Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, Kn∗, exist if and only if nn is odd with n≠15n≠15 and n≠pαn≠pα for pp an odd prime and α≥2α≥2 or n≡2(mod4) with n≠2pαn≠2pα for pp an odd prime and α≥1α≥1. We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2n/2 vertex-independent digons, (Kn−I)∗(Kn−I)∗, exist if and only if n≡0(mod4).