id	sid	tid	token	lemma	pos
ejpam-577	1	1	11_577_germina.dvi	11_577_germina.dvi	NUM
ejpam-577	1	2	european	european	ADJ
ejpam-577	1	3	journal	journal	NOUN
ejpam-577	1	4	of	of	ADP
ejpam-577	1	5	pure	pure	ADJ
ejpam-577	1	6	and	and	CCONJ
ejpam-577	1	7	applied	apply	VERB
ejpam-577	1	8	mathematics	mathematic	NOUN
ejpam-577	1	9	vol	vol	NOUN
ejpam-577	1	10	.	.	PUNCT
ejpam-577	2	1	3	3	NUM
ejpam-577	2	2	,	,	PUNCT
ejpam-577	2	3	no	no	INTJ
ejpam-577	2	4	.	.	NOUN
ejpam-577	2	5	2	2	NUM
ejpam-577	2	6	,	,	PUNCT
ejpam-577	2	7	2010	2010	NUM
ejpam-577	2	8	,	,	PUNCT
ejpam-577	2	9	269	269	NUM
ejpam-577	2	10	-	-	SYM
ejpam-577	2	11	281	281	NUM
ejpam-577	2	12	issn	issn	PROPN
ejpam-577	2	13	1307	1307	NUM
ejpam-577	2	14	-	-	SYM
ejpam-577	2	15	5543	5543	NUM
ejpam-577	2	16	–	–	PUNCT
ejpam-577	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-577	2	18	more	more	ADV
ejpam-577	2	19	on	on	ADP
ejpam-577	2	20	classes	class	NOUN
ejpam-577	2	21	of	of	ADP
ejpam-577	2	22	strongly	strongly	ADV
ejpam-577	2	23	indexable	indexable	ADJ
ejpam-577	2	24	graphs	graph	NOUN
ejpam-577	3	1	k.	k.	PROPN
ejpam-577	3	2	a.	a.	PROPN
ejpam-577	3	3	germina	germina	PROPN
ejpam-577	3	4	p.g	p.g	PROPN
ejpam-577	3	5	.	.	PROPN
ejpam-577	3	6	&	&	CCONJ
ejpam-577	3	7	research	research	PROPN
ejpam-577	3	8	department	department	PROPN
ejpam-577	3	9	of	of	ADP
ejpam-577	3	10	mathematics	mathematics	PROPN
ejpam-577	3	11	,	,	PUNCT
ejpam-577	3	12	mary	mary	PROPN
ejpam-577	3	13	matha	matha	PROPN
ejpam-577	3	14	arts	arts	PROPN
ejpam-577	3	15	&	&	CCONJ
ejpam-577	3	16	science	science	PROPN
ejpam-577	3	17	college	college	PROPN
ejpam-577	3	18	(	(	PUNCT
ejpam-577	3	19	kannur	kannur	PROPN
ejpam-577	3	20	university	university	NOUN
ejpam-577	3	21	)	)	PUNCT
ejpam-577	3	22	,	,	PUNCT
ejpam-577	3	23	mananthavady-670645	mananthavady-670645	PROPN
ejpam-577	3	24	,	,	PUNCT
ejpam-577	3	25	india	india	PROPN
ejpam-577	3	26	.	.	PUNCT
ejpam-577	3	27	abstract	abstract	PROPN
ejpam-577	3	28	.	.	PUNCT
ejpam-577	4	1	given	give	VERB
ejpam-577	4	2	any	any	DET
ejpam-577	4	3	positive	positive	ADJ
ejpam-577	4	4	integer	integer	NOUN
ejpam-577	4	5	k	k	PROPN
ejpam-577	4	6	,	,	PUNCT
ejpam-577	4	7	a	a	DET
ejpam-577	4	8	(	(	PUNCT
ejpam-577	4	9	p	p	NOUN
ejpam-577	4	10	,	,	PUNCT
ejpam-577	4	11	q)-graph	q)-graph	NOUN
ejpam-577	4	12	g	g	NOUN
ejpam-577	4	13	=	=	SYM
ejpam-577	4	14	(	(	PUNCT
ejpam-577	4	15	v	v	NOUN
ejpam-577	4	16	,	,	PUNCT
ejpam-577	4	17	e	e	NOUN
ejpam-577	4	18	)	)	PUNCT
ejpam-577	4	19	is	be	AUX
ejpam-577	4	20	strongly	strongly	ADV
ejpam-577	4	21	k	k	ADJ
ejpam-577	4	22	-	-	ADJ
ejpam-577	4	23	indexable	indexable	ADJ
ejpam-577	4	24	if	if	SCONJ
ejpam-577	4	25	there	there	PRON
ejpam-577	4	26	exists	exist	VERB
ejpam-577	4	27	a	a	DET
ejpam-577	4	28	bijection	bijection	NOUN
ejpam-577	4	29	f	f	NOUN
ejpam-577	4	30	:	:	PUNCT
ejpam-577	4	31	v	v	X
ejpam-577	4	32	→	→	SYM
ejpam-577	4	33	{	{	PUNCT
ejpam-577	4	34	0,1,2	0,1,2	NOUN
ejpam-577	4	35	,	,	PUNCT
ejpam-577	4	36	.	.	PUNCT
ejpam-577	4	37	.	.	PUNCT
ejpam-577	5	1	.	.	PUNCT
ejpam-577	6	1	,	,	PUNCT
ejpam-577	7	1	p	p	NOUN
ejpam-577	7	2	−	−	PROPN
ejpam-577	7	3	1	1	NUM
ejpam-577	7	4	}	}	PUNCT
ejpam-577	7	5	such	such	ADJ
ejpam-577	7	6	that	that	SCONJ
ejpam-577	7	7	f	f	PROPN
ejpam-577	7	8	+	+	ADJ
ejpam-577	7	9	(	(	PUNCT
ejpam-577	7	10	e(g	e(g	PROPN
ejpam-577	7	11	)	)	PUNCT
ejpam-577	7	12	)	)	PUNCT
ejpam-577	8	1	=	=	PRON
ejpam-577	8	2	{	{	PUNCT
ejpam-577	8	3	k	k	NOUN
ejpam-577	8	4	,	,	PUNCT
ejpam-577	8	5	k	k	PROPN
ejpam-577	9	1	+	+	PROPN
ejpam-577	9	2	1	1	NUM
ejpam-577	9	3	,	,	PUNCT
ejpam-577	9	4	k	k	PROPN
ejpam-577	9	5	+	+	PROPN
ejpam-577	9	6	2	2	NUM
ejpam-577	9	7	,	,	PUNCT
ejpam-577	9	8	.	.	PUNCT
ejpam-577	9	9	.	.	PUNCT
ejpam-577	10	1	.	.	PUNCT
ejpam-577	11	1	,	,	PUNCT
ejpam-577	11	2	k	k	PROPN
ejpam-577	12	1	+	+	CCONJ
ejpam-577	12	2	q	q	ADJ
ejpam-577	12	3	−	−	PROPN
ejpam-577	12	4	1	1	NUM
ejpam-577	12	5	,	,	PUNCT
ejpam-577	12	6	}	}	PUNCT
ejpam-577	12	7	where	where	SCONJ
ejpam-577	12	8	f	f	PROPN
ejpam-577	12	9	+	+	ADJ
ejpam-577	12	10	(	(	PUNCT
ejpam-577	12	11	uv	uv	NOUN
ejpam-577	12	12	)	)	PUNCT
ejpam-577	12	13	=	=	SYM
ejpam-577	12	14	f	f	PROPN
ejpam-577	12	15	(	(	PUNCT
ejpam-577	12	16	u)+	u)+	PROPN
ejpam-577	12	17	f	f	PROPN
ejpam-577	12	18	(	(	PUNCT
ejpam-577	12	19	v	v	NOUN
ejpam-577	12	20	)	)	PUNCT
ejpam-577	12	21	for	for	ADP
ejpam-577	12	22	any	any	DET
ejpam-577	12	23	edge	edge	NOUN
ejpam-577	12	24	uv	uv	NOUN
ejpam-577	12	25	∈	∈	NOUN
ejpam-577	12	26	e	e	NOUN
ejpam-577	12	27	;	;	PUNCT
ejpam-577	12	28	in	in	ADP
ejpam-577	12	29	particular	particular	ADJ
ejpam-577	12	30	,	,	PUNCT
ejpam-577	12	31	g	g	PROPN
ejpam-577	12	32	is	be	AUX
ejpam-577	12	33	said	say	VERB
ejpam-577	12	34	to	to	PART
ejpam-577	12	35	be	be	AUX
ejpam-577	12	36	strongly	strongly	ADV
ejpam-577	12	37	indexable	indexable	ADJ
ejpam-577	13	1	when	when	SCONJ
ejpam-577	13	2	k	k	PROPN
ejpam-577	13	3	=	=	SYM
ejpam-577	13	4	1	1	X
ejpam-577	13	5	.	.	X
ejpam-577	13	6	for	for	ADP
ejpam-577	13	7	any	any	DET
ejpam-577	13	8	strongly	strongly	ADV
ejpam-577	13	9	k	k	NOUN
ejpam-577	13	10	-	-	ADJ
ejpam-577	13	11	indexable	indexable	ADJ
ejpam-577	13	12	(	(	PUNCT
ejpam-577	13	13	p	p	NOUN
ejpam-577	13	14	,	,	PUNCT
ejpam-577	13	15	q)-graph	q)-graph	NOUN
ejpam-577	13	16	g	g	NOUN
ejpam-577	13	17	,	,	PUNCT
ejpam-577	13	18	q	q	PROPN
ejpam-577	13	19	≤	≤	NUM
ejpam-577	13	20	2p	2p	NUM
ejpam-577	13	21	−	−	NOUN
ejpam-577	13	22	3	3	NUM
ejpam-577	13	23	and	and	CCONJ
ejpam-577	13	24	if	if	SCONJ
ejpam-577	13	25	,	,	PUNCT
ejpam-577	13	26	in	in	ADP
ejpam-577	13	27	particular	particular	ADJ
ejpam-577	13	28	,	,	PUNCT
ejpam-577	13	29	q	q	NOUN
ejpam-577	13	30	=	=	SYM
ejpam-577	13	31	2p	2p	NUM
ejpam-577	13	32	−	−	NOUN
ejpam-577	13	33	3	3	NUM
ejpam-577	13	34	then	then	ADV
ejpam-577	13	35	g	g	PROPN
ejpam-577	13	36	is	be	AUX
ejpam-577	13	37	called	call	VERB
ejpam-577	13	38	a	a	DET
ejpam-577	13	39	maximal	maximal	ADJ
ejpam-577	13	40	strongly	strongly	ADV
ejpam-577	13	41	indexable	indexable	ADJ
ejpam-577	13	42	graph	graph	NOUN
ejpam-577	13	43	.	.	PUNCT
ejpam-577	14	1	in	in	ADP
ejpam-577	14	2	this	this	DET
ejpam-577	14	3	paper	paper	NOUN
ejpam-577	14	4	,	,	PUNCT
ejpam-577	14	5	our	our	PRON
ejpam-577	14	6	main	main	ADJ
ejpam-577	14	7	focus	focus	NOUN
ejpam-577	14	8	is	be	AUX
ejpam-577	14	9	to	to	PART
ejpam-577	14	10	construct	construct	VERB
ejpam-577	14	11	more	more	ADJ
ejpam-577	14	12	classes	class	NOUN
ejpam-577	14	13	of	of	ADP
ejpam-577	14	14	k	k	ADV
ejpam-577	14	15	-	-	PUNCT
ejpam-577	14	16	strongly	strongly	ADV
ejpam-577	14	17	indexable	indexable	ADJ
ejpam-577	14	18	graphs	graph	NOUN
ejpam-577	14	19	.	.	PUNCT
ejpam-577	15	1	2000	2000	NUM
ejpam-577	15	2	mathematics	mathematic	NOUN
ejpam-577	15	3	subject	subject	NOUN
ejpam-577	15	4	classifications	classification	NOUN
ejpam-577	15	5	:	:	PUNCT
ejpam-577	15	6	05c78	05c78	NUM
ejpam-577	15	7	key	key	ADJ
ejpam-577	15	8	words	word	NOUN
ejpam-577	15	9	and	and	CCONJ
ejpam-577	15	10	phrases	phrase	NOUN
ejpam-577	15	11	:	:	PUNCT
ejpam-577	15	12	strongly	strongly	ADV
ejpam-577	15	13	indexable	indexable	ADJ
ejpam-577	15	14	,	,	PUNCT
ejpam-577	15	15	edge	edge	NOUN
ejpam-577	15	16	-	-	PUNCT
ejpam-577	15	17	magic	magic	NOUN
ejpam-577	15	18	,	,	PUNCT
ejpam-577	15	19	super	super	ADJ
ejpam-577	15	20	-	-	ADJ
ejpam-577	15	21	edge	edge	ADJ
ejpam-577	15	22	-	-	PUNCT
ejpam-577	15	23	magic	magic	NOUN
ejpam-577	15	24	,	,	PUNCT
ejpam-577	15	25	graphs	graph	VERB
ejpam-577	15	26	1	1	NUM
ejpam-577	15	27	.	.	PUNCT
ejpam-577	15	28	introduction	introduction	NOUN
ejpam-577	15	29	unless	unless	SCONJ
ejpam-577	15	30	mentioned	mention	VERB
ejpam-577	15	31	otherwise	otherwise	ADV
ejpam-577	15	32	,	,	PUNCT
ejpam-577	15	33	by	by	ADP
ejpam-577	15	34	a	a	DET
ejpam-577	15	35	graph	graph	NOUN
ejpam-577	15	36	we	we	PRON
ejpam-577	15	37	shall	shall	AUX
ejpam-577	15	38	mean	mean	VERB
ejpam-577	15	39	in	in	ADP
ejpam-577	15	40	this	this	DET
ejpam-577	15	41	paper	paper	NOUN
ejpam-577	15	42	a	a	DET
ejpam-577	15	43	finite	finite	ADJ
ejpam-577	15	44	,	,	PUNCT
ejpam-577	15	45	undirected	undirected	ADJ
ejpam-577	15	46	,	,	PUNCT
ejpam-577	15	47	connected	connected	ADJ
ejpam-577	15	48	graph	graph	NOUN
ejpam-577	15	49	without	without	ADP
ejpam-577	15	50	loops	loop	NOUN
ejpam-577	15	51	or	or	CCONJ
ejpam-577	15	52	multiple	multiple	ADJ
ejpam-577	15	53	edges	edge	NOUN
ejpam-577	15	54	.	.	PUNCT
ejpam-577	16	1	terms	term	NOUN
ejpam-577	16	2	not	not	PART
ejpam-577	16	3	defined	define	VERB
ejpam-577	16	4	here	here	ADV
ejpam-577	16	5	are	be	AUX
ejpam-577	16	6	used	use	VERB
ejpam-577	16	7	in	in	ADP
ejpam-577	16	8	the	the	DET
ejpam-577	16	9	sense	sense	NOUN
ejpam-577	16	10	of	of	ADP
ejpam-577	16	11	harary	harary	NOUN
ejpam-577	16	12	[	[	X
ejpam-577	16	13	11	11	NUM
ejpam-577	16	14	]	]	PUNCT
ejpam-577	16	15	.	.	PUNCT
ejpam-577	17	1	acharya	acharya	PROPN
ejpam-577	17	2	et.al	et.al	PROPN
ejpam-577	18	1	[	[	X
ejpam-577	18	2	2	2	NUM
ejpam-577	18	3	]	]	PUNCT
ejpam-577	18	4	introduced	introduce	VERB
ejpam-577	18	5	the	the	DET
ejpam-577	18	6	concept	concept	NOUN
ejpam-577	18	7	of	of	ADP
ejpam-577	18	8	an	an	DET
ejpam-577	18	9	’	'	PUNCT
ejpam-577	18	10	indexer	indexer	NOUN
ejpam-577	18	11	’	'	PUNCT
ejpam-577	18	12	of	of	ADP
ejpam-577	18	13	a	a	DET
ejpam-577	18	14	graph	graph	NOUN
ejpam-577	18	15	as	as	ADP
ejpam-577	18	16	a	a	DET
ejpam-577	18	17	special	special	ADJ
ejpam-577	18	18	case	case	NOUN
ejpam-577	18	19	of	of	ADP
ejpam-577	18	20	arithmetic	arithmetic	ADJ
ejpam-577	18	21	labelings	labeling	NOUN
ejpam-577	18	22	.	.	PUNCT
ejpam-577	19	1	a	a	DET
ejpam-577	19	2	labeling	labeling	NOUN
ejpam-577	19	3	of	of	ADP
ejpam-577	19	4	a	a	DET
ejpam-577	19	5	graph	graph	NOUN
ejpam-577	19	6	g	g	NOUN
ejpam-577	19	7	=	=	SYM
ejpam-577	19	8	(	(	PUNCT
ejpam-577	19	9	v	v	NOUN
ejpam-577	19	10	,	,	PUNCT
ejpam-577	19	11	e	e	NOUN
ejpam-577	19	12	)	)	PUNCT
ejpam-577	19	13	is	be	AUX
ejpam-577	19	14	an	an	DET
ejpam-577	19	15	assignment	assignment	NOUN
ejpam-577	19	16	f	f	NOUN
ejpam-577	19	17	of	of	ADP
ejpam-577	19	18	distinct	distinct	ADJ
ejpam-577	19	19	nonnegative	nonnegative	ADJ
ejpam-577	19	20	integers	integer	NOUN
ejpam-577	19	21	to	to	ADP
ejpam-577	19	22	the	the	DET
ejpam-577	19	23	vertices	vertex	NOUN
ejpam-577	19	24	of	of	ADP
ejpam-577	19	25	g	g	NOUN
ejpam-577	19	26	;	;	PUNCT
ejpam-577	19	27	it	it	PRON
ejpam-577	19	28	is	be	AUX
ejpam-577	19	29	an	an	DET
ejpam-577	19	30	indexer	indexer	NOUN
ejpam-577	19	31	of	of	ADP
ejpam-577	19	32	g	g	PROPN
ejpam-577	19	33	if	if	SCONJ
ejpam-577	19	34	the	the	DET
ejpam-577	19	35	induced	induced	ADJ
ejpam-577	19	36	’	'	PUNCT
ejpam-577	19	37	edge	edge	NOUN
ejpam-577	19	38	function	function	NOUN
ejpam-577	19	39	’	'	PUNCT
ejpam-577	19	40	f	f	PROPN
ejpam-577	20	1	+	+	CCONJ
ejpam-577	20	2	:	:	PUNCT
ejpam-577	20	3	e(g)→	e(g)→	NOUN
ejpam-577	20	4	n	n	CCONJ
ejpam-577	20	5	,	,	PUNCT
ejpam-577	20	6	from	from	ADP
ejpam-577	20	7	e(g	e(g	PROPN
ejpam-577	20	8	)	)	PUNCT
ejpam-577	20	9	into	into	ADP
ejpam-577	20	10	the	the	DET
ejpam-577	20	11	set	set	NOUN
ejpam-577	20	12	n	n	PROPN
ejpam-577	20	13	of	of	ADP
ejpam-577	20	14	natural	natural	ADJ
ejpam-577	20	15	numbers	number	NOUN
ejpam-577	20	16	,	,	PUNCT
ejpam-577	20	17	defined	define	VERB
ejpam-577	20	18	by	by	ADP
ejpam-577	20	19	the	the	DET
ejpam-577	20	20	rule	rule	NOUN
ejpam-577	20	21	:	:	PUNCT
ejpam-577	20	22	f	f	PROPN
ejpam-577	21	1	+	+	ADJ
ejpam-577	21	2	(	(	PUNCT
ejpam-577	21	3	uv	uv	NOUN
ejpam-577	21	4	)	)	PUNCT
ejpam-577	21	5	=	=	SYM
ejpam-577	21	6	f	f	PROPN
ejpam-577	21	7	(	(	PUNCT
ejpam-577	21	8	u	u	NOUN
ejpam-577	21	9	)	)	PUNCT
ejpam-577	21	10	+	+	NUM
ejpam-577	21	11	f	f	X
ejpam-577	21	12	(	(	PUNCT
ejpam-577	21	13	v	v	NOUN
ejpam-577	21	14	)	)	PUNCT
ejpam-577	21	15	,	,	PUNCT
ejpam-577	21	16	∀	∀	X
ejpam-577	21	17	uv	uv	NOUN
ejpam-577	21	18	∈	∈	PROPN
ejpam-577	21	19	e(g	e(g	PROPN
ejpam-577	21	20	)	)	PUNCT
ejpam-577	21	21	,	,	PUNCT
ejpam-577	21	22	is	be	AUX
ejpam-577	21	23	also	also	ADV
ejpam-577	21	24	injective	injective	ADJ
ejpam-577	21	25	.	.	PUNCT
ejpam-577	22	1	it	it	PRON
ejpam-577	22	2	is	be	AUX
ejpam-577	22	3	known	know	VERB
ejpam-577	22	4	that	that	SCONJ
ejpam-577	22	5	every	every	DET
ejpam-577	22	6	finite	finite	ADJ
ejpam-577	22	7	graph	graph	NOUN
ejpam-577	22	8	has	have	VERB
ejpam-577	22	9	an	an	DET
ejpam-577	22	10	indexer	indexer	NOUN
ejpam-577	22	11	;	;	PUNCT
ejpam-577	22	12	hence	hence	ADV
ejpam-577	22	13	,	,	PUNCT
ejpam-577	22	14	an	an	DET
ejpam-577	22	15	indexer	indexer	NOUN
ejpam-577	22	16	f	f	PROPN
ejpam-577	22	17	is	be	AUX
ejpam-577	22	18	said	say	VERB
ejpam-577	22	19	to	to	PART
ejpam-577	22	20	be	be	AUX
ejpam-577	22	21	optimal	optimal	ADJ
ejpam-577	22	22	if	if	SCONJ
ejpam-577	22	23	f	f	PROPN
ejpam-577	22	24	[	[	X
ejpam-577	22	25	g	g	X
ejpam-577	22	26	]	]	X
ejpam-577	22	27	:	:	PUNCT
ejpam-577	22	28	=	=	SYM
ejpam-577	22	29	maxv∈v	maxv∈v	NOUN
ejpam-577	22	30	(	(	PUNCT
ejpam-577	22	31	g	g	NOUN
ejpam-577	22	32	)	)	PUNCT
ejpam-577	22	33	{	{	PUNCT
ejpam-577	22	34	f	f	X
ejpam-577	22	35	(	(	PUNCT
ejpam-577	22	36	v	v	NOUN
ejpam-577	22	37	)	)	PUNCT
ejpam-577	22	38	}	}	PUNCT
ejpam-577	22	39	has	have	VERB
ejpam-577	22	40	the	the	DET
ejpam-577	22	41	least	least	ADJ
ejpam-577	22	42	possible	possible	ADJ
ejpam-577	22	43	value	value	NOUN
ejpam-577	22	44	υ(g	υ(g	NOUN
ejpam-577	22	45	)	)	PUNCT
ejpam-577	22	46	amongst	amongst	ADP
ejpam-577	22	47	all	all	DET
ejpam-577	22	48	the	the	DET
ejpam-577	22	49	indexers	indexer	NOUN
ejpam-577	22	50	of	of	ADP
ejpam-577	22	51	g.	g.	PROPN
ejpam-577	22	52	clearly	clearly	ADV
ejpam-577	22	53	,	,	PUNCT
ejpam-577	22	54	υ(g	υ(g	NOUN
ejpam-577	22	55	)	)	PUNCT
ejpam-577	22	56	≥	≥	NOUN
ejpam-577	22	57	|v	|v	NOUN
ejpam-577	22	58	(	(	PUNCT
ejpam-577	22	59	g)|	g)|	NOUN
ejpam-577	22	60	for	for	ADP
ejpam-577	22	61	any	any	DET
ejpam-577	22	62	graph	graph	NOUN
ejpam-577	22	63	g	g	NOUN
ejpam-577	22	64	with	with	ADP
ejpam-577	22	65	a	a	DET
ejpam-577	22	66	countable	countable	ADJ
ejpam-577	22	67	number	number	NOUN
ejpam-577	22	68	of	of	ADP
ejpam-577	22	69	vertices	vertex	NOUN
ejpam-577	22	70	.	.	PUNCT
ejpam-577	23	1	for	for	ADP
ejpam-577	23	2	any	any	DET
ejpam-577	23	3	given	give	VERB
ejpam-577	23	4	positive	positive	ADJ
ejpam-577	23	5	integer	integer	NOUN
ejpam-577	23	6	k	k	PROPN
ejpam-577	23	7	,	,	PUNCT
ejpam-577	23	8	an	an	DET
ejpam-577	23	9	indexer	indexer	NOUN
ejpam-577	23	10	f	f	PROPN
ejpam-577	23	11	of	of	ADP
ejpam-577	23	12	g	g	PROPN
ejpam-577	23	13	is	be	AUX
ejpam-577	23	14	called	call	VERB
ejpam-577	23	15	a	a	DET
ejpam-577	23	16	k	k	NOUN
ejpam-577	23	17	-	-	NOUN
ejpam-577	23	18	indexer	indexer	NOUN
ejpam-577	23	19	if	if	SCONJ
ejpam-577	23	20	f	f	PROPN
ejpam-577	23	21	+	+	PROPN
ejpam-577	23	22	(	(	PUNCT
ejpam-577	23	23	e(g	e(g	PROPN
ejpam-577	23	24	)	)	PUNCT
ejpam-577	23	25	)	)	PUNCT
ejpam-577	23	26	:	:	PUNCT
ejpam-577	24	1	=	=	X
ejpam-577	24	2	{	{	PUNCT
ejpam-577	24	3	f	f	PROPN
ejpam-577	24	4	+	+	ADJ
ejpam-577	24	5	(	(	PUNCT
ejpam-577	24	6	uv	uv	NOUN
ejpam-577	24	7	)	)	PUNCT
ejpam-577	24	8	:	:	PUNCT
ejpam-577	24	9	uv	uv	PROPN
ejpam-577	24	10	∈	∈	PROPN
ejpam-577	24	11	e(g	e(g	PROPN
ejpam-577	24	12	)	)	PUNCT
ejpam-577	24	13	}	}	PUNCT
ejpam-577	24	14	=	=	SYM
ejpam-577	24	15	{	{	PUNCT
ejpam-577	24	16	k	k	NOUN
ejpam-577	24	17	,	,	PUNCT
ejpam-577	24	18	k+	k+	NOUN
ejpam-577	24	19	1	1	NUM
ejpam-577	24	20	,	,	PUNCT
ejpam-577	24	21	k+	k+	NOUN
ejpam-577	24	22	2	2	NUM
ejpam-577	24	23	,	,	PUNCT
ejpam-577	24	24	.	.	PUNCT
ejpam-577	24	25	.	.	PUNCT
ejpam-577	24	26	.	.	PUNCT
ejpam-577	25	1	,	,	PUNCT
ejpam-577	25	2	}	}	PUNCT
ejpam-577	25	3	.	.	PUNCT
ejpam-577	26	1	not	not	PART
ejpam-577	26	2	every	every	DET
ejpam-577	26	3	graph	graph	NOUN
ejpam-577	26	4	is	be	AUX
ejpam-577	26	5	k	k	NOUN
ejpam-577	26	6	-	-	ADJ
ejpam-577	26	7	indexable	indexable	ADJ
ejpam-577	26	8	as	as	SCONJ
ejpam-577	26	9	indicated	indicate	VERB
ejpam-577	26	10	by	by	ADP
ejpam-577	26	11	the	the	DET
ejpam-577	26	12	following	following	NOUN
ejpam-577	26	13	theorem	theorem	NOUN
ejpam-577	26	14	for	for	ADP
ejpam-577	26	15	finite	finite	ADJ
ejpam-577	26	16	graphs	graph	NOUN
ejpam-577	26	17	.	.	PUNCT
ejpam-577	27	1	email	email	NOUN
ejpam-577	27	2	address	address	NOUN
ejpam-577	27	3	:	:	PUNCT
ejpam-577	27	4	srgerminaka	srgerminaka	NOUN
ejpam-577	27	5	�	�	NOUN
ejpam-577	27	6	gmail	gmail	NOUN
ejpam-577	27	7	.	.	PUNCT
ejpam-577	28	1	om	om	PROPN
ejpam-577	28	2	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-577	28	3	269	269	NUM
ejpam-577	29	1	c	c	X
ejpam-577	29	2	©	©	PROPN
ejpam-577	29	3	2010	2010	NUM
ejpam-577	29	4	ejpam	ejpam	NOUN
ejpam-577	29	5	all	all	DET
ejpam-577	29	6	rights	right	NOUN
ejpam-577	29	7	reserved	reserve	VERB
ejpam-577	29	8	.	.	PUNCT
ejpam-577	30	1	k.	k.	PROPN
ejpam-577	30	2	a.	a.	PROPN
ejpam-577	30	3	germina	germina	PROPN
ejpam-577	30	4	/	/	SYM
ejpam-577	30	5	eur	eur	PROPN
ejpam-577	30	6	.	.	PUNCT
ejpam-577	31	1	j.	j.	PROPN
ejpam-577	31	2	pure	pure	PROPN
ejpam-577	31	3	appl	appl	PROPN
ejpam-577	31	4	.	.	PROPN
ejpam-577	31	5	math	math	PROPN
ejpam-577	31	6	,	,	PUNCT
ejpam-577	31	7	3	3	NUM
ejpam-577	31	8	(	(	PUNCT
ejpam-577	31	9	2010	2010	NUM
ejpam-577	31	10	)	)	PUNCT
ejpam-577	31	11	,	,	PUNCT
ejpam-577	31	12	269	269	NUM
ejpam-577	31	13	-	-	SYM
ejpam-577	31	14	281	281	NUM
ejpam-577	31	15	270	270	NUM
ejpam-577	31	16	theorem	theorem	NOUN
ejpam-577	31	17	1	1	NUM
ejpam-577	31	18	.	.	PUNCT
ejpam-577	32	1	[	[	X
ejpam-577	32	2	2	2	NUM
ejpam-577	32	3	]	]	PUNCT
ejpam-577	32	4	:	:	PUNCT
ejpam-577	32	5	let	let	VERB
ejpam-577	32	6	g	g	PROPN
ejpam-577	32	7	=	=	SYM
ejpam-577	32	8	(	(	PUNCT
ejpam-577	32	9	v	v	NOUN
ejpam-577	32	10	,	,	PUNCT
ejpam-577	32	11	e	e	NOUN
ejpam-577	32	12	)	)	PUNCT
ejpam-577	32	13	be	be	VERB
ejpam-577	32	14	any	any	DET
ejpam-577	32	15	(	(	PUNCT
ejpam-577	32	16	p	p	NOUN
ejpam-577	32	17	,	,	PUNCT
ejpam-577	32	18	q)-graph	q)-graph	NOUN
ejpam-577	32	19	and	and	CCONJ
ejpam-577	32	20	f	f	PROPN
ejpam-577	32	21	be	be	AUX
ejpam-577	32	22	any	any	DET
ejpam-577	32	23	k	k	NOUN
ejpam-577	32	24	-	-	NOUN
ejpam-577	32	25	indexer	indexer	NOUN
ejpam-577	32	26	of	of	ADP
ejpam-577	32	27	g	g	NOUN
ejpam-577	32	28	,	,	PUNCT
ejpam-577	32	29	where	where	SCONJ
ejpam-577	32	30	k	k	PROPN
ejpam-577	32	31	is	be	AUX
ejpam-577	32	32	odd	odd	ADJ
ejpam-577	32	33	.	.	PUNCT
ejpam-577	33	1	then	then	ADV
ejpam-577	33	2	,	,	PUNCT
ejpam-577	33	3	there	there	PRON
ejpam-577	33	4	exists	exist	VERB
ejpam-577	33	5	an	an	DET
ejpam-577	33	6	’	'	PUNCT
ejpam-577	33	7	equitable	equitable	ADJ
ejpam-577	33	8	partition	partition	NOUN
ejpam-577	33	9	’	'	PUNCT
ejpam-577	33	10	of	of	ADP
ejpam-577	33	11	v	v	NOUN
ejpam-577	33	12	into	into	ADP
ejpam-577	33	13	two	two	NUM
ejpam-577	33	14	subsets	subset	NOUN
ejpam-577	33	15	vo	vo	X
ejpam-577	33	16	and	and	CCONJ
ejpam-577	33	17	ve	ve	VERB
ejpam-577	33	18	such	such	ADJ
ejpam-577	33	19	that	that	SCONJ
ejpam-577	33	20	there	there	PRON
ejpam-577	33	21	are	be	VERB
ejpam-577	33	22	exactly	exactly	ADV
ejpam-577	33	23	⌈	⌈	SYM
ejpam-577	33	24	q+k−1	q+k−1	ADJ
ejpam-577	33	25	2	2	NUM
ejpam-577	33	26	⌉	⌉	PRON
ejpam-577	33	27	edges	edge	VERB
ejpam-577	33	28	each	each	PRON
ejpam-577	33	29	of	of	ADP
ejpam-577	33	30	which	which	PRON
ejpam-577	33	31	joins	join	VERB
ejpam-577	33	32	a	a	DET
ejpam-577	33	33	vertex	vertex	NOUN
ejpam-577	33	34	of	of	ADP
ejpam-577	33	35	vo	vo	NOUN
ejpam-577	33	36	with	with	ADP
ejpam-577	33	37	one	one	NUM
ejpam-577	33	38	of	of	ADP
ejpam-577	33	39	ve	ve	NOUN
ejpam-577	33	40	,	,	PUNCT
ejpam-577	33	41	where	where	SCONJ
ejpam-577	33	42	⌈.⌉	⌈.⌉	VERB
ejpam-577	33	43	denotes	denote	VERB
ejpam-577	33	44	the	the	DET
ejpam-577	33	45	least	least	ADV
ejpam-577	33	46	integer	integer	NOUN
ejpam-577	33	47	function	function	NOUN
ejpam-577	33	48	.	.	PUNCT
ejpam-577	34	1	theorem	theorem	NOUN
ejpam-577	34	2	2	2	NUM
ejpam-577	34	3	.	.	PUNCT
ejpam-577	35	1	[	[	X
ejpam-577	35	2	2	2	NUM
ejpam-577	35	3	]	]	PUNCT
ejpam-577	35	4	:	:	PUNCT
ejpam-577	35	5	for	for	ADP
ejpam-577	35	6	any	any	DET
ejpam-577	35	7	indexable	indexable	ADJ
ejpam-577	35	8	(	(	PUNCT
ejpam-577	35	9	p	p	NOUN
ejpam-577	35	10	,	,	PUNCT
ejpam-577	35	11	q)-graph	q)-graph	NOUN
ejpam-577	35	12	g	g	NOUN
ejpam-577	35	13	,	,	PUNCT
ejpam-577	35	14	q	q	PROPN
ejpam-577	35	15	≤	≤	NUM
ejpam-577	35	16	2p−3	2p−3	NUM
ejpam-577	35	17	,	,	PUNCT
ejpam-577	35	18	calling	call	VERB
ejpam-577	35	19	g	g	PRON
ejpam-577	35	20	a	a	DET
ejpam-577	35	21	maximally	maximally	ADV
ejpam-577	35	22	indexable	indexable	ADJ
ejpam-577	35	23	graph	graph	NOUN
ejpam-577	35	24	if	if	SCONJ
ejpam-577	35	25	q	q	PROPN
ejpam-577	35	26	=	=	SYM
ejpam-577	36	1	2p−	2p−	NUM
ejpam-577	36	2	3	3	NUM
ejpam-577	36	3	.	.	PUNCT
ejpam-577	36	4	acharya	acharya	PROPN
ejpam-577	36	5	and	and	CCONJ
ejpam-577	36	6	germina	germina	PROPN
ejpam-577	37	1	[	[	X
ejpam-577	37	2	3	3	NUM
ejpam-577	37	3	]	]	PUNCT
ejpam-577	37	4	characterized	characterize	VERB
ejpam-577	37	5	the	the	DET
ejpam-577	37	6	classes	class	NOUN
ejpam-577	37	7	of	of	ADP
ejpam-577	37	8	maximal	maximal	ADJ
ejpam-577	37	9	strongly	strongly	ADV
ejpam-577	37	10	indexable	indexable	ADJ
ejpam-577	37	11	graphs	graph	NOUN
ejpam-577	37	12	,	,	PUNCT
ejpam-577	37	13	satisfying	satisfy	VERB
ejpam-577	37	14	q	q	NOUN
ejpam-577	38	1	=	=	PUNCT
ejpam-577	38	2	2p−	2p−	NUM
ejpam-577	38	3	3	3	NUM
ejpam-577	38	4	,	,	PUNCT
ejpam-577	38	5	particularly	particularly	ADV
ejpam-577	38	6	,	,	PUNCT
ejpam-577	38	7	such	such	ADJ
ejpam-577	38	8	outerplanar	outerplanar	NOUN
ejpam-577	38	9	graphs	graph	NOUN
ejpam-577	38	10	.	.	PUNCT
ejpam-577	39	1	we	we	PRON
ejpam-577	39	2	shall	shall	AUX
ejpam-577	39	3	need	need	VERB
ejpam-577	39	4	the	the	DET
ejpam-577	39	5	following	follow	VERB
ejpam-577	39	6	known	know	VERB
ejpam-577	39	7	results	result	NOUN
ejpam-577	39	8	.	.	PUNCT
ejpam-577	40	1	theorem	theorem	VERB
ejpam-577	40	2	3	3	NUM
ejpam-577	40	3	.	.	PUNCT
ejpam-577	41	1	[	[	X
ejpam-577	41	2	1	1	NUM
ejpam-577	41	3	]	]	X
ejpam-577	41	4	:	:	PUNCT
ejpam-577	41	5	for	for	ADP
ejpam-577	41	6	any	any	DET
ejpam-577	41	7	graph	graph	NOUN
ejpam-577	41	8	g	g	NOUN
ejpam-577	41	9	=	=	SYM
ejpam-577	41	10	(	(	PUNCT
ejpam-577	41	11	v	v	NOUN
ejpam-577	41	12	,	,	PUNCT
ejpam-577	41	13	e	e	NOUN
ejpam-577	41	14	)	)	PUNCT
ejpam-577	41	15	and	and	CCONJ
ejpam-577	41	16	for	for	ADP
ejpam-577	41	17	any	any	DET
ejpam-577	41	18	additive	additive	ADJ
ejpam-577	41	19	vertex	vertex	NOUN
ejpam-577	41	20	function	function	NOUN
ejpam-577	41	21	f	f	NOUN
ejpam-577	41	22	:	:	PUNCT
ejpam-577	41	23	v	v	NOUN
ejpam-577	41	24	(	(	PUNCT
ejpam-577	41	25	g)→	g)→	NOUN
ejpam-577	41	26	n	n	CCONJ
ejpam-577	41	27	,	,	PUNCT
ejpam-577	41	28	σe∈e	σe∈e	PROPN
ejpam-577	41	29	f	f	PROPN
ejpam-577	41	30	+	+	ADJ
ejpam-577	41	31	(	(	PUNCT
ejpam-577	41	32	e	e	NOUN
ejpam-577	41	33	)	)	PUNCT
ejpam-577	41	34	=	=	SYM
ejpam-577	42	1	σu∈v	σu∈v	PROPN
ejpam-577	42	2	f	f	PROPN
ejpam-577	42	3	(	(	PUNCT
ejpam-577	42	4	u)d(u	u)d(u	NOUN
ejpam-577	42	5	)	)	PUNCT
ejpam-577	42	6	theorem	theorem	VERB
ejpam-577	42	7	4	4	NUM
ejpam-577	42	8	.	.	PUNCT
ejpam-577	43	1	[	[	X
ejpam-577	43	2	2	2	NUM
ejpam-577	43	3	]	]	PUNCT
ejpam-577	43	4	:	:	PUNCT
ejpam-577	43	5	every	every	DET
ejpam-577	43	6	strongly	strongly	ADV
