Let Gn,p be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let
Gn,n,p be the random bipartite graph on n+n vertices, where each e∈[n]2 appears as an edge independently with probability p. For a graph
G=(V,E), suppose that each edge e∈E is given an independent uniform
exponential rate one cost. Let C(G) denote the random variable equal to the
length of the minimum cost perfect matching, assuming that G contains at
least one. We show that w.h.p. if d=np≫(logn)2 then w.h.p. {\bf
E}[C(G_{n,n,p})] =(1+o(1))\frac{\p^2}{6p}. This generalises the well-known
result for the case G=Kn,n. We also show that w.h.p. {\bf E}[C(G_{n,p})]
=(1+o(1))\frac{\p^2}{12p} along with concentration results for both types of
random graph.Comment: Replaces an earlier paper where G was an arbitrary regular
bipartite grap