Subdivision into i-packings and S-packing chromatic number of some lattices
(1505.07781)Abstract
An $i$-packing in a graph $G$ is a set of vertices at pairwise distance greater than $i$. For a nondecreasing sequence of integers $S=(s_{1},s_{2},\ldots)$, the $S$-packing chromatic number of a graph $G$ is the least integer $k$ such that there exists a coloring of $G$ into $k$ colors where each set of vertices colored $i$, $i=1,\ldots, k$, is an $s_i$-packing. This paper describes various subdivisions of an $i$-packing into $j$-packings ($j\textgreater{}i$) for the hexagonal, square and triangular lattices. These results allow us to bound the $S$-packing chromatic number for these graphs, with more precise bounds and exact values for sequences $S=(s_{i}, i\in\mathbb{N}{*})$, $s_{i}=d+ \lfloor (i-1)/n \rfloor$.
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