Random Oxford graphs

Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G1(m,n,t), the set of bipartite graphs with m left vertices, n right vertices, t edges, and each vertex of degree at least one. We give asymptotic results for the number of such graphs and the number of (i,j) trees they contain. We compute the thresholds for the emergence of a giant component and for the graph to be connected.