On [formula omitted]-packing colourings of distance graphs [formula omitted] and [formula omitted

Abstract

For a non-decreasing sequence of positive integers S=(s1,s2,…), the S-packing chromatic numberχS(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into subsets Xi, i∈{1,2,…,k}, where vertices in Xi are pairwise at distance greater than Si. By an infinite distance graph with distance set D we mean a graph with vertex set Z in which two vertices i,j are adjacent whenever |i−j|∈D. In this paper we investigate the S-packing chromatic number of infinite distance graphs with distance set D={1,t}, t≥2, and D={1,2,t}, t≥3, for sequences S having all elements from {1,2}.