TY - JOUR
AU - Peng,, Sheng-Lung
AB - Abstract The generalized |$k$|-connectivity of a graph |$G$| is a parameter that can measure the reliability of a network |$G$| to connect any |$k$| vertices in |$G$|⁠, which is a generalization of traditional connectivity. Let |$S\subseteq V(G)$| and |$\kappa _{G}(S)$| denote the maximum number |$r$| of edge-disjoint trees |$T_{1}, T_{2}, \cdots , T_{r}$| in |$G$| such that |$V(T_{i})\bigcap V(T_{j})=S$| for any |$i, j \in \{1, 2, \cdots , r\}$| and |$i\neq j$|⁠. For an integer |$k$| with |$2\leq k\leq n$|⁠, the generalized |$k$|-connectivity of a graph |$G$| is defined as |$\kappa _{k}(G)= min\{\kappa _{G}(S)|S\subseteq V(G)$| and |$|S|=k\}$|⁠. In this paper, we introduce a family of regular graph |$G_{n}$| that can be constructed recursively and each vertex with exactly one outside neighbor. The generalized |$3$|-connectivity of the regular graph |$G_{n}$| is studied, which attains a previously proven upper bound on |$\kappa _{3}(G)$|⁠. As applications of the main result, the generalized |$3$|-connectivity of some important networks including some known results such as the alternating group network |$AN_{n}$|⁠, the star graph |$S_{n}$| and the pancake graphs |$P_{n}$| can be obtained directly. 1 Introduction An interconnection network is usually modelled by a connected graph |$G=(V, E)$|⁠, where nodes represent processors and edges represent communication links between processors. One mainly concerns the reliability and fault tolerance of an interconnection network. The connectivity|$\kappa (G)$| of a graph |$G$| is defined as the minimum number of vertices whose deletion results in a disconnected graph, which is an important parameter to evaluate the reliability and fault tolerance of a network. In addition, Whitney [1] defines the connectivity from local point of view. That is for any subset |$S=\{u, v\}\subseteq V(G)$|⁠, let |$\kappa _{G}(S)$| denote the maximum number of internally disjoint paths between |$u$| and |$v$| in |$G$|⁠. Then |$\kappa (G)=min\{\kappa _{G}(S)|S\subseteq V(G)$| and |$|S|=2\}.$| As a generalization of the traditional connectivity, Hager et al. [2] introduced the generalized|$k$|-connectivity in |$1985$|⁠. This parameter can measure the reliability of a network |$G$| to connect any |$k$| vertices in |$G$|⁠. Let |$S\subseteq V(G)$| and |$\kappa _{G}(S)$| denote the maximum number |$r$| of edge-disjoint trees |$T_{1}, T_{2}, \ldots , T_{r}$| in |$G$| such that |$V(T_{i})\bigcap V(T_{j})=S$| for any |$i, j \in \{1, 2, \ldots , r\}$| and |$i\neq j$|⁠. For an integer |$k$| with |$2\leq k\leq n$|⁠, the generalized|$k$|-connectivity of a graph |$G$| is defined as |$\kappa _{k}(G)= min\{\kappa _{G}(S)|S\subseteq V(G)$| and |$|S|=k\}$|⁠. The generalized |$2$|-connectivity is exactly the traditional connectivity, that is |$\kappa _{2}(G)=\kappa (G)$|⁠. Li et al. [3] derived that it is NP-complete for a general graph |$G$| to decide whether there are |$k$| internally disjoint trees connecting |$S$|⁠, where |$k$| is a fixed integer and |$S\subseteq V(G)$|⁠. Later, Li et al. [4] proved the upper bound on |$\kappa _{3}(G),$| and Li et al. [5] proved the lower bound on |$\kappa _{3}(G)$|⁠. In addition, there are some results [6] characterized the minimally |$2$|-connected graphs |$G$| with given generalized connectivity and some results [7, 8] studied the upper and lower bounds of the generalized |$3$|-connectivity of Cartesian product graphs and lexicographic products. Furthermore, there are some classes of graphs that known the exact value of |$\kappa _{k}(G)$| and most of them are about |$k=3$|⁠. For example the complete graphs [9]; the complete bipartite graphs [10]; the star graphs and bubble-sort graphs [11]; the Cayley graph generated by trees and cycles [12], the alternating group graphs and |$(n,k)$|-star graphs [13], the exchanged hypercubes [14] and the hypercubes [15]. In addition, there are many interesting results about the generalized connectivity, and one can refer to [16]. In this paper, the generalized |$3$|-connectivity of the regular graph |$G_{n}$| is studied, and it attains a previously proven upper bound on |$\kappa _{3}(G)$|⁠. As applications of the main result, the generalized |$3$|-connectivity of some important networks including some known results such as the alternating group network |$AN_{n}$|⁠, the star graph |$S_{n}$| and the pancake graphs |$P_{n}$| can be obtained directly. The paper is organized as follows. In Section 2, some notations and definitions are given. In Section 3, the generalized |$3$|-connectivity of the regular graph |$G_{n}$| is determined. In Section 4, as applications of the main result, the generalized |$3$|-connectivity of some networks can be obtained directly. In Section 5, the paper is concluded. 