id	sid	eid	entity	type
ejpam-6851	1	1	european journal	ORG
ejpam-6851	1	2	2025	DATE
ejpam-6851	1	3	18	CARDINAL
ejpam-6851	1	4	4	CARDINAL
ejpam-6851	1	5	6851	CARDINAL
ejpam-6851	1	6	1307-5543	CARDINAL
ejpam-6851	1	7	new york	GPE
ejpam-6851	1	8	jerra mae	PERSON
ejpam-6851	1	9	c. molles1	PERSON
ejpam-6851	2	1	ferdinand p.	ORG
ejpam-6851	3	1	sergio r. canoy	PERSON
ejpam-6851	3	2	jr.1,2,∗ 1	ORG
ejpam-6851	3	3	msu-iligan institute of technology	ORG
ejpam-6851	3	4	iligan city	GPE
ejpam-6851	3	5	2	CARDINAL
ejpam-6851	3	6	msu-iligan institute of technology	ORG
ejpam-6851	3	7	iligan city	GPE
ejpam-6851	3	8	philippines	GPE
ejpam-6851	10	1	γw(g	PERSON
ejpam-6851	13	1	2020	DATE
ejpam-6851	13	2	05c69	CARDINAL
ejpam-6851	13	3	1	CARDINAL
ejpam-6851	15	1	1	CARDINAL
ejpam-6851	17	1	g ∪ h	ORG
ejpam-6851	20	1	∈	ORG
ejpam-6851	23	1	jerramae.molles@g.msuiit.edu.ph	ORG
ejpam-6851	23	2	j. m. molles	PERSON
ejpam-6851	23	3	ferdinand.jamil@g.msuiit.edu.ph	ORG
ejpam-6851	23	4	f. p. jamil	PERSON
ejpam-6851	23	5	sergio.canoy@g.msuiit.edu.ph	ORG
ejpam-6851	23	6	s. r. canoy jr.	PERSON
ejpam-6851	24	1	1	CARDINAL
ejpam-6851	24	2	2025	CARDINAL
ejpam-6851	25	1	j. m. molles	PERSON
ejpam-6851	25	2	f. p. jamil	PERSON
ejpam-6851	25	3	s. r. canoy	PERSON
ejpam-6851	26	1	j. pure	PERSON
ejpam-6851	26	2	18	CARDINAL
ejpam-6851	26	3	4	CARDINAL
ejpam-6851	26	4	2025	CARDINAL
ejpam-6851	26	5	6851 2	DATE
ejpam-6851	26	6	18	CARDINAL
ejpam-6851	26	7	one	CARDINAL
ejpam-6851	26	8	g ⋄ h	ORG
ejpam-6851	26	9	one	CARDINAL
ejpam-6851	26	10	u = u′	ORG
ejpam-6851	26	11	∈ e(h	ORG
ejpam-6851	28	1	∈	ORG
ejpam-6851	29	1	1	CARDINAL
ejpam-6851	29	2	∈	ORG
ejpam-6851	31	1	⊆	CARDINAL
ejpam-6851	32	1	⊆	CARDINAL
ejpam-6851	33	1	γ(g	NORP
ejpam-6851	34	1	2–7	CARDINAL
ejpam-6851	35	1	two	CARDINAL
ejpam-6851	35	2	∈	ORG
ejpam-6851	35	3	∈	ORG
ejpam-6851	36	1	⊆	CARDINAL
ejpam-6851	37	1	∈	ORG
ejpam-6851	38	1	∈	ORG
ejpam-6851	39	1	u ≼g s	ORG
ejpam-6851	41	1	⊆	CARDINAL
ejpam-6851	47	1	∈	ORG
ejpam-6851	48	1	u ≼g s	ORG
ejpam-6851	51	1	γw(g	PERSON
ejpam-6851	53	1	γw(g	PERSON
ejpam-6851	54	1	e. sampathkumar	PERSON
ejpam-6851	54	2	l. pushpa latha	PERSON
ejpam-6851	55	1	8	CARDINAL
ejpam-6851	55	2	1996	DATE
ejpam-6851	56	1	two	CARDINAL
ejpam-6851	56	2	9	CARDINAL
ejpam-6851	56	3	10	CARDINAL
ejpam-6851	57	1	11][12	CARDINAL
ejpam-6851	57	2	13	CARDINAL
ejpam-6851	57	3	14	CARDINAL
ejpam-6851	58	1	15]-[16	CARDINAL
ejpam-6851	59	1	12	CARDINAL
ejpam-6851	59	2	17	CARDINAL
ejpam-6851	60	1	j. m. molles	PERSON
ejpam-6851	60	2	f. p. jamil	PERSON
ejpam-6851	60	3	s. r. canoy	PERSON
ejpam-6851	61	1	j. pure	PERSON
ejpam-6851	61	2	18	CARDINAL
ejpam-6851	61	3	4	CARDINAL
ejpam-6851	61	4	2025	CARDINAL
ejpam-6851	61	5	6851 3	DATE
ejpam-6851	61	6	18	CARDINAL
ejpam-6851	61	7	γw(g	PERSON
ejpam-6851	61	8	18–21	CARDINAL
ejpam-6851	61	9	22–25	CARDINAL
ejpam-6851	62	1	two	CARDINAL
ejpam-6851	63	1	∈	ORG
ejpam-6851	66	1	⊆	CARDINAL
ejpam-6851	68	1	s. hence	PERSON
ejpam-6851	72	1	γw(g	PERSON
ejpam-6851	74	1	γw(g	PERSON
ejpam-6851	76	1	2	CARDINAL
