The super connectivity of the pancake graphs and the super laceability of the star graphs

www.elsevier.com/locate/tcs

The super connectivityof the pancake graphs and

the super laceabilityof the star graphs

Cheng-Kuan Lin

, Hua-Min Huang

, Lih-Hsing Hsu

a_{Department of Mathematics, National Central University, Chung-Li, Taiwan 32054, ROC} b_{Information Engineering Department, Ta Hwa Institute of Technology, Hsinchu, Taiwan 307, ROC}

Received 8 April 2004; received in revised form 31 January2005; accepted 21 February2005

Communicated byD-Z. Du

Abstract

A k-containerC(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) of G is ak∗-container if it contains all the vertices of G. A graph G isk∗-connected if there exists a

k∗_{-container between anytwo distinct vertices. Let
(G) be the connectivityof G. A graph G is super}

connected if G isi∗-connected for all 1
i

(G). A bipartite graph G is k∗-laceable if there exists a

k∗-container between anytwo vertices from different parts of G. A bipartite graph G is super laceable if G isi∗-laceable for all 1
i

(G). In this paper, we prove that the n-dimensional pancake graph

Pnis super connected if and onlyifn = 3 and the n-dimensional star graph S_nis super laceable if and onlyifn = 3.

© 2005 Elsevier B.V. All rights reserved.

Keywords: Hamiltonian; Hamiltonian connected; Hamiltonian laceable; Connectivity

1. Introduction

An interconnection network connects the processors of parallel computers. Its architec-ture can be represented as a graph in which the vertices correspond to processors and the edges correspond to connections. Hence, we use graphs and networks interchangeably. There are manymutuallyconflicting requirements in designing the topologyfor computer ∗_{Corresponding author. Department of Computer and Information Science, National Chiao Tung University,} 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC. Tel.: +886 3 5720659; fax: +886 3 5721490.

E-mail address:[email protected](L.-H. Hsu).

0304-3975/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2005.02.007

networks. The n-cube is one of the most popular topologies[18]. The n-dimensional star networkS_n was proposed in [1] as “an attractive alternative to the n-cube” topologyfor interconnecting processors in parallel computers. Since its introduction, the network has received considerable attention. Akers et al. [1] showed that the star graphs are vertex tran-sitive and edge trantran-sitive. The diameter and fault diameters were computed in [1,17,22,23]. The hamiltonian and hamiltonian laceabilityof star graphs are studied in [12,15,19]. In par-ticular, Fragopoulou and Akl [7,8] studied the embedding of(n − 1) directed edge-disjoint spanning trees on the star networkS_n. These spanning trees are used in communication algorithms for star networks.

Akers et al. [1] also proposed another familyof interesting graphs, the n-dimensional pancake graphP_n. Theyalso showed that the pancake graphs are vertex transitive. Hung et al. [14] studied the hamiltonian connectivityon the faultypancake graphs. The embed-ding of cycles and trees into the pancake graphs where discussed in [6,14,16]. Gates and Papadimitriou [10] studied the diameter of the pancake graphs. Until now, we do not know the exact value of the diameter of the pancake graphs [11].

For the graph definition and notation, we follow [3].G = (V, E) is a graph if V is a finite set and E is a subset of {(a, b) | (a, b) is an unordered pair of V}. We saythat V is the vertex set and E is the edge set. A path of length k from x to y is a sequence of distinct verticesv0, v1, v2, . . . , vk, where x = v0,y = vk, and(vi−1, vi) ∈ E for all 1
i
k. We also write the path v0, v1, . . . , vk as v0, . . . , vi, Q, vj, . . . , vk, where Q is a path fromv_i tov_j. Note that we allow Q to be a path of length zero. We also write the pathv0, v1, v2, . . . , vk as v0, Q1, vi, vi+1, . . . , vj, Q2, vt, . . . , vk, where Q1is the pathv0, v1, . . . , vi and Q2is the pathvj, vj+1, . . . , vt. We use d(u, v) to denote the distance between u andv, i.e., the length of the shortest path joining u and v.

A path of graph G from u tov is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path joining anytwo distinct vertices. A cycle is a path (except that the first vertex is the same as the last vertex) containing at least three vertices. A cycle of G is a hamiltonian cycle if it contains all vertices. A graph is hamiltonian if it has a hamiltonian cycle.

The connectivity of G,
(G), is the minimum number of vertices whose removal leaves the remaining graph disconnected or trivial. It follows from Menger’s Theorem [20] that there are k internal vertex-disjoint (abbreviated as disjoint) paths joining anytwo distinct vertices u andv for any k

(G). A k-container C(u, v) of G is a set of k disjoint paths joining u tov. In this paper, we discuss another type of container. A k-container C(u, v) is ak∗-container if it contains all vertices of G. A graph G isk∗-connected if there exists a k∗_{-container between anytwo distinct vertices. In particular, a graph G is 1}∗_{-connected if}

and onlyif it is hamiltonian connected, and a graph G is 2∗-connected if and onlyif it is hamiltonian. All 1∗-connected graphs except thatK1andK2are 2∗-connected. The studyof k∗_{-connected graphs is motivated bythe globally3}∗_{-connected graphs proposed byAlbert}

et al. [2]. A graph G is super connected if it isi∗-connected for all 1
i

(G). In this paper, we will prove that the pancake graphP_nis super connected if and onlyifn = 3.

A graph G is bipartite if its vertex set can be partitioned into two subsetsV1andV2such that everyedge joins vertices betweenV1andV2. Let G be a k-connected bipartite graph with bipartitionV1 andV2such that|V1||V2|. Suppose that there exists a k∗-container C(u, v) = {P1, P2, . . . , Pk} in a bipartite graph joining u to v with u, v ∈ V1. Obviously,

the number of vertices inP_iis 2k_i+ 1 for some integer k_i. There arek_i− 1 vertices of P_iin V1other than u andv, and k_ivertices ofP_iinV2. As a consequence,|V1| =

_k

i=1(ki−1)+2 and|V2| =

_k

i=1ki. Therefore, anybipartite graph G with
(G)3 is notk∗-connected for any3
k

(G).

