The Generalized 3-Connectivity of Exchanged Folded Hypercubes

Abstract

For $S\subseteq V\left(G\right),{\kappa }_G\left(S\right)$ denotes the maximum number k of edge disjoint trees $T_1,T_2,\dots ,T_k$ in G, such that $V\left(T_i\right)\cap V\left(T_j\right)=S$ for any $i,j\in \{1,2,\dots ,k\}$ and $i\ne j$. For an integer $2\le r\le \left|V\right(G\left)\right|$, the generalized r-connectivity of G is defined as ${\kappa }_r\left(G\right)=min\{{\kappa }_G\left(S\right)|S\subseteq V\left(G\right)\mathrm{and}|S|=r\}$. In fact, ${\kappa }_2\left(G\right)$ is the traditional connectivity of G. Hence, the generalized r-connectivity is an extension of traditional connectivity. The exchanged folded hypercube $EFH(s,t)$, in which $s\ge 1$ and $t\ge 1$ are positive integers, is a variant of the hypercube. In this paper, we find that ${\kappa }_3\left(EFH(s,t)\right)=s+1$ with $3\le s\le t$.

1. Introduction

An interconnection network is usually modeled as a simple graph $G=\left(V\right(G),E(G\left)\right)$, in which $V\left(G\right)$ represents the set of processors and $E\left(G\right)$ represents the set of links. For $v\in V\left(G\right)$, $N\left(v\right)$ is the neighborhood of v in G. $d\left(v\right)=\left|N\right(v\left)\right|$ is the degree of v in G. The minimum degree of G is defined as $\delta \left(G\right)=min\left\{d\right(v\left)\right|v\in V\left(G\right)\}$. For two graphs $G_1$ and $G_2$, $G_1\cong G_2$ means that they are isomorphic. Let $S\subseteq V\left(G\right)$. The subgraph of G, whose vertex set is S and whose edge set is the set of those edges of G that have both ends in S, is called the subgraph of G induced by S and is denoted by $G\left[S\right]$. We say that $G\left[S\right]$ is an induced subgraph of G. $G-S$ means the induced subgraph $G\left[V\right(G)\setminus S]$, where $V\left(G\right)\setminus S$ represents the vertex set obtained from $V\left(G\right)$ by deleting the vertices in S. Let $V\subseteq V\left(G\right)\setminus \left\{v\right\}$. The $(v,V)$ paths is a family of internally disjoint paths whose starting vertex is v and terminal vertices are distinct in V, which is called a fan from v to V. For other terminologies and notations, please refer to [1].

Connectivity is a basic and important metric in measuring the reliability and fault tolerance of networks. A cut set S of G is a vertex set of G, such that $G-S$ is disconnected or it is only one vertex. $\kappa \left(G\right)=min\left\{\right|S\left|\right|S\mathrm{is}\mathrm{a}\mathrm{cut}\mathrm{set}\mathrm{of}G\}$, which is the connectivity of G. In [2], Whitney proposed an equivalent concept of connectivity. For each 2-subset $S=\{u,w\}$ of vertices of G, let ${\kappa }_G\left(S\right)$ be the maximum number of internally disjoint paths from u to w in G. Then, $\kappa \left(G\right)=min\left\{{\kappa }_G\left(S\right)\right|S\subseteq V\left(G\right)\mathrm{and}\left|S\right|=2\}$. As an extension of connectivity, Chartrand et al. [3] showed the concept of generalized k-connectivity in 1984. Let $S\subseteq V\left(G\right)$. A tree T in G is called an S-tree if $S\subseteq V\left(T\right)$. The trees $T_1,T_2,\dots ,T_r$ are called internally edge disjoint S-trees if $V\left(T_i\right)\cap V\left(T_j\right)=S$ and $E\left(T_i\right)\cap E\left(T_j\right)=\varnothing$ for any distinct integers $i,j$ with $1\le i,j\le r$. ${\kappa }_G\left(S\right)$ refers to the maximum number of internally edge disjoint S-trees. For an integer k with $2\le k\le \left|V\right(G\left)\right|$, ${\kappa }_k\left(G\right)=min\{{\kappa }_G\left(S\right)|S\subseteq V\left(G\right)\mathrm{and}|S|=k\}$ is defined as the generalized k-connectivity of G.

In a graph G, an S-tree is also called an S-Steiner tree. Steiner trees have significant applications in computer networks [4]. Internally edge disjoint S-Steiner trees have been applied to VLSI [5]. From the definition of generalized k-connectivity, we can see that the core of generalized k-connectivity is to seek the maximum number of internally edge disjoint S-Steiner trees. The generalized k-connectivity is an extension of traditional connectivity. It can more precisely measure the fault tolerance of networks. To decide whether there exist k internally edge disjoint S-Steiner trees is NP-complete for a graph [6]. The generalized 3-connectivities of augmented cubes, $(n,k)$-bubble-sort graphs, and generalized hypercubes have been obtained in [7,8,9], respectively. The generalized 4-connectivities of hypercubes, crossed cubes, exchanged hypercubes, and hierarchical cubic networks have been obtained in [10,11,12,13], respectively. On the whole, the generalized k-connectivity is known for a small number of graphs and almost all known results are about $k=3$ or 4.

The n-dimensional hypercube is denoted by $Q_n$, whose vertices are the ordered n-tuples of 0’s and 1’s. Two vertices are adjacent if and only if they differ in exactly one dimension. As variants of hypercubes $Q_n$, folded hypercubes $F{Q}_n$ and exchanged hypercubes $EH(s,t)$ were proposed in [14,15], respectively. Based on $EH(s,t)$ and $F{Q}_n$, Qi et al. proposed an interconnection network named exchanged folded hypercube $EFH(s,t)$ in [16]. In this work, we will prove ${\kappa }_3\left(EFH(s,t)\right)=s+1$ for $3\le s\le t$.

2. Definitions and Lemmas

Exchanged hypercubes were defined by Lou et al. [15] as follows. Let $s\ge 1$ and $t\ge 1$ be positive integers. The exchanged hypercubes $EH(s,t)$ are defined as undirected graphs, whose vertex set V is

$$V=\{a_s\cdots a_1{b}_t\cdots b_{1}c|a_i,b_j,c\in \{0,1\}\mathrm{for}i\in [1,s],j\in [1,t]\}.$$

For $u,v\in V$, $u\left[0\right]$ means the c index of u. $u[i:j]$ is the indexes of u from dimension j to dimension i. $H\left(u\right[i:j],v[i:j\left]\right)$ represents the number of different indexes at the same dimension between $u[i:j]$ and $v[i:j]$.

