在[3]中，Dirac在1952年證明，只要簡單圖G中，頂點個數至少3個，頂點的維度都大於或等於頂點個數的一半，就有漢米頓圈。Chvátal-Erdös在1972年證明，簡單圖G中，頂點個數至少3個且κ(G)大於或等於 α(G)，則有漢米頓圈。此兩類的圖，邊的個數都要相當多，而在邊數比較少的圖中是否有漢米頓圈則是一個難解的問題。例如：Odd graph O(n)，頂點集合V為2n-1個元素集合的n-1元子集合，任二頂點A與B相鄰若且唯若A交集 B為空集合 。這種圖有 個頂點，但每點的邊數只有n條。這種圖是否是漢米頓圖就是很難的問題，n=3是Petersen圖，沒有漢米頓圈，但n大於或等於 4時是有名的Kneser 猜想：假設n>3，則O(n) 是漢米頓圖。這個問題大部分的情況到現在仍未解決。
考慮最極端的例子，我們想要檢查一些3-正則圖是否有漢米頓圈。根據Cayley圖是漢米頓圖的猜想，我們有興趣去知道3-正則Cayley圖是否有漢米頓圈。然而要去分析所有的3-正則Cayley圖是一件難事，特別是缺乏邊對稱的圖[8]。本論文將探討下列特殊的3-正則類：1.圈化圖(Cycle connected graph)。2.SEP(n)。

We wnat to check some 3-regular graphs has Hamiltonian cycle.According to the conjucture:Cayley graphs are Hamiltonian cycle,we want to know 3-regular Cayley graphs has Hamiltonian cycle.But it is hard to anlysis all 3-regular Cayley graphs.In this paper,we discuss some specal 3-regular graphs: Cycle connected graph and SEP(n).

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