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Brooks' Theorem (1941) is one of the most famous and fundamental theorems in graph theory – it is mentioned/treated in all general monographs on graph theory. It has sparked research in several directions. This book presents a comprehensive overview of this development and see it in context.
Michael Stiebitz +2 more
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Yet another proof of Brooks' theorem
Discrete Mathematics, 2022 Arguably the simplest variation of this style of proof as we avoid reducing to the cubic case entirely.
Landon Rabern
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An improvement on Brooks' Theorem [PDF]
, 2011 We prove that $ (G) \leq \max { (G), _2(G), (5/6)( (G) + 1)}$ for every graph $G$ with $ (G) \geq 3$. Here $ _2$ is the parameter introduced by Stacho that gives the largest degree that a vertex $v$ can have subject to the condition that $v$ is adjacent to a vertex whose degree is at least as large as its own.
Landon Rabern
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AxiomsWe show how the Brooks–Chacon Biting Lemma can be combined with the Castaing–Saadoune procedure to provide the complete rate of convergence along subsequences when the uniformly boundedness condition is violated.
George Stoica, Deli Li, Liping Liu
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On Brooks' theorem and some related results.
MATHEMATICA SCANDINAVICA, 1983 Helge Tverberg
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Zeros of Convex Combinations of Elementary Families of Harmonic Functions
Mathematics, 2023 Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we
Jennifer Brooks +4 more
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Brooks' Theorem and Beyond [PDF]
Journal of Graph Theory, 2014 AbstractWe 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
Cranston, Daniel W., Rabern, Landon
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Equitable colourings of Borel graphs
Forum of Mathematics, Pi, 2021 Hajnal and Szemerédi proved that if G is a finite graph with maximum degree $\Delta $ , then for every integer $k \geq \Delta +1$ , G has a proper colouring with k colours in which every two colour classes differ in size at most by $1$ ;
Anton Bernshteyn, Clinton T. Conley
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Precoloring Extensions of Brooks' Theorem [PDF]
SIAM Journal on Discrete Mathematics, 2004 Summary: Let \(G\) be a connected graph with maximum degree \(k\) (other than a complete graph or odd cycle), let \(W\) be a precolored set of vertices in \(G\) inducing a subgraph \(F\), and let \(D\) be the minimum distance in \(G\) between components of \(F\).
Albertson, Michael O. +2 more
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Generalized Hypergraph Coloring
Discussiones Mathematicae Graph Theory, 2021 A smooth hypergraph property 𝒫 is a class of hypergraphs that is hereditary and non-trivial, i.e., closed under induced subhypergraphs and it contains a non-empty hypergraph but not all hypergraphs.
Schweser Thomas
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