给定一个图g和一个非负整数g,若图g中存在(边)点集,使得删除该集合后图g不连通并且每个连通分支的点数大于g,所有这样的(边)点集的最小基数,称为g-额外(边)连通度(记作κg(g)(λg(g)).本文将确定由对换树生成的凯莱图的3-额外(边)连通度(记作κ3(λ3).Given a graph G and a non-negative integer g, the g-extra(edge) connectivity of G (written κg(G)(λg(G))is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices.In this paper, we determine 3-extra(edge) connectivity(written κ3(λ3)) of Cayley graphs generated by transposition trees.supportedbyNSFC(No.10671165

孟吉翔

李恒哲

杨卫华

Xiamen University Institutional Repository

第 28卷第 2期2011年 5月新疆大学学报(自然科学版)Journal of Xinjiang University(Natural Science Edition)Vol.28, No.2May, 20113-extraconnectivity of Cayley Graphs Generated byTransposition Generating Trees∗LI Heng-zhe1, MENG Ji-xiang1,†, YANG Wei-hua2(1. Department of Mathematics, Xinjiang University, Urumqi, Xinjiang 830046, China;2. School of Mathematics Science, Xiamen University, Xiamen, Fujian 361005, China)Abstract : Given a graph G and a non-negative integer g, the g-extra(edge) connectivity of G (written κg(G)(λg(G))is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remainingcomponent has more than g vertices. In this paper, we determine 3-extra(edge) connectivity(written κ3(λ3)) of Cayleygraphs generated by transposition trees.Key words : Interconnection networks; Cayley graphs; Conditional connectivity; Transition treeCLC number：O157.5 Document Code：A Article ID：1000-2839(2011)02-0148-04由对换树生成的凯莱图的 3-额外连通度李恒哲1，孟吉翔1，杨卫华2(1. 新疆大学数学与系统科学学院，新疆乌鲁木齐 830046; 2. 厦门大学数学与系统科学学院，福建厦门 361005)摘 要： 给定一个图G和一个非负整数 g,若图 G中存在（边）点集,使得删除该集合后图 G不连通并且每个连通分支的点数大于 g,所有这样的（边）点集的最小基数，称为 g-额外（边）连通度（记作 κg(G)(λg(G)). 本文将确定由对换树生成的凯莱图的 3-额外（边）连通度（记作 κ3(λ3).关键词： 交互网络；凯莱图；条件连通度；对换树0 IntroductionThroughout this paper, only undirected simple graphs without loops and multiple edges are considered. Wefollow[1] for terminology not given here.The traditional connectivity(denote by κ), is an important measure for the networks, which can correctly reflectsthe fault tolerance of system with few processor, but it always underestimates the resilience of large networks. Toovercome such shortcoming, Harary[2] introduced the concept of conditional connectivity by imposing conditions onthe components of G−F, where F is a subset of edges or vertices. Some kind of conditional connectivity introducedin [2] have been received much attention in the literature, such as extra(edge)connectivity.Let g≥ 1 be a integer. Fàbrega and Fiol[3] defined the g−extra(edge)connectivity of G is the minimum cardinalityof a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than gvertices. For convenience, we denote 3-extra(edge)connectivity by κ3(λ3). Next we introduced a useful terminologies,κg − vertex cut(λg − edge cut) of G is a subset S of V(G)(E(G)) such that G−F is disconnected and every remainingcomponent of G−F has more than g vertices. Here we consider κ3(λ3).Let X be a group and S be a subset of X containing no identity element of X. the Cayley digraph Cay(X,S ) is adigraph with vertex set X and arc set {(g,gs) | g ∈ X, s ∈ S }. The arc (g,gs) is labeled by s. Clearly, If S = S −1, whereS −1 = {s−1|s ∈ S }, then Cay(X,S ) is an undirected graph. Denoted by Σn the group of all permutation on {1,2, . . . ,n}.