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\title{\bf Cubic non-Cayley vertex-transitive bi-Cayley graphs over a regular $p$-group}

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\author{Jin-Xin Zhou, Yan-Quan Feng\\
\small Department of Mathematics\\[-0.8ex]
\small  Beijing Jiaotong University\\[-0.8ex]
\small Beijing 100044, P.R. China\\
\small\tt jxzhou@bjtu.edu.cn, yqfeng@bjtu.edu.cn\\
}

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% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Dec 29, 2013}{July 27, 2016}\\
\small Mathematics Subject Classifications: 05C25, 20B25}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
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% principal new results that are to be found in the paper. The abstract
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\begin{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 a prime. As an
application, a classification of cubic non-Cayley
vertex-transitive graphs of order $2p^3$ is given for each prime $p$.


  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} bi-Cayley graph; vertex-transitive; automorphism group; Cayley graph
\end{abstract}

\section{Introduction}

All groups considered in this paper are finite, and all graphs are finite, connected, simple and
undirected. For the group-theoretic and graph-theoretic terminology not defined here we refer the
reader to \cite{BMBook,WI}.

A graph is said to be a {\em bi-Cayley graph} over a group $H$ if it admits $H$ as a semiregular automorphism
group with two orbits of equal size. (Some authors have used the term semi-Cayley instead~\cite{Gao2010,Gao2011,dJ,LM}.
In this paper, we follow~\cite{kovacs} to use the term bi-Cayley.)
Note that every bi-Cayley graph admits the following concrete realization. Let $R,L$ and $S$ be subsets of a
group $H$ such that $R=R^{-1}$, $L=L^{-1}$ and $R\cup L$ does not contain the identity element of $H$. Define
the graph BiCay$(H, R, L, S)$ to have vertex set the union of the {\em right part} $H_0=\{h_0\ |\ h\in H\}$ and
the {\em left part} $H_1=\{h_1\ |\ h\in H\}$, and edge set the union of the {\em right edges}
$\{\{h_0, g_0\}\ |\ gh^{-1}\in R\}$, the {\em left edges} $\{\{h_1, g_1\}\ |\ gh^{-1}\in L\}$ and
the {\em spokes} $\{\{h_0, g_1\}\ |\ gh^{-1}\in S\}$ (Note that some authors label the vertices of a bi-Cayley graph for a group $H$ with ordered pairs $(h, i)$ for $h\in H$ and $i\in\{0, 1\}$, while we are using $h_i$ to denote $(h, i)$).
For the case when $|S|=1$, the bi-Cayley graph BiCay$(H, R, L, S)$ is also called {\em one-matching bi-Cayley graph}
(see~\cite{kovacs}). Also, if $|R|=|L|=s$, then ${\rm BiCay}(H,R,L,S)$,
is said to be an {\em $s$-type bi-Cayley graph}, and if $H$ is abelian, then ${\rm BiCay}(H,R,L,S)$
is simply called an {\em abelian bi-Cayley graph}.

In the study of bi-Cayley graphs, a natural problem is
to characterize or classify bi-Cayley graphs over a given group with certain valency and specific
symmetric property.  Some partial answers for this problem have been obtained. For example,
in~\cite{Pisanski} Pisanski classified cubic bi-Cayley graphs over cyclic
groups, and in~\cite{kovacs},  Kov\'acs et al. gave a description of
arc-transitive one-matching bi-Cayley graphs over abelian groups, from which one can
obtain the classification of cubic arc-transitive one-matching bi-Cayley graphs over abelian groups.
In~\cite{ZF}, the automorphisms of
the bi-Cayley graphs were investigated. In particular, some sufficient conditions
for a bi-Cayley graph being vertex-transitive or Cayley were given, and moreover,
for a one-matching bi-Cayley graph $\G$ over a group $H$, the normalizer of
the group $H$ in $\Aut(\G)$ was determined. By using this, a classification of cubic vertex-transitive bi-Cayley graphs
over abelian groups was given.
The facts listed above provide the motivation for us to consider the following problem.

\begin{problem}\label{prob}
Characterize cubic non-Cayley vertex-transitive bi-Cayley graphs over a
$p$-group for an odd prime $p$.
\end{problem}

Another motivation for us to consider this problem is:
it is also related to the study of non-Cayley vertex-transitive
graphs which is very active in 1980's.
Let $p>3$ be a prime. It is easy to prove that every connected cubic non-Cayley vertex-transitive graph of order $2p^n (n\geq 1)$ is
a bi-Cayley graph over a $p$-group (see Lemma~\ref{isbicayley}).
So the above problem is equivalent to the problem of
characterizing cubic non-Cayley vertex-transitive graphs of order $2p^n$.
By \cite{FK3}, every cubic symmetric graph of order $2p^n$ ($p>5$ is a prime)
is a Cayley graph (see Proposition~\ref{2pnsymmetric}).
Clearly, this is not true
for the case when $p=5$ because
the Petersen graph is non-Cayley. In fact, one may construct infinitely many
cubic non-Cayley symmetric graphs of order $2\cdot5^n$ by considering
the regular coverings of the Petersen graph.

It is known that for a prime $p$, every cubic non-Cayley vertex-transitive graph of order $2p$ or $2p^2$
is a generalized Petersen graph~(see \cite{Marusic,Zhouadv}).
In~\cite{KMZ}, the authors proved that every cubic non-Cayley
vertex-transitive graph of order
$2p^n$, where $p>7$ is a prime and $n\leq p$, 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 $\a\in\Aut(P)$ of
order $4$ such that $a^\a=b$ and $b^\a=a^{-1}$.

