Generalized Color Complements in Graphs: A Characterization

Abstract

The notion of graph complements has been widely generalized to study diverse structural and spectral properties of graphs. In this paper, we introduce and investigate the concept of generalized color complements of graphs with respect to a prescribed vertex partition. Building on earlier work on generalized color complements, we focus on structural properties arising from the interaction between graph coloring and partition-based complement operations. Sufficient conditions are established under which generalized color complements are disconnected, regular, and Eulerian. Explicit expressions are derived for the degree of any vertex in the generalized color complements [Formula: see text], [Formula: see text]. Furthermore, several classes of self-color-complementary graphs are identified for fixed partitions. A collection of illustrative examples is provided to demonstrate and validate the theoretical results. The findings extend existing results on generalized complements to a color-based framework and contribute to a deeper understanding of partition-dependent graph complements.