DOI 10.1007/s10878-007-9113-1
A note on fault-free mutually independent Hamiltonian
cycles in hypercubes with faulty edges
Tz-Liang Kueng· Cheng-Kuan Lin · Tyne Liang · Jimmy J.M. Tan· Lih-Hsing Hsu
Published online: 17 November 2007
© Springer Science+Business Media, LLC 2007
Abstract In the paper “Fault-free Mutually Independent Hamiltonian Cycles in
Hypercubes with Faulty Edges” (J. Comb. Optim. 13:153–162,2007), the authors claimed that an n-dimensional hypercube can be embedded with (n−1−f )-mutually independent Hamiltonian cycles when f ≤ n−2 faulty edges may occur accidentally. However, there are two mistakes in their proof. In this paper, we give examples to ex-plain why the proof is deficient. Then we present a correct proof.
Keywords Interconnection network· Hypercube · Fault tolerance · Hamiltonian
cycle
1 Introduction
In many parallel computer systems, processors are connected on the basis of in-terconnection networks such as hypercubes, star graphs, meshes, bubble-sort net-works, etc. For the sake of simplicity, a network topology is usually represented by a graph, in which vertices correspond to processors and edges correspond to con-nections or communication links. Hence, we use the terms, graph and network, in-terchangeably. Throughout this paper, we concentrate on loopless undirected graphs. For the graph definitions and notations we follow the ones defined in (Bondy and Murty 1980). A graph G consists of a set V (G) and a subset E(G) of {(u, v) |
This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-233-002.
T.-L. Kueng· C.-K. Lin · T. Liang (
)· J.J.M. TanDepartment of Computer Science, National Chiao Tung University, Hsinchu 30050, Taiwan e-mail:[email protected]
L.-H. Hsu
Department of Computer Science and Information Engineering, Providence University, Taichung 43301, Taiwan
(u, v)is an unordered pair of V (G)}. The set V (G) is called the vertex set and E(G) is called the edge set. Two vertices u and v of G are adjacent if (u, v)∈ E(G). A graph H is a subgraph of G if V (H )⊆ V (G) and E(H ) ⊆ E(G). Let S be a nonempty subset of V (G). The subgraph induced by S is the subgraph of G with its vertex set S and with its edge set which consists of those edges joining any two vertices in S. We use G− S to denote the subgraph of G induced by V (G) − S. Anal-ogously, let F be a nonempty subset of E(G). We use G− F to denote the subgraph of G with vertex set V (G) and edge set E(G)− F . The degree of a vertex u in G is the number of edges incident to u. A graph G is k-regular if all its vertices have the same degree k. A graph G is bipartite if its vertex set can be partitioned into two disjoint partite sets V0(G)and V1(G)such that every edge will join a vertex of V0(G) and a vertex of V1(G).
A path P of length k from a vertex x to a vertex y in a graph G is a sequence of distinct verticesv1, v2, . . . , vk+1 such that x = v1, y= vk+1, and (vi, vi+1)∈ E(G) for every 1≤ i ≤ k. For convenience, we write P as v1, . . . , vi, Q, vj, . . . , vk+1 where Q= vi, vi+1, . . . , vj. Note that we allow Q to be a path of length zero. The i-th vertex of P is denoted by P (i); i.e., P (i)= vi. To emphasize the beginning and ending vertices of P , we also write P as P[x, y]. A cycle is a path with at least three vertices such that the last vertex is adjacent to the first one. For clarity, a cycle of length k is represented by v1, v2, . . . , vk, v1. A path of a graph G is a Hamiltonian path if it spans G. Similarly, a cycle of a graph G is a Hamiltonian cycle if it spans G. A bipartite graph is Hamiltonian laceable (Simmons1978) if there is a Hamiltonian path between any two vertices which are in different partite sets. Moreover, a Hamiltonian laceable graph G is hyper Hamiltonian laceable (Lewinter and Widulski1997) if for any vertex v∈ Vi(G)with i∈ {0, 1}, there is a Hamiltonian path of G− {v} between any two vertices of V1−i(G).
