Gallai-Ramsey numbers for a class of graphs with five vertices

TL;DR

This paper determines the Gallai-Ramsey numbers for all connected five-vertex graphs with up to six edges in the specific case of $gr_k(K_3 : H)$, filling gaps in previous research and extending to some unicyclic graphs.

Contribution

It provides complete values of Gallai-Ramsey numbers for a class of small graphs, completing prior partial results and exploring related unicyclic graphs.

Findings

All Gallai-Ramsey numbers for the 13 graphs in the class are determined.

New results for some unicyclic graphs are obtained.

The study extends existing knowledge on Gallai-Ramsey numbers for small graphs.

Abstract

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we consider $gr_k(K_3 : H)$ where $H$ is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai-Ramsey numbers for some of them have been studied step by step in several papers. We determine all the Gallai-Ramsey numbers for the remaining graphs, and we also obtain some related results for a class of unicyclic graphs.