Super Vertex (Edge)-Connectivity of Varietal Hypercube

Abstract

The reliability measure of networks is of significant importance to the design and maintenance of networks. Based on connectivity, many refined quantitative indicators for the reliability of network systems have been introduced. The super vertex edge-connectivity and cyclic edge-connectivity, as important parameters to evaluate the robustness of networks, are explored extensively. As a variant of the hypercube $Q_n$, the varietal hypercube $V{Q}_n$ has better properties than $Q_n$ with the same number of edges and vertices. Wang and Xu have proved that $V{Q}_n$ is super vertex-connected for $n\ge 1$ and is also super edge-connected if $n\ne 2$. In this paper, we use another method to prove these results. Moreover, we also obtain the super restricted connectivity and the cyclic edge-connectivity of the varietal hypercube $V{Q}_n$.

1. Introduction

In the era of internet-of-things (IoT), wireless sensor networks (WSNs) are utilised in many applications, including smart grids (SGs) [1]. When the number of sensors continues to increase, faulty nodes and faulty links are inevitable in wireless sensor networks, which undoubtedly increase the difficulty of network reliability assessment and will adversely affect its fault tolerance and reliability.

Hence, to the design and maintenance purpose of networks, appropriate measures of reliability should be found. Because the topology of network system can usually be represented by a connected graph $G=\left(V\right(G),E(G\left)\right)$, where $V\left(G\right)$ and $E\left(G\right)$ can be represented as the set of processors and the set of communication links between processors, respectively. Connectivity $\kappa \left(G\right)$ and edge-connectivity $\lambda \left(G\right)$ represent the maximum number of faulty vertices and the maximum number of faulty edges that an interconnected network can tolerate in normal communication, respectively. Obviously, the higher the (edge-) connectivity is, the more reliable an interconnection network is [2]. As is known, $\kappa \left(G\right)\le \lambda \left(G\right)\le \delta \left(G\right)$, where $\delta \left(G\right)$ is the minimum degree of G. Boesch and Tindell [3] introduced the concepts of super connectivity and super edge- connectivity. A connected graph G is super connected (resp., edge-connected, in short super-$\kappa$ (resp., super-$\lambda$, ) if any minimum vertex-cut (resp., edge-cut) isolates a vertex of G.

However, the biggest disadvantage of connectivity and edge-connectivity in reliability evaluation and fault-tolerant performance of networks is that they all default to a vertex where all adjacent vertices or adjacent edges fail at the same time, which rarely happens in the real world. To overcome such shortcomings, several new concepts on the (edge-) connectivity of graphs, which are called conditional (edge-) connectivity, were proposed by Harary [4]. Among them are restricted (edge-)connectivity and cyclic edge-connectivity.

Esfahanian and Hakimi proposed the concept of restricted connectivity [5]. For a given non-negative integer h, a vertex-set (resp., edge-set) F is an h-restricted vertex-cut (resp., edge-cut) of a connected graph G if $G-F$ is disconnected, and each component of $G-F$ contains at least $h+1$ vertices. The minimum size of all h-restricted vertex-cuts (resp., edge-cuts), denoted by ${\kappa }^{\left(h\right)}\left(G\right)$ (resp., ${\lambda }^{\left(h\right)}\left(G\right)$), is called h-restricted connectivity (resp., edge-connectivity) of the graph G. In the exceptional cases, G is called super restricted vertex (edge)-connected and denoted by super-${\kappa }^{\prime }$ (resp., super-${\lambda }^{\prime }$), if every minimum restricted edge cut isolates one component of order 2. The h-restricted (edge)-connectivity and super-${\kappa }^{\prime }$ (resp., super-${\lambda }^{\prime }$) graph for some classic interconnection networks have been investigated recently. For instance, Li et al. [3] determine that the h-restricted connectivity of the arrangement graph $A_{n,k}$ is $n(n-1)/2$ for h is odd and $h=n-3$, otherwise the h-restricted connectivity of the arrangement graph $A_{n,k}$ is $(h+2)(n-2)-\lfloor h^2/2+h\rfloor$ for $k=2$ and $0\le h\le n-2$. Chen et al. [6] proved that the hypercube, the twisted cube, the cross cube, the Möbius cube, and the locally twisted cube are super connected and super edge-connected. Futhermore, Xu et al. [7] obtained super edge-connectivity, and the restricted (edge-) connectivity of the hypercube, the twisted cube, the cross cube, Möbius cube, and the locally twisted cube is $2n-2$ for $n\ge 3$. Zhou et al. [8] obtained that the balanced hypercube $B{H}_n$ is super-${\lambda }^{\prime }$ but not super-${\kappa }^{\prime }$ for $n\ge 2$.

