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