Cubic Non-Cayley Vertex-Transitive Bi-Cayley Graphs over a Regular $p$-Group

Jin-Xin Zhou,Yan-Quan Feng

doi:10.37236/3915

Abstract

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, we give a characterization of cubic non-Cayley vertex-transitive bi-Cayley graphs over a regular $p$-group, where $p>5$ is an odd prime. As an application, a classification of cubic non-Cayley vertex-transitive graphs of order $2p^3$ is given for each prime $p$.

Highlights

All groups considered in this paper are finite, and all graphs are finite, connected, simple and undirected
A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size
We give a characterization of cubic nonCayley vertex-transitive bi-Cayley graphs over a regular p-group, where p > 5 is a prime

Summary

Introduction

All groups considered in this paper are finite, and all graphs are finite, connected, simple and undirected. Characterize cubic non-Cayley vertex-transitive bi-Cayley graphs over a p-group for an odd prime p. In [12], the authors proved that every cubic non-Cayley vertex-transitive graph of order 2pn, where p > 7 is a prime and n p, is a bi-Cayley graph over a p-group P generated by two elements a and b of the same order and admitting an automorphism α ∈ Aut(P ) of order 4 such that aα = b and bα = a−1. It is proved that a connected cubic vertex-transitive bi-Cayley graph over a regular p-group P , where p > 5 is a prime, is non-Cayley if and only if Γ = BiCay(P, R, L, {1}) is 2-type, and Cay(P, R ∪ L) is a tetravalent normal arctransitive Cayley graph with Aut(P, R ∪ L) ∼= Z4. A classification of cubic non-Cayley vertex-transitive graphs of order 2p3 is given for each prime p

Preliminaries

Characterization

Cubic non-Cayley vertex-transitive graphs of order 2p3