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We consider the following problem posed by Volkmann in 2007: How close toregular must a c-partite tournament be, to secure a strongly connectedsubtournament of order $c$? We give sufficient conditions on the regularity ofbalanced $c$-partite tournaments to assure the existence of strong maximalsubtournament with minimum degree at least $\left\lfloor\frac{c-2}{4}\right\rfloor+1$. We obtain this result as an application ofcounting the number of subtournaments of order $c$ for which a vertex hasminimum out-degree (resp. in-degree) at most $q\geq 0$.

Conditions on the regularity of balanced $c$-partite tournaments for the existence of strong subtournaments with high minimum degree