ejpam-6851	77	1	n, γ(g	ORG
ejpam-6851	78	1	26	CARDINAL
ejpam-6851	80	1	1	CARDINAL
ejpam-6851	81	1	1	CARDINAL
ejpam-6851	81	2	1	CARDINAL
ejpam-6851	81	3	γw(g	PERSON
ejpam-6851	82	1	1	CARDINAL
ejpam-6851	82	2	γw(g	PERSON
ejpam-6851	82	3	1	CARDINAL
ejpam-6851	83	1	γw(g	PERSON
ejpam-6851	84	1	1	CARDINAL
ejpam-6851	84	2	g = k1,n−1	PERSON
ejpam-6851	84	3	∪kk	DATE
ejpam-6851	85	1	2	CARDINAL
ejpam-6851	87	1	2	CARDINAL
ejpam-6851	87	2	γw(g	PERSON
ejpam-6851	88	1	2	CARDINAL
ejpam-6851	88	2	2	CARDINAL
ejpam-6851	93	1	s. j. m. molles	PERSON
ejpam-6851	93	2	f. p. jamil	PERSON
ejpam-6851	93	3	s. r. canoy	PERSON
ejpam-6851	94	1	j. pure	PERSON
ejpam-6851	94	2	18	CARDINAL
ejpam-6851	94	3	4	CARDINAL
ejpam-6851	94	4	2025	CARDINAL
ejpam-6851	94	5	6851 4	DATE
ejpam-6851	94	6	18	CARDINAL
ejpam-6851	94	7	1	CARDINAL
ejpam-6851	95	1	∈	PRODUCT
ejpam-6851	95	2	∩	WORK_OF_ART
ejpam-6851	95	3	|s∗| ≤	QUANTITY
ejpam-6851	98	1	∈	ORG
ejpam-6851	98	2	|{xv	LOC
ejpam-6851	99	1	∈	ORG
ejpam-6851	99	2	∈	ORG
ejpam-6851	100	1	∈	PRODUCT
ejpam-6851	100	2	|s∗|	CARDINAL
ejpam-6851	101	1	|{xv	LOC
ejpam-6851	102	1	∈	ORG
ejpam-6851	104	1	2	CARDINAL
ejpam-6851	111	1	⊆	CARDINAL
ejpam-6851	111	2	∈	ORG
ejpam-6851	112	1	∈	ORG
ejpam-6851	113	1	/∈ s.	ORG
ejpam-6851	115	1	3	CARDINAL
ejpam-6851	116	1	27	CARDINAL
ejpam-6851	117	1	3	CARDINAL
ejpam-6851	117	2	1(mod 3	DATE
ejpam-6851	117	3	1 +	CARDINAL
ejpam-6851	118	1	3	CARDINAL
ejpam-6851	119	1	1	CARDINAL
ejpam-6851	121	1	1	CARDINAL
ejpam-6851	123	1	1	CARDINAL
ejpam-6851	123	2	2	CARDINAL
ejpam-6851	125	1	1	CARDINAL
ejpam-6851	125	2	1	CARDINAL
ejpam-6851	125	3	2	CARDINAL
ejpam-6851	126	1	j. m. molles	PERSON
ejpam-6851	126	2	f. p. jamil	PERSON
ejpam-6851	126	3	s. r. canoy	PERSON
ejpam-6851	127	1	j. pure	PERSON
ejpam-6851	127	2	18	CARDINAL
ejpam-6851	127	3	4	CARDINAL
ejpam-6851	127	4	2025	CARDINAL
ejpam-6851	127	5	6851 5	DATE
ejpam-6851	127	6	18	CARDINAL
ejpam-6851	128	1	1(i	CARDINAL
ejpam-6851	131	1	2	CARDINAL
ejpam-6851	132	1	2	CARDINAL
ejpam-6851	133	1	∈	ORG
ejpam-6851	134	1	∈	ORG
ejpam-6851	137	1	∈	ORG
ejpam-6851	137	2	∈	ORG
ejpam-6851	138	1	∈	ORG
ejpam-6851	141	1	∈	ORG
ejpam-6851	141	2	1	CARDINAL
ejpam-6851	143	1	2	CARDINAL
ejpam-6851	146	1	2	CARDINAL
ejpam-6851	146	2	∈	PRODUCT
ejpam-6851	149	1	2	CARDINAL
ejpam-6851	151	1	2	CARDINAL
ejpam-6851	151	2	≼gg	ORG
ejpam-6851	151	3	s.	PERSON
ejpam-6851	151	4	∈	ORG
ejpam-6851	151	5	∈	ORG
ejpam-6851	152	1	∈	ORG
ejpam-6851	153	1	w ∈	PERSON
ejpam-6851	153	2	u w ∈	ORG
ejpam-6851	155	1	∈	ORG
ejpam-6851	155	2	∈	ORG
ejpam-6851	159	1	4	CARDINAL
ejpam-6851	161	1	⊆	CARDINAL
ejpam-6851	161	2	sg ⊆	ORG
ejpam-6851	162	1	∈	ORG
ejpam-6851	163	1	∈	ORG
ejpam-6851	165	1	4	DATE
ejpam-6851	165	2	sg ∈	PERSON
ejpam-6851	166	1	1 or	CARDINAL
ejpam-6851	166	2	1	CARDINAL
ejpam-6851	167	1	≼gg	ORG
ejpam-6851	167	2	{kn	PERSON
ejpam-6851	168	1	2	CARDINAL
ejpam-6851	168	2	1 +	CARDINAL
ejpam-6851	168	3	3 ≤ n ≤ 5	QUANTITY
ejpam-6851	168	4	2 +	DATE
ejpam-6851	168	5	3	CARDINAL
ejpam-6851	168	6	n ≥ 6	DATE
ejpam-6851	169	1	3	CARDINAL
