arXiv subjects

Ruy Fabila-Monroy

Publications and source records attributed to Ruy Fabila-Monroy.

At least 37 records · Page 2Linked to original sources

arXiv2020

Empty Rainbow Triangles in $k$-colored Point Sets

Let $S$ be a set of $n$ points in general position in the plane. Suppose that each point of $S$ has been assigned one of $k \ge 3$ possible colors and that there is the same number, $m$, of points of each color class. A polygon with vertices on $S$ is empty if it does not contain points of $S$ in its interior; and it is rainbow if all its vertices have different colors. Let $f(k,m)$ be the minimum number of empty rainbow triangles determined by $S$. In this paper we give tight asymptotic bounds for this function. Furthermore, we show that $S$ may not determine an empty rainbow quadrilateral for some arbitrarily large values of $k$ and $m$.

cs.CG↗

arXiv2020

The Chromatic Number of the Disjointness Graph of the Double Chain

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if $P$ is the point configuration known as the double chain, with $k$ points in the upper convex chain and $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor \sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that this number is sufficient.

math.CO↗

arXiv2020

A Note on Empty Balanced Tetrahedra in Two colored Point sets in $\mathbb{R}^3$

Let $S$ be a set of $n$ red and $n$ blue points in general position in $\mathbb{R}^3$. Let $τ$ be a tetrahedra with vertices on $S$. We say that $τ$ is \emph{empty} if it does not contain any point of $S$ in its interior. We say that $τ$ is \emph{balanced} if it contains two blue vertices and two red vertices. In this paper we show that $S$ spans $Ω(n^{5/2})$ empty balanced tetrahedra.

cs.CG↗

arXiv2020

Chirotopes of Random Points in Space are Realizable on a Small Integer Grid

We prove that with high probability, a uniform sample of $n$ points in a convex domain in $\mathbb{R}^d$ can be rounded to points on a grid of step size proportional to $1/n^{d+1+ε}$ without changing the underlying chirotope (oriented matroid). Therefore, chirotopes of random point sets can be encoded with $O(n\log n)$ bits. This is in stark contrast to the worst case, where the grid may be forced to have step size $1/2^{2^{Ω(n)}}$ even for $d=2$. This result is a high-dimensional generalization of previous results on order types of random planar point sets due to Fabila-Monroy and Huemer (2017) and Devillers, Duchon, Glisse, and Goaoc (2018).

cs.CG↗

arXiv2020

Non-crossing Monotone Paths and Binary Trees in Edge-ordered Complete Geometric Graphs

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$. We say that $P$ is monotone if it is increasing or decreasing. A rooted tree $T$ in an edge-ordered graph is called monotone if either every path from the root of to a leaf is increasing or every path from the root to a leaf is decreasing. Let $G$ be a graph. In a straight-line drawing $D$ of $G$, its vertices are drawn as different points in the plane and its edges are straight line segments. Let $\overlineα(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing path of length $\overlineα(G)$. Let $\overlineτ(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing complete binary tree of size $\overlineτ(G)$. In this paper we show that $\overline α(K_n) = Ω(\log\log n)$, $\overline α(K_n) = O(\log n)$, $\overline τ(K_n) = Ω(\log\log \log n)$ and $\overline τ(K_n) = O(\sqrt{n \log n})$.

math.CO↗

arXiv2019

On the 2-colored crossing number

Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show lower and upper bounds on the rectilinear 2-colored crossing number for the complete graph $K_n$. To obtain this result, we prove that asymptotic bounds can be derived from optimal and near-optimal instances with few vertices. We obtain such instances using a combination of heuristics and integer programming. Second, for any fixed drawing of $K_n$, we improve the bound on the ratio between its rectilinear 2-colored crossing number and its rectilinear crossing number.

cs.CG↗

arXiv2018

On the Number of Order Types in Integer Grids of Small Size

Let $\{p_1,\dots,p_n\}$ and $\{q_1,\dots,q_n\}$ be two sets of $n$ labeled points in general position in the plane. We say that these two point sets have the same order type if for every triple of indices $(i,j,k)$, $p_k$ is above the directed line from $p_i$ to $p_j$ if and only if $q_k$ is above the directed line from $q_i$ to $q_j$. In this paper we give the first non-trivial lower bounds on the number of different order types of $n$ points that can be realized in integer grids of polynomial

cs.CG↗

arXiv2018

Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.

cs.DM↗

arXiv2018

Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $α\in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil αr\rceil$ red points and exactly $\lceil αb \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil αr\rceil+\lceil αb\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$.

