Grundy packing coloring of graphs

Abstract

A map $c:V\left(G\right)\to \{1,\dots ,k\}$ of a graph $G$ is a packing $k$-coloring if every two different vertices of the same color $i\in \{1,\dots ,k\}$ are at distance more than $i$. The packing chromatic number ${\chi }_{\rho }\left(G\right)$ of $G$ is the smallest integer $k$ such that there exists a packing $k$-coloring. In this paper we introduce the notion of Grundy packing chromatic number, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of any graph $G$. We then define the Grundy packing chromatic number ${\Gamma }_{\rho }\left(G\right)$ of a graph $G$ as the maximum value that this algorithm yields in $G$. We present several properties of ${\Gamma }_{\rho }\left(G\right)$, provide results on the complexity of the problem as well as bounds and some exact results for ${\Gamma }_{\rho }\left(G\right)$.