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Fault-tolerant hamiltonicity and
fault-tolerant hamiltonian connectivity of
the folded Petersen cube networks
Cheng-Kuan Lin a , Tung-Yang Ho b , Jimmy J.M. Tan a & Lih-Hsing Hsu c
a
Department of Computer Science , National Chiao Tung University , Hsinchu, Taiwan, Republic of China
b
Department of Industrial Engineering and Management , Ta Hwa Institute of Technology , Hsinchu, Taiwan, Republic of China c
Department of Computer Science and Information Engineering , Providence University , Taichung, Taiwan, Republic of China Published online: 07 Nov 2008.
To cite this article: Cheng-Kuan Lin , Tung-Yang Ho , Jimmy J.M. Tan & Lih-Hsing Hsu (2009) Fault-tolerant hamiltonicity and fault-tolerant hamiltonian connectivity of the folded Petersen cube networks, International Journal of Computer Mathematics, 86:1, 57-66, DOI: 10.1080/00207160701684485
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International Journal of Computer Mathematics
Vol. 86, No. 1, January 2009, 57–66
Fault-tolerant hamiltonicity and fault-tolerant hamiltonian
connectivity of the folded Petersen cube networks
Cheng-kuan Lina, Tung-yang Hob*, Jimmy J.M. Tanaand Lih-Hsing HsucaDepartment of Computer Science, National Chiao Tung University, Hsinchu, Taiwan, Republic of China; bDepartment of Industrial Engineering and Management, Ta Hwa Institute of Technology, Hsinchu,
Taiwan, Republic of China;cDepartment of Computer Science and Information Engineering, Providence University, Taichung, Taiwan, Republic of China
(Received 07 December 2006; revised version received 03 July 2007; second revision received 04 September 2007;
accepted 10 September 2007 )
Some research on the folded Petersen cube networks have been published for the past several years due to its favourite properties. In this paper, we consider the fault-tolerant hamiltonicity and the fault-tolerant hamil-tonian connectivity of the folded Petersen cube networks. We use F P Qn,kto denote the folded Petersen cube networks of parameters n and k. In this paper, we show that F P Qn,k− F remains hamiltonian for any F⊆ V (FP Qn,k)∪ E(FP Qn,k)with|F | ≤ n + 3k − 2 and FP Qn,k− F remains hamiltonian con-nected for any F⊆ V (FP Qn,k)∪ E(FP Qn,k)with|F | ≤ n + 3k − 3 if (n, k) /∈ {(0, 1)} ∪ {(n, 0) | n is a positive integer}. Moreover, this result is optimal.
Keywords: hamiltonian; hamiltonian connected; folded Petersen cube networks
2000 AMS Subject Classification: 05C45; 05C75; 05C85; 05C90; 68M10
1. Introduction
A large number of forms of topology have been proposed and studied for multicomputer inter-connection networks. Such a form of topology is usually modelled as an undirected graph, where the set of vertices represents the processors and the set of edges represents the bidirectional com-munication links between the processors. The existing static interconnection networks include linear arrays, rings, meshes, complete binary trees, X-trees, full-ringed binary trees, tree machines, pyramids, fat trees [16], hypercubes [18], meshes of trees, cube-connected cycles [17], de Bruijn networks [19], and so on. For general surveys on multicomputer networks, refer to [13,14]. Among these networks, the hypercube family has been popular because of such properties as symmetry, regularity, high fault-tolerance, logarithmic degree and diameter, and selfrouting and simple broadcasting schemes. Also, several commercial multicomputer architectures (e.g., Intel
*Corresponding author. Email: hoho@thit.edu.tw ISSN 0020-7160 print/ISSN 1029-0265 online © 2009 Taylor & Francis
DOI: 10.1080/00207160701684485 http://www.informaworld.com
iPSC/2, NCUBE/10, and connection machine CM-2) use the hypercube topology for intercon-necting processors. Nevertheless, new networks are being proposed and analysed with regard to their applicability and enhanced topological or performance properties.
It is almost impossible to design a network that is optimum from all aspects. One has to design a suitable network depending on the requirements of its properties. The hamiltonian property is one of the major requirements in designing the topology of a network. For example, ‘Token Passing’ approach is used in some distributed operation systems. Interconnection network requires the presence of hamiltonian cycles in the structure to meet this approach. Fault-tolerance is another major requirement in the designing of network topology. Thus, the fault-tolerant hamiltonicity is studied [9–12]. The fault-tolerant hamiltonicity of a network and the fault-tolerant hamiltonian connectivity for the same network are also studied [6–8,21].
