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 V1, V2, . . . ,Vc be the partite sets of D. If there exist two vertex disjoint cycles C and C′ in D such that $${V_{\mathrm{i}}\cap(V(C)\cup V(C'))\neq\emptyset}$$ 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 D3,2, D3,2,2 or D3,3,1.

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