Table of Contents
International Journal of Combinatorics
Volume 2012 (2012), Article ID 273416, 9 pages
http://dx.doi.org/10.1155/2012/273416
Research Article

Vertex-Disjoint Subtournaments of Prescribed Minimum Outdegree or Minimum Semidegree: Proof for Tournaments of a Conjecture of Stiebi

Nicolas Lichiardopol, Lycée A. de Craponne, 13651 Salonde Provence Cedex, France

Received 23 January 2011; Revised 26 April 2011; Accepted 30 May 2011

Academic Editor: Charles Semple

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 ).