On orders of automorphisms of vertex-transitive graphs

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d, $d\le 4$, is at most $c_{d}n$ where $c_3=1$ and $c_4=9$. Whether such a constant $c_d$ exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, $\Gamma \ncong K_{3,3}$, has a regular orbit, that is, an orbit of $〈g〉$ of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C.