On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time
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We prove that for k+1≥ 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)-core of G_n,p, if non empty, spans a (near) spanning k-regular subgraph. This improves upon a result of Chan and Molloy and completely resolves a conjecture of Bollobás, Kim and Verstraëte. In addition, we show that w.h.p. such a subgraph can be found in linear time. A substantial element of the proof is the analysis of a randomized algorithm for finding k-matchings in random graphs with minimum degree k+1.
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