Bipartite Graphs and their Degree Sets

The degree set D(G) of a graph G is the set of degrees of G. Given a finite, nonempty set S of positive integers, it is shown that there exists a bipartite graph G such that D(G) = S. In addition, the minimum order of such a graph is determined when it is either unicyclic or connected. Furthermore, for any two sets S and T of positive integers of the same cardinality, it is shown that there exists a bipartite graph B(X, Y) such that D(X) = S, D(Y) = T. Moreover, the minimum of order of such a connected bipartite graph is obtained.