The Packing Coloring of Distance Graphs $D(k,t)$

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than $i$. For $k < t$ we study the packing chromatic number of infinite distance graphs $D(k, t)$, i.e. graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in \{k, t\}$. We generalize results by Ekstein et al. for graphs $D (1, t)$. For sufficiently large $t$ we prove that $\chi_{\rho}(D(k, t)) \leq 30$ for both $k$, $t$ odd, and that $\chi_{\rho}(D(k, t)) \leq 56$ for exactly one of $k$, $t$ odd. We also give some upper and lower bounds for $\chi_{\rho}(D(k, t))$ with small $k$ and $t$. Keywords: distance graph; packing coloring; packing chromatic number