,	PUNCT
american_scientific_journal-1255	25	14	is	be	AUX
american_scientific_journal-1255	25	15	the	the	DET
american_scientific_journal-1255	25	16	minimum	minimum	ADJ
american_scientific_journal-1255	25	17	number	number	NOUN
american_scientific_journal-1255	25	18	of	of	ADP
american_scientific_journal-1255	25	19	colors	color	NOUN
american_scientific_journal-1255	25	20	the	the	DET
american_scientific_journal-1255	25	21	vertices	vertex	NOUN
american_scientific_journal-1255	25	22	of	of	ADP
american_scientific_journal-1255	25	23	𝐺	𝐺	PROPN
american_scientific_journal-1255	25	24	maybe	maybe	ADV
american_scientific_journal-1255	25	25	colored	color	VERB
american_scientific_journal-1255	25	26	such	such	ADJ
american_scientific_journal-1255	25	27	that	that	SCONJ
american_scientific_journal-1255	25	28	any	any	DET
american_scientific_journal-1255	25	29	two	two	NUM
american_scientific_journal-1255	25	30	adjacent	adjacent	ADJ
american_scientific_journal-1255	25	31	vertices	vertex	NOUN
american_scientific_journal-1255	25	32	have	have	VERB
american_scientific_journal-1255	25	33	different	different	ADJ
american_scientific_journal-1255	25	34	colors	color	NOUN
american_scientific_journal-1255	25	35	.	.	PUNCT
american_scientific_journal-1255	26	1	the	the	DET
american_scientific_journal-1255	26	2	edge	edge	NOUN
american_scientific_journal-1255	26	3	-	-	PUNCT
american_scientific_journal-1255	26	4	chromatic	chromatic	ADJ
american_scientific_journal-1255	26	5	number	number	NOUN
american_scientific_journal-1255	26	6	of	of	ADP
american_scientific_journal-1255	26	7	a	a	DET
american_scientific_journal-1255	26	8	graph	graph	NOUN
american_scientific_journal-1255	26	9	𝐺	𝐺	NOUN
american_scientific_journal-1255	26	10	,	,	PUNCT
american_scientific_journal-1255	26	11	denoted	denote	VERB
american_scientific_journal-1255	26	12	by	by	ADP
american_scientific_journal-1255	26	13	𝜒𝑒(𝐺	𝜒𝑒(𝐺	NOUN
american_scientific_journal-1255	26	14	)	)	PUNCT
american_scientific_journal-1255	26	15	,	,	PUNCT
american_scientific_journal-1255	26	16	is	be	AUX
american_scientific_journal-1255	26	17	the	the	DET
american_scientific_journal-1255	26	18	minimum	minimum	ADJ
american_scientific_journal-1255	26	19	number	number	NOUN
american_scientific_journal-1255	26	20	of	of	ADP
american_scientific_journal-1255	26	21	colors	color	NOUN
american_scientific_journal-1255	26	22	the	the	DET
american_scientific_journal-1255	26	23	edges	edge	NOUN
american_scientific_journal-1255	26	24	of	of	ADP
american_scientific_journal-1255	26	25	𝐺	𝐺	PROPN
american_scientific_journal-1255	26	26	maybe	maybe	ADV
american_scientific_journal-1255	26	27	colored	color	VERB
american_scientific_journal-1255	26	28	such	such	ADJ
american_scientific_journal-1255	26	29	that	that	SCONJ
american_scientific_journal-1255	26	30	any	any	DET
american_scientific_journal-1255	26	31	two	two	NUM
american_scientific_journal-1255	26	32	incident	incident	NOUN
american_scientific_journal-1255	26	33	edges	edge	NOUN
american_scientific_journal-1255	26	34	have	have	VERB
american_scientific_journal-1255	26	35	different	different	ADJ
american_scientific_journal-1255	26	36	colors	color	NOUN
american_scientific_journal-1255	26	37	.	.	PUNCT
american_scientific_journal-1255	27	1	known	know	VERB
american_scientific_journal-1255	27	2	result	result	VERB
american_scientific_journal-1255	27	3	1	1	NUM
american_scientific_journal-1255	28	1	[	[	X
american_scientific_journal-1255	28	2	2	2	X
american_scientific_journal-1255	28	3	]	]	PUNCT
american_scientific_journal-1255	28	4	if	if	SCONJ
american_scientific_journal-1255	28	5	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	28	6	is	be	AUX
american_scientific_journal-1255	28	7	a	a	DET
american_scientific_journal-1255	28	8	cycle	cycle	NOUN
american_scientific_journal-1255	28	9	of	of	ADP
american_scientific_journal-1255	28	10	order	order	NOUN
american_scientific_journal-1255	28	11	𝑛	𝑛	VERB
american_scientific_journal-1255	28	12	,	,	PUNCT
american_scientific_journal-1255	28	13	then	then	ADV
american_scientific_journal-1255	28	14	define	define	VERB
american_scientific_journal-1255	28	15	the	the	DET
american_scientific_journal-1255	28	16	ottomar	ottomar	NOUN
american_scientific_journal-1255	28	17	graph	graph	NOUN
american_scientific_journal-1255	28	18	,	,	PUNCT
american_scientific_journal-1255	28	19	denoted	denote	VERB
american_scientific_journal-1255	28	20	by	by	ADP
american_scientific_journal-1255	28	21	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	28	22	,	,	PUNCT
american_scientific_journal-1255	28	23	to	to	PART
american_scientific_journal-1255	28	24	be	be	AUX
american_scientific_journal-1255	28	25	the	the	DET
american_scientific_journal-1255	28	26	graph	graph	NOUN
american_scientific_journal-1255	28	27	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	28	28	,	,	PUNCT
american_scientific_journal-1255	28	29	𝑛	𝑛	PRON
american_scientific_journal-1255	28	30	𝜖ℤ+,𝑛	𝜖ℤ+,𝑛	PROPN
american_scientific_journal-1255	28	31	≥	≥	NOUN
american_scientific_journal-1255	28	32	3	3	NUM
american_scientific_journal-1255	28	33	,	,	PUNCT
american_scientific_journal-1255	28	34	with	with	ADP
american_scientific_journal-1255	28	35	a	a	DET
american_scientific_journal-1255	28	36	vertex	vertex	NOUN
american_scientific_journal-1255	28	37	connected	connect	VERB
american_scientific_journal-1255	28	38	by	by	ADP
american_scientific_journal-1255	28	39	a	a	DET
american_scientific_journal-1255	28	40	path	path	NOUN
american_scientific_journal-1255	28	41	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	28	42	to	to	ADP
american_scientific_journal-1255	28	43	a	a	DET
american_scientific_journal-1255	28	44	vertex	vertex	NOUN
american_scientific_journal-1255	28	45	of	of	ADP
american_scientific_journal-1255	28	46	𝐶𝑚,𝑚	𝐶𝑚,𝑚	PROPN
american_scientific_journal-1255	28	47	𝜖	𝜖	PROPN
american_scientific_journal-1255	28	48	ℤ+,𝑛	ℤ+,𝑛	PROPN
american_scientific_journal-1255	28	49	≥	≥	NOUN
american_scientific_journal-1255	28	50	3	3	NUM
american_scientific_journal-1255	28	51	.	.	PUNCT
american_scientific_journal-1255	29	1	𝐶𝑛	𝐶𝑛	NOUN
american_scientific_journal-1255	29	2	is	be	AUX
american_scientific_journal-1255	29	3	called	call	VERB
american_scientific_journal-1255	29	4	the	the	DET
american_scientific_journal-1255	29	5	heart	heart	NOUN
american_scientific_journal-1255	29	6	while	while	SCONJ
american_scientific_journal-1255	29	7	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	29	8	is	be	AUX
american_scientific_journal-1255	29	9	called	call	VERB
american_scientific_journal-1255	29	10	a	a	DET
american_scientific_journal-1255	29	11	foot	foot	NOUN
american_scientific_journal-1255	29	12	(	(	PUNCT
american_scientific_journal-1255	29	13	feet	foot	NOUN
american_scientific_journal-1255	29	14	for	for	ADP
american_scientific_journal-1255	29	15	plural	plural	ADJ
american_scientific_journal-1255	29	16	)	)	PUNCT
american_scientific_journal-1255	29	17	.	.	PUNCT
american_scientific_journal-1255	30	1	note	note	VERB
american_scientific_journal-1255	30	2	that	that	SCONJ
american_scientific_journal-1255	30	3	there	there	PRON
american_scientific_journal-1255	30	4	are	be	VERB
american_scientific_journal-1255	30	5	𝑛	𝑛	DET
american_scientific_journal-1255	30	6	copies	copy	NOUN
american_scientific_journal-1255	30	7	of	of	ADP
american_scientific_journal-1255	30	8	𝐶𝑚.	𝐶𝑚.	NOUN
american_scientific_journal-1255	30	9	2	2	NUM
american_scientific_journal-1255	30	10	.	.	PUNCT
american_scientific_journal-1255	31	1	identities	identity	NOUN
american_scientific_journal-1255	31	2	of	of	ADP
american_scientific_journal-1255	31	3	chromatic	chromatic	ADJ
american_scientific_journal-1255	31	4	number	number	NOUN
american_scientific_journal-1255	31	5	and	and	CCONJ
american_scientific_journal-1255	31	6	edge	edge	NOUN
american_scientific_journal-1255	31	7	-	-	PUNCT
american_scientific_journal-1255	31	8	chromatic	chromatic	ADJ
american_scientific_journal-1255	31	9	number	number	NOUN
american_scientific_journal-1255	31	10	of	of	ADP
american_scientific_journal-1255	31	11	ottomar	ottomar	NOUN
american_scientific_journal-1255	31	12	graph	graph	NOUN
american_scientific_journal-1255	31	13	theorem	theorem	VERB
american_scientific_journal-1255	31	14	2.1	2.1	NUM
american_scientific_journal-1255	31	15	for	for	ADP
american_scientific_journal-1255	31	16	all	all	DET
american_scientific_journal-1255	31	17	integers	integer	NOUN
american_scientific_journal-1255	31	18	𝑚,𝑛	𝑚,𝑛	VERB
american_scientific_journal-1255	31	19	≥	≥	NOUN
american_scientific_journal-1255	31	20	3,𝜒	3,𝜒	NUM
american_scientific_journal-1255	31	21	�	�	SYM
american_scientific_journal-1255	31	22	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	31	23	�	�	PROPN
american_scientific_journal-1255	31	24	=	=	SYM
american_scientific_journal-1255	31	25	3	3	NUM
american_scientific_journal-1255	31	26	,	,	PUNCT
american_scientific_journal-1255	31	27	if	if	SCONJ
american_scientific_journal-1255	31	28	:	:	PUNCT
american_scientific_journal-1255	31	29	99	99	NUM
american_scientific_journal-1255	31	30	american	american	ADJ
american_scientific_journal-1255	31	31	scientific	scientific	ADJ
american_scientific_journal-1255	31	32	research	research	NOUN
american_scientific_journal-1255	31	33	journal	journal	NOUN
american_scientific_journal-1255	31	34	for	for	ADP
american_scientific_journal-1255	31	35	engineering	engineering	NOUN
american_scientific_journal-1255	31	36	,	,	PUNCT
american_scientific_journal-1255	31	37	technology	technology	NOUN
american_scientific_journal-1255	31	38	,	,	PUNCT
american_scientific_journal-1255	31	39	and	and	CCONJ
american_scientific_journal-1255	31	40	sciences	science	NOUN
american_scientific_journal-1255	31	41	(	(	PUNCT
american_scientific_journal-1255	31	42	asrjets	asrjet	NOUN
american_scientific_journal-1255	31	43	)	)	PUNCT
american_scientific_journal-1255	31	44	(	(	PUNCT
american_scientific_journal-1255	31	45	2016	2016	NUM
american_scientific_journal-1255	31	46	)	)	PUNCT
american_scientific_journal-1255	31	47	volume	volume	NOUN
american_scientific_journal-1255	31	48	16	16	NUM
american_scientific_journal-1255	31	49	,	,	PUNCT
american_scientific_journal-1255	31	50	no	no	DET
american_scientific_journal-1255	31	51	1	1	NUM
american_scientific_journal-1255	31	52	,	,	PUNCT
american_scientific_journal-1255	31	53	pp	pp	ADV
american_scientific_journal-1255	31	54	98	98	NUM
american_scientific_journal-1255	31	55	-	-	SYM
american_scientific_journal-1255	31	56	107	107	NUM
american_scientific_journal-1255	31	57	•	•	NOUN
american_scientific_journal-1255	31	58	case	case	NOUN
american_scientific_journal-1255	31	59	1	1	NUM
american_scientific_journal-1255	31	60	:	:	PUNCT
american_scientific_journal-1255	31	61	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	31	62	are	be	AUX
american_scientific_journal-1255	31	63	both	both	PRON
american_scientific_journal-1255	31	64	odd	odd	ADJ
american_scientific_journal-1255	31	65	•	•	NOUN
american_scientific_journal-1255	31	66	case	case	NOUN
american_scientific_journal-1255	31	67	2	2	NUM
american_scientific_journal-1255	31	68	:	:	PUNCT
american_scientific_journal-1255	31	69	𝑚	𝑚	PROPN
american_scientific_journal-1255	31	70	is	be	AUX
american_scientific_journal-1255	31	71	odd	odd	ADJ
american_scientific_journal-1255	31	72	,	,	PUNCT
american_scientific_journal-1255	31	73	𝑛	𝑛	PRON
american_scientific_journal-1255	31	74	is	be	AUX
american_scientific_journal-1255	31	75	even	even	ADV
american_scientific_journal-1255	31	76	•	•	NUM
american_scientific_journal-1255	31	77	case	case	NOUN
american_scientific_journal-1255	31	78	3	3	NUM
american_scientific_journal-1255	31	79	:	:	PUNCT
american_scientific_journal-1255	31	80	𝑚	𝑚	NOUN
american_scientific_journal-1255	31	81	is	be	AUX
american_scientific_journal-1255	31	82	even	even	ADV
american_scientific_journal-1255	31	83	,	,	PUNCT
american_scientific_journal-1255	31	84	𝑛	𝑛	PRON
american_scientific_journal-1255	31	85	is	be	AUX
american_scientific_journal-1255	31	86	odd	odd	ADJ
american_scientific_journal-1255	31	87	and	and	CCONJ
american_scientific_journal-1255	31	88	•	•	NUM
american_scientific_journal-1255	31	89	𝜒	𝜒	X
american_scientific_journal-1255	31	90	�	�	PROPN
american_scientific_journal-1255	31	91	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	31	92	�	�	PROPN
american_scientific_journal-1255	31	93	=	=	SYM
american_scientific_journal-1255	31	94	2	2	NUM
american_scientific_journal-1255	31	95	if	if	SCONJ
american_scientific_journal-1255	31	96	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	31	97	are	be	AUX
american_scientific_journal-1255	31	98	even	even	ADV
american_scientific_journal-1255	31	99	.	.	PUNCT
american_scientific_journal-1255	32	1	proof	proof	NOUN
american_scientific_journal-1255	32	2	:	:	PUNCT
american_scientific_journal-1255	32	3	•	•	NUM
american_scientific_journal-1255	32	4	case	case	NOUN
american_scientific_journal-1255	32	5	1	1	NUM
american_scientific_journal-1255	32	6	:	:	PUNCT
american_scientific_journal-1255	32	7	𝑚	𝑚	PROPN
american_scientific_journal-1255	32	8	and	and	CCONJ
american_scientific_journal-1255	32	9	𝑛	𝑛	PROPN
american_scientific_journal-1255	32	10	are	be	AUX
american_scientific_journal-1255	32	11	both	both	PRON
american_scientific_journal-1255	32	12	odd	odd	ADJ
american_scientific_journal-1255	32	13	if	if	SCONJ
american_scientific_journal-1255	32	14	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	32	15	are	be	AUX
american_scientific_journal-1255	32	16	odd	odd	ADJ
american_scientific_journal-1255	32	17	,	,	PUNCT
american_scientific_journal-1255	32	18	then	then	ADV
american_scientific_journal-1255	32	19	by	by	ADP
american_scientific_journal-1255	32	20	equation	equation	NOUN
american_scientific_journal-1255	32	21	(	(	PUNCT
american_scientific_journal-1255	32	22	1.1	1.1	NUM
american_scientific_journal-1255	32	23	)	)	PUNCT
american_scientific_journal-1255	32	24	,	,	PUNCT
american_scientific_journal-1255	32	25	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NUM
american_scientific_journal-1255	32	26	)	)	PUNCT
american_scientific_journal-1255	32	27	=	=	SYM
american_scientific_journal-1255	32	28	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NOUN
american_scientific_journal-1255	32	29	)	)	PUNCT
american_scientific_journal-1255	32	30	=	=	SYM
american_scientific_journal-1255	32	31	3	3	NUM
american_scientific_journal-1255	32	32	,	,	PUNCT
american_scientific_journal-1255	32	33	thus	thus	ADV
american_scientific_journal-1255	32	34	𝜒	𝜒	X
american_scientific_journal-1255	32	35	�	�	PROPN
american_scientific_journal-1255	32	36	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	32	37	�	�	PROPN
american_scientific_journal-1255	32	38	≠	≠	PROPN
american_scientific_journal-1255	32	39	2	2	NUM
american_scientific_journal-1255	32	40	.	.	PUNCT
american_scientific_journal-1255	32	41	suppose	suppose	VERB
american_scientific_journal-1255	32	42	the	the	DET
american_scientific_journal-1255	32	43	vertices	vertex	NOUN
american_scientific_journal-1255	32	44	of	of	ADP
american_scientific_journal-1255	32	45	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	32	46	and	and	CCONJ
american_scientific_journal-1255	32	47	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	32	48	are	be	AUX
american_scientific_journal-1255	32	49	colored	color	VERB
american_scientific_journal-1255	32	50	with	with	ADP
american_scientific_journal-1255	32	51	the	the	DET
american_scientific_journal-1255	32	52	same	same	ADJ
american_scientific_journal-1255	32	53	set	set	NOUN
american_scientific_journal-1255	32	54	of	of	ADP
american_scientific_journal-1255	32	55	different	different	ADJ
american_scientific_journal-1255	32	56	colors	color	NOUN
american_scientific_journal-1255	32	57	say	say	VERB
american_scientific_journal-1255	32	58	𝑎1	𝑎1	ADV
american_scientific_journal-1255	32	59	,	,	PUNCT
american_scientific_journal-1255	32	60	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	32	61	,	,	PUNCT
american_scientific_journal-1255	32	62	𝑎3	𝑎3	PROPN
american_scientific_journal-1255	32	63	.	.	PUNCT
american_scientific_journal-1255	33	1	then	then	ADV
american_scientific_journal-1255	33	2	the	the	DET
american_scientific_journal-1255	33	3	path	path	NOUN
american_scientific_journal-1255	33	4	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	33	5	is	be	AUX
american_scientific_journal-1255	33	6	attached	attach	VERB
american_scientific_journal-1255	33	7	to	to	ADP
american_scientific_journal-1255	33	8	a	a	DET
american_scientific_journal-1255	33	9	vertex	vertex	NOUN
american_scientific_journal-1255	33	10	colored	color	VERB
american_scientific_journal-1255	33	11	say	say	VERB
american_scientific_journal-1255	33	12	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	33	13	,	,	PUNCT
american_scientific_journal-1255	33	14	𝑖	𝑖	PROPN
american_scientific_journal-1255	33	15	=	=	NOUN
american_scientific_journal-1255	33	16	1	1	NUM
american_scientific_journal-1255	33	17	,	,	PUNCT
american_scientific_journal-1255	33	18	2	2	NUM
american_scientific_journal-1255	33	19	,	,	PUNCT
american_scientific_journal-1255	33	20	3	3	NUM
american_scientific_journal-1255	33	21	of	of	ADP
american_scientific_journal-1255	33	22	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	33	23	and	and	CCONJ
american_scientific_journal-1255	33	24	the	the	DET
american_scientific_journal-1255	33	25	other	other	ADJ
american_scientific_journal-1255	33	26	end	end	NOUN
american_scientific_journal-1255	33	27	vertex	vertex	NOUN
american_scientific_journal-1255	33	28	is	be	AUX
american_scientific_journal-1255	33	29	attached	attach	VERB
american_scientific_journal-1255	33	30	to	to	ADP
american_scientific_journal-1255	33	31	a	a	DET
american_scientific_journal-1255	33	32	vertex	vertex	NOUN
american_scientific_journal-1255	33	33	colored	color	VERB
american_scientific_journal-1255	33	34	say	say	VERB
american_scientific_journal-1255	33	35	𝑎𝑗	𝑎𝑗	ADV
american_scientific_journal-1255	33	36	,	,	PUNCT
american_scientific_journal-1255	33	37	𝑗	𝑗	PROPN
american_scientific_journal-1255	33	38	=	=	NOUN
american_scientific_journal-1255	33	39	1,2,3	1,2,3	NUM
american_scientific_journal-1255	33	40	of	of	ADP
american_scientific_journal-1255	33	41	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	33	42	,	,	PUNCT
american_scientific_journal-1255	33	43	where	where	SCONJ
american_scientific_journal-1255	33	44	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	33	45	≠	≠	PROPN
american_scientific_journal-1255	33	46	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	33	47	hence	hence	ADV
american_scientific_journal-1255	33	48	,	,	PUNCT
american_scientific_journal-1255	33	49	three	three	NUM
american_scientific_journal-1255	33	50	is	be	AUX
american_scientific_journal-1255	33	51	the	the	DET
american_scientific_journal-1255	33	52	minimum	minimum	ADJ
american_scientific_journal-1255	33	53	number	number	NOUN
american_scientific_journal-1255	33	54	of	of	ADP
american_scientific_journal-1255	33	55	colors	color	NOUN
american_scientific_journal-1255	33	56	to	to	PART
american_scientific_journal-1255	33	57	color	color	VERB
american_scientific_journal-1255	33	58	the	the	DET
american_scientific_journal-1255	33	59	vertices	vertex	NOUN
american_scientific_journal-1255	33	60	of	of	ADP
american_scientific_journal-1255	33	61	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	33	62	,	,	PUNCT
american_scientific_journal-1255	33	63	where	where	SCONJ
american_scientific_journal-1255	33	64	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	33	65	are	be	AUX
american_scientific_journal-1255	33	66	both	both	PRON
american_scientific_journal-1255	33	67	odd	odd	ADJ
american_scientific_journal-1255	33	68	.	.	PUNCT
american_scientific_journal-1255	34	1	consequently	consequently	ADV
american_scientific_journal-1255	34	2	,	,	PUNCT
american_scientific_journal-1255	34	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	34	4	�	�	SYM
american_scientific_journal-1255	34	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	34	6	�	�	PROPN
american_scientific_journal-1255	34	7	=	=	SYM
american_scientific_journal-1255	34	8	3	3	NUM
american_scientific_journal-1255	34	9	.	.	NOUN
american_scientific_journal-1255	34	10	•	•	NUM
american_scientific_journal-1255	34	11	case	case	NOUN
american_scientific_journal-1255	34	12	2	2	NUM
american_scientific_journal-1255	34	13	:	:	PUNCT
american_scientific_journal-1255	34	14	𝑚	𝑚	PROPN
american_scientific_journal-1255	34	15	is	be	AUX
american_scientific_journal-1255	34	16	odd	odd	ADJ
american_scientific_journal-1255	34	17	,	,	PUNCT
american_scientific_journal-1255	34	18	𝑛	𝑛	PRON
american_scientific_journal-1255	34	19	is	be	AUX
american_scientific_journal-1255	34	20	even	even	ADV
american_scientific_journal-1255	34	21	if	if	SCONJ
american_scientific_journal-1255	34	22	𝑚	𝑚	PROPN
american_scientific_journal-1255	34	23	is	be	AUX
american_scientific_journal-1255	34	24	odd	odd	ADJ
american_scientific_journal-1255	34	25	and	and	CCONJ
american_scientific_journal-1255	34	26	𝑛	𝑛	PROPN
american_scientific_journal-1255	34	27	is	be	AUX
american_scientific_journal-1255	34	28	even	even	ADV
american_scientific_journal-1255	34	29	,	,	PUNCT
american_scientific_journal-1255	34	30	then	then	ADV
american_scientific_journal-1255	34	31	by	by	ADP
american_scientific_journal-1255	34	32	equation	equation	NOUN
american_scientific_journal-1255	34	33	(	(	PUNCT
american_scientific_journal-1255	34	34	1.1	1.1	NUM
american_scientific_journal-1255	34	35	)	)	PUNCT
american_scientific_journal-1255	34	36	,	,	PUNCT
american_scientific_journal-1255	34	37	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NOUN
american_scientific_journal-1255	34	38	)	)	PUNCT
american_scientific_journal-1255	34	39	=	=	SYM
american_scientific_journal-1255	34	40	3	3	NUM
american_scientific_journal-1255	34	41	and	and	CCONJ
american_scientific_journal-1255	34	42	𝜒(𝐶𝑛	𝜒(𝐶𝑛	NOUN
american_scientific_journal-1255	34	43	)	)	PUNCT
american_scientific_journal-1255	34	44	=	=	SYM
american_scientific_journal-1255	35	1	2	2	NUM
american_scientific_journal-1255	35	2	,	,	PUNCT
american_scientific_journal-1255	35	3	so	so	ADV
american_scientific_journal-1255	35	4	𝜒	𝜒	ADJ
american_scientific_journal-1255	35	5	�	�	PROPN
american_scientific_journal-1255	35	6	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	35	7	�	�	PROPN
american_scientific_journal-1255	35	8	≠	≠	PROPN
american_scientific_journal-1255	35	9	2	2	NUM
american_scientific_journal-1255	35	10	.	.	PUNCT
american_scientific_journal-1255	35	11	suppose	suppose	VERB
american_scientific_journal-1255	35	12	that	that	SCONJ
american_scientific_journal-1255	35	13	the	the	DET
american_scientific_journal-1255	35	14	vertices	vertex	NOUN
american_scientific_journal-1255	35	15	of	of	ADP
american_scientific_journal-1255	35	16	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	35	17	are	be	AUX
american_scientific_journal-1255	35	18	colored	color	VERB
american_scientific_journal-1255	35	19	with	with	ADP
american_scientific_journal-1255	35	20	two	two	NUM
american_scientific_journal-1255	35	21	of	of	ADP
american_scientific_journal-1255	35	22	the	the	DET
american_scientific_journal-1255	35	23	colors	color	NOUN
american_scientific_journal-1255	35	24	that	that	PRON
american_scientific_journal-1255	35	25	also	also	ADV
american_scientific_journal-1255	35	26	color	color	VERB
american_scientific_journal-1255	35	27	the	the	DET
american_scientific_journal-1255	35	28	vertices	vertex	NOUN
american_scientific_journal-1255	35	29	of	of	ADP
american_scientific_journal-1255	35	30	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	35	31	,	,	PUNCT
american_scientific_journal-1255	35	32	say	say	VERB
american_scientific_journal-1255	35	33	𝑎1	𝑎1	ADV
american_scientific_journal-1255	35	34	,	,	PUNCT
american_scientific_journal-1255	35	35	𝑎2	𝑎2	NOUN
american_scientific_journal-1255	35	36	for	for	ADP
american_scientific_journal-1255	35	37	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	35	38	and	and	CCONJ
american_scientific_journal-1255	35	39	𝑎1	𝑎1	PROPN
american_scientific_journal-1255	35	40	,	,	PUNCT
american_scientific_journal-1255	35	41	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	35	42	,	,	PUNCT
american_scientific_journal-1255	35	43	𝑎3	𝑎3	NOUN
american_scientific_journal-1255	35	44	for	for	ADP
american_scientific_journal-1255	35	45	𝐶𝑚.	𝐶𝑚.	PROPN
american_scientific_journal-1255	35	46	then	then	ADV
american_scientific_journal-1255	35	47	a	a	DET
american_scientific_journal-1255	35	48	path	path	NOUN
american_scientific_journal-1255	35	49	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	35	50	is	be	AUX
american_scientific_journal-1255	35	51	attached	attach	VERB
american_scientific_journal-1255	35	52	to	to	ADP
american_scientific_journal-1255	35	53	vertex	vertex	NOUN
american_scientific_journal-1255	35	54	colored	color	VERB
american_scientific_journal-1255	35	55	say	say	VERB
american_scientific_journal-1255	35	56	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	35	57	,	,	PUNCT
american_scientific_journal-1255	35	58	𝑖	𝑖	PROPN
american_scientific_journal-1255	35	59	=	=	NOUN
american_scientific_journal-1255	35	60	1,2,3	1,2,3	NUM
american_scientific_journal-1255	35	61	of	of	ADP
american_scientific_journal-1255	35	62	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	35	63	and	and	CCONJ
american_scientific_journal-1255	35	64	other	other	ADJ
american_scientific_journal-1255	35	65	end	end	NOUN
american_scientific_journal-1255	35	66	vertex	vertex	NOUN
american_scientific_journal-1255	35	67	is	be	AUX
american_scientific_journal-1255	35	68	attached	attach	VERB
american_scientific_journal-1255	35	69	to	to	ADP
american_scientific_journal-1255	35	70	a	a	DET
american_scientific_journal-1255	35	71	vertex	vertex	NOUN
american_scientific_journal-1255	35	72	colored	color	VERB
american_scientific_journal-1255	35	73	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	35	74	,	,	PUNCT
american_scientific_journal-1255	35	75	𝑗	𝑗	PROPN
american_scientific_journal-1255	35	76	=	=	SYM
american_scientific_journal-1255	35	77	1,2	1,2	NUM
american_scientific_journal-1255	35	78	of	of	ADP
american_scientific_journal-1255	35	79	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	35	80	,	,	PUNCT
american_scientific_journal-1255	35	81	where	where	SCONJ
american_scientific_journal-1255	35	82	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	35	83	≠	≠	PROPN
american_scientific_journal-1255	35	84	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	35	85	this	this	PRON
american_scientific_journal-1255	35	86	means	mean	VERB
american_scientific_journal-1255	35	87	that	that	SCONJ
american_scientific_journal-1255	35	88	the	the	DET
