Geodesic
The s-packing chromatic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 795-806

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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 a_i, and the S-packing chromatic number χ_S(G) 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 χ_S = 2 and determine χ_S for several common families of graphs. We examine χ_S for the infinite path and give some exact values and asymptotic bounds. Finally we consider complexity questions, especially about recognizing graphs with χ_S = 3.
Keywords: graph, coloring, packing, broadcast chromatic number 
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Goddard, Wayne; Xu, Honghai. The s-packing chromatic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 795-806. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a13/