  Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $\pi :
V(H) \longrightarrow \{1,2,\ldots, k\}$, if each color class induces a
$K_2$-free subgraph. For a graph $G$ of order at least $2$, a $G$-free
$k$-coloring of $H$, is a mapping $\pi : V(H) \longrightarrow
\{1,2,\ldots,k\}$, so that the induced subgraph by each color class of $\pi$,
contains no copy of $G$. The $G$-free chromatic number of $H$, is the minimum
number $k$, so that it has a $G$-free $k$-coloring, and denoted by $\chi_G(H)$.
In this paper, we give some bounds and attributes on the $G$-free chromatic
number of graphs, in terms of the number of vertices, maximum degree, minimum
degree, and chromatic number. Our main results are the Nordhaus-Gaddum-type
theorem for the $\G$-free chromatic number of a graph.
