Graph Colouring and the Probabilistic Method

Molloy, Michael; Reed, Bruce

doi:10.1007/978-3-642-04016-0

Cited by 318 publications

(438 citation statements)

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“…However, obtaining sharp enough concentration bounds using Azuma's Inequality is not always easy. Consider the following version of the inequality [23]:…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

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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“…However, obtaining sharp enough concentration bounds using Azuma's Inequality is not always easy. Consider the following version of the inequality [23]:…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

The Randomized Coloring Procedure with Symmetry-Breaking

Pemmaraju

Srinivasan

2008

Automata, Languages and Programming

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“…Labelled Graphs. Labelled graphs have been widely used both in Computer Science and in Mathematics, e.g., [28]. Single-labelled and multi-labelled Temporal Networks.…”

Section: Other Related Workmentioning

confidence: 99%

Ephemeral networks with random availability of links: The case of fast networks

Akrida

Gąsieniec

Mertzios

et al. 2016

Journal of Parallel and Distributed Computing

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights• We define the Temporal Diameter (TD) and the expected TD of temporal networks with random availabilities of links• We define fast and slow random temporal networks, according to their expected TD• Random temporal networks on instances of dense (directed) random graphs G n,p are fast• We introduce the critical availability as a measure of periodic availability of links• We give a lower and an upper bound on the critical availability AbstractWe consider here a model of temporal networks, the links of which are available only at certain moments in time, chosen randomly from a subset of the positive integers. We define the notion of the Temporal Diameter of such networks. We also define fast and slow such temporal networks with respect to the expected value of their temporal diameter. We then provide a partial characterisation of fast random temporal networks. We also define the critical availability as a measure of periodic random availability of the links of a network, required to make the network fast. We finally give a lower bound as well as an upper bound on the (critical) availability.

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“…This is a very standard lemma -it's proof is nearly identical to, eg., that of Lemma 25.2 in [19] -so we omit the details. The key fact needed for this proof is: Consider any collection of t disjoint hyperedges.…”

Section: Algorithmsmentioning

confidence: 99%

“…Similar enough to allow us to show that, with positive probability, no vertex appears β times in any neighbourhood. Our procedure is a version of what is often called the "Rodl Nibble", the "semi-random method" or the "Naive Colouring Procedure" (see [26,19]). …”

Section: Introductionmentioning

confidence: 99%