The edge set consists of three disjoint subsets $E_H,E_R$ and $E_L$, where

$$E_H=\left\{(u,v)\right|u[s+t:1]=v[s+t:1],u\left[0\right]\ne v\left[0\right]\},$$

$$E_R=\left\{(u,v)\right|u[s+t:t+1]=v_2[s+t:t+1],H(u[t:1],v[t:1])=1,u\left[0\right]=v\left[0\right]=1\},$$

$$E_L=\left\{(u,v)\right|u[t:1]=v[t:1],H(u[s+t:t+1],v[s+t:t+1])=1,u\left[0\right]=v\left[0\right]=0\},$$

Figure 1 shows an example of $EH(1,2)$. Based on the concept of $EH(s,t)$, Qi et al. [16] put in a network called an exchanged folded hypercube $EFH(s,t)$. $EFH(s,t)$ and $EH(s,t)$ have the same vertex set. The edge set of $EFH(s,t)$ consists of $E_H,E_R,E_L$ and $E_{\mathrm{comp}}$, where

$$E_{\mathrm{comp}}=\left\{(u,v)\right|H(u[s+t:1],v[s+t:1])=s+t,u\left[0\right]\ne v\left[0\right]\}.$$

The edges in $E_{\mathrm{comp}}$ are called complementary edges of $EFH(s,t)$. From the two definitions, we know that $EFH(s,t)$ can be obtained from $EH(s,t)$ by adding extra $2^{s+t}$ edges. Figure 2 is an example of $EFH(1,2)$. From the definition, we can see that $\left|V\right(EFH(s,t)\left)\right|$=$2^{s+t+1}$. For each vertex $v\in V\left(EFH\right(s,t\left)\right)$, $d\left(v\right)=s+2$ or $t+2$. For simplicity, we always use $EFH$ instead of $EFH(s,t)$. The following results are useful.

Lemma 1.

([16]) $EFH(t,s)\cong EFH(s,t)$.

From the lemma, we always assume $s\le t$ from now on. Then, $\delta \left(EFH\right(s,t\left)\right)=s+2$.

Lemma 2.

([1]) $\kappa \left(Q_n\right)=n$ for $n\ge 2$.

Lemma 3.

([17]) ${\kappa }_3\left(Q_n\right)=n-1$ for $n\ge 2$.

Lemma 4.

([18]) If there are two adjacent vertices of degree $\delta \left(G\right)$ in graph G, then ${\kappa }_k\left(G\right)\le \delta \left(G\right)-1$ for $3\le k\le \left|V\right(G\left)\right|$.

Lemma 5.

( [1]) Let G be a k-connected graph, and let u and v be a pair of distinct vertices in G. Then, there exist k internally disjoint paths in G connecting u and v.

Lemma 6.

(Fan lemma [1]) For a k-connected graph G, let $u\in V\left(G\right)$, and suppose $U\subseteq V\left(G\right)\setminus \left\{u\right\}$ and $\left|U\right|\ge k$. Then, there exists a k-fan in G from u to U, that is, there exists a family of k internally disjoint $(u,U)$ paths whose terminal vertices are distinct in U.

In this work, we will prove the following result.

Theorem 1.

${\kappa }_3\left(EFH(s,t)\right)=s+1$ for $3\le s\le t$.

3. Proof of Theorem 1

We partition $EFH(s,t)$ into two subgraphs $L,R$ and edges between them, in which for $u\in V\left(L\right)$ and $v\in V\left(R\right)$, $u\left[0\right]=0$ and $v\left[0\right]=1$.

In $V\left(L\right)$, each collection of $2^s$ vertices u, with $u[t:1]$ being identical, forms $Q_s$ via the edges in $E_L$. We use $L_i$ to denote these $Q_s$ for $i=1,2,\dots ,2^t$. Similarly, in $V\left(R\right)$, each collection of $2^t$ vertices v, with $v[s+t:t+1]$ being identical, forms $Q_t$ via the edges in $E_R$. We use $R_j$ to denote these $Q_t$ for $j=1,2,\dots ,2^s$.

Each vertex $x\in V\left(L\right)$ has two neighbors in $V\left(R\right)$. One is $x^{\prime }$ with $x{x}^{\prime }\in E_H$. It is called the hypercube neighbor of x. The other is $\overline{x}$ with $x\overline{x}\in E_{\mathrm{comp}}$. It is called the complement neighbor of x. $x^{\prime }$ and $\overline{x}$ are called outside neighbors of x. Similarly, for $y\in V\left(R\right)$, $y^{\prime }$ and $\overline{y}$, the outside neighbors of y, are called the hypercube neighbor and the complement neighbor of y, respectively.

In the following, for each vertex x in a graph, we use $x^{\prime }$ and $\overline{x}$ to denote the hypercube neighbor and the complement neighbor of x, respectively.

Lemma 7.

For $Q_n$ and $EFH(s,t)$, the following results hold.

1. 

Each $L_i\cong Q_s$, $R_j\cong Q_t$ and $|V\left(L_i\right)|=2^s$, $|V\left(R_j\right)|=2^t$ for $i=1,2,\dots ,2^t,j=1,2,\dots ,2^s$.

2. 

There are no edges between any two distinct $L_i$ and $L_k$ for $i,k\in \{1,2,\dots ,2^t\}$. Similarly, there are no edges between any two distinct $R_j$ and $R_h$ for $j,h\in \{1,2,\dots ,2^s\}$.

3. 

For each vertex $x\in V\left(L\right)$, $x^{\prime }$ and $\overline{x}$ belong to distinct $V\left(R_j\right)$ and $V\left(R_h\right)$, where $j,h\in \{1,2,\dots ,2^s\}$. Similarly, for each vertex $w\in V\left(R\right)$, $w^{\prime }$ and $\overline{w}$ belong to distinct $V\left(L_i\right)$ and $V\left(L_k\right)$, where $i,k\in \{1,2,\dots ,2^t\}$.

4. 

For two distinct vertices $x,y\in V\left(L_i\right)$ with $i\in \{1,2,\dots ,2^t\}$, $x^{\prime }$ and $y^{\prime }$ lie in distinct $V\left(R_j\right)$ and $V\left(R_h\right)$, where $j,h\in \{1,2,\dots ,2^s\}$, $\overline{x}$ and $\overline{y}$ lie in distinct $V\left(R_i\right)$ and $V\left(R_k\right)$, where $i,k\in \{1,2,\dots ,2^s\}$. Similar results hold for two distinct vertices $u,v\in V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$.

