id	sid	tid	token	lemma	pos
cana-2702	1	1	pebbling	pebble	VERB
cana-2702	1	2	on	on	ADP
cana-2702	1	3	crisscross	crisscross	ADJ
cana-2702	1	4	sequence	sequence	NOUN
cana-2702	1	5	of	of	ADP
cana-2702	1	6	m	m	PROPN
cana-2702	1	7	complete	complete	ADJ
cana-2702	1	8	graphs	graph	NOUN
cana-2702	1	9	communications	communication	NOUN
cana-2702	1	10	on	on	ADP
cana-2702	1	11	applied	apply	VERB
cana-2702	1	12	nonlinear	nonlinear	ADJ
cana-2702	1	13	analysis	analysis	NOUN
cana-2702	1	14	issn	issn	NOUN
cana-2702	1	15	:	:	PUNCT
cana-2702	1	16	1074	1074	NUM
cana-2702	1	17	-	-	PUNCT
cana-2702	1	18	133x	133x	NUM
cana-2702	1	19	vol	vol	NOUN
cana-2702	1	20	32	32	NUM
cana-2702	1	21	no	no	NOUN
cana-2702	1	22	.	.	PUNCT
cana-2702	2	1	3s	3s	NUM
cana-2702	2	2	(	(	PUNCT
cana-2702	2	3	2025	2025	NUM
cana-2702	2	4	)	)	PUNCT
cana-2702	2	5	648	648	NUM
cana-2702	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	2	7	pebbling	pebble	VERB
cana-2702	2	8	on	on	ADP
cana-2702	2	9	crisscross	crisscross	ADJ
cana-2702	2	10	sequence	sequence	NOUN
cana-2702	2	11	of	of	ADP
cana-2702	2	12	𝒎	𝒎	PROPN
cana-2702	2	13	complete	complete	ADJ
cana-2702	2	14	graphs	graph	NOUN
cana-2702	2	15	1j.jenifer	1j.jenifer	NUM
cana-2702	2	16	steffi	steffi	PROPN
cana-2702	2	17	,	,	PUNCT
cana-2702	2	18	2	2	NUM
cana-2702	2	19	m	m	NOUN
cana-2702	2	20	gayathri	gayathri	PROPN
cana-2702	2	21	lakshmi	lakshmi	PROPN
cana-2702	2	22	,	,	PUNCT
cana-2702	2	23	3dr	3dr	NOUN
cana-2702	2	24	.	.	PUNCT
cana-2702	3	1	d.	d.	PROPN
cana-2702	3	2	vamsi	vamsi	PROPN
cana-2702	3	3	priya	priya	PROPN
cana-2702	3	4	,	,	PUNCT
cana-2702	3	5	4a.k.bhuvaneswari	4a.k.bhuvaneswari	NUM
cana-2702	3	6	,	,	PUNCT
cana-2702	3	7	5m.kannan	5m.kannan	NUM
cana-2702	3	8	,	,	PUNCT
cana-2702	3	9	6j	6j	NUM
cana-2702	3	10	.	.	PUNCT
cana-2702	4	1	juli	juli	PROPN
cana-2702	4	2	amala	amala	PROPN
cana-2702	4	3	rani	rani	PROPN
cana-2702	4	4	,	,	PUNCT
cana-2702	4	5	7abdr.m.elangovan	7abdr.m.elangovan	NUM
cana-2702	4	6	1malla	1malla	NUM
cana-2702	4	7	reddy	reddy	PROPN
cana-2702	4	8	engineering	engineering	PROPN
cana-2702	4	9	college	college	PROPN
cana-2702	4	10	,	,	PUNCT
cana-2702	4	11	hyderabad	hyderabad	PROPN
cana-2702	4	12	,	,	PUNCT
cana-2702	4	13	telangana	telangana	PROPN
cana-2702	4	14	,	,	PUNCT
cana-2702	4	15	india	india	PROPN
cana-2702	4	16	.	.	PUNCT
cana-2702	5	1	1email	1email	NUM
cana-2702	5	2	id:drjenifersteffi@gmail.com	id:drjenifersteffi@gmail.com	X
cana-2702	6	1	2assistant	2assistant	NUM
cana-2702	6	2	professor	professor	NOUN
cana-2702	6	3	,	,	PUNCT
cana-2702	6	4	mathematics	mathematic	NOUN
cana-2702	6	5	,	,	PUNCT
cana-2702	6	6	saveetha	saveetha	PROPN
cana-2702	6	7	engineering	engineering	PROPN
cana-2702	6	8	college	college	PROPN
cana-2702	6	9	,	,	PUNCT
cana-2702	6	10	chennai	chennai	NOUN
cana-2702	6	11	.	.	PUNCT
cana-2702	7	1	2email	2email	NUM
cana-2702	7	2	id:gayathrilakshmi1804@gmail.com	id:gayathrilakshmi1804@gmail.com	SYM
cana-2702	7	3	3associate	3associate	NUM
cana-2702	7	4	professor	professor	NOUN
cana-2702	7	5	,	,	PUNCT
cana-2702	7	6	basic	basic	ADJ
cana-2702	7	7	science	science	NOUN
cana-2702	7	8	and	and	CCONJ
cana-2702	7	9	humanities	humanity	NOUN
cana-2702	7	10	,	,	PUNCT
cana-2702	7	11	vignan	vignan	NOUN
cana-2702	7	12	’s	’s	PART
cana-2702	7	13	institute	institute	PROPN
cana-2702	7	14	of	of	ADP
cana-2702	7	15	information	information	NOUN
cana-2702	7	16	and	and	CCONJ
cana-2702	7	17	technology	technology	NOUN
cana-2702	7	18	,	,	PUNCT
cana-2702	7	19	duvvada	duvvada	PROPN
cana-2702	7	20	,	,	PUNCT
cana-2702	7	21	visakhapatnam	visakhapatnam	PROPN
cana-2702	7	22	.	.	PUNCT
cana-2702	8	1	3email	3email	NUM
cana-2702	8	2	i	i	NOUN
cana-2702	8	3	d	d	NOUN
cana-2702	8	4	:	:	PUNCT
cana-2702	9	1	vamsipriyabagi@gmail.com	vamsipriyabagi@gmail.com	PROPN
cana-2702	9	2	4associate	4associate	NUM
cana-2702	9	3	professor	professor	NOUN
cana-2702	9	4	,	,	PUNCT
cana-2702	9	5	department	department	NOUN
cana-2702	9	6	of	of	ADP
cana-2702	9	7	mathematics	mathematic	NOUN
cana-2702	9	8	,	,	PUNCT
cana-2702	9	9	aarupadai	aarupadai	PROPN
cana-2702	9	10	veedu	veedu	PROPN
cana-2702	9	11	institute	institute	PROPN
cana-2702	9	12	of	of	ADP
cana-2702	9	13	technology	technology	PROPN
cana-2702	9	14	,	,	PUNCT
cana-2702	9	15	vmrf(du),paiyanoor	vmrf(du),paiyanoor	PROPN
cana-2702	9	16	4email	4email	NUM
cana-2702	9	17	id:bhuvanabalaji13@gmail.com	id:bhuvanabalaji13@gmail.com	NUM
cana-2702	9	18	5assistant	5assistant	NUM
cana-2702	9	19	professor	professor	NOUN
cana-2702	9	20	,	,	PUNCT
cana-2702	9	21	department	department	NOUN
cana-2702	9	22	of	of	ADP
cana-2702	9	23	mathematics	mathematics	PROPN
cana-2702	9	24	,	,	PUNCT
cana-2702	9	25	sardar	sardar	PROPN
cana-2702	9	26	vallabhbhai	vallabhbhai	PROPN
cana-2702	9	27	patel	patel	PROPN
cana-2702	9	28	international	international	PROPN
cana-2702	9	29	school	school	NOUN
cana-2702	9	30	of	of	ADP
cana-2702	9	31	textiles	textile	NOUN
cana-2702	9	32	and	and	CCONJ
cana-2702	9	33	management	management	NOUN
cana-2702	9	34	,	,	PUNCT
cana-2702	9	35	coimbatore	coimbatore	NOUN
cana-2702	9	36	.	.	PUNCT
cana-2702	10	1	5email	5email	NUM
cana-2702	10	2	id:kannan8383@gmail.com	id:kannan8383@gmail.com	X
cana-2702	10	3	6assistant	6assistant	NUM
cana-2702	10	4	professor	professor	NOUN
cana-2702	10	5	,	,	PUNCT
cana-2702	10	6	department	department	NOUN
cana-2702	10	7	of	of	ADP
cana-2702	10	8	mathematics	mathematic	NOUN
cana-2702	10	9	,	,	PUNCT
cana-2702	10	10	panimalar	panimalar	ADJ
cana-2702	10	11	engineering	engineering	NOUN
cana-2702	10	12	college	college	NOUN
cana-2702	10	13	,	,	PUNCT
cana-2702	10	14	chennai	chennai	NOUN
cana-2702	10	15	.	.	PUNCT
cana-2702	11	1	6email	6email	NUM
cana-2702	11	2	i	i	PRON
cana-2702	11	3	d	d	NOUN
cana-2702	11	4	:	:	PUNCT
cana-2702	12	1	jesujuli@gmail.com	jesujuli@gmail.com	X
cana-2702	12	2	7adepartment	7adepartment	NUM
cana-2702	12	3	of	of	ADP
cana-2702	12	4	biosciences	bioscience	NOUN
cana-2702	12	5	,	,	PUNCT
cana-2702	12	6	saveetha	saveetha	PROPN
cana-2702	12	7	school	school	NOUN
cana-2702	12	8	of	of	ADP
cana-2702	12	9	engineering	engineering	NOUN
cana-2702	12	10	.	.	PUNCT
cana-2702	13	1	saveetha	saveetha	PROPN
cana-2702	13	2	institute	institute	PROPN
cana-2702	13	3	of	of	ADP
cana-2702	13	4	medical	medical	ADJ
cana-2702	13	5	and	and	CCONJ
cana-2702	13	6	technical	technical	ADJ
cana-2702	13	7	sciences	science	NOUN
cana-2702	13	8	,	,	PUNCT
cana-2702	13	9	chennai	chennai	VERB
cana-2702	13	10	602	602	NUM
cana-2702	13	11	105	105	NUM
cana-2702	13	12	7bapplied	7bapplied	NUM
cana-2702	13	13	science	science	NOUN
cana-2702	13	14	research	research	NOUN
cana-2702	13	15	center	center	NOUN
cana-2702	13	16	.	.	PUNCT
cana-2702	14	1	applied	apply	VERB
cana-2702	14	2	science	science	PROPN
cana-2702	14	3	private	private	ADJ
cana-2702	14	4	university	university	NOUN
cana-2702	14	5	,	,	PUNCT
cana-2702	14	6	amman	amman	PROPN
cana-2702	14	7	,	,	PUNCT
cana-2702	14	8	jordan	jordan	PROPN
cana-2702	14	9	.	.	PUNCT
cana-2702	15	1	7abemail	7abemail	NUM
cana-2702	15	2	id:muniyandy.e@gmail.com	id:muniyandy.e@gmail.com	NUM
cana-2702	15	3	article	article	NOUN
cana-2702	15	4	history	history	NOUN
cana-2702	15	5	:	:	PUNCT
cana-2702	15	6	received	receive	VERB
cana-2702	15	7	:	:	PUNCT
cana-2702	15	8	26	26	NUM
cana-2702	15	9	-	-	SYM
cana-2702	15	10	09	09	NUM
cana-2702	15	11	-	-	PUNCT
cana-2702	15	12	2024	2024	NUM
cana-2702	15	13	revised	revise	VERB
cana-2702	15	14	:	:	PUNCT
cana-2702	15	15	14	14	NUM
cana-2702	15	16	-	-	SYM
cana-2702	15	17	11	11	NUM
cana-2702	15	18	-	-	PUNCT
cana-2702	15	19	2024	2024	NUM
cana-2702	15	20	accepted	accept	VERB
cana-2702	15	21	:	:	PUNCT
cana-2702	15	22	29	29	NUM
cana-2702	15	23	-	-	SYM
cana-2702	15	24	11	11	NUM
cana-2702	15	25	-	-	PUNCT
cana-2702	15	26	2024	2024	NUM
cana-2702	15	27	abstract	abstract	NOUN
cana-2702	15	28	:	:	PUNCT
cana-2702	15	29	this	this	DET
cana-2702	15	30	paper	paper	NOUN
cana-2702	15	31	investigates	investigate	VERB
cana-2702	15	32	the	the	DET
cana-2702	15	33	pebbling	pebble	VERB
cana-2702	15	34	number	number	NOUN
cana-2702	15	35	,	,	PUNCT
cana-2702	15	36	the	the	DET
cana-2702	15	37	two	two	NUM
cana-2702	15	38	pebbling	pebble	VERB
cana-2702	15	39	property	property	NOUN
cana-2702	15	40	,	,	PUNCT
cana-2702	15	41	the	the	DET
cana-2702	15	42	t	t	NOUN
cana-2702	15	43	-	-	PUNCT
cana-2702	15	44	pebbling	pebble	VERB
cana-2702	15	45	number	number	NOUN
cana-2702	15	46	,	,	PUNCT
cana-2702	15	47	and	and	CCONJ
cana-2702	15	48	the	the	DET
cana-2702	15	49	2t	2t	NOUN
cana-2702	15	50	-	-	PUNCT
cana-2702	15	51	pebbling	pebble	VERB
cana-2702	15	52	property	property	NOUN
cana-2702	15	53	of	of	ADP
cana-2702	15	54	a	a	DET
cana-2702	15	55	crisscross	crisscross	NOUN
cana-2702	15	56	sequence	sequence	NOUN
cana-2702	15	57	comprised	comprise	VERB
cana-2702	15	58	of	of	ADP
cana-2702	15	59	m	m	PROPN
cana-2702	15	60	complete	complete	ADJ
cana-2702	15	61	graphs	graph	NOUN
cana-2702	15	62	.	.	PUNCT
cana-2702	16	1	by	by	ADP
cana-2702	16	2	analyzing	analyze	VERB
cana-2702	16	3	these	these	DET
cana-2702	16	4	pebbling	pebble	VERB
cana-2702	16	5	properties	property	NOUN
cana-2702	16	6	,	,	PUNCT
cana-2702	16	7	we	we	PRON
cana-2702	16	8	aim	aim	VERB
cana-2702	16	9	to	to	PART
cana-2702	16	10	gain	gain	VERB
cana-2702	16	11	insights	insight	NOUN
cana-2702	16	12	into	into	ADP
cana-2702	16	13	the	the	DET
cana-2702	16	14	efficiency	efficiency	NOUN
cana-2702	16	15	and	and	CCONJ
cana-2702	16	16	feasibility	feasibility	NOUN
cana-2702	16	17	of	of	ADP
cana-2702	16	18	pebbling	pebble	VERB
cana-2702	16	19	operations	operation	NOUN
cana-2702	16	20	within	within	ADP
cana-2702	16	21	this	this	DET
cana-2702	16	22	specific	specific	ADJ
cana-2702	16	23	graph	graph	NOUN
cana-2702	16	24	structure	structure	NOUN
cana-2702	16	25	.	.	PUNCT
cana-2702	17	1	keywords	keyword	NOUN
cana-2702	17	2	:	:	PUNCT
cana-2702	17	3	graph	graph	NOUN
cana-2702	17	4	pebbling	pebbling	NOUN
cana-2702	17	5	,	,	PUNCT
cana-2702	17	6	graph	graph	NOUN
cana-2702	17	7	theory	theory	NOUN
cana-2702	17	8	,	,	PUNCT
cana-2702	17	9	crisscross	crisscross	VERB
cana-2702	17	10	sequence	sequence	NOUN
cana-2702	17	11	of	of	ADP
cana-2702	17	12	m	m	PROPN
cana-2702	17	13	complete	complete	ADJ
cana-2702	17	14	graphs	graph	NOUN
cana-2702	17	15	,	,	PUNCT
cana-2702	17	16	pebbling	pebble	VERB
cana-2702	17	17	properties	property	NOUN
cana-2702	17	18	,	,	PUNCT
cana-2702	17	19	graph	graph	NOUN
cana-2702	17	20	optimization	optimization	NOUN
cana-2702	17	21	introduction	introduction	NOUN
cana-2702	17	22	graph	graph	NOUN
cana-2702	17	23	pebbling	pebble	VERB
cana-2702	17	24	is	be	AUX
cana-2702	17	25	a	a	DET
cana-2702	17	26	mathematical	mathematical	ADJ
cana-2702	17	27	concept	concept	NOUN
cana-2702	17	28	that	that	PRON
cana-2702	17	29	involves	involve	VERB
cana-2702	17	30	manipulating	manipulate	VERB
cana-2702	17	31	configurations	configuration	NOUN
cana-2702	17	32	of	of	ADP
cana-2702	17	33	pebbles	pebble	NOUN
cana-2702	17	34	on	on	ADP
cana-2702	17	35	the	the	DET
cana-2702	17	36	vertices	vertex	NOUN
cana-2702	17	37	of	of	ADP
cana-2702	17	38	graph	graph	NOUN
cana-2702	17	39	in	in	ADP
cana-2702	17	40	order	order	NOUN
cana-2702	17	41	to	to	PART
cana-2702	17	42	achieve	achieve	VERB
cana-2702	17	43	a	a	DET
cana-2702	17	44	specific	specific	ADJ
cana-2702	17	45	target	target	NOUN
cana-2702	17	46	vertex	vertex	NOUN
cana-2702	17	47	with	with	ADP
cana-2702	17	48	the	the	DET
cana-2702	17	49	desired	desire	VERB
cana-2702	17	50	amount	amount	NOUN
cana-2702	17	51	of	of	ADP
cana-2702	17	52	pebbles	pebble	NOUN
cana-2702	17	53	.	.	PUNCT
cana-2702	18	1	initially	initially	ADV
cana-2702	18	2	,	,	PUNCT
cana-2702	18	3	lagarias	lagarias	PROPN
cana-2702	18	4	and	and	CCONJ
cana-2702	18	5	saks	sak	NOUN
cana-2702	18	6	introduced	introduce	VERB
cana-2702	18	7	graph	graph	NOUN
cana-2702	18	8	pebbling	pebble	VERB
cana-2702	18	9	while	while	SCONJ
cana-2702	18	10	attempting	attempt	VERB
cana-2702	18	11	to	to	PART
cana-2702	18	12	answer	answer	VERB
cana-2702	18	13	a	a	DET
cana-2702	18	14	number	number	NOUN
cana-2702	18	15	theoretic	theoretic	NOUN
cana-2702	18	16	question	question	NOUN
cana-2702	18	17	posed	pose	VERB
cana-2702	18	18	by	by	ADP
cana-2702	18	19	erdos	erdo	NOUN
cana-2702	18	20	and	and	CCONJ
cana-2702	18	21	lemke	lemke	NOUN
cana-2702	18	22	concerning	concern	VERB
cana-2702	18	23	zero	zero	NUM
cana-2702	18	24	-	-	PUNCT
cana-2702	18	25	sum	sum	NOUN
cana-2702	18	26	sequences	sequence	NOUN
cana-2702	18	27	of	of	ADP
cana-2702	18	28	finite	finite	PROPN
cana-2702	18	29	group	group	NOUN
cana-2702	18	30	.	.	PUNCT
cana-2702	19	1	the	the	DET
cana-2702	19	2	graph	graph	NOUN
cana-2702	19	3	pebbling	pebble	VERB
cana-2702	19	4	concept	concept	NOUN
cana-2702	19	5	was	be	AUX
cana-2702	19	6	formally	formally	ADV
cana-2702	19	7	introduced	introduce	VERB
cana-2702	19	8	by	by	ADP
cana-2702	19	9	chung	chung	PROPN
cana-2702	19	10	,	,	PUNCT
cana-2702	19	11	who	who	PRON
cana-2702	19	12	defined	define	VERB
cana-2702	19	13	the	the	DET
cana-2702	19	14	pebbling	pebble	VERB
cana-2702	19	15	number	number	NOUN
cana-2702	19	16	𝑓(𝐺	𝑓(𝐺	ADJ
cana-2702	19	17	)	)	PUNCT
cana-2702	19	18	for	for	ADP
cana-2702	19	19	connected	connected	ADJ
cana-2702	19	20	graph	graph	NOUN
cana-2702	19	21	𝐺.	𝐺.	PROPN
cana-2702	19	22	[	[	X
cana-2702	19	23	11][12]since	11][12]since	NUM
cana-2702	19	24	then	then	ADV
cana-2702	19	25	,	,	PUNCT
cana-2702	19	26	the	the	DET
cana-2702	19	27	field	field	NOUN
cana-2702	19	28	of	of	ADP
cana-2702	19	29	graph	graph	NOUN
cana-2702	19	30	pebbling	pebbling	NOUN
cana-2702	19	31	has	have	AUX
cana-2702	19	32	become	become	VERB
cana-2702	19	33	highly	highly	ADV
cana-2702	19	34	active	active	ADJ
cana-2702	19	35	,	,	PUNCT
cana-2702	19	36	with	with	ADP
cana-2702	19	37	numerous	numerous	ADJ
cana-2702	19	38	open	open	ADJ
cana-2702	19	39	problems	problem	NOUN
cana-2702	19	40	and	and	CCONJ
cana-2702	19	41	conjectures	conjecture	VERB
cana-2702	19	42	awaiting	await	VERB
cana-2702	19	43	resolution	resolution	NOUN
cana-2702	19	44	.	.	PUNCT
cana-2702	20	1	in	in	ADP
cana-2702	20	2	the	the	DET
cana-2702	20	3	context	context	NOUN
cana-2702	20	4	of	of	ADP
cana-2702	20	5	graph	graph	NOUN
cana-2702	20	6	pebbling	pebble	VERB
cana-2702	20	7	,	,	PUNCT
cana-2702	20	8	pebbling	pebble	VERB
cana-2702	20	9	move	move	NOUN
cana-2702	20	10	involves	involve	VERB
cana-2702	20	11	taking	take	VERB
cana-2702	20	12	two	two	NUM
cana-2702	20	13	pebbles	pebble	NOUN
cana-2702	20	14	from	from	ADP
cana-2702	20	15	one	one	NUM
cana-2702	20	16	vertex	vertex	NOUN
cana-2702	20	17	and	and	CCONJ
cana-2702	20	18	placing	place	VERB
cana-2702	20	19	one	one	NUM
cana-2702	20	20	of	of	ADP
cana-2702	20	21	them	they	PRON
cana-2702	20	22	on	on	ADP
cana-2702	20	23	a	a	DET
cana-2702	20	24	neighboring	neighboring	NOUN
cana-2702	20	25	vertex	vertex	NOUN
cana-2702	20	26	.	.	PUNCT
cana-2702	21	1	the	the	DET
cana-2702	21	2	second	second	ADJ
cana-2702	21	3	pebble	pebble	NOUN
cana-2702	21	4	is	be	AUX
cana-2702	21	5	disregarded	disregarded	ADJ
cana-2702	21	6	.	.	PUNCT
cana-2702	22	1	if	if	SCONJ
cana-2702	22	2	there	there	PRON
cana-2702	22	3	are	be	VERB
cana-2702	22	4	𝑤	𝑤	ADP
cana-2702	22	5	pebbles	pebble	NOUN
cana-2702	22	6	distributed	distribute	VERB
cana-2702	22	7	among	among	ADP
cana-2702	22	8	the	the	DET
cana-2702	22	9	vertices	vertex	NOUN
cana-2702	22	10	of	of	ADP
cana-2702	22	11	graph	graph	NOUN
cana-2702	22	12	𝐺	𝐺	PROPN
cana-2702	22	13	,	,	PUNCT
cana-2702	22	14	the	the	DET
cana-2702	22	15	distribution	distribution	NOUN
cana-2702	22	16	is	be	AUX
cana-2702	22	17	considered	consider	VERB
cana-2702	22	18	solvable	solvable	ADJ
cana-2702	22	19	,	,	PUNCT
cana-2702	22	20	if	if	SCONJ
cana-2702	22	21	it	it	PRON
cana-2702	22	22	is	be	AUX
cana-2702	22	23	possible	possible	ADJ
cana-2702	22	24	to	to	PART
cana-2702	22	25	manipulate	manipulate	VERB
cana-2702	22	26	the	the	DET
cana-2702	22	27	pebbles	pebble	NOUN
cana-2702	22	28	through	through	ADP
cana-2702	22	29	a	a	DET
cana-2702	22	30	series	series	NOUN
cana-2702	22	31	of	of	ADP
cana-2702	22	32	moves	move	NOUN
cana-2702	22	33	so	so	SCONJ
cana-2702	22	34	that	that	SCONJ
cana-2702	22	35	any	any	DET
cana-2702	22	36	given	give	VERB
cana-2702	22	37	vertex	vertex	NOUN
cana-2702	22	38	𝑣	𝑣	PROPN
cana-2702	22	39	ends	end	VERB
cana-2702	22	40	up	up	ADP
cana-2702	22	41	with	with	ADP
cana-2702	22	42	at	at	ADV
cana-2702	22	43	least	least	ADV
cana-2702	22	44	one	one	NUM
cana-2702	22	45	pebble	pebble	NOUN
cana-2702	22	46	.	.	PUNCT
cana-2702	23	1	on	on	ADP
cana-2702	23	2	the	the	DET
cana-2702	23	3	other	other	ADJ
cana-2702	23	4	hand	hand	NOUN
cana-2702	23	5	,	,	PUNCT
cana-2702	23	6	if	if	SCONJ
cana-2702	23	7	the	the	DET
cana-2702	23	8	distribution	distribution	NOUN
cana-2702	23	9	can	can	AUX
cana-2702	23	10	not	not	PART
cana-2702	23	11	be	be	AUX
cana-2702	23	12	solved	solve	VERB
cana-2702	23	13	,	,	PUNCT
cana-2702	23	14	it	it	PRON
cana-2702	23	15	is	be	AUX
cana-2702	23	16	referred	refer	VERB
cana-2702	23	17	to	to	ADP
cana-2702	23	18	as	as	ADV
cana-2702	23	19	unsolvable	unsolvable	ADJ
cana-2702	23	20	.	.	PUNCT
cana-2702	24	1	the	the	DET
cana-2702	24	2	pebbling	pebble	VERB
cana-2702	24	3	number	number	NOUN
cana-2702	24	4	,	,	PUNCT
cana-2702	24	5	denoted	denote	VERB
cana-2702	24	6	as	as	ADP
cana-2702	24	7	𝑓(𝐺),represents	𝑓(𝐺),represent	NOUN
cana-2702	24	8	the	the	DET
cana-2702	24	9	smallest	small	ADJ
cana-2702	24	10	value	value	NOUN
cana-2702	24	11	of	of	ADP
cana-2702	24	12	𝑚	𝑚	NOUN
cana-2702	24	13	for	for	ADP
cana-2702	24	14	which	which	PRON
cana-2702	24	15	all	all	DET
cana-2702	24	16	initial	initial	ADJ
cana-2702	24	17	distributions	distribution	NOUN
cana-2702	24	18	of	of	ADP
cana-2702	24	19	𝑚	𝑚	NOUN
cana-2702	24	20	pebbles	pebble	NOUN
cana-2702	24	21	on	on	ADP
cana-2702	24	22	the	the	DET
cana-2702	24	23	graph	graph	NOUN
cana-2702	24	24	𝐺	𝐺	NOUN
cana-2702	24	25	can	can	AUX
cana-2702	24	26	be	be	AUX
cana-2702	24	27	solved	solve	VERB
cana-2702	24	28	.	.	PUNCT
cana-2702	25	1	the	the	DET
cana-2702	25	2	𝑡	𝑡	PROPN
cana-2702	25	3	−pebbling	−pebble	VERB
cana-2702	25	4	number	number	NOUN
cana-2702	25	5	,	,	PUNCT
cana-2702	25	6	𝑓𝑡(𝐺	𝑓𝑡(𝐺	PROPN
cana-2702	25	7	)	)	PUNCT
cana-2702	25	8	,	,	PUNCT
cana-2702	25	9	of	of	ADP
cana-2702	25	10	a	a	DET
cana-2702	25	11	connected	connected	ADJ
cana-2702	25	12	graph	graph	NOUN
cana-2702	25	13	𝐺	𝐺	PROPN
cana-2702	25	14	,	,	PUNCT
cana-2702	25	15	represents	represent	VERB
cana-2702	25	16	the	the	DET
cana-2702	25	17	minimum	minimum	ADJ
cana-2702	25	18	positive	positive	ADJ
cana-2702	25	19	integer	integer	NOUN
cana-2702	25	20	such	such	ADJ
cana-2702	25	21	that	that	SCONJ
cana-2702	25	22	given	give	VERB
cana-2702	25	23	distribution	distribution	NOUN
cana-2702	25	24	of	of	ADP
cana-2702	25	25	𝑓𝑡(𝐺	𝑓𝑡(𝐺	PROPN
cana-2702	25	26	)	)	PUNCT
cana-2702	25	27	pebbles	pebble	NOUN
cana-2702	25	28	,	,	PUNCT
cana-2702	25	29	it	it	PRON
cana-2702	25	30	is	be	AUX
cana-2702	25	31	possible	possible	ADJ
cana-2702	25	32	to	to	PART
cana-2702	25	33	move	move	VERB
cana-2702	25	34	𝑡	𝑡	PROPN
cana-2702	25	35	pebbles	pebble	NOUN
cana-2702	25	36	to	to	ADP
cana-2702	25	37	the	the	DET
cana-2702	25	38	chosen	choose	VERB
cana-2702	25	39	target	target	NOUN
cana-2702	25	40	vertex	vertex	NOUN
cana-2702	25	41	by	by	ADP
cana-2702	25	42	performing	perform	VERB
cana-2702	25	43	series	series	NOUN
cana-2702	25	44	of	of	ADP
cana-2702	25	45	pebbling	pebble	VERB
cana-2702	25	46	moves	move	NOUN
cana-2702	25	47	.	.	PUNCT
cana-2702	26	1	the	the	DET
cana-2702	26	2	connected	connected	ADJ
cana-2702	26	3	graph	graph	NOUN
cana-2702	26	4	𝐺	𝐺	PROPN
cana-2702	26	5	said	say	VERB
cana-2702	26	6	to	to	PART
cana-2702	26	7	possess	possess	VERB
cana-2702	26	8	the	the	DET
cana-2702	26	9	2	2	NUM
cana-2702	26	10	-	-	PUNCT
cana-2702	26	11	pebbling	pebble	VERB
cana-2702	26	12	property	property	NOUN
cana-2702	26	13	if	if	SCONJ
cana-2702	26	14	for	for	ADP
cana-2702	26	15	any	any	DET
cana-2702	26	16	distribution	distribution	NOUN
cana-2702	26	17	of	of	ADP
cana-2702	26	18	pebbles	pebble	NOUN
cana-2702	26	19	in	in	ADP
cana-2702	26	20	𝐺	𝐺	PROPN
cana-2702	26	21	where	where	SCONJ
cana-2702	26	22	the	the	DET
cana-2702	26	23	number	number	NOUN
cana-2702	26	24	of	of	ADP
cana-2702	26	25	pebbles	pebble	NOUN
cana-2702	26	26	exceeds	exceed	VERB
cana-2702	26	27	2𝑓(𝐺	2𝑓(𝐺	NUM
cana-2702	26	28	)	)	PUNCT
cana-2702	26	29	−	−	PROPN
cana-2702	27	1	𝑞	𝑞	PROPN
cana-2702	27	2	,	,	PUNCT
cana-2702	27	3	where	where	SCONJ
cana-2702	27	4	𝑞	𝑞	PROPN
cana-2702	27	5	,	,	PUNCT
cana-2702	27	6	the	the	DET
cana-2702	27	7	number	number	NOUN
cana-2702	27	8	of	of	ADP
cana-2702	27	9	vertices	vertex	NOUN
cana-2702	27	10	with	with	ADP
cana-2702	27	11	at	at	ADV
cana-2702	27	12	least	least	ADV
cana-2702	27	13	one	one	NUM
cana-2702	27	14	pebble	pebble	ADJ
cana-2702	27	15	,	,	PUNCT
cana-2702	27	16	it	it	PRON
cana-2702	27	17	is	be	AUX
cana-2702	27	18	feasible	feasible	ADJ
cana-2702	27	19	,	,	PUNCT
cana-2702	27	20	through	through	ADP
cana-2702	27	21	the	the	DET
cana-2702	27	22	execution	execution	NOUN
cana-2702	27	23	of	of	ADP
cana-2702	27	24	mailto:gayathrilakshmi1804@gmail.com	mailto:gayathrilakshmi1804@gmail.com	NOUN
cana-2702	27	25	mailto:vamsipriyabagi@gmail.com	mailto:vamsipriyabagi@gmail.com	NOUN
cana-2702	27	26	mailto:id%3abhuvanabalaji13@gmail.com	mailto:id%3abhuvanabalaji13@gmail.com	X
cana-2702	27	27	mailto:email%3akannan8383@gmail.com	mailto:email%3akannan8383@gmail.com	X
cana-2702	28	1	mailto:jesujuli@gmail.com	mailto:jesujuli@gmail.com	PROPN
cana-2702	28	2	mailto:muniyandy.e@gmail.com	mailto:muniyandy.e@gmail.com	X
cana-2702	29	1	communications	communication	NOUN
cana-2702	29	2	on	on	ADP
cana-2702	29	3	applied	apply	VERB
cana-2702	29	4	nonlinear	nonlinear	ADJ
cana-2702	29	5	analysis	analysis	NOUN
cana-2702	29	6	issn	issn	NOUN
cana-2702	29	7	:	:	PUNCT
cana-2702	29	8	1074	1074	NUM
cana-2702	29	9	-	-	PUNCT
cana-2702	29	10	133x	133x	NUM
cana-2702	29	11	vol	vol	NOUN
cana-2702	29	12	32	32	NUM
cana-2702	29	13	no	no	NOUN
cana-2702	29	14	.	.	PUNCT
cana-2702	30	1	3s	3s	NUM
cana-2702	30	2	(	(	PUNCT
cana-2702	30	3	2025	2025	NUM
cana-2702	30	4	)	)	PUNCT
cana-2702	30	5	649	649	NUM
cana-2702	30	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	30	7	pebbling	pebble	VERB
cana-2702	30	8	moves	move	NOUN
cana-2702	30	9	,	,	PUNCT
cana-2702	30	10	to	to	PART
cana-2702	30	11	ensure	ensure	VERB
cana-2702	30	12	the	the	DET
cana-2702	30	13	presence	presence	NOUN
cana-2702	30	14	of	of	ADP
cana-2702	30	15	two	two	NUM
cana-2702	30	16	pebbles	pebble	NOUN
cana-2702	30	17	at	at	ADP
cana-2702	30	18	the	the	DET
cana-2702	30	19	given	give	VERB
cana-2702	30	20	vertex	vertex	NOUN
cana-2702	30	21	within	within	ADP
cana-2702	30	22	𝐺.	𝐺.	PROPN
cana-2702	30	23	[	[	X
cana-2702	30	24	15]a	15]a	NUM
cana-2702	30	25	graph	graph	NOUN
cana-2702	30	26	𝐺	𝐺	PROPN
cana-2702	30	27	considered	consider	VERB
cana-2702	30	28	to	to	PART
cana-2702	30	29	possess	possess	VERB
cana-2702	30	30	the	the	DET
cana-2702	30	31	2𝑡	2𝑡	NOUN
cana-2702	30	32	−pebbling	−pebble	VERB
cana-2702	30	33	property	property	NOUN
cana-2702	30	34	if	if	SCONJ
cana-2702	30	35	for	for	ADP
cana-2702	30	36	any	any	DET
cana-2702	30	37	distribution	distribution	NOUN
cana-2702	30	38	of	of	ADP
cana-2702	30	39	pebbles	pebble	NOUN
cana-2702	30	40	in	in	ADP
cana-2702	30	41	𝐺	𝐺	PROPN
cana-2702	30	42	where	where	SCONJ
cana-2702	30	43	the	the	DET
cana-2702	30	44	number	number	NOUN
cana-2702	30	45	of	of	ADP
cana-2702	30	46	pebbles	pebble	NOUN
cana-2702	30	47	exceeds	exceed	VERB
cana-2702	30	48	2𝑓𝑡(𝐺	2𝑓𝑡(𝐺	NUM
cana-2702	30	49	)	)	PUNCT
cana-2702	30	50	−	−	PROPN
cana-2702	31	1	𝑞	𝑞	PROPN
cana-2702	31	2	,	,	PUNCT
cana-2702	31	3	through	through	ADP
cana-2702	31	4	the	the	DET
cana-2702	31	5	execution	execution	NOUN
cana-2702	31	6	of	of	ADP
cana-2702	31	7	pebbling	pebble	VERB
cana-2702	31	8	moves	move	NOUN
cana-2702	31	9	,	,	PUNCT
cana-2702	31	10	it	it	PRON
cana-2702	31	11	is	be	AUX
cana-2702	31	12	possible	possible	ADJ
cana-2702	31	13	to	to	PART
cana-2702	31	14	move	move	VERB
cana-2702	31	15	2𝑡	2𝑡	NUM
cana-2702	31	16	pebbles	pebble	NOUN
cana-2702	31	17	to	to	ADP
cana-2702	31	18	any	any	DET
cana-2702	31	19	arbitrary	arbitrary	ADJ
cana-2702	31	20	vertex	vertex	NOUN
cana-2702	31	21	in	in	ADP
cana-2702	31	22	𝐺.	𝐺.	NOUN
cana-2702	31	23	in	in	ADP
cana-2702	31	24	section	section	NOUN
cana-2702	31	25	2	2	NUM
cana-2702	31	26	,	,	PUNCT
cana-2702	31	27	we	we	PRON
cana-2702	31	28	calculate	calculate	VERB
cana-2702	31	29	the	the	DET
cana-2702	31	30	pebbling	pebble	VERB
cana-2702	31	31	number	number	NOUN
cana-2702	31	32	and	and	CCONJ
cana-2702	31	33	𝑡	𝑡	NOUN
cana-2702	31	34	−pebbling	−pebble	VERB
cana-2702	31	35	number	number	NOUN
cana-2702	31	36	of	of	ADP
cana-2702	31	37	crisscross	crisscross	ADJ
cana-2702	31	38	sequence	sequence	NOUN
cana-2702	31	39	of	of	ADP
cana-2702	31	40	𝑚	𝑚	ADP
cana-2702	31	41	complete	complete	ADJ
cana-2702	31	42	graph	graph	NOUN
cana-2702	31	43	and	and	CCONJ
cana-2702	31	44	in	in	ADP
cana-2702	31	45	section	section	NOUN
cana-2702	31	46	3	3	NUM
cana-2702	31	47	and	and	CCONJ
cana-2702	31	48	section	section	NOUN
cana-2702	31	49	4	4	NUM
cana-2702	31	50	,	,	PUNCT
cana-2702	31	51	we	we	PRON
cana-2702	31	52	demonstrate	demonstrate	VERB
cana-2702	31	53	the	the	DET
cana-2702	31	54	graph	graph	NOUN
cana-2702	31	55	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	31	56	)	)	PUNCT
cana-2702	31	57	exhibits	exhibit	VERB
cana-2702	31	58	both	both	DET
cana-2702	31	59	2	2	NUM
cana-2702	31	60	-	-	PUNCT
cana-2702	31	61	pebbling	pebble	VERB
cana-2702	31	62	property	property	NOUN
cana-2702	31	63	and	and	CCONJ
cana-2702	31	64	2𝑡	2𝑡	NOUN
cana-2702	31	65	−pebbling	−pebble	VERB
cana-2702	31	66	property[12	property[12	NOUN
cana-2702	31	67	]	]	PUNCT
cana-2702	31	68	.	.	PUNCT
cana-2702	32	1	pebbling	pebble	VERB
cana-2702	32	2	on	on	ADP
cana-2702	32	3	crisscross	crisscross	ADJ
cana-2702	32	4	sequence	sequence	NOUN
cana-2702	32	5	of	of	ADP
cana-2702	32	6	𝒎	𝒎	PROPN
cana-2702	32	7	complete	complete	ADJ
cana-2702	32	8	graphs	graph	NOUN
cana-2702	32	9	in	in	ADP
cana-2702	32	10	this	this	DET
cana-2702	32	11	section	section	NOUN
cana-2702	32	12	,	,	PUNCT
cana-2702	32	13	we	we	PRON
cana-2702	32	14	aim	aim	VERB
cana-2702	32	15	to	to	PART
cana-2702	32	16	determine	determine	VERB
cana-2702	32	17	the	the	DET
cana-2702	32	18	pebbling	pebble	VERB
cana-2702	32	19	number	number	NOUN
cana-2702	32	20	and	and	CCONJ
cana-2702	32	21	the	the	DET
cana-2702	32	22	𝑡	𝑡	PROPN
cana-2702	32	23	−pebbling	−pebble	VERB
cana-2702	32	24	number	number	NOUN
cana-2702	32	25	of	of	ADP
cana-2702	32	26	crisscross	crisscross	ADJ
cana-2702	32	27	chain	chain	NOUN
cana-2702	32	28	graph	graph	NOUN
cana-2702	32	29	consisting	consist	VERB
cana-2702	32	30	𝑚	𝑚	ADP
cana-2702	32	31	complete	complete	ADJ
cana-2702	32	32	graphs	graph	NOUN
cana-2702	32	33	.	.	PUNCT
cana-2702	33	1	however	however	ADV
cana-2702	33	2	,	,	PUNCT
cana-2702	33	3	before	before	ADP
cana-2702	33	4	getting	get	VERB
cana-2702	33	5	into	into	ADP
cana-2702	33	6	the	the	DET
cana-2702	33	7	calculations	calculation	NOUN
cana-2702	33	8	,	,	PUNCT
cana-2702	33	9	it	it	PRON
cana-2702	33	10	is	be	AUX
cana-2702	33	11	crucial	crucial	ADJ
cana-2702	33	12	to	to	PART
cana-2702	33	13	comprehend	comprehend	VERB
cana-2702	33	14	the	the	DET
cana-2702	33	15	structure	structure	NOUN
cana-2702	33	16	of	of	ADP
cana-2702	33	17	crisscross	crisscross	ADJ
cana-2702	33	18	chain	chain	NOUN
cana-2702	33	19	graph	graph	NOUN
cana-2702	33	20	of	of	ADP
cana-2702	33	21	𝑚	𝑚	ADP
cana-2702	33	22	complete	complete	ADJ
cana-2702	33	23	graph	graph	NOUN
cana-2702	33	24	.	.	PUNCT
cana-2702	34	1	to	to	PART
cana-2702	34	2	aid	aid	VERB
cana-2702	34	3	in	in	ADP
cana-2702	34	4	this	this	DET
cana-2702	34	5	understanding	understanding	NOUN
cana-2702	34	6	,	,	PUNCT
cana-2702	34	7	we	we	PRON
cana-2702	34	8	introduce	introduce	VERB
cana-2702	34	9	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	34	10	)	)	PUNCT
cana-2702	34	11	with	with	ADP
cana-2702	34	12	the	the	DET
cana-2702	34	13	help	help	NOUN
cana-2702	34	14	of	of	ADP
cana-2702	34	15	zig	zig	VERB
cana-2702	34	16	-	-	PUNCT
cana-2702	34	17	zag	zag	NOUN
cana-2702	34	18	sequence	sequence	NOUN
cana-2702	34	19	of	of	ADP
cana-2702	34	20	𝑛	𝑛	DET
cana-2702	34	21	cycles	cycle	NOUN
cana-2702	34	22	which	which	PRON
cana-2702	34	23	has	have	AUX
cana-2702	34	24	been	be	AUX
cana-2702	34	25	already	already	ADV
cana-2702	34	26	studied	study	VERB
cana-2702	34	27	in	in	ADP
cana-2702	34	28	[	[	X
cana-2702	34	29	8],[9],[10	8],[9],[10	NUM
cana-2702	34	30	]	]	PUNCT
cana-2702	34	31	.	.	PUNCT
cana-2702	35	1	definition	definition	NOUN
cana-2702	35	2	1	1	NUM
cana-2702	35	3	.	.	PUNCT
cana-2702	36	1	[	[	X
cana-2702	36	2	14][15	14][15	X
cana-2702	36	3	]	]	X
cana-2702	36	4	the	the	DET
cana-2702	36	5	zig	zig	NOUN
cana-2702	36	6	-	-	PUNCT
cana-2702	36	7	zag	zag	NOUN
cana-2702	36	8	chain	chain	NOUN
cana-2702	36	9	graph	graph	NOUN
cana-2702	36	10	of	of	ADP
cana-2702	36	11	even	even	ADV
cana-2702	36	12	cycles	cycle	NOUN
cana-2702	36	13	denoted	denote	VERB
cana-2702	36	14	by	by	ADP
cana-2702	36	15	𝑍𝑍𝑛(𝐶2𝑘	𝑍𝑍𝑛(𝐶2𝑘	NOUN
cana-2702	36	16	)	)	PUNCT
cana-2702	36	17	,	,	PUNCT
cana-2702	36	18	is	be	AUX
cana-2702	36	19	a	a	DET
cana-2702	36	20	graph	graph	NOUN
cana-2702	36	21	which	which	PRON
cana-2702	36	22	consists	consist	VERB
cana-2702	36	23	of	of	ADP
cana-2702	36	24	zig	zig	NOUN
cana-2702	36	25	-	-	PUNCT
cana-2702	36	26	zag	zag	NOUN
cana-2702	36	27	sequence	sequence	NOUN
cana-2702	36	28	of	of	ADP
cana-2702	36	29	𝑛	𝑛	DET
cana-2702	36	30	even	even	ADV
cana-2702	36	31	cycles	cycle	NOUN
cana-2702	36	32	,	,	PUNCT
cana-2702	36	33	𝐶2𝑘	𝐶2𝑘	PROPN
cana-2702	36	34	with	with	ADP
cana-2702	36	35	𝑘	𝑘	PRON
cana-2702	36	36	≥	≥	NOUN
cana-2702	36	37	3	3	NUM
cana-2702	36	38	.	.	PUNCT
cana-2702	37	1	we	we	PRON
cana-2702	37	2	have	have	VERB
cana-2702	37	3	the	the	DET
cana-2702	37	4	following	follow	VERB
cana-2702	37	5	vertex	vertex	NOUN
cana-2702	37	6	set	set	NOUN
cana-2702	37	7	and	and	CCONJ
cana-2702	37	8	edge	edge	NOUN
cana-2702	37	9	set	set	NOUN
cana-2702	37	10	of	of	ADP
cana-2702	37	11	𝑍𝑍𝑛(𝐶2𝑘	𝑍𝑍𝑛(𝐶2𝑘	NOUN
cana-2702	37	12	)	)	PUNCT
cana-2702	37	13	for	for	ADP
cana-2702	37	14	𝑛	𝑛	PRON
cana-2702	37	15	even	even	ADV
cana-2702	37	16	as	as	SCONJ
cana-2702	37	17	follows	follow	VERB
cana-2702	37	18	.	.	PUNCT
cana-2702	38	1	𝑉(𝑍𝑍𝑛(𝐶2𝑘	𝑉(𝑍𝑍𝑛(𝐶2𝑘	VERB
cana-2702	38	2	)	)	PUNCT
cana-2702	38	3	)	)	PUNCT
cana-2702	39	1	=	=	PRON
cana-2702	39	2	{	{	PUNCT
cana-2702	39	3	𝑎𝑖	𝑎𝑖	PROPN
cana-2702	39	4	,	,	PUNCT
cana-2702	39	5	𝑏𝑖	𝑏𝑖	ADP
cana-2702	39	6	:	:	PUNCT
cana-2702	39	7	1	1	NUM
cana-2702	39	8	≤	≤	NUM
cana-2702	39	9	𝑖	𝑖	PUNCT
cana-2702	39	10	≤	≤	ADJ
cana-2702	39	11	𝑛(𝑘	𝑛(𝑘	NOUN
cana-2702	39	12	−	−	PROPN
cana-2702	39	13	1	1	NUM
cana-2702	39	14	)	)	PUNCT
cana-2702	39	15	}	}	PUNCT
cana-2702	39	16	∪	∪	X
cana-2702	39	17	{	{	PUNCT
cana-2702	39	18	𝑥	𝑥	NOUN
cana-2702	39	19	,	,	PUNCT
cana-2702	39	20	𝑦	𝑦	NOUN
cana-2702	39	21	}	}	PUNCT
cana-2702	39	22	and	and	CCONJ
cana-2702	39	23	𝐸(𝑍𝑍𝑛(𝐶2𝑘	𝐸(𝑍𝑍𝑛(𝐶2𝑘	NOUN
cana-2702	39	24	)	)	PUNCT
cana-2702	39	25	)	)	PUNCT
cana-2702	40	1	=	=	PRON
cana-2702	40	2	{	{	PUNCT
cana-2702	40	3	𝑎𝑖𝑎𝑖+1	𝑎𝑖𝑎𝑖+1	PROPN
cana-2702	40	4	,	,	PUNCT
cana-2702	40	5	𝑏𝑖𝑏𝑖+1:1	𝑏𝑖𝑏𝑖+1:1	NOUN
cana-2702	40	6	≤≤	≤≤	CCONJ
cana-2702	41	1	𝑛(𝑘	𝑛(𝑘	NOUN
cana-2702	41	2	−	−	NUM
cana-2702	41	3	1	1	NUM
cana-2702	41	4	)	)	PUNCT
cana-2702	41	5	−	−	ADP
cana-2702	41	6	1	1	NUM
cana-2702	41	7	}	}	PUNCT
cana-2702	41	8	∪	∪	X
cana-2702	41	9	{	{	PUNCT
cana-2702	41	10	𝑎(𝑘+1)𝑖−1𝑏(𝑘+1)𝑖−2	𝑎(𝑘+1)𝑖−1𝑏(𝑘+1)𝑖−2	PROPN
cana-2702	41	11	,	,	PUNCT
cana-2702	41	12	𝑎(𝑘+1)𝑗𝑏(𝑘+1)𝑗+1:1	𝑎(𝑘+1)𝑗𝑏(𝑘+1)𝑗+1:1	PROPN
cana-2702	41	13	≤	≤	PROPN
cana-2702	41	14	𝑖	𝑖	SYM
cana-2702	41	15	≤	≤	NUM
cana-2702	41	16	𝑛	𝑛	DET
cana-2702	41	17	2	2	NUM
cana-2702	41	18	,	,	PUNCT
cana-2702	41	19	1	1	NUM
cana-2702	41	20	≤	≤	NUM
cana-2702	41	21	𝑗	𝑗	PRON
cana-2702	41	22	≤	≤	ADJ
cana-2702	41	23	(	(	PUNCT
cana-2702	41	24	𝑛	𝑛	PRON
cana-2702	41	25	2	2	NUM
cana-2702	41	26	−	−	NUM
cana-2702	41	27	1	1	NUM
cana-2702	41	28	)	)	PUNCT
cana-2702	41	29	}	}	PUNCT
cana-2702	41	30	∪	∪	ADJ
cana-2702	41	31	{	{	PUNCT
cana-2702	41	32	𝑥𝑎1	𝑥𝑎1	NOUN
cana-2702	41	33	,	,	PUNCT
cana-2702	41	34	𝑥𝑏1	𝑥𝑏1	NOUN
cana-2702	41	35	,	,	PUNCT
cana-2702	41	36	𝑦𝑎𝑛(𝑘−1	𝑦𝑎𝑛(𝑘−1	PROPN
cana-2702	41	37	)	)	PUNCT
cana-2702	41	38	,	,	PUNCT
cana-2702	41	39	𝑦𝑏𝑛(𝑘−1	𝑦𝑏𝑛(𝑘−1	PROPN
cana-2702	41	40	)	)	PUNCT
cana-2702	41	41	}	}	PUNCT
cana-2702	41	42	for	for	ADP
cana-2702	41	43	𝑛	𝑛	PRON
cana-2702	41	44	odd	odd	ADJ
cana-2702	41	45	,	,	PUNCT
cana-2702	41	46	we	we	PRON
cana-2702	41	47	have	have	VERB
cana-2702	41	48	the	the	DET
cana-2702	41	49	following	follow	VERB
cana-2702	41	50	vertex	vertex	NOUN
cana-2702	41	51	set	set	NOUN
cana-2702	41	52	and	and	CCONJ
cana-2702	41	53	edge	edge	NOUN
cana-2702	41	54	set	set	NOUN
cana-2702	41	55	.	.	PUNCT
cana-2702	42	1	𝑉(𝑍𝑍𝑛(𝐶2𝑘	𝑉(𝑍𝑍𝑛(𝐶2𝑘	VERB
cana-2702	42	2	)	)	PUNCT
cana-2702	42	3	)	)	PUNCT
cana-2702	43	1	=	=	PRON
cana-2702	43	2	{	{	PUNCT
cana-2702	43	3	𝑎𝑖	𝑎𝑖	PROPN
cana-2702	43	4	,	,	PUNCT
cana-2702	43	5	𝑏𝑖	𝑏𝑖	ADP
cana-2702	43	6	:	:	PUNCT
cana-2702	43	7	1	1	NUM
cana-2702	43	8	≤	≤	NUM
cana-2702	43	9	𝑖	𝑖	PUNCT
cana-2702	43	10	≤	≤	ADJ
cana-2702	43	11	𝑛(𝑘	𝑛(𝑘	NOUN
cana-2702	43	12	−	−	PROPN
cana-2702	43	13	1	1	NUM
cana-2702	43	14	)	)	PUNCT
cana-2702	43	15	}	}	PUNCT
cana-2702	43	16	∪	∪	X
cana-2702	43	17	{	{	PUNCT
cana-2702	43	18	𝑥	𝑥	NOUN
cana-2702	43	19	,	,	PUNCT
cana-2702	43	20	𝑦	𝑦	NOUN
cana-2702	43	21	}	}	PUNCT
cana-2702	43	22	and	and	CCONJ
cana-2702	43	23	𝐸(𝑍𝑍𝑛(𝐶2𝑘	𝐸(𝑍𝑍𝑛(𝐶2𝑘	NOUN
cana-2702	43	24	)	)	PUNCT
cana-2702	43	25	)	)	PUNCT
cana-2702	44	1	=	=	PRON
cana-2702	44	2	{	{	PUNCT
cana-2702	44	3	𝑎𝑖𝑎𝑖+1	𝑎𝑖𝑎𝑖+1	PROPN
cana-2702	44	4	,	,	PUNCT
cana-2702	44	5	𝑏𝑖𝑏𝑖+1	𝑏𝑖𝑏𝑖+1	NOUN
cana-2702	44	6	:	:	PUNCT
cana-2702	44	7	1	1	NUM
cana-2702	44	8	≤	≤	NUM
cana-2702	44	9	𝑖	𝑖	PUNCT
cana-2702	44	10	≤	≤	ADJ
cana-2702	44	11	𝑛(𝑘	𝑛(𝑘	NOUN
cana-2702	44	12	−	−	PROPN
cana-2702	44	13	1	1	NUM
cana-2702	44	14	)	)	PUNCT
cana-2702	44	15	−	−	ADP
cana-2702	44	16	1	1	NUM
cana-2702	44	17	}	}	PUNCT
cana-2702	44	18	∪	∪	ADJ
cana-2702	44	19	{	{	PUNCT
cana-2702	44	20	𝑥𝑎1	𝑥𝑎1	NOUN
cana-2702	44	21	,	,	PUNCT
cana-2702	44	22	𝑥𝑏1	𝑥𝑏1	NOUN
cana-2702	44	23	,	,	PUNCT
cana-2702	44	24	𝑦𝑎𝑛(𝑘−1	𝑦𝑎𝑛(𝑘−1	PROPN
cana-2702	44	25	)	)	PUNCT
cana-2702	44	26	,	,	PUNCT
cana-2702	44	27	𝑦𝑏𝑛(𝑘−1	𝑦𝑏𝑛(𝑘−1	PROPN
cana-2702	44	28	)	)	PUNCT
cana-2702	44	29	}	}	PUNCT
cana-2702	44	30	∪	∪	X
cana-2702	44	31	{	{	PUNCT
cana-2702	44	32	𝑎(𝑘+1)𝑖−1𝑏(𝑘+1)𝑖−2	𝑎(𝑘+1)𝑖−1𝑏(𝑘+1)𝑖−2	PROPN
cana-2702	44	33	,	,	PUNCT
cana-2702	44	34	𝑎(𝑘+1)𝑗𝑏(𝑘+1)𝑗+1	𝑎(𝑘+1)𝑗𝑏(𝑘+1)𝑗+1	PROPN
cana-2702	44	35	:	:	PUNCT
cana-2702	44	36	1	1	NUM
cana-2702	44	37	≤	≤	NUM
cana-2702	44	38	𝑖	𝑖	PUNCT
cana-2702	44	39	,	,	PUNCT
cana-2702	44	40	𝑗	𝑗	PROPN
cana-2702	44	41	≤	≤	NUM
cana-2702	44	42	𝑛−1	𝑛−1	NUM
cana-2702	44	43	2	2	NUM
cana-2702	44	44	}	}	PUNCT
cana-2702	44	45	.	.	PUNCT
cana-2702	45	1	the	the	DET
cana-2702	45	2	figure	figure	NOUN
cana-2702	45	3	2.1(𝑎	2.1(𝑎	NUM
cana-2702	45	4	)	)	PUNCT
cana-2702	45	5	depicts	depict	VERB
cana-2702	45	6	the	the	DET
cana-2702	45	7	graph	graph	NOUN
cana-2702	45	8	𝑍𝑍3	𝑍𝑍3	PROPN
cana-2702	45	9	for	for	ADP
cana-2702	45	10	𝑘	𝑘	NOUN
cana-2702	45	11	=	=	SYM
cana-2702	45	12	3	3	X
cana-2702	45	13	.	.	X
cana-2702	45	14	figure	figure	NOUN
cana-2702	45	15	2.1(a	2.1(a	NUM
cana-2702	45	16	)	)	PUNCT
cana-2702	45	17	similarly	similarly	ADV
cana-2702	45	18	,	,	PUNCT
cana-2702	45	19	for	for	SCONJ
cana-2702	45	20	the	the	DET
cana-2702	45	21	graph	graph	NOUN
cana-2702	45	22	zig	zig	VERB
cana-2702	45	23	-	-	PUNCT
cana-2702	45	24	zag	zag	NOUN
cana-2702	45	25	chain	chain	NOUN
cana-2702	45	26	graph	graph	NOUN
cana-2702	45	27	of	of	ADP
cana-2702	45	28	odd	odd	ADJ
cana-2702	45	29	cycles	cycle	NOUN
cana-2702	45	30	,	,	PUNCT
cana-2702	45	31	denoted	denote	VERB
cana-2702	45	32	by	by	ADP
cana-2702	45	33	𝑍𝑍𝑛(𝐶2𝑘+1	𝑍𝑍𝑛(𝐶2𝑘+1	NOUN
cana-2702	45	34	)	)	PUNCT
cana-2702	45	35	has	have	AUX
cana-2702	45	36	defined	define	VERB
cana-2702	45	37	in[4	in[4	NOUN
cana-2702	45	38	]	]	PUNCT
cana-2702	45	39	.	.	PUNCT
cana-2702	46	1	definition	definition	NOUN
cana-2702	46	2	2.[5],[7],[9],[10	2.[5],[7],[9],[10	NUM
cana-2702	46	3	]	]	PUNCT
cana-2702	46	4	.	.	PUNCT
cana-2702	47	1	the	the	DET
cana-2702	47	2	crisscross	crisscross	ADJ
cana-2702	47	3	sequence	sequence	NOUN
cana-2702	47	4	of	of	ADP
cana-2702	47	5	𝑚	𝑚	ADP
cana-2702	47	6	complete	complete	ADJ
cana-2702	47	7	graphs	graph	NOUN
cana-2702	47	8	is	be	AUX
cana-2702	47	9	denoted	denote	VERB
cana-2702	47	10	by	by	ADP
cana-2702	47	11	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	47	12	)	)	PUNCT
cana-2702	47	13	,	,	PUNCT
cana-2702	47	14	is	be	AUX
cana-2702	47	15	a	a	DET
cana-2702	47	16	graph	graph	NOUN
cana-2702	47	17	which	which	PRON
cana-2702	47	18	consists	consist	VERB
cana-2702	47	19	of	of	ADP
cana-2702	47	20	crisscross	crisscross	ADJ
cana-2702	47	21	sequence	sequence	NOUN
cana-2702	47	22	of	of	ADP
cana-2702	47	23	𝑚	𝑚	PROPN
cana-2702	47	24	copies	copy	NOUN
cana-2702	47	25	of	of	ADP
cana-2702	47	26	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	47	27	,	,	PUNCT
cana-2702	47	28	with	with	ADP
cana-2702	47	29	𝑛	𝑛	DET
cana-2702	47	30	≥	≥	NUM
cana-2702	47	31	5	5	NUM
cana-2702	47	32	.	.	PUNCT
cana-2702	48	1	we	we	PRON
cana-2702	48	2	define	define	VERB
cana-2702	48	3	the	the	DET
cana-2702	48	4	graph	graph	NOUN
cana-2702	48	5	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	48	6	)	)	PUNCT
cana-2702	48	7	as	as	SCONJ
cana-2702	48	8	follows	follow	VERB
cana-2702	48	9	:	:	PUNCT
cana-2702	48	10	the	the	DET
cana-2702	48	11	vertex	vertex	NOUN
cana-2702	48	12	set	set	NOUN
cana-2702	48	13	of	of	ADP
cana-2702	48	14	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	48	15	)	)	PUNCT
cana-2702	48	16	is	be	AUX
cana-2702	48	17	𝑉(𝐶𝑚(𝐾𝑛	𝑉(𝐶𝑚(𝐾𝑛	NOUN
cana-2702	48	18	)	)	PUNCT
cana-2702	48	19	)	)	PUNCT
cana-2702	49	1	=	=	SYM
cana-2702	49	2	𝑉(𝑍𝑍𝑚(𝐶𝑛	𝑉(𝑍𝑍𝑚(𝐶𝑛	PROPN
cana-2702	49	3	)	)	PUNCT
cana-2702	49	4	)	)	PUNCT
cana-2702	49	5	and	and	CCONJ
cana-2702	49	6	the	the	DET
cana-2702	49	7	edge	edge	NOUN
cana-2702	49	8	set	set	NOUN
cana-2702	49	9	of	of	ADP
cana-2702	49	10	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	49	11	)	)	PUNCT
cana-2702	49	12	is	be	AUX
cana-2702	49	13	the	the	DET
cana-2702	49	14	union	union	NOUN
cana-2702	49	15	of	of	ADP
cana-2702	49	16	𝐸(𝑍𝑍𝑚(𝐶𝑛	𝐸(𝑍𝑍𝑚(𝐶𝑛	PROPN
cana-2702	49	17	)	)	PUNCT
cana-2702	49	18	)	)	PUNCT
cana-2702	49	19	and	and	CCONJ
cana-2702	49	20	each	each	DET
cana-2702	49	21	vertex	vertex	NOUN
cana-2702	49	22	𝑣	𝑣	ADP
cana-2702	49	23	∈	∈	PROPN
cana-2702	49	24	𝐶𝑖	𝐶𝑖	PROPN
cana-2702	49	25	is	be	AUX
cana-2702	49	26	adjacent	adjacent	ADJ
cana-2702	49	27	to	to	ADP
cana-2702	49	28	all	all	DET
cana-2702	49	29	the	the	DET
cana-2702	49	30	vertices	vertex	NOUN
cana-2702	49	31	of	of	ADP
cana-2702	49	32	𝐶𝑖.	𝐶𝑖.	PROPN
cana-2702	49	33	here	here	ADV
cana-2702	49	34	𝐶1	𝐶1	PROPN
cana-2702	49	35	,	,	PUNCT
cana-2702	49	36	𝐶2	𝐶2	ADJ
cana-2702	49	37	,	,	PUNCT
cana-2702	49	38	.	.	PUNCT
cana-2702	49	39	.	.	PUNCT
cana-2702	50	1	.	.	PUNCT
cana-2702	51	1	,	,	PUNCT
cana-2702	51	2	𝐶𝑚	𝐶𝑚	NOUN
cana-2702	51	3	is	be	AUX
cana-2702	51	4	considered	consider	VERB
cana-2702	51	5	as	as	ADP
cana-2702	51	6	𝑚	𝑚	ADP
cana-2702	51	7	number	number	NOUN
cana-2702	51	8	of	of	ADP
cana-2702	51	9	cells	cell	NOUN
cana-2702	51	10	of	of	ADP
cana-2702	51	11	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	51	12	)	)	PUNCT
cana-2702	51	13	.	.	PUNCT
cana-2702	52	1	the	the	DET
cana-2702	52	2	figure	figure	NOUN
cana-2702	52	3	depicts	depict	VERB
cana-2702	52	4	the	the	DET
cana-2702	52	5	the	the	DET
cana-2702	52	6	crisscross	crisscross	ADJ
cana-2702	52	7	sequence	sequence	NOUN
cana-2702	52	8	of	of	ADP
cana-2702	52	9	𝑚	𝑚	PROPN
cana-2702	52	10	copies	copy	NOUN
cana-2702	52	11	of	of	ADP
cana-2702	52	12	𝐾6	𝐾6	PROPN
cana-2702	52	13	.	.	PUNCT
cana-2702	53	1	communications	communication	NOUN
cana-2702	53	2	on	on	ADP
cana-2702	53	3	applied	apply	VERB
cana-2702	53	4	nonlinear	nonlinear	ADJ
cana-2702	53	5	analysis	analysis	NOUN
cana-2702	53	6	issn	issn	NOUN
cana-2702	53	7	:	:	PUNCT
cana-2702	53	8	1074	1074	NUM
cana-2702	53	9	-	-	PUNCT
cana-2702	53	10	133x	133x	NUM
cana-2702	53	11	vol	vol	NOUN
cana-2702	53	12	32	32	NUM
cana-2702	53	13	no	no	NOUN
cana-2702	53	14	.	.	PUNCT
cana-2702	54	1	3s	3s	NUM
cana-2702	54	2	(	(	PUNCT
cana-2702	54	3	2025	2025	NUM
cana-2702	54	4	)	)	PUNCT
cana-2702	54	5	650	650	NUM
cana-2702	54	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	54	7	figure	figure	NOUN
cana-2702	54	8	2.2(b	2.2(b	NUM
cana-2702	54	9	)	)	PUNCT
cana-2702	54	10	the	the	DET
cana-2702	54	11	reader	reader	NOUN
cana-2702	54	12	can	can	AUX
cana-2702	54	13	easily	easily	ADV
cana-2702	54	14	view	view	VERB
cana-2702	54	15	that	that	SCONJ
cana-2702	54	16	the	the	DET
cana-2702	54	17	figure	figure	NOUN
cana-2702	54	18	2.2(b	2.2(b	NUM
cana-2702	54	19	)	)	PUNCT
cana-2702	54	20	has	have	VERB
cana-2702	54	21	4	4	NUM
cana-2702	54	22	copies	copy	NOUN
cana-2702	54	23	of	of	ADP
cana-2702	54	24	𝐾6	𝐾6	NOUN
cana-2702	54	25	.	.	PUNCT
cana-2702	55	1	for	for	ADP
cana-2702	55	2	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	55	3	)	)	PUNCT
cana-2702	55	4	we	we	PRON
cana-2702	55	5	have	have	VERB
cana-2702	55	6	𝑚	𝑚	ADP
cana-2702	55	7	copies	copy	NOUN
cana-2702	55	8	of	of	ADP
cana-2702	55	9	𝐾6	𝐾6	NOUN
cana-2702	55	10	and	and	CCONJ
cana-2702	55	11	we	we	PRON
cana-2702	55	12	label	label	VERB
cana-2702	55	13	each	each	DET
cana-2702	55	14	𝐾6	𝐾6	NOUN
cana-2702	55	15	as	as	ADP
cana-2702	55	16	𝐶1	𝐶1	PROPN
cana-2702	55	17	,	,	PUNCT
cana-2702	55	18	𝐶2	𝐶2	ADJ
cana-2702	55	19	,	,	PUNCT
cana-2702	55	20	.	.	PUNCT
cana-2702	55	21	.	.	PUNCT
cana-2702	55	22	.	.	PUNCT
cana-2702	56	1	,	,	PUNCT
cana-2702	57	1	𝐶𝑚	𝐶𝑚	NOUN
cana-2702	57	2	in	in	ADP
cana-2702	57	3	order	order	NOUN
cana-2702	57	4	from	from	ADP
cana-2702	57	5	left	leave	VERB
cana-2702	57	6	to	to	ADP
cana-2702	57	7	right	right	NOUN
cana-2702	57	8	.	.	PUNCT
cana-2702	58	1	pebbling	pebble	VERB
cana-2702	58	2	number	number	NOUN
cana-2702	58	3	of	of	ADP
cana-2702	58	4	crisscross	crisscross	ADJ
cana-2702	58	5	sequence	sequence	NOUN
cana-2702	58	6	of	of	ADP
cana-2702	58	7	𝒎	𝒎	PROPN
cana-2702	58	8	complete	complete	ADJ
cana-2702	58	9	graphs	graph	NOUN
cana-2702	58	10	theorem	theorem	VERB
cana-2702	58	11	1	1	NUM
cana-2702	58	12	.	.	PUNCT
cana-2702	59	1	the	the	DET
cana-2702	59	2	pebbling	pebble	VERB
cana-2702	59	3	number	number	NOUN
cana-2702	59	4	of	of	ADP
cana-2702	59	5	the	the	DET
cana-2702	59	6	graph	graph	NOUN
cana-2702	59	7	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	59	8	)	)	PUNCT
cana-2702	59	9	is	be	AUX
cana-2702	59	10	,	,	PUNCT
cana-2702	59	11	𝑓(𝐶2(𝐾𝑛	𝑓(𝐶2(𝐾𝑛	ADJ
cana-2702	59	12	)	)	PUNCT
cana-2702	59	13	)	)	PUNCT
cana-2702	60	1	=	=	SYM
cana-2702	60	2	2𝑛	2𝑛	PROPN
cana-2702	61	1	−	−	NOUN
cana-2702	61	2	2	2	X
cana-2702	61	3	.	.	PUNCT
cana-2702	61	4	proof	proof	NOUN
cana-2702	61	5	.	.	PUNCT
cana-2702	62	1	put	put	VERB
cana-2702	62	2	3	3	NUM
cana-2702	62	3	pebbles	pebble	NOUN
cana-2702	62	4	on	on	ADP
cana-2702	62	5	the	the	DET
cana-2702	62	6	vertex	vertex	NOUN
cana-2702	62	7	𝑥	𝑥	NOUN
cana-2702	62	8	and	and	CCONJ
cana-2702	62	9	one	one	NUM
cana-2702	62	10	pebble	pebble	NOUN
cana-2702	62	11	each	each	PRON
cana-2702	62	12	on	on	ADP
cana-2702	62	13	vertices	vertex	NOUN
cana-2702	62	14	𝑎𝑖	𝑎𝑖	ADV
cana-2702	62	15	and	and	CCONJ
cana-2702	62	16	𝑏𝑖	𝑏𝑖	ADP
cana-2702	62	17	,	,	PUNCT
cana-2702	62	18	excluding	exclude	VERB
cana-2702	62	19	vertices	vertex	NOUN
cana-2702	62	20	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	62	21	and	and	CCONJ
cana-2702	62	22	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	62	23	where	where	SCONJ
cana-2702	62	24	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	62	25	,	,	PUNCT
cana-2702	62	26	𝑏𝑘	𝑏𝑘	PROPN
cana-2702	62	27	∈	∈	PROPN
cana-2702	62	28	𝑉(𝐶1	𝑉(𝐶1	ADJ
cana-2702	62	29	)	)	PUNCT
cana-2702	62	30	∩	∩	NOUN
cana-2702	62	31	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	62	32	)	)	PUNCT
cana-2702	62	33	.	.	PUNCT
cana-2702	63	1	we	we	PRON
cana-2702	63	2	are	be	AUX
cana-2702	63	3	unable	unable	ADJ
cana-2702	63	4	to	to	PART
cana-2702	63	5	transport	transport	VERB
cana-2702	63	6	a	a	DET
cana-2702	63	7	single	single	ADJ
cana-2702	63	8	pebble	pebble	NOUN
cana-2702	63	9	to	to	ADP
cana-2702	63	10	the	the	DET
cana-2702	63	11	vertex	vertex	NOUN
cana-2702	63	12	𝑦.	𝑦.	PROPN
cana-2702	64	1	so	so	ADV
cana-2702	64	2	we	we	PRON
cana-2702	64	3	have	have	VERB
cana-2702	64	4	𝑓(𝐶2(𝐾𝑛	𝑓(𝐶2(𝐾𝑛	VERB
cana-2702	64	5	)	)	PUNCT
cana-2702	64	6	)	)	PUNCT
cana-2702	65	1	≥	≥	NOUN
cana-2702	65	2	2𝑛	2𝑛	NOUN
cana-2702	66	1	−	−	NOUN
cana-2702	66	2	2	2	X
cana-2702	66	3	.	.	X
cana-2702	66	4	think	think	VERB
cana-2702	66	5	of	of	ADP
cana-2702	66	6	a	a	DET
cana-2702	66	7	graph	graph	NOUN
cana-2702	66	8	where	where	SCONJ
cana-2702	66	9	the	the	DET
cana-2702	66	10	vertices	vertex	NOUN
cana-2702	66	11	are	be	AUX
cana-2702	66	12	covered	cover	VERB
cana-2702	66	13	in	in	ADP
cana-2702	66	14	at	at	ADP
cana-2702	66	15	least	least	ADJ
cana-2702	66	16	2𝑛	2𝑛	NOUN
cana-2702	66	17	−	−	PROPN
cana-2702	66	18	2	2	NUM
cana-2702	66	19	pebbles	pebble	NOUN
cana-2702	66	20	.	.	PUNCT
cana-2702	67	1	suppose	suppose	VERB
cana-2702	67	2	𝑣	𝑣	PRON
cana-2702	67	3	is	be	AUX
cana-2702	67	4	any	any	DET
cana-2702	67	5	target	target	NOUN
cana-2702	67	6	vertex	vertex	NOUN
cana-2702	67	7	.	.	PUNCT
cana-2702	68	1	let	let	VERB
cana-2702	68	2	𝑝(𝑣	𝑝(𝑣	PRON
cana-2702	68	3	)	)	PUNCT
cana-2702	69	1	=	=	PUNCT
cana-2702	70	1	0	0	X
cana-2702	70	2	.	.	PUNCT
cana-2702	71	1	we	we	PRON
cana-2702	71	2	consider	consider	VERB
cana-2702	71	3	the	the	DET
cana-2702	71	4	following	follow	VERB
cana-2702	71	5	scenarios	scenario	NOUN
cana-2702	71	6	:	:	PUNCT
cana-2702	71	7	case	case	NOUN
cana-2702	71	8	(	(	PUNCT
cana-2702	71	9	1	1	NUM
cana-2702	71	10	)	)	PUNCT
cana-2702	71	11	𝑣	𝑣	NOUN
cana-2702	71	12	=	=	PUNCT
cana-2702	71	13	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	71	14	or	or	CCONJ
cana-2702	71	15	𝑣	𝑣	PRON
cana-2702	71	16	=	=	PROPN
cana-2702	71	17	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	71	18	.	.	PUNCT
cana-2702	72	1	take	take	VERB
cana-2702	72	2	the	the	DET
cana-2702	72	3	supposition	supposition	NOUN
cana-2702	72	4	that	that	SCONJ
cana-2702	72	5	𝑣	𝑣	X
cana-2702	72	6	=	=	SYM
cana-2702	72	7	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	72	8	without	without	ADP
cana-2702	72	9	losing	lose	VERB
cana-2702	72	10	generality	generality	NOUN
cana-2702	72	11	.	.	PUNCT
cana-2702	73	1	consider	consider	VERB
cana-2702	73	2	the	the	DET
cana-2702	73	3	case	case	NOUN
cana-2702	73	4	where	where	SCONJ
cana-2702	73	5	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	73	6	)	)	PUNCT
cana-2702	73	7	≥	≥	NOUN
cana-2702	73	8	𝑛	𝑛	PROPN
cana-2702	73	9	or	or	CCONJ
cana-2702	73	10	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	73	11	)	)	PUNCT
cana-2702	73	12	≥	≥	NOUN
cana-2702	73	13	𝑛.	𝑛.	NOUN
cana-2702	73	14	after	after	ADP
cana-2702	73	15	that	that	SCONJ
cana-2702	73	16	we	we	PRON
cana-2702	73	17	may	may	AUX
cana-2702	73	18	move	move	VERB
cana-2702	73	19	a	a	DET
cana-2702	73	20	pebble	pebble	NOUN
cana-2702	73	21	to	to	ADP
cana-2702	73	22	𝑎𝑘.	𝑎𝑘.	CCONJ
cana-2702	73	23	considering	consider	VERB
cana-2702	73	24	the	the	DET
cana-2702	73	25	way	way	NOUN
cana-2702	73	26	𝐶1	𝐶1	PROPN
cana-2702	73	27	≅	≅	PROPN
cana-2702	73	28	𝐶2	𝐶2	PROPN
cana-2702	73	29	≅	≅	PROPN
cana-2702	73	30	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	73	31	also	also	ADV
