A Cayley graph X(G;S) where G is a group and , where S is inverse-closed, is defined by taking the vertices of the graph to be the elements of G, with an edge between vertices g and g' iff . A graph has the Cayley Isomorphism property if whenever it can be isomorphically represented as both the Cayley graph X(G;S) and X(G;S'), there is an automorphism of the group G that takes S to S'. A group has the Cayley Isomorphism property if all Cayley graphs on that group have the Cayley isomorphism property.

The Cayley Isomorphism problem is the question of which groups, and which graphs, have the Cayley Isomorphism property. This talk will provide an overview of results in this problem, leading up to the solution of Toida's conjecture: that if G is a cyclic group, and S is a subset of the units of G, then X(G;S) has the Cayley Isomorphism property.