A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph $G$ is at most a given number $k$. In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.

翻译：图 $G$ 的无冲突着色是其顶点的一个（部分）着色，使得每个顶点 $u$ 在其邻域内存在一个邻居，其分配的颜色在该邻域中是唯一的。该着色有两种变体，一种基于开邻域定义，另一种基于闭邻域定义。针对这两种变体，我们研究了判定给定图 $G$ 是否可以用不超过给定数 $k$ 种颜色进行无冲突着色的问题。本文研究了团宽与所需最小颜色数（针对两种变体）之间的关系，并表明这两个参数并不能相互界定。此外，我们考虑了特定图类，特别是有界团宽图和各种交图，例如距离遗传图、区间图以及单位正方形图和圆盘图。我们还考虑了Kneser图和分裂图。我们给出了（通常是紧的）上下界，并确定了这些图类上决策问题的复杂性，改进了文献中的部分结果。特别地，我们解决了区间图在开邻域模型下进行无冲突着色所需颜色数的问题，该问题曾被作为公开问题提出。