(Quasi) Out-arcs Pancyclic-Vertex Questions Of Multipartite Tournaments

This paper is composed of four chapters.The main contents involve two aspects: (1) On cycles through all out-arcs of a given vertex in multipartite tournaments; (2) The number of vertices whose out-arcs are pancyclic in a strong tournament.Firstly, the background and the problem we want to resolve are introduced.In the second chapter,the main contents are preliminaries.Some basic but very important definitions are introduced in detail.The lemmas,theorems and corollaries which are related to the third and the fourth chapter are introduced in this chapter.In the third chapter,we discussed the cycles through all out-arcs of a given vertex in multipartite tournaments. The results on the number of out-arcs pancyclic-vertices in tournaments are very rich.The problem in multipartite tournaments seems much more difficult. In 2004,Guo and Volkmann proved that every vertex of a strong connected semicomplete n-partite(n≥3)digraph D belongs to a cycle Cq and for each q∈{3,…,n}such that V(C3) (?)…(?) V(Cn).We tried to extend the results in tournaments to multipartite tournaments.Now we state the main results:Theorem 3.1:If D is a strong n-partite (n≥3) tournaments with k(D)= 1 and every arc belongs to a 3-cycle of D,then there exists three vertices v1,v2,v3 such that each out-arc of vi(i= 1,2,3)belongs to a cycle Cj which contains vertices from exactly j partite sets for each j∈{3,4,…, n}such that V(C4) (?)…(?)V(Cn)(if n≥4).In the fourth chapter,we discussed the number of vertices whose all out-arcs are pancyclic in a strong tournament with conncetivity 1 and minimum out-degree at least two.Feng proved that a s-strong (s≥3)tournament contains s+1 vertices whose out-arcs are 4-pancyclic and a 3-strong tournament contains at least 3 out-arcs pancyclic-vertices; Guo proved a 2-strong tournament contains at least 3 out-arcs pancyclic-vertices. So the number of out-arcs pancyclic-vertices in tournament with conncetivity 1 was leaved. Now we state the main results in this chapter:Theorem 4.2:Let T be a strong tournament on n vertices with k(T)= 1 and minimum out-degree at least two. Then T contains at least three vertices such that all of whose out-arcs are pancyclic.