MAJORIZATION AND THE NUMBER OF BIPARTITE GRAPHS FOR GIVEN VERTEX DEGREES

The bipartite realisation problem asks for a pair of non-negative, non-increasing integer lists a := (a1,...,a(n)) and b := (b(1),..., bn') if there is a labeled bipartite graph G(U, V, E) (no loops or multiple edges) such that each vertex u(i) is an element of U has degree a(i) and each vertex v(i) is an element of V degree b(i). The Gale-Ryser theorem provides characterisations for the existence of a 'realisation' G(U, V, E) that are strongly related to the concept of majorisation. We prove a generalisation; list pair (a, b) has more realisations than (a', b), if a' majorises a. Furthermore, we give explicitly list pairs which possess the largest number of realisations under all (a, b) with fixed n, n' and m := Sigma(ai)(i=1). We introduce the notion minconvex list pairs for them. If n and n' divide m, minconvex list pairs turn in the special case of two constant lists a = (m/n,...,m/n) and b = (m/n',...,m/n').