Copyright © 2012 Nicolas Lichiardopol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It was proved (Bessy et al., 2010) that for , a tournament with minimum semidegree at least contains at least r vertex-disjoint directed triangles. It was also proved (Lichiardopol, 2010) that for integers and , every tournament with minimum semidegree at least () contains at least r vertex-disjoint directed cycles of length . None information was given on these directed cycles. In this paper, we fill a little this gap. Namely, we prove that for and , every tournament of minimum outdegree at least contains at least vertex-disjoint strongly connected subtournaments of minimum outdegree . Next, we prove for tournaments a conjecture of Stiebitz stating that for integers and , there exists a least number such that every digraph with minimum outdegree at least can be vertex-partitioned into two sets inducing subdigraphs with minimum outdegree at least and at least , respectively. Similar results related to the semidegree will be given. All these results are consequences of two results concerning the maximum order of a tournament of minimum outdegree (of minimum semidegree ) not containing proper subtournaments of minimum outdegree (of minimum semidegree ).