For this reason, a bipartite graph isk∗-laceable if there exists ak∗-container between any two vertices from different partite sets. Obviously, any bipartitek∗-laceable graph withk2 has the equal size of bipartition. A 1∗-laceable graph is also known as hamiltonian laceable graph. Moreover, a graph G is 2∗-laceable if and onlyif it is hamiltonian. All 1∗-laceable graphs except thatK1andK2are 2∗-laceable. A bipartite graph G is super laceable if G is i∗<sub>-laceable for all 1
i

(G). In this paper, we will prove that the star graph Snis super</sub> laceable if and onlyifn = 3.

In the following section, we give the definition of the pancake graphs and discuss some of their properties. In Section 3, we prove that the pancake graphP_nis super connected if and onlyifn = 3. The definition of the star graphs and some of their properties are presented in Section 4. In Section 5, we prove that the star graphS_nis super laceable if and onlyif n = 3. In the final section, we discuss further research.

2. The pancake graphs

Let n be a positive integer. We usen to denote the set {1, 2, . . . , n}. The n-dimensional pancake graph, denoted byP_n, is a graph with the vertex setV (P_n) = {u1u2. . . u_n| u_i ∈ n andu_i = u_jfori = j}. The adjacencyis defined as follows: u1u2. . . u_i. . . u_nis adjacent tov1v2. . . v_i. . . v_nthrough an edge of dimension i with 2
i
n if v_j = u_i−j+1for all 1
j
i and v_j = u_j for alli < j
n. We will use bold face to denote a vertex of P_n. Hence, u1, u2, . . . , undenote a sequence of vertices inPn. In particular, e denotes the vertex

12. . . n. Bydefinition, P_nis an(n − 1)-regular graph with n! vertices.

Let u = u1u2. . . u_n be anyvertex ofP_n. We use (u)_i to denote the ith component ui of u, and useP_n{i}to denote the ith subgraph ofP_n induced bythose vertices u with (u)n = i. Obviously, Pn can be decomposed into n vertex disjoint subgraphs P_n{i} for everyi ∈ n such that each P_n{i}is isomorphic toP_n−1. Thus, the pancake graph can be constructed recursively. LetH ⊆ n, we use P_nH to denote the subgraph ofP_n induced by∪_i∈HV (P_n{i}). Bydefinition, there is exactlyone neighbor v of u such that u and v are adjacent through an i-dimensional edge with 2
i
n. For this reason, we use (u)ito denote the unique i-neighbor of u. We have((u)i)i = u and (u)n ∈ P{(u)1}

n . For 1
i, j
n and i = j, we use Ei,j _{to denote the set of edges betweenP}{i}

n andP_n{j}. The pancake graphs P2,P3, andP4are shown in Fig.1 for illustration.

The following theorem is proved byHung et al. [14].

Theorem 1 (Hung et al.[14]). P_n is 1∗-connected if n = 3, and P_n is 2∗-connected if n3.

4132 3142 2314 4231 4321 3421 3214 a 1432 3412 1324 3241 4213 1423 2413 2431 b 1243 2143 4123 a 123 321 213 231 312 132 12 21 P4 P3 P2 1342 4312 1234 2134 3124 2341 b

Fig. 1. The pancake graphsP2,P3, andP4.

Lemma 2. Let u and v be any two distinct vertices ofP_nwithd(u, v)
2. Then(u)1= (v)1. Moreover,{((u)i)1| 2
i
n − 1} = n − {(u)1, (u)_n} if n3.

Lemma 3. Letn5 andH = {i1, i2, . . . , i_m} be any nonempty subset of n. There is a hamiltonian path ofP_nH joining any vertex u∈ P{i1}

n to any other vertex v∈ P_n{im}. Proof. Note thatP_n{ij} is isomorphic toP_n−1 for every1
j
m. We set x1 = u and

ym = v. ByTheorem1, this theorem holds form = 1. Assume that m2. ByLemma 1, we choose(yj, xj+1) ∈ Eij,ij+1 with yj = xjand ym = xm for every1
j
m − 1. By

Theorem 1, there is a hamiltonian pathQ_jofP_n{ij}joining xjto yjfor every1
j
m. The

pathx1, Q1, y1, x2, Q2, y2, . . . , xm, Qm, ym forms a desired path.  3. The super connectivity of the pancake graphs

Lemma 4. Letn5. Let u and v be any two distinct vertices inP_n{t}for somet ∈ n. If Pn−1isk∗-connected, then there is a(k + 1)∗-container ofP_nbetween u and v.

Proof. SinceP_n{t}is isomorphic toP_n−1, there is ak∗-container{Q1, Q2, . . . , Q_k} of P_n{t} joining u to v. We need to find a(k + 1)∗-container ofP_njoining u to v. We setp = (u)1 andq = (v)1.

Case 1:p = q. Thus, (u)n and(v)n are inP_n{p}. ByLemma3, there is a hamiltonian path Q ofP_n{p}joining (u)n to(v)n. We write Q as(u)n, Q, y, z, (v)n. ByLemma 2, (y)1 = (z)1,(y)1 = t, and (z)1 = t. ByLemma 3, there is a hamiltonian path R of Pnn−{t,p}joining (y)n to(z)n. We setQ_k+1 asu, (u)n, Q, y, (y)n, R, (z)n, z, (v)n, v.

Q1 u P_n{t} (v)n Qk Q2 (u)n Qk+1 P_n{p} v y z P_n<n>-{t,p} (y)n (z)n Q₁ u P_n{t} (v)n Q_k Q₂ (u)n Q_k+1 P_n<n>-{t} v (a) _(b)

Fig. 2. Illustration for Lemma4.

Then{Q1, Q2, . . . , Q_k+1} forms a (k + 1)∗-container ofP_njoining u to v. See Fig.2a for illustration.

Case 2:p = q. Thus, (u)n and(v)n are in different subgraphs P_n{p} andP_n{q}. By Lemma 3, there is a hamiltonian path Q ofP_nn−{t} joining(u)nto(v)n. We setQ_k+1as

u, (u)n_{, Q, (v)}n_{, v. Then {Q}

1, Q2, . . . , Qk+1} forms a (k + 1)∗-container ofPnjoining u to v. See Fig. 2b for illustration.

