Oberwolfach 問題及其推廣的探討

淡江大學機構典藏 > 理學院 > 數學學系暨研究所 > 研究報告 > Item 987654321/102964

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题名:	Oberwolfach 問題及其推廣的探討
其它题名:	The Study of Oberwolfach Problem and Its Generalization
作者:	高金美
贡献者:	淡江大學數學學系
日期:	2012-08
上传时间:	2015-05-12 15:30:39 (UTC+8)
摘要:	一個圖中任兩點都相鄰，稱之為完全圖， m 個點的完全圖記為Km 。每點度數均為 2 的n 個點連通圖稱為n-迴圈，記為Cn。一個2-因子是指Km 中的一個含有m 個點的子圖且每一點的度數都為2。圖的分割是指將圖G 分成數個邊相異的子圖G1, G2, …, Gt, 且 G 的每一邊會落在唯一的一個子圖Gi 上。Oberwolfach problem 是探討一個含有2k+1 個點的完全圖K2k+1 是否能分割成一些邊都相異的2-因子，且每個2-因子中的迴圈均對應相同呢? Hamilton-Waterloo problem 是Oberwolfach problem 的推廣。 Hamilton-Waterloo problem 是探討一個含有2k+1 個點的完全圖K2k+1 是否能分割成一些邊都相異的2-因子，且此2-因子中的迴圈均相同，但是其中共含有兩種2-因子呢? 在此計畫中，我們將針對尚未解決的 Oberwolfach problem, Hamilton-Waterloo problem 及其推廣作一些研究，希望能找到完全圖分割成所想要的兩種或三種2-因子的充分必要條件。 A graph in which any two distinct vertices are adjacent is called a complete graph, Km. An n-cycle Cn is a connected graph with n vertices and all have degree 2. A 2-factor of Km is a subgraph of Km with m vertices and the degree of each vertex is 2. A decomposition of a graph is a collection of edge-disjoint subgraphs G1, G2, …, Gn of G such that every edge of G belongs to exactly one Gi. The Oberwolfach problem is the existence of the decomposition of K2k+1 into 2-factors such that each 2-factors contains the same cycles. The Hamilton-Waterloo problem is the existence of the decomposition of K2k+1 into two kinds of 2-factors such that each kind 2-factor contains exactly only one kind cycle . In this project, we will find some solutions of Oberwolfach problem, Hamilton-Waterloo problem and their generalization. Hope that we can find the necessary and sufficient conditions of the decomposition of complete graph Km into two kinds or three kinds 2-factors.
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