Let $MFVS(G)$ denote the size of minimum feedback vertex set of $G$.

We believe we proved $\chi(G) \ge (|G| - MFVS(\overline{G}))/2$ and this bound is sharp.

Is this known or trivial result?

This is standard notation and to address comments $|G|$ is the number of vertices of $G$, $\overline{G}$ is the complement of $G$ and feedback vertex set $S$ is subset of $V(G)$ such that deleting $S$ from $G$ leaves acyclic graph.