Brooks' Theorem and Beyond

TL;DR

This paper reviews various proofs and extensions of Brooks' Theorem, exploring techniques in graph coloring and discussing stronger variants and related conjectures in the field.

Contribution

It provides a comparative analysis of multiple proofs of Brooks' Theorem and discusses extensions to advanced coloring concepts and open conjectures.

Findings

Different proofs of Brooks' Theorem highlight key techniques.

Extensions to list coloring, online list coloring, and Alon--Tarsi orientations are discussed.

Conjectures like Borodin--Kostochka and Reed's are proposed as stronger variants.

Abstract

We collect some of our favorite proofs of Brooks' Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon--Tarsi orientations, since analogues of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin--Kostochka Conjecture and Reed's Conjecture.