# Trivalent dihedrants and bi-dihedrants

## TL;DR

This paper classifies all trivalent non-arc-transitive dihedrants and vertex-transitive non-Cayley bi-dihedrants, completing previous classifications and generalizing existing theorems in graph theory.

## Contribution

It provides a comprehensive classification of trivalent non-arc-transitive dihedrants and bi-dihedrants, extending prior work and generalizing known theorems.

## Key findings

- Classification of trivalent non-arc-transitive dihedrants
- Complete classification of trivalent vertex-transitive non-Cayley bi-dihedrants
- Generalization of a theorem in combinatorics

## Abstract

A Cayley (resp. bi-Cayley) graph on a dihedral group is called a {\em dihedrant} (resp. {\em bi-dihedrant}). In 2000, a classification of trivalent arc-transitive dihedrants was given by Maru\v si\v c 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].