We prove that for every $m\in \mathbb{N}$ and every $\delta \in (-m,0)$, the chromatic number of the preferential attachment graph $P{A}_t(m,\delta )$ is asymptotically almost surely equal to $m+1$. The proof relies on a combinatorial construction of a family of digraphs of chromatic number $m+1$ followed by a proof that asymptotically almost surely there is a digraph in this family, which is realised as a subgraph of the preferential attachment graph.

中文翻译：

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关于优先依附图的色数

我们证明对于每个 $米\in \mathbb{N}$ 并且每 $\delta \in (-米,0)$, 优先附着图的色数 $磷{\mathrm{一种}}_{吨}(米,\delta )$ 渐近地几乎肯定等于 $米+1$. 证明依赖于色数有向图族的组合构造$米+1$ 随后证明渐近几乎肯定在这个族中有一个有向图，它被实现为优先连接图的一个子图。