On the threshold for k-regular subgraphs of random graphs

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the (k + 2)-core of a random graph G(n, p) asymptotically almost surely has a spanning k-regular subgraph. Thus the threshold for the appearance of a k-regular subgraph of a random graph is at most the threshold for the (k + 2)-core. In particular, this pins down the point of appearance of a k-regular subgraph in G(n, p) to a window for p of width roughly 2/n for large n and moderately large k.