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 C-1 and C-2 such that V(D) = V(C-1) boolean OR V(C-2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C-1 and C-2 such that V(C-1) boolean OR V(C-2) contains vertices of all partite sets of D. The problem of complementary cycles in 2-connected tournaments was completely solved by Reid [4] in 1985 and Z. Song [5] in 1993. They proved that every 2-connected tournament T on at least 8 vertices has complementary cycles of length t and vertical bar V(T)vertical bar - t for all 3 <= t <= vertical bar V(T)vertical bar/2. Recently, Volkmann [8] proved that each regular multipartite tournament D of order vertical bar V(D)vertical bar >= 8 is cycle complementary. In this article, we analyze multipartite tournaments that are weakly cycle complementary. Especially, we will characterize all 3-connected c-partite tournaments with c >= 3 that are weakly cycle complementary.