Ramsey and Gallai-Ramsey numbers for two classes of unicyclic graphs

TL;DR

This paper determines the 2-color Ramsey numbers for specific unicyclic graphs and establishes bounds on their Gallai-Ramsey numbers, advancing understanding of edge colorings in graph theory.

Contribution

It provides exact 2-color Ramsey numbers for two classes of unicyclic graphs and derives bounds on their Gallai-Ramsey numbers, a novel contribution in this graph class.

Findings

Exact 2-color Ramsey numbers for the two unicyclic graph classes.

Derived upper and lower bounds for Gallai-Ramsey numbers of these graphs.

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

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 consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the $2$-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.