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General penny graphs are at most $\frac{43}{18}$-dense

Arsenii Sagdeev
Vol. 14 (2025), No. 1, 75–89
DOI: 10.2140/cnt.2025.14.75
Abstract

We prove that among n points in the plane in general position, the shortest distance occurs at most 43 18n times, improving upon the upper bound of 17 7 n obtained by G. Tóth in 1997.

Keywords
planar graph, minimum distance, penny graph, general position, edge density
Mathematical Subject Classification
Primary: 52C10
Secondary: 05C10, 52C15
Milestones
Received: 15 February 2024
Revised: 3 November 2024
Accepted: 17 November 2024
Published: 17 January 2025
Authors
Arsenii Sagdeev
Alfréd Rényi Institute of Mathematics
Budapest
Hungary
© 2025 MSP (Mathematical Sciences Publishers).