cana-2702	73	32	𝑓(𝐾𝑛	𝑓(𝐾𝑛	VERB
cana-2702	73	33	)	)	PUNCT
cana-2702	74	1	=	=	SYM
cana-2702	74	2	𝑛.	𝑛.	NOUN
cana-2702	74	3	assume	assume	VERB
cana-2702	74	4	that	that	SCONJ
cana-2702	74	5	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	74	6	)	)	PUNCT
cana-2702	74	7	≥	≥	NOUN
cana-2702	75	1	𝑛	𝑛	DET
cana-2702	75	2	−	−	NUM
cana-2702	75	3	1	1	NUM
cana-2702	75	4	and	and	CCONJ
cana-2702	75	5	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	75	6	)	)	PUNCT
cana-2702	75	7	≥	≥	NOUN
cana-2702	75	8	𝑛	𝑛	PRON
cana-2702	75	9	−	−	NOUN
cana-2702	76	1	1	1	NUM
cana-2702	76	2	.	.	PUNCT
cana-2702	77	1	if	if	SCONJ
cana-2702	77	2	𝑝(𝑉(𝐶1	𝑝(𝑉(𝐶1	PROPN
cana-2702	77	3	)	)	PUNCT
cana-2702	77	4	−	−	PROPN
cana-2702	77	5	{	{	PUNCT
cana-2702	77	6	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	77	7	}	}	PUNCT
cana-2702	77	8	)	)	PUNCT
cana-2702	77	9	≥	≥	NOUN
cana-2702	77	10	𝑛	𝑛	DET
cana-2702	77	11	−	−	PROPN
cana-2702	77	12	1	1	NUM
cana-2702	77	13	,	,	PUNCT
cana-2702	77	14	at	at	ADV
cana-2702	77	15	least	least	ADJ
cana-2702	77	16	two	two	NUM
cana-2702	77	17	of	of	ADP
cana-2702	77	18	the	the	DET
cana-2702	77	19	pebbles	pebble	NOUN
cana-2702	77	20	are	be	AUX
cana-2702	77	21	present	present	ADJ
cana-2702	77	22	in	in	ADP
cana-2702	77	23	one	one	NUM
cana-2702	77	24	of	of	ADP
cana-2702	77	25	the	the	DET
cana-2702	77	26	vertices	vertex	NOUN
cana-2702	77	27	of	of	ADP
cana-2702	77	28	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	77	29	)	)	PUNCT
cana-2702	77	30	−	−	PROPN
cana-2702	77	31	{	{	PUNCT
cana-2702	77	32	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	77	33	,	,	PUNCT
cana-2702	77	34	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	77	35	}	}	PUNCT
cana-2702	77	36	.	.	PUNCT
cana-2702	78	1	assume	assume	VERB
cana-2702	78	2	that	that	SCONJ
cana-2702	78	3	𝑝(𝑦	𝑝(𝑦	PROPN
cana-2702	78	4	)	)	PUNCT
cana-2702	78	5	≥	≥	NOUN
cana-2702	78	6	2	2	NUM
cana-2702	78	7	to	to	PART
cana-2702	78	8	maintain	maintain	VERB
cana-2702	78	9	the	the	DET
cana-2702	78	10	generality	generality	NOUN
cana-2702	78	11	.	.	PUNCT
cana-2702	79	1	due	due	ADP
cana-2702	79	2	to	to	ADP
cana-2702	79	3	the	the	DET
cana-2702	79	4	fact	fact	NOUN
cana-2702	79	5	that	that	SCONJ
cana-2702	79	6	𝑦	𝑦	NOUN
cana-2702	79	7	is	be	AUX
cana-2702	79	8	close	close	ADJ
cana-2702	79	9	to	to	ADP
cana-2702	79	10	𝑎𝑘	𝑎𝑘	INTJ
cana-2702	79	11	,	,	PUNCT
cana-2702	79	12	we	we	PRON
cana-2702	79	13	can	can	AUX
cana-2702	79	14	transfer	transfer	VERB
cana-2702	79	15	a	a	DET
cana-2702	79	16	pebble	pebble	ADJ
cana-2702	79	17	there	there	ADV
cana-2702	79	18	.	.	PUNCT
cana-2702	80	1	case	case	NOUN
cana-2702	80	2	(	(	PUNCT
cana-2702	80	3	2	2	NUM
cana-2702	80	4	)	)	PUNCT
cana-2702	80	5	𝑣	𝑣	PRON
cana-2702	80	6	∈	∈	NOUN
cana-2702	80	7	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	80	8	)	)	PUNCT
cana-2702	80	9	−	−	PROPN
cana-2702	80	10	{	{	PUNCT
cana-2702	80	11	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	80	12	}	}	PUNCT
cana-2702	80	13	or	or	CCONJ
cana-2702	80	14	𝑣	𝑣	PRON
cana-2702	80	15	∈	∈	PROPN
cana-2702	80	16	𝑉(𝐶1	𝑉(𝐶1	PROPN
cana-2702	80	17	)	)	PUNCT
cana-2702	80	18	−	−	PROPN
cana-2702	80	19	{	{	PUNCT
cana-2702	80	20	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	80	21	}	}	PUNCT
cana-2702	80	22	.	.	PUNCT
cana-2702	81	1	take	take	VERB
cana-2702	81	2	𝑣	𝑣	PRON
cana-2702	81	3	=	=	PUNCT
cana-2702	81	4	𝑦	𝑦	NOUN
cana-2702	81	5	and	and	CCONJ
cana-2702	81	6	suppose	suppose	VERB
cana-2702	81	7	that	that	SCONJ
cana-2702	81	8	𝑣	𝑣	PRON
cana-2702	81	9	∈	∈	PROPN
cana-2702	81	10	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	81	11	)	)	PUNCT
cana-2702	81	12	−	−	PROPN
cana-2702	81	13	{	{	PUNCT
cana-2702	81	14	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	81	15	}	}	PUNCT
cana-2702	81	16	.	.	PUNCT
cana-2702	82	1	if	if	SCONJ
cana-2702	82	2	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	82	3	)	)	PUNCT
cana-2702	82	4	≥	≥	NOUN
cana-2702	82	5	𝑛	𝑛	NOUN
cana-2702	82	6	,	,	PUNCT
cana-2702	82	7	then	then	ADV
cana-2702	82	8	we	we	PRON
cana-2702	82	9	can	can	AUX
cana-2702	82	10	reach	reach	VERB
cana-2702	82	11	the	the	DET
cana-2702	82	12	vertex	vertex	NOUN
cana-2702	82	13	𝑦	𝑦	NOUN
cana-2702	82	14	with	with	ADP
cana-2702	82	15	one	one	NUM
cana-2702	82	16	pebble	pebble	NOUN
cana-2702	82	17	.	.	PUNCT
cana-2702	83	1	given	give	VERB
cana-2702	83	2	that	that	PRON
cana-2702	83	3	𝐶2	𝐶2	INTJ
cana-2702	83	4	≅	≅	PROPN
cana-2702	83	5	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	83	6	thus	thus	ADV
cana-2702	83	7	we	we	PRON
cana-2702	83	8	suppose	suppose	VERB
cana-2702	83	9	that	that	SCONJ
cana-2702	83	10	𝑝(𝐶2	𝑝(𝐶2	AUX
cana-2702	83	11	)	)	PUNCT
cana-2702	83	12	<	<	X
cana-2702	83	13	𝑛.	𝑛.	NOUN
cana-2702	83	14	assume	assume	VERB
cana-2702	83	15	that	that	SCONJ
cana-2702	83	16	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	83	17	)	)	PUNCT
cana-2702	83	18	−	−	AUX
cana-2702	83	19	{	{	PUNCT
cana-2702	83	20	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	83	21	}	}	PUNCT
cana-2702	83	22	)	)	PUNCT
cana-2702	83	23	≥	≥	NOUN
cana-2702	83	24	𝑛	𝑛	PRON
cana-2702	83	25	−	−	NOUN
cana-2702	83	26	1	1	NUM
cana-2702	83	27	.	.	PUNCT
cana-2702	84	1	after	after	ADP
cana-2702	84	2	that	that	PRON
cana-2702	84	3	,	,	PUNCT
cana-2702	84	4	we	we	PRON
cana-2702	84	5	may	may	AUX
cana-2702	84	6	move	move	VERB
cana-2702	84	7	a	a	DET
cana-2702	84	8	pebble	pebble	NOUN
cana-2702	84	9	to	to	ADP
cana-2702	84	10	the	the	DET
cana-2702	84	11	vertex	vertex	NOUN
cana-2702	84	12	𝑦.	𝑦.	PROPN
cana-2702	84	13	due	due	ADP
cana-2702	84	14	to	to	ADP
cana-2702	84	15	the	the	DET
cana-2702	84	16	fact	fact	NOUN
cana-2702	84	17	that	that	SCONJ
cana-2702	84	18	the	the	DET
cana-2702	84	19	graph	graph	NOUN
cana-2702	84	20	induced	induce	VERB
cana-2702	84	21	by	by	ADP
cana-2702	84	22	the	the	DET
cana-2702	84	23	vertices	vertex	NOUN
cana-2702	84	24	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	84	25	)	)	PUNCT
cana-2702	84	26	−	−	PROPN
cana-2702	84	27	{	{	PUNCT
cana-2702	84	28	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	84	29	}	}	PUNCT
cana-2702	84	30	is	be	AUX
cana-2702	84	31	isomorphic	isomorphic	ADJ
cana-2702	84	32	to	to	ADP
cana-2702	84	33	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-2702	84	34	.	.	PUNCT
cana-2702	85	1	after	after	ADP
cana-2702	85	2	that	that	PRON
cana-2702	85	3	,	,	PUNCT
cana-2702	85	4	we	we	PRON
cana-2702	85	5	presumptively	presumptively	ADV
cana-2702	85	6	know	know	VERB
cana-2702	85	7	that	that	SCONJ
cana-2702	85	8	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	85	9	)	)	PUNCT
cana-2702	85	10	−	−	PROPN
cana-2702	85	11	{	{	PUNCT
cana-2702	85	12	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	85	13	}	}	PUNCT
cana-2702	85	14	)	)	PUNCT
cana-2702	85	15	≤	≤	NUM
cana-2702	86	1	𝑛	𝑛	PRON
cana-2702	86	2	−	−	NOUN
cana-2702	86	3	2	2	NUM
cana-2702	86	4	.	.	PUNCT
cana-2702	87	1	we	we	PRON
cana-2702	87	2	consider	consider	VERB
cana-2702	87	3	the	the	DET
cana-2702	87	4	following	follow	VERB
cana-2702	87	5	subcases	subcase	NOUN
cana-2702	87	6	:	:	PUNCT
cana-2702	87	7	subcase	subcase	PROPN
cana-2702	87	8	(	(	PUNCT
cana-2702	87	9	1a	1a	X
cana-2702	87	10	)	)	PUNCT
cana-2702	87	11	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	87	12	)	)	PUNCT
cana-2702	87	13	−	−	PROPN
cana-2702	88	1	{	{	PUNCT
cana-2702	88	2	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	88	3	}	}	PUNCT
cana-2702	88	4	)	)	PUNCT
cana-2702	89	1	=	=	SYM
cana-2702	89	2	𝑛	𝑛	PRON
cana-2702	89	3	−	−	NUM
cana-2702	89	4	2	2	NUM
cana-2702	89	5	.	.	PUNCT
cana-2702	89	6	based	base	VERB
cana-2702	89	7	on	on	ADP
cana-2702	89	8	our	our	PRON
cana-2702	89	9	assumptions	assumption	NOUN
cana-2702	89	10	,	,	PUNCT
cana-2702	89	11	we	we	PRON
cana-2702	89	12	obtain	obtain	VERB
cana-2702	89	13	𝑝(𝑉(𝐶1	𝑝(𝑉(𝐶1	PROPN
cana-2702	89	14	−	−	PROPN
cana-2702	89	15	{	{	PUNCT
cana-2702	89	16	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	89	17	}	}	PUNCT
cana-2702	89	18	)	)	PUNCT
cana-2702	90	1	=	=	SYM
cana-2702	90	2	𝑛.	𝑛.	NOUN
cana-2702	90	3	then	then	ADV
cana-2702	90	4	,	,	PUNCT
cana-2702	90	5	at	at	ADP
cana-2702	90	6	least	least	ADJ
cana-2702	90	7	any	any	PRON
cana-2702	90	8	of	of	ADP
cana-2702	90	9	the	the	DET
cana-2702	90	10	vertices	vertex	NOUN
cana-2702	90	11	belonging	belong	VERB
cana-2702	90	12	to	to	ADP
cana-2702	90	13	𝑉(𝐶1	𝑉(𝐶1	NUM
cana-2702	90	14	)	)	PUNCT
cana-2702	90	15	−	−	PROPN
cana-2702	90	16	{	{	PUNCT
cana-2702	90	17	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	90	18	}	}	PUNCT
cana-2702	90	19	contain	contain	VERB
cana-2702	90	20	at	at	ADV
cana-2702	90	21	least	least	ADV
cana-2702	90	22	two	two	NUM
cana-2702	90	23	pebbles	pebble	NOUN
cana-2702	90	24	.	.	PUNCT
cana-2702	91	1	assume	assume	VERB
cana-2702	91	2	that	that	SCONJ
cana-2702	91	3	𝑝(𝑥	𝑝(𝑥	PROPN
cana-2702	91	4	)	)	PUNCT
cana-2702	91	5	≥	≥	NOUN
cana-2702	91	6	2	2	NUM
cana-2702	91	7	and	and	CCONJ
cana-2702	91	8	we	we	PRON
cana-2702	91	9	can	can	AUX
cana-2702	91	10	move	move	VERB
cana-2702	91	11	one	one	NUM
cana-2702	91	12	pebble	pebble	NOUN
cana-2702	91	13	to	to	ADP
cana-2702	91	14	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	91	15	.	.	PUNCT
cana-2702	92	1	this	this	PRON
cana-2702	92	2	is	be	AUX
cana-2702	92	3	because	because	SCONJ
cana-2702	92	4	𝑥	𝑥	PROPN
cana-2702	92	5	is	be	AUX
cana-2702	92	6	adjacent	adjacent	ADJ
cana-2702	92	7	to	to	ADP
cana-2702	92	8	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	92	9	.	.	PUNCT
cana-2702	93	1	the	the	DET
cana-2702	93	2	subgraph	subgraph	NOUN
cana-2702	93	3	induced	induce	VERB
cana-2702	93	4	by	by	ADP
cana-2702	93	5	the	the	DET
cana-2702	93	6	vertices	vertex	NOUN
cana-2702	93	7	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	93	8	)	)	PUNCT
cana-2702	93	9	−	−	PROPN
cana-2702	93	10	{	{	PUNCT
cana-2702	93	11	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	93	12	}	}	PUNCT
cana-2702	93	13	contains	contain	VERB
cana-2702	93	14	𝑛	𝑛	DET
cana-2702	93	15	−	−	NUM
cana-2702	93	16	1	1	NUM
cana-2702	93	17	pebbles	pebble	NOUN
cana-2702	93	18	.	.	PUNCT
cana-2702	94	1	therefore	therefore	ADV
cana-2702	94	2	,	,	PUNCT
cana-2702	94	3	the	the	DET
cana-2702	94	4	pebble	pebble	NOUN
cana-2702	94	5	can	can	AUX
cana-2702	94	6	be	be	AUX
cana-2702	94	7	moved	move	VERB
cana-2702	94	8	to	to	PART
cana-2702	94	9	𝑦.	𝑦.	VERB
cana-2702	94	10	the	the	DET
cana-2702	94	11	subgraph	subgraph	NOUN
cana-2702	94	12	induced	induce	VERB
cana-2702	94	13	by	by	ADP
cana-2702	94	14	vertices	vertex	NOUN
cana-2702	94	15	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	94	16	)	)	PUNCT
cana-2702	94	17	−	−	PROPN
cana-2702	94	18	{	{	PUNCT
cana-2702	94	19	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	94	20	}	}	PUNCT
cana-2702	94	21	is	be	AUX
cana-2702	94	22	isomorphic	isomorphic	ADJ
cana-2702	94	23	to	to	ADP
cana-2702	94	24	𝑘𝑛−1	𝑘𝑛−1	PROPN
cana-2702	94	25	.	.	PUNCT
cana-2702	95	1	subcase	subcase	PROPN
cana-2702	95	2	(	(	PUNCT
cana-2702	95	3	1b	1b	NUM
cana-2702	95	4	)	)	PUNCT
cana-2702	95	5	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	PROPN
cana-2702	95	6	)	)	PUNCT
cana-2702	95	7	−	−	PROPN
cana-2702	95	8	{	{	PUNCT
cana-2702	95	9	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	95	10	}	}	PUNCT
cana-2702	95	11	)	)	PUNCT
cana-2702	95	12	<	<	X
cana-2702	96	1	𝑛	𝑛	DET
cana-2702	96	2	−	−	NUM
cana-2702	96	3	2	2	NUM
cana-2702	96	4	.	.	PUNCT
cana-2702	96	5	communications	communication	NOUN
cana-2702	96	6	on	on	ADP
cana-2702	96	7	applied	apply	VERB
cana-2702	96	8	nonlinear	nonlinear	ADJ
cana-2702	96	9	analysis	analysis	NOUN
cana-2702	96	10	issn	issn	NOUN
cana-2702	96	11	:	:	PUNCT
cana-2702	96	12	1074	1074	NUM
cana-2702	96	13	-	-	PUNCT
cana-2702	96	14	133x	133x	NUM
cana-2702	96	15	vol	vol	NOUN
cana-2702	96	16	32	32	NUM
cana-2702	96	17	no	no	NOUN
cana-2702	96	18	.	.	PUNCT
cana-2702	97	1	3s	3s	NUM
cana-2702	97	2	(	(	PUNCT
cana-2702	97	3	2025	2025	NUM
cana-2702	97	4	)	)	PUNCT
cana-2702	97	5	651	651	NUM
cana-2702	97	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	97	7	by	by	ADP
cana-2702	97	8	assumption	assumption	NOUN
cana-2702	97	9	,	,	PUNCT
cana-2702	97	10	𝑝(𝑉(𝐶1	𝑝(𝑉(𝐶1	PROPN
cana-2702	97	11	)	)	PUNCT
cana-2702	97	12	−	−	PROPN
cana-2702	97	13	{	{	PUNCT
cana-2702	97	14	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	97	15	}	}	PUNCT
cana-2702	97	16	)	)	PUNCT
cana-2702	97	17	≥	≥	NOUN
cana-2702	97	18	𝑛	𝑛	DET
cana-2702	97	19	+	+	NUM
cana-2702	97	20	1	1	NUM
cana-2702	97	21	and	and	CCONJ
cana-2702	97	22	any	any	PRON
cana-2702	97	23	of	of	ADP
cana-2702	97	24	the	the	DET
cana-2702	97	25	vertices	vertex	NOUN
cana-2702	97	26	in	in	ADP
cana-2702	97	27	𝑉(𝐶1	𝑉(𝐶1	NOUN
cana-2702	97	28	)	)	PUNCT
cana-2702	97	29	−	−	PROPN
cana-2702	97	30	{	{	PUNCT
cana-2702	97	31	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	97	32	}	}	PUNCT
cana-2702	97	33	has	have	VERB
cana-2702	97	34	more	more	ADJ
cana-2702	97	35	than	than	ADP
cana-2702	97	36	one	one	NUM
cana-2702	97	37	pebble	pebble	NOUN
cana-2702	97	38	.	.	PUNCT
cana-2702	98	1	using	use	VERB
cana-2702	98	2	exactly	exactly	ADV
cana-2702	98	3	two	two	NUM
cana-2702	98	4	pebbles	pebble	NOUN
cana-2702	98	5	,	,	PUNCT
cana-2702	98	6	we	we	PRON
cana-2702	98	7	can	can	AUX
cana-2702	98	8	reach	reach	VERB
cana-2702	98	9	vertex	vertex	NOUN
cana-2702	98	10	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	98	11	with	with	ADP
cana-2702	98	12	one	one	NUM
cana-2702	98	13	pebble	pebble	NOUN
cana-2702	98	14	.	.	PUNCT
cana-2702	99	1	thus	thus	ADV
cana-2702	99	2	,	,	PUNCT
cana-2702	99	3	the	the	DET
cana-2702	99	4	number	number	NOUN
cana-2702	99	5	of	of	ADP
cana-2702	99	6	pebbles	pebble	NOUN
cana-2702	99	7	retained	retain	VERB
cana-2702	99	8	in	in	ADP
cana-2702	99	9	𝑉(𝐶1	𝑉(𝐶1	PRON
cana-2702	99	10	)	)	PUNCT
cana-2702	99	11	−	−	PROPN
cana-2702	99	12	{	{	PUNCT
cana-2702	99	13	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	99	14	}	}	PUNCT
cana-2702	99	15	was	be	AUX
cana-2702	99	16	at	at	ADP
cana-2702	99	17	least	least	ADJ
cana-2702	99	18	𝑛	𝑛	DET
cana-2702	99	19	−	−	NUM
cana-2702	99	20	1	1	NUM
cana-2702	99	21	.	.	PUNCT
cana-2702	100	1	we	we	PRON
cana-2702	100	2	can	can	AUX
cana-2702	100	3	then	then	ADV
cana-2702	100	4	move	move	VERB
cana-2702	100	5	an	an	DET
cana-2702	100	6	additional	additional	ADJ
cana-2702	100	7	pebble	pebble	NOUN
cana-2702	100	8	to	to	ADP
cana-2702	100	9	𝑎𝑘.	𝑎𝑘.	X
cana-2702	100	10	therefore	therefore	ADV
cana-2702	100	11	,	,	PUNCT
cana-2702	100	12	one	one	NUM
cana-2702	100	13	pebble	pebble	NOUN
cana-2702	100	14	can	can	AUX
cana-2702	100	15	be	be	AUX
cana-2702	100	16	moved	move	VERB
cana-2702	100	17	to	to	ADP
cana-2702	100	18	𝑦	𝑦	NOUN
cana-2702	100	19	from	from	ADP
cana-2702	100	20	𝑎𝑘.	𝑎𝑘.	CCONJ
cana-2702	100	21	theorem	theorem	ADJ
cana-2702	100	22	2	2	NUM
cana-2702	100	23	.	.	PUNCT
cana-2702	101	1	the	the	DET
cana-2702	101	2	pebbling	pebble	VERB
cana-2702	101	3	number	number	NOUN
cana-2702	101	4	of	of	ADP
cana-2702	101	5	the	the	DET
cana-2702	101	6	graph	graph	NOUN
cana-2702	101	7	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	101	8	)	)	PUNCT
cana-2702	101	9	is	be	AUX
cana-2702	101	10	𝑓(𝐶3(𝐾𝑛	𝑓(𝐶3(𝐾𝑛	X
cana-2702	101	11	)	)	PUNCT
cana-2702	101	12	)	)	PUNCT
cana-2702	102	1	=	=	PUNCT
cana-2702	102	2	3𝑛	3𝑛	NUM
cana-2702	102	3	−	−	NOUN
cana-2702	102	4	2	2	X
cana-2702	102	5	.	.	PUNCT
cana-2702	102	6	proof	proof	NOUN
cana-2702	102	7	.	.	PUNCT
cana-2702	103	1	put	put	VERB
cana-2702	103	2	seven	seven	NUM
cana-2702	103	3	pebbles	pebble	NOUN
cana-2702	103	4	on	on	ADP
cana-2702	103	5	vertex	vertex	NOUN
cana-2702	103	6	𝑥	𝑥	PROPN
cana-2702	103	7	and	and	CCONJ
cana-2702	103	8	put	put	VERB
cana-2702	103	9	one	one	NUM
cana-2702	103	10	pebble	pebble	NOUN
cana-2702	103	11	on	on	ADP
cana-2702	103	12	each	each	DET
cana-2702	103	13	𝑎𝑖′𝑠	𝑎𝑖′𝑠	PROPN
cana-2702	103	14	and	and	CCONJ
cana-2702	103	15	𝑏𝑖′𝑠	𝑏𝑖′𝑠	PROPN
cana-2702	103	16	except	except	SCONJ
cana-2702	103	17	{	{	PUNCT
cana-2702	103	18	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	103	19	,	,	PUNCT
cana-2702	103	20	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	103	21	,	,	PUNCT
cana-2702	103	22	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	103	23	,	,	PUNCT
cana-2702	103	24	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	103	25	}	}	PUNCT
cana-2702	103	26	.	.	PUNCT
cana-2702	104	1	then	then	ADV
cana-2702	104	2	,	,	PUNCT
cana-2702	104	3	we	we	PRON
cana-2702	104	4	can	can	AUX
cana-2702	104	5	not	not	PART
cana-2702	104	6	move	move	VERB
cana-2702	104	7	the	the	DET
cana-2702	104	8	pebble	pebble	NOUN
cana-2702	104	9	to	to	ADP
cana-2702	104	10	𝑦.	𝑦.	PROPN
cana-2702	104	11	thus	thus	ADV
cana-2702	104	12	,	,	PUNCT
cana-2702	104	13	𝑓(𝐶3(𝐾𝑛	𝑓(𝐶3(𝐾𝑛	NUM
cana-2702	104	14	)	)	PUNCT
cana-2702	104	15	)	)	PUNCT
cana-2702	105	1	≥	≥	X
cana-2702	105	2	3𝑛	3𝑛	NUM
cana-2702	106	1	−	−	NOUN
cana-2702	106	2	2	2	X
cana-2702	106	3	.	.	X
cana-2702	106	4	consider	consider	VERB
cana-2702	106	5	a	a	DET
cana-2702	106	6	graph	graph	NOUN
cana-2702	106	7	with	with	ADP
cana-2702	106	8	3𝑛	3𝑛	NUM
cana-2702	106	9	−	−	NUM
cana-2702	106	10	2	2	NUM
cana-2702	106	11	pebbles	pebble	NOUN
cana-2702	106	12	distributed	distribute	VERB
cana-2702	106	13	at	at	ADP
cana-2702	106	14	its	its	PRON
cana-2702	106	15	vertices	vertex	NOUN
cana-2702	106	16	.	.	PUNCT
cana-2702	107	1	let	let	VERB
cana-2702	107	2	𝑣	𝑣	PART
cana-2702	107	3	be	be	AUX
cana-2702	107	4	a	a	DET
cana-2702	107	5	target	target	NOUN
cana-2702	107	6	vertex	vertex	NOUN
cana-2702	107	7	.	.	PUNCT
cana-2702	108	1	clearly	clearly	ADV
cana-2702	108	2	𝑣	𝑣	PRON
cana-2702	108	3	∈	∈	PROPN
cana-2702	108	4	𝐶𝑖	𝐶𝑖	PROPN
cana-2702	108	5	,	,	PUNCT
cana-2702	108	6	for	for	ADP
cana-2702	108	7	any	any	DET
cana-2702	108	8	𝑖	𝑖	NOUN
cana-2702	108	9	=	=	NOUN
cana-2702	108	10	1,2,3	1,2,3	NUM
cana-2702	108	11	.	.	PUNCT
cana-2702	109	1	we	we	PRON
cana-2702	109	2	consider	consider	VERB
cana-2702	109	3	the	the	DET
cana-2702	109	4	following	follow	VERB
cana-2702	109	5	cases	case	NOUN
cana-2702	109	6	.	.	PUNCT
cana-2702	110	1	case(1	case(1	NOUN
cana-2702	110	2	)	)	PUNCT
cana-2702	111	1	if	if	SCONJ
cana-2702	111	2	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	111	3	)	)	PUNCT
cana-2702	111	4	≥	≥	NOUN
cana-2702	111	5	𝑛	𝑛	NOUN
cana-2702	111	6	+	+	NOUN
cana-2702	111	7	2	2	NUM
cana-2702	111	8	,	,	PUNCT
cana-2702	111	9	then	then	ADV
cana-2702	111	10	the	the	DET
cana-2702	111	11	two	two	NUM
cana-2702	111	12	pebbles	pebble	NOUN
cana-2702	111	13	can	can	AUX
cana-2702	111	14	be	be	AUX
cana-2702	111	15	moved	move	VERB
cana-2702	111	16	to	to	ADP
cana-2702	111	17	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	111	18	and	and	CCONJ
cana-2702	111	19	one	one	NUM
cana-2702	111	20	pebble	pebble	ADJ
cana-2702	111	21	to	to	ADP
cana-2702	111	22	𝑣.	𝑣.	PROPN
cana-2702	111	23	𝐶3	𝐶3	PROPN
cana-2702	111	24	≅	≅	PROPN
cana-2702	111	25	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	111	26	and	and	CCONJ
cana-2702	111	27	the	the	DET
cana-2702	111	28	pebbling	pebble	VERB
cana-2702	111	29	number	number	NOUN
cana-2702	111	30	of	of	ADP
cana-2702	111	31	a	a	DET
cana-2702	111	32	complete	complete	ADJ
cana-2702	111	33	graph	graph	NOUN
cana-2702	112	1	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	112	2	is	be	AUX
cana-2702	112	3	𝑛.	𝑛.	NOUN
cana-2702	112	4	therefore	therefore	ADV
cana-2702	112	5	,	,	PUNCT
cana-2702	112	6	we	we	PRON
cana-2702	112	7	assume	assume	VERB
cana-2702	112	8	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	112	9	)	)	PUNCT
cana-2702	112	10	≤	≤	NOUN
cana-2702	112	11	𝑛	𝑛	PRON
cana-2702	113	1	+	+	NOUN
cana-2702	113	2	1	1	X
cana-2702	113	3	.	.	PUNCT
cana-2702	113	4	this	this	PRON
cana-2702	113	5	implies	imply	VERB
cana-2702	113	6	that	that	SCONJ
cana-2702	113	7	at	at	ADP
cana-2702	113	8	least	least	ADJ
cana-2702	113	9	2𝑛	2𝑛	NOUN
cana-2702	113	10	−	−	NOUN
cana-2702	113	11	3	3	NUM
cana-2702	113	12	pebbles	pebble	NOUN
cana-2702	113	13	were	be	AUX
cana-2702	113	14	retained	retain	VERB
cana-2702	113	15	in	in	ADP
cana-2702	113	16	𝐶1	𝐶1	NUM
cana-2702	113	17	∩	∩	NOUN
cana-2702	113	18	𝐶2	𝐶2	ADJ
cana-2702	113	19	.	.	PUNCT
cana-2702	114	1	if	if	SCONJ
cana-2702	114	2	𝑝(𝑉(𝐶3	𝑝(𝑉(𝐶3	NOUN
cana-2702	114	3	)	)	PUNCT
cana-2702	115	1	−	−	PROPN
cana-2702	115	2	{	{	PUNCT
cana-2702	115	3	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	115	4	,	,	PUNCT
cana-2702	115	5	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	115	6	}	}	PUNCT
cana-2702	115	7	)	)	PUNCT
cana-2702	115	8	≥	≥	PART
cana-2702	116	1	𝑛.	𝑛.	NOUN
cana-2702	116	2	then	then	ADV
cana-2702	116	3	,	,	PUNCT
cana-2702	116	4	the	the	DET
cana-2702	116	5	two	two	NUM
cana-2702	116	6	pebbles	pebble	NOUN
cana-2702	116	7	can	can	AUX
cana-2702	116	8	be	be	AUX
cana-2702	116	9	moved	move	VERB
cana-2702	116	10	to	to	ADP
cana-2702	116	11	𝑦.	𝑦.	NOUN
cana-2702	116	12	we	we	PRON
cana-2702	116	13	can	can	AUX
cana-2702	116	14	reach	reach	VERB
cana-2702	116	15	vertex	vertex	NOUN
cana-2702	116	16	𝑎𝑘+1	𝑎𝑘+1	NOUN
cana-2702	116	17	using	use	VERB
cana-2702	116	18	one	one	NUM
cana-2702	116	19	pebble	pebble	NOUN
cana-2702	116	20	.	.	PUNCT
cana-2702	117	1	this	this	DET
cana-2702	117	2	results	result	VERB
cana-2702	117	3	in	in	ADP
cana-2702	117	4	the	the	DET
cana-2702	117	5	subgraph	subgraph	NOUN
cana-2702	117	6	𝐶1	𝐶1	PROPN
cana-2702	117	7	∩	∩	NOUN
cana-2702	117	8	𝐶2	𝐶2	ADJ
cana-2702	117	9	obtaining	obtain	VERB
cana-2702	117	10	2𝑛	2𝑛	PROPN
cana-2702	117	11	−	−	PROPN
cana-2702	117	12	2	2	NUM
cana-2702	117	13	pebbles	pebble	NOUN
cana-2702	117	14	after	after	ADP
cana-2702	117	15	moving	move	VERB
cana-2702	117	16	one	one	NUM
cana-2702	117	17	pebble	pebble	NOUN
cana-2702	117	18	to	to	ADP
cana-2702	117	19	𝑎𝑘+1	𝑎𝑘+1	PROPN
cana-2702	117	20	.	.	PUNCT
cana-2702	118	1	by	by	ADP
cana-2702	118	2	theorem	theorem	NOUN
cana-2702	118	3	1	1	NUM
cana-2702	118	4	we	we	PRON
cana-2702	118	5	move	move	VERB
cana-2702	118	6	the	the	DET
cana-2702	118	7	pebble	pebble	NOUN
cana-2702	118	8	to	to	ADP
cana-2702	118	9	𝑣.	𝑣.	NOUN
cana-2702	118	10	case	case	NOUN
cana-2702	118	11	(	(	PUNCT
cana-2702	118	12	2	2	X
cana-2702	118	13	)	)	PUNCT
cana-2702	118	14	𝑣	𝑣	PART
cana-2702	118	15	∈	∈	NOUN
cana-2702	118	16	𝐶1	𝐶1	NOUN
cana-2702	118	17	or	or	CCONJ
cana-2702	118	18	𝑣	𝑣	ADP
cana-2702	118	19	∈	∈	PROPN
cana-2702	118	20	𝐶3	𝐶3	PROPN
cana-2702	118	21	.	.	PUNCT
cana-2702	119	1	without	without	ADP
cana-2702	119	2	loss	loss	NOUN
cana-2702	119	3	of	of	ADP
cana-2702	119	4	generality	generality	NOUN
cana-2702	119	5	,	,	PUNCT
cana-2702	119	6	assume	assume	VERB
cana-2702	119	7	that	that	SCONJ
cana-2702	119	8	𝑣	𝑣	ADP
cana-2702	119	9	∈	∈	PROPN
cana-2702	119	10	𝐶3	𝐶3	NOUN
cana-2702	119	11	and	and	CCONJ
cana-2702	119	12	let	let	VERB
cana-2702	119	13	us	we	PRON
cana-2702	119	14	assume	assume	VERB
cana-2702	119	15	𝑣	𝑣	ADP
cana-2702	119	16	=	=	PUNCT
cana-2702	119	17	𝑦.	𝑦.	PROPN
cana-2702	119	18	if	if	SCONJ
cana-2702	119	19	𝑝(𝑉(𝐶3	𝑝(𝑉(𝐶3	NOUN
cana-2702	119	20	)	)	PUNCT
cana-2702	119	21	−	−	PROPN
cana-2702	119	22	{	{	PUNCT
cana-2702	119	23	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	119	24	,	,	PUNCT
cana-2702	119	25	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	119	26	}	}	PUNCT
cana-2702	119	27	)	)	PUNCT
cana-2702	119	28	≥	≥	NOUN
cana-2702	119	29	𝑛	𝑛	DET
cana-2702	119	30	−	−	PROPN
cana-2702	119	31	2	2	NUM
cana-2702	119	32	,	,	PUNCT
cana-2702	119	33	then	then	ADV
cana-2702	119	34	we	we	PRON
cana-2702	119	35	can	can	AUX
cana-2702	119	36	move	move	VERB
cana-2702	119	37	the	the	DET
cana-2702	119	38	pebble	pebble	NOUN
cana-2702	119	39	to	to	PART
cana-2702	119	40	𝑦.	𝑦.	VERB
cana-2702	119	41	the	the	DET
cana-2702	119	42	graph	graph	NOUN
cana-2702	119	43	is	be	AUX
cana-2702	119	44	induced	induce	VERB
cana-2702	119	45	by	by	ADP
cana-2702	119	46	vertices	vertex	NOUN
cana-2702	119	47	⟨𝑉(𝐶3	⟨𝑉(𝐶3	NOUN
cana-2702	119	48	)	)	PUNCT
cana-2702	120	1	−	−	PROPN
cana-2702	120	2	{	{	PUNCT
cana-2702	120	3	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	120	4	,	,	PUNCT
cana-2702	120	5	𝑏2𝑘−1}⟩	𝑏2𝑘−1}⟩	PROPN
cana-2702	120	6	≅	≅	NUM
cana-2702	120	7	𝐾𝑛−2	𝐾𝑛−2	PROPN
cana-2702	120	8	.	.	PUNCT
cana-2702	121	1	therefore	therefore	ADV
cana-2702	121	2	,	,	PUNCT
cana-2702	121	3	we	we	PRON
cana-2702	121	4	assume	assume	VERB
cana-2702	121	5	𝑝(𝑉(𝐶3	𝑝(𝑉(𝐶3	NOUN
cana-2702	121	6	)	)	PUNCT
cana-2702	121	7	−	−	PROPN
cana-2702	121	8	{	{	PUNCT
cana-2702	121	9	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	121	10	,	,	PUNCT
cana-2702	121	11	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	121	12	}	}	PUNCT
cana-2702	121	13	)	)	PUNCT
cana-2702	121	14	≤	≤	NUM
cana-2702	122	1	𝑛	𝑛	DET
cana-2702	122	2	−	−	NOUN
cana-2702	122	3	3	3	NUM
cana-2702	122	4	.	.	PUNCT
cana-2702	123	1	thus	thus	ADV
cana-2702	123	2	,	,	PUNCT
cana-2702	123	3	the	the	DET
cana-2702	123	4	number	number	NOUN
cana-2702	123	5	of	of	ADP
cana-2702	123	6	pebbles	pebble	NOUN
cana-2702	123	7	retained	retain	VERB
cana-2702	123	8	in	in	ADP
cana-2702	123	9	𝑉(𝐶1	𝑉(𝐶1	PRON
cana-2702	123	10	∩	∩	NOUN
cana-2702	123	11	𝐶2	𝐶2	ADJ
cana-2702	123	12	)	)	PUNCT
cana-2702	123	13	is	be	AUX
cana-2702	123	14	at	at	ADP
cana-2702	123	15	least	least	ADJ
cana-2702	123	16	2𝑛	2𝑛	NOUN
cana-2702	124	1	+	+	CCONJ
cana-2702	124	2	1	1	X
cana-2702	124	3	.	.	X
cana-2702	124	4	we	we	PRON
cana-2702	124	5	may	may	AUX
cana-2702	124	6	take	take	VERB
cana-2702	124	7	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	124	8	)	)	PUNCT
cana-2702	124	9	−	−	PROPN
cana-2702	124	10	{	{	PUNCT
cana-2702	124	11	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	124	12	,	,	PUNCT
cana-2702	124	13	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	124	14	}	}	PUNCT
cana-2702	124	15	)	)	PUNCT
cana-2702	124	16	≥	≥	NOUN
cana-2702	125	1	𝑛	𝑛	DET
cana-2702	125	2	−	−	NOUN
cana-2702	125	3	2	2	NUM
cana-2702	125	4	.	.	PUNCT
cana-2702	126	1	we	we	PRON
cana-2702	126	2	can	can	AUX
cana-2702	126	3	then	then	ADV
cana-2702	126	4	move	move	VERB
cana-2702	126	5	the	the	DET
cana-2702	126	6	pebble	pebble	NOUN
cana-2702	126	7	to	to	ADP
cana-2702	126	8	𝑎𝑘.	𝑎𝑘.	CCONJ
cana-2702	126	9	additionally	additionally	ADV
cana-2702	126	10	,	,	PUNCT
cana-2702	126	11	the	the	DET
cana-2702	126	12	pebbles	pebble	NOUN
cana-2702	126	13	retained	retain	VERB
cana-2702	126	14	in	in	ADP
cana-2702	126	15	𝐶1	𝐶1	NUM
cana-2702	126	16	are	be	AUX
cana-2702	126	17	at	at	ADP
cana-2702	126	18	least	least	ADJ
cana-2702	126	19	𝑛	𝑛	DET
cana-2702	126	20	+	+	NOUN
cana-2702	126	21	3	3	X
cana-2702	126	22	.	.	PUNCT
cana-2702	127	1	in	in	ADP
cana-2702	127	2	this	this	DET
cana-2702	127	3	case	case	NOUN
cana-2702	127	4	,	,	PUNCT
cana-2702	127	5	we	we	PRON
cana-2702	127	6	can	can	AUX
cana-2702	127	7	move	move	VERB
cana-2702	127	8	the	the	DET
cana-2702	127	9	two	two	NUM
cana-2702	127	10	pebbles	pebble	NOUN
cana-2702	127	11	to	to	ADP
cana-2702	127	12	𝑎𝑘	𝑎𝑘	VERB
cana-2702	127	13	and	and	CCONJ
cana-2702	127	14	move	move	VERB
cana-2702	127	15	an	an	DET
cana-2702	127	16	additional	additional	ADJ
cana-2702	127	17	pebble	pebble	NOUN
cana-2702	127	18	to	to	ADP
cana-2702	127	19	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	127	20	.	.	PUNCT
cana-2702	128	1	thus	thus	ADV
cana-2702	128	2	,	,	PUNCT
cana-2702	128	3	the	the	DET
cana-2702	128	4	pebble	pebble	NOUN
cana-2702	128	5	can	can	AUX
cana-2702	128	6	be	be	AUX
cana-2702	128	7	moved	move	VERB
cana-2702	128	8	to	to	ADP
cana-2702	128	9	𝑦.	𝑦.	PROPN
cana-2702	128	10	therefore	therefore	ADV
cana-2702	128	11	,	,	PUNCT
cana-2702	128	12	we	we	PRON
cana-2702	128	13	take	take	VERB
cana-2702	128	14	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	128	15	)	)	PUNCT
cana-2702	128	16	≤	≤	NOUN
cana-2702	129	1	𝑛	𝑛	DET
cana-2702	129	2	−	−	PROPN
cana-2702	129	3	4	4	NUM
cana-2702	129	4	,	,	PUNCT
cana-2702	129	5	𝑝(𝑎2𝑘−2	𝑝(𝑎2𝑘−2	NUM
cana-2702	129	6	)	)	PUNCT
cana-2702	129	7	=	=	SYM
cana-2702	129	8	0	0	NUM
cana-2702	129	9	and	and	CCONJ
cana-2702	129	10	𝑝(𝑏2𝑘−1	𝑝(𝑏2𝑘−1	NOUN
cana-2702	129	11	)	)	PUNCT
cana-2702	130	1	=	=	NOUN
cana-2702	130	2	0	0	X
cana-2702	130	3	.	.	PUNCT
cana-2702	131	1	this	this	PRON
cana-2702	131	2	implies	imply	VERB
cana-2702	131	3	that	that	SCONJ
cana-2702	131	4	the	the	DET
cana-2702	131	5	number	number	NOUN
cana-2702	131	6	of	of	ADP
cana-2702	131	7	pebbles	pebble	NOUN
cana-2702	131	8	retained	retain	VERB
cana-2702	131	9	in	in	ADP
cana-2702	131	10	𝑉(𝐶1	𝑉(𝐶1	NUM
cana-2702	131	11	)	)	PUNCT
cana-2702	131	12	was	be	AUX
cana-2702	131	13	at	at	ADP
cana-2702	131	14	least	least	ADJ
cana-2702	131	15	𝑛	𝑛	DET
cana-2702	131	16	+	+	ADJ
cana-2702	131	17	5	5	NUM
cana-2702	131	18	.	.	PUNCT
cana-2702	131	19	suppose	suppose	VERB
cana-2702	131	20	𝑝(𝑎𝑘	𝑝(𝑎𝑘	PROPN
cana-2702	131	21	)	)	PUNCT
cana-2702	131	22	≥	≥	NOUN
cana-2702	131	23	2	2	NUM
cana-2702	131	24	or	or	CCONJ
cana-2702	131	25	𝑝(𝑏𝑘−1	𝑝(𝑏𝑘−1	NOUN
cana-2702	131	26	)	)	PUNCT
cana-2702	131	27	≥	≥	NOUN
cana-2702	131	28	2	2	NUM
cana-2702	131	29	.	.	PUNCT
cana-2702	132	1	then	then	ADV
cana-2702	132	2	,	,	PUNCT
cana-2702	132	3	we	we	PRON
cana-2702	132	4	can	can	AUX
cana-2702	132	5	move	move	VERB
cana-2702	132	6	one	one	NUM
cana-2702	132	7	pebble	pebble	NOUN
cana-2702	132	8	to	to	ADP
cana-2702	132	9	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	132	10	and	and	CCONJ
cana-2702	132	11	use	use	VERB
cana-2702	132	12	exactly	exactly	ADV
cana-2702	132	13	two	two	NUM
cana-2702	132	14	pebbles	pebble	NOUN
cana-2702	132	15	from	from	ADP
cana-2702	132	16	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	132	17	or	or	CCONJ
cana-2702	132	18	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	132	19	.	.	PUNCT
cana-2702	133	1	in	in	ADP
cana-2702	133	2	addition	addition	NOUN
cana-2702	133	3	,	,	PUNCT
cana-2702	133	4	𝑛	𝑛	PRON
cana-2702	133	5	+	+	SYM
cana-2702	133	6	3	3	NUM
cana-2702	133	7	pebbles	pebble	NOUN
cana-2702	133	8	settled	settle	VERB
cana-2702	133	9	in	in	ADP
cana-2702	133	10	𝐶1	𝐶1	PRON
cana-2702	133	11	.	.	PUNCT
cana-2702	134	1	because	because	SCONJ
cana-2702	134	2	𝐶1	𝐶1	PROPN
cana-2702	134	3	≅	≅	PROPN
cana-2702	134	4	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	134	5	and	and	CCONJ
cana-2702	134	6	by	by	ADP
cana-2702	134	7	using	use	VERB
cana-2702	134	8	the	the	DET
cana-2702	134	9	pebbling	pebble	VERB
cana-2702	134	10	number	number	NOUN
cana-2702	134	11	of	of	ADP
cana-2702	134	12	complete	complete	ADJ
cana-2702	134	13	graph	graph	NOUN
cana-2702	134	14	we	we	PRON
cana-2702	134	15	can	can	AUX
cana-2702	134	16	move	move	VERB
cana-2702	134	17	two	two	NUM
cana-2702	134	18	pebbles	pebble	NOUN
cana-2702	134	19	to	to	ADP
cana-2702	134	20	𝑎𝑘	𝑎𝑘	VERB
cana-2702	134	21	and	and	CCONJ
cana-2702	134	22	move	move	VERB
cana-2702	134	23	an	an	DET
cana-2702	134	24	additional	additional	ADJ
cana-2702	134	25	pebble	pebble	NOUN
cana-2702	134	26	to	to	ADP
cana-2702	134	27	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	134	28	.	.	PUNCT
cana-2702	135	1	we	we	PRON
cana-2702	135	2	can	can	AUX
cana-2702	135	3	then	then	ADV
cana-2702	135	4	reach	reach	VERB
cana-2702	135	5	vertex	vertex	NOUN
cana-2702	135	6	𝑦	𝑦	NOUN
cana-2702	135	7	with	with	ADP
cana-2702	135	8	one	one	NUM
cana-2702	135	9	pebble	pebble	NOUN
cana-2702	135	10	.	.	PUNCT
cana-2702	136	1	we	we	PRON
cana-2702	136	2	assume	assume	VERB
cana-2702	136	3	that	that	SCONJ
cana-2702	136	4	𝑝(𝑎𝑘	𝑝(𝑎𝑘	NOUN
cana-2702	136	5	)	)	PUNCT
cana-2702	136	6	≤	≤	NOUN
cana-2702	136	7	1	1	NUM
cana-2702	136	8	and	and	CCONJ
cana-2702	136	9	𝑝(𝑏𝑘−1	𝑝(𝑏𝑘−1	NOUN
cana-2702	136	10	)	)	PUNCT
cana-2702	136	11	≤	≤	NUM
cana-2702	136	12	1	1	NUM
cana-2702	136	13	.	.	PUNCT
cana-2702	137	1	if	if	SCONJ
cana-2702	137	2	𝑝(𝑎𝑘	𝑝(𝑎𝑘	NOUN
cana-2702	137	3	)	)	PUNCT
cana-2702	137	4	=	=	SYM
cana-2702	138	1	1	1	NUM
cana-2702	138	2	,	,	PUNCT
cana-2702	138	3	at	at	ADP
cana-2702	138	4	least𝑛	least𝑛	ADJ
cana-2702	138	5	−	−	PROPN
cana-2702	138	6	3	3	NUM
cana-2702	138	7	pebbles	pebble	NOUN
cana-2702	138	8	are	be	AUX
cana-2702	138	9	distributed	distribute	VERB
cana-2702	138	10	on	on	ADP
cana-2702	138	11	𝑛	𝑛	DET
cana-2702	138	12	−	−	NUM
cana-2702	138	13	1	1	NUM
cana-2702	138	14	vertices	vertex	NOUN
cana-2702	138	15	.	.	PUNCT
cana-2702	139	1	thus	thus	ADV
cana-2702	139	2	,	,	PUNCT
cana-2702	139	3	using	use	VERB
cana-2702	139	4	the	the	DET
cana-2702	139	5	pigeonhole	pigeonhole	NOUN
cana-2702	139	6	principle	principle	NOUN
cana-2702	139	7	,	,	PUNCT
cana-2702	139	8	we	we	PRON
cana-2702	139	9	conclude	conclude	VERB
cana-2702	139	10	that	that	SCONJ
cana-2702	139	11	at	at	ADV
cana-2702	139	12	least	least	ADJ
cana-2702	139	13	one	one	NUM
cana-2702	139	14	of	of	ADP
cana-2702	139	15	those	those	DET
cana-2702	139	16	vertices	vertex	NOUN
cana-2702	139	17	must	must	AUX
cana-2702	139	18	contain	contain	VERB
cana-2702	139	19	two	two	NUM
cana-2702	139	20	pebbles	pebble	NOUN
cana-2702	139	21	,	,	PUNCT
cana-2702	139	22	and	and	CCONJ
cana-2702	139	23	we	we	PRON
cana-2702	139	24	can	can	AUX
cana-2702	139	25	then	then	ADV
cana-2702	139	26	move	move	VERB
cana-2702	139	27	an	an	DET
cana-2702	139	28	additional	additional	ADJ
cana-2702	139	29	pebble	pebble	NOUN
cana-2702	139	30	to	to	ADP
cana-2702	139	31	𝑎𝑘.	𝑎𝑘.	NOUN
cana-2702	139	32	after	after	ADP
cana-2702	139	33	moving	move	VERB
cana-2702	139	34	a	a	DET
cana-2702	139	35	pebble	pebble	NOUN
cana-2702	139	36	to	to	ADP
cana-2702	139	37	𝑎𝑘	𝑎𝑘	PRON
cana-2702	139	38	we	we	PRON
cana-2702	139	39	obtain	obtain	VERB
cana-2702	139	40	that	that	SCONJ
cana-2702	139	41	𝑝(𝑉(𝐶1	𝑝(𝑉(𝐶1	PROPN
cana-2702	139	42	)	)	PUNCT
cana-2702	139	43	−	−	PROPN
cana-2702	139	44	{	{	PUNCT
cana-2702	139	45	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	139	46	}	}	PUNCT
cana-2702	139	47	)	)	PUNCT
cana-2702	139	48	is	be	AUX
cana-2702	139	49	at	at	ADP
cana-2702	139	50	least	least	ADJ
cana-2702	139	51	𝑛	𝑛	PRON
cana-2702	139	52	+	+	NOUN
cana-2702	139	53	1	1	X
cana-2702	139	54	.	.	X
cana-2702	139	55	we	we	PRON
cana-2702	139	56	assume	assume	VERB
cana-2702	139	57	that	that	SCONJ
cana-2702	139	58	𝑝(𝑦	𝑝(𝑦	PROPN
cana-2702	139	59	)	)	PUNCT
cana-2702	139	60	=	=	SYM
cana-2702	140	1	1	1	X
cana-2702	140	2	.	.	PUNCT
cana-2702	141	1	then	then	ADV
cana-2702	141	2	,	,	PUNCT
cana-2702	141	3	𝑛	𝑛	DET
cana-2702	141	4	pebbles	pebble	NOUN
cana-2702	141	5	are	be	AUX
cana-2702	141	6	settled	settle	VERB
cana-2702	141	7	in	in	ADP
cana-2702	141	8	𝑉(𝐶1	𝑉(𝐶1	NUM
cana-2702	141	9	)	)	PUNCT
cana-2702	141	10	−	−	PROPN
cana-2702	141	11	{	{	PUNCT
cana-2702	141	12	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	141	13	,	,	PUNCT
cana-2702	141	14	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	141	15	}	}	PUNCT
cana-2702	141	16	with	with	ADP
cana-2702	141	17	𝑛	𝑛	DET
cana-2702	141	18	−	−	NUM
cana-2702	141	19	1	1	NUM
cana-2702	141	20	vertices	vertex	NOUN
cana-2702	141	21	.	.	PUNCT
cana-2702	142	1	again	again	ADV
cana-2702	142	2	,	,	PUNCT
cana-2702	142	3	using	use	VERB
cana-2702	142	4	the	the	DET
cana-2702	142	5	pigeonhole	pigeonhole	NOUN
cana-2702	142	6	principle	principle	NOUN
cana-2702	142	7	,	,	PUNCT
cana-2702	142	8	we	we	PRON
cana-2702	142	9	can	can	AUX
cana-2702	142	10	move	move	VERB
cana-2702	142	11	the	the	DET
cana-2702	142	12	two	two	NUM
cana-2702	142	13	pebbles	pebble	NOUN
cana-2702	142	14	to	to	ADP
cana-2702	142	15	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	142	16	from	from	ADP
cana-2702	142	17	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	142	18	and	and	CCONJ
cana-2702	142	19	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	142	20	.	.	PUNCT
cana-2702	143	1	therefore	therefore	ADV
cana-2702	143	2	,	,	PUNCT
cana-2702	143	3	we	we	PRON
cana-2702	143	4	assume	assume	VERB
cana-2702	143	5	that	that	SCONJ
cana-2702	143	6	𝑝(𝑎𝑘	𝑝(𝑎𝑘	NOUN
cana-2702	143	7	)	)	PUNCT
cana-2702	143	8	=	=	SYM
cana-2702	143	9	0	0	NUM
cana-2702	143	10	and	and	CCONJ
cana-2702	143	11	𝑝(𝑏𝑘	𝑝(𝑏𝑘	PROPN
cana-2702	143	12	)	)	PUNCT
cana-2702	143	13	=	=	SYM
cana-2702	144	1	0	0	X
cana-2702	144	2	.	.	PUNCT
cana-2702	145	1	then	then	ADV
cana-2702	145	2	,	,	PUNCT
cana-2702	145	3	all	all	DET
cana-2702	145	4	𝑛	𝑛	DET
cana-2702	145	5	+	+	NOUN
cana-2702	145	6	4	4	NUM
cana-2702	145	7	pebbles	pebble	NOUN
cana-2702	145	8	are	be	AUX
cana-2702	145	9	distributed	distribute	VERB
cana-2702	145	10	only	only	ADV
cana-2702	145	11	on	on	ADP
cana-2702	145	12	𝑉(𝐶1	𝑉(𝐶1	NOUN
cana-2702	145	13	)	)	PUNCT
cana-2702	145	14	−	−	PROPN
cana-2702	145	15	{	{	PUNCT
cana-2702	145	16	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	145	17	,	,	PUNCT
cana-2702	145	18	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	145	19	}	}	PUNCT
cana-2702	145	20	.	.	PUNCT
cana-2702	146	1	because	because	SCONJ
cana-2702	146	2	the	the	DET
cana-2702	146	3	graph	graph	NOUN
cana-2702	146	4	induced	induce	VERB
cana-2702	146	5	by	by	ADP
cana-2702	146	6	vertex	vertex	NOUN
cana-2702	146	7	set	set	NOUN
cana-2702	146	8	⟨𝑉(𝐶1	⟨𝑉(𝐶1	PROPN
cana-2702	146	9	)	)	PUNCT
cana-2702	146	10	−	−	PROPN
cana-2702	146	11	{	{	PUNCT
cana-2702	146	12	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	146	13	,	,	PUNCT
cana-2702	146	14	𝑏𝑘−1}⟩	𝑏𝑘−1}⟩	PROPN
cana-2702	146	15	is	be	AUX
cana-2702	146	16	isomorphic	isomorphic	ADJ
cana-2702	146	17	to	to	ADP
cana-2702	146	18	𝐾𝑛−2	𝐾𝑛−2	NUM
cana-2702	146	19	,	,	PUNCT
cana-2702	146	20	by	by	ADP
cana-2702	146	21	referring	refer	VERB
cana-2702	146	22	the	the	DET
cana-2702	146	23	pebbling	pebble	VERB
cana-2702	146	24	number	number	NOUN
cana-2702	146	25	of	of	ADP
cana-2702	146	26	complete	complete	ADJ
cana-2702	146	27	graph	graph	NOUN
cana-2702	146	28	we	we	PRON
cana-2702	146	29	can	can	AUX
cana-2702	146	30	move	move	VERB
cana-2702	146	31	four	four	NUM
cana-2702	146	32	pebbles	pebble	NOUN
cana-2702	146	33	to	to	ADP
cana-2702	146	34	𝑎𝑘	𝑎𝑘	VERB
cana-2702	146	35	and	and	CCONJ
cana-2702	146	36	move	move	VERB
cana-2702	146	37	two	two	NUM
cana-2702	146	38	pebbles	pebble	NOUN
cana-2702	146	39	to	to	ADP
cana-2702	146	40	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	146	41	.	.	PUNCT
cana-2702	147	1	then	then	ADV
cana-2702	147	2	,	,	PUNCT
cana-2702	147	3	we	we	PRON
cana-2702	147	4	move	move	VERB
cana-2702	147	5	the	the	DET
cana-2702	147	6	pebble	pebble	NOUN
cana-2702	147	7	to	to	ADP
cana-2702	147	8	𝑦	𝑦	NOUN
cana-2702	147	9	from	from	ADP
cana-2702	147	10	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	147	11	.	.	PUNCT
cana-2702	148	1	lemma	lemma	PROPN
cana-2702	148	2	1	1	X
cana-2702	148	3	.	.	PUNCT
cana-2702	149	1	let	let	VERB
cana-2702	149	2	𝐺	𝐺	PROPN
cana-2702	149	3	=	=	SYM
cana-2702	149	4	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	PROPN
cana-2702	149	5	)	)	PUNCT
cana-2702	149	6	be	be	VERB
cana-2702	149	7	a	a	DET
cana-2702	149	8	crisscross	crisscross	NOUN
cana-2702	149	9	sequence	sequence	NOUN
cana-2702	149	10	of	of	ADP
cana-2702	149	11	𝑚	𝑚	ADP
cana-2702	149	12	complete	complete	ADJ
cana-2702	149	13	graphs	graph	NOUN
cana-2702	149	14	.	.	PUNCT
cana-2702	150	1	let	let	VERB
cana-2702	150	2	us	we	PRON
cana-2702	150	3	define	define	VERB
cana-2702	150	4	𝑋1	𝑋1	NOUN
cana-2702	150	5	=	=	SYM
cana-2702	150	6	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	150	7	(	(	PUNCT
cana-2702	150	8	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	150	9	)	)	PUNCT
cana-2702	150	10	=	=	SYM
cana-2702	150	11	𝐶1	𝐶1	ADJ
cana-2702	150	12	∪.	∪.	X
cana-2702	150	13	.	.	PUNCT
cana-2702	151	1	.∪	.∪	PROPN
cana-2702	151	2	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	151	3	and	and	CCONJ
cana-2702	151	4	𝑋2	𝑋2	VERB
cana-2702	151	5	=	=	SYM
cana-2702	151	6	𝐶𝑚2	𝐶𝑚2	NOUN
cana-2702	151	7	(	(	PUNCT
cana-2702	151	8	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	151	9	)	)	PUNCT
cana-2702	151	10	=	=	PUNCT
cana-2702	152	1	𝐶𝑚1	𝐶𝑚1	X
cana-2702	152	2	+	+	ADJ
cana-2702	152	3	1	1	NUM
cana-2702	152	4	∪.	∪.	NOUN
cana-2702	152	5	.	.	PUNCT
cana-2702	153	1	.∪	.∪	PUNCT
cana-2702	154	1	𝐶𝑚	𝐶𝑚	NOUN
cana-2702	154	2	be	be	AUX
cana-2702	154	3	two	two	NUM
cana-2702	154	4	subgraphs	subgraph	NOUN
cana-2702	154	5	of	of	ADP
cana-2702	154	6	𝐺	𝐺	PROPN
cana-2702	154	7	where	where	SCONJ
cana-2702	154	8	𝑚1	𝑚1	NOUN
cana-2702	154	9	+	+	CCONJ
cana-2702	154	10	𝑚2	𝑚2	NOUN
cana-2702	154	11	=	=	SYM
cana-2702	154	12	𝑚.	𝑚.	PROPN
cana-2702	154	13	suppose	suppose	VERB
cana-2702	154	14	that	that	SCONJ
cana-2702	154	15	the	the	DET
cana-2702	154	16	number	number	NOUN
cana-2702	154	17	of	of	ADP
cana-2702	154	18	pebbles	pebble	NOUN
cana-2702	154	19	distributed	distribute	VERB
cana-2702	154	20	on	on	ADP
cana-2702	154	21	𝑋2	𝑋2	PROPN
cana-2702	154	22	is	be	AUX
cana-2702	154	23	at	at	ADV
cana-2702	154	24	least	least	ADJ
cana-2702	154	25	2	2	NUM
cana-2702	154	26	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	154	27	−	−	PROPN
cana-2702	154	28	1	1	NUM
cana-2702	154	29	)	)	PUNCT
cana-2702	154	30	+	+	CCONJ
cana-2702	154	31	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	154	32	−	−	PROPN
cana-2702	154	33	4	4	NUM
cana-2702	154	34	)	)	PUNCT
cana-2702	154	35	+	+	CCONJ
cana-2702	154	36	2	2	X
cana-2702	154	37	.	.	PUNCT
cana-2702	154	38	then	then	ADV
cana-2702	154	39	,	,	PUNCT
cana-2702	154	40	move	move	VERB
cana-2702	154	41	two	two	NUM
cana-2702	154	42	pebbles	pebble	NOUN
cana-2702	154	43	to	to	ADP
cana-2702	154	44	𝑎	𝑎	PROPN
cana-2702	154	45	𝑚1	𝑚1	NOUN
cana-2702	154	46	(	(	PUNCT
cana-2702	154	47	𝑛	𝑛	PROPN
cana-2702	154	48	2	2	NUM
cana-2702	154	49	−1	−1	NOUN
cana-2702	154	50	)	)	PUNCT
cana-2702	154	51	,	,	PUNCT
cana-2702	154	52	for	for	ADP
cana-2702	154	53	𝑚1	𝑚1	NOUN
cana-2702	154	54	or	or	CCONJ
cana-2702	154	55	𝑎	𝑎	PROPN
cana-2702	154	56	𝑚1	𝑚1	NOUN
cana-2702	154	57	(	(	PUNCT
cana-2702	154	58	𝑛	𝑛	DET
cana-2702	154	59	2	2	NUM
cana-2702	154	60	−1)+1	−1)+1	NOUN
cana-2702	154	61	,	,	PUNCT
cana-2702	154	62	𝑚1	𝑚1	NOUN
cana-2702	154	63	is	be	AUX
cana-2702	154	64	odd	odd	ADJ
cana-2702	154	65	.	.	PUNCT
cana-2702	155	1	communications	communication	NOUN
cana-2702	155	2	on	on	ADP
cana-2702	155	3	applied	apply	VERB
cana-2702	155	4	nonlinear	nonlinear	ADJ
cana-2702	155	5	analysis	analysis	NOUN
cana-2702	155	6	issn	issn	NOUN
cana-2702	155	7	:	:	PUNCT
cana-2702	155	8	1074	1074	NUM
cana-2702	155	9	-	-	PUNCT
cana-2702	155	10	133x	133x	NUM
cana-2702	155	11	vol	vol	NOUN
cana-2702	155	12	32	32	NUM
cana-2702	155	13	no	no	NOUN
cana-2702	155	14	.	.	PUNCT
cana-2702	156	1	3s	3s	NUM
cana-2702	156	2	(	(	PUNCT
cana-2702	156	3	2025	2025	NUM
cana-2702	156	4	)	)	PUNCT
cana-2702	156	5	652	652	NUM
cana-2702	156	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	156	7	proof	proof	NOUN
cana-2702	156	8	.	.	PUNCT
cana-2702	157	1	consider	consider	VERB
cana-2702	157	2	graph	graph	NOUN
cana-2702	157	3	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	157	4	)	)	PUNCT
cana-2702	157	5	with	with	ADP
cana-2702	157	6	2	2	NUM
cana-2702	157	7	𝑚1(2𝑚2	𝑚1(2𝑚2	NOUN
cana-2702	157	8	−	−	PROPN
cana-2702	157	9	1	1	NUM
cana-2702	157	10	)	)	PUNCT
cana-2702	157	11	+	+	CCONJ
cana-2702	157	12	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	157	13	−	−	PROPN
cana-2702	157	14	4	4	NUM
cana-2702	157	15	)	)	PUNCT
cana-2702	157	16	+	+	CCONJ
cana-2702	157	17	2	2	NUM
cana-2702	157	18	pebbles	pebble	NOUN
cana-2702	157	19	distributed	distribute	VERB
cana-2702	157	20	only	only	ADV
cana-2702	157	21	on	on	ADP
cana-2702	157	22	the	the	DET
cana-2702	157	23	vertices	vertex	NOUN
cana-2702	157	24	of	of	ADP
cana-2702	157	25	𝑋2	𝑋2	ADJ
cana-2702	157	26	.	.	PUNCT
cana-2702	158	1	two	two	NUM
cana-2702	158	2	pebbles	pebble	NOUN
cana-2702	158	3	must	must	AUX
cana-2702	158	4	be	be	AUX
cana-2702	158	5	moved	move	VERB
cana-2702	158	6	to	to	ADP
cana-2702	158	7	𝑎	𝑎	PROPN
cana-2702	158	8	𝑚1	𝑚1	NOUN
cana-2702	158	9	(	(	PUNCT
cana-2702	158	10	𝑛	𝑛	PROPN
cana-2702	158	11	2	2	NUM
cana-2702	158	12	−1	−1	NOUN
cana-2702	158	13	)	)	PUNCT
cana-2702	158	14	,	,	PUNCT
cana-2702	158	15	and	and	CCONJ
cana-2702	158	16	we	we	PRON
cana-2702	158	17	prove	prove	VERB
cana-2702	158	18	this	this	DET
cana-2702	158	19	result	result	NOUN
cana-2702	158	20	through	through	ADP
cana-2702	158	21	induction	induction	NOUN
cana-2702	158	22	on	on	ADP
cana-2702	158	23	𝑚2	𝑚2	NOUN
cana-2702	158	24	.	.	PUNCT
cana-2702	159	1	for	for	ADP
cana-2702	159	2	𝑚2	𝑚2	NOUN
cana-2702	159	3	=	=	SYM
cana-2702	159	4	1	1	NUM
cana-2702	159	5	,	,	PUNCT
cana-2702	159	6	at	at	ADV
cana-2702	159	7	least	least	ADJ
cana-2702	159	8	2	2	NUM
cana-2702	159	9	𝑚1	𝑚1	NOUN
cana-2702	159	10	+	+	CCONJ
cana-2702	159	11	𝑛	𝑛	DET
cana-2702	159	12	−	−	NUM
cana-2702	159	13	2	2	NUM
cana-2702	159	14	pebbles	pebble	NOUN
cana-2702	159	15	were	be	AUX
cana-2702	159	16	distributed	distribute	VERB
cana-2702	159	17	on	on	ADP
cana-2702	159	18	𝑋2	𝑋2	VERB
cana-2702	159	19	.	.	PUNCT
cana-2702	160	1	since	since	SCONJ
cana-2702	160	2	𝑚	𝑚	PROPN
cana-2702	160	3	≥	≥	NUM
cana-2702	160	4	4	4	NUM
cana-2702	160	5	.	.	PUNCT
cana-2702	161	1	this	this	PRON
cana-2702	161	2	implies	imply	VERB
cana-2702	161	3	that	that	SCONJ
cana-2702	161	4	at	at	ADP
cana-2702	161	5	least	least	ADJ
cana-2702	161	6	𝑛	𝑛	DET
cana-2702	161	7	+	+	NUM
cana-2702	161	8	6	6	NUM
cana-2702	161	9	pebbles	pebble	NOUN
cana-2702	161	10	were	be	AUX
cana-2702	161	11	retained	retain	VERB
cana-2702	161	12	in	in	ADP
cana-2702	161	13	𝑋2	𝑋2	PROPN
cana-2702	161	14	.	.	PUNCT
cana-2702	162	1	for	for	ADP
cana-2702	162	2	𝑚2	𝑚2	NOUN
cana-2702	162	3	=	=	SYM
cana-2702	162	4	1	1	NUM
cana-2702	162	5	,	,	PUNCT
cana-2702	162	6	we	we	PRON
cana-2702	162	7	have	have	AUX
cana-2702	162	8	𝑋2	𝑋2	VERB
cana-2702	163	1	≅	≅	PROPN
cana-2702	163	2	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	163	3	in	in	ADP
cana-2702	163	4	addition	addition	NOUN
cana-2702	163	5	,	,	PUNCT
cana-2702	163	6	the	the	DET
cana-2702	163	7	two	two	NUM
cana-2702	163	8	pebbling	pebble	VERB
cana-2702	163	9	numbers	number	NOUN
cana-2702	163	10	of	of	ADP
cana-2702	163	11	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	163	12	are	be	AUX
cana-2702	163	13	𝑓2(𝐾𝑛	𝑓2(𝐾𝑛	NOUN
cana-2702	163	14	)	)	PUNCT
cana-2702	163	15	=	=	SYM
cana-2702	163	16	𝑛	𝑛	PROPN
cana-2702	164	1	+	+	NOUN
cana-2702	164	2	2	2	NUM
cana-2702	164	3	according	accord	VERB
cana-2702	164	4	to	to	ADP
cana-2702	164	5	the	the	DET
cana-2702	164	6	pebbling	pebble	VERB
cana-2702	164	7	number	number	NOUN
cana-2702	164	8	of	of	ADP
cana-2702	164	9	complete	complete	ADJ
cana-2702	164	10	graph	graph	NOUN
cana-2702	164	11	.	.	PUNCT
cana-2702	165	1	thus	thus	ADV
cana-2702	165	2	,	,	PUNCT
cana-2702	165	3	the	the	DET
cana-2702	165	4	two	two	NUM
cana-2702	165	5	pebbles	pebble	NOUN
cana-2702	165	6	can	can	AUX
cana-2702	165	7	move	move	VERB
cana-2702	165	8	2	2	NUM
cana-2702	165	9	pebbles	pebble	NOUN
cana-2702	165	10	to	to	ADP
cana-2702	165	11	𝑎	𝑎	PROPN
cana-2702	165	12	𝑚1	𝑚1	NOUN
cana-2702	165	13	(	(	PUNCT
cana-2702	165	14	𝑛	𝑛	PROPN
cana-2702	165	15	2	2	NUM
cana-2702	165	16	−1	−1	NOUN
cana-2702	165	17	)	)	PUNCT
cana-2702	165	18	.	.	PUNCT
cana-2702	166	1	now	now	ADV
cana-2702	166	2	,	,	PUNCT
cana-2702	166	3	we	we	PRON
cana-2702	166	4	assume	assume	VERB
cana-2702	166	5	that	that	SCONJ
cana-2702	166	6	the	the	DET
cana-2702	166	7	result	result	NOUN
cana-2702	166	8	is	be	AUX
cana-2702	166	9	true	true	ADJ
cana-2702	166	10	for	for	ADP
cana-2702	166	11	all	all	DET
cana-2702	166	12	𝑚′2	𝑚′2	NOUN
cana-2702	166	13	<	<	X
cana-2702	166	14	𝑚2	𝑚2	NOUN
cana-2702	166	15	.	.	PUNCT
cana-2702	167	1	let	let	VERB
cana-2702	167	2	2	2	NUM
cana-2702	167	3	𝑚1(2𝑚2	𝑚1(2𝑚2	ADP
cana-2702	167	4	−	−	PROPN
cana-2702	167	5	1	1	NUM
cana-2702	167	6	)	)	PUNCT
cana-2702	168	1	+	+	CCONJ
cana-2702	169	1	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	169	2	−	−	PROPN
cana-2702	169	3	4	4	NUM
cana-2702	169	4	)	)	PUNCT
cana-2702	169	5	+	+	CCONJ
cana-2702	169	6	2	2	NUM
