  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.
