On solvable groups and Cayley graphs

Let φ be Eulerʹs phi function. Let n be a square-free positive integer such that gcd ( n , φ ( n ) ) = q , q a prime, and if p | n is prime, then q 2 ∤ ( p − 1 ) . We prove that a vertex-transitive graph Γ of order n is isomorphic to a Cayley graph of order n if and only if Aut ( Γ ) contains a transitive solvable subgroup.