arXiv subjects

Ruy Fabila-Monroy

Publications and source records attributed to Ruy Fabila-Monroy.

50 records · Page 3Linked to original sources

arXiv2014

$(n,m)$-Fold Covers of Spheres

A well known consequence of the Borsuk-Ulam theorem is that if the $d$-dimensional sphere $S^d$ is covered with less than $d+2$ open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the $d$-dimensional sphere $n$ times, with the additional property that the northern hemisphere is covered $m > n$ times. We prove that if the open northern hemisphere is to be covered $m$ times then at least $ \lceil \frac{d-1}{2} \rceil+n+m$ and at most $d+n+m$ sets are needed. For the case of $n=1$ and $d \ge 2$, this number is equal to $d+2$ if $m \le \lfloor \frac{d}{2} \rfloor + 1$ and equal to $ \lfloor \frac{d-1}{2} \rfloor + 2 +m$ if $m > \lfloor \frac{d}{2} \rfloor + 1$. If the closed northern hemisphere is to be covered $m$ times then $d+2m-1$ sets are needed, this number is also sufficient. We also present results on a related problem of independent interest. We prove that if $S^d$ is covered $n$ times with open sets, not containing a pair of antipodal points, then there exists a point that is covered at least $ \lceil \frac{d}{2} \rceil +n$ times. Furthermore, we show that there are covers in which no point is covered more than $n+d$ times.

math.CO↗

arXiv2014

On $k$-Gons and $k$-Holes in Point Sets

We consider a variation of the classical Erdős-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position.

cs.DM↗

arXiv2014

Computational search of small point sets with small rectilinear crossing number

Let $\crs(K_n)$ be the minimum number of crossings over all rectilinear drawings of the complete graph on $n$ vertices on the plane. In this paper we prove that $\crs(K_n) < 0.380473\binom{n}{4}+Θ(n^3)$; improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of $K_n$ for small values of $n$, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The "small" sets where found using a simple heuristic detailed in this paper.

math.CO↗

arXiv2013

Drawing the double circle on a grid of minimum size

In 1926, Jarník introduced the problem of drawing a convex $n$-gon with vertices having integer coordinates. He constructed such a drawing in the grid $[1,c\cdot n^{3/2}]^2$ for some constant $c>0$, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem for drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n)-time algorithm to construct such a point set.

cs.CG↗

arXiv2012

Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $Ω(n^{d-k+1+2^{-d}})$ and strengthen this to $Ω(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+Ω(\log n)$ simplices.

math.CO↗

arXiv2012

Hamiltonian Tetrahedralizations with Steiner Points

Let $S$ be a set of $n$ points in 3-dimensional space. A tetrahedralization $\mathcal{T}$ of $S$ is a set of interior disjoint tetrahedra with vertices on $S$, not containing points of $S$ in their interior, and such that their union is the convex hull of $S$. Given $\mathcal{T}$, $D_\mathcal{T}$ is defined as the graph having as vertex set the tetrahedra of $\mathcal{T}$, two of which are adjacent if they share a face. We say that $\mathcal{T}$ is Hamiltonian if $D_\mathcal{T}$ has a Hamiltonian path. Let $m$ be the number of convex hull vertices of $S$. We prove that by adding at most $\lfloor \frac{m-2}{2} \rfloor$ Steiner points to interior of the convex hull of $S$, we can obtain a point set that admits a Hamiltonian tetrahedralization. An $O(m^{3/2}) + O(n \log n)$ time algorithm to obtain these points is given. We also show that all point sets with at most 20 convex hull points admit a Hamiltonian tetrahedralization without the addition of any Steiner points. Finally we exhibit a set of 84 points that does not admit a Hamiltonian tetrahedralization in which all tetrahedra share a vertex.

