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On Brooks' Theorem

Gopalan Sajith, Sanjeev Saxena

TL;DR

The paper studies coloring graphs with maximum degree $Δ ≥ 3$ under Brooks' Theorem, which asserts a $Δ$-coloring unless the graph is complete or an odd cycle. It presents two constructive proofs: a modified MV/Wilson approach that yields a linear-time coloring via vertex deletion and Kempe-chain recolorings, and a Zajac–Bondy–inspired DFS approach that delivers a linear-time algorithm through separation-point analysis and Kempe recolorings. These proofs provide explicit, teachable algorithms and deepen understanding of constructive techniques in coloring. Together, they offer practical methods for efficient $Δ$-coloring and facilitate CS education.

Abstract

In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.

On Brooks' Theorem

TL;DR

The paper studies coloring graphs with maximum degree under Brooks' Theorem, which asserts a -coloring unless the graph is complete or an odd cycle. It presents two constructive proofs: a modified MV/Wilson approach that yields a linear-time coloring via vertex deletion and Kempe-chain recolorings, and a Zajac–Bondy–inspired DFS approach that delivers a linear-time algorithm through separation-point analysis and Kempe recolorings. These proofs provide explicit, teachable algorithms and deepen understanding of constructive techniques in coloring. Together, they offer practical methods for efficient -coloring and facilitate CS education.

Abstract

In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.
Paper Structure (3 sections, 2 theorems)

This paper contains 3 sections, 2 theorems.

Table of Contents

  1. Introduction
  2. The First Proof
  3. The Second Proof

Key Result

Lemma 1

If there is no edge from any proper ancestor of $s$ to any proper descendant of $s$ in the DFS tree, then $s$ is a separation point.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof