On imbalances in oriented tripartite graphs

An oriented tripartite graph is the result of assigning a direction to each edge of a simple tripartite graph. For any vertex x in an oriented tripartite graph D(U,V,W), let d _x ⁺ and d _x ⁻ denote the outdegree and indegree respectively of x. Define $ a_{u_i } = d_{u_i }^ + - d_{u_i }^ - , b_{v_j } = d_{v_j }^ + - d_{v_j }^ - $ a_{u_i } = d_{u_i }^ + - d_{u_i }^ - , b_{v_j } = d_{v_j }^ + - d_{v_j }^ - and $ c_{w_k } = d_{w_k }^ + - d_{w_k }^ - $ c_{w_k } = d_{w_k }^ + - d_{w_k }^ - as the imbalances of the vertices u _i in U, v _j in V and w _k in W respectively. In this paper, we obtain criteria for sequences of integers to be the imbalances of some oriented tripartite graph.