當一個含有 n 個點的簡單圖中，任意兩點皆有邊相連，則稱此圖為完全圖，記為 K_n；當一個含有 n 個點的連通圖，若其每一個點的度數皆為 2，則稱此圖為 n-迴圈，記為 C_n。
  若一個完全圖 K_n可分割成4-迴圈或 n-迴圈是指 K_n中的邊可分割成一些邊均相異的4-迴圈或 n-迴圈，且這些迴圈之邊的聯集即為 K_n中的邊集合。
  在本篇論文中，我們證明了：
(1) 當 n 為奇數且 n≥3 時，設 α,β 為正整數或零，若 4α+βn=(n(n-1))/2  ，則 K_n  可以分割成 α 個4-迴圈以及 β 個漢米爾頓迴圈。
(2) 當 n 為偶數且 n≥4 時，設 α,β 為正整數或零，若 4α+βn=(n(n-2))/2  ，則 K_n-I 可以分割成 α 個4-迴圈以及 β 個漢米爾頓迴圈，其中 I 為 K_n  的一個1-因子。

A complete graph with n vertices is a simple graph in which every pair of distinct vertices is connected by a unique edge, denoted by K_n.
The cycle is a connected graph with n vertices which all vertices are degree 2 and denoted by C_n.
    A complete graph K_n  can be decomposed into 4-cycles and n-cycles if K_n  can be partitioned into edge-disjoint 4-cycles and n-cycles such that the union of edge sets of these cycles is the edge set of K_n.
    In this thesis, we prove that:
(1) For odd integer n, n≥3, if there exists nonnegative integers α and β such that 4α+βn=(n(n-1))/2  , then K_n  can be decomposed into α 4-cycles and β n-cycles.
(2) For even integer n, n≥4, if there exists nonnegative integers α and β such that 4α+βn=(n(n-2))/2  , then K_n-I can be decomposed into α 
4-cycles and β n-cycles, where I is a 1-factor of K_n.

第一章   簡介	1
第二章   預備知識	3
第三章   K_n  分割成 C_4  或 C_n	13
第四章   K_n-I 分割成 C_4  或 C_n	26
參 考 文 獻	32

圖 目 錄
圖2.1	3
圖2.2	3
圖2.3	4
圖2.4	4
圖2.5	5
圖2.6	5
圖2.7	6
圖2.8	6
圖2.9	7
圖2.10	8
圖2.11	9
圖2.12	9
圖2.13	10
圖2.14	10
圖2.15	11
圖2.16	11
圖3.1	14
圖3.2	15
圖3.3	16
圖3.4	16
圖3.5	17
圖3.6	17
圖3.7	19
圖3.8	20
圖3.9	22
圖3.10	24
圖4.1	29

[1] P. Adams, D. Bryant and A. Khodar, 3,5-cycles 
    decompositions, J.Combin. Des. 6 (1998) , 91-110.
[2] B. Alspach, Research Problem 3, Discrete Math. 36 
    (1981), 333.
[3] B. Alspach and H. Gavlas, Cycle decompositions of K_n 
    and K_n-I, J. Combin. Theory Ser. B 81 (2001), 77-99.
[4] P. Balister, Packing Circuits into K_n, Combin. Probab. 
    Comput. 10 (2001), 463-499.
[5] P. Balister, On the Alspach Conjectrue, Combin. Probab. 
    Comput. 10 (2001), 95-125.
[6] J.-C. Bermond, O. Favaron, and M. Matheo, Hamiltonian 
    decomposition of Cayley graphs of degree 4, J. Combin. 
    Theory Ser. B 46 (1989), 142-153.
[7] D. Bryant, Cycle decompositions of complete graphs, 
    Surveys in Cambridge, 2007, London Math. Soc. Lecture 
    Note Ser. , 346, Univ. Press, Cambridge, 2007, 67-97.
[8] D. Bryant and B. Maenhaut, Decompositions of complete 
    graphs into triangles and Hamilton cycles, J. Combin. 
    Des. 12 (2004), 221-232.
[9] H. L. Fu and K. C. Hung, The Hamilton-Waterloo problem 
    for two even cycle factors, Taiwanese Journal of 
    Mathematics Vol. 12, No. 4, 933-940.
[10] H. L. Fu, C. A. Rodger, Four-cycle systems with two 
     regular leaves, Graphs and Combinatorics, 17 (2001), 
     457–461.
[11] K. Heinrich, P. Horak and A. Rosa, On Alspach’s 
     conjecture, Discrete Math. 77 (1989), 97-121.
[12] A. Rosa, Alspach’s conjecture is true for n≤10, 
     Mathematical Reports, McMaster University.
[13] M. Šajna, Cycle Decompositions III: Complete graphs 
     and fixed length cycles. J.
   Combin. Des. 10 (2002), 27-78.