On certain edge-transitive bicirculants

TL;DR

This paper investigates the existence and classification of edge-transitive bicirculant graphs of higher valences, especially valence 6, revealing infinite families including arc- and half-arc-transitive graphs and classifying those with girth 3.

Contribution

It proves the existence of infinite families of edge-transitive bicirculants of valence 6 and classifies those with girth 3, expanding understanding of symmetry in higher valence bicirculants.

Findings

Infinite families of valence 6 edge-transitive bicirculants exist.

Among these, infinitely many are arc-transitive and half-arc-transitive.

Classification of valence 6, girth 3 bicirculants is provided.

Abstract

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Taba\v{c}jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Taba\v{c}jn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained.