Hamiltonian circuit and linear array embeddings in faulty k-ary n-cubes

Hamiltonian circuit and linear array embeddings in faulty k-ary n-cubes

Ming-Chien Yang

, Jimmy J.M. Tan

, Lih-Hsing Hsu

a_{Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC} b_{Department of Information Engineering, Ta Hwa Institute of Technology, Hsinchu County 307, Taiwan, ROC}

Received 7 July 2004; received in revised form 4 August 2005; accepted 6 October 2005 Available online 8 February 2007

Abstract

In this paper, we investigate the fault-tolerant capabilities of the k-ary n-cubes for even integer k with respect to the hamiltonian and hamiltonian-connected properties. The k-ary n-cube is a bipartite graph if and only if k is an even integer. Let F be a faulty set with nodes and/or links, and let k  3 be an odd integer. When |F |  2n − 2, we show that there exists a hamiltonian cycle in a wounded k-ary n-cube. In addition, when|F |2n−3, we prove that, for two arbitrary nodes, there exists a hamiltonian path connecting these two nodes in a wounded k-ary n-cube. Since the k-ary n-cube is regular of degree 2n, the degrees of fault-tolerance 2n − 3 and 2n − 2 respectively, are optimal in the worst case. © 2005 Elsevier Inc. All rights reserved.

Keywords: Cycle embeddings; Hamiltonian; k-ary n-cube; Fault tolerance; Linear array embeddings

1. Introduction

In many parallel computer systems, processors are connected based on an interconnection network. Such networks usually have a regular degree, i.e., every node is incident with the same number of links. Popular instances of interconnection networks include hypercubes, star graphs, meshes, the k-ary n-cubes, etc.

The k-ary n-cube, denoted by Qk_n, is regular of degree 2n, edge symmetric, and vertex symmetric. Several properties of it has been studied in the literature. For example, in [3,4], meshes and hamiltonian cycles are embedded into healthy k-ary n-cubes, and the connectivity of Qkn is shown to be 2n, which equals the degree of each vertex. Furthermore, message routing and single-node broadcasting algorithms are given in [4]. The problem of conditional node connectivity on Qknis investigated in [6]. Cycles are said to be disjoint if they share no edges. In [2], n edge disjoint hamiltonian cycles are found in Qk

n. In [1], Ashir and Stewart studied the problem of hamiltonian cycle embeddings in Qk

n with a possibility of link failures.

Hamiltonian circuit and linear array embeddings are desired properties in an interconnection network [5,9,14]. Many works related to embeddings of longest cycles and paths in various in-terconnection networks have been studied previously, including

∗_{Corresponding author. Fax: +886 35721490.}

E-mail addresses:[email protected](M.-C. Yang), [email protected](J.J.M. Tan),[email protected](L.-H. Hsu). 0743-7315/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jpdc.2005.10.004

hypercubes[5,11], k-ary n-cubes [1], stars [8,14], arrangement graphs [9,12], etc.

Ashir and Stewart [1] showed that, with only edge faults and under the condition that every node is incident with at least two fault-free edges, a wounded k-ary n-cube still has a hamiltonian circuit, provided that there are no more than 4n−5 faulty edges. The situation of having both faulty nodes and faulty links remains unanswered, and the hamiltonian linear array embeddings in Qk

nhave not been discussed yet even in a healthy Qk

n.

Since failures are inevitable, fault-tolerance is an important issue in multiprocessor systems. In this paper, we consider a possibility of both node and link failures, and discuss the fault-tolerant capabilities of the k-ary n-cubes with respect to the hamiltonian and hamiltonian-connected properties. Let

F be a faulty set with nodes and/or links. We observe that Qk_n is bipartite if and only if k is even. When k is even and there is a faulty node, there exists neither a hamiltonian cycle nor a hamiltonian path between two vertices in different par-tite sets in a wounded Qkn. Therefore, throughout this paper, we suppose that k is an odd integer with k 3. Then, a ring of maximum length, or a hamiltonian cycle, in a wounded

Qk_n can be constructed, provided that|F |2n − 2 for n2. On the other hand, if |F |2n − 3 for n2, we provide a construction of a linear array of maximum length, or a hamil-tonian path, connecting two arbitrary vertices in a wounded

The reason is as follows. First, any hamiltonian cycles cannot be found in a wounded Qk_nwhen there are 2n − 1 faulty edges incident to a single node. Second, suppose that there are 2n − 2 edge faults incident to a node x. Let y and z be two nodes of Qkn

incident to x. Then, there is no hamiltonian path connecting y and z when all the edges incident to x are faulty except (x, y) and (x, z).

