Multipartite tournaments with small number of cycles

Weakly Complementary Cycles in 3-Connected Multipartite 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. A digraph D is called cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (D) = V (C1) ∪ V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (C1) ∪ V (C2...

On cycles through two arcs in strong multipartite tournaments

A multipartite tournament is an orientation of a complete c-partite graph. In [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148–1150], Volkmann proved that a strongly connected cpartite tournament with c > 3 contains an arc that belongs to a directed cycle of length m for every m ∈ {3, 4, . . . , c}. He also conjectu...

On two-path convexity in multipartite tournaments

In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds ho...

Determining properties of a multipartite tournament from its lattice of convex subsets

The collection of convex subsets of a multipartite tournament T forms a lattice C(T ). Given a lattice structure for C(T ), we deduce properties of T . In particular, we find conditions under which we can detect clones in T . We also determine conditions on the lattice which will imply that T is bipartite, except for a few cases. We classify the ambiguous cases. Finally, we study a property of ...

Complementary cycles in regular multipartite tournaments