ejpam-6851	169	2	3	CARDINAL
ejpam-6851	169	3	4	DATE
ejpam-6851	169	4	4 ≤ n ≤ 6	QUANTITY
ejpam-6851	169	5	2	CARDINAL
ejpam-6851	169	6	n ≥ 4	DATE
ejpam-6851	169	7	3	CARDINAL
ejpam-6851	169	8	5	CARDINAL
ejpam-6851	169	9	2	CARDINAL
ejpam-6851	170	1	j. m. molles	PERSON
ejpam-6851	170	2	f. p. jamil	PERSON
ejpam-6851	170	3	s. r. canoy	PERSON
ejpam-6851	171	1	j. pure	PERSON
ejpam-6851	171	2	18	CARDINAL
ejpam-6851	171	3	4	CARDINAL
ejpam-6851	171	4	2025	CARDINAL
ejpam-6851	171	5	6851 6	DATE
ejpam-6851	171	6	18	CARDINAL
ejpam-6851	177	1	3 ≤ n ≤ 5	QUANTITY
ejpam-6851	178	1	n ≥ 6	DATE
ejpam-6851	179	1	⊆	CARDINAL
ejpam-6851	179	2	x4	PERSON
ejpam-6851	181	1	xn−1	PERSON
ejpam-6851	183	1	x4	PERSON
ejpam-6851	185	1	xn−1	GPE
ejpam-6851	186	1	4	CARDINAL
ejpam-6851	186	2	∈	ORG
ejpam-6851	187	1	2 +	DATE
ejpam-6851	187	2	2 +	CARDINAL
ejpam-6851	187	3	3	CARDINAL
ejpam-6851	187	4	⊆	CARDINAL
ejpam-6851	188	1	4	CARDINAL
ejpam-6851	189	1	lemma 2	ORG
ejpam-6851	190	1	2	CARDINAL
ejpam-6851	191	1	x2 ∈ sg	PERSON
ejpam-6851	192	1	∈	ORG
ejpam-6851	198	1	2 +	CARDINAL
ejpam-6851	198	2	3	CARDINAL
ejpam-6851	198	3	3 ≤ n ≤ 6	QUANTITY
ejpam-6851	199	1	n ≥ 7	ORG
ejpam-6851	200	1	x3	GPE
ejpam-6851	201	1	3k	CARDINAL
ejpam-6851	201	2	x3	GPE
ejpam-6851	203	1	3	CARDINAL
ejpam-6851	204	1	4	DATE
ejpam-6851	205	1	2 +	CARDINAL
ejpam-6851	205	2	⊆	CARDINAL
ejpam-6851	208	1	n ≥ 7	DATE
ejpam-6851	208	2	2	CARDINAL
ejpam-6851	208	3	2	CARDINAL
ejpam-6851	209	1	3	CARDINAL
ejpam-6851	209	2	2 +	CARDINAL
ejpam-6851	209	3	4 ≤ n ≤ 5	QUANTITY
ejpam-6851	210	1	n ≥ 6. 4	DATE
ejpam-6851	211	1	5	CARDINAL
ejpam-6851	213	1	∈	ORG
ejpam-6851	214	1	∈	ORG
ejpam-6851	215	1	+m. theorem 1	PERSON
ejpam-6851	217	1	⊆	CARDINAL
ejpam-6851	218	1	γs(g+h	NORP
ejpam-6851	219	1	⊆	CARDINAL
ejpam-6851	219	2	m− n. (ii	PERSON
ejpam-6851	219	3	⊆	CARDINAL
ejpam-6851	220	1	one	CARDINAL
ejpam-6851	221	1	u ≼g sg	ORG
ejpam-6851	221	2	∈	ORG
ejpam-6851	222	1	+ n−m	DATE
ejpam-6851	222	2	j. m. molles	PERSON
ejpam-6851	222	3	f. p. jamil	PERSON
ejpam-6851	222	4	s. r. canoy	PERSON
ejpam-6851	223	1	j. pure	PERSON
ejpam-6851	223	2	18	CARDINAL
ejpam-6851	223	3	4	CARDINAL
ejpam-6851	223	4	2025	CARDINAL
ejpam-6851	223	5	6851 7	DATE
ejpam-6851	223	6	18	CARDINAL
ejpam-6851	223	7	∈	ORG
ejpam-6851	224	1	m− n. proof	PERSON
ejpam-6851	225	1	+h	GPE
ejpam-6851	226	1	⊆	CARDINAL
ejpam-6851	228	1	∈	ORG
ejpam-6851	229	1	+ h	GPE
ejpam-6851	229	2	∈	ORG
ejpam-6851	229	3	5(ii	CARDINAL
ejpam-6851	229	4	⊆	CARDINAL
ejpam-6851	231	1	g+h	NORP
ejpam-6851	231	2	\s	ORG
ejpam-6851	232	1	∈	ORG
ejpam-6851	232	2	∈	ORG
ejpam-6851	232	3	u ≼g sg	ORG
ejpam-6851	233	1	∈	ORG
ejpam-6851	233	2	5(ii	CARDINAL
ejpam-6851	233	3	degg(u)+n−m	PRODUCT
ejpam-6851	234	1	∈	ORG
ejpam-6851	236	1	∈	ORG
ejpam-6851	236	2	u ≼g s	ORG
ejpam-6851	236	3	s.	PERSON
ejpam-6851	236	4	∈	ORG
ejpam-6851	237	1	∈	ORG
ejpam-6851	237	2	m− n.	PERSON
ejpam-6851	240	1	γs(g+h	NORP
ejpam-6851	242	1	\s	ORG
ejpam-6851	243	1	u ≼g sg	ORG
ejpam-6851	243	2	s.	PERSON
ejpam-6851	243	3	∈	ORG
ejpam-6851	244	1	∈	ORG
