Complete bipartite graphs without small rainbow stars

The k-edge-colored bipartite Gallai-Ramsey number bgrk(G : H) is defined as the minimum integer n such that n2 > k and for every N > n, every edge-coloring (using all k colors) of complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first study the structural theorem on the complete bipartite graph Kn,n with no rainbow copy of K1,3. Next, we utilize the results to prove the exact values of bgrk(P4 : H), bgrk(P5 : H), bgrk(K1,3 : H), where H is a various union of cycles and paths and stars. & COPY; 2023 Published by Elsevier B.V.