Gallai-Ramsey numbers of $C_{10}$ and $C_{12}$

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. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the Gallai-Ramsey number $GR(H_1, \ldots, H_k)$ is the least integer $n$ such that every Gallai $k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H_i$ in color $i$ for some $i \in {1, \ldots, k}$. When $H = H_1 = \cdots = H_k$, we simply write $GR_k(H)$. We continue to study Gallai-Ramsey numbers of even cycles and paths. For all $n\ge3$ and $k\ge1$, let $G_i=P_{2i+3}$ be a path on $2i+3$ vertices for all $i\in{0,1, \ldots, n-2}$ and $G_{n-1}\in{C_{2n}, P_{2n+1}}$. Let $ i_j\in{0,1,\ldots, n-1}$ for all $j\in{1, \ldots, k}$ with $ i_1\ge i_2\ge\cdots\ge i_k $. Song recently conjectured that $GR(G_{i_1}, \ldots, G_{i_k}) = 3+\min{i_1, n^*-2}+\sum_{j=1}^k i_j$, where $n^* =n$ when $G_{i_1}\ne P_{2n+1}$ and $n^* =n+1$ when $G_{i_1}= P_{2n+1}$. This conjecture has been verified to be true for $n\in{3,4}$ and all $k\ge1$. In this paper, we prove that the aforementioned conjecture holds for $n \in{5, 6}$ and all $k \ge1$. Our result implies that for all $k \ge 1$, $GR_k(C_{2n}) = GR_k(P_{2n}) = (n-1)k+n+1$ for $n\in{5,6}$ and $GR_k(P_{2n+1})= (n-1)k+n+2$ for $1\le n \le6 $.