The Generalized Three-Connectivity of Two Kinds of Cayley Graphs

The Generalized Three-Connectivity of Two Kinds of Cayley Graphs Abstract Let S⊆V(G) and κG(S) denote the maximum number r of edge-disjoint trees T1,T2,…,Tr in G such that V(Ti)∩V(Tj)=S for any i,j∈{1,2,…,r} and i≠j. For an integer k with 2≤k≤n, the generalized k-connectivity of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}. The generalized k-connectivity is a generalization of traditional connectivity. In this paper, we focus on the Cayley graph generated by complete graphs and the Cayley graph generated by wheel graphs, denoted by CTn and WGn, respectively. We study the generalized 3-connectivity of the two kinds of graphs and show that κ3(CTn)=n(n−1)2−1 and κ3(WGn)=2n−3 for n≥3. 1. INTRODUCTION It is well known that an interconnection network is usually modeled by a connected graph G=(V,E), where nodes represent processors and edges represent communication links between processors. For an interconnection network, one major concern is the fault tolerance. The connectivity is one of the important parameters to evaluate the reliability and fault tolerance of a network. The connectivity κ(G) of a graph G is defined as the minimum number of vertices whose deletion results in a disconnected graph. Whitney [1] provides another definition of connectivity. For any subset S={u,v}⊆V(G), let κG(S) denote the maximum number of internally disjoint paths between u and v in G. Then κ(G)=min{κG(S)∣S⊆V(G) and ∣S∣=2}. As a generalization of the traditional connectivity, the generalized k-connectivity was introduced by Chartrand et al. [2] in 1984. It is a parameter that can measure the reliability of a network G to connect any k vertices in G. Let S⊆V(G) and κG(S) denote the maximum number r of edge-disjoint trees T1,T2,…,Tr in G such that V(Ti)∩V(Tj)=S for any i,j∈{1,2,…,r} and i≠j. For an integer k with 2≤k≤n, the generalized k-connectivity of a graph G is defined as κk(G)=min{κG(S)∣S⊆V(G) and ∣S∣=k}. If κk(G)=m, then there are m internally disjoint trees in G with each of them connecting the vertices of S. The generalized 2-connectivity is exactly the traditional connectivity. In [3], Li 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⊆V(G). There are some results about the upper and lower bounds of the generalized connectivity. For example, Li et al. [4] gave the sharp upper and lower bounds of κ3(G) for general graphs G and Li et al. [5] gave a sharp upper bound of generalized k-connectivity. In addition, there are known results on the generalized k-connectivity for some classes of graphs and most of them are about k=3. For example, Chartrand et al. [6] studied the generalized connectivity of complete graphs; Li et al. [7] first studied the generalized 3-connectivity of Cartesian product graphs, then Li et al. [8] studied the generalized 3-connectivity of graph products; Li et al. [9] studied the generalized connectivity of complete bipartite graphs and Lin and Zhang [10] studied the generalized 4-connectivity of hypercubes. As the Cayley graph has some attractive properties when used as an interconnection network, Li et al. [11] studied the generalized 3-connectivity of the star graphs and bubble-sort graphs and Li et al. [12] studied the generalized 3-connectivity of the Cayley graph generated by trees and cycles. In this paper, we study the generalized 3-connectivity κ3(G) of a regular graph G with given properties. As applications, the generalized 3-connectivity of the Cayley graphs generated by complete graphs and the Cayley graph generated by wheel graphs, denoted by CTn and WGn, respectively, are determined. We show that κ3(CTn)=n(n−1)2−1 and κ3(WGn)=2n−3 for n≥3. The paper is organized as follows. In Section 2, some notation and definitions are given. In Section 3, the generalized 3-connectivity of regular graphs with given properties is determined. As applications, in Section 4, the generalized 3-connectivity of the Cayley graph CTn generated by complete graphs and the Cayley graph WGn generated by wheel graphs are determined. In Section 5, the paper is concluded. 2. PRELIMINARY Let G=(V,E) be a simple, undirected graph. Let ∣V(G)∣ be the size of the vertex set and ∣E(G)∣ be the size of the edge set. The subgraph induced by V′ in G, denoted by G[V′], is a graph whose vertex set is V′ and the edge set is the set of all the edges of G with both ends in V′. For a vertex v in G, we denote by NG(v) the neighborhood of the vertex v in G. For a vertex set U⊆V(G), the neighborhood of U in G is defined as NG(U)=∪v∈UNG(v)−U and ∣NG(U)∣ denotes the size of the neighborhood of U in G. Let dG(v) denote the number of edges incident with v and δ(G) denote the minimum degree of the graph G. A graph is said to be k-regular if for any vertex v of G, dG(v)=k. A matching in a graph is a set of pairwise non-adjacent edges. If M is a matching, the two ends of each edge of M are said to be matched under M, and each vertex incident with an edge of M is said to be covered by M. A perfect matching is one which covers every vertex of the graph. Two xy-paths P and Q in G are internally disjoint if they have no common internal vertices, that is V(P)∩V(Q)={x,y}. Let Y⊆V(G) and X⊂V(G)⧹Y. The set of (X,Y)-paths is a family of internally disjoint paths starting at a vertex x∈X, ending at a vertex y∈Y and whose internal vertices belong neither to X nor Y. If X={x}, then the set of (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. For terminologies and notations not defined here we follow [13]. Let Γ be a finite group and S be a subset of Γ, where the identity of the group does not belong to S. The Cayley graph Cay(Γ,S) is a digraph with vertex set Γ and arc set {(g,g.s)∣g∈Γ,s∈S}. If S=S−1, then Cay(Γ,S) is an undirected graph, where S−1={s−1∣s∈S}. In this paper, we consider the Cayley graph Cay(Sym(n),T), where Sym(n) is the symmetric group on {1,2,…,n} and T is a set of transpositions of Sym(n). Let G(T) be the graph on n vertices {1,2,…,n} such that there