Abstract: It is known that the DP-chromatic number (also called correspondence chromatic number), $\chi_{DP}(G)$, and the online chromatic number (also called paintability), $\chi_{P}(G)$, of a graph $G$ are both at least the list chromatic number (also called choosability), $\chi_{\ell}(G)$, and can be significantly larger. The goal of the talk is twofold. First, we present examples of graphs $G$ with $\chi_P(G)>\chi_{DP}(G)$ (but only by $1$). Second, we introduce online DP-coloring of graphs and the online DP-chromatic number, $\chi_{DPP}(G)$. This parameter is an upper bound for both, $\chi_{P}(G)$ and $\chi_{DP}(G)$, but still has good properties of colorings: $\chi_{DPP}(G)$ is at most the degeneracy of $G$ plus $1$, a version of Brooks' Theorem holds for it, and every planar graph is online DP-colorable with $5$ colors.
This is joint work with S.-J. Kim, X. Li and X. Zhu.