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.

## 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) |
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Official URL: | http://www.cdam.lse.ac.uk |

Additional Information: | © 2006 Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics Research centres and groups > Computational, Discrete and Applicable Mathematics@LSE (CDAM) |

Date Deposited: | 09 Oct 2008 13:56 |

Last Modified: | 01 Oct 2010 09:05 |

URI: | http://eprints.lse.ac.uk/id/eprint/13800 |

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