Let G be the class of plane graphs without triangles normally
adjacent to 8β-cycles, without 4-cycles normally adjacent to
6β-cycles, and without normally adjacent 5-cycles. In this paper, it is
showed that every graph in G is 3-choosable. Instead of proving
this result, we directly prove a stronger result in the form of "weakly"
DP-3-coloring. The main theorem improves the results in [J. Combin. Theory
Ser. B 129 (2018) 38--54; European J. Combin. 82 (2019) 102995]. Consequently,
every planar graph without 4-, 6-, 8-cycles is 3-choosable, and every
planar graph without 4-, 5-, 7-, 8-cycles is 3-choosable. In the
third section, it is proved that the vertex set of every graph in G
can be partitioned into an independent set and a set that induces a forest,
which strengthens the result in [Discrete Appl. Math. 284 (2020) 626--630]. In
the final section, tightness is considered.Comment: 19 pages, 3 figures. The result is strengthened, and a new result is
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