ejpam-6851	244	2	γs(g+h	NORP
ejpam-6851	245	1	1	CARDINAL
ejpam-6851	247	1	1	CARDINAL
ejpam-6851	247	2	1 or	CARDINAL
ejpam-6851	248	1	1.	CARDINAL
ejpam-6851	248	2	γ(g	PERSON
ejpam-6851	248	3	2	CARDINAL
ejpam-6851	248	4	γ(h	ORG
ejpam-6851	248	5	2	CARDINAL
ejpam-6851	249	1	γs(g+h	NORP
ejpam-6851	250	1	2	CARDINAL
ejpam-6851	251	1	γs(g+h	NORP
ejpam-6851	251	2	min{γs(g	PERSON
ejpam-6851	251	3	1 +	CARDINAL
ejpam-6851	252	1	k =	PERSON
ejpam-6851	254	1	1(i	CARDINAL
ejpam-6851	255	1	γ(g	NORP
ejpam-6851	255	2	2	CARDINAL
ejpam-6851	255	3	γ(h	ORG
ejpam-6851	255	4	2	CARDINAL
ejpam-6851	255	5	∈	ORG
ejpam-6851	256	1	1	CARDINAL
ejpam-6851	256	2	γs(g+h	NORP
ejpam-6851	257	1	γs(g+h	NORP
ejpam-6851	257	2	2	CARDINAL
ejpam-6851	258	1	first	ORDINAL
ejpam-6851	258	2	⊆	CARDINAL
ejpam-6851	258	3	j. m. molles	PERSON
ejpam-6851	258	4	f. p. jamil	PERSON
ejpam-6851	258	5	s. r. canoy	PERSON
ejpam-6851	259	1	j. pure	PERSON
ejpam-6851	259	2	18	CARDINAL
ejpam-6851	259	3	4	CARDINAL
ejpam-6851	259	4	2025	CARDINAL
ejpam-6851	259	5	6851 8	DATE
ejpam-6851	259	6	18	CARDINAL
ejpam-6851	259	7	2	CARDINAL
ejpam-6851	259	8	∈	ORG
ejpam-6851	261	1	1	CARDINAL
ejpam-6851	261	2	γs(g+h	NORP
ejpam-6851	262	1	γs(g+h	NORP
ejpam-6851	263	1	∈	ORG
ejpam-6851	269	1	1(iii	CARDINAL
ejpam-6851	269	2	γs(g+h	NORP
ejpam-6851	270	1	γs(g+h	NORP
ejpam-6851	270	2	1 +	DATE
ejpam-6851	271	1	1 +	CARDINAL
ejpam-6851	272	1	⊆	CARDINAL
ejpam-6851	272	2	g+h	ORG
ejpam-6851	273	1	lemma 2	ORG
ejpam-6851	274	1	⊆	CARDINAL
ejpam-6851	274	2	1	CARDINAL
ejpam-6851	274	3	+h	GPE
ejpam-6851	278	1	k)\sg	PERSON
ejpam-6851	279	1	∈	ORG
ejpam-6851	279	2	∈	ORG
ejpam-6851	279	3	∈	ORG
ejpam-6851	280	1	∈	ORG
ejpam-6851	281	1	k.	PERSON
ejpam-6851	282	1	1	CARDINAL
ejpam-6851	282	2	+h	GPE
ejpam-6851	283	1	γs(g+h	NORP
ejpam-6851	283	2	min{γs(g	PERSON
ejpam-6851	283	3	1 +	CARDINAL
ejpam-6851	284	1	2	CARDINAL
ejpam-6851	285	1	1	CARDINAL
ejpam-6851	286	1	1 +	CARDINAL
ejpam-6851	287	1	2	CARDINAL
ejpam-6851	287	2	k =	PERSON
ejpam-6851	289	1	2	CARDINAL
ejpam-6851	295	1	•	CARDINAL
ejpam-6851	295	2	1	CARDINAL
ejpam-6851	296	1	2	CARDINAL
ejpam-6851	298	1	⊆	CARDINAL
ejpam-6851	300	1	⊆	CARDINAL
ejpam-6851	300	2	δ(h	PERSON
ejpam-6851	301	1	⊆	CARDINAL
ejpam-6851	301	2	∈	ORG
ejpam-6851	302	1	one	CARDINAL
ejpam-6851	303	1	u ≽g sg	ORG
ejpam-6851	303	2	∈	ORG
ejpam-6851	303	3	+ n−m	DATE
ejpam-6851	303	4	j. m. molles	PERSON
ejpam-6851	303	5	f. p. jamil	PERSON
ejpam-6851	303	6	s. r. canoy	PERSON
ejpam-6851	304	1	j. pure	PERSON
ejpam-6851	304	2	18	CARDINAL
ejpam-6851	304	3	4	CARDINAL
ejpam-6851	304	4	2025	CARDINAL
ejpam-6851	304	5	6851 9	DATE
ejpam-6851	304	6	18	CARDINAL
ejpam-6851	304	7	∈	ORG
ejpam-6851	305	1	2	CARDINAL
ejpam-6851	307	1	γ(g	PERSON
ejpam-6851	307	2	2	CARDINAL
ejpam-6851	307	3	γ(h	ORG
ejpam-6851	308	1	2	CARDINAL
ejpam-6851	308	2	δ(h	PERSON
ejpam-6851	309	1	2	CARDINAL
ejpam-6851	310	1	min{γw(g	PERSON
ejpam-6851	310	2	1 +	CARDINAL
ejpam-6851	310	3	k =	PERSON
ejpam-6851	310	4	u ≼]⟩ and u ∈	ORG
ejpam-6851	310	5	δ(h	PERSON
