Cubic arc-transitive $k$-circulants

TL;DR

This paper investigates cubic arc-transitive k-circulant graphs, proving the existence of infinitely many for even k and establishing an order bound for odd k under certain conditions, advancing understanding of symmetric graph structures.

Contribution

It proves the existence of infinitely many cubic arc-transitive k-circulants for even k and confirms a conjecture on order bounds for odd k when k is squarefree and coprime to 6.

Findings

Infinitely many cubic arc-transitive k-circulants exist for even k.

A conjecture on order bounds for odd k is proven for squarefree, coprime to 6 k.

The paper advances classification of symmetric graphs with cyclic automorphism groups.

Abstract

For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive $k$-circulants. We conjecture that, if $k$ is odd, then a cubic arc-transitive $k$-circulant has order at most $6k^2$. Our main result is a proof of this conjecture when $k$ is squarefree and coprime to $6$.