Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2pq, where p, q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2pq are classified. This depends on the finite simple group classification. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2pq is a non-Cayley number, where 2<q<p and q and p are primes, if and only if one of p = 1 (mod 4), or q = 1 (mod 4), or p = 1 (mod q), or p = 4q - 1, or (p, q) = (11, 7) or (19, 7) holds.