The panpositionable panconnectedness of crossed cubes

In many parallel and distributed multiprocessor systems, the processors are connected based on different types of interconnection networks. The topological structure of an interconnection network is typically modeled as a graph. Among the many kinds of network topologies, the crossed cube is one of the most popular. In this paper, we investigate the panpositionable panconnectedness problem with respect to the crossed cube. A graph G is r-panpositionably panconnected if for any three distinct vertices x, y, z of G and for any integer \(l_1\) satisfying \(r \le l_1 \le |V(G)| - r - 1\), there exists a path \(P = [x, P_1, y, P_2, z]\) in G such that (i) \(P_1\) joins x and y with \(l(P_1) = l_1\) and (ii) \(P_2\) joins y and z with \(l(P_2) = l_2\) for any integer \(l_2\) satisfying \(r \le l_2 \le |V(G)| - l_1 - 1\), where |V(G)| is the total number of vertices in G and \(l(P_1)\) (respectively, \(l(P_2)\)) is the length of path \(P_1\) (respectively, \(P_2\)). By mathematical induction, we demonstrate that the n-dimensional crossed cube \(CQ_n\) is n-panpositionably panconnected. This result indicates that the path embedding of joining x and z with a mediate vertex y in \(CQ_n\) is extremely flexible. Moreover, applying our result, crossed cube problems such as panpositionable pancyclicity, panpositionably Hamiltonian connectedness, and panpositionable Hamiltonicity can be solved.     

Edge-fault-tolerant vertex-pancyclicity of augmented cubes

The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQn lies on a fault-free cycle of every length from 3 to n², even if there are up to n−1 edge faults. We also show that our result is optimal.     

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.     

Improving the panconnectedness property of locally twisted cubes

The n-dimensional locally twisted cube LTQ_n is a promising alternative to the hypercube because of its great properties. Not only is LTQ_n n-connected, but also meshes, torus, and edge-disjoint Hamiltonian cycles can embed in it. Ma and Xu [Panconnectivity of locally twisted cubes, Appl. Math. Lett. 19 (2006), pp. 681–685] investigated the panconnectivity of LTQ_n for flexible routing. In this paper, we combine panconnectivity with Hamiltonian connectedness to define Hamiltonian r-panconnectedness: a graph G of m vertices, m≥3, is Hamiltonian r-panconnected if for any three distinct vertices x, y, and z of G there exists a Hamiltonian path P of G such that P(1)=x, P(l+1)=y, and P(m)=z for every r≤l≤m?1?r, where P(i) denotes the ith vertex of P for 1≤i≤m. Then, we show that LTQ_n is Hamiltonian n-panconnected for n≥5. This property admits the path embedding via an intermediate node at any prescribed position, and our result achieves an improvement over that of Ma and Xu.     

Fault-tolerant cycle-embedding of crossed cubes

The crossed cube CQ_n introduced by Efe has many properties similar to those of the popular hypercube. However, the diameter of CQ_n is about one half of that of the hypercube. Failures of links and nodes in an interconnection network are inevitable. Hence, in this paper, we consider the hybrid fault-tolerant capability of the crossed cube. Letting f_e and f_v be the numbers of faulty edges and vertices in CQ_n, we show that a cycle of length l, for any 4?l?|V(CQ_n)|−f_v, can be embedded into a wounded crossed cube as long as the total number of faults (f_v+f_e) is no more than n−2, and we say that CQ_n is (n−2)-fault-tolerant pancyclic. This result is optimal in the sense that if there are n−1 faults, there is no guarantee of having a cycle of a certain length in it.     

On embedding cycles into faulty twisted cubes

The twisted cube TQ_n is an alternative to the popular hypercube network. Recently, some interesting properties of TQ_n were investigated. In this paper, we study the pancycle problem on faulty twisted cubes. Let f_e and f_v be the numbers of faulty edges and faulty vertices in TQ_n, respectively. We show that, with f_e + f_v ? n − 2, a faulty TQ_n still contains a cycle of length l for every 4 ? l ? ∣V(TQ_n)∣ − f_v and odd integer n ? 3.     