ejpam-577	43	7	indexable	indexable	ADJ
ejpam-577	43	8	finite	finite	ADJ
ejpam-577	43	9	graph	graph	NOUN
ejpam-577	43	10	has	have	VERB
ejpam-577	43	11	at	at	ADP
ejpam-577	43	12	most	most	ADJ
ejpam-577	43	13	one	one	NUM
ejpam-577	43	14	nontrivial	nontrivial	ADJ
ejpam-577	43	15	component	component	NOUN
ejpam-577	43	16	which	which	PRON
ejpam-577	43	17	is	be	AUX
ejpam-577	43	18	either	either	CCONJ
ejpam-577	43	19	a	a	DET
ejpam-577	43	20	star	star	NOUN
ejpam-577	43	21	or	or	CCONJ
ejpam-577	43	22	has	have	VERB
ejpam-577	43	23	a	a	DET
ejpam-577	43	24	triangle	triangle	NOUN
ejpam-577	43	25	.	.	PUNCT
ejpam-577	44	1	lemma	lemma	PROPN
ejpam-577	44	2	1	1	NUM
ejpam-577	44	3	.	.	PUNCT
ejpam-577	45	1	[	[	X
ejpam-577	45	2	4	4	NUM
ejpam-577	45	3	]	]	PUNCT
ejpam-577	45	4	:	:	PUNCT
ejpam-577	45	5	let	let	VERB
ejpam-577	45	6	g	g	PROPN
ejpam-577	45	7	=	=	SYM
ejpam-577	45	8	(	(	PUNCT
ejpam-577	45	9	v	v	NOUN
ejpam-577	45	10	,	,	PUNCT
ejpam-577	45	11	e	e	NOUN
ejpam-577	45	12	)	)	PUNCT
ejpam-577	45	13	be	be	AUX
ejpam-577	45	14	a	a	DET
ejpam-577	45	15	maximal	maximal	ADJ
ejpam-577	45	16	outerplanar	outerplanar	NOUN
ejpam-577	45	17	graph	graph	NOUN
ejpam-577	45	18	with	with	ADP
ejpam-577	45	19	p	p	PROPN
ejpam-577	45	20	>	>	X
ejpam-577	45	21	7	7	X
ejpam-577	45	22	.	.	PUNCT
ejpam-577	46	1	let	let	VERB
ejpam-577	46	2	h	h	NOUN
ejpam-577	46	3	=	=	PUNCT
ejpam-577	46	4	(	(	PUNCT
ejpam-577	46	5	u1,u2,u3	u1,u2,u3	NOUN
ejpam-577	46	6	,	,	PUNCT
ejpam-577	46	7	.	.	PUNCT
ejpam-577	46	8	.	.	PUNCT
ejpam-577	47	1	.	.	PUNCT
ejpam-577	48	1	,	,	PUNCT
ejpam-577	48	2	up	up	ADV
ejpam-577	48	3	)	)	PUNCT
ejpam-577	48	4	be	be	AUX
ejpam-577	48	5	a	a	DET
ejpam-577	48	6	hamiltonian	hamiltonian	ADJ
ejpam-577	48	7	cycle	cycle	NOUN
ejpam-577	48	8	in	in	ADP
ejpam-577	48	9	g.	g.	PROPN
ejpam-577	48	10	let	let	VERB
ejpam-577	48	11	v1	v1	VERB
ejpam-577	48	12	=	=	SYM
ejpam-577	48	13	{	{	PUNCT
ejpam-577	48	14	u1,u2,u3	u1,u2,u3	NOUN
ejpam-577	48	15	,	,	PUNCT
ejpam-577	48	16	.	.	PUNCT
ejpam-577	48	17	.	.	PUNCT
ejpam-577	49	1	.	.	PUNCT
ejpam-577	50	1	,	,	PUNCT
ejpam-577	51	1	u⌊	u⌊	NUM
ejpam-577	51	2	p	p	NOUN
ejpam-577	51	3	2	2	NUM
ejpam-577	51	4	⌋	⌋	NOUN
ejpam-577	51	5	}	}	PUNCT
ejpam-577	51	6	and	and	CCONJ
ejpam-577	51	7	v2	v2	NOUN
ejpam-577	51	8	=	=	SYM
ejpam-577	51	9	{	{	PUNCT
ejpam-577	51	10	u⌊	u⌊	PROPN
ejpam-577	51	11	p	p	NOUN
ejpam-577	51	12	2	2	NUM
ejpam-577	51	13	⌋+1,u⌊	⌋+1,u⌊	PUNCT
ejpam-577	51	14	p	p	NOUN
ejpam-577	51	15	2	2	NUM
ejpam-577	51	16	⌋+2,u⌊	⌋+2,u⌊	PUNCT
ejpam-577	51	17	p	p	VERB
ejpam-577	51	18	2	2	NUM
ejpam-577	51	19	⌋+3	⌋+3	NOUN
ejpam-577	51	20	,	,	PUNCT
ejpam-577	51	21	.	.	PUNCT
ejpam-577	51	22	.	.	PUNCT
ejpam-577	52	1	.	.	PUNCT
ejpam-577	53	1	,	,	PUNCT
ejpam-577	53	2	up	up	ADP
ejpam-577	53	3	}	}	PUNCT
ejpam-577	53	4	constitute	constitute	VERB
ejpam-577	53	5	an	an	DET
ejpam-577	53	6	equitable	equitable	ADJ
ejpam-577	53	7	partition	partition	NOUN
ejpam-577	53	8	of	of	ADP
ejpam-577	53	9	the	the	DET
ejpam-577	53	10	vertex	vertex	NOUN
ejpam-577	53	11	set	set	NOUN
ejpam-577	53	12	of	of	ADP
ejpam-577	53	13	g.	g.	PROPN
ejpam-577	53	14	then	then	ADV
ejpam-577	53	15	,	,	PUNCT
ejpam-577	53	16	no	no	DET
ejpam-577	53	17	chord	chord	NOUN
ejpam-577	53	18	of	of	ADP
ejpam-577	53	19	g	g	PROPN
ejpam-577	53	20	has	have	AUX
ejpam-577	53	21	both	both	DET
ejpam-577	53	22	vertices	vertex	NOUN
ejpam-577	53	23	in	in	ADP
ejpam-577	53	24	v1	v1	NOUN
ejpam-577	53	25	or	or	CCONJ
ejpam-577	53	26	v2	v2	VERB
ejpam-577	53	27	if	if	SCONJ
ejpam-577	54	1	and	and	CCONJ
ejpam-577	54	2	only	only	ADV
ejpam-577	54	3	if	if	SCONJ
ejpam-577	54	4	∆(g	∆(g	NOUN
ejpam-577	54	5	)	)	PUNCT
ejpam-577	54	6	=	=	PUNCT
ejpam-577	55	1	⌊	⌊	AUX
ejpam-577	55	2	p	p	NOUN
ejpam-577	55	3	2	2	NUM
ejpam-577	55	4	⌋+	⌋+	NUM
ejpam-577	55	5	2	2	NUM
ejpam-577	55	6	and	and	CCONJ
ejpam-577	55	7	there	there	PRON
ejpam-577	55	8	exist	exist	VERB
ejpam-577	55	9	exactly	exactly	ADV
ejpam-577	55	10	two	two	NUM
ejpam-577	55	11	vertices	vertex	NOUN
ejpam-577	55	12	of	of	ADP
ejpam-577	55	13	degree	degree	NOUN
ejpam-577	55	14	2	2	NUM
ejpam-577	55	15	.	.	PUNCT
ejpam-577	55	16	theorem	theorem	NOUN
ejpam-577	55	17	5	5	NUM
ejpam-577	55	18	.	.	PUNCT
ejpam-577	56	1	[	[	X
ejpam-577	56	2	4	4	NUM
ejpam-577	56	3	]	]	PUNCT
ejpam-577	56	4	:	:	PUNCT
ejpam-577	56	5	let	let	VERB
ejpam-577	56	6	g	g	PROPN
ejpam-577	56	7	=	=	SYM
ejpam-577	56	8	(	(	PUNCT
ejpam-577	56	9	v	v	NOUN
ejpam-577	56	10	,	,	PUNCT
ejpam-577	56	11	e	e	NOUN
ejpam-577	56	12	)	)	PUNCT
ejpam-577	56	13	be	be	AUX
ejpam-577	56	14	a	a	DET
ejpam-577	56	15	maximal	maximal	ADJ
ejpam-577	56	16	outerplanar	outerplanar	NOUN
ejpam-577	56	17	graph	graph	NOUN
ejpam-577	56	18	with	with	ADP
ejpam-577	56	19	p	p	PROPN
ejpam-577	56	20	>	>	X
ejpam-577	56	21	7	7	NUM
ejpam-577	56	22	.	.	PUNCT
ejpam-577	57	1	then	then	ADV
ejpam-577	57	2	,	,	PUNCT
ejpam-577	57	3	g	g	PROPN
ejpam-577	57	4	is	be	AUX
ejpam-577	57	5	strongly	strongly	ADV
ejpam-577	57	6	indexable	indexable	ADJ
ejpam-577	57	7	if	if	SCONJ
ejpam-577	57	8	and	and	CCONJ
ejpam-577	57	9	only	only	ADV
ejpam-577	57	10	if	if	SCONJ
ejpam-577	57	11	∆(g	∆(g	NOUN
ejpam-577	57	12	)	)	PUNCT
ejpam-577	57	13	=	=	PUNCT
ejpam-577	58	1	⌊	⌊	ADP
ejpam-577	58	2	p	p	NOUN
ejpam-577	58	3	2	2	NUM
ejpam-577	58	4	⌋+	⌋+	NUM
ejpam-577	58	5	2	2	NUM
ejpam-577	58	6	and	and	CCONJ
ejpam-577	58	7	there	there	PRON
ejpam-577	58	8	exist	exist	VERB
ejpam-577	58	9	exactly	exactly	ADV
ejpam-577	58	10	two	two	NUM
ejpam-577	58	11	vertices	vertex	NOUN
ejpam-577	58	12	of	of	ADP
ejpam-577	58	13	degree	degree	NOUN
ejpam-577	58	14	2	2	NUM
ejpam-577	58	15	.	.	NOUN
ejpam-577	58	16	2	2	NUM
ejpam-577	58	17	.	.	X
ejpam-577	59	1	construction	construction	NOUN
ejpam-577	59	2	of	of	ADP
ejpam-577	59	3	strongly	strongly	ADV
ejpam-577	59	4	indexable	indexable	ADJ
ejpam-577	59	5	graphs	graph	NOUN
ejpam-577	59	6	definition	definition	NOUN
ejpam-577	59	7	1	1	X
ejpam-577	59	8	.	.	PUNCT
ejpam-577	60	1	let	let	VERB
ejpam-577	60	2	g1	g1	PROPN
ejpam-577	60	3	=	=	SYM
ejpam-577	60	4	(	(	PUNCT
ejpam-577	60	5	v1	v1	PROPN
ejpam-577	60	6	,	,	PUNCT
ejpam-577	60	7	e1	e1	NOUN
ejpam-577	60	8	)	)	PUNCT
ejpam-577	60	9	and	and	CCONJ
ejpam-577	60	10	g2	g2	PROPN
ejpam-577	60	11	=	=	PUNCT
ejpam-577	60	12	(	(	PUNCT
ejpam-577	60	13	v2	v2	PROPN
ejpam-577	60	14	,	,	PUNCT
ejpam-577	60	15	e2	e2	PROPN
ejpam-577	60	16	)	)	PUNCT
ejpam-577	60	17	be	be	VERB
ejpam-577	60	18	two	two	NUM
ejpam-577	60	19	graphs	graph	NOUN
ejpam-577	60	20	.	.	PUNCT
ejpam-577	61	1	then	then	ADV
ejpam-577	61	2	the	the	DET
ejpam-577	61	3	join	join	NOUN
ejpam-577	61	4	g	g	PROPN
ejpam-577	61	5	=	=	SYM
ejpam-577	61	6	(	(	PUNCT
ejpam-577	61	7	v	v	NOUN
ejpam-577	61	8	,	,	PUNCT
ejpam-577	61	9	e	e	NOUN
ejpam-577	61	10	)	)	PUNCT
ejpam-577	61	11	of	of	ADP
ejpam-577	61	12	g1	g1	PROPN
ejpam-577	61	13	and	and	CCONJ
ejpam-577	61	14	g2	g2	PROPN
ejpam-577	61	15	is	be	AUX
ejpam-577	61	16	defines	define	NOUN
ejpam-577	61	17	as	as	ADP
ejpam-577	61	18	the	the	DET
ejpam-577	61	19	v	v	NOUN
ejpam-577	61	20	=	=	SYM
ejpam-577	61	21	v1	v1	PROPN
ejpam-577	61	22	+	+	X
ejpam-577	61	23	v2	v2	NOUN
ejpam-577	61	24	and	and	CCONJ
ejpam-577	61	25	the	the	DET
ejpam-577	61	26	edge	edge	NOUN
ejpam-577	61	27	set	set	VERB
ejpam-577	61	28	e	e	NOUN
ejpam-577	61	29	of	of	ADP
ejpam-577	61	30	g	g	PROPN
ejpam-577	61	31	is	be	AUX
ejpam-577	61	32	the	the	DET
ejpam-577	61	33	edges	edge	NOUN
ejpam-577	61	34	in	in	ADP
ejpam-577	61	35	g1	g1	NOUN
ejpam-577	61	36	∪g2	∪g2	PROPN
ejpam-577	61	37	and	and	CCONJ
ejpam-577	61	38	all	all	DET
ejpam-577	61	39	edges	edge	NOUN
ejpam-577	61	40	joining	join	VERB
ejpam-577	61	41	g1	g1	PROPN
ejpam-577	61	42	and	and	CCONJ
ejpam-577	61	43	g2	g2	PROPN
ejpam-577	61	44	.	.	PUNCT
ejpam-577	62	1	in	in	ADP
ejpam-577	62	2	this	this	DET
ejpam-577	62	3	section	section	NOUN
ejpam-577	62	4	we	we	PRON
ejpam-577	62	5	study	study	VERB
ejpam-577	62	6	the	the	DET
ejpam-577	62	7	properties	property	NOUN
ejpam-577	62	8	of	of	ADP
ejpam-577	62	9	some	some	DET
ejpam-577	62	10	important	important	ADJ
ejpam-577	62	11	families	family	NOUN
ejpam-577	62	12	of	of	ADP
ejpam-577	62	13	graphs	graph	NOUN
ejpam-577	62	14	such	such	ADJ
ejpam-577	62	15	as	as	ADP
ejpam-577	62	16	fans	fan	NOUN
ejpam-577	62	17	,	,	PUNCT
ejpam-577	62	18	ladders	ladder	NOUN
ejpam-577	62	19	,	,	PUNCT
ejpam-577	62	20	and	and	CCONJ
ejpam-577	62	21	generalized	generalized	ADJ
ejpam-577	62	22	prisms	prism	NOUN
ejpam-577	62	23	that	that	PRON
ejpam-577	62	24	are	be	AUX
ejpam-577	62	25	strongly	strongly	ADV
ejpam-577	62	26	indexable	indexable	ADJ
ejpam-577	62	27	(	(	PUNCT
ejpam-577	62	28	for	for	ADP
ejpam-577	62	29	some	some	DET
ejpam-577	62	30	value	value	NOUN
ejpam-577	62	31	of	of	ADP
ejpam-577	62	32	k	k	NOUN
ejpam-577	62	33	)	)	PUNCT
ejpam-577	62	34	.	.	PUNCT
ejpam-577	63	1	theorem	theorem	VERB
ejpam-577	63	2	6	6	NUM
ejpam-577	63	3	.	.	PUNCT
ejpam-577	64	1	the	the	DET
ejpam-577	64	2	fan	fan	PROPN
ejpam-577	64	3	pn	pn	PROPN
ejpam-577	64	4	+	+	CCONJ
ejpam-577	64	5	k1	k1	PROPN
ejpam-577	64	6	is	be	AUX
ejpam-577	64	7	strongly	strongly	ADV
ejpam-577	64	8	indexable	indexable	ADJ
ejpam-577	64	9	if	if	SCONJ
ejpam-577	64	10	and	and	CCONJ
ejpam-577	64	11	only	only	ADV
ejpam-577	64	12	if	if	SCONJ
ejpam-577	64	13	n	n	PRON
ejpam-577	64	14	∈	∈	PROPN
ejpam-577	64	15	{	{	PUNCT
ejpam-577	64	16	1,2,3,4,5,6	1,2,3,4,5,6	NUM
ejpam-577	64	17	}	}	PUNCT
ejpam-577	64	18	.	.	PUNCT
ejpam-577	65	1	proof	proof	NOUN
ejpam-577	65	2	.	.	PUNCT
ejpam-577	66	1	the	the	DET
ejpam-577	66	2	strongly	strongly	ADV
ejpam-577	66	3	indexable	indexable	ADJ
ejpam-577	66	4	labellings	labelling	NOUN
ejpam-577	66	5	of	of	ADP
ejpam-577	66	6	pn	pn	PROPN
ejpam-577	66	7	+	+	CCONJ
ejpam-577	66	8	k1	k1	X
ejpam-577	66	9	for	for	ADP
ejpam-577	66	10	n	n	PRON
ejpam-577	66	11	∈	∈	NOUN
ejpam-577	66	12	{	{	PUNCT
ejpam-577	66	13	1,2,3,4,5,6	1,2,3,4,5,6	NUM
ejpam-577	66	14	}	}	PUNCT
ejpam-577	66	15	is	be	AUX
ejpam-577	66	16	depicted	depict	VERB
ejpam-577	66	17	in	in	ADP
ejpam-577	66	18	figure	figure	NOUN
ejpam-577	66	19	1	1	NUM
ejpam-577	66	20	conversely	conversely	ADV
ejpam-577	66	21	,	,	PUNCT
ejpam-577	66	22	note	note	VERB
ejpam-577	66	23	that	that	SCONJ
ejpam-577	66	24	pn	pn	PROPN
ejpam-577	66	25	+	+	CCONJ
ejpam-577	66	26	k1	k1	NOUN
ejpam-577	66	27	,	,	PUNCT
ejpam-577	66	28	for	for	ADP
ejpam-577	66	29	n	n	PRON
ejpam-577	66	30	≥	≥	NOUN
ejpam-577	66	31	2	2	NUM
ejpam-577	66	32	is	be	AUX
ejpam-577	66	33	a	a	DET
ejpam-577	66	34	maximal	maximal	ADJ
ejpam-577	66	35	outer	outer	ADJ
ejpam-577	66	36	planar	planar	ADJ
ejpam-577	66	37	graph	graph	NOUN
ejpam-577	66	38	with	with	ADP
ejpam-577	66	39	q	q	PROPN
ejpam-577	67	1	=	=	PUNCT
ejpam-577	67	2	2p−	2p−	NUM
ejpam-577	67	3	3	3	NUM
ejpam-577	68	1	and	and	CCONJ
ejpam-577	68	2	there	there	PRON
ejpam-577	68	3	exists	exist	VERB
ejpam-577	68	4	a	a	DET
ejpam-577	68	5	vertex	vertex	NOUN
ejpam-577	68	6	of	of	ADP
ejpam-577	68	7	full	full	ADJ
ejpam-577	68	8	degree	degree	NOUN
ejpam-577	68	9	.	.	PUNCT
ejpam-577	69	1	hence	hence	ADV
ejpam-577	69	2	invoking	invoke	VERB
ejpam-577	69	3	lemma	lemma	PROPN
ejpam-577	69	4	1	1	NUM
ejpam-577	69	5	(	(	PUNCT
ejpam-577	69	6	see	see	VERB
ejpam-577	69	7	,	,	PUNCT
ejpam-577	69	8	[	[	X
ejpam-577	69	9	4	4	NUM
ejpam-577	69	10	]	]	NUM
ejpam-577	69	11	)	)	PUNCT
ejpam-577	69	12	,	,	PUNCT
ejpam-577	69	13	g	g	PROPN
ejpam-577	69	14	is	be	AUX
ejpam-577	69	15	strongly	strongly	ADV
ejpam-577	69	16	indexable	indexable	ADJ
ejpam-577	69	17	if	if	SCONJ
ejpam-577	69	18	and	and	CCONJ
ejpam-577	69	19	only	only	ADV
ejpam-577	69	20	if	if	SCONJ
ejpam-577	69	21	p	p	PRON
ejpam-577	69	22	≤	≤	ADV
ejpam-577	69	23	7	7	NUM
ejpam-577	69	24	.	.	PUNCT
ejpam-577	70	1	hence	hence	ADV
ejpam-577	70	2	the	the	DET
ejpam-577	70	3	proof	proof	NOUN
ejpam-577	70	4	follows	follow	VERB
ejpam-577	70	5	.	.	PUNCT
ejpam-577	71	1	theorem	theorem	ADJ
ejpam-577	71	2	7	7	NUM
ejpam-577	71	3	.	.	PUNCT
ejpam-577	72	1	pn	pn	PROPN
ejpam-577	72	2	+	+	CCONJ
ejpam-577	72	3	k2	k2	PROPN
ejpam-577	72	4	is	be	AUX
ejpam-577	72	5	strongly	strongly	ADV
ejpam-577	72	6	indexable	indexable	ADJ
ejpam-577	72	7	if	if	SCONJ
ejpam-577	72	8	and	and	CCONJ
ejpam-577	72	9	only	only	ADV
ejpam-577	72	10	if	if	SCONJ
ejpam-577	72	11	n≤	n≤	PRON
ejpam-577	72	12	2	2	NUM
ejpam-577	72	13	.	.	PUNCT
ejpam-577	72	14	k.	k.	PROPN
ejpam-577	72	15	a.	a.	PROPN
ejpam-577	72	16	germina	germina	PROPN
ejpam-577	72	17	/	/	SYM
ejpam-577	72	18	eur	eur	PROPN
ejpam-577	72	19	.	.	PUNCT
ejpam-577	73	1	j.	j.	PROPN
ejpam-577	73	2	pure	pure	PROPN
ejpam-577	73	3	appl	appl	PROPN
ejpam-577	73	4	.	.	PROPN
ejpam-577	73	5	math	math	PROPN
ejpam-577	73	6	,	,	PUNCT
ejpam-577	73	7	3	3	NUM
ejpam-577	73	8	(	(	PUNCT
ejpam-577	73	9	2010	2010	NUM
ejpam-577	73	10	)	)	PUNCT
ejpam-577	73	11	,	,	PUNCT
ejpam-577	73	12	269	269	NUM
ejpam-577	73	13	-	-	SYM
ejpam-577	73	14	281	281	NUM
ejpam-577	73	15	271	271	NUM
ejpam-577	73	16	figure	figure	NOUN
ejpam-577	73	17	1	1	NUM
ejpam-577	73	18	proof	proof	NOUN
ejpam-577	73	19	.	.	PUNCT
ejpam-577	74	1	the	the	DET
ejpam-577	74	2	strongly	strongly	ADV
ejpam-577	74	3	indexable	indexable	ADJ
ejpam-577	74	4	labellings	labelling	NOUN
ejpam-577	74	5	of	of	ADP
ejpam-577	74	6	pn	pn	PROPN
ejpam-577	74	7	+	+	CCONJ
ejpam-577	74	8	k2	k2	PROPN
ejpam-577	74	9	for	for	ADP
ejpam-577	74	10	n=	n=	ADJ
ejpam-577	74	11	1,2	1,2	NUM
ejpam-577	74	12	is	be	AUX
ejpam-577	74	13	depicted	depict	VERB
ejpam-577	74	14	in	in	ADP
ejpam-577	74	15	figure	figure	NOUN
ejpam-577	74	16	2	2	NUM
ejpam-577	74	17	figure	figure	NOUN
ejpam-577	74	18	2	2	NUM
ejpam-577	74	19	converse	converse	NOUN
ejpam-577	74	20	follows	follow	VERB
ejpam-577	74	21	from	from	ADP
ejpam-577	74	22	the	the	DET
ejpam-577	74	23	fact	fact	NOUN
ejpam-577	74	24	that	that	SCONJ
ejpam-577	74	25	for	for	ADP
ejpam-577	74	26	any	any	DET
ejpam-577	74	27	indexable	indexable	ADJ
ejpam-577	74	28	(	(	PUNCT
ejpam-577	74	29	p	p	NOUN
ejpam-577	74	30	,	,	PUNCT
ejpam-577	74	31	q)-graph	q)-graph	NOUN
ejpam-577	74	32	g	g	NOUN
ejpam-577	74	33	,	,	PUNCT
ejpam-577	74	34	q	q	PROPN
ejpam-577	74	35	≤	≤	NUM
ejpam-577	74	36	2p	2p	NUM
ejpam-577	74	37	−	−	NOUN
ejpam-577	74	38	3	3	NUM
ejpam-577	74	39	(	(	PUNCT
ejpam-577	74	40	see	see	VERB
ejpam-577	74	41	[	[	X
ejpam-577	74	42	1	1	NUM
ejpam-577	74	43	,	,	PUNCT
ejpam-577	74	44	2	2	NUM
ejpam-577	74	45	]	]	PUNCT
ejpam-577	74	46	)	)	PUNCT
ejpam-577	74	47	,	,	PUNCT
ejpam-577	74	48	since	since	SCONJ
ejpam-577	74	49	|e(pn+	|e(pn+	ADJ
ejpam-577	74	50	k2)|	k2)|	PROPN
ejpam-577	74	51	>	>	X
ejpam-577	74	52	2|v	2|v	PROPN
ejpam-577	74	53	(	(	PUNCT
ejpam-577	74	54	pn+	pn+	NOUN
ejpam-577	74	55	k2)|	k2)|	PROPN
ejpam-577	74	56	−	−	PROPN
ejpam-577	74	57	3	3	X
ejpam-577	74	58	.	.	PUNCT
ejpam-577	75	1	in	in	ADP
ejpam-577	75	2	general	general	ADJ
ejpam-577	75	3	,	,	PUNCT
ejpam-577	75	4	we	we	PRON
ejpam-577	75	5	have	have	VERB
ejpam-577	75	6	the	the	DET
ejpam-577	75	7	following	follow	VERB
ejpam-577	75	8	theorem	theorem	NOUN
ejpam-577	75	9	theorem	theorem	NOUN
ejpam-577	75	10	8	8	NUM
ejpam-577	75	11	.	.	PUNCT
ejpam-577	76	1	pn	pn	PROPN
ejpam-577	77	1	+	+	CCONJ
ejpam-577	77	2	ki	ki	PROPN
ejpam-577	77	3	is	be	AUX
ejpam-577	77	4	strongly	strongly	ADV
ejpam-577	77	5	indexable	indexable	ADJ
ejpam-577	77	6	if	if	SCONJ
ejpam-577	77	7	and	and	CCONJ
ejpam-577	77	8	only	only	ADV
ejpam-577	77	9	if	if	SCONJ
ejpam-577	77	10	n	n	PRON
ejpam-577	77	11	≤	≤	ADV
ejpam-577	77	12	2	2	NUM
ejpam-577	77	13	,	,	PUNCT
ejpam-577	77	14	when	when	SCONJ
ejpam-577	77	15	i	i	PRON
ejpam-577	77	16	≤	≤	ADV
ejpam-577	77	17	2	2	NUM
ejpam-577	77	18	,	,	PUNCT
ejpam-577	77	19	and	and	CCONJ
ejpam-577	77	20	n	n	PRON
ejpam-577	77	21	≤	≤	NOUN
ejpam-577	77	22	6	6	NUM
ejpam-577	77	23	,	,	PUNCT
ejpam-577	77	24	when	when	SCONJ
ejpam-577	77	25	i	i	PRON
ejpam-577	77	26	=	=	NOUN
ejpam-577	77	27	1	1	NUM
ejpam-577	77	28	lemma	lemma	PROPN
ejpam-577	77	29	2	2	NUM
ejpam-577	77	30	.	.	PUNCT
ejpam-577	78	1	for	for	ADP
ejpam-577	78	2	every	every	DET
ejpam-577	78	3	positive	positive	ADJ
ejpam-577	78	4	integer	integer	NOUN
ejpam-577	78	5	n	n	CCONJ
ejpam-577	78	6	,	,	PUNCT
ejpam-577	78	7	the	the	DET
ejpam-577	78	8	graph	graph	NOUN
ejpam-577	78	9	k2	k2	NOUN
ejpam-577	78	10	+	+	CCONJ
ejpam-577	78	11	nk1	nk1	NOUN
ejpam-577	78	12	is	be	AUX
ejpam-577	78	13	strongly	strongly	ADV
ejpam-577	78	14	indexable	indexable	ADJ
ejpam-577	78	15	.	.	PUNCT
ejpam-577	79	1	proof	proof	NOUN
ejpam-577	79	2	.	.	PUNCT
ejpam-577	80	1	let	let	VERB
ejpam-577	80	2	v	v	NOUN
ejpam-577	80	3	(	(	PUNCT
ejpam-577	80	4	k2	k2	NOUN
ejpam-577	80	5	)	)	PUNCT
ejpam-577	80	6	=	=	SYM
ejpam-577	80	7	{	{	PUNCT
ejpam-577	80	8	v1	v1	NOUN
ejpam-577	80	9	,	,	PUNCT
ejpam-577	80	10	v2	v2	NOUN
ejpam-577	80	11	}	}	PUNCT
ejpam-577	80	12	and	and	CCONJ
ejpam-577	80	13	v	v	NOUN
ejpam-577	80	14	(	(	PUNCT
ejpam-577	80	15	nk1	nk1	NOUN
ejpam-577	80	16	)	)	PUNCT
ejpam-577	80	17	=	=	PRON
ejpam-577	80	18	{	{	PUNCT
ejpam-577	80	19	u1,u2	u1,u2	PROPN
ejpam-577	80	20	,	,	PUNCT
ejpam-577	80	21	.	.	PUNCT
ejpam-577	80	22	.	.	PUNCT
ejpam-577	80	23	.	.	PUNCT
ejpam-577	81	1	,	,	PUNCT
ejpam-577	81	2	un	un	PROPN
ejpam-577	81	3	}	}	PUNCT
ejpam-577	81	4	.	.	PUNCT
ejpam-577	82	1	let	let	VERB
ejpam-577	82	2	f	f	NOUN
ejpam-577	82	3	:	:	PUNCT
ejpam-577	82	4	v	v	X
ejpam-577	82	5	(	(	PUNCT
ejpam-577	82	6	k2	k2	X
ejpam-577	82	7	+	+	CCONJ
ejpam-577	82	8	nk1	nk1	NOUN
ejpam-577	82	9	)	)	PUNCT
ejpam-577	82	10	→	→	SYM
ejpam-577	82	11	{	{	PUNCT
ejpam-577	82	12	0,1,2	0,1,2	NOUN
ejpam-577	82	13	.	.	PUNCT
ejpam-577	82	14	.	.	PUNCT
ejpam-577	82	15	.	.	PUNCT
ejpam-577	83	1	,	,	PUNCT
ejpam-577	83	2	n+	n+	ADP
ejpam-577	83	3	1	1	X
ejpam-577	83	4	}	}	PUNCT
ejpam-577	83	5	defined	define	VERB
ejpam-577	83	6	by	by	ADP
ejpam-577	83	7	f	f	PROPN
ejpam-577	83	8	(	(	PUNCT
ejpam-577	83	9	v1	v1	NOUN
ejpam-577	83	10	)	)	PUNCT
ejpam-577	83	11	=	=	SYM
ejpam-577	83	12	0	0	NUM
ejpam-577	83	13	;	;	PUNCT
ejpam-577	83	14	f	f	PROPN
ejpam-577	83	15	(	(	PUNCT
ejpam-577	83	16	v2	v2	PROPN
ejpam-577	83	17	)	)	PUNCT
ejpam-577	83	18	=	=	PUNCT
ejpam-577	83	19	n+	n+	PUNCT
ejpam-577	83	20	1	1	NUM
ejpam-577	83	21	;	;	PUNCT
ejpam-577	83	22	f	f	PROPN
ejpam-577	83	23	(	(	PUNCT
ejpam-577	83	24	ui	ui	PROPN
ejpam-577	83	25	)	)	PUNCT
ejpam-577	83	26	=	=	SYM
ejpam-577	84	1	i	i	PROPN
ejpam-577	84	2	,	,	PUNCT
ejpam-577	84	3	1≤	1≤	INTJ
ejpam-577	84	4	i	i	PROPN
ejpam-577	84	5	≤	≤	X
ejpam-577	84	6	p−	p−	NOUN
ejpam-577	84	7	2	2	NUM
ejpam-577	84	8	remark	remark	NOUN
ejpam-577	84	9	1	1	NUM
ejpam-577	84	10	.	.	PUNCT
ejpam-577	85	1	lemma	lemma	PROPN
ejpam-577	85	2	2	2	NUM
ejpam-577	85	3	establishes	establish	VERB
ejpam-577	85	4	the	the	DET
ejpam-577	85	5	sharpness	sharpness	NOUN
ejpam-577	85	6	of	of	ADP
ejpam-577	85	7	the	the	DET
ejpam-577	85	8	theorem	theorem	NOUN
ejpam-577	85	9	2	2	NUM
ejpam-577	85	10	and	and	CCONJ
ejpam-577	85	11	hence	hence	ADV
ejpam-577	85	12	we	we	PRON
ejpam-577	85	13	obtain	obtain	VERB
ejpam-577	85	14	a	a	DET
ejpam-577	85	15	sequence	sequence	NOUN
ejpam-577	85	16	of	of	ADP
ejpam-577	85	17	strongly	strongly	ADV
ejpam-577	85	18	indexable	indexable	ADJ
ejpam-577	85	19	graphs	graph	NOUN
ejpam-577	85	20	as	as	SCONJ
ejpam-577	85	21	follows	follow	VERB
ejpam-577	85	22	:	:	PUNCT
ejpam-577	85	23	take	take	VERB
ejpam-577	85	24	the	the	DET
ejpam-577	85	25	labellling	labellling	NOUN
ejpam-577	85	26	f	f	PROPN
ejpam-577	85	27	defined	define	VERB
ejpam-577	85	28	for	for	ADP
ejpam-577	85	29	k2	k2	NOUN
ejpam-577	85	30	+	+	CCONJ
ejpam-577	85	31	nk1	nk1	NOUN
ejpam-577	85	32	.	.	PUNCT
ejpam-577	86	1	remove	remove	VERB
ejpam-577	86	2	the	the	DET
ejpam-577	86	3	edge	edge	NOUN
ejpam-577	86	4	with	with	ADP
ejpam-577	86	5	maximum	maximum	ADJ
ejpam-577	86	6	labelling	labelling	NOUN
ejpam-577	86	7	(	(	PUNCT
ejpam-577	86	8	here	here	ADV
ejpam-577	86	9	2p−	2p−	NUM
ejpam-577	86	10	3	3	NUM
ejpam-577	86	11	)	)	PUNCT
ejpam-577	86	12	and	and	CCONJ
ejpam-577	86	13	continue	continue	VERB
ejpam-577	86	14	this	this	DET
ejpam-577	86	15	process	process	NOUN
ejpam-577	86	16	of	of	ADP
ejpam-577	86	17	removing	remove	VERB
ejpam-577	86	18	the	the	DET
ejpam-577	86	19	edge	edge	NOUN
ejpam-577	86	20	with	with	ADP
ejpam-577	86	21	maximum	maximum	ADJ
ejpam-577	86	22	labelling	labelling	NOUN
ejpam-577	86	23	until	until	SCONJ
ejpam-577	86	24	we	we	PRON
ejpam-577	86	25	arrive	arrive	VERB
ejpam-577	86	26	at	at	ADP
ejpam-577	86	27	k1,n	k1,n	PROPN
ejpam-577	86	28	.	.	PUNCT
ejpam-577	87	1	hence	hence	ADV
ejpam-577	87	2	we	we	PRON
ejpam-577	87	3	are	be	AUX
ejpam-577	87	4	able	able	ADJ
ejpam-577	87	5	to	to	PART
ejpam-577	87	6	characterize	characterize	VERB
ejpam-577	87	7	all	all	DET
ejpam-577	87	8	the	the	DET
ejpam-577	87	9	strongly	strongly	ADV
ejpam-577	87	10	indexable	indexable	ADJ
ejpam-577	87	11	complete	complete	ADJ
ejpam-577	87	12	m	m	NOUN
ejpam-577	87	13	-	-	PUNCT
ejpam-577	87	14	bipartite	bipartite	ADJ
ejpam-577	87	15	graphs	graph	NOUN
ejpam-577	87	16	as	as	SCONJ
ejpam-577	87	17	follows	follow	VERB
ejpam-577	87	18	.	.	PUNCT
ejpam-577	88	1	theorem	theorem	ADJ
ejpam-577	88	2	9	9	NUM
ejpam-577	88	3	.	.	PUNCT
ejpam-577	89	1	the	the	DET
ejpam-577	89	2	only	only	ADJ
ejpam-577	89	3	strongly	strongly	ADV
ejpam-577	89	4	indexable	indexable	ADJ
ejpam-577	89	5	complete	complete	ADJ
ejpam-577	89	6	m	m	ADJ
ejpam-577	89	7	-	-	ADJ
ejpam-577	89	8	partite	partite	ADJ
ejpam-577	89	9	graphs	graph	NOUN
ejpam-577	89	10	are	be	AUX
ejpam-577	89	11	k1,n	k1,n	PROPN
ejpam-577	89	12	and	and	CCONJ
ejpam-577	89	13	k1,1,n	k1,1,n	PROPN
ejpam-577	89	14	,	,	PUNCT
ejpam-577	89	15	for	for	ADP
ejpam-577	89	16	all	all	DET
ejpam-577	89	17	integers	integer	NOUN
ejpam-577	89	18	n≥	n≥	NOUN
ejpam-577	89	19	1	1	NUM
ejpam-577	89	20	k.	k.	NOUN
ejpam-577	89	21	a.	a.	PROPN
ejpam-577	89	22	germina	germina	PROPN
ejpam-577	89	23	/	/	SYM
ejpam-577	89	24	eur	eur	PROPN
ejpam-577	89	25	.	.	PUNCT
ejpam-577	90	1	j.	j.	PROPN
ejpam-577	90	2	pure	pure	PROPN
ejpam-577	90	3	appl	appl	PROPN
ejpam-577	90	4	.	.	PROPN
ejpam-577	90	5	math	math	PROPN
ejpam-577	90	6	,	,	PUNCT
ejpam-577	90	7	3	3	NUM
ejpam-577	90	8	(	(	PUNCT
ejpam-577	90	9	2010	2010	NUM
ejpam-577	90	10	)	)	PUNCT
ejpam-577	90	11	,	,	PUNCT
ejpam-577	90	12	269	269	NUM
ejpam-577	90	13	-	-	SYM
ejpam-577	90	14	281	281	NUM
ejpam-577	90	15	272	272	NUM
ejpam-577	90	16	proof	proof	NOUN
ejpam-577	90	17	.	.	PUNCT
ejpam-577	91	1	it	it	PRON
ejpam-577	91	2	is	be	AUX
ejpam-577	91	3	easy	easy	ADJ
ejpam-577	91	4	to	to	PART
ejpam-577	91	5	see	see	VERB
ejpam-577	91	6	that	that	SCONJ
ejpam-577	91	7	k1,n	k1,n	PROPN
ejpam-577	91	8	is	be	AUX
ejpam-577	91	9	strongly	strongly	ADV
ejpam-577	91	10	indexable	indexable	ADJ
ejpam-577	91	11	by	by	ADP
ejpam-577	91	12	assigning	assign	VERB
ejpam-577	91	13	0	0	NUM
ejpam-577	91	14	to	to	ADP
ejpam-577	91	15	the	the	DET
ejpam-577	91	16	central	central	ADJ
ejpam-577	91	17	vertex	vertex	NOUN
ejpam-577	91	18	and	and	CCONJ
ejpam-577	91	19	the	the	DET
ejpam-577	91	20	integers	integer	NOUN
ejpam-577	91	21	1,2	1,2	NUM
ejpam-577	91	22	,	,	PUNCT
ejpam-577	91	23	.	.	PUNCT
ejpam-577	91	24	.	.	PUNCT
ejpam-577	91	25	.	.	PUNCT
ejpam-577	92	1	n−	n−	NOUN
ejpam-577	92	2	1	1	NUM
ejpam-577	92	3	to	to	ADP
ejpam-577	92	4	the	the	DET
