The Problem For Out-arcs Pancyclic In Tournament And The Panpath Of Some Special Digraphs

This paper is composed of four chapters, mainly discuss the problem of panpath and pancyclic vertex for tournaments, the problem of panpath for special digraphs and the prob-lem of panpath for locally tournaments.Chapter 1 introduce the background of the development of graph and some basic defi-nitions and concepts on graphs.In chapter 2, we introduce the present results about tournaments, and discuss the problems of the out-arcs pancyclic vertices for k-strong tournaments. The result is as follows: a k-strong tournament with k> 4 has two out-arcs pancyclic vertices v1, v2, and if d+(y)> d+(uk-1) for every y in N+(v1)\{v2}, that exit another vertex χ, either χ is out-arcs pancyclic, or is 4- pancyclic.Chapter 3 is about the problems of panpath for directed wheelgraph and directed multi-wheelgraph. We give the number of pan-connectivity vertex pair of directed wheelgraph and the number of pan-path-connectivity vertex pair of directed multiwheelgraph. The result is as follows:(i) Every directed multiwheelgraph â†'/Wk,t has one vertex u, which is out-arcs panpath.(ii) Every directed multiwheelgraph â†'/Wk,t is arcs-panpath.(iii) The number of pan-path-connectivity vertex pair of directed multiwheelgraph â†'/Wk,t is t.(iv) The number of pan-connectivity vertex pair of directed wheelgraph â†'/Wn is n.In chapter 4 we study the problems of panpath for locally tournaments. We show the number of Hamilton path in connected locally tournaments but not strong and pan-connectivity vertex pair in connected locally tournaments but not strong. The results are as follows: (i) Let D be a connnected locally tournament, but not strong. Then D is traceable.(ii)Let D be a connnected locally tournament,but not strong.Let u,v be a pair of distinct vertices in D with d+(u)=n-1,d-(v)=n-1.Then the vertex pair{u,v}.is the pan-connectivity vertex pair in D.