DOI 10.1007/s11227-009-0316-3
A family of Hamiltonian and Hamiltonian connected
graphs with fault tolerance
Y-Chuang Chen· Yong-Zen Huang · Lih-Hsing Hsu· Jimmy J. M. Tan
Published online: 10 July 2009
© Springer Science+Business Media, LLC 2009
Abstract Processor (vertex) faults and link (edge) faults may happen when a network is used, and it is meaningful to consider networks (graphs) with faulty processors and/or links. A k-regular Hamiltonian and Hamiltonian connected graph G is optimal
fault-tolerant Hamiltonian and Hamiltonian connected if G remains Hamiltonian
af-ter removing at most k− 2 vertices and/or edges and remains Hamiltonian connected after removing at most k− 3 vertices and/or edges. In this paper, we investigate in constructing optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamil-tonian connected graphs. Therefore, some of the generalized hypercubes,
twisted-cubes, crossed-twisted-cubes, and Möbius cubes are optimal fault-tolerant Hamiltonian and
optimal fault-tolerant Hamiltonian connected.
Keywords Fault-tolerance· Hamiltonicity · Hamiltonian connectivity · Generalized hypercube
Y-C. Chen (
)Department of Information Management, Ming Hsin University of Science and Technology, Hsin Feng, Hsinchu 304, Taiwan
e-mail:[email protected]
Y.-Z. Huang
Chunghwa Telecom Co., Ltd., 12, Lane 551, Min-Tsu Rd. Sec. 5 Yang-Mei, Taoyuan 326, Taiwan
L.-H. Hsu
Department of Computer Science and Information Engineering, Providence University, Taichung 433, Taiwan
J.J.M. Tan
1 Introduction
The architecture of an interconnection network is usually represented by a graph. There exist conflicting requirements in designing the topology of interconnection networks. It is almost impossible to design a network which is optimum for all con-ditions. One has to design a suitable network depending on the required properties. The Hamiltonian property is one of the major requirements in designing the topology of networks. Fault tolerance is also desirable in massive parallel systems that have relatively high probability of failure. Much research have been proposed on the ring embedding problems in fault-tolerant networks [1,8,21,23,25,27].
For the graph definitions and notations, we follow [4,16]. 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 say that V is the vertex set and E is the edge set. The degree of a vertex v, denoted by deg(v), is the number of edges incident to v. A graph G is k-regular if deg(v)= k for every vertex in G. A path is a Hamiltonian path if its vertices are distinct and they span V . A cycle is a path with at least three vertices such that the first vertex is the same as the last one. A cycle is a Hamiltonian cycle if it traverses every vertex of G exactly once. A graph G is Hamiltonian if it has a Hamiltonian cycle, and G is Hamiltonian connected if there exists a Hamiltonian path joining any two vertices of G.
Since vertex faults and edge faults may happen when a network is used, it is practically meaningful to consider faulty networks. A graph G is called
l-fault-tolerant Hamiltonian (l-fault-l-fault-tolerant Hamiltonian connected respectively) or
sim-ply l-Hamiltonian (l-Hamiltonian connected, respectively) if it remains Hamiltonian (Hamiltonian connected, respectively), after removing at most l vertices and/or edges. The fault-tolerant Hamiltonicity, Hf(G), is defined to be the maximum integer l
such that G− F remains Hamiltonian for every F ⊂ V (G) ∪ E(G) with |F | ≤ l if Gis Hamiltonian, and undefined if otherwise. Obviously,Hf(G)≤ δ(G) − 2, where
δ(G)= minc{deg(v)|v ∈ V (G)}. In establishing their fault-tolerant Hamiltonicity, another parameter called tolerant Hamiltonian connectivity is used. The
fault-tolerant Hamiltonian connectivity,Hκ
f(G), is defined to be the maximum integer l
such that G− F remains Hamiltonian connected for every F ⊂ V (G) ∪ E(G) with
|F | ≤ l if G is Hamiltonian connected, and undefined if otherwise. It is not hard to
see thatHκf(G)≤ δ(G) − 3. A regular graph is called optimal fault-tolerant
Hamil-tonian and optimal fault-tolerant HamilHamil-tonian connected ifHf(G)= δ(G) − 2 and
Hκ
f(G)= δ(G) − 3. Twisted-cubes, crossed-cubes, Möbius cubes, and recursive cir-culant graphs are proved to be optimal tolerant Hamiltonian and optimal
fault-tolerant Hamiltonian connected [6,7,18–20,26]. All these families of graphs have some good properties in common, including that they can all be recursively con-structed.
