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arXiv · 2502.18797

Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles

Zhengjiao Liu·Tao Wang·Xiaojing Yang
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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.

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Zhengjiao Liu, Tao Wang, Xiaojing Yang. 2025-02-26. Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles. https://doi.org/10.1016/j.dam.2024.07.003

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