The n-dimensional hypercube (or n-cube for short) is one of the most popular topologies yet discovered for parallel computation (Leighton 1992). Thus, many attractive properties of hypercubes have been studied in the literature (Akers and Krishnameurthy1989; Chang et al.2004; Johnsson and Ho1989; Leighton1992; Leu and Kuo 1999; Tsai et al.2002; Yang et al. 1994). The formal definition of an n-cube is given as follows. Let u= bn. . . bi. . . b1 be an n-bit binary string. For 1≤ i ≤ n, we use (u)i to denote the binary string bn. . . ¯bi. . . b1. Moreover, we use (u)i to denote the i-th bit bi of u. The Hamming weight of u, denoted by w(u), is |{i | (u)i = 1, 1 ≤ i ≤ n}|. The n-cube Qn consists of all n-bit binary strings representing its vertices. Two vertices u and v are adjacent if and only if v= (u)i with some i and we call the edge (u, (u)i)an i-dimensional edge. Note that Qn is a bipartite graph with partite sets V0(Qn)= {u ∈ V (Qn)| w(u) is even} and V1(Qn)= {u ∈ V (Qn)| w(u) is odd}.
Because the components of a network may fail accidentally, it is demanded to consider the fault-tolerance on a network. The faults in a network may take var-ious forms such as hardware failures, software errors, or even missing of trans-mitted packets. In this paper, faulty edges, one kind of hardware failures, are ad-dressed. More precisely, a set F of faulty edges in a graph G contains those edges which will be removed from G. When all faulty edges are removed, we investigate the properties of the fault-free graph G− F . In particular, we concern the mutu-ally independent Hamiltonian cycles, initimutu-ally proposed by Sun et al. (2006), on a
faulty n-cube. The mutually independent Hamiltonian cycles are defined as follows. Let G be a graph with N vertices. A Hamiltonian cycle C of G is described by
u1, u2, . . . , uN, u1 to emphasize the order of vertices on C. Accordingly, u1 is referred to as the beginning vertex. Two Hamiltonian cycles of G beginning from a given vertex s, namely C1= u1, u2, . . . , uN, u1 and C2= v1, v2, . . . , vN, v1, are independent if u1= v1= s and ui= vi for 2≤ i ≤ N. Two Hamiltonian paths of G, P1= u1, u2, . . . , uN and P2= v1, v2, . . . , vN, are independent if u1= v1, uN= vN, and ui= vi for every 1 < i < N ; P1and P2are fully independent if ui= vi for every 1≤ i ≤ N. We say a set of m Hamiltonian cycles {C1, . . . , Cm} of G, begin-ning from the same vertex, is m-mutually independent if Ci and Cj are independent whenever i= j. A set of m Hamiltonian paths {P1, . . . , Pm} of G are m-mutually independent (resp. m-mutually fully independent) if any two different Hamiltonian paths in the set are independent (resp. fully independent). Moreover, the mutually independent hamiltonicity of G, denoted byIHC(G), is defined as the maximum integer m such that for any vertex u there exist m-mutually independent Hamiltonian cycles of G beginning from u. The concept of mutually independent Hamiltonian cy-cles can be applied in many different areas like those introduced in (Sun et al.2006; Hsieh and Yu2007).
Suppose that Qn denotes an n-cube. Sun et al. (2006) proved that IHC(Qn)= n− 1 if n ∈ {1, 2, 3} and IHC(Qn)= n if n ≥ 4. Later, Hsieh and Yu (2007) fur-ther addressed this issue and claimed that Qn contains (n− 1 − f )-mutually inde-pendent Hamiltonian cycles when f ≤ n − 2 faulty edges may occur accidentally. However, there are two mistakes in (Hsieh and Yu2007); one is related to the proof of “Lemma 2” and the other is related to the proof of “Theorem 2”. In this paper, we give counterexamples to indicate why their argument fails and then we present a correct proof.
The rest of this paper is organized as follows. The basic properties of hypercubes are given in Sect.2. In Sect.3, we explain why the proof in (Hsieh and Yu2007) is deficient. The correct proof is given in Sect.4. Finally, the future work is discussed in Sect.5.
2 Preliminaries
By definition, an n-cube Qnis n-regular. It is well known that Qnhas a recursive con-struction; that is, it can be decomposed into two (n− 1)-dimensional subcubes. Let Qjn be the subgraph of Qn induced by{u ∈ V (Qn)| (u)n= j} for j ∈ {0, 1}. Obvi-ously, Qjnis isomorphic to Qn−1. Then an n-partition of Qnpartitions the Qnalong dimension n into{Q0
n, Q1n}. The set of crossing edges between Q0nand Q1n, denoted by Ec= {(u, v) ∈ E(Qn)| u ∈ V (Q0n), v∈ V (Q1n)}, consists of all n-dimensional edges of Qn. It is also known that Qn is vertex-transitive and edge-transitive. For convenience, we use e to denote the identity vertex 0nof Qn.
