A tight bound for the number of edges of matchstick graphs

TL;DR

This paper proves Harborth's conjecture that the maximum number of edges in a matchstick graph with n vertices is exactly  3n -   12n - 3  for all n  1, using an isoperimetric inequality related to L'Huilier's inequality.

Contribution

The paper provides a proof of Harborth's conjecture on the maximum edges in matchstick graphs for all n  1, employing a novel geometric inequality approach.

Findings

Confirmed the maximum edge count formula for all n  1.

Established a new geometric inequality related to matchstick graphs.

Extended the understanding of planar unit-distance graphs.

Abstract

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.