Recognizing the Hidden Structure of Cayley Graphs

Extending the srudy of triangles and tetrahedra in Cayley graphs of cyclic groups, we define a ki-type as an edge-weighted complete graph on i vertices with nonnegative integer weights respecting a triangular sum condition. The interaction among K4-types is reflected in the structure graph G(2, 4) of K4-types, whose vertices are K4-types and where two vertices are adjacent if they share two K3-types. A natural description of neighborhoods of vertices of G(2,4) took us to a recognition algorithni that allowed us to travel inside G(2, 4) and plot a sketch of its structure. This, in tum, allowed us to establish surprising properties of this graph showing much symmetry and structure, which in tum proved to be useful in the study of the induced tetrahedral subgraphs of Cayley graphs. The recognition algorithm above is an example of the use we have been making of set and number theoretical properties in combinatorial structures to recognize provable properties.