The number of small-degree vertices in matchstick graphs

Lavollée, Jérémy and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2023) The number of small-degree vertices in matchstick graphs. Australasian Journal of Combinatorics, 85 (1). 92 - 99. ISSN 1034-4942

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Abstract

A matchstick graph is a crossing-free unit-distance graph in the plane. Harborth (1981) proposed the problem of determining whether there exists a matchstick graph in which every vertex has degree exactly 5. In 1982, Blokhuis gave a proof of non-existence. A shorter proof was found by Kurz and Pinchasi (2011) using a discharging method. We combine their method with the isoperimetric inequality to show that there are Ω(√ n) vertices in a matchstick graph on n vertices that are of degree at most 4, which is asymptotically tight.

Item Type:	Article
Official URL:	https://ajc.maths.uq.edu.au/
Additional Information:	© 2022 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	01 Nov 2022 10:12
Last Modified:	02 Dec 2024 08:45
URI:	http://eprints.lse.ac.uk/id/eprint/117229

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