When the vertices and edges of a graph $G$ are colored with $k$ colors, an edge is called monochromatic if the edge and the two vertices incident with it all have the same color. The adapted chromatic number of $G$, $\chi _{a}\left( G\right)$, is the least integer $k$ such that for each $k$-edge coloring of $G$ the vertices of $G$ can be colored with the same set of colors without creating any monochromatic edges. It is known that the chromatic number of $G$ is a tight upper bound of the adapted chromatic number. In this talk we present results concerning the lower bound of adapted chromatic

number of a graph $G$ in terms of the chromatic number of $G$.