  It was conjectured by Ohba, and proved by Noel, Reed and Wu that
$k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are chromatic-choosable. This
upper bound on $|V(G)|$ is tight: if $k$ is even, then $K_{3 \star (k/2+1), 1
\star (k/2-1)}$ and $K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $2
k+2$ vertices that are not chromatic-choosable. It was proved in
[arXiv:2201.02060] that these are the only non-$k$-choosable complete
$k$-partite graphs with $2k+2$ vertices. For $G =K_{3 \star (k/2+1), 1 \star
(k/2-1)}$ or $K_{4, 2 \star (k-1)}$, a bad list assignment of $G$ is a $k$-list
assignment $L$ of $G$ such that $G$ is not $L$-colourable. Bad list assignments
for $G=K_{4, 2 \star (k-1)}$ were characterized in [Discrete Mathematics 244
(2002), 55-66]. In this paper, we first give a simpler proof of this result,
and then we characterize bad list assignments for $G=K_{3 \star (k/2+1), 1
\star (k/2-1)}$. Using these results, we characterize all non-$k$-choosable
(non-complete) $k$-partite graphs with $2k+2$ vertices.
