Hidden Cayley graph structures (Q1379825)

Let \(C\) be a cyclic group of odd order \(n\). A totally multicolored triple of \(C\) is a set of three distinct elemets \(a\), \(b\) and \(c\) of \(C\) such that \(a + b = \pm c\). It is shown that the Cayley graph of \(C\) with respect to a totally multicolored triple with \(\text{gcd}(a,b,c,n) = 1\) is a triangulation of a torus. If two totally multicolored triples share two elements, then the corresponding torus embeddings are related by first deleting one orbit of edges, forming a quadrilateral torus embedding, and then adding another orbit of edges comprising the other diagonals of the quadrilaterals. The authors define a graph \(G_n'\) on the set of totally multicolored triples by setting two triples adjacent if they share two elements and show that the diameter of \(G_n'\) is asymptotic to the square root of \(n\).