Chromatic number is Ramsey distinguishing

TL;DR

This paper proves that the chromatic number can distinguish non-Ramsey equivalent graphs and extends this concept to multiple colours, introducing new insights into graph parameters related to Ramsey theory.

Contribution

The paper demonstrates that the chromatic number is a Ramsey distinguishing parameter and identifies another such parameter using similar methods.

Findings

Chromatic number is a Ramsey distinguishing parameter.

Extension of the concept to multi-colour Ramsey problems.

Identification of an additional Ramsey distinguishing parameter.

Abstract

A graph $G$ is Ramsey for a graph $H$ if every colouring of the edges of $G$ in two colours contains a monochromatic copy of $H$. Two graphs $H_1$ and $H_2$ are Ramsey equivalent if any graph $G$ is Ramsey for $H_1$ if and only if it is Ramsey for $H_2$. A graph parameter $s$ is Ramsey distinguishing if $s(H_1)\neq s(H_2)$ implies that $H_1$ and $H_2$ are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multi-colour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.