In this paper, a network is represented as a loopless undirected graph. For graph definitions and notations, we follow [1]. G= (V, E) is a graph if V is a finite set and E is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V is the vertex set and E is the edge set. Two vertices u and v are adjacent if (u, v)∈ E. The degree of a vertex u of G, degG(u), is the
number of edges incident with u. We use δ(G) to denote min{degG(x)| x ∈ V (G)}. A graph G
is k-regular if degG(x)= k for any vertex in G. A path, v0, v1, v2, . . . , vk, is an ordered list of
distinct vertices such that viand vi+1are adjacent for 0≤ i ≤ k − 1. The length of a path P is the
number of edges in P . The distance between two vertices u and v in G is the length of a shortest path joining them. The diameter of a graph G is the distance between the farthest points in G.
A path is a hamiltonian path if its vertices are distinct and span V . A cycle is a path with at least three vertices such that the first vertex is the same as the last vertex. A cycle is a hamiltonian
cycle if it traverses every vertex of G exactly once. A graph is hamiltonian if it has a hamiltonian
cycle. We will use Knto denote the complete graph with n vertices and use Cnto denote the cycle
graph with n vertices.
A hamiltonian graph G is k fault-tolerant hamiltonian if G− F remains hamiltonian for every
F ⊂ V (G) ∪ E(G) with |F | ≤ k. The fault-tolerant hamiltonicity H F (G) is defined to be the
maximum integer k such that G is k fault-tolerant hamiltonian if G is hamiltonian and is undefined otherwise. Clearly, H F (G)≤ δ(G) − 2 if H F (G) is defined.A graph G is hamiltonian connected if there exists a hamiltonian path joining any two vertices of G. All hamiltonian connected graphs except the complete graphs K1and K2are hamiltonian. A graph G is k fault-tolerant hamiltonian
connected if G− F remains hamiltonian connected for every F ⊂ V (G) ∪ E(G) with |F | ≤ k.
The fault-tolerant hamiltonian connectivity H F K(G) is defined to be the maximum integer
k such that G is k fault-tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise. It can be checked that H F K(G)≤ δ(G) − 3 only if H F K(G) is defined and |V (G)| ≥ 4. There are a lot of studies on fault-tolerant hamiltonicity and fault-tolerant hamiltonian connectivity [6–12,21].
In [5], the performance of the hamiltonian property in faulty networks is discussed. Huang et al. [11] introduced the term, fault-tolerant hamiltonian connected. The Petersen graph [1] is a 3-regular graph with 10 vertices and diameter 2 as compared with the three-dimensional hypercube, which is a 3-regular graph with eight vertices and diameter 3. The folded Petersen cube networks are motivated by the Petersen graph. In this paper, we consider the fault-tolerant hamiltonicity and the fault-tolerant hamiltonian connectivity of the folded Petersen cube networks. The k-dimensional folded Petersen network, F Pk, is constructed by an iterative Cartesian
product on the Petersen graph. The folded Petersen network is then generalized into folded Petersen cube network, F P Qn,k. The graph F P Qn,kis defined as a product of F Pkand the n-dimensional
binary hypercube, Qn. The number of vertices in F Pkis 10k, whereas there are 2n× 10kvertices in
F P Qn,k, which is therefore more scalable than F Pk. It turns out that F P Qn,kand even its special
derivations F P Q0,k= FPk and F P Qn−3,1= HPn, called the n-dimensional hyper Petersen
network originally proposed by Das et al. [4], are better than the comparable-size hypercubes and
International Journal of Computer Mathematics 59
several other networks with respect to the usual metrics (such as degree, diameter, connectivity, packing density, or cost) of a multicomputer architecture.
In this paper, we prove that H F (F P Qn,k)= n + 3k − 2 and H F K(FP Qn,k)= n + 3k − 3
if (n, k) /∈ {(0, 1)} ∪ {(n, 0) | n is a positive integer}. Moreover, FP Q0,1is neither hamiltonian nor hamiltonian connected. Furthermore, F P Qn,0is hamiltonian but not hamiltonian connected if n > 1; F P Q1,0is hamiltonian connected but hamiltonian.
In the following section, we give the definition of the folded Petersen cube networks. In Section 3, we present some mathematical preliminary. In Section 4, we prove our main result. In the final section, we give a discussion of our work.