2 Preliminary In this section, some terminologies and notations will be introduced. For terminologies and notations undefined here, one can refer to [17]. 2.1 Terminology and notation Let |$G=(V, E)$| be a simple, undirected graph. Let |$|V(G)|$| be the size of vertex set and |$|E(G)|$| be the size of edge set. For a vertex |$v$| in |$G$|⁠, we denote by |$N_{G}(v)$| the neighborhood of the vertex |$v$| in |$G$| and |$N_{G}[v]=N_{G}(v)\bigcup \{v\}$|⁠. Let |$U \subseteq V(G)$|⁠, denote |$N_G(U)=\bigcup \limits _{v\in U}N_{G}(v)-U$|⁠. Let |$d_{G}(v)$| denote the degree of the vertex |$v$| in |$G$| and |$\delta (G)$| denote the minimum degree of the graph |$G$|⁠. The subgraph induced by |$V^{\prime }$| in |$G$|⁠, denoted by |$G[V^{\prime }]$|⁠, is a graph whose vertex set is |$V^{\prime }$| and the edge set is the set of all the edges of |$G$| with both ends in |$V^{\prime }$|⁠. A graph is said to be |$k$|-regular if for any vertex |$v$| of |$G$|⁠, |$d_{G}(v)=k$|⁠. Two |$xy$|- paths |$P$| and |$Q$| in |$G$| are internally disjoint if they have no common internal vertices, that is |$V(P)\bigcap V(Q)=\{x, y\}$|⁠. Let |$Y\subseteq V(G)$| and |$X\subset V(G)\setminus Y$|⁠, the |$(X, Y)$|-paths is a family of internally disjoint paths starting at a vertex |$x\in X$|⁠, ending at a vertex |$y\in Y$| and whose internal vertices belong to neither |$X$| nor |$Y$|⁠. If |$X=\{x\}$|⁠, the |$(X, Y)$|-paths is a family of internal disjoint paths whose starting vertex is |$x$| and the terminal vertices are distinct in |$Y$|⁠, which is referred to as a |$k$|-fan from |$x$| to |$Y$|⁠. Let |$[n]=\{1,2,\cdots ,n\}$|⁠. Let |$\Gamma $| be a finite group and |$S$| be a subset of |$\Gamma $|⁠, where the identity of the group does not belong to |$S$|⁠. The Cayley graph |$Cay(\Gamma , S)$| is a digraph with vertex set |$\Gamma $| and arc set |$\{(g, g.s)| g\in \Gamma , s\in S\}$|⁠. If |$S= S^{-1}$|⁠, then |$Cay(\Gamma , S)$| is an undirected graph, where |$S^{-1}=\{s^{-1}|s \in S\}$|⁠. Following, we will introduce the definition of the regular graph |$G_{n}$|⁠, which is constructed recursively. Definition 2.1. Let |$n, a, r \geq 1$| be integers. An |$n$|-th regular graph, say |$G_{n}$|⁠, can be constructed recursively as follows: (1) The |$1$|-th regular graph, say |$G_{1}$|⁠, is a |$r$|-regular and |$r$|-connected graph with order |$a$|⁠. (2) For |$n\geq 2$|⁠, the |$n$|-th regular graph, say |$G_{n}$|⁠, is a regular graph that consists of |$p_{n}$| copies of |$G_{n-1}$|⁠, say |$G_{n-1}^{1}, G_{n-1}^{2}, \cdots , G_{n-1}^{p_{n}}$|⁠. (3) For each |$u\in V(G_{n-1}^{i})$|⁠, it has exactly one neighbor outside |$G_{n-1}^{i}$|⁠, which is called the outside neighbor of |$u$|⁠. (4) There are same number of independent edges between |$G_{n-1}^{i}$| and |$G_{n-1}^{j}$| for |$i\neq j$|⁠. That is there are |$\frac{ap_{2}p_{3}\cdots p_{n-1}}{p_{n}-1}$| edges between |$G_{n-1}^{i}$| and |$G_{n-1}^{j}$| for |$i\neq j$| and |$i,j\in [p_{n}]$|⁠. (5) |$\frac{ap_{2}p_{3}\cdots p_{n-1}}{p_{n}-1}> 2(r+n-3)$|⁠. (6) |$G_{n}$| is |$(r+n-1)$|-regular and |$(r+n-1)$|-connected. For convenience, let |$G_{n}=G_{n-1}^{1}\bigoplus G_{n-1}^{2}\bigoplus \cdots \bigoplus G_{n-1}^{p_{n}}$| of order |$N$|⁠. By the definition of |$G_{n}$|⁠, |$|G_{n}|=N=ap_{2}p_{3}\cdots p_{n}$|⁠. 2.2 Some networks can be regarded as the graph |$G_{n}$| 2.2.1 The alternating group network |$AN_{n}$| The alternating group network, denoted by |$AN_{n}$|⁠, was introduced by Y. Ji [18] in |$1998$|⁠. It is defined as follows. Definition 2.2. Let |$A_{n}$| be the alternating group of order |$n$| for |$n\geq 3,$| and let |$S=\{(123), (132),\\ (12)(3i)|4\leq i\leq n\}$|⁠. The alternating group network, denoted by |$AN_{n}$|⁠, is defined as the Cayley graph |$Cay(A_{n}, S)$|⁠. By the definition of |$AN_{n}$|⁠, it is a |$(n-1)$|-regular graph with |$\frac{n!