The base size of vertex-transitive cubic graphs

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