Super Spanning Connectivity of the Folded Divide-and-SwapCube

Abstract

A $k^{*}$-container of a graph G is a set of k disjoint paths between any pair of nodes whose union covers all nodes of G. The spanning connectivity of G, ${\kappa }^{*}\left(G\right)$, is the largest k, such that there exists a $j^{*}$-container between any pair of nodes of G for all $1\le j\le k$. If ${\kappa }^{*}\left(G\right)=\kappa \left(G\right)$, then G is super spanning connected. Spanning connectivity is an important property to measure the fault tolerance of an interconnection network. The divide-and-swap cube $DS{C}_n$ is a newly proposed hypercube variant, which reduces the network cost from $O\left(n^2\right)$ to $O\left(n{log}_{2}n\right)$ compared with the hypercube and other hypercube variants. The folded divide-and-swap cube $FDS{C}_n$ is proposed based on $DS{C}_n$ to reduce the diameter of $DS{C}_n$. Both $DS{C}_n$ and $FDS{C}_n$ possess many better properties than hypercubes. In this paper, we investigate the super spanning connectivity of $FDS{C}_n$ where $n=2^d$ and $d\ge 1$. We show that ${\kappa }^{*}\left(FDS{C}_n\right)=\kappa \left(FDS{C}_n\right)=d+2$, which means there exists an m-DPC(node-disjoint path cover) between any pair of nodes in $FDS{C}_n$ for all $1\le m\le d+2$.

1. Introduction

In a parallel computer system, the interconnection network determines the performance of the system. Generally, an interconnection network is denoted by $G=\left(V\right(G),E(G\left)\right)$, where $E\left(G\right)$ is the edge set and $V\left(G\right)$ is the node set. We say that two nodes $x\in V\left(G\right)$ and $y\in V\left(G\right)$ are adjacent if $(x,y)\in E\left(G\right)$. A path P is a sequence of n nodes in G, denoted by $⟨{\upsilon }_1,{\upsilon }_2,\cdots ,{\upsilon }_n⟩$ where $({\upsilon }_i,{\upsilon }_{i+1})\in E\left(G\right)$ with $1\le i\le n-1$. If the endpoints ${\upsilon }_1$ and ${\upsilon }_n$ are the same, then P is a cycle. A Hamiltonian path consists of all the nodes in G. A Hamiltonian cycle traverses all nodes of G exactly once. G is Hamiltonian connected if and only if there exists a Hamiltonian path connecting any two nodes of G, and G is Hamiltonian if there exists a Hamiltonian cycle in G. Two paths between a and b are node-disjoint if they do not have common nodes except a and b.

Let $\kappa \left(G\right)$ be the connectivity of G. From Menger’s theorem, there exist k node-disjoint paths between any pair of nodes a and b in G if $k\le \kappa \left(G\right)$. The node-disjoint path problem has been studied for many networks, such as alternating group networks [1], balanced hypercubes [2], DCell networks [3], and so on. A set of k node-disjoint paths between a and b is called a k-container, denoted by $C(a,b)$. $C(a,b)$ is called a spanning k-container, denoted by $k^{*}$-container, if it covers all nodes of G. A $k^{*}$-container is also known as a one-to-one disjoint path cover (DPC for short) of G. A disjoint path cover problem of G is to find node-disjoint paths between any two distinct nodes whose union covers all nodes of G [4]. If any two nodes have a $k^{*}$-container between them, then G is said to be $k^{*}$-connected. Obviously, if G is Hamiltonian-connected, it is $1^{*}$-connected, and if G is Hamiltonian, then it is $2^{*}$-connected. The spanning connectivity of a graph G, denoted by ${\kappa }^{*}\left(G\right)$, is the largest k such that G is $j^{*}$-connected for all $1\le j\le k$. If ${\kappa }^{*}\left(G\right)=\kappa \left(G\right)$, then G is super spanning connected.

It is important to study the node-disjoint path problem between any pair of distinct nodes [5,6]. Disjoint paths between nodes can be used to accelerate data transmission by allocating data to different communication paths [7]. Other advantages of adopting such a node-disjoint routing scheme are the improved robustness to node failures, as well as the improved load balancing capability [8,9]. Software testing is another well-known application of multiple disjoint path covers [10]. Node-disjoint path cover or spanning connectivity is an important property to measure the fault tolerance of a network, which has been studied for many networks, such as hypercube-like networks [11], alternating group graphs [12], multi-dimensional toris [13], DCell networks [14], WK-recursive networks [15], generalized Petersen graphs [16], split-star networks [17], and so on.

The n-dimensional divide-and-swap cube, denoted by $DS{C}_n$($n=2^d,d\ge 1$), was proposed by Kim et al. in [18] as a hypercube variant. Compared with a hypercube, $DS{C}_n$ reduces the network cost from $O\left(n^2\right)$ to $O\left(n{log}_{2}n\right)$. Moreover, $DS{C}_n$ possesses many other attractive properties. The diameter upper bound of $DS{C}_n$ is $\frac{5n}{4}-1$ and the degree is ${log}_{2}n+1$. $DS{C}_n$ is Hamiltonian and Hamiltonian-connected. In [19], Ning has shown that the connectivity and edge connectivity are both ${log}_{2}n+1$, and the super connectivity and super edge connectivity are both ${log}_2{n}^2$. In [20], Zhou et al. studied the $𝘍$-structure connectivity and $𝘍$-substructure connectivity for $𝘍\in \{K_1,K_{1,1},K_{1,i}(2\le i\le d+1),C_4\}$. In [21], Zhou et al. showed that the component connectivity $c{\kappa }_{r+1}\left(DS{C}_n\right)=(d-1)r+2$ and the diagnosability $c{t}_{r+1}\left(DS{C}_n\right)=(d-1)r+d+1$ for $1\le r\le d-1$, $d\ge 3$. In [22], Zhao et al. investigated the generalized connectivity ${\kappa }_4\left(DS{C}_n\right)=d$ for $d\ge 1$. The n-dimensional folded divide-and-swap cube, $FDS{C}_n$($n=2^d,d\ge 1$), was also proposed in [18]. $FDS{C}_n$ is constructed based on $DS{C}_n$ by adding one edge to each node of $DS{C}_n$. Obviously, the connectivity of $FDS{C}_n$ is ${log}_{2}n+2=d+2$ [23]. In addition, the diameter upper bound of $FDS{C}_n$ is $n-1$ and the network cost is $O\left(n{log}_{2}n\right)$. $FDS{C}_n$ is also Hamiltonian and Hamiltonian-connected. $FDS{C}_n$ is appropriate as a candidate topology for a data center network. In [24], Chang et al. constructed the dual-CIST for $FDS{C}_n$. In [23], Zhao and Chang investigated the connectivity, super connectivity and generalized 3-connectivity of $FDS{C}_n$. We can see that $DS{C}_n$ and $FDS{C}_n$ have many better properties than the hypercube.

In this work, we investigated the super spanning connectivity of $FDS{C}_n$ with $n=2^d$ and $d\ge 1$. We show that there exists an m-DPC between any pair of nodes in $FDS{C}_n$ for all $1\le m\le \kappa \left(FDS{C}_n\right)$. The remainder of the paper is structured as follows. We formally define $DS{C}_n$ and $FDS{C}_n$ In Section 2, and then provide two basic properties of $FDS{C}_n$. The proof of super spanning connectivity of $FDS{C}_n$ is given in Section 3. Finally, we summarize the paper.

2. Preliminaries

In this section, we give the formal definition of $DS{C}_n$ and $FDS{C}_n$. Each node u in $DS{C}_n$ is denoted by an n-bit binary string, where $u=x_1{x}_2\cdots x_n$ and $x_i\in \{0,1\}$ for $1\le i\le n$. $DS{C}_n$ is defined as follows.

Definition 1

([18]). For $n=2^d$ and $d\ge 1$, $DS{C}_n$ has $2^n$ nodes, each of which is represented by an n-bit binary string, where $u=x_1{x}_2\cdots x_n=s_1{s}_2{s}_3$ and $x_i\in \{0,1\}$ for $1\le i\le n$. For each k with $1\le k\le d$, $s_1=x_1{x}_2\cdots x_{\frac{n}{2^k}}$, $s_2=x_{\frac{n}{2^k}+1}{x}_{\frac{n}{2^k}+2}\cdots x_{\frac{n}{2^{k-1}}}$ and $s_3=x_{\frac{n}{2^{k-1}}+1}{x}_{\frac{n}{2^{k-1}}+2}\cdots x_{\frac{n}{2^{k-2}}}$. If $k=1$, then $u=s_1{s}_2$ and $s_3$ is empty. If node v satisfies one of the following conditions, then $(u,v)\in E\left(DS{C}_n\right)$.

(1) 

$v={\overline{x}}_1{x}_2\cdots x_n$, where ${\overline{x}}_1$ is the complement of $x_1$. Here, $(u,v)$ is called an $e\left(1\right)$-edge.

(2) 

$\left\{\begin{array}{c}{\overline{s}}_1{\overline{s}}_2{s}_{3}if{s}_1=s_2,\hfill \\ s_2{s}_1{s}_{3}if{s}_1\ne s_2\hfill \end{array}\right.$ Here, $(u,v)$ is called an $e\left(\frac{2n}{2^k}\right)$-edge.

$DS{C}_2$ and $DS{C}_4$ are shown in Figure 1. For any integer n with $n=2^d$ and $d\ge 1$, $FDS{C}_n$ is built based on $DS{C}_n$ by adding an edge to each node of $DS{C}_n$.

Figure 1. $DS{C}_2$ and $DS{C}_4$.

Definition 2

([18]). $V\left(FDS{C}_n\right)=V\left(DS{C}_n\right)$, $E\left(FDS{C}_n\right)=E\left(DS{C}_n\right)\cup E_f$, where $E_f=\left\{(u,v)\right|u=x_1{x}_2\cdots x_n$ and $v=x_1{\overline{x}}_2\cdots x_n\}$. An edge in $E_f$ is called an $e\left(f\right)$-edge.

$FDS{C}_n$ can be divided into $2^{\frac{n}{2}}$$FDS{C}_{\frac{n}{2}}$, each of which is represented by $FDS{C}_{\frac{n}{2}}^i$ or $FDS{C}_{2^{d-1}}^i$, with $0\le i\le 2^{\frac{n}{2}}-1$. We call $FDS{C}_{2^{d-1}}^i$ a module of $FDS{C}_n$. Let $u=x_1{x}_2\cdots x_n=AB$ be any node in $FDS{C}_n$. We have $A=x_1{x}_2\cdots x_{\frac{n}{2}}$ is the local address in the module and $B=x_{\frac{n}{2}+1}{x}_{\frac{n}{2}+2}\cdots x_n$ is the module address. Then, a $FDS{C}_{2^{d-1}}$ module can also be represented by $D_B$. For example, node 0100 is in $D_{00}$(or $D_0$ with decimal representation) and node 1111 is in $D_{11}$ (or $D_3$ with decimal representation) in $FDS{C}_4$ (see Figure 2).

Figure 2. $FDS{C}_2$ and $FDS{C}_4$.

Property 1

([18]). For modules $D_{B_i}$ and $D_{B_j}$ with $i\ne j$, if $B_i={\overline{B}}_j$, then there are two edges, $(B_i{B}_i,B_j{B}_j)$ and $(B_j{B}_i,B_i{B}_j)$, between $D_{B_i}$ and $D_{B_j}$; otherwise, there exists only one edge, $(B_j{B}_i,B_i{B}_j)$, between $D_{B_i}$ and $D_{B_j}$.

