Fault-tolerant pancyclicity of augmented cubes

As an enhancement on the hypercube Qn, the augmented cube AQn, prosed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], not only retains some favorable properties of Qn but also possesses some embedding properties that Qn does not. For example, AQn is pancyclic, that is, AQn contains cycles of arbitrary length for n?2. This paper shows that AQn remains pancyclic provided faulty vertices and/or edges do not exceed 2n−3 and n?4.     

The forwarding indices of augmented cubes

For a given connected graph G of order n, a routing R in G is a set of n(n−1) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of G is the maximum number of paths in R passing through any vertex (resp. edge) in G. Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2002) 71-84] proposed a variant of the hypercube Qn, called the augmented cube AQn and presented a minimal routing algorithm. This paper determines the vertex and the edge forwarding indices of AQn as and ²n−1, respectively, which shows that the above algorithm is optimal in view of maximizing the network capacity.     

Vertex-pancyclicity of twisted cubes with maximal faulty edges

The n-dimensional twisted cube, denoted by TQ _n , a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, we show that every vertex in TQ _n lies on a fault-free cycle of every length from 6 to 2 ⁿ , even if there are up to n?2 link faults. We also show that our result is optimal.     

On embedding cycles into faulty dual-cubes

A dual-cube uses low-dimensional hypercubes as basic components such that keeps the main desired properties of the hypercube. Each hypercube component is referred as a cluster. A (n+1)-connected dual-cube DC(n) has ²2n+1 nodes and the number of nodes in a cluster is n². There are two classes with each class consisting of n² clusters. Each node is incident with exactly n+1 links where n is the degree of a cluster, one more link is used for connecting to a node in another cluster. In this paper, we show that every node of DC(n) lies on a cycle of every even length from 4 to ²2n+1 inclusive for n?3, that is, DC(n) is node-bipancyclic for n?3. Furthermore, we show that DC(n), n?3, is bipancyclic even if it has up to n−1 edge faults. The result is optimal with respect to the number of edge faults tolerant.     

Hamilton-connectivity and cycle-embedding of the Möbius cubes

The recently introduced interconnection network, the Möbius cube, is an important variant of the hypercube. This network has several attractive properties compared with the hypercube. In this paper, we show that the n-dimensional Möbius cube M_n is Hamilton-connected when n?3. Then, by using the Hamilton-connectivity of M_n, we also show that any cycle of length l (4?l?2ⁿ) can be embedded into M_n with dilation 1 (n?2). It is a fact that the n-dimensional hypercube Q_n does not possess these two properties.     

Augmented k-ary n-cubes

We define an interconnection network AQ_n,k which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQ_n,k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube AQ_n,k: is a Cayley graph, and so is vertex-symmetric, but not edge-symmetric unless n = 2; has connectivity 4n − 2 and wide-diameter at most max{(n − 1)k − (n − 2), k + 7}; has diameter , when n = 2; and has diameter at most , for n ? 3 and k even, and at most , for n ? 3 and k odd.     

Highly fault-tolerant cycle embeddings of hypercubes

The hypercube Q_n is one of the most popular networks. In this paper, we first prove that the n-dimensional hypercube is 2n − 5 conditional fault-bipancyclic. That is, an injured hypercube with up to 2n − 5 faulty links has a cycle of length l for every even 4 ⩽ l ⩽ 2ⁿ when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n − 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2n − 5 faulty links under all possible fault distributions.     

Conditional edge-fault-tolerant edge-bipancyclicity of hypercubes

The hypercube has been one of the most popular interconnection networks for parallel computer/communication systems. In this paper, we assume that each node is incident with at least two fault-free links. Under this assumption, we show that every fault-free edge lies on a fault-free cycle of every even length from 6 to 2ⁿ inclusive, even if it has up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated.     

Long paths and cycles in hypercubes with faulty vertices

A fault-free path in the n-dimensional hypercube Q_n with f faulty vertices is said to be long if it has length at least 2ⁿ-2f-2. Similarly, a fault-free cycle in Q_n is long if it has length at least 2ⁿ-2f. If all faulty vertices are from the same bipartite class of Q_n, such length is the best possible. We show that for every set of at most 2n-4 faulty vertices in Q_n and every two fault-free vertices u and v satisfying a simple necessary condition on neighbors of u and v, there exists a long fault-free path between u and v. This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most n²/10+n/2+1 faulty vertices in Q_n where n?15 that Q_n has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.     

Edge-fault-tolerant edge-bipancyclicity of hypercubes

In this paper, we consider the problem embedding a cycle into the hypercube Qn with existence of faulty edges and show that for any edge subset F of Qn with |F|?n−1 every edge of Qn−F lies on a cycle of every even length from 6 to n² inclusive provided n?4 and all edges in F are not incident with the same vertex. This result improves some known results.     

Cycles embedding in hypercubes with node failures

The hypercube has been widely used as the interconnection network in parallel computers. The n-dimensional hypercube Qn is a graph having n² vertices each labeled with a distinct n-bit binary strings. Two vertices are linked by an edge if and only if their addresses differ exactly in the one bit position. Let fv denote the number of faulty vertices in Qn. For n?3, in this paper, we prove that every fault-free edge and fault-free vertex of Qn lies on a fault-free cycle of every even length from 4 to n²−2fv inclusive even if fv?n−2. Our results are optimal.     

Linear array and ring embeddings in conditional faulty hypercubes

The n-dimensional hypercube Q_n is a graph having 2ⁿ vertices labeled from 0 to 2ⁿ−1. Two vertices are connected by an edge if their binary labels differ in exactly one bit position. In this paper, we consider the faulty hypercube Q_n with n⩾3 that each vertex of Q_n is incident to at least two nonfaulty edges. Based on this requirement, we prove that Q_n contains a hamiltonian path joining any two different colored vertices even if it has up to 2n−5 edge faults. Moreover, we show that there exists a path of length 2ⁿ−2 between any two the same colored vertices in this faulty Q_n. Furthermore, we also prove that the faulty Q_n still contains a cycle of every even length from 4 to 2ⁿ inclusive.     

