A note on the k-defect number: Vertex Coloring with a Fixed Number of Monochromatic Edges

In this paper, we introduce and study a novel graph parameter called the k-defect number, denoted ϕₖ(G), for a graph G and an integer 0≤ k≤ |E(G)|. Unlike traditional defective colorings that bound the local degree within monochromatic components, the k-defect number represents the smallest number of colors required to achieve a vertex coloring of G having exactly k monochromatic edges (also termed ``bad edges"). This parameter generalizes the well-known chromatic number of a graph, $χ(G)$, which is precisely ϕ₀(G). We establish fundamental properties of the k-defect number and derive bounds on ϕₖ(G) for specific graph classes, including trees, cycles, and wheels. Furthermore, we extend and generalize several classical properties of the chromatic number to this new edge-centric k-defect framework for values of 1≤ k≤ |E(G)|.