A note on Gallai-Ramsey number of even wheels

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 $k\ge 1$, the Gallai-Ramsey number $G{R}_k(H)$ 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 $2n+1\ge 5$ vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and completely determine the exact value of $G{R}_k(W_4)$ for all $k\ge 2$.

This note was submitted to Discrete Mathematics for publication on January 28, 2019. arXiv admin note: text overlap with arXiv:1809.00227, arXiv:1808.09963, arXiv:1808.10282, arXiv:1803.07963