# Planar graphs without normally adjacent short cycles

## TL;DR

This paper proves that certain classes of planar graphs with specific cycle restrictions are 3-choosable, using a stronger form of DP-coloring, and also shows their vertices can be partitioned into an independent set and a forest.

## Contribution

It introduces a new proof technique for 3-choosability of graphs with cycle restrictions and extends previous results, also providing vertex partition results.

## Key findings

- Graphs in class  are 3-choosable.
- Planar graphs without certain cycles are 3-choosable.
- Vertices can be partitioned into an independent set and a forest.

## Abstract

Let $\mathscr{G}$ be the class of plane graphs without triangles normally adjacent to $8^{-}$-cycles, without $4$-cycles normally adjacent to $6^{-}$-cycles, and without normally adjacent $5$-cycles. In this paper, it is shown that every graph in $\mathscr{G}$ is $3$-choosable. Instead of proving this result, we directly prove a stronger result in the form of ``weakly'' DP-$3$-coloring. The main theorem improves the results in [J. Combin. Theory Ser. B 129 (2018) 38--54; European J. Combin. 82 (2019) 102995]. Consequently, every planar graph without $4$-, $6$-, $8$-cycles is $3$-choosable, and every planar graph without $4$-, $5$-, $7$-, $8$-cycles is $3$-choosable. In the third section, using almost the same technique, we prove that the vertex set of every graph in $\mathscr{G}$ can be partitioned into an independent set and a set that induces a forest, which strengthens the result in [Discrete Appl. Math. 284 (2020) 626--630]. In the final section, tightness is discussed.