| 摘要: | 在這篇論文中，我們主要是探討一個完全二分圖扣除一個最大迴圈後是否仍可分割成C2k。
我們先證明了K2n,2n－C4n可以完全被分割成C2k，k＝2,3,4,5，只要n≧k。進而證得，若k為偶數時，只要n≡1(mod k)，K2n,2n－C4n都可分割成C2k。
接著，我們討論在K2n+1,2n+1中的圈分割，為使圖中點的度數為偶數，我們由K2n+1,2n+1中扣除一個最大迴圈及一因子的聯集G2n+1，我們先證明了K2n+1,2n+1－G2n+1可以完全被分割成C2k，k＝2,3,4,5，只要n≧k。進而證得，若 k 為偶數時，只要n≡1(mod k)，K2n+1,2n+1－G2n+1都可分割成C2k。
In this thesis, we try to decompose a graph which is a complete bipartite graph taking away a Hamiltonian cycle into cycles of length 2k.
First, we show that K2n,2n－C4n can be decomposed into C2k, k＝2, 3, 4, 5, for all n≧k. Then we extend the results to all even k, if n≡1(mod k), K2n,2n－C4n can be decomposed into C2k.
After that, we consider the decomposition of K2n+1,2n+1－G2n+1, where G2n+1 is the union of C4n+2 and a 1-factor of K2n+1,2n+1, can be decomposed into C2k as k＝2, 3, 4, 5, for all n≧k. Then we extend the results to all even k, if n≡1(mod k), K2n+1,2n+1－G2n+1 can be decomposed into C2k. |
