Dihedral covers of the complete graph \(K_5\) (Q2921639)

The main result obtained in this paper is a classification of all regular covers of the complete graph \(K_5\) for which the covering transformation group is a dihedral group and whose fibre-preserving subgroup of the automorphism group of the cover graph is arc-transitive. Even though, this is a very specific result. It fits well within the overall body of work in this area, where we already have a complete classification of the dihedral coverings of the complete graph \(K_4\) as well as a number of results devoted to coverings by abelian groups (with or without the additional condition that the fibre-preserving subgroup acts arc-transitively). The class of graphs classified in the paper turns out to be rather limited -- all graphs satisfying the above requirements are obtained from a single voltage assignment of six different elements formed from the generators of the dihedral group (the assignment is even independent of the order of the dihedral group in use). Essentially, the entire paper is taken by a case-by-case proof of the classification.