cana-2702	169	7	pebbles	pebble	NOUN
cana-2702	169	8	be	be	AUX
cana-2702	169	9	distributed	distribute	VERB
cana-2702	169	10	on	on	ADP
cana-2702	169	11	𝑋2	𝑋2	PROPN
cana-2702	169	12	.	.	PUNCT
cana-2702	170	1	suppose	suppose	VERB
cana-2702	170	2	the	the	DET
cana-2702	170	3	number	number	NOUN
cana-2702	170	4	of	of	ADP
cana-2702	170	5	pebbles	pebble	NOUN
cana-2702	170	6	distributed	distribute	VERB
cana-2702	170	7	on	on	ADP
cana-2702	170	8	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	170	9	+	+	NOUN
cana-2702	170	10	1	1	NUM
cana-2702	170	11	is	be	AUX
cana-2702	170	12	at	at	ADP
cana-2702	170	13	least	least	ADJ
cana-2702	170	14	𝑛	𝑛	PRON
cana-2702	170	15	+	+	ADJ
cana-2702	170	16	2	2	NUM
cana-2702	170	17	.	.	PUNCT
cana-2702	170	18	then	then	ADV
cana-2702	170	19	,	,	PUNCT
cana-2702	170	20	the	the	DET
cana-2702	170	21	two	two	NUM
cana-2702	170	22	pebbles	pebble	NOUN
cana-2702	170	23	can	can	AUX
cana-2702	170	24	be	be	AUX
cana-2702	170	25	moved	move	VERB
cana-2702	170	26	to	to	ADP
cana-2702	170	27	𝑎	𝑎	PROPN
cana-2702	170	28	𝑚1	𝑚1	NOUN
cana-2702	170	29	(	(	PUNCT
cana-2702	170	30	𝑛	𝑛	PROPN
cana-2702	170	31	2	2	NUM
cana-2702	170	32	−1	−1	NOUN
cana-2702	170	33	)	)	PUNCT
cana-2702	170	34	.	.	PUNCT
cana-2702	171	1	therefore	therefore	ADV
cana-2702	171	2	,	,	PUNCT
cana-2702	171	3	we	we	PRON
cana-2702	171	4	assume	assume	VERB
cana-2702	171	5	that	that	SCONJ
cana-2702	171	6	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	PROPN
cana-2702	171	7	+	+	NOUN
cana-2702	171	8	1	1	NUM
cana-2702	171	9	)	)	PUNCT
cana-2702	171	10	≤	≤	NOUN
cana-2702	171	11	𝑛	𝑛	DET
cana-2702	171	12	+	+	NOUN
cana-2702	171	13	1	1	X
cana-2702	171	14	.	.	X
cana-2702	172	1	we	we	PRON
cana-2702	172	2	have	have	VERB
cana-2702	172	3	the	the	DET
cana-2702	172	4	following	follow	VERB
cana-2702	172	5	cases	case	NOUN
cana-2702	172	6	.	.	PUNCT
cana-2702	173	1	case	case	NOUN
cana-2702	173	2	(	(	PUNCT
cana-2702	173	3	1	1	NUM
cana-2702	173	4	)	)	PUNCT
cana-2702	173	5	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	PROPN
cana-2702	173	6	+	+	PROPN
cana-2702	173	7	1	1	NUM
cana-2702	173	8	)	)	PUNCT
cana-2702	173	9	≥	≥	NOUN
cana-2702	173	10	𝑛.	𝑛.	NOUN
cana-2702	173	11	based	base	VERB
cana-2702	173	12	on	on	ADP
cana-2702	173	13	our	our	PRON
cana-2702	173	14	assumptions	assumption	NOUN
cana-2702	173	15	,	,	PUNCT
cana-2702	173	16	we	we	PRON
cana-2702	173	17	can	can	AUX
cana-2702	173	18	move	move	VERB
cana-2702	173	19	one	one	NUM
cana-2702	173	20	pebble	pebble	NOUN
cana-2702	173	21	to	to	ADP
cana-2702	173	22	𝑎	𝑎	PROPN
cana-2702	173	23	𝑚1	𝑚1	NOUN
cana-2702	173	24	(	(	PUNCT
cana-2702	173	25	𝑛	𝑛	PROPN
cana-2702	173	26	2	2	NUM
cana-2702	173	27	−1	−1	NOUN
cana-2702	173	28	)	)	PUNCT
cana-2702	173	29	.	.	PUNCT
cana-2702	174	1	this	this	PRON
cana-2702	174	2	implies	imply	VERB
cana-2702	174	3	that	that	SCONJ
cana-2702	174	4	at	at	ADV
cana-2702	174	5	least	least	ADJ
cana-2702	174	6	2	2	NUM
cana-2702	174	7	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	174	8	−	−	PROPN
cana-2702	174	9	1	1	NUM
cana-2702	174	10	)	)	PUNCT
cana-2702	174	11	+	+	CCONJ
cana-2702	174	12	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	174	13	−	−	PROPN
cana-2702	174	14	4	4	NUM
cana-2702	174	15	)	)	PUNCT
cana-2702	174	16	+	+	CCONJ
cana-2702	174	17	2	2	NUM
cana-2702	174	18	−	−	NOUN
cana-2702	174	19	𝑛	𝑛	DET
cana-2702	174	20	pebbles	pebble	NOUN
cana-2702	174	21	are	be	AUX
cana-2702	174	22	retained	retain	VERB
cana-2702	174	23	in	in	ADP
cana-2702	174	24	⟨𝑋2	⟨𝑋2	NOUN
cana-2702	174	25	−	−	NOUN
cana-2702	174	26	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	174	27	+	+	NOUN
cana-2702	174	28	1⟩.	1⟩.	PROPN
cana-2702	174	29	we	we	PRON
cana-2702	174	30	need	need	VERB
cana-2702	174	31	to	to	PART
cana-2702	174	32	make	make	VERB
cana-2702	174	33	the	the	DET
cana-2702	174	34	following	follow	VERB
cana-2702	174	35	claim	claim	NOUN
cana-2702	174	36	:	:	PUNCT
cana-2702	174	37	claim	claim	NOUN
cana-2702	174	38	(	(	PUNCT
cana-2702	174	39	1	1	NUM
cana-2702	174	40	)	)	PUNCT
cana-2702	174	41	𝑝(⟨𝑋2	𝑝(⟨𝑋2	ADJ
cana-2702	174	42	−	−	NOUN
cana-2702	174	43	𝐶𝑚1	𝐶𝑚1	ADV
cana-2702	174	44	+	+	NOUN
cana-2702	174	45	1⟩	1⟩	NUM
cana-2702	174	46	)	)	PUNCT
cana-2702	174	47	≥	≥	NOUN
cana-2702	174	48	2	2	NUM
cana-2702	174	49	𝑚1(2𝑚2−1	𝑚1(2𝑚2−1	NOUN
cana-2702	174	50	−	−	NOUN
cana-2702	174	51	1	1	NUM
cana-2702	174	52	)	)	PUNCT
cana-2702	174	53	+	+	CCONJ
cana-2702	174	54	(	(	PUNCT
cana-2702	174	55	𝑚2	𝑚2	NOUN
cana-2702	174	56	−	−	NOUN
cana-2702	174	57	1)(𝑛	1)(𝑛	NUM
cana-2702	174	58	−	−	NOUN
cana-2702	174	59	4	4	NUM
cana-2702	174	60	)	)	PUNCT
cana-2702	174	61	+	+	NOUN
cana-2702	174	62	2	2	X
cana-2702	174	63	.	.	X
cana-2702	174	64	we	we	PRON
cana-2702	174	65	have	have	VERB
cana-2702	174	66	,	,	PUNCT
cana-2702	174	67	𝑝(⟨𝑋2	𝑝(⟨𝑋2	ADJ
cana-2702	174	68	−	−	NOUN
cana-2702	174	69	𝐶𝑚1	𝐶𝑚1	ADV
cana-2702	174	70	+	+	NOUN
cana-2702	174	71	1⟩	1⟩	NUM
cana-2702	174	72	)	)	PUNCT
cana-2702	174	73	=	=	SYM
cana-2702	174	74	2	2	X
cana-2702	174	75	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	174	76	−	−	PROPN
cana-2702	174	77	1	1	NUM
cana-2702	174	78	)	)	PUNCT
cana-2702	174	79	+	+	CCONJ
cana-2702	174	80	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	174	81	−	−	PROPN
cana-2702	174	82	4	4	NUM
cana-2702	174	83	)	)	PUNCT
cana-2702	174	84	+	+	CCONJ
cana-2702	174	85	2	2	NUM
cana-2702	174	86	−	−	NOUN
cana-2702	174	87	𝑛	𝑛	NOUN
cana-2702	174	88	=	=	SYM
cana-2702	174	89	2	2	NUM
cana-2702	174	90	𝑚1(2.2𝑚2−1	𝑚1(2.2𝑚2−1	NOUN
cana-2702	174	91	)	)	PUNCT
cana-2702	175	1	+	+	CCONJ
cana-2702	175	2	(	(	PUNCT
cana-2702	175	3	𝑚2	𝑚2	NOUN
cana-2702	175	4	−	−	NOUN
cana-2702	175	5	1)(𝑛	1)(𝑛	NUM
cana-2702	175	6	−	−	NOUN
cana-2702	175	7	4	4	NUM
cana-2702	175	8	)	)	PUNCT
cana-2702	176	1	+	+	CCONJ
cana-2702	176	2	(	(	PUNCT
cana-2702	176	3	𝑛	𝑛	DET
cana-2702	176	4	−	−	PROPN
cana-2702	176	5	4	4	NUM
cana-2702	176	6	)	)	PUNCT
cana-2702	176	7	+	+	CCONJ
cana-2702	177	1	2	2	NUM
cana-2702	177	2	−	−	NOUN
cana-2702	177	3	𝑛	𝑛	NOUN
cana-2702	177	4	=	=	SYM
cana-2702	177	5	2	2	NUM
cana-2702	177	6	𝑚1(2𝑚2−1	𝑚1(2𝑚2−1	NOUN
cana-2702	177	7	−	−	NOUN
cana-2702	177	8	1	1	NUM
cana-2702	177	9	)	)	PUNCT
cana-2702	177	10	+	+	CCONJ
cana-2702	177	11	(	(	PUNCT
cana-2702	177	12	𝑚2	𝑚2	NOUN
cana-2702	177	13	−	−	NOUN
cana-2702	177	14	1)(𝑛	1)(𝑛	NUM
cana-2702	177	15	−	−	NOUN
cana-2702	177	16	4	4	NUM
cana-2702	177	17	)	)	PUNCT
cana-2702	177	18	−	−	PROPN
cana-2702	177	19	2	2	NUM
cana-2702	177	20	≥	≥	NOUN
cana-2702	177	21	2	2	NUM
cana-2702	177	22	𝑚1(2𝑚2−1	𝑚1(2𝑚2−1	NOUN
cana-2702	177	23	−	−	NOUN
cana-2702	177	24	1	1	NUM
cana-2702	177	25	)	)	PUNCT
cana-2702	177	26	+	+	CCONJ
cana-2702	177	27	(	(	PUNCT
cana-2702	177	28	𝑚2	𝑚2	NOUN
cana-2702	177	29	−	−	NOUN
cana-2702	178	1	1)(𝑛	1)(𝑛	NUM
cana-2702	179	1	−	−	NOUN
cana-2702	179	2	4	4	NUM
cana-2702	179	3	)	)	PUNCT
cana-2702	179	4	+	+	NUM
cana-2702	179	5	2	2	NUM
cana-2702	179	6	,	,	PUNCT
cana-2702	179	7	since	since	SCONJ
cana-2702	179	8	𝑚1	𝑚1	NOUN
cana-2702	179	9	+	+	CCONJ
cana-2702	179	10	𝑚2	𝑚2	NOUN
cana-2702	179	11	=	=	SYM
cana-2702	179	12	𝑛	𝑛	PRON
cana-2702	179	13	≥	≥	NUM
cana-2702	179	14	4	4	NUM
cana-2702	179	15	.	.	PUNCT
cana-2702	180	1	thus	thus	ADV
cana-2702	180	2	,	,	PUNCT
cana-2702	180	3	claim	claim	NOUN
cana-2702	180	4	(	(	PUNCT
cana-2702	180	5	1	1	X
cana-2702	180	6	)	)	PUNCT
cana-2702	180	7	allows	allow	VERB
cana-2702	180	8	us	we	PRON
cana-2702	180	9	to	to	PART
cana-2702	180	10	move	move	VERB
cana-2702	180	11	two	two	NUM
cana-2702	180	12	pebbles	pebble	NOUN
cana-2702	180	13	to	to	ADP
cana-2702	180	14	𝑎	𝑎	PROPN
cana-2702	180	15	(	(	PUNCT
cana-2702	180	16	𝑚1	𝑚1	NOUN
cana-2702	180	17	+	+	PROPN
cana-2702	180	18	1	1	NUM
cana-2702	180	19	)	)	PUNCT
cana-2702	180	20	(	(	PUNCT
cana-2702	180	21	𝑛	𝑛	DET
cana-2702	180	22	2	2	NUM
cana-2702	180	23	−1)+1	−1)+1	NOUN
cana-2702	180	24	.	.	PUNCT
cana-2702	181	1	through	through	ADP
cana-2702	181	2	induction	induction	NOUN
cana-2702	181	3	,	,	PUNCT
cana-2702	181	4	we	we	PRON
cana-2702	181	5	can	can	AUX
cana-2702	181	6	move	move	VERB
cana-2702	181	7	an	an	DET
cana-2702	181	8	additional	additional	ADJ
cana-2702	181	9	pebble	pebble	NOUN
cana-2702	181	10	to	to	ADP
cana-2702	181	11	𝑎	𝑎	PROPN
cana-2702	181	12	𝑚1	𝑚1	NOUN
cana-2702	181	13	(	(	PUNCT
cana-2702	181	14	𝑛	𝑛	PROPN
cana-2702	181	15	2	2	NUM
cana-2702	181	16	−1	−1	NOUN
cana-2702	181	17	)	)	PUNCT
cana-2702	181	18	.	.	PUNCT
cana-2702	182	1	case	case	NOUN
cana-2702	182	2	(	(	PUNCT
cana-2702	182	3	2	2	NUM
cana-2702	182	4	)	)	PUNCT
cana-2702	182	5	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	PROPN
cana-2702	182	6	+	+	PROPN
cana-2702	182	7	1	1	NUM
cana-2702	182	8	)	)	PUNCT
cana-2702	182	9	<	<	X
cana-2702	182	10	𝑛.suppose	𝑛.suppose	X
cana-2702	182	11	that	that	SCONJ
cana-2702	182	12	any	any	DET
cana-2702	182	13	vertex	vertex	NOUN
cana-2702	182	14	belonging	belong	VERB
cana-2702	182	15	to	to	ADP
cana-2702	182	16	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	182	17	+	+	NOUN
cana-2702	182	18	1	1	NUM
cana-2702	182	19	has	have	AUX
cana-2702	182	20	at	at	ADV
cana-2702	182	21	least	least	ADV
cana-2702	182	22	two	two	NUM
cana-2702	182	23	pebbles	pebble	NOUN
cana-2702	182	24	.	.	PUNCT
cana-2702	183	1	then	then	ADV
cana-2702	183	2	,	,	PUNCT
cana-2702	183	3	we	we	PRON
cana-2702	183	4	can	can	AUX
cana-2702	183	5	move	move	VERB
cana-2702	183	6	a	a	DET
cana-2702	183	7	pebble	pebble	NOUN
cana-2702	183	8	to	to	ADP
cana-2702	183	9	𝑎	𝑎	PROPN
cana-2702	183	10	𝑚1	𝑚1	NOUN
cana-2702	183	11	(	(	PUNCT
cana-2702	183	12	𝑛	𝑛	PROPN
cana-2702	183	13	2	2	NUM
cana-2702	183	14	−1	−1	NOUN
cana-2702	183	15	)	)	PUNCT
cana-2702	183	16	and	and	CCONJ
cana-2702	183	17	by	by	ADP
cana-2702	183	18	claim	claim	NOUN
cana-2702	183	19	(	(	PUNCT
cana-2702	183	20	1	1	NUM
cana-2702	183	21	)	)	PUNCT
cana-2702	183	22	,	,	PUNCT
cana-2702	183	23	we	we	PRON
cana-2702	183	24	can	can	AUX
cana-2702	183	25	move	move	VERB
cana-2702	183	26	an	an	DET
cana-2702	183	27	additional	additional	ADJ
cana-2702	183	28	pebble	pebble	NOUN
cana-2702	183	29	to	to	ADP
cana-2702	183	30	𝑎	𝑎	PROPN
cana-2702	183	31	𝑚1	𝑚1	NOUN
cana-2702	183	32	(	(	PUNCT
cana-2702	183	33	𝑛	𝑛	PROPN
cana-2702	183	34	2	2	NUM
cana-2702	183	35	−1	−1	NOUN
cana-2702	183	36	)	)	PUNCT
cana-2702	183	37	.	.	PUNCT
cana-2702	184	1	therefore	therefore	ADV
cana-2702	184	2	,	,	PUNCT
cana-2702	184	3	we	we	PRON
cana-2702	184	4	assume	assume	VERB
cana-2702	184	5	that	that	SCONJ
cana-2702	184	6	no	no	DET
cana-2702	184	7	vertices	vertex	NOUN
cana-2702	184	8	in	in	ADP
cana-2702	184	9	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	184	10	+	+	NOUN
cana-2702	184	11	1	1	NUM
cana-2702	184	12	contain	contain	VERB
cana-2702	184	13	two	two	NUM
cana-2702	184	14	pebbles	pebble	NOUN
cana-2702	184	15	.	.	PUNCT
cana-2702	185	1	without	without	ADP
cana-2702	185	2	loss	loss	NOUN
cana-2702	185	3	of	of	ADP
cana-2702	185	4	generality	generality	NOUN
cana-2702	185	5	,	,	PUNCT
cana-2702	185	6	we	we	PRON
cana-2702	185	7	assume	assume	VERB
cana-2702	185	8	that	that	SCONJ
cana-2702	185	9	most	most	ADJ
cana-2702	185	10	pebbles	pebble	NOUN
cana-2702	185	11	are	be	AUX
cana-2702	185	12	distributed	distribute	VERB
cana-2702	185	13	on	on	ADP
cana-2702	185	14	the	the	DET
cana-2702	185	15	vertices	vertex	NOUN
cana-2702	185	16	of	of	ADP
cana-2702	185	17	𝐶1	𝐶1	PRON
cana-2702	185	18	.	.	PUNCT
cana-2702	186	1	for	for	ADP
cana-2702	186	2	the	the	DET
cana-2702	186	3	least	least	ADJ
cana-2702	186	4	case	case	NOUN
cana-2702	186	5	scenario	scenario	NOUN
cana-2702	186	6	,	,	PUNCT
cana-2702	186	7	we	we	PRON
cana-2702	186	8	consider	consider	VERB
cana-2702	186	9	that	that	SCONJ
cana-2702	186	10	all	all	DET
cana-2702	186	11	the	the	DET
cana-2702	186	12	𝑎𝑖	𝑎𝑖	NOUN
cana-2702	186	13	’s	’s	NOUN
cana-2702	186	14	and	and	CCONJ
cana-2702	186	15	𝑏𝑖	𝑏𝑖	ADP
cana-2702	186	16	’s	’s	NOUN
cana-2702	186	17	in	in	ADV
cana-2702	186	18	𝑋2	𝑋2	PROPN
cana-2702	186	19	have	have	VERB
cana-2702	186	20	one	one	NUM
cana-2702	186	21	pebble	pebble	NOUN
cana-2702	186	22	on	on	ADP
cana-2702	186	23	each	each	PRON
cana-2702	186	24	and	and	CCONJ
cana-2702	186	25	then	then	ADV
cana-2702	186	26	place	place	VERB
cana-2702	186	27	the	the	DET
cana-2702	186	28	remaining	remain	VERB
cana-2702	186	29	pebbles	pebble	NOUN
cana-2702	186	30	on	on	ADP
cana-2702	186	31	𝑦.	𝑦.	PROPN
cana-2702	186	32	this	this	PRON
cana-2702	186	33	implies	imply	VERB
cana-2702	186	34	at	at	ADV
cana-2702	186	35	least	least	ADV
cana-2702	186	36	2	2	NUM
cana-2702	187	1	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	187	2	−	−	PROPN
cana-2702	187	3	1	1	NUM
cana-2702	187	4	)	)	PUNCT
cana-2702	188	1	+	+	CCONJ
cana-2702	188	2	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	188	3	−	−	PROPN
cana-2702	188	4	4	4	NUM
cana-2702	188	5	)	)	PUNCT
cana-2702	188	6	+	+	CCONJ
cana-2702	188	7	2	2	NUM
cana-2702	188	8	−	−	NOUN
cana-2702	188	9	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	188	10	−	−	PROPN
cana-2702	188	11	4	4	NUM
cana-2702	188	12	)	)	PUNCT
cana-2702	188	13	=	=	SYM
cana-2702	189	1	2	2	X
cana-2702	189	2	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	189	3	−	−	PROPN
cana-2702	189	4	1	1	NUM
cana-2702	189	5	)	)	PUNCT
cana-2702	189	6	+	+	CCONJ
cana-2702	189	7	2	2	NUM
cana-2702	189	8	=	=	SYM
cana-2702	189	9	2	2	NUM
cana-2702	189	10	𝑚1+𝑚2	𝑚1+𝑚2	NOUN
cana-2702	189	11	−	−	PROPN
cana-2702	189	12	2	2	NUM
cana-2702	189	13	𝑚1	𝑚1	NOUN
cana-2702	189	14	+	+	CCONJ
cana-2702	189	15	2	2	NUM
cana-2702	189	16	>	>	SYM
cana-2702	189	17	2.2𝑚2	2.2𝑚2	NUM
cana-2702	189	18	pebbles	pebble	NOUN
cana-2702	189	19	retained	retain	VERB
cana-2702	189	20	in	in	ADP
cana-2702	189	21	the	the	DET
cana-2702	189	22	vertex	vertex	NOUN
cana-2702	189	23	𝑦.	𝑦.	PROPN
cana-2702	189	24	then	then	ADV
cana-2702	189	25	,	,	PUNCT
cana-2702	189	26	using	use	VERB
cana-2702	189	27	2	2	NUM
cana-2702	189	28	𝑚2	𝑚2	NOUN
cana-2702	189	29	pebbles	pebble	NOUN
cana-2702	189	30	twice	twice	ADV
cana-2702	189	31	,	,	PUNCT
cana-2702	189	32	the	the	DET
cana-2702	189	33	two	two	NUM
cana-2702	189	34	pebbles	pebble	NOUN
cana-2702	189	35	can	can	AUX
cana-2702	189	36	move	move	VERB
cana-2702	189	37	to	to	ADP
cana-2702	189	38	𝑎	𝑎	PROPN
cana-2702	189	39	𝑚1	𝑚1	NOUN
cana-2702	189	40	(	(	PUNCT
cana-2702	189	41	𝑛	𝑛	PROPN
cana-2702	189	42	2	2	NUM
cana-2702	189	43	−1	−1	NOUN
cana-2702	189	44	)	)	PUNCT
cana-2702	189	45	.	.	PUNCT
cana-2702	190	1	theorem	theorem	VERB
cana-2702	190	2	3	3	NUM
cana-2702	190	3	.	.	X
cana-2702	190	4	for	for	ADP
cana-2702	190	5	any	any	DET
cana-2702	190	6	𝑛	𝑛	PROPN
cana-2702	190	7	,	,	PUNCT
cana-2702	190	8	𝑚	𝑚	PROPN
cana-2702	190	9	,	,	PUNCT
cana-2702	190	10	the	the	DET
cana-2702	190	11	pebbling	pebble	VERB
cana-2702	190	12	number	number	NOUN
cana-2702	190	13	of	of	ADP
cana-2702	190	14	graph	graph	NOUN
cana-2702	190	15	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	190	16	)	)	PUNCT
cana-2702	190	17	is	be	AUX
cana-2702	190	18	𝑓(𝐶𝑚(𝐾𝑛	𝑓(𝐶𝑚(𝐾𝑛	NUM
cana-2702	190	19	)	)	PUNCT
cana-2702	190	20	)	)	PUNCT
cana-2702	191	1	=	=	SYM
cana-2702	191	2	2	2	NUM
cana-2702	191	3	𝑚	𝑚	NOUN
cana-2702	191	4	+	+	NOUN
cana-2702	191	5	2(𝑛	2(𝑛	NUM
cana-2702	191	6	−	−	NOUN
cana-2702	191	7	3	3	NUM
cana-2702	191	8	)	)	PUNCT
cana-2702	191	9	+	+	CCONJ
cana-2702	191	10	(	(	PUNCT
cana-2702	191	11	𝑚	𝑚	PROPN
cana-2702	191	12	−	−	PROPN
cana-2702	191	13	2)(𝑛	2)(𝑛	NUM
cana-2702	191	14	−	−	NOUN
cana-2702	191	15	4	4	NUM
cana-2702	191	16	)	)	PUNCT
cana-2702	191	17	.	.	PUNCT
cana-2702	192	1	proof	proof	NOUN
cana-2702	192	2	.	.	PUNCT
cana-2702	193	1	set	set	VERB
cana-2702	193	2	2	2	NUM
cana-2702	193	3	𝑚	𝑚	ADP
cana-2702	193	4	−	−	NUM
cana-2702	193	5	1	1	NUM
cana-2702	193	6	pebbles	pebble	NOUN
cana-2702	193	7	on	on	ADP
cana-2702	193	8	vertex	vertex	NOUN
cana-2702	193	9	𝑥	𝑥	PROPN
cana-2702	193	10	and	and	CCONJ
cana-2702	193	11	put	put	VERB
cana-2702	193	12	one	one	NUM
cana-2702	193	13	pebble	pebble	NOUN
cana-2702	193	14	on	on	ADP
cana-2702	193	15	each	each	DET
cana-2702	193	16	𝑎𝑖	𝑎𝑖	NOUN
cana-2702	193	17	and	and	CCONJ
cana-2702	193	18	𝑏𝑖	𝑏𝑖	ADP
cana-2702	193	19	except	except	SCONJ
cana-2702	193	20	for	for	ADP
cana-2702	193	21	the	the	DET
cana-2702	193	22	vertices	vertex	NOUN
cana-2702	193	23	in	in	ADP
cana-2702	193	24	{	{	PUNCT
cana-2702	193	25	𝑎𝑖𝑎𝑖+1	𝑎𝑖𝑎𝑖+1	PROPN
cana-2702	193	26	,	,	PUNCT
cana-2702	193	27	𝑏𝑖𝑏𝑖+1	𝑏𝑖𝑏𝑖+1	NOUN
cana-2702	193	28	:	:	PUNCT
cana-2702	193	29	1	1	NUM
cana-2702	193	30	≤	≤	NUM
cana-2702	193	31	𝑖	𝑖	PUNCT
cana-2702	193	32	≤	≤	ADJ
cana-2702	193	33	𝑛(𝑘	𝑛(𝑘	NOUN
cana-2702	193	34	−	−	PROPN
cana-2702	193	35	1	1	NUM
cana-2702	193	36	)	)	PUNCT
cana-2702	193	37	−	−	PROPN
cana-2702	193	38	1	1	NUM
cana-2702	193	39	}	}	PUNCT
cana-2702	193	40	.	.	PUNCT
cana-2702	194	1	thus	thus	ADV
cana-2702	194	2	,	,	PUNCT
cana-2702	194	3	we	we	PRON
cana-2702	194	4	can	can	AUX
cana-2702	194	5	not	not	PART
cana-2702	194	6	move	move	VERB
cana-2702	194	7	one	one	NUM
cana-2702	194	8	pebble	pebble	NOUN
cana-2702	194	9	to	to	ADP
cana-2702	194	10	vertex	vertex	NOUN
cana-2702	194	11	𝑦.	𝑦.	PROPN
cana-2702	195	1	so	so	ADV
cana-2702	195	2	𝑓(𝐶𝑚(𝐾𝑛	𝑓(𝐶𝑚(𝐾𝑛	NUM
cana-2702	195	3	)	)	PUNCT
cana-2702	195	4	)	)	PUNCT
cana-2702	196	1	≥	≥	NOUN
cana-2702	196	2	2	2	NUM
cana-2702	196	3	𝑚	𝑚	ADP
cana-2702	196	4	+	+	NOUN
cana-2702	196	5	2(𝑛	2(𝑛	NUM
cana-2702	196	6	−	−	NOUN
cana-2702	196	7	3	3	NUM
cana-2702	196	8	)	)	PUNCT
cana-2702	196	9	+	+	CCONJ
cana-2702	196	10	(	(	PUNCT
cana-2702	196	11	𝑚	𝑚	PROPN
cana-2702	196	12	−	−	PROPN
cana-2702	196	13	2)(𝑛	2)(𝑛	NUM
cana-2702	196	14	−	−	NOUN
cana-2702	196	15	4	4	NUM
cana-2702	196	16	)	)	PUNCT
cana-2702	196	17	.	.	PUNCT
cana-2702	197	1	consider	consider	VERB
cana-2702	197	2	a	a	DET
cana-2702	197	3	graph	graph	NOUN
cana-2702	197	4	with	with	ADP
cana-2702	197	5	at	at	ADV
cana-2702	197	6	least	least	ADV
cana-2702	197	7	2	2	NUM
cana-2702	197	8	𝑚	𝑚	NOUN
cana-2702	197	9	+	+	NOUN
cana-2702	197	10	2(𝑛	2(𝑛	NUM
cana-2702	197	11	−	−	NOUN
cana-2702	197	12	3	3	NUM
cana-2702	197	13	)	)	PUNCT
cana-2702	197	14	+	+	CCONJ
cana-2702	197	15	(	(	PUNCT
cana-2702	197	16	𝑚	𝑚	PROPN
cana-2702	197	17	−	−	PROPN
cana-2702	197	18	2)(𝑛	2)(𝑛	NUM
cana-2702	197	19	−	−	NOUN
cana-2702	197	20	4	4	NUM
cana-2702	197	21	)	)	PUNCT
cana-2702	197	22	pebbles	pebble	NOUN
cana-2702	197	23	,	,	PUNCT
cana-2702	197	24	let	let	VERB
cana-2702	197	25	𝑣	𝑣	PRON
cana-2702	197	26	∈	∈	PROPN
cana-2702	197	27	𝐶𝑖	𝐶𝑖	PROPN
cana-2702	197	28	,	,	PUNCT
cana-2702	197	29	𝑖	𝑖	SYM
cana-2702	197	30	∈	∈	PROPN
cana-2702	197	31	{	{	PUNCT
cana-2702	197	32	1,2	1,2	NUM
cana-2702	197	33	,	,	PUNCT
cana-2702	197	34	.	.	PUNCT
cana-2702	197	35	.	.	PUNCT
cana-2702	198	1	.	.	PUNCT
cana-2702	199	1	,	,	PUNCT
cana-2702	199	2	𝑚	𝑚	AUX
cana-2702	199	3	}	}	PUNCT
cana-2702	199	4	be	be	AUX
cana-2702	199	5	any	any	DET
cana-2702	199	6	target	target	NOUN
cana-2702	199	7	vertex	vertex	NOUN
cana-2702	199	8	.	.	PUNCT
cana-2702	200	1	we	we	PRON
cana-2702	200	2	prove	prove	VERB
cana-2702	200	3	this	this	DET
cana-2702	200	4	result	result	NOUN
cana-2702	200	5	by	by	ADP
cana-2702	200	6	induction	induction	NOUN
cana-2702	200	7	on	on	ADP
cana-2702	200	8	𝑚.	𝑚.	NOUN
cana-2702	200	9	for	for	ADP
cana-2702	200	10	𝑚	𝑚	NOUN
cana-2702	200	11	=	=	SYM
cana-2702	200	12	2	2	NUM
cana-2702	200	13	and	and	CCONJ
cana-2702	200	14	𝑚	𝑚	X
cana-2702	200	15	=	=	SYM
cana-2702	200	16	3	3	NUM
cana-2702	200	17	,	,	PUNCT
cana-2702	200	18	the	the	DET
cana-2702	200	19	results	result	NOUN
cana-2702	200	20	follow	follow	VERB
cana-2702	200	21	from	from	ADP
cana-2702	200	22	theorems	theorem	NOUN
cana-2702	200	23	1	1	NUM
cana-2702	200	24	and	and	CCONJ
cana-2702	200	25	2	2	NUM
cana-2702	200	26	respectively	respectively	ADV
cana-2702	200	27	.	.	PUNCT
cana-2702	201	1	we	we	PRON
cana-2702	201	2	assume	assume	VERB
cana-2702	201	3	that	that	SCONJ
cana-2702	201	4	the	the	DET
cana-2702	201	5	result	result	NOUN
cana-2702	201	6	is	be	AUX
cana-2702	201	7	true	true	ADJ
cana-2702	201	8	for	for	SCONJ
cana-2702	201	9	all	all	DET
cana-2702	201	10	𝑚′	𝑚′	NUM
cana-2702	201	11	<	<	X
cana-2702	201	12	𝑚.	𝑚.	NOUN
cana-2702	201	13	we	we	PRON
cana-2702	201	14	prove	prove	VERB
cana-2702	201	15	this	this	DET
cana-2702	201	16	result	result	NOUN
cana-2702	201	17	for	for	ADP
cana-2702	201	18	all	all	DET
cana-2702	201	19	𝑚.	𝑚.	NOUN
cana-2702	201	20	we	we	PRON
cana-2702	201	21	consider	consider	VERB
cana-2702	201	22	the	the	DET
cana-2702	201	23	following	follow	VERB
cana-2702	201	24	cases	case	NOUN
cana-2702	201	25	:	:	PUNCT
cana-2702	201	26	communications	communication	NOUN
cana-2702	201	27	on	on	ADP
cana-2702	201	28	applied	apply	VERB
cana-2702	201	29	nonlinear	nonlinear	ADJ
cana-2702	201	30	analysis	analysis	NOUN
cana-2702	201	31	issn	issn	NOUN
cana-2702	201	32	:	:	PUNCT
cana-2702	201	33	1074	1074	NUM
cana-2702	201	34	-	-	PUNCT
cana-2702	201	35	133x	133x	NUM
cana-2702	201	36	vol	vol	NOUN
cana-2702	201	37	32	32	NUM
cana-2702	202	1	no	no	NOUN
cana-2702	202	2	.	.	PUNCT
cana-2702	203	1	3s	3s	NUM
cana-2702	203	2	(	(	PUNCT
cana-2702	203	3	2025	2025	NUM
cana-2702	203	4	)	)	PUNCT
cana-2702	203	5	653	653	NUM
cana-2702	203	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	203	7	case	case	NOUN
cana-2702	203	8	(	(	PUNCT
cana-2702	203	9	1	1	X
cana-2702	203	10	)	)	PUNCT
cana-2702	203	11	𝑣	𝑣	PRON
cana-2702	203	12	∈	∈	PROPN
cana-2702	203	13	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	203	14	for	for	ADP
cana-2702	203	15	2	2	NUM
cana-2702	203	16	<	<	X
cana-2702	203	17	𝑚1	𝑚1	X
cana-2702	203	18	<	<	X
cana-2702	203	19	𝑚.let	𝑚.let	NOUN
cana-2702	203	20	us	we	PRON
cana-2702	203	21	assume	assume	VERB
cana-2702	203	22	that	that	SCONJ
cana-2702	203	23	𝑋	𝑋	NOUN
cana-2702	203	24	=	=	SYM
cana-2702	203	25	𝐶1	𝐶1	ADJ
cana-2702	203	26	∪.	∪.	X
cana-2702	203	27	.	.	PUNCT
cana-2702	204	1	.∪	.∪	X
cana-2702	204	2	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	204	3	and	and	CCONJ
cana-2702	204	4	𝑌	𝑌	PROPN
cana-2702	204	5	=	=	PUNCT
cana-2702	204	6	𝐶𝑚1	𝐶𝑚1	X
cana-2702	204	7	+	+	ADJ
cana-2702	204	8	1	1	NUM
cana-2702	204	9	∪.	∪.	NOUN
cana-2702	204	10	.	.	PUNCT
cana-2702	205	1	.∪	.∪	PUNCT
cana-2702	206	1	𝐶𝑚	𝐶𝑚	NOUN
cana-2702	206	2	are	be	AUX
cana-2702	206	3	two	two	NUM
cana-2702	206	4	disjoint	disjoint	ADJ
cana-2702	206	5	subgraphs	subgraph	NOUN
cana-2702	206	6	of	of	ADP
cana-2702	206	7	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	206	8	)	)	PUNCT
cana-2702	206	9	.	.	PUNCT
cana-2702	207	1	here	here	ADV
cana-2702	207	2	𝑋	𝑋	PROPN
cana-2702	207	3	=	=	X
cana-2702	207	4	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	207	5	(	(	PUNCT
cana-2702	207	6	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	207	7	)	)	PUNCT
cana-2702	207	8	and	and	CCONJ
cana-2702	207	9	𝑌	𝑌	PROPN
cana-2702	207	10	=	=	PUNCT
cana-2702	207	11	𝐶𝑚2	𝐶𝑚2	NOUN
cana-2702	207	12	(	(	PUNCT
cana-2702	207	13	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	207	14	)	)	PUNCT
cana-2702	207	15	,	,	PUNCT
cana-2702	207	16	where	where	SCONJ
cana-2702	207	17	𝑚	𝑚	NOUN
cana-2702	207	18	=	=	SYM
cana-2702	207	19	𝑚1	𝑚1	NOUN
cana-2702	207	20	+	+	NUM
cana-2702	207	21	𝑚2	𝑚2	NOUN
cana-2702	207	22	.	.	PUNCT
cana-2702	208	1	clearly	clearly	ADV
cana-2702	208	2	𝑣	𝑣	ADP
cana-2702	208	3	∈	∈	PROPN
cana-2702	208	4	𝑋.	𝑋.	PROPN
cana-2702	208	5	suppose	suppose	VERB
cana-2702	208	6	𝑝(𝑋	𝑝(𝑋	PROPN
cana-2702	208	7	)	)	PUNCT
cana-2702	208	8	≥	≥	NOUN
cana-2702	208	9	2	2	NUM
cana-2702	208	10	𝑚1	𝑚1	NOUN
cana-2702	208	11	+	+	CCONJ
cana-2702	208	12	2(𝑛	2(𝑛	NUM
cana-2702	208	13	−	−	NOUN
cana-2702	208	14	3	3	NUM
cana-2702	208	15	)	)	PUNCT
cana-2702	208	16	+	+	CCONJ
cana-2702	208	17	(	(	PUNCT
cana-2702	208	18	𝑚1	𝑚1	NOUN
cana-2702	208	19	−	−	PROPN
cana-2702	208	20	2)(𝑛	2)(𝑛	NUM
cana-2702	208	21	−	−	NOUN
cana-2702	208	22	4	4	NUM
cana-2702	208	23	)	)	PUNCT
cana-2702	208	24	.	.	PUNCT
cana-2702	209	1	then	then	ADV
cana-2702	209	2	,	,	PUNCT
cana-2702	209	3	we	we	PRON
cana-2702	209	4	can	can	AUX
cana-2702	209	5	move	move	VERB
cana-2702	209	6	the	the	DET
cana-2702	209	7	pebble	pebble	NOUN
cana-2702	209	8	to	to	ADP
cana-2702	209	9	𝑣	𝑣	ADP
cana-2702	209	10	by	by	ADP
cana-2702	209	11	induction	induction	NOUN
cana-2702	209	12	on	on	ADP
cana-2702	209	13	𝑚.	𝑚.	NOUN
cana-2702	209	14	therefore	therefore	ADV
cana-2702	209	15	,	,	PUNCT
cana-2702	209	16	we	we	PRON
cana-2702	209	17	assume	assume	VERB
cana-2702	209	18	𝑝(𝑋	𝑝(𝑋	PROPN
cana-2702	209	19	)	)	PUNCT
cana-2702	209	20	<	<	X
cana-2702	209	21	2	2	NUM
cana-2702	209	22	𝑚1	𝑚1	NOUN
cana-2702	209	23	+	+	CCONJ
cana-2702	209	24	2(𝑛	2(𝑛	NUM
cana-2702	209	25	−	−	NOUN
cana-2702	209	26	3	3	NUM
cana-2702	209	27	)	)	PUNCT
cana-2702	209	28	+	+	CCONJ
cana-2702	209	29	(	(	PUNCT
cana-2702	209	30	𝑚1	𝑚1	NOUN
cana-2702	209	31	−	−	PROPN
cana-2702	209	32	2)(𝑛	2)(𝑛	NUM
cana-2702	209	33	−	−	NOUN
cana-2702	209	34	4	4	NUM
cana-2702	209	35	)	)	PUNCT
cana-2702	209	36	−	−	PROPN
cana-2702	209	37	1	1	X
cana-2702	209	38	.	.	PUNCT
cana-2702	209	39	also	also	ADV
cana-2702	209	40	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	X
cana-2702	209	41	)	)	PUNCT
cana-2702	210	1	≤	≤	NOUN
cana-2702	211	1	𝑛	𝑛	DET
cana-2702	211	2	−	−	PROPN
cana-2702	211	3	2	2	NUM
cana-2702	211	4	.	.	PUNCT
cana-2702	211	5	suppose	suppose	VERB
cana-2702	211	6	𝑝(𝑋	𝑝(𝑋	PROPN
cana-2702	211	7	−	−	PROPN
cana-2702	211	8	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	211	9	)	)	PUNCT
cana-2702	211	10	≥	≥	PROPN
cana-2702	211	11	2	2	NUM
cana-2702	211	12	𝑚1	𝑚1	NOUN
cana-2702	211	13	+	+	PROPN
cana-2702	211	14	1(21	1(21	NUM
cana-2702	211	15	−	−	NOUN
cana-2702	211	16	1	1	NUM
cana-2702	211	17	)	)	PUNCT
cana-2702	212	1	+	+	CCONJ
cana-2702	212	2	(	(	PUNCT
cana-2702	212	3	𝑛	𝑛	DET
cana-2702	212	4	−	−	PROPN
cana-2702	212	5	4	4	NUM
cana-2702	212	6	)	)	PUNCT
cana-2702	212	7	+	+	CCONJ
cana-2702	212	8	2	2	X
cana-2702	212	9	.	.	X
cana-2702	212	10	from	from	ADP
cana-2702	212	11	lemma	lemma	PROPN
cana-2702	212	12	1	1	NUM
cana-2702	212	13	we	we	PRON
cana-2702	212	14	can	can	AUX
cana-2702	212	15	move	move	VERB
cana-2702	212	16	a	a	DET
cana-2702	212	17	pebble	pebble	NOUN
cana-2702	212	18	to	to	PART
cana-2702	212	19	𝑣.	𝑣.	VERB
cana-2702	212	20	so	so	ADV
cana-2702	212	21	assume	assume	VERB
cana-2702	212	22	that	that	SCONJ
cana-2702	212	23	𝑝(𝑋	𝑝(𝑋	PROPN
cana-2702	212	24	−	−	PROPN
cana-2702	212	25	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	212	26	)	)	PUNCT
cana-2702	213	1	<	<	X
cana-2702	213	2	2	2	NUM
cana-2702	213	3	𝑚1−1(21	𝑚1−1(21	NOUN
cana-2702	213	4	−	−	NOUN
cana-2702	213	5	1	1	NUM
cana-2702	213	6	)	)	PUNCT
cana-2702	213	7	+	+	CCONJ
cana-2702	213	8	(	(	PUNCT
cana-2702	213	9	𝑛	𝑛	PRON
cana-2702	213	10	−	−	PROPN
cana-2702	213	11	4	4	NUM
cana-2702	213	12	)	)	PUNCT
cana-2702	213	13	+	+	CCONJ
cana-2702	213	14	2	2	NUM
cana-2702	213	15	and	and	CCONJ
cana-2702	213	16	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	X
cana-2702	213	17	)	)	PUNCT
cana-2702	214	1	≤	≤	NOUN
cana-2702	214	2	𝑛	𝑛	DET
cana-2702	214	3	−	−	PROPN
cana-2702	214	4	2	2	NUM
cana-2702	214	5	.	.	PUNCT
cana-2702	214	6	therefore	therefore	ADV
cana-2702	214	7	𝑝(𝑋	𝑝(𝑋	PROPN
cana-2702	214	8	)	)	PUNCT
cana-2702	214	9	≤	≤	NUM
cana-2702	214	10	2	2	NUM
cana-2702	214	11	𝑚1−1	𝑚1−1	NUM
cana-2702	214	12	+	+	CCONJ
cana-2702	214	13	2𝑛	2𝑛	PROPN
cana-2702	214	14	−	−	PROPN
cana-2702	214	15	4	4	X
cana-2702	214	16	.	.	PUNCT
cana-2702	215	1	this	this	PRON
cana-2702	215	2	implies	imply	VERB
cana-2702	215	3	that	that	SCONJ
cana-2702	215	4	the	the	DET
cana-2702	215	5	number	number	NOUN
cana-2702	215	6	of	of	ADP
cana-2702	215	7	pebbles	pebble	NOUN
cana-2702	215	8	distributed	distribute	VERB
cana-2702	215	9	on	on	ADP
cana-2702	215	10	𝑌	𝑌	PROPN
cana-2702	215	11	was	be	AUX
cana-2702	215	12	at	at	ADV
cana-2702	215	13	least	least	ADJ
cana-2702	215	14	2	2	NUM
cana-2702	215	15	𝑚1(2𝑚2	𝑚1(2𝑚2	ADP
cana-2702	215	16	−	−	PROPN
cana-2702	215	17	1/2	1/2	NUM
cana-2702	215	18	)	)	PUNCT
cana-2702	216	1	+	+	CCONJ
cana-2702	216	2	(	(	PUNCT
cana-2702	216	3	𝑚1	𝑚1	NOUN
cana-2702	216	4	+	+	CCONJ
cana-2702	216	5	𝑚2	𝑚2	PROPN
cana-2702	216	6	−	−	PROPN
cana-2702	216	7	2)(𝑛	2)(𝑛	NUM
cana-2702	216	8	−	−	NOUN
cana-2702	216	9	4	4	NUM
cana-2702	216	10	)	)	PUNCT
cana-2702	216	11	−	−	PROPN
cana-2702	217	1	1	1	X
cana-2702	217	2	.	.	PUNCT
cana-2702	218	1	we	we	PRON
cana-2702	218	2	have	have	VERB
cana-2702	218	3	𝑝(𝑌	𝑝(𝑌	PROPN
cana-2702	218	4	)	)	PUNCT
cana-2702	218	5	≥	≥	NOUN
cana-2702	218	6	2	2	NUM
cana-2702	218	7	𝑚1(2𝑚2	𝑚1(2𝑚2	ADP
cana-2702	218	8	−	−	PROPN
cana-2702	218	9	1/2	1/2	NUM
cana-2702	218	10	)	)	PUNCT
cana-2702	219	1	+	+	CCONJ
cana-2702	219	2	(	(	PUNCT
cana-2702	219	3	𝑚1	𝑚1	NOUN
cana-2702	219	4	+	+	CCONJ
cana-2702	219	5	𝑚2	𝑚2	PROPN
cana-2702	219	6	−	−	PROPN
cana-2702	219	7	2)(𝑛	2)(𝑛	NUM
cana-2702	219	8	−	−	NOUN
cana-2702	219	9	4	4	NUM
cana-2702	219	10	)	)	PUNCT
cana-2702	219	11	−	−	PROPN
cana-2702	219	12	1	1	NUM
cana-2702	219	13	≥	≥	NOUN
cana-2702	219	14	2	2	NUM
cana-2702	219	15	𝑚1(2𝑚2	𝑚1(2𝑚2	ADP
cana-2702	219	16	−	−	PROPN
cana-2702	219	17	1	1	NUM
cana-2702	219	18	)	)	PUNCT
cana-2702	219	19	+	+	PUNCT
cana-2702	220	1	+	+	ADP
cana-2702	220	2	𝑚2(𝑛	𝑚2(𝑛	X
cana-2702	220	3	−	−	PROPN
cana-2702	220	4	4	4	NUM
cana-2702	220	5	)	)	PUNCT
cana-2702	220	6	+	+	CCONJ
cana-2702	220	7	2	2	X
cana-2702	220	8	.	.	PUNCT
cana-2702	220	9	again	again	ADV
cana-2702	220	10	,	,	PUNCT
cana-2702	220	11	by	by	ADP
cana-2702	220	12	lemma	lemma	PROPN
cana-2702	220	13	1	1	NUM
cana-2702	220	14	,	,	PUNCT
cana-2702	220	15	two	two	NUM
cana-2702	220	16	pebbles	pebble	NOUN
cana-2702	220	17	can	can	AUX
cana-2702	220	18	be	be	AUX
cana-2702	220	19	moved	move	VERB
cana-2702	220	20	to	to	ADP
cana-2702	220	21	𝑎	𝑎	PROPN
cana-2702	220	22	𝑚1	𝑚1	NOUN
cana-2702	220	23	(	(	PUNCT
cana-2702	220	24	𝑛	𝑛	PROPN
cana-2702	220	25	2	2	NUM
cana-2702	220	26	−1	−1	NOUN
cana-2702	220	27	)	)	PUNCT
cana-2702	220	28	or	or	CCONJ
cana-2702	220	29	𝑎	𝑎	PRON
cana-2702	220	30	𝑚1	𝑚1	NOUN
cana-2702	220	31	(	(	PUNCT
cana-2702	220	32	𝑛	𝑛	DET
cana-2702	220	33	2	2	NUM
cana-2702	220	34	−1)+1	−1)+1	NOUN
cana-2702	220	35	when	when	SCONJ
cana-2702	220	36	𝑚1	𝑚1	NOUN
cana-2702	220	37	is	be	AUX
cana-2702	220	38	even	even	ADV
cana-2702	220	39	and	and	CCONJ
cana-2702	220	40	𝑚1	𝑚1	NOUN
cana-2702	220	41	is	be	AUX
cana-2702	220	42	odd	odd	ADJ
cana-2702	220	43	.	.	PUNCT
cana-2702	221	1	then	then	ADV
cana-2702	221	2	,	,	PUNCT
cana-2702	221	3	we	we	PRON
cana-2702	221	4	can	can	AUX
cana-2702	221	5	move	move	VERB
cana-2702	221	6	one	one	NUM
cana-2702	221	7	pebble	pebble	NOUN
cana-2702	221	8	to	to	ADP
cana-2702	221	9	𝑣.	𝑣.	NOUN
cana-2702	221	10	case	case	NOUN
cana-2702	221	11	(	(	PUNCT
cana-2702	221	12	2	2	X
cana-2702	221	13	)	)	PUNCT
cana-2702	221	14	𝑣	𝑣	PART
cana-2702	221	15	∈	∈	NOUN
cana-2702	221	16	𝐶1	𝐶1	NOUN
cana-2702	221	17	or	or	CCONJ
cana-2702	221	18	𝑣	𝑣	PRON
cana-2702	221	19	∈	∈	NOUN
cana-2702	221	20	𝐶𝑚.	𝐶𝑚.	NOUN
cana-2702	221	21	without	without	ADP
cana-2702	221	22	a	a	DET
cana-2702	221	23	loss	loss	NOUN
cana-2702	221	24	of	of	ADP
cana-2702	221	25	generality	generality	NOUN
cana-2702	221	26	,	,	PUNCT
cana-2702	221	27	we	we	PRON
cana-2702	221	28	assume	assume	VERB
cana-2702	221	29	that	that	SCONJ
cana-2702	221	30	𝑣	𝑣	PRON
cana-2702	221	31	∈	∈	PROPN
cana-2702	221	32	𝐶𝑚	𝐶𝑚	PROPN
cana-2702	221	33	and	and	CCONJ
cana-2702	221	34	𝑣	𝑣	X
cana-2702	221	35	=	=	PUNCT
cana-2702	221	36	𝑦.	𝑦.	PROPN
cana-2702	221	37	also	also	ADV
cana-2702	221	38	𝑝(𝐶𝑚	𝑝(𝐶𝑚	NOUN
cana-2702	221	39	)	)	PUNCT
cana-2702	221	40	≤	≤	NOUN
cana-2702	222	1	𝑛	𝑛	DET
cana-2702	222	2	−	−	PROPN
cana-2702	222	3	2	2	NUM
cana-2702	222	4	.	.	PUNCT
cana-2702	222	5	now	now	ADV
cana-2702	222	6	take	take	VERB
cana-2702	222	7	𝑋1	𝑋1	NOUN
cana-2702	222	8	=	=	PUNCT
cana-2702	223	1	𝐶𝑚	𝐶𝑚	PROPN
cana-2702	223	2	≅	≅	PROPN
cana-2702	223	3	𝐶1(𝐾𝑛	𝐶1(𝐾𝑛	PROPN
cana-2702	223	4	)	)	PUNCT
cana-2702	224	1	and	and	CCONJ
cana-2702	224	2	𝑋2	𝑋2	VERB
cana-2702	224	3	=	=	SYM
cana-2702	224	4	𝐶1	𝐶1	ADJ
cana-2702	224	5	∪.	∪.	X
cana-2702	224	6	.	.	PUNCT
cana-2702	225	1	.∪	.∪	PROPN
cana-2702	225	2	𝐶𝑚−1	𝐶𝑚−1	PROPN
cana-2702	225	3	≅	≅	PROPN
cana-2702	225	4	𝐶𝑚−1(𝐾𝑛	𝐶𝑚−1(𝐾𝑛	NUM
cana-2702	225	5	)	)	PUNCT
cana-2702	225	6	.	.	PUNCT
cana-2702	226	1	claim	claim	NOUN
cana-2702	226	2	(	(	PUNCT
cana-2702	226	3	1	1	X
cana-2702	226	4	)	)	PUNCT
cana-2702	226	5	𝑝(𝑋2	𝑝(𝑋2	PROPN
cana-2702	226	6	)	)	PUNCT
cana-2702	226	7	≥	≥	NOUN
cana-2702	227	1	2(2𝑚−1	2(2𝑚−1	NUM
cana-2702	227	2	−	−	NOUN
cana-2702	227	3	1	1	NUM
cana-2702	227	4	)	)	PUNCT
cana-2702	227	5	+	+	CCONJ
cana-2702	227	6	(	(	PUNCT
cana-2702	227	7	𝑚	𝑚	PROPN
cana-2702	227	8	−	−	PROPN
cana-2702	227	9	1)(𝑛	1)(𝑛	NUM
cana-2702	227	10	−	−	NOUN
cana-2702	227	11	4	4	NUM
cana-2702	227	12	)	)	PUNCT
cana-2702	227	13	+	+	NOUN
cana-2702	227	14	2	2	X
cana-2702	227	15	.	.	X
cana-2702	227	16	we	we	PRON
cana-2702	227	17	have	have	VERB
cana-2702	227	18	to	to	PART
cana-2702	227	19	prove	prove	VERB
cana-2702	227	20	𝑝(𝑋2	𝑝(𝑋2	PROPN
cana-2702	227	21	)	)	PUNCT
cana-2702	227	22	−	−	PROPN
cana-2702	228	1	2(2𝑚−1	2(2𝑚−1	NUM
cana-2702	229	1	−	−	NOUN
cana-2702	229	2	1	1	NUM
cana-2702	229	3	)	)	PUNCT
cana-2702	229	4	+	+	CCONJ
cana-2702	229	5	(	(	PUNCT
cana-2702	229	6	𝑚	𝑚	PROPN
cana-2702	229	7	−	−	PROPN
cana-2702	229	8	1)(𝑛	1)(𝑛	NUM
cana-2702	229	9	−	−	NOUN
cana-2702	229	10	4	4	NUM
cana-2702	229	11	)	)	PUNCT
cana-2702	229	12	+	+	CCONJ
cana-2702	229	13	2	2	NUM
cana-2702	229	14	≥	≥	NOUN
cana-2702	229	15	0	0	NUM
cana-2702	229	16	=	=	SYM
cana-2702	229	17	2	2	NUM
cana-2702	229	18	𝑚	𝑚	NOUN
cana-2702	229	19	+	+	NOUN
cana-2702	229	20	2(𝑛	2(𝑛	NUM
cana-2702	229	21	−	−	NOUN
cana-2702	229	22	3	3	NUM
cana-2702	229	23	)	)	PUNCT
cana-2702	229	24	+	+	CCONJ
cana-2702	229	25	(	(	PUNCT
cana-2702	229	26	𝑚	𝑚	PROPN
cana-2702	229	27	−	−	PROPN
cana-2702	229	28	2)(𝑛	2)(𝑛	NUM
cana-2702	229	29	−	−	NOUN
cana-2702	229	30	4	4	NUM
cana-2702	229	31	)	)	PUNCT
cana-2702	229	32	−	−	NOUN
cana-2702	229	33	𝑛	𝑛	PROPN
cana-2702	229	34	+	+	NOUN
cana-2702	229	35	2	2	NUM
cana-2702	229	36	−	−	NOUN
cana-2702	229	37	2	2	NUM
cana-2702	229	38	𝑚	𝑚	NOUN
cana-2702	229	39	+	+	NOUN
cana-2702	229	40	2	2	NUM
cana-2702	229	41	−	−	NOUN
cana-2702	229	42	(	(	PUNCT
cana-2702	229	43	𝑚	𝑚	PROPN
cana-2702	229	44	−	−	PROPN
cana-2702	229	45	1)(𝑛	1)(𝑛	NUM
cana-2702	229	46	−	−	NOUN
cana-2702	229	47	4	4	NUM
cana-2702	229	48	)	)	PUNCT
cana-2702	229	49	−	−	PROPN
cana-2702	229	50	2	2	NUM
cana-2702	229	51	=	=	SYM
cana-2702	229	52	2(𝑛	2(𝑛	NUM
cana-2702	229	53	−	−	NUM
cana-2702	229	54	3	3	NUM
cana-2702	229	55	)	)	PUNCT
cana-2702	229	56	−	−	PROPN
cana-2702	229	57	(	(	PUNCT
cana-2702	229	58	𝑛	𝑛	PRON
cana-2702	229	59	−	−	PROPN
cana-2702	229	60	4	4	NUM
cana-2702	229	61	)	)	PUNCT
cana-2702	229	62	−	−	NOUN
cana-2702	229	63	𝑛	𝑛	PROPN
cana-2702	229	64	+	+	CCONJ
cana-2702	229	65	2	2	NUM
cana-2702	229	66	=	=	SYM
cana-2702	229	67	0	0	NUM
cana-2702	229	68	.	.	PUNCT
cana-2702	230	1	from	from	ADP
cana-2702	230	2	claim	claim	NOUN
cana-2702	230	3	(	(	PUNCT
cana-2702	230	4	1	1	NUM
cana-2702	230	5	)	)	PUNCT
cana-2702	230	6	,	,	PUNCT
cana-2702	230	7	we	we	PRON
cana-2702	230	8	obtain	obtain	VERB
cana-2702	230	9	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	230	10	)	)	PUNCT
cana-2702	230	11	≥	≥	NOUN
cana-2702	231	1	2(2𝑚−1	2(2𝑚−1	NUM
cana-2702	231	2	−	−	NOUN
cana-2702	231	3	1	1	NUM
cana-2702	231	4	)	)	PUNCT
cana-2702	231	5	+	+	CCONJ
cana-2702	231	6	(	(	PUNCT
cana-2702	231	7	𝑚	𝑚	PROPN
cana-2702	231	8	−	−	PROPN
cana-2702	231	9	1)(𝑛	1)(𝑛	NUM
cana-2702	231	10	−	−	NOUN
cana-2702	231	11	4	4	NUM
cana-2702	231	12	)	)	PUNCT
cana-2702	231	13	+	+	CCONJ
cana-2702	231	14	2	2	NUM
cana-2702	231	15	pebbles	pebble	NOUN
cana-2702	231	16	.	.	PUNCT
cana-2702	232	1	again	again	ADV
cana-2702	232	2	,	,	PUNCT
cana-2702	232	3	by	by	ADP
cana-2702	232	4	lemma	lemma	PROPN
cana-2702	232	5	1	1	NUM
cana-2702	232	6	,	,	PUNCT
cana-2702	232	7	two	two	NUM
cana-2702	232	8	pebbles	pebble	NOUN
cana-2702	232	9	can	can	AUX
cana-2702	232	10	be	be	AUX
cana-2702	232	11	moved	move	VERB
cana-2702	232	12	to	to	ADP
cana-2702	232	13	𝑎	𝑎	PROPN
cana-2702	232	14	(	(	PUNCT
cana-2702	232	15	𝑚−1	𝑚−1	NOUN
cana-2702	232	16	)	)	PUNCT
cana-2702	232	17	(	(	PUNCT
cana-2702	232	18	𝑛	𝑛	PROPN
cana-2702	232	19	2	2	NUM
cana-2702	232	20	+1	+1	NOUN
cana-2702	232	21	)	)	PUNCT
cana-2702	232	22	or	or	CCONJ
cana-2702	232	23	𝑎	𝑎	X
cana-2702	232	24	(	(	PUNCT
cana-2702	232	25	𝑚−1	𝑚−1	NOUN
cana-2702	232	26	)	)	PUNCT
cana-2702	232	27	(	(	PUNCT
cana-2702	232	28	𝑛	𝑛	PROPN
cana-2702	232	29	2	2	NUM
cana-2702	232	30	+1)+1	+1)+1	NOUN
cana-2702	232	31	when	when	SCONJ
cana-2702	232	32	𝑚	𝑚	PROPN
cana-2702	232	33	is	be	AUX
cana-2702	232	34	odd	odd	ADJ
cana-2702	232	35	or	or	CCONJ
cana-2702	232	36	even	even	ADV
cana-2702	232	37	,	,	PUNCT
cana-2702	232	38	respectively	respectively	ADV
cana-2702	232	39	.	.	PUNCT
cana-2702	233	1	we	we	PRON
cana-2702	233	2	can	can	AUX
cana-2702	233	3	then	then	ADV
cana-2702	233	4	move	move	VERB
cana-2702	233	5	the	the	DET
cana-2702	233	6	pebble	pebble	NOUN
cana-2702	233	7	to	to	PART
cana-2702	233	8	𝑣.	𝑣.	VERB
cana-2702	233	9	the	the	DET
cana-2702	233	10	𝒕pebbling	𝒕pebble	VERB
cana-2702	233	11	number	number	NOUN
cana-2702	233	12	of	of	ADP
cana-2702	233	13	crisscross	crisscross	ADJ
cana-2702	233	14	sequence	sequence	NOUN
cana-2702	233	15	of	of	ADP
cana-2702	233	16	𝒎	𝒎	PROPN
cana-2702	233	17	complete	complete	ADJ
cana-2702	233	18	graphs	graph	NOUN
cana-2702	233	19	theorem	theorem	VERB
cana-2702	233	20	4	4	NUM
cana-2702	233	21	.	.	PUNCT
cana-2702	234	1	the	the	DET
cana-2702	234	2	t	t	NOUN
cana-2702	234	3	-	-	PUNCT
cana-2702	234	4	pebbling	pebble	VERB
cana-2702	234	5	number	number	NOUN
cana-2702	234	6	of	of	ADP
cana-2702	234	7	the	the	DET
cana-2702	234	8	graph	graph	NOUN
cana-2702	234	9	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	234	10	)	)	PUNCT
cana-2702	234	11	is	be	AUX
cana-2702	234	12	𝑓𝑡(𝐶2(𝐾𝑛	𝑓𝑡(𝐶2(𝐾𝑛	NUM
cana-2702	234	13	)	)	PUNCT
cana-2702	234	14	)	)	PUNCT
cana-2702	235	1	=	=	NOUN
cana-2702	235	2	4𝑡	4𝑡	NOUN
cana-2702	236	1	+	+	CCONJ
cana-2702	236	2	2(𝑛	2(𝑛	NUM
cana-2702	236	3	−	−	NOUN
cana-2702	236	4	3	3	NUM
cana-2702	236	5	)	)	PUNCT
cana-2702	236	6	.	.	PUNCT
cana-2702	237	1	proof	proof	NOUN
cana-2702	237	2	.	.	PUNCT
cana-2702	238	1	set	set	VERB
cana-2702	238	2	4𝑡	4𝑡	NUM
cana-2702	238	3	−	−	PROPN
cana-2702	238	4	1	1	NUM
cana-2702	238	5	pebbles	pebble	NOUN
cana-2702	238	6	on	on	ADP
cana-2702	238	7	vertex	vertex	NOUN
cana-2702	238	8	𝑥	𝑥	PROPN
cana-2702	238	9	and	and	CCONJ
cana-2702	238	10	one	one	NUM
cana-2702	238	11	pebble	pebble	NOUN
cana-2702	238	12	on	on	ADP
cana-2702	238	13	each	each	DET
cana-2702	238	14	𝑎𝑖	𝑎𝑖	NOUN
cana-2702	238	15	and	and	CCONJ
cana-2702	238	16	𝑏𝑖	𝑏𝑖	ADP
cana-2702	238	17	except	except	SCONJ
cana-2702	238	18	for	for	ADP
cana-2702	238	19	vertices	vertex	NOUN
cana-2702	238	20	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	238	21	and	and	CCONJ
cana-2702	238	22	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	238	23	where	where	SCONJ
cana-2702	238	24	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	238	25	,	,	PUNCT
cana-2702	238	26	𝑏𝑘	𝑏𝑘	PROPN
cana-2702	238	27	∈	∈	PROPN
cana-2702	238	28	𝑉(𝐶1	𝑉(𝐶1	VERB
cana-2702	238	29	)	)	PUNCT
cana-2702	238	30	∪	∪	ADP
cana-2702	238	31	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	238	32	)	)	PUNCT
cana-2702	238	33	.	.	PUNCT
cana-2702	239	1	therefore	therefore	ADV
cana-2702	239	2	,	,	PUNCT
cana-2702	239	3	we	we	PRON
cana-2702	239	4	can	can	AUX
cana-2702	239	5	not	not	PART
cana-2702	239	6	move	move	VERB
cana-2702	239	7	𝑡	𝑡	NOUN
cana-2702	239	8	pebbles	pebble	NOUN
cana-2702	239	9	to	to	ADP
cana-2702	239	10	the	the	DET
cana-2702	239	11	vertex	vertex	NOUN
cana-2702	239	12	𝑦.	𝑦.	PROPN
cana-2702	239	13	therefore	therefore	ADV
cana-2702	239	14	𝑓𝑡(𝐶2(𝐾𝑛	𝑓𝑡(𝐶2(𝐾𝑛	NUM
cana-2702	239	15	)	)	PUNCT
cana-2702	239	16	)	)	PUNCT
cana-2702	239	17	≥	≥	NOUN
cana-2702	239	18	4𝑡	4𝑡	NUM
cana-2702	240	1	+	+	CCONJ
cana-2702	240	2	2𝑛	2𝑛	PROPN
cana-2702	240	3	−	−	PROPN
cana-2702	240	4	6	6	X
cana-2702	240	5	.	.	PUNCT
cana-2702	240	6	consider	consider	VERB
cana-2702	240	7	a	a	DET
cana-2702	240	8	graph	graph	NOUN
cana-2702	240	9	with	with	ADP
cana-2702	240	10	at	at	ADV
cana-2702	240	11	least	least	ADJ
cana-2702	240	12	4𝑡	4𝑡	NOUN
cana-2702	241	1	+	+	CCONJ
cana-2702	241	2	2𝑛	2𝑛	PROPN
cana-2702	241	3	−	−	NUM
cana-2702	241	4	6	6	NUM
cana-2702	241	5	pebbles	pebble	NOUN
cana-2702	241	6	distributed	distribute	VERB
cana-2702	241	7	at	at	ADP
cana-2702	241	8	the	the	DET
cana-2702	241	9	vertices	vertex	NOUN
cana-2702	241	10	of	of	ADP
cana-2702	241	11	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	241	12	)	)	PUNCT
cana-2702	241	13	.	.	PUNCT
cana-2702	242	1	let	let	VERB
cana-2702	242	2	𝑣	𝑣	PART
cana-2702	242	3	be	be	AUX
cana-2702	242	4	a	a	DET
cana-2702	242	5	target	target	NOUN
cana-2702	242	6	vertex	vertex	NOUN
cana-2702	242	7	and	and	CCONJ
cana-2702	242	8	𝑝(𝑣	𝑝(𝑣	NUM
cana-2702	242	9	)	)	PUNCT
cana-2702	243	1	=	=	PUNCT
cana-2702	244	1	0	0	X
cana-2702	244	2	.	.	PUNCT
cana-2702	245	1	we	we	PRON
cana-2702	245	2	prove	prove	VERB
cana-2702	245	3	this	this	DET
cana-2702	245	4	result	result	NOUN
cana-2702	245	5	by	by	ADP
cana-2702	245	6	induction	induction	NOUN
cana-2702	245	7	of	of	ADP
cana-2702	245	8	𝑡.	𝑡.	NOUN
cana-2702	245	9	for	for	ADP
cana-2702	245	10	𝑡	𝑡	PROPN
cana-2702	245	11	=	=	SYM
cana-2702	245	12	1	1	NUM
cana-2702	245	13	,	,	PUNCT
cana-2702	245	14	the	the	DET
cana-2702	245	15	result	result	NOUN
cana-2702	245	16	follows	follow	VERB
cana-2702	245	17	from	from	ADP
cana-2702	245	18	theorem	theorem	NOUN
cana-2702	245	19	1	1	NUM
cana-2702	245	20	.	.	PUNCT
cana-2702	246	1	we	we	PRON
cana-2702	246	2	assume	assume	VERB
cana-2702	246	3	that	that	SCONJ
cana-2702	246	4	this	this	DET
cana-2702	246	5	result	result	NOUN
cana-2702	246	6	is	be	AUX
cana-2702	246	7	true	true	ADJ
cana-2702	246	8	for	for	ADP
cana-2702	246	9	all	all	DET
cana-2702	246	10	𝑡′	𝑡′	PUNCT
cana-2702	246	11	<	<	X
cana-2702	246	12	𝑡.	𝑡.	NOUN
cana-2702	246	13	without	without	ADP
cana-2702	246	14	a	a	DET
cana-2702	246	15	loss	loss	NOUN
cana-2702	246	16	of	of	ADP
cana-2702	246	17	generality	generality	NOUN
cana-2702	246	18	,	,	PUNCT
cana-2702	246	19	we	we	PRON
cana-2702	246	20	assume	assume	VERB
cana-2702	246	21	that	that	SCONJ
cana-2702	246	22	𝑣	𝑣	PRON
cana-2702	246	23	∈	∈	PROPN
cana-2702	246	24	𝐶2	𝐶2	INTJ
cana-2702	246	25	and	and	CCONJ
cana-2702	246	26	𝑣	𝑣	X
cana-2702	246	27	=	=	PUNCT
cana-2702	246	28	𝑦.	𝑦.	PROPN
cana-2702	246	29	we	we	PRON
cana-2702	246	30	consider	consider	VERB
cana-2702	246	31	the	the	DET
cana-2702	246	32	following	follow	VERB
cana-2702	246	33	cases	case	NOUN
cana-2702	246	34	:	:	PUNCT
cana-2702	246	35	case	case	NOUN
cana-2702	246	36	(	(	PUNCT
cana-2702	246	37	1	1	X
cana-2702	246	38	)	)	PUNCT
cana-2702	246	39	𝑛	𝑛	DET
cana-2702	246	40	≤	≤	NOUN
cana-2702	247	1	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	247	2	)	)	PUNCT
cana-2702	247	3	<	<	X
cana-2702	247	4	2𝑡	2𝑡	NUM
cana-2702	247	5	+	+	CCONJ
cana-2702	247	6	𝑛	𝑛	DET
cana-2702	247	7	−	−	PROPN
cana-2702	247	8	2	2	NUM
cana-2702	247	9	.	.	PUNCT
cana-2702	248	1	if	if	SCONJ
cana-2702	248	2	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	248	3	)	)	PUNCT
cana-2702	248	4	≥	≥	NOUN
cana-2702	248	5	2𝑡	2𝑡	NOUN
cana-2702	248	6	+	+	CCONJ
cana-2702	248	7	𝑛	𝑛	PRON
cana-2702	248	8	−	−	PROPN
cana-2702	248	9	2	2	NUM
cana-2702	248	10	.	.	PUNCT
cana-2702	248	11	by	by	ADP
cana-2702	248	12	referring	refer	VERB
cana-2702	248	13	the	the	DET
cana-2702	248	14	pebbling	pebble	VERB
cana-2702	248	15	number	number	NOUN
cana-2702	248	16	of	of	ADP
cana-2702	248	17	complete	complete	ADJ
cana-2702	248	18	graph	graph	NOUN
cana-2702	248	19	,	,	PUNCT
cana-2702	248	20	we	we	PRON
cana-2702	248	21	can	can	AUX
cana-2702	248	22	move	move	VERB
cana-2702	248	23	𝑡	𝑡	PROPN
cana-2702	248	24	pebbles	pebble	NOUN
cana-2702	248	25	to	to	ADP
cana-2702	248	26	𝑦.	𝑦.	PROPN
cana-2702	248	27	therefore	therefore	ADV
cana-2702	248	28	,	,	PUNCT
cana-2702	248	29	we	we	PRON
cana-2702	248	30	assume	assume	VERB
cana-2702	248	31	that	that	SCONJ
cana-2702	248	32	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	248	33	)	)	PUNCT
cana-2702	248	34	≥	≥	X
cana-2702	248	35	𝑛.	𝑛.	NOUN
cana-2702	248	36	we	we	PRON
cana-2702	248	37	used	use	VERB
cana-2702	248	38	𝑛	𝑛	DET
cana-2702	248	39	pebbles	pebble	NOUN
cana-2702	248	40	for	for	ADP
cana-2702	248	41	𝑦.	𝑦.	PROPN
cana-2702	248	42	also	also	ADV
cana-2702	248	43	𝑝(𝐶2	𝑝(𝐶2	PROPN
cana-2702	248	44	)	)	PUNCT
cana-2702	248	45	≤	≤	NOUN
cana-2702	248	46	2𝑡	2𝑡	NOUN
cana-2702	248	47	+	+	CCONJ
cana-2702	248	48	𝑛	𝑛	DET
cana-2702	248	49	−	−	PROPN
cana-2702	248	50	2	2	NUM
cana-2702	248	51	.	.	PUNCT
cana-2702	248	52	suppose	suppose	VERB
cana-2702	248	53	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	248	54	)	)	PUNCT
cana-2702	248	55	≤	≤	NOUN
cana-2702	248	56	4𝑡	4𝑡	NOUN
cana-2702	248	57	+	+	CCONJ
cana-2702	249	1	𝑛	𝑛	DET
cana-2702	249	2	−	−	NUM
cana-2702	249	3	5	5	NUM
cana-2702	249	4	.	.	PUNCT
cana-2702	249	5	then	then	ADV
cana-2702	249	6	𝑝(𝐶2	𝑝(𝐶2	X
cana-2702	249	7	)	)	PUNCT
cana-2702	249	8	=	=	SYM
cana-2702	249	9	𝑛	𝑛	PROPN
cana-2702	250	1	+	+	NOUN
cana-2702	250	2	1	1	X
cana-2702	250	3	.	.	PUNCT
cana-2702	251	1	this	this	PRON
cana-2702	251	2	was	be	AUX
cana-2702	251	3	because	because	SCONJ
cana-2702	251	4	𝐶2	𝐶2	INTJ
cana-2702	251	5	≅	≅	PROPN
cana-2702	251	6	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	251	7	therefore	therefore	ADV
cana-2702	251	8	,	,	PUNCT
cana-2702	251	9	any	any	PRON
cana-2702	251	10	of	of	ADP
cana-2702	251	11	the	the	DET
cana-2702	251	12	vertices	vertex	NOUN
cana-2702	251	13	in	in	ADP
cana-2702	251	14	𝐶2	𝐶2	PROPN
cana-2702	251	15	contains	contain	VERB
cana-2702	251	16	two	two	NUM
cana-2702	251	17	pebbles	pebble	NOUN
cana-2702	251	18	.	.	PUNCT
cana-2702	252	1	using	use	VERB
cana-2702	252	2	exactly	exactly	ADV
cana-2702	252	3	two	two	NUM
cana-2702	252	4	pebbles	pebble	NOUN
cana-2702	252	5	,	,	PUNCT
cana-2702	252	6	we	we	PRON
cana-2702	252	7	can	can	AUX
cana-2702	252	8	move	move	VERB
cana-2702	252	9	a	a	DET
cana-2702	252	10	pebble	pebble	NOUN
cana-2702	252	11	to	to	PART
cana-2702	252	12	𝑦.	𝑦.	VERB
cana-2702	252	13	this	this	DET
cana-2702	252	14	leave	leave	NOUN
cana-2702	252	15	at	at	ADP
cana-2702	252	16	least	least	ADJ
cana-2702	252	17	4𝑡	4𝑡	NOUN
cana-2702	253	1	+	+	CCONJ
cana-2702	253	2	2𝑛	2𝑛	PROPN
cana-2702	253	3	−	−	NUM
cana-2702	253	4	8	8	NUM
cana-2702	253	5	pebbles	pebble	NOUN
cana-2702	253	6	retained	retain	VERB
cana-2702	253	7	in	in	ADP
cana-2702	253	8	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	253	9	)	)	PUNCT
cana-2702	253	10	.	.	PUNCT
cana-2702	254	1	since	since	SCONJ
cana-2702	254	2	4𝑡	4𝑡	PRON
cana-2702	254	3	+	+	CCONJ
cana-2702	254	4	2𝑛	2𝑛	PROPN
cana-2702	254	5	−	−	PROPN
cana-2702	254	6	8	8	NUM
cana-2702	254	7	=	=	SYM
cana-2702	254	8	4(𝑡	4(𝑡	NUM
cana-2702	254	9	−	−	NUM
cana-2702	254	10	1	1	NUM
cana-2702	254	11	)	)	PUNCT
cana-2702	254	12	+	+	NUM
cana-2702	254	13	2𝑛	2𝑛	PROPN
cana-2702	254	14	−	−	PROPN
cana-2702	254	15	6	6	NUM
cana-2702	254	16	.	.	PUNCT
cana-2702	254	17	so	so	ADV
cana-2702	254	18	by	by	ADP
cana-2702	254	19	induction	induction	NOUN
cana-2702	254	20	we	we	PRON
cana-2702	254	21	can	can	AUX
cana-2702	254	22	move	move	VERB
cana-2702	254	23	additional	additional	ADJ
cana-2702	254	24	𝑡	𝑡	NOUN
cana-2702	254	25	−	−	NUM
cana-2702	254	26	1	1	NUM
cana-2702	254	27	pebbles	pebble	NOUN
cana-2702	254	28	to	to	ADP
cana-2702	254	29	𝑦.	𝑦.	PROPN
cana-2702	254	30	case2:𝑝(𝐶2	case2:𝑝(𝐶2	PROPN
cana-2702	254	31	)	)	PUNCT
cana-2702	254	32	<	<	X
cana-2702	255	1	𝑛.it	𝑛.it	PRON
cana-2702	255	2	is	be	AUX
cana-2702	255	3	assumed	assume	VERB
cana-2702	255	4	that	that	SCONJ