The following results are fault-tolerant properties on hypercubes.
Theorem 1 (Tsai et al.2002) Let n≥ 3. Suppose that F ⊆ E(Qn)is a set of at most n− 2 faulty edges. Then Qn− F is Hamiltonian laceable.
Theorem 2 (Tsai et al.2002) Let n≥ 3. Suppose that F ⊆ E(Qn)is a set of at most n− 3 faulty edges. Then Qn− F is hyper Hamiltonian laceable.
Lemma 1 (Sun et al.2006) Let n≥ 4. Suppose that x and y are any two vertices from different partite sets of Qn. Then Qn− {x, y} is Hamiltonian laceable.
3 Mistakes in the previous work
As Hsieh and Yu (2007) claimed, Qn can be embedded with (n− 1 − f )-mutually independent Hamiltonian cycles when f ≤ n−2 faulty edges may occur accidentally. Their proof is by induction on n and mainly relies on “Lemma 2” of (Hsieh and Yu
2007). However, there is a major mistake in their proof of “Lemma 2”. To be precise, this mistake corresponds to the statements within lines 3–12 of p. 159 in (Hsieh and Yu2007):
We prove this Lemma by induction on n. The base case where n= 3 clearly holds. We now consider an n-cube for n≥ 4. Let dl = |{(bi, wi): (bi, wi)be an edge of dimension l, 1≤ i ≤ δ}|. Without loss of generality, we assume that d1 ≥ d2 ≥ . . . ≥ dn. Obviously, dn = 0. We then execute an n-partition of Qn to obtain Q0n−1 and Q1n−1. Note that each (bi, wi) is in either Q0n−1 or Q1n−1. Let r0= |{(bi, wi)∈ E(Q0n−1): 1 ≤ i ≤ δ}| and r1= |{(bi, wi)∈ E(Q1n−1): 1 ≤ i ≤ δ}|. Clearly, r0+ r1= δ. Without loss of generality, we assume that{(b1, w1), (b2, w2), . . . , (br0, wr0)} ⊂ E(Q0n−1) and {(br0+1, wr0+1), (br0+2, wr0+2), . . . , (bδ, wδ)} ⊂ E(Q
1
n−1). Since |F0| ≤
|F | − 1 and r0 ≤ n − |F | − 1 ≤ (n − 1) − |F0| − 1, by the inductive hypothesis, there exist r0-mutually fully independent Hamiltonian paths P1[b1, w1], P2[b2, w2], . . ., Pr0[br0, wr0] in Q
0
n−1− F0.
Once an n-partition is executed on Qn, the proof provided by Hsieh and Yu (2007) is merely fitted to the special case when both Q0n and Q1n contain n− 3 or less faulty edges; i.e., the requirements|F0| ≤ |F | − 1 and |F1| ≤ |F | − 1 need to be satisfied. In this case, Q0n− F0 and Q1n− F1 still contain (n− 1 − |F |)-mutually fully independent Hamiltonian paths. For example, let F = {(00000, 10000)} and A= {(00011, 00010), (00110, 00100), (00101, 00001)}. Obviously, F contains a 5-dimensional edge. Moreover, the edges of A are 1-5-dimensional, 2-5-dimensional, and 3-dimensional, respectively. According to the proof of “Lemma 2” in (Hsieh and Yu
2007), we may partition Q5into{Q05, Q15} along dimension 5. Since the faulty edge is 5-dimensional, both Q05and Q15are fault-free. Therefore, we have r0= 3, r1= 0, and δ= n − 1 − |F | = n − 2 − |F0| = 3. Since Q05is isomorphic to Q4, the inductive hypothesis guarantees that Q05has 3-mutually fully independent Hamiltonian paths, namely P1[00011, 00010], P2[00110, 00100], P3[00101, 00001].
