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A simple solution to the k-core problem

Janson, Svante and Luczak, Malwina J. (2006) A simple solution to the k-core problem. CDAM research report series 2006, LSE-CDAM-2006-13. Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science, London, UK.

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Abstract

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n ! 1. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the k-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer andWormald [19] on the existence and size of a k-core in G(n, p) and G(n,m), see also Molloy [17] and Cooper [3]. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.

Item Type: Monograph (Report)
Official URL: http://www.cdam.lse.ac.uk
Additional Information: © 2006 Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Mathematics
Research centres and groups > Computational, Discrete and Applicable Mathematics@LSE (CDAM)
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Identification Number: LSE-CDAM-2006-13
Date Deposited: 09 Oct 2008 13:56
URL: http://eprints.lse.ac.uk/13800/

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