Symmetric cubic graphs with solvable automorphism groups

A cubic graph Gamma is G-arc-transitive if G <= Aut( Gamma) acts transitively on the set of arcs of Gamma, and G-basic if Gamma is G-arc-transitive and G has no non-trivial normal subgroup with more than two orbits. Let G be a solvable group. In this paper, we first classify all connected G-basic cubic graphs and determine the group structure for every G. Then, combining covering techniques, we prove that a connected cubic G-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type 2(2), that is, a 2-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order 6. (C) 2014 Elsevier Ltd. All rights reserved.