We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no $n^{1/2}$ curves (lines or circles) lying on an algebraic surface of degree at most two, then the number of these intersection points is $O(n^{3/2})$.
An arrangement of circles in which circles intersect only in angles of $π/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is planar. The same result holds for arrangement of circles that intersect in an angle of at most $π/2$. For the general case we prove that the maximal number of edges in an intersection graph of an arrangement of orthogonal circles lies in between $4n - O\left(\sqrt{n}\right)$ and $\left(4+\frac{5}{11}\right)n$, for $n$ being the number of circles. Based on the …
Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szabó [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szabó [Combinatorica, 2012] (and by Elekes and Rónyai [J. Combin. Theory Ser. A, 2000]).
Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a planar map? An equivalent question, under a different guise, is the following: is the arrangement graph of great circles on the sphere always vertex 3-colorable? (The arrangement graph has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points.) Assume that no three circles meet at a point, so that this arrangement graph is 4-regular. In this note we have shown that all arrangement graphs defined as above are 3-colorable.
We study a new discrete-time dynamical system on circle patterns with the combinatorics of the square grid. This dynamics, called Miquel dynamics, relies on Miquel's six circles theorem. We provide a coordinatization of the appropriate space of circle patterns on which the dynamics acts and use it to derive local recurrence formulas. Isoradial circle patterns arise as periodic points of Miquel dynamics. Furthermore, we prove that certain signed sums of intersection angles are preserved by the dynamics. Finally, when the initial circle pattern is spatially biperiodic with a fundamental domain o…
An interval graph is the intersection graph of a finite set of intervals on a line and a circular-arc graph is the intersection graph of a finite set of arcs on a circle. While a forbidden induced subgraph characterization of interval graphs was found fifty years ago, finding an analogous characterization for circular-arc graphs is a long-standing open problem. In this work, we study the intersection graphs of finite sets of arcs on a circle no three of which cover the circle, known as normal Helly circular-arc graphs. Those circular-arc graphs which are minimal forbidden induced subgraphs for…
A graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n\}$ is an intersection graph of segments if there are segments $s_1,\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\{v_i,v_j\}$ is an edge of~$G$. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments. It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by $\exists\mathbb{R}$. This class consists of problems that can be reduced, in polynomial time, t…
Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of th…
Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result states that $| \mathcal{Q}_1\cap \mathcal{Q}_2|\le 4q^{n-2}+π_{n-3}$. This result inspires us to establish the conjecture on the number of points of an algebraic set $X\subset {\mathbb{P}}^n(\mathbb{F}_q)$ of dimension $s$ and degree $d$: $|X(\mathbb{F}_q)|\le dq^s+π_{s-1}$.
Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time $O((n+m)α(n+m))$ where $n$ is the number of vertices, $m$ is the number of edges and $α$ is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time $O(nm)$ of the previous algorithm [Hsu, 1995] based on a similar approach.
In this paper we consider families of compact convex sets in $\mathbb R^d$ such that any subfamily of size at most $d$ has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg's theorem for such families.
By extending the definition of boxicity, we extend a Helly-type result given by Danzer and Grumbaum on 2-piercings of family of boxes in $d$-dimensional Euclidian space by lowering the dimension of the boxes in the ambient space.
We prove that there are intersection graphs of axis-aligned boxes in $\mathbb{R}^3$ and intersection graphs of straight lines in $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number.
Let $V$ be a finite-dimensional representation of the complex circle $\mathbb{C}^\times$ determined by a weight vector $\mathbf{a}\in\mathbb{Z}^n$. We study the Hilbert series $\operatorname{Hilb}_{\mathbf{a}}(t)$ of the graded algebra $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ of polynomial $\mathbb{C}^\times$-invariants in terms of the weight vector $\mathbf{a}$ of the $\mathbb{C}^\times$-action. In particular, we give explicit formulas for $\operatorname{Hilb}_{\mathbf{a}}(t)$ as well as the first four coefficients of the Laurent expansion of $\operatorname{Hilb}_{\mathbf{a}}(t)$ at $…
Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that $K_1\subseteq \mathbb R^2$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\mathbb R^2\to\mathbb R^2$. Kira Adaricheva and Madina Bolat have recently proved that if $K_0$ is a disk and both $K_0$ and $K_1$ are included in a triangle with vertices $A_0$, $A_1$, and $A_2$, then there exist a $j\in \{0,1,2\}$ and a $k\in\{0,1\}$ such that $K_{1-k}$ is included in the convex hull of $K_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. Here we prove that this property char…
We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb S^{d-2}$-bundles over $\mathbb S^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is non-orientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $β_1\neq 0$ and $β_2=0$, where $β_i$'s are the Betti numbers, computed with coefficients in $\mathbb Q$.
Circle graphs are intersection graphs of chords in a circle and $k$-polygon graphs are intersection graphs of chords in a convex $k$-sided polygon where each chord has its endpoints on distinct sides. The $k$-polygon graphs, for $k \ge 2$, form an infinite chain of graph classes, each of which contains the class of permutation graphs. The union of all of those graph classes is the class of circle graphs. The polygon number $\gp(G)$ of a circle graph $G$ is the minimum $k$ such that $G$ is a $k$-polygon graph. Given a circle graph $G$ and an integer $k$, determining whether $\gp(G) \le k$ is NP…
Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least $\min(|S||P|^{1/4-\varepsilon},|S|^{2/3}|P|^{2/3},|S|^2,|P|^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.
Let $m>1$ be an integer, and let $I(\mathbb{Z}_m)^*$ be the set of all non-zero proper ideals of $\mathbb{Z}_m$. The intersection graph of ideals of $\mathbb{Z}_m$, denoted by $G(\mathbb{Z}_m)$, is a graph with vertices $I(\mathbb{Z}_m)^*$ and two distinct vertices $I,J\in I(\mathbb{Z}_m)^*$ are adjacent if and only if $I\cap J\neq 0$. Let $n>1$ be an integer and $\mathbb{Z}_n$ be a $\mathbb{Z}_m$-module. In this paper, we introduce and study a kind of graph structure of $\mathbb{Z}_m$, denoted by $G_n(\mathbb{Z}_m)$. It is the undirected graph with the vertex set $I(\mathbb{Z}_m)^*$, and two …
Let $C_1, \dots, C_n$ denote the $1/n-$neighborhood of $n$ great circles on $\mathbb{S}^2$. We are interested in how much these areas have to overlap and prove the sharp bounds $$ \sum_{i, j = 1 \atop i \neq j}^{n}{|C_i \cap C_j|^s} \gtrsim_s \begin{cases} n^{2 - 2s} \qquad &\mbox{if}~0 \leq s < 2 \\ n^{-2} \log{n} \qquad &\mbox{if}~s = 2\\ n^{1- 3s/2} \qquad &\mbox{if}~s > 2. \end{cases} .$$ For $s=1$ there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in $\mathbb{R}^2$ the lower bound is $\gtrsim \log{n}$) and there are strong connection…
