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

Abstract

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák 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.