Planar graphs having no cycle of length $4$, $6$ or $8$ are DP-3-colorable

TL;DR

This paper proves that planar graphs without cycles of lengths 4, 6, or 8 are DP-3-colorable, confirming a stronger conjecture and extending previous results on graph colorability.

Contribution

It establishes that such planar graphs are DP-3-colorable, a stronger result than previously known 3-choosability, confirming a conjecture by Dvořák and Postle.

Findings

Planar graphs with no cycles of length 4, 6, or 8 are DP-3-colorable.

The result extends the understanding of graph colorability under cycle restrictions.

Confirms a stronger conjecture on DP-coloring for these graphs.

Abstract

The concept of DP-coloring of graphs was introduced by Dvo\v{r}\'{a}k and Postle, and was used to prove that planar graphs without cycles of length from $4$ to $8$ are $3$-choosable. In the same paper, they proposed a more natural and stronger claim that such graphs are DP-$3$-colorable. This paper confirms that claim by proving a stronger result that planar graphs having no cycle of length $4$, $6$ or $8$ are DP-3-colorable.