Abstract: | Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs and , the -colored Gallai–Ramsey number is defined to be the minimum integer such that and for every , every rainbow -free coloring (using all colors) of the complete graph contains a monochromatic copy of . In this paper, we first provide some exact values and bounds of . Moreover, we define the -colored bipartite Gallai–Ramsey number as the minimum integer such that and for every , every rainbow -free coloring (using all colors) of the complete bipartite graph contains a monochromatic copy of . Furthermore, we describe the structures of complete bipartite graph with no rainbow and , respectively. Finally, we find the exact values of (), (where is a subgraph of ), and by using the structural results. |