Introduction to dominated edge chromatic number of a graph

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by ${\chi }_{dom}^{\mathrm{\prime }}(G)$. We obtain some properties of ${\chi }_{dom}^{\mathrm{\prime }}(G)$ and compute it for specific graphs. Also we examine the effects on ${\chi }_{dom}^{\mathrm{\prime }}(G)$ when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.