Generalized 4-connectivity of hierarchical star networks

Junzhen Wang; Jinyu Zou; Shumin Zhang

doi:10.1515/math-2022-0490

Open Access Published by De Gruyter Open Access October 17, 2022

Generalized 4-connectivity of hierarchical star networks

Junzhen Wang , Jinyu Zou and Shumin Zhang

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0490

Abstract

The connectivity is an important measurement for the fault-tolerance of a network. The generalized connectivity is a natural generalization of the classical connectivity. An S -tree of a connected graph G is a tree T = ( V ′ , E ′ ) that contains all the vertices in S subject to S ⊆ V ( G ) . Two S -trees T and T ′ are internally disjoint if and only if E ( T ) ∩ E ( T ′ ) = ∅ and V ( T ) ∩ V ( T ′ ) = S . Denote by κ ( S ) the maximum number of internally disjoint S -trees in graph G . The generalized k -connectivity is defined as κ k ( G ) = min { κ ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = k } . Clearly, κ 2 ( G ) = κ ( G ) . In this article, we show that κ 4 ( H S n ) = n − 1 , where H S n is the hierarchical star network.

Keywords: hierarchical star networks; fault-tolerance; generalized connectivity; disjoint S-trees

MSC 2010: 05C05; 05C40; 05C76

1 Introduction

The graphs considered in this article are simple undirected finite graphs. For graph theory symbols and terms that are covered but not mentioned in this article, please refer to [1]. Let G = ( V , E ) be a connected graph with vertex set V ( G ) and edge set E ( G ) . Let u ∈ V ( G ) , and N G ( u ) = { v ∈ V ( G ) \ u ∣ u v ∈ E ( G ) } be the neighbour set of u in the graph G , and N G [ u ] = N G ( u ) ∪ { u } . Let d G ( u ) = ∣ N G ( u ) ∣ be the degree of u in G . If d G ( v ) = k for any vertex v ∈ V ( G ) , then the graph G is k -regular. For any vertices u , v ∈ V ( G ) , an ( u , v ) -path starts at u and ends at v . Any two ( u , v ) -paths P and Q are internally disjoint if and only if V ( P ) ∩ V ( Q ) = { u , v } .

The classic connectivity of graph G , denoted as κ ( G ) , is an important parameter to measure the reliability and fault-tolerance of the network. The connectivity κ ( G ) is larger, the reliability of the network is higher. There are two versions to define κ ( G ) . The version definition of “cut” is that deleting the minimum number of vertices disconnects the graph G . The version of “path” is defined as follows: for any vertex set S = { u , v } , κ G ( S ) represents the maximum number of internally disjoint paths joining u and v in G , and κ ( G ) = min { κ G ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = 2 } .

Although there are fruitful research results in the study of classical connectivity, classical connectivity itself has certain limitations, which lead to large defects in evaluating the reliability of the network. For example, in the actual application of an interconnection network, all processors connected to the same processor are less likely to fail at the same time, so this parameter is not accurate enough to measure network reliability and fault tolerance. In view of this, Chartrand et al. [2] generalized classical connectivity and proposed the concept of generalized connectivity. Let G be a connected graph with S ⊆ V ( G ) . An S -tree of graph G is a tree T = ( V ′ , E ′ ) that contains all the vertices in S subject to S ⊆ V ( G ) . Two S -trees T and T ′ are internally disjoint if and only if E ( T ) ∩ E ( T ′ ) = ∅ and V ( T ) ∩ V ( T ′ ) = S . Denote κ ( S ) as the maximum number of internally disjoint S -trees in graph G . The generalized k-connectivity is defined as κ k ( G ) = min { κ ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = k } , clearly, κ 2 ( G ) = κ ( G ) .

The bounds of generalized connectivity and the relationship between connectivity and generalized connectivity have been extensively investigated [3,4,5]. In addition, the generalized 3-connectivity of many networks has been studied, for example, complete graphs [6], card product graphs [7], product graphs [8], complete bipartite graph [9], star graphs, and alternating group graphs [10]. However, there are few results about generalized 4-connectivity, such as hypercubes [11], exchanged hypercubes [12], dual cube [13], and hierarchical cubic networks [14]. For more works and results on generalized connectivity, please refer to [15,16, 17,18].

In this article, we study the generalized 4-connectivity of the hierarchical star networks.

2 Hierarchical star networks and their properties

In this section, we give the definition, structure of the hierarchical star networks, and some lemmas.

Definition 1

[19] An n -dimensional star graph, denoted by the graph S n , is defined as an undirected graph with each vertex representing a distinct permutation of [ n ] and two vertices are adjacent if and only if their labels differ only in the first and another position, that is, two vertices u = u 1 u 2 … u n , v = v 1 v 2 … v n are adjacent if and only if v = u i u 2 u 3 … u i − 1 u 1 u i + 1 … u n for some i ∈ [ n ] ⧹ { 1 } , where [ n ] = 1 , 2 , … , n . In this situation, ( u , v ) is an i -edge.

The star graph S 4 is shown in Figure 1.

