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

TL;DR

This paper investigates Gallai-Ramsey numbers involving rainbow 4-paths, providing exact values and bounds for certain graphs H, advancing understanding of edge-coloring Ramsey problems.

Contribution

It offers new exact values and bounds for Gallai-Ramsey numbers with rainbow 4-paths and specific graph classes, extending previous results.

Findings

Derived exact Gallai-Ramsey numbers for specific graphs H

Established bounds for Gallai-Ramsey numbers involving rainbow 4-paths

Extended known results to new classes of graphs

Abstract

Given two non-empty graphs $G,H$ and a positive integer $k$, the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $N$ such that for all $n\geq N$, 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 $\operatorname{gr}_k(P_5:H) \ (k\geq 3)$ if $H$ is a general graph or a star with extra independent edges or a pineapple.