In this paper, we solve the above problem for the case when $P$
is a regular $p$-group where $p>5$ is a prime. 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 $\G={\rm BiCay}(P,R,L,\{1\})$ is $2$-type,
and $\Cay(P,R\cup L)$ is a tetravalent normal arc-transitive Cayley graph with $\Aut(P,R\cup L)\cong\mz_4$.
As an
application, a classification of cubic non-Cayley
vertex-transitive graphs of order $2p^3$ is given for each prime $p$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
In this section, we shall introduce some notations, terminology and preliminary results.
Let $n$ be a positive integer. Denote by $\mz_n$ the cyclic group
of order $n$, by
$\mz_n^*$ the multiplicative group of $\mz_n$ consisting of numbers
coprime to $n$, and by $D_{2n}$ the dihedral group of order $2n$, respectively. %by $S_n$ the symmetric group of degree $n$

For a finite, simple and undirected graph $X$, we use $V(X)$,
$E(X)$, $A(X)$ and $\Aut(X)$ to denote its vertex set, edge set, arc
set and full automorphism group, respectively. For $u,v\in V(X)$,
$u\sim v$ means that $u$ is adjacent to $v$ and denote by $\{u,v\}$
the edge incident to $u$ and $v$ in $X$. For any subset $B$ of $V(X)$, the subgraph of $X$ induced by $B$
will be denoted by $X[B]$. A graph $X$ is said to be
{\em vertex-transitive},  and {\em arc-transitive} (or {\em
symmetric}) if $\Aut(X)$ acts transitively on $V(X)$ and $A(X)$,
respectively.

Let $G$ be a permutation group on a set $\Omega$ and $\a\in \Omega$.
Denote by $G_\a$ the stabilizer of $\a$ in $G$, that is, the
subgroup of $G$ fixing the point $\a$. We say that $G$ is {\em
semiregular} on $\Omega$ if $G_\a=1$ for every $\a\in \Omega$ and
{\em regular} if $G$ is transitive and semiregular. Given a finite
group $G$ and an inverse closed subset $S\subseteq G\setminus\{1\}$,
the {\em Cayley graph} $\Cay(G,S)$ on $G$ with respect to $S$ is
defined to have vertex set $G$ and edge set $\{\{g,sg\}\mid g\in
G,s\in S\}$. A Cayley graph $\Cay(G,S)$ is connected if and only if
$S$ generates $G$. Given a $g\in G$, define the permutation $R(g)$
on $G$ by $x\mapsto xg, x\in G$. Then $R(G)=\{R(g)\ |\ g\in G\}$,
called the {\em right regular representation} of $G$, is a
permutation group isomorphic to $G$. It is well-known that
$R(G)\leq\Aut(\Cay(G,S))$. So, $\Cay(G,S)$ is vertex-transitive. In
general, a vertex-transitive graph $X$ is isomorphic to a Cayley
graph on a group $G$ if and only if its automorphism group has a
subgroup isomorphic to $G$, acting regularly on the vertex set of
$X$ (see \cite[Lemma~16.3]{B}).


A Cayley graph $\Cay(G, S)$ is
said to be {\em normal} if $R(G)$ is normal in $\Aut(\Cay(G, S))$ (see~\cite{X1}).
%Let $\Cay(G,S)$ be a Cayley graph on a group $G$ with respect to a
%subset $S$ of $G$.
Set $A=\Aut(\Cay(G,S))$ and $\Aut(G,S)=\{\a\in
\Aut(G)\ |\ S^\a=S\}$.

\begin{proposition} {\rm\cite[Proposition~1.5]{X1}}  \label{stabi}
The Cayley graph $\Cay(G,S)$ is normal if and only if
$A_1=\Aut(G,S)$, where $A_1$ is the stabilizer of the identity $1$
of $G$ in $A$.
\end{proposition}

Let $p$ be a prime. A finite $p$-group $P$ is called a {\em regular $p$-group} if for any two elements
$x$ and $y$ in $P$, there exist $c_1, c_2, \cdots, c_r$ in the derived group of $\lg x, y\rg$ such that
$(xy)^p=x^py^pc_1^pc_2^p\cdots c_r^p$.

\begin{proposition}{\rm\cite[Theorem~3.1]{FXregular}}\label{regular-p}
Let $p$ be a prime and $G$ a regular $p$-group with $p\neq 2, 5$. Let $X=\Cay(G, S)$
be a connected tetravalent Cayley graph on $G$. Then we have $\Aut(\Cay(G, S))= R(G)\rtimes
\Aut(G, S)$.
\end{proposition}


\begin{proposition}{\rm \cite[Corollary~3.4]{FK3}}\label{2pnsymmetric}
Let $p>5$ be a prime. Then every connected cubic symmetric graph of order $2p^n$
is a Cayley graph.
\end{proposition}

The following proposition lists all of the tetravalent connected
arc-transitive Cayley graphs of order $p^3$ for each prime $p$.