ejpam-6851	312	1	1	CARDINAL
ejpam-6851	313	1	n ≥ 4	DATE
ejpam-6851	313	2	1	CARDINAL
ejpam-6851	313	3	2	CARDINAL
ejpam-6851	313	4	2	CARDINAL
ejpam-6851	313	5	2	CARDINAL
ejpam-6851	313	6	3	CARDINAL
ejpam-6851	313	7	1	CARDINAL
ejpam-6851	313	8	1	CARDINAL
ejpam-6851	313	9	0	CARDINAL
ejpam-6851	313	10	2	CARDINAL
ejpam-6851	313	11	3	CARDINAL
ejpam-6851	314	1	2	CARDINAL
ejpam-6851	315	1	2	CARDINAL
ejpam-6851	315	2	5	CARDINAL
ejpam-6851	315	3	3	CARDINAL
ejpam-6851	316	1	⊆	CARDINAL
ejpam-6851	317	1	∪v∈v (g)sv	ORG
ejpam-6851	317	2	1	CARDINAL
ejpam-6851	317	3	⊆	CARDINAL
ejpam-6851	317	4	sv ⊆	ORG
ejpam-6851	317	5	∈	ORG
ejpam-6851	317	6	sv ∈	PERSON
ejpam-6851	320	1	∈	ORG
ejpam-6851	321	1	1	CARDINAL
ejpam-6851	324	1	∈	ORG
ejpam-6851	325	1	j. m. molles	PERSON
ejpam-6851	325	2	f. p. jamil	PERSON
ejpam-6851	325	3	s. r. canoy	PERSON
ejpam-6851	326	1	j. pure	PERSON
ejpam-6851	326	2	18	CARDINAL
ejpam-6851	326	3	4	CARDINAL
ejpam-6851	326	4	2025	CARDINAL
ejpam-6851	326	5	6851 10	DATE
ejpam-6851	329	1	w ∈ s	PERSON
ejpam-6851	329	2	u ≼g◦h w.	ORG
ejpam-6851	329	3	w ̸=	PERSON
ejpam-6851	329	4	w ∈ sv.	PERSON
ejpam-6851	331	1	sv ∈	PERSON
ejpam-6851	332	1	1	CARDINAL
ejpam-6851	333	1	w ∈	PERSON
ejpam-6851	334	1	w ∈	PERSON
ejpam-6851	335	1	w ∈	PERSON
ejpam-6851	335	2	w ≼g◦	GPE
ejpam-6851	335	3	w ∈	PERSON
ejpam-6851	335	4	w	GPE
ejpam-6851	335	5	w ≼g◦h a.	GPE
ejpam-6851	335	6	w ≼g◦h s.	ORG
ejpam-6851	337	1	⊆	CARDINAL
ejpam-6851	338	1	∈	MONEY
ejpam-6851	338	2	∪v∈v (g)sv	ORG
ejpam-6851	338	3	⊆	CARDINAL
ejpam-6851	338	4	sv ∈	PERSON
ejpam-6851	340	1	∈	PRODUCT
ejpam-6851	341	1	∪v∈v (g)sv	ORG
ejpam-6851	342	1	∈	ORG
ejpam-6851	344	1	y ∈ s	PERSON
ejpam-6851	344	2	≽g◦h y.	ORG
ejpam-6851	344	3	≽hv	DATE
ejpam-6851	344	4	sv ∈	PERSON
ejpam-6851	345	1	∪v∈v (g)sv	ORG
ejpam-6851	345	2	⊆	CARDINAL
ejpam-6851	345	3	sv ∈	PERSON
ejpam-6851	347	1	∈	ORG
ejpam-6851	348	1	sv ̸=	PERSON
ejpam-6851	348	2	w ∈ sv.	ORG
ejpam-6851	349	1	≽g◦h w.	ORG
ejpam-6851	349	2	∈	ORG
ejpam-6851	350	1	≽hv	DATE
ejpam-6851	351	1	3	CARDINAL
ejpam-6851	352	1	n ≥ 2	ORG
ejpam-6851	354	1	25	CARDINAL
ejpam-6851	354	2	γw(g	PERSON
ejpam-6851	355	1	⊆	CARDINAL
ejpam-6851	355	2	g ◦ h	ORG
ejpam-6851	355	3	⊆	CARDINAL
ejpam-6851	355	4	sv ⊆	ORG
ejpam-6851	355	5	3	CARDINAL
ejpam-6851	356	1	3(ii	CARDINAL
ejpam-6851	356	2	1	CARDINAL
ejpam-6851	356	3	∈	ORG
ejpam-6851	358	1	5.1	CARDINAL
ejpam-6851	359	1	g ⋄ h.	ORG
ejpam-6851	360	1	huv	GPE
ejpam-6851	361	1	j. m. molles	PERSON
ejpam-6851	361	2	f. p. jamil	PERSON
ejpam-6851	361	3	s. r. canoy	PERSON
ejpam-6851	362	1	j. pure	PERSON
ejpam-6851	362	2	18	CARDINAL
ejpam-6851	362	3	4	CARDINAL
ejpam-6851	362	4	2025	CARDINAL
ejpam-6851	362	5	6851 11	DATE
ejpam-6851	362	6	18	CARDINAL
ejpam-6851	363	1	2	CARDINAL
ejpam-6851	363	2	2	CARDINAL
ejpam-6851	363	3	⊆	CARDINAL
ejpam-6851	364	1	2	CARDINAL
ejpam-6851	364	2	⊆	CARDINAL
ejpam-6851	364	3	⊆	CARDINAL
ejpam-6851	364	4	huv	ORG
ejpam-6851	364	5	∈	ORG
ejpam-6851	364	6	∈	ORG
ejpam-6851	366	1	∈	ORG
ejpam-6851	367	1	2	CARDINAL
ejpam-6851	368	1	∈	ORG
