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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 minimumnumber $k$, 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 chromaticnumber of graphs, in terms of the number of vertices, maximum degree, minimumdegree, and chromatic number. Our main results are the Nordhaus-Gaddum-typetheorem for the $\G$-free chromatic number of a graph.

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