| Abstract: | 一個具有n個點的圖中，若任意兩點皆有邊相連，我們稱此圖為n個點的完全圖，記作Kn。一個具有n個點的連通圖，其每一點的度數皆為2，我們稱此圖為n-迴圈，記作Cn。令C5的頂點分別為v1、v2、v3、v4和v5，並在C5外面加入5個頂點為w1、w2、w3、w4和w5及5條邊{vi,wi}，1≦i≦5，則稱此圖為5-太陽圖，記作S(C5)。
設G為一個簡單圖，G1, G2, …, Gt為G的子圖，若滿足下列條件：
(1) E(G1)∪E(G2)∪...∪E(Gt)= E(G)
(2) 對於1≦i≠j≦t，E(Gi)∩E(Gj)= Ø
則稱G可分割為G1, G2,…, Gt。若G1, G2,…, Gt都與H同構，則稱G可分割成H。
在本論文中，我們證明了：
(1) 當n≡1 (mod 20)時，Kn可分割成5-太陽圖。
(2) 當n≡0 (mod 20)時，Kn可分割成5-太陽圖。
(3) 當n≡5 (mod 20)時，Kn可分割成5-太陽圖。
(4) 當n=16,36時，Kn可分割成5-太陽圖。
A graph with n vertices in which every pair of distinct vertices is connected by a unique edge is called a complete graph with n vertices, denoted by Kn. A graph with n vertices in which every vertex has degree 2 is called a cycle, denoted by Cn. Let {v1, v2, v3, v4, v5} be the vertex set of C5. If we add another five vertices w1, w2, w3, w4, w5 and five edges {vi, wi}, 1≦i≦5, then we call this graph a 5-sun graph, denoted by S(C5). Let G be a simple graph and G1,G2,…,Gt be subgraphs of G.
If G1,G2,…,Gt satisfy the following conditions：
(1) E(G1)∪E(G2)∪...∪E(Gt)= E(G)
(2) ∀1≦i≠j≦t，E(Gi)∩E(Gj)= Ø
, then we call G can be decomposed into G1, G2, …, Gt. If G1, G2, …, Gt are isomorphic to H, then we call G can be decomposed into H.
In this thesis, we prove that：
(1) if n≡1 (mod 20), then Kn can be decomposed into 5-sun graphs.
(2) if n≡0 (mod 20), then Kn can be decomposed into 5-sun graphs.
(3) if n≡5 (mod 20), then Kn can be decomposed into 5-sun graphs.
(4) if n=16, 36, then Kn can be decomposed into 5-sun graphs. |
