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arXiv · 2406.06035

Degree-truncated choosability of planar graphs

Yiting Jiang·Huijuan Xu·Xinbo Xu·Xuding Zhu
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Abstract

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. Answering a question of Richter, it was proved in [Zhou,Zhu,Zhu, Degree-truncated choice number of graphs, arXiv:2308.15853] that there exists a 3-connected non-complete planar graph that is not degree-truncated 7-choosable, and every 3-connected non-complete planar graph is degree-truncated 16-choosable. This paper improves the bounds, and proves that there exists a 3-connected non-complete planar graph that is not degree-truncated 8-choosable, and that every 3-connected non-complete planar graph is degree-truncated $12$-choosable.

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Yiting Jiang, Huijuan Xu, Xinbo Xu, Xuding Zhu. 2024-06-10. Degree-truncated choosability of planar graphs. https://arxiv.org/abs/2406.06035

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