Cycles through a given arc and certain partite sets in strong multipartite tournaments

Moon [J. Combin. Inform. System Sci. 19 (1994), 207–214] showed that every strong tournament contains a Hamiltonian cycle through at least three pancyclic arcs. In this paper, we extend the result of Moon and prove that if D is a strong c-partite tournament with c ≥ 3, then D contains a cycle C containing vertices from exactly c partite sets such that C contains at least three arcs, each of which belongs to a cycle containing vertices from exactly l partite sets for each l ∈{ 3, 4,...,c}. In addition, this bound is best possible.