In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows:let m = m(n) be a positive integer-valued function on n and ζ(n,m;{pk}) the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_(ai),b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_(ai),b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_(c,d,A), the number of vertices in A with degree between c and d of G_(n,m)∈ζ(n, m;{pk}) has asymptotically Poisson distribution, and answer the following two questions about the space ζ(n,m;{pk}) with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D(n) such that almost every random multigraph G_(n,m)∈ζ(n,m;{pk}) has maximum degree D(n)in A? under which condition for {pk} has almost every multigraph G(n,m)∈ζ(n,m;{pk}) a unique vertex of maximum degree in A?
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