Cubic arc-transitive k-circulants

For a positive integer k, a graph is called a k-circulant if its automorphism group contains a cyclic semiregular subgroup with k orbits on the vertices. In this talk, I am going to discuss the following question: for which k there exist infinitely many cubic arc-transitive k-circulants? This is based on a joint work with Michael Giudici, Cai Heng Li and Gabriel Verret.