However, one should notice that the faulty edges may occur accidentally. For ex-ample, we suppose the faulty edge is 1-dimensional rather than a 5-dimensional edge. Then either Q05or Q15is no longer fault-free, so the final result cannot be directly de-rived from the inductive hypothesis. For clarity, we give a counterexample against the argument given in (Hsieh and Yu2007).
Example 1 Let F = {(00000, 00001)} consist of a 1-dimensional faulty edge in Q5 and let A= {(00011, 00010), (00110, 00100), (00101, 00001)}. Obviously, the edges of A are 1-dimensional, 2-dimensional, and 3-dimensional, respectively. Accord-ing to the proof of “Lemma 2” in (Hsieh and Yu2007), we may partition Q5 into
{Q0
5, Q15} along dimension 5. Then we have F ∪A ⊂ E(Q 0
5); that is, we have F0= F , δ= r0= 3, and r1= 0. Since Q05is isomorphic to Q4, Q05− F0has at most 2, not 3, mutually fully independent Hamiltonian paths by the inductive hypothesis. Indeed, no matter which dimension is used to partition Q5, the inductive argument proposed in (Hsieh and Yu2007) always fails.
In summary, Hsieh and Yu (2007) did not ever consider the case when all faulty edges are unfortunately located in the same (n− 1)-dimensional subcube of Qn. In this case, the final result cannot be derived directly from the inductive hypothesis since Q0n− F0has at most r0− 1, instead of r0, mutually fully independent Hamil-tonian paths. More generally, we give the next example to show this deficiency. Example 2 Let n be a multiple of 5. Suppose that F= {(0n,0n−i10i−1)| 1 ≤ i ≤n5} consists of n5 faulty edges in Qn. Moreover, suppose that wi = 0n−1−i110i−1and bi= 0n−1−i10i for 1≤ i ≤4n5 − 2. Besides, let w4n
5−1= 0 n 5+1104n5−31 and b4n 5−1= 0n−11. Then A= {(wi, bi)| 1 ≤ i ≤4n5 − 1} is a set of 4n
5 − 1 edges with no shared endpoints. Obviously, the edges of A are over dimensions 1, 2, . . . , (4n5 − 1), respec-tively. According to the proof of “Lemma 2” in (Hsieh and Yu2007), we may parti-tion Qninto{Q0n, Q1n} along dimension n. Then we have F ∪ A ⊂ E(Q0n); that is, we have F0= F , δ = r0=4n5 − 1, and r1= 0. By the inductive hypothesis, Q0n− F0has at most (4n5 − 2), instead of (4n5 − 1), mutually fully independent Hamiltonian paths. In addition to this mistake, another minor one corresponds to the proof of “The-orem 2” in (Hsieh and Yu2007). From line 10 to line 13 of page 160, the authors claimed that there exist δ edges (Ci(t ), Ci(t+ 1(mod 2n−1))) for all 1≤ i ≤ δ such that (Ci(t ), (Ci(t ))d)and (Ci(t+ 1(mod 2n−1)), (Ci(t+ 1(mod 2n−1)))d)are fault-free. They further mentioned that if these edges do not exist, then |F | ≥ |Fc| ≥ 2n−2> n− 2 for n > 3. However, this argument is wrong because every faulty edge of Fcrepeats δ times. In contrast, it should be argued that if such edges do not exist, we will have δ|Fc| ≥ 2n−2; that is, |Fc| ≥ 2n−2/δ >|F | will lead to an immediate contradiction for n≥ 3.
4 Fault-free mutually independent Hamiltonian cycles of faulty Qn
To derive the main theorem of this paper, we need the following results.
Lemma 2 (Sun et al.2006) Let Qnbe an n-cube for n≥ 2. Suppose that {(wi, bi)∈ E(Qn)| wi ∈ V0(Qn), bi∈ V1(Qn),1≤ i ≤ n − 1} consists of n − 1 distinct edges with no shared endpoints. Then Qn contains (n− 1)-mutually fully independent Hamiltonian paths P1[w1, b1], . . . , Pn−1[wn−1, bn−1].
Theorem 3 (Sun et al.2006) IHC(Qn)= n − 1 if n ∈ {1, 2, 3} and IHC(Qn)= n if n≥ 4.
