$S$packing colorings of distance graphs $G(\mathbb{Z},\{2,t\})$
Abstract
Given a graph $G$ and a nondecreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, the mapping $f:V(G) \rightarrow \{1,\ldots,k\}$ is an $S$packing $k$coloring of $G$ if for any distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=i$ the distance between $u$ and $v$ in $G$ is greater than $a_i$. The smallest $k$ such that $G$ has an $S$packing $k$coloring is the $S$packing chromatic number, $\chi_S(G)$, of $G$. In this paper, we consider the distance graphs $G(\mathbb{Z},\{2,t\})$, where $t>1$ is an odd integer, which has $\mathbb{Z}$ as its vertex set, and $i,j\in\mathbb{Z}$ are adjacent if $ij\in\{2,t\}$. We determine the $S$packing chromatic numbers of the graphs $G(\mathbb{Z},\{2,t\})$, where $S$ is any sequence with $a_i\in\{1,2\}$ for all $i$. In addition, we give lower and upper bounds for the $d$distance chromatic numbers of the distance graphs $G(\mathbb{Z},\{2,t\})$, which in the cases $d\ge t3$ give the exact values. Implications for the corresponding $S$packing chromatic numbers of the circulant graphs are also discussed.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.10491
 Bibcode:
 2020arXiv200510491B
 Keywords:

 Mathematics  Combinatorics;
 05C15;
 05C12
 EPrint:
 21 pages, 3 figures