}{2}$| vertices and |$\frac{n!(n-1)}{4}$| edges. Let |$A_{n}^{i}$| be the subset of |$A_{n}$| that consists of all even permutations with element |$i$| in the rightmost position and let |$AN_{n-1}^{i}$| be the subgraph of |$AN_{n}$| induced by |$A_{n}^{i}$| for |$i\in [n]$|⁠. Then |$AN_{n-1}^{i}$| is isomorphic to |$AN_{n-1}$|⁠, and we call |$AN_{n-1}^{i}$| a copy of |$AN_{n-1}$|⁠. Thus, |$AN_{n}$| can be decomposed into |$n$| copies of |$AN_{n-1}$|⁠, namely, |$AN_{n-1}^{1}, AN_{n-1}^{2}, \cdots , AN_{n-1}^{n}$|⁠. For convenience, let |$AN_{n}=AN_{n-1}^{1}\bigoplus AN_{n-1}^{2}\bigoplus \cdots \bigoplus AN_{n-1}^{n}$|⁠, where |$\bigoplus $| just denotes the corresponding decomposition of |$AN_{n}$|⁠. For each vertex |$u\in V(AN_{n-1}^{i})$|⁠, it has |$n-2$| neighbors in |$AN_{n-1}^{i}$| and one neighbor outside |$AN_{n-1}^{i}$|⁠, which is called the outside neighbor of |$u$|⁠. Clearly, |$AN_{3}$| is a triangle and |$AN_{4}$| consists of four copies of |$AN_{3}$|⁠. The following result is about the properties of |$AN_{n}$|⁠. Lemma 2.1. ([19]) Let |$AN_{n}=AN_{n-1}^{1}\bigoplus AN_{n-1}^{2}\bigoplus \ldots \bigoplus AN_{n-1}^{n}$| for |$n\geq 3$|⁠. Then the following results hold. (1) For any vertex |$u$| of |$AN_{n-1}^{i}$|⁠, it has exactly one outside neighbor. (2) For any copy |$AN_{n-1}^{i}$|⁠, no two vertices in |$AN_{n-1}^{i}$| have a common outside neighbor. In addition, |$|N(AN_{n-1}^{i})|=\frac{(n-1)!}{2}$| and |$|N(AN_{n-1}^{i})\bigcap V(AN_{n-1}^{j})|=\frac{(n-2)!}{2}$| for |$i\neq j$|⁠. The following lemma is about the connectivity of |$AN_{n}$|⁠. Lemma 2.2. ([18]) |$\kappa (AN_{n})=n-1$| for |$n\geq 3$|⁠. Remark 2.1. By Definition 2.2, Lemma 2.1 and Lemma 2.2, |$AN_{n}$| can be regarded as the special regular graph |$G_{n-4}$| with |$G_{1}=AN_{5}$|⁠, |$a=60$|⁠, |$r=4$|⁠, |$p_{n-4}=n$| and |$N=ap_{2}p_{3}\cdots p_{n-4}=\frac{n!}{2}$|⁠. 2.2.2 The star graph |$S_{n}$| The star graph, denoted by |$S_{n}$|⁠, was introduced by S. Akers [20]. It is defined as follows. Definition 2.3. Let |$Sym(n)$| be the symmetric group on |$[n]$|⁠. The star graph, denoted by |$S_{n}$|⁠, is defined as the Cayley graph |$Cay(Sym(n), S)$| with |$S=\{(1i)|1<i\leq n\}$|⁠. By the definition of |$S_{n}$|⁠, it is a |$(n-1)$|-regular graph with |$n!$| vertices. For an integer |$i\in [n]$|⁠, let |$S_{n-1}^{i}$| denote the subgraph induced by the vertex set |$\{p_{1}p_{2}\cdots p_{n-1}i\}$|⁠, where |$p_{1}p_{2}\cdots p_{n-1}$| ranges over all the permutations of |$\{1,2,\cdots , i-1,i+1,\cdots ,n\}$|⁠. Then |$S_{n-1}^{i}$| is isomorphic to |$S_{n-1}$| for each |$i\in [n],$| and we call such an |$S_{n-1}^{i}$| a copy of |$S_{n-1}$|⁠. One can see that |$S_{n}$| can be decomposed into |$n$| copies of |$S_{n-1}$|⁠, say |$S_{n-1}^{1}, S_{n-1}^{2}, \cdots , S_{n-1}^{n}$|⁠. For convenience, let |$S_{n}=S_{n-1}^{1}\bigoplus S_{n-1}^{2}\bigoplus \cdots \bigoplus S_{n-1}^{n}$|⁠, where |$\bigoplus $| just denotes the corresponding decomposition of |$S_{n}$|⁠. For each vertex |$u\in V(S_{n-1}^{i})$|⁠, it has |$n-2$| neighbors in |$S_{n-1}^{i}$| and one neighbor outside |$S_{n-1}^{i}$|⁠, which is called the outside neighbor of |$u$|⁠. The following result is about the properties of |$S_{n}$|⁠. Lemma 2.3. ([11]) Let |$S_{n}=S_{n-1}^{1}\bigoplus S_{n-1}^{2}\bigoplus \ldots \bigoplus S_{n-1}^{n}$| for |$n\geq 3$|⁠. Then the following results hold. (1) For any vertex |$u$| of |$S_{n-1}^{i}$|⁠, it has exactly one outside neighbor. (2) For any copy |$S_{n-1}^{i}$|⁠, no two vertices in |$S_{n-1}^{i}$| have a common outside neighbor. In addition, |$|N(S_{n-1}^{i})|=(n-1)!$| and |$|N(S_{n-1} ^{i})\bigcap V(S_{n-1}^{j})|=(n-2)!$| for |$i\neq j$|⁠. The following lemma is about the connectivity of |$S_{n}$|⁠. Lemma 2.4. ([21, 22]) |$\kappa (S_{n})=n-1$| for |$n\geq 3$|⁠. Remark 2.2. By Definition 2.3, Lemma 2.3 and Lemma 2.4, |$S_{n}$| can be regarded as the special regular graph |$G_{n-3}$| with |$G_{1}=S_{4}$|⁠, |$a=24$|⁠, |$r=3$|⁠, |$p_{n-3}=n$| and |$N=ap_{2}p_{3}\cdots p_{n-3}= n!$|⁠. 2.2.3 The pancake graph |$P_{n}$| The pancake graph, denoted by |$P_{n}$|⁠, was introduced by S. Akers [23] in |$1989$|⁠. It is defined as follows. Definition 2.4. The |$n$|-dimensional pancake graph |$P_{n}$| has |$n!