Thus, the theorem is proved. 

Lemma 5. Letn5 and k be any positive integer with 3
k
n − 1. Let u be any vertex in Pn{s}and v be any vertex inP_n{t}such thats = t. Suppose that P_n−1isk∗-connected. Then there is ak∗-container ofP_nbetween u and v not using the edge(u, v) if (u, v) ∈ E(P_n). Proof. Since|Es,t| = (n − 2)!6, we can choose a vertex y inP_n{s}− {u} and a vertex z inP_n{t}− {v} with (y, z) ∈ Es,t. Note thatP_n{s}andP_n{t}are both isomorphic toP_n−1. Let{R1, R2, . . . , R_k} be a k∗-container ofP_n{s}joining u to y, and{H1, H2, . . . , H_k} be a k∗_{-container ofP}{t}

n joining z to v. We writeR_i = u, R_i, yi, y and Hi = z, zi, H_i, v.

(Note that yi = u if the length of R_i is zero and zi = v if the length of H_i is zero.)

Let I = {yi| 1
i
k} and J = {zi| 1
i
k}. Note that (yi)1 = (y)j for somej ∈

{2, 3, . . . , n − 1}, and (y)l = (y)mifl = m. ByLemma2,{(yi)1| 1
i
k} ∩ {s, t} = ∅. Similarly,{(zi)1| 1
i
k} ∩ {s, t} = ∅. Let A = {yi| yi ∈ I and there exists an element zj∈ J such that (yi)1 = (zj)1}. Then we relabel the indices of I and J such that (yi)1 = (zi)1for 1
i
|A|. We set X as {(yi)1| 1
i
k − 2} ∪ {(zi)1| 1
i
k − 2} ∪ {s, t}. By Lemma 3, there is a hamiltonian pathT_i of P{(yi)1,(zi)1}

n joining (yi)n to(zi)n for every

1
i
k − 2, and there is a hamiltonian path T_k−1 ofP_nn−X joining (yk−1)n to(zk)n.

(Note that{(yi)1, (zi)1} = {(yi)1} if (yi)1= (zi)1.) We set

Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
k − 2,

Qk−1= u, Rk−1 , yk−1, (yk−1)n, Tk−1, (zk)n, zk, H_k, v, and

Qk= u, Rk, yk, y, z, zk−1, H_k−1 , v.

It is easyto check that{Q1, Q2, . . . , Qk} forms a k∗-container ofPn joining u to v not using the edge(u, v) if (u, v) ∈ E(P_n). See Fig. 3 for illustration. 

z v Hk-1' Pn {t} zk-1 y u Rk-1' Pn {s} yk-1 Tk-1 (zk)n Rk' y_k z_k Hk' Pn<n>-X (yk-1)n y1 yk-2 T1 (z1)n (y1) Tk-2 (zk-2)n (yk-2)n z1 H1' zk-2 H_k-2' Pn {(y 1)1,(z1)1} Pn {(y_k-2)₁,(z_k-2)₁} R1' Rk-2'

Fig. 3. Illustration for Lemma5.

Theorem 2. P_nis(n − 1)∗-connected ifn2.

Proof. It is easyto see thatP2 is 1∗-connected andP3is 2∗-connected. Since theP4is vertex transitive, we claim thatP4is 3∗-connected bylisting all 3∗-containers from 1234 to anyvertex as follows: (1234), (2134), (4312) (1234), (3214), (4123), (2143), (3412), (4312) (1234), (4321), (2341), (1432), (4132), (2314), (1324), (3124), (4213), (1243), (3421), (2431), (4231), (3241), (1423), (2413), (3142), (1342), (4312) (1234), (2134), (3124), (1324), (4231), (2431), (1342) (1234), (3214), (2314), (4132), (1432), (2341), (3241), (1423), (4123), (2143), (3412), (4312), (1342) (1234), (4321), (3421), (1243), (4213), (2413), (3142), (1342) (1234), (2134), (3124), (4213), (2413), (1423), (3241), (2341), (1432), (3412), (4312), (1342), (3142), (4132), (2314), (1324), (4231), (2431), (3421) (1234), (3214), (4123), (2143), (1243), (3421) (1234), (4321), (3421) (1234), (2134), (3124), (1324), (4231), (2431), (3421), (4321) (1234), (3214), (2314), (4132), (1432), (3412), (4312), (1342), (3142), (2413), (4213), (1243), (2143), (4123), (1423), (3241), (2341), (4321) (1234), (4321) (1234), (2134), (3124), (1324), (4231), (2431) (1234), (3214), (2314), (4132), (3142), (2413), (4213), (1243), (2143), (4123), (1423), (3241), (2341), (1432), (3412), (4312), (1342), (2431) (1234), (4321), (3421), (2431) (1234), (2134), (3124), (4213), (1243), (2143), (3412), (4312), (1342), (3142), (2413), (1423) (1234), (3214), (4123), (1423) (1234), (4321), (3421), (2431), (4231), (1324), (2314), (4132), (1432), (2341), (3241), (1423) (1234), (2134), (3124), (4213), (2413), (3142), (1342), (4312), (3412), (1432), (4132), (2314), (1324), (4231), (2431), (3421), (1243), (2143), (4123) (1234), (3214), (4123) (1234), (4321), (2341), (3241), (1423), (4123) (1234), (2134), (3124), (4213), (2413), (1423), (3241), (4231) (1234), (3214), (4123), (2143), (1243), (3421), (2431), (4231) (1234), (4321), (2341), (1432), (3412), (4312), (1342), (3142), (4132), (2314), (1324), (4231) (1234), (2134), (3124), (1324), (2314), (4132), (3142), (1342), (4312), (3412), (1432), (2341), (3241) (1234), (3214), (4123), (2143), (1243), (4213), (2413), (1423), (3241) (1234), (4321), (3421), (2431), (4231), (3241)

