Gallai-Ramsey numbers for fans

TL;DR

This paper investigates Gallai-Ramsey numbers for fan graphs, establishing bounds and exact values for specific cases, advancing understanding of edge colorings avoiding rainbow triangles and monochromatic fans.

Contribution

The paper provides general bounds and exact values for Gallai-Ramsey numbers for fans, specifically for cases where m=2 and m=3 with even k.

Findings

Established bounds for Gallai-Ramsey numbers of fans.

Proved exact values for fans with m=2.

Proved exact values for fans with m=3 when k is even.

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 $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for fans $F_{m} = K_{1} + mK_{2}$ and prove the sharp result for $m = 2$ and for $m = 3$ with $k$ even.