Abstract
The packing chromatic number chi(rho)(G) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X-1.....X-k where vertices in X-i 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 epsilon Z are adjacent if and only if vertical bar i - j vertical bar epsilon 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 chi(rho)(G(Z, D)) {= 35 for sufficiently large odd t and chi(rho)(G(Z, D)) {= 56 for sufficiently large even t.
We also give a lower bound 12 for t }= 9 and tighten several gaps for chi(rho)(G(Z. D)) with small t.