Quasi-Hamiltonian cycles in k-strong multipartite tournaments

A c-partite tournament is an orientation of a complete c-partite graph. Recently, M. Lu, et al., introduced the concept of quasi-Hamiltonian cycles, that is to say, cycles containing vertices from each partite set, in multipartite tournaments. W.D. Goddard and O.R. Oellermann established that every strong multipartite tournament contains a quasi-Hamiltonian cycle. In this paper, we show that every k-strong (or k-arc-strong) multipartite tournament contains at least k quasi-Hamiltonian cycles. To that end, we prove the following stronger result: Every strong multipartite tournament contains a vertex whose all out-arcs are contained in a quasi-Hamiltonian cycle. Our results include and extend corresponding ones concerning tournaments due to C. Thomassen, as well as M. Goldberg and J.W. Moon. (C) 2014 Elsevier B.V. All rights reserved.