當一個含有n個點的圖中，任兩個點都有邊相連，我們稱此圖為n點的完全圖，記為Kn。若任兩個點都有兩個邊相連，我們稱此圖為n點的二重完全圖，記為2Kn。假設Cn = (v1, v2, v3, ..., vn)，在Cn外面加入n個點w1, w2,w3, ..., wn及n條邊{vi, wi}, 1≦i≦n，所形成的圖稱為Cn的太陽圖，記為S(Cn)。設G為一個簡單圖，且G1, G2, G3, ..., Gt為G的子圖，若滿足下列條件：
    (1) E(G1)∪E(G2)∪E(G3)∪...∪E(Gt) = E(G)
    (2)對於1≦i, j≦t, i不等於j,E(Gi)∩E(Gj) = Ø
則稱G可分割成G1, G2, G3, ..., Gt。若G1, G2, G3, ..., Gt均與圖H同構，則稱G可分割成圖H。
在此論文中，我們證明了：
(1)當n ≡ 1 or 3 (mod 6)時，2Kn可分割成循環3-太陽圖。
(2)當n ≡ 0 or 4 (mod 6)時，2Kn可分割成1-旋轉3太陽圖。

A graph with n vertices such that every two vertices are joined by an edge is called a complete graph with n vertices, denoted by Kn. If every two vertices are joined by two edges, then we call this graph a 2-fold complete graph with n vertices, denoted by 2Kn. Let (v1, v2, v3, ..., vn) be an n-cycle Cn. If we add another n vertices w1, w2, w3, ..., wn and n edges {vi, wi}, 1≦i≦n, then we call this graph an n-sun graph,denoted by S(Cn). Let G be a simple graph and G1, G2, G3, ..., Gt be subgraphs of G. If G1, G2, G3, ..., Gt satisfy the following conditions：
    (1) E(G1)∪E(G2)∪E(G3)∪...∪E(Gt) = E(G)
    (2) 1≦i, j≦t, i is not j,E(Gi)∩E(Gj) = Ø
Then we call G be decomposed into G1, G2, G3, ..., Gt. If G1, G2, G3, ..., Gt are isomorphic to H, then we call G can decomposed into H.
In this thesis, we have the following results.
  (1) n ≡ 1 or 3 (mod 6), 2Kn can be decomposed into cyclic 3-sun graphs.
  (2) n ≡ 0 or 4 (mod 6), 2Kn can be decomposed into 1-rotational 3-sun graphs.

第一章 簡介................................................................................................1
第二章 預備知識..........................................................................................3
第三章 2Kn可分割成循環或1-旋轉3-太陽圖...................................................12
 第一節 n ≡ 1 (mod 6)，則2Kn可分割成循環3-太陽圖.....................................13
 第二節 n ≡ 0 (mod 6)，則2Kn可分割成1-旋轉3-太陽圖...................................21
 第三節 n ≡ 3 (mod 6)，則2Kn可分割成循環3-太陽圖.....................................27
 第四節 n ≡ 4 (mod 6)，則2Kn可分割成1-旋轉3-太陽圖...................................39
參考文獻...................................................................................................51

圖表目錄

圖2.1    P5.....................................................................................................3
圖2.2    C6 = (1,2,3,4,5,6)...............................................................................4
圖2.3    K6.....................................................................................................5
圖2.4    H為G的子圖.......................................................................................5
圖2.5    K4可分割成G1, G2.............................................................................6
圖2.6    H為G的一個1-因子..............................................................................7
圖2.7    S(C3).................................................................................................7
圖2.8    △(K4)................................................................................................8
圖2.9    3-太陽圖差序列...................................................................................8
圖2.10   G+2為圖G的一個平移........................................................................9
圖3.1    2K7的初始3-太陽圖...........................................................................14
圖3.2    2K13的初始3-太陽圖.........................................................................15
圖3.3    2K19的初始3-太陽圖.........................................................................16
圖3.4    2K37的初始3-太陽圖....................................................................17, 18
圖3.5    2K12的初始3-太陽圖.........................................................................22
圖3.6    2K18的初始3-太陽圖.........................................................................23
圖3.7    2K36的初始3-太陽圖.........................................................................25
圖3.8    2K9的初始3-太陽圖...........................................................................28
圖3.9    2K15的初始3-太陽圖.........................................................................30
圖3.10   2K21的初始3-太陽圖..................................................................31, 32
圖3.11   2K33的初始3-太陽圖........................................................................33
圖3.12   2K39的初始3-太陽圖........................................................................35
圖3.13   2K10的初始3-太陽圖........................................................................40
圖3.14   2K16的初始3-太陽圖........................................................................42
圖3.15   2K22的初始3-太陽圖........................................................................44
圖3.16   2K34的初始3-太陽圖..................................................................45, 46
圖3.17               2K40的初始3-太陽圖........................................................................48

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