All regular multipartite tournaments that are cycle complementary

Abstract

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. A digraph D is cycle complementary if there exist two vertex-disjoint directed cycles spanning the vertex set V(D) of D. In this paper we prove that each regular c-partite tournament D of order |V(D)|⩾6 with c⩾3 is cycle complementary, unless D is isomorphic to T₇ or to D_3,2, where T₇ is a 3-regular tournament of order 7, and D_3,2 is a 2-regular 3-partite tournament such that there are exactly two vertices in each partite set.