id	sid	tid	token	lemma	pos
ejpam-6059	1	1	european	european	PROPN
ejpam-6059	1	2	journal	journal	PROPN
ejpam-6059	1	3	of	of	ADP
ejpam-6059	1	4	pure	pure	ADJ
ejpam-6059	1	5	and	and	CCONJ
ejpam-6059	1	6	applied	applied	ADJ
ejpam-6059	1	7	mathematics	mathematic	NOUN
ejpam-6059	1	8	2025	2025	NUM
ejpam-6059	1	9	,	,	PUNCT
ejpam-6059	1	10	vol	vol	NOUN
ejpam-6059	1	11	.	.	PROPN
ejpam-6059	1	12	18	18	NUM
ejpam-6059	1	13	,	,	PUNCT
ejpam-6059	1	14	issue	issue	NOUN
ejpam-6059	1	15	2	2	NUM
ejpam-6059	1	16	,	,	PUNCT
ejpam-6059	1	17	article	article	NOUN
ejpam-6059	1	18	number	number	NOUN
ejpam-6059	1	19	6059	6059	NUM
ejpam-6059	1	20	issn	issn	VERB
ejpam-6059	1	21	1307	1307	NUM
ejpam-6059	1	22	-	-	SYM
ejpam-6059	1	23	5543	5543	NUM
ejpam-6059	1	24	–	–	PUNCT
ejpam-6059	1	25	ejpam.com	ejpam.com	X
ejpam-6059	1	26	published	publish	VERB
ejpam-6059	1	27	by	by	ADP
ejpam-6059	1	28	new	new	PROPN
ejpam-6059	1	29	york	york	PROPN
ejpam-6059	1	30	business	business	PROPN
ejpam-6059	1	31	global	global	PROPN
ejpam-6059	1	32	lower	low	ADJ
ejpam-6059	1	33	and	and	CCONJ
ejpam-6059	1	34	upper	upper	ADJ
ejpam-6059	1	35	acyclicities	acyclicitie	NOUN
ejpam-6059	1	36	on	on	ADP
ejpam-6059	1	37	unitary	unitary	ADJ
ejpam-6059	1	38	cayley	cayley	ADJ
ejpam-6059	1	39	graphs	graph	NOUN
ejpam-6059	1	40	of	of	ADP
ejpam-6059	1	41	finite	finite	PROPN
ejpam-6059	1	42	commutative	commutative	ADJ
ejpam-6059	1	43	rings	ring	NOUN
ejpam-6059	1	44	denpong	denpong	PROPN
ejpam-6059	1	45	pongpipat1	pongpipat1	PROPN
ejpam-6059	1	46	,	,	PUNCT
ejpam-6059	1	47	nuttawoot	nuttawoot	PROPN
ejpam-6059	1	48	nupo1,∗	nupo1,∗	PROPN
ejpam-6059	1	49	1	1	NUM
ejpam-6059	1	50	department	department	NOUN
ejpam-6059	1	51	of	of	ADP
ejpam-6059	1	52	mathematics	mathematic	NOUN
ejpam-6059	1	53	,	,	PUNCT
ejpam-6059	1	54	faculty	faculty	NOUN
ejpam-6059	1	55	of	of	ADP
ejpam-6059	1	56	science	science	NOUN
ejpam-6059	1	57	,	,	PUNCT
ejpam-6059	1	58	khon	khon	PROPN
ejpam-6059	1	59	kaen	kaen	PROPN
ejpam-6059	1	60	university	university	PROPN
ejpam-6059	1	61	,	,	PUNCT
ejpam-6059	1	62	khon	khon	PROPN
ejpam-6059	1	63	kaen	kaen	PROPN
ejpam-6059	1	64	,	,	PUNCT
ejpam-6059	1	65	40002	40002	NUM
ejpam-6059	1	66	,	,	PUNCT
ejpam-6059	1	67	thailand	thailand	PROPN
ejpam-6059	1	68	abstract	abstract	PROPN
ejpam-6059	1	69	.	.	PUNCT
ejpam-6059	2	1	a	a	DET
ejpam-6059	2	2	unitary	unitary	ADJ
ejpam-6059	2	3	cayley	cayley	NOUN
ejpam-6059	2	4	graph	graph	NOUN
ejpam-6059	2	5	γn	γn	NOUN
ejpam-6059	2	6	of	of	ADP
ejpam-6059	2	7	a	a	DET
ejpam-6059	2	8	finite	finite	ADJ
ejpam-6059	2	9	cyclic	cyclic	PROPN
ejpam-6059	2	10	ring	ring	NOUN
ejpam-6059	2	11	zn	zn	PROPN
ejpam-6059	2	12	is	be	AUX
ejpam-6059	2	13	a	a	DET
ejpam-6059	2	14	graph	graph	NOUN
ejpam-6059	2	15	with	with	ADP
ejpam-6059	2	16	vertex	vertex	NOUN
ejpam-6059	2	17	set	set	VERB
ejpam-6059	2	18	zn	zn	PROPN
ejpam-6059	2	19	and	and	CCONJ
ejpam-6059	2	20	two	two	NUM
ejpam-6059	2	21	vertices	vertex	NOUN
ejpam-6059	2	22	x	x	PUNCT
ejpam-6059	2	23	and	and	CCONJ
ejpam-6059	2	24	y	y	PROPN
ejpam-6059	2	25	are	be	AUX
ejpam-6059	2	26	adjacent	adjacent	ADJ
ejpam-6059	2	27	if	if	SCONJ
ejpam-6059	2	28	and	and	CCONJ
ejpam-6059	2	29	only	only	ADV
ejpam-6059	2	30	if	if	SCONJ
ejpam-6059	2	31	x−y	x−y	PROPN
ejpam-6059	2	32	is	be	AUX
ejpam-6059	2	33	a	a	DET
ejpam-6059	2	34	unit	unit	NOUN
ejpam-6059	2	35	in	in	ADP
ejpam-6059	2	36	zn	zn	PROPN
ejpam-6059	2	37	or	or	CCONJ
ejpam-6059	2	38	equivalently	equivalently	ADV
ejpam-6059	2	39	,	,	PUNCT
ejpam-6059	2	40	gcd(x−y	gcd(x−y	NOUN
ejpam-6059	2	41	,	,	PUNCT
ejpam-6059	2	42	n	n	CCONJ
ejpam-6059	2	43	)	)	PUNCT
ejpam-6059	3	1	=	=	SYM
ejpam-6059	3	2	1	1	X
ejpam-6059	3	3	.	.	PUNCT
ejpam-6059	4	1	a	a	DET
ejpam-6059	4	2	nonempty	nonempty	NOUN
ejpam-6059	4	3	subset	subset	VERB
ejpam-6059	4	4	a	a	PRON
ejpam-6059	4	5	of	of	ADP
ejpam-6059	4	6	zn	zn	PROPN
ejpam-6059	4	7	is	be	AUX
ejpam-6059	4	8	called	call	VERB
ejpam-6059	4	9	an	an	DET
ejpam-6059	4	10	acyclic	acyclic	ADJ
ejpam-6059	4	11	set	set	NOUN
ejpam-6059	4	12	of	of	ADP
ejpam-6059	4	13	γn	γn	PRON
ejpam-6059	4	14	if	if	SCONJ
ejpam-6059	4	15	a	a	DET
ejpam-6059	4	16	subgraph	subgraph	NOUN
ejpam-6059	4	17	of	of	ADP
ejpam-6059	4	18	γn	γn	NOUN
ejpam-6059	4	19	induced	induce	VERB
ejpam-6059	4	20	by	by	ADP
ejpam-6059	4	21	a	a	DET
ejpam-6059	4	22	contains	contain	NOUN
ejpam-6059	4	23	no	no	DET
ejpam-6059	4	24	cycles	cycle	NOUN
ejpam-6059	4	25	.	.	PUNCT
ejpam-6059	5	1	the	the	DET
ejpam-6059	5	2	maximum	maximum	PROPN
ejpam-6059	5	3	cardinality	cardinality	NOUN
ejpam-6059	5	4	among	among	ADP
ejpam-6059	5	5	the	the	DET
ejpam-6059	5	6	acyclic	acyclic	ADJ
ejpam-6059	5	7	sets	set	NOUN
ejpam-6059	5	8	of	of	ADP
ejpam-6059	5	9	γn	γn	NOUN
ejpam-6059	5	10	is	be	AUX
ejpam-6059	5	11	called	call	VERB
ejpam-6059	5	12	the	the	DET
ejpam-6059	5	13	upper	upper	ADJ
ejpam-6059	5	14	acyclic	acyclic	ADJ
ejpam-6059	5	15	number	number	NOUN
ejpam-6059	5	16	of	of	ADP
ejpam-6059	5	17	γn	γn	NOUN
ejpam-6059	5	18	and	and	CCONJ
ejpam-6059	5	19	is	be	AUX
ejpam-6059	5	20	denoted	denote	VERB
ejpam-6059	5	21	by	by	ADP
ejpam-6059	5	22	λ(γn	λ(γn	NOUN
ejpam-6059	5	23	)	)	PUNCT
ejpam-6059	5	24	.	.	PUNCT
ejpam-6059	6	1	moreover	moreover	ADV
ejpam-6059	6	2	,	,	PUNCT
ejpam-6059	6	3	the	the	DET
ejpam-6059	6	4	maximum	maximum	ADJ
ejpam-6059	6	5	number	number	NOUN
ejpam-6059	6	6	k	k	PROPN
ejpam-6059	6	7	of	of	ADP
ejpam-6059	6	8	vertices	vertex	NOUN
ejpam-6059	6	9	of	of	ADP
ejpam-6059	6	10	γn	γn	NOUN
ejpam-6059	6	11	in	in	ADP
ejpam-6059	6	12	which	which	PRON
ejpam-6059	6	13	every	every	DET
ejpam-6059	6	14	subgraph	subgraph	NOUN
ejpam-6059	6	15	of	of	ADP
ejpam-6059	6	16	γn	γn	NOUN
ejpam-6059	6	17	induced	induce	VERB
ejpam-6059	6	18	by	by	ADP
ejpam-6059	6	19	k	k	PROPN
ejpam-6059	6	20	vertices	vertex	NOUN
ejpam-6059	6	21	contains	contain	VERB
ejpam-6059	6	22	no	no	DET
ejpam-6059	6	23	cycles	cycle	NOUN
ejpam-6059	6	24	is	be	AUX
ejpam-6059	6	25	called	call	VERB
ejpam-6059	6	26	the	the	DET
ejpam-6059	6	27	lower	low	ADJ
ejpam-6059	6	28	acyclic	acyclic	ADJ
ejpam-6059	6	29	number	number	NOUN
ejpam-6059	6	30	of	of	ADP
ejpam-6059	6	31	γn	γn	NOUN
ejpam-6059	6	32	and	and	CCONJ
ejpam-6059	6	33	denoted	denote	VERB
ejpam-6059	6	34	by	by	ADP
ejpam-6059	6	35	λ(γn	λ(γn	NOUN
ejpam-6059	6	36	)	)	PUNCT
ejpam-6059	6	37	.	.	PUNCT
ejpam-6059	7	1	in	in	ADP
ejpam-6059	7	2	this	this	DET
ejpam-6059	7	3	paper	paper	NOUN
ejpam-6059	7	4	,	,	PUNCT
ejpam-6059	7	5	we	we	PRON
ejpam-6059	7	6	determine	determine	VERB
ejpam-6059	7	7	the	the	DET
ejpam-6059	7	8	lower	low	ADJ
ejpam-6059	7	9	and	and	CCONJ
ejpam-6059	7	10	upper	upper	ADJ
ejpam-6059	7	11	acyclic	acyclic	ADJ
ejpam-6059	7	12	numbers	number	NOUN
ejpam-6059	7	13	for	for	ADP
ejpam-6059	7	14	unitary	unitary	ADJ
ejpam-6059	7	15	cayley	cayley	ADJ
ejpam-6059	7	16	graphs	graph	NOUN
ejpam-6059	7	17	of	of	ADP
ejpam-6059	7	18	zn	zn	PROPN
ejpam-6059	7	19	and	and	CCONJ
ejpam-6059	7	20	their	their	PRON
ejpam-6059	7	21	complements	complement	NOUN
ejpam-6059	7	22	.	.	PUNCT
ejpam-6059	8	1	2020	2020	NUM
ejpam-6059	8	2	mathematics	mathematic	NOUN
ejpam-6059	8	3	subject	subject	NOUN
ejpam-6059	8	4	classifications	classification	NOUN
ejpam-6059	8	5	:	:	PUNCT
ejpam-6059	8	6	05c25	05c25	NUM
ejpam-6059	8	7	,	,	PUNCT
ejpam-6059	8	8	05c69	05c69	NUM
ejpam-6059	8	9	,	,	PUNCT
ejpam-6059	8	10	05c99	05c99	NOUN
ejpam-6059	8	11	key	key	ADJ
ejpam-6059	8	12	words	word	NOUN
ejpam-6059	8	13	and	and	CCONJ
ejpam-6059	8	14	phrases	phrase	NOUN
ejpam-6059	8	15	:	:	PUNCT
ejpam-6059	8	16	acyclic	acyclic	ADJ
ejpam-6059	8	17	sets	set	NOUN
ejpam-6059	8	18	,	,	PUNCT
ejpam-6059	8	19	lower	low	ADJ
ejpam-6059	8	20	acyclic	acyclic	ADJ
ejpam-6059	8	21	numbers	number	NOUN
ejpam-6059	8	22	,	,	PUNCT
ejpam-6059	8	23	upper	upper	ADJ
ejpam-6059	8	24	acyclic	acyclic	ADJ
ejpam-6059	8	25	numbers	number	NOUN
ejpam-6059	8	26	,	,	PUNCT
ejpam-6059	8	27	unitary	unitary	ADJ
ejpam-6059	8	28	cayley	cayley	ADJ
ejpam-6059	8	29	graphs	graph	NOUN
ejpam-6059	8	30	1	1	NUM
ejpam-6059	8	31	.	.	PUNCT
ejpam-6059	9	1	introduction	introduction	NOUN
ejpam-6059	9	2	in	in	ADP
ejpam-6059	9	3	algebraic	algebraic	ADJ
ejpam-6059	9	4	graph	graph	NOUN
ejpam-6059	9	5	theory	theory	NOUN
ejpam-6059	9	6	,	,	PUNCT
ejpam-6059	9	7	the	the	DET
ejpam-6059	9	8	structure	structure	NOUN
ejpam-6059	9	9	of	of	ADP
ejpam-6059	9	10	algebraic	algebraic	ADJ
ejpam-6059	9	11	methods	method	NOUN
ejpam-6059	9	12	are	be	AUX
ejpam-6059	9	13	studied	study	VERB
ejpam-6059	9	14	and	and	CCONJ
ejpam-6059	9	15	then	then	ADV
ejpam-6059	9	16	applied	apply	VERB
ejpam-6059	9	17	to	to	ADP
ejpam-6059	9	18	problems	problem	NOUN
ejpam-6059	9	19	about	about	ADP
ejpam-6059	9	20	graphs	graph	NOUN
ejpam-6059	9	21	.	.	PUNCT
ejpam-6059	10	1	an	an	DET
ejpam-6059	10	2	interesting	interesting	ADJ
ejpam-6059	10	3	topic	topic	NOUN
ejpam-6059	10	4	is	be	AUX
ejpam-6059	10	5	to	to	PART
ejpam-6059	10	6	study	study	VERB
ejpam-6059	10	7	properties	property	NOUN
ejpam-6059	10	8	of	of	ADP
ejpam-6059	10	9	graphs	graph	NOUN
ejpam-6059	10	10	in	in	ADP
ejpam-6059	10	11	connection	connection	NOUN
ejpam-6059	10	12	to	to	ADP
ejpam-6059	10	13	algebraic	algebraic	ADJ
ejpam-6059	10	14	systems	system	NOUN
ejpam-6059	10	15	.	.	PUNCT
ejpam-6059	11	1	a	a	DET
ejpam-6059	11	2	well	well	ADV
ejpam-6059	11	3	-	-	PUNCT
ejpam-6059	11	4	known	know	VERB
ejpam-6059	11	5	connection	connection	NOUN
ejpam-6059	11	6	between	between	ADP
ejpam-6059	11	7	graphs	graph	NOUN
ejpam-6059	11	8	and	and	CCONJ
ejpam-6059	11	9	algebraic	algebraic	ADJ
ejpam-6059	11	10	system	system	NOUN
ejpam-6059	11	11	is	be	AUX
ejpam-6059	11	12	the	the	DET
ejpam-6059	11	13	construction	construction	NOUN
ejpam-6059	11	14	of	of	ADP
ejpam-6059	11	15	graphs	graph	NOUN
ejpam-6059	11	16	from	from	ADP
ejpam-6059	11	17	algebras	algebra	NOUN
ejpam-6059	11	18	.	.	PUNCT
ejpam-6059	12	1	algebraic	algebraic	ADJ
ejpam-6059	12	2	tools	tool	NOUN
ejpam-6059	12	3	can	can	AUX
ejpam-6059	12	4	be	be	AUX
ejpam-6059	12	5	used	use	VERB
ejpam-6059	12	6	to	to	PART
ejpam-6059	12	7	give	give	VERB
ejpam-6059	12	8	elegant	elegant	ADJ
ejpam-6059	12	9	proofs	proof	NOUN
ejpam-6059	12	10	of	of	ADP
ejpam-6059	12	11	graph	graph	NOUN
ejpam-6059	12	12	theoretic	theoretic	ADJ
ejpam-6059	12	13	facts	fact	NOUN
ejpam-6059	12	14	.	.	PUNCT
ejpam-6059	13	1	for	for	ADP
ejpam-6059	13	2	each	each	DET
ejpam-6059	13	3	n	n	PRON
ejpam-6059	13	4	≥	≥	NOUN
ejpam-6059	13	5	2	2	NUM
ejpam-6059	13	6	,	,	PUNCT
ejpam-6059	13	7	let	let	VERB
ejpam-6059	13	8	γn	γn	PRON
ejpam-6059	13	9	denote	denote	VERB
ejpam-6059	13	10	the	the	DET
ejpam-6059	13	11	unitary	unitary	ADJ
ejpam-6059	13	12	cayley	cayley	ADJ
ejpam-6059	13	13	graph	graph	NOUN
ejpam-6059	13	14	of	of	ADP
ejpam-6059	13	15	a	a	DET
ejpam-6059	13	16	ring	ring	NOUN
ejpam-6059	13	17	zn	zn	PROPN
ejpam-6059	13	18	,	,	PUNCT
ejpam-6059	13	19	the	the	DET
ejpam-6059	13	20	ring	ring	NOUN
ejpam-6059	13	21	of	of	ADP
ejpam-6059	13	22	integers	integer	NOUN
ejpam-6059	13	23	modulo	modulo	PROPN
ejpam-6059	13	24	n	n	PRON
ejpam-6059	13	25	,	,	PUNCT
ejpam-6059	13	26	whose	whose	DET
ejpam-6059	13	27	vertex	vertex	NOUN
ejpam-6059	13	28	set	set	NOUN
ejpam-6059	13	29	is	be	AUX
ejpam-6059	13	30	zn	zn	PROPN
ejpam-6059	13	31	itself	itself	PRON
ejpam-6059	13	32	and	and	CCONJ
ejpam-6059	13	33	two	two	NUM
ejpam-6059	13	34	vertices	vertex	NOUN
ejpam-6059	13	35	x	x	PUNCT
ejpam-6059	13	36	and	and	CCONJ
ejpam-6059	13	37	y	y	PROPN
ejpam-6059	13	38	are	be	AUX
ejpam-6059	13	39	joined	join	VERB
ejpam-6059	13	40	by	by	ADP
ejpam-6059	13	41	edge	edge	NOUN
ejpam-6059	13	42	if	if	SCONJ
ejpam-6059	13	43	x	x	PRON
ejpam-6059	13	44	−	−	PROPN
ejpam-6059	13	45	y	y	PROPN
ejpam-6059	13	46	is	be	AUX
ejpam-6059	13	47	a	a	DET
ejpam-6059	13	48	unit	unit	NOUN
ejpam-6059	13	49	in	in	ADP
ejpam-6059	13	50	the	the	DET
ejpam-6059	13	51	ring	ring	NOUN
ejpam-6059	14	1	zn	zn	X
ejpam-6059	14	2	.	.	PUNCT
ejpam-6059	15	1	let	let	VERB
ejpam-6059	15	2	us	we	PRON
ejpam-6059	15	3	denote	denote	VERB
ejpam-6059	15	4	all	all	DET
ejpam-6059	15	5	elements	element	NOUN
ejpam-6059	15	6	in	in	ADP
ejpam-6059	15	7	zn	zn	NUM
ejpam-6059	15	8	by	by	ADP
ejpam-6059	15	9	integers	integer	NOUN
ejpam-6059	15	10	0	0	NUM
ejpam-6059	15	11	,	,	PUNCT
ejpam-6059	15	12	1	1	NUM
ejpam-6059	15	13	,	,	PUNCT
ejpam-6059	15	14	2	2	NUM
ejpam-6059	15	15	,	,	PUNCT
ejpam-6059	15	16	.	.	PUNCT
ejpam-6059	15	17	.	.	PUNCT
ejpam-6059	16	1	.	.	PUNCT
ejpam-6059	17	1	,	,	PUNCT
ejpam-6059	18	1	n	n	CCONJ
ejpam-6059	18	2	−	−	PROPN
ejpam-6059	18	3	1	1	X
ejpam-6059	18	4	.	.	PUNCT
ejpam-6059	19	1	it	it	PRON
ejpam-6059	19	2	is	be	AUX
ejpam-6059	19	3	well	well	ADV
ejpam-6059	19	4	known	know	VERB
ejpam-6059	19	5	that	that	SCONJ
ejpam-6059	19	6	all	all	DET
ejpam-6059	19	7	units	unit	NOUN
ejpam-6059	19	8	in	in	ADP
ejpam-6059	19	9	the	the	DET
ejpam-6059	19	10	ring	ring	NOUN
ejpam-6059	19	11	zn	zn	PROPN
ejpam-6059	19	12	are	be	AUX
ejpam-6059	19	13	the	the	DET
ejpam-6059	19	14	integers	integer	NOUN
ejpam-6059	20	1	a	a	PRON
ejpam-6059	20	2	in	in	ADP
ejpam-6059	20	3	which	which	PRON
ejpam-6059	20	4	gcd(a	gcd(a	PROPN
ejpam-6059	20	5	,	,	PUNCT
ejpam-6059	20	6	n	n	CCONJ
ejpam-6059	20	7	)	)	PUNCT
ejpam-6059	20	8	=	=	SYM
ejpam-6059	21	1	1	1	X
ejpam-6059	21	2	.	.	PUNCT
ejpam-6059	21	3	therefore	therefore	ADV
ejpam-6059	21	4	,	,	PUNCT
ejpam-6059	21	5	the	the	DET
ejpam-6059	21	6	edge	edge	NOUN
ejpam-6059	21	7	set	set	NOUN
ejpam-6059	21	8	of	of	ADP
ejpam-6059	21	9	γn	γn	NOUN
ejpam-6059	21	10	can	can	AUX
ejpam-6059	21	11	be	be	AUX
ejpam-6059	21	12	expressed	express	VERB
ejpam-6059	21	13	as	as	ADP
ejpam-6059	21	14	e(γn	e(γn	NUM
ejpam-6059	21	15	)	)	PUNCT
ejpam-6059	21	16	=	=	SYM
ejpam-6059	21	17	{	{	PUNCT
ejpam-6059	21	18	{	{	PUNCT
ejpam-6059	21	19	x	x	NOUN
ejpam-6059	21	20	,	,	PUNCT
ejpam-6059	21	21	y	y	PROPN
ejpam-6059	21	22	}	}	PUNCT
ejpam-6059	21	23	:	:	PUNCT
ejpam-6059	21	24	x	x	X
ejpam-6059	21	25	,	,	PUNCT
ejpam-6059	21	26	y	y	PROPN
ejpam-6059	21	27	∈	∈	PROPN
ejpam-6059	21	28	zn	zn	PROPN
ejpam-6059	21	29	and	and	CCONJ
ejpam-6059	21	30	gcd(x	gcd(x	PROPN
ejpam-6059	21	31	−	−	PROPN
ejpam-6059	21	32	y	y	PROPN
ejpam-6059	21	33	,	,	PUNCT
ejpam-6059	21	34	n	n	CCONJ
ejpam-6059	21	35	)	)	PUNCT
ejpam-6059	21	36	=	=	SYM
ejpam-6059	21	37	1	1	NUM
ejpam-6059	21	38	}	}	PUNCT
ejpam-6059	21	39	.	.	PUNCT
ejpam-6059	22	1	clearly	clearly	ADV
ejpam-6059	22	2	,	,	PUNCT
ejpam-6059	22	3	if	if	SCONJ
ejpam-6059	22	4	p	p	NOUN
ejpam-6059	22	5	is	be	AUX
ejpam-6059	22	6	prime	prime	ADJ
ejpam-6059	22	7	,	,	PUNCT
ejpam-6059	22	8	then	then	ADV
ejpam-6059	22	9	γp	γp	PROPN
ejpam-6059	22	10	is	be	AUX
ejpam-6059	22	11	a	a	DET
ejpam-6059	22	12	complete	complete	ADJ
ejpam-6059	22	13	graph	graph	NOUN
ejpam-6059	22	14	.	.	PUNCT
ejpam-6059	23	1	moreover	moreover	ADV
ejpam-6059	23	2	,	,	PUNCT
ejpam-6059	23	3	all	all	DET
ejpam-6059	23	4	unitary	unitary	ADJ
ejpam-6059	23	5	cayley	cayley	ADJ
ejpam-6059	23	6	graphs	graph	NOUN
ejpam-6059	23	7	of	of	ADP
ejpam-6059	23	8	order	order	NOUN
ejpam-6059	23	9	greater	great	ADJ
ejpam-6059	23	10	than	than	ADP
ejpam-6059	23	11	2	2	NUM
ejpam-6059	23	12	always	always	ADV
ejpam-6059	23	13	contain	contain	VERB
ejpam-6059	23	14	cycles	cycle	NOUN
ejpam-6059	23	15	and	and	CCONJ
ejpam-6059	23	16	their	their	PRON
ejpam-6059	23	17	structures	structure	NOUN
ejpam-6059	23	18	are	be	AUX
ejpam-6059	23	19	highly	highly	ADV
ejpam-6059	23	20	symmetric	symmetric	ADJ
ejpam-6059	23	21	.	.	PUNCT
ejpam-6059	24	1	moreover	moreover	ADV
ejpam-6059	24	2	,	,	PUNCT
ejpam-6059	24	3	there	there	PRON
ejpam-6059	24	4	are	be	VERB
ejpam-6059	24	5	∗corresponding	∗corresponde	VERB
ejpam-6059	24	6	author	author	NOUN
ejpam-6059	24	7	.	.	PUNCT
ejpam-6059	25	1	doi	doi	NOUN
ejpam-6059	25	2	:	:	PUNCT
ejpam-6059	25	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6059	https://doi.org/10.29020/nybg.ejpam.v18i2.6059	NUM
ejpam-6059	25	4	email	email	NOUN
ejpam-6059	25	5	addresses	address	NOUN
ejpam-6059	25	6	:	:	PUNCT
ejpam-6059	25	7	denpong	denpong	NUM
ejpam-6059	25	8	p@kkumail.com	p@kkumail.com	X
ejpam-6059	25	9	(	(	PUNCT
ejpam-6059	25	10	d.	d.	NOUN
ejpam-6059	25	11	pongpipat	pongpipat	PROPN
ejpam-6059	25	12	)	)	PUNCT
ejpam-6059	25	13	,	,	PUNCT
ejpam-6059	25	14	nuttanu@kku.ac.th	nuttanu@kku.ac.th	PROPN
ejpam-6059	25	15	(	(	PUNCT
ejpam-6059	25	16	n.	n.	NOUN
ejpam-6059	25	17	nupo	nupo	NOUN
ejpam-6059	25	18	)	)	PUNCT
ejpam-6059	25	19	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6059	25	20	1	1	NUM
ejpam-6059	25	21	copyright	copyright	NOUN
ejpam-6059	25	22	:	:	PUNCT
ejpam-6059	26	1	©	©	PROPN
ejpam-6059	26	2	2025	2025	NUM
ejpam-6059	26	3	the	the	DET
ejpam-6059	26	4	author(s	author(s	NOUN
ejpam-6059	26	5	)	)	PUNCT
ejpam-6059	26	6	.	.	PUNCT
ejpam-6059	27	1	(	(	PUNCT
ejpam-6059	27	2	cc	cc	NOUN
ejpam-6059	27	3	by	by	ADP
ejpam-6059	27	4	-	-	PUNCT
ejpam-6059	27	5	nc	nc	PROPN
ejpam-6059	27	6	4.0	4.0	NUM
ejpam-6059	27	7	)	)	PUNCT
ejpam-6059	27	8	d.	d.	NOUN
ejpam-6059	27	9	pongpipat	pongpipat	PROPN
ejpam-6059	27	10	,	,	PUNCT
ejpam-6059	27	11	n.	n.	NOUN
ejpam-6059	27	12	nupo	nupo	PROPN
ejpam-6059	27	13	/	/	SYM
ejpam-6059	27	14	eur	eur	PROPN
ejpam-6059	27	15	.	.	PUNCT
ejpam-6059	28	1	j.	j.	PROPN
ejpam-6059	28	2	pure	pure	PROPN
ejpam-6059	28	3	appl	appl	PROPN
ejpam-6059	28	4	.	.	PROPN
ejpam-6059	28	5	math	math	PROPN
ejpam-6059	28	6	,	,	PUNCT
ejpam-6059	28	7	18	18	NUM
ejpam-6059	28	8	(	(	PUNCT
ejpam-6059	28	9	2	2	NUM
ejpam-6059	28	10	)	)	PUNCT
ejpam-6059	28	11	(	(	PUNCT
ejpam-6059	28	12	2025	2025	NUM
ejpam-6059	28	13	)	)	PUNCT
ejpam-6059	28	14	,	,	PUNCT
ejpam-6059	28	15	6059	6059	NUM
ejpam-6059	28	16	2	2	NUM
ejpam-6059	28	17	of	of	ADP
ejpam-6059	28	18	11	11	NUM
ejpam-6059	28	19	some	some	DET
ejpam-6059	28	20	remarkable	remarkable	ADJ
ejpam-6059	28	21	properties	property	NOUN
ejpam-6059	28	22	between	between	ADP
ejpam-6059	28	23	algebraic	algebraic	ADJ
ejpam-6059	28	24	graph	graph	NOUN
ejpam-6059	28	25	theory	theory	NOUN
ejpam-6059	28	26	and	and	CCONJ
ejpam-6059	28	27	number	number	NOUN
ejpam-6059	28	28	theory	theory	NOUN
ejpam-6059	28	29	.	.	PUNCT
ejpam-6059	29	1	some	some	DET
ejpam-6059	29	2	prominent	prominent	ADJ
ejpam-6059	29	3	results	result	NOUN
ejpam-6059	29	4	of	of	ADP
ejpam-6059	29	5	unitary	unitary	ADJ
ejpam-6059	29	6	cayley	cayley	ADJ
ejpam-6059	29	7	graphs	graph	NOUN
ejpam-6059	29	8	were	be	AUX
ejpam-6059	29	9	studied	study	VERB
ejpam-6059	29	10	by	by	ADP
ejpam-6059	29	11	several	several	ADJ
ejpam-6059	29	12	researchers	researcher	NOUN
ejpam-6059	29	13	.	.	PUNCT
ejpam-6059	30	1	in	in	ADP
ejpam-6059	30	2	1995	1995	NUM
ejpam-6059	30	3	,	,	PUNCT
ejpam-6059	30	4	dejter	dejter	NOUN
ejpam-6059	30	5	and	and	CCONJ
ejpam-6059	30	6	giudici	giudici	NOUN
ejpam-6059	30	7	[	[	X
ejpam-6059	30	8	1	1	NUM
ejpam-6059	30	9	]	]	PUNCT
ejpam-6059	30	10	showed	show	VERB
ejpam-6059	30	11	that	that	SCONJ
ejpam-6059	30	12	unitary	unitary	ADJ
ejpam-6059	30	13	cayley	cayley	ADJ
ejpam-6059	30	14	graphs	graph	NOUN
ejpam-6059	30	15	are	be	AUX
ejpam-6059	30	16	unions	union	NOUN
ejpam-6059	30	17	of	of	ADP
ejpam-6059	30	18	disjoint	disjoint	NOUN
ejpam-6059	30	19	hamiltonian	hamiltonian	ADJ
ejpam-6059	30	20	cycles	cycle	NOUN
ejpam-6059	30	21	and	and	CCONJ
ejpam-6059	30	22	presented	present	VERB
ejpam-6059	30	23	the	the	DET
ejpam-6059	30	24	sufficient	sufficient	ADJ
ejpam-6059	30	25	condition	condition	NOUN
ejpam-6059	30	26	for	for	ADP
ejpam-6059	30	27	being	be	AUX
ejpam-6059	30	28	bipartite	bipartite	ADJ
ejpam-6059	30	29	graphs	graph	NOUN
ejpam-6059	30	30	.	.	PUNCT
ejpam-6059	31	1	in	in	ADP
ejpam-6059	31	2	2007	2007	NUM
ejpam-6059	31	3	,	,	PUNCT
ejpam-6059	31	4	klotz	klotz	PROPN
ejpam-6059	31	5	and	and	CCONJ
ejpam-6059	31	6	sander	sander	VERB
ejpam-6059	31	7	[	[	X
ejpam-6059	31	8	2	2	NUM
ejpam-6059	31	9	]	]	PUNCT
ejpam-6059	31	10	determined	determine	VERB
ejpam-6059	31	11	some	some	DET
ejpam-6059	31	12	invariant	invariant	ADJ
ejpam-6059	31	13	properties	property	NOUN
ejpam-6059	31	14	of	of	ADP
ejpam-6059	31	15	unitary	unitary	ADJ
ejpam-6059	31	16	cayley	cayley	ADJ
ejpam-6059	31	17	graphs	graph	NOUN
ejpam-6059	31	18	and	and	CCONJ
ejpam-6059	31	19	studied	study	VERB
ejpam-6059	31	20	their	their	PRON
ejpam-6059	31	21	perfectness	perfectness	NOUN
ejpam-6059	31	22	.	.	PUNCT
ejpam-6059	32	1	in	in	ADP
ejpam-6059	32	2	2012	2012	NUM
ejpam-6059	32	3	,	,	PUNCT
ejpam-6059	32	4	kiani	kiani	PROPN
ejpam-6059	32	5	and	and	CCONJ
ejpam-6059	32	6	aghaei	aghaei	PROPN
ejpam-6059	32	7	[	[	X
ejpam-6059	32	8	3	3	NUM
ejpam-6059	32	9	]	]	PUNCT
ejpam-6059	32	10	provided	provide	VERB
ejpam-6059	32	11	isomorphism	isomorphism	NOUN
ejpam-6059	32	12	theorems	theorem	NOUN
ejpam-6059	32	13	for	for	ADP
ejpam-6059	32	14	unitary	unitary	ADJ
ejpam-6059	32	15	cayley	cayley	ADJ
ejpam-6059	32	16	graphs	graph	NOUN
ejpam-6059	32	17	of	of	ADP
ejpam-6059	32	18	rings	ring	NOUN
ejpam-6059	32	19	associated	associate	VERB
ejpam-6059	32	20	with	with	ADP
ejpam-6059	32	21	jacobson	jacobson	PROPN
ejpam-6059	32	22	radicals	radical	NOUN
ejpam-6059	32	23	.	.	PUNCT
ejpam-6059	33	1	in	in	ADP
ejpam-6059	33	2	2014	2014	NUM
ejpam-6059	33	3	,	,	PUNCT
ejpam-6059	33	4	naghipour	naghipour	NOUN
ejpam-6059	34	1	[	[	X
ejpam-6059	34	2	4	4	NUM
ejpam-6059	34	3	]	]	PUNCT
ejpam-6059	34	4	considered	consider	VERB
ejpam-6059	34	5	some	some	DET
ejpam-6059	34	6	properties	property	NOUN
ejpam-6059	34	7	of	of	ADP
ejpam-6059	34	8	induced	induced	ADJ
ejpam-6059	34	9	subgraphs	subgraph	NOUN
ejpam-6059	34	10	of	of	ADP
ejpam-6059	34	11	unitary	unitary	ADJ
ejpam-6059	34	12	cayley	cayley	ADJ
ejpam-6059	34	13	graphs	graph	NOUN
ejpam-6059	34	14	of	of	ADP
ejpam-6059	34	15	commutative	commutative	ADJ
ejpam-6059	34	16	rings	ring	NOUN
ejpam-6059	34	17	.	.	PUNCT
ejpam-6059	35	1	one	one	NUM
ejpam-6059	35	2	of	of	ADP
ejpam-6059	35	3	interesting	interesting	ADJ
ejpam-6059	35	4	parameters	parameter	NOUN
ejpam-6059	35	5	of	of	ADP
ejpam-6059	35	6	graphs	graph	NOUN
ejpam-6059	35	7	is	be	AUX
ejpam-6059	35	8	the	the	DET
ejpam-6059	35	9	acyclic	acyclic	ADJ
ejpam-6059	35	10	number	number	NOUN
ejpam-6059	35	11	which	which	PRON
ejpam-6059	35	12	is	be	AUX
ejpam-6059	35	13	the	the	DET
ejpam-6059	35	14	maximum	maximum	ADJ
ejpam-6059	35	15	number	number	NOUN
ejpam-6059	35	16	of	of	ADP
ejpam-6059	35	17	vertices	vertex	NOUN
ejpam-6059	35	18	that	that	SCONJ
ejpam-6059	35	19	the	the	DET
ejpam-6059	35	20	subgraph	subgraph	NOUN
ejpam-6059	35	21	induced	induce	VERB
ejpam-6059	35	22	by	by	ADP
ejpam-6059	35	23	such	such	ADJ
ejpam-6059	35	24	vertices	vertex	NOUN
ejpam-6059	35	25	contains	contain	VERB
ejpam-6059	35	26	no	no	DET
ejpam-6059	35	27	cycles	cycle	NOUN
ejpam-6059	35	28	.	.	PUNCT
ejpam-6059	36	1	there	there	PRON
ejpam-6059	36	2	are	be	VERB
ejpam-6059	36	3	many	many	ADJ
ejpam-6059	36	4	discussions	discussion	NOUN
ejpam-6059	36	5	of	of	ADP
ejpam-6059	36	6	acyclic	acyclic	ADJ
ejpam-6059	36	7	numbers	number	NOUN
ejpam-6059	36	8	,	,	PUNCT
ejpam-6059	36	9	for	for	ADP
ejpam-6059	36	10	instance	instance	NOUN
ejpam-6059	36	11	,	,	PUNCT
ejpam-6059	36	12	in	in	ADP
ejpam-6059	36	13	2009	2009	NUM
ejpam-6059	36	14	,	,	PUNCT
ejpam-6059	36	15	samodivkin	samodivkin	NOUN
ejpam-6059	36	16	[	[	X
ejpam-6059	36	17	5	5	NUM
ejpam-6059	36	18	]	]	PUNCT
ejpam-6059	36	19	investigated	investigate	VERB
ejpam-6059	36	20	the	the	DET
ejpam-6059	36	21	acyclic	acyclic	ADJ
ejpam-6059	36	22	number	number	NOUN
ejpam-6059	36	23	of	of	ADP
ejpam-6059	36	24	graphs	graph	NOUN
ejpam-6059	36	25	with	with	ADP
ejpam-6059	36	26	cut	cut	NOUN
ejpam-6059	36	27	-	-	PUNCT
ejpam-6059	36	28	vertices	vertex	NOUN
ejpam-6059	36	29	.	.	PUNCT
ejpam-6059	37	1	later	later	ADV
ejpam-6059	37	2	in	in	ADP
ejpam-6059	37	3	2017	2017	NUM
ejpam-6059	37	4	,	,	PUNCT
ejpam-6059	37	5	petrusevski	petrusevski	VERB
ejpam-6059	37	6	and	and	CCONJ
ejpam-6059	37	7	skrekovski	skrekovski	VERB
ejpam-6059	38	1	[	[	X
ejpam-6059	38	2	6	6	NUM
ejpam-6059	38	3	]	]	PUNCT
ejpam-6059	38	4	proposed	propose	VERB
ejpam-6059	38	5	a	a	DET
ejpam-6059	38	6	conjecture	conjecture	NOUN
ejpam-6059	38	7	on	on	ADP
ejpam-6059	38	8	this	this	DET
ejpam-6059	38	9	parameter	parameter	NOUN
ejpam-6059	38	10	.	.	PUNCT
ejpam-6059	39	1	moreover	moreover	ADV
ejpam-6059	39	2	,	,	PUNCT
ejpam-6059	39	3	they	they	PRON
ejpam-6059	39	4	provided	provide	VERB
ejpam-6059	39	5	some	some	DET
ejpam-6059	39	6	conditions	condition	NOUN
ejpam-6059	39	7	that	that	PRON
ejpam-6059	39	8	make	make	VERB
ejpam-6059	39	9	such	such	DET
ejpam-6059	39	10	the	the	DET
ejpam-6059	39	11	conjecture	conjecture	NOUN
ejpam-6059	39	12	weaker	weak	ADJ
ejpam-6059	39	13	for	for	ADP
ejpam-6059	39	14	planar	planar	ADJ
ejpam-6059	39	15	graphs	graph	NOUN
ejpam-6059	39	16	.	.	PUNCT
ejpam-6059	40	1	throughout	throughout	ADP
ejpam-6059	40	2	the	the	DET
ejpam-6059	40	3	paper	paper	NOUN
ejpam-6059	40	4	,	,	PUNCT
ejpam-6059	40	5	the	the	DET
ejpam-6059	40	6	notation	notation	NOUN
ejpam-6059	40	7	γn	γn	NOUN
ejpam-6059	40	8	stands	stand	VERB
ejpam-6059	40	9	for	for	ADP
ejpam-6059	40	10	the	the	DET
ejpam-6059	40	11	unitary	unitary	ADJ
ejpam-6059	40	12	cayley	cayley	ADJ
ejpam-6059	40	13	graph	graph	NOUN
ejpam-6059	40	14	of	of	ADP
ejpam-6059	40	15	a	a	DET
ejpam-6059	40	16	finite	finite	ADJ