$| vertices, each of which has the form |$x=x_{1}x_{2}\cdots x_{n}$|⁠, where |$x_{i}\in [n]$| for |$i\in [n]$| and |$x_{i}\neq x_{j}$| for |$i \neq j$| and |$i,j\in [n]$|⁠. Two vertices |$x=x_{1}x_{2}\cdots x_{n}$| and |$y=y_{1}y_{2}\cdots y_{n}$| are adjacent if and only if there exists an integer |$j$| with |$2\leq j\leq n$|⁠, such that |$y_{1}=x_{j}, y_{2}=x_{j-1},\cdots , y_{j}=x_{1}$| and |$x_{l}=y_{l}$| for all |$l\in [n]\setminus [j]$|⁠. By the definition of |$P_{n}$|⁠, it is a |$(n-1)$|-regular graph with |$n!$| vertices. For an integer |$i\in [n]$|⁠, let |$P_{n-1}^{i}$| denote the subgraph induced by the vertex set |$\{p_{1}p_{2}\cdots p_{n-1}i\}$|⁠, where |$p_{1}p_{2}\cdots p_{n-1}$| ranges over all the permutations of |$\{1,2,\cdots , i-1,i+1,\cdots ,n\}$|⁠. Then |$P_{n-1}^{i}$| is isomorphic to |$P_{n-1}$| for each |$i\in [n]$| and we call such an |$P_{n-1}^{i}$| a copy of |$P_{n-1}$|⁠. One can see that |$P_{n}$| can be decomposed into |$n$| copies of |$P_{n-1}$|⁠, say |$P_{n-1}^{1}, P_{n-1}^{2}, \cdots , P_{n-1}^{n}$|⁠. For convenience, let |$P_{n}=P_{n-1}^{1}\bigoplus P_{n-1}^{2}\bigoplus \cdots \bigoplus P_{n-1}^{n}$|⁠, where |$\bigoplus $| just denotes the corresponding decomposition of |$P_{n}$|⁠. For each vertex |$u\in V(P_{n-1}^{i})$|⁠, it has |$n-2$| neighbors in |$P_{n-1}^{i}$| and one neighbor outside |$P_{n-1}^{i}$|⁠, which is called the outside neighbor of |$u$|⁠. By the definition of the pancake graphs, the following lemma can be obtained directly. Lemma 2.5. Let |$P_{n}=P_{n-1}^{1}\bigoplus P_{n-1}^{2}\bigoplus \ldots \bigoplus P_{n-1}^{n}$| for |$n\geq 3$|⁠. Then the following results hold. (1) For any vertex |$u$| of |$P_{n-1}^{i}$|⁠, it has exactly one outside neighbor. (2) For any copy |$P_{n-1}^{i}$|⁠, no two vertices in |$P_{n-1}^{i}$| have a common outside neighbor. In addition, |$|N(P_{n-1}^{i})|=(n-1)!$| and |$|N(P_{n-1} ^{i})\bigcap V(P_{n-1}^{j})|=(n-2)!$| for |$i\neq j$|⁠. The following lemma is about the connectivity of |$P_{n}$|⁠. Lemma 2.6. ([23]) |$\kappa (P_{n})=n-1$| for |$n\geq 3$|⁠. Remark 2.3. By Definition 2.4, Lemma 2.5 and Lemma 2.6, |$P_{n}$| can be regarded as the special regular graph |$G_{n-3}$| with |$G_{1}=P_{4}$|⁠, |$a=24$|⁠, |$r=3$|⁠, |$p_{n-3}=n$| and |$N=ap_{2}p_{3}\cdots p_{n-3}= n!$|⁠. 3 The generalized |$3$|-connectivity of the regular graph |$G_{n}$| In this section, we will study the generalized |$3$|-connectivity of the regular graph |$G_{n}$|⁠. To prove the main result, the following lemmas are useful. In [4], the following upper bound on |$\kappa _{3}(G)$| is obtained. Lemma 3.1. ([4]) Let |$G$| be a connected graph and |$\delta $| be its minimum degree. Then |$\kappa _{3}(G)\leq \delta $|⁠. Further, if there are two adjacent vertices of degree |$\delta $|⁠, then |$\kappa _{3}(G)\leq \delta -1$|⁠. In [4], Li et al. studied the relationship between |$\kappa (G)$| and |$\kappa _{3}(G)$|⁠. Lemma 3.2. ([4]) Let |$G$| be a connected graph with |$n$| vertices. If |$\kappa (G)=4k+r,$| where |$k$| and |$r$| are two integers with |$k\geq 0$| and |$r\in \{0, 1, 2, 3\},$| then |$\kappa _{3}(G)\geq 3k+\lceil \frac{r}{2}\rceil $|⁠. Moreover, the lower bound is sharp. The following two lemmas are well-known results about |$k$|-connected graphs. Lemma 3.3. ([17]) Let |$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$|⁠. Lemma 3.4. ([17]) Let |$G=(V, E)$| be a |$k$|-connected graph, let |$x$| be a vertex of |$G$| and let |$Y\subseteq V\setminus \{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$| internally disjoint |$(x, Y)$|-paths whose terminal vertices are distinct in |$Y$|⁠. Following, we will study the connectivity of the subgraph of |$G_{n}$|⁠, which is useful to the main result. Lemma 3.5. Let |$G_{n}$| be the |$n$|-th regular graph which is the same as Definition |$1$|⁠. Let |$G_{n}=G_{n-1}^{1}\bigoplus G_{n-1}^{2}\bigoplus \ldots \bigoplus G_{n-1}^{p_{n}}$| and |$H=G_{n-1}^{i_{1}}\bigoplus G_{n-1}^{i_{2}}\bigoplus \ldots \bigoplus G_{n-1}^{i_{l}}$| be the induced subgraph of |$G_{n}$| on |$\bigcup _{m=1}^{l}V(G_{n-1}^{i_{m}})$| for |$2\leq l \leq p_{n}-1$|⁠, then |$\kappa (H)\geq r+n-2$|⁠. Proof. Without loss of generality, let |$H=G_{n-1}^{1}\bigoplus G_{n-1} ^{2}\bigoplus \ldots \bigoplus G_{n-1}^{l}$|⁠. To prove the result, we just need to show that there are |$r+n-2$| internally