Semisymmetric graphs

Abstract

Let ℘ N : X ˜ → X be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection ℘ N is called p-elementary abelian. The projection ℘ N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of the automorphism group of X lifts along ℘ N , and semisymmetric if it is edge- but not vertex-transitive. The projection ℘ N is minimal semisymmetric if it cannot be written as a composition ℘ N = ℘ ℘ M of two (nontrivial) regular covering projections, where ℘ M is semisymmetric. Malnič et al. [Semisymmetric elementary abelian covers of the Möbius–Kantor graph, Discrete Math. 307 (2007) 2156–2175] determined all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius–Kantor graph, the Generalized Petersen graph GP ( 8 , 3 ) , by explicitly giving the corresponding voltage rules generating the covering projections. It was remarked at the end of the above paper that the covering graphs arising from these covering projections need not themselves be semisymmetric (a graph with regular valency is said to be semisymmetric if its automorphism group is edge- but not vertex-transitive). In this paper it is shown that all these covering graphs are indeed semisymmetric.