$Figure 1 The graph of S 4 {S}_{4} .$

Figure 1

The graph of S 4 .

Let Γ n be a permutation group on the set [ n ] and S = { ( 1 , 2 ) , ( 1 , 3 ) , … , ( 1 , n ) } , where ( 1 , i ) is a transposition of Γ n . Then S n is the undirected Cayley graph Cay ( Γ n , S ) . For a permutation x ∈ Γ n , the permutation by interchanging the first element with i th element of x is denoted as x ( 1 , i ) for i ∈ [ n ] ⧹ { 1 } .

For two integers i , j ∈ [ n ] , denoted by S n j : i the subgraph of S n induced by all the vertices with the j th element being i . For a fixed dimension j ∈ [ n ] \ 1 , S n can be partitioned into n subgraphs S n j : i , which is isomorphic to S n − 1 for each i ∈ [ n ] .

Lemma 2.1

[19] For any integer n ≥ 3 , S n is ( n − 1 ) -regular and ( n − 1 ) -connected, vertex transitive, edge transitive, bipartite graph with girth 6. Any two vertices have at most one common neighbor in S n .

Definition 2

[20] An n -dimensional hierarchical star network, H S n , is made of n ! n-dimensional star graphs S n , called copies. Each vertex of H S n is denoted by a two-tuple address ⟨ a , b ⟩ , where both a and b are arbitrary permutation of n distinct symbols. The first n -bit permutation a identifies the copy of a and the second n -bit permutation b identifies the position of b inside its copy. Two vertices ⟨ a , b ⟩ and ⟨ a ^ , b ^ ⟩ in H S n are adjacent, if one of the following three conditions holds:

a = a ^ and ( b , b ^ ) ∈ E ( S n ) . That is, ⟨ a , b ⟩ is adjacent to ⟨ a , b ^ ⟩ if ( b , b ^ ) ∈ E ( S n ) ;
a ≠ a ^ , a = b , and a ^ = b ^ = a ( 1 , n ) . That is, ⟨ a , a ⟩ is adjacent to ⟨ a ( 1 , n ) , a ( 1 , n ) ⟩ ;
a ≠ a ^ , a ≠ b , a = b ^ , and b = a ^ . That is, ⟨ a , b ⟩ is adjacent to ⟨ b , a ⟩ if a ≠ b .

The hierarchical star networks H S 2 and H S 3 are shown in Figure 2.

$Figure 2 The graph of H S 2 H{S}_{2} and H S 3 H{S}_{3} .$

Figure 2

The graph of H S 2 and H S 3 .

Remark 2.1

[21] Each node in H S n is assigned a label ⟨ a , b ⟩ = ⟨ a 1 a 2 … a n , b 1 b 2 … b n ⟩ , where a 1 a 2 … a n and b 1 b 2 … b n are permutations of n distinct symbols (not necessarily distinct from each other). The edges of the H S n are defined by the following n generators:

h 1 ( ⟨ a , b ⟩ ) = ⟨ a ( 1 , n ) , b ( 1 , n ) ⟩ , a = b ; ⟨ b , a ⟩ , a ≠ b ,

and h i ( ⟨ a , b ⟩ ) = ⟨ a , b ( 1 , i ) ⟩ for i ∈ [ n ] ⧹ { 1 } .

Let ⟨ a , b ⟩ be a vertex of H S n . The neighbor set of ⟨ a , b ⟩ is exactly { h i ( ⟨ a , b ⟩ ) ∣ i ∈ [ n ] } . Furthermore, h 1 ( ⟨ a , b ⟩ ) is called the external neighbor of ⟨ a , b ⟩ and h i ( ⟨ a , b ⟩ ) is called the internal neighbor of ⟨ a , b ⟩ for i ∈ [ n ] ⧹ { 1 } . We denote by H S n a the subgraph induced by the vertex set { ⟨ a , b ⟩ ∈ V ( H S n ) ∣ b ∈ V ( S n ) } , which is isomorphic to an n -dimensional star graph S n identified by a . Moreover, we define H S n a , b as a subgraph of H S n induced by the vertex set V ( H S n a ) ∪ V ( H S n b ) and H S n Γ n − { a } as a subgraph of H S n induced by the vertex set V ( H S n ) \ V ( H S n a ) . For a vertex x ∈ V ( H S n ) , we use x ′ to denote the external neighbor of x .

Remark 2.2

[21] Any vertex has exactly one external neighbor in H S n , that is, every vertex ⟨ a , b ⟩ in H S n a is exactly incident one cross edge ( ⟨ a , b ⟩ , h 1 ( ⟨ a , b ⟩ ) ) . There is one or two cross edges between any pair of copies. Moreover, for a fixed copy H S n a , there are two cross edges between H S n a and H S n a ( 1 , n ) ; there is only one cross edge between H S n a and H S n b , where b ∈ Γ n \ { a , a ( 1 , n ) } .

Lemma 2.2

[20] For any integer n ≥ 3 , H S n is an n -regular n-connected graph, and its girth is 4. Any two vertices have at most two common neighbors in H S n .

