Geodesic
Tripartite multidigraphs and imbalances
Kragujevac Journal of Mathematics, Tome 36 (2012) no. 1, p. 109

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

A tripartite $r$-digraph $(r \geq 1)$ is an orientation of a tripartite multigraph that is without loops and contains at most $r$ edges between any pair of vertices from distinct parts. For any vertex $x$ in a tripartite $r$-digraph $D(U,V,W)$, let $d_{_{x}}^{+}$ and $d_{_{x}}^{-}$ denote the outdegree and indegree respectively of $x$. Define $a_{_{u_{_{i}}}}= d_{_{u_{_{i}}}}^{+} - d_{_{u_{_{i}}}}^{-}$, $b_{_{v_{_{j}}}}= d_{_{v_{_{j}}}}^{+} - d_{_{v_{_{j}}}}^{-}$ and $c_{_{w_{_{k}}}}= d_{_{w_{_{k}}}}^{+} - d_{_{w_{_{k}}}}^{-}$ as the $r$-imbalances of the vertices $u_{_{i}} \in U$, $v_{_{j}} \in V$ and $w_{_{k}} \in W$ respectively. We characterize $r$-imbalances in tripartite $r$-digraphs and obtain necessary and sufficient conditions for three sequences of integers to be $r$-imbalance sequences of some tripartite $r$-digraph.
Classification : 05C20 05C65
Keywords: Digraph, tripartite digraph, imbalance sequence, imbalance set 
Shariefuddin Pirzada; Koko K. Kayibi; Nasir A. Shah. Tripartite multidigraphs and imbalances. Kragujevac Journal of Mathematics, Tome 36 (2012) no. 1, p. 109 . http://geodesic.mathdoc.fr/item/KJM_2012_36_1_a11/
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     title = {Tripartite multidigraphs and imbalances},
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     year = {2012},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2012_36_1_a11/}
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