On imbalances in oriented bipartite graphs

An oriented bipartite graph is the result of assigning a direction to each edge of a simple bipartite graph. For any vertex $x$ in an oriented bipartite graph $D(U,V)$, let $d_{x}^{+}$ and $d_{x}^{-}$ respectively denote the outdegree and indegree of $x$. Define $a_{u_{i}}=d_{u_{i}}^{+}-d_{u_{i}}^{-}$ and $b_{v_{j}}=d_{v_{j}}^{+}-d_{v_{j}}^{-}$ respectively as the imbalances of vertices $u_i$ in $U$ and $v_j$ in $V$. In this paper, we obtain constructive and existence criteria for a pair of sequences of integers to be the imbalances of some oriented bipartite graph. We also show the existence of a bipartite oriented graph with given imbalance set.