The λ-Fold Spectrum Problem for the Oriented Pentagons

A D-decomposition of a directed graph G is a collection of arc-disjoint subgraphs of G, each isomorphic to D, such that every arc of G belongs to exactly one subgraph. The λ-fold spectrum problem for a directed graph D asks for the set of all integers v such that the λ-fold complete symmetric directed graph K*λKv* admits a D-decomposition. A five-cycle (pentagon) has 4 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented pentagons. In this paper, we provide a complete solution for each of the remaining three orientations, proving that the necessary and sufficient condition is 5∣λv(v−1) in all cases.