ejpam-6059	40	17	commutative	commutative	ADJ
ejpam-6059	40	18	ring	ring	NOUN
ejpam-6059	40	19	zn	zn	PROPN
ejpam-6059	40	20	.	.	PUNCT
ejpam-6059	41	1	in	in	ADP
ejpam-6059	41	2	addition	addition	NOUN
ejpam-6059	41	3	,	,	PUNCT
ejpam-6059	41	4	let	let	VERB
ejpam-6059	41	5	us	we	PRON
ejpam-6059	41	6	denote	denote	VERB
ejpam-6059	41	7	by	by	ADP
ejpam-6059	41	8	γn	γn	PRON
ejpam-6059	41	9	the	the	DET
ejpam-6059	41	10	complement	complement	NOUN
ejpam-6059	41	11	of	of	ADP
ejpam-6059	41	12	γn	γn	NUM
ejpam-6059	41	13	which	which	PRON
ejpam-6059	41	14	is	be	AUX
ejpam-6059	41	15	a	a	DET
ejpam-6059	41	16	simple	simple	ADJ
ejpam-6059	41	17	graph	graph	NOUN
ejpam-6059	41	18	obtained	obtain	VERB
ejpam-6059	41	19	from	from	ADP
ejpam-6059	41	20	γn	γn	NUM
ejpam-6059	41	21	by	by	ADP
ejpam-6059	41	22	deleting	delete	VERB
ejpam-6059	41	23	all	all	DET
ejpam-6059	41	24	edges	edge	NOUN
ejpam-6059	41	25	in	in	ADP
ejpam-6059	41	26	e(γn	e(γn	NOUN
ejpam-6059	41	27	)	)	PUNCT
ejpam-6059	41	28	and	and	CCONJ
ejpam-6059	41	29	then	then	ADV
ejpam-6059	41	30	adding	add	VERB
ejpam-6059	41	31	all	all	DET
ejpam-6059	41	32	edges	edge	NOUN
ejpam-6059	41	33	outside	outside	ADP
ejpam-6059	41	34	e(γn	e(γn	NUM
ejpam-6059	41	35	)	)	PUNCT
ejpam-6059	41	36	.	.	PUNCT
ejpam-6059	42	1	in	in	ADP
ejpam-6059	42	2	this	this	DET
ejpam-6059	42	3	paper	paper	NOUN
ejpam-6059	42	4	,	,	PUNCT
ejpam-6059	42	5	we	we	PRON
ejpam-6059	42	6	study	study	VERB
ejpam-6059	42	7	some	some	DET
ejpam-6059	42	8	properties	property	NOUN
ejpam-6059	42	9	on	on	ADP
ejpam-6059	42	10	unitary	unitary	ADJ
ejpam-6059	42	11	cayley	cayley	ADJ
ejpam-6059	42	12	graphs	graph	NOUN
ejpam-6059	42	13	of	of	ADP
ejpam-6059	42	14	finite	finite	PROPN
ejpam-6059	42	15	commutative	commutative	ADJ
ejpam-6059	42	16	rings	ring	NOUN
ejpam-6059	42	17	.	.	PUNCT
ejpam-6059	43	1	furthermore	furthermore	ADV
ejpam-6059	43	2	,	,	PUNCT
ejpam-6059	43	3	we	we	PRON
ejpam-6059	43	4	present	present	VERB
ejpam-6059	43	5	the	the	DET
ejpam-6059	43	6	construction	construction	NOUN
ejpam-6059	43	7	of	of	ADP
ejpam-6059	43	8	cycles	cycle	NOUN
ejpam-6059	43	9	of	of	ADP
ejpam-6059	43	10	lengths	length	NOUN
ejpam-6059	43	11	3	3	NUM
ejpam-6059	43	12	and	and	CCONJ
ejpam-6059	43	13	4	4	NUM
ejpam-6059	43	14	which	which	PRON
ejpam-6059	43	15	is	be	AUX
ejpam-6059	43	16	useful	useful	ADJ
ejpam-6059	43	17	for	for	ADP
ejpam-6059	43	18	finding	find	VERB
ejpam-6059	43	19	the	the	DET
ejpam-6059	43	20	maximum	maximum	ADJ
ejpam-6059	43	21	cardinality	cardinality	NOUN
ejpam-6059	43	22	among	among	ADP
ejpam-6059	43	23	acyclic	acyclic	ADJ
ejpam-6059	43	24	sets	set	NOUN
ejpam-6059	43	25	of	of	ADP
ejpam-6059	43	26	γn	γn	NOUN
ejpam-6059	43	27	and	and	CCONJ
ejpam-6059	43	28	γn	γn	NOUN
ejpam-6059	43	29	,	,	PUNCT
ejpam-6059	43	30	such	such	ADJ
ejpam-6059	43	31	the	the	DET
ejpam-6059	43	32	number	number	NOUN
ejpam-6059	43	33	is	be	AUX
ejpam-6059	43	34	called	call	VERB
ejpam-6059	43	35	the	the	DET
ejpam-6059	43	36	upper	upper	ADJ
ejpam-6059	43	37	acyclic	acyclic	ADJ
ejpam-6059	43	38	number	number	NOUN
ejpam-6059	43	39	.	.	PUNCT
ejpam-6059	44	1	moreover	moreover	ADV
ejpam-6059	44	2	,	,	PUNCT
ejpam-6059	44	3	we	we	PRON
ejpam-6059	44	4	provide	provide	VERB
ejpam-6059	44	5	some	some	DET
ejpam-6059	44	6	characterization	characterization	NOUN
ejpam-6059	44	7	of	of	ADP
ejpam-6059	44	8	the	the	DET
ejpam-6059	44	9	structure	structure	NOUN
ejpam-6059	44	10	of	of	ADP
ejpam-6059	44	11	γn	γn	NOUN
ejpam-6059	44	12	and	and	CCONJ
ejpam-6059	44	13	γn	γn	NOUN
ejpam-6059	44	14	,	,	PUNCT
ejpam-6059	44	15	where	where	SCONJ
ejpam-6059	44	16	n	n	PRON
ejpam-6059	44	17	is	be	AUX
ejpam-6059	44	18	a	a	DET
ejpam-6059	44	19	power	power	NOUN
ejpam-6059	44	20	of	of	ADP
ejpam-6059	44	21	a	a	DET
ejpam-6059	44	22	prime	prime	ADJ
ejpam-6059	44	23	number	number	NOUN
ejpam-6059	44	24	,	,	PUNCT
ejpam-6059	44	25	to	to	PART
ejpam-6059	44	26	find	find	VERB
ejpam-6059	44	27	the	the	DET
ejpam-6059	44	28	maximum	maximum	ADJ
ejpam-6059	44	29	number	number	NOUN
ejpam-6059	44	30	k	k	PROPN
ejpam-6059	44	31	of	of	ADP
ejpam-6059	44	32	vertices	vertex	NOUN
ejpam-6059	44	33	of	of	ADP
ejpam-6059	44	34	γn	γn	NOUN
ejpam-6059	44	35	in	in	ADP
ejpam-6059	44	36	which	which	PRON
ejpam-6059	44	37	every	every	DET
ejpam-6059	44	38	subgraph	subgraph	NOUN
ejpam-6059	44	39	of	of	ADP
ejpam-6059	44	40	γn	γn	NOUN
ejpam-6059	44	41	induced	induce	VERB
ejpam-6059	44	42	by	by	ADP
ejpam-6059	44	43	k	k	PROPN
ejpam-6059	44	44	vertices	vertex	NOUN
ejpam-6059	44	45	contains	contain	VERB
ejpam-6059	44	46	no	no	DET
ejpam-6059	44	47	cycles	cycle	NOUN
ejpam-6059	44	48	.	.	PUNCT
ejpam-6059	45	1	this	this	DET
ejpam-6059	45	2	cardinality	cardinality	NOUN
ejpam-6059	45	3	is	be	AUX
ejpam-6059	45	4	called	call	VERB
ejpam-6059	45	5	the	the	DET
ejpam-6059	45	6	lower	low	ADJ
ejpam-6059	45	7	acyclic	acyclic	ADJ
ejpam-6059	45	8	number	number	NOUN
ejpam-6059	45	9	,	,	PUNCT
ejpam-6059	45	10	and	and	CCONJ
ejpam-6059	45	11	all	all	DET
ejpam-6059	45	12	sets	set	NOUN
ejpam-6059	45	13	mentioned	mention	VERB
ejpam-6059	45	14	in	in	ADP
ejpam-6059	45	15	this	this	DET
ejpam-6059	45	16	research	research	NOUN
ejpam-6059	45	17	are	be	AUX
ejpam-6059	45	18	considered	consider	VERB
ejpam-6059	45	19	to	to	PART
ejpam-6059	45	20	be	be	AUX
ejpam-6059	45	21	finite	finite	ADJ
ejpam-6059	45	22	sets	set	NOUN
ejpam-6059	45	23	.	.	PUNCT
ejpam-6059	46	1	2	2	X
ejpam-6059	46	2	.	.	NUM
ejpam-6059	46	3	preliminaries	preliminary	NOUN
ejpam-6059	46	4	a	a	DET
ejpam-6059	46	5	graph	graph	NOUN
ejpam-6059	46	6	g	g	NOUN
ejpam-6059	46	7	is	be	AUX
ejpam-6059	46	8	a	a	DET
ejpam-6059	46	9	pair	pair	NOUN
ejpam-6059	46	10	(	(	PUNCT
ejpam-6059	46	11	v	v	NOUN
ejpam-6059	46	12	(	(	PUNCT
ejpam-6059	46	13	g	g	NOUN
ejpam-6059	46	14	)	)	PUNCT
ejpam-6059	46	15	,	,	PUNCT
ejpam-6059	46	16	e(g	e(g	PROPN
ejpam-6059	46	17	)	)	PUNCT
ejpam-6059	46	18	)	)	PUNCT
ejpam-6059	46	19	where	where	SCONJ
ejpam-6059	46	20	v	v	X
ejpam-6059	46	21	(	(	PUNCT
ejpam-6059	46	22	g	g	NOUN
ejpam-6059	46	23	)	)	PUNCT
ejpam-6059	46	24	is	be	AUX
ejpam-6059	46	25	the	the	DET
ejpam-6059	46	26	vertex	vertex	NOUN
ejpam-6059	46	27	set	set	NOUN
ejpam-6059	46	28	of	of	ADP
ejpam-6059	46	29	g	g	PROPN
ejpam-6059	46	30	and	and	CCONJ
ejpam-6059	46	31	e(g	e(g	PROPN
ejpam-6059	46	32	)	)	PUNCT
ejpam-6059	46	33	is	be	AUX
ejpam-6059	46	34	an	an	DET
ejpam-6059	46	35	edge	edge	NOUN
ejpam-6059	46	36	set	set	NOUN
ejpam-6059	46	37	of	of	ADP
ejpam-6059	46	38	g.	g.	PROPN
ejpam-6059	46	39	an	an	DET
ejpam-6059	46	40	edge	edge	NOUN
ejpam-6059	46	41	of	of	ADP
ejpam-6059	46	42	g	g	NOUN
ejpam-6059	46	43	joining	join	VERB
ejpam-6059	46	44	between	between	ADP
ejpam-6059	46	45	vertices	vertex	NOUN
ejpam-6059	46	46	u	u	NOUN
ejpam-6059	46	47	,	,	PUNCT
ejpam-6059	46	48	v	v	NOUN
ejpam-6059	46	49	∈	∈	PROPN
ejpam-6059	46	50	v	v	NOUN
ejpam-6059	46	51	(	(	PUNCT
ejpam-6059	46	52	g	g	NOUN
ejpam-6059	46	53	)	)	PUNCT
ejpam-6059	46	54	is	be	AUX
ejpam-6059	46	55	written	write	VERB
ejpam-6059	46	56	as	as	ADP
ejpam-6059	46	57	{	{	PUNCT
ejpam-6059	46	58	u	u	NOUN
ejpam-6059	46	59	,	,	PUNCT
ejpam-6059	46	60	v	v	NOUN
ejpam-6059	46	61	}	}	PUNCT
ejpam-6059	46	62	,	,	PUNCT
ejpam-6059	46	63	that	that	ADV
ejpam-6059	46	64	is	is	ADV
ejpam-6059	46	65	,	,	PUNCT
ejpam-6059	46	66	{	{	PUNCT
ejpam-6059	46	67	u	u	NOUN
ejpam-6059	46	68	,	,	PUNCT
ejpam-6059	46	69	v	v	NOUN
ejpam-6059	46	70	}	}	PUNCT
ejpam-6059	46	71	∈	∈	PROPN
ejpam-6059	46	72	e(g	e(g	PROPN
ejpam-6059	46	73	)	)	PUNCT
ejpam-6059	46	74	means	mean	VERB
ejpam-6059	46	75	that	that	SCONJ
ejpam-6059	46	76	u	u	NOUN
ejpam-6059	46	77	is	be	AUX
ejpam-6059	46	78	adjacent	adjacent	ADJ
ejpam-6059	46	79	to	to	ADP
ejpam-6059	46	80	v	v	NOUN
ejpam-6059	46	81	in	in	ADP
ejpam-6059	46	82	g.	g.	PROPN
ejpam-6059	47	1	let	let	VERB
ejpam-6059	47	2	v	v	ADP
ejpam-6059	47	3	∈	∈	PROPN
ejpam-6059	47	4	v	v	NOUN
ejpam-6059	47	5	(	(	PUNCT
ejpam-6059	47	6	g	g	NOUN
ejpam-6059	47	7	)	)	PUNCT
ejpam-6059	47	8	.	.	PUNCT
ejpam-6059	48	1	the	the	DET
ejpam-6059	48	2	degree	degree	NOUN
ejpam-6059	48	3	of	of	ADP
ejpam-6059	48	4	v	v	NOUN
ejpam-6059	48	5	,	,	PUNCT
ejpam-6059	48	6	denoted	denote	VERB
ejpam-6059	48	7	by	by	ADP
ejpam-6059	48	8	deg(v	deg(v	NOUN
ejpam-6059	48	9	)	)	PUNCT
ejpam-6059	48	10	,	,	PUNCT
ejpam-6059	48	11	is	be	AUX
ejpam-6059	48	12	the	the	DET
ejpam-6059	48	13	number	number	NOUN
ejpam-6059	48	14	of	of	ADP
ejpam-6059	48	15	vertices	vertex	NOUN
ejpam-6059	48	16	adjacent	adjacent	ADJ
ejpam-6059	48	17	to	to	ADP
ejpam-6059	48	18	v	v	NOUN
ejpam-6059	48	19	in	in	ADP
ejpam-6059	48	20	g.	g.	PROPN
ejpam-6059	48	21	furthermore	furthermore	ADV
ejpam-6059	48	22	,	,	PUNCT
ejpam-6059	48	23	let	let	VERB
ejpam-6059	48	24	c	c	PRON
ejpam-6059	48	25	be	be	AUX
ejpam-6059	48	26	a	a	DET
ejpam-6059	48	27	sequence	sequence	NOUN
ejpam-6059	48	28	v1	v1	NOUN
ejpam-6059	48	29	,	,	PUNCT
ejpam-6059	48	30	v2	v2	NOUN
ejpam-6059	48	31	,	,	PUNCT
ejpam-6059	48	32	.	.	PUNCT
ejpam-6059	48	33	.	.	PUNCT
ejpam-6059	49	1	.	.	PUNCT
ejpam-6059	50	1	,	,	PUNCT
ejpam-6059	50	2	vk	vk	INTJ
ejpam-6059	50	3	of	of	ADP
ejpam-6059	50	4	distinct	distinct	ADJ
ejpam-6059	50	5	k	k	PROPN
ejpam-6059	50	6	vertices	vertex	NOUN
ejpam-6059	50	7	of	of	ADP
ejpam-6059	50	8	g	g	PROPN
ejpam-6059	50	9	where	where	SCONJ
ejpam-6059	50	10	k	k	PROPN
ejpam-6059	50	11	≥	≥	NUM
ejpam-6059	50	12	3	3	X
ejpam-6059	50	13	.	.	PUNCT
ejpam-6059	51	1	if	if	SCONJ
ejpam-6059	51	2	vi	vi	PROPN
ejpam-6059	51	3	and	and	CCONJ
ejpam-6059	51	4	vi+1	vi+1	NUM
ejpam-6059	51	5	are	be	AUX
ejpam-6059	51	6	adjacent	adjacent	ADJ
ejpam-6059	51	7	in	in	ADP
ejpam-6059	51	8	g	g	PROPN
ejpam-6059	51	9	for	for	ADP
ejpam-6059	51	10	all	all	DET
ejpam-6059	51	11	i	i	PRON
ejpam-6059	51	12	=	=	NOUN
ejpam-6059	51	13	1	1	NUM
ejpam-6059	51	14	,	,	PUNCT
ejpam-6059	51	15	2	2	NUM
ejpam-6059	51	16	,	,	PUNCT
ejpam-6059	51	17	.	.	PUNCT
ejpam-6059	51	18	.	.	PUNCT
ejpam-6059	52	1	.	.	PUNCT
ejpam-6059	53	1	,	,	PUNCT
ejpam-6059	54	1	k	k	PROPN
ejpam-6059	55	1	−	−	PROPN
ejpam-6059	55	2	1	1	NUM
ejpam-6059	55	3	and	and	CCONJ
ejpam-6059	55	4	v1	v1	NOUN
ejpam-6059	55	5	is	be	AUX
ejpam-6059	55	6	adjacent	adjacent	ADJ
ejpam-6059	55	7	to	to	PART
ejpam-6059	55	8	vk	vk	VERB
ejpam-6059	55	9	,	,	PUNCT
ejpam-6059	55	10	then	then	ADV
ejpam-6059	55	11	c	c	PROPN
ejpam-6059	55	12	is	be	AUX
ejpam-6059	55	13	called	call	VERB
ejpam-6059	55	14	a	a	DET
ejpam-6059	55	15	cycle	cycle	NOUN
ejpam-6059	55	16	in	in	ADP
ejpam-6059	55	17	g	g	NOUN
ejpam-6059	55	18	with	with	ADP
ejpam-6059	55	19	length	length	NOUN
ejpam-6059	55	20	k	k	PROPN
ejpam-6059	55	21	and	and	CCONJ
ejpam-6059	55	22	denoted	denote	VERB
ejpam-6059	55	23	by	by	ADP
ejpam-6059	55	24	ck	ck	PROPN
ejpam-6059	55	25	.	.	PUNCT
ejpam-6059	56	1	moreover	moreover	ADV
ejpam-6059	56	2	,	,	PUNCT
ejpam-6059	56	3	if	if	SCONJ
ejpam-6059	56	4	k	k	PROPN
ejpam-6059	56	5	is	be	AUX
ejpam-6059	56	6	odd	odd	ADJ
ejpam-6059	56	7	(	(	PUNCT
ejpam-6059	56	8	even	even	ADV
ejpam-6059	56	9	)	)	PUNCT
ejpam-6059	56	10	,	,	PUNCT
ejpam-6059	56	11	then	then	ADV
ejpam-6059	56	12	ck	ck	INTJ
ejpam-6059	56	13	is	be	AUX
ejpam-6059	56	14	said	say	VERB
ejpam-6059	56	15	to	to	PART
ejpam-6059	56	16	be	be	AUX
ejpam-6059	56	17	an	an	DET
ejpam-6059	56	18	odd	odd	ADJ
ejpam-6059	56	19	(	(	PUNCT
ejpam-6059	56	20	even	even	ADJ
ejpam-6059	56	21	)	)	PUNCT
ejpam-6059	56	22	cycle	cycle	NOUN
ejpam-6059	56	23	.	.	PUNCT
ejpam-6059	57	1	in	in	ADP
ejpam-6059	57	2	particular	particular	ADJ
ejpam-6059	57	3	,	,	PUNCT
ejpam-6059	57	4	if	if	SCONJ
ejpam-6059	57	5	k	k	PROPN
ejpam-6059	57	6	=	=	SYM
ejpam-6059	57	7	3	3	NUM
ejpam-6059	57	8	,	,	PUNCT
ejpam-6059	57	9	then	then	ADV
ejpam-6059	57	10	the	the	DET
ejpam-6059	57	11	cycle	cycle	NOUN
ejpam-6059	57	12	c3	c3	PROPN
ejpam-6059	57	13	is	be	AUX
ejpam-6059	57	14	sometimes	sometimes	ADV
ejpam-6059	57	15	called	call	VERB
ejpam-6059	57	16	a	a	DET
ejpam-6059	57	17	triangle	triangle	NOUN
ejpam-6059	57	18	.	.	PUNCT
ejpam-6059	58	1	a	a	DET
ejpam-6059	58	2	graph	graph	NOUN
ejpam-6059	58	3	g	g	NOUN
ejpam-6059	58	4	is	be	AUX
ejpam-6059	58	5	said	say	VERB
ejpam-6059	58	6	to	to	PART
ejpam-6059	58	7	be	be	AUX
ejpam-6059	58	8	complete	complete	ADJ
ejpam-6059	58	9	if	if	SCONJ
ejpam-6059	58	10	{	{	PUNCT
ejpam-6059	58	11	u	u	NOUN
ejpam-6059	58	12	,	,	PUNCT
ejpam-6059	58	13	v	v	NOUN
ejpam-6059	58	14	}	}	PUNCT
ejpam-6059	58	15	∈	∈	PROPN
ejpam-6059	58	16	e(g	e(g	PROPN
ejpam-6059	58	17	)	)	PUNCT
ejpam-6059	58	18	for	for	ADP
ejpam-6059	58	19	all	all	DET
ejpam-6059	58	20	u	u	NOUN
ejpam-6059	58	21	,	,	PUNCT
ejpam-6059	58	22	v	v	NOUN
ejpam-6059	58	23	∈	∈	PROPN
ejpam-6059	58	24	v	v	NOUN
ejpam-6059	58	25	(	(	PUNCT
ejpam-6059	58	26	g	g	NOUN
ejpam-6059	58	27	)	)	PUNCT
ejpam-6059	58	28	.	.	PUNCT
ejpam-6059	59	1	in	in	ADP
ejpam-6059	59	2	addition	addition	NOUN
ejpam-6059	59	3	.	.	PUNCT
ejpam-6059	60	1	a	a	DET
ejpam-6059	60	2	graph	graph	NOUN
ejpam-6059	60	3	g	g	NOUN
ejpam-6059	60	4	will	will	AUX
ejpam-6059	60	5	be	be	AUX
ejpam-6059	60	6	called	call	VERB
ejpam-6059	60	7	a	a	DET
ejpam-6059	60	8	complete	complete	ADJ
ejpam-6059	60	9	p	p	ADJ
ejpam-6059	60	10	-	-	PUNCT
ejpam-6059	60	11	partite	partite	ADJ
ejpam-6059	60	12	graph	graph	NOUN
ejpam-6059	60	13	if	if	SCONJ
ejpam-6059	60	14	v	v	X
ejpam-6059	60	15	(	(	PUNCT
ejpam-6059	60	16	g	g	NOUN
ejpam-6059	60	17	)	)	PUNCT
ejpam-6059	60	18	is	be	AUX
ejpam-6059	60	19	partitioned	partition	VERB
ejpam-6059	60	20	into	into	ADP
ejpam-6059	60	21	p	p	NOUN
ejpam-6059	60	22	sets	set	NOUN
ejpam-6059	60	23	,	,	PUNCT
ejpam-6059	60	24	called	call	VERB
ejpam-6059	60	25	a	a	DET
ejpam-6059	60	26	partite	partite	ADJ
ejpam-6059	60	27	set	set	NOUN
ejpam-6059	60	28	,	,	PUNCT
ejpam-6059	60	29	provided	provide	VERB
ejpam-6059	60	30	that	that	SCONJ
ejpam-6059	60	31	vertices	vertex	NOUN
ejpam-6059	60	32	in	in	ADP
ejpam-6059	60	33	the	the	DET
ejpam-6059	60	34	same	same	ADJ
ejpam-6059	60	35	partite	partite	ADJ
ejpam-6059	60	36	set	set	NOUN
ejpam-6059	60	37	are	be	AUX
ejpam-6059	60	38	not	not	PART
ejpam-6059	60	39	mutually	mutually	ADV
ejpam-6059	60	40	adjacent	adjacent	ADJ
ejpam-6059	60	41	and	and	CCONJ
ejpam-6059	60	42	every	every	DET
ejpam-6059	60	43	two	two	NUM
ejpam-6059	60	44	vertices	vertex	NOUN
ejpam-6059	60	45	from	from	ADP
ejpam-6059	60	46	different	different	ADJ
ejpam-6059	60	47	partite	partite	ADJ
ejpam-6059	60	48	sets	set	NOUN
ejpam-6059	60	49	must	must	AUX
ejpam-6059	60	50	be	be	AUX
ejpam-6059	60	51	adjacent	adjacent	ADJ
ejpam-6059	60	52	.	.	PUNCT
ejpam-6059	61	1	let	let	VERB
ejpam-6059	61	2	h	h	NOUN
ejpam-6059	61	3	=	=	PUNCT
ejpam-6059	61	4	(	(	PUNCT
ejpam-6059	61	5	v	v	NOUN
ejpam-6059	61	6	(	(	PUNCT
ejpam-6059	61	7	h	h	NOUN
ejpam-6059	61	8	)	)	PUNCT
ejpam-6059	61	9	,	,	PUNCT
ejpam-6059	61	10	e(h	e(h	PROPN
ejpam-6059	61	11	)	)	PUNCT
ejpam-6059	61	12	)	)	PUNCT
ejpam-6059	61	13	be	be	AUX
ejpam-6059	61	14	a	a	DET
ejpam-6059	61	15	graph	graph	NOUN
ejpam-6059	61	16	.	.	PUNCT
ejpam-6059	62	1	the	the	DET
ejpam-6059	62	2	graph	graph	NOUN
ejpam-6059	62	3	h	h	NOUN
ejpam-6059	62	4	is	be	AUX
ejpam-6059	62	5	called	call	VERB
ejpam-6059	62	6	a	a	DET
ejpam-6059	62	7	subgraph	subgraph	NOUN
ejpam-6059	62	8	of	of	ADP
ejpam-6059	62	9	g	g	PROPN
ejpam-6059	62	10	if	if	SCONJ
ejpam-6059	62	11	v	v	X
ejpam-6059	62	12	(	(	PUNCT
ejpam-6059	62	13	h	h	NOUN
ejpam-6059	62	14	)	)	PUNCT
ejpam-6059	62	15	⊆	⊆	NUM
ejpam-6059	62	16	v	v	NOUN
ejpam-6059	62	17	(	(	PUNCT
ejpam-6059	62	18	g	g	NOUN
ejpam-6059	62	19	)	)	PUNCT
ejpam-6059	62	20	and	and	CCONJ
ejpam-6059	62	21	e(h	e(h	NOUN
ejpam-6059	62	22	)	)	PUNCT
ejpam-6059	62	23	⊆	⊆	NUM
ejpam-6059	62	24	e(g	e(g	PROPN
ejpam-6059	62	25	)	)	PUNCT
ejpam-6059	62	26	.	.	PUNCT
ejpam-6059	63	1	a	a	DET
ejpam-6059	63	2	decomposition	decomposition	NOUN
ejpam-6059	63	3	of	of	ADP
ejpam-6059	63	4	g	g	PROPN
ejpam-6059	63	5	is	be	AUX
ejpam-6059	63	6	a	a	DET
ejpam-6059	63	7	collection	collection	NOUN
ejpam-6059	63	8	of	of	ADP
ejpam-6059	63	9	edge	edge	NOUN
ejpam-6059	63	10	disjoint	disjoint	NOUN
ejpam-6059	63	11	subgraphs	subgraph	NOUN
ejpam-6059	63	12	of	of	ADP
ejpam-6059	63	13	g	g	NOUN
ejpam-6059	63	14	in	in	ADP
ejpam-6059	63	15	which	which	PRON
ejpam-6059	63	16	every	every	DET
ejpam-6059	63	17	edge	edge	NOUN
ejpam-6059	63	18	d.	d.	PROPN
ejpam-6059	63	19	pongpipat	pongpipat	PROPN
ejpam-6059	63	20	,	,	PUNCT
ejpam-6059	63	21	n.	n.	NOUN
ejpam-6059	63	22	nupo	nupo	PROPN
ejpam-6059	63	23	/	/	SYM
ejpam-6059	63	24	eur	eur	PROPN
ejpam-6059	63	25	.	.	PUNCT
ejpam-6059	64	1	j.	j.	PROPN
ejpam-6059	64	2	pure	pure	PROPN
ejpam-6059	64	3	appl	appl	PROPN
ejpam-6059	64	4	.	.	PROPN
ejpam-6059	64	5	math	math	PROPN
ejpam-6059	64	6	,	,	PUNCT
ejpam-6059	64	7	18	18	NUM
ejpam-6059	64	8	(	(	PUNCT
ejpam-6059	64	9	2	2	NUM
ejpam-6059	64	10	)	)	PUNCT
ejpam-6059	64	11	(	(	PUNCT
ejpam-6059	64	12	2025	2025	NUM
ejpam-6059	64	13	)	)	PUNCT
ejpam-6059	64	14	,	,	PUNCT
ejpam-6059	64	15	6059	6059	NUM
ejpam-6059	64	16	3	3	NUM
ejpam-6059	64	17	of	of	ADP
ejpam-6059	64	18	11	11	NUM
ejpam-6059	64	19	of	of	ADP
ejpam-6059	64	20	g	g	PROPN
ejpam-6059	64	21	belongs	belong	VERB
ejpam-6059	64	22	to	to	ADP
ejpam-6059	64	23	exactly	exactly	ADV
ejpam-6059	64	24	one	one	NUM
ejpam-6059	64	25	subgraph	subgraph	NOUN
ejpam-6059	64	26	.	.	PUNCT
ejpam-6059	65	1	next	next	ADV
ejpam-6059	65	2	,	,	PUNCT
ejpam-6059	65	3	let	let	VERB
ejpam-6059	65	4	w	w	NOUN
ejpam-6059	65	5	be	be	AUX
ejpam-6059	65	6	a	a	DET
ejpam-6059	65	7	nonempty	nonempty	ADJ
ejpam-6059	65	8	subset	subset	NOUN
ejpam-6059	65	9	of	of	ADP
ejpam-6059	65	10	v	v	NOUN
ejpam-6059	65	11	(	(	PUNCT
ejpam-6059	65	12	g	g	NOUN
ejpam-6059	65	13	)	)	PUNCT
ejpam-6059	65	14	.	.	PUNCT
ejpam-6059	66	1	a	a	DET
ejpam-6059	66	2	subgraph	subgraph	NOUN
ejpam-6059	66	3	of	of	ADP
ejpam-6059	66	4	g	g	PROPN
ejpam-6059	66	5	induced	induce	VERB
ejpam-6059	66	6	by	by	ADP
ejpam-6059	66	7	w	w	PROPN
ejpam-6059	66	8	,	,	PUNCT
ejpam-6059	66	9	simply	simply	ADV
ejpam-6059	66	10	called	call	VERB
ejpam-6059	66	11	an	an	DET
ejpam-6059	66	12	induced	induced	ADJ
ejpam-6059	66	13	subgraph	subgraph	NOUN
ejpam-6059	66	14	and	and	CCONJ
ejpam-6059	66	15	denoted	denote	VERB
ejpam-6059	66	16	by	by	ADP
ejpam-6059	66	17	g[w	g[w	PROPN
ejpam-6059	66	18	]	]	PUNCT
ejpam-6059	66	19	,	,	PUNCT
ejpam-6059	66	20	is	be	AUX
ejpam-6059	66	21	a	a	DET
ejpam-6059	66	22	subgraph	subgraph	NOUN
ejpam-6059	66	23	of	of	ADP
ejpam-6059	66	24	g	g	NOUN
ejpam-6059	66	25	satisfying	satisfy	VERB
ejpam-6059	66	26	the	the	DET
ejpam-6059	66	27	condition	condition	NOUN
ejpam-6059	66	28	that	that	SCONJ
ejpam-6059	66	29	if	if	SCONJ
ejpam-6059	66	30	u	u	NOUN
ejpam-6059	66	31	,	,	PUNCT
ejpam-6059	66	32	v	v	ADP
ejpam-6059	66	33	∈	∈	PROPN
ejpam-6059	66	34	w	w	NOUN
ejpam-6059	66	35	and	and	CCONJ
ejpam-6059	66	36	{	{	PUNCT
ejpam-6059	66	37	u	u	NOUN
ejpam-6059	66	38	,	,	PUNCT
ejpam-6059	66	39	v	v	NOUN
ejpam-6059	66	40	}	}	PUNCT
ejpam-6059	66	41	∈	∈	PROPN
ejpam-6059	66	42	e(g	e(g	PROPN
ejpam-6059	66	43	)	)	PUNCT
ejpam-6059	66	44	,	,	PUNCT
ejpam-6059	66	45	then	then	ADV
ejpam-6059	66	46	{	{	PUNCT
ejpam-6059	66	47	u	u	NOUN
ejpam-6059	66	48	,	,	PUNCT
ejpam-6059	66	49	v	v	NOUN
ejpam-6059	66	50	}	}	PUNCT
ejpam-6059	66	51	∈	∈	NOUN
ejpam-6059	66	52	e(g[w	e(g[w	NOUN
ejpam-6059	66	53	]	]	X
ejpam-6059	66	54	)	)	PUNCT
ejpam-6059	66	55	,	,	PUNCT
ejpam-6059	66	56	as	as	ADV
ejpam-6059	66	57	well	well	ADV
ejpam-6059	66	58	.	.	PUNCT
ejpam-6059	67	1	throughout	throughout	ADP
ejpam-6059	67	2	the	the	DET
ejpam-6059	67	3	paper	paper	NOUN
ejpam-6059	67	4	,	,	PUNCT
ejpam-6059	67	5	all	all	DET
ejpam-6059	67	6	sets	set	NOUN
ejpam-6059	67	7	are	be	AUX
ejpam-6059	67	8	finite	finite	ADJ
ejpam-6059	67	9	sets	set	NOUN
ejpam-6059	67	10	and	and	CCONJ
ejpam-6059	67	11	all	all	DET
ejpam-6059	67	12	graphs	graph	NOUN
ejpam-6059	67	13	are	be	AUX
ejpam-6059	67	14	simple	simple	ADJ
ejpam-6059	67	15	.	.	PUNCT
ejpam-6059	68	1	more	more	ADJ
ejpam-6059	68	2	information	information	NOUN
ejpam-6059	68	3	about	about	ADP
ejpam-6059	68	4	graph	graph	NOUN
ejpam-6059	68	5	theory	theory	NOUN
ejpam-6059	68	6	can	can	AUX
ejpam-6059	68	7	be	be	AUX
ejpam-6059	68	8	found	find	VERB
ejpam-6059	68	9	in	in	ADP
ejpam-6059	68	10	[	[	X
ejpam-6059	68	11	7	7	NUM
ejpam-6059	68	12	]	]	PUNCT
ejpam-6059	68	13	.	.	PUNCT
ejpam-6059	69	1	we	we	PRON
ejpam-6059	69	2	now	now	ADV
ejpam-6059	69	3	provide	provide	VERB
ejpam-6059	69	4	some	some	DET
ejpam-6059	69	5	definitions	definition	NOUN
ejpam-6059	69	6	and	and	CCONJ
ejpam-6059	69	7	prominent	prominent	ADJ
ejpam-6059	69	8	facts	fact	NOUN
ejpam-6059	69	9	which	which	PRON
ejpam-6059	69	10	play	play	VERB
ejpam-6059	69	11	a	a	DET
ejpam-6059	69	12	crucial	crucial	ADJ
ejpam-6059	69	13	role	role	NOUN
ejpam-6059	69	14	in	in	ADP
ejpam-6059	69	15	the	the	DET
ejpam-6059	69	16	paper	paper	NOUN
ejpam-6059	69	17	.	.	PUNCT
ejpam-6059	70	1	definition	definition	NOUN
ejpam-6059	70	2	1	1	NUM
ejpam-6059	70	3	.	.	PUNCT
ejpam-6059	71	1	[	[	X
ejpam-6059	71	2	8	8	NUM
ejpam-6059	71	3	]	]	PUNCT
ejpam-6059	71	4	let	let	VERB
ejpam-6059	71	5	g	g	PRON
ejpam-6059	71	6	be	be	AUX
ejpam-6059	71	7	a	a	DET
ejpam-6059	71	8	group	group	NOUN
ejpam-6059	71	9	.	.	PUNCT
ejpam-6059	72	1	a	a	DET
ejpam-6059	72	2	nonempty	nonempty	ADV
ejpam-6059	72	3	subset	subset	VERB
ejpam-6059	72	4	h	h	NOUN
ejpam-6059	72	5	of	of	ADP
ejpam-6059	72	6	g	g	PROPN
ejpam-6059	72	7	is	be	AUX
ejpam-6059	72	8	called	call	VERB
ejpam-6059	72	9	a	a	DET
ejpam-6059	72	10	subgroup	subgroup	NOUN
ejpam-6059	72	11	of	of	ADP
ejpam-6059	72	12	g	g	PROPN
ejpam-6059	72	13	if	if	SCONJ
ejpam-6059	72	14	h	h	PROPN
ejpam-6059	72	15	itself	itself	PRON
ejpam-6059	72	16	is	be	AUX
ejpam-6059	72	17	a	a	DET
ejpam-6059	72	18	group	group	NOUN
ejpam-6059	72	19	under	under	ADP
ejpam-6059	72	20	the	the	DET
ejpam-6059	72	21	group	group	NOUN
ejpam-6059	72	22	operation	operation	NOUN
ejpam-6059	72	23	of	of	ADP
ejpam-6059	72	24	g	g	PROPN
ejpam-6059	72	25	restricted	restrict	VERB
ejpam-6059	72	26	to	to	ADP
ejpam-6059	72	27	h.	h.	PROPN
ejpam-6059	72	28	moreover	moreover	ADV
ejpam-6059	72	29	,	,	PUNCT
ejpam-6059	72	30	a	a	DET
ejpam-6059	72	31	left	left	ADJ
ejpam-6059	72	32	coset	coset	NOUN
ejpam-6059	72	33	of	of	ADP
ejpam-6059	72	34	a	a	DET
ejpam-6059	72	35	subgroup	subgroup	NOUN
ejpam-6059	72	36	h	h	NOUN
ejpam-6059	72	37	of	of	ADP
ejpam-6059	72	38	g	g	PROPN
ejpam-6059	72	39	is	be	AUX
ejpam-6059	72	40	a	a	DET
ejpam-6059	72	41	set	set	NOUN
ejpam-6059	72	42	of	of	ADP
ejpam-6059	72	43	the	the	DET
ejpam-6059	72	44	form	form	NOUN
ejpam-6059	72	45	gh	gh	PROPN
ejpam-6059	72	46	:	:	PUNCT
ejpam-6059	72	47	=	=	SYM
ejpam-6059	72	48	{	{	PUNCT
ejpam-6059	72	49	gh	gh	PROPN
ejpam-6059	72	50	:	:	PUNCT
ejpam-6059	72	51	h	h	PROPN
ejpam-6059	72	52	∈	∈	PROPN
ejpam-6059	72	53	h	h	NOUN
ejpam-6059	72	54	}	}	PUNCT
ejpam-6059	72	55	.	.	PUNCT
ejpam-6059	73	1	the	the	DET
ejpam-6059	73	2	set	set	NOUN
ejpam-6059	73	3	of	of	ADP
ejpam-6059	73	4	all	all	DET
ejpam-6059	73	5	cosets	coset	NOUN
ejpam-6059	73	6	of	of	ADP
ejpam-6059	73	7	h	h	NOUN
ejpam-6059	73	8	is	be	AUX
ejpam-6059	73	9	denoted	denote	VERB
ejpam-6059	73	10	by	by	ADP
ejpam-6059	73	11	g	g	PROPN
ejpam-6059	73	12	/	/	SYM
ejpam-6059	73	13	h	h	NOUN
ejpam-6059	73	14	,	,	PUNCT
ejpam-6059	73	15	that	that	ADV
ejpam-6059	73	16	is	is	ADV
ejpam-6059	73	17	,	,	PUNCT
ejpam-6059	73	18	g	g	PROPN
ejpam-6059	73	19	/	/	SYM
ejpam-6059	73	20	h	h	NOUN
ejpam-6059	73	21	:	:	PUNCT
ejpam-6059	73	22	=	=	SYM
ejpam-6059	73	23	{	{	PUNCT
ejpam-6059	73	24	gh	gh	PROPN
ejpam-6059	73	25	:	:	PUNCT
ejpam-6059	73	26	g	g	PROPN
ejpam-6059	73	27	∈	∈	PROPN
ejpam-6059	73	28	g	g	PROPN
ejpam-6059	73	29	}	}	PUNCT
ejpam-6059	73	30	.	.	PUNCT
ejpam-6059	74	1	definition	definition	NOUN
ejpam-6059	74	2	2	2	NUM
ejpam-6059	74	3	.	.	PUNCT
ejpam-6059	75	1	[	[	X
ejpam-6059	75	2	2	2	X
ejpam-6059	75	3	]	]	X
ejpam-6059	75	4	let	let	VERB
ejpam-6059	75	5	n	n	PRON
ejpam-6059	75	6	≥	≥	X
ejpam-6059	75	7	2	2	NUM
ejpam-6059	75	8	be	be	AUX
ejpam-6059	75	9	a	a	DET
ejpam-6059	75	10	positive	positive	ADJ
ejpam-6059	75	11	integer	integer	NOUN
ejpam-6059	75	12	.	.	PUNCT
ejpam-6059	76	1	the	the	DET
ejpam-6059	76	2	unitary	unitary	ADJ
ejpam-6059	76	3	cayley	cayley	NOUN
ejpam-6059	76	4	graph	graph	NOUN
ejpam-6059	76	5	γn	γn	NOUN
ejpam-6059	76	6	=	=	SYM
ejpam-6059	76	7	(	(	PUNCT
ejpam-6059	76	8	zn	zn	PROPN
ejpam-6059	76	9	,	,	PUNCT
ejpam-6059	76	10	un	un	PROPN
ejpam-6059	76	11	)	)	PUNCT
ejpam-6059	76	12	is	be	AUX
ejpam-6059	76	13	defined	define	VERB
ejpam-6059	76	14	by	by	ADP
ejpam-6059	76	15	the	the	DET
ejpam-6059	76	16	additive	additive	ADJ
ejpam-6059	76	17	group	group	NOUN
ejpam-6059	76	18	of	of	ADP
ejpam-6059	76	19	the	the	DET
ejpam-6059	76	20	ring	ring	NOUN
ejpam-6059	76	21	zn	zn	PROPN
ejpam-6059	76	22	and	and	CCONJ
ejpam-6059	76	23	the	the	DET
ejpam-6059	76	24	multiplicative	multiplicative	PROPN
ejpam-6059	76	25	group	group	NOUN
ejpam-6059	76	26	un	un	PROPN
ejpam-6059	76	27	of	of	ADP
ejpam-6059	76	28	its	its	PRON
ejpam-6059	76	29	units	unit	NOUN
ejpam-6059	76	30	such	such	ADJ
ejpam-6059	76	31	that	that	PRON
ejpam-6059	76	32	v	v	NOUN
ejpam-6059	76	33	(	(	PUNCT
ejpam-6059	76	34	γn	γn	NOUN
ejpam-6059	76	35	)	)	PUNCT
ejpam-6059	76	36	=	=	SYM
ejpam-6059	76	37	zn	zn	NOUN
ejpam-6059	76	38	and	and	CCONJ
ejpam-6059	76	39	e(γn	e(γn	NUM
