  This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of
$G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable,
and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$
contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the
other hand, we prove that every planar graph $G$ contains a forest $F$ such
that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$
is 3-paintable and 3-choosable.