Let F be a set of faulty edges of Qn. Suppose that Qnis partitioned along dimen-sion n into{Q0n, Q1n} and Ecis the set of crossing edges between Q0nand Q1n. Then we define F0= F ∩ E(Q0n), F1= F ∩ E(Q1n)and Fc= F ∩ Ec. Moreover, we set δ= n − 1 − |F | in the remainder of this paper. To tolerate faulty edges in hypercubes, we have the next lemma.
Lemma 3 Let F ⊆ E(Qn)be a set of at most n− 2 faulty edges for n ≥ 3. Sup-pose that A= {(wi, bi)∈ E(Qn)| wi∈ V0(Qn), bi∈ V1(Qn),1≤ i ≤ δ} consists of δ distinct edges with no shared endpoints. Then Qn− F contains δ-mutually fully independent Hamiltonian paths P1[w1, b1], . . . , Pδ[wδ, bδ].
Proof This proof proceeds by induction on n. First suppose|F | = 0. Then this case follows from Lemma2. Suppose|F | = n − 2. Then we have δ = n − 1 − (n − 2) = 1. By Theorem1, Qn− F has a Hamiltonian path between any two vertices from differ-ent partite sets. Obviously, the statemdiffer-ent holds for Q3, as the induction basis. In what follows, we only consider 1≤ |F | ≤ n − 3 and n ≥ 4. As the inductive hypothesis, suppose that the statement is true for Qn−1.
Since δ+ |F | = n − 1 < n, there must exist a dimension d of {1, 2, . . . , n} such that A∪ F contains no d-dimensional edges. Since Qn is edge-transitive, we can assume d= n. Then we partition Qn into{Q0n, Q1n} along dimension n. Thus, each edge of A∪ F is in either Q0nor Q1n. Let r0= |{(wi, bi)∈ E(Q0n)| 1 ≤ i ≤ δ}| and r1= |{(wi, bi)∈ E(Q1n)| 1 ≤ i ≤ δ}|. Clearly, r0+ r1= δ. Without loss of generality, we assume{(w1, b1), . . . , (wr0, br0)} ⊂ E(Q
0
n)and{(wr0+1, br0+1), . . . , (wδ, bδ)} ⊂
E(Q1n). Since n− 1 = δ + |F | = r0+ r1+ |F0| + |F1|, we have ri+ |Fj| ≤ n − 1 for any i, j∈ {0, 1}. Then we have to take the following cases into account.
Case 1: Suppose ri + |Fj| ≤ n − 2 for any i, j ∈ {0, 1}. Since r0+ |F0| ≤ n − 2, r0≤ n − 2 − |F0| = (n − 1) − 1 − |F0|. By the inductive hypothesis, Q0n− F0has r0-mutually fully independent Hamiltonian paths Hi[wi, bi], 1 ≤ i ≤ r0. Obviously, Hi[wi, bi] can be represented as wi, Hi , ui, bi, in which ui is some vertex adja-cent to bi. Similarly, Q1
n− F1has r1-mutually fully independent Hamiltonian paths Hi[wi, bi] = wi, Hi , ui, bi, r0+ 1 ≤ i ≤ δ.
Next, we construct r0paths in Q1n−F1to incorporate the previously established r0 paths of Q0n−F0. Since r0+|F1| ≤ n−2, we have r0≤ n−2−|F1|. By the inductive hypothesis, Q1n− F1also contains r0-mutually fully independent Hamiltonian paths R1[(u1)n, (b1)n], . . . , Rr0[(ur0)
n, (b r0)
n]. Similarly, Q0
n−F0also contains r1-mutual-ly fulr1-mutual-ly independent Hamiltonian paths Rr0+1[(ur0+1)
n, (b r0+1)
n], . . . , R δ[(uδ)n, (bδ)n]. Accordingly, we set Pi[wi, bi] = wi, Hi , ui, (ui)n, Ri, (bi)n, bi for every 1≤ i ≤ δ. Thus, {P1, . . . , Pδ} forms a set of δ-mutually fully independent Hamil-tonian paths in Qn− F . See Fig.1(a) for illustration.
