Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2006-13

October 2006

A Simple Solution to the k-Core Problem

Svante Janson and Malwina J. Luczak

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. We let \ntoo. 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 and Wormald \cite{psw96} on the existence and size of a $k$-core in $G(n,p)$ and $G(n,m)$, see also Molloy~\cite{Molloy05} and Cooper~\cite{c04}.

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