Edge-transitive cubic graphs: Cataloguing and Enumeration
Marston Conder, Primož Potočnik
TL;DR
The paper advances the study of cubic edge-transitive graphs by systematizing the 22 amalgam types (7 arc-transitive Djoković–Miller and 15 semisymmetric Goldschmidt), proving every type can be realized and that infinitely many graphs exist for each type via regular coverings. It provides a detailed analysis of inclusions among amalgams, constructs all edge-transitive cubic graphs up to order 10{,}000 (totaling 4858 graphs), and presents explicit examples across all types, including the smallest representatives and notable omissions. A central methodological contribution is a lifting framework: given any amalgam type, almost all primes yield regular coverings whose automorphism group is the lift of the original group, guaranteeing infinite families for every type. The paper also establishes sharp asymptotic bounds for the number of edge-transitive graphs of a fixed type up to order $n$, showing growth of the form $n^{a\log n}$, and discusses order-density implications, challenging expectations about the rarity of semisymmetric graphs. These results fuse combinatorial enumeration, group amalgams, and covering theory to map the landscape of cubic edge-transitive graphs and their automorphism groups.
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 Djoković 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 $7$ Djoković-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 $\ n\ge n_0$.