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

TL;DR

This paper establishes conditions on the regularity of balanced c-partite tournaments that guarantee the existence of strongly connected subtournaments with high minimum degree, addressing a problem posed by Volkmann in 2007.

Contribution

It provides new sufficient conditions on the regularity of balanced c-partite tournaments to ensure the existence of strongly connected subtournaments with high minimum degree.

Findings

Identifies regularity thresholds for strong c-partite subtournaments

Uses counting arguments for subtournaments with degree constraints

Addresses a longstanding open problem from 2007

Abstract

We consider the following problem posed by Volkmann in 2007: How close to regular must a c-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 assure the existence of strong maximal subtournament with minimum degree at least $\left\lfloor \frac{c-2}{4}\right\rfloor+1$. We obtain this result as an application of counting the number of subtournaments of order $c$ for which a vertex has minimum out-degree (resp. in-degree) at most $q\geq 0$.