# Planar graphs without 7-cycles and butterflies are DP-4-colorable

## TL;DR

This paper proves that planar graphs lacking 7-cycles and butterflies are DP-4-colorable, extending known results for graphs without smaller cycles and clusters of triangles, using a proof adaptable to other forbidden structures.

## Contribution

It establishes a new sufficient condition for DP-4-colorability of planar graphs by forbidding 7-cycles and butterflies, broadening the class of graphs known to be DP-4-colorable.

## Key findings

- Planar graphs without 7-cycles and butterflies are DP-4-colorable.
- The proof method can be adapted to other forbidden cluster conditions.
- Extends previous results on DP-colorability for graphs without smaller cycles.

## Abstract

DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvo\v{r}\'ak and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP-$4$-colorable. In particular, for each $k \in \{3, 4, 5, 6\}$, every planar graph without $k$-cycles is DP-$4$-colorable. In this paper, we prove that every planar graph without $7$-cycles and butterflies is DP-$4$-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters formed by many triangles.