On some domination colorings of graphs

In this paper, we are interested in four proper vertex colorings of graphs, with additional domination property. In the dominator colorings, strong colorings and strict strong colorings of a graph G, every vertex has to dominate at least one color class. Conversely, in the dominated colorings of G, every color class has to be dominated by at least one vertex. We study arbitrary graphs as well as P4-sparse graphs, P5-free graphs, bounded treewidth graphs and claw-free graphs.