For example, modules $D_{00}$ and $D_{11}$ are connected by two edges $(0000,1111)$ and $(1100,0011)$ in $FDS{C}_4$, where $B_i=00$ and $B_j=11$. Modules $D_{00}$ and $D_{01}$ are connected by one edge, $(0100,0001)$ in $FDS{C}_4$, where $B_i=00$ and $B_j=01$. If $k=1$, then $u=s_1{s}_2=AB$ is in $D_{s_2}$. By Definition 1, $(0000,1111)$, $(1100,0011)$, and $(0100,0001)$ are all $e\left(n\right)$-edges with $n=4$. An $e\left(n\right)$-edge of node u is denoted by $e_n^u$, which is an external edge of u. Obviously, each module has only one node $u_1=s_1{s}_2$ with $s_1=s_2$. Let $u_2={\overline{s}}_1{s}_2$ and $S=V\left(D_{s_2}\right)-\left\{u_1\right\}$. Since each node in the same module has the same module address and the different local address, then each node in S is connected to a different module of $FDS{C}_n$ and $u_2$ is connected to $s_2{\overline{s}}_1$ in module $D_{{\overline{s}}_1}$. Since $u_1$ is connected to node ${\overline{s}}_1{\overline{s}}_2={\overline{s}}_1{\overline{s}}_1$, then $u_1$ and $u_2$ are both connected to module $D_{{\overline{s}}_1}$. Let $h=2^{\frac{n}{2}}$. According to Property 1, we have the following property.

Property 2.

Let $D_0,D_1,\cdots ,D_{h-1}$ be the h modules of $FDS{C}_n$ each of which has h nodes. Let $u=s_1{s}_2$ be the node with $s_1=s_2$ in $D_i$ with $0\le i\le h-1$. Then each node in $V\left(D_i\right)-\left\{u\right\}$ is connected to a different $D_j$ with $0\le i\ne j\le h-1$ with an $e\left(n\right)$-edge. Let $v={\overline{s}}_1{s}_2$. Then both u and v are connected to $D_{{\overline{s}}_1}$ with an $e\left(n\right)$-edge.

If the local addresses of two nodes u and v are complemented, $e_n^u$ and $e_n^v$ are connected to the same module of $FDS{C}_n$.

3. The Super Spanning Connectivity of ${\mathit{FDSC}}_{\mathit{n}}$

In this section, we will give some terminologies of $DS{C}_n$ and $FDS{C}_n$. We list some symbols in Table 1. Throughout this paper, let $h=2^{\frac{n}{2}}$.

Table 1. Acronyms and Notation.

Symbol	Definition
$DS{C}_n$	the n-dimensional divide-and-swap cube where $n=2^d$ and d is any positive integer
$FDS{C}_n$	the n-dimensional folded divide-and-swap cube
$DPC$	disjoint path cover
$D_i$	the ith module of $FDS{C}_n$ with $0\le i\le 2^{\frac{n}{2}}-1$
${\kappa }^{*}\left(G\right)$	the spanning connectivity of a graph G

Lemma 1

([18]). $DS{C}_n$ is Hamiltonian-connected for $n=2^d$ and $d\ge 1$.

Lemma 2

([18]). $DS{C}_n$ is Hamiltonian for $n=2^d$ and $d\ge 1$.

Based on the definition of $FDS{C}_n$, we have two lemmas below.

Lemma 3.

$FDS{C}_n$ is Hamiltonian-connected for $n=2^d$ and $d\ge 1$.

Lemma 4

([18]). $FDS{C}_n$ is Hamiltonian for $n=2^d$ and $d\ge 1$.

By Lemmas 3 and 4, we have Lemma 5.

Lemma 5.

For any integer n with $n=2^d$ and $d\ge 1$, there is a 1-DPC and a 2-DPC between any pair of nodes μ and ν.

Lemma 6.

Given $FDS{C}_n$ and its h modules $D_0,D_1,\cdots ,D_{h-1}$ where $n=2^d$ and $d\ge 2$, let $T={\bigcup }_{i=1}^r{D}_{B_i}$ where $2\le r\le h-1$ and $0\le B_i\le h-1$, which consists of r modules and the $e\left(n\right)$-edges between them. If u and v are two nodes in different modules of T with $e_n^u,e_n^v\notin E\left(T\right)$, then there exists a Hamiltonian path between u and v, which covers all the nodes of T.

Proof. 

We present the cases as follows.

Case 1. $r=2$.

Suppose $u\in V\left(D_{B_1}\right)$ and $v\in V\left(D_{B_2}\right)$. Since $e_n^u,e_n^v\notin E\left(T\right)$, by Property 2, we can find a node $x\in V\left(D_{B_1}\right)\ne u$ and a node $y\in V\left(D_{B_2}\right)\ne v$ such that $(x,y)\in E\left(T\right)$. Since $D_{B_1}$ and $D_{B_2}$ are both Hamiltonian-connected, there exists a Hamiltonian path $⟨u,P_1,x⟩$ in $D_{B_1}$ and a Hamiltonian path $⟨y,P_2,v⟩$ in $D_{B_2}$. Combining the paths constructed above, we obtain the required Hamiltonian path $S=⟨u,P_1,x,y,P_2,v⟩$.

Case 2. $r\ge 3$.

Suppose $u\in V\left(D_{B_2}\right)$ and $v\in V\left(D_{B_1}\right)$. For each module $D_{B_j}$ with $3\le j\le r-1$, we can find two nodes $x_j$ and $y_j$ such that $x_j$ is connected to $y_{j-1}$ in $D_{B_{j-1}}$ and $y_j$ is connected to $x_{j+1}$ in $D_{B_{j+1}}$. For module $D_{B_r}$, we can find two nodes, $x_r$ and $y_r$, such that $y_r$ is connected to $x_1$ in $D_{B_1}$. Hence, we can construct the required Hamiltonian path, $S=⟨u,P_2,y_2,x_3,P_3,y_3,\cdots ,x_r,P_r,y_r,x_1,P_1,v⟩$ (see Figure 3). □

Figure 3. Illustration for Case 2 of Lemma 6.

Lemma 7.

