On the connectivity of close to regular multipartite tournaments

• Authors: Lutz Volkmann;Stefan Winzen
• Venue: Discrete Applied Mathematics - Special issue: 2nd cologne/twente workshop on graphs and combinatorial optimization (CTW 2003)
• Year: 2006

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{d^+(x),d^-(x)}-min{d^+(y),d^-(y)} over all vertices x and y of D (including x=y) and the local irregularity of a digraph D is i"l(D)=max|d^+(x)-d^-(x)| over all vertices x of D. Clearly, i"l(D)==|V(D)|-|V"c|-2i"l(D)3for each c-partite tournament D, where @k(D) is the connectivity of D. Using Yeo's proof, we will present the structure of those multipartite tournaments, which fulfill the last inequality with equality. These investigations yield the better bound@k(D)=|V(D)|-|V"c|-2i"l(D)+13in the case that |V"c| is odd. Especially, we obtain a 1980 result by Thomassen for tournaments of arbitrary (global) irregularity. Furthermore, we will give a shorter proof of the recent result of Volkmann that@k(D)=|V(D)|-|V"c|+13for all regular multipartite tournaments with exception of a well-determined family of regular (3q+1)-partite tournaments. Finally we will characterize all almost regular tournaments with this property.