(1234), (2134), (4312), (1342), (2431), (3421), (1243), (4213), (3124), (1324), (4231), (3241), (2341) (1234), (3214), (2314), (4132), (3142), (2413), (1423), (4123), (2143), (3412), (1432), (2341) (1234), (4321), (2341) (1234), (2134), (3124), (4213), (2413) (1234), (3214), (4123), (2143), (1243), (3421), (2431), (4231), (1324), (2314), (4132), (1432), (3412), (4312), (1342), (3142), (2413) (1234), (4321), (2341), (3241), (1423), (2413) (1234), (2134), (3124), (1324), (4231), (2431), (3421), (1243) (1234), (3214), (2314), (4132), (1432), (3412), (4312), (1342), (3142), (2413), (4213), (1243) (1234), (4321), (2341), (3241), (1423), (4123), (2143), (1243) (1234), (2134), (3124), (1324), (2314), (3214) (1234), (3214) (1234), (4321), (3421), (2431), (4231), (3241), (2341), (1432), (4132), (3142), (1342), (4312), (3412), (2143), (1243), (4213), (2413), (1423), (4123), (3214) (1234), (2134), (3124), (4213), (2413), (3142), (4132), (2314) (1234), (3214), (2314) (1234), (4321), (3421), (1243), (2143), (4123), (1423), (3241), (2341), (1432), (3412), (4312), (1342), (2431), (4231), (1324), (2314) (1234), (2134), (4312), (3412), (1432), (4132), (3142), (1342), (2431), (3421), (1243), (2143), (4123), (1423), (2413), (4213), (3124), (1324) (1234), (3214), (2314), (1324) (1234), (4321), (2341), (3241), (4231), (1324) (1234), (2134), (3124) (1234), (3214), (4123), (1423), (3241), (4231), (2431), (3421), (1243), (2143), (3412), (4312), (1342), (3142), (2413), (4213), (3124) (1234), (4321), (2341), (1432), (4132), (2314), (1324), (3124) (1234), (2134) (1234), (3214), (2314), (1324), (3124), (2134) (1234), (4321), (2341), (3241), (4231), (2431), (3421), (1243), (4213), (2413), (1423), (4123), (2143), (3412), (1432), (4132), (3142), (1342), (4312), (2134) (1234), (2134), (4312), (1342), (3142) (1234), (3214), (4123), (1423), (2413), (3142) (1234), (4321), (3421), (2431), (4231), (3241), (2341), (1432), (3412), (2143), (1243), (4213), (3124), (1324), (2314), (4132), (3142) (1234), (2134), (3124), (1324), (2314), (4132) (1234), (3214), (4123), (1423), (2413), (4213), (1243), (2143), (3412), (4312), (1342), (3142), (4132) (1234), (4321), (3421), (2431), (4231), (3241), (2341), (1432), (4132) (1234), (2134), (4312), (1342), (3142), (2413), (1423), (3241), (4231), (2431), (3421), (1243), (4213), (3124), (1324), (2314), (4132), (1432) (1234), (3214), (4123), (2143), (3412), (1432) (1234), (4321), (2341), (1432) (1234), (2134), (3124), (1324), (2314), (4132), (3142), (1342), (4312), (3412) (1234), (3214), (4123), (1423), (2413), (4213), (1243), (2143), (3412) (1234), (4321), (3412), (2431), (4231), (3241), (2341), (1432), (3412) (1234), (2134), (4312), (1342), (3142), (4132), (2314), (1324), (3124), (4213) (1234), (3214), (4123), (1423), (2413), (4213) (1234), (4321), (3421), (2431), (4231), (3241), (2341), (1432), (3412), (2143), (1243), (4213) (1234), (2134), (4312), (1342), (3142), (4132), (1432), (3412), (2143) (1234), (3214), (2314), (1324), (3124), (4213), (2413), (1423), (4123), (2143) (1234), (4321), (2341), (3241), (4231), (2431), (3421), (1243), (2143)

Assume thatP_k is(k − 1)∗-connected for every4
k
n − 1. Let u and v be anytwo distinct vertices ofP_nwith u∈ P_n{s}and v∈ P_n{t}. We need to find an(n − 1)∗-container between u and v ofP_n. Suppose thats = t. ByLemma4, there is an(n − 1)∗-container of Pnjoining u to v. Thus, we assume thats = t. We set p = (u)1andq = (v)1.

Case 1:p = t and q = s. Thus, (u)n∈ P_n{t}and(v)n∈ P_n{s}.

Subcase 1.1: u= (v)n. Thus,(u, v) ∈ E(P_n). ByLemma 5, there is an (n−2)∗-container

{Q1, Q2, . . . , Qn−2} of Pnjoining u to v not using the edge(u, v). We set Qn−1asu, v. Then{Q1, Q2, . . . , Qn−1} forms an (n − 1)∗-container ofPnjoining u to v.

Subcase 1.2: u = (v)n. We set y = (v)n and z= (u)n. Let{R1, R2, . . . , Rn−2} be an (n − 2)∗_{-container of P}{s}

n joining u to y, and let {H1, H2, . . . , Hn−2} be an (n − 2)∗ -container ofP_n{t}joining z to v. We writeR_i = u, R_i, yi, y and Hi = z, zi, H_i, v. We set

I = {(yi)1| 1
i
n − 2} and J = {(zi)1| 1
i
n − 2}. Note that (yi)1= (y)j for some j ∈ {2, 3, . . . , n − 1}, and (y)k= (y)lifk = l. ByLemma 2, I = {(y)_i| 2
i
n − 1} =

Fig. 4. Illustration for Theorem2.

n − {s, t}. Similarly, J = n − {s, t}. We have I = J . Without loss of generality, we assume that(yi)1= (zi)1for every1
i
n − 2. ByLemma3, there is a hamiltonian path Ti ofP{(yi)1}

n joining(yi)nto(zi)nfor every1
i
n − 4, and there is a hamiltonian path

Tn−3ofP{(yn−3)1,(yn−2)1} n joining(yn−3)nto(zn−2)n. We set Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
n − 4, Qn−3= u, Rn−3 , yn−3, (yn−3)n, Tn−3, (zn−2)n, zn−2, H_n−2 , v, Qn−2= u, z, zn−3, H_n−3 , v, and Qn−1= u, Rn−2 , yn−2, y, v.