The rest of this paper is organized as follows. We give some definitions, notation, and terminology in Section 2. Using the recursive structure of the k-ary n-cubes, we construct rings and linear arrays, respectively, traversing all the nodes in wounded

k-ary n-cubes in Section 3. Finally, in Section 4, we present the

conclusion.

2. Preliminaries

Throughout this paper, an interconnection network is rep-resented by an undirected simple graph G. Given a graph

G, we denote the vertex set and the edge set as V (G) and E(G), respectively. A path, denoted by v1, v2, . . . , vk, is a

sequence of adjacent vertices where all the vertices are distinct except possibly v1 = vk. We say that a path is a hamiltonian

path if it traverses all the vertices of G exactly once. A cy-cle is a path that begins and ends with the same vertex. A hamiltonian cycle is a cycle which includes all the vertices

of G. A graph is hamiltonian if it has a hamiltonian cycle. A graph G is hamiltonian connected if, for any two arbitrary vertices x and y in G, there is a hamiltonian path connecting x and y.

We consider the fault-tolerance of a graph G in the following. Let F be a faulty set which may contain both vertices and edges. Let Fv = F



V (G) and Fe = F



E(G). G − F

denotes the subgraph of G − Fe induced by V (G) − Fv. Let k

be a positive integer. A graph G is k-fault-tolerant hamiltonian (abbreviated as k-hamiltonian) if G − F is hamiltonian for every F with|F |k. A graph G is k-fault-tolerant hamiltonian

connected (abbreviated as k-hamiltonian connected) if G − F

is hamiltonian connected for every F with|F |k.

The k-ary n-cube Qk_n is a graph consisting of kn vertices labeled by the integers from 0 to kn− 1 for k 3 and n1. Two vertices are adjacent if and only if the representations of their labels in base k differ by one (modulo k) in exactly one position. We refer to (x, y) ∈ E(Qkn) where x differs from y in

the dth position, for 0d n − 1, as an edge of dimension d. We say that Qk

n is divided into Qkn[0], Qkn[1], . . . , Qkn[k − 1]

(abbreviated as Q[0], Q[1], . . . , Q[k − 1], if there are no am-biguities) along dimension d for some 0d n − 1 if Q[l], for every 0l k − 1, is a subgraph of Qk

n induced by the

vertices labeled by xn−1. . . xd+1lxd−1. . . x0(see Fig.1). It is

clear that each Q[l] is isomorphic to Qk_n−1 for 0l k − 1. Note that Qkn can be divided into k copies of Qkn−1 along

n different dimensions. For 0i, j k − 1, we use [i, j] to

denote a set of integers: [i, j] = {l | i l j} if i j, and [i, j] = {l | i l k − 1 or 0l j} if i > j. Qk

n[i, j] (abbreviated as Q[i, j ] if there is no ambiguity) denotes the subgraph of Qk

n which is induced by {u | u ∈ V (Q[l]);

l ∈ [i, j ]}.

Q[0] Q[1] Q[k-2] Q[k-1]

Fig. 1. Qknis divided into Q[0], Q[1], . . . , Q[k − 1].

3. Hamiltonian path and cycle embeddings

Let k be an odd integer with k 3, and let n2 be an integer. Let F ⊆ V (Qk_n)E(Qk_n) be the set of faulty vertices and/or

edges in Qk_n. Let Qk_nbe divided into Q[0], Q[1], . . . , Q[k − 1] along some dimension, and let Fl= F(V (Q[l])E(Q[l]))

for every 0l k − 1. We refer to an edge (x, y) ∈ E(Qk_n)

where all of x, y, and (x, y) are fault-free, as a safe

crossing-edge.

In the following lemmas, namely Lemmas 1–3, we shall construct hamiltonian paths in faulty Q[i, j ] for every i, j ∈ [0, k−1] when each faulty Q[l] is hamiltonian connected for l ∈ [i, j]. These preliminaries will be useful for further discussions.

As a first step, we shall construct a hamiltonian path between two arbitrary vertices belonging to Q[i] in a faulty Q[i, j ] (see Fig. 2(a)).

Lemma 1. Let i, j ∈ [0, k − 1], and let F ⊆ V (Q[i, j ])E

(Q[i, j ]) be a faulty set with |F |2n−3. If Q[l]−Flis hamil-tonian connected for every l ∈ [i, j ], there exists a hamilhamil-tonian path connecting every two vertices ui and vi ∈ V (Q[i] − Fi)

in Q[i, j ] − F for every n3 and odd k 3.

