Strongly 4-path-connectivity in almost regular multipartite tournaments

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 ig(D)=max{d+(x),d-(x)}-min{d+(y),d-(y)} over all vertices x and y of D (including x=y). If ig(D)=0, then D is regular and if ig(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.