令G為含有q個邊的簡單圖，若存在一個函數f，其中f: V(G)→{0, 1, 2, ..., q}且f為一對一函數。若由f所衍生出的函數g，g：E(G)→{1, 2, ..., q}，∀e={u,v}∈E(G)，g(e) =│f(u)−f(v)│且g為一對一且映成函數，則稱f為G的一個優美標號，此圖G為優美圖。
若將n-迴圈Cn的每一個點都各自黏上一條路徑Pm所得的圖，我們以Cn⊙Pm表示之；若將n-迴圈Cn中的一個點黏上一條路徑Pu，其餘n－1個點各自黏上一條路徑Pm所得的圖，我們以Cn⊙[(n－1)Pm∪Pu]表示之。
在本論文中，我們證明了：
(1) 當n≡0或3(mod 4)，且m為正整數時，Cn⊙Pm為優美圖。
(2) 當n≡1或2(mod 4)，且m為正整數時，Cn⊙P2m為優美圖。
(3) 當n≡0或3(mod 4)，且m、u為正整數時，Cn⊙[(n－1)Pm∪Pu]為優美圖。

Let G be a simple graph with q edges. If there exists a function f from V(G) to {0, 1, 2, ..., q} and f is one-to-one. If from f we can get a function g, g : E(G)→{1, 2, ..., q} defined by g(e) =│ f (u) − f (v)│for every edge e = {u, v}∈E(G), and g is a bijective function, then we call  f is a graceful labeling of G and the graph G is a graceful graph. 
Let Cn⊙Pm be the graph obtained by attaching a path Pm to each vertex of an n-cycle Cn. Let Cn⊙[(n－1)Pm∪Pu] be the graph obtained by attaching a path Pu to a vertex of an n-cycle Cn and attaching a path Pm to the other vertices.
In this thesis, we obtain the following results.
(1) Let m be a positive integer. If n≡0, 3(mod 4), then Cn⊙Pm is a graceful graph.
(2) Let m be a positive integer. If n≡1, 2(mod 4), then Cn⊙P2m is a graceful graph.
(3) Let m and u be a positive integers. If n≡0, 3(mod 4), then Cn⊙[(n－1)Pm∪Pu] is a graceful graph.

目錄..............................i
圖目錄............................ii
第一章  簡介......................1
第二章  預備知識..................3
第三章  Cn⊙Pm優美圖的探討........11
第一節  n ≡ 0 or 3(mod 4)........11
第二節  n ≡ 1 or 2(mod 4)........21
第四章  圖Cn⊙[(n－1)Pm∪Pu]......28
第五章  結論......................41
參考文獻..........................42
圖目錄
圖2.1 簡單圖G....................3
圖2.2 u,a,b,c,v步行..............4
圖2.3 步跡，迴路.................4
圖2.4 P6.........................5
圖2.5 C4.........................5
圖2.6 連通圖G和非連通圖H.........6
圖2.7 毛毛蟲圖...................6
圖2.8 圖的聯集...................6
圖2.9 C7⊙P3.....................7
圖2.10 C5⊙[4P2∪P1].............8
圖2.11 C6⊙[5P2∪P3].............8
圖2.12 三種圖的標號..............8
圖2.13 彼得森圖(Petersen Graph)的優美標號........9
圖2.14 P4和P5的優美標號...........9
圖2.15 毛毛蟲圖的優美標號........10
圖3.1 C3⊙P2及C4⊙P2的優美標號...11
圖3.2 C3⊙P3及C4⊙P3的優美標號...12
圖3.3 C4k⊙Pm....................13
圖3.4 C8⊙P4的優美標號...........16
圖3.5 C4k-1⊙Pm..................17
圖3.6 C7⊙P4的優美標號...........20
圖3.7 C7⊙P5的優美標號...........20
圖3.8 C5⊙P2及C6⊙P2的優美標號...21
圖3.9 C5⊙P4及C6⊙P4的優美標號...21
圖3.10 C4k＋2⊙Pm................22
圖3.11 C10⊙P4的優美標號.........24
圖3.12 C4k＋1⊙Pm................25
圖3.13 C9⊙P4的優美標號..........27
圖4.1 C3⊙[2P3∪P2]及C3⊙[2P2∪P3]的優美標號.......28
圖4.2 C4⊙[3P2∪P3]及C4⊙[3P3∪P2]的優美標號.......29
圖4.3 C4k⊙[(4k－1)Pm∪Pu].......30
圖4.4 C8⊙[7P3∪P2]的優美標號....34
圖4.5 C8⊙[7P2∪P4]的優美標號....35
圖4.6 C4k-1⊙[(4k－2)Pm∪Pu].....36
圖4.7 C7⊙[6P2∪P3]的優美標號....40
圖4.8 C7⊙[6P4∪P2]的優美標號....40

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