Proof. If i = j , this lemma holds. So we suppose that i = j .

We may assume without loss of generality that i = 0 in the fol-lowing discussion. Since Q[l] − Fl is hamiltonian connected for every l ∈ [0, j ], there is a hamiltonian path, say P0(u0, v0)

(u0= u_i and v0= v_i), in Q[0]− F0(see Fig.3(a)). The length

of P0(u0, v0) = |V (Q[0]−F0)|−1kn−1−|F0|−1, and the

number of faults outside Q[0] is at most (2n−3)−|F0_{|. When}

n and k 3, kn−1−|F₂ 0|−1 3n−1−|F₂ 0|−1>(2n−3)−|F0|.

Hence, we can find two consecutive vertices, say w0and z0, on

P0(u0, v0) such that (w0, w1) and (z0, z1) are safe

crossing-edges where w1and z1are the neighbors of w0and z0in Q[1],

respectively. Let u0, P0,1(u0, w0), w0, z0, P0,2(z0, v0), v0 =

P0(u0, v0), and let P1(w1, z1) be a hamiltonian path in

Q[1] − F1. u0, P0,1(u0, w0), w0, w1, P1(w1, z1), z1, z0,

P0,2(z0, v0), v0 forms a hamiltonian path in Q[0, 1] − F .

Repeating the above construction, we have a hamiltonian path in Q[0, j ] − F . 

Q[i] Q[j] ui uj Q[i+1, j-1] Q[i] u_i v_i Q[i+1, j] Q[i] ui Q[i+1, j] us (a) (c) (b)

Fig. 2. Hamiltonian paths in faulty Q[i, j ]. (a) Lemma 1; (b) Lemma 2; (c) Lemma 3.

In the following lemma, we shall construct a hamiltonian path between two arbitrary vertices ui ∈ V (Q[i] − Fi) and

uj ∈ V (Q[j] − Fj) in a faulty Q[i, j ] (see Fig.2(b)). Note

that Q[i, j ] can tolerate 2n − 2 faults in this lemma, which is the maximum degree of the fault-tolerance of hamiltonian cycle embeddings. In addition, we want all the vertices in Q[j ] −

Fj _{to form a subpath on this hamiltonian path for proving} Lemma 3.

Lemma 2. Let i, j ∈ [0, k − 1], and let F ⊆ V (Q[i, j ])E

(Q[i, j ]) be a faulty set with |F |2n−2. If Q[l]−Flis hamil-tonian connected for every l ∈ [i, j ], there exists a hamilhamil-tonian path connecting two arbitrary vertices ui ∈ V (Q[i] − Fi) and

uj ∈ V (Q[j] − Fj) in Q[i, j ] − F such that all the vertices in

Q[j ] − Fj form a subpath on this hamiltonian path for every n3 and odd k 3.

Proof. If i = j , the statement follows. Hence, we suppose that

i = j . Without loss of generality, we may assume that i = 0

Q[0] u₀ v0 Q[1] w1 z₁ w0 z₀ P_0,1(u₀,w₀) P_0,2(z₀,v₀) P₀(u₀,v₀) P1(w1,z1) Q[0] Q[j] u₀ v₀ v_j u_j Q[1, j-1] v_j-1 v₁ P₀(u₀,v₀) R(u₀,v_j-1 ) S(v_j,u_j) (b) (a)

Fig. 3. (a,b) The proofs of Lemmas 1 and 2.

(see Fig. 3(b)). Note that |F | = (2n − 2) and |V (Q[0])| =

kn−1. Since kn−1 − (2n − 2)9 − 4 = 5 for every n3 and odd k 3, there exists a safe crossing-edge, say (v0, v1),

where v0 = u0, v0 ∈ V (Q[0] − F0), v1 = u_j, and v1 ∈

V (Q[1] − F1). By assumption, Q[l] − Flis hamiltonian

con-nected for every l ∈ [0, j ], so we have a hamiltonian path, say P0(u0, v0), in Q[0] − F0. Continuing this process, we can

join all hamiltonian paths in Q[l] − Fl_{, for all l ∈ [0, j − 1],} to form a hamiltonian path, namely R(u0, vj −1), in Q[0, j − 1] − F such that (v_{j −1}, vj) is a safe crossing-edge where

vj −1 = u0, vj −1 ∈ V (Q(j − 1) − Fj −1), vj = uj, and

vj ∈ V (Q[j] − Fj). Let S(vj, uj) be a hamiltonian path in

Q[j ]. u0, R(u0, vj −1), vj −1, vj, S(vj, uj), uj is a hamilto-nian path in Q[0, j ] − F , and S(vj, uj) contains all vertices in

Q[j ] − Fj. 