Then{Q1, Q2, . . . , Qn−1} forms an (n − 1)∗-container ofPnjoining u to v. See Fig. 4a for illustration.

Case 2:p = t and q ∈ n − {s, t}. Since |Es,q| = (n − 2)!6, we can choose a vertex y inP_n{s}− {u} with (y)n ∈ P_n{q}. We set z= (u)n∈ P_n{t}. Let{R1, R2, . . . , Rn−2} be an(n − 2)∗-container ofP_n{s}joining u to y, and{H1, H2, . . . , Hn−2} be an (n − 2)∗ -container ofP_n{t}joining z to v. We writeR_i = u, R_i, yi, y and Hi = z, zi, H_i, v. We

have{(yi)1| 1
i
n − 2} = {(y)i| 2
i
n − 1}. ByLemma 2, {(yi)1| 1
i
n − 2} =

assume that(yi)1= (zi)1for every1
i
n−3, (yn−2)1= t, and (zn−2)1= q. ByLemma 3, there is a hamiltonian pathT_iofP{(yi)1}

n joining(yi)nto(zi)nfor every1
i
n − 3, and

there is a hamiltonian pathT_n−2ofP_n{q}joining(y)nto(v)n. We set Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
n − 3,

Qn−2= u, Rn−2 , yn−2, y, (y)n, Tn−2, (v)n, v, and

Qn−1= u, z, zn−2, H_n−2 , v.

Then{Q1, Q2, . . . , Qn−1} forms an (n − 1)∗-container ofPn joining u to v. See Fig. 4b for illustration.

Case 3:p, q ∈ n − {s, t}. Since |Es,t| = (n − 2)!6, there exists an edge(y, z) in Es,t with y∈ P_n{s}−{u} and z ∈ P_n{t}−{v}. Let {R1, R2, . . . , Rn−2} be an (n−2)∗-container of Pn{s}joining u to y, and let{H1, H2, . . . , Hn−2} be an (n−2)∗-container ofP_n{t}joining z to

v. We writeR_i = u, R_i, yi, y and Hi = z, zi, H_i, v. We set I = {(yi)1| 1
i
n−2} and

J = {(zi)1| 1
i
n−2}. We have I = {(y)i| 2
i
n−1}. ByLemma 2, I = n−{s, t}. Similarly,J = n − {s, t}. We have I = J . Without loss of generality, we assume that (yi)1= (zi)1for every1
i
n − 2 with (yn−2)1= p.

Subcase 3.1:p = q. ByLemma 3, there is a hamiltonian path T_iofP{(yi)1}

n joining(yi)n

to(zi)nfor everyi ∈ n − 3, and there is a hamiltonian path Tn−2inPn{p}joining(u)nto

(v)n_{. We set}

Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
n − 3,

Qn−2= u, Rn−2 , yn−2, y, z, zn−2, H_n−2 , v, and

Qn−1= u, (u)n, Tn−2, (v)n, v.

Then{Q1, Q2, . . . , Qn−1} forms an (n − 1)∗-container ofPnjoining u and v. See Fig. 4c for illustration.

Subcase 3.2:p = q. Without loss of generality, we assume that (yn−3)1= q. ByTheorem 1, there is a hamiltonian pathT_i of P{(yi)1}

n joining(yi)n to(zi)n for every1
i
n − 4,

there is a hamiltonian pathT_n−3ofP_n{q}joining(yn−3)nto(v)n, and there is a hamiltonian

pathT_n−2ofP_n{p}joining(u)nto(zn−2)n. We set

Qi= u, Ri, yi, (yi)n, Ti, (zi)n, H_i, v for 1
i
n − 4,

Qn−3= u, Rn−3 , yn−3, (yn−3)n, Tn−3, (v)n, v,

Qn−2= u, (u)n, Tn−2, (zn−2)n, zn−2, H_n−2 , v, and

Qn−1= u, Rn−2 , yn−2, y, z, zn−3, H_n−3 , v.

It is easyto check that{Q1, Q2, . . . , Qn−1} is an (n − 1)∗-container ofPnfrom u to v. See Fig. 4d for illustration.

Thus, the theorem is proved. 

Theorem 3. P_nis super connected if and only ifn = 3.

Proof. We prove this theorem byinduction. Obviously, this theorem is true forP1andP2. SinceP3is isomorphic to a cycle with six vertices,P3is not 1∗-connected. Thus,P3 is

not super connected. ByTheorems1 and 2, this theorem holds onP4. Assume thatPk is super connected for every4
k
n − 1. ByTheorems 1 and 2, P_nisk∗-connected for any k ∈ {1, 2, n − 1}. Thus, we still need to construct a k∗_{-container ofPnbetween anytwo}

distinct vertices u∈ P_n{s}and v∈ P_n{t}for every3
k
n − 2.

Suppose thats = t. Byinduction, P_n−1is(k − 1)∗-connected. ByLemma 4, there is a k∗_{-container ofPnjoining u to v. Suppose thats = t. Byinduction, Pn−1isk}∗_-connected.

ByLemma 5, there is ak∗-container ofP_njoining u to v. Hence, the theorem is proved. 

4. The star graphs

The n-dimensional star graph, denoted byS_n, is a graph with the vertex setV (S_n) =

{u1u2. . . un| ui ∈ n and ui = uj for i = j}. The adjacencyis defined as follows: u1u2. . . ui. . . un is adjacent to v1v2. . . vi. . . vn through an edge of dimension i with 2
i
n if v_j = u_j for j /∈ {1, i}, v1 = ui, andvi = u1. Again, we use bold face to denote a vertex ofS_n. Hence, u1, u2, . . . , un denote a sequence of vertices ofSn. In

par-ticular, e denotes the vertex 12. . . n. Bydefinition, S_nis an(n − 1)-regular graph with n! vertices.