Lemma 2.3

Let H S n 1 , H S n 2 , … , H S n n ! be the n ! copies of H S n , and H = H S n [ ⋃ j i = 1 k V ( H S n j i ) ] for j i ∈ [ n ! ] , k ≥ 1 , and n ≥ 3 , 1 ≤ i ≤ n ! , then H is connected.

Proof

Without loss of generality, suppose H = H S n [ ⋃ j i = 1 k V ( H S n j i ) ] . By Remark 2.2, there is at least one cross edge between any two distinct copies of H S n . Thus, H is connected.□

Lemma 2.4

For j i ∈ [ n ! ] , 1 ≤ i ≤ n ! , let v ∈ V ( H S n j i ) with v = ⟨ x , y ⟩ and x = y . The external neighbors of different vertices in V ( H S n j i ) \ { v } belong to different copies of H S n . Moreover, if u ∈ V ( H S n j i ) \ { v } with u = ⟨ x , y ( 1 , n ) ⟩ , then v ′ , u ′ belong to the same copy of H S n .

Proof

Let v 1 , v 2 ∈ V ( H S n j i ) \ { v } with v 1 = ⟨ x 1 , y 1 ⟩ , v 2 = ⟨ x 2 , y 2 ⟩ . By the definition of H S n , x 1 ≠ y 1 , x 2 ≠ y 2 , and y 1 ≠ y 2 , v ′ = ⟨ x ( 1 , n ) , y ( 1 , n ) ⟩ . Clearly, v 1 ′ = ⟨ y 1 , x 1 ⟩ and v 2 ′ = ⟨ y 2 , x 2 ⟩ . Since y 1 ≠ y 2 , v 1 ′ and v 2 ′ belong to different copies of H S n . Let u ∈ V ( H S n j i ) \ { v } , x ≠ y ( 1 , n ) , and u ′ = ⟨ y ( 1 , n ) , x ⟩ . Since x ( 1 , n ) = y ( 1 , n ) , v ′ , u ′ belong to the same copy of H S n .□

3 Generalized 4-connectivity of H S n

In this section, we will study the generalized 4-connectivity of hierarchical star networks. To prove the main result, the following results are useful.

Lemma 3.1

[4] If there are two adjacent vertices of degree δ ( G ) , then κ k ( G ) ≤ δ ( G ) − 1 for 3 ≤ k ≤ ∣ V ( G ) ∣ .

Lemma 3.2

[1] Let G be a k -connected graph, and let x and y be a pair of distinct vertices in G . Then there exist k internally disjoint paths P 1 , P 2 , … , P k in G connecting x and y .

Lemma 3.3

[1] Let G = ( V , E ) be a k -connected graph, let x be a vertex of G , and let Y ⊆ V \ { x } be a set of at least k vertices of G. Then there exists a k -fan in G from x to Y. That is, there exists a family of k internally vertex-disjoint ( x , Y ) -paths whose terminal vertices are distinct in Y .

Lemma 3.4

[1] Let G = ( V , E ) be a k -connected graph, and let X and Y be subsets of V ( G ) of cardinality at least k . Then there exists a family of k pairwise disjoint ( X , Y ) -paths in G .

Theorem 3.1

[22] κ 3 ( S n ) = n − 2 , for n ≥ 3 .

Theorem 3.2

[23] κ 4 ( S n ) = n − 2 , for n ≥ 3 .

Lemma 3.5

For any three vertices x , y , z ∈ V ( S n ) , n ≥ 3 , there exist n − 2 internally disjoint trees T ^ 1 , … , T ^ n − 2 connecting x , y , z and each T ^ i contains a vertex u i such that u i is distinct from x , y , z and u i ≠ u j , for 1 ≤ i ≠ j ≤ n − 2 .

Proof

By Theorem 3.2, κ 4 ( S n ) = n − 2 . That is to say, for any four distinct vertices x , y , z , u ∈ V ( S n ) , there are n − 2 internally disjoint trees T 1 ′ , … , T n − 2 ′ connecting x , y , z , u in S n . There exists some integer i such that ∣ V ( T i ′ ) ∣ = 4 and ∣ V ( T j ′ ) ∣ ≥ 5 , where 1 ≤ i ≠ j ≤ n − 2 .

Without loss of generality, suppose ∣ V ( T 1 ′ ) ∣ = 4 . By Lemma 2.1, d S n ( u ) = n − 1 .

Case 1. d T 1 ′ ( u ) = 2 .

Obviously, for T j ′ , 2 ≤ j ≤ n − 2 , d T j ′ ( u ) = 1 . Let T ^ 1 = T 1 ′ and T ^ j = T j ′ − u for 1 ≤ i ≠ j ≤ n − 2 . Then T ^ 1 , … , T ^ n − 2 are the desired trees.

Case 2. d T 1 ′ ( u ) = 1 .

Case 2.1. For 2 ≤ j ≤ n − 2 , d T j ′ ( u ) = 1 .

The proof is similar to that of Case 1.

Case 2.2. For 2 ≤ j ≤ n − 2 , there exists some T j ′ subject to d T j ′ ( u ) = 2 .

