| 摘要: | 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. |
