On imbalances in multipartite multidigraphs
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-dig...
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| Main Authors: | , |
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| Format: | EJournal Article |
| Published: |
GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB,
2018-04-03.
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| Subjects: | |
| Online Access: | Get Fulltext |
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| Summary: | 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. |
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| Item Description: | https://www.ejgta.org/index.php/ejgta/article/view/482 |