Classifying cubic symmetric graphs of order or

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 of the classification, all cubic symmetric graphs of order 8 p or 8 p 2 are classified for each prime p , as a continuation of the first two authors’ work, in Y.-Q. Feng, J.H. Kwak [Cubic symmetric graphs of order a small number times a prime or a prime square (submitted for publication)] in which all cubic symmetric graphs of order 4 p , 4 p 2 , 6 p or 6 p 2 are classified and of Cheng and Oxley’s classification of symmetric graphs of order 2 p , in Y. Cheng, J. Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42 (1987) 196–211].