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 Tabač 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 Tabač 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.
@article{10_37236_7588,
author = {Robert Jajcay and \v{S}tefko Miklavi\v{c} and Primo\v{z} \v{S}parl and Gorazd Vasiljevi\'c},
title = {On certain edge-transitive bicirculants},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {2},
doi = {10.37236/7588},
zbl = {1409.05138},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7588/}
}
TY - JOUR
AU - Robert Jajcay
AU - Štefko Miklavič
AU - Primož Šparl
AU - Gorazd Vasiljević
TI - On certain edge-transitive bicirculants
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/7588/
DO - 10.37236/7588
ID - 10_37236_7588
ER -
%0 Journal Article
%A Robert Jajcay
%A Štefko Miklavič
%A Primož Šparl
%A Gorazd Vasiljević
%T On certain edge-transitive bicirculants
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/7588/
%R 10.37236/7588
%F 10_37236_7588
Robert Jajcay; Štefko Miklavič; Primož Šparl; Gorazd Vasiljević. On certain edge-transitive bicirculants. The electronic journal of combinatorics, Tome 26 (2019) no. 2. doi: 10.37236/7588
