Embedding spanning disjoint cycles in augmented cube networks with prescribed vertices in each cycle

How many triangles and quadrilaterals are there in an n-dimensional augmented cube?

Deriving an effective VLSI layout for interconnected network is important, since it increases the cost-effectiveness of parallel architectures. Graph embedding is the key to solving the problems of parallel structure simulation and layout design of VLSI. Wirelength is a criterion measuring the quality for graph embedding. And it is extensively used for VLSI design. Owing to the limitation of the chip area, the total wirelength of embedded network becomes a key issue affecting the network-on-chip communication performance. \(AQ_{n}\), the n-dimensional augmented cube, is an important interconnection network topology proposed for parallel computers. In this paper, we first study the minimum wirelength of embedding augmented cube into a linear array based on the maximum induced subgraph problem. Furthermore, we obtain the exact wirelength of embedding augmented cubes into grids and propose a linear embedding algorithm to prepare for further study of efficient layout areas.

Constructing spanning trees in augmented cubes

Letter to the Editor

One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,…,ek of G, there exist k vertex-disjoint cycles C1,C2,…,Ck in G such that V(C1)∪V(C2)∪⋯∪V(Ck)=V(G) and ei∈E(Ci) for all 1≤i≤k. According to the definition, the problem of finding hamiltonian cycle focuses on k=1. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Qn is spanning k-edge-cyclable for 1≤k≤n−1 and n≥2. This is the best possible result, in the sense that the n-dimensional hypercube Qn is not spanning n-edge-cyclable.

Cycle embedding of augmented cubes

Edge-independent spanning trees in augmented cubes

Packing internally disjoint Steiner trees to compute the κ3-connectivity in augmented cubes

Conditional edge-fault pancyclicity of augmented cubes

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

Finding a Hamiltonian cycle in a graph is one of the classical NP-complete problems. Complexity of the Hamiltonian problem in permutation graphs has been a well-known open problem. In this paper the authors settle the complexity of the Hamiltonian problem in the more general class of cocomparability graphs. It is shown that the Hamiltonian cycle existence problem for cocomparability graphs is in P. A polynomial time algorithm for constructing a Hamiltonian path and cycle is also presented. The approach is based on exploiting the relationship between the Hamiltonian problem in a cocomparability graph and the bump number problem in a partial order corresponding to the transitive orientation of its complementary graph.

The augmented cube was introduced as a better interconnection network than the hypercube. An interconnection network needs to have good structural properties beyond simple measures such as connectivity. There are many different measures of structural integrity of interconnection networks. In this paper we prove that if vertices are deleted from an augmented cube of dimension n (where g is a quadratic function), the resulting graph will either be connected or will have a large component and small components having at most vertices in total. Additional results on the cyclic vertex-connectivity and the restricted vertex-connectivity of the augmented cubes will also be given.

The interconnection network is an essential component of a distributed system or of a supercomputer based on large-sale parallel processing. Because in distributed systems the communication between processors is based on message exchange, the network topology is of a great importance. The interconnection network can be seen as a graph and the properties of a network can be studied using combinatorics and graph theory. A number of interconnection network topologies have been studied. The Extended Fibonacci Cube, EFC, is a topology which provides good properties for an interconnection network regarding diameter, node degree, recursive decomposition, embeddability and communication algorithms. In this research we present some properties of the Extended Fibonacci Cubes, we define a Gray code for extended Fibonacci cubes and show how a hamiltonian path, a hamiltonian cycle and a 2D mesh can be embedded in an Extended Fibonacci Cube.

The foundation of information society is computer interconnection network, and the key of information exchange is communication algorithm. Finding interconnection networks with simple routing algorithm and high fault-tolerant performance is the premise of realizing various communication algorithms and protocols. Nowadays, people can build complex interconnection networks by using very large scale integration (VLSI) technology. Locally exchanged twisted cubes, denoted by (s + t + 1)-dimensional LeTQs,t, which combines the merits of the exchanged hypercube and the locally twisted cube. It has been proved that the LeTQs,t has many excellent properties for interconnection networks, such as fewer edges, lower overhead and smaller diameter. Embeddability is an important indicator to measure the performance of interconnection networks. We mainly study the fault tolerant Hamiltonian properties of a faulty locally exchanged twisted cube, LeTQs,t − (fv + fe), with faulty vertices fv and faulty edges fe. Firstly, we prove that an LeTQs,t can tolerate up to s − 1 faulty vertices and edges when embedding a Hamiltonian cycle, for s ⩾ 2, t ⩾ 3, and s ⩽ t. Furthermore, we also prove another result that there is a Hamiltonian path between any two distinct fault-free vertices in a faulty LeTQs,t with up to (s − 2) faulty vertices and edges. That is, we show that LeTQs,t is (s − 1)-Hamiltonian and (s − 2)-Hamiltonian-connected. The results are proved to be optimal in this paper with at most (s − 1)-fault-tolerant Hamiltonicity and (s − 2) fault-tolerant Hamiltonian connectivity of LeTQs,t.

In this chapter, the concepts of Hamiltonian paths and Hamiltonian cycles are discussed. In the first section, the history of Hamiltonian graphs is described, and then some concepts such as Hamiltonian paths, Hamiltonian cycles, traceable graphs, and Hamiltonian graphs are defined. Also some most known Hamiltonian graph problems such as travelling salesman problem (TSP), Kirkman’s cell of a bee, Icosian game, and knight’s tour problem are presented. In addition, necessary and (or) sufficient conditions for existence of a Hamiltonian cycle are investigated. Furthermore, in order to solve Hamiltonian cycle problems, some algorithms are introduced in the last section.