DP-3-Coloring of Planar Graphs Without 4, 9-Cycles and Cycles of Two Lengths from $\{6,7,8\}$

Abstract

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle (Comb Theory Ser B 129:38–54, 2018). 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 (Liu and Li, Discrete Math 342:623–627, 2019; Liu et al., Discrete Math 342(1):178–189, 2019; Kim and Ozeki, A note on a Brooks type theorem for DP-coloring, arXiv:1709.09807, 2019; Kim and Yu, Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, arXiv:1712.08999, 2019; Sittitrai and Nakprasit, Every planar graph without i-cycles adjacent simultaneously to j-cycles and k-cycles is DP-4-colorable when $\{i,j,k\}=\{3,4,5\}$, arXiv:1801.06760, 2019; Yin and Yu, Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable, arXiv:1809.00925, 2019). We note that list-coloring results do not always extend to DP-coloring results, as shown in Bernshteyn and Kostochka (On differences between DP-coloring and list coloring, arXiv:1705.04883, 2019). 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 (Shen and Wang, Inf Process Lett 104:146–151, 2007; Wang and Shen, Discrete Appl Math 159:232–239, 2011; Whang et al., Inf Process Lett 105:206–211, 2008) on 3-choosability of planar graphs.