Score sets in oriented bipartite graphs

S. Pirzada, T. A. Pirzada, T. A. Chishti

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 $|A|=1,2,3$, or when $A$ is a geometric or arithmetic progression.