Without loss of generality, suppose d T 2 ′ ( u ) = 2 . Let T ^ 1 = T 1 ′ , T ^ 2 = T 2 ′ , T ^ j = T j ′ − u for 3 ≤ j ≤ n − 2 . Then T ^ 1 , … , T ^ n − 2 are the desired trees.□

Theorem 3.3

κ 4 ( H S n ) = n − 1 for n ≥ 2 .

By Lemma 3.1, κ 4 ( H S n ) ≤ δ ( H S n ) − 1 = n − 1 . To obtain Theorem 3.3, it suffices to establish the following claim.

Claim. For any given vertex set S = { x , y , z , w } ⊆ V ( H S n ) , there exist n − 1 internally disjoint S -trees in H S n .

Proof

It proceeds by induction on n . Obviously, H S 2 is connected, there is a tree connecting x , y , z , w in H S 2 . Suppose that n ≥ 3 and the theorem holds for n − 1 .□

Recall that H S n consists of n ! copies isomorphic to S n , denoted by H S n ℓ , ℓ ∈ Γ n . We consider the following cases.

Case 1. The vertices of S are distributed in one copy of H S n .

Without loss of generality, suppose x , y , z , w ∈ V ( H S n α ) , where α ∈ Γ n . By Theorem 3.2, there exist n − 2 internally disjoint S -trees connecting x , y , z , w in H S n α , say T 1 , T 2 , … , T n − 2 . Let H = H S n Γ n − { α } , by Lemma 2.3, H is connected. Clearly, x ′ , y ′ , z ′ , w ′ ∈ V ( H ) , then there exists a tree T ^ n − 1 connecting x ′ , y ′ , z ′ , w ′ in H . Let T n − 1 = T ^ n − 1 ∪ x x ′ ∪ y y ′ ∪ z z ′ ∪ w w ′ . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 2. The vertices of S are distributed in two distinct copies of H S n .

Without loss of generality, suppose x , y , z ∈ V ( H S n α ) and w ∈ V ( H S n β ) , where α , β ∈ Γ n , α ≠ β . By Lemma 3.5, there are n − 2 internally disjoint trees connecting x , y , z , say T ^ i , and u i ∈ V ( T ^ i ) is distinct with x , y , z for i ∈ { 1 , … , n − 2 } .

Case 2.1. There is no cross edge between { T ^ 1 , … , T ^ n − 2 } and H S n β .

Without loss of generality, suppose u i ′ ∈ V ( H S n j i ) , where j i ∈ Γ n − { α , β } , i ∈ { 1 , … , n − 2 } . First, we suppose that different vertices of { u 1 ′ , … , u n − 2 ′ } belong to different copies.

Case 2.1.1. x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) for i ∈ { 1 , … , n − 2 } .

$Figure 3 The illustration of Case 2.1.1.$

Figure 3

The illustration of Case 2.1.1.

Clearly, there exists a path P i joining u i ′ and m i in H S n j i such that m i ′ ∈ V ( H S n β ) . Let W = { m 1 ′ , … , m n − 2 ′ , m } . By Lemma 3.3, there are n − 1 internally disjoint paths joining w and W , say P 1 ′ , … , P n − 1 ′ , m i ′ ∈ V ( P i ′ ) , m ∈ V ( P n − 1 ′ ) , m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , for i ∈ { 1 , … , n − 2 } . Clearly, x ′ , y ′ , z ′ , m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , and so there is a tree T ^ n − 1 connecting x ′ , y ′ , z ′ , m ′ . Let T i = T ^ i ∪ u i u i ′ ∪ P i ∪ m i m i ′ ∪ P i ′ for i ∈ { 1 , … , n − 2 } and T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 3).

Case 2.1.2. One of { x ′ , y ′ , z ′ } belongs to some H S n j i for i ∈ { 1 , … , n − 2 } .

Suppose x ′ belongs to the copy H S n j n − 2 , clearly u n − 2 ′ also belongs to H S n j n − 2 . Let m n − 2 , h ∈ V ( H S n j n − 2 ) , X = { u n − 2 ′ , x ′ } , and Y = { m n − 2 , h } such that m n − 2 ′ ∈ V ( H S n β ) , h ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . By Lemma 3.4, there are two internally disjoint paths between X and Y , say P n − 2 and P n − 1 , where P n − 2 is the path joining u n − 2 ′ and m n − 2 and P n − 1 is the path joining x ′ and h . Let W = { m 1 ′ , … , m n − 2 ′ , m } , by Lemma 3.3, there are n − 1 internally disjoint paths between w and W , say P 1 ′ , … , P n − 1 ′ , and m i ′ ∈ V ( P i ′ ) , m ∈ V ( P n − 1 ′ ) , such that m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . In view of y ′ , z ′ , m ′ , h ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , there exists a tree connecting y ′ , z ′ , m ′ , h ′ , say T ^ n − 1 . Let T i = T ^ i ∪ u i u i ′ ∪ P i ∪ m i m i ′ ∪ P i ′ for i ∈ { 1 , … , n − 2 } and T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ P n − 1 ∪ h h ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 4).

