The k-Ramsey number of two five cycles

Abstract

Given any two graphs F and H, the Ramsey number R(F, H) is defined as the smallest positive integer n such that every red-blue coloring of the edges of the complete graph Kn of order n, there will be a subgraph of Kn isomorphic to F whose edges are all colored red (a red F) or a subgraph of Kn isomorphic to H whose edges are all colored blue (a blue H). If F and H are bipartite graphs, then the k-Ramsey number R k ( F , H ) is defined as the smallest positive integer n such that for any red-blue coloring of the edges of the complete k-partite graph of order n in which each partite set is of order ⌊ n k ⌋ or ⌈ n k ⌉ there will be a subgraph isomorphic to F whose edges are all colored red (a red F) or a subgraph isomorphic to H whose edges are all colored blue (a blue H). Andrews, Chartrand, Lumduanhom and Zhang found the k-Ramsey number R k ( F , H ) for F = H = C 4 , and for F = K 1 , t and H = K 1 , s where s , t ≥ 2 . We continue their work by investigating the case where the graphs F and H are both C 5.