Gallai-Ramsey numbers for books

TL;DR

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

Contribution

It provides new bounds and exact results for Gallai-Ramsey numbers of books, especially for cases where m ≤ 5, extending previous knowledge in graph Ramsey theory.

Findings

Established general upper and lower bounds for Gallai-Ramsey numbers of books.

Proved exact Gallai-Ramsey numbers for books with m ≤ 5.

Extended the understanding of rainbow and monochromatic structures in edge-colored complete graphs.

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) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for books $B_{m} = K_{2} + \overline{K_{m}}$ and prove sharp results for $m \leq 5$.