  Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let
$G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in
[n]^2$ appears as an edge independently with probability $p$. For a graph
$G=(V,E)$, suppose that each edge $e\in E$ 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 $d=np\gg(\log n)^2$ 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 $G=K_{n,n}$. 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.
