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 $\phi_{k}(G)$, for a graph $G$ and an integer $0\leq k\leq |E(G)|$. 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 \emph{$k$ monochromatic edges (also termed ``bad edges")}. This parameter generalizes the well-known chromatic number of a graph, $\chi (G)$, which is precisely $\phi _{0}(G)$. We establish fundamental properties of the $k$-defect number and derive bounds on $\phi _{k}(G)$ 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 $1\leq k\leq |E(G)|$.