# DP-colorings of graphs with high chromatic number

## TL;DR

This paper proves that for large graphs with chromatic number nearly equal to the number of vertices, the DP-chromatic number matches the chromatic number, highlighting a key difference from list coloring.

## Contribution

It establishes a tight bound showing DP-coloring equals chromatic number for graphs with high chromatic number, extending understanding of DP-coloring's behavior.

## Key findings

- DP-chromatic number equals chromatic number for graphs with hi(G) e n - O( n)
- Lower bound on hi_{DP}(G) is tight up to constant factors
- Contrast with list coloring where the bound is not tight

## Abstract

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 $\chi(G)\geq n-O(\sqrt{n})$, 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.