Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

摘要

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).