$Figure 4 The illustration of Case 2.1.2.$

Figure 4

The illustration of Case 2.1.2.

Next, we suppose two of { u 1 ′ , … , u n − 2 ′ } belong to the same copy. Let X ′ = ( X \ { x ′ } ) ∪ { u n − 3 ′ } , Y ′ = ( Y \ { h } ) ∪ { m n − 3 } , and m n − 3 ′ ∈ V ( H S n β ) . We obtain n − 3 internally disjoint S -tees T 1 , T 2 , … , T n − 4 , T n − 2 , similar to that of Case 2.1.2 by replacing X with X ′ and Y with Y ′ . In view of x ′ , y ′ , z ′ , m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) , there is a tree connecting x ′ , y ′ , z ′ , m ′ say T ^ n − 1 . Let T n − 3 = T ^ n − 3 ∪ u n − 3 u n − 3 ′ ∪ P n − 3 ∪ m n − 3 m n − 3 ′ ∪ P n − 3 ′ , T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 2.2. There is one cross edge between { T ^ 1 , … , T ^ n − 2 } and H S n β .

Consequently, we just consider the case different vertices of { u 1 ′ , … , u n − 2 ′ } belong to different copies and the case that two of { u 1 ′ , … , u n − 2 ′ } in the same copy is similar.

Case 2.2.1. For some i ∈ { 1 , … , n − 2 } , u i ′ ∈ V ( H S n β ) .

Without loss of generality, suppose u n − 2 ′ ∈ V ( H S n β ) , u i ′ ∈ V ( H S n j i ) for i ∈ { 1 , … , n − 3 } .

When x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) , we set W ′ = ( W \ { m n − 2 ′ } ) ∪ { u n − 2 ′ } . We obtain n − 2 internally disjoint S -trees T 1 , T 2 , … , T n − 3 , T n − 1 , similar to that of Case 2.1.1 by replacing W with W ′ . And T n − 2 = T ^ n − 2 ∪ P n − 2 ′ ∪ u n − 2 u n − 2 ′ (Figure 5[a]).

$Figure 5 The illustrations of Case 2.2.1: (a) x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) x^{\prime} ,y^{\prime} ,z^{\prime} \in V\left(H{S}_{n}^{{\Gamma }_{n}-\left\{\alpha ,\beta ,{\bigcup }_{i=1}^{n-3}{j}_{i}\right\}}\phantom{\rule[-0.45em]{}{0ex}}\right) ; and (b) x ′ , u n − 3 ′ ∈ V ( H S n j n − 3 ) x^{\prime} ,{u}_{n-3}^{^{\prime} }\in V\left(H{S}_{n}^{{j}_{n-3}}) .$

Figure 5

The illustrations of Case 2.2.1: (a) x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) ; and (b) x ′ , u n − 3 ′ ∈ V ( H S n j n − 3 ) .

When one of { x ′ , y ′ , z ′ } and u i ′ belong to the same copy, for some i ∈ { 1 , … , n − 3 } , we suppose that x ′ , u n − 3 ′ belong to the same copy H S n j n − 3 . Let X ′ = ( X \ { u n − 2 ′ } ) ∪ { u n − 3 ′ } , Y ′ = ( Y \ { m n − 2 } ) ∪ { m n − 3 } , and W ′ = ( W \ { m n − 2 ′ } ) ∪ { u n − 2 ′ } . We obtain n − 2 internally disjoint S -trees T 1 , T 2 , … , T n − 3 , T n − 1 , similar to that of Case 2.1.2 by replacing X with X ′ , Y with Y ′ , W with W ′ . And T n − 2 = T ^ n − 2 ∪ P n − 2 ′ ∪ u n − 2 u n − 2 ′ (Figure 5[b]).

Case 2.2.2. One of { x ′ , y ′ , z ′ } belongs to H S n β .

Without loss of generality, suppose z ′ ∈ V ( H S n β ) , u i ′ ∈ V ( H S n j i ) , where j i ∈ Γ n − { α , β } , i ∈ { 1 , … , n − 2 } . There is a path joining u i ′ and m i in H S n j i , say P i , and m i ′ ∈ V ( H S n β ) , for i ∈ { 1 , … , n − 2 } .

Case 2.2.2.1. x ′ , y ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) .

When z ′ ∈ N H S n β ( w ) , we set T ^ n − 1 be the tree connecting x ′ , y ′ , m ′ . We obtain n − 2 internally disjoint S -trees T 1 , T 2 , … , T n − 2 , similar to that of Case 2.1.1. And T n − 1 = T ^ n − 1 ∪ P n − 1 ′ ∪ x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ .

$Figure 6 The illustrations of z ′ ∈ N { H S n β } ( w ) z^{\prime} \in {N}_{\left\{H{S}_{n}^{\beta }\right\}}\left(w) : (a) Case 2.2.2.1 and (b) Case 2.2.2.2.$

Figure 6

The illustrations of z ′ ∈ N { H S n β } ( w ) : (a) Case 2.2.2.1 and (b) Case 2.2.2.2.

