The Out-arcs Pancyclic Property Of Tournaments

This paper is composed of three chapters. In this paper, the problems of the out-arcs pancyclic vertices for tournaments and a few kinds of strong tournaments are discussed under some constrained conditions.In the first chapter, the background and the related contents are introduced.In the second chapter, the main contents are preliminaries. Some important and basic definitions which like tournaments, k-strong tournaments and out-arcs pancyclic , etc are introduced in detail. The lemmas and theorems which are related of the paper are intronduced in this paper, too.In the third chapter, all of important results are given and proved. Two sufficient conditions that the tournaments which have strong decomposition have three out-arcs pancyclic vertices are described under some constrained conditions in section 3.1. Yeo conjectured that there are three out-arcs pancyclic verteices in a 2-strong tournaments in bibliography[3]. In section 3.2, we prove that a 2-strong tournament T contains a vertex v₃ which is not same to v₁ and v₂ if v₁ and v₂ are out-arcs pancyclic vertices of T and T—{v₁, v₂} is not strong. In section 3.3, the out-arcs pancyclic problems of a 3-strong tournament are discussed. In the section, we use path-contraction to prove that a 3-stong tournament T contains at least three vertices v₁, v₂, v₃ and all of out-arcs of v₃ are 4-pancyclic, the rest vertices are out-arcs pancyclic vertices. Another result that a 3-strong tournament T contains a out-arcs pancyclic vertex v₃ which is not same to v₁ and v₂ if the out-degrees of v₁ are minimum in V(T) and the out-degrees of v₂ are minimum in N⁺(v₁) is presented in this section. In section 3.4, we discuss principally a sufficient condition for a 4-strong tournament which includes three out-arcs pancyclic vertices under degree constrained conditons.