A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

Kierstead, H. A.; Kostochka, Alexandr

doi:10.1017/s0963548307008619

Cited by 93 publications

(69 citation statements)

References 10 publications

(17 reference statements)

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“…In a pivotal result, Hajnal and Szemerédi [9] showed that every graph G with maximum degree at most ∆ has an equitable k-coloring for every k ≥ ∆ + 1. Recently, Kierstead and Kostochka [17] have presented a short proof of this result, along with a polynomial-time algorithm for computing such a coloring. Equitable colorings naturally arise in scheduling, partitioning, and load balancing [1,3,12,20,25,26].…”

Section: Weighted Equitablementioning

confidence: 95%

The Randomized Coloring Procedure with Symmetry-Breaking

Pemmaraju

Srinivasan

2008

Automata, Languages and Programming

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Abstract. A basic randomized coloring procedure has been used in probabilistic proofs to obtain remarkably strong results on graph coloring. These results include the asymptotic version of the List Coloring Conjecture due to Kahn, the extensions of Brooks' Theorem to sparse graphs due to Kim and Johansson, and Luby's fast parallel and distributed algorithms for graph coloring. The most challenging aspect of a typical probabilistic proof is showing adequate concentration bounds for key random variables. In this paper, we present a simple symmetry-breaking augmentation to the randomized coloring procedure that works well in conjunction with Azuma's Martingale Inequality to easily yield the requisite concentration bounds. We use this approach to obtain a number of results in two areas: frugal coloring and weighted equitable coloring. A β-frugal coloring of a graph G is a proper vertex-coloring of G in which no color appears more than β times in any neighborhood. Let G = (V, E) be a vertex-weighted graph with weight function w : V → [0, 1] and let W = v∈V w(v). A weighted equitable coloring of G is a proper k-coloring such that the total weight of every color class is "large", i.e., "not much smaller" than W/k; this notion is useful in obtaining tail bounds for sums of dependent random variables.

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Section: Weighted Equitablementioning

confidence: 95%

The Randomized Coloring Procedure with Symmetry-Breaking

Pemmaraju

Srinivasan

2008

Automata, Languages and Programming

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“…A shorter proof appeared in [38]. Then Kierstead, Kostochka, Mydlarz, and Szemerédi [45] devised an algorithm that in time O(rn 2 ) finds an equitable (r+1)-coloring for any n-vertex graph with maximum degree at most r. It is based on a modification of the proof of Theorem 5.1 in [38].…”

Section: Equitable Coloringmentioning

confidence: 99%

Extremal graph packing problems: Ore-type versus Dirac-type

Kierstead¹,

Kostochka²,

Yu³

2009

Surveys in Combinatorics 2009

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“…Only recently, an alternative algorithmic proof of Theorem 2.4 has been obtained by Mydlarz and Szemerédi [67]. Subsequently, an alternative simple and algorithmic proof has been obtained by Kirstead and Kostochka [47]. Their algorithm employs a clever discharging technique and runs in O(n 5 ) time.…”

Section: H-factors In Graphs With Sufficiently Large Minimum Degreementioning

confidence: 99%

Combinatorial and computational aspects of graph packing and graph decomposition

Yuster

2007

Computer Science Review

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Packing and decomposition of combinatorial objects such as graphs, digraphs, and hypergraphs by smaller objects are central problems in combinatorics and combinatorial optimization. Their study combines probabilistic, combinatorial, and algebraic methods. In addition to being among the most fascinating purely combinatorial problems, they are often motivated by algorithmic applications. There are a considerable number of intriguing fundamental problems and results in this area, and the goal of this paper is to survey the state of the art.

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A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

Cited by 93 publications

References 10 publications

The Randomized Coloring Procedure with Symmetry-Breaking

The Randomized Coloring Procedure with Symmetry-Breaking

Extremal graph packing problems: Ore-type versus Dirac-type

Combinatorial and computational aspects of graph packing and graph decomposition

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