A tight bound for the number of edges of matchstick graphs
Abstract
A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on $n$ vertices is $\lfloor 3n-\sqrt{12n-3} \rfloor$. In this paper we prove this conjecture for all $n\geq 1$. The main geometric ingredient of the proof is an isoperimetric inequality related to L'Huilier's inequality.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2022
- DOI:
- 10.48550/arXiv.2209.09800
- arXiv:
- arXiv:2209.09800
- Bibcode:
- 2022arXiv220909800L
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Computational Geometry;
- Mathematics - Metric Geometry;
- Primary 52C10. Secondary 05C10
- E-Print:
- 12 pages, 3 figures. Minor typos corrected