ejpam-577	92	5	non	non	ADJ
ejpam-577	92	6	-	-	ADJ
ejpam-577	92	7	central	central	ADJ
ejpam-577	92	8	vertices	vertex	NOUN
ejpam-577	92	9	in	in	ADP
ejpam-577	92	10	a	a	DET
ejpam-577	92	11	one	one	NUM
ejpam-577	92	12	-	-	PUNCT
ejpam-577	92	13	one	one	NUM
ejpam-577	92	14	manner	manner	NOUN
ejpam-577	92	15	.	.	PUNCT
ejpam-577	93	1	furthermore	furthermore	ADV
ejpam-577	93	2	,	,	PUNCT
ejpam-577	93	3	the	the	DET
ejpam-577	93	4	complete	complete	ADJ
ejpam-577	93	5	tripartite	tripartite	ADJ
ejpam-577	93	6	graph	graph	NOUN
ejpam-577	93	7	k1,1,n	k1,1,n	PROPN
ejpam-577	93	8	∼=	∼=	PROPN
ejpam-577	93	9	k2	k2	NOUN
ejpam-577	93	10	+	+	CCONJ
ejpam-577	93	11	nk1	nk1	NOUN
ejpam-577	93	12	is	be	AUX
ejpam-577	93	13	strongly	strongly	ADV
ejpam-577	93	14	indexable	indexable	ADJ
ejpam-577	93	15	by	by	ADP
ejpam-577	93	16	lemma	lemma	PROPN
ejpam-577	93	17	2	2	NUM
ejpam-577	93	18	.	.	PUNCT
ejpam-577	93	19	for	for	ADP
ejpam-577	93	20	the	the	DET
ejpam-577	93	21	uniqueness	uniqueness	NOUN
ejpam-577	93	22	of	of	ADP
ejpam-577	93	23	k1,1,n	k1,1,n	PROPN
ejpam-577	93	24	let	let	VERB
ejpam-577	93	25	g	g	NOUN
ejpam-577	93	26	∼=	∼=	PROPN
ejpam-577	93	27	kn1,n2,n3	kn1,n2,n3	NOUN
ejpam-577	93	28	be	be	VERB
ejpam-577	93	29	a	a	DET
ejpam-577	93	30	complete	complete	ADJ
ejpam-577	93	31	tripartite	tripartite	ADJ
ejpam-577	93	32	graph	graph	NOUN
ejpam-577	93	33	with	with	ADP
ejpam-577	93	34	n1	n1	NOUN
ejpam-577	93	35	,	,	PUNCT
ejpam-577	93	36	n2	n2	NOUN
ejpam-577	93	37	,	,	PUNCT
ejpam-577	93	38	n3	n3	VERB
ejpam-577	93	39	≥	≥	NUM
ejpam-577	93	40	1	1	NUM
ejpam-577	93	41	.	.	PUNCT
ejpam-577	94	1	now	now	ADV
ejpam-577	94	2	,	,	PUNCT
ejpam-577	94	3	assume	assume	VERB
ejpam-577	94	4	the	the	DET
ejpam-577	94	5	contrary	contrary	NOUN
ejpam-577	94	6	that	that	SCONJ
ejpam-577	94	7	n2	n2	ADJ
ejpam-577	94	8	≥	≥	NUM
ejpam-577	94	9	2	2	NUM
ejpam-577	94	10	and	and	CCONJ
ejpam-577	94	11	g	g	NOUN
ejpam-577	94	12	is	be	AUX
ejpam-577	94	13	strongly	strongly	ADV
ejpam-577	94	14	indexable	indexable	ADJ
ejpam-577	94	15	.	.	PUNCT
ejpam-577	95	1	the	the	DET
ejpam-577	95	2	order	order	NOUN
ejpam-577	95	3	of	of	ADP
ejpam-577	95	4	g	g	PROPN
ejpam-577	95	5	is	be	AUX
ejpam-577	95	6	n1	n1	ADJ
ejpam-577	95	7	+	+	CCONJ
ejpam-577	95	8	n2	n2	ADJ
ejpam-577	95	9	+	+	CCONJ
ejpam-577	95	10	n3	n3	NOUN
ejpam-577	95	11	and	and	CCONJ
ejpam-577	95	12	the	the	DET
ejpam-577	95	13	size	size	NOUN
ejpam-577	95	14	of	of	ADP
ejpam-577	95	15	g	g	PROPN
ejpam-577	95	16	is	be	AUX
ejpam-577	95	17	n1n2	n1n2	PUNCT
ejpam-577	95	18	+	+	X
ejpam-577	95	19	n1n3	n1n3	ADJ
ejpam-577	95	20	+	+	CCONJ
ejpam-577	95	21	n2n3	n2n3	X
ejpam-577	95	22	.	.	PROPN
ejpam-577	95	23	since	since	SCONJ
ejpam-577	95	24	kn1,n2,n3	kn1,n2,n3	PROPN
ejpam-577	95	25	is	be	AUX
ejpam-577	95	26	strongly	strongly	ADV
ejpam-577	95	27	indexable	indexable	ADJ
ejpam-577	95	28	by	by	ADP
ejpam-577	95	29	assumption	assumption	NOUN
ejpam-577	95	30	,	,	PUNCT
ejpam-577	95	31	n1n2	n1n2	PUNCT
ejpam-577	95	32	+	+	X
ejpam-577	95	33	n1n3	n1n3	ADJ
ejpam-577	95	34	+	+	ADJ
ejpam-577	95	35	n2n3	n2n3	X
ejpam-577	95	36	≤	≤	NUM
ejpam-577	95	37	2(n1	2(n1	NUM
ejpam-577	95	38	+	+	SYM
ejpam-577	95	39	n2	n2	ADJ
ejpam-577	95	40	+	+	CCONJ
ejpam-577	95	41	n3)−	n3)−	PROPN
ejpam-577	95	42	3	3	NUM
ejpam-577	95	43	,	,	PUNCT
ejpam-577	95	44	which	which	PRON
ejpam-577	95	45	in	in	ADP
ejpam-577	95	46	turn	turn	NOUN
ejpam-577	95	47	implies	imply	VERB
ejpam-577	95	48	n1n3	n1n3	PUNCT
ejpam-577	95	49	<	<	X
ejpam-577	95	50	2n2−	2n2−	NUM
ejpam-577	95	51	3	3	NUM
ejpam-577	95	52	,	,	PUNCT
ejpam-577	95	53	since	since	SCONJ
ejpam-577	95	54	n2	n2	ADJ
ejpam-577	95	55	≥	≥	NOUN
ejpam-577	95	56	2	2	NUM
ejpam-577	95	57	.	.	PUNCT
ejpam-577	95	58	hence	hence	ADV
ejpam-577	95	59	n2n3	n2n3	X
ejpam-577	95	60	≤	≤	NOUN
ejpam-577	95	61	2n2	2n2	NUM
ejpam-577	95	62	,	,	PUNCT
ejpam-577	95	63	which	which	PRON
ejpam-577	95	64	implies	imply	VERB
ejpam-577	95	65	2−	2−	NUM
ejpam-577	95	66	n3	n3	NOUN
ejpam-577	95	67	>	>	X
ejpam-577	95	68	0	0	PROPN
ejpam-577	95	69	,	,	PUNCT
ejpam-577	95	70	from	from	ADP
ejpam-577	95	71	which	which	PRON
ejpam-577	95	72	we	we	PRON
ejpam-577	95	73	conclude	conclude	VERB
ejpam-577	95	74	that	that	PRON
ejpam-577	95	75	n3	n3	NOUN
ejpam-577	95	76	=	=	NOUN
ejpam-577	95	77	1	1	X
ejpam-577	95	78	.	.	PUNCT
ejpam-577	95	79	by	by	ADP
ejpam-577	95	80	similar	similar	ADJ
ejpam-577	95	81	argument	argument	NOUN
ejpam-577	95	82	we	we	PRON
ejpam-577	95	83	get	get	VERB
ejpam-577	95	84	n1	n1	NOUN
ejpam-577	95	85	=	=	SYM
ejpam-577	95	86	1	1	X
ejpam-577	95	87	.	.	PUNCT
ejpam-577	95	88	now	now	ADV
ejpam-577	95	89	to	to	PART
ejpam-577	95	90	show	show	VERB
ejpam-577	95	91	that	that	SCONJ
ejpam-577	95	92	there	there	PRON
ejpam-577	95	93	are	be	VERB
ejpam-577	95	94	no	no	DET
ejpam-577	95	95	strongly	strongly	ADV
ejpam-577	95	96	indexable	indexable	ADJ
ejpam-577	95	97	complete	complete	ADJ
ejpam-577	95	98	m	m	ADJ
ejpam-577	95	99	-	-	ADJ
ejpam-577	95	100	partite	partite	ADJ
ejpam-577	95	101	graphs	graph	NOUN
ejpam-577	95	102	for	for	ADP
ejpam-577	95	103	m	m	PROPN
ejpam-577	95	104	≥	≥	NOUN
ejpam-577	95	105	4	4	NUM
ejpam-577	95	106	,	,	PUNCT
ejpam-577	95	107	observe	observe	VERB
ejpam-577	95	108	that	that	SCONJ
ejpam-577	95	109	k1,1,1,n	k1,1,1,n	PROPN
ejpam-577	95	110	is	be	AUX
ejpam-577	95	111	such	such	ADJ
ejpam-577	95	112	that	that	PRON
ejpam-577	95	113	|e(k1,1,1,n)|	|e(k1,1,1,n)|	NOUN
ejpam-577	95	114	>	>	X
ejpam-577	96	1	2|v	2|v	PROPN
ejpam-577	97	1	(	(	PUNCT
ejpam-577	97	2	k1,1,1,n)|	k1,1,1,n)|	NOUN
ejpam-577	97	3	−	−	NOUN
ejpam-577	97	4	3	3	X
ejpam-577	97	5	.	.	PUNCT
ejpam-577	97	6	(	(	PUNCT
ejpam-577	97	7	|v	|v	PROPN
ejpam-577	97	8	(	(	PUNCT
ejpam-577	97	9	k1,1,1,n|	k1,1,1,n|	PROPN
ejpam-577	97	10	=	=	SYM
ejpam-577	97	11	3	3	NUM
ejpam-577	97	12	+	+	CCONJ
ejpam-577	97	13	n	n	NOUN
ejpam-577	97	14	and	and	CCONJ
ejpam-577	97	15	|e(k1,1,1,n|	|e(k1,1,1,n|	NOUN
ejpam-577	97	16	=	=	PUNCT
ejpam-577	97	17	3n+	3n+	NUM
ejpam-577	97	18	3	3	NUM
ejpam-577	97	19	)	)	PUNCT
ejpam-577	97	20	.	.	PUNCT
ejpam-577	98	1	definition	definition	NOUN
ejpam-577	98	2	2	2	NUM
ejpam-577	98	3	.	.	PUNCT
ejpam-577	99	1	let	let	VERB
ejpam-577	99	2	g1	g1	PROPN
ejpam-577	99	3	=	=	SYM
ejpam-577	99	4	(	(	PUNCT
ejpam-577	99	5	v1	v1	PROPN
ejpam-577	99	6	,	,	PUNCT
ejpam-577	99	7	e1	e1	NOUN
ejpam-577	99	8	)	)	PUNCT
ejpam-577	99	9	and	and	CCONJ
ejpam-577	99	10	g2	g2	PROPN
ejpam-577	99	11	=	=	PUNCT
ejpam-577	99	12	(	(	PUNCT
ejpam-577	99	13	v2	v2	PROPN
ejpam-577	99	14	,	,	PUNCT
ejpam-577	99	15	e2	e2	PROPN
ejpam-577	99	16	)	)	PUNCT
ejpam-577	99	17	be	be	VERB
ejpam-577	99	18	two	two	NUM
ejpam-577	99	19	graphs	graph	NOUN
ejpam-577	99	20	.	.	PUNCT
ejpam-577	100	1	then	then	ADV
ejpam-577	100	2	the	the	DET
ejpam-577	100	3	cartesian	cartesian	ADJ
ejpam-577	100	4	product	product	NOUN
ejpam-577	100	5	g	g	NOUN
ejpam-577	100	6	=	=	SYM
ejpam-577	100	7	(	(	PUNCT
ejpam-577	100	8	v	v	NOUN
ejpam-577	100	9	,	,	PUNCT
ejpam-577	100	10	e	e	NOUN
ejpam-577	100	11	)	)	PUNCT
ejpam-577	100	12	of	of	ADP
ejpam-577	100	13	g1	g1	PROPN
ejpam-577	100	14	and	and	CCONJ
ejpam-577	100	15	g2	g2	PROPN
ejpam-577	100	16	is	be	AUX
ejpam-577	100	17	defined	define	VERB
ejpam-577	100	18	as	as	SCONJ
ejpam-577	100	19	:	:	PUNCT
ejpam-577	100	20	consider	consider	VERB
ejpam-577	100	21	any	any	DET
ejpam-577	100	22	two	two	NUM
ejpam-577	100	23	nodes	node	NOUN
ejpam-577	100	24	u	u	NOUN
ejpam-577	100	25	=	=	X
ejpam-577	100	26	u1u2	u1u2	X
ejpam-577	100	27	and	and	CCONJ
ejpam-577	100	28	v1v2	v1v2	X
ejpam-577	100	29	in	in	ADP
ejpam-577	100	30	v	v	NOUN
ejpam-577	100	31	=	=	SYM
ejpam-577	100	32	v1×v2	v1×v2	PROPN
ejpam-577	100	33	.	.	PUNCT
ejpam-577	101	1	then	then	ADV
ejpam-577	101	2	u	u	PROPN
ejpam-577	101	3	and	and	CCONJ
ejpam-577	101	4	v	v	NOUN
ejpam-577	101	5	are	be	AUX
ejpam-577	101	6	adjacent	adjacent	ADJ
ejpam-577	101	7	in	in	ADP
ejpam-577	101	8	g1	g1	PROPN
ejpam-577	101	9	×	×	PROPN
ejpam-577	101	10	g2	g2	PROPN
ejpam-577	101	11	,	,	PUNCT
ejpam-577	101	12	whenever	whenever	SCONJ
ejpam-577	101	13	u1	u1	NOUN
ejpam-577	101	14	=	=	SYM
ejpam-577	101	15	v1	v1	PROPN
ejpam-577	101	16	and	and	CCONJ
ejpam-577	101	17	u2v2	u2v2	PROPN
ejpam-577	101	18	∈	∈	PROPN
ejpam-577	101	19	e(g2	e(g2	X
ejpam-577	101	20	)	)	PUNCT
ejpam-577	101	21	or	or	CCONJ
ejpam-577	101	22	u2	u2	NOUN
ejpam-577	101	23	=	=	PUNCT
ejpam-577	101	24	v2	v2	PROPN
ejpam-577	101	25	and	and	CCONJ
ejpam-577	101	26	u1v1	u1v1	NOUN
ejpam-577	101	27	∈	∈	PROPN
ejpam-577	101	28	e(g1	e(g1	NOUN
ejpam-577	101	29	)	)	PUNCT
ejpam-577	101	30	.	.	PUNCT
ejpam-577	102	1	the	the	DET
ejpam-577	102	2	ladder	ladder	NOUN
ejpam-577	102	3	ln	ln	NOUN
ejpam-577	102	4	∼=	∼=	PROPN
ejpam-577	102	5	pn	pn	PROPN
ejpam-577	102	6	×	×	NOUN
ejpam-577	102	7	p2	p2	NOUN
ejpam-577	102	8	is	be	AUX
ejpam-577	102	9	not	not	PART
ejpam-577	102	10	strongly	strongly	ADV
ejpam-577	102	11	indexable	indexable	ADJ
ejpam-577	102	12	for	for	ADP
ejpam-577	102	13	all	all	DET
ejpam-577	102	14	n	n	DET
ejpam-577	102	15	≥	≥	NOUN
ejpam-577	102	16	2	2	NUM
ejpam-577	102	17	,	,	PUNCT
ejpam-577	102	18	since	since	SCONJ
ejpam-577	102	19	l2	l2	NOUN
ejpam-577	102	20	contains	contain	VERB
ejpam-577	102	21	no	no	DET
ejpam-577	102	22	triangle	triangle	NOUN
ejpam-577	102	23	.	.	PUNCT
ejpam-577	103	1	however	however	ADV
ejpam-577	103	2	there	there	PRON
ejpam-577	103	3	exists	exist	VERB
ejpam-577	103	4	an	an	DET
ejpam-577	103	5	integer	integer	NOUN
ejpam-577	103	6	k	k	PROPN
ejpam-577	103	7	such	such	ADJ
ejpam-577	103	8	that	that	DET
ejpam-577	103	9	l2	l2	NOUN
ejpam-577	103	10	is	be	AUX
ejpam-577	103	11	k	k	NOUN
ejpam-577	103	12	-	-	PUNCT
ejpam-577	103	13	strongly	strongly	ADV
ejpam-577	103	14	indexable	indexable	ADJ
ejpam-577	103	15	.	.	PUNCT
ejpam-577	104	1	theorem	theorem	ADJ
ejpam-577	104	2	10	10	NUM
ejpam-577	104	3	.	.	PUNCT
ejpam-577	105	1	the	the	DET
ejpam-577	105	2	ladder	ladder	NOUN
ejpam-577	105	3	ln	ln	NOUN
ejpam-577	105	4	∼=	∼=	PROPN
ejpam-577	105	5	pn	pn	PROPN
ejpam-577	105	6	×	×	PROPN
ejpam-577	105	7	p2	p2	PROPN
ejpam-577	105	8	is	be	AUX
ejpam-577	105	9	⌈	⌈	NUM
ejpam-577	105	10	n	n	PRON
ejpam-577	105	11	2	2	NUM
ejpam-577	105	12	⌉-strongly	⌉-strongly	ADV
ejpam-577	105	13	indexable	indexable	ADJ
ejpam-577	105	14	,	,	PUNCT
ejpam-577	105	15	if	if	SCONJ
ejpam-577	105	16	n	n	PRON
ejpam-577	105	17	is	be	AUX
ejpam-577	105	18	odd	odd	ADJ
ejpam-577	105	19	.	.	PUNCT
ejpam-577	106	1	proof	proof	NOUN
ejpam-577	106	2	.	.	PUNCT
ejpam-577	107	1	let	let	VERB
ejpam-577	107	2	v	v	NOUN
ejpam-577	107	3	(	(	PUNCT
ejpam-577	107	4	p2	p2	PROPN
ejpam-577	107	5	)	)	PUNCT
ejpam-577	107	6	=	=	SYM
ejpam-577	107	7	{	{	PUNCT
ejpam-577	107	8	v1	v1	NOUN
ejpam-577	107	9	,	,	PUNCT
ejpam-577	107	10	v2	v2	NOUN
ejpam-577	107	11	}	}	PUNCT
ejpam-577	107	12	and	and	CCONJ
ejpam-577	107	13	v	v	NOUN
ejpam-577	107	14	(	(	PUNCT
ejpam-577	107	15	pn	pn	NOUN
ejpam-577	107	16	)	)	PUNCT
ejpam-577	107	17	=	=	PRON
ejpam-577	108	1	{	{	PUNCT
ejpam-577	108	2	ui	ui	NOUN
ejpam-577	108	3	:	:	PUNCT
ejpam-577	108	4	1	1	NUM
ejpam-577	108	5	≤	≤	NUM
ejpam-577	108	6	i	i	PRON
ejpam-577	108	7	≤	≤	NOUN
ejpam-577	108	8	n	n	CCONJ
ejpam-577	108	9	}	}	PUNCT
ejpam-577	108	10	.	.	PUNCT
ejpam-577	109	1	define	define	VERB
ejpam-577	109	2	f	f	X
ejpam-577	109	3	:	:	PUNCT
ejpam-577	109	4	v	v	X
ejpam-577	109	5	(	(	PUNCT
ejpam-577	109	6	pn	pn	PROPN
ejpam-577	109	7	×	×	NOUN
ejpam-577	109	8	p2	p2	NOUN
ejpam-577	109	9	)	)	PUNCT
ejpam-577	109	10	→	→	SYM
ejpam-577	109	11	{	{	PUNCT
ejpam-577	109	12	0,1,2	0,1,2	NOUN
ejpam-577	109	13	,	,	PUNCT
ejpam-577	109	14	.	.	PUNCT
ejpam-577	109	15	.	.	PUNCT
ejpam-577	109	16	.	.	PUNCT
ejpam-577	110	1	,	,	PUNCT
ejpam-577	110	2	2n	2n	X
ejpam-577	110	3	}	}	PUNCT
ejpam-577	110	4	defined	define	VERB
ejpam-577	110	5	by	by	ADP
ejpam-577	110	6	f	f	PROPN
ejpam-577	110	7	(	(	PUNCT
ejpam-577	110	8	ui	ui	PROPN
ejpam-577	110	9	,	,	PUNCT
ejpam-577	110	10	v1	v1	NOUN
ejpam-577	110	11	)	)	PUNCT
ejpam-577	110	12	=	=	PUNCT
ejpam-577	110	13	¨	¨	NOUN
ejpam-577	110	14	i−1	i−1	PROPN
ejpam-577	110	15	2	2	NUM
ejpam-577	110	16	1≤	1≤	NUM
ejpam-577	110	17	i	i	NOUN
ejpam-577	110	18	≤	≤	PROPN
ejpam-577	110	19	n	n	CCONJ
ejpam-577	110	20	,	,	PUNCT
ejpam-577	110	21	i	i	PRON
ejpam-577	110	22	odd	odd	ADJ
ejpam-577	110	23	n+i−1	n+i−1	PROPN
ejpam-577	110	24	2	2	NUM
ejpam-577	110	25	1≤	1≤	NUM
ejpam-577	110	26	i	i	NOUN
ejpam-577	110	27	≤	≤	PROPN
ejpam-577	110	28	n	n	CCONJ
ejpam-577	110	29	,	,	PUNCT
ejpam-577	110	30	i	i	PRON
ejpam-577	110	31	even	even	ADV
ejpam-577	110	32	and	and	CCONJ
ejpam-577	110	33	f	f	PROPN
ejpam-577	110	34	(	(	PUNCT
ejpam-577	110	35	ui	ui	PROPN
ejpam-577	110	36	,	,	PUNCT
ejpam-577	110	37	v2	v2	PROPN
ejpam-577	110	38	)	)	PUNCT
ejpam-577	110	39	=	=	PUNCT
ejpam-577	110	40	¨	¨	NOUN
ejpam-577	110	41	f	f	X
ejpam-577	110	42	(	(	PUNCT
ejpam-577	110	43	un−1v1	un−1v1	ADJ
ejpam-577	110	44	)	)	PUNCT
ejpam-577	111	1	+	+	CCONJ
ejpam-577	111	2	i	i	PRON
ejpam-577	111	3	2	2	NUM
ejpam-577	111	4	1≤	1≤	NUM
ejpam-577	111	5	i	i	NOUN
ejpam-577	111	6	≤	≤	PROPN
ejpam-577	111	7	n	n	CCONJ
ejpam-577	111	8	,	,	PUNCT
ejpam-577	111	9	i	i	PRON
ejpam-577	111	10	even	even	ADV
ejpam-577	111	11	f	f	PROPN
ejpam-577	111	12	(	(	PUNCT
ejpam-577	111	13	un−1v1	un−1v1	ADJ
ejpam-577	111	14	)	)	PUNCT
ejpam-577	111	15	+	+	CCONJ
ejpam-577	111	16	n+i	n+i	NUM
ejpam-577	111	17	2	2	NUM
ejpam-577	111	18	1≤	1≤	NUM
ejpam-577	111	19	i	i	NOUN
ejpam-577	111	20	≤	≤	PROPN
ejpam-577	111	21	n	n	CCONJ
ejpam-577	111	22	,	,	PUNCT
ejpam-577	111	23	i	i	PRON
ejpam-577	111	24	odd	odd	ADJ
ejpam-577	111	25	remark	remark	NOUN
ejpam-577	111	26	2	2	NUM
ejpam-577	111	27	.	.	PUNCT
ejpam-577	112	1	the	the	DET
ejpam-577	112	2	converse	converse	NOUN
ejpam-577	112	3	of	of	ADP
ejpam-577	112	4	theorem	theorem	NOUN
ejpam-577	112	5	10	10	NUM
ejpam-577	112	6	is	be	AUX
ejpam-577	112	7	not	not	PART
ejpam-577	112	8	true	true	ADJ
ejpam-577	112	9	.	.	PUNCT
ejpam-577	113	1	l2	l2	NOUN
ejpam-577	113	2	∼=	∼=	NOUN
ejpam-577	113	3	c4	c4	NOUN
ejpam-577	113	4	is	be	AUX
ejpam-577	113	5	not	not	PART
ejpam-577	113	6	k	k	ADJ
ejpam-577	113	7	-	-	PUNCT
ejpam-577	113	8	strongly	strongly	ADV
ejpam-577	113	9	indexable	indexable	ADJ
ejpam-577	113	10	.	.	PUNCT
ejpam-577	114	1	however	however	ADV
ejpam-577	114	2	p4	p4	ADJ
ejpam-577	114	3	×	×	NOUN
ejpam-577	114	4	p2	p2	NOUN
ejpam-577	114	5	and	and	CCONJ
ejpam-577	114	6	p6	p6	VERB
ejpam-577	114	7	×	×	NOUN
ejpam-577	114	8	p2	p2	NOUN
ejpam-577	114	9	are	be	AUX
ejpam-577	114	10	3	3	NUM
ejpam-577	114	11	-	-	PUNCT
ejpam-577	114	12	strongly	strongly	ADV
ejpam-577	114	13	indexable	indexable	ADJ
ejpam-577	114	14	and	and	CCONJ
ejpam-577	114	15	4	4	NUM
ejpam-577	114	16	-	-	PUNCT
ejpam-577	114	17	strongly	strongly	ADV
ejpam-577	114	18	indexable	indexable	ADJ
ejpam-577	114	19	respectively	respectively	ADV
ejpam-577	114	20	.	.	PUNCT
ejpam-577	115	1	(	(	PUNCT
ejpam-577	115	2	see	see	VERB
ejpam-577	115	3	figure	figure	NOUN
ejpam-577	115	4	3	3	NUM
ejpam-577	115	5	)	)	PUNCT
ejpam-577	115	6	theorem	theorem	NOUN
ejpam-577	115	7	11	11	NUM
ejpam-577	115	8	.	.	PUNCT
ejpam-577	116	1	k3×	k3×	PROPN
ejpam-577	116	2	pn	pn	PROPN
ejpam-577	116	3	is	be	AUX
ejpam-577	116	4	strongly	strongly	ADV
ejpam-577	116	5	indexable	indexable	ADJ
ejpam-577	116	6	proof	proof	NOUN
ejpam-577	116	7	.	.	PUNCT
ejpam-577	117	1	let	let	VERB
ejpam-577	117	2	v	v	NOUN
ejpam-577	117	3	(	(	PUNCT
ejpam-577	117	4	k3	k3	PROPN
ejpam-577	117	5	)	)	PUNCT
ejpam-577	117	6	=	=	SYM
ejpam-577	118	1	{	{	PUNCT
ejpam-577	118	2	ui	ui	NOUN
ejpam-577	118	3	:	:	PUNCT
ejpam-577	118	4	1≤	1≤	INTJ
ejpam-577	119	1	i	i	X
ejpam-577	119	2	≤	≤	VERB
ejpam-577	119	3	3	3	NUM
ejpam-577	119	4	}	}	PUNCT
ejpam-577	119	5	and	and	CCONJ
ejpam-577	119	6	v	v	X
ejpam-577	119	7	(	(	PUNCT
ejpam-577	119	8	pn	pn	NOUN
ejpam-577	119	9	)	)	PUNCT
ejpam-577	119	10	=	=	PRON
ejpam-577	119	11	{	{	PUNCT
ejpam-577	119	12	x	x	PUNCT
ejpam-577	119	13	i	i	NOUN
ejpam-577	119	14	:	:	PUNCT
ejpam-577	119	15	1≤	1≤	INTJ
ejpam-577	119	16	i	i	NOUN
ejpam-577	119	17	≤	≤	PROPN
ejpam-577	119	18	n	n	CCONJ
ejpam-577	119	19	}	}	PUNCT
ejpam-577	119	20	.	.	PUNCT
ejpam-577	120	1	define	define	VERB
ejpam-577	120	2	f	f	X
ejpam-577	120	3	:	:	PUNCT
ejpam-577	120	4	v	v	PROPN
ejpam-577	120	5	(	(	PUNCT
ejpam-577	120	6	k3×	k3×	NOUN
ejpam-577	120	7	pn)→	pn)→	NOUN
ejpam-577	120	8	{	{	PUNCT
ejpam-577	120	9	0,1	0,1	NUM
ejpam-577	120	10	,	,	PUNCT
ejpam-577	120	11	,	,	PUNCT
ejpam-577	120	12	2	2	NUM
ejpam-577	120	13	,	,	PUNCT
ejpam-577	120	14	.	.	PUNCT
ejpam-577	120	15	.	.	PUNCT
ejpam-577	120	16	.	.	PUNCT
ejpam-577	121	1	,	,	PUNCT
ejpam-577	121	2	3n	3n	NUM
ejpam-577	121	3	}	}	PUNCT
ejpam-577	121	4	defined	define	VERB
ejpam-577	121	5	by	by	ADP
ejpam-577	121	6	f	f	PROPN
ejpam-577	121	7	(	(	PUNCT
ejpam-577	121	8	u1	u1	PROPN
ejpam-577	121	9	x	x	SYM
ejpam-577	121	10	i	i	NOUN
ejpam-577	121	11	)	)	PUNCT
ejpam-577	121	12	=	=	PRON
ejpam-577	121	13	{	{	PUNCT
ejpam-577	121	14	0,4,6,10,12,16,18,22,24,30,32,36	0,4,6,10,12,16,18,22,24,30,32,36	PROPN
ejpam-577	121	15	,	,	PUNCT
ejpam-577	121	16	.	.	PUNCT
ejpam-577	121	17	.	.	PUNCT
ejpam-577	121	18	.	.	PUNCT
ejpam-577	121	19	}	}	PUNCT
ejpam-577	122	1	f	f	X
ejpam-577	122	2	(	(	PUNCT
ejpam-577	122	3	u2	u2	NOUN
ejpam-577	122	4	x	x	PROPN
ejpam-577	122	5	i	i	PROPN
ejpam-577	122	6	)	)	PUNCT
ejpam-577	122	7	=	=	PRON
ejpam-577	122	8	{	{	PUNCT
ejpam-577	122	9	2,3,8,9,14,15,20,21,26,27,32	2,3,8,9,14,15,20,21,26,27,32	NUM
ejpam-577	122	10	,	,	PUNCT
ejpam-577	122	11	.	.	PUNCT
ejpam-577	122	12	.	.	PUNCT
ejpam-577	122	13	.	.	PUNCT
ejpam-577	122	14	}	}	PUNCT
ejpam-577	123	1	and	and	CCONJ
ejpam-577	123	2	f	f	X
ejpam-577	123	3	(	(	PUNCT
ejpam-577	123	4	u3	u3	NOUN
ejpam-577	123	5	x	x	SYM
ejpam-577	123	6	i	i	PROPN
ejpam-577	123	7	)	)	PUNCT
ejpam-577	123	8	=	=	PRON
ejpam-577	123	9	{	{	PUNCT
ejpam-577	123	10	1,5,7,11,13,17,19,23,25,29,31,35	1,5,7,11,13,17,19,23,25,29,31,35	PROPN
ejpam-577	123	11	,	,	PUNCT
ejpam-577	123	12	.	.	PUNCT
ejpam-577	123	13	.	.	PUNCT
ejpam-577	123	14	.	.	PUNCT
ejpam-577	124	1	}	}	PUNCT
ejpam-577	124	2	figure	figure	VERB
ejpam-577	124	3	4	4	NUM
ejpam-577	124	4	illustrates	illustrate	VERB
ejpam-577	124	5	the	the	DET
ejpam-577	124	6	strongly	strongly	ADV
ejpam-577	124	7	indexable	indexable	ADJ
ejpam-577	124	8	labelling	labelling	NOUN
ejpam-577	124	9	of	of	ADP
ejpam-577	124	10	k3	k3	ADJ
ejpam-577	124	11	×	×	PROPN
ejpam-577	124	12	p4	p4	PROPN
ejpam-577	124	13	k.	k.	PROPN
ejpam-577	124	14	a.	a.	PROPN
ejpam-577	124	15	germina	germina	PROPN
ejpam-577	124	16	/	/	SYM
ejpam-577	124	17	eur	eur	PROPN
ejpam-577	124	18	.	.	PUNCT
ejpam-577	125	1	j.	j.	PROPN
ejpam-577	125	2	pure	pure	PROPN
ejpam-577	125	3	appl	appl	PROPN
ejpam-577	125	4	.	.	PROPN
ejpam-577	125	5	math	math	PROPN
ejpam-577	125	6	,	,	PUNCT
ejpam-577	125	7	3	3	NUM
ejpam-577	125	8	(	(	PUNCT
ejpam-577	125	9	2010	2010	NUM
ejpam-577	125	10	)	)	PUNCT
ejpam-577	125	11	,	,	PUNCT
ejpam-577	125	12	269	269	NUM
ejpam-577	125	13	-	-	SYM
ejpam-577	125	14	281	281	NUM
ejpam-577	125	15	273	273	NUM
ejpam-577	125	16	figure	figure	NOUN
ejpam-577	125	17	3	3	NUM
ejpam-577	125	18	figure	figure	NOUN
ejpam-577	125	19	4	4	NUM
ejpam-577	125	20	theorem	theorem	NOUN
ejpam-577	125	21	12	12	NUM
ejpam-577	125	22	.	.	PUNCT
ejpam-577	126	1	in	in	ADP
ejpam-577	126	2	general	general	ADJ
ejpam-577	126	3	kn×	kn×	PROPN
ejpam-577	126	4	pk	pk	NOUN
ejpam-577	126	5	is	be	AUX
ejpam-577	126	6	strongly	strongly	ADV
ejpam-577	126	7	indexable	indexable	ADJ
ejpam-577	126	8	if	if	SCONJ
ejpam-577	126	9	and	and	CCONJ
ejpam-577	126	10	only	only	ADV
ejpam-577	126	11	n=	n=	ADJ
ejpam-577	126	12	3	3	NUM
ejpam-577	126	13	.	.	PUNCT
ejpam-577	127	1	proof	proof	NOUN
ejpam-577	127	2	.	.	PUNCT
ejpam-577	128	1	necessary	necessary	ADJ
ejpam-577	128	2	part	part	NOUN
ejpam-577	128	3	follows	follow	VERB
ejpam-577	128	4	from	from	ADP
ejpam-577	128	5	theorem	theorem	ADJ
ejpam-577	128	6	11	11	NUM
ejpam-577	128	7	.	.	PUNCT
ejpam-577	129	1	converse	converse	NOUN
ejpam-577	129	2	follows	follow	VERB
ejpam-577	129	3	from	from	ADP
ejpam-577	129	4	the	the	DET
ejpam-577	129	5	fact	fact	NOUN
ejpam-577	129	6	that	that	SCONJ
ejpam-577	129	7	|e(kn×	|e(kn×	VERB
ejpam-577	129	8	pk)|	pk)|	PROPN
ejpam-577	129	9	>	>	X
ejpam-577	130	1	2|v	2|v	PROPN
ejpam-577	131	1	(	(	PUNCT
ejpam-577	131	2	kn×	kn×	PROPN
ejpam-577	131	3	pk)|	pk)|	PROPN
ejpam-577	131	4	−	−	PROPN
ejpam-577	131	5	3	3	X
ejpam-577	131	6	.	.	PUNCT
ejpam-577	131	7	theorem	theorem	VERB
ejpam-577	131	8	13	13	NUM
ejpam-577	131	9	.	.	PUNCT
ejpam-577	132	1	cm×	cm×	PROPN
ejpam-577	133	1	pn	pn	PROPN
ejpam-577	133	2	is	be	AUX
ejpam-577	133	3	2	2	NUM
ejpam-577	133	4	-	-	PUNCT
ejpam-577	133	5	strongly	strongly	ADV
ejpam-577	133	6	indexable	indexable	ADJ
ejpam-577	133	7	if	if	SCONJ
ejpam-577	133	8	m	m	NOUN
ejpam-577	133	9	is	be	AUX
ejpam-577	133	10	odd	odd	ADJ
ejpam-577	133	11	and	and	CCONJ
ejpam-577	133	12	n≥	n≥	ADJ
ejpam-577	133	13	2	2	NUM
ejpam-577	133	14	proof	proof	NOUN
ejpam-577	133	15	.	.	PUNCT
ejpam-577	134	1	assume	assume	VERB
ejpam-577	134	2	v	v	X
ejpam-577	134	3	(	(	PUNCT
ejpam-577	134	4	cm	cm	NOUN
ejpam-577	134	5	)	)	PUNCT
ejpam-577	134	6	=	=	PRON
ejpam-577	134	7	{	{	PUNCT
ejpam-577	134	8	vi	vi	NOUN
ejpam-577	134	9	,	,	PUNCT
ejpam-577	134	10	1	1	NUM
ejpam-577	134	11	≤	≤	NUM
ejpam-577	134	12	i	i	X
ejpam-577	134	13	≤	≤	NOUN
ejpam-577	134	14	m	m	VERB
ejpam-577	134	15	}	}	PUNCT
ejpam-577	134	16	and	and	CCONJ
ejpam-577	134	17	v	v	X
ejpam-577	134	18	(	(	PUNCT
ejpam-577	134	19	pn	pn	NOUN
ejpam-577	134	20	)	)	PUNCT
ejpam-577	134	21	=	=	PUNCT
ejpam-577	134	22	{	{	PUNCT
ejpam-577	134	23	u	u	X
ejpam-577	134	24	j	j	PROPN
ejpam-577	134	25	,	,	PUNCT
ejpam-577	134	26	1	1	NUM
ejpam-577	134	27	≤	≤	NUM
ejpam-577	134	28	j	j	PROPN
ejpam-577	134	29	≤	≤	PROPN
ejpam-577	134	30	n.	n.	NOUN
ejpam-577	134	31	}	}	PUNCT
ejpam-577	134	32	now	now	ADV
ejpam-577	134	33	,	,	PUNCT
ejpam-577	134	34	the	the	DET
ejpam-577	134	35	2	2	NUM
ejpam-577	134	36	-	-	PUNCT
ejpam-577	134	37	strongly	strongly	ADV
ejpam-577	134	38	indexable	indexable	ADJ
ejpam-577	134	39	labeling	labeling	NOUN
ejpam-577	134	40	f	f	PROPN
ejpam-577	134	41	is	be	AUX
ejpam-577	134	42	defined	define	VERB
ejpam-577	134	43	as	as	SCONJ
ejpam-577	134	44	follows	follow	VERB
ejpam-577	134	45	f	f	PROPN
ejpam-577	134	46	(	(	PUNCT
ejpam-577	134	47	vi	vi	PROPN
ejpam-577	134	48	,	,	PUNCT
ejpam-577	134	49	u	u	NOUN
ejpam-577	134	50	j	j	NOUN
ejpam-577	134	51	)	)	PUNCT
ejpam-577	134	52	=	=	PUNCT
ejpam-577	134	53			PROPN
ejpam-577	134	54			VERB
ejpam-577	134	55			PROPN
ejpam-577	134	56			NOUN
ejpam-577	134	57			PROPN
ejpam-577	134	58			PROPN
ejpam-577	134	59			NOUN
ejpam-577	134	60	i+m−1	i+m−1	NOUN
ejpam-577	134	61	2	2	NUM
ejpam-577	134	62	1≤	1≤	NUM
ejpam-577	134	63	i	i	X
ejpam-577	134	64	≤	≤	PROPN
ejpam-577	134	65	m	m	VERB
ejpam-577	134	66	,	,	PUNCT
ejpam-577	135	1	i	i	PRON
ejpam-577	135	2	even	even	ADV
ejpam-577	135	3	j	j	PROPN
ejpam-577	135	4	=	=	SYM
ejpam-577	135	5	1	1	NUM
ejpam-577	135	6	i−1+m(2	i−1+m(2	PROPN
ejpam-577	135	7	j−1	j−1	PROPN
ejpam-577	135	8	)	)	PUNCT
ejpam-577	135	9	2	2	NUM
ejpam-577	135	10	2≤	2≤	NUM
ejpam-577	135	11	j	j	PROPN
ejpam-577	135	12	≤	≤	PROPN