Given $FDS{C}_n$ and its h modules $D_0,D_1,\cdots ,D_{h-1}$, where $n=2^d$ and $d\ge 2$, let $T={\bigcup }_{i=1}^r{D}_{B_i}$, where $\frac{h}{2}<r\le h-1$ and $0\le B_i\le h-1$, which consists of r modules and the $e\left(n\right)$-edges between them. If u and v are two nodes in the same module of T with $e_n^u,e_n^v\notin E\left(T\right)$, then there exists a Hamiltonian path between u and v, which covers all the nodes of T.

Proof. 

Suppose $u,v\in V\left(D_{B_1}\right)$. Since $r>\frac{h}{2}$ and $e_n^u,e_n^v\notin E\left(T\right)$, by Property 2, we can build a Hamiltonian path $⟨u,P_0,x,y,P_1,v⟩$ in $D_{B_1}$ such that x and y are connected to different modules of T and $(x,y)\in E\left(T\right)$. Suppose x is connected to $x^{\prime }$ in $D_{B_i}$ and y is connected to $y^{\prime }$ in $D_{B_j}$, where $2\le i\ne j\le r$. By Lemma 6, there is a Hamiltonian path $⟨x^{\prime },S,y^{\prime }⟩$, which covers all the nodes of $V\left(T\right)-V\left(D_{B_1}\right)$. Combining the paths constructed above, we obtain the required Hamiltonian path, $S=⟨u,P_0,x,x^{\prime },S,y^{\prime },y,P_1,v⟩$ (see Figure 4). □

Figure 4. Illustration for Lemma 7.

Lemma 8.

Let μ be any node in $FDS{C}_4$. We can find a node set $T=\left\{t_i\right|1\le i\le r\}$ with $1\le r\le 4$ such that there exists a one-to-many node disjoint path cover between μ and T in $FDS{C}_4$ and the node labels in $T+\left\{\mu \right\}$ are not complementary with each other.

Proof. 

For $r=1$, since $FDS{C}_4$ is Hamiltonian connected, it is easy to construct a one-to-many node disjoint path cover between $\mu$ and $T=\left\{t_1\right\}$ in $FDS{C}_4$ where $t_1$ is not complementary with $\mu$. For $2\le r\le 4$, W.L.O.G., suppose $\mu$ is in $D_0$. Let $T_0=\left\{t_i\right|1\le i\le m,t_i\in V\left(D_0\right)\}$ where $1\le m\le 3$. It is obvious the node labels are not complementary with each other in $T_0+\left\{\mu \right\}$. Since $D_0$ is a complete graph with four nodes, then there is a one-to-many node disjoint path cover between $\mu$ and $T_0$ in $D_0$. Suppose $\mu$ is connected to x in $D_i$ with $1\le i\le 3$. We can find at least one node $y\in V\left(D_j\right)$, of which the label is not complementary with those in $T_0+\left\{\mu \right\}$, where $1\le i\ne j\le 3$. By Lemma 6, there exists a Hamiltonian path between x and y, which covers all the nodes of $V\left(FDS{C}_4\right)-V\left(D_0\right)$. Hence, $T=T_0+\left\{y\right\}$ is the required node set for $2\le r\le 4$. □

Lemma 9.

Let μ be any node in $FDS{C}_n$ for $d\ge 2$ and $n=2^d$. We can find a node set $T=\left\{t_i\right|1\le i\le r\}$ with $1\le r\le d+2$, such that there exists a one-to-many node disjoint path cover between μ and T in $FDS{C}_n$ and the node labels are not complementary with each other in $T+\left\{\mu \right\}$.

Proof. 

To prove this lemma, we proceed by induction on d. According to Lemma 8, this lemma holds for $d=2$ and $n=4$. Suppose that this lemma holds for $FDS{C}_{2^{d-1}}$ for $d\ge 3$, we will prove that this lemma holds for $FDS{C}_{2^d}$.

For $r=1$, it is easy to find a node set $T=\left\{t_1\right\}$ to meet the condition this lemma holds. For $2\le r\le d+2$, W.L.O.G., suppose $\mu$ is in $D_0$. According to the hypothesis, there is a one-to-many node disjoint path cover between $\mu$ and $T_0=\left\{t_i\right|1\le i\le m\}$ in $D_0$ where $1\le m\le d+1$. Suppose that $\mu$ is connected to x in $D_i$ with $1\le i\le h-1$. Let y be any vertex in $D_j$ where $1\le i\ne j\le h-1$ and y is not complementary with each node in $T_0+\left\{\mu \right\}$. By Lemma 6, we can construct a path between x and y covering all the nodes of $V\left(FDS{C}_n\right)-V\left(D_0\right)$. Hence, $T=T_0\cup \left\{y\right\}$ is the required node set for $2\le r\le d+2$ (see Figure 5). □

Figure 5. Illustration for Lemma 9.

Lemma 10.

Let μ and ν be any two nodes in $FDS{C}_n$ for integer $d\in \{1,2\}$ and $n=2^d$. There exists an m-DPC between μ and ν in $FDS{C}_n$ where $1\le m\le d+2$.

Proof. 

By Lemma 5, there exists an m-DPC between $\mu$ and $\nu$ in $FDS{C}_n$ for $1\le m\le 2$. For $d=1$ and $n=2$, $FDS{C}_2$ is a complete graph. Obviously, there exists a 3-DPC between any two distinct nodes in $FDS{C}_2$. For $FDS{C}_4$, we need to prove there exists an m-DPC between $\mu$ and $\nu$ for $3\le m\le 4$. We consider the following three cases.

Case 1. $\mu$, $\nu \in V\left(D_i\right)$ where $0\le i\le 3$.

Suppose that $\mu$, $\nu \in V\left(D_0\right)$. Since $D_0$ is a complete graph, there exists an m-DPC between $\mu$ and $\nu$ in $D_0$ with $2\le m\le 3$.

Case 1.1. $e_4^{\mu }$ and $e_4^{\nu }$ are connected to the same $D_j$ with $1\le j\le 3$.

