Abstract
Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs G and H, the k-colored Gallai–Ramsey number grk(G:H) is defined to be the minimum integer n such that n2≥k and for every N≥n, every rainbow G-free coloring (using all k colors) of the complete graph KN contains a monochromatic copy of H. In this paper, we first provide some exact values and bounds of grk(P5:Kt). Moreover, we define the k-colored bipartite Gallai–Ramsey number bgrk(G:H) as the minimum integer n such that n2≥k and for every N≥n, every rainbow G-free coloring (using all k colors) of the complete bipartite graph KN,N contains a monochromatic copy of H. Furthermore, we describe the structures of complete bipartite graph Kn,n with no rainbow P4 and P5, respectively. Finally, we find the exact values of bgrk(P4:Ks,t) (1≤s≤t), bgrk(P4:F) (where F is a subgraph of Ks,t), bgrk(P5:K1,t) and bgrk(P5:K2,t) by using the structural results.
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