american_scientific_journal-1255	35	89	minimum	minimum	ADJ
american_scientific_journal-1255	35	90	number	number	NOUN
american_scientific_journal-1255	35	91	of	of	ADP
american_scientific_journal-1255	35	92	colors	color	NOUN
american_scientific_journal-1255	35	93	to	to	PART
american_scientific_journal-1255	35	94	color	color	VERB
american_scientific_journal-1255	35	95	the	the	DET
american_scientific_journal-1255	35	96	vertices	vertex	NOUN
american_scientific_journal-1255	35	97	of	of	ADP
american_scientific_journal-1255	35	98	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	35	99	,	,	PUNCT
american_scientific_journal-1255	35	100	where	where	SCONJ
american_scientific_journal-1255	35	101	𝑛	𝑛	PROPN
american_scientific_journal-1255	35	102	is	be	AUX
american_scientific_journal-1255	35	103	even	even	ADV
american_scientific_journal-1255	35	104	and	and	CCONJ
american_scientific_journal-1255	35	105	𝑚	𝑚	PROPN
american_scientific_journal-1255	35	106	is	be	AUX
american_scientific_journal-1255	35	107	odd	odd	ADJ
american_scientific_journal-1255	35	108	is	be	AUX
american_scientific_journal-1255	35	109	three	three	NUM
american_scientific_journal-1255	35	110	.	.	PUNCT
american_scientific_journal-1255	36	1	thus	thus	ADV
american_scientific_journal-1255	36	2	,	,	PUNCT
american_scientific_journal-1255	36	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	36	4	�	�	PROPN
american_scientific_journal-1255	36	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	36	6	�	�	PROPN
american_scientific_journal-1255	36	7	=	=	SYM
american_scientific_journal-1255	36	8	3	3	NUM
american_scientific_journal-1255	36	9	.	.	NOUN
american_scientific_journal-1255	36	10	•	•	NUM
american_scientific_journal-1255	36	11	case	case	NOUN
american_scientific_journal-1255	36	12	3	3	NUM
american_scientific_journal-1255	36	13	:	:	PUNCT
american_scientific_journal-1255	36	14	𝑚	𝑚	NOUN
american_scientific_journal-1255	36	15	is	be	AUX
american_scientific_journal-1255	36	16	even	even	ADV
american_scientific_journal-1255	36	17	,	,	PUNCT
american_scientific_journal-1255	36	18	𝑛	𝑛	PRON
american_scientific_journal-1255	36	19	is	be	AUX
american_scientific_journal-1255	36	20	odd	odd	ADJ
american_scientific_journal-1255	36	21	proof	proof	NOUN
american_scientific_journal-1255	36	22	of	of	ADP
american_scientific_journal-1255	36	23	this	this	DET
american_scientific_journal-1255	36	24	case	case	NOUN
american_scientific_journal-1255	36	25	is	be	AUX
american_scientific_journal-1255	36	26	similar	similar	ADJ
american_scientific_journal-1255	36	27	to	to	ADP
american_scientific_journal-1255	36	28	case	case	NOUN
american_scientific_journal-1255	36	29	2	2	NUM
american_scientific_journal-1255	36	30	.	.	NOUN
american_scientific_journal-1255	36	31	•	•	NUM
american_scientific_journal-1255	36	32	case	case	NOUN
american_scientific_journal-1255	36	33	4	4	NUM
american_scientific_journal-1255	36	34	:	:	PUNCT
american_scientific_journal-1255	36	35	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	36	36	are	be	AUX
american_scientific_journal-1255	36	37	both	both	PRON
american_scientific_journal-1255	36	38	even	even	ADV
american_scientific_journal-1255	36	39	if	if	SCONJ
american_scientific_journal-1255	36	40	𝑚,𝑛	𝑚,𝑛	NOUN
american_scientific_journal-1255	36	41	are	be	AUX
american_scientific_journal-1255	36	42	even	even	ADV
american_scientific_journal-1255	36	43	then	then	ADV
american_scientific_journal-1255	36	44	by	by	ADP
american_scientific_journal-1255	36	45	equation	equation	NOUN
american_scientific_journal-1255	36	46	(	(	PUNCT
american_scientific_journal-1255	36	47	1.1	1.1	NUM
american_scientific_journal-1255	36	48	)	)	PUNCT
american_scientific_journal-1255	36	49	,	,	PUNCT
american_scientific_journal-1255	36	50	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NOUN
american_scientific_journal-1255	36	51	)	)	PUNCT
american_scientific_journal-1255	36	52	=	=	SYM
american_scientific_journal-1255	36	53	𝜒(𝐶𝑛	𝜒(𝐶𝑛	X
american_scientific_journal-1255	36	54	)	)	PUNCT
american_scientific_journal-1255	36	55	=	=	SYM
american_scientific_journal-1255	36	56	2	2	X
american_scientific_journal-1255	36	57	.	.	PUNCT
american_scientific_journal-1255	36	58	suppose	suppose	VERB
american_scientific_journal-1255	36	59	the	the	DET
american_scientific_journal-1255	36	60	vertices	vertex	NOUN
american_scientific_journal-1255	36	61	of	of	ADP
american_scientific_journal-1255	36	62	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	36	63	and	and	CCONJ
american_scientific_journal-1255	36	64	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	36	65	are	be	AUX
american_scientific_journal-1255	36	66	colored	color	VERB
american_scientific_journal-1255	36	67	with	with	ADP
american_scientific_journal-1255	36	68	the	the	DET
american_scientific_journal-1255	36	69	same	same	ADJ
american_scientific_journal-1255	36	70	set	set	NOUN
american_scientific_journal-1255	36	71	of	of	ADP
american_scientific_journal-1255	36	72	different	different	ADJ
american_scientific_journal-1255	36	73	colors	color	NOUN
american_scientific_journal-1255	36	74	,	,	PUNCT
american_scientific_journal-1255	36	75	say	say	VERB
american_scientific_journal-1255	36	76	𝑎1	𝑎1	ADV
american_scientific_journal-1255	36	77	,	,	PUNCT
american_scientific_journal-1255	36	78	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	36	79	.	.	PUNCT
american_scientific_journal-1255	37	1	then	then	ADV
american_scientific_journal-1255	37	2	a	a	DET
american_scientific_journal-1255	37	3	path	path	NOUN
american_scientific_journal-1255	37	4	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	37	5	is	be	AUX
american_scientific_journal-1255	37	6	attached	attach	VERB
american_scientific_journal-1255	37	7	to	to	ADP
american_scientific_journal-1255	37	8	a	a	DET
american_scientific_journal-1255	37	9	vertex	vertex	NOUN
american_scientific_journal-1255	37	10	colored	color	VERB
american_scientific_journal-1255	37	11	say	say	VERB
american_scientific_journal-1255	37	12	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	37	13	,	,	PUNCT
american_scientific_journal-1255	37	14	𝑖	𝑖	PROPN
american_scientific_journal-1255	37	15	=	=	SYM
american_scientific_journal-1255	37	16	1,2	1,2	NUM
american_scientific_journal-1255	37	17	of	of	ADP
american_scientific_journal-1255	37	18	100	100	NUM
american_scientific_journal-1255	37	19	american	american	ADJ
american_scientific_journal-1255	37	20	scientific	scientific	ADJ
american_scientific_journal-1255	37	21	research	research	NOUN
american_scientific_journal-1255	37	22	journal	journal	NOUN
american_scientific_journal-1255	37	23	for	for	ADP
american_scientific_journal-1255	37	24	engineering	engineering	NOUN
american_scientific_journal-1255	37	25	,	,	PUNCT
american_scientific_journal-1255	37	26	technology	technology	NOUN
american_scientific_journal-1255	37	27	,	,	PUNCT
american_scientific_journal-1255	37	28	and	and	CCONJ
american_scientific_journal-1255	37	29	sciences	science	NOUN
american_scientific_journal-1255	37	30	(	(	PUNCT
american_scientific_journal-1255	37	31	asrjets	asrjet	NOUN
american_scientific_journal-1255	37	32	)	)	PUNCT
american_scientific_journal-1255	37	33	(	(	PUNCT
american_scientific_journal-1255	37	34	2016	2016	NUM
american_scientific_journal-1255	37	35	)	)	PUNCT
american_scientific_journal-1255	37	36	volume	volume	NOUN
american_scientific_journal-1255	37	37	16	16	NUM
american_scientific_journal-1255	37	38	,	,	PUNCT
american_scientific_journal-1255	37	39	no	no	DET
american_scientific_journal-1255	37	40	1	1	NUM
american_scientific_journal-1255	37	41	,	,	PUNCT
american_scientific_journal-1255	37	42	pp	pp	ADV
american_scientific_journal-1255	37	43	98	98	NUM
american_scientific_journal-1255	37	44	-	-	SYM
american_scientific_journal-1255	37	45	107	107	NUM
american_scientific_journal-1255	38	1	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	38	2	and	and	CCONJ
american_scientific_journal-1255	38	3	the	the	DET
american_scientific_journal-1255	38	4	other	other	ADJ
american_scientific_journal-1255	38	5	end	end	NOUN
american_scientific_journal-1255	38	6	vertex	vertex	NOUN
american_scientific_journal-1255	38	7	is	be	AUX
american_scientific_journal-1255	38	8	attached	attach	VERB
american_scientific_journal-1255	38	9	to	to	ADP
american_scientific_journal-1255	38	10	a	a	DET
american_scientific_journal-1255	38	11	vertex	vertex	NOUN
american_scientific_journal-1255	38	12	colored	color	VERB
american_scientific_journal-1255	38	13	say	say	VERB
american_scientific_journal-1255	38	14	𝑎𝑗	𝑎𝑗	ADV
american_scientific_journal-1255	38	15	,	,	PUNCT
american_scientific_journal-1255	38	16	𝑗	𝑗	PROPN
american_scientific_journal-1255	38	17	=	=	SYM
american_scientific_journal-1255	38	18	1,2	1,2	NUM
american_scientific_journal-1255	38	19	of	of	ADP
american_scientific_journal-1255	38	20	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	38	21	,	,	PUNCT
american_scientific_journal-1255	38	22	where	where	SCONJ
american_scientific_journal-1255	38	23	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	38	24	≠	≠	PROPN
american_scientific_journal-1255	38	25	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	38	26	hence	hence	ADV
american_scientific_journal-1255	38	27	,	,	PUNCT
american_scientific_journal-1255	38	28	two	two	NUM
american_scientific_journal-1255	38	29	is	be	AUX
american_scientific_journal-1255	38	30	the	the	DET
american_scientific_journal-1255	38	31	minimum	minimum	ADJ
american_scientific_journal-1255	38	32	number	number	NOUN
american_scientific_journal-1255	38	33	of	of	ADP
american_scientific_journal-1255	38	34	colors	color	NOUN
american_scientific_journal-1255	38	35	to	to	ADP
american_scientific_journal-1255	38	36	color	color	NOUN
american_scientific_journal-1255	38	37	vertices	vertex	NOUN
american_scientific_journal-1255	38	38	of	of	ADP
american_scientific_journal-1255	38	39	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	38	40	,	,	PUNCT
american_scientific_journal-1255	38	41	where	where	SCONJ
american_scientific_journal-1255	38	42	,	,	PUNCT
american_scientific_journal-1255	38	43	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	38	44	are	be	AUX
american_scientific_journal-1255	38	45	both	both	PRON
american_scientific_journal-1255	38	46	even	even	ADV
american_scientific_journal-1255	38	47	.	.	PUNCT
american_scientific_journal-1255	39	1	consequently	consequently	ADV
american_scientific_journal-1255	39	2	,	,	PUNCT
american_scientific_journal-1255	39	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	39	4	�	�	SYM
american_scientific_journal-1255	39	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	39	6	�	�	PROPN
american_scientific_journal-1255	39	7	=	=	SYM
american_scientific_journal-1255	39	8	2	2	X
american_scientific_journal-1255	39	9	.	.	PUNCT
american_scientific_journal-1255	39	10	∎	∎	PROPN
american_scientific_journal-1255	39	11	theorem	theorem	VERB
american_scientific_journal-1255	39	12	2.2	2.2	NUM
american_scientific_journal-1255	39	13	for	for	ADP
american_scientific_journal-1255	39	14	all	all	DET
american_scientific_journal-1255	39	15	integers	integer	NOUN
american_scientific_journal-1255	39	16	𝑚,𝑛	𝑚,𝑛	VERB
american_scientific_journal-1255	39	17	,	,	PUNCT
american_scientific_journal-1255	39	18	𝑚,𝑛	𝑚,𝑛	X
american_scientific_journal-1255	39	19	≥	≥	NUM
american_scientific_journal-1255	39	20	3	3	NUM
american_scientific_journal-1255	39	21	,	,	PUNCT
american_scientific_journal-1255	39	22	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	39	23	�	�	PROPN
american_scientific_journal-1255	39	24	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	39	25	�	�	PROPN
american_scientific_journal-1255	39	26	=	=	SYM
american_scientific_journal-1255	39	27	3	3	X
american_scientific_journal-1255	39	28	.	.	X
american_scientific_journal-1255	39	29	proof	proof	NOUN
american_scientific_journal-1255	39	30	:	:	PUNCT
american_scientific_journal-1255	39	31	to	to	PART
american_scientific_journal-1255	39	32	prove	prove	VERB
american_scientific_journal-1255	39	33	this	this	DET
american_scientific_journal-1255	39	34	theorem	theorem	NOUN
american_scientific_journal-1255	39	35	,	,	PUNCT
american_scientific_journal-1255	39	36	we	we	PRON
american_scientific_journal-1255	39	37	consider	consider	VERB
american_scientific_journal-1255	39	38	the	the	DET
american_scientific_journal-1255	39	39	following	follow	VERB
american_scientific_journal-1255	39	40	cases	case	NOUN
american_scientific_journal-1255	39	41	:	:	PUNCT
american_scientific_journal-1255	39	42	•	•	NUM
american_scientific_journal-1255	39	43	case	case	NOUN
american_scientific_journal-1255	39	44	1	1	NUM
american_scientific_journal-1255	39	45	:	:	PUNCT
american_scientific_journal-1255	39	46	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	39	47	are	be	AUX
american_scientific_journal-1255	39	48	both	both	PRON
american_scientific_journal-1255	39	49	odd	odd	ADJ
american_scientific_journal-1255	39	50	if	if	SCONJ
american_scientific_journal-1255	39	51	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	39	52	are	be	AUX
american_scientific_journal-1255	39	53	both	both	PRON
american_scientific_journal-1255	39	54	odd	odd	ADJ
american_scientific_journal-1255	39	55	,	,	PUNCT
american_scientific_journal-1255	39	56	then	then	ADV
american_scientific_journal-1255	39	57	by	by	ADP
american_scientific_journal-1255	39	58	equation	equation	NOUN
american_scientific_journal-1255	39	59	(	(	PUNCT
american_scientific_journal-1255	39	60	1.2	1.2	NUM
american_scientific_journal-1255	39	61	)	)	PUNCT
american_scientific_journal-1255	39	62	,	,	PUNCT
american_scientific_journal-1255	39	63	𝜒𝑒(𝐶𝑚	𝜒𝑒(𝐶𝑚	PROPN
american_scientific_journal-1255	39	64	)	)	PUNCT
american_scientific_journal-1255	39	65	=	=	SYM
american_scientific_journal-1255	39	66	𝜒𝑒(𝐶𝑛	𝜒𝑒(𝐶𝑛	PROPN
american_scientific_journal-1255	39	67	)	)	PUNCT
american_scientific_journal-1255	39	68	=	=	SYM
american_scientific_journal-1255	40	1	3	3	X
american_scientific_journal-1255	40	2	.	.	PUNCT
american_scientific_journal-1255	40	3	thus	thus	ADV
american_scientific_journal-1255	40	4	,	,	PUNCT
american_scientific_journal-1255	40	5	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	40	6	�	�	PROPN
american_scientific_journal-1255	40	7	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	40	8	�	�	PROPN
american_scientific_journal-1255	40	9	≠	≠	PROPN
american_scientific_journal-1255	40	10	2	2	NUM
american_scientific_journal-1255	40	11	since	since	SCONJ
american_scientific_journal-1255	40	12	there	there	PRON
american_scientific_journal-1255	40	13	exist	exist	VERB
american_scientific_journal-1255	40	14	three	three	NUM
american_scientific_journal-1255	40	15	incident	incident	NOUN
american_scientific_journal-1255	40	16	edges	edge	NOUN
american_scientific_journal-1255	40	17	.	.	PUNCT
american_scientific_journal-1255	41	1	suppose	suppose	VERB
american_scientific_journal-1255	41	2	the	the	DET
american_scientific_journal-1255	41	3	edges	edge	NOUN
american_scientific_journal-1255	41	4	of	of	ADP
american_scientific_journal-1255	41	5	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	41	6	and	and	CCONJ
american_scientific_journal-1255	41	7	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	41	8	are	be	AUX
american_scientific_journal-1255	41	9	colored	color	VERB
american_scientific_journal-1255	41	10	with	with	ADP
american_scientific_journal-1255	41	11	the	the	DET
american_scientific_journal-1255	41	12	same	same	ADJ
american_scientific_journal-1255	41	13	set	set	NOUN
american_scientific_journal-1255	41	14	of	of	ADP
american_scientific_journal-1255	41	15	different	different	ADJ
american_scientific_journal-1255	41	16	colors	color	NOUN
american_scientific_journal-1255	41	17	,	,	PUNCT
american_scientific_journal-1255	41	18	say	say	VERB
american_scientific_journal-1255	41	19	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	41	20	,	,	PUNCT
american_scientific_journal-1255	41	21	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	41	22	,	,	PUNCT
american_scientific_journal-1255	41	23	𝑏𝑘.	𝑏𝑘.	PROPN
american_scientific_journal-1255	41	24	then	then	ADV
american_scientific_journal-1255	41	25	a	a	DET
american_scientific_journal-1255	41	26	path	path	NOUN
american_scientific_journal-1255	41	27	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	41	28	connecting	connect	VERB
american_scientific_journal-1255	41	29	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	41	30	and	and	CCONJ
american_scientific_journal-1255	41	31	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	41	32	must	must	AUX
american_scientific_journal-1255	41	33	have	have	VERB
american_scientific_journal-1255	41	34	an	an	DET
american_scientific_journal-1255	41	35	edge	edge	NOUN
american_scientific_journal-1255	41	36	colored	color	VERB
american_scientific_journal-1255	41	37	with	with	ADP
american_scientific_journal-1255	41	38	one	one	NUM
american_scientific_journal-1255	41	39	of	of	ADP
american_scientific_journal-1255	41	40	the	the	DET
american_scientific_journal-1255	41	41	colors	color	NOUN
american_scientific_journal-1255	41	42	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	41	43	,	,	PUNCT
american_scientific_journal-1255	41	44	𝑏2	𝑏2	PROPN
american_scientific_journal-1255	41	45	,	,	PUNCT
american_scientific_journal-1255	41	46	𝑏3	𝑏3	NOUN
american_scientific_journal-1255	41	47	,	,	PUNCT
american_scientific_journal-1255	41	48	say	say	VERB
american_scientific_journal-1255	41	49	𝑏𝑖	𝑏𝑖	ADV
american_scientific_journal-1255	41	50	,	,	PUNCT
american_scientific_journal-1255	41	51	𝑖	𝑖	NOUN
american_scientific_journal-1255	41	52	=	=	NOUN
american_scientific_journal-1255	41	53	1,2,3	1,2,3	NUM
american_scientific_journal-1255	41	54	,	,	PUNCT
american_scientific_journal-1255	41	55	where	where	SCONJ
american_scientific_journal-1255	41	56	𝑏𝑖	𝑏𝑖	PROPN
american_scientific_journal-1255	41	57	is	be	AUX
american_scientific_journal-1255	41	58	incident	incident	NOUN
american_scientific_journal-1255	41	59	to	to	ADP
american_scientific_journal-1255	41	60	edges	edge	NOUN
american_scientific_journal-1255	41	61	colored	color	VERB
american_scientific_journal-1255	41	62	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	41	63	and	and	CCONJ
american_scientific_journal-1255	41	64	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	41	65	of	of	ADP
american_scientific_journal-1255	41	66	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	41	67	and	and	CCONJ
american_scientific_journal-1255	41	68	is	be	AUX
american_scientific_journal-1255	41	69	also	also	ADV
american_scientific_journal-1255	41	70	incident	incident	NOUN
american_scientific_journal-1255	41	71	to	to	ADP
american_scientific_journal-1255	41	72	edges	edge	NOUN
american_scientific_journal-1255	41	73	colored	color	VERB
american_scientific_journal-1255	41	74	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	41	75	and	and	CCONJ
american_scientific_journal-1255	41	76	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	41	77	of	of	ADP
american_scientific_journal-1255	41	78	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	41	79	,	,	PUNCT
american_scientific_journal-1255	41	80	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	41	81	≠	≠	PROPN
american_scientific_journal-1255	41	82	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	41	83	≠	≠	PROPN
american_scientific_journal-1255	41	84	𝑏𝑘.	𝑏𝑘.	X
american_scientific_journal-1255	41	85	hence	hence	ADV
american_scientific_journal-1255	41	86	,	,	PUNCT
american_scientific_journal-1255	41	87	three	three	NUM
american_scientific_journal-1255	41	88	is	be	AUX
american_scientific_journal-1255	41	89	the	the	DET
american_scientific_journal-1255	41	90	minimum	minimum	ADJ
american_scientific_journal-1255	41	91	number	number	NOUN
american_scientific_journal-1255	41	92	of	of	ADP
american_scientific_journal-1255	41	93	colors	color	NOUN
american_scientific_journal-1255	41	94	to	to	PART
american_scientific_journal-1255	41	95	color	color	VERB
american_scientific_journal-1255	41	96	the	the	DET
american_scientific_journal-1255	41	97	edges	edge	NOUN
american_scientific_journal-1255	41	98	of	of	ADP
american_scientific_journal-1255	41	99	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	41	100	,	,	PUNCT
american_scientific_journal-1255	41	101	where	where	SCONJ
american_scientific_journal-1255	41	102	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	41	103	are	be	AUX
american_scientific_journal-1255	41	104	both	both	PRON
american_scientific_journal-1255	41	105	odd	odd	ADJ
american_scientific_journal-1255	41	106	.	.	PUNCT
american_scientific_journal-1255	42	1	consequently	consequently	ADV
american_scientific_journal-1255	42	2	,	,	PUNCT
american_scientific_journal-1255	42	3	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	42	4	�	�	PROPN
american_scientific_journal-1255	42	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	42	6	�	�	PROPN
american_scientific_journal-1255	42	7	=	=	SYM
american_scientific_journal-1255	42	8	3	3	NUM
american_scientific_journal-1255	42	9	.	.	NOUN
american_scientific_journal-1255	42	10	•	•	NUM
american_scientific_journal-1255	42	11	case	case	NOUN
american_scientific_journal-1255	42	12	2	2	NUM
american_scientific_journal-1255	42	13	:	:	PUNCT
american_scientific_journal-1255	42	14	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	42	15	are	be	AUX
american_scientific_journal-1255	42	16	both	both	PRON
american_scientific_journal-1255	42	17	even	even	ADV
american_scientific_journal-1255	42	18	if	if	SCONJ
american_scientific_journal-1255	42	19	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	42	20	are	be	AUX
american_scientific_journal-1255	42	21	both	both	PRON
american_scientific_journal-1255	42	22	even	even	ADV
american_scientific_journal-1255	42	23	,	,	PUNCT
american_scientific_journal-1255	42	24	then	then	ADV
american_scientific_journal-1255	42	25	by	by	ADP
american_scientific_journal-1255	42	26	equation	equation	NOUN
american_scientific_journal-1255	42	27	(	(	PUNCT
american_scientific_journal-1255	42	28	1.2	1.2	NUM
american_scientific_journal-1255	42	29	)	)	PUNCT
american_scientific_journal-1255	42	30	,	,	PUNCT
american_scientific_journal-1255	42	31	𝜒𝑒(𝐶𝑚	𝜒𝑒(𝐶𝑚	PROPN
american_scientific_journal-1255	42	32	)	)	PUNCT
american_scientific_journal-1255	42	33	=	=	SYM
american_scientific_journal-1255	42	34	𝜒𝑒(𝐶𝑛	𝜒𝑒(𝐶𝑛	PROPN
american_scientific_journal-1255	42	35	)	)	PUNCT
american_scientific_journal-1255	42	36	=	=	SYM
american_scientific_journal-1255	42	37	2	2	X
american_scientific_journal-1255	42	38	.	.	PUNCT
american_scientific_journal-1255	42	39	thus	thus	ADV
american_scientific_journal-1255	42	40	,	,	PUNCT
american_scientific_journal-1255	42	41	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	42	42	�	�	PROPN
american_scientific_journal-1255	42	43	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	42	44	�	�	PROPN
american_scientific_journal-1255	42	45	≠	≠	PROPN
american_scientific_journal-1255	42	46	2	2	NUM
american_scientific_journal-1255	42	47	since	since	SCONJ
american_scientific_journal-1255	42	48	there	there	PRON
american_scientific_journal-1255	42	49	exist	exist	VERB
american_scientific_journal-1255	42	50	three	three	NUM
american_scientific_journal-1255	42	51	incident	incident	NOUN
american_scientific_journal-1255	42	52	edges	edge	NOUN
american_scientific_journal-1255	42	53	.	.	PUNCT
american_scientific_journal-1255	43	1	without	without	ADP
american_scientific_journal-1255	43	2	a	a	DET
american_scientific_journal-1255	43	3	loss	loss	NOUN
american_scientific_journal-1255	43	4	of	of	ADP
american_scientific_journal-1255	43	5	generality	generality	NOUN
american_scientific_journal-1255	43	6	,	,	PUNCT
american_scientific_journal-1255	43	7	suppose	suppose	VERB
american_scientific_journal-1255	43	8	the	the	DET
american_scientific_journal-1255	43	9	edges	edge	NOUN
american_scientific_journal-1255	43	10	of	of	ADP
american_scientific_journal-1255	43	11	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	43	12	and	and	CCONJ
american_scientific_journal-1255	43	13	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	43	14	are	be	AUX
american_scientific_journal-1255	43	15	colored	color	VERB
american_scientific_journal-1255	43	16	with	with	ADP
american_scientific_journal-1255	43	17	same	same	ADJ
american_scientific_journal-1255	43	18	set	set	NOUN
american_scientific_journal-1255	43	19	of	of	ADP
american_scientific_journal-1255	43	20	different	different	ADJ
american_scientific_journal-1255	43	21	colors	color	NOUN
american_scientific_journal-1255	43	22	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	43	23	,	,	PUNCT
american_scientific_journal-1255	43	24	𝑏2	𝑏2	PROPN
american_scientific_journal-1255	43	25	.	.	PUNCT
american_scientific_journal-1255	44	1	then	then	ADV
american_scientific_journal-1255	44	2	a	a	DET
american_scientific_journal-1255	44	3	path	path	NOUN
american_scientific_journal-1255	44	4	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	44	5	connecting	connect	VERB
american_scientific_journal-1255	44	6	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	44	7	and	and	CCONJ
american_scientific_journal-1255	44	8	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	44	9	where	where	SCONJ
american_scientific_journal-1255	44	10	its	its	PRON
american_scientific_journal-1255	44	11	edge	edge	NOUN
american_scientific_journal-1255	44	12	is	be	AUX
american_scientific_journal-1255	44	13	incident	incident	NOUN
american_scientific_journal-1255	44	14	to	to	ADP
american_scientific_journal-1255	44	15	colors	color	NOUN
american_scientific_journal-1255	44	16	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	44	17	,	,	PUNCT
american_scientific_journal-1255	44	18	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	44	19	edges	edge	NOUN
american_scientific_journal-1255	44	20	of	of	ADP
american_scientific_journal-1255	44	21	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	44	22	and	and	CCONJ
american_scientific_journal-1255	44	23	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	44	24	,	,	PUNCT
american_scientific_journal-1255	44	25	must	must	AUX
american_scientific_journal-1255	44	26	have	have	VERB
american_scientific_journal-1255	44	27	an	an	DET
american_scientific_journal-1255	44	28	edge	edge	NOUN
american_scientific_journal-1255	44	29	colored	color	VERB
american_scientific_journal-1255	44	30	with	with	ADP
american_scientific_journal-1255	44	31	𝑏3	𝑏3	NOUN
american_scientific_journal-1255	44	32	such	such	ADJ
american_scientific_journal-1255	44	33	that	that	DET
american_scientific_journal-1255	44	34	𝑏1	𝑏1	ADJ
american_scientific_journal-1255	44	35	≠	≠	PROPN
american_scientific_journal-1255	44	36	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	44	37	≠	≠	PROPN
