Centre for Discrete and Applicable Mathematics |
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CDAM Research Report, LSE-CDAM-2006-13October 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}.A PDF file (215 kB) with the full contents of this report can be downloaded by clicking here.
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