On imbalances in multipartite multidigraphs - Uma Tul Samee; Shariefuddin Pirzada (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2018)
A k-partite r-digraph(multipartite multidigraph) (or briefly MMD)(k ≥ 3, r ≥ 1) is the result of assigning a direction to each edge of a k-partite multigraph that is without loops and contains at most r edges between any pair of vertices from distinct parts. Let D(X1, X2, ⋯, Xk) be a k-partite r-digraph with parts Xi = xi1, xi2, ⋯, xini, 1 ≤ i ≤ k. Let dxij + and dxij − be respectively the outdegree and indegree of a vertex xij in Xi. Define axij (or simply aij) as aij = dxij + − dxij − as the imbalance of the vertex xij, 1 ≤ j ≤ ni. In this paper, we characterize the imbalances of k-partite r-digraphs and give a constructive and existence criteria for sequences of integers to be the imbalances of some k-partite r-digraph. Also, we show the existence of a k-partite r-digraph with the given imbalance set.