Zobrazeno 1 - 10
of 134
pro vyhledávání: '"Swanepoel, Konrad J."'
Autor:
Lavollée, Jérémy, Swanepoel, Konrad J.
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-ex
Externí odkaz:
http://arxiv.org/abs/2206.03956
Autor:
Naszódi, Márton, Swanepoel, Konrad J.
Publikováno v:
Journal of Computational Geometry 13 (2022), 471--483
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
Externí odkaz:
http://arxiv.org/abs/2201.12097
Autor:
Lavollée, Jérémy, Swanepoel, Konrad J.
We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper bound for t
Externí odkaz:
http://arxiv.org/abs/2108.07522
Autor:
Swanepoel, Konrad J.
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
Externí odkaz:
http://arxiv.org/abs/1908.02230
Autor:
Swanepoel, Konrad J.
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, Erd\H{o}s and Pach (1988) introduced the extremal quantity $f
Externí odkaz:
http://arxiv.org/abs/1907.08402
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 an
Externí odkaz:
http://arxiv.org/abs/1903.07172
Publikováno v:
Journal of Combinatorial Theory, Ser. A, 171 (2020), article 105146
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 quest
Externí odkaz:
http://arxiv.org/abs/1802.03092
Publikováno v:
Combinator. Probab. Comp. 28 (2019) 280-286
A set of points in d-dimensional Euclidean space is almost equidistant if among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in $\mathbb{R}^d$ has cardinality $O(d^{4/3})$.
Comment: 6 pages
Comment: 6 pages
Externí odkaz:
http://arxiv.org/abs/1708.01590
Autor:
Naszódi, Márton, Swanepoel, Konrad J.
Publikováno v:
Contributions to Discrete Mathematics 13 (2018), 116--123
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 arrang
Externí odkaz:
http://arxiv.org/abs/1705.09253
Autor:
Swanepoel, Konrad J.
Publikováno v:
New Trends in Intuitive Geometry, Bolyai Soc. Math. Studies 27, Springer, 2018. pp.407--458
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 span
Externí odkaz:
http://arxiv.org/abs/1702.00066