Related papers: Penny graphs in the hyperbolic plane
Let $h(n)$ denote the maximum number of triangles with angles between $59^\circ$ and $61^\circ$ in any $n$-element planar set. Our main result is an exact formula for $h(n)$. We also prove $h(n)= n^3/24+ O(n \log n)$ as $n\to \infty$.…
A Euclidean noncrossing Steiner $(1+\epsilon)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+\epsilon$ times the Euclidean…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the…
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…
We provide a complete classification of possible configurations of mutually pairwise touching infinite cylinders in Euclidian 3D space. It turns out that there is a maximum number of such cylinders possible in 3D independently on the shape…
We study 3-plane drawings, that is, drawings of graphs in which every edge has at most three crossings. We show how the recently developed Density Formula for topological drawings of graphs (KKKRSU GD 2024) can be used to count the…
A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have.…
Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the…
We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as…
A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…
We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same…
The degree diameter problem asks for the maximum possible number of vertices in a graph of maximum degree $\Delta$ and diameter $D$. In this paper, we focus on planar graphs of diameter $3$. Fellows, Hell and Seyffarth (1995) proved that…
We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of $N$ points in the plane. More…
We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number $n$, every set of $n$ points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line…
We study the problem of finding a tour of $n$ points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as…
Motivated by classical work of Alon and F\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) :=…
An arrangement of circles in which circles intersect only in angles of $\pi/2$ 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…
A double-normal pair of a finite set $S$ of points from $R^d$ is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ perpendicular to $pq$. A double-normal pair $pq$ is…
We improve the lower bound on the $d$-dimensional rectilinear crossing number of the complete $d$-uniform hypergraph having $2d$ vertices to $\Omega\left(\dfrac{(4\sqrt{2}/3^{3/4})^d}{d}\right)$ from $\Omega(2^d \sqrt{d})$. We also…