Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909511900463104 |
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| author | Liu, Zhengjiao Wang, Tao Yang, Xiaojing |
| author_facet | Liu, Zhengjiao Wang, Tao Yang, Xiaojing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles Liu, Zhengjiao Wang, Tao Yang, Xiaojing Combinatorics 05C15 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. |
| title | Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2502.18797 |