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

L VOLKMANN
2004 Discrete Mathematics  
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 deÿned 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) 6 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
more » ... 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.
doi:10.1016/s0012-365x(04)00158-x fatcat:5g4o2it55vfjtgdz65dlo3amde