american_scientific_journal-1255	44	38	𝑏3	𝑏3	NOUN
american_scientific_journal-1255	44	39	.	.	PUNCT
american_scientific_journal-1255	45	1	hence	hence	ADV
american_scientific_journal-1255	45	2	,	,	PUNCT
american_scientific_journal-1255	45	3	three	three	NUM
american_scientific_journal-1255	45	4	is	be	AUX
american_scientific_journal-1255	45	5	the	the	DET
american_scientific_journal-1255	45	6	minimum	minimum	ADJ
american_scientific_journal-1255	45	7	number	number	NOUN
american_scientific_journal-1255	45	8	of	of	ADP
american_scientific_journal-1255	45	9	colors	color	NOUN
american_scientific_journal-1255	45	10	to	to	PART
american_scientific_journal-1255	45	11	color	color	VERB
american_scientific_journal-1255	45	12	the	the	DET
american_scientific_journal-1255	45	13	edges	edge	NOUN
american_scientific_journal-1255	45	14	of	of	ADP
american_scientific_journal-1255	45	15	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	45	16	,	,	PUNCT
american_scientific_journal-1255	45	17	where	where	SCONJ
american_scientific_journal-1255	45	18	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	45	19	are	be	AUX
american_scientific_journal-1255	45	20	both	both	PRON
american_scientific_journal-1255	45	21	even	even	ADV
american_scientific_journal-1255	45	22	.	.	PUNCT
american_scientific_journal-1255	46	1	consequently	consequently	ADV
american_scientific_journal-1255	46	2	,	,	PUNCT
american_scientific_journal-1255	46	3	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	46	4	�	�	PROPN
american_scientific_journal-1255	46	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	46	6	�	�	PROPN
american_scientific_journal-1255	46	7	=	=	SYM
american_scientific_journal-1255	46	8	3	3	NUM
american_scientific_journal-1255	46	9	.	.	NOUN
american_scientific_journal-1255	46	10	•	•	NUM
american_scientific_journal-1255	46	11	case	case	NOUN
american_scientific_journal-1255	46	12	3	3	NUM
american_scientific_journal-1255	46	13	:	:	PUNCT
american_scientific_journal-1255	46	14	𝑚	𝑚	PROPN
american_scientific_journal-1255	46	15	is	be	AUX
american_scientific_journal-1255	46	16	odd	odd	ADJ
american_scientific_journal-1255	46	17	,	,	PUNCT
american_scientific_journal-1255	46	18	𝑛	𝑛	PRON
american_scientific_journal-1255	46	19	is	be	AUX
american_scientific_journal-1255	46	20	even	even	ADV
american_scientific_journal-1255	46	21	if	if	SCONJ
american_scientific_journal-1255	46	22	𝑚	𝑚	PROPN
american_scientific_journal-1255	46	23	is	be	AUX
american_scientific_journal-1255	46	24	odd	odd	ADJ
american_scientific_journal-1255	46	25	,	,	PUNCT
american_scientific_journal-1255	46	26	𝑛	𝑛	PRON
american_scientific_journal-1255	46	27	is	be	AUX
american_scientific_journal-1255	46	28	even	even	ADV
american_scientific_journal-1255	46	29	,	,	PUNCT
american_scientific_journal-1255	46	30	then	then	ADV
american_scientific_journal-1255	46	31	by	by	ADP
american_scientific_journal-1255	46	32	equation	equation	NOUN
american_scientific_journal-1255	46	33	(	(	PUNCT
american_scientific_journal-1255	46	34	1.2	1.2	NUM
american_scientific_journal-1255	46	35	)	)	PUNCT
american_scientific_journal-1255	46	36	,	,	PUNCT
american_scientific_journal-1255	46	37	𝜒𝑒(𝐶𝑚	𝜒𝑒(𝐶𝑚	PROPN
american_scientific_journal-1255	46	38	)	)	PUNCT
american_scientific_journal-1255	46	39	=	=	SYM
american_scientific_journal-1255	46	40	3	3	NUM
american_scientific_journal-1255	46	41	and	and	CCONJ
american_scientific_journal-1255	46	42	𝜒𝑒(𝐶𝑛	𝜒𝑒(𝐶𝑛	PROPN
american_scientific_journal-1255	46	43	)	)	PUNCT
american_scientific_journal-1255	46	44	=	=	SYM
american_scientific_journal-1255	47	1	2	2	X
american_scientific_journal-1255	47	2	.	.	PUNCT
american_scientific_journal-1255	47	3	thus	thus	ADV
american_scientific_journal-1255	47	4	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	47	5	�	�	PROPN
american_scientific_journal-1255	47	6	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	47	7	�	�	PROPN
american_scientific_journal-1255	47	8	≠	≠	PROPN
american_scientific_journal-1255	47	9	2	2	NUM
american_scientific_journal-1255	47	10	.	.	PUNCT
american_scientific_journal-1255	47	11	without	without	ADP
american_scientific_journal-1255	47	12	a	a	DET
american_scientific_journal-1255	47	13	loss	loss	NOUN
american_scientific_journal-1255	47	14	of	of	ADP
american_scientific_journal-1255	47	15	generalization	generalization	NOUN
american_scientific_journal-1255	47	16	,	,	PUNCT
american_scientific_journal-1255	47	17	suppose	suppose	VERB
american_scientific_journal-1255	47	18	that	that	SCONJ
american_scientific_journal-1255	47	19	the	the	DET
american_scientific_journal-1255	47	20	edges	edge	NOUN
american_scientific_journal-1255	47	21	of	of	ADP
american_scientific_journal-1255	47	22	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	47	23	are	be	AUX
american_scientific_journal-1255	47	24	colored	color	VERB
american_scientific_journal-1255	47	25	with	with	ADP
american_scientific_journal-1255	47	26	two	two	NUM
american_scientific_journal-1255	47	27	of	of	ADP
american_scientific_journal-1255	47	28	the	the	DET
american_scientific_journal-1255	47	29	colors	color	NOUN
american_scientific_journal-1255	47	30	that	that	PRON
american_scientific_journal-1255	47	31	also	also	ADV
american_scientific_journal-1255	47	32	color	color	VERB
american_scientific_journal-1255	47	33	the	the	DET
american_scientific_journal-1255	47	34	edges	edge	NOUN
american_scientific_journal-1255	47	35	of	of	ADP
american_scientific_journal-1255	47	36	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	47	37	,	,	PUNCT
american_scientific_journal-1255	47	38	say	say	VERB
american_scientific_journal-1255	47	39	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	47	40	,	,	PUNCT
american_scientific_journal-1255	47	41	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	47	42	colors	color	NOUN
american_scientific_journal-1255	47	43	of	of	ADP
american_scientific_journal-1255	47	44	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	47	45	and	and	CCONJ
american_scientific_journal-1255	47	46	𝑏1	𝑏1	ADJ
american_scientific_journal-1255	47	47	,	,	PUNCT
american_scientific_journal-1255	47	48	𝑏2	𝑏2	PROPN
american_scientific_journal-1255	47	49	,	,	PUNCT
american_scientific_journal-1255	47	50	𝑏3	𝑏3	NOUN
american_scientific_journal-1255	47	51	colors	color	NOUN
american_scientific_journal-1255	47	52	of	of	ADP
american_scientific_journal-1255	47	53	𝐶𝑚.	𝐶𝑚.	NOUN
american_scientific_journal-1255	47	54	then	then	ADV
american_scientific_journal-1255	47	55	a	a	DET
american_scientific_journal-1255	47	56	path	path	NOUN
american_scientific_journal-1255	47	57	𝑃2	𝑃2	NOUN
american_scientific_journal-1255	47	58	connecting	connect	VERB
american_scientific_journal-1255	47	59	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	47	60	and	and	CCONJ
american_scientific_journal-1255	47	61	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	47	62	must	must	AUX
american_scientific_journal-1255	47	63	have	have	VERB
american_scientific_journal-1255	47	64	an	an	DET
american_scientific_journal-1255	47	65	edge	edge	NOUN
american_scientific_journal-1255	47	66	colored	color	VERB
american_scientific_journal-1255	47	67	with	with	ADP
american_scientific_journal-1255	47	68	𝑏3	𝑏3	NOUN
american_scientific_journal-1255	47	69	and	and	CCONJ
american_scientific_journal-1255	47	70	must	must	AUX
american_scientific_journal-1255	47	71	also	also	ADV
american_scientific_journal-1255	47	72	be	be	AUX
american_scientific_journal-1255	47	73	incident	incident	NOUN
american_scientific_journal-1255	47	74	to	to	AUX
american_scientific_journal-1255	47	75	edges	edge	VERB
american_scientific_journal-1255	47	76	colored	colored	ADJ
american_scientific_journal-1255	47	77	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	47	78	and	and	CCONJ
american_scientific_journal-1255	47	79	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	47	80	of	of	ADP
american_scientific_journal-1255	47	81	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	47	82	and	and	CCONJ
american_scientific_journal-1255	47	83	must	must	AUX
american_scientific_journal-1255	47	84	also	also	ADV
american_scientific_journal-1255	47	85	be	be	AUX
american_scientific_journal-1255	47	86	incident	incident	NOUN
american_scientific_journal-1255	47	87	to	to	AUX
american_scientific_journal-1255	47	88	edges	edge	VERB
american_scientific_journal-1255	47	89	colored	colored	ADJ
american_scientific_journal-1255	47	90	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	47	91	and	and	CCONJ
american_scientific_journal-1255	47	92	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	47	93	of	of	ADP
american_scientific_journal-1255	47	94	𝐶𝑛.	𝐶𝑛.	PROPN
american_scientific_journal-1255	47	95	hence	hence	ADV
american_scientific_journal-1255	47	96	,	,	PUNCT
american_scientific_journal-1255	47	97	three	three	NUM
american_scientific_journal-1255	47	98	is	be	AUX
american_scientific_journal-1255	47	99	the	the	DET
american_scientific_journal-1255	47	100	minimum	minimum	ADJ
american_scientific_journal-1255	47	101	number	number	NOUN
american_scientific_journal-1255	47	102	of	of	ADP
american_scientific_journal-1255	47	103	colors	color	NOUN
american_scientific_journal-1255	47	104	to	to	PART
american_scientific_journal-1255	47	105	color	color	VERB
american_scientific_journal-1255	47	106	the	the	DET
american_scientific_journal-1255	47	107	edges	edge	NOUN
american_scientific_journal-1255	47	108	of	of	ADP
american_scientific_journal-1255	47	109	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	47	110	,	,	PUNCT
american_scientific_journal-1255	47	111	where	where	SCONJ
american_scientific_journal-1255	47	112	𝑚	𝑚	PROPN
american_scientific_journal-1255	47	113	is	be	AUX
american_scientific_journal-1255	47	114	odd	odd	ADJ
american_scientific_journal-1255	47	115	,	,	PUNCT
american_scientific_journal-1255	47	116	𝑛	𝑛	PRON
american_scientific_journal-1255	47	117	is	be	AUX
american_scientific_journal-1255	47	118	even	even	ADV
american_scientific_journal-1255	47	119	.	.	PUNCT
american_scientific_journal-1255	48	1	consequently	consequently	ADV
american_scientific_journal-1255	48	2	,	,	PUNCT
american_scientific_journal-1255	48	3	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	48	4	�	�	PROPN
american_scientific_journal-1255	48	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	48	6	�	�	PROPN
american_scientific_journal-1255	48	7	=	=	SYM
american_scientific_journal-1255	48	8	3	3	NUM
american_scientific_journal-1255	48	9	.	.	NOUN
american_scientific_journal-1255	48	10	•	•	NUM
american_scientific_journal-1255	48	11	case	case	NOUN
american_scientific_journal-1255	48	12	4	4	NUM
american_scientific_journal-1255	48	13	:	:	PUNCT
american_scientific_journal-1255	48	14	𝑚	𝑚	NOUN
american_scientific_journal-1255	48	15	is	be	AUX
american_scientific_journal-1255	48	16	even	even	ADV
american_scientific_journal-1255	48	17	,	,	PUNCT
american_scientific_journal-1255	48	18	𝑛	𝑛	PRON
american_scientific_journal-1255	48	19	is	be	AUX
american_scientific_journal-1255	48	20	odd	odd	ADJ
american_scientific_journal-1255	48	21	if	if	SCONJ
american_scientific_journal-1255	48	22	𝑚	𝑚	PROPN
american_scientific_journal-1255	48	23	is	be	AUX
american_scientific_journal-1255	48	24	even	even	ADV
american_scientific_journal-1255	48	25	,	,	PUNCT
american_scientific_journal-1255	48	26	𝑛	𝑛	PRON
american_scientific_journal-1255	48	27	is	be	AUX
american_scientific_journal-1255	48	28	odd	odd	ADJ
american_scientific_journal-1255	48	29	,	,	PUNCT
american_scientific_journal-1255	48	30	then	then	ADV
american_scientific_journal-1255	48	31	the	the	DET
american_scientific_journal-1255	48	32	proof	proof	NOUN
american_scientific_journal-1255	48	33	of	of	ADP
american_scientific_journal-1255	48	34	this	this	DET
american_scientific_journal-1255	48	35	case	case	NOUN
american_scientific_journal-1255	48	36	is	be	AUX
american_scientific_journal-1255	48	37	similar	similar	ADJ
american_scientific_journal-1255	48	38	to	to	PART
american_scientific_journal-1255	48	39	case	case	NOUN
american_scientific_journal-1255	48	40	3	3	NUM
american_scientific_journal-1255	48	41	.	.	PUNCT
american_scientific_journal-1255	49	1	∎	∎	PROPN
american_scientific_journal-1255	49	2	corollary	corollary	ADJ
american_scientific_journal-1255	49	3	2.1	2.1	NUM
american_scientific_journal-1255	49	4	101	101	NUM
american_scientific_journal-1255	49	5	american	american	ADJ
american_scientific_journal-1255	49	6	scientific	scientific	ADJ
american_scientific_journal-1255	49	7	research	research	NOUN
american_scientific_journal-1255	49	8	journal	journal	NOUN
american_scientific_journal-1255	49	9	for	for	ADP
american_scientific_journal-1255	49	10	engineering	engineering	NOUN
american_scientific_journal-1255	49	11	,	,	PUNCT
american_scientific_journal-1255	49	12	technology	technology	NOUN
american_scientific_journal-1255	49	13	,	,	PUNCT
american_scientific_journal-1255	49	14	and	and	CCONJ
american_scientific_journal-1255	49	15	sciences	science	NOUN
american_scientific_journal-1255	49	16	(	(	PUNCT
american_scientific_journal-1255	49	17	asrjets	asrjet	NOUN
american_scientific_journal-1255	49	18	)	)	PUNCT
american_scientific_journal-1255	49	19	(	(	PUNCT
american_scientific_journal-1255	49	20	2016	2016	NUM
american_scientific_journal-1255	49	21	)	)	PUNCT
american_scientific_journal-1255	49	22	volume	volume	NOUN
american_scientific_journal-1255	49	23	16	16	NUM
american_scientific_journal-1255	49	24	,	,	PUNCT
american_scientific_journal-1255	49	25	no	no	DET
american_scientific_journal-1255	49	26	1	1	NUM
american_scientific_journal-1255	49	27	,	,	PUNCT
american_scientific_journal-1255	49	28	pp	pp	ADV
american_scientific_journal-1255	49	29	98	98	NUM
american_scientific_journal-1255	49	30	-	-	SYM
american_scientific_journal-1255	49	31	107	107	NUM
american_scientific_journal-1255	49	32	for	for	ADP
american_scientific_journal-1255	49	33	all	all	DET
american_scientific_journal-1255	49	34	integers	integer	NOUN
american_scientific_journal-1255	49	35	𝑚,𝑛	𝑚,𝑛	VERB
american_scientific_journal-1255	49	36	≥	≥	NUM
american_scientific_journal-1255	49	37	3	3	NUM
american_scientific_journal-1255	49	38	𝜒	𝜒	NUM
american_scientific_journal-1255	49	39	�	�	PROPN
american_scientific_journal-1255	49	40	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	49	41	�	�	PROPN
american_scientific_journal-1255	49	42	≤	≤	PROPN
american_scientific_journal-1255	49	43	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	49	44	�	�	PROPN
american_scientific_journal-1255	49	45	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	49	46	�	�	PROPN
american_scientific_journal-1255	49	47	proof	proof	NOUN
american_scientific_journal-1255	49	48	:	:	PUNCT
american_scientific_journal-1255	49	49	note	note	VERB
american_scientific_journal-1255	49	50	that	that	SCONJ
american_scientific_journal-1255	49	51	for	for	ADP
american_scientific_journal-1255	49	52	the	the	DET
american_scientific_journal-1255	49	53	cases	case	NOUN
american_scientific_journal-1255	49	54	where	where	SCONJ
american_scientific_journal-1255	49	55	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	49	56	are	be	AUX
american_scientific_journal-1255	49	57	both	both	PRON
american_scientific_journal-1255	49	58	odd	odd	ADJ
american_scientific_journal-1255	49	59	,	,	PUNCT
american_scientific_journal-1255	49	60	𝑚	𝑚	PROPN
american_scientific_journal-1255	49	61	is	be	AUX
american_scientific_journal-1255	49	62	odd	odd	ADJ
american_scientific_journal-1255	49	63	,	,	PUNCT
american_scientific_journal-1255	49	64	𝑛	𝑛	PRON
american_scientific_journal-1255	49	65	is	be	AUX
american_scientific_journal-1255	49	66	even	even	ADV
american_scientific_journal-1255	49	67	,	,	PUNCT
american_scientific_journal-1255	49	68	and	and	CCONJ
american_scientific_journal-1255	49	69	𝑚	𝑚	PROPN
american_scientific_journal-1255	49	70	is	be	AUX
american_scientific_journal-1255	49	71	even	even	ADV
american_scientific_journal-1255	49	72	,	,	PUNCT
american_scientific_journal-1255	49	73	𝑛	𝑛	PRON
american_scientific_journal-1255	49	74	is	be	AUX
american_scientific_journal-1255	49	75	odd	odd	ADJ
american_scientific_journal-1255	49	76	,	,	PUNCT
american_scientific_journal-1255	49	77	by	by	ADP
american_scientific_journal-1255	49	78	theorem	theorem	NOUN
american_scientific_journal-1255	49	79	2.1	2.1	NUM
american_scientific_journal-1255	49	80	,	,	PUNCT
american_scientific_journal-1255	49	81	𝜒	𝜒	PROPN
american_scientific_journal-1255	49	82	�	�	PROPN
american_scientific_journal-1255	49	83	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	49	84	�	�	PROPN
american_scientific_journal-1255	49	85	=	=	SYM
american_scientific_journal-1255	49	86	3	3	NUM
american_scientific_journal-1255	49	87	and	and	CCONJ
american_scientific_journal-1255	49	88	by	by	ADP
american_scientific_journal-1255	49	89	theorem	theorem	ADJ
american_scientific_journal-1255	49	90	2.2	2.2	NUM
american_scientific_journal-1255	49	91	,	,	PUNCT
american_scientific_journal-1255	49	92	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	49	93	�	�	PROPN
american_scientific_journal-1255	49	94	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	49	95	�	�	PROPN
american_scientific_journal-1255	49	96	=	=	SYM
american_scientific_journal-1255	49	97	3	3	NUM
american_scientific_journal-1255	49	98	.	.	PUNCT
american_scientific_journal-1255	50	1	thus	thus	ADV
american_scientific_journal-1255	50	2	,	,	PUNCT
american_scientific_journal-1255	50	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	50	4	�	�	PROPN
american_scientific_journal-1255	50	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	50	6	�	�	PROPN
american_scientific_journal-1255	50	7	≤	≤	PROPN
american_scientific_journal-1255	50	8	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	50	9	�	�	PROPN
american_scientific_journal-1255	50	10	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	50	11	�	�	PROPN
american_scientific_journal-1255	50	12	.	.	PUNCT
american_scientific_journal-1255	51	1	similarly	similarly	ADV
american_scientific_journal-1255	51	2	,	,	PUNCT
american_scientific_journal-1255	51	3	for	for	ADP
american_scientific_journal-1255	51	4	cases	case	NOUN
american_scientific_journal-1255	51	5	where	where	SCONJ
american_scientific_journal-1255	51	6	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	51	7	are	be	AUX
american_scientific_journal-1255	51	8	both	both	PRON
american_scientific_journal-1255	51	9	even	even	ADV
american_scientific_journal-1255	51	10	,	,	PUNCT
american_scientific_journal-1255	51	11	by	by	ADP
american_scientific_journal-1255	51	12	theorem	theorem	NOUN
american_scientific_journal-1255	51	13	2.1	2.1	NUM
american_scientific_journal-1255	51	14	,	,	PUNCT
american_scientific_journal-1255	51	15	𝜒	𝜒	PROPN
american_scientific_journal-1255	51	16	�	�	PROPN
american_scientific_journal-1255	51	17	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	51	18	�	�	PROPN
american_scientific_journal-1255	51	19	=	=	SYM
american_scientific_journal-1255	51	20	2	2	NUM
american_scientific_journal-1255	51	21	and	and	CCONJ
american_scientific_journal-1255	51	22	by	by	ADP
american_scientific_journal-1255	51	23	theorem	theorem	ADJ
american_scientific_journal-1255	51	24	2.2	2.2	NUM
american_scientific_journal-1255	51	25	,	,	PUNCT
american_scientific_journal-1255	51	26	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	51	27	�	�	PROPN
american_scientific_journal-1255	51	28	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	51	29	�	�	PROPN
american_scientific_journal-1255	51	30	=	=	SYM
american_scientific_journal-1255	51	31	3	3	NUM
american_scientific_journal-1255	51	32	.	.	PUNCT
american_scientific_journal-1255	52	1	thus	thus	ADV
american_scientific_journal-1255	52	2	,	,	PUNCT
american_scientific_journal-1255	52	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	52	4	�	�	PROPN
american_scientific_journal-1255	52	5	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	52	6	�	�	PROPN
american_scientific_journal-1255	52	7	≤	≤	PROPN
american_scientific_journal-1255	52	8	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	52	9	�	�	PROPN
american_scientific_journal-1255	52	10	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	52	11	�	�	PROPN
american_scientific_journal-1255	52	12	.	.	PUNCT
american_scientific_journal-1255	53	1	therefore	therefore	ADV
american_scientific_journal-1255	53	2	,	,	PUNCT
american_scientific_journal-1255	53	3	in	in	ADP
american_scientific_journal-1255	53	4	all	all	DET
american_scientific_journal-1255	53	5	cases	case	NOUN
american_scientific_journal-1255	53	6	,	,	PUNCT
american_scientific_journal-1255	53	7	𝜒	𝜒	PROPN
american_scientific_journal-1255	53	8	�	�	SYM
american_scientific_journal-1255	53	9	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	53	10	�	�	PROPN
american_scientific_journal-1255	53	11	≤	≤	PROPN
american_scientific_journal-1255	53	12	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	53	13	�	�	PROPN
american_scientific_journal-1255	53	14	𝑂𝑛,𝑚	𝑂𝑛,𝑚	PROPN
american_scientific_journal-1255	53	15	�	�	PROPN
american_scientific_journal-1255	53	16	.	.	PUNCT
american_scientific_journal-1255	54	1	∎	∎	PROPN
american_scientific_journal-1255	54	2	remark	remark	VERB
american_scientific_journal-1255	54	3	2.1	2.1	NUM
american_scientific_journal-1255	54	4	for	for	ADP
american_scientific_journal-1255	54	5	all	all	DET
american_scientific_journal-1255	54	6	integers	integer	NOUN
american_scientific_journal-1255	54	7	𝑘	𝑘	DET
american_scientific_journal-1255	54	8	≥	≥	NOUN
american_scientific_journal-1255	54	9	3	3	NUM
american_scientific_journal-1255	54	10	,	,	PUNCT
american_scientific_journal-1255	54	11	𝜒(𝑃𝑘	𝜒(𝑃𝑘	NOUN
american_scientific_journal-1255	54	12	)	)	PUNCT
american_scientific_journal-1255	54	13	=	=	SYM
american_scientific_journal-1255	54	14	𝜒𝑒(𝑃𝑘	𝜒𝑒(𝑃𝑘	NOUN
american_scientific_journal-1255	54	15	)	)	PUNCT
american_scientific_journal-1255	54	16	=	=	SYM
american_scientific_journal-1255	54	17	2	2	NUM
american_scientific_journal-1255	54	18	.	.	NOUN
american_scientific_journal-1255	54	19	3	3	NUM
american_scientific_journal-1255	54	20	.	.	NUM
american_scientific_journal-1255	54	21	generalized	generalize	VERB
american_scientific_journal-1255	54	22	ottomar	ottomar	NOUN
american_scientific_journal-1255	54	23	graph	graph	NOUN
american_scientific_journal-1255	54	24	define	define	VERB
american_scientific_journal-1255	54	25	the	the	DET
american_scientific_journal-1255	54	26	generalized	generalized	ADJ
american_scientific_journal-1255	54	27	ottomar	ottomar	NOUN
american_scientific_journal-1255	54	28	graph	graph	NOUN
american_scientific_journal-1255	54	29	,	,	PUNCT
american_scientific_journal-1255	54	30	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	54	31	,	,	PUNCT
american_scientific_journal-1255	54	32	is	be	AUX
american_scientific_journal-1255	54	33	graph	graph	NOUN
american_scientific_journal-1255	54	34	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	54	35	,	,	PUNCT
american_scientific_journal-1255	54	36	𝑛	𝑛	PRON
american_scientific_journal-1255	54	37	∈	∈	PROPN
american_scientific_journal-1255	54	38	ℤ+,𝑛	ℤ+,𝑛	NOUN
american_scientific_journal-1255	54	39	≥	≥	NOUN
american_scientific_journal-1255	54	40	3	3	NUM
american_scientific_journal-1255	54	41	,	,	PUNCT
american_scientific_journal-1255	54	42	with	with	ADP
american_scientific_journal-1255	54	43	each	each	DET
american_scientific_journal-1255	54	44	vertex	vertex	NOUN
american_scientific_journal-1255	54	45	connected	connect	VERB
american_scientific_journal-1255	54	46	by	by	ADP
american_scientific_journal-1255	54	47	a	a	DET
american_scientific_journal-1255	54	48	path	path	NOUN
american_scientific_journal-1255	54	49	𝑃𝑘	𝑃𝑘	PROPN
american_scientific_journal-1255	54	50	,	,	PUNCT
american_scientific_journal-1255	54	51	𝑘	𝑘	PROPN
american_scientific_journal-1255	54	52	∈	∈	PROPN
american_scientific_journal-1255	54	53	ℤ+	ℤ+	PROPN
american_scientific_journal-1255	54	54	,	,	PUNCT
american_scientific_journal-1255	54	55	𝑘	𝑘	DET
american_scientific_journal-1255	54	56	≥	≥	NOUN
american_scientific_journal-1255	54	57	3	3	NUM
american_scientific_journal-1255	54	58	to	to	ADP
american_scientific_journal-1255	54	59	a	a	DET
american_scientific_journal-1255	54	60	vertex	vertex	NOUN
american_scientific_journal-1255	54	61	of	of	ADP
american_scientific_journal-1255	54	62	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	54	63	,	,	PUNCT
american_scientific_journal-1255	54	64	𝑚	𝑚	PROPN
american_scientific_journal-1255	54	65	∈	∈	PROPN
american_scientific_journal-1255	54	66	ℤ+,𝑚	ℤ+,𝑚	X
american_scientific_journal-1255	54	67	≥	≥	NOUN
american_scientific_journal-1255	54	68	3	3	NUM
american_scientific_journal-1255	54	69	.	.	PUNCT
american_scientific_journal-1255	55	1	𝐶𝑛	𝐶𝑛	NOUN
american_scientific_journal-1255	55	2	is	be	AUX
american_scientific_journal-1255	55	3	called	call	VERB
american_scientific_journal-1255	55	4	a	a	DET
american_scientific_journal-1255	55	5	heart	heart	NOUN
american_scientific_journal-1255	55	6	while	while	SCONJ
american_scientific_journal-1255	55	7	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	55	8	is	be	AUX
american_scientific_journal-1255	55	9	called	call	VERB
american_scientific_journal-1255	55	10	a	a	DET
american_scientific_journal-1255	55	11	foot	foot	NOUN
american_scientific_journal-1255	55	12	(	(	PUNCT
american_scientific_journal-1255	55	13	feet	foot	NOUN
american_scientific_journal-1255	55	14	for	for	ADP
american_scientific_journal-1255	55	15	plural	plural	ADJ
american_scientific_journal-1255	55	16	)	)	PUNCT
american_scientific_journal-1255	55	17	.	.	PUNCT
american_scientific_journal-1255	56	1	note	note	VERB
american_scientific_journal-1255	56	2	that	that	SCONJ
american_scientific_journal-1255	56	3	there	there	PRON
american_scientific_journal-1255	56	4	are	be	VERB
american_scientific_journal-1255	56	5	𝑛	𝑛	DET
american_scientific_journal-1255	56	6	copies	copy	NOUN
american_scientific_journal-1255	56	7	of	of	ADP
american_scientific_journal-1255	56	8	𝐶𝑚.	𝐶𝑚.	NOUN
american_scientific_journal-1255	56	9	theorem	theorem	VERB
american_scientific_journal-1255	56	10	3.1	3.1	NUM
american_scientific_journal-1255	56	11	for	for	ADP
american_scientific_journal-1255	56	12	all	all	DET
american_scientific_journal-1255	56	13	integers	integer	NOUN
american_scientific_journal-1255	56	14	𝑘	𝑘	X
american_scientific_journal-1255	56	15	=	=	SYM
american_scientific_journal-1255	56	16	3	3	NUM
american_scientific_journal-1255	56	17	,	,	PUNCT
american_scientific_journal-1255	56	18	4	4	NUM
american_scientific_journal-1255	56	19	,	,	PUNCT
american_scientific_journal-1255	56	20	𝜒	𝜒	NOUN
american_scientific_journal-1255	56	21	�	�	PROPN
american_scientific_journal-1255	56	22	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	56	23	�	�	NOUN
