On differences between DP-coloring and list coloring

DP-Coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle [12]. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq2$ is always at least $d+1$.