The Packing Chromatic Number of the Infinite Square Grid Is at Least 14

Authors Bernardo Subercaseaux , Marijn J.H. Heule

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Bernardo Subercaseaux
  • Carnegie Mellon University, Pittsburgh, PA, USA
Marijn J.H. Heule
  • Carnegie Mellon University, Pittsburgh, PA, USA


Authors thank the Pittsburgh Supercomputing Center for allowing us to use Bridges2 [Brown et al., 2021] in our experiments. The first author thanks Dylan Pizzo for his intriguing post on a Math Facebook group in 2019, which inspired the journey leading to this article.

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Bernardo Subercaseaux and Marijn J.H. Heule. The Packing Chromatic Number of the Infinite Square Grid Is at Least 14. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A packing k-coloring of a graph G = (V, E) is a mapping from V to {1, ..., k} such that any pair of vertices u, v that receive the same color c must be at distance greater than c in G. Arguably the most fundamental problem regarding packing colorings is to determine the packing chromatic number of the infinite square grid. A sequence of previous works has proved this number to be between 13 and 15. Our work improves the lower bound to 14. Moreover, we present a new encoding that is asymptotically more compact than the previously used ones.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • packing coloring
  • SAT solvers
  • encodings


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