s-Regular cyclic coverings of the three-dimensional hypercube Q3

A graph is s-regular if its automorphism group acts regularly on the set of s-arcs. An infinite family of cubic 1-regular graphs was constructed in European J. Combin. 23 (2002) 559, as cyclic coverings of the three-dimensional hypercube Q3. In this paper, we classify the s-regular cyclic coverings of Q3 for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.