Packing chromatic number of distance graphs

The packing chromatic number ? ? ( 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 ? 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 ? ? ( G ( Z , D ) ) ? 35 for sufficiently large odd t and ? ? ( G ( Z , D ) ) ? 56 for sufficiently large even t . We also give a lower bound 12 for t ? 9 and tighten several gaps for ? ? ( G ( Z , D ) ) with small t .

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