The complete graph Knis a fully connected network with high performance, but
high cost. It has many desirable properties, such as small fixed diameter, maximum connectivity, shortest path routing, parallel paths, optimal fault-tolerant Hamiltonic-ity, and optimal fault-tolerant Hamiltonian connectivHamiltonic-ity, due to the high cost with fully connected links/edges between processors/vertices [14]. To reduce the cost and preserve the optimal fault-tolerant Hamiltonian and fault-tolerant Hamiltonian con-nected properties of the complete graph are the purposes of this paper.
Fig. 1 (a) The hypercube Q3; (b) the twisted-cube TQ3; (c) the crossed-cube CQ3; (d) the Möbius cube
0-MQ3; and (e) the Möbius cube 1-MQ3
Based on the recursively constructed graphs, the hypercube Qn is an important
one since it has a simple structure and easy to implement [5,24]. There are many im-portant variants of the hypercubes appearing in literature, such as twisted-cubes [12, 13,20], crossed-cubes [15,19,22], and Möbius cubes [9,18]. These variants pos-sess some desirable features of the hypercubes, and even better. For example, the diameter of these variants is around half of that of the hypercube. The ring and path structure embedding into these variants have also been heavily discussed. Many peo-ple have studied the problem of the existence of cycles and paths of arbitrary lengths in these networks [2,6,11–13,15,17–20,22]. The recursive structures of the hyper-cubes, twisted-hyper-cubes, crossed-hyper-cubes, and Möbius cubes are briefly illustrated below. Let Xn be any one of the n-dimensional hypercube, twisted-cube, crossed-cube, or
Möbius cube. An Xn is composed of two copies of Xn−1s and a specific perfect
matching between the two copies of Xn−1s. Figure1shows the 3-dimensional
hyper-cube, twisted-hyper-cube, crossed-hyper-cube, and Möbius cube.
The generalized hypercube [3] is another variant of the hypercube and also has several good topologies. First, the design of generalized hypercubes is based on the allowable diameter of the network. If the diameter can be increased, a structure with a lower degree of a vertex can be obtained. Secondly, the structures are very general in nature. Single loop, Boolean n-cubes, nearest neighbor mesh hypercubes, and fully
connected systems can be considered as a part of this generalized structure. Finally,
the structures are highly fault-tolerant and they possess a small average message dis-tance and a low traffic density. A generalized hypercube has been recursively defined in [10] as the following. Let G(mr, mr−1, . . . , m1)denote a generalized hypercube
graph of size mr × mr−1× · · · × m1, where mi ≥ 2 for all 1 ≤ i ≤ r. There are
N= mr× mr−1× · · · × m1vertices in G(mr, mr−1, . . . , m1)which are assigned
r-digit identifiers xrxr−1· · · x1, where xi ∈ {0, 1, . . . , mi − 1} for all 1 ≤ i ≤ r. Two
vertices in G(mr, mr−1, . . . , m1)are adjacent if and only if their identifiers differ at exactly one digit position. In Fig.2, the structure of generalized hypercube G(4, 2, 2) is depicted for illustration. It is clear that G(mr, mr−1, . . . , m1) with mi = 2 for
Fig. 2 The generalized hypercube G(4, 2, 2)
1≤ i ≤ r is isomorphic to the hypercube Qr. In this paper, the twisted-cube, crossed-cube, Möbius crossed-cube, and generalized hypercube are elements of the graph family gen-erated by the proposed construction scheme in the next section. The graph family preserves the optimal fault-tolerant Hamiltonian and fault-tolerant Hamiltonian con-nected properties, and it has lower cost than the complete graph.
The rest of this paper is organized as follows. In Sect.2, we give a recursively con-structed scheme for optimal fault-tolerant Hamiltonian and Hamiltonian connected graphs; and introduce the notations and terminology. In Sect.3, the optimal fault-tolerant Hamiltonicity and optimal fault-fault-tolerant Hamiltonian connectivity are dis-cussed. Section4shows the proof of optimal fault-tolerant Hamiltonicity and Sect.5 concludes the brief contribution.
