The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

中文翻译：

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Clique 色数的严格界限

图的团色数是为其顶点着色所需的最小颜色数，以便不是孤立顶点的包含方式最大团是单色的。我们证明每个最大度数 $\Delta$ 的图都有团色数 $O\left(\frac{\Delta}{\log~\Delta}\right)$。我们得到作为推论，每个 $n$-顶点图都有团色数 $O\left(\sqrt{\frac{n}{\log ~n}}\right)$。这两个结果都是紧的。