On the threshold for k-regular subgraphs of random graphs
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The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor.
Mathematics Subject Classification (2000)05C80 05C70 05D40
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