In 1978, Bollobás indicated all multigraphs without two vertex-disjoint cycles [9]. So, it is necessary to further study the cyclically separable graphs. An edge set S is a cyclic edge-cut if $G-S$ is disconnected and at least two of its components contain cycles. Obviously, a graph G has a cyclic edge-cut if and only if it has two vertex-disjoint cycles and is a cyclically separable graphs. For a cyclically separable graph G, the cyclic edge-connectivity of G, denoted by ${\lambda }_c\left(G\right)$, is defined as the cardinality of the minimum cyclic edge-cut of G [10]. Let $\omega \left(X\right)$ be the number of edges with one end in X and another end in $V\left(G\right)-X$, and $\xi \left(G\right)=min\left\{\omega \right(X\left)\right|X\mathrm{induces}\mathrm{a}\mathrm{shortest}\mathrm{cycle}\mathrm{in}G\}$. A cyclically separable graph G with ${\lambda }_c\left(G\right)=\xi \left(G\right)$ is called cyclically optimal. Moreover, if removing any minimum cyclic edge-cut of the graph G results in the shortest cycle in a component of the graph G, then the graph G is said to be super cyclically edge-connected [11], in short, super-${\lambda }_c$. There are some results of the super cyclically edge-connectivity for some classes of the graphs. For example, Zhang [12] showed that the cyclic edge-connectivity of a strongly regular connected graph G (not $K_{3,3}$) of degree $k\ge 3$ with girth g is equal to $(k-2)g$, where $g=3,4\mathrm{or}5$. Wang and Zhang [13] showed that any k-regular vertex-transitive connected graph with $k\ge 4$ and girth of at least 5 is cyclically optimal, and any edge-transitive connected graph with minimum degree of at least 4 and order of at least 6 is cyclically optimal.

Hypercube $Q_n$, one of the most important and attractive network topologies so far, is widely used due to its excellent characteristics. In 1994, Cheng and Chuang designed a network topology capable of obtaining an interconnection network that is both efficient and cost-effective, this is, the varietal hypercube $V{Q}_n$. In recent years, $V{Q}_n$ has received considerable attention and many of its properties have been studied. For example, Zhou [14] showed that $V{Q}_n$ is a Cayley graph. Wang et al. [15] also determined the full automorphism group of the varietal hypercube $V{Q}_n$. More relevant results on the varietal hypercube $V{Q}_n$ are shown in [16,17,18,19,20].

Our work in this paper concerns the n-dimensional varietal hypercube $V{Q}_n$. We use another method to prove that $V{Q}_n$ is super vertex-connected for any $n\ge 1$ and super edge-connected for $n\ne 2$. We also obtain that $V{Q}_n$ is super-${\lambda }_c$ for $n\ge 3$, super-${\lambda }^{\prime }$ for $n\ge 3$ and $n\ne 4$, and the cyclic edge-connectivity of $V{Q}_n$ is $4(n-2)$ for $n\ge 3$.

The paper is organized as follows. In Section 2, we give some propositions related to the vertex-transitive and edge-transitive graph. In Section 3, we introduce the necessary definitions and notations. In Section 4, we use the symmetry properties of the n-dimensional varietal hypercube to discuss some reliability measures of the n-dimensional varietal hypercube. In Section 5, we add some concluding remarks.

2. Related Works

We use a number of known results about some propositions of the vertex-transitive and edge-transitive graphs in the following.

Proposition 1

([21,22]). Let G be a connected graph that is both vertex-transitive and edge-transitive. Then $\kappa \left(G\right)=\delta \left(G\right)$, and moreover, G is not super-κ if and only if $G\cong C_n\circ m{K}_1(n\ge 6)\mathrm{or}L\left(Q_3\right)\circ m{K}_1$, where $L\left(Q_3\right)$ is the line graph of the three dimensional hypercube $Q_3$.

