On size tripartite Ramsey numbers of P3 versus mK1,n

Saputro

2017

ejgta

Self Cite

Let K l×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G 1 and G 2 , and integer j ≥ 2, the size multipartite Ramsey number m j (G 1 , G 2 ) is the smallest integer t such that every factorization of the graph K j×t := F 1 ⊕ F 2 satisfies the following condition: either F 1 contains G 1 or F 2 contains G 2 . In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths P n versus stars, for n = 2, 3 only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths P n versus stars, for n = 3, 4, 5, 6. In this paper, we investigate the size tripartite Ramsey numbers of paths P n versus stars, with all n ≥ 2. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers m 2 (K 1,m , C n ) of stars versus cycles, for n ≥ 3, m ≥ 2.

Section: Stars Versus Pathsmentioning

confidence: 99%

On size multipartite Ramsey numbers for stars versus paths and cycles

Lusiani

Saputro

2017

ejgta

Self Cite

“…In [8], Lusiani et al determined the size multipartite Ramsey numbers m j (G, K 1,2 ), for j = 2, 3, where G is a union of two, three, or four distinct stars. In [6,7], Lusiani et al determined the size multipartite Ramsey numbers m j (mK 1,n , H) where H = K 1,2 or H = K 1,3 . In this paper, we shall consider multipartite Ramsey numbers for the union of stars in a more general setting.…”

Section: Introductionmentioning

confidence: 99%

Multipartite Ramsey Numbers of Union of Stars

Anuwiksa

Simanjuntak

2022

EJGTA

Let s and k be positive integers with k ≥ 2 and G 1 , G 2 , . . . , G k be simple graphs. The set multipartite Ramsey number, denoted by M s (G 1 , G 2 , . . . , G k ), is the smallest positive integer c such that any k-coloring of the edges of K c×s contains a monochromatic copy of G i in color i for some i ∈ {1, 2, . . . , k}. The size multipartite Ramsey number, denoted by m c (G 1 , G 2 , . . . , G k ), is the smallest positive integer s such that any k-coloring of the edges of K c×s contains a monochromatic copy of G i in color i for some i ∈ {1, 2, . . . , k}. In this paper, we establish some lower and upper bounds, and some exact values of multipartite Ramsey numbers for the union of stars.

“…Then, Surahmat et al [9] gave the size tripartite Ramsey numbers for stars versus P n , for 3 ≤ n ≤ 6. Furthermore, we gave the size multipartite Ramsey numbers for stars versus cycles [5] and the size tripartite Ramsey numbers for a disjoint union of m copies of a star K 1,n versus P 3 [6]. In 2017, Jayawardene et al [4] and Effendi et al [2] determined the size multipartite Ramsey numbers for stars versus paths.…”

Section: Introductionmentioning

confidence: 99%

On size multipartite Ramsey numbers for stars

Lusiani

Saputro

2020

IJC

Self Cite

Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs mj(Kaxb,Kcxd), for natural numbers a,b,c,d and j, where a,c >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers mj(Kaxb,Kcxd). Syafrizal et al. generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number mj(G,H) as the smallest natural number t such that any red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers mj(G,H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers mj(K1,m, K1,n) for all integers m,n >= 1 and j = 2,3, where K1,m is a star of order m+1. In addition, we also determine the lower bound of m3(kK1,m, C3), where kK1,m is a disjoint union of k copies of a star K1,m and C3 is a cycle of order 3.