2 Construction schemes of fault-tolerant Hamiltonian graphs and some notations
Now, we construct a more generalized graph G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j)
based on the generalized hypercube and complete graph. Let G1, G2, . . . , Gnbe
k-regular graphs with the same number of vertices. The graph H= G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j) is defined as follows. Graph H has vertex set V (H )= V (G1)∪
V (G2)∪ · · · ∪ V (Gn), and edge set E(H )= E(G1)∪ E(G2)∪ · · · ∪ E(Gn)∪
1≤i<j≤nMi,j, where Mi,j is an arbitrary perfect matching between the vertices
of Gi and Gj. See Fig. 3. We call each Gi a component for every i. Considering
each component Gi as a vertex and each perfect matching Mi,j as an edge, then
G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j)is reduced to a complete graph Kn. For
con-venience, we shall abbreviate G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j)as G(1..n). As an
example, generalized hypercubes are essentially constructed in this way. The G(1..n) is a (k+ n − 1)-regular graph with |V |2 × (k + n − 1) edges, which has lower cost than that of the complete graph Knwith |V |2 × (|V | − 1) edges.
In this paper, we show that if all of G1, G2, . . . , Gn are optimal fault-tolerant
Hamiltonian and optimal fault-tolerant Hamiltonian connected, then G(1..n) is also optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected for any arbitrary perfect matchings, Mi,j, provided k≥ 5.
All the proofs of fault-tolerant Hamiltonicity and Hamiltonian connectivity of the aforementioned families of graph are done by induction. We observe that there are certain common phenomena behind the recursive structures so that we may construct
Fig. 3 A schematic diagram of graph G(1..4)
other optimal fault-tolerant graphs. In this paper, we try to investigate these phenom-ena and establish a construction scheme for optimal fault-tolerant Hamiltonian and Hamiltonian connected graphs.
For ease of exposition, we make some convention about our notations. Consider the graph G(1..n). For each component Gi, we use small letters with subscript i to
denote the vertices in Gi, e.g., ui, vi, etc. For example, u1 is a vertex in G1, and
u2is a vertex in G2. Let G(i..j )be the graph G(Gi, Gi+1, . . . , Gj,i≤p<q≤jMp,q).
For each G(i..j ), we use small letters with subscript (i..j ) to denote the vertices in
G(i..j ), e.g., u(i..j ), v(i..j ), etc. For example, u(1..2) is a vertex in G(1..2), and u(2..5) is a vertex in G(2..5). A perfect matching Mi,j connecting the vertices of Gi and
Gj in pairs, such pairs of vertices are called matching vertices, and these edges are
called matching edges. We shall use the same letter with different subscripts to denote matching vertices of each other; e.g., uiand ujare the matching vertices of each other
in components Gi and Gjif there is a perfect matching between Gi and Gj.
We shall also consider graphs with some faults. Our objective is to find a fault free Hamiltonian cycle (Hamiltonian path, respectively) and each fault can be a faulty vertex or a faulty edge. If a vertex v is not faulty, we say v is a healthy vertex. If an edge e is not faulty, we say e is a healthy edge. We call an edge e (respectively, a matching edge e) super healthy if both edge e and its two endpoints are not faulty. We use Fito denote the set of faults in Gi, F(i..j )to denote the set of faults in G(i..j ).
Let fi= |Fi| and f(i..j )= |F(i..j )|. Given two distinct healthy vertices x and y, we
use the x-y Hamiltonian path to call a fault free Hamiltonian path joining x and y,
HPito denote a fault free Hamiltonian path in Gi− Fi, and HP(i..j )to denote a fault
free Hamiltonian path in G(i..j )− F(i..j ) where i≤ j. A fault free x-y Hamiltonian
path in Gi− Fi can be written asx, HPi, y and a fault free x-y Hamiltonian path
in G(i..j )− F(i..j )can be written asx, HP(i..j ), y. In addition, path x, HPi, y and
pathx, HP(i..j ), y are cycles if x = y.
3 Hamiltonian and Hamiltonian connected graphs with fault tolerance
This section shows optimal fault-tolerant Hamiltonicity and optimal fault-tolerant
Hamiltonian connectivity of the graph G(1..n).