disjoint paths for any two distinct vertices of |$H$|⁠. Let |$v_{1}, v_{2}\in V(H)$| and |$v_{1}\neq v_{2}$|⁠, then the following two cases are considered. Case 1. |$v_{1}$| and |$v_{2}$| belong to the same copy of |$G_{n-1}$|⁠. Without loss of generality, let |$v_{1}, v_{2}\in V(G_{n-1}^{1})$|⁠. By (6) of Definition 2.1, |$\kappa (G_{n-1}^{1})=r+n-2$|⁠. Then there are |$r+n-2$| internally disjoint paths between |$v_{1}$| and |$v_{2}$| in |$G_{n-1}^{1}$| and the result is desired. Case 2. |$v_{1}$| and |$v_{2}$| belong to two different copies of |$G_{n-1}$|⁠. Without loss of generality, let |$v_{1}\in V(G_{n-1}^{1})$| and |$v_{2}\in V(G_{n-1}^{2})$|⁠. Select |$r+n-2$| vertices |$u_{1}, u_{2}, u_{3},\cdots ,$||$u_{r+n-2}$| from |$G_{n-1}^{1}$| such that the outside neighbor |$u_{i}^{\prime }$| of |$u_{i}$| belongs to |$G_{n-1}^{2}$| for each |$i\in \{1,2,\cdots , r+n-2\}$|⁠. By (5) of Definition 2.1, this can be done. Let |$S=\{u_{1}, u_{2}, u_{3},\cdots , u_{r+n-2}\}$| and |$S^{\prime }=\{u_{1}^{\prime }, u_{2}^{\prime }, u_{3}^{\prime },\cdots , u_{r+n-2}^{\prime }\}$|⁠. By (6) of Definition 2.1, |$\kappa (G_{n-1}^{1})=\kappa (G_{n-1}^{2})=r+n-2$|⁠. By Lemma 3.4, there exists a family of |$r+n-2$| internally disjoint |$(v_{1}, S)$|-paths |$P_{1}, P_{2}, \cdots , P_{r+n-2}$| such that the terminal vertex of |$P_{i}$| is |$u_{i}$|⁠. Note that at most one of them is a singleton. If so, say |$P_{1}$|⁠. Similarly, there exists a family of |$r+n-2$| internally disjoint |$(v_{2}, S^{\prime })$| paths |$P_{1}^{\prime }, P_{2}^{\prime }, \cdots , P_{r+n-2}^{\prime }$| such that the terminal vertex of |$P_{i}^{\prime }$| is |$u_{i}^{\prime }$|⁠. Note that at most one of them is a singleton. If so, say |$P_{1}^{\prime }$|⁠. Let |$\widehat{P_{i}}=P_{i}\bigcup u_{i}u_{i}^{\prime }\bigcup P_{i}^{\prime }$| for each |$i\in \{1,2,\cdots ,r+n-2\}$|⁠, then |$r+n-2$| disjoint paths between |$v_{1}$| and |$v_{2}$| are obtained in |$H$|⁠. Finally, we will study the generalized |$3$|-connectivity of |$G_{n}$|⁠, which is the main result. Theorem 3.6. Let |$G_{n}$| be the |$n$|-th regular graph which is the same as Definition |$1$| and let |$G_{n}=G_{n-1}^{1}\bigoplus G_{n-1}^{2}\bigoplus \ldots \bigoplus G_{n-1}^{p_{n}}$|⁠. Suppose all of the following conditions hold: (1) The outside neighbors of vertices in |$N_{G_{n-1}^{i}}[v]$| belong to different copies of |$G_{n-1}$|⁠. (2) |$\kappa _{3}(G_{1})=\delta (G_{1})-1=r-1$|⁠. (3) |$r+n=p_{n}$|⁠. Then, |$\kappa _{3}(G_{n})=r+n-2$|⁠. Proof. By (6) of Definition 2.1, |$G_{n}$| is |$(r+n-1)$|-regular. By Lemma 3.1, |$\kappa _{3}(G_{n})\leq \delta -1=r+n-2$|⁠. To prove the result, we just need to show |$\kappa _{3}(G_{n})\geq r+n-2$|⁠. We prove the result by induction on |$n$|⁠. By |$(2)$|⁠, |$\kappa _{3}(G_{1})=r-1$|⁠. Thus, the result holds for |$n=1$|⁠. Next we assume that |$n\geq 2$|⁠. Let |$G_{n}=G_{n-1}^{1}\bigoplus G_{n-1}^{2}$||$\bigoplus \ldots \bigoplus G_{n-1}^{p_{n}}$| and |$v_{1}, v_{2}, v_{3}$| be any three distinct vertices of |$G_{n}$|⁠. For convenience, let |$S=\{v_{1}, v_{2}, v_{3}\}$|⁠. We prove the result by considering the following three cases. Case 1. |$v_{1}, v_{2}$| and |$v_{3}$| belong to the same copy of |$G_{n-1}$|⁠. Without loss of generality, let |$v_{1}, v_{2}, v_{3} \in V(G_{n-1}^{1})$|⁠. By the inductive hypothesis, |$\kappa _{3}(G_{n-1}^{1})\geq r+n-3$|⁠. That is there are |$r+n-3$| internally disjoint trees |$T_{1}, T_{2}\cdots , T_{r+n-3}$| connecting |$S$| in |$G_{n-1}^{1}$|⁠. Let |$v_{1}^{\prime }, v_{2}^{\prime }$| and |$v_{3}^{\prime }$| be the outside neighbors of |$v_{1}, v_{2}$| and |$v_{3}$|⁠, respectively. By Lemma 3.5, |$G_{n}[V(G_{n}\setminus G_{n-1}^{1})]$| is connected, then there is a tree |$T$| connecting |$v_{1}^{\prime }, v_{2}^{\prime }$| and |$v_{3}^{\prime }$| in |$G_{n}[V(G_{n}\setminus G_{n-1}^{1})]$|⁠. Let |$T_{r+n-2}=T\bigcup v_{1}v_{1}^{\prime }\bigcup v_{2}v_{2}^{\prime }\bigcup v_{3}v_{3}^{\prime }$|⁠, then it is a tree connecting |$S$| and |$V(T_{r+n-2})\bigcap V(G_{n-1}^{1})=S$|⁠. Hence, there are |$r+n-2$| internally disjoint trees connecting |$S$| in |$G_{n}$| and the result is desired. Case 2. |$v_{1}, v_{2}$| and |$v_{3}$| belong to two different copies of |$G_{n-1}$|⁠. Without loss of generality, let |$v_{1}, v_{2}\in