american_scientific_journal-1255	56	24	=	=	SYM
american_scientific_journal-1255	56	25	3	3	NUM
american_scientific_journal-1255	56	26	if	if	SCONJ
american_scientific_journal-1255	56	27	,	,	PUNCT
american_scientific_journal-1255	56	28	•	•	NOUN
american_scientific_journal-1255	56	29	𝑚,𝑛	𝑚,𝑛	NOUN
american_scientific_journal-1255	56	30	are	be	AUX
american_scientific_journal-1255	56	31	both	both	PRON
american_scientific_journal-1255	56	32	odd	odd	ADJ
american_scientific_journal-1255	56	33	•	•	NOUN
american_scientific_journal-1255	56	34	𝑚	𝑚	NOUN
american_scientific_journal-1255	56	35	is	be	AUX
american_scientific_journal-1255	56	36	odd	odd	ADJ
american_scientific_journal-1255	56	37	,	,	PUNCT
american_scientific_journal-1255	56	38	𝑛	𝑛	PRON
american_scientific_journal-1255	56	39	is	be	AUX
american_scientific_journal-1255	56	40	even	even	ADV
american_scientific_journal-1255	56	41	•	•	NUM
american_scientific_journal-1255	56	42	𝑚	𝑚	NOUN
american_scientific_journal-1255	56	43	is	be	AUX
american_scientific_journal-1255	56	44	even	even	ADV
american_scientific_journal-1255	56	45	,	,	PUNCT
american_scientific_journal-1255	56	46	𝑛	𝑛	PRON
american_scientific_journal-1255	56	47	is	be	AUX
american_scientific_journal-1255	56	48	odd	odd	ADJ
american_scientific_journal-1255	56	49	and	and	CCONJ
american_scientific_journal-1255	56	50	𝜒	𝜒	NOUN
american_scientific_journal-1255	56	51	�	�	PROPN
american_scientific_journal-1255	56	52	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	56	53	�	�	NOUN
american_scientific_journal-1255	56	54	=	=	SYM
american_scientific_journal-1255	56	55	2	2	NUM
american_scientific_journal-1255	56	56	,	,	PUNCT
american_scientific_journal-1255	56	57	𝑖𝑓	𝑖𝑓	ADP
american_scientific_journal-1255	56	58	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	56	59	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
american_scientific_journal-1255	56	60	𝑏𝑜𝑡ℎ	𝑏𝑜𝑡ℎ	PROPN
american_scientific_journal-1255	56	61	𝑒𝑣𝑒𝑛.	𝑒𝑣𝑒𝑛.	PROPN
american_scientific_journal-1255	56	62	proof	proof	NOUN
american_scientific_journal-1255	56	63	:	:	PUNCT
american_scientific_journal-1255	56	64	•	•	NUM
american_scientific_journal-1255	56	65	case	case	NOUN
american_scientific_journal-1255	56	66	1	1	NUM
american_scientific_journal-1255	56	67	:	:	PUNCT
american_scientific_journal-1255	56	68	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	56	69	are	be	AUX
american_scientific_journal-1255	56	70	both	both	PRON
american_scientific_journal-1255	56	71	odd	odd	ADJ
american_scientific_journal-1255	56	72	if	if	SCONJ
american_scientific_journal-1255	56	73	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	56	74	are	be	AUX
american_scientific_journal-1255	56	75	both	both	PRON
american_scientific_journal-1255	56	76	odd	odd	ADJ
american_scientific_journal-1255	56	77	,	,	PUNCT
american_scientific_journal-1255	56	78	then	then	ADV
american_scientific_journal-1255	56	79	by	by	ADP
american_scientific_journal-1255	56	80	equation	equation	NOUN
american_scientific_journal-1255	56	81	(	(	PUNCT
american_scientific_journal-1255	56	82	1.1	1.1	NUM
american_scientific_journal-1255	56	83	)	)	PUNCT
american_scientific_journal-1255	56	84	,	,	PUNCT
american_scientific_journal-1255	56	85	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NOUN
american_scientific_journal-1255	56	86	)	)	PUNCT
american_scientific_journal-1255	56	87	=	=	SYM
american_scientific_journal-1255	57	1	𝜒(𝐶𝑛	𝜒(𝐶𝑛	X
american_scientific_journal-1255	57	2	)	)	PUNCT
american_scientific_journal-1255	57	3	=	=	SYM
american_scientific_journal-1255	58	1	3	3	X
american_scientific_journal-1255	58	2	.	.	PUNCT
american_scientific_journal-1255	58	3	thus	thus	ADV
american_scientific_journal-1255	58	4	,	,	PUNCT
american_scientific_journal-1255	58	5	𝜒	𝜒	PROPN
american_scientific_journal-1255	58	6	�	�	PROPN
american_scientific_journal-1255	58	7	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	58	8	�	�	PROPN
american_scientific_journal-1255	58	9	≥	≥	NUM
american_scientific_journal-1255	58	10	3	3	NUM
american_scientific_journal-1255	58	11	.	.	PUNCT
american_scientific_journal-1255	58	12	suppose	suppose	VERB
american_scientific_journal-1255	58	13	the	the	DET
american_scientific_journal-1255	58	14	vertices	vertex	NOUN
american_scientific_journal-1255	58	15	of	of	ADP
american_scientific_journal-1255	58	16	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	58	17	and	and	CCONJ
american_scientific_journal-1255	58	18	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	58	19	are	be	AUX
american_scientific_journal-1255	58	20	colored	color	VERB
american_scientific_journal-1255	58	21	with	with	ADP
american_scientific_journal-1255	58	22	the	the	DET
american_scientific_journal-1255	58	23	same	same	ADJ
american_scientific_journal-1255	58	24	set	set	NOUN
american_scientific_journal-1255	58	25	of	of	ADP
american_scientific_journal-1255	58	26	different	different	ADJ
american_scientific_journal-1255	58	27	colors	color	NOUN
american_scientific_journal-1255	58	28	say	say	VERB
american_scientific_journal-1255	58	29	𝑎1	𝑎1	ADV
american_scientific_journal-1255	58	30	,	,	PUNCT
american_scientific_journal-1255	58	31	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	58	32	,	,	PUNCT
american_scientific_journal-1255	58	33	𝑎3	𝑎3	PROPN
american_scientific_journal-1255	58	34	.	.	PUNCT
american_scientific_journal-1255	58	35	consider	consider	VERB
american_scientific_journal-1255	58	36	the	the	DET
american_scientific_journal-1255	58	37	following	follow	VERB
american_scientific_journal-1255	58	38	subcases	subcase	NOUN
american_scientific_journal-1255	58	39	where	where	SCONJ
american_scientific_journal-1255	58	40	𝑘	𝑘	PROPN
american_scientific_journal-1255	58	41	=	=	SYM
american_scientific_journal-1255	58	42	3	3	NUM
american_scientific_journal-1255	58	43	(	(	PUNCT
american_scientific_journal-1255	58	44	odd	odd	ADJ
american_scientific_journal-1255	58	45	)	)	PUNCT
american_scientific_journal-1255	58	46	and	and	CCONJ
american_scientific_journal-1255	58	47	𝑘	𝑘	X
american_scientific_journal-1255	58	48	=	=	SYM
american_scientific_journal-1255	58	49	4	4	NUM
american_scientific_journal-1255	58	50	(	(	PUNCT
american_scientific_journal-1255	58	51	even	even	ADV
american_scientific_journal-1255	58	52	):	):	PUNCT
american_scientific_journal-1255	58	53	•	•	NUM
american_scientific_journal-1255	58	54	subcase	subcase	NOUN
american_scientific_journal-1255	58	55	1.1	1.1	NUM
american_scientific_journal-1255	58	56	:	:	PUNCT
american_scientific_journal-1255	58	57	if	if	SCONJ
american_scientific_journal-1255	58	58	𝑘	𝑘	X
american_scientific_journal-1255	58	59	=	=	SYM
american_scientific_journal-1255	58	60	3	3	NUM
american_scientific_journal-1255	58	61	then	then	ADV
american_scientific_journal-1255	58	62	a	a	DET
american_scientific_journal-1255	58	63	path	path	NOUN
american_scientific_journal-1255	58	64	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	58	65	with	with	ADP
american_scientific_journal-1255	58	66	vertices	vertex	NOUN
american_scientific_journal-1255	58	67	colored	color	VERB
american_scientific_journal-1255	58	68	with	with	ADP
american_scientific_journal-1255	58	69	two	two	NUM
american_scientific_journal-1255	58	70	from	from	ADP
american_scientific_journal-1255	58	71	the	the	DET
american_scientific_journal-1255	58	72	same	same	ADJ
american_scientific_journal-1255	58	73	set	set	NOUN
american_scientific_journal-1255	58	74	of	of	ADP
american_scientific_journal-1255	58	75	different	different	ADJ
american_scientific_journal-1255	58	76	colors	color	NOUN
american_scientific_journal-1255	58	77	𝑎1	𝑎1	PROPN
american_scientific_journal-1255	58	78	,	,	PUNCT
american_scientific_journal-1255	58	79	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	58	80	,	,	PUNCT
american_scientific_journal-1255	58	81	𝑎3	𝑎3	PROPN
american_scientific_journal-1255	58	82	,	,	PUNCT
american_scientific_journal-1255	58	83	is	be	AUX
american_scientific_journal-1255	58	84	attached	attach	VERB
american_scientific_journal-1255	58	85	to	to	ADP
american_scientific_journal-1255	58	86	a	a	DET
american_scientific_journal-1255	58	87	102	102	NUM
american_scientific_journal-1255	58	88	american	american	ADJ
american_scientific_journal-1255	58	89	scientific	scientific	ADJ
american_scientific_journal-1255	58	90	research	research	NOUN
american_scientific_journal-1255	58	91	journal	journal	NOUN
american_scientific_journal-1255	58	92	for	for	ADP
american_scientific_journal-1255	58	93	engineering	engineering	NOUN
american_scientific_journal-1255	58	94	,	,	PUNCT
american_scientific_journal-1255	58	95	technology	technology	NOUN
american_scientific_journal-1255	58	96	,	,	PUNCT
american_scientific_journal-1255	58	97	and	and	CCONJ
american_scientific_journal-1255	58	98	sciences	science	NOUN
american_scientific_journal-1255	58	99	(	(	PUNCT
american_scientific_journal-1255	58	100	asrjets	asrjet	NOUN
american_scientific_journal-1255	58	101	)	)	PUNCT
american_scientific_journal-1255	58	102	(	(	PUNCT
american_scientific_journal-1255	58	103	2016	2016	NUM
american_scientific_journal-1255	58	104	)	)	PUNCT
american_scientific_journal-1255	58	105	volume	volume	NOUN
american_scientific_journal-1255	58	106	16	16	NUM
american_scientific_journal-1255	58	107	,	,	PUNCT
american_scientific_journal-1255	58	108	no	no	DET
american_scientific_journal-1255	58	109	1	1	NUM
american_scientific_journal-1255	58	110	,	,	PUNCT
american_scientific_journal-1255	58	111	pp	pp	ADV
american_scientific_journal-1255	58	112	98	98	NUM
american_scientific_journal-1255	58	113	-	-	SYM
american_scientific_journal-1255	58	114	107	107	NUM
american_scientific_journal-1255	58	115	vertex	vertex	NOUN
american_scientific_journal-1255	58	116	colored	color	VERB
american_scientific_journal-1255	58	117	say	say	VERB
american_scientific_journal-1255	58	118	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	58	119	,	,	PUNCT
american_scientific_journal-1255	58	120	𝑖	𝑖	PROPN
american_scientific_journal-1255	59	1	=	=	NOUN
american_scientific_journal-1255	59	2	1,2,3	1,2,3	NUM
american_scientific_journal-1255	59	3	of	of	ADP
american_scientific_journal-1255	59	4	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	59	5	and	and	CCONJ
american_scientific_journal-1255	59	6	the	the	DET
american_scientific_journal-1255	59	7	other	other	ADJ
american_scientific_journal-1255	59	8	end	end	NOUN
american_scientific_journal-1255	59	9	vertex	vertex	NOUN
american_scientific_journal-1255	59	10	is	be	AUX
american_scientific_journal-1255	59	11	also	also	ADV
american_scientific_journal-1255	59	12	connected	connect	VERB
american_scientific_journal-1255	59	13	to	to	ADP
american_scientific_journal-1255	59	14	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	59	15	of	of	ADP
american_scientific_journal-1255	59	16	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	59	17	,	,	PUNCT
american_scientific_journal-1255	59	18	such	such	ADJ
american_scientific_journal-1255	59	19	that	that	SCONJ
american_scientific_journal-1255	59	20	the	the	DET
american_scientific_journal-1255	59	21	second	second	ADJ
american_scientific_journal-1255	59	22	(	(	PUNCT
american_scientific_journal-1255	59	23	middle	middle	ADJ
american_scientific_journal-1255	59	24	)	)	PUNCT
american_scientific_journal-1255	59	25	vertex	vertex	NOUN
american_scientific_journal-1255	59	26	of	of	ADP
american_scientific_journal-1255	59	27	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	59	28	is	be	AUX
american_scientific_journal-1255	59	29	𝑎𝑗	𝑎𝑗	PROPN
american_scientific_journal-1255	59	30	,	,	PUNCT
american_scientific_journal-1255	59	31	𝑗	𝑗	PROPN
american_scientific_journal-1255	59	32	=	=	NOUN
american_scientific_journal-1255	59	33	1,2,3	1,2,3	NUM
american_scientific_journal-1255	59	34	,	,	PUNCT
american_scientific_journal-1255	59	35	where	where	SCONJ
american_scientific_journal-1255	59	36	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	59	37	≠	≠	PROPN
american_scientific_journal-1255	59	38	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	59	39	thus	thus	ADV
american_scientific_journal-1255	59	40	three	three	NUM
american_scientific_journal-1255	59	41	is	be	AUX
american_scientific_journal-1255	59	42	the	the	DET
american_scientific_journal-1255	59	43	minimum	minimum	ADJ
american_scientific_journal-1255	59	44	number	number	NOUN
american_scientific_journal-1255	59	45	of	of	ADP
american_scientific_journal-1255	59	46	colors	color	NOUN
american_scientific_journal-1255	59	47	to	to	PART
american_scientific_journal-1255	59	48	color	color	VERB
american_scientific_journal-1255	59	49	the	the	DET
american_scientific_journal-1255	59	50	vertices	vertex	NOUN
american_scientific_journal-1255	59	51	of	of	ADP
american_scientific_journal-1255	59	52	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	59	53	,	,	PUNCT
american_scientific_journal-1255	59	54	where	where	SCONJ
american_scientific_journal-1255	59	55	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	59	56	are	be	AUX
american_scientific_journal-1255	59	57	both	both	PRON
american_scientific_journal-1255	59	58	odd	odd	ADJ
american_scientific_journal-1255	59	59	.	.	PUNCT
american_scientific_journal-1255	60	1	consequently	consequently	ADV
american_scientific_journal-1255	60	2	,	,	PUNCT
american_scientific_journal-1255	60	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	60	4	�	�	NOUN
american_scientific_journal-1255	60	5	𝑂3	𝑂3	VERB
american_scientific_journal-1255	60	6	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	60	7	�	�	X
american_scientific_journal-1255	60	8	=	=	SYM
american_scientific_journal-1255	60	9	3	3	NUM
american_scientific_journal-1255	60	10	.	.	NOUN
american_scientific_journal-1255	60	11	•	•	NUM
american_scientific_journal-1255	60	12	subcase	subcase	NOUN
american_scientific_journal-1255	60	13	1.2	1.2	NUM
american_scientific_journal-1255	60	14	:	:	PUNCT
american_scientific_journal-1255	60	15	if	if	SCONJ
american_scientific_journal-1255	60	16	𝑘	𝑘	X
american_scientific_journal-1255	60	17	=	=	SYM
american_scientific_journal-1255	60	18	4	4	NUM
american_scientific_journal-1255	60	19	note	note	VERB
american_scientific_journal-1255	60	20	that	that	SCONJ
american_scientific_journal-1255	60	21	by	by	ADP
american_scientific_journal-1255	60	22	remark	remark	NOUN
american_scientific_journal-1255	60	23	2.1	2.1	NUM
american_scientific_journal-1255	60	24	,	,	PUNCT
american_scientific_journal-1255	60	25	𝜒(𝑃4	𝜒(𝑃4	PROPN
american_scientific_journal-1255	60	26	)	)	PUNCT
american_scientific_journal-1255	60	27	=	=	SYM
american_scientific_journal-1255	60	28	2	2	X
american_scientific_journal-1255	60	29	.	.	PUNCT
american_scientific_journal-1255	60	30	suppose	suppose	VERB
american_scientific_journal-1255	60	31	further	far	ADV
american_scientific_journal-1255	60	32	that	that	DET
american_scientific_journal-1255	60	33	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	60	34	is	be	AUX
american_scientific_journal-1255	60	35	colored	color	VERB
american_scientific_journal-1255	60	36	with	with	ADP
american_scientific_journal-1255	60	37	two	two	NUM
american_scientific_journal-1255	60	38	from	from	ADP
american_scientific_journal-1255	60	39	the	the	DET
american_scientific_journal-1255	60	40	same	same	ADJ
american_scientific_journal-1255	60	41	set	set	NOUN
american_scientific_journal-1255	60	42	of	of	ADP
american_scientific_journal-1255	60	43	different	different	ADJ
american_scientific_journal-1255	60	44	colors	color	NOUN
american_scientific_journal-1255	60	45	that	that	PRON
american_scientific_journal-1255	60	46	color	color	VERB
american_scientific_journal-1255	60	47	the	the	DET
american_scientific_journal-1255	60	48	vertices	vertex	NOUN
american_scientific_journal-1255	60	49	of	of	ADP
american_scientific_journal-1255	60	50	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	60	51	and	and	CCONJ
american_scientific_journal-1255	60	52	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	60	53	,	,	PUNCT
american_scientific_journal-1255	60	54	say	say	VERB
american_scientific_journal-1255	60	55	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	60	56	,	,	PUNCT
american_scientific_journal-1255	60	57	𝑎𝑗	𝑎𝑗	INTJ
american_scientific_journal-1255	60	58	,	,	PUNCT
american_scientific_journal-1255	60	59	𝑖	𝑖	PROPN
american_scientific_journal-1255	60	60	,	,	PUNCT
american_scientific_journal-1255	60	61	𝑗	𝑗	NOUN
american_scientific_journal-1255	60	62	=	=	NOUN
american_scientific_journal-1255	60	63	1,2,3	1,2,3	NUM
american_scientific_journal-1255	60	64	.	.	PUNCT
american_scientific_journal-1255	61	1	then	then	ADV
american_scientific_journal-1255	61	2	,	,	PUNCT
american_scientific_journal-1255	61	3	the	the	DET
american_scientific_journal-1255	61	4	first	first	ADJ
american_scientific_journal-1255	61	5	vertex	vertex	NOUN
american_scientific_journal-1255	61	6	of	of	ADP
american_scientific_journal-1255	61	7	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	61	8	is	be	AUX
american_scientific_journal-1255	61	9	colored	color	VERB
american_scientific_journal-1255	61	10	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	61	11	of	of	ADP
american_scientific_journal-1255	61	12	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	61	13	and	and	CCONJ
american_scientific_journal-1255	61	14	the	the	DET
american_scientific_journal-1255	61	15	last	last	ADJ
american_scientific_journal-1255	61	16	vertex	vertex	NOUN
american_scientific_journal-1255	61	17	of	of	ADP
american_scientific_journal-1255	61	18	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	61	19	,	,	PUNCT
american_scientific_journal-1255	61	20	colored	color	VERB
american_scientific_journal-1255	61	21	𝑎𝑗	𝑎𝑗	NOUN
american_scientific_journal-1255	61	22	is	be	AUX
american_scientific_journal-1255	61	23	attached	attach	VERB
american_scientific_journal-1255	61	24	to	to	ADP
american_scientific_journal-1255	61	25	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	61	26	of	of	ADP
american_scientific_journal-1255	61	27	𝐶𝑛.	𝐶𝑛.	PROPN
american_scientific_journal-1255	61	28	the	the	DET
american_scientific_journal-1255	61	29	other	other	ADJ
american_scientific_journal-1255	61	30	vertices	vertex	NOUN
american_scientific_journal-1255	61	31	of	of	ADP
american_scientific_journal-1255	61	32	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	61	33	are	be	AUX
american_scientific_journal-1255	61	34	colored	color	VERB
american_scientific_journal-1255	61	35	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	61	36	,	,	PUNCT
american_scientific_journal-1255	61	37	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	61	38	such	such	ADJ
american_scientific_journal-1255	61	39	that	that	SCONJ
american_scientific_journal-1255	61	40	no	no	DET
american_scientific_journal-1255	61	41	two	two	NUM
american_scientific_journal-1255	61	42	adjacent	adjacent	ADJ
american_scientific_journal-1255	61	43	vertices	vertex	NOUN
american_scientific_journal-1255	61	44	have	have	VERB
american_scientific_journal-1255	61	45	the	the	DET
american_scientific_journal-1255	61	46	same	same	ADJ
american_scientific_journal-1255	61	47	color	color	NOUN
american_scientific_journal-1255	61	48	.	.	PUNCT
american_scientific_journal-1255	62	1	thus	thus	ADV
american_scientific_journal-1255	62	2	,	,	PUNCT
american_scientific_journal-1255	62	3	three	three	NUM
american_scientific_journal-1255	62	4	is	be	AUX
american_scientific_journal-1255	62	5	the	the	DET
american_scientific_journal-1255	62	6	minimum	minimum	ADJ
american_scientific_journal-1255	62	7	number	number	NOUN
american_scientific_journal-1255	62	8	of	of	ADP
american_scientific_journal-1255	62	9	colors	color	NOUN
american_scientific_journal-1255	62	10	to	to	PART
american_scientific_journal-1255	62	11	color	color	VERB
american_scientific_journal-1255	62	12	the	the	DET
american_scientific_journal-1255	62	13	vertices	vertex	NOUN
american_scientific_journal-1255	62	14	of	of	ADP
american_scientific_journal-1255	62	15	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	62	16	,	,	PUNCT
american_scientific_journal-1255	62	17	where	where	SCONJ
american_scientific_journal-1255	62	18	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	62	19	are	be	AUX
american_scientific_journal-1255	62	20	both	both	PRON
american_scientific_journal-1255	62	21	odd	odd	ADJ
american_scientific_journal-1255	62	22	.	.	PUNCT
american_scientific_journal-1255	63	1	consequently	consequently	ADV
american_scientific_journal-1255	63	2	,	,	PUNCT
american_scientific_journal-1255	63	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	63	4	�	�	NOUN
american_scientific_journal-1255	63	5	𝑂4	𝑂4	NOUN
american_scientific_journal-1255	63	6	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	63	7	�	�	PROPN
american_scientific_journal-1255	63	8	=	=	SYM
american_scientific_journal-1255	63	9	3	3	NUM
american_scientific_journal-1255	63	10	.	.	NOUN
american_scientific_journal-1255	63	11	•	•	NUM
american_scientific_journal-1255	63	12	case	case	NOUN
american_scientific_journal-1255	63	13	2	2	NUM
american_scientific_journal-1255	63	14	:	:	PUNCT
american_scientific_journal-1255	63	15	𝑚	𝑚	PROPN
american_scientific_journal-1255	63	16	is	be	AUX
american_scientific_journal-1255	63	17	odd	odd	ADJ
american_scientific_journal-1255	63	18	,	,	PUNCT
american_scientific_journal-1255	63	19	𝑛	𝑛	PRON
american_scientific_journal-1255	63	20	is	be	AUX
american_scientific_journal-1255	63	21	even	even	ADV
american_scientific_journal-1255	63	22	if	if	SCONJ
american_scientific_journal-1255	63	23	𝑚	𝑚	PROPN
american_scientific_journal-1255	63	24	is	be	AUX
american_scientific_journal-1255	63	25	odd	odd	ADJ
american_scientific_journal-1255	63	26	,	,	PUNCT
american_scientific_journal-1255	63	27	𝑛	𝑛	PRON
american_scientific_journal-1255	63	28	is	be	AUX
american_scientific_journal-1255	63	29	even	even	ADV
american_scientific_journal-1255	63	30	,	,	PUNCT
american_scientific_journal-1255	63	31	then	then	ADV
american_scientific_journal-1255	63	32	by	by	ADP
american_scientific_journal-1255	63	33	equation	equation	NOUN
american_scientific_journal-1255	63	34	(	(	PUNCT
american_scientific_journal-1255	63	35	1.1	1.1	NUM
american_scientific_journal-1255	63	36	)	)	PUNCT
american_scientific_journal-1255	63	37	,	,	PUNCT
american_scientific_journal-1255	63	38	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NOUN
american_scientific_journal-1255	63	39	)	)	PUNCT
american_scientific_journal-1255	63	40	=	=	SYM
american_scientific_journal-1255	63	41	3	3	NUM
american_scientific_journal-1255	63	42	and	and	CCONJ
american_scientific_journal-1255	63	43	𝜒(𝐶𝑛	𝜒(𝐶𝑛	NOUN
american_scientific_journal-1255	63	44	)	)	PUNCT
american_scientific_journal-1255	63	45	=	=	SYM
american_scientific_journal-1255	63	46	2	2	X
american_scientific_journal-1255	63	47	.	.	PUNCT
american_scientific_journal-1255	64	1	thus	thus	ADV
american_scientific_journal-1255	64	2	,	,	PUNCT
american_scientific_journal-1255	64	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	64	4	�	�	PROPN
american_scientific_journal-1255	64	5	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	64	6	�	�	PROPN
american_scientific_journal-1255	64	7	≥	≥	NUM
american_scientific_journal-1255	64	8	3	3	NUM
american_scientific_journal-1255	64	9	.	.	PUNCT
american_scientific_journal-1255	64	10	suppose	suppose	VERB
american_scientific_journal-1255	64	11	that	that	SCONJ
american_scientific_journal-1255	64	12	the	the	DET
american_scientific_journal-1255	64	13	vertices	vertex	NOUN
american_scientific_journal-1255	64	14	of	of	ADP
american_scientific_journal-1255	64	15	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	64	16	are	be	AUX
american_scientific_journal-1255	64	17	colored	color	VERB
american_scientific_journal-1255	64	18	with	with	ADP
american_scientific_journal-1255	64	19	two	two	NUM
american_scientific_journal-1255	64	20	of	of	ADP
american_scientific_journal-1255	64	21	the	the	DET
american_scientific_journal-1255	64	22	different	different	ADJ
american_scientific_journal-1255	64	23	colors	color	NOUN
american_scientific_journal-1255	64	24	that	that	PRON
american_scientific_journal-1255	64	25	also	also	ADV
american_scientific_journal-1255	64	26	color	color	VERB
american_scientific_journal-1255	64	27	the	the	DET
american_scientific_journal-1255	64	28	vertices	vertex	NOUN
american_scientific_journal-1255	64	29	of	of	ADP
american_scientific_journal-1255	64	30	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	64	31	,	,	PUNCT
american_scientific_journal-1255	64	32	say	say	VERB
american_scientific_journal-1255	64	33	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	64	34	,	,	PUNCT
american_scientific_journal-1255	64	35	𝑎2	𝑎2	NOUN
american_scientific_journal-1255	64	36	colors	color	NOUN
american_scientific_journal-1255	64	37	for	for	ADP
american_scientific_journal-1255	64	38	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	64	39	and	and	CCONJ
american_scientific_journal-1255	64	40	𝑎1	𝑎1	PROPN
american_scientific_journal-1255	64	41	,	,	PUNCT
american_scientific_journal-1255	64	42	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	64	43	,	,	PUNCT
american_scientific_journal-1255	64	44	𝑎3	𝑎3	PROPN
american_scientific_journal-1255	64	45	colors	color	NOUN
american_scientific_journal-1255	64	46	for	for	ADP
american_scientific_journal-1255	64	47	𝐶𝑚.	𝐶𝑚.	NOUN
american_scientific_journal-1255	64	48	consider	consider	VERB
american_scientific_journal-1255	64	49	the	the	DET
american_scientific_journal-1255	64	50	following	follow	VERB
american_scientific_journal-1255	64	51	subcases	subcase	NOUN
american_scientific_journal-1255	64	52	where	where	SCONJ
american_scientific_journal-1255	64	53	𝑘	𝑘	PROPN
american_scientific_journal-1255	64	54	=	=	SYM
american_scientific_journal-1255	64	55	3	3	NUM
american_scientific_journal-1255	64	56	(	(	PUNCT
american_scientific_journal-1255	64	57	odd	odd	ADJ
american_scientific_journal-1255	64	58	)	)	PUNCT
american_scientific_journal-1255	64	59	and	and	CCONJ
american_scientific_journal-1255	64	60	𝑘	𝑘	X
american_scientific_journal-1255	64	61	=	=	SYM
american_scientific_journal-1255	64	62	4	4	NUM
american_scientific_journal-1255	64	63	(	(	PUNCT
american_scientific_journal-1255	64	64	even	even	ADV
american_scientific_journal-1255	64	65	):	):	PUNCT
american_scientific_journal-1255	64	66	•	•	NUM
american_scientific_journal-1255	64	67	subcase	subcase	NOUN
american_scientific_journal-1255	64	68	2.1	2.1	NUM
american_scientific_journal-1255	64	69	:	:	PUNCT
american_scientific_journal-1255	64	70	if	if	SCONJ
american_scientific_journal-1255	64	71	𝑘	𝑘	X
american_scientific_journal-1255	64	72	=	=	SYM
american_scientific_journal-1255	64	73	3	3	NUM
american_scientific_journal-1255	64	74	then	then	ADV
american_scientific_journal-1255	64	75	a	a	DET