It is known thatS_nis a bipartite graph with one partite set containing all odd permutations and the other partite set containing all even permutations. For convenience, we refer an even permutation as a white vertex, and refer an odd permutation as a black vertex. Let

u= u1u2. . . unbe anyvertex ofSn. We use(u)i to denote the ith componentui of u and Sn{i}to denote the ith subgraph ofS_ninduced bythose vertices u with(u)_n = i. Obviously, Sncan be decomposed into n vertex disjoint subgraphsS_n{i}for 1
i
n, such that each S_n{i} is isomorphic toS_n−1. Thus, the star graph can be constructed recursively. LetH ⊆ n. We useS_nH to denote the subgraph ofS_ninduced by∪_i∈HV (S_n{i}). Bythe definition of S_n, there is exactlyone neighbor v of u such that u and v are adjacent through an i-dimensional edge with 2
i
n. For this reason, we use (u)i to denote the unique i-neighbor of u. We have((u)i)i = u and (u)n∈ S{(u)1}

n . For 1
i, j
n and i = j, we use Ei,j to denote the set of edges betweenS{i}_n andS_n{j}. The star graphsS2,S3, andS4are shown in Fig. 5 for illustration.

The following theorem is proved byHsieh et al. [12].

Theorem 4 (Hsieh et al.[12]). S_nis 1∗-laceable ifn = 3, and S_nis 2∗-connected ifn3. Lemma 6. Assume thatn3.|Ei,j| = (n − 2)! for any 1
i = j
n. Moreover, there are

(n−2)!

2 edges joining black vertices ofS {i}

n to white vertices ofS_n{j}.

Lemma 7. Let u and v be any two distinct vertices ofS_nwithd(u, v)
2. Then(u)1= (v)1. Moreover,{((u)i)1| 2
i
n − 1} = n − {(u)1, (u)_n} if n3.

Lemma 8. Letn5 andH = {i1, i2, . . . , i_m} be any nonempty subset of n. There is a hamiltonian path ofSH_n joining any white vertex u∈ S{i1}

1423 2413 4213 3214 2134 1234 1342 3142 4132 c d f g 2314 1324 3124 a b e 4123 2143 1243 4231 3241 2431 3412 4312 1432 b c e f 2341 4321 3421 a d _g 123 213 321 312 231 132 12 21 S4 S3 S2

Fig. 5. The star graphsS2,S3, andS4.

Proof. Note thatS_n{ij}is isomorphic toS_n−1for every1
j
m. We set x1= u and ym = v.

ByTheorem4, this theorem holds form = 1. Assume that m2. ByLemma 6, we choose (yj, xj+1) ∈ Eij,ij+1 with yjis a black vertex ofSn{j}and xj+1is a white vertex ofSn{j+1}

for every1
j
m − 1. ByTheorem 4, there is a hamiltonian path Q_j ofS_n{ij}joining xj

to yj. The pathx1, Q1, y1, x2, Q2, y2, . . . , xm, Qm, ym forms a desired path.  5. The super laceability of the star graphs

In this section, we are going to prove thatS_nis super laceable if and onlyifn = 3. As you will observe, the proof is very similar to the proof thatP_n is super connected if and onlyifn = 3.

Lemma 9. Letn5 and k be any positive integer with 3
k
n − 1. Let u be any white vertex and v be any black vertex ofS_n. Suppose thatS_n−1isk∗-laceable. Then there is a k∗_{-container ofSnbetween u and v not using the edge(u, v) if (u, v) ∈ E(Sn).}

Proof. SinceS_n is edge transitive, we mayassume that u ∈ S_n{n}and v ∈ S_n{n−1}. By Lemma6, there are (n−2)!₂ 3 edges joining black vertices ofS_n{n} to white vertices of Sn{n−1}. We can choose an edge(y, z) ∈ En−1,n where y is a black vertex inS_n{n} and z is a white vertex inS_n{n−1}. Byinduction, there is ak∗-container{R1, R2, . . . , Rk} of Sn{n} joining u to y, and there is ak∗-container{H1, H2, . . . , Hk} of S{n−1}n joining z to v. We writeR_i = u, R_i, yi, y and Hi = z, zi, H_i, v. Note that yiis a white vertex and zi is

a black vertex for every1
i
k. Let I = {yi| (yi, y) ∈ E(Ri) and 1
i
k}, and J =

{zi| (zi, x) ∈ E(Hi) and 1
i
k}. Note that (yi)1= (y)j for somej ∈ {2, 3, . . . , n − 1}, and(y)_l = (y)_mifl = m. ByLemma 7, {(yi)1| 1
i
k} ∩ {n − 1, n} = ∅. Similarly,

such that(yi)1= (zj)1}. Then we relabel the indices of I and J such that (yi)1= (zi)1for

1
i
|A|. We set X as {(yi)1| 1
i
k −2}∪{(zi)1| 1
i
k −2}∪{n−1, n}. ByLemma 8, there is a hamiltonian pathT_i ofS{(yi)1,(zi)1}

n joining the black vertex(yi)nto the white

vertex(zi)nfor every1
i
k − 2, and there is a hamiltonian path Tk−1ofSnn−Xjoining

the black vertex(yk−1)n to the white vertex (zk)n. (Note that{(yi)1, (zi)1} = {(yi)1} if (yi)1= (zi)1.) We set

Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
k − 2,

Qk−1= u, Rk−1 , yk−1, (yk−1)n, Tk−1, (zk)n, zk, H_k, v, and

Qk= u, Rk, yk, y, z, zk−1, H_k−1 , v.

It is easyto check that{Q1, Q2, . . . , Qk} forms a k∗-container ofSn joining u to v not using the edge(u, v) if (u, v) ∈ E(S_n). 

Theorem 5. S_nis(n − 1)∗-laceable ifn2.