Proposition 2

([23,24,25,26,27]). Let G be an edge-transitive k-regular graph where $k\ge 3$. Then

(1) 

G is super-λ.

(2) 

${\lambda }^{\prime }\left(G\right)=2k-2$.

(3) 

If G is not isomorphic to $K_4$, $K_5$ or $K_{3,3}$, then ${\lambda }_c\left(G\right)=g(k-2)$, where g is the girth of G.

(4) 

G is not super-${\lambda }^{\prime }$ if and only if G is isomorphic to the three dimensional hypercube $Q_3$ or to an edge-transitive 4-regular graph of girth 4.

Let ${\mathbb{Z}}_n$ be denoted the residues modulo n for any positive integer n. We denote by ${\mathcal{G}}_{16}$ a graph with vertex set $\{a_i,b_i|i\in {\mathbb{Z}}_8\}$, and for each $i\in {\mathbb{Z}}_8$, if i is odd then $a_i{b}_{i+j}\in E\left({\mathcal{G}}_{16}\right)$ with $0\le j\le 2$ and $b_i{a}_{i+j}\in E\left({\mathcal{G}}_{16}\right)$ with $0\le j\le 2$, and if i is even then $a_i{b}_{i+j}\in E\left({\mathcal{G}}_{16}\right)$ with $0\le j\le 3$ and $b_i{a}_{i+j}\in E\left({\mathcal{G}}_{16}\right)$ with $0\le j\le 3$. If $n\ge 3$ and $1\le t<n/2$, the generalized petersen graph $P(n,t)$ is a graph with vertex set $\{x_i,y_i|i\in {\mathbb{Z}}_n\}$ and edge set $\{\{x_i,x_{i+1}\},\{x_i,y_i\},\{y_i,y_{i+t}\}|i\in {\mathbb{Z}}_n\}$. For a 4-regular graph H, the subdivided double $\mathcal{D}\left(H\right)$ of H is the bipartite graph with the vertex set $(V\left(H\right)\times {\mathbb{Z}}_2)\cup E\left(H\right)$ and an edge between vertices $(v,i)\in V\left(H\right)\times {\mathbb{Z}}_2$ and $e\in E\left(H\right)$ whenever v is incident with e in H, where $(V\left(H\right)\times {\mathbb{Z}}_2)=\{(u,0),(u,1)|u\in V\left(H\right)\}$. The graph $G-m{K}_2$ is obtained by removing m vertex disjoint edges from G.

In [28], Zhou and Feng have classified all ${\lambda }_c$-optimal nonsuper-${\lambda }_c$ edge-transitive regular graphs and obtained some properties of ${\lambda }_c$-optimal nonsuper-${\lambda }_c$ edge-transitive regular graphs.

Proposition 3

([28]). Let G be an edge-transitive k-regular graph of girth g where $k\ge 3$. Suppose that G is not isomorphic to $K_4$, $K_5$, or $K_{3,3}$. Then, G is not super-${\lambda }_c$ if and only if at least one of the following holds.

(1) 

$g=6$, $k=3$ and $G\cong P(10,7)$ or $P(8,3)$.

(2) 

$g=4$ and $k=4$, G is isomorphic to $Q_4$ (the 4-cube), $K_{5.5}-5K_2$, $C_n\circ 2K_1(n\ge 4)$ or $\mathcal{D}\left(H\right)$ for some arc-transitive 4-regular graph H.

(3) 

$g=4$, $k=5$ and $G\cong K_{6.6}-6K_2$.

(4) 

$g=4$, $k=6$ and $G\cong {\mathcal{G}}_{16}$.

(5) 

$g=3$, $k=6$ and $G\cong K_8-4K_2$ or G is isomorphic to the line graph of an arc-transitive 4-regular triangle-free graph.

3. Preliminaries

In this section, we give some definitions and notations. If definitions and notations are not defined here, we follow [29].

