Global dominator coloring of the graph [Formula: see text] is the proper coloring of [Formula: see text] such that every vertex of [Formula: see text] dominates atleast one color class as well as anti-dominates atleast one color class. The minimum number of colors required for global dominator coloring of [Formula: see text] is called global dominator chromatic number of [Formula: see text] denoted by [Formula: see text]. In this paper, we characterize trees [Formula: see text] of order [Formula: see text] [Formula: see text] such that [Formula: see text] and also establish a strict upper bound for [Formula: see text] for a tree of even order [Formula: see text] [Formula: see text]. We construct some family of graphs [Formula: see text] with [Formula: see text] and prove some results on [Formula: see text]-partitions of [Formula: see text] when [Formula: see text].

For a graph G = (V (G), E(G)), a dominating set D is a vertex subset of V (G) in which every vertex of V (G) \ D is adjacent to a vertex in D. The domination number of G is the minimum cardinality of a dominating set of G and is denoted by γ(G). A coloring of G is a partition C = (V 1 , ..., V k ) such that each of V i in an independent set. The chromatic number is the smallest k among all colorings C = (V 1 , ..., V k ) of G and is denoted by χ(G). A coloring C = (V 1 , ..., V k ) is said to be dominator if, for all V i , every vertex v ∈ V i is singleton in V i or is adjacent to every vertex of V j . The dominator chromatic number of G is the minimum k of all dominator colorings of G and is denoted byIn this paper, for n ≥ 4k − 3, we prove that there always exists a D(k) graph of order n. We further prove that there is no planar D(k) graph when k ∈ {3, 4}. Namely, we prove that, for a non-trivial planar graph G, the graph G is D(k) if and only if G is K 2,q where q ≥ 2.

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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.