On the connectivity of close to 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 { 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 ) ⩽ i g ( D ) . If i g ( D ) = 0 , then D is regular and if i g ( D ) ⩽ 1 , then D is almost regular. A c-partite tournament is an orientation of a complete c-partite graph. Let V 1 , V 2 , … , V c be the partite sets of a c-partite tournament such that | V 1 | ⩽ | V 2 | ⩽ ⋯ ⩽ | V c | . In 1998, Yeo proved κ ( D ) ⩾ | V ( D ) | - | V c | - 2 i l ( D ) 3 for each c-partite tournament D, where κ ( 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 κ ( D ) ⩾ | V ( D ) | - | V c | - 2 i l ( D ) + 1 3 in 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 κ ( D ) ⩾ | V ( D ) | - | V c | + 1 3 for all regular multipartite tournaments with exception of a well-determined family of regular ( 3 q + 1 ) -partite tournaments. Finally we will characterize all almost regular tournaments with this property.