DP-colorings of graphs with high chromatic number

DP-coloring is a generalization of list coloring introduced recently by Dvo\v{r}'ak and Postle. We prove that for every $n$-vertex graph $G$ whose chromatic number $\chi (G)$ is “close” to $n$, the DP-chromatic number of $G$ equals $\chi (G)$. “Close” here means $\chi (G)\ge n-O(\sqrt{n})$, and we also show that this lower bound is best possible (up to the constant factor in front of $\sqrt{n}$), in contrast to the case of list coloring.