In the following lemma, we construct a hamiltonian path between two arbitrary vertices ui ∈ V (Q[i] − Fi) and us ∈

V (Q[s] − Fs) with s ∈ [i, j ] in a faulty Q[i, j ] (see Fig. 2(c)).

Note that Q[i, j ] can tolerate 2n−3 faults in this lemma, which is the maximum degree of the fault-tolerance of hamiltonian path embeddings.

Lemma 3. Let i, j ∈ [0, k − 1], and let F ⊆ V (Q[i, j ])E

(Q[i, j ]) be a faulty set with |F |2n−3. If Q[l]−Flis hamil-tonian connected for every l ∈ [i, j ], there exists a hamilhamil-tonian path connecting every two vertices ui ∈ V (Q[i] − Fi) and

us ∈ V (Q[s] − Fs) in Q[i, j ] − F with s ∈ [i, j ] for every

n3 and odd k 3.

Proof. If i = j , the statement is true. Therefore, we assume

that i = j . By Lemma2, there exists a hamiltonian path, say

R(ui, us), in Q[i, s]− F such that all the vertices in Q[s]− Fs form a subpath on R(ui, us). Using the counting argument in the proof of Lemma 1, we can find two consecutive vertices, say us and vs ∈ V (Q[s]), on R(ui, us) such that (us, us+1) and (vs, vs+1) are safe crossing-edges where us+1and vs+1∈

V (Q[s + 1]). By Lemma 1, there is a hamiltonian path, namely S(us+1, vs+1), in Q[s + 1, j ] − F . Let ui, R1(ui, us), us, vs,

Q[0] u0 Q[1, k-1] R(u1,v1) P0(u0,v0) u1 v₀ v1 Q[0] Q[1, i-1] P0(u0,v0) v0 ui-1 Q[i] Q[i+1, k-1] Pi(ui,vi) vi u0 u1 ui R(u₁,u_i-1) v_k-1 vi+1 C0 Ci S(vi+1,vk-1) (a) (b)

Fig. 4. Cases of Theorem6. (a) Case 1; (b) Case 2 (when Q[i]-F is not hamiltonian connected).

R2(vs, vi), vi = R(ui, us). Then, ui, R1(ui, us), us, us+1,

S(us+1, vs+1), vs+1, vs, R2(v_s, v_i), v_i forms a hamiltonian

path in Q[j ] − F . 

The m×n torus is a graph of mn vertices labeled as ab where

a and b are integers with 0a m − 1 and 0bn − 1. Two

vertices ab and cd are adjacent if and only if either a = c and

b = d ± 1(mod n)or b = d and a = c ± 1(mod m). Therefore, Qk2

is a k × k torus for every k 3 by the definition. The following theorem related to the fault-tolerant hamiltonicity of the m × n torus is proved in[10].

Theorem 4 (Kim and Park[10]). If m3, n3, and n is odd,

the m × n torus is 2-hamiltonian and 1-hamiltonian connected.

The following corollary immediately follows by Theorem4.

Corollary 5. If k is odd with k 3, Qk2 is 2-hamiltonian and

1-hamiltonian connected.

Using the fault-tolerant hamiltonian and hamiltonian con-nected properties of Qk_n−1, we shall show the fault-tolerant hamiltonian property of Qkn.

Theorem 6. Let k be an odd integer with k 3. If Qk_n−1 is

(2n − 4)-hamiltonian and (2n − 5)-hamiltonian connected for some n3, then Qk

n is (2n − 2)-hamiltonian.

Proof. Let F ⊆ V (Qk_n)E(Qk_n) be the set of faulty

ver-tices and/or edges in Qk_n with |F |2n − 2. We claim that we can divide Qk_n into Q[0], Q[1], . . . , Q[k − 1] along some dimension such that |Fl|2n − 3 for every 0l k − 1. If |F |2n − 3, it is done. So we assume that |F | = 2n − 2. Then, if there is a faulty edge, we can divide Qk_n along the di-mension of this faulty edge. On the other hand, suppose that

F ⊆ V (Qk_n). Since |F |4, for every n3, picking arbitrarily

two faulty vertices in Qkn, we can divide Qknalong some

dimen-sion such that these two faulty vertices are in different Qk_n−1’s. Hence, the claim follows. Furthermore, without loss of gener-ality, we may assume that|F0_||Fl_{| for every l ∈ [0, k − 1].}

We discuss the existence of a hamiltonian cycle in the following three cases.

Case 1:|F0| = 2n − 3 (see Fig.4(a)).

By assumption, Qk_n−1 is (2n − 4)-hamiltonian. Therefore, there is a hamiltonian path, namely P0(u0, v0), in Q[0] − F0.

Let u1 and v1 be the neighbors of u0 and v0 in Q[1],

re-spectively, and let uk−1 and vk−1 be the neighbors of u0

and v0 in Q[k − 1], respectively. Since there is at most

one fault outside Q[0], either the two edges (u0, u1) and

(v0, v1) are safe crossing-edges or the two edges (u0, u_k−1)

and (v0, vk−1) are safe crossing-edges. Without loss of

gen-erality, we may assume that (u0, u1) and (v0, v1) are safe

crossing-edges. By assumption, Qk_n−1is (2n − 5)-hamiltonian connected and 2n − 51 for n3, so Q[l] − Fl is hamil-tonian connected for every l ∈ [1, k − 1] and n3. Since 1 < 2n − 3 for n3, by Lemma 3, there is a hamiltonian path, namely R(u1, v1), in Q[1, k − 1] − F . Therefore, u0,

P0(u0, v0), v0, v1, R(v1, u1), u1, u0 forms a hamiltonian cycle

in Qk n− F .

Case 2:|F0| = 2n − 4.

By assumption, Qk_n−1 is (2n − 4)-hamiltonian. Therefore, there is a hamiltonian cycle, say C0, in Q[0] − F0. Since

there are at most two faults outside Q[0], we can find two consecutive vertices, namely u0 and v0, on C0 for n3 such

that (u0, u1) and (v0, v1) are safe crossing-edges, where u1

and v1 are the neighbors of u0 and v0 in Q[1] respectively.

Note that Q[l] − Fl is hamiltonian-connected for every l ∈ [1, k − 1] and n4. In this situation, the proof is similar to Case 1.

When n = 3, it is possible that in addition to Q[0], there ex-ists another copy of Qk_n−1, say Q[i], which contains two faults (if all other copies contain at most 1 fault then by proceeding as above we are done). Hence, both of Q[0] − F0and Q[i] − Fi are not necessarily hamiltonian connected, but both are hamil-tonian. There is a hamiltonian cycle, say Ci, in Q[i] (see Fig. 4(b)). Note that there is no fault outside Q[0] and Q[i], and

Q[l]−Flis hamiltonian connected for every l /∈ {0, i}. We may

assume without loss of generality that i = k − 1. We can find a safe crossing-edge, say (ui−1, ui), where ui−1 ∈ Q[i − 1] and ui ∈ Q[i]. By Lemma 3, there is a hamiltonian path, namely R(u1, ui−1), in Q[1, i − 1] (if i = 1, then (ui−1, ui) =

(u0, u1), and there is no R(u1, ui−1)). Let vk−1∈ V (Q[k −1]) be a neighbor of v1. Let vi be adjacent to ui on Ci such that

vi+1, the neighbor of vi in Q[i + 1], = vk−1. By Lemma 3, there exists a hamiltonian path, namely S(vi+1, vk−1), in

Q[i, k-1] S(uk-1,y) uk-1 y Q[1, i-1] Q[0] v0 u0 x w0 w1 v1 R(v1,w1) P0(u0,v0) P0,1(u0,w0) P_0,2(x,v₀) Q[1, k-2] y Q[0] v₀ u0 w0 Q[k-1] vk-1 x w1 z0 z1 R(w1,z1) P0(u0,v0) P0,1(u0,w0) P0,2(z0,v0) Q[1, i-1] y Q[0] v₀ u0 Q[i, k-1] vk-1 x P0(u0,v0) T(u₁,y) Pk-1(x,vk-1) _S(x,v k-1) u₁ (a) (b) (c)

Fig. 5. Case 1 of Theorem7.

ui, Pi(ui, vi), vi = Ci. Then,u0, u1, R(u1, ui−1), ui−1, ui,

Pi(ui, vi), vi, vi+1, S(vi+1, vk−1), vk−1, v0, P0(v0, u0), u0 is

a hamiltonian cycle in Qk3− F .

Case 3:|F0|2n − 5.

Since kn−1 _{> 2n − 2 for k 3 and n3, we can find}

a safe crossing-edge, say (u0, uk−1), where u0 ∈ Q[1] and

uk−1∈ Q[k − 1]. |F0|2n − 5, and, by assumption, Qk_n−1is

(2n−5)-hamiltonian connected. Therefore, Q[l]−Flis hamil-tonian connected for every 0l k − 1. By Lemma 2, there is a hamiltonian path, namely P (u0, uk−1), in Q[0, k − 1]. Therefore,u0, P (u0, uk−1), uk−1, u0 is a hamiltonian cycle

in Qk

n− F . 

