Strong subtournaments containing a given vertex in regular multipartite tournaments

Abstract

If x is a vertex of a digraph D , then we denote by d + ( x ) and d − ( x ) the outdegree and the indegree of x , respectively. The global irregularity of a digraph D is defined by i g ( D ) = max x ∈ V ( D ) { d + ( x ) , d − ( x ) } − min y ∈ V ( D ) { d + ( y ) , d − ( y ) } . If i g ( D ) = 0 , then D is regular and if i g ( D ) ≤ 1 , then D is called almost regular. A c -partite tournament is an orientation of a complete c -partite graph. Recently, Volkmann and Winzen [L. Volkmann, S. Winzen, Almost regular c -partite tournaments contain a strong subtournament of order c when c ≥ 5 , Discrete Math. (2007), 10.1016/j.disc.2006.10.019] showed that every almost regular c -partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈ { 3 , 4 , … , c } . In this paper for the class of regular multipartite tournaments we will consider the more difficult question for the existence of strong subtournaments containing a given vertex. We will prove that each vertex of a regular multipartite tournament D with c ≥ 7 partite sets is contained in a strong subtournament of order p for every p ∈ { 3 , 4 , … , c − 4 } .