cana-2702	255	5	vertices	vertice	VERB
cana-2702	255	6	adjacent	adjacent	ADJ
cana-2702	255	7	to	to	ADP
cana-2702	255	8	𝑦	𝑦	NOUN
cana-2702	255	9	have	have	VERB
cana-2702	255	10	at	at	ADP
cana-2702	255	11	most	most	ADV
cana-2702	255	12	one	one	NUM
cana-2702	255	13	pebble	pebble	NOUN
cana-2702	255	14	.	.	PUNCT
cana-2702	256	1	suppose	suppose	VERB
cana-2702	256	2	𝑝(𝑎𝑘	𝑝(𝑎𝑘	NOUN
cana-2702	256	3	)	)	PUNCT
cana-2702	256	4	=	=	SYM
cana-2702	256	5	1	1	NUM
cana-2702	256	6	or	or	CCONJ
cana-2702	256	7	𝑝(𝑏𝑘	𝑝(𝑏𝑘	PROPN
cana-2702	256	8	)	)	PUNCT
cana-2702	256	9	=	=	SYM
cana-2702	256	10	1	1	X
cana-2702	256	11	.	.	PUNCT
cana-2702	256	12	by	by	ADP
cana-2702	256	13	using	use	VERB
cana-2702	256	14	at	at	ADV
cana-2702	256	15	least	least	ADV
cana-2702	256	16	two	two	NUM
cana-2702	256	17	pebbles	pebble	NOUN
cana-2702	256	18	from	from	ADP
cana-2702	256	19	𝐶1	𝐶1	PRON
cana-2702	256	20	and	and	CCONJ
cana-2702	256	21	move	move	VERB
cana-2702	256	22	an	an	DET
cana-2702	256	23	additional	additional	ADJ
cana-2702	256	24	pebble	pebble	NOUN
cana-2702	256	25	to	to	ADP
cana-2702	256	26	𝑎𝑘.	𝑎𝑘.	NOUN
cana-2702	256	27	then	then	ADV
cana-2702	256	28	,	,	PUNCT
cana-2702	256	29	we	we	PRON
cana-2702	256	30	move	move	VERB
cana-2702	256	31	the	the	DET
cana-2702	256	32	pebble	pebble	NOUN
cana-2702	256	33	to	to	PART
cana-2702	256	34	𝑦.	𝑦.	VERB
cana-2702	256	35	this	this	PRON
cana-2702	256	36	left	leave	VERB
cana-2702	256	37	at	at	ADV
cana-2702	256	38	least	least	ADJ
cana-2702	256	39	4(𝑡	4(𝑡	NUM
cana-2702	256	40	−	−	NUM
cana-2702	256	41	1	1	NUM
cana-2702	256	42	)	)	PUNCT
cana-2702	257	1	+	+	NUM
cana-2702	257	2	2𝑛	2𝑛	NUM
cana-2702	257	3	−	−	NUM
cana-2702	257	4	6	6	NUM
cana-2702	257	5	pebbles	pebble	NOUN
cana-2702	257	6	on	on	ADP
cana-2702	257	7	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	257	8	)	)	PUNCT
cana-2702	257	9	.	.	PUNCT
cana-2702	258	1	thus	thus	ADV
cana-2702	258	2	,	,	PUNCT
cana-2702	258	3	by	by	ADP
cana-2702	258	4	induction	induction	NOUN
cana-2702	258	5	,	,	PUNCT
cana-2702	258	6	we	we	PRON
cana-2702	258	7	can	can	AUX
cana-2702	258	8	move	move	VERB
cana-2702	258	9	an	an	DET
cana-2702	258	10	additional	additional	ADJ
cana-2702	258	11	𝑡	𝑡	NOUN
cana-2702	258	12	−	−	NOUN
cana-2702	258	13	1	1	NUM
cana-2702	258	14	pebbles	pebble	NOUN
cana-2702	258	15	to	to	PART
cana-2702	258	16	𝑦.	𝑦.	VERB
cana-2702	258	17	so	so	PROPN
cana-2702	258	18	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	258	19	)	)	PUNCT
cana-2702	258	20	≤	≤	NOUN
cana-2702	259	1	𝑛	𝑛	DET
cana-2702	259	2	−	−	NOUN
cana-2702	259	3	3	3	NUM
cana-2702	259	4	.	.	PUNCT
cana-2702	260	1	this	this	PRON
cana-2702	260	2	implies	imply	VERB
cana-2702	260	3	that	that	SCONJ
cana-2702	260	4	the	the	DET
cana-2702	260	5	number	number	NOUN
cana-2702	260	6	of	of	ADP
cana-2702	260	7	unused	unused	ADJ
cana-2702	260	8	communications	communication	NOUN
cana-2702	260	9	on	on	ADP
cana-2702	260	10	applied	apply	VERB
cana-2702	260	11	nonlinear	nonlinear	ADJ
cana-2702	260	12	analysis	analysis	NOUN
cana-2702	260	13	issn	issn	NOUN
cana-2702	260	14	:	:	PUNCT
cana-2702	260	15	1074	1074	NUM
cana-2702	260	16	-	-	PUNCT
cana-2702	260	17	133x	133x	NUM
cana-2702	260	18	vol	vol	NOUN
cana-2702	260	19	32	32	NUM
cana-2702	260	20	no	no	NOUN
cana-2702	260	21	.	.	PUNCT
cana-2702	261	1	3s	3s	NUM
cana-2702	261	2	(	(	PUNCT
cana-2702	261	3	2025	2025	NUM
cana-2702	261	4	)	)	PUNCT
cana-2702	261	5	654	654	NUM
cana-2702	261	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	261	7	pebbles	pebble	NOUN
cana-2702	261	8	retained	retain	VERB
cana-2702	261	9	in	in	ADP
cana-2702	261	10	𝐶1	𝐶1	PROPN
cana-2702	261	11	was	be	AUX
cana-2702	261	12	at	at	ADP
cana-2702	261	13	least	least	ADJ
cana-2702	261	14	4𝑡	4𝑡	NOUN
cana-2702	262	1	+	+	CCONJ
cana-2702	262	2	𝑛	𝑛	DET
cana-2702	262	3	−	−	PROPN
cana-2702	262	4	2	2	NUM
cana-2702	262	5	.	.	PUNCT
cana-2702	262	6	thus	thus	ADV
cana-2702	262	7	by	by	ADP
cana-2702	262	8	𝑓(𝐾𝑛	𝑓(𝐾𝑛	NOUN
cana-2702	262	9	)	)	PUNCT
cana-2702	262	10	=	=	SYM
cana-2702	262	11	𝑛	𝑛	NOUN
cana-2702	262	12	,	,	PUNCT
cana-2702	262	13	we	we	PRON
cana-2702	262	14	can	can	AUX
cana-2702	262	15	move	move	VERB
cana-2702	262	16	2𝑡	2𝑡	NUM
cana-2702	262	17	pebbles	pebble	NOUN
cana-2702	262	18	to	to	ADP
cana-2702	262	19	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	262	20	and	and	CCONJ
cana-2702	262	21	then	then	ADV
cana-2702	262	22	move	move	VERB
cana-2702	262	23	𝑡	𝑡	PROPN
cana-2702	262	24	pebbles	pebble	NOUN
cana-2702	262	25	to	to	ADP
cana-2702	262	26	𝑦.	𝑦.	PROPN
cana-2702	262	27	theorem	theorem	PROPN
cana-2702	262	28	5	5	NUM
cana-2702	262	29	.	.	PUNCT
cana-2702	263	1	the	the	DET
cana-2702	263	2	t	t	NOUN
cana-2702	263	3	-	-	PUNCT
cana-2702	263	4	pebbling	pebble	VERB
cana-2702	263	5	number	number	NOUN
cana-2702	263	6	of	of	ADP
cana-2702	263	7	the	the	DET
cana-2702	263	8	graph	graph	NOUN
cana-2702	263	9	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	263	10	)	)	PUNCT
cana-2702	263	11	is	be	AUX
cana-2702	263	12	𝑓𝑡(𝐶3(𝐾𝑛	𝑓𝑡(𝐶3(𝐾𝑛	NUM
cana-2702	263	13	)	)	PUNCT
cana-2702	263	14	)	)	PUNCT
cana-2702	264	1	=	=	SYM
cana-2702	264	2	8𝑡	8𝑡	NOUN
cana-2702	265	1	+	+	CCONJ
cana-2702	265	2	3𝑛	3𝑛	NUM
cana-2702	265	3	−	−	NOUN
cana-2702	265	4	10	10	NUM
cana-2702	265	5	.	.	PUNCT
cana-2702	266	1	proof	proof	NOUN
cana-2702	266	2	.	.	PUNCT
cana-2702	267	1	set	set	VERB
cana-2702	267	2	8𝑡	8𝑡	NOUN
cana-2702	267	3	−	−	NOUN
cana-2702	267	4	1	1	NUM
cana-2702	267	5	pebbles	pebble	NOUN
cana-2702	267	6	on	on	ADP
cana-2702	267	7	vertex	vertex	NOUN
cana-2702	267	8	𝑥	𝑥	PROPN
cana-2702	267	9	and	and	CCONJ
cana-2702	267	10	place	place	VERB
cana-2702	267	11	one	one	NUM
cana-2702	267	12	pebble	pebble	NOUN
cana-2702	267	13	on	on	ADP
cana-2702	267	14	each	each	DET
cana-2702	267	15	𝑎𝑖	𝑎𝑖	NOUN
cana-2702	267	16	and	and	CCONJ
cana-2702	267	17	𝑏𝑖	𝑏𝑖	ADP
cana-2702	267	18	except	except	SCONJ
cana-2702	267	19	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	267	20	,	,	PUNCT
cana-2702	267	21	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	267	22	,	,	PUNCT
cana-2702	267	23	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	267	24	and	and	CCONJ
cana-2702	267	25	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	267	26	.	.	PUNCT
cana-2702	268	1	thus	thus	ADV
cana-2702	268	2	,	,	PUNCT
cana-2702	268	3	we	we	PRON
cana-2702	268	4	can	can	AUX
cana-2702	268	5	not	not	PART
cana-2702	268	6	move	move	VERB
cana-2702	268	7	𝑡	𝑡	NOUN
cana-2702	268	8	pebbles	pebble	NOUN
cana-2702	268	9	to	to	PART
cana-2702	268	10	vertex	vertex	VERB
cana-2702	268	11	𝑦.	𝑦.	PROPN
cana-2702	269	1	so	so	SCONJ
cana-2702	269	2	𝑓𝑡(𝐶3(𝐾𝑛	𝑓𝑡(𝐶3(𝐾𝑛	NUM
cana-2702	269	3	)	)	PUNCT
cana-2702	269	4	)	)	PUNCT
cana-2702	270	1	≥	≥	NOUN
cana-2702	270	2	8𝑡	8𝑡	NOUN
cana-2702	270	3	+	+	CCONJ
cana-2702	270	4	3𝑛	3𝑛	NUM
cana-2702	270	5	−	−	NOUN
cana-2702	270	6	10	10	NUM
cana-2702	270	7	.	.	PUNCT
cana-2702	271	1	consider	consider	VERB
cana-2702	271	2	graph	graph	NOUN
cana-2702	271	3	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	271	4	)	)	PUNCT
cana-2702	271	5	with	with	ADP
cana-2702	271	6	at	at	ADV
cana-2702	271	7	least	least	ADJ
cana-2702	271	8	8𝑡	8𝑡	NOUN
cana-2702	271	9	+	+	CCONJ
cana-2702	271	10	3𝑛	3𝑛	NUM
cana-2702	271	11	−	−	NUM
cana-2702	271	12	10	10	NUM
cana-2702	271	13	pebbles	pebble	NOUN
cana-2702	271	14	distributed	distribute	VERB
cana-2702	271	15	on	on	ADP
cana-2702	271	16	its	its	PRON
cana-2702	271	17	vertices	vertex	NOUN
cana-2702	271	18	.	.	PUNCT
cana-2702	272	1	let	let	VERB
cana-2702	272	2	𝑣	𝑣	PRON
cana-2702	272	3	∈	∈	PROPN
cana-2702	272	4	𝐶𝑖	𝐶𝑖	PROPN
cana-2702	272	5	for	for	ADP
cana-2702	272	6	𝑖	𝑖	PRON
cana-2702	272	7	=	=	NOUN
cana-2702	272	8	1,2,3	1,2,3	NUM
cana-2702	272	9	be	be	AUX
cana-2702	272	10	any	any	DET
cana-2702	272	11	target	target	NOUN
cana-2702	272	12	vertex	vertex	NOUN
cana-2702	272	13	.	.	PUNCT
cana-2702	273	1	we	we	PRON
cana-2702	273	2	consider	consider	VERB
cana-2702	273	3	the	the	DET
cana-2702	273	4	following	follow	VERB
cana-2702	273	5	cases	case	NOUN
cana-2702	273	6	:	:	PUNCT
cana-2702	273	7	case	case	NOUN
cana-2702	273	8	(	(	PUNCT
cana-2702	273	9	1	1	X
cana-2702	273	10	)	)	PUNCT
cana-2702	273	11	𝑣	𝑣	PRON
cana-2702	273	12	∈	∈	PROPN
cana-2702	274	1	𝐶2	𝐶2	PROPN
cana-2702	274	2	.	.	PUNCT
cana-2702	274	3	suppose	suppose	VERB
cana-2702	274	4	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	274	5	)	)	PUNCT
cana-2702	274	6	≥	≥	NOUN
cana-2702	274	7	4𝑡	4𝑡	VERB
cana-2702	275	1	+	+	CCONJ
cana-2702	276	1	𝑛	𝑛	DET
cana-2702	276	2	−	−	NOUN
cana-2702	276	3	2	2	NUM
cana-2702	276	4	.	.	PUNCT
cana-2702	277	1	we	we	PRON
cana-2702	277	2	can	can	AUX
cana-2702	277	3	move	move	VERB
cana-2702	277	4	2𝑡	2𝑡	NUM
cana-2702	277	5	pebbles	pebble	NOUN
cana-2702	277	6	to	to	ADP
cana-2702	277	7	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	277	8	and	and	CCONJ
cana-2702	277	9	move	move	VERB
cana-2702	277	10	𝑡	𝑡	PROPN
cana-2702	277	11	pebbles	pebble	NOUN
cana-2702	277	12	to	to	ADP
cana-2702	277	13	𝑣	𝑣	NOUN
cana-2702	277	14	by	by	ADP
cana-2702	277	15	using	use	VERB
cana-2702	277	16	pebbling	pebble	VERB
cana-2702	277	17	number	number	NOUN
cana-2702	277	18	of	of	ADP
cana-2702	277	19	complete	complete	ADJ
cana-2702	277	20	graph	graph	NOUN
cana-2702	277	21	.	.	PUNCT
cana-2702	278	1	so	so	ADV
cana-2702	278	2	assume	assume	VERB
cana-2702	278	3	that	that	SCONJ
cana-2702	278	4	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	278	5	)	)	PUNCT
cana-2702	278	6	≤	≤	NOUN
cana-2702	278	7	4𝑡	4𝑡	NOUN
cana-2702	279	1	+	+	CCONJ
cana-2702	279	2	𝑛	𝑛	PRON
cana-2702	279	3	−	−	NOUN
cana-2702	279	4	3	3	NUM
cana-2702	279	5	.	.	PUNCT
cana-2702	280	1	this	this	DET
cana-2702	280	2	results	result	VERB
cana-2702	280	3	in	in	ADP
cana-2702	280	4	at	at	ADV
cana-2702	280	5	least	least	ADJ
cana-2702	280	6	8𝑡	8𝑡	NOUN
cana-2702	280	7	+	+	CCONJ
cana-2702	280	8	3𝑛	3𝑛	NUM
cana-2702	280	9	−	−	PROPN
cana-2702	280	10	10	10	NUM
cana-2702	280	11	−	−	PROPN
cana-2702	280	12	(	(	PUNCT
cana-2702	280	13	4𝑡	4𝑡	NOUN
cana-2702	280	14	+	+	CCONJ
cana-2702	280	15	𝑛	𝑛	PRON
cana-2702	280	16	−	−	NOUN
cana-2702	280	17	3	3	NUM
cana-2702	280	18	)	)	PUNCT
cana-2702	280	19	=	=	NOUN
cana-2702	280	20	4𝑡	4𝑡	NOUN
cana-2702	281	1	+	+	CCONJ
cana-2702	281	2	2𝑛	2𝑛	PROPN
cana-2702	281	3	−	−	PROPN
cana-2702	282	1	7	7	X
cana-2702	282	2	.	.	X
cana-2702	282	3	we	we	PRON
cana-2702	282	4	have	have	VERB
cana-2702	282	5	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	282	6	∪	∪	ADJ
cana-2702	282	7	𝐶2	𝐶2	PROPN
cana-2702	282	8	)	)	PUNCT
cana-2702	282	9	≥	≥	NOUN
cana-2702	282	10	4𝑡	4𝑡	NUM
cana-2702	283	1	+	+	CCONJ
cana-2702	283	2	2𝑛	2𝑛	PROPN
cana-2702	283	3	−	−	PROPN
cana-2702	284	1	7	7	X
cana-2702	284	2	.	.	PUNCT
cana-2702	285	1	if	if	SCONJ
cana-2702	285	2	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	285	3	∪	∪	X
cana-2702	285	4	𝐶2	𝐶2	PROPN
cana-2702	285	5	)	)	PUNCT
cana-2702	285	6	≥	≥	NOUN
cana-2702	285	7	4𝑡	4𝑡	NUM
cana-2702	286	1	+	+	CCONJ
cana-2702	286	2	2𝑛	2𝑛	PROPN
cana-2702	286	3	−	−	PROPN
cana-2702	286	4	6	6	NUM
cana-2702	286	5	,	,	PUNCT
cana-2702	286	6	by	by	ADP
cana-2702	286	7	previous	previous	ADJ
cana-2702	286	8	theorem	theorem	NOUN
cana-2702	286	9	we	we	PRON
cana-2702	286	10	can	can	AUX
cana-2702	286	11	move	move	VERB
cana-2702	286	12	𝑡	𝑡	PROPN
cana-2702	286	13	pebbles	pebble	NOUN
cana-2702	286	14	to	to	PART
cana-2702	286	15	𝑣.	𝑣.	VERB
cana-2702	286	16	therefore	therefore	ADV
cana-2702	286	17	,	,	PUNCT
cana-2702	286	18	we	we	PRON
cana-2702	286	19	assume	assume	VERB
cana-2702	286	20	that	that	SCONJ
cana-2702	286	21	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	286	22	∪	∪	ADP
cana-2702	286	23	𝐶2	𝐶2	NOUN
cana-2702	286	24	)	)	PUNCT
cana-2702	286	25	≤	≤	NOUN
cana-2702	286	26	4𝑡	4𝑡	NOUN
cana-2702	287	1	+	+	CCONJ
cana-2702	287	2	2𝑛	2𝑛	PROPN
cana-2702	287	3	−	−	PROPN
cana-2702	288	1	7	7	X
cana-2702	288	2	.	.	PUNCT
cana-2702	289	1	if	if	SCONJ
cana-2702	289	2	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	289	3	∪	∪	X
cana-2702	289	4	𝐶2	𝐶2	NOUN
cana-2702	289	5	)	)	PUNCT
cana-2702	289	6	<	<	X
cana-2702	289	7	4𝑡	4𝑡	PROPN
cana-2702	290	1	+	+	CCONJ
cana-2702	290	2	2𝑛	2𝑛	PROPN
cana-2702	290	3	−	−	PROPN
cana-2702	290	4	7	7	NUM
cana-2702	290	5	,	,	PUNCT
cana-2702	290	6	then	then	ADV
cana-2702	290	7	we	we	PRON
cana-2702	290	8	get	get	VERB
cana-2702	290	9	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	290	10	)	)	PUNCT
cana-2702	290	11	≥	≥	NOUN
cana-2702	290	12	4𝑡	4𝑡	NOUN
cana-2702	291	1	+	+	CCONJ
cana-2702	291	2	3𝑛	3𝑛	NUM
cana-2702	291	3	−	−	NOUN
cana-2702	291	4	3	3	X
cana-2702	291	5	.	.	PUNCT
cana-2702	292	1	otherwise	otherwise	ADV
cana-2702	292	2	,	,	PUNCT
cana-2702	292	3	this	this	PRON
cana-2702	292	4	contradicts	contradict	VERB
cana-2702	292	5	the	the	DET
cana-2702	292	6	total	total	ADJ
cana-2702	292	7	number	number	NOUN
cana-2702	292	8	of	of	ADP
cana-2702	292	9	pebbles	pebble	NOUN
cana-2702	292	10	distributed	distribute	VERB
cana-2702	292	11	on	on	ADP
cana-2702	292	12	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	292	13	)	)	PUNCT
cana-2702	292	14	.	.	PUNCT
cana-2702	293	1	so	so	ADV
cana-2702	293	2	we	we	PRON
cana-2702	293	3	take	take	VERB
cana-2702	293	4	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	293	5	∪	∪	ADJ
cana-2702	293	6	𝐶2	𝐶2	NOUN
cana-2702	293	7	)	)	PUNCT
cana-2702	294	1	=	=	NOUN
cana-2702	294	2	4𝑡	4𝑡	NUM
cana-2702	295	1	−	−	NOUN
cana-2702	295	2	2𝑛	2𝑛	NOUN
cana-2702	295	3	−	−	PROPN
cana-2702	295	4	7	7	NUM
cana-2702	295	5	and	and	CCONJ
cana-2702	295	6	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	295	7	)	)	PUNCT
cana-2702	295	8	≥	≥	NOUN
cana-2702	295	9	4𝑡	4𝑡	VERB
cana-2702	296	1	+	+	CCONJ
cana-2702	296	2	𝑛	𝑛	PRON
cana-2702	296	3	−	−	NUM
cana-2702	296	4	3	3	NUM
cana-2702	296	5	≥	≥	NOUN
cana-2702	296	6	2(2(𝑡	2(2(𝑡	NUM
cana-2702	296	7	−	−	NOUN
cana-2702	296	8	1	1	NUM
cana-2702	296	9	)	)	PUNCT
cana-2702	296	10	)	)	PUNCT
cana-2702	297	1	+	+	CCONJ
cana-2702	298	1	𝑛	𝑛	DET
cana-2702	298	2	−	−	NUM
cana-2702	298	3	3	3	NUM
cana-2702	298	4	=	=	NOUN
cana-2702	298	5	4𝑡	4𝑡	NOUN
cana-2702	298	6	+	+	CCONJ
cana-2702	298	7	𝑛	𝑛	PRON
cana-2702	298	8	−	−	NOUN
cana-2702	298	9	7	7	NUM
cana-2702	298	10	.	.	PUNCT
cana-2702	298	11	again	again	ADV
cana-2702	298	12	,	,	PUNCT
cana-2702	298	13	by	by	ADP
cana-2702	298	14	using	use	VERB
cana-2702	298	15	𝑓(𝐾𝑛	𝑓(𝐾𝑛	NOUN
cana-2702	298	16	)	)	PUNCT
cana-2702	298	17	=	=	SYM
cana-2702	298	18	𝑛	𝑛	ADP
cana-2702	298	19	we	we	PRON
cana-2702	298	20	can	can	AUX
cana-2702	298	21	move	move	VERB
cana-2702	298	22	2(𝑡	2(𝑡	NUM
cana-2702	298	23	−	−	NOUN
cana-2702	298	24	1	1	NUM
cana-2702	298	25	)	)	PUNCT
cana-2702	298	26	pebbles	pebble	NOUN
cana-2702	298	27	to	to	ADP
cana-2702	298	28	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	298	29	and	and	CCONJ
cana-2702	298	30	𝑡	𝑡	PROPN
cana-2702	298	31	−	−	PROPN
cana-2702	298	32	1	1	NUM
cana-2702	298	33	pebble	pebble	ADJ
cana-2702	298	34	to	to	ADP
cana-2702	298	35	𝑣.	𝑣.	NOUN
cana-2702	298	36	also	also	ADV
cana-2702	298	37	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	298	38	∪	∪	ADP
cana-2702	298	39	𝐶2	𝐶2	NOUN
cana-2702	298	40	)	)	PUNCT
cana-2702	299	1	=	=	NOUN
cana-2702	299	2	4𝑡	4𝑡	NOUN
cana-2702	300	1	+	+	CCONJ
cana-2702	300	2	2𝑛	2𝑛	PROPN
cana-2702	300	3	−	−	PROPN
cana-2702	300	4	7	7	NUM
cana-2702	300	5	≥	≥	NOUN
cana-2702	300	6	2𝑛	2𝑛	NOUN
cana-2702	300	7	−	−	PROPN
cana-2702	300	8	2	2	NUM
cana-2702	300	9	,	,	PUNCT
cana-2702	300	10	since	since	SCONJ
cana-2702	300	11	𝑡	𝑡	PROPN
cana-2702	300	12	≥	≥	NOUN
cana-2702	300	13	2	2	NUM
cana-2702	300	14	and	and	CCONJ
cana-2702	300	15	𝐶1	𝐶1	PROPN
cana-2702	300	16	∪	∪	X
cana-2702	300	17	𝐶2	𝐶2	PROPN
cana-2702	300	18	≅	≅	PROPN
cana-2702	300	19	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	PROPN
cana-2702	300	20	)	)	PUNCT
cana-2702	300	21	.	.	PUNCT
cana-2702	301	1	thus	thus	ADV
cana-2702	301	2	,	,	PUNCT
cana-2702	301	3	from	from	ADP
cana-2702	301	4	theorem	theorem	NOUN
cana-2702	301	5	1	1	NUM
cana-2702	301	6	we	we	PRON
cana-2702	301	7	can	can	AUX
cana-2702	301	8	move	move	VERB
cana-2702	301	9	an	an	DET
cana-2702	301	10	additional	additional	ADJ
cana-2702	301	11	pebble	pebble	NOUN
cana-2702	301	12	to	to	ADP
cana-2702	301	13	𝑣.	𝑣.	NOUN
cana-2702	301	14	case	case	NOUN
cana-2702	301	15	(	(	PUNCT
cana-2702	301	16	2	2	X
cana-2702	301	17	)	)	PUNCT
cana-2702	301	18	𝑣	𝑣	PART
cana-2702	301	19	∈	∈	NOUN
cana-2702	301	20	𝐶1	𝐶1	NOUN
cana-2702	301	21	or	or	CCONJ
cana-2702	301	22	𝑣	𝑣	ADP
cana-2702	301	23	∈	∈	PROPN
cana-2702	301	24	𝐶3	𝐶3	NOUN
cana-2702	301	25	.	.	PUNCT
cana-2702	302	1	fix	fix	VERB
cana-2702	302	2	𝑣	𝑣	ADP
cana-2702	302	3	∈	∈	NOUN
cana-2702	302	4	𝐶3	𝐶3	NOUN
cana-2702	302	5	and	and	CCONJ
cana-2702	302	6	𝑣	𝑣	X
cana-2702	303	1	=	=	PUNCT
cana-2702	303	2	𝑦.	𝑦.	PROPN
cana-2702	303	3	we	we	PRON
cana-2702	303	4	assume	assume	VERB
cana-2702	303	5	that	that	SCONJ
cana-2702	303	6	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	303	7	)	)	PUNCT
cana-2702	303	8	≥	≥	NOUN
cana-2702	303	9	𝑛.	𝑛.	NOUN
cana-2702	303	10	subsequently	subsequently	ADV
cana-2702	303	11	,	,	PUNCT
cana-2702	303	12	at	at	ADV
cana-2702	303	13	least	least	ADJ
cana-2702	303	14	one	one	NUM
cana-2702	303	15	of	of	ADP
cana-2702	303	16	the	the	DET
cana-2702	303	17	vertices	vertex	NOUN
cana-2702	303	18	in	in	ADP
cana-2702	303	19	𝐶3	𝐶3	PROPN
cana-2702	303	20	contains	contain	VERB
cana-2702	303	21	at	at	ADV
cana-2702	303	22	least	least	ADV
cana-2702	303	23	two	two	NUM
cana-2702	303	24	pebbles	pebble	NOUN
cana-2702	303	25	.	.	PUNCT
cana-2702	304	1	using	use	VERB
cana-2702	304	2	exactly	exactly	ADV
cana-2702	304	3	two	two	NUM
cana-2702	304	4	pebbles	pebble	NOUN
cana-2702	304	5	,	,	PUNCT
cana-2702	304	6	we	we	PRON
cana-2702	304	7	can	can	AUX
cana-2702	304	8	move	move	VERB
cana-2702	304	9	one	one	NUM
cana-2702	304	10	pebble	pebble	NOUN
cana-2702	304	11	to	to	AUX
cana-2702	304	12	𝑦.	𝑦.	VERB
cana-2702	304	13	this	this	DET
cana-2702	304	14	results	result	NOUN
cana-2702	304	15	in	in	ADP
cana-2702	304	16	at	at	ADV
cana-2702	304	17	least	least	ADJ
cana-2702	304	18	8𝑡	8𝑡	NOUN
cana-2702	304	19	+	+	CCONJ
cana-2702	304	20	3𝑛	3𝑛	NUM
cana-2702	304	21	−	−	PROPN
cana-2702	304	22	12	12	NUM
cana-2702	304	23	≥	≥	NOUN
cana-2702	304	24	8(𝑡	8(𝑡	NUM
cana-2702	304	25	−	−	NUM
cana-2702	304	26	1	1	NUM
cana-2702	304	27	)	)	PUNCT
cana-2702	304	28	+	+	NUM
cana-2702	304	29	3𝑛	3𝑛	NUM
cana-2702	304	30	−	−	NUM
cana-2702	304	31	10	10	NUM
cana-2702	304	32	pebbles	pebble	NOUN
cana-2702	304	33	in	in	ADP
cana-2702	304	34	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	304	35	)	)	PUNCT
cana-2702	304	36	.	.	PUNCT
cana-2702	305	1	by	by	ADP
cana-2702	305	2	induction	induction	NOUN
cana-2702	305	3	,	,	PUNCT
cana-2702	305	4	we	we	PRON
cana-2702	305	5	can	can	AUX
cana-2702	305	6	move	move	VERB
cana-2702	305	7	an	an	DET
cana-2702	305	8	additional	additional	ADJ
cana-2702	305	9	𝑡	𝑡	NOUN
cana-2702	305	10	−	−	NOUN
cana-2702	305	11	1	1	NUM
cana-2702	305	12	pebbles	pebble	NOUN
cana-2702	305	13	to	to	PART
cana-2702	305	14	𝑦.	𝑦.	PROPN
cana-2702	305	15	thus	thus	ADV
cana-2702	305	16	,	,	PUNCT
cana-2702	305	17	no	no	DET
cana-2702	305	18	vertices	vertex	NOUN
cana-2702	305	19	in	in	ADP
cana-2702	305	20	𝐶3	𝐶3	NOUN
cana-2702	305	21	contained	contain	VERB
cana-2702	305	22	two	two	NUM
cana-2702	305	23	pebbles	pebble	NOUN
cana-2702	305	24	.	.	PUNCT
cana-2702	306	1	thus	thus	ADV
cana-2702	306	2	𝑝(𝑉(𝐶3	𝑝(𝑉(𝐶3	NOUN
cana-2702	306	3	)	)	PUNCT
cana-2702	306	4	−	−	PROPN
cana-2702	306	5	{	{	PUNCT
cana-2702	306	6	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	306	7	,	,	PUNCT
cana-2702	306	8	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	306	9	}	}	PUNCT
cana-2702	306	10	)	)	PUNCT
cana-2702	306	11	≤	≤	NUM
cana-2702	307	1	𝑛	𝑛	DET
cana-2702	307	2	−	−	NOUN
cana-2702	307	3	3	3	NUM
cana-2702	307	4	.	.	PUNCT
cana-2702	308	1	this	this	PRON
cana-2702	308	2	implies	imply	VERB
cana-2702	308	3	that	that	SCONJ
cana-2702	308	4	at	at	ADV
cana-2702	308	5	least	least	ADJ
cana-2702	308	6	8𝑡	8𝑡	NOUN
cana-2702	308	7	+	+	CCONJ
cana-2702	308	8	3𝑛	3𝑛	NUM
cana-2702	308	9	−	−	PROPN
cana-2702	308	10	10	10	NUM
cana-2702	308	11	−	−	NOUN
cana-2702	308	12	𝑛	𝑛	PROPN
cana-2702	308	13	+	+	NOUN
cana-2702	308	14	3	3	NUM
cana-2702	308	15	=	=	NOUN
cana-2702	308	16	8𝑡	8𝑡	NOUN
cana-2702	308	17	+	+	CCONJ
cana-2702	308	18	2𝑛	2𝑛	NUM
cana-2702	308	19	−	−	NOUN
cana-2702	308	20	7	7	NUM
cana-2702	308	21	pebbles	pebble	NOUN
cana-2702	308	22	are	be	AUX
cana-2702	308	23	retained	retain	VERB
cana-2702	308	24	in	in	ADP
cana-2702	308	25	𝐶1	𝐶1	NUM
cana-2702	308	26	∪	∪	NOUN
cana-2702	308	27	𝐶2	𝐶2	ADJ
cana-2702	308	28	because	because	SCONJ
cana-2702	308	29	8𝑡	8𝑡	NOUN
cana-2702	308	30	+	+	CCONJ
cana-2702	308	31	3𝑛	3𝑛	NUM
cana-2702	308	32	−	−	NUM
cana-2702	308	33	7	7	NUM
cana-2702	308	34	≥	≥	NOUN
cana-2702	308	35	8𝑡	8𝑡	NOUN
cana-2702	309	1	+	+	CCONJ
cana-2702	309	2	2𝑛	2𝑛	PROPN
cana-2702	309	3	−	−	PROPN
cana-2702	309	4	6	6	NUM
cana-2702	309	5	for	for	ADP
cana-2702	309	6	𝑛	𝑛	PRON
cana-2702	309	7	≥	≥	NUM
cana-2702	309	8	4	4	NUM
cana-2702	309	9	.	.	PUNCT
cana-2702	310	1	thus	thus	ADV
cana-2702	310	2	,	,	PUNCT
cana-2702	310	3	from	from	ADP
cana-2702	310	4	theorem	theorem	ADJ
cana-2702	310	5	4	4	NUM
cana-2702	310	6	,	,	PUNCT
cana-2702	310	7	we	we	PRON
cana-2702	310	8	can	can	AUX
cana-2702	310	9	move	move	VERB
cana-2702	310	10	2𝑡	2𝑡	NUM
cana-2702	310	11	pebbles	pebble	NOUN
cana-2702	310	12	to	to	ADP
cana-2702	310	13	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	310	14	and	and	CCONJ
cana-2702	310	15	then	then	ADV
cana-2702	310	16	move	move	VERB
cana-2702	310	17	𝑡	𝑡	PROPN
cana-2702	310	18	pebbles	pebble	NOUN
cana-2702	310	19	to	to	ADP
cana-2702	310	20	𝑦.	𝑦.	PROPN
cana-2702	310	21	theorem	theorem	PROPN
cana-2702	310	22	6	6	NUM
cana-2702	310	23	.	.	PUNCT
cana-2702	311	1	the	the	DET
cana-2702	311	2	t	t	NOUN
cana-2702	311	3	-	-	PUNCT
cana-2702	311	4	pebbling	pebble	VERB
cana-2702	311	5	number	number	NOUN
cana-2702	311	6	of	of	ADP
cana-2702	311	7	the	the	DET
cana-2702	311	8	graph	graph	NOUN
cana-2702	311	9	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	311	10	)	)	PUNCT
cana-2702	311	11	is	be	AUX
cana-2702	311	12	𝑓𝑡(𝐶𝑚(𝐾𝑛	𝑓𝑡(𝐶𝑚(𝐾𝑛	PROPN
cana-2702	311	13	)	)	PUNCT
cana-2702	311	14	)	)	PUNCT
cana-2702	312	1	=	=	PUNCT
cana-2702	312	2	𝑡2𝑚	𝑡2𝑚	PROPN
cana-2702	312	3	+	+	CCONJ
cana-2702	312	4	2(𝑛	2(𝑛	NUM
cana-2702	312	5	−	−	NOUN
cana-2702	312	6	3	3	NUM
cana-2702	312	7	)	)	PUNCT
cana-2702	312	8	+	+	CCONJ
cana-2702	312	9	(	(	PUNCT
cana-2702	312	10	𝑚	𝑚	PROPN
cana-2702	312	11	−	−	PROPN
cana-2702	312	12	2)(𝑛	2)(𝑛	NUM
cana-2702	312	13	−	−	NOUN
cana-2702	312	14	4	4	NUM
cana-2702	312	15	)	)	PUNCT
cana-2702	312	16	.	.	PUNCT
cana-2702	313	1	proof	proof	NOUN
cana-2702	313	2	.	.	PUNCT
cana-2702	314	1	set	set	VERB
cana-2702	314	2	𝑡2𝑚	𝑡2𝑚	PROPN
cana-2702	314	3	−	−	PROPN
cana-2702	314	4	1	1	NUM
cana-2702	314	5	pebbles	pebble	NOUN
cana-2702	314	6	on	on	ADP
cana-2702	314	7	vertex	vertex	NOUN
cana-2702	314	8	𝑥	𝑥	PROPN
cana-2702	314	9	and	and	CCONJ
cana-2702	314	10	put	put	VERB
cana-2702	314	11	one	one	NUM
cana-2702	314	12	pebble	pebble	NOUN
cana-2702	314	13	on	on	ADP
cana-2702	314	14	each	each	DET
cana-2702	314	15	𝑎𝑖	𝑎𝑖	NOUN
cana-2702	314	16	and	and	CCONJ
cana-2702	314	17	𝑏𝑖	𝑏𝑖	ADP
cana-2702	314	18	except	except	SCONJ
cana-2702	314	19	for	for	ADP
cana-2702	314	20	the	the	DET
cana-2702	314	21	vertices	vertex	NOUN
cana-2702	314	22	in	in	ADP
cana-2702	314	23	{	{	PUNCT
cana-2702	314	24	𝑎𝑖𝑎𝑖+1	𝑎𝑖𝑎𝑖+1	PROPN
cana-2702	314	25	,	,	PUNCT
cana-2702	314	26	𝑏𝑖𝑏𝑖+1	𝑏𝑖𝑏𝑖+1	NOUN
cana-2702	314	27	:	:	PUNCT
cana-2702	314	28	1	1	NUM
cana-2702	314	29	≤	≤	NUM
cana-2702	314	30	𝑖	𝑖	PUNCT
cana-2702	314	31	≤	≤	ADJ
cana-2702	314	32	𝑛(𝑘	𝑛(𝑘	NOUN
cana-2702	314	33	−	−	PROPN
cana-2702	314	34	1	1	NUM
cana-2702	314	35	)	)	PUNCT
cana-2702	314	36	−	−	PROPN
cana-2702	314	37	1	1	NUM
cana-2702	314	38	}	}	PUNCT
cana-2702	314	39	.	.	PUNCT
cana-2702	315	1	thus	thus	ADV
cana-2702	315	2	,	,	PUNCT
cana-2702	315	3	we	we	PRON
cana-2702	315	4	can	can	AUX
cana-2702	315	5	not	not	PART
cana-2702	315	6	move	move	VERB
cana-2702	315	7	𝑡	𝑡	PROPN
cana-2702	315	8	pebbles	pebble	NOUN
cana-2702	315	9	to	to	PART
cana-2702	315	10	𝑦.	𝑦.	VERB
cana-2702	315	11	so	so	ADV
cana-2702	315	12	𝑓𝑡(𝐾𝑛	𝑓𝑡(𝐾𝑛	PROPN
cana-2702	315	13	)	)	PUNCT
cana-2702	315	14	≥	≥	NOUN
cana-2702	315	15	𝑡.	𝑡.	NOUN
cana-2702	315	16	2𝑚	2𝑚	NOUN
cana-2702	315	17	+	+	CCONJ
cana-2702	315	18	2(𝑛	2(𝑛	NUM
cana-2702	315	19	−	−	NOUN
cana-2702	315	20	3	3	NUM
cana-2702	315	21	)	)	PUNCT
cana-2702	315	22	+	+	CCONJ
cana-2702	315	23	(	(	PUNCT
cana-2702	315	24	𝑚	𝑚	PROPN
cana-2702	315	25	−	−	PROPN
cana-2702	315	26	2)(𝑛	2)(𝑛	NUM
cana-2702	315	27	−	−	NOUN
cana-2702	315	28	4	4	NUM
cana-2702	315	29	)	)	PUNCT
cana-2702	315	30	.	.	PUNCT
cana-2702	316	1	consider	consider	VERB
cana-2702	316	2	a	a	DET
cana-2702	316	3	graph	graph	NOUN
cana-2702	316	4	with	with	ADP
cana-2702	316	5	at	at	ADP
cana-2702	316	6	least	least	ADJ
cana-2702	316	7	𝑡2𝑚	𝑡2𝑚	NOUN
cana-2702	316	8	+	+	CCONJ
cana-2702	316	9	2(𝑛	2(𝑛	NUM
cana-2702	316	10	−	−	NOUN
cana-2702	316	11	3	3	NUM
cana-2702	316	12	)	)	PUNCT
cana-2702	316	13	+	+	CCONJ
cana-2702	317	1	(	(	PUNCT
cana-2702	317	2	𝑚	𝑚	PROPN
cana-2702	317	3	−	−	PROPN
cana-2702	317	4	2)(𝑛	2)(𝑛	NUM
cana-2702	317	5	−	−	NOUN
cana-2702	317	6	4	4	NUM
cana-2702	317	7	)	)	PUNCT
cana-2702	317	8	pebbles	pebble	NOUN
cana-2702	317	9	distributed	distribute	VERB
cana-2702	317	10	at	at	ADP
cana-2702	317	11	the	the	DET
cana-2702	317	12	vertices	vertex	NOUN
cana-2702	317	13	of	of	ADP
cana-2702	317	14	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	317	15	)	)	PUNCT
cana-2702	317	16	.	.	PUNCT
cana-2702	318	1	let	let	VERB
cana-2702	318	2	𝑣	𝑣	PRON
cana-2702	318	3	∈	∈	PROPN
cana-2702	318	4	𝐶𝑖	𝐶𝑖	PROPN
cana-2702	318	5	for	for	ADP
cana-2702	318	6	𝑖	𝑖	PRON
cana-2702	318	7	=	=	SYM
cana-2702	318	8	1,2	1,2	NUM
cana-2702	318	9	,	,	PUNCT
cana-2702	318	10	.	.	PUNCT
cana-2702	318	11	.	.	PUNCT
cana-2702	319	1	.	.	PUNCT
cana-2702	320	1	,	,	PUNCT
cana-2702	320	2	𝑚	𝑚	X
cana-2702	320	3	be	be	AUX
cana-2702	320	4	any	any	DET
cana-2702	320	5	target	target	NOUN
cana-2702	320	6	vertex	vertex	NOUN
cana-2702	320	7	.	.	PUNCT
cana-2702	321	1	we	we	PRON
cana-2702	321	2	prove	prove	VERB
cana-2702	321	3	this	this	DET
cana-2702	321	4	result	result	NOUN
cana-2702	321	5	by	by	ADP
cana-2702	321	6	induction	induction	NOUN
cana-2702	321	7	of	of	ADP
cana-2702	321	8	𝑡	𝑡	PROPN
cana-2702	321	9	and	and	CCONJ
cana-2702	321	10	𝑚.	𝑚.	ADJ
cana-2702	321	11	for	for	ADP
cana-2702	321	12	𝑡	𝑡	PROPN
cana-2702	321	13	=	=	SYM
cana-2702	321	14	1	1	NUM
cana-2702	321	15	,	,	PUNCT
cana-2702	321	16	𝑚	𝑚	NOUN
cana-2702	321	17	=	=	SYM
cana-2702	321	18	2	2	NUM
cana-2702	321	19	and	and	CCONJ
cana-2702	321	20	𝑚	𝑚	X
cana-2702	321	21	=	=	SYM
cana-2702	321	22	3	3	NUM
cana-2702	321	23	,	,	PUNCT
cana-2702	321	24	the	the	DET
cana-2702	321	25	results	result	NOUN
cana-2702	321	26	follow	follow	VERB
cana-2702	321	27	from	from	ADP
cana-2702	321	28	theorems	theorem	NOUN
cana-2702	321	29	3	3	NUM
cana-2702	321	30	,	,	PUNCT
cana-2702	321	31	4	4	NUM
cana-2702	321	32	and	and	CCONJ
cana-2702	321	33	2	2	NUM
cana-2702	321	34	.	.	X
cana-2702	322	1	we	we	PRON
cana-2702	322	2	assume	assume	VERB
cana-2702	322	3	that	that	SCONJ
cana-2702	322	4	this	this	DET
cana-2702	322	5	result	result	NOUN
cana-2702	322	6	is	be	AUX
cana-2702	322	7	true	true	ADJ
cana-2702	322	8	for	for	SCONJ
cana-2702	322	9	all	all	DET
cana-2702	322	10	𝑡′	𝑡′	PUNCT
cana-2702	322	11	<	<	X
cana-2702	322	12	𝑡.	𝑡.	X
cana-2702	322	13	we	we	PRON
cana-2702	322	14	prove	prove	VERB
cana-2702	322	15	this	this	DET
cana-2702	322	16	result	result	NOUN
cana-2702	322	17	for	for	SCONJ
cana-2702	322	18	all	all	DET
cana-2702	322	19	𝑡.	𝑡.	NOUN
cana-2702	322	20	let	let	VERB
cana-2702	322	21	𝑣	𝑣	DET
cana-2702	322	22	∈	∈	PROPN
cana-2702	322	23	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	322	24	.	.	PUNCT
cana-2702	323	1	let	let	VERB
cana-2702	323	2	𝑋1	𝑋1	NOUN
cana-2702	323	3	=	=	SYM
cana-2702	323	4	𝐶1	𝐶1	ADJ
cana-2702	323	5	∪.	∪.	X
cana-2702	323	6	.	.	PUNCT
cana-2702	324	1	.∪	.∪	PROPN
cana-2702	324	2	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	324	3	and	and	CCONJ
cana-2702	324	4	𝑋2	𝑋2	VERB
cana-2702	324	5	=	=	SYM
cana-2702	324	6	𝐶𝑚1	𝐶𝑚1	X
cana-2702	324	7	+	+	ADJ
cana-2702	324	8	1	1	NUM
cana-2702	324	9	∪.	∪.	NOUN
cana-2702	324	10	.	.	PUNCT
cana-2702	325	1	.∪	.∪	PROPN
cana-2702	326	1	𝐶𝑚.	𝐶𝑚.	VERB
cana-2702	326	2	clearly	clearly	ADV
cana-2702	326	3	𝑋1	𝑋1	PROPN
cana-2702	326	4	∪	∪	ADV
cana-2702	326	5	𝑋2	𝑋2	VERB
cana-2702	326	6	≅	≅	PROPN
cana-2702	326	7	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	PROPN
cana-2702	326	8	)	)	PUNCT
cana-2702	326	9	.	.	PUNCT
cana-2702	327	1	suppose	suppose	VERB
cana-2702	327	2	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	327	3	)	)	PUNCT
cana-2702	327	4	≥	≥	NOUN
cana-2702	327	5	2	2	NUM
cana-2702	327	6	𝑚1(2𝑚2	𝑚1(2𝑚2	ADP
cana-2702	327	7	−	−	PROPN
cana-2702	327	8	1	1	NUM
cana-2702	327	9	)	)	PUNCT
cana-2702	327	10	+	+	CCONJ
cana-2702	327	11	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	327	12	−	−	PROPN
cana-2702	327	13	4	4	NUM
cana-2702	327	14	)	)	PUNCT
cana-2702	327	15	+	+	CCONJ
cana-2702	327	16	2	2	X
cana-2702	327	17	.	.	PUNCT
cana-2702	327	18	then	then	ADV
cana-2702	327	19	,	,	PUNCT
cana-2702	327	20	the	the	DET
cana-2702	327	21	two	two	NUM
cana-2702	327	22	pebbles	pebble	NOUN
cana-2702	327	23	can	can	AUX
cana-2702	327	24	be	be	AUX
cana-2702	327	25	moved	move	VERB
cana-2702	327	26	to	to	ADP
cana-2702	327	27	𝑎𝑚1	𝑎𝑚1	PROPN
cana-2702	327	28	(	(	PUNCT
cana-2702	327	29	𝑛	𝑛	PROPN
cana-2702	327	30	2	2	NUM
cana-2702	327	31	−	−	NUM
cana-2702	327	32	1	1	NUM
cana-2702	327	33	)	)	PUNCT
cana-2702	327	34	and	and	CCONJ
cana-2702	327	35	move	move	VERB
cana-2702	327	36	a	a	DET
cana-2702	327	37	pebble	pebble	NOUN
cana-2702	327	38	to	to	PART
cana-2702	327	39	𝑣.	𝑣.	VERB
cana-2702	327	40	this	this	PRON
cana-2702	327	41	implies	imply	VERB
cana-2702	327	42	that	that	SCONJ
cana-2702	327	43	the	the	DET
cana-2702	327	44	number	number	NOUN
cana-2702	327	45	of	of	ADP
cana-2702	327	46	pebbles	pebble	NOUN
cana-2702	327	47	retained	retain	VERB
cana-2702	327	48	in	in	ADP
cana-2702	327	49	𝑋1	𝑋1	PROPN
cana-2702	327	50	was	be	AUX
cana-2702	327	51	at	at	ADP
cana-2702	327	52	least	least	ADJ
cana-2702	327	53	𝑡2𝑚	𝑡2𝑚	NOUN
cana-2702	328	1	+	+	CCONJ
cana-2702	328	2	2(𝑛	2(𝑛	NUM
cana-2702	328	3	−	−	NOUN
cana-2702	328	4	3	3	NUM
cana-2702	328	5	)	)	PUNCT
cana-2702	328	6	+	+	CCONJ
cana-2702	328	7	(	(	PUNCT
cana-2702	328	8	𝑚	𝑚	PROPN
cana-2702	328	9	−	−	PROPN
cana-2702	328	10	2)(𝑛	2)(𝑛	NUM
cana-2702	328	11	−	−	NOUN
cana-2702	328	12	4	4	NUM
cana-2702	328	13	)	)	PUNCT
cana-2702	328	14	−	−	PROPN
cana-2702	329	1	[	[	X
cana-2702	329	2	2𝑚1(2𝑚2	2𝑚1(2𝑚2	NUM
cana-2702	329	3	−	−	NOUN
cana-2702	329	4	1	1	NUM
cana-2702	329	5	)	)	PUNCT
cana-2702	330	1	+	+	NOUN
cana-2702	331	1	𝑚1(𝑛	𝑚1(𝑛	PUNCT
cana-2702	331	2	−	−	NUM
cana-2702	331	3	4	4	NUM
cana-2702	331	4	)	)	PUNCT
cana-2702	331	5	+	+	CCONJ
cana-2702	331	6	2	2	X
cana-2702	331	7	]	]	SYM
cana-2702	331	8	≥	≥	X
cana-2702	331	9	(	(	PUNCT
cana-2702	331	10	𝑡	𝑡	NOUN
cana-2702	331	11	−	−	PROPN
cana-2702	331	12	1)2𝑚	1)2𝑚	NOUN
cana-2702	331	13	+	+	CCONJ
cana-2702	331	14	2(𝑛	2(𝑛	NUM
cana-2702	331	15	−	−	NOUN
cana-2702	331	16	3	3	NUM
cana-2702	331	17	)	)	PUNCT
cana-2702	331	18	+	+	CCONJ
cana-2702	331	19	(	(	PUNCT
cana-2702	331	20	𝑚	𝑚	PROPN
cana-2702	331	21	−	−	PROPN
cana-2702	331	22	2)(𝑛	2)(𝑛	NUM
cana-2702	331	23	−	−	NOUN
cana-2702	331	24	4	4	NUM
cana-2702	331	25	)	)	PUNCT
cana-2702	331	26	.	.	PUNCT
cana-2702	332	1	thus	thus	ADV
cana-2702	332	2	,	,	PUNCT
cana-2702	332	3	by	by	ADP
cana-2702	332	4	induction	induction	NOUN
cana-2702	332	5	,	,	PUNCT
cana-2702	332	6	we	we	PRON
cana-2702	332	7	can	can	AUX
cana-2702	332	8	move	move	VERB
cana-2702	332	9	an	an	DET
cana-2702	332	10	additional	additional	ADJ
cana-2702	332	11	𝑡	𝑡	NOUN
cana-2702	332	12	−	−	NOUN
cana-2702	332	13	1	1	NUM
cana-2702	332	14	pebbles	pebble	NOUN
cana-2702	332	15	to	to	PART
cana-2702	332	16	𝑣.	𝑣.	VERB
cana-2702	332	17	so	so	ADV
cana-2702	332	18	assume	assume	VERB
cana-2702	332	19	that	that	SCONJ
cana-2702	332	20	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	332	21	)	)	PUNCT
cana-2702	332	22	<	<	X
cana-2702	332	23	2	2	NUM
cana-2702	332	24	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	332	25	−	−	PROPN
cana-2702	332	26	1	1	NUM
cana-2702	332	27	)	)	PUNCT
cana-2702	332	28	+	+	CCONJ
cana-2702	332	29	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	332	30	−	−	PROPN
cana-2702	332	31	4	4	NUM
cana-2702	332	32	)	)	PUNCT
cana-2702	332	33	+	+	CCONJ
cana-2702	332	34	2	2	X
cana-2702	332	35	.	.	X
cana-2702	332	36	claim	claim	NOUN
cana-2702	332	37	(	(	PUNCT
cana-2702	332	38	1	1	X
cana-2702	332	39	)	)	PUNCT
cana-2702	332	40	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NUM
cana-2702	332	41	)	)	PUNCT
cana-2702	332	42	)	)	PUNCT
cana-2702	333	1	−	−	PROPN
cana-2702	333	2	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	333	3	)	)	PUNCT
cana-2702	333	4	≥	≥	NOUN
cana-2702	333	5	𝑓𝑡	𝑓𝑡	PROPN
cana-2702	333	6	(	(	PUNCT
cana-2702	333	7	𝐶𝑚1	𝐶𝑚1	X
cana-2702	333	8	(	(	PUNCT
cana-2702	333	9	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	333	10	)	)	PUNCT
cana-2702	333	11	)	)	PUNCT
cana-2702	333	12	.	.	PUNCT
cana-2702	334	1	communications	communication	NOUN
cana-2702	334	2	on	on	ADP
cana-2702	334	3	applied	apply	VERB
cana-2702	334	4	nonlinear	nonlinear	ADJ
cana-2702	334	5	analysis	analysis	NOUN
cana-2702	334	6	issn	issn	NOUN
cana-2702	334	7	:	:	PUNCT
cana-2702	334	8	1074	1074	NUM
cana-2702	334	9	-	-	PUNCT
cana-2702	334	10	133x	133x	NUM
cana-2702	334	11	vol	vol	NOUN
cana-2702	334	12	32	32	NUM
cana-2702	334	13	no	no	NOUN
cana-2702	334	14	.	.	PUNCT
cana-2702	335	1	3s	3s	NUM
cana-2702	335	2	(	(	PUNCT
cana-2702	335	3	2025	2025	NUM
cana-2702	335	4	)	)	PUNCT
cana-2702	335	5	655	655	NUM
cana-2702	335	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	335	7	we	we	PRON
cana-2702	335	8	have	have	VERB
cana-2702	335	9	,	,	PUNCT
cana-2702	335	10	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NUM
cana-2702	335	11	)	)	PUNCT
cana-2702	335	12	)	)	PUNCT
cana-2702	336	1	−	−	PROPN
cana-2702	336	2	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	336	3	)	)	PUNCT
cana-2702	336	4	=	=	PUNCT
cana-2702	337	1	[	[	X
cana-2702	337	2	𝑡.	𝑡.	NOUN
cana-2702	337	3	2𝑚	2𝑚	NOUN
cana-2702	337	4	+	+	CCONJ
cana-2702	337	5	2(𝑛	2(𝑛	NUM
cana-2702	337	6	−	−	NOUN
cana-2702	337	7	3	3	NUM
cana-2702	337	8	)	)	PUNCT
cana-2702	337	9	+	+	CCONJ
cana-2702	337	10	(	(	PUNCT
cana-2702	337	11	𝑚	𝑚	PROPN
cana-2702	337	12	−	−	PROPN
cana-2702	337	13	2)(𝑛	2)(𝑛	NUM
cana-2702	337	14	−	−	NOUN
cana-2702	337	15	4	4	NUM
cana-2702	337	16	)	)	PUNCT
cana-2702	337	17	]	]	PUNCT
cana-2702	338	1	−	−	PROPN
cana-2702	339	1	[	[	X
cana-2702	339	2	2𝑚1(2𝑚2	2𝑚1(2𝑚2	NUM
cana-2702	339	3	−	−	NOUN
cana-2702	339	4	1	1	NUM
cana-2702	339	5	)	)	PUNCT
cana-2702	339	6	+	+	CCONJ
cana-2702	339	7	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	339	8	−	−	PROPN
cana-2702	339	9	4	4	NUM
cana-2702	339	10	)	)	PUNCT
cana-2702	339	11	+	+	CCONJ
cana-2702	339	12	1	1	X
cana-2702	339	13	]	]	PUNCT
cana-2702	339	14	=	=	PUNCT
cana-2702	339	15	𝑡.	𝑡.	NOUN
cana-2702	339	16	2𝑚	2𝑚	NOUN
cana-2702	339	17	+	+	CCONJ
cana-2702	339	18	2(𝑛	2(𝑛	NUM
cana-2702	339	19	−	−	NOUN
cana-2702	339	20	3	3	NUM
cana-2702	339	21	)	)	PUNCT
cana-2702	339	22	+	+	CCONJ
cana-2702	339	23	(	(	PUNCT
cana-2702	339	24	𝑚	𝑚	PROPN
cana-2702	339	25	−	−	PROPN
cana-2702	339	26	2)(𝑛	2)(𝑛	NUM
cana-2702	339	27	−	−	NOUN
cana-2702	339	28	4	4	NUM
cana-2702	339	29	)	)	PUNCT
cana-2702	339	30	−	−	PROPN
cana-2702	339	31	2	2	NUM
cana-2702	339	32	𝑚	𝑚	PRON
cana-2702	339	33	+	+	SYM
cana-2702	339	34	2	2	NUM
cana-2702	339	35	𝑚1	𝑚1	NOUN
cana-2702	339	36	−	−	PROPN
cana-2702	339	37	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	339	38	−	−	PROPN
cana-2702	339	39	4	4	NUM
cana-2702	339	40	)	)	PUNCT
cana-2702	339	41	−	−	PROPN
cana-2702	339	42	1	1	NUM
cana-2702	339	43	=	=	PUNCT
cana-2702	339	44	𝑡.	𝑡.	VERB
cana-2702	339	45	2𝑚1(2𝑚2	2𝑚1(2𝑚2	NUM
cana-2702	339	46	−	−	NOUN
cana-2702	339	47	1	1	X
cana-2702	339	48	)	)	PUNCT
cana-2702	339	49	+	+	CCONJ
cana-2702	339	50	𝑡.	𝑡.	VERB
cana-2702	339	51	2𝑚1	2𝑚1	NUM
cana-2702	339	52	+	+	CCONJ
cana-2702	339	53	2(𝑛	2(𝑛	NUM
cana-2702	339	54	−	−	NOUN
cana-2702	339	55	3	3	NUM
cana-2702	339	56	)	)	PUNCT
cana-2702	339	57	+	+	CCONJ
cana-2702	339	58	(	(	PUNCT
cana-2702	339	59	𝑚1	𝑚1	NOUN
cana-2702	339	60	−	−	PROPN
cana-2702	339	61	2)(𝑛	2)(𝑛	NUM
cana-2702	339	62	−	−	NOUN
cana-2702	339	63	4	4	NUM
cana-2702	339	64	)	)	PUNCT
cana-2702	339	65	−	−	NOUN
cana-2702	340	1	1	1	NUM
cana-2702	340	2	=	=	SYM
cana-2702	340	3	𝑓𝑡	𝑓𝑡	NOUN
cana-2702	340	4	(	(	PUNCT
cana-2702	340	5	𝐶𝑚1	𝐶𝑚1	X
cana-2702	340	6	(	(	PUNCT
cana-2702	340	7	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	340	8	)	)	PUNCT
cana-2702	340	9	)	)	PUNCT
cana-2702	341	1	+	+	CCONJ
cana-2702	341	2	𝑡.	𝑡.	NOUN
cana-2702	341	3	2𝑚	2𝑚	NOUN
cana-2702	341	4	−	−	NOUN
cana-2702	341	5	𝑡.	𝑡.	NOUN
cana-2702	341	6	2𝑚1	2𝑚1	NUM
cana-2702	341	7	−	−	PROPN
cana-2702	341	8	2	2	NUM
cana-2702	341	9	𝑚	𝑚	NOUN
cana-2702	341	10	−	−	PROPN
cana-2702	341	11	1	1	NUM
cana-2702	341	12	=	=	SYM
cana-2702	341	13	𝑓𝑡	𝑓𝑡	NOUN
cana-2702	341	14	(	(	PUNCT
cana-2702	341	15	𝐶𝑚1	𝐶𝑚1	X
cana-2702	341	16	(	(	PUNCT
cana-2702	341	17	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	341	18	)	)	PUNCT
cana-2702	341	19	)	)	PUNCT
cana-2702	342	1	+	+	CCONJ
cana-2702	342	2	2	2	NUM
cana-2702	342	3	𝑚1(𝑡.	𝑚1(𝑡.	NOUN
cana-2702	342	4	2𝑚2	2𝑚2	NUM
cana-2702	342	5	−	−	NOUN
cana-2702	342	6	𝑡	𝑡	NOUN
cana-2702	342	7	−	−	NOUN
cana-2702	342	8	1	1	NUM
cana-2702	342	9	)	)	PUNCT
cana-2702	342	10	−	−	PROPN
cana-2702	342	11	1	1	NUM
cana-2702	342	12	≥	≥	NOUN
cana-2702	342	13	𝑓𝑡	𝑓𝑡	NOUN
cana-2702	342	14	(	(	PUNCT
cana-2702	342	15	𝐶𝑚1	𝐶𝑚1	X
cana-2702	342	16	(	(	PUNCT
cana-2702	342	17	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	342	18	)	)	PUNCT
cana-2702	342	19	)	)	PUNCT
cana-2702	342	20	,	,	PUNCT
cana-2702	342	21	since	since	SCONJ
cana-2702	342	22	𝑡	𝑡	PROPN
cana-2702	342	23	≥	≥	VERB
cana-2702	342	24	2	2	NUM
cana-2702	342	25	and	and	CCONJ
cana-2702	342	26	𝑚	𝑚	X
cana-2702	342	27	≥	≥	NUM
cana-2702	342	28	4	4	NUM
cana-2702	342	29	.	.	PUNCT
cana-2702	342	30	by	by	ADP
cana-2702	342	31	above	above	ADP
cana-2702	342	32	claim	claim	NOUN
cana-2702	342	33	(	(	PUNCT
cana-2702	342	34	1	1	NUM
cana-2702	342	35	)	)	PUNCT
cana-2702	342	36	,	,	PUNCT
cana-2702	342	37	𝑝(𝑋1	𝑝(𝑋1	PROPN
cana-2702	342	38	)	)	PUNCT
cana-2702	342	39	≥	≥	NOUN
cana-2702	342	40	𝑓𝑡	𝑓𝑡	PROPN
cana-2702	342	41	(	(	PUNCT
cana-2702	342	42	𝐶𝑚1	𝐶𝑚1	X
cana-2702	342	43	(	(	PUNCT
cana-2702	342	44	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	342	45	)	)	PUNCT
cana-2702	342	46	)	)	PUNCT
cana-2702	342	47	,	,	PUNCT
cana-2702	342	48	and	and	CCONJ
cana-2702	342	49	by	by	ADP
cana-2702	342	50	induction	induction	NOUN
cana-2702	342	51	on	on	ADP
cana-2702	342	52	𝑚	𝑚	X
cana-2702	342	53	we	we	PRON
cana-2702	342	54	can	can	AUX
cana-2702	342	55	move	move	VERB
cana-2702	342	56	𝑡	𝑡	PROPN
cana-2702	342	57	pebbles	pebble	NOUN
cana-2702	342	58	to	to	PART
cana-2702	342	59	𝑣.	𝑣.	VERB
cana-2702	342	60	in	in	ADP
cana-2702	342	61	the	the	DET
cana-2702	342	62	next	next	ADJ
cana-2702	342	63	sections	section	NOUN
cana-2702	342	64	,	,	PUNCT
cana-2702	342	65	we	we	PRON
cana-2702	342	66	show	show	VERB
cana-2702	342	67	that	that	SCONJ
cana-2702	342	68	the	the	DET
cana-2702	342	69	graph	graph	NOUN
cana-2702	342	70	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	342	71	)	)	PUNCT
cana-2702	342	72	has	have	VERB
cana-2702	342	73	both	both	CCONJ
cana-2702	342	74	the	the	DET
cana-2702	342	75	2	2	NUM
cana-2702	342	76	-	-	PUNCT
cana-2702	342	77	pebbling	pebble	VERB
cana-2702	342	78	and	and	CCONJ
cana-2702	342	79	2𝑡	2𝑡	NOUN
cana-2702	342	80	−pebbling	−pebble	VERB
cana-2702	342	81	properties	property	NOUN
cana-2702	342	82	.	.	PUNCT
cana-2702	343	1	the	the	DET
cana-2702	343	2	2	2	NUM
cana-2702	343	3	-	-	PUNCT
cana-2702	343	4	pebbling	pebble	VERB
cana-2702	343	5	property	property	NOUN
cana-2702	343	6	of	of	ADP
cana-2702	343	7	crisscross	crisscross	ADJ
cana-2702	343	8	sequence	sequence	NOUN
cana-2702	343	9	of	of	ADP
cana-2702	343	10	𝒎	𝒎	PROPN
cana-2702	343	11	−	−	PROPN
cana-2702	343	12	complete	complete	ADJ
cana-2702	343	13	graphs	graph	NOUN
cana-2702	343	14	theorem	theorem	VERB
cana-2702	343	15	7	7	NUM
cana-2702	343	16	.	.	PUNCT
cana-2702	344	1	the	the	DET
cana-2702	344	2	graph	graph	NOUN
cana-2702	344	3	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	PROPN
cana-2702	344	4	)	)	PUNCT
cana-2702	344	5	exhibits	exhibit	VERB
cana-2702	344	6	the	the	DET
cana-2702	344	7	two	two	NUM
cana-2702	344	8	-	-	PUNCT
cana-2702	344	9	pebbling	pebble	VERB
cana-2702	344	10	property	property	NOUN
cana-2702	344	11	.	.	PUNCT
cana-2702	345	1	proof	proof	NOUN
cana-2702	345	2	.	.	PUNCT
cana-2702	346	1	consider	consider	VERB
cana-2702	346	2	a	a	DET
cana-2702	346	3	graph	graph	NOUN
cana-2702	346	4	with	with	ADP
cana-2702	346	5	at	at	ADV
cana-2702	346	6	least	least	ADJ
cana-2702	346	7	2(2𝑛	2(2𝑛	NUM
cana-2702	346	8	−	−	NOUN
cana-2702	346	9	2	2	NUM
cana-2702	346	10	)	)	PUNCT
cana-2702	346	11	−	−	ADP
cana-2702	347	1	𝑞	𝑞	PROPN
cana-2702	347	2	+	+	NOUN
cana-2702	347	3	1	1	NUM
cana-2702	347	4	pebbles	pebble	NOUN
cana-2702	347	5	scattered	scatter	VERB
cana-2702	347	6	at	at	ADP
cana-2702	347	7	the	the	DET
cana-2702	347	8	vertices	vertex	NOUN
cana-2702	347	9	of	of	ADP
cana-2702	347	10	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	347	11	)	)	PUNCT
cana-2702	347	12	.	.	PUNCT
cana-2702	348	1	a	a	DET
cana-2702	348	2	minimum	minimum	NOUN
cana-2702	348	3	of	of	ADP
cana-2702	348	4	two	two	NUM
cana-2702	348	5	pebbles	pebble	NOUN
cana-2702	348	6	must	must	AUX
cana-2702	348	7	be	be	AUX
cana-2702	348	8	pushed	push	VERB
cana-2702	348	9	to	to	ADP
cana-2702	348	10	the	the	DET
cana-2702	348	11	target	target	NOUN
cana-2702	348	12	vertex	vertex	NOUN
cana-2702	348	13	.	.	PUNCT
cana-2702	349	1	we	we	PRON
cana-2702	349	2	assume	assume	VERB
cana-2702	349	3	that	that	SCONJ
cana-2702	349	4	𝑣	𝑣	PRON
cana-2702	349	5	∈	∈	NOUN
cana-2702	349	6	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	349	7	)	)	PUNCT
cana-2702	349	8	without	without	ADP
cana-2702	349	9	losing	lose	VERB
cana-2702	349	10	generality	generality	NOUN
cana-2702	349	11	.	.	PUNCT
cana-2702	350	1	we	we	PRON
cana-2702	350	2	look	look	VERB
cana-2702	350	3	at	at	ADP
cana-2702	350	4	the	the	DET
cana-2702	350	5	following	following	ADJ
cana-2702	350	6	scenarios	scenario	NOUN
cana-2702	350	7	:	:	PUNCT
cana-2702	350	8	case	case	NOUN
cana-2702	350	9	(	(	PUNCT
cana-2702	350	10	1	1	NUM
cana-2702	350	11	)	)	PUNCT
cana-2702	350	12	𝑣	𝑣	NOUN
cana-2702	350	13	=	=	PUNCT
cana-2702	350	14	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	350	15	or	or	CCONJ
cana-2702	350	16	𝑣	𝑣	PRON
cana-2702	350	17	=	=	PROPN
cana-2702	350	18	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	350	19	.	.	PUNCT
cana-2702	351	1	we	we	PRON
cana-2702	351	2	assume	assume	VERB
cana-2702	351	3	that	that	SCONJ
cana-2702	351	4	𝑣	𝑣	X
cana-2702	351	5	=	=	SYM
cana-2702	351	6	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	351	7	without	without	ADP
cana-2702	351	8	losing	lose	VERB
cana-2702	351	9	generality	generality	NOUN
cana-2702	351	10	.	.	PUNCT
cana-2702	352	1	because	because	SCONJ
cana-2702	352	2	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	352	3	∈	∈	PROPN
cana-2702	352	4	𝑉(𝐶1	𝑉(𝐶1	ADJ
cana-2702	352	5	)	)	PUNCT
cana-2702	352	6	∩	∩	NOUN
cana-2702	352	7	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	352	8	)	)	PUNCT
cana-2702	352	9	.	.	PUNCT
cana-2702	353	1	we	we	PRON
cana-2702	353	2	have	have	VERB
cana-2702	353	3	2𝑓(𝐶2(𝐾𝑛	2𝑓(𝐶2(𝐾𝑛	NUM
cana-2702	353	4	)	)	PUNCT
cana-2702	353	5	)	)	PUNCT
cana-2702	354	1	−	−	PROPN
cana-2702	355	1	+1	+1	PROPN
cana-2702	355	2	≥	≥	NOUN
cana-2702	355	3	4𝑛	4𝑛	NOUN
cana-2702	355	4	−	−	NOUN
cana-2702	355	5	4	4	NUM
cana-2702	355	6	−	−	PROPN
cana-2702	355	7	(	(	PUNCT
cana-2702	355	8	2𝑛	2𝑛	PROPN
cana-2702	355	9	−	−	PROPN
cana-2702	355	10	2	2	NUM
cana-2702	355	11	)	)	PUNCT
cana-2702	355	12	+	+	CCONJ
cana-2702	355	13	1	1	NUM
cana-2702	355	14	=	=	SYM
cana-2702	355	15	2𝑛	2𝑛	NOUN
cana-2702	355	16	−	−	PROPN
cana-2702	355	17	1	1	NUM
cana-2702	355	18	≥	≥	NOUN
cana-2702	355	19	|𝑉(𝐺)|	|𝑉(𝐺)|	NOUN
cana-2702	355	20	.	.	PUNCT
cana-2702	356	1	according	accord	VERB
cana-2702	356	2	to	to	ADP
cana-2702	356	3	the	the	DET
cana-2702	356	4	pigeonhole	pigeonhole	NOUN
cana-2702	356	5	principle	principle	NOUN
cana-2702	356	6	,	,	PUNCT
cana-2702	356	7	one	one	NUM
cana-2702	356	8	of	of	ADP
cana-2702	356	9	the	the	DET
cana-2702	356	10	vertices	vertex	NOUN
cana-2702	356	11	in	in	ADP
cana-2702	356	12	𝑉(𝐶2(𝐾𝑛	𝑉(𝐶2(𝐾𝑛	NOUN
cana-2702	356	13	)	)	PUNCT
cana-2702	356	14	)	)	PUNCT
cana-2702	356	15	,	,	PUNCT
cana-2702	356	16	say	say	VERB
cana-2702	356	17	𝑥	𝑥	NOUN
cana-2702	356	18	,	,	PUNCT
cana-2702	356	19	must	must	AUX
cana-2702	356	20	have	have	VERB
cana-2702	356	21	at	at	ADV
cana-2702	356	22	least	least	ADV
cana-2702	356	23	two	two	NUM
cana-2702	356	24	pebbles	pebble	NOUN
cana-2702	356	25	.	.	PUNCT
cana-2702	357	1	thus	thus	ADV
cana-2702	357	2	,	,	PUNCT
cana-2702	357	3	we	we	PRON
cana-2702	357	4	can	can	AUX
cana-2702	357	5	move	move	VERB
cana-2702	357	6	one	one	NUM
cana-2702	357	7	pebble	pebble	NOUN
cana-2702	357	8	to	to	ADP
cana-2702	357	9	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	357	10	using	use	VERB
cana-2702	357	11	precisely	precisely	ADV
cana-2702	357	12	two	two	NUM
cana-2702	357	13	pebbles	pebble	NOUN
cana-2702	357	14	.	.	PUNCT
cana-2702	358	1	this	this	PRON
cana-2702	358	2	is	be	AUX
cana-2702	358	3	due	due	ADJ
cana-2702	358	4	to	to	ADP
cana-2702	358	5	the	the	DET
cana-2702	358	6	fact	fact	NOUN
cana-2702	358	7	that	that	SCONJ
cana-2702	358	8	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	358	9	is	be	AUX
cana-2702	358	10	adjacent	adjacent	ADJ
cana-2702	358	11	to	to	ADP
cana-2702	358	12	all	all	PRON
cana-2702	358	13	of	of	ADP
cana-2702	358	14	the	the	DET
cana-2702	358	15	graph	graph	NOUN
cana-2702	358	16	’s	’s	PART
cana-2702	358	17	vertices	vertex	NOUN
cana-2702	358	18	.	.	PUNCT
cana-2702	359	1	this	this	PRON
cana-2702	359	2	means	mean	VERB
cana-2702	359	3	that	that	SCONJ
cana-2702	359	4	as	as	ADP
cana-2702	359	5	𝑞	𝑞	PROPN
cana-2702	359	6	≤	≤	PROPN
cana-2702	359	7	2𝑛	2𝑛	PROPN
cana-2702	359	8	−	−	PROPN
cana-2702	359	9	3	3	NUM
cana-2702	359	10	,	,	PUNCT
cana-2702	359	11	at	at	ADP
cana-2702	359	12	least	least	ADJ
cana-2702	359	13	2(2𝑛	2(2𝑛	NUM
cana-2702	359	14	−	−	NOUN
cana-2702	359	15	2	2	NUM
cana-2702	359	16	)	)	PUNCT
cana-2702	359	17	−	−	ADP
cana-2702	359	18	𝑞	𝑞	X
cana-2702	359	19	+	+	NOUN
cana-2702	359	20	1	1	NUM
cana-2702	359	21	−	−	NUM
cana-2702	359	22	2	2	NUM
cana-2702	359	23	pebbles	pebble	NOUN
cana-2702	359	24	are	be	AUX
cana-2702	359	25	maintained	maintain	VERB
cana-2702	359	26	in	in	ADP
cana-2702	359	27	𝑉(𝐶2(𝐾𝑛	𝑉(𝐶2(𝐾𝑛	NOUN
cana-2702	359	28	)	)	PUNCT
cana-2702	359	29	)	)	PUNCT
cana-2702	359	30	.	.	PUNCT
cana-2702	360	1	as	as	ADP
cana-2702	360	2	a	a	DET
cana-2702	360	3	result	result	NOUN
cana-2702	360	4	,	,	PUNCT
cana-2702	360	5	2(2𝑛	2(2𝑛	NUM
cana-2702	360	6	−	−	NOUN
cana-2702	360	7	2	2	NUM
cana-2702	360	8	)	)	PUNCT
cana-2702	360	9	−	−	NOUN
cana-2702	360	10	𝑞	𝑞	NOUN
cana-2702	360	11	−	−	PROPN
cana-2702	360	12	1	1	NUM
cana-2702	360	13	≥	≥	NOUN
cana-2702	360	14	2𝑛	2𝑛	NOUN
cana-2702	360	15	−	−	PROPN
cana-2702	360	16	2	2	NUM