Suppose that $\mu$ is connected to x and $\nu$ is connected to y in $D_j$. By Lemma 7, there exists a $(x,y)$-Hamiltonian path covering all the nodes of $V\left(FDS{C}_4\right)-V\left(D_0\right)$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ where $3\le m+1\le 4$.

Case 1.2. $e_4^{\mu }$ is connected to $D_j$ and $e_4^{\nu }$ is connected to $D_k$ with $1\le j\ne k\le 3$.

Suppose that $\mu$ is connected to x in $D_j$ and $\nu$ is connected to y in $D_k$. By Lemma 6, there exists a $(x,y)$-Hamiltonian path covering all the nodes of $V\left(FDS{C}_4\right)-V\left(D_0\right)$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ where $3\le m+1\le 4$.

Case 2. $\mu \in V\left(D_i\right)$ and $\nu \in V\left(D_j\right)$ and $D_i$ and $D_j$ are connected with two $e\left(4\right)$-edges where $0\le i\ne j\le 3$.

We can suppose $i=0$, $j=3$, and $1\le k\ne l\le 2$. Let $H=FDS{C}_4-V\left(D_0\right)-V\left(D_3\right)$. furthermore let $U=\left\{s_a\right|1\le a\le m\}\subseteq V\left(D_i\right)$ and $V=\left\{t_a\right|1\le a\le m\}\subseteq V\left(D_j\right)$ with $2\le m\le 3$. Since $D_i$ and $D_j$ are complete graphs, there exists a one-to-many disjoint path cover between $\mu$ and U in $D_i$, and a one-to-many disjoint path cover between $\nu$ and V in $D_j$.

Case 2.1. $e_4^{\mu }$ and $e_4^{\nu }$ are connected to ${\mu }^{\prime }\in V\left(D_k\right)$ and ${\nu }^{\prime }\in V\left(D_k\right)$, respectively.

For $m=2$, we can have $(s_1,t_1)\in E\left(FDS{C}_4\right)$, $s_2$ and $t_2$ are connected to $s_2^{\prime }\in V\left(D_l\right)$ and $t_2^{\prime }\in V\left(D_l\right)$, respectively. Then we construct the 3-DPC as follows: $P_1=⟨\mu ,S_1,s_1,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,s_2^{\prime },Q_l,t_2^{\prime },t_2,T_2,\nu ⟩$, and $P_3=⟨\mu ,{\mu }^{\prime },Q_k,{\nu }^{\prime },\nu ⟩$.

For $m=3$, we can have $(s_1,t_1),(s_2,t_2)\in E\left(FDS{C}_4\right)$, $s_3$ and $t_3$ are connected to $s_3^{\prime }\in V\left(D_l\right)$ and $t_3^{\prime }\in V\left(D_l\right)$, respectively. Then we construct the 4-DPC as follows: $P_1=⟨\mu ,S_1,s_1,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,t_2,T_2,\nu ⟩$, $P_3=⟨\mu ,S_3,s_3,s_3^{\prime },Q_l,t_3^{\prime },t_3,T_3,\nu ⟩$, and $P_4=⟨\mu ,{\mu }^{\prime },Q_k,{\nu }^{\prime },\nu ⟩$.

Case 2.2. $e_4^{\mu }$ and $e_4^{\nu }$ are connected to ${\mu }^{\prime }\in V\left(D_k\right)$ and ${\nu }^{\prime }\in V\left(D_l\right)$, respectively.

By Lemma 6, there is a path $P_3=⟨\mu ,{\mu }^{\prime },Q,{\nu }^{\prime },\nu ⟩$ which covers all the nodes of $V\left(H\right)$. For $m=2$, we can have $(s_1,t_1),(s_2,t_2)\in E\left(FDS{C}_4\right)$. Then we construct the 3-DPC as follows: $P_1=⟨\mu ,S_1,s_1,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,t_2,T_2,\nu ⟩$, and $P_3$.

For $m=3$, we can have $(s_1,t_1),(s_2,t_2)\in E\left(FDS{C}_4\right)$, $s_3$ and $t_3$ are connected to $s_3^{\prime }\in V\left(D_l\right)$ and $t_3^{\prime }\in V\left(D_k\right)$, respectively. Then we construct the 4-DPC as follows: $P_1=⟨\mu ,S_1,s_1,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,t_2,T_2,\nu ⟩$, $P_3=⟨\mu ,{\mu }^{\prime },Q_k,t_3^{\prime },t_3,T_3,\nu ⟩$, and $P_4=⟨\mu ,S_3,s_3,s_3^{\prime },Q_l,{\nu }^{\prime },\nu ⟩$.

Case 2.3. $e_4^{\mu }$ and $e_4^{\nu }$ are connected to $t_1\in V\left(D_j\right)$ and ${\nu }^{\prime }\in V\left(D_k\right)$, respectively.

For $m=2$, we can have $(s_2,t_2)\in E\left(FDS{C}_4\right)$, $s_1$ is connected to $s_1^{\prime }\in V\left(D_l\right)$. By Lemma 6, there is a path $P_3=⟨\mu ,S_1,s_1,s_1^{\prime },Q,{\nu }^{\prime },\nu ⟩$ which covers all the nodes of $V\left(H\right)$. Then we construct the 3-DPC as follows: $P_1=⟨\mu ,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,t_2,T_2,\nu ⟩$, and $P_3$.

For $m=3$, we can have $(s_2,t_2)\in E\left(FDS{C}_4\right)$, $s_1$ is connected to $s_1^{\prime }\in V\left(D_k\right)$, $s_3$ and $t_3$ are connected to $s_3^{\prime }\in V\left(D_l\right)$ and $t_3^{\prime }\in V\left(D_l\right)$, respectively. Then we construct the 4-DPC as follows: $P_1=⟨\mu ,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,t_2,T_2,\nu ⟩$, $P_3=⟨\mu ,S_3,s_3,s_3^{\prime },Q_l,t_3^{\prime },t_3,T_3,\nu ⟩$, and $P_4=⟨\mu ,S_1,s_1,s_1^{\prime },Q_k,{\nu }^{\prime },\nu ⟩$.

