Brooks’ Theorem via the Alon–Tarsi Theorem

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).

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“…In both cases, the Alon–Tarsi Theorem shows that G is degree‐choosable. This is essentially the proof given by Hladký, Král, and Schauz .…”

Section: Degree‐choosable Graphsmentioning

confidence: 55%

Brooks' Theorem and Beyond

Cranston

Rabern²

2014

Journal of Graph Theory

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“…So the set of all connected graphs that are not degree-choosable are precisely the Gallai trees. Hladký, Krá ' l, and Schauz [7] extended this characterization to the setting of Alon-Tarsi orientations.…”

Section: Introductionmentioning

confidence: 92%

Beyond Degree Choosability

Cranston

Rabern

2017

Electron. J. Combin.

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Let G be a connected graph with maximum degree ∆. Brooks' theorem states that G has a ∆-coloring unless G is a complete graph or an odd cycle. A graph G is degreechoosable if G can be properly colored from its lists whenever each vertex v gets a list of d(v) colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph G is degree-choosable unless each block of G is a complete graph or an odd cycle; such a graph G is a Gallai tree.This degree-choosability result was further strengthened to Alon-Tarsi orientations; these are orientations of G in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph G is degree-AT if G has an Alon-Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if G is degree-AT, then G is also degree-choosable. Hladký, Krá ' l, and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs (G, x) where G is a connected graph and x is some specified vertex in V (G). We characterize pairs such that G has no Alon-Tarsi orientation in which each vertex has indegree at least 1 and x has indegree at least 2. When G is 2-connected, the characterization is simple to state.

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“…This paper was inspired by an early version of [4], which gives a different algebraic proof of Brooks' theorem for listcolouring. However, its roots go back more than 50 years.…”

Section: Historical Remarks and A Questionmentioning

confidence: 99%

“…Upon seeing [4], it was natural to see whether my old algebraic approach would work also for the list version of Brooks' theorem. It was clear that now Alon's theorem would have to be used, and it seemed then that a different Lovász-type lemma had to be used.…”

Section: Historical Remarks and A Questionmentioning

confidence: 99%