Arcs Belonging To Cycles In Multipartite Tournaments

Multipartite tournaments are without doubt the important studied class of directed graphs and they have been intensively studied. A tournament is a c-partite tournament with exactly c vertices. The investigation on directed paths and cycles in tournaments has become intensive and productive. It is a useful way of investigation on multipartite tournaments that popularizing the results on tournaments to multipartite tournaments. A cycle in a digraph D is Hamiltonian if it includes all vertices of D. A digraph D is called to be pancyclic, if it contains a cycle of length from 3 to |V(D))|. A vertex (an arc) is said to be pancyclic in a digraph D, if it belongs to an l-cycle for all 3≤l≤|V(D)|. We call a digraph D is vertex pancyclic (arc pancyclic), if each vertex (each arc) of D is pancyclic. At present, there are many results on pancyclicity, vertex pancyclicity and arc pancyclicity for tournaments. In 2000, Yao, Guo and Zhang discussed out-arc pancyclicity of vertices in tournaments and proved the existence of an out-arc pancyclic vertex in strong tournaments. For multipartite tournaments, we can not affirm the existence of an Hamiltonian cycle. Therefor, For c-partite tournaments, we consider the vertex (arc) that belonging to a l-cycle (3≤l≤c), and the vertex (arc) that belonging to a cycle with vertices from exactly l partite sets (3≤l≤c). In 1994, Moon obtained that every strong tournament contains at least three pancyclic arcs. In connected wit this result of Moon, in 2007, Volkmann gave a conjecture that if D is a strong c-partite tournament with c≥3, then D contains at least three arcs that belong to an m-cycle for each m∈{3,4,..., c}. This paper mainly investigates the existence of ares as above in strong multipartite tournaments, prove this conjecture of Volkmann and extend the result of Moon. In 2008, Volkmann and Winzen on regular multipartite tournament gave a conjecture:each vertex of a regular multipartite tournament with c≥5 partite sets is contained in a strong subtournament of order p for every p∈{3,4,…,c}. we obtain a sufficient condition of existence of strong subtournament with order c in strong c-partite tournaments. As a corollary of this conjecture, we prove the condition above of Volkmann and Winzen when c≥16.