Gallai Ramsey number for double stars

TL;DR

This paper investigates the Gallai-Ramsey numbers for double star graphs, establishing bounds and exact values in certain cases, advancing understanding of edge colorings in complete graphs.

Contribution

It provides general bounds and exact results for Gallai-Ramsey numbers of double star graphs, a new class of graphs in this context.

Findings

Established upper and lower bounds for Gallai-Ramsey numbers of double stars.

Derived sharp results for specific cases of double star graphs.

Enhanced understanding of rainbow and monochromatic subgraph occurrences in edge colorings.

Abstract

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for double stars $S(n,m)$, where $S(n,m)$ is the graph obtained from the union of two stars $K_{1,n}$ and $K_{1,m}$ by adding an edge between their centers. We also provide the sharp result in some cases.