ejpam-577	135	13	n	n	CCONJ
ejpam-577	135	14	,	,	PUNCT
ejpam-577	135	15	j	j	PROPN
ejpam-577	135	16	odd	odd	ADJ
ejpam-577	135	17	,	,	PUNCT
ejpam-577	136	1	1≤	1≤	INTJ
ejpam-577	136	2	i	i	PRON
ejpam-577	137	1	≤	≤	PUNCT
ejpam-577	137	2	m	m	VERB
ejpam-577	137	3	i−2+m(2	i−2+m(2	NUM
ejpam-577	137	4	j−1	j−1	PROPN
ejpam-577	137	5	)	)	PUNCT
ejpam-577	137	6	2	2	NUM
ejpam-577	137	7	2≤	2≤	NUM
ejpam-577	137	8	j	j	PROPN
ejpam-577	137	9	≤	≤	PROPN
ejpam-577	137	10	n	n	CCONJ
ejpam-577	137	11	,	,	PUNCT
ejpam-577	137	12	j	j	PROPN
ejpam-577	137	13	even	even	ADV
ejpam-577	137	14	,	,	PUNCT
ejpam-577	137	15	1≤	1≤	INTJ
ejpam-577	137	16	i	i	X
ejpam-577	137	17	≤	≤	PUNCT
ejpam-577	137	18	m	m	VERB
ejpam-577	137	19	,	,	PUNCT
ejpam-577	137	20	i	i	PRON
ejpam-577	137	21	odd	odd	ADJ
ejpam-577	137	22	i−2+m(2	i−2+m(2	NUM
ejpam-577	137	23	j−2	j−2	PROPN
ejpam-577	137	24	)	)	PUNCT
ejpam-577	137	25	2	2	NUM
ejpam-577	137	26	2≤	2≤	NUM
ejpam-577	137	27	j	j	PROPN
ejpam-577	137	28	≤	≤	PROPN
ejpam-577	137	29	n	n	CCONJ
ejpam-577	137	30	,	,	PUNCT
ejpam-577	137	31	j	j	PROPN
ejpam-577	137	32	even	even	ADV
ejpam-577	137	33	1≤	1≤	NUM
ejpam-577	138	1	i	i	PRON
ejpam-577	138	2	≤	≤	PROPN
ejpam-577	138	3	m	m	VERB
ejpam-577	138	4	,	,	PUNCT
ejpam-577	138	5	i	i	PRON
ejpam-577	138	6	even	even	ADV
ejpam-577	138	7	remark	remark	VERB
ejpam-577	138	8	3	3	NUM
ejpam-577	138	9	.	.	PUNCT
ejpam-577	139	1	even	even	ADV
ejpam-577	139	2	though	though	SCONJ
ejpam-577	139	3	cm×pn	cm×pn	NOUN
ejpam-577	139	4	is	be	AUX
ejpam-577	139	5	not	not	PART
ejpam-577	139	6	strongly	strongly	ADV
ejpam-577	139	7	indexable	indexable	ADJ
ejpam-577	139	8	we	we	PRON
ejpam-577	139	9	can	can	AUX
ejpam-577	139	10	generate	generate	VERB
ejpam-577	139	11	infinitely	infinitely	ADV
ejpam-577	139	12	many	many	ADJ
ejpam-577	139	13	classes	class	NOUN
ejpam-577	139	14	of	of	ADP
ejpam-577	139	15	strongly	strongly	ADV
ejpam-577	139	16	indexable	indexable	ADJ
ejpam-577	139	17	graph	graph	NOUN
ejpam-577	139	18	g	g	NOUN
ejpam-577	139	19	by	by	ADP
ejpam-577	139	20	adjoining	adjoin	VERB
ejpam-577	139	21	two	two	NUM
ejpam-577	139	22	vertices	vertex	NOUN
ejpam-577	139	23	say	say	VERB
ejpam-577	139	24	u	u	NOUN
ejpam-577	139	25	and	and	CCONJ
ejpam-577	139	26	v	v	NOUN
ejpam-577	139	27	with	with	ADP
ejpam-577	139	28	the	the	DET
ejpam-577	139	29	vertex	vertex	NOUN
ejpam-577	139	30	assignments	assignment	NOUN
ejpam-577	139	31	0	0	NUM
ejpam-577	140	1	and	and	CCONJ
ejpam-577	140	2	1	1	NUM
ejpam-577	140	3	respectively	respectively	ADV
ejpam-577	140	4	.	.	PUNCT
ejpam-577	141	1	hence	hence	ADV
ejpam-577	141	2	cm×	cm×	VERB
ejpam-577	142	1	pn∪{uv	pn∪{uv	NOUN
ejpam-577	142	2	}	}	PUNCT
ejpam-577	142	3	,	,	PUNCT
ejpam-577	142	4	where	where	SCONJ
ejpam-577	142	5	u	u	NOUN
ejpam-577	142	6	and	and	CCONJ
ejpam-577	142	7	v	v	ADP
ejpam-577	142	8	having	have	VERB
ejpam-577	142	9	the	the	DET
ejpam-577	142	10	vertex	vertex	NOUN
ejpam-577	142	11	assignments	assignment	NOUN
ejpam-577	142	12	0	0	NUM
ejpam-577	142	13	and	and	CCONJ
ejpam-577	142	14	1	1	NUM
ejpam-577	142	15	are	be	AUX
ejpam-577	142	16	classes	class	NOUN
ejpam-577	142	17	of	of	ADP
ejpam-577	142	18	strongly	strongly	ADV
ejpam-577	142	19	indexable	indexable	ADJ
ejpam-577	142	20	graphs	graph	NOUN
ejpam-577	142	21	.	.	PUNCT
ejpam-577	143	1	in	in	ADP
ejpam-577	143	2	fact	fact	NOUN
ejpam-577	143	3	,	,	PUNCT
ejpam-577	143	4	this	this	DET
ejpam-577	143	5	constriction	constriction	NOUN
ejpam-577	143	6	of	of	ADP
ejpam-577	143	7	adjoining	adjoin	VERB
ejpam-577	143	8	an	an	DET
ejpam-577	143	9	edge	edge	NOUN
ejpam-577	143	10	uv	uv	INTJ
ejpam-577	143	11	where	where	SCONJ
ejpam-577	143	12	,	,	PUNCT
ejpam-577	143	13	f	f	PROPN
ejpam-577	143	14	(	(	PUNCT
ejpam-577	143	15	u	u	NOUN
ejpam-577	143	16	)	)	PUNCT
ejpam-577	144	1	=	=	SYM
ejpam-577	144	2	0and	0and	PROPN
ejpam-577	144	3	f	f	X
ejpam-577	144	4	(	(	PUNCT
ejpam-577	144	5	v	v	NOUN
ejpam-577	144	6	)	)	PUNCT
ejpam-577	144	7	=	=	SYM
ejpam-577	144	8	1	1	NUM
ejpam-577	144	9	of	of	ADP
ejpam-577	144	10	a	a	DET
ejpam-577	144	11	2	2	NUM
ejpam-577	144	12	-	-	PUNCT
ejpam-577	144	13	strongly	strongly	ADV
ejpam-577	144	14	indexable	indexable	ADJ
ejpam-577	144	15	graphs	graph	NOUN
ejpam-577	144	16	results	result	VERB
ejpam-577	144	17	in	in	ADP
ejpam-577	144	18	to	to	ADP
ejpam-577	144	19	a	a	DET
ejpam-577	144	20	strongly	strongly	ADV
ejpam-577	144	21	indexable	indexable	ADJ
ejpam-577	144	22	graph	graph	NOUN
ejpam-577	144	23	.	.	PUNCT
ejpam-577	145	1	theorem	theorem	NOUN
ejpam-577	145	2	14	14	NUM
ejpam-577	145	3	.	.	PUNCT
ejpam-577	146	1	given	give	VERB
ejpam-577	146	2	any	any	DET
ejpam-577	146	3	k	k	NOUN
ejpam-577	146	4	-	-	PUNCT
ejpam-577	146	5	strongly	strongly	ADV
ejpam-577	146	6	indexable	indexable	ADJ
ejpam-577	146	7	graph	graph	NOUN
ejpam-577	146	8	g	g	PROPN
ejpam-577	146	9	=	=	PUNCT
ejpam-577	146	10	(	(	PUNCT
ejpam-577	146	11	p	p	X
ejpam-577	146	12	,	,	PUNCT
ejpam-577	146	13	q	q	NOUN
ejpam-577	146	14	)	)	PUNCT
ejpam-577	146	15	,	,	PUNCT
ejpam-577	146	16	there	there	PRON
ejpam-577	146	17	exists	exist	VERB
ejpam-577	146	18	a	a	DET
ejpam-577	146	19	strongly	strongly	ADV
ejpam-577	146	20	indexable	indexable	ADJ
ejpam-577	146	21	graph	graph	NOUN
ejpam-577	146	22	h	h	NOUN
ejpam-577	146	23	=	=	PUNCT
ejpam-577	146	24	(	(	PUNCT
ejpam-577	146	25	p	p	NOUN
ejpam-577	146	26	,	,	PUNCT
ejpam-577	146	27	q+	q+	ADV
ejpam-577	146	28	k−	k−	PROPN
ejpam-577	146	29	1	1	NUM
ejpam-577	146	30	)	)	PUNCT
ejpam-577	146	31	graph	graph	NOUN
ejpam-577	146	32	,	,	PUNCT
ejpam-577	146	33	with	with	ADP
ejpam-577	146	34	g	g	PROPN
ejpam-577	146	35	a	a	DET
ejpam-577	146	36	spanning	span	VERB
ejpam-577	146	37	subgraph	subgraph	NOUN
ejpam-577	146	38	of	of	ADP
ejpam-577	146	39	h.	h.	PROPN
ejpam-577	146	40	k.	k.	PROPN
ejpam-577	146	41	a.	a.	PROPN
ejpam-577	146	42	germina	germina	PROPN
ejpam-577	146	43	/	/	SYM
ejpam-577	146	44	eur	eur	PROPN
ejpam-577	146	45	.	.	PUNCT
ejpam-577	147	1	j.	j.	PROPN
ejpam-577	147	2	pure	pure	PROPN
ejpam-577	147	3	appl	appl	PROPN
ejpam-577	147	4	.	.	PROPN
ejpam-577	147	5	math	math	PROPN
ejpam-577	147	6	,	,	PUNCT
ejpam-577	147	7	3	3	NUM
ejpam-577	147	8	(	(	PUNCT
ejpam-577	147	9	2010	2010	NUM
ejpam-577	147	10	)	)	PUNCT
ejpam-577	147	11	,	,	PUNCT
ejpam-577	147	12	269	269	NUM
ejpam-577	147	13	-	-	SYM
ejpam-577	147	14	281	281	NUM
ejpam-577	147	15	274	274	NUM
ejpam-577	147	16	proof	proof	NOUN
ejpam-577	147	17	.	.	PUNCT
ejpam-577	148	1	let	let	VERB
ejpam-577	148	2	g	g	NOUN
ejpam-577	148	3	=	=	PUNCT
ejpam-577	148	4	(	(	PUNCT
ejpam-577	148	5	p	p	X
ejpam-577	148	6	,	,	PUNCT
ejpam-577	148	7	q	q	NOUN
ejpam-577	148	8	)	)	PUNCT
ejpam-577	148	9	,	,	PUNCT
ejpam-577	148	10	be	be	AUX
ejpam-577	148	11	k	k	NOUN
ejpam-577	148	12	-	-	ADJ
ejpam-577	148	13	strongly	strongly	ADV
ejpam-577	148	14	indexable	indexable	ADJ
ejpam-577	148	15	and	and	CCONJ
ejpam-577	148	16	let	let	VERB
ejpam-577	148	17	f	f	PRON
ejpam-577	148	18	be	be	AUX
ejpam-577	148	19	the	the	DET
ejpam-577	148	20	k	k	ADJ
ejpam-577	148	21	-	-	PUNCT
ejpam-577	148	22	strong	strong	ADJ
ejpam-577	148	23	indexer	indexer	NOUN
ejpam-577	148	24	of	of	ADP
ejpam-577	148	25	g.	g.	PROPN
ejpam-577	148	26	hence	hence	ADV
ejpam-577	148	27	,	,	PUNCT
ejpam-577	148	28	f	f	PROPN
ejpam-577	148	29	(	(	PUNCT
ejpam-577	148	30	v	v	X
ejpam-577	148	31	(	(	PUNCT
ejpam-577	148	32	g	g	NOUN
ejpam-577	148	33	)	)	PUNCT
ejpam-577	148	34	=	=	SYM
ejpam-577	148	35	{	{	PUNCT
ejpam-577	148	36	0,1,2	0,1,2	NOUN
ejpam-577	148	37	,	,	PUNCT
ejpam-577	148	38	.	.	PUNCT
ejpam-577	148	39	.	.	PUNCT
ejpam-577	148	40	.	.	PUNCT
ejpam-577	149	1	,	,	PUNCT
ejpam-577	150	1	p−	p−	NOUN
ejpam-577	150	2	1	1	NUM
ejpam-577	150	3	}	}	PUNCT
ejpam-577	150	4	and	and	CCONJ
ejpam-577	150	5	f	f	PROPN
ejpam-577	150	6	+	+	PROPN
ejpam-577	150	7	(	(	PUNCT
ejpam-577	150	8	e(g	e(g	PROPN
ejpam-577	150	9	)	)	PUNCT
ejpam-577	150	10	=	=	PRON
ejpam-577	150	11	{	{	PUNCT
ejpam-577	150	12	k	k	NOUN
ejpam-577	150	13	,	,	PUNCT
ejpam-577	150	14	k+	k+	NOUN
ejpam-577	150	15	1	1	NUM
ejpam-577	150	16	,	,	PUNCT
ejpam-577	150	17	.	.	PUNCT
ejpam-577	150	18	.	.	PUNCT
ejpam-577	150	19	.	.	PUNCT
ejpam-577	151	1	,	,	PUNCT
ejpam-577	151	2	k+	k+	NOUN
ejpam-577	151	3	q−	q−	PROPN
ejpam-577	151	4	1	1	NUM
ejpam-577	151	5	}	}	PUNCT
ejpam-577	151	6	.	.	PUNCT
ejpam-577	152	1	let	let	VERB
ejpam-577	152	2	u	u	PRON
ejpam-577	152	3	∈	∈	PROPN
ejpam-577	152	4	v	v	ADP
ejpam-577	152	5	(	(	PUNCT
ejpam-577	152	6	g	g	NOUN
ejpam-577	152	7	)	)	PUNCT
ejpam-577	152	8	be	be	AUX
ejpam-577	152	9	such	such	ADJ
ejpam-577	152	10	that	that	SCONJ
ejpam-577	152	11	f	f	PROPN
ejpam-577	152	12	(	(	PUNCT
ejpam-577	152	13	u	u	NOUN
ejpam-577	152	14	)	)	PUNCT
ejpam-577	152	15	=	=	SYM
ejpam-577	152	16	0	0	PUNCT
ejpam-577	152	17	and	and	CCONJ
ejpam-577	152	18	let	let	VERB
ejpam-577	152	19	ui	ui	NOUN
ejpam-577	152	20	,	,	PUNCT
ejpam-577	152	21	1	1	NUM
ejpam-577	152	22	≤	≤	NUM
ejpam-577	152	23	i	i	PROPN
ejpam-577	152	24	≤	≤	PROPN
ejpam-577	152	25	k−	k−	PROPN
ejpam-577	152	26	1	1	NUM
ejpam-577	152	27	be	be	AUX
ejpam-577	152	28	the	the	DET
ejpam-577	152	29	vertices	vertex	NOUN
ejpam-577	152	30	of	of	ADP
ejpam-577	152	31	g	g	NOUN
ejpam-577	152	32	with	with	ADP
ejpam-577	152	33	f	f	PROPN
ejpam-577	152	34	(	(	PUNCT
ejpam-577	152	35	ui	ui	PROPN
ejpam-577	152	36	)	)	PUNCT
ejpam-577	152	37	=	=	SYM
ejpam-577	153	1	i	i	PROPN
ejpam-577	153	2	,	,	PUNCT
ejpam-577	153	3	1≤	1≤	INTJ
ejpam-577	153	4	i	i	PROPN
ejpam-577	153	5	≤	≤	PROPN
ejpam-577	153	6	k−	k−	PROPN
ejpam-577	153	7	1	1	NUM
ejpam-577	153	8	.	.	PUNCT
ejpam-577	153	9	now	now	ADV
ejpam-577	153	10	construct	construct	VERB
ejpam-577	153	11	the	the	DET
ejpam-577	153	12	edges	edge	NOUN
ejpam-577	153	13	by	by	ADP
ejpam-577	153	14	joining	join	VERB
ejpam-577	153	15	uui	uui	PRON
ejpam-577	154	1	so	so	SCONJ
ejpam-577	154	2	that	that	SCONJ
ejpam-577	154	3	f	f	PROPN
ejpam-577	154	4	(	(	PUNCT
ejpam-577	154	5	uui	uui	PROPN
ejpam-577	154	6	)	)	PUNCT
ejpam-577	154	7	=	=	PUNCT
ejpam-577	154	8	{	{	PUNCT
ejpam-577	154	9	1,2	1,2	NUM
ejpam-577	154	10	,	,	PUNCT
ejpam-577	154	11	.	.	PUNCT
ejpam-577	154	12	.	.	PUNCT
ejpam-577	154	13	.	.	PUNCT
ejpam-577	154	14	,	,	PUNCT
ejpam-577	155	1	k	k	PROPN
ejpam-577	156	1	−	−	PROPN
ejpam-577	157	1	1	1	NUM
ejpam-577	157	2	}	}	PUNCT
ejpam-577	157	3	.	.	PUNCT
ejpam-577	158	1	the	the	DET
ejpam-577	158	2	new	new	ADJ
ejpam-577	158	3	graph	graph	NOUN
ejpam-577	158	4	h	h	NOUN
ejpam-577	158	5	constructed	construct	VERB
ejpam-577	158	6	is	be	AUX
ejpam-577	158	7	a	a	DET
ejpam-577	158	8	(	(	PUNCT
ejpam-577	158	9	p	p	X
ejpam-577	158	10	,	,	PUNCT
ejpam-577	158	11	q+k−1	q+k−1	NOUN
ejpam-577	158	12	)	)	PUNCT
ejpam-577	158	13	graph	graph	NOUN
ejpam-577	158	14	with	with	ADP
ejpam-577	158	15	f	f	PROPN
ejpam-577	158	16	(	(	PUNCT
ejpam-577	158	17	v	v	NOUN
ejpam-577	158	18	(	(	PUNCT
ejpam-577	158	19	h	h	NOUN
ejpam-577	158	20	)	)	PUNCT
ejpam-577	158	21	=	=	SYM
ejpam-577	158	22	{	{	PUNCT
ejpam-577	158	23	0,1,2	0,1,2	PROPN
ejpam-577	158	24	,	,	PUNCT
ejpam-577	158	25	p−1	p−1	PROPN
ejpam-577	158	26	}	}	PUNCT
ejpam-577	158	27	and	and	CCONJ
ejpam-577	158	28	f	f	PROPN
ejpam-577	158	29	+	+	PROPN
ejpam-577	158	30	(	(	PUNCT
ejpam-577	158	31	e(h	e(h	PROPN
ejpam-577	158	32	)	)	PUNCT
ejpam-577	158	33	=	=	PUNCT
ejpam-577	158	34	{	{	PUNCT
ejpam-577	158	35	1,2	1,2	NUM
ejpam-577	158	36	,	,	PUNCT
ejpam-577	158	37	.	.	PUNCT
ejpam-577	158	38	.	.	PUNCT
ejpam-577	159	1	.	.	PUNCT
ejpam-577	160	1	,	,	PUNCT
ejpam-577	160	2	k+	k+	NOUN
ejpam-577	160	3	q−	q−	PROPN
ejpam-577	160	4	1	1	NUM
ejpam-577	160	5	}	}	PUNCT
ejpam-577	160	6	and	and	CCONJ
ejpam-577	160	7	hence	hence	ADV
ejpam-577	160	8	f	f	PROPN
ejpam-577	160	9	is	be	AUX
ejpam-577	160	10	a	a	DET
ejpam-577	160	11	strong	strong	ADJ
ejpam-577	160	12	indexer	indexer	NOUN
ejpam-577	160	13	of	of	ADP
ejpam-577	160	14	h.	h.	PROPN
ejpam-577	160	15	figure	figure	NOUN
ejpam-577	160	16	5	5	NUM
ejpam-577	160	17	and	and	CCONJ
ejpam-577	160	18	figure	figure	VERB
ejpam-577	160	19	6	6	NUM
ejpam-577	160	20	gives	give	VERB
ejpam-577	160	21	the	the	DET
ejpam-577	160	22	strongly	strongly	ADV
ejpam-577	160	23	indexable	indexable	ADJ
ejpam-577	160	24	labellings	labelling	NOUN
ejpam-577	160	25	of	of	ADP
ejpam-577	160	26	c5×	c5×	NOUN
ejpam-577	160	27	pn	pn	PROPN
ejpam-577	160	28	and	and	CCONJ
ejpam-577	160	29	c5×	c5×	PROPN
ejpam-577	160	30	pn∪{uv	pn∪{uv	PROPN
ejpam-577	160	31	}	}	PUNCT
ejpam-577	160	32	figure	figure	NOUN
ejpam-577	160	33	5	5	NUM
ejpam-577	160	34	figure	figure	NOUN
ejpam-577	160	35	6	6	NUM
ejpam-577	160	36	definition	definition	NOUN
ejpam-577	160	37	3	3	NUM
ejpam-577	160	38	.	.	PUNCT
ejpam-577	161	1	for	for	ADP
ejpam-577	161	2	three	three	NUM
ejpam-577	161	3	or	or	CCONJ
ejpam-577	161	4	more	more	ADJ
ejpam-577	161	5	disjoint	disjoint	NOUN
ejpam-577	161	6	graph	graph	NOUN
ejpam-577	161	7	g1	g1	NOUN
ejpam-577	161	8	,	,	PUNCT
ejpam-577	161	9	g2	g2	PROPN
ejpam-577	161	10	,	,	PUNCT
ejpam-577	161	11	.	.	PUNCT
ejpam-577	161	12	.	.	PUNCT
ejpam-577	161	13	.	.	PUNCT
ejpam-577	162	1	,	,	PUNCT
ejpam-577	162	2	gk	gk	PROPN
ejpam-577	162	3	sequential	sequential	ADJ
ejpam-577	162	4	join	join	VERB
ejpam-577	162	5	g1	g1	PROPN
ejpam-577	162	6	+	+	CCONJ
ejpam-577	162	7	g2	g2	PROPN
ejpam-577	162	8	+	+	X
ejpam-577	162	9	.	.	PUNCT
ejpam-577	162	10	.	.	PUNCT
ejpam-577	162	11	.	.	PUNCT
ejpam-577	163	1	gk	gk	PROPN
ejpam-577	163	2	is	be	AUX
ejpam-577	163	3	the	the	DET
ejpam-577	163	4	graph	graph	NOUN
ejpam-577	163	5	(	(	PUNCT
ejpam-577	163	6	g1	g1	X
ejpam-577	163	7	+	+	CCONJ
ejpam-577	163	8	g2)∪	g2)∪	PROPN
ejpam-577	163	9	(	(	PUNCT
ejpam-577	163	10	g2	g2	PROPN
ejpam-577	163	11	+	+	CCONJ
ejpam-577	163	12	g3)∪	g3)∪	PROPN
ejpam-577	163	13	·	·	PUNCT
ejpam-577	163	14	·	·	PUNCT
ejpam-577	163	15	·	·	PUNCT
ejpam-577	163	16	∪	∪	X
ejpam-577	163	17	(	(	PUNCT
ejpam-577	163	18	gk−1	gk−1	PROPN
ejpam-577	163	19	+	+	CCONJ
ejpam-577	163	20	gn	gn	PROPN
ejpam-577	163	21	)	)	PUNCT
ejpam-577	163	22	lemma	lemma	PROPN
ejpam-577	163	23	3	3	X
ejpam-577	163	24	.	.	PUNCT
ejpam-577	164	1	let	let	VERB
ejpam-577	164	2	gi	gi	VERB
ejpam-577	164	3	∼=	∼=	NOUN
ejpam-577	164	4	k1	k1	NOUN
ejpam-577	164	5	,	,	PUNCT
ejpam-577	164	6	1≤	1≤	NUM
ejpam-577	165	1	i	i	PROPN
ejpam-577	165	2	≤	≤	PROPN
ejpam-577	166	1	n.	n.	NOUN
ejpam-577	166	2	then	then	ADV
ejpam-577	166	3	the	the	DET
ejpam-577	166	4	sequential	sequential	ADJ
ejpam-577	166	5	join	join	NOUN
ejpam-577	166	6	(	(	PUNCT
ejpam-577	166	7	g1+g2)∪(g2+g3)∪	g1+g2)∪(g2+g3)∪	PROPN
ejpam-577	166	8	·	·	PUNCT
ejpam-577	166	9	·	·	PUNCT
ejpam-577	166	10	·	·	PUNCT
ejpam-577	166	11	∪(gn−1	∪(gn−1	NUM
ejpam-577	166	12	+	+	NUM
ejpam-577	166	13	gn	gn	X
ejpam-577	166	14	)	)	PUNCT
ejpam-577	166	15	is	be	AUX
ejpam-577	166	16	strongly	strongly	ADV
ejpam-577	166	17	indexable	indexable	ADJ
ejpam-577	166	18	if	if	SCONJ
ejpam-577	167	1	and	and	CCONJ
ejpam-577	167	2	only	only	ADV
ejpam-577	167	3	if	if	SCONJ
ejpam-577	167	4	n≤	n≤	PRON
ejpam-577	167	5	3	3	X
ejpam-577	167	6	.	.	PUNCT
ejpam-577	168	1	proof	proof	NOUN
ejpam-577	168	2	.	.	PUNCT
ejpam-577	169	1	the	the	DET
ejpam-577	169	2	proof	proof	NOUN
ejpam-577	169	3	follows	follow	VERB
ejpam-577	169	4	from	from	ADP
ejpam-577	169	5	the	the	DET
ejpam-577	169	6	fact	fact	NOUN
ejpam-577	169	7	that	that	SCONJ
ejpam-577	169	8	pn	pn	PROPN
ejpam-577	169	9	is	be	AUX
ejpam-577	169	10	strongly	strongly	ADV
ejpam-577	169	11	indexable	indexable	ADJ
ejpam-577	169	12	if	if	SCONJ
ejpam-577	169	13	and	and	CCONJ
ejpam-577	170	1	only	only	ADV
ejpam-577	170	2	if	if	SCONJ
ejpam-577	170	3	n≤	n≤	PRON
ejpam-577	170	4	3	3	X
ejpam-577	170	5	.	.	PUNCT
ejpam-577	170	6	lemma	lemma	PROPN
ejpam-577	170	7	4	4	X
ejpam-577	170	8	.	.	PUNCT
ejpam-577	171	1	let	let	VERB
ejpam-577	171	2	gi	gi	VERB
ejpam-577	171	3	∼=	∼=	NOUN
ejpam-577	171	4	k1	k1	NOUN
ejpam-577	171	5	,	,	PUNCT
ejpam-577	171	6	1≤	1≤	NUM
ejpam-577	172	1	i	i	PROPN
ejpam-577	172	2	≤	≤	PROPN
ejpam-577	173	1	n.	n.	NOUN
ejpam-577	173	2	then	then	ADV
ejpam-577	173	3	the	the	DET
ejpam-577	173	4	sequential	sequential	ADJ
ejpam-577	173	5	join	join	NOUN
ejpam-577	173	6	(	(	PUNCT
ejpam-577	173	7	g1+g2)∪(g2+g3)∪	g1+g2)∪(g2+g3)∪	PROPN
ejpam-577	173	8	·	·	PUNCT
ejpam-577	173	9	·	·	PUNCT
ejpam-577	173	10	·	·	PUNCT
ejpam-577	173	11	∪(gn−1	∪(gn−1	NUM
ejpam-577	173	12	+	+	NUM
ejpam-577	173	13	gn	gn	X
ejpam-577	173	14	)	)	PUNCT
ejpam-577	173	15	is	be	AUX
ejpam-577	173	16	⌈	⌈	NUM
ejpam-577	173	17	n	n	CCONJ
ejpam-577	173	18	2	2	NUM
ejpam-577	173	19	⌉strongly	⌉strongly	ADV
ejpam-577	173	20	indexable	indexable	ADJ
ejpam-577	173	21	for	for	ADP
ejpam-577	173	22	all	all	DET
ejpam-577	173	23	n.	n.	NOUN
ejpam-577	173	24	proof	proof	NOUN
ejpam-577	173	25	.	.	PUNCT
ejpam-577	174	1	the	the	DET
ejpam-577	174	2	proof	proof	NOUN
ejpam-577	174	3	follows	follow	VERB
ejpam-577	174	4	from	from	ADP
ejpam-577	174	5	the	the	DET
ejpam-577	174	6	fact	fact	NOUN
ejpam-577	174	7	that	that	SCONJ
ejpam-577	174	8	pn	pn	PROPN
ejpam-577	174	9	is	be	AUX
ejpam-577	174	10	⌈	⌈	NUM
ejpam-577	174	11	n	n	PRON
ejpam-577	174	12	2	2	NUM
ejpam-577	174	13	⌉-strongly	⌉-strongly	ADV
ejpam-577	174	14	indexable	indexable	ADJ
ejpam-577	174	15	for	for	ADP
ejpam-577	174	16	all	all	DET
ejpam-577	174	17	n	n	CCONJ
ejpam-577	174	18	,	,	PUNCT
ejpam-577	174	19	where	where	SCONJ
ejpam-577	174	20	the	the	DET
ejpam-577	174	21	⌈	⌈	NUM
ejpam-577	174	22	n	n	PRON
ejpam-577	174	23	2	2	NUM
ejpam-577	174	24	⌉-strongly	⌉-strongly	ADV
ejpam-577	174	25	indexable	indexable	ADJ
ejpam-577	174	26	labelling	labelling	NOUN
ejpam-577	174	27	of	of	ADP
ejpam-577	174	28	pn	pn	PROPN
ejpam-577	174	29	is	be	AUX
ejpam-577	174	30	as	as	SCONJ
ejpam-577	174	31	follows	follow	VERB
ejpam-577	174	32	k.	k.	PROPN
ejpam-577	174	33	a.	a.	PROPN
ejpam-577	174	34	germina	germina	PROPN
ejpam-577	174	35	/	/	SYM
ejpam-577	174	36	eur	eur	PROPN
ejpam-577	174	37	.	.	PUNCT
ejpam-577	175	1	j.	j.	PROPN
ejpam-577	175	2	pure	pure	PROPN
ejpam-577	175	3	appl	appl	PROPN
ejpam-577	175	4	.	.	PROPN
ejpam-577	175	5	math	math	PROPN
ejpam-577	175	6	,	,	PUNCT
ejpam-577	175	7	3	3	NUM
ejpam-577	175	8	(	(	PUNCT
ejpam-577	175	9	2010	2010	NUM
ejpam-577	175	10	)	)	PUNCT
ejpam-577	175	11	,	,	PUNCT
ejpam-577	175	12	269	269	NUM
ejpam-577	175	13	-	-	SYM
ejpam-577	175	14	281	281	NUM
ejpam-577	175	15	275	275	NUM
ejpam-577	175	16	define	define	VERB
ejpam-577	175	17	f	f	NOUN
ejpam-577	175	18	:	:	PUNCT
ejpam-577	175	19	v	v	NOUN
ejpam-577	175	20	(	(	PUNCT
ejpam-577	175	21	pm)→	pm)→	X
ejpam-577	175	22	{	{	PUNCT
ejpam-577	175	23	0,1,2	0,1,2	NOUN
ejpam-577	175	24	,	,	PUNCT
ejpam-577	175	25	.	.	PUNCT
ejpam-577	175	26	.	.	PUNCT
ejpam-577	175	27	.	.	PUNCT
ejpam-577	176	1	,	,	PUNCT
ejpam-577	176	2	m−	m−	PROPN
ejpam-577	176	3	1	1	NUM
ejpam-577	176	4	}	}	PUNCT
ejpam-577	176	5	defined	define	VERB
ejpam-577	176	6	by	by	ADP
ejpam-577	176	7	f	f	PROPN
ejpam-577	176	8	(	(	PUNCT
ejpam-577	176	9	ui	ui	PROPN
ejpam-577	176	10	)	)	PUNCT
ejpam-577	176	11	=	=	PUNCT
ejpam-577	176	12			PROPN
ejpam-577	176	13			VERB
ejpam-577	176	14			DET
ejpam-577	176	15			ADJ
ejpam-577	176	16			NOUN
ejpam-577	176	17	i−1	i−1	PROPN
ejpam-577	176	18	2	2	NUM
ejpam-577	176	19	1≤	1≤	NUM
ejpam-577	176	20	i	i	X
ejpam-577	176	21	≤	≤	PROPN
ejpam-577	176	22	m	m	VERB
ejpam-577	176	23	,	,	PUNCT
ejpam-577	176	24	i	i	PRON
ejpam-577	176	25	odd	odd	ADJ
ejpam-577	176	26	,	,	PUNCT
ejpam-577	176	27	m	m	VERB
ejpam-577	176	28	odd	odd	ADJ
ejpam-577	176	29	i−1	i−1	PROPN
ejpam-577	176	30	2	2	NUM
ejpam-577	176	31	1≤	1≤	NUM
ejpam-577	176	32	i	i	PRON
ejpam-577	176	33	≤	≤	PUNCT
ejpam-577	176	34	m−	m−	PROPN
ejpam-577	176	35	1	1	NUM
ejpam-577	176	36	,	,	PUNCT
ejpam-577	176	37	i	i	PRON
ejpam-577	176	38	odd	odd	ADJ
ejpam-577	176	39	,	,	PUNCT
ejpam-577	176	40	m	m	VERB
ejpam-577	176	41	even	even	ADV
ejpam-577	176	42	n−i−1	n−i−1	PROPN
ejpam-577	176	43	2	2	NUM
ejpam-577	176	44	2≤	2≤	NUM
ejpam-577	176	45	i	i	PRON
ejpam-577	176	46	≤	≤	VERB
ejpam-577	176	47	m−	m−	PROPN
ejpam-577	176	48	1	1	NUM
ejpam-577	176	49	,	,	PUNCT
ejpam-577	176	50	i	i	PRON
ejpam-577	176	51	even	even	ADV
ejpam-577	176	52	,	,	PUNCT
ejpam-577	176	53	m	m	VERB
ejpam-577	176	54	odd	odd	ADJ
ejpam-577	176	55	n−i−2	n−i−2	PROPN
ejpam-577	176	56	2	2	NUM
ejpam-577	176	57	2≤	2≤	NUM
ejpam-577	176	58	j	j	PROPN
ejpam-577	176	59	≤	≤	ADV
ejpam-577	176	60	m−	m−	PROPN
ejpam-577	176	61	1	1	NUM
ejpam-577	176	62	,	,	PUNCT
ejpam-577	176	63	i	i	PRON
ejpam-577	176	64	even	even	ADV
ejpam-577	176	65	,	,	PUNCT
ejpam-577	176	66	m	m	VERB
ejpam-577	176	67	even	even	ADV
ejpam-577	176	68	lemma	lemma	PROPN
ejpam-577	176	69	5	5	NUM
ejpam-577	176	70	.	.	PUNCT
ejpam-577	177	1	let	let	VERB
ejpam-577	177	2	g1	g1	VERB
ejpam-577	177	3	∼=	∼=	PROPN
ejpam-577	177	4	k1	k1	NOUN
ejpam-577	177	5	,	,	PUNCT
ejpam-577	177	6	and	and	CCONJ
ejpam-577	177	7	gi	gi	VERB
ejpam-577	177	8	∼=	∼=	PROPN
ejpam-577	177	9	k2	k2	NOUN
ejpam-577	177	10	,	,	PUNCT
ejpam-577	177	11	2	2	NUM
ejpam-577	177	12	≤	≤	NUM
ejpam-577	177	13	i	i	PRON
ejpam-577	177	14	≤	≤	PROPN
ejpam-577	177	15	n.	n.	VERB
ejpam-577	177	16	the	the	DET
ejpam-577	177	17	sequential	sequential	ADJ
ejpam-577	177	18	join	join	NOUN
ejpam-577	177	19	(	(	PUNCT
ejpam-577	177	20	g1	g1	PROPN
ejpam-577	177	21	+	+	CCONJ
ejpam-577	177	22	g2	g2	PROPN
ejpam-577	177	23	)	)	PUNCT
ejpam-577	177	24	∪	∪	NOUN
ejpam-577	177	25	(	(	PUNCT
ejpam-577	177	26	g2	g2	PROPN
ejpam-577	177	27	+	+	CCONJ
ejpam-577	177	28	g3)∪	g3)∪	PROPN
ejpam-577	177	29	·	·	PUNCT
ejpam-577	177	30	·	·	PUNCT
ejpam-577	177	31	·	·	PUNCT
ejpam-577	177	32	∪(gn−1+gn	∪(gn−1+gn	X
ejpam-577	177	33	)	)	PUNCT
ejpam-577	177	34	is	be	AUX
ejpam-577	177	35	strongly	strongly	ADV
ejpam-577	177	36	indexable	indexable	ADJ
ejpam-577	177	37	if	if	SCONJ
ejpam-577	178	1	and	and	CCONJ
ejpam-577	178	2	only	only	ADV
ejpam-577	178	3	if	if	SCONJ
ejpam-577	178	4	n=	n=	ADJ
ejpam-577	178	5	2	2	NUM
ejpam-577	178	6	.	.	PUNCT
ejpam-577	178	7	however	however	ADV
ejpam-577	178	8	,	,	PUNCT
ejpam-577	178	9	pn	pn	PROPN
ejpam-577	178	10	is	be	AUX
ejpam-577	178	11	⌈	⌈	NUM
ejpam-577	178	12	n	n	PRON
ejpam-577	178	13	2	2	NUM
ejpam-577	178	14	⌉-strongly	⌉-strongly	ADV
ejpam-577	178	15	indexable	indexable	ADJ
ejpam-577	178	16	.	.	PUNCT
ejpam-577	179	1	proof	proof	NOUN
ejpam-577	179	2	.	.	PUNCT
ejpam-577	180	1	the	the	DET
ejpam-577	180	2	proof	proof	NOUN
ejpam-577	180	3	follows	follow	VERB
ejpam-577	180	4	from	from	ADP
ejpam-577	180	5	the	the	DET
ejpam-577	180	6	fact	fact	NOUN
ejpam-577	180	7	that	that	SCONJ
ejpam-577	180	8	pn	pn	PROPN
ejpam-577	180	9	is	be	AUX
ejpam-577	180	10	strongly	strongly	ADV
ejpam-577	180	11	indexable	indexable	ADJ
ejpam-577	180	12	if	if	SCONJ
ejpam-577	180	13	and	and	CCONJ
ejpam-577	180	14	only	only	ADV
ejpam-577	180	15	if	if	SCONJ
ejpam-577	180	16	n≤	n≤	PRON
ejpam-577	180	17	3	3	NUM
ejpam-577	180	18	and	and	CCONJ
ejpam-577	180	19	that	that	SCONJ
ejpam-577	180	20	pn	pn	PROPN
ejpam-577	180	21	is	be	AUX
ejpam-577	180	22	⌈	⌈	NUM
ejpam-577	180	23	n	n	PRON
ejpam-577	180	24	2	2	NUM
ejpam-577	180	25	⌉-strongly	⌉-strongly	ADV
ejpam-577	180	26	indexable	indexable	ADJ
ejpam-577	180	27	theorem	theorem	ADJ
ejpam-577	180	28	15	15	NUM
ejpam-577	180	29	.	.	PUNCT
ejpam-577	180	30	k1,n+	k1,n+	PROPN
ejpam-577	180	31	ki	ki	PROPN
