Gallai-Ramsey number for the union of stars

TL;DR

This paper determines the exact Gallai-Ramsey numbers for the union of two stars in many cases, advancing understanding of monochromatic subgraphs in edge-colored complete graphs.

Contribution

It provides the first exact values and bounds for Gallai-Ramsey numbers involving disconnected graphs, specifically unions of two stars.

Findings

Exact Gallai-Ramsey numbers for union of two stars in many cases

Bounds established for remaining cases

First results for disconnected graphs in this context

Abstract

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain the exact value of the Gallai-Ramsey numbers for the union of two stars in many cases and bounds in other cases. This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.