5. 

For two distinct vertices $x,y\in V\left(L_i\right)$ with $i\in \{1,2,\dots ,2^t\}$, if $x^{\prime },\overline{y}\in V\left(R_j\right)$ for some $j\in \{1,2,\dots ,2^s\}$, then $\overline{x},y^{\prime }\in V\left(R_k\right)$ for some $k\in \{1,2,\dots ,2^s\}$ with $k\ne j$. A similar result holds for two distinct vertices $u,v\in V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$.

Proof. 

The first and second results are obvious. For two distinct vertices $x,y\in V\left(L_i\right)$ with $i\in \{1,2,\dots ,2^t\}$, there exists at least one index m for which x and y differ. Let $x=a_s\cdots a_m\cdots a_1{b}_t\cdots b_{1}0$, $y=a_s^{\prime }\cdots {\overline{a}}_m\cdots a_1^{\prime }{b}_t\cdots b_{1}0$ in same $V\left(L_i\right)$ with some $m\in \{1,2,\dots ,s\}$. Then, $x^{\prime }=a_s\cdots a_m\cdots a_1{b}_t\cdots b_{1}1$, $\overline{x}={\overline{a}}_s\cdots {\overline{a}}_m\dots {\overline{a}}_1{\overline{b}}_t\cdots {\overline{b}}_{1}1$, $y^{\prime }=a_s^{\prime }\cdots {\overline{a}}_m\cdots a_1^{\prime }{b}_t\cdots b_{1}1$. $\overline{y}={\overline{a^{\prime }}}_s\cdots a_m\cdots {\overline{a^{\prime }}}_1{\overline{b}}_t\cdots {\overline{b}}_{1}1$, where ${\overline{a}}_i=1-a_i,{\overline{a^{\prime }}}_i=1-a_i^{\prime },{\overline{b}}_j=1-b_j$ (Figure 3).

$x^{\prime }$ and $\overline{x}$ belong to distinct $V\left(R_j\right)$ and $V\left(R_h\right)$ where $j,h\in \{1,2,\dots ,2^s\}$ since $a_i\ne {\overline{a}}_i$ for $i=1,2,\dots ,s$. Similarly, we can prove that, for any vertex $w\in V\left(R\right)$, $w^{\prime }$ and $\overline{w}$ belong to distinct $V\left(L_i\right)$ and $V\left(L_k\right)$, where $i,k\in \{1,2,\dots ,2^t\}$. Hence, the third result holds.

Since $a_m\ne {\overline{a}}_m$ for some $m\in \{1,2,\dots ,s\}$, $x^{\prime }$ and $y^{\prime }$ lie in different $V\left(R_j\right)$ and $V\left(R_h\right)$, where $j,h\in \{1,2,\dots ,2^s\}$, $\overline{x}$ and $\overline{y}$ lie in different $V\left(R_i\right)$ and $V\left(R_k\right)$, where $i,k\in \{1,2,\dots ,2^s\}$. We can prove that similar results for any distinct vertices $u,v\in V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$. Hence, the fourth result holds.

If $x^{\prime },\overline{y}\in V\left(R_j\right)$ for some $j\in \{1,2,\dots ,2^s\}$, then $a_j={\overline{a^{\prime }}}_j$ for $j=1,\dots ,m-1,$$m+1,$$\dots ,s$. Hence, ${\overline{a}}_j=a_j^{\prime }$ for $j=1,\dots ,m-1,m+1,\dots ,s$. This implies that $\overline{x},y^{\prime }\in V\left(R_k\right)$ for some $k\in \{1,2,\dots ,2^s\}$ with $k\ne j$. We can prove that a similar result for any distinct vertices $u,v\in V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$. Hence, the fifth result holds. □

Proof of Theorem 1.

By Lemma 7, for any vertex $u\in V\left(L_1\right)$, $d\left(u\right)=s+2$. Since $\delta \left(EFH\right(s,t\left)\right)=s+2$, ${\kappa }_3\left(EFH(s,t)\right)\le s+1$ by Lemma 4. In the following, we will prove ${\kappa }_3\left(EFH(s,t)\right)\ge s+1$. Take any three distinct vertices $x,y$, and z in $EFH$ and let $S=\{x,y,z\}$. If we can prove that there are $s+1$ internally edge disjoint S-trees in $EFH$, we are done.

Case 1.

$x,y,z\in V\left(L_i\right)$ for some $i\in \{1,2,\dots ,2^t\}$.

Without loss of generality, let $x,y,z\in V\left(L_1\right)$. By Lemma 3, there exist $s-1$ internally edge disjoint S-trees $T_1,T_2,\dots ,T_{s-1}$ in $L_1$. Without loss of generality, suppose $x^{\prime }\in V\left(R_1\right),y^{\prime }\in V\left(R_2\right)$, and $z^{\prime }\in V\left(R_3\right)$ by Lemma 7(4).

If $\{\overline{x},\overline{y},\overline{z}\}\cap (V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right))=\varnothing$, we can assume $\overline{x}\in V\left(R_4\right),\overline{y}\in V\left(R_5\right),\overline{z}\in V\left(R_6\right)$. By Lemma 7(4), $EFH[V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_2\right)]$ is connected. Hence, there exists a tree ${\overline{T}}_s$ containing $x^{\prime },y^{\prime }$, and $z^{\prime }$ in $EFH[V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_2\right)]$. Take $T_s={\overline{T}}_s\cup x{x}^{\prime }\cup y{y}^{\prime }\cup z{z}^{\prime }$. Since $EFH[V\left(R_4\right)\cup V\left(R_5\right)\cup V\left(R_6\right)\cup V\left(L_3\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{x},\overline{y}$, and $\overline{z}$ in $EFH[V\left(R_4\right)\cup V\left(R_5\right)\cup V\left(R_6\right)\cup V\left(L_3\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup y\overline{y}\cup z\overline{z}$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

If $\{\overline{x},\overline{y},\overline{z}\}\cap (V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right))\ne \varnothing$, without loss of generality, noting that $\overline{x}\notin V\left(R_1\right)$ by Lemma 7(3), let $\overline{x}\in V\left(R_2\right)$. By Lemma 7(5), $\overline{y}\in V\left(R_1\right)$. By Lemma 7(3)(4), we can let $\overline{z}\in V\left(R_4\right)$. Since $EFH[V\left(R_1\right)\cup V\left(R_3\right)\cup V\left(L_2\right)]$ is connected, there exists a tree ${\overline{T}}_s$ containing $x^{\prime },\overline{y}$, and $z^{\prime }$ in $EFH[V\left(R_1\right)\cup V\left(R_3\right)\cup V\left(L_2\right)]$. Take $T_s={\overline{T}}_s\cup x{x}^{\prime }\cup y\overline{y}\cup z{z}^{\prime }$. Since $EFH[V\left(R_2\right)\cup V\left(R_4\right)\cup V\left(L_3\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{x},y^{\prime }$, and $\overline{z}$ in $EFH[V\left(R_2\right)\cup V\left(R_4\right)\cup V\left(L_3\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup y{y}^{\prime }\cup z\overline{z}$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

By symmetry and $t\ge s$, if $x,y,z\in V\left(R_j\right)$ for some $j\in \{1,2,\dots ,2^s\}$, we can also obtain ${\kappa }_3\left(EFH\right)\ge s+1$.

Case 2.

$x,y\in V\left(L_i\right)$ for some $i\in \{1,2,\dots ,2^t\}$. $z\in V\left(L_j\right)$ for some $j\in \{1,2,\dots ,2^t\}$ and $i\ne j$ or $z\in V\left(R_k\right)$ for some $k\in \{1,2,\dots ,2^s\}$.

Without loss of generality, we let $x,y\in V\left(L_1\right)$. By Lemmas 2 and 5, there exist s internally disjoint paths $P_1,P_2,\dots ,P_s$ from x to y in $L_1$. Let $x_i\in V\left(P_i\right)$, such that $x{x}_i\in E\left(P_i\right)$ for $i=1,2,\dots ,s$. In the following, we will show that for any two distinct vertices $x_i$ and $x_j$ with $i,j\in \{1,2,\dots ,s\}$, $x^{\prime },x_i^{\prime },x_j^{\prime },\overline{x},{\overline{x}}_i,{\overline{x}}_j$ lie in distinct $V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$. Without loss of generality, let $x=a_s\cdots a_2{a}_1{b}_t\cdots b_{1}0$, $x_i=a_s\cdots a_2{\overline{a}}_1{b}_t\cdots b_{1}0$, and $x_j=a_s\cdots {\overline{a}}_2{a}_1{b}_t\cdots b_{1}0$. Then, $x^{\prime }=a_s\cdots a_2{a}_1{b}_t\cdots b_{1}1$, $\overline{x}={\overline{a}}_s\cdots {\overline{a}}_2{\overline{a}}_1{\overline{b}}_t\cdots {\overline{b}}_{1}1$, $x_i^{\prime }=a_s\cdots a_2{\overline{a}}_1{b}_t\cdots b_{1}1$, ${\overline{x}}_i={\overline{a}}_s\cdots {\overline{a}}_2{a}_1{\overline{b}}_t\cdots {\overline{b}}_{1}1$, $x_j^{\prime }=a_s\cdots {\overline{a}}_2{a}_1{b}_t\cdots b_{1}1$, ${\overline{x}}_j={\overline{a}}_s\cdots a_2{\overline{a}}_1{\overline{b}}_t\cdots {\overline{b}}_{1}1$. By $s\ge 3$ and the definition of $R_k$, we can show that $x^{\prime },x_i^{\prime },x_j^{\prime },\overline{x},{\overline{x}}_i,{\overline{x}}_j$ lie in different $V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$, where $i,j\in \{1,2,\dots ,s\}$ and $i\ne j$. This implies that $x^{\prime },x_1^{\prime },x_2^{\prime },\dots ,x_s^{\prime },\overline{x},{\overline{x}}_1,{\overline{x}}_2,\dots ,{\overline{x}}_s$ lie in distinct $V\left(R_k\right)$ for $k\in \{1,2,\dots ,2^s\}$.

Subcase 2.1.

$z\in V\left(R_k\right)$ for some $k\in \{1,2,\dots ,2^s\}$.

Let $z\in V\left(R_1\right)$. We know that $\{x^{\prime },x_1^{\prime },x_2^{\prime },\dots ,x_s^{\prime }\}\cap V\left(R_1\right)=\varnothing$ or $\{\overline{x},{\overline{x}}_1,{\overline{x}}_2,\dots ,{\overline{x}}_s\}\cap V\left(R_1\right)=\varnothing$. Without loss of generality, let $\{\overline{x},{\overline{x}}_1,{\overline{x}}_2,\dots ,{\overline{x}}_s\}\cap V\left(R_1\right)=\varnothing$. Suppose $\overline{x}\in V\left(R_4\right)$ and ${\overline{x}}_i\in V\left(R_{i+4}\right)$ for $i=1,2,\dots ,s$.

Subcase 2.1.1.

$y=x_i$ for some $i\in \{1,2,\dots ,s\}$.

Without loss of generality, let $y=x_s$. Then, $y^{\prime }\notin V\left(R_{i+4}\right)$ for $i=0,1,2,\dots ,s$ by the above discussion. We can let $y^{\prime }\in V\left(R_1\right)$ or $y^{\prime }\in V\left(R_2\right)$.

First, we consider $y^{\prime }\in V\left(R_2\right)$ (Figure 4). By Lemma 7(3), $z^{\prime }\notin V\left(L_1\right)$ or $\overline{z}\notin V\left(L_1\right)$. Without loss of generality, let $\overline{z}\notin V\left(L_1\right)$. Suppose $\overline{z}\in V\left(L_2\right)$. Take s vertices $z_1,z_2,\dots ,z_s$ in $V\left(R_1\right)$, such that ${\overline{z}}_i\in V\left(L_{i+4}\right)$ for $i=1,2,\dots ,s$. Let $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6, there exist s internally disjoint paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $M_i$ be the path from z to $z_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i$ and ${\overline{z}}_i$ in $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ for $i=1,2,\dots ,s$. Take $T_i={\overline{T}}_i\cup P_i\cup M_i\cup x_i{\overline{x}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(L_2\right)\cup V\left(R_2\right)\cup V\left(R_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{z},y^{\prime }$, and $\overline{x}$ in $EFH[V\left(L_2\right)\cup V\left(R_2\right)\cup V\left(R_4\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup z\overline{z}\cup y{y}^{\prime }$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

Now, we consider $y^{\prime }\in V\left(R_1\right)$.

If $y^{\prime }=z$, then $\overline{z}\notin V\left(L_1\right)$. Let $\overline{z}\in V\left(L_2\right)$. Taking $T_1,T_2,\dots ,T_s$ to be the same as above, since $EFH[V\left(L_2\right)\cup V\left(R_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{z}$ and $\overline{x}$ in $EFH[V\left(L_2\right)\cup V\left(R_4\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup yz\overline{z}$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

Let $y^{\prime }\ne z$ (Figure 5). By Lemma 7(4), $z^{\prime }\notin V\left(L_1\right)$. Suppose $z^{\prime }\in V\left(L_2\right)$. Take $s-1$ vertices $z_1,z_2,\dots ,z_{s-1}$ in $V\left(R_1\right)$, such that ${\overline{z}}_i\in V\left(L_{i+4}\right)$ for $i=1,2,\dots ,s-1$. Let $Z=\{z_1,z_2,\dots ,z_{s-1},y^{\prime }\}$. By Lemma 6, there exist s internally disjoint paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $M_i$ be the path from z to $z_i$ for $i=1,2,\dots ,s-1$ and $M_s$ be the path from z to $y^{\prime }$. Since $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i$ and ${\overline{z}}_i$ in $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ for $i=1,2,\dots ,s-1$. Take $T_i={\overline{T}}_i\cup P_i\cup M_i\cup x_i{\overline{x}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s-1$. Noting that $y=x_s$, then $\overline{y}\in V\left(R_{s+4}\right)$. Since $EFH[V\left(L_2\right)\cup V\left(R_{s+4}\right)\cup V\left(R_4\right)]$ is connected, there exists a tree ${\overline{T}}_s$ containing $z^{\prime },\overline{y}$ and $\overline{x}$ in $EFH[V\left(L_2\right)\cup V\left(R_{s+4}\right)\cup V\left(R_4\right)]$. Take $T_s={\overline{T}}_s\cup z{z}^{\prime }\cup y\overline{y}\cup x\overline{x}$ and $T_{s+1}=P_s\cup y{y}^{\prime }\cup M_s$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

Subcase 2.1.2.

$y\ne x_i$ for each $i=1,2,\dots ,s$.

By Lemma 7(4), we can show $\overline{y}\notin V\left(R_{i+4}\right)$ for $i=0,1,\dots ,s$. Without loss of generality, let $\overline{y}\in V\left(R_1\right)\cup V\left(R_2\right)$.

First, we let $\overline{y}\in V\left(R_2\right)$. By Lemma 7(3), $z^{\prime }\notin V\left(L_1\right)$ or $\overline{z}\notin V\left(L_1\right)$. Without loss of generality, let $\overline{z}\notin V\left(L_1\right)$. Suppose $\overline{z}\in V\left(L_2\right)$. Take s vertices $z_1,z_2,\dots ,z_s$ in $V\left(R_1\right)$, such that ${\overline{z}}_i\in V\left(L_{i+4}\right)$ for $i=1,2,\dots ,s$. Let $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6, there exist s internally disjoint paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $M_i$ be the path from z to $z_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i$ and ${\overline{z}}_i$ in $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ for $i=1,2,\dots ,s$. Take $T_i={\overline{T}}_i\cup P_i\cup M_i\cup x_i{\overline{x}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(L_2\right)\cup V\left(R_2\right)\cup V\left(R_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{z},\overline{y}$ and $\overline{x}$ in $EFH[V\left(L_2\right)\cup V\left(R_2\right)\cup V\left(R_4\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup z\overline{z}\cup y\overline{y}$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

Now, we let $\overline{y}\in V\left(R_1\right)$.

If $\overline{y}=z$, then $z^{\prime }\notin V\left(L_1\right)$. We can let $z^{\prime }\in V\left(L_2\right)$. Taking $T_1,T_2,\dots ,T_s$ to be the same as above, since $EFH[V\left(L_2\right)\cup V\left(R_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{x}$ and $z^{\prime }$ in $EFH[V\left(L_2\right)\cup V\left(R_4\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup yz{z}^{\prime }$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

If $\overline{y}\ne z$. By Lemma 7(3), suppose ${\overline{y}}^{\prime }\in V\left(L_2\right)$, where ${\overline{y}}^{\prime }$ is the hypercube neighbor of $\overline{y}$. By Lemma 7(4), $\overline{z}\notin V\left(L_1\right)$. Without loss of generality, let $\overline{z}\in V\left(L_2\right)\cup V\left(L_3\right)$. Take $s-1$ vertices $z_1,z_2,\dots ,z_{s-1}$ in $V\left(R_1\right)$, such that ${\overline{z}}_i\in V\left(L_{i+4}\right)$ for $i=1,2,\dots ,s-1$. Let $Z=\{z_1,z_2,\dots ,z_{s-1},\overline{y}\}$. By Lemma 6, there exist s internally disjoint paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $M_i$ be the path from z to $z_i$ for $i=1,2,\dots ,s-1$ and $M_s$ be the path from z to $\overline{y}$. Since $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i$ and ${\overline{z}}_i$ in $EFH[V\left(L_{i+4}\right)\cup V\left(R_{i+4}\right)]$ for $i=1,2,\dots ,s-1$. Take $T_i={\overline{T}}_i\cup P_i\cup M_i\cup x_i{\overline{x}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s-1$. If $\overline{z}\in V\left(L_3\right)$ (Figure 6), noting that ${\overline{x}}_s\in V\left(R_{s+4}\right)$, since $EFH[V\left(L_3\right)\cup V\left(R_{s+4}\right)]$ is connected, there exists a tree ${\overline{T}}_s$ containing $\overline{z}$ and ${\overline{x}}_s$ in $EFH[V\left(L_3\right)\cup V\left(R_{s+4}\right)]$. Take $T_s={\overline{T}}_s\cup P_s\cup x_s{\overline{x}}_s\cup z\overline{z}$. Since $EFH[V\left(L_2\right)\cup V\left(R_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{x}$ and ${\overline{y}}^{\prime }$ in $EFH[V\left(L_2\right)\cup V\left(R_4\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup y\overline{y}{\overline{y}}^{\prime }\cup M_s$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. If $\overline{z}\in V\left(L_2\right)$ (Figure 7), since $\overline{y}\ne z$, then ${\overline{y}}^{\prime }\ne \overline{z}$ by Lemma 7(3). Since $L_2\cong Q_s$, we can partition $L_2$ into $L_{21}$ and $L_{22}$, such that $L_{21}\cong Q_{s-1},L_{22}\cong Q_{s-1}$ and ${\overline{y}}^{\prime }\in V\left(L_{21}\right),\overline{z}\in V\left(L_{22}\right)$. In $L_{21}$, there exists a spanning tree $T_{21}$ containing ${\overline{y}}^{\prime }$. Since $|V\left(T_{21}\right)|=|V\left(L_{21}\right)|=2^{s-1}\ge s+1$ for $s\ge 3$, there exists a vertex $u\in V\left(L_{21}\right)$, such that $u^{\prime }\notin V\left(R_1\right)\cup V\left(R_{i+4}\right)$ for $i=1,\dots ,s$ by Lemma 7(4). Let $u^{\prime }\in V\left(R_2\right)\cup V\left(R_4\right)$. Similarly, there exists a spanning tree $T_{22}$ containing $\overline{z}$ in $L_{22}$. Since $|V\left(T_{22}\right)|=|V\left(L_{22}\right)|=2^{s-1}\ge s+1$ for $s\ge 3$, there exists a vertex $v\in V\left(L_{22}\right)$, such that $v^{\prime }\notin V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_{i+4}\right)$ for $i=0,1,\dots ,s-1$ by Lemma 7(4). Let $v^{\prime }\in V\left(R_3\right)\cup V\left(R_{s+4}\right)$. Since $EFH[V\left(R_2\right)\cup V\left(R_4\right)\cup V\left(L_3\right)]$ is connected, there exists a tree ${\overline{T}}_s$ containing $u^{\prime }$ and $\overline{x}$. Take $T_s={\overline{T}}_s\cup x\overline{x}\cup T_{21}\cup u{u}^{\prime }\cup y\overline{y}{\overline{y}}^{\prime }\cup M_s$. Since $EFH[V\left(R_3\right)\cup V\left(R_{s+4}\right)\cup V\left(L_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $v^{\prime }$ and ${\overline{x}}_s$. Take $T_{s+1}={\overline{T}}_{s+1}\cup v{v}^{\prime }\cup T_{22}\cup z\overline{z}\cup P_s\cup x_s{\overline{x}}_s$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

