Nomadic decompositions of bidirected complete graphs

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