ejpam-6851	369	1	≼g⋄h	NORP
ejpam-6851	369	2	2	CARDINAL
ejpam-6851	369	3	2	CARDINAL
ejpam-6851	369	4	∈	ORG
ejpam-6851	370	1	∈	ORG
ejpam-6851	373	1	w ≼g⋄h z. clearly	PERSON
ejpam-6851	373	2	∈	ORG
ejpam-6851	373	3	w ≼	ORG
ejpam-6851	374	1	huv	PRODUCT
ejpam-6851	375	1	∈	ORG
ejpam-6851	376	1	xy ∈	PERSON
ejpam-6851	376	2	∈	ORG
ejpam-6851	377	1	first	ORDINAL
ejpam-6851	377	2	∈	ORG
ejpam-6851	378	1	∈	ORG
ejpam-6851	378	2	y /∈ a.	PERSON
ejpam-6851	379	1	w ∈	PERSON
ejpam-6851	379	2	w.	PERSON
ejpam-6851	380	1	5	CARDINAL
ejpam-6851	380	2	1	CARDINAL
ejpam-6851	381	1	2	CARDINAL
ejpam-6851	458	1	• • •	DATE
ejpam-6851	458	2	2	CARDINAL
ejpam-6851	458	3	6	CARDINAL
ejpam-6851	460	1	⊆	CARDINAL
ejpam-6851	461	1	γw(g	PRODUCT
ejpam-6851	461	2	⊆	CARDINAL
ejpam-6851	461	3	∈	ORG
ejpam-6851	464	1	∈	ORG
ejpam-6851	466	1	huv	ORG
ejpam-6851	467	1	j. m. molles	PERSON
ejpam-6851	467	2	f. p. jamil	PERSON
ejpam-6851	467	3	s. r. canoy	PERSON
ejpam-6851	468	1	j. pure	PERSON
ejpam-6851	468	2	18	CARDINAL
ejpam-6851	468	3	4	CARDINAL
ejpam-6851	468	4	2025	CARDINAL
ejpam-6851	468	5	6851 12	DATE
ejpam-6851	468	6	18	CARDINAL
ejpam-6851	470	1	∈	ORG
ejpam-6851	471	1	∈	PRODUCT
ejpam-6851	473	1	∈	ORG
ejpam-6851	474	1	∈	PRODUCT
ejpam-6851	474	2	≽g⋄h	DATE
ejpam-6851	474	3	w ∈	ORG
ejpam-6851	475	1	∈	ORG
ejpam-6851	475	2	w ∈	ORG
ejpam-6851	475	3	w.	PERSON
ejpam-6851	475	4	w.	PERSON
ejpam-6851	475	5	γw(g	PRODUCT
ejpam-6851	476	1	⊆	CARDINAL
ejpam-6851	476	2	ae	ORG
ejpam-6851	476	3	∈	ORG
ejpam-6851	480	1	|ae|	CARDINAL
ejpam-6851	480	2	∈	ORG
ejpam-6851	481	1	γw(g	PERSON
ejpam-6851	485	1	∈	ORG
ejpam-6851	487	1	2	CARDINAL
ejpam-6851	488	1	1	CARDINAL
ejpam-6851	488	2	n ≥ 3	DATE
ejpam-6851	489	1	1	CARDINAL
ejpam-6851	489	2	2	CARDINAL
ejpam-6851	489	3	2	CARDINAL
ejpam-6851	489	4	∈	ORG
ejpam-6851	490	1	one	CARDINAL
ejpam-6851	492	1	∈	ORG
ejpam-6851	492	2	∈	ORG
ejpam-6851	493	1	5	CARDINAL
ejpam-6851	493	2	∈	ORG
ejpam-6851	493	3	|ae|	CARDINAL
ejpam-6851	496	1	⊆	CARDINAL
ejpam-6851	497	1	5	CARDINAL
ejpam-6851	497	2	∈	ORG
ejpam-6851	498	1	2	CARDINAL
ejpam-6851	498	2	1	CARDINAL
ejpam-6851	499	1	∈	ORG
ejpam-6851	501	1	∈	ORG
ejpam-6851	501	2	∈	ORG
ejpam-6851	501	3	∈	ORG
ejpam-6851	506	1	∈	ORG
ejpam-6851	506	2	|ae|	CARDINAL
ejpam-6851	506	3	∈	PRODUCT
ejpam-6851	506	4	1	CARDINAL
ejpam-6851	508	1	a\end(g))∪	ORG
ejpam-6851	511	1	∈	ORG
ejpam-6851	512	1	γs(g	PERSON
ejpam-6851	512	2	|ae|	CARDINAL
ejpam-6851	513	1	j. m. molles	PERSON
ejpam-6851	513	2	f. p. jamil	PERSON
ejpam-6851	513	3	s. r. canoy	PERSON
ejpam-6851	514	1	j. pure	PERSON
ejpam-6851	514	2	18	CARDINAL
ejpam-6851	514	3	4	CARDINAL
ejpam-6851	514	4	2025	CARDINAL
ejpam-6851	514	5	6851 13	DATE
ejpam-6851	514	6	18	CARDINAL
ejpam-6851	514	7	⊆	CARDINAL
ejpam-6851	515	1	∈	ORG
ejpam-6851	516	1	∈	ORG
ejpam-6851	519	1	∈	ORG
ejpam-6851	521	1	γs(g	PERSON
ejpam-6851	521	2	∑ uv∈ae |suv|+ ∑	PERSON
ejpam-6851	521	3	|ae|	CARDINAL
ejpam-6851	522	1	6	CARDINAL
ejpam-6851	522	2	⊆	CARDINAL
ejpam-6851	522	3	g ⋄ h	ORG
ejpam-6851	522	4	g ⋄ h	ORG
ejpam-6851	523	1	∈	ORG