By symmetry and $t\ge s$, if $x,y\in V\left(R_i\right),z\in V\left(L_j\right)$ for some $i\in \{1,2,\dots ,2^s\}$ and some $j\in \{1,2,\dots ,2^t\}$, we can also obtain ${\kappa }_3\left(EFH\right)\ge s+1$.

Subcase 2.2.

$z\in V\left(L_j\right)$ for some $j\in \{2,\dots ,2^t\}$.

Without loss of generality, we let $z\in V\left(L_2\right)$ (Figure 8), and suppose $\overline{x}\in V\left(R_3\right),{\overline{x}}_i\in V\left(R_{i+3}\right),x^{\prime }\in V\left(R_{s+4}\right),x_i^{\prime }\in V\left(R_{s+i+4}\right)$ for $i=1,2,\dots ,s$. Then, $z^{\prime }\notin V\left(R_{i+3}\right)$ or $\overline{z}\notin V\left(R_{i+3}\right)$ or $z^{\prime }\notin V\left(R_{s+i+4}\right)$ or $\overline{z}\notin V\left(R_{s+i+4}\right)$ for $i=0,1,\dots ,s$. Without loss of generality, let $\overline{z}\notin V\left(R_{i+3}\right)$ for $i=0,1,\dots ,s$. Suppose $\overline{z}\in V\left(R_2\right)$. If $y=x_i$ for some $i\in \{1,2,\dots ,s\}$, then $\overline{y}={\overline{x}}_i$ for some $i\in \{1,2,\dots ,s\}$. Then, $y^{\prime }\notin V\left(R_{i+3}\right)$ for $i=0,1,\dots ,s$. If $y\ne x_i$ for each $i=1,2,\dots ,s$, then $\overline{y}\notin V\left(R_{i+3}\right)$ for $i=0,1,\dots ,s$ by Lemma 7(4). Without loss of generality, let $\overline{y}\notin V\left(R_{i+3}\right)$ for $i=0,1,\dots ,s$. Suppose $\overline{y}\in V\left(R_1\right)\cup V\left(R_2\right)$. Choose s vertices $z_1,z_2,\dots ,z_s$ in $V\left(L_2\right)$, such that ${\overline{z}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6, there exist s internally disjoint paths $M_1,M_2,\dots ,M_s$ from z to Z in $L_2$. Let $M_i$ be the path from z to $z_i$ for $i=1,2,\dots ,s$. Since $R_{i+3}$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i$ and ${\overline{z}}_i$ in $R_{i+3}$ for $i=1,2,\dots ,s$. Take $T_i={\overline{T}}_i\cup P_i\cup M_i\cup x_i{\overline{x}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_3\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{x},\overline{y}$ and $\overline{z}$ in $EFH[V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_3\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup y\overline{y}\cup z\overline{z}$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

By symmetry and $t\ge s$, if $x,y\in V\left(R_i\right),z\in V\left(R_j\right)$ for some $i,j\in \{1,2,\dots ,2^s\}$ and $i\ne j$, we can also obtain ${\kappa }_3\left(EFH\right)\ge s+1$.

Case 3.

$x\in V\left(L_i\right),y\in V\left(L_j\right)$, and $z\in V\left(R_k\right)$ for some $i,j\in \{1,2,\dots ,2^t\}$ with $i\ne j$ and some $k\in \{1,2,\dots ,2^s\}$.

Without loss of generality, let $x\in V\left(L_1\right),y\in V\left(L_2\right),z\in V\left(R_1\right)$.

Subcase 3.1.

$z^{\prime },\overline{z}\in V\left(L_1\right)\cup V\left(L_2\right)$.

By Lemma 7(3), without loss of generality, let $\overline{z}\in V\left(L_1\right)$, $z^{\prime }\in V\left(L_2\right)$.

