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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## 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 |
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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: | 17 May 2024 00:06 |

URI: | http://eprints.lse.ac.uk/id/eprint/113476 |

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