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The Complexity of Grid Coloring

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Abstract

A c-coloring of the grid GN,M = [N] × [M] is a mapping of GN,M into [c] such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed to find a 4-coloring of G17,17. Though a coloring was produced, finding it proved to be difficult. This raises the question of whether there is some complexity lower bound. Consider the following problem: given a partial c-coloring of the GN,M grid, can it be extended to a full c-coloring? We show that this problem is NP-complete. We also give a Fixed Parameter Tractable algorithm for this problem with parameter c.

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Acknowledgements

We thank Amy Apon, Doug Chen, Jacob Gilbert, Matt Kovacs-Deak, Stasys Junka, Jon Katz, Clyde Kruskal, Nathan Hayes, Erika Melder, Erik Metz, and Rishab Pallepati for proofreading and discussion.

We thank Wing Ning Li for pointing out that the case of N,M binary, while it seems to not be in NP, is actually unknown.

We thank Jacob Gilbert, David Harris, and Daniel Marx for pointing out many improvements in the fixed parameter algorithm which we subsequently used.

We thank Tucker Bane, Richard Chang, Peter Fontana, David Harris, Jared Marx-Kuo, Jessica Shi, and Marius Zimand, for listening to Bill present these results and hence clarifying them.

We thank the referee for many helpful comments including a complete reworking of the proof of Theorem 3.

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Authors and Affiliations

  1. University of Maryland at College Park, College Park, MD, Baltimore, 20742, USA

    Daniel Apon & William Gasarch

  2. Permanent, Berkeley, 94710, CA, USA

    Kevin Lawler

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  1. Daniel Apon
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  2. William Gasarch
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Correspondence to William Gasarch.

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Apon, D., Gasarch, W. & Lawler, K. The Complexity of Grid Coloring. Theory Comput Syst 67, 521–547 (2023). https://doi.org/10.1007/s00224-022-10098-5

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