On the existence of 4-regular matchstick graphs

Abstract

A matchstick graph is a planar unit-distance graph. We call it \emph{4-regular} if every vertex has degree 4. While examples of 4-regular matchstick graphs with fewer than 63 vertices are known only for $n\in \{52,54,57,60\}$, we prove the existence of such graphs for every integer $n\ge 63$.

8 pages, 5 figures, 3 tables