ejpam-6059	76	40	)	)	PUNCT
ejpam-6059	76	41	=	=	SYM
ejpam-6059	76	42	{	{	PUNCT
ejpam-6059	76	43	{	{	PUNCT
ejpam-6059	76	44	a	a	PROPN
ejpam-6059	76	45	,	,	PUNCT
ejpam-6059	76	46	b	b	NOUN
ejpam-6059	76	47	}	}	PUNCT
ejpam-6059	76	48	:	:	PUNCT
ejpam-6059	76	49	a	a	X
ejpam-6059	76	50	,	,	PUNCT
ejpam-6059	76	51	b	b	PROPN
ejpam-6059	76	52	∈	∈	PROPN
ejpam-6059	76	53	zn	zn	PROPN
ejpam-6059	76	54	and	and	CCONJ
ejpam-6059	76	55	a	a	DET
ejpam-6059	76	56	−	−	PROPN
ejpam-6059	76	57	b	b	PROPN
ejpam-6059	76	58	∈	∈	PROPN
ejpam-6059	76	59	un	un	PROPN
ejpam-6059	76	60	}	}	PUNCT
ejpam-6059	76	61	or	or	CCONJ
ejpam-6059	76	62	equivalently	equivalently	ADV
ejpam-6059	76	63	,	,	PUNCT
ejpam-6059	76	64	e(γn	e(γn	NUM
ejpam-6059	76	65	)	)	PUNCT
ejpam-6059	77	1	=	=	SYM
ejpam-6059	77	2	{	{	PUNCT
ejpam-6059	77	3	{	{	PUNCT
ejpam-6059	77	4	a	a	PROPN
ejpam-6059	77	5	,	,	PUNCT
ejpam-6059	77	6	b	b	NOUN
ejpam-6059	77	7	}	}	PUNCT
ejpam-6059	77	8	:	:	PUNCT
ejpam-6059	77	9	a	a	X
ejpam-6059	77	10	,	,	PUNCT
ejpam-6059	77	11	b	b	PROPN
ejpam-6059	77	12	∈	∈	PROPN
ejpam-6059	77	13	zn	zn	PROPN
ejpam-6059	77	14	and	and	CCONJ
ejpam-6059	77	15	gcd(a−	gcd(a−	PROPN
ejpam-6059	77	16	b	b	PROPN
ejpam-6059	77	17	,	,	PUNCT
ejpam-6059	77	18	n	n	CCONJ
ejpam-6059	77	19	)	)	PUNCT
ejpam-6059	77	20	=	=	SYM
ejpam-6059	77	21	1	1	NUM
ejpam-6059	77	22	}	}	PUNCT
ejpam-6059	77	23	.	.	PUNCT
ejpam-6059	78	1	definition	definition	NOUN
ejpam-6059	78	2	3	3	NUM
ejpam-6059	78	3	.	.	PUNCT
ejpam-6059	79	1	[	[	X
ejpam-6059	79	2	9	9	NUM
ejpam-6059	79	3	]	]	PUNCT
ejpam-6059	79	4	in	in	ADP
ejpam-6059	79	5	number	number	NOUN
ejpam-6059	79	6	theory	theory	NOUN
ejpam-6059	79	7	,	,	PUNCT
ejpam-6059	79	8	the	the	DET
ejpam-6059	79	9	euler	euler	NOUN
ejpam-6059	79	10	’s	’s	PART
ejpam-6059	79	11	totient	totient	PROPN
ejpam-6059	79	12	function	function	NOUN
ejpam-6059	79	13	counts	count	VERB
ejpam-6059	79	14	the	the	DET
ejpam-6059	79	15	positive	positive	ADJ
ejpam-6059	79	16	integers	integer	NOUN
ejpam-6059	79	17	up	up	ADP
ejpam-6059	79	18	to	to	ADP
ejpam-6059	79	19	a	a	DET
ejpam-6059	79	20	given	give	VERB
ejpam-6059	79	21	integer	integer	NOUN
ejpam-6059	79	22	n	n	PROPN
ejpam-6059	79	23	that	that	PRON
ejpam-6059	79	24	are	be	AUX
ejpam-6059	79	25	relatively	relatively	ADV
ejpam-6059	79	26	prime	prime	ADJ
ejpam-6059	79	27	to	to	PART
ejpam-6059	79	28	n.	n.	VERB
ejpam-6059	79	29	generally	generally	ADV
ejpam-6059	79	30	,	,	PUNCT
ejpam-6059	79	31	a	a	DET
ejpam-6059	79	32	well	well	ADV
ejpam-6059	79	33	-	-	PUNCT
ejpam-6059	79	34	known	know	VERB
ejpam-6059	79	35	notation	notation	NOUN
ejpam-6059	79	36	written	write	VERB
ejpam-6059	79	37	for	for	ADP
ejpam-6059	79	38	the	the	DET
ejpam-6059	79	39	euler	euler	PROPN
ejpam-6059	79	40	’s	’s	PART
ejpam-6059	79	41	totient	totient	PROPN
ejpam-6059	79	42	function	function	NOUN
ejpam-6059	79	43	is	be	AUX
ejpam-6059	79	44	φ(n	φ(n	VERB
ejpam-6059	79	45	)	)	PUNCT
ejpam-6059	79	46	and	and	CCONJ
ejpam-6059	79	47	may	may	AUX
ejpam-6059	79	48	also	also	ADV
ejpam-6059	79	49	be	be	AUX
ejpam-6059	79	50	called	call	VERB
ejpam-6059	79	51	the	the	DET
ejpam-6059	79	52	euler	euler	NOUN
ejpam-6059	79	53	’s	’s	PART
ejpam-6059	79	54	phi	phi	NOUN
ejpam-6059	79	55	function	function	NOUN
ejpam-6059	79	56	.	.	PUNCT
ejpam-6059	80	1	in	in	ADP
ejpam-6059	80	2	other	other	ADJ
ejpam-6059	80	3	words	word	NOUN
ejpam-6059	80	4	,	,	PUNCT
ejpam-6059	80	5	it	it	PRON
ejpam-6059	80	6	is	be	AUX
ejpam-6059	80	7	defined	define	VERB
ejpam-6059	80	8	as	as	ADP
ejpam-6059	80	9	the	the	DET
ejpam-6059	80	10	number	number	NOUN
ejpam-6059	80	11	of	of	ADP
ejpam-6059	80	12	integers	integer	NOUN
ejpam-6059	80	13	k	k	X
ejpam-6059	81	1	such	such	ADJ
ejpam-6059	81	2	that	that	SCONJ
ejpam-6059	81	3	1	1	NUM
ejpam-6059	81	4	≤	≤	NUM
ejpam-6059	81	5	k	k	NOUN
ejpam-6059	81	6	≤	≤	PROPN
ejpam-6059	81	7	n	n	PROPN
ejpam-6059	81	8	and	and	CCONJ
ejpam-6059	81	9	gcd(k	gcd(k	PROPN
ejpam-6059	81	10	,	,	PUNCT
ejpam-6059	81	11	n	n	CCONJ
ejpam-6059	81	12	)	)	PUNCT
ejpam-6059	81	13	=	=	SYM
ejpam-6059	81	14	1	1	X
ejpam-6059	81	15	.	.	X
ejpam-6059	81	16	theorem	theorem	NOUN
ejpam-6059	81	17	1	1	NUM
ejpam-6059	81	18	.	.	PUNCT
ejpam-6059	82	1	[	[	X
ejpam-6059	82	2	10	10	NUM
ejpam-6059	82	3	]	]	PUNCT
ejpam-6059	82	4	let	let	VERB
ejpam-6059	82	5	n	n	PRON
ejpam-6059	82	6	be	be	AUX
ejpam-6059	82	7	an	an	DET
ejpam-6059	82	8	even	even	ADV
ejpam-6059	82	9	positive	positive	ADJ
ejpam-6059	82	10	integer	integer	NOUN
ejpam-6059	82	11	such	such	DET
ejpam-6059	82	12	that	that	SCONJ
ejpam-6059	82	13	n	n	CCONJ
ejpam-6059	82	14	≥	≥	NUM
ejpam-6059	82	15	8	8	NUM
ejpam-6059	82	16	.	.	PUNCT
ejpam-6059	83	1	then	then	ADV
ejpam-6059	83	2	φ(n	φ(n	NOUN
ejpam-6059	83	3	)	)	PUNCT
ejpam-6059	83	4	≥	≥	NOUN
ejpam-6059	83	5	4	4	NUM
ejpam-6059	83	6	.	.	PUNCT
ejpam-6059	83	7	remark	remark	NOUN
ejpam-6059	83	8	1	1	NUM
ejpam-6059	83	9	.	.	PUNCT
ejpam-6059	84	1	[	[	X
ejpam-6059	84	2	2	2	X
ejpam-6059	84	3	]	]	PUNCT
ejpam-6059	84	4	let	let	VERB
ejpam-6059	84	5	γn	γn	PART
ejpam-6059	84	6	be	be	AUX
ejpam-6059	84	7	the	the	DET
ejpam-6059	84	8	unitary	unitary	ADJ
ejpam-6059	84	9	cayley	cayley	ADJ
ejpam-6059	84	10	graph	graph	NOUN
ejpam-6059	84	11	with	with	ADP
ejpam-6059	84	12	n	n	ADP
ejpam-6059	84	13	vertices	vertex	NOUN
ejpam-6059	84	14	.	.	PUNCT
ejpam-6059	85	1	then	then	ADV
ejpam-6059	85	2	|e(g)|	|e(g)|	PROPN
ejpam-6059	85	3	=	=	NOUN
ejpam-6059	85	4	n(φ(n	n(φ(n	PROPN
ejpam-6059	85	5	)	)	PUNCT
ejpam-6059	85	6	)	)	PUNCT
ejpam-6059	85	7	2	2	NUM
ejpam-6059	85	8	and	and	CCONJ
ejpam-6059	85	9	the	the	DET
ejpam-6059	85	10	degree	degree	NOUN
ejpam-6059	85	11	of	of	ADP
ejpam-6059	85	12	a	a	PRON
ejpam-6059	85	13	vertex	vertex	NOUN
ejpam-6059	85	14	v	v	ADP
ejpam-6059	85	15	∈	∈	PROPN
ejpam-6059	85	16	v	v	NOUN
ejpam-6059	85	17	(	(	PUNCT
ejpam-6059	85	18	γn	γn	NOUN
ejpam-6059	85	19	)	)	PUNCT
ejpam-6059	85	20	is	be	AUX
ejpam-6059	85	21	given	give	VERB
ejpam-6059	85	22	by	by	ADP
ejpam-6059	85	23	deg(v	deg(v	PROPN
ejpam-6059	85	24	)	)	PUNCT
ejpam-6059	85	25	=	=	SYM
ejpam-6059	85	26	φ(n	φ(n	NOUN
ejpam-6059	85	27	)	)	PUNCT
ejpam-6059	85	28	.	.	PUNCT
ejpam-6059	86	1	theorem	theorem	NOUN
ejpam-6059	86	2	2	2	NUM
ejpam-6059	86	3	.	.	PUNCT
ejpam-6059	87	1	[	[	X
ejpam-6059	87	2	2	2	NUM
ejpam-6059	87	3	]	]	X
ejpam-6059	87	4	let	let	VERB
ejpam-6059	87	5	n	n	NOUN
ejpam-6059	87	6	=	=	SYM
ejpam-6059	87	7	pk	pk	NOUN
ejpam-6059	87	8	be	be	AUX
ejpam-6059	87	9	such	such	ADJ
ejpam-6059	87	10	that	that	SCONJ
ejpam-6059	87	11	p	p	NOUN
ejpam-6059	87	12	is	be	AUX
ejpam-6059	87	13	prime	prime	ADJ
ejpam-6059	87	14	and	and	CCONJ
ejpam-6059	87	15	k	k	PROPN
ejpam-6059	87	16	∈	∈	PROPN
ejpam-6059	87	17	n	n	PRON
ejpam-6059	87	18	\	\	NOUN
ejpam-6059	87	19	{	{	PUNCT
ejpam-6059	87	20	1	1	NUM
ejpam-6059	87	21	}	}	PUNCT
ejpam-6059	87	22	.	.	PUNCT
ejpam-6059	88	1	then	then	ADV
ejpam-6059	88	2	γn	γn	INTJ
ejpam-6059	88	3	is	be	AUX
ejpam-6059	88	4	a	a	DET
ejpam-6059	88	5	complete	complete	ADJ
ejpam-6059	88	6	p	p	ADJ
ejpam-6059	88	7	-	-	PUNCT
ejpam-6059	88	8	partite	partite	ADJ
ejpam-6059	88	9	graph	graph	NOUN
ejpam-6059	88	10	where	where	SCONJ
ejpam-6059	88	11	each	each	DET
ejpam-6059	88	12	partite	partite	ADJ
ejpam-6059	88	13	set	set	NOUN
ejpam-6059	88	14	has	have	VERB
ejpam-6059	88	15	size	size	NOUN
ejpam-6059	88	16	pk−1	pk−1	PROPN
ejpam-6059	88	17	.	.	PUNCT
ejpam-6059	88	18	corollary	corollary	ADJ
ejpam-6059	88	19	1	1	NUM
ejpam-6059	88	20	.	.	PUNCT
ejpam-6059	89	1	[	[	X
ejpam-6059	89	2	1	1	X
ejpam-6059	89	3	]	]	X
ejpam-6059	89	4	if	if	SCONJ
ejpam-6059	89	5	n	n	PRON
ejpam-6059	89	6	is	be	AUX
ejpam-6059	89	7	an	an	DET
ejpam-6059	89	8	even	even	ADV
ejpam-6059	89	9	positive	positive	ADJ
ejpam-6059	89	10	integer	integer	NOUN
ejpam-6059	89	11	,	,	PUNCT
ejpam-6059	89	12	then	then	ADV
ejpam-6059	89	13	the	the	DET
ejpam-6059	89	14	unitary	unitary	ADJ
ejpam-6059	89	15	cayley	cayley	ADJ
ejpam-6059	89	16	graph	graph	NOUN
ejpam-6059	89	17	γn	γn	NOUN
ejpam-6059	89	18	has	have	VERB
ejpam-6059	89	19	no	no	DET
ejpam-6059	89	20	odd	odd	ADJ
ejpam-6059	89	21	cycles	cycle	NOUN
ejpam-6059	89	22	.	.	PUNCT
ejpam-6059	90	1	in	in	ADP
ejpam-6059	90	2	particular	particular	ADJ
ejpam-6059	90	3	,	,	PUNCT
ejpam-6059	90	4	γn	γn	PRON
ejpam-6059	90	5	has	have	VERB
ejpam-6059	90	6	no	no	DET
ejpam-6059	90	7	triangles	triangle	NOUN
ejpam-6059	90	8	.	.	PUNCT
ejpam-6059	91	1	definition	definition	NOUN
ejpam-6059	91	2	4	4	NUM
ejpam-6059	91	3	.	.	PUNCT
ejpam-6059	92	1	let	let	VERB
ejpam-6059	92	2	γn	γn	PART
ejpam-6059	92	3	be	be	AUX
ejpam-6059	92	4	the	the	DET
ejpam-6059	92	5	unitary	unitary	ADJ
ejpam-6059	92	6	cayley	cayley	ADJ
ejpam-6059	92	7	graph	graph	NOUN
ejpam-6059	92	8	of	of	ADP
ejpam-6059	92	9	zn	zn	PROPN
ejpam-6059	92	10	.	.	PUNCT
ejpam-6059	93	1	the	the	DET
ejpam-6059	93	2	complement	complement	NOUN
ejpam-6059	93	3	γn	γn	ADP
ejpam-6059	93	4	of	of	ADP
ejpam-6059	93	5	γn	γn	NOUN
ejpam-6059	93	6	is	be	AUX
ejpam-6059	93	7	the	the	DET
ejpam-6059	93	8	graph	graph	NOUN
ejpam-6059	93	9	in	in	ADP
ejpam-6059	93	10	which	which	PRON
ejpam-6059	93	11	v	v	ADP
ejpam-6059	93	12	(	(	PUNCT
ejpam-6059	93	13	γn	γn	NOUN
ejpam-6059	93	14	)	)	PUNCT
ejpam-6059	93	15	=	=	SYM
ejpam-6059	93	16	zn	zn	NOUN
ejpam-6059	93	17	and	and	CCONJ
ejpam-6059	93	18	e(γn	e(γn	NUM
ejpam-6059	93	19	)	)	PUNCT
ejpam-6059	93	20	=	=	SYM
ejpam-6059	93	21	{	{	PUNCT
ejpam-6059	93	22	{	{	PUNCT
ejpam-6059	93	23	a	a	PROPN
ejpam-6059	93	24	,	,	PUNCT
ejpam-6059	93	25	b	b	NOUN
ejpam-6059	93	26	}	}	PUNCT
ejpam-6059	93	27	:	:	PUNCT
ejpam-6059	93	28	a	a	X
ejpam-6059	93	29	,	,	PUNCT
ejpam-6059	93	30	b	b	PROPN
ejpam-6059	93	31	∈	∈	PROPN
ejpam-6059	93	32	zn	zn	PROPN
ejpam-6059	93	33	and	and	CCONJ
ejpam-6059	93	34	gcd(a−	gcd(a−	PROPN
ejpam-6059	93	35	b	b	PROPN
ejpam-6059	93	36	,	,	PUNCT
ejpam-6059	93	37	n	n	CCONJ
ejpam-6059	93	38	)	)	PUNCT
ejpam-6059	93	39	̸=	̸=	PROPN
ejpam-6059	93	40	1	1	NUM
ejpam-6059	93	41	}	}	PUNCT
ejpam-6059	93	42	.	.	PUNCT
ejpam-6059	94	1	theorem	theorem	NOUN
ejpam-6059	94	2	3	3	NUM
ejpam-6059	94	3	.	.	PUNCT
ejpam-6059	95	1	[	[	X
ejpam-6059	95	2	11	11	NUM
ejpam-6059	95	3	]	]	PUNCT
ejpam-6059	95	4	let	let	VERB
ejpam-6059	95	5	n	n	NOUN
ejpam-6059	95	6	=	=	SYM
ejpam-6059	95	7	pk	pk	NOUN
ejpam-6059	95	8	be	be	AUX
ejpam-6059	95	9	such	such	ADJ
ejpam-6059	95	10	that	that	SCONJ
ejpam-6059	95	11	p	p	NOUN
ejpam-6059	95	12	is	be	AUX
ejpam-6059	95	13	prime	prime	ADJ
ejpam-6059	95	14	and	and	CCONJ
ejpam-6059	95	15	k	k	PROPN
ejpam-6059	95	16	∈	∈	PROPN
ejpam-6059	95	17	n	n	PRON
ejpam-6059	95	18	\	\	NOUN
ejpam-6059	95	19	{	{	PUNCT
ejpam-6059	95	20	1	1	NUM
ejpam-6059	95	21	}	}	PUNCT
ejpam-6059	95	22	.	.	PUNCT
ejpam-6059	96	1	then	then	ADV
ejpam-6059	96	2	γn	γn	NOUN
ejpam-6059	96	3	is	be	AUX
ejpam-6059	96	4	decomposed	decompose	VERB
ejpam-6059	96	5	into	into	ADP
ejpam-6059	96	6	p	p	NOUN
ejpam-6059	96	7	complete	complete	ADJ
ejpam-6059	96	8	graphs	graph	NOUN
ejpam-6059	96	9	of	of	ADP
ejpam-6059	96	10	order	order	NOUN
ejpam-6059	96	11	pk−1	pk−1	PROPN
ejpam-6059	96	12	.	.	PUNCT
ejpam-6059	97	1	lemma	lemma	PROPN
ejpam-6059	97	2	1	1	NUM
ejpam-6059	97	3	.	.	PUNCT
ejpam-6059	98	1	[	[	X
ejpam-6059	98	2	7	7	X
ejpam-6059	98	3	]	]	X
ejpam-6059	98	4	if	if	SCONJ
ejpam-6059	98	5	every	every	DET
ejpam-6059	98	6	vertex	vertex	NOUN
ejpam-6059	98	7	of	of	ADP
ejpam-6059	98	8	a	a	DET
ejpam-6059	98	9	finite	finite	ADJ
ejpam-6059	98	10	simple	simple	ADJ
ejpam-6059	98	11	graph	graph	NOUN
ejpam-6059	98	12	g	g	NOUN
ejpam-6059	98	13	has	have	VERB
ejpam-6059	98	14	degree	degree	NOUN
ejpam-6059	98	15	at	at	ADV
ejpam-6059	98	16	least	least	ADJ
ejpam-6059	98	17	2	2	NUM
ejpam-6059	98	18	,	,	PUNCT
ejpam-6059	98	19	then	then	ADV
ejpam-6059	98	20	g	g	PROPN
ejpam-6059	98	21	contains	contain	VERB
ejpam-6059	98	22	a	a	DET
ejpam-6059	98	23	cycle	cycle	NOUN
ejpam-6059	98	24	.	.	PUNCT
ejpam-6059	99	1	definition	definition	NOUN
ejpam-6059	99	2	5	5	NUM
ejpam-6059	99	3	.	.	PUNCT
ejpam-6059	100	1	[	[	X
ejpam-6059	100	2	7	7	X
ejpam-6059	100	3	]	]	X
ejpam-6059	100	4	a	a	DET
ejpam-6059	100	5	nonempty	nonempty	NOUN
ejpam-6059	100	6	subset	subset	VERB
ejpam-6059	100	7	x	x	PUNCT
ejpam-6059	100	8	of	of	ADP
ejpam-6059	100	9	v	v	NOUN
ejpam-6059	100	10	(	(	PUNCT
ejpam-6059	100	11	γn	γn	NOUN
ejpam-6059	100	12	)	)	PUNCT
ejpam-6059	100	13	is	be	AUX
ejpam-6059	100	14	an	an	DET
ejpam-6059	100	15	independent	independent	ADJ
ejpam-6059	100	16	set	set	NOUN
ejpam-6059	100	17	of	of	ADP
ejpam-6059	100	18	γn	γn	NOUN
ejpam-6059	100	19	if	if	SCONJ
ejpam-6059	100	20	every	every	DET
ejpam-6059	100	21	pair	pair	NOUN
ejpam-6059	100	22	of	of	ADP
ejpam-6059	100	23	vertices	vertex	NOUN
ejpam-6059	100	24	in	in	ADP
ejpam-6059	100	25	x	x	PROPN
ejpam-6059	100	26	is	be	AUX
ejpam-6059	100	27	not	not	PART
ejpam-6059	100	28	adjacent	adjacent	ADJ
ejpam-6059	100	29	in	in	ADP
ejpam-6059	100	30	γn	γn	NUM
ejpam-6059	100	31	.	.	PUNCT
ejpam-6059	101	1	d.	d.	PROPN
ejpam-6059	101	2	pongpipat	pongpipat	PROPN
ejpam-6059	101	3	,	,	PUNCT
ejpam-6059	101	4	n.	n.	NOUN
ejpam-6059	101	5	nupo	nupo	PROPN
ejpam-6059	101	6	/	/	SYM
ejpam-6059	101	7	eur	eur	PROPN
ejpam-6059	101	8	.	.	PUNCT
ejpam-6059	102	1	j.	j.	PROPN
ejpam-6059	102	2	pure	pure	PROPN
ejpam-6059	102	3	appl	appl	PROPN
ejpam-6059	102	4	.	.	PROPN
ejpam-6059	102	5	math	math	PROPN
ejpam-6059	102	6	,	,	PUNCT
ejpam-6059	102	7	18	18	NUM
ejpam-6059	102	8	(	(	PUNCT
ejpam-6059	102	9	2	2	NUM
ejpam-6059	102	10	)	)	PUNCT
ejpam-6059	102	11	(	(	PUNCT
ejpam-6059	102	12	2025	2025	NUM
ejpam-6059	102	13	)	)	PUNCT
ejpam-6059	102	14	,	,	PUNCT
ejpam-6059	102	15	6059	6059	NUM
ejpam-6059	102	16	4	4	NUM
ejpam-6059	102	17	of	of	ADP
ejpam-6059	102	18	11	11	NUM
ejpam-6059	102	19	3	3	NUM
ejpam-6059	102	20	.	.	PUNCT
ejpam-6059	103	1	lower	low	ADJ
ejpam-6059	103	2	acyclic	acyclic	ADJ
ejpam-6059	103	3	numbers	number	NOUN
ejpam-6059	103	4	of	of	ADP
ejpam-6059	103	5	γn	γn	NOUN
ejpam-6059	103	6	and	and	CCONJ
ejpam-6059	103	7	γn	γn	ADP
ejpam-6059	103	8	this	this	DET
ejpam-6059	103	9	section	section	NOUN
ejpam-6059	103	10	begins	begin	VERB
ejpam-6059	103	11	with	with	ADP
ejpam-6059	103	12	the	the	DET
ejpam-6059	103	13	definition	definition	NOUN
ejpam-6059	103	14	of	of	ADP
ejpam-6059	103	15	a	a	DET
ejpam-6059	103	16	lower	low	ADJ
ejpam-6059	103	17	acyclic	acyclic	ADJ
ejpam-6059	103	18	number	number	NOUN
ejpam-6059	103	19	of	of	ADP
ejpam-6059	103	20	the	the	DET
ejpam-6059	103	21	graph	graph	NOUN
ejpam-6059	103	22	g.	g.	NOUN
ejpam-6059	103	23	definition	definition	NOUN
ejpam-6059	103	24	6	6	NUM
ejpam-6059	103	25	.	.	PUNCT
ejpam-6059	104	1	a	a	DET
ejpam-6059	104	2	lower	lower	ADV
ejpam-6059	104	3	acyclic	acyclic	ADJ
ejpam-6059	104	4	number	number	NOUN
ejpam-6059	104	5	of	of	ADP
ejpam-6059	104	6	g	g	NOUN
ejpam-6059	104	7	,	,	PUNCT
ejpam-6059	104	8	denoted	denote	VERB
ejpam-6059	104	9	by	by	ADP
ejpam-6059	104	10	λ(g	λ(g	PROPN
ejpam-6059	104	11	)	)	PUNCT
ejpam-6059	104	12	,	,	PUNCT
ejpam-6059	104	13	is	be	AUX
ejpam-6059	104	14	the	the	DET
ejpam-6059	104	15	maximum	maximum	ADJ
ejpam-6059	104	16	number	number	NOUN
ejpam-6059	104	17	of	of	ADP
ejpam-6059	104	18	k	k	PROPN
ejpam-6059	104	19	vertices	vertex	NOUN
ejpam-6059	104	20	in	in	ADP
ejpam-6059	104	21	which	which	PRON
ejpam-6059	104	22	every	every	DET
ejpam-6059	104	23	induced	induced	ADJ
ejpam-6059	104	24	subgraph	subgraph	NOUN
ejpam-6059	104	25	of	of	ADP
ejpam-6059	104	26	k	k	PROPN
ejpam-6059	104	27	vertices	vertex	NOUN
ejpam-6059	104	28	contains	contain	VERB
ejpam-6059	104	29	no	no	DET
ejpam-6059	104	30	cycles	cycle	NOUN
ejpam-6059	104	31	.	.	PUNCT
ejpam-6059	104	32	example	example	NOUN
ejpam-6059	105	1	1	1	NUM
ejpam-6059	105	2	.	.	PUNCT
ejpam-6059	105	3	let	let	VERB
ejpam-6059	105	4	g	g	PRON
ejpam-6059	105	5	be	be	AUX
ejpam-6059	105	6	a	a	DET
ejpam-6059	105	7	graph	graph	NOUN
ejpam-6059	105	8	with	with	ADP
ejpam-6059	105	9	v	v	NOUN
ejpam-6059	105	10	(	(	PUNCT
ejpam-6059	105	11	g	g	NOUN
ejpam-6059	105	12	)	)	PUNCT
ejpam-6059	105	13	=	=	NOUN
ejpam-6059	105	14	{	{	PUNCT
ejpam-6059	105	15	a	a	PRON
ejpam-6059	105	16	,	,	PUNCT
ejpam-6059	105	17	b	b	NOUN
ejpam-6059	105	18	,	,	PUNCT
ejpam-6059	105	19	c	c	NOUN
ejpam-6059	105	20	,	,	PUNCT
ejpam-6059	105	21	d	d	NOUN
ejpam-6059	105	22	,	,	PUNCT
ejpam-6059	105	23	e	e	NOUN
ejpam-6059	105	24	,	,	PUNCT
ejpam-6059	105	25	f	f	PROPN
ejpam-6059	105	26	,	,	PUNCT
ejpam-6059	105	27	g	g	PROPN
ejpam-6059	105	28	,	,	PUNCT
ejpam-6059	105	29	h	h	NOUN
ejpam-6059	105	30	,	,	PUNCT
ejpam-6059	105	31	i	i	PROPN
ejpam-6059	105	32	}	}	PUNCT
ejpam-6059	105	33	and	and	CCONJ
ejpam-6059	105	34	let	let	VERB
ejpam-6059	105	35	e(g	e(g	PROPN
ejpam-6059	105	36	)	)	PUNCT
ejpam-6059	105	37	be	be	AUX
ejpam-6059	105	38	defined	define	VERB
ejpam-6059	105	39	as	as	ADP
ejpam-6059	105	40	the	the	DET
ejpam-6059	105	41	following	follow	VERB
ejpam-6059	105	42	diagram	diagram	NOUN
ejpam-6059	105	43	:	:	PUNCT
ejpam-6059	105	44	figure	figure	NOUN
ejpam-6059	105	45	1	1	NUM
ejpam-6059	105	46	:	:	PUNCT
ejpam-6059	105	47	a	a	DET
ejpam-6059	105	48	graph	graph	NOUN
ejpam-6059	105	49	g	g	NOUN
ejpam-6059	106	1	we	we	PRON
ejpam-6059	106	2	observe	observe	VERB
ejpam-6059	106	3	that	that	SCONJ
ejpam-6059	106	4	an	an	DET
ejpam-6059	106	5	induced	induced	ADJ
ejpam-6059	106	6	subgraph	subgraph	NOUN
ejpam-6059	106	7	g[{a	g[{a	NOUN
ejpam-6059	106	8	,	,	PUNCT
ejpam-6059	106	9	d	d	X
ejpam-6059	106	10	,	,	PUNCT
ejpam-6059	106	11	g	g	NOUN
ejpam-6059	106	12	}	}	PUNCT
ejpam-6059	106	13	]	]	PUNCT
ejpam-6059	106	14	forms	form	VERB
ejpam-6059	106	15	a	a	DET
ejpam-6059	106	16	cycle	cycle	NOUN
ejpam-6059	106	17	of	of	ADP
ejpam-6059	106	18	length	length	NOUN
ejpam-6059	106	19	3	3	NUM
ejpam-6059	106	20	.	.	PUNCT
ejpam-6059	107	1	therefore	therefore	ADV
ejpam-6059	107	2	,	,	PUNCT
ejpam-6059	107	3	λ(g	λ(g	PROPN
ejpam-6059	107	4	)	)	PUNCT
ejpam-6059	107	5	=	=	SYM
ejpam-6059	108	1	2	2	X
ejpam-6059	108	2	.	.	X
ejpam-6059	108	3	lemma	lemma	PROPN
ejpam-6059	108	4	2	2	X
ejpam-6059	108	5	.	.	PUNCT
ejpam-6059	109	1	let	let	VERB
ejpam-6059	109	2	n	n	PRON
ejpam-6059	109	3	be	be	AUX
ejpam-6059	109	4	an	an	DET
ejpam-6059	109	5	even	even	ADV
ejpam-6059	109	6	positive	positive	ADJ
ejpam-6059	109	7	integer	integer	NOUN
ejpam-6059	109	8	such	such	DET
ejpam-6059	109	9	that	that	SCONJ
ejpam-6059	109	10	n	n	CCONJ
ejpam-6059	109	11	≥	≥	NUM
ejpam-6059	109	12	8	8	NUM
ejpam-6059	109	13	.	.	PUNCT
ejpam-6059	110	1	then	then	ADV
ejpam-6059	110	2	the	the	DET
ejpam-6059	110	3	unitary	unitary	ADJ
ejpam-6059	110	4	cayley	cayley	ADJ
ejpam-6059	110	5	graph	graph	NOUN
ejpam-6059	110	6	γn	γn	NOUN
ejpam-6059	110	7	contains	contain	VERB
ejpam-6059	110	8	a	a	DET
ejpam-6059	110	9	cycle	cycle	NOUN
ejpam-6059	110	10	c4	c4	NOUN
ejpam-6059	110	11	of	of	ADP
ejpam-6059	110	12	length	length	NOUN
ejpam-6059	110	13	4	4	NUM
ejpam-6059	110	14	as	as	ADP
ejpam-6059	110	15	a	a	DET
ejpam-6059	110	16	subgraph	subgraph	NOUN
ejpam-6059	110	17	.	.	PUNCT
ejpam-6059	111	1	proof	proof	NOUN
ejpam-6059	111	2	.	.	PUNCT
ejpam-6059	112	1	let	let	VERB
ejpam-6059	112	2	φn	φn	VERB
ejpam-6059	112	3	=	=	PUNCT
ejpam-6059	112	4	{	{	PUNCT
ejpam-6059	112	5	k	k	PROPN
ejpam-6059	112	6	∈	∈	PROPN
ejpam-6059	112	7	n	n	CCONJ
ejpam-6059	112	8	:	:	PUNCT
ejpam-6059	112	9	1	1	NUM
ejpam-6059	112	10	≤	≤	NUM
ejpam-6059	112	11	k	k	NOUN
ejpam-6059	112	12	≤	≤	PROPN
ejpam-6059	112	13	n	n	PROPN
ejpam-6059	112	14	and	and	CCONJ
ejpam-6059	112	15	gcd(k	gcd(k	PROPN
ejpam-6059	112	16	,	,	PUNCT
ejpam-6059	112	17	n	n	CCONJ
ejpam-6059	112	18	)	)	PUNCT
ejpam-6059	112	19	=	=	SYM
ejpam-6059	113	1	1	1	NUM
ejpam-6059	113	2	}	}	PUNCT
ejpam-6059	113	3	.	.	PUNCT
ejpam-6059	114	1	then	then	ADV
ejpam-6059	114	2	|φn|	|φn|	PROPN
ejpam-6059	114	3	=	=	PUNCT
ejpam-6059	114	4	φ(n	φ(n	NOUN
ejpam-6059	114	5	)	)	PUNCT
ejpam-6059	114	6	.	.	PUNCT
ejpam-6059	115	1	for	for	ADP
ejpam-6059	115	2	convenience	convenience	NOUN
ejpam-6059	115	3	,	,	PUNCT
ejpam-6059	115	4	we	we	PRON
ejpam-6059	115	5	write	write	VERB
ejpam-6059	115	6	φn	φn	ADP
ejpam-6059	115	7	=	=	PUNCT
ejpam-6059	115	8	{	{	PUNCT
ejpam-6059	115	9	1	1	NUM
ejpam-6059	115	10	,	,	PUNCT
ejpam-6059	115	11	a1	a1	NOUN
ejpam-6059	115	12	,	,	PUNCT
ejpam-6059	115	13	a2	a2	PROPN
ejpam-6059	115	14	,	,	PUNCT
ejpam-6059	115	15	.	.	PUNCT
ejpam-6059	115	16	.	.	PUNCT
ejpam-6059	115	17	.	.	PUNCT
ejpam-6059	116	1	,	,	PUNCT
ejpam-6059	116	2	aφ(n)−1	aφ(n)−1	NOUN
ejpam-6059	116	3	}	}	PUNCT
ejpam-6059	116	4	,	,	PUNCT
ejpam-6059	116	5	where	where	SCONJ
ejpam-6059	116	6	ai	ai	VERB
ejpam-6059	116	7	<	<	X
ejpam-6059	116	8	aj	aj	PROPN
ejpam-6059	116	9	and	and	CCONJ
ejpam-6059	116	10	1	1	NUM
ejpam-6059	116	11	≤	≤	NUM
ejpam-6059	117	1	i	i	PRON
ejpam-6059	117	2	<	<	X
ejpam-6059	117	3	j	j	PROPN
ejpam-6059	117	4	≤	≤	PROPN
ejpam-6059	117	5	φ(n	φ(n	NOUN
ejpam-6059	117	6	)	)	PUNCT
ejpam-6059	117	7	−	−	PROPN
ejpam-6059	118	1	1	1	X
ejpam-6059	118	2	.	.	PUNCT
ejpam-6059	119	1	we	we	PRON
ejpam-6059	119	2	claim	claim	VERB
ejpam-6059	119	3	that	that	SCONJ
ejpam-6059	119	4	c4	c4	NOUN
ejpam-6059	119	5	is	be	AUX
ejpam-6059	119	6	contained	contain	VERB
ejpam-6059	119	7	in	in	ADP
ejpam-6059	119	8	γn	γn	NUM
ejpam-6059	119	9	.	.	PUNCT
ejpam-6059	120	1	firstly	firstly	ADV
ejpam-6059	120	2	,	,	PUNCT
ejpam-6059	120	3	since	since	SCONJ
ejpam-6059	120	4	gcd(1	gcd(1	VERB
ejpam-6059	120	5	−	−	PROPN
ejpam-6059	120	6	0	0	NUM
ejpam-6059	120	7	,	,	PUNCT
ejpam-6059	120	8	n	n	CCONJ
ejpam-6059	120	9	)	)	PUNCT
ejpam-6059	120	10	=	=	PUNCT
ejpam-6059	120	11	gcd(1	gcd(1	NOUN
ejpam-6059	120	12	,	,	PUNCT
ejpam-6059	120	13	n	n	CCONJ
ejpam-6059	120	14	)	)	PUNCT
ejpam-6059	120	15	=	=	SYM
ejpam-6059	120	16	1	1	NUM
ejpam-6059	120	17	,	,	PUNCT
ejpam-6059	120	18	there	there	PRON
ejpam-6059	120	19	exists	exist	VERB
ejpam-6059	120	20	an	an	DET
ejpam-6059	120	21	edge	edge	NOUN
ejpam-6059	120	22	between	between	ADP
ejpam-6059	120	23	vertices	vertex	NOUN
ejpam-6059	120	24	0	0	NUM
ejpam-6059	120	25	and	and	CCONJ
ejpam-6059	120	26	1	1	NUM
ejpam-6059	120	27	in	in	ADP
ejpam-6059	120	28	γn	γn	NUM
ejpam-6059	120	29	.	.	PUNCT
ejpam-6059	121	1	consider	consider	VERB
ejpam-6059	121	2	a1	a1	NOUN
ejpam-6059	121	3	∈	∈	PROPN
ejpam-6059	121	4	φn	φn	NOUN
ejpam-6059	121	5	.	.	PUNCT
ejpam-6059	122	1	we	we	PRON
ejpam-6059	122	2	have	have	VERB
ejpam-6059	122	3	gcd(a1	gcd(a1	NOUN
ejpam-6059	122	4	−	−	PROPN
ejpam-6059	122	5	0	0	NUM
ejpam-6059	122	6	,	,	PUNCT
ejpam-6059	122	7	n	n	CCONJ
ejpam-6059	122	8	)	)	PUNCT
ejpam-6059	122	9	=	=	SYM
ejpam-6059	122	10	gcd(a1	gcd(a1	NOUN
ejpam-6059	122	11	,	,	PUNCT
ejpam-6059	122	12	n	n	CCONJ
ejpam-6059	122	13	)	)	PUNCT
ejpam-6059	122	14	=	=	SYM
ejpam-6059	123	1	1	1	X
ejpam-6059	123	2	.	.	PUNCT
ejpam-6059	124	1	then	then	ADV
ejpam-6059	124	2	,	,	PUNCT
ejpam-6059	124	3	there	there	PRON
ejpam-6059	124	4	exists	exist	VERB
ejpam-6059	124	5	an	an	DET
ejpam-6059	124	6	edge	edge	NOUN
ejpam-6059	124	7	between	between	ADP
ejpam-6059	124	8	vertices	vertex	NOUN
ejpam-6059	124	9	0	0	PUNCT
ejpam-6059	124	10	and	and	CCONJ
ejpam-6059	124	11	a1	a1	VERB
ejpam-6059	124	12	in	in	ADP
ejpam-6059	124	13	γn	γn	NUM
ejpam-6059	124	14	.	.	PUNCT
ejpam-6059	125	1	next	next	ADV
ejpam-6059	125	2	,	,	PUNCT
ejpam-6059	125	3	by	by	ADP
ejpam-6059	125	4	theorem	theorem	NOUN
ejpam-6059	125	5	1	1	NUM
ejpam-6059	125	6	,	,	PUNCT
ejpam-6059	125	7	we	we	PRON
ejpam-6059	125	8	get	get	VERB
ejpam-6059	125	9	that	that	DET
ejpam-6059	125	10	a1	a1	NOUN
ejpam-6059	125	11	+	+	CCONJ
ejpam-6059	125	12	1	1	NUM
ejpam-6059	125	13	<	<	X
ejpam-6059	125	14	n.	n.	NOUN
ejpam-6059	125	15	then	then	ADV
ejpam-6059	125	16	a1	a1	VERB
ejpam-6059	125	17	+	+	CCONJ
ejpam-6059	125	18	1	1	NUM
ejpam-6059	125	19	∈	∈	PROPN
ejpam-6059	125	20	zn	zn	X
ejpam-6059	125	21	and	and	CCONJ
ejpam-6059	125	22	gcd((a1	gcd((a1	PROPN
ejpam-6059	126	1	+	+	CCONJ
ejpam-6059	126	2	1	1	X
ejpam-6059	126	3	)	)	PUNCT
ejpam-6059	126	4	−	−	NOUN
ejpam-6059	126	5	a1	a1	NOUN
ejpam-6059	126	6	,	,	PUNCT
ejpam-6059	126	7	n	n	CCONJ
ejpam-6059	126	8	)	)	PUNCT
ejpam-6059	126	9	=	=	PUNCT
ejpam-6059	127	1	gcd(1	gcd(1	NOUN
ejpam-6059	127	2	,	,	PUNCT
ejpam-6059	127	3	n	n	CCONJ
ejpam-6059	127	4	)	)	PUNCT
ejpam-6059	127	5	=	=	SYM
ejpam-6059	128	1	1	1	X
ejpam-6059	128	2	.	.	PUNCT
ejpam-6059	129	1	hence	hence	ADV
ejpam-6059	129	2	,	,	PUNCT
ejpam-6059	129	3	there	there	PRON
ejpam-6059	129	4	is	be	VERB
ejpam-6059	129	5	an	an	DET
ejpam-6059	129	6	edge	edge	NOUN
ejpam-6059	129	7	between	between	ADP
ejpam-6059	129	8	vertices	vertex	NOUN
ejpam-6059	129	9	a1	a1	NOUN
ejpam-6059	129	10	and	and	CCONJ
ejpam-6059	129	11	a1	a1	NOUN
ejpam-6059	129	12	+	+	CCONJ
ejpam-6059	129	13	1	1	NUM
ejpam-6059	129	14	in	in	ADP
ejpam-6059	129	15	γn	γn	NUM
ejpam-6059	129	16	.	.	PUNCT
ejpam-6059	130	1	finally	finally	ADV
ejpam-6059	130	2	,	,	PUNCT
ejpam-6059	130	3	we	we	PRON
ejpam-6059	130	4	have	have	VERB
ejpam-6059	130	5	gcd((a1	gcd((a1	PROPN
ejpam-6059	130	6	+	+	CCONJ
ejpam-6059	130	7	1	1	NUM
ejpam-6059	130	8	)	)	PUNCT
ejpam-6059	130	9	−	−	PROPN
ejpam-6059	130	10	1	1	NUM
ejpam-6059	130	11	,	,	PUNCT
ejpam-6059	130	12	n	n	CCONJ
ejpam-6059	130	13	)	)	PUNCT
ejpam-6059	130	14	=	=	SYM
ejpam-6059	130	15	gcd(a1	gcd(a1	NOUN
ejpam-6059	130	16	,	,	PUNCT
ejpam-6059	130	17	n	n	CCONJ
ejpam-6059	130	18	)	)	PUNCT
