Gallai–Ramsey numbers for graphs with chromatic number three

Given a graph H and an integer k ≥ 1 , the Gallai–Ramsey number G R k ( H ) is defined to be the minimum integer n such that every k -edge coloring of the complete graph K n 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 K m for m ≥ 2 , where K m is a kipas with m + 1 vertices obtained from the join of K 1 and P m , 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 G R k ( K m ) . Then we give a unified proof to determine the Gallai–Ramsey numbers for many graphs in ℋ and obtain the exact value of G R k ( K 4 ) for k ≥ 1 . Our outcomes not only indicate that the conjecture on G R k ( K m ) is true for m ≤ 4 , but also imply several results on G R k ( H ) for some H ∈ ℋ which are proved individually in different papers.