# Arc-transitive graphs of valency twice a prime admit a semiregular automorphism

## TL;DR

This paper proves that all finite arc-transitive graphs with valency twice a prime have a nontrivial automorphism acting regularly on the vertices, supporting the broader Polycirculant Conjecture.

## Contribution

It establishes the existence of semiregular automorphisms in a specific class of highly symmetric graphs, advancing the understanding of automorphism structures.

## Key findings

- Every finite arc-transitive graph of valency twice a prime admits a semiregular automorphism.
- Supports the Polycirculant Conjecture for a new class of graphs.
- Provides structural insights into automorphisms of symmetric graphs.

## Abstract

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.