Hamilto-connected derangement graphs on S n

We consider the derangements graph in which the vertices are permutations of f1 : : : ng. Two vertices are joined by an edge if the corresponding permutations diier 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 2 k n and k 6 = 3, to generate permutations of f1 : : : ng so that successive permutations diier in k consecutive positions. In fact, the associated k-consecutive derangements graph is also Hamilton-connected.