For a given $\varepsilon > 0$, we say that a graph $G$ is $\epsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of lists of size $k$ on $V(G)$, if a preferred color is requested at any set $R$ of vertices, then at least $\epsilon |R|$ of these requests may be satisfied by some $L$-coloring. We consider flexible list colorings in several graph classes with certain special degeneracy conditions. We characterize the graphs of maximum degree $\Delta$ that are $\epsilon$-flexibly $\Delta$-choosable for some $\epsilon = \epsilon(\Delta) > 0$, which answers a question of Dvo\v{r}\'ak, Norin, and Postle [List coloring with requests, JGT 2019]. We also show that graphs of treewidth $2$ are $\frac{1}{3}$-flexibly $3$-choosable, answering a question of I. Choi et al. [arXiv 2020], and we give conditions for list assignments by which graphs of treewidth $k$ are $\frac{1}{k+1}$-flexibly $(k+1)$-choosable. We show furthermore that graphs of treedepth $k$ are $\frac{1}{k}$-flexibly $k$-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, three-connected non-regular graphs of maximum degree $\Delta$ are flexibly $(\Delta - 1)$-degenerate.

中文翻译：

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具有特殊简并条件的图形中的灵活列表着色

对于给定的 $\varepsilon > 0$，我们说一个图 $G$ 是 $\epsilon$-灵活的 $k$-choosable，如果满足以下条件：对于 $k$ 上的大小为 $k$ 的列表的任何赋值 $L$ V(G)$，如果在顶点的任何集合 $R$ 处请求首选颜色，那么这些请求中至少 $\epsilon |R|$ 可以通过一些 $L$ 着色来满足。我们在具有某些特殊简并条件的几个图类中考虑灵活的列表着色。我们刻画了最大度数 $\Delta$ 的图，对于某些 $\epsilon = \epsilon(\Delta) > 0$，$\epsilon$-flexible $\Delta$-choosable，这回答了一个问题 Dvo\v{ r}\'ak、Norin 和 Postle [带请求的列表着色，JGT 2019]。我们还表明，树宽 $2$ 的图是 $\frac{1}{3}$-flexably $3$-choosable，回答了 I. Choi 等人的问题。[arXiv 2020]，并且我们给出了列表分配的条件，其中树宽 $k$ 的图是 $\frac{1}{k+1}$-flexably $(k+1)$-choosable。我们进一步表明，树深度 $k$ 的图是 $\frac{1}{k}$-灵活的 $k$-choosable。最后，我们引入了灵活简并的概念，它增强了灵活的选择性，并且我们表明，除了一类很好理解的异常之外，最大度 $\Delta$ 的三连通非正则图是灵活的 $(\Delta - 1)$-退化。