Summary: We consider the following problem posed by L. Volkmann [Discrete Math. 307, No. 24, 3097–3129 (2007; Zbl 1134.05033)]: How close to regular must ac-partite tournament be, to secure a strongly connected subtournament of order \(c\)? We give sufficient conditions on the regularity of balanced \(c\)-partite tournaments to ensure the existence of a strong maximal subtournament with minimum degree at least \(\lfloor\frac{c-2}{4}\rfloor+1\). We obtain 4 this result as an application of counting the number of subtournaments of order \(c\) for which a vertex has minimum out-degree (respectively, indegree) at most \(q\geq 0\).