THE SPLIT (D,C) NUMBER OF A GRAPH AND ITS IMPLICATIONS

THE SPLIT (D,C) NUMBER OF A GRAPH AND ITS IMPLICATIONS

In this paper, we introduce the integrated color variable, called the domination chromatic number, or the (D, C) - number, for connected graphs. We explore the concept of dominating chromatic sets in the graph G, known as (D, C) - sets for G, and split dominating chromatic sets, called split (D, C) - sets, for various connected graphs. A set S ⊆ V of vertices in the graph G is called a (D, C) - set for the graph G if it is both a dominating set and a chromatic set of the graph G. The smallest size of such a set is called the (D, C) - number for G, represented as : γ_χ(G) = min{|S| : S is a (D, C)- set of G}. A set S ⊆ V of vertices in the graph G is called a split (D, C) - set for G if it is both a (D, C) - set and the induced subgraph ⟨V \ S⟩ is disconnected. The smallest size of a split (D, C) - set is called the split (D, C) - number, represented as: γ_χS(G) = min{|S| : S is a split (D, C)- set of G}. This paper also discusses the characterization of this parameter and optimized dominating sets. We identify the split (D, C) - number for some standard graphs and examine the realization problem for K - coloring a graph G. For any two positive integers λ and p where 2 ≤ λ ≤ p, there exists a connected graph G with order p such that γ_χS(G) = λ.