Problem Of Componentwise Complementary Cycles In 2-Equilibrium Multipartite Tournaments

Reid have proved that every 2-connected tournament D on n 6 vertices contains two complementary cycles of lengths 3 and n - 3 respectively, unless it is isomorphic to a tournament T71 (see figure in ?). Song extended the result of Reid and proved that a 2-connected tournament D on at least 6 vertices, for every t satisfying 3 t n - 3 contains two complementary cycles of lengths t and n - t, respectively, unless it is isomorphic to T71 (see figure in ?). The two theorems above give a complete solution of the problem of complementary cycles in tournaments. However, for multipartite tournaments which are not tournaments, the problem of complementary cycles is still open. In this paper, we extend the theorem of Song to multipartite tournaments and prove that every 2-connected and 2- equilibrium multipartite tournament contains a pair of componentwise complementary cycles, unless it is isomorphic to T71 (see figure in ?).