Bounding the number of edges of matchstick graphs

08/17/2021

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by   Jérémy Lavollée, et al.

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We show that a matchstick graph with n vertices has no more than 3n-c√(n-1/4) edges, where c=1/2(√(12) + √(2π√(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.