is an edge ij in G(T) if and only if transposition (ij)∈T [14]. The graph G(T) is the transposition generating graph of Cay(Sym(n),T). If G(T) is a tree, then Cay(Sym(n),T) is denoted by Γn. It was introduced in [15] that Γn can be decomposed into smaller Cayley graphs of the same type by considering the leaves of G(T). It is well known that if G(T)≅K1,n−1, then Cay(Sym(n),T) is an n-dimensional star graph and denoted by Sn; if G(T)≅Pn, then Cay(Sym(n),T) is an n-dimensional bubble-sort graph and denoted by Bn. If G(T) is a cycle with length n, then Cay(Sym(n),T) is denoted by MBn, which is the modified bubble-sort graph. Similarly, if G(T) is a cycle with length n−1, then Cay(Sym(n),T) is denoted by MBn−1. In [12], Li, Shi and Tu determined the generalized 3-connectivity of Cayley graphs on Symmetric Groups generated by trees and cycles. If G(T) is a complete graph with n vertices, then Cay(Sym(n),T) is denoted by CTn. For convenience, we call CTn an n-dimensional complete-transposition graph. If G(T) is a wheel graph Wn, which is a graph with n vertices (n≥4), obtained by connecting a single vertex to all vertices of an (n−1)-cycle, then Cay(Sym(n),T) is denoted by WGn. The Cayley graph WGn was introduced by Tu, Zhou and Su in [16], where it is called the n-dimensional wheel-transposition graph. It is known that CT4=WG4. The Cayley graphs CT3 and CT4 or (WG4) are depicted in Figs 1 and 2, respectively. Figure 1. View largeDownload slide The complete-transposition graph CT3. Figure 1. View largeDownload slide The complete-transposition graph CT3. Figure 2. View largeDownload slide The illustration of WG4 (or CT4). Figure 2. View largeDownload slide The illustration of WG4 (or CT4). First, we describe the hierarchical structure of the complete-transposition graph CTn. The transposition generating graph G(T) of CTn is a complete graph Kn. If deleting n in G(T), then G(T′)=G(T)−n is a complete graph on vertex set {1,2,…,n−1}. The graph CTn can be decomposed into n copies of CTn−1 as follows: for each i∈{1,2,…,n}, let CTn−1i be the subgraph of CTn induced by vertex set {(p1,p2,…,pn−1,i)∣(p1,p2,…,pn−1) ranges over all permutations of {1,2,…,n}⧹{i}}. Then CTn−1i≅Cay(Sym(n−1),T′) is a Cayley graph generated by a complete graph Kn−1, where it is a copy of CTn−1 for 1≤i≤n. By the definition of CTn, one has that CTn can be decomposed into n copies, which is denoted by CTn−11,CTn−12,…,CTn−1n, of CTn−1. For i∈{1,2,…,n}, each vertex u∈V(CTn−1i) has n−1 neighbors outside CTn−1i, called outside neighbors of u. For simple representation, this decomposition is denoted by CTn=CTn−11⊕CTn−12⊕⋯⊕CTn−1n, where ⊕ denotes the corresponding decomposition of CTn. By the definition of the complete-transposition graph CTn, it is a n(n−1)2-regular graph with n! vertices. The Cayley graph CTn has many properties which are desirable in an interconnection network such as hierarchical structure (hierarchical structure means that CTn can be decomposed into copies of CTn−1), high connectivity n(n−1)2 and small diameter n−1 [17, 18]. For the Cayley graph, vertex symmetry makes it possible to use the same routing protocols and communication schemes at all nodes; hierarchical structure facilitates recursive constructions; and high fault tolerance implies robustness, among others. Next, we describe the hierarchical structure of the wheel-transposition graph WGn. If the center vertex n of Wn is deleted, G(T′)=G(T)−n is a cycle on the vertex set {1,2,3,…,n−1}. Hence, the graph WGn can be decomposed into n copies of MBn−1 (where MBn−1 is the (n−1)-dimension modified bubble-sort graph) as follows: for each i∈{1,2,…,n}, let MBn−1i be the subgraph of WGn induced by the vertex set {(p1,p2,…,pn−1,i)∣(p1,p2,…,pn−1) ranges over all permutations of {1,2,…,n}⧹{i}}. Then MBn−1i≅Cay(Sym(n−1),T′) is a Cayley graph generated by a cycle with n−1 vertices, which is a copy of MBn−1. In this case, WGn can be decomposed into n copies of MBn−1, which are denoted by MBn−11,MBn−12,…,MBn−1n, and for any i∈{1,2,…,n}, each vertex u∈V(MBn−1i) has n−1 neighbors outside MBn−1i, called outside neighbors of u. The decomposition is denoted by WGn=MBn−11⊕MBn−12⊕⋯⊕MBn−1n, where ⊕ denotes the corresponding decomposition of WGn. By the definition of WGn, it is a 2(n−1)-regular graph with n! vertices. 3. THE GENERALIZED 3-CONNECTIVITY OF REGULAR GRAPHS WITH GIVEN PROPERTIES Lemma 1 (Lemma 3.1 [4]) Let Gbe a connected graph and δbe its minimum degree. Then κ3(G)≤δ. Further, if there are two adjacent vertices of degree δ, then κ3(G)≤δ−1. Lemma 2 (Proposition 9.4 [13]) Let G=(V,E)be a k-connected graph, and let Xand Ybe subsets of V(G)of cardinality at least k. Then there exists a family of kpairwise disjoint (X,Y)-paths in G. Lemma 3 (Proposition 9.5 [13]) Let G=(V,E)be a k-connected graph, let xbe a vertex of G, and let Y⊆V⧹{x}be a set of at least kvertices of G. Then there exists a k-fan in Gfrom xto Y, that is, there exists a family of kinternally vertex-disjoint (x,Y)-paths whose terminal vertices are distinct in Y. Lemma 4 Let Gmbe a k-regular graph with κ(Gm)=kfor each m∈{1,2,…,n}. Let Hnbe a graph obtained from G1,G2,…,Gnby adding some edges, where Hnis denoted by Hn=G1⊕G2⊕⋯⊕Gnfor n≥3. Let H=Gi⊕Gjbe the induced subgraph of Hnon V(Gi)∪V(Gj)for distinct i,j∈{1,2,…,n}. If there is exactly a perfect matching between Giand Gjin Hand ∣V(Gi)∣=∣V(Gj)∣≥k+1, then the following results hold: For any vertex vof H, dH(v)=k+1. Let x∈V(H), Y⊆V(H)⧹{x}with ∣Y∣=k+1such that ∣Y∩V(Gi)∣≤kand ∣Y∩V(Gj)∣≤k. Then there exists a (k+1)-fan in Hfrom xto Y, that is, there exists a family of k+1internally disjoint paths from xto Ysuch that the terminal vertices are distinct in Y. There exists a family of k+1internally disjoint paths joining any two distinct vertices of H. Proof Without loss of generality, let H=G1⊕G2. (1) As Gm is a k-regular graph for each m∈{1,2,…,n}, then G1 and G2 are both k-regular. As there is exactly a perfect matching between G1 and G2 in H, then dH(v)=k+1 holds for any vertex