A Census of Edge-transitive Surfaces

TL;DR

This paper classifies edge-transitive surfaces, identifying four main types and five sub-types, and provides a census of such surfaces with up to 5000 faces using graph-theoretic constructions.

Contribution

It offers a comprehensive classification of edge-transitive surfaces and computational enumeration up to 5000 faces, combining theoretical and algorithmic methods.

Findings

Four main types of edge-transitive surfaces identified

A total of five sub-types of these surfaces classified

Census of surfaces with up to 5000 faces computed

Abstract

In this paper, we study edge-transitive surfaces, i.e. triangulated 2-dimensional manifolds whose automorphism groups act transitively on the edges of these triangulated surfaces. We show that there exist four types of edge-transitive surfaces, splitting up further into a total of five sub-types. We exploit our theoretical results to compute a census of edge-transitive surfaces with up to 5000 faces by constructing suitable cycle double covers of edge-transitive cubic graphs.