∗ Received Date：2009-11-11Foundation Item：The research is supported by NSFC (No.10671165).Biography：LI Hengzhe(1982-), male, master student. Email: lihengzhe2009@126.com.† Corresponding author：Email: mjx@xju.edu.cn.第 2期 LI Heng-zhe, et al.: 3-extraconnectivity of Cayley Graphs Generated by Transposition Generating Trees 149For convenience, we use (p1 p2 · · · pn) to denote the permutation1 2 . . .np1 p2 · · · pn and (i j), which is called a transposition,denotes the permutation that swaps the objects at position i and j, that is (p1 · · · pi · · · p j · · · pn)(i j) = (p1 · · · p j · · · pi · · · pn).Let M be a minimal set of generators for Σn. In this paper, we only consider the case that the minimal set M ofgenerators for Σn consists of transpositions. For the convenience of discussion, we describe the the minimal set M ofgenerators for Σn by a graph which has vertex set {1,2, . . . ,n} and edge set {(x,y) | (xy) ∈M}. Clearly, this graph is a tree,written TM (See [2] for the detail), and is called a transposition tree. It is well known that the K1,n−1 generates the stargraphs and Pn generates the bubble-sort graph. In the following we use Γn to denote the graph Cay(Γn,M), where M isminimal set of generators for Σn consisting of transposition, S n to denote the star graph whose transposition tree TMis isomorphic to K1,n−1 and Gn to denote the graph whose transposition tree TM is not isomorphic to K1,n−1. Clearly, Γnare (n−1)-regular bipartite graph and have n! vertices, see [4]. The networks basic Γn have gained much attention inthe area of interconnection networks. Some recent research in this area includes [2,5∼10].1 PreliminariesG is a graph and H is a subgraph of G. Let NG(x) and NH(x) denote the neighborhood of the vertex x in G and inH, respectively. Sometimes, for convenience of writing, we replace NG(x) with N(x).For any vertex x ∈V(Γn), we denote the ith position of the permutation x by x[i]. For example, if x = 4312 ∈V(Γ4),then, x[1] = 4, x[2] = 3, x[3] = 1, x[4] = 2, respectively. The following useful properties of Γn are reported in [5∼7,9,10].Property 1 If n is a leaf of TM, then Γn can be decomposed into n disjoint copies of Γn−1, say Γn(1),Γn(2), . . . ,Γn(n),where Γn(i) is a induced subgraph of Γn by vertex set {x | x[n] = i} = {p1 p2 · · · pn−1i | p1 p2 · · · pn−1 ranges over the per-mutation on {1,2, · · · , i − 1, i + 1, · · · ,n}}. We denote this decomposition by Γn = Γn(1)⊕Γn(2)⊕ · · ·⊕Γn and saythis decomposition is one by leaf n. Similarly, we denote Gn and S n by Gn = Gn(1)⊕Gn(2)⊕ · · ·⊕Gn(n) and S n =S n(1)⊕S n(2)⊕ · · ·⊕S n(n). Sometimes, we denote V0(Γn)⋂V(Γn(i)) by V0(Γn(i)), V1(Γn)⋂V(Γn(i)) by V1(Γn(i)), respectively.In the following arguments, we always assume that n is a leaf of TM.Property 2 Let (tn) ∈ E(TM) be an leaf edge of TM. For any vertex u of Γn(i), u(tn) is the unique neighborof u outside Γn(i), is