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On Rainbow Quadrilaterals in Colored Point Sets

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Abstract

Let S be a set of n points on the plane in general position whose elements have been colored with k colors. A rainbow polygon of S is a polygon such that all of its vertices are elements of S and have different colors. In this paper we give \(O(k n^2)\)-time algorithms to solve the following problems: find a minimum(maximum)-area rainbow triangle, and a minimum(maximum)-area rainbow quadrilateral of S, \(k \ge 3\). We also present an \(O(n^2)\)-time algorithm to determine if a 4-colored point set contains a convex rainbow quadrilateral, and an \(O(n^3)\)-time algorithm to determine if a 4-colored point set contains an empty rainbow quadrilateral, whether convex or not.

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Funding

A. Arévalo, R. Chávez-Jiménez, A. Hernández-Mora, R. López-López and N. Marín are supported by SEP-CONACyT of Mexico; A. Ramírez-Vigueras and J. Urrutia are partially supported by PAPIIT IN105221, Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM.

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Authors and Affiliations

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

    A. Arévalo, R. Chávez-Jiménez, A. Hernández-Mora, R. López-López & N. Marín

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

    A. Ramírez-Vigueras & J. Urrutia

  3. Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico

    O. Solé-Pi

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  1. A. Arévalo
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  2. R. Chávez-Jiménez
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  3. A. Hernández-Mora
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  5. N. Marín
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  8. J. Urrutia
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Correspondence to N. Marín.

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A. Arévalo, R. Chávez-Jiménez, A. Hernández-Mora, R. López-López and N. Marín are supported by SEP-CONACyT of Mexico. A. Ramírez-Vigueras and J. Urrutia are partially supported by PAPIIT IN105221, Programa de Apoyo a la Investigación e Innovación Tecnologíca UNAM.

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Arévalo, A., Chávez-Jiménez, R., Hernández-Mora, A. et al. On Rainbow Quadrilaterals in Colored Point Sets. Graphs and Combinatorics 38, 152 (2022). https://doi.org/10.1007/s00373-022-02559-y

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  • DOI: https://doi.org/10.1007/s00373-022-02559-y

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