Case 2: Suppose ri+ |Fi| = n − 1 for some i ∈ {0, 1}. Without loss of generality, we assume r0+ |F0| = n − 1. Since r0= n − 1 − |F0| ≥ n − 1 − |F | = δ, we must have r0= δ and |F0| = |F | ≤ n − 3. Note that r0− 1 = δ − 1 = n − 2 − |F | = (n− 1) − 1 − |F0|. By the inductive hypothesis, Q0n− F0 has (r0− 1)-mutually
(a) Case 1
(b) Subcase 2.1 of Case 2
(c) Subcase 2.2 of Case 2
(d) Case 3
fully independent Hamiltonian paths Hi[wi, bi], 2 ≤ i ≤ r0. Again, Hi[wi, bi] can be represented aswi, Hi , ui, bi, in which ui is some vertex adjacent to bi.
Subcase 2.1: Suppose n= 4. Thus, we have r0= 2. By Theorem 2, Q04− F0 has a Hamiltonian path H1[w1, b1] = w1, u1, H1 , (b1)j, b1, in which u1 is a ver-tex adjacent to w1 and j is some integer of {1, 2, 3, 4}. Let X = {((u1)4, (u2)4)}. Similarly, there are two Hamiltonian paths R1[(w1)4, (u1)4] and R2[(u2)4, (b2)4] in Q14 − X. Obviously, one can see that R1(7) = R2(1) and R1(8)= R2(2). Then we set P1[w1, b1] = w1, (w1)4, R1, (u1)4, u1, H1 , (b1)j, b1 and P2[w2, b2] =
w2, H2 , u2, (u2)4, R2, (b2)4, b2. Consequently, {P1, P2} forms a set of 2-mutually fully independent Hamiltonian paths in Q4− F . See Fig.1(b) for illustration.
Subcase 2.2: Suppose n≥ 5. We first consider |F0| ≤ n − 4. By the induc-tive hypothesis, Q1n has (r0 − 1)-mutually fully independent Hamiltonian paths Ri[(ui)n, (bi)n], 2 ≤ i ≤ r0. Then we can choose an integer j of{1, . . . , n − 1} such that both (b1)j= w
1and ((b1)j)n∈ {R/ i(2n−1− 1) | 2 ≤ i ≤ r0} are satisfied. Since r0= n − 1 − |F | ≤ n − 2, such an integer exists. By Theorem2, Q0n− (F0∪ {b1}) has a Hamiltonian path H1[w1, (b1)j] = w1, u1, H1 , (b1)j, in which u1 is some ver-tex adjacent to w1. By Lemma1, there exists a Hamiltonian path R1[(w1)n, (u1)n] in Q1n− {(b1)n, ((b1)j)n}. Then we set P1[w1, b1] = w1, (w1)n, R1, (u1)n, u1, H1 , (b1)j, ((b1)j)n, (b1)n, b1 and Pi[wi, bi] = wi, Hi , ui, (ui)n, Ri, (bi)n, bi for 2 ≤ i≤ r0. As a result,{P1, . . . , Pr0} forms a set of r0-mutually fully independent
Hamil-tonian paths in Qn− F . See Fig.1(c) for illustration.
Next, we consider|F0| = n − 3. Thus, we have r0= 2. By Theorem1, Q0n− F0 has a Hamiltonian path H1[w1, b1] = w1, u1, H1 , (b1)j, b1, in which u1is a vertex adjacent to w1 and j is some integer of{1, 2, . . . , n − 1}. By Lemma1, there ex-ists a Hamiltonian path R1[(w1)n, (u1)n] in Qn1− {(b1)n, ((b1)j)n}. By the inductive hypothesis, Q1n− {((b2)n, ((b1)j)n)} has a Hamiltonian path R2[(u2)n, (b2)n]. Obvi-ously, we have R2(2n−1− 1) = ((b1)j)n. Again, we set P1[w1, b1] = w1, (w1)n, R1, (u1)n, u1, H1 , (b1)j, ((b1)j)n, (b1)n, b1 and P2[w2, b2] = w2, H2 , u2, (u2)n, R2, (b2)n, b2. Hence, {P1, P2} forms a set of 2-mutually fully independent Hamil-tonian paths in Qn− F . See Fig.1(c).
