On the packing chromatic number of square and hexagonal lattice

Danilo Korže, Aleksander Vesel

Abstract


The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set V(G) can be partitioned into disjoint classes X1, …, Xk, with the condition that vertices in Xi 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 P6□ 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□ P2. The results are partially obtained by a computer search.


Keywords


Packing chromatic number, hexagonal lattice, square lattice, computer search

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DOI: https://doi.org/10.26493/1855-3974.255.88d

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications