| 摘要: | 在本論文中，我們討論全部可追蹤圖的性質，並獲得利用米希爾斯基建構法建構出的圖M(G)中，只要圖G是一個迴圈或是漢米爾頓圖，則圖M(G)就會是一個是全部可追蹤的圖。
接著，我們定義了π(G)為圖G中，可追蹤的邊數佔所有邊數的比例，即當G有b個邊，其中a個邊為可追蹤邊的個數，則π(G)=a/b。並獲得下面的結果：
(1) 任給兩個正整數a, b，只要 a≦b≦[(a2 +4)/4]，則存在一個邊數為b的圖G中含有a個可追蹤的邊，即π(G)=a/b。
(2) 任給正整數n，則對於所有介於n – 1和1+(n – 1)2/4 之間的整數b，存在一個點數為n，邊數為b的圖G，使得π(G)=(n – 1)/b。
(3) 任給大於等於5的正整數n，則對於所有介於 和(n2–5n+14)/2之間的整數b，存在一個點數為n，邊數為b的圖G，使得π(G)=(b – 1)/b。
In this thesis, we discuss the properties of a totally traceable graph, and using Mycielski’s construction to construct a sequence of totally traceable graphs M(G) as G is a Hamiltonian graph.
Next, we define π(G) to be the ratio of traceable edges and total edges of G. That is if G has b edges which contains a traceable edges, then π(G) = a/b. We obtains the following results：
(1) Given any two positive integers a, b and a≦b≦[(a2 +4)/4]， then there is a graph G with b edges such that π(G) = a/b.
(2) Given any positive integer n, then there is a graph G with n vertices and b edges such that π(G) = (n – 1)/b, for each b, n–1≦b≦1+(n – 1)2/4.
(3) Given any positive integer n, then there is a graph G with n vertices and b edges such that π(G) = (b – 1)/b, for each b, ≦b≦(n2–5n+14)/2. |