$S$-packing colorings of distance graphs $G(\mathbb{Z},\{2,t\})$

Abstract

Given a graph G and a non-decreasing sequence S = (a 1 , a 2 , . . .) of positive integers, the mapping f :In this paper, we consider the distance graphs G(Z, {2, t}), where t > 1 is an odd integer, which has Z as its vertex set, and i, j ∈ Z are adjacent if |i − j| ∈ {2, t}. We determine the S-packing chromatic numbers of the graphs G(Z, {2, t}), where S is any sequence with a i ∈ {1, 2} for all i. In addition, we give lower and upper bounds for the d-distance chromatic numbers of the distance graphs G(Z, {2, t}), which in the cases d ≥ t − 3 give the exact values. Implications for the corresponding S-packing chromatic numbers of the circulant graphs are also discussed.