# On differences between DP-coloring and list coloring

## TL;DR

This paper explores the differences between DP-coloring and list coloring, highlighting unique properties of DP-coloring through examples and bounds, and establishing new lower bounds for edge-DP-chromatic numbers.

## Contribution

It identifies properties that distinguish DP-coloring from list coloring and provides new bounds for the edge-DP-chromatic number of regular graphs.

## Key findings

- Existence of a planar bipartite graph with DP-chromatic number 4
- Edge-DP-chromatic number of a d-regular graph is at least d+1 for d ≥ 2
- DP-coloring exhibits properties not shared with list coloring

## Abstract

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'{a}k and Postle. 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 unusual 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 $d\geq 2$ is always at least $d+1$.