Theorem 1 For n≥ 2 and k ≥ 5, let G1, G2, . . . , Gn be k-regular, (k −
2)-Hamiltonian, and (k− 3)-Hamiltonian connected graphs with the same number
of vertices. Then G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j)= G(1..n) is (k− 2 + n −
Proof We prove this theorem by inducting on n. It is proved in [6,7] that for n= 2 and n= 3, the result holds for G(1..2)and G(1..3). So G(1..2)is (k−2+1)-Hamiltonian and (k− 3 + 1)-Hamiltonian connected for k ≥ 5. And G(1..3) is (k − 2 + 2)-Hamiltonian and (k− 3 + 2)-Hamiltonian connected for k ≥ 5. For the induction
step, we divide the proof into the following two lemmas.
Lemma 1 For n≥ 3 and k ≥ 5, let G1, G2, . . . , Gn be k-regular, (k − 2)-Hamiltonian, and (k− 3)-Hamiltonian connected graphs with the same number of vertices. Then graph G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j)= G(1..n)is (k
−2+n−1)-Hamiltonian.
Lemma 2 For n≥ 3 and k ≥ 5, let G1, G2, . . . , Gn be k-regular, (k − 2)-Hamiltonian, and (k− 3)-Hamiltonian connected graphs with the same number of vertices. Then graph G(G1, G2, . . . , Gn,
1≤i<j≤nMi,j)= G(1..n)is (k
−3+n−1)-Hamiltonian connected.
The proof of Lemma1 is left in Sect.4. The proof of Lemma2 is rather long and similar to the proof of Lemma1; we omit it here. With the above two lemmas, Theorem1is proved.
4 Proof of Lemma1
Suppose n≥ 3. Assume that G(1..d) is (k− 2 + d − 1)-Hamiltonian and (k − 3 + d− 1)-Hamiltonian connected for all d ≤ n where k ≥ 5. We shall show that graph G(1..n+1)is (k− 2 + n)-Hamiltonian where k ≥ 5.
To show that the fault-tolerant HamiltonicityHf(G)of G(1..n+1)is (k− 2 + n) for k≥ 5, it suffices to show that G(1..n+1)− F(1..n+1)is Hamiltonian for any faulty set F(1..n+1)⊂ V (G(1..n+1))∪E(G(1..n+1))with|F(1..n+1)| = f(1..n+1)= k −2+n. If f(1..n+1)< (k− 2 + n), we may arbitrary choose (k − 2 + n) − f(1..n+1)healthy edges and designate them as faulty, then the total number of faults is exactly k− 2 + n. In G(1..n+1), every component Gi is adjacent to n perfect matchings,
1≤i =j≤n+1Mi,j.
For each Gi∪
1≤i =j≤n+1Mi,j for 1≤ i ≤ n + 1, we may without loss of
general-ity assume that Gn+1∪
1≤j≤nMn+1,j contains the most number of faults. So the
number of faults of it must be greater or equal to k−2+nn+1 ≥ 5−2+nn+1 ≥ 2. Thus, f(1..n)≤ (k − 2 + n) − 2 = k − 3 + n − 1. By induction, G(1..n)− F(1..n) is Hamil-tonian connected. Let fn+1be the number of faults of Gn+1, this lemma is proved by
three cases.
Case 1: fn+1≤ k − 3.
Gn+1is k-regular, so the number of vertices of Gn+1is k+ 1 at least. The number
of matching edges between Gn+1− Fn+1and G(1..n)is at least n(k+ 1 − fn+1), and the total number of faults is k− 2 + n, so there exists a super healthy matching edge (un+1, u(1..n)). Suppose not, n(k+1−fn+1)≤ (f(1..n+1)−fn+1), n(k+1−fn+1)≤ ((k− 2 + n) − fn+1), and (n− 1)(k − fn+1)+ 2 ≤ 0, which is a contradiction. Besides, we shall find another super healthy matching edge (vn+1, v(1..n))such that
Fig. 4 Case 1: fn+1≤ k − 3
Fig. 5 Case 2: fn+1= k − 2
un+1 = vn+1and u(1..n) = v(1..n). Suppose not, the number of matching edges be-tween Gn+1− (Fn+1∪ {un+1}) and G(1..n)is at least n((k+ 1) − fn+1− 1). Thus, n((k+ 1) − fn+1− 1) ≤ (f(1..n+1)− fn+1), n((k+ 1) − fn+1− 1) ≤ ((k − 2 + n) − fn+1), and (n−1)(k −fn+1−1)+1 ≤ 0, which is a contradiction to our assumption.