called the outgoing neighbor of u, written u′, and any two distinct vertices of Γn(i) have dif-ferent outgoing neighbors. There are exactly (n − 2)! independent edges between Γn(i) and Γn( j) if i , j, that is,|N(Γn(i))⋂V(Γn( j))|= (n−2)! if i, j.Property 3 Gn has grith 4 and S n has grith 6.Lemma 1[10] For any two distinct vertices of Γn, they have at most two common neighbors if they have any.Lemma 2[5,6,9,10] κ(Γn) = λ(Γn) = n−1, κ1(Γn) = λ1(Γn) = 2n−4 for n≥ 3, κ2(Gn) = 3n−8, κ2(S n) = 3n−7,λ2(Γn) = 3n−7for n≥ 4.Another two important connectivity, super connected and super edge− connected were proposed in [11,12]. Agraph G is super connected, super−κ for short (resp. super edge−connected, super−λ, for short) if every minimumvertex-cut (edge-cut) isolates a vertex of G. Xu et al.[13] shown that G is super-κ if and only if κ1 > κ, and super-λ ifand only if λ1 >λ. So we have the following corollary.Corollary 1 Γn is super-κ and super-λ.Lemma 3 Let B be any subgraph in Gn(i), i ∈ {1,2, . . . ,n}. If x ∈NGn(i)(B), then |NGn (B⋃{x})| ≥ |NGn (B)|.Proof Note that NGn (B) = NGn (B)\{x}+{x} and NGn (B)\{x} ∈NGn (B⋃{x}). It is sufficient to show that there existsa vertex y ∈ NGn (x) which is not contained in NGn (B). Clearly, the outgoing neighbor of x satisfies the condition byProperty 2.Lemma 4 For Gn with n≥ 6, there exists a decomposition Gn = Gn(1)⊕Gn(2)⊕ · · ·⊕Gn(n)such that C4 ∈Gn(i)for any integer i ∈ {1,2, . . . ,n}. Furthermore, N(C4) is a κ3-vertex cut of Gn.Proof Such decomposition clearly exists. Pick a C4 in Gn(i), where C4 is cycle of length 4. Assume thatV(C4) = {x1, x2, x3, x4} and xi and x j are adjacent if i− j ≡ ±1 mod 4. To prove N(C4) is a κ3-vertex cut of Gn, clearly, itis sufficient to prove that Gn−N(C4)−V(C4) is connected. Assume that xi has outgoing neighbor x′i for 1≤ i≤ 4. Note150 新疆大学学报(自然科学版) 2011年that |N(C4)⋂V(Gn( j))| ≤ 4 for j, i. So Gn( j)−N(C4) is connected since κ(Gn( j)) = (n−1)−1 = n−2≥ 5> 4 for n≥ 7. Inparticular, for n = 6, if |N(C4)⋂V(Gn( j))|= 4 and Gn( j)−N(C4) is disconnected, then, it is easy to see that {x′1, x′2, x′3, x′4}induce a C4. Since G6( j) is super-κ, {x′1, x′2, x′3, x′4} = NG6( j)(u) for some vertex u ∈ V(G6( j)). Thus there exists a triangleinduced by {u, x′1, x′2}, a contradiction.Let Ḡ =⋃j,i(Gn( j)−N(C4)). We will show that Ḡ is connected. Let j,k be integers different from i. Note that|N(Gn( j))⋂V(Gn(k))| = (n−2)! if j , k, |N(C4)| = 4(n−1)−8 = 4n−12, and (n−2)! > 4n−12 for n ≥ 6. That is, thereexists u ∈V(Gn( j))−N(C4),v ∈V(Gn(k))−N(C4) such that (u,v) ∈ E(Gn). Hence Gn( j)−N(C4) connects to Gn(k)−N(C4).Clearly, for any vertex u ∈ V(Gn(i))−N(C4)−V(C4), its outgoing neighbor u′ is contained in Ḡ by Property 2. ThusGn−N(C4)−V(C4) is connected. Thus N(C4) is a κ3-vertex cut of Gn.Remark 1 Let P4 ∈ S n be a path of length 4, it is not different to see that N(P4) is a κ3-vertex cut of S n for n≥ 6.Furthermore, ω(C4)∗ and ω(P4) are λ3-edge cut of Gn and S n, respectively.2 R3-extra(edge)connectivity of ΓnIn this section, we shall determine