  We prove that for every $m\in \mathbb N$ and every $\delta\in (-m,0)$, the
chromatic number of the preferential attachment graph $PA_t(m, \delta)$ 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.
