Nordhaus-Gaddum problem in term of $G$-free coloring

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.