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

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

2010-03-11

arxiv.org/abs/1003.2291v1

Computer Science

Discrete Mathematics

3 pages

Scientific paper

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.

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