Case 2.4. $e_4^{\mu }$ and $e_4^{\nu }$ are connected to ${\mu }^{\prime }\in V\left(D_k\right)$ and $s_1\in V\left(D_i\right)$ respectively.

The proof of this case is similar to Case 2.3.

Case 2.5. $(\mu ,\nu )\in E\left(FDS{C}_4\right)$.

For $m=2$, we can have $(s_1,t_1)\in E\left(FDS{C}_4\right)$, $s_2$ and $t_2$ are connected to $s_2^{\prime }\in V\left(D_l\right)$ and $t_2^{\prime }\in V\left(D_k\right)$, respectively. Then we construct the 3-DPC as follows: $P_1=⟨\mu ,S_1,s_1,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,s_2^{\prime },Q,t_2^{\prime },t_2,T_2,\nu ⟩$, and $P_3=⟨\mu ,\nu ⟩$.

For $m=3$, we can have $(s_1,t_1)\in E\left(FDS{C}_4\right)$, $s_2$ and $t_2$ are connected to $s_2^{\prime }\in V\left(D_k\right)$ and $t_2^{\prime }\in V\left(D_k\right)$, $s_3$ and $t_3$ are connected to $s_3^{\prime }\in V\left(D_l\right)$ and $t_3^{\prime }\in V\left(D_l\right)$, respectively. Then we construct the 4-DPC as follows: $P_1=⟨\mu ,S_1,s_1,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,s_2^{\prime },Q_2,t_2^{\prime },t_2,T_2,\nu ⟩$, $P_3=⟨\mu ,S_3,s_3,s_3^{\prime },Q_3,t_3^{\prime },t_3,T_3,\nu ⟩$, and $P_4=⟨\mu ,\nu ⟩$.

Case 2.6. $e_4^{\mu }$ and $e_4^{\nu }$ are connected to $t_1\in V\left(D_j\right)$ and $s_1\in V\left(D_i\right)$, respectively.

For $m=2$, we can have $s_2$ and $t_2$ are connected to $s_2^{\prime }\in V\left(D_l\right)$ and $t_2^{\prime }\in V\left(D_k\right)$, respectively. Then we construct the 3-DPC as follows: $P_1=⟨\mu ,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,s_2^{\prime },Q,t_2^{\prime },t_2,T_2,\nu ⟩$, and $P_3=⟨\mu ,S_1,s_1,\nu ⟩$.

For $m=3$, we can have $s_2$ and $t_2$ are connected to $s_2^{\prime }\in V\left(D_k\right)$ and $t_2^{\prime }\in V\left(D_k\right)$, $s_3$ and $t_3$ are connected to $s_3^{\prime }\in V\left(D_l\right)$ and $t_3^{\prime }\in V\left(D_l\right)$, respectively. Then we construct the 4-DPC as follows: $P_1=⟨\mu ,t_1,T_1,\nu ⟩$, $P_2=⟨\mu ,S_2,s_2,s_2^{\prime },Q_k,t_2^{\prime },t_2,T_2,\nu ⟩$, $P_3=⟨\mu ,S_3,s_3,s_3^{\prime },Q_l,t_3^{\prime },t_3,T_3,\nu ⟩$, and $P_3=⟨\mu ,S_1,s_1,\nu ⟩$.

Case 3. $\mu \in V\left(D_i\right)$ and $\nu \in V\left(D_j\right)$ and $D_i$ and $D_j$ are connected with one $e\left(4\right)$-edge where $0\le i\ne j\le 3$.

We can suppose $i=0$ and $j=1$. When $\nu =0001$, we list the 3-DPC and 4-DPC between $\mu \in V\left(D_0\right)$ and $\nu$ in Table 2. Similarly, there is a 3-DPC and a 4-DPC between every two nodes $\mu \in V\left(D_i\right)$ and $\nu \in V\left(D_j\right)$.

Table 2. The 3-DPC and 4-DPC between $\mu$ and $\nu$.

	3-DPC	4-DPC
$\mu =0100$	$⟨0100,0001⟩$	$⟨0100,0001⟩$
$\nu =0001$	$⟨0100,1100,0011,1111,1011,0111,1101,0101,0001⟩$	$⟨0100,1100,0011,0111,1101,0001⟩$
	$⟨0100,0000,1000,0010,1110,1010,0110,1001,0001⟩$	$⟨0100,1000,0010,0110,1001,0001⟩$
		$⟨0100,0000,1111,1011,1110,1010,0101,0001⟩$
$\mu =1100$	$⟨1100,0100,0001⟩$	$⟨1100,0100,0001⟩$
$\nu =0001$	$⟨1100,0011,1111,1011,0111,1101,0101,0001⟩$	$⟨1100,0011,0111,1101,0001⟩$
	$⟨1100,0000,1000,0010,1110,1010,0110,1001,0001⟩$	$⟨1100,1000,0010,0110,1001,0001⟩$
		$⟨1100,0000,1111,1011,1110,1010,0101,0001⟩$
$\mu =1000$	$⟨1000,0100,0001⟩$	$⟨1000,0100,0001⟩$
$\nu =0001$	$⟨1000,0000,1100,0011,1111,1011,0111,1101,0001⟩$	$⟨1000,1100,0011,0111,1101,0001⟩$
	$⟨1000,0010,1110,1010,0110,1001,0101,0001⟩$	$⟨1000,0010,0110,1001,0001⟩$
		$⟨1000,0000,1111,1011,1110,1010,0101,0001⟩$
$\mu =0000$	$⟨0000,0100,0001⟩$	$⟨0000,0100,0001⟩$
$\nu =0001$	$⟨0000,1100,0011,1111,1011,0111,1101,0001⟩$	$⟨0000,1100,0011,0111,1101,0001⟩$
	$⟨0000,1000,0010,1110,1010,0110,1001,0101,0001⟩$	$⟨0000,1000,0010,0110,1001,0001⟩$
		$⟨0000,1111,1011,1110,1010,0101,0001⟩$

Hence, there exists an m-DPC between $\mu$ and $\nu$ for $1\le m\le d+2$ in $FDS{C}_n$ for integer $d\in \{1,2\}$ and $n=2^d$. □

Lemma 11.