ejpam-577	180	32	is	be	AUX
ejpam-577	180	33	not	not	PART
ejpam-577	180	34	strongly	strongly	ADV
ejpam-577	180	35	indexable	indexable	ADJ
ejpam-577	180	36	for	for	ADP
ejpam-577	180	37	n≥	n≥	NOUN
ejpam-577	180	38	2	2	NUM
ejpam-577	180	39	,	,	PUNCT
ejpam-577	180	40	i	i	PRON
ejpam-577	180	41	≥	≥	VERB
ejpam-577	180	42	1	1	NUM
ejpam-577	180	43	.	.	PUNCT
ejpam-577	181	1	proof	proof	NOUN
ejpam-577	181	2	.	.	PUNCT
ejpam-577	182	1	the	the	DET
ejpam-577	182	2	proof	proof	NOUN
ejpam-577	182	3	follows	follow	VERB
ejpam-577	182	4	from	from	ADP
ejpam-577	182	5	the	the	DET
ejpam-577	182	6	fact	fact	NOUN
ejpam-577	182	7	that	that	SCONJ
ejpam-577	182	8	|e(k1,n+	|e(k1,n+	X
ejpam-577	182	9	ki)|	ki)|	NOUN
ejpam-577	182	10	>	>	X
ejpam-577	182	11	2v	2v	PROPN
ejpam-577	182	12	(	(	PUNCT
ejpam-577	182	13	k1,n+	k1,n+	PROPN
ejpam-577	182	14	ki)|	ki)|	PROPN
ejpam-577	182	15	−	−	PROPN
ejpam-577	182	16	3	3	NUM
ejpam-577	182	17	.	.	PUNCT
ejpam-577	182	18	theorem	theorem	NOUN
ejpam-577	182	19	16	16	NUM
ejpam-577	182	20	.	.	PUNCT
ejpam-577	183	1	let	let	AUX
ejpam-577	183	2	gi	gi	AUX
ejpam-577	183	3	∼=	∼=	PROPN
ejpam-577	183	4	k1,n	k1,n	PROPN
ejpam-577	183	5	,	,	PUNCT
ejpam-577	183	6	1	1	NUM
ejpam-577	183	7	≤	≤	NUM
ejpam-577	183	8	i	i	PRON
ejpam-577	183	9	≤	≤	PROPN
ejpam-577	183	10	n.	n.	VERB
ejpam-577	183	11	the	the	DET
ejpam-577	183	12	sequential	sequential	ADJ
ejpam-577	183	13	join	join	VERB
ejpam-577	183	14	g	g	PROPN
ejpam-577	183	15	∼=	∼=	PROPN
ejpam-577	183	16	(	(	PUNCT
ejpam-577	183	17	g1	g1	PROPN
ejpam-577	183	18	+	+	CCONJ
ejpam-577	183	19	g2	g2	PROPN
ejpam-577	183	20	)	)	PUNCT
ejpam-577	183	21	∪	∪	NOUN
ejpam-577	183	22	(	(	PUNCT
ejpam-577	183	23	g2	g2	PROPN
ejpam-577	183	24	+	+	CCONJ
ejpam-577	183	25	g3	g3	PROPN
ejpam-577	183	26	)	)	PUNCT
ejpam-577	183	27	∪	∪	X
ejpam-577	183	28	·	·	PUNCT
ejpam-577	183	29	·	·	PUNCT
ejpam-577	183	30	·	·	PUNCT
ejpam-577	183	31	∪	∪	X
ejpam-577	183	32	(	(	PUNCT
ejpam-577	183	33	gn−1	gn−1	PROPN
ejpam-577	183	34	+	+	CCONJ
ejpam-577	183	35	gn	gn	X
ejpam-577	183	36	)	)	PUNCT
ejpam-577	183	37	is	be	AUX
ejpam-577	183	38	strongly	strongly	ADV
ejpam-577	183	39	indexable	indexable	ADJ
ejpam-577	183	40	if	if	SCONJ
ejpam-577	183	41	and	and	CCONJ
ejpam-577	183	42	only	only	ADV
ejpam-577	183	43	if	if	SCONJ
ejpam-577	183	44	,	,	PUNCT
ejpam-577	183	45	either	either	CCONJ
ejpam-577	183	46	i	i	PRON
ejpam-577	183	47	=	=	SYM
ejpam-577	183	48	n	n	PROPN
ejpam-577	183	49	=	=	SYM
ejpam-577	183	50	1	1	NUM
ejpam-577	183	51	or	or	CCONJ
ejpam-577	183	52	i	i	PRON
ejpam-577	183	53	=	=	NOUN
ejpam-577	183	54	2	2	NUM
ejpam-577	183	55	and	and	CCONJ
ejpam-577	183	56	n	n	CCONJ
ejpam-577	183	57	=	=	SYM
ejpam-577	183	58	1	1	NUM
ejpam-577	183	59	or	or	CCONJ
ejpam-577	183	60	i	i	PRON
ejpam-577	183	61	=	=	NOUN
ejpam-577	183	62	1	1	NUM
ejpam-577	183	63	,	,	PUNCT
ejpam-577	183	64	n=	n=	ADJ
ejpam-577	183	65	3	3	NUM
ejpam-577	183	66	.	.	PUNCT
ejpam-577	183	67	proof	proof	NOUN
ejpam-577	183	68	.	.	PUNCT
ejpam-577	184	1	let	let	VERB
ejpam-577	184	2	i	i	PRON
ejpam-577	184	3	=	=	SYM
ejpam-577	184	4	n	n	PROPN
ejpam-577	184	5	=	=	SYM
ejpam-577	184	6	1	1	NUM
ejpam-577	184	7	,	,	PUNCT
ejpam-577	184	8	then	then	ADV
ejpam-577	184	9	g	g	PROPN
ejpam-577	184	10	∼=	∼=	PROPN
ejpam-577	184	11	p2	p2	NOUN
ejpam-577	184	12	,	,	PUNCT
ejpam-577	184	13	which	which	PRON
ejpam-577	184	14	is	be	AUX
ejpam-577	184	15	strongly	strongly	ADV
ejpam-577	184	16	indexable	indexable	ADJ
ejpam-577	184	17	and	and	CCONJ
ejpam-577	184	18	when	when	SCONJ
ejpam-577	184	19	i	i	PRON
ejpam-577	184	20	=	=	SYM
ejpam-577	184	21	2	2	NUM
ejpam-577	184	22	and	and	CCONJ
ejpam-577	184	23	n	n	CCONJ
ejpam-577	184	24	=	=	SYM
ejpam-577	184	25	1	1	NUM
ejpam-577	184	26	or	or	CCONJ
ejpam-577	184	27	i	i	PRON
ejpam-577	184	28	=	=	NOUN
ejpam-577	184	29	1	1	NUM
ejpam-577	184	30	,	,	PUNCT
ejpam-577	184	31	n=	n=	ADJ
ejpam-577	184	32	3	3	NUM
ejpam-577	184	33	,	,	PUNCT
ejpam-577	184	34	g	g	PROPN
ejpam-577	184	35	∼=	∼=	PROPN
ejpam-577	184	36	p3	p3	NOUN
ejpam-577	184	37	which	which	PRON
ejpam-577	184	38	is	be	AUX
ejpam-577	184	39	again	again	ADV
ejpam-577	184	40	strongly	strongly	ADV
ejpam-577	184	41	indexable	indexable	ADJ
ejpam-577	184	42	.	.	PUNCT
ejpam-577	185	1	converse	converse	NOUN
ejpam-577	185	2	follows	follow	VERB
ejpam-577	185	3	from	from	ADP
ejpam-577	185	4	the	the	DET
ejpam-577	185	5	fact	fact	NOUN
ejpam-577	185	6	that	that	SCONJ
ejpam-577	185	7	,	,	PUNCT
ejpam-577	185	8	whenever	whenever	SCONJ
ejpam-577	185	9	i	i	PRON
ejpam-577	185	10	=	=	SYM
ejpam-577	185	11	n	n	CCONJ
ejpam-577	185	12	>	>	SYM
ejpam-577	185	13	1	1	NUM
ejpam-577	185	14	or	or	CCONJ
ejpam-577	185	15	i	i	PRON
ejpam-577	185	16	>	>	X
ejpam-577	185	17	2	2	NUM
ejpam-577	185	18	and	and	CCONJ
ejpam-577	185	19	n	n	NOUN
ejpam-577	185	20	>	>	ADP
ejpam-577	185	21	1	1	NUM
ejpam-577	185	22	or	or	CCONJ
ejpam-577	185	23	i	i	PRON
ejpam-577	185	24	>	>	X
ejpam-577	185	25	1	1	NUM
ejpam-577	185	26	,	,	PUNCT
ejpam-577	185	27	n	n	CCONJ
ejpam-577	185	28	>	>	X
ejpam-577	185	29	3	3	NUM
ejpam-577	185	30	,	,	PUNCT
ejpam-577	185	31	|e(g)|	|e(g)|	PROPN
ejpam-577	185	32	>	>	X
ejpam-577	185	33	|v	|v	X
ejpam-577	185	34	(	(	PUNCT
ejpam-577	185	35	g)−	g)−	PROPN
ejpam-577	185	36	3	3	NUM
ejpam-577	185	37	.	.	PUNCT
ejpam-577	185	38	definition	definition	NOUN
ejpam-577	185	39	4	4	NUM
ejpam-577	185	40	.	.	PUNCT
ejpam-577	186	1	let	let	VERB
ejpam-577	186	2	g1	g1	PROPN
ejpam-577	186	3	=	=	SYM
ejpam-577	186	4	(	(	PUNCT
ejpam-577	186	5	v1	v1	PROPN
ejpam-577	186	6	,	,	PUNCT
ejpam-577	186	7	e1	e1	NOUN
ejpam-577	186	8	)	)	PUNCT
ejpam-577	186	9	and	and	CCONJ
ejpam-577	186	10	g2	g2	PROPN
ejpam-577	186	11	=	=	PUNCT
ejpam-577	186	12	(	(	PUNCT
ejpam-577	186	13	v2	v2	PROPN
ejpam-577	186	14	,	,	PUNCT
ejpam-577	186	15	e2	e2	PROPN
ejpam-577	186	16	)	)	PUNCT
ejpam-577	186	17	be	be	VERB
ejpam-577	186	18	two	two	NUM
ejpam-577	186	19	graphs	graph	NOUN
ejpam-577	186	20	.	.	PUNCT
ejpam-577	187	1	then	then	ADV
ejpam-577	187	2	the	the	DET
ejpam-577	187	3	union	union	NOUN
ejpam-577	187	4	g	g	PROPN
ejpam-577	187	5	=	=	SYM
ejpam-577	187	6	(	(	PUNCT
ejpam-577	187	7	v	v	NOUN
ejpam-577	187	8	,	,	PUNCT
ejpam-577	187	9	e	e	NOUN
ejpam-577	187	10	)	)	PUNCT
ejpam-577	187	11	of	of	ADP
ejpam-577	187	12	g1	g1	PROPN
ejpam-577	187	13	and	and	CCONJ
ejpam-577	187	14	g2	g2	PROPN
ejpam-577	187	15	is	be	AUX
ejpam-577	187	16	defines	define	NOUN
ejpam-577	187	17	as	as	ADP
ejpam-577	187	18	the	the	DET
ejpam-577	187	19	v	v	NOUN
ejpam-577	187	20	=	=	SYM
ejpam-577	187	21	v1	v1	NOUN
ejpam-577	187	22	∪	∪	NOUN
ejpam-577	187	23	v2	v2	NOUN
ejpam-577	187	24	and	and	CCONJ
ejpam-577	187	25	e	e	NOUN
ejpam-577	187	26	=	=	NOUN
ejpam-577	187	27	e1	e1	PROPN
ejpam-577	187	28	∪	∪	PROPN
ejpam-577	187	29	e2	e2	PROPN
ejpam-577	187	30	.	.	PUNCT
ejpam-577	188	1	theorem	theorem	VERB
ejpam-577	188	2	17	17	NUM
ejpam-577	188	3	.	.	PUNCT
ejpam-577	189	1	for	for	ADP
ejpam-577	189	2	any	any	DET
ejpam-577	189	3	integer	integer	NOUN
ejpam-577	189	4	n≥	n≥	NOUN
ejpam-577	189	5	3	3	NUM
ejpam-577	189	6	,	,	PUNCT
ejpam-577	189	7	the	the	DET
ejpam-577	189	8	linear	linear	PROPN
ejpam-577	189	9	forest	forest	NOUN
ejpam-577	189	10	p1	p1	PROPN
ejpam-577	189	11	∪	∪	X
ejpam-577	189	12	pn	pn	PROPN
ejpam-577	189	13	is	be	AUX
ejpam-577	189	14	strongly	strongly	ADV
ejpam-577	189	15	indexable	indexable	ADJ
ejpam-577	189	16	if	if	SCONJ
ejpam-577	189	17	and	and	CCONJ
ejpam-577	189	18	only	only	ADV
ejpam-577	189	19	if	if	SCONJ
ejpam-577	189	20	n≤	n≤	PRON
ejpam-577	189	21	3	3	X
ejpam-577	189	22	.	.	PUNCT
ejpam-577	190	1	proof	proof	NOUN
ejpam-577	190	2	.	.	PUNCT
ejpam-577	191	1	clearly	clearly	ADV
ejpam-577	191	2	,	,	PUNCT
ejpam-577	191	3	for	for	ADP
ejpam-577	191	4	n	n	PRON
ejpam-577	191	5	∈	∈	NOUN
ejpam-577	191	6	{	{	PUNCT
ejpam-577	191	7	0,1,2	0,1,2	NOUN
ejpam-577	191	8	}	}	PUNCT
ejpam-577	191	9	,	,	PUNCT
ejpam-577	191	10	p1	p1	NOUN
ejpam-577	191	11	∪	∪	NOUN
ejpam-577	191	12	pn	pn	PROPN
ejpam-577	191	13	is	be	AUX
ejpam-577	191	14	strongly	strongly	ADV
ejpam-577	191	15	indexable	indexable	ADJ
ejpam-577	191	16	.	.	PUNCT
ejpam-577	192	1	conversely	conversely	ADV
ejpam-577	192	2	,	,	PUNCT
ejpam-577	192	3	since	since	SCONJ
ejpam-577	192	4	pn	pn	PROPN
ejpam-577	192	5	,	,	PUNCT
ejpam-577	192	6	n	n	PRON
ejpam-577	192	7	≥	≥	NOUN
ejpam-577	192	8	4	4	NUM
ejpam-577	192	9	is	be	AUX
ejpam-577	192	10	not	not	PART
ejpam-577	192	11	strongly	strongly	ADV
ejpam-577	192	12	indexable	indexable	ADJ
ejpam-577	192	13	since	since	SCONJ
ejpam-577	192	14	we	we	PRON
ejpam-577	192	15	can	can	AUX
ejpam-577	192	16	not	not	PART
ejpam-577	192	17	have	have	VERB
ejpam-577	192	18	a	a	DET
ejpam-577	192	19	strongly	strongly	ADV
ejpam-577	192	20	indexable	indexable	ADJ
ejpam-577	192	21	labelling	labelling	NOUN
ejpam-577	192	22	of	of	ADP
ejpam-577	192	23	pn	pn	PROPN
ejpam-577	192	24	.	.	PROPN
ejpam-577	192	25	theorem	theorem	PROPN
ejpam-577	192	26	18	18	NUM
ejpam-577	192	27	.	.	PUNCT
ejpam-577	193	1	for	for	ADP
ejpam-577	193	2	any	any	DET
ejpam-577	193	3	integer	integer	NOUN
ejpam-577	193	4	n≥	n≥	NOUN
ejpam-577	193	5	3	3	NUM
ejpam-577	193	6	,	,	PUNCT
ejpam-577	193	7	the	the	DET
ejpam-577	193	8	linear	linear	PROPN
ejpam-577	193	9	forest	forest	NOUN
ejpam-577	193	10	p1	p1	PROPN
ejpam-577	193	11	∪	∪	X
ejpam-577	193	12	pn	pn	PROPN
ejpam-577	193	13	is	be	AUX
ejpam-577	193	14	⌈	⌈	NUM
ejpam-577	193	15	n	n	PRON
ejpam-577	193	16	2	2	NUM
ejpam-577	193	17	⌉-strongly	⌉-strongly	ADV
ejpam-577	193	18	indexable	indexable	ADJ
ejpam-577	193	19	.	.	PUNCT
ejpam-577	194	1	proof	proof	NOUN
ejpam-577	194	2	.	.	PUNCT
ejpam-577	195	1	the	the	DET
ejpam-577	195	2	proof	proof	NOUN
ejpam-577	195	3	is	be	AUX
ejpam-577	195	4	immediate	immediate	ADJ
ejpam-577	195	5	as	as	ADP
ejpam-577	195	6	pn	pn	PROPN
ejpam-577	195	7	,	,	PUNCT
ejpam-577	195	8	n≥	n≥	PROPN
ejpam-577	195	9	4	4	NUM
ejpam-577	195	10	is	be	AUX
ejpam-577	195	11	⌈	⌈	NUM
ejpam-577	195	12	n	n	PRON
ejpam-577	195	13	2	2	NUM
ejpam-577	195	14	⌉-strongly	⌉-strongly	ADV
ejpam-577	195	15	indexable	indexable	ADJ
ejpam-577	195	16	.	.	PUNCT
ejpam-577	196	1	theorem	theorem	NOUN
ejpam-577	196	2	19	19	NUM
ejpam-577	196	3	.	.	PUNCT
ejpam-577	197	1	the	the	DET
ejpam-577	197	2	linear	linear	PROPN
ejpam-577	197	3	forest	forest	NOUN
ejpam-577	197	4	p2	p2	PROPN
ejpam-577	197	5	∪	∪	ADP
ejpam-577	197	6	pn	pn	PROPN
ejpam-577	197	7	is	be	AUX
ejpam-577	197	8	not	not	PART
ejpam-577	197	9	strongly	strongly	ADV
ejpam-577	197	10	indexable	indexable	ADJ
ejpam-577	197	11	.	.	PUNCT
ejpam-577	198	1	however	however	SCONJ
ejpam-577	198	2	p2	p2	PROPN
ejpam-577	198	3	∪	∪	ADP
ejpam-577	198	4	pn	pn	PROPN
ejpam-577	198	5	is	be	AUX
ejpam-577	198	6	⌈	⌈	NUM
ejpam-577	198	7	n+3	n+3	PROPN
ejpam-577	198	8	2	2	NUM
ejpam-577	198	9	⌉strongly	⌉strongly	ADV
ejpam-577	198	10	indexable	indexable	ADJ
ejpam-577	198	11	.	.	PUNCT
ejpam-577	199	1	k.	k.	PROPN
ejpam-577	199	2	a.	a.	PROPN
ejpam-577	199	3	germina	germina	PROPN
ejpam-577	199	4	/	/	SYM
ejpam-577	199	5	eur	eur	PROPN
ejpam-577	199	6	.	.	PUNCT
ejpam-577	200	1	j.	j.	PROPN
ejpam-577	200	2	pure	pure	PROPN
ejpam-577	200	3	appl	appl	PROPN
ejpam-577	200	4	.	.	PROPN
ejpam-577	200	5	math	math	PROPN
ejpam-577	200	6	,	,	PUNCT
ejpam-577	200	7	3	3	NUM
ejpam-577	200	8	(	(	PUNCT
ejpam-577	200	9	2010	2010	NUM
ejpam-577	200	10	)	)	PUNCT
ejpam-577	200	11	,	,	PUNCT
ejpam-577	200	12	269	269	NUM
ejpam-577	200	13	-	-	SYM
ejpam-577	200	14	281	281	NUM
ejpam-577	200	15	276	276	NUM
ejpam-577	200	16	proof	proof	NOUN
ejpam-577	200	17	.	.	PUNCT
ejpam-577	201	1	let	let	VERB
ejpam-577	201	2	p2	p2	PROPN
ejpam-577	201	3	∪	∪	ADJ
ejpam-577	201	4	pn	pn	X
ejpam-577	201	5	be	be	AUX
ejpam-577	201	6	a	a	DET
ejpam-577	201	7	linear	linear	ADJ
ejpam-577	201	8	forest	forest	NOUN
ejpam-577	201	9	.	.	PUNCT
ejpam-577	202	1	let	let	VERB
ejpam-577	202	2	v	v	NOUN
ejpam-577	202	3	(	(	PUNCT
ejpam-577	202	4	p2	p2	PROPN
ejpam-577	202	5	∪	∪	X
ejpam-577	202	6	pn	pn	NOUN
ejpam-577	202	7	)	)	PUNCT
ejpam-577	202	8	=	=	PRON
ejpam-577	202	9	{	{	PUNCT
ejpam-577	202	10	u1,u2	u1,u2	PROPN
ejpam-577	202	11	}	}	PUNCT
ejpam-577	202	12	∪	∪	ADJ
ejpam-577	202	13	{	{	PUNCT
ejpam-577	202	14	vi	vi	NOUN
ejpam-577	202	15	,	,	PUNCT
ejpam-577	202	16	1	1	NUM
ejpam-577	202	17	≤	≤	NUM
ejpam-577	202	18	i	i	PRON
ejpam-577	202	19	≤	≤	NOUN
ejpam-577	202	20	n	n	CCONJ
ejpam-577	202	21	}	}	PUNCT
ejpam-577	202	22	so	so	SCONJ
ejpam-577	202	23	that	that	SCONJ
ejpam-577	202	24	e(p2	e(p2	ADJ
ejpam-577	202	25	∪	∪	X
ejpam-577	202	26	pn	pn	NOUN
ejpam-577	202	27	)	)	PUNCT
ejpam-577	202	28	=	=	PRON
ejpam-577	202	29	{	{	PUNCT
ejpam-577	202	30	u1u2	u1u2	NOUN
ejpam-577	202	31	}	}	PUNCT
ejpam-577	202	32	∪	∪	ADJ
ejpam-577	202	33	{	{	PUNCT
ejpam-577	202	34	vi	vi	NOUN
ejpam-577	202	35	vi+1	vi+1	NOUN
ejpam-577	202	36	,	,	PUNCT
ejpam-577	203	1	1≤	1≤	NUM
ejpam-577	203	2	i	i	PRON
ejpam-577	203	3	≤	≤	VERB
ejpam-577	203	4	n−	n−	NOUN
ejpam-577	203	5	1	1	NUM
ejpam-577	203	6	}	}	PUNCT
ejpam-577	203	7	.	.	PUNCT
ejpam-577	204	1	now	now	ADV
ejpam-577	204	2	,	,	PUNCT
ejpam-577	204	3	v	v	INTJ
ejpam-577	204	4	(	(	PUNCT
ejpam-577	204	5	p2	p2	PROPN
ejpam-577	204	6	∪	∪	X
ejpam-577	204	7	pn	pn	NOUN
ejpam-577	204	8	)	)	PUNCT
ejpam-577	204	9	=	=	PUNCT
ejpam-577	204	10	n+	n+	ADP
ejpam-577	204	11	2	2	NUM
ejpam-577	204	12	and	and	CCONJ
ejpam-577	204	13	e(p2	e(p2	ADJ
ejpam-577	204	14	∪	∪	X
ejpam-577	204	15	pn	pn	NOUN
ejpam-577	204	16	)	)	PUNCT
ejpam-577	204	17	=	=	VERB
ejpam-577	204	18	n.	n.	AUX
ejpam-577	204	19	define	define	VERB
ejpam-577	204	20	f	f	X
ejpam-577	204	21	:	:	PUNCT
ejpam-577	204	22	v	v	X
ejpam-577	204	23	(	(	PUNCT
ejpam-577	204	24	p2	p2	PROPN
ejpam-577	204	25	∪	∪	VERB
ejpam-577	204	26	pn)→	pn)→	X
ejpam-577	204	27	{	{	PUNCT
ejpam-577	204	28	0,1,2	0,1,2	NOUN
ejpam-577	204	29	,	,	PUNCT
ejpam-577	204	30	.	.	PUNCT
ejpam-577	204	31	.	.	PUNCT
ejpam-577	204	32	.	.	PUNCT
ejpam-577	205	1	,	,	PUNCT
ejpam-577	205	2	n+	n+	ADP
ejpam-577	205	3	1	1	X
ejpam-577	205	4	}	}	PUNCT
ejpam-577	205	5	defined	define	VERB
ejpam-577	205	6	in	in	ADP
ejpam-577	205	7	the	the	DET
ejpam-577	205	8	following	follow	VERB
ejpam-577	205	9	cases	case	NOUN
ejpam-577	205	10	.	.	PUNCT
ejpam-577	206	1	case	case	NOUN
ejpam-577	206	2	1	1	NUM
ejpam-577	206	3	n≡	n≡	NUM
ejpam-577	206	4	0(mod4	0(mod4	NUM
ejpam-577	206	5	)	)	PUNCT
ejpam-577	206	6	f	f	PROPN
ejpam-577	206	7	(	(	PUNCT
ejpam-577	206	8	u1	u1	PROPN
ejpam-577	206	9	)	)	PUNCT
ejpam-577	206	10	=	=	SYM
ejpam-577	206	11	0	0	NUM
ejpam-577	206	12	;	;	PUNCT
ejpam-577	206	13	f	f	PROPN
ejpam-577	206	14	(	(	PUNCT
ejpam-577	206	15	u2	u2	PROPN
ejpam-577	206	16	)	)	PUNCT
ejpam-577	206	17	=	=	SYM
ejpam-577	207	1	n	n	PRON
ejpam-577	207	2	2	2	NUM
ejpam-577	208	1	+	+	SYM
ejpam-577	208	2	2	2	NUM
ejpam-577	208	3	f	f	NOUN
ejpam-577	208	4	(	(	PUNCT
ejpam-577	208	5	v	v	PROPN
ejpam-577	208	6	j	j	NOUN
ejpam-577	208	7	)	)	PUNCT
ejpam-577	208	8	=	=	PUNCT
ejpam-577	208	9			PROPN
ejpam-577	208	10			X
ejpam-577	208	11			PROPN
ejpam-577	208	12			PROPN
ejpam-577	208	13			PROPN
ejpam-577	208	14			PROPN
ejpam-577	208	15			NOUN
ejpam-577	208	16			PROPN
ejpam-577	208	17			PROPN
ejpam-577	208	18			PROPN
ejpam-577	208	19			PROPN
ejpam-577	208	20			PROPN
ejpam-577	208	21			NOUN
ejpam-577	208	22	n	n	ADV
ejpam-577	208	23	2	2	NUM
ejpam-577	209	1	+	+	CCONJ
ejpam-577	209	2	1	1	NUM
ejpam-577	209	3	if	if	SCONJ
ejpam-577	209	4	j	j	PROPN
ejpam-577	209	5	=	=	NOUN
ejpam-577	209	6	1	1	NUM
ejpam-577	209	7	n	n	NUM
ejpam-577	209	8	2	2	NUM
ejpam-577	210	1	+	+	CCONJ
ejpam-577	210	2	3	3	NUM
ejpam-577	210	3	if	if	SCONJ
ejpam-577	210	4	j	j	PROPN
ejpam-577	210	5	=	=	SYM
ejpam-577	210	6	3	3	NUM
ejpam-577	210	7	2i−	2i−	NUM
ejpam-577	210	8	1	1	NUM
ejpam-577	211	1	if	if	SCONJ
ejpam-577	211	2	j	j	PROPN
ejpam-577	211	3	=	=	SYM
ejpam-577	211	4	4i	4i	PROPN
ejpam-577	211	5	and	and	CCONJ
ejpam-577	211	6	1≤	1≤	NUM
ejpam-577	212	1	i	i	PRON
ejpam-577	212	2	≤	≤	ADJ
ejpam-577	212	3	n	n	PRON
ejpam-577	212	4	4	4	NUM
ejpam-577	212	5	n	n	DET
ejpam-577	212	6	2	2	NUM
ejpam-577	212	7	+	+	CCONJ
ejpam-577	212	8	2i+	2i+	NUM
ejpam-577	212	9	3	3	NUM
ejpam-577	212	10	if	if	SCONJ
ejpam-577	212	11	j	j	PROPN
ejpam-577	212	12	=	=	SYM
ejpam-577	212	13	4i+	4i+	NUM
ejpam-577	212	14	2	2	NUM
ejpam-577	212	15	and	and	CCONJ
ejpam-577	212	16	1≤	1≤	NUM
ejpam-577	213	1	i	i	PRON
ejpam-577	213	2	≤	≤	VERB
ejpam-577	213	3	n−4	n−4	PROPN
ejpam-577	213	4	4	4	NUM
ejpam-577	213	5	2i+	2i+	NUM
ejpam-577	213	6	2	2	NUM
ejpam-577	213	7	if	if	SCONJ
ejpam-577	213	8	j	j	PROPN
ejpam-577	213	9	=	=	SYM
ejpam-577	213	10	4i	4i	PROPN
ejpam-577	213	11	and	and	CCONJ
ejpam-577	213	12	1≤	1≤	NUM
ejpam-577	214	1	i	i	PRON
ejpam-577	214	2	≤	≤	VERB
ejpam-577	214	3	n−4	n−4	PROPN
ejpam-577	214	4	4	4	NUM
ejpam-577	214	5	n	n	DET
ejpam-577	214	6	2	2	NUM
ejpam-577	214	7	+	+	CCONJ
ejpam-577	214	8	2i+	2i+	NUM
ejpam-577	214	9	2	2	NUM
ejpam-577	214	10	if	if	SCONJ
ejpam-577	214	11	j	j	PROPN
ejpam-577	214	12	=	=	SYM
ejpam-577	214	13	4i+	4i+	NUM
ejpam-577	214	14	3	3	NUM
ejpam-577	215	1	and	and	CCONJ
ejpam-577	215	2	1≤	1≤	NUM
ejpam-577	216	1	i	i	PRON
ejpam-577	216	2	≤	≤	VERB
ejpam-577	216	3	n−4	n−4	NUM
ejpam-577	216	4	4	4	NUM
ejpam-577	216	5	case	case	NOUN
ejpam-577	216	6	2	2	NUM
ejpam-577	216	7	n≡	n≡	ADP
ejpam-577	216	8	1(mod4	1(mod4	NUM
ejpam-577	216	9	)	)	PUNCT
ejpam-577	216	10	f	f	PROPN
ejpam-577	216	11	(	(	PUNCT
ejpam-577	216	12	u1	u1	PROPN
ejpam-577	216	13	)	)	PUNCT
ejpam-577	216	14	=	=	SYM
ejpam-577	216	15	0	0	NUM
ejpam-577	216	16	;	;	PUNCT
ejpam-577	216	17	f	f	PROPN
ejpam-577	216	18	(	(	PUNCT
ejpam-577	216	19	u2	u2	PROPN
ejpam-577	216	20	)	)	PUNCT
ejpam-577	216	21	=	=	PUNCT
ejpam-577	216	22	n+	n+	PUNCT
ejpam-577	216	23	1	1	NUM
ejpam-577	216	24	f	f	X
ejpam-577	216	25	(	(	PUNCT
ejpam-577	216	26	v	v	PROPN
ejpam-577	216	27	j	j	NOUN
ejpam-577	216	28	)	)	PUNCT
ejpam-577	216	29	=	=	PUNCT
ejpam-577	217	1			PROPN
ejpam-577	217	2			X
ejpam-577	217	3			NOUN
ejpam-577	217	4	n+2	n+2	ADV
ejpam-577	217	5	j+1	j+1	ADV
ejpam-577	217	6	4	4	NUM
ejpam-577	217	7	if	if	SCONJ
ejpam-577	217	8	j	j	PROPN
ejpam-577	217	9	is	be	AUX
ejpam-577	217	10	odd	odd	ADJ
ejpam-577	217	11	and	and	CCONJ
ejpam-577	217	12	1≤	1≤	NUM
ejpam-577	217	13	j	j	PROPN
ejpam-577	217	14	≤	≤	PROPN
ejpam-577	217	15	n	n	CCONJ
ejpam-577	217	16	3n+2	3n+2	PROPN
ejpam-577	217	17	j+1	j+1	ADV
ejpam-577	217	18	4	4	NUM
ejpam-577	217	19	+	+	SYM
ejpam-577	217	20	3	3	NUM
ejpam-577	217	21	if	if	SCONJ
ejpam-577	217	22	j	j	PROPN
ejpam-577	217	23	is	be	AUX
ejpam-577	217	24	even	even	ADV
ejpam-577	217	25	and	and	CCONJ
ejpam-577	217	26	2≤	2≤	NUM
ejpam-577	217	27	j	j	PROPN
ejpam-577	218	1	≤	≤	PROPN
ejpam-577	218	2	n−1	n−1	PROPN
ejpam-577	218	3	2	2	NUM
ejpam-577	218	4	2	2	NUM
ejpam-577	218	5	j−n+1	j−n+1	PROPN
ejpam-577	218	6	4	4	NUM
ejpam-577	218	7	if	if	SCONJ
ejpam-577	218	8	j	j	PROPN
ejpam-577	218	9	is	be	AUX
ejpam-577	218	10	even	even	ADV
ejpam-577	218	11	and	and	CCONJ
ejpam-577	218	12	n+3	n+3	PROPN
ejpam-577	218	13	2	2	NUM
ejpam-577	218	14	≤	≤	NUM
ejpam-577	218	15	j	j	PROPN
ejpam-577	219	1	≤	≤	PROPN
ejpam-577	220	1	n−1	n−1	PROPN
ejpam-577	220	2	2	2	NUM
ejpam-577	220	3	case	case	NOUN
ejpam-577	220	4	3	3	NUM
ejpam-577	220	5	n≡	n≡	ADP
ejpam-577	220	6	2(mod4	2(mod4	NUM
ejpam-577	220	7	)	)	PUNCT
ejpam-577	220	8	f	f	PROPN
ejpam-577	220	9	(	(	PUNCT
ejpam-577	220	10	u1	u1	PROPN
ejpam-577	220	11	)	)	PUNCT
ejpam-577	220	12	=	=	SYM
ejpam-577	220	13	0	0	NUM
ejpam-577	220	14	;	;	PUNCT
ejpam-577	220	15	f	f	PROPN
ejpam-577	220	16	(	(	PUNCT
ejpam-577	220	17	u2	u2	PROPN
ejpam-577	220	18	)	)	PUNCT
ejpam-577	220	19	=	=	SYM
ejpam-577	220	20	n	n	PRON
ejpam-577	220	21	2	2	NUM
ejpam-577	220	22	+	+	SYM
ejpam-577	220	23	1	1	NUM
ejpam-577	220	24	f	f	X
ejpam-577	220	25	(	(	PUNCT
ejpam-577	220	26	v	v	PROPN
ejpam-577	220	27	j	j	NOUN
ejpam-577	220	28	)	)	PUNCT
ejpam-577	220	29	=	=	PUNCT
ejpam-577	220	30			PROPN
ejpam-577	220	31			X
ejpam-577	220	32			PROPN
ejpam-577	220	33			PROPN
ejpam-577	220	34			PROPN
ejpam-577	220	35			PROPN
ejpam-577	220	36			PROPN
ejpam-577	220	37			PROPN
ejpam-577	220	38			NOUN
ejpam-577	220	39			PROPN
ejpam-577	220	40			PROPN
ejpam-577	220	41			PROPN
ejpam-577	220	42			PROPN
ejpam-577	220	43			PROPN
ejpam-577	220	44			PROPN
ejpam-577	220	45			PROPN
ejpam-577	220	46			NOUN
ejpam-577	220	47	n+	n+	PUNCT
ejpam-577	220	48	1	1	NUM
ejpam-577	221	1	if	if	SCONJ
ejpam-577	221	2	j	j	PROPN
ejpam-577	221	3	=	=	NOUN
ejpam-577	221	4	1	1	NUM
ejpam-577	221	5	n−	n−	NOUN
ejpam-577	221	6	1	1	NUM
ejpam-577	221	7	if	if	SCONJ
ejpam-577	221	8	j	j	PROPN
ejpam-577	221	9	=	=	SYM
ejpam-577	221	10	3	3	NUM
ejpam-577	221	11	n	n	NOUN
ejpam-577	221	12	if	if	SCONJ
ejpam-577	221	13	j	j	PROPN
ejpam-577	221	14	=	=	SYM
ejpam-577	221	15	n	n	PROPN
ejpam-577	221	16	n	n	CCONJ
ejpam-577	221	17	2	2	NUM
ejpam-577	221	18	−	−	PROPN
ejpam-577	221	19	2i+	2i+	NUM
ejpam-577	221	20	1	1	NUM
ejpam-577	221	21	if	if	SCONJ
ejpam-577	221	22	j	j	PROPN
ejpam-577	221	23	=	=	SYM
ejpam-577	221	24	4i	4i	PROPN
ejpam-577	221	25	and	and	CCONJ
ejpam-577	221	26	1≤	1≤	NUM
ejpam-577	221	27	i	i	PRON
ejpam-577	221	28	≤	≤	PUNCT
ejpam-577	221	29	n−2	n−2	PROPN
ejpam-577	221	30	4	4	NUM
ejpam-577	221	31	n−	n−	NOUN
ejpam-577	221	32	2i−	2i−	NUM
ejpam-577	221	33	1	1	NUM
ejpam-577	221	34	if	if	SCONJ
ejpam-577	221	35	j	j	PROPN
ejpam-577	221	36	=	=	SYM
ejpam-577	221	37	4i+	4i+	NUM
ejpam-577	221	38	1	1	NUM
ejpam-577	221	39	and	and	CCONJ
ejpam-577	221	40	1≤	1≤	NUM
ejpam-577	221	41	i	i	PRON
ejpam-577	221	42	≤	≤	X
ejpam-577	221	43	n−2	n−2	PROPN
ejpam-577	221	44	4	4	NUM
ejpam-577	221	45	n	n	NUM
ejpam-577	221	46	2	2	NUM
ejpam-577	221	47	−	−	NOUN
ejpam-577	221	48	2i−	2i−	NUM
ejpam-577	221	49	2	2	NUM
ejpam-577	221	50	if	if	SCONJ
ejpam-577	221	51	j	j	PROPN
ejpam-577	221	52	=	=	SYM
ejpam-577	221	53	4i+	4i+	NUM
ejpam-577	221	54	2	2	NUM
ejpam-577	221	55	and	and	CCONJ
ejpam-577	221	56	0≤	0≤	NUM
ejpam-577	222	1	i	i	PRON
ejpam-577	222	2	≤	≤	PUNCT
ejpam-577	222	3	n−6	n−6	PROPN
ejpam-577	222	4	4	4	NUM
ejpam-577	222	5	n−	n−	NOUN
ejpam-577	222	6	2i	2i	NUM
ejpam-577	222	7	if	if	SCONJ
ejpam-577	222	8	j	j	PROPN
ejpam-577	222	9	=	=	SYM
ejpam-577	222	10	4i+	4i+	NUM
ejpam-577	222	11	3	3	NUM
ejpam-577	222	12	and	and	CCONJ
ejpam-577	222	13	1≤	1≤	NUM
ejpam-577	223	1	i	i	PRON
ejpam-577	223	2	≤	≤	PUNCT
ejpam-577	223	3	n−6	n−6	PROPN
ejpam-577	223	4	4	4	NUM
ejpam-577	223	5	case	case	NOUN
ejpam-577	223	6	4	4	NUM
ejpam-577	223	7	n≡	n≡	ADP
ejpam-577	223	8	3(mod4	3(mod4	NUM
ejpam-577	223	9	)	)	PUNCT
ejpam-577	223	10	f	f	PROPN
ejpam-577	223	11	(	(	PUNCT
ejpam-577	223	12	u1	u1	PROPN
ejpam-577	223	13	)	)	PUNCT
ejpam-577	223	14	=	=	SYM
ejpam-577	224	1	0	0	NUM
ejpam-577	224	2	;	;	PUNCT
ejpam-577	224	3	f	f	PROPN
ejpam-577	224	4	(	(	PUNCT
ejpam-577	224	5	u2	u2	PROPN
ejpam-577	224	6	)	)	PUNCT
ejpam-577	224	7	=	=	PUNCT
ejpam-577	224	8	n+	n+	PUNCT
ejpam-577	224	9	1	1	NUM
ejpam-577	224	10	k.	k.	NOUN
ejpam-577	224	11	a.	a.	PROPN
ejpam-577	224	12	germina	germina	PROPN
ejpam-577	224	13	/	/	SYM
ejpam-577	224	14	eur	eur	PROPN
ejpam-577	224	15	.	.	PUNCT
ejpam-577	225	1	j.	j.	PROPN
ejpam-577	225	2	pure	pure	PROPN
ejpam-577	225	3	appl	appl	PROPN
ejpam-577	225	4	.	.	PROPN
ejpam-577	225	5	math	math	PROPN
ejpam-577	225	6	,	,	PUNCT
