The Two Circles In The Tournament Results And Problems

In this paper, the Hamiltonian qualities in Bipartite tournament are studied. On thebasis of two new su?cient conditions brought up by Wang Jian-zhong[4]that Bipartite tour-nament would be directed Hamiltonian graph and a su?cient and essential condition broughtup by an associate professor Li Gui-rongXu [69] that Bipartite tournament would be directedHamiltonian graph and results of other forfathers, the su?cient conditions about Bipartitetournament with directed Hamiltonian graph are shown. In addition, we try to find andprove some characters on the paths and cycles in almost regular Bipartite tournament, somesatisfied results are gained.This thesis consists of 4 chapters. In the first chapter ,we will introduce the backgroundas the basic concepts, and main results obtained in this thesis is liven.In the second chapter, the terminology and notations on graph theory are introduced.In the third chapter, Hamilton problems are discussed. We prove a su?cient conditionfor Hamiltonian cycles in Bipartite tournaments, which are listed as follows: if an n×nbipartite tournament T satisfies the conditions: (i) W(n - 2), if | Q | + | R |=n - 1;(ii)W(n - 3), if | Q | + | R |= n - 1; then T is Hamiltonian, except for three exceptionalgraphs.In the last chapter, we find and prove some characters on the paths and cycles in almost regular Bipartite tournament, which are listed as follows:If an n×n almost regularBipartite tournament T satisfies d_T~- (u) + d_T~+ (v) = n for every arc uv∈A, then T has the kbypass property for k = 3,5,7,···,n - 5,n - 1, unless it is isomorphic to one of the graphsF(k - 1,k - 1,k,k),k > 1.we mainly study the traceability of quasi-claw-free graphs.