A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on $n$ vertices is $\lfloor 3n-\sqrt{12n-3} \rfloor$. In this paper we prove this conjecture for all $n\geq 1$. The main geometric ingredient of the proof is an isoperimetric inequality related to L'Huilier's inequality.

翻译：火柴棍图是一种平面图，其边被绘制为单位距离的线段。Harborth于1981年引入此类图，并猜想在$n$个顶点的火柴棍图中，边数的最大值为$\lfloor 3n-\sqrt{12n-3} \rfloor$。本文证明了该猜想对所有$n\geq 1$成立。证明的主要几何工具是与L'Huilier不等式相关的等周不等式。