On hexavalent half-arc-transitive bi-dihedrants

A graph is half-arc-transitive if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is half-arc-regular if its full automorphism group acts regularly on its edges. A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called bi-dihedrant. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in [8, Theorem 6.7] completes the classification of edge-regular bi-dihedrants of valency 6.