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.