The number of small-degree vertices in matchstick graphs

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 discharging method. We combine their method with the isoperimetric inequality to show that there are Ω(√ n) vertices in a matchstick graph on n vertices that are of degree at most 4, which is asymptotically tight.