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 C _i (1 ≤ i ≤ k ) and an ordered partition Π = ( C ₁ , …, C _k ) of V ( G ), the dominator code of a vertex v ∈ V ( G ) with respect to Π is defined as the k ‐tuple dc _Π ( v ) = ( d ( v , C ₁ ), d ( v , C ₂ ), …, d ( v , C _k )), where d ( v , C _i ) = min{ d ( v , x ) | x ∈ C _i } 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 . 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 χ _d ( G ), the locating chromatic number χ _L ( G ), and the locating‐dominator chromatic number .