A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the '(partial) conflict-free chromatic number' of $\mathcal{H}$. It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ with maximum degree $Δ$ is at most $O(\ln^2 Δ)$. In this paper, we extend the $O(\ln^2 Δ)$ upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers.

翻译：超图 $\mathcal{H}$ 的"（部分）冲突避免着色"是指对 $\mathcal{H}$ 的顶点集（的子集）进行颜色分配，使得 $\mathcal{H}$ 中的每条超边均存在一个顶点，其颜色与该超边内所有其他顶点的颜色均不相同。实现此类着色所需的最小颜色数称为 $\mathcal{H}$ 的"（部分）冲突避免色数"。易见超图的冲突避免色数至多比其部分冲突避免色数大1。冲突避免着色问题亦在给定图的开/闭邻域超图框架下被研究，称为开/闭邻域冲突避免着色。本文研究冲突避免着色的部分列表变体与全列表变体，其中对每个顶点 $v$ 给定可接受颜色列表 $L_v$，$v$ 仅允许从 $L_v$ 中选取颜色。Bhyravarapu、Kalyanasundaram 与 Mathew [《图论杂志》, 2021] 证明：任意最大度为 $\Delta$ 的图 $G$ 的闭邻域冲突避免色数至多为 $O(\ln^2 \Delta)$。本文将 $O(\ln^2 \Delta)$ 上界推广至闭邻域冲突避免色数的部分列表变体。进一步，我们建立了关于列表开/闭邻域冲突避免色数的计算复杂性结果。