\begin{proposition}{\rm\cite[Theorem~4.1]{FXp3}}\label{p3}
Let $p$ be a prime and let $X=\Cay(G,S)$ be a tetravalent
connected arc-transitive Cayley graph of order $p^3$.
Then one of the following holds.
\begin{enumerate}
  \item [{\rm (1)}]\ $G=\mz_{p^3}=\lg a\rg$, $S=\{a,a^{-1},a^\ld,a^{-\ld}\}$ $(\ld^2\equiv-1\ (\mod p^3))$.
  \item [{\rm (2)}]\ $G=\mz_{p^2}\times\mz_p=\lg a\rg\times\lg b\rg$, $S=\{a,a^{-1},a^\ld b,(a^\ld b)^{-1}\}$ $(\ld^2\equiv-1\ (\mod p))$.
  \item [{\rm (3)}]\  $G=\mz_{p^2}\times\mz_p=\lg a\rg\times\lg b\rg$, $S=\{a,a^{-1},ab,(ab)^{-1}\}$.
  \item [{\rm (4)}]\ $G=\lg a,b,c\ |\ a^p=b^p=c^p=1, c=[a,b], [a,c]=[b,c]=1\rg$, $S=\{a,a^{-1},b,b^{-1}\}$.
\end{enumerate}
\end{proposition}

To end this section, we give some results regarding the bi-Cayley graphs.
For the proof of these results, one may see ~\cite{ZF}.
Let $\G=$ BiCay$(H,R,L,S)$ be a
connected bi-Cayley graph over a group $H$. The following proposition
gives some basic properties of $\G$.

\begin{proposition}\label{basicprop}
The following hold.
\begin{enumerate}
  \item [{\rm (1)}]\  $H$ is generated by $R\cup L\cup S$.
  \item  [{\rm (2)}]\ If $S\neq\emptyset$, then $S$ can be chosen to contain the identity element of $H$.
  \item [{\rm (3)}]\ For any automorphism $\a$ of $H$, {\rm BiCay}$(H,R,L,S)\cong${\rm BiCay}$(H,R^\a,L^\a,S^\a)$.
\end{enumerate}
\end{proposition}


Let $R(H)$ denote the right regular representation of $H$. Then
$R(H)$ can be regarded as a group of
automorphisms of BiCay$(H, R, L, S)$ acting on its vertices by the rule
$$h_i^{R(g)}=(hg)_i, \forall i\in\mz_2, h,g\in H.$$
For an automorphism $\a$ of $H$, define two permutations on $V(\G)=H_0\cup H_1$ as follows:
\begin{equation}\label{eq-auto}
\begin{array}{ll}
\d_\a:& h_0\mapsto (h^\a)_1, h_1\mapsto (h^\a)_0, \forall h\in H,\\
\s_\a:& h_0\mapsto (h^\a)_0, h_1\mapsto (h^\a)_1, \forall h\in H.
\end{array}
\end{equation}
Set \begin{equation}\label{eq-autoset}
\begin{array}{lll}
{\rm I}&=& \{\d_\a\ |\ \a\in\Aut(H)\ s.t.\ R^\a=L, L^\a=R, S^\a=S^{-1}\},\\
{\rm F} &=&\lg \s_\a\ |\ \a\in\Aut(H)\ s.t.\ R^\a=R, L^\a=L, S^\a=S\rg.
\end{array}
\end{equation}

\begin{proposition}\label{automorphism2}
Each element in ${\rm I}\cup{\rm F}$ is an automorphism of $\G$.
Furthermore, for any
$\d_\a\in{\rm I}$, if $\a$ has order $2$, then  $\lg R(H),\d_\a\rg=R(H)\rtimes\lg \d_\a\rg$ acts regularly
on $V(\G)$.
\end{proposition}

\begin{proposition}\label{one-matching}
Let $\G=${\rm BiCay}$(H,R,L,\{1\})$ be a connected one-matching bi-Cayley graph over the group $H$.
Then $\Aut(\G)$ contains a regular subgroup containing $R(H)$ if and only if there exists an automorphism $\a\in\Aut(H)$ of order
at most $2$ such that $R^\a=L$.
\end{proposition}

\section{Characterization}

\begin{lemma}\label{isbicayley}
Let $p>3$ be a prime. Then every connected cubic non-Cayley vertex-transitive graph of order $2p^n (n\geq 1)$ is
a bi-Cayley graph over a $p$-group.
\end{lemma}

\begin{proof} Let $\G$ be a cubic non-Cayley vertex-transitive graph of order $2p^n$ with $p>3$ a prime.
Set $A=\Aut(\G)$. By Proposition~\ref{2pnsymmetric}, either $\G$ is non-symmetric or $\G$ is symmetric and $p=5$.
Let $P$ be a Sylow $p$-subgroup of $A$. If $\G$ is non-symmetric, then the vertex-stabilizer
$A_v$ of $v\in V(\G)$ is a $2$-group, and so $P$ is semiregular on $V(\G)$. If $\G$ is symmetric and $p=5$,
then $A_v$ is a $\{2,3\}$-group, and so $P$ is also semiregular on $V(\G)$. Consequently, $P$
is always semiregular and has two orbits of equal size.
This implies that
$\G$ must be a bi-Cayley graph over $P$.
\end{proof}

Now we introduce the concept of quotient graph which will be used in
the proof of the following lemma.
Let $\G$ be a connected vertex-transitive
graph, and let $G\leq \Aut(\G)$ be vertex-transitive on $\G$. For a
$G$-invariant partition $\B$ of $V(\G)$, the {\em quotient graph}
$\G_\B$ is defined as the graph with vertex set $\B$ such that, for
any two vertices $B,C\in \B$, $B$ is adjacent to $C$ if and only if
there exist $u\in B$ and $v\in C$ which are adjacent in $\G$. Let $N$
be a normal subgroup of $G$. Then the set $\B$ of orbits of $N$ in
$V(\G)$ is a $G$-invariant partition of $V(\G)$. In this case, the
symbol $\G_\B$ will be replaced by $\G_N$.

