On the partial Grundy coloring of graphs

In a proper vertex coloring of a graph, a Grundy vertex is a vertex colored with color c and adjacent to any color 1 to c-1. A partial Grundy coloring of a graph is a proper coloring of its vertices where any color admits at least one Grundy vertex. The partial Grundy number of a graph is the maximum number of colors used in a partial Grundy coloring of this graph. In this article we consider this parameter for some classes of graphs, in particular $K_{1,s}$-free graphs, regular graphs, Cartesian and direct products of graphs.