New proof of brooks' theorem

Mel'nikov, L. S.; Vizing, V. G.

doi:10.1016/s0021-9800(69)80057-8

Cited by 17 publications

(11 citation statements)

References 1 publication

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“…Kempe changes were introduced in 1879 by Kempe in his attempted proof of the Four Colour Theorem [20]. Though this proof was fallacious, the Kempe change technique has proved useful in, for example, the proof of the Five Colour Theorem and a short proof of Brooks' Theorem [23]. We briefly review the purely graph theoretical studies of Kempe equivalence.…”

Section: Introductionmentioning

confidence: 99%

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Bonamy

Bousquet

Feghali

et al. 2019

Journal of Combinatorial Theory, Series B

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Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes.A conjecture of Mohar (2007) asserts that, for k ≥ 3, all k-colourings of a k-regular graph that is not complete are Kempe equivalent. It was later shown that all 3-colourings of a cubic graph that is neither K 4 nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each k ≥ 4. We also report the implications of this result on the validity of the Wang-Swendsen-Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.

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Section: Introductionmentioning

confidence: 99%

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Bonamy

Bousquet

Feghali

et al. 2019

Journal of Combinatorial Theory, Series B

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“…We begin by considering the case where the color

Δ + 1

appears on only one vertex. The proof of the following lemma is inspired by a proof of Brooks' Theorem but also uses some new arguments. Lemma Let

G = (V, E)

be a connected graph on n vertices with maximum degree

Δ \geq 3

, and let α be a

(Δ + 1)

‐coloring of G with exactly one vertex v colored

Δ + 1

.…”

Section: The Proof Of Theoremmentioning

confidence: 99%

“…We begin by considering the case where the color + 1 appears on only one vertex. The proof of the following lemma is inspired by a proof of Brooks' Theorem [31] but also uses some new arguments. Proof.…”

Section: The Proof Of Theoremmentioning

confidence: 99%

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

Feghali

Johnson

Paulusma

2015

Journal of Graph Theory

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Let G be a simple undirected connected graph on n vertices with maximum degree . Brooks' Theorem states that G has a proper -coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k-coloring, k > , a -coloring of G can be obtained by a sequence of O(n 2 ) recolorings using only the original k colors unless -G is a complete graph or a cycle with an odd number of vertices, or -k = + 1, G is -regular and, for each vertex v in G, no two neighbors of v are colored alike.

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“…In 1969 Mel'nikov and Vizing used Kempe chains to give the following elegant proof of Brooks' Theorem. We phrase this proof in terms of a minimal counterexample G , and for an arbitrary vertex v , we color

G - v

by minimality.…”

Section: Kempe Chainsmentioning

confidence: 99%

Brooks' Theorem and Beyond

Cranston

Rabern²

2014

Journal of Graph Theory

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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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New proof of brooks' theorem

Cited by 17 publications

References 1 publication

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

Brooks' Theorem and Beyond

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