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arXiv · 1003.2291

The packing chromatic number of the square lattice is at least 12

Jan Ekstein·Jiří Fiala·Přemysl Holub·Bernard Lidický
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Abstract

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set $V(G)$ can be partitioned into disjoint classes $X_1, ..., X_k$, where vertices in $X_i$ have pairwise distance greater than $i$. For the 2-dimensional square lattice $\mathbb{Z}^2$ it is proved that $χ_ρ(\mathbb{Z}^2) \geq 12$, which improves the previously known lower bound 10.

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Jan Ekstein, Jiří Fiala, Přemysl Holub, Bernard Lidický. 2010-03-11. The packing chromatic number of the square lattice is at least 12. https://arxiv.org/abs/1003.2291

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