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Konrad J. Swanepoel

Publications and source records attributed to Konrad J. Swanepoel.

At least 19 recordsLinked to original sources
arXiv2025

Penny graphs in the hyperbolic plane

We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a spiral construction of the tiling of the plane with equilateral triangles (Harborth 1974), with a similar result in the hyperbolic plane for the values of $d$ corresponding to the order-$k$ triangular tilings (Bowen 2000). We present various upper and lower bounds for $e_d(n)$ for all values of $d > 0$. In particular, we prove that if $d > 0.66114\dots$ except for $d=0.76217\dots$, then the number of touching pairs is less than the one coming from a spiral construction in the order-$7$ triangular tiling, which we conjecture to be extremal. We also give a lower bound $e_d(n) > (2+\varepsilon_d)n$ where $\varepsilon_d > 1$ for all $d > 0$.

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arXiv2025

Regular matchstick graphs on the sphere

We show that the $5$-regular matchstick graphs on the sphere are exactly the five $5$-regular contact graphs of congruent caps on the sphere found by R. M. Robinson (1969).

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arXiv2022

The number of small-degree vertices in matchstick graphs

A matchstick graph is a crossing-free unit-distance graph in the plane. Harborth (1981) proposed the problem of determining whether there exists a matchstick graph in which every vertex has degree exactly $5$. In 1982, Blokhuis gave a proof of non-existence. A shorter proof was found by Kurz and Pinchasi (2011) using a charging method. We combine their method with the isoperimetric inequality to show that there are $Ω(\sqrt{n})$ vertices in a matchstick graph on $n$ vertices that are of degree at most $4$, which is asymptotically tight.

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arXiv2022

Contacts in totally separable packings in the plane and in high dimensions

We study the contact structure of totally separable} packings of translates of a convex body $K$ in $\mathbb{R}^d$, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number $H_{\text{sep}}(K)$ of $K$ is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of $K$. We show that for each $d\geq 8$, there exists a smooth and strictly convex $K$ in $\mathbb{R}^d$ with $H_{\text{sep}}(K)>2d$, and asymptotically, $H_{\text{sep}}(K)=Ω\bigl((3/\sqrt{8})^d\bigr)$. We show that Alon's packing of Euclidean unit balls such that each translate touches at least $2^{\sqrt{d}}$ others whenever $d$ is a power of $4$, can be adapted to give a totally separable packing of translates of the $\ell_1$-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of $n$ translates of any planar convex body $K$. We prove that the maximum equals $\lfloor 2n-2\sqrt{n}\rfloor$ if and only if $K$ is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.

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arXiv2021

Bounding the number of edges of matchstick graphs

We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2π\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper bound for the number of edges in terms of $n$ and the number of non-triangular faces. We also find a sharp upper bound for the number of triangular faces in a matchstick graph.

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arXiv2020

Shortest directed networks in the plane

Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms.

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arXiv2019

Outer linear measure of connected sets via Steiner trees

We resurrect an old definition of the linear measure of a metric continuum in terms of Steiner trees, independently due to Menger (1930) and Choquet (1938). We generalise it to any metric space and provide a proof of a little-known theorem of Choquet that it coincides with the outer linear measure for any connected metric space. As corollaries we obtain simple proofs of Gołąb's theorem (1928) on the lower semicontinuity of linear measure of continua and a theorem of Bognár (1989) on the linear measure of the closure of a set. We do not use any measure theory apart from the definition of outer linear measure.

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arXiv2019

Favourite distances in 3-space

Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\in S$ a distance $r(x)>0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erdős and Pach (1988) introduced the extremal quantity $f_3(n)=\max\sum_{x\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of 3-space and all assignments $r\colon S\to(0,\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\mathcal{C}$ and the axis of symmetry $\mathcal{L}$ of $\mathcal{C}$, and $r(x)$ equals the distance from $x$ to $C$ for each $x\in S\cap\mathcal{L}$. This, together with a new construction, implies that $f_3(n)=n^2/4 + 5n/2 + O(1)$.

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arXiv2018

Embedding graphs in Euclidean space

The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances). Answering a question of Erdős and Simonovits [Ars Combin. 9 (1980) 229--246], we show that any graph with less than $\binom{d+2}{2}$ edges has dimension at most $d$. Improving their result, we prove that that the dimension of a graph with maximum degree $d$ is at most $d$. We show the following Ramsey result: if each edge of the complete graph on $2d$ vertices is coloured red or blue, then either the red graph or the blue graph can be embedded in Euclidean $d$-space. We also derive analogous results for embeddings of graphs into the $(d-1)$-dimensional sphere of radius $1/\sqrt{2}$.

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arXiv2017

Arrangements of homothets of a convex body II

A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most $2\cdot 3^d$ members. This improves a result of Polyanskii (arXiv:1610.04400). Using similar ideas, we also give a proof the following result of Polyanskii: Let $K_1,\dots,K_n$ be a sequence of homothets of the o-symmetric convex body $K$, such that for any $i<j$, the center of $K_j$ lies on the boundary of $K_i$. Then $n\leq O(3^d d)$.

