Decompositions of Uniform Complete Directed Multigraphs Into Each of the Orientations of a \(4\)-Cycle

We present necessary and sufficient conditions for the decomposition of \(\lambda\) times the complete directed digraph, \(D_v^{\lambda}\), into each of the orientations of a \(4\)-cycle. In our constructions, we also give necessary and sufficient conditions for such decompositions which admit cyclic or rotational automorphisms.