Improved Upper Bounds for Gallai-Ramsey Numbers of Odd Cycles

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 an integer $k\ge1$ and a graph $H$, the Gallai-Ramsey number $GR_k(H)$ is the least positive integer $n$ such that every Gallai $k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. Gy\'{a}rf\'{a}s, S\'{a}rk\"{o}zy, Seb\H{o} and Selkow proved in 2010 that $GR_k (H)$ is exponential in $k$ if $H$ is not bipartite, linear in $k$ if $H$ is bipartite but not a star, and constant (does not depend on $k$) when $H$ is a star. Hence, $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial, even when $|V(H)|$ is small. In this paper, we first improve the existing upper bounds for Gallai-Ramsey numbers of odd cycles by showing that $GR_k(C_{2n+1}) \le (n\ln n) \cdot 2^k -(k+1)n+1$ for all $k \ge 3$ and $n \ge 8$. We then prove that $GR_k( C_{13})= 6\cdot 2^k+1$ and $GR_k( C_{15})= 7\cdot 2^k+1$ for all $k\ge1$.