令K_n是一個具有n個點的完全圖。 令C_k為一個長度為k的迴圈，S_k為一個具有K個邊的圖。 若k = 4，則我們稱C_4為一個4-迴圈，S_3為一個3-星圖。 設 λ 為正整數，λG 代表一個 λ重邊圖其中G中相鄰的任兩點均有 λ 個邊相連。G 的一個分割是指將G分成邊相異的子圖(不需要是相異的圖)。 若 H_1, H_2, …, H_t 是G的子圖，其邊均相異且E(G) = E(H_1)∪E(H_2)∪…∪E(H_t)及Σ_(i=1)^t |E(H_i)|=|E(G)|，則稱G可分割成H_1, H_2, …, H_t。 若對於i = 1, 2, …, t，每個H_i 都與H同構則稱G有一個H-分割。 設p和q為兩個非負的整數，若G 可分割成t個子圖H_1, H_2, …, H_t且其中p個是與H_1同構，剩餘的q個與H_2同構，則稱G可分割成p個H_1和q個H_2。 
在這篇論文中，我們將證明假設p和q為兩個非負的整數，對於所有可能的p和q，λK_n均可分割成p個C_4和q個S_3 的充分必要條件為 4p +3q =λ(n choose 2) 及 
(1) λ 是偶數，n≧4。 
(2) λ = 1，n≧6，當n是奇數時， q≠1, 2；當n是偶數時， 
q ≧ ceil(n/4) 。 
(3) λ 是奇數，λ≧3，n≧5，當n是奇數時， q ≠ 1；當n是偶數時，q ≧ ceil(n/4)。

Let K_n be the complete graph on n vertices. Let C_k be a cycle of length k and S_k be a star with k edges. If k = 4, then we call C_4 a 4-cycle and S_3 a 3-star. For any positive integer λ, let λG denote the λ-fold multigraph with λ edges between any two adjacent vertices of G. A decomposition of G is a partition of G into edge-disjoint subgraphs (not necessary distinct) of G. If H_1, H_2, …, H_t are edge-disjoint subgraphs of G such that E(G) = E(H_1)∪E(H_2)∪…∪E(H_t) and Σ_(i=1)^t |E(H_i)|=|E(G)| , then we say that H_1, H_2, …, H_t decompose G. Furthermore, G is said to have a H-decomposition if Hi is isomorphic to H for each i = 1, 2, …, t. G can be decomposed into p copies of H_1 and q copies of H_2 for some nonnegative integers p and q if p of H_1, H_2, …, H_t are isomorphic to H1 and the rest q of them are isomorphic to H_2.
In this thesis, we prove that for nonnegative integers p and q, there exists a decomposition of λK_n into p copies of C_4 and q copies of S_3 for all possible 
values of p and q if and only if 4p +3q = λ(n choose 2) and
(1) λ is even, n≧4. 
(2) λ = 1, n≧6, q≠1,2 for odd n and q ≧ ceil(n/4) for even n.
(3) λ is odd, λ≧3, n≧5, q≠1 for odd n and q ≧ ceil(n/4) for even n.

Chapter 1. Introduction..............................................1
1.1  Definitions and Preliminaries........................1
1.2  Known results........................................4
1.3  Main works...........................................6
Chapter 2. Decomposition of K_n .............................8
2.1  Some small cases ....................................9
2.2  Main theorem for odd n..............................13
2.3  Main theorem for even n.............................20
Chapter 3. Decomposition of 2K_n ............................32
3.1  Some small cases....................................32
3.2  Main theorem .......................................37
Chapter 4. Decomposition of λK_n ............................44
4.1 λ = 3 …………………………….....................................44
4.2 λ = 4 …………………............................................49
4.3 λ = 5 ……………………………........................................51
4.4 6≦λ≦13 ……………...……………......................................52
4.5  General cases ......................................56
Reference................................................61
Appendix.................................................64

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