Extremal results on $G$-free colorings of graphs

Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $\pi : V(H) \longrightarrow \{1,2,\ldots, k\}$ so that 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 subgraph of $H$ induced by each color class of $\pi$ is $G$-free, i.e. contains no copy of $G$. The $G$-free chromatic number of $H$ is the minimum number $k$ so that there is a $G$-free $k$-coloring of $H$, denoted by $\chi_G(H)$. A graph $H$ is uniquely $k$-$G$-free colouring if $\chi_G(H)=k$ and every $k$-$G$-free colouring of $H$ produces the same color classes. A graph $H$ is minimal with respect to $G$-free, or $G$-free-minimal, if for every edges of $E(H)$ we have $\chi_G(H\setminus\{e\})= \chi_G(H)-1$. In this paper we give some bounds and attribute about uniquely $k$-$G$-free colouring and $k$-$G$-free-minimal.