Regular matchstick graphs

Abstract A match-stick graph is a plane geometric graph in which every edge has length 1 and no two edges cross each other. It was conjectured that no 5-regular match-stick graph exists. In this paper we prove this conjecture

Fast regocnition of planar non unit distance graphs

We study criteria attesting that a given graph can not be embedded in the plane so that neighboring vertices are at unit distance apart and the straight line edges do not cross.Comment: 9 pages, 1 table, 5 figure

No finite 55-regular matchstick graph exists

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.Comment: 15 pages, 12 figures, 2 table