Let $Aut\left(G\right)$ be the automorphism group of the graph G. We say that G is vertex-transitive if for any two vertices $u,v\in V\left(G\right)$ there exists an automorphism $\pi \in Aut\left(G\right)$ such that $\pi \left(u\right)=v$. If for any pair of edges (resp. arcs), there exists an automorphism that transforms one into the other, then G is edge-transitive (resp. arc-transitive) graph. Clearly, if a graph is arc-transitive, then it must be vertex-transitive and edge-transitive. Let u and v be two vertices of G, and the length of a shortest path is called the distance and denoted by $d_G(u,v)$. The maximum distance between any pair of vertices of a connected graph G is called the diameter of the graph G and denoted by $diam\left(G\right)$.

Suppose that G and H are two graphs, the lexicographic product $G\circ H$ is defined as the graph with vertex set $V\left(G\right)\times V\left(H\right)$, two vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent if and only if either $x_1{x}_2\in E\left(G\right)$, or $x_1=x_2$ and $y_1{y}_2\in E\left(H\right)$. The graph showed in Figure 1 is $C_6\circ 2K_1$.

Figure 1. The lexicographic product of $C_6\circ 2K_1$.

Let n be a positive integer. The definitions of the n-dimensional hypercube $Q_n$ and the n-dimensional varietal hypercube $V{Q}_n$ are stated as follows.

Definition 1

([30]). The n-dimensional hypercube is a connected graph with $2^n$ vertices and denoted by $Q_n$. The vertex set $V\left(Q_n\right)=\{x_1{x}_2\cdots x_n:x_i=0\mathrm{or}1,1\le i\le n\}$. Two vertices $u=u_1{u}_2\cdots u_n$ and $v=v_1{v}_2\cdots v_n$ in $Q_n$ are adjacent if and only if they differ in exact one position.

Definition 2

([17]). The n-dimensional varietal hypercube, denoted by $V{Q}_n$, has $2^n$ vertices, each labeled by an n-bit binary string and $V\left(V{Q}_n\right)=\{x_n{x}_{n-1}\cdots x_2{x}_1:x_i=0or1,i=1,2,\dots ,n\}$. $V{Q}_1$ is a complete graph $K_2$ of two vertices labeled with 0 and 1, respectively. For $n\ge 2$, $V{Q}_n$ can be recursively constructed from two copies of $V{Q}_{n-1}$, denoted by $V{Q}_{n-1}^0$ and $V{Q}_{n-1}^1$, and by adding $2^{n-1}$ edges between $V{Q}_{n-1}^0$ and $V{Q}_{n-1}^1$, where $V\left(V{Q}_{n-1}^0\right)=\{0x_{n-1}\cdots x_2{x}_1:x_i=0or1,i=1,2,\dots ,n-1\}$, $V\left(V{Q}_{n-1}^1\right)=\{1x_{n-1}\cdots x_2{x}_1:x_i=0or1,i=1,2,\dots ,n-1\}$. The vertex $x=0x_{n-1}{x}_{n-2}{x}_{n-3}\cdots x_2{x}_1\in V\left(V{Q}_{n-1}^0\right)$ is adjacent to the vertex $y=1y_{n-1}{y}_{n-2}{y}_{n-3}\dots y_2{y}_1\in V\left(V{Q}_{n-1}^1\right)$ if and only if

$$x_{n-1}{x}_{n-2}{x}_{n-3}\cdots x_2{x}_1=y_{n-1}{y}_{n-2}{y}_{n-3}\cdots y_2{y}_{1}if3\nmid n,or$$

(1)

$$x_{n-3}\cdots x_2{x}_1=y_{n-3}\cdots y_2{y}_{1}and(x_{n-1}{x}_{n-2},y_{n-1}{y}_{n-2})\in \{(00,00),(01,01),(10,11),(11,10)\}if3\mid n.$$

(2)

Obviously, $V{Q}_n$ is an n-regular and its girth is 4. Moreover, it contains circles of length 5 when $n\ge 3$ [18,19]. The varietal hypercubes $V{Q}_1$, $V{Q}_2$, and $V{Q}_3$ are illustrated in Figure 2.

Figure 2. The varietal hypercubes $V{Q}_1$, $V{Q}_2$, $V{Q}_3$, and $V{Q}_4$.

