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 Walkupʹs 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.