一個圖的點集合如果可分為兩個互不相交的非空集合，且同一集合中的任意兩點皆不相連，則稱此圖為ㄧ個二分圖，若在不同的集合中的任意兩點都有邊相連，則稱此圖為完全二分圖，並用Km,n來表示。
    ㄧ個完全二分圖Km,n能分割成一些子圖是指Km,n中的邊可以分成為一些邊均相異的子圖，且這些子圖的點集合的聯集為Km,n的點集合，邊集合的聯集為Km,n的邊集合。
    在本篇論文中，首先我們證明當2<m<12、2<n<10 時，對於每個非負整數p、q、r、s，只要4p+6q+8r+10s = 4mn，K2m,2n都可以分割成p個4-迴圈、q個6-迴圈、r個8-迴圈、s個10-迴圈。接著利用這些結果我們獲得對於m,n>2及非負整數p、q、r、s，若4p+6q+8r+10s = 4mn，則我們可將完全二分圖K2m,2n分割成p個4-迴圈、q個6-迴圈、r個8-迴圈、s個10-迴圈。

A graph is called a bipartite graph if the vertex set of the graph can be partitioned into two disjoint nonempty sets, and any two vertices in the same set are not adjacency. Moreover, if any two vertices in the different set are adjacency, then this bipartite graph is called a complete bipartite graph, and denoted by Km,n.
    A complete bipartite graph Km,n can be decomposed into some subgraphs if Km,n can be partitioned into edge-disjoint subgraphs, such that the union of vertex sets of these subgraphs is the vertex set of Km,n, and the union of edge sets is the edge set of Km,n.
    In this thesis, we show that when 2<m<12、2<n<10, if 4p+6q+8r+10s = 4mn, for all non-negative integers p, q, r, s, then K2m,2n can be decomposed into  p 4-cycle, q 6-cycle, r 8-cycle, and s 10-cycle.By using above results, we obtain the following result. For all m,n>2 and non-negative integets p, q, r, s, if 4p+6q+8r+10s = 4mn, then K2m,2n can be decomposed into  p 4-cycle, q 6-cycle, r 8-cycle, or s 10-cycle.

目     錄
第一章   簡介  ……………………………  1
第二章   預備知識  ………………………  3
第三章   分割成4、6、8、10迴圈 ………  6
参考書目 …………………………………… 25
附錄一   …………………………………… 26
附錄二   …………………………………… 31
附錄三   …………………………………… 43

[1] P. Adams, D. E. Bryant, and A. Khodkar, (1998) (3,5)-cycle decompositions, J. Combin. Designs, 6, 91-110.
[2] B. Alspach and H.Gavlas, Cycle decompositions of Kn and Km,n - I,
 J. Combin. Theory Ser. B (2001), 77-99.
[3] B. Alspach and S. Marshall, (1994) Even cycle decompositions of complete graphs minus a 1-factor, J. Combin. Designs 2, 441-458.
[4] B. Alspach, (1980) Research Problem 3, Discrete Math. 36, 333-334.
[5] D. E. Bryant, A. Khodkar and H. L. Fu, (m,n)-cycle systems. J. Statist. Planning ang Inference 74 (1998), 365-370.
[6] C. C. Chou, C. M. Fu and W. C. Huang, (2000) Decomposition of 
2Km,n into short cycles, Utilitas Mathematica 58, pp. 3-10
[7] Dorit Dor, Michael Tarsi, (1997) Graph Decomposition is NP-Complete: A Complete Proof of Holyer's Conjecture, SIAM Journal on Computing, Volume 26, Number 4, 1166-1187
[8] C. M. Fu and W. C. Huang, (1998) Decompositions of Km,n into 4-cycles and 8-cycles, Tamkang Journal of Math. Vol. 29, No. 1, 69-72
[9] R.Haggkvis, (1985) A lemma on cycle decomposition, Ann, Discrete Math., 27, 227-232.
[10] K. Heinrich, P. Horak and A. Rosa, (1989) On Alspach’s conjecture. Discrete Math., 77, 97-121.
[11] Y. S. Huang, (2003) The study of decomposing K2m,2n and  K2m+1,2n+1 F into cycles of length 4 or 2t.
[12] A. Rosa, (1989) Alspach’s conjecture is true for  , Mathematical reports, McMaster University.
[13] D. Sotteau, (1981) Decomposition of Km,n (K*m,n) into cycles(circuit) of length  , J. Combin. Theory Ser. B, 30, 75-81.