The classification of half-arc-regular bi-circulants of valency 6

A graph with a cyclic group of automorphisms acting semiregularly on the vertices with two orbits is called a bi-circulant. A graph is half-arc-transitive if it is both vertex-transitive and edge-transitive but not arc-transitive. Then we say that a half-arc-transitive graph is half-arc-regular if its full automorphism group acts regularly on its edges. It is known that the smallest possible valency of a half-arc-transitive bi-circulant is 6. In this paper, a classification is given of connected half-arc-regular bi-circulants of valency 6. As byproduct, we construct the first known infinite family of half-arc-transitive bi-circulants of valency 6.