Strongly [formula omitted]-path-connectivity in almost regular multipartite tournaments

Abstract

If x is a vertex of a digraph D , 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 ). If i g ( D ) = 0 , then D is regular and if i g ( D ) ⩽ 1 , then D is almost regular. A digraph D is said to be strongly k-path-connected if for any two vertices x , y ∈ V ( D ) there is an ( x , y ) -path of order k and a ( y , x ) -path of order k in D . In this paper we show that an almost regular c -partite tournament with c ⩾ 8 is strongly 4 -path-connected. Examples show that the condition c ⩾ 8 is best possible.