Let μ and ν be any two nodes in $FDS{C}_n$ for any integer $d\ge 1$ and $n=2^d$. There exists an m-DPC between μ and ν in $FDS{C}_n$ with $1\le m\le d+2$.

Proof. 

To prove this lemma, we proceed by induction on d. The base case is given in Lemma 10 for $d\in \{1,2\}$. Suppose this lemma holds for $FDS{C}_{2^{d-1}}$ when $d\ge 3$, we will prove that this lemma holds for $FDS{C}_{2^d}$ with $d\ge 3$. By Lemma 5, there exists an m-DPC between $\mu$ and $\nu$ in $FDS{C}_n$ for $1\le m\le 2$, so we need to prove this lemma holds for $3\le m\le d+2$. Considering the locations of $\mu$ and $\nu$ in $FDS{C}_n$, we have the cases as follows.

Case 1. $\mu$, $\nu$$\in V\left(D_k\right)$ with $0\le k\le h-1$.

W.L.O.G., suppose $\mu$ and $\nu$ are in $D_0$. According to the hypothesis, there exists an m-DPC between $\mu$ and $\nu$ in $D_0$ where $2\le m\le d+1$.

Case 1.1. $e_n^{\mu }$ and $e_n^{\nu }$ are connected to different modules of $FDS{C}_n$.

Suppose that $\mu$ is connected to $x\in V\left(D_i\right)$ and $\nu$ is connected to $y\in V\left(D_j\right)$, where $1\le i\ne j\le h-1$. By Lemma 6, there exists a Hamiltonian path covering all nodes of $V\left(FDS{C}_n\right)-V\left(D_0\right)$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ in $FDS{C}_n$ where $3\le m+1\le d+2$ (see Figure 6).

Figure 6. Illustration for Case 1.1 of Lemma 11.

Case 1.2. $\mu$ and $\nu$ are connected to the same module of $FDS{C}_n$.

Suppose that $\mu$ and $\nu$ are connected to $D_i$ with $1\le i\le h-1$. By Lemma 7, there exists a Hamiltonian path covering all nodes of $V\left(FDS{C}_n\right)-V\left(D_0\right)$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ in $FDS{C}_n$ where $3\le m+1\le d+2$.

Case 2. $\mu \in V\left(D_i\right)$ and $\nu \in V\left(D_j\right)$ with $0\le i\ne j\le h-1$.

Let $H=FDS{C}_n-V\left(D_i\right)-V\left(D_j\right)$.

Case 2.1. $e_n^{\mu }$ and $e_n^{\nu }$ are connected to H.

By Property 1, we can find two nodes $s\in V\left(D_i\right)$ and $t\in V\left(D_j\right)$ such that $(s,t)\in E\left(FDS{C}_n\right)$. According to the hypothesis, there exists an m-DPC between $\mu$ and s in $D_i$ and an m-DPC between $\nu$ and t in $D_j$ where $2\le m\le d+1$. Let $S=\{s_1,s_2,\cdots ,s_m\}$ and $T=\{t_1,t_2,\cdots ,t_m\}$ be the neighbor set of s and t in the m-DPC. By Property 1, $D_i$ and $D_j$ have at most two edges between them. Let $(x,y)$ be another edge between $D_i$ and $D_j$ if it exists. Suppose that x is in path $⟨\mu ,S_1,s_1,s⟩$ and y is in path $⟨\nu ,T_1,t_1,t⟩$. We can construct a path $⟨\mu ,S_1,s_1,s,t,t_1,T_1,\nu ⟩$ between $\mu$ and $\nu$ whether or not x, y exist. Then, for each k with $2\le k\le m$, $t_k$(resp. $s_k$) is connected to a different module in H. We can find $m-1$ node pairs between $\left\{s_k\right|2\le k\le m\}$ and $\left\{t_l\right|2\le l\le m\}$ to build paths $⟨\mu ,S_k,s_k,s_k^{\prime },P_{k,l},t_l^{\prime },t_l,T_l,\nu ⟩$ which covers 1 or 2 module(s) of H. Let M be the module set consisting of modules covered by the paths between $\left\{s_k\right|2\le k\le m\}$ and $\left\{t_l\right|2\le l\le m\}$. Then $\left|M\right|\le 2(m-1)\le 2d<\frac{h}{2}$. By Lemmas 6 and 7, there exists a path $⟨\mu ,{\mu }^{\prime },P,{\mu }^{\prime },\nu ⟩$ which covers all the modules of $H-M$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ where $3\le m+1\le d+2$ (see Figure 7).

Figure 7. Illustration for Case 2.1 of Lemma 11.

Case 2.2. $e_n^{\mu }$ is connected to $D_j$ and $e_n^{\nu }$ is connected to H.

