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    Title: 完全二分圖上二分解集之4迴圈的個數
    The number of 4-cycles in 2-factorizations of Kn,n
    Authors: 徐育鋒
    Contributors: 淡江大學數學學系
    Keywords: 完全二分圖;二分解集;迴圈個數;complete bipartite graph;2-factorization;number of cycles
    Date: 2005
    Issue Date: 2016-09-30 10:09:54 (UTC+8)
    Abstract: n階完全二分圖(Kn,n)的二分解為其2-正則衍生子圖。Kn,n的二分解集為其邊集合之分割使其成為二分解。當然,Kn,n的二分解集存在若且為若n為偶數且二分解集中4-迴圈的最大個數為n2/4。

    當n為奇數時,Kn,n為不可二分解的。因此,Kn,n的二分解集修正為Kn,n\F的二分解集,其中F為Kn,n的一分解,且二分解集中4-迴圈的最大個數為(n-1)(n-3)/4。

    我們在第3節討論,當n ≡ 1 (mod 2)時,計算Kn,n\F的二分解集之4-迴圈個數的建構方式。並且在第4節討論,當n ≡ 0 (mod 2)時,計算Kn,n的二分解集中所包含之4-迴圈個數的遞迴方法。

    令Q(n)為所有x的集合,使得Kn,n的二分解集包含x個4-迴圈。
    最後,我們可以得到結果如下:
    Q(4) = {0, 4}; Q(5) = {0, 1};Q(6) = {0, 1, 2, 3, 5, 9}; Q(7) = {0, 1, …, 6}; Q(8) = {0, 1, …, 10, 12, 16}; Q(9) = {0, 1, …, 12}; Q(11) ={0, 1, …,20};
    當n為偶數且n≧10,則Q(n) = {0, 1, …, n2/4-2, n2/4};
    當n為奇數且n≧13,則Q(n) = {0, 1, …, (n-1)(n-3)/4-6, (n-1)(n-3)/4-4,
    (n-1)(n-3)/4}。
    Appears in Collections:[數學學系暨研究所] 學位論文

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