No finite $5$-regular matchstick graph exists

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.

15 pages, 12 figures, 2 tables

References in corpus (2)

• Minimal Polynomials for the Coordinates of the Harborth Graph
• Regular finite planar maps with equal edges