Power Dominator Equitable Coloring of Graphs: Bounds and Exact Values for Special Graph Classes

Abstract

This paper introduces the concept of the power dominator equitable coloring, which combines the monitoring capability of power dominator coloring with the balanced partition requirement of equitable coloring. A proper coloring is called a power dominator equitable coloring if every vertex power dominates at least one color class and the sizes of any two color classes differ by at most one. The minimum number of colors required for such a coloring is called the power dominator equitable chromatic number and is denoted by \(\chi_{pde} (G)\). We establish several fundamental bounds for this new parameter and characterize graphs attaining equality. Exact values are obtained for complete multipartite graphs, disconnected graphs, forests and circulant graphs \(C_n (1,2)\). In addition, relationships between the power dominator equitable chromatic number and the equitable chromatic number are investigated, together with sufficient conditions under which they coincide. These results provide a theoretical foundation for studying graph coloring problems involving both monitoring capability and balanced resource allocation.