GENERALIZED COLOR COMPLEMENTS IN GRAPHS

Let Gc= (V,E) be a color graph, and P ={V1, V2,....,Vk} be a partition of V or order k >= 1. The k and k(i)-color complement of Gc is defined as follows: For all Vi and Vj in P, i \neq j, remove the edges between Vi and Vj and add the edges which are not in Gc such that end vertices have different colors. For each subset Vr in the partition P, remove the edges Gc that exist within Vr and add the edges of Gc joining the vertices of Vr. The resulting graph (Gc)^P_{k(i)} is known as k(i)-color complement of Gc with respect to the partition P of V. This paper establishes connectivity conditions for the k-color complement and k(i)-color complement of a connected graph based on specific vertex partitioning and color assignments. Additionally, the relationship between clique numbers and independence numbers in the generalized color complements is explored with respect to same color class partitions, and the number of edges is determined for certain graph families.