We first consider $\overline{z}=x$ or $z^{\prime }=y$. Without loss of generality, let $\overline{z}=x$. By Lemma 7(3), we can let $x^{\prime }\in V\left(R_2\right)$ and $y^{\prime }\notin V\left(R_1\right)$ or $\overline{y}\notin V\left(R_1\right)$. Suppose $\overline{y}\notin V\left(R_1\right)$. Then, put $\overline{y}\in V\left(R_2\right)\cup V\left(R_3\right)$. Choose $x_1,x_2,\dots ,x_s$ in $V\left(L_1\right)\setminus \left\{x\right\}$, such that ${\overline{x}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $X=\{x_1,x_2,\dots ,x_s\}$. Choose $y_1,y_2,\dots ,y_s$ in $V\left(L_2\right)\setminus \left\{y\right\}$, such that ${\overline{y}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $Y=\{y_1,y_2,\dots ,y_s\}$. Choose $z_1,z_2,\dots ,z_s$ in $V\left(R_1\right)\setminus \left\{z\right\}$, such that ${\overline{z}}_i\in V\left(L_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6, there exist s paths $P_1,P_2,\dots ,P_s$ from x to X in $L_1$, s paths $N_1,N_2,\dots ,N_s$ from y to Y in $L_2$, s paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $P_i,N_i,M_i$ be the paths from x to $x_i$, from y to $y_i$, and from z to $z_i$, respectively, for $i=1,2,\dots ,s$. Since $EFH[V\left(L_{i+3}\right)\cup V\left(R_{i+3}\right)]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i,{\overline{y}}_i$, and ${\overline{z}}_i$ in $EFH[V\left(L_{i+3}\right)\cup V\left(R_{i+3}\right)]$ for $i=1,2,\dots ,s$. Take $T_i={\overline{T}}_i\cup P_i\cup N_i\cup M_i\cup x_i{\overline{x}}_i\cup y_i{\overline{y}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_3\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $x^{\prime },\overline{y}$ in $EFH[V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_3\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup y\overline{y}\cup x{x}^{\prime }\cup xz$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

Now, we consider $\overline{z}\ne x$ and $z^{\prime }\ne y$ (Figure 9). Since $L_1\cong Q_s$ and $L_2\cong Q_s$, we can partition $L_1$ into $L_{11}$ and $L_{12}$, such that $L_{11}\cong Q_{s-1}$, $L_{12}\cong Q_{s-1}$ and $\overline{z}\in V\left(L_{11}\right)$, $x\in V\left(L_{12}\right)$. Similarly, we partition $L_2$ into $L_{21}$ and $L_{22}$, such that $L_{21}\cong Q_{s-1}$, $L_{22}\cong Q_{s-1}$ and $z^{\prime }\in V\left(L_{21}\right),y\in V\left(L_{22}\right)$. By Lemma 7(4), we can let $\overline{x}\in V\left(R_2\right)$ and $y^{\prime }\in V\left(R_2\right)\cup V\left(R_3\right)$. Choose $x_1,x_2,\dots ,x_{s-1}$ in $V\left(L_{12}\right)\setminus \left\{x\right\}$ such that ${\overline{x}}_i\notin V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)$ for $i=1,2,\dots ,s-1$. This can be performed since $2^{s-1}-1\ge 3$ with $s\ge 3$. Let ${\overline{x}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s-1$. Denote $X=\{x_1,x_2,\dots ,x_{s-1}\}$. Choose $y_1,y_2,\dots ,y_{s-1}$ in $V\left(L_{22}\right)\setminus \left\{y\right\}$, such that $y_i^{\prime }\notin V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)$ for $i=1,2,\dots ,s-1$. Without loss of generality, for simplicity of description, we can let $y_1^{\prime }\in V\left(R_4\right)$ and $y_i^{\prime }\in V\left(R_{s+i+1}\right)$ for $i=2,\dots ,s-1$. Note that ${\overline{x}}_1\in V\left(R_4\right)$ and ${\overline{x}}_i\in V\left(R_{i+3}\right)$ for $i=2,\dots ,s-1$. Denote $Y=\{y_1,y_2,\dots ,y_{s-1}\}$. Choose $z_1,z_2,\dots ,z_s\in V\left(R_1\right)\setminus \left\{z\right\}$ such that ${\overline{z}}_i\in V\left(L_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6 and $\kappa \left(L_{12}\right)=\kappa \left(L_{22}\right)=s-1,\kappa \left(R_1\right)=s$, there exist $s-1$ paths $P_1,P_2,\dots ,P_{s-1}$ from x to X in $L_{12}$, $s-1$ paths $N_1,N_2,\dots ,N_{s-1}$ from y to Y in $L_{22}$, s paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $P_i,N_i,M_i$ be the paths from x to $x_i$, from y to $y_i$, and from z to $z_i$, respectively, for $i=1,2,\dots ,s-1$ and $M_s$ be the path from z to $z_s$. Since $EFH[V\left(R_4\right)\cup V\left(L_4\right)]$ is connected, there exists a tree ${\overline{T}}_1$ containing ${\overline{x}}_1,y_1^{\prime }$ and ${\overline{z}}_1$ in $EFH[V\left(R_4\right)\cup V\left(L_4\right)]$. Take $T_1={\overline{T}}_1\cup P_1\cup N_1\cup M_1\cup x_1{\overline{x}}_1\cup y_1{y}_1^{\prime }\cup z_1{\overline{z}}_1$. Since $EFH[V\left(R_{i+3}\right)\cup V\left(R_{s+i+1}\right)\cup V\left(L_{i+3}\right)]$ is connected for $i=2,3,\dots ,s-1$, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i,y_i^{\prime }$ and ${\overline{z}}_i$ in $EFH[V\left(R_{i+3}\right)\cup V\left(R_{s+i+1}\right)\cup V\left(L_{i+3}\right)]$ for $i=2,3,\dots ,s-1$. Take $T_i={\overline{T}}_i\cup P_i\cup N_i\cup M_i\cup x_i{\overline{x}}_i\cup y_i{y}_i^{\prime }\cup z_i{\overline{z}}_i$ for $i=2,3,\dots ,s-1$. Since $EFH[V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_{s+3}\right)]$ is connected, there exists a tree ${\overline{T}}_s$ containing $\overline{x},y^{\prime }$ and ${\overline{z}}_s$ in $EFH[V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_{s+3}\right)]$. Take $T_s={\overline{T}}_s\cup M_s\cup x\overline{x}\cup y{y}^{\prime }\cup z_s{\overline{z}}_s$. Let u be the neighbor of x in $V\left(L_{11}\right)$ and v be the neighbor of y in $V\left(L_{21}\right)$. Suppose that $T_{11}$ is a spanning tree of $L_{11}$ and $T_{21}$ is a spanning tree of $L_{21}$. Take $T_s=T_{11}\cup T_{21}\cup ux\cup vy\cup z\overline{z}\cup z{z}^{\prime }$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

Subcase 3.2.

$z^{\prime }\notin V\left(L_1\right)\cup V\left(L_2\right)$ or $\overline{z}\notin V\left(L_1\right)\cup V\left(L_2\right)$.

