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博碩士論文 etd-0124111-164137 詳細資訊
Title page for etd-0124111-164137
論文名稱
Title
無向圖及有向圖的漢米爾頓值
The hamiltonian numbers of graphs and digraphs
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
59
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2011-01-19
繳交日期
Date of Submission
2011-01-24
關鍵字
Keywords
漢米爾頓圈、漢米爾頓值、雙迴圈網路
hamiltonian number, hamiltonian cycle, double loop network
統計
Statistics
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The thesis/dissertation has been browsed 5762 times, has been downloaded 1317 times.
中文摘要
漢米爾頓值的討論是經由漢米爾頓圈的問題衍生而來。而漢米爾頓圈的問題眾所皆知是NP-complete。當然,計算漢米爾頓值也是NP-complete。 另一方面,漢米爾頓值的問題也是旅行推銷員問題其中的一種。
 
漢米爾頓道路是指一條長度最小的封閉生成道路。不是所有的圖都有漢米爾頓圈,但只要是連通圖必定會有一條漢米爾頓道路。而這條道路的長度,我們稱為漢米爾頓值。在無向圖的部分,我們決定了漢米爾頓值的上界及改善了[14]中的結果,並且決定出h(G)<e(G)的充分必要條件,此處的e(G)是指長度最小且經過所有邊的封閉道路。同時也證明了當二個不相鄰的點,它們的度數和不小於頂點數時,增加這條邊不會影響到漢米爾頓值。這個結果推廣了Bondy及Chv′atal在漢米爾頓圈上的定理。最後,我們證明了在一個有n個頂點的二連通圖中,如果任取三個都不相鄰的頂點,他們的度數和不小於(3n-k-2)/2時,則此圖的漢米爾頓值會不大於n+k,此處的k是介於0和n-2之間。這個結果是從Bondy的定理延伸而來的。

在有向圖的部分,我們也給予了漢米爾頓值的上界。對於一個強連通的有向圖而言,它的漢米爾頓值是介於n和⌊(n+1)^2/4⌋之間。在論文中構造了一些圖,它們的漢米爾頓值能對應到這個區間的所有數字。最後,針對莫氏雙迴圈網路的漢米爾頓值來討論。莫氏雙迴圈網路包含了所有強連通的雙迴圈網路,我們決定了它的上界以及提供了漢米爾頓值等於n、n+1、n+2的充分必要條件。

關鍵字: 漢米爾頓圈、漢米爾頓圈、雙迴圈網路。
Abstract
The hamiltonian number problem is a generalization of hamiltonian cycle problem in graph theory. It is well known that the hamiltonian cycle problem in graph theory is NP-complete [16]. So the hamiltonian number problem is also NP-complete. On the other hand, the hamiltonian number problem is the traveling salesman problem with each edge having weight 1.

A hamiltonian walk of a graph G is a closed spanning walk of minimum length. The length of a hamiltonian walk in G is called the hamiltonian number. For graphs, we give some bounds for hamiltonian numbers of graphs. First, we improve some results in [14] and give a necessary and sufficient condition for h(G) < e(G) where e(G) is the minimum length of a closed walk passing through all edges of G. Next, we prove that if two nonadjacent vertices u and v satisfying that deg(u)+deg(v) &#8805; |G|, then h(G) = h(G + uv). This result generalizes a theorem of Bondy and Chv′atal for the hamiltonian cycle. Finally, we show that if 0 &#8804; k &#8804; n &#8722; 2 and G is a 2-connected graph of order n satisfying deg(u) + deg(v) + deg(w) &#8805; 3n&#8722;k&#8722;2 for every independent set {u, v,w} of three vertices in G, then h(G) &#8804; n+k. It is a generalization of a Bondy’s result.

For digraphs, we give some bounds for hamiltonian numbers of digraphs first. We prove that if a digraph D of order n is strongly connected, thenn &#8804; h(D) &#8804; &#8970;(n+1)^2/4 &#8971;. Next, we also present some digraphs of order n &#8805; 5 which have hamiltonian number k for n &#8804; k &#8804; &#8970;(n+1)^2/4 &#8971;. Finally, we study hamiltonian numbers of M‥obius double loop networks. We introduce M‥obius double loop network and every strongly connected double loop network is isomorphic to some M‥obius double loop network. Next, we give an upper bound for the hamiltonian numbers of M‥obius double loop networks. Then, we find some necessary and sufficient conditions for M‥obius double loop networks MDL(d, m, &#8467;) to have hamiltonian numbers dm, dm + 1 or dm + 2.
目次 Table of Contents
Abstract. . . . . . . . . . . . . . . . i
Contents. . . . . . . . . . . . . . . . ii
List of Figures . . . . . . . . . . . . . . . . iv
1 Introduction . . . . . . . . . . . . . . . . 1
1.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 An alternating version of hamiltonian number . . . . . . . . . 6
1.3 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . 7
2 Hamiltonian numbers of graphs 8
2.1 Some properties of hamiltonian numbers in graphs . . . . . . . 8
2.2 Relations between hamiltonian and Eulerian walk . . . . . . . 10
2.3 Existence of hamiltonian walks in graphs . . . . . . . . . . . . 12
3 Hamiltonian numbers of digraphs 18
3.1 Some properties of hamiltonian numbers in digraphs . . . . . . 18
3.2 Existence of hamiltonian walks in digraphs . . . . . . . . . . . 21
3.3 The hamiltonian spectrum of graphs . . . . . . . . . . . . . . 23
4 Hamiltonian numbers of M&#246;bius double loop networks 26
4.1 Basic definitions and related concepts . . . . . . . . . . . . . . 26
4.2 The relation between DL(n; a, b) and MDL(d, m, &#8467;) . . . . 29
4.3 Hamiltonian numbers of MDL(d, m, &#8467;) . . . . . . . . . . . . 32
5 Conclusions and Further Research 42
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