Degree-truncated choosability of planar graphs

TL;DR

This paper advances the understanding of degree-truncated choosability in planar graphs by improving bounds on the choosability number for 3-connected non-complete planar graphs.

Contribution

It improves previous bounds, showing a 3-connected non-complete planar graph can fail degree-truncated 8-choosability and that all such graphs are degree-truncated 12-choosable.

Findings

Existence of a 3-connected non-complete planar graph not degree-truncated 8-choosable

All 3-connected non-complete planar graphs are degree-truncated 12-choosable

Improved bounds from previous results on degree-truncated choosability

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.