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論文名稱 Title |
圖具單一元素漢米爾頓譜之研究 Graphs with a singleton Hamiltonian spectrum |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
35 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2015-09-24 |
繳交日期 Date of Submission |
2015-10-19 |
關鍵字 Keywords |
漢米爾頓值、定向、漢米爾頓譜 orientation, Hamiltonian spectrum, Hamiltonian number |
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統計 Statistics |
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中文摘要 |
一個強連通有向圖的漢米爾頓道路所指的是一條長度最小並且通過有向圖所有點的封閉道路,而一個強連通有向圖的漢米爾頓值所代表的是此有向圖之漢米爾頓道路的長度。Chang與Tong在[2]中證明了強連通有向圖的漢米爾頓值介於n與⌊(n+1)^2/4⌋之間,並且描述了點數n大於等於5、漢米爾頓值為⌊(n+1)^2/4⌋之強連通有向圖的特徵。另外Chang與Tong在[2]中亦介紹了簡單圖之漢米爾頓譜的定義,一個簡單圖(G)的漢米爾頓譜(S_h(G)),是一個包含所有此簡單圖之強連通定向之漢米爾頓值的集合;也就是說,S_h(G) ={h(D) : D是一個G的強連通定向}。在此論文中,我們延續Chang與Tong的研究,描述了點數n大於等於9、漢米爾頓值為⌊(n+1)^2/4⌋-1之強連通有向圖的特徵,並且描述了漢米爾頓譜各為單一元素n、n+1 與n+2 時,所對應之簡單圖的特徵。 |
Abstract |
A closed walk in a strongly connected digraph is called a Hamiltonian walk if it passes through all vertices with minimum length. The Hamiltonian number h(D) of a strongly connected digraph D is the length of a Hamiltonian walk in D. Chang and Tong proved that if a digraph D of order n is strongly connected, then n≤h(D)≤ ⌊(n+1)^2/4⌋, and characterized the strongly connected digraphs of order n≥5 with h(D) =⌊(n+1)^2/4⌋ in [2]. Also, the Hamiltonian spectrum of a graph was introduced by them in [2]. The Hamiltonian spectrum S_h(G) of a simple graph G is a set of Hamiltonian numbers among all strongly connected orientation of it; i.e. S_h(G) = {h(D) : D is a strongly connected orientation of G}. In this thesis, we just go on with their work. We characterize digraphs D of order n≥9 with Hamiltonian number ⌊(n+1)^2/4⌋-1, and characterize simple graphs G of order n with S_h(G)={n}; S_h(G)={n+1} and S_h(G)={n+2}. |
目次 Table of Contents |
摘要 i Abstract ii Contents iii List of Figures iv 1 Introduction 1 2 Previous Results 4 3 Main Results 6 4 Conclusion 28 Reference 29 |
參考文獻 References |
[1] S. E. Goodman and S. T. Hedetniemi, On hamiltonian walks in graphs, SIAM J. Comput. 3 (1974), 214-221. [2] T. P. Chang and L. D. Tong, The hamiltonian numbers in digraphs, Journal of Combinatorial Optimization. 25 (2013), 694-701. [3] G. Chartrand, V. Saenpholphat, T. Thomas and P. Zhang, A new look at Hamiltonian walks, Bull. ICA. 42 (2004), 37-52. [4] T. Asano, T .Nishizeli and T. watanabe, An upper bound on the length of a Hamiltonian walk of a maximal planar graph, J. Graph Theory. 4 (1980), 315-336. [5] T. Asano, T. Nishizeli and T. watanabe, An approximation algorithm for the Hamiltonian walk problems on maximal planar graphs, Discrete Appl. Math. 5 (1983), 211-222. [6] P. Vacek, On open Hamiltonian walks in graphs, Arch, Math. 27A (1991), 105-111. [7] P. Camion, Chemins et circuits hamiltoniens des graphes complets. C. R. Acad. Sci. Paris, 249 (1959), 2151–2152. [8] M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5 (1976), 704-714. |
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