Using the fault-tolerant hamiltonian and hamiltonian con-nected properties of Qk_n−1 again, we shall prove the fault-tolerant hamiltonian connected property of Qk

nas follows.

Theorem 7. Let k be an odd integer with k 3. If Qk_n−1 is

(2n − 4)-hamiltonian and (2n − 5)-hamiltonian connected for some n3, Qk_n is (2n − 3)-hamiltonian connected.

Proof. We want to prove that there exists a hamiltonian path

connecting every two vertices x and y in Qk_n− F for every

F with |F |2n − 3. Since x = y, we can divide Qk_n into

Q[0], Q[1] . . . , Q[k − 1] along some dimension such that x

and y are in different Qk_n−1’s. Furthermore, without loss of generality, we may assume that|F0||Fl| for every 0l 

k−1. We discuss the existence of a hamiltonian path connecting x and y in the following three cases. 

Case 1:|F0| = 2n − 3.

By assumption, Qk_n−1is (2n−4)-hamiltonian. Hence, there is a hamiltonian path, namely P0(u0, v0), in Q[0]− F0. Note that

there is no fault outside Q[0]. So Q[l] is hamiltonian connected

for every l ∈ [1, k − 1]. We divide this case further into two subcases, Case 1.1 and Case 1.2, as follows.

Case 1.1: x ∈ V (Q[0] − F0) and y ∈ V (Q[i] − Fi) where i = 0 (see Fig.5(a)).

We may assume that the distance from x to u0 is at

least as far as the distance from x to v0 on P0(u0, v0).

Let u0, P0,1(u0, w0), w0, x, P0,2(x, v0), v0 = P0(u0, v0).

|V (P0(u0, v0))|kn−1−(2n−3)32−3 = 6 for k and n3,

so w0= u0and w0= x. Without loss of generality, we may

as-sume that i = 1. Then, let v1and w1be the neighbors of v0and

w0 in Q[1], respectively. Furthermore, let uk−1 be the neigh-bor of u0 in Q[k − 1]. First, we consider the case y = uk−1. By Lemma 3, there is a hamiltonian path R(v1, w1) in

Q[1, i − 1]. By Lemma 3, there exists a hamiltonian path

S(uk−1, y) in Q[i, k − 1]. Then, x, P0,2(x, v0), v0, v1,

R(v1, w1), w1, w0, P0,1(w0, u0), u0, u_k−1, S(u_k−1, y), y

forms a hamiltonian path in Qkn − F . Next, we consider the case y = uk−1. Since n3, by Lemma 3, there is a hamilto-nian path R(v1, w1) in Q[1, k − 1] − y. Then, x, P0,2(x, v0),

v0, v1, R(v1, w1), w1, w0, P0,1(w0, u0), u0, y forms a

hamil-tonian path in Qk n− F .

Case 1.2: x ∈ V (Q[i] − Fi_{) and y ∈ V (Q[j ] − F}j_{) where}

i, j = 0.

We may assume that i > j . Suppose that both of x and y are neighbors of u0 (or v0). So, x ∈ Q[k − 1] and y ∈ Q[1]

(see Fig. 5(b)). Let vk−1 be the neighbor of v0 in Q[k − 1].

Since there is no fault in Q[k − 1], by assumption, there ex-ists a hamiltonian path, say Pk−1(x, vk−1), in Q[k − 1]. Let

w0and z0be two consecutive vertices on P0(u0, v0). Also, let

w1and z1be the neighbors of w0and z0in Q[1], respectively.

By Lemma 3, there is a hamiltonian path, namely R(w1, z1),

in Q[1, k − 2] − y. Let u0, P0,1(u0, w0), w0, z0, P0,2(z0, v0),

v0 = P0(u0, v0). y, u0, P0,1(u0, w0), w0, w1, R(w1, z1),

z1, z0, P0,2(z0, v0), v0, vk−1, Pk−1(vk−1, x), x is a hamilto-nian path connecting x and y in Qk

Q[1, k-1] Q[0] u₀ Q[1, i-1] y Q[0] v₀ u₀ Q[i, k-1] x P₀(u₀,v₀) R(v₁,y) P₀(x,u₀) x u₁ v₁ R(u₁,y) y S(x,u_k-1) u_k-1 C₀ C₀ (a) (b)

Fig. 6. Case 2 of Theorem7. (a) Case 2.1; (b) Case 2.2.

either x or y is not a neighbor of u0(or v0). Let u1∈ Q[1] and

vk−1 ∈ Q[k − 1] be neighbors of u0and v0, respectively (see

Fig.5(c)). We may assume without loss of generality that u1=

y and vk−1 = x. By Lemma 3, there exist hamiltonian paths,

say S(x, vk−1) and T (u1, y), in Q[i, k − 1] and Q[1, i − 1],

respectively. As a result,x, S(x, v_k−1), vk−1, v0, P0(v0, u0),

u0, u1, T (u1, y), y is a hamiltonian path connecting x and y

in Qk_n− F .