american_scientific_journal-1255	64	76	path	path	NOUN
american_scientific_journal-1255	64	77	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	64	78	is	be	AUX
american_scientific_journal-1255	64	79	attached	attach	VERB
american_scientific_journal-1255	64	80	to	to	ADP
american_scientific_journal-1255	64	81	vertex	vertex	NOUN
american_scientific_journal-1255	64	82	colored	color	VERB
american_scientific_journal-1255	64	83	say	say	VERB
american_scientific_journal-1255	64	84	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	64	85	,	,	PUNCT
american_scientific_journal-1255	64	86	𝑖	𝑖	PROPN
american_scientific_journal-1255	64	87	=	=	NOUN
american_scientific_journal-1255	64	88	1	1	NUM
american_scientific_journal-1255	64	89	,	,	PUNCT
american_scientific_journal-1255	64	90	2	2	NUM
american_scientific_journal-1255	64	91	,	,	PUNCT
american_scientific_journal-1255	64	92	3	3	NUM
american_scientific_journal-1255	64	93	of	of	ADP
american_scientific_journal-1255	64	94	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	64	95	and	and	CCONJ
american_scientific_journal-1255	64	96	the	the	DET
american_scientific_journal-1255	64	97	other	other	ADJ
american_scientific_journal-1255	64	98	end	end	NOUN
american_scientific_journal-1255	64	99	vertex	vertex	NOUN
american_scientific_journal-1255	64	100	is	be	AUX
american_scientific_journal-1255	64	101	also	also	ADV
american_scientific_journal-1255	64	102	connected	connect	VERB
american_scientific_journal-1255	64	103	to	to	ADP
american_scientific_journal-1255	64	104	a	a	DET
american_scientific_journal-1255	64	105	vertex	vertex	NOUN
american_scientific_journal-1255	64	106	colored	color	VERB
american_scientific_journal-1255	64	107	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	64	108	of	of	ADP
american_scientific_journal-1255	64	109	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	64	110	,	,	PUNCT
american_scientific_journal-1255	64	111	such	such	ADJ
american_scientific_journal-1255	64	112	that	that	SCONJ
american_scientific_journal-1255	64	113	the	the	DET
american_scientific_journal-1255	64	114	second	second	ADJ
american_scientific_journal-1255	64	115	(	(	PUNCT
american_scientific_journal-1255	64	116	middle	middle	ADJ
american_scientific_journal-1255	64	117	)	)	PUNCT
american_scientific_journal-1255	64	118	vertex	vertex	NOUN
american_scientific_journal-1255	64	119	of	of	ADP
american_scientific_journal-1255	64	120	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	64	121	is	be	AUX
american_scientific_journal-1255	64	122	𝑎𝑗	𝑎𝑗	PROPN
american_scientific_journal-1255	64	123	,	,	PUNCT
american_scientific_journal-1255	64	124	𝑗	𝑗	PROPN
american_scientific_journal-1255	64	125	=	=	SYM
american_scientific_journal-1255	64	126	1	1	NUM
american_scientific_journal-1255	64	127	,	,	PUNCT
american_scientific_journal-1255	64	128	2	2	NUM
american_scientific_journal-1255	64	129	,	,	PUNCT
american_scientific_journal-1255	64	130	3	3	NUM
american_scientific_journal-1255	64	131	,	,	PUNCT
american_scientific_journal-1255	64	132	where	where	SCONJ
american_scientific_journal-1255	64	133	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	64	134	≠	≠	PROPN
american_scientific_journal-1255	64	135	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	64	136	thus	thus	ADV
american_scientific_journal-1255	64	137	,	,	PUNCT
american_scientific_journal-1255	64	138	three	three	NUM
american_scientific_journal-1255	64	139	is	be	AUX
american_scientific_journal-1255	64	140	the	the	DET
american_scientific_journal-1255	64	141	minimum	minimum	ADJ
american_scientific_journal-1255	64	142	number	number	NOUN
american_scientific_journal-1255	64	143	of	of	ADP
american_scientific_journal-1255	64	144	colors	color	NOUN
american_scientific_journal-1255	64	145	to	to	PART
american_scientific_journal-1255	64	146	color	color	VERB
american_scientific_journal-1255	64	147	the	the	DET
american_scientific_journal-1255	64	148	vertices	vertex	NOUN
american_scientific_journal-1255	64	149	of	of	ADP
american_scientific_journal-1255	64	150	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	64	151	where	where	SCONJ
american_scientific_journal-1255	64	152	𝑚	𝑚	PROPN
american_scientific_journal-1255	64	153	is	be	AUX
american_scientific_journal-1255	64	154	odd	odd	ADJ
american_scientific_journal-1255	64	155	and	and	CCONJ
american_scientific_journal-1255	64	156	𝑛	𝑛	PROPN
american_scientific_journal-1255	64	157	is	be	AUX
american_scientific_journal-1255	64	158	even	even	ADV
american_scientific_journal-1255	64	159	.	.	PUNCT
american_scientific_journal-1255	65	1	consequently	consequently	ADV
american_scientific_journal-1255	65	2	,	,	PUNCT
american_scientific_journal-1255	65	3	𝜒	𝜒	NOUN
american_scientific_journal-1255	65	4	�	�	NOUN
american_scientific_journal-1255	65	5	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	65	6	�	�	PROPN
american_scientific_journal-1255	65	7	=	=	SYM
american_scientific_journal-1255	65	8	3	3	NUM
american_scientific_journal-1255	65	9	.	.	NOUN
american_scientific_journal-1255	65	10	•	•	NUM
american_scientific_journal-1255	65	11	subcase	subcase	NOUN
american_scientific_journal-1255	65	12	2.2	2.2	NUM
american_scientific_journal-1255	65	13	:	:	PUNCT
american_scientific_journal-1255	65	14	if	if	SCONJ
american_scientific_journal-1255	65	15	𝑘	𝑘	X
american_scientific_journal-1255	65	16	=	=	SYM
american_scientific_journal-1255	65	17	4	4	NUM
american_scientific_journal-1255	65	18	note	note	VERB
american_scientific_journal-1255	65	19	that	that	SCONJ
american_scientific_journal-1255	65	20	by	by	ADP
american_scientific_journal-1255	65	21	remark	remark	NOUN
american_scientific_journal-1255	65	22	2.1	2.1	NUM
american_scientific_journal-1255	65	23	,	,	PUNCT
american_scientific_journal-1255	65	24	𝜒(𝑃4	𝜒(𝑃4	PROPN
american_scientific_journal-1255	65	25	)	)	PUNCT
american_scientific_journal-1255	65	26	=	=	SYM
american_scientific_journal-1255	65	27	2	2	X
american_scientific_journal-1255	65	28	.	.	PUNCT
american_scientific_journal-1255	65	29	suppose	suppose	VERB
american_scientific_journal-1255	65	30	further	far	ADV
american_scientific_journal-1255	65	31	that	that	DET
american_scientific_journal-1255	65	32	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	65	33	is	be	AUX
american_scientific_journal-1255	65	34	colored	color	VERB
american_scientific_journal-1255	65	35	with	with	ADP
american_scientific_journal-1255	65	36	two	two	NUM
american_scientific_journal-1255	65	37	from	from	ADP
american_scientific_journal-1255	65	38	the	the	DET
american_scientific_journal-1255	65	39	same	same	ADJ
american_scientific_journal-1255	65	40	set	set	NOUN
american_scientific_journal-1255	65	41	of	of	ADP
american_scientific_journal-1255	65	42	different	different	ADJ
american_scientific_journal-1255	65	43	colors	color	NOUN
american_scientific_journal-1255	65	44	that	that	PRON
american_scientific_journal-1255	65	45	color	color	VERB
american_scientific_journal-1255	65	46	the	the	DET
american_scientific_journal-1255	65	47	vertices	vertex	NOUN
american_scientific_journal-1255	65	48	of	of	ADP
american_scientific_journal-1255	65	49	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	65	50	and	and	CCONJ
american_scientific_journal-1255	65	51	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	65	52	,	,	PUNCT
american_scientific_journal-1255	65	53	say	say	VERB
american_scientific_journal-1255	65	54	𝑎1	𝑎1	ADV
american_scientific_journal-1255	65	55	,	,	PUNCT
american_scientific_journal-1255	65	56	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	65	57	.	.	PUNCT
american_scientific_journal-1255	66	1	then	then	ADV
american_scientific_journal-1255	66	2	,	,	PUNCT
american_scientific_journal-1255	66	3	the	the	DET
american_scientific_journal-1255	66	4	first	first	ADJ
american_scientific_journal-1255	66	5	vertex	vertex	NOUN
american_scientific_journal-1255	66	6	of	of	ADP
american_scientific_journal-1255	66	7	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	66	8	is	be	AUX
american_scientific_journal-1255	66	9	attached	attach	VERB
american_scientific_journal-1255	66	10	to	to	ADP
american_scientific_journal-1255	66	11	a	a	DET
american_scientific_journal-1255	66	12	vertex	vertex	NOUN
american_scientific_journal-1255	66	13	colored	color	VERB
american_scientific_journal-1255	66	14	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	66	15	,	,	PUNCT
american_scientific_journal-1255	66	16	𝑖	𝑖	PROPN
american_scientific_journal-1255	66	17	=	=	NOUN
american_scientific_journal-1255	66	18	1,2	1,2	NUM
american_scientific_journal-1255	66	19	of	of	ADP
american_scientific_journal-1255	66	20	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	66	21	and	and	CCONJ
american_scientific_journal-1255	66	22	is	be	AUX
american_scientific_journal-1255	66	23	adjacent	adjacent	ADJ
american_scientific_journal-1255	66	24	to	to	ADP
american_scientific_journal-1255	66	25	a	a	DET
american_scientific_journal-1255	66	26	vertex	vertex	NOUN
american_scientific_journal-1255	66	27	colored	color	VERB
american_scientific_journal-1255	66	28	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	66	29	,	,	PUNCT
american_scientific_journal-1255	66	30	𝑗	𝑗	NOUN
american_scientific_journal-1255	66	31	=	=	SYM
american_scientific_journal-1255	66	32	1,2	1,2	NUM
american_scientific_journal-1255	66	33	,	,	PUNCT
american_scientific_journal-1255	66	34	which	which	PRON
american_scientific_journal-1255	66	35	is	be	AUX
american_scientific_journal-1255	66	36	the	the	DET
american_scientific_journal-1255	66	37	second	second	ADJ
american_scientific_journal-1255	66	38	vertex	vertex	NOUN
american_scientific_journal-1255	66	39	of	of	ADP
american_scientific_journal-1255	66	40	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	66	41	,	,	PUNCT
american_scientific_journal-1255	66	42	and	and	CCONJ
american_scientific_journal-1255	66	43	the	the	DET
american_scientific_journal-1255	66	44	last	last	ADJ
american_scientific_journal-1255	66	45	vertex	vertex	NOUN
american_scientific_journal-1255	66	46	is	be	AUX
american_scientific_journal-1255	66	47	then	then	ADV
american_scientific_journal-1255	66	48	connected	connect	VERB
american_scientific_journal-1255	66	49	to	to	ADP
american_scientific_journal-1255	66	50	a	a	DET
american_scientific_journal-1255	66	51	vertex	vertex	NOUN
american_scientific_journal-1255	66	52	colored	color	VERB
american_scientific_journal-1255	66	53	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	66	54	,	,	PUNCT
american_scientific_journal-1255	66	55	𝑗	𝑗	PROPN
american_scientific_journal-1255	66	56	=	=	SYM
american_scientific_journal-1255	66	57	1,2	1,2	NUM
american_scientific_journal-1255	66	58	of	of	ADP
american_scientific_journal-1255	66	59	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	66	60	,	,	PUNCT
american_scientific_journal-1255	66	61	where	where	SCONJ
american_scientific_journal-1255	66	62	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	66	63	≠	≠	PROPN
american_scientific_journal-1255	66	64	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	66	65	note	note	VERB
american_scientific_journal-1255	66	66	that	that	SCONJ
american_scientific_journal-1255	66	67	the	the	DET
american_scientific_journal-1255	66	68	vertices	vertex	NOUN
american_scientific_journal-1255	66	69	of	of	ADP
american_scientific_journal-1255	66	70	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	66	71	are	be	AUX
american_scientific_journal-1255	66	72	colored	color	VERB
american_scientific_journal-1255	66	73	{	{	PUNCT
american_scientific_journal-1255	66	74	𝑎1	𝑎1	PROPN
american_scientific_journal-1255	66	75	,	,	PUNCT
american_scientific_journal-1255	66	76	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	66	77	,	,	PUNCT
american_scientific_journal-1255	66	78	𝑎3	𝑎3	PROPN
american_scientific_journal-1255	66	79	}	}	PUNCT
american_scientific_journal-1255	66	80	.	.	PUNCT
american_scientific_journal-1255	67	1	thus	thus	ADV
american_scientific_journal-1255	67	2	,	,	PUNCT
american_scientific_journal-1255	67	3	three	three	NUM
american_scientific_journal-1255	67	4	is	be	AUX
american_scientific_journal-1255	67	5	the	the	DET
american_scientific_journal-1255	67	6	minimum	minimum	ADJ
american_scientific_journal-1255	67	7	number	number	NOUN
american_scientific_journal-1255	67	8	of	of	ADP
american_scientific_journal-1255	67	9	colors	color	NOUN
american_scientific_journal-1255	67	10	to	to	PART
american_scientific_journal-1255	67	11	color	color	VERB
american_scientific_journal-1255	67	12	the	the	DET
american_scientific_journal-1255	67	13	vertices	vertex	NOUN
american_scientific_journal-1255	67	14	of	of	ADP
american_scientific_journal-1255	67	15	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	67	16	,	,	PUNCT
american_scientific_journal-1255	67	17	where	where	SCONJ
american_scientific_journal-1255	67	18	𝑚	𝑚	PROPN
american_scientific_journal-1255	67	19	is	be	AUX
american_scientific_journal-1255	67	20	odd	odd	ADJ
american_scientific_journal-1255	67	21	,	,	PUNCT
american_scientific_journal-1255	67	22	𝑛	𝑛	PRON
american_scientific_journal-1255	67	23	is	be	AUX
american_scientific_journal-1255	67	24	even	even	ADV
american_scientific_journal-1255	67	25	.	.	PUNCT
american_scientific_journal-1255	68	1	consequently	consequently	ADV
american_scientific_journal-1255	68	2	,	,	PUNCT
american_scientific_journal-1255	68	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	68	4	�	�	NOUN
american_scientific_journal-1255	68	5	𝑂4	𝑂4	NOUN
american_scientific_journal-1255	68	6	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	68	7	�	�	PROPN
american_scientific_journal-1255	68	8	=	=	SYM
american_scientific_journal-1255	68	9	3	3	NUM
american_scientific_journal-1255	68	10	.	.	NOUN
american_scientific_journal-1255	68	11	•	•	NUM
american_scientific_journal-1255	68	12	case	case	NOUN
american_scientific_journal-1255	68	13	3	3	NUM
american_scientific_journal-1255	68	14	:	:	PUNCT
american_scientific_journal-1255	68	15	𝑚	𝑚	NOUN
american_scientific_journal-1255	68	16	is	be	AUX
american_scientific_journal-1255	68	17	even	even	ADV
american_scientific_journal-1255	68	18	,	,	PUNCT
american_scientific_journal-1255	68	19	𝑛	𝑛	PRON
american_scientific_journal-1255	68	20	is	be	AUX
american_scientific_journal-1255	68	21	odd	odd	ADJ
american_scientific_journal-1255	68	22	if	if	SCONJ
american_scientific_journal-1255	68	23	𝑚	𝑚	PROPN
american_scientific_journal-1255	68	24	is	be	AUX
american_scientific_journal-1255	68	25	even	even	ADV
american_scientific_journal-1255	68	26	,	,	PUNCT
american_scientific_journal-1255	68	27	𝑛	𝑛	PRON
american_scientific_journal-1255	68	28	is	be	AUX
american_scientific_journal-1255	68	29	odd	odd	ADJ
american_scientific_journal-1255	68	30	,	,	PUNCT
american_scientific_journal-1255	68	31	then	then	ADV
american_scientific_journal-1255	68	32	the	the	DET
american_scientific_journal-1255	68	33	proof	proof	NOUN
american_scientific_journal-1255	68	34	of	of	ADP
american_scientific_journal-1255	68	35	this	this	DET
american_scientific_journal-1255	68	36	case	case	NOUN
american_scientific_journal-1255	68	37	is	be	AUX
american_scientific_journal-1255	68	38	similar	similar	ADJ
american_scientific_journal-1255	68	39	to	to	ADP
american_scientific_journal-1255	68	40	case	case	NOUN
american_scientific_journal-1255	68	41	2	2	NUM
american_scientific_journal-1255	68	42	.	.	NOUN
american_scientific_journal-1255	68	43	•	•	NUM
american_scientific_journal-1255	68	44	case	case	NOUN
american_scientific_journal-1255	68	45	4	4	NUM
american_scientific_journal-1255	68	46	:	:	PUNCT
american_scientific_journal-1255	68	47	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	68	48	are	be	AUX
american_scientific_journal-1255	68	49	both	both	PRON
american_scientific_journal-1255	68	50	even	even	ADV
american_scientific_journal-1255	68	51	if	if	SCONJ
american_scientific_journal-1255	68	52	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	68	53	are	be	AUX
american_scientific_journal-1255	68	54	both	both	PRON
american_scientific_journal-1255	68	55	even	even	ADV
american_scientific_journal-1255	68	56	,	,	PUNCT
american_scientific_journal-1255	68	57	then	then	ADV
american_scientific_journal-1255	68	58	by	by	ADP
american_scientific_journal-1255	68	59	equation	equation	NOUN
american_scientific_journal-1255	68	60	(	(	PUNCT
american_scientific_journal-1255	68	61	1.1	1.1	NUM
american_scientific_journal-1255	68	62	)	)	PUNCT
american_scientific_journal-1255	68	63	,	,	PUNCT
american_scientific_journal-1255	68	64	𝜒(𝐶𝑚	𝜒(𝐶𝑚	NOUN
american_scientific_journal-1255	68	65	)	)	PUNCT
american_scientific_journal-1255	68	66	=	=	SYM
american_scientific_journal-1255	68	67	𝜒(𝐶𝑛	𝜒(𝐶𝑛	X
american_scientific_journal-1255	68	68	)	)	PUNCT
american_scientific_journal-1255	68	69	=	=	SYM
american_scientific_journal-1255	68	70	2	2	X
american_scientific_journal-1255	68	71	.	.	PUNCT
american_scientific_journal-1255	68	72	suppose	suppose	VERB
american_scientific_journal-1255	68	73	the	the	DET
american_scientific_journal-1255	68	74	vertices	vertex	NOUN
american_scientific_journal-1255	68	75	of	of	ADP
american_scientific_journal-1255	68	76	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	68	77	and	and	CCONJ
american_scientific_journal-1255	68	78	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	68	79	are	be	AUX
american_scientific_journal-1255	68	80	103	103	NUM
american_scientific_journal-1255	68	81	american	american	ADJ
american_scientific_journal-1255	68	82	scientific	scientific	ADJ
american_scientific_journal-1255	68	83	research	research	NOUN
american_scientific_journal-1255	68	84	journal	journal	NOUN
american_scientific_journal-1255	68	85	for	for	ADP
american_scientific_journal-1255	68	86	engineering	engineering	NOUN
american_scientific_journal-1255	68	87	,	,	PUNCT
american_scientific_journal-1255	68	88	technology	technology	NOUN
american_scientific_journal-1255	68	89	,	,	PUNCT
american_scientific_journal-1255	68	90	and	and	CCONJ
american_scientific_journal-1255	68	91	sciences	science	NOUN
american_scientific_journal-1255	68	92	(	(	PUNCT
american_scientific_journal-1255	68	93	asrjets	asrjet	NOUN
american_scientific_journal-1255	68	94	)	)	PUNCT
american_scientific_journal-1255	68	95	(	(	PUNCT
american_scientific_journal-1255	68	96	2016	2016	NUM
american_scientific_journal-1255	68	97	)	)	PUNCT
american_scientific_journal-1255	68	98	volume	volume	NOUN
american_scientific_journal-1255	68	99	16	16	NUM
american_scientific_journal-1255	68	100	,	,	PUNCT
american_scientific_journal-1255	68	101	no	no	DET
american_scientific_journal-1255	68	102	1	1	NUM
american_scientific_journal-1255	68	103	,	,	PUNCT
american_scientific_journal-1255	68	104	pp	pp	ADV
american_scientific_journal-1255	68	105	98	98	NUM
american_scientific_journal-1255	68	106	-	-	SYM
american_scientific_journal-1255	68	107	107	107	NUM
american_scientific_journal-1255	68	108	colored	color	VERB
american_scientific_journal-1255	68	109	with	with	ADP
american_scientific_journal-1255	68	110	the	the	DET
american_scientific_journal-1255	68	111	same	same	ADJ
american_scientific_journal-1255	68	112	set	set	NOUN
american_scientific_journal-1255	68	113	of	of	ADP
american_scientific_journal-1255	68	114	different	different	ADJ
american_scientific_journal-1255	68	115	colors	color	NOUN
american_scientific_journal-1255	68	116	say	say	VERB
american_scientific_journal-1255	68	117	𝑎1	𝑎1	ADV
american_scientific_journal-1255	68	118	,	,	PUNCT
american_scientific_journal-1255	68	119	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	68	120	.	.	PUNCT
american_scientific_journal-1255	69	1	consider	consider	VERB
american_scientific_journal-1255	69	2	the	the	DET
american_scientific_journal-1255	69	3	following	follow	VERB
american_scientific_journal-1255	69	4	subcases	subcase	NOUN
american_scientific_journal-1255	69	5	where	where	SCONJ
american_scientific_journal-1255	69	6	𝑘	𝑘	PROPN
american_scientific_journal-1255	69	7	=	=	SYM
american_scientific_journal-1255	69	8	3	3	NUM
american_scientific_journal-1255	69	9	(	(	PUNCT
american_scientific_journal-1255	69	10	odd	odd	ADJ
american_scientific_journal-1255	69	11	)	)	PUNCT
american_scientific_journal-1255	69	12	and	and	CCONJ
american_scientific_journal-1255	69	13	𝑘	𝑘	X
american_scientific_journal-1255	69	14	=	=	SYM
american_scientific_journal-1255	69	15	4	4	NUM
american_scientific_journal-1255	69	16	(	(	PUNCT
american_scientific_journal-1255	69	17	even	even	ADV
american_scientific_journal-1255	69	18	):	):	PUNCT
american_scientific_journal-1255	69	19	•	•	NUM
american_scientific_journal-1255	69	20	subcase	subcase	NOUN
american_scientific_journal-1255	69	21	4.1	4.1	NUM
american_scientific_journal-1255	69	22	:	:	PUNCT
american_scientific_journal-1255	69	23	if	if	SCONJ
american_scientific_journal-1255	69	24	𝑘	𝑘	X
american_scientific_journal-1255	69	25	=	=	SYM
american_scientific_journal-1255	69	26	3	3	NUM
american_scientific_journal-1255	69	27	then	then	ADV
american_scientific_journal-1255	69	28	a	a	DET
american_scientific_journal-1255	69	29	path	path	NOUN
american_scientific_journal-1255	69	30	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	69	31	with	with	ADP
american_scientific_journal-1255	69	32	vertices	vertex	NOUN
american_scientific_journal-1255	69	33	colored	color	VERB
american_scientific_journal-1255	69	34	with	with	ADP
american_scientific_journal-1255	69	35	the	the	DET
american_scientific_journal-1255	69	36	same	same	ADJ
american_scientific_journal-1255	69	37	set	set	NOUN
american_scientific_journal-1255	69	38	of	of	ADP
american_scientific_journal-1255	69	39	different	different	ADJ
american_scientific_journal-1255	69	40	colors	color	NOUN
american_scientific_journal-1255	69	41	𝑎1	𝑎1	PROPN
american_scientific_journal-1255	69	42	,	,	PUNCT
american_scientific_journal-1255	69	43	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	69	44	,	,	PUNCT
american_scientific_journal-1255	69	45	is	be	AUX
american_scientific_journal-1255	69	46	attached	attach	VERB
american_scientific_journal-1255	69	47	to	to	ADP
american_scientific_journal-1255	69	48	vertex	vertex	NOUN
american_scientific_journal-1255	69	49	colored	color	VERB
american_scientific_journal-1255	69	50	say	say	VERB
american_scientific_journal-1255	69	51	𝑎𝑖	𝑎𝑖	INTJ
american_scientific_journal-1255	69	52	,	,	PUNCT
american_scientific_journal-1255	69	53	𝑖	𝑖	PROPN
american_scientific_journal-1255	69	54	=	=	NOUN
american_scientific_journal-1255	69	55	1	1	NUM
american_scientific_journal-1255	69	56	,	,	PUNCT
american_scientific_journal-1255	69	57	2	2	NUM
american_scientific_journal-1255	69	58	,	,	PUNCT
american_scientific_journal-1255	69	59	3	3	NUM
american_scientific_journal-1255	69	60	of	of	ADP
american_scientific_journal-1255	69	61	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	69	62	and	and	CCONJ
american_scientific_journal-1255	69	63	the	the	DET
american_scientific_journal-1255	69	64	other	other	ADJ
american_scientific_journal-1255	69	65	end	end	NOUN
american_scientific_journal-1255	69	66	vertex	vertex	NOUN
american_scientific_journal-1255	69	67	is	be	AUX
american_scientific_journal-1255	69	68	also	also	ADV
american_scientific_journal-1255	69	69	connected	connect	VERB
american_scientific_journal-1255	69	70	to	to	ADP
american_scientific_journal-1255	69	71	a	a	DET
american_scientific_journal-1255	69	72	vertex	vertex	NOUN
american_scientific_journal-1255	69	73	colored	color	VERB
american_scientific_journal-1255	69	74	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	69	75	of	of	ADP
american_scientific_journal-1255	69	76	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	69	77	,	,	PUNCT
american_scientific_journal-1255	69	78	such	such	ADJ
american_scientific_journal-1255	69	79	that	that	SCONJ
american_scientific_journal-1255	69	80	the	the	DET
american_scientific_journal-1255	69	81	second	second	ADJ
american_scientific_journal-1255	69	82	(	(	PUNCT
american_scientific_journal-1255	69	83	middle	middle	ADJ
american_scientific_journal-1255	69	84	)	)	PUNCT
american_scientific_journal-1255	69	85	vertex	vertex	NOUN
american_scientific_journal-1255	69	86	of	of	ADP
american_scientific_journal-1255	69	87	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	69	88	is	be	AUX
american_scientific_journal-1255	69	89	𝑎𝑗	𝑎𝑗	PROPN
american_scientific_journal-1255	69	90	,	,	PUNCT
american_scientific_journal-1255	69	91	𝑗	𝑗	PROPN
american_scientific_journal-1255	69	92	=	=	SYM
american_scientific_journal-1255	69	93	1	1	NUM
american_scientific_journal-1255	69	94	,	,	PUNCT
american_scientific_journal-1255	69	95	2	2	NUM
american_scientific_journal-1255	69	96	,	,	PUNCT
american_scientific_journal-1255	69	97	3	3	NUM
american_scientific_journal-1255	69	98	,	,	PUNCT
american_scientific_journal-1255	69	99	where	where	SCONJ
american_scientific_journal-1255	69	100	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	69	101	≠	≠	PROPN
american_scientific_journal-1255	69	102	𝑎𝑗.	𝑎𝑗.	NOUN
american_scientific_journal-1255	69	103	thus	thus	ADV
american_scientific_journal-1255	69	104	,	,	PUNCT
american_scientific_journal-1255	69	105	two	two	NUM
american_scientific_journal-1255	69	106	is	be	AUX
american_scientific_journal-1255	69	107	the	the	DET
american_scientific_journal-1255	69	108	minimum	minimum	ADJ
american_scientific_journal-1255	69	109	number	number	NOUN
american_scientific_journal-1255	69	110	of	of	ADP
american_scientific_journal-1255	69	111	colors	color	NOUN
american_scientific_journal-1255	69	112	to	to	PART
american_scientific_journal-1255	69	113	color	color	VERB
american_scientific_journal-1255	69	114	the	the	DET
american_scientific_journal-1255	69	115	vertices	vertex	NOUN
american_scientific_journal-1255	69	116	of	of	ADP
american_scientific_journal-1255	69	117	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	69	118	where	where	SCONJ
american_scientific_journal-1255	69	119	𝑚	𝑚	PROPN
american_scientific_journal-1255	69	120	is	be	AUX
american_scientific_journal-1255	69	121	odd	odd	ADJ
american_scientific_journal-1255	69	122	and	and	CCONJ
american_scientific_journal-1255	69	123	𝑛	𝑛	PROPN
american_scientific_journal-1255	69	124	is	be	AUX
american_scientific_journal-1255	69	125	even	even	ADV
american_scientific_journal-1255	69	126	.	.	PUNCT
american_scientific_journal-1255	70	1	consequently	consequently	ADV
american_scientific_journal-1255	70	2	,	,	PUNCT
american_scientific_journal-1255	70	3	𝜒	𝜒	NOUN
american_scientific_journal-1255	70	4	�	�	NOUN
american_scientific_journal-1255	70	5	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	70	6	�	�	PROPN
american_scientific_journal-1255	70	7	=	=	SYM
american_scientific_journal-1255	70	8	2	2	NUM
american_scientific_journal-1255	70	9	.	.	NOUN
american_scientific_journal-1255	70	10	•	•	NOUN
