A conjecture on Gallai-Ramsey numbers of even cycles and paths

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 at most $k$ 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 $i \in {1,2, \ldots, k}$. When $H = H_1 = \cdots = H_k$, we simply write $GR_k(H)$. We study Gallai-Ramsey numbers of even cycles and paths. For all $n\ge3$ and $k\ge2$, 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 $j\in{1,2, \ldots, k}$ with $ i_1\ge i_2\ge\cdots\ge i_k $. The first author recently conjectured that $ GR(G_{i_1}, G_{i_2}, \ldots, G_{i_k}) = |G_{i_1}|+\sum_{j=2}^k i_j$. The truth of this conjecture implies that $GR_k(C_{2n})=GR_k(P_{2n})=(n-1)k+n+1$ for all $n\ge3$ and $k\ge1$, and $GR_k(P_{2n+1})=(n-1)k+n+2$ for all $n\ge1$ and $k\ge1$. In this paper, we prove that the aforementioned conjecture holds for $n\in{3,4}$ and all $k\ge2$. Our proof relies only on Gallai's result and the classical Ramsey numbers $R(H_1, H_2)$, where $H_1, H_2\in{C_8, C_6, P_7, P_5, P_3}$. We believe the recoloring method we developed here will be very useful for solving subsequent cases, and perhaps the conjecture.