We construct orientations of rook graphs (whose underlying graphs are claw-free) that contain no directed $C_3$ but have unbounded dichromatic number. This disproves a conjecture of Aboulker, Charbit and Naserasr and improves a result of Carbonero, Koerts, Moore and Spirkl.

翻译：我们构造了车图（其底层图是爪形无向图）的定向，这些定向不包含有向 $C_3$ 但具有无界二色数。这推翻了 Aboulker、Charbit 和 Naserasr 的一个猜想，并改进了 Carbonero、Koerts、Moore 和 Spirkl 的一个结果。