Gallai–Ramsey numbers for graphs with chromatic number three

Abstract

Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) is defined to be the minimum integer n such that every k-edge coloring of the complete graph Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of H. In this paper, we study Gallai–Ramsey numbers for graphs with chromatic number three such as K̂m for m≥2, where K̂m is a kipas with m+1 vertices obtained from the join of K1 and Pm, and a class of graphs with five vertices, denoted by ℋ. We first study the general lower bound of such graphs and propose a conjecture for the exact value of GRk(K̂m). Then we give a unified proof to determine the Gallai–Ramsey numbers for many graphs in ℋ and obtain the exact value of GRk(K̂4) for k≥1. Our outcomes not only indicate that the conjecture on GRk(K̂m) is true for m≤4, but also imply several results on GRk(H) for some H∈ℋ which are proved individually in different papers.