The number of small-degree vertices in matchstick graphs

TL;DR

This paper proves that in large matchstick graphs, a linear number of vertices have degree at most four, using a novel combination of charging methods and isoperimetric inequalities.

Contribution

It introduces a new approach combining charging methods with isoperimetric inequalities to analyze vertex degrees in matchstick graphs.

Findings

At least Ω(√n) vertices have degree ≤ 4 in large matchstick graphs

The bound on low-degree vertices is asymptotically tight

Provides insights into the structure of crossing-free unit-distance 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.