On the Chromatic Number and Connectivity of Delta-Color Complement of Graphs

Let Ģ = (V, E) be a finite, simple, colored graph of order n and size m. In this study, we focus on the coloring characteristics of δ-color complement of graphs. Specifically, we establish both lower and upper bounds for the product and the sum involving χ and χ_δc, drawing parallels with the classical Nordhaus-Gaddum type inequalities. We also identify graph families that attain these bounds. Compute the δ_c and δ^′c-chromatic numbers of certain graphs. Inspired by the foundational work of Nordhaus and Gaddum, we derive bounds on maximum degree, minimum degree, vertex connectivity, and edge connectivity of a graph Ģ and its δ-color complement.