Variable degeneracy of planar graphs without chorded 6-cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929731450961920 |
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| author | Fang, Huihui Huang, Danjun Wang, Tao Wang, Weifan |
| author_facet | Fang, Huihui Huang, Danjun Wang, Tao Wang, Weifan |
| contents | A cover of a graph $G$ is a graph $H$ with vertex set $V(H) = \bigcup_{v \in V(G)} L_{v}$, where $L_{v} = \{v\} \times [s]$, and the edge set $M = \bigcup_{uv \in E(G)} M_{uv}$, where $M_{uv}$ is a matching between $L_{u}$ and $L_{v}$. A vertex set $T \subseteq V(H)$ is a transversal of $H$ if $|T \cap L_{v}| = 1$ for each $v \in V(G)$. Let $f$ be a nonnegative integer valued function on the vertex-set of $H$. If for any nonempty subgraph $Γ$ of $H[T]$, there exists a vertex $x \in V(H)$ such that $d(x) < f(x)$, then $T$ is called a strictly $f$-degenerate transversal. In this paper, we give a sufficient condition for the existence of strictly $f$-degenerate transversal for planar graphs without chorded $6$-cycles. As a consequence, every planar graph without subgraphs isomorphic to the configurations in Fig. 4 is DP-$4$-colorable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_18089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variable degeneracy of planar graphs without chorded 6-cycles Fang, Huihui Huang, Danjun Wang, Tao Wang, Weifan Combinatorics 05C15 A cover of a graph $G$ is a graph $H$ with vertex set $V(H) = \bigcup_{v \in V(G)} L_{v}$, where $L_{v} = \{v\} \times [s]$, and the edge set $M = \bigcup_{uv \in E(G)} M_{uv}$, where $M_{uv}$ is a matching between $L_{u}$ and $L_{v}$. A vertex set $T \subseteq V(H)$ is a transversal of $H$ if $|T \cap L_{v}| = 1$ for each $v \in V(G)$. Let $f$ be a nonnegative integer valued function on the vertex-set of $H$. If for any nonempty subgraph $Γ$ of $H[T]$, there exists a vertex $x \in V(H)$ such that $d(x) < f(x)$, then $T$ is called a strictly $f$-degenerate transversal. In this paper, we give a sufficient condition for the existence of strictly $f$-degenerate transversal for planar graphs without chorded $6$-cycles. As a consequence, every planar graph without subgraphs isomorphic to the configurations in Fig. 4 is DP-$4$-colorable. |
| title | Variable degeneracy of planar graphs without chorded 6-cycles |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2502.18089 |