  For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the
chromatic number of $(T, \mathcal{B})$ is the maximum of $\chi(G[B])$, taken
over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the
minimum chromatic number of all tree-decompositions $(T, \mathcal{B})$ of $G$.
The path-chromatic number of $G$ is defined analogously. In this paper, we
introduce an operation that always increases the path-chromatic number of a
graph. As an easy corollary of our construction, we obtain an infinite family
of graphs whose path-chromatic number and tree-chromatic number are different.
This settles a question of Seymour. Our results also imply that the
path-chromatic numbers of the Mycielski graphs are unbounded.
