Relaxed coloring of a graph

Let P be a hereditary family of graphs. A relaxed coloring of a graph G = (V, E) with respect to P is an assignment of colors to vertices of G so that each color class induces a graph which is the disjoint union of members of B. The P-chromatic number chi(P)(G) of G is the minimum number of colors in a relaxed coloring of G with respect to P. We study the relation between the girth and the P-chromatic number of a graph, and the P-chromatic number of product graphs. Our results generalize some results of M.L. Weaver and D.B. West in [10], and answer some questions in that paper.