The s-packing chromatic number of a graph

Let S = (a₁, a₂, ...) be an infinite nondecreasing sequence of positive integers. An S-packing k-coloring of a graph G is a mapping from V(G) to {1,2,...,k} such that vertices with color i have pairwise distance greater than ${\displaystyle a_i}$, and the S-packing chromatic number ${\displaystyle {\chi }_S\left(G\right)}$ of G is the smallest integer k such that G has an S-packing k-coloring. This concept generalizes the concept of proper coloring (when S = (1,1,1,...)) and broadcast coloring (when S = (1,2,3,4,...)). In this paper, we consider bounds on the parameter and its relationship with other parameters. We characterize the graphs with ${\displaystyle {\chi }_S=2}$ and determine ${\displaystyle {\chi }_S}$ for several common families of graphs. We examine ${\displaystyle {\chi }_S}$ for the infinite path and give some exact values and asymptotic bounds. Finally we consider complexity questions, especially about recognizing graphs with ${\displaystyle {\chi }_S=3}$.