The chromatic gap and its extremes

TL;DR

This paper explores the maximum difference between the chromatic number and clique number of graphs with n vertices, providing an almost exact formula linked to Ramsey numbers through theoretical insights.

Contribution

It introduces a novel connection between the chromatic gap and Ramsey theory, offering a simple formula for extremal graphs based on these concepts.

Findings

Derived an almost exact formula for ap(n) in terms of Ramsey numbers.

Established a theoretical link between the chromatic gap and Ramsey theory.

Provided insights into the structure of extremal graphs for the chromatic gap.

Abstract

The {\em chromatic gap} is the difference between the chromatic number and the clique number of a graph. Here we investigate $\gap(n)$, the maximum chromatic gap over graphs on $n$ vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for $\gap(n)$ in terms of Ramsey numbers.