Gallai-Ramsey numbers of $C_9$ with multiple colors

Abstract

We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\cdot 2^k+1$ for all $k\ge1$. This new result provides partial evidence for the first open case 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 edge-colorings of complete graphs without rainbow triangles. We believe the method we developed can be used to determine the exact values of $gr_k(K_3, C_n)$ for odd integers $n\ge11$.