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 $m$.

中文翻译：

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3 选关键图的多重列表着色

如果$G$ 不是$2$-choosable 但任何适当的子图$2$-choosable，则图$G$ 被称为$3$-choice critical。Voigt 在 [On list Colorings and Choosability of Graphs, Habilitationsschrift, Tu Ilmenau (1998)] 中给出了 $3$-choice 临界图的特征。Voigt 推测，如果 $G$ 是一个二部 $3$-choice 临界图，则 $G$ 对于每个整数 $m$ 都是 $(4m, 2m)$-choosable。Meng、Puleo 和 Zhu 在 [On (4, 2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)] 中反驳了这一猜想。他们表明，如果 $G=\Theta_{r,s,t}$ where $r,s,t$ 具有相同的奇偶性并且 $\min\{r,s,t\} \ge 3$，或 $G =\Theta_{2,2,2,2p}$ 与 $p \ge 2$，则 $G$ 是二分 $3$-choice 关键，但不是 $(4,2)$-choosable。另一方面，所有其他二分 3-choice 临界图都是 $(4,2)$-choosable。