Grundy Packing Coloring of Graphs

Abstract: A map $c:V(G)\rightarrow{1,\dots,k}$ 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 \textit{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 $G$. We then define the Grundy packing chromatic number $\Gamma_{\rho}(G)$ of a graph $G$ as the maximum value that this algorithm yields in a graph $G$. We present several properties of $\Gamma_{\rho}(G)$, provide results on the complexity of the problem as well as bounds and some exact results for $\Gamma_{\rho}(G)$.