A heuristic approach for searching ▫$(d, n)$▫-packing colorings of infinite lattices
Korže, Danilo ; Markuš, Žiga ; Vesel, Aleksander
A ▫$(d, n)$▫-packing ▫$k$▫-coloring of a graph ▫$G$▫ for integers ▫$d$▫ and ▫$n$▫ is a mapping from ▫$V(G)$▫ to the set ▫$\{1, 2, \ldots, k \}$▫ such that vertices with color ▫$i \in \{1, 2, \ldots, ...k \}$▫ have pairwise distance greater than ▫$d + \lfloor \frac{i - 1}{n} \rfloor$▫. The smallest integer ▫$k$▫ for which there exists a ▫$(d, n)$▫-packing coloring of a graph ▫$G$▫ is called the ▫$(d, n)$▫-packing chromatic number of ▫$G$▫. We propose a new heuristic approach to find ▫$(d, n)$▫-packing colorings of infinite lattices. The proposed algorithms have been able to obtain several ▫$(d, n)$▫-packing colorings of the infinite square, hexagonal, triangular, eight-regular and octagonal lattice. The obtained results improve upper bounds on the corresponding ▫$(d, n)$▫-packing chromatic numbers for these graphs.
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