Case 3: Suppose that ri+ |F1−i| = n − 1 for some i ∈ {0, 1}. Without loss of gen-erality, we assume r1+ |F0| = n − 1. Since r1= n − 1 − |F0| ≥ n − 1 − |F | = δ, we have r1= δ and F0= F . By the inductive hypothesis, Q1n has (r1− 1)-mutually fully independent Hamiltonian paths Hi[wi, bi] = wi, Hi , ui, bi, in which ui is some vertex adjacent to bi with 1≤ i ≤ r1− 1. Since r1− 1 = δ − 1 = n− 2 − |F | = (n − 1) − 1 − |F0|, Q0n− F0 has (r1− 1)-mutually fully indepen-dent Hamiltonian paths Ri[(ui)n, (bi)n], 1 ≤ i ≤ r1− 1. Then we set Pi[wi, bi] =
wi, Hi , ui, (ui)n, Ri, (bi)n, bi with 1 ≤ i ≤ r1− 1. Next, we have to choose a ver-tex v of V0(Q0n)and construct a Hamiltonian path Rr1[(wr1)
n, v] in Q0
n− F0such that v= Ri(2) and Rr1(2
n−1− 1) = (u
i)n for every 1≤ i ≤ r1− 1. We distinguish the following subcases.
Subcase 3.1: Suppose n= 5 or |F | > 1. One can see that (u1)n, . . . , (ur1−1)
nhave at most (r1− 1)(n − 1) neighbors in Q0
n. Since|V0(Q0n)| = 2n−2> (r1− 1)(n − 1) = (n− 2 − |F |)(n − 1) in this subcase, we can choose v other than all neighbors of (u1)n, . . . , (ur1−1)
n. Obviously, we have v= R
i(2) for 1≤ i ≤ r1−1. By Theorem1, there is a Hamiltonian path Rr1[(wr1)
n, v] in Q0
any node of{(u1)n, . . . , (ur1−1)
n}, we have R r1(2
n−1− 1) = (u
i)nfor every 1≤ i ≤ r1− 1. By Theorem2, there is a Hamiltonian path Hr1[(v)
n, b r1] in Q 1 n− {wr1}. Then we set Pr1= wr1, (wr1) n, Rr 1, v, (v) n, Hr 1, br1. Consequently, {P1, . . . , Pr1} forms
a set of r1-mutually fully independent Hamiltonian paths in Qn− F . See Fig.1(d) for illustration.
In the following, we consider n= 5 and |F | = 1; that is, r1= 3.
Subcase 3.2: For n= 5 and |F | = 1, suppose that (u1)n and (u2)n have at least one common neighbor. Since |V0(Q0n)| = 2n−2= 8 > 7 = (r1− 1)(n − 1) − 1, we still can choose a vertex v from V0(Q0n)other than all neighbors of (u1)n and (u2)n. Obviously, we have v= Ri(2) for 1≤ i ≤ r1− 1. By Theorem1, there is a Hamiltonian path Rr1[(wr1)
n, v] of Q0
n− F0 such that Rr1(2
n−1− 1) = (u i)n for every 1≤ i ≤ r1− 1. By Theorem 2, there is a Hamiltonian path Hr1[(v)
n, b r1] in Q1n− {wr1}. Similarly, we set Pr1 = wr1, (wr1) n, R r1, v, (v) n, H r1, br1. Then {P1, . . . , Pr1} forms a set of r1-mutually fully independent Hamiltonian paths in
Qn− F . See Fig.1(d).
Subcase 3.3: For n= 5 and |F | = 1, suppose that (u1)n and (u2)n have no common neighbors. Then we assign the vertex v as the one that is adjacent to (u1)n but is not identical to R1(2). Obviously, we have v= Ri(2) for 1≤ i ≤ r1− 1. By Theorem 1, Q0n − (F0∪ {(v, (u1)n)}) remains Hamiltonian laceable. Thus, there is a Hamiltonian path Rr1[(wr1)
n, v] of Q0
n − (F0 ∪ {(v, (u1)n)}) such that Rr1(2n−1− 1) = (u
i)n for every 1≤ i ≤ r1− 1. By Theorem 2, there is a Hamiltonian path Hr1[(v)n, br 1] in Q 1 n − {wr1}. Similarly, we set Pr1 = wr1, (wr1) n, R r1, v, (v) n, H
r1, br1. Then {P1, . . . , Pr1} forms a set of r1-mutually
fully independent Hamiltonian paths in Qn− F . See Fig.1(d). With Lemma3, the next theorem can be easily derived.
Theorem 4 Let n≥ 3. Suppose that F ⊆ E(Qn)consists of at most n− 2 faulty edges. Then Qn−F contains (n−1−|F |)-mutually independent Hamiltonian cycles beginning from any vertex.