V(G_{n-1}^{1})$| and |$v_{3}\in V(G_{n-1}^{2})$|⁠. By (6) of Definition 2.1, |$\kappa (G_{n-1}^{1})=r+n-2$|⁠. Hence, there are |$r+n-2$| internally disjoint paths |$P_{1}, P_{2}, \ldots , P_{r+n-2}$| between |$v_{1}$| and |$v_{2}$| in |$G_{n-1}^{1}$|⁠. Choose |$r+n-2$| distinct vertices |$x_{1}, x_{2}, \ldots , x_{r+n-2}$| from |$P_{1}, P_{2}, \ldots , P_{r+n-2}$| such that |$x_{i}\in V(P_{i})$| for each |$i\in \{1,2,\cdots ,r+n-2\}$|⁠. Note that at most one of these paths has length |$1$|⁠. If so, say |$P_{1}$| and let |$x_{1}=v_{1}$|⁠. Let |$x_{i}^{\prime }$| be the outside neighbor of |$x_{i}$| for each |$i\in \{1,2,\cdots ,r+n-2\}$|⁠. Let |$X^{\prime }=\{x_{1}^{\prime }, x_{2}^{\prime }, \cdots , x_{r+n-2}^{\prime }\}$|⁠. By (3) of Definition 2.1, |$|X^{\prime }|=r+n-2$|⁠. By Lemma 3.5, |$G_{n}\setminus G_{n-1}^{1}$| is |$r+n-2$| connected. By Lemma 3.4, there are |$r+n-2$| internally disjoint paths |$P_{1}^{\prime }, P_{2}^{\prime }, \ldots , P_{r+n-2}^{\prime }$| between |$v_{3}$| and |$X^{\prime }$| such that the terminal vertex of |$P_{i}^{\prime }$| is |$x_{i}^{\prime }$| for |$1\leq i\leq r+n-2$|⁠. Note that at most one of these paths has length |$0$|⁠. If so, let |$P_{r+n-2}^{\prime }=v_{3}$|⁠. Let |$T_{i}=P_{i}\bigcup x_{i}x_{i}^{\prime }\bigcup P_{i}^{\prime }$| for each |$i\in \{1,2,\cdots ,r+n-2\}$|⁠, then |$r+n-2$| internally disjoint trees connecting |$S$| in |$G_{n}$| are obtained. Case 3. |$v_{1}, v_{2}$| and |$v_{3}$| belong to three different copies of |$G_{n-1}$|⁠, respectively. Without loss of generality, let |$v_{1} \in V(G_{n-1}^{1}), v_{2}\in V(G_{n-1}^{2})$| and |$v_{3}\in V(G_{n-1}^{3})$|⁠. Let |$N_{G_{n-1}^{i}}[v_{i}]=N_{G_{n-1}^{i}}(v_{i})\bigcup \{v_{i}\}$| for |$i=1,2,3$|⁠, then |$|N_{G_{n-1}^{i}}[v_{i}]|=r+n-1$|⁠. By (3), |$r+n-1=p_{n}-1$|⁠. Then by |$(1)$|⁠, there is a vertex in |$N_{G_{n-1}^{i}}[v_{i}]$|⁠, say |$u_{i}^{j}$|⁠, such that the outside neighbor |$(u_{i}^{j})^{\prime }$| of |$u_{i}^{j}$| belongs to |$G_{n-1}^{j}$| for each |$i\in \{1,2,3\}$| and |$4\leq j\leq p_{n}$|⁠. As |$G_{n-1}^{j}$| is connected, we can find a tree |$\widehat{T}_{j}$| connecting |$(u_{1}^{j})^{\prime }, (u_{2}^{j})^{\prime }$| and |$(u_{3}^{j})^{\prime }$| for each |$j\in \{4,5,\cdots , p_{n}\}$|⁠. Let |$T_{j}=\widehat{T}_{j}\bigcup u_{1}^{j}(u_{1}^{j})^{\prime }\bigcup u_{2}^{j}(u_{2}^{j})^{\prime }\bigcup u_{3}^{j}(u_{3}^{j})^{\prime }\bigcup v_{1}u_{1}^{j}\bigcup v_{2}u_{2}^{j}\bigcup v_{3}u_{3}^{j}$|⁠, then |$r+n-3$| internally disjoint trees connecting |$S$| are obtained. Let |$\widehat{G}_{n-1}^{i}=G_{n-1}^{i}-(\{u_{i}^{4}, u_{i}^{5},\cdots , u_{i}^{r+n}\}\setminus \{v_{i}\})$| for |$1\leq i\leq 3$|⁠. Then there are at most |$r+n-3$| vertices deleted from |$G_{n-1}^{i}$| for each |$i\in \{1,2,3\}$|⁠. As |$G_{n-1}^{i}$| is |$r+n-2$| connected, |$\widehat{G}_{n-1}^{i}$| is still connected. By (4) of Definition 2.1, the number of edges between |$G_{n-1}^{i}$| and |$G_{n-1}^{j}$| is |$\frac{ap_{2}p_{3}\cdots p_{n-1}}{p_{n}-1}> 2(r+n-3)$| for |$i,j \in \{1,2,3\}$| and |$i\neq j$|⁠. Thus, |$G_{n}[\bigcup _{i=1}^{3}V(\widehat{G}_{n}^{i})]$| is connected and there is a tree |$T_{r+n-2}$| connecting |$S$|⁠. Hence, there exist |$r+n-2$| internally disjoint trees connecting |$S$| in |$G_{n}$| and the result is desired. 4 Applications In this section, we will demonstrate the usefulness of our main result. As applications, the generalized |$3$|-connectivity of the following networks can be obtained directly. 4.1 Application to the alternating group network |$AN_{n}$| Lemma 4.1. ([19]) Let |$AN_{n}$| be the |$n$|-dimensional alternating group network, let |$AN_{n}=AN_{n-1}^{1}\bigoplus AN_{n-1}^{2}\bigoplus \ldots \bigoplus AN_{n-1}^{n}$| for |$n\geq 3$| and let |$u\in V(AN_{n-1}^{i})$| for |$1\leq i\leq n$|⁠. Then the outside neighbors of vertices in |$N_{AN_{n-1}^{i}}[u]$| belong to different |$AN_{n-1}^{j}s$| for |$j\neq i$|⁠. Corollary 4.2. |$\kappa _{3}(AN_{n})=n-2$| for |$n\geq 5$|⁠. Proof. By Lemma 4.1, condition |$(1)$| of Theorem |$1$| holds. Note that |$AN_{n}$| can be regarded as the regular graph |$G_{n-4}$| for |$n\geq 5$|⁠, |$G_{1}=AN_{5}, r=4$| and |$p_{n-4}=n$|⁠. By Lemma 2.2, |$\kappa (AN_{5})=4$|⁠. By Lemma 3.1, |$\kappa _{3}(AN_{5})\leq 3$|⁠. By Lemma 3.2, |$\kappa _{3}(AN_{5})\geq 3$|⁠. Thus, |$\kappa _{3}(AN_{5})=3$| and condition |$(2)$| of Theorem |$1$| holds. For |$AN_{n}$| and |$n\geq 5$|⁠, recall that |$r=4$| and |$p_{n-4}=n$|⁠, then |$r+n=p_{n}$| and condition |$(3)$| of Theorem |$1$| holds. Thus, |$\kappa _{3}(AN_{n})=n-2$| for |$n\geq 5$|⁠. 