ejpam-6059	130	19	=	=	SYM
ejpam-6059	131	1	1	1	X
ejpam-6059	131	2	.	.	PUNCT
ejpam-6059	131	3	then	then	ADV
ejpam-6059	131	4	there	there	PRON
ejpam-6059	131	5	exists	exist	VERB
ejpam-6059	131	6	an	an	DET
ejpam-6059	131	7	edge	edge	NOUN
ejpam-6059	131	8	between	between	ADP
ejpam-6059	131	9	vertices	vertex	NOUN
ejpam-6059	131	10	a1	a1	NOUN
ejpam-6059	131	11	+	+	CCONJ
ejpam-6059	131	12	1	1	NUM
ejpam-6059	131	13	and	and	CCONJ
ejpam-6059	131	14	1	1	NUM
ejpam-6059	131	15	in	in	ADP
ejpam-6059	131	16	γn	γn	NUM
ejpam-6059	131	17	.	.	PUNCT
ejpam-6059	132	1	hence	hence	ADV
ejpam-6059	132	2	,	,	PUNCT
ejpam-6059	132	3	0	0	NUM
ejpam-6059	132	4	,	,	PUNCT
ejpam-6059	132	5	1	1	NUM
ejpam-6059	132	6	,	,	PUNCT
ejpam-6059	132	7	a1	a1	NOUN
ejpam-6059	132	8	,	,	PUNCT
ejpam-6059	132	9	and	and	CCONJ
ejpam-6059	132	10	a1	a1	NOUN
ejpam-6059	132	11	+	+	CCONJ
ejpam-6059	132	12	1	1	NUM
ejpam-6059	132	13	form	form	NOUN
ejpam-6059	132	14	a	a	DET
ejpam-6059	132	15	cycle	cycle	NOUN
ejpam-6059	132	16	of	of	ADP
ejpam-6059	132	17	length	length	NOUN
ejpam-6059	132	18	4	4	NUM
ejpam-6059	132	19	in	in	ADP
ejpam-6059	132	20	γn	γn	NUM
ejpam-6059	132	21	.	.	PUNCT
ejpam-6059	133	1	before	before	SCONJ
ejpam-6059	133	2	we	we	PRON
ejpam-6059	133	3	present	present	VERB
ejpam-6059	133	4	the	the	DET
ejpam-6059	133	5	lower	lower	ADV
ejpam-6059	133	6	acyclic	acyclic	ADJ
ejpam-6059	133	7	number	number	NOUN
ejpam-6059	133	8	of	of	ADP
ejpam-6059	133	9	γn	γn	NOUN
ejpam-6059	133	10	where	where	SCONJ
ejpam-6059	133	11	n	n	PRON
ejpam-6059	133	12	is	be	AUX
ejpam-6059	133	13	even	even	ADV
ejpam-6059	133	14	,	,	PUNCT
ejpam-6059	133	15	the	the	DET
ejpam-6059	133	16	following	follow	VERB
ejpam-6059	133	17	example	example	NOUN
ejpam-6059	133	18	is	be	AUX
ejpam-6059	133	19	needed	need	VERB
ejpam-6059	133	20	for	for	ADP
ejpam-6059	133	21	n	n	NOUN
ejpam-6059	133	22	=	=	SYM
ejpam-6059	133	23	4	4	NUM
ejpam-6059	133	24	,	,	PUNCT
ejpam-6059	133	25	6	6	NUM
ejpam-6059	133	26	.	.	PUNCT
ejpam-6059	133	27	further	further	ADJ
ejpam-6059	133	28	results	result	NOUN
ejpam-6059	133	29	for	for	ADP
ejpam-6059	133	30	n	n	X
ejpam-6059	133	31	>	>	X
ejpam-6059	133	32	6	6	NUM
ejpam-6059	133	33	will	will	AUX
ejpam-6059	133	34	be	be	AUX
ejpam-6059	133	35	proved	prove	VERB
ejpam-6059	133	36	in	in	ADP
ejpam-6059	133	37	theorem	theorem	NOUN
ejpam-6059	133	38	5	5	NUM
ejpam-6059	133	39	.	.	PUNCT
ejpam-6059	133	40	d.	d.	PROPN
ejpam-6059	133	41	pongpipat	pongpipat	PROPN
ejpam-6059	133	42	,	,	PUNCT
ejpam-6059	133	43	n.	n.	NOUN
ejpam-6059	133	44	nupo	nupo	PROPN
ejpam-6059	133	45	/	/	SYM
ejpam-6059	133	46	eur	eur	PROPN
ejpam-6059	133	47	.	.	PUNCT
ejpam-6059	134	1	j.	j.	PROPN
ejpam-6059	134	2	pure	pure	PROPN
ejpam-6059	134	3	appl	appl	PROPN
ejpam-6059	134	4	.	.	PROPN
ejpam-6059	134	5	math	math	PROPN
ejpam-6059	134	6	,	,	PUNCT
ejpam-6059	134	7	18	18	NUM
ejpam-6059	134	8	(	(	PUNCT
ejpam-6059	134	9	2	2	NUM
ejpam-6059	134	10	)	)	PUNCT
ejpam-6059	134	11	(	(	PUNCT
ejpam-6059	134	12	2025	2025	NUM
ejpam-6059	134	13	)	)	PUNCT
ejpam-6059	134	14	,	,	PUNCT
ejpam-6059	134	15	6059	6059	NUM
ejpam-6059	134	16	5	5	NUM
ejpam-6059	134	17	of	of	ADP
ejpam-6059	134	18	11	11	NUM
ejpam-6059	134	19	example	example	NOUN
ejpam-6059	134	20	2	2	NUM
ejpam-6059	134	21	.	.	PUNCT
ejpam-6059	135	1	the	the	DET
ejpam-6059	135	2	unitary	unitary	ADJ
ejpam-6059	135	3	cayley	cayley	NOUN
ejpam-6059	135	4	graphs	graph	NOUN
ejpam-6059	135	5	γ4	γ4	NOUN
ejpam-6059	135	6	and	and	CCONJ
ejpam-6059	135	7	γ6	γ6	PROPN
ejpam-6059	135	8	are	be	AUX
ejpam-6059	135	9	shown	show	VERB
ejpam-6059	135	10	as	as	SCONJ
ejpam-6059	135	11	follows	follow	VERB
ejpam-6059	135	12	.	.	PUNCT
ejpam-6059	136	1	figure	figure	VERB
ejpam-6059	136	2	2	2	NUM
ejpam-6059	136	3	:	:	PUNCT
ejpam-6059	136	4	the	the	DET
ejpam-6059	136	5	unitary	unitary	ADJ
ejpam-6059	136	6	cayley	cayley	NOUN
ejpam-6059	136	7	graphs	graph	NOUN
ejpam-6059	136	8	γ4	γ4	NOUN
ejpam-6059	136	9	and	and	CCONJ
ejpam-6059	136	10	γ6	γ6	NOUN
ejpam-6059	136	11	we	we	PRON
ejpam-6059	136	12	see	see	VERB
ejpam-6059	136	13	that	that	SCONJ
ejpam-6059	136	14	γ4	γ4	NOUN
ejpam-6059	136	15	is	be	AUX
ejpam-6059	136	16	a	a	DET
ejpam-6059	136	17	cycle	cycle	NOUN
ejpam-6059	136	18	of	of	ADP
ejpam-6059	136	19	length	length	NOUN
ejpam-6059	136	20	4	4	NUM
ejpam-6059	136	21	and	and	CCONJ
ejpam-6059	136	22	γ6	γ6	PROPN
ejpam-6059	136	23	is	be	AUX
ejpam-6059	136	24	a	a	DET
ejpam-6059	136	25	cycle	cycle	NOUN
ejpam-6059	136	26	of	of	ADP
ejpam-6059	136	27	length	length	NOUN
ejpam-6059	136	28	6	6	NUM
ejpam-6059	136	29	.	.	PUNCT
ejpam-6059	137	1	therefore	therefore	ADV
ejpam-6059	137	2	,	,	PUNCT
ejpam-6059	137	3	λ(γ4	λ(γ4	ADJ
ejpam-6059	137	4	)	)	PUNCT
ejpam-6059	137	5	=	=	SYM
ejpam-6059	137	6	3	3	NUM
ejpam-6059	137	7	and	and	CCONJ
ejpam-6059	137	8	λ(γ6	λ(γ6	NOUN
ejpam-6059	137	9	)	)	PUNCT
ejpam-6059	137	10	=	=	SYM
ejpam-6059	138	1	5	5	X
ejpam-6059	138	2	.	.	PUNCT
ejpam-6059	138	3	lemma	lemma	PROPN
ejpam-6059	138	4	3	3	X
ejpam-6059	138	5	.	.	PUNCT
ejpam-6059	139	1	let	let	VERB
ejpam-6059	139	2	n	n	PRON
ejpam-6059	139	3	be	be	AUX
ejpam-6059	139	4	an	an	DET
ejpam-6059	139	5	odd	odd	ADJ
ejpam-6059	139	6	positive	positive	ADJ
ejpam-6059	139	7	integer	integer	NOUN
ejpam-6059	140	1	such	such	DET
ejpam-6059	140	2	that	that	SCONJ
ejpam-6059	140	3	n	n	NUM
ejpam-6059	140	4	≥	≥	NOUN
ejpam-6059	140	5	3	3	NUM
ejpam-6059	140	6	.	.	PUNCT
ejpam-6059	141	1	then	then	ADV
ejpam-6059	141	2	the	the	DET
ejpam-6059	141	3	unitary	unitary	ADJ
ejpam-6059	141	4	cayley	cayley	ADJ
ejpam-6059	141	5	graph	graph	NOUN
ejpam-6059	141	6	γn	γn	NOUN
ejpam-6059	141	7	contains	contain	VERB
ejpam-6059	141	8	a	a	DET
ejpam-6059	141	9	cycle	cycle	NOUN
ejpam-6059	141	10	c3	c3	NOUN
ejpam-6059	141	11	of	of	ADP
ejpam-6059	141	12	length	length	NOUN
ejpam-6059	141	13	3	3	NUM
ejpam-6059	141	14	as	as	ADP
ejpam-6059	141	15	a	a	DET
ejpam-6059	141	16	subgraph	subgraph	NOUN
ejpam-6059	141	17	.	.	PUNCT
ejpam-6059	142	1	proof	proof	NOUN
ejpam-6059	142	2	.	.	PUNCT
ejpam-6059	143	1	let	let	VERB
ejpam-6059	143	2	n	n	PRON
ejpam-6059	143	3	be	be	AUX
ejpam-6059	143	4	an	an	DET
ejpam-6059	143	5	odd	odd	ADJ
ejpam-6059	143	6	positive	positive	ADJ
ejpam-6059	143	7	integer	integer	NOUN
ejpam-6059	143	8	.	.	PUNCT
ejpam-6059	144	1	we	we	PRON
ejpam-6059	144	2	claim	claim	VERB
ejpam-6059	144	3	that	that	SCONJ
ejpam-6059	144	4	c3	c3	PROPN
ejpam-6059	144	5	is	be	AUX
ejpam-6059	144	6	contained	contain	VERB
ejpam-6059	144	7	in	in	ADP
ejpam-6059	144	8	γn	γn	NUM
ejpam-6059	144	9	.	.	PUNCT
ejpam-6059	145	1	since	since	SCONJ
ejpam-6059	145	2	gcd(1	gcd(1	VERB
ejpam-6059	145	3	−	−	PROPN
ejpam-6059	145	4	0	0	NUM
ejpam-6059	145	5	,	,	PUNCT
ejpam-6059	145	6	n	n	CCONJ
ejpam-6059	145	7	)	)	PUNCT
ejpam-6059	145	8	=	=	SYM
ejpam-6059	145	9	1	1	NUM
ejpam-6059	145	10	and	and	CCONJ
ejpam-6059	145	11	gcd(2	gcd(2	NOUN
ejpam-6059	145	12	−	−	PROPN
ejpam-6059	145	13	1	1	NUM
ejpam-6059	145	14	,	,	PUNCT
ejpam-6059	145	15	n	n	CCONJ
ejpam-6059	145	16	)	)	PUNCT
ejpam-6059	145	17	=	=	SYM
ejpam-6059	145	18	1	1	NUM
ejpam-6059	145	19	,	,	PUNCT
ejpam-6059	145	20	there	there	PRON
ejpam-6059	145	21	exist	exist	VERB
ejpam-6059	145	22	an	an	DET
ejpam-6059	145	23	edge	edge	NOUN
ejpam-6059	145	24	between	between	ADP
ejpam-6059	145	25	vertices	vertex	NOUN
ejpam-6059	145	26	0	0	NUM
ejpam-6059	145	27	and	and	CCONJ
ejpam-6059	145	28	1	1	NUM
ejpam-6059	145	29	in	in	ADP
ejpam-6059	145	30	γn	γn	NUM
ejpam-6059	145	31	,	,	PUNCT
ejpam-6059	145	32	and	and	CCONJ
ejpam-6059	145	33	an	an	DET
ejpam-6059	145	34	edge	edge	NOUN
ejpam-6059	145	35	between	between	ADP
ejpam-6059	145	36	vertices	vertex	NOUN
ejpam-6059	145	37	1	1	NUM
ejpam-6059	145	38	and	and	CCONJ
ejpam-6059	145	39	2	2	NUM
ejpam-6059	145	40	in	in	ADP
ejpam-6059	145	41	γn	γn	NUM
ejpam-6059	145	42	.	.	PUNCT
ejpam-6059	146	1	next	next	ADV
ejpam-6059	146	2	,	,	PUNCT
ejpam-6059	146	3	we	we	PRON
ejpam-6059	146	4	show	show	VERB
ejpam-6059	146	5	that	that	SCONJ
ejpam-6059	146	6	gcd(2	gcd(2	NOUN
ejpam-6059	146	7	,	,	PUNCT
ejpam-6059	146	8	n	n	CCONJ
ejpam-6059	146	9	)	)	PUNCT
ejpam-6059	146	10	=	=	SYM
ejpam-6059	147	1	1	1	X
ejpam-6059	147	2	.	.	X
ejpam-6059	148	1	as	as	ADP
ejpam-6059	148	2	the	the	DET
ejpam-6059	148	3	fact	fact	NOUN
ejpam-6059	148	4	that	that	SCONJ
ejpam-6059	148	5	n	n	PRON
ejpam-6059	148	6	is	be	AUX
ejpam-6059	148	7	an	an	DET
ejpam-6059	148	8	odd	odd	ADJ
ejpam-6059	148	9	positive	positive	ADJ
ejpam-6059	148	10	integer	integer	NOUN
ejpam-6059	148	11	,	,	PUNCT
ejpam-6059	148	12	then	then	ADV
ejpam-6059	148	13	there	there	PRON
ejpam-6059	148	14	exists	exist	VERB
ejpam-6059	148	15	an	an	DET
ejpam-6059	148	16	integer	integer	NOUN
ejpam-6059	148	17	k	k	PROPN
ejpam-6059	148	18	such	such	ADJ
ejpam-6059	148	19	that	that	SCONJ
ejpam-6059	148	20	n	n	NOUN
ejpam-6059	148	21	=	=	SYM
ejpam-6059	148	22	2k+1	2k+1	PROPN
ejpam-6059	148	23	.	.	PUNCT
ejpam-6059	149	1	this	this	PRON
ejpam-6059	149	2	means	mean	VERB
ejpam-6059	149	3	that	that	SCONJ
ejpam-6059	149	4	n	n	PRON
ejpam-6059	149	5	is	be	AUX
ejpam-6059	149	6	not	not	PART
ejpam-6059	149	7	divisible	divisible	ADJ
ejpam-6059	149	8	by	by	ADP
ejpam-6059	149	9	2	2	NUM
ejpam-6059	149	10	.	.	PUNCT
ejpam-6059	150	1	since	since	SCONJ
ejpam-6059	150	2	the	the	DET
ejpam-6059	150	3	only	only	ADJ
ejpam-6059	150	4	positive	positive	ADJ
ejpam-6059	150	5	divisors	divisor	NOUN
ejpam-6059	150	6	of	of	ADP
ejpam-6059	150	7	2	2	NUM
ejpam-6059	150	8	are	be	AUX
ejpam-6059	150	9	1	1	NUM
ejpam-6059	150	10	and	and	CCONJ
ejpam-6059	150	11	2	2	NUM
ejpam-6059	150	12	,	,	PUNCT
ejpam-6059	150	13	and	and	CCONJ
ejpam-6059	150	14	n	n	PRON
ejpam-6059	150	15	is	be	AUX
ejpam-6059	150	16	odd	odd	ADJ
ejpam-6059	150	17	,	,	PUNCT
ejpam-6059	150	18	2	2	NUM
ejpam-6059	150	19	does	do	AUX
ejpam-6059	150	20	not	not	PART
ejpam-6059	150	21	divide	divide	VERB
ejpam-6059	150	22	n.	n.	NOUN
ejpam-6059	151	1	so	so	SCONJ
ejpam-6059	151	2	the	the	DET
ejpam-6059	151	3	only	only	ADJ
ejpam-6059	151	4	common	common	ADJ
ejpam-6059	151	5	positive	positive	ADJ
ejpam-6059	151	6	divisor	divisor	NOUN
ejpam-6059	151	7	of	of	ADP
ejpam-6059	151	8	2	2	NUM
ejpam-6059	151	9	and	and	CCONJ
ejpam-6059	151	10	n	n	NOUN
ejpam-6059	151	11	is	be	AUX
ejpam-6059	151	12	1	1	NUM
ejpam-6059	151	13	.	.	PUNCT
ejpam-6059	152	1	it	it	PRON
ejpam-6059	152	2	follows	follow	VERB
ejpam-6059	152	3	that	that	SCONJ
ejpam-6059	152	4	1	1	NUM
ejpam-6059	152	5	=	=	NOUN
ejpam-6059	152	6	gcd(2	gcd(2	NOUN
ejpam-6059	152	7	,	,	PUNCT
ejpam-6059	152	8	n	n	CCONJ
ejpam-6059	152	9	)	)	PUNCT
ejpam-6059	152	10	=	=	NOUN
ejpam-6059	152	11	gcd(2	gcd(2	NOUN
ejpam-6059	152	12	−	−	PROPN
ejpam-6059	152	13	0	0	NUM
ejpam-6059	152	14	,	,	PUNCT
ejpam-6059	152	15	n	n	CCONJ
ejpam-6059	152	16	)	)	PUNCT
ejpam-6059	152	17	.	.	PUNCT
ejpam-6059	153	1	then	then	ADV
ejpam-6059	153	2	there	there	PRON
ejpam-6059	153	3	is	be	VERB
ejpam-6059	153	4	an	an	DET
ejpam-6059	153	5	edge	edge	NOUN
ejpam-6059	153	6	between	between	ADP
ejpam-6059	153	7	vertices	vertex	NOUN
ejpam-6059	153	8	0	0	NUM
ejpam-6059	153	9	and	and	CCONJ
ejpam-6059	153	10	2	2	NUM
ejpam-6059	153	11	in	in	ADP
ejpam-6059	153	12	γn	γn	NUM
ejpam-6059	153	13	.	.	PUNCT
ejpam-6059	154	1	hence	hence	ADV
ejpam-6059	154	2	,	,	PUNCT
ejpam-6059	154	3	0	0	NUM
ejpam-6059	154	4	,	,	PUNCT
ejpam-6059	154	5	1	1	NUM
ejpam-6059	154	6	and	and	CCONJ
ejpam-6059	154	7	2	2	NUM
ejpam-6059	154	8	form	form	NOUN
ejpam-6059	154	9	a	a	DET
ejpam-6059	154	10	cycle	cycle	NOUN
ejpam-6059	154	11	of	of	ADP
ejpam-6059	154	12	length	length	NOUN
ejpam-6059	154	13	3	3	NUM
ejpam-6059	154	14	in	in	ADP
ejpam-6059	154	15	γn	γn	NUM
ejpam-6059	154	16	.	.	PUNCT
ejpam-6059	155	1	in	in	ADP
ejpam-6059	155	2	order	order	NOUN
ejpam-6059	155	3	to	to	PART
ejpam-6059	155	4	complete	complete	VERB
ejpam-6059	155	5	our	our	PRON
ejpam-6059	155	6	results	result	NOUN
ejpam-6059	155	7	of	of	ADP
ejpam-6059	155	8	the	the	DET
ejpam-6059	155	9	part	part	NOUN
ejpam-6059	155	10	of	of	ADP
ejpam-6059	155	11	acyclic	acyclic	ADJ
ejpam-6059	155	12	numbers	number	NOUN
ejpam-6059	155	13	of	of	ADP
ejpam-6059	155	14	γn	γn	NUM
ejpam-6059	155	15	,	,	PUNCT
ejpam-6059	155	16	we	we	PRON
ejpam-6059	155	17	present	present	VERB
ejpam-6059	155	18	such	such	DET
ejpam-6059	155	19	the	the	DET
ejpam-6059	155	20	numbers	number	NOUN
ejpam-6059	155	21	as	as	SCONJ
ejpam-6059	155	22	follows	follow	VERB
ejpam-6059	155	23	.	.	PUNCT
ejpam-6059	156	1	theorem	theorem	ADJ
ejpam-6059	156	2	4	4	NUM
ejpam-6059	156	3	.	.	PUNCT
ejpam-6059	157	1	let	let	VERB
ejpam-6059	157	2	n	n	PRON
ejpam-6059	157	3	be	be	AUX
ejpam-6059	157	4	an	an	DET
ejpam-6059	157	5	odd	odd	ADJ
ejpam-6059	157	6	positive	positive	ADJ
ejpam-6059	157	7	integer	integer	NOUN
ejpam-6059	157	8	such	such	DET
ejpam-6059	157	9	that	that	SCONJ
ejpam-6059	157	10	n	n	NUM
ejpam-6059	157	11	≥	≥	NOUN
ejpam-6059	157	12	3	3	NUM
ejpam-6059	157	13	.	.	PUNCT
ejpam-6059	158	1	then	then	ADV
ejpam-6059	158	2	λ(γn	λ(γn	NOUN
ejpam-6059	158	3	)	)	PUNCT
ejpam-6059	158	4	=	=	SYM
ejpam-6059	158	5	2	2	X
ejpam-6059	158	6	.	.	PUNCT
ejpam-6059	158	7	proof	proof	NOUN
ejpam-6059	158	8	.	.	PUNCT
ejpam-6059	159	1	by	by	ADP
ejpam-6059	159	2	lemma	lemma	PROPN
ejpam-6059	159	3	3	3	NUM
ejpam-6059	159	4	,	,	PUNCT
ejpam-6059	159	5	the	the	DET
ejpam-6059	159	6	statement	statement	NOUN
ejpam-6059	159	7	holds	hold	VERB
ejpam-6059	159	8	.	.	PUNCT
ejpam-6059	160	1	theorem	theorem	NOUN
ejpam-6059	160	2	5	5	NUM
ejpam-6059	160	3	.	.	PUNCT
ejpam-6059	161	1	let	let	VERB
ejpam-6059	161	2	n	n	PRON
ejpam-6059	161	3	be	be	AUX
ejpam-6059	161	4	an	an	DET
ejpam-6059	161	5	even	even	ADV
ejpam-6059	161	6	positive	positive	ADJ
ejpam-6059	161	7	integer	integer	NOUN
ejpam-6059	161	8	such	such	DET
ejpam-6059	161	9	that	that	SCONJ
ejpam-6059	161	10	n	n	CCONJ
ejpam-6059	161	11	≥	≥	NOUN
ejpam-6059	161	12	8	8	NUM
ejpam-6059	161	13	and	and	CCONJ
ejpam-6059	161	14	n	n	NOUN
ejpam-6059	161	15	is	be	AUX
ejpam-6059	161	16	not	not	PART
ejpam-6059	161	17	prime	prime	ADJ
ejpam-6059	161	18	.	.	PUNCT
ejpam-6059	162	1	then	then	ADV
ejpam-6059	162	2	λ(γn	λ(γn	NOUN
ejpam-6059	162	3	)	)	PUNCT
ejpam-6059	162	4	=	=	SYM
ejpam-6059	162	5	3	3	X
ejpam-6059	162	6	.	.	PUNCT
ejpam-6059	162	7	proof	proof	NOUN
ejpam-6059	162	8	.	.	PUNCT
ejpam-6059	163	1	by	by	ADP
ejpam-6059	163	2	corollary	corollary	ADJ
ejpam-6059	163	3	1	1	NUM
ejpam-6059	163	4	,	,	PUNCT
ejpam-6059	163	5	γn	γn	PROPN
ejpam-6059	163	6	does	do	AUX
ejpam-6059	163	7	not	not	PART
ejpam-6059	163	8	contain	contain	VERB
ejpam-6059	163	9	a	a	DET
ejpam-6059	163	10	triangle	triangle	NOUN
ejpam-6059	163	11	.	.	PUNCT
ejpam-6059	164	1	it	it	PRON
ejpam-6059	164	2	follows	follow	VERB
ejpam-6059	164	3	that	that	SCONJ
ejpam-6059	164	4	λ(γn	λ(γn	NOUN
ejpam-6059	164	5	)	)	PUNCT
ejpam-6059	164	6	≥	≥	NOUN
ejpam-6059	164	7	3	3	NUM
ejpam-6059	164	8	.	.	PUNCT
ejpam-6059	164	9	by	by	ADP
ejpam-6059	164	10	lemma	lemma	PROPN
ejpam-6059	164	11	2	2	NUM
ejpam-6059	164	12	,	,	PUNCT
ejpam-6059	164	13	we	we	PRON
ejpam-6059	164	14	can	can	AUX
ejpam-6059	164	15	construct	construct	VERB
ejpam-6059	164	16	c4	c4	NOUN
ejpam-6059	164	17	which	which	PRON
ejpam-6059	164	18	is	be	AUX
ejpam-6059	164	19	contained	contain	VERB
ejpam-6059	164	20	in	in	ADP
ejpam-6059	164	21	γn	γn	NUM
ejpam-6059	164	22	.	.	PUNCT
ejpam-6059	165	1	then	then	ADV
ejpam-6059	165	2	λ(γn	λ(γn	NOUN
ejpam-6059	165	3	)	)	PUNCT
ejpam-6059	165	4	≤	≤	NUM
ejpam-6059	165	5	3	3	NUM
ejpam-6059	165	6	.	.	PUNCT
ejpam-6059	166	1	therefore	therefore	ADV
ejpam-6059	166	2	,	,	PUNCT
ejpam-6059	166	3	λ(γn	λ(γn	NOUN
ejpam-6059	166	4	)	)	PUNCT
ejpam-6059	166	5	=	=	SYM
ejpam-6059	167	1	3	3	X
ejpam-6059	167	2	.	.	X
ejpam-6059	168	1	for	for	ADP
ejpam-6059	168	2	more	more	ADJ
ejpam-6059	168	3	results	result	NOUN
ejpam-6059	168	4	related	relate	VERB
ejpam-6059	168	5	to	to	ADP
ejpam-6059	168	6	the	the	DET
ejpam-6059	168	7	acyclic	acyclic	ADJ
ejpam-6059	168	8	numbers	number	NOUN
ejpam-6059	168	9	of	of	ADP
ejpam-6059	168	10	the	the	DET
ejpam-6059	168	11	complement	complement	NOUN
ejpam-6059	168	12	γn	γn	NUM
ejpam-6059	168	13	,	,	PUNCT
ejpam-6059	168	14	we	we	PRON
ejpam-6059	168	15	present	present	VERB
ejpam-6059	168	16	these	these	DET
ejpam-6059	168	17	numbers	number	NOUN
ejpam-6059	168	18	in	in	ADP
ejpam-6059	168	19	the	the	DET
ejpam-6059	168	20	following	follow	VERB
ejpam-6059	168	21	discussion	discussion	NOUN
ejpam-6059	168	22	.	.	PUNCT
ejpam-6059	169	1	in	in	ADP
ejpam-6059	169	2	particular	particular	ADJ
ejpam-6059	169	3	,	,	PUNCT
ejpam-6059	169	4	we	we	PRON
ejpam-6059	169	5	explore	explore	VERB
ejpam-6059	169	6	their	their	PRON
ejpam-6059	169	7	properties	property	NOUN
ejpam-6059	169	8	,	,	PUNCT
ejpam-6059	169	9	characteristics	characteristic	NOUN
ejpam-6059	169	10	,	,	PUNCT
ejpam-6059	169	11	and	and	CCONJ
ejpam-6059	169	12	the	the	DET
ejpam-6059	169	13	conditions	condition	NOUN
ejpam-6059	169	14	under	under	ADP
ejpam-6059	169	15	which	which	PRON
ejpam-6059	169	16	they	they	PRON
ejpam-6059	169	17	arise	arise	VERB
ejpam-6059	169	18	.	.	PUNCT
ejpam-6059	170	1	this	this	PRON
ejpam-6059	170	2	allows	allow	VERB
ejpam-6059	170	3	for	for	ADP
ejpam-6059	170	4	a	a	DET
ejpam-6059	170	5	deeper	deep	ADJ
ejpam-6059	170	6	understanding	understanding	NOUN
ejpam-6059	170	7	of	of	ADP
ejpam-6059	170	8	d.	d.	PROPN
ejpam-6059	170	9	pongpipat	pongpipat	PROPN
ejpam-6059	170	10	,	,	PUNCT
ejpam-6059	170	11	n.	n.	NOUN
ejpam-6059	170	12	nupo	nupo	PROPN
ejpam-6059	170	13	/	/	SYM
ejpam-6059	170	14	eur	eur	PROPN
ejpam-6059	170	15	.	.	PUNCT
ejpam-6059	171	1	j.	j.	PROPN
ejpam-6059	171	2	pure	pure	PROPN
ejpam-6059	171	3	appl	appl	PROPN
ejpam-6059	171	4	.	.	PROPN
ejpam-6059	171	5	math	math	PROPN
ejpam-6059	171	6	,	,	PUNCT
ejpam-6059	171	7	18	18	NUM
ejpam-6059	171	8	(	(	PUNCT
ejpam-6059	171	9	2	2	NUM
ejpam-6059	171	10	)	)	PUNCT
ejpam-6059	171	11	(	(	PUNCT
ejpam-6059	171	12	2025	2025	NUM
ejpam-6059	171	13	)	)	PUNCT
ejpam-6059	171	14	,	,	PUNCT
ejpam-6059	171	15	6059	6059	NUM
ejpam-6059	171	16	6	6	NUM
ejpam-6059	171	17	of	of	ADP
ejpam-6059	171	18	11	11	NUM
ejpam-6059	171	19	how	how	SCONJ
ejpam-6059	171	20	acyclic	acyclic	ADJ
ejpam-6059	171	21	numbers	number	NOUN
ejpam-6059	171	22	behave	behave	VERB
ejpam-6059	171	23	in	in	ADP
ejpam-6059	171	24	the	the	DET
ejpam-6059	171	25	context	context	NOUN
ejpam-6059	171	26	of	of	ADP
ejpam-6059	171	27	the	the	DET
ejpam-6059	171	28	complement	complement	NOUN
ejpam-6059	171	29	graph	graph	NOUN
ejpam-6059	171	30	and	and	CCONJ
ejpam-6059	171	31	their	their	PRON
ejpam-6059	171	32	significance	significance	NOUN
ejpam-6059	171	33	in	in	ADP
ejpam-6059	171	34	graph	graph	NOUN
ejpam-6059	171	35	theory	theory	NOUN
ejpam-6059	171	36	.	.	PUNCT
ejpam-6059	172	1	the	the	DET
ejpam-6059	172	2	results	result	NOUN
ejpam-6059	172	3	presented	present	VERB
ejpam-6059	172	4	here	here	ADV
ejpam-6059	172	5	provide	provide	VERB
ejpam-6059	172	6	a	a	DET
ejpam-6059	172	7	foundation	foundation	NOUN
ejpam-6059	172	8	for	for	ADP
ejpam-6059	172	9	further	further	ADJ
ejpam-6059	172	10	investigation	investigation	NOUN
ejpam-6059	172	11	and	and	CCONJ
ejpam-6059	172	12	possible	possible	ADJ
ejpam-6059	172	13	extensions	extension	NOUN
ejpam-6059	172	14	of	of	ADP
ejpam-6059	172	15	this	this	DET
ejpam-6059	172	16	concept	concept	NOUN
ejpam-6059	172	17	.	.	PUNCT
ejpam-6059	173	1	example	example	NOUN
ejpam-6059	174	1	3	3	NUM
ejpam-6059	174	2	.	.	PUNCT
ejpam-6059	175	1	the	the	DET
ejpam-6059	175	2	complements	complement	NOUN
ejpam-6059	175	3	γ4	γ4	VERB
ejpam-6059	175	4	and	and	CCONJ
ejpam-6059	175	5	γ6	γ6	PROPN
ejpam-6059	175	6	of	of	ADP
ejpam-6059	175	7	the	the	DET
ejpam-6059	175	8	unitary	unitary	ADJ
ejpam-6059	175	9	cayley	cayley	NOUN
ejpam-6059	175	10	graphs	graph	NOUN
ejpam-6059	175	11	γ4	γ4	NOUN
ejpam-6059	175	12	and	and	CCONJ
ejpam-6059	175	13	γ6	γ6	PROPN
ejpam-6059	175	14	are	be	AUX
ejpam-6059	175	15	shown	show	VERB
ejpam-6059	175	16	as	as	SCONJ
ejpam-6059	175	17	follows	follow	VERB
ejpam-6059	175	18	.	.	PUNCT
ejpam-6059	176	1	figure	figure	VERB
ejpam-6059	176	2	3	3	NUM
ejpam-6059	176	3	:	:	PUNCT
ejpam-6059	176	4	the	the	DET
ejpam-6059	176	5	complements	complement	NOUN
ejpam-6059	176	6	γ4	γ4	VERB
ejpam-6059	176	7	and	and	CCONJ
ejpam-6059	176	8	γ6	γ6	NOUN
ejpam-6059	176	9	we	we	PRON
ejpam-6059	176	10	observe	observe	VERB
ejpam-6059	176	11	that	that	SCONJ
ejpam-6059	176	12	γ4	γ4	NOUN
ejpam-6059	176	13	does	do	AUX
ejpam-6059	176	14	not	not	PART
ejpam-6059	176	15	contain	contain	VERB
ejpam-6059	176	16	any	any	DET
ejpam-6059	176	17	cycle	cycle	NOUN
ejpam-6059	176	18	.	.	PUNCT
ejpam-6059	177	1	for	for	ADP
ejpam-6059	177	2	γ6	γ6	PROPN
ejpam-6059	177	3	,	,	PUNCT
ejpam-6059	177	4	it	it	PRON
ejpam-6059	177	5	is	be	AUX
ejpam-6059	177	6	obvious	obvious	ADJ
ejpam-6059	177	7	that	that	SCONJ
ejpam-6059	177	8	γ6	γ6	PROPN
ejpam-6059	177	9	contains	contain	VERB
ejpam-6059	177	10	triangles	triangle	NOUN
ejpam-6059	177	11	.	.	PUNCT
ejpam-6059	178	1	therefore	therefore	ADV
ejpam-6059	178	2	,	,	PUNCT
ejpam-6059	178	3	λ(γ4	λ(γ4	ADJ
ejpam-6059	178	4	)	)	PUNCT
ejpam-6059	178	5	=	=	SYM
ejpam-6059	178	6	4	4	NUM
ejpam-6059	178	7	and	and	CCONJ
ejpam-6059	178	8	λ(γ6	λ(γ6	NOUN
ejpam-6059	178	9	)	)	PUNCT
ejpam-6059	178	10	=	=	SYM
ejpam-6059	179	1	2	2	X
ejpam-6059	179	2	.	.	X
ejpam-6059	179	3	lemma	lemma	PROPN
ejpam-6059	179	4	4	4	X
ejpam-6059	179	5	.	.	PUNCT
ejpam-6059	180	1	let	let	VERB
ejpam-6059	180	2	n	n	PRON
ejpam-6059	180	3	be	be	AUX
ejpam-6059	180	4	a	a	DET
ejpam-6059	180	5	positive	positive	ADJ
ejpam-6059	180	6	integer	integer	NOUN
ejpam-6059	180	7	such	such	ADJ
ejpam-6059	180	8	that	that	SCONJ
ejpam-6059	180	9	n	n	CCONJ
ejpam-6059	180	10	≥	≥	NOUN
ejpam-6059	180	11	8	8	NUM
ejpam-6059	180	12	and	and	CCONJ
ejpam-6059	180	13	n	n	NOUN
ejpam-6059	180	14	is	be	AUX
ejpam-6059	180	15	not	not	PART
ejpam-6059	180	16	prime	prime	ADJ
ejpam-6059	180	17	.	.	PUNCT
ejpam-6059	181	1	then	then	ADV
ejpam-6059	181	2	the	the	DET
ejpam-6059	181	3	complement	complement	NOUN
ejpam-6059	181	4	γn	γn	ADP
ejpam-6059	181	5	of	of	ADP
ejpam-6059	181	6	the	the	DET
ejpam-6059	181	7	unitary	unitary	ADJ
ejpam-6059	181	8	cayley	cayley	ADJ
ejpam-6059	181	9	graph	graph	NOUN
ejpam-6059	181	10	γn	γn	NOUN
ejpam-6059	181	11	contains	contain	VERB
ejpam-6059	181	12	a	a	DET
ejpam-6059	181	13	cycle	cycle	NOUN
ejpam-6059	181	14	c3	c3	NOUN
ejpam-6059	181	15	of	of	ADP
ejpam-6059	181	16	length	length	NOUN
ejpam-6059	181	17	3	3	NUM
ejpam-6059	181	18	as	as	ADP
ejpam-6059	181	19	a	a	DET
ejpam-6059	181	20	subgraph	subgraph	NOUN
ejpam-6059	181	21	.	.	PUNCT
ejpam-6059	182	1	proof	proof	NOUN
ejpam-6059	182	2	.	.	PUNCT
ejpam-6059	183	1	let	let	VERB
ejpam-6059	183	2	n	n	PRON
ejpam-6059	183	3	be	be	AUX
ejpam-6059	183	4	a	a	DET
ejpam-6059	183	5	positive	positive	ADJ
ejpam-6059	183	6	integer	integer	NOUN
ejpam-6059	183	7	such	such	ADJ
ejpam-6059	183	8	that	that	SCONJ
ejpam-6059	183	9	n	n	CCONJ
ejpam-6059	183	10	≥	≥	NOUN
ejpam-6059	183	11	8	8	NUM
ejpam-6059	183	12	and	and	CCONJ
ejpam-6059	183	13	n	n	NOUN
ejpam-6059	183	14	is	be	AUX
ejpam-6059	183	15	not	not	PART
ejpam-6059	183	16	prime	prime	ADJ
ejpam-6059	183	17	.	.	PUNCT
ejpam-6059	184	1	case	case	NOUN
ejpam-6059	184	2	1	1	NUM
ejpam-6059	184	3	:	:	PUNCT
ejpam-6059	184	4	n	n	PRON
ejpam-6059	184	5	is	be	AUX
ejpam-6059	184	6	an	an	DET
ejpam-6059	184	7	even	even	ADV
ejpam-6059	184	8	integer	integer	NOUN
ejpam-6059	184	9	.	.	PUNCT
ejpam-6059	185	1	then	then	ADV
ejpam-6059	185	2	there	there	PRON
ejpam-6059	185	3	exists	exist	VERB
ejpam-6059	185	4	an	an	DET
ejpam-6059	185	5	integer	integer	NOUN
ejpam-6059	185	6	k	k	PROPN
ejpam-6059	185	7	such	such	ADJ
ejpam-6059	185	8	that	that	SCONJ
ejpam-6059	185	9	n	n	NOUN
ejpam-6059	185	10	=	=	SYM
ejpam-6059	185	11	2k	2k	NUM
ejpam-6059	185	12	.	.	PUNCT
ejpam-6059	186	1	since	since	SCONJ
ejpam-6059	186	2	gcd(4	gcd(4	ADJ
ejpam-6059	186	3	−	−	PROPN
ejpam-6059	186	4	2	2	NUM
ejpam-6059	186	5	,	,	PUNCT
ejpam-6059	186	6	2k	2k	NUM
ejpam-6059	186	7	)	)	PUNCT
ejpam-6059	186	8	=	=	SYM
ejpam-6059	186	9	gcd(2	gcd(2	NOUN
ejpam-6059	186	10	,	,	PUNCT
ejpam-6059	186	11	2k	2k	NUM
ejpam-6059	186	12	)	)	PUNCT
ejpam-6059	186	13	̸=	̸=	PROPN
ejpam-6059	186	14	1	1	NUM
ejpam-6059	186	15	,	,	PUNCT
ejpam-6059	186	16	gcd(6	gcd(6	ADV
ejpam-6059	186	17	−	−	PROPN
ejpam-6059	186	18	4	4	NUM
ejpam-6059	186	19	,	,	PUNCT
ejpam-6059	186	20	2k	2k	NUM
ejpam-6059	186	21	)	)	PUNCT
ejpam-6059	187	1	=	=	SYM
ejpam-6059	187	2	gcd(2	gcd(2	NOUN
ejpam-6059	187	3	,	,	PUNCT
ejpam-6059	187	4	2k	2k	NUM
ejpam-6059	187	5	)	)	PUNCT
ejpam-6059	187	6	̸=	̸=	PROPN
ejpam-6059	187	7	1	1	NUM
ejpam-6059	187	8	and	and	CCONJ
ejpam-6059	187	9	gcd(6	gcd(6	ADJ
ejpam-6059	187	10	−	−	PROPN
ejpam-6059	187	11	2	2	NUM
ejpam-6059	187	12	,	,	PUNCT
ejpam-6059	187	13	2k	2k	NUM
ejpam-6059	187	14	)	)	PUNCT
ejpam-6059	187	15	=	=	SYM
ejpam-6059	187	16	gcd(2(2	gcd(2(2	PROPN
ejpam-6059	187	17	)	)	PUNCT
ejpam-6059	187	18	,	,	PUNCT
ejpam-6059	187	19	2k	2k	NUM
ejpam-6059	187	20	)	)	PUNCT
ejpam-6059	187	21	̸=	̸=	PROPN
ejpam-6059	187	22	1	1	NUM
ejpam-6059	187	23	,	,	PUNCT
ejpam-6059	187	24	there	there	PRON
ejpam-6059	187	25	exist	exist	VERB
ejpam-6059	187	26	an	an	DET
ejpam-6059	187	27	edge	edge	NOUN