\begin{lemma}\label{normal}
Let $\G={\rm BiCay}(P,R,L,S)$ be a
connected cubic non-Cayley vertex-transitive bi-Cayley graph over a regular $p$-group $P$, where $p>5$ is a prime.
Let $A=\Aut(\G)$. Then $R(P)$ is a normal Sylow $p$-subgroup of $A$.
\end{lemma}

\begin{proof} By Proposition~\ref{2pnsymmetric}, $\G$ must be non-symmetric.
It follows that the vertex-stabilizer $A_v$ of any $v\in V(\G)$ in $A$
is a $2$-group. This implies that
$A_v/A_v^*\leq\mz_2$, where $A_v^*$ is the kernel of $A_v$ acting on the neighborhood of $v$. Since $\G$ is non-Cayley, one has $A_v>1$.
If $A_v/A_v^*=1$, then $A_v(=A_v^*)$ fixes all neighbors of $v$, and by the vertex-transitivity
and connectedness of $\G$, we get that $A_v$ fixes all vertices of $\G$,
forcing that $A_v=1$, a contradiction. Thus, $A_v/A_v^*\cong\mz_2$, and so
there is one and only one neighbor, say $u$, of $v$ such that $A_u=A_v$.
By the arbitrariness of $v$, the following set
$$\B=\{\{u,v\}\ |\ u,v\in V(\G)\ {\rm such\ that}\ A_u=A_v\}.$$
is a $1$-factor of $\G$. Clearly, for any $g\in A$, $A_{u^g}=A^g_u=A^g_v=A_{v^g}$.
It follows that
$\B$ is also an $A$-invariant partition of $V(\G)$. Consider the quotient graph $\G_\B$
of $\G$ relative to $\B$, and let $K$ be the kernel of $A$ acting on $V(\G_\B)$.
Then $A/K$ is vertex-transitive on $\G_\B$ and so $\G_\B$ has regular valency.
Since $\G$ is cubic, $\G_\B$ has valency at most $4$, and since $|\G_\B|=|P|$ is odd,
the valency of $\G_\B$ is $2$ or $4$. Since $p$ is odd and $K$ is a $2$-group,
$R(P)\cong R(P)K/K$ must be regular on $V(\G_\B)$.

Suppose $K\neq 1$. Then, $\B$ is the set of orbits of $K$. Since each orbit of $K$ is just an edge of $\G$,
the quotient graph $\G_\B$ must be a cycle of length $p^n$, where $p^n=|P|$, and moreover, since $\G$ has valency $3$, the edges between any two adjacent orbits of $K$ form a perfect matching. It follows that
the neighbors of any $v\in V(\G)$ are in three different orbits of $K$.
Thus, $K_v$ fixes each neighbor of $v$. By the connectedness of $\G$, we obtain that $K_v=1$
and hence $K\cong\mz_2$. As $R(P)K/K$ is regular on $V(\G_\B)$, $R(P)K=R(P)\times K$
is regular on $V(\G)$, implying that $\G$ is a Cayley graph, a contradiction.

Now we know that $K=1$. Then $A$ acts faithfully on $\B$, and so $A\leq\Aut(\G_\B)$.
Since $R(P)=R(P)K/K$ acts regularly on $V(\G_\B)$, $\G_\B$ can be viewed as a
Cayley graph on $P$. If $\G_\B$ has valency $2$, then $\G_\B$ is a cycle of length $p^n$ and
$\Aut(\G_\B)\cong D_{2p^n}$. It follows that $|A|=2p^n=2|P|$, contradicting that
$A$ is not regular on $V(\G)$. Thus, $\G_\B$ has valency $4$. Since $P$ is a regular $p$-group with $p\neq 2,5$,
by Proposition~\ref{regular-p}, $\G_\B$ is a normal Cayley graph on $P$. It follows that
$R(P)\unlhd\Aut(\G_\B)$, and since $A\leq \Aut(\G_\B)$, one has $R(P)\unlhd A$, as required.
\end{proof}

By Huppert \cite[III, Theorem 10.2]{H}, a $p$-group of order $p^n$ with $n\leq p$ is regular.
By Lemma~\ref{isbicayley}, the
following corollary is straightforward.

\begin{corollary}{\rm \cite[Lemma~4.2]{KMZ}}
Let $\G$ be a connected cubic vertex-transitive graph of order $2p^n$, where
$p>7$ is a prime and $n\leq p$. Then a Sylow $p$-subgroup of $\Aut(\G)$ is normal.
\end{corollary}

The following is the main result of this section.

