Hamilton Powers of Eulerian Digraphs

Powers of graphs are well-studied combinatorial objects; it is well-known that the cube of an undirected graph is Hamiltonian, and Fleischner's theorem states that the square of a two-connected undirected graph is also Hamiltonian, resolving a conjecture of Plummer-Nash-Williams. In this note, we prove that the 1/2 √n log₂² n ᵗʰ power of a connected n-vertex Eulerian digraph is Hamiltonian, and provide an infinite family of digraphs for which the √n/2 ᵗʰ power is not.