ejpam-6059	187	28	between	between	ADP
ejpam-6059	187	29	vertices	vertex	NOUN
ejpam-6059	187	30	2	2	NUM
ejpam-6059	187	31	and	and	CCONJ
ejpam-6059	187	32	4	4	NUM
ejpam-6059	187	33	in	in	ADP
ejpam-6059	187	34	γn	γn	NUM
ejpam-6059	187	35	,	,	PUNCT
ejpam-6059	187	36	an	an	DET
ejpam-6059	187	37	edge	edge	NOUN
ejpam-6059	187	38	between	between	ADP
ejpam-6059	187	39	vertices	vertex	NOUN
ejpam-6059	187	40	4	4	NUM
ejpam-6059	187	41	and	and	CCONJ
ejpam-6059	187	42	6	6	NUM
ejpam-6059	187	43	in	in	ADP
ejpam-6059	187	44	γn	γn	NUM
ejpam-6059	187	45	,	,	PUNCT
ejpam-6059	187	46	and	and	CCONJ
ejpam-6059	187	47	an	an	DET
ejpam-6059	187	48	edge	edge	NOUN
ejpam-6059	187	49	between	between	ADP
ejpam-6059	187	50	vertices	vertex	NOUN
ejpam-6059	187	51	2	2	NUM
ejpam-6059	187	52	and	and	CCONJ
ejpam-6059	187	53	6	6	NUM
ejpam-6059	187	54	in	in	ADP
ejpam-6059	187	55	γn	γn	NUM
ejpam-6059	187	56	.	.	PUNCT
ejpam-6059	188	1	so	so	ADV
ejpam-6059	188	2	2	2	NUM
ejpam-6059	188	3	,	,	PUNCT
ejpam-6059	188	4	4	4	NUM
ejpam-6059	188	5	and	and	CCONJ
ejpam-6059	188	6	6	6	NUM
ejpam-6059	188	7	induce	induce	VERB
ejpam-6059	188	8	a	a	DET
ejpam-6059	188	9	cycle	cycle	NOUN
ejpam-6059	188	10	c3	c3	NOUN
ejpam-6059	188	11	in	in	ADP
ejpam-6059	188	12	γn	γn	NOUN
ejpam-6059	188	13	as	as	ADP
ejpam-6059	188	14	a	a	DET
ejpam-6059	188	15	subgraph	subgraph	NOUN
ejpam-6059	188	16	.	.	PUNCT
ejpam-6059	189	1	case	case	NOUN
ejpam-6059	189	2	2	2	NUM
ejpam-6059	189	3	:	:	PUNCT
ejpam-6059	189	4	n	n	PRON
ejpam-6059	189	5	is	be	AUX
ejpam-6059	189	6	an	an	DET
ejpam-6059	189	7	odd	odd	ADJ
ejpam-6059	189	8	integer	integer	NOUN
ejpam-6059	189	9	.	.	PUNCT
ejpam-6059	190	1	assume	assume	VERB
ejpam-6059	190	2	that	that	SCONJ
ejpam-6059	190	3	n	n	PRON
ejpam-6059	190	4	=	=	SYM
ejpam-6059	190	5	pm1	pm1	PROPN
ejpam-6059	190	6	1	1	NUM
ejpam-6059	190	7	·	·	PUNCT
ejpam-6059	190	8	pm2	pm2	NOUN
ejpam-6059	190	9	2	2	NUM
ejpam-6059	190	10	·	·	PUNCT
ejpam-6059	190	11	·	·	PUNCT
ejpam-6059	190	12	·	·	PUNCT
ejpam-6059	191	1	pmt	pmt	PROPN
ejpam-6059	191	2	t	t	PROPN
ejpam-6059	191	3	where	where	SCONJ
ejpam-6059	191	4	pi	pi	NOUN
ejpam-6059	191	5	and	and	CCONJ
ejpam-6059	191	6	mi	mi	PROPN
ejpam-6059	191	7	are	be	AUX
ejpam-6059	191	8	prime	prime	ADJ
ejpam-6059	191	9	and	and	CCONJ
ejpam-6059	191	10	positive	positive	ADJ
ejpam-6059	191	11	,	,	PUNCT
ejpam-6059	191	12	respectively	respectively	ADV
ejpam-6059	191	13	such	such	ADJ
ejpam-6059	191	14	that	that	SCONJ
ejpam-6059	191	15	i	i	PRON
ejpam-6059	191	16	=	=	NOUN
ejpam-6059	191	17	1	1	NUM
ejpam-6059	191	18	,	,	PUNCT
ejpam-6059	191	19	2	2	NUM
ejpam-6059	191	20	,	,	PUNCT
ejpam-6059	191	21	.	.	PUNCT
ejpam-6059	191	22	.	.	PUNCT
ejpam-6059	192	1	.	.	PUNCT
ejpam-6059	193	1	,	,	PUNCT
ejpam-6059	193	2	t	t	PROPN
ejpam-6059	193	3	and	and	CCONJ
ejpam-6059	193	4	pi	pi	NOUN
ejpam-6059	193	5	<	<	X
ejpam-6059	193	6	pj	pj	PROPN
ejpam-6059	193	7	for	for	ADP
ejpam-6059	193	8	i	i	PRON
ejpam-6059	193	9	<	<	X
ejpam-6059	193	10	j.	j.	PROPN
ejpam-6059	193	11	since	since	SCONJ
ejpam-6059	193	12	gcd(p1−0	gcd(p1−0	PROPN
ejpam-6059	193	13	,	,	PUNCT
ejpam-6059	193	14	n	n	CCONJ
ejpam-6059	193	15	)	)	PUNCT
ejpam-6059	194	1	=	=	VERB
ejpam-6059	194	2	gcd(p1	gcd(p1	NOUN
ejpam-6059	194	3	,	,	PUNCT
ejpam-6059	194	4	p	p	PROPN
ejpam-6059	194	5	m1	m1	PROPN
ejpam-6059	194	6	1	1	NUM
ejpam-6059	194	7	·	·	PUNCT
ejpam-6059	194	8	pm2	pm2	NOUN
ejpam-6059	194	9	2	2	NUM
ejpam-6059	194	10	·	·	PUNCT
ejpam-6059	194	11	·	·	PUNCT
ejpam-6059	194	12	·	·	PUNCT
ejpam-6059	195	1	pmt	pmt	PROPN
ejpam-6059	195	2	t	t	PROPN
ejpam-6059	195	3	)	)	PUNCT
ejpam-6059	195	4	̸=	̸=	PROPN
ejpam-6059	195	5	1	1	NUM
ejpam-6059	195	6	,	,	PUNCT
ejpam-6059	195	7	gcd(2p1−p1	gcd(2p1−p1	PROPN
ejpam-6059	195	8	,	,	PUNCT
ejpam-6059	195	9	n	n	CCONJ
ejpam-6059	195	10	)	)	PUNCT
ejpam-6059	195	11	=	=	VERB
ejpam-6059	196	1	gcd(p1	gcd(p1	NOUN
ejpam-6059	196	2	,	,	PUNCT
ejpam-6059	196	3	p	p	PROPN
ejpam-6059	196	4	m1	m1	PROPN
ejpam-6059	196	5	1	1	NUM
ejpam-6059	196	6	·	·	PUNCT
ejpam-6059	196	7	pm2	pm2	NOUN
ejpam-6059	196	8	2	2	NUM
ejpam-6059	196	9	·	·	PUNCT
ejpam-6059	196	10	·	·	PUNCT
ejpam-6059	196	11	·	·	PUNCT
ejpam-6059	197	1	pmt	pmt	PROPN
ejpam-6059	197	2	t	t	PROPN
ejpam-6059	197	3	)	)	PUNCT
ejpam-6059	197	4	̸=	̸=	PROPN
ejpam-6059	197	5	1	1	NUM
ejpam-6059	197	6	,	,	PUNCT
ejpam-6059	197	7	and	and	CCONJ
ejpam-6059	197	8	gcd(2p1	gcd(2p1	NOUN
ejpam-6059	197	9	−	−	PROPN
ejpam-6059	197	10	0	0	NUM
ejpam-6059	197	11	,	,	PUNCT
ejpam-6059	197	12	n	n	CCONJ
ejpam-6059	197	13	)	)	PUNCT
ejpam-6059	197	14	=	=	SYM
ejpam-6059	197	15	gcd(2p1	gcd(2p1	NOUN
ejpam-6059	197	16	,	,	PUNCT
ejpam-6059	197	17	p	p	PROPN
ejpam-6059	197	18	m1	m1	PROPN
ejpam-6059	197	19	1	1	NUM
ejpam-6059	197	20	·	·	PUNCT
ejpam-6059	197	21	pm2	pm2	NOUN
ejpam-6059	197	22	2	2	NUM
ejpam-6059	197	23	·	·	PUNCT
ejpam-6059	197	24	·	·	PUNCT
ejpam-6059	197	25	·	·	PUNCT
ejpam-6059	198	1	pmt	pmt	PROPN
ejpam-6059	198	2	t	t	PROPN
ejpam-6059	198	3	)	)	PUNCT
ejpam-6059	198	4	̸=	̸=	PROPN
ejpam-6059	198	5	1	1	NUM
ejpam-6059	198	6	,	,	PUNCT
ejpam-6059	198	7	so	so	SCONJ
ejpam-6059	198	8	there	there	PRON
ejpam-6059	198	9	exist	exist	VERB
ejpam-6059	198	10	an	an	DET
ejpam-6059	198	11	edge	edge	NOUN
ejpam-6059	198	12	between	between	ADP
ejpam-6059	198	13	vertices	vertex	NOUN
ejpam-6059	198	14	0	0	PUNCT
ejpam-6059	198	15	and	and	CCONJ
ejpam-6059	198	16	p1	p1	PROPN
ejpam-6059	198	17	in	in	ADP
ejpam-6059	198	18	γn	γn	NUM
ejpam-6059	198	19	,	,	PUNCT
ejpam-6059	198	20	an	an	DET
ejpam-6059	198	21	edge	edge	NOUN
ejpam-6059	198	22	between	between	ADP
ejpam-6059	198	23	vertices	vertex	NOUN
ejpam-6059	198	24	p1	p1	NOUN
ejpam-6059	198	25	and	and	CCONJ
ejpam-6059	198	26	2p1	2p1	NUM
ejpam-6059	198	27	in	in	ADP
ejpam-6059	198	28	γn	γn	NUM
ejpam-6059	198	29	,	,	PUNCT
ejpam-6059	198	30	and	and	CCONJ
ejpam-6059	198	31	an	an	DET
ejpam-6059	198	32	edge	edge	NOUN
ejpam-6059	198	33	between	between	ADP
ejpam-6059	198	34	vertices	vertex	NOUN
ejpam-6059	198	35	0	0	NUM
ejpam-6059	198	36	and	and	CCONJ
ejpam-6059	198	37	2p1	2p1	NUM
ejpam-6059	198	38	in	in	ADP
ejpam-6059	198	39	γn	γn	NUM
ejpam-6059	198	40	,	,	PUNCT
ejpam-6059	198	41	respectively	respectively	ADV
ejpam-6059	198	42	.	.	PUNCT
ejpam-6059	199	1	thus	thus	ADV
ejpam-6059	199	2	,	,	PUNCT
ejpam-6059	199	3	0	0	NUM
ejpam-6059	199	4	,	,	PUNCT
ejpam-6059	199	5	p1	p1	NOUN
ejpam-6059	199	6	and	and	CCONJ
ejpam-6059	199	7	2p1	2p1	NUM
ejpam-6059	199	8	induce	induce	VERB
ejpam-6059	199	9	a	a	DET
ejpam-6059	199	10	cycle	cycle	NOUN
ejpam-6059	199	11	c3	c3	NOUN
ejpam-6059	199	12	in	in	ADP
ejpam-6059	199	13	γn	γn	PROPN
ejpam-6059	199	14	.	.	PUNCT
ejpam-6059	200	1	theorem	theorem	NOUN
ejpam-6059	200	2	6	6	NUM
ejpam-6059	200	3	.	.	PUNCT
ejpam-6059	201	1	let	let	VERB
ejpam-6059	201	2	n	n	PRON
ejpam-6059	201	3	be	be	AUX
ejpam-6059	201	4	a	a	DET
ejpam-6059	201	5	positive	positive	ADJ
ejpam-6059	201	6	integer	integer	NOUN
ejpam-6059	201	7	such	such	ADJ
ejpam-6059	201	8	that	that	SCONJ
ejpam-6059	201	9	n	n	CCONJ
ejpam-6059	201	10	≥	≥	NUM
ejpam-6059	201	11	8	8	NUM
ejpam-6059	201	12	.	.	PUNCT
ejpam-6059	202	1	then	then	ADV
ejpam-6059	202	2	λ(γn	λ(γn	NOUN
ejpam-6059	202	3	)	)	PUNCT
ejpam-6059	202	4	=	=	PRON
ejpam-6059	202	5	{	{	PUNCT
ejpam-6059	202	6	2	2	NUM
ejpam-6059	202	7	if	if	SCONJ
ejpam-6059	202	8	n	n	PRON
ejpam-6059	202	9	is	be	AUX
ejpam-6059	202	10	not	not	PART
ejpam-6059	202	11	prime	prime	ADJ
ejpam-6059	202	12	;	;	PUNCT
ejpam-6059	202	13	n	n	CCONJ
ejpam-6059	202	14	if	if	SCONJ
ejpam-6059	202	15	n	n	PRON
ejpam-6059	202	16	is	be	AUX
ejpam-6059	202	17	prime	prime	ADJ
ejpam-6059	202	18	.	.	PUNCT
ejpam-6059	203	1	proof	proof	NOUN
ejpam-6059	203	2	.	.	PUNCT
ejpam-6059	204	1	we	we	PRON
ejpam-6059	204	2	consider	consider	VERB
ejpam-6059	204	3	the	the	DET
ejpam-6059	204	4	following	follow	VERB
ejpam-6059	204	5	two	two	NUM
ejpam-6059	204	6	cases	case	NOUN
ejpam-6059	204	7	.	.	PUNCT
ejpam-6059	205	1	case	case	NOUN
ejpam-6059	205	2	1	1	NUM
ejpam-6059	205	3	:	:	PUNCT
ejpam-6059	205	4	n	n	PRON
ejpam-6059	205	5	is	be	AUX
ejpam-6059	205	6	not	not	PART
ejpam-6059	205	7	prime	prime	ADJ
ejpam-6059	205	8	.	.	PUNCT
ejpam-6059	206	1	by	by	ADP
ejpam-6059	206	2	lemma	lemma	PROPN
ejpam-6059	206	3	4	4	NUM
ejpam-6059	206	4	,	,	PUNCT
ejpam-6059	206	5	it	it	PRON
ejpam-6059	206	6	follows	follow	VERB
ejpam-6059	206	7	that	that	SCONJ
ejpam-6059	206	8	λ(γn	λ(γn	NOUN
ejpam-6059	206	9	)	)	PUNCT
ejpam-6059	206	10	=	=	SYM
ejpam-6059	206	11	2	2	X
ejpam-6059	206	12	.	.	X
ejpam-6059	206	13	case	case	NOUN
ejpam-6059	206	14	2	2	NUM
ejpam-6059	206	15	:	:	PUNCT
ejpam-6059	206	16	n	n	PRON
ejpam-6059	206	17	is	be	AUX
ejpam-6059	206	18	prime	prime	ADJ
ejpam-6059	206	19	.	.	PUNCT
ejpam-6059	207	1	then	then	ADV
ejpam-6059	207	2	γn	γn	PRON
ejpam-6059	207	3	is	be	AUX
ejpam-6059	207	4	a	a	DET
ejpam-6059	207	5	complete	complete	ADJ
ejpam-6059	207	6	graph	graph	NOUN
ejpam-6059	207	7	which	which	PRON
ejpam-6059	207	8	implies	imply	VERB
ejpam-6059	207	9	that	that	SCONJ
ejpam-6059	207	10	γn	γn	ADJ
ejpam-6059	207	11	is	be	AUX
ejpam-6059	207	12	an	an	DET
ejpam-6059	207	13	empty	empty	ADJ
ejpam-6059	207	14	graph	graph	NOUN
ejpam-6059	207	15	.	.	PUNCT
ejpam-6059	208	1	then	then	ADV
ejpam-6059	208	2	λ(γn	λ(γn	NOUN
ejpam-6059	208	3	)	)	PUNCT
ejpam-6059	208	4	=	=	SYM
ejpam-6059	208	5	n	n	CCONJ
ejpam-6059	208	6	,	,	PUNCT
ejpam-6059	208	7	immediately	immediately	ADV
ejpam-6059	208	8	.	.	PUNCT
ejpam-6059	209	1	d.	d.	PROPN
ejpam-6059	209	2	pongpipat	pongpipat	PROPN
ejpam-6059	209	3	,	,	PUNCT
ejpam-6059	209	4	n.	n.	NOUN
ejpam-6059	209	5	nupo	nupo	PROPN
ejpam-6059	209	6	/	/	SYM
ejpam-6059	209	7	eur	eur	PROPN
ejpam-6059	209	8	.	.	PUNCT
ejpam-6059	210	1	j.	j.	PROPN
ejpam-6059	210	2	pure	pure	PROPN
ejpam-6059	210	3	appl	appl	PROPN
ejpam-6059	210	4	.	.	PROPN
ejpam-6059	210	5	math	math	PROPN
ejpam-6059	210	6	,	,	PUNCT
ejpam-6059	210	7	18	18	NUM
ejpam-6059	210	8	(	(	PUNCT
ejpam-6059	210	9	2	2	NUM
ejpam-6059	210	10	)	)	PUNCT
ejpam-6059	210	11	(	(	PUNCT
ejpam-6059	210	12	2025	2025	NUM
ejpam-6059	210	13	)	)	PUNCT
ejpam-6059	210	14	,	,	PUNCT
ejpam-6059	210	15	6059	6059	NUM
ejpam-6059	210	16	7	7	NUM
ejpam-6059	210	17	of	of	ADP
ejpam-6059	210	18	11	11	NUM
ejpam-6059	210	19	4	4	NUM
ejpam-6059	210	20	.	.	PUNCT
ejpam-6059	211	1	upper	upper	ADJ
ejpam-6059	211	2	acyclic	acyclic	ADJ
ejpam-6059	211	3	numbers	number	NOUN
ejpam-6059	211	4	of	of	ADP
ejpam-6059	211	5	γn	γn	NOUN
ejpam-6059	211	6	and	and	CCONJ
ejpam-6059	211	7	γn	γn	ADP
ejpam-6059	211	8	this	this	DET
ejpam-6059	211	9	section	section	NOUN
ejpam-6059	211	10	begins	begin	VERB
ejpam-6059	211	11	with	with	ADP
ejpam-6059	211	12	the	the	DET
ejpam-6059	211	13	definition	definition	NOUN
ejpam-6059	211	14	of	of	ADP
ejpam-6059	211	15	an	an	DET
ejpam-6059	211	16	upper	upper	ADJ
ejpam-6059	211	17	acyclic	acyclic	ADJ
ejpam-6059	211	18	number	number	NOUN
ejpam-6059	211	19	of	of	ADP
ejpam-6059	211	20	the	the	DET
ejpam-6059	211	21	graph	graph	NOUN
ejpam-6059	211	22	g.	g.	NOUN
ejpam-6059	211	23	definition	definition	NOUN
ejpam-6059	211	24	7	7	NUM
ejpam-6059	211	25	.	.	PUNCT
ejpam-6059	212	1	a	a	DET
ejpam-6059	212	2	nonempty	nonempty	NOUN
ejpam-6059	212	3	subset	subset	VERB
ejpam-6059	212	4	a	a	PRON
ejpam-6059	212	5	of	of	ADP
ejpam-6059	212	6	the	the	DET
ejpam-6059	212	7	vertex	vertex	NOUN
ejpam-6059	212	8	set	set	VERB
ejpam-6059	212	9	v	v	NOUN
ejpam-6059	212	10	(	(	PUNCT
ejpam-6059	212	11	g	g	NOUN
ejpam-6059	212	12	)	)	PUNCT
ejpam-6059	212	13	of	of	ADP
ejpam-6059	212	14	a	a	DET
ejpam-6059	212	15	graph	graph	NOUN
ejpam-6059	212	16	g	g	NOUN
ejpam-6059	212	17	is	be	AUX
ejpam-6059	212	18	called	call	VERB
ejpam-6059	212	19	an	an	DET
ejpam-6059	212	20	acyclic	acyclic	ADJ
ejpam-6059	212	21	set	set	NOUN
ejpam-6059	212	22	of	of	ADP
ejpam-6059	212	23	g	g	NOUN
ejpam-6059	212	24	induced	induce	VERB
ejpam-6059	212	25	by	by	ADP
ejpam-6059	212	26	a	a	DET
ejpam-6059	212	27	contains	contain	NOUN
ejpam-6059	212	28	no	no	DET
ejpam-6059	212	29	cycles	cycle	NOUN
ejpam-6059	212	30	.	.	PUNCT
ejpam-6059	213	1	an	an	DET
ejpam-6059	213	2	upper	upper	ADJ
ejpam-6059	213	3	acyclic	acyclic	ADJ
ejpam-6059	213	4	number	number	NOUN
ejpam-6059	213	5	of	of	ADP
ejpam-6059	213	6	a	a	DET
ejpam-6059	213	7	graph	graph	NOUN
ejpam-6059	213	8	g	g	NOUN
ejpam-6059	213	9	,	,	PUNCT
ejpam-6059	213	10	denoted	denote	VERB
ejpam-6059	213	11	by	by	ADP
ejpam-6059	213	12	λ(g	λ(g	PROPN
ejpam-6059	213	13	)	)	PUNCT
ejpam-6059	213	14	,	,	PUNCT
ejpam-6059	213	15	is	be	AUX
ejpam-6059	213	16	the	the	DET
ejpam-6059	213	17	maximum	maximum	ADJ
ejpam-6059	213	18	cardinality	cardinality	NOUN
ejpam-6059	213	19	among	among	ADP
ejpam-6059	213	20	acyclic	acyclic	ADJ
ejpam-6059	213	21	sets	set	NOUN
ejpam-6059	213	22	of	of	ADP
ejpam-6059	213	23	g	g	NOUN
ejpam-6059	213	24	,	,	PUNCT
ejpam-6059	213	25	that	that	ADV
ejpam-6059	213	26	is	is	ADV
ejpam-6059	213	27	,	,	PUNCT
ejpam-6059	213	28	λ(g	λ(g	PROPN
ejpam-6059	213	29	)	)	PUNCT
ejpam-6059	214	1	=	=	PRON
ejpam-6059	214	2	max{|a|	max{|a|	X
ejpam-6059	214	3	:	:	PUNCT
ejpam-6059	214	4	a	a	PRON
ejpam-6059	214	5	is	be	AUX
ejpam-6059	214	6	an	an	DET
ejpam-6059	214	7	acyclic	acyclic	ADJ
ejpam-6059	214	8	set	set	NOUN
ejpam-6059	214	9	of	of	ADP
ejpam-6059	214	10	g	g	NOUN
ejpam-6059	214	11	}	}	PUNCT
ejpam-6059	214	12	.	.	PUNCT
ejpam-6059	215	1	example	example	NOUN
ejpam-6059	216	1	4	4	NUM
ejpam-6059	216	2	.	.	PUNCT
ejpam-6059	217	1	in	in	ADP
ejpam-6059	217	2	figure	figure	NOUN
ejpam-6059	217	3	1	1	NUM
ejpam-6059	217	4	,	,	PUNCT
ejpam-6059	217	5	we	we	PRON
ejpam-6059	217	6	observe	observe	VERB
ejpam-6059	217	7	that	that	SCONJ
ejpam-6059	217	8	the	the	DET
ejpam-6059	217	9	set	set	NOUN
ejpam-6059	217	10	a	a	X
ejpam-6059	217	11	=	=	X
ejpam-6059	217	12	{	{	PUNCT
ejpam-6059	217	13	a	a	PROPN
ejpam-6059	217	14	,	,	PUNCT
ejpam-6059	217	15	b	b	NOUN
ejpam-6059	217	16	,	,	PUNCT
ejpam-6059	217	17	c	c	NOUN
ejpam-6059	217	18	,	,	PUNCT
ejpam-6059	217	19	d	d	NOUN
ejpam-6059	217	20	,	,	PUNCT
ejpam-6059	217	21	e	e	NOUN
ejpam-6059	217	22	,	,	PUNCT
ejpam-6059	217	23	f	f	X
ejpam-6059	217	24	}	}	PUNCT
ejpam-6059	217	25	is	be	AUX
ejpam-6059	217	26	an	an	DET
ejpam-6059	217	27	acyclic	acyclic	ADJ
ejpam-6059	217	28	set	set	NOUN
ejpam-6059	217	29	of	of	ADP
ejpam-6059	217	30	g	g	NOUN
ejpam-6059	217	31	with	with	ADP
ejpam-6059	217	32	the	the	DET
ejpam-6059	217	33	largest	large	ADJ
ejpam-6059	217	34	cardinality	cardinality	NOUN
ejpam-6059	217	35	.	.	PUNCT
ejpam-6059	218	1	thus	thus	ADV
ejpam-6059	218	2	,	,	PUNCT
ejpam-6059	218	3	λ(g	λ(g	PROPN
ejpam-6059	218	4	)	)	PUNCT
ejpam-6059	218	5	=	=	PUNCT
ejpam-6059	219	1	6	6	NUM
ejpam-6059	219	2	.	.	PUNCT
ejpam-6059	220	1	next	next	ADV
ejpam-6059	220	2	,	,	PUNCT
ejpam-6059	220	3	we	we	PRON
ejpam-6059	220	4	find	find	VERB
ejpam-6059	220	5	upper	upper	ADJ
ejpam-6059	220	6	acyclic	acyclic	ADJ
ejpam-6059	220	7	numbers	number	NOUN
ejpam-6059	220	8	of	of	ADP
ejpam-6059	220	9	unitary	unitary	ADJ
ejpam-6059	220	10	cayley	cayley	ADJ
ejpam-6059	220	11	graphs	graph	NOUN
ejpam-6059	220	12	and	and	CCONJ
ejpam-6059	220	13	their	their	PRON
ejpam-6059	220	14	complements	complement	NOUN
ejpam-6059	220	15	.	.	PUNCT
ejpam-6059	221	1	the	the	DET
ejpam-6059	221	2	following	follow	VERB
ejpam-6059	221	3	two	two	NUM
ejpam-6059	221	4	examples	example	NOUN
ejpam-6059	221	5	illustrate	illustrate	VERB
ejpam-6059	221	6	results	result	NOUN
ejpam-6059	221	7	for	for	ADP
ejpam-6059	221	8	γn	γn	NOUN
ejpam-6059	221	9	and	and	CCONJ
ejpam-6059	221	10	γn	γn	ADP
ejpam-6059	221	11	where	where	SCONJ
ejpam-6059	221	12	n	n	NOUN
ejpam-6059	221	13	=	=	SYM
ejpam-6059	221	14	4	4	NUM
ejpam-6059	221	15	,	,	PUNCT
ejpam-6059	221	16	6	6	NUM
ejpam-6059	221	17	.	.	PUNCT
ejpam-6059	221	18	other	other	ADJ
ejpam-6059	221	19	results	result	NOUN
ejpam-6059	221	20	will	will	AUX
ejpam-6059	221	21	be	be	AUX
ejpam-6059	221	22	presented	present	VERB
ejpam-6059	221	23	in	in	ADP
ejpam-6059	221	24	the	the	DET
ejpam-6059	221	25	sequel	sequel	NOUN
ejpam-6059	221	26	.	.	PUNCT
ejpam-6059	222	1	example	example	NOUN
ejpam-6059	222	2	5	5	NUM
ejpam-6059	222	3	.	.	PUNCT
ejpam-6059	223	1	the	the	DET
ejpam-6059	223	2	set	set	NOUN
ejpam-6059	223	3	{	{	PUNCT
ejpam-6059	223	4	0	0	NUM
ejpam-6059	223	5	,	,	PUNCT
ejpam-6059	223	6	1	1	NUM
ejpam-6059	223	7	,	,	PUNCT
ejpam-6059	223	8	2	2	NUM
ejpam-6059	223	9	}	}	PUNCT
ejpam-6059	223	10	of	of	ADP
ejpam-6059	223	11	vertices	vertex	NOUN
ejpam-6059	223	12	in	in	ADP
ejpam-6059	223	13	the	the	DET
ejpam-6059	223	14	left	left	ADJ
ejpam-6059	223	15	diagram	diagram	NOUN
ejpam-6059	223	16	of	of	ADP
ejpam-6059	223	17	figure	figure	NOUN
ejpam-6059	223	18	2	2	NUM
ejpam-6059	223	19	and	and	CCONJ
ejpam-6059	223	20	the	the	DET
ejpam-6059	223	21	set	set	NOUN
ejpam-6059	223	22	{	{	PUNCT
ejpam-6059	223	23	0	0	NUM
ejpam-6059	223	24	,	,	PUNCT
ejpam-6059	223	25	1	1	NUM
ejpam-6059	223	26	,	,	PUNCT
ejpam-6059	223	27	2	2	NUM
ejpam-6059	223	28	,	,	PUNCT
ejpam-6059	223	29	3	3	NUM
ejpam-6059	223	30	}	}	PUNCT
ejpam-6059	223	31	of	of	ADP
ejpam-6059	223	32	vertices	vertex	NOUN
ejpam-6059	223	33	in	in	ADP
ejpam-6059	223	34	the	the	DET
ejpam-6059	223	35	left	left	ADJ
ejpam-6059	223	36	diagram	diagram	NOUN
ejpam-6059	223	37	of	of	ADP
ejpam-6059	223	38	figure	figure	NOUN
ejpam-6059	223	39	3	3	NUM
ejpam-6059	223	40	are	be	AUX
ejpam-6059	223	41	acyclic	acyclic	ADJ
ejpam-6059	223	42	sets	set	NOUN
ejpam-6059	223	43	of	of	ADP
ejpam-6059	223	44	γ4	γ4	NOUN
ejpam-6059	223	45	and	and	CCONJ
ejpam-6059	223	46	γ4	γ4	NOUN
ejpam-6059	223	47	,	,	PUNCT
ejpam-6059	223	48	respectively	respectively	ADV
ejpam-6059	223	49	.	.	PUNCT
ejpam-6059	224	1	therefore	therefore	ADV
ejpam-6059	224	2	,	,	PUNCT
ejpam-6059	224	3	it	it	PRON
ejpam-6059	224	4	is	be	AUX
ejpam-6059	224	5	not	not	PART
ejpam-6059	224	6	hard	hard	ADJ
ejpam-6059	224	7	to	to	PART
ejpam-6059	224	8	conclude	conclude	VERB
ejpam-6059	224	9	that	that	DET
ejpam-6059	224	10	λ(γ4	λ(γ4	NOUN
ejpam-6059	224	11	)	)	PUNCT
ejpam-6059	224	12	=	=	SYM
ejpam-6059	224	13	3	3	NUM
ejpam-6059	224	14	and	and	CCONJ
ejpam-6059	224	15	λ(γ4	λ(γ4	ADJ
ejpam-6059	224	16	)	)	PUNCT
ejpam-6059	224	17	=	=	SYM
ejpam-6059	224	18	4	4	NUM
ejpam-6059	224	19	.	.	NOUN
ejpam-6059	224	20	example	example	NOUN
ejpam-6059	224	21	6	6	NUM
ejpam-6059	224	22	.	.	PUNCT
ejpam-6059	225	1	it	it	PRON
ejpam-6059	225	2	is	be	AUX
ejpam-6059	225	3	easy	easy	ADJ
ejpam-6059	225	4	to	to	PART
ejpam-6059	225	5	see	see	VERB
ejpam-6059	225	6	that	that	SCONJ
ejpam-6059	225	7	the	the	DET
ejpam-6059	225	8	set	set	NOUN
ejpam-6059	225	9	{	{	PUNCT
ejpam-6059	225	10	0	0	NUM
ejpam-6059	225	11	,	,	PUNCT
ejpam-6059	225	12	1	1	NUM
ejpam-6059	225	13	,	,	PUNCT
ejpam-6059	225	14	2	2	NUM
ejpam-6059	225	15	,	,	PUNCT
ejpam-6059	225	16	3	3	NUM
ejpam-6059	225	17	,	,	PUNCT
ejpam-6059	225	18	4	4	NUM
ejpam-6059	225	19	}	}	PUNCT
ejpam-6059	225	20	of	of	ADP
ejpam-6059	225	21	vertices	vertex	NOUN
ejpam-6059	225	22	in	in	ADP
ejpam-6059	225	23	the	the	DET
ejpam-6059	225	24	right	right	ADJ
ejpam-6059	225	25	diagram	diagram	NOUN
ejpam-6059	225	26	of	of	ADP
ejpam-6059	225	27	figure	figure	NOUN
ejpam-6059	225	28	2	2	NUM
ejpam-6059	225	29	is	be	AUX
ejpam-6059	225	30	an	an	DET
ejpam-6059	225	31	acyclic	acyclic	ADJ
ejpam-6059	225	32	set	set	NOUN
ejpam-6059	225	33	of	of	ADP
ejpam-6059	225	34	γ6	γ6	PROPN
ejpam-6059	225	35	with	with	ADP
ejpam-6059	225	36	maximum	maximum	ADJ
ejpam-6059	225	37	cardinality	cardinality	NOUN
ejpam-6059	225	38	.	.	PUNCT
ejpam-6059	226	1	furthermore	furthermore	ADV
ejpam-6059	226	2	,	,	PUNCT
ejpam-6059	226	3	the	the	DET
ejpam-6059	226	4	set	set	NOUN
ejpam-6059	226	5	{	{	PUNCT
ejpam-6059	226	6	0	0	NUM
ejpam-6059	226	7	,	,	PUNCT
ejpam-6059	226	8	1	1	NUM
ejpam-6059	226	9	,	,	PUNCT
ejpam-6059	226	10	2	2	NUM
ejpam-6059	226	11	,	,	PUNCT
ejpam-6059	226	12	3	3	NUM
ejpam-6059	226	13	}	}	PUNCT
ejpam-6059	226	14	of	of	ADP
ejpam-6059	226	15	vertices	vertex	NOUN
ejpam-6059	226	16	in	in	ADP
ejpam-6059	226	17	the	the	DET
ejpam-6059	226	18	right	right	ADJ
ejpam-6059	226	19	diagram	diagram	NOUN
ejpam-6059	226	20	of	of	ADP
ejpam-6059	226	21	figure	figure	NOUN
ejpam-6059	226	22	3	3	NUM
ejpam-6059	226	23	is	be	AUX
ejpam-6059	226	24	an	an	DET
ejpam-6059	226	25	acyclic	acyclic	ADJ
ejpam-6059	226	26	set	set	NOUN
ejpam-6059	226	27	of	of	ADP
ejpam-6059	226	28	γ6	γ6	PROPN
ejpam-6059	226	29	with	with	ADP
ejpam-6059	226	30	maximum	maximum	ADJ
ejpam-6059	226	31	cardinality	cardinality	NOUN
ejpam-6059	226	32	.	.	PUNCT
ejpam-6059	227	1	therefore	therefore	ADV
ejpam-6059	227	2	,	,	PUNCT
ejpam-6059	227	3	λ(γ6	λ(γ6	NOUN
ejpam-6059	227	4	)	)	PUNCT
ejpam-6059	227	5	=	=	SYM
ejpam-6059	227	6	5	5	NUM
ejpam-6059	227	7	and	and	CCONJ
ejpam-6059	227	8	λ(γ6	λ(γ6	NOUN
ejpam-6059	227	9	)	)	PUNCT
ejpam-6059	228	1	=	=	SYM
ejpam-6059	228	2	4	4	X
ejpam-6059	228	3	.	.	PUNCT
ejpam-6059	228	4	theorem	theorem	VERB
ejpam-6059	228	5	7	7	NUM
ejpam-6059	228	6	.	.	PUNCT
ejpam-6059	229	1	let	let	VERB
ejpam-6059	229	2	p	p	PRON
ejpam-6059	229	3	be	be	AUX
ejpam-6059	229	4	a	a	DET
ejpam-6059	229	5	prime	prime	ADJ
ejpam-6059	229	6	number	number	NOUN
ejpam-6059	229	7	such	such	ADJ
ejpam-6059	229	8	that	that	SCONJ
ejpam-6059	229	9	p	p	NOUN
ejpam-6059	229	10	≥	≥	NUM
ejpam-6059	229	11	3	3	NUM
ejpam-6059	229	12	.	.	PUNCT
ejpam-6059	230	1	then	then	ADV
ejpam-6059	230	2	λ(γp	λ(γp	NUM
ejpam-6059	230	3	)	)	PUNCT
ejpam-6059	230	4	=	=	SYM
ejpam-6059	230	5	2	2	NUM
ejpam-6059	230	6	and	and	CCONJ
ejpam-6059	230	7	λ(γp	λ(γp	NUM
ejpam-6059	230	8	)	)	PUNCT
ejpam-6059	230	9	=	=	SYM
ejpam-6059	231	1	p.	p.	NOUN
ejpam-6059	231	2	proof	proof	NOUN
ejpam-6059	231	3	.	.	PUNCT
ejpam-6059	232	1	let	let	VERB
ejpam-6059	232	2	p	p	PRON
ejpam-6059	232	3	be	be	AUX
ejpam-6059	232	4	a	a	DET
ejpam-6059	232	5	prime	prime	ADJ
ejpam-6059	232	6	number	number	NOUN
ejpam-6059	232	7	such	such	ADJ
ejpam-6059	233	1	that	that	SCONJ
ejpam-6059	233	2	p	p	NOUN
ejpam-6059	233	3	≥	≥	NUM
ejpam-6059	233	4	3	3	NUM
ejpam-6059	233	5	.	.	PUNCT
ejpam-6059	233	6	then	then	ADV
ejpam-6059	233	7	γp	γp	PROPN
ejpam-6059	233	8	is	be	AUX
ejpam-6059	233	9	a	a	DET
ejpam-6059	233	10	complete	complete	ADJ
ejpam-6059	233	11	graph	graph	NOUN
ejpam-6059	233	12	and	and	CCONJ
ejpam-6059	233	13	γp	γp	PROPN
ejpam-6059	233	14	is	be	AUX
ejpam-6059	233	15	an	an	DET
ejpam-6059	233	16	empty	empty	ADJ
ejpam-6059	233	17	graph	graph	NOUN
ejpam-6059	233	18	.	.	PUNCT
ejpam-6059	234	1	it	it	PRON
ejpam-6059	234	2	follows	follow	VERB
ejpam-6059	234	3	that	that	SCONJ
ejpam-6059	234	4	λ(γp	λ(γp	NUM
ejpam-6059	234	5	)	)	PUNCT
ejpam-6059	234	6	=	=	SYM
ejpam-6059	234	7	2	2	NUM
ejpam-6059	234	8	and	and	CCONJ
ejpam-6059	234	9	λ(γp	λ(γp	NUM
ejpam-6059	234	10	)	)	PUNCT
ejpam-6059	234	11	=	=	SYM
ejpam-6059	234	12	p	p	NOUN
ejpam-6059	234	13	,	,	PUNCT
ejpam-6059	234	14	respectively	respectively	ADV
ejpam-6059	234	15	.	.	PUNCT
ejpam-6059	235	1	theorem	theorem	VERB
ejpam-6059	235	2	8	8	NUM
ejpam-6059	235	3	.	.	PUNCT
ejpam-6059	236	1	let	let	VERB
ejpam-6059	236	2	n	n	NOUN
ejpam-6059	236	3	=	=	SYM
ejpam-6059	236	4	pk	pk	NOUN
ejpam-6059	236	5	be	be	AUX
ejpam-6059	236	6	such	such	ADJ
ejpam-6059	236	7	that	that	SCONJ
ejpam-6059	236	8	p	p	NOUN
ejpam-6059	236	9	is	be	AUX
ejpam-6059	236	10	prime	prime	ADJ
ejpam-6059	236	11	and	and	CCONJ
ejpam-6059	236	12	k	k	PROPN
ejpam-6059	236	13	∈	∈	PROPN
ejpam-6059	236	14	n\{1	n\{1	PROPN
ejpam-6059	236	15	}	}	PUNCT
ejpam-6059	236	16	.	.	PUNCT
ejpam-6059	237	1	then	then	ADV
ejpam-6059	237	2	λ(γn	λ(γn	ADJ
ejpam-6059	237	3	)	)	PUNCT
ejpam-6059	237	4	=	=	SYM
ejpam-6059	238	1	pk−1	pk−1	PROPN
ejpam-6059	238	2	+	+	NOUN
ejpam-6059	238	3	1	1	NUM
ejpam-6059	238	4	.	.	PUNCT
ejpam-6059	239	1	proof	proof	NOUN
ejpam-6059	239	2	.	.	PUNCT
ejpam-6059	240	1	let	let	VERB
ejpam-6059	240	2	n	n	PRON
ejpam-6059	240	3	=	=	SYM
ejpam-6059	240	4	pk	pk	NOUN
ejpam-6059	240	5	be	be	AUX
ejpam-6059	240	6	such	such	ADJ
ejpam-6059	240	7	that	that	SCONJ
ejpam-6059	240	8	p	p	NOUN
ejpam-6059	240	9	is	be	AUX
ejpam-6059	240	10	prime	prime	ADJ
ejpam-6059	240	11	and	and	CCONJ
ejpam-6059	240	12	k	k	PROPN
ejpam-6059	240	13	∈	∈	PROPN
ejpam-6059	240	14	n	n	PRON
ejpam-6059	240	15	\	\	NOUN
ejpam-6059	240	16	{	{	PUNCT
ejpam-6059	240	17	1	1	NUM
ejpam-6059	240	18	}	}	PUNCT
ejpam-6059	240	19	.	.	PUNCT
ejpam-6059	241	1	by	by	ADP
ejpam-6059	241	2	theorem	theorem	NOUN
ejpam-6059	241	3	2	2	NUM
ejpam-6059	241	4	,	,	PUNCT
ejpam-6059	241	5	we	we	PRON
ejpam-6059	241	6	obtain	obtain	VERB
ejpam-6059	241	7	that	that	SCONJ
ejpam-6059	241	8	γn	γn	ADJ
ejpam-6059	241	9	is	be	AUX
ejpam-6059	241	10	a	a	DET
ejpam-6059	241	11	complete	complete	ADJ
ejpam-6059	241	12	p	p	ADJ
ejpam-6059	241	13	-	-	PUNCT
ejpam-6059	241	14	partite	partite	ADJ
ejpam-6059	241	15	graph	graph	NOUN
ejpam-6059	241	16	such	such	ADJ
ejpam-6059	241	17	that	that	SCONJ
ejpam-6059	241	18	each	each	DET
ejpam-6059	241	19	partite	partite	ADJ
ejpam-6059	241	20	set	set	NOUN
ejpam-6059	241	21	has	have	VERB
ejpam-6059	241	22	size	size	NOUN
ejpam-6059	241	23	pk−1	pk−1	PROPN
ejpam-6059	241	24	.	.	PUNCT
ejpam-6059	242	1	let	let	VERB
ejpam-6059	242	2	p	p	PRON
ejpam-6059	242	3	be	be	AUX
ejpam-6059	242	4	a	a	DET
ejpam-6059	242	5	partite	partite	ADJ
ejpam-6059	242	6	set	set	NOUN
