For a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u,v)$-paths over all distinct pairs of vertices $(u,v)$ in $G$. Stiebitz and Toft (2018) proved a Brooks-type theorem for graphs with maximum local edge-connectivity $k$, showing that a graph with maximum local edge-connectivity $k$ is not $k$-colourable if and only if it has a block in $\mathcal{H}_k$, which is the class of graphs that can be obtained by taking Hajós joins of copies of $K_{k+1}$ and, when $k=3$, odd wheels. We prove that a $2$-connected graph with maximum local edge-connectivity $k$ is $k$-choosable if and only if it is not in $\mathcal{H}_k$. On the other hand, deciding $k$-choosability when restricted to graphs with maximum local edge-connectivity $k$ (that might not be $2$-connected) is $Π_2$-complete. To prove the former result, we first prove several generalisations of a well-known characterisation of degree-choosability; these may be of independent interest.

翻译：对于至少包含两个顶点的图$G$，其最大局部边连通度定义为$G$中所有不同顶点对$(u,v)$之间边不交$(u,v)$-路径的最大数量。Stiebitz与Toft（2018）证明了关于最大局部边连通度为$k$的图的一种Brooks型定理，指出此类图不可$k$-着色当且仅当其包含一个属于$\mathcal{H}_k$的块，其中$\mathcal{H}_k$是由$K_{k+1}$的副本（当$k=3$时还包括奇轮图）通过Hajós并运算所得的图类。我们证明：最大局部边连通度为$k$的$2$-连通图是$k$-可选择的当且仅当其不属于$\mathcal{H}_k$。另一方面，当限制在（可能非$2$-连通的）最大局部边连通度为$k$的图时，判定其$k$-可选择性问题是$\Pi_2$-完全的。为证明前一结论，我们首先推广了关于度可选择性的一种著名刻画，这些推广结果本身可能具有独立意义。