As a variant of the hypercube, the n-dimensional varietal hypercubes $V{Q}_n$, which has the same number of vertices and edges as $Q_n$, not only has the most ideal characteristics of $Q_n$, including some characteristics such as recursive structure, strong connectivity, and symmetry but also has a smaller diameter than $Q_n$, and its average distance is smaller than the hypercube [17].

Lemma 1

([17]). $diam\left(V{Q}_n\right)=\lceil 2n/3\rceil$ for $n\ge 3$.

Lemma 2

([14]). ${\kappa }^{\left(2\right)}\left(V{Q}_n\right)>{\kappa }^{\left(1\right)}\left(V{Q}_n\right)$ for $n\ge 9$.

Lemma 3

([15,20,31]). $V{Q}_n$ is vertex-transitive and edge-transitive.

Lemma 4

([32]). Let G and H be two graphs. If G is non-trivial, non-complete, and connected, then $\kappa (G\circ H)=\kappa \left(G\right)\left|V\right(H\left)\right|$.

Lemma 5

([33]). Suppose $(x_1,y_1)$ and $(x_2,y_2)$ are two vertices of $G\circ H$. Then, the distance between $(x_1,y_1)$ and $(x_2,y_2)$ is

$$d_{G\circ H}((x_1,y_1),(x_2,y_2))=\left\{\begin{array}{cc}{d}_G(x_1,x_2)\hfill & \mathit{if}{x}_1\ne x_2,\hfill \\ d_H(y_1,y_2)\hfill & \mathit{if}{x}_1=x_2\mathit{and}{d}_G\left(x_1\right)=0,\hfill \\ min\{d_H(y_1,y_2),2\}& \mathit{if}{x}_1=x_2\mathit{and}{d}_G\left(x_1\right)\ne 0.\hfill \end{array}\right.$$

(3)

From the above lemma, we can obtain that the diameter of $G\circ H$ is

$$diam(G\circ H)=max\{d_{G\circ H}((x_1,y_1),(x_2,y_2)):(x_1,y_1),(x_2,y_2)\in V\left(G\right)\times V\left(H\right)\}.$$

4. Reliability Evaluation of Varietal Hypercubes

In this section, we will study some reliability measures in varietal hypercubes, such as restricted (vertex-)connectivity and cyclic connectivity. Although Xu et al. proved that $V{Q}_n$ is super vertex-connected for any $n\ge 1$ and super edge-connected for $n\ne 2$ in [19], we prove these results and obtain other results such as $V{Q}_n$ is super-${\lambda }_c$ for $n\ge 3$, super-${\lambda }^{\prime }$ for $n\ge 3$, and $n\ne 4$ through symmetric properties of $V{Q}_n$.

Theorem 1.

$V{Q}_n$ is super-κ for $n\ge 1$.

Proof. 

If $n=1$, then $V{Q}_n$ is a complete graph $K_2$. Obviously, it is super-$\kappa$.

If $n=2$, then $V{Q}_2$ is cycle $C_4$, and clearly it is super-$\kappa$.

Assume that $n\ge 3$. Suppose to the contrary that $V{Q}_n$ is not super-$\kappa$. Since $V{Q}_n$ is both vertex-transitive and edge-transitive, it follows that $V{Q}_n\cong C_l\circ m{K}_1(l\ge 6)\mathrm{or}L\left(Q_3\right)\circ m{K}_1$ from Proposition 1.

First suppose that $V{Q}_n\cong C_l\circ m{K}_1(l\ge 6)$; we know that $\kappa \left(V{Q}_n\right)=n$ and $\kappa (C_l\circ m{K}_1)=2m$ by Lemma 4, $n=2m$ and $m>1$. Moreover, since $|V(C_l\circ m{K}_1)|=|V\left(V{Q}_n\right)|$, it follows that $2^n=ml$. Thus, m and l are even. According to Lemma 1 and Lemma 5, we know that $diam\left(V{Q}_n\right)=\lceil 2n/3\rceil$ and $diam(C_l\circ m{K}_1)=l/2$, respectively. Now, $l/2=\lceil 2n/3\rceil =\lceil 4m/3\rceil$ and $l=2\lceil 4m/3\rceil$. Now, $2^n=ml=2m\lceil 4m/3\rceil$, and we have $2^{n-1}=m\lceil 4m/3\rceil$.

