We generalize bistellar operations (often called flips) on simplicial manifolds to a notion of general flips on PL-spheres. We provide methods for computing the cd-index of these general flips, which is the change in the cd-index of any sphere to which the flip is applied. We provide formulas and relations among flips in certain classes, paying special attention to the classic case of bistellar flips. We also consider questions of "flip-connecticity", that is, we show that any two polytopes in certain classes can be connected via a sequence of flips in an appropriate class.

Includes bibliography. / 281 p ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--Dept. of Pure Mathematics, University of Adelaide, 1978

In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory. We first consider tilings of $\mathbb{Z}^n$. In this setting a tile $T$ is just a finite subset of $\mathbb{Z}^n$. We say that $T$ tiles $\mathbb{Z}^n$ if the latter set admits a partition into isometric copies of $T$. Chalcraft observed that there exist $T$ that do not tile $\mathbb{Z}^n$ but tile $\mathbb{Z}^{d}$ for some $d > n$. He conjectured that such $d$ exists for any given tile. We prove this conjecture in Chapter 2. In Chapter 3 we prove a conjecture of Lonc, stating that for any poset $P$ of size a power of $2$, if $P$ has a greatest and a least element, then there is a positive integer $k$ such that $[2]^k$ can be partitioned into copies of $P$. The third tiling problem is about vertex-partitions of the hypercube graph $Q_n$. Offner asked: if $G$ is a subgraph of $Q_n$ such $|G|$ is a power of $2$, must $V(Q_d)$, for some $d$, admit a partition into isomorphic copies of $G$? In Chapter 4 we answer this question in the affirmative. We follow up with a question in combinatorial geometry. A line in a planar set $P$ is a maximal collinear subset of $P$. P\'or and Wood considered colourings of finite $P$ without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that $|P|$ is large. They conjectured that for all $k,l \ge 2$ there exists an $n \ge 2$ such that if $|P| \ge n$ and $P$ does not contain a line of cardinality larger than $l$, then every colouring of $P$ with $k$ colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case $k=l=3$. We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with $n$ vertices and $m$ edges? For sufficiently large $n$ we prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6. Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an $r$-uniform graph $H$ is to assign for each of its edges one of the $r!$ possible orderings of its elements. Then, for any $p$-set of vertices $A$ and any $p$-set of indices $I \subset [r]$, we define the $I$-degree of $A$ to be the number of edges containing vertices $A$ in precisely the positions labelled by $I$. Caro and Hansberg were interested in determining whether a given $r$-uniform hypergraph admits an orientation where every set of $p$ vertices has some $I$-degree equal to $0$. They conjectured that a certain Hall-type condition is sufficient. We show that this is true for $r$ large, but false in general.

A configuration is a finite set of points in the plane. Two configurations have the same order type if there exists a bijection between them that preserves the orientation of every ordered triple. A configuration A contains a copy of a configuration B if some subset of A has the same order type of B and we denote this by B \\subset A. For a configuration B and a integer N, the extremal number ex(N,B)=max{|A|: B ot \\subset A \\subset [N]^2} is the maximum size of a subset of [N]^2 without a copy of B. We give an upper bound for general and convex cases. A random N-set is a set obtained by randomly choosing N points uniformly and independently in the unit square. A configuration is n-universal if contains all order types in general position of size n. We obtain the threshold for the n-universal property up to a log log factor, that is, we obtain integers N_0 and N_1 with log log N_1=O(log log N_0) such that if N >> N_1 (N << N_0), then a random N-set is n-universal with probability tending to 1 (tending to 0). We also determine a bound for the probability of obtaining a random set without a copy of a fixed configuration. / Uma configuração é um conjunto finito de pontos no plano. Duas configurações possuem o mesmo o-tipo se existe uma bijeção entre elas que preserva a orientação de toda tripla orientada. Uma configuração A contém uma cópia da configuração B se algum subconjunto de A possui o mesmo o-tipo que B e denotamos este fato por B \\subset A. Para uma configuração B e um inteiro N, o número extremal ex(N,B)=max{|A|: B ot \\subset A \\subset [N]^2} é o maior tamanho de um subconjunto de [N]^2 sem uma cópia de B. Neste trabalho, determinamos cotas superiores para o caso geral e para o caso convexo. Um N-conjunto aleatório é um conjunto obtido escolhendo N pontos uniformemente e independentemente ao acaso do quadrado unitário. Uma configuração é n-universal se contém todos os o-tipos de tamanho n. Determinamos o limiar da propriedade de um N-conjunto aleatório ser n-universal a menos de erros da ordem de log log, isto é, determinamos inteiros N_0 e N_1 com log log N_0=O(log log N_1) tais que se N >> N_1 (N << N_0), então o N-conjunto aleatório é n-universal com probabilidade tendendo a 1 (tendendo a 0). Também obtivemos cotas para a probabilidade de um conjunto aleatório não possuir determinado o-tipo.

