The 3-Good Neighbor Edge-Fault-Tolerant Strong Menger Edge Connectivity on the Class of BC Networks

A graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text] for short) if the number of disjoint paths between any two of its vertices equals the minimum degree of these two vertices. In this paper, we focus on the maximally edge-fault-tolerant of the class of BC-networks (contain hypercubes, twisted cubes, Möbius cubes, crossed cubes, etc.) concerning the SM-[Formula: see text] property. Under the restricted condition that each vertex is incident with at least three fault-free edges, we show that even if there are [Formula: see text] faulty edges, all BC-networks still have SM-[Formula: see text] property and the bound [Formula: see text] is sharp.

The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees

The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.

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.

Unpaired Many-to-Many Disjoint Path Cover of Balanced Hypercubes

A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC) of a graph [Formula: see text] is a set of [Formula: see text] vertex-disjoint paths joining [Formula: see text] distinct pairs of source and sink in which each vertex of [Formula: see text] is contained exactly once in a path. The balanced hypercube [Formula: see text], a variant of the hypercube, was introduced as a desired interconnection network topology. Let [Formula: see text] and [Formula: see text] be any two sets of vertices in different partite sets of [Formula: see text] ([Formula: see text]). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of [Formula: see text] when [Formula: see text]. In this paper, we prove that there exists unpaired many-to-many [Formula: see text]-disjoint path cover of [Formula: see text] ([Formula: see text]) from [Formula: see text] to [Formula: see text], which has improved some known results. The upper bound [Formula: see text] is best possible in terms of the number of disjoint paths in unpaired many-to-many [Formula: see text]-DPC of [Formula: see text].

The Linkedness of Cubical Polytopes: The Cube

The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is $k$-linked if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is $k$-linked if its graph is $k$-linked. We establish that the $d$-dimensional cube is $\lfloor (d+1)/2 \rfloor$-linked, for every $d\ne 3$; this is the maximum possible linkedness of a $d$-polytope. This result implies that, for every $d\geqslant 1$, a cubical $d$-polytope is $\lfloor d/2\rfloor$-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph $G$ is strongly $k$-linked if it has at least $2k+1$ vertices and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We show that cubical 4-polytopes are strongly $2$-linked and that, for each $d\geqslant 1$, $d$-dimensional cubes are strongly $\lfloor d/2\rfloor$-linked.

A novel design layout of three disjoint paths multistage interconnection network & its reliability analysis

Purpose The purpose of this paper is to design a efficient layout of Multistage interconnection network which has cost effective solution with high reliability and fault-tolerence capability. For parallel computation, various multistage interconnection networks (MINs) have been discussed hitherto in the literature, however, these networks always required further improvement in reliability and fault-tolerance capability. The fault-tolerance capability of the network can be achieved by increasing the number of disjoint paths as a result the reliability of the interconnection networks is also improved. Design/methodology/approach This proposed design is a modification of gamma interconnection network (GIN) and three disjoint path gamma interconnection network (3-DGIN). It has a total seven number of paths for all tag values which is uniform out of these seven paths, three paths are disjoint paths which increase the fault tolerance capability by two faults. Due to the presence of more paths than the GIN and 3-DGIN, this proposed design is more reliable. Findings In this study, a new design layout of a MIN has been proposed which provides three disjoint paths and uniformity in terms of an equal number of paths for all source-destination (S-D) pairs. The new layout contains fewer nodes as compared to GIN and 3-DGIN. This design provides a symmetrical structure, low cost, better terminal reliability and provides an equal number of paths for all tag values (|S-D|) when compared with existing MINs of this class. Originality/value A new design layout of MINs has been purposed and its two terminal reliability is calculated with the help of the reliability block diagram technique.

Interference-free Walks in Time: Temporally Disjoint Paths

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk version" is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The "path version" remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counterintuitively, we find NP-hardness in general but also identify natural tractable cases.