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

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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 
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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/