Multicolor Gallai-Ramsey numbers of $C_9$ and $C_{11}$

Abstract: A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. We study Ramsey-type problems in Gallai colorings. 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 on $n$ vertices contains a monochromatic copy of $H$. It turns out that $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial. In this paper, we study Gallai-Ramsey numbers of odd cycles. We prove that for $n\in{4,5}$ and all $k\ge1$, $GR_k(C_{2n+1})= n\cdot 2^k+1$. This new result provides partial evidence for the first two open cases of the Triple Odd Cycle Conjecture of Bondy and Erd\H{o}s from 1973. Our technique relies heavily on the structural result of Gallai on Gallai colorings of complete graphs. We believe the method we developed can be used to determine the exact values of $GR_k(C_{2n+1})$ for all $n\ge6$.