  The $k$th power $G^k$ of a graph $G$ is the graph defined on $V(G)$ such that
two vertices $u$ and $v$ are adjacent in $G^k$ if the distance between $u$ and
$v$ in $G$ is at most $k$. Let $\chi(H)$ and $\chi_l(H)$ be the chromatic
number and the list chromatic number of $H$, respectively. A graph $H$ is
called {\em chromatic-choosable} if $\chi_l (H) = \chi(H)$. It is an
interesting problem to find graphs that are chromatic-choosable. A natural
question raised by Xuding Zhu (2012) is whether there exists a constant integer
$k$ such that $G^k$ is chromatic-choosable for every graph $G$. Motivated by
the List Total Coloring Conjecture, Kostochka and Woodall (2001) asked whether
$G^2$ is chromatic-choosable for every graph $G$. Kim and Park (2013) answered
the Kostochka and Woodall's question in the negative by finding a family of
graphs whose squares are complete multipartite graphs with partite sets of
equal and unbounded size. In this paper, we answer Zhu's question by showing
that for every integer $k \geq 2$, there exists a graph $G$ such that $G^k$ is
not chromatic-choosable. Moreover, for any fixed $k$ we show that the value
$\chi_l(G^k) - \chi(G^k)$ can be arbitrarily large.