v∈V(H). (2) Let x∈V(H) and Y⊆V(H)⧹{x} with ∣Y∣=k+1 such that ∣Y∩V(Gi)∣≤k for i=1,2. Without loss of generality, let x∈V(G1) and let the neighbor of x in G2 be x′. Let Y∩V(Gi)=Ai and ∣Ai∣=ki for i=1,2. Then ki≤k and k1+k2=k+1. As k1≤k, then k2≥1. Let M1⊆V(G1) such that ∣M1∣=k2−1 and M1∩(A1∪{x})=ø. This can be done as ∣V(G1)∣≥k+1. Let M=A1∪M1, then ∣M∣=k. As κ(Gm)=k for each m∈{1,2,…,n}, then κ(G1)=κ(G2)=k. By Lemma 3, there is a F1 from x to A1 and a fan F2 from x to M1 in G1, respectively, so that the only common vertex to F1 and F2 is x. As there is exactly a perfect matching between G1 and G2 in H, then any vertex v∈V(G1) has exactly one neighbor in G2. Let M2={y′∣y′ is the neighbor of y in G2 for each y∈M1} and E2={yy′∈E(H)∣y∈M1 and y′∈M2}. Let M′=M2∪{x′}. Then ∣M′∣=k2. As κ(G2)=k≥k2, by Lemma 2, there exists a family of k2 pairwise disjoint (M′,A2)-paths F2′ in G2. It is easy to see that combining the two fans F1,F2, the edge set E2, the edge xx′ and the set of paths F2′, we can obtain a (k+1)-fan in H from x to Y. (3) Let x,y∈V(H) be any two distinct vertices of H. We prove the result by considering the following two cases. Case 1. x and y are non-adjacent. Without loss of generality, let x∈V(G1) and Y=NH(y). Then x∉Y, otherwise x and y are adjacent. By Lemma 4(1), ∣Y∣=k+1. If y∈V(G1), then ∣Y∩V(G1)∣=k and ∣Y∩V(G2)∣=1. If y∈V(G2), then ∣Y∩V(G1)∣=1 and ∣Y∩V(G2)∣=k. By (2), there exists a family of k+1 internally disjoint paths in H from x to Y. Combining the edges from y to Y, we obtain k+1 internally disjoint paths between x and y in H. Case 2. x and y are adjacent. Let Y=(NH(y)∪{y})⧹{x}; then ∣Y∣=k+1. Without loss of generality, let y∈V(G1). If x∈V(G1), then ∣Y∩V(G1)∣=k and ∣Y∩V(G2)∣=1. By (2), there exists a family of k+1 internally disjoint paths in H from x to Y in H. Combining the edges from y to Y⧹{y} and the edge xy, we can obtain k+1 internally disjoint paths from x to y in H. If x∈V(G2), then ∣(Y⧹{y})∩V(G1)∣=k. Let yi∈NG1(y) and yi′ be the neighbor of yi in G2 for 1≤i≤k. Let Y′={y1′,y2′,…,yk′}. Then ∣Y′∣=k. As there is a perfect matching between G1 and G2, then x∉Y′. As κ(G2)=k, by Lemma 3, there exists a k-fan from x to Y′ in G2. That is, there are k internally disjoint paths P1,P2,…,Pk from x to Y′ in G2. Let Ti=yyi∪yiyi′∪Pi for each i∈{1,2,…,k}. Combining the edge xy and Tis for 1≤i≤k, then k+1 internally disjoint paths from x to y are obtained in H. □ Theorem 1 Let Hn=G1⊕G2⊕⋯⊕Gnbe the same as that in Lemma4, where Gmis a k-regular graph with κ(Gm)=kand ∣V(Gm)∣≥k+1for each m∈{1,2,…,n}. Assume that there is exactly a perfect matching between Giand Gjin Hn[V(Gi)∪V(Gj)]for distinct i,j∈{1,2,…,n}. If κ3(Gm)=k−1for each m∈{1,2,…,n}, then κ3(Hn)=n+k−2for n≥3. Proof Let Hn=G1⊕G2⊕⋯⊕Gn. As there is exactly a perfect matching between Gi and Gj in H, where H=Hn[V(Gi)∪V(Gj)] for distinct i,j∈{1,2,…,n}, then for any vertex v of Gm, it has exactly (n−1) neighbors outside Gm for each m∈{1,2,…,n}. As Gm is k-regular for each m∈{1,2,…,n}, then δ=dHn(v)=k+n−1 for any v∈V(Hn). By Lemma 1, κ3(Hn)≤δ−1=k+n−2. To prove the result, we just need to show that κ3(Hn)≥k+n−2. Let v1,v2 and v3 be any three vertices of Hn. For convenience, let S={v1,v2,v3}. We prove the result by considering the following three cases. Case 1. v1,v2 and v3 belong to the same Gi for some i∈{1,2,…,n}. Without loss of generality, assume that v1,v2 and v3 belong to G1. As κ3(Gm)=k−1 for each m∈{1,2,…,n}, then there exist k−1 internally disjoint trees T1,T2,…,Tk−1 connecting S in G1. As there is exactly a perfect matching between Gi and Gj in Hn[V(Gi)∪V(Gj)] for distinct i,j∈{1,2,…,n}, then any vertex v of G1 has exactly one outside neighbor in G2,G3,…, Gn, respectively. Let v1j,v2j and v3j be the neighbor of v1,v2 and v3 in Gj, respectively, where 2≤j≤n. As Gj is connected, then there is a tree Tj^ connecting the three vertices v1j,v2j and v3j in Gj. Let Aj=Tj^∪v1v1j∪v2v2j∪v3v3j, then Aj is a tree connecting S for each j with 2≤j≤n. Thus, Ajs for 2≤j≤n and Tis for 1≤i≤k−1 are n+k−2 internally disjoint trees with each of them connecting S in Hn. Thus, in this case, κ3(Hn)≥n+k−2. Case 2. v1,v2 and v3 belong to two different Gis. Without loss of generality, assume that v1,v2∈V(G1) and v3∈V(G2). As κ(Gm)=k for each m∈{1,2,…,n}, then κ(G1)=κ(G2)=k and there exist k pairwise disjoint paths P1,P2,…,Pk between v1 and v2 in G1. Choose k distinct vertices x1,x2,…,xk from P1,P2,…,Pk such that xi∈V(Pi) for each i∈{1,2,…,k}. As there is exactly a perfect matching between Gi and Gj in Hn[V(Gi)∪V(Gj)] for distinct i,j∈{1,2,…,n}, then there is exactly a perfect matching between G1 and G2 in Hn[V(G1)∪V(G2)]. Let xi′ be the neighbor of xi in G2 and X′={x1′,x2′,…,xk′}. As κ(G2)=k. By Lemma 3, there exist k internally disjoint paths P1′,P2′,…,Pk′ between v3 and X′. If xi′=v3 for some 1≤i≤k, then let Pi′=v3. Let Ti=Pi∪xixi′∪Pi′ for each i∈{1,2,…,k}. Then Tis for 1≤i≤k are k pairwise disjoint trees with each of them connecting S. Similar as Case 1, let v1j,v2j and v3j be the neighbors of v1,v2 and v3 in Gj for 3≤j≤n, respectively. As Gj is connected, we can find a tree Tj^ connecting the three vertices v1j,v2j and v3j in Gj. Let Bj=Tj^∪v1v1j∪v2v2j∪v3v3j, then Bj is a tree connecting S for 3≤j≤n. Hence, Bjs for 3≤j≤n and Tis for 1≤j≤k are n+k−2 internally disjoint trees in Hn with each of them connecting S. Thus, in this case, κ3(Hn)≥n+k−2. Case 3. v1,v2 and v3 belong to three different Gis. Without loss of generality, assume that v1∈V(G1),v2∈V(G2) and v3∈V(G3). Let H=G1⊕G2. By Lemma 4(3), there exist (k+1) internally disjoint paths P1,P2,…,Pk+1 between v1 and v2 in H. Let v3′ be the neighbor of v3 in G2 (it is possible that v3′=v2). As H is connected, then there exists a path P˜ connecting v3′ and v1 in H. Let t1 be the first vertex of the path P˜ which is in ∪i=1k+1Pi. Without loss of generality, assume t1∈V(Pk+1). Choose k distinct vertices x1,x2,…,xk from P1,P2,…,Pk such that xi∈V(Pi), where 1≤i≤k. Let xi′ be the neighbor of xi in G3, where 1≤i≤k. As there is