Nomadic decompositions of bidirected complete graphs

We use Kn* to denote the bidirected complete graph on n   vertices. A nomadic Hamiltonian decomposition of Kn* 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-1n-1, rather than length n  . Bondy asked whether these decompositions of Kn* exist for all n  . We show that Kn* admits a nomadic near-Hamiltonian decomposition when n≢2mod4.