The super connectivity of augmented cubes

The augmented cube AQn, proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a (2n−1)-regular (2n−1)-connected graph (n≠3). This paper determines that the super connectivity of AQn is 4n−8 for n?6 and the super edge-connectivity is 4n−4 for n?5. That is, for n?6 (respectively, n?5), at least 4n−8 vertices (respectively, 4n−4 edges) of AQn are removed to get a disconnected graph that contains no isolated vertices. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

Path bipancyclicity of hypercubes

A bipartite graph is vertex-bipancyclic (respectively, edge-bipancyclic) if every vertex (respectively, edge) lies in a cycle of every even length from 4 to |V(G)| inclusive. It is easy to see that every connected edge-bipancyclic graph is vertex-bipancyclic. An n-dimensional hypercube, or n-cube denoted by Qn, is well known as bipartite and one of the most efficient networks for parallel computation. In this paper, we study a stronger bipancyclicity of hypercubes. We prove that every n-dimensional hypercube is (2n−4)-path-bipancyclic for n?3. That is, for any path P of length k with 1?k?2n−4 and any integer l with max{2,k}?l?²n−1, an even cycle C of length 2l can be found in Qn such that the path P is included in C for n?3.     

Embedding Hamiltonian Cycles into Folded Hypercubes with Faulty Links

It has been known that an n-dimensional hypercube (n-cube for short) can always embed a Hamiltonian cycle when the n-cube has no more than n−2 faulty links. In this paper, we study the link-fault tolerant embedding of a Hamiltonian cycle into the folded hypercube, which is a variant of the hypercube, obtained by adding a link to every pair of nodes with complementary addresses. We will show that a folded n-cube can tolerate up to n−1 faulty links when embedding a Hamiltonian cycle. We present an algorithm, FT_HAMIL, that finds a Hamiltonian cycle while avoiding any set of faulty links F provided that |F|⩽n−1. An operation, called bit-flip, on links of hyper-cube is introduced. Simple yet elegant, bit-flip will be employed by FT_HAMIL as a basic operation to generate a new Hamiltonian cycle from an old one (that contains faulty links). It is worth pointing out that the algorithm is optimal in the sense that for a folded n-cube, n−1 is the maximum number for |F| that can be tolerated, F being an arbitrary set of faulty links.     

Paths in Möbius cubes and crossed cubes

The Möbius cube MQn and the crossed cube CQn are two important variants of the hypercube Qn. This paper shows that for any two different vertices u and v in G∈{MQn,CQn} with n?3, there exists a uv-path of every length from dG(u,v)+2 to n²−1 except for a shortest uv-path, where dG(u,v) is the distance between u and v in G. This result improves some known results.     

Pancyclicity and bipancyclicity of conditional faulty folded hypercubes

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network’s topology can simulate rings of various lengths. An n-dimensional folded hypercube FQ_n is an attractive variant of an n-dimensional hypercube Q_n that is obtained by establishing some extra edges between the vertices of Q_n. FQ_n for any odd n is known to be bipartite. In this paper, we explore the pancyclicity and bipancyclicity of FQ_n. For any FQ_n (n ? 2) with at most 2n − 3 faulty edges, where each vertex is incident to at least two fault-free edges, we prove that there exists a fault-free cycle of every even length from 4 to 2ⁿ; and when n ? 2 is even, we prove there also exists a fault-free cycle of every odd length from n + 1 to 2ⁿ − 1. The result is optimal with respect to the number of faulty edges tolerated.     

The panconnectivity and the pancycle-connectivity of the generalized base-<Emphasis Type="Italic">b</Emphasis> hypercube

The interconnection network considered in this paper is the generalized base-b hypercube that is an attractive variant of the well-known hypercube. The generalized base-b hypercube is superior to the hypercube in many criteria, such as diameter, connectivity, and fault diameter. In this paper, we study the panconnectivity and pancycle-connectivity of the generalized base-b hypercube. We show that a generalized base-b hypercube is panconnected for b≥3. That is, for each pair of distinct vertices x and y of the n-dimensional generalized base-b hypercube GH(b,n) and for any integer l, where Dist(x,y)≤l≤N−1, there exists a path of the length l joining x and y, where N is the order of the graph GH(b,n) and Dist(x,y) is the distance between x and y. We also show that a generalized base-b hypercube is pancycle-connected for b≥3. That is, every two distinct vertices x and y of the graph GH(b,n) are contained by a cycle of every length ranging from the length of the smallest cycle that contains x and y to N.     

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u,v∈V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC(u,v)=max{dC(x,y)|x,y∈V(C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to |V(G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices u,v∈V(G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG(u,v),3}?l?|V(G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249-1251] proved that the n-dimensional Augmented cube, AQn, is pancyclic in the sense that a cycle of length l exists, 3?l?|V(AQn)|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n?2.     

Matrix Representation of Graph Embedding in a Hypercube

The purpose of this paper is to demonstrate the use of matrices for the representation of graph embedding in a hypercube. We denote the image of an embedding (which is a subgraph of the hypercube) as a matrix. With this representation, we are able to simplify, unify, generalize, or improve existing results regarding multigraph embedding in a hypercube. A class of trees called regular binary-reflected trees is identified, which includes linear paths, binomial trees, and many others. We show that for any regular binary-reflected tree T, n copies of T can be simultaneously embedded in an n-cube with congestion = 2. Embeddings of binary trees and meshes are also discussed.