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 $\text{G}(n,p)$ 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 $\text{G}(n,p)$ to a window for $p$ of width roughly $2\mathrm{/}n$ for large $n$ and moderately large $k$.