| Digraphs occupy a very high position in graph theory,and the tournaments are the most important kind of graphs.Thus,the tournaments are concerned by many researchers.The pancyclic problem is also a hot issue in graph theory,there are many aspects about pancyclic,such as vertex-pancyclic,arc-pancyclic,pancyclic out-arcs of a vertex and so on.Among them,the problem of arc-pancyclicity is important,and more and more scholars have also carried on the thorough research to the arc-pancyclicity.An arc uv in a digraph D is called pancyclic if there is a path from v to u of length k for every 2≤k ≤|V(D)|-1.An arc uv in a digraph D is called anti-pancyclic if there is a path from u to v of length k for every 2 ≤k ≤|V(-D)|-1.In 1994,Moon showed that every non-trivial strong tournament contains at least three pancyclic arcs and characterized the tournaments that attain this lower bound.In 1997,Guo proved that every arc of a 3-strong and arc-3-cyclic tournament T is k-anticyclic for every k≥4,unless T is isomorphic to one of two specific tournaments,each of which has exactly 8 vertices.In this thesis,we investigate the existence of anti-pancyclic arcs in tournaments,and investigate the number of anti-pan cyclic arcs in strong tournaments.The thesis consists of three sections.Chapter 1 is the preface.The background,the development and basic concepts are introduced.And the main content of this thesis is proposed.In Chapter 2,we study the anti-pancyclic arcs in tournaments.We investigate the existence of anti-pancyclic in tournaments and characterize all tournaments with at least one anti-pancyclic arc.And this can be proved in two ways:1.Every non-strong tournament contains at least one anti-pancyclic arc.2.Every strong tournament contains at least one anti-pancyclic arc unless it is isomor-phic to one of four specific tournametns.In Chapter 3,we investigate the number of anti-pancyclic arcs in strong tournaments.We prove the conclusions:Every strong tournament with order n≥6 contains at least four anti-pancyclic arcs unless it is isomorphic to one of five specific strong tournament,each of which has exactly 6 vertices. |
