A Census of Edge-transitive Surfaces

TL;DR

This work classifies edge-transitive triangulations of surfaces through cycle double covers of edge-transitive cubic graphs, proving four edge-transitive surface types (with five subtypes) and developing constructive CDC-based methods to realize them from face graphs. By introducing the face-edge type invariant $\mathrm{fe}(X)$ and leveraging the injective map $\lambda_X:\mathrm{Aut}(X)\to\mathrm{Aut}(\mathcal{F}(X))$, the authors provide case-by-case constructions for each fe-type, yielding explicit surfaces and algorithms to enumerate all edge-transitive surfaces up to $5{,}000$ faces. The paper reports a comprehensive census of $2185$ edge-transitive surfaces (up to isomorphism), including $2002$ orientable and $183$ non-orientable examples, with detailed minimal examples for the $(2,2)$ and $(2,1)$ cases. All algorithms and the census data are implemented and made available, enabling systematic exploration of edge-transitive triangulations and advancing understanding of CDC realizations in topological graph theory.

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.

Figures (2)

• Figure 1: The umbrella $u(v)$ of $X$ with $\mathop{\mathrm{Aut}}\nolimits(X)$-orbits on $X_2$ coloured in blue and yellow
• Figure 2: The umbrella $u(v)$ of $X$ with $\mathop{\mathrm{Aut}}\nolimits(X)$-orbits on the faces coloured in blue and yellow

Theorems & Definitions (26)

• Theorem
• Theorem
• Definition 2.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof