On size multipartite Ramsey numbers for stars versus paths and cycles

Lusiani, Anie; Baskoro, Edy Tri; Saputro, Suhadi Wido

doi:10.5614/ejgta.2017.5.1.5

Cited by 11 publications

(17 citation statements)

References 5 publications

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“…In 2017, Jayawardene et al [4] and Effendi et al [2] determined the size multipartite Ramsey numbers for stars versus paths. Then, we also gave the size multipartite Ramsey numbers for stars versus paths and cycles [7], that complete the previous results given by Syafrizal and Surahmat. Recently, we determined m j (mK 1,n , H), where…”

Section: Introductionsupporting

confidence: 79%

On size multipartite Ramsey numbers for stars

Lusiani

Baskoro

Saputro

2020

IJC

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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.

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Section: Introductionsupporting

confidence: 79%

On size multipartite Ramsey numbers for stars

Lusiani

Baskoro

Saputro

2020

IJC

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“…In [3], Burgeet et al studied the M-R-number m j (G 1 , G 2 ), where both G 1 and G 2 are a complete multipartite graphs. Recently the M-R-number m j (G 1 , G 2 ) has been studied for special classes of graphs, see [4,6,11,13] and its references, which can be naturally extent to several colors, see [9,10,17,[19][20][21]. The M-R-number m j (K 1,m , G), for j = 2, 3 where G is a P n or a C n is determined in [12].…”

Section: Introductionmentioning

confidence: 99%

Multipartite Ramsey number $m_j(K_m, nK_2)$

Rowshan¹

2022

Preprint

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Assume that Kj×n be a complete, multipartite graph consisting of j partite sets and n vertices in each partite set. For given graphs G1, G2, . . . , Gn, the multipartite Ramsey number (M-R-number) mj(G1, G2, . . . , Gn) is the smallest integer t such that for any n-edge-coloring (G 1 , G 2 , . . . , G n ) of the edges of Kj×t, G i contains a monochromatic copy of Gi for at least one i. The size of M-R-number mj(nK2, Cm) for j, n ≥ 2 and 4 ≤ m ≤ 6, the size of M-R-number mj(nK2, C7) for j ≥ 2 and n ≥ 2, the size of M-R-number mj (nK2, K3), for each j, n ≥ 2, the size of M-R-number mj(K3, K3, n1K2, n2K, . . . , niK2) for j ≤ 6 and i, ni ≥ 1 and the size of M-Rnumber mj(K3, K3, nK2) for j ≥ 2 and n ≥ 1 have been computed in several papers up to now. In this article we obtain the values of M-R-number mj(Km, nK2), for each j, n ≥ 2 and each m ≥ 4.

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“…The size of the multipartite Ramsey numbers of small paths versus certain classes of graphs have been studied in [8][9][10]. The size of the multipartite Ramsey numbers of stars versus certain classes of graphs have been studied in [11,12]. In [13,14], Burger, Stipp, Vuuren, and Grobler investigated the multipartite Ramsey numbers m j (G 1 , G 2 ), where G 1 and G 2 are in a completely balanced multipartite graph, which can be naturally extended to several colors.…”

Section: Introductionmentioning

confidence: 99%

“…In [12], Lusiani et al determined the size of the multipartite Ramsey numbers of m j (K 1,m , H), for j = 2, 3, where H is a path or a cycle on n vertices, and K 1,m is a star of order m + 1. In this paper, we computed the size of the multipartite Ramsey numbers m j (K 1,2 , P 4 , nK 2 ) for n, j ≥ 2 and m j (nK 2 , C 7 ), for j ≤ 4 and n ≥ 2 which are the new results of multipartite Ramsey numbers.…”

Section: Introductionmentioning

confidence: 99%

The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

2021

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For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

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On size multipartite Ramsey numbers for stars versus paths and cycles

Cited by 11 publications

References 5 publications

On size multipartite Ramsey numbers for stars

On size multipartite Ramsey numbers for stars

Multipartite Ramsey number $m_j(K_m, nK_2)$

The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

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