Gallai-Ramsey numbers for graphs with chromatic number three

Abstract

Given a graph H and an integer k ≥ 1, the Gallai-Ramsey number GR k (H) is defined to be the minimum integer n such that every k-edge coloring of 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 GR 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 GR k ( K 4 ) for k ≥ 1. Our outcomes not only indicate that the conjecture on GR k ( K m ) is true for m ≤ 4, but also imply several results on GR k (H) for some H ∈ H which are proved individually in different papers.

Gallai-Ramsey numbers for graphs with five vertices and eight edges

“…The Gallai-Ramsey numbers gr k (K 3 : H) for all the graphs H on five vertices and at most seven edges are obtained (see [10,17,14,16]). There are two graphs on five vertices and eight edges, one of which is a wheel graph W 4 , and the other is a graph B + 3 obtained from the book graph B 3 by adding an edge between two vertices with degree two.…”

Gallai-Ramsey numbers for graphs with five vertices of chromatic number four

“…It is known that there are twenty three isolated-free graphs with five vertices, denoted by H . The Gallai-Ramsey numbers for the graphs in H with chromatic number two and three were completely determined in [3,11,14,15,[18][19][20][21][22][23]. In this paper, we obtain the Gallai-Ramsey numbers for graphs in H with chromatic number four as follows.…”