We discuss the value of m, as follows.

Case 1.$m=3k$ where k is a positive integer.

Then, $2^{n-1}=m\lceil 4m/3\rceil =4mk=12k^2$. Now, we obtain $2^{n-3}=3k^2$, a contradiction.

Case 2.$m=3k+1$ where k is a positive integer.

Then, $2^{n-1}=m\lceil 4m/3\rceil =(3k+1)\lceil (12k+4)/3\rceil =(3k+1)(4k+2)$. So, we have $2^{n-2}=(3k+1)(2k+1)$. In this case, $(2k+1)|2^{n-2}$; hence, we have $k=0$ and $m=1$. This contradicts the fact that m is even.

Case 3.$m=3k+2$ where k is a positive integer.

Then, $2^{n-1}=m\lceil 4m/3\rceil =(3k+2)\lceil (12k+8)/3\rceil =(3k+2)(4k+3)$. So, we have $2^{n-1}=(3k+2)(4k+3)$ and $(4k+3)|2^{n-1}$, a contradiction.

Now, suppose that $V{Q}_n\cong L\left(Q_3\right)\circ m{K}_1$, then the number of vertices of these graphs is same and we obtain $2^n=12m$, so we have $2^{n-2}=3m$ and $3|2^{n-2}$ where $n\ge 3$, a contradiction. □

Theorem 2.

$V{Q}_n$ is super-λ for $n\ne 2$.

Proof. 

Since $V{Q}_1$ is a complete graph $K_2$, it is super-$\lambda$. If $n=2$, $V{Q}_n$ is a cycle $C_4$. Clearly, it is not super-$\lambda$. Suppose that $V{Q}_n$ is not super-$\lambda$ for $n\ge 3$. $V{Q}_n$ is edge-transitive k-regular graph where $k\ge 3$, so the theorem follows from Proposition 2(1). □

In [19], Xu et al. proved that 1-restricted edge-connectivity of $V{Q}_n$ is $2n-2$. Here, we present another proof of this result.

Theorem 3.

${\lambda }^{\prime }\left(V{Q}_n\right)=2n-2$ for $n\ge 2$.

Proof. 

If $n=2$, then $V{Q}_n$ is a cycle $C_4$. Clearly, ${\lambda }^{\prime }\left(V{Q}_n\right)=2$. Thus, we may suppose that $n\ge 3$. $V{Q}_n$ is an edge-transitive k-regular graph where $k\ge 3$, so the theorem follows from Proposition 2(2). □

In the following theorem, we show that every minimum edge-cut of $V{Q}_n$ for $n\ge 3$ and $n\ne 4$ isolates an edge.

Theorem 4.

$V{Q}_n$ is super-${\lambda }^{\prime }$ for $n\ge 3$ and $n\ne 4$.

Proof. 

Assume that $V{Q}_n$ is not super-${\lambda }^{\prime }$ for $n\ge 3$ and $n\ne 4$. According to $V{Q}_n$ is edge-transitive and Proposition 2(4), we know that $V{Q}_n$ is not super-${\lambda }^{\prime }$ if and only if $V{Q}_n$ is isomorphic to $Q_3$ or to an edge-transitive 4-regular graph of girth 4.

If $n=3$, $Q_3$ does not have cycle of length 5 but $V{Q}_3$ has, then $V{Q}_3$ is not isomorphic to $Q_3$.

If $n\ge 5$, since $V{Q}_n$ is edge-transitive k-regular graph where $k\ge 5$, then it implies that $V{Q}_n$ is super-${\lambda }^{\prime }$ when $n\ge 5$.

We have accordingly proved the theorem. □

In the following theorem, we prove that if we remove $4(n-2)$ edges from the $V{Q}_n$, then we will obtain a disconnected graph with at least two components containing cycle for $n\ge 3$.

Theorem 5.

${\lambda }_c\left(V{Q}_n\right)=4(n-2)$ for $n\ge 3$.

Proof. 

