Brooks' Theorem and Beyond

Abstract

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.Brooks' Theorem is among the most fundamental results in graph coloring. In short, it characterizes the (very few) connected graphs for which an obvious upper bound on the chromatic number holds with equality. It has been proved and reproved using a wide range of techniques, and the different proofs generalize and extend in many directions. In this paper we share some of our favorite proofs. In addition to surveying Brooks' Theorem, we aim to illustrate many of the standard techniques in vertex coloring 1 ; furthermore, we prove versions of Brooks' Theorem for standard strengthenings of vertex coloring, including list coloring, online list coloring, and Alon-Tarsi orientations. We present the proofs roughly in order of increasing complexity, but each section is self-contained and the proofs can be read in any order. Before we state the theorem, we need a little background.A proper coloring assigns colors, denoted by positive integers, to the vertices of a graph so that endpoints of each edge get different colors. A graph G is k-colorable if it has a proper coloring with at most k colors, and its chromatic number χ(G) is the minimum value k such that G is k-colorable. If a graph G has maximum degree ∆, then χ(G) ≤ ∆ + 1, since we can repeatedly color an uncolored vertex with the smallest color not already used on its neighbors. Since the proof of this upper bound is so easy, it is natural to ask whether we can strengthen it. The answer is yes, nearly always.A clique is a subset of vertices that are pairwise adjacent. If G contains a clique K k on k vertices, then χ(G) ≥ k, since all clique vertices need distinct colors. Similarly, if G contains an odd length cycle, then χ(G) ≥ 3, even when ∆ = 2. In 1941, Brooks [10] proved that these are the only two cases in which we cannot strengthen our trivial upper bound on χ(G).

Improved distributed $$\Delta $$-coloring

“…A T -node is a node with two neighbors that have the same color. In a partially colored graph, node u is a T -node of v if u is a T -node and there is an uncolored path from u to v. For several (centralized) proofs of Brooks' Theorem and further work on Gallai trees and degree choosability we refer to [15]. We also want to point out that degree choosable components recently became important for the distributed coloring of sparse and planar graphs [1,14].…”

Chromatic Ramsey number of acyclic hypergraphs

“…, t} for v ∈ V (F ) \ Z, F is L-choosable. We use the reduction argument present in many coloring proofs (see, for example, the very recent survey paper [6]).…”

“…Likewise we may assume d F ′ (v) = t for all v ∈ V (F ′ ) \ Z. By the degreechoosability version of Brooks' theorem (see [11], Lemma 1 or [6], Theorem 11), F ′ is a Gallai tree: a graph whose blocks are complete graphs or odd cycles.…”

Brooks’ Theorem for Measurable Colorings

“…Brooks's theorem is a fundamental result of graph coloring which has been generalized in a variety of different settings. See [8] for a recent survey. This paper examines measurable generalizations of Brooks's Theorem.…”