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.