κ3(Γn)((λ3(Γn)).Theorem 1 κ3(Gn) = 4n−12, κ3(S n) = 4n−10 for n≥ 6.Proof Take the decomposition by leaf n and assume that all outgoing edges have the same label ((n−1)n). Weonly show that κ3(Gn) = 4n−12 for n≥ 6. κ3(S n) = 4n−10 for n≥ 6 can be prove with a similar argument.By Lemma 4, N(C4) is a κ3-vertex cut of Gn. So κ3(Gn)≤ |N(C4)|= 4n−12. Now we show that κ3(Gn)≥ 4n−12.Suppose by the way of contradiction that κ3(Gn) ≤ 4n − 13. Let F be a minimum κ3-vertex cut of Gn, then|F| ≤ 4n−13. For convenience, we denote F ⋂V(Gn(i)) by Fi. Let I = {i |Gn(i)−Fi is disconnected}. Since |F| ≤ 4n−13and κ(Gn(i)) = n−2, |I| ≤ 3. We consider the following four cases.Case 1 |I|= 0.Gn(i)−Fi is no empty graph, since (n−1)!> 4n−13 for n≥ 6. Note that |N(Gn(i))⋂V(Gn( j))|= (n−2)!> 4n−13for n ≥ 6, where i , j, that is That is, there exists u ∈ V(Gn(i))−Fi,v ∈ V(Gn( j))−F j such that (u,v) ∈ E(Gn). Hence,Gn(i)−Fi connects to Gn( j)−F j for i, j. Thus Gn−F is connected, a contradiction.Case 2 |I|= 1.Without loss of generality, we assume that Gn(1)− F1 is disconnected. By a similar argument of Case 1. Ḡ =⋃j,1(Gn( j)− F j) is connected. Note that F is a κ3-vertex cut of Gn, so Gn(1)− F1 contains a component H of size≥ 4 of Gn − F. Let B be a connected subgraph of size 4 in H. For any x ∈ NGn(1)(B), let x′ be outgoing neighborof x. Clearly, at least one of x and x′ is contained in F, otherwise, B is connected to Ḡ, a contradiction. Note that|NGn(1)(B)| ≥ 4(n − 2) − 8 = 4n − 16 and V(B) has four outgoing neighbors which are contained in F. Thus F ≥|NGn(1)(B)|+4≥ 4n−16+4 = 4n−12> 4n−13, a contradiction.Case 3 |I|= 2.Without loss of generality, we assume that Gn(1)−F1 and Gn(2)−F2 is disconnected. By a similar argument ofCase 1. Ḡ =⋃j,1,2(Gn( j)−F j) is connected. Clearly, there exists no component of Gn−F in Gn(1) and Gn(2). Otherwise,we can derive a contradiction by a similar argument of Case 2. Let B be a component of Gn−F in Gn(1)⋃Gn(2). Forconvenience, we denote B⋂Gn(i) by Bi for i = 1,2.If |V(B)|= 4, |F| ≥ |NGn (B)| ≥ 4n−12> 4n−13, a contradiction.If there exists i, i = 1,2, such that |V(Bi)| ≥ 5. By Lemma 1 and Lemma 3, |NGn (Bi)| ≥ 5n−17≥ 4n−11 for n≥ 6.Claim 1 Let F be a vertex-cut of graph Gn, B ∈ Gn(i)⋃Gn( j) be a connected subgraph of G − F, then |F| ≥|NGn (Bi)|, |F| ≥ |NGn (B j)|.Note that NGn (Bi) =⋃1≤k≤n NGn(k)(Bi),⋃k, j NGn(k)(Bi) ⊆ F and NGn( j)(Bi)\V(B j) ⊆ F. For any x ∈ NGn( j)(Bi)⋂V(B j),pick a neighbor of x in Gn( j), say y = x(l(n− 1)), l < {n− 1,n}. Clearly, the outgoing neighbor y′ = y((n− 1)n) of y isnot contained in Gn(i), since x[n−1] , y[n−1]. We claim that at least one of y and y′ is contained in F. Otherwise, Bconnects to Ḡ, a contradiction. Thus, |F| ≥ |⋃k, j NGn(k)(Bi)|+ |NGn( j)(Bi)\V(B j)|+ |NGn( j)(Bi)⋂V(B j)| ≥ |NGn (Bi)|.By above Claim, |F| ≥ |NGn (Bi)| ≥ 5n−17≥ 4n−11 for n≥ 6, a contradiction.∗For a subgraph H of G, ω(H) denotes an edge set whose edge has exactly one endpoint in V(H).