Edge-transitive cubic graphs: Cataloguing and Enumeration

TL;DR

This paper extends the catalog of finite cubic edge-transitive graphs up to 10,000 vertices, analyzes their group structures, provides examples for each type, and discusses their asymptotic enumeration.

Contribution

It significantly expands the known list of such graphs, analyzes their automorphism groups, and establishes bounds on their asymptotic growth.

Findings

Extended catalog of cubic edge-transitive graphs up to 10,000 vertices.

Identified embeddings among different group amalgams.

Proved bounds on the asymptotic number of such graphs.

Abstract

This paper deals with finite cubic ($3$-regular) graphs whose automorphism group acts transitively on the edges of the graph. Such graphs split into two broad classes, namely arc-transitive and semisymmetric cubic graphs, and then these divide respectively into $7$ types (according to a classification by Djokovi\'c and Miller (1980)) and $15$ types (according to a classification by Goldschmidt(1980)), in terms of certain group amalgams. Such graphs of small order were previously known up to orders $2048$ and $768$, respectively, and we have extended each of the two lists of all such graphs up to order $10000$. Before describing how we did that, we carry out an analysis of the $22$ amalgams, to show which of the finitely-presented groups associated with the $15$ Goldschmidt amalgams can be faithfully embedded in one or more of the other $21$ (as subgroups of finite index), complementing what is already known about such embeddings of the $7$ Djokovi\'c-Miller groups in each other. We also give an example of a graph of each of the $22$ types, and in most cases, describe the smallest such graph, and we then use regular coverings to prove that there are infinitely many examples of each type. Finally, we discuss the asymptotic enumeration of the graph orders, proving that if $f_{\mathcal C}(n)$ is the number of cubic edge-transitive graphs of type ${\mathcal C}$ on at most $n$ vertices, then there exist positive real constants $a$ and $b$ and a positive integer $n_0$ such that $n^{a \log(n)} \le f_{\mathcal C}(n) \le n^{b \log(n)}$ for all $\ n\ge n_0$.