When z ′ ∉ N H S n β ( w ) , we set P n − 1 ′ be the path from w to z ′ , then there exists a vertex a distinct from z ′ and w in P n − 1 ′ . First, we suppose a ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . The proof is similar to the case of z ′ ∈ N H S n β ( w ) . Next, we suppose a ′ ∈ V ( H S n α , j i ) , for some i ∈ { 1 , … , n − 2 } . Clearly, in this case, w ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , the proof is similar to the case of a ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) (Figure 6[a]).

Case 2.2.2.2. For some i ∈ { 1 , … , n − 2 } , one of { x ′ , y ′ } belongs to V ( H S n j i ) .

Without loss of generality, suppose x ′ , u n − 2 ′ ∈ V ( H S n j n − 2 ) .

When z ′ ∈ N H S n β ( w ) , we set T ^ n − 1 be the tree connecting h ′ , y ′ , m ′ . We can obtain n − 2 internally disjoint S -trees T 1 , T 2 , … , T n − 2 , similar to Case 2.1.2. And T n − 1 = T ^ n − 1 ∪ P n − 1 ∪ P n − 1 ′ ∪ x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ ∪ h h ′ (Figure 6[b]).

When z ′ ∉ N H S n β ( w ) , there exists a vertex a distinct from z ′ and w in P n − 1 ′ . First, we suppose a ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . The proof is similar to the case of z ′ ∈ N H S n β ( w ) . Next, we suppose a ′ ∈ V ( H S n α , j i ) for some i ∈ { 1 , … , n − 2 } . Clearly, in this case, w ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . The proof is similar to the case of a ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) .

Case 2.3. There are two cross edges between { T ^ 1 , … , T ^ n − 2 } and H S n β .

Case 2.3.1. One of { x ′ , y ′ , z ′ } belongs to H S n β , and some u i ′ ∈ V ( H S n β ) for i ∈ { 1 , … , n − 2 } .

Without loss of generality, suppose z ′ , u n − 2 ′ ∈ V ( H S n β ) , and u i ′ ∈ V ( H S n j i ) for i ∈ { 1 , … , n − 3 } . There exists a path joining u i ′ and m i in H S n j i , say P i , and m i ′ ∈ V ( H S n β ) , for i ∈ { 1 , … , n − 3 } .

Let W = { m 1 ′ , … , m n − 3 ′ , u n − 2 ′ , z ′ } . By Lemma 3.3, there are n − 1 internally disjoint paths between w and W , say P 1 ′ , … , P n − 1 ′ , and m i ′ ∈ V ( P i ′ ) for i ∈ { 1 , … , n − 3 } , u n − 2 ′ ∈ V ( P n − 2 ′ ) , z ′ ∈ V ( P n − 1 ′ ) . If w ∉ W , clearly, x ′ , y ′ , w ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) , there exists a tree connecting x ′ , y ′ , and w ′ , say T ^ n − 1 . Let

T i = T ^ i ∪ u i u i ′ ∪ P i ∪ m i m i ′ ∪ P i ′ , i ∈ { 1 , … , n − 3 } T n − 2 = T ^ n − 2 ∪ u n − 2 u n − 2 ′ ∪ P n − 2 ′ T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ w w ′ ∪ T ^ n − 1

Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 7[a]). If w ∈ W , the proof is similar to the case of w ∉ W .

$Figure 7 The illustrations of (a) Case 2.3.1 and (b) Case 2.3.2.$

Figure 7

The illustrations of (a) Case 2.3.1 and (b) Case 2.3.2.

Case 2.3.2. Two of { x ′ , y ′ , z ′ } belong to H S n β .

Without loss of generality, suppose x ′ , y ′ ∈ V ( H S n β ) , clearly, x ′ is adjacent to y ′ . Let W ′ = ( W \ { u n − 2 ′ , z ′ } ) ∪ { x ′ , y ′ } , we obtain n − 3 internally disjoint S -trees T 1 , T 2 , … , T n − 3 , similar to that of Case 2.3.1 by replacing W with W ′ . Let x ′ ∈ V ( P n − 2 ′ ) , y ′ ∈ V ( P n − 1 ′ ) . Since the girth of H S n β is 6, there is a vertex m in P n − 2 ′ or P n − 1 ′ such that m is distinct with x ′ , y ′ , and w . Without loss of generality, suppose m ∈ V ( P n − 2 ′ ) . Since there are two cross edges between H S n α and H S n β , m ′ and w ′ belong to different copies, z ′ and u n − 2 ′ belong to different copies. We consider the case that m ′ belongs to different copies with z ′ or u n − 2 ′ , and the case that m ′ belongs to the same copy with z ′ or u n − 2 ′ is similar. Suppose z ′ ∈ V ( H S n j n − 2 ) , m ′ ∈ V ( H S n j n − 1 ) . By Lemma 2.3, H S n j n − 2 , j n − 1 is connected, then there is a path joining z ′ and m ′ , say Q . Obviously, u n − 2 ′ , w ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) . By Lemma 2.3, H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } is connected, then there is a path joining u n − 2 ′ and w ′ , say R . Let T n − 2 = T ^ n − 2 ∪ u n − 2 u n − 2 ′ ∪ R ∪ w w ′ , T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ P n − 2 ′ ∪ P n − 1 ′ ∪ m m ′ ∪ Q . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 7[b]).

