Compelling colorings: A generalization of the dominator chromatic number

We define a P-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property P. The P-compelling chromatic number is the minimum number of colors in such a coloring. We show that this notion generalizes the dominator and total dominator chromatic numbers, and provide some general bounds and algorithmic results. We also investigate the specific cases where P is that the subset contains at least one edge or that the subset is connected.