本文藉由矩形棋盤著色探討完全二分圖K_{m,n}由兩種顏色任意塗邊，使得此兩色著邊之完全二分圖$K_{m,n}$會包含單色子圖K_{s,2}、K_{s,3}與K_{s,t} （s大於或等於 2），我們將討論參數n與s之間滿必須滿足何種關係。 本文也將介紹處理棋盤著色問題的一般方法與技巧，以及透過棋盤如何將棋盤問題轉化為圖論問題，並且將它推廣。

在這篇論文中，我們針對2-連通的最大外平面圖而且是二分圖的圖形，討論 其容忍表示法，並找到它的所有禁止子圖H1、H2、H3、H4。 / In this thesis, we prove a 2-connected graph G which is maximal outerplanar and bipartite is a tolerance graph if and only if there is no induced subgraphs H1; H2; H3 and H4 of G.

中文摘要 有限離散條件分配相容性問題可依相容性檢驗、唯一性檢驗以及找出所有的聯合機率分配三層次來討論。目前的文獻資料有幾種研究方法，本文僅分析、比較其中的比值矩陣法和圖形法。 二維中，我們發現簡化二分圖的分支與IBD矩陣中的對角塊狀矩陣有密切的對應關係。在檢驗相容性時，圖形法只需檢驗簡化二分圖中的每個分支，正如同比值矩陣法只需檢驗IBD矩陣中的每一個對角塊狀矩陣即可。在檢驗唯一性時，圖形法只需檢驗簡化二分圖中的分支數是否唯一，正如同比值矩陣法只需檢驗IBD矩陣中的對角塊狀數是否唯一即可。在求所有的聯合機率分配時，運用比值矩陣法可推算出所有的聯合機率分配，但是圖形法則無法求出。 三維中，本文提出了修正比值矩陣法，將比值數組按照某種索引方式在平面上有規則地呈現，可降低所需處理矩陣的大小。此外，我們也發現修正比值矩陣中的橫直縱迴路和簡化二分圖中的迴路有對應的關係，因此可觀察出兩種方法所獲致某些結論的關聯性。在檢驗唯一性時，圖形法是檢驗簡化二分圖中的分支數是否唯一，而修正比值矩陣法是檢驗兩個修正比值矩陣是否分別有唯一的GROPE矩陣。修正比值矩陣法可推算出所有的聯合機率分配。 圖形法可用於任何維度中，修正比值矩陣法也可推廣到任何維度中，但在應用上，修正比值矩陣法比圖形法較為可行。 / The issue of the compatibility of finite discrete conditional distributions could be discussed hierarchically according to the compatibility, the uniqueness, and finding all possible joint probability distributions. There are several published methods, but only the Ratio Matrix Method and the Graphical Method are analyzed and compared in this thesis. In bivariate case, a close correspondence between the components of the reduced bipartite graph and the diagonal block matrices of the IBD matrix can be found. When we examine the compatibility, just as simply each diagonal block matrix of the IBD matrix needs to be examined using the Ratio Matrix Method, so does each component of the reduced bipartite graph using the Graphical Method. When we examine the uniqueness, just as whether the number of the diagonal blocks of the IBD matrix is unique needs to be examined, so does the number of the components of the reduced bipartite graph. The Ratio Matrix Method can provide all possible joint probability distributions, but the Graphical Method cannot. In trivariate case, this thesis proposes a Revised Ratio Matrix Method, in which we can present the ratio array regularly in the plane according to the index and reduce the corresponding matrix size. It is also found that each circuit in the revised ratio matrix corresponds to a circuit in the reduced bipartite graph. Therefore, the relation between the results of the two methods can be observed. When we examine the uniqueness with the Graphical Method, we examine whether the number of the components in the reduced bipartite graph is unique. But with the Revised Ratio Matrix Method, we examine whether each revised ratio matrix has a unique GROPE matrix. All possible joint probability distributions can be derived through the Revised Ratio Matrix Method. The Graphical Method can be applied to the higher dimensional cases, so can the Revised Ratio Matrix Method. But the Revised Ratio Matrix Method is more feasible than the Graphical Method in application.