american_scientific_journal-1255	70	11	subcase	subcase	NOUN
american_scientific_journal-1255	70	12	4.2	4.2	NUM
american_scientific_journal-1255	70	13	:	:	PUNCT
american_scientific_journal-1255	70	14	if	if	SCONJ
american_scientific_journal-1255	70	15	𝑘	𝑘	X
american_scientific_journal-1255	70	16	=	=	SYM
american_scientific_journal-1255	70	17	4	4	NUM
american_scientific_journal-1255	70	18	note	note	VERB
american_scientific_journal-1255	70	19	that	that	SCONJ
american_scientific_journal-1255	70	20	by	by	ADP
american_scientific_journal-1255	70	21	remark	remark	NOUN
american_scientific_journal-1255	70	22	2.1	2.1	NUM
american_scientific_journal-1255	70	23	,	,	PUNCT
american_scientific_journal-1255	70	24	𝜒(𝑃4	𝜒(𝑃4	PROPN
american_scientific_journal-1255	70	25	)	)	PUNCT
american_scientific_journal-1255	70	26	=	=	SYM
american_scientific_journal-1255	70	27	2	2	X
american_scientific_journal-1255	70	28	.	.	PUNCT
american_scientific_journal-1255	70	29	suppose	suppose	VERB
american_scientific_journal-1255	70	30	further	far	ADV
american_scientific_journal-1255	70	31	that	that	DET
american_scientific_journal-1255	70	32	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	70	33	is	be	AUX
american_scientific_journal-1255	70	34	colored	color	VERB
american_scientific_journal-1255	70	35	with	with	ADP
american_scientific_journal-1255	70	36	two	two	NUM
american_scientific_journal-1255	70	37	from	from	ADP
american_scientific_journal-1255	70	38	the	the	DET
american_scientific_journal-1255	70	39	same	same	ADJ
american_scientific_journal-1255	70	40	set	set	NOUN
american_scientific_journal-1255	70	41	of	of	ADP
american_scientific_journal-1255	70	42	different	different	ADJ
american_scientific_journal-1255	70	43	colors	color	NOUN
american_scientific_journal-1255	70	44	that	that	PRON
american_scientific_journal-1255	70	45	color	color	VERB
american_scientific_journal-1255	70	46	the	the	DET
american_scientific_journal-1255	70	47	vertices	vertex	NOUN
american_scientific_journal-1255	70	48	of	of	ADP
american_scientific_journal-1255	70	49	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	70	50	and	and	CCONJ
american_scientific_journal-1255	70	51	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	70	52	,	,	PUNCT
american_scientific_journal-1255	70	53	say	say	VERB
american_scientific_journal-1255	70	54	𝑎1	𝑎1	ADV
american_scientific_journal-1255	70	55	,	,	PUNCT
american_scientific_journal-1255	70	56	𝑎2	𝑎2	PROPN
american_scientific_journal-1255	70	57	.	.	PUNCT
american_scientific_journal-1255	71	1	then	then	ADV
american_scientific_journal-1255	71	2	,	,	PUNCT
american_scientific_journal-1255	71	3	the	the	DET
american_scientific_journal-1255	71	4	first	first	ADJ
american_scientific_journal-1255	71	5	vertex	vertex	NOUN
american_scientific_journal-1255	71	6	of	of	ADP
american_scientific_journal-1255	71	7	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	71	8	is	be	AUX
american_scientific_journal-1255	71	9	attached	attach	VERB
american_scientific_journal-1255	71	10	to	to	ADP
american_scientific_journal-1255	71	11	a	a	DET
american_scientific_journal-1255	71	12	vertex	vertex	NOUN
american_scientific_journal-1255	71	13	colored	color	VERB
american_scientific_journal-1255	71	14	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	71	15	,	,	PUNCT
american_scientific_journal-1255	71	16	𝑖	𝑖	PROPN
american_scientific_journal-1255	71	17	=	=	NOUN
american_scientific_journal-1255	71	18	1,2	1,2	NUM
american_scientific_journal-1255	71	19	of	of	ADP
american_scientific_journal-1255	71	20	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	71	21	and	and	CCONJ
american_scientific_journal-1255	71	22	is	be	AUX
american_scientific_journal-1255	71	23	adjacent	adjacent	ADJ
american_scientific_journal-1255	71	24	to	to	ADP
american_scientific_journal-1255	71	25	a	a	DET
american_scientific_journal-1255	71	26	vertex	vertex	NOUN
american_scientific_journal-1255	71	27	colored	color	VERB
american_scientific_journal-1255	71	28	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	71	29	,	,	PUNCT
american_scientific_journal-1255	71	30	𝑗	𝑗	NOUN
american_scientific_journal-1255	71	31	=	=	SYM
american_scientific_journal-1255	71	32	1,2	1,2	NUM
american_scientific_journal-1255	71	33	,	,	PUNCT
american_scientific_journal-1255	71	34	which	which	PRON
american_scientific_journal-1255	71	35	is	be	AUX
american_scientific_journal-1255	71	36	the	the	DET
american_scientific_journal-1255	71	37	second	second	ADJ
american_scientific_journal-1255	71	38	vertex	vertex	NOUN
american_scientific_journal-1255	71	39	of	of	ADP
american_scientific_journal-1255	71	40	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	71	41	,	,	PUNCT
american_scientific_journal-1255	71	42	and	and	CCONJ
american_scientific_journal-1255	71	43	the	the	DET
american_scientific_journal-1255	71	44	second	second	ADJ
american_scientific_journal-1255	71	45	vertex	vertex	NOUN
american_scientific_journal-1255	71	46	is	be	AUX
american_scientific_journal-1255	71	47	adjacent	adjacent	ADJ
american_scientific_journal-1255	71	48	to	to	ADP
american_scientific_journal-1255	71	49	a	a	DET
american_scientific_journal-1255	71	50	vertex	vertex	NOUN
american_scientific_journal-1255	71	51	colored	color	VERB
american_scientific_journal-1255	71	52	𝑎𝑖	𝑎𝑖	ADP
american_scientific_journal-1255	71	53	,	,	PUNCT
american_scientific_journal-1255	71	54	𝑖	𝑖	PROPN
american_scientific_journal-1255	71	55	=	=	SYM
american_scientific_journal-1255	71	56	1,2	1,2	NUM
american_scientific_journal-1255	71	57	,	,	PUNCT
american_scientific_journal-1255	71	58	which	which	PRON
american_scientific_journal-1255	71	59	is	be	AUX
american_scientific_journal-1255	71	60	the	the	DET
american_scientific_journal-1255	71	61	third	third	ADJ
american_scientific_journal-1255	71	62	vertex	vertex	NOUN
american_scientific_journal-1255	71	63	of	of	ADP
american_scientific_journal-1255	71	64	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	71	65	,	,	PUNCT
american_scientific_journal-1255	71	66	and	and	CCONJ
american_scientific_journal-1255	71	67	the	the	DET
american_scientific_journal-1255	71	68	last	last	ADJ
american_scientific_journal-1255	71	69	vertex	vertex	NOUN
american_scientific_journal-1255	71	70	is	be	AUX
american_scientific_journal-1255	71	71	then	then	ADV
american_scientific_journal-1255	71	72	connected	connect	VERB
american_scientific_journal-1255	71	73	to	to	ADP
american_scientific_journal-1255	71	74	a	a	DET
american_scientific_journal-1255	71	75	vertex	vertex	NOUN
american_scientific_journal-1255	71	76	colored	color	VERB
american_scientific_journal-1255	71	77	𝑎𝑗	𝑎𝑗	ADP
american_scientific_journal-1255	71	78	,	,	PUNCT
american_scientific_journal-1255	71	79	𝑗	𝑗	PROPN
american_scientific_journal-1255	71	80	=	=	SYM
american_scientific_journal-1255	71	81	1,2	1,2	NUM
american_scientific_journal-1255	71	82	of	of	ADP
american_scientific_journal-1255	71	83	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	71	84	,	,	PUNCT
american_scientific_journal-1255	71	85	where	where	SCONJ
american_scientific_journal-1255	71	86	𝑎𝑖	𝑎𝑖	DET
american_scientific_journal-1255	71	87	≠	≠	PROPN
american_scientific_journal-1255	71	88	𝑎𝑗	𝑎𝑗	PROPN
american_scientific_journal-1255	71	89	.	.	PUNCT
american_scientific_journal-1255	72	1	thus	thus	ADV
american_scientific_journal-1255	72	2	,	,	PUNCT
american_scientific_journal-1255	72	3	two	two	NUM
american_scientific_journal-1255	72	4	is	be	AUX
american_scientific_journal-1255	72	5	the	the	DET
american_scientific_journal-1255	72	6	minimum	minimum	ADJ
american_scientific_journal-1255	72	7	number	number	NOUN
american_scientific_journal-1255	72	8	of	of	ADP
american_scientific_journal-1255	72	9	colors	color	NOUN
american_scientific_journal-1255	72	10	to	to	PART
american_scientific_journal-1255	72	11	color	color	VERB
american_scientific_journal-1255	72	12	the	the	DET
american_scientific_journal-1255	72	13	vertices	vertex	NOUN
american_scientific_journal-1255	72	14	of	of	ADP
american_scientific_journal-1255	72	15	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	72	16	,	,	PUNCT
american_scientific_journal-1255	72	17	where	where	SCONJ
american_scientific_journal-1255	72	18	𝑚	𝑚	PROPN
american_scientific_journal-1255	72	19	is	be	AUX
american_scientific_journal-1255	72	20	odd	odd	ADJ
american_scientific_journal-1255	72	21	,	,	PUNCT
american_scientific_journal-1255	72	22	𝑛	𝑛	PRON
american_scientific_journal-1255	72	23	is	be	AUX
american_scientific_journal-1255	72	24	even	even	ADV
american_scientific_journal-1255	72	25	.	.	PUNCT
american_scientific_journal-1255	73	1	consequently	consequently	ADV
american_scientific_journal-1255	73	2	,	,	PUNCT
american_scientific_journal-1255	73	3	𝜒	𝜒	PROPN
american_scientific_journal-1255	73	4	�	�	NOUN
american_scientific_journal-1255	73	5	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	73	6	�	�	PROPN
american_scientific_journal-1255	73	7	=	=	PROPN
american_scientific_journal-1255	73	8	2	2	X
american_scientific_journal-1255	73	9	.	.	PUNCT
american_scientific_journal-1255	74	1	∎	∎	NOUN
american_scientific_journal-1255	74	2	it	it	PRON
american_scientific_journal-1255	74	3	is	be	AUX
american_scientific_journal-1255	74	4	easy	easy	ADJ
american_scientific_journal-1255	74	5	to	to	PART
american_scientific_journal-1255	74	6	prove	prove	VERB
american_scientific_journal-1255	74	7	that	that	SCONJ
american_scientific_journal-1255	74	8	the	the	DET
american_scientific_journal-1255	74	9	next	next	ADJ
american_scientific_journal-1255	74	10	corollaries	corollary	NOUN
american_scientific_journal-1255	74	11	hold	hold	VERB
american_scientific_journal-1255	74	12	.	.	PUNCT
american_scientific_journal-1255	75	1	proofs	proof	NOUN
american_scientific_journal-1255	75	2	are	be	AUX
american_scientific_journal-1255	75	3	similar	similar	ADJ
american_scientific_journal-1255	75	4	to	to	AUX
american_scientific_journal-1255	75	5	theorem	theorem	VERB
american_scientific_journal-1255	75	6	3.1	3.1	NUM
american_scientific_journal-1255	75	7	.	.	PUNCT
american_scientific_journal-1255	75	8	corollary	corollary	ADJ
american_scientific_journal-1255	75	9	3.1	3.1	NUM
american_scientific_journal-1255	75	10	for	for	ADP
american_scientific_journal-1255	75	11	all	all	DET
american_scientific_journal-1255	75	12	integers	integer	NOUN
american_scientific_journal-1255	75	13	𝑘	𝑘	DET
american_scientific_journal-1255	75	14	≥	≥	NOUN
american_scientific_journal-1255	75	15	3	3	NUM
american_scientific_journal-1255	75	16	,	,	PUNCT
american_scientific_journal-1255	75	17	𝑘	𝑘	PRON
american_scientific_journal-1255	75	18	is	be	AUX
american_scientific_journal-1255	75	19	odd	odd	ADJ
american_scientific_journal-1255	75	20	,	,	PUNCT
american_scientific_journal-1255	75	21	𝜒	𝜒	NOUN
american_scientific_journal-1255	75	22	�	�	PROPN
american_scientific_journal-1255	75	23	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	75	24	�	�	NOUN
american_scientific_journal-1255	75	25	=	=	SYM
american_scientific_journal-1255	75	26	3	3	NUM
american_scientific_journal-1255	75	27	𝑖𝑓	𝑖𝑓	ADP
american_scientific_journal-1255	75	28	:	:	PUNCT
american_scientific_journal-1255	75	29	•	•	NUM
american_scientific_journal-1255	75	30	𝑚,𝑛	𝑚,𝑛	NOUN
american_scientific_journal-1255	75	31	are	be	AUX
american_scientific_journal-1255	75	32	both	both	PRON
american_scientific_journal-1255	75	33	odd	odd	ADJ
american_scientific_journal-1255	75	34	•	•	NOUN
american_scientific_journal-1255	75	35	𝑚	𝑚	NOUN
american_scientific_journal-1255	75	36	is	be	AUX
american_scientific_journal-1255	75	37	odd	odd	ADJ
american_scientific_journal-1255	75	38	,	,	PUNCT
american_scientific_journal-1255	75	39	𝑛	𝑛	PRON
american_scientific_journal-1255	75	40	is	be	AUX
american_scientific_journal-1255	75	41	even	even	ADV
american_scientific_journal-1255	75	42	•	•	NUM
american_scientific_journal-1255	75	43	𝑚	𝑚	NOUN
american_scientific_journal-1255	75	44	is	be	AUX
american_scientific_journal-1255	75	45	even	even	ADV
american_scientific_journal-1255	75	46	,	,	PUNCT
american_scientific_journal-1255	75	47	𝑛	𝑛	PRON
american_scientific_journal-1255	75	48	is	be	AUX
american_scientific_journal-1255	75	49	odd	odd	ADJ
american_scientific_journal-1255	75	50	and	and	CCONJ
american_scientific_journal-1255	75	51	𝜒	𝜒	NOUN
american_scientific_journal-1255	75	52	�	�	PROPN
american_scientific_journal-1255	75	53	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	75	54	�	�	NOUN
american_scientific_journal-1255	75	55	=	=	SYM
american_scientific_journal-1255	75	56	2	2	NUM
american_scientific_journal-1255	75	57	𝑖𝑓	𝑖𝑓	NOUN
american_scientific_journal-1255	75	58	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	75	59	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
american_scientific_journal-1255	75	60	𝑏𝑜𝑡ℎ	𝑏𝑜𝑡ℎ	PROPN
american_scientific_journal-1255	75	61	𝑒𝑣𝑒𝑛.	𝑒𝑣𝑒𝑛.	PROPN
american_scientific_journal-1255	75	62	corollary	corollary	ADJ
american_scientific_journal-1255	75	63	3.2	3.2	NUM
american_scientific_journal-1255	75	64	for	for	ADP
american_scientific_journal-1255	75	65	all	all	DET
american_scientific_journal-1255	75	66	integers	integer	NOUN
american_scientific_journal-1255	75	67	𝑘	𝑘	DET
american_scientific_journal-1255	75	68	≥	≥	NOUN
american_scientific_journal-1255	75	69	2	2	NUM
american_scientific_journal-1255	75	70	,	,	PUNCT
american_scientific_journal-1255	75	71	𝑘	𝑘	PRON
american_scientific_journal-1255	75	72	is	be	AUX
american_scientific_journal-1255	75	73	even	even	ADV
american_scientific_journal-1255	75	74	,	,	PUNCT
american_scientific_journal-1255	75	75	𝜒	𝜒	X
american_scientific_journal-1255	75	76	�	�	PROPN
american_scientific_journal-1255	75	77	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	75	78	�	�	NOUN
american_scientific_journal-1255	75	79	=	=	SYM
american_scientific_journal-1255	75	80	2	2	X
american_scientific_journal-1255	75	81	.	.	PUNCT
american_scientific_journal-1255	75	82	theorem	theorem	VERB
american_scientific_journal-1255	75	83	3.2	3.2	NUM
american_scientific_journal-1255	75	84	for	for	ADP
american_scientific_journal-1255	75	85	all	all	DET
american_scientific_journal-1255	75	86	integers	integer	NOUN
american_scientific_journal-1255	75	87	𝑚,𝑛	𝑚,𝑛	VERB
american_scientific_journal-1255	75	88	≥	≥	NOUN
american_scientific_journal-1255	75	89	3	3	NUM
american_scientific_journal-1255	75	90	and	and	CCONJ
american_scientific_journal-1255	75	91	for	for	ADP
american_scientific_journal-1255	75	92	integers	integer	NOUN
american_scientific_journal-1255	75	93	𝑘	𝑘	X
american_scientific_journal-1255	75	94	=	=	SYM
american_scientific_journal-1255	75	95	3	3	NUM
american_scientific_journal-1255	75	96	,	,	PUNCT
american_scientific_journal-1255	75	97	4	4	NUM
american_scientific_journal-1255	75	98	,	,	PUNCT
american_scientific_journal-1255	75	99	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	75	100	�	�	PROPN
american_scientific_journal-1255	75	101	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	75	102	�	�	PROPN
american_scientific_journal-1255	75	103	=	=	SYM
american_scientific_journal-1255	75	104	3	3	NUM
american_scientific_journal-1255	75	105	.	.	NUM
american_scientific_journal-1255	75	106	104	104	NUM
american_scientific_journal-1255	75	107	american	american	PROPN
american_scientific_journal-1255	75	108	scientific	scientific	ADJ
american_scientific_journal-1255	75	109	research	research	NOUN
american_scientific_journal-1255	75	110	journal	journal	NOUN
american_scientific_journal-1255	75	111	for	for	ADP
american_scientific_journal-1255	75	112	engineering	engineering	NOUN
american_scientific_journal-1255	75	113	,	,	PUNCT
american_scientific_journal-1255	75	114	technology	technology	NOUN
american_scientific_journal-1255	75	115	,	,	PUNCT
american_scientific_journal-1255	75	116	and	and	CCONJ
american_scientific_journal-1255	75	117	sciences	science	NOUN
american_scientific_journal-1255	75	118	(	(	PUNCT
american_scientific_journal-1255	75	119	asrjets	asrjet	NOUN
american_scientific_journal-1255	75	120	)	)	PUNCT
american_scientific_journal-1255	75	121	(	(	PUNCT
american_scientific_journal-1255	75	122	2016	2016	NUM
american_scientific_journal-1255	75	123	)	)	PUNCT
american_scientific_journal-1255	75	124	volume	volume	NOUN
american_scientific_journal-1255	75	125	16	16	NUM
american_scientific_journal-1255	75	126	,	,	PUNCT
american_scientific_journal-1255	75	127	no	no	DET
american_scientific_journal-1255	75	128	1	1	NUM
american_scientific_journal-1255	75	129	,	,	PUNCT
american_scientific_journal-1255	75	130	pp	pp	ADV
american_scientific_journal-1255	75	131	98	98	NUM
american_scientific_journal-1255	75	132	-	-	SYM
american_scientific_journal-1255	75	133	107	107	NUM
american_scientific_journal-1255	75	134	proof	proof	NOUN
american_scientific_journal-1255	75	135	:	:	PUNCT
american_scientific_journal-1255	75	136	•	•	NUM
american_scientific_journal-1255	75	137	case	case	NOUN
american_scientific_journal-1255	75	138	1	1	NUM
american_scientific_journal-1255	75	139	:	:	PUNCT
american_scientific_journal-1255	75	140	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	75	141	are	be	AUX
american_scientific_journal-1255	75	142	both	both	PRON
american_scientific_journal-1255	75	143	odd	odd	ADJ
american_scientific_journal-1255	75	144	if	if	SCONJ
american_scientific_journal-1255	75	145	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	75	146	are	be	AUX
american_scientific_journal-1255	75	147	both	both	PRON
american_scientific_journal-1255	75	148	odd	odd	ADJ
american_scientific_journal-1255	75	149	,	,	PUNCT
american_scientific_journal-1255	75	150	then	then	ADV
american_scientific_journal-1255	75	151	by	by	ADP
american_scientific_journal-1255	75	152	equation	equation	NOUN
american_scientific_journal-1255	75	153	(	(	PUNCT
american_scientific_journal-1255	75	154	1.2	1.2	NUM
american_scientific_journal-1255	75	155	)	)	PUNCT
american_scientific_journal-1255	75	156	,	,	PUNCT
american_scientific_journal-1255	75	157	𝜒𝑒(𝐶𝑚	𝜒𝑒(𝐶𝑚	PROPN
american_scientific_journal-1255	75	158	)	)	PUNCT
american_scientific_journal-1255	75	159	=	=	SYM
american_scientific_journal-1255	75	160	𝜒𝑒(𝐶𝑛	𝜒𝑒(𝐶𝑛	PROPN
american_scientific_journal-1255	75	161	)	)	PUNCT
american_scientific_journal-1255	75	162	=	=	SYM
american_scientific_journal-1255	75	163	3	3	X
american_scientific_journal-1255	75	164	.	.	PUNCT
american_scientific_journal-1255	76	1	thus	thus	ADV
american_scientific_journal-1255	76	2	,	,	PUNCT
american_scientific_journal-1255	76	3	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	76	4	�	�	PROPN
american_scientific_journal-1255	76	5	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	76	6	�	�	PROPN
american_scientific_journal-1255	76	7	≥	≥	NUM
american_scientific_journal-1255	76	8	3	3	NUM
american_scientific_journal-1255	76	9	.	.	PUNCT
american_scientific_journal-1255	76	10	suppose	suppose	VERB
american_scientific_journal-1255	76	11	the	the	DET
american_scientific_journal-1255	76	12	edges	edge	NOUN
american_scientific_journal-1255	76	13	of	of	ADP
american_scientific_journal-1255	76	14	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	76	15	and	and	CCONJ
american_scientific_journal-1255	76	16	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	76	17	are	be	AUX
american_scientific_journal-1255	76	18	colored	color	VERB
american_scientific_journal-1255	76	19	with	with	ADP
american_scientific_journal-1255	76	20	the	the	DET
american_scientific_journal-1255	76	21	same	same	ADJ
american_scientific_journal-1255	76	22	set	set	NOUN
american_scientific_journal-1255	76	23	of	of	ADP
american_scientific_journal-1255	76	24	different	different	ADJ
american_scientific_journal-1255	76	25	colors	color	NOUN
american_scientific_journal-1255	76	26	,	,	PUNCT
american_scientific_journal-1255	76	27	say	say	VERB
american_scientific_journal-1255	76	28	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	76	29	,	,	PUNCT
american_scientific_journal-1255	76	30	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	76	31	,	,	PUNCT
american_scientific_journal-1255	76	32	𝑏3	𝑏3	PROPN
american_scientific_journal-1255	76	33	.	.	NOUN
american_scientific_journal-1255	76	34	•	•	NUM
american_scientific_journal-1255	76	35	subcase	subcase	NOUN
american_scientific_journal-1255	76	36	1.1	1.1	NUM
american_scientific_journal-1255	76	37	:	:	PUNCT
american_scientific_journal-1255	76	38	if	if	SCONJ
american_scientific_journal-1255	76	39	𝑘	𝑘	X
american_scientific_journal-1255	76	40	=	=	SYM
american_scientific_journal-1255	76	41	3	3	NUM
american_scientific_journal-1255	76	42	note	note	NOUN
american_scientific_journal-1255	76	43	that	that	SCONJ
american_scientific_journal-1255	76	44	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	76	45	has	have	VERB
american_scientific_journal-1255	76	46	two	two	NUM
american_scientific_journal-1255	76	47	edges	edge	NOUN
american_scientific_journal-1255	76	48	and	and	CCONJ
american_scientific_journal-1255	76	49	suppose	suppose	VERB
american_scientific_journal-1255	76	50	we	we	PRON
american_scientific_journal-1255	76	51	color	color	VERB
american_scientific_journal-1255	76	52	its	its	PRON
american_scientific_journal-1255	76	53	edges	edge	NOUN
american_scientific_journal-1255	76	54	with	with	ADP
american_scientific_journal-1255	76	55	two	two	NUM
american_scientific_journal-1255	76	56	from	from	ADP
american_scientific_journal-1255	76	57	the	the	DET
american_scientific_journal-1255	76	58	set	set	NOUN
american_scientific_journal-1255	76	59	of	of	ADP
american_scientific_journal-1255	76	60	different	different	ADJ
american_scientific_journal-1255	76	61	colors	color	NOUN
american_scientific_journal-1255	76	62	that	that	PRON
american_scientific_journal-1255	76	63	color	color	VERB
american_scientific_journal-1255	76	64	the	the	DET
american_scientific_journal-1255	76	65	edges	edge	NOUN
american_scientific_journal-1255	76	66	of	of	ADP
american_scientific_journal-1255	76	67	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	76	68	and	and	CCONJ
american_scientific_journal-1255	76	69	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	76	70	,	,	PUNCT
american_scientific_journal-1255	76	71	say	say	VERB
american_scientific_journal-1255	76	72	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	76	73	,	,	PUNCT
american_scientific_journal-1255	76	74	𝑏𝑗.	𝑏𝑗.	ADP
american_scientific_journal-1255	76	75	then	then	ADV
american_scientific_journal-1255	76	76	𝑏𝑖	𝑏𝑖	SCONJ
american_scientific_journal-1255	76	77	color	color	NOUN
american_scientific_journal-1255	76	78	of	of	ADP
american_scientific_journal-1255	76	79	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	76	80	is	be	AUX
american_scientific_journal-1255	76	81	attached	attach	VERB
american_scientific_journal-1255	76	82	to	to	ADP
american_scientific_journal-1255	76	83	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	76	84	and	and	CCONJ
american_scientific_journal-1255	76	85	is	be	AUX
american_scientific_journal-1255	76	86	incident	incident	NOUN
american_scientific_journal-1255	76	87	to	to	ADP
american_scientific_journal-1255	76	88	edges	edge	NOUN
american_scientific_journal-1255	76	89	colored	color	VERB
american_scientific_journal-1255	76	90	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	76	91	and	and	CCONJ
american_scientific_journal-1255	76	92	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	76	93	of	of	ADP
american_scientific_journal-1255	76	94	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	76	95	,	,	PUNCT
american_scientific_journal-1255	76	96	while	while	SCONJ
american_scientific_journal-1255	76	97	the	the	DET
american_scientific_journal-1255	76	98	other	other	ADJ
american_scientific_journal-1255	76	99	color	color	NOUN
american_scientific_journal-1255	76	100	of	of	ADP
american_scientific_journal-1255	76	101	the	the	DET
american_scientific_journal-1255	76	102	edge	edge	NOUN
american_scientific_journal-1255	76	103	of	of	ADP
american_scientific_journal-1255	76	104	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	76	105	say	say	VERB
american_scientific_journal-1255	76	106	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	76	107	is	be	AUX
american_scientific_journal-1255	76	108	attached	attach	VERB
american_scientific_journal-1255	76	109	to	to	ADP
american_scientific_journal-1255	76	110	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	76	111	and	and	CCONJ
american_scientific_journal-1255	76	112	is	be	AUX
american_scientific_journal-1255	76	113	incident	incident	NOUN
american_scientific_journal-1255	76	114	to	to	ADP
american_scientific_journal-1255	76	115	edges	edge	NOUN
american_scientific_journal-1255	76	116	colored	color	VERB
american_scientific_journal-1255	76	117	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	76	118	and	and	CCONJ
american_scientific_journal-1255	76	119	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	76	120	of	of	ADP
american_scientific_journal-1255	76	121	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	76	122	,	,	PUNCT
american_scientific_journal-1255	76	123	where	where	SCONJ
american_scientific_journal-1255	76	124	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	76	125	≠	≠	PROPN
american_scientific_journal-1255	76	126	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	76	127	≠	≠	PROPN
american_scientific_journal-1255	76	128	𝑏𝑘.	𝑏𝑘.	X
american_scientific_journal-1255	76	129	hence	hence	ADV
american_scientific_journal-1255	76	130	,	,	PUNCT
american_scientific_journal-1255	76	131	three	three	NUM
american_scientific_journal-1255	76	132	is	be	AUX
american_scientific_journal-1255	76	133	the	the	DET
american_scientific_journal-1255	76	134	minimum	minimum	ADJ
american_scientific_journal-1255	76	135	number	number	NOUN
american_scientific_journal-1255	76	136	of	of	ADP
american_scientific_journal-1255	76	137	colors	color	NOUN
american_scientific_journal-1255	76	138	that	that	PRON
american_scientific_journal-1255	76	139	color	color	VERB
american_scientific_journal-1255	76	140	the	the	DET
american_scientific_journal-1255	76	141	edges	edge	NOUN
american_scientific_journal-1255	76	142	of	of	ADP
american_scientific_journal-1255	76	143	𝑂3𝑛,𝑚.	𝑂3𝑛,𝑚.	NOUN
american_scientific_journal-1255	76	144	consequently	consequently	ADV
american_scientific_journal-1255	76	145	,	,	PUNCT
american_scientific_journal-1255	76	146	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	76	147	�	�	PROPN
american_scientific_journal-1255	76	148	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	76	149	�	�	PROPN
american_scientific_journal-1255	76	150	=	=	SYM
american_scientific_journal-1255	76	151	3	3	NUM
american_scientific_journal-1255	76	152	.	.	NOUN
american_scientific_journal-1255	76	153	•	•	NUM
american_scientific_journal-1255	76	154	subcase	subcase	NOUN
american_scientific_journal-1255	76	155	2.1	2.1	NUM
