Gallai-Ramsey numbers of odd cycles

Abstract

Given two graphs $G$ and $H$ and a positive integer $k$, the $k$-color Gallai-Ramsey number, denoted by $g{r}_k(G:H)$, is the minimum integer $N$ such that for all $n\ge N$, every $k$-coloring of the edges of $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. We prove that $g{r}_k(K_3:C_{2\ell +1})=\ell \cdot 2^k+1$ for all $k\ge 1$ and $\ell \ge 3$.