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;,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 >= 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 e above the threshold. In contrast to G(n,lambda/n), this fraction tends to 0 as epsilon -> 0.