The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by ${\chi }_{CN}\left(G\right)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once in its closed neighborhood. Pach and Tardos showed that ${\chi }_{CN}\left(G\right)=O\left({\log }^{2+\epsilon }\mathrm{\Delta }\right)$, for any $\epsilon >0$, where $\mathrm{\Delta }$ is the maximum degree. In 2014, Glebov et al. showed existence of graphs $G$ with ${\chi }_{CN}\left(G\right)=\mathrm{\Omega }\left({\log }^2\mathrm{\Delta }\right)$. In this article, we bridge the gap between the two bounds by showing that ${\chi }_{CN}\left(G\right)=O\left({\log }^2\mathrm{\Delta }\right)$.

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关于有界度图的封闭邻域上的无冲突着色的简短说明

在封闭的邻里无冲突色数图的$G$，表示为 ${\chi }_{CN}\left(G\right)$, 是为顶点着色所需的最少颜色数 $G$这样对于每个顶点，都有一种颜色在其封闭邻域中只出现一次。Pach 和 Tardos 表明${\chi }_{CN}\left(G\right)=哦\left({\mathrm{日志}}^{2+\epsilon }\mathrm{\Delta }\right)$，对于任何 $\epsilon >0$， 在哪里 $\mathrm{\Delta }$是最大程度。2014 年，Glebov 等人。显示图形的存在$G$ 和 ${\chi }_{CN}\left(G\right)=\mathrm{\Omega }\left({\mathrm{日志}}^2\mathrm{\Delta }\right)$. 在本文中，我们通过证明来弥合这两个界限之间的差距${\chi }_{CN}\left(G\right)=哦\left({\mathrm{日志}}^2\mathrm{\Delta }\right)$.