Case 2:|F0| = 2n − 4.

By assumption, Q[0] is (2n − 4)-hamiltonian. So there is a hamiltonian cycle, namely C0, in Q[0]−F0. Note that there is at

most one fault outside Q[0]. Therefore, Q[l]−Flis hamiltonian connected for every l ∈ [1, k − 1]. We divide this case further into two subcases Case 2.1 and Case 2.2 as follows.

Case 2.1: x ∈ V (Q[0] − F0) and y ∈ V (Q[i] − Fi) where

i = 0 (see Fig. 6(a)).

Let u0 ∈ V (C0) be adjacent to x on C0 such that u0is not

a neighbor of y. Let u1∈ V (Q[1] − F1) be a neighbor of u0.

Since there is at most one fault outside Q[0], we may assume without loss of generality that (u0, u1) is a safe

crossing-edge. By Lemma 3, there is a hamiltonian path, namely

R(u1, y), in Q[1, k − 1] − F . Let x, P0(x, u0), u0, x = C0.

x, P0(x, u0), u0, u1, R(u1, y), y forms a hamiltonian path

connecting x and y in Qkn− F .

Case 2.2: x ∈ V (Q[i] − Fi) and y ∈ V (Q[j ] − Fj) where

i, j = 0 (see Fig. 6(b)).

We may assume that i > j . Since there is at most one fault outside Q[0], we can choose two adjacent vertices, say u0and

v0, on C0such that (u0, uk−1) and (v0, v1) are safe

crossing-edges, uk−1 = x, and v1 = y where uk−1 ∈ Q[k − 1] and

v1∈ Q[1] are neighbors of u0and v0, respectively. By Lemma

3, there exists a hamiltonian path, namely R(v1, y), in Q[1, i −

1] − F , and also, a hamiltonian path, namely S(x, uk−1), in

Q[i, k − 1] − F . Let u0, P0(u0, v0), v0, u0 = C0. Then,

x, S(x, uk−1), uk−1, u0, P0(u0, v0), v0, v1, R(v1, y), y is a

hamiltonian path in Qk_n− F .

Case 3:|F0|2n − 5.

As a result, Q[l] − Fl is hamiltonian connected for every

l ∈ [0, k − 1]. We may assume without loss of generality that

x ∈ V (Q[0] − F0). Since |F |2n − 3, by Lemma 3, there

is a hamiltonian path connecting x and y in Q[0, k − 1] −

F . Hence there exists a hamiltonian path connecting x and y

in Qk n− F .

In conclusion, the fault-tolerant hamiltonicity of Qk

nis given

in the following theorem.

Theorem 8. If k is odd with k 3 and n2, Qkn is (2n −

2)-hamiltonian and (2n − 3)-2)-hamiltonian connected.

Proof. By Corollary 5, Theorems 6, 7, and a simple

mathe-matical induction, this theorem is proved.  4. Conclusion

We have shown how to find a hamiltonian cycle and a hamil-tonian path joining two arbitrary vertices in a wounded k-ary

n-cube. When k is an odd integer, Qk_nis (2n − 2)-hamiltonian

and (2n − 3)-hamiltonian connected. Furthermore, our results are optimal (explained in Section 1). For even integer k, Qk_nis a bipartite graph. It is easy to see that Qkncontains a hamilto-nian cycle. However, with one single vertex fault, the remaining network does not contain any hamiltonian cycle. Therefore, for the fault-tolerant hamiltonian and hamiltonian-connected prop-erties of Qk

n, with k even, we can only consider edge faults. Let

Fe⊆ E(Qkn) be the set of faulty edges in Qknwith|Fe|2n−2 (not 2n − 3). For even integer k, we intend in future to show that Qk

n− Fe has a hamiltonian path connecting two arbitrary vertices belonging to different partite sets and a path of max-imum length, kn− 2, connecting two arbitrary vertices in the same partite set for every n2 and even k 4. This problem has not yet been resolved.

