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.