Tripartite multidigraphs and imbalances

A tripartite r-digraph (r ≥ 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+ and d- denote the outdegree and indegree respectively of x. Definene a(ui)=d+(ui)-d-(ui), b(vj)=d+(vj)-d-(vj) and c(wk)=d+(wk)-d-(wk) as the r-imbalances of the vertices u(i) ∈U, v(j) ∈V, w(k) ∈W respectively. We characterize r-imbalances in tripartite r-digraphs and obtain necessary and suficient conditions for three sequences of integers to be r-imbalance sequences of some tripartite r-digraph.