The fault-tolerant hamiltonian and hamiltonian-connected properties are fundamental tools for exploring further prop-erties concerning cycle or path embedding problems. For example, a graph G is pancyclic if a cycle of length l can be embedded into G for 4l |V (G)|. In [7], fault-tolerant pancyclicity of Möbius cubes was studied by using the fault-tolerant hamiltonian and hamiltonian-connected properties of Möbius cubes. In addition, by employing hamiltonian cycles and paths in faulty hypercubes, linear array and cycle embed-dings in conditional faulty hypercubes were investigated [13]. Acknowledgements

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 90-2213-E-009-149.

References

[1]Y.A. Ashir, I.A. Stewart, Fault-tolerant embeddings of hamiltonian circuits in k-ary n-cubes, SIAM J. Discrete Math. 15 (3) (2002) 317–328.

[2]M.M. Bae, B. Bose, Edge disjoint hamiltonian cycles in k-ary n-cubes and hypercubes, IEEE Trans. Comput. 52 (10) (2003) 1271–1284.

[3]S. Bettayeb, On the k-ary hypercube, Theoret. Comput. Sci. 140 (1995) 333–339.

[4]B. Bose, B. Broeg, Y. Kwon, Y. Ashir, Lee distance and topological properties of k-ary n-cubes, IEEE Trans. Comput. 33 (1995) 1021–1030.

[5]M.Y. Chan, S.-J. Lee, On the existence of hamiltonian circuits in faulty hypercubes, SIAM J. Discrete Math. 4 (1991) 511–527.

[6]K. Day, The conditional node connectivity of the k-ary n-cube, J. Interconnection Networks 5 (1) (2004) 13–26.

[7]S.Y. Hsieh, C.H. Chen, Pancyclicity on Möbius cubes with maximal edge faults, Parallel Comput. 30 (2004) 407–421.

[8]S.Y. Hsieh, G.H. Chen, C.W. Ho, Longest fault-free paths in star graphs with edge faults, IEEE Trans. Comput. 50 (9) (2001) 960–971.

[9]H.C. Hsu, T.K. Li, J.M. Tan, L.H. Hsu, Fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs, IEEE Trans. Comput. 53 (1) (2004) 39–53.

[10]H.-C. Kim, J.-H. Park, Fault hamiltonicity of two-dimensional torus networks, in: Proceedings of the Workshop on Algorithms and Computation WAAC’00, Tokyo, Japan, 2000, pp. 110–117.

[11]S. Latifi, S.Q. Zheng, N. Bagherzadeh, Optimal ring embedding in hypercubes with faulty links, Proc. IEEE Symp. Fault-Tolerant Comput. 42 (1992) 178–184.

[12]R.S. Lo, G.H. Chen, Embedding hamiltonian paths in faulty arrangement graphs with the backtracking method, IEEE Trans. Parallel Distrib. Systems 12 (2) (2001) 209–221.

[13]C.H. Tsai, Linear array and ring embeddings in conditional faulty hypercubes, Theoret. Comput. Sci. 314 (2004) 431–443.

[14]Y.C. Tseng, S.H. Chang, J.P. Sheu, Fault-tolerant ring embedding in a star graph with both link and node failures, IEEE Trans. Parallel Distrib. Systems 8 (12) (1997) 1185–1195.

Ming-Chien Yang received his B.S. degree in

Computer Science and Information Engineering from the Tamkang University, Taiwan, Repub-lic of China, in 1999. He received his M.S. and Ph.D. degrees in Computer and Information Sci-ence from the National Chiao Tung University in 2001 and 2005, respectively. Since 2005, he has been an engineer in the Industrial Technol-ogy and Research Institute, Taiwan, Republic of China. His research interests include intercon-nection networks, graph theory, algorithms, and digital home.

Jimmy J. M. Tan received his B.S. and M.S.

degrees in Mathematics from the National Tai-wan University in 1970 and 1973, respectively, and his Ph.D. degree from the Carleton Uni-versity, Ottawa, Canada, in 1981. He has been on the faculty of the Department of Computer and Information Science, National Chiao Tung University, since 1983. His research interests in-clude design and analysis of algorithms, combi-natorial optimization, interconnection networks, and graph theory.

Lih-Hsing Hsu received his B.S. degree in

Mathematics from the Chung Yuan Christian University, Taiwan, Republic of China, in 1975, and his Ph.D. degree in Mathematics from the State University of New York at Stony Brook in 1981. He is currently a Chairman in the Depart-ment of Computer Science and Information En-gineering, Providence University, Taiwan, Re-public of China. His research interests include interconnection networks, algorithms, graph the-ory, and VLSI layout.