Trivalent dihedrants and bi-dihedrants

A Cayley (resp. bi-Cayley) graph on a dihedral group is called a dihedrant (resp. bidihedrant). In 2000, a classification of trivalent arc-transitive dihedrants was given by Marusic and Pisanski, and several years later, trivalent non-arc-transitive dihedrants of order 4p or 8p (p a prime) were classified by Feng et al. As a generalization of these results, our first result presents a classification of trivalent non-arc-transitive dihedrants. Using this, a complete classification of trivalent vertex-transitive non-Cayley bi-dihedrants is given, thus completing the study of trivalent bi-dihedrants initiated in our previous paper [Discrete Math. 340 (2017) 1757-1772]. As a by-product, we generalize a theorem in [The Electronic Journal of Combinatorics 19 (2012) #P53].