Gallai-Ramsey numbers involving a rainbow $4$-path

Abstract

Given two non-empty graphs $G,H$ and a positive integer $k$, the Gallai-Ramsey number ${\mathrm{gr}}_k(G:H)$ is defined as the minimum integer $N$ such that for all $n\ge N$, every $k$-edge-coloring of $K_n$ contains either a rainbow colored copy of $G$ or a monochromatic copy of $H$. In this paper, we got some exact values or bounds for ${\mathrm{gr}}_k(P_5:H)(k\ge 3)$ if $H$ is a general graph or a star with extra independent edges or a pineapple.

16 pages