Gallai-Ramsey number of an 8-cycle

Abstract

Given graphs $G$ and $H$ and a positive integer $k$, the Gallai-Ramsey number $g{r}_k(G:H)$ is the minimum integer $N$ such that for any integer $n\ge N$, every $k$-edge-coloring of $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. These numbers have recently been studied for the case when $G=K_3$, where still only a few precise numbers are known for all $k$. In this paper, we extend the known precise Gallai-Ramsey numbers to include $H=C_8$ for all $k$.