\begin{theorem}\label{characterize}
Let $\G={\rm BiCay}(P,R,L,S)$ be a
connected cubic vertex-transitive bi-Cayley graph over a regular $p$-group $P$, where $p>5$ is a prime. Then $\G$ is non-Cayley if and only if $\G={\rm BiCay}(P,R,L,\{1\})$ is $2$-type, and
$\Cay(P,R\cup L)$ is a tetravalent normal arc-transitive Cayley graph with $\Aut(P,R\cup L)\cong\mz_4$.
\end{theorem}

\begin{proof}
We first prove the sufficiency. Since $R=R^{-1}$ and $L=L^{-1}$, we may assume that
$R=\{a,a^{-1}\}$ and $L=\{b,b^{-1}\}$. Since $\Aut(P,R\cup L)\cong\mz_4$, we may assume that
$\Aut(P,R\cup L)=\lg \a\rg$ such that $a^\a=b, b^\a=a^{-1}$. In particular, $\a$ interchanges $R$ and
$L$. By the definition of $\d_\a$ (see Eq~\ref{eq-auto})), $\d_\a$ interchanges the two orbits of $R(P)$.
It follows from Proposition~\ref{automorphism2} that $\G$ is vertex-transitive. Suppose that $\G$ is Cayley.
Since $\G={\rm BiCay}(P,R,L,\{1\})$ is $2$-type, by Proposition~\ref{one-matching} there is an automorphism $\b\in\Aut(P)$ of order
at most $2$ such that $R^\b=L$. This implies that $\b\in\Aut(P,R\cup L)=\lg \a\rg$. Clearly, $R\neq L$, so $\b$ has order $2$.
It follows that $\b=\a^2$. However, $R^{\a^2}=R$ and $L^{\a^2}=L$, a contradiction. Thus, $\G$ is non-Cayley.

For the necessity, since $p$ is odd, the subgraph induced by each orbit of $R(P)$ must have even valency.
It follows that $\G$ is $0$- or $2$-type.
\medskip

\f{\bf Case~1}\ $\G$ is $0$-type.

By Lemma~\ref{normal}, $R(P)\unlhd\Aut(\G)$. Since $R(P)$ has two orbits on $V(\G)$,
the quotient graph $\G_{R(P)}$ of $\G$ relative to $R(P)$ is the $3$-dipole Dip$_3$ (see Fig.~(\ref{fig-1})).
\begin{figure}[ht]
\begin{center}
\unitlength 4mm
\begin{picture}(10,6)
\put(2, 3){\circle*{0.4}}\put(8, 3){\circle*{0.4}}
\qbezier(2, 3)(5, 6)(8, 3)\qbezier(2, 3)(5, 0)(8, 3)\put(2,3){\line(1, 0){6}}
\end{picture}
\end{center}\vspace{-.5cm}
\caption{The $3$-dipole Dip$_3$} \label{fig-1}
\end{figure}
This implies that $\G$ is a regular cover of Dip$_3$, and so $\Aut(\G)$ can project to a subgroup of
$\Aut(\G_{R(P)})$. Since $\Aut($Dip$_3)\cong S_3\times\mz_2$, $\Aut(\G)/R(P)\leq\mz_2\times\mz_2$
because $\G$ is not symmetric by Proposition~\ref{2pnsymmetric}. Since $p>2$, $\Aut(\G)=R(P)\rtimes Q$, where $1<Q\leq\mz_2\times\mz_2$
is a Sylow $2$-subgroup of $\Aut(\G)$.
As $\G$ is vertex-transitive, there exists a $2$-element, say
$g$, such that $g$ interchanges the
two orbits of $R(P)$. Since $Q\leq\mz_2\times\mz_2$, $g$ must be an involution.
Therefore, $R(P)\rtimes\lg g\rg$ acts regularly on $V(\G)$, and so
$\G$ is Cayley. A contradiction occurs.\medskip

\f{\bf Case~2}\ $\G$ is $2$-type.

Recall that $V(\G)=P_0\cup P_1$ with $P_0=\{g_0\ |\ g\in P\}$ and $P_1=\{g_1\ |\ g\in P\}$.
By Proposition~\ref{basicprop}~(2), we may assume that $S=\{1\}$.
It follows that $\{g_0,g_1\}\in E(\G)$ for each $g\in P$.
Set $A=\Aut(\G)$. Then $A$ is not regular on $V(\G)$.
So, for each $g\in P$, we have $A_{g_0}\neq 1$.
Since $R(P)\unlhd A$, $A$ fixes the partition $V(\G)=P_0\cup P_1$, and since $g_1$ is the unique neighbor of $g_0$
in $P_1$, it follows that $A_{g_0}$ fixes $g_1$.
This implies that for any $\a\in A$, either $\{g_0,g_1\}^\a=\{g_0,g_1\}$ or
$\{g_0,g_1\}^\a\cap \{g_0,g_1\}=\emptyset$.
Set $\B=\{\{g_0,g_1\}\ |\ g\in P\}$. Then $A$ acts transitively on $\B$.
Let $K$ be the kernel of $A$ acting on $\B$.

Suppose $K\neq 1$.
Clearly, $\B$ is the set of orbits of $K$. Since each orbit of $K$ contains exactly one edge, the
quotient graph $\G_\B$ of $\G$ relative to $\B$ must be a cycle of length $p^n$, where $p^n=|P|$.
It follows that the neighbors of any $v\in V(\G)$ are in three different orbits of $K$.
Thus, $K_v$ fixes each neighbor of $v$. By the connectedness of $\G$, we obtain that $K_v=1$
and hence $K\cong\mz_2$. Then $R(P)\times K$ is regular on $V(\G)$, and so
$\G$ can be viewed as a Cayley graph on $R(P)\times K$. This is impossible.

Now assume that $K=1$. Then $A$ acts faithfully on $\B$. It follows that $A\leq\Aut(\G_\B)$.
It is easy to see that
$R(P)$ is regular on $\B$, and so $\G_\B$ can be viewed as a Cayley graph on $P$.
Recall that $\G={\rm BiCay}(P,R,L,\{1\})$. Set $R=\{a,b\}$ and $L=\{x,y\}$.

