Every planar graph without $i$-cycles adjacent simultaneously to $j$-cycles and $k$-cycles is DP-$4$-colorable when $\{i,j,k\}=\{3,4,5\}$

TL;DR

This paper proves that certain planar graphs with restrictions on cycle adjacencies are DP-4-colorable, extending previous results on list coloring and DP-coloring for graphs without specific cycle configurations.

Contribution

It establishes DP-4-colorability for planar graphs without $i$-cycles adjacent to $j$- and $k$-cycles when $oxed{i,j,k=3,4,5}$, generalizing prior work.

Findings

Planar graphs without specified cycle adjacencies are DP-4-colorable.

Extends previous results on list coloring to DP-coloring.

Generalizes earlier theorems on cycle restrictions in planar graphs.

Abstract

DP-coloring is a generalization of a list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that planar graphs without $k$-cycles where $k=3,4,5,$ or $6$ are DP-$4$-colorable. Recently, Kim and Yu extended the result on $3$- and $4$-cycles by showing that planar graphs without triangles adjacent to $4$-cycles are DP-$4$-colorable. Xu and Wu showed that planar graphs without $5$-cycles adjacent simultaneously to $3$-cycles and $4$-cycles are $4$-choosable. In this paper, we extend the result on $5$-cycles and triangles adjacent to $4$-cycles by showing that planar graphs without $i$-cycles adjacent simultaneously to $j$-cycles and $k$-cycles are DP-$4$-colorable when $\{i,j,k\}=\{3,4,5\}.$ This also generalizes the result of Xu and Wu.