2. Folded Petersen cube networks
Let G1= (V1, E1)and G2= (V2, E2)be two graphs. The cartesian product of G1 and G2, denoted by G1× G2, is the graph with vertex set V1× V2such that (u1, v1)is joined to (u2, v2)if and only if either u1= u2and v1is joined to v2in G2or v1 = v2and u1is joined to u2in G1. For any graph G and any positive integer k, we define Gk= G if k = 1 and Gk= Gk−1× G if k > 1.
For any positive integer, the n-dimensional hypercube Qnis defined as Q1= K2 and Qn=
Qn−1× Q1. Thus, Qn= K2n. The Petersen graph P , shown in Figure 1(a), is a graph with 10 ver-tices having an outer 5-cycle, an inner 5-cycle, and five spokes joining them. The folded Petersen cube network F P Qn,kis defined as Qn× Pk. In particular, F P Q0,k= Pkand F P Qn,0 = Qn.
The graph F P Q0,2= P2is shown in Figure 1(b).
The topology of an interconnection network plays an important role in the performance of a distributed system. Various network topologies are in use and have their respective advantages and disadvantages. A good network topology should be one with a small diameter. Regularity and symmetry are some other qualities, which a good network topology is expected to possess. Some of the well-known network topologies in use are the ring topology, the hypercube, the cube-connected cycles, and so on. The Petersen graph is a 3-regular graph with 10 vertices and of diameter 2. Compared with this graph, the three-dimensional hypercube is a 3-regular graph with
Figure 1. The Petersen graph P and (b) a schematic representation of P .
eight vertices and of diameter 3. It has more vertices as compared with the three-dimensional hypercube and of a smaller diameter. We call it the simple Petersen graph. As an extension, Öhring and Das [15] introduce the k-dimensional folded Petersen graph, F Pk, to be F P Q0,k. It is observed that F Pkpossesses qualities of a good network topology for distributed systems with
large number of sites since it accommodates 10kvertices and is a symmetric, 3k-regular graph of
diameter 2k. Being an iterative Cartesian product on the Petersen graph, it is scalable. Moreover, Öhring and Das [15] define the folded Petersen cube networks F P Qn,kand show that a number
of standard topologies like linear arrays, rings, meshes, hypercubes, and so on can be embedded into it. Recently, some research on the folded Petersen cube networks have been published for the past several years due to its favourite properties [4,15,20].
Yet, to our knowledge, there is no study on the hamiltonian property on folded Petersen cube networks. Perhaps, the difficulty is on the known result that the Petersen graph is not hamiltonian. With the result of this paper, there are only two non-hamiltonian graphs in the family of folded Petersen cube networks, namely F P Q1,0and F P Q0,1.
3. Preliminary
We say a k-regular graph is super fault-tolerant hamiltonian if H F (G)= k − 2 and H F K(G) =
k− 3. Some interesting families of interconnection networks are proved to be super fault-tolerant
[9–11]. Chen et al. [2] observed the insight of the proofs of the aforementioned results and proposed the following construction scheme.
Let G1and G2be two graphs with the same number of vertices. Let M be an arbitrary perfect
matching between the vertices of G1and G2; i.e., M is a set of edges connecting the vertices of
G1and G2in a one to one fashion. For convenience, G1and G2 are called components. Then,
G(G1, G2; M) is the graph with vertex set V (G(G1, G2; M)) = V (G1)∪ V (G2)and edge set
E(G(G1, G2; M)) = E(G1)∪ E(G2)∪ M (see Figure 2 for an illustration).
Let H be a graph. Obviously, the Cartesian product H× K2can be viewed as G(H, H; M) for some matching M. The Petersen graph P can be viewed as G(C5, C5; M) for some matching
Mas shown in Figure 3.
Furthermore, P× C5can be viewed as G(C5× C5, C5× C5; M) for some matching M (see Figure 4 for an illustration).
The following two theorems are proved in [2].
THEOREM1 Assume k≥ 4. Let G1 and G2be two k-regular super fault-tolerant hamiltonian
graphs and|V (G1)| = |V (G2)|. Then graph G(G1, G2; M) is (k − 1) fault-tolerant
hamilto-nian[2].
Figure 2. G(G1, G2; M).
International Journal of Computer Mathematics 61
Figure 3. (a) A copy of C5, (b) another copy of C5, (c) the matching M, and (d) the Petersen graph P .
Figure 4. (a) A copy of C5× C5, (b) another copy of C5× C5, (c) the matching M, and (d) P× C5.
