Efficient Open Dominator Coloring in Cycles and Paths

In 2006, Gera introduced and studied the dominator chromatic number. [4]. A dominator coloring of a graph G is a proper coloring in which each vertex of G dominates all the vertices of at least one color class. The dominator chromatic number χd(G) is the minimum number of colors required for a dominator coloring of G. In this paper we introduced, a proper coloring of G is called open dominator coloring, if N(v) contains at least one color class for each v ɛ V(G). An open dominator coloring of G is called efficient open dominator coloring, if N(v) contains exactly one color class for each v ɛ V(G). An open dominator coloring of G is called efficient k-open dominator coloring, if N(v) contains exactly k color classes for each v ɛ V(G) Also, we characterized some classes of graphs which admits efficient open dominator coloring of complete graphs, complete bipartite graphs, stars, wheels, paths, cycles, gear graphs.