In this paper, we introduce and study a novel graph parameter called the k-defect number, denoted ϕ₊ (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 ϕ₊ (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) |.
Mphako-Banda et al. (Wed,) studied this question.