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The s-packing chromatic number of a graph

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Goddard W., Xu H.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 4

795-806

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 $χ_S = 3$.

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1 Discussiones Mathematicae Graph Theory

1 Goddard W.

1 Xu H.

1 2012

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The s-packing chromatic number of a graph