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Regular subgraphs of random graphs

In this paper, we prove that there exists a function ρk = (4 + o(1))k such that G(n,ρ/n) contains a k−regular graph with high probability whenever ρ > ρk. In the case of k = 3, it is also shown that G(n,ρ/n) contains a 3−regular graph with high probability whenever ρ > λ ≈︁ 5.1494. These are the first constant bounds on the average degree in G(n,p) for the existence of a k−regular subgraph. We also discuss the appearance of 3−regular subgraphs in cores of random graphs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006