On the order of semiregular automorphisms of cubic vertex-transitive graphs

We prove that, if $\mathrm{\Gamma }$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\mathrm{\Gamma }$ of order at least $6$, or the number of vertices of $\mathrm{\Gamma }$ is bounded above by an absolute constant.