On the Number of Pseudo-Triangulations of Certain Point Sets

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically $12^n n^{\Theta(1)}$ pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.Comment: 31 pages, 11 figures, 4 tables. Not much technical changes with respect to v1, except some proofs and statements are slightly more precise and some expositions more clear. This version has been accepted in J. Combin. Th. A. The increase in number of pages from v1 is mostly due to formatting the paper with "elsart.cls" for Elsevie

Notes on Shapes of Polyhedra

These are course notes I wrote for my Fall 2013 graduate topics course on geometric structures, taught at ICERM. The notes rework many of proofs in William P. Thurston's beautiful but hard-to-understand paper, "Shapes of Polyhedra". A number of people, both in and out of the class, found these notes very useful and so I decided to put them on the arXiv.Comment: This is a 21 page expository pape

Cambrian triangulations and their tropical realizations

This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical realizations. For any signature $\varepsilon \in \{\pm\}^n$, we consider a family of $\varepsilon$-trees in bijection with the triangulations of the $\varepsilon$-polygon. These $\varepsilon$-trees define a flag regular triangulation $\mathcal{T}^\varepsilon$ of the subpolytope $\operatorname{conv} \{(\mathbf{e}_{i_\bullet}, \mathbf{e}_{j_\circ}) \, | \, 0 \le i_\bullet < j_\circ \le n+1 \}$ of the product of simplices $\triangle_{\{0_\bullet, \dots, n_\bullet\}} \times \triangle_{\{1_\circ, \dots, (n+1)_\circ\}}$. The oriented dual graph of the triangulation $\mathcal{T}^\varepsilon$ is the Hasse diagram of the (type $A$) $\varepsilon$-Cambrian lattice of N. Reading. For any $I_\bullet \subseteq \{0_\bullet, \dots, n_\bullet\}$ and $J_\circ \subseteq \{1_\circ, \dots, (n+1)_\circ\}$, we consider the restriction $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ of the triangulation $\mathcal{T}^\varepsilon$ to the face $\triangle_{I_\bullet} \times \triangle_{J_\circ}$. Its dual graph is naturally interpreted as the increasing flip graph on certain $(\varepsilon, I_\bullet, J_\circ)$-trees, which is shown to be a lattice generalizing in particular the $\nu$-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ as a polyhedral complex induced by a tropical hyperplane arrangement.Comment: 16 pages, 11 figure

On the number of drawings of a combinatorial triangulations

Aquesta tesi explora la relació entre triangulacions combinatòries i geomètriques en geometria discreta i combinatòria. Per triangulacions combinatòries ens referim a grafs, mentre que per triangulacions geomètriques ens referim a dibuixos de grafs com a plans maximals amb línies rectes com a arestes sobre un conjunt de punts fixat al pla. Estudiem de quantes maneres es pot traçar una triangulació combinatòria com a triangulació geomètrica sobre un conjunt de punts donat. La nostra contribució central és demostrar que una triangulació combinatòria fixa amb n vèrtexs es pot dibuixar d'almenys 1,31^n maneres en un conjunt de n punts com a diferents triangulacions geomètriques. També analitzem els límits superiors i una versió acolorida daquest problema. L'enfocament suggerit pot ajudar a avançar en la resolució del problema obert per limitar el nombre de triangulacions geomètriques. A més, aprofundim en fonaments històrics, com el treball de Tutte, que proporciona el nombre exacte de triangulacions combinatòries amb n vèrtexs.Esta tesis explora la relación entre triangulaciones combinatorias y geométricas en geometría discreta y combinatoria. Con triangulaciones combinatorias nos referimos a grafos, mientras que con triangulaciones geométricas nos referimos a dibujos de grafos como planos maximales con líneas rectas como aristas sobre un conjunto de puntos fijado en el plano. Estudiamos de cuántas maneras se puede trazar una triangulación combinatoria como triangulación geométrica sobre un conjunto de puntos dado. Nuestra contribución central es demostrar que una triangulación combinatoria fija con n vértices se puede dibujar de al menos 1,31^n maneras en un conjunto de n puntos como diferentes triangulaciones geométricas. También analizamos los límites superiores y una versión coloreada de este problema. El enfoque sugerido puede ayudar a avanzar en la resolución del problema abierto para limitar el número de triangulaciones geométricas. Además, profundizamos en fundamentos históricos, como el trabajo de Tutte, que proporciona el número exacto de triangulaciones combinatorias con n vértices.This thesis explores the intricate relationship between combinatorial and geometric triangulations in discrete and combinatorial geometry. With combinatorial triangulations we refer to graphs, while with geometric triangulations we refer to maximal planar straight-line drawings on a point set in the plane. We study in how many ways can a combinatorial triangulation be drawn as geometric triangulation on a given point set. Our central contribution is proving that a fixed combinatorial triangulation with n vertices can be drawn in at least 1.31^n ways in a set of n points as different geometric triangulations. We also discuss upper bounds and a colored version of this problem. The suggested approach may help to advance the resolution of the open problem to bound the number of geometric triangulations. Additionally, we delve into historical foundations, such as Tutte's work, which provides the exact number of combinatorial triangulations with n vertices

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them