ejpam-577	225	7	3	3	NUM
ejpam-577	225	8	(	(	PUNCT
ejpam-577	225	9	2010	2010	NUM
ejpam-577	225	10	)	)	PUNCT
ejpam-577	225	11	,	,	PUNCT
ejpam-577	225	12	269	269	NUM
ejpam-577	225	13	-	-	SYM
ejpam-577	225	14	281	281	NUM
ejpam-577	225	15	277	277	NUM
ejpam-577	225	16	f	f	NOUN
ejpam-577	225	17	(	(	PUNCT
ejpam-577	225	18	v	v	PROPN
ejpam-577	225	19	j	j	NOUN
ejpam-577	225	20	)	)	PUNCT
ejpam-577	225	21	=	=	PUNCT
ejpam-577	226	1			PROPN
ejpam-577	226	2			VERB
ejpam-577	226	3			PROPN
ejpam-577	226	4			NOUN
ejpam-577	226	5			ADJ
ejpam-577	226	6			PROPN
ejpam-577	226	7			NOUN
ejpam-577	226	8	j+1	j+1	ADV
ejpam-577	226	9	2	2	NUM
ejpam-577	226	10	if	if	SCONJ
ejpam-577	226	11	j	j	PROPN
ejpam-577	226	12	is	be	AUX
ejpam-577	226	13	odd	odd	ADJ
ejpam-577	226	14	and	and	CCONJ
ejpam-577	226	15	1≤	1≤	NUM
ejpam-577	226	16	j	j	PROPN
ejpam-577	226	17	≤	≤	PROPN
ejpam-577	226	18	n−1	n−1	PROPN
ejpam-577	226	19	2	2	NUM
ejpam-577	226	20	j+n	j+n	ADJ
ejpam-577	226	21	2	2	NUM
ejpam-577	226	22	+	+	CCONJ
ejpam-577	226	23	3	3	NUM
ejpam-577	226	24	if	if	SCONJ
ejpam-577	226	25	j	j	PROPN
ejpam-577	226	26	is	be	AUX
ejpam-577	226	27	odd	odd	ADJ
ejpam-577	226	28	and	and	CCONJ
ejpam-577	226	29	n+3	n+3	PROPN
ejpam-577	226	30	2	2	NUM
ejpam-577	226	31	≤	≤	NUM
ejpam-577	226	32	j	j	PROPN
ejpam-577	226	33	≤	≤	PROPN
ejpam-577	226	34	n	n	CCONJ
ejpam-577	226	35	j+n+1	j+n+1	VERB
ejpam-577	226	36	2	2	NUM
ejpam-577	226	37	if	if	SCONJ
ejpam-577	226	38	j	j	PROPN
ejpam-577	226	39	is	be	AUX
ejpam-577	226	40	even	even	ADV
ejpam-577	226	41	and	and	CCONJ
ejpam-577	226	42	2≤	2≤	NUM
ejpam-577	226	43	j	j	PROPN
ejpam-577	226	44	≤	≤	ADV
ejpam-577	226	45	n−3	n−3	PROPN
ejpam-577	226	46	2	2	NUM
ejpam-577	226	47	j+2	j+2	ADP
ejpam-577	226	48	2	2	NUM
ejpam-577	226	49	+	+	CCONJ
ejpam-577	226	50	3	3	NUM
ejpam-577	226	51	if	if	SCONJ
ejpam-577	226	52	j	j	PROPN
ejpam-577	226	53	is	be	AUX
ejpam-577	226	54	even	even	ADV
ejpam-577	226	55	and	and	CCONJ
ejpam-577	226	56	n+1	n+1	PROPN
ejpam-577	226	57	2	2	NUM
ejpam-577	226	58	≤	≤	NUM
ejpam-577	226	59	j	j	PROPN
ejpam-577	226	60	≤	≤	ADJ
ejpam-577	226	61	n−	n−	NOUN
ejpam-577	226	62	1	1	NUM
ejpam-577	226	63	in	in	ADP
ejpam-577	226	64	all	all	DET
ejpam-577	226	65	the	the	DET
ejpam-577	226	66	cases	case	NOUN
ejpam-577	226	67	f	f	PROPN
ejpam-577	226	68	extends	extend	VERB
ejpam-577	226	69	a	a	DET
ejpam-577	226	70	⌈	⌈	NUM
ejpam-577	226	71	n+3	n+3	PROPN
ejpam-577	226	72	2	2	NUM
ejpam-577	226	73	⌉-strongly	⌉-strongly	ADV
ejpam-577	226	74	indexable	indexable	ADJ
ejpam-577	226	75	labelling	labelling	NOUN
ejpam-577	226	76	,	,	PUNCT
ejpam-577	226	77	as	as	SCONJ
ejpam-577	226	78	the	the	DET
ejpam-577	226	79	edge	edge	NOUN
ejpam-577	226	80	values	value	NOUN
ejpam-577	226	81	are	be	AUX
ejpam-577	226	82	consecutive	consecutive	ADJ
ejpam-577	226	83	integers	integer	NOUN
ejpam-577	226	84	from	from	ADP
ejpam-577	226	85	⌈	⌈	NOUN
ejpam-577	226	86	n+3	n+3	PROPN
ejpam-577	226	87	2	2	NUM
ejpam-577	226	88	⌉	⌉	X
ejpam-577	226	89	to	to	ADP
ejpam-577	226	90	⌈3n+1	⌈3n+1	PROPN
ejpam-577	226	91	2	2	NUM
ejpam-577	226	92	⌉.	⌉.	ADV
ejpam-577	226	93	theorem	theorem	ADJ
ejpam-577	226	94	20	20	NUM
ejpam-577	226	95	.	.	PUNCT
ejpam-577	227	1	for	for	ADP
ejpam-577	227	2	an	an	DET
ejpam-577	227	3	integer	integer	NOUN
ejpam-577	227	4	m	m	VERB
ejpam-577	227	5	≥	≥	NOUN
ejpam-577	227	6	2	2	NUM
ejpam-577	227	7	,	,	PUNCT
ejpam-577	227	8	p3	p3	PROPN
ejpam-577	227	9	∪	∪	X
ejpam-577	227	10	mp2	mp2	PROPN
ejpam-577	227	11	is	be	AUX
ejpam-577	227	12	k	k	NOUN
ejpam-577	227	13	-	-	PUNCT
ejpam-577	227	14	strongly	strongly	ADV
ejpam-577	227	15	indexable	indexable	ADJ
ejpam-577	227	16	,	,	PUNCT
ejpam-577	228	1	where	where	SCONJ
ejpam-577	228	2	k	k	PROPN
ejpam-577	228	3	=	=	PUNCT
ejpam-577	228	4	3m+3	3m+3	NUM
ejpam-577	228	5	2	2	NUM
ejpam-577	228	6	if	if	SCONJ
ejpam-577	228	7	m	m	NOUN
ejpam-577	228	8	is	be	AUX
ejpam-577	228	9	odd	odd	ADJ
ejpam-577	228	10	and	and	CCONJ
ejpam-577	228	11	k	k	NOUN
ejpam-577	229	1	=	=	SYM
ejpam-577	229	2	3m+2	3m+2	NOUN
ejpam-577	229	3	2	2	NUM
ejpam-577	229	4	if	if	SCONJ
ejpam-577	229	5	m	m	NOUN
ejpam-577	229	6	is	be	AUX
ejpam-577	229	7	even	even	ADV
ejpam-577	229	8	.	.	PUNCT
ejpam-577	230	1	proof	proof	NOUN
ejpam-577	230	2	.	.	PUNCT
ejpam-577	231	1	case	case	NOUN
ejpam-577	231	2	1	1	NUM
ejpam-577	231	3	m	m	NOUN
ejpam-577	231	4	is	be	AUX
ejpam-577	231	5	odd	odd	ADJ
ejpam-577	231	6	define	define	VERB
ejpam-577	231	7	f	f	PROPN
ejpam-577	231	8	:	:	PUNCT
ejpam-577	231	9	v	v	X
ejpam-577	231	10	(	(	PUNCT
ejpam-577	231	11	p3	p3	NOUN
ejpam-577	231	12	∪mp2)→	∪mp2)→	X
ejpam-577	231	13	{	{	PUNCT
ejpam-577	231	14	0,1,2	0,1,2	NOUN
ejpam-577	231	15	,	,	PUNCT
ejpam-577	231	16	.	.	PUNCT
ejpam-577	231	17	.	.	PUNCT
ejpam-577	231	18	.	.	PUNCT
ejpam-577	232	1	,	,	PUNCT
ejpam-577	232	2	2m+	2m+	NUM
ejpam-577	232	3	2	2	NUM
ejpam-577	232	4	}	}	PUNCT
ejpam-577	232	5	defined	define	VERB
ejpam-577	232	6	by	by	ADP
ejpam-577	232	7	f	f	PROPN
ejpam-577	232	8	(	(	PUNCT
ejpam-577	232	9	x	x	NOUN
ejpam-577	232	10	)	)	PUNCT
ejpam-577	232	11	=	=	SYM
ejpam-577	232	12			PROPN
ejpam-577	232	13			X
ejpam-577	232	14			PROPN
ejpam-577	232	15			PROPN
ejpam-577	232	16			PROPN
ejpam-577	232	17			PROPN
ejpam-577	232	18			PROPN
ejpam-577	232	19			PROPN
ejpam-577	232	20			NOUN
ejpam-577	232	21			PROPN
ejpam-577	232	22			PROPN
ejpam-577	232	23			PROPN
ejpam-577	232	24			PROPN
ejpam-577	232	25			PROPN
ejpam-577	232	26			PROPN
ejpam-577	232	27			PROPN
ejpam-577	232	28			NOUN
ejpam-577	232	29	3m−5	3m−5	NUM
ejpam-577	232	30	2	2	NUM
ejpam-577	232	31	if	if	SCONJ
ejpam-577	232	32	x	x	NOUN
ejpam-577	232	33	=	=	PUNCT
ejpam-577	232	34	u	u	NOUN
ejpam-577	232	35	m+	m+	NUM
ejpam-577	232	36	1	1	NUM
ejpam-577	232	37	if	if	SCONJ
ejpam-577	232	38	x	x	X
ejpam-577	232	39	=	=	SYM
ejpam-577	232	40	v	v	X
ejpam-577	232	41	m+3	m+3	SYM
ejpam-577	232	42	2	2	NUM
ejpam-577	232	43	if	if	SCONJ
ejpam-577	232	44	x	x	PROPN
ejpam-577	232	45	=	=	PUNCT
ejpam-577	233	1	w	w	NOUN
ejpam-577	233	2	i	i	PRON
ejpam-577	233	3	−	−	NOUN
ejpam-577	233	4	1	1	NUM
ejpam-577	234	1	if	if	SCONJ
ejpam-577	234	2	x	x	PRON
ejpam-577	234	3	=	=	PRON
ejpam-577	234	4	ui	ui	PROPN
ejpam-577	234	5	and	and	CCONJ
ejpam-577	234	6	m−1	m−1	PROPN
ejpam-577	234	7	2	2	NUM
ejpam-577	234	8	≤	≤	NUM
ejpam-577	235	1	i	i	PRON
ejpam-577	235	2	≤	≤	ADV
ejpam-577	235	3	m+3	m+3	SYM
ejpam-577	235	4	2	2	NUM
ejpam-577	235	5	i	i	PRON
ejpam-577	235	6	if	if	SCONJ
ejpam-577	235	7	x	x	PRON
ejpam-577	235	8	=	=	VERB
ejpam-577	235	9	ui	ui	PROPN
ejpam-577	236	1	and	and	CCONJ
ejpam-577	236	2	m+5	m+5	PROPN
ejpam-577	236	3	2	2	NUM
ejpam-577	236	4	≤	≤	NUM
ejpam-577	236	5	i	i	PRON
ejpam-577	237	1	≤	≤	NOUN
ejpam-577	237	2	m	m	VERB
ejpam-577	238	1	i	i	PRON
ejpam-577	238	2	+	+	NOUN
ejpam-577	238	3	3m−2	3m−2	NUM
ejpam-577	238	4	2	2	NUM
ejpam-577	238	5	+	+	SYM
ejpam-577	238	6	2	2	NUM
ejpam-577	238	7	if	if	SCONJ
ejpam-577	238	8	x	x	PROPN
ejpam-577	238	9	=	=	SYM
ejpam-577	238	10	vi	vi	PROPN
ejpam-577	238	11	and	and	CCONJ
ejpam-577	238	12	1≤	1≤	NUM
ejpam-577	239	1	i	i	PRON
ejpam-577	239	2	≤	≤	VERB
ejpam-577	239	3	m+3	m+3	SYM
ejpam-577	239	4	2	2	NUM
ejpam-577	240	1	i	i	NOUN
ejpam-577	240	2	+	+	CCONJ
ejpam-577	240	3	m−1	m−1	PROPN
ejpam-577	240	4	2	2	NUM
ejpam-577	240	5	if	if	SCONJ
ejpam-577	240	6	x	x	X
ejpam-577	240	7	=	=	SYM
ejpam-577	240	8	vi	vi	PROPN
ejpam-577	240	9	and	and	CCONJ
ejpam-577	240	10	m+5	m+5	SYM
ejpam-577	240	11	2	2	NUM
ejpam-577	240	12	≤	≤	NUM
ejpam-577	240	13	i	i	PRON
ejpam-577	240	14	≤	≤	NOUN
ejpam-577	240	15	m	m	VERB
ejpam-577	240	16	clearly	clearly	ADV
ejpam-577	240	17	f	f	PROPN
ejpam-577	240	18	is	be	AUX
ejpam-577	240	19	infective	infective	ADJ
ejpam-577	240	20	and	and	CCONJ
ejpam-577	240	21	the	the	DET
ejpam-577	240	22	edge	edge	NOUN
ejpam-577	240	23	values	value	NOUN
ejpam-577	240	24	are	be	AUX
ejpam-577	240	25	consecutive	consecutive	ADJ
ejpam-577	240	26	numbers	number	NOUN
ejpam-577	240	27	from	from	ADP
ejpam-577	240	28	3m+3	3m+3	PROPN
ejpam-577	240	29	2	2	NUM
ejpam-577	240	30	to	to	ADP
ejpam-577	240	31	5m−1	5m−1	NUM
ejpam-577	240	32	2	2	NUM
ejpam-577	240	33	.	.	PUNCT
ejpam-577	241	1	case	case	NOUN
ejpam-577	241	2	2	2	NUM
ejpam-577	241	3	m	m	NOUN
ejpam-577	241	4	is	be	AUX
ejpam-577	241	5	even	even	ADV
ejpam-577	241	6	define	define	VERB
ejpam-577	241	7	f	f	PROPN
ejpam-577	241	8	:	:	PUNCT
ejpam-577	241	9	v	v	X
ejpam-577	241	10	(	(	PUNCT
ejpam-577	241	11	p3	p3	NOUN
ejpam-577	241	12	∪mp2)→	∪mp2)→	X
ejpam-577	241	13	{	{	PUNCT
ejpam-577	241	14	0,1,2	0,1,2	NOUN
ejpam-577	241	15	,	,	PUNCT
ejpam-577	241	16	.	.	PUNCT
ejpam-577	241	17	.	.	PUNCT
ejpam-577	241	18	.	.	PUNCT
ejpam-577	242	1	,	,	PUNCT
ejpam-577	242	2	2m+	2m+	NUM
ejpam-577	242	3	2	2	NUM
ejpam-577	242	4	}	}	PUNCT
ejpam-577	242	5	defined	define	VERB
ejpam-577	242	6	by	by	ADP
ejpam-577	242	7	f	f	PROPN
ejpam-577	242	8	(	(	PUNCT
ejpam-577	242	9	x	x	NOUN
ejpam-577	242	10	)	)	PUNCT
ejpam-577	242	11	=	=	SYM
ejpam-577	242	12			PROPN
ejpam-577	242	13			X
ejpam-577	242	14			PROPN
ejpam-577	242	15			PROPN
ejpam-577	242	16			PROPN
ejpam-577	242	17			PROPN
ejpam-577	242	18			PROPN
ejpam-577	242	19			NOUN
ejpam-577	242	20			PROPN
ejpam-577	242	21			PROPN
ejpam-577	242	22			PROPN
ejpam-577	242	23			PROPN
ejpam-577	242	24			PROPN
ejpam-577	242	25			PROPN
ejpam-577	242	26			NOUN
ejpam-577	242	27	m+	m+	NUM
ejpam-577	242	28	1	1	NUM
ejpam-577	242	29	if	if	SCONJ
ejpam-577	242	30	x	x	NOUN
ejpam-577	242	31	=	=	PUNCT
ejpam-577	242	32	u	u	X
ejpam-577	242	33	m	m	VERB
ejpam-577	242	34	if	if	SCONJ
ejpam-577	242	35	x	x	SYM
ejpam-577	242	36	=	=	SYM
ejpam-577	242	37	v	v	NUM
ejpam-577	242	38	m+	m+	NUM
ejpam-577	242	39	2	2	NUM
ejpam-577	242	40	if	if	SCONJ
ejpam-577	242	41	x	x	PROPN
ejpam-577	242	42	=	=	PUNCT
ejpam-577	243	1	w	w	NOUN
ejpam-577	243	2	i	i	PRON
ejpam-577	243	3	−	−	NOUN
ejpam-577	243	4	1	1	NUM
ejpam-577	244	1	if	if	SCONJ
ejpam-577	244	2	x	x	PRON
ejpam-577	244	3	=	=	PRON
ejpam-577	244	4	ui	ui	PROPN
ejpam-577	244	5	and	and	CCONJ
ejpam-577	244	6	1≤	1≤	NUM
ejpam-577	245	1	i	i	X
ejpam-577	245	2	≤	≤	VERB
ejpam-577	245	3	m+2	m+2	ADP
ejpam-577	245	4	2	2	NUM
ejpam-577	245	5	i	i	PRON
ejpam-577	245	6	if	if	SCONJ
ejpam-577	245	7	x	x	PRON
ejpam-577	245	8	=	=	VERB
ejpam-577	245	9	ui	ui	PROPN
ejpam-577	245	10	and	and	CCONJ
ejpam-577	245	11	m	m	PROPN
ejpam-577	245	12	2	2	NUM
ejpam-577	245	13	≤	≤	NUM
ejpam-577	245	14	i	i	PRON
ejpam-577	246	1	≤	≤	NOUN
ejpam-577	246	2	m	m	VERB
ejpam-577	247	1	i	i	PRON
ejpam-577	247	2	+	+	NOUN
ejpam-577	247	3	3	3	NUM
ejpam-577	247	4	m	m	NOUN
ejpam-577	247	5	2	2	NUM
ejpam-577	247	6	+	+	NUM
ejpam-577	247	7	2	2	NUM
ejpam-577	247	8	if	if	SCONJ
ejpam-577	247	9	x	x	PROPN
ejpam-577	247	10	=	=	SYM
ejpam-577	247	11	vi	vi	PROPN
ejpam-577	248	1	and	and	CCONJ
ejpam-577	248	2	1≤	1≤	NUM
ejpam-577	249	1	i	i	PRON
ejpam-577	249	2	≤	≤	NUM
ejpam-577	249	3	m	m	VERB
ejpam-577	249	4	2	2	NUM
ejpam-577	250	1	i	i	NOUN
ejpam-577	250	2	+	+	NOUN
ejpam-577	250	3	m	m	VERB
ejpam-577	250	4	2	2	NUM
ejpam-577	250	5	+	+	NUM
ejpam-577	250	6	2	2	NUM
ejpam-577	250	7	if	if	SCONJ
ejpam-577	250	8	x	x	PROPN
ejpam-577	250	9	=	=	SYM
ejpam-577	250	10	vi	vi	PROPN
ejpam-577	250	11	and	and	CCONJ
ejpam-577	250	12	m+2	m+2	ADP
ejpam-577	250	13	2	2	NUM
ejpam-577	250	14	≤	≤	NUM
ejpam-577	251	1	i	i	PRON
ejpam-577	251	2	≤	≤	NOUN
ejpam-577	251	3	m	m	VERB
ejpam-577	251	4	clearly	clearly	ADV
ejpam-577	251	5	f	f	PROPN
ejpam-577	251	6	is	be	AUX
ejpam-577	251	7	infective	infective	ADJ
ejpam-577	251	8	and	and	CCONJ
ejpam-577	251	9	the	the	DET
ejpam-577	251	10	edge	edge	NOUN
ejpam-577	251	11	values	value	NOUN
ejpam-577	251	12	are	be	AUX
ejpam-577	251	13	consecutive	consecutive	ADJ
ejpam-577	251	14	numbers	number	NOUN
ejpam-577	251	15	from	from	ADP
ejpam-577	251	16	3m+2	3m+2	PROPN
ejpam-577	251	17	2	2	NUM
ejpam-577	251	18	to	to	ADP
ejpam-577	251	19	5	5	NUM
ejpam-577	251	20	m	m	NOUN
ejpam-577	251	21	2	2	NUM
ejpam-577	251	22	+	+	CCONJ
ejpam-577	251	23	2	2	NUM
ejpam-577	251	24	.	.	X
ejpam-577	251	25	theorem	theorem	NOUN
ejpam-577	251	26	21	21	NUM
ejpam-577	251	27	.	.	PUNCT
ejpam-577	252	1	k1,n	k1,n	PROPN
ejpam-577	252	2	∪	∪	PROPN
ejpam-577	252	3	k1,n+1	k1,n+1	PROPN
ejpam-577	252	4	,	,	PUNCT
ejpam-577	252	5	n≥	n≥	NOUN
ejpam-577	252	6	1	1	NUM
ejpam-577	252	7	is	be	AUX
ejpam-577	252	8	strongly	strongly	ADV
ejpam-577	252	9	3	3	NUM
ejpam-577	252	10	-	-	PUNCT
ejpam-577	252	11	indexable	indexable	ADJ
ejpam-577	252	12	proof	proof	NOUN
ejpam-577	252	13	.	.	PUNCT
ejpam-577	253	1	let	let	VERB
ejpam-577	253	2	v	v	NOUN
ejpam-577	253	3	(	(	PUNCT
ejpam-577	253	4	k1,n∪k1,n+1	k1,n∪k1,n+1	NOUN
ejpam-577	253	5	)	)	PUNCT
ejpam-577	253	6	=	=	PRON
ejpam-577	253	7	{	{	PUNCT
ejpam-577	253	8	u	u	PROPN
ejpam-577	253	9	,	,	PUNCT
ejpam-577	253	10	ui	ui	PROPN
ejpam-577	253	11	,	,	PUNCT
ejpam-577	253	12	1≤	1≤	INTJ
ejpam-577	254	1	i	i	PROPN
ejpam-577	254	2	≤	≤	PROPN
ejpam-577	254	3	n}∪{v	n}∪{v	PROPN
ejpam-577	254	4	,	,	PUNCT
ejpam-577	254	5	vi	vi	PROPN
ejpam-577	254	6	,	,	PUNCT
ejpam-577	254	7	1≤	1≤	NUM
ejpam-577	255	1	i	i	PRON
ejpam-577	255	2	≤	≤	PUNCT
ejpam-577	255	3	n+1	n+1	VERB
ejpam-577	255	4	}	}	PUNCT
ejpam-577	255	5	and	and	CCONJ
ejpam-577	255	6	e(k1,n∪k1,n+1	e(k1,n∪k1,n+1	ADJ
ejpam-577	255	7	)	)	PUNCT
ejpam-577	255	8	=	=	PRON
ejpam-577	256	1	{	{	PUNCT
ejpam-577	256	2	uui	uui	NOUN
ejpam-577	256	3	:	:	PUNCT
ejpam-577	256	4	1≤	1≤	NUM
ejpam-577	256	5	i	i	NOUN
ejpam-577	256	6	≤	≤	ADJ
ejpam-577	256	7	n	n	CCONJ
ejpam-577	256	8	}	}	PUNCT
ejpam-577	256	9	∪	∪	ADJ
ejpam-577	256	10	{	{	PUNCT
ejpam-577	256	11	vvi	vvi	NOUN
ejpam-577	256	12	:	:	PUNCT
ejpam-577	256	13	1≤	1≤	NUM
ejpam-577	256	14	i	i	PROPN
ejpam-577	256	15	≤	≤	PUNCT
ejpam-577	256	16	n+	n+	PUNCT
ejpam-577	256	17	1	1	NUM
ejpam-577	256	18	}	}	PUNCT
ejpam-577	256	19	.	.	PUNCT
ejpam-577	257	1	k.	k.	PROPN
ejpam-577	257	2	a.	a.	PROPN
ejpam-577	257	3	germina	germina	PROPN
ejpam-577	257	4	/	/	SYM
ejpam-577	257	5	eur	eur	PROPN
ejpam-577	257	6	.	.	PUNCT
ejpam-577	258	1	j.	j.	PROPN
ejpam-577	258	2	pure	pure	PROPN
ejpam-577	258	3	appl	appl	PROPN
ejpam-577	258	4	.	.	PROPN
ejpam-577	258	5	math	math	PROPN
ejpam-577	258	6	,	,	PUNCT
ejpam-577	258	7	3	3	NUM
ejpam-577	258	8	(	(	PUNCT
ejpam-577	258	9	2010	2010	NUM
ejpam-577	258	10	)	)	PUNCT
ejpam-577	258	11	,	,	PUNCT
ejpam-577	258	12	269	269	NUM
ejpam-577	258	13	-	-	SYM
ejpam-577	258	14	281	281	NUM
ejpam-577	258	15	278	278	NUM
ejpam-577	258	16	define	define	NOUN
ejpam-577	258	17	f	f	NOUN
ejpam-577	258	18	:	:	PUNCT
ejpam-577	258	19	v	v	X
ejpam-577	258	20	(	(	PUNCT
ejpam-577	258	21	k1,n	k1,n	PROPN
ejpam-577	258	22	∪	∪	VERB
ejpam-577	258	23	k1,n+1)→	k1,n+1)→	PUNCT
ejpam-577	258	24	{	{	PUNCT
ejpam-577	258	25	0,1,2	0,1,2	NOUN
ejpam-577	258	26	,	,	PUNCT
ejpam-577	258	27	.	.	PUNCT
ejpam-577	258	28	.	.	PUNCT
ejpam-577	259	1	.	.	PUNCT
ejpam-577	260	1	,	,	PUNCT
ejpam-577	260	2	2n+	2n+	NUM
ejpam-577	260	3	2	2	NUM
ejpam-577	260	4	}	}	PUNCT
ejpam-577	260	5	defined	define	VERB
ejpam-577	260	6	by	by	ADP
ejpam-577	260	7	f	f	PROPN
ejpam-577	260	8	(	(	PUNCT
ejpam-577	260	9	u	u	NOUN
ejpam-577	260	10	)	)	PUNCT
ejpam-577	260	11	=	=	SYM
ejpam-577	260	12	0	0	NUM
ejpam-577	260	13	;	;	PUNCT
ejpam-577	260	14	f	f	PROPN
ejpam-577	260	15	(	(	PUNCT
ejpam-577	260	16	ui	ui	PROPN
ejpam-577	260	17	)	)	PUNCT
ejpam-577	260	18	=	=	SYM
ejpam-577	260	19	2(i+	2(i+	NUM
ejpam-577	260	20	1	1	NUM
ejpam-577	260	21	)	)	PUNCT
ejpam-577	260	22	,	,	PUNCT
ejpam-577	260	23	1≤	1≤	NUM
ejpam-577	261	1	i	i	ADV
ejpam-577	261	2	≤	≤	PUNCT
ejpam-577	262	1	n	n	PRON
ejpam-577	262	2	f	f	PROPN
ejpam-577	262	3	(	(	PUNCT
ejpam-577	262	4	v	v	NOUN
ejpam-577	262	5	)	)	PUNCT
ejpam-577	262	6	=	=	SYM
ejpam-577	262	7	2	2	NUM
ejpam-577	262	8	;	;	PUNCT
ejpam-577	262	9	f	f	PROPN
ejpam-577	262	10	(	(	PUNCT
ejpam-577	262	11	vi	vi	NOUN
ejpam-577	262	12	)	)	PUNCT
ejpam-577	262	13	=	=	SYM
ejpam-577	263	1	2i−	2i−	NUM
ejpam-577	263	2	1	1	NUM
ejpam-577	263	3	,	,	PUNCT
ejpam-577	263	4	1≤	1≤	NUM
ejpam-577	263	5	i	i	PROPN
ejpam-577	263	6	≤	≤	PUNCT
ejpam-577	263	7	n+	n+	PUNCT
ejpam-577	263	8	1	1	X
ejpam-577	263	9	.	.	PUNCT
ejpam-577	264	1	now	now	ADV
ejpam-577	264	2	,	,	PUNCT
ejpam-577	264	3	clearly	clearly	ADV
ejpam-577	264	4	f	f	X
ejpam-577	264	5	(	(	PUNCT
ejpam-577	264	6	v	v	NOUN
ejpam-577	264	7	(	(	PUNCT
ejpam-577	264	8	k1,n	k1,n	PROPN
ejpam-577	264	9	∪	∪	PROPN
ejpam-577	264	10	k1,n+1	k1,n+1	NOUN
ejpam-577	264	11	)	)	PUNCT
ejpam-577	264	12	)	)	PUNCT
ejpam-577	265	1	=	=	PRON
ejpam-577	265	2	{	{	PUNCT
ejpam-577	265	3	0,1,2	0,1,2	NOUN
ejpam-577	265	4	,	,	PUNCT
ejpam-577	265	5	.	.	PUNCT
ejpam-577	265	6	.	.	PUNCT
ejpam-577	265	7	.	.	PUNCT
ejpam-577	266	1	2n+	2n+	NUM
ejpam-577	267	1	2	2	NUM
ejpam-577	267	2	}	}	PUNCT
ejpam-577	267	3	.	.	PUNCT
ejpam-577	268	1	also	also	ADV
ejpam-577	268	2	,	,	PUNCT
ejpam-577	268	3	the	the	DET
ejpam-577	268	4	minimum	minimum	ADJ
ejpam-577	268	5	and	and	CCONJ
ejpam-577	268	6	maximum	maximum	ADJ
ejpam-577	268	7	edge	edge	NOUN
ejpam-577	268	8	value	value	NOUN
ejpam-577	268	9	induced	induce	VERB
ejpam-577	268	10	at	at	ADP
ejpam-577	268	11	the	the	DET
ejpam-577	268	12	edges	edge	NOUN
ejpam-577	268	13	are	be	AUX
ejpam-577	268	14	f	f	PROPN
ejpam-577	268	15	+	+	ADJ
ejpam-577	268	16	(	(	PUNCT
ejpam-577	268	17	vv1	vv1	NOUN
ejpam-577	268	18	)	)	PUNCT
ejpam-577	268	19	=	=	SYM
ejpam-577	269	1	2	2	NUM
ejpam-577	269	2	+	+	NUM
ejpam-577	269	3	f	f	X
ejpam-577	269	4	(	(	PUNCT
ejpam-577	269	5	v1	v1	NOUN
ejpam-577	269	6	)	)	PUNCT
ejpam-577	269	7	=	=	SYM
ejpam-577	269	8	3	3	NUM
ejpam-577	269	9	and	and	CCONJ
ejpam-577	269	10	f	f	PROPN
ejpam-577	270	1	+	+	ADJ
ejpam-577	270	2	(	(	PUNCT
ejpam-577	270	3	vvn+1	vvn+1	NOUN
ejpam-577	270	4	)	)	PUNCT
ejpam-577	270	5	=	=	SYM
ejpam-577	271	1	2	2	NUM
ejpam-577	271	2	+	+	NUM
ejpam-577	271	3	f	f	X
ejpam-577	271	4	(	(	PUNCT
ejpam-577	271	5	vn+1	vn+1	PROPN
ejpam-577	271	6	)	)	PUNCT
ejpam-577	271	7	=	=	SYM
ejpam-577	272	1	2n+	2n+	NUM
ejpam-577	272	2	3	3	NUM
ejpam-577	272	3	;	;	PUNCT
ejpam-577	272	4	also	also	ADV
ejpam-577	272	5	note	note	VERB
ejpam-577	272	6	that	that	SCONJ
ejpam-577	272	7	f	f	PROPN
ejpam-577	272	8	(	(	PUNCT
ejpam-577	272	9	u	u	NOUN
ejpam-577	272	10	)	)	PUNCT
ejpam-577	273	1	+	+	NUM
ejpam-577	273	2	f	f	X
ejpam-577	273	3	(	(	PUNCT
ejpam-577	273	4	ui	ui	NOUN
ejpam-577	273	5	)	)	PUNCT
ejpam-577	273	6	is	be	AUX
ejpam-577	273	7	always	always	ADV
ejpam-577	273	8	even	even	ADV
ejpam-577	273	9	and	and	CCONJ
ejpam-577	273	10	f	f	PROPN
ejpam-577	273	11	(	(	PUNCT
ejpam-577	273	12	v	v	NOUN
ejpam-577	273	13	)	)	PUNCT
ejpam-577	273	14	+	+	NUM
ejpam-577	273	15	f	f	X
ejpam-577	273	16	(	(	PUNCT
ejpam-577	273	17	vi	vi	NOUN
ejpam-577	273	18	)	)	PUNCT
ejpam-577	273	19	is	be	AUX
ejpam-577	273	20	always	always	ADV
ejpam-577	273	21	odd	odd	ADJ
ejpam-577	273	22	.	.	PUNCT
ejpam-577	274	1	again	again	ADV
ejpam-577	274	2	,	,	PUNCT
ejpam-577	274	3	f	f	PROPN
ejpam-577	274	4	(	(	PUNCT
ejpam-577	274	5	u	u	NOUN
ejpam-577	274	6	)	)	PUNCT
ejpam-577	274	7	+	+	NUM
ejpam-577	274	8	f	f	X
ejpam-577	274	9	(	(	PUNCT
ejpam-577	274	10	ui	ui	PROPN
ejpam-577	274	11	)	)	PUNCT
ejpam-577	274	12	6=	6=	ADP
ejpam-577	275	1	f	f	PROPN
ejpam-577	275	2	(	(	PUNCT
ejpam-577	275	3	u	u	NOUN
ejpam-577	275	4	)	)	PUNCT
ejpam-577	276	1	+	+	NUM
ejpam-577	276	2	f	f	X
ejpam-577	276	3	(	(	PUNCT
ejpam-577	276	4	u	u	NOUN
ejpam-577	276	5	j	j	PROPN
ejpam-577	276	6	)	)	PUNCT
ejpam-577	276	7	for	for	ADP
ejpam-577	276	8	all	all	DET
ejpam-577	276	9	i	i	PRON
ejpam-577	276	10	6=	6=	PROPN
ejpam-577	276	11	j	j	PROPN
ejpam-577	276	12	,	,	PUNCT
ejpam-577	276	13	and	and	CCONJ
ejpam-577	276	14	f	f	PROPN
ejpam-577	276	15	(	(	PUNCT
ejpam-577	276	16	v	v	NOUN
ejpam-577	276	17	)	)	PUNCT
ejpam-577	276	18	+	+	NUM
ejpam-577	276	19	f	f	X
ejpam-577	276	20	(	(	PUNCT
ejpam-577	276	21	vi	vi	PROPN
ejpam-577	276	22	)	)	PUNCT
ejpam-577	276	23	6=	6=	NUM
ejpam-577	276	24	f	f	PROPN
ejpam-577	276	25	(	(	PUNCT
ejpam-577	276	26	v	v	NOUN
ejpam-577	276	27	)	)	PUNCT
ejpam-577	277	1	+	+	NUM
ejpam-577	277	2	f	f	X
ejpam-577	277	3	(	(	PUNCT
ejpam-577	277	4	v	v	PROPN
ejpam-577	277	5	j	j	NOUN
ejpam-577	277	6	)	)	PUNCT
ejpam-577	277	7	for	for	ADP
ejpam-577	277	8	all	all	DET
ejpam-577	277	9	i	i	PRON
ejpam-577	277	10	6=	6=	NUM
ejpam-577	277	11	j.	j.	PROPN
ejpam-577	277	12	hence	hence	PROPN
ejpam-577	277	13	,	,	PUNCT
ejpam-577	277	14	the	the	DET
ejpam-577	277	15	induced	induced	ADJ
ejpam-577	277	16	edge	edge	NOUN
ejpam-577	277	17	values	value	NOUN
ejpam-577	277	18	are	be	AUX
ejpam-577	277	19	consecutive	consecutive	ADJ
ejpam-577	277	20	integers	integer	NOUN
ejpam-577	277	21	from	from	ADP
ejpam-577	277	22	3	3	NUM
ejpam-577	277	23	to	to	ADP
ejpam-577	277	24	2n+	2n+	NUM
ejpam-577	277	25	2	2	NUM
ejpam-577	277	26	,	,	PUNCT
ejpam-577	277	27	which	which	PRON
ejpam-577	277	28	implies	imply	VERB
ejpam-577	277	29	f	f	PROPN
ejpam-577	277	30	is	be	AUX
ejpam-577	277	31	a	a	DET
ejpam-577	277	32	3	3	NUM
ejpam-577	277	33	-	-	PUNCT
ejpam-577	277	34	strong	strong	ADJ
ejpam-577	277	35	indexer	indexer	NOUN
ejpam-577	277	36	of	of	ADP
ejpam-577	277	37	k1,n	k1,n	PROPN
ejpam-577	277	38	∪	∪	PROPN
ejpam-577	277	39	k1,n+1	k1,n+1	PROPN
ejpam-577	277	40	.	.	PUNCT
ejpam-577	277	41	theorem	theorem	PROPN
ejpam-577	277	42	21	21	NUM
ejpam-577	277	43	can	can	AUX
ejpam-577	277	44	be	be	AUX
ejpam-577	277	45	expanded	expand	VERB
ejpam-577	277	46	to	to	ADP
ejpam-577	277	47	the	the	DET
ejpam-577	277	48	following	follow	VERB
ejpam-577	277	49	form	form	NOUN
ejpam-577	277	50	theorem	theorem	VERB
ejpam-577	277	51	22	22	NUM
ejpam-577	277	52	.	.	PUNCT
ejpam-577	278	1	for	for	ADP
ejpam-577	278	2	the	the	DET
ejpam-577	278	3	integers	integer	NOUN
ejpam-577	278	4	m	m	PRON
ejpam-577	278	5	,	,	PUNCT
ejpam-577	278	6	n	n	CCONJ
ejpam-577	278	7	,	,	PUNCT
ejpam-577	278	8	mk1,n	mk1,n	NOUN
ejpam-577	278	9	,	,	PUNCT
ejpam-577	278	10	is	be	AUX
ejpam-577	278	11	3m−1	3m−1	NUM
ejpam-577	278	12	2	2	NUM
ejpam-577	278	13	-strongly	-strongly	NOUN
ejpam-577	278	14	indexable	indexable	ADJ
ejpam-577	278	15	,	,	PUNCT
ejpam-577	278	16	whenever	whenever	SCONJ
ejpam-577	278	17	m	m	VERB
ejpam-577	278	18	is	be	AUX
ejpam-577	278	19	odd	odd	ADJ
ejpam-577	278	20	.	.	PUNCT
ejpam-577	279	1	proof	proof	NOUN
ejpam-577	279	2	.	.	PUNCT
ejpam-577	280	1	let	let	VERB
ejpam-577	280	2	v	v	NOUN
ejpam-577	280	3	(	(	PUNCT
ejpam-577	280	4	mk1,n	mk1,n	NOUN
ejpam-577	280	5	)	)	PUNCT
ejpam-577	280	6	=	=	PRON
ejpam-577	281	1	{	{	PUNCT
ejpam-577	281	2	ui	ui	NOUN
ejpam-577	281	3	:	:	PUNCT
ejpam-577	281	4	1≤	1≤	NUM
ejpam-577	282	1	i	i	X
ejpam-577	282	2	≤	≤	PUNCT
ejpam-577	282	3	m	m	VERB
ejpam-577	282	4	}	}	PUNCT
ejpam-577	282	5	∪	∪	ADJ
ejpam-577	282	6	{	{	PUNCT
ejpam-577	282	7	vi	vi	PROPN
ejpam-577	282	8	,	,	PUNCT
ejpam-577	282	9	j	j	NOUN
ejpam-577	282	10	:	:	PUNCT
ejpam-577	282	11	1≤	1≤	NUM
ejpam-577	282	12	i	i	X
ejpam-577	282	13	≤	≤	PUNCT
ejpam-577	282	14	m	m	ADP
ejpam-577	282	15	,	,	PUNCT
ejpam-577	282	16	1≤	1≤	PROPN
ejpam-577	282	17	j	j	PROPN
ejpam-577	282	18	≤	≤	PROPN
ejpam-577	282	19	n	n	CCONJ
ejpam-577	282	20	}	}	PUNCT
ejpam-577	282	21	and	and	CCONJ
ejpam-577	282	22	e(mk1,n	e(mk1,n	NOUN
ejpam-577	282	23	)	)	PUNCT
ejpam-577	283	1	=	=	PRON
ejpam-577	283	2	{	{	PUNCT
ejpam-577	283	3	ui	ui	PROPN
