Year: 2014
Type: preprint
Abstract: A graph $G=(V,E)$ is called a unit-distance graph in the plane if there is an injective embedding of $V$ in the plane such that every pair of adjacent vertices are at unit distance apart. If additionally the corresponding edges are non-crossing and all vertices have the same degree $r$ we talk of a regular matchstick graph. Due to Euler's polyhedron formula we have $r\le 5$. The smallest known $4$-regular matchstick graph is the so called Harborth graph consisting of $52$ vertices. In this article we prove that no finite $5$-regular matchstick graph exists.
Source: arXiv (Cornell University)
Author: Kurz, Sascha
Institution: University of Bayreuth
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1635360885