Conditional edge-fault Hamiltonicity of augmented cubes

The augmented cube is a variation of hypercubes, it possesses many superior properties. In this paper, we show that, for any n-dimensional augmented cube (n?3) with faulty edges up to 4n-8 in which each vertex is incident to at least two fault-free edges, there exists a fault-free Hamiltonian cycle. Our result is optimal with respect to the number of faulty edges tolerated.     

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.     

Complete path embeddings in crossed cubes

Crossed cubes are popular variants of hypercubes. In this paper, we study path embeddings between any two distinct nodes in crossed cubes. We prove two important results in the n-dimensional crossed cube: (a) for any two nodes, all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1; (b) for any two integers n ? 2 and l with , there always exist two nodes x and y whose distance is l, such that no path of length l + 1 can be embedded between x and y with dilation 1. The obtained results are optimal in the sense that the dilations of path embeddings are all 1. The results are also complete, because the embeddings of paths of all possible lengths between any two nodes are considered.     

Constructing the nearly shortest path in crossed cubes

A graph G is panconnected if each pair of distinct vertices u,v∈V(G) are joined by a path of length l for all d_G(u,v)?l?|V(G)|-1, where d_G(u,v) is the length of a shortest path joining u and v in G. Recently, Fan et. al. [J. Fan, X. Lin, X. Jia, Optimal path embedding in crossed cubes, IEEE Trans. Parall. Distrib. Syst. 16 (2) (2005) 1190-1200, J. Fan, X. Jia, X. Lin, Complete path embeddings in crossed cubes, Inform. Sci. 176 (22) (2006) 3332-3346] and Xu et. al. [J.M. Xu, M.J. Ma, M. Lu, Paths in Möbius cubes and crossed cubes, Inform. Proc. Lett. 97 (3) (2006) 94-97] both proved that n-dimensional crossed cube, CQ_n, is almost panconnected except the path of length d_CQn(u,v)+1 for any two distinct vertices u,v∈V(CQ_n). In this paper, we give a necessary and sufficient condition to check for the existence of paths of length d_CQn(u,v)+1, called the nearly shortest paths, for any two distinct vertices u,v in CQ_n. Moreover, we observe that only some pair of vertices have no nearly shortest path and we give a construction scheme for the nearly shortest path if it exists.     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Hamiltonian paths and cycles passing through a prescribed path in hypercubes

Assume that P is any path in a bipartite graph G of length k with 2?k?h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. In this paper, the following result is obtained: The n-dimensional hypercube Qn with n?3 is (2n−3)-path bipancyclic but is not (2n−2)-path bipancyclic, moreover, a path P of length k with 2?k?2n−3 lies in a cycle of length 2k−2 if and only if P contains two edges of the same dimension. In order to prove the above result we first show that any path of length at most 2n−1 is a subpath of a Hamiltonian path in Qn with n?2, moreover, the upper bound 2n−1 is sharp when n?4.     

A sufficient condition for pancyclic graphs

In 2005, Rahman and Kaykobad proved that if G is a connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for each pair x, y of distinct nonadjacent vertices in G, where d(x,y) is the length of a shortest path between x and y in G, then G has a Hamiltonian path [Inform. Process. Lett. 94 (2005) 37–41]. In 2006 Li proved that if G is a 2-connected graph of order n3 such that d(x)+d(y)+d(x,y)n+2 for each pair x,y of nonadjacent vertices in G, then G is pancyclic or G=K_n/2,n/2 where n4 is an even integer [Inform. Process. Lett. 98 (2006) 159–161]. In this work we prove that if G is a 2-connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for all pairs x, y of distinct nonadjacent vertices in G, then G is pancyclic or G belongs to one of four specified families of graphs.     

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.     