THEOREM2 Assume k≥ 5. Let G1 and G2 be two k-regular super fault-tolerant hamiltonian
graphs and|V (G1)| = |V (G2)|. Then graph G(G1, G2; M) is (k − 2) fault-tolerant hamiltonian
connected[2].
Combining Theorems 1 and 2, we have the following corollary.
COROLLARY1 Assume k≥ 5. Let G1and G2be two k-regular super fault-tolerant hamiltonian
graphs and |V (G1)| = |V (G2)|. Then graph G(G1, G2; M) is (k + 1)-regular super
fault-tolerant hamiltonian connected.
Chen et al. [3] further extended their work by considering another construction scheme. Let r and t be positive integers with r≥ 3. Assume that G0, G1, . . . , Gr−1are graphs with|V (Gi)| = t
for 0≤ i ≤ r − 1. We define H = G(G0, G1, . . . , Gr−1; M) with V (H ) =
r−1
i=0V (Gi)and
E(H )= M ∪ri=0−1E(Gi), whereM =
r−1
i=0Mi,i+1 (mod r)with Mi,i+1 (mod r)is any arbitrary
perfect matching between V (Gi)and V (Gi+1 (mod r)), (see Figure 5 for an illustration). Let H be
any graph. Obviously, the Cartesian product H× C4can be viewed as G(H, H, H, H; M). The following theorem is proved in [3].
THEOREM3 Assume that G0, G1, . . . , Gn−1are k-regular super fault-tolerant hamiltonian with
the same number of vertices where n≥ 3 and k ≥ 5. Then G(G0, G1, . . . , Gn−1; M) is a (k +
2)-regular super fault-tolerant hamiltonian [3].
Theorems 1, 2, and 3 are useful to construct super fault-tolerant hamiltonian graphs. Yet, one drawback of these theorems is that k= 4 is excluded. However, we have some difficulty to improve Theorem 2 by including the case k= 4. For this reason, in this paper, we introduce the concept of extendable 4-regular super fault-tolerant hamiltonian graph. A 4-regular super fault-tolerant hamiltonian graph H is extendable if H− {x, y} remains hamiltonian connected for any x and y such that (x, y)∈ E(H ).
The following lemma is proved by brute force. Here, we just state the result to reduce the complexity.
Figure 5. H= G(G0, G1, . . . , Gr−1; M).
International Journal of Computer Mathematics 63
LEMMA1 Both P× Q1and C5× C5are extendable 4-regular super fault-tolerant hamiltonian
graphs. However, C5× C4 is a 4-regular super fault-tolerant hamiltonian but not extendable
graph.
THEOREM 4 Assume that G1 and G2 are extendable 4-regular super fault-tolerant
hamilto-nian graphs with |V (G1)| = |V (G2)|. Then G(G1, G2; M) is a 5-regular super fault-tolerant
hamiltonian.
Proof Since Giis a 4-regular,|V (Gi)| ≥ 5. Obviously, G(G1, G2; M) is a 5-regular graph. By Theorem 1, G(G1, G2; M) is a 3 fault-tolerant hamiltonian.
Now, we need to prove G(G1, G2; M) is 2 fault-tolerant hamiltonian connected. Let F be any subset of V (G(G1, G2; M)) ∪ E(G(G1, G2; M)) with |F | ≤ 2. We need to find a hamiltonian path in G(G1, G2; M) − F between all pairs of vertices in V (G(G1, G2; M)) − F .
We use Fi to denote F∩ (V (Gi)∪ E(Gi)) for i= 1, 2. Moreover, we use F0 to denote
F ∩ M. Obviously, |F | = |F0| + |F1| + |F2|. Let x be any vertex in V (G(G1, G2; M)). We use x to denote the vertex of V (G(G1, G2; M)) matched under M. Thus, x in V (Gi)if and only
if x ∈ V (G3−i), for i= 1, 2. By considering the symmetric role of G1 and G2, we have the following cases.
Case 1 |Fi| ≤ 1 for i = 1, 2.
(a) u∈ V (G1)and v∈ V (G2). Since|V (Gi)| ≥ 5 and |F | ≤ 2, we can find an edge (x, x )∈
M− F with x ∈ V (G1)− (F1∪ {u}) and x ∈ V (G2)− (F2∪ {v}). Since G1 and G2 are 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R1of G1− F1 join-ing u to x and there exists a hamiltonian path R2 of G2− F2 joining x to v. Obviously, u, R1, x, x , R2, v forms a hamiltonian path of G(G1, G2; M) − F joining u to v (see Figure 6(a) for an illustration).
