Global dominated coloring of graphs

In this paper, we initiate a study of global dominated coloring of graphs as a variation of dominated colorings. A global dominated coloring of a graph $G$ {is a proper coloring such that for each color class there are at least two vertices, one of which is adjacent to all the vertices of this class while the other one is not adjacent to any vertex of the class. The global dominated chromatic number of $G$ is the minimum number of colors used among all global dominated colorings of $G.$ In this paper, we establish various bounds on the global dominated chromatic number of a graph in terms of some graph invariants including the order, dominated chromatic number, domination number and total domination number. Moreover, characterizations of extremal graphs attaining some of these bounds are provided.} We also discuss the global dominated coloring in trees and split graphs.