Suppose $|R\cap L|=2$. Then $R=L$.
In this case,
it is easy to see that the permutation $\a=\prod_{g\in P}(g_0\ g_1)$ is an automorphism
of $\G$. Furthermore, $\a$ commutes with $R(P)$.
So, $R(P)\times\lg\a\rg$ acts regularly on $V(\G)$, a contradiction.

Suppose $|R\cap L|=1$. Without loss of generality, assume that $a=x$. Then $P=\lg a,b,y\rg$.
Since $R^{-1}=R$ and $L^{-1}=L$, all $a,b,x,y$ are involutions. This is clearly impossible because
$P$ is a $p$-group with $p>2$.

Suppose $|R\cap L|=0$. In this case, the neighbors of $\{1_0,1_1\}$ in $\G_\B$ are
$\{a_0,a_1\},$ $\{b_0,b_1\},$ $\{x_0,x_1\}$ and $\{y_0,y_1\}$.
So, $\G_\B$ is a tetravalent Cayley graph on $P$.
Note that $A_{1_0}=A_{1_1}$.
Since $A$ is not regular on $V(\G)$,
$A_{1_0}$ interchanges $a_0$ and $b_0$, and also interchanges $x_1$ and $y_1$.
Since $\{1_0,1_1\}$ is a block of $A$, $A$ contains an element interchanging $1_0$ and
$1_1$. This implies that $A_{\{1_0,1_1\}}$ is transitive on the neighborhood of $\{1_0,1_1\}$ in $\G_\B$.
So, $A$ is an arc-transitive automorphism group of $\G_\B$.
Let $X=\Cay(P, R\cup L)$. It is easy to verify that the following map
$$f: g\mapsto \{g_0,g_1\}, \forall g\in P$$
is an isomorphism from $X$ to $\G_\B$. So, $X\cong\G_\B$. Since $\G_\B$
is arc-transitive, $X$ is also arc-transitive.
Since $P$ is a $p$-group with $p>2$, we may assume that $R=\{a,a^{-1}\}$ and
$L=\{x,x^{-1}\}$. By Proposition~\ref{regular-p}, $X$ is a normal Cayley graph, and so
$\Aut(\G_\B)=R(P)\rtimes\Aut(P, R\cup L)$. Clearly, $\Aut(P, R\cup L)$
acts faithfully on $R\cup L$. Since $p>2$, it is easy to see that
$R,L$ are blocks of $\Aut(P, R\cup L)$ on $R\cup L$. It follows that $\Aut(P, R\cup L)\leq D_8$,
and so $A\leq \Aut(\G_\B)\leq R(P)\rtimes D_8$. Also, as $R,L$ are blocks of $\Aut(P, R\cup L)$ on $R\cup L$,
by Proposition~\ref{automorphism2}, each element in $\Aut(P, R\cup L)$ can also induce an automorphism of $\G$.
This implies that $A=\Aut(\G_\B)$.
Since $\G$ is arc-transitive, $\Aut(P, R\cup L)$ is isomorphic to $\mz_2\times\mz_2, \mz_4,$ or $D_8$.

If $\Aut(P, R\cup L)\cong\mz_2\times\mz_2$ or $D_8$, then there exists an involution, say $\a$,
in $\Aut(P, R\cup L)$ which interchanges $R$ and $L$. By Proposition~\ref{automorphism2}, $\a$ can induce an automorphism,
say $\d_\a$, of $\G$ of order $2$ such that $R(P)\rtimes\lg \d_\a\rg$ is regular on $V(\G)$,
and so $\G$ is Cayley, a contradiction.
Thus, $\Aut(P, R\cup L)\cong\mz_4$.
\end{proof}



\section{Cubic non-Cayley vertex-transitive graphs of order $2p^3$}

Let $p$ be an odd prime. We first introduce some cubic connected non-Cayley vertex-transitive
graphs of order $2p^3$. It is well known that $\mz_{p^n}^*$ is cyclic and has order $p^{n-1}(p-1)$.
So, if $4\ |\ (p-1)$ then $\mz_{p^n}^*$ has a unique subgroup of order $4$. Clearly, if $\ld$ is
an element of order $4$ in $\mz_{p^n}^*$, then $\{1,-1,\ld,-\ld\}$ is the unique subgroup of order $4$
in the cyclic group $\mz_{p^n}^*$.

\begin{example}\label{exam-0}
Let $p$ be a prime such that $p-1$ is divisible by $4$ and let $\ld$ be an element of
order $4$ in $\mz_{p^3}^*$. The graph $\mathcal{NC}_{2p^3}^0$ is defined to be the
bi-Cayley graph ${\rm BiCay}(\mz_{p^3}, R, L,\{1\})$,
where $\mz_{p^3}=\lg a\rg$, $R=\{a,a^{-1}\}$ and $L=\{a^\ld, a^{-\ld}\}$.
\end{example}

By the uniqueness of the subgroup of order $4$ in $\mz_{p^3}$, the graph $\mathcal{NC}_{2p^3}^0$
is independent of the choice of $\ld$. Let $\a$ be the automorphism of $\mz_{p^3}$ induced by the map $a\mapsto a^\ld$. Then $\a$ swaps $R$ and $L$, and by Proposition~\ref{automorphism2}, $\d_\a\in\Aut(\mathcal{NC}_{2p^3}^0)$ (see Equations (1)-(2) for the definition of $\d_\a$) and so $\mathcal{NC}_{2p^3}^0$ is vertex-transitive because $\d_\a$ swaps the two orbits of $\mz_{p^3}$ on $V(\mathcal{NC}_{2p^3}^0)$.

