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Uma Tul Samee
Department of Mathematics, Islamia College for Science and Commerce, Srinagar, Kashmir, India
Electrical & Electronics Engineering

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On imbalances in multipartite multidigraphs Uma Tul Samee; Shariefuddin Pirzada
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 6, No 1 (2018): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2018.6.1.6

Abstract

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.
Co-Authors Shariefuddin Pirzada