Hidden Cayley graph structures

A contribution to the study of the structure of complete Cayley graphs is given by means of a method of construction of graphs whose vertices are labeled by toral subgraphs induced by equally colored K3's. These graphs shed some light on the mentioned structure because the traversal of each one of its edges from one of its end vertices into the other one represents a transformation between corresponding toral embeddings. As a result, a family of labeled graphs indexed on the odd integers appear whose diameters are asymptotically of the order of the square root of the number of vertices. This family can be obtained by modular reduction from a graph arising from the Cayley graph of the group of integers with the natural numbers as set of generators, which have remarkable local symmetry properties.