Hamilton Powers of Eulerian Digraphs

In this note, we prove that the $\lceil \tfrac{1}{2} \sqrt{n} \log_2^2 n \rceil^{th}$ power of a connected $n$-vertex Eulerian digraph is Hamiltonian, and provide an infinite family of digraphs for which the $\lfloor \sqrt{n}/2 \rfloor^{th}$ power is not.