Cycles through a given arc and certain partite sets in almost 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). 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. In 1998, Guo and Kwak showed that, if D is a regular c-partite tournament with c⩾4, then every arc of D is in a directed cycle, which contains vertices from exactly m partite sets for all m∈{4,5,…, c}. In this paper we shall extend this theorem to almost regular c-partite tournaments, which have at least two vertices in each partite set. An example will show that there are almost regular c-partite tournaments with arbitrary large c such that not all arcs are in directed cycles through exactly 3 partite sets. Another example will show that the result is not valid for the case that c=4 and there is only one vertex in a partite set.