The number of small-degree vertices in matchstick graphs

06/08/2022

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by   Jérémy Lavollée, et al.

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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 Ω(√(n)) vertices in a matchstick graph on n vertices that are of degree at most 4, which is asymptotically tight.