An improvement on Brooks' Theorem

TL;DR

This paper presents a new upper bound on the chromatic number of graphs, extending Brooks' Theorem by incorporating the parameter Δ₂ and the maximum degree, applicable to graphs with degree at least 3.

Contribution

It introduces a generalized upper bound on graph chromatic number that unifies and extends previous theorems like Brooks' and Ore-degree versions.

Findings

New upper bound on χ(G) involving Δ, ω, and Δ₂

Generalizes Brooks' Theorem and Ore-degree bounds

Applicable to graphs with Δ(G) ≥ 3

Abstract

We prove that $\chi(G) \leq \max {\omega(G), \Delta_2(G), (5/6)(\Delta(G) + 1)}$ for every graph $G$ with $\Delta(G) \geq 3$. Here $\Delta_2$ is the parameter introduced by Stacho that gives the largest degree that a vertex $v$ can have subject to the condition that $v$ is adjacent to a vertex whose degree is at least as large as its own. This upper bound generalizes both Brooks' Theorem and the Ore-degree version of Brooks' Theorem.