Matchings and cycle covers in random digraphs

Consider a random bipartite graph in which each of n white vertices is adjacent to exactly r of the n black vertices. The authors study the degree distribution of the black vertices and they find that if r=log(nωn) where ωn⇒•, then almost all of these graphs have no isolated black vertices, are connected and have a perfect matching. A consequence of this result is that a random digraph that is regular of outdegree r almost surely has a cycle cover.