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    Title: 一些特殊圖的優美標號
    Other Titles: Graceful labelings of some special graphs
    Authors: 許炎午;Hsu, Yen-Wu
    Contributors: 淡江大學中等學校教師在職進修數學教學碩士學位班
    高金美
    Keywords: 迴圈;路徑;優美標號;優美圖;cycle;path;graceful labeling;graceful graph
    Date: 2015
    Issue Date: 2016-01-22 14:52:19 (UTC+8)
    Abstract: 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.
    Appears in Collections:[數學學系暨研究所] 學位論文

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