Locating-Dominator Coloring of Some Certain Graphs

Let G be a graph. A dominator coloring of G is a proper vertex coloring such that every vertex dominates all vertices of at least one color class. Given a dominator k-coloring c with color classes Ci1≤i≤k and an ordered partition Π=C1,…,Ck of VG, the dominator code of a vertex v∈VG with respect to Πis defined as the k-tuple dcΠv=dv,C1,dv,C2,…,dv,Ck, where dv,Ci=mindv,x∣x∈Ci for 1≤i≤k. A coloring c is called a locating-dominator k-coloring of G if distinct vertices have distinct dominator codes. The minimum number of colors required for such a coloring is the locating-dominator chromatic number of G, denoted by χdLG. In this paper, we determine the locating-dominator chromatic number for several well-known families of graphs, including bistars, wheels, Brooms, and various types of Helm graphs. Additionally, we establish comparative results between the dominator chromatic number χdG, the locating chromatic number χLG, and the locating-dominator chromatic number χdLG.