Hamilton Powers of Eulerian Digraphs

In this note, we prove that the $\lceil \frac{1}{2}\sqrt{n}{\log }_2^{2}n{\rceil }^{th}$ power of a connected $n$-vertex Eulerian digraph is Hamiltonian, and provide an infinite family of digraphs for which the $\lfloor \sqrt{n}\mathrm{/}2{\rfloor }^{th}$ power is not.