american_scientific_journal-1255	76	156	:	:	PUNCT
american_scientific_journal-1255	76	157	if	if	SCONJ
american_scientific_journal-1255	76	158	𝑘	𝑘	X
american_scientific_journal-1255	76	159	=	=	SYM
american_scientific_journal-1255	76	160	4	4	NUM
american_scientific_journal-1255	76	161	note	note	VERB
american_scientific_journal-1255	76	162	that	that	SCONJ
american_scientific_journal-1255	76	163	by	by	ADP
american_scientific_journal-1255	76	164	remark	remark	NOUN
american_scientific_journal-1255	76	165	3.1	3.1	NUM
american_scientific_journal-1255	76	166	,	,	PUNCT
american_scientific_journal-1255	76	167	𝜒𝑒(𝑃4	𝜒𝑒(𝑃4	NUM
american_scientific_journal-1255	76	168	)	)	PUNCT
american_scientific_journal-1255	76	169	=	=	SYM
american_scientific_journal-1255	76	170	2	2	NUM
american_scientific_journal-1255	76	171	and	and	CCONJ
american_scientific_journal-1255	76	172	suppose	suppose	VERB
american_scientific_journal-1255	76	173	the	the	DET
american_scientific_journal-1255	76	174	edges	edge	NOUN
american_scientific_journal-1255	76	175	of	of	ADP
american_scientific_journal-1255	76	176	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	76	177	are	be	AUX
american_scientific_journal-1255	76	178	colored	color	VERB
american_scientific_journal-1255	76	179	with	with	ADP
american_scientific_journal-1255	76	180	two	two	NUM
american_scientific_journal-1255	76	181	from	from	ADP
american_scientific_journal-1255	76	182	the	the	DET
american_scientific_journal-1255	76	183	same	same	ADJ
american_scientific_journal-1255	76	184	set	set	NOUN
american_scientific_journal-1255	76	185	of	of	ADP
american_scientific_journal-1255	76	186	different	different	ADJ
american_scientific_journal-1255	76	187	colors	color	NOUN
american_scientific_journal-1255	76	188	that	that	PRON
american_scientific_journal-1255	76	189	color	color	VERB
american_scientific_journal-1255	76	190	the	the	DET
american_scientific_journal-1255	76	191	edges	edge	NOUN
american_scientific_journal-1255	76	192	of	of	ADP
american_scientific_journal-1255	76	193	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	76	194	and	and	CCONJ
american_scientific_journal-1255	76	195	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	76	196	,	,	PUNCT
american_scientific_journal-1255	76	197	say	say	VERB
american_scientific_journal-1255	76	198	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	76	199	,	,	PUNCT
american_scientific_journal-1255	76	200	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	76	201	.	.	PUNCT
american_scientific_journal-1255	77	1	since	since	SCONJ
american_scientific_journal-1255	77	2	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	77	3	has	have	VERB
american_scientific_journal-1255	77	4	three	three	NUM
american_scientific_journal-1255	77	5	edges	edge	NOUN
american_scientific_journal-1255	77	6	,	,	PUNCT
american_scientific_journal-1255	77	7	suppose	suppose	VERB
american_scientific_journal-1255	77	8	that	that	SCONJ
american_scientific_journal-1255	77	9	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	77	10	is	be	AUX
american_scientific_journal-1255	77	11	colored	color	VERB
american_scientific_journal-1255	77	12	with	with	ADP
american_scientific_journal-1255	77	13	𝑏𝑖′𝑠	𝑏𝑖′𝑠	PROPN
american_scientific_journal-1255	77	14	and	and	CCONJ
american_scientific_journal-1255	77	15	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	77	16	,	,	PUNCT
american_scientific_journal-1255	77	17	𝑖	𝑖	PROPN
american_scientific_journal-1255	77	18	=	=	NOUN
american_scientific_journal-1255	77	19	1,2,3	1,2,3	NUM
american_scientific_journal-1255	77	20	,	,	PUNCT
american_scientific_journal-1255	77	21	𝑗	𝑗	NOUN
american_scientific_journal-1255	77	22	=	=	NOUN
american_scientific_journal-1255	77	23	1,2,3	1,2,3	NUM
american_scientific_journal-1255	77	24	such	such	ADJ
american_scientific_journal-1255	77	25	that	that	SCONJ
american_scientific_journal-1255	77	26	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	77	27	is	be	AUX
american_scientific_journal-1255	77	28	the	the	DET
american_scientific_journal-1255	77	29	middle	middle	ADJ
american_scientific_journal-1255	77	30	edge	edge	NOUN
american_scientific_journal-1255	77	31	and	and	CCONJ
american_scientific_journal-1255	77	32	the	the	DET
american_scientific_journal-1255	77	33	two	two	NUM
american_scientific_journal-1255	77	34	𝑏𝑖′𝑠	𝑏𝑖′𝑠	PROPN
american_scientific_journal-1255	77	35	are	be	AUX
american_scientific_journal-1255	77	36	first	first	ADJ
american_scientific_journal-1255	77	37	and	and	CCONJ
american_scientific_journal-1255	77	38	the	the	DET
american_scientific_journal-1255	77	39	last	last	ADJ
american_scientific_journal-1255	77	40	edges	edge	NOUN
american_scientific_journal-1255	77	41	.	.	PUNCT
american_scientific_journal-1255	78	1	then	then	ADV
american_scientific_journal-1255	78	2	,	,	PUNCT
american_scientific_journal-1255	78	3	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	78	4	of	of	ADP
american_scientific_journal-1255	78	5	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	78	6	is	be	AUX
american_scientific_journal-1255	78	7	connected	connect	VERB
american_scientific_journal-1255	78	8	to	to	ADP
american_scientific_journal-1255	78	9	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	78	10	and	and	CCONJ
american_scientific_journal-1255	78	11	is	be	AUX
american_scientific_journal-1255	78	12	incident	incident	NOUN
american_scientific_journal-1255	78	13	to	to	ADP
american_scientific_journal-1255	78	14	edges	edge	NOUN
american_scientific_journal-1255	78	15	colored	color	VERB
american_scientific_journal-1255	78	16	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	78	17	and	and	CCONJ
american_scientific_journal-1255	78	18	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	78	19	of	of	ADP
american_scientific_journal-1255	78	20	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	78	21	,	,	PUNCT
american_scientific_journal-1255	78	22	while	while	SCONJ
american_scientific_journal-1255	78	23	the	the	DET
american_scientific_journal-1255	78	24	other	other	ADJ
american_scientific_journal-1255	78	25	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	78	26	of	of	ADP
american_scientific_journal-1255	78	27	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	78	28	.	.	PUNCT
american_scientific_journal-1255	79	1	hence	hence	ADV
american_scientific_journal-1255	79	2	,	,	PUNCT
american_scientific_journal-1255	79	3	three	three	NUM
american_scientific_journal-1255	79	4	is	be	AUX
american_scientific_journal-1255	79	5	the	the	DET
american_scientific_journal-1255	79	6	minimum	minimum	ADJ
american_scientific_journal-1255	79	7	number	number	NOUN
american_scientific_journal-1255	79	8	of	of	ADP
american_scientific_journal-1255	79	9	colors	color	NOUN
american_scientific_journal-1255	79	10	that	that	PRON
american_scientific_journal-1255	79	11	color	color	VERB
american_scientific_journal-1255	79	12	the	the	DET
american_scientific_journal-1255	79	13	edges	edge	NOUN
american_scientific_journal-1255	79	14	of	of	ADP
american_scientific_journal-1255	79	15	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	79	16	.	.	PUNCT
american_scientific_journal-1255	80	1	consequently	consequently	ADV
american_scientific_journal-1255	80	2	,	,	PUNCT
american_scientific_journal-1255	80	3	𝜒𝑒	𝜒𝑒	PROPN
american_scientific_journal-1255	80	4	�	�	PROPN
american_scientific_journal-1255	80	5	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	80	6	�	�	PROPN
american_scientific_journal-1255	80	7	=	=	SYM
american_scientific_journal-1255	80	8	3	3	NUM
american_scientific_journal-1255	80	9	.	.	NOUN
american_scientific_journal-1255	80	10	•	•	NUM
american_scientific_journal-1255	80	11	case	case	NOUN
american_scientific_journal-1255	80	12	2	2	NUM
american_scientific_journal-1255	80	13	:	:	PUNCT
american_scientific_journal-1255	80	14	𝑚	𝑚	PROPN
american_scientific_journal-1255	80	15	is	be	AUX
american_scientific_journal-1255	80	16	odd	odd	ADJ
american_scientific_journal-1255	80	17	,	,	PUNCT
american_scientific_journal-1255	80	18	𝑛	𝑛	PRON
american_scientific_journal-1255	80	19	is	be	AUX
american_scientific_journal-1255	80	20	even	even	ADV
american_scientific_journal-1255	80	21	if	if	SCONJ
american_scientific_journal-1255	80	22	𝑚	𝑚	PROPN
american_scientific_journal-1255	80	23	is	be	AUX
american_scientific_journal-1255	80	24	odd	odd	ADJ
american_scientific_journal-1255	80	25	,	,	PUNCT
american_scientific_journal-1255	80	26	𝑛	𝑛	PRON
american_scientific_journal-1255	80	27	is	be	AUX
american_scientific_journal-1255	80	28	even	even	ADV
american_scientific_journal-1255	80	29	,	,	PUNCT
american_scientific_journal-1255	80	30	then	then	ADV
american_scientific_journal-1255	80	31	by	by	ADP
american_scientific_journal-1255	80	32	equation	equation	NOUN
american_scientific_journal-1255	80	33	(	(	PUNCT
american_scientific_journal-1255	80	34	1.2	1.2	NUM
american_scientific_journal-1255	80	35	)	)	PUNCT
american_scientific_journal-1255	80	36	,	,	PUNCT
american_scientific_journal-1255	80	37	𝜒𝑒(𝐶𝑚	𝜒𝑒(𝐶𝑚	PROPN
american_scientific_journal-1255	80	38	)	)	PUNCT
american_scientific_journal-1255	80	39	=	=	SYM
american_scientific_journal-1255	80	40	3	3	NUM
american_scientific_journal-1255	80	41	and	and	CCONJ
american_scientific_journal-1255	80	42	𝜒𝑒(𝐶𝑛	𝜒𝑒(𝐶𝑛	PROPN
american_scientific_journal-1255	80	43	)	)	PUNCT
american_scientific_journal-1255	81	1	=	=	SYM
american_scientific_journal-1255	81	2	2	2	X
american_scientific_journal-1255	81	3	.	.	PUNCT
american_scientific_journal-1255	81	4	suppose	suppose	VERB
american_scientific_journal-1255	81	5	the	the	DET
american_scientific_journal-1255	81	6	edges	edge	NOUN
american_scientific_journal-1255	81	7	of	of	ADP
american_scientific_journal-1255	81	8	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	81	9	are	be	AUX
american_scientific_journal-1255	81	10	colored	color	VERB
american_scientific_journal-1255	81	11	with	with	ADP
american_scientific_journal-1255	81	12	two	two	NUM
american_scientific_journal-1255	81	13	from	from	ADP
american_scientific_journal-1255	81	14	the	the	DET
american_scientific_journal-1255	81	15	same	same	ADJ
american_scientific_journal-1255	81	16	set	set	NOUN
american_scientific_journal-1255	81	17	of	of	ADP
american_scientific_journal-1255	81	18	different	different	ADJ
american_scientific_journal-1255	81	19	colors	color	NOUN
american_scientific_journal-1255	81	20	that	that	PRON
american_scientific_journal-1255	81	21	color	color	VERB
american_scientific_journal-1255	81	22	the	the	DET
american_scientific_journal-1255	81	23	edges	edge	NOUN
american_scientific_journal-1255	81	24	of	of	ADP
american_scientific_journal-1255	81	25	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	81	26	,	,	PUNCT
american_scientific_journal-1255	81	27	say	say	VERB
american_scientific_journal-1255	81	28	𝑏1	𝑏1	NOUN
american_scientific_journal-1255	81	29	,	,	PUNCT
american_scientific_journal-1255	81	30	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	81	31	for	for	ADP
american_scientific_journal-1255	81	32	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	81	33	and	and	CCONJ
american_scientific_journal-1255	81	34	𝑏1	𝑏1	ADJ
american_scientific_journal-1255	81	35	,	,	PUNCT
american_scientific_journal-1255	81	36	𝑏2	𝑏2	NOUN
american_scientific_journal-1255	81	37	,	,	PUNCT
american_scientific_journal-1255	81	38	𝑏3	𝑏3	NOUN
american_scientific_journal-1255	81	39	for	for	ADP
american_scientific_journal-1255	81	40	𝐶𝑚.	𝐶𝑚.	NOUN
american_scientific_journal-1255	81	41	by	by	ADP
american_scientific_journal-1255	81	42	this	this	PRON
american_scientific_journal-1255	81	43	,	,	PUNCT
american_scientific_journal-1255	81	44	the	the	DET
american_scientific_journal-1255	81	45	entire	entire	ADJ
american_scientific_journal-1255	81	46	proof	proof	NOUN
american_scientific_journal-1255	81	47	follows	follow	VERB
american_scientific_journal-1255	81	48	from	from	ADP
american_scientific_journal-1255	81	49	case	case	NOUN
american_scientific_journal-1255	81	50	1	1	NUM
american_scientific_journal-1255	81	51	.	.	NOUN
american_scientific_journal-1255	82	1	•	•	NUM
american_scientific_journal-1255	82	2	case	case	NOUN
american_scientific_journal-1255	82	3	3	3	NUM
american_scientific_journal-1255	82	4	:	:	PUNCT
american_scientific_journal-1255	82	5	𝑚	𝑚	NOUN
american_scientific_journal-1255	82	6	is	be	AUX
american_scientific_journal-1255	82	7	even	even	ADV
american_scientific_journal-1255	82	8	,	,	PUNCT
american_scientific_journal-1255	82	9	𝑛	𝑛	PRON
american_scientific_journal-1255	82	10	is	be	AUX
american_scientific_journal-1255	82	11	odd	odd	ADJ
american_scientific_journal-1255	82	12	if	if	SCONJ
american_scientific_journal-1255	82	13	𝑚	𝑚	PROPN
american_scientific_journal-1255	82	14	is	be	AUX
american_scientific_journal-1255	82	15	even	even	ADV
american_scientific_journal-1255	82	16	,	,	PUNCT
american_scientific_journal-1255	82	17	𝑛	𝑛	PRON
american_scientific_journal-1255	82	18	is	be	AUX
american_scientific_journal-1255	82	19	odd	odd	ADJ
american_scientific_journal-1255	82	20	,	,	PUNCT
american_scientific_journal-1255	82	21	then	then	ADV
american_scientific_journal-1255	82	22	the	the	DET
american_scientific_journal-1255	82	23	proof	proof	NOUN
american_scientific_journal-1255	82	24	of	of	ADP
american_scientific_journal-1255	82	25	this	this	DET
american_scientific_journal-1255	82	26	case	case	NOUN
american_scientific_journal-1255	82	27	is	be	AUX
american_scientific_journal-1255	82	28	similar	similar	ADJ
american_scientific_journal-1255	82	29	to	to	ADP
american_scientific_journal-1255	82	30	case	case	NOUN
american_scientific_journal-1255	82	31	2	2	NUM
american_scientific_journal-1255	82	32	.	.	NOUN
american_scientific_journal-1255	82	33	•	•	NUM
american_scientific_journal-1255	82	34	case	case	NOUN
american_scientific_journal-1255	82	35	4	4	NUM
american_scientific_journal-1255	82	36	:	:	PUNCT
american_scientific_journal-1255	82	37	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	82	38	are	be	AUX
american_scientific_journal-1255	82	39	both	both	PRON
american_scientific_journal-1255	82	40	even	even	ADV
american_scientific_journal-1255	82	41	if	if	SCONJ
american_scientific_journal-1255	82	42	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	82	43	are	be	AUX
american_scientific_journal-1255	82	44	both	both	PRON
american_scientific_journal-1255	82	45	even	even	ADV
american_scientific_journal-1255	82	46	,	,	PUNCT
american_scientific_journal-1255	82	47	then	then	ADV
american_scientific_journal-1255	82	48	by	by	ADP
american_scientific_journal-1255	82	49	equation	equation	NOUN
american_scientific_journal-1255	82	50	(	(	PUNCT
american_scientific_journal-1255	82	51	1.2	1.2	NUM
american_scientific_journal-1255	82	52	)	)	PUNCT
american_scientific_journal-1255	82	53	,	,	PUNCT
american_scientific_journal-1255	82	54	𝜒𝑒(𝐶𝑚	𝜒𝑒(𝐶𝑚	PROPN
american_scientific_journal-1255	82	55	)	)	PUNCT
american_scientific_journal-1255	82	56	=	=	PUNCT
american_scientific_journal-1255	82	57	𝜒𝑒(𝐶𝑛	𝜒𝑒(𝐶𝑛	PROPN
american_scientific_journal-1255	82	58	)	)	PUNCT
american_scientific_journal-1255	83	1	=	=	SYM
american_scientific_journal-1255	83	2	2	2	X
american_scientific_journal-1255	83	3	.	.	PUNCT
american_scientific_journal-1255	83	4	suppose	suppose	VERB
american_scientific_journal-1255	83	5	the	the	DET
american_scientific_journal-1255	83	6	edges	edge	NOUN
american_scientific_journal-1255	83	7	of	of	ADP
american_scientific_journal-1255	83	8	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	83	9	are	be	AUX
american_scientific_journal-1255	83	10	colored	color	VERB
american_scientific_journal-1255	83	11	by	by	ADP
american_scientific_journal-1255	83	12	a	a	DET
american_scientific_journal-1255	83	13	set	set	NOUN
american_scientific_journal-1255	83	14	of	of	ADP
american_scientific_journal-1255	83	15	different	different	ADJ
american_scientific_journal-1255	83	16	colors	color	NOUN
american_scientific_journal-1255	83	17	say	say	VERB
american_scientific_journal-1255	83	18	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	83	19	,	,	PUNCT
american_scientific_journal-1255	83	20	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	83	21	and	and	CCONJ
american_scientific_journal-1255	83	22	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	83	23	is	be	AUX
american_scientific_journal-1255	83	24	colored	color	VERB
american_scientific_journal-1255	83	25	with	with	ADP
american_scientific_journal-1255	83	26	the	the	DET
american_scientific_journal-1255	83	27	set	set	NOUN
american_scientific_journal-1255	83	28	of	of	ADP
american_scientific_journal-1255	83	29	different	different	ADJ
american_scientific_journal-1255	83	30	colors	color	NOUN
american_scientific_journal-1255	83	31	,	,	PUNCT
american_scientific_journal-1255	83	32	say	say	VERB
american_scientific_journal-1255	83	33	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	83	34	,	,	PUNCT
american_scientific_journal-1255	83	35	𝑏𝑘	𝑏𝑘	NOUN
american_scientific_journal-1255	83	36	,	,	PUNCT
american_scientific_journal-1255	83	37	where	where	SCONJ
american_scientific_journal-1255	83	38	𝑖	𝑖	X
american_scientific_journal-1255	83	39	,	,	PUNCT
american_scientific_journal-1255	83	40	𝑗	𝑗	INTJ
american_scientific_journal-1255	83	41	,	,	PUNCT
american_scientific_journal-1255	83	42	𝑘	𝑘	X
american_scientific_journal-1255	83	43	=	=	SYM
american_scientific_journal-1255	83	44	1	1	NUM
american_scientific_journal-1255	83	45	,	,	PUNCT
american_scientific_journal-1255	83	46	2	2	NUM
american_scientific_journal-1255	83	47	,	,	PUNCT
american_scientific_journal-1255	83	48	3	3	NUM
american_scientific_journal-1255	83	49	and	and	CCONJ
american_scientific_journal-1255	83	50	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	83	51	≠	≠	PROPN
american_scientific_journal-1255	83	52	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	83	53	≠	≠	PROPN
american_scientific_journal-1255	83	54	𝑏𝑘.	𝑏𝑘.	X
american_scientific_journal-1255	83	55	•	•	NUM
american_scientific_journal-1255	83	56	subcase	subcase	NOUN
american_scientific_journal-1255	83	57	4.1	4.1	NUM
american_scientific_journal-1255	83	58	:	:	PUNCT
american_scientific_journal-1255	83	59	if	if	SCONJ
american_scientific_journal-1255	83	60	𝑘	𝑘	X
american_scientific_journal-1255	83	61	=	=	SYM
american_scientific_journal-1255	83	62	3	3	NUM
american_scientific_journal-1255	83	63	105	105	NUM
american_scientific_journal-1255	83	64	american	american	ADJ
american_scientific_journal-1255	83	65	scientific	scientific	ADJ
american_scientific_journal-1255	83	66	research	research	NOUN
american_scientific_journal-1255	83	67	journal	journal	NOUN
american_scientific_journal-1255	83	68	for	for	ADP
american_scientific_journal-1255	83	69	engineering	engineering	NOUN
american_scientific_journal-1255	83	70	,	,	PUNCT
american_scientific_journal-1255	83	71	technology	technology	NOUN
american_scientific_journal-1255	83	72	,	,	PUNCT
american_scientific_journal-1255	83	73	and	and	CCONJ
american_scientific_journal-1255	83	74	sciences	science	NOUN
american_scientific_journal-1255	83	75	(	(	PUNCT
american_scientific_journal-1255	83	76	asrjets	asrjet	NOUN
american_scientific_journal-1255	83	77	)	)	PUNCT
american_scientific_journal-1255	83	78	(	(	PUNCT
american_scientific_journal-1255	83	79	2016	2016	NUM
american_scientific_journal-1255	83	80	)	)	PUNCT
american_scientific_journal-1255	83	81	volume	volume	NOUN
american_scientific_journal-1255	83	82	16	16	NUM
american_scientific_journal-1255	83	83	,	,	PUNCT
american_scientific_journal-1255	83	84	no	no	DET
american_scientific_journal-1255	83	85	1	1	NUM
american_scientific_journal-1255	83	86	,	,	PUNCT
american_scientific_journal-1255	83	87	pp	pp	ADV
american_scientific_journal-1255	83	88	98	98	NUM
american_scientific_journal-1255	83	89	-	-	SYM
american_scientific_journal-1255	83	90	107	107	NUM
american_scientific_journal-1255	83	91	note	note	NOUN
american_scientific_journal-1255	83	92	that	that	SCONJ
american_scientific_journal-1255	83	93	by	by	ADP
american_scientific_journal-1255	83	94	remark	remark	NOUN
american_scientific_journal-1255	83	95	2.1	2.1	NUM
american_scientific_journal-1255	83	96	,	,	PUNCT
american_scientific_journal-1255	83	97	𝜒𝑒(𝑃3	𝜒𝑒(𝑃3	X
american_scientific_journal-1255	83	98	)	)	PUNCT
american_scientific_journal-1255	83	99	=	=	SYM
american_scientific_journal-1255	84	1	2	2	X
american_scientific_journal-1255	84	2	.	.	PUNCT
american_scientific_journal-1255	84	3	clearly	clearly	ADV
american_scientific_journal-1255	84	4	,	,	PUNCT
american_scientific_journal-1255	84	5	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	84	6	�	�	PROPN
american_scientific_journal-1255	84	7	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	PROPN
american_scientific_journal-1255	84	8	�	�	PROPN
american_scientific_journal-1255	84	9	≥	≥	NUM
american_scientific_journal-1255	84	10	3	3	NUM
american_scientific_journal-1255	84	11	since	since	SCONJ
american_scientific_journal-1255	84	12	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	84	13	has	have	VERB
american_scientific_journal-1255	84	14	three	three	NUM
american_scientific_journal-1255	84	15	edges	edge	NOUN
american_scientific_journal-1255	84	16	incident	incident	NOUN
american_scientific_journal-1255	84	17	to	to	ADP
american_scientific_journal-1255	84	18	each	each	DET
american_scientific_journal-1255	84	19	other	other	ADJ
american_scientific_journal-1255	84	20	at	at	ADP
american_scientific_journal-1255	84	21	the	the	DET
american_scientific_journal-1255	84	22	endpoints	endpoint	NOUN
american_scientific_journal-1255	84	23	of	of	ADP
american_scientific_journal-1255	84	24	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	84	25	.	.	PUNCT
american_scientific_journal-1255	85	1	then	then	ADV
american_scientific_journal-1255	85	2	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	85	3	color	color	NOUN
american_scientific_journal-1255	85	4	of	of	ADP
american_scientific_journal-1255	85	5	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	85	6	is	be	AUX
american_scientific_journal-1255	85	7	attached	attach	VERB
american_scientific_journal-1255	85	8	to	to	ADP
american_scientific_journal-1255	85	9	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	85	10	and	and	CCONJ
american_scientific_journal-1255	85	11	is	be	AUX
american_scientific_journal-1255	85	12	incident	incident	NOUN
american_scientific_journal-1255	85	13	to	to	ADP
american_scientific_journal-1255	85	14	edges	edge	NOUN
american_scientific_journal-1255	85	15	colored	color	VERB
american_scientific_journal-1255	85	16	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	85	17	and	and	CCONJ
american_scientific_journal-1255	85	18	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	85	19	of	of	ADP
american_scientific_journal-1255	85	20	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	85	21	,	,	PUNCT
american_scientific_journal-1255	85	22	while	while	SCONJ
american_scientific_journal-1255	85	23	the	the	DET
american_scientific_journal-1255	85	24	other	other	ADJ
american_scientific_journal-1255	85	25	color	color	NOUN
american_scientific_journal-1255	85	26	of	of	ADP
american_scientific_journal-1255	85	27	the	the	DET
american_scientific_journal-1255	85	28	edge	edge	NOUN
american_scientific_journal-1255	85	29	of	of	ADP
american_scientific_journal-1255	85	30	𝑃3	𝑃3	NOUN
american_scientific_journal-1255	85	31	say	say	VERB
american_scientific_journal-1255	85	32	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	85	33	is	be	AUX
american_scientific_journal-1255	85	34	attached	attach	VERB
american_scientific_journal-1255	85	35	to	to	ADP
american_scientific_journal-1255	85	36	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	85	37	and	and	CCONJ
american_scientific_journal-1255	85	38	is	be	AUX
american_scientific_journal-1255	85	39	incident	incident	NOUN
american_scientific_journal-1255	85	40	to	to	ADP
american_scientific_journal-1255	85	41	edges	edge	NOUN
american_scientific_journal-1255	85	42	colored	color	VERB
american_scientific_journal-1255	85	43	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	85	44	and	and	CCONJ
american_scientific_journal-1255	85	45	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	85	46	of	of	ADP
american_scientific_journal-1255	85	47	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	85	48	and	and	CCONJ
american_scientific_journal-1255	85	49	is	be	AUX
american_scientific_journal-1255	85	50	incident	incident	NOUN
american_scientific_journal-1255	85	51	to	to	ADP
american_scientific_journal-1255	85	52	edges	edge	NOUN
american_scientific_journal-1255	85	53	colored	color	VERB
american_scientific_journal-1255	85	54	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	85	55	and	and	CCONJ
american_scientific_journal-1255	85	56	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	85	57	of	of	ADP
american_scientific_journal-1255	85	58	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	85	59	,	,	PUNCT
american_scientific_journal-1255	85	60	where	where	SCONJ
american_scientific_journal-1255	85	61	𝑏𝑖	𝑏𝑖	ADP
american_scientific_journal-1255	85	62	≠	≠	PROPN
american_scientific_journal-1255	85	63	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	85	64	≠	≠	PROPN
american_scientific_journal-1255	85	65	𝑏𝑘.	𝑏𝑘.	X
american_scientific_journal-1255	85	66	hence	hence	ADV
american_scientific_journal-1255	85	67	,	,	PUNCT
american_scientific_journal-1255	85	68	three	three	NUM
american_scientific_journal-1255	85	69	is	be	AUX
american_scientific_journal-1255	85	70	the	the	DET
american_scientific_journal-1255	85	71	minimum	minimum	ADJ
american_scientific_journal-1255	85	72	number	number	NOUN
american_scientific_journal-1255	85	73	of	of	ADP
american_scientific_journal-1255	85	74	colors	color	NOUN
american_scientific_journal-1255	85	75	that	that	PRON
american_scientific_journal-1255	85	76	color	color	VERB
american_scientific_journal-1255	85	77	the	the	DET
american_scientific_journal-1255	85	78	edges	edge	NOUN
american_scientific_journal-1255	85	79	of	of	ADP
american_scientific_journal-1255	85	80	𝑂3𝑛,𝑚.	𝑂3𝑛,𝑚.	NOUN
american_scientific_journal-1255	85	81	consequently	consequently	ADV
american_scientific_journal-1255	85	82	,	,	PUNCT
american_scientific_journal-1255	85	83	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	85	84	�	�	PROPN
american_scientific_journal-1255	85	85	𝑂3𝑛,𝑚	𝑂3𝑛,𝑚	NOUN
american_scientific_journal-1255	85	86	�	�	PROPN
american_scientific_journal-1255	85	87	=	=	SYM
american_scientific_journal-1255	86	1	3	3	NUM
american_scientific_journal-1255	86	2	.	.	NOUN
american_scientific_journal-1255	86	3	•	•	NUM
american_scientific_journal-1255	86	4	subcase	subcase	NOUN
american_scientific_journal-1255	86	5	4.2	4.2	NUM
american_scientific_journal-1255	86	6	:	:	PUNCT
