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Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems

Authors Alfredo Hubard, Andrew Suk

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Disjoint faces
  • simple topological graphs
  • topological Heilbronn problems

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Abstract

Given a complete simple topological graph G, a k-face generated by G is the open bounded region enclosed by the edges of a non-self-intersecting k-cycle in G. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete n-vertex simple topological graph generates at least Ω(n^{1/3}) pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on n vertices drawn in the unit square generates a 4-face with area at most O(n^{-1/3}). Finally, we investigate a ℤ₂ variant of Heilbronn’s triangle problem for not necessarily simple complete topological graphs.

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Alfredo Hubard and Andrew Suk. Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.41

Author Details

Alfredo Hubard
  • LIGM, Université Gustave Eiffel, CNRS, ESIEE Paris, F-77454 Marne-la-Vallée, France
Andrew Suk
  • Department of Mathematics, University of California San Diego, CA, USA

Funding

  • Hubard, Alfredo: Supported by the projects SOS ANR-17-CE40-0033 and Min Max ANR-19-CE40-0033. This work was mostly done in a year of deélégation funded by the CNRS at the Laboratoire Solomon Lefschetz (LaSol) at the Instituto de Matematicas-UNAM-Ciudad Universitaria.
  • Suk, Andrew: Supported by NSF CAREER award DMS-1800746 and NSF award DMS-1952786.

Acknowledgements

AH thanks Dominic Dotterer for sharing his ideas on Heilbronn’s triangle problem.

References

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