cs.CG↗

arXiv2012

The Erdős-Sós Conjecture for Geometric Graphs

Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that $(1/2)\frac{n^2}{k-1}-\frac{n}{2}\le f(n,k) \le 2 \frac{n(n-2)}{k-2}$. For the case when $k=n$, we show that $2 \le f(n,n) \le 3$. For the case when $k=n$ and $G$ is a geometric graph on a set of points in convex position, we show that at least three edges must be removed.

math.CO↗

arXiv2012

On the number of radial orderings of planar point sets

Given a set $S$ of $n$ points in the plane, a \emph{radial ordering} of $S$ with respect to a point $p$ (not in $S$) is a clockwise circular ordering of the elements in $S$ by angle around $p$. If $S$ is two-colored, a \emph{colored radial ordering} is a radial ordering of $S$ in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of $S$. We assume a strong general position on $S$, not three points are collinear and not three lines---each passing through a pair of points in $S$---intersect in a point of $\R^2\setminus S$. In the colored case, $S$ is a set of $2n$ points partitioned into $n$ red and $n$ blue points, and $n$ is even. We prove that: the number of distinct radial orderings of $S$ is at most $O(n^4)$ and at least $Ω(n^3)$; the number of colored radial orderings of $S$ is at most $O(n^4)$ and at least $Ω(n)$; there exist sets of points with $Θ(n^4)$ colored radial orderings and sets of points with only $O(n^2)$ colored radial orderings.

cs.CG↗

arXiv2012

Decompositions of Complete Multipartite Graphs into Complete Graphs

Let $k\geq\ell\geq1$ and $n\geq 1$ be integers. Let $G(k,n)$ be the complete $k$-partite graph with $n$ vertices in each colour class. An $\ell$-decomposition of $G(k,n)$ is a set $X$ of copies of $K_k$ in $G(k,n)$ such that each copy of $K_\ell$ in $G(k,n)$ is a subgraph of exactly one copy of $K_k$ in $X$. This paper asks: when does $G(k,n)$ have an $\ell$-decomposition? The answer is well known for the $\ell=2$ case. In particular, $G(k,n)$ has a 2-decomposition if and only if there exists $k-2$ mutually orthogonal Latin squares of order $n$. For general $\ell$, we prove that $G(k,n)$ has an $\ell$-decomposition if and only if there are $k-\ell$ Latin cubes of dimension $\ell$ and order $n$, with an additional property that we call mutually invertible. This property is stronger than being mutually orthogonal. An $\ell$-decomposition of $G(k,n)$ is then constructed whenever no prime less than $k$ divides $n$.

math.CO↗

arXiv2012

On the Connectedness and Diameter of a Geometric Johnson Graph

Let $P$ be a set of $n$ points in general position in the plane. A subset $I$ of $P$ is called an \emph{island} if there exists a convex set $C$ such that $I = P \cap C$. In this paper we define the \emph{generalized island Johnson graph} of $P$ as the graph whose vertex consists of all islands of $P$ of cardinality $k$, two of which are adjacent if their intersection consists of exactly $l$ elements. We show that for large enough values of $n$, this graph is connected, and give upper and lower bounds on its diameter.

math.CO↗

arXiv2011

Colouring the Triangles Determined by a Point Set

Let P be a set of n points in general position in the plane. We study the chromatic number of the intersection graph of the open triangles determined by P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary P, the chromatic number is at most n^3/19.259+O(n^2).

math.CO↗

arXiv2011

The chromatic number of the convex segment disjointness graph

Let $P$ be a set of $n$ points in general and convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo et al. [\emph{CGTA}, 2005]. The previous best bounds are $\frac{3n}{4}\leqχ(D_n) <n-\sqrt{\frac{n}{2}}$ (ignoring lower order terms). In this paper we improve the lower bound to $χ(D_n)\geq n-\sqrt{2n}$, to conclude a near-tight bound on $χ(D_n)$.

math.CO↗

arXiv2011

Rooted $K_4$-Minors

Let $a,b,c,d$ be four vertices in a graph $G$. A \emph{$K_4$-minor rooted} at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

math.CO↗

arXiv2009

Token Graphs

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.

math.CO↗