ejpam-6851	524	1	γw(g ⋄ h	ORG
ejpam-6851	525	1	γ(h	ORG
ejpam-6851	526	1	1	CARDINAL
ejpam-6851	527	1	γw(g	PERSON
ejpam-6851	530	1	lemma 2	ORG
ejpam-6851	531	1	∈	ORG
ejpam-6851	532	1	∈	ORG
ejpam-6851	533	1	suv	ORG
ejpam-6851	535	1	4	CARDINAL
ejpam-6851	535	2	3	CARDINAL
ejpam-6851	535	3	c6.	ORG
ejpam-6851	537	1	2	CARDINAL
ejpam-6851	538	1	2	CARDINAL
ejpam-6851	538	2	3	CARDINAL
ejpam-6851	539	1	25	CARDINAL
ejpam-6851	540	1	1)γs(g	CARDINAL
ejpam-6851	540	2	mγw(g	ORG
ejpam-6851	541	1	6	CARDINAL
ejpam-6851	541	2	degg[h]((u	GPE
ejpam-6851	543	1	⊆	CARDINAL
ejpam-6851	543	2	∃y ∈	PERSON
ejpam-6851	551	1	j. m. molles	PERSON
ejpam-6851	551	2	f. p. jamil	PERSON
ejpam-6851	551	3	s. r. canoy	PERSON
ejpam-6851	552	1	j. pure	PERSON
ejpam-6851	552	2	18	CARDINAL
ejpam-6851	552	3	4	CARDINAL
ejpam-6851	552	4	2025	CARDINAL
ejpam-6851	552	5	6851 14	DATE
ejpam-6851	552	6	18	CARDINAL
ejpam-6851	554	1	∪x∈a	TIME
ejpam-6851	554	2	⊆	CARDINAL
ejpam-6851	554	3	tx	ORG
ejpam-6851	554	4	⊆	PRODUCT
ejpam-6851	555	1	∈	ORG
ejpam-6851	555	2	tx	ORG
ejpam-6851	555	3	∈	PRODUCT
ejpam-6851	557	1	tx	ORG
ejpam-6851	559	1	∪x∈a	TIME
ejpam-6851	560	1	∈	ORG
ejpam-6851	561	1	∈	ORG
ejpam-6851	562	1	∩	ORG
ejpam-6851	563	1	∈	ORG
ejpam-6851	565	1	tx	ORG
ejpam-6851	565	2	∈	PRODUCT
ejpam-6851	567	1	tx	ORG
ejpam-6851	569	1	∈	ORG
ejpam-6851	571	1	n ≤ n[degg(x)− degg(u	FAC
ejpam-6851	573	1	∈ tx ∩nh(y	ORG
ejpam-6851	573	2	tx	ORG
ejpam-6851	573	3	∈	PRODUCT
ejpam-6851	575	1	∪x∈a	TIME
ejpam-6851	575	2	⊆	CARDINAL
ejpam-6851	575	3	tx	ORG
ejpam-6851	576	1	∈	ORG
ejpam-6851	576	2	tx	ORG
ejpam-6851	576	3	tx	ORG
ejpam-6851	576	4	∈	PRODUCT
ejpam-6851	577	1	∈	ORG
ejpam-6851	580	1	1	CARDINAL
ejpam-6851	583	1	lemma 2	ORG
ejpam-6851	584	1	∈	ORG
ejpam-6851	585	1	w	GPE
ejpam-6851	585	2	∈	ORG
ejpam-6851	589	1	2	CARDINAL
ejpam-6851	592	1	3	CARDINAL
ejpam-6851	593	1	tx	ORG
ejpam-6851	593	2	∈	PRODUCT
ejpam-6851	594	1	w ∈ tx	PERSON
ejpam-6851	594	2	w.	PERSON
ejpam-6851	594	3	∈	ORG
ejpam-6851	596	1	3	CARDINAL
ejpam-6851	596	2	∈	ORG
ejpam-6851	597	1	two	CARDINAL
ejpam-6851	597	2	j. m. molles	PERSON
ejpam-6851	597	3	f. p. jamil	PERSON
ejpam-6851	597	4	s. r. canoy	PERSON
ejpam-6851	598	1	j. pure	PERSON
ejpam-6851	598	2	18	CARDINAL
ejpam-6851	598	3	4	CARDINAL
ejpam-6851	598	4	2025	CARDINAL
ejpam-6851	598	5	6851 15	DATE
ejpam-6851	598	6	18	CARDINAL
ejpam-6851	600	1	∪x∈a	TIME
ejpam-6851	600	2	⊆	CARDINAL
ejpam-6851	600	3	tx	ORG
ejpam-6851	600	4	⊆	PRODUCT
ejpam-6851	601	1	∈	PRODUCT
ejpam-6851	601	2	tx ∈	ORG
ejpam-6851	602	1	10	CARDINAL
ejpam-6851	603	1	∪x∈a	TIME
ejpam-6851	603	2	⊆	CARDINAL
ejpam-6851	603	3	tx	ORG
ejpam-6851	604	1	∈	PRODUCT
ejpam-6851	604	2	tx	ORG
ejpam-6851	604	3	δ(h	PERSON
ejpam-6851	604	4	tx ∈	ORG
ejpam-6851	606	1	11	CARDINAL
ejpam-6851	608	1	∈	ORG
ejpam-6851	608	2	∪x∈a	TIME
ejpam-6851	608	3	∈	ORG
ejpam-6851	608	4	|s∗|	PRODUCT
ejpam-6851	608	5	y ∈ tx	PERSON
ejpam-6851	610	1	∈	ORG
ejpam-6851	611	1	∈	ORG
ejpam-6851	611	2	∈	ORG
ejpam-6851	612	1	tx	ORG
ejpam-6851	612	2	•	CARDINAL
ejpam-6851	612	3	•	CARDINAL
