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

Authors Alfredo Hubard, Andrew Suk

Thumbnail PDF


  • Filesize: 0.92 MB
  • 15 pages

Document Identifiers

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


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

Cite AsGet BibTex

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)


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.

Subject Classification

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. O. Aichholzer, A. García, J. Tejel, B. Vogtenhuber, and A. Weinberger. Twisted ways to find plane structures in simple drawings of complete graphs. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 5:1-5:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  2. A. Arroyo, M. Derka, and I. Parada. Extending simple drawings. In 27th International Symposium on Graph Drawing and Network Visualization (GD), volume 11904 of Lecture Notes in Computer Science, pages 230-243. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_18.
  3. R. Courant and H. Robbins. What is Mathematics? Oxford University Press, 1941. Google Scholar
  4. P. Erdos. Problems and results in combinatorial geometry. Annals of the New York Academy of Sciences, 440(1):1-11, 1985. Google Scholar
  5. R. Fulek and A.J. Ruiz-Vargas. Topological graphs: empty triangles and disjoint matchings. In Proceedings of the 29th Annual Symposium on Computational Geometry (SoCG’13), pages 259-266, 2013. Google Scholar
  6. V. Gullemin and A. Pollock. Differentiable Geometry. American Mathematical Society. Chelsea Publishing, 1974. Google Scholar
  7. H. Harborth and I. Mengersen. Drawings of the complete graph with maximum number of crossings. Congr. Numer., 88:225-228, 1992. Google Scholar
  8. M. Hoffmann, CH. Liu, M.M. Reddy, and C.D. Tóth. Simple topological drawings of k-planar graphs. In Graph Drawing and Network Visualization, volume 12590 of Lecture Notes in Computer Science. Springer, 2020. Google Scholar
  9. J. Komlós, J. Pintz, and E. Szemerédi. On Heilbronn’s triangle problem. J. London Math. Soc., 24:385-396, 1981. Google Scholar
  10. J. Komlós, J. Pintz, and E. Szemerédi. A lower bound for Heilbronn’s problem. J. London Math. Soc., 25:13-24, 1982. Google Scholar
  11. H. Lefmann. Distributions of points in the unit square and large k-gons. European J. Combin., 29:946-965, 2008. Google Scholar
  12. A. Marcus and G. Tardos. Intersection reverse sequences and geometric applications. J. Comb. Theory, Ser. A, 113:675-691, 2006. Google Scholar
  13. J. Pach, J. Solymosi, and G. Tóth. Unavoidable configurations in complete topological graphs. Disc. Comput. Geom., 30:311-320, 2003. Google Scholar
  14. R. Pinchasi and R. Radoičić. On the number of edges in geometric graphs with no self-intersecting cycle of length 4. Towards a Theory of Geometric Graphs, Contemporary Mathematics (J. Pach, ed.), 342, 2004. Google Scholar
  15. V. V. Prasolov. Elements of homology theory. American Mathematical Society, Vol. 81, 2007. Google Scholar
  16. K. F. Roth. On a problem of Heilbronn. J. London Math. Soc., 26:198-204, 1951. Google Scholar
  17. K. F. Roth. On a problem of Heilbronn II. J. London Math. Soc., 25:193-212, 1972. Google Scholar
  18. K. F. Roth. On a problem of Heilbronn III. J. London Math. Soc., 25:543-549, 1972. Google Scholar
  19. K. F. Roth. Developments in Heilbronn’s triangle problem. Adv. Math., 22:364-385, 1976. Google Scholar
  20. A.J. Ruiz-Vargas. Empty triangles in complete topological graphs. Discrete Comput. Geom., 53:703-712, 2015. Google Scholar
  21. W. M. Schmid. On a problem of Heilbronn. J. London Math. Soc., 2:545-550, 1971/72. Google Scholar
  22. A. Suk and J. Zeng. Unavoidable patterns in complete simple topological graphs. Graph Drawing and Network Visualization, GD 2022, 2022. Google Scholar