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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Text (the number of small-degree vertices in matchstickgraphs)
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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 |
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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: | 20 Feb 2025 17:44 |
URI: | http://eprints.lse.ac.uk/id/eprint/117229 |
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