In addition, both Gn+1− Fn+1and G(1..n)− F(1..n) are Hamiltonian connected. So we have a fault free Hamiltonian cycleun+1,HPn+1, vn+1, v(1..n),HP(1..n), u(1..n),
un+1 in G(1..n)− F(1..n). See Fig.4. Case 2: fn+1= k − 2.
There is a fault free Hamiltonian cycle, say HC, in Gn+1 − Fn+1. Since |F(1..n+1)| − |Fn+1| = n, we may without loss of generality assume that the number of faults in G1∪ M1,n+1is at most 1. The length of the fault free Hamiltonian cycle in Gn+1− Fn+1is at least 3, so there is an edge (un+1, vn+1)in HC, such that both
matching edges (un+1, u1)and (vn+1, v1)are super healthy. By induction, G(1..n) is (k− 3 + n − 1)-Hamiltonian connected where k ≥ 5. So there is a u1-v1Hamiltonian path in G(1..n)− F(1..n)since f(1..n)= n. Therefore, we have a fault free Hamiltonian cycleun+1,HPn+1, vn+1, v1,HP(1..n), u1, un+1 in G(1..n)− F(1..n). See Fig.5. Case 3: k− 2 + 1 ≤ fn+1≤ k − 2 + n.
Fig. 6 Case 3: k− 2 + 1 ≤ fn+1≤ k − 2 + n
While fn+1= k−2, Gn+1−Fn+1has a fault free Hamiltonian cycle. Now, fn+1=
k− 2 + i where 1 ≤ i ≤ n, the fault free Hamiltonian cycle in Gn+1may be cut into j paths, where j≤ i, say u1n+1, . . . , u2n+1, u3n+1, . . . , u4n+1, . . . , u2jn+1−1, . . . , u2jn+1.
Now, there are n− i faults in G(1..n+1)− Gn+1. We choose j components in G(1..n), such that the j components have no fault, and the matching edges between
the j components and Gn+1 are healthy. We may without loss of generality say
that the j components are G1, G2, . . . , Gj. Then we choose 2j matching edges
(u2n+1, u21), (u3n+1, u31), (u4n+1, u42), (u5n+1, u25), . . . , (u2jn+1−2, uj2j−1−2), (u2jn+1−1, u2jj−1−1), (u1n+1, u1j), and (u2jn+1, u2jj ). There are n− i faults in G(j..n) at most, by induction, G(j..n) is ((n− (j − 1)) + k − 3) = (n − j − 2 + k)-Hamiltonian connected. And
n− j − 2 + k > n − i since j ≤ i and k ≥ 5. So G(j..n) is fault free Hamiltonian connected. In addition, each component G1, . . . , Gj−1is Hamiltonian connected. As
a result, we have a fault free Hamiltonian cycle as shown in the case of Fig.6.
This completes the proof of this lemma.
Corollary 1 Let G1, G2, . . . , Gn be k-regular optimal fault-tolerant Hamiltonian and Hamiltonian connected graphs with the same number of vertices where k≥ 5 and n≥ 2. Then graph G(1..n) is also an optimal fault-tolerant Hamiltonian and an
optimal fault-tolerant Hamiltonian connected graph.
As for the case of k < 5 in Corollary1, we conjecture that there have similar results. But the proof is rather long, we left it as an open problem here.
5 Conclusions
The fault-tolerant Hamiltonicity and the fault-tolerant Hamiltonian connectivity are essential parameters of an interconnection network. In this paper, we propose that k-regular Hamiltonian and Hamiltonian connected graphs are optimal fault-tolerant
Hamiltonian and Hamiltonian connected if graph G remains Hamiltonian after re-moving at most k−2 vertices/edges and remains Hamiltonian connected after remov-ing at most k− 3 vertices/edges. We investigate in constructing optimal fault-tolerant Hamiltonian and Hamiltonian connected graphs with flexibility.
There are many popular interconnection networks which are k-regular graphs. Some of them, e.g., twisted-cubes, crossed-cubes, Möbius cubes, and generalized hypercubes, can be recursively constructed using our construction scheme and, there-fore, they are in fact a subclass of our proposed family of graphs, and some of them are optimal fault-tolerant Hamiltonian and Hamiltonian connected.
Acknowledgements The authors would like to express their gratitude to the anonymous referees for their kind suggestions and useful comments on the original manuscript, which have greatly improved the quality of the paper.
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