The Packing Chromatic Number of the Infinite Square Grid is 15

TL;DR

This paper proves that the packing chromatic number of the infinite square grid is 15, resolving a long-standing open problem by significantly improving computational methods with new encoding and symmetry-breaking techniques.

Contribution

The authors establish the exact packing chromatic number of the infinite square grid as 15, introducing advanced encoding and symmetry-breaking methods for this problem.

Findings

Packing chromatic number of the infinite square grid is 15

New propositional encoding improves computational efficiency

Developed a symmetry-breaking method for better problem solving

Abstract

A packing $k$-coloring is a natural variation on the standard notion of graph $k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, and any two vertices assigned a common color $c \in \{1, \ldots, k\}$ need to be at a distance greater than $c$ (as opposed to $1$, in standard graph colorings). Despite a sequence of incremental work, determining the packing chromatic number of the infinite square grid has remained an open problem since its introduction in 2002. We culminate the search by proving this number to be 15. We achieve this result by improving the best-known method for this problem by roughly two orders of magnitude. The most important technique to boost performance is a novel and surprisingly effective propositional encoding. Additionally, we developed a new symmetry-breaking method. Since both new techniques are more complex than existing techniques for this problem, a verified approach is required to trust them. We include both techniques in a proof of unsatisfiability, reducing the trusted core to the correctness of the direct encoding.