Hamilton Connected Derangement Graphs on Sn

We consider the derangements graph in which the vertices are permutations of f ng Two vertices are joined by an edge if the corresponding permuta tions di er in every position The derangements graph is known to be hamil tonian and it follows from a recent result of Jung that every pair of vertices is joined by a Hamilton path We use this result to settle an open question by showing that it is possible for any n and k satisfying k n and k to generate permutations of f ng so that successive permutations di er in k consecutive positions In fact the associated k consecutive derangements graph is also Hamilton connected