cana-2702	360	17	=	=	SYM
cana-2702	360	18	𝑓(𝐶2(𝐾𝑛	𝑓(𝐶2(𝐾𝑛	NUM
cana-2702	360	19	)	)	PUNCT
cana-2702	360	20	)	)	PUNCT
cana-2702	360	21	.	.	PUNCT
cana-2702	361	1	we	we	PRON
cana-2702	361	2	can	can	AUX
cana-2702	361	3	transfer	transfer	VERB
cana-2702	361	4	n	n	PRON
cana-2702	361	5	more	more	ADJ
cana-2702	361	6	pebbles	pebble	NOUN
cana-2702	361	7	to	to	ADP
cana-2702	361	8	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	361	9	using	use	VERB
cana-2702	361	10	theorem	theorem	NOUN
cana-2702	361	11	1	1	NUM
cana-2702	361	12	.	.	PUNCT
cana-2702	361	13	case	case	NOUN
cana-2702	361	14	(	(	PUNCT
cana-2702	361	15	2	2	NUM
cana-2702	361	16	)	)	PUNCT
cana-2702	361	17	𝑣	𝑣	PRON
cana-2702	361	18	∈	∈	PROPN
cana-2702	361	19	𝑉(𝐶1	𝑉(𝐶1	PROPN
cana-2702	361	20	)	)	PUNCT
cana-2702	361	21	−	−	PROPN
cana-2702	361	22	{	{	PUNCT
cana-2702	361	23	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	361	24	}	}	PUNCT
cana-2702	361	25	or	or	CCONJ
cana-2702	361	26	𝑣	𝑣	PRON
cana-2702	361	27	∈	∈	NOUN
cana-2702	361	28	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	361	29	)	)	PUNCT
cana-2702	361	30	−	−	PROPN
cana-2702	361	31	{	{	PUNCT
cana-2702	361	32	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	361	33	}	}	PUNCT
cana-2702	361	34	.	.	PUNCT
cana-2702	362	1	we	we	PRON
cana-2702	362	2	assume	assume	VERB
cana-2702	362	3	that	that	SCONJ
cana-2702	362	4	𝑣	𝑣	PRON
cana-2702	362	5	∈	∈	NOUN
cana-2702	362	6	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	362	7	)	)	PUNCT
cana-2702	362	8	−	−	PROPN
cana-2702	362	9	{	{	PUNCT
cana-2702	362	10	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	362	11	}	}	PUNCT
cana-2702	362	12	and	and	CCONJ
cana-2702	362	13	𝑣	𝑣	ADP
cana-2702	362	14	=	=	PUNCT
cana-2702	362	15	𝑦	𝑦	PROPN
cana-2702	362	16	and	and	CCONJ
cana-2702	362	17	𝑣	𝑣	X
cana-2702	362	18	=	=	PUNCT
cana-2702	362	19	𝑦	𝑦	PROPN
cana-2702	362	20	without	without	ADP
cana-2702	362	21	losing	lose	VERB
cana-2702	362	22	generality	generality	NOUN
cana-2702	362	23	.	.	PUNCT
cana-2702	363	1	by	by	ADP
cana-2702	363	2	the	the	DET
cana-2702	363	3	pigeonhole	pigeonhole	NOUN
cana-2702	363	4	principle	principle	NOUN
cana-2702	363	5	,	,	PUNCT
cana-2702	363	6	if	if	SCONJ
cana-2702	363	7	𝑝((𝐶2	𝑝((𝐶2	PROPN
cana-2702	363	8	)	)	PUNCT
cana-2702	363	9	−	−	PROPN
cana-2702	363	10	{	{	PUNCT
cana-2702	363	11	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	363	12	}	}	PUNCT
cana-2702	363	13	)	)	PUNCT
cana-2702	363	14	≥	≥	NOUN
cana-2702	363	15	𝑛	𝑛	DET
cana-2702	363	16	−	−	PROPN
cana-2702	363	17	1	1	NUM
cana-2702	363	18	,	,	PUNCT
cana-2702	363	19	at	at	ADV
cana-2702	363	20	least	least	ADV
cana-2702	363	21	one	one	NUM
cana-2702	363	22	vertex	vertex	NOUN
cana-2702	363	23	,	,	PUNCT
cana-2702	363	24	say	say	VERB
cana-2702	363	25	𝑎𝑘	𝑎𝑘	ADP
cana-2702	363	26	+	+	NOUN
cana-2702	363	27	1	1	NUM
cana-2702	363	28	,	,	PUNCT
cana-2702	363	29	contains	contain	VERB
cana-2702	363	30	two	two	NUM
cana-2702	363	31	pebbles	pebble	NOUN
cana-2702	363	32	,	,	PUNCT
cana-2702	363	33	and	and	CCONJ
cana-2702	363	34	then	then	ADV
cana-2702	363	35	pushes	push	VERB
cana-2702	363	36	a	a	DET
cana-2702	363	37	pebble	pebble	NOUN
cana-2702	363	38	to	to	ADP
cana-2702	363	39	𝑦	𝑦	NOUN
cana-2702	363	40	from	from	ADP
cana-2702	363	41	𝑎𝑘+1	𝑎𝑘+1	NOUN
cana-2702	363	42	using	use	VERB
cana-2702	363	43	precisely	precisely	ADV
cana-2702	363	44	two	two	NUM
cana-2702	363	45	pebbles	pebble	NOUN
cana-2702	363	46	.	.	PUNCT
cana-2702	364	1	as	as	ADP
cana-2702	364	2	a	a	DET
cana-2702	364	3	result	result	NOUN
cana-2702	364	4	,	,	PUNCT
cana-2702	364	5	we	we	PRON
cana-2702	364	6	suppose	suppose	VERB
cana-2702	364	7	that	that	SCONJ
cana-2702	364	8	𝑝((𝑉2	𝑝((𝑉2	PROPN
cana-2702	364	9	)	)	PUNCT
cana-2702	364	10	−	−	PROPN
cana-2702	364	11	{	{	PUNCT
cana-2702	364	12	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	364	13	}	}	PUNCT
cana-2702	364	14	)	)	PUNCT
cana-2702	364	15	≤	≤	NUM
cana-2702	365	1	𝑛	𝑛	DET
cana-2702	365	2	−	−	PROPN
cana-2702	365	3	2	2	NUM
cana-2702	365	4	.	.	PUNCT
cana-2702	366	1	the	the	DET
cana-2702	366	2	following	follow	VERB
cana-2702	366	3	subcases	subcase	NOUN
cana-2702	366	4	must	must	AUX
cana-2702	366	5	be	be	AUX
cana-2702	366	6	present	present	ADJ
cana-2702	366	7	:	:	PUNCT
cana-2702	366	8	subcase	subcase	PROPN
cana-2702	366	9	(	(	PUNCT
cana-2702	366	10	1a	1a	X
cana-2702	366	11	)	)	PUNCT
cana-2702	366	12	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	366	13	)	)	PUNCT
cana-2702	366	14	−	−	PROPN
cana-2702	366	15	{	{	PUNCT
cana-2702	366	16	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	366	17	}	}	PUNCT
cana-2702	366	18	)	)	PUNCT
cana-2702	366	19	=	=	SYM
cana-2702	367	1	𝑛	𝑛	DET
cana-2702	367	2	−	−	NOUN
cana-2702	367	3	2	2	NUM
cana-2702	367	4	.	.	PUNCT
cana-2702	367	5	as	as	ADP
cana-2702	367	6	by	by	ADP
cana-2702	367	7	our	our	PRON
cana-2702	367	8	assumption	assumption	NOUN
cana-2702	367	9	𝑝(𝑉(𝐶1	𝑝(𝑉(𝐶1	PROPN
cana-2702	367	10	)	)	PUNCT
cana-2702	367	11	−	−	PROPN
cana-2702	367	12	{	{	PUNCT
cana-2702	367	13	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	367	14	}	}	PUNCT
cana-2702	367	15	)	)	PUNCT
cana-2702	367	16	≥	≥	NOUN
cana-2702	368	1	2(2𝑛	2(2𝑛	NUM
cana-2702	368	2	−	−	NOUN
cana-2702	368	3	2	2	NUM
cana-2702	368	4	)	)	PUNCT
cana-2702	368	5	−	−	ADP
cana-2702	368	6	𝑞	𝑞	X
cana-2702	368	7	+	+	NOUN
cana-2702	368	8	1	1	NUM
cana-2702	368	9	−	−	NOUN
cana-2702	368	10	𝑛	𝑛	DET
cana-2702	368	11	+	+	NOUN
cana-2702	368	12	2	2	NUM
cana-2702	368	13	=	=	SYM
cana-2702	368	14	2(2𝑛	2(2𝑛	NUM
cana-2702	368	15	−	−	NOUN
cana-2702	368	16	2	2	NUM
cana-2702	368	17	)	)	PUNCT
cana-2702	368	18	−	−	PROPN
cana-2702	368	19	𝑞	𝑞	NOUN
cana-2702	368	20	−	−	PROPN
cana-2702	368	21	𝑛	𝑛	PROPN
cana-2702	368	22	+	+	NOUN
cana-2702	368	23	3	3	NUM
cana-2702	368	24	=	=	SYM
cana-2702	368	25	3𝑛	3𝑛	NUM
cana-2702	368	26	−	−	NOUN
cana-2702	368	27	𝑞	𝑞	X
cana-2702	368	28	−	−	PROPN
cana-2702	368	29	1	1	NUM
cana-2702	368	30	≥	≥	NOUN
cana-2702	368	31	3𝑛	3𝑛	NUM
cana-2702	368	32	−	−	PROPN
cana-2702	368	33	(	(	PUNCT
cana-2702	368	34	2𝑛	2𝑛	PROPN
cana-2702	368	35	−	−	PROPN
cana-2702	368	36	3	3	NUM
cana-2702	368	37	)	)	PUNCT
cana-2702	368	38	−	−	PROPN
cana-2702	368	39	1	1	NUM
cana-2702	368	40	=	=	SYM
cana-2702	368	41	𝑛	𝑛	PROPN
cana-2702	368	42	+	+	NOUN
cana-2702	368	43	2	2	X
cana-2702	368	44	.	.	X
cana-2702	369	1	the	the	DET
cana-2702	369	2	pigeonhole	pigeonhole	NOUN
cana-2702	369	3	principle	principle	NOUN
cana-2702	369	4	states	state	VERB
cana-2702	369	5	that	that	SCONJ
cana-2702	369	6	one	one	NUM
cana-2702	369	7	of	of	ADP
cana-2702	369	8	the	the	DET
cana-2702	369	9	vertices	vertex	NOUN
cana-2702	369	10	in	in	ADP
cana-2702	369	11	𝑉(𝐶1	𝑉(𝐶1	NOUN
cana-2702	369	12	)	)	PUNCT
cana-2702	369	13	−	−	PROPN
cana-2702	369	14	{	{	PUNCT
cana-2702	369	15	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	369	16	}	}	PUNCT
cana-2702	369	17	has	have	VERB
cana-2702	369	18	at	at	ADV
cana-2702	369	19	least	least	ADV
cana-2702	369	20	two	two	NUM
cana-2702	369	21	pebbles	pebble	NOUN
cana-2702	369	22	,	,	PUNCT
cana-2702	369	23	i.e.	i.e.	X
cana-2702	369	24	,	,	PUNCT
cana-2702	369	25	𝑝(𝑎1	𝑝(𝑎1	NOUN
cana-2702	369	26	)	)	PUNCT
cana-2702	369	27	=	=	SYM
cana-2702	370	1	2	2	X
cana-2702	370	2	.	.	X
cana-2702	370	3	as	as	ADP
cana-2702	370	4	a	a	DET
cana-2702	370	5	result	result	NOUN
cana-2702	370	6	,	,	PUNCT
cana-2702	370	7	we	we	PRON
cana-2702	370	8	can	can	AUX
cana-2702	370	9	transfer	transfer	VERB
cana-2702	370	10	one	one	NUM
cana-2702	370	11	pebble	pebble	NOUN
cana-2702	370	12	to	to	ADP
cana-2702	370	13	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	370	14	with	with	ADP
cana-2702	370	15	only	only	ADV
cana-2702	370	16	two	two	NUM
cana-2702	370	17	pebbles	pebble	NOUN
cana-2702	370	18	.	.	PUNCT
cana-2702	371	1	hence	hence	ADV
cana-2702	371	2	,	,	PUNCT
cana-2702	371	3	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	PROPN
cana-2702	371	4	)	)	PUNCT
cana-2702	371	5	−	−	PROPN
cana-2702	371	6	{	{	PUNCT
cana-2702	371	7	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	371	8	}	}	PUNCT
cana-2702	371	9	)	)	PUNCT
cana-2702	372	1	=	=	SYM
cana-2702	372	2	𝑛	𝑛	PRON
cana-2702	372	3	−	−	NUM
cana-2702	373	1	1	1	NUM
cana-2702	373	2	.	.	PUNCT
cana-2702	374	1	this	this	PRON
cana-2702	374	2	is	be	AUX
cana-2702	374	3	due	due	ADJ
cana-2702	374	4	to	to	ADP
cana-2702	374	5	the	the	DET
cana-2702	374	6	fact	fact	NOUN
cana-2702	374	7	that	that	SCONJ
cana-2702	374	8	|𝑉(𝐶2	|𝑉(𝐶2	ADJ
cana-2702	374	9	)	)	PUNCT
cana-2702	374	10	−	−	PROPN
cana-2702	374	11	{	{	PUNCT
cana-2702	374	12	𝑏𝑘−1}|	𝑏𝑘−1}|	PROPN
cana-2702	374	13	=	=	PROPN
cana-2702	374	14	𝑛	𝑛	PROPN
cana-2702	374	15	−	−	NUM
cana-2702	374	16	1	1	NUM
cana-2702	374	17	and	and	CCONJ
cana-2702	374	18	𝑝(𝑦	𝑝(𝑦	PROPN
cana-2702	374	19	)	)	PUNCT
cana-2702	374	20	=	=	NOUN
cana-2702	375	1	0	0	X
cana-2702	375	2	.	.	PUNCT
cana-2702	376	1	as	as	ADP
cana-2702	376	2	a	a	DET
cana-2702	376	3	result	result	NOUN
cana-2702	376	4	,	,	PUNCT
cana-2702	376	5	one	one	NUM
cana-2702	376	6	of	of	ADP
cana-2702	376	7	the	the	DET
cana-2702	376	8	vertices	vertex	NOUN
cana-2702	376	9	in	in	ADP
cana-2702	376	10	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	376	11	)	)	PUNCT
cana-2702	376	12	−	−	PROPN
cana-2702	376	13	{	{	PUNCT
cana-2702	376	14	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	376	15	}	}	PUNCT
cana-2702	376	16	has	have	VERB
cana-2702	376	17	two	two	NUM
cana-2702	376	18	pebbles	pebble	NOUN
cana-2702	376	19	.	.	PUNCT
cana-2702	377	1	then	then	ADV
cana-2702	377	2	,	,	PUNCT
cana-2702	377	3	with	with	ADP
cana-2702	377	4	precisely	precisely	ADV
cana-2702	377	5	two	two	NUM
cana-2702	377	6	pebbles	pebble	NOUN
cana-2702	377	7	,	,	PUNCT
cana-2702	377	8	we	we	PRON
cana-2702	377	9	can	can	AUX
cana-2702	377	10	move	move	VERB
cana-2702	377	11	one	one	NUM
cana-2702	377	12	pebble	pebble	NOUN
cana-2702	377	13	to	to	ADP
cana-2702	377	14	𝑦.	𝑦.	PROPN
cana-2702	377	15	let	let	VERB
cana-2702	377	16	𝑞1	𝑞1	PROPN
cana-2702	377	17	represent	represent	VERB
cana-2702	377	18	the	the	DET
cana-2702	377	19	number	number	NOUN
cana-2702	377	20	of	of	ADP
cana-2702	377	21	occupied	occupy	VERB
cana-2702	377	22	vertices	vertex	NOUN
cana-2702	377	23	in	in	ADP
cana-2702	377	24	𝑉(𝐶1	𝑉(𝐶1	NOUN
cana-2702	377	25	)	)	PUNCT
cana-2702	378	1	−	−	PROPN
cana-2702	378	2	{	{	PUNCT
cana-2702	378	3	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	378	4	}	}	PUNCT
cana-2702	378	5	and	and	CCONJ
cana-2702	378	6	𝑞2	𝑞2	NOUN
cana-2702	378	7	,	,	PUNCT
cana-2702	378	8	and	and	CCONJ
cana-2702	378	9	𝑞2	𝑞2	NOUN
cana-2702	378	10	represent	represent	VERB
cana-2702	378	11	the	the	DET
cana-2702	378	12	number	number	NOUN
cana-2702	378	13	of	of	ADP
cana-2702	378	14	occupied	occupy	VERB
cana-2702	378	15	vertices	vertex	NOUN
cana-2702	378	16	in	in	ADP
cana-2702	378	17	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	378	18	)	)	PUNCT
cana-2702	378	19	−	−	PROPN
cana-2702	378	20	{	{	PUNCT
cana-2702	378	21	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	378	22	}	}	PUNCT
cana-2702	378	23	.	.	PUNCT
cana-2702	379	1	communications	communication	NOUN
cana-2702	379	2	on	on	ADP
cana-2702	379	3	applied	apply	VERB
cana-2702	379	4	nonlinear	nonlinear	ADJ
cana-2702	379	5	analysis	analysis	NOUN
cana-2702	379	6	issn	issn	NOUN
cana-2702	379	7	:	:	PUNCT
cana-2702	379	8	1074	1074	NUM
cana-2702	379	9	-	-	PUNCT
cana-2702	379	10	133x	133x	NUM
cana-2702	379	11	vol	vol	NOUN
cana-2702	379	12	32	32	NUM
cana-2702	379	13	no	no	NOUN
cana-2702	379	14	.	.	PUNCT
cana-2702	380	1	3s	3s	NUM
cana-2702	380	2	(	(	PUNCT
cana-2702	380	3	2025	2025	NUM
cana-2702	380	4	)	)	PUNCT
cana-2702	380	5	656	656	NUM
cana-2702	380	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	380	7	claim	claim	NOUN
cana-2702	380	8	(	(	PUNCT
cana-2702	380	9	1	1	X
cana-2702	380	10	)	)	PUNCT
cana-2702	380	11	𝑝(𝐶2(𝐾𝑛	𝑝(𝐶2(𝐾𝑛	NOUN
cana-2702	380	12	)	)	PUNCT
cana-2702	380	13	)	)	PUNCT
cana-2702	380	14	−	−	PROPN
cana-2702	381	1	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	PROPN
cana-2702	381	2	)	)	PUNCT
cana-2702	381	3	−	−	PROPN
cana-2702	381	4	{	{	PUNCT
cana-2702	381	5	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	381	6	}	}	PUNCT
cana-2702	381	7	)	)	PUNCT
cana-2702	381	8	−	−	PROPN
cana-2702	381	9	𝑞2	𝑞2	NOUN
cana-2702	381	10	−	−	PROPN
cana-2702	381	11	2	2	NUM
cana-2702	381	12	≥	≥	NOUN
cana-2702	381	13	2𝑓(𝐾𝑛	2𝑓(𝐾𝑛	NUM
cana-2702	381	14	)	)	PUNCT
cana-2702	381	15	−	−	ADP
cana-2702	381	16	𝑞2	𝑞2	NOUN
cana-2702	381	17	+	+	X
cana-2702	381	18	1	1	X
cana-2702	381	19	.	.	X
cana-2702	382	1	we	we	PRON
cana-2702	382	2	have	have	VERB
cana-2702	382	3	𝑝(𝐶2(𝐾𝑛	𝑝(𝐶2(𝐾𝑛	NOUN
cana-2702	382	4	)	)	PUNCT
cana-2702	382	5	)	)	PUNCT
cana-2702	382	6	−	−	PROPN
cana-2702	383	1	[	[	X
cana-2702	383	2	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	383	3	)	)	PUNCT
cana-2702	383	4	−	−	PROPN
cana-2702	383	5	{	{	PUNCT
cana-2702	383	6	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	383	7	}	}	PUNCT
cana-2702	383	8	)	)	PUNCT
cana-2702	383	9	−	−	ADP
cana-2702	383	10	𝑞2	𝑞2	NOUN
cana-2702	383	11	]	]	PUNCT
cana-2702	383	12	−	−	PROPN
cana-2702	383	13	2	2	NUM
cana-2702	383	14	=	=	SYM
cana-2702	383	15	2(2𝑛	2(2𝑛	NUM
cana-2702	383	16	−	−	NOUN
cana-2702	383	17	2	2	NUM
cana-2702	383	18	)	)	PUNCT
cana-2702	383	19	−	−	ADP
cana-2702	383	20	𝑞	𝑞	X
cana-2702	383	21	+	+	NOUN
cana-2702	383	22	1	1	NUM
cana-2702	383	23	−	−	NOUN
cana-2702	383	24	(	(	PUNCT
cana-2702	383	25	𝑛	𝑛	PROPN
cana-2702	383	26	−	−	PROPN
cana-2702	383	27	2	2	NUM
cana-2702	383	28	)	)	PUNCT
cana-2702	383	29	+	+	NUM
cana-2702	383	30	𝑞2	𝑞2	NOUN
cana-2702	383	31	−	−	PROPN
cana-2702	383	32	2	2	NUM
cana-2702	383	33	=	=	SYM
cana-2702	383	34	4𝑛	4𝑛	NOUN
cana-2702	383	35	−	−	NOUN
cana-2702	383	36	4	4	NUM
cana-2702	383	37	−	−	NOUN
cana-2702	383	38	𝑞1	𝑞1	ADJ
cana-2702	383	39	−	−	PROPN
cana-2702	383	40	𝑛	𝑛	NOUN
cana-2702	383	41	+	+	NUM
cana-2702	383	42	𝑞2	𝑞2	NOUN
cana-2702	383	43	+	+	CCONJ
cana-2702	383	44	1	1	NUM
cana-2702	383	45	=	=	SYM
cana-2702	383	46	3𝑛	3𝑛	NUM
cana-2702	383	47	−	−	PROPN
cana-2702	383	48	𝑞3	𝑞3	NOUN
cana-2702	383	49	−	−	PROPN
cana-2702	383	50	3	3	NUM
cana-2702	383	51	=	=	SYM
cana-2702	383	52	2𝑛	2𝑛	PROPN
cana-2702	383	53	−	−	PROPN
cana-2702	384	1	𝑞1	𝑞1	PROPN
cana-2702	384	2	+	+	CCONJ
cana-2702	384	3	1	1	NUM
cana-2702	384	4	+	+	CCONJ
cana-2702	384	5	𝑛	𝑛	PRON
cana-2702	384	6	−	−	PROPN
cana-2702	384	7	4	4	NUM
cana-2702	384	8	≥	≥	NOUN
cana-2702	384	9	2𝑛	2𝑛	PROPN
cana-2702	384	10	−	−	PROPN
cana-2702	385	1	𝑞1	𝑞1	PROPN
cana-2702	385	2	+	+	CCONJ
cana-2702	385	3	1	1	NUM
cana-2702	385	4	,	,	PUNCT
cana-2702	385	5	since	since	SCONJ
cana-2702	385	6	𝑛	𝑛	PRON
cana-2702	385	7	≥	≥	NUM
cana-2702	385	8	5	5	NUM
cana-2702	385	9	=	=	SYM
cana-2702	385	10	2𝑓(𝐾𝑛	2𝑓(𝐾𝑛	NUM
cana-2702	385	11	)	)	PUNCT
cana-2702	385	12	−	−	PROPN
cana-2702	386	1	𝑞1	𝑞1	PROPN
cana-2702	386	2	+	+	CCONJ
cana-2702	386	3	1	1	X
cana-2702	386	4	.	.	X
cana-2702	386	5	claim	claim	NOUN
cana-2702	386	6	(	(	PUNCT
cana-2702	386	7	1	1	X
cana-2702	386	8	)	)	PUNCT
cana-2702	386	9	states	state	VERB
cana-2702	386	10	that	that	SCONJ
cana-2702	386	11	at	at	ADV
cana-2702	386	12	least	least	ADJ
cana-2702	386	13	2𝑓(𝐾𝑛	2𝑓(𝐾𝑛	NUM
cana-2702	386	14	)	)	PUNCT
cana-2702	386	15	−	−	PROPN
cana-2702	386	16	𝑞1	𝑞1	ADJ
cana-2702	386	17	+	+	CCONJ
cana-2702	386	18	1	1	NUM
cana-2702	386	19	pebbles	pebble	NOUN
cana-2702	386	20	are	be	AUX
cana-2702	386	21	scattered	scatter	VERB
cana-2702	386	22	in	in	ADP
cana-2702	386	23	𝐶1	𝐶1	PRON
cana-2702	386	24	.	.	PUNCT
cana-2702	387	1	this	this	PRON
cana-2702	387	2	is	be	AUX
cana-2702	387	3	due	due	ADJ
cana-2702	387	4	to	to	ADP
cana-2702	387	5	the	the	DET
cana-2702	387	6	fact	fact	NOUN
cana-2702	387	7	that	that	SCONJ
cana-2702	387	8	𝐶1	𝐶1	PROPN
cana-2702	387	9	≅	≅	NUM
cana-2702	387	10	𝐾	𝐾	PROPN
cana-2702	387	11	−	−	PROPN
cana-2702	387	12	𝑛	𝑛	PROPN
cana-2702	387	13	and	and	CCONJ
cana-2702	387	14	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	387	15	satisfy	satisfy	VERB
cana-2702	387	16	the	the	DET
cana-2702	387	17	2	2	NUM
cana-2702	387	18	-	-	PUNCT
cana-2702	387	19	pebbling	pebble	VERB
cana-2702	387	20	property	property	NOUN
cana-2702	387	21	.	.	PUNCT
cana-2702	388	1	because	because	SCONJ
cana-2702	388	2	𝑎𝑘	𝑎𝑘	PRON
cana-2702	388	3	is	be	AUX
cana-2702	388	4	next	next	ADJ
cana-2702	388	5	to	to	ADP
cana-2702	388	6	𝑦	𝑦	NUM
cana-2702	388	7	,	,	PUNCT
cana-2702	388	8	we	we	PRON
cana-2702	388	9	can	can	AUX
cana-2702	388	10	transfer	transfer	VERB
cana-2702	388	11	two	two	NUM
cana-2702	388	12	pebbles	pebble	NOUN
cana-2702	388	13	to	to	ADP
cana-2702	388	14	𝑎𝑘	𝑎𝑘	NOUN
cana-2702	388	15	and	and	CCONJ
cana-2702	388	16	one	one	NUM
cana-2702	388	17	more	more	ADV
cana-2702	388	18	pebble	pebble	ADJ
cana-2702	388	19	to	to	ADP
cana-2702	388	20	𝑦	𝑦	NOUN
cana-2702	388	21	using	use	VERB
cana-2702	388	22	the	the	DET
cana-2702	388	23	pebbling	pebble	VERB
cana-2702	388	24	number	number	NOUN
cana-2702	388	25	of	of	ADP
cana-2702	388	26	complete	complete	ADJ
cana-2702	388	27	graph	graph	NOUN
cana-2702	388	28	.	.	PUNCT
cana-2702	389	1	subcase	subcase	PROPN
cana-2702	389	2	(	(	PUNCT
cana-2702	389	3	1b	1b	NUM
cana-2702	389	4	)	)	PUNCT
cana-2702	389	5	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	PROPN
cana-2702	389	6	)	)	PUNCT
cana-2702	389	7	−	−	PROPN
cana-2702	389	8	{	{	PUNCT
cana-2702	389	9	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	389	10	}	}	PUNCT
cana-2702	389	11	)	)	PUNCT
cana-2702	389	12	<	<	X
cana-2702	389	13	𝑛	𝑛	PRON
cana-2702	389	14	−	−	NUM
cana-2702	389	15	2	2	X
cana-2702	389	16	.	.	PUNCT
cana-2702	389	17	assume	assume	VERB
cana-2702	389	18	that	that	SCONJ
cana-2702	389	19	at	at	ADP
cana-2702	389	20	most	most	ADJ
cana-2702	389	21	𝑛	𝑛	DET
cana-2702	389	22	−	−	NUM
cana-2702	389	23	3	3	NUM
cana-2702	389	24	pebbles	pebble	NOUN
cana-2702	389	25	are	be	AUX
cana-2702	389	26	spread	spread	VERB
cana-2702	389	27	on	on	ADP
cana-2702	389	28	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	389	29	)	)	PUNCT
cana-2702	389	30	−	−	PROPN
cana-2702	389	31	{	{	PUNCT
cana-2702	389	32	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	389	33	}	}	PUNCT
cana-2702	389	34	without	without	ADP
cana-2702	389	35	losing	lose	VERB
cana-2702	389	36	generality	generality	NOUN
cana-2702	389	37	.	.	PUNCT
cana-2702	390	1	claim	claim	NOUN
cana-2702	390	2	(	(	PUNCT
cana-2702	390	3	2	2	X
cana-2702	390	4	)	)	PUNCT
cana-2702	390	5	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	390	6	)	)	PUNCT
cana-2702	390	7	≥	≥	NOUN
cana-2702	390	8	2𝑓2(𝐾𝑛	2𝑓2(𝐾𝑛	NUM
cana-2702	390	9	)	)	PUNCT
cana-2702	390	10	−	−	PROPN
cana-2702	391	1	𝑞1	𝑞1	X
cana-2702	391	2	+	+	CCONJ
cana-2702	391	3	1.since	1.since	NUM
cana-2702	391	4	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	391	5	)	)	PUNCT
cana-2702	391	6	−	−	NOUN
cana-2702	391	7	𝑞2	𝑞2	NOUN
cana-2702	391	8	=	=	SYM
cana-2702	391	9	0	0	NUM
cana-2702	391	10	,	,	PUNCT
cana-2702	391	11	then	then	ADV
cana-2702	391	12	we	we	PRON
cana-2702	391	13	have	have	VERB
cana-2702	391	14	,	,	PUNCT
cana-2702	391	15	2(2𝑛	2(2𝑛	NUM
cana-2702	391	16	−	−	NOUN
cana-2702	391	17	2	2	NUM
cana-2702	391	18	)	)	PUNCT
cana-2702	391	19	−	−	PROPN
cana-2702	391	20	𝑞1	𝑞1	PROPN
cana-2702	391	21	−	−	PROPN
cana-2702	391	22	𝑞2	𝑞2	NOUN
cana-2702	391	23	+	+	CCONJ
cana-2702	391	24	1	1	NUM
cana-2702	391	25	=	=	SYM
cana-2702	391	26	4𝑛	4𝑛	NOUN
cana-2702	392	1	−	−	NOUN
cana-2702	393	1	4	4	NUM
cana-2702	393	2	−	−	NOUN
cana-2702	393	3	𝑞1	𝑞1	NOUN
cana-2702	393	4	+	+	CCONJ
cana-2702	393	5	1	1	NUM
cana-2702	393	6	−	−	NOUN
cana-2702	393	7	(	(	PUNCT
cana-2702	393	8	𝑛	𝑛	PROPN
cana-2702	393	9	−	−	NOUN
cana-2702	393	10	3	3	NUM
cana-2702	393	11	)	)	PUNCT
cana-2702	393	12	3𝑛	3𝑛	NUM
cana-2702	393	13	−	−	PROPN
cana-2702	393	14	𝑞1	𝑞1	PROPN
cana-2702	393	15	≥	≥	PROPN
cana-2702	393	16	2𝑛	2𝑛	PROPN
cana-2702	393	17	−	−	PROPN
cana-2702	394	1	𝑞1	𝑞1	PROPN
cana-2702	394	2	+	+	CCONJ
cana-2702	394	3	5,since	5,since	PROPN
cana-2702	394	4	𝑛	𝑛	PRON
cana-2702	394	5	≥	≥	NOUN
cana-2702	394	6	5,2(𝑛	5,2(𝑛	NUM
cana-2702	394	7	+	+	CCONJ
cana-2702	394	8	2	2	NUM
cana-2702	394	9	)	)	PUNCT
cana-2702	394	10	−	−	PROPN
cana-2702	395	1	𝑞1	𝑞1	ADJ
cana-2702	395	2	+	+	CCONJ
cana-2702	395	3	1	1	NUM
cana-2702	395	4	=	=	SYM
cana-2702	395	5	2𝑓2(𝐾𝑛	2𝑓2(𝐾𝑛	NUM
cana-2702	395	6	)	)	PUNCT
cana-2702	395	7	−	−	PROPN
cana-2702	396	1	𝑞1	𝑞1	PROPN
cana-2702	396	2	+	+	CCONJ
cana-2702	396	3	1	1	X
cana-2702	396	4	.	.	PUNCT
cana-2702	397	1	because	because	SCONJ
cana-2702	397	2	𝐶1	𝐶1	PROPN
cana-2702	397	3	≅	≅	PROPN
cana-2702	397	4	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	397	5	and	and	CCONJ
cana-2702	397	6	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	397	7	)	)	PUNCT
cana-2702	397	8	≥	≥	NOUN
cana-2702	397	9	2𝑓2(𝐾𝑛	2𝑓2(𝐾𝑛	NUM
cana-2702	397	10	)	)	PUNCT
cana-2702	397	11	−	−	PROPN
cana-2702	397	12	𝑞1	𝑞1	PROPN
cana-2702	397	13	+	+	CCONJ
cana-2702	397	14	1	1	X
cana-2702	397	15	.	.	PUNCT
cana-2702	398	1	then	then	ADV
cana-2702	398	2	we	we	PRON
cana-2702	398	3	may	may	AUX
cana-2702	398	4	shift	shift	VERB
cana-2702	398	5	four	four	NUM
cana-2702	398	6	pebbles	pebble	NOUN
cana-2702	398	7	to	to	ADP
cana-2702	398	8	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	398	9	and	and	CCONJ
cana-2702	398	10	two	two	NUM
cana-2702	398	11	pebbles	pebble	NOUN
cana-2702	398	12	to	to	ADP
cana-2702	398	13	𝑦.	𝑦.	PROPN
cana-2702	398	14	theorem	theorem	PROPN
cana-2702	398	15	8	8	NUM
cana-2702	398	16	.	.	PUNCT
cana-2702	399	1	the	the	DET
cana-2702	399	2	graph	graph	NOUN
cana-2702	399	3	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	399	4	)	)	PUNCT
cana-2702	399	5	exhibits	exhibit	VERB
cana-2702	399	6	the	the	DET
cana-2702	399	7	two	two	NUM
cana-2702	399	8	-	-	PUNCT
cana-2702	399	9	pebbling	pebble	VERB
cana-2702	399	10	property[9][10	property[9][10	PROPN
cana-2702	399	11	]	]	PUNCT
cana-2702	399	12	..	..	PUNCT
cana-2702	399	13	proof	proof	NOUN
cana-2702	399	14	.	.	PUNCT
cana-2702	400	1	consider	consider	VERB
cana-2702	400	2	a	a	DET
cana-2702	400	3	graph	graph	NOUN
cana-2702	400	4	containing	contain	VERB
cana-2702	400	5	at	at	ADV
cana-2702	400	6	least	least	ADJ
cana-2702	400	7	2(3𝑛	2(3𝑛	NUM
cana-2702	400	8	−	−	NUM
cana-2702	400	9	2	2	NUM
cana-2702	400	10	)	)	PUNCT
cana-2702	400	11	−	−	ADP
cana-2702	401	1	𝑞	𝑞	X
cana-2702	401	2	+	+	NOUN
cana-2702	401	3	1	1	NUM
cana-2702	401	4	pebbles	pebble	NOUN
cana-2702	401	5	at	at	ADP
cana-2702	401	6	each	each	DET
cana-2702	401	7	vertex	vertex	NOUN
cana-2702	401	8	.	.	PUNCT
cana-2702	402	1	the	the	DET
cana-2702	402	2	two	two	NUM
cana-2702	402	3	pebbles	pebble	NOUN
cana-2702	402	4	must	must	AUX
cana-2702	402	5	be	be	AUX
cana-2702	402	6	moved	move	VERB
cana-2702	402	7	to	to	ADP
cana-2702	402	8	any	any	DET
cana-2702	402	9	vertex	vertex	NOUN
cana-2702	402	10	of	of	ADP
cana-2702	402	11	the	the	DET
cana-2702	402	12	target	target	NOUN
cana-2702	402	13	.	.	PUNCT
cana-2702	403	1	let	let	VERB
cana-2702	403	2	𝑣	𝑣	PRON
cana-2702	403	3	∈	∈	PROPN
cana-2702	403	4	𝐶𝑖	𝐶𝑖	PROPN
cana-2702	403	5	be	be	AUX
cana-2702	403	6	the	the	DET
cana-2702	403	7	target	target	NOUN
cana-2702	403	8	vertex	vertex	NOUN
cana-2702	403	9	for	for	ADP
cana-2702	403	10	𝑖	𝑖	NOUN
cana-2702	403	11	=	=	NOUN
cana-2702	403	12	1,2,3	1,2,3	X
cana-2702	403	13	.	.	PUNCT
cana-2702	404	1	we	we	PRON
cana-2702	404	2	look	look	VERB
cana-2702	404	3	at	at	ADP
cana-2702	404	4	the	the	DET
cana-2702	404	5	following	following	ADJ
cana-2702	404	6	scenarios	scenario	NOUN
cana-2702	404	7	:	:	PUNCT
cana-2702	404	8	case	case	NOUN
cana-2702	404	9	(	(	PUNCT
cana-2702	404	10	1	1	X
cana-2702	404	11	)	)	PUNCT
cana-2702	404	12	𝑣	𝑣	PRON
cana-2702	404	13	∈	∈	PROPN
cana-2702	404	14	𝐶2	𝐶2	PROPN
cana-2702	404	15	.	.	PUNCT
cana-2702	405	1	we	we	PRON
cana-2702	405	2	will	will	AUX
cana-2702	405	3	suppose	suppose	VERB
cana-2702	405	4	that	that	SCONJ
cana-2702	405	5	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	405	6	)	)	PUNCT
cana-2702	405	7	≥	≥	NOUN
cana-2702	405	8	𝑛	𝑛	NOUN
cana-2702	406	1	+	+	NOUN
cana-2702	406	2	2	2	X
cana-2702	406	3	.	.	X
cana-2702	406	4	we	we	PRON
cana-2702	406	5	can	can	AUX
cana-2702	406	6	transfer	transfer	VERB
cana-2702	406	7	two	two	NUM
cana-2702	406	8	pebbles	pebble	NOUN
cana-2702	406	9	to	to	ADP
cana-2702	406	10	𝑣	𝑣	NOUN
cana-2702	406	11	using	use	VERB
cana-2702	406	12	the	the	DET
cana-2702	406	13	pebbling	pebble	VERB
cana-2702	406	14	number	number	NOUN
cana-2702	406	15	of	of	ADP
cana-2702	406	16	complete	complete	ADJ
cana-2702	406	17	graph	graph	NOUN
cana-2702	406	18	.	.	PUNCT
cana-2702	407	1	as	as	ADP
cana-2702	407	2	a	a	DET
cana-2702	407	3	result	result	NOUN
cana-2702	407	4	,	,	PUNCT
cana-2702	407	5	we	we	PRON
cana-2702	407	6	suppose	suppose	VERB
cana-2702	407	7	that	that	SCONJ
cana-2702	407	8	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	407	9	)	)	PUNCT
cana-2702	407	10	<	<	X
cana-2702	407	11	𝑛	𝑛	PROPN
cana-2702	408	1	+	+	NOUN
cana-2702	408	2	2	2	X
cana-2702	408	3	.	.	X
cana-2702	408	4	if	if	SCONJ
cana-2702	408	5	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	408	6	)	)	PUNCT
cana-2702	408	7	≥	≥	NOUN
cana-2702	408	8	𝑛	𝑛	NOUN
cana-2702	408	9	,	,	PUNCT
cana-2702	408	10	we	we	PRON
cana-2702	408	11	can	can	AUX
cana-2702	408	12	transfer	transfer	VERB
cana-2702	408	13	a	a	DET
cana-2702	408	14	pebble	pebble	NOUN
cana-2702	408	15	to	to	ADP
cana-2702	408	16	𝑣	𝑣	PRON
cana-2702	408	17	for	for	ADP
cana-2702	408	18	no	no	DET
cana-2702	408	19	more	more	ADJ
cana-2702	408	20	than	than	ADP
cana-2702	408	21	two	two	NUM
cana-2702	408	22	pebbles	pebble	NOUN
cana-2702	408	23	.	.	PUNCT
cana-2702	409	1	because	because	SCONJ
cana-2702	409	2	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	409	3	)	)	PUNCT
cana-2702	409	4	=	=	SYM
cana-2702	409	5	0	0	NUM
cana-2702	409	6	and	and	CCONJ
cana-2702	409	7	𝑞(𝐶2	𝑞(𝐶2	PROPN
cana-2702	409	8	)	)	PUNCT
cana-2702	409	9	≤	≤	NUM
cana-2702	409	10	𝑛	𝑛	PRON
cana-2702	409	11	−	−	PROPN
cana-2702	409	12	1	1	NUM
cana-2702	409	13	,	,	PUNCT
cana-2702	409	14	one	one	NUM
cana-2702	409	15	of	of	ADP
cana-2702	409	16	the	the	DET
cana-2702	409	17	vertices	vertex	NOUN
cana-2702	409	18	of	of	ADP
cana-2702	409	19	𝐶2	𝐶2	PROPN
cana-2702	409	20	holds	hold	NOUN
cana-2702	409	21	at	at	ADV
cana-2702	409	22	least	least	ADV
cana-2702	409	23	two	two	NUM
cana-2702	409	24	pebbles	pebble	NOUN
cana-2702	409	25	.	.	PUNCT
cana-2702	410	1	this	this	PRON
cana-2702	410	2	suggests	suggest	VERB
cana-2702	410	3	that	that	SCONJ
cana-2702	410	4	𝑉(𝐶3(𝐾𝑛	𝑉(𝐶3(𝐾𝑛	NOUN
cana-2702	410	5	)	)	PUNCT
cana-2702	410	6	)	)	PUNCT
cana-2702	410	7	contained	contain	VERB
cana-2702	410	8	at	at	ADP
cana-2702	410	9	least	least	ADJ
cana-2702	410	10	2(3𝑛	2(3𝑛	NUM
cana-2702	410	11	−	−	NUM
cana-2702	410	12	2	2	NUM
cana-2702	410	13	)	)	PUNCT
cana-2702	410	14	−	−	ADP
cana-2702	410	15	𝑞	𝑞	X
cana-2702	410	16	+	+	NOUN
cana-2702	410	17	1	1	NUM
cana-2702	410	18	−	−	NUM
cana-2702	410	19	2	2	NUM
cana-2702	410	20	pebbles	pebble	NOUN
cana-2702	410	21	.	.	PUNCT
cana-2702	411	1	we	we	PRON
cana-2702	411	2	have	have	VERB
cana-2702	411	3	2(3𝑛	2(3𝑛	NUM
cana-2702	411	4	−	−	NUM
cana-2702	411	5	2	2	NUM
cana-2702	411	6	)	)	PUNCT
cana-2702	411	7	−	−	PROPN
cana-2702	412	1	(	(	PUNCT
cana-2702	412	2	3𝑛	3𝑛	NUM
cana-2702	412	3	−	−	NOUN
cana-2702	412	4	6	6	NUM
cana-2702	412	5	)	)	PUNCT
cana-2702	412	6	=	=	PUNCT
cana-2702	412	7	3𝑛	3𝑛	NUM
cana-2702	412	8	+	+	CCONJ
cana-2702	412	9	2	2	NUM
cana-2702	412	10	≥	≥	NOUN
cana-2702	412	11	3𝑛	3𝑛	NUM
cana-2702	412	12	−	−	PROPN
cana-2702	412	13	2	2	NUM
cana-2702	412	14	=	=	SYM
cana-2702	412	15	𝑓(𝐶3(𝐾𝑛	𝑓(𝐶3(𝐾𝑛	NUM
cana-2702	412	16	)	)	PUNCT
cana-2702	412	17	)	)	PUNCT
cana-2702	412	18	since	since	SCONJ
cana-2702	412	19	𝑞	𝑞	X
cana-2702	412	20	≤	≤	X
cana-2702	412	21	3𝑛	3𝑛	NUM
cana-2702	412	22	−	−	ADP
cana-2702	412	23	5	5	NUM
cana-2702	412	24	.	.	PUNCT
cana-2702	413	1	after	after	ADP
cana-2702	413	2	that	that	PRON
cana-2702	413	3	,	,	PUNCT
cana-2702	413	4	the	the	DET
cana-2702	413	5	pebble	pebble	NOUN
cana-2702	413	6	may	may	AUX
cana-2702	413	7	be	be	AUX
cana-2702	413	8	transferred	transfer	VERB
cana-2702	413	9	to	to	ADP
cana-2702	413	10	𝑣.	𝑣.	NOUN
cana-2702	413	11	assume	assume	VERB
cana-2702	413	12	that	that	SCONJ
cana-2702	413	13	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	413	14	)	)	PUNCT
cana-2702	413	15	−	−	PROPN
cana-2702	413	16	{	{	PUNCT
cana-2702	413	17	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	413	18	,	,	PUNCT
cana-2702	413	19	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	413	20	,	,	PUNCT
cana-2702	413	21	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	413	22	,	,	PUNCT
cana-2702	413	23	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	413	24	}	}	PUNCT
cana-2702	413	25	)	)	PUNCT
cana-2702	413	26	≥	≥	NOUN
cana-2702	414	1	𝑛	𝑛	DET
cana-2702	414	2	−	−	NOUN
cana-2702	414	3	4	4	NUM
cana-2702	414	4	.	.	PUNCT
cana-2702	415	1	we	we	PRON
cana-2702	415	2	can	can	AUX
cana-2702	415	3	then	then	ADV
cana-2702	415	4	transfer	transfer	VERB
cana-2702	415	5	a	a	DET
cana-2702	415	6	pebble	pebble	NOUN
cana-2702	415	7	to	to	PART
cana-2702	415	8	𝑣.	𝑣.	VERB
cana-2702	415	9	as	as	ADP
cana-2702	415	10	a	a	DET
cana-2702	415	11	result	result	NOUN
cana-2702	415	12	,	,	PUNCT
cana-2702	415	13	we	we	PRON
cana-2702	415	14	suppose	suppose	VERB
cana-2702	415	15	that	that	SCONJ
cana-2702	415	16	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	415	17	)	)	PUNCT
cana-2702	415	18	−	−	PROPN
cana-2702	415	19	{	{	PUNCT
cana-2702	415	20	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	415	21	,	,	PUNCT
cana-2702	415	22	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	415	23	,	,	PUNCT
cana-2702	415	24	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	415	25	,	,	PUNCT
cana-2702	415	26	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	415	27	}	}	PUNCT
cana-2702	415	28	)	)	PUNCT
cana-2702	415	29	≤	≤	NUM
cana-2702	416	1	𝑛	𝑛	PRON
cana-2702	416	2	−	−	PROPN
cana-2702	416	3	5	5	NUM
cana-2702	416	4	,	,	PUNCT
cana-2702	416	5	which	which	PRON
cana-2702	416	6	implies	imply	VERB
cana-2702	416	7	that	that	SCONJ
cana-2702	417	1	at	at	ADP
cana-2702	417	2	least	least	ADJ
cana-2702	417	3	2(3𝑛	2(3𝑛	NUM
cana-2702	417	4	−	−	NUM
cana-2702	417	5	2	2	NUM
cana-2702	417	6	)	)	PUNCT
cana-2702	417	7	−	−	NOUN
cana-2702	417	8	𝑞	𝑞	X
cana-2702	417	9	−	−	PROPN
cana-2702	417	10	1	1	NUM
cana-2702	417	11	−	−	PROPN
cana-2702	417	12	(	(	PUNCT
cana-2702	417	13	(	(	PUNCT
cana-2702	417	14	𝑛	𝑛	PRON
cana-2702	417	15	−	−	NOUN
cana-2702	417	16	5	5	NUM
cana-2702	417	17	)	)	PUNCT
cana-2702	417	18	−	−	NOUN
cana-2702	417	19	𝑞2	𝑞2	NOUN
cana-2702	417	20	−	−	PROPN
cana-2702	417	21	1	1	NUM
cana-2702	417	22	)	)	PUNCT
cana-2702	417	23	pebbles	pebble	NOUN
cana-2702	417	24	are	be	AUX
cana-2702	417	25	scattered	scatter	VERB
cana-2702	417	26	on	on	ADP
cana-2702	417	27	both	both	PRON
cana-2702	417	28	𝐶1	𝐶1	NOUN
cana-2702	417	29	and	and	CCONJ
cana-2702	417	30	𝐶3	𝐶3	NOUN
cana-2702	417	31	.	.	PUNCT
cana-2702	418	1	𝑞	𝑞	X
cana-2702	418	2	=	=	PROPN
cana-2702	418	3	𝑞1	𝑞1	PROPN
cana-2702	418	4	+	+	NUM
cana-2702	418	5	𝑞2	𝑞2	NOUN
cana-2702	418	6	+	+	CCONJ
cana-2702	418	7	𝑞3	𝑞3	PROPN
cana-2702	418	8	,	,	PUNCT
cana-2702	418	9	where	where	SCONJ
cana-2702	418	10	𝑞1	𝑞1	PROPN
cana-2702	418	11	=	=	SYM
cana-2702	418	12	𝑞(𝐶1	𝑞(𝐶1	PROPN
cana-2702	418	13	)	)	PUNCT
cana-2702	418	14	,	,	PUNCT
cana-2702	418	15	𝑞2	𝑞2	NOUN
cana-2702	418	16	=	=	SYM
cana-2702	418	17	𝑞(𝐶2	𝑞(𝐶2	PROPN
cana-2702	418	18	−	−	PROPN
cana-2702	418	19	{	{	PUNCT
cana-2702	418	20	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	418	21	,	,	PUNCT
cana-2702	418	22	𝑏𝑘	𝑏𝑘	PROPN
cana-2702	418	23	,	,	PUNCT
cana-2702	418	24	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	418	25	,	,	PUNCT
cana-2702	418	26	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	418	27	}	}	PUNCT
cana-2702	418	28	)	)	PUNCT
cana-2702	418	29	,	,	PUNCT
cana-2702	418	30	and	and	CCONJ
cana-2702	418	31	𝑞3	𝑞3	PROPN
cana-2702	418	32	=	=	PROPN
cana-2702	418	33	𝑞(𝐶3	𝑞(𝐶3	PROPN
cana-2702	418	34	)	)	PUNCT
cana-2702	418	35	.	.	PUNCT
cana-2702	419	1	as	as	ADP
cana-2702	419	2	a	a	DET
cana-2702	419	3	result	result	NOUN
cana-2702	419	4	,	,	PUNCT
cana-2702	419	5	in	in	ADP
cana-2702	419	6	𝑉(𝐶1	𝑉(𝐶1	NOUN
cana-2702	419	7	)	)	PUNCT
cana-2702	419	8	or	or	CCONJ
cana-2702	419	9	𝑉(𝐶3	𝑉(𝐶3	NUM
cana-2702	419	10	)	)	PUNCT
cana-2702	419	11	,	,	PUNCT
cana-2702	419	12	which	which	PRON
cana-2702	419	13	include	include	VERB
cana-2702	419	14	at	at	ADV
cana-2702	419	15	least	least	ADJ
cana-2702	419	16	𝑛	𝑛	DET
cana-2702	419	17	+	+	ADJ
cana-2702	419	18	2	2	NUM
cana-2702	419	19	pebbles	pebble	NOUN
cana-2702	419	20	,	,	PUNCT
cana-2702	419	21	at	at	ADP
cana-2702	419	22	least	least	ADJ
cana-2702	419	23	6𝑛	6𝑛	NUM
cana-2702	419	24	−	−	NOUN
cana-2702	419	25	4	4	NUM
cana-2702	419	26	−	−	NOUN
cana-2702	419	27	𝑞1	𝑞1	PROPN
cana-2702	419	28	−	−	PROPN
cana-2702	419	29	𝑞2	𝑞2	NOUN
cana-2702	419	30	−	−	PROPN
cana-2702	419	31	𝑞3	𝑞3	PROPN
cana-2702	419	32	+	+	NOUN
cana-2702	419	33	1	1	NUM
cana-2702	419	34	−	−	NOUN
cana-2702	419	35	𝑛	𝑛	DET
cana-2702	419	36	+	+	NUM
cana-2702	419	37	5	5	NUM
cana-2702	419	38	+	+	NUM
cana-2702	419	39	𝑞2	𝑞2	NOUN
cana-2702	419	40	+	+	CCONJ
cana-2702	419	41	1	1	NUM
cana-2702	419	42	=	=	NOUN
cana-2702	419	43	5𝑛	5𝑛	NUM
cana-2702	419	44	+	+	CCONJ
cana-2702	419	45	3	3	NUM
cana-2702	419	46	−	−	NOUN
cana-2702	419	47	𝑞1	𝑞1	PROPN
cana-2702	419	48	−	−	ADP
cana-2702	419	49	𝑞3	𝑞3	PROPN
cana-2702	419	50	pebbles	pebble	NOUN
cana-2702	419	51	are	be	AUX
cana-2702	419	52	kept	keep	VERB
cana-2702	419	53	.	.	PUNCT
cana-2702	420	1	assume	assume	VERB
cana-2702	420	2	that	that	SCONJ
cana-2702	420	3	𝑝(𝑉(𝐶1	𝑝(𝑉(𝐶1	PROPN
cana-2702	420	4	)	)	PUNCT
cana-2702	420	5	)	)	PUNCT
cana-2702	420	6	≥	≥	X
cana-2702	420	7	𝑛	𝑛	PRON
cana-2702	421	1	+	+	NOUN
cana-2702	421	2	2	2	NUM
cana-2702	421	3	.	.	PUNCT
cana-2702	422	1	the	the	DET
cana-2702	422	2	two	two	NUM
cana-2702	422	3	pebbles	pebble	NOUN
cana-2702	422	4	may	may	AUX
cana-2702	422	5	then	then	ADV
cana-2702	422	6	be	be	AUX
cana-2702	422	7	moved	move	VERB
cana-2702	422	8	to	to	ADP
cana-2702	422	9	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	422	10	,	,	PUNCT
cana-2702	422	11	and	and	CCONJ
cana-2702	422	12	the	the	DET
cana-2702	422	13	pebble	pebble	NOUN
cana-2702	422	14	can	can	AUX
cana-2702	422	15	be	be	AUX
cana-2702	422	16	moved	move	VERB
cana-2702	422	17	to	to	ADP
cana-2702	422	18	𝑣𝑖𝑛𝐶2	𝑣𝑖𝑛𝐶2	PROPN
cana-2702	422	19	.	.	PUNCT
cana-2702	423	1	this	this	PRON
cana-2702	423	2	replaces	replace	VERB
cana-2702	423	3	at	at	ADP
cana-2702	423	4	least	least	ADJ
cana-2702	423	5	5𝑛	5𝑛	NUM
cana-2702	423	6	+	+	CCONJ
cana-2702	423	7	3	3	NUM
cana-2702	423	8	−	−	NOUN
cana-2702	423	9	𝑞1	𝑞1	PROPN
cana-2702	423	10	−	−	PROPN
cana-2702	424	1	𝑞3	𝑞3	PROPN
cana-2702	424	2	−	−	PROPN
cana-2702	424	3	(	(	PUNCT
cana-2702	424	4	𝑛	𝑛	PROPN
cana-2702	424	5	+	+	NOUN
cana-2702	424	6	2	2	NUM
cana-2702	424	7	)	)	PUNCT
cana-2702	424	8	=	=	PUNCT
cana-2702	425	1	4𝑛	4𝑛	NOUN
cana-2702	425	2	+	+	CCONJ
cana-2702	425	3	1	1	NUM
cana-2702	425	4	−	−	NOUN
cana-2702	425	5	𝑞1	𝑞1	PROPN
cana-2702	425	6	−	−	ADP
cana-2702	426	1	𝑞3	𝑞3	PROPN
cana-2702	426	2	pebbles	pebble	NOUN
cana-2702	426	3	left	leave	VERB
cana-2702	426	4	in	in	ADP
cana-2702	426	5	𝐶1	𝐶1	PRON
cana-2702	426	6	and	and	CCONJ
cana-2702	426	7	𝐶3	𝐶3	PROPN
cana-2702	426	8	.	.	PROPN
cana-2702	427	1	assume	assume	VERB
cana-2702	427	2	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	427	3	)	)	PUNCT
cana-2702	427	4	≥	≥	NOUN
cana-2702	427	5	2𝑛	2𝑛	PROPN
cana-2702	427	6	−	−	PROPN
cana-2702	428	1	𝑞1	𝑞1	PROPN
cana-2702	428	2	+	+	CCONJ
cana-2702	428	3	1	1	X
cana-2702	428	4	.	.	PUNCT
cana-2702	429	1	then	then	ADV
cana-2702	429	2	we	we	PRON
cana-2702	429	3	may	may	AUX
cana-2702	429	4	relocate	relocate	VERB
cana-2702	429	5	the	the	DET
cana-2702	429	6	two	two	NUM
cana-2702	429	7	pebbles	pebble	NOUN
cana-2702	429	8	to	to	ADP
cana-2702	429	9	𝑎𝑘	𝑎𝑘	VERB
cana-2702	429	10	and	and	CCONJ
cana-2702	429	11	add	add	VERB
cana-2702	429	12	another	another	DET
cana-2702	429	13	pebble	pebble	NOUN
cana-2702	429	14	to	to	PART
cana-2702	429	15	𝑣.	𝑣.	VERB
cana-2702	429	16	this	this	PRON
cana-2702	429	17	was	be	AUX
cana-2702	429	18	due	due	ADJ
cana-2702	429	19	to	to	ADP
cana-2702	429	20	𝐶1	𝐶1	PROPN
cana-2702	429	21	≅	≅	PROPN
cana-2702	429	22	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	429	23	if	if	SCONJ
cana-2702	429	24	not	not	PART
cana-2702	429	25	,	,	PUNCT
cana-2702	429	26	at	at	ADP
cana-2702	429	27	least	least	ADJ
cana-2702	429	28	2𝑛	2𝑛	NOUN
cana-2702	430	1	−	−	PROPN
cana-2702	431	1	𝑞3	𝑞3	PROPN
cana-2702	431	2	+	+	NOUN
cana-2702	431	3	1	1	NUM
cana-2702	431	4	pebbles	pebble	NOUN
cana-2702	431	5	remained	remain	VERB
cana-2702	431	6	in	in	ADP
cana-2702	431	7	𝐶3	𝐶3	PROPN
cana-2702	431	8	.	.	PUNCT
cana-2702	432	1	this	this	PRON
cana-2702	432	2	was	be	AUX
cana-2702	432	3	due	due	ADJ
cana-2702	432	4	to	to	ADP
cana-2702	432	5	𝐶3	𝐶3	PROPN
cana-2702	432	6	≅	≅	PROPN
cana-2702	432	7	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	432	8	using	use	VERB
cana-2702	432	9	the	the	DET
cana-2702	432	10	pebbling	pebble	VERB
cana-2702	432	11	number	number	NOUN
cana-2702	432	12	of	of	ADP
cana-2702	432	13	the	the	DET
cana-2702	432	14	entire	entire	ADJ
cana-2702	432	15	graph	graph	NOUN
cana-2702	432	16	,	,	PUNCT
cana-2702	432	17	we	we	PRON
cana-2702	432	18	can	can	AUX
cana-2702	432	19	add	add	VERB
cana-2702	432	20	another	another	DET
cana-2702	432	21	pebble	pebble	NOUN
cana-2702	432	22	to	to	ADP
cana-2702	432	23	𝑣	𝑣	PROPN
cana-2702	432	24	in	in	ADP
cana-2702	432	25	𝐶2	𝐶2	PROPN
cana-2702	432	26	.	.	PUNCT
cana-2702	433	1	as	as	ADP
cana-2702	433	2	a	a	DET
cana-2702	433	3	result	result	NOUN
cana-2702	433	4	,	,	PUNCT
cana-2702	433	5	we	we	PRON
cana-2702	433	6	suppose	suppose	VERB
cana-2702	433	7	that	that	SCONJ
cana-2702	433	8	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	433	9	)	)	PUNCT
cana-2702	433	10	≤	≤	NOUN
cana-2702	433	11	𝑛	𝑛	PRON
cana-2702	434	1	+	+	NOUN
cana-2702	434	2	1	1	X
cana-2702	434	3	.	.	PUNCT
cana-2702	435	1	this	this	PRON
cana-2702	435	2	means	mean	VERB
cana-2702	435	3	that	that	SCONJ
cana-2702	435	4	at	at	ADV
cana-2702	435	5	least	least	ADJ
cana-2702	435	6	𝑝	𝑝	NOUN
cana-2702	435	7	(	(	PUNCT
cana-2702	435	8	𝑉(𝐶3(𝐾𝑛	𝑉(𝐶3(𝐾𝑛	PROPN
cana-2702	435	9	)	)	PUNCT
cana-2702	435	10	)	)	PUNCT
cana-2702	435	11	)	)	PUNCT
cana-2702	436	1	−	−	PROPN
cana-2702	437	1	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	437	2	)	)	PUNCT
cana-2702	437	3	=	=	SYM
cana-2702	437	4	6𝑛	6𝑛	NOUN
cana-2702	437	5	−	−	NOUN
cana-2702	437	6	4	4	NUM
cana-2702	437	7	−	−	NOUN
cana-2702	437	8	𝑞	𝑞	NOUN
cana-2702	437	9	+	+	PROPN
cana-2702	437	10	1	1	NUM
cana-2702	437	11	−	−	NOUN
cana-2702	437	12	(	(	PUNCT
cana-2702	437	13	𝑛	𝑛	PROPN
cana-2702	437	14	+	+	NOUN
cana-2702	437	15	1	1	NUM
cana-2702	437	16	)	)	PUNCT
cana-2702	437	17	−	−	NOUN
cana-2702	437	18	𝑞15𝑛	𝑞15𝑛	PRON
cana-2702	437	19	−	−	PROPN
cana-2702	437	20	(	(	PUNCT
cana-2702	437	21	𝑞2	𝑞2	PROPN
cana-2702	437	22	+	+	CCONJ
cana-2702	437	23	𝑞3	𝑞3	NOUN
cana-2702	437	24	)	)	PUNCT
cana-2702	437	25	−	−	PROPN
cana-2702	437	26	4	4	NUM
cana-2702	437	27	is	be	AUX
cana-2702	437	28	required	require	VERB
cana-2702	437	29	that	that	PRON
cana-2702	437	30	equals	equal	VERB
cana-2702	437	31	𝑓(𝐶2(𝐾𝑛	𝑓(𝐶2(𝐾𝑛	NOUN
cana-2702	437	32	)	)	PUNCT
cana-2702	437	33	)	)	PUNCT
cana-2702	438	1	−	−	ADP
cana-2702	438	2	𝑞2	𝑞2	NOUN
cana-2702	438	3	−	−	PROPN
cana-2702	438	4	𝑞3	𝑞3	PROPN
cana-2702	438	5	+	+	NOUN
cana-2702	438	6	1	1	X
cana-2702	438	7	.	.	PUNCT
cana-2702	438	8	because	because	SCONJ
cana-2702	438	9	𝐶2	𝐶2	INTJ
cana-2702	438	10	∪	∪	ADP
cana-2702	438	11	𝐶3	𝐶3	PROPN
cana-2702	438	12	≅	≅	PROPN
cana-2702	438	13	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	PROPN
cana-2702	438	14	)	)	PUNCT
cana-2702	438	15	.	.	PUNCT
cana-2702	439	1	we	we	PRON
cana-2702	439	2	can	can	AUX
cana-2702	439	3	transfer	transfer	VERB
cana-2702	439	4	two	two	NUM
cana-2702	439	5	pebbless	pebbless	NOUN
cana-2702	439	6	to	to	ADP
cana-2702	439	7	𝑣𝑖𝑛𝐶2	𝑣𝑖𝑛𝐶2	NOUN
cana-2702	439	8	using	use	VERB
cana-2702	439	9	theorem	theorem	ADJ
cana-2702	439	10	7	7	NUM
cana-2702	439	11	.	.	NOUN
cana-2702	439	12	case	case	NOUN
cana-2702	439	13	(	(	PUNCT
cana-2702	439	14	2	2	NUM
cana-2702	439	15	)	)	PUNCT
cana-2702	439	16	𝑣	𝑣	PRON
cana-2702	439	17	∈	∈	NOUN
cana-2702	439	18	𝐶1	𝐶1	ADJ
cana-2702	439	19	−	−	PROPN
cana-2702	439	20	{	{	PUNCT
cana-2702	439	21	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	439	22	,	,	PUNCT
cana-2702	439	23	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	439	24	}	}	PUNCT
cana-2702	439	25	or	or	CCONJ
cana-2702	439	26	𝑣	𝑣	ADP
cana-2702	439	27	∈	∈	PROPN
cana-2702	439	28	𝐶3	𝐶3	PROPN
cana-2702	439	29	−	−	PROPN
cana-2702	439	30	{	{	PUNCT
cana-2702	439	31	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	439	32	,	,	PUNCT
cana-2702	439	33	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	439	34	}	}	PUNCT
cana-2702	439	35	.	.	PUNCT
cana-2702	440	1	communications	communication	NOUN
cana-2702	440	2	on	on	ADP
cana-2702	440	3	applied	apply	VERB
cana-2702	440	4	nonlinear	nonlinear	ADJ
cana-2702	440	5	analysis	analysis	NOUN
cana-2702	440	6	issn	issn	NOUN
cana-2702	440	7	:	:	PUNCT
cana-2702	440	8	1074	1074	NUM
cana-2702	440	9	-	-	PUNCT
cana-2702	440	10	133x	133x	NUM
cana-2702	440	11	vol	vol	NOUN
cana-2702	440	12	32	32	NUM
cana-2702	440	13	no	no	NOUN
cana-2702	440	14	.	.	PUNCT
cana-2702	441	1	3s	3s	NUM
cana-2702	441	2	(	(	PUNCT
cana-2702	441	3	2025	2025	NUM
cana-2702	441	4	)	)	PUNCT
cana-2702	441	5	657	657	NUM
cana-2702	441	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	442	1	we	we	PRON
cana-2702	442	2	assume	assume	VERB
cana-2702	442	3	𝑣	𝑣	ADP
cana-2702	442	4	∈	∈	PROPN
cana-2702	442	5	𝐶3	𝐶3	NOUN
cana-2702	442	6	and	and	CCONJ
cana-2702	442	7	take	take	VERB
cana-2702	442	8	𝑣	𝑣	PRON
cana-2702	442	9	=	=	PUNCT
cana-2702	442	10	𝑦	𝑦	PROPN
cana-2702	442	11	without	without	ADP
cana-2702	442	12	losing	lose	VERB
cana-2702	442	13	generality	generality	NOUN
cana-2702	442	14	.	.	PUNCT
cana-2702	443	1	let	let	VERB
cana-2702	443	2	𝑞	𝑞	X
cana-2702	443	3	=	=	SYM
cana-2702	443	4	𝑞1	𝑞1	PROPN
cana-2702	443	5	+	+	NUM
cana-2702	443	6	𝑞2	𝑞2	NOUN
cana-2702	443	7	+	+	CCONJ
cana-2702	443	8	𝑞3	𝑞3	PROPN
cana-2702	443	9	−	−	PROPN
cana-2702	443	10	4	4	NUM
cana-2702	443	11	and	and	CCONJ
cana-2702	443	12	assume	assume	VERB
cana-2702	443	13	that	that	SCONJ
cana-2702	443	14	all	all	DET
cana-2702	443	15	vertices	vertex	NOUN
cana-2702	443	16	of	of	ADP
cana-2702	443	17	{	{	PUNCT
cana-2702	443	18	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	443	19	,	,	PUNCT
cana-2702	443	20	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	443	21	,	,	PUNCT
cana-2702	443	22	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	443	23	,	,	PUNCT
cana-2702	443	24	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	443	25	}	}	PUNCT
cana-2702	443	26	contain	contain	VERB
cana-2702	443	27	at	at	ADV
cana-2702	443	28	least	least	ADV
cana-2702	443	29	one	one	NUM
cana-2702	443	30	pebble	pebble	NOUN
cana-2702	443	31	.	.	PUNCT
cana-2702	444	1	if	if	SCONJ
cana-2702	444	2	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	444	3	)	)	PUNCT
cana-2702	444	4	≥	≥	NOUN
cana-2702	444	5	2(𝑛	2(𝑛	NUM
cana-2702	444	6	+	+	CCONJ
cana-2702	444	7	2	2	NUM
cana-2702	444	8	)	)	PUNCT
cana-2702	444	9	−	−	PROPN
cana-2702	445	1	𝑞1	𝑞1	ADJ
cana-2702	445	2	+	+	CCONJ
cana-2702	445	3	1	1	NUM
cana-2702	445	4	,	,	PUNCT
cana-2702	445	5	the	the	DET
cana-2702	445	6	four	four	NUM
cana-2702	445	7	pebbles	pebble	NOUN
cana-2702	445	8	can	can	AUX
cana-2702	445	9	transfer	transfer	VERB
cana-2702	445	10	to	to	ADP
cana-2702	445	11	𝑎𝑘.	𝑎𝑘.	CCONJ
cana-2702	445	12	the	the	DET
cana-2702	445	13	number	number	NOUN
cana-2702	445	14	of	of	ADP
cana-2702	445	15	pebbles	pebble	NOUN
cana-2702	445	16	maintained	maintain	VERB
cana-2702	445	17	in	in	ADP
cana-2702	445	18	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	445	19	)	)	PUNCT
cana-2702	445	20	must	must	AUX
cana-2702	445	21	thus	thus	ADV
cana-2702	445	22	be	be	AUX
cana-2702	446	1	at	at	ADP
cana-2702	446	2	least	least	ADJ
cana-2702	446	3	2(3𝑛	2(3𝑛	NUM
cana-2702	446	4	−	−	NUM
cana-2702	446	5	2	2	NUM
cana-2702	446	6	)	)	PUNCT
cana-2702	446	7	−	−	ADP
cana-2702	446	8	𝑞	𝑞	X
cana-2702	446	9	+	+	NOUN
cana-2702	446	10	1	1	NUM
cana-2702	446	11	−	−	NOUN
cana-2702	446	12	(	(	PUNCT
cana-2702	446	13	2(𝑛	2(𝑛	NUM
cana-2702	446	14	+	+	CCONJ
cana-2702	446	15	2	2	NUM
cana-2702	446	16	)	)	PUNCT
cana-2702	446	17	−	−	PROPN
cana-2702	447	1	𝑞1	𝑞1	ADJ
cana-2702	447	2	+	+	CCONJ
cana-2702	447	3	1	1	NUM
cana-2702	447	4	)	)	PUNCT
cana-2702	447	5	≥	≥	NOUN
cana-2702	447	6	𝑓(𝐶2(𝐾𝑛	𝑓(𝐶2(𝐾𝑛	NUM
cana-2702	447	7	)	)	PUNCT
cana-2702	447	8	)	)	PUNCT
cana-2702	447	9	,	,	PUNCT
cana-2702	447	10	and	and	CCONJ
cana-2702	447	11	we	we	PRON
cana-2702	447	12	may	may	AUX
cana-2702	447	13	transfer	transfer	VERB
cana-2702	447	14	an	an	DET
cana-2702	447	15	extra	extra	ADJ
cana-2702	447	16	pebble	pebble	NOUN
cana-2702	447	17	to	to	ADP
cana-2702	447	18	𝑦	𝑦	NOUN
cana-2702	447	19	using	use	VERB
cana-2702	447	20	theorem	theorem	NOUN
cana-2702	447	21	1	1	NUM
cana-2702	447	22	.	.	PUNCT
cana-2702	448	1	as	as	ADP
cana-2702	448	2	a	a	DET
cana-2702	448	3	result	result	NOUN
cana-2702	448	4	,	,	PUNCT
cana-2702	448	5	we	we	PRON
cana-2702	448	6	suppose	suppose	VERB
cana-2702	448	7	that	that	SCONJ
cana-2702	448	8	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	448	9	)	)	PUNCT
cana-2702	448	10	<	<	X
cana-2702	448	11	2(𝑛	2(𝑛	NUM
cana-2702	448	12	+	+	CCONJ
cana-2702	448	13	2	2	NUM
cana-2702	448	14	)	)	PUNCT
cana-2702	448	15	−	−	NOUN
cana-2702	449	1	𝑞1	𝑞1	ADJ
cana-2702	449	2	.	.	PUNCT
cana-2702	450	1	if	if	SCONJ
cana-2702	450	2	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	450	3	)	)	PUNCT
cana-2702	450	4	≥	≥	NOUN
cana-2702	450	5	𝑛	𝑛	NOUN
cana-2702	450	6	+	+	NOUN
cana-2702	450	7	2	2	NUM
cana-2702	450	8	,	,	PUNCT
cana-2702	450	9	two	two	NUM
cana-2702	450	10	pebbles	pebble	NOUN
cana-2702	450	11	can	can	AUX
cana-2702	450	12	be	be	AUX
cana-2702	450	13	transferred	transfer	VERB
cana-2702	450	14	to	to	ADP
cana-2702	450	15	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	450	16	and	and	CCONJ
cana-2702	450	17	one	one	NUM
cana-2702	450	18	pebble	pebble	NOUN
cana-2702	450	19	to	to	ADP
cana-2702	450	20	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	450	21	.	.	PUNCT
cana-2702	451	1	then	then	ADV
cana-2702	451	2	,	,	PUNCT
cana-2702	451	3	using	use	VERB
cana-2702	451	4	at	at	ADV
cana-2702	451	5	least	least	ADV
cana-2702	451	6	two	two	NUM
cana-2702	451	7	pebbles	pebble	NOUN
cana-2702	451	8	from	from	ADP
cana-2702	451	9	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	451	10	)	)	PUNCT
cana-2702	451	11	or	or	CCONJ
cana-2702	451	12	𝑉(𝐶3	𝑉(𝐶3	NUM
cana-2702	451	13	)	)	PUNCT
cana-2702	451	14	,	,	PUNCT
cana-2702	451	15	we	we	PRON
cana-2702	451	16	may	may	AUX
cana-2702	451	17	transfer	transfer	VERB
cana-2702	451	18	another	another	DET
cana-2702	451	19	pebble	pebble	NOUN
cana-2702	451	20	to	to	ADP
cana-2702	451	21	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	451	22	and	and	CCONJ
cana-2702	451	23	one	one	NUM
cana-2702	451	24	to	to	PART
cana-2702	451	25	𝑦.	𝑦.	VERB
cana-2702	451	26	this	this	PRON
cana-2702	451	27	means	mean	VERB
cana-2702	451	28	that	that	SCONJ
cana-2702	451	29	in	in	ADP
cana-2702	451	30	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	451	31	)	)	PUNCT
cana-2702	451	32	and	and	CCONJ
cana-2702	451	33	𝑉(𝐶3	𝑉(𝐶3	NUM
cana-2702	451	34	)	)	PUNCT
cana-2702	451	35	,	,	PUNCT
cana-2702	451	36	at	at	ADP
