Characterization of Generalized Color Complements of Gm,n Graph

In this paper, we study the generalized color complement of the graph Gm,n for specific values of m and n. The graph Gm,n = (V,E) is an undirected, simple graph defined on a finite subset of natural numbers. The vertex set of Gm,n is V = I_n = {1, 2, . . . , n}, and two distinct vertices a, b ∈ V are adjacent if and only if a not equal to b and a + b is not divisible by m, where m ∈ N and m > 1. The paper specifically focuses on the generalized color complement of Gm,n graphs with respect to the equal degree partition. Additionally, we determine the 2-color complement of Gm,n for m = 2 and for both even and odd values of n. Our aim is to analyze the color complement of the graphs with respect to the equaldegree partition for certain values of m, n ∈ N. We determine conditions under which the generalized color complement of Gm,n results in specific types of graphs, such as a path graph and a disjoint union of K_2 and K_1.