Pancyclic Out-Arcs Of Vertex In Local Tournaments

Graph Theory, originated in the eighteenth century, is one of the most important and popular subjects in discrete mathematics. There has been two hundred years of history. With the deepening development of scientific research, the practical applications of Graph Theory are more widely. Graph Theory played a significant role not only in promoting to the research of the mathematical theory, but also it was more closely with other mathematical disciplines. They learned for each other, so that they had received a huge development potential and development. Learning Graph Theory can improve and exercise students' comprehensive thinking ability. They also can describe and solve practical problems better by the mathematical tools.The graph of the No 2-local semi-complete directed graph is local tournament. The local semi-complete directed graph was put forward firstly by J. Bang-Jensen. It is a class of very interesting and important figure. It has promoted the concepts of semi-complete directed graph and tournaments. Meanwhile, he also put a half to plan and complete a tournament in the nature of the expansion to such figure. One round decomposable digraph is locally semi-complete directed graph in a class of very important, figure, this paper, we mainly directed graph through the circle to study the structure of local tournaments outside the arc pancyclicity. Before this tournament we first summed up this issue on a number of related conclusions, Tournaments pancyclicity on the degree of arc constraints, necessary and sufficient condition and so on. These conclusions can be extended to local tournaments?This article is divided into three chapters:In the first chapter, we introduce some paper that will be used on the basic concepts of graph theory and its notation.In the second chapter, we mainly reviewed some relevant results of the tournamentsIn the third chapter, the above two chapters introduce the theory and the corresponding results of analysis and research to further expand the issues of pancyclic out-arcs of vertex in the local tournament and conclusions have been further than the promotion and discussion. Through research and discussion of previous, by Guo theoretical results, Yao's results are extended to the local tournaments and in the process we draw some new conclusions. We conclude a local tournament T of strongly but not round decomposable must contain a vertex v, such that all of outside arcs of the v are pancircle. Entirely, by the Study of the local semicomplete digraph and the structure of circle digraph, we have a conclusion of the local tournament on the pancyclicity of the arc:Let T is a connected and arc 3-circle but not 2-strong of local tournament on the pancycic, then it is isomorphic to C [T1, T2,{v}], where every Ti(i= 1,2) is a tournament of the arc 3-circle, and v is an isolated vertex.