A note on Reed's Conjecture about $\omega$, $\Delta$ and $\chi$ with respect to vertices of high degree

TL;DR

This paper investigates Reed's Conjecture relating chromatic number, clique number, and maximum degree, introducing an algorithm to analyze potential counterexamples and identifying graph classes where the conjecture holds.

Contribution

An algorithm is developed to analyze hypothetical counterexamples to Reed's Conjecture, revealing structural properties and identifying specific graph classes where the conjecture is valid.

Findings

Algorithm uncovers structures related to high-degree vertices.

Reed's Conjecture holds for graphs with high-degree vertices forming a stable set.

Reed's Conjecture holds for graphs where all odd cycles contain a low-degree vertex.

Abstract

Reed conjectured that for every graph, $\chi \leq \left \lceil \frac{\Delta + \omega + 1}{2} \right \rceil$ holds, where $\chi$, $\omega$ and $\Delta$ denote the chromatic number, clique number and maximum degree of the graph, respectively. We develop an algorithm which takes a hypothetical counterexample as input. The output discloses some hidden structures closely related to high vertex degrees. Consequently, we deduce two graph classes where Reed's Conjecture holds: One contains all graphs in which the vertices of degree at least $5$ form a stable set. The other contains all graphs in which every induced cycle of odd length contains a vertex of at most degree 3.