Classifying cubic symmetric graphs of order 10p or 10p2

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ⩾ 1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s-regular cubic graphs of order 10p or 10p ² are also classified for each s ⩾ 1 and each prime p, of which the proof depends on the classification of finite simple groups.