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Trotter, William T.; Committee Member: Duke, Richard A.; Committee Member: Randall, Dana; Committee Member: Thomas, Robin; Committee Member: Yu, Xingxing

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Orientadores: Cid Carvalho de Souza, Pedro Jussieu de Rezende / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-17T02:29:56Z (GMT). No. of bitstreams: 1 Couto_MarceloCastilho_M.pdf: 3682547 bytes, checksum: 899151df78f8e6950ce90ea8215ded91 (MD5) Previous issue date: 2010 / Resumo: Nesta dissertação, faz-se um amplo estudo multidisciplinar sobre duas variantes de um problema geométrico NP-DIFÍCIL, o Problema da Galeria de Arte, que é analisado tanto pela ótica geométrica quanto combinatória. O objetivo consiste em minimizar o número de guardas suficientes para cobrir todo o interior de uma galeria de arte, representada por um polígono simples. Dentre as muitas variantes desse problema, o foco foi dado àquela onde os guardas são estacionários e restritos aos vértices do polígono, ortogonal ou simples, sem obstáculos. Propõe-se neste trabalho um algoritmo iterativo exato que é capaz de resolver ambas as variantes do problema. Nesse algoritmo, o problema original é discretizado, reduzido a um problema combinatório, o Problema da Cobertura de Conjuntos, e modelado por programação linear inteira. A redução entre os problemas que assegura a corretude do algoritmo e as provas de exatidão e convergência para uma solução ótima do problema original são detalhadas. Apresenta-se também uma extensa experimentação de uma implementação desse algoritmo com o intuito de validar seu uso prático e analisar as várias estratégias propostas aqui para a discretização inicial da galeria. São dados resultados para instâncias com até 2500 vértices, mais de dez vezes o tamanho das maiores instâncias resolvidas exatamente na literatura pré-existente. Mostra-se que o número de iterações executadas pelo algoritmo está extremamente relacionada com o modo como a galeria é inicialmente discretizada. Considerando a estratégia de discretização com o melhor desempenho geral, tem-se que, na prática, o algoritmo converge para uma solução ótima para o problema original em um baixo tempo computacional e em um número de iterações que é ordens de grandeza aquém do limite teórico resultante da análise de pior caso / Abstract: In this dissertation, a broad multidisciplinary study is done on two variants of a geometrical NP-HARD problem, the Art Gallery Problem, which is approached both from geometrical and combinatorics perspectives. The goal is to minimize the number of guards sufficient to cover the interior of an art gallery whose boundary is represented by a simple polygon. Among the many variants of the problem, the focus was on one where the guards are stationary and are restricted to vertices of the polygon, orthogonal or simple, without holes. We propose an iterative exact algorithm to solve both variants of the problem. In this algorithm, the original problem is discretized, reduced to a combinatorial problem, the Set Cover Problem, and modeled as an integer linear program. The reduction between the problems, which ensures the correctness of the algorithm, and the proofs of its exactness and convergence to an optimal solution are detailed. We also present an extensive experimentation of an implementation of this algorithm in order to validate its practical use and analyze the various strategies proposed here for the initial discretization of the gallery. Results are given for instances with up to 2500 vertices, more than ten times the size of the largest instances solved to optimality in prior literature. It is shown that the number of iterations performed by the algorithm is highly related to how the gallery is initially discretized. Considering the discretization strategy with the best performance in practice, the algorithm converges to an optimal solution for the original problem in a low computation time and in a number of iterations that is orders of magnitude below the theoretical bound arising from the worst case analysis / Mestrado / Geometria Computacional e Otimização Combinatória / Mestre em Ciência da Computação

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This thesis is a contribution to a classical problem in computational and combinatorial geometry: the study of the k-sets of a set V of n points in the plane. First we introduce the notion of convex inclusion chain that is an ordering of the points of V such that no point is inside the convex hull of the points that precede it. Every k-set of an initial sub-sequence of the chain is called a k-set of the chain. We prove that the number of these k-sets is an invariant of V and is equal to the number of regions in the order-k Voronoi diagram of V. We then deduce an online algorithm for the construction of the k-sets of the vertices of a simple polygonal line such that every vertex of this line is outside the convex hull of all its preceding vertices on the line. If c is the total number of k-sets built with this algorithm, the complexity of our algorithm is in O(n log n + c log^2k) and is equal, per constructed k-set, to the complexity of the best algorithm known. Afterward, we prove that the classical divide and conquer algorithmic method can be adapted to the construction of the k-sets of V. The algorithm has a complexity of O(n log n + c log^2k log(n/k)), where c is the maximum number of k-sets of a set of n points. We finally prove that the centers of gravity of the k-sets of a convex inclusion chain are the vertices of a triangulation belonging to the family of so-called centroid triangulations. This family notably contains the dual of the order-k Voronoi diagram. We give an algorithm that builds particular centroid triangulations in O(n log n + k(n- k) log^2 k) time, which is more efficient than all the currently known algorithms.

The thesis consist of three independent parts. In the first part, we investigate the height sequence of an element of a partially ordered set. Let $x$ be an element of the partially ordered set $P$. Then $h_i(x)$ is the number of linear extensions of $P$ in which $x$ is in the $i$th lowest position. The sequence ${h_i(x)}$ is called the height sequence of $x$ in $P$. Stanley proved in 1981 that the height sequence is log-concave, but no combinatorial proof has been found, and Stanley's proof does not reveal anything about the deeper structure of the height sequence. In this part of the thesis, we provide a combinatorial proof of a special case of Stanley's theorem. The proof of the inequality uses the Ahlswede--Daykin Four Functions Theorem. In the second part, we study two classes of segment orders introduced by Shahrokhi. Both classes are natural generalizations of interval containment orders and interval orders. We prove several properties of the classes, and inspired by the observation, that the classes seem to be very similar, we attempt to find out if they actually contain the same partially ordered sets. We prove that the question is equivalent to a stretchability question involving certain sets of pseudoline arrangements. We also prove several facts about continuous universal functions that would transfer segment orders of the first kind into segments orders of the second kind. In the third part, we consider the lattice whose elements are the subsets of ${1,2,ldots,n}$. Trotter and Felsner asked whether this subset lattice always contains a monotone Hamiltonian path. We make progress toward answering this question by constructing a path for all $n$ that satisfies the monotone properties and covers every set of size at most $3$. This portion of thesis represents joint work with David M.~Howard.