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Coloring decompositions of complete geometric graphs

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Abstract

A decomposition of a non-empty simple graph G is a pair [G,P] such that P is a set of non-empty induced subgraphs of G, and every edge of G belongs to exactly one subgraph in P. The chromatic index \(\chi'([G,P])\) of a decomposition [G,P] is the smallest number k for which there exists a k-coloring of the elements of P in such a way that for every element of P all of its edges have the same color, and if two members of P share at least one vertex, then they have different colors. A long standing conjecture of Erdős–Faber–Lovász states that every decomposition \([K_{n}, P]\) of the complete graph \(K_n\) satisfies \(\chi'([K_{n}, P])\leq n\). In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case when the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.

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Acknowledgement

We thank the anonymous referee for helpful suggestions.

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    C. Huemer

  2. Departamento de Computación, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico, Mexico

    D. Lara

  3. División de Matemáticas e Ingeniería, FES Acatlán, Universidad Nacional Autónoma de México, Mexico, Mexico

    C. Rubio-Montiel

  4. UMI LAFMIA 3175 CNRS at CINVESTAV-IPN, Mexico, Mexico

    C. Rubio-Montiel

  5. Department of Algebra, Comenius University, Bratislava, Slovakia

    C. Rubio-Montiel

Authors
  1. C. Huemer
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  2. D. Lara
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  3. C. Rubio-Montiel
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Corresponding author

Correspondence to D. Lara.

Additional information

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922.

C. Huemer was supported by projects MINECO MTM2015-63791-R and Gen. Cat. DGR 2017SGR1336.

C.Rubio-Montiel was partially supported by a CONACyT-México Postdoctoral fellowship, by the National scholarship programme of the Slovak Republic and PAIDI/007/19.

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Huemer, C., Lara, D. & Rubio-Montiel, C. Coloring decompositions of complete geometric graphs. Acta Math. Hungar. 159, 429–446 (2019). https://doi.org/10.1007/s10474-019-00963-0

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  • DOI: https://doi.org/10.1007/s10474-019-00963-0

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