Proof. It is easyto see thatS2is 1∗-laceable andS3is 2∗-laceable. Since theS4is vertex transitive, we claim thatS4is 3∗-laceable bylisting all 3∗-containers from the white vertex 1234 to anyblack vertex as follows:

(1234), (2134) (1234), (3214), (2314), (4312), (1342), (2341), (4321), (1324), (3124), (2134) (1234), (4231), (3241), (1243), (4213), (2413), (3412), (1432), (2431), (3421), (1423), (4123), (2143), (3142), (4132), (2134) (1234), (3214) (1234), (4231), (3241), (2341), (1342), (3142), (2143), (1243), (4213), (3214) (1234), (2134), (4132), (1432), (2431), (3421), (4321), (1324), (3124), (4123), (1423), (2413), (3412), (4312), (2314), (3214) (1234), (4231) (1234), (2134), (4132), (3142), (1342), (4312), (3412), (1432), (2431), (4231) (1234), (3214), (2314), (1324), (3124), (4123), (2143), (1243), (4213), (2413), (1423), (3421), (4321), (2341), (3241), (4231) (1234), (2134), (3124), (1324), (2314), (4312), (1342), (3142), (4132), (1432), (3412), (2413), (1423), (4123), (2143), (1243) (1234), (3214), (4213), (1243) (1234), (4231), (2431), (3421), (4321), (2341), (3241), (1243) (1234), (2134), (4132), (1432) (1234), (3214), (2314), (1324), (3124), (4123), (1423), (2413), (4213), (1243), (2143), (3142), (1342), (4312), (3412), (1432) (1234), (4231), (3241), (2341), (4321), (3421), (2431), (1432) (1234), (2134), (4132), (3142), (1342), (4312), (3412), (1432), (2431), (3421), (1423), (2413), (4213), (1243), (2143), (4123), (3124), (1324) (1234), (3214), (2314), (1324) (1234), (4231), (3241), (2341), (4321), (1324) (1234), (2134), (3124), (1324), (2314), (4312), (3412), (1432), (4132), (3142), (1342), (2341) (1234), (3214), (4213), (1423), (2413), (4123), (2143), (1243), (3241), (2341) (1234), (4231), (2431), (3421), (4321), (2341) (1234), (2134), (4132), (3142), (1342), (4312), (3412), (1432), (2431), (3421) (1234), (3214), (2314), (1324), (3124), (4123), (2143), (1243), (4213), (2413), (1423), (3421) (1234), (4231), (3241), (2341), (4321), (3421) (1234), (2134), (3124), (1324), (2314), (4312) (1234), (3214), (4213), (1243), (2143), (4123), (1423), (2413), (3412), (4312) (1234), (4231), (3241), (2341), (4321), (3421), (2431), (1432), (4132), (3142), (1342), (4312) (1234), (2134), (4132), (1432), (3412), (4312), (1342), (3142), (2143), (4123) (1234), (3214), (2314), (1324), (3124), (4123) (1234), (4231), (2431), (3421), (4321), (2341), (3241), (1243), (4213), (2413), (1423), (4123) (1234), (2134), (4132), (3142) (1234), (3214), (2314), (1324), (3124), (4123), (1423), (2413), (4213), (1243), (2143), (3142) (1234), (4231), (3241), (2341), (4321), (3421), (2431), (1432), (3412), (4312), (1342), (3142) (1234), (2134), (3124), (1324), (2314), (4312), (1342), (3142), (4132), (1432), (3412), (2413) (1234), (3214), (4213), (2413) (1234), (4231), (2431), (3421), (4321), (2341), (3241), (1243), (2143), (4123), (1423), (2413)

Assume thatS_k is(k − 1)∗-laceable for every4
k
n − 1. We need to construct an (n − 1)∗_{-container ofSnbetween anywhite vertex u to anyblack vertex v.}

Case 1:d(u, v) = 1. We have (u, v) ∈ E(S_n). Byinduction, S_n−1is(n − 2)∗-laceable. ByLemma 9, there exists a(n − 2)∗-container{Q1, Q2, . . . , Qn−2} of Sn joining u to

v not using the edge(u, v). We set Q_n−1 asu, v. Then {Q1, Q2, . . . , Qn−1} forms an

(n − 1)∗_{-container ofSnjoining u to v.}

Case 2:d(u, v)3. We have star graph is edge transitive. Without loss of generality, we mayassume that u∈ S_n{n}and v∈ S_n{n−1}with(u)1 = n − 1 and (v)1= n. ByLemma 6, there are (n−2)!₂ 3 edges joining black vertices ofS_n{n}to white vertices ofS_n{n−1}. We can choose an edge(y, z) ∈ En−1,n where y is a black vertex inS_n{n} and z is a white vertex inS_n{n−1}. Let{R1, R2, . . . , Rn−2} be an (n − 2)∗-container ofSn{n}joining u to y, and let{H1, H2, . . . , Hn−2} be an (n − 2)∗-container ofSn{n−1}joining z to v. We write Ri = u, Ri, yi, y and Hi = z, zi, H_i, v. Note that yiis a white vertex and ziis a black

vertex for every1
i
n − 2. We have {(yi)1| 1
i
n − 2} = {(zi)1| 1
i
n − 2} =

n − 2. Without loss of generality, we assume that (yi)1 = (zi)1for every1
i
n − 2 with(yn−2)1= (u)1.

Subcase 2.1:(u)1= (v)1. ByTheorem 4, there is a hamiltonian pathTiofSn{(yi)1}joining the black vertex(yi)nto the white vertex(zi)nfor everyi ∈ n−3, and there is a hamiltonian

path H ofS{(yn−2)1}

n joining the black vertex(u)nto the white vertex(v)n. We set Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
n − 3,

Qn−1= u, Rn−2 , yn−2, y, z, zn−2, H_n−2 , v, and

Qn−2= u, (u)n, H, (v)n, v, .

Then{Q1, Q2, . . . , Qn−1} forms an (n − 1)∗-container ofSnjoining u and v.

Subcase 2.2:(u)1 = (v)1. Without loss of generality, we assume that(yn−3)1 = (v)1. ByTheorem 4, there is a hamiltonian pathT_i of S{(yi)1}

n joining(yi)n to(zi)n for every

i ∈ n − 4, there is a hamiltonian path H of S{(yn−3)1}

n joining the black vertex(yn−3)nto

the white vertex(v)n, and there is a hamiltonian path P ofS_n{(yn−2)1}joining the black vertex (u)n_{to the white vertex(z}

n−2)n. We set

Qi= u, Ri, yi, (yi)n, Ti, (zi)n, zi, H_i, v for 1
i
n − 4,

Qn−3= u, Rn−3 , yn−3, (yn−3)n, H, (v)n, v, ,

Qn−2= u, (u)n, P, (zn−2)n, zn−2, H_n−2 , v, , and

Qn−1= u, Rn−2 , yn−2, y, z, zn−3, H_n−3 , v.