ejpam-577	283	4	vi	vi	PROPN
ejpam-577	283	5	,	,	PUNCT
ejpam-577	283	6	j	j	NOUN
ejpam-577	283	7	:	:	PUNCT
ejpam-577	283	8	1≤	1≤	NUM
ejpam-577	283	9	i	i	X
ejpam-577	284	1	≤	≤	PUNCT
ejpam-577	284	2	m	m	ADP
ejpam-577	284	3	,	,	PUNCT
ejpam-577	284	4	1≤	1≤	PROPN
ejpam-577	284	5	j	j	PROPN
ejpam-577	284	6	≤	≤	PROPN
ejpam-577	284	7	n	n	CCONJ
ejpam-577	284	8	}	}	PUNCT
ejpam-577	284	9	.	.	PUNCT
ejpam-577	285	1	define	define	VERB
ejpam-577	285	2	f	f	X
ejpam-577	285	3	:	:	PUNCT
ejpam-577	285	4	v	v	X
ejpam-577	285	5	(	(	PUNCT
ejpam-577	285	6	mk1,n)→	mk1,n)→	NOUN
ejpam-577	285	7	{	{	PUNCT
ejpam-577	285	8	0,1,2	0,1,2	NOUN
ejpam-577	285	9	.	.	PUNCT
ejpam-577	285	10	.	.	PUNCT
ejpam-577	285	11	.	.	PUNCT
ejpam-577	286	1	,	,	PUNCT
ejpam-577	286	2	m(n+	m(n+	PROPN
ejpam-577	287	1	1)−	1)−	PROPN
ejpam-577	287	2	1	1	NUM
ejpam-577	287	3	}	}	PUNCT
ejpam-577	287	4	defined	define	VERB
ejpam-577	287	5	by	by	ADP
ejpam-577	287	6	f	f	PROPN
ejpam-577	287	7	(	(	PUNCT
ejpam-577	287	8	ui	ui	PROPN
ejpam-577	287	9	)	)	PUNCT
ejpam-577	287	10	=	=	VERB
ejpam-577	288	1	i	i	PRON
ejpam-577	288	2	−	−	PROPN
ejpam-577	288	3	1	1	NUM
ejpam-577	288	4	,	,	PUNCT
ejpam-577	288	5	1≤	1≤	NUM
ejpam-577	288	6	i	i	PRON
ejpam-577	288	7	≤	≤	PROPN
ejpam-577	288	8	m	m	VERB
ejpam-577	288	9	;	;	PUNCT
ejpam-577	288	10	f	f	PROPN
ejpam-577	288	11	(	(	PUNCT
ejpam-577	288	12	vi,1	vi,1	PROPN
ejpam-577	288	13	)	)	PUNCT
ejpam-577	288	14	=	=	PUNCT
ejpam-577	289	1	¨	¨	NOUN
ejpam-577	289	2	i+	i+	NUM
ejpam-577	289	3	3m−3	3m−3	NUM
ejpam-577	289	4	2	2	NUM
ejpam-577	289	5	if	if	SCONJ
ejpam-577	289	6	1≤	1≤	NUM
ejpam-577	289	7	i	i	VERB
ejpam-577	289	8	≤	≤	VERB
ejpam-577	290	1	m+1	m+1	NUM
ejpam-577	290	2	2	2	NUM
ejpam-577	290	3	i+	i+	NUM
ejpam-577	290	4	m−3	m−3	NOUN
ejpam-577	290	5	2	2	NUM
ejpam-577	290	6	if	if	SCONJ
ejpam-577	290	7	m+1	m+1	PRON
ejpam-577	290	8	2	2	NUM
ejpam-577	290	9	<	<	X
ejpam-577	290	10	i	i	NOUN
ejpam-577	290	11	≤	≤	PUNCT
ejpam-577	290	12	m	m	VERB
ejpam-577	290	13	and	and	CCONJ
ejpam-577	290	14	f	f	PROPN
ejpam-577	290	15	(	(	PUNCT
ejpam-577	290	16	vi	vi	PROPN
ejpam-577	290	17	,	,	PUNCT
ejpam-577	290	18	j	j	NOUN
ejpam-577	290	19	)	)	PUNCT
ejpam-577	290	20	=	=	SYM
ejpam-577	290	21	f	f	PROPN
ejpam-577	290	22	(	(	PUNCT
ejpam-577	290	23	vi,1	vi,1	PROPN
ejpam-577	290	24	)	)	PUNCT
ejpam-577	291	1	+	+	NOUN
ejpam-577	291	2	m	m	PROPN
ejpam-577	291	3	(	(	PUNCT
ejpam-577	291	4	j−	j−	PROPN
ejpam-577	291	5	1	1	NUM
ejpam-577	291	6	)	)	PUNCT
ejpam-577	291	7	,	,	PUNCT
ejpam-577	291	8	1≤	1≤	NUM
ejpam-577	292	1	i	i	X
ejpam-577	292	2	≤	≤	PUNCT
ejpam-577	292	3	m	m	ADP
ejpam-577	292	4	,	,	PUNCT
ejpam-577	292	5	2≤	2≤	NUM
ejpam-577	292	6	j	j	PROPN
ejpam-577	292	7	≤	≤	PROPN
ejpam-577	292	8	n	n	CCONJ
ejpam-577	292	9	;	;	PUNCT
ejpam-577	292	10	clearly	clearly	ADV
ejpam-577	292	11	,	,	PUNCT
ejpam-577	292	12	f	f	PROPN
ejpam-577	292	13	is	be	AUX
ejpam-577	292	14	a	a	DET
ejpam-577	292	15	3m−1	3m−1	NUM
ejpam-577	292	16	2	2	NUM
ejpam-577	292	17	-strong	-strong	NOUN
ejpam-577	292	18	indexer	indexer	NOUN
ejpam-577	292	19	of	of	ADP
ejpam-577	292	20	mk1,n	mk1,n	NOUN
ejpam-577	292	21	.	.	PUNCT
ejpam-577	293	1	figure	figure	NOUN
ejpam-577	293	2	7	7	NUM
ejpam-577	293	3	gives	give	VERB
ejpam-577	293	4	the	the	DET
ejpam-577	293	5	strong	strong	ADJ
ejpam-577	293	6	indexer	indexer	NOUN
ejpam-577	293	7	of	of	ADP
ejpam-577	293	8	5k4	5k4	NUM
ejpam-577	293	9	.	.	PUNCT
ejpam-577	294	1	figure	figure	NOUN
ejpam-577	294	2	7	7	NUM
ejpam-577	294	3	theorem	theorem	NOUN
ejpam-577	294	4	23	23	NUM
ejpam-577	294	5	.	.	PUNCT
ejpam-577	295	1	the	the	DET
ejpam-577	295	2	graph	graph	NOUN
ejpam-577	295	3	mcn	mcn	PROPN
ejpam-577	295	4	is	be	AUX
ejpam-577	295	5	m⌊	m⌊	PROPN
ejpam-577	295	6	n	n	ADV
ejpam-577	295	7	2	2	NUM
ejpam-577	295	8	⌋-strongly	⌋-strongly	ADV
ejpam-577	295	9	indexable	indexable	ADJ
ejpam-577	295	10	for	for	ADP
ejpam-577	295	11	all	all	DET
ejpam-577	295	12	m	m	PROPN
ejpam-577	295	13	≥	≥	NOUN
ejpam-577	295	14	1	1	NUM
ejpam-577	295	15	and	and	CCONJ
ejpam-577	295	16	n≥	n≥	ADJ
ejpam-577	295	17	3	3	NUM
ejpam-577	295	18	.	.	PUNCT
ejpam-577	296	1	proof	proof	NOUN
ejpam-577	296	2	.	.	PUNCT
ejpam-577	297	1	let	let	VERB
ejpam-577	297	2	both	both	DET
ejpam-577	297	3	m	m	PROPN
ejpam-577	297	4	and	and	CCONJ
ejpam-577	297	5	n	n	ADV
ejpam-577	297	6	be	be	AUX
ejpam-577	297	7	odd	odd	ADJ
ejpam-577	297	8	integers	integer	NOUN
ejpam-577	297	9	.	.	PUNCT
ejpam-577	298	1	when	when	SCONJ
ejpam-577	298	2	m	m	VERB
ejpam-577	298	3	=	=	VERB
ejpam-577	298	4	1	1	NUM
ejpam-577	298	5	the	the	DET
ejpam-577	298	6	graph	graph	NOUN
ejpam-577	298	7	is	be	AUX
ejpam-577	298	8	an	an	DET
ejpam-577	298	9	odd	odd	ADJ
ejpam-577	298	10	cycle	cycle	NOUN
ejpam-577	298	11	which	which	PRON
ejpam-577	298	12	is	be	AUX
ejpam-577	298	13	k	k	NOUN
ejpam-577	298	14	-	-	PUNCT
ejpam-577	298	15	strongly	strongly	ADV
ejpam-577	298	16	indexable	indexable	ADJ
ejpam-577	298	17	.	.	PUNCT
ejpam-577	299	1	let	let	VERB
ejpam-577	299	2	v	v	X
ejpam-577	299	3	(	(	PUNCT
ejpam-577	299	4	cn	cn	PROPN
ejpam-577	299	5	)	)	PUNCT
ejpam-577	299	6	=	=	SYM
ejpam-577	299	7	{	{	PUNCT
ejpam-577	299	8	0,1,2	0,1,2	NOUN
ejpam-577	299	9	,	,	PUNCT
ejpam-577	299	10	.	.	PUNCT
ejpam-577	299	11	.	.	PUNCT
ejpam-577	300	1	.	.	PUNCT
ejpam-577	301	1	,	,	PUNCT
ejpam-577	301	2	n−	n−	NOUN
ejpam-577	301	3	1	1	NUM
ejpam-577	301	4	}	}	PUNCT
ejpam-577	301	5	define	define	VERB
ejpam-577	301	6	the	the	DET
ejpam-577	301	7	strong	strong	ADJ
ejpam-577	301	8	indexer	indexer	NOUN
ejpam-577	301	9	of	of	ADP
ejpam-577	301	10	cn	cn	PROPN
ejpam-577	301	11	as	as	SCONJ
ejpam-577	301	12	follows	follow	VERB
ejpam-577	301	13	:	:	PUNCT
ejpam-577	301	14	k.	k.	PROPN
ejpam-577	301	15	a.	a.	PROPN
ejpam-577	301	16	germina	germina	PROPN
ejpam-577	301	17	/	/	SYM
ejpam-577	301	18	eur	eur	PROPN
ejpam-577	301	19	.	.	PUNCT
ejpam-577	302	1	j.	j.	PROPN
ejpam-577	302	2	pure	pure	PROPN
ejpam-577	302	3	appl	appl	PROPN
ejpam-577	302	4	.	.	PROPN
ejpam-577	302	5	math	math	PROPN
ejpam-577	302	6	,	,	PUNCT
ejpam-577	302	7	3	3	NUM
ejpam-577	302	8	(	(	PUNCT
ejpam-577	302	9	2010	2010	NUM
ejpam-577	302	10	)	)	PUNCT
ejpam-577	302	11	,	,	PUNCT
ejpam-577	302	12	269	269	NUM
ejpam-577	302	13	-	-	SYM
ejpam-577	302	14	281	281	NUM
ejpam-577	302	15	279	279	NUM
ejpam-577	302	16	f	f	NOUN
ejpam-577	302	17	(	(	PUNCT
ejpam-577	302	18	vi	vi	NOUN
ejpam-577	302	19	)	)	PUNCT
ejpam-577	302	20	=	=	PUNCT
ejpam-577	303	1	¨	¨	NOUN
ejpam-577	303	2	i−2	i−2	VERB
ejpam-577	303	3	2	2	NUM
ejpam-577	303	4	if	if	SCONJ
ejpam-577	303	5	1≡	1≡	NUM
ejpam-577	303	6	0(mod2	0(mod2	NUM
ejpam-577	303	7	)	)	PUNCT
ejpam-577	303	8	n+i−2	n+i−2	NOUN
ejpam-577	303	9	2	2	NUM
ejpam-577	303	10	if	if	SCONJ
ejpam-577	303	11	i	i	PRON
ejpam-577	303	12	≡	≡	PROPN
ejpam-577	303	13	1(mod2	1(mod2	NUM
ejpam-577	303	14	)	)	PUNCT
ejpam-577	303	15	then	then	ADV
ejpam-577	303	16	f	f	PROPN
ejpam-577	303	17	is	be	AUX
ejpam-577	303	18	a	a	DET
ejpam-577	303	19	⌊	⌊	PROPN
ejpam-577	303	20	n	n	ADV
ejpam-577	303	21	2	2	NUM
ejpam-577	303	22	⌋-strong	⌋-strong	NOUN
ejpam-577	303	23	indexer	indexer	NOUN
ejpam-577	303	24	of	of	ADP
ejpam-577	303	25	cn	cn	PROPN
ejpam-577	303	26	.	.	PUNCT
ejpam-577	304	1	now	now	ADV
ejpam-577	304	2	,	,	PUNCT
ejpam-577	304	3	let	let	VERB
ejpam-577	304	4	m≥	m≥	NOUN
ejpam-577	304	5	3	3	X
ejpam-577	304	6	.	.	PUNCT
ejpam-577	305	1	let	let	VERB
ejpam-577	305	2	v	v	X
ejpam-577	305	3	(	(	PUNCT
ejpam-577	305	4	mcn	mcn	PROPN
ejpam-577	305	5	)	)	PUNCT
ejpam-577	305	6	=	=	PRON
ejpam-577	305	7	{	{	PUNCT
ejpam-577	305	8	ui	ui	PROPN
ejpam-577	305	9	,	,	PUNCT
ejpam-577	305	10	j	j	PROPN
ejpam-577	305	11	:	:	PUNCT
ejpam-577	305	12	1≤	1≤	NUM
ejpam-577	306	1	i	i	X
ejpam-577	306	2	≤	≤	PUNCT
ejpam-577	306	3	m	m	ADP
ejpam-577	306	4	,	,	PUNCT
ejpam-577	306	5	1≤	1≤	PROPN
ejpam-577	306	6	j	j	PROPN
ejpam-577	306	7	≤	≤	PROPN
ejpam-577	306	8	n	n	CCONJ
ejpam-577	306	9	}	}	PUNCT
ejpam-577	306	10	and	and	CCONJ
ejpam-577	306	11	e(mcn	e(mcn	PROPN
ejpam-577	306	12	)	)	PUNCT
ejpam-577	306	13	=	=	PRON
ejpam-577	306	14	{	{	PUNCT
ejpam-577	306	15	ui	ui	PROPN
ejpam-577	306	16	,	,	PUNCT
ejpam-577	306	17	jui	jui	NOUN
ejpam-577	306	18	,	,	PUNCT
ejpam-577	306	19	j+1	j+1	PROPN
ejpam-577	306	20	:	:	PUNCT
ejpam-577	306	21	1≤	1≤	NUM
ejpam-577	306	22	i	i	X
ejpam-577	306	23	≤	≤	PUNCT
ejpam-577	306	24	m	m	ADP
ejpam-577	306	25	,	,	PUNCT
ejpam-577	306	26	1≤	1≤	PROPN
ejpam-577	306	27	j	j	PROPN
ejpam-577	306	28	≤	≤	PROPN
ejpam-577	306	29	n−	n−	PROPN
ejpam-577	306	30	1	1	NUM
ejpam-577	306	31	}	}	PUNCT
ejpam-577	306	32	∪	∪	ADJ
ejpam-577	306	33	{	{	PUNCT
ejpam-577	306	34	ui	ui	PROPN
ejpam-577	306	35	,	,	PUNCT
ejpam-577	306	36	nui,1	nui,1	NOUN
ejpam-577	306	37	:	:	PUNCT
ejpam-577	306	38	1≤	1≤	NUM
ejpam-577	307	1	i	i	X
ejpam-577	307	2	≤	≤	PUNCT
ejpam-577	307	3	m	m	VERB
ejpam-577	307	4	}	}	PUNCT
ejpam-577	307	5	.	.	PUNCT
ejpam-577	308	1	define	define	VERB
ejpam-577	308	2	f	f	X
ejpam-577	308	3	:	:	PUNCT
ejpam-577	308	4	v	v	NOUN
ejpam-577	308	5	(	(	PUNCT
ejpam-577	308	6	mcn)→	mcn)→	NOUN
ejpam-577	308	7	{	{	PUNCT
ejpam-577	308	8	0,1,2	0,1,2	NOUN
ejpam-577	308	9	,	,	PUNCT
ejpam-577	308	10	.	.	PUNCT
ejpam-577	308	11	.	.	PUNCT
ejpam-577	308	12	.	.	PUNCT
ejpam-577	309	1	,	,	PUNCT
ejpam-577	309	2	mn	mn	PROPN
ejpam-577	309	3	}	}	PUNCT
ejpam-577	309	4	defined	define	VERB
ejpam-577	309	5	by	by	ADP
ejpam-577	309	6	f	f	PROPN
ejpam-577	309	7	(	(	PUNCT
ejpam-577	309	8	ui,1	ui,1	PROPN
ejpam-577	309	9	)	)	PUNCT
ejpam-577	309	10	=	=	PUNCT
ejpam-577	310	1	i−	i−	PROPN
ejpam-577	310	2	1	1	NUM
ejpam-577	310	3	,	,	PUNCT
ejpam-577	310	4	1≤	1≤	NUM
ejpam-577	310	5	i	i	PRON
ejpam-577	310	6	≤	≤	PUNCT
ejpam-577	310	7	m	m	VERB
ejpam-577	310	8	and	and	CCONJ
ejpam-577	310	9	f	f	PROPN
ejpam-577	310	10	(	(	PUNCT
ejpam-577	310	11	vi	vi	PROPN
ejpam-577	310	12	,	,	PUNCT
ejpam-577	310	13	j	j	NOUN
ejpam-577	310	14	)	)	PUNCT
ejpam-577	310	15	=	=	PUNCT
ejpam-577	310	16			PROPN
ejpam-577	310	17			VERB
ejpam-577	310	18			PROPN
ejpam-577	310	19			NOUN
ejpam-577	310	20			PROPN
ejpam-577	310	21			PROPN
ejpam-577	310	22			NOUN
ejpam-577	310	23	m(⌈	m(⌈	NOUN
ejpam-577	310	24	n	n	PRON
ejpam-577	310	25	2	2	NUM
ejpam-577	310	26	⌉+	⌉+	NOUN
ejpam-577	310	27	j−2	j−2	PROPN
ejpam-577	310	28	2	2	NUM
ejpam-577	310	29	)	)	PUNCT
ejpam-577	310	30	+	+	CCONJ
ejpam-577	311	1	2i+m−1	2i+m−1	NUM
ejpam-577	311	2	2	2	NUM
ejpam-577	311	3	if	if	SCONJ
ejpam-577	311	4	1≤	1≤	NUM
ejpam-577	312	1	i	i	NOUN
ejpam-577	312	2	≤	≤	PUNCT
ejpam-577	313	1	m−1	m−1	PROPN
ejpam-577	313	2	2	2	NUM
ejpam-577	313	3	and	and	CCONJ
ejpam-577	313	4	j	j	PROPN
ejpam-577	313	5	even	even	ADV
ejpam-577	313	6	m(⌈	m(⌈	PROPN
ejpam-577	313	7	n	n	CCONJ
ejpam-577	313	8	2	2	NUM
ejpam-577	313	9	⌉+	⌉+	NOUN
ejpam-577	313	10	j−2	j−2	PROPN
ejpam-577	313	11	2	2	NUM
ejpam-577	313	12	)	)	PUNCT
ejpam-577	313	13	+	+	CCONJ
ejpam-577	313	14	2i−m−1	2i−m−1	NUM
ejpam-577	313	15	2	2	NUM
ejpam-577	313	16	if	if	SCONJ
ejpam-577	313	17	m+1	m+1	PRON
ejpam-577	313	18	2	2	NUM
ejpam-577	313	19	≤	≤	NUM
ejpam-577	313	20	i	i	PRON
ejpam-577	313	21	≤	≤	NOUN
ejpam-577	313	22	m	m	VERB
ejpam-577	313	23	and	and	CCONJ
ejpam-577	313	24	j	j	PROPN
ejpam-577	313	25	even	even	ADV
ejpam-577	313	26	m	m	PROPN
ejpam-577	313	27	(	(	PUNCT
ejpam-577	313	28	j−i	j−i	PROPN
ejpam-577	313	29	2	2	NUM
ejpam-577	313	30	+	+	PROPN
ejpam-577	313	31	1)−	1)−	NUM
ejpam-577	313	32	2i	2i	NUM
ejpam-577	313	33	if	if	SCONJ
ejpam-577	313	34	1≤	1≤	NUM
ejpam-577	313	35	i	i	NOUN
ejpam-577	313	36	≤	≤	PUNCT
ejpam-577	313	37	m−1	m−1	PROPN
ejpam-577	313	38	2	2	NUM
ejpam-577	313	39	and	and	CCONJ
ejpam-577	313	40	j	j	PROPN
ejpam-577	313	41	6=	6=	NUM
ejpam-577	313	42	1	1	NUM
ejpam-577	313	43	is	be	AUX
ejpam-577	313	44	odd	odd	ADJ
ejpam-577	313	45	m	m	PROPN
ejpam-577	313	46	(	(	PUNCT
ejpam-577	313	47	j−i	j−i	PROPN
ejpam-577	313	48	2	2	NUM
ejpam-577	314	1	+	+	SYM
ejpam-577	314	2	2)−	2)−	NUM
ejpam-577	314	3	2i	2i	NOUN
ejpam-577	314	4	if	if	SCONJ
ejpam-577	314	5	m+1	m+1	PRON
ejpam-577	314	6	2	2	NUM
ejpam-577	314	7	≤	≤	NUM
ejpam-577	314	8	i	i	PRON
ejpam-577	314	9	≤	≤	NOUN
ejpam-577	314	10	m	m	PROPN
ejpam-577	314	11	and	and	CCONJ
ejpam-577	314	12	j	j	PROPN
ejpam-577	314	13	6=	6=	NUM
ejpam-577	314	14	1	1	NUM
ejpam-577	314	15	is	be	AUX
ejpam-577	314	16	odd	odd	ADJ
ejpam-577	314	17	it	it	PRON
ejpam-577	314	18	is	be	AUX
ejpam-577	314	19	not	not	PART
ejpam-577	314	20	difficult	difficult	ADJ
ejpam-577	314	21	to	to	PART
ejpam-577	314	22	check	check	VERB
ejpam-577	314	23	that	that	SCONJ
ejpam-577	314	24	f	f	PROPN
ejpam-577	314	25	is	be	AUX
ejpam-577	314	26	a	a	DET
ejpam-577	314	27	strong	strong	ADJ
ejpam-577	314	28	indexer	indexer	NOUN
ejpam-577	314	29	of	of	ADP
ejpam-577	314	30	mcn	mcn	PROPN
ejpam-577	314	31	figure	figure	NOUN
ejpam-577	314	32	8	8	NUM
ejpam-577	314	33	is	be	AUX
ejpam-577	314	34	strongly	strongly	ADV
ejpam-577	314	35	indexable	indexable	ADJ
ejpam-577	314	36	labelling	labelling	NOUN
ejpam-577	314	37	of	of	ADP
ejpam-577	314	38	7c5	7c5	NUM
ejpam-577	314	39	figure	figure	NOUN
ejpam-577	314	40	8	8	NUM
ejpam-577	314	41	remark	remark	NOUN
ejpam-577	314	42	4	4	NUM
ejpam-577	314	43	.	.	PUNCT
ejpam-577	314	44	invoking	invoke	VERB
ejpam-577	314	45	theorem	theorem	NOUN
ejpam-577	314	46	3	3	NUM
ejpam-577	314	47	due	due	ADP
ejpam-577	314	48	to	to	ADP
ejpam-577	314	49	acharya	acharya	NOUN
ejpam-577	314	50	[	[	X
ejpam-577	314	51	1	1	X
ejpam-577	314	52	]	]	PUNCT
ejpam-577	314	53	which	which	PRON
ejpam-577	314	54	state	state	NOUN
ejpam-577	314	55	that	that	SCONJ
ejpam-577	314	56	“	"	PUNCT
ejpam-577	314	57	if	if	SCONJ
ejpam-577	314	58	g	g	PROPN
ejpam-577	314	59	is	be	AUX
ejpam-577	314	60	r	r	NOUN
ejpam-577	314	61	-	-	ADJ
ejpam-577	314	62	regular	regular	ADJ
ejpam-577	314	63	k	k	NOUN
ejpam-577	314	64	-	-	PUNCT
ejpam-577	314	65	strongly	strongly	ADV
ejpam-577	314	66	indexable	indexable	ADJ
ejpam-577	314	67	(	(	PUNCT
ejpam-577	314	68	p	p	NOUN
ejpam-577	314	69	,	,	PUNCT
ejpam-577	314	70	q)-graph	q)-graph	NOUN
ejpam-577	314	71	(	(	PUNCT
ejpam-577	314	72	r	r	NOUN
ejpam-577	314	73	≥	≥	NOUN
ejpam-577	314	74	1	1	NUM
ejpam-577	314	75	)	)	PUNCT
ejpam-577	314	76	,	,	PUNCT
ejpam-577	314	77	then	then	ADV
ejpam-577	314	78	q	q	X
ejpam-577	314	79	is	be	AUX
ejpam-577	314	80	odd	odd	ADJ
ejpam-577	314	81	”	"	PUNCT
ejpam-577	314	82	we	we	PRON
ejpam-577	314	83	see	see	VERB
ejpam-577	314	84	that	that	SCONJ
ejpam-577	314	85	the	the	DET
ejpam-577	314	86	converse	converse	NOUN
ejpam-577	314	87	of	of	ADP
ejpam-577	314	88	theorem	theorem	NOUN
ejpam-577	314	89	23	23	NUM
ejpam-577	314	90	also	also	ADV
ejpam-577	314	91	holds	hold	VERB
ejpam-577	314	92	good	good	ADJ
ejpam-577	314	93	.	.	PUNCT
ejpam-577	315	1	from	from	ADP
ejpam-577	315	2	theorem	theorem	ADJ
ejpam-577	315	3	23	23	NUM
ejpam-577	315	4	and	and	CCONJ
ejpam-577	315	5	remark	remark	VERB
ejpam-577	315	6	4	4	NUM
ejpam-577	315	7	we	we	PRON
ejpam-577	315	8	have	have	VERB
ejpam-577	315	9	the	the	DET
ejpam-577	315	10	following	follow	VERB
ejpam-577	315	11	theorems	theorem	NOUN
ejpam-577	315	12	theorem	theorem	VERB
ejpam-577	315	13	24	24	NUM
ejpam-577	315	14	.	.	PUNCT
ejpam-577	316	1	the	the	DET
ejpam-577	316	2	2	2	NUM
ejpam-577	316	3	-	-	PUNCT
ejpam-577	316	4	regular	regular	ADJ
ejpam-577	316	5	graph	graph	NOUN
ejpam-577	316	6	mcn	mcn	PROPN
ejpam-577	316	7	is	be	AUX
ejpam-577	316	8	k	k	ADJ
ejpam-577	316	9	-	-	ADJ
ejpam-577	316	10	strongly	strongly	ADV
ejpam-577	316	11	indexable	indexable	ADJ
ejpam-577	316	12	if	if	SCONJ
ejpam-577	316	13	and	and	CCONJ
ejpam-577	316	14	only	only	ADV
ejpam-577	316	15	if	if	SCONJ
ejpam-577	316	16	m	m	PROPN
ejpam-577	316	17	≥	≥	VERB
ejpam-577	316	18	1	1	NUM
ejpam-577	316	19	and	and	CCONJ
ejpam-577	316	20	n	n	PRON
ejpam-577	316	21	≥	≥	NOUN
ejpam-577	316	22	3	3	NUM
ejpam-577	316	23	are	be	AUX
ejpam-577	316	24	odd	odd	ADJ
ejpam-577	316	25	theorem	theorem	ADJ
ejpam-577	316	26	25	25	NUM
ejpam-577	316	27	.	.	PUNCT
ejpam-577	317	1	any	any	DET
ejpam-577	317	2	3	3	NUM
ejpam-577	317	3	-	-	PUNCT
ejpam-577	317	4	regular	regular	ADJ
ejpam-577	317	5	graph	graph	NOUN
ejpam-577	317	6	g	g	NOUN
ejpam-577	317	7	=	=	PUNCT
ejpam-577	317	8	(	(	PUNCT
ejpam-577	317	9	p	p	X
ejpam-577	317	10	,	,	PUNCT
ejpam-577	317	11	q	q	NOUN
ejpam-577	317	12	)	)	PUNCT
ejpam-577	317	13	is	be	AUX
ejpam-577	317	14	k	k	NOUN
ejpam-577	317	15	-	-	ADJ
ejpam-577	317	16	strongly	strongly	ADV
ejpam-577	317	17	indexable	indexable	ADJ
ejpam-577	317	18	then	then	ADV
ejpam-577	317	19	p	p	PROPN
ejpam-577	317	20	≡	≡	PROPN
ejpam-577	317	21	2(mod4	2(mod4	NUM
ejpam-577	317	22	)	)	PUNCT
ejpam-577	317	23	proof	proof	NOUN
ejpam-577	317	24	.	.	PUNCT
ejpam-577	318	1	assume	assume	VERB
ejpam-577	318	2	g	g	PROPN
ejpam-577	318	3	=	=	PUNCT
ejpam-577	318	4	(	(	PUNCT
ejpam-577	318	5	p	p	X
ejpam-577	318	6	,	,	PUNCT
ejpam-577	318	7	q	q	NOUN
ejpam-577	318	8	)	)	PUNCT
ejpam-577	318	9	be	be	AUX
ejpam-577	318	10	3	3	NUM
ejpam-577	318	11	-	-	PUNCT
ejpam-577	318	12	regular	regular	ADJ
ejpam-577	318	13	k	k	NOUN
ejpam-577	318	14	-	-	PUNCT
ejpam-577	318	15	strongly	strongly	ADV
ejpam-577	318	16	indexable	indexable	ADJ
ejpam-577	318	17	.	.	PUNCT
ejpam-577	319	1	since	since	SCONJ
ejpam-577	319	2	g	g	PROPN
ejpam-577	319	3	is	be	AUX
ejpam-577	319	4	3	3	NUM
ejpam-577	319	5	-	-	PUNCT
ejpam-577	319	6	regular	regular	ADJ
ejpam-577	319	7	,	,	PUNCT
ejpam-577	319	8	p	p	NOUN
ejpam-577	319	9	should	should	AUX
ejpam-577	319	10	necessarily	necessarily	ADV
ejpam-577	319	11	be	be	AUX
ejpam-577	319	12	even	even	ADV
ejpam-577	319	13	so	so	SCONJ
ejpam-577	319	14	that	that	SCONJ
ejpam-577	319	15	either	either	CCONJ
ejpam-577	319	16	p	p	PROPN
ejpam-577	319	17	≡	≡	PROPN
ejpam-577	319	18	0(mod4	0(mod4	NUM
ejpam-577	319	19	)	)	PUNCT
ejpam-577	319	20	or	or	CCONJ
ejpam-577	319	21	p	p	PROPN
ejpam-577	319	22	≡	≡	PROPN
ejpam-577	319	23	2(mod4	2(mod4	NUM
ejpam-577	319	24	)	)	PUNCT
ejpam-577	319	25	.	.	PUNCT
ejpam-577	320	1	references	reference	NOUN
ejpam-577	320	2	280	280	NUM
ejpam-577	320	3	when	when	SCONJ
ejpam-577	320	4	p	p	PROPN
ejpam-577	320	5	≡	≡	PROPN
ejpam-577	320	6	0(mod4	0(mod4	NUM
ejpam-577	320	7	)	)	PUNCT
ejpam-577	320	8	,	,	PUNCT
ejpam-577	320	9	p	p	X
ejpam-577	320	10	=	=	NOUN
ejpam-577	320	11	4	4	NUM
ejpam-577	320	12	t	t	NOUN
ejpam-577	320	13	say	say	VERB
ejpam-577	320	14	,	,	PUNCT
ejpam-577	320	15	for	for	ADP
ejpam-577	320	16	some	some	DET
ejpam-577	320	17	positive	positive	ADJ
ejpam-577	320	18	integer	integer	NOUN
ejpam-577	320	19	t	t	NOUN
ejpam-577	320	20	so	so	SCONJ
ejpam-577	320	21	that	that	SCONJ
ejpam-577	320	22	q	q	NOUN
ejpam-577	321	1	=	=	NUM
ejpam-577	321	2	12	12	NUM
ejpam-577	321	3	t	t	NOUN
ejpam-577	321	4	2	2	NUM
ejpam-577	321	5	=	=	SYM
ejpam-577	321	6	6	6	NUM
ejpam-577	321	7	t	t	NOUN
ejpam-577	321	8	if	if	SCONJ
ejpam-577	321	9	f	f	PROPN
ejpam-577	321	10	is	be	AUX
ejpam-577	321	11	the	the	DET
ejpam-577	321	12	strong	strong	ADJ
ejpam-577	321	13	indexer	indexer	NOUN
ejpam-577	321	14	of	of	ADP
ejpam-577	321	15	g	g	NOUN
ejpam-577	321	16	,	,	PUNCT
ejpam-577	321	17	then	then	ADV
ejpam-577	321	18	by	by	ADP
ejpam-577	321	19	theorem	theorem	NOUN
ejpam-577	321	20	3	3	NUM
ejpam-577	321	21	[	[	X
ejpam-577	321	22	1	1	NUM
ejpam-577	321	23	]	]	PUNCT
ejpam-577	321	24	σ	σ	PROPN
ejpam-577	321	25	p−1	p−1	PROPN
ejpam-577	321	26	i=0	i=0	PROPN
ejpam-577	321	27	id(ui	id(ui	PROPN
ejpam-577	321	28	)	)	PUNCT
ejpam-577	322	1	=	=	PUNCT
ejpam-577	322	2	k+	k+	X
ejpam-577	322	3	k+	k+	NOUN
ejpam-577	322	4	1	1	NUM
ejpam-577	322	5	+	+	NUM
ejpam-577	322	6	·	·	PUNCT
ejpam-577	322	7	·	·	PUNCT
ejpam-577	322	8	·	·	PUNCT
ejpam-577	323	1	+	+	NUM
ejpam-577	323	2	k+	k+	NOUN
ejpam-577	323	3	q−	q−	PROPN
ejpam-577	323	4	1	1	NUM
ejpam-577	323	5	hence	hence	ADV
ejpam-577	323	6	,	,	PUNCT
ejpam-577	323	7	3p(p−1	3p(p−1	NUM
ejpam-577	323	8	)	)	PUNCT
ejpam-577	323	9	2	2	NUM
ejpam-577	323	10	=	=	SYM
ejpam-577	323	11	kq+	kq+	X
ejpam-577	323	12	q(q−1	q(q−1	PROPN
ejpam-577	323	13	)	)	PUNCT
ejpam-577	323	14	2	2	NUM
ejpam-577	323	15	,	,	PUNCT
ejpam-577	323	16	applying	apply	VERB
ejpam-577	323	17	p	p	NOUN
ejpam-577	323	18	=	=	NOUN
ejpam-577	323	19	4	4	NUM
ejpam-577	323	20	t	t	NOUN
ejpam-577	323	21	,	,	PUNCT
ejpam-577	323	22	q	q	NOUN
ejpam-577	323	23	=	=	PUNCT
ejpam-577	323	24	6	6	NUM
ejpam-577	323	25	t	t	NOUN
ejpam-577	323	26	⇒	⇒	NOUN
ejpam-577	323	27	21t(4	21t(4	PROPN
ejpam-577	323	28	t	t	NOUN
ejpam-577	323	29	−	−	NOUN
ejpam-577	323	30	1	1	NUM
ejpam-577	323	31	)	)	PUNCT
ejpam-577	323	32	=	=	SYM
ejpam-577	324	1	12tk+	12tk+	NUM
ejpam-577	324	2	6t(6	6t(6	NUM
ejpam-577	324	3	t	t	NOUN
ejpam-577	324	4	−	−	PROPN
ejpam-577	324	5	1)⇒	1)⇒	NUM
ejpam-577	324	6	t	t	NOUN
ejpam-577	324	7	=	=	SYM
ejpam-577	324	8	2k+1	2k+1	NOUN
ejpam-577	324	9	2	2	NUM
ejpam-577	324	10	,	,	PUNCT
ejpam-577	324	11	a	a	DET
ejpam-577	324	12	contradiction	contradiction	NOUN
ejpam-577	324	13	.	.	PUNCT
ejpam-577	325	1	hence	hence	ADV
ejpam-577	325	2	,	,	PUNCT
ejpam-577	325	3	p	p	PROPN
ejpam-577	325	4	≡	≡	PROPN
ejpam-577	325	5	2(mod4	2(mod4	NUM
ejpam-577	325	6	)	)	PUNCT
ejpam-577	325	7	.	.	PUNCT
ejpam-577	326	1	3	3	X
ejpam-577	326	2	.	.	X
ejpam-577	326	3	conclusion	conclusion	NOUN
ejpam-577	326	4	and	and	CCONJ
ejpam-577	326	5	scope	scope	NOUN
ejpam-577	326	6	graph	graph	NOUN
ejpam-577	326	7	labelings	labeling	NOUN
ejpam-577	326	8	,	,	PUNCT
ejpam-577	326	9	where	where	SCONJ
ejpam-577	326	10	the	the	DET
ejpam-577	326	11	vertices	vertex	NOUN
ejpam-577	326	12	and	and	CCONJ
ejpam-577	326	13	edges	edge	NOUN
ejpam-577	326	14	are	be	AUX
ejpam-577	326	15	assigned	assign	VERB
ejpam-577	326	16	,	,	PUNCT
ejpam-577	326	17	real	real	ADJ
ejpam-577	326	18	values	value	NOUN
ejpam-577	326	19	subject	subject	ADJ
ejpam-577	326	20	to	to	ADP
ejpam-577	326	21	certain	certain	ADJ
ejpam-577	326	22	conditions	condition	NOUN
ejpam-577	326	23	,	,	PUNCT
ejpam-577	326	24	have	have	AUX
ejpam-577	326	25	often	often	ADV
ejpam-577	326	26	been	be	AUX
ejpam-577	326	27	motivated	motivate	VERB
ejpam-577	326	28	by	by	ADP
ejpam-577	326	29	their	their	PRON
ejpam-577	326	30	utility	utility	NOUN
ejpam-577	326	31	to	to	ADP
ejpam-577	326	32	various	various	ADJ
ejpam-577	326	33	applied	apply	VERB
ejpam-577	326	34	fields	field	NOUN
ejpam-577	326	35	and	and	CCONJ
ejpam-577	326	36	their	their	PRON
ejpam-577	326	37	intrinsic	intrinsic	ADJ
ejpam-577	326	38	mathematical	mathematical	ADJ
ejpam-577	326	39	interest	interest	NOUN
ejpam-577	326	40	(	(	PUNCT
ejpam-577	326	41	logico	logico	PROPN
ejpam-577	326	42	mathematical	mathematical	PROPN
ejpam-577	326	43	)	)	PUNCT
ejpam-577	326	44	.	.	PUNCT
ejpam-577	327	1	graph	graph	NOUN
ejpam-577	327	2	labelings	labeling	NOUN
ejpam-577	327	3	are	be	AUX
ejpam-577	327	4	applied	apply	VERB
ejpam-577	327	5	in	in	ADP
ejpam-577	327	6	determination	determination	NOUN
ejpam-577	327	7	of	of	ADP
ejpam-577	327	8	crystal	crystal	NOUN
ejpam-577	327	9	structure	structure	NOUN
ejpam-577	327	10	from	from	ADP
ejpam-577	327	11	x	x	ADJ
ejpam-577	327	12	-	-	NOUN
ejpam-577	327	13	ray	ray	NOUN
ejpam-577	327	14	diffraction	diffraction	NOUN
ejpam-577	327	15	data	datum	NOUN
ejpam-577	327	16	[	[	X
ejpam-577	327	17	6	6	NUM
ejpam-577	327	18	,	,	PUNCT
ejpam-577	327	19	10	10	NUM
ejpam-577	327	20	,	,	PUNCT
ejpam-577	327	21	14	14	NUM
