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Graphs and Combinatorics

, Volume 31, Issue 2, pp 335–345 | Cite as

Empty Triangles in Good Drawings of the Complete Graph

  • Oswin Aichholzer
  • Thomas Hackl
  • Alexander Pilz
  • Pedro Ramos
  • Vera Sacristán
  • Birgit VogtenhuberEmail author
Original Paper
  • 136 Downloads

Abstract

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph \(K_n\) with \(n\) vertices is at least \(n\).

Keywords

Good drawings Empty triangles Erdős–Szekeres type problems 

Notes

Acknowledgments

This work was initiated during a research visit of Pedro Ramos and Vera Sacristán in May 2013 in Graz, Austria.

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Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Thomas Hackl
    • 1
  • Alexander Pilz
    • 1
  • Pedro Ramos
    • 2
  • Vera Sacristán
    • 3
  • Birgit Vogtenhuber
    • 1
    Email author
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Departamento de MatemáticasUniversidad de AlcaláMadridSpain
  3. 3.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain

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