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, K n ∗ , exist if and only if n is odd with n ≠ 15 and n ≠ p α for p an odd prime and α ≥ 2 or n ≡ 2 ( mod 4 ) with n ≠ 2 p α for p an odd prime and α ≥ 1 . We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n / 2 vertex-independent digons, ( K n − I ) ∗ , exist if and only if n ≡ 0 ( mod 4 ) .