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

The packing chromatic number $\chi_\rho(G)$ 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 $\chi_\rho(\mathbb{Z}^2) \geq 12$, which improves the previously known lower bound 10.