A note on Gallai-Ramsey number of even wheels

TL;DR

This paper investigates the Gallai-Ramsey numbers for even wheel graphs, providing exact values for the case of $W_4$ across all color counts, advancing understanding of edge-coloring properties in complete graphs.

Contribution

It determines the exact Gallai-Ramsey numbers for the even wheel graph $W_4$ for all numbers of colors, filling a gap in combinatorial graph theory.

Findings

Exact values of $GR_k(W_4)$ for all $k \\geq 2$

Complete characterization of Gallai-Ramsey numbers for $W_{2n}$

Insights into monochromatic subgraph existence in Gallai colorings

Abstract

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $W_{2n} $ denote an even wheel on $2n+1\ge5$ vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and completely determine the exact value of $GR_k(W_4)$ for all $k\ge2$.