Weakly Cycle Complementary 3-Partite Tournaments

The vertex set of a digraph D is denoted by V(D). A c-partite tournament is an orientation of a complete c-partite graph. Let V ₁, V ₂, . . . ,V _c be the partite sets of D. If there exist two vertex disjoint cycles C and C′ in D such that V_i?(V(C)èV(C￠)) 1 ?{V_{mathrm{i}}cap(V(C)cup V(C'))neqemptyset} for all i = 1, 2, . . . , c, then D is weakly cycle complementary. In 2008, Volkmann and Winzen gave the above definition of weakly complementary cycles and proved that all 3-connected c-partite tournaments with c ≥ 3 are weakly cycle complementary. In this paper, we characterize multipartite tournaments are weakly cycle complementary. Especially, we show that all 2-connected 3-partite tournaments that are weakly cycle complementary, unless D is isomorphic to D _3,2, D _3,2,2 or D _3,3,1.