Gallai–Ramsey numbers for graphs with chromatic number three

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