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Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

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Abstract

We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or noninduced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic.

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Authors and Affiliations

  1. Department of Computer Science, University of Durham, South Road, Durham DH1 3LE, England

    Hajo Broersma

  2. Institutt for Informatikk, University of Bergen, PO Box 7800, 5020 Bergen, Norway

    Fedor V. Fomin

  3. Department of Applied Mathematics, Charles University, Malostranske nam. 25, 118 00 Praha 1, Czech Republic

    Jan Kratochvil

  4. Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MD Eindhoven, The Netherlands

    Gerhard J. Woeginger

Authors
  1. Hajo Broersma
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  2. Fedor V. Fomin
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  3. Jan Kratochvil
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  4. Gerhard J. Woeginger
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Corresponding authors

Correspondence to Hajo Broersma, Fedor V. Fomin, Jan Kratochvil or Gerhard J. Woeginger.

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Broersma, H., Fomin, F., Kratochvil, J. et al. Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult. Algorithmica 44, 343–361 (2006). https://doi.org/10.1007/s00453-005-1176-8

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  • DOI: https://doi.org/10.1007/s00453-005-1176-8

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