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.