Case 2.3.3. Two of { u 1 ′ , … , u n − 2 ′ } belong to H S n β .

Let W ′ = ( W \ { m n − 3 ′ , z ′ } ) ∪ { u n − 3 ′ , m } , and m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 4 j i } ) . We obtain n − 3 internally disjoint S -trees T 1 , T 2 , … , T n − 4 , T n − 2 , similar to that of Case 2.3.1 by replacing W with W ′ . Let T ^ n − 1 be the tree connecting x ′ , y ′ , z ′ , m ′ , T n − 3 = T ^ n − 3 ∪ u n − 3 u n − 3 ′ ∪ P n − 3 ′ , T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 3. The vertices of S are distributed equally in two distinct copies of H S n .

Without loss of generality, suppose x , y ∈ V ( H S n α ) and z , w ∈ V ( H S n β ) , where α , β ∈ Γ n , α ≠ β . Since H S n α is isomorphic to S n , by Lemma 3.2, there are n − 1 internally disjoint paths joining x and y in H S n α , say P 1 , … , P n − 1 . Similarly, there are n − 1 internally disjoint paths joining z and w in H S n β , say P 1 ′ , … , P n − 1 ′ . Let x i ∈ N ( x ) ∩ V ( P i ) , w i ∈ N ( w ) ∩ V ( P i ′ ) for i ∈ { 1 , … , n − 1 } . By Lemma 2.4, different vertices of { x 1 ′ , … , x n − 1 ′ } belong to distinct copies and different vertices of { w 1 ′ , … , w n − 1 ′ } belong to distinct copies. Obviously, at most one x i ′ ∈ V ( H S n β ) , and at most one w i ′ ∈ V ( H S n α ) for i ∈ { 1 , … , n − 1 } . We suppose x i ′ , w i ′ belong to different copies, where i ∈ { 1 , … , n − 1 } , and the case that for some i ∈ { 1 , … , n − 1 } , x i ′ , w i ′ belong to the same copy is similar.

Case 3.1. None of { x 1 ′ , … , x n − 1 ′ } belongs to H S n β , and none of { w 1 ′ , … , w n − 1 ′ } belongs to H S n α for i ∈ { 1 , … , n − 1 } .

As n ! − 2 ≥ 2 n − 2 ( n ≥ 3 ), we can suppose x i ′ ∈ V ( H S n γ i ) , w i ′ ∈ V ( H S n η i ) , and γ i , η i ∈ Γ n − { α , β } , γ i ≠ η i for i ∈ { 1 , … , n − 1 } . By Lemma 2.3, H S n γ i , η i is connected, there exists a path joining x i ′ and w i ′ , say P ^ i , for i ∈ { 1 , … , n − 1 } . Let T i = P i ∪ P i ′ ∪ x i x i ′ ∪ w i w i ′ ∪ P ^ i , where i ∈ { 1 , … , n − 1 } . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 3.2. One of { x 1 ′ , … , x n − 1 ′ } belongs to H S n β , and none of { w 1 ′ , … , w n − 1 ′ } belongs to H S n α for i ∈ { 1 , … , n − 1 } .

Without loss of generality, suppose x n − 1 ′ ∈ V ( H S n β ) .

Case 3.2.1. x n − 1 ′ ∈ V ( P i ′ ) , where i ∈ { 1 , … , n − 1 } .

Without loss of generality, suppose x n − 1 ′ ∈ V ( P n − 1 ′ ) . We can obtain n − 2 internally disjoint S -trees T 1 , … , T n − 2 similar to that of Case 3.1. Let T n − 1 = P n − 1 ∪ P n − 1 ′ ∪ x n − 1 x n − 1 ′ . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 3.2.2. x n − 1 ′ ∉ V ( P i ′ ) , where i ∈ { 1 , … , n − 1 } .

Let W = { w 1 , … , w n − 1 } . By Lemma 3.3, there are n − 1 internally disjoint paths Q 1 , … , Q n − 1 between x n − 1 ′ and W such that w i ∈ V ( Q i ) , where i ∈ { 1 , … , n − 1 } . It is necessary to consider the following situations.

When ∣ V ( Q i ) ∩ V ( P i ′ ) ∣ = n − 1 , that is, V ( Q i ) ∩ V ( P i ′ ) = W , where i ∈ { 1 , … , n − 1 } . We can obtain n − 2 internally disjoint S -trees T 1 , … , T n − 2 similar to that of Case 3.1. Let T n − 1 = P n − 1 ∪ P n − 1 ′ ∪ Q n − 1 ∪ x n − 1 x n − 1 ′ . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

When ∣ V ( Q i ) ∩ V ( P i ′ ) ∣ ≥ n , where i ∈ { 1 , … , n − 1 } . The proof is similar to that of Case 3.2.1.

