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

We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.

The packing chromatic number χ ρ (G) of a graph G is the smallest integer p such that vertices of G can be partitioned into disjoint classes X 1 , ..., X p where vertices in X i have pairwise distance 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 by Ekstein et al. for graphs D(1, t). For sufficiently large t we prove that χ ρ (D(k, t)) ≤ 30 for both k, t odd, and that χ ρ (D(k, t)) ≤ 56 for exactly one of k, t odd. We also give some upper and lower bounds for χ ρ (D(k, t)) with small k and t.

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The square of a graph G, denoted G 2