Recently, Dvo\v{r}\'ak, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [JGT 19']. In this new setting, each vertex $v$ in some subset of $V(G)$ has a request for a certain color $r(v)$ in its list of colors $L(v)$. The goal is to find an $L$ coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant $\epsilon >0$ such that any graph $G$ in some graph class $\mathcal{C}$ satisfies at least $\epsilon$ proportion of the requests. More formally, for $k > 0$ the goal is to prove that for any graph $G \in \mathcal{C}$ on vertex set $V$, with any list assignment $L$ of size $k$ for each vertex, and for every $R \subseteq V$ and a request vector $(r(v): v\in R, ~r(v) \in L(v))$, there exists an $L$-coloring of $G$ satisfying at least $\epsilon|R|$ requests. If this is true, then $\mathcal{C}$ is called $\epsilon$-flexible for lists of size $k$. Choi et al. [arXiv 20'] introduced the notion of weak flexibility, where $R = V$. We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer $b$ there exists $\epsilon(b)>0$ so that the class of planar graphs without $K_4, C_5 , C_6 , C_7, B_b$ is weakly $\epsilon(b)$-flexible for lists of size $4$ (here $K_n$, $C_n$ and $B_n$ are the complete graph, a cycle, and a book on $n$ vertices, respectively). We also show that the class of planar graphs without $K_4, C_5 , C_6 , C_7, B_5$ is $\epsilon$-flexible for lists of size $4$. The results are tight as these graph classes are not even 3-colorable.

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关于平面图中的弱灵活性

最近，Dvo\v{r}\'ak、Norin 和 Postle 引入了灵活性，作为图上列表着色的扩展 [JGT 19']。在这个新设置中，$V(G)$ 某个子集中的每个顶点 $v$ 在其颜色列表 $L(v)$ 中都有对特定颜色 $r(v)$ 的请求。目标是找到满足许多（但不一定是所有）请求的 $L$ 着色。主要研究的问题是是否存在一个通用常数 $\epsilon >0$ 使得某个图类 $\mathcal{C}$ 中的任何图 $G$ 至少满足 $\epsilon$ 比例的请求。更正式地说，对于 $k > 0$ 的目标是证明对于顶点集 $V$ 上的任何图 $G \in \mathcal{C}$，每个顶点的任何列表分配 $L$ 大小为 $k$ ，并且对于每个 $R \subseteq V$ 和一个请求向量 $(r(v): v\in R, ~r(v) \in L(v))$，存在满足至少 $\epsilon|R|$ 请求的 $G$ 的 $L$-coloring。如果这是真的，则 $\mathcal{C}$ 被称为 $\epsilon$-flexible 对于大小为 $k$ 的列表。崔等人。[arXiv 20'] 引入了弱灵活性的概念，其中 $R = V$。我们通过引入一种处理弱灵活性的工具来进一步发展这个方向。我们通过证明对于每个正整数 $b$ 都存在 $\epsilon(b)>0$ 来演示这个新工具，因此没有 $K_4, C_5 , C_6 , C_7, B_b$ 的平面图类是弱 $\epsilon (b)$-flexible 对于大小为 $4$ 的列表（这里 $K_n$、$C_n$ 和 $B_n$ 分别是完整的图、一个循环和一本关于 $n$ 顶点的书）。我们还表明，对于大小为 $4$ 的列表，没有 $K_4、C_5、C_6、C_7、B_5$ 的平面图类是 $\epsilon$-flexible。