ejpam-6059	242	7	of	of	ADP
ejpam-6059	242	8	γn	γn	NOUN
ejpam-6059	242	9	.	.	PUNCT
ejpam-6059	243	1	thus	thus	ADV
ejpam-6059	243	2	the	the	DET
ejpam-6059	243	3	induced	induced	ADJ
ejpam-6059	243	4	subgraph	subgraph	NOUN
ejpam-6059	243	5	γn[p	γn[p	PROPN
ejpam-6059	243	6	]	]	PUNCT
ejpam-6059	243	7	is	be	AUX
ejpam-6059	243	8	an	an	DET
ejpam-6059	243	9	empty	empty	ADJ
ejpam-6059	243	10	graph	graph	NOUN
ejpam-6059	243	11	.	.	PUNCT
ejpam-6059	244	1	let	let	VERB
ejpam-6059	244	2	v	v	NUM
ejpam-6059	244	3	∈	∈	PROPN
ejpam-6059	244	4	v	v	NOUN
ejpam-6059	244	5	(	(	PUNCT
ejpam-6059	244	6	γn	γn	NOUN
ejpam-6059	244	7	)	)	PUNCT
ejpam-6059	244	8	\	\	PROPN
ejpam-6059	245	1	p	p	NOUN
ejpam-6059	245	2	.	.	PUNCT
ejpam-6059	246	1	hence	hence	ADV
ejpam-6059	246	2	γn[p	γn[p	VERB
ejpam-6059	246	3	∪	∪	ADJ
ejpam-6059	246	4	{	{	PUNCT
ejpam-6059	246	5	v	v	NOUN
ejpam-6059	246	6	}	}	PUNCT
ejpam-6059	246	7	]	]	PUNCT
ejpam-6059	246	8	is	be	AUX
ejpam-6059	246	9	a	a	DET
ejpam-6059	246	10	tree	tree	NOUN
ejpam-6059	246	11	.	.	PUNCT
ejpam-6059	247	1	it	it	PRON
ejpam-6059	247	2	follows	follow	VERB
ejpam-6059	247	3	that	that	SCONJ
ejpam-6059	247	4	p	p	PROPN
ejpam-6059	247	5	∪	∪	X
ejpam-6059	247	6	{	{	PUNCT
ejpam-6059	247	7	v	v	NOUN
ejpam-6059	247	8	}	}	PUNCT
ejpam-6059	247	9	is	be	AUX
ejpam-6059	247	10	an	an	DET
ejpam-6059	247	11	acyclic	acyclic	ADJ
ejpam-6059	247	12	set	set	NOUN
ejpam-6059	247	13	of	of	ADP
ejpam-6059	247	14	γn	γn	NUM
ejpam-6059	247	15	which	which	PRON
ejpam-6059	247	16	leads	lead	VERB
ejpam-6059	247	17	to	to	ADP
ejpam-6059	247	18	λ(γn	λ(γn	NOUN
ejpam-6059	247	19	)	)	PUNCT
ejpam-6059	247	20	≥	≥	PROPN
ejpam-6059	247	21	|p	|p	PROPN
ejpam-6059	247	22	∪	∪	X
ejpam-6059	247	23	{	{	PUNCT
ejpam-6059	247	24	v}|	v}|	NOUN
ejpam-6059	247	25	=	=	PUNCT
ejpam-6059	247	26	pk−1	pk−1	NOUN
ejpam-6059	248	1	+	+	NOUN
ejpam-6059	248	2	1	1	X
ejpam-6059	248	3	.	.	PUNCT
ejpam-6059	248	4	we	we	PRON
ejpam-6059	248	5	now	now	ADV
ejpam-6059	248	6	suppose	suppose	VERB
ejpam-6059	248	7	that	that	SCONJ
ejpam-6059	248	8	there	there	PRON
ejpam-6059	248	9	is	be	VERB
ejpam-6059	248	10	an	an	DET
ejpam-6059	248	11	acyclic	acyclic	ADJ
ejpam-6059	248	12	set	set	NOUN
ejpam-6059	248	13	,	,	PUNCT
ejpam-6059	248	14	say	say	VERB
ejpam-6059	248	15	a	a	PRON
ejpam-6059	248	16	,	,	PUNCT
ejpam-6059	248	17	of	of	ADP
ejpam-6059	248	18	γn	γn	NOUN
ejpam-6059	248	19	in	in	ADP
ejpam-6059	248	20	which	which	PRON
ejpam-6059	248	21	|a|	|a|	PROPN
ejpam-6059	248	22	>	>	X
ejpam-6059	248	23	pk−1	pk−1	PROPN
ejpam-6059	248	24	+	+	NOUN
ejpam-6059	248	25	1	1	X
ejpam-6059	248	26	.	.	PUNCT
ejpam-6059	249	1	then	then	ADV
ejpam-6059	249	2	there	there	PRON
ejpam-6059	249	3	exist	exist	VERB
ejpam-6059	249	4	x	x	NOUN
ejpam-6059	249	5	,	,	PUNCT
ejpam-6059	249	6	y	y	PROPN
ejpam-6059	249	7	,	,	PUNCT
ejpam-6059	249	8	z	z	PROPN
ejpam-6059	249	9	∈	∈	PROPN
ejpam-6059	249	10	a	a	DET
ejpam-6059	249	11	such	such	ADJ
ejpam-6059	249	12	that	that	SCONJ
ejpam-6059	249	13	x	x	SYM
ejpam-6059	249	14	∈	∈	PROPN
ejpam-6059	249	15	p1	p1	NOUN
ejpam-6059	249	16	and	and	CCONJ
ejpam-6059	249	17	y	y	PROPN
ejpam-6059	249	18	,	,	PUNCT
ejpam-6059	249	19	z	z	NOUN
ejpam-6059	249	20	/∈	/∈	PUNCT
ejpam-6059	250	1	p1	p1	PROPN
ejpam-6059	250	2	where	where	SCONJ
ejpam-6059	250	3	p1	p1	PROPN
ejpam-6059	250	4	is	be	AUX
ejpam-6059	250	5	a	a	DET
ejpam-6059	250	6	partite	partite	ADJ
ejpam-6059	250	7	set	set	NOUN
ejpam-6059	250	8	of	of	ADP
ejpam-6059	250	9	γn	γn	NUM
ejpam-6059	250	10	.	.	PUNCT
ejpam-6059	251	1	we	we	PRON
ejpam-6059	251	2	now	now	ADV
ejpam-6059	251	3	consider	consider	VERB
ejpam-6059	251	4	the	the	DET
ejpam-6059	251	5	following	follow	VERB
ejpam-6059	251	6	two	two	NUM
ejpam-6059	251	7	cases	case	NOUN
ejpam-6059	251	8	.	.	PUNCT
ejpam-6059	252	1	case	case	NOUN
ejpam-6059	252	2	1	1	NUM
ejpam-6059	252	3	:	:	PUNCT
ejpam-6059	252	4	y	y	NOUN
ejpam-6059	252	5	,	,	PUNCT
ejpam-6059	252	6	z	z	NOUN
ejpam-6059	252	7	∈	∈	NOUN
ejpam-6059	252	8	p2	p2	NOUN
ejpam-6059	252	9	for	for	ADP
ejpam-6059	252	10	some	some	PRON
ejpam-6059	252	11	a	a	DET
ejpam-6059	252	12	partite	partite	ADJ
ejpam-6059	252	13	set	set	NOUN
ejpam-6059	252	14	p2	p2	NOUN
ejpam-6059	252	15	in	in	ADP
ejpam-6059	252	16	which	which	PRON
ejpam-6059	252	17	p1	p1	NOUN
ejpam-6059	252	18	̸=	̸=	PROPN
ejpam-6059	252	19	p2	p2	NOUN
ejpam-6059	252	20	.	.	PUNCT
ejpam-6059	253	1	since	since	SCONJ
ejpam-6059	253	2	|a|	|a|	PROPN
ejpam-6059	253	3	>	>	X
ejpam-6059	253	4	pk−1	pk−1	PROPN
ejpam-6059	253	5	+	+	PROPN
ejpam-6059	253	6	1	1	NUM
ejpam-6059	253	7	where	where	SCONJ
ejpam-6059	253	8	p	p	NOUN
ejpam-6059	253	9	is	be	AUX
ejpam-6059	253	10	prime	prime	ADJ
ejpam-6059	253	11	and	and	CCONJ
ejpam-6059	253	12	k	k	PROPN
ejpam-6059	253	13	≥	≥	NUM
ejpam-6059	253	14	2	2	NUM
ejpam-6059	253	15	,	,	PUNCT
ejpam-6059	253	16	there	there	PRON
ejpam-6059	253	17	exists	exist	VERB
ejpam-6059	253	18	u	u	PROPN
ejpam-6059	253	19	∈	∈	PROPN
ejpam-6059	253	20	a	a	DET
ejpam-6059	253	21	\	\	NOUN
ejpam-6059	253	22	{	{	PUNCT
ejpam-6059	253	23	x	x	NOUN
ejpam-6059	253	24	,	,	PUNCT
ejpam-6059	253	25	y	y	PROPN
ejpam-6059	253	26	,	,	PUNCT
ejpam-6059	253	27	z	z	NOUN
ejpam-6059	253	28	}	}	PUNCT
ejpam-6059	253	29	such	such	ADJ
ejpam-6059	253	30	that	that	SCONJ
ejpam-6059	253	31	u	u	NOUN
ejpam-6059	253	32	/∈	/∈	NOUN
ejpam-6059	253	33	p2	p2	PROPN
ejpam-6059	253	34	.	.	PUNCT
ejpam-6059	254	1	if	if	SCONJ
ejpam-6059	254	2	u	u	PROPN
ejpam-6059	254	3	∈	∈	PROPN
ejpam-6059	254	4	p1	p1	NOUN
ejpam-6059	254	5	,	,	PUNCT
ejpam-6059	254	6	then	then	ADV
ejpam-6059	254	7	the	the	DET
ejpam-6059	254	8	sequence	sequence	NOUN
ejpam-6059	254	9	of	of	ADP
ejpam-6059	254	10	edges	edge	NOUN
ejpam-6059	254	11	uy	uy	ADP
ejpam-6059	254	12	,	,	PUNCT
ejpam-6059	254	13	yx	yx	PROPN
ejpam-6059	254	14	,	,	PUNCT
ejpam-6059	254	15	xz	xz	PROPN
ejpam-6059	254	16	,	,	PUNCT
ejpam-6059	254	17	zu	zu	NOUN
ejpam-6059	254	18	forms	form	VERB
ejpam-6059	254	19	a	a	DET
ejpam-6059	254	20	cycle	cycle	NOUN
ejpam-6059	254	21	of	of	ADP
ejpam-6059	254	22	length	length	NOUN
ejpam-6059	254	23	4	4	NUM
ejpam-6059	254	24	in	in	ADP
ejpam-6059	254	25	γn[a	γn[a	PROPN
ejpam-6059	254	26	]	]	PUNCT
ejpam-6059	254	27	(	(	PUNCT
ejpam-6059	254	28	see	see	VERB
ejpam-6059	254	29	figure	figure	NOUN
ejpam-6059	254	30	4	4	NUM
ejpam-6059	254	31	.	.	PUNCT
ejpam-6059	254	32	)	)	PUNCT
ejpam-6059	255	1	since	since	SCONJ
ejpam-6059	255	2	γn	γn	PRON
ejpam-6059	255	3	is	be	AUX
ejpam-6059	255	4	a	a	DET
ejpam-6059	255	5	complete	complete	ADJ
ejpam-6059	255	6	p	p	ADJ
ejpam-6059	255	7	-	-	PUNCT
ejpam-6059	255	8	partite	partite	ADJ
ejpam-6059	255	9	graph	graph	NOUN
ejpam-6059	255	10	.	.	PUNCT
ejpam-6059	256	1	this	this	PRON
ejpam-6059	256	2	contradicts	contradict	VERB
ejpam-6059	256	3	to	to	ADP
ejpam-6059	256	4	the	the	DET
ejpam-6059	256	5	acyclicity	acyclicity	NOUN
ejpam-6059	256	6	of	of	ADP
ejpam-6059	256	7	a.	a.	NOUN
ejpam-6059	256	8	on	on	ADP
ejpam-6059	256	9	the	the	DET
ejpam-6059	256	10	other	other	ADJ
ejpam-6059	256	11	hand	hand	NOUN
ejpam-6059	256	12	,	,	PUNCT
ejpam-6059	256	13	if	if	SCONJ
ejpam-6059	256	14	u	u	PROPN
ejpam-6059	256	15	∈	∈	PROPN
ejpam-6059	256	16	p3	p3	NOUN
ejpam-6059	256	17	where	where	SCONJ
ejpam-6059	256	18	p3	p3	PROPN
ejpam-6059	256	19	̸=	̸=	PROPN
ejpam-6059	256	20	p1	p1	NOUN
ejpam-6059	256	21	,	,	PUNCT
ejpam-6059	256	22	then	then	ADV
ejpam-6059	256	23	the	the	DET
ejpam-6059	256	24	sequence	sequence	NOUN
ejpam-6059	256	25	of	of	ADP
ejpam-6059	256	26	edges	edge	NOUN
ejpam-6059	256	27	d.	d.	PROPN
ejpam-6059	256	28	pongpipat	pongpipat	PROPN
ejpam-6059	256	29	,	,	PUNCT
ejpam-6059	256	30	n.	n.	NOUN
ejpam-6059	256	31	nupo	nupo	PROPN
ejpam-6059	256	32	/	/	SYM
ejpam-6059	256	33	eur	eur	PROPN
ejpam-6059	256	34	.	.	PUNCT
ejpam-6059	257	1	j.	j.	PROPN
ejpam-6059	257	2	pure	pure	PROPN
ejpam-6059	257	3	appl	appl	PROPN
ejpam-6059	257	4	.	.	PROPN
ejpam-6059	257	5	math	math	PROPN
ejpam-6059	257	6	,	,	PUNCT
ejpam-6059	257	7	18	18	NUM
ejpam-6059	257	8	(	(	PUNCT
ejpam-6059	257	9	2	2	NUM
ejpam-6059	257	10	)	)	PUNCT
ejpam-6059	257	11	(	(	PUNCT
ejpam-6059	257	12	2025	2025	NUM
ejpam-6059	257	13	)	)	PUNCT
ejpam-6059	257	14	,	,	PUNCT
ejpam-6059	257	15	6059	6059	NUM
ejpam-6059	257	16	8	8	NUM
ejpam-6059	257	17	of	of	ADP
ejpam-6059	257	18	11	11	NUM
ejpam-6059	257	19	figure	figure	NOUN
ejpam-6059	257	20	4	4	NUM
ejpam-6059	257	21	:	:	PUNCT
ejpam-6059	257	22	two	two	NUM
ejpam-6059	257	23	possible	possible	ADJ
ejpam-6059	257	24	cases	case	NOUN
ejpam-6059	257	25	of	of	ADP
ejpam-6059	257	26	the	the	DET
ejpam-6059	257	27	sequence	sequence	NOUN
ejpam-6059	257	28	of	of	ADP
ejpam-6059	257	29	edges	edge	NOUN
ejpam-6059	257	30	on	on	ADP
ejpam-6059	257	31	cycles	cycle	NOUN
ejpam-6059	257	32	of	of	ADP
ejpam-6059	257	33	length	length	NOUN
ejpam-6059	257	34	4	4	NUM
ejpam-6059	257	35	and	and	CCONJ
ejpam-6059	257	36	3	3	NUM
ejpam-6059	257	37	.	.	X
ejpam-6059	257	38	ux	ux	PROPN
ejpam-6059	257	39	,	,	PUNCT
ejpam-6059	257	40	xy	xy	PROPN
ejpam-6059	257	41	,	,	PUNCT
ejpam-6059	257	42	yu	yu	PROPN
ejpam-6059	257	43	forms	form	VERB
ejpam-6059	257	44	a	a	DET
ejpam-6059	257	45	cycle	cycle	NOUN
ejpam-6059	257	46	of	of	ADP
ejpam-6059	257	47	length	length	NOUN
ejpam-6059	257	48	3	3	NUM
ejpam-6059	257	49	in	in	ADP
ejpam-6059	257	50	γn[a	γn[a	PROPN
ejpam-6059	257	51	]	]	PUNCT
ejpam-6059	257	52	(	(	PUNCT
ejpam-6059	257	53	see	see	VERB
ejpam-6059	257	54	figure	figure	NOUN
ejpam-6059	257	55	4	4	NUM
ejpam-6059	257	56	.	.	PUNCT
ejpam-6059	257	57	)	)	PUNCT
ejpam-6059	257	58	which	which	PRON
ejpam-6059	257	59	is	be	AUX
ejpam-6059	257	60	also	also	ADV
ejpam-6059	257	61	a	a	DET
ejpam-6059	257	62	contradiction	contradiction	NOUN
ejpam-6059	257	63	.	.	PUNCT
ejpam-6059	258	1	case	case	NOUN
ejpam-6059	258	2	2	2	NUM
ejpam-6059	258	3	:	:	PUNCT
ejpam-6059	258	4	y	y	PROPN
ejpam-6059	258	5	∈	∈	PROPN
ejpam-6059	258	6	p2	p2	PROPN
ejpam-6059	258	7	and	and	CCONJ
ejpam-6059	258	8	z	z	NOUN
ejpam-6059	258	9	∈	∈	PROPN
ejpam-6059	258	10	p3	p3	NOUN
ejpam-6059	258	11	for	for	ADP
ejpam-6059	258	12	some	some	DET
ejpam-6059	258	13	partite	partite	ADJ
ejpam-6059	258	14	sets	set	NOUN
ejpam-6059	258	15	p2	p2	NOUN
ejpam-6059	258	16	,	,	PUNCT
ejpam-6059	258	17	p3	p3	PROPN
ejpam-6059	258	18	in	in	ADP
ejpam-6059	258	19	which	which	PRON
ejpam-6059	258	20	p1	p1	NOUN
ejpam-6059	258	21	,	,	PUNCT
ejpam-6059	258	22	p2	p2	NOUN
ejpam-6059	258	23	,	,	PUNCT
ejpam-6059	258	24	p3	p3	PROPN
ejpam-6059	258	25	are	be	AUX
ejpam-6059	258	26	pairwise	pairwise	NOUN
ejpam-6059	258	27	disjoint	disjoint	NOUN
ejpam-6059	258	28	.	.	PUNCT
ejpam-6059	259	1	we	we	PRON
ejpam-6059	259	2	can	can	AUX
ejpam-6059	259	3	observe	observe	VERB
ejpam-6059	259	4	that	that	SCONJ
ejpam-6059	259	5	this	this	DET
ejpam-6059	259	6	case	case	NOUN
ejpam-6059	259	7	will	will	AUX
ejpam-6059	259	8	generate	generate	VERB
ejpam-6059	259	9	a	a	DET
ejpam-6059	259	10	cycle	cycle	NOUN
ejpam-6059	259	11	of	of	ADP
ejpam-6059	259	12	length	length	NOUN
ejpam-6059	259	13	3	3	NUM
ejpam-6059	259	14	with	with	ADP
ejpam-6059	259	15	the	the	DET
ejpam-6059	259	16	sequence	sequence	NOUN
ejpam-6059	259	17	of	of	ADP
ejpam-6059	259	18	edges	edge	NOUN
ejpam-6059	259	19	xy	xy	PROPN
ejpam-6059	259	20	,	,	PUNCT
ejpam-6059	259	21	yz	yz	PROPN
ejpam-6059	259	22	,	,	PUNCT
ejpam-6059	259	23	zx	zx	PROPN
ejpam-6059	259	24	in	in	ADP
ejpam-6059	259	25	γn[a	γn[a	PROPN
ejpam-6059	259	26	]	]	PUNCT
ejpam-6059	259	27	similar	similar	ADJ
ejpam-6059	259	28	to	to	PART
ejpam-6059	259	29	figure	figure	VERB
ejpam-6059	259	30	4	4	NUM
ejpam-6059	259	31	.	.	PUNCT
ejpam-6059	260	1	this	this	PRON
ejpam-6059	260	2	also	also	ADV
ejpam-6059	260	3	contradicts	contradict	VERB
ejpam-6059	260	4	to	to	ADP
ejpam-6059	260	5	the	the	DET
ejpam-6059	260	6	acyclicity	acyclicity	NOUN
ejpam-6059	260	7	of	of	ADP
ejpam-6059	260	8	a.	a.	NOUN
ejpam-6059	260	9	from	from	ADP
ejpam-6059	260	10	the	the	DET
ejpam-6059	260	11	above	above	ADJ
ejpam-6059	260	12	two	two	NUM
ejpam-6059	260	13	cases	case	NOUN
ejpam-6059	260	14	,	,	PUNCT
ejpam-6059	260	15	we	we	PRON
ejpam-6059	260	16	can	can	AUX
ejpam-6059	260	17	conclude	conclude	VERB
ejpam-6059	260	18	that	that	SCONJ
ejpam-6059	260	19	λ(γn	λ(γn	NOUN
ejpam-6059	260	20	)	)	PUNCT
ejpam-6059	261	1	=	=	SYM
ejpam-6059	261	2	pk−1	pk−1	NOUN
ejpam-6059	261	3	+	+	NOUN
ejpam-6059	261	4	1	1	NUM
ejpam-6059	261	5	.	.	X
ejpam-6059	261	6	theorem	theorem	NOUN
ejpam-6059	261	7	9	9	NUM
ejpam-6059	261	8	.	.	PUNCT
ejpam-6059	262	1	let	let	VERB
ejpam-6059	262	2	n	n	PRON
ejpam-6059	262	3	be	be	AUX
ejpam-6059	262	4	a	a	DET
ejpam-6059	262	5	positive	positive	ADJ
ejpam-6059	262	6	integer	integer	NOUN
ejpam-6059	262	7	such	such	ADJ
ejpam-6059	262	8	that	that	SCONJ
ejpam-6059	262	9	n	n	CCONJ
ejpam-6059	262	10	≥	≥	NUM
ejpam-6059	262	11	8	8	NUM
ejpam-6059	262	12	.	.	PUNCT
ejpam-6059	263	1	if	if	SCONJ
ejpam-6059	263	2	p	p	NOUN
ejpam-6059	263	3	is	be	AUX
ejpam-6059	263	4	the	the	DET
ejpam-6059	263	5	least	least	ADJ
ejpam-6059	263	6	prime	prime	ADJ
ejpam-6059	263	7	divisor	divisor	NOUN
ejpam-6059	263	8	of	of	ADP
ejpam-6059	263	9	n	n	CCONJ
ejpam-6059	263	10	,	,	PUNCT
ejpam-6059	263	11	then	then	ADV
ejpam-6059	263	12	n	n	PRON
ejpam-6059	263	13	p	p	NOUN
ejpam-6059	264	1	+	+	CCONJ
ejpam-6059	264	2	1	1	NUM
ejpam-6059	264	3	≤	≤	NUM
ejpam-6059	264	4	λ(γn	λ(γn	NOUN
ejpam-6059	264	5	)	)	PUNCT
ejpam-6059	264	6	≤	≤	NUM
ejpam-6059	264	7	n−	n−	NOUN
ejpam-6059	264	8	(	(	PUNCT
ejpam-6059	264	9	φ(n	φ(n	ADJ
ejpam-6059	264	10	)	)	PUNCT
ejpam-6059	264	11	2	2	NUM
ejpam-6059	264	12	+	+	SYM
ejpam-6059	264	13	1	1	NUM
ejpam-6059	264	14	)	)	PUNCT
ejpam-6059	264	15	.	.	PUNCT
ejpam-6059	265	1	proof	proof	NOUN
ejpam-6059	265	2	.	.	PUNCT
ejpam-6059	266	1	let	let	VERB
ejpam-6059	266	2	n	n	PRON
ejpam-6059	266	3	be	be	AUX
ejpam-6059	266	4	a	a	DET
ejpam-6059	266	5	positive	positive	ADJ
ejpam-6059	266	6	integer	integer	NOUN
ejpam-6059	266	7	such	such	ADJ
ejpam-6059	266	8	that	that	SCONJ
ejpam-6059	266	9	n	n	CCONJ
ejpam-6059	266	10	≥	≥	NUM
ejpam-6059	266	11	8	8	NUM
ejpam-6059	266	12	.	.	PUNCT
ejpam-6059	267	1	assume	assume	VERB
ejpam-6059	267	2	that	that	SCONJ
ejpam-6059	267	3	p	p	NOUN
ejpam-6059	267	4	is	be	AUX
ejpam-6059	267	5	the	the	DET
ejpam-6059	267	6	least	least	ADJ
ejpam-6059	267	7	prime	prime	ADJ
ejpam-6059	267	8	divisor	divisor	NOUN
ejpam-6059	267	9	of	of	ADP
ejpam-6059	267	10	n.	n.	NOUN
ejpam-6059	267	11	to	to	PART
ejpam-6059	267	12	prove	prove	VERB
ejpam-6059	267	13	the	the	DET
ejpam-6059	267	14	lower	low	ADJ
ejpam-6059	267	15	bound	bind	VERB
ejpam-6059	267	16	of	of	ADP
ejpam-6059	267	17	λ(γn	λ(γn	NOUN
ejpam-6059	267	18	)	)	PUNCT
ejpam-6059	267	19	,	,	PUNCT
ejpam-6059	267	20	consider	consider	VERB
ejpam-6059	267	21	the	the	DET
ejpam-6059	267	22	set	set	NOUN
ejpam-6059	267	23	a	a	DET
ejpam-6059	267	24	=	=	PUNCT
ejpam-6059	267	25	{	{	PUNCT
ejpam-6059	267	26	0	0	NUM
ejpam-6059	267	27	,	,	PUNCT
ejpam-6059	267	28	p	p	X
ejpam-6059	267	29	,	,	PUNCT
ejpam-6059	267	30	2p	2p	NOUN
ejpam-6059	267	31	,	,	PUNCT
ejpam-6059	267	32	.	.	PUNCT
ejpam-6059	267	33	.	.	PUNCT
ejpam-6059	267	34	.	.	PUNCT
ejpam-6059	268	1	,	,	PUNCT
ejpam-6059	268	2	n−p	n−p	PROPN
ejpam-6059	268	3	}	}	PUNCT
ejpam-6059	268	4	.	.	PUNCT
ejpam-6059	269	1	it	it	PRON
ejpam-6059	269	2	is	be	AUX
ejpam-6059	269	3	clear	clear	ADJ
ejpam-6059	269	4	that	that	SCONJ
ejpam-6059	269	5	|a|	|a|	PROPN
ejpam-6059	269	6	=	=	PROPN
ejpam-6059	269	7	n	n	PRON
ejpam-6059	269	8	p	p	NOUN
ejpam-6059	269	9	.	.	PUNCT
ejpam-6059	270	1	for	for	ADP
ejpam-6059	270	2	each	each	DET
ejpam-6059	270	3	{	{	PUNCT
ejpam-6059	270	4	x	x	PROPN
ejpam-6059	270	5	,	,	PUNCT
ejpam-6059	270	6	y	y	PROPN
ejpam-6059	270	7	}	}	PUNCT
ejpam-6059	270	8	∈	∈	PROPN
ejpam-6059	270	9	a	a	PRON
ejpam-6059	270	10	,	,	PUNCT
ejpam-6059	270	11	we	we	PRON
ejpam-6059	270	12	have	have	VERB
ejpam-6059	270	13	x	x	PART
ejpam-6059	271	1	−	−	PROPN
ejpam-6059	271	2	y	y	PROPN
ejpam-6059	271	3	is	be	AUX
ejpam-6059	271	4	the	the	DET
ejpam-6059	271	5	multiple	multiple	NOUN
ejpam-6059	271	6	of	of	ADP
ejpam-6059	271	7	p	p	PRON
ejpam-6059	271	8	which	which	PRON
ejpam-6059	271	9	directly	directly	ADV
ejpam-6059	271	10	implies	imply	VERB
ejpam-6059	271	11	that	that	SCONJ
ejpam-6059	271	12	gcd(x	gcd(x	PROPN
ejpam-6059	271	13	−	−	PROPN
ejpam-6059	271	14	y	y	PROPN
ejpam-6059	271	15	,	,	PUNCT
ejpam-6059	271	16	n	n	CCONJ
ejpam-6059	271	17	)	)	PUNCT
ejpam-6059	271	18	̸=	̸=	PROPN
ejpam-6059	271	19	1	1	NUM
ejpam-6059	271	20	,	,	PUNCT
ejpam-6059	271	21	that	that	ADV
ejpam-6059	271	22	is	is	ADV
ejpam-6059	271	23	,	,	PUNCT
ejpam-6059	271	24	{	{	PUNCT
ejpam-6059	271	25	x	x	NOUN
ejpam-6059	271	26	,	,	PUNCT
ejpam-6059	271	27	y	y	PROPN
ejpam-6059	271	28	}	}	PUNCT
ejpam-6059	271	29	/∈	/∈	PUNCT
ejpam-6059	271	30	e(γn	e(γn	NUM
ejpam-6059	271	31	)	)	PUNCT
ejpam-6059	271	32	.	.	PUNCT
ejpam-6059	272	1	it	it	PRON
ejpam-6059	272	2	follows	follow	VERB
ejpam-6059	272	3	that	that	SCONJ
ejpam-6059	272	4	γn[a	γn[a	PROPN
ejpam-6059	272	5	]	]	PUNCT
ejpam-6059	272	6	is	be	AUX
ejpam-6059	272	7	an	an	DET
ejpam-6059	272	8	empty	empty	ADJ
ejpam-6059	272	9	graph	graph	NOUN
ejpam-6059	272	10	.	.	PUNCT
ejpam-6059	273	1	now	now	ADV
ejpam-6059	273	2	,	,	PUNCT
ejpam-6059	273	3	let	let	VERB
ejpam-6059	273	4	v	v	ADP
ejpam-6059	273	5	∈	∈	PROPN
ejpam-6059	273	6	zn	zn	PROPN
ejpam-6059	273	7	\	\	PROPN
ejpam-6059	273	8	a.	a.	NOUN
ejpam-6059	274	1	it	it	PRON
ejpam-6059	274	2	is	be	AUX
ejpam-6059	274	3	not	not	PART
ejpam-6059	274	4	hard	hard	ADJ
ejpam-6059	274	5	to	to	PART
ejpam-6059	274	6	verify	verify	VERB
ejpam-6059	274	7	that	that	SCONJ
ejpam-6059	274	8	γn[a	γn[a	PROPN
ejpam-6059	274	9	∪	∪	ADV
ejpam-6059	274	10	{	{	PUNCT
ejpam-6059	274	11	v	v	NOUN
ejpam-6059	274	12	}	}	PUNCT
ejpam-6059	274	13	]	]	PUNCT
ejpam-6059	274	14	contains	contain	VERB
ejpam-6059	274	15	no	no	DET
ejpam-6059	274	16	cycles	cycle	NOUN
ejpam-6059	274	17	.	.	PUNCT
ejpam-6059	275	1	therefore	therefore	ADV
ejpam-6059	275	2	,	,	PUNCT
ejpam-6059	275	3	a	a	DET
ejpam-6059	275	4	∪	∪	ADJ
ejpam-6059	275	5	{	{	PUNCT
ejpam-6059	275	6	v	v	NOUN
ejpam-6059	275	7	}	}	PUNCT
ejpam-6059	275	8	is	be	AUX
ejpam-6059	275	9	an	an	DET
ejpam-6059	275	10	acyclic	acyclic	ADJ
ejpam-6059	275	11	set	set	VERB
ejpam-6059	275	12	in	in	ADP
ejpam-6059	275	13	γn	γn	NUM
ejpam-6059	275	14	.	.	PUNCT
ejpam-6059	276	1	hence	hence	ADV
ejpam-6059	276	2	λ(γn	λ(γn	NOUN
ejpam-6059	276	3	)	)	PUNCT
ejpam-6059	276	4	≥	≥	NOUN
ejpam-6059	276	5	|a	|a	VERB
ejpam-6059	276	6	∪	∪	X
ejpam-6059	276	7	{	{	PUNCT
ejpam-6059	276	8	v}|	v}|	NOUN
ejpam-6059	276	9	=	=	PUNCT
ejpam-6059	276	10	n	n	NOUN
ejpam-6059	276	11	p	p	NOUN
ejpam-6059	277	1	+	+	NOUN
ejpam-6059	277	2	1	1	X
ejpam-6059	277	3	.	.	X
ejpam-6059	277	4	for	for	ADP
ejpam-6059	277	5	proving	prove	VERB
ejpam-6059	277	6	the	the	DET
ejpam-6059	277	7	upper	upper	ADJ
ejpam-6059	277	8	bound	bound	NOUN
ejpam-6059	277	9	of	of	ADP
ejpam-6059	277	10	λ(γn	λ(γn	NOUN
ejpam-6059	277	11	)	)	PUNCT
ejpam-6059	277	12	,	,	PUNCT
ejpam-6059	277	13	let	let	VERB
ejpam-6059	277	14	b	b	X
ejpam-6059	277	15	be	be	AUX
ejpam-6059	277	16	any	any	DET
ejpam-6059	277	17	subset	subset	NOUN
ejpam-6059	277	18	of	of	ADP
ejpam-6059	277	19	zn	zn	PROPN
ejpam-6059	277	20	containing	contain	VERB
ejpam-6059	278	1	at	at	ADV
ejpam-6059	278	2	least	least	ADJ
ejpam-6059	278	3	n	n	CCONJ
ejpam-6059	278	4	−	−	ADP
ejpam-6059	278	5	φ(n	φ(n	ADJ
ejpam-6059	278	6	)	)	PUNCT
ejpam-6059	278	7	2	2	NUM
ejpam-6059	278	8	elements	element	NOUN
ejpam-6059	278	9	.	.	PUNCT
ejpam-6059	279	1	by	by	ADP
ejpam-6059	279	2	remark	remark	NOUN
ejpam-6059	279	3	1	1	NUM
ejpam-6059	279	4	,	,	PUNCT
ejpam-6059	279	5	deg(u	deg(u	PROPN
ejpam-6059	279	6	)	)	PUNCT
ejpam-6059	279	7	=	=	SYM
ejpam-6059	279	8	φ(n	φ(n	NOUN
ejpam-6059	279	9	)	)	PUNCT
ejpam-6059	279	10	for	for	ADP
ejpam-6059	279	11	all	all	DET
ejpam-6059	279	12	u	u	PROPN
ejpam-6059	279	13	∈	∈	PROPN
ejpam-6059	279	14	v	v	NOUN
ejpam-6059	279	15	(	(	PUNCT
ejpam-6059	279	16	γn	γn	NUM
ejpam-6059	279	17	)	)	PUNCT
ejpam-6059	279	18	,	,	PUNCT
ejpam-6059	279	19	we	we	PRON
ejpam-6059	279	20	obtain	obtain	VERB
ejpam-6059	279	21	that	that	DET
ejpam-6059	279	22	deg	deg	PROPN
ejpam-6059	279	23	γn[b](w	γn[b](w	PROPN
ejpam-6059	279	24	)	)	PUNCT
ejpam-6059	279	25	≥	≥	NOUN
ejpam-6059	279	26	φ(n	φ(n	VERB
ejpam-6059	279	27	)	)	PUNCT
ejpam-6059	279	28	2	2	NUM
ejpam-6059	279	29	for	for	ADP
ejpam-6059	279	30	all	all	DET
ejpam-6059	279	31	w	w	PROPN
ejpam-6059	279	32	∈	∈	PROPN
ejpam-6059	279	33	v	v	NOUN
ejpam-6059	279	34	(	(	PUNCT
ejpam-6059	279	35	γn[b	γn[b	PROPN
ejpam-6059	279	36	]	]	PUNCT
ejpam-6059	279	37	)	)	PUNCT
ejpam-6059	279	38	.	.	PUNCT
ejpam-6059	280	1	since	since	SCONJ
ejpam-6059	280	2	n	n	PROPN
ejpam-6059	280	3	≥	≥	NOUN
ejpam-6059	280	4	8	8	NUM
ejpam-6059	280	5	,	,	PUNCT
ejpam-6059	280	6	we	we	PRON
ejpam-6059	280	7	conclude	conclude	VERB
ejpam-6059	280	8	by	by	ADP
ejpam-6059	280	9	theorem	theorem	NOUN
ejpam-6059	280	10	1	1	NUM
ejpam-6059	280	11	,	,	PUNCT
ejpam-6059	280	12	that	that	DET
ejpam-6059	280	13	φ(n	φ(n	NOUN
ejpam-6059	280	14	)	)	PUNCT
ejpam-6059	280	15	≥	≥	NUM
ejpam-6059	280	16	4	4	NUM
ejpam-6059	280	17	which	which	PRON
ejpam-6059	280	18	leads	lead	VERB
ejpam-6059	280	19	to	to	PART
ejpam-6059	280	20	deg	deg	VERB
ejpam-6059	280	21	γn[b](w	γn[b](w	PROPN
ejpam-6059	280	22	)	)	PUNCT
ejpam-6059	280	23	≥	≥	NOUN
ejpam-6059	280	24	2	2	NUM
ejpam-6059	280	25	for	for	ADP
ejpam-6059	280	26	all	all	DET
ejpam-6059	280	27	w	w	PROPN
ejpam-6059	280	28	∈	∈	PROPN
ejpam-6059	280	29	v	v	NOUN
ejpam-6059	280	30	(	(	PUNCT
ejpam-6059	280	31	γn[b	γn[b	PROPN
ejpam-6059	280	32	]	]	PUNCT
ejpam-6059	280	33	)	)	PUNCT
ejpam-6059	280	34	.	.	PUNCT
ejpam-6059	281	1	by	by	ADP
ejpam-6059	281	2	lemma	lemma	PROPN
ejpam-6059	281	3	1	1	NUM
ejpam-6059	281	4	,	,	PUNCT
ejpam-6059	281	5	we	we	PRON
ejpam-6059	281	6	get	get	VERB
ejpam-6059	281	7	that	that	DET
ejpam-6059	281	8	γn[b	γn[b	NOUN
ejpam-6059	281	9	]	]	PUNCT
ejpam-6059	281	10	contains	contain	VERB
ejpam-6059	281	11	a	a	DET
ejpam-6059	281	12	cycle	cycle	NOUN
ejpam-6059	281	13	.	.	PUNCT
ejpam-6059	282	1	it	it	PRON
ejpam-6059	282	2	follows	follow	VERB
ejpam-6059	282	3	that	that	SCONJ
ejpam-6059	282	4	b	b	NOUN
ejpam-6059	282	5	is	be	AUX
ejpam-6059	282	6	not	not	PART
ejpam-6059	282	7	an	an	DET
ejpam-6059	282	8	acyclic	acyclic	ADJ
ejpam-6059	282	9	set	set	NOUN
ejpam-6059	282	10	of	of	ADP
ejpam-6059	282	11	γn	γn	NOUN
ejpam-6059	282	12	.	.	PUNCT
ejpam-6059	283	1	since	since	SCONJ
ejpam-6059	283	2	b	b	PROPN
ejpam-6059	283	3	is	be	AUX
ejpam-6059	283	4	arbitrary	arbitrary	ADJ
ejpam-6059	283	5	,	,	PUNCT
ejpam-6059	283	6	we	we	PRON
ejpam-6059	283	7	have	have	VERB
ejpam-6059	283	8	that	that	DET
ejpam-6059	283	9	λ(γn	λ(γn	NOUN
ejpam-6059	283	10	)	)	PUNCT
ejpam-6059	283	11	≤	≤	NOUN
ejpam-6059	283	12	|b|	|b|	VERB
ejpam-6059	283	13	−	−	PROPN
ejpam-6059	283	14	1	1	NUM
ejpam-6059	283	15	≤	≤	NOUN
ejpam-6059	283	16	(	(	PUNCT
ejpam-6059	283	17	n−	n−	NOUN
ejpam-6059	283	18	φ(n	φ(n	ADJ
ejpam-6059	283	19	)	)	PUNCT
ejpam-6059	283	20	2	2	NUM
ejpam-6059	283	21	)	)	PUNCT
ejpam-6059	284	1	−	−	NOUN
ejpam-6059	285	1	1	1	NUM
ejpam-6059	286	1	=	=	SYM
ejpam-6059	287	1	n−	n−	PROPN
ejpam-6059	288	1	(	(	PUNCT
ejpam-6059	289	1	φ(n	φ(n	ADJ
ejpam-6059	289	2	)	)	PUNCT
ejpam-6059	289	3	2	2	NUM
ejpam-6059	290	1	+	+	SYM
ejpam-6059	290	2	1	1	NUM
ejpam-6059	290	3	)	)	PUNCT
ejpam-6059	290	4	.	.	PUNCT
ejpam-6059	291	1	then	then	ADV
ejpam-6059	291	2	the	the	DET
ejpam-6059	291	3	statement	statement	NOUN
ejpam-6059	291	4	is	be	AUX
ejpam-6059	291	5	proved	prove	VERB
ejpam-6059	291	6	,	,	PUNCT
ejpam-6059	291	7	as	as	SCONJ
ejpam-6059	291	8	required	require	VERB
ejpam-6059	291	9	.	.	PUNCT
ejpam-6059	292	1	d.	d.	PROPN
ejpam-6059	292	2	pongpipat	pongpipat	PROPN
ejpam-6059	292	3	,	,	PUNCT
ejpam-6059	292	4	n.	n.	NOUN
ejpam-6059	292	5	nupo	nupo	PROPN
ejpam-6059	292	6	/	/	SYM
ejpam-6059	292	7	eur	eur	PROPN
ejpam-6059	292	8	.	.	PUNCT
ejpam-6059	293	1	j.	j.	PROPN
ejpam-6059	293	2	pure	pure	PROPN
ejpam-6059	293	3	appl	appl	PROPN
ejpam-6059	293	4	.	.	PROPN
ejpam-6059	293	5	math	math	PROPN
ejpam-6059	293	6	,	,	PUNCT
ejpam-6059	293	7	18	18	NUM
ejpam-6059	293	8	(	(	PUNCT
ejpam-6059	293	9	2	2	NUM
ejpam-6059	293	10	)	)	PUNCT
ejpam-6059	293	11	(	(	PUNCT
ejpam-6059	293	12	2025	2025	NUM
ejpam-6059	293	13	)	)	PUNCT
ejpam-6059	293	14	,	,	PUNCT
ejpam-6059	293	15	6059	6059	NUM
ejpam-6059	293	16	9	9	NUM
ejpam-6059	293	17	of	of	ADP
ejpam-6059	293	18	11	11	NUM
ejpam-6059	293	19	we	we	PRON
ejpam-6059	293	20	now	now	ADV
ejpam-6059	293	21	consider	consider	VERB
ejpam-6059	293	22	the	the	DET
ejpam-6059	293	23	unitary	unitary	ADJ
ejpam-6059	293	24	cayley	cayley	NOUN
ejpam-6059	293	25	graph	graph	NOUN
ejpam-6059	293	26	γ8	γ8	NOUN
ejpam-6059	293	27	as	as	SCONJ
ejpam-6059	293	28	shown	show	VERB
ejpam-6059	293	29	in	in	ADP
ejpam-6059	293	30	the	the	DET
ejpam-6059	293	31	following	follow	VERB
ejpam-6059	293	32	example	example	NOUN
ejpam-6059	293	33	.	.	PUNCT
ejpam-6059	294	1	actually	actually	ADV
ejpam-6059	294	2	,	,	PUNCT
ejpam-6059	294	3	the	the	DET
ejpam-6059	294	4	lower	lower	ADV
ejpam-6059	294	5	bound	bind	VERB
ejpam-6059	294	6	and	and	CCONJ
ejpam-6059	294	7	upper	upper	ADJ
