A census of Cayley graphs

Given positive integers k and n , we present methods to construct all groups of order at most n that contain a Cayley set of size k , and to enumerate the Cayley sets of order k in a given group, up to the action of the automorphism group. We use these methods to generate complete lists of pairwise nonisomorphic 3-valent Cayley graphs with at most 5,000 vertices and 4-valent Cayley graphs with at most 1,025 vertices.