4.2 Application to the star graph |$S_{n}$| Lemma 4.3. ([11]) Let |$S_{n}$| be the |$n$|-dimensional star graph, let |$S_{n}=S_{n-1}^{1}\bigoplus S_{n-1}^{2}\bigoplus \ldots \bigoplus S_{n-1}^{n}$| for |$n\geq 3$| and let |$u\in V(S_{n-1}^{i})$| for |$1\leq i\leq n$|⁠. Then the outside neighbors of vertices in |$N_{S_{n-1}^{i}}[u]$| belong to different |$S_{n-1}^{j}s$| for |$j\neq i$|⁠. Corollary 4.4. |$\kappa _{3}(S_{n})=n-2$| for |$n\geq 4$|⁠. Proof. By Lemma 4.3, condition |$(1)$| of Theorem |$1$| holds. Note that |$S_{n}$| can be regarded as the regular graph |$G_{n-3}$| for |$n\geq 4$|⁠, |$G_{1}=S_{4}, r=3$| and |$p_{n-3}=n$|⁠. By Lemma 2.4, |$\kappa (S_{4})=3$|⁠. By Lemma 3.1, |$\kappa _{3}(S_{4})\leq 2$|⁠. By Lemma 3.2, |$\kappa _{3}(S_{4})\geq 2$|⁠. Thus, |$\kappa _{3}(S_{4})=2$| and condition |$(2)$| of Theorem |$1$| holds. For |$S_{n}$| and |$n\geq 4$|⁠, recall that |$r=3$| and |$p_{n-3}=n$|⁠, then |$r+n=p_{n}$| and condition |$(3)$| of Theorem |$1$| holds. Thus, |$\kappa _{3}(S_{n})=n-2$| for |$n\geq 4$|⁠. 4.3 Application to the pancake graph |$P_{n}$| Lemma 4.5. Let |$P_{n}$| be the |$n$|-dimensional pancake graph, let |$P_{n}=P_{n-1}^{1}\bigoplus P_{n-1}^{2}\bigoplus \cdots \bigoplus P_{n-1}^{n}$| and let |$v\in V(P_{n-1}^{i})$| for |$1\leq i\leq n$| and |$n\geq 3$|⁠. Then the outside neighbors of vertices in |$N_{P_{n-1}^{i}}[v]$| belong to different copies of |$P_{n-1}^{j}s$| for |$j\neq i$|⁠. Proof. Without loss of generality, let |$v=v_{1}v_{2}\cdots v_{n}\in V(P_{n-1}^{i})$|⁠. Then |$N_{P_{n-1}^{i}}[v]=\{v_{1}v_{2}\cdots v_{n}, v_{2}v_{1}v_{3}\cdots v_{n}, \cdots , v_{n-1}v_{n-2}\cdots v_{2}v_{1}v_{n}\}$|⁠. Let the outside neighbor set of vertices in |$N_{P_{n-1}^{i}}[v]$| be |$N_{P_{n-1}^{i}}^{\prime }[v]$|⁠. Then |$N_{P_{n-1}^{i}}^{\prime }[v]=\{v_{n}\cdots v_{2}v_{1}, v_{n}\cdots v_{3}v_{1}v_{2}, \cdots , v_{n}v_{1}v_{2}\cdots v_{n-2}v_{n-1}\}$|⁠. As |$v_{i}\neq v_{j}$| for each |$i,j\in [n]$| and |$i\neq j$|⁠, then the vertices in |$N_{P_{n-1}^{i}}^{\prime }[v]$| belong to different copies of |$P_{n-1}$| and the result holds. Corollary 4.6. |$\kappa _{3}(P_{n})=n-2$| for |$n\geq 4$|⁠. Proof. By Lemma 4.5, condition |$(1)$| of Theorem |$1$| holds. Note that |$P_{n}$| can be regarded as the regular graph |$G_{n-3}$| for |$n\geq 4$|⁠, |$G_{1}=P_{4}, r=3$| and |$p_{n-3}=n$|⁠. By Lemma 2.6, |$\kappa (P_{4})=3$|⁠. By Lemma 3.1, |$\kappa _{3}(P_{4})\leq 2$|⁠. By Lemma 3.2, |$\kappa _{3}(P_{4})\geq 2$|⁠. Thus, |$\kappa _{3}(P_{4})=2$| and condition |$(2)$| of Theorem |$1$| holds. For |$P_{n}$| and |$n\geq 4$|⁠, recall that |$r=3$| and |$p_{n-3}=n$|⁠, then |$r+n=p_{n}$| and condition |$(3)$| of Theorem |$1$| holds. Thus, |$\kappa _{3}(P_{n})=n-2$| for |$n\geq 4$|⁠. 5 Concluding remarks The generalized |$k$|-connectivity is a generalization of traditional connectivity. In this paper, we introduce a family of regular graph |$G_{n}$| which can be constructed recursively and each vertex contains exactly one outside neighbor. The regular graph |$G_{n}$| contains many important networks. In this paper, the generalized |$3$|-connectivity of the regular graph |$G_{n}$| is studied. As applications of the main result, the generalized |$3$|-connectivity of many important networks including some known results such as the alternating group network |$AN_{n}$|⁠, the star graph |$S_{n}$| and the pancake graphs |$P_{n}$| can be obtained directly. We believe that the generalized |$3$|-connectivity of many other networks can be obtained by this main result. In the future, we would like to study the generalized |$3$|-connectivity of the regular graph |$G_{n}$| with at least two outside neighbors at each vertex. Acknowledgments The authors express their sincere thanks to the editor and the anonymous referees for their valuable suggestions which greatly improved the original manuscript. Funding The Fundamental Research Funds for the Central Universities (No. 2019YJS192). References 1 Whitney , H. ( 1932 ) Congruent graphs and the connectivity of graphs . Am. J. Math. , 