ejpam-6851	612	4	tx	ORG
ejpam-6851	612	5	∈	ORG
ejpam-6851	612	6	•	CARDINAL
ejpam-6851	612	7	tx	ORG
ejpam-6851	613	1	∪x∈a	TIME
ejpam-6851	613	2	∈	ORG
ejpam-6851	615	1	5	CARDINAL
ejpam-6851	618	1	∈	ORG
ejpam-6851	620	1	≼)|	ORG
ejpam-6851	620	2	∈	PRODUCT
ejpam-6851	621	1	j. m. molles	PERSON
ejpam-6851	621	2	f. p. jamil	PERSON
ejpam-6851	621	3	s. r. canoy	PERSON
ejpam-6851	622	1	j. pure	PERSON
ejpam-6851	622	2	18	CARDINAL
ejpam-6851	622	3	4	CARDINAL
ejpam-6851	622	4	2025	CARDINAL
ejpam-6851	622	5	6851 16	DATE
ejpam-6851	622	6	18	CARDINAL
ejpam-6851	624	1	∈	ORG
ejpam-6851	625	1	∈	ORG
ejpam-6851	625	2	∈	ORG
ejpam-6851	625	3	tx	ORG
ejpam-6851	625	4	tx	ORG
ejpam-6851	625	5	⊆	PRODUCT
ejpam-6851	625	6	∪x∈a	TIME
ejpam-6851	625	7	∈	ORG
ejpam-6851	628	1	∪x∈a	TIME
ejpam-6851	628	2	g[h	ORG
ejpam-6851	629	1	proposition 11	LAW
ejpam-6851	629	2	∪x∈a	TIME
ejpam-6851	629	3	⊆	CARDINAL
ejpam-6851	629	4	tx	ORG
ejpam-6851	629	5	⊆	PRODUCT
ejpam-6851	629	6	∈	ORG
ejpam-6851	629	7	tx ̸=	ORG
ejpam-6851	629	8	tx	ORG
ejpam-6851	629	9	∈	PRODUCT
ejpam-6851	632	1	6	CARDINAL
ejpam-6851	633	1	1	CARDINAL
ejpam-6851	634	1	∈	ORG
ejpam-6851	635	1	≼)|	PERSON
ejpam-6851	635	2	∈	PRODUCT
ejpam-6851	636	1	philippines	GPE
ejpam-6851	636	2	msu-iligan institute of technology	ORG
ejpam-6851	637	1	1	CARDINAL
ejpam-6851	637	2	f. buckley	PERSON
ejpam-6851	637	3	f. harary	PERSON
ejpam-6851	639	1	addison-wesley	ORG
ejpam-6851	639	2	redwood city	GPE
ejpam-6851	639	3	1990	DATE
ejpam-6851	640	1	2	CARDINAL
ejpam-6851	640	2	claude berge	PERSON
ejpam-6851	641	1	théorie des graphes	ORG
ejpam-6851	642	1	dunod	PERSON
ejpam-6851	642	2	paris	GPE
ejpam-6851	642	3	1958	DATE
ejpam-6851	643	1	english	NORP
ejpam-6851	643	2	methuen	GPE
ejpam-6851	643	3	london	GPE
ejpam-6851	643	4	andwiley	PERSON
ejpam-6851	643	5	new york	GPE
ejpam-6851	643	6	1962	DATE
ejpam-6851	644	1	3	CARDINAL
ejpam-6851	644	2	e. cockayne	PERSON
ejpam-6851	644	3	s. hedetniemi	PERSON
ejpam-6851	646	1	7(3):247–261	CARDINAL
ejpam-6851	646	2	1977	DATE
ejpam-6851	647	1	4	CARDINAL
ejpam-6851	647	2	w. haynes	PERSON
ejpam-6851	647	3	stephen t. hedetniemi	PERSON
ejpam-6851	647	4	peter j. slater	PERSON
ejpam-6851	649	1	marcel dekker	PERSON
ejpam-6851	649	2	inc.	ORG
ejpam-6851	649	3	new york	GPE
ejpam-6851	649	4	1998	DATE
ejpam-6851	650	1	5	CARDINAL
ejpam-6851	650	2	f. p. jamil	PERSON
ejpam-6851	650	3	r. p. malalay	PERSON
ejpam-6851	652	1	quaestiones mathematicae	ORG
ejpam-6851	652	2	43(2):149–168	CARDINAL
ejpam-6851	652	3	2020	DATE
ejpam-6851	653	1	j. m. molles	PERSON
ejpam-6851	653	2	f. p. jamil	PERSON
ejpam-6851	653	3	s. r. canoy	PERSON
ejpam-6851	654	1	j. pure	PERSON
ejpam-6851	654	2	18	CARDINAL
ejpam-6851	654	3	4	CARDINAL
ejpam-6851	654	4	2025	CARDINAL
ejpam-6851	654	5	6851 17	DATE
ejpam-6851	654	6	18	CARDINAL
ejpam-6851	654	7	6	CARDINAL
ejpam-6851	654	8	h. m. nuenay	PERSON
ejpam-6851	654	9	f. p. jamil	PERSON
ejpam-6851	656	1	35(3):403–418	CARDINAL
ejpam-6851	656	2	2015	DATE
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