A Counterexample on the Conjecture and bounds on χgd-number of Mycielskian of a graph

A coloring C = (V1; : : : ; Vk) of G partitions the vertex set V (G) into independent sets Vi which are said to be color classes with respect to the coloring C. A vertex v is said to have a dominator (dom) color class in C if there is color class Vi such that v is adjacent to all the vertices of Vi and v is said to have an anti-dominator (anti-dom) color class in C if there is color class Vj such that v is not adjacent to any vertex of Vj . Dominator coloring of G is a coloring C of G such that every vertex has a dom color class. The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G, denoted by d(G). Global Dominator coloring of G is a coloring C of G such that every vertex has a dom color class and an anti-dom color class. The minimum number of colors required for a global dominator coloring of G is called the global dominator chromatic number of G, denoted by gd(G). In this paper, we give a counterexample for the conjecture posed in [I. Sahul Hamid, M.Rajeswari, Global dominator coloring of graphs, Discuss. Math. Graph Theory 39 (2019), 325{339] that for a graph G, if gd(G) = 2 d(G), then G is a complete multipartite graph. We deduce upper and lower bound for the global dominator chromatic number of Mycielskian of the graph G in terms of dominator chromatic number of G.