License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.41
URN: urn:nbn:de:0030-drops-178917
URL: https://drops.dagstuhl.de/opus/volltexte/2023/17891/
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Hubard, Alfredo ; Suk, Andrew

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

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LIPIcs-SoCG-2023-41.pdf (0.9 MB)


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.

BibTeX - Entry

@InProceedings{hubard_et_al:LIPIcs.SoCG.2023.41,
  author =	{Hubard, Alfredo and Suk, Andrew},
  title =	{{Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{41:1--41:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17891},
  URN =		{urn:nbn:de:0030-drops-178917},
  doi =		{10.4230/LIPIcs.SoCG.2023.41},
  annote =	{Keywords: Disjoint faces, simple topological graphs, topological Heilbronn problems}
}

Keywords: Disjoint faces, simple topological graphs, topological Heilbronn problems
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023


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