Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

(n − 2)-Fault-Tolerant Edge-Pancyclicity of Crossed Cubes CQn

As one of the most fundamental networks for parallel and distributed computation, cycle is suitable for developing simple algorithms with low communication cost. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant edge-pancyclic if after deleting any faulty set [Formula: see text] of [Formula: see text] vertices and/or edges from [Formula: see text], every correct edge in the resulting graph lies in a cycle of every length from [Formula: see text] to [Formula: see text], inclusively, where [Formula: see text] is the girth of [Formula: see text], the length of a shortest cycle in [Formula: see text]. The [Formula: see text]-dimensional crossed cube [Formula: see text] is an important variant of the hypercube [Formula: see text], which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of [Formula: see text], and shows that if [Formula: see text] contains at most [Formula: see text] faulty vertices and/or edges then, for any fault-free edge [Formula: see text] and every length [Formula: see text] from [Formula: see text] to [Formula: see text] except [Formula: see text], there is a fault-free cycle of length [Formula: see text] containing the edge [Formula: see text]. The result is optimal in some senses.

An Exchanged 3-Ary n-Cube Interconnection Network for Parallel Computation

The interconnetion network plays an important role in a parallel system. To avoid the edge number of the interconnect network scaling rapidly with the increase of dimension and achieve a good balance of hardware costs and properties, this paper presents a new interconnection network called exchanged [Formula: see text]-ary [Formula: see text]-cube ([Formula: see text]). Compared with the [Formula: see text]-ary [Formula: see text]-cube structures, [Formula: see text] shows better performance in terms of many metrics such as small degree and fewer links. In this paper, we first introduce the structure of [Formula: see text] and present some properties of [Formula: see text]; then, we propose a routing algorithm and obtain the diameter of [Formula: see text]. Finally, we analyze the diagnosis of [Formula: see text] and give the diagnosibility under PMC model and MM* model.

Subgraph-based Strong Menger Connectivity of Hypercube and Exchanged Hypercube

Large scale multiprocessor systems or multicomputer systems, taking interconnection networks as underlying topologies, have been widely used in the big data era. Fault tolerance is becoming an essential attribute in multiprocessor systems as the number of processors is getting larger. A connected graph [Formula: see text] is called strong Menger (edge) connected if, for any two distinct vertices [Formula: see text] and [Formula: see text], there are [Formula: see text] vertex (edge)-disjoint paths between them. Exchanged hypercube [Formula: see text], as a variant of hypercube [Formula: see text], remains lots of preferable fault tolerant properties of hypercube. In this paper, we show that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are strong Menger (edge) connected, respectively. Moreover, as a by-product, for dual cube [Formula: see text], one popular generalization of hypercube, [Formula: see text] is also showed to be strong Menger (edge) connected, where [Formula: see text].

Super Ck and Sub-Ck Connectivity of k-Ary n-Cube Networks

Let [Formula: see text] be an undirected graph. An H-structure-cut (resp. H-substructure-cut) of [Formula: see text] is a set of subgraphs of [Formula: see text], if any, whose deletion disconnects [Formula: see text], where the subgraphs deleted are isomorphic to a certain graph [Formula: see text] (resp. where for any [Formula: see text] of the subgraphs deleted, there is a subgraph [Formula: see text] of [Formula: see text], isomorphic to [Formula: see text], such that [Formula: see text] is a subgraph of [Formula: see text]). [Formula: see text] is super [Formula: see text]-connected (resp. super sub-[Formula: see text]-connected) if the deletion of an arbitrary minimum [Formula: see text]-structure-cut (resp. minimum [Formula: see text]-substructure-cut) isolates a component isomorphic to a certain graph [Formula: see text]. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that [Formula: see text] with [Formula: see text] is super sub-[Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd, and super [Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd.

Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube

The problem “Given a Boolean function [Formula: see text] of [Formula: see text] variables by its truth table vector. Find (if exists) a vector [Formula: see text] of maximal (or minimal) weight, such that [Formula: see text].” is considered here. It is closely related to the problem of computing the algebraic degree of Boolean functions which is an important cryptographic parameter. To solve this problem efficiently, we explore the orders of the vectors of the [Formula: see text]-dimensional Boolean cube [Formula: see text] according to their weights. The notion of “[Formula: see text]th layer” of [Formula: see text] is involved in the definition and examination of the “weight order” relation. It is compared with the known relation “precedes”. Several enumeration problems concerning these relations are solved and the relevant notes were added to three sequences in the on-line encyclopedia of integer sequences (OEIS). One special weight order is defined and examined in detail. In it, the lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called weight-lexicographic order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. The results obtained by them were used in creating two new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed — the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results from numerous tests confirm the efficiency of these algorithms. Other applications of the obtained algorithms are also discussed — when representing, generating and ranking other combinatorial objects.

Fault-Tolerant Maximal Local-Edge-Connectivity of Augmented Cubes

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A connected graph [Formula: see text] is said to be maximally local-edge-connected if each pair of vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are connected by [Formula: see text] pairwise edge-disjoint paths. In this paper, we show that the [Formula: see text]-dimensional augmented cube [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected and the bound [Formula: see text] is sharp; under the restricted condition that each vertex has at least three fault-free adjacent vertices, [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected and the bound [Formula: see text] is sharp; and under the restricted condition that each vertex has at least [Formula: see text] fault-free adjacent vertices, [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected. Furthermore, we show that a [Formula: see text]-regular graph [Formula: see text] is [Formula: see text]-fault-tolerant one-to-many maximally local-connected if [Formula: see text] does not contain [Formula: see text] and is super [Formula: see text]-vertex-connected of order 1, a [Formula: see text]-regular graph [Formula: see text] is [Formula: see text]-fault-tolerant one-to-many maximally local-connected if [Formula: see text] does not contain [Formula: see text] and is super [Formula: see text]-vertex-connected of order 1.

Fault-Tolerant Panconnectivity of Augmented Cubes AQn

The augmented cube [Formula: see text] is a variation of the hypercube [Formula: see text]. This paper considers the fault-tolerant Panconnectivity of [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. We prove that for any two fault-free vertices [Formula: see text] and [Formula: see text] with distance [Formula: see text] in [Formula: see text], there exists a fault-free path [Formula: see text] of each length from [Formula: see text] to [Formula: see text] in [Formula: see text] if [Formula: see text], where [Formula: see text] is the number of faulty vertices in [Formula: see text]. Moreover, the bound is sharp.

A Remark on Rainbow 6-Cycles in Hypercubes

We call an edge-coloring of a graph [Formula: see text] a rainbow coloring if the edges of [Formula: see text] are colored with distinct colors. For every even positive integer [Formula: see text], let [Formula: see text] denote the minimum number of colors required to color the edges of the [Formula: see text]-dimensional cube [Formula: see text], so that every copy of [Formula: see text] is rainbow. Faudree et al. [6] proved that [Formula: see text] for [Formula: see text] or [Formula: see text]. Mubayi et al. [8] showed that [Formula: see text]. In this note, we show that [Formula: see text]. Moreover, we obtain the number of 6-cycles of [Formula: see text].

A note on minimum linear arrangement for BC graphs

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, [Formula: see text]-cubes, etc., as the subfamilies.