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arXiv2017

Combinatorial distance geometry in normed spaces

We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces.

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arXiv2016

Approximate Euclidean Steiner Trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error e from 120 degrees.This notion arises in numerical approximations of minimum Steiner trees (W. D. Smith, Algorithmica, 7 (1992), 137--177). We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology.Rubinstein, Weng and Wormald (J. Global Optim. 35 (2006), 573--592) conjectured that this relative error is at most linear in $e$, independent of the number of terminals. We verify their conjecture for the two-dimensional case as long as the error $e$ is sufficiently small in terms of the number of terminals. We derive a lower bound linear in $e$ for the relative error in the two-dimensional case when $e$ is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error for larger values of $e$, and calculate exact values in the plane for three and four terminals.

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arXiv2014

Equilateral sets and a Schütte Theorem for the 4-norm

A well-known theorem of Schütte (1963) gives a sharp lower bound for the ratio between the maximum distance and minimum distance between n+2 points in n-dimensional Euclidean space. In this brief note we adapt Bárány's elegant proof of this theorem to the space $\ell_4^n$. This gives a new proof that the largest cardinality of an equilateral set in $\ell_4^n$ is n+1, and gives a constructive bound for an interval $(4-ε_n,4+ε_n)$ of values of p close to 4 for which it is guaranteed that the largest cardinality of an equilateral set in $\ell_p^n$ is n+1.

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arXiv2014

Double-normal pairs in the plane and on the sphere

A double-normal pair of a finite set $S$ of points from Euclidean space is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ that are perpendicular to $pq$. A double-normal pair $pq$ is strict if $S\setminus\{p,q\}$ lies in the open strip. We answer a question of Martini and Soltan (2006) by showing that a set of $n\geq 3$ points in the plane has at most $3\lfloor n/2\rfloor$ double-normal pairs. This bound is sharp for each $n\geq 3$. In a companion paper, we have asymptotically determined this maximum for points in $R^3$. Here we show that if the set lies on some 2-sphere, it has at most $17n/4 - 6$ double-normal pairs. This bound is attained for infinitely many values of $n$. We also establish tight bounds for the maximum number of strict double-normal pairs in a set of $n$ points in the plane and on the sphere.

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arXiv2014

Sets of unit vectors with small subset sums

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a $k$-collapsing family of unit vectors in a $d$\dimensional Banach space, where the maximum is taken over all spaces of dimension $d$. Similarly, let $CB(k,d)$ denote the maximum cardinality if we require in addition that $\sum_{i=1}^m x_i=o$. The case $k=2$ was considered by Füredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawlor and Morgan (1994) on geometric shortest networks in smooth finite-dimensional Banach spaces. We show that $CB(k,d)=\max{k+1,2d}$ for all $k,d\geq 2$. The behaviour of $C(k,d)$ is not as simple, and we derive various upper and lower bounds for various ranges of $k$ and $d$. These include the exact values $C(k,d)=\max{k+1,2d}$ in certain cases. We use a variety of tools from graph theory, convexity and linear algebra in the proofs: in particular the Hajnal-Szemerédi Theorem, the Brunn-Minkowski inequality, and lower bounds for the rank of a perturbation of the identity matrix.

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arXiv2013

Maximal equilateral sets

A subset of a normed space $X$ is called equilateral if the distance between any two points is the same. Let $m(X)$ be the smallest possible size of an equilateral subset of $X$ maximal with respect to inclusion. We first observe that Petty's construction of a $d$-dimensional $X$ of any finite dimension $d\geq 4$ with $m(X)=4$ can be generalised to give $m(X\oplus_1\mathbb{R})=4$ for any $X$ of dimension at least 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $Γ$, $m(\ell_p(Γ))$ is finite and bounded above by a function of $p$, for all $1\leq p<2$. Also, for all $p\in[1,\infty)$ and $d\in\mathbb{N}$ there exists $c=c(p,d)>1$ such that $m(X)\leq d+1$ for all $d$-dimensional $X$ with Banach-Mazur distance less than $c$ from $\ell_p^d$. Using Brouwer's fixed-point theorem we show that $m(X)\leq d+1$ for all $d$-dimensional $X$ with Banach-Mazur distance less than 3/2 from $\ell_\infty^d$. A graph-theoretical argument furthermore shows that $m(\ell_\infty^d)=d+1$. The above results lead us to conjecture that $m(X)\leq 1+\dim X$ for all finite-dimensional normed spaces $X$.

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arXiv2013

Generalised k-Steiner Tree Problems in Normed Planes

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within $O(n^2)$ time. In this paper we generalise their approach in order to solve the $k$-Steiner tree problem, in which the Steiner minimum tree may contain up to $k$ Steiner points for a given constant $k$. We also extend their approach further to encompass other normed planes, and to solve a much wider class of problems, including the $k$-bottleneck Steiner tree problem and other generalised $k$-Steiner tree problems. We show that, for any fixed $k$, such problems can be solved in $O(n^{2k})$ time.

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