On imbalances in multipartite multidigraphs

Uma Tul Samee (Department of Mathematics, Islamia College for Science and Commerce, Srinagar, Kashmir, India)
Shariefuddin Pirzada (Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India.)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 6, No 1 (2018): Electronic Journal of Graph Theory and Applications

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.