Antidirected paths in 5-chromatic digraphs

Let $T_5$ be the regular 5-tournament. B. Grünbaum proved that $T_5$ is the only 5-tournament which contains no copy of the antidirected path $P_4$. In this Note, we prove that, except for $T_5$, any connected 5-chromatic oriented digraph in which each vertex has out-degree at least two contains a copy of $P_4$. It will be shown, by an example, that the condition that each vertex has out-degree at least two is indispensable. To cite this article: A. El Sahili, C. R. Acad. Sci. Paris, Ser. I 339 (2004).