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arXiv · 2502.18089

Variable degeneracy of planar graphs without chorded 6-cycles

Huihui Fang·Danjun Huang·Tao Wang·Weifan Wang
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Abstract

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 $\Gamma$ 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.

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Huihui Fang, Danjun Huang, Tao Wang, Weifan Wang. 2025-02-25. Variable degeneracy of planar graphs without chorded 6-cycles. https://doi.org/10.1007/s10114-024-2245-8

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