Cycles of length four through a given arc in almost regular multipartite tournaments.

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). If i(g)(D) = 0, then D is regular and if i(g)(D) less than or equal to 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-partite graph. It is easy to see that there exist regular c-partite tournaments with arbitrary large c which contain arcs that do not belong to a directed cycle of length 3. In this paper we show, however, that every arc of an almost regular c-partite tournament is contained in a directed cycle of length four, when c greater than or equal to 8. Examples show that the condition c greater than or equal to 8 is best possible.