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) intoindependent sets vi which are said to be color classes with respect to the coloringc. a vertex v is said to have a dominator (dom) color class in c if there is colorclass vi such that v is adjacent to all the vertices of vi and v is said to have ananti-dominator (anti-dom) color class in c if there is color class vj such that v is notadjacent to any vertex of vj . dominator coloring of g is a coloring c of g such thatevery vertex has a dom color class. the minimum number of colors required for adominator coloring of g is called the dominator chromatic number of g, denoted byd(g). global dominator coloring of g is a coloring c of g such that every vertex hasa dom color class and an anti-dom color class. the minimum number of colors requiredfor a global dominator coloring of g is called the global dominator chromatic numberof g, denoted by gd(g). in this paper, we give a counterexample for the conjectureposed 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) = 2d(g), theng is a complete multipartite graph. we deduce upper and lower bound for the globaldominator chromatic number of mycielskian of the graph g in terms of dominatorchromatic number of g.