Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles
Zhengjiao Liu, Tao Wang, Xiaojing Yang
TL;DR
This work studies planar graphs excluding certain cycles and their normal adjacencies, proving they are both $(\mathcal{I},\mathcal{F})$-partitionable and weakly $2$-degenerate. It combines a local structural lemma proved via discharging with a global framework based on valued/canonical covers and strictly $f$-degenerate transversals to construct the required partition. The results extend prior findings on cycle-restricted planar graphs and connect degeneracy bounds to graph-coloring notions through a chain of parameters related to DP-coloring. The methods provide a blueprint for deriving partitionability and weak degeneracy in other restricted planar classes.
Abstract
A graph is \emph{$(\mathcal{I}, \mathcal{F})$-partitionable} if its vertex set can be partitioned into two parts such that one part $\mathcal{I}$ is an independent set, and the other $\mathcal{F}$ induces a forest. A graph is \emph{$k$-degenerate} if every subgraph $H$ contains a vertex of degree at most $k$ in $H$. Bernshteyn and Lee defined a generalization of $k$-degenerate graphs, which is called \emph{weakly $k$-degenerate}. In this paper, we show that planar graphs without $4$-, $7$-, $9$-cycles, and $5$-cycles normally adjacent to $3$-cycles are both $(\mathcal{I}, \mathcal{F})$-partitionable and weakly $2$-degenerate.