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Bounding the number of edges of matchstick graphs

Lavollée, Jérémy and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2022) Bounding the number of edges of matchstick graphs. SIAM Journal on Discrete Mathematics, 36 (1). 777 - 785. ISSN 0895-4801

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Identification Number: 10.1137/21M1441134

Abstract

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.

Item Type: Article
Official URL: https://epubs.siam.org/journal/sjdmec
Additional Information: © 2022 SIAM
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Jan 2022 14:27
Last Modified: 07 Jun 2022 10:33
URI: http://eprints.lse.ac.uk/id/eprint/113476

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