exactly a perfect matching between G2 and G3 in Hn[V(G2)∪V(G3)], this can be done. Let X′={x1′,x2′,…,xk′}. As κ(G3)=k, by Lemma 3, there exist k internally disjoint paths P1′,P2′,…,Pk′ in G3 from v3 to X′. If v3∈X′, assume v3=x1′, then let P1′=v3. Let Ti=Pi∪xixi′∪Pi′ for 1≤i≤k and Tk+1=P˜∪v3v3′∪Pk+1 (if v3′=v2, then Tk+1=v3v2∪Pk+1). Hence, Tis for 1≤i≤k+1 are k+1 internally disjoint trees with each of them connecting S. Similar as Case 1, let v1j,v2j and v3j be the outside neighbors of v1,v2 and v3 in Gj for 4≤j≤n, respectively. As Gj is connected, we can find a tree Tj^ connecting the three vertices v1j,v2j and v3j in Gj. Let Cj=Tj^∪v1v1j∪v2v2j∪v3v3j, then Cj is a tree connecting S for each j∈{4,…,n}. Hence, Cjs for 4≤j≤n and Tis for 1≤i≤k+1 are n+k−2 internally disjoint trees in Hn with each of them connecting S. Thus, in this case, κ3(Hn)≥n+k−2. In conclusion, κ3(Hn)≥n+k−2. The proof is complete. □ 4. THE GENERALIZED 3-CONNECTIVITY OF TWO KINDS OF CAYLEY GRAPHS As applications of Theorem 1, in this section, we will study the generalized 3-connectivity of Cayley graphs generated by complete graphs and wheel graphs. 4.1. The generalized 3-connectivity of the Cayley graph generated by complete graphs To prove the main result, we need the following lemmas. Lemma 5 (Lemma 2.5 [19]) Let CTn=CTn−11⊕CTn−12⊕⋯⊕CTn−1nfor n≥3. Then the following results hold: For any vertex uof CTn, the outside neighbors of uare in different copies. For any copy CTn−1i, no two distinct vertices in CTn−1ihave a common outside neighbor, and thus ∣N(V(CTn−1i))∣=(n−1)(n−1)!. For any i≠j, ∣N(V(CTn−1i))∩V(CTn−1j)∣=(n−1)!. Lemma 6 (Theorem 2.3 [4]). Let Gbe a connected graph with nvertices. If κ(G)=4k+r,where kand rare two integers with k≥0and r∈{0,1,2,3}, then κ3(G)≥3k+⌈r2⌉. Moreover, the lower bound is sharp. In [20], Lakshmivarahan et al. have shown that CTn is both vertex and edge transitive. It has been shown that every edge transitive graph is maximally connected [13]. Thus, we have the following lemma. Lemma 7 Let CTnbe an n-dimensional complete-transposition graph, then κ(CTn)=n(n−1)2. Theorem 2 Let CTnbe an n-dimensional complete-transposition graph, then κ3(CTn)=n(n−1)2−1for n≥3. Proof Since CTn=CTn−11⊕CTn−12⊕⋯⊕CTn−1n for n≥3. We prove the result by induction on n. For n=3, note that CT3 is 3-regular. By Lemma 6, κ3(CT3)≥⌈32⌉=2. By Lemma 1, κ3(CT3)≤δ−1=2. Hence, κ3(CT3)=2 and the result holds. Next, suppose that n≥4 and the result holds for n−1. That is, κ3(CTn−1)=(n−1)(n−2)2−1. By CTn−1i≅CTn−1, one has that κ3(CTn−1i)=(n−1)(n−2)2−1 for each i∈{1,2,…,n}. By Lemma 7, κ(CTn−1i)=(n−1)(n−2)2 for each i∈{1,2,…,n}. By (2) and (3) of Lemma 5, there is a perfect matching between CTn−1i and CTn−1j in CTn[V(CTn−1i)∪V(CTn−1j)] for distinct i,j∈{1,2,…,n}. As ∣V(CTn−1i)∣=(n−1)!≥(n−1)(n−2)2+1, by Theorem 1 for k=(n−1)(n−2)2, one has that κ3(CTn)=n+k−2=n(n−1)2−1. □ 4.2. The generalized 3-connectivity of the Cayley graph generated by wheel graphs In this section, we will study the generalized 3-connectivity of the Cayley graph generated by wheel graphs. To prove the main result, we need the following Lemmas. Lemma 8 (Lemma 3.4 [16]) Let WGn=MBn−11⊕MBn−12⊕⋯⊕MBn−1nfor n≥3, where MBn−1iis a copy of MBn−1for 1≤i≤n. Then the following result holds: For any vertex uof MBn−1i, the outside neighbors of uare in different MBn−1js. For any copy MBn−1i, no vertices in MBn−1ihave a common outside neighbor, and thus ∣N(V(MBn−1i))∣=(n−1)(n−1)!. For any i≠j, ∣N(V(MBn−1i)∩V(MBn−1j)∣=(n−1)!. Lemma 9 (Lemma 12 [12]) κ(MBn)=n for n≥3. Lemma 10 (Theorem 13 [12]) κ3(MBn)=n−1for any integer n≥3. Theorem 3 Let WGnbe an n-dimensional wheel-transposition graph, then κ3(WGn)=2n−3for n≥3. Proof Since WGn=MBn−11⊕MBn−12⊕⋯⊕MBn−1n, where MBn−1i is a copy of MBn−1 for each i∈{1,2,…,n}. By Lemma 9, κ(MBn−1i)=n−1 for each i∈{1,2,…,n}. By (2) and (3) of Lemma 8, there is exactly a perfect matching between MBn−1i and MBn−1j in WGn[V(MBn−1i)∪V(MBn−1j)] for any i,j∈{1,2,…,n}. By Lemma 10, κ3(MBn−1i)=n−2 for each i∈{1,2,…,n}. Thus, all conditions of Theorem 1 hold. By Theorem 1, one has that κ3(WGn)=2n−3. □ 5. CONCLUDING REMARKS In this paper, the generalized 3-connectivity of regular graphs with given properties are obtained. As the Cayley graph has some attractive properties when used as an interconnection network, as applications of the Theorem 1, we focus on the Cayley graph CTn generated by complete graphs and the Cayley graph WGn generated by wheel graphs. We give the exact values of the generalized 3-connectivity of these two classes of graphs as direct results of Theorem 1. We believe that Theorem 1 can be applied to some other networks. As we all know, almost all known results of generalized k-connectivity are about k=3. We are interested in this topic for k≥4 and we would like to study in this direction to show the corresponding results for these two kinds of Cayley graphs and other classical networks such as arrangement graphs, balanced graphs, (n,k)-star networks and so on. FUNDING This work was supported by the National Natural Science Foundation of China (No. 11731002), the Fundamental Research Funds for the Central Universities (Nos. 2016JBM071 and 2016JBZ012). ACKNOWLEDGEMENTS The authors express their sincere thanks to the editor and the anonymous referees for their valuable suggestions which greatly improved the original manuscript. REFERENCES 1 Whitney , H. ( 1932 ) Congruent graphs and the connectivity of graphs . Am. J. Math , 54 , 150 – 168 . Google Scholar CrossRef Search ADS 2 Chartrand , G. , Kapoor , S. , Lesniak , L. and Lick , D. ( 1984 ) Generalized connectivity in graphs . Bull. Bombay Math. Colloq , 2 , 1 – 6 . 3 Li , S. and Li , X. 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Google Scholar CrossRef Search ADS Author notes Handling editor: Iain Stewart © The British Computer Society 2018. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Computer Journal Oxford University Press

The Generalized Three-Connectivity of Two Kinds of Cayley Graphs

