The packing coloring of distance graphs [formula omitted

Abstract

The packing chromatic number χρ(G) of a graph G is the smallest integer p such that vertices of G can be partitioned into disjoint classes X1,…,Xp where vertices in Xi have pairwise distance between them 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∈Z are adjacent if and only if |i−j|∈{k,t}.We generalize results given by Ekstein et al. for graphs D(1,t). For sufficiently large t we prove that χρ(D(k,t))≤30 for both k and t odd, and that χρ(D(k,t))≤56 for exactly one of k and t odd. We also give some upper and lower bounds for χρ(D(k,t)) with small k and t.