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We prove that for $k+1\geq 3$ and $c>(k+1)/2$ w.h.p. the random graph on $n$vertices, $cn$ edges and minimum degree $k+1$ contains a (near) perfect$k$-matching. As an immediate consequence we get that w.h.p. the $(k+1)$-coreof $G_{n,p}$, if non empty, spans a (near) spanning $k$-regular subgraph. Thisimproves upon a result of Chan and Molloy and completely resolves a conjectureof Bollob\'as, Kim and Verstra\"{e}te. In addition, we show that w.h.p. such asubgraph can be found in linear time. A substantial element of the proof is theanalysis of a randomized algorithm for finding $k$-matchings in random graphswith minimum degree $k+1$.

On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time