Gallai-Ramsey Numbers for $\ell$-Connected Graphs

TL;DR

This paper investigates Gallai-Ramsey numbers for specific graphs, providing exact values and bounds for cases involving $ ext{ell}$-connected graphs, especially for paths and stars, advancing understanding of rainbow and monochromatic subgraph conditions.

Contribution

The paper derives exact Gallai-Ramsey numbers and bounds for $ ext{ell}$-connected graphs, focusing on $P_5$ and $K_{1,3}$, expanding known results in graph coloring theory.

Findings

Exact Gallai-Ramsey numbers for $P_5$ and $K_{1,3}$.

General lower and upper bounds for $ ext{ell}$-connected graphs.

New insights into rainbow and monochromatic subgraph occurrences.

Abstract

Given a nonempty graph $G$, a collection of nonempty graphs $\cal{H}$, and a positive integer $k$, the Gallai-Ramsey number $\mathrm{gr}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact $k$-edge-coloring of a complete graph $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of some element in $\mathcal{H}$. In this paper, we obtain some exact values and general lower and upper bounds for $\mathrm{gr}_k(G:\mathcal{F}^\ell)$, where $\mathcal{F}^\ell$ is the set of $\ell$-connected graphs and $G\in\{P_5, K_{1,3}\}$.