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A graph is 1-regular if its automorphism group acts regularly on the set of its arcs. Miller [J. 138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs with girth 3 and Alspach et al. constructed infinitely many tetravalent half-transitive graphs with girth 3. Using these results, Miller's construction can be generalized… (More)

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… (More)

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… (More)

By a regular embedding of a graph into an orientable surface we mean a 2-cell embedding with the automorphism group acting regularly on arcs. In 1997 Nedela andŠkoviera [Europ. J. Comb. 18, 807-823] presented a construction giving for each solution of the congruence e 2 ≡ 1(mod n) a regular embedding M e of the hypercube Q n. It was conjectured that all… (More)