Acta Mathematica Scientia 2011,31B(3):1155–1166 http://actams.wipm.ac.cn THE MAXIMUM AND MINIMUM DEGREES OF RANDOM BIPARTITE MULTIGRAPHS ∗ Chen Ailian ( ) College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China E-mail: elian1425@sina.com Zhang Fuji ( ) School of Mathematical Sciences, Xiamen University, Xiamen 361005, China E-mail: fjzhang@xmu.edu.cn Li Hao ( ) Laboratoire de Recherche en Informatique, UMR 8623, C. N. R. S. -Universit´ e de Paris-sud, 91405-Orsay cedex, France E-mail: Hao.Li@lri.fr Abstract 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 G(n, m; {p k }) the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A = {a1,a2, ··· ,an} and B = {b1,b2, ··· ,bm}, in which the numbers t a i ,b j of the edges between any two vertices ai ∈ A and bj ∈ B are identically distributed independent random variables with distribution P {t a i ,b j = k} = p k , k =0, 1, 2, ··· , where p k ≥ 0 and ∞ ∑ k=0 p k = 1. They obtain that X c,d,A , the number of vertices in A with degree between c and d of Gn,m ∈G(n, m; {p k }) has asymptotically Poisson distribution, and answer the following two questions about the space G(n, m; {p k }) with {p k } having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {p k } can there be a function D(n) such that almost every random multigraph Gn,m ∈G(n, m; {p k }) has maximum degree D(n) in A? under which condition for {p k } has almost every multigraph Gn,m ∈G(n, m; {p k }) a unique vertex of maximum degree in A? Key words maximum degree; minimum degree; degree distribution; random bipartite multigraphs 2000 MR Subject Classification 05C80 * Received June 29, 2008; revised March 5, 2010. The work was supported by NSFC (10671162; 10831001; 10871046).