It is easyto check that{Q1, Q2, . . . , Qn−1} is an (n − 1)∗-container ofSnjoining u to v. Thus, this theorem is proved. 

Theorem 6. S_nis super laceable if and only ifn = 3.

Proof. It is easyto see that this theorem is true forS1andS2. SinceS3is isomorphic to a cycle with six vertices,S3is not 1∗-laceable. Thus,S3is not super laceable. ByTheorems 4 and 5, this theorem holds onS4. Assume thatSkis super laceable for every4
k
n − 1. ByTheorems 4 and 5,S_nisk∗-laceable for anyk ∈ {1, 2, n − 1}. Thus, we still need to

construct ak∗-container ofS_nbetween anywhite vertex u and anyblack vertex v for every 3
k
n − 2. Byinduction, S_n−1isk∗-laceable. ByLemma9, there is ak∗-container of Snjoining u to v. 

6. Further study

In this paper, we prove that the pancake graphP_nis super connected forn = 3 and the star graphsS_nis super laceable forn = 3. We believe that there are other super connected and super laceable graphs. It would be veryinteresting to classifysuch graphs.

We mayalso studythe fault tolerantk∗-connectivityfor anysuper connected graph. For example, letF ⊂ V (P_n) ∪ E(P_n) with |F | = f
n − 3. Obviously, P_n− F is (n − 1 − f ) connected. However, we believe thatP_n− F is (n − 1 − f )∗-connected. Similarly, we can studythe fault tolerantk∗-laceabilityfor anysuper laceable graph. For example, let F ⊂ E(Sn) with |F | = f
n − 3. Obviously, Sn− F is (n − 1 − f ) connected. However,

we believe thatS_n− F is (n − 1 − f )∗-connected.

Assume that G isk∗-connected. We mayalso define thek∗-connected distance between anytwo vertices u andv, denoted by d_ks(u, v), which is the minimum length among all k∗ -containers between u andv. The k∗-diameter of G, denote byDs_k(G), is max{d_ks(u, v) | u and v are two different vertices of G}. In particular, we are intrigued in Ds
_(G)(G) and Ds

2(G). Similarly, we define the ksL-laceable distance on bipartite graph between anytwo vertices u andv from different partite sets, denoted by d_ksL(u, v), which is the minimum length among allk∗-containers between u andv. The ksL_{-diameter of G, denoted byD}sL

k (G),

is max{d_ksL(u, v) | u and v are vertices from different partite sets}. Again, we are intrigued inD
_(G)sL (G) and D₂sL(G).

References

[1]S.B. Akers, B. Krishnameurthy, A group-theoretic model for symmetric interconnection networks, IEEE

Trans. Comput. 38 (1989) 555–566.

[2]M. Albert, R.E.L. Aldred, D. Holton, On 3∗-connected graphs, Australasian J. Combin. 24 (2001) 193–208.

[3]J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North Holland, New York, 1980.

[4]A. Bouabdallah, M.C. Heydemann, J. Opatmy, D. Sotteau, Embedding complete binary trees into star and

pancake graphs, Theoryof Comput. Syst. 31 (1998) 279–305.

[5]K. Day, A. Tripathi, A comparative study of topological properties, IEEE Trans. Parallel and Distributed Syst.

5 (1994) 31–38.

[6]W.C. Fang, C.C. Hsu, On the fault-tolerant embedding of complete binarytree in the pancake graph

interconnection network, Inform. Sci. 126 (2000) 191–204.

[7]P. Fragopoulou, S.G. Akl, Optimal communication algorithms on the star graphs using spanning tree

constructions, J. Parallel and Distributed Comput. 23 (1995) 55–71.

[8]P. Fragopoulou, S.G. Akl, Edge-disjoint spanning trees on the star networks with applications to fault

tolerance, IEEE Trans. Comput. 45 (1996) 174–185.

[9]L. Gargano, U. Vaccaro, A. Vozella, Fault tolerant routing in the star and pancake interconnection networks,

Inform. Process. Lett. 45 (1993) 315–320.

[10]W.H. Gates, C.H. Papadimitriou, Bounds for sorting byprefix reversal, Discrete Math. 27 (1979) 47–57.

[11]M.H. Heydari, I.H. Sudborough, On the diameter of the pancake network, J. Algorithms 25 (1997) 67–94.

[13]D.F. Hsu, On container width and length in graphs, groups, and networks, IEICE Trans. Fundamentals of Electron. E 77A (1994) 668–680.

[14]C.N. Hung, H.C. Hsu, K.Y. Liang, L.H. Hsu, Ring embedding in faultypancake graphs, Inform. Process.

Lett. 86 (2003) 271–275.

[15]J.S. Jwo, S. Lakshmivarahan, S.K. Dhall, Embedding of cycles and grids in star graphs, J. Circuits, Systems,

and Comput. 1 (1991) 43–74.

[16]A. Kanevsky, C. Feng, On the embedding of cycles in pancake graphs, Parallel Comput. 21 (1995) 923–936.

[17]S. Latifi, On the fault-diameter of the star graph, Inform. Process. Lett. 46 (1993) 143–150.

[18]F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays. Trees. Hypercubes, Morgan

Kaufmann, San Mateo, CA, 1992.

[19]T.K. Li, J.M. Tan, L.H. Hsu, Hyper Hamiltonian Laceability on the Edge Fault Star Graph, Inform. Sci. 165

(2004) 59–71.

[20]K. Menger, Zur allgemeinen Kurventheorie, Fundamenta Math. 10 (1927) 95–115.

[21]K. Qiu, H. Meijer, S.G. Akl, Decomposing a star graph into disjoint cycles, Inform. Process. Lett. 39 (1991)

125–129.

[22]Y. Rouskov, S. Latifi, P.K. Srimani, Conditional fault diameter of star graph networks, J. Parallel and

Distributed Comput. 33 (1996) 91–97.