Cycles through a given arc and certain partite sets in almost regular multipartite tournaments Original Research Article

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 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 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.