(b) {u, v} ⊂ V (G1). Since G1is 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R1of G1− F1joining u to v. Thus, the length of R1is at least|V (G1− F1)| − 1. Suppose that we can find an edge (x, y) in R1such that{x , y , (x, x ), (y, y )} ∩ F = ∅. We can rewrite R1asu, R11, x, y, R12, v. Note the length of R11is 0 if u= x and the length of R12is 0 if y= v. Since G2is 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R2 of G2− F2joining x to y . Obviously,u, R11, x, x , R2, y , y, R12, v forms a hamiltonian path of G(G1, G2; M) − F joining u to v (see Figure 6(b) for an illustration).
Suppose that we cannot find an edge (x, y) in R1such that{x , y , (x, x ), (y, y )} ∩ F = ∅. Then R1can be written asu = x1, x2, x3, x4, x5 = v and F is one of the following cases: (1) {(x2, x2 ), (x4, x4 )}, (2) {x 2, (x4, x4 )}, or (3) {(x2, x2 ), x4 }. In this case, |V (G1)| = 5. Hence, G1 and G2are isomorphic to the complete graph K5. Moreover, G(G1, G2; M) is isomorphic to the Cartesian product of K5and K2. In this case, we can find another hamiltonian path R 1joining u and
vasu = x1, x2, x4, x3, x5= v. Now, the edge (x3, x5)in R1 satisfies{x3 , x 5, (x3, x3 ), (x5, x5 )} ∩
F = ∅. As in the previous case, we can find a hamiltonian path of G(G1, G2; M) − F joining u to v (see Figure 6(c) for an illustration).
Case 2 |F1| = 2. Thus, |F0| = |F2| = 0.
(a) {u, v} ⊂ V (G1). Choose an element f ∈ F . Set F = F − {f }. Thus, |F | = 1. Since G1is 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R1of G1− F joining
uto v. Note that f is in R1if f is a vertex. However, f may not be in R1if f is an edge. We can write R1asu, R11, x, f, y, R12, u if f is in R1. Note that the length of R11is 0 if u= x and the length of R2
1is 0 if y= v. We can also write R1asu, R11, x, y, R21, u by picking any
Figure 6. Illustration for Theorem 4.
edge (x, y)∈ R1if f is not in R1. Since G2is 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R2of G2joining x to y . Obviously,u, R11, x, x , R2, y , y, R12, v forms a hamiltonian path of G(G1, G2; M) − F joining u to v (see Figure 6(d),(e), and (f) for an illustration).
(b) u∈ V (G1)and v∈ V (G2). Since G1is a 2 fault-tolerant hamiltonian, there exists a hamilto-nian cycle C of G1− F . Since C can be traversed backward and forward, we can write
C as u, R1, x, u such that x = v. Since G2 is 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R2 of G2 joining x to v. Obviously,u, R1, x, x , R2, v forms a hamiltonian path of G(G1, G2; M) − F joining u to v (see Figure 6(g) for an illustration).
(c) {u, v} ⊂ V (G2). Suppose u ∈ F . Since G/ 1 is a 2 fault-tolerant hamiltonian, there exists a hamiltonian cycle C of G1− F . Since C can be traversed backward and forward, we can write
Casu , R1, x, u such that x = v. Since G2is 1 fault-tolerant hamiltonian connected, there exists a hamiltonian path R2of G2− {u} joining x to v. Obviously,u, u , R1, x, x , R2, v forms a hamiltonian path of G(G1, G2; M) − F joining u to v (see Figure 6(h) for an illustration).
Suppose u ∈ F . Since G2 is 4-regular, there exists a neighbour x of u in G2− {v} such that x ∈ F . Since G/ 1 is a 2 fault-tolerant hamiltonian, there exists a hamiltonian cycle C of
G1− F . Since C can be traversed backward and forward, we can write C as x , R1, y , x such that y = v. Since u ∈ F , y = u. Since G2 is extendable, there exists a hamiltonian path R2of
G2− {u, x} joining y to v. Obviously, u, x, x , R1, y , y, R2, v forms a hamiltonian path of
G(G1, G2; M) − F joining u to v (see Figure 6(i) for an illustration).
The theorem is proved.
International Journal of Computer Mathematics 65
4. Main result
LEMMA2 Pkis a (3k)-regular super fault-tolerant hamiltonian if and only if k≥ 2.
