Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 95).

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains iii, 34 p. : ill. Includes abstract. Includes bibliographical references (p. 33-34).

Thesis (Ph. D.)--Lehigh University, 2004. / Includes vita. Includes bibliographical references (leaves 113-118).

Thesis (Ph. D.)--University of Louisville, 2009. / Department of Mathematics. Vita. "May 2009." Includes bibliographical references (leaves 67-69) and index.

Alon 和Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) 證明了如果一個圖是ε -非二部圖，那麼階數為Ỡ(1/ε) 的隨機導出于圖以很大概率是非二部圖。我們進一步猜想，這個導出子圖以很大概率是Ω(ε)-非二部圖。Gonen 和Ron (RANDOM 2007) 證明了當圖的最大度不超過O (εn )時猜想成立。我們將對更一般的情形給出證明，對於任意d，所有d 正則(或幾乎d 正則)的圖猜想成立。 / 假設猜想成立的情況下，我們證明二分屬性是可以被單側誤差在O(1/ε^c ) 時間內檢驗的，其中c 是一個嚴格小於2 的常數，而這個結果也改進了Alon 和Krivelevich 的檢驗算法。由於己知對二分屬性的非適應性的檢驗算法需要Ω(1 /ε²) 的複雜性(Bogdanov 和Trevisan ， CCC 2004) ，我們的結果也得出，假設猜想成立，適應性對檢驗二分屬性是有幫助的。 / 另外一個有很多屬性檢驗問題被廣泛研究的領域是布爾函數。對布爾函數單調性的檢驗也是屬性檢驗的經典問題。給定對布爾函數f: {0,1}{U+207F} → {0,1} 的訪問，在[18]中，證明了檢驗算法複雜性的下界是Ω(√n) 。另一方面，在[21]中，作者們構造了一個複雜性為O(n) 的算法。在本文中，我們刻畫一些單調布爾函數的本質，設計算法并分析其對於一些困難例子的效果。最近，在[14] 中， 一個類似的算法被證明是非適應性，單側誤差，複雜性為Ỡ (n⁵/⁶ ε⁻⁵/³) 的算法。 / Alon and Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ε-far from bipartite, then the subgraph induced by a random subset of Ỡ (1/ε) vertices is not bipartite with high probability. We conjecture that the induced subgraph is Ω(ε)-far from bipartite with high probability. Gonen and Ron (RANDOM 2007) proved this conjecture in the case when the degrees of all vertices are at most O(εn). We give a more general proof that works for any d-regular (or almost d-regular) graph for arbitrary degree d. / Assuming this conjecture, we prove that bipartiteness is testable with one-sided error in time O(1=ε{U+1D9C}), where c is a constant strictly smaller than two, improving upon the tester of Alon and Krivelevich. As it is known that non-adaptive testers for bipartiteness require Ω(1/ε²) queries (Bogdanov and Trevisan, CCC2004), our result shows, assuming the conjecture, that adaptivity helps in testing bipartiteness. / The other area in which various properties have been well studied is boolean function. Testing monotonicity of Boolean functions is a classical question in property testing. Given oracle access to a Boolean function f : {0, 1}{U+207F} →{0, 1}, in [18], it is shown a lower bound of testing is Ω(√n). On the other hand, in [21], the authors introduced an algorithm to test monotonicity using O(n) queries. In this paper, we characterize some nature of monotone functions, design a tester and analyze the performance on some generalizations of the hard case. Recently, in [14], a similar tester is shown to be a non-adaptive, one-sided error tester making Ỡ (n⁵/⁶ ε⁻⁵/³) queries. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Fan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 76-79). / Abstracts also in Chinese. / Abstract --- p.i / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Property Testing --- p.1 / Chapter 1.2 --- Testing Bipartiteness --- p.4 / Chapter 1.3 --- Testing Monotonicity --- p.7 / Chapter 2 --- Testing Bipartiteness --- p.11 / Chapter 2.1 --- Background --- p.11 / Chapter 2.1.1 --- Our result --- p.11 / Chapter 2.1.2 --- The algorithms of Gonen and Ron --- p.13 / Chapter 2.1.3 --- Our proof --- p.16 / Chapter 2.1.4 --- Notation --- p.19 / Chapter 2.2 --- Splitting the vertices by degree --- p.19 / Chapter 2.3 --- The algorithm for high degree vertices --- p.20 / Chapter 2.4 --- Eliminating the high degree vertices --- p.22 / Chapter 2.5 --- From an XOR game to a bipartite graph --- p.33 / Chapter 2.6 --- Proof of the main theorem --- p.35 / Chapter 2.7 --- Proof of the conjecture for regular graphs --- p.37 / Chapter 3 --- Testing Monotonicity --- p.40 / Chapter 3.1 --- Towards an improved tester --- p.40 / Chapter 3.1.1 --- Properties of Distribution D --- p.42 / Chapter 3.1.2 --- An Alternative Representation of D --- p.46 / Chapter 3.1.3 --- Performance of D on Decreasing Functions --- p.51 / Chapter 3.1.4 --- Functions Containing Ω(2{U+207F}) Disjoint Violating Edges --- p.54 / Chapter 3.2 --- A o(n) Monotonicity Tester [14] and Some Improvements --- p.62 / Chapter 3.2.1 --- A o(n) Monotonicity Tester [14] --- p.62 / Chapter 3.2.2 --- An Alternative Proof of Theorem 3.2.2 --- p.65 / Chapter 4 --- Other Related Results --- p.67 / Chapter 4.1 --- Short Odd Cycles in Graphs that are Far From Bipartiteness --- p.67 / Chapter 4.2 --- Fourier Analysis on Boolean Functions --- p.69 / Bibliography --- p.76

