  A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any
proper subgraph is $2$-choosable. A characterization of $3$-choice critical
graphs was given by Voigt in [On list Colourings and Choosability of Graphs,
Habilitationsschrift, Tu Ilmenau(1998)]. Voigt conjectured that if $G$ is a
bipartite $3$-choice critical graph, then $G$ is $(4m, 2m)$-choosable for every
integer $m$. This conjecture was disproved by Meng, Puleo and Zhu in [On (4,
2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)]. They showed
that if $G=\Theta_{r,s,t}$ where $r,s,t$ have the same parity and
$\min\{r,s,t\} \ge 3$, or $G=\Theta_{2,2,2,2p}$ with $p \ge 2$, then $G$ is
bipartite $3$-choice critical, but not $(4,2)$-choosable. On the other hand,
all the other bipartite 3-choice critical graphs are $(4,2)$-choosable. This
paper strengthens the result of Meng, Puleo and Zhu and shows that all the
other bipartite $3$-choice critical graphs are $(4m,2m)$-choosable for every
integer $m$.