In view of \cite[Theorem~1]{X}, we have
$\Cay(\mz_{p^3}, R\cup L)$ is a tetravalent normal arc-transitive Cayley graph and
$\Aut(\mz_{p^3}, R\cup L)\cong\mz_4$. If $p=5$, then by Magma~\cite{BCP}, $\mathcal{NC}_{2p^3}^0$
is non-Cayley vertex-transitive and $|\Aut(\mathcal{NC}_{2p^3}^0)|=4p^3$, and if $p>5$, then by Theorem~\ref{characterize}, again we have $\mathcal{NC}_{2p^3}^0$
is non-Cayley vertex-transitive.

\begin{example}\label{exam-1}
Let $p$ be a prime such that $p-1$ is divisible by $4$ and let $\ld$ be an element of
order $4$ in $\mz_{p}^*$. The graph $\mathcal{NC}_{2p^3}^1$ is defined to be the
bi-Cayley graph ${\rm BiCay}(\mz_{p^2}\times\mz_p, R, L,\{1\})$,
where $\mz_{p^2}\times\mz_p=\lg a\rg\times\lg b\rg$, $R=\{a,a^{-1}\}$ and $L=\{(ab)^\ld, (ab)^{-\ld}\}$.
\end{example}

By the uniqueness of the subgroup of order $4$ in $\mz_{p}$, the graph $\mathcal{NC}_{2p^3}^1$
is independent of the choice of $\ld$. Let $\b$ be the automorphism of $\mz_{p^2}\times\mz_p$ induced by the map $a\mapsto (ab)^\ld, b\mapsto a^{\ld^3+\ld}b^{-\ld}$. Then $\b$ swaps $R$ and $L$, and by Proposition~\ref{automorphism2}, $\d_\b\in\Aut(\mathcal{NC}_{2p^3}^1)$ and so $\mathcal{NC}_{2p^3}^1$ is vertex-transitive because $\d_\b$ swaps the two orbits of $\mz_{p^2}\rtimes\mz_p$ on $V(\mathcal{NC}_{2p^3}^1)$.

In view of \cite[Proposition~3.3]{xx}, we have
$\Cay(\mz_{p^2}\times\mz_p, R\cup L)$ is a tetravalent normal arc-transitive Cayley graph and
$\Aut(\mz_{p^2}\times\mz_p, R\cup L)\cong\mz_4$. If $p=5$, then by Magma~\cite{BCP}, $\mathcal{NC}_{2p^3}^1$
is non-Cayley vertex-transitive and $|\Aut(\mathcal{NC}_{2p^3}^1)|=4p^3$, and if $p>5$, then by Theorem~\ref{characterize}, again we have  $\mathcal{NC}_{2p^3}^1$
is non-Cayley vertex-transitive.

Also, note that $\Aut(\mathcal{NC}_{2p^3}^0)\cong\mz_{p^3}\rtimes\mz_4$ and
$\Aut(\mathcal{NC}_{2p^3}^1)\cong(\mz_{p^2}\rtimes\mz_p)\rtimes\mz_4$. It follows that $\mathcal{NC}_{2p^3}^0$
and $\mathcal{NC}_{2p^3}^1$ are not isomorphic to each other.

\begin{theorem}\label{classification}
Let $p$ be a prime. Then a cubic vertex-transitive graph of order $2p^3$ is non-Cayley if and only if
it is isomorphic to $\mathcal{NC}_{2p^3}^0$ or $\mathcal{NC}_{2p^3}^1$.
\end{theorem}

\begin{proof}
By~\cite{McKay}, all connected cubic vertex-transitive graphs of order $16$
are Cayley. By~\cite{PSV}, if $p=3$, then all connected cubic vertex-transitive graphs of order $54$ are Cayley, and if $p=5$, then up to isomorphism, there are
exactly two non-Cayley vertex-transitive graphs of order $2\cdot 5^3$,
and so $\G\cong$ $\mathcal{NC}_{2\cdot 5^3}^0$ or $\mathcal{NC}_{2\cdot 5^3}^1$.

In what follows, assume that $p>5$. By Lemma~\ref{isbicayley}, $\G$ is a bi-Cayley graph over a group $P$, where $P$ is a Sylow $p$-subgroup of $\Aut(\G)$. By Theorem~\ref{characterize}, $\G={\rm BiCay}(P,$ $ R,$ $ L,$ $ \{1\})$ and $\Cay(P,$ $ R\cup L)$ is a tetravalent normal arc-transitive Cayley graph such that $\Aut(P, R\cup L)\cong\mz_4$.
Since $|\G|=2p^3$, one has $|P|=p^3$. Noting that $R=R^{-1}$ and $L=L^{-1}$, by Proposition~\ref{p3}, one of the following happens:
$$
\begin{array}{ll}
(1)&P=\mz_{p^3}=\lg a\rg, R=\{a, a^{-1}\}, L=\{a^\ld, a^{-\ld}\} (\ld^2\equiv-1\ (\mod p^3));\\
(2)&P=\mz_{p^2}\times\mz_p=\lg a\rg\times\lg b\rg, R=\{a, a^{-1}\}, L=\{(ab)^\ld, (ab)^{-\ld}\} (\ld^2\equiv-1\ (\mod p));\\
(3)&P=\mz_{p^2}\times\mz_p=\lg a\rg\times\lg b\rg, R=\{a, a^{-1}\}, L=\{ab, (ab)^{-1}\};\\
(4)&P=\lg a, b, c\ |\ a^p=b^p=c^p=1, c=[a, b], [a, c]=[b, c]=1\rg,  \\
&R=\{a, a^{-1}\}, L=\{b, b^{-1}\}.
\end{array}
$$
If $(1)$ happens, then $\G\cong\mathcal{NC}_{2p^3}^0$, and if $(2)$ happens, then $\G\cong\mathcal{NC}_{2p^3}^1$.
If $(3)$ happens, then in view of \cite[Proposition~3.3]{xx}, we have
$\Aut(P, R\cup L)\cong\mz_2\times\mz_2$. This is impossible by Theorem~\ref{characterize}.
If $(3)$ happens, then $P=\lg a, b, c\ |\ a^p=b^p=c^p=1, c=[a, b], [a, c]=[b, c]=1\rg$, and any two
elements generating $P$ have the same relation as $a$ and $b$. It follows that
$\Aut(P, R\cup L)\cong D_8$, and by Theorem~\ref{characterize}, $\G$ is not non-Cayley, a contradiction.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
This work was supported by the National Natural Science Foundation of China $(11271012,$ $ 11571035,$ $11231008),$ the Fundamental
Research Funds for the Central Universities\\ $(2015$JBM$110)$ and the 111 Project of China (B16002). The author also would like to thank the anonymous referee for the valuable comments and suggestions.