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Abstract Let S⊆V(G) and κG(S) denote the maximum number r of edge-disjoint trees T1,T2,…,Tr in G such that V(Ti)∩V(Tj)=S for any i,j∈{1,2,…,r} and i≠j. For an integer k with 2≤k≤n, the generalized k-connectivity of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}. The generalized k-connectivity is a generalization of traditional connectivity. In this paper, we focus on the Cayley graph generated by complete graphs and the Cayley graph generated by wheel graphs, denoted by CTn and WGn, respectively. We study the generalized 3-connectivity of the two kinds of graphs and show that κ3(CTn)=n(n−1)2−1 and κ3(WGn)=2n−3 for n≥3. 1. INTRODUCTION It is well known that an interconnection network is usually modeled by a connected graph G=(V,E), where nodes represent processors and edges represent communication links between processors. For an interconnection network, one major concern is the fault tolerance. The connectivity is one of the important parameters to evaluate the reliability and fault tolerance of a network. The connectivity κ(G) of a graph G is defined as the minimum number of vertices whose deletion results in a disconnected graph. Whitney [1] provides another definition of connectivity. For any subset S={u,v}⊆V(G), let κG(S) denote the maximum number of internally disjoint paths between u and v in G. Then κ(G)=min{κG(S)∣S⊆V(G) and ∣S∣=2}. As a generalization of the traditional connectivity, the generalized k-connectivity was introduced by Chartrand et al. [2] in 1984. It is a parameter that can measure the reliability of a network G to connect any k vertices in G. Let S⊆V(G) and κG(S) denote the maximum number r of edge-disjoint trees T1,T2,…,Tr in G such that V(Ti)∩V(Tj)=S for any i,j∈{1,2,…,r} and i≠j. For an integer k with 2≤k≤n, the generalized k-connectivity of a graph G is defined as κk(G)=min{κG(S)∣S⊆V(G) and ∣S∣=k}. If κk(G)=m, then there are m internally disjoint trees in G with each of them connecting the vertices of S. The generalized 2-connectivity is exactly the traditional connectivity. In [3], Li 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⊆V(G). There are some results about the upper and lower bounds of the generalized connectivity. For example, Li et al. [4] gave the sharp upper and lower bounds of κ3(G) for general graphs G and Li et al. [5] gave a sharp upper bound of generalized k-connectivity. In addition, there are known results on the generalized k-connectivity for some classes of graphs and most of them are about k=3. For example, Chartrand et al. [6] studied the generalized connectivity of complete graphs; Li et al. [7] first studied the generalized 3-connectivity of Cartesian product graphs, then Li et al. [8] studied the generalized 3-connectivity of graph products; Li et al. [9] studied the generalized connectivity of complete bipartite graphs and Lin and Zhang [10] studied the generalized 4-connectivity of hypercubes. As the Cayley graph has some attractive properties when used as an interconnection network, Li et al. [11] studied the generalized 3-connectivity of the star graphs and bubble-sort graphs and Li et al. [12] studied the generalized 3-connectivity of the Cayley graph generated by trees and cycles. In this paper, we study the generalized 3-connectivity κ3(G) of a regular graph G with given properties. As applications, the generalized 3-connectivity of the Cayley graphs generated by complete graphs and the Cayley graph generated by wheel graphs, denoted by CTn and WGn, respectively, are determined. We show that κ3(CTn)=n(n−1)2−1 and κ3(WGn)=2n−3 for n≥3. The paper is organized as follows. In Section 2, some notation and definitions are given. In Section 3, the generalized 3-connectivity of regular graphs with given properties is determined. As applications, in Section 4, the generalized 3-connectivity of the Cayley graph CTn generated by complete graphs and the Cayley graph WGn generated by wheel graphs are determined. In Section 5, the paper is concluded. 2. PRELIMINARY Let G=(V,E) be a simple, undirected graph. Let ∣V(G)∣ be the size of the vertex set and ∣E(G)∣ be the size of the edge set. The subgraph induced by V′ in G, denoted by G[V′], is a graph whose vertex set is V′ and the edge set is the set of all the edges of G with both ends in V′. For a vertex v in G, we denote by NG(v) the neighborhood of the vertex v in G. For a vertex set U⊆V(G), the neighborhood of U in G is defined as NG(U)=∪v∈UNG(v)−U and ∣NG(U)∣ denotes the size of the neighborhood of U in G. Let dG(v) denote the number of edges incident with v and δ(G) denote the minimum degree of the graph G. A graph is said to be k-regular if for any vertex v of G, dG(v)=k. A matching in a graph is a set of pairwise non-adjacent edges. If M is a matching, the two ends of each edge of M are said to be matched under M, and each vertex incident with an edge of M is said to be covered by M. A perfect matching is one which covers every vertex of the graph. Two xy-paths P and Q in G are internally disjoint if they have no common internal vertices, that is V(P)∩V(Q)={x,y}. Let Y⊆V(G) and X⊂V(G)⧹Y. The set of (X,Y)-paths is a family of internally disjoint paths starting at a vertex x∈X, ending at a vertex y∈Y and whose internal vertices belong neither to X nor Y. If X={x}, then the set of (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. For terminologies and notations not defined here we follow [13]. Let Γ be a finite group and S be a subset of Γ, where the identity of the group does not belong to S. The Cayley graph Cay(Γ,S) is a digraph with vertex set Γ and arc set {(g,g.s)∣g∈Γ,s∈S}. If S=S−1, then Cay(Γ,S) is an undirected graph, where S−1={s−1∣s∈S}. In this paper, we consider the Cayley graph Cay(Sym(n),T), where Sym(n) is the symmetric group on {1,2,…,n} and T is a set of transpositions of Sym(n). Let G(T) be the graph on n vertices {1,2,…,n} such that there is an edge ij in G(T) if and only if transposition (ij)∈T [14]. The graph