Proof It is known that P is not hamiltonian. Hence, it is not hamiltonian connected. It is observed that P2 can be viewed as G(P × C
5, P × C5; M). Moreover, P × C5can be viewed as G(C5× C5, C5× C5; M). By Lemma 1, C5× C5 is an extendable 4-regular super fault-tolerant hamiltonian. By Theorem 4, P × C5is a 5-regular super fault-tolerant hamiltonian. By Corollary 1, P2is a 6-regular super fault-tolerant hamiltonian. Assume that k is a positive inte-ger with k≥ 3. Again, Pk can be viewed as G(Pk−1× C
5, Pk−1× C5; M). By Theorem 3,
Pk−1× C
5is a (3k− 1)-regular super fault-tolerant hamiltonian. By Theorems 1 and 2, Pkis a
(3k)-regular super fault-tolerant hamiltonian. Thus, the lemma is proved. THEOREM 5 H F (F P Qn,k)= n + 3k − 2 and H F K(FP Qn,k)= n + 3k − 3 if (n, k) /∈
{(0, 1)} ∪ {(n, 0) | n is a positive integer}. Moreover, FP Q0,1is neither hamiltonian nor
hamil-tonian connected. Furthermore, F P Qn,0is hamiltonian but not hamiltonian connected if n > 1;
F P Q1,0is hamiltonian connected but not hamiltonian.
Proof By Lemma 2, H F (F P Q0,k)= 3k − 2 and H F K(FP Q0,k)= 3k − 3 if k ≥ 2. By Lemma 1, H F (F P Q1,1)= 2 and HF K(FP Q1,1)= 1. By Lemma 1 and Theorem 4,
H F (F P Q2,1)= 3 and HF K(FP Q2,1)= 2. By Corollary 1, H F (FP Qn,k)= n + 3k − 2 and
H F K(F P Qn,k)= n + 3k − 3 if n ≥ 3 and k ≥ 1. Thus, H F (FP Qn,k)= n + 3k − 2 and
H F K(F P Qn,k)= n + 3k − 3 if (n, k) /∈ {(0, 1)} ∪ {(n, 0) | n is a positive integer}. Obviously,
F P Q0,1is neither hamiltonian nor hamiltonian connected. Note that F P Qn,0is isomorphic to
Qn. It is known that Q1is hamiltonian connected but not hamiltonian. Moreover, Qnis
hamil-tonian and there is no hamilhamil-tonian path of Qnjoining any two vertices in the same partite set if
n≥ 2. Thus, FP Qn,0 is hamiltonian but not hamiltonian connected if n > 1. Since F P Q1,0is isomorphic to K2, F P Q1,0is hamiltonian connected but not hamiltonian. We believe that our approach of this paper can be applied to the same problem on other interconnection networks. Definitely, we can repeatedly apply Theorems 1, 2, and 3 to obtain the result in this paper. However, we need a lot of efforts to check the base cases for the requirement
k≥ 5 in Theorems 2 and 3. By introducing the concept of extendable 4-regular super fault-tolerant
hamiltonian graph, all difficulties have been overcome. Thus, it would be a great improvement if Theorems 1, 2, and 3 remain true for smaller k.
Let H be the graph shown in Figure 7(a). Obviously, H is a 3-regular graph. By brute force, we can check that H is a 3-regular super fault-tolerant hamiltonian. Let G1and G2be two copies of
H. Let G be the graph shown in Figure 7(b). Obviously, G= (G1, G2; M) for some matching M.
Figure 7. (a) The graph H and (b) the graph G.
By brute force, we can prove that G− {2, 12} is not hamiltonian. Thus, G is not a 2 fault-tolerant hamiltonian.
Hence, the bound of k in Theorem 1 is optimal. Yet, we believe the bounds of k in Theorems 2 and 3 are optimal.
5. Conclusion
In this paper, we study the fault-tolerant hamiltonicity and fault-tolerant hamiltonian connec-tivity on the folded Petersen cube graph. We prove that, H F (F P Qn,k) = n + 3k − 2 and
H F K(F P Qn,k)= n + 3k − 3 if (n, k) /∈ {(0, 1)} ∪ {(n, 0) | n is a positive integer}. We believe
that the approach of this paper can be used to obtain the fault-tolerant hamiltonicity of other interconnection networks. Moreover, it is interesting to prove that the bounds of k in Theorems 2 and 3 are optimal.
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