Decompositions and Packings of Digraphs with Orientations of a \(4\)-Cycle

We present necessary and sufficient conditions for the decomposition of the complete symmetric bipartite digraph into each of the orientations of a \(4\)-cycle (in the cases for which such decompositions are not already known). We use these results to find optimal packings of the complete symmetric digraph with each of the orientations of a \(4\)-cycle. Finally, we give necessary and sufficient conditions for the existence of a decomposition of the complete symmetric digraph on \(v\) vertices with a hole of size \(w\) into each of the orientations of a \(4\)-cycle.