| Abstract: | 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. |