ejpam-577	327	22	,	,	PUNCT
ejpam-577	327	23	15	15	NUM
ejpam-577	327	24	,	,	PUNCT
ejpam-577	327	25	16	16	NUM
ejpam-577	327	26	]	]	PUNCT
ejpam-577	327	27	,	,	PUNCT
ejpam-577	327	28	the	the	DET
ejpam-577	327	29	design	design	NOUN
ejpam-577	327	30	of	of	ADP
ejpam-577	327	31	certain	certain	ADJ
ejpam-577	327	32	important	important	ADJ
ejpam-577	327	33	classes	class	NOUN
ejpam-577	327	34	of	of	ADP
ejpam-577	327	35	good	good	ADJ
ejpam-577	327	36	non	non	ADJ
ejpam-577	327	37	periodic	periodic	ADJ
ejpam-577	327	38	codes	code	NOUN
ejpam-577	327	39	for	for	ADP
ejpam-577	327	40	pulse	pulse	NOUN
ejpam-577	327	41	radar	radar	NOUN
ejpam-577	327	42	and	and	CCONJ
ejpam-577	327	43	missile	missile	NOUN
ejpam-577	327	44	guidance	guidance	NOUN
ejpam-577	327	45	[	[	X
ejpam-577	327	46	7	7	NUM
ejpam-577	327	47	]	]	PUNCT
ejpam-577	327	48	,	,	PUNCT
ejpam-577	327	49	and	and	CCONJ
ejpam-577	327	50	in	in	ADP
ejpam-577	327	51	the	the	DET
ejpam-577	327	52	problem	problem	NOUN
ejpam-577	327	53	in	in	ADP
ejpam-577	327	54	radio	radio	NOUN
ejpam-577	327	55	-	-	PUNCT
ejpam-577	327	56	astronomy	astronomy	NOUN
ejpam-577	327	57	that	that	SCONJ
ejpam-577	327	58	a	a	DET
ejpam-577	327	59	few	few	ADJ
ejpam-577	327	60	movable	movable	ADJ
ejpam-577	327	61	antennae	antennae	NOUN
ejpam-577	327	62	are	be	AUX
ejpam-577	327	63	required	require	VERB
ejpam-577	327	64	to	to	PART
ejpam-577	327	65	be	be	AUX
ejpam-577	327	66	located	locate	VERB
ejpam-577	327	67	in	in	ADP
ejpam-577	327	68	several	several	ADJ
ejpam-577	327	69	successive	successive	ADJ
ejpam-577	327	70	array	array	NOUN
ejpam-577	327	71	configurations	configuration	NOUN
ejpam-577	327	72	to	to	PART
ejpam-577	327	73	receive	receive	VERB
ejpam-577	327	74	various	various	ADJ
ejpam-577	327	75	spatial	spatial	ADJ
ejpam-577	327	76	frequencies	frequency	NOUN
ejpam-577	327	77	relative	relative	ADJ
ejpam-577	327	78	to	to	ADP
ejpam-577	327	79	some	some	DET
ejpam-577	327	80	area	area	NOUN
ejpam-577	327	81	of	of	ADP
ejpam-577	327	82	the	the	DET
ejpam-577	327	83	sky	sky	NOUN
ejpam-577	328	1	[	[	X
ejpam-577	328	2	5	5	NUM
ejpam-577	328	3	]	]	PUNCT
ejpam-577	328	4	.	.	PUNCT
ejpam-577	329	1	harper	harper	NOUN
ejpam-577	329	2	formulated	formulate	VERB
ejpam-577	329	3	this	this	DET
ejpam-577	329	4	design	design	NOUN
ejpam-577	329	5	optimization	optimization	NOUN
ejpam-577	329	6	problem	problem	NOUN
ejpam-577	329	7	in	in	ADP
ejpam-577	329	8	graph	graph	NOUN
ejpam-577	329	9	labeling	labeling	NOUN
ejpam-577	329	10	terms	term	NOUN
ejpam-577	329	11	and	and	CCONJ
ejpam-577	329	12	solved	solve	VERB
ejpam-577	329	13	some	some	DET
ejpam-577	329	14	cases	case	NOUN
ejpam-577	329	15	using	use	VERB
ejpam-577	329	16	this	this	DET
ejpam-577	329	17	technic	technic	NOUN
ejpam-577	329	18	for	for	ADP
ejpam-577	329	19	minimum	minimum	ADJ
ejpam-577	329	20	-	-	PUNCT
ejpam-577	329	21	confusion	confusion	NOUN
ejpam-577	329	22	code	code	NOUN
ejpam-577	329	23	design	design	NOUN
ejpam-577	329	24	[	[	X
ejpam-577	329	25	12	12	NUM
ejpam-577	329	26	]	]	PUNCT
ejpam-577	329	27	.	.	PUNCT
ejpam-577	330	1	k	k	X
ejpam-577	330	2	-	-	PUNCT
ejpam-577	330	3	strongly	strongly	ADV
ejpam-577	330	4	indexable	indexable	ADJ
ejpam-577	330	5	graphs	graph	NOUN
ejpam-577	330	6	are	be	AUX
ejpam-577	330	7	used	use	VERB
ejpam-577	330	8	in	in	ADP
ejpam-577	330	9	the	the	DET
ejpam-577	330	10	construction	construction	NOUN
ejpam-577	330	11	of	of	ADP
ejpam-577	330	12	polygons	polygon	NOUN
ejpam-577	330	13	of	of	ADP
ejpam-577	330	14	same	same	ADJ
ejpam-577	330	15	internal	internal	ADJ
ejpam-577	330	16	angle	angle	NOUN
ejpam-577	330	17	and	and	CCONJ
ejpam-577	330	18	distinct	distinct	ADJ
ejpam-577	330	19	sides	side	NOUN
ejpam-577	330	20	:	:	PUNCT
ejpam-577	330	21	using	use	VERB
ejpam-577	330	22	strongly	strongly	ADV
ejpam-577	330	23	k	k	ADJ
ejpam-577	330	24	-	-	ADJ
ejpam-577	330	25	indexable	indexable	ADJ
ejpam-577	330	26	labelings	labeling	NOUN
ejpam-577	330	27	of	of	ADP
ejpam-577	330	28	a	a	DET
ejpam-577	330	29	cycle	cycle	NOUN
ejpam-577	330	30	c2n+1	c2n+1	PROPN
ejpam-577	330	31	,	,	PUNCT
ejpam-577	330	32	one	one	PRON
ejpam-577	330	33	can	can	AUX
ejpam-577	330	34	construct	construct	VERB
ejpam-577	330	35	a	a	DET
ejpam-577	330	36	polygon	polygon	NOUN
ejpam-577	330	37	p4n+2	p4n+2	NOUN
ejpam-577	330	38	with	with	ADP
ejpam-577	330	39	4n+2	4n+2	PROPN
ejpam-577	330	40	sides	side	NOUN
ejpam-577	330	41	such	such	ADJ
ejpam-577	330	42	that	that	SCONJ
ejpam-577	330	43	all	all	DET
ejpam-577	330	44	the	the	DET
ejpam-577	330	45	internal	internal	ADJ
ejpam-577	330	46	angles	angle	NOUN
ejpam-577	330	47	are	be	AUX
ejpam-577	330	48	equal	equal	ADJ
ejpam-577	330	49	and	and	CCONJ
ejpam-577	330	50	lengths	length	NOUN
ejpam-577	330	51	of	of	ADP
ejpam-577	330	52	the	the	DET
ejpam-577	330	53	sides	side	NOUN
ejpam-577	330	54	are	be	AUX
ejpam-577	330	55	distinct	distinct	ADJ
ejpam-577	331	1	[	[	PUNCT
ejpam-577	331	2	13	13	NUM
ejpam-577	331	3	]	]	PUNCT
ejpam-577	331	4	.	.	PUNCT
ejpam-577	332	1	acknowledgements	acknowledgement	NOUN
ejpam-577	332	2	the	the	DET
ejpam-577	332	3	author	author	NOUN
ejpam-577	332	4	is	be	AUX
ejpam-577	332	5	thankful	thankful	ADJ
ejpam-577	332	6	to	to	ADP
ejpam-577	332	7	the	the	DET
ejpam-577	332	8	department	department	PROPN
ejpam-577	332	9	of	of	ADP
ejpam-577	332	10	science	science	PROPN
ejpam-577	332	11	&	&	CCONJ
ejpam-577	332	12	technology	technology	NOUN
ejpam-577	332	13	,	,	PUNCT
ejpam-577	332	14	government	government	NOUN
ejpam-577	332	15	of	of	ADP
ejpam-577	332	16	india	india	PROPN
ejpam-577	332	17	for	for	ADP
ejpam-577	332	18	supporting	support	VERB
ejpam-577	332	19	this	this	DET
ejpam-577	332	20	research	research	NOUN
ejpam-577	332	21	under	under	ADP
ejpam-577	332	22	the	the	DET
ejpam-577	332	23	project	project	NOUN
ejpam-577	333	1	no	no	INTJ
ejpam-577	333	2	.	.	PUNCT
ejpam-577	334	1	sr	sr	PROPN
ejpam-577	334	2	/	/	SYM
ejpam-577	334	3	s4	s4	PROPN
ejpam-577	334	4	/	/	SYM
ejpam-577	334	5	ms:277/06	ms:277/06	PROPN
ejpam-577	334	6	.	.	PUNCT
ejpam-577	335	1	references	reference	NOUN
ejpam-577	335	2	[	[	X
ejpam-577	335	3	1	1	NUM
ejpam-577	335	4	]	]	X
ejpam-577	335	5	b.d	b.d	PROPN
ejpam-577	335	6	.	.	PROPN
ejpam-577	335	7	acharya	acharya	PROPN
ejpam-577	335	8	.	.	PUNCT
ejpam-577	336	1	on	on	ADP
ejpam-577	336	2	the	the	DET
ejpam-577	336	3	construction	construction	NOUN
ejpam-577	336	4	of	of	ADP
ejpam-577	336	5	graphs	graph	NOUN
ejpam-577	336	6	with	with	ADP
ejpam-577	336	7	given	give	VERB
ejpam-577	336	8	constant	constant	ADJ
ejpam-577	336	9	valencedifference(s	valencedifference(s	NOUN
ejpam-577	336	10	)	)	PUNCT
ejpam-577	336	11	on	on	ADP
ejpam-577	336	12	each	each	PRON
ejpam-577	336	13	of	of	ADP
ejpam-577	336	14	their	their	PRON
ejpam-577	336	15	lines	line	NOUN
ejpam-577	336	16	,	,	PUNCT
ejpam-577	336	17	wiss	wiss	PROPN
ejpam-577	336	18	.	.	PUNCT
ejpam-577	337	1	z.	z.	PROPN
ejpam-577	337	2	th	th	PROPN
ejpam-577	337	3	.	.	PROPN
ejpam-577	337	4	ilmenau	ilmenau	PROPN
ejpam-577	337	5	,	,	PUNCT
ejpam-577	337	6	23	23	NUM
ejpam-577	337	7	,	,	PUNCT
ejpam-577	337	8	33	33	NUM
ejpam-577	337	9	-	-	SYM
ejpam-577	337	10	60	60	NUM
ejpam-577	337	11	,	,	PUNCT
ejpam-577	337	12	1977	1977	NUM
ejpam-577	337	13	.	.	PUNCT
ejpam-577	338	1	[	[	X
ejpam-577	338	2	2	2	NUM
ejpam-577	338	3	]	]	X
ejpam-577	338	4	b.d	b.d	PROPN
ejpam-577	338	5	.	.	PROPN
ejpam-577	338	6	acharya	acharya	PROPN
ejpam-577	338	7	and	and	CCONJ
ejpam-577	338	8	s.m	s.m	PROPN
ejpam-577	338	9	.	.	PROPN
ejpam-577	338	10	hegde	hegde	PROPN
ejpam-577	338	11	.	.	PUNCT
ejpam-577	339	1	arithmetic	arithmetic	ADJ
ejpam-577	339	2	graphs	graph	NOUN
ejpam-577	339	3	,	,	PUNCT
ejpam-577	339	4	j.	j.	PROPN
ejpam-577	339	5	graph	graph	PROPN
ejpam-577	339	6	theory	theory	NOUN
ejpam-577	339	7	,	,	PUNCT
ejpam-577	339	8	14	14	NUM
ejpam-577	339	9	,	,	PUNCT
ejpam-577	339	10	275	275	NUM
ejpam-577	339	11	-	-	SYM
ejpam-577	339	12	299	299	NUM
ejpam-577	339	13	,	,	PUNCT
ejpam-577	339	14	1990	1990	NUM
ejpam-577	339	15	.	.	PUNCT
ejpam-577	340	1	[	[	X
ejpam-577	340	2	3	3	X
ejpam-577	340	3	]	]	X
ejpam-577	340	4	b.d	b.d	PROPN
ejpam-577	340	5	.	.	PROPN
ejpam-577	340	6	acharya	acharya	PROPN
ejpam-577	340	7	and	and	CCONJ
ejpam-577	340	8	germina	germina	PROPN
ejpam-577	340	9	k.a	k.a	PROPN
ejpam-577	340	10	.	.	PUNCT
ejpam-577	341	1	strongly	strongly	ADV
ejpam-577	341	2	k	k	ADJ
ejpam-577	341	3	-	-	ADJ
ejpam-577	341	4	indexable	indexable	ADJ
ejpam-577	341	5	unicyclic	unicyclic	ADJ
ejpam-577	341	6	graphs	graph	NOUN
ejpam-577	341	7	,	,	PUNCT
ejpam-577	341	8	graph	graph	NOUN
ejpam-577	341	9	theory	theory	NOUN
ejpam-577	341	10	notes	note	NOUN
ejpam-577	341	11	of	of	ADP
ejpam-577	341	12	new	new	ADJ
ejpam-577	341	13	york,45	york,45	NOUN
ejpam-577	341	14	-	-	SYM
ejpam-577	341	15	49	49	NUM
ejpam-577	341	16	,	,	PUNCT
ejpam-577	341	17	lv:2008	lv:2008	NOUN
ejpam-577	341	18	.	.	PUNCT
ejpam-577	342	1	[	[	X
ejpam-577	342	2	4	4	X
ejpam-577	342	3	]	]	X
ejpam-577	342	4	b.d	b.d	PROPN
ejpam-577	342	5	.	.	PROPN
ejpam-577	342	6	acharya	acharya	PROPN
ejpam-577	342	7	and	and	CCONJ
ejpam-577	342	8	germina	germina	PROPN
ejpam-577	342	9	k.a	k.a	PROPN
ejpam-577	342	10	.	.	PROPN
ejpam-577	342	11	maximal	maximal	ADJ
ejpam-577	342	12	strongly	strongly	ADV
ejpam-577	342	13	indexable	indexable	ADJ
ejpam-577	342	14	graphs	graph	NOUN
ejpam-577	342	15	,	,	PUNCT
ejpam-577	342	16	ars	ar	NOUN
ejpam-577	342	17	.	.	PUNCT
ejpam-577	343	1	combinatorics	combinatoric	NOUN
ejpam-577	343	2	,	,	PUNCT
ejpam-577	343	3	to	to	PART
ejpam-577	343	4	appear	appear	VERB
ejpam-577	343	5	.	.	PUNCT
ejpam-577	344	1	[	[	X
ejpam-577	344	2	5	5	NUM
ejpam-577	344	3	]	]	X
ejpam-577	344	4	j.c	j.c	PROPN
ejpam-577	344	5	.	.	PROPN
ejpam-577	344	6	bermond	bermond	PROPN
ejpam-577	344	7	,	,	PUNCT
ejpam-577	344	8	a.	a.	NOUN
ejpam-577	344	9	kotzig	kotzig	PROPN
ejpam-577	344	10	,	,	PUNCT
ejpam-577	344	11	and	and	CCONJ
ejpam-577	344	12	j.	j.	PROPN
ejpam-577	344	13	turgeon	turgeon	PROPN
ejpam-577	344	14	.	.	PUNCT
ejpam-577	345	1	on	on	ADP
ejpam-577	345	2	a	a	DET
ejpam-577	345	3	combinatorial	combinatorial	ADJ
ejpam-577	345	4	problem	problem	NOUN
ejpam-577	345	5	of	of	ADP
ejpam-577	345	6	antennas	antenna	NOUN
ejpam-577	345	7	in	in	ADP
ejpam-577	345	8	radio	radio	NOUN
ejpam-577	345	9	-	-	PUNCT
ejpam-577	345	10	astronomy	astronomy	NOUN
ejpam-577	345	11	.	.	PUNCT
ejpam-577	346	1	colloquium	colloquium	NOUN
ejpam-577	346	2	of	of	ADP
ejpam-577	346	3	the	the	DET
ejpam-577	346	4	mathematical	mathematical	ADJ
ejpam-577	346	5	society	society	NOUN
ejpam-577	346	6	,	,	PUNCT
ejpam-577	346	7	janos	janos	PROPN
ejpam-577	346	8	bolyai	bolyai	VERB
ejpam-577	346	9	18	18	NUM
ejpam-577	346	10	,	,	PUNCT
ejpam-577	346	11	combinatorics	combinatoric	NOUN
ejpam-577	346	12	,	,	PUNCT
ejpam-577	346	13	hungary	hungary	PROPN
ejpam-577	346	14	,	,	PUNCT
ejpam-577	346	15	135	135	NUM
ejpam-577	346	16	-	-	SYM
ejpam-577	346	17	149	149	NUM
ejpam-577	346	18	,	,	PUNCT
ejpam-577	346	19	1976	1976	NUM
ejpam-577	346	20	.	.	PUNCT
ejpam-577	347	1	references	reference	NOUN
ejpam-577	347	2	281	281	NUM
ejpam-577	347	3	[	[	SYM
ejpam-577	347	4	6	6	NUM
ejpam-577	347	5	]	]	X
ejpam-577	347	6	g.s	g.s	PROPN
ejpam-577	347	7	.	.	PROPN
ejpam-577	347	8	bloom	bloom	PROPN
ejpam-577	347	9	.	.	PUNCT
ejpam-577	348	1	numbered	number	VERB
ejpam-577	348	2	undirected	undirected	ADJ
ejpam-577	348	3	graphs	graph	NOUN
ejpam-577	348	4	and	and	CCONJ
ejpam-577	348	5	their	their	PRON
ejpam-577	348	6	uses	use	NOUN
ejpam-577	348	7	:	:	PUNCT
ejpam-577	348	8	a	a	DET
ejpam-577	348	9	survey	survey	NOUN
ejpam-577	348	10	of	of	ADP
ejpam-577	348	11	unifying	unify	VERB
ejpam-577	348	12	scientific	scientific	ADJ
ejpam-577	348	13	and	and	CCONJ
ejpam-577	348	14	engineering	engineering	NOUN
ejpam-577	348	15	concepts	concept	NOUN
ejpam-577	348	16	and	and	CCONJ
ejpam-577	348	17	its	its	PRON
ejpam-577	348	18	use	use	NOUN
ejpam-577	348	19	in	in	ADP
ejpam-577	348	20	developing	develop	VERB
ejpam-577	348	21	a	a	DET
ejpam-577	348	22	theory	theory	NOUN
ejpam-577	348	23	of	of	ADP
ejpam-577	348	24	non	non	ADJ
ejpam-577	348	25	-	-	ADJ
ejpam-577	348	26	redundant	redundant	ADJ
ejpam-577	348	27	homometric	homometric	ADJ
ejpam-577	348	28	sets	set	NOUN
ejpam-577	348	29	relating	relate	VERB
ejpam-577	348	30	to	to	ADP
ejpam-577	348	31	some	some	DET
ejpam-577	348	32	ambiguities	ambiguity	NOUN
ejpam-577	348	33	in	in	ADP
ejpam-577	348	34	x	x	ADJ
ejpam-577	348	35	-	-	NOUN
ejpam-577	348	36	ray	ray	NOUN
ejpam-577	348	37	diffraction	diffraction	NOUN
ejpam-577	348	38	analysis	analysis	NOUN
ejpam-577	348	39	,	,	PUNCT
ejpam-577	348	40	ph	ph	PROPN
ejpam-577	348	41	.	.	PROPN
ejpam-577	348	42	d.	d.	PROPN
ejpam-577	348	43	,	,	PUNCT
ejpam-577	348	44	dissertation	dissertation	NOUN
ejpam-577	348	45	,	,	PUNCT
ejpam-577	348	46	univ	univ	PROPN
ejpam-577	348	47	.	.	PROPN
ejpam-577	348	48	of	of	ADP
ejpam-577	348	49	southern	southern	PROPN
ejpam-577	348	50	california	california	PROPN
ejpam-577	348	51	,	,	PUNCT
ejpam-577	348	52	los	los	PROPN
ejpam-577	348	53	angeles	angeles	PROPN
ejpam-577	348	54	,	,	PUNCT
ejpam-577	348	55	1975	1975	NUM
ejpam-577	349	1	[	[	X
ejpam-577	349	2	7	7	X
ejpam-577	349	3	]	]	X
ejpam-577	349	4	a.r	a.r	PROPN
ejpam-577	349	5	.	.	PROPN
ejpam-577	349	6	eckler	eckler	PROPN
ejpam-577	349	7	.	.	PUNCT
ejpam-577	350	1	the	the	DET
ejpam-577	350	2	construction	construction	NOUN
ejpam-577	350	3	of	of	ADP
ejpam-577	350	4	missile	missile	NOUN
ejpam-577	350	5	guidance	guidance	NOUN
ejpam-577	350	6	codes	code	NOUN
ejpam-577	350	7	resistant	resistant	ADJ
ejpam-577	350	8	to	to	ADP
ejpam-577	350	9	random	random	ADJ
ejpam-577	350	10	interference	interference	NOUN
ejpam-577	350	11	,	,	PUNCT
ejpam-577	350	12	bell.syst	bell.syst	NOUN
ejpam-577	350	13	.	.	PUNCT
ejpam-577	351	1	tech	tech	NOUN
ejpam-577	351	2	.	.	PUNCT
ejpam-577	351	3	,	,	PUNCT
ejpam-577	351	4	j	j	PROPN
ejpam-577	351	5	,	,	PUNCT
ejpam-577	351	6	vol	vol	NOUN
ejpam-577	351	7	.	.	PROPN
ejpam-577	351	8	39	39	NUM
ejpam-577	351	9	,	,	PUNCT
ejpam-577	351	10	973	973	NUM
ejpam-577	351	11	-	-	SYM
ejpam-577	351	12	994	994	NUM
ejpam-577	351	13	,	,	PUNCT
ejpam-577	351	14	1960	1960	NUM
ejpam-577	351	15	[	[	X
ejpam-577	351	16	8	8	NUM
ejpam-577	351	17	]	]	X
ejpam-577	351	18	h.	h.	PROPN
ejpam-577	351	19	enomoto	enomoto	NOUN
ejpam-577	351	20	.	.	PUNCT
ejpam-577	352	1	a.s	a.s	PROPN
ejpam-577	352	2	.	.	PROPN
ejpam-577	352	3	llado	llado	PROPN
ejpam-577	352	4	,	,	PUNCT
ejpam-577	352	5	t.	t.	PROPN
ejpam-577	352	6	nakamigawa	nakamigawa	PROPN
ejpam-577	352	7	,	,	PUNCT
ejpam-577	352	8	and	and	CCONJ
ejpam-577	352	9	g.	g.	PROPN
ejpam-577	352	10	ringel	ringel	NOUN
ejpam-577	352	11	,	,	PUNCT
ejpam-577	352	12	super	super	ADJ
ejpam-577	352	13	edge	edge	NOUN
ejpam-577	352	14	-	-	PUNCT
ejpam-577	352	15	magic	magic	NOUN
ejpam-577	352	16	graphs	graph	NOUN
ejpam-577	352	17	,	,	PUNCT
ejpam-577	352	18	sut	sut	PROPN
ejpam-577	352	19	j.	j.	PROPN
ejpam-577	352	20	math	math	PROPN
ejpam-577	352	21	.	.	PROPN
ejpam-577	352	22	,	,	PUNCT
ejpam-577	352	23	34,105	34,105	NUM
ejpam-577	352	24	-	-	SYM
ejpam-577	352	25	109	109	NUM
ejpam-577	352	26	,	,	PUNCT
ejpam-577	352	27	1998	1998	NUM
ejpam-577	352	28	.	.	PUNCT
ejpam-577	353	1	[	[	X
ejpam-577	353	2	9	9	NUM
ejpam-577	353	3	]	]	PUNCT
ejpam-577	353	4	r.	r.	PROPN
ejpam-577	353	5	figueroa	figueroa	PROPN
ejpam-577	353	6	-	-	PUNCT
ejpam-577	353	7	centeno	centeno	PROPN
ejpam-577	353	8	,	,	PUNCT
ejpam-577	353	9	r.	r.	PROPN
ejpam-577	353	10	ichishima	ichishima	PROPN
ejpam-577	353	11	,	,	PUNCT
ejpam-577	353	12	and	and	CCONJ
ejpam-577	353	13	f.	f.	PROPN
ejpam-577	353	14	muntaner	muntaner	PROPN
ejpam-577	353	15	-	-	PUNCT
ejpam-577	353	16	batle	batle	PROPN
ejpam-577	353	17	.	.	PUNCT
ejpam-577	354	1	the	the	DET
ejpam-577	354	2	place	place	NOUN
ejpam-577	354	3	of	of	ADP
ejpam-577	354	4	super	super	ADJ
ejpam-577	354	5	-	-	ADJ
ejpam-577	354	6	edge	edge	ADJ
ejpam-577	354	7	-	-	PUNCT
ejpam-577	354	8	magic	magic	NOUN
ejpam-577	354	9	labellings	labelling	NOUN
ejpam-577	354	10	among	among	ADP
ejpam-577	354	11	other	other	ADJ
ejpam-577	354	12	classes	class	NOUN
ejpam-577	354	13	of	of	ADP
ejpam-577	354	14	labellings	labelling	NOUN
ejpam-577	354	15	,	,	PUNCT
ejpam-577	354	16	discrete	discrete	ADJ
ejpam-577	354	17	math	math	NOUN
ejpam-577	354	18	.	.	PUNCT
ejpam-577	354	19	,	,	PUNCT
ejpam-577	354	20	231	231	NUM
ejpam-577	354	21	,	,	PUNCT
ejpam-577	354	22	153	153	NUM
ejpam-577	354	23	-	-	SYM
ejpam-577	354	24	168	168	NUM
ejpam-577	354	25	,	,	PUNCT
ejpam-577	354	26	2001	2001	NUM
ejpam-577	354	27	.	.	PUNCT
ejpam-577	355	1	[	[	X
ejpam-577	355	2	10	10	NUM
ejpam-577	355	3	]	]	X
ejpam-577	355	4	j.n	j.n	PROPN
ejpam-577	355	5	.	.	PROPN
ejpam-577	355	6	franklin	franklin	PROPN
ejpam-577	355	7	.	.	PUNCT
ejpam-577	356	1	ambiguities	ambiguity	NOUN
ejpam-577	356	2	in	in	ADP
ejpam-577	356	3	the	the	DET
ejpam-577	356	4	x	x	NOUN
ejpam-577	356	5	-	-	NOUN
ejpam-577	356	6	ray	ray	NOUN
ejpam-577	356	7	analysis	analysis	NOUN
ejpam-577	356	8	of	of	ADP
ejpam-577	356	9	crystal	crystal	NOUN
ejpam-577	356	10	structures	structure	NOUN
ejpam-577	356	11	,	,	PUNCT
ejpam-577	356	12	acta	acta	PROPN
ejpam-577	356	13	cryst	cryst	PROPN
ejpam-577	356	14	.	.	PUNCT
ejpam-577	357	1	,	,	PUNCT
ejpam-577	357	2	vol	vol	NOUN
ejpam-577	357	3	.	.	PUNCT
ejpam-577	358	1	a	a	DET
ejpam-577	358	2	30	30	NUM
ejpam-577	358	3	,	,	PUNCT
ejpam-577	358	4	698	698	NUM
ejpam-577	358	5	-	-	SYM
ejpam-577	358	6	702	702	NUM
ejpam-577	358	7	,	,	PUNCT
ejpam-577	358	8	nov	nov	PROPN
ejpam-577	358	9	.	.	PROPN
ejpam-577	358	10	1974	1974	NUM
ejpam-577	358	11	.	.	PUNCT
ejpam-577	359	1	[	[	X
ejpam-577	359	2	11	11	NUM
ejpam-577	359	3	]	]	X
ejpam-577	359	4	f.	f.	PROPN
ejpam-577	359	5	harary	harary	PROPN
ejpam-577	359	6	.	.	PUNCT
ejpam-577	360	1	graph	graph	NOUN
ejpam-577	360	2	theory	theory	NOUN
ejpam-577	360	3	,	,	PUNCT
ejpam-577	360	4	addison	addison	PROPN
ejpam-577	360	5	wesley	wesley	PROPN
ejpam-577	360	6	,	,	PUNCT
ejpam-577	360	7	reading	reading	NOUN
ejpam-577	360	8	,	,	PUNCT
ejpam-577	360	9	massachusetts	massachusetts	PROPN
ejpam-577	360	10	,	,	PUNCT
ejpam-577	360	11	1969	1969	NUM
ejpam-577	360	12	.	.	PUNCT
ejpam-577	361	1	[	[	X
ejpam-577	361	2	12	12	NUM
ejpam-577	361	3	]	]	PUNCT
ejpam-577	361	4	l.	l.	PROPN
ejpam-577	361	5	h.	h.	PROPN
ejpam-577	361	6	harper	harper	PROPN
ejpam-577	361	7	.	.	PUNCT
ejpam-577	362	1	dsif	dsif	NOUN
ejpam-577	362	2	integrated	integrate	VERB
ejpam-577	362	3	circuit	circuit	NOUN
ejpam-577	362	4	layout	layout	NOUN
ejpam-577	362	5	and	and	CCONJ
ejpam-577	362	6	iso	iso	NOUN
ejpam-577	362	7	-	-	PUNCT
ejpam-577	362	8	perimetric	perimetric	ADJ
ejpam-577	362	9	problems	problem	NOUN
ejpam-577	362	10	,	,	PUNCT
ejpam-577	362	11	jpl	jpl	PROPN
ejpam-577	362	12	space	space	NOUN
ejpam-577	362	13	program	program	NOUN
ejpam-577	362	14	summary	summary	NOUN
ejpam-577	362	15	37	37	NUM
ejpam-577	362	16	-	-	SYM
ejpam-577	362	17	66	66	NUM
ejpam-577	362	18	vol	vol	NOUN
ejpam-577	362	19	.	.	PROPN
ejpam-577	363	1	2	2	NUM
ejpam-577	363	2	,	,	PUNCT
ejpam-577	363	3	pp	pp	ADJ
ejpam-577	363	4	.	.	PUNCT
ejpam-577	364	1	37	37	NUM
ejpam-577	364	2	-	-	SYM
ejpam-577	364	3	42	42	NUM
ejpam-577	364	4	,	,	PUNCT
ejpam-577	364	5	sept	sept	PROPN
ejpam-577	364	6	.	.	PROPN
ejpam-577	364	7	1970	1970	NUM
ejpam-577	365	1	[	[	X
ejpam-577	365	2	13	13	NUM
ejpam-577	365	3	]	]	SYM
ejpam-577	365	4	s.m	s.m	PROPN
ejpam-577	365	5	.	.	PROPN
ejpam-577	365	6	hegde	hegde	PROPN
ejpam-577	365	7	and	and	CCONJ
ejpam-577	365	8	sudhakar	sudhakar	PROPN
ejpam-577	365	9	shetty	shetty	PROPN
ejpam-577	365	10	.	.	PUNCT
ejpam-577	366	1	strongly	strongly	ADV
ejpam-577	366	2	indexable	indexable	ADJ
ejpam-577	366	3	graphs	graph	NOUN
ejpam-577	366	4	and	and	CCONJ
ejpam-577	366	5	applications	application	NOUN
ejpam-577	366	6	,	,	PUNCT
ejpam-577	366	7	discrete	discrete	ADJ
ejpam-577	366	8	mathematics	mathematic	NOUN
ejpam-577	366	9	,	,	PUNCT
ejpam-577	366	10	to	to	PART
ejpam-577	366	11	appear	appear	VERB
ejpam-577	366	12	.	.	PUNCT
ejpam-577	367	1	[	[	X
ejpam-577	367	2	14	14	NUM
ejpam-577	367	3	]	]	X
ejpam-577	367	4	j.	j.	PROPN
ejpam-577	367	5	leech	leech	PROPN
ejpam-577	367	6	.	.	PUNCT
ejpam-577	368	1	on	on	ADP
ejpam-577	368	2	the	the	DET
ejpam-577	368	3	representation	representation	NOUN
ejpam-577	368	4	of	of	ADP
ejpam-577	368	5	1,2	1,2	NUM
ejpam-577	368	6	,	,	PUNCT
ejpam-577	368	7	.	.	PUNCT
ejpam-577	368	8	.	.	PUNCT
ejpam-577	368	9	.	.	PUNCT
ejpam-577	369	1	,	,	PUNCT
ejpam-577	369	2	n	n	CCONJ
ejpam-577	369	3	by	by	ADP
ejpam-577	369	4	differences	difference	NOUN
ejpam-577	369	5	,	,	PUNCT
ejpam-577	369	6	j.	j.	PROPN
ejpam-577	369	7	london	london	PROPN
ejpam-577	369	8	math	math	PROPN
ejpam-577	369	9	.	.	PUNCT
ejpam-577	370	1	soc	soc	PROPN
ejpam-577	370	2	.	.	PUNCT
ejpam-577	371	1	,	,	PUNCT
ejpam-577	371	2	vol	vol	NOUN
ejpam-577	371	3	.	.	PROPN
ejpam-577	371	4	31	31	NUM
ejpam-577	371	5	,	,	PUNCT
ejpam-577	371	6	160	160	NUM
ejpam-577	371	7	-	-	SYM
ejpam-577	371	8	169	169	NUM
ejpam-577	371	9	,	,	PUNCT
ejpam-577	371	10	apr	apr	NOUN
ejpam-577	371	11	.	.	PUNCT
ejpam-577	372	1	1956	1956	NUM
ejpam-577	373	1	[	[	X
ejpam-577	373	2	15	15	NUM
ejpam-577	373	3	]	]	X
ejpam-577	373	4	j.c	j.c	PROPN
ejpam-577	373	5	.	.	PROPN
ejpam-577	373	6	p.	p.	PROPN
ejpam-577	373	7	miller	miller	PROPN
ejpam-577	373	8	,	,	PUNCT
ejpam-577	373	9	differences	difference	VERB
ejpam-577	373	10	basis	basis	NOUN
ejpam-577	373	11	,	,	PUNCT
ejpam-577	373	12	three	three	NUM
ejpam-577	373	13	problems	problem	NOUN
ejpam-577	373	14	in	in	ADP
ejpam-577	373	15	additive	additive	ADJ
ejpam-577	373	16	number	number	NOUN
ejpam-577	373	17	theory	theory	NOUN
ejpam-577	373	18	,	,	PUNCT
ejpam-577	373	19	a.d.l	a.d.l	PROPN
ejpam-577	373	20	.	.	PROPN
ejpam-577	373	21	atkin	atkin	PROPN
ejpam-577	373	22	and	and	CCONJ
ejpam-577	373	23	b.j	b.j	PROPN
ejpam-577	373	24	.	.	PROPN
ejpam-577	373	25	birch	birch	PROPN
ejpam-577	373	26	,	,	PUNCT
ejpam-577	373	27	eds	eds	PROPN
ejpam-577	373	28	,	,	PUNCT
ejpam-577	373	29	london	london	PROPN
ejpam-577	373	30	,	,	PUNCT
ejpam-577	373	31	academic	academic	ADJ
ejpam-577	373	32	press,299	press,299	PROPN
ejpam-577	373	33	-	-	PUNCT
ejpam-577	373	34	322	322	NUM
ejpam-577	373	35	,	,	PUNCT
ejpam-577	373	36	1971	1971	NUM
ejpam-577	373	37	.	.	PUNCT
ejpam-577	374	1	[	[	X
ejpam-577	374	2	16	16	NUM
ejpam-577	374	3	]	]	X
ejpam-577	374	4	g.	g.	NOUN
ejpam-577	374	5	ringel	ringel	NOUN
ejpam-577	374	6	.	.	PUNCT
ejpam-577	375	1	labeling	labeling	NOUN
ejpam-577	375	2	problems	problem	NOUN
ejpam-577	375	3	,	,	PUNCT
ejpam-577	375	4	proceedings	proceeding	NOUN
ejpam-577	375	5	of	of	ADP
ejpam-577	375	6	the	the	DET
ejpam-577	375	7	eighth	eighth	ADJ
ejpam-577	375	8	international	international	ADJ
ejpam-577	375	9	conference	conference	NOUN
ejpam-577	375	10	on	on	ADP
ejpam-577	375	11	graph	graph	NOUN
ejpam-577	375	12	theory	theory	NOUN
ejpam-577	375	13	,	,	PUNCT
ejpam-577	375	14	combinatorics	combinatoric	NOUN
ejpam-577	375	15	,	,	PUNCT
ejpam-577	375	16	algorithms	algorithm	NOUN
ejpam-577	375	17	and	and	CCONJ
ejpam-577	375	18	applications	application	NOUN
ejpam-577	375	19	.	.	PUNCT
ejpam-577	376	1	1996	1996	NUM