Trivalent vertex-transitive bi-dihedrants

Abstract

A Cayley (resp. bi-Cayley) graph on a dihedral group is called a dihedrant (resp. bi-dihedrant). In 2000, a classification of trivalent arc-transitive dihedrants was given by Marušič 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 byproduct, we generalize a theorem in [The Electronic Journal of Combinatorics 19 (2012) #P53].

Trivalent dihedrants and bi-dihedrants

“…In [20], we gave a classification of trivalent arc-transitive bi-dihedrants, and we also proved that every trivalent vertex-transitive 0or 1-type bi-dihedrant is a Cayley graph, and gave a classification of trivalent vertex-transitive non-Cayley bi-dihedrants of order 4n with n odd. The goal of this paper is to complete the classification of trivalent vertex-transitive non-Cayley bi-dihedrants.…”

“…The following lemma given in [20] shows that the group G must be solvable. The case where H 0 and H 1 are blocks of imprimitivity of G has been considered in [20], and the main result is the following proposition. (1) (R, L, S) ≡ ({b, ba +1 }, {ba, ba 2 + +1 }, {1}), where n ≥ 5, 3…”

“…In particular, it is proved in [20] that if n is odd and Γ is not a Cayley graph, then H 0 and H 1 are blocks of imprimitivity of G on V (Γ). Consequently, we can get a classification of trivalent vertex-transitive non-Cayley bi-Cayley graphs over a dihedral group D 2n with n odd.…”

“…In [20] we initiated a program to investigate the automorphism groups of the trivalent vertex-transitive bi-dihedrants. This was partially motivated by the following facts.…”

Bipartite edge-transitive bi-p-metacirculants

“…Kovács et al [31] gave a description of arc-transitive one-matching bi-Cayley graphs over abelian groups. All cubic vertex-transitive bi-Cayley graphs over cyclic groups, abelian groups or dihedral groups were determined in [39,52,54]. Recently, Conder et al [11] investigated bi-Cayley graphs over abelian groups, dihedral groups and metacyclic p-groups, and using these results, a complete classification of connected trivalent edge-transitive graphs of girth at most 6 was obtained.…”