Bounding the number of edges of matchstick graphs

TL;DR

This paper establishes an upper bound on the number of edges in matchstick graphs with n vertices, using geometric and combinatorial tools, and also determines a sharp bound on the number of triangular faces.

Contribution

It introduces a new upper bound on edges in matchstick graphs and provides a sharp limit on the number of triangular faces, advancing geometric graph theory.

Findings

Maximum edges in matchstick graphs are bounded by 3n - c√(n-1/4).

Derived a sharp upper bound for the number of triangular faces.

Utilized Euler formula and isoperimetric inequality in proofs.

Abstract

We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper bound for the number of edges in terms of $n$ and the number of non-triangular faces. We also find a sharp upper bound for the number of triangular faces in a matchstick graph.