| This thesis mainly studies the relationship between toughness and hamiltonian index. The whole thesis is divided into four chapters. In the following, let me explain explicitly what I have done.Chapter 1 is the foundation of the whole thesis. It mainly introduces the basic concepts and symbols, the research background and the development of graph theory, and the related knowledge of toughness and hamiltonian index of a graph.Chapter 2 mainly studies the relationship between vertex toughness (hereinafter referred to as the toughness) and hamiltonian index. The toughness of graph G, denoted by t(G), and t(G)=min{|S|ω(G-S):S∈V(G),ω(G-S)>1}, where ω(G-S) denotes the number of components of G-S. For nontrivial simple graph G, the thesis proves that if the toughness t(G)> 1, then the hamiltonian index h(G)< 2; if t(G)> 3/2, then h(G)<1. Moreover, the thesis constructs two new graphs to prove the bounds in above results are sharp.Chapter 3 mainly studies the relationship between edge toughness and hamiltonian index. The definition of edge toughness given by Chvatal in 1973 denoted by t’(G), and t’(G)=min{|X|/ω(G-X):X∈E(G),ω(G-X)>1}, where ω(G-X) denotes the number of components of G-X. For nontrivial simple graph G, the thesis proves that if the edge toughness t’(G)> 1, then the hamiltonian index h(G)≤2; if t’(G)> 3/2, then h(G)≤ 1. In 1997, Katona gave another definition of edge toughness, denoted by te(G). In this new definition edges and vertices can be deleted simultaneously. For nontrivial simple graph G, the thesis proves that if the edge toughness te(G)> 1, then the hamiltonian index h(G)≤2; if te(G)> 3/2, then h(G)< 1. The results are same for the two different definitions of edge toughness.Chapter 4 is the conclusion, including the main results of the thesis, the innovations of the thesis and what should be research further in future. |