cs.CG↗

arXiv2018

The exact chromatic number of the convex segment disjointness graph

Let $P$ be a set of $n$ points in strictly 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, Dumitrescu, Hurtado, Noy, and Urrutia [2005] and then by Dujmović and Wood [2007]. Improving on their estimates, we prove the following exact formula: $$χ(D_n) = n - \left\lfloor \sqrt{2n + \tfrac{1}{4}} - \tfrac{1}{2}\right\rfloor.$$

math.CO↗

arXiv2018

Colored ray configurations

We study the cyclic color sequences induced at infinity by colored rays with apices being a given balanced finite bichromatic point set. We first study the case in which the rays are required to be pairwise disjoint. We derive a lower bound on the number of color sequences that can be realized from any such fixed point set and examine color sequences that can be realized regardless of the point set, exhibiting negative examples as well. We also provide a tight upper bound on the number of configurations that can be realized from a point set, and point sets for which there are asymptotically less configurations than that number. In addition, we provide algorithms to decide whether a color sequence is realizable from a given point set in a line or in general position. We address afterwards the variant of the problem where the rays are allowed to intersect. We prove that for some configurations and point sets, the number of ray crossings must be $Θ(n^2)$ and study then configurations that can be realized by rays that pairwise cross. We show that there are point sets for which the number of configurations that can be realized by pairwise-crossing rays is asymptotically smaller than the number of configurations realizable by pairwise-disjoint rays. We provide also point sets from which any configuration can be realized by pairwise-crossing rays and show that there is no configuration that can be realized by pairwise-crossing rays from every point set.

cs.CG↗

arXiv2017

Updating the Number of Crossings in Rectilinear Drawings of the Complete Graph

Let $S$ be a set of $n$ points in general position in the plane. Join every pair of points in $S$ with a straight line segment. Let $\overline{cr}(S)$ be number of pairs of these edges that intersect in their interior. Suppose that this number is known. In this paper we consider the problem of computing $\overline{cr}(S')$, where $S'$ comes from adding, deleting or moving a point from $S$.

cs.CG↗

arXiv2017

Point Sets with Small Integer Coordinates and with Small Convex Polygons

In 1935, Erdős and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a set of $n:=2^{t-2}$ points in general position in the plane, such that every convex polygon with vertices in this set has at most $\log_2(n)+1$ vertices. In this paper we show how to realize their construction in an integer grid of size $O(n^2 \log_2(n)^3)$.

math.CO↗

arXiv2017

Caratheodory's Theorem in Depth

Let $X$ be a finite set of points in $\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $τ_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carathéodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $τ_X(q)$) and pairwise disjoint sets $X_1,\dots, X_{d+1} \subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\dots, x_{d+1}$. We also prove depth versions of Helly's and Kirchberger's theorems.

cs.CG↗

arXiv2016

Regularity and Planarity of Token Graphs

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper we characterize precisely, for each value of $k$, which graphs have a regular $k$-token graph and which connected graphs have a planar $k$-token graph.

math.CO↗

arXiv2016

Drawing the Almost Convex Set in an Integer Grid of Minimum Size

In 2001, Károlyi, Pach and Tóth introduced a family of point sets to solve an Erdős-Szekeres type problem; which have been used to solve several other Edős-Szekeres type problems. In this paper we refer to these sets as nested almost convex sets. A nested almost convex set $\mathcal{X}$ has the property that the interior of every triangle determined by three points in the same convex layer of $\mathcal{X}$, contains exactly one point of $\mathcal{X}$. In this paper, we introduce a characterization of nested almost convex sets. Our characterization implies that there exists at most one (up to order type) nested almost convex set of $n$ points. We use our characterization to obtain a linear time algorithm to construct nested almost convex sets of $n$ points, with integer coordinates of absolute values at most $O(n^{\log_2 5})$. Finally, we use our characterization to obtain an $O(n\log n)$-time algorithm to determine whether a set of points is a nested almost convex set.

cs.CG↗

arXiv2015

Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant.

cs.CG↗

arXiv2015

Modem Illumination of Monotone Polygons

We study a generalization of the classical problem of the illumination of polygons. Instead of modeling a light source we model a wireless device whose radio signal can penetrate a given number $k$ of walls. We call these objects $k$-modems and study the minimum number of $k$-modems sufficient and sometimes necessary to illuminate monotone and monotone orthogonal polygons. We show that every monotone polygon with $n$ vertices can be illuminated with $\big\lceil \frac{n-2}{2k+3} \big\rceil$ $k$-modems. In addition, we exhibit examples of monotone polygons requiring at least $\lceil \frac {n-2} {2k+3}\rceil$ $k$-modems to be illuminated. For monotone orthogonal polygons with $n$ vertices we show that for $k=1$ and for even $k$, every such polygon can be illuminated with $\big\lceil \frac{n-2}{2k+4} \big\rceil$ $k$-modems, while for odd $k\geq3$, $\big\lceil \frac{n-2}{2k+6} \big\rceil$ $k$-modems are always sufficient. Further, by presenting according examples of monotone orthogonal polygons, we show that both bounds are tight.

cs.CG↗