  A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of
$k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that
assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of
deciding whether a given graph is $k$-choosable for some constant $k$. In
particular, it is shown that deciding whether a given planar graph is
4-choosable is NP-hard, and so is the problem of deciding whether a given
planar triangle-free graph is 3-choosable. We also obtain simple constructions
of a planar graph which is not 4-choosable and a planar triangle-free graph
which is not 3-choosable.