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2009. / Includes bibliographical references.

<p>This thesis consists of two papers.</p><p>The first paper is a study of the structure of the k-assignment polytope, whose vertices are the <em>m x n</em> (0; 1)-matrices with exactly <em>k</em> 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. Two equivalent representations of the faces are given, one as (0; 1)-matrices and one as ear decompositions of bipartite graphs. These tools are used to describe properties of the polytope, especially a complete description of the cover relation in the face lattice of the polytope and an exact expression for the diameter.</p><p>The second paper studies the edge-product space <em>Є(X)</em> for trees on <em>X</em>. This space is generated by the set of edge-weighted finite trees on <em>X</em>, and arises by multiplying the weights of edges on paths in trees. These spaces are closely connected to tree-indexed Markov processes in molecular evolutionary biology. It is known that <em>Є(X)</em> has a natural <em>CW</em>-complex structure, and a combinatorial description of the associated face poset exists which is a poset <em>S(X)</em> of <em>X</em>-forests. In this paper it is shown that the edge-product space is a regular cell complex. One important part in showing that is to conclude that all intervals <em>[Ô, Г], Г </em>Є<em> S(X),</em> have recursive coatom orderings.</p> / Report code: LiU-TEK-LIC-2004:46.

Algorithms will be domonstrated for how to embed complete bipartite graphs onto 2xn type grids, where the imimum grid cutwidth is attained.

This thesis consists of two papers. The first paper is a study of the structure of the k-assignment polytope, whose vertices are the m x n (0; 1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. Two equivalent representations of the faces are given, one as (0; 1)-matrices and one as ear decompositions of bipartite graphs. These tools are used to describe properties of the polytope, especially a complete description of the cover relation in the face lattice of the polytope and an exact expression for the diameter. The second paper studies the edge-product space Є(X) for trees on X. This space is generated by the set of edge-weighted finite trees on X, and arises by multiplying the weights of edges on paths in trees. These spaces are closely connected to tree-indexed Markov processes in molecular evolutionary biology. It is known that Є(X) has a natural CW-complex structure, and a combinatorial description of the associated face poset exists which is a poset S(X) of X-forests. In this paper it is shown that the edge-product space is a regular cell complex. One important part in showing that is to conclude that all intervals [Ô, Г], Г Є S(X), have recursive coatom orderings.

Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3. The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.