Since $V{Q}_n$ has at least 8 vertices for $n\ge 3$, it is not isomorphic to $K_4$, $K_5$ or $K_{3,3}$. According to the fact that $V{Q}_n$ has girth 4, it follows that ${\lambda }_c\left(V{Q}_n\right)=4(n-2)$ by Proposition 2(3). □

The following theorem shows that every minimum cyclic edge-cut of $V{Q}_n$ isolates a shortest cycle for $n\ge 3$.

Theorem 6.

$V{Q}_n$ is super-${\lambda }_c$ for $n\ge 3$.

Proof. 

Since $V{Q}_n$ is an edge-transitive n-regular graph of girth $g=4$ with $2^n$ vertices. By the definition of $V{Q}_n$, we obtain that $V{Q}_n$ is not isomorphic to $K_4$, $K_5$ or $K_{3,3}$. If $n=3$, then it is obvious that $V{Q}_n$ is super-${\lambda }_c$.

It is assumed that $V{Q}_n$ is not super-${\lambda }_c$ for $n\ge 4$. So if $n=4$, then $V{Q}_n$ is edge-transitive 4-regular graph. According to Proposition 3(3), it follows that $\Gamma$ is isomorphic to $Q_4$ (the 4-cube), $K_{5.5}-5K_2$, $C_n\circ 2K_1(n\ge 4)$, or $\mathcal{D}\left(H\right)$, for some arc-transitive 4-regular graph H. Since $V{Q}_4$ has odd cycle of length 5 and 16 vertices, it is not difficult to check that $V{Q}_4$ is not isomorphic to $Q_4$, $K_{5.5}-5K_2$, or $\mathcal{D}\left(H\right)$, for some 4-regular arc-transitive graph H. Furthermore, if $V{Q}_n\cong C_l\circ 2K_1$ for $n=4$ and $l\ge 4$, then $|V(C_l\circ 2K_1]\left)\right|=|V\left(V{Q}_n\right)|$, it follows that $2l=2^4=16$ and $l=8$. However, in this case, $diam\left(V{Q}_n\right)=\lceil 2n/3\rceil =3$ and $diam(C_l\circ 2K_1)=\lfloor l/2\rfloor =4$ by Lemma 1 and Lemma 5. So, $V{Q}_n$ is not isomorphic to $C_l\circ 2K_1$, a contradiction. Now, by Proposition 3(3) and (4), $V{Q}_5$ or $V{Q}_6$ has 12 or 16 vertices, a contradiction.

Thus, we conclude that $V{Q}_n$ is super-${\lambda }_c$ for $n\ge 3$. □

The next theorem shows that every minimum vertex-cut of $V{Q}_n$ isolates an edge for $n\ge 9$.

Theorem 7.

$V{Q}_n$ is super-${\kappa }^{\prime }$ for $n\ge 9$.

Proof. 

By Lemma 2, we know that ${\kappa }^{\left(2\right)}\left(V{Q}_n\right)>{\kappa }^{\left(1\right)}\left(V{Q}_n\right)$ for $n\ge 9$. It is assumed that $V{Q}_n$ is not super-${\kappa }^{\prime }$ for $n\ge 9$. Hence, there is an 1-restricted vertex-cut S and $\left|S\right|={\kappa }^{\left(1\right)}\left(V{Q}_n\right)$, but the cut is not the neighborhood of any edge. That is, S is also a two-restricted vertex-cut and $\left|S\right|\ge {\kappa }^{\left(2\right)}\left(V{Q}_n\right)$. So, we have $\left|S\right|>{\kappa }^{\left(1\right)}\left(V{Q}_n\right)$, and this leads to a contradiction. □

5. Concluding Remarks

In order to better measure the performance of the network, the concepts of restricted connectivity, restricted edge-connectivity, and cyclic edge-connectivity have been successively proposed. The research in this paper is based on these concepts and uses some properties of vertex-transitive and edge-transitive graph to get the main result that the varietal hypercube is super restricted vertex-connected, super restricted edge-connected, and super cyclically edge-connected. Moreover, the cyclic edge-connectivity of $V{Q}_n$ is $4(n-2)$ for $n\ge 3$. In the future, we will continue to research the other properties of the varietal hypercube.