Size multipartite Ramsey numbers for small paths versus books (original) (raw)
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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 d...
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On size multipartite Ramsey numbers for paths versus stars
International Journal of Mathematical Analysis, 2016
For given two graphs G 1 and G 2 , and integer j ≥ 2, the size multipartite Ramsey numbers m j (G 1 , G 2) = t is the smallest integer such that every factorization of graph K j×t := F 1 ⊕ F 2 satisfies the following condition: either F 1 contains G 1 as a subgraph or F 2 contains G 2 as a subgraph. In this paper, we determine that m j (K 1,m , P l) for l ≥ 2 where K 1,n is a star on n vertices and P l is a path on l vertices.
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Size multipartite Ramsey numbers for stripes versus small cycles
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For simple graphs G 1 and G 2 , the size Ramsey multipartite number m j (G 1 , G 2) is defined as the smallest natural number s such that any arbitrary two coloring of the graph K j×s using the colors red and blue, contains a red G 1 or a blue G 2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers m j (nK 2 , C m) for j ≥ 2 and m ∈ {3, 4, 5, 6}.
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For graphs G 1 , G 2 , G 3 , the three-color Ramsey number R(G 1 , G 2 , G 3) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it contains a monochromatic copy of G i in color i, for some 1 ≤ i ≤ 3. First, we prove that the conjectured equality R(C 2n , C 2n , C 2n) = 4n, if true, implies that R(P 2n+1 , P 2n+1 , P 2n+1) = 4n + 1 for all n ≥ 3. We also obtain two new exact values R(P 8 , P 8 , P 8) = 14 and R(P 9 , P 9 , P 9) = 17, furthermore we do so without help of computer algorithms. Our results agree with a formula R(P n , P n , P n) = 2n−2+(n mod 2) which was proved for sufficiently large n by Gyárfás, Ruszinkó, Sárközy, and Szemerédi in 2007. This provides more evidence for the conjecture that the latter holds for all n ≥ 1.
On size multipartite Ramsey numbers for stars versus paths and cycles
Electronic Journal of Graph Theory and Applications
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.