Spanning Directed Triangles Paths And Cycles Containing Given Arcs In Tournaments

This paper is composed of three chapters. In this paper, The problems of spanning directed triangles in the regularity tournaments and paths and cycles containing given arcs in multipartite tournaments are discussed.In the first chapter, we introduce some useful basic concepts which will be used in the paper.In the second chapter, We study the problem of spanning directed triangles in the regularity tournaments. The main results are as follows:(1) Let T is a regular tournaments of order 5, for any vertex x in V(T), there exist 2 triangles Ti such that:V(Ti)∩V(Tj)=χfor 1≤i＜j≤2.(2) Let T is a regular tournaments of order 7, for any vertex x in V(T), there exist 3 triangles Ti such that:V(Ti)∩V(Tj)=χfor 1≤i＜j＜3.(3) Let T is a regular tournaments of order 9, for any vertex x in V(T), there exist n triangles Ti such that:V(Ti)∩V(Tj)=χfor 1≤i＜j≤4.In the third chapter, We study the problem of paths and cycles containing given arcs in multipartite tournaments. The main results are as follows:(1) Let D is an c—partite tournament with order n,χ, y are different vertexs. if c≥5 and|V(D)|＞105ig(D)+2790, then there exists (χ, y)-path of length l, for all 42≤l≤|V(D)|-1 are right.(2) Let D is an c—partite tournament with order n and c≥5, P is a path of length l in D. if|V(D)|＞105ig(D)+106l+2684, then there exists a H—cycle containing path P in D.(3) Let D is an c—partite tournament with order n and c≥5, A=｛e1,e2…ek｝is an k—path extendible, if c≥5 and |V(D)|＞105ig(D)+2366+424k, then there exists a Hamilton-cycle containing A in D.