ejpam-6059	294	8	bound	bind	VERB
ejpam-6059	294	9	in	in	ADP
ejpam-6059	294	10	theorem	theorem	ADJ
ejpam-6059	294	11	9	9	NUM
ejpam-6059	294	12	are	be	AUX
ejpam-6059	294	13	sharp	sharp	ADJ
ejpam-6059	294	14	.	.	PUNCT
ejpam-6059	294	15	example	example	NOUN
ejpam-6059	295	1	7	7	X
ejpam-6059	295	2	.	.	X
ejpam-6059	295	3	consider	consider	VERB
ejpam-6059	295	4	the	the	DET
ejpam-6059	295	5	unitary	unitary	ADJ
ejpam-6059	295	6	cayley	cayley	NOUN
ejpam-6059	295	7	graph	graph	NOUN
ejpam-6059	295	8	γ8	γ8	NOUN
ejpam-6059	295	9	follows	follow	VERB
ejpam-6059	295	10	.	.	PUNCT
ejpam-6059	296	1	figure	figure	VERB
ejpam-6059	296	2	5	5	NUM
ejpam-6059	296	3	:	:	PUNCT
ejpam-6059	296	4	the	the	DET
ejpam-6059	296	5	unitary	unitary	ADJ
ejpam-6059	296	6	cayley	cayley	NOUN
ejpam-6059	296	7	graph	graph	NOUN
ejpam-6059	296	8	γ8	γ8	NOUN
ejpam-6059	296	9	by	by	ADP
ejpam-6059	296	10	theorem	theorem	NOUN
ejpam-6059	296	11	8	8	NUM
ejpam-6059	296	12	,	,	PUNCT
ejpam-6059	296	13	we	we	PRON
ejpam-6059	296	14	obtain	obtain	VERB
ejpam-6059	296	15	that	that	DET
ejpam-6059	296	16	λ(γ8	λ(γ8	NOUN
ejpam-6059	296	17	)	)	PUNCT
ejpam-6059	297	1	=	=	SYM
ejpam-6059	298	1	23−1	23−1	NUM
ejpam-6059	298	2	+1	+1	NOUN
ejpam-6059	298	3	=	=	SYM
ejpam-6059	298	4	5	5	NUM
ejpam-6059	298	5	=	=	SYM
ejpam-6059	298	6	8	8	NUM
ejpam-6059	298	7	2	2	NUM
ejpam-6059	298	8	+	+	SYM
ejpam-6059	298	9	1	1	NUM
ejpam-6059	298	10	which	which	PRON
ejpam-6059	298	11	achieves	achieve	VERB
ejpam-6059	298	12	the	the	DET
ejpam-6059	298	13	lower	lower	ADV
ejpam-6059	298	14	bound	bind	VERB
ejpam-6059	298	15	in	in	ADP
ejpam-6059	298	16	theorem	theorem	NOUN
ejpam-6059	298	17	9	9	NUM
ejpam-6059	298	18	.	.	PUNCT
ejpam-6059	299	1	moreover	moreover	ADV
ejpam-6059	299	2	,	,	PUNCT
ejpam-6059	299	3	we	we	PRON
ejpam-6059	299	4	can	can	AUX
ejpam-6059	299	5	observe	observe	VERB
ejpam-6059	299	6	that	that	DET
ejpam-6059	299	7	λ(γ8	λ(γ8	NOUN
ejpam-6059	299	8	)	)	PUNCT
ejpam-6059	299	9	=	=	SYM
ejpam-6059	299	10	5	5	NUM
ejpam-6059	299	11	=	=	SYM
ejpam-6059	299	12	8−	8−	PROPN
ejpam-6059	299	13	(	(	PUNCT
ejpam-6059	299	14	φ(8	φ(8	NOUN
ejpam-6059	299	15	)	)	PUNCT
ejpam-6059	299	16	2	2	NUM
ejpam-6059	300	1	+	+	CCONJ
ejpam-6059	300	2	1	1	NUM
ejpam-6059	300	3	)	)	PUNCT
ejpam-6059	300	4	which	which	PRON
ejpam-6059	300	5	attains	attain	VERB
ejpam-6059	300	6	the	the	DET
ejpam-6059	300	7	upper	upper	ADJ
ejpam-6059	300	8	bound	bind	VERB
ejpam-6059	300	9	mentioned	mention	VERB
ejpam-6059	300	10	in	in	ADP
ejpam-6059	300	11	theorem	theorem	NOUN
ejpam-6059	300	12	9	9	NUM
ejpam-6059	300	13	.	.	PUNCT
ejpam-6059	300	14	theorem	theorem	NOUN
ejpam-6059	300	15	10	10	NUM
ejpam-6059	300	16	.	.	PUNCT
ejpam-6059	301	1	let	let	VERB
ejpam-6059	301	2	n	n	NOUN
ejpam-6059	301	3	=	=	SYM
ejpam-6059	301	4	pk	pk	NOUN
ejpam-6059	301	5	be	be	AUX
ejpam-6059	301	6	such	such	ADJ
ejpam-6059	301	7	that	that	SCONJ
ejpam-6059	301	8	p	p	NOUN
ejpam-6059	301	9	is	be	AUX
ejpam-6059	301	10	prime	prime	ADJ
ejpam-6059	301	11	and	and	CCONJ
ejpam-6059	301	12	k	k	PROPN
ejpam-6059	301	13	∈	∈	PROPN
ejpam-6059	301	14	n	n	PRON
ejpam-6059	301	15	\	\	NOUN
ejpam-6059	301	16	{	{	PUNCT
ejpam-6059	301	17	1	1	NUM
ejpam-6059	301	18	}	}	PUNCT
ejpam-6059	301	19	.	.	PUNCT
ejpam-6059	302	1	then	then	ADV
ejpam-6059	302	2	λ(γn	λ(γn	NOUN
ejpam-6059	302	3	)	)	PUNCT
ejpam-6059	302	4	=	=	SYM
ejpam-6059	302	5	2p	2p	NOUN
ejpam-6059	302	6	.	.	PUNCT
ejpam-6059	303	1	proof	proof	NOUN
ejpam-6059	303	2	.	.	PUNCT
ejpam-6059	304	1	by	by	ADP
ejpam-6059	304	2	theorem	theorem	NOUN
ejpam-6059	304	3	3	3	NUM
ejpam-6059	304	4	,	,	PUNCT
ejpam-6059	304	5	we	we	PRON
ejpam-6059	304	6	obtain	obtain	VERB
ejpam-6059	304	7	that	that	SCONJ
ejpam-6059	304	8	γn	γn	ADJ
ejpam-6059	304	9	is	be	AUX
ejpam-6059	304	10	decomposed	decompose	VERB
ejpam-6059	304	11	into	into	ADP
ejpam-6059	304	12	p	p	NOUN
ejpam-6059	304	13	complete	complete	ADJ
ejpam-6059	304	14	graphs	graph	NOUN
ejpam-6059	304	15	of	of	ADP
ejpam-6059	304	16	order	order	NOUN
ejpam-6059	304	17	pk−1	pk−1	PROPN
ejpam-6059	304	18	.	.	PUNCT
ejpam-6059	305	1	choose	choose	VERB
ejpam-6059	305	2	two	two	NUM
ejpam-6059	305	3	vertices	vertex	NOUN
ejpam-6059	305	4	of	of	ADP
ejpam-6059	305	5	each	each	DET
ejpam-6059	305	6	complete	complete	ADJ
ejpam-6059	305	7	graph	graph	NOUN
ejpam-6059	305	8	.	.	PUNCT
ejpam-6059	306	1	we	we	PRON
ejpam-6059	306	2	get	get	VERB
ejpam-6059	306	3	that	that	SCONJ
ejpam-6059	306	4	those	those	DET
ejpam-6059	306	5	vertices	vertex	NOUN
ejpam-6059	306	6	form	form	VERB
ejpam-6059	306	7	an	an	DET
ejpam-6059	306	8	acyclic	acyclic	ADJ
ejpam-6059	306	9	set	set	NOUN
ejpam-6059	306	10	of	of	ADP
ejpam-6059	306	11	γn	γn	NUM
ejpam-6059	306	12	.	.	PUNCT
ejpam-6059	307	1	moreover	moreover	ADV
ejpam-6059	307	2	,	,	PUNCT
ejpam-6059	307	3	we	we	PRON
ejpam-6059	307	4	can	can	AUX
ejpam-6059	307	5	easily	easily	ADV
ejpam-6059	307	6	observe	observe	VERB
ejpam-6059	307	7	that	that	SCONJ
ejpam-6059	307	8	there	there	PRON
ejpam-6059	307	9	is	be	VERB
ejpam-6059	307	10	no	no	DET
ejpam-6059	307	11	acyclic	acyclic	ADJ
ejpam-6059	307	12	set	set	NOUN
ejpam-6059	307	13	containing	contain	VERB
ejpam-6059	307	14	more	more	ADJ
ejpam-6059	307	15	than	than	ADP
ejpam-6059	307	16	two	two	NUM
ejpam-6059	307	17	vertices	vertex	NOUN
ejpam-6059	307	18	from	from	ADP
ejpam-6059	307	19	one	one	NUM
ejpam-6059	307	20	complete	complete	ADJ
ejpam-6059	307	21	graph	graph	NOUN
ejpam-6059	307	22	.	.	PUNCT
ejpam-6059	308	1	therefore	therefore	ADV
ejpam-6059	308	2	,	,	PUNCT
ejpam-6059	308	3	λ(γn	λ(γn	NOUN
ejpam-6059	308	4	)	)	PUNCT
ejpam-6059	308	5	=	=	SYM
ejpam-6059	308	6	2p	2p	NOUN
ejpam-6059	308	7	.	.	PUNCT
ejpam-6059	309	1	we	we	PRON
ejpam-6059	309	2	mention	mention	VERB
ejpam-6059	309	3	a	a	DET
ejpam-6059	309	4	certain	certain	ADJ
ejpam-6059	309	5	prominent	prominent	ADJ
ejpam-6059	309	6	fact	fact	NOUN
ejpam-6059	309	7	that	that	SCONJ
ejpam-6059	309	8	we	we	PRON
ejpam-6059	309	9	will	will	AUX
ejpam-6059	309	10	use	use	VERB
ejpam-6059	309	11	in	in	ADP
ejpam-6059	309	12	the	the	DET
ejpam-6059	309	13	next	next	ADJ
ejpam-6059	309	14	theorem	theorem	NOUN
ejpam-6059	309	15	.	.	PUNCT
ejpam-6059	310	1	in	in	ADP
ejpam-6059	310	2	combinatorics	combinatoric	NOUN
ejpam-6059	310	3	,	,	PUNCT
ejpam-6059	310	4	the	the	DET
ejpam-6059	310	5	pigeonhole	pigeonhole	NOUN
ejpam-6059	310	6	principle	principle	NOUN
ejpam-6059	310	7	states	state	NOUN
ejpam-6059	310	8	that	that	SCONJ
ejpam-6059	310	9	if	if	SCONJ
ejpam-6059	310	10	n	n	NUM
ejpam-6059	310	11	items	item	NOUN
ejpam-6059	310	12	are	be	AUX
ejpam-6059	310	13	put	put	VERB
ejpam-6059	310	14	into	into	ADP
ejpam-6059	310	15	m	m	PROPN
ejpam-6059	310	16	containers	container	NOUN
ejpam-6059	310	17	with	with	ADP
ejpam-6059	310	18	n	n	PROPN
ejpam-6059	310	19	>	>	X
ejpam-6059	310	20	m	m	PROPN
ejpam-6059	310	21	,	,	PUNCT
ejpam-6059	310	22	then	then	ADV
ejpam-6059	310	23	at	at	ADV
ejpam-6059	310	24	least	least	ADV
ejpam-6059	310	25	one	one	NUM
ejpam-6059	310	26	container	container	NOUN
ejpam-6059	310	27	must	must	AUX
ejpam-6059	310	28	contain	contain	VERB
ejpam-6059	310	29	more	more	ADJ
ejpam-6059	310	30	than	than	ADP
ejpam-6059	310	31	one	one	NUM
ejpam-6059	310	32	item	item	NOUN
ejpam-6059	310	33	.	.	PUNCT
ejpam-6059	311	1	theorem	theorem	NOUN
ejpam-6059	311	2	11	11	NUM
ejpam-6059	311	3	.	.	PUNCT
ejpam-6059	312	1	let	let	VERB
ejpam-6059	312	2	n	n	PRON
ejpam-6059	312	3	be	be	AUX
ejpam-6059	312	4	an	an	DET
ejpam-6059	312	5	even	even	ADV
ejpam-6059	312	6	positive	positive	ADJ
ejpam-6059	312	7	integer	integer	NOUN
ejpam-6059	312	8	such	such	DET
ejpam-6059	312	9	that	that	SCONJ
ejpam-6059	312	10	n	n	CCONJ
ejpam-6059	312	11	≥	≥	NOUN
ejpam-6059	312	12	4	4	NUM
ejpam-6059	312	13	.	.	PUNCT
ejpam-6059	313	1	then	then	ADV
ejpam-6059	313	2	λ(γn	λ(γn	NOUN
ejpam-6059	313	3	)	)	PUNCT
ejpam-6059	313	4	=	=	SYM
ejpam-6059	313	5	4	4	X
ejpam-6059	313	6	.	.	X
ejpam-6059	313	7	proof	proof	NOUN
ejpam-6059	313	8	.	.	PUNCT
ejpam-6059	314	1	let	let	VERB
ejpam-6059	314	2	n	n	PRON
ejpam-6059	314	3	be	be	AUX
ejpam-6059	314	4	an	an	DET
ejpam-6059	314	5	even	even	ADV
ejpam-6059	314	6	positive	positive	ADJ
ejpam-6059	314	7	integer	integer	NOUN
ejpam-6059	314	8	such	such	DET
ejpam-6059	314	9	that	that	SCONJ
ejpam-6059	314	10	n	n	CCONJ
ejpam-6059	314	11	≥	≥	NOUN
ejpam-6059	314	12	4	4	NUM
ejpam-6059	314	13	.	.	PUNCT
ejpam-6059	314	14	further	far	ADV
ejpam-6059	314	15	,	,	PUNCT
ejpam-6059	314	16	let	let	VERB
ejpam-6059	314	17	a	a	PRON
ejpam-6059	314	18	=	=	PUNCT
ejpam-6059	314	19	{	{	PUNCT
ejpam-6059	314	20	0	0	NUM
ejpam-6059	314	21	,	,	PUNCT
ejpam-6059	314	22	1	1	NUM
ejpam-6059	314	23	,	,	PUNCT
ejpam-6059	314	24	2	2	NUM
ejpam-6059	314	25	,	,	PUNCT
ejpam-6059	314	26	3	3	NUM
ejpam-6059	314	27	}	}	PUNCT
ejpam-6059	314	28	.	.	PUNCT
ejpam-6059	315	1	clearly	clearly	ADV
ejpam-6059	315	2	,	,	PUNCT
ejpam-6059	315	3	γn[a	γn[a	PROPN
ejpam-6059	315	4	]	]	PUNCT
ejpam-6059	315	5	contains	contain	VERB
ejpam-6059	315	6	no	no	DET
ejpam-6059	315	7	cycles	cycle	NOUN
ejpam-6059	315	8	.	.	PUNCT
ejpam-6059	316	1	hence	hence	ADV
ejpam-6059	316	2	,	,	PUNCT
ejpam-6059	316	3	a	a	PRON
ejpam-6059	316	4	is	be	AUX
ejpam-6059	316	5	an	an	DET
ejpam-6059	316	6	acyclic	acyclic	ADJ
ejpam-6059	316	7	set	set	NOUN
ejpam-6059	316	8	of	of	ADP
ejpam-6059	316	9	γn	γn	NUM
ejpam-6059	316	10	which	which	PRON
ejpam-6059	316	11	implies	imply	VERB
ejpam-6059	316	12	that	that	SCONJ
ejpam-6059	316	13	λ(γn	λ(γn	NOUN
ejpam-6059	316	14	)	)	PUNCT
ejpam-6059	316	15	≥	≥	NOUN
ejpam-6059	316	16	4	4	NUM
ejpam-6059	316	17	.	.	PUNCT
ejpam-6059	316	18	consider	consider	VERB
ejpam-6059	316	19	the	the	DET
ejpam-6059	316	20	set	set	NOUN
ejpam-6059	316	21	of	of	ADP
ejpam-6059	316	22	any	any	DET
ejpam-6059	316	23	five	five	NUM
ejpam-6059	316	24	vertices	vertex	NOUN
ejpam-6059	316	25	of	of	ADP
ejpam-6059	316	26	γn	γn	NUM
ejpam-6059	316	27	,	,	PUNCT
ejpam-6059	316	28	say	say	VERB
ejpam-6059	316	29	b	b	NOUN
ejpam-6059	316	30	=	=	NOUN
ejpam-6059	316	31	{	{	PUNCT
ejpam-6059	316	32	v1	v1	PROPN
ejpam-6059	316	33	,	,	PUNCT
ejpam-6059	316	34	v2	v2	PROPN
ejpam-6059	316	35	,	,	PUNCT
ejpam-6059	316	36	v3	v3	PROPN
ejpam-6059	316	37	,	,	PUNCT
ejpam-6059	316	38	v4	v4	PROPN
ejpam-6059	316	39	,	,	PUNCT
ejpam-6059	316	40	v5	v5	PROPN
ejpam-6059	316	41	}	}	PUNCT
ejpam-6059	316	42	.	.	PUNCT
ejpam-6059	317	1	by	by	ADP
ejpam-6059	317	2	the	the	DET
ejpam-6059	317	3	pigeonhole	pigeonhole	PROPN
ejpam-6059	317	4	principle	principle	NOUN
ejpam-6059	317	5	,	,	PUNCT
ejpam-6059	317	6	there	there	PRON
ejpam-6059	317	7	exist	exist	VERB
ejpam-6059	317	8	at	at	ADV
ejpam-6059	317	9	least	least	ADV
ejpam-6059	317	10	three	three	NUM
ejpam-6059	317	11	vertices	vertex	NOUN
ejpam-6059	317	12	vi	vi	PROPN
ejpam-6059	317	13	,	,	PUNCT
ejpam-6059	317	14	vj	vj	INTJ
ejpam-6059	317	15	,	,	PUNCT
ejpam-6059	317	16	vk	vk	ADP
ejpam-6059	317	17	∈	∈	PROPN
ejpam-6059	317	18	b	b	NOUN
ejpam-6059	317	19	such	such	ADJ
ejpam-6059	317	20	that	that	SCONJ
ejpam-6059	317	21	they	they	PRON
ejpam-6059	317	22	are	be	AUX
ejpam-6059	317	23	all	all	ADV
ejpam-6059	317	24	even	even	ADV
ejpam-6059	317	25	or	or	CCONJ
ejpam-6059	317	26	odd	odd	ADJ
ejpam-6059	317	27	.	.	PUNCT
ejpam-6059	318	1	hence	hence	ADV
ejpam-6059	318	2	the	the	DET
ejpam-6059	318	3	difference	difference	NOUN
ejpam-6059	318	4	between	between	ADP
ejpam-6059	318	5	any	any	DET
ejpam-6059	318	6	two	two	NUM
ejpam-6059	318	7	vertices	vertex	NOUN
ejpam-6059	318	8	of	of	ADP
ejpam-6059	318	9	vi	vi	PROPN
ejpam-6059	318	10	,	,	PUNCT
ejpam-6059	318	11	vj	vj	INTJ
ejpam-6059	318	12	,	,	PUNCT
ejpam-6059	318	13	vk	vk	PROPN
ejpam-6059	318	14	must	must	AUX
ejpam-6059	318	15	be	be	AUX
ejpam-6059	318	16	even	even	ADV
ejpam-6059	318	17	.	.	PUNCT
ejpam-6059	319	1	then	then	ADV
ejpam-6059	319	2	{	{	PUNCT
ejpam-6059	319	3	vl	vl	PROPN
ejpam-6059	319	4	,	,	PUNCT
ejpam-6059	319	5	vm	vm	NOUN
ejpam-6059	319	6	}	}	PUNCT
ejpam-6059	319	7	/∈	/∈	PUNCT
ejpam-6059	319	8	e(γn	e(γn	NOUN
ejpam-6059	319	9	)	)	PUNCT
ejpam-6059	319	10	for	for	ADP
ejpam-6059	319	11	all	all	DET
ejpam-6059	319	12	l	l	NOUN
ejpam-6059	319	13	,	,	PUNCT
ejpam-6059	319	14	m	m	VERB
ejpam-6059	319	15	∈	∈	ADJ
ejpam-6059	319	16	{	{	PUNCT
ejpam-6059	319	17	i	i	PROPN
ejpam-6059	319	18	,	,	PUNCT
ejpam-6059	319	19	j	j	PROPN
ejpam-6059	319	20	,	,	PUNCT
ejpam-6059	319	21	k	k	NOUN
ejpam-6059	319	22	}	}	PUNCT
ejpam-6059	319	23	.	.	PUNCT
ejpam-6059	320	1	thus	thus	ADV
ejpam-6059	320	2	,	,	PUNCT
ejpam-6059	320	3	{	{	PUNCT
ejpam-6059	320	4	vl	vl	PROPN
ejpam-6059	320	5	,	,	PUNCT
ejpam-6059	320	6	vm	vm	NOUN
ejpam-6059	320	7	}	}	PUNCT
ejpam-6059	320	8	∈	∈	NOUN
ejpam-6059	320	9	e(γn	e(γn	NOUN
ejpam-6059	320	10	)	)	PUNCT
ejpam-6059	320	11	for	for	ADP
ejpam-6059	320	12	all	all	DET
ejpam-6059	320	13	l	l	NOUN
ejpam-6059	320	14	,	,	PUNCT
ejpam-6059	320	15	m	m	VERB
ejpam-6059	320	16	∈	∈	ADJ
ejpam-6059	320	17	{	{	PUNCT
ejpam-6059	320	18	i	i	PROPN
ejpam-6059	320	19	,	,	PUNCT
ejpam-6059	320	20	j	j	PROPN
ejpam-6059	320	21	,	,	PUNCT
ejpam-6059	320	22	k	k	NOUN
ejpam-6059	320	23	}	}	PUNCT
ejpam-6059	320	24	.	.	PUNCT
ejpam-6059	321	1	it	it	PRON
ejpam-6059	321	2	follows	follow	VERB
ejpam-6059	321	3	that	that	SCONJ
ejpam-6059	321	4	γn[{vi	γn[{vi	PUNCT
ejpam-6059	321	5	,	,	PUNCT
ejpam-6059	321	6	vj	vj	INTJ
ejpam-6059	321	7	,	,	PUNCT
ejpam-6059	321	8	vk	vk	PROPN
ejpam-6059	321	9	}	}	PUNCT
ejpam-6059	321	10	]	]	PUNCT
ejpam-6059	321	11	is	be	AUX
ejpam-6059	321	12	a	a	DET
ejpam-6059	321	13	cycle	cycle	NOUN
ejpam-6059	321	14	of	of	ADP
ejpam-6059	321	15	length	length	NOUN
ejpam-6059	321	16	3	3	NUM
ejpam-6059	321	17	contained	contain	VERB
ejpam-6059	321	18	in	in	ADP
ejpam-6059	321	19	γn[b	γn[b	PROPN
ejpam-6059	321	20	]	]	PUNCT
ejpam-6059	321	21	.	.	PUNCT
ejpam-6059	322	1	therefore	therefore	ADV
ejpam-6059	322	2	,	,	PUNCT
ejpam-6059	322	3	b	b	PROPN
ejpam-6059	322	4	is	be	AUX
ejpam-6059	322	5	not	not	PART
ejpam-6059	322	6	acyclic	acyclic	ADJ
ejpam-6059	322	7	in	in	ADP
ejpam-6059	322	8	γn	γn	NUM
ejpam-6059	322	9	.	.	PUNCT
ejpam-6059	323	1	since	since	SCONJ
ejpam-6059	323	2	b	b	PROPN
ejpam-6059	323	3	is	be	AUX
ejpam-6059	323	4	arbitrary	arbitrary	ADJ
ejpam-6059	323	5	,	,	PUNCT
ejpam-6059	323	6	we	we	PRON
ejpam-6059	323	7	can	can	AUX
ejpam-6059	323	8	conclude	conclude	VERB
ejpam-6059	323	9	that	that	SCONJ
ejpam-6059	323	10	λ(γn	λ(γn	NOUN
ejpam-6059	323	11	)	)	PUNCT
ejpam-6059	323	12	<	<	X
ejpam-6059	323	13	|b|	|b|	PROPN
ejpam-6059	323	14	=	=	SYM
ejpam-6059	323	15	5	5	X
ejpam-6059	323	16	.	.	PUNCT
ejpam-6059	323	17	thus	thus	ADV
ejpam-6059	323	18	λ(γn	λ(γn	VERB
ejpam-6059	323	19	)	)	PUNCT
ejpam-6059	323	20	=	=	SYM
ejpam-6059	323	21	4	4	NUM
ejpam-6059	323	22	,	,	PUNCT
ejpam-6059	323	23	as	as	SCONJ
ejpam-6059	323	24	required	require	VERB
ejpam-6059	323	25	.	.	PUNCT
ejpam-6059	324	1	d.	d.	PROPN
ejpam-6059	324	2	pongpipat	pongpipat	PROPN
ejpam-6059	324	3	,	,	PUNCT
ejpam-6059	324	4	n.	n.	NOUN
ejpam-6059	324	5	nupo	nupo	PROPN
ejpam-6059	324	6	/	/	SYM
ejpam-6059	324	7	eur	eur	PROPN
ejpam-6059	324	8	.	.	PUNCT
ejpam-6059	325	1	j.	j.	PROPN
ejpam-6059	325	2	pure	pure	PROPN
ejpam-6059	325	3	appl	appl	PROPN
ejpam-6059	325	4	.	.	PROPN
ejpam-6059	325	5	math	math	PROPN
ejpam-6059	325	6	,	,	PUNCT
ejpam-6059	325	7	18	18	NUM
ejpam-6059	325	8	(	(	PUNCT
ejpam-6059	325	9	2	2	NUM
ejpam-6059	325	10	)	)	PUNCT
ejpam-6059	325	11	(	(	PUNCT
ejpam-6059	325	12	2025	2025	NUM
ejpam-6059	325	13	)	)	PUNCT
ejpam-6059	325	14	,	,	PUNCT
ejpam-6059	325	15	6059	6059	NUM
ejpam-6059	325	16	10	10	NUM
ejpam-6059	325	17	of	of	ADP
ejpam-6059	325	18	11	11	NUM
ejpam-6059	325	19	theorem	theorem	NOUN
ejpam-6059	325	20	12	12	NUM
ejpam-6059	325	21	.	.	PUNCT
ejpam-6059	326	1	let	let	VERB
ejpam-6059	326	2	n	n	PRON
ejpam-6059	326	3	be	be	AUX
ejpam-6059	326	4	an	an	DET
ejpam-6059	326	5	odd	odd	ADJ
ejpam-6059	326	6	positive	positive	ADJ
ejpam-6059	326	7	integer	integer	NOUN
ejpam-6059	326	8	such	such	DET
ejpam-6059	326	9	that	that	SCONJ
ejpam-6059	326	10	n	n	NUM
ejpam-6059	326	11	≥	≥	NOUN
ejpam-6059	326	12	5	5	NUM
ejpam-6059	326	13	and	and	CCONJ
ejpam-6059	326	14	n	n	NOUN
ejpam-6059	326	15	is	be	AUX
ejpam-6059	326	16	not	not	PART
ejpam-6059	326	17	prime	prime	ADJ
ejpam-6059	326	18	.	.	PUNCT
ejpam-6059	327	1	if	if	SCONJ
ejpam-6059	327	2	p	p	NOUN
ejpam-6059	327	3	is	be	AUX
ejpam-6059	327	4	the	the	DET
ejpam-6059	327	5	least	least	ADJ
ejpam-6059	327	6	prime	prime	ADJ
ejpam-6059	327	7	divisor	divisor	NOUN
ejpam-6059	327	8	of	of	ADP
ejpam-6059	327	9	n	n	CCONJ
ejpam-6059	327	10	,	,	PUNCT
ejpam-6059	327	11	then	then	ADV
ejpam-6059	327	12	p+	p+	VERB
ejpam-6059	327	13	3	3	NUM
ejpam-6059	327	14	≤	≤	NUM
ejpam-6059	327	15	λ(γn	λ(γn	NOUN
ejpam-6059	327	16	)	)	PUNCT
ejpam-6059	327	17	≤	≤	NUM
ejpam-6059	327	18	2p	2p	NOUN
ejpam-6059	327	19	.	.	PUNCT
ejpam-6059	328	1	proof	proof	NOUN
ejpam-6059	328	2	.	.	PUNCT
ejpam-6059	329	1	let	let	VERB
ejpam-6059	329	2	n	n	PRON
ejpam-6059	329	3	be	be	AUX
ejpam-6059	329	4	an	an	DET
ejpam-6059	329	5	odd	odd	ADJ
ejpam-6059	329	6	positive	positive	ADJ
ejpam-6059	329	7	integer	integer	NOUN
ejpam-6059	329	8	such	such	DET
ejpam-6059	329	9	that	that	SCONJ
ejpam-6059	329	10	n	n	PROPN
ejpam-6059	329	11	>	>	SYM
ejpam-6059	329	12	5	5	NUM
ejpam-6059	329	13	and	and	CCONJ
ejpam-6059	329	14	n	n	NOUN
ejpam-6059	329	15	is	be	AUX
ejpam-6059	329	16	not	not	PART
ejpam-6059	329	17	prime	prime	ADJ
ejpam-6059	329	18	.	.	PUNCT
ejpam-6059	330	1	assume	assume	VERB
ejpam-6059	330	2	that	that	SCONJ
ejpam-6059	330	3	n	n	PRON
ejpam-6059	330	4	=	=	SYM
ejpam-6059	330	5	pm1	pm1	PROPN
ejpam-6059	330	6	1	1	NUM
ejpam-6059	330	7	·	·	PUNCT
ejpam-6059	330	8	pm2	pm2	NOUN
ejpam-6059	330	9	2	2	NUM
ejpam-6059	330	10	·	·	PUNCT
ejpam-6059	330	11	·	·	PUNCT
ejpam-6059	330	12	·	·	PUNCT
ejpam-6059	331	1	pmt	pmt	PROPN
ejpam-6059	331	2	t	t	PROPN
ejpam-6059	331	3	where	where	SCONJ
ejpam-6059	331	4	pi	pi	NOUN
ejpam-6059	331	5	and	and	CCONJ
ejpam-6059	331	6	mi	mi	PROPN
ejpam-6059	331	7	are	be	AUX
ejpam-6059	331	8	prime	prime	ADJ
ejpam-6059	331	9	and	and	CCONJ
ejpam-6059	331	10	positive	positive	ADJ
ejpam-6059	331	11	numbers	number	NOUN
ejpam-6059	331	12	,	,	PUNCT
ejpam-6059	331	13	respectively	respectively	ADV
ejpam-6059	331	14	such	such	ADJ
ejpam-6059	331	15	that	that	SCONJ
ejpam-6059	331	16	i	i	PRON
ejpam-6059	331	17	=	=	NOUN
ejpam-6059	331	18	1	1	NUM
ejpam-6059	331	19	,	,	PUNCT
ejpam-6059	331	20	2	2	NUM
ejpam-6059	331	21	,	,	PUNCT
ejpam-6059	331	22	.	.	PUNCT
ejpam-6059	331	23	.	.	PUNCT
ejpam-6059	332	1	.	.	PUNCT
ejpam-6059	333	1	,	,	PUNCT
ejpam-6059	333	2	t	t	PROPN
ejpam-6059	333	3	and	and	CCONJ
ejpam-6059	333	4	pi	pi	NOUN
ejpam-6059	333	5	<	<	X
ejpam-6059	333	6	pj	pj	PROPN
ejpam-6059	333	7	for	for	ADP
ejpam-6059	333	8	i	i	PRON
ejpam-6059	333	9	<	<	X
ejpam-6059	333	10	j.	j.	PROPN
ejpam-6059	333	11	let	let	VERB
ejpam-6059	333	12	p	p	PRON
ejpam-6059	333	13	be	be	AUX
ejpam-6059	333	14	the	the	DET
ejpam-6059	333	15	least	least	ADJ
ejpam-6059	333	16	prime	prime	ADJ
ejpam-6059	333	17	divisor	divisor	NOUN
ejpam-6059	333	18	of	of	ADP
ejpam-6059	333	19	n	n	PROPN
ejpam-6059	333	20	and	and	CCONJ
ejpam-6059	333	21	a	a	DET
ejpam-6059	333	22	=	=	X
ejpam-6059	333	23	{	{	PUNCT
ejpam-6059	333	24	0	0	NUM
ejpam-6059	333	25	,	,	PUNCT
ejpam-6059	333	26	1	1	NUM
ejpam-6059	333	27	,	,	PUNCT
ejpam-6059	333	28	2	2	NUM
ejpam-6059	333	29	,	,	PUNCT
ejpam-6059	333	30	.	.	PUNCT
ejpam-6059	333	31	.	.	PUNCT
ejpam-6059	334	1	.	.	PUNCT
ejpam-6059	335	1	,	,	PUNCT
ejpam-6059	335	2	p	p	X
ejpam-6059	335	3	+	+	NOUN
ejpam-6059	335	4	1	1	NUM
ejpam-6059	335	5	,	,	PUNCT
ejpam-6059	335	6	p	p	X
ejpam-6059	335	7	+	+	NOUN
ejpam-6059	335	8	2	2	NUM
ejpam-6059	335	9	}	}	PUNCT
ejpam-6059	335	10	.	.	PUNCT
ejpam-6059	336	1	we	we	PRON
ejpam-6059	336	2	show	show	VERB
ejpam-6059	336	3	that	that	SCONJ
ejpam-6059	336	4	γn[a	γn[a	PROPN
ejpam-6059	336	5	]	]	PUNCT
ejpam-6059	336	6	contains	contain	VERB
ejpam-6059	336	7	no	no	DET
ejpam-6059	336	8	cycles	cycle	NOUN
ejpam-6059	336	9	.	.	PUNCT
ejpam-6059	337	1	suppose	suppose	VERB
ejpam-6059	337	2	that	that	SCONJ
ejpam-6059	337	3	γn[a	γn[a	PROPN
ejpam-6059	337	4	]	]	PUNCT
ejpam-6059	337	5	contains	contain	VERB
ejpam-6059	337	6	a	a	DET
ejpam-6059	337	7	cycle	cycle	NOUN
ejpam-6059	337	8	ck	ck	INTJ
ejpam-6059	337	9	:	:	PUNCT
ejpam-6059	337	10	=	=	SYM
ejpam-6059	337	11	{	{	PUNCT
ejpam-6059	337	12	v1	v1	PROPN
ejpam-6059	337	13	,	,	PUNCT
ejpam-6059	337	14	v2	v2	PROPN
ejpam-6059	337	15	,	,	PUNCT
ejpam-6059	337	16	.	.	PUNCT
ejpam-6059	337	17	.	.	PUNCT
ejpam-6059	338	1	.	.	PUNCT
ejpam-6059	339	1	,	,	PUNCT
ejpam-6059	339	2	vk	vk	INTJ
ejpam-6059	339	3	,	,	PUNCT
ejpam-6059	339	4	v1	v1	NOUN
ejpam-6059	339	5	}	}	PUNCT
ejpam-6059	339	6	such	such	ADJ
ejpam-6059	339	7	that	that	DET
ejpam-6059	339	8	v1	v1	NOUN
ejpam-6059	339	9	,	,	PUNCT
ejpam-6059	339	10	v2	v2	NOUN
ejpam-6059	339	11	,	,	PUNCT
ejpam-6059	339	12	.	.	PUNCT
ejpam-6059	339	13	.	.	PUNCT
ejpam-6059	340	1	.	.	PUNCT
ejpam-6059	341	1	,	,	PUNCT
ejpam-6059	341	2	vk	vk	AUX
ejpam-6059	341	3	∈	∈	PROPN
ejpam-6059	341	4	a	a	PRON
ejpam-6059	341	5	and	and	CCONJ
ejpam-6059	341	6	3	3	NUM
ejpam-6059	341	7	≤	≤	NUM
ejpam-6059	341	8	k	k	PROPN
ejpam-6059	341	9	≤	≤	PROPN
ejpam-6059	341	10	p+	p+	VERB
ejpam-6059	341	11	3	3	NUM
ejpam-6059	341	12	.	.	NOUN
ejpam-6059	341	13	without	without	ADP
ejpam-6059	341	14	loss	loss	NOUN
ejpam-6059	341	15	of	of	ADP
ejpam-6059	341	16	generality	generality	NOUN
ejpam-6059	341	17	,	,	PUNCT
ejpam-6059	341	18	we	we	PRON
ejpam-6059	341	19	can	can	AUX
ejpam-6059	341	20	assume	assume	VERB
ejpam-6059	341	21	that	that	SCONJ
ejpam-6059	341	22	v1	v1	VERB
ejpam-6059	341	23	<	<	X
ejpam-6059	341	24	vj	vj	INTJ
ejpam-6059	341	25	for	for	ADP
ejpam-6059	341	26	2	2	NUM
ejpam-6059	341	27	≤	≤	NUM
ejpam-6059	341	28	j	j	PROPN
ejpam-6059	341	29	≤	≤	PROPN
ejpam-6059	341	30	k.	k.	PROPN
ejpam-6059	342	1	let	let	VERB
ejpam-6059	342	2	x	x	PRON
ejpam-6059	342	3	:	:	PUNCT
ejpam-6059	342	4	=	=	SYM
ejpam-6059	342	5	{	{	PUNCT
ejpam-6059	342	6	x1	x1	PROPN
ejpam-6059	342	7	,	,	PUNCT
ejpam-6059	342	8	x2	x2	PROPN
ejpam-6059	342	9	,	,	PUNCT
ejpam-6059	342	10	.	.	PUNCT
ejpam-6059	342	11	.	.	PUNCT
ejpam-6059	342	12	.	.	PUNCT
ejpam-6059	343	1	,	,	PUNCT
ejpam-6059	343	2	xk	xk	AUX
ejpam-6059	343	3	}	}	PUNCT
ejpam-6059	343	4	be	be	AUX
ejpam-6059	343	5	such	such	ADJ
ejpam-6059	343	6	that	that	SCONJ
ejpam-6059	344	1	x1	x1	PROPN
ejpam-6059	344	2	=	=	SYM
ejpam-6059	344	3	0	0	PROPN
ejpam-6059	344	4	and	and	CCONJ
ejpam-6059	344	5	xl	xl	PROPN
ejpam-6059	344	6	=	=	NOUN
ejpam-6059	344	7	vl	vl	PROPN
ejpam-6059	344	8	−	−	PROPN
ejpam-6059	344	9	v1	v1	NOUN
ejpam-6059	344	10	for	for	ADP
ejpam-6059	344	11	all	all	DET
ejpam-6059	344	12	l	l	NOUN
ejpam-6059	344	13	=	=	SYM
ejpam-6059	344	14	2	2	NUM
ejpam-6059	344	15	,	,	PUNCT
ejpam-6059	344	16	3	3	NUM
ejpam-6059	344	17	,	,	PUNCT
ejpam-6059	344	18	.	.	PUNCT
ejpam-6059	344	19	.	.	PUNCT
ejpam-6059	344	20	.	.	PUNCT
ejpam-6059	345	1	,	,	PUNCT
ejpam-6059	345	2	k.	k.	PROPN
ejpam-6059	345	3	then	then	ADV
ejpam-6059	345	4	x1	x1	NUM
ejpam-6059	345	5	,	,	PUNCT
ejpam-6059	345	6	x2	x2	PROPN
ejpam-6059	345	7	,	,	PUNCT
ejpam-6059	345	8	.	.	PUNCT
ejpam-6059	345	9	.	.	PUNCT
ejpam-6059	345	10	.	.	PUNCT
ejpam-6059	346	1	,	,	PUNCT
ejpam-6059	346	2	xk	xk	PROPN
ejpam-6059	346	3	,	,	PUNCT
ejpam-6059	346	4	x1	x1	PROPN
ejpam-6059	346	5	form	form	VERB
ejpam-6059	346	6	a	a	DET
ejpam-6059	346	7	cycle	cycle	NOUN
ejpam-6059	346	8	in	in	ADP
ejpam-6059	346	9	γn[x	γn[x	PROPN
ejpam-6059	346	10	]	]	PUNCT
ejpam-6059	346	11	.	.	PUNCT
ejpam-6059	347	1	we	we	PRON
ejpam-6059	347	2	now	now	ADV
ejpam-6059	347	3	consider	consider	VERB
ejpam-6059	347	4	the	the	DET
ejpam-6059	347	5	following	follow	VERB
ejpam-6059	347	6	two	two	NUM
ejpam-6059	347	7	cases	case	NOUN
ejpam-6059	347	8	.	.	PUNCT
ejpam-6059	348	1	case	case	NOUN
ejpam-6059	348	2	1	1	NUM
ejpam-6059	348	3	:	:	PUNCT
ejpam-6059	348	4	(	(	PUNCT
ejpam-6059	348	5	p+	p+	NOUN
ejpam-6059	348	6	2)|n	2)|n	NUM
ejpam-6059	348	7	.	.	PUNCT
ejpam-6059	349	1	consider	consider	VERB
ejpam-6059	349	2	{	{	PUNCT
ejpam-6059	349	3	x2	x2	ADJ
ejpam-6059	349	4	,	,	PUNCT
ejpam-6059	349	5	x3	x3	ADJ
ejpam-6059	349	6	}	}	PUNCT
ejpam-6059	349	7	∈	∈	NOUN
ejpam-6059	349	8	e(γn[x	e(γn[x	NOUN
ejpam-6059	349	9	]	]	PUNCT
ejpam-6059	349	10	)	)	PUNCT
ejpam-6059	349	11	,	,	PUNCT
ejpam-6059	349	12	if	if	SCONJ
ejpam-6059	349	13	x2	x2	PROPN
ejpam-6059	349	14	<	<	X
ejpam-6059	349	15	x3	x3	PROPN
ejpam-6059	349	16	,	,	PUNCT
ejpam-6059	349	17	then	then	ADV
ejpam-6059	349	18	x3	x3	VERB
ejpam-6059	349	19	=	=	PUNCT
ejpam-6059	349	20	2p	2p	NUM
ejpam-6059	349	21	or	or	CCONJ
ejpam-6059	349	22	x3	x3	NOUN
ejpam-6059	349	23	=	=	NOUN
ejpam-6059	349	24	2p+	2p+	NUM
ejpam-6059	349	25	2	2	NUM
ejpam-6059	349	26	and	and	CCONJ
ejpam-6059	349	27	so	so	ADV
ejpam-6059	349	28	x3	x3	ADJ
ejpam-6059	349	29	/∈	/∈	PUNCT
ejpam-6059	350	1	x	x	X
ejpam-6059	350	2	,	,	PUNCT
ejpam-6059	350	3	a	a	DET
ejpam-6059	350	4	contradiction	contradiction	NOUN
ejpam-6059	350	5	.	.	PUNCT
ejpam-6059	351	1	if	if	SCONJ
ejpam-6059	351	2	x3	x3	ADJ
ejpam-6059	351	3	<	<	X
ejpam-6059	351	4	x2	x2	PROPN
ejpam-6059	351	5	,	,	PUNCT
ejpam-6059	351	6	then	then	ADV
ejpam-6059	351	7	x3	x3	VERB
ejpam-6059	351	8	=	=	SYM