cana-2702	451	37	least	least	ADJ
cana-2702	451	38	2𝑛	2𝑛	NOUN
cana-2702	451	39	−	−	NOUN
cana-2702	451	40	2	2	NUM
cana-2702	451	41	pebbles	pebble	NOUN
cana-2702	451	42	were	be	AUX
cana-2702	451	43	kept	keep	VERB
cana-2702	451	44	.	.	PUNCT
cana-2702	452	1	as	as	ADP
cana-2702	452	2	a	a	DET
cana-2702	452	3	result	result	NOUN
cana-2702	452	4	,	,	PUNCT
cana-2702	452	5	we	we	PRON
cana-2702	452	6	can	can	AUX
cana-2702	452	7	add	add	VERB
cana-2702	452	8	another	another	DET
cana-2702	452	9	pebble	pebble	NOUN
cana-2702	452	10	to	to	PART
cana-2702	452	11	𝑦.	𝑦.	VERB
cana-2702	452	12	as	as	ADP
cana-2702	452	13	a	a	DET
cana-2702	452	14	result	result	NOUN
cana-2702	452	15	,	,	PUNCT
cana-2702	452	16	we	we	PRON
cana-2702	452	17	suppose	suppose	VERB
cana-2702	452	18	that	that	SCONJ
cana-2702	452	19	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	452	20	)	)	PUNCT
cana-2702	452	21	<	<	X
cana-2702	452	22	𝑛	𝑛	PROPN
cana-2702	453	1	+	+	NOUN
cana-2702	453	2	2	2	X
cana-2702	453	3	.	.	X
cana-2702	453	4	claim	claim	NOUN
cana-2702	453	5	(	(	PUNCT
cana-2702	453	6	1	1	X
cana-2702	453	7	)	)	PUNCT
cana-2702	453	8	𝑝(𝐶3(𝐾𝑛	𝑝(𝐶3(𝐾𝑛	NOUN
cana-2702	453	9	)	)	PUNCT
cana-2702	453	10	)	)	PUNCT
cana-2702	454	1	−	−	PROPN
cana-2702	454	2	(	(	PUNCT
cana-2702	454	3	𝑛	𝑛	PROPN
cana-2702	454	4	+	+	NOUN
cana-2702	454	5	1	1	NUM
cana-2702	454	6	)	)	PUNCT
cana-2702	454	7	+	+	CCONJ
cana-2702	455	1	𝑞1	𝑞1	ADJ
cana-2702	455	2	≥	≥	NUM
cana-2702	455	3	2𝑓(𝐶2(𝐾𝑛	2𝑓(𝐶2(𝐾𝑛	NUM
cana-2702	455	4	)	)	PUNCT
cana-2702	455	5	)	)	PUNCT
cana-2702	456	1	−	−	PROPN
cana-2702	456	2	(	(	PUNCT
cana-2702	456	3	𝑞2	𝑞2	PROPN
cana-2702	456	4	+	+	CCONJ
cana-2702	456	5	𝑞3	𝑞3	PROPN
cana-2702	456	6	)	)	PUNCT
cana-2702	456	7	+	+	CCONJ
cana-2702	456	8	1	1	X
cana-2702	456	9	.	.	X
cana-2702	456	10	𝑝(𝐶3(𝐾𝑛	𝑝(𝐶3(𝐾𝑛	NOUN
cana-2702	456	11	)	)	PUNCT
cana-2702	456	12	)	)	PUNCT
cana-2702	457	1	−	−	PROPN
cana-2702	457	2	(	(	PUNCT
cana-2702	457	3	𝑛	𝑛	PROPN
cana-2702	457	4	+	+	NOUN
cana-2702	457	5	1	1	NUM
cana-2702	457	6	)	)	PUNCT
cana-2702	457	7	+	+	CCONJ
cana-2702	458	1	𝑞1	𝑞1	PROPN
cana-2702	458	2	=	=	SYM
cana-2702	458	3	2(3𝑛	2(3𝑛	NUM
cana-2702	459	1	−	−	NOUN
cana-2702	459	2	2	2	NUM
cana-2702	459	3	)	)	PUNCT
cana-2702	459	4	−	−	ADP
cana-2702	459	5	𝑞	𝑞	X
cana-2702	459	6	+	+	NOUN
cana-2702	459	7	1	1	NUM
cana-2702	459	8	−	−	NOUN
cana-2702	459	9	𝑛	𝑛	PRON
cana-2702	459	10	−	−	PROPN
cana-2702	459	11	1	1	NUM
cana-2702	459	12	−	−	PROPN
cana-2702	459	13	𝑞1	𝑞1	PROPN
cana-2702	459	14	=	=	PUNCT
cana-2702	459	15	(	(	PUNCT
cana-2702	459	16	4𝑛	4𝑛	NOUN
cana-2702	459	17	−	−	PROPN
cana-2702	459	18	4	4	NUM
cana-2702	459	19	)	)	PUNCT
cana-2702	459	20	−	−	PROPN
cana-2702	459	21	(	(	PUNCT
cana-2702	459	22	𝑞2	𝑞2	PROPN
cana-2702	459	23	+	+	CCONJ
cana-2702	459	24	𝑞3	𝑞3	PROPN
cana-2702	459	25	)	)	PUNCT
cana-2702	459	26	+	+	CCONJ
cana-2702	459	27	4	4	NUM
cana-2702	459	28	+	+	NUM
cana-2702	459	29	𝑛	𝑛	PRON
cana-2702	459	30	−	−	NUM
cana-2702	459	31	1	1	NUM
cana-2702	459	32	≥	≥	NOUN
cana-2702	459	33	(	(	PUNCT
cana-2702	459	34	4𝑛	4𝑛	NOUN
cana-2702	459	35	−	−	PROPN
cana-2702	459	36	4	4	NUM
cana-2702	459	37	)	)	PUNCT
cana-2702	459	38	−	−	PROPN
cana-2702	459	39	(	(	PUNCT
cana-2702	459	40	𝑞2	𝑞2	PROPN
cana-2702	459	41	+	+	CCONJ
cana-2702	459	42	𝑞3	𝑞3	PROPN
cana-2702	459	43	)	)	PUNCT
cana-2702	459	44	+	+	CCONJ
cana-2702	459	45	1	1	NUM
cana-2702	459	46	=	=	SYM
cana-2702	459	47	2(2𝑛	2(2𝑛	NUM
cana-2702	459	48	−	−	NOUN
cana-2702	459	49	2	2	NUM
cana-2702	459	50	)	)	PUNCT
cana-2702	459	51	−	−	PROPN
cana-2702	459	52	(	(	PUNCT
cana-2702	459	53	𝑞2	𝑞2	PROPN
cana-2702	459	54	+	+	CCONJ
cana-2702	459	55	𝑞3	𝑞3	PROPN
cana-2702	459	56	)	)	PUNCT
cana-2702	459	57	+	+	CCONJ
cana-2702	459	58	1	1	NUM
cana-2702	459	59	=	=	SYM
cana-2702	459	60	2𝑓(𝐶2(𝐾𝑛	2𝑓(𝐶2(𝐾𝑛	NUM
cana-2702	459	61	)	)	PUNCT
cana-2702	459	62	)	)	PUNCT
cana-2702	460	1	−	−	PROPN
cana-2702	460	2	(	(	PUNCT
cana-2702	460	3	𝑞2	𝑞2	PROPN
cana-2702	460	4	+	+	CCONJ
cana-2702	460	5	𝑞3	𝑞3	PROPN
cana-2702	460	6	)	)	PUNCT
cana-2702	460	7	+	+	NOUN
cana-2702	461	1	1	1	X
cana-2702	461	2	.	.	X
cana-2702	461	3	we	we	PRON
cana-2702	461	4	can	can	AUX
cana-2702	461	5	transfer	transfer	VERB
cana-2702	461	6	two	two	NUM
cana-2702	461	7	pebbles	pebble	NOUN
cana-2702	461	8	to	to	ADP
cana-2702	461	9	𝑦	𝑦	NOUN
cana-2702	461	10	using	use	VERB
cana-2702	461	11	the	the	DET
cana-2702	461	12	following	following	NOUN
cana-2702	461	13	theorem	theorem	NOUN
cana-2702	461	14	7	7	NUM
cana-2702	461	15	.	.	PUNCT
cana-2702	461	16	because	because	SCONJ
cana-2702	461	17	𝐶2	𝐶2	INTJ
cana-2702	461	18	∪	∪	ADP
cana-2702	461	19	𝐶3	𝐶3	PROPN
cana-2702	461	20	≅	≅	PROPN
cana-2702	461	21	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	PROPN
cana-2702	461	22	)	)	PUNCT
cana-2702	461	23	.	.	PUNCT
cana-2702	462	1	then	then	ADV
cana-2702	462	2	we	we	PRON
cana-2702	462	3	’re	’re	AUX
cana-2702	462	4	finished	finish	VERB
cana-2702	462	5	.	.	PUNCT
cana-2702	463	1	lemma	lemma	PROPN
cana-2702	463	2	2	2	X
cana-2702	463	3	.	.	PUNCT
cana-2702	464	1	let	let	VERB
cana-2702	464	2	𝐺	𝐺	PROPN
cana-2702	464	3	=	=	SYM
cana-2702	464	4	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	PROPN
cana-2702	464	5	)	)	PUNCT
cana-2702	464	6	be	be	VERB
cana-2702	464	7	a	a	DET
cana-2702	464	8	crisscross	crisscross	NOUN
cana-2702	464	9	sequence	sequence	NOUN
cana-2702	464	10	of	of	ADP
cana-2702	464	11	𝑚	𝑚	ADP
cana-2702	464	12	complete	complete	ADJ
cana-2702	464	13	graphs	graph	NOUN
cana-2702	464	14	.	.	PUNCT
cana-2702	465	1	let	let	VERB
cana-2702	465	2	𝑋1	𝑋1	NOUN
cana-2702	465	3	=	=	SYM
cana-2702	465	4	𝐶𝑚1	𝐶𝑚1	X
cana-2702	465	5	(	(	PUNCT
cana-2702	465	6	𝐾𝑛	𝐾𝑛	PROPN
cana-2702	465	7	)	)	PUNCT
cana-2702	465	8	=	=	SYM
cana-2702	465	9	𝐶1	𝐶1	ADJ
cana-2702	465	10	∪.	∪.	X
cana-2702	465	11	.	.	PUNCT
cana-2702	466	1	.∪	.∪	PROPN
cana-2702	466	2	𝐶𝑚−1	𝐶𝑚−1	PROPN
cana-2702	466	3	and	and	CCONJ
cana-2702	466	4	𝑋2	𝑋2	VERB
cana-2702	466	5	=	=	SYM
cana-2702	466	6	𝐶𝑚2	𝐶𝑚2	NOUN
cana-2702	466	7	(	(	PUNCT
cana-2702	466	8	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	466	9	)	)	PUNCT
cana-2702	466	10	=	=	PUNCT
cana-2702	466	11	𝐶𝑚1	𝐶𝑚1	X
cana-2702	466	12	+	+	ADJ
cana-2702	466	13	1	1	NUM
cana-2702	466	14	∪	∪	ADP
cana-2702	466	15	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	466	16	+	+	NOUN
cana-2702	466	17	2	2	NUM
cana-2702	466	18	∪.	∪.	NOUN
cana-2702	466	19	.	.	PUNCT
cana-2702	467	1	.∪	.∪	PUNCT
cana-2702	468	1	𝐶𝑚	𝐶𝑚	NOUN
cana-2702	468	2	be	be	AUX
cana-2702	468	3	two	two	NUM
cana-2702	468	4	subgraphs	subgraph	NOUN
cana-2702	468	5	of	of	ADP
cana-2702	468	6	𝐺	𝐺	NOUN
cana-2702	468	7	,	,	PUNCT
cana-2702	468	8	where	where	SCONJ
cana-2702	468	9	𝑚1	𝑚1	NOUN
cana-2702	468	10	+	+	CCONJ
cana-2702	468	11	𝑚2	𝑚2	NOUN
cana-2702	468	12	=	=	SYM
cana-2702	468	13	𝑚.	𝑚.	NOUN
cana-2702	468	14	assume	assume	VERB
cana-2702	468	15	the	the	DET
cana-2702	468	16	number	number	NOUN
cana-2702	468	17	of	of	ADP
cana-2702	468	18	pebbles	pebble	NOUN
cana-2702	468	19	scattered	scatter	VERB
cana-2702	468	20	on	on	ADP
cana-2702	468	21	𝑋2	𝑋2	ADJ
cana-2702	468	22	is	be	AUX
cana-2702	468	23	more	more	ADJ
cana-2702	468	24	than	than	ADP
cana-2702	468	25	2(2𝑚1(2𝑚2	2(2𝑚1(2𝑚2	NUM
cana-2702	468	26	−	−	NUM
cana-2702	468	27	1	1	NUM
cana-2702	468	28	)	)	PUNCT
cana-2702	468	29	+	+	CCONJ
cana-2702	468	30	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	468	31	−	−	PROPN
cana-2702	468	32	4	4	NUM
cana-2702	468	33	)	)	PUNCT
cana-2702	468	34	+	+	CCONJ
cana-2702	468	35	2	2	X
cana-2702	468	36	)	)	PUNCT
cana-2702	468	37	−	−	NOUN
cana-2702	468	38	𝑞2	𝑞2	NOUN
cana-2702	468	39	+	+	X
cana-2702	468	40	1	1	X
cana-2702	468	41	.	.	PUNCT
cana-2702	469	1	then	then	ADV
cana-2702	469	2	we	we	PRON
cana-2702	469	3	may	may	AUX
cana-2702	469	4	relocate	relocate	VERB
cana-2702	469	5	four	four	NUM
cana-2702	469	6	pebbles	pebble	NOUN
cana-2702	469	7	to	to	ADP
cana-2702	469	8	𝑎	𝑎	PROPN
cana-2702	469	9	𝑚1	𝑚1	NOUN
cana-2702	469	10	(	(	PUNCT
cana-2702	469	11	𝑛	𝑛	PROPN
cana-2702	469	12	2	2	NUM
cana-2702	469	13	−1	−1	NOUN
cana-2702	469	14	)	)	PUNCT
cana-2702	469	15	for	for	ADP
cana-2702	469	16	𝑚1	𝑚1	NOUN
cana-2702	469	17	is	be	AUX
cana-2702	469	18	an	an	DET
cana-2702	469	19	even	even	ADJ
cana-2702	469	20	number	number	NOUN
cana-2702	469	21	,	,	PUNCT
cana-2702	469	22	or	or	CCONJ
cana-2702	469	23	𝑎	𝑎	PRON
cana-2702	469	24	𝑚1	𝑚1	NOUN
cana-2702	469	25	(	(	PUNCT
cana-2702	469	26	𝑛	𝑛	DET
cana-2702	469	27	2	2	NUM
cana-2702	469	28	−1)+1	−1)+1	NOUN
cana-2702	469	29	,	,	PUNCT
cana-2702	469	30	if	if	SCONJ
cana-2702	469	31	𝑚1	𝑚1	NOUN
cana-2702	469	32	is	be	AUX
cana-2702	469	33	an	an	DET
cana-2702	469	34	odd	odd	ADJ
cana-2702	469	35	number	number	NOUN
cana-2702	469	36	.	.	PUNCT
cana-2702	470	1	here	here	ADV
cana-2702	470	2	,	,	PUNCT
cana-2702	470	3	𝑞	𝑞	PROPN
cana-2702	470	4	=	=	SYM
cana-2702	470	5	𝑞1	𝑞1	PROPN
cana-2702	470	6	+	+	NUM
cana-2702	470	7	𝑞2	𝑞2	NOUN
cana-2702	470	8	,	,	PUNCT
cana-2702	470	9	with	with	ADP
cana-2702	470	10	𝑞1	𝑞1	PROPN
cana-2702	470	11	representing	represent	VERB
cana-2702	470	12	the	the	DET
cana-2702	470	13	number	number	NOUN
cana-2702	470	14	of	of	ADP
cana-2702	470	15	occupied	occupy	VERB
cana-2702	470	16	vertices	vertex	NOUN
cana-2702	470	17	of	of	ADP
cana-2702	470	18	𝑋1	𝑋1	NOUN
cana-2702	470	19	and	and	CCONJ
cana-2702	470	20	𝑞2	𝑞2	NOUN
cana-2702	470	21	representing	represent	VERB
cana-2702	470	22	the	the	DET
cana-2702	470	23	number	number	NOUN
cana-2702	470	24	of	of	ADP
cana-2702	470	25	occupied	occupy	VERB
cana-2702	470	26	vertices	vertex	NOUN
cana-2702	470	27	of	of	ADP
cana-2702	470	28	𝑋2	𝑋2	ADJ
cana-2702	470	29	.	.	PUNCT
cana-2702	471	1	proof	proof	NOUN
cana-2702	471	2	.	.	PUNCT
cana-2702	472	1	consider	consider	VERB
cana-2702	472	2	the	the	DET
cana-2702	472	3	graph	graph	NOUN
cana-2702	472	4	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	472	5	)	)	PUNCT
cana-2702	472	6	with	with	ADP
cana-2702	472	7	the	the	DET
cana-2702	472	8	values	value	NOUN
cana-2702	472	9	2	2	NUM
cana-2702	472	10	𝑚1(2𝑚2	𝑚1(2𝑚2	ADV
cana-2702	472	11	−	−	PROPN
cana-2702	472	12	1	1	NUM
cana-2702	472	13	)	)	PUNCT
cana-2702	473	1	+	+	CCONJ
cana-2702	473	2	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	473	3	−	−	PROPN
cana-2702	473	4	4	4	NUM
cana-2702	473	5	)	)	PUNCT
cana-2702	473	6	+	+	CCONJ
cana-2702	473	7	2	2	NUM
cana-2702	473	8	−	−	NOUN
cana-2702	473	9	𝑞2	𝑞2	NOUN
cana-2702	473	10	+	+	CCONJ
cana-2702	473	11	1	1	NUM
cana-2702	473	12	pebbles	pebble	NOUN
cana-2702	473	13	are	be	AUX
cana-2702	473	14	exclusively	exclusively	ADV
cana-2702	473	15	found	find	VERB
cana-2702	473	16	on	on	ADP
cana-2702	473	17	the	the	DET
cana-2702	473	18	vertices	vertex	NOUN
cana-2702	473	19	of	of	ADP
cana-2702	473	20	𝑋2	𝑋2	ADJ
cana-2702	473	21	.	.	PUNCT
cana-2702	474	1	we	we	PRON
cana-2702	474	2	must	must	AUX
cana-2702	474	3	relocate	relocate	VERB
cana-2702	474	4	the	the	DET
cana-2702	474	5	four	four	NUM
cana-2702	474	6	pebbles	pebble	NOUN
cana-2702	474	7	to	to	ADP
cana-2702	474	8	𝑎	𝑎	PROPN
cana-2702	474	9	𝑚1	𝑚1	NOUN
cana-2702	474	10	(	(	PUNCT
cana-2702	474	11	𝑛	𝑛	PROPN
cana-2702	474	12	2	2	NUM
cana-2702	474	13	−1	−1	NOUN
cana-2702	474	14	)	)	PUNCT
cana-2702	474	15	.	.	PUNCT
cana-2702	475	1	we	we	PRON
cana-2702	475	2	demonstrate	demonstrate	VERB
cana-2702	475	3	this	this	DET
cana-2702	475	4	lemma	lemma	PROPN
cana-2702	475	5	using	use	VERB
cana-2702	475	6	induction	induction	NOUN
cana-2702	475	7	on	on	ADP
cana-2702	475	8	𝑚2	𝑚2	NOUN
cana-2702	475	9	.	.	PUNCT
cana-2702	476	1	at	at	ADV
cana-2702	476	2	least	least	ADJ
cana-2702	476	3	2(2𝑚1	2(2𝑚1	NOUN
cana-2702	476	4	+	+	CCONJ
cana-2702	476	5	𝑛	𝑛	DET
cana-2702	476	6	−	−	PROPN
cana-2702	476	7	2	2	NUM
cana-2702	476	8	)	)	PUNCT
cana-2702	476	9	−	−	NOUN
cana-2702	476	10	𝑞2	𝑞2	NOUN
cana-2702	476	11	+	+	CCONJ
cana-2702	476	12	1	1	NUM
cana-2702	476	13	pebbles	pebble	NOUN
cana-2702	476	14	dispersed	disperse	VERB
cana-2702	476	15	on	on	ADP
cana-2702	476	16	𝑋2	𝑋2	VERB
cana-2702	476	17	for	for	ADP
cana-2702	476	18	𝑚2	𝑚2	NOUN
cana-2702	476	19	=	=	PUNCT
cana-2702	476	20	1	1	X
cana-2702	476	21	.	.	PUNCT
cana-2702	476	22	because	because	SCONJ
cana-2702	476	23	𝑚	𝑚	PROPN
cana-2702	476	24	≥	≥	NUM
cana-2702	476	25	4	4	NUM
cana-2702	476	26	,	,	PUNCT
cana-2702	476	27	at	at	ADV
cana-2702	476	28	least	least	ADV
cana-2702	476	29	2(𝑛	2(𝑛	NUM
cana-2702	476	30	+	+	CCONJ
cana-2702	476	31	6	6	NUM
cana-2702	476	32	)	)	PUNCT
cana-2702	476	33	−	−	NOUN
cana-2702	477	1	𝑞2	𝑞2	NOUN
cana-2702	477	2	+	+	CCONJ
cana-2702	477	3	1	1	NUM
cana-2702	477	4	pebbles	pebble	NOUN
cana-2702	477	5	must	must	AUX
cana-2702	477	6	be	be	AUX
cana-2702	477	7	kept	keep	VERB
cana-2702	477	8	in	in	ADP
cana-2702	477	9	𝑋2	𝑋2	ADJ
cana-2702	477	10	.	.	PUNCT
cana-2702	478	1	if	if	SCONJ
cana-2702	478	2	𝑚2	𝑚2	NOUN
cana-2702	478	3	=	=	SYM
cana-2702	478	4	1	1	NUM
cana-2702	478	5	,	,	PUNCT
cana-2702	478	6	we	we	PRON
cana-2702	478	7	get	get	AUX
cana-2702	478	8	𝑋2	𝑋2	VERB
cana-2702	478	9	≅	≅	PROPN
cana-2702	478	10	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2702	478	11	also	also	ADV
cana-2702	478	12	2(𝑛	2(𝑛	NUM
cana-2702	478	13	+	+	CCONJ
cana-2702	478	14	6	6	NUM
cana-2702	478	15	)	)	PUNCT
cana-2702	478	16	−	−	NOUN
cana-2702	478	17	𝑞2	𝑞2	NOUN
cana-2702	479	1	+	+	CCONJ
cana-2702	480	1	1	1	NUM
cana-2702	480	2	≥	≥	NOUN
cana-2702	480	3	2𝑓2(𝐾𝑛	2𝑓2(𝐾𝑛	NUM
cana-2702	480	4	)	)	PUNCT
cana-2702	480	5	−	−	NOUN
cana-2702	480	6	𝑞2	𝑞2	NOUN
cana-2702	480	7	+	+	CCONJ
cana-2702	480	8	1	1	NUM
cana-2702	480	9	.	.	X
cana-2702	480	10	four	four	NUM
cana-2702	480	11	pebbles	pebble	NOUN
cana-2702	480	12	can	can	AUX
cana-2702	480	13	be	be	AUX
cana-2702	480	14	moved	move	VERB
cana-2702	480	15	to	to	ADP
cana-2702	480	16	𝑎	𝑎	PROPN
cana-2702	480	17	𝑚1	𝑚1	NOUN
cana-2702	480	18	(	(	PUNCT
cana-2702	480	19	𝑛	𝑛	PROPN
cana-2702	480	20	2	2	NUM
cana-2702	480	21	−1	−1	NOUN
cana-2702	480	22	)	)	PUNCT
cana-2702	480	23	.	.	PUNCT
cana-2702	481	1	this	this	DET
cana-2702	481	2	result	result	NOUN
cana-2702	481	3	is	be	AUX
cana-2702	481	4	assumed	assume	VERB
cana-2702	481	5	to	to	PART
cana-2702	481	6	be	be	AUX
cana-2702	481	7	true	true	ADJ
cana-2702	481	8	for	for	ADP
cana-2702	481	9	every	every	DET
cana-2702	481	10	𝑚2′	𝑚2′	NOUN
cana-2702	481	11	<	<	X
cana-2702	481	12	𝑚2	𝑚2	NOUN
cana-2702	481	13	.	.	PUNCT
cana-2702	482	1	we	we	PRON
cana-2702	482	2	demonstrate	demonstrate	VERB
cana-2702	482	3	the	the	DET
cana-2702	482	4	findings	finding	NOUN
cana-2702	482	5	for	for	ADP
cana-2702	482	6	all	all	DET
cana-2702	482	7	𝑚2	𝑚2	NOUN
cana-2702	482	8	.	.	PUNCT
cana-2702	483	1	allow	allow	VERB
cana-2702	483	2	2(2𝑚1(2𝑚2	2(2𝑚1(2𝑚2	NUM
cana-2702	483	3	−	−	NUM
cana-2702	483	4	1	1	NUM
cana-2702	483	5	)	)	PUNCT
cana-2702	484	1	+	+	CCONJ
cana-2702	484	2	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	484	3	−	−	PROPN
cana-2702	484	4	4	4	NUM
cana-2702	484	5	)	)	PUNCT
cana-2702	484	6	+	+	CCONJ
cana-2702	484	7	2	2	X
cana-2702	484	8	)	)	PUNCT
cana-2702	484	9	−	−	NOUN
cana-2702	484	10	𝑞2	𝑞2	NOUN
cana-2702	484	11	+	+	CCONJ
cana-2702	484	12	1	1	NUM
cana-2702	484	13	pebbles	pebble	NOUN
cana-2702	484	14	to	to	PART
cana-2702	484	15	be	be	AUX
cana-2702	484	16	dispersed	disperse	VERB
cana-2702	484	17	on	on	ADP
cana-2702	484	18	𝑉(𝑋2	𝑉(𝑋2	PROPN
cana-2702	484	19	)	)	PUNCT
cana-2702	484	20	.	.	PUNCT
cana-2702	485	1	assume	assume	VERB
cana-2702	485	2	the	the	DET
cana-2702	485	3	number	number	NOUN
cana-2702	485	4	of	of	ADP
cana-2702	485	5	pebbles	pebble	NOUN
cana-2702	485	6	dispersed	disperse	VERB
cana-2702	485	7	on	on	ADP
cana-2702	485	8	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	485	9	+	+	NOUN
cana-2702	485	10	1	1	NUM
cana-2702	485	11	is	be	AUX
cana-2702	485	12	more	more	ADJ
cana-2702	485	13	than	than	ADP
cana-2702	485	14	2(𝑛	2(𝑛	NUM
cana-2702	485	15	+	+	CCONJ
cana-2702	485	16	2	2	NUM
cana-2702	485	17	)	)	PUNCT
cana-2702	485	18	−	−	PRON
cana-2702	485	19	𝑞𝑚1	𝑞𝑚1	NOUN
cana-2702	486	1	+	+	ADJ
cana-2702	486	2	1	1	NUM
cana-2702	486	3	+	+	SYM
cana-2702	486	4	1	1	NUM
cana-2702	486	5	.	.	PUNCT
cana-2702	487	1	the	the	DET
cana-2702	487	2	four	four	NUM
cana-2702	487	3	pebbles	pebble	NOUN
cana-2702	487	4	can	can	AUX
cana-2702	487	5	then	then	ADV
cana-2702	487	6	be	be	AUX
cana-2702	487	7	transferred	transfer	VERB
cana-2702	487	8	to	to	ADP
cana-2702	487	9	𝑎	𝑎	PROPN
cana-2702	487	10	𝑚1	𝑚1	NOUN
cana-2702	487	11	(	(	PUNCT
cana-2702	487	12	𝑛	𝑛	PROPN
cana-2702	487	13	2	2	NUM
cana-2702	487	14	−1	−1	NOUN
cana-2702	487	15	)	)	PUNCT
cana-2702	487	16	.	.	PUNCT
cana-2702	488	1	as	as	ADP
cana-2702	488	2	a	a	DET
cana-2702	488	3	result	result	NOUN
cana-2702	488	4	,	,	PUNCT
cana-2702	488	5	we	we	PRON
cana-2702	488	6	suppose	suppose	VERB
cana-2702	488	7	that	that	SCONJ
cana-2702	488	8	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	PROPN
cana-2702	488	9	+	+	NOUN
cana-2702	488	10	1	1	NUM
cana-2702	488	11	)	)	PUNCT
cana-2702	488	12	is	be	AUX
cana-2702	488	13	no	no	DET
cana-2702	488	14	more	more	ADJ
cana-2702	488	15	than	than	ADP
cana-2702	488	16	2(𝑛	2(𝑛	NUM
cana-2702	488	17	+	+	CCONJ
cana-2702	488	18	2	2	NUM
cana-2702	488	19	)	)	PUNCT
cana-2702	488	20	−	−	PRON
cana-2702	488	21	𝑞𝑚1	𝑞𝑚1	NOUN
cana-2702	489	1	+	+	ADJ
cana-2702	489	2	1	1	NUM
cana-2702	489	3	.	.	PUNCT
cana-2702	490	1	case	case	NOUN
cana-2702	490	2	(	(	PUNCT
cana-2702	490	3	1	1	NUM
cana-2702	490	4	)	)	PUNCT
cana-2702	490	5	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	PROPN
cana-2702	491	1	+	+	PROPN
cana-2702	491	2	1	1	NUM
cana-2702	491	3	)	)	PUNCT
cana-2702	491	4	≥	≥	NOUN
cana-2702	491	5	2𝑛	2𝑛	NUM
cana-2702	491	6	−	−	NOUN
cana-2702	491	7	𝑞	𝑞	X
cana-2702	491	8	+	+	NOUN
cana-2702	491	9	1	1	X
cana-2702	491	10	.	.	PUNCT
cana-2702	491	11	we	we	PRON
cana-2702	491	12	may	may	AUX
cana-2702	491	13	relocate	relocate	VERB
cana-2702	491	14	the	the	DET
cana-2702	491	15	two	two	NUM
cana-2702	491	16	pebbles	pebble	NOUN
cana-2702	491	17	to	to	ADP
cana-2702	491	18	𝑎	𝑎	PROPN
cana-2702	491	19	𝑚1	𝑚1	NOUN
cana-2702	491	20	(	(	PUNCT
cana-2702	491	21	𝑛	𝑛	PROPN
cana-2702	491	22	2	2	NUM
cana-2702	491	23	−1	−1	NOUN
cana-2702	491	24	)	)	PUNCT
cana-2702	491	25	based	base	VERB
cana-2702	491	26	on	on	ADP
cana-2702	491	27	this	this	DET
cana-2702	491	28	assumption	assumption	NOUN
cana-2702	491	29	.	.	PUNCT
cana-2702	492	1	this	this	PRON
cana-2702	492	2	means	mean	VERB
cana-2702	492	3	that	that	SCONJ
cana-2702	492	4	at	at	ADV
cana-2702	492	5	least	least	ADJ
cana-2702	492	6	2	2	NUM
cana-2702	492	7	𝑚1(2𝑚2−1	𝑚1(2𝑚2−1	NOUN
cana-2702	492	8	)	)	PUNCT
cana-2702	492	9	+	+	CCONJ
cana-2702	492	10	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	492	11	−	−	PROPN
cana-2702	492	12	4	4	NUM
cana-2702	492	13	)	)	PUNCT
cana-2702	492	14	+	+	CCONJ
cana-2702	492	15	2	2	NUM
cana-2702	492	16	−	−	NOUN
cana-2702	492	17	𝑞2	𝑞2	NOUN
cana-2702	492	18	−	−	PROPN
cana-2702	492	19	2(𝑛	2(𝑛	NUM
cana-2702	492	20	)	)	PUNCT
cana-2702	493	1	+	+	CCONJ
cana-2702	493	2	𝑞𝑚1	𝑞𝑚1	X
cana-2702	493	3	+	+	ADJ
cana-2702	493	4	1	1	NUM
cana-2702	493	5	−	−	SYM
cana-2702	493	6	1	1	NUM
cana-2702	493	7	pebbles	pebble	NOUN
cana-2702	493	8	are	be	AUX
cana-2702	493	9	maintained	maintain	VERB
cana-2702	493	10	in	in	ADP
cana-2702	493	11	⟨𝑋2	⟨𝑋2	NOUN
cana-2702	493	12	−	−	NOUN
cana-2702	493	13	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	493	14	+	+	PROPN
cana-2702	493	15	1⟩.	1⟩.	PROPN
cana-2702	493	16	communications	communication	NOUN
cana-2702	493	17	on	on	ADP
cana-2702	493	18	applied	apply	VERB
cana-2702	493	19	nonlinear	nonlinear	ADJ
cana-2702	493	20	analysis	analysis	NOUN
cana-2702	493	21	issn	issn	NOUN
cana-2702	493	22	:	:	PUNCT
cana-2702	493	23	1074	1074	NUM
cana-2702	493	24	-	-	PUNCT
cana-2702	493	25	133x	133x	NUM
cana-2702	493	26	vol	vol	NOUN
cana-2702	493	27	32	32	NUM
cana-2702	493	28	no	no	NOUN
cana-2702	493	29	.	.	PUNCT
cana-2702	494	1	3s	3s	NUM
cana-2702	494	2	(	(	PUNCT
cana-2702	494	3	2025	2025	NUM
cana-2702	494	4	)	)	PUNCT
cana-2702	494	5	658	658	NUM
cana-2702	494	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	494	7	claim(1	claim(1	NOUN
cana-2702	494	8	)	)	PUNCT
cana-2702	495	1	𝑝(𝑋2	𝑝(𝑋2	VERB
cana-2702	495	2	−	−	NOUN
cana-2702	495	3	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	495	4	+	+	NOUN
cana-2702	495	5	1	1	NUM
cana-2702	495	6	)	)	PUNCT
cana-2702	495	7	≥	≥	NOUN
cana-2702	496	1	2(2𝑚1(2𝑚2−1	2(2𝑚1(2𝑚2−1	NUM
cana-2702	496	2	−	−	NOUN
cana-2702	496	3	1	1	NUM
cana-2702	496	4	)	)	PUNCT
cana-2702	496	5	+	+	CCONJ
cana-2702	496	6	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	496	7	−	−	PROPN
cana-2702	496	8	4	4	NUM
cana-2702	496	9	)	)	PUNCT
cana-2702	496	10	+	+	CCONJ
cana-2702	496	11	2	2	X
cana-2702	496	12	)	)	PUNCT
cana-2702	496	13	−	−	NOUN
cana-2702	496	14	𝑞	𝑞	X
cana-2702	496	15	+	+	NOUN
cana-2702	496	16	1	1	X
cana-2702	496	17	.	.	X
cana-2702	496	18	we	we	PRON
cana-2702	496	19	have	have	VERB
cana-2702	496	20	𝑝(𝑋2	𝑝(𝑋2	VERB
cana-2702	496	21	−	−	NOUN
cana-2702	496	22	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	496	23	+	+	NOUN
cana-2702	496	24	1	1	NUM
cana-2702	496	25	)	)	PUNCT
cana-2702	496	26	=	=	SYM
cana-2702	497	1	2(2𝑚1(2𝑚2	2(2𝑚1(2𝑚2	NUM
cana-2702	497	2	−	−	NUM
cana-2702	497	3	1	1	NUM
cana-2702	497	4	)	)	PUNCT
cana-2702	497	5	+	+	CCONJ
cana-2702	497	6	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	497	7	−	−	PROPN
cana-2702	497	8	4	4	NUM
cana-2702	497	9	)	)	PUNCT
cana-2702	497	10	+	+	CCONJ
cana-2702	497	11	2	2	X
cana-2702	497	12	)	)	PUNCT
cana-2702	497	13	−	−	NOUN
cana-2702	497	14	𝑞	𝑞	X
cana-2702	497	15	+	+	NOUN
cana-2702	497	16	1	1	NUM
cana-2702	497	17	−	−	PROPN
cana-2702	497	18	2𝑛	2𝑛	PROPN
cana-2702	497	19	+	+	CCONJ
cana-2702	497	20	𝑞𝑚1	𝑞𝑚1	PROPN
cana-2702	497	21	+	+	ADJ
cana-2702	497	22	1	1	NUM
cana-2702	497	23	−	−	SYM
cana-2702	497	24	1	1	NUM
cana-2702	497	25	=	=	SYM
cana-2702	497	26	2	2	NUM
cana-2702	497	27	(	(	PUNCT
cana-2702	497	28	2𝑚1(2𝑚2−1	2𝑚1(2𝑚2−1	NUM
cana-2702	497	29	−	−	NOUN
cana-2702	497	30	1	1	NUM
cana-2702	497	31	)	)	PUNCT
cana-2702	497	32	+	+	CCONJ
cana-2702	497	33	2	2	NUM
cana-2702	497	34	𝑚1(2𝑚2−1)+(𝑚2−1)(𝑛−4)+2	𝑚1(2𝑚2−1)+(𝑚2−1)(𝑛−4)+2	NUM
cana-2702	497	35	)	)	PUNCT
cana-2702	497	36	−	−	ADP
cana-2702	497	37	𝑞	𝑞	X
cana-2702	497	38	+	+	NOUN
cana-2702	497	39	1	1	NUM
cana-2702	497	40	−	−	PROPN
cana-2702	497	41	2𝑛	2𝑛	PROPN
cana-2702	497	42	−	−	PROPN
cana-2702	497	43	𝑞𝑚1	𝑞𝑚1	PROPN
cana-2702	498	1	+	+	PROPN
cana-2702	498	2	1	1	NUM
cana-2702	498	3	−	−	SYM
cana-2702	498	4	1	1	NUM
cana-2702	498	5	=	=	SYM
cana-2702	498	6	2(2𝑚1(2𝑚2	2(2𝑚1(2𝑚2	NUM
cana-2702	498	7	−	−	NOUN
cana-2702	498	8	1	1	NUM
cana-2702	498	9	)	)	PUNCT
cana-2702	499	1	+	+	CCONJ
cana-2702	499	2	(	(	PUNCT
cana-2702	499	3	𝑚2	𝑚2	NOUN
cana-2702	499	4	−	−	NOUN
cana-2702	499	5	1)(𝑛	1)(𝑛	NUM
cana-2702	499	6	−	−	NOUN
cana-2702	499	7	4	4	NUM
cana-2702	499	8	)	)	PUNCT
cana-2702	499	9	+	+	CCONJ
cana-2702	499	10	2	2	X
cana-2702	499	11	)	)	PUNCT
cana-2702	499	12	−	−	NOUN
cana-2702	499	13	𝑞	𝑞	X
cana-2702	499	14	+	+	NOUN
cana-2702	499	15	1	1	NUM
cana-2702	499	16	+	+	CCONJ
cana-2702	499	17	𝑞𝑚1	𝑞𝑚1	NOUN
cana-2702	499	18	+	+	ADJ
cana-2702	499	19	1	1	NUM
cana-2702	499	20	+	+	SYM
cana-2702	499	21	2	2	NUM
cana-2702	499	22	𝑚1+𝑚2	𝑚1+𝑚2	PROPN
cana-2702	499	23	−	−	NOUN
cana-2702	499	24	𝑞	𝑞	X
cana-2702	499	25	>	>	X
cana-2702	499	26	2(2𝑚1(2𝑚2−1	2(2𝑚1(2𝑚2−1	NUM
cana-2702	499	27	−	−	NUM
cana-2702	499	28	1	1	NUM
cana-2702	499	29	)	)	PUNCT
cana-2702	499	30	+	+	CCONJ
cana-2702	499	31	(	(	PUNCT
cana-2702	499	32	𝑚2	𝑚2	NOUN
cana-2702	499	33	−	−	NOUN
cana-2702	500	1	1)(𝑛	1)(𝑛	NUM
cana-2702	501	1	−	−	NOUN
cana-2702	501	2	4	4	NUM
cana-2702	501	3	)	)	PUNCT
cana-2702	501	4	+	+	CCONJ
cana-2702	501	5	2	2	X
cana-2702	501	6	)	)	PUNCT
cana-2702	501	7	−	−	NOUN
cana-2702	501	8	𝑞	𝑞	PROPN
cana-2702	501	9	+	+	NOUN
cana-2702	501	10	1	1	NUM
cana-2702	501	11	,	,	PUNCT
cana-2702	501	12	since	since	SCONJ
cana-2702	501	13	𝑚1	𝑚1	NOUN
cana-2702	501	14	+	+	CCONJ
cana-2702	501	15	𝑚2	𝑚2	NOUN
cana-2702	501	16	=	=	SYM
cana-2702	501	17	𝑛	𝑛	PRON
cana-2702	501	18	≥	≥	NUM
cana-2702	501	19	4	4	NUM
cana-2702	502	1	and	and	CCONJ
cana-2702	502	2	we	we	PRON
cana-2702	502	3	get	get	VERB
cana-2702	502	4	the	the	DET
cana-2702	502	5	term	term	NOUN
cana-2702	502	6	𝑞𝑚1	𝑞𝑚1	PROPN
cana-2702	503	1	+	+	ADJ
cana-2702	503	2	1	1	NUM
cana-2702	503	3	+	+	SYM
cana-2702	503	4	2	2	NUM
cana-2702	503	5	𝑚1+𝑚2	𝑚1+𝑚2	PROPN
cana-2702	503	6	−	−	PROPN
cana-2702	503	7	𝑞.	𝑞.	NOUN
cana-2702	503	8	we	we	PRON
cana-2702	503	9	can	can	AUX
cana-2702	503	10	move	move	VERB
cana-2702	503	11	four	four	NUM
cana-2702	503	12	pebbles	pebble	NOUN
cana-2702	503	13	to	to	PART
cana-2702	503	14	𝑎𝑚1	𝑎𝑚1	VERB
cana-2702	503	15	+	+	PROPN
cana-2702	503	16	1	1	NUM
cana-2702	503	17	(	(	PUNCT
cana-2702	503	18	𝑛	𝑛	PROPN
cana-2702	503	19	2	2	NUM
cana-2702	503	20	+	+	NUM
cana-2702	503	21	1	1	NUM
cana-2702	503	22	)	)	PUNCT
cana-2702	503	23	using	use	VERB
cana-2702	503	24	claim(1	claim(1	NOUN
cana-2702	503	25	)	)	PUNCT
cana-2702	503	26	.	.	PUNCT
cana-2702	504	1	using	use	VERB
cana-2702	504	2	induction	induction	NOUN
cana-2702	504	3	on	on	ADP
cana-2702	504	4	𝑚2	𝑚2	NOUN
cana-2702	504	5	,	,	PUNCT
cana-2702	504	6	two	two	NUM
cana-2702	504	7	pebbles	pebble	NOUN
cana-2702	504	8	may	may	AUX
cana-2702	504	9	be	be	AUX
cana-2702	504	10	relocated	relocate	VERB
cana-2702	504	11	to	to	ADP
cana-2702	504	12	𝑎𝑚1	𝑎𝑚1	PROPN
cana-2702	504	13	(	(	PUNCT
cana-2702	504	14	𝑛	𝑛	PROPN
cana-2702	504	15	2	2	NUM
cana-2702	504	16	−	−	NOUN
cana-2702	504	17	1	1	NUM
cana-2702	504	18	)	)	PUNCT
cana-2702	504	19	.	.	PUNCT
cana-2702	505	1	case	case	NOUN
cana-2702	505	2	(	(	PUNCT
cana-2702	505	3	2	2	NUM
cana-2702	505	4	)	)	PUNCT
cana-2702	505	5	𝑝(𝐶𝑚1	𝑝(𝐶𝑚1	PROPN
cana-2702	505	6	+	+	PROPN
cana-2702	505	7	1	1	NUM
cana-2702	505	8	)	)	PUNCT
cana-2702	505	9	<	<	X
cana-2702	505	10	2𝑛	2𝑛	PROPN
cana-2702	505	11	−	−	PROPN
cana-2702	505	12	𝑞𝑚1	𝑞𝑚1	PROPN
cana-2702	505	13	+	+	PROPN
cana-2702	505	14	1	1	NUM
cana-2702	505	15	.	.	PUNCT
cana-2702	506	1	if	if	SCONJ
cana-2702	506	2	at	at	ADV
cana-2702	506	3	least	least	ADV
cana-2702	506	4	two	two	NUM
cana-2702	506	5	pebbles	pebble	NOUN
cana-2702	506	6	are	be	AUX
cana-2702	506	7	present	present	ADJ
cana-2702	506	8	at	at	ADP
cana-2702	506	9	the	the	DET
cana-2702	506	10	two	two	NUM
cana-2702	506	11	vertices	vertex	NOUN
cana-2702	506	12	in	in	ADP
cana-2702	506	13	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	506	14	+	+	NOUN
cana-2702	506	15	1	1	NUM
cana-2702	506	16	,	,	PUNCT
cana-2702	506	17	they	they	PRON
cana-2702	506	18	can	can	AUX
cana-2702	506	19	be	be	AUX
cana-2702	506	20	transferred	transfer	VERB
cana-2702	506	21	to	to	ADP
cana-2702	506	22	𝑎	𝑎	PROPN
cana-2702	506	23	𝑚1	𝑚1	NOUN
cana-2702	506	24	(	(	PUNCT
cana-2702	506	25	𝑛	𝑛	PROPN
cana-2702	506	26	2	2	NUM
cana-2702	506	27	−1	−1	NOUN
cana-2702	506	28	)	)	PUNCT
cana-2702	506	29	.	.	PUNCT
cana-2702	507	1	we	we	PRON
cana-2702	507	2	can	can	AUX
cana-2702	507	3	move	move	VERB
cana-2702	507	4	two	two	NUM
cana-2702	507	5	additional	additional	ADJ
cana-2702	507	6	pebbles	pebble	NOUN
cana-2702	507	7	to	to	ADP
cana-2702	507	8	𝑎𝑚1	𝑎𝑚1	PROPN
cana-2702	507	9	(	(	PUNCT
cana-2702	507	10	𝑛	𝑛	PROPN
cana-2702	507	11	2	2	NUM
cana-2702	507	12	−	−	NOUN
cana-2702	507	13	1	1	NUM
cana-2702	507	14	)	)	PUNCT
cana-2702	507	15	using	use	VERB
cana-2702	507	16	claim(1	claim(1	NOUN
cana-2702	507	17	)	)	PUNCT
cana-2702	507	18	.	.	PUNCT
cana-2702	508	1	assume	assume	VERB
cana-2702	508	2	,	,	PUNCT
cana-2702	508	3	for	for	ADP
cana-2702	508	4	the	the	DET
cana-2702	508	5	sake	sake	NOUN
cana-2702	508	6	of	of	ADP
cana-2702	508	7	argument	argument	NOUN
cana-2702	508	8	,	,	PUNCT
cana-2702	508	9	that	that	SCONJ
cana-2702	508	10	no	no	DET
cana-2702	508	11	two	two	NUM
cana-2702	508	12	pebbles	pebble	NOUN
cana-2702	508	13	in	in	ADP
cana-2702	508	14	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	508	15	+	+	NOUN
cana-2702	508	16	1	1	NUM
cana-2702	508	17	have	have	VERB
cana-2702	508	18	two	two	NUM
cana-2702	508	19	pebbles	pebble	NOUN
cana-2702	508	20	on	on	ADP
cana-2702	508	21	each	each	PRON
cana-2702	508	22	.	.	PUNCT
cana-2702	509	1	we	we	PRON
cana-2702	509	2	investigate	investigate	VERB
cana-2702	509	3	the	the	DET
cana-2702	509	4	simplest	simple	ADJ
cana-2702	509	5	example	example	NOUN
cana-2702	509	6	,	,	PUNCT
cana-2702	509	7	in	in	ADP
cana-2702	509	8	which	which	PRON
cana-2702	509	9	the	the	DET
cana-2702	509	10	majority	majority	NOUN
cana-2702	509	11	of	of	ADP
cana-2702	509	12	the	the	DET
cana-2702	509	13	pebbles	pebble	NOUN
cana-2702	509	14	are	be	AUX
cana-2702	509	15	placed	place	VERB
cana-2702	509	16	on	on	ADP
cana-2702	509	17	the	the	DET
cana-2702	509	18	vertices	vertex	NOUN
cana-2702	509	19	of	of	ADP
cana-2702	509	20	𝐶1	𝐶1	PRON
cana-2702	509	21	.	.	PUNCT
cana-2702	510	1	𝑦	𝑦	X
cana-2702	510	2	,	,	PUNCT
cana-2702	510	3	in	in	ADP
cana-2702	510	4	particular	particular	ADJ
cana-2702	510	5	,	,	PUNCT
cana-2702	510	6	had	have	VERB
cana-2702	510	7	a	a	DET
cana-2702	510	8	higher	high	ADJ
cana-2702	510	9	amount	amount	NOUN
cana-2702	510	10	of	of	ADP
cana-2702	510	11	pebbles	pebble	NOUN
cana-2702	510	12	.	.	PUNCT
cana-2702	511	1	all	all	DET
cana-2702	511	2	other	other	ADJ
cana-2702	511	3	𝑎𝑖	𝑎𝑖	ADV
cana-2702	511	4	and	and	CCONJ
cana-2702	511	5	𝑏𝑖	𝑏𝑖	PART
cana-2702	511	6	have	have	VERB
cana-2702	511	7	no	no	PRON
cana-2702	511	8	more	more	ADJ
cana-2702	511	9	than	than	ADP
cana-2702	511	10	one	one	NUM
cana-2702	511	11	pebble	pebble	NOUN
cana-2702	511	12	.	.	PUNCT
cana-2702	512	1	so	so	ADV
cana-2702	512	2	the	the	DET
cana-2702	512	3	vertex	vertex	NOUN
cana-2702	512	4	𝑦	𝑦	NOUN
cana-2702	512	5	has	have	VERB
cana-2702	512	6	2(2𝑚1(2𝑚2	2(2𝑚1(2𝑚2	NUM
cana-2702	512	7	−	−	NUM
cana-2702	512	8	1	1	NUM
cana-2702	512	9	)	)	PUNCT
cana-2702	513	1	+	+	CCONJ
cana-2702	513	2	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	513	3	−	−	NOUN
cana-2702	513	4	4	4	NUM
cana-2702	513	5	)	)	PUNCT
cana-2702	513	6	)	)	PUNCT
cana-2702	514	1	−	−	PROPN
cana-2702	515	1	𝑞	𝑞	X
cana-2702	515	2	+	+	NOUN
cana-2702	515	3	1	1	NUM
cana-2702	515	4	−	−	NOUN
cana-2702	515	5	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	515	6	−	−	PROPN
cana-2702	515	7	4	4	NUM
cana-2702	515	8	)	)	PUNCT
cana-2702	515	9	≥	≥	NOUN
cana-2702	515	10	4(2𝑚2	4(2𝑚2	NUM
cana-2702	515	11	)	)	PUNCT
cana-2702	515	12	pebbles	pebble	NOUN
cana-2702	515	13	,	,	PUNCT
cana-2702	515	14	since	since	SCONJ
cana-2702	515	15	𝑚1	𝑚1	NOUN
cana-2702	515	16	=	=	SYM
cana-2702	515	17	1	1	X
cana-2702	515	18	.	.	PUNCT
cana-2702	516	1	this	this	PRON
cana-2702	516	2	means	mean	VERB
cana-2702	516	3	that	that	SCONJ
cana-2702	516	4	at	at	ADV
cana-2702	516	5	least	least	ADJ
cana-2702	516	6	4(2𝑚2	4(2𝑚2	NUM
cana-2702	516	7	)	)	PUNCT
cana-2702	516	8	pebbles	pebble	NOUN
cana-2702	516	9	were	be	AUX
cana-2702	516	10	kept	keep	VERB
cana-2702	516	11	in	in	ADP
cana-2702	516	12	𝑦.	𝑦.	PROPN
cana-2702	516	13	as	as	ADP
cana-2702	516	14	a	a	DET
cana-2702	516	15	result	result	NOUN
cana-2702	516	16	,	,	PUNCT
cana-2702	516	17	the	the	DET
cana-2702	516	18	four	four	NUM
cana-2702	516	19	pebbles	pebble	NOUN
cana-2702	516	20	can	can	AUX
cana-2702	516	21	shift	shift	VERB
cana-2702	516	22	to	to	ADP
cana-2702	516	23	𝑎𝑚1	𝑎𝑚1	PROPN
cana-2702	516	24	(	(	PUNCT
cana-2702	516	25	𝑛	𝑛	PROPN
cana-2702	516	26	2	2	NUM
cana-2702	516	27	−	−	NUM
cana-2702	516	28	1	1	NUM
cana-2702	516	29	)	)	PUNCT
cana-2702	516	30	.	.	PUNCT
cana-2702	517	1	theorem	theorem	VERB
cana-2702	517	2	9	9	NUM
cana-2702	517	3	.	.	PUNCT
cana-2702	518	1	the	the	DET
cana-2702	518	2	graph	graph	NOUN
cana-2702	518	3	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	518	4	)	)	PUNCT
cana-2702	518	5	satisfies	satisfy	VERB
cana-2702	518	6	two	two	NUM
cana-2702	518	7	-	-	PUNCT
cana-2702	518	8	pebbling	pebble	VERB
cana-2702	518	9	property	property	NOUN
cana-2702	518	10	.	.	PUNCT
cana-2702	519	1	proof	proof	NOUN
cana-2702	519	2	.	.	PUNCT
cana-2702	520	1	consider	consider	VERB
cana-2702	520	2	a	a	DET
cana-2702	520	3	graph	graph	NOUN
cana-2702	520	4	that	that	PRON
cana-2702	520	5	has	have	VERB
cana-2702	520	6	at	at	ADV
cana-2702	520	7	least	least	ADJ
cana-2702	520	8	2	2	NUM
cana-2702	520	9	𝑚	𝑚	NOUN
cana-2702	520	10	+	+	NOUN
cana-2702	520	11	2(𝑛	2(𝑛	NUM
cana-2702	520	12	−	−	NOUN
cana-2702	520	13	3	3	NUM
cana-2702	520	14	)	)	PUNCT
cana-2702	520	15	+	+	CCONJ
cana-2702	521	1	(	(	PUNCT
cana-2702	521	2	𝑚	𝑚	PROPN
cana-2702	521	3	−	−	PROPN
cana-2702	521	4	2)(𝑛	2)(𝑛	NUM
cana-2702	521	5	−	−	NOUN
cana-2702	521	6	4	4	NUM
cana-2702	521	7	)	)	PUNCT
cana-2702	521	8	−	−	ADP
cana-2702	522	1	𝑞	𝑞	X
cana-2702	522	2	+	+	NOUN
cana-2702	522	3	1	1	NUM
cana-2702	522	4	pebbles	pebble	NOUN
cana-2702	522	5	scattered	scatter	VERB
cana-2702	522	6	across	across	ADP
cana-2702	522	7	its	its	PRON
cana-2702	522	8	vertices	vertex	NOUN
cana-2702	522	9	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	522	10	)	)	PUNCT
cana-2702	522	11	.	.	PUNCT
cana-2702	523	1	the	the	DET
cana-2702	523	2	remaining	remain	VERB
cana-2702	523	3	two	two	NUM
cana-2702	523	4	pebbles	pebble	NOUN
cana-2702	523	5	must	must	AUX
cana-2702	523	6	be	be	AUX
cana-2702	523	7	moved	move	VERB
cana-2702	523	8	to	to	ADP
cana-2702	523	9	any	any	DET
cana-2702	523	10	desired	desire	VERB
cana-2702	523	11	vertex	vertex	NOUN
cana-2702	523	12	.	.	PUNCT
cana-2702	524	1	for	for	ADP
cana-2702	524	2	1	1	NUM
cana-2702	524	3	≤	≤	NUM
cana-2702	524	4	𝑖	𝑖	SYM
cana-2702	524	5	≤	≤	NOUN
cana-2702	524	6	𝑚	𝑚	ADP
cana-2702	524	7	,	,	PUNCT
cana-2702	524	8	let	let	VERB
cana-2702	524	9	𝑣	𝑣	PART
cana-2702	524	10	∈	∈	PROPN
cana-2702	524	11	𝐶𝑖.	𝐶𝑖.	PROPN
cana-2702	524	12	we	we	PRON
cana-2702	524	13	demonstrate	demonstrate	VERB
cana-2702	524	14	this	this	DET
cana-2702	524	15	result	result	NOUN
cana-2702	524	16	using	use	VERB
cana-2702	524	17	induction	induction	NOUN
cana-2702	524	18	on	on	ADP
cana-2702	524	19	𝑚.	𝑚.	ADJ
cana-2702	524	20	theorems	theorem	NOUN
cana-2702	524	21	7	7	NUM
cana-2702	524	22	and	and	CCONJ
cana-2702	524	23	8	8	NUM
cana-2702	524	24	provide	provide	VERB
cana-2702	524	25	the	the	DET
cana-2702	524	26	following	follow	VERB
cana-2702	524	27	findings	finding	NOUN
cana-2702	524	28	for	for	ADP
cana-2702	524	29	𝑚	𝑚	NOUN
cana-2702	524	30	=	=	SYM
cana-2702	524	31	2	2	NUM
cana-2702	524	32	and	and	CCONJ
cana-2702	524	33	𝑚	𝑚	X
cana-2702	524	34	=	=	SYM
cana-2702	524	35	3	3	NUM
cana-2702	524	36	,	,	PUNCT
cana-2702	524	37	respectively	respectively	ADV
cana-2702	524	38	.	.	PUNCT
cana-2702	525	1	we	we	PRON
cana-2702	525	2	suppose	suppose	VERB
cana-2702	525	3	the	the	DET
cana-2702	525	4	result	result	NOUN
cana-2702	525	5	holds	hold	VERB
cana-2702	525	6	true	true	ADJ
cana-2702	525	7	for	for	ADP
cana-2702	525	8	every	every	DET
cana-2702	525	9	𝑚′	𝑚′	NUM
cana-2702	525	10	<	<	X
cana-2702	525	11	𝑚.	𝑚.	NOUN
cana-2702	525	12	we	we	PRON
cana-2702	525	13	demonstrate	demonstrate	VERB
cana-2702	525	14	the	the	DET
cana-2702	525	15	findings	finding	NOUN
cana-2702	525	16	for	for	ADP
cana-2702	525	17	all	all	DET
cana-2702	525	18	𝑚.	𝑚.	NOUN
cana-2702	525	19	we	we	PRON
cana-2702	525	20	have	have	VERB
cana-2702	525	21	the	the	DET
cana-2702	525	22	following	follow	VERB
cana-2702	525	23	situations	situation	NOUN
cana-2702	525	24	:	:	PUNCT
cana-2702	525	25	case	case	NOUN
cana-2702	525	26	(	(	PUNCT
cana-2702	525	27	1	1	X
cana-2702	525	28	)	)	PUNCT
cana-2702	525	29	𝑣	𝑣	PRON
cana-2702	525	30	∈	∈	PROPN
cana-2702	525	31	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	525	32	for	for	ADP
cana-2702	525	33	2	2	NUM
cana-2702	525	34	<	<	X
cana-2702	525	35	𝑚1	𝑚1	X
cana-2702	525	36	<	<	X
cana-2702	525	37	𝑚.	𝑚.	PROPN
cana-2702	525	38	as	as	ADP
cana-2702	525	39	the	the	DET
cana-2702	525	40	two	two	NUM
cana-2702	525	41	subgraphs	subgraph	NOUN
cana-2702	525	42	of	of	ADP
cana-2702	525	43	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	525	44	)	)	PUNCT
cana-2702	525	45	,	,	PUNCT
cana-2702	525	46	let	let	VERB
cana-2702	525	47	us	we	PRON
cana-2702	525	48	define	define	VERB
cana-2702	525	49	𝑊	𝑊	NOUN
cana-2702	525	50	=	=	SYM
cana-2702	525	51	𝐶1	𝐶1	NOUN
cana-2702	525	52	∪.	∪.	X
cana-2702	525	53	.	.	PUNCT
cana-2702	526	1	.∪	.∪	ADV
cana-2702	526	2	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	526	3	and	and	CCONJ
cana-2702	526	4	𝑍	𝑍	PROPN
cana-2702	526	5	=	=	SYM
cana-2702	526	6	𝐶𝑚1	𝐶𝑚1	X
cana-2702	526	7	+	+	NOUN
cana-2702	526	8	1	1	NUM
cana-2702	526	9	∪.	∪.	NOUN
cana-2702	526	10	.	.	PUNCT
cana-2702	527	1	.∪	.∪	PROPN
cana-2702	527	2	𝐶𝑚.	𝐶𝑚.	PROPN
cana-2702	527	3	assume	assume	VERB
cana-2702	527	4	𝑝(𝑊	𝑝(𝑊	NUM
cana-2702	527	5	)	)	PUNCT
cana-2702	527	6	≥	≥	NOUN
cana-2702	527	7	2(2𝑚1	2(2𝑚1	NUM
cana-2702	528	1	+	+	CCONJ
cana-2702	528	2	2(𝑛	2(𝑛	NUM
cana-2702	528	3	−	−	NOUN
cana-2702	528	4	3	3	NUM
cana-2702	528	5	)	)	PUNCT
cana-2702	528	6	+	+	CCONJ
cana-2702	528	7	(	(	PUNCT
cana-2702	528	8	𝑚1	𝑚1	NOUN
cana-2702	528	9	−	−	PROPN
cana-2702	528	10	2)(𝑛	2)(𝑛	NUM
cana-2702	528	11	−	−	NOUN
cana-2702	528	12	4	4	NUM
cana-2702	528	13	)	)	PUNCT
cana-2702	528	14	)	)	PUNCT
cana-2702	529	1	−	−	PROPN
cana-2702	530	1	𝑞1	𝑞1	ADJ
cana-2702	530	2	+	+	CCONJ
cana-2702	530	3	1	1	X
cana-2702	530	4	.	.	PUNCT
cana-2702	531	1	the	the	DET
cana-2702	531	2	two	two	NUM
cana-2702	531	3	pebbles	pebble	NOUN
cana-2702	531	4	can	can	AUX
cana-2702	531	5	then	then	ADV
cana-2702	531	6	be	be	AUX
cana-2702	531	7	inducted	induct	VERB
cana-2702	531	8	to	to	ADP
cana-2702	531	9	𝑣.	𝑣.	NOUN
cana-2702	531	10	assume	assume	PROPN
cana-2702	531	11	𝑝(𝑊	𝑝(𝑊	PROPN
cana-2702	531	12	)	)	PUNCT
cana-2702	531	13	<	<	X
cana-2702	531	14	2(2𝑚1	2(2𝑚1	NUM
cana-2702	531	15	+	+	CCONJ
cana-2702	531	16	2(𝑛	2(𝑛	NUM
cana-2702	531	17	−	−	NOUN
cana-2702	531	18	3	3	NUM
cana-2702	531	19	)	)	PUNCT
cana-2702	532	1	+	+	CCONJ
cana-2702	532	2	(	(	PUNCT
cana-2702	532	3	𝑚1	𝑚1	NOUN
cana-2702	532	4	−	−	PROPN
cana-2702	532	5	2)(𝑛	2)(𝑛	NUM
cana-2702	532	6	−	−	NOUN
cana-2702	532	7	4	4	NUM
cana-2702	532	8	)	)	PUNCT
cana-2702	532	9	−	−	PROPN
cana-2702	532	10	𝑞1	𝑞1	ADJ
cana-2702	532	11	)	)	PUNCT
cana-2702	532	12	.	.	PUNCT
cana-2702	533	1	if	if	SCONJ
cana-2702	533	2	𝑝(𝑊	𝑝(𝑊	PROPN
cana-2702	533	3	−	−	PROPN
cana-2702	533	4	𝐶𝑚1	𝐶𝑚1	ADJ
cana-2702	533	5	)	)	PUNCT
cana-2702	533	6	≥	≥	PROPN
cana-2702	533	7	2	2	NUM
cana-2702	533	8	(	(	PUNCT
cana-2702	533	9	2𝑚1−1(21	2𝑚1−1(21	NUM
cana-2702	533	10	−	−	NOUN
cana-2702	533	11	1	1	NUM
cana-2702	533	12	)	)	PUNCT
cana-2702	533	13	+	+	CCONJ
cana-2702	533	14	(	(	PUNCT
cana-2702	533	15	𝑛	𝑛	DET
cana-2702	533	16	−	−	PROPN
cana-2702	533	17	4	4	NUM
cana-2702	533	18	+	+	NOUN
cana-2702	533	19	2	2	NUM
cana-2702	533	20	)	)	PUNCT
cana-2702	533	21	)	)	PUNCT
cana-2702	533	22	−	−	PROPN
cana-2702	534	1	𝑞1	𝑞1	ADJ
cana-2702	534	2	+	+	CCONJ
cana-2702	534	3	1	1	NUM
cana-2702	534	4	,	,	PUNCT
cana-2702	534	5	we	we	PRON
cana-2702	534	6	can	can	AUX
cana-2702	534	7	shift	shift	VERB
cana-2702	534	8	four	four	NUM
cana-2702	534	9	pebbles	pebble	NOUN
cana-2702	534	10	to	to	ADP
cana-2702	534	11	𝑎	𝑎	PROPN
cana-2702	534	12	(	(	PUNCT
cana-2702	534	13	𝑚1−1	𝑚1−1	NUM
cana-2702	534	14	)	)	PUNCT
cana-2702	534	15	(	(	PUNCT
cana-2702	534	16	𝑛	𝑛	PROPN
cana-2702	534	17	2	2	NUM
cana-2702	534	18	−1	−1	NOUN
cana-2702	534	19	)	)	PUNCT
cana-2702	534	20	or	or	CCONJ
cana-2702	534	21	𝑎	𝑎	X
cana-2702	534	22	(	(	PUNCT
cana-2702	534	23	𝑚1−1	𝑚1−1	NUM
cana-2702	534	24	)	)	PUNCT
cana-2702	534	25	(	(	PUNCT
cana-2702	534	26	𝑛	𝑛	DET
cana-2702	534	27	2	2	NUM
cana-2702	534	28	−1)+1	−1)+1	NOUN
cana-2702	534	29	and	and	CCONJ
cana-2702	534	30	then	then	ADV
cana-2702	534	31	two	two	NUM
cana-2702	534	32	pebbles	pebble	NOUN
cana-2702	534	33	to	to	ADP
cana-2702	534	34	𝑣	𝑣	ADP
cana-2702	534	35	using	use	VERB
cana-2702	534	36	lemma	lemma	PROPN
cana-2702	534	37	2	2	NUM
cana-2702	534	38	.	.	PUNCT
cana-2702	535	1	so	so	ADV
cana-2702	535	2	suppose	suppose	VERB
cana-2702	535	3	𝑝(𝑊	𝑝(𝑊	X
cana-2702	535	4	−	−	NOUN
cana-2702	535	5	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	535	6	)	)	PUNCT
cana-2702	535	7	<	<	X
cana-2702	535	8	2(2𝑚1−1(21	2(2𝑚1−1(21	NOUN
cana-2702	535	9	−	−	NOUN
cana-2702	535	10	1	1	NUM
cana-2702	535	11	)	)	PUNCT
cana-2702	535	12	−	−	PROPN
cana-2702	536	1	(	(	PUNCT
cana-2702	536	2	𝑛	𝑛	PRON
cana-2702	536	3	−	−	NOUN
cana-2702	536	4	4	4	NUM
cana-2702	536	5	)	)	PUNCT
cana-2702	536	6	+	+	CCONJ
cana-2702	536	7	2	2	X
cana-2702	536	8	)	)	PUNCT
cana-2702	536	9	−	−	PROPN
cana-2702	537	1	𝑞1	𝑞1	ADJ
cana-2702	537	2	+	+	CCONJ
cana-2702	537	3	1	1	X
cana-2702	537	4	.	.	PUNCT
cana-2702	538	1	this	this	PRON
cana-2702	538	2	means	mean	VERB
cana-2702	538	3	that	that	SCONJ
cana-2702	538	4	there	there	PRON
cana-2702	538	5	were	be	VERB
cana-2702	538	6	at	at	ADP
cana-2702	538	7	least	least	ADJ
cana-2702	538	8	2(2𝑚	2(2𝑚	NUM
cana-2702	538	9	+	+	CCONJ
cana-2702	538	10	2(𝑛	2(𝑛	NUM
cana-2702	538	11	−	−	NOUN
cana-2702	538	12	3	3	NUM
cana-2702	538	13	)	)	PUNCT
cana-2702	538	14	+	+	CCONJ
cana-2702	538	15	(	(	PUNCT
cana-2702	538	16	𝑚	𝑚	PROPN
cana-2702	538	17	−	−	PROPN
cana-2702	538	18	2(𝑛	2(𝑛	NUM
cana-2702	538	19	−	−	NOUN
cana-2702	538	20	4	4	NUM
cana-2702	538	21	)	)	PUNCT
cana-2702	538	22	)	)	PUNCT
cana-2702	539	1	+	+	CCONJ
cana-2702	539	2	2	2	X
cana-2702	539	3	)	)	PUNCT
cana-2702	539	4	−	−	NOUN
cana-2702	539	5	𝑞2	𝑞2	NOUN
cana-2702	539	6	+	+	CCONJ
cana-2702	539	7	1	1	NUM
cana-2702	539	8	pebbles	pebble	NOUN
cana-2702	539	9	dispersed	disperse	VERB
cana-2702	539	10	on	on	ADP
cana-2702	539	11	𝑍.	𝑍.	PROPN
cana-2702	539	12	we	we	PRON
cana-2702	539	13	can	can	AUX
cana-2702	539	14	shift	shift	VERB
cana-2702	539	15	four	four	NUM
cana-2702	539	16	pebbles	pebble	NOUN
cana-2702	539	17	using	use	VERB
cana-2702	539	18	lemma	lemma	PROPN
cana-2702	539	19	2	2	NUM
cana-2702	539	20	to	to	ADP
cana-2702	539	21	𝑎	𝑎	PROPN
cana-2702	539	22	𝑚1	𝑚1	NOUN
cana-2702	539	23	(	(	PUNCT
cana-2702	539	24	𝑛	𝑛	PROPN
cana-2702	539	25	2	2	NUM
cana-2702	539	26	−1	−1	NOUN
cana-2702	539	27	)	)	PUNCT
cana-2702	539	28	or	or	CCONJ
cana-2702	539	29	𝑎	𝑎	PRON
cana-2702	539	30	𝑚1	𝑚1	NOUN
cana-2702	539	31	(	(	PUNCT
cana-2702	539	32	𝑛	𝑛	DET
cana-2702	539	33	2	2	NUM
cana-2702	539	34	−1)+1	−1)+1	NOUN
cana-2702	539	35	.	.	PUNCT
cana-2702	540	1	the	the	DET
cana-2702	540	2	two	two	NUM
cana-2702	540	3	pebbles	pebble	NOUN
cana-2702	540	4	can	can	AUX
cana-2702	540	5	then	then	ADV
cana-2702	540	6	be	be	AUX
cana-2702	540	7	relocated	relocate	VERB
cana-2702	540	8	to	to	ADP
cana-2702	540	9	𝑣.	𝑣.	NOUN
cana-2702	540	10	case	case	NOUN
cana-2702	540	11	(	(	PUNCT
cana-2702	540	12	2	2	X
cana-2702	540	13	)	)	PUNCT
cana-2702	540	14	𝑣	𝑣	PART
cana-2702	540	15	∈	∈	NOUN
cana-2702	540	16	𝐶1	𝐶1	NOUN
cana-2702	540	17	or	or	CCONJ
cana-2702	540	18	𝑣	𝑣	PRON
cana-2702	540	19	∈	∈	NOUN
cana-2702	540	20	𝐶𝑚.	𝐶𝑚.	NOUN
cana-2702	540	21	communications	communication	NOUN
cana-2702	540	22	on	on	ADP
cana-2702	540	23	applied	apply	VERB
cana-2702	540	24	nonlinear	nonlinear	ADJ
cana-2702	540	25	analysis	analysis	NOUN
cana-2702	540	26	issn	issn	NOUN
cana-2702	540	27	:	:	PUNCT
cana-2702	540	28	1074	1074	NUM
cana-2702	540	29	-	-	PUNCT
cana-2702	540	30	133x	133x	NUM
cana-2702	540	31	vol	vol	NOUN
cana-2702	540	32	32	32	NUM
cana-2702	540	33	no	no	NOUN
cana-2702	540	34	.	.	PUNCT
cana-2702	541	1	3s	3s	NUM
cana-2702	541	2	(	(	PUNCT
cana-2702	541	3	2025	2025	NUM
cana-2702	541	4	)	)	PUNCT
cana-2702	541	5	659	659	NUM
cana-2702	541	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	541	7	we	we	PRON
cana-2702	541	8	assume	assume	VERB
cana-2702	541	9	that	that	SCONJ
cana-2702	541	10	𝑣	𝑣	PRON
cana-2702	541	11	∈	∈	PROPN
cana-2702	541	12	𝐶𝑚	𝐶𝑚	PROPN
cana-2702	541	13	and	and	CCONJ
cana-2702	541	14	𝑣	𝑣	X
cana-2702	541	15	=	=	PUNCT
cana-2702	541	16	𝑦	𝑦	PROPN
cana-2702	541	17	without	without	ADP
cana-2702	541	18	losing	lose	VERB
cana-2702	541	19	generality	generality	NOUN
cana-2702	541	20	.	.	PUNCT
cana-2702	542	1	also	also	ADV
cana-2702	542	2	,	,	PUNCT
cana-2702	542	3	𝑝(𝐶𝑚	𝑝(𝐶𝑚	NOUN
cana-2702	542	4	)	)	PUNCT
cana-2702	542	5	≤	≤	NOUN
cana-2702	542	6	𝑛	𝑛	DET
cana-2702	542	7	+	+	NOUN
cana-2702	542	8	1	1	X
cana-2702	542	9	.	.	X
cana-2702	543	1	we	we	PRON
cana-2702	543	2	assume	assume	VERB
cana-2702	543	3	𝑊	𝑊	PROPN
cana-2702	543	4	=	=	SYM
cana-2702	543	5	𝐶𝑚	𝐶𝑚	PROPN
cana-2702	543	6	≅	≅	PROPN
cana-2702	543	7	𝐶1(𝐾𝑛	𝐶1(𝐾𝑛	PROPN
cana-2702	543	8	)	)	PUNCT
cana-2702	543	9	and	and	CCONJ
cana-2702	543	10	𝑍	𝑍	PROPN
cana-2702	543	11	=	=	SYM
cana-2702	543	12	𝐶1	𝐶1	X
cana-2702	543	13	∪.	∪.	X
cana-2702	543	14	.	.	PUNCT
cana-2702	544	1	.∪	.∪	PROPN
cana-2702	544	2	𝐶𝑚−1	𝐶𝑚−1	PROPN
cana-2702	544	3	≅	≅	PROPN
cana-2702	544	4	𝐶𝑚−1(𝐾𝑛	𝐶𝑚−1(𝐾𝑛	NUM
cana-2702	544	5	)	)	PUNCT
cana-2702	544	6	.	.	PUNCT
cana-2702	545	1	claim(1	claim(1	NOUN
cana-2702	545	2	)	)	PUNCT
cana-2702	545	3	𝑝(𝑍	𝑝(𝑍	NUM
cana-2702	545	4	)	)	PUNCT
cana-2702	545	5	≥	≥	NOUN
cana-2702	545	6	2(2(2𝑚−1	2(2(2𝑚−1	NUM
cana-2702	545	7	−	−	NUM
cana-2702	545	8	1	1	NUM
cana-2702	545	9	)	)	PUNCT
cana-2702	545	10	+	+	CCONJ
cana-2702	545	11	(	(	PUNCT
cana-2702	545	12	𝑚	𝑚	PROPN
cana-2702	545	13	−	−	PROPN
cana-2702	545	14	1)(𝑛	1)(𝑛	NUM
cana-2702	545	15	−	−	NOUN
cana-2702	545	16	4	4	NUM
cana-2702	545	17	)	)	PUNCT
cana-2702	545	18	+	+	CCONJ
cana-2702	545	19	2	2	X
cana-2702	545	20	)	)	PUNCT
cana-2702	545	21	−	−	NOUN
cana-2702	545	22	𝑞2	𝑞2	NOUN
cana-2702	545	23	+	+	X
cana-2702	545	24	1	1	X
cana-2702	545	25	.	.	X
cana-2702	545	26	we	we	PRON
cana-2702	545	27	have	have	VERB
cana-2702	545	28	to	to	PART
cana-2702	545	29	prove	prove	VERB
cana-2702	545	30	𝑝(𝑍	𝑝(𝑍	ADJ
cana-2702	545	31	)	)	PUNCT
cana-2702	545	32	−	−	PUNCT
cana-2702	546	1	[	[	X
cana-2702	546	2	2(2(2𝑚−1	2(2(2𝑚−1	NUM
cana-2702	546	3	−	−	NUM
cana-2702	546	4	1	1	NUM
cana-2702	546	5	)	)	PUNCT
cana-2702	546	6	+	+	CCONJ
cana-2702	546	7	(	(	PUNCT
cana-2702	546	8	𝑚	𝑚	PROPN
cana-2702	546	9	−	−	PROPN
cana-2702	546	10	1)(𝑛	1)(𝑛	NUM
cana-2702	546	11	−	−	NOUN
cana-2702	546	12	4	4	NUM
cana-2702	546	13	)	)	PUNCT
cana-2702	546	14	+	+	CCONJ
cana-2702	546	15	2	2	X
cana-2702	546	16	)	)	PUNCT
cana-2702	546	17	−	−	NOUN
cana-2702	546	18	𝑞2	𝑞2	NOUN
cana-2702	546	19	+	+	CCONJ
cana-2702	546	20	1	1	NUM
cana-2702	546	21	]	]	PUNCT
cana-2702	546	22	≥	≥	X
cana-2702	546	23	0	0	NUM
cana-2702	546	24	=	=	SYM
cana-2702	546	25	2(2𝑚	2(2𝑚	NUM
cana-2702	546	26	+	+	CCONJ
cana-2702	546	27	2(𝑛	2(𝑛	NUM
cana-2702	546	28	−	−	NOUN
cana-2702	546	29	3	3	NUM
cana-2702	546	30	)	)	PUNCT
cana-2702	546	31	+	+	CCONJ
cana-2702	546	32	(	(	PUNCT
cana-2702	546	33	𝑚	𝑚	PROPN
cana-2702	546	34	−	−	PROPN
cana-2702	546	35	2)(𝑛	2)(𝑛	NUM
cana-2702	546	36	−	−	NOUN
cana-2702	546	37	4	4	NUM
cana-2702	546	38	)	)	PUNCT
cana-2702	546	39	)	)	PUNCT
cana-2702	546	40	−	−	PROPN
cana-2702	547	1	𝑞	𝑞	X
cana-2702	547	2	+	+	NOUN
cana-2702	547	3	1	1	NUM
cana-2702	547	4	−	−	NOUN
cana-2702	548	1	[	[	X
cana-2702	548	2	2(2(2𝑚−1	2(2(2𝑚−1	NUM
cana-2702	548	3	−	−	NUM
cana-2702	548	4	1	1	NUM
cana-2702	548	5	)	)	PUNCT
cana-2702	548	6	+	+	CCONJ
cana-2702	548	7	(	(	PUNCT
cana-2702	548	8	𝑚	𝑚	PROPN
cana-2702	548	9	−	−	PROPN
cana-2702	548	10	1)(𝑛	1)(𝑛	NUM
cana-2702	548	11	−	−	NOUN
cana-2702	548	12	4	4	NUM
cana-2702	548	13	)	)	PUNCT
cana-2702	548	14	+	+	CCONJ
cana-2702	548	15	2	2	X
cana-2702	548	16	)	)	PUNCT
cana-2702	548	17	−	−	NOUN
cana-2702	548	18	𝑞2	𝑞2	NOUN
cana-2702	548	19	+	+	CCONJ
cana-2702	548	20	1	1	NUM
cana-2702	548	21	]	]	PUNCT
cana-2702	548	22	=	=	SYM
cana-2702	548	23	4(𝑛	4(𝑛	NUM
cana-2702	548	24	−	−	NOUN
cana-2702	548	25	3	3	NUM
cana-2702	548	26	)	)	PUNCT
cana-2702	548	27	+	+	CCONJ
cana-2702	549	1	2(𝑚	2(𝑚	NUM
cana-2702	549	2	−	−	NOUN
cana-2702	549	3	2)(𝑛	2)(𝑛	NUM
cana-2702	549	4	−	−	NOUN
cana-2702	549	5	4	4	NUM
cana-2702	549	6	)	)	PUNCT
cana-2702	549	7	−	−	NOUN
cana-2702	549	8	𝑞	𝑞	PROPN
cana-2702	549	9	+	+	NOUN
cana-2702	549	10	1	1	NUM
cana-2702	549	11	−	−	NOUN
cana-2702	549	12	2(𝑚	2(𝑚	NUM