Without loss of generality, let $z^{\prime }\notin V\left(L_1\right)\cup V\left(L_2\right)$. Suppose $z^{\prime }\in V\left(L_3\right)$. By Lemma 7(3), $x^{\prime }\notin V\left(R_1\right)$ or $\overline{x}\notin V\left(R_1\right)$, $y^{\prime }\notin V\left(R_1\right)$ or $\overline{y}\notin V\left(R_1\right)$. Without loss of generality, we can let $\overline{x}\in V\left(R_2\right)$, $y^{\prime }\in V\left(R_2\right)\cup V\left(R_3\right)$. Choose $x_1,x_2,\dots ,x_s\in V\left(L_1\right)\setminus \left\{x\right\}$, such that ${\overline{x}}_i\notin V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)$ for $i=1,2,\dots ,s$. Suppose ${\overline{x}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $X=\{x_1,x_2,\dots ,x_s\}$. Choose $y_1,y_2,\dots ,y_s\in V\left(L_2\right)$, such that ${\overline{y}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $Y=\{y_1,y_2,\dots ,y_s\}$. Choose $z_1,z_2,\dots ,z_s\in V\left(R_1\right)\setminus \left\{z\right\}$, such that $z_i^{\prime }\in V\left(L_{i+3}\right)$ for $i=1,2,\dots ,s$. Denote $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6, there exist s paths $P_1,P_2,\dots ,P_s$ from x to X in $L_1$, s paths $N_1,N_2,\dots ,N_s$ from y to Y in $L_2$, s paths $M_1,M_2,\dots ,M_s$ from z to Z in $R_1$. Let $P_i,N_i,M_i$ be the paths from x to $x_i$, from y to $y_i$, and from z to $z_i$, respectively, for $i=1,2,\dots ,s$. Note that if $y=y_i$ for some $i\in \{1,2,\dots ,s\}$, we regard $N_i$ as the vertex y. Since $EFH[V\left(L_{i+3}\right)\cup V\left(R_{i+3}\right)]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i,{\overline{y}}_i$ and $z_i^{\prime }$ in $EFH[V\left(L_{i+3}\right)\cup V\left(R_{i+3}\right)]$ for $i=1,2,\dots ,s$. Take $T_i={\overline{T}}_i\cup P_i\cup N_i\cup M_i\cup x_i{\overline{x}}_i\cup y_i{\overline{y}}_i\cup z_i{z}_i^{\prime }$ for $i=1,2,\dots ,s$. Since $EFH[V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_3\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $z^{\prime },\overline{x},y^{\prime }$ in $EFH[V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_3\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup y{y}^{\prime }\cup z{z}^{\prime }$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

By symmetry and $t\ge s$, if $x\in V\left(R_i\right),y\in V\left(R_j\right),z\in V\left(L_k\right)$ for some $i,j\in \{1,2,\dots ,2^s\}$ with $i\ne j$ and some $k\in \{1,2,\dots ,2^t\}$, we can also obtain ${\kappa }_3\left(EFH\right)\ge s+1$.

Case 4.

$x\in V\left(L_i\right),y\in V\left(L_j\right)$, and $z\in V\left(L_k\right)$ for some $i,j,k\in \{1,2,\dots ,2^t\}$ with $i\ne j\ne k$.

Let $x\in V\left(L_1\right),y\in V\left(L_2\right)$, and $z\in V\left(L_3\right)$ (Figure 10). Without loss of generality, suppose $\overline{x},\overline{y},\overline{z}\in V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)$. Choose $x_i\in V\left(L_1\right)\setminus \left\{x\right\},y_i\in V\left(L_2\right)\setminus \left\{y\right\},z_i\in V\left(L_3\right)\setminus \left\{z\right\}$, such that ${\overline{x}}_i,{\overline{y}}_i,{\overline{z}}_i\in V\left(R_{i+3}\right)$ for $i=1,2,\dots ,s$. Let $X=\{x_1,x_2,\dots ,x_s\}$, $Y=\{y_1,y_2,\dots ,y_s\}$ and $Z=\{z_1,z_2,\dots ,z_s\}$. By Lemma 6, there exist s paths $P_1,P_2,\dots ,P_s$ from x to X in $L_1$, s paths $N_1,N_2,\dots ,N_s$ from y to Y in $L_2$, s paths $M_1,M_2,\dots ,M_s$ from z to Z in $L_3$. Let $P_i,N_i,M_i$ be the paths from x to $x_i$, from y to $y_i$, and from z to $z_i$, respectively, for $i=1,2,\dots ,s$. Since $EFH\left[V\left(R_{i+3}\right)\right]$ is connected, there exists a tree ${\overline{T}}_i$ containing ${\overline{x}}_i,{\overline{y}}_i$ and ${\overline{z}}_i$ in $EFH\left[V\left(R_{i+3}\right)\right]$ for $i=1,2,\dots ,s$. Take $T_i={\overline{T}}_i\cup P_i\cup N_i\cup M_i\cup x_i{\overline{x}}_i\cup y_i{\overline{y}}_i\cup z_i{\overline{z}}_i$ for $i=1,2,\dots ,s$. Since $EFH[V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_4\right)]$ is connected, there exists a tree ${\overline{T}}_{s+1}$ containing $\overline{x},\overline{y}$ and $\overline{z}$ in $EFH[V\left(R_1\right)\cup V\left(R_2\right)\cup V\left(R_3\right)\cup V\left(L_4\right)]$. Take $T_{s+1}={\overline{T}}_{s+1}\cup x\overline{x}\cup y\overline{y}\cup z\overline{z}$. Then, $T_1,T_2,\dots ,T_{s+1}$ are $s+1$ internally edge disjoint S-trees. Thus, ${\kappa }_3\left(EFH\right)\ge s+1$.

By symmetry and $t\ge s$, if $x\in V\left(R_i\right),y\in V\left(R_j\right),z\in V\left(R_k\right)$ for some $i,j,k\in \{1,2,\dots ,2^s\}$ with $i\ne j\ne k$, we can also obtain ${\kappa }_3\left(EFH\right)\ge s+1$.

We have completed the proof. □

4. Conclusions

The exchanged folded hypercube is a variant of the hypercube and denoted by $EFH(s,t)$. It has many attractive properties to design interconnection networks. The generalized k-connectivity is an extension of the traditional connectivity. In this paper, we computed the generalized 3-connectivity of the exchanged folded hypercube. The study of the generalized k-connectivity of the exchanged folded hypercube for $k\ge 4$ is a meaningful and challenging problem.