A simple solution to the k-core problem

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n -> infinity. 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 (J Combinator Theory 67 (1996), 111-151) on the existence and size of a k-core in G(n,p) and G(n, m), see also Molloy (Random Struct Algor 27 (2005), 124-135) and Cooper (Random Struct Algor 25 (2004), 353-375). Our method is based on the properties of empirical distributions of independent random variables and leads to simple proofs. (c) 2006 Wiley Periodicals, Inc.