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.
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