S-packing colorings of distance graphs with distance sets of cardinality 2

Abstract

For a non-decreasing sequence 𝑆 = (𝑠1, 𝑠2,…) of positive integers, a partition of the vertex set of a graph 𝐺 into subsets 𝑋1,…,𝑋𝓁, such that vertices in 𝑋𝑖 are pairwise at distance greater than 𝑠𝑖 for every 𝑖 ∈ {1,…,𝓁}, is called an 𝑆-packing 𝓁-coloring of 𝐺. The minimum 𝓁 for which 𝐺 admits an 𝑆-packing 𝓁-coloring is called the 𝑆-packing chromatic number of 𝐺. In this paper, we consider 𝑆-packing colorings of the integer distance graphs with respect two positive integers 𝑘 and 𝑡, which are the graphs whose vertex set is ℤ, and two vertices 𝑥, 𝑦 ∈ ℤ are adjacent whenever |𝑥−𝑦| ∈ {𝑘,𝑡}. We complement partial results from two earlier papers, thus determining all values of the 𝑆-packing chromatic numbers of these distance graphs for all sequence 𝑆 such that 𝑠𝑖 ≤ 2 for all 𝑖. In particular, if 𝑆 = (1, 1, 2, 2,…), then the 𝑆-packing chromatic number is 2 if 𝑘 + 𝑡 is even, and 4 otherwise, while if 𝑆 = (1, 2, 2,…), then the 𝑆-packing chromatic number is 5, unless {𝑘,𝑡} = {2, 3} when it is 6; when 𝑆 = (2, 2, 2,…), the corresponding formula is more complex.