54 , 150 – 168 . Google Scholar Crossref Search ADS WorldCat 2 Hager , M. ( 1985 ) Pendant tree-connectivity . J. Combin. Theory Ser , 38 , 179 – 189 . Google Scholar Crossref Search ADS WorldCat 3 Li , S. and Li , X. ( 2012 ) Note on the hardness of generalized connectivity . J. Comb. Optim. , 24 , 389 – 396 . Google Scholar Crossref Search ADS WorldCat 4 Li , S. , Li , X. and Zhou , W. ( 2010 ) Sharp bounds for the generalized connectivity |${\kappa }\_3(G)$| . Discret. Math. , 310 , 2147 – 2163 . Google Scholar Crossref Search ADS WorldCat 5 Li , H. , Li , X. , Mao , Y. and Sun , Y. ( 2014 ) Note on the generalized connectivity . Ars Comb. , 114 , 193 – 202 . WorldCat 6 Li , S. , Li , W. , Shi , Y. and Sun , H. ( 2017 ) On the minimally 2-connected graphs with generalized connectivity |${\kappa }\_3=2$| . J. Comb. Optim. , 34 , 141 – 164 . Google Scholar Crossref Search ADS WorldCat 7 Li , H. , Li , X. and Sun , Y. ( 2012 ) The generalized 3-connectivity of Cartesian product graphs . Discret. Math. , 14 , 43 – 54 . WorldCat 8 Li , H. , Ma , Y. , Yang , W. and Wang , Y. ( 2017 ) The generalized 3-connectivity of graph products . Appl. Math Comput. , 295 , 77 – 83 . Google Scholar Crossref Search ADS WorldCat 9 Chartrand , G. , Okamoto , F. and Zhang , P. ( 2010 ) Rainbow trees in graphs and generalized connectivity . Networks , 55 , 360 – 367 . WorldCat 10 Li , S. , Li , W. and Li , X. ( 2012 ) The generalized connectivity of complete bipartite graphs . Ars Comb. , 104 , 65 – 79 . WorldCat 11 Li , S. , Tu , J. and Yu , C. ( 2016 ) The generalized 3-connectivity of star graphs and bubble-sort graphs . Appl. Math Comput. , 274 , 41 – 46 . WorldCat 12 Li , S. , Shi , Y. and Tu , J. ( 2017 ) The generalized 3-connectivity of Cayley graphs on symmetric groups generated by trees and cycles . Graph. Combinator. , 33 , 1195 – 1209 . Google Scholar Crossref Search ADS WorldCat 13 Zhao , S. and Hao , R. ( 2018 ) The generalized connectivity of alternating group graphs and |$\left (n,k\right )$|-star graphs . Discrete Appl. Math. , 251 , 310 – 321 . Google Scholar Crossref Search ADS WorldCat 14 Zhao , S. and Hao , R. ( 2019 ) The generalized 4-connectivity of exchanged hypercubes . Appl. Math Comput. , 347 , 342 – 353 . WorldCat 15 Lin , S. and Zhang , Q. ( 2017 ) The generalized 4-connectivity of hypercubes . Discrete Appl. Math. , 220 , 60 – 67 . Google Scholar Crossref Search ADS WorldCat 16 Li , X. and Mao , Y. ( 2016 ) Generalized Connectivity of Graphs . Springer Briefs in Mathematics . Springer , Switzerland . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC 17 Bondy , J.A. and Murty , U.S.R. ( 2008 ) Graph Theory . Springer , Berlin . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC 18 Ji , Y. ( 1998 ) A class of Cayley networks based on the alternating groups . Adv. Math. (Chinese) , 4 , 361 – 362 . WorldCat 19 Chang , J.M. , Pai , K.J. , Yang , J.S. and Wu , R.Y. ( 2018 ) Two Kinds of Generalized 3-Connectivities of Alternating Group Networks . International Workshop on Frontiers in Algorithmics . Springer, Cham, pp. 3 – 14 . 20 Akers , S.B. , Harel , D. and Krishnamurthy , B. ( 1994 ) The star graph: an attractive alternative to the n-cube . Interconnection networks for high-performance parallel computers . IEEE Computer Society Press, pp. 145 – 152 . 21 Cheng , E. , Liptcutek , L. and Shawash , N. ( 2008 ) Orienting Cayley graphs generated by transposition trees . Comput. Math. Appl. , 55 , 2662 – 2672 . Google Scholar Crossref Search ADS WorldCat 22 Yang , W.H. , Li , H.Z. and Meng , J.X. ( 2010 ) Conditional connectivity of Cayley graphs generated by transposition trees . Inform. Process. Lett. , 110 , 1027 – 1030 . Google Scholar Crossref Search ADS WorldCat 23 Akers , S.B. and Krishnamurthy , B. ( 1989 ) A group-theoretic model for symmetric interconnection networks . IEEE Trans. Comput. , 38 , 555 – 566 . Google Scholar Crossref Search ADS WorldCat © The Authors 2019. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Reliability Assessment of Some Regular Networks
JF - The Computer Journal
DO - 10.1093/comjnl/bxz116
DA - 2002-10-01
UR - https://www.deepdyve.com/lp/oxford-university-press/reliability-assessment-of-some-regular-networks-9fUIbeN09q
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -