Brooks' Theorem and Beyond

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; 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 Kk 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). Date: 3 March 2014. Virginia Commonwealth University, Richmond, VA. [email protected]. LBD Data Solutions, Orange, CT. [email protected]. One common coloring approach notably absent from this paper is the probabilistic method. Although the so-called naive coloring procedure has been remarkably effective in proving numerous coloring conjectures asymptotically, we are not aware of any probabilistic proof of Brooks’ Theorem. For the reader interested in this technique, we highly recommend the monograph of Molloy and Reed “Graph Colouring and the Probabilistic Method” [48].