ejpam-6059	351	9	0	0	PUNCT
ejpam-6059	352	1	=	=	SYM
ejpam-6059	352	2	x1	x1	PROPN
ejpam-6059	352	3	,	,	PUNCT
ejpam-6059	352	4	a	a	DET
ejpam-6059	352	5	contradiction	contradiction	NOUN
ejpam-6059	352	6	as	as	ADP
ejpam-6059	352	7	xi	xi	PROPN
ejpam-6059	352	8	̸=	̸=	PROPN
ejpam-6059	352	9	xj	xj	PROPN
ejpam-6059	352	10	for	for	ADP
ejpam-6059	352	11	i	i	PRON
ejpam-6059	352	12	̸=	̸=	PROPN
ejpam-6059	352	13	j.	j.	PROPN
ejpam-6059	352	14	case	case	NOUN
ejpam-6059	352	15	2	2	NUM
ejpam-6059	352	16	:	:	PUNCT
ejpam-6059	352	17	(	(	PUNCT
ejpam-6059	352	18	p+	p+	NOUN
ejpam-6059	352	19	2	2	NUM
ejpam-6059	352	20	)	)	PUNCT
ejpam-6059	352	21	∤	∤	SYM
ejpam-6059	352	22	n.	n.	PROPN
ejpam-6059	352	23	since	since	SCONJ
ejpam-6059	352	24	{	{	PUNCT
ejpam-6059	352	25	x1	x1	PROPN
ejpam-6059	352	26	,	,	PUNCT
ejpam-6059	352	27	xk	xk	ADJ
ejpam-6059	352	28	}	}	PUNCT
ejpam-6059	352	29	∈	∈	PROPN
ejpam-6059	352	30	e(γn[x	e(γn[x	NOUN
ejpam-6059	352	31	]	]	PUNCT
ejpam-6059	352	32	)	)	PUNCT
ejpam-6059	352	33	,	,	PUNCT
ejpam-6059	352	34	it	it	PRON
ejpam-6059	352	35	follows	follow	VERB
ejpam-6059	352	36	that	that	SCONJ
ejpam-6059	352	37	xk	xk	PROPN
ejpam-6059	353	1	=	=	PUNCT
ejpam-6059	353	2	p	p	X
ejpam-6059	353	3	=	=	PUNCT
ejpam-6059	353	4	x2	x2	PROPN
ejpam-6059	353	5	,	,	PUNCT
ejpam-6059	353	6	a	a	DET
ejpam-6059	353	7	contradiction	contradiction	NOUN
ejpam-6059	353	8	.	.	PUNCT
ejpam-6059	354	1	hence	hence	ADV
ejpam-6059	354	2	γn[a	γn[a	PROPN
ejpam-6059	354	3	]	]	PUNCT
ejpam-6059	354	4	contains	contain	VERB
ejpam-6059	354	5	no	no	DET
ejpam-6059	354	6	cycles	cycle	NOUN
ejpam-6059	354	7	.	.	PUNCT
ejpam-6059	355	1	that	that	PRON
ejpam-6059	355	2	is	be	AUX
ejpam-6059	355	3	,	,	PUNCT
ejpam-6059	355	4	λ(γn	λ(γn	NOUN
ejpam-6059	355	5	)	)	PUNCT
ejpam-6059	355	6	≥	≥	NOUN
ejpam-6059	356	1	|a|	|a|	NOUN
ejpam-6059	356	2	=	=	NOUN
ejpam-6059	356	3	p	p	PROPN
ejpam-6059	356	4	+	+	NOUN
ejpam-6059	356	5	3	3	X
ejpam-6059	356	6	.	.	PUNCT
ejpam-6059	357	1	now	now	ADV
ejpam-6059	357	2	,	,	PUNCT
ejpam-6059	357	3	let	let	VERB
ejpam-6059	357	4	b	b	NOUN
ejpam-6059	357	5	:	:	PUNCT
ejpam-6059	357	6	=	=	SYM
ejpam-6059	357	7	{	{	PUNCT
ejpam-6059	357	8	v0	v0	NOUN
ejpam-6059	357	9	,	,	PUNCT
ejpam-6059	357	10	v1	v1	NOUN
ejpam-6059	357	11	,	,	PUNCT
ejpam-6059	357	12	.	.	PUNCT
ejpam-6059	357	13	.	.	PUNCT
ejpam-6059	358	1	.	.	PUNCT
ejpam-6059	359	1	,	,	PUNCT
ejpam-6059	359	2	v2p	v2p	PROPN
ejpam-6059	359	3	}	}	PUNCT
ejpam-6059	359	4	be	be	AUX
ejpam-6059	359	5	arbitrary	arbitrary	ADJ
ejpam-6059	359	6	.	.	PUNCT
ejpam-6059	360	1	let	let	VERB
ejpam-6059	360	2	br	br	PRON
ejpam-6059	360	3	be	be	AUX
ejpam-6059	360	4	the	the	DET
ejpam-6059	360	5	set	set	NOUN
ejpam-6059	360	6	of	of	ADP
ejpam-6059	360	7	elements	element	NOUN
ejpam-6059	360	8	in	in	ADP
ejpam-6059	360	9	b	b	NOUN
ejpam-6059	360	10	having	have	VERB
ejpam-6059	360	11	r	r	NOUN
ejpam-6059	360	12	as	as	ADP
ejpam-6059	360	13	the	the	DET
ejpam-6059	360	14	remainder	remainder	NOUN
ejpam-6059	360	15	from	from	ADP
ejpam-6059	360	16	dividing	divide	VERB
ejpam-6059	360	17	by	by	ADP
ejpam-6059	360	18	p	p	PRON
ejpam-6059	360	19	where	where	SCONJ
ejpam-6059	360	20	0	0	NUM
ejpam-6059	360	21	≤	≤	NUM
ejpam-6059	360	22	r	r	NOUN
ejpam-6059	360	23	≤	≤	NOUN
ejpam-6059	361	1	p	p	PRON
ejpam-6059	361	2	−	−	PROPN
ejpam-6059	361	3	1	1	NUM
ejpam-6059	361	4	.	.	PUNCT
ejpam-6059	362	1	we	we	PRON
ejpam-6059	362	2	show	show	VERB
ejpam-6059	362	3	that	that	SCONJ
ejpam-6059	362	4	γn[b	γn[b	NOUN
ejpam-6059	362	5	]	]	PUNCT
ejpam-6059	362	6	contains	contain	VERB
ejpam-6059	362	7	a	a	DET
ejpam-6059	362	8	cycle	cycle	NOUN
ejpam-6059	362	9	.	.	PUNCT
ejpam-6059	363	1	now	now	ADV
ejpam-6059	363	2	,	,	PUNCT
ejpam-6059	363	3	we	we	PRON
ejpam-6059	363	4	can	can	AUX
ejpam-6059	363	5	place	place	VERB
ejpam-6059	363	6	v0	v0	NOUN
ejpam-6059	363	7	,	,	PUNCT
ejpam-6059	363	8	v1	v1	NOUN
ejpam-6059	363	9	,	,	PUNCT
ejpam-6059	363	10	.	.	PUNCT
ejpam-6059	363	11	.	.	PUNCT
ejpam-6059	364	1	.	.	PUNCT
ejpam-6059	365	1	,	,	PUNCT
ejpam-6059	365	2	v2p	v2p	VERB
ejpam-6059	365	3	in	in	ADP
ejpam-6059	365	4	such	such	ADJ
ejpam-6059	365	5	p	p	NOUN
ejpam-6059	365	6	sets	set	NOUN
ejpam-6059	365	7	and	and	CCONJ
ejpam-6059	365	8	by	by	ADP
ejpam-6059	365	9	the	the	DET
ejpam-6059	365	10	pigeonhole	pigeonhole	NOUN
ejpam-6059	365	11	principle	principle	NOUN
ejpam-6059	365	12	,	,	PUNCT
ejpam-6059	365	13	we	we	PRON
ejpam-6059	365	14	obtain	obtain	VERB
ejpam-6059	365	15	that	that	PRON
ejpam-6059	365	16	one	one	NUM
ejpam-6059	365	17	of	of	ADP
ejpam-6059	365	18	those	those	DET
ejpam-6059	365	19	sets	set	NOUN
ejpam-6059	365	20	must	must	AUX
ejpam-6059	365	21	contain	contain	VERB
ejpam-6059	365	22	at	at	ADV
ejpam-6059	365	23	least	least	ADV
ejpam-6059	365	24	three	three	NUM
ejpam-6059	365	25	vertices	vertex	NOUN
ejpam-6059	365	26	x	x	X
ejpam-6059	365	27	,	,	PUNCT
ejpam-6059	365	28	y	y	PROPN
ejpam-6059	365	29	,	,	PUNCT
ejpam-6059	365	30	z.	z.	PROPN
ejpam-6059	365	31	without	without	ADP
ejpam-6059	365	32	loss	loss	NOUN
ejpam-6059	365	33	of	of	ADP
ejpam-6059	365	34	generality	generality	NOUN
ejpam-6059	365	35	,	,	PUNCT
ejpam-6059	365	36	assume	assume	VERB
ejpam-6059	365	37	that	that	SCONJ
ejpam-6059	365	38	x	x	PROPN
ejpam-6059	365	39	,	,	PUNCT
ejpam-6059	365	40	y	y	PROPN
ejpam-6059	365	41	,	,	PUNCT
ejpam-6059	365	42	z	z	PROPN
ejpam-6059	365	43	∈	∈	NOUN
ejpam-6059	365	44	bs	bs	NOUN
ejpam-6059	365	45	and	and	CCONJ
ejpam-6059	365	46	x	x	X
ejpam-6059	365	47	>	>	X
ejpam-6059	365	48	y	y	PROPN
ejpam-6059	365	49	>	>	X
ejpam-6059	365	50	z	z	NOUN
ejpam-6059	365	51	where	where	SCONJ
ejpam-6059	365	52	0	0	NUM
ejpam-6059	365	53	≤	≤	NOUN
ejpam-6059	365	54	s	s	PART
ejpam-6059	365	55	≤	≤	NOUN
ejpam-6059	366	1	p	p	PRON
ejpam-6059	366	2	−	−	PROPN
ejpam-6059	366	3	1	1	NUM
ejpam-6059	366	4	.	.	PUNCT
ejpam-6059	367	1	then	then	ADV
ejpam-6059	367	2	x	x	X
ejpam-6059	367	3	=	=	PUNCT
ejpam-6059	367	4	m1p	m1p	X
ejpam-6059	367	5	+	+	X
ejpam-6059	367	6	s	s	X
ejpam-6059	367	7	,	,	PUNCT
ejpam-6059	367	8	y	y	PROPN
ejpam-6059	367	9	=	=	PUNCT
ejpam-6059	367	10	m2p	m2p	PROPN
ejpam-6059	368	1	+	+	SYM
ejpam-6059	368	2	s	s	X
ejpam-6059	368	3	and	and	CCONJ
ejpam-6059	368	4	z	z	NOUN
ejpam-6059	368	5	=	=	PUNCT
ejpam-6059	368	6	m3p+	m3p+	NOUN
ejpam-6059	368	7	s	s	NOUN
ejpam-6059	368	8	for	for	ADP
ejpam-6059	368	9	some	some	DET
ejpam-6059	368	10	m1,m2,m3	m1,m2,m3	ADJ
ejpam-6059	368	11	∈	∈	PROPN
ejpam-6059	368	12	z.	z.	PROPN
ejpam-6059	369	1	so	so	ADV
ejpam-6059	369	2	x−	x−	PROPN
ejpam-6059	369	3	y	y	PROPN
ejpam-6059	369	4	=	=	PRON
ejpam-6059	370	1	(	(	PUNCT
ejpam-6059	370	2	m1	m1	PROPN
ejpam-6059	370	3	−m2)p	−m2)p	PROPN
ejpam-6059	370	4	,	,	PUNCT
ejpam-6059	370	5	y	y	PROPN
ejpam-6059	370	6	−	−	PROPN
ejpam-6059	370	7	z	z	NOUN
ejpam-6059	370	8	=	=	SYM
ejpam-6059	370	9	(	(	PUNCT
ejpam-6059	370	10	m2	m2	PROPN
ejpam-6059	370	11	−m3)p	−m3)p	PROPN
ejpam-6059	370	12	and	and	CCONJ
ejpam-6059	370	13	x−z	x−z	X
ejpam-6059	370	14	=	=	SYM
ejpam-6059	370	15	(	(	PUNCT
ejpam-6059	370	16	m1−m3)p	m1−m3)p	PROPN
ejpam-6059	370	17	,	,	PUNCT
ejpam-6059	370	18	we	we	PRON
ejpam-6059	370	19	obtain	obtain	VERB
ejpam-6059	370	20	that	that	DET
ejpam-6059	370	21	gcd(x−y	gcd(x−y	NOUN
ejpam-6059	370	22	,	,	PUNCT
ejpam-6059	370	23	n	n	CCONJ
ejpam-6059	370	24	)	)	PUNCT
ejpam-6059	370	25	̸=	̸=	PROPN
ejpam-6059	370	26	1	1	NUM
ejpam-6059	370	27	,	,	PUNCT
ejpam-6059	370	28	gcd(y−z	gcd(y−z	NOUN
ejpam-6059	370	29	,	,	PUNCT
ejpam-6059	370	30	n	n	CCONJ
ejpam-6059	370	31	)	)	PUNCT
ejpam-6059	370	32	̸=	̸=	PROPN
ejpam-6059	370	33	1	1	NUM
ejpam-6059	370	34	and	and	CCONJ
ejpam-6059	370	35	gcd(z−x	gcd(z−x	NOUN
ejpam-6059	370	36	,	,	PUNCT
ejpam-6059	370	37	n	n	CCONJ
ejpam-6059	370	38	)	)	PUNCT
ejpam-6059	370	39	̸=	̸=	PROPN
ejpam-6059	370	40	1	1	NUM
ejpam-6059	370	41	,	,	PUNCT
ejpam-6059	370	42	respectively	respectively	ADV
ejpam-6059	370	43	.	.	PUNCT
ejpam-6059	371	1	hence	hence	ADV
ejpam-6059	371	2	x	x	SYM
ejpam-6059	371	3	,	,	PUNCT
ejpam-6059	371	4	y	y	PROPN
ejpam-6059	371	5	,	,	PUNCT
ejpam-6059	371	6	z	z	PROPN
ejpam-6059	371	7	,	,	PUNCT
ejpam-6059	371	8	x	x	PRON
ejpam-6059	371	9	form	form	VERB
ejpam-6059	371	10	a	a	DET
ejpam-6059	371	11	cycle	cycle	NOUN
ejpam-6059	371	12	.	.	PUNCT
ejpam-6059	372	1	it	it	PRON
ejpam-6059	372	2	follows	follow	VERB
ejpam-6059	372	3	that	that	SCONJ
ejpam-6059	372	4	γn[b	γn[b	PROPN
ejpam-6059	372	5	]	]	PUNCT
ejpam-6059	372	6	contains	contain	VERB
ejpam-6059	372	7	a	a	DET
ejpam-6059	372	8	cycle.therefore	cycle.therefore	NOUN
ejpam-6059	372	9	,	,	PUNCT
ejpam-6059	372	10	λ(γn	λ(γn	NOUN
ejpam-6059	372	11	)	)	PUNCT
ejpam-6059	372	12	≤	≤	NOUN
ejpam-6059	372	13	|b|	|b|	VERB
ejpam-6059	373	1	−	−	PROPN
ejpam-6059	373	2	1	1	NUM
ejpam-6059	373	3	=	=	SYM
ejpam-6059	373	4	2p.then	2p.then	PROPN
ejpam-6059	373	5	the	the	DET
ejpam-6059	373	6	statement	statement	NOUN
ejpam-6059	373	7	is	be	AUX
ejpam-6059	373	8	proved	prove	VERB
ejpam-6059	373	9	,	,	PUNCT
ejpam-6059	373	10	as	as	SCONJ
ejpam-6059	373	11	required	require	VERB
ejpam-6059	373	12	.	.	PUNCT
ejpam-6059	374	1	5	5	X
ejpam-6059	374	2	.	.	X
ejpam-6059	374	3	conclusion	conclusion	NOUN
ejpam-6059	374	4	in	in	ADP
ejpam-6059	374	5	this	this	DET
ejpam-6059	374	6	paper	paper	NOUN
ejpam-6059	374	7	,	,	PUNCT
ejpam-6059	374	8	we	we	PRON
ejpam-6059	374	9	have	have	AUX
ejpam-6059	374	10	provided	provide	VERB
ejpam-6059	374	11	certain	certain	ADJ
ejpam-6059	374	12	structural	structural	ADJ
ejpam-6059	374	13	properties	property	NOUN
ejpam-6059	374	14	and	and	CCONJ
ejpam-6059	374	15	some	some	DET
ejpam-6059	374	16	invariant	invariant	ADJ
ejpam-6059	374	17	properties	property	NOUN
ejpam-6059	374	18	of	of	ADP
ejpam-6059	374	19	unitary	unitary	ADJ
ejpam-6059	374	20	cayley	cayley	ADJ
ejpam-6059	374	21	graphs	graph	NOUN
ejpam-6059	374	22	γn	γn	ADP
ejpam-6059	374	23	of	of	ADP
ejpam-6059	374	24	finite	finite	PROPN
ejpam-6059	374	25	commutative	commutative	PROPN
ejpam-6059	374	26	rings	ring	NOUN
ejpam-6059	374	27	zn	zn	PROPN
ejpam-6059	374	28	.	.	PUNCT
ejpam-6059	375	1	such	such	ADJ
ejpam-6059	375	2	invariant	invariant	ADJ
ejpam-6059	375	3	properties	property	NOUN
ejpam-6059	375	4	consist	consist	VERB
ejpam-6059	375	5	of	of	ADP
ejpam-6059	375	6	the	the	DET
ejpam-6059	375	7	lower	low	ADJ
ejpam-6059	375	8	acyclic	acyclic	ADJ
ejpam-6059	375	9	number	number	NOUN
ejpam-6059	375	10	and	and	CCONJ
ejpam-6059	375	11	the	the	DET
ejpam-6059	375	12	upper	upper	ADJ
ejpam-6059	375	13	acyclic	acyclic	ADJ
ejpam-6059	375	14	number	number	NOUN
ejpam-6059	375	15	.	.	PUNCT
ejpam-6059	376	1	in	in	ADP
ejpam-6059	376	2	the	the	DET
ejpam-6059	376	3	process	process	NOUN
ejpam-6059	376	4	of	of	ADP
ejpam-6059	376	5	the	the	DET
ejpam-6059	376	6	d.	d.	PROPN
ejpam-6059	376	7	pongpipat	pongpipat	PROPN
ejpam-6059	376	8	,	,	PUNCT
ejpam-6059	376	9	n.	n.	NOUN
ejpam-6059	376	10	nupo	nupo	PROPN
ejpam-6059	376	11	/	/	SYM
ejpam-6059	376	12	eur	eur	PROPN
ejpam-6059	376	13	.	.	PUNCT
ejpam-6059	377	1	j.	j.	PROPN
ejpam-6059	377	2	pure	pure	PROPN
ejpam-6059	377	3	appl	appl	PROPN
ejpam-6059	377	4	.	.	PROPN
ejpam-6059	377	5	math	math	PROPN
ejpam-6059	377	6	,	,	PUNCT
ejpam-6059	377	7	18	18	NUM
ejpam-6059	377	8	(	(	PUNCT
ejpam-6059	377	9	2	2	NUM
ejpam-6059	377	10	)	)	PUNCT
ejpam-6059	377	11	(	(	PUNCT
ejpam-6059	377	12	2025	2025	NUM
ejpam-6059	377	13	)	)	PUNCT
ejpam-6059	377	14	,	,	PUNCT
ejpam-6059	377	15	6059	6059	NUM
ejpam-6059	377	16	11	11	NUM
ejpam-6059	377	17	of	of	ADP
ejpam-6059	377	18	11	11	NUM
ejpam-6059	377	19	study	study	NOUN
ejpam-6059	377	20	results	result	NOUN
ejpam-6059	378	1	,	,	PUNCT
ejpam-6059	378	2	we	we	PRON
ejpam-6059	378	3	have	have	AUX
ejpam-6059	378	4	found	find	VERB
ejpam-6059	378	5	that	that	SCONJ
ejpam-6059	378	6	the	the	DET
ejpam-6059	378	7	decomposition	decomposition	NOUN
ejpam-6059	378	8	of	of	ADP
ejpam-6059	378	9	graphs	graph	NOUN
ejpam-6059	378	10	is	be	AUX
ejpam-6059	378	11	useful	useful	ADJ
ejpam-6059	378	12	and	and	CCONJ
ejpam-6059	378	13	plays	play	VERB
ejpam-6059	378	14	an	an	DET
ejpam-6059	378	15	important	important	ADJ
ejpam-6059	378	16	role	role	NOUN
ejpam-6059	378	17	for	for	ADP
ejpam-6059	378	18	determining	determine	VERB
ejpam-6059	378	19	those	those	DET
ejpam-6059	378	20	invariant	invariant	ADJ
ejpam-6059	378	21	parameters	parameter	NOUN
ejpam-6059	378	22	.	.	PUNCT
ejpam-6059	379	1	finally	finally	ADV
ejpam-6059	379	2	,	,	PUNCT
ejpam-6059	379	3	we	we	PRON
ejpam-6059	379	4	have	have	AUX
ejpam-6059	379	5	presented	present	VERB
ejpam-6059	379	6	the	the	DET
ejpam-6059	379	7	results	result	NOUN
ejpam-6059	379	8	of	of	ADP
ejpam-6059	379	9	invariant	invariant	ADJ
ejpam-6059	379	10	parameters	parameter	NOUN
ejpam-6059	379	11	in	in	ADP
ejpam-6059	379	12	the	the	DET
ejpam-6059	379	13	unitary	unitary	ADJ
ejpam-6059	379	14	cayley	cayley	NOUN
ejpam-6059	379	15	graphs	graph	NOUN
ejpam-6059	379	16	and	and	CCONJ
ejpam-6059	379	17	their	their	PRON
ejpam-6059	379	18	complements	complement	NOUN
ejpam-6059	379	19	.	.	PUNCT
ejpam-6059	380	1	for	for	ADP
ejpam-6059	380	2	further	further	ADJ
ejpam-6059	380	3	works	work	NOUN
ejpam-6059	380	4	,	,	PUNCT
ejpam-6059	380	5	one	one	PRON
ejpam-6059	380	6	can	can	AUX
ejpam-6059	380	7	investigate	investigate	VERB
ejpam-6059	380	8	those	those	DET
ejpam-6059	380	9	invariant	invariant	ADJ
ejpam-6059	380	10	parameters	parameter	NOUN
ejpam-6059	380	11	for	for	ADP
ejpam-6059	380	12	various	various	ADJ
ejpam-6059	380	13	types	type	NOUN
ejpam-6059	380	14	of	of	ADP
ejpam-6059	380	15	algebraic	algebraic	ADJ
ejpam-6059	380	16	graphs	graph	NOUN
ejpam-6059	380	17	and	and	CCONJ
ejpam-6059	380	18	their	their	PRON
ejpam-6059	380	19	complements	complement	NOUN
ejpam-6059	380	20	.	.	PUNCT
ejpam-6059	381	1	acknowledgements	acknowledgement	NOUN
ejpam-6059	381	2	the	the	DET
ejpam-6059	381	3	authors	author	NOUN
ejpam-6059	381	4	are	be	AUX
ejpam-6059	381	5	grateful	grateful	ADJ
ejpam-6059	381	6	to	to	ADP
ejpam-6059	381	7	the	the	DET
ejpam-6059	381	8	referee(s	referee(s	NOUN
ejpam-6059	381	9	)	)	PUNCT
ejpam-6059	381	10	for	for	ADP
ejpam-6059	381	11	suggestions	suggestion	NOUN
ejpam-6059	381	12	on	on	ADP
ejpam-6059	381	13	the	the	DET
ejpam-6059	381	14	manuscript	manuscript	NOUN
ejpam-6059	381	15	.	.	PUNCT
ejpam-6059	382	1	the	the	DET
ejpam-6059	382	2	first	first	ADJ
ejpam-6059	382	3	author	author	NOUN
ejpam-6059	382	4	would	would	AUX
ejpam-6059	382	5	like	like	VERB
ejpam-6059	382	6	to	to	PART
ejpam-6059	382	7	thank	thank	VERB
ejpam-6059	382	8	the	the	DET
ejpam-6059	382	9	science	science	NOUN
ejpam-6059	382	10	achievement	achievement	NOUN
ejpam-6059	382	11	scholarship	scholarship	NOUN
ejpam-6059	382	12	of	of	ADP
ejpam-6059	382	13	thailand	thailand	PROPN
ejpam-6059	382	14	(	(	PUNCT
ejpam-6059	382	15	sast	sast	NOUN
ejpam-6059	382	16	)	)	PUNCT
ejpam-6059	382	17	.	.	PUNCT
ejpam-6059	383	1	the	the	DET
ejpam-6059	383	2	corresponding	corresponding	ADJ
ejpam-6059	383	3	author	author	NOUN
ejpam-6059	383	4	also	also	ADV
ejpam-6059	383	5	thanks	thank	VERB
ejpam-6059	383	6	the	the	DET
ejpam-6059	383	7	faculty	faculty	NOUN
ejpam-6059	383	8	of	of	ADP
ejpam-6059	383	9	science	science	NOUN
ejpam-6059	383	10	,	,	PUNCT
ejpam-6059	383	11	khon	khon	PROPN
ejpam-6059	383	12	kaen	kaen	PROPN
ejpam-6059	383	13	university	university	PROPN
ejpam-6059	383	14	.	.	PUNCT
ejpam-6059	384	1	conflict	conflict	NOUN
ejpam-6059	384	2	of	of	ADP
ejpam-6059	384	3	interest	interest	NOUN
ejpam-6059	384	4	the	the	DET
ejpam-6059	384	5	authors	author	NOUN
ejpam-6059	384	6	declare	declare	VERB
ejpam-6059	384	7	that	that	SCONJ
ejpam-6059	384	8	there	there	PRON
ejpam-6059	384	9	are	be	VERB
ejpam-6059	384	10	no	no	DET
ejpam-6059	384	11	conflicts	conflict	NOUN
ejpam-6059	384	12	of	of	ADP
ejpam-6059	384	13	interest	interest	NOUN
ejpam-6059	384	14	.	.	PUNCT
ejpam-6059	385	1	references	reference	NOUN
ejpam-6059	385	2	[	[	X
ejpam-6059	385	3	1	1	NUM
ejpam-6059	385	4	]	]	PUNCT
ejpam-6059	385	5	i.	i.	NOUN
ejpam-6059	385	6	dejter	dejter	PROPN
ejpam-6059	385	7	and	and	CCONJ
ejpam-6059	385	8	r.	r.	PROPN
ejpam-6059	385	9	e.	e.	PROPN
ejpam-6059	385	10	giudici	giudici	PROPN
ejpam-6059	385	11	.	.	PUNCT
ejpam-6059	386	1	on	on	ADP
ejpam-6059	386	2	unitary	unitary	ADJ
ejpam-6059	386	3	cayley	cayley	ADJ
ejpam-6059	386	4	graphs	graph	NOUN
ejpam-6059	386	5	.	.	PUNCT
ejpam-6059	387	1	journal	journal	NOUN
ejpam-6059	387	2	of	of	ADP
ejpam-6059	387	3	combinatorial	combinatorial	ADJ
ejpam-6059	387	4	mathematics	mathematic	NOUN
ejpam-6059	387	5	and	and	CCONJ
ejpam-6059	387	6	combinatorial	combinatorial	ADJ
ejpam-6059	387	7	computing	computing	NOUN
ejpam-6059	387	8	,	,	PUNCT
ejpam-6059	387	9	18:121–124	18:121–124	NUM
ejpam-6059	387	10	,	,	PUNCT
ejpam-6059	387	11	1995	1995	NUM
ejpam-6059	387	12	.	.	PUNCT
ejpam-6059	388	1	[	[	X
ejpam-6059	388	2	2	2	X
ejpam-6059	388	3	]	]	PUNCT
ejpam-6059	388	4	w.	w.	PROPN
ejpam-6059	388	5	klotz	klotz	PROPN
ejpam-6059	388	6	and	and	CCONJ
ejpam-6059	388	7	t.	t.	PROPN
ejpam-6059	388	8	sander	sander	NOUN
ejpam-6059	388	9	.	.	PUNCT
ejpam-6059	389	1	some	some	DET
ejpam-6059	389	2	properties	property	NOUN
ejpam-6059	389	3	of	of	ADP
ejpam-6059	389	4	unitary	unitary	ADJ
ejpam-6059	389	5	cayley	cayley	ADJ
ejpam-6059	389	6	graphs	graph	NOUN
ejpam-6059	389	7	.	.	PUNCT
ejpam-6059	390	1	electronic	electronic	ADJ
ejpam-6059	390	2	journal	journal	NOUN
ejpam-6059	390	3	of	of	ADP
ejpam-6059	390	4	combinatorics	combinatoric	NOUN
ejpam-6059	390	5	,	,	PUNCT
ejpam-6059	390	6	14(1):r45	14(1):r45	NUM
ejpam-6059	390	7	,	,	PUNCT
ejpam-6059	390	8	2007	2007	NUM
ejpam-6059	390	9	.	.	PUNCT
ejpam-6059	391	1	[	[	X
ejpam-6059	391	2	3	3	X
ejpam-6059	391	3	]	]	X
ejpam-6059	391	4	d.	d.	PROPN
ejpam-6059	391	5	kiani	kiani	PROPN
ejpam-6059	391	6	and	and	CCONJ
ejpam-6059	391	7	m.	m.	PROPN
ejpam-6059	391	8	m.	m.	PROPN
ejpam-6059	391	9	h.	h.	PROPN
ejpam-6059	391	10	aghaei	aghaei	PROPN
ejpam-6059	391	11	.	.	PUNCT
ejpam-6059	392	1	on	on	ADP
ejpam-6059	392	2	the	the	DET
ejpam-6059	392	3	unitary	unitary	ADJ
ejpam-6059	392	4	cayley	cayley	ADJ
ejpam-6059	392	5	graph	graph	NOUN
ejpam-6059	392	6	of	of	ADP
ejpam-6059	392	7	a	a	DET
ejpam-6059	392	8	ring	ring	NOUN
ejpam-6059	392	9	.	.	PUNCT
ejpam-6059	393	1	electronic	electronic	ADJ
ejpam-6059	393	2	journal	journal	NOUN
ejpam-6059	393	3	of	of	ADP
ejpam-6059	393	4	combinatorics	combinatoric	NOUN
ejpam-6059	393	5	,	,	PUNCT
ejpam-6059	393	6	19(2):p10	19(2):p10	NUM
ejpam-6059	393	7	,	,	PUNCT
ejpam-6059	393	8	2012	2012	NUM
ejpam-6059	393	9	.	.	PUNCT
ejpam-6059	394	1	[	[	X
ejpam-6059	394	2	4	4	NUM
ejpam-6059	394	3	]	]	PUNCT
ejpam-6059	394	4	a.	a.	NOUN
ejpam-6059	394	5	naghipour	naghipour	PROPN
ejpam-6059	394	6	.	.	PUNCT
ejpam-6059	395	1	the	the	DET
ejpam-6059	395	2	induced	induced	ADJ
ejpam-6059	395	3	subgraph	subgraph	NOUN
ejpam-6059	395	4	of	of	ADP
ejpam-6059	395	5	the	the	DET
ejpam-6059	395	6	unitary	unitary	ADJ
ejpam-6059	395	7	cayley	cayley	ADJ
ejpam-6059	395	8	graph	graph	NOUN
ejpam-6059	395	9	of	of	ADP
ejpam-6059	395	10	a	a	DET
ejpam-6059	395	11	commutative	commutative	ADJ
ejpam-6059	395	12	ring	ring	NOUN
ejpam-6059	395	13	over	over	ADP
ejpam-6059	395	14	regular	regular	ADJ
ejpam-6059	395	15	elements	element	NOUN
ejpam-6059	395	16	.	.	PUNCT
ejpam-6059	396	1	miskolc	miskolc	ADJ
ejpam-6059	396	2	mathematical	mathematical	ADJ
ejpam-6059	396	3	notes	note	NOUN
ejpam-6059	396	4	,	,	PUNCT
ejpam-6059	396	5	17(2):965–977	17(2):965–977	PROPN
ejpam-6059	396	6	,	,	PUNCT
ejpam-6059	396	7	2016	2016	NUM
ejpam-6059	396	8	.	.	PUNCT
ejpam-6059	397	1	[	[	X
ejpam-6059	397	2	5	5	X
ejpam-6059	397	3	]	]	PUNCT
ejpam-6059	397	4	v.	v.	CCONJ
ejpam-6059	397	5	samodivkin	samodivkin	NOUN
ejpam-6059	397	6	.	.	PUNCT
ejpam-6059	398	1	acyclic	acyclic	ADJ
ejpam-6059	398	2	number	number	NOUN
ejpam-6059	398	3	of	of	ADP
ejpam-6059	398	4	graphs	graph	NOUN
ejpam-6059	398	5	.	.	PUNCT
ejpam-6059	399	1	acta	acta	PROPN
ejpam-6059	399	2	mathematica	mathematica	PROPN
ejpam-6059	399	3	academiae	academiae	PROPN
ejpam-6059	399	4	paedagogicae	paedagogicae	VERB
ejpam-6059	399	5	nýıregyháziensis	nýıregyháziensis	NOUN
ejpam-6059	399	6	,	,	PUNCT
ejpam-6059	399	7	25(1):1–7	25(1):1–7	NUM
ejpam-6059	399	8	,	,	PUNCT
ejpam-6059	399	9	2009	2009	NUM
ejpam-6059	399	10	.	.	PUNCT
ejpam-6059	400	1	[	[	X
ejpam-6059	400	2	6	6	NUM
ejpam-6059	400	3	]	]	PUNCT
ejpam-6059	400	4	m.	m.	NOUN
ejpam-6059	400	5	petrusevski	petrusevski	NOUN
ejpam-6059	400	6	and	and	CCONJ
ejpam-6059	400	7	r.	r.	PROPN
ejpam-6059	400	8	škrekovski	škrekovski	PROPN
ejpam-6059	400	9	.	.	PUNCT
ejpam-6059	401	1	a	a	DET
ejpam-6059	401	2	note	note	NOUN
ejpam-6059	401	3	on	on	ADP
ejpam-6059	401	4	acyclic	acyclic	ADJ
ejpam-6059	401	5	number	number	NOUN
ejpam-6059	401	6	of	of	ADP
ejpam-6059	401	7	planar	planar	ADJ
ejpam-6059	401	8	graphs	graph	NOUN
ejpam-6059	401	9	.	.	PUNCT
ejpam-6059	402	1	ars	ars	PROPN
ejpam-6059	402	2	mathematica	mathematica	PROPN
ejpam-6059	402	3	contemporanea	contemporanea	PROPN
ejpam-6059	402	4	,	,	PUNCT
ejpam-6059	402	5	13(2):317–322	13(2):317–322	PROPN
ejpam-6059	402	6	,	,	PUNCT
ejpam-6059	402	7	2017	2017	NUM
ejpam-6059	402	8	.	.	PUNCT
ejpam-6059	403	1	[	[	X
ejpam-6059	403	2	7	7	X
ejpam-6059	403	3	]	]	X
ejpam-6059	403	4	d.	d.	PROPN
ejpam-6059	403	5	b.	b.	PROPN
ejpam-6059	403	6	west	west	PROPN
ejpam-6059	403	7	.	.	PUNCT
ejpam-6059	404	1	introduction	introduction	NOUN
ejpam-6059	404	2	to	to	AUX
ejpam-6059	404	3	graph	graph	NOUN
ejpam-6059	404	4	theory	theory	NOUN
ejpam-6059	404	5	.	.	PUNCT
ejpam-6059	405	1	pearson	pearson	PROPN
ejpam-6059	405	2	education	education	PROPN
ejpam-6059	405	3	,	,	PUNCT
ejpam-6059	405	4	singapore	singapore	PROPN
ejpam-6059	405	5	,	,	PUNCT
ejpam-6059	405	6	2	2	NUM
ejpam-6059	405	7	edition	edition	NOUN
ejpam-6059	405	8	,	,	PUNCT
ejpam-6059	405	9	2002	2002	NUM
ejpam-6059	405	10	.	.	PUNCT
ejpam-6059	406	1	[	[	X
ejpam-6059	406	2	8	8	NUM
ejpam-6059	406	3	]	]	X
ejpam-6059	406	4	y.	y.	NOUN
ejpam-6059	406	5	meemark	meemark	PROPN
ejpam-6059	406	6	.	.	PUNCT
ejpam-6059	407	1	abstract	abstract	ADJ
ejpam-6059	407	2	algebra	algebra	PROPN
ejpam-6059	407	3	.	.	PUNCT
ejpam-6059	408	1	danex	danex	PROPN
ejpam-6059	408	2	intercorporation	intercorporation	PROPN
ejpam-6059	408	3	co.	co.	PROPN
ejpam-6059	408	4	,	,	PUNCT
ejpam-6059	408	5	ltd	ltd	PROPN
ejpam-6059	408	6	.	.	PROPN
ejpam-6059	408	7	,	,	PUNCT
ejpam-6059	408	8	bangkok	bangkok	PROPN
ejpam-6059	408	9	,	,	PUNCT
ejpam-6059	408	10	thailand	thailand	PROPN
ejpam-6059	408	11	,	,	PUNCT
ejpam-6059	408	12	2015	2015	NUM
ejpam-6059	408	13	.	.	PUNCT
ejpam-6059	409	1	[	[	X
ejpam-6059	409	2	9	9	NUM
ejpam-6059	409	3	]	]	X
ejpam-6059	409	4	h.	h.	PROPN
ejpam-6059	409	5	e.	e.	PROPN
ejpam-6059	409	6	rose	rise	VERB
ejpam-6059	409	7	.	.	PUNCT
ejpam-6059	410	1	a	a	DET
ejpam-6059	410	2	course	course	NOUN
ejpam-6059	410	3	in	in	ADP
ejpam-6059	410	4	number	number	NOUN
ejpam-6059	410	5	theory	theory	NOUN
ejpam-6059	410	6	.	.	PUNCT
ejpam-6059	411	1	oxford	oxford	PROPN
ejpam-6059	411	2	science	science	PROPN
ejpam-6059	411	3	publications	publication	NOUN
ejpam-6059	411	4	.	.	PUNCT
ejpam-6059	412	1	oxford	oxford	PROPN
ejpam-6059	412	2	university	university	PROPN
ejpam-6059	412	3	press	press	NOUN
ejpam-6059	412	4	,	,	PUNCT
ejpam-6059	412	5	oxford	oxford	PROPN
ejpam-6059	412	6	,	,	PUNCT
ejpam-6059	412	7	england	england	PROPN
ejpam-6059	412	8	,	,	PUNCT
ejpam-6059	412	9	1994	1994	NUM
ejpam-6059	412	10	.	.	PUNCT
ejpam-6059	413	1	[	[	X
ejpam-6059	413	2	10	10	NUM
ejpam-6059	413	3	]	]	X
ejpam-6059	413	4	d.	d.	PROPN
ejpam-6059	413	5	tipyotha	tipyotha	PROPN
ejpam-6059	413	6	.	.	PUNCT
ejpam-6059	414	1	number	number	NOUN
ejpam-6059	414	2	theory	theory	NOUN
ejpam-6059	414	3	.	.	PUNCT
ejpam-6059	415	1	chulalongkorn	chulalongkorn	PROPN
ejpam-6059	415	2	university	university	NOUN
ejpam-6059	415	3	press	press	NOUN
ejpam-6059	415	4	,	,	PUNCT
ejpam-6059	415	5	bangkok	bangkok	PROPN
ejpam-6059	415	6	,	,	PUNCT
ejpam-6059	415	7	thailand	thailand	PROPN
ejpam-6059	415	8	,	,	PUNCT
ejpam-6059	415	9	2012	2012	NUM
ejpam-6059	415	10	.	.	PUNCT
ejpam-6059	416	1	[	[	X
ejpam-6059	416	2	11	11	NUM
ejpam-6059	416	3	]	]	X
ejpam-6059	416	4	d.	d.	NOUN
ejpam-6059	416	5	pongpipat	pongpipat	PROPN
ejpam-6059	416	6	and	and	CCONJ
ejpam-6059	416	7	n.	n.	PROPN
ejpam-6059	416	8	nupo	nupo	NOUN
ejpam-6059	416	9	.	.	PUNCT
ejpam-6059	417	1	nordhaus	nordhaus	NOUN
ejpam-6059	417	2	-	-	PUNCT
ejpam-6059	417	3	gaddum	gaddum	PROPN
ejpam-6059	417	4	type	type	NOUN
ejpam-6059	417	5	inequalities	inequality	NOUN
ejpam-6059	417	6	for	for	ADP
ejpam-6059	417	7	tree	tree	NOUN
ejpam-6059	417	8	covering	cover	VERB
ejpam-6059	417	9	numbers	number	NOUN
ejpam-6059	417	10	on	on	ADP
ejpam-6059	417	11	unitary	unitary	ADJ
ejpam-6059	417	12	cayley	cayley	ADJ
ejpam-6059	417	13	graphs	graph	NOUN
ejpam-6059	417	14	of	of	ADP
ejpam-6059	417	15	finite	finite	ADJ
ejpam-6059	417	16	rings	ring	NOUN
ejpam-6059	417	17	.	.	PUNCT
ejpam-6059	418	1	transactions	transaction	NOUN
ejpam-6059	418	2	on	on	ADP
ejpam-6059	418	3	combinatorics	combinatoric	NOUN
ejpam-6059	418	4	,	,	PUNCT
ejpam-6059	418	5	11(2):111–122	11(2):111–122	PROPN
ejpam-6059	418	6	,	,	PUNCT
ejpam-6059	418	7	2022	2022	NUM
ejpam-6059	418	8	.	.	PUNCT