Bounding the number of edges of matchstick graphs

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