Nomadic decompositions of bidirected complete graphs

We use K∗ n to denote the bidirected complete graph on n vertices. A nomadic Hamiltonian decomposition of K∗ n is a Hamiltonian decomposition, with the additional property that “nomads” walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length n − 1, rather than length n. J.A. Bondy asked whether these decompositions of K∗ n exist for all n. We show that K ∗ n admits a nomadic near-Hamiltonian decomposition when n 6≡ 2 mod 4.