G(T) is the transposition generating graph of Cay(Sym(n),T). If G(T) is a tree, then Cay(Sym(n),T) is denoted by Γn. It was introduced in [15] that Γn can be decomposed into smaller Cayley graphs of the same type by considering the leaves of G(T). It is well known that if G(T)≅K1,n−1, then Cay(Sym(n),T) is an n-dimensional star graph and denoted by Sn; if G(T)≅Pn, then Cay(Sym(n),T) is an n-dimensional bubble-sort graph and denoted by Bn. If G(T) is a cycle with length n, then Cay(Sym(n),T) is denoted by MBn, which is the modified bubble-sort graph. Similarly, if G(T) is a cycle with length n−1, then Cay(Sym(n),T) is denoted by MBn−1. In [12], Li, Shi and Tu determined the generalized 3-connectivity of Cayley graphs on Symmetric Groups generated by trees and cycles. If G(T) is a complete graph with n vertices, then Cay(Sym(n),T) is denoted by CTn. For convenience, we call CTn an n-dimensional complete-transposition graph. If G(T) is a wheel graph Wn, which is a graph with n vertices (n≥4), obtained by connecting a single vertex to all vertices of an (n−1)-cycle, then Cay(Sym(n),T) is denoted by WGn. The Cayley graph WGn was introduced by Tu, Zhou and Su in [16], where it is called the n-dimensional wheel-transposition graph. It is known that CT4=WG4. The Cayley graphs CT3 and CT4 or (WG4) are depicted in Figs 1 and 2, respectively. Figure 1. View largeDownload slide The complete-transposition graph CT3. Figure 1. View largeDownload slide The complete-transposition graph CT3. Figure 2. View largeDownload slide The illustration of WG4 (or CT4). Figure 2. View largeDownload slide The illustration of WG4 (or CT4). First, we describe the hierarchical structure of the complete-transposition graph CTn. The transposition generating graph G(T) of CTn is a complete graph Kn. If deleting n in G(T), then G(T′)=G(T)−n is a complete graph on vertex set {1,2,…,n−1}. The graph CTn can be decomposed into n copies of CTn−1 as follows: for each i∈{1,2,…,n}, let CTn−1i be the subgraph of CTn induced by vertex set {(p1,p2,…,pn−1,i)∣(p1,p2,…,pn−1) ranges over all permutations of {1,2,…,n}⧹{i}}. Then CTn−1i≅Cay(Sym(n−1),T′) is a Cayley graph generated by a complete graph Kn−1, where it is a copy of CTn−1 for 1≤i≤n. By the definition of CTn, one has that CTn can be decomposed into n copies, which is denoted by CTn−11,CTn−12,…,CTn−1n, of CTn−1. For i∈{1,2,…,n}, each vertex u∈V(CTn−1i) has n−1 neighbors outside CTn−1i, called outside neighbors of u. For simple representation, this decomposition is denoted by CTn=CTn−11⊕CTn−12⊕⋯⊕CTn−1n, where ⊕ denotes the corresponding decomposition of CTn. By the definition of the complete-transposition graph CTn, it is a n(n−1)2-regular graph with n! vertices. The Cayley graph CTn has many properties which are desirable in an interconnection network such as hierarchical structure (hierarchical structure means that CTn can be decomposed into copies of CTn−1), high connectivity n(n−1)2 and small diameter n−1 [17, 18]. For the Cayley graph, vertex symmetry makes it possible to use the same routing protocols and communication schemes at all nodes; hierarchical structure facilitates recursive constructions; and high fault tolerance implies robustness, among others. Next, we describe the hierarchical structure of the wheel-transposition graph WGn. If the center vertex n of Wn is deleted, G(T′)=G(T)−n is a cycle on the vertex set {1,2,3,…,n−1}. Hence, the graph WGn can be decomposed into n copies of MBn−1 (where MBn−1 is the (n−1)-dimension modified bubble-sort graph) as follows: for each i∈{1,2,…,n}, let MBn−1i be the subgraph of WGn induced by the vertex set {(p1,p2,…,pn−1,i)∣(p1,p2,…,pn−1) ranges over all permutations of {1,2,…,n}⧹{i}}. Then MBn−1i≅Cay(Sym(n−1),T′) is a Cayley graph generated by a cycle with n−1 vertices, which is a copy of MBn−1. In this case, WGn can be decomposed into n copies of MBn−1, which are denoted by MBn−11,MBn−12,…,MBn−1n, and for any i∈{1,2,…,n}, each vertex u∈V(MBn−1i) has n−1 neighbors outside MBn−1i, called outside neighbors of u. The decomposition is denoted by WGn=MBn−11⊕MBn−12⊕⋯⊕MBn−1n, where ⊕ denotes the corresponding decomposition of WGn. By the definition of WGn, it is a 2(n−1)-regular graph with n! vertices. 3. THE GENERALIZED 3-CONNECTIVITY OF REGULAR GRAPHS WITH GIVEN PROPERTIES Lemma 1 (Lemma 3.1 [4]) Let Gbe a connected graph and δbe its minimum degree. Then κ3(G)≤δ. Further, if there are two adjacent vertices of degree δ, then κ3(G)≤δ−1. Lemma 2 (Proposition 9.4 [13]) Let G=(V,E)be a k-connected graph, and let Xand Ybe subsets of V(G)of cardinality at least k. Then there exists a family of kpairwise disjoint (X,Y)-paths in G. Lemma 3 (Proposition 9.5 [13]) Let G=(V,E)be a k-connected graph, let xbe a vertex of G, and let Y⊆V⧹{x}be a set of at least kvertices of G. Then there exists a k-fan in Gfrom xto Y, that is, there exists a family of kinternally vertex-disjoint (x,Y)-paths whose terminal vertices are distinct in Y. Lemma 4 Let Gmbe a k-regular graph with κ(Gm)=kfor each m∈{1,2,…,n}. Let Hnbe a graph obtained from G1,G2,…,Gnby adding some edges, where Hnis denoted by Hn=G1⊕G2⊕⋯⊕Gnfor n≥3. Let H=Gi⊕Gjbe the induced subgraph of Hnon V(Gi)∪V(Gj)for distinct i,j∈{1,2,…,n}. If there is exactly a perfect matching between Giand Gjin Hand ∣V(Gi)∣=∣V(Gj)∣≥k+1, then the following results hold: For any vertex vof H, dH(v)=k+1. Let x∈V(H), Y⊆V(H)⧹{x}with ∣Y∣=k+1such that ∣Y∩V(Gi)∣≤kand ∣Y∩V(Gj)∣≤k. Then there exists a (k+1)-fan in Hfrom xto Y, that is, there exists a family of k+1internally disjoint paths from xto Ysuch that the terminal vertices are distinct in Y. There exists a family of k+1internally disjoint paths joining any two distinct vertices of H. Proof Without loss of generality, let H=G1⊕G2. (1) As Gm is a k-regular graph for each m∈{1,2,…,n}, then G1 and G2 are both k-regular. As there is exactly a perfect matching between G1 and G2 in H, then dH(v)=k+1 holds for any vertex v∈V(H). (2) Let x∈V(H) and Y⊆V(H)⧹{x} with ∣Y∣=k+1 such that ∣Y∩V(Gi)∣≤k