第 2期 LI Heng-zhe, et al.: 3-extraconnectivity of Cayley Graphs Generated by Transposition Generating Trees 151In the following argument of this case, we assume that |V(Bi)| ≤ 4 and 5 ≤ |V(B)| ≤ 8. Furthermore, assume that|V(Bi)| ≥ |V(B j)|. Clearly, |V(Bi)| ≥ 3.Suppose |V(Bi)|= 3.If |V(B j)|= 2,|F| ≥ |NGn(i)(Bi)|+ |NGn( j)(B j)| ≥ 3n−11+2n−6 = 5n−17≥ 4n−11> 4n−13for n≥ 6, a contradiction.If |V(B j)|= 3,|F| ≥ |NGn(i)(Bi)|+ |NGn( j)(B j)| ≥ 3n−11+3n−11 = 6n−22≥ 4n−10> 4n−13for n≥ 6, a contradiction.Suppose |V(Bi)|= 4.If |V(B j)|= 1, then |V(B)|= 5,|NGn (B)| ≥ 5n−17≥ 4n−11> 4n−13for n≥ 6, a contradiction.If |V(B j)|= 2,|F| ≥ |NGn(i)(Bi)|+ |NGn( j)(B j)| ≥ 4n−16+2n−6 = 6n−22≥ 4n−10> 4n−13for n≥ 6, a contradiction.If |V(B j)|= 3,|F| ≥ |NGn(i)(Bi)|+ |NGn( j)(B j)| ≥ 4n−16+3n−11 = 7n−27≥ 4n−9> 4n−13for n≥ 6, a contradiction.If |V(B j)|= 4,|F| ≥ |NGn(i)(Bi)|+ |NGn( j)(B j)| ≥ 4n−16+4n−16 = 8n−32≥ 4n−8> 4n−13for n≥ 6, a contradiction.Case 4 |I|= 3.Without loss of generality, we assume that I = {1,2,3}. By a similar argument of Case 1. Ḡ = ⋃ j,1,2,3(Gn( j)−F j) isconnected. For any component B of Gn−F, we denote B⋂Gn(i) by Bi. Clearly, there exists no component B of Gn−Fexcepting Ḡ such that Bi = ∅ for i ∈ {1,2,3}. Otherwise, we can derive a contradiction by a similar arguments of Case 2and Case 3. If there exists a component B of Gn−F excepting Ḡ such that |V(B)|= 4, |F| ≥ |NGn (B)| ≥ 4n−12> 4n−13,a contradiction. In the following argument of this case, assume that for any component B of Gn − F excepting Ḡ,|V(B)|> 4. We consider the following two subcases.Subcase 4.1 If there exists integer i such that |Bi|= 1 for some component B of Gn−F contained in Gn(1)⋃Gn(2)⋃Gn(3). Without loss of generality, we assume that |V(B1)| = 1 and V(B1) = {y}. By a similar arguments of Case 3,we know that |NGn(2)(B2)|+ |NGn(3)(B3)| ≥ 4n− 12, furthermore, equality holds if and only if |V(B2)|+ |V(B3)| = 4. Ifthe equality holds. |NGn (B)| ≥ 5n− 17 ≥ 4n− 11 for n ≥ 6, a contradiction. If the equality does not hold, clearly,|F| ≥ |NGn(2)(B2)|+ |NGn(3)(B3)|−1> 4n−12−1> 4n−13, a contradiction.Subcase 4.2 If there exists no integer i such that |Bi|= 1 for any component of Gn−F contained in Gn(1)⋃Gn(2)⋃Gn(3).Claim 2 If there exists a component B of Gn−F in Gn(1)⋃Gn(2)⋃Gn(3) such that Bi is disconnected for someinteger i, then |Fi| ≥ 2n−7.If Bi is disconnected, we say that there exists at least three components in Gn(i)−Fi. Otherwise, there exists twocomponents Bi1 ,Bi2 in Gn(i)−Fi, clearly, Bi1 ,Bi2 is the subgraph of Bi. Since |N(Gn(i))⋂V(Gn(4))| = (n−2)! > 4n−13for n≥ 6, there exists x ∈V(Gn(i))−F,y ∈V(Gn(4))−F such that (x,y) ∈ E(Gn). So B is connected to Ḡ by edge (x,y), acontradiction. Thus there exists at least three components in Gn(i)−Fi.If there exists at least two trivial component in Gn(i)−Fi. Then |Fi| ≥ n−2+n−2−2 = 2n−6.(下转第 200页)200 新疆大学学报(自然科学版) 2011年参考文献：[1] Xu Quan hua. Inegalites pour les matrtingales de Hardy et renormage des espaces quasi-normes[J]. C R Acad paris,1988,306:601-604.[2] Xu Quan hua. Convexites uniformes et inegalites de martingales[J]. Mathematica Annual, 1990, 287:193-211.[3] 刘培德，Turdebek N Bekjan. 