Let Γn​ be the complete undirected Cayley graph of the odd cyclic
group Zn​. Connected graphs whose vertices are rainbow tetrahedra in
Γn​ are studied, with any two such vertices adjacent if and only if they
share (as tetrahedra) precisely two distinct triangles. This yields graphs G
of largest degree 6, asymptotic diameter ∣V(G)∣1/3 and almost all vertices
with degree: {\bf(a)} 6 in G; {\bf(b)} 4 in exactly six connected subgraphs
of the (3,6,3,6)-semi-regular tessellation; and {\bf(c)} 3 in exactly four
connected subgraphs of the {6,3}-regular hexagonal tessellation. These
vertices have as closed neighborhoods the union (in a fixed way) of closed
neighborhoods in the ten respective resulting tessellations. Generalizing
asymptotic results are discussed as well.Comment: 21 pages, 7 figure