Bad list assignments for non-$k$-choosable $k$-chromatic graphs with $2k+2$-vertices

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.