Case 3.3. One of { x 1 ′ , … , x n − 1 ′ } belongs to H S n β , and one of { w 1 ′ , … , w n − 1 ′ } belongs to H S n α , where i ∈ { 1 , … , n − 1 } .

The proof is similar to that of Case 3.2.

Case 4. The vertices of S are distributed in three distinct copies of H S n .

Without loss of generality, suppose x , y ∈ V ( H S n α ) and z ∈ V ( H S n β ) , w ∈ V ( H S n γ ) where α , β , γ ∈ Γ n , α ≠ β ≠ γ . Since H S n α is isomorphic to S n , by Lemma 3.2, there exist n − 1 internally disjoint paths joining x and y in H S n α , say P 1 , … , P n − 1 . Let N ( x ) = { x 1 , … , x n − 1 } and x i ∈ N ( x ) ∩ V ( P i ) , by Lemma 2.4, the external neighbors of N ( x ) belong to different copies. Consequently, we just consider the case y ∉ N ( x ) and the proof of the case y ∈ N ( x ) is similar.

Case 4.1. There are three cross edges between N [ x ] and H S n β , γ .

Suppose x i ′ ∈ V ( H S n j i ) , x n − 2 ′ ∈ V ( H S n β ) , x n − 1 ′ ∈ V ( H S n γ ) , where j i ∈ Γ n − { α , β , γ } , i ∈ { 1 , … , n − 3 } , and let y ′ ∈ V ( H S n j n − 2 ) . By Lemma 2.4, x ′ ∈ V ( H S n β , γ ) .

When x ′ ∈ V ( H S n β ) , there is a tree connecting x i ′ , a i , b i in H S n j i , say T ^ i , and a i ′ ∈ V ( H S n β ) , b i ′ ∈ V ( H S n γ ) for i ∈ { 1 , … , n − 3 } . There is a path joining y ′ and b n − 2 in H S n j n − 2 , say P , and b n − 2 ′ ∈ V ( H S n γ ) . Let Z = { a 1 ′ , … , a n − 3 ′ , x n − 2 ′ , x ′ } , W = { b 1 ′ , … , b n − 2 ′ , x n − 1 ′ } . By Lemma 3.3, there are n − 1 internally disjoint paths between z and Z , say P 1 ′ , … , P n − 1 ′ , n − 1 internally disjoint paths between w and W , say P ^ 1 , … , P ^ n − 1 , a i ′ ∈ V ( P i ′ ) for i ∈ { 1 , … , n − 3 } , x n − 2 ′ ∈ V ( P n − 2 ′ ) , x ′ ∈ V ( P n − 1 ′ ) , b i ′ ∈ V ( P ^ i ) for i ∈ { 1 , … , n − 2 } , x n − 1 ′ ∈ V ( P ^ n − 1 ) . Let

T i = P i ∪ P i ′ ∪ P ^ i ∪ T ^ i ∪ x i x i ′ ∪ a i a i ′ ∪ b i b i ′ , i ∈ { 1 , … , n − 3 } T n − 2 = P n − 2 ∪ P n − 2 ′ ∪ P ^ n − 2 ∪ x n − 2 x n − 2 ′ ∪ y y ′ ∪ P ∪ b n − 2 b n − 2 ′ T n − 1 = P n − 1 ∪ P n − 1 ′ ∪ P ^ n − 1 ∪ x x ′ ∪ x n − 1 x n − 1 ′ .

Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 8).

$Figure 8 The illustration of x ′ ∈ V ( H S n β ) x^{\prime} \in V\left(H{S}_{n}^{\beta }) .$

Figure 8

The illustration of x ′ ∈ V ( H S n β ) .

When x ′ ∈ V ( H S n γ ) , the proof is similar to the case of x ′ ∈ V ( H S n β ) .

Case 4.2. There are two cross edges between N [ x ] and H S n β , γ .

Case 4.2.1. For two integers 1 ≤ j ≠ k ≤ n − 1 , x j ′ ∈ V ( H S n β ) , x ′ ∈ V ( H S n γ ) .

Without loss of generality, suppose x n − 2 ′ ∈ V ( H S n β ) , x n − 1 ′ ∈ V ( H S n γ ) , x i ′ ∈ V ( H S n j i ) , for j i ∈ Γ n − { α , β , γ } , i ∈ { 1 , … , n − 3 } .

Case 4.2.1.1. For some i ∈ { 1 , … , n − 3 } , x ′ ∈ V ( H S n j i ) .

$Figure 9 The illustration of Case 4.2.1.1.$

Figure 9

The illustration of Case 4.2.1.1.

Without loss of generality, suppose x ′ , x n − 3 ′ ∈ V ( H S n j n − 3 ) , and let y ′ ∈ V ( H S n j n − 2 ) . Let u , v ∈ V ( H S n j n − 3 ) , and u ′ ∈ V ( H S n j n − 1 ) , v ′ ∈ V ( H S n γ ) . Let A = { x n − 3 ′ , x ′ } , B = { u , v } . By Lemma 3.4, there are two disjoint paths between A and B , say R 1 and R 2 , R 1 is the path joining x n − 3 ′ and u , and R 2 is the path joining x ′ and