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