A unified proof of Brooks' theorem and Catlin's theorem

TL;DR

This paper presents a new proof that simultaneously establishes Brooks' theorem and its strengthening, Catlin's theorem, providing a unified approach to these graph coloring results.

Contribution

The authors introduce a novel proof technique that unifies and strengthens the proofs of Brooks' theorem and Catlin's theorem.

Findings

Unified proof of Brooks' and Catlin's theorems

Immediate implications for graph coloring theory

Strengthening of existing bounds in graph coloring

Abstract

We give a new proof of Brooks' theorem that immediately implies a strengthening of Brooks' theorem, known as Catlin's theorem.