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

TL;DR

This paper proves that the packing chromatic number of the 2D square lattice is at least 12, improving the previous lower bound of 10, which advances understanding of graph coloring in lattice structures.

Contribution

It establishes a new lower bound of 12 for the packing chromatic number of the square lattice, surpassing prior known bounds.

Findings

Lower bound of 12 for _(^2)

Improves previous lower bound of 10

Advances knowledge of graph coloring in lattice graphs

Abstract

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.