for i=1,2. Without loss of generality, let x∈V(G1) and let the neighbor of x in G2 be x′. Let Y∩V(Gi)=Ai and ∣Ai∣=ki for i=1,2. Then ki≤k and k1+k2=k+1. As k1≤k, then k2≥1. Let M1⊆V(G1) such that ∣M1∣=k2−1 and M1∩(A1∪{x})=ø. This can be done as ∣V(G1)∣≥k+1. Let M=A1∪M1, then ∣M∣=k. As κ(Gm)=k for each m∈{1,2,…,n}, then κ(G1)=κ(G2)=k. By Lemma 3, there is a F1 from x to A1 and a fan F2 from x to M1 in G1, respectively, so that the only common vertex to F1 and F2 is x. As there is exactly a perfect matching between G1 and G2 in H, then any vertex v∈V(G1) has exactly one neighbor in G2. Let M2={y′∣y′ is the neighbor of y in G2 for each y∈M1} and E2={yy′∈E(H)∣y∈M1 and y′∈M2}. Let M′=M2∪{x′}. Then ∣M′∣=k2. As κ(G2)=k≥k2, by Lemma 2, there exists a family of k2 pairwise disjoint (M′,A2)-paths F2′ in G2. It is easy to see that combining the two fans F1,F2, the edge set E2, the edge xx′ and the set of paths F2′, we can obtain a (k+1)-fan in H from x to Y. (3) Let x,y∈V(H) be any two distinct vertices of H. We prove the result by considering the following two cases. Case 1. x and y are non-adjacent. Without loss of generality, let x∈V(G1) and Y=NH(y). Then x∉Y, otherwise x and y are adjacent. By Lemma 4(1), ∣Y∣=k+1. If y∈V(G1), then ∣Y∩V(G1)∣=k and ∣Y∩V(G2)∣=1. If y∈V(G2), then ∣Y∩V(G1)∣=1 and ∣Y∩V(G2)∣=k. By (2), there exists a family of k+1 internally disjoint paths in H from x to Y. Combining the edges from y to Y, we obtain k+1 internally disjoint paths between x and y in H. Case 2. x and y are adjacent. Let Y=(NH(y)∪{y})⧹{x}; then ∣Y∣=k+1. Without loss of generality, let y∈V(G1). If x∈V(G1), then ∣Y∩V(G1)∣=k and ∣Y∩V(G2)∣=1. By (2), there exists a family of k+1 internally disjoint paths in H from x to Y in H. Combining the edges from y to Y⧹{y} and the edge xy, we can obtain k+1 internally disjoint paths from x to y in H. If x∈V(G2), then ∣(Y⧹{y})∩V(G1)∣=k. Let yi∈NG1(y) and yi′ be the neighbor of yi in G2 for 1≤i≤k. Let Y′={y1′,y2′,…,yk′}. Then ∣Y′∣=k. As there is a perfect matching between G1 and G2, then x∉Y′. As κ(G2)=k, by Lemma 3, there exists a k-fan from x to Y′ in G2. That is, there are k internally disjoint paths P1,P2,…,Pk from x to Y′ in G2. Let Ti=yyi∪yiyi′∪Pi for each i∈{1,2,…,k}. Combining the edge xy and Tis for 1≤i≤k, then k+1 internally disjoint paths from x to y are obtained in H. □ Theorem 1 Let Hn=G1⊕G2⊕⋯⊕Gnbe the same as that in Lemma4, where Gmis a k-regular graph with κ(Gm)=kand ∣V(Gm)∣≥k+1for each m∈{1,2,…,n}. Assume that there is exactly a perfect matching between Giand Gjin Hn[V(Gi)∪V(Gj)]for distinct i,j∈{1,2,…,n}. If κ3(Gm)=k−1for each m∈{1,2,…,n}, then κ3(Hn)=n+k−2for n≥3. Proof Let Hn=G1⊕G2⊕⋯⊕Gn. As there is exactly a perfect matching between Gi and Gj in H, where H=Hn[V(Gi)∪V(Gj)] for distinct i,j∈{1,2,…,n}, then for any vertex v of Gm, it has exactly (n−1) neighbors outside Gm for each m∈{1,2,…,n}. As Gm is k-regular for each m∈{1,2,…,n}, then δ=dHn(v)=k+n−1 for any v∈V(Hn). By Lemma 1, κ3(Hn)≤δ−1=k+n−2. To prove the result, we just need to show that κ3(Hn)≥k+n−2. Let v1,v2 and v3 be any three vertices of Hn. For convenience, let S={v1,v2,v3}. We prove the result by considering the following three cases. Case 1. v1,v2 and v3 belong to the same Gi for some i∈{1,2,…,n}. Without loss of generality, assume that v1,v2 and v3 belong to G1. As κ3(Gm)=k−1 for each m∈{1,2,…,n}, then there exist k−1 internally disjoint trees T1,T2,…,Tk−1 connecting S in G1. As there is exactly a perfect matching between Gi and Gj in Hn[V(Gi)∪V(Gj)] for distinct i,j∈{1,2,…,n}, then any vertex v of G1 has exactly one outside neighbor in G2,G3,…, Gn, respectively. Let v1j,v2j and v3j be the neighbor of v1,v2 and v3 in Gj, respectively, where 2≤j≤n. As Gj is connected, then there is a tree Tj^ connecting the three vertices v1j,v2j and v3j in Gj. Let Aj=Tj^∪v1v1j∪v2v2j∪v3v3j, then Aj is a tree connecting S for each j with 2≤j≤n. Thus, Ajs for 2≤j≤n and Tis for 1≤i≤k−1 are n+k−2 internally disjoint trees with each of them connecting S in Hn. Thus, in this case, κ3(Hn)≥n+k−2. Case 2. v1,v2 and v3 belong to two different Gis. Without loss of generality, assume that v1,v2∈V(G1) and v3∈V(G2). As κ(Gm)=k for each m∈{1,2,…,n}, then κ(G1)=κ(G2)=k and there exist k pairwise disjoint paths P1,P2,…,Pk between v1 and v2 in G1. Choose k distinct vertices x1,x2,…,xk from P1,P2,…,Pk such that xi∈V(Pi) for each i∈{1,2,…,k}. As there is exactly a perfect matching between Gi and Gj in Hn[V(Gi)∪V(Gj)] for distinct i,j∈{1,2,…,n}, then there is exactly a perfect matching between G1 and G2 in Hn[V(G1)∪V(G2)]. Let xi′ be the neighbor of xi in G2 and X′={x1′,x2′,…,xk′}. As κ(G2)=k. By Lemma 3, there exist k internally disjoint paths P1′,P2′,…,Pk′ between v3 and X′. If xi′=v3 for some 1≤i≤k, then let Pi′=v3. Let Ti=Pi∪xixi′∪Pi′ for each i∈{1,2,…,k}. Then Tis for 1≤i≤k are k pairwise disjoint trees with each of them connecting S. Similar as Case 1, let v1j,v2j and v3j be the neighbors of v1,v2 and v3 in Gj for 3≤j≤n, respectively. As Gj is connected, we can find a tree Tj^ connecting the three vertices v1j,v2j and v3j in Gj. Let Bj=Tj^∪v1v1j∪v2v2j∪v3v3j, then Bj is a tree connecting S for 3≤j≤n. Hence, Bjs for 3≤j≤n and Tis for 1≤j≤k are n+k−2 internally disjoint trees in Hn with each of them connecting S. Thus, in this case, κ3(Hn)≥n+k−2. Case 3. v1,v2 and v3 belong to three different Gis. Without loss of generality, assume that v1∈V(G1),v2∈V(G2) and v3∈V(G3). Let H=G1⊕G2. By Lemma 4(3), there exist (k+1) internally disjoint paths P1,P2,…,Pk+1 between v1 and v2 in H. Let v3′ be the neighbor of v3 in G2 (it is possible that v3′=v2). As H is connected, then there exists a path P˜ connecting v3′ and v1 in H. Let t1 be the first vertex of the path P˜ which is in ∪i=1k+1Pi. Without loss of generality, assume t1∈V(Pk+1). Choose k distinct vertices x1,x2,…,xk from P1,P2,…,Pk such that xi∈V(Pi), where 1≤i≤k. Let xi′ be the neighbor of xi in G3, where 1≤i≤k. As there is exactly a perfect matching between G2 and G3 in Hn[V(G2)∪V(G3)], this