Strong subtournaments of close to regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d + (x) and d − (x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) =max{d + (x),d − (x)}−min{d + (y),d − (y)} over all vertices x and y of D (including x = y). If ig(D) =0,thenD is regular and if ig(D) ≤ 1, then D is called almost regular. Recently, L. Volkmann and S. Winzen showed that every almost regular c-partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈{3, 4,...,c}. In this paper we will investigate multipartite tournaments with ig(D) ≤ l and l ≥ 2. Treating a problem of L. Volkmann (Australas. J. Combin. 20 (1999), 189–196) we will prove that, if D is a c-partite tournament with at least three vertices in each partite set, ig(D) ≤ l and c ≥ l +2with l ≥ 2, then D contains a strongly connected subtournament of order p for every p ∈{3, 4,...,c − l +1}