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

For a non-decreasing sequence S = ( s 1 , s 2 , … ) of positive integers, a partition of the vertex set of a graph G into subsets X 1 , … , X ℓ , such that vertices in X i are pairwise at distance greater than s i 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 s i ≤ 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. • Studies S-packing colorings for graphs using non-decreasing sequences of integers. • Analyses integer distance graphs based on two positive integers. • Determines S-packing chromatic numbers for cases where sequence values are at most 2. • Complements prior research on S-packing chromatic numbers for specific sequences.