Hamiltonicity of digraphs for universal cycles of permutations

The digraphs P (n, k) have vertices corresponding to length k permutations of an n set and arcs corresponding to (k + 1) permutations. Answering a question of Starling, Klerlein, Kier and Carr we show that these digraphs are Hamiltonian for k ≤ n − 3. We do this using restricted Eulerian cycles and the fact that P (n, k) is nearly the line digraph of P (n, k−1). We also show that the digraphs P (n, n−2) are not Hamiltonian for n ≥ 4 using a result of Rankin on Cayley digraphs.