The number of small-degree vertices in matchstick graphs

Abstract

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-existence. A shorter proof was found by Kurz and Pinchasi (2011) using a charging method. We combine their method with the isoperimetric inequality to show that there are $\Omega (\sqrt{n})$ vertices in a matchstick graph on $n$ vertices that are of degree at most $4$, which is asymptotically tight.