can be done. Let X′={x1′,x2′,…,xk′}. As κ(G3)=k, by Lemma 3, there exist k internally disjoint paths P1′,P2′,…,Pk′ in G3 from v3 to X′. If v3∈X′, assume v3=x1′, then let P1′=v3. Let Ti=Pi∪xixi′∪Pi′ for 1≤i≤k and Tk+1=P˜∪v3v3′∪Pk+1 (if v3′=v2, then Tk+1=v3v2∪Pk+1). Hence, Tis for 1≤i≤k+1 are k+1 internally disjoint trees with each of them connecting S. Similar as Case 1, let v1j,v2j and v3j be the outside neighbors of v1,v2 and v3 in Gj for 4≤j≤n, respectively. As Gj is connected, we can find a tree Tj^ connecting the three vertices v1j,v2j and v3j in Gj. Let Cj=Tj^∪v1v1j∪v2v2j∪v3v3j, then Cj is a tree connecting S for each j∈{4,…,n}. Hence, Cjs for 4≤j≤n and Tis for 1≤i≤k+1 are n+k−2 internally disjoint trees in Hn with each of them connecting S. Thus, in this case, κ3(Hn)≥n+k−2. In conclusion, κ3(Hn)≥n+k−2. The proof is complete. □ 4. THE GENERALIZED 3-CONNECTIVITY OF TWO KINDS OF CAYLEY GRAPHS As applications of Theorem 1, in this section, we will study the generalized 3-connectivity of Cayley graphs generated by complete graphs and wheel graphs. 4.1. The generalized 3-connectivity of the Cayley graph generated by complete graphs To prove the main result, we need the following lemmas. Lemma 5 (Lemma 2.5 [19]) Let CTn=CTn−11⊕CTn−12⊕⋯⊕CTn−1nfor n≥3. Then the following results hold: For any vertex uof CTn, the outside neighbors of uare in different copies. For any copy CTn−1i, no two distinct vertices in CTn−1ihave a common outside neighbor, and thus ∣N(V(CTn−1i))∣=(n−1)(n−1)!. For any i≠j, ∣N(V(CTn−1i))∩V(CTn−1j)∣=(n−1)!. Lemma 6 (Theorem 2.3 [4]). Let Gbe a connected graph with nvertices. If κ(G)=4k+r,where kand rare two integers with k≥0and r∈{0,1,2,3}, then κ3(G)≥3k+⌈r2⌉. Moreover, the lower bound is sharp. In [20], Lakshmivarahan et al. have shown that CTn is both vertex and edge transitive. It has been shown that every edge transitive graph is maximally connected [13]. Thus, we have the following lemma. Lemma 7 Let CTnbe an n-dimensional complete-transposition graph, then κ(CTn)=n(n−1)2. Theorem 2 Let CTnbe an n-dimensional complete-transposition graph, then κ3(CTn)=n(n−1)2−1for n≥3. Proof Since CTn=CTn−11⊕CTn−12⊕⋯⊕CTn−1n for n≥3. We prove the result by induction on n. For n=3, note that CT3 is 3-regular. By Lemma 6, κ3(CT3)≥⌈32⌉=2. By Lemma 1, κ3(CT3)≤δ−1=2. Hence, κ3(CT3)=2 and the result holds. Next, suppose that n≥4 and the result holds for n−1. That is, κ3(CTn−1)=(n−1)(n−2)2−1. By CTn−1i≅CTn−1, one has that κ3(CTn−1i)=(n−1)(n−2)2−1 for each i∈{1,2,…,n}. By Lemma 7, κ(CTn−1i)=(n−1)(n−2)2 for each i∈{1,2,…,n}. By (2) and (3) of Lemma 5, there is a perfect matching between CTn−1i and CTn−1j in CTn[V(CTn−1i)∪V(CTn−1j)] for distinct i,j∈{1,2,…,n}. As ∣V(CTn−1i)∣=(n−1)!≥(n−1)(n−2)2+1, by Theorem 1 for k=(n−1)(n−2)2, one has that κ3(CTn)=n+k−2=n(n−1)2−1. □ 4.2. The generalized 3-connectivity of the Cayley graph generated by wheel graphs In this section, we will study the generalized 3-connectivity of the Cayley graph generated by wheel graphs. To prove the main result, we need the following Lemmas. Lemma 8 (Lemma 3.4 [16]) Let WGn=MBn−11⊕MBn−12⊕⋯⊕MBn−1nfor n≥3, where MBn−1iis a copy of MBn−1for 1≤i≤n. Then the following result holds: For any vertex uof MBn−1i, the outside neighbors of uare in different MBn−1js. For any copy MBn−1i, no vertices in MBn−1ihave a common outside neighbor, and thus ∣N(V(MBn−1i))∣=(n−1)(n−1)!. For any i≠j, ∣N(V(MBn−1i)∩V(MBn−1j)∣=(n−1)!. Lemma 9 (Lemma 12 [12]) κ(MBn)=n for n≥3. Lemma 10 (Theorem 13 [12]) κ3(MBn)=n−1for any integer n≥3. Theorem 3 Let WGnbe an n-dimensional wheel-transposition graph, then κ3(WGn)=2n−3for n≥3. Proof Since WGn=MBn−11⊕MBn−12⊕⋯⊕MBn−1n, where MBn−1i is a copy of MBn−1 for each i∈{1,2,…,n}. By Lemma 9, κ(MBn−1i)=n−1 for each i∈{1,2,…,n}. By (2) and (3) of Lemma 8, there is exactly a perfect matching between MBn−1i and MBn−1j in WGn[V(MBn−1i)∪V(MBn−1j)] for any i,j∈{1,2,…,n}. By Lemma 10, κ3(MBn−1i)=n−2 for each i∈{1,2,…,n}. Thus, all conditions of Theorem 1 hold. By Theorem 1, one has that κ3(WGn)=2n−3. □ 5. CONCLUDING REMARKS In this paper, the generalized 3-connectivity of regular graphs with given properties are obtained. As the Cayley graph has some attractive properties when used as an interconnection network, as applications of the Theorem 1, we focus on the Cayley graph CTn generated by complete graphs and the Cayley graph WGn generated by wheel graphs. We give the exact values of the generalized 3-connectivity of these two classes of graphs as direct results of Theorem 1. We believe that Theorem 1 can be applied to some other networks. As we all know, almost all known results of generalized k-connectivity are about k=3. We are interested in this topic for k≥4 and we would like to study in this direction to show the corresponding results for these two kinds of Cayley graphs and other classical networks such as arrangement graphs, balanced graphs, (n,k)-star networks and so on. FUNDING This work was supported by the National Natural Science Foundation of China (No. 11731002), the Fundamental Research Funds for the Central Universities (Nos. 2016JBM071 and 2016JBZ012). ACKNOWLEDGEMENTS The authors express their sincere thanks to the editor and the anonymous referees for their valuable suggestions which greatly improved the original manuscript. REFERENCES 1 Whitney , H. ( 1932 ) Congruent graphs and the connectivity of graphs . Am. J. Math , 54 , 150 – 168 . Google Scholar CrossRef Search ADS 2 Chartrand , G. , Kapoor , S. , Lesniak , L. and Lick , D. ( 1984 ) Generalized connectivity in graphs . Bull. Bombay Math. Colloq , 2 , 1 – 6 . 3 Li , S. and Li , X. 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Google Scholar CrossRef Search ADS Author notes Handling editor: Iain Stewart © The British Computer Society 2018. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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The Computer JournalOxford University Press

Published: May 26, 2018

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