Bounding the Number of Edges of Matchstick Graphs

A matchstick graph is a crossing-free unit-distance in the plane. Harborth conjectured 1981 that maximum number of edges with $n$ vertices $\lfloor 3n-\sqrt{12n-3}\rfloor$. Using Euler formula and isoperimetric inequality, it can be shown has no more than $3n-\sqrt{2\pi\sqrt{3}\cdot n}+O(1)$ edges. We improve this upper bound to $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\sqrt{3}})$. The main tool proof new for takes into account nontriangular faces. also find sharp triangular faces graph.