Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between dominator chromatic number, chromat...

The total dominator coloring of a graph is the such that each object (vertex or edge) adjacent incident to every some color class. minimum number classes called chromatic graph. In (A.P. Kazemi, F. Kazemnejad and S. Moradi, Contrib. Discrete Math. (2022).), authors initiated study found useful results, presented problems. Finding numbers cycles paths were two them which we consider here.

A total global dominator coloring of a graph is proper vertex with respect to which every in dominates color class, not containing and does dominate another class. The minimum number colors required such called the chromatic number, denoted by . In this paper, trees unicyclic graphs are explored.

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

A total dominator coloring of a graph G is proper in which each vertex the adjacent to every some color class. The chromatic number minimum classes coloring. In this paper, we study on middle graphs by giving several bounds for case general and trees. Moreover, calculate explicitly known families graphs.

A dominator coloring is a proper of the vertices graph such that each vertex dominates all at least one color class (possibly its own class). The chromatic number G minimum classes in . In this paper, we determine exact value subclass forests which call, generalized caterpillars forest, where every degree three support vertex.