Existence of a perfect matching in a random (1+e-1)--out bipartite graph

Structural properties of a random bipartite graph with bipartition (V1, V2), (|V1| = |V2|) = n), are studied. The graph is generated via two rounds of potential mates selections. In the first round every vertex in Vi chooses uniformly at random a vertex from Vj, j ≠ i, i = 1,2. In the second round each of the "unpopular" vertices, i.e. neglected completely in the first round, is allowed to make another random selection of a vertex. The resulting graph is "sandwiched" between Bn(1), the first-round graph, and Bn(2), the graph obtained by allowing every vertex to make two random selections of a mate. It seems natural to denote our graph Bn(1 + e-1), as the expected number of selections per vertex is 1 + e-1 in the limit. We prove that, asymptotically almost surely (a.a.s.), Bn(1 + e-1) contains a perfect matching, thus strengthening a well-known Walkup's theorem on a.a.s, existence of a perfect matching in the graph Bn(2). We demonstrate also that a.a.s. Bn(1 + e-1) consists of a giant component and several one-cycle components of a total size bounded in probability, and that Bn(1 + e-1) is connected with the limiting probability √1 - exp(-2(1 + e-1)) = 0.96703... perfect.