Packing chromatic number of distance graphs

Abstract

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X1,…,Xk where vertices in Xi have pairwise distance greater than i. We study the packing chromatic number of infinite distance graphs G(Z,D), i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i,j∈Z are adjacent if and only if |i−j|∈D.In this paper we focus on distance graphs with D={1,t}. We improve some results of Togni who initiated the study. It is shown that χρ(G(Z,D))≤35 for sufficiently large odd t and χρ(G(Z,D))≤56 for sufficiently large even t. We also give a lower bound 12 for t≥9 and tighten several gaps for χρ(G(Z,D)) with small t.