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arXiv · 2506.01589

On the number of edges of restricted matchstick graphs

Panna Gehér·János Pach·Konrad Swanepoel·Géza Tóth
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Abstract

A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges of such graphs in two different scenarios. First, we show that there is a constant $c>0$ such that every triangle-free matchstick graph on $n$ vertices has at most $2n-c\sqrt{n}$ edges. This statement is not true for any $c>\sqrt2.$ We also prove that for every $r>0$, there is a constant $\varepsilon(r)>0$ with the property that every matchstick graph on $n$ vertices contained in a disk of radius $r$ has at most $(2-\varepsilon(r))n$ edges.

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Panna Gehér, János Pach, Konrad Swanepoel, Géza Tóth. 2025-06-02. On the number of edges of restricted matchstick graphs. https://arxiv.org/abs/2506.01589

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