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Large Cliques in a Power-Law Random Graph

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  14 July 2016

Svante Janson ,
Tomasz Łuczak  and
Ilkka Norros
Svante Janson*
Affiliation:
Uppsala University
Tomasz Łuczak*
Affiliation:
Adam Mickiewicz University
Ilkka Norros*
Affiliation:
VTT Technical Research Centre of Finland
*
∗Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: svante.janson@math.uu.se
∗∗Postal address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland. Email address: tomasz@amu.edu.pl
∗∗∗Postal address: VTT Technical Research Centre of Finland, PO Box 1000, 02044 VTT, Finland. Email address: ilkka.norros@vtt.fi
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Abstract

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In this paper we study the size of the largest clique ω(G(n, α)) in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α > 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 − o(1))ω(G(n, α)) in polynomial time.

Keywords

Power-law random graphcliquegreedy algorithm

MSC classification

Secondary: 05C80: Random graphs 05C69: Dominating sets, independent sets, cliques 60C05: Combinatorial probability
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1124 - 1135
Copyright
Copyright © Applied Probability Trust 2010 

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