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Francisco Santos

Publications and source records attributed to Francisco Santos.

At least 73 records · Page 4Linked to original sources
arXiv2011

Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The author recently showed in arXiv:1006.2814 that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases of $Q$.

math.CO↗
arXiv2011

Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases.

math.CO↗
arXiv2010

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

math.MG↗
arXiv2010

Sobre un contraejemplo a la conjetura de Hirsch

This is an expository paper (in Spanish) describing the origin and history of the Hirsch Conjecture about the maximum diameter of graphs of polytopes, and the ideas that led to the counter-example to it recently announced by the author in arxiv:1006.2814

math.CO↗
arXiv2010

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups

In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by the second author and D. Bochis, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 "full" cubic groups cannot have more than 25 facets. Here we study the remaining "quarter" cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets.

math.CO↗
arXiv2009

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.

math.CO↗
arXiv2009

Maximizing Maximal Angles for Plane Straight-Line Graphs

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $ϕ$-open if each vertex has an incident angle of size at least $ϕ$. In this paper we study the following type of question: What is the maximum angle $ϕ$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $ϕ$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.

cs.CG↗
arXiv2009

Graphs of Transportation Polytopes

This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$.

math.CO↗
arXiv2008

The Gromov Norm of the Product of Two Surfaces

We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.

math.GT↗
arXiv2008

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups

We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. The problem for non-cubic groups was studied in previous papers by D. Bochis and the second author (Discrete Comput. Geom. 25:3 (2001), 419-444, and Beitr. Algebra Geom., 47:1 (2006), 89-120). This paper deals with ''full'' cubic groups, while ''quarter'' cubic groups are left for a subsequent paper. Here, ''full'' and ''quarter'' refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston (math.MG/9911185, Beitr. Algebra Geom. 42.2 (2001), 475-507). Our main result in this paper is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups.

math.CO↗
arXiv2007

Lattice points in Minkowski sums

Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P.

math.CO↗
arXiv2007

Pseudo-Triangulations - a Survey

A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.

math.CO↗
arXiv2007

Multi-triangulations as complexes of star polygons

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.

math.CO↗
arXiv2007

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 $12^n n^{Θ(1)}$ pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.

math.CO↗
arXiv2007

Triple-loop networks with arbitrarily many minimum distance diagrams

Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form "L-shape''. In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.

math.CO↗
arXiv2006

Geometric bistellar flips. The setting, the context and a construction

We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected.

math.CO↗
arXiv2005

Combinatorial pseudo-Triangulations

We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of math.CO/0307347 which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain ``generalized Laman property'' is a necessary and sufficient condition for a CPT to be ``stretchable'', and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of combinatorial pseudo-triangulations on closed surfaces.

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