A simple solution to the k-core problem

Abstract

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n tend to 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 on the existence and size of a k-core in G(n,p) and G(n,m). Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.