Proof Since Qnis vertex-transitive, we only need to construct δ-mutually indepen-dent Hamiltonian cycles beginning from e= 0n. Suppose|F | = 0. Then the statement follows from Theorem3. Thus, we only consider the situation that F is nonempty. Furthermore, since Qnis edge-transitive, we assume that at least one faulty edge is an n-dimensional edge.
The proof idea is based on the partition of Qn. As discussed previously, Qncan be partitioned into{Q0n, Q1n}. Obviously, e is located in Q0n. Recall that F0and F1 denote the sets of faulty edges in Q0n and Q1n, respectively. Then the proof idea is outlined as follows:
(1) We first build δ-mutually independent Hamiltonian cycles C1, C2, . . . , Cδ begin-ning from e in Q0n− F0.
(2) Next, we have to claim that there must exist an integer t , 1≤ t ≤ 2n−2, so that the crossing edges (Ci(2t− 1), (Ci(2t− 1))n)and (Ci(2t), (Ci(2t))n)are fault-free for all 1≤ i ≤ δ. For convenience, let xi= Ci(2t− 1) and yi= Ci(2t).
Fig. 2 Illustration for the proof of Theorem4. Without loss of generality, we assume xi∈ V0(Qn)for
1≤ i ≤ δ
(3) By Lemma3, Q1n− F1contains δ-mutually fully independent Hamiltonian paths R1[(x1)n, (y1)n], . . . , Rδ[(xδ)n, (yδ)n].
(4) Finally, we obtain the desired Hamiltonian cycles from combining Ci and Ri, 1≤ i ≤ δ. See Fig.2for illustration.
More precisely, the proof is by induction on n. It is trivial that the statement holds for Q3, as the induction basis. When n≥ 4, we assume that the statement holds for Qn−1. Now we consider how to build δ-mutually independent Hamil-tonian cycles in Qn − F . Since we assume there is at least one n-dimensional faulty edge, we partition Qn into {Q0n, Q1n} along dimension n. Accordingly, we have|F0| ≤ |F | − 1 ≤ n − 3, |F1| ≤ |F | − 1 ≤ n − 3, and (n − 1) − 1 − |F0| ≥ (n− 1) − 1 − (|F | − 1) = n − 1 − |F | = δ. Thus, by the inductive hypothesis, Q0n− F0contains δ-mutually independent Hamiltonian cycles C1, C2, . . . , Cδ begin-ning from e. For convenience, we assume that the vertices on each cycle are indexed sequentially from 1 to 2n−1; that is, the beginning vertex e has index 1. Next, we claim that there must exist an integer t , 1≤ t ≤ 2n−2, so that the crossing edges (Ci(2t− 1), (Ci(2t− 1))n)and (Ci(2t), (Ci(2t))n)are fault-free for all 1≤ i ≤ δ. If such edges do not exist, then we have|F | ≥ |Fc| ≥ 2n−2/δ >|F | for n ≥ 3, leading to an immediate contradiction. Let xi= Ci(2t− 1) and yi= Ci(2t). Accordingly, Ci can be represented ase, Pi, xi, yi, Hi,e, 1 ≤ i ≤ δ. By the definition of hyper-cubes, (xi)nand (yi)nare adjacent in Qn1. By Lemma3, Q1n−F1contains δ-mutually fully independent Hamiltonian paths R1[(x1)n, (y1)n], . . . , Rδ[(xδ)n, (yδ)n]. There-fore,{e, Pi, xi, (xi)n, Ri, (yi)n, yi, Hi,e | 1 ≤ i ≤ δ} forms a set of δ-mutually
in-dependent Hamiltonian cycles beginning from e.
5 Conclusion
In this paper, we concentrate on the problem of embedding mutually independent Hamiltonian cycles in a faulty n-cube, as previously addressed by Hsieh and Yu
(Hsieh and Yu2007). However, there are two mistakes in (Hsieh and Yu2007). There-fore, we first point out why their proof is deficient. Then we prove that Qncontains (n− 1 − f )-mutually independent Hamiltonian cycles when f ≤ n − 2 faulty edges may occur accidentally. Indeed, we believe this result can be further refined; that is, we would like to show Qn can be embedded with (n− f )-mutually independent Hamiltonian cycles beginning from any vertex when f ≤ n − 2 faulty edges occur.
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