Score sets in oriented bipartite graphs

Abstract

The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U, V) such that score set of D(U, V) is A? We conjecture that there is an affirmative answer, and verify this conjecture when $\mid A\mid$ = 1, 2, 3, or when A is a geometric or arithmetic progression.