Arc-transitive elementary abelian covers of the Pappus graph

In this paper, all pairwise non-isomorphic p-elementary abelian covering projections admitting a lift of an arc-transitive subgroup of the full automorphism group of the Pappus graph F 18 , the unique connected cubic symmetric graph of order 18, are constructed. The number of such covering projections is equal to 5, 19, 9, 11 and 5 if p = 2 , p = 3 , p ≡ 7 ( mod 12 ) , p ≡ 1 ( mod 12 ) and p ≡ 5 ( mod 6 ) , respectively. As results, three infinite families of cubic s -regular graphs for s = 1 , 2 and 3 are constructed, and a classification of the cubic s -regular graphs of order 18 p for each s ≥ 1 and each prime p is given. From the classification of cubic s -regular graphs of order 18 p we have the following: (1) apart from the two 5-regular graphs F 90 and F 234 B and the 2-regular graph F 54 , all of these graphs are 1-regular with p ≡ 1 ( mod 6 ) ; (2) apart from F 90 and F 234 B all of these graphs are of girth 6; (3) apart from F 234 B all of these graphs are bipartite.