DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$

TL;DR

This paper proves that certain planar graphs without specific cycle lengths are DP-3-colorable, extending known list-coloring results to the more general DP-coloring framework.

Contribution

It establishes the DP-3-colorability of planar graphs lacking cycles of lengths 4, 9, and two from 5, 6, 7, 8, broadening previous list-coloring findings.

Findings

Proves DP-3-colorability for specified cycle-restricted planar graphs

Extends list-coloring results to DP-coloring setting

Identifies new classes of graphs with guaranteed DP-3-colorability

Abstract

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvo\v{r}\'{a}k and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring. We note that list-coloring results do not always extend to DP-coloring results. Our main result in this paper is to prove that every planar graph without cycles of length $\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8\}$ is DP-$3$-colorable, extending three existing results on $3$-choosability of planar graphs.