  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 $6$.

Combinatorics;Group Theory