Complete bipartite graphs without small rainbow subgraphs

Abstract

Motivated by bipartite Gallai–Ramsey type problems, we consider edge-colorings of complete bipartite graphs without rainbow tree and matching. Given two graphs G and H, and a positive integer k, define the bipartite Gallai–Ramsey numberbgrk(G:H) as the minimum number of vertices n such that n2≥k and for every N≥n, any coloring (using all k colors) of the complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first describe the structures of a complete bipartite graph Kn,n without rainbow P4+ and 3K2, respectively, where P4+ is the graph consisting of a P4 with one extra edge incident with an interior vertex. Furthermore, we determine the exact values or upper and lower bounds on bgrk(G:H) when G is a 3-matching or a 4-path or P4+, and H is a bipartite graph.