Complete bipartite graphs without small rainbow subgraphs

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 number bgr(k)(G : H) as the minimum number of vertices n such that n(2) >= 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 K-n,K-n without rainbow P-4(+) and 3K(2), respectively, where P-4(+) is the graph consisting of a P-4 with one extra edge incident with an interior vertex. Furthermore, we determine the exact values or upper and lower bounds on bgr(k)(G : H) when G is a 3-matching or a 4-path or P-4(+), and H is a bipartite graph. (c) 2023 Elsevier B.V. All rights reserved.