Characterization of Generalized Color Complements of G_{m,n} Graph

Abstract

In this paper, we study the generalized color complement of the graph G_{m,n} for specific
values of m and n. The graph G_{m,n} = (V,E) is an undirected, simple graph defined on
a finite subset of natural numbers. The vertex set of G_{m,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 G_{m,n} graphs with respect to the equal degree partition. Additionally, we
determine the 2−color complement of G_{m,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 G_{m,n} results in specific types of graphs, such as a path
graph and a disjoint union of K_2 and K_1.