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, then D 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 + 2 with l ≥ 2, then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c− l + 1}. 1 Terminology and introduction In this paper all digraphs are finite without loops and multiple arcs. The vertex set and arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we write x → y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y . Furthermore, X Y denotes the fact that there is no arc leading from Y to X . For the number of arcs from X to Y we write d(X, Y ), i.e., d(X, Y ) = |{xy ∈ E(D) : x ∈ X, y ∈ Y }|. If D is a digraph, then the out-neighborhood N D (x) = N (x) of a vertex x is the set of vertices dominated by x and the inneighborhood N− D (x) = N −(x) is the set of vertices dominating x. Therefore, if there is the arc xy ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor of y. The numbers dD(x) = d (x) = |N+(x)| and dD(x) = d−(x) = |N−(x)| are