复空间的凸性与Hardy鞅不等式[J]. Acta Mathematica Sinica, 1997,40(1):133-143.[4] Bekjan Turdebek N. 复空间的PL凸性与解析鞅不等式[J]. Mathematica Acta Scientia, 1997, 17(1):64-73.[5] LIU Pei-de, HOU You-liang, WANG Mao-fa. Weak Orlicz Space and Its Application to the Martingale Theory[J]. ScienceChina (Mathematics), 2010,53(4):905-916.[6] 吐尔德·别克．Hardy鞅空间[J]. Journal of Xinjiang University, 1999, 16(4):1-5.[7] 吐尔德·别克，波拉提汗．Hardy鞅空间上若干算子的有界性[R]. Proceedings of the Third Annual Conference of XinjiangYoung Academics, 1998, 1337-1338.责任编辑：王宪清(上接第 151页)If there exists one trivial component in Gn(i)−Fi. Assume that the trivial component is x. Then Fi⋃{x}is a κ1-cut of Gn(i). By Lemma 2, κ1(Gn(i))= 2(n−1)−4= 2n−6. So |Fi| ≥ 2n−6−1= 2n−7.If there exists no trivial component in Gn(i)−Fi. Then Fi is a κ1-cut of Gn(i). By Lemma 2, κ1(Gn(i))=2(n−1)−4= 2n−6.Thus, |Fi| ≥ 2n−7.Claim 3 If there exists no component B of Gn−F in Gn(1)⋃Gn(2)⋃Gn(3) such that Bi is disconnectedfor any i=1,2,3, then |Fi| ≥ 2n−6 for i=1,2,3.Clearly, Fi is a κ1-cut of Gn(i). By Lemma 2, κ1(Gn(i))= 2(n−1)−4= 2n−6.By Claim 2 and 3, |Fi| ≥ 2n−7. Thus |F | ≥ |F1|+|F2|+|F3| ≥ 3(2n−7)= 6n−21≥ 4n−9, a contradiction.Theorem 2 λ3(Gn)= 4n−12, λ3(Sn)= 4n−10 for n≥ 6.proof By a similar argument of Theorem 1, we can prove this theorem.References：[1] Bondy J A, Murty U S R. Graph Theory with Applications[M]. London: Macmillan press, 1976.[2] Haray F. Conditional connectivity[J]. Networks, 1983, 13:346-357.[3] Fàbrega J, Fiol M A. On the extraconnectivity of graghs[J]. Discrete Mathematics, 1996, 155:49-57.[4] Godsil C, Royle G. Algebraic graph theory[M]. Berlin: Springer press, 2004.[5] Cheng E, Liptàk L, Shawash N. Orienting Cayley graphs generated by transposition trees[J]. Computer & mathematicswith applications, 2008, 55:2662-2672.[6] Cheng E, Lipman M, Liptàk L. Strong structural properties of unidirectional star graphs[J]. Discrete Applied Mathematics,2008, 5:1-11.[7] Cheng E, Liptàk L. Linearly many faults in Cayley graphs generated by transposition trees[J]. Information Sciences, 2007,177:4877-4882.[8] Meng J, Ji Y. On a king of restricted edge connectivity of graphs[J]. Discrete Apple Math, 2002, 177: 183-193.[9] Wan M, Zhang Z. A kind of conditional vertex connectivity of a star graphs[J]. Applied Mathematics Letters, 2008,doi:10,1016/j.aml.2008. 03.021.[10] Yang W, Meng J. Conditional connectivity of Cayley graphs generated by transposition trees[J]. Received by Aplliedmathematics and computation.[11] Boesch F. Synthesis of networks――a survey[J], IEEE Trans. Reliability 1986, 35:240-246.[12] Boesch F, Tindell R. Circulant and their connectivities[J]. Graph Theory, 1984, 8:487-499.[13] Xu J M . Super connectivity of line graph[J]. Information Processing Letters, 2005, 94:191-195.责任编辑：王宪清

3-extraconnectivity of Cayley Graphs Generated by Transposition Generating Trees

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3-extraconnectivity of Cayley Graphs Generated by Transposition Generating Trees

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