On the packing chromatic number of square and hexagonal lattice

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 , with the condition that vertices in X i have pairwise distance greater than i. We show that the packing chromatic number for the hexagonal lattice H is 7. We also investigate the packing chromatic number for infinite subgraphs of the square lattice Z 2 with up to 13 rows. In particular, we establish the packing chromatic number for P 6 Z and provide new upper bounds on these numbers for the other subgraphs of interest. Finally, we explore the packing chromatic number for some infinite subgraphs of Z 2 P 2. The results are partially obtained by a computer search.

Modeling the packing coloring problem of graphs

“…The results are presented in Table 5. The exception are values marked with a, which had been already established in [12].…”

“…4. h It was showed in [10] that v q ðGÃZÞ < 1 for any finite graph G. It follows that v q ðP n ÃP m ÃZÞ < 1 for any m; n. Some results on the packing chromatic number of P 2 ÃP 3 ÃP k are presented in [12], while the proposed integer linear programming models and satisfiability test model enables the computation for some larger families of graphs. The results are presented in Table 5.…”

“…The proposed models outperform other exact methods such as a backtracking and dynamic algorithm [12]. In particular, the packing chromatic numbers and improved bounds have been found for the Cartesian products of paths and cycles as presented in Section 3.…”

“…The upper bound was improved to 17 by Holub and Soukal [8], while Ekstein et al [9] showed that a packing 11-coloring of the square lattice cannot exist. Some finite and infinite subgraphs of the square lattice as well as some related graphs, in particular the Cartesian products of paths and cycles, have been also extensively explored [3,[9][10][11]8,12].…”

“…A value marked with # denotes an upper bound; values marked with a had been already established in[12]. dð1; 2; .…”