The base size of vertex-transitive cubic graphs

TL;DR

This paper establishes bounds and conditions on the automorphism groups of finite connected vertex-transitive cubic graphs, identifying specific graph structures and symmetry properties.

Contribution

It proves a size bound for such graphs, characterizes split Praeger–Xu graphs, and describes automorphism fixing properties, advancing understanding of their symmetry structures.

Findings

If the graph has more than 90 vertices, it is either a split Praeger–Xu graph or has trivial automorphism fixing two vertices.

The paper identifies a size threshold for vertex-transitive cubic graphs.

It characterizes automorphism groups fixing pairs of vertices in these graphs.

Abstract

We prove that if $\Gamma$ is a finite connected vertex-transitive cubic graph, then either $|V\Gamma| \le 90$, or $\Gamma$ is a split Praeger--Xu graph, or there exist two vertices $\alpha$ and $\beta$ such that the identity is the only automorphism of $\Gamma$ fixing both $\alpha$ and $\beta$.