Cubic symmetric graphs of order a small number times a prime or a prime square

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K 3 , 3 and the s-regular cyclic or elementary abelian coverings of the complete graph K 4 for each s ⩾ 1 are classified when the fibre-preserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regular graphs with girth 12 is found, in which the smallest one has order 2058. As an interesting application, a complete list of pairwise non-isomorphic s-regular cubic graphs of order 4p, 6p, 4 p 2 or 6 p 2 is given for each s ⩾ 1 and each prime p.