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% \bibliographystyle{plain}
% \bibliography{myBibFile}
% If you use BibTeX to create a bibliography
% then copy and past the contents of your .bbl file into your .tex file

\begin{thebibliography}{10}

\bibitem{B}
N. Biggs, Algebraic Graph Theory, Second ed, { Cambridge
University Press}, Cambridge, 1993.

\bibitem{BMBook}
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, New York: Elsevier North Holland, 1976.

\bibitem{BCP}
W. Bosma, C. Cannon, C. Playoust, The MAGMA algebra system I: The
user language, J. Symbolic Comput. 24 (1997) 235--265.

\bibitem{dJ}
M.J. de Resmini, D. Jungnickel, Strongly regular semi-Cayley graphs, J. Algebraic Combin. 1 (1992) 171--195.

\bibitem{FK3}
Y.-Q. Feng, J.H. Kwak, Cubic symmetric graphs of order twice an odd
prime power, J. Aust. Math. Soc. 81 (2006), 153--164.


\bibitem{FXregular}
Y.-Q. Feng, M.Y. Xu, Automorphism groups of tetravalent Cayley graphs on regular $p$-groups,
Discrete Math. 305 (2005) 354--360.

\bibitem{FXp3}
Y.-Q. Feng, M.Y. Xu, Normality of tetravalent Cayley graphs of odd
prime-cube order and its application, Acta Math. Sin., English Ser. 21 (2005)
903--912.


\bibitem{Gao2011}
X. Gao, W. Liu, Y. Luo,
On the extendability of certain semi-Cayley graphs of finite abelian groups, Discrete Math.
311 (2011) 1978--1987.

\bibitem{Gao2010}
X. Gao, Y. Luo, The spectrum of semi-Cayley graphs over abelian groups,
Linear Algebra Appl. 432 (2010) 2974--2983.



\bibitem{H}
B. Huppert, Eudliche Gruppen I, Springer-Verlag, Berlin, 1967.

\bibitem{kovacs}
I. Kov\'acs, A. Malni\v c, D. Maru\v si\v c, \v S. Miklavi\v c,
One-matching bi-Cayley graphs over abelian groups, European J. Combin. 30 (2009) 602--616.


\bibitem{KMZ}
K. Kutnar, D. Maru\v si\v c, C. Zhang, On cubic non-Cayley vertex-transitive graphs, J. Graph Theory 69 (2012)
77--95.

\bibitem{LM}
K.H. Leung, S.L. Ma, Partial difference triples, J. Algebraic Combin. 2 (1993) 397--409.


\bibitem{Marusic}
D. Maru\v si\v c, On vertex symmetric digraphs, Discrete Math. 36
(1981) 69--81.


\bibitem{McKay}
B.D. McKay,  Transitive graphs with fewer than $20$ vertices, Math.
Comp. 33 (1979) 1101--1121.


\bibitem{Pisanski}
T. Pisanski, A classication of cubic bicirculants, Discrete Math. 307 (2007) 567--578.

\bibitem{PSV}
P. Poto\v cnik, P. Spiga, G. Verret, A census of small connected cubic vertex-transitive
graphs, http://www.matapp.unimib.it/$\sim$spiga/.


\bibitem{WI}
H. Wielandt, Finite Permutation Groups, Academic Press, New York,
1964.

\bibitem{X1}
M.Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs,
Discrete Math. 182 (1998) 309--319.

\bibitem{X}
M.Y. Xu, A note on one-regular graphs, Chinese Sci. Bull. 45 (2000)
2160--2162.

\bibitem{xx}
J. Xu, M.Y. Xu, Arc-transitive Cayley graphs of valency at most four on
abelian groups, Southeast Asian Bull. Math. 25 (2001) 355--363.

\bibitem{Zhouadv}J.-X. Zhou, Cubic vertex-transitive graphs of order $2p^2$,
Advances in Math. (China) 37 (2008) 605--609.


\bibitem{ZF}
J.-X. Zhou, Yan-Quan Feng, Cubic bi-Cayley graphs over abelian groups,
European J. Combin. 36 (2014) 679-693.

\end{thebibliography}

\end{document}
