The k-core and branching processes

The k-core of a graph G is the maximal subgraph of G having minimum degree at
least k. In 1996, Pittel, Spencer and Wormald found the threshold $\lambda_c$
for the emergence of a non-trivial k-core in the random graph $G(n,\lambda/n)$,
and the asymptotic size of the k-core above the threshold. We give a new proof
of this result using a local coupling of the graph to a suitable branching
process. This proof extends to a general model of inhomogeneous random graphs
with independence between the edges. As an example, we study the k-core in a
certain power-law or `scale-free' graph with a parameter c controlling the
overall density of edges. For each k at least 3, we find the threshold value of
c at which the k-core emerges, and the fraction of vertices in the k-core when
c is \epsilon above the threshold. In contrast to $G(n,\lambda/n)$, this
fraction tends to 0 as \epsilon tends to 0.