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On the Number of Edges in Geometric Graphs Without Empty Triangles

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Abstract

In this paper we study the extremal type problem arising from the question: What is the maximum number ET(S) of edges that a geometric graph G on a planar point set S can have such that it does not contain empty triangles? We prove: \({{n \choose 2} - O(n \log n) \leq ET(n) \leq {n \choose 2} - \left(n - 2 + \left\lfloor \frac{n}{8} \right\rfloor \right)}\) .

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Authors and Affiliations

  1. Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Mexico City, Mexico

    C. Bautista-Santiago

  2. Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México, Mexico City, Mexico

    M. A. Heredia & A. Ramírez-Vigueras

  3. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain

    C. Huemer

  4. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    C. Seara

  5. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    J. Urrutia

Authors
  1. C. Bautista-Santiago
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  2. M. A. Heredia
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  3. C. Huemer
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  4. A. Ramírez-Vigueras
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  5. C. Seara
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  6. J. Urrutia
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Corresponding author

Correspondence to J. Urrutia.

Additional information

C. Bautista-Santiago, M. A. Heredia and A. Ramírez-Vigueras partially supported by SEP-CONACYT of Mexico. C. Huemer was partially supported by projects MEC MTM2009-07242, Gen. Cat. DGR 2009SGR1040, and the ESF EUROCORES programme EuroGIGA, CRP ComPoSe, MICINN Project EUI-EURC-2011-4306. C. Seara was partially supported by projects MTM2009-07242, Gen. Cat. DGR2009GR1040, and the ESF EUROCORES programme EuroGIGA-ComPoSe IP04-MICINN Project EUI-EURC-2011-4306. J. Urrutia was partially supported by projects MTM2006-03909 (Spain) and SEP-CONACYT of México, Proyecto 80268.

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Bautista-Santiago, C., Heredia, M.A., Huemer, C. et al. On the Number of Edges in Geometric Graphs Without Empty Triangles. Graphs and Combinatorics 29, 1623–1631 (2013). https://doi.org/10.1007/s00373-012-1220-9

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  • DOI: https://doi.org/10.1007/s00373-012-1220-9

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