american_scientific_journal-1255	86	7	if	if	SCONJ
american_scientific_journal-1255	86	8	𝑘	𝑘	X
american_scientific_journal-1255	86	9	=	=	SYM
american_scientific_journal-1255	86	10	4	4	NUM
american_scientific_journal-1255	86	11	note	note	NOUN
american_scientific_journal-1255	86	12	that	that	SCONJ
american_scientific_journal-1255	86	13	𝑃4	𝑃4	NOUN
american_scientific_journal-1255	86	14	has	have	VERB
american_scientific_journal-1255	86	15	three	three	NUM
american_scientific_journal-1255	86	16	edges	edge	NOUN
american_scientific_journal-1255	86	17	and	and	CCONJ
american_scientific_journal-1255	86	18	by	by	ADP
american_scientific_journal-1255	86	19	remark	remark	NOUN
american_scientific_journal-1255	86	20	3.1	3.1	NUM
american_scientific_journal-1255	86	21	,	,	PUNCT
american_scientific_journal-1255	86	22	𝜒𝑒(𝑃4	𝜒𝑒(𝑃4	NUM
american_scientific_journal-1255	86	23	)	)	PUNCT
american_scientific_journal-1255	86	24	=	=	SYM
american_scientific_journal-1255	87	1	2	2	X
american_scientific_journal-1255	87	2	.	.	PUNCT
american_scientific_journal-1255	87	3	clearly	clearly	ADV
american_scientific_journal-1255	87	4	,	,	PUNCT
american_scientific_journal-1255	87	5	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	87	6	�	�	PROPN
american_scientific_journal-1255	87	7	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	PROPN
american_scientific_journal-1255	87	8	�	�	PROPN
american_scientific_journal-1255	87	9	≥	≥	NUM
american_scientific_journal-1255	87	10	3	3	NUM
american_scientific_journal-1255	87	11	since	since	SCONJ
american_scientific_journal-1255	87	12	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	PROPN
american_scientific_journal-1255	87	13	has	have	VERB
american_scientific_journal-1255	87	14	three	three	NUM
american_scientific_journal-1255	87	15	edges	edge	NOUN
american_scientific_journal-1255	87	16	incident	incident	NOUN
american_scientific_journal-1255	87	17	to	to	ADP
american_scientific_journal-1255	87	18	each	each	DET
american_scientific_journal-1255	87	19	other	other	ADJ
american_scientific_journal-1255	87	20	at	at	ADP
american_scientific_journal-1255	87	21	the	the	DET
american_scientific_journal-1255	87	22	endpoints	endpoint	NOUN
american_scientific_journal-1255	87	23	of	of	ADP
american_scientific_journal-1255	87	24	𝑃𝑘.	𝑃𝑘.	PROPN
american_scientific_journal-1255	87	25	suppose	suppose	VERB
american_scientific_journal-1255	87	26	that	that	SCONJ
american_scientific_journal-1255	87	27	there	there	PRON
american_scientific_journal-1255	87	28	are	be	VERB
american_scientific_journal-1255	87	29	two	two	NUM
american_scientific_journal-1255	87	30	𝑏𝑖′𝑠	𝑏𝑖′𝑠	PROPN
american_scientific_journal-1255	87	31	,	,	PUNCT
american_scientific_journal-1255	87	32	𝑖	𝑖	SYM
american_scientific_journal-1255	87	33	=	=	NOUN
american_scientific_journal-1255	87	34	1	1	NUM
american_scientific_journal-1255	87	35	,	,	PUNCT
american_scientific_journal-1255	87	36	2	2	NUM
american_scientific_journal-1255	87	37	,	,	PUNCT
american_scientific_journal-1255	87	38	3	3	NUM
american_scientific_journal-1255	87	39	and	and	CCONJ
american_scientific_journal-1255	87	40	one	one	NUM
american_scientific_journal-1255	87	41	𝑏𝑗	𝑏𝑗	NOUN
american_scientific_journal-1255	87	42	,	,	PUNCT
american_scientific_journal-1255	87	43	𝑗	𝑗	PROPN
american_scientific_journal-1255	87	44	=	=	ADJ
american_scientific_journal-1255	87	45	1	1	NUM
american_scientific_journal-1255	87	46	,	,	PUNCT
american_scientific_journal-1255	87	47	2	2	NUM
american_scientific_journal-1255	87	48	,	,	PUNCT
american_scientific_journal-1255	87	49	3	3	NUM
american_scientific_journal-1255	87	50	is	be	AUX
american_scientific_journal-1255	87	51	the	the	DET
american_scientific_journal-1255	87	52	color	color	NOUN
american_scientific_journal-1255	87	53	of	of	ADP
american_scientific_journal-1255	87	54	the	the	DET
american_scientific_journal-1255	87	55	middle	middle	ADJ
american_scientific_journal-1255	87	56	edge	edge	NOUN
american_scientific_journal-1255	87	57	,	,	PUNCT
american_scientific_journal-1255	87	58	while	while	SCONJ
american_scientific_journal-1255	87	59	the	the	DET
american_scientific_journal-1255	87	60	two	two	NUM
american_scientific_journal-1255	87	61	𝑏𝑖′𝑠	𝑏𝑖′𝑠	PROPN
american_scientific_journal-1255	87	62	,	,	PUNCT
american_scientific_journal-1255	87	63	𝑖	𝑖	SYM
american_scientific_journal-1255	87	64	=	=	NOUN
american_scientific_journal-1255	87	65	1	1	NUM
american_scientific_journal-1255	87	66	,	,	PUNCT
american_scientific_journal-1255	87	67	2	2	NUM
american_scientific_journal-1255	87	68	,	,	PUNCT
american_scientific_journal-1255	87	69	3	3	NUM
american_scientific_journal-1255	87	70	are	be	AUX
american_scientific_journal-1255	87	71	the	the	DET
american_scientific_journal-1255	87	72	colors	color	NOUN
american_scientific_journal-1255	87	73	of	of	ADP
american_scientific_journal-1255	87	74	the	the	DET
american_scientific_journal-1255	87	75	first	first	ADJ
american_scientific_journal-1255	87	76	and	and	CCONJ
american_scientific_journal-1255	87	77	the	the	DET
american_scientific_journal-1255	87	78	last	last	ADJ
american_scientific_journal-1255	87	79	edges	edge	NOUN
american_scientific_journal-1255	87	80	.	.	PUNCT
american_scientific_journal-1255	88	1	then	then	ADV
american_scientific_journal-1255	88	2	the	the	DET
american_scientific_journal-1255	88	3	first	first	ADJ
american_scientific_journal-1255	88	4	𝑏𝑖	𝑏𝑖	PROPN
american_scientific_journal-1255	88	5	color	color	NOUN
american_scientific_journal-1255	88	6	is	be	AUX
american_scientific_journal-1255	88	7	attached	attach	VERB
american_scientific_journal-1255	88	8	to	to	ADP
american_scientific_journal-1255	88	9	𝐶𝑚	𝐶𝑚	NOUN
american_scientific_journal-1255	88	10	and	and	CCONJ
american_scientific_journal-1255	88	11	is	be	AUX
american_scientific_journal-1255	88	12	incident	incident	NOUN
american_scientific_journal-1255	88	13	to	to	ADP
american_scientific_journal-1255	88	14	edges	edge	NOUN
american_scientific_journal-1255	88	15	colored	color	VERB
american_scientific_journal-1255	88	16	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	88	17	and	and	CCONJ
american_scientific_journal-1255	88	18	𝑏𝑘	𝑏𝑘	PROPN
american_scientific_journal-1255	88	19	colors	color	NOUN
american_scientific_journal-1255	88	20	𝐶𝑚	𝐶𝑚	PROPN
american_scientific_journal-1255	88	21	,	,	PUNCT
american_scientific_journal-1255	88	22	while	while	SCONJ
american_scientific_journal-1255	88	23	the	the	DET
american_scientific_journal-1255	88	24	other	other	ADJ
american_scientific_journal-1255	88	25	𝑏𝑖	𝑏𝑖	NOUN
american_scientific_journal-1255	88	26	is	be	AUX
american_scientific_journal-1255	88	27	connected	connect	VERB
american_scientific_journal-1255	88	28	to	to	ADP
american_scientific_journal-1255	88	29	𝐶𝑛	𝐶𝑛	PROPN
american_scientific_journal-1255	88	30	and	and	CCONJ
american_scientific_journal-1255	88	31	is	be	AUX
american_scientific_journal-1255	88	32	also	also	ADV
american_scientific_journal-1255	88	33	incident	incident	NOUN
american_scientific_journal-1255	88	34	to	to	PART
american_scientific_journal-1255	88	35	edges	edge	VERB
american_scientific_journal-1255	88	36	𝑏𝑗	𝑏𝑗	PROPN
american_scientific_journal-1255	88	37	and	and	CCONJ
american_scientific_journal-1255	88	38	𝑏𝑘	𝑏𝑘	NOUN
american_scientific_journal-1255	88	39	colors	color	NOUN
american_scientific_journal-1255	88	40	of	of	ADP
american_scientific_journal-1255	88	41	𝐶𝑛.	𝐶𝑛.	PROPN
american_scientific_journal-1255	88	42	hence	hence	ADV
american_scientific_journal-1255	88	43	,	,	PUNCT
american_scientific_journal-1255	88	44	three	three	NUM
american_scientific_journal-1255	88	45	is	be	AUX
american_scientific_journal-1255	88	46	the	the	DET
american_scientific_journal-1255	88	47	minimum	minimum	ADJ
american_scientific_journal-1255	88	48	number	number	NOUN
american_scientific_journal-1255	88	49	of	of	ADP
american_scientific_journal-1255	88	50	colors	color	NOUN
american_scientific_journal-1255	88	51	that	that	PRON
american_scientific_journal-1255	88	52	color	color	VERB
american_scientific_journal-1255	88	53	the	the	DET
american_scientific_journal-1255	88	54	edges	edge	NOUN
american_scientific_journal-1255	88	55	of	of	ADP
american_scientific_journal-1255	88	56	𝑂4𝑛,𝑚.	𝑂4𝑛,𝑚.	PROPN
american_scientific_journal-1255	88	57	consequently	consequently	ADV
american_scientific_journal-1255	88	58	,	,	PUNCT
american_scientific_journal-1255	88	59	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	88	60	�	�	PROPN
american_scientific_journal-1255	88	61	𝑂4𝑛,𝑚	𝑂4𝑛,𝑚	NOUN
american_scientific_journal-1255	88	62	�	�	PROPN
american_scientific_journal-1255	88	63	=	=	SYM
american_scientific_journal-1255	88	64	3	3	X
american_scientific_journal-1255	88	65	.	.	PUNCT
american_scientific_journal-1255	88	66	therefore	therefore	ADV
american_scientific_journal-1255	88	67	for	for	ADP
american_scientific_journal-1255	88	68	any	any	DET
american_scientific_journal-1255	88	69	cases	case	NOUN
american_scientific_journal-1255	88	70	,	,	PUNCT
american_scientific_journal-1255	88	71	the	the	DET
american_scientific_journal-1255	88	72	proof	proof	NOUN
american_scientific_journal-1255	88	73	follows	follow	VERB
american_scientific_journal-1255	88	74	.	.	PUNCT
american_scientific_journal-1255	89	1	similar	similar	ADJ
american_scientific_journal-1255	89	2	arguments	argument	NOUN
american_scientific_journal-1255	89	3	will	will	AUX
american_scientific_journal-1255	89	4	hold	hold	VERB
american_scientific_journal-1255	89	5	for	for	ADP
american_scientific_journal-1255	89	6	the	the	DET
american_scientific_journal-1255	89	7	last	last	ADJ
american_scientific_journal-1255	89	8	theorem	theorem	NOUN
american_scientific_journal-1255	89	9	.	.	PUNCT
american_scientific_journal-1255	89	10	theorem	theorem	VERB
american_scientific_journal-1255	89	11	3.3	3.3	NUM
american_scientific_journal-1255	89	12	for	for	ADP
american_scientific_journal-1255	89	13	all	all	DET
american_scientific_journal-1255	89	14	integers	integer	NOUN
american_scientific_journal-1255	89	15	m	m	PRON
american_scientific_journal-1255	89	16	,	,	PUNCT
american_scientific_journal-1255	89	17	n	n	PRON
american_scientific_journal-1255	89	18	≥	≥	NOUN
american_scientific_journal-1255	89	19	3	3	NUM
american_scientific_journal-1255	89	20	,	,	PUNCT
american_scientific_journal-1255	89	21	and	and	CCONJ
american_scientific_journal-1255	89	22	for	for	ADP
american_scientific_journal-1255	89	23	all	all	DET
american_scientific_journal-1255	89	24	integers	integer	NOUN
american_scientific_journal-1255	89	25	k	k	X
american_scientific_journal-1255	89	26	≥	≥	NUM
american_scientific_journal-1255	89	27	3	3	NUM
american_scientific_journal-1255	89	28	,	,	PUNCT
american_scientific_journal-1255	89	29	χe	χe	PROPN
american_scientific_journal-1255	89	30	�	�	PROPN
american_scientific_journal-1255	89	31	ok	ok	NOUN
american_scientific_journal-1255	89	32	n	n	CCONJ
american_scientific_journal-1255	89	33	,	,	PUNCT
american_scientific_journal-1255	89	34	m	m	VERB
american_scientific_journal-1255	89	35	�	�	NOUN
american_scientific_journal-1255	89	36	=	=	SYM
american_scientific_journal-1255	89	37	3	3	X
american_scientific_journal-1255	89	38	.	.	PUNCT
american_scientific_journal-1255	89	39	corollary	corollary	ADJ
american_scientific_journal-1255	89	40	3.3	3.3	NUM
american_scientific_journal-1255	89	41	for	for	ADP
american_scientific_journal-1255	89	42	all	all	DET
american_scientific_journal-1255	89	43	integers	integer	NOUN
american_scientific_journal-1255	89	44	m	m	PRON
american_scientific_journal-1255	89	45	,	,	PUNCT
american_scientific_journal-1255	89	46	n	n	PRON
american_scientific_journal-1255	89	47	≥	≥	NOUN
american_scientific_journal-1255	89	48	3	3	NUM
american_scientific_journal-1255	89	49	and	and	CCONJ
american_scientific_journal-1255	89	50	for	for	ADP
american_scientific_journal-1255	89	51	any	any	DET
american_scientific_journal-1255	89	52	integers	integer	NOUN
american_scientific_journal-1255	89	53	k	k	X
american_scientific_journal-1255	89	54	≥	≥	NUM
american_scientific_journal-1255	89	55	3	3	NUM
american_scientific_journal-1255	89	56	,	,	PUNCT
american_scientific_journal-1255	89	57	𝜒	𝜒	NOUN
american_scientific_journal-1255	89	58	�	�	ADP
american_scientific_journal-1255	89	59	𝑂𝑘	𝑂𝑘	PROPN
american_scientific_journal-1255	89	60	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	89	61	�	�	PROPN
american_scientific_journal-1255	89	62	≤	≤	NUM
american_scientific_journal-1255	89	63	𝜒𝑒	𝜒𝑒	NUM
american_scientific_journal-1255	89	64	�	�	PROPN
american_scientific_journal-1255	89	65	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	X
american_scientific_journal-1255	89	66	�	�	PROPN
american_scientific_journal-1255	89	67	≤	≤	PROPN
american_scientific_journal-1255	89	68	3	3	NUM
american_scientific_journal-1255	89	69	.	.	PUNCT
american_scientific_journal-1255	90	1	proof	proof	NOUN
american_scientific_journal-1255	90	2	:	:	PUNCT
american_scientific_journal-1255	90	3	note	note	VERB
american_scientific_journal-1255	90	4	that	that	SCONJ
american_scientific_journal-1255	90	5	for	for	ADP
american_scientific_journal-1255	90	6	cases	case	NOUN
american_scientific_journal-1255	90	7	where	where	SCONJ
american_scientific_journal-1255	90	8	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	90	9	are	be	AUX
american_scientific_journal-1255	90	10	both	both	PRON
american_scientific_journal-1255	90	11	odd	odd	ADJ
american_scientific_journal-1255	90	12	,	,	PUNCT
american_scientific_journal-1255	90	13	𝑚	𝑚	PROPN
american_scientific_journal-1255	90	14	is	be	AUX
american_scientific_journal-1255	90	15	odd	odd	ADJ
american_scientific_journal-1255	90	16	,	,	PUNCT
american_scientific_journal-1255	90	17	𝑛	𝑛	PRON
american_scientific_journal-1255	90	18	is	be	AUX
american_scientific_journal-1255	90	19	even	even	ADV
american_scientific_journal-1255	90	20	and	and	CCONJ
american_scientific_journal-1255	90	21	𝑚	𝑚	PROPN
american_scientific_journal-1255	90	22	is	be	AUX
american_scientific_journal-1255	90	23	even	even	ADV
american_scientific_journal-1255	90	24	,	,	PUNCT
american_scientific_journal-1255	90	25	𝑛	𝑛	PRON
american_scientific_journal-1255	90	26	is	be	AUX
american_scientific_journal-1255	90	27	odd	odd	ADJ
american_scientific_journal-1255	90	28	,	,	PUNCT
american_scientific_journal-1255	90	29	by	by	ADP
american_scientific_journal-1255	90	30	theorem	theorem	ADJ
american_scientific_journal-1255	90	31	3.1	3.1	NUM
american_scientific_journal-1255	90	32	,	,	PUNCT
american_scientific_journal-1255	90	33	𝜒	𝜒	NOUN
american_scientific_journal-1255	90	34	�	�	ADP
american_scientific_journal-1255	90	35	𝑂𝑘	𝑂𝑘	PROPN
american_scientific_journal-1255	90	36	𝑛,𝑚	𝑛,𝑚	NOUN
american_scientific_journal-1255	90	37	�	�	NOUN
american_scientific_journal-1255	90	38	=	=	SYM
american_scientific_journal-1255	90	39	3	3	NUM
american_scientific_journal-1255	90	40	and	and	CCONJ
american_scientific_journal-1255	90	41	by	by	ADP
american_scientific_journal-1255	90	42	theorem	theorem	NOUN
american_scientific_journal-1255	90	43	3.2	3.2	NUM
american_scientific_journal-1255	90	44	,	,	PUNCT
american_scientific_journal-1255	90	45	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	90	46	�	�	PROPN
american_scientific_journal-1255	90	47	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	ADJ
american_scientific_journal-1255	90	48	�	�	PROPN
american_scientific_journal-1255	90	49	=	=	SYM
american_scientific_journal-1255	90	50	3	3	NUM
american_scientific_journal-1255	90	51	.	.	PUNCT
american_scientific_journal-1255	90	52	thus	thus	ADV
american_scientific_journal-1255	90	53	,	,	PUNCT
american_scientific_journal-1255	90	54	𝜒	𝜒	PROPN
american_scientific_journal-1255	90	55	�	�	PROPN
american_scientific_journal-1255	90	56	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	90	57	�	�	PROPN
american_scientific_journal-1255	90	58	≤	≤	NUM
american_scientific_journal-1255	90	59	𝜒𝑒	𝜒𝑒	NUM
american_scientific_journal-1255	90	60	�	�	PROPN
american_scientific_journal-1255	90	61	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	90	62	�	�	PROPN
american_scientific_journal-1255	90	63	.	.	PUNCT
american_scientific_journal-1255	91	1	similarly	similarly	ADV
american_scientific_journal-1255	91	2	,	,	PUNCT
american_scientific_journal-1255	91	3	for	for	ADP
american_scientific_journal-1255	91	4	cases	case	NOUN
american_scientific_journal-1255	91	5	where	where	SCONJ
american_scientific_journal-1255	91	6	𝑚,𝑛	𝑚,𝑛	PROPN
american_scientific_journal-1255	91	7	are	be	AUX
american_scientific_journal-1255	91	8	both	both	PRON
american_scientific_journal-1255	91	9	even	even	ADV
american_scientific_journal-1255	91	10	,	,	PUNCT
american_scientific_journal-1255	91	11	by	by	ADP
american_scientific_journal-1255	91	12	theorem	theorem	ADJ
american_scientific_journal-1255	91	13	3.1	3.1	NUM
american_scientific_journal-1255	91	14	,	,	PUNCT
american_scientific_journal-1255	91	15	𝜒	𝜒	NOUN
american_scientific_journal-1255	91	16	�	�	PROPN
american_scientific_journal-1255	91	17	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NOUN
american_scientific_journal-1255	91	18	�	�	NOUN
american_scientific_journal-1255	91	19	=	=	SYM
american_scientific_journal-1255	91	20	2	2	NUM
american_scientific_journal-1255	91	21	and	and	CCONJ
american_scientific_journal-1255	91	22	by	by	ADP
american_scientific_journal-1255	91	23	theorem	theorem	NOUN
american_scientific_journal-1255	91	24	3.2	3.2	NUM
american_scientific_journal-1255	91	25	,	,	PUNCT
american_scientific_journal-1255	91	26	𝜒𝑒	𝜒𝑒	NOUN
american_scientific_journal-1255	91	27	�	�	PROPN
american_scientific_journal-1255	91	28	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	ADJ
american_scientific_journal-1255	91	29	�	�	PROPN
american_scientific_journal-1255	91	30	=	=	SYM
american_scientific_journal-1255	91	31	3	3	NUM
american_scientific_journal-1255	91	32	.	.	PUNCT
american_scientific_journal-1255	91	33	thus	thus	ADV
american_scientific_journal-1255	91	34	,	,	PUNCT
american_scientific_journal-1255	91	35	𝜒	𝜒	PROPN
american_scientific_journal-1255	91	36	�	�	PROPN
american_scientific_journal-1255	91	37	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	91	38	�	�	PROPN
american_scientific_journal-1255	91	39	≤	≤	NUM
american_scientific_journal-1255	91	40	𝜒𝑒	𝜒𝑒	NUM
american_scientific_journal-1255	91	41	�	�	PROPN
american_scientific_journal-1255	91	42	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	91	43	�	�	PROPN
american_scientific_journal-1255	91	44	.	.	PUNCT
american_scientific_journal-1255	92	1	therefore	therefore	ADV
american_scientific_journal-1255	92	2	,	,	PUNCT
american_scientific_journal-1255	92	3	in	in	ADP
american_scientific_journal-1255	92	4	all	all	DET
american_scientific_journal-1255	92	5	cases	case	NOUN
american_scientific_journal-1255	92	6	,	,	PUNCT
american_scientific_journal-1255	92	7	𝜒	𝜒	X
american_scientific_journal-1255	92	8	�	�	PROPN
american_scientific_journal-1255	92	9	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	92	10	�	�	PROPN
american_scientific_journal-1255	92	11	≤	≤	NUM
american_scientific_journal-1255	92	12	𝜒𝑒	𝜒𝑒	NUM
american_scientific_journal-1255	92	13	�	�	PROPN
american_scientific_journal-1255	92	14	𝑂𝑘𝑛,𝑚	𝑂𝑘𝑛,𝑚	NUM
american_scientific_journal-1255	92	15	�	�	PROPN
american_scientific_journal-1255	92	16	.	.	PUNCT
american_scientific_journal-1255	93	1	acknowledgements	acknowledgement	VERB
american_scientific_journal-1255	93	2	the	the	DET
american_scientific_journal-1255	93	3	authors	author	NOUN
american_scientific_journal-1255	93	4	would	would	AUX
american_scientific_journal-1255	93	5	like	like	VERB
american_scientific_journal-1255	93	6	to	to	PART
american_scientific_journal-1255	93	7	thank	thank	VERB
american_scientific_journal-1255	93	8	the	the	DET
american_scientific_journal-1255	93	9	referee	referee	NOUN
american_scientific_journal-1255	93	10	for	for	ADP
american_scientific_journal-1255	93	11	the	the	DET
american_scientific_journal-1255	93	12	helpful	helpful	ADJ
american_scientific_journal-1255	93	13	suggestions	suggestion	NOUN
american_scientific_journal-1255	93	14	and	and	CCONJ
american_scientific_journal-1255	93	15	the	the	DET
american_scientific_journal-1255	93	16	members	member	NOUN
american_scientific_journal-1255	93	17	of	of	ADP
american_scientific_journal-1255	93	18	asrjets	asrjet	NOUN
american_scientific_journal-1255	93	19	.	.	PUNCT
american_scientific_journal-1255	94	1	special	special	ADJ
american_scientific_journal-1255	94	2	thanks	thank	NOUN
american_scientific_journal-1255	94	3	to	to	ADP
american_scientific_journal-1255	94	4	authors	author	NOUN
american_scientific_journal-1255	94	5	’	’	PART
american_scientific_journal-1255	94	6	families	family	NOUN
american_scientific_journal-1255	94	7	and	and	CCONJ
american_scientific_journal-1255	94	8	friends	friend	NOUN
american_scientific_journal-1255	94	9	.	.	PUNCT
american_scientific_journal-1255	95	1	references	reference	NOUN
american_scientific_journal-1255	95	2	[	[	X
american_scientific_journal-1255	95	3	1	1	NUM
american_scientific_journal-1255	95	4	]	]	PUNCT
american_scientific_journal-1255	95	5	.	.	PUNCT
american_scientific_journal-1255	96	1	e.	e.	PROPN
american_scientific_journal-1255	96	2	arugay	arugay	PROPN
american_scientific_journal-1255	96	3	.	.	PUNCT
american_scientific_journal-1255	97	1	“	"	PUNCT
american_scientific_journal-1255	97	2	path	path	NOUN
american_scientific_journal-1255	97	3	chromatic	chromatic	ADJ
american_scientific_journal-1255	97	4	number	number	NOUN
american_scientific_journal-1255	97	5	of	of	ADP
american_scientific_journal-1255	97	6	a	a	DET
american_scientific_journal-1255	97	7	graph	graph	NOUN
american_scientific_journal-1255	97	8	.	.	PUNCT
american_scientific_journal-1255	97	9	”	"	PUNCT
american_scientific_journal-1255	98	1	in	in	ADP
american_scientific_journal-1255	98	2	dissertation	dissertation	NOUN
american_scientific_journal-1255	98	3	conference	conference	NOUN
american_scientific_journal-1255	98	4	.	.	PUNCT
american_scientific_journal-1255	99	1	ateneo	ateneo	PROPN
american_scientific_journal-1255	99	2	de	de	PROPN
american_scientific_journal-1255	99	3	manila	manila	PROPN
american_scientific_journal-1255	99	4	university	university	PROPN
american_scientific_journal-1255	99	5	,	,	PUNCT
american_scientific_journal-1255	99	6	1990	1990	NUM
american_scientific_journal-1255	99	7	.	.	PUNCT
american_scientific_journal-1255	100	1	106	106	NUM
american_scientific_journal-1255	100	2	american	american	ADJ
american_scientific_journal-1255	100	3	scientific	scientific	ADJ
american_scientific_journal-1255	100	4	research	research	NOUN
american_scientific_journal-1255	100	5	journal	journal	NOUN
american_scientific_journal-1255	100	6	for	for	ADP
american_scientific_journal-1255	100	7	engineering	engineering	NOUN
american_scientific_journal-1255	100	8	,	,	PUNCT
american_scientific_journal-1255	100	9	technology	technology	NOUN
american_scientific_journal-1255	100	10	,	,	PUNCT
american_scientific_journal-1255	100	11	and	and	CCONJ
american_scientific_journal-1255	100	12	sciences	science	NOUN
american_scientific_journal-1255	100	13	(	(	PUNCT
american_scientific_journal-1255	100	14	asrjets	asrjet	NOUN
american_scientific_journal-1255	100	15	)	)	PUNCT
american_scientific_journal-1255	100	16	(	(	PUNCT
american_scientific_journal-1255	100	17	2016	2016	NUM
american_scientific_journal-1255	100	18	)	)	PUNCT
american_scientific_journal-1255	100	19	volume	volume	NOUN
american_scientific_journal-1255	100	20	16	16	NUM
american_scientific_journal-1255	100	21	,	,	PUNCT
american_scientific_journal-1255	100	22	no	no	DET
american_scientific_journal-1255	100	23	1	1	NUM
american_scientific_journal-1255	100	24	,	,	PUNCT
american_scientific_journal-1255	100	25	pp	pp	ADV
american_scientific_journal-1255	100	26	98	98	NUM
american_scientific_journal-1255	100	27	-	-	SYM
american_scientific_journal-1255	100	28	107	107	NUM
american_scientific_journal-1255	101	1	[	[	X
american_scientific_journal-1255	101	2	2	2	NUM
american_scientific_journal-1255	101	3	]	]	PUNCT
american_scientific_journal-1255	101	4	.	.	PUNCT
american_scientific_journal-1255	102	1	r.	r.	PROPN
american_scientific_journal-1255	102	2	balakrishnan	balakrishnan	PROPN
american_scientific_journal-1255	102	3	and	and	CCONJ
american_scientific_journal-1255	102	4	k.	k.	PROPN
american_scientific_journal-1255	102	5	ranganathan	ranganathan	PROPN
american_scientific_journal-1255	102	6	.	.	PUNCT
american_scientific_journal-1255	103	1	a	a	DET
american_scientific_journal-1255	103	2	textbook	textbook	NOUN
american_scientific_journal-1255	103	3	of	of	ADP
american_scientific_journal-1255	103	4	graph	graph	NOUN
american_scientific_journal-1255	103	5	theory	theory	NOUN
american_scientific_journal-1255	103	6	,	,	PUNCT
american_scientific_journal-1255	103	7	2nd	2nd	ADJ
american_scientific_journal-1255	103	8	ed	ed	NOUN
american_scientific_journal-1255	103	9	.	.	PROPN
american_scientific_journal-1255	103	10	,	,	PUNCT
american_scientific_journal-1255	103	11	springer	springer	NOUN
american_scientific_journal-1255	103	12	,	,	PUNCT
american_scientific_journal-1255	103	13	2012	2012	NUM
american_scientific_journal-1255	103	14	.	.	PUNCT
american_scientific_journal-1255	104	1	[	[	X
american_scientific_journal-1255	104	2	3	3	NUM
american_scientific_journal-1255	104	3	]	]	PUNCT
american_scientific_journal-1255	104	4	.	.	PUNCT
american_scientific_journal-1255	105	1	t.	t.	PROPN
american_scientific_journal-1255	105	2	harju	harju	PROPN
american_scientific_journal-1255	105	3	.	.	PUNCT
american_scientific_journal-1255	106	1	lecture	lecture	NOUN
american_scientific_journal-1255	106	2	notes	note	NOUN
american_scientific_journal-1255	106	3	on	on	ADP
american_scientific_journal-1255	106	4	graph	graph	NOUN
american_scientific_journal-1255	106	5	theory	theory	NOUN
american_scientific_journal-1255	106	6	.	.	PUNCT
american_scientific_journal-1255	107	1	department	department	NOUN
american_scientific_journal-1255	107	2	of	of	ADP
american_scientific_journal-1255	107	3	mathematics	mathematics	PROPN
american_scientific_journal-1255	107	4	,	,	PUNCT
american_scientific_journal-1255	107	5	university	university	PROPN
american_scientific_journal-1255	107	6	of	of	ADP
american_scientific_journal-1255	107	7	turku	turku	PROPN
american_scientific_journal-1255	107	8	,	,	PUNCT
american_scientific_journal-1255	107	9	fin-20014	fin-20014	NOUN
american_scientific_journal-1255	107	10	,	,	PUNCT
american_scientific_journal-1255	107	11	turku	turku	PROPN
american_scientific_journal-1255	107	12	,	,	PUNCT
american_scientific_journal-1255	107	13	finland	finland	PROPN
american_scientific_journal-1255	107	14	,	,	PUNCT
american_scientific_journal-1255	107	15	2012	2012	NUM
american_scientific_journal-1255	107	16	.	.	PUNCT
american_scientific_journal-1255	108	1	107	107	NUM