Strong subtournaments of multipartite tournaments

An orientation of a complete graph is a tournament, and an orientation of a complete n-partite graph is an n-partite tournament. For each n 2:: 4, there exist examples of strongly connected n-partite tournament without any strongly connected subtournaments of order p 2:: 4. If D is a digraph, then let d+ (x) be the out degree and d- (x) the indegree of the vertex x in D. The minimum (maximum) out degree and the minimum (maximum) in degree of D are denoted by J+ (~+) and J- (~-), respectively. Furthermore, we define J = mini J+, J-} and ~ = maxi ~+, ~-}. A digraph D is almost regular, if ~- 8::; 1. If Vi, Vz,..., Vn are the partite sets of an n-partite tournament D, then we define "((D) = minl:s;i$n{IYiI}. In this paper we prove that every almost regular n-partite tournament with n 2:: 4 contains a strongly connected subtournament of order p for each p E {3, 4,...,;z,- I}. Examples show that this result is best possible for n = 4. If in addition, "((D) < 3n/2- 6, for an almost regular n-partite tournament D with n 2:: 5, then D even contains a strong subtournament of order n