ON GRAPHS OF PRIME VALENCY ADMITTING A SOLVABLE ARC-TRANSITIVE GROUP

Let X be a simple, connected, p-valent, G-arc-transitive graph, where the subgroup G <= Aut(X) is solvable and p >= 3 is a prime. We prove that X is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that G is at most 3-arc-transitive on X and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all G-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.