cana-2702	549	13	−	−	PROPN
cana-2702	549	14	1)(𝑛	1)(𝑛	NUM
cana-2702	549	15	−	−	NOUN
cana-2702	549	16	4	4	NUM
cana-2702	549	17	)	)	PUNCT
cana-2702	549	18	≥	≥	NOUN
cana-2702	549	19	4(𝑛	4(𝑛	NUM
cana-2702	549	20	−	−	NUM
cana-2702	549	21	3	3	NUM
cana-2702	549	22	+	+	CCONJ
cana-2702	549	23	1	1	NUM
cana-2702	549	24	)	)	PUNCT
cana-2702	549	25	>	>	X
cana-2702	549	26	0	0	X
cana-2702	549	27	.	.	PUNCT
cana-2702	549	28	according	accord	VERB
cana-2702	549	29	to	to	ADP
cana-2702	549	30	claim(1	claim(1	NOUN
cana-2702	549	31	)	)	PUNCT
cana-2702	549	32	,	,	PUNCT
cana-2702	549	33	𝑝(𝑍	𝑝(𝑍	NOUN
cana-2702	549	34	)	)	PUNCT
cana-2702	549	35	≥	≥	NOUN
cana-2702	549	36	2(2(2𝑚−1	2(2(2𝑚−1	NUM
cana-2702	549	37	−	−	NUM
cana-2702	549	38	1	1	NUM
cana-2702	549	39	)	)	PUNCT
cana-2702	549	40	+	+	CCONJ
cana-2702	549	41	(	(	PUNCT
cana-2702	549	42	𝑚	𝑚	PROPN
cana-2702	549	43	−	−	PROPN
cana-2702	549	44	1)(𝑛	1)(𝑛	NUM
cana-2702	549	45	−	−	NOUN
cana-2702	549	46	4	4	NUM
cana-2702	549	47	)	)	PUNCT
cana-2702	549	48	+	+	CCONJ
cana-2702	549	49	2	2	X
cana-2702	549	50	)	)	PUNCT
cana-2702	549	51	−	−	NOUN
cana-2702	549	52	𝑞	𝑞	X
cana-2702	549	53	+	+	NOUN
cana-2702	549	54	1	1	X
cana-2702	549	55	.	.	PUNCT
cana-2702	549	56	again	again	ADV
cana-2702	549	57	,	,	PUNCT
cana-2702	549	58	using	use	VERB
cana-2702	549	59	lemma	lemma	PROPN
cana-2702	549	60	2	2	NUM
cana-2702	549	61	,	,	PUNCT
cana-2702	549	62	we	we	PRON
cana-2702	549	63	can	can	AUX
cana-2702	549	64	move	move	VERB
cana-2702	549	65	four	four	NUM
cana-2702	549	66	pebbles	pebble	NOUN
cana-2702	549	67	to	to	ADP
cana-2702	549	68	𝑎	𝑎	PROPN
cana-2702	549	69	(	(	PUNCT
cana-2702	549	70	𝑚−1	𝑚−1	NOUN
cana-2702	549	71	)	)	PUNCT
cana-2702	549	72	(	(	PUNCT
cana-2702	549	73	𝑛	𝑛	PROPN
cana-2702	549	74	2	2	NUM
cana-2702	549	75	+1	+1	NOUN
cana-2702	549	76	)	)	PUNCT
cana-2702	549	77	or	or	CCONJ
cana-2702	549	78	𝑎	𝑎	X
cana-2702	549	79	(	(	PUNCT
cana-2702	549	80	𝑚−1	𝑚−1	NOUN
cana-2702	549	81	)	)	PUNCT
cana-2702	549	82	(	(	PUNCT
cana-2702	549	83	𝑛	𝑛	PROPN
cana-2702	549	84	2	2	NUM
cana-2702	549	85	+1)+1	+1)+1	NOUN
cana-2702	549	86	,	,	PUNCT
cana-2702	549	87	and	and	CCONJ
cana-2702	549	88	we	we	PRON
cana-2702	549	89	may	may	AUX
cana-2702	549	90	then	then	ADV
cana-2702	549	91	transfer	transfer	VERB
cana-2702	549	92	two	two	NUM
cana-2702	549	93	pebbles	pebble	NOUN
cana-2702	549	94	to	to	PART
cana-2702	549	95	𝑣.	𝑣.	VERB
cana-2702	549	96	the	the	DET
cana-2702	549	97	𝟐𝒕	𝟐𝒕	NOUN
cana-2702	549	98	−pebbling	−pebble	VERB
cana-2702	549	99	property	property	NOUN
cana-2702	549	100	of	of	ADP
cana-2702	549	101	𝑪𝒎(𝑲𝒏	𝑪𝒎(𝑲𝒏	PROPN
cana-2702	549	102	)	)	PUNCT
cana-2702	549	103	theorem	theorem	VERB
cana-2702	549	104	10	10	NUM
cana-2702	549	105	.	.	PUNCT
cana-2702	550	1	the	the	DET
cana-2702	550	2	graph	graph	NOUN
cana-2702	550	3	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	PROPN
cana-2702	550	4	)	)	PUNCT
cana-2702	550	5	exhibits	exhibit	VERB
cana-2702	550	6	2𝑡	2𝑡	NOUN
cana-2702	550	7	−	−	ADP
cana-2702	550	8	pebbling	pebble	VERB
cana-2702	550	9	property	property	NOUN
cana-2702	550	10	.	.	PUNCT
cana-2702	551	1	proof	proof	NOUN
cana-2702	551	2	.	.	PUNCT
cana-2702	552	1	consider	consider	VERB
cana-2702	552	2	a	a	DET
cana-2702	552	3	graph	graph	NOUN
cana-2702	552	4	with	with	ADP
cana-2702	552	5	at	at	ADP
cana-2702	552	6	least	least	ADJ
cana-2702	552	7	2(4𝑡	2(4𝑡	NUM
cana-2702	552	8	+	+	CCONJ
cana-2702	552	9	2𝑛	2𝑛	PROPN
cana-2702	552	10	−	−	PROPN
cana-2702	552	11	6	6	NUM
cana-2702	552	12	)	)	PUNCT
cana-2702	552	13	−	−	NOUN
cana-2702	553	1	𝑞	𝑞	X
cana-2702	553	2	+	+	NOUN
cana-2702	553	3	1	1	NUM
cana-2702	553	4	pebbles	pebble	NOUN
cana-2702	553	5	scattered	scatter	VERB
cana-2702	553	6	at	at	ADP
cana-2702	553	7	the	the	DET
cana-2702	553	8	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	553	9	)	)	PUNCT
cana-2702	553	10	vertices	vertex	NOUN
cana-2702	553	11	.	.	PUNCT
cana-2702	554	1	assume	assume	VERB
cana-2702	554	2	𝑣	𝑣	ADP
cana-2702	554	3	∈	∈	PROPN
cana-2702	554	4	𝐶2	𝐶2	INTJ
cana-2702	554	5	and	and	CCONJ
cana-2702	554	6	𝑣	𝑣	X
cana-2702	555	1	=	=	PUNCT
cana-2702	555	2	𝑦.	𝑦.	PROPN
cana-2702	555	3	we	we	PRON
cana-2702	555	4	demonstrate	demonstrate	VERB
cana-2702	555	5	this	this	DET
cana-2702	555	6	result	result	NOUN
cana-2702	555	7	using	use	VERB
cana-2702	555	8	induction	induction	NOUN
cana-2702	555	9	on	on	ADP
cana-2702	555	10	𝑡.	𝑡.	NOUN
cana-2702	555	11	theorem	theorem	VERB
cana-2702	555	12	7	7	NUM
cana-2702	555	13	yields	yield	NOUN
cana-2702	555	14	the	the	DET
cana-2702	555	15	following	following	ADJ
cana-2702	555	16	result	result	NOUN
cana-2702	555	17	for	for	ADP
cana-2702	555	18	𝑡	𝑡	PROPN
cana-2702	555	19	=	=	SYM
cana-2702	555	20	1	1	X
cana-2702	555	21	.	.	PUNCT
cana-2702	555	22	assume	assume	VERB
cana-2702	555	23	that	that	SCONJ
cana-2702	555	24	the	the	DET
cana-2702	555	25	outcome	outcome	NOUN
cana-2702	555	26	holds	hold	VERB
cana-2702	555	27	true	true	ADJ
cana-2702	555	28	for	for	SCONJ
cana-2702	555	29	every	every	DET
cana-2702	555	30	𝑡′	𝑡′	PUNCT
cana-2702	555	31	<	<	X
cana-2702	555	32	𝑡.	𝑡.	NOUN
cana-2702	555	33	we	we	PRON
cana-2702	555	34	take	take	VERB
cana-2702	555	35	into	into	ADP
cana-2702	555	36	account	account	NOUN
cana-2702	555	37	the	the	DET
cana-2702	555	38	following	following	NOUN
cana-2702	555	39	.	.	PUNCT
cana-2702	556	1	case	case	NOUN
cana-2702	556	2	(	(	PUNCT
cana-2702	556	3	1	1	X
cana-2702	556	4	)	)	PUNCT
cana-2702	556	5	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	556	6	)	)	PUNCT
cana-2702	556	7	=	=	PUNCT
cana-2702	557	1	0	0	X
cana-2702	557	2	.	.	PUNCT
cana-2702	557	3	to	to	PART
cana-2702	557	4	prove	prove	VERB
cana-2702	557	5	case(1	case(1	NOUN
cana-2702	557	6	)	)	PUNCT
cana-2702	557	7	,	,	PUNCT
cana-2702	557	8	we	we	PRON
cana-2702	557	9	investigate	investigate	VERB
cana-2702	557	10	the	the	DET
cana-2702	557	11	following	follow	VERB
cana-2702	557	12	subcases	subcase	NOUN
cana-2702	557	13	.	.	PUNCT
cana-2702	558	1	subcase	subcase	PROPN
cana-2702	558	2	(	(	PUNCT
cana-2702	558	3	1	1	X
cana-2702	558	4	)	)	PUNCT
cana-2702	558	5	𝑣	𝑣	PRON
cana-2702	558	6	∈	∈	PROPN
cana-2702	558	7	{	{	PUNCT
cana-2702	558	8	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	558	9	,	,	PUNCT
cana-2702	558	10	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	558	11	}	}	PUNCT
cana-2702	558	12	.	.	PUNCT
cana-2702	559	1	without	without	ADP
cana-2702	559	2	loss	loss	NOUN
cana-2702	559	3	of	of	ADP
cana-2702	559	4	generality	generality	NOUN
cana-2702	559	5	,	,	PUNCT
cana-2702	559	6	assume	assume	VERB
cana-2702	559	7	that	that	SCONJ
cana-2702	559	8	𝑣	𝑣	DET
cana-2702	559	9	∈	∈	PROPN
cana-2702	559	10	𝑎𝑘.	𝑎𝑘.	NOUN
cana-2702	559	11	since	since	SCONJ
cana-2702	559	12	2(4𝑡	2(4𝑡	NUM
cana-2702	559	13	+	+	CCONJ
cana-2702	559	14	2𝑛	2𝑛	PROPN
cana-2702	559	15	−	−	PROPN
cana-2702	559	16	6	6	NUM
cana-2702	559	17	)	)	PUNCT
cana-2702	559	18	−	−	NOUN
cana-2702	559	19	𝑞	𝑞	PROPN
cana-2702	559	20	+	+	NOUN
cana-2702	559	21	1	1	NUM
cana-2702	559	22	−	−	NUM
cana-2702	559	23	4	4	NUM
cana-2702	559	24	≥	≥	NOUN
cana-2702	559	25	8𝑡	8𝑡	NOUN
cana-2702	559	26	+	+	CCONJ
cana-2702	559	27	4𝑛	4𝑛	NOUN
cana-2702	560	1	−	−	NOUN
cana-2702	560	2	12	12	NUM
cana-2702	560	3	−	−	PROPN
cana-2702	561	1	(	(	PUNCT
cana-2702	561	2	2𝑛	2𝑛	PROPN
cana-2702	561	3	−	−	PROPN
cana-2702	561	4	2	2	NUM
cana-2702	561	5	)	)	PUNCT
cana-2702	561	6	−	−	NOUN
cana-2702	561	7	3	3	NUM
cana-2702	561	8	=	=	SYM
cana-2702	561	9	8𝑡	8𝑡	NOUN
cana-2702	561	10	+	+	CCONJ
cana-2702	561	11	2𝑛	2𝑛	PROPN
cana-2702	561	12	−	−	PROPN
cana-2702	561	13	13	13	NUM
cana-2702	561	14	≥	≥	NOUN
cana-2702	561	15	5	5	NUM
cana-2702	561	16	,	,	PUNCT
cana-2702	561	17	since	since	SCONJ
cana-2702	561	18	𝑡	𝑡	PROPN
cana-2702	561	19	≥	≥	NUM
cana-2702	561	20	2	2	NUM
cana-2702	561	21	and	and	CCONJ
cana-2702	561	22	𝑛	𝑛	PRON
cana-2702	561	23	≥	≥	NUM
cana-2702	561	24	5	5	NUM
cana-2702	561	25	.	.	PUNCT
cana-2702	562	1	thus	thus	ADV
cana-2702	562	2	,	,	PUNCT
cana-2702	562	3	we	we	PRON
cana-2702	562	4	may	may	AUX
cana-2702	562	5	transfer	transfer	VERB
cana-2702	562	6	two	two	NUM
cana-2702	562	7	pebbles	pebble	NOUN
cana-2702	562	8	to	to	ADP
cana-2702	562	9	𝑎𝑘	𝑎𝑘	PRON
cana-2702	562	10	by	by	ADP
cana-2702	562	11	utilising	utilise	VERB
cana-2702	562	12	precisely	precisely	ADV
cana-2702	562	13	four	four	NUM
cana-2702	562	14	pebbles	pebble	NOUN
cana-2702	562	15	from	from	ADP
cana-2702	562	16	either	either	PRON
cana-2702	562	17	𝐶1	𝐶1	PROPN
cana-2702	562	18	or	or	CCONJ
cana-2702	562	19	𝐶2	𝐶2	ADJ
cana-2702	562	20	since	since	SCONJ
cana-2702	562	21	𝑎𝑘	𝑎𝑘	PRON
cana-2702	562	22	is	be	AUX
cana-2702	562	23	adjacent	adjacent	ADJ
cana-2702	562	24	to	to	ADP
cana-2702	562	25	all	all	DET
cana-2702	562	26	the	the	DET
cana-2702	562	27	other	other	ADJ
cana-2702	562	28	vertices	vertex	NOUN
cana-2702	562	29	of	of	ADP
cana-2702	562	30	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	NOUN
cana-2702	562	31	)	)	PUNCT
cana-2702	562	32	.	.	PUNCT
cana-2702	563	1	this	this	DET
cana-2702	563	2	yields	yield	NOUN
cana-2702	563	3	at	at	ADP
cana-2702	563	4	least	least	ADJ
cana-2702	563	5	2(4𝑡	2(4𝑡	NUM
cana-2702	564	1	+	+	CCONJ
cana-2702	564	2	2𝑛	2𝑛	PROPN
cana-2702	564	3	−	−	PROPN
cana-2702	564	4	6	6	NUM
cana-2702	564	5	)	)	PUNCT
cana-2702	565	1	−	−	NOUN
cana-2702	565	2	𝑞	𝑞	PROPN
cana-2702	565	3	+	+	NOUN
cana-2702	565	4	1	1	NUM
cana-2702	565	5	−	−	NOUN
cana-2702	565	6	4	4	NUM
cana-2702	565	7	,	,	PUNCT
cana-2702	565	8	which	which	PRON
cana-2702	565	9	is	be	AUX
cana-2702	565	10	greater	great	ADJ
cana-2702	565	11	than	than	ADP
cana-2702	565	12	2(4(𝑡	2(4(𝑡	NUM
cana-2702	565	13	−	−	PROPN
cana-2702	565	14	1	1	NUM
cana-2702	565	15	)	)	PUNCT
cana-2702	566	1	+	+	NUM
cana-2702	566	2	2𝑛	2𝑛	PROPN
cana-2702	566	3	−	−	PROPN
cana-2702	566	4	6	6	NUM
cana-2702	566	5	)	)	PUNCT
cana-2702	566	6	−	−	NOUN
cana-2702	566	7	𝑞	𝑞	X
cana-2702	566	8	+	+	NOUN
cana-2702	566	9	5	5	NUM
cana-2702	566	10	.	.	PUNCT
cana-2702	566	11	as	as	ADP
cana-2702	566	12	a	a	DET
cana-2702	566	13	result	result	NOUN
cana-2702	566	14	,	,	PUNCT
cana-2702	566	15	we	we	PRON
cana-2702	566	16	can	can	AUX
cana-2702	566	17	induct	induct	VERB
cana-2702	566	18	an	an	DET
cana-2702	566	19	extra	extra	ADJ
cana-2702	566	20	2(𝑡	2(𝑡	NUM
cana-2702	566	21	−	−	NOUN
cana-2702	566	22	1	1	NUM
cana-2702	566	23	)	)	PUNCT
cana-2702	566	24	pebbles	pebble	NOUN
cana-2702	566	25	to	to	ADP
cana-2702	566	26	𝑎𝑘.	𝑎𝑘.	NOUN
cana-2702	566	27	as	as	ADP
cana-2702	566	28	a	a	DET
cana-2702	566	29	result	result	NOUN
cana-2702	566	30	of	of	ADP
cana-2702	566	31	induction	induction	NOUN
cana-2702	566	32	,	,	PUNCT
cana-2702	566	33	we	we	PRON
cana-2702	566	34	can	can	AUX
cana-2702	566	35	transfer	transfer	VERB
cana-2702	566	36	one	one	NUM
cana-2702	566	37	extra	extra	NOUN
cana-2702	566	38	2(𝑡	2(𝑡	NUM
cana-2702	566	39	−	−	NOUN
cana-2702	566	40	1	1	NUM
cana-2702	566	41	)	)	PUNCT
cana-2702	566	42	pebble	pebble	NOUN
cana-2702	566	43	to	to	ADP
cana-2702	566	44	𝑎𝑘.	𝑎𝑘.	CCONJ
cana-2702	566	45	subcase	subcase	NOUN
cana-2702	566	46	(	(	PUNCT
cana-2702	566	47	2	2	NUM
cana-2702	566	48	)	)	PUNCT
cana-2702	566	49	𝑣	𝑣	PART
cana-2702	566	50	∈	∈	NOUN
cana-2702	566	51	𝐶1	𝐶1	NOUN
cana-2702	566	52	or	or	CCONJ
cana-2702	566	53	𝑣	𝑣	ADP
cana-2702	566	54	∈	∈	PROPN
cana-2702	566	55	𝐶2	𝐶2	PROPN
cana-2702	566	56	.	.	PUNCT
cana-2702	567	1	we	we	PRON
cana-2702	567	2	assume	assume	VERB
cana-2702	567	3	that	that	SCONJ
cana-2702	567	4	𝑣	𝑣	PRON
cana-2702	567	5	∈	∈	PROPN
cana-2702	567	6	𝐶2	𝐶2	INTJ
cana-2702	567	7	and	and	CCONJ
cana-2702	567	8	𝑣	𝑣	X
cana-2702	567	9	=	=	PUNCT
cana-2702	567	10	𝑦	𝑦	PROPN
cana-2702	567	11	without	without	ADP
cana-2702	567	12	losing	lose	VERB
cana-2702	567	13	generality	generality	NOUN
cana-2702	567	14	.	.	PUNCT
cana-2702	568	1	assume	assume	VERB
cana-2702	568	2	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	568	3	)	)	PUNCT
cana-2702	568	4	≥	≥	NOUN
cana-2702	568	5	𝑛	𝑛	NOUN
cana-2702	568	6	+	+	NOUN
cana-2702	568	7	2	2	X
cana-2702	568	8	.	.	PUNCT
cana-2702	568	9	at	at	ADV
cana-2702	568	10	least	least	ADV
cana-2702	568	11	two	two	NUM
cana-2702	568	12	pebbles	pebble	NOUN
cana-2702	568	13	must	must	AUX
cana-2702	568	14	be	be	AUX
cana-2702	568	15	present	present	ADJ
cana-2702	568	16	on	on	ADP
cana-2702	568	17	each	each	PRON
cana-2702	568	18	of	of	ADP
cana-2702	568	19	its	its	PRON
cana-2702	568	20	two	two	NUM
cana-2702	568	21	vertices	vertex	NOUN
cana-2702	568	22	.	.	PUNCT
cana-2702	569	1	as	as	ADP
cana-2702	569	2	a	a	DET
cana-2702	569	3	result	result	NOUN
cana-2702	569	4	,	,	PUNCT
cana-2702	569	5	we	we	PRON
cana-2702	569	6	can	can	AUX
cana-2702	569	7	move	move	VERB
cana-2702	569	8	two	two	NUM
cana-2702	569	9	pebbles	pebble	NOUN
cana-2702	569	10	to	to	ADP
cana-2702	569	11	𝑦	𝑦	NOUN
cana-2702	569	12	using	use	VERB
cana-2702	569	13	precisely	precisely	ADV
cana-2702	569	14	four	four	NUM
cana-2702	569	15	pebbles	pebble	NOUN
cana-2702	569	16	.	.	PUNCT
cana-2702	570	1	this	this	PRON
cana-2702	570	2	means	mean	VERB
cana-2702	570	3	that	that	SCONJ
cana-2702	570	4	in	in	ADP
cana-2702	570	5	𝑉(𝐶2(𝐾𝑛	𝑉(𝐶2(𝐾𝑛	PROPN
cana-2702	570	6	)	)	PUNCT
cana-2702	570	7	)	)	PUNCT
cana-2702	570	8	,	,	PUNCT
cana-2702	570	9	at	at	ADP
cana-2702	570	10	least	least	ADJ
cana-2702	570	11	2(4(𝑡	2(4(𝑡	NUM
cana-2702	570	12	−	−	NOUN
cana-2702	570	13	1	1	NUM
cana-2702	570	14	)	)	PUNCT
cana-2702	570	15	+	+	NUM
cana-2702	570	16	2𝑛	2𝑛	PROPN
cana-2702	570	17	−	−	PROPN
cana-2702	570	18	6	6	NUM
cana-2702	570	19	)	)	PUNCT
cana-2702	570	20	−	−	NOUN
cana-2702	570	21	𝑞	𝑞	X
cana-2702	570	22	+	+	NOUN
cana-2702	570	23	1	1	NUM
cana-2702	570	24	pebbles	pebble	NOUN
cana-2702	570	25	were	be	AUX
cana-2702	570	26	kept	keep	VERB
cana-2702	570	27	.	.	PUNCT
cana-2702	571	1	so	so	ADV
cana-2702	571	2	suppose	suppose	VERB
cana-2702	571	3	𝑝(𝐶2	𝑝(𝐶2	X
cana-2702	571	4	)	)	PUNCT
cana-2702	571	5	=	=	SYM
cana-2702	571	6	𝑛	𝑛	PROPN
cana-2702	571	7	or	or	CCONJ
cana-2702	571	8	𝑝(𝐶2	𝑝(𝐶2	NOUN
cana-2702	571	9	)	)	PUNCT
cana-2702	571	10	=	=	SYM
cana-2702	571	11	𝑛	𝑛	PROPN
cana-2702	572	1	+	+	NOUN
cana-2702	572	2	1	1	X
cana-2702	572	3	.	.	X
cana-2702	572	4	we	we	PRON
cana-2702	572	5	can	can	AUX
cana-2702	572	6	transfer	transfer	VERB
cana-2702	572	7	one	one	NUM
cana-2702	572	8	pebble	pebble	NOUN
cana-2702	572	9	to	to	ADP
cana-2702	572	10	𝑣	𝑣	PRON
cana-2702	572	11	for	for	ADP
cana-2702	572	12	no	no	DET
cana-2702	572	13	more	more	ADJ
cana-2702	572	14	than	than	ADP
cana-2702	572	15	two	two	NUM
cana-2702	572	16	pebbles	pebble	NOUN
cana-2702	572	17	from	from	ADP
cana-2702	572	18	𝑉(𝐶2	𝑉(𝐶2	NOUN
cana-2702	572	19	)	)	PUNCT
cana-2702	572	20	.	.	PUNCT
cana-2702	573	1	as	as	ADP
cana-2702	573	2	a	a	DET
cana-2702	573	3	result	result	NOUN
cana-2702	573	4	,	,	PUNCT
cana-2702	573	5	the	the	DET
cana-2702	573	6	number	number	NOUN
cana-2702	573	7	of	of	ADP
cana-2702	573	8	pebbles	pebble	NOUN
cana-2702	573	9	kept	keep	VERB
cana-2702	573	10	in	in	ADP
cana-2702	573	11	𝑉(𝐶1	𝑉(𝐶1	PRON
cana-2702	573	12	)	)	PUNCT
cana-2702	573	13	must	must	AUX
cana-2702	573	14	be	be	AUX
cana-2702	573	15	more	more	ADJ
cana-2702	573	16	than	than	ADP
cana-2702	573	17	2(4𝑡	2(4𝑡	NUM
cana-2702	573	18	+	+	CCONJ
cana-2702	573	19	2𝑛	2𝑛	PROPN
cana-2702	573	20	−	−	PROPN
cana-2702	573	21	6	6	NUM
cana-2702	573	22	)	)	PUNCT
cana-2702	573	23	−	−	NOUN
cana-2702	573	24	𝑞	𝑞	PROPN
cana-2702	573	25	+	+	NOUN
cana-2702	573	26	1	1	NUM
cana-2702	573	27	−	−	NOUN
cana-2702	573	28	(	(	PUNCT
cana-2702	573	29	𝑛	𝑛	PROPN
cana-2702	573	30	+	+	NOUN
cana-2702	573	31	1	1	NUM
cana-2702	573	32	)	)	PUNCT
cana-2702	573	33	>	>	X
cana-2702	573	34	𝑛	𝑛	PROPN
cana-2702	574	1	+	+	NOUN
cana-2702	574	2	5	5	NUM
cana-2702	574	3	.	.	PUNCT
cana-2702	575	1	this	this	PRON
cana-2702	575	2	means	mean	VERB
cana-2702	575	3	that	that	SCONJ
cana-2702	575	4	any	any	DET
cana-2702	575	5	two	two	NUM
cana-2702	575	6	of	of	ADP
cana-2702	575	7	the	the	DET
cana-2702	575	8	vertices	vertex	NOUN
cana-2702	575	9	must	must	AUX
cana-2702	575	10	have	have	VERB
cana-2702	575	11	at	at	ADV
cana-2702	575	12	least	least	ADV
cana-2702	575	13	two	two	NUM
cana-2702	575	14	pebbles	pebble	NOUN
cana-2702	575	15	,	,	PUNCT
cana-2702	575	16	or	or	CCONJ
cana-2702	575	17	one	one	NUM
cana-2702	575	18	of	of	ADP
cana-2702	575	19	them	they	PRON
cana-2702	575	20	must	must	AUX
cana-2702	575	21	have	have	VERB
cana-2702	575	22	four	four	NUM
cana-2702	575	23	pebbles	pebble	NOUN
cana-2702	575	24	.	.	PUNCT
cana-2702	576	1	then	then	ADV
cana-2702	576	2	,	,	PUNCT
cana-2702	576	3	using	use	VERB
cana-2702	576	4	precisely	precisely	ADV
cana-2702	576	5	four	four	NUM
cana-2702	576	6	pebbles	pebble	NOUN
cana-2702	576	7	,	,	PUNCT
cana-2702	576	8	two	two	NUM
cana-2702	576	9	pebbles	pebble	NOUN
cana-2702	576	10	may	may	AUX
cana-2702	576	11	be	be	AUX
cana-2702	576	12	relocated	relocate	VERB
cana-2702	576	13	to	to	ADP
cana-2702	576	14	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	576	15	,	,	PUNCT
cana-2702	576	16	and	and	CCONJ
cana-2702	576	17	the	the	DET
cana-2702	576	18	number	number	NOUN
cana-2702	576	19	of	of	ADP
cana-2702	576	20	pebbles	pebble	NOUN
cana-2702	576	21	kept	keep	VERB
cana-2702	576	22	in	in	ADP
cana-2702	576	23	𝑉(𝐶2(𝐾𝑛	𝑉(𝐶2(𝐾𝑛	NOUN
cana-2702	576	24	)	)	PUNCT
cana-2702	576	25	)	)	PUNCT
cana-2702	577	1	is	be	AUX
cana-2702	577	2	at	at	ADP
cana-2702	577	3	least	least	ADJ
cana-2702	577	4	2(4𝑡	2(4𝑡	NUM
cana-2702	577	5	+	+	CCONJ
cana-2702	577	6	2𝑛	2𝑛	PROPN
cana-2702	578	1	−	−	PROPN
cana-2702	578	2	6	6	NUM
cana-2702	578	3	)	)	PUNCT
cana-2702	578	4	−	−	NOUN
cana-2702	578	5	𝑞	𝑞	PROPN
cana-2702	578	6	+	+	NOUN
cana-2702	578	7	1	1	NUM
cana-2702	578	8	−	−	NUM
cana-2702	578	9	6	6	NUM
cana-2702	578	10	≥	≥	NOUN
cana-2702	578	11	2(4(𝑡	2(4(𝑡	NUM
cana-2702	578	12	−	−	NOUN
cana-2702	578	13	1	1	NUM
cana-2702	578	14	)	)	PUNCT
cana-2702	578	15	+	+	NUM
cana-2702	578	16	2𝑛	2𝑛	PROPN
cana-2702	578	17	−	−	PROPN
cana-2702	578	18	6	6	NUM
cana-2702	578	19	)	)	PUNCT
cana-2702	578	20	−	−	NOUN
cana-2702	578	21	𝑞	𝑞	X
cana-2702	578	22	+	+	ADP
cana-2702	578	23	3	3	NUM
cana-2702	578	24	>	>	SYM
cana-2702	578	25	2(4(𝑡	2(4(𝑡	NUM
cana-2702	578	26	−	−	NOUN
cana-2702	578	27	1	1	NUM
cana-2702	578	28	)	)	PUNCT
cana-2702	578	29	+	+	NUM
cana-2702	578	30	2𝑛	2𝑛	PROPN
cana-2702	578	31	−	−	PROPN
cana-2702	578	32	6	6	NUM
cana-2702	578	33	)	)	PUNCT
cana-2702	578	34	−	−	NOUN
cana-2702	578	35	𝑞	𝑞	PROPN
cana-2702	578	36	+	+	NOUN
cana-2702	578	37	1	1	X
cana-2702	578	38	.	.	PUNCT
cana-2702	578	39	as	as	ADP
cana-2702	578	40	a	a	DET
cana-2702	578	41	result	result	NOUN
cana-2702	578	42	of	of	ADP
cana-2702	578	43	induction	induction	NOUN
cana-2702	578	44	,	,	PUNCT
cana-2702	578	45	we	we	PRON
cana-2702	578	46	can	can	AUX
cana-2702	578	47	transfer	transfer	VERB
cana-2702	578	48	more	more	ADJ
cana-2702	578	49	2(𝑡	2(𝑡	NUM
cana-2702	578	50	−	−	NOUN
cana-2702	578	51	1	1	NUM
cana-2702	578	52	)	)	PUNCT
cana-2702	578	53	pebbles	pebble	NOUN
cana-2702	578	54	to	to	ADP
cana-2702	578	55	𝑣.	𝑣.	NOUN
cana-2702	578	56	case	case	NOUN
cana-2702	578	57	(	(	PUNCT
cana-2702	578	58	2	2	NUM
cana-2702	578	59	)	)	PUNCT
cana-2702	578	60	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	578	61	)	)	PUNCT
cana-2702	579	1	=	=	SYM
cana-2702	579	2	𝑥	𝑥	PROPN
cana-2702	579	3	for	for	ADP
cana-2702	579	4	𝑥	𝑥	PRON
cana-2702	579	5	≥	≥	NUM
cana-2702	579	6	1	1	NUM
cana-2702	579	7	.	.	PUNCT
cana-2702	580	1	we	we	PRON
cana-2702	580	2	take	take	VERB
cana-2702	580	3	into	into	ADP
cana-2702	580	4	account	account	NOUN
cana-2702	580	5	the	the	DET
cana-2702	580	6	following	follow	VERB
cana-2702	580	7	subcases	subcase	NOUN
cana-2702	580	8	:	:	PUNCT
cana-2702	580	9	subcase	subcase	PROPN
cana-2702	580	10	(	(	PUNCT
cana-2702	580	11	2a	2a	NUM
cana-2702	580	12	)	)	PUNCT
cana-2702	580	13	𝑥	𝑥	PRON
cana-2702	580	14	is	be	AUX
cana-2702	580	15	even	even	ADV
cana-2702	580	16	.	.	PUNCT
cana-2702	581	1	communications	communication	NOUN
cana-2702	581	2	on	on	ADP
cana-2702	581	3	applied	apply	VERB
cana-2702	581	4	nonlinear	nonlinear	ADJ
cana-2702	581	5	analysis	analysis	NOUN
cana-2702	581	6	issn	issn	NOUN
cana-2702	581	7	:	:	PUNCT
cana-2702	581	8	1074	1074	NUM
cana-2702	581	9	-	-	PUNCT
cana-2702	581	10	133x	133x	NUM
cana-2702	581	11	vol	vol	NOUN
cana-2702	581	12	32	32	NUM
cana-2702	581	13	no	no	NOUN
cana-2702	581	14	.	.	PUNCT
cana-2702	582	1	3s	3s	NUM
cana-2702	582	2	(	(	PUNCT
cana-2702	582	3	2025	2025	NUM
cana-2702	582	4	)	)	PUNCT
cana-2702	582	5	660	660	NUM
cana-2702	582	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	582	7	assume	assume	VERB
cana-2702	582	8	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	582	9	)	)	PUNCT
cana-2702	583	1	=	=	PUNCT
cana-2702	584	1	𝑥	𝑥	NOUN
cana-2702	585	1	=	=	SYM
cana-2702	585	2	2𝑥′.	2𝑥′.	NUM
cana-2702	585	3	we	we	PRON
cana-2702	585	4	need	need	VERB
cana-2702	585	5	to	to	PART
cana-2702	585	6	transport	transport	VERB
cana-2702	585	7	more	more	ADJ
cana-2702	585	8	2(𝑡	2(𝑡	NUM
cana-2702	585	9	−	−	PROPN
cana-2702	585	10	𝑥′	𝑥′	NUM
cana-2702	585	11	)	)	PUNCT
cana-2702	585	12	pebbles	pebble	NOUN
cana-2702	585	13	to	to	PART
cana-2702	585	14	𝑣.	𝑣.	VERB
cana-2702	585	15	because	because	SCONJ
cana-2702	585	16	we	we	PRON
cana-2702	585	17	have	have	VERB
cana-2702	585	18	𝑝(𝐶2(𝐾𝑛	𝑝(𝐶2(𝐾𝑛	NOUN
cana-2702	585	19	)	)	PUNCT
cana-2702	585	20	)	)	PUNCT
cana-2702	586	1	−	−	PROPN
cana-2702	586	2	2𝑥′	2𝑥′	NUM
cana-2702	586	3	=	=	SYM
cana-2702	586	4	2(4𝑡	2(4𝑡	NUM
cana-2702	586	5	+	+	NUM
cana-2702	586	6	2𝑛	2𝑛	PROPN
cana-2702	586	7	−	−	PROPN
cana-2702	586	8	6	6	NUM
cana-2702	586	9	)	)	PUNCT
cana-2702	586	10	−	−	NOUN
cana-2702	586	11	𝑞	𝑞	PROPN
cana-2702	586	12	+	+	NOUN
cana-2702	586	13	1	1	NUM
cana-2702	586	14	−	−	PROPN
cana-2702	586	15	2𝑥′	2𝑥′	NUM
cana-2702	586	16	=	=	SYM
cana-2702	586	17	2(4(𝑡	2(4(𝑡	NUM
cana-2702	586	18	−	−	NUM
cana-2702	586	19	𝑥′	𝑥′	NUM
cana-2702	586	20	)	)	PUNCT
cana-2702	587	1	+	+	CCONJ
cana-2702	587	2	4𝑥′	4𝑥′	NUM
cana-2702	588	1	+	+	NUM
cana-2702	588	2	2𝑛	2𝑛	PROPN
cana-2702	588	3	−	−	PROPN
cana-2702	588	4	6	6	NUM
cana-2702	588	5	)	)	PUNCT
cana-2702	588	6	−	−	NOUN
cana-2702	588	7	𝑞	𝑞	PROPN
cana-2702	588	8	+	+	NOUN
cana-2702	588	9	1	1	NUM
cana-2702	588	10	−	−	PROPN
cana-2702	588	11	2𝑥′	2𝑥′	NUM
cana-2702	588	12	=	=	SYM
cana-2702	588	13	2(4(𝑡	2(4(𝑡	NUM
cana-2702	588	14	−	−	NUM
cana-2702	588	15	𝑥′	𝑥′	NUM
cana-2702	588	16	)	)	PUNCT
cana-2702	589	1	+	+	CCONJ
cana-2702	589	2	2𝑛	2𝑛	PROPN
cana-2702	589	3	−	−	PROPN
cana-2702	589	4	6	6	NUM
cana-2702	589	5	)	)	PUNCT
cana-2702	589	6	−	−	NOUN
cana-2702	589	7	𝑞	𝑞	PROPN
cana-2702	589	8	+	+	NOUN
cana-2702	589	9	1	1	NUM
cana-2702	589	10	+	+	CCONJ
cana-2702	589	11	6𝑥′	6𝑥′	NUM
cana-2702	589	12	>	>	X
cana-2702	589	13	2𝑓𝑡−𝑥′(𝐶2(𝐾𝑛	2𝑓𝑡−𝑥′(𝐶2(𝐾𝑛	NUM
cana-2702	589	14	)	)	PUNCT
cana-2702	589	15	)	)	PUNCT
cana-2702	590	1	−	−	PROPN
cana-2702	591	1	𝑞	𝑞	X
cana-2702	591	2	+	+	NOUN
cana-2702	591	3	1	1	X
cana-2702	591	4	.	.	X
cana-2702	591	5	we	we	PRON
cana-2702	591	6	can	can	AUX
cana-2702	591	7	transfer	transfer	VERB
cana-2702	591	8	more	more	ADJ
cana-2702	591	9	2(𝑡	2(𝑡	NUM
cana-2702	591	10	−	−	PROPN
cana-2702	591	11	𝑥′	𝑥′	NUM
cana-2702	591	12	)	)	PUNCT
cana-2702	591	13	pebbles	pebble	NOUN
cana-2702	591	14	to	to	ADP
cana-2702	591	15	𝑣	𝑣	NOUN
cana-2702	591	16	via	via	ADP
cana-2702	591	17	induction	induction	NOUN
cana-2702	591	18	.	.	PUNCT
cana-2702	592	1	subcase	subcase	PROPN
cana-2702	592	2	(	(	PUNCT
cana-2702	592	3	2b	2b	NUM
cana-2702	592	4	)	)	PUNCT
cana-2702	592	5	𝑥	𝑥	PROPN
cana-2702	592	6	is	be	AUX
cana-2702	592	7	odd	odd	ADJ
cana-2702	592	8	.	.	PUNCT
cana-2702	593	1	assume	assume	VERB
cana-2702	593	2	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	593	3	)	)	PUNCT
cana-2702	594	1	=	=	PUNCT
cana-2702	594	2	𝑥	𝑥	NOUN
cana-2702	595	1	=	=	SYM
cana-2702	595	2	2𝑥′	2𝑥′	NUM
cana-2702	595	3	+	+	CCONJ
cana-2702	595	4	1	1	X
cana-2702	595	5	.	.	X
cana-2702	595	6	we	we	PRON
cana-2702	595	7	need	need	VERB
cana-2702	595	8	to	to	PART
cana-2702	595	9	transport	transport	VERB
cana-2702	595	10	more	more	ADJ
cana-2702	595	11	2(𝑡	2(𝑡	NUM
cana-2702	595	12	−	−	PROPN
cana-2702	596	1	𝑥′	𝑥′	NUM
cana-2702	596	2	)	)	PUNCT
cana-2702	597	1	−	−	PROPN
cana-2702	597	2	1	1	NUM
cana-2702	597	3	pebbles	pebble	NOUN
cana-2702	597	4	to	to	PART
cana-2702	597	5	𝑣.	𝑣.	VERB
cana-2702	597	6	since	since	SCONJ
cana-2702	597	7	𝑝(𝐶2(𝐾𝑛	𝑝(𝐶2(𝐾𝑛	NOUN
cana-2702	597	8	)	)	PUNCT
cana-2702	597	9	)	)	PUNCT
cana-2702	598	1	−	−	PROPN
cana-2702	599	1	2𝑥′	2𝑥′	NUM
cana-2702	599	2	−	−	NOUN
cana-2702	599	3	1	1	NUM
cana-2702	599	4	=	=	SYM
cana-2702	599	5	2(4𝑡	2(4𝑡	NUM
cana-2702	599	6	+	+	NUM
cana-2702	599	7	2𝑛	2𝑛	PROPN
cana-2702	599	8	−	−	PROPN
cana-2702	599	9	6	6	NUM
cana-2702	599	10	)	)	PUNCT
cana-2702	599	11	−	−	NOUN
cana-2702	599	12	𝑞	𝑞	PROPN
cana-2702	599	13	+	+	NOUN
cana-2702	599	14	1	1	NUM
cana-2702	599	15	−	−	NUM
cana-2702	599	16	2𝑥′	2𝑥′	NUM
cana-2702	599	17	−	−	NOUN
cana-2702	599	18	1	1	NUM
cana-2702	599	19	=	=	SYM
cana-2702	599	20	2(4(𝑡	2(4(𝑡	NUM
cana-2702	599	21	−	−	NUM
cana-2702	599	22	𝑥′	𝑥′	NUM
cana-2702	599	23	)	)	PUNCT
cana-2702	600	1	+	+	CCONJ
cana-2702	600	2	4𝑥′	4𝑥′	NUM
cana-2702	601	1	+	+	NUM
cana-2702	601	2	2𝑛	2𝑛	PROPN
cana-2702	601	3	−	−	PROPN
cana-2702	601	4	6	6	NUM
cana-2702	601	5	)	)	PUNCT
cana-2702	601	6	−	−	NOUN
cana-2702	601	7	𝑞	𝑞	PROPN
cana-2702	601	8	−	−	PROPN
cana-2702	601	9	2𝑥′	2𝑥′	NUM
cana-2702	601	10	=	=	SYM
cana-2702	601	11	2(4(𝑡	2(4(𝑡	NUM
cana-2702	601	12	−	−	NUM
cana-2702	601	13	𝑥′	𝑥′	NUM
cana-2702	601	14	)	)	PUNCT
cana-2702	602	1	+	+	CCONJ
cana-2702	602	2	2𝑛	2𝑛	PROPN
cana-2702	602	3	−	−	PROPN
cana-2702	602	4	6	6	NUM
cana-2702	602	5	)	)	PUNCT
cana-2702	602	6	−	−	PROPN
cana-2702	602	7	𝑞	𝑞	PROPN
cana-2702	602	8	+	+	NOUN
cana-2702	602	9	6𝑥	6𝑥	NUM
cana-2702	603	1	′	′	NUM
cana-2702	603	2	>	>	X
cana-2702	603	3	2𝑓𝑡−𝑥′(𝐶2(𝐾𝑛	2𝑓𝑡−𝑥′(𝐶2(𝐾𝑛	NUM
cana-2702	603	4	)	)	PUNCT
cana-2702	603	5	)	)	PUNCT
cana-2702	604	1	−	−	PROPN
cana-2702	605	1	𝑞	𝑞	X
cana-2702	605	2	+	+	NOUN
cana-2702	605	3	1since𝑥′	1since𝑥′	NUM
cana-2702	605	4	≥	≥	NOUN
cana-2702	605	5	1	1	NUM
cana-2702	605	6	so	so	SCONJ
cana-2702	605	7	we	we	PRON
cana-2702	605	8	can	can	AUX
cana-2702	605	9	add	add	VERB
cana-2702	605	10	more	more	ADJ
cana-2702	605	11	2(𝑡	2(𝑡	NUM
cana-2702	605	12	−	−	PROPN
cana-2702	606	1	𝑥′	𝑥′	NUM
cana-2702	606	2	)	)	PUNCT
cana-2702	607	1	−	−	PROPN
cana-2702	607	2	1	1	NUM
cana-2702	607	3	pebbles	pebble	NOUN
cana-2702	607	4	to	to	PART
cana-2702	607	5	𝑣.	𝑣.	NOUN
cana-2702	607	6	theorem	theorem	PROPN
cana-2702	607	7	11	11	NUM
cana-2702	607	8	.	.	PUNCT
cana-2702	608	1	the	the	DET
cana-2702	608	2	graph	graph	NOUN
cana-2702	608	3	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	608	4	)	)	PUNCT
cana-2702	608	5	exhibits	exhibit	VERB
cana-2702	608	6	2𝑡	2𝑡	NOUN
cana-2702	608	7	−pebbling	−pebble	VERB
cana-2702	608	8	property[13][16	property[13][16	PROPN
cana-2702	608	9	]	]	PUNCT
cana-2702	608	10	.	.	PUNCT
cana-2702	609	1	proof	proof	NOUN
cana-2702	609	2	.	.	PUNCT
cana-2702	610	1	consider	consider	VERB
cana-2702	610	2	the	the	DET
cana-2702	610	3	graph	graph	NOUN
cana-2702	610	4	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	PROPN
cana-2702	610	5	)	)	PUNCT
cana-2702	610	6	,	,	PUNCT
cana-2702	610	7	which	which	PRON
cana-2702	610	8	has	have	VERB
cana-2702	610	9	at	at	ADP
cana-2702	610	10	least	least	ADJ
cana-2702	610	11	2(8𝑡	2(8𝑡	NUM
cana-2702	610	12	+	+	CCONJ
cana-2702	610	13	3𝑛	3𝑛	NUM
cana-2702	610	14	−	−	NOUN
cana-2702	610	15	10	10	NUM
cana-2702	610	16	)	)	PUNCT
cana-2702	610	17	−	−	ADP
cana-2702	611	1	𝑞	𝑞	X
cana-2702	611	2	+	+	NOUN
cana-2702	611	3	1	1	NUM
cana-2702	611	4	pebbles	pebble	NOUN
cana-2702	611	5	on	on	ADP
cana-2702	611	6	its	its	PRON
cana-2702	611	7	vertices	vertex	NOUN
cana-2702	611	8	.	.	PUNCT
cana-2702	612	1	we	we	PRON
cana-2702	612	2	must	must	AUX
cana-2702	612	3	relocate	relocate	VERB
cana-2702	612	4	2𝑡	2𝑡	NUM
cana-2702	612	5	pebbles	pebble	NOUN
cana-2702	612	6	to	to	ADP
cana-2702	612	7	any	any	DET
cana-2702	612	8	target	target	NOUN
cana-2702	612	9	vertex	vertex	NOUN
cana-2702	612	10	.	.	PUNCT
cana-2702	613	1	for	for	ADP
cana-2702	613	2	𝑖	𝑖	PRON
cana-2702	613	3	=	=	NOUN
cana-2702	613	4	1,2,3	1,2,3	NUM
cana-2702	613	5	,	,	PUNCT
cana-2702	613	6	let	let	VERB
cana-2702	613	7	𝑣	𝑣	PART
cana-2702	613	8	∈	∈	PROPN
cana-2702	613	9	𝐶𝑖.	𝐶𝑖.	PROPN
cana-2702	613	10	we	we	PRON
cana-2702	613	11	look	look	VERB
cana-2702	613	12	at	at	ADP
cana-2702	613	13	the	the	DET
cana-2702	613	14	following	following	ADJ
cana-2702	613	15	scenarios	scenario	NOUN
cana-2702	613	16	:	:	PUNCT
cana-2702	613	17	case	case	NOUN
cana-2702	613	18	(	(	PUNCT
cana-2702	613	19	1	1	X
cana-2702	613	20	)	)	PUNCT
cana-2702	613	21	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	613	22	)	)	PUNCT
cana-2702	614	1	=	=	SYM
cana-2702	614	2	0	0	X
cana-2702	614	3	.	.	PUNCT
cana-2702	614	4	theorem	theorem	VERB
cana-2702	614	5	8	8	NUM
cana-2702	614	6	yields	yield	NOUN
cana-2702	614	7	the	the	DET
cana-2702	614	8	following	following	ADJ
cana-2702	614	9	result	result	NOUN
cana-2702	614	10	for	for	ADP
cana-2702	614	11	𝑡	𝑡	PROPN
cana-2702	614	12	=	=	SYM
cana-2702	614	13	1	1	X
cana-2702	614	14	.	.	PUNCT
cana-2702	614	15	assume	assume	VERB
cana-2702	614	16	that	that	SCONJ
cana-2702	614	17	the	the	DET
cana-2702	614	18	outcome	outcome	NOUN
cana-2702	614	19	holds	hold	VERB
cana-2702	614	20	true	true	ADJ
cana-2702	614	21	for	for	ADP
cana-2702	614	22	every	every	DET
cana-2702	614	23	𝑡′	𝑡′	PUNCT
cana-2702	614	24	<	<	X
cana-2702	614	25	𝑡.	𝑡.	NOUN
cana-2702	614	26	we	we	PRON
cana-2702	614	27	take	take	VERB
cana-2702	614	28	into	into	ADP
cana-2702	614	29	account	account	NOUN
cana-2702	614	30	the	the	DET
cana-2702	614	31	following	follow	VERB
cana-2702	614	32	subcases	subcase	NOUN
cana-2702	614	33	:	:	PUNCT
cana-2702	615	1	subcase	subcase	PROPN
cana-2702	615	2	(	(	PUNCT
cana-2702	615	3	1a)𝑣	1a)𝑣	PROPN
cana-2702	615	4	∈	∈	PROPN
cana-2702	615	5	𝐶2	𝐶2	PROPN
cana-2702	615	6	.	.	PUNCT
cana-2702	615	7	consider	consider	VERB
cana-2702	615	8	that	that	DET
cana-2702	615	9	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	NOUN
cana-2702	615	10	)	)	PUNCT
cana-2702	615	11	−	−	PROPN
cana-2702	616	1	{	{	PUNCT
cana-2702	616	2	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	616	3	,	,	PUNCT
cana-2702	616	4	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	616	5	,	,	PUNCT
cana-2702	616	6	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	616	7	,	,	PUNCT
cana-2702	616	8	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	616	9	}	}	PUNCT
cana-2702	616	10	)	)	PUNCT
cana-2702	616	11	≥	≥	NOUN
cana-2702	616	12	𝑛	𝑛	PRON
cana-2702	616	13	−	−	NOUN
cana-2702	616	14	2	2	NUM
cana-2702	616	15	.	.	PUNCT
cana-2702	617	1	then	then	ADV
cana-2702	617	2	we	we	PRON
cana-2702	617	3	may	may	AUX
cana-2702	617	4	transfer	transfer	VERB
cana-2702	617	5	two	two	NUM
cana-2702	617	6	pebbles	pebble	NOUN
cana-2702	617	7	to	to	ADP
cana-2702	617	8	𝑣	𝑣	PRON
cana-2702	617	9	for	for	ADP
cana-2702	617	10	a	a	DET
cana-2702	617	11	total	total	NOUN
cana-2702	617	12	of	of	ADP
cana-2702	617	13	four	four	NUM
cana-2702	617	14	pebbles	pebble	NOUN
cana-2702	617	15	.	.	PUNCT
cana-2702	618	1	we	we	PRON
cana-2702	618	2	have	have	VERB
cana-2702	618	3	𝑝(𝑉(𝐶2	𝑝(𝑉(𝐶2	ADJ
cana-2702	618	4	)	)	PUNCT
cana-2702	618	5	−	−	PROPN
cana-2702	619	1	{	{	PUNCT
cana-2702	619	2	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	619	3	,	,	PUNCT
cana-2702	619	4	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	619	5	,	,	PUNCT
cana-2702	619	6	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	619	7	,	,	PUNCT
cana-2702	619	8	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	619	9	}	}	PUNCT
cana-2702	619	10	)	)	PUNCT
cana-2702	620	1	≅	≅	PROPN
cana-2702	620	2	𝐾𝑛−4	𝐾𝑛−4	PROPN
cana-2702	620	3	.	.	PUNCT
cana-2702	621	1	this	this	PRON
cana-2702	621	2	indicates	indicate	VERB
cana-2702	621	3	that	that	SCONJ
cana-2702	621	4	𝑉(𝐶3(𝐾𝑛	𝑉(𝐶3(𝐾𝑛	NOUN
cana-2702	621	5	)	)	PUNCT
cana-2702	621	6	)	)	PUNCT
cana-2702	621	7	has	have	VERB
cana-2702	621	8	at	at	ADV
cana-2702	621	9	least	least	ADJ
cana-2702	621	10	𝑝(𝐶3(𝐾𝑛	𝑝(𝐶3(𝐾𝑛	NOUN
cana-2702	621	11	)	)	PUNCT
cana-2702	621	12	)	)	PUNCT
cana-2702	622	1	−	−	ADP
cana-2702	622	2	4	4	NUM
cana-2702	622	3	≥	≥	NOUN
cana-2702	622	4	2(8(𝑡	2(8(𝑡	NUM
cana-2702	622	5	−	−	NOUN
cana-2702	622	6	1	1	NUM
cana-2702	622	7	)	)	PUNCT
cana-2702	622	8	+	+	NUM
cana-2702	622	9	3𝑛	3𝑛	NUM
cana-2702	622	10	−	−	PROPN
cana-2702	622	11	10)𝑞	10)𝑞	NUM
cana-2702	622	12	+	+	CCONJ
cana-2702	622	13	1	1	NUM
cana-2702	622	14	pebbles	pebble	NOUN
cana-2702	622	15	.	.	PUNCT
cana-2702	623	1	we	we	PRON
cana-2702	623	2	can	can	AUX
cana-2702	623	3	transfer	transfer	VERB
cana-2702	623	4	more	more	ADJ
cana-2702	623	5	2(𝑡	2(𝑡	NUM
cana-2702	623	6	−	−	NOUN
cana-2702	623	7	1	1	NUM
cana-2702	623	8	)	)	PUNCT
cana-2702	623	9	pebbles	pebble	NOUN
cana-2702	623	10	to	to	ADP
cana-2702	623	11	𝑣	𝑣	NOUN
cana-2702	623	12	via	via	ADP
cana-2702	623	13	induction	induction	NOUN
cana-2702	623	14	.	.	PUNCT
cana-2702	624	1	we	we	PRON
cana-2702	624	2	now	now	ADV
cana-2702	624	3	suppose	suppose	VERB
cana-2702	624	4	that	that	SCONJ
cana-2702	624	5	𝑝(𝑉2(𝐶2	𝑝(𝑉2(𝐶2	NOUN
cana-2702	624	6	)	)	PUNCT
cana-2702	624	7	−	−	PROPN
cana-2702	625	1	{	{	PUNCT
cana-2702	625	2	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	625	3	,	,	PUNCT
cana-2702	625	4	𝑏𝑘−1	𝑏𝑘−1	PROPN
cana-2702	625	5	,	,	PUNCT
cana-2702	625	6	𝑎2𝑘−2	𝑎2𝑘−2	PROPN
cana-2702	625	7	,	,	PUNCT
cana-2702	625	8	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	625	9	}	}	PUNCT
cana-2702	625	10	)	)	PUNCT
cana-2702	625	11	≤	≤	NUM
cana-2702	626	1	𝑛	𝑛	DET
cana-2702	626	2	−	−	NOUN
cana-2702	626	3	3	3	NUM
cana-2702	626	4	.	.	PUNCT
cana-2702	626	5	as	as	ADP
cana-2702	626	6	a	a	DET
cana-2702	626	7	result	result	NOUN
cana-2702	626	8	,	,	PUNCT
cana-2702	626	9	the	the	DET
cana-2702	626	10	total	total	ADJ
cana-2702	626	11	number	number	NOUN
cana-2702	626	12	of	of	ADP
cana-2702	626	13	pebbles	pebble	NOUN
cana-2702	626	14	maintained	maintain	VERB
cana-2702	626	15	in	in	ADP
cana-2702	626	16	𝐶1	𝐶1	PRON
cana-2702	626	17	and	and	CCONJ
cana-2702	626	18	𝐶2	𝐶2	ADJ
cana-2702	626	19	was	be	AUX
cana-2702	626	20	𝑝(𝐶3(𝐾𝑛	𝑝(𝐶3(𝐾𝑛	NOUN
cana-2702	626	21	)	)	PUNCT
cana-2702	627	1	−	−	PROPN
cana-2702	627	2	(	(	PUNCT
cana-2702	627	3	𝑛	𝑛	PROPN
cana-2702	627	4	−	−	NOUN
cana-2702	627	5	3	3	NUM
cana-2702	627	6	)	)	PUNCT
cana-2702	627	7	+	+	NUM
cana-2702	627	8	𝑞2	𝑞2	NOUN
cana-2702	627	9	)	)	PUNCT
cana-2702	627	10	≥	≥	NOUN
cana-2702	627	11	4𝑛	4𝑛	NOUN
cana-2702	628	1	+	+	CCONJ
cana-2702	629	1	12	12	NUM
cana-2702	629	2	.	.	PUNCT
cana-2702	630	1	then	then	ADV
cana-2702	630	2	,	,	PUNCT
cana-2702	630	3	at	at	ADV
cana-2702	630	4	least	least	ADJ
cana-2702	630	5	one	one	NUM
cana-2702	630	6	𝐶1	𝐶1	NOUN
cana-2702	630	7	or	or	CCONJ
cana-2702	630	8	𝐶3	𝐶3	NOUN
cana-2702	630	9	contains	contain	VERB
cana-2702	630	10	at	at	ADP
cana-2702	630	11	least	least	ADJ
cana-2702	630	12	2𝑛	2𝑛	NOUN
cana-2702	630	13	+	+	CCONJ
cana-2702	630	14	6	6	NUM
cana-2702	630	15	pebbles	pebble	NOUN
cana-2702	630	16	.	.	PUNCT
cana-2702	631	1	make	make	VERB
cana-2702	631	2	sure	sure	ADJ
cana-2702	631	3	𝑝(𝐶1	𝑝(𝐶1	NOUN
cana-2702	631	4	)	)	PUNCT
cana-2702	631	5	≥	≥	NOUN
cana-2702	631	6	2𝑛	2𝑛	NOUN
cana-2702	632	1	+	+	CCONJ
cana-2702	633	1	6	6	X
cana-2702	633	2	.	.	X
cana-2702	633	3	we	we	PRON
cana-2702	633	4	can	can	AUX
cana-2702	633	5	transfer	transfer	VERB
cana-2702	633	6	four	four	NUM
cana-2702	633	7	pebbles	pebble	NOUN
cana-2702	633	8	to	to	ADP
cana-2702	633	9	𝑎𝑘	𝑎𝑘	PROPN
cana-2702	633	10	and	and	CCONJ
cana-2702	633	11	two	two	NUM
cana-2702	633	12	pebbles	pebble	NOUN
cana-2702	633	13	to	to	ADP
cana-2702	633	14	𝑣	𝑣	PRON
cana-2702	633	15	∈	∈	PROPN
cana-2702	633	16	𝐶2	𝐶2	INTJ
cana-2702	633	17	using	use	VERB
cana-2702	633	18	at	at	ADV
cana-2702	633	19	least	least	ADV
cana-2702	633	20	eight	eight	NUM
cana-2702	633	21	pebbles	pebble	NOUN
cana-2702	633	22	from	from	ADP
cana-2702	633	23	𝐶1	𝐶1	PRON
cana-2702	633	24	.	.	PUNCT
cana-2702	634	1	then	then	ADV
cana-2702	634	2	we	we	PRON
cana-2702	634	3	have	have	VERB
cana-2702	634	4	2(8𝑡	2(8𝑡	NUM
cana-2702	634	5	+	+	NUM
cana-2702	634	6	3𝑛	3𝑛	NUM
cana-2702	634	7	−	−	NOUN
cana-2702	634	8	10	10	NUM
cana-2702	634	9	)	)	PUNCT
cana-2702	634	10	−	−	ADP
cana-2702	634	11	𝑞	𝑞	X
cana-2702	634	12	+	+	NOUN
cana-2702	634	13	1	1	NUM
cana-2702	634	14	−	−	NOUN
cana-2702	634	15	8	8	NUM
cana-2702	634	16	>	>	SYM
cana-2702	634	17	2(8(𝑡	2(8(𝑡	NUM
cana-2702	634	18	−	−	NOUN
cana-2702	634	19	1	1	NUM
cana-2702	634	20	)	)	PUNCT
cana-2702	634	21	+	+	NUM
cana-2702	634	22	3𝑛	3𝑛	NUM
cana-2702	634	23	−	−	NUM
cana-2702	634	24	10	10	NUM
cana-2702	634	25	)	)	PUNCT
cana-2702	634	26	−	−	ADP
cana-2702	634	27	𝑞	𝑞	X
cana-2702	634	28	+	+	NOUN
cana-2702	634	29	1	1	X
cana-2702	634	30	.	.	X
cana-2702	635	1	we	we	PRON
cana-2702	635	2	can	can	AUX
cana-2702	635	3	transfer	transfer	VERB
cana-2702	635	4	more	more	ADJ
cana-2702	635	5	2(𝑡	2(𝑡	NUM
cana-2702	635	6	−	−	NOUN
cana-2702	635	7	1	1	NUM
cana-2702	635	8	)	)	PUNCT
cana-2702	635	9	pebbles	pebble	NOUN
cana-2702	635	10	to	to	ADP
cana-2702	635	11	𝑣	𝑣	NOUN
cana-2702	635	12	via	via	ADP
cana-2702	635	13	induction	induction	NOUN
cana-2702	635	14	.	.	PUNCT
cana-2702	636	1	subcase	subcase	PROPN
cana-2702	636	2	(	(	PUNCT
cana-2702	636	3	1b	1b	NUM
cana-2702	636	4	)	)	PUNCT
cana-2702	636	5	𝑣	𝑣	PRON
cana-2702	636	6	∈	∈	PROPN
cana-2702	636	7	𝐶1	𝐶1	NOUN
cana-2702	636	8	or	or	CCONJ
cana-2702	636	9	𝑣	𝑣	PRON
cana-2702	636	10	∈	∈	PROPN
cana-2702	636	11	𝐶3	𝐶3	PROPN
cana-2702	636	12	.	.	PUNCT
cana-2702	636	13	allow	allow	VERB
cana-2702	636	14	𝑣	𝑣	DET
cana-2702	636	15	∈	∈	PROPN
cana-2702	636	16	𝐶3	𝐶3	NOUN
cana-2702	636	17	and	and	CCONJ
cana-2702	636	18	set	set	VERB
cana-2702	636	19	𝑣	𝑣	PRON
cana-2702	636	20	=	=	PUNCT
cana-2702	636	21	𝑦.	𝑦.	PROPN
cana-2702	636	22	assume	assume	VERB
cana-2702	636	23	that	that	SCONJ
cana-2702	636	24	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	636	25	−	−	PROPN
cana-2702	636	26	{	{	PUNCT
cana-2702	636	27	𝑎2𝑘−2,𝑏2𝑘−1	𝑎2𝑘−2,𝑏2𝑘−1	NOUN
cana-2702	636	28	}	}	PUNCT
cana-2702	636	29	)	)	PUNCT
cana-2702	636	30	≥	≥	PROPN
cana-2702	636	31	𝑛.	𝑛.	NOUN
cana-2702	636	32	then	then	ADV
cana-2702	636	33	,	,	PUNCT
cana-2702	636	34	for	for	ADP
cana-2702	636	35	the	the	DET
cana-2702	636	36	expense	expense	NOUN
cana-2702	636	37	of	of	ADP
cana-2702	636	38	four	four	NUM
cana-2702	636	39	pebbles	pebble	NOUN
cana-2702	636	40	,	,	PUNCT
cana-2702	636	41	we	we	PRON
cana-2702	636	42	may	may	AUX
cana-2702	636	43	shift	shift	VERB
cana-2702	636	44	two	two	NUM
cana-2702	636	45	pebbles	pebble	NOUN
cana-2702	636	46	to	to	PART
cana-2702	636	47	𝑣.	𝑣.	VERB
cana-2702	636	48	the	the	DET
cana-2702	636	49	number	number	NOUN
cana-2702	636	50	of	of	ADP
cana-2702	636	51	pebbles	pebble	NOUN
cana-2702	636	52	kept	keep	VERB
cana-2702	636	53	in	in	ADP
cana-2702	636	54	𝑉(𝐶3(𝐾𝑛	𝑉(𝐶3(𝐾𝑛	NOUN
cana-2702	636	55	)	)	PUNCT
cana-2702	636	56	)	)	PUNCT
cana-2702	636	57	was	be	AUX
cana-2702	636	58	obviously	obviously	ADV
cana-2702	636	59	at	at	ADP
cana-2702	636	60	least	least	ADJ
cana-2702	636	61	2𝑓𝑡−1(𝐶3(𝐾𝑛	2𝑓𝑡−1(𝐶3(𝐾𝑛	NUM
cana-2702	636	62	)	)	PUNCT
cana-2702	636	63	)	)	PUNCT
cana-2702	637	1	−	−	PROPN
cana-2702	638	1	𝑞	𝑞	X
cana-2702	638	2	+	+	NOUN
cana-2702	638	3	1	1	X
cana-2702	638	4	.	.	PUNCT
cana-2702	638	5	with	with	ADP
cana-2702	638	6	induction	induction	NOUN
cana-2702	638	7	,	,	PUNCT
cana-2702	638	8	we	we	PRON
cana-2702	638	9	can	can	AUX
cana-2702	638	10	transfer	transfer	VERB
cana-2702	638	11	2(𝑡	2(𝑡	NUM
cana-2702	638	12	−	−	NOUN
cana-2702	638	13	1	1	NUM
cana-2702	638	14	)	)	PUNCT
cana-2702	638	15	additional	additional	ADJ
cana-2702	638	16	pebbles	pebble	NOUN
cana-2702	638	17	to	to	PART
cana-2702	638	18	𝑦.	𝑦.	VERB
cana-2702	638	19	as	as	ADP
cana-2702	638	20	a	a	DET
cana-2702	638	21	result	result	NOUN
cana-2702	638	22	,	,	PUNCT
cana-2702	638	23	we	we	PRON
cana-2702	638	24	can	can	AUX
cana-2702	638	25	shift	shift	VERB
cana-2702	638	26	zero	zero	NUM
cana-2702	638	27	pebbles	pebble	NOUN
cana-2702	638	28	to	to	ADP
cana-2702	638	29	𝑦	𝑦	NOUN
cana-2702	638	30	by	by	ADP
cana-2702	638	31	employing	employ	VERB
cana-2702	638	32	pebbles	pebble	NOUN
cana-2702	638	33	in	in	ADP
cana-2702	638	34	𝑉(𝐶3	𝑉(𝐶3	ADJ
cana-2702	638	35	)	)	PUNCT
cana-2702	638	36	−	−	PROPN
cana-2702	638	37	{	{	PUNCT
cana-2702	638	38	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	638	39	,	,	PUNCT
cana-2702	638	40	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	638	41	}	}	PUNCT
cana-2702	638	42	.	.	PUNCT
cana-2702	639	1	this	this	PRON
cana-2702	639	2	implies	imply	VERB
cana-2702	639	3	that	that	SCONJ
cana-2702	639	4	𝑝(𝑉(𝐶3	𝑝(𝑉(𝐶3	NOUN
cana-2702	639	5	)	)	PUNCT
cana-2702	639	6	−	−	PROPN
cana-2702	640	1	{	{	PUNCT
cana-2702	640	2	𝑎2𝑘−2	𝑎2𝑘−2	NOUN
cana-2702	640	3	,	,	PUNCT
cana-2702	640	4	𝑏2𝑘−1	𝑏2𝑘−1	PROPN
cana-2702	640	5	}	}	PUNCT
cana-2702	640	6	)	)	PUNCT
cana-2702	640	7	≤	≤	NUM
cana-2702	641	1	𝑛	𝑛	DET
cana-2702	641	2	−	−	NOUN
cana-2702	641	3	4	4	NUM
cana-2702	641	4	.	.	PUNCT
cana-2702	641	5	claim(1	claim(1	NOUN
cana-2702	641	6	)	)	PUNCT
cana-2702	642	1	𝑝(𝐶3(𝐾𝑛	𝑝(𝐶3(𝐾𝑛	NOUN
cana-2702	642	2	)	)	PUNCT
cana-2702	642	3	)	)	PUNCT
cana-2702	643	1	−	−	PROPN
cana-2702	643	2	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	643	3	)	)	PUNCT
cana-2702	643	4	≥	≥	NOUN
cana-2702	643	5	2𝑓2𝑡(𝐶2(𝐾𝑛	2𝑓2𝑡(𝐶2(𝐾𝑛	NUM
cana-2702	643	6	)	)	PUNCT
cana-2702	643	7	)	)	PUNCT
cana-2702	644	1	−	−	PROPN
cana-2702	645	1	𝑞	𝑞	X
cana-2702	645	2	+	+	NOUN
cana-2702	645	3	1.we	1.we	NUM
cana-2702	645	4	have	have	AUX
cana-2702	645	5	𝑝(𝐶3(𝐾	𝑝(𝐶3(𝐾	VERB
cana-2702	645	6	−	−	NUM
cana-2702	645	7	𝑛	𝑛	NOUN
cana-2702	645	8	)	)	PUNCT
cana-2702	645	9	)	)	PUNCT
cana-2702	646	1	−	−	PROPN
cana-2702	647	1	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	647	2	)	)	PUNCT
cana-2702	647	3	=	=	SYM
cana-2702	648	1	2(8𝑡	2(8𝑡	NUM
cana-2702	649	1	+	+	NUM
cana-2702	649	2	3𝑛	3𝑛	NUM
cana-2702	649	3	−	−	NOUN
cana-2702	649	4	10	10	NUM
cana-2702	649	5	)	)	PUNCT
cana-2702	649	6	−	−	ADP
cana-2702	649	7	𝑞	𝑞	X
cana-2702	649	8	+	+	NOUN
cana-2702	649	9	1	1	NUM
cana-2702	649	10	−	−	NOUN
cana-2702	649	11	𝑛	𝑛	DET
cana-2702	649	12	+	+	NOUN
cana-2702	649	13	4	4	NUM
cana-2702	649	14	=	=	SYM
cana-2702	649	15	2(8𝑡	2(8𝑡	NUM
cana-2702	649	16	+	+	NUM
cana-2702	649	17	2𝑛	2𝑛	PROPN
cana-2702	649	18	+	+	CCONJ
cana-2702	649	19	𝑛	𝑛	PRON
cana-2702	649	20	−	−	PROPN
cana-2702	649	21	10	10	NUM
cana-2702	649	22	)	)	PUNCT
cana-2702	649	23	−	−	ADP
cana-2702	649	24	𝑞	𝑞	X
cana-2702	649	25	+	+	NOUN
cana-2702	649	26	1	1	NUM
cana-2702	649	27	−	−	NOUN
cana-2702	649	28	𝑛	𝑛	DET
cana-2702	649	29	+	+	NOUN
cana-2702	649	30	4	4	NUM
cana-2702	649	31	=	=	SYM
cana-2702	649	32	2(8𝑡	2(8𝑡	NUM
cana-2702	649	33	+	+	CCONJ
cana-2702	649	34	2𝑛	2𝑛	PROPN
cana-2702	649	35	−	−	PROPN
cana-2702	649	36	6	6	NUM
cana-2702	649	37	)	)	PUNCT
cana-2702	649	38	−	−	NOUN
cana-2702	649	39	𝑞	𝑞	X
cana-2702	649	40	+	+	NOUN
cana-2702	649	41	1	1	NUM
cana-2702	649	42	+	+	NUM
cana-2702	649	43	2𝑛	2𝑛	NUM
cana-2702	649	44	−	−	PROPN
cana-2702	649	45	8	8	NUM
cana-2702	649	46	−	−	NOUN
cana-2702	649	47	𝑛	𝑛	DET
cana-2702	649	48	+	+	NOUN
cana-2702	649	49	4	4	NUM
cana-2702	649	50	=	=	SYM
cana-2702	649	51	2(8𝑡	2(8𝑡	NUM
cana-2702	649	52	+	+	CCONJ
cana-2702	649	53	2𝑛	2𝑛	PROPN
cana-2702	649	54	−	−	PROPN
cana-2702	649	55	6	6	NUM
cana-2702	649	56	)	)	PUNCT
cana-2702	649	57	−	−	NOUN
cana-2702	649	58	𝑞	𝑞	X
cana-2702	649	59	+	+	NOUN
cana-2702	649	60	1	1	NUM
cana-2702	649	61	+	+	CCONJ
cana-2702	649	62	𝑛	𝑛	PRON
cana-2702	649	63	−	−	PROPN
cana-2702	649	64	4	4	NUM
cana-2702	649	65	>	>	SYM
cana-2702	649	66	2(8𝑡	2(8𝑡	NUM
cana-2702	649	67	+	+	CCONJ
cana-2702	649	68	2𝑛	2𝑛	PROPN
cana-2702	649	69	−	−	PROPN
cana-2702	649	70	6	6	NUM
cana-2702	649	71	)	)	PUNCT
cana-2702	649	72	−	−	ADP
cana-2702	650	1	𝑞	𝑞	SYM
cana-2702	650	2	+	+	NOUN
cana-2702	650	3	1,since𝑛	1,since𝑛	NUM
cana-2702	650	4	≥	≥	NOUN
cana-2702	650	5	5	5	NUM
cana-2702	650	6	=	=	SYM
cana-2702	650	7	2𝑓2𝑡(𝐶(𝐾𝑛	2𝑓2𝑡(𝐶(𝐾𝑛	NOUN
cana-2702	650	8	)	)	PUNCT
cana-2702	650	9	)	)	PUNCT
cana-2702	651	1	−	−	PROPN
cana-2702	652	1	𝑞	𝑞	X
cana-2702	652	2	+	+	NOUN
cana-2702	652	3	1	1	NUM
cana-2702	652	4	=	=	SYM
cana-2702	652	5	𝑝(𝐶1	𝑝(𝐶1	PROPN
cana-2702	652	6	∪	∪	ADJ
cana-2702	652	7	𝐶2	𝐶2	NOUN
cana-2702	652	8	)	)	PUNCT
cana-2702	652	9	.	.	PUNCT
cana-2702	653	1	communications	communication	NOUN
cana-2702	653	2	on	on	ADP
cana-2702	653	3	applied	apply	VERB
cana-2702	653	4	nonlinear	nonlinear	ADJ
cana-2702	653	5	analysis	analysis	NOUN
cana-2702	653	6	issn	issn	NOUN
cana-2702	653	7	:	:	PUNCT
cana-2702	653	8	1074	1074	NUM
cana-2702	653	9	-	-	PUNCT
cana-2702	653	10	133x	133x	NUM
cana-2702	653	11	vol	vol	NOUN
cana-2702	653	12	32	32	NUM
cana-2702	653	13	no	no	NOUN
cana-2702	653	14	.	.	PUNCT
cana-2702	654	1	3s	3s	NUM
cana-2702	654	2	(	(	PUNCT
cana-2702	654	3	2025	2025	NUM
cana-2702	654	4	)	)	PUNCT
cana-2702	654	5	661	661	NUM
cana-2702	654	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2702	654	7	according	accord	VERB
cana-2702	654	8	to	to	ADP
cana-2702	654	9	claim	claim	NOUN
cana-2702	654	10	,	,	PUNCT
cana-2702	654	11	we	we	PRON
cana-2702	654	12	can	can	AUX
cana-2702	654	13	transfer	transfer	VERB
cana-2702	654	14	4𝑡	4𝑡	NUM
cana-2702	654	15	pebbles	pebble	NOUN
cana-2702	654	16	to	to	ADP
cana-2702	654	17	𝑎2𝑘	𝑎2𝑘	NOUN
cana-2702	654	18	using	use	VERB
cana-2702	654	19	theorem	theorem	NOUN
cana-2702	654	20	4	4	NUM
cana-2702	654	21	because	because	SCONJ
cana-2702	654	22	𝐶3(𝐾𝑛	𝐶3(𝐾𝑛	NOUN
cana-2702	654	23	)	)	PUNCT
cana-2702	655	1	−	−	PROPN
cana-2702	656	1	𝐶3	𝐶3	PROPN
cana-2702	656	2	≅	≅	PROPN
cana-2702	656	3	𝐶1	𝐶1	PROPN
cana-2702	656	4	∪	∪	X
cana-2702	656	5	𝐶2	𝐶2	PROPN
cana-2702	656	6	≅	≅	PROPN
cana-2702	656	7	𝐶2(𝐾𝑛	𝐶2(𝐾𝑛	PROPN
cana-2702	656	8	)	)	PUNCT
cana-2702	656	9	and	and	CCONJ
cana-2702	656	10	then	then	ADV
cana-2702	656	11	move	move	VERB
cana-2702	656	12	2𝑡	2𝑡	NOUN
cana-2702	656	13	pebble	pebble	ADJ
cana-2702	656	14	to	to	ADP
cana-2702	656	15	𝑦.	𝑦.	NOUN
cana-2702	656	16	case	case	NOUN
cana-2702	656	17	(	(	PUNCT
cana-2702	656	18	2	2	X
cana-2702	656	19	)	)	PUNCT
cana-2702	656	20	𝑥	𝑥	NOUN
cana-2702	656	21	is	be	AUX
cana-2702	656	22	even	even	ADV
cana-2702	656	23	.	.	PUNCT
cana-2702	657	1	we	we	PRON
cana-2702	657	2	may	may	AUX
cana-2702	657	3	express	express	VERB
cana-2702	657	4	this	this	PRON
cana-2702	657	5	as	as	ADP
cana-2702	657	6	𝑥	𝑥	PROPN
cana-2702	657	7	=	=	SYM
cana-2702	657	8	2𝑥′.	2𝑥′.	NUM
cana-2702	657	9	we	we	PRON
cana-2702	657	10	need	need	VERB
cana-2702	657	11	to	to	PART
cana-2702	657	12	transport	transport	VERB
cana-2702	657	13	2(𝑡	2(𝑡	NUM
cana-2702	658	1	−	−	PROPN
cana-2702	658	2	𝑥′	𝑥′	NUM
cana-2702	658	3	)	)	PUNCT
cana-2702	658	4	more	more	ADJ
cana-2702	658	5	pebbles	pebble	NOUN
cana-2702	658	6	to	to	PART
cana-2702	658	7	𝑣.	𝑣.	VERB
cana-2702	658	8	since	since	SCONJ
cana-2702	658	9	,	,	PUNCT
cana-2702	658	10	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	658	11	)	)	PUNCT
cana-2702	658	12	−	−	PROPN
cana-2702	658	13	2𝑥′	2𝑥′	NUM
cana-2702	658	14	=	=	SYM
cana-2702	658	15	2(8𝑡	2(8𝑡	NUM
cana-2702	658	16	+	+	NUM
cana-2702	658	17	3𝑛	3𝑛	NUM
cana-2702	658	18	−	−	NOUN
cana-2702	658	19	10	10	NUM
cana-2702	658	20	)	)	PUNCT
cana-2702	658	21	−	−	ADP
cana-2702	658	22	𝑞	𝑞	X
cana-2702	658	23	+	+	NOUN
cana-2702	658	24	1	1	NUM
cana-2702	658	25	−	−	PROPN
cana-2702	658	26	2𝑥′	2𝑥′	NUM
cana-2702	658	27	=	=	SYM
cana-2702	658	28	2(8(𝑡	2(8(𝑡	NUM
cana-2702	658	29	−	−	PROPN
cana-2702	658	30	𝑥′	𝑥′	NUM
cana-2702	658	31	)	)	PUNCT
cana-2702	659	1	+	+	CCONJ
cana-2702	659	2	8𝑥′	8𝑥′	NUM
cana-2702	659	3	+	+	SYM
cana-2702	659	4	3𝑛	3𝑛	NUM
cana-2702	659	5	−	−	NUM
cana-2702	659	6	10	10	NUM
cana-2702	659	7	)	)	PUNCT
cana-2702	659	8	−	−	ADP
