Decompositions of Uniform Complete Directed Multigraphs Into Each of the Orientations of a $4$-Cycle

Abstract

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.