Arc-transitive elementary abelian covers of the Pappus graph

AbstractIn 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 F18, 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(mod12), p≡1(mod12) and p≡5(mod6), 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 18p for each s≥1 and each prime p is given. From the classification of cubic s-regular graphs of order 18p we have the following: (1) apart from the two 5-regular graphs F90 and F234B and the 2-regular graph F54, all of these graphs are 1-regular with p≡1(mod6); (2) apart from F90 and F234B all of these graphs are of girth 6; (3) apart from F234B all of these graphs are bipartite