cana-2702	659	9	𝑞	𝑞	X
cana-2702	659	10	+	+	NOUN
cana-2702	659	11	1	1	NUM
cana-2702	659	12	−	−	PROPN
cana-2702	659	13	2𝑥′	2𝑥′	NUM
cana-2702	659	14	>	>	PUNCT
cana-2702	659	15	2(8(𝑡	2(8(𝑡	NUM
cana-2702	660	1	−	−	PROPN
cana-2702	660	2	𝑥′	𝑥′	NUM
cana-2702	660	3	)	)	PUNCT
cana-2702	661	1	+	+	CCONJ
cana-2702	661	2	3𝑛	3𝑛	NUM
cana-2702	661	3	−	−	NUM
cana-2702	661	4	10	10	NUM
cana-2702	661	5	)	)	PUNCT
cana-2702	661	6	−	−	ADP
cana-2702	661	7	𝑞	𝑞	X
cana-2702	661	8	+	+	NOUN
cana-2702	661	9	1	1	NUM
cana-2702	661	10	+	+	NUM
cana-2702	661	11	16𝑥	16𝑥	NUM
cana-2702	662	1	′	′	NUM
cana-2702	662	2	−	−	PROPN
cana-2702	662	3	2𝑥	2𝑥	PROPN
cana-2702	662	4	>	>	SYM
cana-2702	662	5	2𝑓𝑡−𝑥′(𝐶3(𝐾𝑛	2𝑓𝑡−𝑥′(𝐶3(𝐾𝑛	NUM
cana-2702	662	6	)	)	PUNCT
cana-2702	662	7	)	)	PUNCT
cana-2702	663	1	−	−	PROPN
cana-2702	664	1	𝑞	𝑞	X
cana-2702	664	2	+	+	NOUN
cana-2702	664	3	1	1	X
cana-2702	664	4	.	.	PUNCT
cana-2702	664	5	then	then	ADV
cana-2702	664	6	we	we	PRON
cana-2702	664	7	may	may	AUX
cana-2702	664	8	add	add	VERB
cana-2702	664	9	another	another	DET
cana-2702	664	10	2(𝑡	2(𝑡	NUM
cana-2702	664	11	−	−	PROPN
cana-2702	664	12	𝑥′	𝑥′	NUM
cana-2702	664	13	)	)	PUNCT
cana-2702	664	14	pebbles	pebble	NOUN
cana-2702	664	15	to	to	ADP
cana-2702	664	16	𝑣.	𝑣.	PROPN
cana-2702	664	17	subcase	subcase	PROPN
cana-2702	664	18	(	(	PUNCT
cana-2702	664	19	2a	2a	NUM
cana-2702	664	20	)	)	PUNCT
cana-2702	665	1	𝑥	𝑥	PROPN
cana-2702	665	2	is	be	AUX
cana-2702	665	3	odd	odd	ADJ
cana-2702	665	4	.	.	PUNCT
cana-2702	666	1	we	we	PRON
cana-2702	666	2	may	may	AUX
cana-2702	666	3	express	express	VERB
cana-2702	666	4	this	this	PRON
cana-2702	666	5	as	as	ADP
cana-2702	666	6	𝑥	𝑥	PROPN
cana-2702	666	7	=	=	SYM
cana-2702	666	8	2𝑥′	2𝑥′	NUM
cana-2702	666	9	+	+	CCONJ
cana-2702	666	10	1	1	X
cana-2702	666	11	.	.	X
cana-2702	666	12	we	we	PRON
cana-2702	666	13	need	need	VERB
cana-2702	666	14	to	to	PART
cana-2702	666	15	transport	transport	VERB
cana-2702	666	16	another	another	DET
cana-2702	666	17	2𝑡	2𝑡	NOUN
cana-2702	667	1	−	−	ADP
cana-2702	667	2	2𝑥′	2𝑥′	NUM
cana-2702	667	3	−	−	NOUN
cana-2702	667	4	1	1	NUM
cana-2702	667	5	pebble	pebble	ADJ
cana-2702	667	6	to	to	ADP
cana-2702	667	7	𝑣.	𝑣.	VERB
cana-2702	667	8	because	because	SCONJ
cana-2702	667	9	𝑝(𝐶3(𝐾𝑛	𝑝(𝐶3(𝐾𝑛	NOUN
cana-2702	667	10	)	)	PUNCT
cana-2702	667	11	−	−	PROPN
cana-2702	667	12	𝑝(𝐶3	𝑝(𝐶3	PROPN
cana-2702	667	13	)	)	PUNCT
cana-2702	667	14	)	)	PUNCT
cana-2702	667	15	≥	≥	NOUN
cana-2702	667	16	2𝑓𝑡−𝑥′(𝐶2(𝐾𝑛	2𝑓𝑡−𝑥′(𝐶2(𝐾𝑛	NUM
cana-2702	667	17	)	)	PUNCT
cana-2702	667	18	)	)	PUNCT
cana-2702	668	1	−	−	PROPN
cana-2702	669	1	𝑞	𝑞	X
cana-2702	669	2	+	+	NOUN
cana-2702	669	3	1	1	X
cana-2702	669	4	.	.	X
cana-2702	669	5	we	we	PRON
cana-2702	669	6	can	can	AUX
cana-2702	669	7	transfer	transfer	VERB
cana-2702	669	8	more	more	ADJ
cana-2702	669	9	2(𝑡	2(𝑡	NUM
cana-2702	669	10	−	−	PROPN
cana-2702	669	11	𝑥′	𝑥′	NUM
cana-2702	669	12	)	)	PUNCT
cana-2702	669	13	pebbles	pebble	NOUN
cana-2702	669	14	to	to	ADP
cana-2702	669	15	𝑣	𝑣	NOUN
cana-2702	669	16	via	via	ADP
cana-2702	669	17	induction	induction	NOUN
cana-2702	669	18	.	.	PUNCT
cana-2702	670	1	theorem	theorem	NOUN
cana-2702	670	2	12	12	NUM
cana-2702	670	3	.	.	PUNCT
cana-2702	671	1	the	the	DET
cana-2702	671	2	graph	graph	NOUN
cana-2702	671	3	𝐶𝑚(𝐾𝑛	𝐶𝑚(𝐾𝑛	NOUN
cana-2702	671	4	)	)	PUNCT
cana-2702	671	5	exhibits	exhibit	VERB
cana-2702	671	6	the	the	DET
cana-2702	671	7	2𝑡	2𝑡	NOUN
cana-2702	671	8	−	−	ADP
cana-2702	671	9	pebbling	pebble	VERB
cana-2702	671	10	property	property	NOUN
cana-2702	671	11	.	.	PUNCT
cana-2702	672	1	proof	proof	NOUN
cana-2702	672	2	.	.	PUNCT
cana-2702	673	1	consider	consider	VERB
cana-2702	673	2	a	a	DET
cana-2702	673	3	graph	graph	NOUN
cana-2702	673	4	with	with	ADP
cana-2702	673	5	at	at	ADP
cana-2702	673	6	least	least	ADJ
cana-2702	673	7	2(𝑡2𝑚	2(𝑡2𝑚	NUM
cana-2702	673	8	+	+	CCONJ
cana-2702	673	9	2(𝑛	2(𝑛	NUM
cana-2702	673	10	−	−	NUM
cana-2702	673	11	3	3	NUM
cana-2702	673	12	)	)	PUNCT
cana-2702	673	13	+	+	CCONJ
cana-2702	673	14	(	(	PUNCT
cana-2702	673	15	𝑚	𝑚	PROPN
cana-2702	673	16	−	−	PROPN
cana-2702	673	17	2)(𝑛	2)(𝑛	NUM
cana-2702	673	18	−	−	NOUN
cana-2702	673	19	4	4	NUM
cana-2702	673	20	)	)	PUNCT
cana-2702	673	21	)	)	PUNCT
cana-2702	673	22	−	−	PROPN
cana-2702	674	1	𝑞	𝑞	X
cana-2702	674	2	+	+	CCONJ
cana-2702	674	3	1	1	NUM
cana-2702	674	4	pebbles	pebble	NOUN
cana-2702	674	5	distributed	distribute	VERB
cana-2702	674	6	at	at	ADP
cana-2702	674	7	its	its	PRON
cana-2702	674	8	vertices	vertex	NOUN
cana-2702	674	9	.	.	PUNCT
cana-2702	675	1	the	the	DET
cana-2702	675	2	2𝑡	2𝑡	NUM
cana-2702	675	3	pebbles	pebble	NOUN
cana-2702	675	4	must	must	AUX
cana-2702	675	5	be	be	AUX
cana-2702	675	6	moved	move	VERB
cana-2702	675	7	to	to	ADP
cana-2702	675	8	any	any	DET
cana-2702	675	9	target	target	NOUN
cana-2702	675	10	vertex	vertex	NOUN
cana-2702	675	11	.	.	PUNCT
cana-2702	676	1	for	for	ADP
cana-2702	676	2	1	1	NUM
cana-2702	676	3	≤	≤	NUM
cana-2702	676	4	𝑖	𝑖	SYM
cana-2702	676	5	≤	≤	NOUN
cana-2702	676	6	𝑚	𝑚	NOUN
cana-2702	676	7	,	,	PUNCT
cana-2702	676	8	consider	consider	VERB
cana-2702	676	9	𝑣	𝑣	DET
cana-2702	676	10	∈	∈	PROPN
cana-2702	676	11	𝐶𝑖.	𝐶𝑖.	PROPN
cana-2702	676	12	we	we	PRON
cana-2702	676	13	have	have	AUX
cana-2702	676	14	identified	identify	VERB
cana-2702	676	15	the	the	DET
cana-2702	676	16	following	following	ADJ
cana-2702	676	17	cases	case	NOUN
cana-2702	676	18	.	.	PUNCT
cana-2702	677	1	case	case	NOUN
cana-2702	677	2	(	(	PUNCT
cana-2702	677	3	1	1	X
cana-2702	677	4	)	)	PUNCT
cana-2702	677	5	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	677	6	)	)	PUNCT
cana-2702	677	7	=	=	SYM
cana-2702	678	1	0	0	X
cana-2702	678	2	.	.	PUNCT
cana-2702	678	3	by	by	ADP
cana-2702	678	4	demonstrating	demonstrate	VERB
cana-2702	678	5	that	that	SCONJ
cana-2702	678	6	this	this	DET
cana-2702	678	7	conclusion	conclusion	NOUN
cana-2702	678	8	is	be	AUX
cana-2702	678	9	true	true	ADJ
cana-2702	678	10	for	for	ADP
cana-2702	678	11	all	all	DET
cana-2702	678	12	values	value	NOUN
cana-2702	678	13	of	of	ADP
cana-2702	678	14	𝑡	𝑡	PROPN
cana-2702	678	15	and	and	CCONJ
cana-2702	678	16	𝑚	𝑚	PROPN
cana-2702	678	17	,	,	PUNCT
cana-2702	678	18	we	we	PRON
cana-2702	678	19	can	can	AUX
cana-2702	678	20	verify	verify	VERB
cana-2702	678	21	it	it	PRON
cana-2702	678	22	.	.	PUNCT
cana-2702	679	1	due	due	ADP
cana-2702	679	2	to	to	ADP
cana-2702	679	3	the	the	DET
cana-2702	679	4	fact	fact	NOUN
cana-2702	679	5	that	that	SCONJ
cana-2702	679	6	it	it	PRON
cana-2702	679	7	follows	follow	VERB
cana-2702	679	8	from	from	ADP
cana-2702	679	9	theorems	theorem	NOUN
cana-2702	679	10	10	10	NUM
cana-2702	679	11	and	and	CCONJ
cana-2702	679	12	11	11	NUM
cana-2702	679	13	,	,	PUNCT
cana-2702	679	14	we	we	PRON
cana-2702	679	15	know	know	VERB
cana-2702	679	16	the	the	DET
cana-2702	679	17	conclusion	conclusion	NOUN
cana-2702	679	18	is	be	AUX
cana-2702	679	19	true	true	ADJ
cana-2702	679	20	for	for	ADP
cana-2702	679	21	𝑡	𝑡	PROPN
cana-2702	679	22	=	=	SYM
cana-2702	679	23	1	1	NUM
cana-2702	679	24	,	,	PUNCT
cana-2702	679	25	𝑚	𝑚	X
cana-2702	679	26	=	=	SYM
cana-2702	679	27	2	2	NUM
cana-2702	679	28	,	,	PUNCT
cana-2702	679	29	and	and	CCONJ
cana-2702	679	30	𝑚	𝑚	X
cana-2702	679	31	=	=	SYM
cana-2702	679	32	3	3	X
cana-2702	679	33	.	.	X
cana-2702	680	1	for	for	ADP
cana-2702	680	2	any	any	DET
cana-2702	680	3	values	value	NOUN
cana-2702	680	4	of	of	ADP
cana-2702	680	5	𝑡′	𝑡′	PUNCT
cana-2702	680	6	that	that	PRON
cana-2702	680	7	are	be	AUX
cana-2702	680	8	smaller	small	ADJ
cana-2702	680	9	than	than	ADP
cana-2702	680	10	𝑡	𝑡	PROPN
cana-2702	680	11	,	,	PUNCT
cana-2702	680	12	we	we	PRON
cana-2702	680	13	assume	assume	VERB
cana-2702	680	14	that	that	SCONJ
cana-2702	680	15	the	the	DET
cana-2702	680	16	conclusion	conclusion	NOUN
cana-2702	680	17	is	be	AUX
cana-2702	680	18	true	true	ADJ
cana-2702	680	19	.	.	PUNCT
cana-2702	681	1	then	then	ADV
cana-2702	681	2	,	,	PUNCT
cana-2702	681	3	we	we	PRON
cana-2702	681	4	demonstrate	demonstrate	VERB
cana-2702	681	5	that	that	SCONJ
cana-2702	681	6	the	the	DET
cana-2702	681	7	conclusion	conclusion	NOUN
cana-2702	681	8	must	must	AUX
cana-2702	681	9	be	be	AUX
cana-2702	681	10	true	true	ADJ
cana-2702	681	11	for	for	ADP
cana-2702	681	12	𝑡	𝑡	NOUN
cana-2702	681	13	if	if	SCONJ
cana-2702	681	14	it	it	PRON
cana-2702	681	15	is	be	AUX
cana-2702	681	16	true	true	ADJ
cana-2702	681	17	for	for	ADP
cana-2702	681	18	all	all	DET
cana-2702	681	19	values	value	NOUN
cana-2702	681	20	of	of	ADP
cana-2702	681	21	𝑡′	𝑡′	PUNCT
cana-2702	681	22	that	that	PRON
cana-2702	681	23	are	be	AUX
cana-2702	681	24	smaller	small	ADJ
cana-2702	681	25	than	than	SCONJ
cana-2702	681	26	𝑡.	𝑡.	NOUN
cana-2702	681	27	take	take	VERB
cana-2702	681	28	the	the	DET
cana-2702	681	29	value	value	NOUN
cana-2702	681	30	𝑚	𝑚	NOUN
cana-2702	681	31	=	=	SYM
cana-2702	681	32	𝑚1	𝑚1	NOUN
cana-2702	681	33	+	+	NUM
cana-2702	681	34	𝑚2	𝑚2	NOUN
cana-2702	681	35	.	.	PUNCT
cana-2702	682	1	undoubtedly	undoubtedly	ADV
cana-2702	682	2	,	,	PUNCT
cana-2702	682	3	𝑣	𝑣	DET
cana-2702	682	4	∈	∈	PROPN
cana-2702	682	5	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	682	6	.	.	PUNCT
cana-2702	683	1	establish	establish	VERB
cana-2702	683	2	the	the	DET
cana-2702	683	3	definitions	definition	NOUN
cana-2702	683	4	of	of	ADP
cana-2702	683	5	𝑋1	𝑋1	NOUN
cana-2702	683	6	=	=	SYM
cana-2702	683	7	𝐶1	𝐶1	ADJ
cana-2702	683	8	∪.	∪.	X
cana-2702	683	9	.	.	PUNCT
cana-2702	684	1	.∪	.∪	PROPN
cana-2702	684	2	𝐶𝑚1	𝐶𝑚1	NOUN
cana-2702	684	3	and	and	CCONJ
cana-2702	684	4	𝑋2	𝑋2	VERB
cana-2702	684	5	=	=	SYM
cana-2702	684	6	𝐶𝑚1	𝐶𝑚1	X
cana-2702	684	7	+	+	ADJ
cana-2702	684	8	1	1	NUM
cana-2702	684	9	∪.	∪.	NOUN
cana-2702	684	10	.	.	PUNCT
cana-2702	685	1	.∪	.∪	PROPN
cana-2702	685	2	𝐶𝑚.	𝐶𝑚.	PROPN
cana-2702	685	3	consider	consider	VERB
cana-2702	685	4	the	the	DET
cana-2702	685	5	scenario	scenario	NOUN
cana-2702	685	6	where	where	SCONJ
cana-2702	685	7	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	685	8	)	)	PUNCT
cana-2702	685	9	≥	≥	NOUN
cana-2702	685	10	2(2𝑚1(2𝑚2	2(2𝑚1(2𝑚2	NUM
cana-2702	686	1	−	−	ADP
cana-2702	686	2	1	1	NUM
cana-2702	686	3	)	)	PUNCT
cana-2702	686	4	+	+	CCONJ
cana-2702	686	5	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	686	6	−	−	PROPN
cana-2702	686	7	4	4	NUM
cana-2702	686	8	)	)	PUNCT
cana-2702	686	9	+	+	CCONJ
cana-2702	686	10	2	2	X
cana-2702	686	11	)	)	PUNCT
cana-2702	686	12	−	−	NOUN
cana-2702	686	13	𝑞2	𝑞2	NOUN
cana-2702	686	14	+	+	CCONJ
cana-2702	686	15	1	1	X
cana-2702	686	16	.	.	PUNCT
cana-2702	686	17	based	base	VERB
cana-2702	686	18	on	on	ADP
cana-2702	686	19	lemma	lemma	PROPN
cana-2702	686	20	1	1	NUM
cana-2702	686	21	,	,	PUNCT
cana-2702	686	22	four	four	NUM
cana-2702	686	23	pebbles	pebble	NOUN
cana-2702	686	24	can	can	AUX
cana-2702	686	25	move	move	VERB
cana-2702	686	26	to	to	ADP
cana-2702	686	27	either	either	CCONJ
cana-2702	686	28	𝑎	𝑎	PRON
cana-2702	686	29	𝑚1	𝑚1	NOUN
cana-2702	686	30	(	(	PUNCT
cana-2702	686	31	𝑛	𝑛	PROPN
cana-2702	686	32	2	2	NUM
cana-2702	686	33	−1	−1	NOUN
cana-2702	686	34	)	)	PUNCT
cana-2702	686	35	or	or	CCONJ
cana-2702	686	36	𝑎	𝑎	PRON
cana-2702	686	37	𝑚1	𝑚1	NOUN
cana-2702	686	38	(	(	PUNCT
cana-2702	686	39	𝑛	𝑛	PROPN
cana-2702	686	40	2	2	NUM
cana-2702	686	41	)	)	PUNCT
cana-2702	686	42	+1	+1	PROPN
cana-2702	686	43	,	,	PUNCT
cana-2702	686	44	from	from	ADP
cana-2702	686	45	which	which	PRON
cana-2702	686	46	we	we	PRON
cana-2702	686	47	can	can	AUX
cana-2702	686	48	move	move	VERB
cana-2702	686	49	two	two	NUM
cana-2702	686	50	pebbles	pebble	NOUN
cana-2702	686	51	to	to	PART
cana-2702	686	52	𝑣.	𝑣.	VERB
cana-2702	686	53	as	as	ADP
cana-2702	686	54	a	a	DET
cana-2702	686	55	result	result	NOUN
cana-2702	686	56	,	,	PUNCT
cana-2702	686	57	we	we	PRON
cana-2702	686	58	must	must	AUX
cana-2702	686	59	transfer	transfer	VERB
cana-2702	686	60	an	an	DET
cana-2702	686	61	extra	extra	ADJ
cana-2702	686	62	2(𝑡	2(𝑡	NUM
cana-2702	686	63	−	−	NOUN
cana-2702	686	64	1	1	NUM
cana-2702	686	65	)	)	PUNCT
cana-2702	686	66	pebbles	pebble	NOUN
cana-2702	686	67	to	to	PART
cana-2702	686	68	𝑣.	𝑣.	NOUN
cana-2702	686	69	claim	claim	NOUN
cana-2702	686	70	(	(	PUNCT
cana-2702	686	71	1	1	X
cana-2702	686	72	)	)	PUNCT
cana-2702	686	73	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NUM
cana-2702	686	74	)	)	PUNCT
cana-2702	686	75	)	)	PUNCT
cana-2702	687	1	−	−	PROPN
cana-2702	687	2	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	687	3	)	)	PUNCT
cana-2702	687	4	≥	≥	NOUN
cana-2702	687	5	2((𝑡	2((𝑡	NUM
cana-2702	687	6	−	−	X
cana-2702	687	7	1)2𝑚1	1)2𝑚1	NUM
cana-2702	688	1	+	+	CCONJ
cana-2702	688	2	2(𝑛	2(𝑛	NUM
cana-2702	688	3	−	−	NOUN
cana-2702	688	4	3	3	NUM
cana-2702	688	5	)	)	PUNCT
cana-2702	688	6	+	+	CCONJ
cana-2702	688	7	(	(	PUNCT
cana-2702	688	8	𝑚1	𝑚1	NOUN
cana-2702	688	9	−	−	PROPN
cana-2702	688	10	2)(𝑛	2)(𝑛	NUM
cana-2702	688	11	−	−	NOUN
cana-2702	688	12	4	4	NUM
cana-2702	688	13	)	)	PUNCT
cana-2702	688	14	)	)	PUNCT
cana-2702	689	1	−	−	PROPN
cana-2702	690	1	𝑞1	𝑞1	ADJ
cana-2702	690	2	+	+	CCONJ
cana-2702	690	3	1	1	X
cana-2702	690	4	.	.	X
cana-2702	690	5	we	we	PRON
cana-2702	690	6	have	have	VERB
cana-2702	690	7	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NOUN
cana-2702	690	8	)	)	PUNCT
cana-2702	690	9	)	)	PUNCT
cana-2702	691	1	−	−	PROPN
cana-2702	692	1	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	692	2	)	)	PUNCT
cana-2702	692	3	=	=	SYM
cana-2702	692	4	2	2	NUM
cana-2702	692	5	(	(	PUNCT
cana-2702	692	6	(	(	PUNCT
cana-2702	692	7	𝑡	𝑡	NOUN
cana-2702	692	8	−	−	NOUN
cana-2702	692	9	1)2𝑚1+𝑚2	1)2𝑚1+𝑚2	NUM
cana-2702	692	10	+	+	CCONJ
cana-2702	692	11	2	2	NUM
cana-2702	692	12	𝑚1	𝑚1	NOUN
cana-2702	692	13	+	+	CCONJ
cana-2702	692	14	2(𝑛	2(𝑛	NUM
cana-2702	692	15	−	−	NOUN
cana-2702	692	16	3	3	NUM
cana-2702	692	17	)	)	PUNCT
cana-2702	692	18	+	+	CCONJ
cana-2702	692	19	(	(	PUNCT
cana-2702	692	20	𝑚1	𝑚1	NOUN
cana-2702	692	21	+	+	CCONJ
cana-2702	692	22	𝑚2	𝑚2	PROPN
cana-2702	692	23	−	−	PROPN
cana-2702	692	24	2)(𝑛	2)(𝑛	NUM
cana-2702	692	25	−	−	NOUN
cana-2702	692	26	4	4	NUM
cana-2702	692	27	)	)	PUNCT
cana-2702	692	28	−	−	NOUN
cana-2702	692	29	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	692	30	−	−	PROPN
cana-2702	692	31	4	4	NUM
cana-2702	692	32	)	)	PUNCT
cana-2702	692	33	−	−	ADP
cana-2702	692	34	2	2	NUM
cana-2702	692	35	)	)	PUNCT
cana-2702	692	36	−	−	PROPN
cana-2702	692	37	𝑞1	𝑞1	X
cana-2702	692	38	>	>	X
cana-2702	692	39	2((𝑡	2((𝑡	NUM
cana-2702	693	1	−	−	X
cana-2702	693	2	1)2𝑚1	1)2𝑚1	NUM
cana-2702	694	1	+	+	CCONJ
cana-2702	694	2	2(𝑛	2(𝑛	NUM
cana-2702	694	3	−	−	NOUN
cana-2702	694	4	3	3	NUM
cana-2702	694	5	)	)	PUNCT
cana-2702	694	6	+	+	CCONJ
cana-2702	694	7	(	(	PUNCT
cana-2702	694	8	𝑚1	𝑚1	NOUN
cana-2702	694	9	−	−	PROPN
cana-2702	694	10	2)(𝑛	2)(𝑛	NUM
cana-2702	694	11	−	−	NOUN
cana-2702	694	12	4	4	NUM
cana-2702	694	13	)	)	PUNCT
cana-2702	694	14	)	)	PUNCT
cana-2702	695	1	−	−	PROPN
cana-2702	696	1	𝑞1	𝑞1	ADJ
cana-2702	696	2	+	+	CCONJ
cana-2702	696	3	1	1	NUM
cana-2702	696	4	,	,	PUNCT
cana-2702	696	5	since	since	SCONJ
cana-2702	696	6	𝑚1	𝑚1	PROPN
cana-2702	696	7	≥	≥	PROPN
cana-2702	696	8	2	2	NUM
cana-2702	696	9	>	>	SYM
cana-2702	696	10	2𝑓𝑡−1	2𝑓𝑡−1	NUM
cana-2702	696	11	(	(	PUNCT
cana-2702	696	12	𝐶𝑚1	𝐶𝑚1	X
cana-2702	696	13	(	(	PUNCT
cana-2702	696	14	𝐾𝑛	𝐾𝑛	NOUN
cana-2702	696	15	)	)	PUNCT
cana-2702	696	16	)	)	PUNCT
cana-2702	697	1	−	−	PROPN
cana-2702	698	1	𝑞1	𝑞1	ADJ
cana-2702	698	2	+	+	CCONJ
cana-2702	698	3	1	1	X
cana-2702	698	4	.	.	X
cana-2702	698	5	we	we	PRON
cana-2702	698	6	may	may	AUX
cana-2702	698	7	transfer	transfer	VERB
cana-2702	698	8	the	the	DET
cana-2702	698	9	2(𝑡	2(𝑡	NUM
cana-2702	698	10	−	−	NOUN
cana-2702	698	11	1	1	NUM
cana-2702	698	12	)	)	PUNCT
cana-2702	698	13	pebbles	pebble	NOUN
cana-2702	698	14	to	to	ADP
cana-2702	698	15	𝑣	𝑣	ADP
cana-2702	698	16	from	from	ADP
cana-2702	698	17	claim	claim	NOUN
cana-2702	698	18	(	(	PUNCT
cana-2702	698	19	1	1	NUM
cana-2702	698	20	)	)	PUNCT
cana-2702	698	21	.	.	PUNCT
cana-2702	699	1	as	as	ADP
cana-2702	699	2	a	a	DET
cana-2702	699	3	result	result	NOUN
cana-2702	699	4	,	,	PUNCT
cana-2702	699	5	we	we	PRON
cana-2702	699	6	presume	presume	VERB
cana-2702	699	7	that	that	SCONJ
cana-2702	699	8	𝑝(𝑋2	𝑝(𝑋2	NOUN
cana-2702	699	9	)	)	PUNCT
cana-2702	699	10	≤	≤	NOUN
cana-2702	699	11	2(2𝑚1(2𝑚2−1	2(2𝑚1(2𝑚2−1	NUM
cana-2702	699	12	)	)	PUNCT
cana-2702	700	1	+	+	CCONJ
cana-2702	700	2	𝑚2(𝑛	𝑚2(𝑛	NUM
cana-2702	700	3	−	−	PROPN
cana-2702	700	4	4	4	NUM
cana-2702	700	5	)	)	PUNCT
cana-2702	700	6	+	+	CCONJ
cana-2702	700	7	2	2	X
cana-2702	700	8	)	)	PUNCT
cana-2702	700	9	−	−	NOUN
cana-2702	700	10	𝑞2	𝑞2	NOUN
cana-2702	700	11	,	,	PUNCT
cana-2702	700	12	which	which	PRON
cana-2702	700	13	results	result	VERB
cana-2702	700	14	in	in	ADP
cana-2702	700	15	the	the	DET
cana-2702	700	16	claim	claim	NOUN
cana-2702	700	17	that	that	PRON
cana-2702	700	18	follows	follow	VERB
cana-2702	700	19	below	below	ADV
cana-2702	700	20	.	.	PUNCT
cana-2702	701	1	claim(2)𝑝(𝐶𝑚(𝐾𝑛	claim(2)𝑝(𝐶𝑚(𝐾𝑛	NOUN
cana-2702	701	2	)	)	PUNCT
cana-2702	701	3	)	)	PUNCT
cana-2702	702	1	−	−	PROPN
cana-2702	702	2	𝑝(𝑋2	𝑝(𝑋2	PROPN
cana-2702	702	3	)	)	PUNCT
cana-2702	702	4	≥	≥	NOUN
cana-2702	702	5	2𝑓𝑡	2𝑓𝑡	NOUN
cana-2702	702	6	(	(	PUNCT
cana-2702	702	7	𝐶𝑚1	𝐶𝑚1	X
cana-2702	702	8	(	(	PUNCT
cana-2702	702	9	𝐾𝑛)).we	𝐾𝑛)).we	NOUN
cana-2702	702	10	have	have	VERB
cana-2702	702	11	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NOUN
cana-2702	702	12	)	)	PUNCT
cana-2702	702	13	)	)	PUNCT
cana-2702	703	1	−	−	PROPN
cana-2702	703	2	𝑝	𝑝	NOUN
cana-2702	703	3	=	=	SYM
cana-2702	703	4	2(𝑡2𝑚1+𝑚2	2(𝑡2𝑚1+𝑚2	NUM
cana-2702	703	5	−	−	NOUN
cana-2702	703	6	2	2	NUM
cana-2702	703	7	𝑚1+𝑚2	𝑚1+𝑚2	PROPN
cana-2702	703	8	+	+	CCONJ
cana-2702	703	9	2	2	NUM
cana-2702	703	10	𝑚1	𝑚1	NOUN
cana-2702	703	11	+	+	CCONJ
cana-2702	703	12	2(𝑛	2(𝑛	NUM
cana-2702	703	13	−	−	NOUN
cana-2702	703	14	3	3	NUM
cana-2702	703	15	)	)	PUNCT
cana-2702	703	16	+	+	CCONJ
cana-2702	703	17	(	(	PUNCT
cana-2702	703	18	𝑚1	𝑚1	NOUN
cana-2702	703	19	−	−	PROPN
cana-2702	703	20	2)(𝑛	2)(𝑛	NUM
cana-2702	703	21	−	−	NOUN
cana-2702	703	22	4	4	NUM
cana-2702	703	23	)	)	PUNCT
cana-2702	703	24	−	−	ADP
cana-2702	703	25	2	2	NUM
cana-2702	703	26	)	)	PUNCT
cana-2702	703	27	−	−	PROPN
cana-2702	704	1	𝑞1	𝑞1	ADJ
cana-2702	704	2	+	+	CCONJ
cana-2702	704	3	1	1	NUM
cana-2702	704	4	>	>	SYM
cana-2702	704	5	2(𝑡2𝑚1	2(𝑡2𝑚1	NUM
cana-2702	705	1	+	+	CCONJ
cana-2702	705	2	2(𝑛	2(𝑛	NUM
cana-2702	705	3	−	−	NOUN
cana-2702	705	4	3	3	NUM
cana-2702	705	5	)	)	PUNCT
cana-2702	705	6	+	+	CCONJ
cana-2702	705	7	(	(	PUNCT
cana-2702	705	8	𝑚1	𝑚1	NOUN
cana-2702	705	9	−	−	PROPN
cana-2702	705	10	2)(𝑛	2)(𝑛	NUM
cana-2702	705	11	−	−	NOUN
cana-2702	705	12	4	4	NUM
cana-2702	705	13	)	)	PUNCT
cana-2702	705	14	)	)	PUNCT
cana-2702	706	1	−	−	PROPN
cana-2702	707	1	𝑞1	𝑞1	ADJ
cana-2702	707	2	+	+	CCONJ
cana-2702	707	3	1	1	NUM
cana-2702	707	4	>	>	SYM
cana-2702	707	5	2𝑓𝑡(𝑋1	2𝑓𝑡(𝑋1	NUM
cana-2702	707	6	)	)	PUNCT
cana-2702	707	7	−	−	PROPN
cana-2702	708	1	𝑞1	𝑞1	ADJ
cana-2702	708	2	+	+	CCONJ
cana-2702	708	3	1	1	X
cana-2702	708	4	.	.	X
cana-2702	708	5	we	we	PRON
cana-2702	708	6	can	can	AUX
cana-2702	708	7	transfer	transfer	VERB
cana-2702	708	8	2𝑡	2𝑡	NOUN
cana-2702	708	9	pebbles	pebble	NOUN
cana-2702	708	10	to	to	ADP
cana-2702	708	11	𝑣	𝑣	NOUN
cana-2702	708	12	through	through	ADP
cana-2702	708	13	induction	induction	NOUN
cana-2702	708	14	.	.	PUNCT
cana-2702	709	1	we	we	PRON
cana-2702	709	2	are	be	AUX
cana-2702	709	3	then	then	ADV
cana-2702	709	4	done	do	VERB
cana-2702	709	5	.	.	PUNCT
cana-2702	710	1	case	case	NOUN
cana-2702	710	2	(	(	PUNCT
cana-2702	710	3	2	2	NUM
cana-2702	710	4	)	)	PUNCT
cana-2702	710	5	𝑝(𝑣	𝑝(𝑣	PROPN
cana-2702	710	6	)	)	PUNCT
cana-2702	711	1	=	=	SYM
cana-2702	712	1	𝑥	𝑥	NOUN
cana-2702	712	2	,	,	PUNCT
cana-2702	712	3	for	for	ADP
cana-2702	712	4	𝑥	𝑥	PRON
cana-2702	712	5	≥	≥	NUM
cana-2702	712	6	1	1	NUM
cana-2702	712	7	.	.	PUNCT
cana-2702	713	1	communications	communication	NOUN
cana-2702	713	2	on	on	ADP
cana-2702	713	3	applied	apply	VERB
cana-2702	713	4	nonlinear	nonlinear	ADJ
cana-2702	713	5	analysis	analysis	NOUN
cana-2702	713	6	issn	issn	NOUN
cana-2702	713	7	:	:	PUNCT
cana-2702	713	8	1074	1074	NUM
cana-2702	713	9	-	-	PUNCT
cana-2702	713	10	133x	133x	NUM
cana-2702	713	11	vol	vol	NOUN
cana-2702	713	12	32	32	NUM
cana-2702	713	13	no	no	NOUN
cana-2702	713	14	.	.	PUNCT
cana-2702	714	1	3s	3s	NUM
cana-2702	714	2	(	(	PUNCT
cana-2702	714	3	2025	2025	NUM
cana-2702	714	4	)	)	PUNCT
cana-2702	714	5	662	662	NUM
cana-2702	714	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2702	714	7	more	more	ADJ
cana-2702	714	8	2𝑡	2𝑡	NOUN
cana-2702	714	9	−	−	NOUN
cana-2702	714	10	𝑥	𝑥	PRON
cana-2702	714	11	pebbles	pebble	NOUN
cana-2702	714	12	need	need	VERB
cana-2702	714	13	to	to	PART
cana-2702	714	14	be	be	AUX
cana-2702	714	15	moved	move	VERB
cana-2702	714	16	to	to	PART
cana-2702	714	17	𝑣.	𝑣.	VERB
cana-2702	714	18	the	the	DET
cana-2702	714	19	following	follow	VERB
cana-2702	714	20	subcases	subcase	NOUN
cana-2702	714	21	exist	exist	VERB
cana-2702	714	22	:	:	PUNCT
cana-2702	714	23	subcase	subcase	PROPN
cana-2702	714	24	(	(	PUNCT
cana-2702	714	25	2a	2a	NUM
cana-2702	714	26	)	)	PUNCT
cana-2702	715	1	𝑥	𝑥	PRON
cana-2702	715	2	is	be	AUX
cana-2702	715	3	even	even	ADV
cana-2702	715	4	.	.	PUNCT
cana-2702	716	1	now	now	ADV
cana-2702	716	2	,	,	PUNCT
cana-2702	716	3	𝑥	𝑥	PROPN
cana-2702	716	4	can	can	AUX
cana-2702	716	5	be	be	AUX
cana-2702	716	6	written	write	VERB
cana-2702	716	7	as	as	ADP
cana-2702	716	8	2𝑥′.	2𝑥′.	NUM
cana-2702	716	9	moving	move	VERB
cana-2702	716	10	the	the	DET
cana-2702	716	11	2(𝑡	2(𝑡	NUM
cana-2702	716	12	−	−	PROPN
cana-2702	716	13	𝑥′	𝑥′	NUM
cana-2702	716	14	)	)	PUNCT
cana-2702	717	1	pebbles	pebble	NOUN
cana-2702	717	2	to	to	ADP
cana-2702	717	3	𝑣	𝑣	PROPN
cana-2702	717	4	is	be	AUX
cana-2702	717	5	necessary	necessary	ADJ
cana-2702	717	6	.	.	PUNCT
cana-2702	718	1	it	it	PRON
cana-2702	718	2	is	be	AUX
cana-2702	718	3	evident	evident	ADJ
cana-2702	718	4	that	that	SCONJ
cana-2702	718	5	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NUM
cana-2702	718	6	)	)	PUNCT
cana-2702	718	7	)	)	PUNCT
cana-2702	718	8	−	−	PROPN
cana-2702	719	1	2𝑥′	2𝑥′	NUM
cana-2702	719	2	≥	≥	NOUN
cana-2702	719	3	2𝑓𝑡−𝑥′(𝐶𝑚(𝐾𝑛	2𝑓𝑡−𝑥′(𝐶𝑚(𝐾𝑛	NUM
cana-2702	719	4	)	)	PUNCT
cana-2702	719	5	)	)	PUNCT
cana-2702	720	1	−	−	PROPN
cana-2702	721	1	𝑞	𝑞	X
cana-2702	721	2	+	+	NOUN
cana-2702	721	3	1	1	X
cana-2702	721	4	.	.	PUNCT
cana-2702	722	1	in	in	ADP
cana-2702	722	2	order	order	NOUN
cana-2702	722	3	to	to	PART
cana-2702	722	4	use	use	VERB
cana-2702	722	5	induction	induction	NOUN
cana-2702	722	6	𝑡	𝑡	NOUN
cana-2702	722	7	,	,	PUNCT
cana-2702	722	8	we	we	PRON
cana-2702	722	9	can	can	AUX
cana-2702	722	10	relocate	relocate	VERB
cana-2702	722	11	the	the	DET
cana-2702	722	12	2(𝑡	2(𝑡	NUM
cana-2702	722	13	−	−	PROPN
cana-2702	722	14	𝑥′	𝑥′	NUM
cana-2702	722	15	)	)	PUNCT
cana-2702	722	16	pebbles	pebble	NOUN
cana-2702	722	17	to	to	ADP
cana-2702	722	18	𝑣.	𝑣.	PROPN
cana-2702	722	19	subcase	subcase	PROPN
cana-2702	722	20	(	(	PUNCT
cana-2702	722	21	2b	2b	NUM
cana-2702	722	22	)	)	PUNCT
cana-2702	722	23	𝑥	𝑥	PROPN
cana-2702	722	24	is	be	AUX
cana-2702	722	25	odd	odd	ADJ
cana-2702	722	26	.	.	PUNCT
cana-2702	723	1	let	let	VERB
cana-2702	723	2	𝑥	𝑥	PRON
cana-2702	723	3	may	may	AUX
cana-2702	723	4	be	be	AUX
cana-2702	723	5	expressed	express	VERB
cana-2702	723	6	as	as	ADP
cana-2702	723	7	2𝑥′	2𝑥′	NUM
cana-2702	723	8	+	+	CCONJ
cana-2702	723	9	1	1	NUM
cana-2702	723	10	.	.	X
cana-2702	724	1	moving	move	VERB
cana-2702	724	2	2𝑡	2𝑡	NOUN
cana-2702	724	3	−	−	ADP
cana-2702	724	4	2𝑥′	2𝑥′	NUM
cana-2702	724	5	−	−	NUM
cana-2702	724	6	1	1	NUM
cana-2702	724	7	pebbles	pebble	NOUN
cana-2702	724	8	to	to	ADP
cana-2702	724	9	𝑣	𝑣	PROPN
cana-2702	724	10	is	be	AUX
cana-2702	724	11	necessary	necessary	ADJ
cana-2702	724	12	.	.	PUNCT
cana-2702	725	1	we	we	PRON
cana-2702	725	2	obtain	obtain	VERB
cana-2702	725	3	𝑝(𝐶𝑚(𝐾𝑛	𝑝(𝐶𝑚(𝐾𝑛	NOUN
cana-2702	725	4	)	)	PUNCT
cana-2702	725	5	)	)	PUNCT
cana-2702	726	1	−	−	PROPN
cana-2702	726	2	2𝑥′	2𝑥′	NUM
cana-2702	726	3	−	−	NUM
cana-2702	726	4	1	1	NUM
cana-2702	726	5	≥	≥	NOUN
cana-2702	726	6	2𝑓𝑡−𝑥′(𝐶𝑚(𝐾𝑛	2𝑓𝑡−𝑥′(𝐶𝑚(𝐾𝑛	NUM
cana-2702	726	7	)	)	PUNCT
cana-2702	726	8	)	)	PUNCT
cana-2702	727	1	−	−	PROPN
cana-2702	728	1	𝑞	𝑞	X
cana-2702	728	2	+	+	PROPN
cana-2702	728	3	1	1	NUM
cana-2702	728	4	without	without	ADP
cana-2702	728	5	a	a	DET
cana-2702	728	6	doubt	doubt	NOUN
cana-2702	728	7	.	.	PUNCT
cana-2702	729	1	therefore	therefore	ADV
cana-2702	729	2	,	,	PUNCT
cana-2702	729	3	we	we	PRON
cana-2702	729	4	may	may	AUX
cana-2702	729	5	transfer	transfer	VERB
cana-2702	729	6	2𝑥′	2𝑥′	NUM
cana-2702	729	7	−	−	NUM
cana-2702	729	8	1	1	NUM
cana-2702	729	9	more	more	ADJ
cana-2702	729	10	pebbles	pebble	NOUN
cana-2702	729	11	to	to	ADP
cana-2702	729	12	𝑣.	𝑣.	NOUN
cana-2702	729	13	conclusion	conclusion	NOUN
cana-2702	729	14	:	:	PUNCT
cana-2702	729	15	in	in	ADP
cana-2702	729	16	conclusion	conclusion	NOUN
cana-2702	729	17	,	,	PUNCT
cana-2702	729	18	this	this	DET
cana-2702	729	19	paper	paper	NOUN
cana-2702	729	20	delivers	deliver	VERB
cana-2702	729	21	a	a	DET
cana-2702	729	22	comprehensive	comprehensive	ADJ
cana-2702	729	23	exploration	exploration	NOUN
cana-2702	729	24	of	of	ADP
cana-2702	729	25	the	the	DET
cana-2702	729	26	pebbling	pebble	VERB
cana-2702	729	27	number	number	NOUN
cana-2702	729	28	,	,	PUNCT
cana-2702	729	29	the	the	DET
cana-2702	729	30	twopebbling	twopebble	VERB
cana-2702	729	31	property	property	NOUN
cana-2702	729	32	,	,	PUNCT
cana-2702	729	33	the	the	DET
cana-2702	729	34	t	t	NOUN
cana-2702	729	35	-	-	PUNCT
cana-2702	729	36	pebbling	pebble	VERB
cana-2702	729	37	number	number	NOUN
cana-2702	729	38	,	,	PUNCT
cana-2702	729	39	and	and	CCONJ
cana-2702	729	40	the	the	DET
cana-2702	729	41	2t	2t	NOUN
cana-2702	729	42	-	-	PUNCT
cana-2702	729	43	pebbling	pebble	VERB
cana-2702	729	44	property	property	NOUN
cana-2702	729	45	inside	inside	ADP
cana-2702	729	46	a	a	DET
cana-2702	729	47	crisscross	crisscross	NOUN
cana-2702	729	48	sequence	sequence	NOUN
cana-2702	729	49	of	of	ADP
cana-2702	729	50	m	m	NOUN
cana-2702	729	51	-	-	ADJ
cana-2702	729	52	complete	complete	ADJ
cana-2702	729	53	graphs	graph	NOUN
cana-2702	729	54	.	.	PUNCT
cana-2702	730	1	by	by	ADP
cana-2702	730	2	investigating	investigate	VERB
cana-2702	730	3	these	these	DET
cana-2702	730	4	properties	property	NOUN
cana-2702	730	5	,	,	PUNCT
cana-2702	730	6	we	we	PRON
cana-2702	730	7	have	have	AUX
cana-2702	730	8	expanded	expand	VERB
cana-2702	730	9	valuable	valuable	ADJ
cana-2702	730	10	perceptions	perception	NOUN
cana-2702	730	11	into	into	ADP
cana-2702	730	12	the	the	DET
cana-2702	730	13	productivity	productivity	NOUN
cana-2702	730	14	of	of	ADP
cana-2702	730	15	pebbling	pebble	VERB
cana-2702	730	16	operations	operation	NOUN
cana-2702	730	17	in	in	ADP
cana-2702	730	18	this	this	DET
cana-2702	730	19	structure	structure	NOUN
cana-2702	730	20	.	.	PUNCT
cana-2702	731	1	the	the	DET
cana-2702	731	2	study	study	NOUN
cana-2702	731	3	highlights	highlight	VERB
cana-2702	731	4	the	the	DET
cana-2702	731	5	complications	complication	NOUN
cana-2702	731	6	and	and	CCONJ
cana-2702	731	7	variations	variation	NOUN
cana-2702	731	8	in	in	ADP
cana-2702	731	9	pebbling	pebble	VERB
cana-2702	731	10	dynamics	dynamic	NOUN
cana-2702	731	11	,	,	PUNCT
cana-2702	731	12	offering	offer	VERB
cana-2702	731	13	a	a	DET
cana-2702	731	14	profounder	profounder	NOUN
cana-2702	731	15	understanding	understanding	NOUN
cana-2702	731	16	of	of	ADP
cana-2702	731	17	how	how	SCONJ
cana-2702	731	18	these	these	DET
cana-2702	731	19	properties	property	NOUN
cana-2702	731	20	impact	impact	VERB
cana-2702	731	21	the	the	DET
cana-2702	731	22	behavior	behavior	NOUN
cana-2702	731	23	of	of	ADP
cana-2702	731	24	the	the	DET
cana-2702	731	25	graph	graph	NOUN
cana-2702	731	26	.	.	PUNCT
cana-2702	732	1	this	this	DET
cana-2702	732	2	research	research	NOUN
cana-2702	732	3	contributes	contribute	VERB
cana-2702	732	4	to	to	ADP
cana-2702	732	5	the	the	DET
cana-2702	732	6	broader	broad	ADJ
cana-2702	732	7	field	field	NOUN
cana-2702	732	8	of	of	ADP
cana-2702	732	9	graph	graph	NOUN
cana-2702	732	10	theory	theory	NOUN
cana-2702	732	11	,	,	PUNCT
cana-2702	732	12	precisely	precisely	ADV
cana-2702	732	13	in	in	ADP
cana-2702	732	14	understanding	understand	VERB
cana-2702	732	15	resource	resource	NOUN
cana-2702	732	16	distribution	distribution	NOUN
cana-2702	732	17	and	and	CCONJ
cana-2702	732	18	optimization	optimization	NOUN
cana-2702	732	19	in	in	ADP
cana-2702	732	20	networked	networked	ADJ
cana-2702	732	21	systems	system	NOUN
cana-2702	732	22	.	.	PUNCT
cana-2702	733	1	references	reference	NOUN
cana-2702	733	2	:	:	PUNCT
cana-2702	734	1	[	[	X
cana-2702	734	2	1	1	NUM
cana-2702	734	3	]	]	PUNCT
cana-2702	734	4	alcón	alcón	PROPN
cana-2702	734	5	,	,	PUNCT
cana-2702	734	6	liliana	liliana	PROPN
cana-2702	734	7	,	,	PUNCT
cana-2702	734	8	and	and	CCONJ
cana-2702	734	9	glenn	glenn	PROPN
cana-2702	734	10	hurlbert	hurlbert	PROPN
cana-2702	734	11	.	.	PUNCT
cana-2702	735	1	"	"	PUNCT
cana-2702	735	2	pebbling	pebble	VERB
cana-2702	735	3	in	in	ADP
cana-2702	735	4	powers	power	NOUN
cana-2702	735	5	of	of	ADP
cana-2702	735	6	paths	path	NOUN
cana-2702	735	7	.	.	PUNCT
cana-2702	735	8	"	"	PUNCT
cana-2702	736	1	discrete	discrete	ADJ
cana-2702	736	2	mathematics	mathematic	NOUN
cana-2702	736	3	346.5	346.5	NUM
cana-2702	736	4	(	(	PUNCT
cana-2702	736	5	2023	2023	NUM
cana-2702	736	6	):	):	PUNCT
cana-2702	736	7	113315	113315	NUM
cana-2702	736	8	.	.	PUNCT
cana-2702	737	1	[	[	X
cana-2702	737	2	2	2	NUM
cana-2702	737	3	]	]	PUNCT
cana-2702	737	4	alcón	alcón	PROPN
cana-2702	737	5	,	,	PUNCT
cana-2702	737	6	liliana	liliana	PROPN
cana-2702	737	7	,	,	PUNCT
cana-2702	737	8	marisa	marisa	PROPN
cana-2702	737	9	gutierrez	gutierrez	PROPN
cana-2702	737	10	,	,	PUNCT
cana-2702	737	11	and	and	CCONJ
cana-2702	737	12	glenn	glenn	PROPN
cana-2702	737	13	hurlbert	hurlbert	PROPN
cana-2702	737	14	.	.	PUNCT
cana-2702	738	1	"	"	PUNCT
cana-2702	738	2	pebbling	pebble	VERB
cana-2702	738	3	in	in	ADP
cana-2702	738	4	split	split	ADJ
cana-2702	738	5	graphs	graph	NOUN
cana-2702	738	6	.	.	PUNCT
cana-2702	738	7	"	"	PUNCT
cana-2702	739	1	siam	siam	ADJ
cana-2702	739	2	journal	journal	NOUN
cana-2702	739	3	on	on	ADP
cana-2702	739	4	discrete	discrete	ADJ
cana-2702	739	5	mathematics	mathematic	NOUN
cana-2702	739	6	28.3	28.3	NUM
cana-2702	739	7	(	(	PUNCT
cana-2702	739	8	2014	2014	NUM
cana-2702	739	9	):	):	PUNCT
cana-2702	739	10	1449	1449	NUM
cana-2702	739	11	-	-	SYM
cana-2702	739	12	1466	1466	NUM
cana-2702	739	13	.	.	PUNCT
cana-2702	740	1	[	[	X
cana-2702	740	2	3	3	NUM
cana-2702	740	3	]	]	X
cana-2702	740	4	behzad	behzad	PROPN
cana-2702	740	5	,	,	PUNCT
cana-2702	740	6	mehdi	mehdi	PROPN
cana-2702	740	7	,	,	PUNCT
cana-2702	740	8	gary	gary	PROPN
cana-2702	740	9	chartrand	chartrand	PROPN
cana-2702	740	10	,	,	PUNCT
cana-2702	740	11	and	and	CCONJ
cana-2702	740	12	john	john	PROPN
cana-2702	740	13	k.	k.	PROPN
cana-2702	740	14	cooper	cooper	PROPN
cana-2702	740	15	jr	jr	PROPN
cana-2702	740	16	.	.	PUNCT
cana-2702	741	1	"	"	PUNCT
cana-2702	741	2	the	the	DET
cana-2702	741	3	colour	colour	NOUN
cana-2702	741	4	numbers	number	NOUN
cana-2702	741	5	of	of	ADP
cana-2702	741	6	complete	complete	ADJ
cana-2702	741	7	graphs	graph	NOUN
cana-2702	741	8	.	.	PUNCT
cana-2702	741	9	"	"	PUNCT
cana-2702	742	1	journal	journal	NOUN
cana-2702	742	2	of	of	ADP
cana-2702	742	3	the	the	DET
cana-2702	742	4	london	london	PROPN
cana-2702	742	5	mathematical	mathematical	ADJ
cana-2702	742	6	society	society	NOUN
cana-2702	742	7	1.1	1.1	NUM
cana-2702	742	8	(	(	PUNCT
cana-2702	742	9	1967	1967	NUM
cana-2702	742	10	):	):	PUNCT
cana-2702	742	11	226	226	NUM
cana-2702	742	12	-	-	SYM
cana-2702	742	13	228	228	NUM
cana-2702	742	14	.	.	PUNCT
cana-2702	743	1	[	[	X
cana-2702	743	2	4	4	NUM
cana-2702	743	3	]	]	X
cana-2702	743	4	berven	berven	ADJ
cana-2702	743	5	,	,	PUNCT
cana-2702	743	6	r.	r.	PROPN
cana-2702	743	7	j.	j.	PROPN
cana-2702	743	8	"	"	PUNCT
cana-2702	743	9	cardium	cardium	NOUN
cana-2702	743	10	sandstone	sandstone	NOUN
cana-2702	743	11	bodies	body	NOUN
cana-2702	743	12	,	,	PUNCT
cana-2702	743	13	crossfield	crossfield	NOUN
cana-2702	743	14	-	-	PUNCT
cana-2702	743	15	garrington	garrington	NOUN
cana-2702	743	16	area	area	NOUN
cana-2702	743	17	,	,	PUNCT
cana-2702	743	18	alberta	alberta	PROPN
cana-2702	743	19	.	.	PUNCT
cana-2702	743	20	"	"	PUNCT
cana-2702	744	1	bulletin	bulletin	NOUN
cana-2702	744	2	of	of	ADP
cana-2702	744	3	canadian	canadian	ADJ
cana-2702	744	4	petroleum	petroleum	NOUN
cana-2702	744	5	geology	geology	NOUN
cana-2702	744	6	14.2	14.2	NUM
cana-2702	744	7	(	(	PUNCT
cana-2702	744	8	1966	1966	NUM
cana-2702	744	9	):	):	PUNCT
cana-2702	744	10	208	208	NUM
cana-2702	744	11	-	-	SYM
cana-2702	744	12	240	240	NUM
cana-2702	744	13	.	.	PUNCT
cana-2702	745	1	[	[	X
cana-2702	745	2	5	5	NUM
cana-2702	745	3	]	]	PUNCT
cana-2702	745	4	bianchi	bianchi	NOUN
cana-2702	745	5	,	,	PUNCT
cana-2702	745	6	mariagrazia	mariagrazia	PROPN
cana-2702	745	7	,	,	PUNCT
cana-2702	745	8	et	et	PROPN
cana-2702	745	9	al	al	PROPN
cana-2702	745	10	.	.	PUNCT
cana-2702	746	1	"	"	PUNCT
cana-2702	746	2	character	character	NOUN
cana-2702	746	3	degree	degree	NOUN
cana-2702	746	4	graphs	graph	NOUN
cana-2702	746	5	that	that	PRON
cana-2702	746	6	are	be	AUX
cana-2702	746	7	complete	complete	ADJ
cana-2702	746	8	graphs	graph	NOUN
cana-2702	746	9	.	.	PUNCT
cana-2702	746	10	"	"	PUNCT
cana-2702	747	1	proceedings	proceeding	NOUN
cana-2702	747	2	of	of	ADP
cana-2702	747	3	the	the	DET
cana-2702	747	4	american	american	PROPN
cana-2702	747	5	mathematical	mathematical	PROPN
cana-2702	747	6	society	society	NOUN
cana-2702	747	7	135.3	135.3	NUM
cana-2702	747	8	(	(	PUNCT
cana-2702	747	9	2007	2007	NUM
cana-2702	747	10	):	):	PUNCT
cana-2702	747	11	671	671	NUM
cana-2702	747	12	-	-	SYM
cana-2702	747	13	676	676	NUM
cana-2702	747	14	.	.	PUNCT
cana-2702	748	1	[	[	X
cana-2702	748	2	6	6	NUM
cana-2702	748	3	]	]	X
cana-2702	748	4	crull	crull	NOUN
cana-2702	748	5	,	,	PUNCT
cana-2702	748	6	betsy	betsy	PROPN
cana-2702	748	7	,	,	PUNCT
cana-2702	748	8	et	et	PROPN
cana-2702	748	9	al	al	PROPN
cana-2702	748	10	.	.	PUNCT
cana-2702	749	1	"	"	PUNCT
cana-2702	749	2	the	the	DET
cana-2702	749	3	cover	cover	NOUN
cana-2702	749	4	pebbling	pebble	VERB
cana-2702	749	5	number	number	NOUN
cana-2702	749	6	of	of	ADP
cana-2702	749	7	graphs	graph	NOUN
cana-2702	749	8	.	.	PUNCT
cana-2702	749	9	"	"	PUNCT
cana-2702	750	1	discrete	discrete	ADJ
cana-2702	750	2	mathematics	mathematic	NOUN
cana-2702	750	3	296.1	296.1	NUM
cana-2702	750	4	(	(	PUNCT
cana-2702	750	5	2005	2005	NUM
cana-2702	750	6	):	):	PUNCT
cana-2702	750	7	15	15	NUM
cana-2702	750	8	-	-	SYM
cana-2702	750	9	23	23	NUM
cana-2702	750	10	.	.	PUNCT
cana-2702	751	1	[	[	X
cana-2702	751	2	7	7	NUM
cana-2702	751	3	]	]	SYM
cana-2702	751	4	erdös	erdös	PROPN
cana-2702	751	5	,	,	PUNCT
cana-2702	751	6	paul	paul	PROPN
cana-2702	751	7	,	,	PUNCT
cana-2702	751	8	frank	frank	PROPN
cana-2702	751	9	harary	harary	PROPN
cana-2702	751	10	,	,	PUNCT
cana-2702	751	11	and	and	CCONJ
cana-2702	751	12	maria	maria	PROPN
cana-2702	751	13	klawe	klawe	PROPN
cana-2702	751	14	.	.	PUNCT
cana-2702	752	1	"	"	PUNCT
cana-2702	752	2	residually	residually	ADV
cana-2702	752	3	-	-	PUNCT
cana-2702	752	4	complete	complete	ADJ
cana-2702	752	5	graphs	graph	NOUN
cana-2702	752	6	.	.	PUNCT
cana-2702	752	7	"	"	PUNCT
cana-2702	753	1	annals	annal	NOUN
cana-2702	753	2	of	of	ADP
cana-2702	753	3	discrete	discrete	ADJ
cana-2702	753	4	mathematics	mathematic	NOUN
cana-2702	753	5	.	.	PUNCT
cana-2702	754	1	vol	vol	NOUN
cana-2702	754	2	.	.	PROPN
cana-2702	755	1	6	6	NUM
cana-2702	755	2	.	.	PUNCT
cana-2702	756	1	elsevier	elsevier	NOUN
cana-2702	756	2	,	,	PUNCT
cana-2702	756	3	1980	1980	NUM
cana-2702	756	4	.	.	PUNCT
cana-2702	757	1	117	117	NUM
cana-2702	757	2	-	-	SYM
cana-2702	757	3	123	123	NUM
cana-2702	757	4	.	.	PUNCT
cana-2702	758	1	[	[	X
cana-2702	758	2	8	8	NUM
cana-2702	758	3	]	]	X
cana-2702	758	4	feng	feng	X
cana-2702	758	5	,	,	PUNCT
cana-2702	758	6	rongquan	rongquan	ADJ
cana-2702	758	7	,	,	PUNCT
cana-2702	758	8	and	and	CCONJ
cana-2702	758	9	ju	ju	PROPN
cana-2702	758	10	young	young	ADJ
cana-2702	758	11	kim	kim	PROPN
cana-2702	758	12	.	.	PUNCT
cana-2702	759	1	"	"	PUNCT
cana-2702	759	2	pebbling	pebble	VERB
cana-2702	759	3	numbers	number	NOUN
cana-2702	759	4	of	of	ADP
cana-2702	759	5	some	some	DET
cana-2702	759	6	graphs	graph	NOUN
cana-2702	759	7	.	.	PUNCT
cana-2702	759	8	"	"	PUNCT
cana-2702	760	1	science	science	NOUN
cana-2702	760	2	in	in	ADP
cana-2702	760	3	china	china	PROPN
cana-2702	760	4	series	series	PROPN
cana-2702	760	5	a	a	PRON
cana-2702	760	6	:	:	PUNCT
cana-2702	760	7	mathematics	mathematic	NOUN
cana-2702	760	8	45	45	NUM
cana-2702	760	9	(	(	PUNCT
cana-2702	760	10	2002	2002	NUM
cana-2702	760	11	):	):	PUNCT
cana-2702	760	12	470	470	NUM
cana-2702	760	13	-	-	SYM
cana-2702	760	14	478	478	NUM
cana-2702	760	15	.	.	PUNCT
cana-2702	761	1	[	[	X
cana-2702	761	2	9	9	NUM
cana-2702	761	3	]	]	PUNCT
cana-2702	761	4	folkman	folkman	NOUN
cana-2702	761	5	,	,	PUNCT
cana-2702	761	6	jon	jon	PROPN
cana-2702	761	7	.	.	PUNCT
cana-2702	762	1	"	"	PUNCT
cana-2702	762	2	graphs	graph	NOUN
cana-2702	762	3	with	with	ADP
cana-2702	762	4	monochromatic	monochromatic	ADJ
cana-2702	762	5	complete	complete	ADJ
cana-2702	762	6	subgraphs	subgraph	NOUN
cana-2702	762	7	in	in	ADP
cana-2702	762	8	every	every	DET
cana-2702	762	9	edge	edge	NOUN
cana-2702	762	10	coloring	coloring	NOUN
cana-2702	762	11	.	.	PUNCT
cana-2702	762	12	"	"	PUNCT
cana-2702	763	1	siam	siam	ADJ
cana-2702	763	2	journal	journal	NOUN
cana-2702	763	3	on	on	ADP
cana-2702	763	4	applied	apply	VERB
cana-2702	763	5	mathematics	mathematic	NOUN
cana-2702	763	6	18.1	18.1	NUM
cana-2702	763	7	(	(	PUNCT
cana-2702	763	8	1970	1970	NUM
cana-2702	763	9	):	):	PUNCT
cana-2702	763	10	19	19	NUM
cana-2702	763	11	-	-	SYM
cana-2702	763	12	24	24	NUM
cana-2702	763	13	.	.	PUNCT
cana-2702	764	1	[	[	X
cana-2702	764	2	10	10	NUM
cana-2702	764	3	]	]	PUNCT
cana-2702	764	4	he	he	PRON
cana-2702	764	5	,	,	PUNCT
cana-2702	764	6	yuguo	yuguo	PROPN
cana-2702	764	7	.	.	PUNCT
cana-2702	765	1	"	"	PUNCT
cana-2702	765	2	k	k	PROPN
cana-2702	765	3	variables	variable	NOUN
cana-2702	765	4	are	be	AUX
cana-2702	765	5	needed	need	VERB
cana-2702	765	6	to	to	PART
cana-2702	765	7	define	define	VERB
cana-2702	765	8	k	k	NOUN
cana-2702	765	9	-	-	NOUN
cana-2702	765	10	clique	clique	NOUN
cana-2702	765	11	in	in	ADP
cana-2702	765	12	first	first	ADJ
cana-2702	765	13	-	-	PUNCT
cana-2702	765	14	order	order	NOUN
cana-2702	765	15	logic	logic	NOUN
cana-2702	765	16	.	.	PUNCT
cana-2702	765	17	"	"	PUNCT
cana-2702	766	1	arxiv	arxiv	PROPN
cana-2702	766	2	preprint	preprint	NOUN
cana-2702	766	3	arxiv:1501.04572	arxiv:1501.04572	PROPN
cana-2702	766	4	(	(	PUNCT
cana-2702	766	5	2015	2015	NUM
cana-2702	766	6	)	)	PUNCT
cana-2702	766	7	.	.	PUNCT
cana-2702	767	1	[	[	X
cana-2702	767	2	11	11	NUM
cana-2702	767	3	]	]	X
cana-2702	767	4	kungumaraj	kungumaraj	NOUN
cana-2702	767	5	,	,	PUNCT
cana-2702	767	6	e.	e.	PROPN
cana-2702	767	7	,	,	PUNCT
cana-2702	767	8	et	et	PROPN
cana-2702	767	9	al	al	PROPN
cana-2702	767	10	.	.	PUNCT
cana-2702	767	11	"	"	PUNCT
cana-2702	767	12	efficiency	efficiency	NOUN
cana-2702	767	13	enhancement	enhancement	NOUN
cana-2702	767	14	in	in	ADP
cana-2702	767	15	heptagonal	heptagonal	ADJ
cana-2702	767	16	fuzzy	fuzzy	ADJ
cana-2702	767	17	transportation	transportation	NOUN
cana-2702	767	18	problems	problem	NOUN
cana-2702	767	19	.	.	PUNCT
cana-2702	767	20	"	"	PUNCT
cana-2702	768	1	international	international	ADJ
cana-2702	768	2	conference	conference	NOUN
cana-2702	768	3	on	on	ADP
cana-2702	768	4	intelligent	intelligent	ADJ
cana-2702	768	5	and	and	CCONJ
cana-2702	768	6	fuzzy	fuzzy	ADJ
cana-2702	768	7	systems	system	NOUN
cana-2702	768	8	.	.	PUNCT
cana-2702	769	1	cham	cham	PROPN
cana-2702	769	2	:	:	PUNCT
cana-2702	769	3	springer	springer	NOUN
cana-2702	769	4	nature	nature	PROPN
cana-2702	769	5	switzerland	switzerland	PROPN
cana-2702	769	6	,	,	PUNCT
cana-2702	769	7	2024	2024	NUM
cana-2702	769	8	.	.	PUNCT
cana-2702	770	1	[	[	X
cana-2702	770	2	12	12	NUM
cana-2702	770	3	]	]	X
cana-2702	770	4	kungumaraj	kungumaraj	NOUN
cana-2702	770	5	,	,	PUNCT
cana-2702	770	6	e.	e.	PROPN
cana-2702	770	7	,	,	PUNCT
cana-2702	770	8	et	et	PROPN
cana-2702	770	9	al	al	PROPN
cana-2702	770	10	.	.	PUNCT
cana-2702	771	1	"	"	PUNCT
cana-2702	771	2	topologized	topologize	VERB
cana-2702	771	3	graphical	graphical	ADJ
cana-2702	771	4	method	method	NOUN
cana-2702	771	5	in	in	ADP
cana-2702	771	6	solving	solve	VERB
cana-2702	771	7	fuzzy	fuzzy	ADJ
cana-2702	771	8	transportation	transportation	NOUN
cana-2702	771	9	problem	problem	NOUN
cana-2702	771	10	with	with	ADP
cana-2702	771	11	computational	computational	ADJ
cana-2702	771	12	techniques	technique	NOUN
cana-2702	771	13	.	.	PUNCT
cana-2702	771	14	"	"	PUNCT
cana-2702	772	1	international	international	ADJ
cana-2702	772	2	conference	conference	NOUN
cana-2702	772	3	on	on	ADP
cana-2702	772	4	intelligent	intelligent	ADJ
cana-2702	772	5	and	and	CCONJ
cana-2702	772	6	fuzzy	fuzzy	ADJ
cana-2702	772	7	systems	system	NOUN
cana-2702	772	8	.	.	PUNCT
cana-2702	773	1	cham	cham	PROPN
cana-2702	773	2	:	:	PUNCT
cana-2702	773	3	springer	springer	NOUN
cana-2702	773	4	nature	nature	PROPN
cana-2702	773	5	switzerland	switzerland	PROPN
cana-2702	773	6	,	,	PUNCT
cana-2702	773	7	2024	2024	NUM
cana-2702	773	8	.	.	PUNCT
cana-2702	774	1	[	[	X
cana-2702	774	2	13	13	NUM
cana-2702	774	3	]	]	X
cana-2702	774	4	koesoemadinata	koesoemadinata	PROPN
cana-2702	774	5	,	,	PUNCT
cana-2702	774	6	r.	r.	PROPN
cana-2702	774	7	p.	p.	PROPN
cana-2702	774	8	,	,	PUNCT
cana-2702	774	9	and	and	CCONJ
cana-2702	774	10	th	th	X
cana-2702	774	11	matasak	matasak	NOUN
cana-2702	774	12	.	.	PUNCT
cana-2702	775	1	"	"	PUNCT
cana-2702	775	2	stratigraphy	stratigraphy	NOUN
cana-2702	775	3	and	and	CCONJ
cana-2702	775	4	sedimentation	sedimentation	NOUN
cana-2702	775	5	:	:	PUNCT
cana-2702	775	6	ombilin	ombilin	PROPN
cana-2702	775	7	basin	basin	PROPN
cana-2702	775	8	,	,	PUNCT
cana-2702	775	9	central	central	ADJ
cana-2702	775	10	sumatra	sumatra	PROPN
cana-2702	775	11	(	(	PUNCT
cana-2702	775	12	west	west	PROPN
cana-2702	775	13	sumatra	sumatra	PROPN
cana-2702	775	14	province	province	PROPN
cana-2702	775	15	)	)	PUNCT
cana-2702	775	16	.	.	PUNCT
cana-2702	775	17	"	"	PUNCT
cana-2702	776	1	(	(	PUNCT
cana-2702	776	2	1981	1981	NUM
cana-2702	776	3	):	):	PUNCT
cana-2702	776	4	217	217	NUM
cana-2702	776	5	-	-	SYM
cana-2702	776	6	249	249	NUM
cana-2702	776	7	.	.	PUNCT
cana-2702	777	1	[	[	X
cana-2702	777	2	14	14	NUM
cana-2702	777	3	]	]	PUNCT
cana-2702	777	4	moews	moews	PROPN
cana-2702	777	5	,	,	PUNCT
cana-2702	777	6	david	david	PROPN
cana-2702	777	7	.	.	PUNCT
cana-2702	778	1	"	"	PUNCT
cana-2702	778	2	pebbling	pebble	VERB
cana-2702	778	3	graphs	graph	NOUN
cana-2702	778	4	.	.	PUNCT
cana-2702	778	5	"	"	PUNCT
cana-2702	779	1	journal	journal	NOUN
cana-2702	779	2	of	of	ADP
cana-2702	779	3	combinatorial	combinatorial	ADJ
cana-2702	779	4	theory	theory	NOUN
cana-2702	779	5	,	,	PUNCT
cana-2702	779	6	series	series	NOUN
cana-2702	779	7	b	b	PROPN
cana-2702	779	8	55.2	55.2	NUM
cana-2702	779	9	(	(	PUNCT
cana-2702	779	10	1992	1992	NUM
cana-2702	779	11	):	):	PUNCT
cana-2702	779	12	244	244	NUM
cana-2702	779	13	-	-	SYM
cana-2702	779	14	252	252	NUM
cana-2702	779	15	.	.	PUNCT
cana-2702	780	1	[	[	X
cana-2702	780	2	15	15	NUM
cana-2702	780	3	]	]	X
cana-2702	780	4	santhi	santhi	ADJ
cana-2702	780	5	,	,	PUNCT
cana-2702	780	6	r.	r.	PROPN
cana-2702	780	7	,	,	PUNCT
cana-2702	780	8	and	and	CCONJ
cana-2702	780	9	e.	e.	PROPN
cana-2702	780	10	kungumaraj	kungumaraj	PROPN
cana-2702	780	11	.	.	PUNCT
cana-2702	781	1	"	"	PUNCT
cana-2702	781	2	indian	indian	ADJ
cana-2702	781	3	journal	journal	NOUN
cana-2702	781	4	of	of	ADP
cana-2702	781	5	information	information	NOUN
cana-2702	781	6	sciences	sciences	PROPN
cana-2702	781	7	and	and	CCONJ
cana-2702	781	8	computer	computer	NOUN
cana-2702	781	9	application	application	NOUN
cana-2702	781	10	issn	issn	PROPN
cana-2702	781	11	2349	2349	NUM
cana-2702	781	12	–	–	PUNCT
cana-2702	781	13	042x	042x	NOUN
cana-2702	781	14	volume	volume	NOUN
cana-2702	781	15	6	6	NUM
cana-2702	781	16	,	,	PUNCT
cana-2702	781	17	number	number	NOUN
cana-2702	781	18	1	1	NUM
cana-2702	781	19	(	(	PUNCT
cana-2702	781	20	2019	2019	NUM
cana-2702	781	21	)	)	PUNCT
cana-2702	781	22	,	,	PUNCT
cana-2702	781	23	pp	pp	ADJ
cana-2702	781	24	.	.	PUNCT
cana-2702	782	1	1	1	NUM
cana-2702	782	2	-	-	SYM
cana-2702	782	3	6	6	NUM
cana-2702	782	4	©	©	PROPN
cana-2702	782	5	gbs	gbs	PROPN
cana-2702	782	6	publishers	publisher	NOUN
cana-2702	782	7	&	&	CCONJ
cana-2702	782	8	distributors	distributor	NOUN
cana-2702	782	9	(	(	PUNCT
cana-2702	782	10	i	i	NOUN
cana-2702	782	11	)	)	PUNCT
cana-2702	782	12	http://www	http://www	PROPN
cana-2702	782	13	.	.	PROPN
cana-2702	782	14	gbspublisher	gbspublisher	PROPN
cana-2702	782	15	.	.	PUNCT
cana-2702	782	16	com	com	PROPN
cana-2702	782	17	.	.	PUNCT
cana-2702	782	18	"	"	PUNCT
cana-2702	783	1	[	[	X
cana-2702	783	2	16	16	NUM
cana-2702	783	3	]	]	PUNCT
cana-2702	783	4	tanselle	tanselle	NOUN
cana-2702	783	5	,	,	PUNCT
cana-2702	783	6	g.	g.	PROPN
cana-2702	783	7	thomas	thomas	PROPN
cana-2702	783	8	.	.	PUNCT
cana-2702	784	1	"	"	PUNCT
cana-2702	784	2	the	the	DET
cana-2702	784	3	bibliographical	bibliographical	ADJ
cana-2702	784	4	description	description	NOUN
cana-2702	784	5	of	of	ADP
cana-2702	784	6	patterns	pattern	NOUN
cana-2702	784	7	.	.	PUNCT
cana-2702	784	8	"	"	PUNCT
cana-2702	785	1	studies	study	NOUN
cana-2702	785	2	in	in	ADP
cana-2702	785	3	bibliography	bibliography	NOUN
cana-2702	785	4	23	23	NUM
cana-2702	785	5	(	(	PUNCT
cana-2702	785	6	1970	1970	NUM
cana-2702	785	7	):	):	PUNCT
cana-2702	785	8	71	71	NUM
cana-2702	785	9	-	-	SYM
cana-2702	785	10	102	102	NUM
cana-2702	785	11	.	.	PUNCT
cana-2702	786	1	[	[	X
cana-2702	786	2	17	17	NUM
cana-2702	786	3	]	]	X
cana-2702	786	4	zhu	zhu	PROPN
cana-2702	786	5	,	,	PUNCT
cana-2702	786	6	yuanjun	yuanjun	PROPN
cana-2702	786	7	,	,	PUNCT
cana-2702	786	8	and	and	CCONJ
cana-2702	786	9	mingan	mingan	PROPN
cana-2702	786	10	shao	shao	PROPN
cana-2702	786	11	.	.	PUNCT
cana-2702	787	1	"	"	PUNCT
cana-2702	787	2	spatial	spatial	ADJ
cana-2702	787	3	distribution	distribution	NOUN
cana-2702	787	4	of	of	ADP
cana-2702	787	5	surface	surface	NOUN
cana-2702	787	6	rock	rock	NOUN
cana-2702	787	7	fragment	fragment	NOUN
cana-2702	787	8	on	on	ADP
cana-2702	787	9	hill	hill	NOUN
cana-2702	787	10	-	-	PUNCT
cana-2702	787	11	slopes	slope	NOUN
cana-2702	787	12	in	in	ADP
cana-2702	787	13	a	a	DET
cana-2702	787	14	small	small	ADJ
cana-2702	787	15	catchment	catchment	NOUN
cana-2702	787	16	in	in	ADP
cana-2702	787	17	wind	wind	NOUN
cana-2702	787	18	-	-	PUNCT
cana-2702	787	19	water	water	NOUN
cana-2702	787	20	erosion	erosion	NOUN
cana-2702	787	21	crisscross	crisscross	NOUN
cana-2702	787	22	region	region	NOUN
cana-2702	787	23	of	of	ADP
cana-2702	787	24	the	the	DET
cana-2702	787	25	loess	loess	NOUN
cana-2702	787	26	plateau	plateau	NOUN
cana-2702	787	27	.	.	PUNCT
cana-2702	787	28	"	"	PUNCT
cana-2702	788	1	science	science	NOUN
cana-2702	788	2	in	in	ADP
cana-2702	788	3	china	china	PROPN
cana-2702	788	4	series	series	PROPN
cana-2702	788	5	d	d	PROPN
cana-2702	788	6	:	:	PUNCT
cana-2702	788	7	earth	earth	NOUN
cana-2702	788	8	sciences	science	NOUN
cana-2702	788	9	51.6	51.6	NUM
cana-2702	788	10	(	(	PUNCT
cana-2702	788	11	2008	2008	NUM
cana-2702	788	12	):	):	PUNCT
cana-2702	788	13	862	862	NUM
cana-2702	788	14	-	-	SYM
cana-2702	788	15	870	870	NUM
cana-2702	788	16	.	.	PUNCT