We consider the following list coloring with separation problem of graphs: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|\le a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$ for any vertex $v$ and $\varphi(u)\cap \varphi(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth $g\ge 5$.

中文翻译：

---

循环和外平面图分离的可选择性

我们考虑以下带有图分离问题的列表着色： 给定一个图 $G$ 和整数 $a,b$，找到最大的整数 $c$，使得对于 $G$ 的任何列表赋值 $L$ 和 $|L (v)|\le a$ 对于任何顶点 $v$ 和 $|L(u)\cap L(v)|\le c$ 对于 $G$ 的任何边 $uv$，存在赋值 $\varphi $ 到 $G$ 的顶点的整数集，使得 $\varphi(u)\subset L(u)$ 和 $|\varphi(v)|=b$ 对于任何顶点 $v$ 和 $\varphi( u)\cap \varphi(v)=\emptyset$ 对于任何边 $uv$。这样的 $c$ 值称为 $(G,a,b)$ 的分离数。我们还研究了称为自由分离数的变体，它的定义类似，但假设一个任意顶点是预着色的。我们确定循环的分离数和自由分离数，并从中推导出仙人掌的自由分离数。