On rainbow tetrahedra in Cayley graphs

TL;DR

This paper investigates the structure and properties of graphs formed by rainbow tetrahedra in Cayley graphs of cyclic groups, revealing their degree, diameter, and tessellation subgraph characteristics.

Contribution

It introduces a novel graph construction based on rainbow tetrahedra in Cayley graphs and analyzes their degree, diameter, and tessellation subgraph structures.

Findings

Graphs have maximum degree 6 and asymptotic diameter proportional to the cube root of the number of vertices.

Vertices exhibit degrees 3, 4, or 6, corresponding to specific tessellation subgraphs.

Vertices' neighborhoods are unions of neighborhoods in particular tessellations.

Abstract

Let $\Gamma_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $\Gamma_n$ 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.