The conflict-free closed neighborhood (CFCN$^*$) chromatic number of a graph $G = (V,E)$ is the smallest positive integer $k$ for which there exists a coloring of a subset of vertices using $k$ colors such that, for every vertex in $V$, there exists a color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) chromatic number is defined analogously. In this paper, we study `list variants' of the above-mentioned coloring parameters. The conflict-free closed neighborhood (CFCN$^*$) choice number of a graph $G = (V,E)$ is the smallest positive integer $k$ such that for every assignment of lists of size $k$ to its vertices, there exists a coloring of a subset of vertices, say $V'$, in which (i) every vertex in $V'$ receives a color from its list, and (ii) for every vertex in $V$ there exists some color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) choice number is defined analogously. Dębski and Przybyło [Journal of Graph Theory, 2022] showed that for any graph $G$ with maximum degree $Δ$, the CFCN$^*$ chromatic number of its line graph is $O(\ln Δ)$. This result was later extended to claw-free graphs by Bhyravarapu et al. [Journal of Graph Theory, 2025], who proved that every $K_{1,k}$-free graph $G$ admits a CFCN$^*$ coloring using $O(k\ln Δ)$ colors. In this paper, we generalize this result to the list setting and show that every $K_{1,k}$-free graph $G$ has a CFCN$^*$ choice number of $O(k\ln Δ)$. Further, we answer some questions concerning the hardness of computing CFCN$^*$/CFON$^*$ choice numbers posed by Gupta and Mathew [SOFSEM, 2026]; in particular, we show that it is NP-hard to determine whether the CFCN$^*$/CFON$^*$ choice number a graph is equal to $k$, for $k=1,2$.

翻译：图 $G = (V,E)$ 的无冲突闭邻域（CFCN$^*$）色数是指存在一个用 $k$ 种颜色对部分顶点进行着色，使得 $V$ 中每个顶点的闭邻域内恰好有一种颜色出现一次的最小正整数 $k$。无冲突开邻域（CFON$^*$）色数的定义类似。本文研究上述着色参数的“列表变体”。图 $G = (V,E)$ 的无冲突闭邻域（CFCN$^*$）列数是指最小正整数 $k$，使得对每个顶点分配大小为 $k$ 的列表后，存在一个部分顶点子集 $V'$ 的着色满足：(i) $V'$ 中每个顶点从其列表中选取一种颜色，且 (ii) 对 $V$ 中每个顶点，其闭邻域内恰好有一种颜色出现一次。无冲突开邻域（CFON$^*$）列数的定义类似。Dębski 与 Przybyło [Journal of Graph Theory, 2022] 证明，对任意最大度为 $Δ$ 的图 $G$，其线图的 CFCN$^*$ 色数为 $O(\ln Δ)$。该结果后被 Bhyravarapu 等人 [Journal of Graph Theory, 2025] 推广至无爪图，他们证明每个 $K_{1,k}$-自由图 $G$ 均存在用 $O(k\ln Δ)$ 种颜色的 CFCN$^*$ 着色。本文将该结果推广至列表情形，证明每个 $K_{1,k}$-自由图 $G$ 的 CFCN$^*$ 列数为 $O(k\ln Δ)$。此外，我们回应了 Gupta 与 Mathew [SOFSEM, 2026] 提出的关于计算 CFCN$^*$/CFON$^*$ 列数难度的若干问题，特别地，证明了判断图的 CFCN$^*$/CFON$^*$ 列数是否等于 $k$（$k=1,2$）是 NP-困难的。