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The k-Core and Branching Processes

Published online by Cambridge University Press:  01 January 2008

OLIVER RIORDAN
OLIVER RIORDAN*
Affiliation:
Royal Society Research Fellow, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK (e-mail: O.M.Riordan@dpmms.cam.ac.uk)
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Abstract

The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold λc for the emergence of a non-trivial k-core in the random graph G(n, λ/n), and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k-core in a certain power-law or ‘scale-free’ graph with a parameter c controlling the overall density of edges. For each k ≥ 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k-core when c is ϵ above the threshold. In contrast to G(n, λ/n), this fraction tends to 0 as ϵ→0.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 1 , January 2008 , pp. 111 - 136
Copyright
Copyright © Cambridge University Press 2007

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