Grundy Packing Coloring of Graphs

A map $c:V(G)\to \{1,\dots ,k\}$ 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 ${\chi }_{\rho }(G)$ of $G$ is the smallest integer $k$ such that there exists a packing $k$-coloring. In this paper we introduce the notion of $\mathit{\text{Grundy}}\mathit{\text{packing}}\mathit{\text{chromatic}}\mathit{\text{number}}$, 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 $G$. We then define the Grundy packing chromatic number ${\mathrm{\Gamma }}_{\rho }(G)$ of a graph $G$ as the maximum value that this algorithm yields in a graph $G$. We present several properties of ${\mathrm{\Gamma }}_{\rho }(G)$, provide results on the complexity of the problem as well as bounds and some exact results for ${\mathrm{\Gamma }}_{\rho }(G)$.