On orders of automorphisms of vertex-transitive graphs

TL;DR

This paper studies automorphisms of finite vertex-transitive graphs, establishing bounds on their orders, properties of their orbits, and the structure of such automorphisms, especially in 3-valent cases.

Contribution

It provides new bounds on automorphism orders for graphs with valence up to 4 and characterizes automorphisms in 3-valent graphs, including the existence of regular orbits and orbit structure.

Findings

Automorphism order is at most c_d n for valence d ≤ 4, with specific constants c_3=1 and c_4=9.

Every automorphism in a 3-valent vertex-transitive graph (except K_{3,3}) has a regular orbit.

At least 5/12 of vertices belong to a regular orbit of an automorphism in 3-valent graphs.

Abstract

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 $\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $\Gamma$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ 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 $\Gamma$ in terms of the number of vertices of $\Gamma$ and the length of a longest orbit of $C$.