Let $(\mu ,t)\in E\left(FDS{C}_n\right)$ and $t\in V\left(D_j\right)$. According to the hypothesis, there exists an m-DPC between $\nu$ and t in $D_j$ where $2\le m\le d+1$. We have $T=\{t_1,t_2,\cdots ,t_m\}$ be the neighbor set of t in the m-DPC. By Lemma 9, we can find a vertex set $S=\{s_1,s_2,\cdots ,s_m\}$ such that there exists a one-to-many node disjoint path cover between $\mu$ and S in $D_i$ and the vertex labels are not complementary with each other in $S+\left\{\mu \right\}$. Hence, each $s_i$ with $1\le i\le m$ is connected to a different module of H. Let $y\in V\left(D_j\right)$ be connected to $D_i$ if exists. Suppose that y is in path $⟨\nu ,T_1,t_1,t⟩$. We can construct a path $⟨\mu ,t,t_1,T_1,\nu ⟩$ between $\mu$ and $\nu$ whether or not y exists. For $\left\{s_k\right|2\le k\le m\}$ and $\left\{t_l\right|2\le l\le m\}$, we can construct $m-1$ paths $⟨\mu ,S_k,s_k,s_k^{\prime },P_{k,l},t_l^{\prime },t_l,T_l,\nu ⟩$ between them, the union of which covers module set M. Then $\left|M\right|\le 2(m-1)\le 2d<\frac{h}{2}$. By Lemmas 6 and 7, there exists a path $⟨\mu ,S_1,s_1,s_1^{\prime },P,{\nu }^{\prime },\nu ⟩$ which covers all the modules of $H-M$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ where $3\le m+1\le d+2$ (see Figure 8).

Figure 8. Illustration for Case 2.2 of Lemma 11.

Case 2.3. $e_n^{\nu }$ is connected to $D_i$ and $e_n^{\mu }$ is connected to H.

This case is similar to Case 2.2.

Case 2.4. ($\mu$, $\nu$)$\in E\left(FDS{C}_n\right)$.

By Lemma 9, we can find a vertex set $S=\{s_1,s_2,\cdots ,s_m\}$ such that there exists a one-to-many node-disjoint path cover between $\mu$ and S in $D_i$ and the vertex labels are not complementary with each other in $S+\left\{\mu \right\}$. Similarly, we can find a vertex set $T=\{t_1,t_2,\cdots ,t_m\}$ such that there exists a one-to-many node disjoint path cover between $\nu$ and T in $D_j$, and the vertex labels are not complementary with each other in $T+\left\{\nu \right\}$. We can construct $m-1$ paths $⟨\mu ,S_k,s_k,s_k^{\prime },P_{k,l},t_l^{\prime },t_l,T_l,\nu ⟩$ between $\left\{s_k\right|2\le k\le m\}$ and $\left\{t_l\right|2\le l\le m\}$ the union of which covers module set M. By Lemmas 6 and 7, there exists a path $⟨\mu ,S_1,s_1,s_1^{\prime },P_1,t_1^{\prime },t_1,T_1,\nu ⟩$, which covers all the modules of $H-M$. Including path $⟨\mu ,\nu ⟩$, we get a $(m+1)$-DPC between $\mu$ and $\nu$(see Figure 9).

Figure 9. Illustration for Case 2.4 of Lemma 11.

Case 2.5. $(\mu ,t),(\nu ,s)\in E\left(FDS{C}_n\right)$, $t\in V\left(D_j\right)$ and $s\in V\left(D_i\right)$.

According to the hypothesis, there exists an m-DPC between $\mu$ and s in $D_i$ and an m-DPC between $\nu$ and t in $D_j$ where $2\le m\le d+1$. Let $S=\{s_1,s_2,\cdots ,s_m\}$ and $T=\{t_1,t_2,\cdots ,t_m\}$ be the neighbor set of s and t in the m-DPC. We have two paths, $⟨\mu ,S_1,s_1,s,\nu ⟩$ and $⟨\mu ,t,t_1,T_1,\nu ⟩$, between $\mu$ and $\nu$. Then, for each k with $2\le k\le m$, $t_k$(resp. $s_k$) is connected to a different module in H. We can find $m-2$ node pairs between $\left\{s_k\right|2\le k\le m-1\}$ and $\left\{t_l\right|2\le l\le m-1\}$ to build paths $⟨\mu ,S_k,s_k,s_k^{\prime },P_{k,l},t_l^{\prime },t_l,T_l,\nu ⟩$ whose union covers module set M. Then $\left|M\right|\le 2(m-2)\le 2(d-1)<\frac{h}{2}$. By Lemma 6 and Lemma 7, there exists a path $⟨\mu ,S_m,s_m,s_m^{\prime },P_m,t_m^{\prime },t_m,T_m,\nu ⟩$ which covers all the modules of $H-M$. Hence, there exists a $(m+1)$-DPC between $\mu$ and $\nu$ where $3\le m+1\le d+2$ (see Figure 10).

Figure 10. Illustration for Case 2.5 of Lemma 11.

Hence, there exists an m-DPC between $\mu$ and $\nu$ in $FDS{C}_n$ where $1\le m\le d+2$. □

For all $1\le j\le d+2$, $FDS{C}_n$ is $j^{*}$-connected, then $FDS{C}_n$ is super spanning connected. We have Theorem 1.

Theorem 1.

${\kappa }^{*}\left(FDS{C}_n\right)=\kappa \left(FDS{C}_n\right)=d+2$ for $d\ge 1$ and $n=2^d$.

With Theorem 1, we show that there exist m disjoint paths between any pair of nodes for all $1\le m\le d+2$. Since each node of $FDS{C}_n$ has $d+2$ neighbors, the result is optimal.

4. Conclusions

$DS{C}_n$ and $FDS{C}_n$ are newly proposed hypercube variants, which reduce the network cost from $O\left(n^2\right)$ to $O\left(n{log}_{2}n\right)$ compared with the hypercube and other hypercube variants. Both $DS{C}_n$ and $FDS{C}_n$ possess many superior properties than hypercubes. In this work, we investigated the super spanning connectivity of $FDS{C}_n$. We show that ${\kappa }^{*}\left(FDS{C}_n\right)=\kappa \left(FDS{C}_n\right)=d+2$, which means there exists an m-DPC between any pair of nodes in $FDS{C}_n$ for all $1\le m\le d+2$. Since the degree of $FDS{C}_n$ is $d+2$, then $d+2$ is the maximal integer of node-disjoint paths that can be built in $FDS{C}_n$. In future work, the one-to-many node-disjoint path cover problem of $FDS{C}_n$ could be considered. In addition, the diagnosability [25,26,27] of $FDS{C}_n$ is another research topic that could be considered.