A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, we classify the s-regular elementary Abelian coverings of the three-dimensional hypercube for each s ≥ 1 whose fibre-preserving automorphism subgroups act arc-transitively. This gives a new infinite family of cubic 1-regular graphs, in which the smallest one has order 19 208. As an application...

Let S be a set of transpositions generating the symmetric group Sn, where n ≥ 3. It is shown that if the girth of the transposition graph of S is at least 5, then the automorphism group of the Cayley graph Cay(Sn, S) is the direct product Sn×Aut(T (S)), where T (S) is the transposition graph of S; the direct factors are the right regular representation of Sn and the image of the left regular ac...