Summary: In this article, a new graph coloring, called the \(b_q\)-coloring, is introduced. A \(b_q\)-coloring of a graph \(G\) is a proper vertex coloring of \(G\) with \(k\) colors such that every color class \(c\) admits a set of vertices \(S\) of size at most \(q\) provided that every color except \(c\) appears in the neighborhood of \(S\). The aim of this coloring is to generalize the domination constraint given in the \(b\)-coloring of a graph where every color admits only one dominating vertex (adjacent to every other color). The largest positive integer \(k\) for which a graph has a \(b_q\)-coloring using \(k\) colors is the \(b_q\)-chromatic number. Some classes of graphs for which the \(b_q\)-chromatic number has maximum value are presented. Also, the exact values of this parameter for paths and cycles are given. Furthermore, some bounds for Cartesian products of graphs are presented.