Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $\lambda_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le \epsilon_n0$, a random induced subgraph has a.s. a unique largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where $\wp(\epsilon_n)$ is the survival probability of a specific branching process.Comment: 18 pages, 1 figur

Fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees

AbstractA bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every pair of vertices that are in different parts of the graph. It is well known that Cay(Sn,B) is Hamiltonian laceable, where Sn is the symmetric group on {1,2,…,n} and B is a generating set consisting of transpositions of Sn. In this paper, we show that for any F⊆E(Cay(Sn,B)), if |F|≤n−3 and n≥4, then there exists a Hamiltonian path in Cay(Sn,B)−F joining every pair of vertices that are in different parts of the graph. The result is optimal with respect to the number of edge faults

Investigation of the robustness of star graph networks

The star interconnection network has been known as an attractive alternative to n-cube for interconnecting a large number of processors. It possesses many nice properties, such as vertex/edge symmetry, recursiveness, sublogarithmic degree and diameter, and maximal fault tolerance, which are all desirable when building an interconnection topology for a parallel and distributed system. Investigation of the robustness of the star network architecture is essential since the star network has the potential of use in critical applications. In this study, three different reliability measures are proposed to investigate the robustness of the star network. First, a constrained two-terminal reliability measure referred to as Distance Reliability (DR) between the source node u and the destination node I with the shortest distance, in an n-dimensional star network, Sn, is introduced to assess the robustness of the star network. A combinatorial analysis on DR especially for u having a single cycle is performed under different failure models (node, link, combined node/link failure). Lower bounds on the special case of the DR: antipode reliability, are derived, compared with n-cube, and shown to be more fault-tolerant than n-cube. The degradation of a container in a Sn having at least one operational optimal path between u and I is also examined to measure the system effectiveness in the presence of failures under different failure models. The values of MTTF to each transition state are calculated and compared with similar size containers in n-cube. Meanwhile, an upper bound under the probability fault model and an approximation under the fixed partitioning approach on the ( n-1)-star reliability are derived, and proved to be similarly accurate and close to the simulations results. Conservative comparisons between similar size star networks and n-cubes show that the star network is more robust than n-cube in terms of ( n-1)-network reliability

Matching preclusion and conditional matching preclusion for bipartite interconnection networks II: Cayley graphs generated by transposition trees and hyper‐stars

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. It is natural to look for obstruction sets beyond those induced by a single vertex. The conditional matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph with no isolated vertices that has no perfect matchings. In this companion paper of Cheng et al. (Networks (NET 1554)), we find these numbers for a number of popular interconnection networks including hypercubes, star graphs, Cayley graphs generated by transposition trees and hyper‐stars. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/91319/1/20441_ftp.pd

Embedding Schemes for Interconnection Networks.

Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth

Disjoint Hamilton cycles in transposition graphs

Most network topologies that have been studied have been subgraphs of transposition graphs. Edge-disjoint Hamilton cycles are important in network topologies for improving fault-tolerance and distribution of messaging traffic over the network. Not much was known about edge-disjoint Hamilton cycles in general transposition graphs until recently Hung produced a construction of 4 edge-disjoint Hamilton cycles in the 5-dimensional transposition graph and showed how edge-disjoint Hamilton cycles in (n + 1)-dimensional transposition graphs can be constructed inductively from edge-disjoint Hamilton cycles in n-dimensional transposition graphs. In the same work it was conjectured that n-dimensional transposition graphs have n − 1 edge-disjoint Hamilton cycles for all n greater than or equal to 5. In this paper we provide an edge-labelling for transposition graphs and, by considering known Hamilton cycles in labelled star subgraphs of transposition graphs, are able to provide an extra edge-disjoint Hamilton cycle at the inductive step from dimension n to n + 1, and thereby prove the conjecture

Properties and algorithms of the hyper-star graph and its related graphs

The hyper-star interconnection network was proposed in 2002 to overcome the drawbacks of the hypercube and its variations concerning the network cost, which is defined by the product of the degree and the diameter. Some properties of the graph such as connectivity, symmetry properties, embedding properties have been studied by other researchers, routing and broadcasting algorithms have also been designed. This thesis studies the hyper-star graph from both the topological and algorithmic point of view. For the topological properties, we try to establish relationships between hyper-star graphs with other known graphs. We also give a formal equation for the surface area of the graph. Another topological property we are interested in is the Hamiltonicity problem of this graph. For the algorithms, we design an all-port broadcasting algorithm and a single-port neighbourhood broadcasting algorithm for the regular form of the hyper-star graphs. These algorithms are both optimal time-wise. Furthermore, we prove that the folded hyper-star, a variation of the hyper-star, to be maixmally fault-tolerant

Properties and algorithms of the (n, k)-arrangement graphs

The (n, k)-arrangement interconnection topology was first introduced in 1992. The (n, k )-arrangement graph is a class of generalized star graphs. Compared with the well known n-star, the (n, k )-arrangement graph is more flexible in degree and diameter. However, there are few algorithms designed for the (n, k)-arrangement graph up to present. In this thesis, we will focus on finding graph theoretical properties of the (n, k)- arrangement graph and developing parallel algorithms that run on this network. The topological properties of the arrangement graph are first studied. They include the cyclic properties. We then study the problems of communication: broadcasting and routing. Embedding problems are also studied later on. These are very useful to develop efficient algorithms on this network. We then study the (n, k )-arrangement network from the algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms such as prefix sums computation, sorting, merging and basic geometry computation: finding convex hull on the (n, k )-arrangement graph. A literature review of the state-of-the-art in relation to the (n, k)-arrangement network is also provided, as well as some open problems in this area

Linearly many faults in 2-tree-generated networks

In this article we consider a class of Cayley graphs that are generated by certain 3-cycles on the alternating group A n . These graphs are generalizations of the alternating group graph A G n . We look at the case when the 3-cycles form a “tree-like structure,” and analyze its fault resiliency. We present a number of structural theorems and prove that even with linearly many vertices deleted, the remaining graph has a large connected component containing almost all vertices. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64908/1/20319_ftp.pd