Embedding a family of meshes into twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The twisted cube is an important variation of the hypercube. Let TQn denote the n-dimensional twisted cube. In this paper, we consider embedding a family of 2-dimensional meshes into a twisted cube. The main results obtained in this paper are: (1) For any odd integer n?1, there exists a mesh of size 2ײn−1 that can be embedded in the TQn with unit dilation and unit expansion. (2) For any odd integer n?5, there exists a mesh of size 4ײn−2 that can be embedded in the TQn with dilation 2 and unit expansion. (3) For any odd integer n?5, a family of two disjoint meshes of size 4ײn−3 can be embedded into the TQn with unit dilation and unit expansion. Results (1) and (3) are optimal in the sense that the dilations and expansions of the embeddings are unit values.     

Embedding multi-dimensional meshes into twisted cubes

The twisted cube is an important variant of the most popular hypercube network for parallel processing. In this paper, we consider the problem of embedding multi-dimensional meshes into twisted cubes in a systematic way. We present a recursive method for embedding a family of disjoint multi-dimensional meshes into a twisted cube with dilation 1 and expansion 1. We also prove that a single multi-dimensional mesh can be embedded into a twisted cube with dilation 2 and expansion 1. Our work extends some previously known results.     

Embedding two edge-disjoint Hamiltonian cycles into locally twisted cubes

The n-dimensional hypercube network Q_n is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n-dimensional locally twisted cube LTQ_n, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as Q_n. One advantage of LTQ_n is that the diameter is only about half of the diameter of Q_n. Recently, some interesting properties of LTQ_n have been investigated in the literature. The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the interconnection network. The existence of two edge-disjoint Hamiltonian cycles in locally twisted cubes has remained unknown. In this paper, we prove that the locally twisted cube LTQ_n with n?4 contains two edge-disjoint Hamiltonian cycles. Based on the proof of existence, we further provide an O(n2ⁿ)-linear time algorithm to construct two edge-disjoint Hamiltonian cycles in an n-dimensional locally twisted cube LTQ_n with n?4, where LTQ_n contains 2ⁿ nodes and n2ⁿ⁻¹ edges.     

The triangular pyramid: Routing and topological properties

In this paper, a new topology for multicomputer interconnection networks, based on triangular mesh, is proposed. The new network, referred to as the triangular pyramid (or tripy for short), has L levels of triangular mesh. We study some basic important properties of the proposed network as well as introduce a routing algorithm for the tripy network based on the routing of triangular meshes. We prove that this form of pyramidal network is Hamiltonian, Hamiltonian-connected, and pancyclic. We also prove that the proposed network is 6-colorable and conduct a brief comparison of the tripy and its traditional pyramid counterpart. Our results show that the proposed network has higher scalability, connectivity, and total network bandwidth while preserving the important properties of the traditional pyramid network.     

A note on path bipancyclicity of hypercubes

In [C.H. Tsai, S.Y. Jiang, Path bipancyclicity of hypercubes, Inform. Process. Lett. 101 (2007) 93–97], the authors showed that any path in an n-cube with length of k, 2k2n−4, lies on a cycle of every even length from 2k to 2ⁿ inclusive. Base on Lemma 5 of that paper, they proved the subcase 2.2.1 of the main theorem of that paper. However, the lemma is false, therefore, we propose a lemma to replace that lemma. Therefore, the main result of [C.H. Tsai, S.Y. Jiang, Path bipancyclicity of hypercubes, Inform. Process. Lett. 101 (2007) 93–97] is still correct.     

Geodesic pancyclicity of twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. An n-dimensional twisted cube, TQ_n, is an important variation of the hypercube Q_n and preserves many of its desirable properties. The problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years. The geodesic cycle embedding problem is for any two distinct vertices, to find all the possible lengths of cycles that include a shortest path joining them. In this paper, we prove that TQ_n is geodesic 2-pancyclic for each odd integer n ? 3. This result implies that TQ_n is edge-pancyclic for each odd integer n ? 3. Moreover, TQ_n × K₂ is also demonstrated to be geodesic 4-pancyclic.