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.
@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/entities/document/10.4230/LIPIcs.SoCG.2023.41}, URN = {urn:nbn:de:0030-drops-178917}, doi = {10.4230/LIPIcs.SoCG.2023.41}, annote = {Keywords: Disjoint faces, simple topological graphs, topological Heilbronn problems} }
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