Hamiltonicity of digraphs for universal cycles of permutations

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